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+{"text":"\\section{Introduction}\n\\label{S1}\n\nIn this paper we study the singular points of suitable weak solutions \nto the three dimensional incompressible Navier--Stokes equations.\nAlthough general boundary and geometric conditions are important, \nwe consider merely an initial value problem of the Navier-Stokes equations in the whole space $\\Omega_T=\\mathbb{R}^3 \\times (0,T)$:\n\\begin{equation}\n\\label{E11}\n\\begin{split}\n(\\partial_t - \\nu\\Delta) v + \\divg (v\\otimes v) +\\nabla p &= 0 \\\\\n\\divg v &= 0\n\\end{split}\n\\end{equation}\nwith the initial data $v(x,0)= v_0(x)$, \nwhere $v$ and $v_0$ are three dimensional solenoidal vector fields \nand the pressure $p$ is a scalar field. \nIn this paper, we let the viscosity $\\nu=1$ since it is not important in our regularity analysis. \nWe denote by $z=(x,t)$ space-time points, space balls by $B(x,r)=\\set{y\\in\\mathbb{R}^3 : |y-x| <r}$, and parabolic cylinders by\n\\[Q(z,r)=B(x,r)\\times (t-r^2,t+r^2).\\]\nWe always assume that parabolic cylinders are in the space-time domain and suppress reference points $z$ in various expressions when it can be understood obviously in the context.\nThe precise definitions will be given in the next section.\n\nIt is not known whether the solution stays regular for all time although the global existence of weak solutions was proved by Leray \\cite{L} long ago.\nTo study the regularity problem Scheffer \\cite{Sch} introduced the concept of suitable weak solutions and proved a partial regularity result. \nFrom Scheffer's result, one can conclude that the Minkowski dimension of the singular set is not greater than 2.\nCaffarelli--Kohn--Nirenberg \\cite{CKN} proved that the Hausdorff dimension of the singular set is not greater than 1 using a regularity criterion based on a scaled invariant functional corresponding to the velocity gradient.\nLin \\cite{MR1488514} gave a simplified short proof by a blowup argument.\nLadyzhenskaya and Seregin \\cite{MR1738171} gave a clear presentation of the H\\\"older regularity. \nChoe and Lewis \\cite{MR1780481} studied the singular set by using a generalized Hausdorff measure.\nGustafson, Kang and Tsai \\cite{MR2308753} unified several known criteria.\nOne of the most important conditions to guarantee the regularity of weak solutions is the so-called Ladyzhenskaya--Prodi--Serrin \\cite{MR0236541,MR0126088,MR0150444} condition, that is,\n\\[u \\in L^l(0,T;L^s(\\mathbb{R}^3))\\]\nfor some $s$ and $l$ satisfying $\\frac{3}{s} + \\frac{2}{l} = 1$ and $3 < s \\le \\infty$.\nEscauriaza--Sergin--\\v{S}ver\\'{a}k \\cite{MR2005639} resolved the regularity problem for the marginal case $v \\in L^\\infty(0,T;L^3(\\mathbb{R}^3))$.\nThere are many variations and generalizations of the LPS condition including the Lorentz spaces.\nBut, most of them dealt with the case that the space integrability exponent is greater than 3.\nThe marginal case $v \\in L^\\infty(0,T;L^{3,w}(\\Omega))$ of the Lorentz spaces was studied by Kozono \\cite{Ko} and Kim--Kozono \\cite{KK}, where $L^{3,w}(\\Omega)$ denotes the standard weak Lebesgue space.\nThey obtained some regularity results when the norm $\\norm{v}_{L^\\infty(0,T;L^{3,w}(\\Omega))}$ is sufficiently small.\nRecently, there arise many interests on this much weaker case $v \\in L^\\infty(0,T;L^{3,w}(\\Omega))$ related to essential singularities without any smallness condition.\n\nIn studying the regularity problem of the Navier--Stokes equations, many papers dealt with the quantities in terms of gradients of solution s like $\\int_Q |\\nabla v|^2$ or $\\int_Q |\\omega|^2$ where $\\omega$ denote the vorticity $\\omega=\\nabla\\times v$.\nIn this paper, we study the regularity problem based on  local kinetic energy $\\int_{B(x,R)} |v(y,t)|^2 dy$.\nUnder the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded, we show that a reverse H\\\"older inequality of $\\nabla v$ with increasing support holds.\nHere is our first theorem.\n\n\\begin{thm}\n\\label{T11}\nSuppose there exists a number $M \\geq 1$ such that for all $Q(z,R) \\subset \\Omega_T$\n\\begin{equation}\n\\label{E12}\n\\sup_{|s-t|<R^2} R^{-1} \\int_{B(x,R)} |v(y,s)|^2 dy \\leq M.\n\\end{equation}\nThen there exist positive constants $C$ and $\\delta_0 = \\frac{2}{CM^{5\/9}-1}$ such that for any $0 \\le \\delta < \\delta_0$ \n\\begin{equation}\n\\label{E13}\n\\norm{\\nabla v}_{L^{2+\\delta}(Q(z,1))} \\leq \\overline{C} (\\norm{\\nabla v}_{L^{2}(Q(z,2))} +1)\n\\end{equation}\nwhere $\\overline{C} = (1+\\frac{\\delta}{2}-\\frac{\\delta}{2}CM^{5\/9})^{-1}$.\n\\end{thm}\n\nWe say that a space-time point $z=(x,t)$ is singular if $v$ is locally unbounded at $z$.\nWe denote by $\\mathcal{S}$ the set of all singular points of $v$.\nIt is well-known that $\\mathcal{S}$ is a compact set.\nUsing this theorem we can gain better information about the distribution of the singular set $\\mathcal{S}$ if a weak solution $v$ of the Navier--Stokes equations belongs to $L^\\infty(0,T;L^{3,w}(\\Omega))$.\nIn a recent paper \\cite{CWY}, it is studied that if a weak solution to the Navier--Stokes equations belongs to $L^\\infty(0,T; L^{3,w}(\\Omega))$, then the number of possible singular points at a singular time is finite.\nIn this paper, we focus on the fractal dimension of the singular set.\nThe following theorem is actually an immediate corollary.\n\n\\begin{thm}\n\\label{T12}\nIf $v \\in L^\\infty(0,T; L^{3,w}(\\Omega))$ is a weak solution to the Navier--Stokes equations, then the Hausdorff dimension of $\\mathcal{S}$ is also at most $1-2\\delta_0$ \n\\begin{equation}\n\\label{E14}\n\\dim_{\\mathcal{H}}(\\mathcal{S}) \\le 1-2\\delta_0\n\\end{equation}\n\\end{thm}\n\nThere are several different ways measuring lower dimensional sets like $\\mathcal{S}$. \nThe concept of the Hausdorff dimension is widely used because the Hausdorff measure is\na natural generalization of the Euclidean Lebesgue measures.\nAnother important one is the Minkowski (box-counting) dimension. \nIn general, to improve the Minkowski dimension is harder than the Hausdorff dimension. \nFor the usual suitable weak solutions, the bound on the Hausdorff dimension is 1.\nBut, the bound on the Minkowski dimension remains significantly greater than 1.\nThere are several attempts to lower the Minkowski dimension of $\\mathcal{S}$, for example, \\cite{MR2967124,Ku,MR2864801,KY}.\nOne of the main result of this paper is to show that if a weak solution $v$ belongs to $L^\\infty(0,T; L^{3,w}(\\Omega))$, then we have a better bound for the Minkowski dimension of singular set of $v$.\nHere is our last theorem.\n\n\\begin{thm}\n\\label{T13}\nIf $v \\in L^\\infty(0,T; L^{3,w}(\\Omega))$ is a weak solution to the Navier--Stokes equations, then the parabolic upper Minkowski dimension \n\\begin{equation}\n\\label{E15}\n\\overline{\\dim}_{\\mathcal{M}}(\\mathcal{S}) \\le \\frac{9-18\\delta_0}{7-6\\delta_0}\n\\end{equation}\nwhere $\\delta_0$ is the same constant in Theorem \\ref{T11}.\n\\end{thm}\n\nWe remark that it is very hard to find the optimal constant $\\delta_0$ in Theorem \\ref{T11}.\nIf the number $\\delta_0$ is very small, then the bound in Theorem \\eqref{T13} is close to $9\/7$.\n\nWe briefly explain the idea of proofs.\nIf a weak solution $v$ is in $L^\\infty(0,T; L^{3,w}(\\Omega))$, then the local kinetic energy of $v$ is uniformly bounded.\nThus, we apply Theorem \\ref{T11} to obtain the reverse H\\\"older inequality of $\\nabla v$, which immediately implies the improvement of the Hausdorff dimension of the singular set.\nTo obtain a bound for the Minkowski dimension of the singular set, we \nneed an interpolation argument to get the local higher integrability of $v$.\nCombining the pressure decomposition and a covering argument, we finally obtain a bound for the Minkowski dimension of $\\mathcal{S}$.\n\n\\section{Preliminaries}\n\\label{S2}\n\nIn this section we recall basic definitions and set up our notations.\nWe denote by $\\mathbb{N}$ the set of natural numbers and by $\\mathbb{R}$ the set of real numbers.\n\n\\begin{defn}[Function spaces]\nFor $0<p\\le\\infty$ we denote by $L^p(\\Omega)$ the standard Lebesgue spaces and by $W^{1,p}(\\Omega)$ the standard Sobolev space with the norm \n\\[\\norm{f}_{W^{1,p}(\\Omega)} = \\left(\\int_{\\Omega} |f|^p +|\\nabla f|^p dx\\right)^{1\/p}.\\]\nWe denote $H^1(\\Omega) = W^{1,2}(\\Omega)$ and define $H_0^1(\\Omega)$ to be the $H^1$ closure of $C^\\infty_0(\\Omega)$.\nWe denote the parabolic space  \n\\begin{equation}\nV(\\Omega_T)= L^{\\infty}(0,T:L^2(\\mathbb{R}^3)) \\cap L^2(0,T:H^1_0(\\mathbb{R}^3)).\n\\end{equation}\nwhere \n\\begin{align*}\nL^{\\infty}(0,T: L^2(\\Omega)) &= \\set{f : \\ess \\sup_{0<t<T} \\norm{f}_{L^2(\\Omega)}(t) < \\infty} \\\\\nL^q(0,T:W^{1,p}(\\Omega)) &= \\set{f : \\int_0^T \\norm{f}^q_{W^{1,p}(\\Omega)}(t)dt <\\infty}.\n\\end{align*}\n\\end{defn}\n\nWe now recall the definition of the weak Lebesgue spaces.\nFor a measurable function $f$ on $\\mathbb{R}^3$, its level set with the height $h$ is denoted by\n\\begin{equation}\n\\label{E21}\nE(h) = \\set{x \\in \\mathbb{R}^3:|f(x)|>h}.\n\\end{equation}\nThe Lebesgue integral can be expressed by the Riemann integral of such level sets.\nIn particular, for $0<q<\\infty$\n\\begin{equation}\n\\label{E22}\n\\int |f(x)|^q dx = \\int_{0}^{\\infty} qh^{q-1} \\boldsymbol m(E(h)) dh.\n\\end{equation}\n\n\\begin{defn}[Weak Lebesgue space]\nThe weak Lebesgue space $L^{q,w}(\\mathbb{R}^3)$ is the set of all measurable function such that the quantity\n\\begin{equation}\n\\label{E23}\n\\norm{f}_{q,w} := \\sup_{h>0} \\big[h \\boldsymbol m(E(h))^{1\/q}\\big]\n\\end{equation}\nis finite.\n\\end{defn}\n\nWe recall the definition of suitable weak solutions (see also \\cite{CKN} and \\cite{MR1488514}).\n\n\\begin{defn}[Suitable weak solutions]\nLet $v_0 \\in L^2(\\mathbb{R}^3)$ denote a given initial data, which is weakly divergence free vector field.\nWe say that $(v,p)\\in V(\\Omega_T) \\times L^{3\/2}(\\Omega_T)$ is a suitable weak solution to the initial value problem if for all $\\phi\\in C^\\infty_0(\\Omega_T)$\n\\begin{equation}\n\\int \\left(v\\cdot \\partial_t\\phi - \\nabla v:\\nabla\\phi + v\\otimes v : \\nabla \\phi + p \\nabla\\cdot\\phi\\right) dz = 0\n\\end{equation}\nwhere $dz=dxdt$.\nThe vector field $v$ is weakly divergence free for almost all time and satisfies\nthe localized energy inequality \n\\begin{equation}\n\\label{E24}\n\\begin{split}\n&\\int |v(x,t)|^2 \\phi dx + 2\\int_0^t \\int |\\nabla v|^2\\phi dz \\\\\n&\\le \\int_0^t \\int |v|^2 (\\partial_t\\phi+\\Delta\\phi) dz\n+ \\int_0^t \\int (|v|^2+2p) v \\cdot \\nabla \\phi dz\n\\end{split}\n\\end{equation}\nfor almost all $0 < t \\le T$ and for all non-negative test functions $\\phi \\in C_0^{\\infty}(\\Omega_T)$ and\n\\[\\int |v(x,t)-v_0(x)|^2 dx \\to 0 \\quad \\text{as} \\quad t \\to 0.\\]\n\\end{defn}\n\n\\begin{defn}[The parabolic Hausdorff dimension]\nFor fixed $\\rho>0$ and $S\\subset\\mathbb{R}^3\\times\\mathbb{R}$, let $\\mathcal{C}(S,\\rho)$ be the family of all coverings of parabolic cylinders $\\set{Q(z_j,r_j)}$ that covers $S$ with $0<r_j \\le \\rho$.\nThe $\\alpha$ dimensional parabolic Hausdorff measure is defined as\n\\[\\mathcal{H}^\\alpha(S) = \\lim_{\\rho\\to0} \\inf_{\\mathcal{C}(E,\\rho)} \\sum_j r_j^\\alpha.\\]\nThe parabolic Hausdorff dimension of the set $S$ is defined as\n\\[\\dim_{\\mathcal{H}}(S) = \\inf\\set{\\alpha : \\mathcal{H}^\\alpha(S)=0}.\\]\n\\end{defn}\n\n\\begin{defn}[Singular points]\nWe call a space-time point $z=(x,t)$ is a singular point of a suitable weak solution $(v,p)$ if $v$ is not essentially bounded in any neighbourhood $Q(z,r)$.\nWe denote the set of all singular points by $\\mathcal{S}$.\n\\end{defn}\n\n\\begin{defn}[The parabolic upper Minkowski dimension]\nLet $N(S;r)$ denote the minimum number of parabolic cylinders $Q(z,r)$ required to cover the set $S$.\nThen the parabolic upper Minkowski dimension of the set $S$ is defined as\n\\begin{equation}\n\\label{E25}\n\\overline{\\dim}_{\\mathcal{M}}(S) = \\limsup_{r\\to0} \\frac{\\log N(S;r)}{-\\log r}.\n\\end{equation}\n\\end{defn}\n\nIn general, different fractal dimensions reflect the geometric structure of the set.\nThe upper Minkowski dimension is strongly control the Hausdorff dimension.\nIndeed, from the definition it is easy to see that \n\\begin{equation}\n\\dim_\\mathcal{H}(S) \\le \\overline{\\dim}_\\mathcal{M}(S).\n\\end{equation}\nWe refer the reader to Falconer's monograph \\cite{MR3236784} for the comprehensive introduction of the fractal geometry.\n\n\\begin{nota}\n\\begin{itemize}\n\\item\nWe denote $\\mathbb{R}_+=\\set{x\\in\\mathbb{R}:0<x<\\infty}$.\n\\item\nWe use the following shorthand notation for balls and cylinders\n\\[RB = B(0,R), \\quad RQ = Q(0,R).\\]\n\\item\nThe average value of $f$ over $X$ is denoted by $(f)_E = \\fint_E f d\\boldsymbol m = \\boldsymbol m(E)^{-1} \\int_X f d\\boldsymbol m$ where $\\boldsymbol m(E)$ denotes the Lebesgue measure of the set $E$.\nWhen the center of ball is obvious in the context, we write \n\\[(f)_R = (f)_{B(x,R)}.\\]\n\\item\nWe write $X \\lesssim Y$ if there is a generic positive constant $C$ such that $|X| \\le C|Y|$. \n\\end{itemize}\n\\end{nota}\n\n\n\n\n\n\\section{Auxiliary lemmas}\n\\label{S3}\n\nIt is easy to see that the pressure satisfies\n\\[-\\Delta p = \\partial x_i\\partial x_j (v_i v_j)\\]\nin the sense of distribution.\nFrom this pressure equation, we have the following explicit decomposition of the localized pressure, which was presented in \\cite{CKN}.\n\nWe choose $\\psi$ satisfying $\\psi(y) = 1$ for $y \\in B(x,(\\rho+r)\/2)$ and $\\psi(y) = 0$ for $y \\notin B(x,r)$ so that  $p_3$ is harmonic in $B(x,\\rho)$. \nLet $\\overline{v} = v - (v)_r$.\nWe have the decomposition\n\\begin{equation}\n\\label{E31}\np\\psi = p_1 + p_2 + p_3\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{E32}\n\\begin{split}\np_1(x,t)\n&=|\\overline{v}|^2\\psi(x,t)+ \\frac{3}{4\\pi} \\int \\partial_{y_i} \\partial_{y_j} \\left(\\frac{1}{|x-y|}\\right)(\\overline{v}_i \\overline{v}_j\\psi)(y,t) dy, \\\\\np_2(x,t) &= \\frac{3}{2\\pi} \\int \\frac{x_i-y_i}{|x-y|^3}\n(\\overline{v}_i\\overline{v}_j\\partial_j\\psi)(y,t) dy\n+ \\frac{3}{4\\pi} \\int \\frac{1}{|x-y|} (\\overline{v}_i\\overline{v}_j\\partial_i\\partial_j\\psi)(y,t) dy, \\\\\np_3(x,t) &= \\frac{3}{4\\pi} \\int \\frac{1}{|x-y|} (p\\Delta \\psi)(y,t) dy \n+ \\frac{3}{2\\pi} \\int \\frac{x_i-y_i}{|x-y|^3} (p\\partial_i\\psi)(y,t) dy.\n\\end{split}\n\\end{equation}\nWe notice that $p_1 + p_2$ depends only on $v$ and that there is no improvement in integrability of $p_3$ with respect to time even if $v$ has higher integrability in time, as is observed in Serrin's example in \\cite{SERR}.\n\n\\begin{lem}\n\\label{T31}\nIf $v \\in L^{\\infty}(t-r^2,t+r^2;L^{2q}(B(x,r)))$  for some $q \\in\n(1,\\infty)$, then\n\\[p_1 + p_2 \\in L^{\\infty} (t-r^2,t+r^2;L^{q}(B(x,r)))\\]\nsatisfies\n\\begin{equation}\n\\label{E33}\n\\sup_{|t-t_0|< r^2} \\norm{p_1 + p_2}_{L^q(B_r)} \\lesssim  \\sup_{|t-t_0|<r^2} \\norm{\\overline{v}}^2_{L^{2q}(B_r)}\n\\end{equation}\nand $p_3$ satisfies\n\\begin{equation}\n\\label{E34}\n\\begin{split}\n&\\norm{p_3}_{L^{3\/2}(t_0-\\rho^2,t_0+\\rho^2; L^{\\infty}(B_\\rho))} + (r-\\rho) \\norm{\\nabla p_3}_{L^{3\/2}(t_0-\\rho^2,t_0+\\rho^2; L^{\\infty}(B_\\rho) )} \\\\\n&\\lesssim \\frac{1}{|r-\\rho|^3}\n\\norm{p}_{L^{3\/2} (t_0-r^2,t_0+r^2; L^{1}(B_r\\setminus B_\\rho))}.\n\\end{split}\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nThe estimate \\eqref{E33} follows from the Calderon--Zygmund estimate for $p_1$ and potential estimates for $p_2$.\nOn the contrary, since $p_3$ is harmonic in $B(x,(\\rho+r)\/2)$, the mean value property gives \n\\[\\norm{p_3}_{L^{\\infty}(B(x,\\rho))} \n+ (r-\\rho) \\norm{\\nabla p_3}_{L^{\\infty}(B(x,\\rho))}\n\\lesssim \\frac{1}{|r-\\rho|^3} \\norm{p}_{L^{1}(B(x,r) \\setminus B(x,\\rho))}.\\]\nIntegrating with respect to time yields the estimate \\eqref{E34}.\n\\end{proof}\n\nWe modify localized energy inequality in terms of average free\nvelocity like Lemma 2.1 in Seregin\\cite{SER}, which was also known\nin parabolic equations (see also Choe \\cite{C} and Giaquinta--Struwe \\cite{GS}).\n\n\\begin{lem}\n\\label{T32}\nLet $\\phi \\in C^\\infty_{0}(B_r(x_0))$ and $\\theta\\in C^\\infty_0(t_0-r^2,t_0+r^2)$ be non-negative cutoff functions with $\\int\\phi dx =1$.\nIf we denote the weighted space average $[v]_{r}(t) = \\int  v(x,t)\\phi(x)dx$  and $\\bar{v} = v-[v]_r$, then \n\\begin{equation}\n\\label{E35}\n\\begin{split}\n&\\sup_t \\int |\\bar{v}|^2 \\phi\\theta dx \n+ 2 \\int |\\nabla v|^2 \\phi\\theta dxdt \\\\\n&\\le \\int |\\bar{v}|^2(\\Delta\\phi\\theta+\\phi\\theta_t) dxdt \n+ \\int v\\cdot\\nabla\\phi|\\bar{v}|^2 \\theta dxdt \n+ 2 \\int p \\bar{v}\\cdot\\nabla\\phi\\theta dxdt.\n\\end{split}\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nWe may assume $x_0=0$.\nIntegrating in space we find that\n\\[\\frac{d}{dt} [v]_r(t) =- \\int \\nabla v \\cdot \\nabla\\phi dx +\\int\nv\\cdot\\nabla\\phi \\bar{v} dx +\\int p \\nabla\\phi dx.\\]\nIn the localized energy inequality \\eqref{E24} we take a cutoff\nfunction $\\phi(x)\\theta(t)$. \nThen \n\\begin{align}\n\\label{E36}\n\\int |v|^2 \\phi\\theta(t_1) dx \n&= \\int_{B_r} |\\bar{v}|^2\\theta dx + |[v]_r|^2\\theta(t_1) \\\\\n\\label{E37}\n\\int_{t_0-r^2}^{t_1}\\int |v|^2 \\phi \\theta_t dxdt\n&= \\int_{t_0-r^2}^{t_1}\\int |\\bar{v}|^2 \\phi \\theta_t dxdt\n+ \\int_{t_0-r^2}^{t_1} |[v]_r|^2  \\theta_t dt.\n\\end{align}\nIntegrating by parts, we also have\n\\begin{equation}\n\\label{E38}\n\\begin{split}\n\\int_{t_0-r^2}^{t_1} |[v]_r|^2 \\theta_t dt \n&= |[v]_r|^2 \\theta(t_1) - 2 \\int_{t_0-r^2}^{t_1} [v]_r\\cdot\\frac{d}{dt}[v]_r \\theta dt \\\\\n&= |[v]_r|^2 \\theta(t_1) + 2 \\int_{t_0-r^2}^{t_1} \\int [v]_r \\cdot\\nabla {v}\\cdot \\nabla\\phi \\theta dxdt \\\\\n&\\quad - 2 \\int_{t_0-r^2}^{t_1} \\int v\\cdot \\nabla \\phi \\bar{v} \\cdot  [v]_r \\theta dx dt \n- 2 \\int_{t_0-r^2}^{t_1} \\int p [v]_r \\cdot \\nabla \\phi \\theta dx dt.\\end{split}\n\\end{equation}\nSince $t_1$ is arbitrary, combining \\eqref{E24}, \\eqref{E36}, \\eqref{E37} and \\eqref{E38}, we get an average free localized energy inequality.\n \\end{proof}\n\\begin{rem} In the weighted average free localized inequality \\eqref{E35}, we can replace\nthe weighted space average $[v]_r$ by the space average $(v)_r$.\n\\end{rem}\n\n\n\n\\section{Proof of Theorem \\ref{T11}} \n\\label{S4}\n\nWe begin by recalling the following iteration lemma, which can be found in M. Giaquinta \\cite{G}.\n\n\\begin{lem}\n\\label{T41}\nLet $f(r)$ be a non-negative bounded function on $[R_0, R_1] \\subset \\mathbb{R}_+$.\nIf there are non-negative constants $A, B, C$ and positive exponents $b<a$ and a parameter $\\theta \\in (0,1)$ such that for all $R_0 \\le r < R \\le R_1$\n\\[f(r) \\le \\theta f(R) + A(R-r)^{-a} + B(R-r)^{-b} + C,\\]\nthen for all $R_0 \\le r < R \\le R_1$\n\\[f(r) \\lesssim_{a,\\theta} A(R-r)^{-a} + B(R-r)^{-b} + C.\\]\n\\end{lem}\n\n\\begin{proof}\nFor reader's convenience, we give a sketch of the proof.\nFix $r$ and $R$.\nDefine $r_0=r$ and \n\\[r_{n+1} = r_n + (R-r) d_n\\]\nwhere $d_n$ is a sequence of positive numbers with $\\sum_{n=0}^{\\infty} d_n=1$ so that \n$\\lim_{n\\to\\infty} r_n = R$.\nFrom the condition \n\\[\\theta^n f(r_n) - \\theta^{n+1} f(r_{n+1}) \\le \\set{A(R-r)^{-a} d_n^{-a} + B(R-r)^{-b} d_n^{-b} + C} \\theta^n.\\]\nSince $d_n\\le1$ and $b<a$, we have $1 \\le d_n^{-b} \\le d_n^{-a}$ and so\n\\[f(r) \\le \\set{A(R-r)^{-a} + B(R-r)^{-b} + C} \\sum_{n=0}^{\\infty} \\theta^n d_n^{-a}.\\]\nIt suffices to find $d_n$ such that $\\sum_{n=0}^{\\infty} d_n=1$ and \n\\[\\sum_{n=0}^{\\infty} \\theta^n d_n^{-a} < \\infty.\\]\nFor example, we can choose $\\tau$ satisfying $\\theta < \\tau^a < 1$ and set $d_n = (1-\\tau) \\tau^n$.\n\\end{proof}\n\nIn order to neatly describe an iteration scheme, we introduce the following functionals. \n\n\\begin{defn} \nLet $z=(x,t)$ and $\\hat{v} = v - (v)_{R}$ and define \n\\begin{align*}\n\\mathcal{A}(z,R) &= \\sup_{|t|<R^2} \\frac{1}{R^5} \\int_{B(x,R)}|\\hat{v}|^2 dx \\\\\n\\mathcal{E}(z,R) &= \\fint_{Q(z,R)} |\\nabla v|^2 dz, \\\\\n\\mathcal{F}(z,R) &=\\left(\\fint_{Q(z,R)}|\\nabla v|^{9\/5} dz\\right)^{10\/9} \\\\\n\\mathcal{P}(z,R) &= \\fint_{Q(z,R)} |\\nabla p_3 \\cdot v| dz.\n\\end{align*}\n\\end{defn}\n\nNow, we are ready to prove Theorem \\ref{T11}.\nWe divide its proof into severl steps.\n\\begin{enumerate}[\\bf{Step} 1)]\n\\item\nWe may assume the reference point $z=0$ and suppress it for the notational convenience because one can easily see that all the estimates in the following proof does not depend on the reference point $z$.\nAssuming \\eqref{E12} for all $Q(z,r) \\subset \\Omega_T$ we shall prove first that $\\nabla v$ satisfies a reverse H\\\"older inequality.\nMore precisely, under the assumption \\eqref{E12} with $M\\ge1$, there exists an absolute positive constant $C$ such that\n\\begin{equation}\n\\label{E41}\n\\fint_{Q\/2}|\\nabla v|^2 dz\n\\le C M^{5\/9} \\left(\\fint_{Q} |\\nabla v|^{9\/5} dz\\right)^{10\/9}\n+ C \\fint_{Q} |\\nabla p_3\\cdot{v}| dz\n\\end{equation}\nfor every cube $Q \\subset \\Omega_T$.\nUsing the shorthand notation for functionals we rewrite it as \n\\begin{equation}\n\\label{E42}\n\\mathcal{E}(R\/2) \\le CM^{5\/9} \\mathcal{F}(R) + C\\mathcal{P}(R).\n\\end{equation}\nThe constant $C$ does not depend on $M$, $R$, and the suppressed reference point $z$.\n\\item\nFix $0<r<R$.\nChoose a smooth cutoff function $\\phi$ satisfying $\\phi(z)=1$ for $z \\in rQ$, $\\phi(z)=0$ for $z\\notin RQ$, and for all $k \\in \\set{0} \\cup \\mathbb{N}$\n\\[|\\nabla^k\\phi| \\lesssim (R-r)^{-k}, \\quad\n|\\nabla^k \\partial_t \\phi| \\lesssim (R-r)^{-k-2}.\\]\nFrom the localized energy inequality \\eqref{E35} replacing the weighted space average $[v]_r$ to space average $(v)_r$, we obtain\nthat\n\\begin{equation}\n\\label{E43}\n\\begin{split}\n\\mathcal{A}(r) + \\mathcal{E}(r)\n&\\lesssim (R-r)^{-2} \\fint_{RQ} |\\hat{v}|^2 dz\n+ (R-r)^{-1} \\fint_{RQ} |v| |\\hat{v}|^2 dz \\\\\n&\\quad + (R-r)^{-1} \\fint_{RQ} |p_1 + p_2| |\\hat{v}| dz\n+ \\fint_{RQ} |\\nabla p_3\\cdot\\hat{v}| dz \\\\\n&=: I_1 + I_2 + I_3 + \\mathcal{P}(R).\n\\end{split}\n\\end{equation}\n\\item\nWe estimate the term $I_1$.\nWe use the Sobolev inequality to get \n\\begin{align*}\nI_1 \n&= (R-r)^{-2} \\fint_{RQ} |\\hat{v}|^2 dz \\\\\n&\\lesssim (R-r)^{-2} R^{-1\/2} R^{-9\/2} \\int_{R^2I} \\left(\\int_{RB} |\\hat{v}|^2 dx\\right)^{1\/10} \\left(\\int_{RB} |\\hat{v}|^2 dx\\right)^{9\/10} dt \\\\\n&\\lesssim (R-r)^{-2} \\mathcal{A}(R)^{1\/10} R^{-1\/2} \\int_{R^2I} \\left(\\int_{RB} |\\nabla v|^{6\/5} dx\\right)^{3\/2} dt.\n\\end{align*}\nBy the Jensen inequality \n\\[\\int_{R^2I} \\left(\\int_{RB} |\\nabla v|^{6\/5} dx\\right)^{3\/2} dt \n\\lesssim R^{3\/2} \\int_{RQ} |\\nabla v|^{9\/5} dx \n\\lesssim R^{13\/2} \\mathcal{F}(R)^{9\/10}.\\]\nHence, we get by the Young inequality \n\\begin{equation}\n\\label{E44}\n\\begin{split}\nI_1 \n&\\lesssim (R-r)^{-2} R^2 \\mathcal{A}(R)^{1\/10} \\mathcal{F}(R)^{9\/10} \\\\\n&\\lesssim \\varepsilon \\mathcal{A}(R) + \\varepsilon^{-1\/9}  (R\/(R-r))^{20\/9} \\mathcal{F}(R).\n\\end{split}\n\\end{equation}\n\\item\nWe estimate the term $I_2$.\nUsing the assumption \\eqref{E12} and the Sobolev inequality, we obtain\n\\begin{align*}\n\\int_{RB} |v| |\\hat{v}|^2 dx\n&\\le \\left(\\int_{RB} |v|^2 dx\\right)^{1\/2} \\left(\\int_{RB} |\\hat{v}|^2 dx\\right)^{1\/10}\n\\left(\\int_{RB} |\\hat{v}|^{9\/2} dx\\right)^{2\/5} \\\\\n&\\lesssim (R M)^{1\/2} \\left(\\int_{RB} |\\hat{v}|^2\ndx\\right)^{1\/10} \\int_{RB} |\\nabla v|^{9\/5} dx.\n\\end{align*}\nIntegrating in time, we get\n\\begin{align*}\nI_2 \n&= (R-r)^{-1} \\fint_{RQ} |v| |\\hat{v}|^2 dz \\\\\n&\\lesssim (R-r)^{-1} R M^{1\/2} \\mathcal{A}(R)^{1\/10} \\fint_{RQ} |\\nabla v|^{9\/5} dz \\\\\n&\\lesssim (R-r)^{-1} R M^{1\/2} \\mathcal{A}(R)^{1\/10} \\mathcal{F}(R)^{9\/10}.\n\\end{align*}\nHence, we get, by the Young inequality, \n\\begin{equation}\n\\label{E45}\nI_2 \\lesssim \\epsilon \\mathcal{A}(R) + \\epsilon^{-1\/9} (R\/(R-r))^{10\/9} M^{5\/9} \\mathcal{F}(R).\n\\end{equation}\n\\item\nWe estimate the term $I_3$.\nUsing the inequality \\eqref{E33} in Lemma \\ref{T31} in which $v$ is replaced by $\\hat{v}$, we obtain\n\\begin{align*}\nI_3\n&\\lesssim (R-r)^{-1} \\left(\\fint_{RQ} |p_1 + p_2|^{3\/2} dz\\right)^{2\/3}\n\\left(\\fint_{RQ} |\\hat{v}|^3 dz\\right)^{1\/3} \\\\\n&\\lesssim (R-r)^{-1} \\fint_{RQ} |\\hat{v}|^3 dz.\n\\end{align*}\nHence, the estimate for $I_3$ is exactly the same as the estimate for $I_2$.\n\\item\nCombining the estimates \\eqref{E44} and \\eqref{E45} for $I_1$, $I_2$, and $I_3$, we obtain that \n\\begin{align*}\n\\mathcal{A}(r) + \\mathcal{E}(r)\n&\\lesssim  \\epsilon \\mathcal{A}(R) + \\varepsilon^{-1\/9}  (R\/(R-r))^{20\/9} \\mathcal{F}(R) \\\\\n&\\quad + \\epsilon^{-1\/9} (R\/(R-r))^{10\/9} M^{5\/9} \\mathcal{F}(R) + \\mathcal{P}(R).\n\\end{align*}\nBy Lemma \\ref{T41} we conclude that for all $0<r<R \\le 2$\n\\begin{align*}\n\\mathcal{A}(r) + \\mathcal{E}(r)\n&\\lesssim (R\/(R-r))^{20\/9} \\mathcal{F}(R) \\\\\n&\\quad + (R\/(R-r))^{10\/9} M^{5\/9} \\mathcal{F}(R) + \\mathcal{P}(R).\n\\end{align*}\nSince $M \\ge 1$, we get the result \\eqref{E42} by choosing $r=R\/2$.\n\\item\nWe improve integrability of $\\nabla v$ using the method in Kinnunen--Lewis \\cite{KL}. \nThe Calderon--Zygmund stopping time argument plays an important role in proving the reverse H\\\"older inequality.\nWe refer the reader Stein's book \\cite{S}.\nWe take the cutoff function $\\psi\\equiv 1$ in $7\/4B$ in the decomposition of pressure $p=p_1 + p_2+p_3$ of \\eqref{E24}.\nIf we set\n\\[h(t)=\\int_{\\{|\\nabla v|^{9\/5}>t\\}\\cap Q_2} \\dist(z,\\partial 3\/2Q)^{9\/2}|\\nabla v|^{9\/5}dz\\]\nfor the modified distance function\n\\[\\dist(z,\\partial 3\/2Q) = \\inf_{w\\in\\partial 3\/2Q} \\set{|z-w|,1\/4},\\]\nwith \\eqref{E42} and $q=10\/9$ in Proposition 5.1 of\nGiaquinta--Modica\\cite{GM}, we obtain\n\\[\\norm{\\nabla v}_{L^{2+\\delta}(Q)}\n\\leq \\overline{C} \\norm{\\nabla v}_{L^{2}(3\/2Q)} +\n\\overline{C}\\left(\\int_{3\/2Q} |\\nabla p_3\\cdot v|^{(2+\\delta)\/2}\ndz\\right)^{1\/(2+\\delta)}.\\] Since $p_3$ is harmonic in $3\/2B$, the mean value property gives\n\\[\\sup_{x\\in 3\/2B} |\\nabla p_3| \\lesssim \\fint_{2B} |p_3| dx.\\]\nBy Jensen's inequality and Young's inequality we obtain that\n\\begin{align*}\n\\int_{3\/2Q} |\\nabla p_3\\cdot v|^{(2+\\delta)\/2} dz\n&\\le \\int_{-9\/4}^{9\/4} \\sup_{x\\in 3\/2B} |\\nabla p_3|^{(2+\\delta)\/2}\n\\int_{3\/2B} |v|^{(2+\\delta)\/2} dx dt \\\\\n&\\lesssim \\int_{-4}^4 \\left(\\fint_{2B} |p_3| dx\\right)^{(2+\\delta)\/2} \\left(\\fint_{2B} |v|^{(2+\\delta)\/2} dx\\right) dt \\\\\n&\\lesssim \\int_{-4}^4 \\left(\\fint_{2B} |p_3|^{3\/2} dx\\right)^{(2+\\delta)\/3} \\left(\\fint_{2B} |v|^{3(2+\\delta)\/2(1-\\delta)} dx\\right)^{(1-\\delta)\/3} dt \\\\\n&\\lesssim \\norm{p}_{L^{3\/2}(2Q)}^{1+\\delta\/2}\n\\norm{v}_{L^{3(2+\\delta)\/2(1-\\delta)}(2Q)}^{1+\\delta\/2}.\n\\end{align*}\n\nRecall that $p\\in L^{3\/2}$ and $v\\in L^{10\/3}$. If\n$\\delta<\\frac{2}{29}$, then we have\n\\[\\frac{3(2+\\delta)}{2(1-\\delta)} < \\frac{10}{3}\\]\nand the righthand side is bounded. \n\\end{enumerate}\n\nThis completes the proof of Theorem \\ref{T11}.\n\n\\begin{rem}\nSeregin \\cite{SER} obtained a different version of reverse H\\\"older inequalities under the assumption $v = \\divg b$ and $b \\in L^{\\infty}(0,T; \\BMO)$. \n\\end{rem}\n\n\n\n\n\\section{Proof of Theorem \\ref{T12}}\n\\label{S5}\n\nWe begin by recalling a very well-known lemma about the Hausdorff measure of an upper density of a locally integrable function.\n\n\\begin{lem}\n\\label{T51}\nLet $f \\in L_{\\loc}^1(\\mathbb{R}^d)$ and $0 < \\alpha < d$.\nDenote \n\\[E_\\alpha(x,r) = r^{-\\alpha} \\int_{B(x,r)} |f| dy,\\]\nThen \n\\[\\mathcal{H}^\\alpha\\set{\\limsup_{r\\to0} E_\\alpha(x,r) > 0}=0.\\]\n\\end{lem}\n\n\\begin{proof}\nFor reader's convenience, we give a sketch of the proof.\nFix a compact set $K$ in an open unit cube $Q$ and $n \\in \\mathbb{N}$.\nSet\n\\[F_n = K \\cap \\set{\\limsup_{r\\to0} E_\\alpha(x,r) > 1\/n}.\\]\nFix $\\delta<\\dist(K,Q^c)$.\nFor each $x \\in F_n$ there exists $r<\\delta\/5$ such that \n\\[E_\\alpha(x,r) > 1\/(2n).\\]\nBy Vitali's covering lemma there exists countable disjoint balls $B(x_j,r_j)$ such that $F_n \\subset \\bigcup B(x_j,5r_j)$.  \nSince\n\\[\\sum r_j^\\alpha \\lesssim \\int_{\\bigcup B(x_j,r_j)} |u| dy\\]\nand \n\\[\\sum r_j^d \\lesssim \\delta^{d-\\alpha} \\sum r_j^\\alpha \\lesssim \\delta^{d-\\alpha} \\int_Q |u| dy,\\]\nwe have $\\mathcal{H}^\\alpha(F_n) = 0$.\nSince $K$ and $n$ are arbitrary, we get the result.\n\\end{proof}\n\nWe need one more elementary lemma.\n\n\\begin{lem}\n\\label{T52}\nThe condition \n\\begin{equation}\n\\label{E51}\nN=\\norm{v}_{L^\\infty(0,T; L^{3,w}(\\Omega))}<\\infty,\n\\end{equation}\nimplies that for almost all $0 \\le t \\le T$ and for all $0 < q < 3$, $x \\in \\mathbb{R}^3$, and $0<R<\\infty$\n\\begin{equation}\n\\label{E52}\nR^{q-3} \\int_{B(x,R)} |v(y,t)|^q dy \\lesssim \\left(\\frac{q}{3-q}\\right)^{q\/3} N^q.\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nWe denote the level set of $v$ at the height $h \\in \\mathbb{R}_+$ by \n\\[E_t(h) = \\set{y\\in\\Omega:h<|v(y,t)|}.\\]\nFrom the condition \\eqref{E51} we have for almost all $0 \\le t \\le T$ and all $h \\in \\mathbb{R}_+$ \n\\[h^3 \\boldsymbol m(E_t(h)) \\le N^3.\\]\nSince we have for all $H \\in \\mathbb{R}_+$\n\\begin{align*}\n\\int_{B(x,R)} |v(y,t)|^q dy \n&= q \\int_{0}^{\\infty} h^{q-1} \\boldsymbol m [B(x,R) \\cap E_t(h)] dh \\\\\n&\\lesssim q \\int_0^H h^{q-1} R^3 dh + q \\int_H^\\infty h^{q-4} N^3 dh \\\\\n&\\lesssim R^3 H^q + \\frac{q}{3-q} N^3 H^{q-3},\n\\end{align*}\nwe obtain the estimate \\eqref{E52} by taking \n\\[H = \\left(\\frac{q}{3-q}\\right)^{1\/3} NR^{-1}.\\]\n\\end{proof}\n\nNow, we are ready to prove Theorem \\ref{T12}, which is, in fact, a direct consequence of Lemma \\ref{T51} and Lemma \\ref{T52}.\nFrom the estimate \\eqref{E52} with $q=2$, we can apply Theorem \\ref{T11} so that  \n\\begin{equation}\n\\label{E53}\n\\nabla v \\in L_{\\loc}^{2+\\delta}(\\Omega_T)\n\\end{equation}\nfor some $\\delta>0$.\nBy H\\\"older's inequality we have \n\\[r^{-1} \\int_{Q(z,r)} |\\nabla v|^2 dz \\lesssim \\left(r^{-1+2\\delta} \\int_{Q(z,r)} |\\nabla v|^{2+\\delta} dz\\right)^{2\/(2+\\delta)}.\\]\nThe Caffarelli--Kohn--Nirenberg regularity theorem \\cite{CKN} implies that \n\\[\\mathcal{S} \\subset \\set{z : \\limsup_{r\\to0} r^{-1+2\\delta} \\int_{Q(z,r)} |\\nabla v|^{2+\\delta} dz > 0}.\\]\nTherefore, $\\mathcal{H}^{1-2\\delta}(\\mathcal{S})=0$ by Lemma \\ref{T51}.\nThis completes the proof of Theorem \\ref{T12}.\n\n\n\\section{Proof of Theorem \\ref{T13}}\n\\label{S6}\n\nWe divide the proof several steps.\n\n\\begin{enumerate}[\\bf{Step} 1)]\n\\item\nWe first claim that for any $\\gamma \\in (0,\\delta\/10)$ and $z=(x,t) \\in \\Omega_T$\n\\begin{equation}\n\\label{E61}\nv \\in L^{4+2\\delta-\\gamma}(Q(z,R)).\n\\end{equation}\nMore precisely, we shall show that \n\\begin{equation}\n\\label{E62}\n\\int_{Q(z,R)} |v|^{4+2\\delta-\\gamma} dz \\lesssim_{\\delta,\\gamma} R^{\\gamma} N^{2+\\delta-\\gamma} \\int_{Q(z,R)} |\\nabla v|^{2+\\delta} dz + R^{1-2\\delta+\\gamma} N^{4+2\\delta-\\gamma}\n\\end{equation}\nwhere the implied constant can be found explicitly.\nIn order to see this, we use H\\\"older's inequality to write \n\\begin{align*}\n&\\int_{B(x,R)} |v|^{4+2\\delta-\\gamma} dy \n=\\int_{B(x,R)} |v|^{2+\\delta-\\gamma} |v|^{2+\\delta} dy \\\\\n&\\le \\left(\\int_{B(x,R)} |v|^{3-3\\gamma\/(2+\\delta)} dy\\right)^{(2+\\delta)\/3}\n\\left(\\int_{B(x,R)} |v|^{3(2+\\delta)\/(1-\\delta)} dy\\right)^{(1-\\delta)\/3}.\n\\end{align*}\nUsing \\eqref{E52} with $q=3-3\\gamma\/(2+\\delta)$, we estimate the first integral on the right as \n\\[\\left(\\int_{B(x,R)} |v|^{3-3\\gamma\/(2+\\delta)} dy\\right)^{(2+\\delta)\/3} \\lesssim \\left(\\frac{2+\\delta}{\\gamma}+\\frac{\\gamma}{2+\\delta}-2\\right)^{(2+\\delta)\/3} R^{\\gamma} N^{2+\\delta-\\gamma}.\\]\nThus, we have \n\\[\\int_{B(x,R)} |v|^{4+2\\delta-\\gamma} dy \\lesssim R^{\\gamma} N^{2+\\delta-\\gamma} \\left(\\int_{B(x,R)} |v|^{3(2+\\delta)\/(1-\\delta)} dy\\right)^{(1-\\delta)\/3}.\\]\nSince $3(2+\\delta)\/(1-\\delta)$ is the Sobolev exponent of $2+\\delta$, we apply the Sobolev--Poincar\\'e inequality to get \n\\begin{align*}\n&\\left(\\int_{B(x,R)} |v|^{3(2+\\delta)\/(1-\\delta)} dy\\right)^{(1-\\delta)\/3} \\\\\n&\\lesssim \\left(\\int_{B(x,R)} |v-(v)_R|^{3(2+\\delta)\/(1-\\delta)} dy\\right)^{(1-\\delta)\/3} + R^{1-\\delta} |(v)_R|^{2+\\delta} \\\\\n&\\lesssim \\int_{B(x,R)} |\\nabla v|^{2+\\delta} dy + R^{1-\\delta} |(v)_R|^{2+\\delta}\n\\end{align*}\nwhere $(v)_R = \\fint_{B(x,R)} v(y,t) dy$.\nFinally, using the Jensen inequality and \\eqref{E52} with $q=2+\\delta$, we obtain \n\\[R^{1-\\delta} |(v)_R|^{2+\\delta} \\lesssim \\left(\\frac{2+\\delta}{1-\\delta}\\right)^{(2+\\delta)\/3} N^{2+\\delta} R^{-1-2\\delta}.\\]\nIntegrating in time yields the estimate \\eqref{E62}.\n\\item\nWe fix $\\psi$ satisfying $\\psi(y) = 1$ for $y \\in B(x,R\/2)$ and $\\psi(y) = 0$ for $y \\notin B(x,R)$.\nWe now use the decomposition of a localized pressure \\eqref{E31} and \\eqref{E32} with this $\\psi$, that is,\n\\[p\\psi = p_1 + p_2 + p_3\\]\nwhere \n\\begin{align*}\np_1(x,t)\n&=|\\overline{v}|^2\\psi(x,t)+ \\frac{3}{4\\pi} \\int \\partial_{y_i} \\partial_{y_j} \\left(\\frac{1}{|x-y|}\\right)(\\overline{v}_i \\overline{v}_j\\psi)(y,t) dy, \\\\\np_2(x,t) &= \\frac{3}{2\\pi} \\int \\frac{x_i-y_i}{|x-y|^3}\n(\\overline{v}_i\\overline{v}_j\\partial_j\\psi)(y,t) dy\n+ \\frac{3}{4\\pi} \\int \\frac{1}{|x-y|} (\\overline{v}_i\\overline{v}_j\\partial_i\\partial_j\\psi)(y,t) dy, \\\\\np_3(x,t) &= \\frac{3}{4\\pi} \\int \\frac{1}{|x-y|} (p\\Delta \\psi)(y,t) dy \n+ \\frac{3}{2\\pi} \\int \\frac{x_i-y_i}{|x-y|^3} (p\\partial_i\\psi)(y,t) dy.\n\\end{align*}\nWe notice that $p_1$ and $p_2$ involve $v$ only.\nSince we have $v \\in L^{4+2\\delta-\\gamma}(Q(z,R))$, \\eqref{E61} in the previous step, we see that\n\\begin{equation}\n\\label{E63}\np_1 + p_2 \\in L^{2+\\delta-\\gamma\/2}(Q(z,R\/5))\n\\end{equation}\nby a direct consequence of $L^q$-continuity of singular integral operators and potential estimates. \nOn the contrary, the representation of $p_3$ strongly depends on the outward data of $p$, so we don't expect that $p_3$ gain such a higher integrability in time.\nBut, since $p_3$ is harmonic in $B(x,R\/4)$, we have \n\\begin{equation}\n\\label{E64}\np_3 \\in L^{5\/3}(t-R^2\/25,t+R^2\/25 : L^\\infty(B(x,R\/5))).\n\\end{equation}\n\\item\nIn the previous steps we showed that the weak solution is locally higher integrable.\nConsidering scaled functional with higher exponent, we can get a better bound for the size of the singular set.\nIndeed, it is a consequence of scaling structure of weak solutions.\nWe have for all $0 < r < R\/5$, by H\\\"older's inequality, \n\\begin{align*}\nr^{-2} \\int_{Q(z,r)} |v|^3 dz \n&\\lesssim \\left(r^{-1+2\\delta-\\gamma} \\int_{Q(z,r)} |v|^{4+2\\delta-\\gamma} dz\\right)^{3\/(4+2\\delta-\\gamma)}, \\\\\nr^{-2} \\int_{Q(z,r)} |p_1+p_2|^{3\/2} dz \n&\\lesssim \\left(r^{-1+2\\delta-\\gamma} \\int_{Q(z,r)} |p_1+p_2|^{2+\\delta-\\gamma\/2} dz\\right)^{3\/(4+2\\delta-\\gamma)}, \n\\end{align*}\nand, by Jensen's inequality,  \n\\begin{align*}\nr^3 \\fint_{Q(z,r)} |p_3|^{3\/2} dz \n&\\lesssim r^3 \\fint_{t-r^2}^{t+r^2} \\left(\\fint_{B(x,r)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/10} dt \\\\\n&\\lesssim r^3 \\left(\\fint_{t-r^2}^{t+r^2} \\left(\\fint_{B(x,r)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/9} dt\\right)^{9\/10} \\\\\n&= \\left(r^{-1+2\\delta-\\gamma} \\int_{t-r^2}^{t+r^2} \\left(\\int_{B(x,r)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/9} dt\\right)^{9\/10}.\n\\end{align*}\nCombining those estimates, we obtain \n\\begin{align*}\n&r^{-2} \\int_{Q(z,r)} |v|^3 +|p|^{3\/2} dz \\\\\n&\\lesssim \\left(r^{-1+2\\delta-\\gamma} \\int_{Q(z,r)} |v|^{4+2\\delta-\\gamma} + |p_1+p_2|^{2+\\delta-\\gamma\/2} dz\\right)^{3\/(4+2\\delta-\\gamma)} \\\\\n&\\quad + \\left(r^{-1+2\\delta-\\gamma} \\int_{t-r^2}^{t+r^2} \\left(\\int_{B(x,r)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/9} dt\\right)^{9\/10}.\n\\end{align*}\nThere exists a positive number $\\varepsilon$ such that if \n\\[\\left(\\int_{Q(z,r)} |v|^{4+2\\delta-\\gamma} + |p|^{2+\\delta-\\gamma\/2} dz + \\int_{t-r^2}^{t+r^2} \\left(\\int_{B(x,r)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/9} dt\\right) < \\varepsilon r^{1-2\\delta+\\gamma}\\]\nfor some $r < R\/5$, then $z$ is a regular point by the well-known $L^3$-regularity criterion.\n\\item\nIn the previous setp, we obtain that for each $z=(x,t) \\in \\mathcal{S} \\cap Q(z_0,R\/10)$ we should have for all $0 < r < R\/50$ \n\\begin{equation}\n\\label{E65}\n\\begin{split}\n\\varepsilon r^{1-2\\delta+\\gamma} \n&\\le \\int_{Q(z,r)} |v|^{4+2\\delta-\\gamma} + |p|^{2+\\delta-\\gamma\/2} dz \\\\\n&\\quad + \\int_{t-r^2}^{t+r^2} \\left(\\int_{B(x,r)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/9} dt.\n\\end{split}\n\\end{equation}\nFix $r < R\/50$ and consider the covering $\\set{Q(z,r) : z \\in \\mathcal{S} \\cap Q(z_0,R\/10)}$.\nAdapting the argument in \\cite{MR2864801}, we can choose a finite disjoint sub-family \n\\[\\set{Q(z_j,r) : j=1,2,\\dots,J(r)}\\]\nsuch that $\\mathcal{S} \\cap Q(z_0,R\/10) \\subset \\bigcup Q(z_j,5r)$ by the Vitali covering lemma.\nWe need to pay more attention in order to use the disjointness of the subcovering.\nSumming the inequality \\eqref{E65} at $z_j$ for $j=1,2,\\dots,J$ yields\n\\begin{align*}\nJ \\varepsilon r^{1-2\\delta+\\gamma}\n&\\le \\sum_{j=1}^{J} \\int_{Q(z_j,r)} |v|^{4+2\\delta-\\gamma} + |p|^{2+\\delta-\\gamma\/2} dz \\\\\n&\\quad + \\sum_{j=1}^{J} \\int_{t_0-R^2\/25}^{t_0+R^2\/25} \\left(\\int_{B(x_j,r)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/9} dt \\\\\n&\\le \\int_{Q(z_0,R\/5)} |v|^{4+2\\delta-\\gamma} + |p|^{2+\\delta-\\gamma\/2} dz \\\\\n&\\quad + \\int_{t_0-R^2\/25}^{t_0+R^2\/25} \\sum_{j=1}^{J} \\left(\\int_{B(x,r)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/9} dt \n\\end{align*}\nFor the last sum, we use Young's inequality, that is, for $\\theta \\in [0,1]$ \n\\[\\sum_{j=1}^{J} a_j^\\theta \\le J^{1-\\theta} \\left(\\sum_{j=1}^{J} a_j\\right)^\\theta.\\]\nDue to the fact that $p_3$ is harmonic in $B(z_0,R\/5)$ and the estimates \\eqref{E63} and \\eqref{E64}, there are positive numbers $A_1, A_2 \\in \\mathbb{R}_+$ such that \n\\begin{equation}\n\\label{E66}\n\\begin{split}\nJ \\varepsilon r^{1-2\\delta+\\gamma}\n&\\le \\int_{Q(z_0,R\/5)} |v|^{4+2\\delta-\\gamma} + |p|^{2+\\delta-\\gamma\/2} dz \\\\\n&\\quad + J^{(2+6\\delta-3\\gamma)\/9} \\int_{t_0-R^2\/25}^{t_0+R^2\/25} \\left(\\int_{B(x_0,R\/5)} |p_3|^{15\/(7-6\\delta+3\\gamma)} dx\\right)^{(7-6\\delta+3\\gamma)\/9} dt \\\\\n&=: A_1+A_2 J^{(2+6\\delta-3\\gamma)\/9}.\n\\end{split}\n\\end{equation}\n\\item\nThe minimum number of parabolic cylinders $Q(z,r)$ required to cover the set $\\mathcal{S} \\cap Q(z,R\/5)$ is less than or equal to $J$.\nWe rewrite \\eqref{E66} as \n\\[J \\varepsilon r^{C_1} \\le A_1 + A_2 J^{C_2}\\]\nwhere $C_1=1-2\\delta+\\gamma$ and $C_2=(2+6\\delta-3\\gamma)\/9$.\nApplying Young's ineaulity to $A_2 J^{C_2}$, we get \n\\[J \\varepsilon r^{C_1} \\le A_1 + (1-C_2) [A_2 (\\varepsilon r^{C_1})^{-C_2}]^{1\/(1-C_2)} + C_2 J \\varepsilon r^{C_1}\\]\nand so \n\\[J(r) \\lesssim A_1 \\varepsilon^{-1} r^{-C_1} + A_2^{1\/(1-C_2)} \\varepsilon^{-1\/(1-C_2)} r^{-C_1\/(1-C_2)}.\\]\nTherefore, by an elementary calculation we get \n\\[\\limsup_{r\\to0} \\frac{\\log J(r)}{-\\log r} = \\frac{C_1}{1-C_2} = \\frac{9-18\\delta+9\\gamma}{7-6\\delta+3\\gamma}.\\]\nSince $\\delta$ and $\\gamma$ can be arbitrarily close to $\\delta_0$ and $0$ respectively, we conclude the assertion \\eqref{E15}.\n\\end{enumerate}\nThis completes the proof of Theorem \\ref{T13}.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nObservations at astronomical and cosmological scales indicate that a majority of the matter content of our Universe is in the form of non-relativistic, long-lived, and non-luminous dark matter~\\cite{PDC, GalaxyFormation, LargeScaleStructure, Clowe}. Extensions of the Standard Model favour a candidate for dark matter in the form of a Weakly Interacting Massive Particle (WIMP)~\\cite{Steigman, Jungman}. Its interaction with normal matter can be probed directly via elastic scattering off target nuclei, thus motivating searches through direct detection \\cite{Goodman:1984dc}. The XENON collaboration has constructed and commissioned the first ton-scale liquid-xenon dark matter detector, aiming to observe primarily low-energy nuclear recoils of WIMPs with unprecedented sensitivity.\n\nXENON1T, a dual-phase time projection chamber (TPC)~\\cite{Bolozdynya, Xe1T_TPC}, was designed to improve the sensitivity of its predecessor, XENON100~\\cite{xenon_xe100-SI, xenon_xe100-SD}, by two orders of magnitude for the spin-independent WIMP-nucleon interaction cross section. The increased sensitivity is achieved through reducing the background and increasing the target mass, i.e. the amount of liquid xenon (LXe) enclosed by the TPC, by a factor of 32, allowing for a sensitive volume, after fiducialization, of $\\sim$1 ton. In rare-event searches, understanding and minimizing background events that occur within the sensitive volume of the detector is of utmost importance. This necessitates the use of construction materials with low intrinsic radioactivity, passive and active detector shielding, and sophisticated analysis techniques in order to prevent background events within the parameter space where a WIMP signal is expected. \n \nThe XENON1T radioassay program addresses backgrounds that come from radioactive impurities within detector construction materials. Radioassay of candidate materials provides information about the type and amount of expected emissions, thus allowing for selection and strategic placement of the most radiopure materials within the detector. The measured results provide the material-induced radiogenic component to the overall background model of XENON1T. Through Monte Carlo simulations using the activities from each component, accurate predictions of the detector sensitivity were performed~\\cite{xenon_xe1t-sensitivity}.\n\nHere we present an overview of the screening and material selection process for XENON1T. Section~\\ref{sec:bkgrd} describes expected background sources and reduction methods. The techniques and instruments used to identify and estimate radioimpurities of each sample are detailed in Section~\\ref{sec:techniques}. Section~\\ref{sec:results} describes the various materials and components that were screened with respect to the decay chains and isotopes that are of greatest concern. For each relevant decay chain and single-line emission, the radioassay results are presented in Table~\\ref{table:Samples}. We summarize the results in Section~\\ref{sec:impact} with a discussion of the impact on the XENON1T sensitivity with respect to the materials measured in this study.\n\n\n\n\\section{Background expectation and minimization}\n\\label{sec:bkgrd}\n\n\nParticle interactions with either atomic electrons or nuclei of the xenon target result in electronic recoil (ER) events or nuclear recoil (NR) events, respectively. The nuclear recoil background, predominantly from neutrons, is the most dangerous, as the signature of a WIMP is a single-scatter, NR event. Background discrimination and rejection techniques include removing multiple-scatter events, fiducializing the target volume through event vertex reconstruction, and exploiting the difference in energy loss per unit track length between ER and NR events~\\cite{xe100analysis}. However, ER events that occur within the fiducial volume can leak into the NR region of the WIMP discrimination parameter space as well as obscure other rare event searches that are otherwise possible in the ER channel. The aim, therefore, is to mitigate sources of both types of backgrounds and to accurately estimate the number of expected background events within the WIMP search region. \n\nExternal background from cosmic rays, i.e.~hadronic components and muon-induced neutrons, is suppressed by operating the detector at an average depth of 3600 meters water equivalent in the Laboratori Nazionali del Gran Sasso (LNGS), thus reducing the muon flux by a factor of 10$^{6}$ relative to a flat overburden~\\cite{LNGS_muonflux}. A water shield instrumented with veto PMTs surrounds the detector by at least 4 meters on all sides to provide passive shielding and to reject coincident events detected via Cherenkov radiation~\\cite{xenon_xe1t-MuonVeto}. Solar neutrinos are another potential source of external background, both ER and NR, the latter from coherent neutrino-nucleus scattering. \n\nSources of ER background that are intrinsic to the xenon target, e.g. the beta emitter $^{85}$Kr, and the double-beta emitter $^{136}$Xe, are expected to be uniformly distributed throughout the xenon, thus cannot be reduced through fiducialization. However $^{85}$Kr can be significantly reduced through distillation from $^{\\mathrm{nat}}$Kr to $<0.2$ ppt~\\cite{Kr_column}, and $^{136}$Xe, comprising 8.9$\\%$ of natural xenon, has a subdominant contribution of $<2\\%$ to the total ER background~\\cite{xenon_xe1t-sensitivity}. Although not natively intrinsic to the scintillator, the noble gas $^{222}$Rn (T$_{1\/2}=$3.8 d), originating from the long-lived $^{226}$Ra (T$_{1\/2}=$1600 yr), mixes with the xenon and becomes homogeneously distributed within the target. Beta decays of its daughter isotopes are the dominant source of ER background. Moreover, $^{214}$Pb and the daughter isotopes from its decay to ground state adhere to material surfaces (plate-out) and can lead to \\mbox{($\\alpha$, n)} reactions within the target volume. Because of plate-out effects from both parent and daughter isotopes of $^{222}$Rn, the level of contamination for this isotope is determined by directly measuring its emanation from construction materials. This technique will be described in a separate publication~\\cite{RnEmanation}. Additionally, a significant reduction in radon by online purification has recently been demonstrated by the XENON collaboration through the use of a cryogenic distillation technique~\\cite{RnDistillation}.\n\nThe radioassay program described in this paper targets the background from radionuclei present as residual traces in the detector components. The most common radioactive contaminants are long-lived (T$_{1\/2} >$ 1 yr) primordial radionuclei within the $^{238}$U and $^{232}$Th decay chains and the single isotope $^{40}$K. The latter isotope as well as several isotopes within the primordial chains decay via high-energy gamma emissions that cannot be completely removed through fiducialization. Additionally, several isotopes belonging to these chains release neutrons either directly through spontaneous fission of heavy nuclei or indirectly via \\mbox{($\\alpha$, n)} reactions following alpha decays within the chains. \n\nIn addition to primordial radioisotopes, anthropogenic radioisotopes, such as $^{137}$Cs and $^{110\\mathrm{m}}$Ag, can be found in some detector materials. These isotopes are either manufactured for medical or industrial use or are generated from nuclear power plant waste, nuclear accidents, or military testing. Cosmogenic isotopes, such as $^{54}$Mn, $^{46}$Sc, and $^{56-58}$Co, can be found mainly in metal components as a result of activation from exposure to cosmic rays~\\cite{cosmogenic}. An additional common radionuclide is $^{60}$Co, which is primarily anthropogenic in origin in stainless steel and cosmogenically induced in copper. Most of the listed radionuclei, including many of the isotopes within the primordial decay chains, can be detected with high sensitivity by the XENON1T radioassay techniques described in Section~\\ref{sec:techniques}.\n\n\n\\section{Techniques and measurements}\n\\label{sec:techniques}\n\nTo determine the amount and isotopic composition of radionuclides present in the XENON1T materials, gamma-ray spectroscopy and mass spectrometry methods were used. The former provides a non-destructive technique sensitive to almost every relevant gamma emitter and allows to detect a break of secular equilibrium within the primordial decay chains. To reach the detection sensitivity required by current dark matter search experiments, e.g. at or below the $\\mu$Bq\/kg level for some materials, large sample masses ($\\sim10$$-$$20\\,$kg) and long counting times ($\\sim15$$-$$20\\,$d) are usually necessary. This is particularly the case for low-energy gamma emitters due to self-absorption within the sample volume.\n\nMass spectrometry, in particular, Inductively Coupled Plasma Mass Spectrometry (ICP-MS) and Glow Discharge Mass Spectrometry (GDMS) were used to assess the composition of a given sample through separation and measurement of individual isotopes. This is particularly useful in determining the amount of $^{238}$U and $^{232}$Th within materials. Because ICP-MS and GDMS require just a few grams of sample material and short measurement times, they are also used when the mass of the sample is too small or the available measurement time too short to achieve the desired sensitivity in an HPGe spectrometer.\n\n \n\\subsection{Germanium spectrometers}\n\\label{sec:Ge}\n\nThe XENON collaboration utilizes several of the world's most sensitive germanium spectrometers, the Gator~\\cite{gator} detector and the four GeMPI detectors~\\cite{gempi}, that are located in ultra-low background facilities at LNGS at the same depth as the XENON1T detector. These spectrometers have an excellent energy resolution over the energy range of interest (\\mbox{$\\sim50 - 2650\\,$ keV} with, e.~g. \\mbox{$<3$ keV} FWHM at \\mbox{1332 keV}) and can reach sensitivities down to the $\\mu$Bq\/kg level. All detectors are p-type, intrinsically pure germanium crystals in a coaxial configuration, with masses between \\mbox{2.2$-$2.3 kg} and enclosed in a low-background cryostat housing. The sensitive region of the cryostat protrudes into an inner chamber made of electro-refined, Oxygen-Free High-Conductivity (OFHC) copper, with a material sample capacity of several liters in volume. The inner chamber is constantly purged with pure nitrogen to suppress the influx of radon. The copper is surrounded by a \\mbox{15$-$25} cm thick lead shield, where the innermost layer of \\mbox{2$-$5 cm} has a low level of $^{210}$Pb contamination. Radon-free nitrogen-flushed glove boxes are located on top of each detector.\n\nThe radioassay program used three additional spectrometers, Corrado, Bruno, and GIOVE~\\cite{giove}, that are operated underground in the Low-Level Laboratory at Max Planck Institut f{\\\"u}r Kernphysik (MPIK) in Heidelberg. The laboratory has an overburden of 15 meters water equivalent that reduces the muon flux by a factor of 2$-$3 and the hadronic background component by a factor of 1000 as compared to sea level. The spectrometers are surrounded by copper and lead shielding. Additionally, an active muon veto and polyethylene to moderate neutrons surround the Giove detector. These three detectors are p-type germanium crystals with masses between 0.9$-$1.8 kg that can reach sensitivities between 0.1$-$1 mBq\/kg. \n\nGiven the higher background and lower detection sensitivity with respect to the spectrometers operated at LNGS, the MPIK detectors were mostly employed for radioassay of components that are far from the sensitive volume of the XENON1T TPC, such as the tank for the water shield as well as the support structures and calibration systems within the shield. Most of the materials from components closest to the active volume of the TPC were screened with the GeMPI or Gator detectors at LNGS. For several smaller samples, additional detectors at the LNGS underground low-background facility STELLA (SubTerranean Low Level Assay) were used~\\cite{GePaolo}.\n\n\nSamples were cleaned using the same techniques as in the final detector construction when possible. Otherwise, the standard procedure was to clean each sample with a mild acid soap (Elma EC70), followed by rinsing with deionized water and immersion in high-purity ethanol \\mbox{($>95$\\%)}. Each step utilized an ultrasonic bath for 20 minutes. Samples where acid soaps or immersion in liquids should be avoided, such as photomultiplier tubes and cables, were cleaned by wiping the surface with ethanol. During transport of the samples to the detector glovebox, they were either enclosed in clean plastic bags or wrapped in plastic foils in order to prevent the plate-out of $^{222}$Rn daughters from the environment. The samples were then stored in an outer glovebox of the detector prior to their measurement in order to let the radon and its daughters decay.\n\n\nFor every measured sample, a Monte Carlo simulation based on the GEANT4 toolkit~\\cite{Agostinelli} was used to calculate the detection efficiencies for each emitted gamma line. The efficiencies were used in combination with the sample mass, measurement time, and branching ratio of each characteristic \\mbox{gamma-ray} line to determine the specific activities or detection upper limits of each radioisotope. Further details on analysis procedures for the HPGe detectors can be found in~\\cite{gempi, gator}.\n\n\n\\subsection{Inductively Coupled Plasma Mass Spectrometry}\n\nInductively Coupled Plasma Mass Spectrometry is one of the most sensitive analytical techniques for trace element analysis. The intrinsic radioactivity of materials can be deduced by measuring the concentration of long-lived radionuclides. The sample (fractions of a gram are enough for a measurement) is introduced as an aqueous solution through a peristaltic pump, nebulized in a spray chamber, then atomized and ionized in a plasma. The ions are extracted into an ultra-high vacuum system and separated according to their mass-to-charge ratio by the mass analyzer. The concentration of the ions is calculated based on the response of reference standard solutions. Depending on the nature of the sample, sensitivities on the order of 10$^{-11}$ to 10$^{-13}$ g\/g for $^{238}$U and $^{232}$Th and 10$^{-7}$ to 10$^{-8}$ g\/g for $^{39}$K can be reached. This corresponds to activities of 1$-$100 $\\mu$Bq\/kg and a few mBq\/kg, respectively. The uncertainty of measurement is between 20$-30\\%$ and accounts for several factors, such as the instrumental precision, the single replicate measurement and the single-point calibration curve.\n\nFor this radioassay campaign, measurements were performed with a 7500a ICP-MS from Agilent Technologies and an Element II HR-ICP-MS from ThermoFisher Scientific. Both instruments are located in a ISO6 clean room at the Chemistry Laboratory of LNGS.\n\n \n\\subsection{Glow-discharge mass spectrometry}\n\nThe Glow Discharge Mass Spectrometry measurements for XENON1T were performed at EAG Laboratories~\\cite{EAG}. Rather than being introduced as an aerosol as in ICP-MS, a negative bias is applied to the solid sample material while exposed to an argon-based plasma in order to induce sputtering via ion-target collisions. Once the material is sputtered into the plasma and subsequently ionized, an ion beam is extracted and focused through a high resolution mass spectrometer. Ions are separated according to their \\mbox{mass-to-charge} ratio. Sensitivity of \\mbox{sub-ppb} level or 10$^{-10}$ g\/g ($\\sim$1 mBq\/kg) can be reached with an uncertainty between 20$-30\\%$. Electrical conductivity of the sample is needed for stable and reproducible glow formation, thus the reliability and sensitivity of GDMS may vary depending upon properties of the target material. As with ICP-MS, GDMS is particularly useful in determining the $^{238}$U and $^{232}$Th concentrations. Although ICP-MS provides a better sensitivity than GDMS, the choice to use GDMS was primarily due to the availability and location of the measurement facilities.\\\\\n\n\n\n\n\\section{Radioassay results}\n\\label{sec:results}\n\nResults obtained through the radioassay program are shown in Table~\\ref{table:Samples}. Throughout the text, the samples are identified by their unique item numbers (``$\\#$\"). The detector is shown in Figs. \\ref{fig:TPC_section} and \\ref{fig:TPC_exploded} to introduce the most relevant subgroups and components. These are given in the ``XENON1T Use\" column of Table~\\ref{table:Samples} in the case where the material or component was chosen for detector construction. \n\nSupplier information is provided where applicable. For measurements conducted with the HPGe spectrometers, the sample mass and measurement duration are noted. Uncertainties, including both statistical and systematic, the latter primarily from efficiency simulations, are given in parentheses as $\\pm$ 1$\\sigma$ of detected activities or at 95$\\%$ confidence level for upper limits. Unless otherwise specified, the uncertainties of ICP-MS and GDMS measurements are 30$\\%$ and are primarily systematic, as described in Section 3.2.\n\nThe upper part of the $^{232}$Th decay chain is measured directly by mass spectrometry methods but is only detectable from gamma-ray spectroscopy following the $^{228}$Ra decay \\mbox{(T$_{1\/2}= 5.7$ yr)}. Rather than assuming secular equilibrium in the upper part of this chain, the two results are presented together with an indication of which part of the chain was measured. It is worth noting that, with one exception (copper $\\#$4), all samples for which both $^{232}$Th with ICP$-$MS and $^{228}$Ra with HPGe spectroscopy were measured show consistent results, thus indicating no break in secular equilibrium at $^{228}$Ra for these samples.\n\nIn addition to the decay chain and radioisotope activities listed in Table~\\ref{table:Samples}, Table~\\ref{table:metalother} shows results from cosmogenic radionuclei with \\mbox{short-to-moderate} \\mbox{half-lives} (T$_{1\/2} <$ 1 yr), as detected with HPGe spectrometers. \n\nAll results reported here will be made available via the radioassay community database at http:\/\/www.radiopurity.org.~\\cite{persephone}. Further details on many of the XENON1T samples and their measurements can be found in~\\cite{Piastra}.\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics*[width=0.9\\linewidth]{figures\/scaledsectionview.png}\n\\end{center}\n\\caption\n{\\label{fig:TPC_section}The XENON1T TPC with cryostat, section view, subgroups are indicated with reference to the ``XENON1T Use\" column of Table~\\ref{table:Samples}.}\n\\setlength{\\belowcaptionskip}{0pt}\n\\end{figure}\n\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics*[width=.7\\linewidth]{figures\/explodedviewall.png}\n\\end{center}\n\\caption\n{\\label{fig:TPC_exploded}The XENON1T TPC with cryostat, subgroups are indicated with reference to the ``XENON1T Use\" column of Table~\\ref{table:Samples}.}\n\\setlength{\\belowcaptionskip}{0pt}\n\\end{figure}\n\n\n\n\\subsection{Metal samples}\n\\label{sec:metal_samples}\n\nCommercially available Oxygen-Free High Conductivity or Oxygen-Free Electrolytic copper (OFHC or OFE copper) from primarily two different distributors was used for several major components of the TPC: the 74 field-shaping rings that surround the TPC ($\\#$1), the top and bottom PMT array support structures ($\\#$2, $\\#$3), and the bottom structural ring of the field cage ($\\#$4), comprising $\\sim$190 kg of the detector mass. Copper is intrinsically radiopure, with detected activities of the natural decay chains at the ppt level (see Table~\\ref{table:Samples}). One can see that the $^{60}$Co activity from cosmogenic activation varies from batch to batch, depending on the storage and shipment of the material~\\cite{cosmometal}. Because of its relative purity, copper was used as a substitute for stainless steel wherever possible. A sample of copper plated with 2 $\\mu$m thick gold ($\\#$6) was considered for the field-shaping rings to minimize radon emanation, however the samples showed significantly higher $^{40}$K activity that was most likely introduced as part of the electrochemical plating process. \n\nThe radiopurity of stainless steel can vary between batches, depending upon the source of the raw material, the method of heating and forming the material, as well as the location and duration of storage of the metal (cosmogenic activation). In order to minimize emissions from stainless steel components near the sensitive volume, the cleanest batches of available material were required. \nTherefore many batches of 304 and 316 stainless steel from six different manufacturers (17 samples in total) were screened for radiopurity. The samples originated from different melts and consisted of varying thicknesses. The NIRONIT Edelstahlhandel GmbH $\\&$ Co. samples ($\\#$8$-$10) that were particularly low in $^{226}$Ra, $^{232}$Th, and $^{60}$Co were selected for production of various TPC components and the cryostat vessels. Materials for components that are in direct contact with the liquid xenon, such as the cryostat pipe ($\\#$11), were selected for low $^{226}$Ra contamination in order to minimize emanation of $^{222}$Ra that can mix with the xenon and end up in the fiducial volume. \n\nIn many of the stainless steel batches, a depletion in $^{226}$Ra with respect to the upper half of the $^{238}$U chain is observed, thus indicating a clear break in secular equilibrium that most likely occurred during processing of the raw materials. In particular, a disequilibrium of more than a factor of 10 can be seen for items $\\#$17, $\\#$23, and $\\#$30. Additionally, a break between the upper and lower parts of the $^{232}$Th chain is observed in some of the HPGe measurements ($\\#$12$-$14, $\\#$17, $\\#$25, $\\#$30). Because these results were obtained through HPGe spectroscopy, the $^{232}$Th activity was not measured directly, thus the break in this chain is a possible indication of depletion in $^{228}$Ra. \n\nThe induced background from the XENON1T structural components, such as the water tank and outer support structures ($\\#$24$-$27, in addition to many screened samples not listed), was shown to be negligible in the Monte Carlo simulations due to their distance from the sensitive volume~\\cite{xenon_xe1t-sensitivity}. The screened stainless steel hardware ($\\#$28, $\\#$29) used for critical internal components, such as for the resistor chain and electrode fasteners, also had a negligible background contribution as the total mass used in the final construction was less than 1 kg. \n\nTitanium was considered as a potential cryostat material because of its high tensile strength as compared to copper and potentially lower radioactivity as compared to stainless steel. It has previously been used in the LUX experiment~\\cite{LUX_titanium} and investigated for use in the upcoming LZ experiment~\\cite{LZ_titanium}. Three different grades of titanium from four different suppliers were measured. The measured contamination of the titanium samples ($\\#$32$-$40) showed roughly a factor of 10 higher activity in the uranium chain as compared to the stainless steel used for the cryostat ($\\#$9). The other difference in contamination between the two material types was with respect to $^{60}$Co, which is subdominant in titanium but of concern in stainless steel, and $^{46}$Sc, a prominent cosmogenic isotope in titanium, as shown in Tables~\\ref{table:Samples} and~\\ref{table:metalother}, respectively. Additionally, the lower mechanical strength of titanium as compared to stainless steel would have required a thicker cryostat. When taking this into account in the Monte Carlo simulations, the neutron background from a titanium cryostat was considerably higher than for its stainless steel counterpart, therefore the latter material was chosen to construct the XENON1T cryostat.\n\n\n\\subsection{Plastic samples}\n\\label{sec:plastic_samples}\n\nDue to its good VUV reflectivity ($>$95$\\%$), a dielectric constant similar to liquid xenon, low-outgassing properties as compared to other plastics, and machinability, polytetrafluoroethylene (PTFE) is the material of choice for reflective surfaces within the field cage. Because it directly encloses the LXe sensitive volume, its radioactive content must be sufficiently low and also precisely measured to achieve an accurate background estimate. All of the PTFE samples were measured using ICP-MS for better quantification of the primordial chain progenitor isotopes and to complement the HPGe measurements where available, showing levels typically at the tens of ppt or $\\mu$Bq\/kg level ($\\#$46$-$50). PTFE doped with 15$\\%$ quartz to increase the reflectivity ($\\#$51) was also measured, however showed gross contamination in all of the natural chains as seen in Table~\\ref{table:Samples}. The primordial chains, due to alpha decays, are of particular concern for PTFE, as neutrons can be generated in the material via $^{19}$F($\\alpha$, n) reactions~\\cite{Norman}. Thus efforts were also made to minimize the total amount of this material used in construction. \n\nPolyamide-imide (PAI, in this case Torlon 4203L) was investigated for use as an insulating, structural material as it has a high dielectric constant, good mechanical strength, low thermal contraction, and allows for high-precision machining. Radioassay results ($\\#$53, $\\#$54) showed activities from the primordial chains to be a factor of 10 to 100 higher than its structural counterpart, PTFE. However, due to the absence of fluorine, neutron emission via ($\\alpha$, n) is not an issue with PAI. It was used for small but critical components, e.g. as insulating spacers. \n\nCommercially available PEEK (polyether ether ketone) screws were used at locations inside the TPC that required a high dielectric constant but with limited load-bearing requirements. Only one PEEK sample was measured ($\\#$52), yielding results on the order of 1$-$10 mBq\/kg, comparable to that of PAI.\n\nFor all of the plastic samples, no clear break in secular equilibrium is observed in the primordial decay chains. However the case of equilibrium is inconclusive, as only upper limits were measured for most samples. One exception is the PTFE doped with quartz ($\\#$51), that shows a clear break in the $^{232}$Th chain.\n\n\n\n\\subsection{Photomultiplier tubes and related components}\n\\label{sec:pmts}\n \nThe radioactive budget of the Hamamatsu R11410 3-inch diameter photomultiplier tubes was initially estimated through screening of the raw materials used in fabrication. Subsequently, several versions of PMTs were produced and screened with the goal of minimizing the total radioactivity of the tube to arrive at the final version, R11410-21. Of this version, the averaged activities of 320 PMTs measured with Gator and 40 PMTs measured with GeMPI I are reported in Table~\\ref{table:Samples} ($\\#$69, $\\#$70). Where only an upper limit was found, no entry is provided. Further details on the specific material contributions and the development of these low-background photomultipliers are given in~\\cite{LowRadPMTs}.\n\nSeveral samples of cables for the PMTs were screened to find clean batches. The detected activities for the signal and high-voltage cabling ($\\#$55$-$56, and $\\#$57$-$58, respectively) that were selected for final construction were typically at the tens or lower mBq\/kg level, with the exception of the considerably higher presence of $^{40}$K, particularly in the high-voltage (kapton) cables. The remaining PTFE ($\\#$59$-$61) and kapton coaxial and flat cables ($\\#$62$-$64) were not used due to higher levels from the primordial decay chains. \n\nThe connectors for the PMT signal and high-voltage cables, respectively, consisted of male\/female pairs of micro-miniature coaxial (MMCX) connectors made from a copper-zinc alloy ($\\#$66) and of subminiature-D (D-sub) pins made from either a copper-beryllium alloy ($\\#$65, $\\#$68) or a gold-plated copper alloy ($\\#$67). Due to the minimal total mass and the locations of the connector assemblies relative to the sensitive volume (in the cryostat pipe and on top of the diving bell), their radioimpurities are considered to have a negligible contribution to the overall background budget. Therefore all of the screened batches were used in the final construction. Additionally, a measurement of a representative sample of the high-voltage connectors mounted in custom-made PTFE holders, as produced for the final assembly, was performed with a new HPGe spectrometer, GeMSE, and showed consistent results~\\cite{GeMSE}.\n\nConnected directly to the base of each PMT is a voltage divider network that consists of a Printed Circuit Board (PCB, $\\#$94) with sockets ($\\#$93), solder ($\\#$99), resistors ($\\#81-84$), and capacitors ($\\#$86, $\\#$88). Several batches of the same types of components were screened, as there was some variation among batches and with respect to different PCB materials. The final PCBs assembled with components (referred to as the PMT base in Table~\\ref{table:Samples}) used the cleanest components where possible and then screened with an HPGe spectrometer ($\\#$100). The activity per assembled base was measured to be about a factor of 10 lower than the activity from the PMT itself.\n\nSeveral of the components for the PMTs show a clear break in secular equilibrium in the $^{238}$U chain indicating a depletion of $^{226}$Ra, particularly the connectors ($\\#$66), the sockets ($\\#$93), many of the resistors, and, consequently, the assembled bases ($\\#$100).\n\n\n\\subsection{Other samples}\n\\label{sec:other_samples}\n\n\nSeveral components that were composites of different materials, such as insulated conductors for electrode high voltage (a copper rod inserted into a PTFE insulator, $\\#$103) and capacitive sensors to measure the LXe level (``levelmeters\", $\\#$101$-$102) were screened post-fabrication and showed acceptable activities. The remaining components listed under ``Miscellaneous'' showed high levels in the primordial decay chains. However, these components are used in the calibration or leveling systems that are located within or outside of the Cherenkov detector and quite far from the TPC sensitive volume, therefore have negligible contributions to the background. \n\n\n\n\n\n\n\n\\subsection{Summary of material placement}\n\\label{sec:instrument}\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics*[width=0.7\\linewidth]{figures\/FC_template.png}\n\\end{center}\n\\caption\n{\\label{fig:TPC_items}The XENON1T TPC with material item numbers as given in Table~\\ref{table:Samples}.}\n\\setlength{\\belowcaptionskip}{0pt}\n\\end{figure}\n\n\nThe contribution from each material to the background depends upon its total mass and proximity to the sensitive volume as well as its type and energy of emission. The locations of screened materials used for the major components of the XENON1T TPC are indicated in Fig~\\ref{fig:TPC_items} by item number. The radioassay results from Table~\\ref{table:Samples} in combination with the material distribution within the instrument informed the XENON1T background predictions, as described in~\\cite{xenon_xe1t-sensitivity}. \n\nThe field cage of the XENON1T TPC consists of PTFE reflector panels and support pillars ($\\#$50), the latter hold and maintain separation between the 74 high-purity copper field-shaping rings ($\\#$1). The bottom ends of the PTFE pillars are mounted to a copper ring ($\\#$4) and are supported on the top by a stainless steel ring ($\\#$10). Bottom and top arrays of photomultiplier tubes ($\\#$69, $\\#$70, $\\#$100) face the target liquid-xenon volume enclosed by the field cage. The bottom array consists of a copper support plate ($\\#$2, $\\#$3) with a PTFE layer underneath ($\\#$49) for electrical insulation and a polished PTFE surface ($\\#$49) at a stand-off distance above the Cu plate in order to reflect the VUV light from the surfaces surrounding the PMT photocathodes. The top array consists of the same layers as the bottom array, mounted upside-down inside of the stainless steel diving bell that controls the LXe level ($\\#$10, shown in Fig~\\ref{fig:TPC_exploded}). In front of the photocathode surfaces of each PMT array are stainless steel screening electrodes ($\\#$10, not indicated in Fig~\\ref{fig:TPC_items}) to protect the PMTs from the field cage high voltage, small PTFE reflectors ($\\#$48), and the three electrodes ($\\#$10) that provide the electric field across the TPC (cathode below the target, gate and anode electrodes above the target). \n\nComponents not shown in Fig~\\ref{fig:TPC_items} include small parts such as the 5 G$\\mathrm{\\Omega}$ resistors ($\\#$71) that connect neighbouring copper field-shaping rings, PMT cabling and connectors ($\\#$55$-$58, $\\#$65$-$68), and small copper ($\\#$1), PEEK ($\\#$52), and stainless steel screws ($\\#$8, $\\#$28, $\\#$29) that were used throughout the TPC. Also not shown are components mounted onto or near the top stainless steel ring such as PAI ($\\#$53, $\\#$54) and PTFE ($\\#$50, $\\#$103) insulating spacers, and the levelmeters ($\\#$102) which are used to precisely measure the vertical position of the xenon liquid\/gas interface. \n\nOther TPC components (not shown in Fig~\\ref{fig:TPC_items}) are two long levelmeters ($\\#$101) which are used during LXe filling and a stainless steel with polyethylene high-voltage feedthrough (made from $\\#$8, $\\#$108) inside of a PTFE insulator ($\\#$47) that span the length of the field cage. The PMT signal and high-voltage cables ($\\#$55, $\\#$57) extend from the bottom PMT array along the length of the field cage and from the top PMT array inside the diving bell. The cables are then routed over the diving bell and connect to the cables ($\\#$56, $\\#$58, connected by $\\#$65$-$68) that arrive from the data acquisition room via the cryostat pipe ($\\#$11, partially shown in Figs~\\ref{fig:TPC_section} and~\\ref{fig:TPC_exploded}). \n\nThe cryostat ($\\#$9), shown in Figs~\\ref{fig:TPC_section} and~\\ref{fig:TPC_exploded}, consists of an inner stainless steel vessel that encloses the TPC and liquid xenon, nested inside an outer vessel that is evacuated for thermal insulation. The cryostat vessels and their domes ($\\#$8, $\\#$9) are covered by mylar insulation ($\\#$41) to reduce heat losses. Not shown are components outside of the cryostat, such as the calibration systems ($\\#$104, $\\#$105, $\\#$107) and stainless steel support structures ($\\#$25$-$27) within the water shield, and the 10 meter high, 9.6 meter diameter stainless steel tank that contains the water shield ($\\#$24).\n\n\n\n\n\n\\section{Discussion and impact on the XENON1T background}\\label{sec:impact}\n\nThe results from the radioassay campaign were used as source terms in the detector Monte Carlo simulations. The detected radioactive isotopes and decay chains were uniformly distributed within each component of the mass model according to their measured radioactivities. Each background source, before ER\/NR descrimination, is given in terms of an event rate over a \\mbox{1 ton} super-ellipsoid fiducial volume with respect to the energy region of interest (ROI). As electronic recoils and nuclear recoils induce a different response in liquid xenon, nuclear recoils in the (4, 50) keV$_{\\mathrm{nr}}$ interval yield the same signal intensities from scintillation as ER events in the (1, 12) keV$_{\\mathrm{ee}}$ (electron equivalent) energy ROI. The simulation and analysis details are given in~\\cite{xenon_xe1t-sensitivity}. \n\nFigure \\ref{fig:counts_All}, top, shows the relative expected contributions to the total ER background events for external background sources (i.e.~solar neutrinos), sources of intrinsic backgrounds ($^{136}$Xe, $^{85}$Kr, and $^{222}$Rn), and for each of the main XENON1T components. Thanks to the material selection campaign described in this work, the material-induced gamma-ray background is negligible within the (1, 12) keV$_{\\mathrm{ee}}$ WIMP search region compared to the contribution from $^{222}$Rn emanation. The dominant intrinsic $^{222}$Rn contamination was estimated to be 10 $\\mu$Bq\/kg in the liquid xenon target, however this can be further reduced through online purification~\\cite{RnDistillation}. A more detailed comparison with respect to the energy and select fiducial volumes can be found in~\\cite{xenon_xe1t-sensitivity}.\n\n\n\n\\begin{figure}[h]\n\n\\begin{center}\n\\includegraphics*[width=1.0\\linewidth]{figures\/ER_all_counts_break.pdf}\n\\includegraphics*[width=1.0\\linewidth]{figures\/NR_all_counts_v4.pdf}\n\\end{center}\n\n\\caption\n{\\label{fig:counts_All}Electronic recoil (top) and nuclear recoil (bottom) background contributions from materials (red) and from intrinsic and external sources (blue). The number of events per year in a 1$-$ton fiducial target is shown in the electron equivalent (1, 12) keV$_{\\mathrm{ee}}$ region of interest for electronic recoil events, corresponding to a nuclear recoil energy interval of (4, 50) keV$_{\\mathrm{nr}}$.}\n\\end{figure}\n\n\nThe expected contributions to the nuclear recoil background in XENON1T are shown in Fig. \\ref{fig:counts_All}, bottom. Most of the NR background comes from materials, as there are no significant intrinsic sources. Considering materials only, the stainless steel components (cryostat, TPC) are the dominant source, in total contributing 40$\\%$. The PMTs contribute 28$\\%$, primarily due to the high concentration of $^{238}$U and $^{232}$Th and their daughter isotopes in the ceramic stem of each PMT. Because of the proximity of the PTFE reflectors to the sensitive volume, the presence of heavier nuclei and their daughters contribute 22$\\%$ by the mechanisms described in Section~\\ref{sec:plastic_samples}. Coherent neutrino-nucleus scattering (CNNS) is subdominant, with a contribution similar to the TPC copper ($\\sim3\\%$ of total). The muon-induced nuclear recoil background is also subdominant due to effective coincidence-tagging with the Cherenkov muon-veto detector~\\cite{xenon_xe1t-MuonVeto}. \n\nAfter conversion into observable signals, ER\/NR discrimination was applied to all background events. Assuming an ER rejection efficiency of \\mbox{99.75$\\%$} at an NR acceptance of 40$\\%$, the total expected NR background in XENON1T for a 1 ton~$\\times$~2 year exposure is expected to be \\textless1 event in the (4, 50)\\,keV$_{\\mathrm{nr}}$ energy range. This corresponds to a best sensitivity to the spin-independent WIMP-nucleon cross section of \\mbox{$\\sigma_\\mathrm{SI} \\lesssim 10^{-47}~\\mathrm{cm}^{2}$} at a WIMP mass of \\mbox{m$_{\\chi}$= 50 GeV\/c $^{2}$}~\\cite{xenon_xe1t-sensitivity}.\n\nIn the planned upgrade of XENON1T to XENONnT, the LXe target mass will increase to a total of $\\sim$6 tons. This will require a \\mbox{$\\sim$40$\\%$} increase in the linear dimensions of the TPC and nearly double the number of PMTs. The larger detector will improve the sensitivity by another order of magnitude, reaching \\mbox{$\\sigma_\\mathrm{SI} \\lesssim 10^{-48}~\\mathrm{cm}^{2}$} at \\mbox{m$_{\\chi}$= 50 GeV\/c $^{2}$}~\\cite{xenon_xe1t-sensitivity}, assuming a negligible contribution from materials and a total exposure of 20 ton$\\cdot$years.\n\nMost of the existing sub-systems for XENON1T were designed to be reused for XENONnT, however the upgrade requires the construction of a new TPC and inner cryostat. As material-induced ER backgrounds are expected to be even lower than in XENON1T, the screening effort and material selection is focused on reducing the nuclear recoil background. This is being addressed particularly through continued efforts to identify low-activity stainless steel and by pursuing viable alternatives to PTFE, where possible.\n \n \n\\section{Acknowledgments}\n\nWe gratefully acknowledge support from the National Science Foundation, Swiss National Science Foundation, Deutsche Forschungsgemeinschaft, Max Planck Gesellschaft, German Ministry for Education and Research, Netherlands Organisation for Scientific Research (NWO), Weizmann Institute of Science, I-CORE, Initial Training Network Invisibles (Marie Curie Actions, PITNGA-2011-289442), Fundacao para a Ciencia e a Tecnologia, Region des Pays de la Loire, Knut and Alice Wallenberg Foundation, Kavli Foundation, and Istituto Nazionale di Fisica Nucleare. We are grateful to Laboratori Nazionali del Gran Sasso for hosting and supporting the XENON project.\n\n\n\n\n\\clearpag\n\\thispagestyle{empty\n\\newgeometry{margin=3cm}\n\n\\begin{sidewaystable*}[tp]\n\\resizebox{\\textwidth}{!}{\n\\tabcolsep=4pt\n\\centering\n\\begin{tabular}{ cllllccccccccccccc}\n\n\\toprule\n\\\\\n\\textbf{Item} &  \\textbf{Sample}  & \\textbf{Supplier} &  \\textbf{XENON1T Use}  &  \\textbf{Facility}  & \\textbf{Mass [kg]}  &  \\textbf{Time [d]}  &  \\textbf{Units}  &  \\textbf{$^{235}$U}  &  \\textbf{$^{238}$U}  &  \\textbf{$^{226}$Ra}  &  \\textbf{$^{228}$Ra ($^{232}$Th)$^{\\dagger}$}  &  \\textbf{$^{228}$Th}  &  \\textbf{$^{40}$K}  &  \\textbf{$^{60}$Co}  &  \\textbf{$^{137}$Cs} \\\\\n\\hline\n\\hline\n\\\\\n\\multicolumn{16}{l}{\\textbf{Copper}} \\\\\n\\hline\n\\input{tables\/Cu.tex}\n\\\\\n\\\\\n\\multicolumn{16}{l}{\\textbf{Stainless steel}} \\\\\n\\hline\n\\input{tables\/SS.tex}\n\\input{tables\/SS2.tex}\n\\input{tables\/structureSS.tex}\n\\\\\n\\\\\n\\multicolumn{16}{l}{\\textbf{Titanium}} \\\\\n\\hline\n\\input{tables\/titanium.tex}\n\\\\\n\\hline\n\\\\\n& &  &  &    &    &     &    &  &     &      &     &     &   & \\textbf{(Continued)}           \\\\\n\n\\end{tabular}}\n\\end{sidewaystable*}\n\n\n\n\\clearpage\n\\thispagestyle{empty}\n\\newgeometry{margin=3cm}\n\n\n\\begin{sidewaystable*}[tp]\n\\resizebox{\\textwidth}{!}{\n\\tabcolsep=4pt\n\\centering\n\\begin{tabular}{ cllllccccccccccccc}\n\n\\multicolumn{16}{l}{\\textbf{(Continuation)}} \\\\\n\\toprule\n\\\\\n\\textbf{Item} &  \\textbf{Sample}  & \\textbf{Supplier} &  \\textbf{XENON1T Use}  &  \\textbf{Facility}  & \\textbf{Mass [kg]}  &  \\textbf{Time [d]}  &  \\textbf{Units}  &  \\textbf{$^{235}$U}  &  \\textbf{$^{238}$U}  &  \\textbf{$^{226}$Ra}  &  \\textbf{$^{228}$Ra ($^{232}$Th)$^{\\dagger}$}  &  \\textbf{$^{228}$Th}  &  \\textbf{$^{40}$K}  &  \\textbf{$^{60}$Co}  &  \\textbf{$^{137}$Cs}  \\\\\n\\hline\n\\hline\n\\\\\n\\multicolumn{16}{l}{\\textbf{Cryostat insulation}} \\\\\n\\hline\n\\input{tables\/Superinsulation.tex}\n\\\\\n\\\\\n\\multicolumn{16}{l}{\\textbf{Plastics}} \\\\\n\\hline\n\\input{tables\/Plastics.tex}\n\\\\\n\\\\\n\\multicolumn{16}{l}{\\textbf{Cables}} \\\\\n\\hline\n\\input{tables\/Cables.tex}\\\\\n\\\\\n\\\\\n\\multicolumn{16}{l}{\\textbf{Connectors}} \\\\\n\\hline\n\\input{tables\/Connectors.tex}\\\\\n\\hline\n\\\\\n& &  &  &    &    &     &    &  &     &      &     &     &   & \\textbf{(Continued)}           \\\\\n\n\n\\end{tabular}}\n\\end{sidewaystable*}\n\n\n\n\n\\clearpage\n\\thispagestyle{empty}\n\\newgeometry{margin=3cm}\n\n\\begin{sidewaystable*}[tp]\n\\resizebox{\\textwidth}{!}{\n\\tabcolsep=4pt\n\\centering\n\\begin{tabular}{ cllllccccccccccccc}\n\n\n\\multicolumn{16}{l}{\\textbf{(Continuation)}} \\\\\n\\toprule\n\\\\\n\\textbf{Item} &  \\textbf{Sample}  & \\textbf{Supplier} &  \\textbf{XENON1T Use}  &  \\textbf{Facility}  & \\textbf{Mass [kg]}  &  \\textbf{Time [d]}  &  \\textbf{Units}  &  \\textbf{$^{235}$U}  &  \\textbf{$^{238}$U}  &  \\textbf{$^{226}$Ra}  &  \\textbf{$^{228}$Ra ($^{232}$Th)$^{\\dagger}$}  &  \\textbf{$^{228}$Th}  &  \\textbf{$^{40}$K}  &  \\textbf{$^{60}$Co}  &  \\textbf{$^{137}$Cs}  \\\\\n\\hline\n\\hline\n\\\\\n\\multicolumn{16}{l}{\\textbf{Photosensors (weighted average)}} \\\\\n\\hline\n\\input{tables\/PMTs.tex}\n\\\\\n\\\\\n\\multicolumn{16}{l}{\\textbf{Electronics}} \\\\\n\\hline\n\\input{tables\/electronics.tex}\n\\\\\n\\\\\n\\multicolumn{16}{l}{\\textbf{Miscellaneous}} \\\\\n\\hline\n\\input{tables\/misc_tpc.tex}\n\\\\\n\\bottomrule\n\n\n\\end{tabular}}\n\\caption{\\small{Measured activities of material samples for the XENON1T radioassay program. For HPGe spectrometer measurements, uncertainties are $\\pm1\\sigma$ for detected lines (in parentheses) and 95$\\%$ C.L. for upper limits. Mass spectrometry measurement uncertainty is 30$\\%$ unless otherwise noted. Listed activities for $^{238}$U that are denoted with ``*\" were inferred assuming natural abundance~\\cite{UraniumRatio}. Direct mass spectrometry measurements of $^{232}$Th are denoted with ``$^{\\dagger}$\". The ``XENON1T Use\" column refers to materials that were used in the final assembly, with reference to the subgroups labeled in Figs \\ref{fig:TPC_section} and \\ref{fig:TPC_exploded}. Dash marks in the table indicate that the sample was not used in construction (``XENON1T Use\"), or that an attribute (``Mass\", ``Time\", or a particular isotope) was either unknown or not applicable to the measurement.}}\n\\label{table:Samples}\n\n\n\n\\end{sidewaystable*}\n\n\\restoregeometry\n\n\n\n\n\\clearpage\n\\thispagestyle{empty}\n\n\\begin{table*}[ht]\n\\centering\n\\begin{tabular}{clccccccc}\n\n\\toprule\n\\textbf{Item} &  \\textbf{Sample}  & \\textbf{Units}  &  \\textbf{$^{56}$Co}  &  \\textbf{$^{57}$Co}  &  \\textbf{$^{58}$Co}  &  \\textbf{$^{54}$Mn}  &  \\textbf{$^{46}$Sc} \\\\  \n\\midrule[0.06em]\n1 &  Copper, CW009A  &  mBq\/kg  &  0.06(2)  &  0.2(1)  &  0.36(4)  & $<$ 0.027 &  $-$  & \\\\\n2 &  Copper, C10100 &  mBq\/kg  &  0.31(3)  &  0.4(1)  &  1.8(2)  & 0.22(3) &  0.08(2)  & \\\\\n3 &  Copper, C10100  &  mBq\/kg  &  $-$  &  0.40(1)  &  0.35(4)  & 0.15(2) &  $-$  & \\\\\n4 &  Copper,  C10100 &  mBq\/kg  &  0.15(2)  &  0.7(2)  &  1.1(1)  & 0.35(4) &  $-$  & \\\\\n8 &  Stainless steel, AISI 316Ti  &  mBq\/kg  &  $<$ 0.8  &  $<$ 7.4  &  $<$ 1.5  & 1.2(5) &  $-$  & \\\\\n9 &  Stainless steel, AISI 304L  &  mBq\/kg  & $-$ & $-$ & $<$ 0.6 &  0.5(2) & $-$ & \\\\\n10 &  Stainless steel, AISI 304  &  mBq\/kg  & $-$ & $-$  & $-$  & 1.1(3) &  $-$ & \\\\\n11 &  Stainless steel, AISI 316L  &  mBq\/kg  & $-$ & $-$ & $-$ & 1.4(3) & $-$ &\\\\\n32 &  Titanium, grade 1  &  mBq\/kg  & $-$ & $-$ & $-$ & $-$ & 2.15(3) &\\\\\n33 &  Titanium, grade 1 &  mBq\/kg  & $-$ & $-$ & $-$ & $-$ & 1.9(2) &\\\\\n34 &  Titanium, grade 4 &  mBq\/kg  & $-$ & $-$ & $-$ & $-$ & 1.9(2) &\\\\\n35 &  Titanium, grade 2  &  mBq\/kg  & $-$ & $-$ & $-$ & $-$ & 2.7(3) &\\\\\n38 &  Titanium, grade 1 &  mBq\/kg  & $-$ & $-$ & $-$ & $-$ & 1.8(2) &\\\\\n39 &  Titanium, grade 1 ($\\#$38, welded) &  mBq\/kg  & $-$ & $-$ & $-$ & $-$ & 1.0(1) &\\\\\n40 &  Titanium, grade 1  &  mBq\/kg  & $-$ & $-$ & $-$ & $-$ & 2.2(3) &\\\\\n \\bottomrule\n \n\\end{tabular}\n\\caption{Cosmogenic radioisotopes detected in metal samples. The ``Item\" numbers are cross-referenced with those in Table \\ref{table:Samples}. Uncertainties are $\\pm1\\sigma$ for detected lines (in parentheses) and 95$\\%$ C.L. for upper limits.}\n\\label{table:metalother}\n\\end{table*}\n\n\n\n\n\n\\clearpage\n\\restoregeometry\n\\twocolumn\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\n\nThe arithmetic intersection theory of toric varieties with respect to toric line bundles equipped with their canonical metric was first studied by Maillot in \\cite{Mail}. Later, the systematic extension of the toric dictionary to Arakelov geometry was carried out by Burgos Gil, Philippon and Sombra in \\cite{BPS}. It turns out from their study that suitably metrized toric line bundles can be expressed in terms of families of concave functions on convex polytopes and that the height of the toric variety with respect to this choice is related to the integral of such functions. Their theory allows to treat a large spectrum of height functions, namely the ones arising from toric line bundles equipped with toric metrics; this includes the canonical heights studied by Maillot and the Fubini-Study height. On the other hand, the techniques developed in \\cite{BPS} only apply to the computation of the height of toric subvarieties and do not solve, for instance, the problem of determining the height of a general cycle of codimension $1$. This question was answered in a very special case by \\cite{Mail}, where a relation between the canonical height of a hypersurface in a smooth projective toric variety and the Mahler measure of the corresponding polynomial was given. Other computations have been performed by Cassaigne and Maillot in \\cite{CM} for the Fubini-Study height of hypersurfaces in projective spaces. Extending the techniques of \\cite{BPS}, we give here a combinatorial formula for the height of a $1$-codimensional cycle in a toric variety for a much more general choice of metrics. \n\\vspace{\\baselineskip}\n\\\\For the sake of simplicity, we restrict for the moment to the case of an ambient proper toric variety $X_\\Sigma$ of dimension $n$ over $\\mathbb{Q}$, leaving the treatment of the case of an arbitrary base adelic field to the body of the paper. Let $\\mathfrak{M}$ stand for the set of places of $\\mathbb{Q}$. As usual in toric geometry, we denote by $M$ the lattice of characters of the torus of $X_\\Sigma$, by $N$ its dual lattice, and by $M_\\mathbb{R}$ and $N_\\mathbb{R}$ the corresponding real vector spaces. We are interested in a combinatorial expression for the height of a cycle of codimension $1$ in $X_\\Sigma$ with respect to a suitable choice of a metrized (Cartier) divisor. By the linearity of height functions, we can restrict to the case of an irreducible hypersurface $Y$. Moreover, since irreducible hypersurfaces in $X_\\Sigma$ not intersecting its dense open torus have to coincide with $1$-codimensional toric orbits, whose height has already been calculated in \\cite[Proposition 5.1.11]{BPS}, we can assume that the generic point of $Y$ lies in the dense open orbit of $X_\\Sigma$. Under this assumption, $Y$ is described by an irreducible Laurent polynomial $f$ with rational coefficients. Its Newton polytope $\\NP(f)$ is a nonempty subset of $M_\\mathbb{R}$ capturing enough information for the intersection theoretical properties of $Y$. For instance, \\hyperref[degree of an hypersurface]{Proposition \\ref*{degree of an hypersurface}} implies that the degree of $Y$ with respect to a toric divisor $D$ on $X_\\Sigma$ generated by its global sections is given by \\[\\deg_D(Y)=\\MV_M(\\Delta,\\dots,\\Delta,\\NP(f)),\\]where $\\Delta$ is the polytope in $M_\\mathbb{R}$ associated to $D$ and $\\MV_M$ denotes the mixed volume of convex bodies in $M_\\mathbb{R}$ with respect to a suitably normalized Haar measure.\n\\\\The height of a cycle in $X$ is the arithmetic counterpart of its degree with respect to a divisor $D$. Its definition requires as an extra datum the choice of an adelic semipositive metric on $D$, see \\hyperref[section about Arakelov on toric varieties]{section \\ref*{section about Arakelov on toric varieties}} for a precise definition. To have a combinatorial description of heights in toric varieties, it is necessary to ask $D$ to be a toric divisor (with associated polytope $\\Delta$) and the metric on it to be ``toric invariant'' in some sense. In such a situation, Burgos Gil, Philippon and Sombra have shown that a combinatorial description is possible, translating the additional information of the metric into an extra dimension on the convex geometrical side: an adelic semipositive toric metric on $D$ is associated to a family $(\\vartheta_v)_{v\\in\\mathfrak{M}}$ of continuous concave functions on $\\Delta$, called the \\emph{roof functions} of the metric, such that $\\vartheta_v=0$ for all but finitely many $v$. We show how the height of $Y$ with respect to the adelic semipositive toric metrized divisor $\\overline{D}$ can be expressed using such an extra dimensional representation, in a spirit analogous to the formula for its degree mentioned above. The key idea consists in associating to the polynomial $f$ defining $Y$, for every place $v$ of $\\mathbb{Q}$, a suitable function which we call the \\emph{$v$-adic Ronkin function} of $f$ and denote by $\\rho_{f,v}$. It is a concave function on $N_\\mathbb{R}$ whose value at $u$ can be interpreted as an average of $-\\log|f|$ on the fiber of the tropicalization map over $u$. When $v$ is archimedean, it is the Ronkin function studied by Passare and Rullg{\\aa}rd among others, while for non-archimedean places it coincides with the $v$-adic tropicalization of the polynomial $f$. Its Legendre-Fenchel dual $\\rho_{f,v}^\\vee$ is a concave function on $M_\\mathbb{R}$ which is supported on the Newton polytope of $f$. Recall now that Philippon and Sombra have introduced in \\cite{PS1} a polarized version of the integration of a concave function with bounded support, called the \\emph{mixed integral}. For the choice of a suitably normalized Haar measure on the vector space $M_\\mathbb{R}$, it is a multilinear symmetric real valued function $\\MI_M$ taking as entries $n+1$ concave functions supported on convex bodies in $M_\\mathbb{R}$.\n\n\\begin{introtheorem}\\label{main theorem introduction}\nThe height of $Y$ with respect to $\\overline{D}$ is given by \\[h_{\\overline{D}}(Y)=\\sum_{v\\in\\mathfrak{M}}\\MI_M\\big(\\vartheta_v,\\dots,\\vartheta_v,\\rho_{f,v}^\\vee\\big).\\]\n\\end{introtheorem}\n\nDespite the complexity of the computation of the archimedean Ronkin function, the formula in the previous theorem clarifies the relation between the defining polynomial of an irreducible hypersurface and its height with respect to an adelic semipositive toric metrized divisor. \nIt is easy to specialize it to the case of the canonical metric on $D$, where it reduces to the equality proved in \\cite{Mail}, or of the Fubini-Study metric in the projective setting. We hope that a better understanding of the properties of mixed integrals and archimedean Ronkin functions could be used to deduce both lower and upper bounds for the height of $Y$. More importantly, our result asserts that the collection of the $v$-adic Ronkin functions of a hypersurface contains enough information to determine its height; we wonder whether other arithmetical properties of $Y$ might be read in terms of such functions.\n\\vspace{\\baselineskip}\n\\\\To show the stated result, we prove more precise formulas for the local height and the toric local height of a $1$-codimensional cycle. We also show some new properties of mixed integrals and we propose a more uniform definition and study of $v$-adic Ronkin functions which is independent on whether the place $v$ is archimedean or not. The obtained formulas for the height extend to the case of admissible adelic toric metrized divisors as alternated sums of mixed integrals, as in \\cite[Remark 5.1.10]{BPS}.\n\\\\For an arbitrary adelic base field $K$, we remark that one needs to prove that the global height of $Y$ with respect to an adelic semipositive toric metrized divisor is a finite sum and hence well-defined. This is automatic if $K$ is a global field, because of \\cite[Proposition 1.5.14 and Theorem 4.9.3]{BPS}. We show it here for an arbitrary adelic field $K$ with product formula, in which case the formulas for the height stay true. In the more general setting of an adelic field $K$ not satisfying the product formula, it is easy to verify that the same equality for the global height holds up to the sum by the \\textit{defect} of $K$, see \\cite[Definition 1.5.9]{BPS}. Finally, the recent work \\cite{GH} suggests that similar statements might hold for a base $M$-field.\n\\vspace{\\baselineskip}\n\\\\We now briefly summarize the content of each section. \n\\\\In \\hyperref[section about convex geometry]{section \\ref*{section about convex geometry}}, we recall the tools from convex geometry which are needed throughout all the paper: Legendre-Fenchel duality of concave functions, real Monge-Amp\\`ere measures and mixed integrals. In particular, we re-interpret the recursive formula for mixed integrals proved by Philippon and Sombra in terms of mixed real Monge-Amp\\`ere measures. We then make use of it to deduce two elementary, though useful, properties of such operators. Finally, we describe what happens when one of the functions appearing in the mixed integral is the indicator function of a line segment.\n\\\\\\hyperref[section about Ronkin functions]{Section \\ref*{section about Ronkin functions}} deals with the key object of our work: $v$-adic Ronkin functions of Laurent polynomials, which are introduced and described after recalling the needed preliminaries in tropical and non-archimedean geometry. In this context, the discussion of a notion of minimal boundaries allows to treat the archimedean and non-archimedean cases homogeneously. \n\\\\In \\hyperref[section about Arakelov on toric varieties]{section \\ref*{section about Arakelov on toric varieties}} we briefly recall the general adelic Arakelov framework and focus then on the results obtained by Burgos Gil, Philippon and Sombra in the toric setting. To keep the treatment of archimedean and non-archimedean places on equal footing, we rephrase their description of the Chambert-Loir measure of semipositive toric metrized divisors in terms of minimal boundaries of tropical fibers.\n\\\\As a needed step for the main proof, we combinatorially describe the Weil divisor of the rational function defined by a Laurent polynomial on a toric variety. This result can be of independent interest and has then been set aside in \\hyperref[section divisor of rational functions]{section \\ref*{section divisor of rational functions}}.\n\\\\\\hyperref[section hypersurface]{Section \\ref*{section hypersurface}} is dedicated to the proofs of our main results \\hyperref[toric local height of hypersurfaces]{Theorem \\ref*{toric local height of hypersurfaces}} and \\hyperref[hypersurfaces are integrable and their global height]{Theorem \\ref*{hypersurfaces are integrable and their global height}}, which are formulas for the local height and the toric local height of cycles of codimension $1$ in toric varieties. We then make use of them to prove the integrability statement and a formula for their global heights, with respect to the choice of adelic semipositive toric metrized divisors.\n\\\\For binomial hypersurfaces, such a formula is compatible with the one deduced from \\cite{BPS}. This is shown in \\hyperref[section about examples]{section \\ref*{section about examples}}, where we also apply our results to some other particular cases. We provide convex geometrical formulas for the canonical height of $1$-codimensional cycles, obtaining the quoted result by Maillot, and for the Fubini-Study height of a projective hypersurface. We also propose a new height function, the $\\rho$-height, for which we give a compact formula. We do not know any application of such a height, which could be anyway worth studying.\n\\vspace{\\baselineskip}\n\\\\\\textbf{Terminology and notations.} A \\emph{variety} $X$ is assumed to be a reduced and irreducible separated scheme of finite type over a field. By an \\emph{irreducible hypersurface} in it we mean a closed integral subscheme of codimension $1$ in $X$. A \\emph{divisor} on $X$ is a Cartier divisor, unless otherwise stated. Toric varieties are assumed to be normal; whenever the choice of the base field $K$ is clear from the context, the notation $X_\\Sigma$ will refer to the toric variety over $K$ associated to the fan $\\Sigma$.\n\\\\The term \\emph{measure} on a topological space stands for a signed Borel measure on it; in particular, measures admit a well-defined push-forward via continuous mappings. A measure which only takes non-negative real values on Borel subsets is called a \\emph{positive measure}.\n\\vspace{\\baselineskip}\n\\\\\\textbf{Acknowledgments.} The author wishes to thank his PhD supervisors Mart\\'{\\i}n Sombra and Alain Yger for their guidance and many fruitful discussions, as well as C\\'esar Mart\\'{\\i}nez for the interest shown in a preliminary version of this text. Also, he is grateful to the two anonymous referees for the careful reading and the valuable remarks, which highly contributed to improve the quality of the text. The work has been prepared at the Universit\\'e de Bordeaux and at the Universitat de Barcelona as part of the author's PhD project, and partially supported by the MINECO research project MTM2015-65361-P and the CNRS project PICS 6381 ``G\\'eom\\'etrie diophantienne et calcul formel''.\n\n\n\\section{Preliminaries in convex geometry}\\label{section about convex geometry}\n\nThis section is devoted to recalling notions from convex geometry that will be useful in the sequel. We follow the conventions and notations of \\cite[Chapter 2]{BPS}, referring to \\cite[\\S12]{R} for a more complete treatment of the subject. We refer to these two sources for the proofs of the statements we make here.\n\\\\For the whole section, let $N$ be a lattice of rank $n$ and $M:=\\hom(N,\\mathbb{Z})$ its dual lattice. Denote by $N_\\mathbb{R}=N\\otimes_\\mathbb{Z}\\mathbb{R}$ and by $M_\\mathbb{R}=M\\otimes_\\mathbb{Z}\\mathbb{R}$ the corresponding $n$-dimensional real vector spaces.\n\\\\By a \\emph{polyhedron} in $N_\\mathbb{R}$ we mean a convex subset of $N_\\mathbb{R}$ obtained as the intersection of finitely many closed halfspaces $\\{u\\in N_\\mathbb{R}:\\langle x,u\\rangle+c\\geq0\\}$, with $x\\in M_\\mathbb{R}$ and $c\\in\\mathbb{R}$. If all the slopes $x$ can be chosen in $M$, the polyhedron is said to be \\emph{rational}. A \\emph{polytope} is a bounded polyhedron. A polytope in $N_\\mathbb{R}$ whose vertices all lie in $N$ is called a \\emph{lattice polytope}; it is in particular a rational polytope. A compact convex subset of $N_\\mathbb{R}$ is called a \\emph{convex body}.\n\n\n\\subsection{Legendre-Fenchel duality}\n\nA function $f:N_\\mathbb{R}\\to\\mathbb{R}\\cup\\{-\\infty\\}$ is said to be \\emph{concave} if it is not identically $-\\infty$ and for every $u_1,u_2\\in N_\\mathbb{R}$ and for every $t\\in\\left[0,1\\right]$, one has the inequality \\[f(tu_1+(1-t)u_2)\\geq tf(u_1)+(1-t)f(u_2).\\] The \\emph{effective domain} of a concave function $f$ is the set on which the function takes values different from $-\\infty$ and it is denoted by $\\dom(f)$: it is a convex subset of $N_\\mathbb{R}$. A concave function is said to be \\emph{closed} if it is upper semicontinuous. Every concave function with closed effective domain and continuous on it is closed. The \\emph{recession function} of a closed concave function $f$ is the concave conical function which takes on $u\\in N_\\mathbb{R}$ the value \\[\\rec(f)(u):=\\lim_{\\lambda\\to\\infty}\\frac{f(v_0+\\lambda u)}{\\lambda}\\in\\mathbb{R}\\cup\\{-\\infty\\},\\] for any $v_0\\in\\dom(f)$, see \\cite[Theorem 8.5]{R}. Finally, a concave function $f$ with effective domain a polyhedron in $N_\\mathbb{R}$ is \\emph{piecewise affine} if \\[f(u)=\\min_{\\alpha\\in S}(\\langle\\alpha,u\\rangle+c_\\alpha)\\] for every $u\\in\\dom(f)$, with $S$ a finite subset of $M_\\mathbb{R}$ and $c_\\alpha\\in\\mathbb{R}$ for every $\\alpha\\in S$. \n\\\\To each concave function $f$ on $N_\\mathbb{R}$, one can associate its \\emph{Legendre-Fenchel dual}, which is the closed concave function $f^\\vee$ on $M_\\mathbb{R}$ defined as \\[f^\\vee(x):=\\inf_{u\\in N_\\mathbb{R}}(\\langle x,u\\rangle-f(u)),\\]for every $x\\in M_\\mathbb{R}$, see \\cite[\\S 2.2]{BPS}. If $f$ is closed, $(f^\\vee)^\\vee=f$. The effective domain of $f^\\vee$ is a convex subset of $M_\\mathbb{R}$, which one calls the \\emph{stability set of $f$} and denotes by $\\stab(f)$.\n\\\\The following example is classical and will play a role later on.\n\n\\begin{ex}\\label{indicator and support function}\nAny nonempty convex body $B$ in $M_\\mathbb{R}$ induces a concave function $\\Psi_B$ on $N_\\mathbb{R}$, called the \\emph{support function} of $B$ and defined as \\[\\Psi_B(u):=\\min_{x\\in B}\\langle x,u\\rangle\\]for every $u\\in N_\\mathbb{R}$. Its Legendre-Fenchel dual is the \\emph{indicator function $\\iota_B$ of $B$}, which is the function taking the value $0$ on $B$ and $-\\infty$ elsewhere. Hence, $\\dom(\\Psi_B)=N_\\mathbb{R}$ and $\\stab(\\Psi_B)=B$. Notice that, whenever $B$ is a polytope, $\\Psi_B$ is a conic piecewise affine concave function.\n\\end{ex}\n\nWe also recall that there exists a number of operations that one can define on concave functions, in addition to the usual pointwise sum and scalar multiplication. Among these, the \\emph{sup-convolution} of two concave functions $f$ and $g$ on $N_\\mathbb{R}$ with non-disjoint stability sets is defined as\n\\[(f\\boxplus g)(v):=\\sup_{u_1+u_2=v}(f(u_1)+g(u_2))\\]\nand the \\emph{right scalar multiplication} of $f$ by $\\lambda\\in\\mathbb{R}_{\\geq0}$ as\n\\[(f\\lambda)(u):=\\lambda f(u\/\\lambda).\\]\nAlso, the \\emph{translate} of a concave function $f$ on $N_\\mathbb{R}$ by a point $u_0\\in N_\\mathbb{R}$ is set to be \\[(\\tau_{u_0}f)(u):=f(u-u_0).\\]These operations are dual, via Legendre-Fenchel duality, to the usual pointwise addition, scalar multiplication and sum by a linear function, respectively, see \\cite[Proposition 2.3.1 and Proposition 2.3.3]{BPS}.\n\n\n\n\\subsection{Real Monge-Amp\\`ere measures}\\label{subsection about real Monge-Ampere measures}\n\nFor any closed concave function $f$ on $N_\\mathbb{R}$ with $\\dom(f)=N_\\mathbb{R}$ and for any Haar measure $\\mu$ on $M_\\mathbb{R}$, one can define a corresponding \\emph{real Monge-Amp\\`ere measure} $\\mathcal{M}_\\mu(f)$, as in \\cite[\\S2.7]{BPS}. It is a measure on $N_\\mathbb{R}$, of total mass $\\mu(\\stab(f))$, being supported on finitely many points if $f$ is piecewise affine. The Monge-Amp\\`ere operator, associating to each closed concave function $f$ with $\\dom(f)=N_\\mathbb{R}$ the corresponding measure $\\mathcal{M}_\\mu(f)$ on $N_\\mathbb{R}$ is homogeneous of degree $n$ with respect to pointwise scalar multiplication. It was shown in \\cite{PR} that such an operator admits a polarization: for $f_1,\\dots,f_n$ closed concave functions with effective domain $N_\\mathbb{R}$, their \\emph{mixed real Monge-Amp\\`ere measure} is defined as the measure\n\\begin{equation}\\label{mixed Monge-Ampere measure}\n\\MM_\\mu(f_1,\\dots,f_n):=\\sum_{k=1}^n(-1)^{n-k}\\sum_{1\\leq i_1<\\dots<i_k\\leq n}\\mathcal{M}_\\mu(f_{i_1}+\\dots+f_{i_k}).\n\\end{equation}\nNotice that this definition differs from \\cite[formula (14)]{PR} and \\cite[Definition 2.7.12]{BPS} by a multiplicative constant. It follows from \\cite[\\S5]{PR} that the measure in \\eqref{mixed Monge-Ampere measure} is in fact a positive measure. The so obtained mixed Monge-Amp\\`ere operator is, by definition, symmetric and multilinear in its entries (with respect to pointwise sum) and it satisfies \\[\\MM_\\mu(f,\\dots,f)=n!\\mathcal{M}_\\mu(f)\\] for every closed concave function $f$. In particular, if $f_1,\\dots,f_n$ are closed concave functions with convex bodies as stability sets, their mixed real Monge-Amp\\`ere measure is a finite measure on $N_\\mathbb{R}$ of total mass $\\MV_\\mu(\\stab(f_1),\\dots,\\stab(f_n))$. Here, $\\MV_\\mu$ denotes the mixed volume of convex bodies with respect to the measure $\\mu$, normalized in such a way that $\\MV_\\mu(Q,\\dots,Q)=n!\\mu(Q)$ for every convex body $Q$ in $M_\\mathbb{R}$.\n\n\\begin{rem}\\label{lattice volume}\nThe integral structure on $M_\\mathbb{R}$ coming from the subjacent lattice $M$ gives a distinguished measure $\\vol_M$ on $M_\\mathbb{R}$, which is the unique Haar measure for which the volume of a fundamental domain of $M$ is $1$ (this does not depend on the choice of a basis of $M$). To lighten the notation, the corresponding (mixed) real Monge-Amp\\`ere operator will be denoted by $\\MM_M$.\n\\end{rem}\n\n\n\\subsection{Mixed integrals}\\label{subsection about mixed integrals}\n\nFor any Haar measure $\\mu$ on $M_\\mathbb{R}$, the application mapping a compactly supported concave function $g$ on $M_\\mathbb{R}$ to its Lebesgue integral with respect to $\\mu$ is homogeneous of degree $(n+1)$ with respect to right scalar multiplication. As a consequence of \\cite[Proposition 4.5]{PS1}, there exists a polarized operator: for $g_0,\\dots,g_n$, concave functions on $M_\\mathbb{R}$ with respective domains the convex bodies $Q_0,\\dots,Q_n$, their \\emph{mixed integral} with respect to $\\mu$ is defined as \\[\\MI_\\mu(g_0,\\dots,g_n):=\\sum_{k=0}^n(-1)^{n-k}\\sum_{0\\leq i_0<\\dots<i_k\\leq n}\\int_{Q_{i_0}+\\dots+Q_{i_k}}(g_{i_0}\\boxplus\\dots\\boxplus g_{i_k})\\ d\\mu.\\]\nThis notion was introduced and studied in \\cite{PS1} and \\cite{PS2}. The mixed integral operator is symmetric and multilinear in its entries (with respect to sup-convolution) and it satisfies \\[\\MI_\\mu(g,\\dots,g)=(n+1)!\\int_Q g\\ d\\mu\\] for every concave function $g$ on a convex body $Q$. As for the mixed Monge-Amp\\`ere operator, we will denote by $\\MI_M$ the mixed integral computed with respect to the Haar measure $\\vol_M$ on $M_\\mathbb{R}$.\n\n\\vspace{\\baselineskip}\nThe following proposition describes the behaviour of mixed integrals with respect to translation of the entries.\n\n\\begin{proposition}\\label{translation in mixed integral does not change the value}\nLet $g_i$ be a concave function defined on a convex body $Q_i$ in $M_\\mathbb{R}$, for $i=0,\\dots,n$. Let also $x_0\\in M_\\mathbb{R}$. Then, \\[\\MI_\\mu(\\tau_{x_0}g_0,g_1,\\dots,g_n)=\\MI_\\mu(g_0,\\dots,g_n).\\]\n\\end{proposition}\n\\begin{proof}\nFor any subset $\\{i_1,\\dots,i_r\\}\\subseteq\\{1,\\dots,n\\}$ one has, directly by definition, that\\[((\\tau_{x_0}g_0)\\boxplus g_{i_1}\\boxplus\\dots\\boxplus g_{i_r})(x+x_0)=(g_0\\boxplus g_{i_1}\\boxplus\\dots\\boxplus g_{i_r})(x)\\]for every $x\\in M_\\mathbb{R}$. The change of variables formula implies then that the integrals appearing in the definitions of $\\MI_\\mu(\\tau_{x_0}g_0,g_1,\\dots,g_n)$ and of $\\MI_\\mu(g_0,\\dots,g_n)$ are pairwise equal, from which the statement follows trivially.\n\\end{proof}\n\nWe now focus on a recursive formula for mixed integrals. For any closed convex subset $C$ in $M_\\mathbb{R}$, and for every $u\\in N_\\mathbb{R}$, one can consider the subset \\[C^u:=\\bigg\\{x\\in C:\\langle x,u\\rangle=\\inf_{y\\in C}\\langle y,u\\rangle\\bigg\\}\\]of $C$. For $u\\neq0$, this subset is contained in an affine subspace of $M_\\mathbb{R}$ of codimension 1 and parallel to $u^\\perp:=\\{x\\in M_\\mathbb{R}:\\langle x,u\\rangle=0\\}$. It is immediate from the definition that for every pair of convex bodies $C_1$ and $C_2$ in $M_\\mathbb{R}$, and for every $u\\in N_\\mathbb{R}$, $C_1^u+C_2^u=(C_1+C_2)^u$, where the plus sign denotes the Minkowski sum of convex sets. When $u\\in N_\\mathbb{Q}\\setminus\\{0\\}$, the intersection $M(u):=M\\cap u^\\perp$ is a lattice of rank $n-1$ spanning the linear space $u^\\perp$ and hence it induces a normalized Haar measure $\\vol_{M(u)}$ on $u^\\perp$, as in \\hyperref[lattice volume]{Remark \\ref*{lattice volume}}.\n\n\\begin{rem}\nLet $B_1,\\dots, B_n$ be $n$ convex bodies in $M_\\mathbb{R}$ and let $u\\in N_\\mathbb{R}$. The invariancy under translation of $\\vol_{M(u)}$ allows to consistently consider the mixed volume of $B_1^u,\\dots,B_n^u$ as convex bodies in $u^\\perp$.\n\\\\Similarly, let $g_0,\\dots,g_n$ be concave functions defined on convex bodies $B_0,\\dots,B_n$ in $M_\\mathbb{R}$, respectively. For every $u\\in N_\\mathbb{Q}\\setminus\\{0\\}$, \\hyperref[translation in mixed integral does not change the value]{Proposition \\ref*{translation in mixed integral does not change the value}} allows to consider the mixed integral with respect to $\\vol_{M(u)}$ of $g_0|_{B_0^u},\\dots,g_n|_{B_n^u}$, as functions defined on convex subsets of $u^\\perp$.\n\\end{rem}\n\nFor a concave function $g$ with effective domain a polytope $Q$ in $M_\\mathbb{R}$, its \\emph{hypograph} is the closed convex set \\[\\Gamma(g):=\\{(x,t):x\\in Q, t\\leq g(x)\\}\\subseteq M_\\mathbb{R}\\times\\mathbb{R}.\\]\nWith the pairing between $N_\\mathbb{R}\\times\\mathbb{R}$ and $M_\\mathbb{R}\\times\\mathbb{R}$ given by $\\langle(x,t),(u,\\lambda)\\rangle:=\\langle x,u\\rangle+t\\lambda$ for every $(u,\\lambda)\\in N_\\mathbb{R}\\times\\mathbb{R}$ and $(x,t)\\in M_\\mathbb{R}\\times\\mathbb{R}$, one can consider the subset $\\Gamma(g)^{(u,\\lambda)}$ of $\\Gamma(g)$ for every $(u,\\lambda)\\in N_\\mathbb{R}\\times\\mathbb{R}$. It is empty when $\\lambda>0$.\n\\\\The mixed real Monge-Amp\\`ere measure of piecewise affine concave functions can be made explicit in terms of the hypographs of their Legendre-Fenchel duals. We denote by $\\delta_v$ the Dirac measure supported on $v$.\n\n\\begin{proposition}\\label{real Monge-Ampere measure of piecewise affine functions}\nFor $i=1,\\dots,n$, let $g_i$ be a piecewise affine concave function with effective domain a polytope $Q_i\\subset M_\\mathbb{R}$, $\\Gamma_i$ the hypograph of $g_i$. Denote by $\\pi:M_\\mathbb{R}\\times\\mathbb{R}\\to M_\\mathbb{R}$ the projection onto the first factor. For any choice of a Haar measure $\\mu$ on $M_\\mathbb{R}$ one has \\[\\MM_\\mu\\big(g_1^\\vee,\\dots,g_n^\\vee\\big)=\\sum_{v\\in N_\\mathbb{R}}\\MV_\\mu\\bigg(\\pi\\Big(\\Gamma_1^{(v,-1)}\\Big),\\dots,\\pi\\Big(\\Gamma_n^{(v,-1)}\\Big)\\bigg)\\ \\delta_v,\\] and the sum is finite.\n\\end{proposition}\n\\begin{proof}\nFor a piecewise affine concave function $g$ with bounded domain in $M_\\mathbb{R}$, its Legendre-Fenchel dual is a piecewise affine concave function with domain $N_\\mathbb{R}$ and \\cite[Proposition 2.7.4]{BPS} affirms that \\[\\mathcal{M}_\\mu\\big(g^\\vee\\big)=\\sum_{v\\in N_\\mathbb{R}}\\mu(v^*)\\delta_v,\\]with $v^*=\\{x\\in M_\\mathbb{R}: g^\\vee(v)=\\langle x,v\\rangle-g(x)\\}$. From the definition of the Legendre-Fenchel duality, one has hence that\n\\begin{equation*}\n\\begin{split}\nv^*&=\\bigg\\{x\\in M_\\mathbb{R}:\\langle x,v\\rangle-g(x)=\\min_{y\\in M_\\mathbb{R}}(\\langle y,v\\rangle-g(y))\\bigg\\}\\\\&=\\Big\\{x\\in M_\\mathbb{R}:(x,g(x))\\in\\Gamma(g)^{(v,-1)}\\Big\\},\n\\end{split}\n\\end{equation*}\nand so \\[\\mathcal{M}_\\mu\\big(g^\\vee\\big)=\\sum_{v\\in N_\\mathbb{R}}\\mu\\Big(\\pi\\Big(\\Gamma(g)^{(v,-1)}\\Big)\\Big)\\delta_v.\\]\nThe sum is moreover supported on finitely many $v\\in N_\\mathbb{R}$, corresponding to the directions of the finitely many exposed faces of $\\Gamma(g)$.\n\\\\By \\cite[\\S2]{AW}, the relation \\[\\Gamma(g_i\\boxplus g_j)=\\Gamma(g_i)+\\Gamma(g_j)\\] on the hypographs of $g_i$ and $g_j$ holds for any $i,j\\in\\{1,\\dots,n\\}$. As a consequence, for every subset $\\{i_1,\\dots,i_k\\}\\subseteq\\{1,\\dots,n\\}$, \\cite[Proposition 2.3.1]{BPS} and the linearity of $\\pi$ yield\n\\begin{equation*}\n\\begin{split}\n\\mathcal{M}_\\mu\\big(g_{i_1}^\\vee+\\dots+g_{i_k}^\\vee\\big)&=\\mathcal{M}_\\mu\\big((g_{i_1}\\boxplus\\dots\\boxplus g_{i_k})^\\vee\\big)\\\\&=\\sum_{v\\in N_\\mathbb{R}}\\mu\\Big(\\pi\\Big(\\Gamma_{i_1}^{(v,-1)}+\\dots+\\Gamma_{i_k}^{(v,-1)}\\Big)\\Big)\\delta_v\\\\&=\\sum_{v\\in N_\\mathbb{R}}\\mu\\Big(\\pi\\Big(\\Gamma_{i_1}^{(v,-1)}\\Big)+\\dots+\\pi\\Big(\\Gamma_{i_k}^{(v,-1)}\\Big)\\Big)\\delta_v,\n\\end{split}\n\\end{equation*}\nand the sum is finite. The statement follows then from the definition of the mixed real Monge-Amp\\`ere measure, rearranging the terms.\n\\end{proof}\n\nWe can now prove a recursive formula relating the notions of the mixed real Monge-Amp\\`ere measure and the mixed integral of concave functions, via Legendre-Fenchel duality. A vector $u\\in N$ is said to be \\emph{primitive} if it is nonzero and there is no other element $u^\\prime\\in N$ such that $ku^\\prime=u$ for some positive integer $k$.\n\n\\begin{theorem}\\label{recursive formula with mixed integral and mixed real Monge-Ampere measure}\nFor $i=0,\\dots,n$, let $g_i$ be a continuous concave function on a rational polytope $Q_i$ in $M_\\mathbb{R}$. Then \n\\begin{multline*}\n\\MI_M(g_0,\\dots,g_n)=-\\sum_{\\substack{u\\in N\\\\\\primitive}}\\Psi_{Q_0}(u)\\MI_{M(u)}\\big(g_1|_{Q_1^{u}},\\dots,g_n|_{Q_n^{u}}\\big)\\\\-\\int_{N_\\mathbb{R}}g_0^\\vee\\ d\\MM_{M}\\big(g_1^\\vee,\\dots,g_n^\\vee\\big),\n\\end{multline*}\nthe first sum being finite.\n\\\\In particular, if $g_i$ is a piecewise affine concave function on $Q_i$ with hypograph $\\Gamma_i$ for any $i=0,\\dots,n$, denoting by $\\pi:M_\\mathbb{R}\\times\\mathbb{R}\\to M_\\mathbb{R}$ the projection onto the first factor, one has\n\\begin{multline*}\n\\MI_M(g_0,\\dots,g_n)=-\\sum_{\\substack{u\\in N\\\\\\primitive}}\\Psi_{Q_0}(u)\\MI_{M(u)}\\big(g_1|_{Q_1^{u}},\\dots,g_n|_{Q_n^{u}}\\big)\\\\-\\sum_{v\\in N_\\mathbb{R}}g_0^\\vee(v)\\MV_{M}\\bigg(\\pi\\Big(\\Gamma_1^{(v,-1)}\\Big),\\dots,\\pi\\Big(\\Gamma_n^{(v,-1)}\\Big)\\bigg).\n\\end{multline*}\n\\end{theorem}\n\\begin{proof}\nBy \\cite[Proposition 2.5.23 (1)]{BPS}, any continuous concave function on a polytope can be approximated, with respect to uniform convergence, by a sequence of piecewise affine concave functions on the polytope itself. On the other hand, the Legendre-Fenchel duality and the real Monge-Amp\\`ere operator are continuous with respect to uniform limits of concave functions, see \\cite[Proposition 2.2.3]{BPS} and \\cite[\\S3]{RT}, respectively. It is not difficult to show that the same holds for mixed integrals. Thanks to \\hyperref[real Monge-Ampere measure of piecewise affine functions]{Proposition \\ref*{real Monge-Ampere measure of piecewise affine functions}}, it is hence enough to prove the formula in the particular case of $g_0,\\dots,g_n$ being piecewise affine concave functions.\n\\\\Let hence $g_i$ be a concave piecewise affine function on the rational polytope $Q_i$ in $M_\\mathbb{R}$, $\\Gamma_i$ its hypograph, for $i=0,\\dots,n$. The choice of a basis of $N$ (and of the dual basis of $M$) endows $N_\\mathbb{R}$ and $M_\\mathbb{R}$ with an euclidean structure, allowing to consider the sets \\[\\mathbb{S}^{n-1}:=\\{w\\in N_\\mathbb{R}:\\|w\\|=1\\}\\subseteq N_\\mathbb{R}\\]and\\[\\mathbb{S}^n_-:=\\{(v,t)\\in N_\\mathbb{R}\\times\\mathbb{R}:\\|(v,t)\\|=1,t<0\\}\\subseteq N_\\mathbb{R}\\times\\mathbb{R}.\\] After a change of sign due to the use of a different notation, \\cite[Proposition 8.5]{PS2} affirms that\n\\begin{multline}\\label{equation in the proof of the recursive formula for mixed integrals}\n\\MI_M(g_0,\\dots,g_n)=-\\sum_{w\\in\\mathbb{S}^{n-1}}\\Psi_{Q_0}(w)\\MI_{n-1}\\big(g_1|_{Q_1^w},\\dots,g_n|_{Q_n^w}\\big)\\\\-\\sum_{r\\in\\mathbb{S}^n_-}\\Psi_{\\Gamma_0}(r)\\ \\MV_n(\\Gamma^r_1,\\dots,\\Gamma^r_n),\n\\end{multline}\nwhere, on the right hand side, one refers to the mixed integral with respect to the measure obtained restricting $\\vol_M$ to $w^\\perp$ and to the mixed volume with respect to the restriction of $\\vol_{M\\oplus\\mathbb{Z}}$ to $r^\\perp$.\n\\\\Concerning the first sum on the right hand side of \\eqref{equation in the proof of the recursive formula for mixed integrals}, if a term in the sum is different from zero, then there exists a subset $I\\subset\\{1,\\dots,n\\}$ such that the Minkowski sum of $Q_i^w$, with $i\\in I$, is of dimension $n-1$; in particular, denoting $Q:=Q_1+\\dots+Q_n$, $Q^w=Q_1^w+\\dots+Q_n^w$ needs to be of dimension $n-1$. As a consequence, one can restrict the sum to the set of vectors $w\\in\\mathbb{S}^{n-1}$ for which $Q^w$ is a $(n-1)$-dimensional face of $Q$. This set is included in the set of vectors of unitary length which are perpendicular to a $(n-1)$-dimensional face of $Q$, hence it is finite since $Q$ is a polytope. Moreover, since $Q$ is rational, the ray spanned by such a vector $w$ contains a unique primitive vector $u\\in N$. The linearity of $\\Psi_{Q_0}$ yields hence the equality\n\\[\\sum_{w\\in\\mathbb{S}^{n-1}}\\Psi_{Q_0}(w)\\MI_{n-1}\\big(g_1|_{Q_1^w},\\dots,g_n|_{Q_n^w}\\big)=\\sum_{\\substack{u\\in N\\\\\\primitive}}\\frac{\\Psi_{Q_0}(u)}{\\|u\\|}\\MI_{n-1}\\big(g_1|_{Q_1^u},\\dots,g_n|_{Q_n^u}\\big).\n\\]\nThe fact that the restriction of $\\vol_M$ to $u^\\perp$ is equal to the measure $\\vol_{M(u)}$ multiplied by $\\|u\\|$, see \\cite[proof of Corollary 2.7.10]{BPS}, allows to conclude that the first sum in \\eqref{equation in the proof of the recursive formula for mixed integrals} coincides with the desired one.\n\\\\Regarding the second sum in \\eqref{equation in the proof of the recursive formula for mixed integrals}, there exists an obvious bijection between $\\mathbb{S}^n_-$ and $N_\\mathbb{R}$ given by associating to each $r\\in\\mathbb{S}^n_-$ the only vector $v\\in N_\\mathbb{R}$ such that $(v,-1)$ lies on the line spanned by $r$. Hence,\n\\[\\sum_{r\\in\\mathbb{S}^n_-}\\Psi_{\\Gamma_0}(r)\\ \\MV_n(\\Gamma^r_1,\\dots,\\Gamma^r_n)=\\sum_{v\\in N_\\mathbb{R}}\\frac{\\Psi_{\\Gamma_0}(v,-1)}{\\|(v,-1)\\|}\\ \\MV_n\\Big(\\Gamma^{(v,-1)}_1,\\dots,\\Gamma^{(v,-1)}_n\\Big).\\]Directly by the definition of the Legendre-Fenchel duality, one has that $\\Psi_{\\Gamma_0}(v,-1)=g_0^\\vee(v)$. The statement follows then from the fact that for every Borel set $E$ in $(v,-1)^\\perp$, the measure of $E$ with respect to the restriction of $\\vol_{M\\oplus\\mathbb{Z}}$ to $(v,-1)^\\perp$ equals $\\|(v,-1)\\|\\cdot\\vol_M(\\pi(E))$, again by \\cite[proof of Corollary 2.7.10]{BPS}.\n\\end{proof}\n\n\\begin{rem}\\label{recursive formula with sum over the faces}\nFor a rational polytope $P$ of full dimension $n$ in $M_\\mathbb{R}$, every facet $F$ of $P$, that is a face of dimension $n-1$, admits a distinguished orthogonal vector: it is the unique primitive vector $v_F\\in N$ which satisfies $P^{v_F}=F$. Under the additional assumption that the Minkowski sum $Q:=Q_1+\\dots+Q_n$ is of dimension $n$ in $M_\\mathbb{R}$, the formula in \\hyperref[recursive formula with mixed integral and mixed real Monge-Ampere measure]{Theorem \\ref*{recursive formula with mixed integral and mixed real Monge-Ampere measure}} can be written as\n\\begin{multline*}\n\\MI_M(g_0,\\dots,g_n)=-\\sum_{F}\\Psi_{Q_0}(v_F)\\MI_{M(v_F)}\\big(g_1|_{Q_1^{v_F}},\\dots,g_n|_{Q_n^{v_F}}\\big)\\\\-\\int_{N_\\mathbb{R}}g_0^\\vee\\ d\\MM_{M}\\big(g_1^\\vee,\\dots,g_n^\\vee\\big),\n\\end{multline*}\nthe first sum being over the finite set of facets of the polytope $Q$. Indeed, in such a situation the application $F\\mapsto v_F$ realizes a bijection between the set of facets of $Q$ and the set of primitive vectors $u\\in N$ for which $Q^u$ is a $(n-1)$-dimensional face of $Q$, which are the only vectors for which the term of the sum in the statement of the theorem does not vanish.\n\\end{rem}\n\n\\begin{rem}\\label{remark for the Legendre-Fenchel dual version of the recursive formula for mixed integrals}\nThe statement of \\hyperref[recursive formula with mixed integral and mixed real Monge-Ampere measure]{Theorem \\ref*{recursive formula with mixed integral and mixed real Monge-Ampere measure}} can be reformulated in terms of Legendre-Fenchel duality. For $i=0,\\dots,n$, let $f_i$ be a concave function on $N_\\mathbb{R}$ with stability set a rational polytope $Q_i$ in $M_\\mathbb{R}$. Under the assumption that $Q_1+\\dots+Q_n$ is of dimension $n$ in $M_\\mathbb{R}$, \\hyperref[recursive formula with sum over the faces]{Remark \\ref*{recursive formula with sum over the faces}} yields\n\\begin{multline}\\label{version of recursive formula in terms of Legendre-Fenchel duals}\n\\MI_M(f^\\vee_0,\\dots,f^\\vee_n)=-\\sum_{F}\\Psi_{Q_0}(v_F)\\MI_{M(v_F)}\\big(f^\\vee_1|_{Q_1^{v_F}},\\dots,f^\\vee_n|_{Q_n^{v_F}}\\big)\\\\-\\int_{N_\\mathbb{R}}f_0\\ d\\MM_{M}\\big(f_1,\\dots,f_n\\big).\n\\end{multline}\nIndeed, it is sufficient to readily apply the previous theorem to the functions $f_0^\\vee,\\dots,f_n^\\vee$, which are continuous on their domain and satisfy the equality $(f_i^\\vee)^\\vee=f_i$ for each $i=0,\\dots,n$ by concavity and closedness. It is easy to verify that the choice $f_0=\\dots=f_n=f$ in \\eqref{version of recursive formula in terms of Legendre-Fenchel duals} yields the formula in \\cite[Corollary 2.7.10]{BPS}.\n\\end{rem}\n\nWe present now two applications of the recursive formula proved above. The first one concerns the computation of the mixed integral when all except one entry are indicator functions in the sense of \\hyperref[indicator and support function]{Example \\ref*{indicator and support function}}.\n\n\\begin{corollary}\\label{mixed integral with indicator functions}\nLet $Q_1,\\dots,Q_n$ be rational polytopes in $M_\\mathbb{R}$ and $f$ a concave function on $N_\\mathbb{R}$ with stability set a rational polytope. Then\n\\[\\MI_M\\big(\\iota_{Q_1},\\dots,\\iota_{Q_n},f^\\vee\\big)=-\\MV_M(Q_1,\\dots,Q_n)\\cdot f(0).\\]\n\\end{corollary}\n\\begin{proof}\nBy symmetry, one can develop the recursive formula in \\hyperref[remark for the Legendre-Fenchel dual version of the recursive formula for mixed integrals]{Remark \\ref*{remark for the Legendre-Fenchel dual version of the recursive formula for mixed integrals}} with respect to $f^\\vee$ to obtain\n\\[\\MI_M\\big(\\iota_{Q_1},\\dots,\\iota_{Q_n},f^\\vee\\big)=-\\int_{N_\\mathbb{R}}f\\ d\\MM_M\\Big(\\iota_{Q_1}^\\vee,\\dots,\\iota_{Q_n}^\\vee\\Big),\\]\nthe indicator functions $\\iota_{Q_1},\\dots,\\iota_{Q_n}$ being zero where defined. The duality in \\hyperref[indicator and support function]{Example \\ref*{indicator and support function}} and the fact that\n\\[\\MM_M\\big(\\Psi_{Q_1},\\dots,\\Psi_{Q_n}\\big)=\\MV_M(Q_1,\\dots,Q_n)\\delta_0\\]\nbecause of \\hyperref[real Monge-Ampere measure of piecewise affine functions]{Proposition \\ref*{real Monge-Ampere measure of piecewise affine functions}} conclude the proof.\n\\end{proof}\n\nThe second application explains how the mixed integral behaves with respect to pointwise sum by a constant in one entry.\n\n\\begin{corollary}\\label{mixed integral with pointwise sum by a constant}\nLet $g_i$ be a concave function defined on a rational polytope $Q_i\\subseteq M_\\mathbb{R}$, for $i=0,\\dots,n$, and $c\\in\\mathbb{R}$. Then\n\\[\\MI_M(g_0,\\dots,g_{n-1},g_n+c)=\\MI_M(g_0,\\dots,g_n)+c\\cdot\\MV_M(Q_0,\\dots,Q_{n-1}).\\]\n\\end{corollary}\n\\begin{proof}\nDenoting by $c\\delta_0$ the concave function which has value $c$ at $0$ and $-\\infty$ otherwise, it follows from the definitions that $g_n+c=g_n\\boxplus c\\delta_0$. The multilinearity of mixed integrals implies then that \\[\\MI_M(g_0,\\dots,g_{n-1},g_n+c)=\\MI_M(g_0,\\dots,g_n)+\\MI_M(g_0,\\dots,g_{n-1},c\\delta_0).\\]\nUsing the fact that $(c\\delta_0)^\\vee=-c$, the recursive formula in \\hyperref[recursive formula with mixed integral and mixed real Monge-Ampere measure]{Theorem \\ref*{recursive formula with mixed integral and mixed real Monge-Ampere measure}}, developed with respect to $c\\delta_0$, yields\n\\[\\MI_M(g_0,\\dots,g_{n-1},c\\delta_0)=\\int_{N_\\mathbb{R}}c\\ d\\MM_M(g_0^\\vee,\\dots,g_{n-1}^\\vee)=c\\cdot\\MM_M(g_0^\\vee,\\dots,g_{n-1}^\\vee)(N_\\mathbb{R}).\\]\nThe statement follows then from the fact that the total volume of the mixed Monge-Amp\\`ere measure of $g_0^\\vee,\\dots,g_{n-1}^\\vee$ is equal to $\\MV_M(\\dom(g_0),\\dots,\\dom(g_{n-1}))$ by \\cite[Proposition 3(iv)]{PR}.\n\\end{proof}\n\nWe conclude the section by proving a formula expressing the mixed integral of a $(n+1)$-tuple of concave functions on $M_\\mathbb{R}$ where one of them is the indicator function of a line segment. \\\\Let $m$ be a primitive vector of $M$ and consider the quotient $P:=M\/\\mathbb{Z}m$. Since $m$ is primitive, $P$ is a lattice of rank $n-1$. By abuse of notation, let $\\pi$ denote both the projection from $M$ to $P$ and the induced linear map from $M_\\mathbb{R}$ to $P_\\mathbb{R}$. For each closed concave function $g$ defined on a compact subset $B$ of $M_\\mathbb{R}$, let\n\\begin{equation}\\label{definition of direct image of concave functions}\n\\pi_*g:\\pi(B)\\to\\mathbb{R},\\quad x\\mapsto\\max_{y\\in \\pi^{-1}(x)}g(y)\n\\end{equation}\nbe the \\emph{direct image} of $g$ by $\\pi$. It is a well defined closed concave function with domain a bounded subset of $P_\\mathbb{R}$, see \\cite[Theorem 5.7 and Theorem 9.2]{R}. Finally, for $x_1,x_2\\in M_\\mathbb{R}$, denote by $\\overline{x_1x_2}$ the line segment in $M_\\mathbb{R}$ with extremal points $x_1$ and $x_2$. The following lemma is a generalization of \\cite[exercise 3 at page 128]{Ewald} and seems to be well-known to experts, though we could not find an adequate reference in the literature for its proof.\n\n\\begin{lemma}\\label{lemma for mixed volumes and projections}\nIn the above hypotheses and notations and for $n\\geq2$, let $Q_1,\\dots,Q_{n-1}$ be polytopes in $M_\\mathbb{R}$. Then,\n\\[\\MV_M\\big(\\overline{0m},Q_1,\\dots,Q_{n-1}\\big)=\\MV_P\\big(\\pi(Q_1),\\dots,\\pi(Q_{n-1})\\big).\\]\n\\end{lemma}\n\\begin{proof}\nThe vector $m$ being primitive, it can be extended to a basis of the lattice $M$, see for instance \\cite[Theorem 5 at page 21]{Lekk}. We suppose fixed throughout the proof such a basis $(m_1,\\dots,m_{n-1},m)$ of $M$ and the induced isomorphism $M_\\mathbb{R}\\simeq\\mathbb{R}^n$; under this identification, the normalized volume $\\vol_M$ corresponds to the Lebesgue measure $ \\vol_n$ on $\\mathbb{R}^n$. Since $(\\pi(m_1),\\dots,\\pi(m_{n-1}))$ is a basis of $P$, such a lattice is isomorphic to the span of $m_1,\\dots,m_{n-1}$ in $M$ and hence it is identified with the linear subspace $\\mathbb{R}^{n-1}\\times\\{0\\}$ of $\\mathbb{R}^n$. Moreover, $\\vol_P$ corresponds to the $(n-1)$-dimensional Lebesgue measure $\\vol_{n-1}$ on $\\mathbb{R}^{n-1}\\times\\{0\\}$ and the map $\\pi$ to the vertical projection.\n\\\\The claim reduces then to the particular case of a family of polytopes $Q_1,\\dots,Q_{n-1}$ in $\\mathbb{R}^n$, $m=(0,\\dots,0,1)$ and $\\pi$ the vertical projection. Denoting by $S$ the vertical segment of unitary length and rearranging the terms in the definition of the mixed volume given for instance in \\cite[Definition 2.7.14]{BPS} one obtains\n\\begin{multline*}\n\\MV_n(S,Q_1,\\dots,Q_{n-1})=\\\\\\sum_{k=1}^{n-1}(-1)^{n-1-k}\\sum_{1\\leq i_1<\\dots<i_k\\leq n-1}\\big(\\vol_n(S+Q_{i_1}+\\dots+Q_{i_k}\\big)-\\vol_n(Q_{i_1}+\\dots+Q_{i_k})\\big)\n\\end{multline*}\nsince the $n$-dimensional volume of a line segment vanishes for $n\\geq2$. To prove the claim it is hence enough to show that for each polytope $Q$ in $\\mathbb{R}^n$ the equality\n\\[\\vol_n(S+Q)-\\vol_n(Q)=\\vol_{n-1}(\\pi(Q))\\]\nholds. But $Q\\subset S+Q$ and the difference of their volumes coincides with the integral over $\\pi(Q)=\\pi(S+Q)$ of the difference between the concave functions parametrizing the roof of the polytope $S+Q$ and $Q$ respectively. Such a difference being constantly equal to $1$ on $\\pi(Q)$, the claim follows from the definition of the Lebesgue integral, concluding the proof.\n\\end{proof}\n\n\\begin{proposition}\\label{mixed integral with indicator function of a segment}\nIn the above hypotheses and notations, let $g_i$ be a continuous concave function defined on a polytope $Q_i$ in $M_\\mathbb{R}$, for $i=1,\\dots,n$. Then, \\[\\MI_M\\big(\\iota_{\\overline{0m}},g_1,\\dots,g_n\\big)=\\MI_P(\\pi_*g_1,\\dots,\\pi_*g_n).\\]\n\\end{proposition}\n\\begin{proof}\nFor $n=1$, the claim follows from \\hyperref[mixed integral with indicator functions]{Corollary \\ref*{mixed integral with indicator functions}}. Assume hence $n\\geq2$. Choose for each $i=1,\\dots,n$ a nonpositive real number $\\gamma_i$ such that $\\gamma_i\\leq\\min_{x\\in Q_i} g_i(x)$ and consider the convex body\\[Q_{g_i,\\gamma_i}:=\\{(x,t)\\in Q_i\\times\\mathbb{R}: \\gamma_i\\leq t\\leq g_i(x)\\}\\]in $M_\\mathbb{R}\\times\\mathbb{R}$. The formula in \\cite[Proposition IV.5 (d)]{PS1} implies that\n\\begin{multline*}\n\\MI_M\\big(\\iota_{\\overline{0m}},g_1,\\dots,g_n\\big)=\\MV_{M\\oplus\\mathbb{Z}}\\Big(\\overline{(0,0)(m,0)},Q_{g_1,\\gamma_1},\\dots,Q_{g_n,\\gamma_n}\\Big)\\\\+\\sum_{i=1}^n\\gamma_i\\MV_M\\big(\\overline{0m},Q_1,\\dots,Q_{i-1},Q_{i+1},\\dots,Q_n\\big).\n\\end{multline*}\nBy the hypotheses on $m$, $(m,0)$ is a nonzero primitive vector of the lattice $M\\oplus\\mathbb{Z}$. The map $\\pi^\\prime:=\\pi\\times\\id_\\mathbb{Z}:M\\oplus\\mathbb{Z}\\to P\\oplus\\mathbb{Z}$ is a surjective group homomorphism, giving $(M\\oplus\\mathbb{Z})\/\\mathbb{Z}(m,0)\\simeq P\\oplus\\mathbb{Z}$. By \\hyperref[lemma for mixed volumes and projections]{Lemma \\ref*{lemma for mixed volumes and projections}},\n\\begin{multline*}\n\\MI_M\\big(\\iota_{\\overline{0m}},g_1,\\dots,g_n\\big)=\\MV_{P\\oplus\\mathbb{Z}}\\big(\n\\pi^\\prime(Q_{g_1,\\gamma_1}),\\dots,\\pi^\\prime(Q_{g_n,\\gamma_n})\\big)\\\\+\\sum_{i=1}^n\\gamma_i\\MV_P\\big(\\pi(Q_1),\\dots,\\pi(Q_{i-1}),\\pi(Q_{i+1}),\\dots,\\pi(Q_n)\\big).\n\\end{multline*}\nThe statement follows hence from the equality in \\cite[Proposition IV.5 (d)]{PS1} applied to the concave functions $\\pi_*g_1,\\dots,\\pi_*g_n$, the direct image of $g_i$ by $\\pi$ being a concave function defined on $\\pi(Q_i)$ and satisfying\n\\begin{equation*}\n\\begin{split}\n\\pi^\\prime(Q_{g_i,\\gamma_i})&=\\{(\\pi(y),t)\\in \\pi(Q_i)\\times\\mathbb{R}: \\gamma_i\\leq t\\leq g_i(y)\\}\\\\&=\\big\\{(x,t)\\in \\pi(Q_i)\\times\\mathbb{R}: \\gamma_i\\leq t\\leq (\\pi_*g_i)(x)\\big\\}\n\\end{split}\n\\end{equation*}\nfor every $i=1,\\dots,n$.\n\\end{proof}\n\n\n\n\\section{Ronkin functions}\\label{section about Ronkin functions}\n\nFix for the whole section an algebraically closed complete field $(K,|\\cdot|)$, that is a pair of an algebraically closed field and an absolute value with respect to which the field is complete. Depending on the nature of the absolute value, $(K,|\\cdot|)$ is said to be archimedean or non-archimedean. As a consequence of Gelfand-Mazur theorem, the only archimedean algebraically closed complete field is $\\mathbb{C}$ endowed with a power of the usual absolute value. When the choice of the absolute value is clear from the context, an algebraically closed complete field will be simply denoted by $K$.\n\n\n\\subsection{Tropicalization}\n\nGiven an affine variety $X=\\spec A$ over an algebraically closed complete field $K$, let $\\iota:K\\to A$ be the corresponding ring homomorphism. Berkovich's construction allows to define a locally ringed space $(X^{\\an},\\mathcal{O}_{X^{\\an}})$, called the \\emph{analytification} of $X$; as a set\n\\[X^{\\an}:=\\{\\|\\cdot\\|_x \\text{ multiplicative seminorm on }A: \\|\\iota(k)\\|_x=|k| \\text{ for all }k\\in K\\}.\\]\nWe refer to \\cite[\\S1.5]{Ber} (or also to \\cite[\\S1.2]{BPS} for a more concise treatment) for the definition of the topology and the structure sheaf on $X^{\\an}$. By a gluing argument, one can then extend the definition to any variety over $K$.\n\n\\begin{rem}\nWhen $K=\\mathbb{C}$ endowed with the usual absolute value, the Gelfand-Mazur theorem implies that the locally ringed space produced by Berkovich's construction agrees with the standard complex analytification of a variety over $\\spec \\mathbb{C}$.\n\\end{rem}\n\nLet $\\mathbb{T}$ denote a split torus over $K$ of dimension $n$ and $M$ its character lattice, in such a way that $\\mathbb{T}=\\spec K[M]$. A basis of the $K$-algebra $K[M]$ will be denoted, accordingly to \\cite[beginning of \\S1.3]{Ful}, by $(\\chi^m)_{m\\in M}$ and the elements of $K[M]$ will be called \\emph{Laurent polynomials} over $K$. Let $N$ be the dual lattice of $M$ and denote by $N_\\mathbb{R}=N\\otimes\\mathbb{R}$ the associated real vector space.\n\n\\begin{defn}\\label{definition of tropicalization}\nThe \\emph{tropicalization map} $\\trop:\\mathbb{T}^{\\an}\\to N_\\mathbb{R}=\\hom(M,\\mathbb{R})$ is the application defined by \\[\\left(\\trop(\\|\\cdot\\|_x)\\right)(m):=-\\log\\|\\chi^m\\|_x\\] for every $\\|\\cdot\\|_x\\in\\mathbb{T}^{\\an}$.\n\\end{defn}\n\nThe tropicalization map turns out to be a continuous application with respect to the Berkovich topology of $\\mathbb{T}^{\\an}$ and the Euclidean topology of $N_\\mathbb{R}$. Moreover, in the archimedean case, the choice of a basis of $M$ allows to write such map in the more familiar form \\[\\trop((z_1,\\dots,z_n))=(-\\log|z_1|,\\dots,-\\log|z_n|).\\] \n\n\\begin{rem}\nThe tropicalization map coincides with the valuation map $\\val$ used by Burgos Gil, Philippon and Sombra, see \\cite[Equation 4.1.2]{BPS}.\n\\end{rem}\n\nWhen the absolute value on $K$ is non-archimedean, one can construct a suitable section of $\\trop$. For each $u\\in N_\\mathbb{R}$, consider the map which associates to a Laurent polynomial $f=\\sum c_m\\chi^m$ the real value \\[\\|f\\|_{\\kappa(u)}:=\\max_m |c_m|e^{-\\langle m,u\\rangle}.\\] One can verify that $\\|\\cdot\\|_{\\kappa(u)}$ is a multiplicative seminorm on $K[M]$ extending the absolute value on $K$, and hence it corresponds to a point $\\kappa(u)\\in\\mathbb{T}^{\\an}$, which one calls the \\emph{Gauss point} over $u$. The application $\\kappa:N_\\mathbb{R}\\to\\mathbb{T}^{\\an}$ defined by \\[\\kappa:u\\to\\|\\cdot\\|_{\\kappa(u)}\\] is proved to be a continuous section of $\\trop$, see for instance \\cite[Proposition-Definition 4.2.12]{BPS}, $\\kappa$ coinciding with $\\theta_0\\circ\\mathbf{e}$ in the cited reference.\n\n\\vspace{\\baselineskip}\nIt is easily seen that for any closed subvariety $X$ of $\\mathbb{T}$, there is a natural inclusion of sets $X^{\\an}\\subseteq\\mathbb{T}^{\\an}$, making the following definition meaningful.\n\n\\begin{defn}\nLet $f$ be a nonzero Laurent polynomial in $K[M]$ and $V(f)$ the associated closed subvariety of $\\mathbb{T}$. The subset \\[\\mathcal{A}_f:=\\trop\\left(V(f)^{\\an}\\right)\\subseteq N_\\mathbb{R}\\]is called the \\emph{amoeba} of $f$.\n\\end{defn}\n\nThe so-defined set has been widely studied in the literature. In the archimedean case, it coincides (up to a change of sign) with the notion of amoeba studied by Gelfand, Kapranov and Zelevinsky in \\cite{GKZ} and by Passare and Rullg\\aa rd in \\cite{PR}. In the non-archimedean case, the amoeba of a nonzero Laurent polynomial $f$ coincides with the corner locus of the associated tropical polynomial $f^{\\trop}$ (see \\cite[Theorem 2.1.1]{EKL}).\n\n\n\\subsection{Minimal boundaries}\\label{section on boundaries}\n\nTo keep the treatment of the archimedean and non-archimedean settings uniform, the following notion results to be crucial.\n\n\\begin{defn}\nLet $K$ be an algebraically closed complete field and $A$ a $K$-algebra. For any set $\\mathcal{S}$ of multiplicative seminorms on $A$ extending the absolute value of $K$, a \\emph{boundary} of $\\mathcal{S}$ is a subset $\\mathcal{B}\\subseteq\\mathcal{S}$ such that\\[\\max_{x\\in\\mathcal{S}}\\|a\\|_x=\\max_{x\\in\\mathcal{B}}\\|a\\|_x\\]for every $a\\in A$. \n\\end{defn}\n\nIn other words, any boundary of $\\mathcal{S}$ contains the whole information about the maximal values that the seminorms in $\\mathcal{S}$ can attain. As a trivial example, $\\mathcal{S}$ is a boundary of $\\mathcal{S}$.\n\\\\We are especially interested in the boundary of the fibers of the tropicalization map. Keeping the notation of the previous subsection, for a point $u\\in N_\\mathbb{R}$, it is immediate to show that \\[\\trop^{-1}(u)=\\left\\{\\|\\cdot\\|_x\\in\\mathbb{T}^{\\an}:\\|\\chi^m\\|_x=e^{-\\langle m,u\\rangle}\\text{ for all }m\\in M\\right\\}.\\]In the complex case, the existence of a unique minimal boundary for an algebra of functions on a compact space has been proved by Shilov. The following result, which equally holds in the non-archimedean case, is well-known by experts.\n\n\\begin{proposition}\\label{minimal boundary}\nThe set $\\trop^{-1}(u)$ has a unique minimal boundary. In the archime\\-dean case, it coincides with the whole $\\trop^{-1}(u)$, while in the non-archimedean case it consists of the Gauss point over $u$.\n\\end{proposition}\n\\begin{proof}\nIn the archimedean case, after having chosen a basis of $M$, one can treat $\\mathbb{T}^{\\an}$ as $(\\mathbb{C}^*)^n$, by identifying a point $z=(z_1,\\dots,z_n)$ of $(\\mathbb{C}^*)^n$ with the seminorm $\\|\\cdot\\|_z$ defined as \\[\\|f\\|_z=|f(z)|\\] for every $f\\in K[T_1,\\dots,T_n]$. Assuming that the coordinates of $u$ in the basis dual to the chosen one are $(u_1,\\dots,u_n)$, the set $\\trop^{-1}(u)$ corresponds to the subset of $(\\mathbb{C}^*)^n$ consisting of the points $(z_1,\\dots,z_n)$ satisfying $-\\log|z_i|=u_i$ for every $i=1,\\dots,n$. A point $\\widetilde{z}$ of $\\trop^{-1}(u)$ is then of the form \\[\\widetilde{z}:=(e^{-u_1+i\\theta_1},\\dots,e^{-u_n+i\\theta_n}),\\]with $\\theta_i\\in[0,2\\pi)$ for every $i=1,\\dots,n$. It is easy to check that such a point is the only one in $\\trop^{-1}(u)$ for which \\[\\left\\|(T_1+e^{-u_1+i\\theta_1})\\dots(T_n+e^{-u_n+i\\theta_n})\\right\\|_{z}\\] is maximal. As a consequence, $\\widetilde{z}$ must belong to any boundary of $\\trop^{-1}(u)$. This proves that the only boundary of $\\trop^{-1}(u)$ is the set itself.\n\\\\In the non-archimedean case, for every $\\|\\cdot\\|_x\\in\\trop^{-1}(u)$ and for every $f=\\sum c_m\\chi^m\\in K[M]$ one has (by definition of non-archimedean absolute value) that:\\[\\|f\\|_x\\leq\\max_m|c_m|\\|\\chi^m\\|_x=\\max_m|c_m|e^{-\\langle m,u\\rangle}.\\]This trivially implies that the Gauss norm $\\|f\\|_{\\kappa(u)}=\\max_m|c_m|e^{-\\langle m,u\\rangle}$ is a boundary for $\\trop^{-1}(u)$.\n\\end{proof}\n\nFor $u\\in N_\\mathbb{R}$, one denotes the unique minimal boundary of $\\trop^{-1}(u)$ described in the previous proposition by $\\mathcal{B}(u)$. If not otherwise mentioned, such a set is considered to be endowed with the topology induced from $\\mathbb{T}^{\\an}$. If one sets\n\\[\n\\mathcal{B}_K:=\n\\begin{cases}\n\\big(\\mathbb{S}^1\\big)^n  & \\text{if }K\\text{ is archimedean}\\\\\n\\{1\\} & \\text{otherwise}\n\\end{cases},\n\\]\nthe boundary $\\mathcal{B}(u)$ is homeomorphic to the compact group $\\mathcal{B}_K$ for every $u\\in N_\\mathbb{R}$. This allows to define the measure \\[\\sigma_u:=\\Haar_{\\mathcal{B}(u)}\\] on $\\trop^{-1}(u)$, which is the Haar measure on the compact group $\\mathcal{B}(u)$ normalized to have total mass $1$; it is a finite measure on $\\trop^{-1}(u)$, supported on $\\mathcal{B}(u)$ and distributing homogeneously on this set. In the non-archimedean case it coincides with the Dirac delta at the Gauss point over $u$.\n\\\\There exists an embedding $\\iota:N_\\mathbb{R}\\times\\mathcal{B}_K\\to\\mathbb{T}^{\\an}$, fitting in the commutative diagram\n\\begin{equation}\\label{diagram}\n\\begin{tikzcd}\nN_\\mathbb{R}\\times\\mathcal{B}_K \\arrow[r,hook,\"\\iota\"] \\arrow[d] & \\mathbb{T}^{\\an} \\arrow[dl,\"\\trop\"]\\\\\nN_\\mathbb{R}\n\\end{tikzcd}\n\\end{equation}\nwith the vertical arrow being the projection onto the first factor. In the archimedean case, it is determined by the choice of a homeomorphism $\\big(\\mathbb{C}^*\\big)^n\\simeq N_\\mathbb{R}\\times\\big(\\mathbb{S}^1\\big)^n$, while in the non-archimedean case it coincides with the map $(u,1)\\mapsto\\kappa(u)$. The image of $\\iota$ is homeomorphic to $N_\\mathbb{R}\\times\\mathcal{B}_K$ and it is a deformation retract of the analytic torus, coinciding with it if $K$ is archimedean.\n\n\n\\subsection{Ronkin functions}\n\nThe terminology and notations introduced in the previous subsection allow to define Ronkin functions in the archimedean and non-archimedean case simultaneously.\n\n\\begin{defn}\nLet $f$ be a nonzero Laurent polynomial over $K$. The \\emph{Ronkin function} of $f$ is the map $\\rho_f:N_\\mathbb{R}\\to\\mathbb{R}$ defined as \\[\\rho_f(u):=\\int_{\\trop^{-1}(u)}-\\log\\|f\\|_x\\ d\\sigma_u(x)\\]for every $u\\in N_\\mathbb{R}$.\n\\end{defn}\n\nThe integral in the previous definition is finite. Indeed, logarithmic singularities are integrable in the archimedean case, and the Gauss norm of a nonzero Laurent polynomial is positive in the non-archimedean case.\n\n\\begin{rem}\\label{remark about known version of Ronkin functions}\nIn the archimedean case, $\\rho_f(u)=-N_f(-u)$, where $N_f$ is the classical Ronkin function associated to a complex Laurent polynomial (refer for instance to \\cite{PR}). In the non-archimedean case, it is easily checked that $\\rho_f=f^{\\trop}$, where the tropicalization of a Laurent polynomial $f=\\sum c_m\\chi^m$ is defined by \\[f^{\\trop}(u)=\\min_m(\\langle m,u\\rangle-\\log|c_m|)\\]for every $u\\in N_\\mathbb{R}$.\n\\end{rem}\n\nThe following property of the Ronkin function follows immediately from its definition.\n\n\\begin{proposition}\\label{Ronkin function of a product}\nFor every pair of nonzero Laurent polynomials $f$ and $g$, $\\rho_{f\\cdot g}=\\rho_f+\\rho_g$. For every $\\lambda\\in K$, moreover, $\\rho_{\\lambda\\cdot f}=-\\log|\\lambda|+\\rho_f$.\n\\end{proposition}\n\nRecall that for any Laurent polynomial $f=\\sum_mc_m\\chi^m$, we denote by $\\NP(f)$ the \\emph{Newton polytope of $f$}, that is the convex hull in $M_\\mathbb{R}$ of the set $\\{m\\in M:c_m\\neq0\\}$. Its \\emph{support function}, as defined in \\hyperref[indicator and support function]{Example \\ref*{indicator and support function}}, is denoted by $\\Psi_{\\NP(f)}$.\n\n\\begin{proposition}\\label{properties of Ronkin functions}\nLet $f$ be a nonzero Laurent polynomial. Then:\n\\begin{enumerate}\n\\item $\\rho_f$ is a continuous concave function on $N_\\mathbb{R}$ (in particular it is closed) and it is affine on each connected component of the complement of the amoeba of $f$\n\\item $|\\rho_f-\\Psi_{\\NP(f)}|$ is bounded on $N_\\mathbb{R}$\n\\item the stability set of $\\rho_f$ coincides with $\\NP(f)$ and $\\rec(\\rho_f)=\\Psi_{\\NP(f)}$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nThe statements in (1) are trivial in the non-archimedean case. In the archime\\-dean case, the concavity of $\\rho_f$ and its affinity outside the amoeba are shown in \\cite[Theorem 1]{PR}. As a consequence of concavity, $\\rho_f$ is continuous on $N_\\mathbb{R}$, hence closed. \n\\\\To prove $(2)$, suppose that $f=\\sum_mc_m\\chi^m$ and let $\\gamma_K(f)$ be the number of nonzero coefficients of $f$ if $K$ is archimedean, $1$ otherwise. For any $x\\in\\trop^{-1}(u)$, the inequality $\\|f\\|_x\\leq \\gamma_K(f)\\cdot\\max_m(|c_m|\\|\\chi^m\\|_x)$ implies \\[-\\log\\|f\\|_x\\geq \\min_m(\\langle m,u\\rangle-\\log|c_m|)-\\log \\gamma_K(f)\\geq \\Psi_{\\NP(f)}(u)-\\max_m\\log|c_m|-\\log \\gamma_K(f)\\]and hence \\[\\rho_f(u)\\geq \\Psi_{\\NP(f)}(u)-\\log(\\gamma_K(f)\\max_m|c_m|)\\] for every $u\\in N_\\mathbb{R}$. For a reverse inequality, denote by $\\mathcal{V}(f)$ the set of vertices of $\\NP(f)$. Then, for every $m\\in\\mathcal{V}(f)$ the Ronkin function of $f$ coincides with $\\langle m,u\\rangle-\\log|c_m|$ in a nonempty open subset of $N_\\mathbb{R}$ (this follows from \\cite[Proposition 2]{PR} in the archimedean setting and from \\cite[Corollary 2.1.2]{EKL} in the non-archimedean one). By the concavity of $\\rho_f$, one deduces hence that \\[\\rho_f(u)\\leq \\Psi_{\\NP(f)}(u)-\\log\\min_{m\\in\\mathcal{V}(f)}|c_m|,\\] for every $u\\in N_\\mathbb{R}$, concluding the proof of (2).\n\\\\The statements in (3) follows directly from (2). Indeed, since $\\big|\\rho_f-\\Psi_{\\NP(f)}\\big|$ is bounded, $\\stab(\\rho_f)=\\stab\\big(\\Psi_{\\NP(f)}\\big)=\\NP(f)$; the last equality follows then from \\cite[Theorem 13.3]{R}.\n\\end{proof}\n\nThe calculation of the Ronkin function of a nonzero Laurent polynomial is typically very difficult. Anyway, an explicit expression for it is available in the following two simple situations.\n\n\\begin{ex}\\label{example Ronkin of a monomial}\nIt follows from the definition that the Ronkin function of the monomial $\\chi^m$ coincides with the linear function $m$ on $N_\\mathbb{R}$ for every $m\\in M$.\n\\end{ex}\n\n\\begin{ex}\\label{example Ronkin of a binomial}\nFor any $m,m^\\prime\\in M$ with $m\\neq m^\\prime$, the Ronkin function of the binomial $f=\\chi^m-\\chi^{m^\\prime}$ coincides with the support function of the segment $\\overline{mm^\\prime}$, that is \\[\\rho_f(u)=\\min(\\langle m,u\\rangle, \\langle m^\\prime,u\\rangle)\\]for every $u\\in N_\\mathbb{R}$. To prove this, remark first that one can restrict to the case of the binomial $f=\\chi^m-1$ with $m\\neq0$ because of \\hyperref[Ronkin function of a product]{Proposition \\ref*{Ronkin function of a product}} and \\hyperref[example Ronkin of a monomial]{Example \\ref*{example Ronkin of a monomial}} and by factoring with a monomial. In the non-archimedean case the statement follows immediately from \\hyperref[remark about known version of Ronkin functions]{Remark \\ref*{remark about known version of Ronkin functions}}. In the archimedean one, the choice of a basis for $M$ allows to write\n\\[\\rho_f(u)=-\\frac{1}{(2\\pi)^n}\\int_{\\theta_1,\\dots,\\theta_n\\in[0,2\\pi]}\\log\\big|e^{-m_1u_1+im_1\\theta_1}\\cdot\\dots\\cdot e^{-m_nu_n+im_n\\theta_n}-1\\big|\\ d\\theta_1\\dots d\\theta_n,\\]\nwith $m_1,\\dots,m_n$ being the coordinates of $m$ in such a basis and $u_1,\\dots,u_n$ the coordinates of $u$ in the dual one. Assuming that $m_1>0$, which is always possible since $m\\neq0$, Jensen's formula yields, for every $\\theta_2,\\dots,\\theta_n$,\n\\begin{equation}\\label{equality in the proof of the Ronkin function of a binomial}\n\\int_{\\theta_1\\in[0,2\\pi]}\\log\\big|e^{-m_1u_1+im_1\\theta_1}\\cdot\\dots\\cdot e^{-m_nu_n+im_n\\theta_n}-1\\big|\\ d\\theta_1=-2\\pi\\sum_{j=1}^k\\log\\frac{|\\alpha_j|}{e^{-m_1u_1}}\n\\end{equation}\nwith $\\alpha_1,\\dots,\\alpha_k$ being the zeros of the univariate polynomial \\[\\big(e^{-m_2u_2+im_2\\theta_2}\\cdot\\dots\\cdot e^{-m_nu_n+im_n\\theta_n}\\big)T-1\\] lying inside the closed disk of radius $e^{-m_1u_1}$, repeated according to multiplicity. The only complex zero of the above polynomial has modulus $e^{m_2u_2+\\dots+m_nu_n}$; the integral in \\eqref{equality in the proof of the Ronkin function of a binomial} is then zero if $m_1u_1+\\dots+m_nu_n>0$, otherwise it equals $-2\\pi(m_1u_1+\\dots+m_nu_n)$. It follows that\n\\[\\rho_f(u)=\\min(m_1u_1+\\dots+m_nu_n,0),\\]\nhence the claim.\n\\end{ex}\n\n\n\\section{Heights of toric varieties}\\label{section about Arakelov on toric varieties}\n\nA well-suited framework to develop Arakelov geometry is provided by the study of varieties over adelic fields. In this setting, local and global heights of cycles of arbitrary dimension can be defined following \\cite{Z}, \\cite{G0} and \\cite{C-L}. A more general approach involving $M$-fields has been suggested in \\cite{G1}. Even if the theory is often phrased in terms of line bundles, we adopt here the equivalent point of view of divisors, which turns out to be more convenient in the toric case, see for instance \\cite{BPRS}.\n\n\n\\subsection{Adelic fields}\n\nBy a \\textit{place} on a field $K$, we mean an equivalence class of absolute values on $K$, that could be either archimedean or non-archimedean. Whenever $\\mathfrak{M}$ is a collection of places on $K$, the subset of archimedean places in $\\mathfrak{M}$ is denoted by $\\mathfrak{M}_\\infty$.\n\n\\begin{defn}\\label{adelic family}\nLet $K$ be a field. A family of places $\\mathfrak{M}$ on $K$ is said to be \\emph{adelic} if it satisfies the following properties:\n\\begin{enumerate}\n\\item for every $v\\in\\mathfrak{M}\\setminus\\mathfrak{M}_\\infty$, one (and hence all) absolute value in the class of $v$ is associated to a nontrivial discrete valuation\n\\item for each $\\alpha\\in K^*$, the set of places $v$ for which $|\\alpha|_v\\neq1$ for any $|\\cdot|_v\\in v$ is finite.\n\\end{enumerate}\n\\end{defn}\n\nIt is clear that the two conditions of the previous definition do not depend on the choice of the representant of the class $v$.\n\n\\begin{defn}\\label{adelic field}\nAn \\emph{adelic field} is a field $K$ together with an adelic family of places $\\mathfrak{M}$ on $K$ and a choice of an absolute value $|\\cdot|_v$ and of a real positive number $n_v$ for each place $v\\in\\mathfrak{M}$. An adelic field $\\left(K,(|\\cdot|_v,n_v)_{v\\in\\mathfrak{M}}\\right)$ is said to satisfy the \\emph{product formula} if for every $\\alpha\\in K^*$ \\[\\sum_{v\\in\\mathfrak{M}}n_v\\log|\\alpha|_v=0.\\]\n\\end{defn}\n\nWhenever there is no ambiguity on its adelic structure, an adelic field will be simply denoted by $K$.\n\\\\The following property is an easy, though fundamental, consequence of the definition.\n\n\\begin{lemma}\\label{adelic fields have finitely many archimedean places}\nAny adelic field $K$ only admits finitely many archimedean places.\n\\end{lemma}\n\\begin{proof}\nSince an absolute value on $K$ is non-archimedean if and only if it is bounded on the image of $\\mathbb{Z}$ in $K$, a field with positive characteristic has no archimedean absolute values. Suppose hence that $K$ has characteristic zero. In this case it contains a copy of $\\mathbb{Q}$ and any archimedean absolute value $|\\cdot|_v$ on $K$ restricts to an archimedean absolute value on $\\mathbb{Q}$. By Ostrowski's theorem, one has $|2|_v>1$. The second axiom in \\hyperref[adelic field]{Definition \\ref*{adelic family}} allows hence to conclude the claim.\n\\end{proof}\n\nFor an adelic field $K$ and a finite field extension $F$ of $K$, there exists a canonical way of endowing $F$ with the structure of an adelic field, see \\cite[Remark 2.5]{G1} and \\cite[\\S3]{MS} for the detailed construction. With this induced adelic structure, $F$ satisfies the product formula whenever $K$ does.\n\n\\begin{ex}\\label{examples of adelic fields}\nThe archetypical example of an adelic field satisfying the product formula is given by the field $\\mathbb{Q}$, together with the collection of all its nontrivial places, the standard normalized absolute value for each of them and weights equal to $1$.\n\\end{ex}\n\nMore generally, any \\emph{global field}, that is a number field or the function field of a smooth projective curve over a field $k$ with the structure described in \\cite[Example 1.5.4]{BPS}, is an adelic field satisfying the product formula.\n\n\n\\subsection{Local and global heights}\n\nLet $K$ be an adelic field satisfying the product formula and $X$ a proper variety of dimension $n$ over $K$. For every place $v\\in\\mathfrak{M}$, denote by $K_v$ the completion of $K$ with respect to $|\\cdot|_v$ and by $\\mathbb{C}_v$ the completion of an algebraic closure of $K_v$ with respect to the unique extension of the absolute value. It is a well-known fact that $\\mathbb{C}_v$ is algebraically closed; moreover, $\\mathbb{C}_v$ comes with an absolute value that one denotes, with abuse of notation, by $|\\cdot|_v$. The pair $(\\mathbb{C}_v,|\\cdot|_v)$ is hence an algebraically closed complete field as in \\hyperref[section about Ronkin functions]{section \\ref*{section about Ronkin functions}}. \n\\\\The base change $X_{\\mathbb{C}_v}$ is a scheme of finite type over $\\spec \\mathbb{C}_v$ to which one associates its Berkovich analytification $(X_v^{\\an},\\mathscr{O}_{X_v^{\\an}})$, whose underlying topological space is compact because of the properness of $X$. To stress its dependence on the choice of the place $v$, $X_v^{\\an}$ is called the \\emph{$v$-adic analytification} of $X$. Similarly, one can consider the base change $X_{K_v}$ of $X$ over $\\spec K_v$ and consider its Berkovich analytification $X_{K_v}^{\\an}$. The two spaces are related by the isomorphism\\[X_{K_v}^{\\an}\\simeq X_v^{\\an}\/\\Gal(\\overline{K}_v^{\\sep}\/K_v),\\] as shown in \\cite[Proposition 1.3.5]{Ber}. Moreover, there exists a surjective morphism of locally ringed space $\\pi_v:X_v^{\\an}\\to X_{\\mathbb{C}_v}$.\n\n\\begin{rem}\nBy Ostrowski's and Gelfand-Mazur theorems, if $v$ is an archimedean absolute value on $K$, $\\mathbb{C}_v$ is isometric to the field $\\mathbb{C}$ endowed with a power of the usual absolute value. In this case, the Berkovich space $(X_v^{\\an},\\mathscr{O}_{X_v^{\\an}})$ is isomorphic to the usual complex analytification of $X_\\mathbb{C}$.\n\\end{rem}\n\nFor any line bundle $L$ on $X$, its $v$-adic analytification is the analytic line bundle \\[L_v^{\\an}:=\\pi_v^*L_{\\mathbb{C}_v}\\] on $X_v^{\\an}$. Continuous metrics on $L_v^{\\an}$ are defined as in \\cite[\\S1.1.1]{C-L1}, unregardingly on the nature of the place $v$. Relevant classes of metrics on $L_v^{\\an}$ are \\emph{smooth} metrics in the archimedean case and \\emph{algebraic} (or, equivalently, formal, see \\cite[Proposition 8.13]{GK}) metrics when $v$ is non-archimedean, see for example \\cite[\\S1]{C-L1} and \\cite[8.8 and 8.12]{GK} for the precise definitions. A divisor $D$ on $X$ together with a continuous $\\Gal(\\overline{K}_v^{\\sep}\/K_v)$-invariant metric $\\|\\cdot\\|_v$ on the analytic line bundle $\\mathscr{O}(D)_v^{\\an}$ is called a \\emph{$v$-adic metrized divisor} and it is denoted by $\\overline{D}_v$ or also by $(\\mathscr{O}(D),\\|\\cdot\\|_v)$. Sums and pull-backs of $v$-adic metrized divisors can be defined as in \\cite[\\S1.2]{C-L1}.\n\\\\A $v$-adic metrized divisor $\\overline{D}_v$ is said to be \\emph{semipositive} if the corresponding metric can be approximated by semipositive smooth (when $v$ is archimedean) or algebraic (when $v$ is non-archimedean) semipositive metrics in the sense of \\cite[\\S1.4]{BPS}. For any $d$-dimensional subvariety $Y$ of $X$ and for any $d$-tuple of $v$-adic semipositive metrized divisors $\\overline{D}_{0,v},\\dots,\\overline{D}_{d-1,v}$ on $X$, there exists a positive measure \n\\begin{equation}\\label{measure for semipositive metrized line bundles}\nc_1(\\overline{D}_{0,v})\\wedge\\dots\\wedge c_1(\\overline{D}_{d-1,v})\\wedge\\delta_{Y}\n\\end{equation}\non $X_v^{\\an}$, which was first introduced in \\cite[D\\'efinition 2.4 and Proposition 2.7 b)]{C-L} in the non-archimedean setting and extended in \\cite[\\S3.8]{G2} under weaker assumptions. The suggestive notation for the measure in \\hyperref[measure for semipositive metrized line bundles]{(\\ref*{measure for semipositive metrized line bundles})} is compatible with the wedge product of first Chern forms in the smooth archimedean case, while it is justified by the recent advances in the theory of forms and currents on Berkovich spaces otherwise, as shown in \\cite[\\S6.9]{CLD} and \\cite[Theorem 10.5]{GK}.\n\\\\Recall also that for a $d$-dimensional cycle $Z$ in $X$ and a family $(D_0,s_0),\\dots,(D_d,s_d)$ of divisors on $X$ with rational sections of the associated line bundles, one says that $s_0,\\dots,s_d$ \\emph{meet $Z$ properly} if for every $J\\subseteq\\{0,\\dots,d\\}$, each irreducible component of $|Z|\\cap\\bigcap_{i\\in J}|\\divisor(s_i)|$ has dimension $d-\\#J$, $|\\cdot|$ denoting here the support of a cycle.\n\n\\begin{defn}\\label{local height definition}\nLet $Z$ be a $d$-dimensional cycle in $X$ and $\\big(\\overline{D}_{0,v},s_0\\big),\\dots,\\big(\\overline{D}_{d,v},s_d\\big)$ a collection of $v$-adic semipositive metrized divisors on $X$ with rational sections of the corresponding line bundles, with $s_0,\\dots,s_d$ meeting $Z$ properly. The \\emph{$v$-adic local height of $Z$ in $X$ with respect to $\\big(\\overline{D}_{i,v},s_i\\big)$} for $i=0,\\dots,d$ is defined, linearly in its irreducible components, by the recursive formula \n\\begin{multline*}\nh_{\\overline{D}_{0,v},\\dots,\\overline{D}_{d,v}}(Z;s_0,\\dots,s_d):=h_{\\overline{D}_{0,v},\\dots,\\overline{D}_{d-1,v}}(Z\\cdot\\divisor(s_d);s_0,\\dots,s_{d-1})\\\\-\\int_{X_v^{\\an}}\\log\\|s_d\\|_{d,v}\\ c_1(\\overline{D}_{0,v})\\wedge\\dots\\wedge c_1(\\overline{D}_{d-1,v})\\wedge\\delta_{Z},\n\\end{multline*}\nwhere $\\|\\cdot\\|_{d,v}$ denotes the metric of $\\overline{D}_{d,v}$ and one sets the height of the zero cycle to be zero.\n\\end{defn}\n\nThe integrals appearing in the previous definition are well-defined, as shown in \\cite[Th\\'eor\\`eme 4.1]{CLT} in both the archimedean and non-archimedean setting, and \\cite[Theorem 1.4.3]{GH} for the case of non-archimedean valuations which are not necessarily discrete. The $v$-adic local height function is moreover symmetric and multilinear with respect to sums of metrized divisors with rational sections of the associated line bundles, see \\cite[Proposition 3.4 and Remark 9.3]{G}.\n\\vspace{\\baselineskip}\n\\\\The adelic structure on the field $K$ allows to define a \\emph{semipositive metrized divisor} $\\overline{D}$ on $X$ by the choice, for every place $v\\in\\mathfrak{M}$, of a continuous semipositive metric on $\\mathscr{O}(D)_v^{\\an}$. This global definition induces a notion of a $v$-adic local height function at each place of $K$. Some care has to be taken when defining global heights as sums of such $v$-adic local heights, since they do not need to be well-defined in general. \n\n\\begin{defn}\\label{global heights of integrable cycles}\nA $d$-dimensional irreducible subvariety $Y$ of $X$ is said to be \\emph{integrable} with respect to the choice of $d+1$ semipositive metrized divisors $\\overline{D}_0,\\dots,\\overline{D}_d$ if there exists a birational proper map $\\varphi:Y^\\prime\\to Y$, with $Y^\\prime$ projective, and sections $s_i$ of $\\varphi^*\\mathscr{O}(D_i)$ for each $i=0,\\dots,d$, meeting $Y^\\prime$ properly, such that the $v$-adic local height \\[h_{\\varphi^*\\overline{D}_{0,v},\\dots,\\varphi^*\\overline{D}_{d,v}}(Y^\\prime;s_0,\\dots,s_d)\\] is zero for all but finitely many places $v\\in\\mathfrak{M}$. A $d$-dimensional cycle is said to be \\emph{integrable} if each of its irreducible components is. If $Y$ is an integrable $d$-dimensional irreducible subvariety, the \\emph{global height of $Y$ in $X$ with respect to $\\overline{D}_0,\\dots,\\overline{D}_d$} is defined as \\[h_{\\overline{D}_0,\\dots,\\overline{D}_d}(Y):=\\sum_{v\\in\\mathfrak{M}}n_v\\ h_{\\varphi^*\\overline{D}_{0,v},\\dots,\\varphi^*\\overline{D}_{d,v}}(Y^\\prime;s_0,\\dots,s_d).\\]\nThe \\emph{global height} of integrable cycles is defined by linearity.\n\\end{defn} \n\nThe previous definition does not depend on the choice of the projective resolution $Y^\\prime$ of $Y$ nor of the sections $s_0,\\dots,s_d$, as a consequence of \\cite[Proposition 3.6 and Remark 9.3]{G}, \\cite[Theorem 1.4.17 (3)]{BPS} and the product formula on $K$. As its local counterparts, the global height is symmetric and multilinear with respect to sums of metrized divisors. Moreover, it is well-behaved under proper transformations, in the sense of the next proposition.\n\n\\begin{proposition}\\label{heights and proper morphisms}\nLet $\\varphi:X^\\prime\\to X$ be a dominant morphism of proper varieties over $K$, $\\overline{D}_0,\\dots,\\overline{D}_d$ semipositive metrized divisors over $X$ and $Z^\\prime$ a $d$-dimensional cycle in $X^\\prime$. The cycle $\\varphi_*Z^\\prime$ is integrable with respect to $\\overline{D}_0,\\dots,\\overline{D}_d$ if and only if $Z^\\prime$ is integrable with respect to $\\varphi^*\\overline{D}_0,\\dots,\\varphi^*\\overline{D}_d$ and in this case \\[h_{\\overline{D}_0,\\dots,\\overline{D}_d}(\\varphi_*Z^\\prime)=h_{\\varphi^*\\overline{D}_0,\\dots,\\varphi^*\\overline{D}_d}(Z^\\prime).\\]\n\\end{proposition}\n\\begin{proof}\nThe statement about integrability is \\cite[Proposition 1.5.8 (2)]{BPS}, while the equality of the global heights follows from the same property on local heights, as proved in \\cite[Proposition 3.6 and Remark 9.3]{G} in the more general context of pseudo-divisors.\n\\end{proof}\n\n\n\\subsection{Heights on toric varieties}\\label{section height on toric varieties}\n\nIn this section the basic constructions and results of the arithmetic geometry of toric varieties are recalled, following the treatment of \\cite{BPS}. Let hence $X_\\Sigma$ be a proper toric variety of dimension $n$ over an adelic field $K$, with torus $\\mathbb{T}$ and dense open orbit $X_0$. Denote by $N$ and $M$ the character and cocharacter groups of $\\mathbb{T}$ and by $N_\\mathbb{R}$ and $M_\\mathbb{R}$ the associated (reciprocally dual) real vector spaces. The toric variety $X_\\Sigma$ is associated with a complete fan $\\Sigma$ in $N_\\mathbb{R}$, whose collection of $k$-dimensional cones is written $\\Sigma^{(k)}$. For any $\\tau\\in\\Sigma^{(1)}$, denote by $v_\\tau$ its minimal nonzero integral vector and by $V(\\tau)$ its associated orbit closure, consistently with \\cite[\\S3.1]{Ful}.\n\\\\Divisors on $X_\\Sigma$ which are invariant under the torus action are called \\emph{toric divisors} and admit a nice combinatorial description, as follows. By \\cite[\\S I.2, Theorem 9]{KKMS}, there exists a bijection between the set of toric Cartier divisors on $X_\\Sigma$ and the set of \\emph{virtual support functions} on $\\Sigma$, that is piecewise linear real-valued functions on the support of $\\Sigma$, with integral slope on each cone of $\\Sigma$. The toric Cartier divisor constructed from the virtual support function $\\Psi$ is denoted by $D_\\Psi$ and it defines a distinguished rational section $s_\\Psi$ of $\\mathscr{O}(D_\\Psi)$ satisfying $\\divisor(s_\\Psi)=D_\\Psi$. The corresponding Weil divisor is given by\n\\begin{equation}\\label{divisor of toric section}\n[D_\\Psi]=\\sum_{\\tau\\in\\Sigma^{(1)}}-\\Psi(v_\\tau)V(\\tau).\n\\end{equation}\nIn particular, the rational section $s_\\Psi$ is regular and nowhere vanishing on $X_0$. A toric divisor $D_\\Psi$ also determines a polyhedron \\[\\Delta_\\Psi:=\\{x\\in M_\\mathbb{R}:x-\\Psi\\geq0\\}\\] in $M_\\mathbb{R}$, which is bounded because of the properness of $X_\\Sigma$, see \\cite[Proposition at page 67]{Ful}, and coincides with the stability set of $\\Psi$ if $\\Psi$ is concave. Many algebro-geometric properties of a toric divisor are read from its associated virtual support function: for instance, $D_\\Psi$ is generated by global sections if and only if $\\Psi$ is concave, and it is ample if and only if $\\Psi$ is concave and restricts to different linear functions on different maximal cones of $\\Sigma$.\n\\\\Regarding the arithmetic part of the toric dictionary, let $D_\\Psi$ be a toric divisor on $X_\\Sigma$, with associated virtual support function $\\Psi$. The continuous metrics on $\\mathscr{O}(D_\\Psi)$ admitting a combinatorial description are the ones which are invariant under the action of a certain compact torus, see \\cite[\\S4.2]{BPS} for more details about this notion. In concrete terms, a continuous metric $\\|\\cdot\\|_v$ on $\\mathscr{O}(D_\\Psi)_v^{\\an}$ is called a \\emph{$v$-adic toric metric} if the map\n\\[(X_0)_v^{\\an}\\to\\mathbb{R},\\quad p\\mapsto\\|s_\\Psi(p)\\|_v\\]\nis constant along the fibers of the $v$-adic tropicalization map $\\trop_v:(X_0)_v^{\\an}\\to N_\\mathbb{R}$ introduced in \\hyperref[definition of tropicalization]{Definition \\ref*{definition of tropicalization}}. A toric divisor $D$ together with a $v$-adic toric metric on $\\mathscr{O}(D)$ is called a \\emph{$v$-adic toric metrized divisor}. To a $v$-adic toric metrized divisor $\\overline{D}_v$ one can associate the real-valued map $\\psi_{\\overline{D}_v}$ on $N_\\mathbb{R}$ satisfying the equality\n\\begin{equation}\\label{definition of the metric function}\n\\psi_{\\overline{D}_v}\\circ\\trop_v=\\log\\|s_D\\|_v\n\\end{equation}\non the analytic torus $(X_0)_v^{\\an}$, $s_D$ being the distinguished rational section of $\\mathscr{O}(D)$.\n\\\\The map $\\psi_{\\overline{D}_v}$, which will be referred to as the \\emph{metric function} of $\\overline{D}_v$, has been introduced by Burgos Gil, Philippon and Sombra in their study of Arakelov geometry of toric varieties to encode many arithmetic properties of $\\overline{D}_v$, see \\cite[Chapter 4]{BPS}. For instance, it is smooth in the archimedean case if the metric is smooth, while in the non-archimedean setting it is rational piecewise affine if the metric is algebraic, see \\cite[Theorem 4.5.10 (1)]{BPS} and \\cite[Proposition 2.5.5]{GH}. Also, the semipositivity of $\\overline{D}_v$ is translated into the concavity of its corresponding metric function.\n\n\\begin{theorem}\\label{characterization of toric metrics}\nLet $D$ be the toric divisor associated to the virtual support function $\\Psi$. The assignment $\\|\\cdot\\|_v\\mapsto\\psi_{\\overline{D}_v}$ is a bijection between the space of $v$-adic semipositive toric metrics on $\\mathscr{O}(D)_v^{\\an}$ and the space of concave functions $\\psi$ on $N_\\mathbb{R}$ such that $|\\psi-\\Psi|$ is bounded.\n\\end{theorem}\n\\begin{proof}\nThis is \\cite[Theorem 4.8.1 (1)]{BPS}. The extension to the general non-archime\\-dean case is \\cite[Theorem 2.5.8]{GH}.\n\\end{proof}\n\nIf $\\overline{D}_v$ is a $v$-adic semipositive toric metrized divisor, the Legendre-Fenchel dual of the metric function of $\\overline{D}_v$ is called the \\emph{roof function} of $\\overline{D}_v$ and denoted by $\\vartheta_{\\overline{D}_v}$: it is a concave function on $M_\\mathbb{R}$ with effective domain the polytope $\\Delta_\\Psi$.\n\\\\A toric divisor admits a $v$-adic semipositive metric if and only if it is generated by global sections, as proved in \\cite[Corollary 4.8.5]{BPS}. For such divisors, moreover, there exists a distinguished choice of a $v$-adic semipositive metric.\n\n\\begin{defn}\\label{canonical metric definition}\nLet $D$ be a toric divisor generated by global sections, $\\Psi$ its associated virtual support function. The \\emph{$v$-adic canonical metric} on $D$ is the semipositive toric metric on $\\mathscr{O}(D)_v^{\\an}$ corresponding to $\\Psi$ in the bijection of \\hyperref[characterization of toric metrics]{Theorem \\ref*{characterization of toric metrics}}.\n\\end{defn}\n\nIn the non-archimedean case, the canonical metric on $D$ coincides with the algebraic metric induced by the canonical model of $X_\\Sigma$ and $D$, see \\cite[Example 4.5.4]{BPS}.\n\\\\For semipositive $v$-adic toric metrized divisors, the measure in \\eqref{measure for semipositive metrized line bundles} can be expressed in terms of the associated metric functions. Indeed, recall from \\hyperref[section on boundaries]{subsection \\ref*{section on boundaries}} that there exists an embedding \\[\\iota_v:N_\\mathbb{R}\\times\\mathcal{B}_{\\mathbb{C}_v}\\to X_{0,v}^{\\an}\\]which fits into the \\hyperref[diagram]{commutative diagram (\\ref*{diagram})}, and denote by $\\Haar_{\\mathcal{B}_{\\mathbb{C}_v}}$ the Haar measure on $\\mathcal{B}_{\\mathbb{C}_v}$ normalized to have total mass $1$.\n\n\\begin{theorem}\\label{Chambert-Loir measure in the toric case}\nFor $i=0,\\dots,n-1$, let $\\overline{D}_{i,v}$ be a semipositive $v$-adic toric metrized divisor on $X_\\Sigma$, $\\Psi_i$ the virtual support function associated to $D_i$ and $\\psi_{i,v}$ the metric function of $\\overline{D}_{i,v}$. Then, the positive measure \\[c_1(\\overline{D}_{0,v})\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1,v})\\wedge\\delta_{X_\\Sigma}\\]is zero outside $X_{0,v}^{\\an}$ and \\[c_1(\\overline{D}_{0,v})\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1,v})\\wedge\\delta_{X_\\Sigma}|_{X_{0,v}^{\\an}}=(\\iota_v)_*\\big(\\MM_M(\\psi_{0,v},\\dots,\\psi_{n-1,v})\\times\\Haar_{\\mathcal{B}_{\\mathbb{C}_v}}\\big).\\]\nIn particular, \\[(\\trop_v)_*\\big(c_1(\\overline{D}_{0,v})\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1,v})\\wedge\\delta_{X_\\Sigma}|_{X_{0,v}^{\\an}}\\big)=\\MM_M(\\psi_{0,v},\\dots,\\psi_{n-1,v})\\]as measures on $N_\\mathbb{R}$. \n\\end{theorem}\n\\begin{proof}\nThe first statement follows from \\cite[Theorem 1.4.10 (1)]{BPS} and \\cite[Corollary 1.4.5]{GH}. The expression for the measure in the archimedean and the discrete non-archimedean case is obtained from \\cite[Theorem 4.8.11]{BPS} and multilinearity; the general non-archimedean case is deduced from \\cite[Theorem 2.5.10]{GH}. The last assertion is an easy consequence of the commutativity of the \\hyperref[diagram]{diagram (\\ref*{diagram})}.\n\\end{proof}\n\nMoving to the global case, a (\\emph{semipositive}) \\emph{toric metric} on a toric divisor $D$ is a choice, for each place $v\\in\\mathfrak{M}$, of a (semipositive) $v$-adic toric metric on the line bundle $\\mathscr{O}(D)$. The toric divisor $D$ together with a (semipositive) toric metric is called a (\\emph{semipositive}) \\emph{toric metrized divisor} and it is denoted by $\\overline{D}$. From the point of view of convex geometry, the semipositive toric metrized divisor $\\overline{D}$ is completely described by the collection $(\\psi_v)_{v\\in\\mathfrak{M}}$ of its metric functions or, equivalently, by the collection $(\\vartheta_v)_{v\\in\\mathfrak{M}}$ of its roof functions.\n\\\\A notion of well-behaving toric metrics was defined in \\cite[Definition 4.9.1]{BPS}.\n\n\\begin{defn}\\label{definition of adelic toric metric}\nA toric metric $(\\|\\cdot\\|_v)_{v\\in\\mathfrak{M}}$ on a toric divisor is said to be \\emph{adelic} if for all but finitely many $v\\in\\mathfrak{M}$ the $v$-adic toric metric $\\|\\cdot\\|_v$ is the canonical one, in the sense of \\hyperref[canonical metric definition]{Definition \\ref*{canonical metric definition}}.\n\\end{defn}\n\nIn convex terms, a toric metric on the toric divisor $D$ associated to the virtual support function $\\Psi$ is adelic if and only if the family $(\\psi_v)_v$ of its metric functions satisfies $\\psi_v=\\Psi$ for all but finitely many $v\\in\\mathfrak{M}$.\n\\\\It follows from \\cite[Theorem 5.2.4]{BPS} that any toric subvariety of $X_\\Sigma$ is integrable with respect to the choice of adelic semipositive toric metrized divisors. In particular, one can compute the global height of the $n$-dimensional cycle $X_\\Sigma$ with respect to such choices.\n\n\\begin{theorem}\\label{height of a toric variety}\nLet $\\overline{D}_0,\\dots,\\overline{D}_n$ be toric divisors over $X_\\Sigma$, equipped with adelic semipositive toric metrics. Then\\[h_{\\overline{D}_0,\\dots,\\overline{D}_n}(X_\\Sigma)=\\sum_{v\\in\\mathfrak{M}}n_v\\MI_M(\\vartheta_{0,v},\\dots,\\vartheta_{n,v}),\\] where $\\vartheta_{i,v}$ is the roof function of $\\overline{D}_{i,v}$, for every $i=0,\\dots,n$ and $v\\in\\mathfrak{M}$.\n\\end{theorem}\n\\begin{proof}\nThis is \\cite[Theorem 5.2.5]{BPS}.\n\\end{proof}\n\n\n\\section{Divisors of rational functions}\\label{section divisor of rational functions}\n\nThe present section focuses on the combinatorial description of the Weil divisor on a toric variety of the rational function coming from a Laurent polynomial. This result will be used in the proof of the main theorems in the next section.\n\\\\To fix notation, let $X_\\Sigma$ be a proper smooth toric variety of dimension $n$ over a field $K$, $M$ the character lattice of its torus $\\mathbb{T}$ and $N_\\mathbb{R}$ the associated dual vector space. The toric variety $X_\\Sigma$ has a dense open orbit $X_0$ isomorphic to $\\mathbb{T}$ and hence the function field of $X_\\Sigma$ coincides with $K(M)$. In particular, any Laurent polynomial $f=\\sum c_m\\chi^m$ gives rise to a rational function on $X_\\Sigma$, which is regular and coincides with $f$ on $X_0$. To avoid confusion, one will denote by $\\tilde{f}$ such a rational function.\n\\\\Recall from \\cite[Theorem 3.1.19 (a)]{CLS} that the toric variety $X_\\Sigma$ is smooth if and only if each cone of $\\Sigma$ is a smooth cone, that is it is generated by a part of a basis of the lattice $N$. For each cone $\\tau$ in $\\Sigma$ of dimension $1$, denote by $v_\\tau$ its minimal nonzero integral vector, which generates $\\tau\\cap N$ as a monoid. If $\\sigma$ is a smooth cone of dimension $n$ in $N_\\mathbb{R}$, the collection $(v_\\tau)_\\tau$, with $\\tau$ ranging in the set of one dimensional faces of $\\sigma$, is a basis of $N$ and hence gives a dual basis $(v_\\tau^\\vee)_\\tau$ of the lattice $M$.\n\n\\begin{lemma}\\label{lemma prime ideal of the orbit}\nLet $\\sigma$ be a strongly convex polyhedral rational cone in $N_\\mathbb{R}$. For every face $\\tau$ of $\\sigma$ of dimension $1$, the orbit closure $V(\\tau)$ in the affine toric variety $X_\\sigma$ is the subvariety corresponding to the prime ideal \\[\\mathfrak{p}=\\big(\\chi^m:m\\in\\sigma^\\vee\\cap M,m\\notin\\tau^\\perp\\big)\\]of $\\mathscr{O}(X_\\sigma)=K[\\sigma^\\vee\\cap M]$. Moreover, if $\\sigma$ is smooth and of maximal dimension in $N_\\mathbb{R}$, $\\mathfrak{p}$ is principal and generated by $\\chi^{v_\\tau^\\vee}$.\n\\end{lemma}\n\\begin{proof}\nRecall for example from \\cite[\\S 3.1]{Ful} that the orbit closure $V(\\tau)$ is the toric variety $\\spec K[\\sigma^\\vee\\cap\\tau^\\perp\\cap M]$ and can be embedded in $X_\\sigma=\\spec K[\\sigma^\\vee\\cap M]$ via the surjection of rings sending $\\chi^m$ to itself if $m\\in\\tau^\\perp$, and to $0$ otherwise. Then $V(\\tau)$ is seen as the subvariety of $X_\\sigma$ corresponding to the kernel of such homomorphism, that is\n\\[\\mathfrak{p}=\\bigoplus_{\\substack{m\\in\\sigma^\\vee\\cap M\\\\m\\notin\\tau^\\perp}} K\\chi^m=\\big(\\chi^m:m\\in \\sigma^\\vee\\cap M,m\\notin\\tau^\\perp\\big),\\]\nproving the first statement.\n\\\\Suppose now that $\\sigma$ is smooth and of dimension $n$ in $N_\\mathbb{R}$; denote by $v_1,v_2,\\dots,v_n$ the basis of $N$ given by the minimal integral vectors of the rays of $\\sigma$, with the assumption that $v_1=v_\\tau$. By definition, \\[\\sigma=\\mathbb{R}_{\\geq0}v_1+\\dots+\\mathbb{R}_{\\geq0}v_n.\\]As a result, denoting by $(v_i^\\vee)_{i=1,\\dots,n}$ the basis of $M$ dual to $(v_i)_{i=1,\\dots,n}$, one has that \\[\\langle v_i^\\vee,u\\rangle=\\lambda_i\\geq0\\]for every $i\\in\\{1,\\dots,n\\}$ and for every $u=\\sum_i\\lambda_iv_i\\in\\sigma$. In particular, $v_\\tau^\\vee\\in\\sigma^\\vee$. It is easy to check that $v_\\tau^\\vee$ is integrally valued on each element of $N$ and hence it belongs to $M$. It follows then from $\\langle v_\\tau^\\vee,v_\\tau\\rangle=1$ that \\[\\Big(\\chi^{v_\\tau^\\vee}\\Big)\\subseteq\\mathfrak{p}.\\]\nFor the reverse inclusion, consider $m\\in \\sigma^\\vee\\cap M$ with $m\\notin\\tau^\\perp$. By assumption, $\\langle m,v_\\tau\\rangle\\in\\mathbb{Z}$ and $\\langle m,v_\\tau\\rangle\\geq0$; moreover, since $m\\notin\\tau^\\perp$, one has $\\langle m,v_\\tau\\rangle\\geq1$. For each $u=\\sum_i\\lambda_iv_i\\in\\sigma$ one has \\[\\langle m-v_\\tau^\\vee,u\\rangle=\\lambda_1\\langle m-v_\\tau^\\vee,v_\\tau\\rangle+\\sum_{i\\geq2}\\lambda_i\\langle m,v_i\\rangle\\geq\\lambda_1\\left(\\langle m,v_\\tau\\rangle-1\\right)\\geq0.\\]As a result, $m-v_\\tau^\\vee\\in\\sigma^\\vee\\cap M$ and hence $\\chi^m=\\chi^{v_\\tau^\\vee}\\cdot\\chi^{m-v_\\tau^\\vee}\\in\\big(\\chi^{v_\\tau^\\vee}\\big)$, completing the proof.\n\\end{proof}\n\n\\begin{rem}\nThe last statement of the previous lemma is not true for a general strongly convex polyhedral rational cone $\\sigma$ of maximal dimension in $N$. For example, if $\\sigma$ has more than $n$ faces of dimension $1$, the divisor class group of $X_\\sigma$, which is generated by the classes of the orbit closures associated to the rays, turns out to be nontrivial, as a consequence of \\cite[Proposition at page 63]{Ful}. \n\\end{rem}\n\nFor a nonzero Laurent polynomial $f\\in K[M]$, the subset $V(f)$ of zeros of $f$ in $X_0$ is a closed subscheme of the dense open orbit. Its closure in $X_\\Sigma$ is a closed subscheme of $X_\\Sigma$, denoted by $\\overline{V(f)}$. Taking into account multiplicities, one can consider the associated Weil divisor $\\Big[\\overline{V(f)}\\Big]$. It is the zero cycle when $f$ is a monomial.\n\n\\begin{theorem}\\label{divisor of Laurent polynomial}\nLet $f$ be a nonzero Laurent polynomial, $\\tilde{f}$ the rational function on $X_\\Sigma$ arising from $f$. Then, \\[\\divisor(\\tilde{f})=\\Big[\\overline{V(f)}\\Big]+\\sum_{\\tau\\in\\Sigma^{(1)}}\\Psi_{\\NP(f)}(v_\\tau)V(\\tau),\\]\nwhere $\\Psi_{\\NP(f)}$ denotes the support function of the Newton polytope of $f$. In particular, $\\Big[\\overline{V(f)}\\Big]$ is rationally equivalent to the cycle $-\\sum_{\\tau\\in\\Sigma^{(1)}}\\Psi_{\\NP(f)}(v_\\tau)V(\\tau)$ on $X_\\Sigma$.\n\\end{theorem}\n\\begin{proof}\nBy \\cite[formula at page 55]{Ful}, the irreducible components of $X_\\Sigma\\setminus X_0$ are exactly the orbit closures $V(\\tau)$, with $\\tau$ ranging in the set of $1$ dimensional cones of $\\Sigma$. Since moreover the restriction of $\\tilde{f}$ to $X_0$ is the regular function $f$, it follows from the classical theory of divisors that \\[\\divisor(\\tilde{f})=\\Big[\\overline{V(f)}\\Big]+\\sum_\\tau \\nu_\\tau(\\tilde{f})V(\\tau),\\] where $\\nu_\\tau(\\tilde{f})\\in\\mathbb{Z}$ is the order of vanishing of $\\tilde{f}$ along $V(\\tau)$. The statement of the theorem then follows from the fact that, for every $\\tau\\in\\Sigma^{(1)}$, such an order equals $\\Psi_{\\NP(f)}(v_\\tau)$.\n\\\\This claim can be proved locally; fix a ray $\\tau\\in\\Sigma^{(1)}$ and let $\\sigma$ be any maximal dimensional cone of $\\Sigma$ containing $\\tau$. The fan being complete and consisting of smooth cones, such a $\\sigma$ exists and the minimal integral vectors $v_1,\\dots,v_n$ of its rays are a basis of $N$. Assume moreover that $v_1=v_\\tau$ and, for simplicity, denote by $R:=K[\\sigma^\\vee\\cap M]$ the ring of regular functions over $X_\\sigma$. The order of vanishing of $\\tilde{f}$ along $V(\\tau)$ is computed as the valuation of $\\tilde{f}$ determined by the valuation ring $R_{\\mathfrak{p}}$, the localization of $R$ at the prime ideal $\\mathfrak{p}$ corresponding to the subvariety $V(\\tau)$ in $X_\\sigma$. By \\hyperref[lemma prime ideal of the orbit]{Lemma \\ref*{lemma prime ideal of the orbit}}, the cone $\\sigma$ being smooth and maximal dimensional, one has that $\\mathfrak{p}=\\big(\\chi^{v_\\tau^\\vee}\\big)$. The maximal ideal $\\mathfrak{p}R_\\mathfrak{p}$ of $R_\\mathfrak{p}$ is hence the principal ideal generated by $\\chi^{v_\\tau^\\vee}$.\n\\\\Suppose first that $\\tilde{f}=\\sum_m c_m\\chi^m$ lies in $R$, that is every $m$ appearing in $\\tilde{f}$ belongs to $\\sigma^\\vee\\cap M$. By definition of the valuation in $R_\\mathfrak{p}$,\n\\begin{equation*}\n\\begin{split}\n\\nu_\\tau(\\tilde{f})&=\\max\\ \\left\\{l\\in\\mathbb{N}:\\tilde{f}\\in(\\mathfrak{p}R_\\mathfrak{p})^l\\right\\}=\\max\\ \\left\\{l\\in\\mathbb{N}:\\tilde{f}\\in\\big(\\chi^{lv_\\tau^\\vee}\\big)\\right\\}\\\\&=\\max\\ \\left\\{l\\in\\mathbb{N}:\\chi^{m-lv_\\tau^\\vee}\\in R_\\mathfrak{p}\\text{ for all }m\\text{ with } c_m\\neq0\\right\\}.\n\\end{split}\n\\end{equation*}\nThe condition $\\chi^{m-lv_\\tau^\\vee}\\in R_\\mathfrak{p}$ is equivalent to the fact that $\\langle m,v_\\tau\\rangle\\geq l$. Indeed, if the first is true, then \\[\\langle m,v_\\tau\\rangle-l=\\langle m-lv_\\tau^\\vee,v_\\tau\\rangle\\geq0.\\] Conversely, for each $u=\\sum_i\\lambda_iv_i\\in\\sigma$ one has\\[\\langle m-lv_\\tau^\\vee,u\\rangle=\\lambda_1(\\langle m,v_\\tau\\rangle-l)+\\sum_{i\\geq2}\\lambda_i\\langle m,v_i\\rangle\\geq0,\\]and so $m-lv_\\tau^\\vee\\in\\sigma^\\vee\\cap M$. As a consequence,\n\\begin{equation*}\n\\begin{split}\n\\nu_\\tau(\\tilde{f})&=\\max\\ \\left\\{l\\in\\mathbb{N}:\\langle m,v_\\tau\\rangle\\geq l\\text{ for all }m\\text{ with } c_m\\neq0\\right\\}\\\\&=\\min\\ \\{\\langle m,v_\\tau\\rangle: m\\text{ with }c_m\\neq0\\}=\\Psi_{\\NP(f)}(v_\\tau).\n\\end{split}\n\\end{equation*}\nFor a general $\\tilde{f}=\\sum_m c_m\\chi^m$, the fact that $\\sigma^\\vee$ has dimension $n$ in $M_\\mathbb{R}$ ($\\sigma$ is indeed strongly convex) assures that there exists a big enough vector $m_0\\in\\sigma^\\vee\\cap M$ for which $m+m_0\\in\\sigma^\\vee\\cap M$ for each $m$ such that $c_m\\neq0$. Hence\\[\\tilde{f}=\\frac{\\sum_mc_m\\chi^{m+m_0}}{\\chi^{m_0}},\\]with both the numerator and the denominator belonging to $R$. Applying the result for such elements one deduces\\[\\nu_\\tau(\\tilde{f})=\\nu_\\tau\\left(\\sum_mc_m\\chi^{m+m_0}\\right)-\\nu_\\tau(\\chi^{m_0})=\\Psi_{\\NP(f)+m_0}(v_\\tau)-\\langle m_0,v_\\tau\\rangle=\\Psi_{\\NP(f)}(v_\\tau),\\]concluding the proof.\n\\end{proof}\n\n\n\\section{Local and global heights of hypersurfaces}\\label{section hypersurface}\n\nFix for the whole section an adelic field $\\left(K,(|\\cdot|_v,n_v)_{v\\in\\mathfrak{M}}\\right)$ satisfying the product formula. Let $X_\\Sigma$ be a proper toric variety over $K$, of dimension $n$, with torus $\\mathbb{T}=\\spec K[M]$ and dense open orbit $X_0$. For an effective cycle $Z$ on $X_\\Sigma$ of pure codimension $1$, whose prime components intersect $X_0$, we present a series of results concerning its integrability and its local and global height with respect to suitable choices of metrized divisors on $X_\\Sigma$.\n\n\n\\subsection{Degrees}\n\nWith the notations and assumptions given above, the effective cycle $Z$ can be written as \\[Z=\\sum_{i=1}^r\\ell_i Y_i\\] for prime divisors $Y_1,\\dots,Y_r$ intersecting $X_0$. For every $i=1,\\dots,r$, the closed irreducible subvariety of $X_0$ obtained as the intersection between $Y_i$ and $X_0$ is associated to a prime ideal of height one in $K[M]$, which is principal since $K[M]$ is a unique factorization domain; denote by $f_i$ an irreducible Laurent polynomial generating such an ideal. The Laurent polynomial $f=f_1^{\\ell_1}\\cdot\\dots\\cdot f_r^{\\ell_r}$ is called a \\emph{defining polynomial} for the cycle $Z$ and is uniquely defined up to multiplication by an invertible element of $K[M]$, that means by a monomial. Moreover,\n\\begin{equation}\\label{cycle of defining polynomial}\n\\Big[\\overline{V(f)}\\Big]=Z,\n\\end{equation}\nthat is the cycle associated to the closure of the subscheme $V(f)$ in $X_\\Sigma$ agrees with $Z$. Let \\[\\Psi_f:=\\Psi_{\\NP(f)}\\] be the support function, in the sense of \\hyperref[indicator and support function]{Example \\ref*{indicator and support function}}, of the Newton polytope $\\NP(f)$ of $f$; it is a piecewise linear function on $N_\\mathbb{R}$. It is not necessarily a virtual support function on the fan $\\Sigma$, but it can always be made such after a suitable refinement of the fan.\n\n\\begin{lemma}\\label{a suitable resolution exists}\nFor any proper toric variety $X_\\Sigma$ there exist a smooth projective toric variety $X_{\\Sigma^\\prime}$ with fan $\\Sigma^\\prime$ in $N_\\mathbb{R}$ and a proper toric morphism $\\pi:X_{\\Sigma^\\prime}\\to X_\\Sigma$ satisfying:\n\\begin{enumerate}\n\\item $\\pi$ restricts to the identity on the dense open orbit of $X_{\\Sigma^\\prime}$ and $X_\\Sigma$\n\\item $\\Psi_f$ is a virtual support function on $\\Sigma^\\prime$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nOne can always refine the complete fan $\\Sigma$ to a fan $\\Sigma^\\prime$ in such a way that $\\Psi_f$ is a virtual support function on $\\Sigma^\\prime$. After possibly refining again, one can suppose that $\\Sigma^\\prime$ is the fan of a projective toric variety (because of the toric Chow lemma, see \\cite[Theorem 6.1.18]{CLS}) and that each of its cones is generated by a part of a basis of $N$ (see \\cite[\\S2.6]{Ful}). The associated toric variety $X_{\\Sigma^\\prime}$ is smooth, projective and it satisfies (2). Finally, since $\\Sigma^\\prime$ is a refinement of $\\Sigma$, the toric morphism $\\pi$ given by \\cite[Theorem 3.3.4]{CLS} is proper and restricts to the identity on the dense open orbit of $X_{\\Sigma^\\prime}$.\n\\end{proof}\n\nThe previous lemma, together with the fact that intersection theoretical properties are stable under birational transformations, allows to compute the degree of a cycle of codimension $1$ in a toric variety with respect to a family of toric divisors generated by global sections.\n\n\\begin{proposition}\\label{degree of an hypersurface}\nLet $D_{\\Psi_1},\\dots,D_{\\Psi_{n-1}}$ be toric divisors on $X_\\Sigma$ generated by global sections, $Z$ an effective cycle on $X_\\Sigma$ of pure codimension $1$ and prime components intersecting $X_0$, with defining polynomial $f$. Then \\[\\deg_{D_{\\Psi_1},\\dots,D_{\\Psi_{n-1}}}(Z)=\\MV_M(\\Delta_{\\Psi_1},\\dots,\\Delta_{\\Psi_{n-1}},\\NP(f)),\\]where $\\MV_M$ denotes the mixed volume function associated to the measure $\\vol_M$ (see \\hyperref[lattice volume]{Remark \\ref*{lattice volume}}) and $\\Delta_{\\Psi_i}$ the polytope associated to the toric divisor $D_{\\Psi_i}$, for each $i=1,\\dots,n-1$.\n\\end{proposition}\n\\begin{proof}\nConsider the smooth projective toric variety $X_{\\Sigma^\\prime}$ and the proper toric morphism $\\pi:X_{\\Sigma^\\prime}\\to X_\\Sigma$ given by \\hyperref[a suitable resolution exists]{Lemma \\ref*{a suitable resolution exists}}. Since the support function $\\Psi_f$ is a virtual support function on $\\Sigma^\\prime$, one can consider the corresponding toric divisor $D_f$ on $X_{\\Sigma^\\prime}$ and the associated distinguished rational section $s_f$ of $\\mathscr{O}(D_f)$. The product $\\tilde{f}s_f$ is a rational section of $\\mathscr{O}(D_f)$, with associated Weil divisor satisfying\n\\begin{equation*}\n\\pi_*\\divisor\\big(\\tilde{f} s_f\\big)=\\pi_*(\\divisor(\\tilde{f})+\\divisor(s_f))=Z\n\\end{equation*}\nby \\hyperref[divisor of Laurent polynomial]{Theorem \\ref*{divisor of Laurent polynomial}}, \\eqref{divisor of toric section}, \\eqref{cycle of defining polynomial} and the definition of $\\pi$. The projection formula in \\cite[Proposition 2.3 (c)]{Ful2} and B\\'ezout's theorem yield\n\\[\\deg_{D_{\\Psi_1},\\dots,D_{\\Psi_{n-1}}}(Z)=\\deg_{\\pi^*D_{\\Psi_1},\\dots,\\pi^*D_{\\Psi_n-1},D_f}(X_{\\Sigma^\\prime}).\\]\nThe function $\\Psi_f$ being concave, $D_f$ is generated by global sections. Moreover, the virtual support functions associated to the toric divisor $\\pi^*D_{\\Psi_i}$ on $X_{\\Sigma^\\prime}$ agrees with $\\Psi_i$, for every $i=1,\\dots,n-1$. The combinatorial description in \\cite[Proposition 2.10]{O} of the degree of a toric variety with respect to toric divisor generated by global sections concludes then the proof.\n\\end{proof}\n\n\\begin{rem}\\label{remark for degree of toric divisor}\nBy \\cite[formula at page 55]{Ful}, the irreducible components of $X_\\Sigma\\setminus X_0$ are the orbit closures $V(\\tau)$, with $\\tau$ ranging in the set of $1$ dimensional cones of $\\Sigma$. It follows that if $Z$ is a prime divisor of $X_\\Sigma$ not intersecting $X_0$, it coincides with $V(\\tau)$ for some $\\tau\\in\\Sigma^{(1)}$. In such a case, the degree of $Z$ with respect to a collection $D_{\\Psi_1},\\dots,D_{\\Psi_{n-1}}$ of toric divisors on $X_\\Sigma$ generated by global sections is given by\n\\[\\deg_{D_{\\Psi_1},\\dots,D_{\\Psi_{n-1}}}(V(\\tau))=\\MV_{M(v_\\tau)}\\big(\\Delta_{\\Psi_1}^{v_\\tau},\\dots,\\Delta_{\\Psi_{n-1}}^{v_\\tau}\\big),\\]\nwhere $v_\\tau$ is the minimal nonzero integral vector of $\\tau$, see \\cite[formul{\\ae} (3.4.1) and (3.4.4)]{BPS}.\n\\end{rem}\n\n\\begin{rem}\\label{can reduce to the case of a toric variety compatible with the polynomial}\nThe reduction to the case of a smooth projective toric variety employed in the proof of \\hyperref[degree of an hypersurface]{Proposition \\ref*{degree of an hypersurface}} equally works when computing the local height of the cycle $Z$ with respect to a family of $v$-adic semipositive toric metrized divisors $\\overline{D}_{0,v},\\dots,\\overline{D}_{n-1,v}$. Indeed, let $f$ be a defining polynomial for $Z$, $X_{\\Sigma^\\prime}$ and $\\pi$ as in the statement of \\hyperref[a suitable resolution exists]{Lemma \\ref*{a suitable resolution exists}}. For every family of rational sections $s_0,\\dots,s_{n-1}$ of $\\mathscr{O}(D_0),\\dots,\\mathscr{O}(D_{n-1})$ respectively for which the following local heights are well-defined, the local arithmetic projection formula in \\cite[Theorem 1.4.17 (2)]{BPS} asserts that\\[h_{\\overline{D}_{0,v},\\dots,\\overline{D}_{n-1,v}}(Z;s_0,\\dots,s_{n-1})=h_{\\pi^*\\overline{D}_{0,v},\\dots,\\pi^*\\overline{D}_{n-1,v}}(Z^\\prime;\\pi^*s_0,\\dots,\\pi^*s_{n-1}),\\] where $Z^\\prime$ is the cycle in $X_{\\Sigma^\\prime}$ associated to the subscheme obtained as the closure of $V(f)$ and has hence $f$ as a defining polynomial. Because of \\cite[Proposition 4.3.19]{BPS}, the pull-back of $\\overline{D}_{i,v}$ via $\\pi$ is a $v$-adic semipositive toric metrized divisor on $X_{\\Sigma^\\prime}$ whose metric function coincides with the one of $\\overline{D}_{i,v}$, for every $i=0,\\dots,n-1$ and $v\\in\\mathfrak{M}$. It follows that any combinatorial formula for the local height of $Z^\\prime$ in $X_{\\Sigma^\\prime}$ with respect to $\\pi^*\\overline{D}_{0,v},\\dots,\\pi^*\\overline{D}_{n-1,v}$ only involving the defining polynomial of $Z$ and the metric functions of the metrized divisors equally holds for the local height of $Z$ in $X_\\Sigma$ with respect to $\\overline{D}_{0,v},\\dots,\\overline{D}_{n-1,v}$. Similarly, the reduction step can be adopted when dealing with the integrability and the global height of $Z$, because of \\hyperref[heights and proper morphisms]{Proposition \\ref*{heights and proper morphisms}}.\n\\end{rem}\n\n\n\\subsection{Local heights}\\label{section about local heights}\n\nLet $f$ be a defining polynomial for the cycle $Z$ and, as in the previous subsection, denote by $\\Psi_f$ the support function of its Newton polytope. Under the assumption that $\\Psi_f$ is a virtual support function on the fan of $X_\\Sigma$, it determines a toric divisor $D_f$ on $X_\\Sigma$ together with a distinguished rational section $s_f$ of $\\mathscr{O}(D_f)$, as in \\hyperref[section height on toric varieties]{subsection \\ref*{section height on toric varieties}}.\n\n\\begin{defn}\\label{v-adic Ronkin metric}\nIn the notation above, and for a place $v\\in\\mathfrak{M}$, the \\emph{$v$-adic Ronkin metric} on $D_f$ is the $v$-adic semipositive toric metric on $\\mathscr{O}(D_f)_v^{\\an}$ corresponding to the $v$-adic Ronkin function $\\rho_{f,v}$ via \\hyperref[characterization of toric metrics]{Theorem \\ref*{characterization of toric metrics}}.\n\\end{defn}\n\nThe previous definition makes sense since, for every $v\\in\\mathfrak{M}$, the $v$-adic Ronkin function of $f$ is concave on $N_\\mathbb{R}$ and has bounded difference from $\\Psi_f$ because of \\hyperref[properties of Ronkin functions]{Proposition \\ref*{properties of Ronkin functions}}. If not otherwise specified, $\\overline{D}_{f,v}$ will denote the divisor $D_f$ equipped with the $v$-adic Ronkin metric $\\|\\cdot\\|_{f,v}$ defined above. By definition,\n\\begin{equation}\\label{Ronkin metric definition}\n\\log\\|s_f\\|_{f,v}=\\rho_{f,v}\\circ\\trop_v\n\\end{equation}\non $X_{0,v}^{\\an}$. To lighten the notation, we will drop the subscript $v$ whenever the choice of the place is clear from the context.\n\n\\begin{proposition}\nLet $f$ and $g$ be two nonzero Laurent polynomials and assume that $\\Psi_f$ and $\\Psi_g$ are virtual support functions on the fan of $X_\\Sigma$. Then, $\\overline{D}_f+\\overline{D}_g=\\overline{D}_{f\\cdot g}$.\n\\end{proposition}\n\\begin{proof}\nThe equality $\\NP(f\\cdot g)=\\NP(f)+\\NP(g)$ implies that $\\Psi_{f\\cdot g}=\\Psi_f+\\Psi_g$. In particular, $\\Psi_{f\\cdot g}$ is a virtual support function on the fan $\\Sigma$ and then defines a toric divisor $D_{f\\cdot g}$ on $X_\\Sigma$ which satisfies $D_{f\\cdot g}=D_f+D_g$ because of \\cite[\\S3.4]{Ful}. The statement follows now from \\cite[Proposition 4.3.14 (1)]{BPS} and \\hyperref[Ronkin function of a product]{Proposition \\ref*{Ronkin function of a product}}.\n\\end{proof}\n\nThe key property of the Ronkin metric is given in the following proposition.\n\n\\begin{proposition}\\label{local height of hypersurfaces}\nLet $X_\\Sigma$ be a smooth projective toric variety, $Z$ an effective cycle on $X_\\Sigma$ of pure codimension $1$ and prime components intersecting $X_0$. Let $f$ be a defining polynomial for $Z$ and $\\tilde{f}$ the associated rational function on $X_\\Sigma$. Assume moreover that $\\Psi_f$ is a virtual support function on the fan $\\Sigma$. For a fixed place $v$ of $K$, let $\\overline{D}_0,\\dots,\\overline{D}_{n-1}$ be toric divisors on $X_\\Sigma$, equipped with $v$-adic semipositive toric metrics. Then\n\\begin{equation}\\label{local height of hypersurfaces equation}\nh_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z;s_0,\\dots,s_{n-1})=h_{\\overline{D}_0,\\dots,\\overline{D}_{n-1},\\overline{D}_f}\\big(X_\\Sigma;s_0,\\dots,s_{n-1},\\tilde{f}s_f\\big),\n\\end{equation}\nfor every choice of rational sections $s_0,\\dots,s_{n-1}$ of $\\mathscr{O}(D_0),\\dots,\\mathscr{O}(D_{n-1})$ respectively with $\\divisor(s_0),\\dots,\\divisor(s_{n-1}),Z$ intersecting properly. \n\\end{proposition}\n\\begin{proof}\nThe product $\\tilde{f}s_f$ is a rational section of $\\mathscr{O}(D_f)$ on $X_\\Sigma$ with associated Weil divisor\n\\begin{equation*}\n\\divisor\\big(\\tilde{f} s_f\\big)=\\divisor(\\tilde{f})+\\divisor(s_f)=Z\n\\end{equation*}\nby \\hyperref[divisor of Laurent polynomial]{Theorem \\ref*{divisor of Laurent polynomial}}, \\eqref{divisor of toric section} and \\eqref{cycle of defining polynomial}. Hence, the sections $s_0,\\dots,s_{n-1},\\tilde{f}s_f$ meet $X_\\Sigma$ properly and the right hand side term in \\eqref{local height of hypersurfaces equation} is well defined.\n\\\\\\hyperref[local height definition]{Definition \\ref*{local height definition}} stating that\n\\begin{multline*}\nh_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z;s_0,\\dots,s_{n-1})=h_{\\overline{D}_0,\\dots,\\overline{D}_{n-1},\\overline{D}_f}\\big(X_\\Sigma;s_0,\\dots,s_{n-1},\\tilde{f}s_f\\big)\\\\+\\int_{X_\\Sigma^{\\an}}\\log\\|\\tilde{f}s_f\\|_f\\ c_1(\\overline{D}_0)\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1}),\n\\end{multline*}\nthe proposition follows from the vanishing of the integral on the right hand side. Indeed, thanks to \\hyperref[Chambert-Loir measure in the toric case]{Theorem \\ref*{Chambert-Loir measure in the toric case}}, such an integral is supported on the analytification of the dense open orbit of $X_\\Sigma$, where the rational function $\\tilde{f}$ coincides with the regular function $f$. Together with the definition of the Ronkin metric in \\eqref{Ronkin metric definition}, this yields\n\\begin{multline*}\n\\int_{X_\\Sigma^{\\an}}\\log\\|\\tilde{f}s_f\\|_f\\ c_1(\\overline{D}_0)\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1})=\\int_{X_0^{\\an}}\\log|f|\\ c_1(\\overline{D}_0)\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1})\\\\+\\int_{X_0^{\\an}}(\\rho_{f,v}\\circ\\trop_v)\\ c_1(\\overline{D}_0)\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1}).\n\\end{multline*}\nFor every $i=0,\\dots,n-1$, denote by $\\psi_i$ the metric function of $\\overline{D}_i$. The tropicalization map being continuous, the change of variables formula and \\hyperref[Chambert-Loir measure in the toric case]{Theorem \\ref*{Chambert-Loir measure in the toric case}} imply on the one hand that\n\\begin{equation*}\n\\int_{X_0^{\\an}}(\\rho_{f,v}\\circ\\trop_v)\\ c_1(\\overline{D}_0)\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1})=\\int_{N_\\mathbb{R}}\\rho_{f,v}\\ d\\MM_M(\\psi_0,\\dots,\\psi_{n-1}).\n\\end{equation*}\nOn the other hand, \\hyperref[Chambert-Loir measure in the toric case]{Theorem \\ref*{Chambert-Loir measure in the toric case}}, together with the change of variables formula and Fubini's theorem, gives\n\\begin{multline*}\n\\int_{X_0^{\\an}}\\log|f|\\ c_1(\\overline{D}_0)\\wedge\\dots\\wedge c_1(\\overline{D}_{n-1})=\\\\\\int_{N_\\mathbb{R}}\\bigg(\\int_{\\mathcal{B}_{\\mathbb{C}_v}}(\\log|f|\\circ\\iota_v)\\ d\\Haar_{\\mathcal{B}_{\\mathbb{C}_v}}\\bigg)\\ d\\MM_M(\\psi_0,\\dots,\\psi_{n-1}).\n\\end{multline*}\nThe definition of the maps $\\iota_v$ and $\\rho_{f,v}$ assures that the inner integral coincides with the opposite of the $v$-adic Ronkin function of $f$, concluding the proof.\n\\end{proof}\n\n\n\\subsection{Toric local heights}\n\nRecall from \\hyperref[canonical metric definition]{Definition \\ref*{canonical metric definition}} that any toric divisor generated by global sections admits a distinguished $v$-adic semipositive toric metric, the canonical metric. This allows to define a local height with respect to toric divisors that is independent of the choice of the sections. Such a notion was introduced in \\cite[\\S5.1]{BPS} as a key step in the proof of the formula for the global height of a toric variety. \n\n\\begin{defn}\nFor a place $v$ of $K$, let $\\overline{D}_0,\\dots,\\overline{D}_d$ be toric divisors on $X_\\Sigma$, endowed with $v$-adic semipositive toric metrics. Denote by $\\overline{D}_0^{\\can},\\dots,\\overline{D}_d^{\\can}$ the same divisors equipped with their $v$-adic canonical metric. Let $Y$ be an irreducible $d$-dimensional subvariety of $X_\\Sigma$ and $\\varphi:Y^\\prime\\to Y$ a birational morphism, with $Y^\\prime$ projective. The \\emph{$v$-adic toric local height} of $Y$ with respect to $\\overline{D}_0,\\dots,\\overline{D}_{d}$ is defined as\n\\[h^{\\tor}_{\\overline{D}_0,\\dots,\\overline{D}_d}(Y):=h_{\\varphi^*\\overline{D}_0,\\dots,\\varphi^*\\overline{D}_d}(Y^\\prime;s_0,\\dots,s_d)-h_{\\varphi^*\\overline{D}_0^{\\can},\\dots,\\varphi^*\\overline{D}_d^{\\can}}(Y^\\prime;s_0,\\dots,s_d),\\]\nwhere $s_i$ is a rational section of $\\varphi^*\\mathscr{O}(D_i)$, for every $i=0,\\dots,d$ and $s_0,\\dots,s_d$ meet $Y^\\prime$ properly. The definition extends by linearity to any cycle of dimension $d$.\n\\end{defn}\n\nThe toric local height of a cycle does neither depend on the choice of the sections $s_0,\\dots,s_d$, nor on the birational model $Y^\\prime$ of $Y$ because of \\cite[Theorem 1.4.17 (2) and (3)]{BPS}. Moreover, the definition is nonempty: Chow's lemma provides $Y$ with a projective birational model, while the moving lemma assures the existence of rational sections meeting $Y^\\prime$ properly.\n\\\\We prove here a formula for the toric local height of an effective cycle $Z$ on $X_\\Sigma$ of pure codimension $1$ and prime components intersecting $X_0$.\n\n\\begin{theorem}\\label{toric local height of hypersurfaces}\nLet $X_\\Sigma$ be a proper toric variety, $Z$ an effective cycle on $X_\\Sigma$ of pure codimension $1$ and prime components intersecting $X_0$. For a place $v$ of $K$, let $\\overline{D}_0,\\dots,\\overline{D}_{n-1}$ be toric divisors on $X_\\Sigma$, equipped with $v$-adic semipositive toric metrics. Then\n\\[h^{\\tor}_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z)=\\MI_M\\big(\\vartheta_0,\\dots,\\vartheta_{n-1},\\rho_{f,v}^\\vee\\big)+\\deg_{D_0,\\dots,D_{n-1}}(X_\\Sigma)\\cdot\\rho_{f,v}(0),\\]\nwhere $f$ is a defining polynomial for $Z$ and $\\vartheta_i$ is the roof function of $\\overline{D}_i$, for $i=0,\\dots,n-1$.\n\\end{theorem}\n\\begin{proof}\nBecause of \\hyperref[can reduce to the case of a toric variety compatible with the polynomial]{Remark \\ref*{can reduce to the case of a toric variety compatible with the polynomial}}, one can assume that $X_\\Sigma$ is a smooth projective toric variety on whose fan $\\Psi_f$ is a virtual support function. Thanks to the moving lemma, one can choose rational sections $s_0,\\dots,s_{n-1}$ of $\\mathscr{O}(D_0),\\dots,\\mathscr{O}(D_{n-1})$ respectively such that $\\divisor(s_0),\\dots,\\divisor(s_{n-1}),Z$ intersect properly. \\hyperref[local height of hypersurfaces]{Proposition \\ref*{local height of hypersurfaces}} implies then that\n\\begin{multline*}\nh^{\\tor}_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z)=h_{\\overline{D}_0,\\dots,\\overline{D}_{n-1},\\overline{D}_f}\\big(X_\\Sigma;s_0,\\dots,s_{n-1},\\tilde{f}s_f\\big)\\\\-h_{\\overline{D}_0^{\\can},\\dots,\\overline{D}_{n-1}^{\\can},\\overline{D}_f}\\big(X_\\Sigma;s_0,\\dots,s_{n-1},\\tilde{f}s_f\\big).\n\\end{multline*}\nBy adding and subtracting the quantity\n\\[h_{\\overline{D}_0^{\\can},\\dots,\\overline{D}_{n-1}^{\\can},\\overline{D}_f^{\\can}}\\big(X_\\Sigma;s_0,\\dots,s_{n-1},\\tilde{f}s_f\\big)\\]\non the right hand side, one obtains that\n\\[h^{\\tor}_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z)=h^{\\tor}_{\\overline{D}_0,\\dots,\\overline{D}_{n-1},\\overline{D}_f}(X_\\Sigma)-h^{\\tor}_{\\overline{D}_0^{\\can},\\dots,\\overline{D}_{n-1}^{\\can},\\overline{D}_f}(X_\\Sigma).\\]\nDenote by $\\Psi_i$ the virtual support function on $\\Sigma$ associated to the toric divisor $D_i$, for every $i=0,\\dots,n-1$. Thanks to \\cite[Corollary 5.1.9]{BPS}, the previous equality yields\n\\[h^{\\tor}_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z)=\\MI_M\\big(\\vartheta_0,\\dots,\\vartheta_{n-1},\\rho_{f,v}^\\vee\\big)-\\MI_M\\big(\\Psi_0^\\vee,\\dots,\\Psi_{n-1}^\\vee,\\rho_{f,v}^\\vee\\big).\\]\nSince they admit by hypothesis a semipositive toric metric, the toric divisors $D_0,\\dots,\\allowbreak D_{n-1}$ are generated by global sections. For every $i=0,\\dots,n-1$, the function $\\Psi_i$ is hence concave and conic and so it is the support function of the polytope $\\Delta_i:=\\stab(\\Psi_i)\\subseteq M_\\mathbb{R}$. The statement of the theorem follows from a combination of \\hyperref[indicator and support function]{Example \\ref*{indicator and support function}}, \\hyperref[mixed integral with indicator functions]{Corollary \\ref*{mixed integral with indicator functions}} and the combinatorial expression for the degree of a toric variety with respect to toric divisors generated by their global sections, see for example \\cite[Proposition 2.10]{O}.\n\\end{proof}\n\n\n\\subsection{Global heights}\n\nTo state the result concerning the global case, recall that for a defining polynomial $f$ for $Z$, the support function of its Newton polytope is denoted by $\\Psi_f$; whenever it is linear on each cone of a complete fan $\\Sigma$, it defines a toric divisor $D_f$ on the toric variety $X_\\Sigma$, together with a distinguished rational section $s_f$ of $\\mathscr{O}(D_f)$.\n\n\\begin{defn}\\label{definition of Ronkin metric}\nIn the above assumptions, the \\emph{Ronkin metric} on $D_f$ is the choice, for every place $v\\in\\mathfrak{M}$, of the $v$-adic Ronkin metric on $D_f$ defined in \\hyperref[v-adic Ronkin metric]{Definition \\ref*{v-adic Ronkin metric}}.\n\\end{defn}\n\nUnless otherwise stated, $\\overline{D}_f$ will denote the toric divisor $D_f$ equipped with its Ronkin metric. By definition, it is a semipositive toric metrized divisor.\n\n\\begin{lemma}\\label{Ronkin metric is adelic}\nThe Ronkin metric on $D_f$ is adelic.\n\\end{lemma}\n\\begin{proof}\nFor a non-archimedean place $v\\in\\mathfrak{M}$, the function $\\rho_{f,v}$ coincides with the tropicalization of the Laurent polynomial $f$, as claimed in \\hyperref[remark about known version of Ronkin functions]{Remark \\ref*{remark about known version of Ronkin functions}}. The fact that $f$ has finitely many nonzero coefficients and the second axiom in \\hyperref[adelic family]{Definition \\ref*{adelic family}} imply that $\\rho_{f,v}=\\Psi_f$ for all but finitely many non-archimedean places. The statement follows then from \\hyperref[adelic fields have finitely many archimedean places]{Lemma \\ref*{adelic fields have finitely many archimedean places}}.\n\\end{proof}\n\nThe definition of such a toric metrized divisor and the study of the local height of $Z$ in \\hyperref[section about local heights]{section \\ref*{section about local heights}} allow to answer the question of the integrability of $Z$ and to give a formula for its global height, implying \\hyperref[main theorem introduction]{Theorem \\ref*{main theorem introduction}} in the introduction.\n\n\\begin{theorem}\\label{hypersurfaces are integrable and their global height}\nLet $X_\\Sigma$ be a proper toric variety, $Z$ an effective cycle on $X_\\Sigma$ of pure codimension $1$ and prime components intersecting $X_0$. Let $\\overline{D}_0,\\dots,\\overline{D}_{n-1}$ be toric divisors on $X_\\Sigma$, equipped with adelic semipositive toric metrics. Then, $Z$ is integrable with respect to $\\overline{D}_0,\\dots,\\overline{D}_{n-1}$ and its global height is given by\n\\[h_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z)=\\sum_{v\\in\\mathfrak{M}}n_v\\MI_M\\big(\\vartheta_{0,v},\\dots,\\vartheta_{n-1,v},\\rho_{f,v}^\\vee\\big),\\]\nwhere $f$ is a defining polynomial for $Z$ and $\\vartheta_{i,v}$ is the roof function of $\\overline{D}_{i,v}$, for every $i=0,\\dots,n-1$ and $v\\in\\mathfrak{M}$.\n\\end{theorem}\n\\begin{proof}\nBecause of \\hyperref[can reduce to the case of a toric variety compatible with the polynomial]{Remark \\ref*{can reduce to the case of a toric variety compatible with the polynomial}}, one can assume that $X_\\Sigma$ is a smooth projective toric variety on whose fan $\\Psi_f$ is a virtual support function. Let hence $s_0,\\dots,s_{n-1}$ be rational sections of $\\mathscr{O}(D_0),\\dots,\\mathscr{O}(D_{n-1})$ respectively such that $\\divisor(s_0),\\dots,\\divisor(s_{n-1}),Z$ intersect properly. Because of \\hyperref[local height of hypersurfaces]{Proposition \\ref*{local height of hypersurfaces}}, the $v$-adic local height of $Z$ with respect to the above choice of sections is given by\n\\begin{equation}\\label{key equality in the proof of global heights}\nh_{\\overline{D}_{0,v},\\dots,\\overline{D}_{n-1,v}}(Z;s_0,\\dots,s_{n-1})=h_{\\overline{D}_{0,v},\\dots,\\overline{D}_{n-1,v},\\overline{D}_{f,v}}\\big(X_\\Sigma;s_0,\\dots,s_{n-1},\\tilde{f}s_f\\big).\n\\end{equation}\nBecause of \\hyperref[Ronkin metric is adelic]{Lemma \\ref*{Ronkin metric is adelic}}, each member of the family $\\overline{D}_0,\\dots,\\overline{D}_{n-1},\\overline{D}_f$ is an adelic semipositive toric metrized divisor on $X_\\Sigma$. As a consequence of the first assertion in \\cite[Proposition 5.2.4]{BPS}, $X_\\Sigma$ is integrable with respect to such a choice of metrized divisors and hence \\cite[Proposition 1.5.8 (1)]{BPS} allows to conclude that \\[h_{\\overline{D}_{0,v},\\dots,\\overline{D}_{n-1,v},\\overline{D}_{f,v}}\\big(X_\\Sigma;s_0,\\dots,s_{n-1},\\tilde{f}s_f\\big)=0\\] for all but finitely many places $v\\in\\mathfrak{M}$. Comparing with \\eqref{key equality in the proof of global heights}, one deduces that $Z$ is integrable with respect to $\\overline{D}_0,\\dots,\\overline{D}_{n-1}$.\n\\\\From the same equality, the global height of $Z$ is seen to satisfy\n\\[h_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z)=h_{\\overline{D}_0,\\dots,\\overline{D}_{n-1},\\overline{D}_f}(X_\\Sigma).\\]The formula for the global height of $Z$ follows then from \\hyperref[height of a toric variety]{Theorem \\ref*{height of a toric variety}}.\n\\end{proof}\n\n\\begin{rem}\nAs in \\hyperref[remark for degree of toric divisor]{Remark \\ref*{remark for degree of toric divisor}}, if $Z$ is an irreducible hypersurface on $X_\\Sigma$ not intersecting $X_0$ it coincides with $V(\\tau)$ for a $1$ dimensional cone $\\tau$ of the fan $\\Sigma$. In such a case, $Z$ is integrable with respect to a family of adelic semipositive toric metrized divisors $\\overline{D}_0,\\dots,\\overline{D}_{n-1}$ on $X_\\Sigma$ and its global height is given by\n\\[h_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(V(\\tau))=\\sum_{v\\in\\mathfrak{M}}n_v\\MI_{M(v_\\tau)}\\big(\\vartheta_{0,v}|_{\\Delta_0^{v_\\tau}},\\dots,\\vartheta_{n-1,v}|_{\\Delta_{n-1}^{v_\\tau}}\\big),\\]\nsee \\cite[Proposition 5.1.11 and Proposition 5.2.4]{BPS}. In the previous formula, $\\Delta_i$ is the polytope associated to the divisor $D_i$ and $\\vartheta_{i,v}$ is the roof function of $\\overline{D}_{i,v}$ for every $i=0,\\dots,n-1$ and $v\\in\\mathfrak{M}$, while $v_\\tau$ is the minimal nonzero integral vector of $\\tau$.\n\\end{rem}\n\n\\begin{rem}\nLocal and global heights of cycles are symmetric and multilinear with respect to sums of semipositive metrized divisors, provided that all terms are defined. The formulas obtained for $1$-codimensional cycles in toric varieties are consistent with these properties, the sum of semipositive toric metrized divisors corresponding to the sup-convolution of the associated roof functions, as shown in \\cite[4.3.14 (1)]{BPS}.\n\\end{rem}\n\n\n\\section{Examples}\\label{section about examples}\n\nFor a fixed adelic field $\\left(K,(|\\cdot|_v,n_v)_{v\\in\\mathfrak{M}}\\right)$ satisfying the product formula and a proper toric variety $X_\\Sigma$ over $K$, of dimension $n$, with torus $\\mathbb{T}=\\spec K[M]$ and dense open orbit $X_0$, we apply in this section the formula in \\hyperref[hypersurfaces are integrable and their global height]{Theorem \\ref*{hypersurfaces are integrable and their global height}} to four particular cases. In the first one, we focus on specific hypersurfaces of $X_\\Sigma$, while in the following three we made relevant choices of the metrized divisors.\n\n\n\\subsection{Binomial hypersurfaces}\n\n\nFor a primitive vector $m$ in $M$ one can consider the Laurent binomial $f=\\chi^m-1$; it is irreducible in $K[M]$ as can be verified by considering its Newton polytope. Hence the closure $Z$ in $X_\\Sigma$ of the subvariety $V(f)$ of the torus $\\spec K[M]$ is an irreducible hypersurface of $X_\\Sigma$ with defining polynomial $f$.\n\\\\Let $\\overline{D}_0,\\dots,\\overline{D}_{n-1}$ be toric divisors on $X_\\Sigma$, equipped with adelic semipositive toric metrics, with $\\vartheta_{i,v}$ the roof function of $\\overline{D}_{i,v}$, for every $i=0,\\dots,n-1$ and $v\\in\\mathfrak{M}$. By \\hyperref[example Ronkin of a binomial]{Example \\ref*{example Ronkin of a binomial}}, $\\rho_{f,v}$ coincides for every $v\\in\\mathfrak{M}$ with the support function of the segment $\\overline{0m}$ in $M_\\mathbb{R}$. The formula in \\hyperref[hypersurfaces are integrable and their global height]{Theorem \\ref*{hypersurfaces are integrable and their global height}} implies then that $Z$ is integrable with respect to $\\overline{D}_0,\\dots,\\overline{D}_{n-1}$ and that its global height is given by\n\\[h_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z)=\\sum_{v\\in\\mathfrak{M}}n_v\\MI_M\\big(\\vartheta_{0,v},\\dots,\\vartheta_{n-1,v},\\iota_{\\overline{0m}}\\big),\\] because of \\hyperref[indicator and support function]{Example \\ref*{indicator and support function}}. Considering, as at the end of \\hyperref[subsection about mixed integrals]{subsection \\ref*{subsection about mixed integrals}}, the quotient lattice $P:=M\/\\mathbb{Z}m$ and the associated projection $\\pi:M\\to P$, \\hyperref[mixed integral with indicator function of a segment]{Proposition \\ref*{mixed integral with indicator function of a segment}} allows to deduce\n\\begin{equation}\\label{height of toric subvariety}\nh_{\\overline{D}_0,\\dots,\\overline{D}_{n-1}}(Z)=\\sum_{v\\in\\mathfrak{M}}n_v\\MI_P\\big(\\pi_*\\vartheta_{0,v},\\dots,\\pi_*\\vartheta_{n-1,v}\\big),\n\\end{equation}\nwith $\\pi_*\\vartheta_{i,v}$ denoting the direct image of $\\vartheta_{i,v}$ by $\\pi$ for every $i=0,\\dots,n-1$ and $v\\in\\mathfrak{M}$, see \\eqref{definition of direct image of concave functions}.\n\n\\begin{rem}\nLet $Q$ be the dual lattice of $P=M\/\\mathbb{Z}m$. The projection $\\pi:M\\to P$ induces an injective dual map $Q\\to N$, with image $m^\\perp\\cap N$. By identifying $Q$ with such an image, which is a saturated sublattice of $N$, one can consider the restriction of the fan $\\Sigma$ to $Q_\\mathbb{R}$; its corresponding toric variety $X_{\\Sigma_Q}$ is proper and has torus $\\spec K[P]$. It also comes with a toric morphism $\\varphi:X_{\\Sigma_Q}\\to X_\\Sigma$, whose restriction to the dense open orbit coincides with the closed immersion of split tori $\\spec K[P]\\to\\spec K[M]$ given by the surjection $\\pi:M\\to P$, see \\cite[pages 81-83]{BPS}. Finally, the push-forward of the cycle $X_{\\Sigma_Q}$ by $\\varphi$ is the cycle $Z$ associated to the hypersurface defined by $\\chi^m-1$. Indeed, the image of $\\varphi$ coincides by properness with the closure in $X_\\Sigma$ of the image of $\\spec K[P]\\to\\spec K[M]$, which is an irreducible $(n-1)$-dimensional subscheme of $\\spec K[M]$ contained in $V(\\chi^m-1)$ as $\\chi^{\\pi(m)}-1=0$.\n\\\\Hence, equality \\eqref{height of toric subvariety} can also be obtained from \\hyperref[height of a toric variety]{Theorem \\ref*{height of a toric variety}}, the arithmetic projection formula and the fact that $\\pi_*\\vartheta_{i,v}$ is the roof function of the pull-back of $\\overline{D}_{i,v}$ via $\\pi$, for every $i=0,\\dots,n-1$ and $v\\in\\mathfrak{M}$, because of \\cite[Proposition 4.3.19 and Proposition 2.3.8 (3)]{BPS}.\n\\end{rem}\n\n\n\n\\subsection{The canonical height}\n\nA toric divisor $D$ on $X_\\Sigma$ generated by global sections admits by \\hyperref[canonical metric definition]{Definition \\ref*{canonical metric definition}} a distinguished semipositive toric metric at any place. The metrized divisor obtained by the choice of such a family of $v$-adic canonical metrics is denoted by $\\overline{D}^{\\can}$; it is an adelic semipositive toric metrized divisor.\n\\\\For a cycle $Z$ of dimension $d$ in $X_\\Sigma$, the \\emph{canonical global height} of $Z$ with respect to a family $D_0,\\dots,D_d$ of toric divisors on $X_\\Sigma$ generated by global sections is defined to be its global height with respect to $\\overline{D}^{\\can}_0,\\dots,\\overline{D}^{\\can}_d$ and it is also denoted by $h^{\\can}_{D_0,\\dots,D_d}(Z)$. The machinery developed in the previous sections allows to express the canonical global height of an effective cycle on $X_\\Sigma$ of pure codimension $1$ via convex geometry.\n\n\\begin{proposition}\\label{canonical height of hypersurfaces}\nLet $Z$ be an effective cycle on $X_\\Sigma$ of pure codimension $1$ and prime components intersecting $X_0$ and $D_0,\\dots,D_{n-1}$ a family of toric divisors on $X_\\Sigma$ generated by global sections. The canonical global height of $Z$ with respect to $D_0,\\dots,D_{n-1}$ is given by \\[h^{\\can}_{D_0,\\dots,D_{n-1}}(Z)=-\\deg_{D_0,\\dots,D_{n-1}}(X_\\Sigma)\\cdot\\sum_{v\\in\\mathfrak{M}}n_v\\rho_{f,v}(0)\\] for any choice of a defining polynomial $f$ for $Z$.\n\\end{proposition}\n\\begin{proof}\nDenoting for any $i=0,\\dots,n-1$ by $\\Psi_i$ the function associated to $D_i$, the property of being globally generated implies that $\\Psi_i$ is the support function of the lattice polytope $\\Delta_i:=\\stab(\\Psi_i)\\subseteq M_\\mathbb{R}$. The roof function of $\\overline{D}^{\\can}_{i,v}$ is hence $\\iota_{\\Delta_i}$ for every $i=0,\\dots,n-1$ and for every $v\\in\\mathfrak{M}$, because of \\hyperref[indicator and support function]{Example \\ref*{indicator and support function}}. It follows from \\hyperref[hypersurfaces are integrable and their global height]{Theorem \\ref*{hypersurfaces are integrable and their global height}} and \\hyperref[mixed integral with indicator functions]{Corollary \\ref*{mixed integral with indicator functions}} that\n\\[h^{\\can}_{D_0,\\dots,D_{n-1}}(Z)=-\\MV_M(\\Delta_0,\\dots,\\Delta_{n-1})\\cdot\\sum_{v\\in\\mathfrak{M}} n_v\\rho_{f,v}(0),\\]\nwith $f$ any defining polynomial for $Z$. To conclude, recall that the degree of $X_\\Sigma$ with respect to $D_0,\\dots,D_{n-1}$ is given by the mixed volume of the associated polytopes, as proved in \\cite[Proposition 2.10]{O}.\n\\end{proof}\n\nThe case of the base field $\\mathbb{Q}$ with the adelic structure described in \\hyperref[examples of adelic fields]{Example \\ref*{examples of adelic fields}} is particularly interesting for arithmetic purposes. For a Laurent polynomial $f$ in $n$ variables and complex coefficients, one defines its (\\emph{logarithmic}) \\emph{Mahler measure} to be \\[m(f):=\\frac{1}{(2\\pi)^n}\\int_{\\theta_1,\\dots,\\theta_n\\in[0,2\\pi]}\\log\\left|f\\big(e^{i\\theta_1},\\dots,e^{i\\theta_n}\\big)\\right|\\ d\\theta_1\\dots d\\theta_n.\\]Such a quantity is notoriously difficult to compute and is sometimes related to special values of $L$-functions, see \\cite{Smyth}, \\cite{Dening}, \\cite{Boyd} and \\cite{Lalin}.\n\\\\In \\cite[Proposition 7.2.1]{Mail}, Maillot expressed the canonical height of a hypersurface in a toric variety over $\\mathbb{Q}$ in terms of the Mahler measure of the associated section. While its proof relies on the study of the arithmetic Chow ring of the ambient toric variety, we here deduce his result from \\hyperref[canonical height of hypersurfaces]{Proposition \\ref*{canonical height of hypersurfaces}}.\n\n\\begin{corollary}[Maillot]\\label{canonical height over the rationals}\nIn the hypotheses and notations  of \\hyperref[canonical height of hypersurfaces]{Proposition \\ref*{canonical height of hypersurfaces}}, assume moreover that the base adelic field is $\\mathbb{Q}$ with its usual adelic structure. Let $f$ be a defining polynomial for $Z$ having as coefficients integer numbers with greatest common divisor $1$. Then,\\[h^{\\can}_{D_0,\\dots,D_{n-1}}(Z)=\\deg_{D_0,\\dots,D_{n-1}}(X_\\Sigma)\\cdot m(f).\\]\n\\end{corollary}\n\\begin{proof}\nLet $f$ be a defining polynomial for $Z$ satisfying the assumptions. Because of \\hyperref[remark about known version of Ronkin functions]{Remark \\ref*{remark about known version of Ronkin functions}}, for every non-archimedean place $v$ of $\\mathbb{Q}$\\[\\rho_{f,v}(0)=f^{\\trop}(0)=0.\\] At the unique archimedean place $v$ of $\\mathbb{Q}$ one has by definition that $\\rho_{f,v}(0)=-m(f)$. The statement follows then directly from \\hyperref[canonical height of hypersurfaces]{Proposition \\ref*{canonical height of hypersurfaces}}.\n\\end{proof}\n\n\n\\subsection{The $\\rho$-height}\n\nThe strategy adopted in the \\hyperref[section hypersurface]{previous section} to prove the main results of the paper suggests the introduction of a distinguished height function. Let $Z$ be an effective cycle on $X_\\Sigma$ of pure codimension $1$ and prime components intersecting $X_0$ and assume that the support function of the Newton polytope of a defining polynomial for $Z$ is a virtual support function on the fan $\\Sigma$. By \\hyperref[a suitable resolution exists]{Lemma \\ref*{a suitable resolution exists}}, this is always the case up to a birational toric transformation. In this setting, the choice of a defining polynomial $f$ for $Z$ determines a toric divisor $D_f$ on $X_\\Sigma$ and a distinguished toric metric on it, the Ronkin metric, as introduced in \\hyperref[definition of Ronkin metric]{Definition \\ref*{definition of Ronkin metric}}. The so-obtained metrized divisor, which is denoted by $\\overline{D}_f$, is an adelic semipositive toric metrized divisor by \\hyperref[Ronkin metric is adelic]{Lemma \\ref*{Ronkin metric is adelic}}.\n\n\\begin{defn}\nIn the above hypotheses and notations, the \\emph{$\\rho$-height} of $Z$, denoted by $h_{\\rho}(Z)$, is defined as its global height with respect to $\\overline{D}_f,\\dots,\\overline{D}_f$, for a choice of a defining polynomial $f$ for $Z$.\n\\end{defn}\n\nAs shown below, the $\\rho$-height of $Z$ is independent of the choice of the defining polynomial $f$. Even if it is not clear whether such a height has a significant geometrical interpretation or arithmetical application, it is straightforward to give a combinatorial formula for it.\n\n\\begin{proposition}\\label{formula for the rho height}\nIn the above hypotheses and notations, the $\\rho$-height of $Z$ is given by \\[h_\\rho(Z)=(n+1)!\\sum_{v\\in\\mathfrak{M}}n_v\\int_{\\NP(f)}\\rho_{f,v}^\\vee\\ d\\vol_M,\\]where $f$ is a defining polynomial for $Z$ and $\\NP(f)$ is its Newton polytope.\n\\end{proposition}\n\\begin{proof}\nThe statement follows trivially from \\hyperref[hypersurfaces are integrable and their global height]{Theorem \\ref*{hypersurfaces are integrable and their global height}}, \\hyperref[properties of Ronkin functions]{Proposition \\ref*{properties of Ronkin functions}} (3) and the properties of mixed integrals.\n\\end{proof}\n\n\\begin{rem}\nThe equality in \\hyperref[formula for the rho height]{Proposition \\ref*{formula for the rho height}} shows that the $\\rho$-height of $Z$ does not depend on the choice of a defining polynomial for it. Indeed, if $f^\\prime$ is another such polynomial, it must satisfy $f^\\prime=c\\cdot \\chi^m\\cdot f$ for some nonzero monomial $c\\cdot \\chi^m\\in K[M]$. For every $v\\in\\mathfrak{M}$, one has then that \\[\\rho_{f^\\prime,v}=-\\log|c|_v+m+\\rho_{f,v}\\] by \\hyperref[Ronkin function of a product]{Proposition \\ref*{Ronkin function of a product}} and \\hyperref[example Ronkin of a monomial]{Example \\ref*{example Ronkin of a monomial}}. The stated independence follows hence from the relation \\[\\rho_{f^\\prime,v}^\\vee=\\tau_m\\rho_{f,v}^\\vee+\\log|c|_v\\] obtained using \\cite[Proposition 2.3.3]{BPS} and from the product formula on $K$.\n\\end{rem}\n\nIt is significant to stress that the formula in \\hyperref[formula for the rho height]{Proposition \\ref*{formula for the rho height}}, though compact, is difficult to evaluate because of the complexity of the archimedean Ronkin function.\n\n\n\\subsection{The Fubini-Study height}\n\nAs a last example, consider the ambient toric variety $X_\\Sigma$ to be the $n$-dimensional projective space over $K$. Denote by $D_\\infty$ the toric divisor on $\\mathbb{P}^n_K$ whose associated Weil divisor is the hyperplane at infinity; the corresponding sheaf is the universal line bundle $\\mathscr{O}(1)$ on $\\mathbb{P}^n_K$. If not otherwise specified, the notation $\\overline{D}_\\infty$ will refer to $D_\\infty$ equipped with the Fubini-Study metric at archimedean places, see \\cite[Example 1.1.2]{BPS}, and the canonical one at non-archimedean places, in the sense of \\hyperref[canonical metric definition]{Definition \\ref*{canonical metric definition}}. It turns out that $\\overline{D}_\\infty$ is an adelic semipositive toric metrized divisor. Thanks to \\hyperref[hypersurfaces are integrable and their global height]{Theorem \\ref*{hypersurfaces are integrable and their global height}}, any effective cycle $Z$ on $\\mathbb{P}^n_K$ of pure codimension $1$ is then integrable with respect to $\\overline{D}_\\infty,\\dots,\\overline{D}_\\infty$ and the corresponding global height \\[h_{\\FS}(Z):=h_{\\overline{D}_\\infty,\\dots,\\overline{D}_\\infty}(Z)\\]is called the \\emph{Fubini-Study height} of $Z$.\n\n\\begin{rem}\nThe Fubini-Study height defined here coincides with the one introduced in \\cite{Fal2} and studied in \\cite{PhilIII}. Examples of the computation of such height for projective hypersurfaces can be found in \\cite{CM}.\n\\end{rem}\n\nSpecializing \\hyperref[hypersurfaces are integrable and their global height]{Theorem \\ref*{hypersurfaces are integrable and their global height}}, one can write the Fubini-Study height of a projective hypersurface in terms of convex geometry. To do so, denote by $\\mathfrak{M}_\\infty$ the collection of archimedean places of $K$, which is a finite set by \\hyperref[adelic fields have finitely many archimedean places]{Lemma \\ref*{adelic fields have finitely many archimedean places}}. After fixing an isomorphism $M\\simeq\\mathbb{Z}^n$, consider the standard simplex \\[\\Delta^n:=\\big\\{(x_1,\\dots,x_n): x_1+\\dots+x_n\\leq1,x_i\\geq0\\text{ for all }i=1,\\dots,n\\big\\}\\] in $M_\\mathbb{R}\\simeq\\mathbb{R}^n$ and, agreeing that $x_0:=1-\\sum_{i=1}^nx_i$, set the function $\\vartheta_{\\FS}:\\Delta^n\\to\\mathbb{R}$ to be \\[\\vartheta_{\\FS}(x):=-\\frac{1}{2}\\sum_{i=0}^nx_i\\log x_i,\\]which is defined on the boundary of $\\Delta^n$ by continuity.\n\n\\begin{proposition}\\label{Fubini-Study height of a hypersurface}\nLet $Z$ be an effective cycle on $\\mathbb{P}^n_K$ of pure codimension $1$ and prime components intersecting $X_0$. The Fubini-Study height of $Z$ is given by\\[h_{\\FS}(Z)=\\sum_{v\\in\\mathfrak{M}_\\infty}n_v\\MI_M\\big(\\vartheta_{\\FS},\\dots,\\vartheta_{\\FS},\\rho_{f,v}^\\vee\\big)-\\sum_{v\\in\\mathfrak{M}\\setminus\\mathfrak{M}_\\infty}n_v\\rho_{f,v}(0),\\]where $f$ is a defining polynomial for $Z$.\n\\end{proposition}\n\\begin{proof}\nThe roof functions of the metrized divisor $\\overline{D}_\\infty$ are given by the function $\\vartheta_{\\FS}$ at archimedean places, as remarked in \\cite[Example 2.4.3 and Example 4.3.9 (2)]{BPS} and by the indicator function of $\\Delta^n$ at non-archimedean places, by \\cite[Example 4.3.9 (1)]{BPS} and \\hyperref[indicator and support function]{Example \\ref*{indicator and support function}}. The statement follows then from \\hyperref[hypersurfaces are integrable and their global height]{Theorem \\ref*{hypersurfaces are integrable and their global height}} and \\hyperref[mixed integral with indicator functions]{Corollary \\ref*{mixed integral with indicator functions}}, together with the fact that $\\MV_{M}(\\Delta^n,\\dots,\\Delta^n)=1$ because of the conventions introduced in \\hyperref[subsection about real Monge-Ampere measures]{subsection \\ref*{subsection about real Monge-Ampere measures}} and \\hyperref[lattice volume]{Remark \\ref*{lattice volume}}.\n\\end{proof}\n\nThe non-archimedean contributions to the Fubini-Study height are easily computable, since for every Laurent polynomial $f$ with set of coefficients $\\Gamma(f)$,\n\\[-\\rho_{f,v}(0)=\\log\\max_{c\\in\\Gamma(f)}|c|_v\\]\nif $v\\in\\mathfrak{M}\\setminus\\mathfrak{M}_\\infty$. From this equality one easily obtains the following special case.\n\n\\begin{corollary}\\label{FS height over the rational}\nAssume the base adelic field to be $\\mathbb{Q}$ with its usual adelic structure. The Fubini-Study height of an effective cycle $Z$ on $\\mathbb{P}^n_\\mathbb{Q}$ of pure codimension $1$ and prime components intersecting $X_0$ is given by \\[h_{\\FS}(Z)=\\MI_M\\big(\\vartheta_{\\FS},\\dots,\\vartheta_{\\FS},\\rho_{f,\\infty}^\\vee\\big),\\]where $f$ is a defining polynomial for $Z$ whose coefficients are integer numbers with greatest common divisor $1$.\n\\end{corollary}\n\nBecause of the presence of an archimedean Ronkin function, the formula in \\hyperref[FS height over the rational]{Corollary \\ref*{FS height over the rational}} appears arduous to evaluate. It would anyway be interesting to use it to study arithmetical properties of projective hypersurfaces or recover similar results to the ones obtained in \\cite{CM}.\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThere exists a very old problem constantly met in various aspects of applied\nmathematics, which can be formulated as follows. Very often realistic problems \nare so complicated that they do not allow for exact solutions. It is standard\nfor such problems to use some kind of perturbation theory\n\\cite{Bog1961, Gia1972, Nay1973}. Then one gets answers in terms of expansions\nin powers of a small parameter, or a small variable, say for $x \\ra 0$. However,\noften the problem of interest corresponds not to a small variable, but, rather \nthe opposite, to large values of this variable; very often it is the infinite \nlimit $x \\ra \\infty$ that is of the most interest \\cite{Kle2006}. One could find \nthis limit, provided the general formula of expansion terms would be given and \nthe derived expansion would produce convergent series. None of these conditions \nis usually valid. As a rule, only a few expansion terms can be derived. \nAdditionally, the resulting series are divergent, being only asymptotic \n\\cite{Erd1955, Har1949}. Then the question arises: how, from the knowledge of \nseveral terms of an asymptotic expansion at a variable $x \\ra 0$ could one find \nthe limit corresponding to $x \\ra \\infty$?\n\nOne often extrapolates small-variable expansions by means of Pad\\'{e} \napproximants \\cite{Bak1996}. However, the straightforward use of these  \napproximants yields\n$$\nP_{M\/N}(x) \\sim x^{M-N} \\qquad (x\\ra\\infty) \\;   ,\n$$\nwhich, depending on the relation between $M$ and $N$, can tend to:\n\n\\begin{itemize} \n\n\\item \ninfinity (when $M > N$), \n\n\\item\nzero (when $M < N$), \n\n\\item\na constant (if $M = N$).\n\n\\end{itemize}\n\nIn that sense, the limit $x \\ra \\infty$ is not defined. \n\nWhen the character of the large-variable limit is known, one can invoke the \ntwo-point Pad\\'{e} approximants \\cite{Bak1996}. However the accuracy of the \nlatter is not high and one confronts several difficulties: \n\n\\begin{enumerate}\n\n\\item\nFirst of all, when constructing these approximants, one often obtains spurious \npoles yielding unphysical singularities \\cite{Bak1996}, sometimes a large \nnumber of poles \\cite{Saf1976}. \n\n\\item\nSecond, there are the cases when Pad\\'{e} approximants are not able to sum \nperturbation series even for small values of an expansion parameter \\cite{Sim1991}. \n\n\\item\nThird, in the majority of cases, to reach a reasonable accuracy, one needs \nto have tens of terms in perturbative expansions \\cite{Bak1996}, while often \ninteresting problems provide only a few terms. \n\n\\item\nFourth, defining the two-point Pad\\'{e} approximants, one always meets an \nambiguity in distributing the coefficients for deciding which of these must\nreproduce the left-side expansion and which the right-side series. This \nambiguity worsens with the increase of the approximants' orders, making it \ndifficult to compose two-point Pad\\'{e} tables. For the case of a few terms, \nthis ambiguity makes the two-point Pad\\'{e} approximants practically \ninapplicable. For example, it has been shown \\cite{Sel1989} that, for the \nsame problem, one may construct different two-point Pad\\'{e} approximants, \nall having correct left and right-side limits, but differing from each other \nin the intermediate region by a factor of 40, which gives 1000$\\%$ uncertainty. \nThis demonstrates that in the case of short series the two-point Pad\\'{e} \napproximants do not allow one to get a reliable description. \n\n\\item\nFifth, the two-point Pad\\'{e} approximants cannot always be used for \ninterpolating between two different expansions, but only when these two \nexpansions have compatible variables \\cite{Bak1996}. When these expansions\nhave incompatible variables, the two-point Pad\\'{e} approximants cannot be \ndefined in principle.\n\n\\item\nFinally, interpolating between two points, one of which is finite and another\nis at infinity, one is able to characterize the large-variable limit of only \nrational powers \\cite{Bak1996}. \n\n\\end{enumerate}\n\nAnother method that allows for the extrapolation of divergent series is \noptimized perturbation theory, based on the introduction of control functions\ndefined by an optimization condition and guaranteeing the transformation of \ndivergent series into convergent series \\cite{Yuk1976a, Yuk1976b, Yuk2002}. \nSince 1976, when optimized perturbation theory was introduced \n\\cite{Yuk1976a, Yuk1976b}, a number of variants of different control functions \n(see discussion in (\\cite{Yuk1999, Yuk2002}) have been put forward.  \nKleinert \\cite{Kle1993, Kle2006} variational perturbation theory, where control \nfunctions are introduced through a variable transformation and variational \noptimization conditions, is particularly worth mentioning. This method provides \ngood accuracy for the extrapolation of weak-coupling expansions to the \nstrong-coupling limit, especially when a number of terms in the weak-coupling \nperturbation theory are available \\cite{Jan1995}.        \n\nIn the present paper, we address the problem of extrapolating small-variable \nasymptotic expansions to their effective strong-coupling limits by employing\nanother approach, based on self-similar approximation theory \n\\cite{Yuk1990a, Yuk1990b, Yuk1991, Yuk1992, Yuk1993, Yuk1994, Yuk1996}.\nThe main difference of this approach from optimized perturbation theory\nis that we possess the approximation methods without introducing control functions, \nwhich makes calculations essentially simpler. Self-similar approximation theory \ncan be combined with Kleinert variational perturbation theory \\cite{Kle2005}.\nThis, however, also requires the introduction of variational control functions. \nIn the present paper, however, we pay most attention to considering simpler ways \nnot involving control functions. \n\nThere exists a principal problem, when one accomplishes an extrapolation in the \ncase for which the exact solution is not known and only a few terms of \nweak-coupling perturbation theory are available. This is the problem of the \nreliability of the obtained extrapolation. In such a case, it is important to \nbe able to do the extrapolation by several methods, comparing their \nresults. If these results yield close values, this suggests that the \nextrapolation is reliable.  \n\nIn line with this idea, we aim at employing different variants of self-similar \napproximations, applying them to the same problems and comparing the results. \nIf the approximants for a problem, obtained by different methods, are close to \neach other, this would suggest that the derived values are reliable.  \n\nWe consider several variants of self-similar approximants for each problem and \nshow that they really are close to each other, hence they can successfully \nextrapolate asymptotic expansions, valid at $x \\ra 0$, to their effective limits \nof $x \\ra \\infty$. We especially concentrate on the strong-coupling limit, where\napproximate methods usually are the least accurate, leading to the maximal errors.\nWe show that, even in this least favourable situation, with just a few perturbative \nterms available, the self-similar extrapolation methods provide reasonable accuracy.\nFor completeness, we also show that the self-similar methods allow\nus to construct the approximants displaying good accuracy in the whole region of\nthe studied variable. For instance, effective equations of state can be derived,\nthese being in good agreement with experimental data.    \n\nThe difference of the present paper from our previous publications is in the \nfollowing.\n\n(i) We study several types of self-similar approximants and compare their accuracy,\nwhich allows us to draw conclusions on the reliability of the method.\n\n(ii) A large set of examples of different natures is analyzed, demonstrating\nthe generality of the method of self-similar approximants and their \neffectiveness for extrapolating different functions met in various problems\nof applied mathematics.\n\n(iii) We consider a new type of approximants resulting from a double\nself-similar renormalization and show how they improve the accuracy as compared\nwith exact results, when these are available.\n\n(iv) We show an effective way for calculating the large-variable critical \nexponents.    \n\n(v) The method is shown to provide good accuracy for the whole range of the \nvariable. This is demonstrated by constructing the equation of state that exactly\nreproduces a phenomenological equation for quantum hard spheres. \n \n\n\\section{Formulation of extrapolation problem}\n\nSuppose we are interested in the behaviour of a real function $f(x)$ of a \nreal variable $x \\in [0,\\infty)$. Also let this function be defined by a \ncomplicated problem that does not allow for an explicit derivation of the\nform of $f(x)$. What can only be done is use some kind of perturbation theory \nyielding asymptotic expansions representing the function\n\\be\n\\label{2.1}\nf(x) \\simeq f_k(x) \\qquad (x \\ra 0)\n\\ee\nat small values of the variable $x \\ra 0$, with $k=0,1,\\ldots$ being the \nperturbation order. The perturbative series of $k$-th order can be written as \nan expansion in powers of $x$ as\n\\be\n\\label{2.2}\n f_k(x) = f_0(x) \\left ( 1 + \\sum_{n=1}^k a_n x^n \\right ) \\; ,\n\\ee\nwhere $f_0(x)$ is chosen so that the series in the brackets would start \nwith the term one. It is convenient to define the reduced expression\n\\be\n\\label{2.3}\n \\overline f_k(x) \\equiv \\frac{f_k(x)}{f_0(x)} = 1 + \n\\sum_{n=1}^k a_n x^n \\;  ,\n\\ee\nwhich will be subject to self-similar renormalization. \n\nNote that practically any perturbative series can be represented in \nform (2.2). For instance, if we have a Laurent-type series\n$$\n f_{m+k}(x) = \\sum_{n=-m}^k c_n x^n \\;  ,\n$$\nit can be transformed to (2.2) by rewriting it as  \n$$\nf_{m+k}(x) = \\frac{c_{-m}}{x^m} \\left (\n 1 +  \\sum_{n=1}^{m+k} a_n x^n  \\right ) \\;  .\n$$ \n\nHere we consider the series in integer powers, or those that can be \nreduced to such, since this is the most frequent type of perturbation-theory\nexpansions. Thus, the Puiseaux expansion \\cite{Pui1850} of the type\n$$\nf_k(t) = \\sum_{n=n_0}^k c_n t^{n\/m} \\;   ,\n$$  \nwhere $n_0$ is an integer and $m$ is a nonzero natural number, can be \nreduced to form (1.2) by the change of the variable $ t = x^m$. It is\npossible to generalize the approach to the series of the type\n$$\n f_k(x) = \\sum_{n}^k c_n x^{\\al_n} \\qquad ( \\al_n < \\al_{n+1} ) \\;  ,\n$$\nwith arbitrary real powers $\\alpha_n$ arranged in an ascending order.\nWhen $\\alpha_n$ pertains to an ordered group, the latter expression \ncorresponds to the Hahn series \\cite{Ked2001, Mac1939}.\n\nAs is known, the most difficult region for approximating is that of the \nlarge variable, where approximants are usually the least accurate. This \nis why our main interest here will be the large-variable behaviour of the \nfunction, where its asymptotic form is\n\\be\n\\label{2.4}\n f(x) \\simeq Bx^\\bt \\qquad (x\\ra\\infty) \\; .\n\\ee\nThe constant $B$ is called the critical amplitude and the power $\\beta$\nis the critical exponent.\n\nAfter employing the self-similar renormalization for the reduced function\n(2.3), we get a self-similar approximant $\\overline f_k^*(x)$, which gives a\nself-similar approximant\n\\be\n\\label{2.5}\nf_k^*(x) = f_0(x) \\overline f_k^*(x)\n\\ee\nfor the sought function $f(x)$. Considering for the latter the limit \n$x \\ra \\infty$, we find the related approximation for the critical \namplitude and critical exponent. In many cases the exponent is known from \nother arguments. Then, we need to find only the critical amplitude.  \n\n\n\\section{Variants of self-similar approximants}\n\nIn the cases, when one can compare the derived approximants with known \nexpressions, one can easily evaluate the accuracy of the approximants. \nBut how could we trust the approximants, when no exact expression for the \nsought function is available? In that case, it would be desirable to have\nto hand several variants of approximants in order to compare them with \neach other. If all of them give close results, this would suggest that \nthe method is reliable.   \n \nSeveral types of approximants, based on self-similar approximation theory,  \nhave been derived. We shall not repeat their derivation here. This can \nbe found, along with all the details, in our previous publications. We \nshall just present the corresponding expressions and explain how they \nwill be used for the problem of extrapolation to infinity.\n\n\n\\subsection{Self-similar factor approximants}\n\nSelf-similar factor approximants have been introduced in Refs. \n\\cite{Glu2003, Yuk2003}. For the reduced expansion (2.3), the $k$th order\nself-similar factor approximant reads as\n\\be\n\\label{3.1}\n \\overline f_k^*(x) = \\prod_{i=1}^{N_k} ( 1 + A_i x)^{n_i} \\;  ,\n\\ee\nwhere\n\\begin{eqnarray}\n\\label{3.2}\nN_k = \\left\\{ \\begin{array}{ll}\nk\/2 , & ~ k = 2,4,\\ldots \\\\\n(k+1)\/2 , & ~ k =3,5,\\ldots\n\\end{array} \\right. \n\\end{eqnarray}\nand the parameters $A_i$ and $n_i$ are defined from the accuracy-through-order\nprocedure, by expanding expression (3.1) in powers of $x$, comparing the \nlatter expansion with the given sum (2.3), and equating the like terms in \nthese expansions. When the approximation order $k=2p$ is even, the above \nprocedure uniquely defines all $2p$ parameters. When the approximation order \n$k=2p+1$ is odd, the number of equations in the accuracy-through-order \nprocedure is $2p$ which is by one smaller than the number of parameters. Then, \nusing the scale invariance arguments \\cite{Yuk2007}, one sets $A_1=1$, \nthus, uniquely defining all parameters. Another way is to find one of the \ncoefficients $A_i$ from the variational optimization of the approximant \n\\cite{Yuk2009b}. Both these approaches give close results, though the scaling \nprocedure of setting $A_1$ to one is simpler.   \n\nWith approximant (3.1), the self-similar approximant for the sought \nfunction (2.5) becomes\n\\be\n\\label{3.3}\n f^*_k(x) = f_0(x) \\prod_{i=1}^{N_k} ( 1 + A_i x)^{n_i} \\; .\n\\ee\nIf the zero-order factor has the large-variable form\n\\be\n\\label{3.4}\n f_0(x) \\simeq A x^\\al \\qquad (x\\ra\\infty) \\;  ,\n\\ee\nthen approximant (3.3) behaves as \n\\be\n\\label{3.5} \n f_k^*(x) \\simeq B_k x^\\bt \\qquad (x\\ra\\infty) \\;  .\n\\ee\nUnder a given exponent $\\beta$, the powers $n_i$ must satisfy the equality\n\\be\n\\label{3.6}\n \\bt = \\al + \\sum_{i=1}^{N_k} n_i \\;  ,\n\\ee\nwhile the critical amplitude $B$ is approximated by\n\\be\n\\label{3.7}\n B_k = A \\prod_{i=1}^{N_k} A_i^{n_i} \\;  .\n\\ee\n  \nIt is worth stressing that the factor $f_0(x)$ in Eq. (3.3) is explicitly defined\nby the perturbative expansion (2.2), so it is known. The factor approximants (3.3)\nmay have singularities when some $A_i$ and $n_i$ are negative. This makes it \npossible to associate such singularities with critical points and phase transitions.\nInvestigation of the critical points and the related critical exponents, by means \nof the factor approximants, has been done in our previous publications \n\\cite{Glu2003, Yuk2003, Yuk2004, Yuk2007, Yuk2009b}. \n\n\n\\subsection{Self-similar root approximants}\n\nThe derivation of the self-similar root approximants can be found in\nRefs. \\cite{Glu1998, Yuk1998, Yuk2002}. The self-similar renormalization\nof the reduced expansion (2.3) yields\n\\be\n\\label{3.8}\n R_k(x) = \\left ( \\left ( \\left ( \\ldots ( 1 + A_1 x )^{n_1} +\nA_2x^2 \\right )^{n_2} + A_3x^3 \\right )^{n_3} + \\ldots +\nA_k x^k \\right )^{n_k} \\;  .\n\\ee\nThe $k$th order approximant for the sought function then becomes\n\\be\n\\label{3.9}\n f_k^*(x) = f_0(x) R_k(x) \\;  .\n\\ee\nIn Ref. \\cite{Yuk2002}, it has been rigorously proved that the parameters \n$A_i$ and $n_i$ are uniquely defined, provided that $k$ terms of the \nlarge-variable expansion at $x \\ra \\infty$ are known, and the condition\n$pn_p - p + 1 = const$ holds for $p = 1,2,\\ldots, k-1$. Then expression (3.8) \nleads to\n\\be\n\\label{3.10}\n R_k(x) \\simeq A_k^{n_k} x^{kn_k} \\qquad (x\\ra\\infty ) \\;  .\n\\ee\nWith the given exponent $\\beta$, the power $n_k$ satisfies the relation\n\\be\n\\label{3.11}\n\\bt = \\al + kn_k    \n\\ee\nand the $k$th order approximation for the critical amplitude is\n\\be\n\\label{3.12}\n B_k = A A_k^{n_k} \\;  .\n\\ee\n   \n\n\\subsection{Iterated root approximants}\n\nSelf-similar root approximants are uniquely defined when their parameters \nare prescribed by the large-variable behaviour of the sought function. However, \nif we try to find these parameters from the small-variable expansion (2.2), \nthen we meet the problem of multiple solutions \\cite{Yuk2004}. To avoid this \nproblem, one has to impose additional conditions on the parameters. Such a \nstraightforward condition would be the requirement that all $k$ terms in root\n(3.8) would contribute to the large-variable amplitude \\cite{Glu2010}. For \nthis, it is necessary and sufficient that the internal powers $n_j$ be \ndefined as\n\\be\n\\label{3.13}\n n_j = \\frac{j+1}{j} \\qquad ( 1 \\leq j \\leq k-1 ) \\;  ,\n\\ee\nwith the external power related to the exponent $\\beta$ as\n\\be\n\\label{3.14}\n n_k = \\frac{\\gm}{k} \\qquad (\\gm = \\bt - \\al ) \\;  .\n\\ee\nThen expression (3.8) becomes the iterated root approximant\n\\be\n\\label{3.15}\n R_k(x) = \\left ( \\left ( \\left ( \\ldots ( 1 + A_1 x )^2 +\nA_2x^2 \\right )^{3\/2} + A_3x^3 \\right )^{4\/3} + \\ldots +\nA_k x^k \\right )^{\\gm\/k} \\;  ,\n\\ee\nwhere all parameters $A_j$ are uniquely defined by the accuracy-through-order \nprocedure.  \n\nIn the large-variable limit, Eq. (3.15) yields\n\\be\n\\label{3.16}\n R_k \\simeq \\frac{B_k}{A} \\; x^\\gm \\qquad (x\\ra\\infty) \\;  ,\n\\ee\nwith the critical amplitude\n\\be\n\\label{3.17}\nB_k = A \\left ( \\left ( \\ldots \\left ( A_1^2 + A_2 \\right )^{3\/2}\n+ A_3 \\right )^{4\/3} + \\ldots + A_k \\right )^{\\gm\/k} \\; .\n\\ee\n\nIt may happen that the iterated root approximants are well defined up to an \norder $k$, after which they do not exist because some of the parameters $A_p$ \nare negative. At the same time, the higher-order terms of perturbation-theory\nexpansion can be available up to an order $k+p$. How then could we use these\nadditional terms for constructing the higher-order approximants? \n\n\n\\subsection{Corrected root approximants}\n\nCorrections to the iterated root approximants (3.15), employing the higher-order \nterms, can be constructed \\cite{Glu2010} by defining the corrected root \napproximants\n\\be\n\\label{3.18}\n \\widetilde R_{k\/p}(x) = R_k(x) C_{k\/p}(x) \\;  ,\n\\ee\nwith the correction function \n\\be\n\\label{3.19}\n C_{k\/p}(x) = 1 + d_{k+1} x^{k+1} \\left ( \\left ( \\left (\n\\dots ( 1 + b_1 x )^2 + b_2 x^2 \\right )^{3\/2} + \nb_3 x^3 \\right )^{4\/3} + \\ldots + b_{p-1} x^{p-1} \n\\right )^{-(k+1)\/(p-1)} \\; ,\n\\ee\nwhere $p>2$ and all parameters are defined from the accuracy-through-order \nprocedure, when the terms of the expansion of form (3.18) are equated with\nthe corresponding terms of the perturbation theory expansion. Here, the critical \nexponent is defined by the iterated root approximant (3.16), so that the \nlimit $x \\ra \\infty$ of the correction function is finite:\n\\be\n\\label{3.20}\nC_{k\/p}(\\infty) = 1 + d_{k+1} \\left ( \\left (\n\\dots \\left ( b_1^2 + b_2 \\right )^{3\/2} + b_3 \\right )^{4\/3} + \n \\ldots + b_{p-1}  \\right )^{-(k+1)\/(p-1)} \\;   .\n\\ee\n\nThe corresponding approximation for the sought function takes the form\n\\be\n\\label{3.21}\n f_{k\/p}^*(x) = f_0(x) \\widetilde R_{k\/p}(x) \\;  .\n\\ee\nIts large-variable behaviour is\n\\be\n\\label{3.22}\n f_{k\/p}^*(x) \\simeq B_{k\/p} x^\\bt \\qquad (x\\ra\\infty) \\;  ,\n\\ee\nwith the corrected critical amplitude\n\\be\n\\label{3.23}\nB_{k\/p} = A B_k C_{k\/p}(\\infty) \\;   .\n\\ee\n\n\n\\subsection{Self-similar power transforms}\n\nIt is possible to get improvement of approximants by employing power \ntransforms \\cite{Glu2006}. For this purpose, we define the power transform \nof the reduced expansion (2.3) as\n\\be\n\\label{3.24}\nP_k(x,m) \\equiv \\overline f_k^m(x) \\;   ,\n\\ee\nwhich is expanded in powers of $x$ giving\n\\be\n\\label{3.25}\n P_k(x,m) \\cong \\sum_{n=0}^k b_n(m) x^n \\;  .\n\\ee\nAfter the self-similar renormalization of expansion (3.25), we get a \nself-similar approximant $P_k^*(x,m)$. We then accomplish the inverse\ntransformation \n\\be\n\\label{3.26}\n \\overline F_k(x,m) = \\left [ P_k^*(x,m) \\right ]^{1\/m} \\;  .\n\\ee\nThe powers $m_k =m_k(x)$ are defined by the variational condition\n\\be\n\\label{3.27}\n \\frac{\\prt\\overline F_k(x,m)}{\\prt m} = 0 \\;  .\n\\ee\nFinally, the corresponding approximation for the sought function is given by\n\\be\n\\label{3.28}\n f_k^*(x) = f_0(x) \\overline F_k(x,m_k) \\; .\n\\ee\nWhen we are interested in the large-variable limit, condition (3.27) reduces \nto the differentiation of only critical amplitude. \n\n\n\\subsection{Double self-similar approximants}\n\nAnother way of improving the accuracy is by employing the procedure of \nself-similar renormalization twice. The fact that the accuracy does improve can \nbe illustrated by those examples for which exact solutions are known. \n\nThe double renormalization is accomplished as follows. First, renormalizing \nthe reduced expansion (2.3), we construct the self-similar approximants (2.5). \nThe approximants $\\overline f_k^*(x)$ form the approximation sequence \n$\\{\\overline f_k^*(x)\\}$. Introducing the expansion function $x(\\vp)$ by the equation\n\\be\n\\label{3.29} \n \\overline f_1^*(x) = \\vp \\; , \\qquad x = x(\\vp) \\;  ,\n\\ee\nwe define\n\\be\n\\label{3.30}\n y_k(\\vp) =  \\overline f_k^*( x(\\vp) ) \\; .\n\\ee\nBy this definition, the sequence $\\{y_k(\\vp)\\}$ is bijective to the sequence \n$\\{\\overline f_k^*(x)\\}$. In view of Eq. (3.29), we have\n\\be\n\\label{3.31}\n  y_1(\\vp) = \\vp \\; .\n\\ee\n\nConsider the sequence $\\{y_k(\\vp)\\}$ as the trajectory of a dynamical system\nin discrete time, that is, of a cascade, with the initial condition (3.31). \nEmbed this approximation cascade into an approximation flow:\n\\be\n\\label{3.32}\n \\{ y_k(\\vp) : \\; k \\in \\mathbb{Z}_+ \\} \\subset\n\\{ y(t,\\vp) : \\; t \\in \\mathbb{R}_+ \\} \\; ,\n\\ee\nwhere\n$$\n \\mathbb{Z}_+ \\equiv \\{ 0, 1, 2, \\ldots \\} \\; , \\qquad\n\\mathbb{R}_+ \\equiv [0, \\infty) \\; ,\n$$\nso that the flow trajectory passes through all points of the cascade trajectory,\n\\be\n\\label{3.33}\n  y(t,\\vp) = y_k(\\vp) \\qquad ( t = k) \\;  .\n\\ee\nThe evolution equation for the flow reads as\n\\be\n\\label{3.34}\n\\frac{\\prt}{\\prt t} \\; y(t,\\vp) = v(y) \\;   ,\n\\ee\nwith $v(y)$ being the flow velocity.\n\nIntegrating the evolution equation (3.34) gives\n\\be\n\\label{3.35}\n \\int_{y_k}^{y_k^*} \\frac{dy}{v(y)} = \\tau_k \\;  ,\n\\ee\nwhere $y_k = y_k(\\vp)$ and $\\tau_k$ is the minimal effective time necessary \nfor reaching the approximate fixed point $y_k^*(\\vp)$. The latter, \naccording to definition (3.30), is a twice renormalized self-similar \napproximant\n\\be\n\\label{3.36}\n y_k^*(\\vp) = \\overline f_k^{**}(x(\\vp) ) \\;  .\n\\ee\nKeeping in mind definition (3.30) also allows us to rewrite integral (3.35) \nas\n\\be\n\\label{3.37}\n  \\int_{\\overline f_k^*}^{\\overline f_k^{**}} \\frac{d\\vp}{v_k(\\vp)} = \n\\tau_k \\;  ,\n\\ee\nwhere\n$$\n \\overline f_k^{*} = \\overline f_k^{*}(x) \\; , \\qquad\n\\overline f_k^{**} = \\overline f_k^{**}(x) \\;  .\n$$\nAssuming that we reach the quasi-fixed point in one step, we may set \n$\\tau_k =1$.  \n\nEmploying in the evolution integral (3.37) the Euler discretization for \nthe velocity\n\\be\n\\label{3.38}\n v_k(\\vp) = y_k(\\vp) - \\vp = \\overline f_k^{*}(x(\\vp)) -\n\\overline f_1^{*}(x(\\vp))     \n\\ee\nand calculating this integral gives the twice renormalized approximant \nfor the sought function\n\\be\n\\label{3.39}\n \\overline f_k^{**}(x) = f_0(x) \\overline f_k^{**}(x) \\;  .\n\\ee\nThe large-variable limit of the latter \n\\be\n\\label{3.40}\n\\overline f_k^{**}(x) \\simeq B_k^* x^\\bt \\qquad \n(x \\ra \\infty)\n\\ee\ndefines the approximate expression for the critical amplitude $B_k^*$. \nUsually, integral (3.37) can be calculated only numerically.   \n\nIn the following sections, the above methods of extrapolation will be\nillustrated by a number of examples of different nature, with the emphasis\non the large-variable limit $x \\ra \\infty$. Analyzing these examples, \nwe shall pay most attention to the possibility of obtaining accurate\napproximate expressions by taking just a few terms in the small-variable \nexpansions, bearing in mind that complicated realistic problems usually \nprovide us with only a small number of terms of perturbation theory. \n\n\n\\section{Explicitly defined functions}\n\nIn order to clearly demonstrate how the method works and to show that it\nreally provides good accuracy, it is illustrative to start with functions \nwhose explicit form is given. This will allow us to easily evaluate the \naccuracy of approximants. The consideration of such simpler cases is \nnecessary before considering the complicated problems whose exact solutions \nare not known, since only then it is possible to explicitly demonstrate \nthe efficiency of the method and to evaluate what accuracy of the used \napproximants should be expected.\n\nThe variable $x$ will be varying in the range \n$[0, \\infty)$.\n\n\\subsection{Function-1}\n\nConsider a function\n\\be\n\\label{4.1}\nf(x) = \\frac{1}{2} \\left ( \\sqrt{4+x} \\; - \\; 1 \\right ) \\;   ,\n\\ee\nwhich is of importance because of giving the golden ratio\n$$\n\\frac{1}{f(1)} = 1 + f(1) = 1.618034  \\;  .\n$$\nIn its small-variable expansion \n\\be\n\\label{4.2}\nf_k(x) = \\sum_{n=0}^k c_n x^n\n\\ee\nthe first five coefficients are\n$$\n c_0 = \\frac{1}{2} \\; , \\qquad c_1 = \\frac{1}{8} \\; , \\qquad\nc_2 = -\\; \\frac{1}{128} \\; , \\qquad c_3 = \\frac{1}{1024} \\; , \n\\qquad  c_4 = -\\;\\frac{5}{32768} \\; .\n$$\nHere $f_0 = c_0$.\n\nDespite its simplicity, this function expansion is not trivial, since the \nfirst two coefficients are positive, after which they start alternating.\n\nThe large-variable behaviour \n\\be\n\\label{4.3}\nf(x) \\simeq B x^\\bt = 0.5 \\sqrt{x}\n\\ee\nshows that \n$$\n B = 0.5 \\; , \\qquad \\bt = 0.5 \\;  .\n$$\nUsing the approximants, described above, we fix the exponent $\\beta$,\nconcentrating on the accuracy of calculating the critical amplitude.\n\nThe method of factor approximants of Sec. 3.1 yields $B_4 = 0.440$.\nPower transforms of Sec. 3.5, with the factor approximants, do not \nprovide essential improvement. The optimization condition (3.27) \nresults in two solutions for $m$, which yields for the amplitudes\nthe values $0.416$ and $0.455$. The iterated root approximants of \nSec. 3.3 give $B_2 = 0.374, B_3 = 0.385, B_4 = 0.393$. The corrected \niterated roots of Sec. 3.4 give $B_{2\/2} = 0.422$. Power transforms, \nwith iterated roots again yield two solutions for $B_2$, with the values\n$0.404$ and $0.433$. All these results are close to the Pad\\'{e} \napproximant $P_{2\/2} = 0.433$. Essential improvement of accuracy is \nachieved by the double approximants of Sec. 3.6 on the basis of the \niterated roots, giving $B_4^* = 0.476$. \n\n\n\\subsection{Function-2}\n\nLet us take a more complicated function\n\\be\n\\label{4.4} \n f(x) = \\frac{2}{\\pi} \\; {\\rm arccot}(-x) \\exp \\left ( 1\n- \\; \\frac{1}{1+x} \\right ) \\;  .\n\\ee\nIn expansion (4.2), using the value ${\\rm arccot}(0) = \\pi\/2$, we have \n$$\nc_0 = 1 \\; , \\qquad c_1 = 1.637 \\; , \\qquad\nc_2 = 0.137 \\; , \\qquad c_3 = -0.364 \\; , \n\\qquad  c_4 = -0.064 \\;   .\n$$\nAgain $f_0 = c_0$. Here the first three coefficients are positive, while the \nnext two are negative. The limit at infinity is\n\\be\n\\label{4.5}\n f(\\infty) = 2e = 5.437 \\;  ,\n\\ee\nwhere the equality ${\\rm arccot}(-\\infty) = \\pi$ is used. \n\nThe irregularity in the coefficient signs makes the extrapolation more \ndifficult. The factor approximants give $f_4^*(\\infty) = 9.049$. Power\ntransforms, with the factor approximants, improve the result yielding\nthe limit $5.192$. Iterated roots give $R_3(\\infty) = 3.399$, \n$R_4(\\infty) = 3.547$. Corrected iterated roots are close to the latter \nvalues: $R_{2\/2}(\\infty) = 3.424$. Power transforms, with iterated roots, \ngive two values: $3.547$ and $4.535$. As we see, the power-transformed factor \napproximants are the most accurate. \n\n\n\\subsection{Function-3}\n\nExpanding the function\n\\be\n\\label{4.6}\n f(x) = \\frac{{\\rm arccot}(-x)}{1+ e^{-x} } \\; ,\n\\ee\nwe get the coefficients\n$$\n c_0 = \\frac{\\pi}{4} \\; , \\qquad \nc_1 = \\frac{1}{2}\\left ( 1 + \\frac{\\pi}{4} \\right ) \\; , \n\\qquad\nc_2 = \\frac{1}{4} \\; , \\qquad \nc_3 = -\\; \\frac{1}{6 } \\left ( 1 + \\frac{\\pi}{16} \\right )\\; , \n\\qquad  c_4 = -\\;\\frac{5}{48} \\;  .\n$$\nHere $f_0 = c_0$. Again, the first three coefficients are positive, while the \nnext two are negative. The limit at infinity is\n\\be\n\\label{4.7}\n f(\\infty) = \\pi \\;  .\n\\ee\n\nAs in the previous case, the irregularity in the coefficient signs makes\nextrapolation difficult. For instance, Pad\\'{e} approximants fail, the best\nof them giving $1.414$, which is rather far from limit (4.7). The factor\napproximants give $f_4^*(\\infty) = 4.759$. Power-transformed factor \napproximants are more accurate, yielding the limit $3.142$. Iterated roots\nare not good, with the limit $1.698$. Power-transformed iterated roots give\ntwo solutions: $3.742$ and $2.267$. Thus, the power-transformed factor \napproximant, with the value $3.142$, is the best.\n\n\n\\subsection{Debye-H\\\"{u}kel function}\n\n\nThe Debye-H\\\"{u}kel function\n\\be\n\\label{4.8}\n  D(x) = \\frac{2}{x} - \n\\frac{2}{x^2} \\left ( 1 - e^{-x} \\right ) \n\\ee\nappears in the theory of strong electrolytes \\cite{Lan2000}. Its expansion\ngives the sign-alternating coefficients\n$$\nc_0 = 1 \\; , \\qquad c_1 = - \\; \\frac{1}{3} \\; , \n\\qquad c_2 = \\frac{1}{12} \\; \\qquad c_3 = - \\; \\frac{1}{60} \\; ,\n$$\n$$\nc_4 = \\frac{1}{360} \\; , \n\\qquad c_5 = -\\; \\frac{1}{2520} \\; \\qquad c_6 = \\frac{1}{20160} \\; .\n$$\nHere $f_0 = c_0$.\n\nThe large-variable behaviour is\n\\be\n\\label{4.9}\n  D(x) \\simeq \\frac{2}{x} \\qquad ( x \\ra \\infty) \\; .\n\\ee\n\nFactor approximants give $B_4 = 1.640$. Power-transformed factor approximants\nresult in $B_5 = 1.779$. Corrected factor approximants yield $B_{2\/2} = 1.642$. \nIterated roots result in $B_2 = 2.449$, $B_3 = 2.229$, $B_4 = 2.127$. For \ncorrected iterated roots, we have $B_{1\/2} = 1.611$, $B_{1\/3} = 1.841$, \n$B_{1\/4} = 1.934$, $B_{2\/2} = 1.130$, $B_{2\/3} = 1.712$, $B_{2\/4} = 1.811$. \nPower-transformed iterated roots in the fourth order give two solutions: \n$1.993$ and $2.049$. The best two-point Pad\\'{e} approximant $P_{2\/2}$ \ngives the critical amplitude $1.333$, which is much worse than the self-similar\napproximants of the same fourth order.\n\n\n\\subsection{Stirling function}\n\nThe Stirling series expansion for the function\n\\be\n\\label{4.10}\n f(x) = \\frac{1}{\\sqrt{2\\pi}} \\; e^{1\/x} x^{1\/x} \\Gm\\left ( 1 +\n\\frac{1}{x} \\right )  \n\\ee\ncan be written as\n\\be\n\\label{4.11}\nf_k(x) = \\frac{1}{\\sqrt{x}} \\left ( 1 + \n\\sum_{n=1}^k a_n x^n \\right ) \\;  ,\n\\ee\nwith the coefficients\n$$\n a_1 = \\frac{1}{12} \\; , \\qquad a_2 = \\frac{1}{288} \\; \n\\qquad a_3 = -\\; \\frac{139}{51840} \\;  ,\n$$\n$$\n a_4 = -\\;\\frac{571}{2488320} \\; , \\qquad \na_5 = \\frac{163879}{209018880} \\; \\qquad \na_6 = \\frac{5246819}{75246796800} \\;  ,\n$$\n$$\n a_7 = -\\;\\frac{534703531}{902961561600} \\; , \\qquad  \na_8 = -\\; \\frac{4483131259}{86684309913600} \\; .\n$$\nHere $f_0 = 1\/ \\sqrt{x}$.\n\nThe limit at infinity is\n\\be\n\\label{4.12}\nf(\\infty) = \\frac{1}{\\sqrt{2\\pi}} = 0.398942 \\; .\n\\ee\n\nFactor approximants yield the limit $f_6^*(\\infty) = 0.454$. \nPower-transformed factor approximants improve the accuracy, giving \n$f_5^*(\\infty) = 0.406$. Iterated roots result in $B_2 = 0.485$, $B_3 = 0.422$, \nbut the fourth-order approximant is complex. Corrected iterated roots give \nthe limit $B_{2\/1} = 0.408$, $B_{2\/2} = 0.312$, $B_{2\/3} = 0.405$. Pad\\'{e}\napproximants are essentially worse.  \n\n\n\\section{Functions defined through integrals}\n\nMany functions are defined by means of integral representations. Expansions\nof such functions often result in strongly divergent series. However, \nself-similar approximants provide rather accurate extrapolation from the zero\nvariable to its infinite limit. \n\n\n\\subsection{Integral-1}\n\nConsider the integral\n\\be\n\\label{5.1}\nf(x) = ( 1 + 2x) \\int_0^\\infty \\frac{e^{-t}}{1+x^2t^2} \\; dt \\; .\n\\ee\nIts expansion in powers of $x$ contains the coefficients\n$$\nc_0 = 1 \\; , \\qquad c_1 = 2 \\; , \\qquad\nc_2 = -2 \\; , \\qquad c_3 = -4 \\; , \\qquad  c_4 = 24 \\; ,\n$$\n$$\nc_5 = 48 \\; , \\qquad c_6 = -720 \\; , \\qquad\nc_7 = -1440 \\; , \\qquad c_8 = 40320 \\; , \\qquad  c_9 = 80640 \\;  .\n$$\nThe general expressions for the latter are\n$$\n c_{2n} = (-1)^n (2n)! \\; , \\qquad c_{2n+1} = (-1)^n 2 (2n)! \\;  .\n$$\nThe limit of Eq. (5.1) at infinity is\n\\be\n\\label{5.2}\n f(\\infty) = \\pi \\;  .\n\\ee\n\nFactor approximants yield $f_4^*(\\infty) = 1.965$, $f_5^*(\\infty) = 2.015$,\ndemonstrating good numerical convergence, e.g., giving in the ninth order\nthe limit $3.113$. Iterated roots lead to $R_2(\\infty) = 1.754$, \n$R_3(\\infty) = 2.071$, but the higher-order approximants are complex. \nPower-transformed iterated roots in fourth order give two solutions, \n$1.971$ and $2.071$.  Corrected iterated roots in the fourth order\ngive $2.582$ and display good numerical convergence in higher orders. \nPad\\'{e} approximants of the same order are less accurate, for instance, \n$P_{2\/2} = 1.875$.\n \n\n\\subsection{Complimentary error function}\n\nThe complimentary error function\n\\be\n\\label{5.3}\nf(x) = {\\rm erfc}(-x)\n\\ee\nis expressed through the error function as\n$$\n {\\rm erfc}(x) \\equiv 1 - {\\rm erf}(x) \\;  ,\n$$\nthe error function being\n$$\n {\\rm erf}(x) \\equiv \\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2} dt \\;  .\n$$\nHence, function (5.3) is defined by means of the integral\n$$\n {\\rm erfc}(x) \\equiv \n\\frac{2}{\\sqrt{\\pi}} \\int_x^\\infty e^{-t^2} dt \\;   .\n$$\nExpanding Eq. (5.3), we get the coefficients\n$$\nc_0 = 1 \\; , \\qquad c_1 = 1.12838 \\; , \\qquad\nc_2 = 0 \\; , \\qquad c_3 = -0.37613 \\; , \\qquad  c_4 = 0 \\; .\n$$\nThe limit at infinity is\n\\be\n\\label{5.4}\nf(\\infty) = 2 \\;  .\n\\ee\n\nAll self-similar approximants give close results. Thus, factor \napproximants yield $f_4^*(\\infty) = 3.772$. Iterated roots give \n$R_3(\\infty) = 2.382$. Power-transformed iterated roots of fourth order \nhave two solutions $2.305$ and $3.739$. Taking into account more \nexpansion terms results in better accuracy. Thus, $f_5^*(\\infty)=2.629$.\n\n\n\\subsection{Integral-2}\n\nThe function\n\\be\n\\label{5.5}\nf(x) = \\frac{{\\rm erfc}(-x)}{1+e^{-x}}\n\\ee\nis defined through the integral representation for the complimentary \nerror function considered in the previous subsection. The coefficients \nof the corresponding expansion are\n$$\n c_0 = \\frac{1}{2} \\; , \\qquad  c_1 = 1 \\; , \\qquad\nc_2 = 1.62838 \\; , \\qquad c_3 = -0.41779 \\; , \\qquad  \nc_4 = -0.23508 \\;  .\n$$\nThe limit at infinity is\n\\be\n\\label{5.6}\n f(\\infty) = 2\\;  .\n\\ee\n\nFactor approximants overestimate the limit, yielding \n$f_3^*(\\infty) = 5.052$, $f_5^*(\\infty) = 3.286$. Power-transformed factor \napproximants, on the other hand, underestimate it, giving to fourth order \n$1.392$. Iterated root approximants lead to $B_3 = 1.371$, $B_4 = 1.893$. \nPower-transformed iterated roots to fourth order give the limit $1.684$. \nIn the same order, Pad\\'{e} approximants give $1.027$. Iterated root \napproximants here are the most accurate. \n\nThe large-variable behaviour of functions (5.3) and (5.5) involves \nexponentials. Therefore the accuracy of approximations can be essentially \nimproved by employing exponential self-similar approximants \\cite{YukGlu1998}. \nHowever, here we limit ourselves by the analysis of approximants described \nin Sec. 3.\n\n\\subsection{Mittag-Leffler function}\n\nA particular case of the Mittag-Leffler function\n\\be\n\\label{5.7}\n E(x) = e^{x^2} {\\rm erfc}(x) \\;  ,\n\\ee\nwhich is expressed through the complimentary error function, appears in\nthe model of anomalous diffusion \\cite{Pir2005}. The small-variable \nexpansion yields the coefficients\n$$\n c_0 = 1 \\; , \\qquad  c_1 = -\\; \\frac{2}{\\sqrt{\\pi}} \\; , \\qquad\nc_2 = 1 \\; , \\qquad c_3 = -\\; \\frac{4}{3\\sqrt{\\pi}} \\; , \\qquad  \nc_4 = \\frac{1}{2} \\;  .\n$$\nIn the large-variable limit, one has\n\\be\n\\label{5.8}\n E(x) \\simeq \\frac{B}{x} \\qquad (x \\ra \\infty) \\;  ,\n\\ee\nwith the critical amplitude \n\\be\n\\label{5.9}\n B = \\frac{1}{\\sqrt{\\pi}} = 0.56419 \\;  .\n\\ee\n\nFactor approximants give in fourth order $B_4 = 0.511$. The same result \nholds for the corrected factor approximants $B_{2\/2} = 0.511$. \nPower-transformed factors yield, in fourth order, the amplitude $0.541$. \nIterated roots lead to $B_1 = 0.886$, $B_2 = 0.741$, $B_3 = 0.680$, \n$B_4 = 0.650$. Corrected iterated roots give in fourth order $0.403$. \nPower-transformed iterated roots yield three solutions, all being close \nto $0.641$. The accuracy improves, when more terms in the expansion are \ntaken into account. For instance, the factor approximants in sixth order \ngive $B_6 = 0.532$.     \n\n\n\\section{Anharmonic and nonlinear models}\n\n\nDivergent series often appear in applying perturbation theory to \nanharmonic and nonlinear models that are typical for many problems in \nphysics and chemistry. In these problems, perturbation theory is usually \ndone with respect to a parameter called the {\\it coupling parameter} which \ncharacterizes the strength of interactions or anharmonicity of an external\nfield.\n\n\n\\subsection{Zero-dimensional anharmonic model}\n\n\nThis is one of the simplest models that, at the same time, demonstrates \nmathematical features typical of many problems in chemistry and physics.\nThe partition function of this model reads as\n\\be\n\\label{6.1} \n Z(g) = \\frac{1}{\\sqrt{\\pi}} \\int_{-\\infty}^\\infty\n\\exp\\left ( - x^2 - gx^4 \\right ) \\; dx \\;  ,\n\\ee\nwhere $g \\in [0, \\infty)$ is a dimensionless coupling parameter. Weak-coupling\nperturbation theory yields the series \n\\be\n\\label{6.2}\n Z_k(g) = \\sum_{n=0}^k c_n g^n \\;  ,\n\\ee\nwith the coefficients\n$$\n c_n = \\frac{(-1)^n}{\\sqrt{\\pi}\\; n!} \\; \\Gm\\left ( 2n +\n\\frac{1}{2} \\right ) \\;  .\n$$\nExplicitly, the first few coefficients are\n$$\n c_0 = 1 \\; , \\qquad c_1 = -\\; \\frac{3}{4} \\; , \\qquad\nc_2 = \\frac{105}{32} \\; ,  \n$$\n$$\nc_3 = -\\; \\frac{3465}{128} \\; , \\qquad\nc_4 = \\frac{675675}{2048} \\; .\n$$\nIn the strong-coupling limit,\n\\be\n\\label{6.3}\n Z(g) = \\simeq B g^{-1\/4} \\qquad (g\\ra\\infty ) \\; ,\n\\ee\nwith \n\\be\n\\label{6.4}\n B = 1.022765 \\;  .\n\\ee\n\nFixing the exponent $\\beta$, we calculate the critical amplitude $B_k$, \ncomparing it with the known exact value from Eq. (6.4). Factor approximants\ngive to fourth order $B_4 = 0.838$. Corrected factor approximants, to the \nsame order, yield $B_{2\/2} = 1.131$. Iterated root approximants give \n$B_2 = 0.760$, but the higher-order approximants are complex. Corrected \niterated roots result in $B_{2\/2} = 0.678$. Power-transformed iterated \nroots of fourth order produce two solutions, $0.879$ and $0.971$. As we see,\nthe best accuracy is provided by the corrected factor approximants and \npower-transformed iterated roots.       \n\n\n\\subsection{One-dimensional anharmonic oscillator}\n\n\nThe anharmonic oscillator is described by the Hamiltonian\n\\be\n\\label{6.5}\n \\hat H = - \\; \\frac{1}{2} \\; \\frac{d^2}{dx^2} +\n\\frac{1}{2} \\; x^2 + gx^4 \\;  ,\n\\ee\nin which $x \\in (-\\infty, +\\infty)$ and $g$ is a positive anharmonicity \nparameter. Perturbation theory for the ground-state energy yields \n\\cite{Hio1978} the series\n\\be\n\\label{6.6}\n E_k(g) = \\sum_{n=0}^k c_n g^n \\;  ,\n\\ee\nwith the coefficients\n$$\nc_0 = \\frac{1}{2} \\; , \\qquad c_1 = \\frac{3}{4} \\; , \\qquad\nc_2 = -\\; \\frac{21}{8} \\;  \\qquad c_3 = \\frac{333}{16} \\; , \\qquad\nc_4 = -\\; \\frac{30885}{128} \\; .\n$$\nThe strong-coupling limit is\n\\be\n\\label{6.7}\n E(g) \\simeq 0.667986 g^{1\/3} \\qquad\n(g \\ra \\infty ) \\;  .\n\\ee\n\nFactor approximants give $B_3 = 0.750$, $B_5 = 0.725$, $B_7 = 0.712$.\nCorrected factor approximants yield $B_{3\/4} = 0.728$. The power-transformed\nfactor approximant of fourth order gives $0.681$. Iterated root \napproximants result in $B_2 = 0.572$, $B_3 = 0.855$, but the fourth-order\napproximant is complex. Corrected iterated roots give $B_4 = 0.587$, and\npower-transformed iterated roots, $0.665$. The latter value is the closest\nto the exact amplitude in Eq. (6.7).\n\nComparing these results with those obtained by means of the Kleinert variational\nperturbation theory \\cite{Jan1995}, we see that the latter provides better \naccuracy. However, we would like to recall that our main aim in the present \npaper is to test the methods of self-similar approximation theory, without \ninvolving the introduction of variational or other control functions, and based \non just a few initial terms of perturbation theory. Although, in our case, the \naccuracy is lower than in the Kleinert method, the calculations are much \nsimpler. \n\n\n\\subsection{Scalar field theory}\n\n\nConsider the so-called $m \\phi^2$ quantum field theory on a $d$-dimensional\ncubic lattice with lattice spacing $a$. The free energy of the system can be\nexpressed \\cite{Ben1994} as the integral\n\\be\n\\label{6.8}\nf(x) = x \\exp \\left \\{ 2 \\int_0^\\infty e^{-t} \\ln \\left [\ne^{-xt} I_0(xt) \\right ] dt \\right \\} \\;   ,\n\\ee\nwhere $I_0(\\cdot)$ is a modified Bessel function of zero order and \n$x = 1\/ m a^2$. Expanding the integral in powers of the variable $x$ yields \nthe series \n\\be\n\\label{6.9}\n f_k(x) = x \\left ( 1 + \\sum_{n=1}^k a_n x^n \\right ) \\;  ,\n\\ee\nwith the coefficients\n$$\na_1 = -2 \\; , \\qquad a_2 = 3 \\; , \\qquad a_3 = - \\; \\frac{10}{3} \\; ,\n\\qquad a_4 = \\frac{29}{12} \\; ,\n$$\n$$\na_5 = - \\; \\frac{11}{10} \\; ,\\qquad a_6 = \\frac{391}{180} \\; \\qquad\na_7 = - \\; \\frac{2389}{630} \\;  .\n$$\nWhen passing to continuous space, one takes the limit $a \\ra 0$, which means\nthat $x \\ra \\infty$. The sought continuous-space limit is\n\\be\n\\label{6.10}\n  f(\\infty) = \\frac{e^\\gm}{2\\pi} = 0.28347 \\; .\n\\ee\n\nFactor approximants of fourth order give the limit $0.322$ and \npower-transformed factor approximants, $0.333$. Iterated root approximants \nyield $f^*_2(\\infty) = 0.408$, $f^*_3(\\infty) = 0.377$, $f^*_4(\\infty) = 0.365$. \nTheir accuracy can be improved by taking more terms in expansion (6.9), e.g.,  \n$f^*_{12}(\\infty) = 0.280$. Corrected iterated roots give $f^*_{2\/2}(\\infty) = 0.266$, \nand power-transformed iterated roots of fourth order lead to $0.356$ and $0.347$. \nThe best Pad\\'{e} approximant, up to fifth order, gives $P_{2\/3} = 0.326$. \nFor these low orders, the most accurate is the corrected root approximant \n$f^*_{2\/2}(\\infty) = 0.266$.   \n\n\n\\subsection{Nonlinear Schr\\\"{o}dinger equation}\n\n\nThe nonlinear Schr\\\"{o}dinger equation serves as a basic tool for modelling \nseveral different problems, such as those of waves on the surface of a deep \nfluid \\cite{Zak1968}, electromagnetic waves in fibre optics \\cite{Kar2011}, \nand Bose-Einstein condensates \\cite{Pet2008, Yuk2009a, Yuk2011}. For the last \ncase, it is often called the Gross-Pitaevskii equation, although Bogolubov was\nthe first to write down this equation for Bose systems in his famous Lectures \non Quantum Statistics published in 1949 \\cite{Bog1949} and wrote on it many \ntimes since (see, e.g., Refs. \\cite{Bog1967, Bog1970}). This equation for \nnonequilibrium superfluids was also studied in \\cite{Bog1963}. The one-dimensional \nstationary nonlinear Schr\\\"{o}dinger equation for Bose condensed atoms in a \nharmonic trap reads \n\\be\n\\label{6.11}\n\\hat H_{NLS} \\psi = E\\psi \\;  ,\n\\ee\nwith the nonlinear Hamiltonian\n\\be\n\\label{6.12}\n \\hat H_{NLS} = - \\; \\frac{1}{2} \\; \\frac{d^2}{dx^2} +\n\\frac{1}{2} \\; x^2 + g| \\psi|^2 \\;  .\n\\ee\nHere $g$ is a dimensionless coupling parameter. The energy levels can be \nrepresented in the form\n\\be\n\\label{6.13}\n  E(g) = \\left ( n + \\frac{1}{2} \\right ) f(g) \\; ,\n\\ee\nwhere $n = 0,1,2,\\ldots$ is a quantum index labelling the eigenvalues. \nEmploying the optimized perturbation theory for the function $f(g)$,\nas in \\cite{Yuk1998}, gives the expansion\n\\be\n\\label{6.14}\n f_k(g) = 1 + \\sum_{n=1}^k a_n z^n  \n\\ee\nin powers of the effective coupling \n$$\nz \\equiv \\frac{J_n}{n+1\/2} \\; g \\;  ,\n$$\nin which\n$$\n J_n \\equiv \\frac{1}{2^n\\pi n!} \\int_{-\\infty}^\\infty\n\\exp \\left ( -2x^2 \\right ) H_n^4(x) \\; dx \\; ,\n$$\nwith $H_n(\\cdot)$ being a Hermite polynomial. The coefficients in \nexpansion (6.14) are\n$$\n a_1 = 1 \\; , \\qquad a_2 = - \\; \\frac{1}{8} \\; , \\qquad \na_3 = \\frac{1}{32} \\; , \\qquad  a_4 = - \\; \\frac{1}{128} \\; .\n$$\nThen for the strong-coupling limit we have\n\\be\n\\label{6.15}\n f(g) \\simeq \\frac{3}{2} \\; z^{2\/3} \\qquad ( z\\ra \\infty) \\;  .\n\\ee\nHence the critical amplitude is $B = 3\/2$.\n\nFactor approximants give $B_4 = 1.496$, which is very close to $1.5$.\nCorrected factor approximants, to fourth order, yield $1.451$ and \npower-transformed factor approximants, $1.477$. Iterated roots result \nin $B_2 = 1.379$, $B_3 = 1.415$, $B_4 = 1.435$. Corrected iterated roots\ngive $B_{2\/2} = 1.492$ and power-transformed iterated roots, $1.426$. \nFor the double self-similar approximant, based on iterated roots, we \nget $B_4^* = 1.498$. The latter is slightly better than the \nvalue $B_4 = 1.496$, given by the factor approximant, but calculating \nthe doubly renormalized approximants is essentially more complicated.   \nOf course, calculations, employing any of the self-similar approximants, \nare much less time consuming than the direct solution of the nonlinear \ndifferential equation (6.11). \n\n\\section{Problems in many-body theory}\n\nPerturbation theory in many-body problems is usually accomplished with \nrespect to the coupling parameter characterizing the interaction \nstrength. However, this coupling parameter is often rather large. \nMoreover, perturbative expansions practically always yield divergent \nseries for any finite value of the coupling parameter. Another difficulty\nis that the many-body problems, as a rule, are so much complicated that\nthey allow one to calculate only a few low-order terms of perturbation \ntheory. We show here that self-similar approximants allow for an \neffective extrapolation of such short series, giving good accuracy even \nin the extreme case of infinitely strong coupling.  \n\n\n\\subsection{Lieb-Liniger Bose gas}\n\n\nLieb and Liniger \\cite{Lie1963} have considered a one-dimensional Bose \ngas with contact interactions. The ground-state energy of the gas can be \nwritten as an expansion with respect to the coupling parameter as\n\\be\n\\label{7.1}\n E(g) \\simeq g - \\; \\frac{4}{3\\pi} \\; g^{3\/2} +\n\\frac{1.29}{2\\pi^2} \\; g^2 - 0.017201 g^{5\/2} \\;  .\n\\ee \nIn the strong-coupling limit, we have the Tonks-Girardeau expression\n\\be\n\\label{7.2}\n E(\\infty) = \\frac{\\pi^2}{3} = 3.289868 \\; .\n\\ee\nBy the change of the variables\n\\be\n\\label{7.3}\ne(x)  \\equiv E\\left ( x^2 \\right ) \\; , \\qquad\ng \\equiv x^2 \\;\n\\ee\nexpansion (7.1) reduces to the form\n\\be\n\\label{7.4}\n  e(x) \\simeq x^2 \\left ( 1 + a_1 x + a_2 x^2\n+ a_3 x^3 \\right ) \\; ,\n\\ee\nin which\n$$\na_1 = - \\; \\frac{4}{3\\pi}=- 0.424413 \\; , \\qquad\na_2 = \\frac{1.29}{2\\pi^2} = 0.065352 \\; , \\qquad\na_3=-0.017201 \\;   .\n$$\nThe fourth-order term can be set as having $a_4 = 0$.\n\nDifferent self-similar approximants yield close results. The most accurate\namong them correspond to iterated root approximants displaying fast \nnumerical convergence: $E^*_2(\\infty) = 8.713$, $E^*_3(\\infty) = 4.765$,\n$E^*_4(\\infty) = 3.2924$. The last expression provides very good accuracy,\nwhen compared with the exact value (7.2). \n\n\n\\subsection{Bose-Einstein condensation temperature}\n\n\nThe Bose-Einstein condensation temperature of ideal uniform Bose gas in\nthree-dimensional space is known to be\n\\be\n\\label{7.5}\n T_0 = \\frac{2\\pi\\hbar^2}{mk_B} \\left [\n\\frac{\\rho}{\\zeta(3\/2)} \\right ]^{2\/3} \\;  ,\n\\ee\nwhere $m$ is atomic mass and $\\rho$, gas density. The ideal gas is, however, \nunstable below the condensation temperature \\cite{Yuk2011}. Atomic \ninteractions stabilize the system and shift the transition temperature\nby the amount\n\\be\n\\label{7.6}\n \\Dlt T_c \\equiv T_c - T_0 \\;  .\n\\ee\nThis shift, at asymptotically small gas parameter\n\\be\n\\label{7.7}\n \\gm \\equiv \\rho^{1\/3} a_s \\;  ,\n\\ee\nin which $a_s$ is atomic scattering length, behaves as\n\\be\n\\label{7.8}\n \\frac{\\Dlt T_c}{T_0} \\simeq c_1 \\gm \\qquad ( \\gm \\ra 0 ) \\;  .\n\\ee\nMonte Carlo simulations \\cite{Arn2001a, Arn2001b, Kas2001, Pro2001, Nho2004} \ngive \n\\be\n\\label{7.9}\n c_1 = 1.3. \\pm 0.05 \\;  .\n\\ee\n\nAt the same time, the coefficient $c_1$ can be defined \n\\cite{Kas2004a, Kas2004b, Kas2004c} as the strong-coupling limit \n\\be\n\\label{7.10}\nc_1 = \\lim_{g\\ra\\infty} c_1(g) \\equiv B \n\\ee\nof a function $c_1(g)$ that is available only as an expansion in an \neffective coupling parameter,\n\\be\n\\label{7.11}\n c_1(g) \\simeq b_1 g + b_2 g^2 + b_3 g^3 + b_4 g^4 + b_5 g^5 \\;  ,\n\\ee\nwhere  \n$$\n b_1=0.223286\\; , \\qquad b_2=-0.0661032 \\; , \\qquad\nb_3=0.026446 \\; , $$\n$$\nb_4=-0.0129177 \\; , \\qquad b_5=0.00729073 \\;  .\n$$\nExpansion (7.11) can be represented as\n\\be\n\\label{7.12}\nc_1(g) \\simeq b_1 g \\left ( 1 +  a_1 g + a_2 g^2 + \na_3 g^3 + a_4 g^4 \\right ) \\;  ,\n\\ee\nwith the coefficients\n$$\n a_n \\equiv \\frac{b_{n+1}}{b_1} \\qquad ( n = 1,2,3,4 ) \\;  .\n$$\n\nPad\\'{e} approximants do not provide good accuracy, the best of them \ngives $c_1(\\infty) = 0.985$. Factor approximants, to third order, yield \n$B_3 = 1.025$. At fourth order, factor approximants give $B_4 = 1.096$, \nif one of the parameters $A_i$ is set to one, and $1.446$, if it is \ndefined by the variational procedure. On average, the latter values \ngive $B_4 = 1.271$. Iterated roots result in $B_2 = 1.383$ to second order \nand $B_3 = 0.854$ to third order; the fourth-order approximant is\ncomplex. Corrected iterated roots give \n$B_{1\/2} = 0.924$, $B_{1\/3} = 1.289$, $B_{2\/2} = 1.309$. Power-transformed \niterated roots give two solutions, $1.227$ and $1.388$, which on average\nmakes $1.308$. The corrected iterated root $B_{2\/2} = 1.309$ produces the \nmost accurate result, practically coinciding with that found by the\nMonte Carlo simulations \\cite{Arn2001a, Arn2001b, Kas2001, Pro2001, Nho2004}.\nKastening \\cite{Kas2004a, Kas2004b, Kas2004c}, using the Kleinert variational \nperturbation theory involving seven loops, found the value $1.27 \\pm 0.11$, \nwhich is close to our results. \n   \n\n\\subsection{Unitary Fermi gas}\n\nThe ground-state energy of a dilute Fermi gas can be obtained by means \nof perturbation theory \\cite{Bak1999, Ket2008} with respect to the effective \ncoupling parameter\n\\be\n\\label{7.13}\ng \\equiv | k_F a_s | \\;   ,\n\\ee\nwhere $k_F$ is a Fermi wave number, and $a_s$, atomic scattering length.\nThis perturbation theory yields the expansion\n\\be\n\\label{7.14}\n E(g) \\simeq c_0 + c_1 g + c_2 g^2 + c_3 g^3 + c_4 g^4 \\; ,\n\\ee\nwith the coefficients\n$$\n c_0 = \\frac{3}{10} \\; , \\qquad c_1 = - \\; \\frac{1}{3\\pi} \\; , \\qquad\nc_2 = 0.055661 \\;  ,\n$$\n$$\nc_3 = -0.00914 \\; , \\qquad c_4 = -0.018604 \\; .\n$$\n \nThe scattering length, and, respectively, the effective coupling parameter\n(7.13), can be varied by means of Feshbach resonance techniques in a rather \nwide range, including $g \\ra \\infty$. The latter limit corresponds to the\nsystem called a unitary Fermi gas. Numerical calculations \\cite{Car2003, Ast2004}\nyield\n\\be\n\\label{7.15}\n E(\\infty) = 0.132 \\;  .\n\\ee\n\nExpansion (7.14) can be rewritten in the form\n\\be\n\\label{7.16}\n E(g) \\simeq c_0 \\left ( 1 + a_1 g + a_2 g^2 + a_3 g^3 + a_4 g^4 \n\\right ) \\;  ,\n\\ee\nin which\n$$\na_n \\equiv \\frac{c_n}{c_0} \\qquad ( n = 1,2,3,4 ) \\;   .\n$$\n  \nFactor approximants give $E^*_4(\\infty) = 0.174$ and corrected factor \napproximants, $0.143$. Power-transformed factor approximants yield $0.162$.\nIterated roots give $E^*_3(\\infty) = 0.169$, $E^*_4(\\infty) = 0.163$. Corrected\niterated roots result in $E^*_{1\/2}(\\infty) = 0.103$ and power-transformed\niterated roots, in $0.163$. Doubly renormalized iterated roots improve the\nlimit to $0.146$. Pad\\'{e} approximants are not accurate, the best of them \ngiving $P_{2\/2} = 0.170$.\n\n\n\\subsection{One-dimensional Heisenberg antiferromagnet}\n\n\nThe ground-state energy of an equilibrium one-dimensional Heisenberg \nantiferromagnet can be represented \\cite{Hor1984} as the infinite time \nlimit for the energy $E(t)$ of a nonequilibrium antiferromagnet. At small \ntime $t \\ra 0$, one has an expansion\n\\be\n\\label{7.17}\n E(g) \\simeq -\\; \\frac{1}{4} \\left ( 1 + \\sum_{n=1}^4 a_n t^n \n\\right ) \\;  ,\n\\ee\nwith the coefficients\n$$\n a_1 = 4 \\; , \\qquad a_2 = -8 \\; , \\qquad \na_3 = -\\; \\frac{16}{3} \\; , \\qquad a_4 = 64 \\;  .\n$$\nIn the other limit, this ground-state energy was calculated by Hulthen \n\\cite{Hul1938} exactly:\n\\be\n\\label{7.18}\nE = E(\\infty) = - 0.4431 \\;   .\n\\ee\nWe apply the self-similar approximations to extrapolate the small-time\nexpansion (7.17) to the infinite time limit $t \\ra \\infty$ determining\n$E(\\infty)$.\n\nFactor approximants yield $E^*_4(\\infty) = - 0.570$, with power-transformed \nfactor approximants resulting in practically the same value. Corrected \nfactor approximants give $E^*_{2\/2}(\\infty) = - 0.211$. Corrected iterated \nroots also underestimate the limit, giving $- 0.254$. Iterated roots give\n$E^*_3(\\infty) = - 0.511$, $E^*_4(\\infty) = - 0.482$. Power-transformed \niterated roots yield $- 0.475$. The best Pad\\'{e} approximant is\n$P_{2\/2} = - 0.329$. The most accurate here is the power-transformed \niterated root approximant $E^*_4(\\infty) = - 0.475$.\n \n\n\n\\subsection{Fr\\\"{o}hlich optical polaron}\n\n\nThe ground-state energy of the Fr\\\"{o}hlich optical polaron, in the \nweak-coupling perturbation theory \\cite{Koc1982, Sel1989} reads as\n\\be\n\\label{7.19}\n E(g) \\simeq - g \\left ( 1 + a_1 g + a_2 g^2 \\right ) \\;  ,\n\\ee\nwith the coefficients\n$$\n a_1 = 1.591962 \\times 10^{-2} \\; , \\qquad\na_2 = 0.806070 \\times 10^{-3} \\;  .\n$$\nIn the strong-coupling limit, the asymptotic behaviour of the \nground-state energy has been found by Miyake \\cite{Miy1975, Miy1976}\nin the form\n\\be\n\\label{7.20}\n E(g) \\simeq Bg^2 \\qquad ( g\\ra \\infty) \\;  ,\n\\ee\nwith the amplitude\n\\be\n\\label{7.21}\n B = -0.108513 \\;  .\n\\ee\n\nSince just a few terms in the perturbative expansion are available, the\nPad\\'{e} approximants are not applicable at all, yielding unreasonable \nvalues for the amplitude, by many orders differing from Eq. (7.21).  \nSelf-similar approximants give more realistic values. Thus, factor \napproximants give for the amplitude $B$ the value $0.061$ and iterated \nroots, $0.049$. The doubly renormalized iterated roots improve the \naccuracy, giving the value $0.1287$ for the amplitude. \n\n\n\\section{Characteristics of polymer systems}\n\n\nPolymers are rather complicated molecules and are highly important in many\nbranches of physics and chemistry. As a rule, their characteristics \nare calculated by means of perturbation theory with respect to a small \nparameter, although in reality this parameter can be quite large. Self-similar\napproximants can successfully extrapolate these characteristics to arbitrary\nvalues of the parameters, including asymptotically large values.  \n\n\n\\subsection{Randomly branched polymers}\n\n\nMany characteristics of polymers are expressed through their structure \nfactors. The structure factor of three-dimensional branched polymers is \ngiven \\cite{Lam1990, Mil1991} by the confluent hypergeometric function\n\\be\n\\label{8.1}\n S(x) = F_1 \\left ( 1\\; ; \\frac{3}{2}\\; ; \\frac{3}{2}\\; x \n\\right ) \\;  ,\n\\ee\nin which $x$ is a dimensionless wave-vector modulus. The long-wave \nexpansion \n\\be\n\\label{8.2}\nS(x) \\simeq c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4 x^4 \n\\ee\ncontains the coefficients\n$$\n c_0 = 1 \\; , \\qquad c_1 = -1 \\; , \\qquad c_2 = 0.6 \\; , \\qquad\nc_3 = -0.257143 \\; , \\qquad c_4 = 0.085714 \\; .\n$$\nIn the short-wave limit, one has\n\\be\n\\label{8.3}\n S(x) \\simeq \\frac{B}{x} \\qquad ( x \\ra \\infty ) \\;  ,\n\\ee\nwith the amplitude\n\\be\n\\label{8.4}\n B = \\frac{1}{3} \\;  .\n\\ee\n\nThe reconstruction of the short-wave amplitude by Pad\\'{e} approximants\nleads to senseless negative values. Factor approximants give $B_4= 0.097$, \nand the power-transformed factors yield two solutions, $0.179$ and $0.329$.\nIterated roots, at low orders, overestimate the amplitude, giving\n$B_2 = 0.745$, $B_3 = 0.642$, and $B_4 = 0.590$. The same happens for the \npower-transformed roots yielding the values close to $0.6$. However, the higher \norders of the iterated roots converge to value (8.4). For instance, the \nseventh-order iterated root approximant gives a very good accuracy, \nwith $B_7 = 0.330$. \n\n\n\\subsection{Fluctuating fluid string}\n\n\nThere exists an important class of systems, called fluid membranes \n\\cite{Sei1997}, which finds wide applications in chemistry, biology, medicine,\nand in a variety of technological applications. First, let us consider a model \nof a fluid string that is a cartoon of a one-dimensional membrane oscillating \nbetween two rigid walls \\cite{Edw1965, Doi2001}. The free energy of the \nstring coincides with the ground-state energy of a quantum particle in a \none-dimensional rigid potential \\cite{Kle1999, Kas2002}. This energy, as a \nfunction of a finite wall stiffness $g$, can be represented as\n\\be\n\\label{8.5}\n E(g) = \\frac{\\pi^2}{8g^2} \\left ( 1 + \\frac{g^2}{32} +\n\\frac{g}{4} \\; \\sqrt{ 1 + \\frac{g^2}{64} } \\right ) \\;  .\n\\ee\nThe low-stiffness expansion results in\n\\be\n\\label{8.6}\n E_k(g) = \\frac{\\pi^2}{8g^2} \\left ( 1 + \\sum_{n=1}^k a_n g^n \n\\right ) \\;  ,\n\\ee\nwith the coefficients\n$$\na_1 = \\frac{1}{4} \\; , \\qquad a_2 = \\frac{1}{32} \\; , \\qquad\na_3 = \\frac{1}{512} \\; , \\qquad a_4 = 0 \\; ,\n$$\n$$\na_5 = - \\; \\frac{1}{131072} \\; , \\qquad a_6 = 0 \\; , \\qquad\na_7 = \\frac{1}{16777216} \\;   .\n$$\nThe case of interest corresponds to rigid walls, when the stiffness tends \nto infinity. For such rigid walls, the energy is\n\\be\n\\label{8.7}\n E(\\infty) = \\frac{\\pi^2}{128} = 0.077106 \\;  .\n\\ee\n\nPad\\'{e} approximants are not applicable for this problem, giving negative \nvalues of the large-stiffness energy. Factor approximants give positive \nvalues, although overestimating the energy, e.g., $E^*_4(\\infty) = 0.15$. Iterated \nroots yield $E^*_2(\\infty) = 0.039$, $E^*_3(\\infty) = 0.051$, and \n$E^*_4(\\infty) = 0.058$. Corrected iterated roots give \n$E^*_{2\/2} (\\infty) = 0.169$ and power-transformed iterated roots, \n$E^*_4(\\infty) = 0.065$. Taking more terms in the expansion improves the \naccuracy. Thus, iterated roots of higher orders yield $E^*_5(\\infty) = 0.062$, \n$E^*_6(\\infty) = 0.065$, and $E^*_7(\\infty) = 0.067$. The most accurate result \nis obtained by employing the doubly renormalized iterated roots,\ngiving $E_2^{**}(\\infty) = 0.07237$. The variational perturbation theory, to \nsixth order, gives \\cite{Kas2006} the value $0.076991$.     \n\n\n\\subsection{Fluctuating fluid membrane}\n\nIn the case of a two-dimensional membrane, its pressure can be calculated by\nperturbation theory with respect to the wall stiffness \\cite{Kas2006}, which \nyields\n\\be\n\\label{8.8}\n p_k(g) = \\frac{\\pi^2}{8g^2} \\left ( 1 + \n\\sum_{n=1}^k a_n g^n \\right ) \\;  ,\n\\ee\nwith the coefficients\n$$\na_1 = \\frac{1}{4} \\; , \\qquad a_2 = \\frac{1}{32} \\; , \\qquad\na_3=2.176347\\times 10^{-3} \\; ,\n$$\n$$\n a_4=0.552721\\times 10^{-4} \\; , \\qquad \na_5=-0.721482\\times 10^{-5} \\; , \\qquad \na_6=-1.777848\\times 10^{-6} \\; .\n$$\nThe rigid-wall limit, calculated by means of the Monte Carlo simulations \n\\cite{Gom1989} is found to be\n\\be\n\\label{8.9}\n p(\\infty) =0.0798 \\pm 0.0003 \\;  .\n\\ee\n\nPad\\'{e} approximants are again not applicable, resulting in negative values\nof pressure. Factor approximants of low orders overestimate the limit, e.g., \nthe fourth order giving $0.312$. To higher orders, factor approximants become \nslightly better, but still overestimating the pressure. Iterated roots of low \norders give $p^*_2(\\infty)=0.039$,  $p^*_3(\\infty)=0.053$, and \n$p^*_4(\\infty)=0.061$ and power-transformed iterated roots in fourth order, \n$0.068$. Taking into account all available coefficients improves the results. \nFor instance, in the case of the iterated roots, we have $p^*_5(\\infty)=0.067$, \n$p^*_6(\\infty)=0.071$. Doubly renormalized iterated roots give \n$p^{**}_3(\\infty) = 0.0792$, which is the most accurate result. This is to be \ncompared with the value of $0.0821$ from the variational perturbation theory\n\\cite{Kas2006}, which overestimates the Monte Carlo result (8.9).  \n\n\n\\subsection{Two-dimensional polymer chain} \n\n\nAn important characteristic of polymer chains is their expansion factor, \nthat is, the ratio of the mean-square end-to-end distance of the chain, with \ninteractions between its segments, to the value of the mean-square end-to-end\ndistance of the chain, without such interactions. Two-dimensional polymers\nare often met in chemistry and biology. For such polymers, perturbation theory \nwith respect to weak interactions can be developed \\cite{Mut1984, Mut1987} and, \nin a certain limiting case, can be reduced to a series in a single \ndimensionless interaction parameter $g$. For a two-dimensional polymer chain, \nperturbation theory results \\cite{Mut1984} in the expansion factor \n\\be\n\\label{8.10}\n F(g) \\simeq 1 + \\sum_{n=1}^4 a_n g^n \\;  ,\n\\ee\nwith the coefficients\n$$\n a_1 = \\frac{1}{2} \\; , \\qquad a_2 = -0.12154525 \\; , \\qquad\na_3=0.02663136 \\; , \\qquad a_4=-0.13223603 \\;  .\n$$\nIn the strong-interaction limit \\cite{Li1995}, one has\n\\be\n\\label{8.11}\nF(g) \\simeq Bg^\\bt \\qquad (g \\ra \\infty ) \\;   ,\n\\ee\nwith the critical exponent\n\\be\n\\label{8.12}\n  \\bt = 1 \\;  .\n\\ee\nOne also considers the critical index\n\\be\n\\label{8.13}\n  \\nu \\equiv \\frac{1}{2} \\left ( 1 + \\frac{\\bt}{2} \\right ) \\; ,\n\\ee\nwhich here is $\\nu = 0.75$.\n\nCalculating the critical amplitude, we have the following. Factor \napproximants are complex, but the power-transformed factor approximant\nat fourth order gives $0.31$. Iterated roots yield $B_2 = 0.08$, with\nthe higher orders being complex. The corrected iterated roots yield\n$B_{2\/2} = 0.09$. The exact value of the amplitude $B$ is not known, \nbecause of which we cannot evaluate the accuracy of the approximants.\nBut, as we see, all approximants give the values of order $0.1$. \n\n\n\\subsection{Three-dimensional polymer coil}\n\nIn the case of a three-dimensional polymer coil, perturbation theory\n\\cite{Mut1984} for the expansion factor leads to series (8.10), however with\nthe coefficients\n$$\na_1 = \\frac{4}{3} \\; , \\qquad a_2 =-2.075385396 \\; , \\qquad\na_3 = 6.296879676 \\; ,\n$$\n$$\na_4 = -25.05725072 \\; , \\qquad a_5=116.134785 \\; , \\qquad\na_6 = - 594.71663 \\; .\n$$\nThe strong-coupling limit \\cite{Mut1987} is\n\\be\n\\label{8.14}\n F(g) \\simeq 1.531 g^{0.3544} \\qquad ( g \\ra \\infty) \\;  ,\n\\ee\nwhich yields for the critical index (8.13) $\\nu = 0.5866$. Numerical fitting\n\\cite{Mut1987} for the whole range of interactions results in the formula\n\\be\n\\label{8.15}\n F(g) = \\left ( 1 + 7.524 g + 11.06 g^2 \\right )^{0.1772} \\;  .\n\\ee\n\nEmploying four terms in the weak-coupling expansion gives for the factor \napproximants the amplitude $B_4 = 1.548$ and for power-transformed factor \napproximants, $1.535$. Iterated roots yield \n$B_2 = 1.543$, $B_3 = 1.549$, $B_4 = 1.538$. Corrected iterated roots result\nin $B_{2\/2} = 1.544$ and power-transformed iterated roots, in $B_4 = 1.535$.\nDoubly renormalized iterated roots give $1.530$. Higher-order approximants\nimprove the results, but already at fourth order all these approximants are\nclose to the numerical value $B = 1.531$. The accuracy of Pad\\'{e} \napproximants is several orders worse \\cite{Glu2003}.\n\n\n\n\\section{Calculation of critical exponents}\n\nIn the previous sections, we have concentrated on the calculation of \ncritical amplitudes, with known critical exponents, by extrapolating \nthe small-variable perturbative expansions to the large-variable limit, \nemploying the techniques of self-similar approximants. Now we show how the\ncritical exponents can also be found by using these techniques. \n\n\n\\subsection{Scheme of general approach}\n\n\nWhen a function, for asymptotically large variable, behaves as\n\\be\n\\label{9.1}\n f(x) \\simeq B x^\\bt \\qquad ( x \\ra \\infty ) \\;  ,\n\\ee\nthen the critical exponent can be represented by the limit\n\\be\n\\label{9.2}\n \\bt = \\lim_{x\\ra\\infty} x \\; \\frac{d}{dx} \\; \\ln f(x) \\;  .\n\\ee\n\nAssuming that the small-variable expansion for the function is given by the\nsum $f_k(x)$, as in Eq. (2.2), we have the corresponding small-variable \nexpression for the critical exponent\n\\be\n\\label{9.3}\n \\bt_k(x) = x \\; \\frac{d}{dx} \\; \\ln f_k(x) \\;  ,\n\\ee\nwhich can be expanded in powers of $x$, leading to \n\\be\n\\label{9.4}\n\\bt_k(x) = \\sum_{n=0}^k b_n x^n \\; .\n\\ee\n \nApplying the method of self-similar approximants to expansion (9.4), as has \nbeen done above, we get a self-similar approximant\n$\\beta_k^*(x)$ whose limit, being by definition finite,\n$$\n \\bt_k^*(x) \\ra const \\qquad ( x \\ra \\infty ) \\;  ,\n$$\ngives us the sought approximate expression for the critical exponent \n\\be\n\\label{9.5}\n \\bt_k^* = \\lim_{x\\ra\\infty} \\bt_k^*(x) \\;  .\n\\ee\n\nNote that the value of the critical amplitude $B$ does not need to be\nconsidered at all. Below, we illustrate this method of calculating the\ncritical exponents by concrete examples. \n\n\n\\subsection{One-dimensional anharmonic oscillator}\n\n\nLet us consider, as in Sec. 6.2, the model of the one-dimensional anharmonic \noscillator whose mathematical structure is typical for many applied \nproblems, yielding strongly divergent perturbation series.\n\nThe exact critical exponent, as follows from Eq. (6.7), is\n$$\n \\bt = \\frac{1}{3} \\;  .\n$$\nIn addition to the coefficients $c_n$ of Sec. 6.2, we shall analyze the \nhigher-order terms of sum (6.6), with the coefficients\n$$\nc_5 = \\frac{916731}{256} \\; , \\qquad \nc_6 = - \\; \\frac{65518401}{1024} \\; , \\qquad\nc_7 = \\frac{2723294673}{2048} \\; ,\n$$\n$$\n c_8 = -\\; \\frac{1030495099053}{32786} \\; , \\qquad\nc_9 = \\frac{54626982511455}{65536} \\; , \\qquad\nc_{10} = -24478940702.8  \\; .\n$$\n\nEmploying the scheme of Sec. 9.1, we find, for the critical exponent, the \nfactor approximants $\\beta_4^*=0.241$, $\\beta_7^*=0.303$, and $\\beta_8^*=0.282$. \nIterated roots result in $\\beta_2^*=0.397$, $\\beta_3^*=0.181$, but $\\beta_4^*$\nis complex. Corrected iterated roots yield $\\beta_{2\/2}^* = 0.307$, \n$\\beta_{2\/3}^* = 0.328$, $\\beta_{2\/4}^* = 0.310$, $\\beta_{2\/5}^* = 0.346$, and \n$\\beta_{2\/6}^* = 0.305$. Power-transformed roots give two solutions, $0.156$ \nand $0.238$. Doubly renormalized iterated roots of second order lead to \n$0.319$. As we see, the self-similar approximants are rather accurate, being \nclose to $0.3$. \n\n\n\\subsection{Three-dimensional polymer coil}\n\n\nAs another example, we consider the three-dimensional polymer coil of Sec. 8.5.\nThe exponent found numerically, according to Eq. (8.14), is \n$$\n \\bt=0.3544 \\;  .\n$$\n\nFollowing the scheme of Sec. 9.1, we obtain the self-similar approximants \nfor the critical exponent. Factor approximants yield $\\beta_3^* = 0.343$, \n$\\beta_4^* = 0.346$, and $\\beta_5^* = 0.349$. Iterated roots result in \n$\\beta_2^* = 0.345$, $\\beta_3^* = 0.343$, $\\beta_4^* = 0.351$, and \n$\\beta_5^* = 0.349$. Power-transformed iterated roots give two solutions, \n$0.285$ and $0.349$ and corrected iterated roots give $\\beta_{1\/4}^* = 0.348$, \n$\\beta_{2\/2}^* = 0.345$, $\\beta_{3\/2}^* = 0.349$. Doubly-renormalized \niterated roots yield $\\beta_4^{**} = 0.353$, $\\beta_5^{**} = 0.355$. All \nthese approximants are close to the numerical value $\\beta = 0.3544$. \n\n\n\\section{Equation of state}\n\nThe problems, considered in previous sections, were related to the cases\nwhen it was necessary to find the large-variable behaviour of the studied \nfunctions. However, generally, self-similar approximation theory allows us to\nderive approximants valid for the whole range of the variable. To illustrate \nthis, we show below how it is possible to construct an equation of state, \nproviding a good description in the whole region of densities.\n\nLet us consider a system of quantum hard spheres \\cite{Kel1996} characterized\nby the $s$-wave scattering length $a_s$ corresponding to the diameter of a\nhard sphere. The ground-state energy, in the limit of low density $\\rho \\ra 0$, \nis given \\cite{Lee1957} by the asymptotic expression\n\\be\n\\label{10.1} \n \\frac{E}{N} \\simeq 2\\pi\\; \\frac{\\rho a_s}{m} \\left ( 1 +\n\\frac{128}{15\\sqrt{\\pi}} \\; \\sqrt{\\rho a_s^3 } \\right ) \\;  ,\n\\ee\nwhere $m$ is a sphere mass. The density can increase up to the value $\\rho_0$,\nwhen the system of the spheres becomes close packed. For a primitive hexagonal\nclose packing, such as producing a face-centred cubic arrangement, \n\\be\n\\label{10.2}\n  \\rho_0 = \\frac{\\sqrt{2}}{a_s^3} \\; .\n\\ee\nIn the close-packed limit, the energy behaves as\n\\be\n\\label{10.3}\n  \\frac{E}{N} \\simeq \\frac{B}{2m} \\left ( \\rho^{-1\/3} - \\rho_0^{-1\/3}\n\\right ) \\; ,\n\\ee\nwith the experimental value\n\\be\n\\label{10.4}\nB \\equiv 2^{2\/3} \\pi^2\n\\ee\nfound by Cole \\cite{Col1967}.\n\nTo rewrite the low-density asymptotic expression in a more convenient way, we\nintroduce the variable $x$ by the relation\n\\be\n\\label{10.5}\n \\frac{\\rho}{\\rho_0} = \\frac{x^6}{(1+x^2)^3} \\;  .\n\\ee\nAs is seen, $x \\ra 0$ when $\\rho \\ra 0$ and $x \\ra \\infty$ when $\\rho \\ra \\rho_0$. \nWith the new variable, expansion (10.1) for $x \\ra 0$ takes the form\n\\be\n\\label{10.6}\n \\frac{E}{N} \\simeq  2\\pi\\; \\frac{\\rho_0 a_s}{m} \\; x^6 \\left ( 1 -\n3x^2 + \\frac{128}{15\\sqrt{\\pi}} \\; \\sqrt{\\rho_0 a_s^3 }\\; x^3 + 6x^4 - \\;\n\\frac{192}{5} \\; \\sqrt{\\rho_0 a_s^3} \\; x^5 \\right ) \\; ,\n\\ee\nwhile the close-packed limit reads as\n\\be\n\\label{10.7}\n  \\frac{E}{N} \\simeq  \\frac{\\pi^2}{ma_s^2} \\; x^4 \\qquad (x\\ra\\infty) \\; .\n\\ee\nUsing the iterated root approximant of second order for expansion (10.6), we get\n\\be\n\\label{10.8}\n \\frac{E^*_2}{N} = 2\\pi\\; \\frac{\\rho_0 a_s}{m} \\; x^6 \\left ( \n1 + A_2 x^2 \\right )^{-1} \\;   ,\n\\ee\nwith $A_2 = 2 \\sqrt{2} \/ \\pi$ corresponding to limit (10.7). Inverting the change\nof the variable (10.5), we return to the initial variable, that is, to density,\nobtaining the equation of state\n\\be\n\\label{10.9}\n \\frac{E^*_2}{N} = 2\\pi\\; \\frac{\\rho a_s}{m} \\left [ 1 - \\left (\n\\frac{\\rho}{\\rho_0} \\right )^{1\/3} \\right ]^{-2}\n\\left [ 1 + b \\left (\\frac{\\rho}{\\rho_0} \\right )^{1\/3} \\right ]^{-1} \\; ,\n\\ee\nin which\n\\be\n\\label{10.10}\nb = \\frac{2\\sqrt{2}}{\\pi} - 1 \\;   .\n\\ee\nThis equation {\\it exactly} coincides with the empirical equation called the \nmodified London equation \\cite{Sol1994} that is in very good agreement with the\nGreen function Monte Carlo computer simulations for the many-body hard-sphere\nfluid \\cite{Kal1974}. Higher orders of the self-similar iterated root approximants,\nas we have checked, do not essentially change the accuracy of the equation of \nstate (10.9) that already gives a perfect agreement with computer simulations.  \n  \n  \n\n\\section{Conclusion}\n\n\nWe have considered the problem of extrapolating perturbation-theory expansions,\nobtained for asymptotically small variable $x \\ra 0$, to the large-variable\nlimit $x \\ra \\infty$. For this purpose, we have applied the theory of \nself-similar approximations, concentrating on six different variants, resulting \nin self-similar factor approximants (Sec. 3.1), self-similar root approximants \n(Sec. 3.2), iterated root approximants (Sec. 3.3), corrected root approximants\n(Sec. 3.4), self-similar power-transformed approximants (Sec. 3.5), and doubly\nrenormalized self-similar approximants (Sec. 3.6).\n\nPad\\'{e} approximants are shown to be much less accurate than the self-similar \napproximants, and often not applicable at all. In some cases, more refined \ntechniques, such as the Kleinert variational perturbation theory, employing \ncontrol functions introduced through a variable transformation, can give better \naccuracy, although they are essentially more complicated. However, our main aim \nhere has been the analysis of the validity of the approximants that could provide \ngood accuracy, at the same time being sufficiently simple for calculations and \nyielding explicit analytical formulas. \n \nIn order to demonstrate the wide applicability of the self-similar approximants,\nwe treated a number of examples of rather different nature. In the majority \nof cases, the approximants yield close results and provide good accuracy of \nextrapolation. In general, their accuracy is essentially higher than that of \nPad\\'{e} approximants. In some cases, the latter are not applicable at all, \ngiving qualitatively wrong results, while self-similar approximants do work in\nsuch cases. \n\nComparing different variants of the analyzed self-similar approximants, we see\nthat power-transformed approximants often lead to multiple solutions for the \nsought parameters, because of which they are less convenient than other \napproximants enjoying unique solutions. The doubly renormalized approximants, \nalthough improving the final results, are cumbersome allowing only for their \ncomplicated numerical calculation. The self-similar factor approximants and \niterated root approximants seem to be the most convenient for the purpose of\nthe considered extrapolation. \n\nHaving to hand several methods of self-similar extrapolation is important\nbecause of the following reason. A problem under consideration can be so \ncomplicated that the exact answer is not known and only a few terms of \nperturbation theory are available, then it is rather difficult to judge the \naccuracy of the approximation used. However, if different methods give close \nresults, this serves as an argument that the obtained approximations are \nreliable. \n\nFinally, we have considered problems whose large-variable behaviour is\nof power-law type. We are aware that there exists another class of problems\npossessing exponential behaviour and also demonstrating the Stokes phenomenon.\nFor the problems of this class, it is necessary to use another variant of the\nself-similar approximation theory, involving self-similar exponential\napproximants \\cite{YukGlu1998, Yuk1998}. These, as has been demonstrated \nin the cited papers, make it possible to derive accurate approximations for\nthe functions of exponential behaviour as well as to treat problems \naccompanied by the Stokes phenomenon. We do not address such problems here \nbut they have been studied in our previous publications \\cite{YukGlu1998, Yuk1998}.\n\n\n\\vskip 1cm\n{\\bf Acknowledgement}\n\n\\vskip 3mm\n\nOne of the authors (V.I.Y.) is grateful for useful discussions to E.P. \nYukalova and for financial support, to the Russian Foundation for Basic \nResearch.\n\n\n\n\\newpage\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Amoroso  distribution family}\nThe {\\bf Amoroso}  (generalized gamma, Stacy-Mihram) distribution~\\cite{Amoroso1925,Johnson1994} is a four parameter,  continuous, univariate, unimodal probability density, with semi-infinite range. The functional form in the most straightforward parameterization is\n\\begin{align}\n\\label{Amoroso}  \n \\text{Amoroso}(x| a, \\theta, \\alpha, \\beta) \n&=\n\\frac{1}{\\Gamma(\\alpha)} \n\\left|\\frac{\\beta}{\\theta}\\right|\n\\left(\\frac{x-a}{\\theta}\\right)^{\\alpha \\beta -1}\n\\exp \\left\\{\n-  \\left(\\frac{x-a}{\\theta}\\right)^{\\beta}\n\\right\\}\n\\\\ \\notag\n& \\text{for } x,\\ a,\\ \\theta,\\ \\alpha,\\ \\beta\\  \\text{in } \\mathbb{R}, \n\\ \\alpha>0, \\ \n\\\\ \\notag\n& \\text{support } x \\geq a \\ \\text{if}\\ \\theta > 0,  \\ x\\leq a  \\ \\text{if}\\  \\theta < 0 .\n\\end{align}\n\nThe Amoroso distribution was originally developed to model lifetimes~\\cite{Amoroso1925}. It occurs as the Weibullization of the standard gamma distribution (\\ref{Gamma}) and, with integer $\\alpha$, in extreme value statistics (\\ref{GenFisherTippett}). The Amoroso distribution is itself a limiting form of various more general distributions, most notable the generalized beta and generalized beta prime distributions~\\cite{McDonald1984}.\n\nA useful and important property of the Amoroso distribution is that many common and interesting probability distributions are special cases or limiting forms  (See Table~\\ref{AmorosoTable}). (Informally, an ``interesting distribution'' is one that has acquired a name, which generally indicates that the distribution is the solution to one or more interesting problems.) This provides a convenient method for systemizing  a significant fraction of the probability distributions that are encountered in practice, provides a consistent parameterization for those distributions, and to obviates the need to enumerate the properties (mean, mode, variance, entropy and so on) of each and every specialization. \n\n\n\\paragraph*{Notation:} The four real parameters of the Amoroso distribution consist of a location parameter~$a$, \na scale parameter~$\\theta$,  and two shape parameters,~$\\alpha$ and~$\\beta$. Whenever these symbols appears in special cases or limiting forms, they refer directly to the parameters of the Amoroso distribution.\nThe shape parameter $\\alpha$ is positive, and in many specializations an integer, $\\alpha=n$, or half-integer, $\\alpha=\\tfrac{k}{2}$. The negation of a standard parameter is indicated by a bar, e.g. $\\bar{\\beta} = -\\beta$. The chi, chi-squared and related distributions are traditionally parameterized with the scale parameter $\\sigma$, where $\\theta= (2\\sigma^2)^{1\/{\\beta}}$, and $\\sigma$ is the standard deviation of a related normal distribution.  Additional alternative parameters are introduced as necessary. \n  \nWe write  $\\text{Amoroso}(x| a, \\theta, \\alpha, \\beta)$ for a density function,  $\\text{Amoroso}(a, \\theta, \\alpha, \\beta)$ for the corresponding random variables,  and  $X\\sim\\text{Amoroso}(a, \\theta, \\alpha, \\beta)$ to indicate that two random variables have the same probability distribution~\\cite{Gelman2004}. Recall that a  ``random variable'' is an unbound function from events to values, whereas the probability density function maps from values to probabilities. \n\n\n\n\n\n\n\\begin{table}[tph!]\n\\label{AmorosoTable}\n\\caption{The Amoroso family of distributions.}\n\\begin{align*}\n \\text{Amoroso}(x| a, \\theta, \\alpha, \\beta) \n= &\n\\frac{1}{\\Gamma(\\alpha)} \n\\left|\\frac{\\beta}{\\theta}\\right|\n\\left(\\frac{x-a}{\\theta}\\right)^{\\alpha \\beta -1}\n\\exp \\left\\{\n-  \\left(\\frac{x-a}{\\theta}\\right)^{\\beta}\n\\right\\}\n\\\\ \\notag\n& \\text{for } x,\\ a,\\ \\theta,\\ \\alpha,\\ \\beta\\  \\text{in } \\mathbb{R}, \n\\ \\alpha>0, \\ \n\\\\ \\notag & k,\\ n\\ \\text{positive integers}\n\\\\ \\notag\n& \\text{support } x \\geq a \\ \\text{if}\\ \\theta > 0,  \\ x\\leq a  \\ \\text{if}\\  \\theta < 0 .\n\\end{align*}\n\n\n\\begin{tabular}{llccccl}\n(\\ref{Amoroso}) &Amoroso & $a$ & $\\theta$ & $\\alpha$ & $\\beta$\n\\\\ \\hline\n(\\ref{Stacy}) & Stacy & $0$ & . & . & . \\\\\n(\\ref{GenFisherTippett}) & gen. Fisher-Tippett  & . & . & $n$ & .  \\\\\n(\\ref{FisherTippett}) &  Fisher-Tippett & . & . & 1 & .  \\\\\n(\\ref{Frechet}) &Fr\\'{e}chet  & . & . & 1 &  $<\\!\\!0$  \\\\\n(\\ref{GenFrechet}) &  generalized Fr\\'{e}chet & . & . & $n$ & $<\\!\\!0$ \\\\\n(\\ref{ScaledInvChi}) &scaled inverse chi& 0 & . & $\\tfrac{1}{2}k$  & -2  \\\\\n(\\ref{InvChi}) & inverse chi  & 0 & $\\frac{1}{\\sqrt{2}}$ & $\\tfrac{1}{2}k$ & -2 \\\\\n(\\ref{InvRayleigh}) &  inverse Rayleigh  & $0$ & . & $1$ & -2 \\\\\n(\\ref{PearsonV}) & Pearson type V & . & . & . & -1 \\\\\n(\\ref{InvGamma}) & inverse gamma & 0 & . & . & -1 \\\\\n(\\ref{ScaledInvChiSqr}) & scaled inverse chi-square & 0 & . & $\\tfrac{1}{2}k$ & -1 \\\\\n(\\ref{InvChiSqr}) & inverse chi-square & 0 & $\\frac{1}{2}$ & $\\tfrac{1}{2}k$ & -1 \\\\\n(\\ref{Levy}) & L\\'{e}vy &  . & . & $\\frac{1}{2}$ & -1 \\\\\n(\\ref{InvExp}) &  inverse exponential & 0  & . & 1 & -1 \\\\\n(\\ref{PearsonIII}) &Pearson type III  & . & . & . & 1 \\\\\n(\\ref{Gamma}) &gamma & $0$ & . & . & $1$ \\\\\n(\\ref{Gamma}) & Erlang & $0$ & $>\\!\\!0$ & $n$ & $1$ \\\\\n(\\ref{StdGamma}) &standard gamma & 0 & 1 & . & 1  \\\\ \n(\\ref{ScaledChiSqr}) & scaled chi-square & 0 & . & $\\tfrac{1}{2}k$ & 1 \\\\\n(\\ref{ChiSqr}) & chi-square & 0 & 2 & $\\tfrac{1}{2}k$ & 1 \\\\\n(\\ref{ShiftExp}) & shifted exponential & . & . & 1 & 1  \\\\ \n(\\ref{Exp}) & exponential & $0$ & . & $1$ & $1$ \\\\\n(\\ref{Exp}) &standard exponential & 0 & 1 & 1 & 1 \\\\ \n(\\ref{Gamma}) & Wien & 0 & . & 4& 1 \\\\\n(\\ref{Nakagami}) & Nakagami & . & . & . & $2$ \\\\\n(\\ref{ScaledChi}) &scaled chi& 0 & . & $\\tfrac{1}{2}k$  & 2  \\\\\n(\\ref{Chi}) & chi & 0 & $\\sqrt{2}$ & $\\tfrac{1}{2}k$ & 2 \\\\\n(\\ref{HalfNormal}) & half-normal & 0 & . & $\\tfrac{1}{2}$ & 2 & \\\\  \n(\\ref{Rayleigh}) & Rayleigh & 0 & . & 1 & 2  \\\\\n(\\ref{Maxwell}) & Maxwell& 0 & . & $\\frac{3}{2}$  & 2  \\\\\n(\\ref{WilsonHilferty}) &Wilson-Hilferty& 0 & . & .  & 3  \\\\\n(\\ref{GenWeibull}) & generalized Weibull  & . & . & $n$ & $>\\!\\!0$  \\\\\n(\\ref{Weibull}) & Weibull & . & . & 1 &  $>\\!\\!0$  \\\\\n(\\ref{PseudoWeibull}) & pseudo-Weibull & . & . & $1$+$\\tfrac{1}{\\beta}$ &  $>\\!\\!0$  \\\\\n(\\ref{StretchedExp}) & stretched exponential & 0 & . & 1 & $>\\!\\!0$ & \\\\\n\n\n\n\\\\\n & \\underline{Limits} \\\\\n(\\ref{LogGamma}) & log-gamma  &. & . & . & . &$\\lim_{\\beta\\rightarrow\\infty}$  \\\\\n(\\ref{PowerLaw}) & power law &.&.& $\\tfrac{1-p}{\\beta}$& . &$\\lim_{\\beta\\rightarrow0}$  \\\\\n(\\ref{LogNormal}) & log-normal  & . & . &$\\tfrac{1}{(\\beta\\sigma)^2}$  & . &$\\lim_{\\beta\\rightarrow0}$  \\\\\n(\\ref{Normal}) & normal  & . &  . & . & 1 &$\\lim_{\\alpha\\rightarrow\\infty}$  \\\\\n\\\\\n\\\\\n\\\\\n\\\\\n\\\\\n\\end{tabular} \n\n\\end{table}\n\n\n\n\n\n\\subsection{Special cases: Miscellaneous}\n\n\n\\dist{Stacy} (hyper gamma, generalized Weibull, Nukiyama-Tanasawa, generalized gamma, generalized semi-normal, hydrograph, Leonard hydrograph, transformed gamma)  distribution~\\cite{Stacy1962,Dadpay2007}:\n\\begin{align}\n\\label{Stacy}\n\\text{Stacy}(x | \\theta, \\alpha, \\beta) \n=& \\frac{1}{\\Gamma(\\alpha)} \\left|\\frac{\\beta}{\\theta}\\right| \\left(\\frac{x}{\\theta}\\right)^{\\alpha\\beta-1} \n\\exp \\left\\{ -\\left(\\frac{x}{\\theta}\\right)^{\\beta} \\right\\}\n\\\\=&  \\text{Amoroso}(x| 0, \\theta, \\alpha, \\beta) \\notag\n\\end{align}\nIf we drop the location parameter from $\\text{Amoroso}$, then we obtain the \nStacy, or generalized gamma distribution, the parent of the gamma family of distributions.\nIf $\\beta$ is negative then the distribution is  {\\bf generalized inverse gamma}, the parent of various inverse distributions, including the inverse gamma (\\ref{InvGamma}) and inverse chi (\\ref{InvChi}). \n\nThe Stacy distribution is obtained as the positive even powers,  modulus, and powers of the modulus of a centered, normal random variable (\\ref{Normal}), \n\\[\n\\text{Stacy}\\left((2\\sigma^2)^{\\tfrac{1}{\\beta}} ,\\tfrac{1}{2}, \\beta\\right) \\sim \\Big|\\text{Normal}(0,\\sigma)\\Big|^{\\tfrac{2}{\\beta}}\n\\]\nand as powers of the sum of squares of $k$ centered, normal random variables. \n\\[\n\\text{Stacy}\\left( (2\\sigma^2)^{\\tfrac{1}{\\beta}} ,\\tfrac{1}{2}k, \\beta\\right) \\sim  \\left( \\sum_{i=1}^{k} \\Big(\\text{Normal}(0,\\sigma)\\Big)^2\\right)^{\\tfrac{1}{\\beta}}\n\\]\n\n\\dist{Stretched exponential} distribution~\\cite{Laherrere1998}:\n\\begin{align}\n\\label{StretchedExp}\n\\text{StretchedExp}(x|\\theta,\\beta) = & \n\\frac{\\beta}{|\\theta|}\n\\left(\\frac{x}{\\theta}\\right)^{\\beta -1}\n\\exp\\left\\{\n-  \\left(\\frac{x}{\\theta}\\right)^{\\beta}\n\\right\\}\n\\\\ \\notag & \\text {for } \\beta>0\n\\\\= & \\text{Weibull}(x| 0, \\theta, \\beta) \\notag  \n \\\\= & \\text{Amoroso}(x| 0, \\theta,1, \\beta) \\notag \n\\end{align}\nStretched exponentials are an alternative to power laws for modeling fat tailed distributions. For $\\beta=1$ we recover the exponential distribution (\\ref{Exp}), and $\\beta=0$ a power law distribution  (\\ref{PowerLaw}). \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{pdfStretchedExpPDF}\n\\end{center}\n\\caption{$\\text{\\rm Amoroso}(x|0,1,1,\\beta)$, stretched exponential}\n\\end{figure}\n\n\n\n\\dist{Pseudo-Weibull} distribution~\\cite{Voda1989}:\n\\begin{align}\n\\label{PseudoWeibull}\n\\text{PseudoWeibull}(x|\\theta,\\beta)  \n=& \\frac{1}{\\Gamma(1+\\tfrac{1}{\\beta})} \\frac{\\beta}{|\\theta|} \\left(\\frac{x}{\\theta}\\right)^{\\beta} \n\\exp \\left\\{ -\\left(\\frac{x}{\\theta}\\right)^{\\beta} \\right\\}\n\\\\ & \\text{for } \\beta>0 \\notag \n\\\\=&  \\text{Amoroso}(x| 0, \\theta, 1+\\tfrac{1}{\\beta}, \\beta) \\notag\n\\end{align}\nProposed as another model of failure times. \n\n\n\n\n\n\\subsection{Special cases: Positive integer $\\beta$}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{pdfAmorosoBetaPDF}\n\\end{center}\n\\caption{$\\text{\\rm Amoroso}(x|0,1,2,\\beta)$}\n\\end{figure}\n\n\n\n\\dist{Gamma} ($\\Gamma$)  distribution~\\cite{Pearson1893, Pearson1895, Johnson1994} : \n\\begin{align}\n\\label{Gamma}\n\\text{Gamma}(x | \\theta, \\alpha) \n&=  \\frac{1}{\\Gamma(\\alpha)|\\theta|} \\left(\\frac{x}{\\theta}\\right)^{\\alpha-1} \\exp\\left\\{-\\frac{x}{\\theta}\\right\\} \\\\\n&=\\text{PearsonIII}(x | 0, \\theta, \\alpha) \\notag \\\\\n&=\\text{Stacy}(x| \\theta, \\alpha,1) \\notag \n\\\\&=  \\text{Amoroso}(x| 0, \\theta, \\alpha, 1) \\notag \n\\end{align}\nThe name of this distribution derives from the normalization constant.\nThe gamma distribution often appear as a solution to problems in statistical physics. For example, the energy density of a classical ideal gas, or the {\\bf Wien} (Vienna) distribution $\\text{Wien}(x|T)=\\text{Gamma}(x|T,4)$, an approximation to the relative intensity of black body radiation as a function of the frequency. The {\\bf Erlang} (m-Erlang) distribution~\\cite{Erlang1909} is a gamma distribution with integer $\\alpha$, which models the waiting time to observe $\\alpha$ events from a Poisson process with rate $1\/\\theta$ ($\\theta>0$).\n\n\n\nGamma distributions obey an addition property:\n\\begin{align*}\n\\text{Gamma}(\\theta, \\alpha_1) +  \\text{Gamma}(\\theta,\\alpha_2) \\sim \\text{Gamma}(\\theta,\\alpha_1+\\alpha_2)\n\\end{align*}\nThe  sum of two independent, gamma distributed random variables (with common $\\theta$'s, but possibly different $\\alpha$'s) is again a gamma random variable~\\cite{Johnson1994}.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{pdfGammaPDF}\n\\end{center}\n\\caption{$\\text{\\rm Gamma}(x|\\tfrac{1}{\\alpha},\\alpha)$ (unit variance) }\n\\end{figure}\n\n\n\n\n\n\\dist{Standard gamma} (standard Amoroso) distribution~\\cite{Johnson1994}: \n\\begin{equation}\n\\text{StdGamma}(x|\\alpha) = \\frac{1}{\\Gamma(\\alpha)} x^{\\alpha-1} e^{-x}\n\\label{StdGamma}\n\\end{equation}\nThe Amoroso distribution can be obtained from the standard gamma distribution by the Weibull change of variables, $x \\mapsto \\left(\\tfrac{x-a}{\\theta}\\right)^\\beta$.\n\\[\n\\text{Amoroso}(a ,\\theta,\\alpha,\\beta) \\sim\na+\\theta \\Big[{\\text{StdGamma}}(\\alpha)\\Big]^{1\/\\beta} \n\\]\n  \n\n\n\n\\dist{Pearson type III} \\noindent distribution~\\cite{Pearson1895, Johnson1994}:\n\\begin{align}\n\\label{PearsonIII}\n\\text{PearsonIII}(x | a , \\theta, \\alpha) \n=& \\frac{1}{\\Gamma(\\alpha)|\\theta|} \\left(\\frac{x-a }{\\theta}\\right)^{\\alpha-1} \n\\exp\\left\\{-\\left(\\frac{x-a }{\\theta}\\right)\\right\\}\n\\\\\n=&  \\text{Amoroso}(x| a  , \\theta, \\alpha, 1) \\notag\n\\end{align}\nThe gamma distribution with a location parameter.\n\n\n\n\n\n\\dist{Exponential} (Pearson type X, waiting time, negative exponential) distribution~\\cite{Johnson1994}:\n\\begin{align}\n\\label{Exp}\n\\text{Exp}(x | \\theta) \n&= \\frac{1}{|\\theta|} \\exp\\left\\{-\\frac{x}{\\theta}\\right\\} \\\\\n&=  \\text{Gamma}(x| \\theta, 1) \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\theta, 1, 1)  \\notag\n\\end{align}\nAn important property of the exponential distribution is that it is memoryless: the conditional probability given that $x>c$, where $c$ is a positive content, is again an exponential distribution with the same scale parameter. The only other distribution with this property is the geometric distribution~\\cite{Evans2000}, the discrete analog of the exponential distribution. With $\\theta=1$ we obtain a {\\bf standard exponential} distribution. See also shifted exponential (\\ref{ShiftExp}), stretched exponential (\\ref{StretchedExp}) and inverse exponential (\\ref{InvExp}).\n\n\\dist{Shifted exponential} distribution~\\cite{Johnson1994}:\n\\begin{align}\n\\label{ShiftExp}\n\\text{ShiftExp}(x |a , \\theta) \n&=  \\frac{1}{|\\theta|}  \\exp\\left\\{-\\left(\\frac{x-a }{\\theta}\\right)\\right\\}\\\\\n&= \\text{PearsonIII}(x | a , \\theta, 1) \\notag  \\\\\n&=  \\text{Amoroso}(x| a ,\\theta,1,1)  \\notag \n\\end{align}\nThe exponential distribution with a location parameter. \n\n\n\n\\dist{Nakagami} (generalized normal, Nakagami-m) distribution~\\cite{Nakagami1960}:\n\\begin{align}\n\\label{Nakagami}\n \\text{Nakagami}(x | a , \\theta, m) \n& =\n \\frac{2}{\\Gamma(\\tfrac{m}{2}) |\\theta| }\n\\left(\\frac{x-a }{\\theta}\\right)^{m -1}\n\\exp \\left\\{\n-  \\left(\\frac{x-a }{\\theta}\\right)^{2}\n\\right\\}\n\\\\ \\notag\n& = \\text{Amoroso}(x|a,\\theta, \\tfrac{m}{2} ,2)\n\\notag\n\\end{align}\nUsed to model attenuation of radio signals that reach a receiver by multiple paths~\\cite{Nakagami1960}.\n\n\n\n\n\\dist{Half-normal} (semi-normal, positive definite normal, one-sided normal) distribution~\\cite{Johnson1994}:\n\\begin{align}\n\\label{HalfNormal}\n\\text{HalfNormal}(x | \\sigma ) \n&= \\frac{2}{\\sqrt{2\\pi \\sigma^2}} \n\\exp\\left\\{-\\left( \\frac{x^2}{2\\sigma^2}\\right) \\right\\} \\\\\n&=\\text{ScaledChi}(x |  \\sigma, 1) \\notag \\\\\n&=  \\text{Stacy}(x| \\sqrt{2\\sigma^2} ,\\tfrac{1}{2},2) \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\sqrt{2\\sigma^2} , \\tfrac{1}{2}, 2) \\notag \n\\end{align}\nThe modulus of a normal distribution with zero mean and variance $\\sigma^2$.\n\n\n\n\n\\dist{Chi-square} ($\\chi^2$)  distribution~\\cite{Fisher1924,Johnson1994}:\n\\begin{align}\n\\label{ChiSqr}\n\\text{ChiSqr}(x | k) \n&= \\frac{1}{2\\Gamma(\\tfrac{k}{2})} \\left(\\frac{x}{2}\\right)^{\\tfrac{k}{2}-1} \n\\exp\\left\\{-\\left(\\frac{x}{2}\\right)\\right\\} \\\\\n& \\qquad \\text{for positive integer } k \\notag \\\\\n&=  \\text{Gamma}(x| 2,\\tfrac{k}{2}) \\notag \\\\\n&=  \\text{Stacy}(x|2, \\tfrac{k}{2},1) \\notag \\\\\n&=  \\text{Amoroso}(x| 0, 2, \\tfrac{k}{2}, 1) \\notag \n\\end{align}\nThe distribution of a sum of squares of $k$ independent standard normal random variables.  The chi-square distribution is important for statistical hypothesis testing in the frequentist approach to statistical inference.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{pdfChiSqr}\n\\end{center}\n\\caption{$\\text{\\rm ChiSqr}(x|k)$}\n\\end{figure}\n\n\n\n\n\\dist{Scaled chi-square}  distribution~\\cite{Lee2009}:\n\\begin{align}\n\\label{ScaledChiSqr}\n\\text{ScaledChiSqr}(x | \\sigma, k) \n&= \\frac{1}{2\\sigma^2\\Gamma(\\tfrac{k}{2})} \\left(\\frac{x}{2\\sigma^2}\\right)^{\\tfrac{k}{2}-1} \n\\exp\\left\\{-\\left(\\frac{x}{2\\sigma^2} \\right)\\right\\} \\\\\n& \\qquad \\text{for positive integer } k \\notag \\\\\n&=  \\text{Stacy}(x|2\\sigma^2, \\tfrac{k}{2},1) \\notag \\\\\n&=\\text{Gamma}(x|2\\sigma^2, \\tfrac{k}{2}) \\notag \\\\\n&=  \\text{Amoroso}(x| 0, 2\\sigma^2, \\tfrac{k}{2}, 1) \\notag \n\\end{align}\nThe distribution of a sum of squares of $k$ independent normal random variables with variance $\\sigma^2$.\n\n\\dist{Chi} ($\\chi$) distribution~\\cite{Johnson1994}:\n\\begin{align}\n\\label{Chi}\n\\text{Chi}(x | k) \n&= \\frac{ \\sqrt{2}}{\\Gamma(\\tfrac{k}{2})} { \\left(\\frac{x}{\\sqrt{2}}\\right)}^{k-1} \n\\exp\\left\\{ -\\left( \\frac{x^2}{2}    \\right)\\right\\} \n\\\\\n& \\qquad \\text{for positive integer } k \\notag \\\\\n& = \\text{ScaledChi}(x|1,k) \\notag \\\\\n&=  \\text{Stacy}(x|\\sqrt{2}, \\tfrac{k}{2}, 2)  \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\sqrt{2} , \\tfrac{k}{2}, 2) \\notag \n\\end{align}\nThe root-mean-square of $k$ independent standard normal variables, or the square root of a chi-square random variable.\n\\[\n\\text{Chi}(k) \\sim \\sqrt{\\text{ChiSqr}(k)}\n\\]\n\n\\dist{Scaled chi} (generalized Rayleigh) distribution~\\cite{Miller1964,Johnson1994}:\n\\begin{align}\n\\text{ScaledChi}(x | \\sigma, k) \n&= \\frac{2}{\\Gamma(\\tfrac{k}{2}) \\sqrt{2\\sigma^2}} { \\left(\\frac{x}{\\sqrt{2\\sigma^2}}\\right)}^{k-1} \n\\exp\\left\\{-\\left(\\frac{x^2}{2\\sigma^2}\\right)\\right\\} \n\\notag\n\\\\\n& \\qquad \\text{for positive integer } k \\notag \\\\\n&=  \\text{Stacy}(x|\\sqrt{2\\sigma^2}, \\tfrac{k}{2},2) \n\\label{ScaledChi}\n\\\\\n&=  \\text{Amoroso}(x | 0, \\sqrt{2\\sigma^2}, \\tfrac{k}{2}, 2) \n\\notag \n\\end{align}\nThe root-mean-square of $k$ independent and identically distributed normal variables with zero mean and variance~$\\sigma^2$. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{pdfBeta2PDF}\n\\end{center}\n\\caption{$\\text{\\rm Amoroso}(x|0,1,\\alpha,2)$}\n\\end{figure}\n\n\n\n\\dist{Rayleigh} distribution~\\cite{Strutt1880,Johnson1994}:\n\\begin{align}\n\\label{Rayleigh}\n\\text{Rayleigh}(x | \\sigma) \n&= \\frac{1}{\\sigma^2 }\\ x\\  \\exp\\left\\{-\\left(\\frac{x^2}{2 \\sigma^2}\\right)\\right\\} \n\\\\\n&=\\text{ScaledChi}(x | \\sigma, 2) \\notag \\\\\n&=  \\text{Stacy}(x| \\sqrt{2\\sigma^2} ,1,2) \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\sqrt{2\\sigma^2} , 1, 2) \\notag \n\\end{align}\n The root-mean-square of two independent and identically distributed normal variables with zero mean and variance $\\sigma^2$. \n For instance, wind speeds are approximately Rayleigh distributed, since the horizontal components of the velocity are approximately normal, and the vertical component is typically small~\\cite{Justus1978}. \n\n\n\\dist{Maxwell} (Maxwell-Boltzmann, Maxwell speed) distribution~\\cite{Maxwell1860, Abramowitz1965}:\n\\begin{align}\n\\label{Maxwell}\n\\text{Maxwell}(x | \\sigma) \n&= \\frac{\\sqrt{2}}{\\sqrt{\\pi} \\sigma^3}\\ x^2 \\exp\\left\\{-\\left(\\frac{x^2}{2\\sigma^2}\\right)\\right\\}  \n \\\\\n&=\\text{ScaledChi}(x | \\sigma, 3) \\notag \\\\\n&=  \\text{Stacy}(x| \\sqrt{2\\sigma^2} ,\\tfrac{3}{2},2) \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\sqrt{2\\sigma^2} , \\tfrac{3}{2}, 2) \\notag \n\\end{align}\nThe speed distribution of molecules in thermal equilibrium. The root-mean-square of three independent and identically distributed normal variables with zero mean and variance $\\sigma^2$.\n\n\n\n\\dist{Wilson-Hilferty} distribution~\\cite{Wilson1931,Johnson1994}:\n\\begin{align}\n\\label{WilsonHilferty}\n\\text{WilsonHilferty}(x | \\theta, \\alpha) \n&= \\frac{3}{\\Gamma(\\alpha)|\\theta|} \\left(\\frac{x}{\\theta}\\right)^{3 \\alpha-1} \\exp\\left\\{-\\left(\\frac{x}{\\theta}\\right)^{3}\\right\\}\n\\\\ \n&=  \\text{Stacy}(x|\\theta, \\alpha, 3) \n\\notag \n\\\\ &=  \\text{Amoroso}(x| 0, \\theta, \\alpha, 3) \\notag\n\\end{align}\nThe cube root of a gamma variable follows the Wilson-Hilferty distribution~\\cite{Wilson1931}, which has been used to approximate a normal distribution if $\\alpha$ is not too small.\nA related approximation using quartic roots of gamma variables~\\cite{Hawkins1986} leads to   $\\text{Amoroso}(x| 0, \\theta, \\alpha, 4)$.\n\n\n\n\\subsection{Special cases: Negative integer $\\beta$}\n\n\n\n\n\n\n\\dist{Pearson type V} distribution~\\cite{Pearson1901}:\n\\begin{align}\n\\label{PearsonV}\n\\text{PearsonV}(x |a, \\theta, \\alpha) \n=&\n\\frac{1}{\\Gamma(\\alpha)\\left|{\\theta}\\right|} \n\\left(\\frac{\\theta}{x-a }\\right)^{\\alpha +1}\n \\exp\\left\\{-\\left( \\frac{\\theta}{x-a }\n\\right)\\right\\} \n\\notag \n\\\\\n=&  \\text{Amoroso}(x| a , \\theta, \\alpha, -1) \n\\end{align}\nWith negative $\\beta$ we obtain various ``inverse'' distributions related to distributions with positive $\\beta$ by the reciprocal transformation $ (\\tfrac{x-a}{\\theta} ) \\mapsto (\\tfrac{\\theta}{x-a} )$.\nPearson's type V is the inverse of Pearson's type III distribution.\n\n\n\n\\dist{Inverse gamma} (Vinci) distribution~\\cite{Johnson1994}:\n\\begin{align}\n\\label{InvGamma}\n\\text{InvGamma}(x | \\theta, \\alpha) \n&= \\frac{1}{\\Gamma(\\alpha) |\\theta|} \\left(\\frac{\\theta}{x}\\right)^{\\alpha+1} \n \\exp\\left\\{-\\left( \\frac{\\theta}{x}   \\right)\\right\\} \n\\\\\n&=  \\text{Stacy}(x| \\theta, \\alpha,-1)\\notag \\\\\n& = \\text{PearsonV}(x|0,\\theta,\\alpha)  \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\theta, \\alpha, -1) \\notag \n\\end{align}\nOccurs as the conjugate prior for an exponential distribution's scale parameter~\\cite{Johnson1994}, or the prior for variance of a  normal distribution with known mean~\\cite{Gelman2004}.\n\n\\dist{Inverse exponential} distribution~\\cite{Kleiber2003}:\n\\begin{align}\n\\label{InvExp}\n\\text{InvExp}(x | \\theta) \n&= \\frac{|\\theta|}{x^2 }  \\exp\\left\\{-\\left( \\frac{\\theta}{x}   \\right)\\right\\}   \\\\\n&=  \\text{InvGamma}(x| \\theta, 1) \\notag \\\\\n&=  \\text{Stacy}(x|\\theta,1,-1) \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\theta, 1, -1) \\notag \n\\end{align}\nNote that the name ``inverse exponential'' is occasionally used for the ordinary exponential distribution (\\ref{Exp}).\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{pdfAmorosoBetaNegPDF}\n\\end{center}\n\\caption{$\\text{\\rm Amoroso}(x|0,1,2,\\beta)$, negative $\\beta$.}\n\\end{figure}\n\n\\dist{L\\'{e}vy} distribution (van der Waals profile)~\\cite{Feller1971}: \n\\begin{align}\n\\label{Levy}\n\\text{L\\'{e}vy}(x | a, c) \n&= \\sqrt{\\frac{c}{2\\pi}} \\frac{1}{(x-a)^{3\/2}}  \\exp\\left\\{-\\frac{c}{2(x-a)}\\right\\}  \n\\\\\n&= \\text{PearsonV}(x|a,\\tfrac{c}{2},\\tfrac{1}{2})  \\notag \\\\\n&=  \\text{Amoroso}(x| a, \\tfrac{c}{2}, \\tfrac{1}{2}, -1) \\notag \n\\end{align}\nThe L\\'{e}vy distribution is notable for being stable:  a linear combination of identically distributed  L\\'{e}vy distributions is again a  L\\'{e}vy distribution. The other stable distributions with analytic forms are the normal distribution (\\ref{Normal}), which is also a limit of the Amoroso distribution, and the Cauchy distribution~\\cite{Johnson1994}, which is not. L\\'{e}vy distributions describe first passage times in one dimensional Brownian diffusion~\\cite{Feller1971}.\n\n\n\\dist{Scaled inverse chi-square}  distribution~\\cite{Gelman2004}:\n\\begin{align}\n\\label{ScaledInvChiSqr}\n\\text{ScaledInvChiSqr}(x | \\sigma, k) \n=& \\frac{2 \\sigma^2}{\\Gamma(\\tfrac{k}{2}) } \\left(\\frac{1}{2 \\sigma^2x}\\right)^{\\frac{k}{2}+1} \n\\exp\\left\\{-\\left( \\frac{1}{2 \\sigma^2x}   \\right)\\right\\}\n\\\\\n&\\qquad  \\text{for positive integer } k \\notag \\\\\n&=  \\text{InvGamma}(x| \\tfrac{1}{2 \\sigma^2}, \\tfrac{k}{2}) \\notag \\\\\n&= \\text{PearsonV}(x|0,\\tfrac{1}{2 \\sigma ^2},\\tfrac{k}{2})  \\notag  \\\\\n&= \\text{Stacy}(x| \\tfrac{1}{2 \\sigma ^2},\\tfrac{k}{2}, -1)  \\notag  \\\\\n&=  \\text{Amoroso}(x| 0, \\tfrac{1}{2 \\sigma ^2}, \\tfrac{k}{2}, -1) \\notag \n\\end{align}\nA special case of the inverse gamma distribution with half-integer $\\alpha$. Used as a prior for variance parameters in normal models~\\cite{Gelman2004}.\n\n\n\n\n\\dist{Inverse chi-square} distribution~\\cite{Gelman2004}: \n\\begin{align}\n\\label{InvChiSqr}\n\\text{InvChiSqr}(x | k) \n=& \\frac{2}{\\Gamma(\\tfrac{k}{2}) } \\left(\\frac{1}{2x}\\right)^{\\frac{k}{2}+1} \\exp\\left\\{-\\left( \\frac{1}{2x}  \\right)\\right\\}\n  \\\\\n&\\qquad  \\text{for positive integer } k \\notag \\\\\n& = \\text{ScaledInvChiSqr}(x|1,k)\\notag \\\\\n&=  \\text{InvGamma}(x| \\tfrac{1}{2}, \\tfrac{k}{2}) \\notag \\\\\n&= \\text{PearsonV}(x|0,\\tfrac{1}{2},\\tfrac{k}{2})  \\notag \\\\\n&=  \\text{Stacy}(x|\\tfrac{1}{2}, \\tfrac{k}{2},-1) \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\tfrac{1}{2}, \\tfrac{k}{2}, -1) \\notag \n\\end{align}\nA standard scaled inverse chi-square distribution.\n\n\n\n\\dist{Scaled inverse chi} distribution~\\cite{Lee2009}:\n\\begin{align}\n\\label{ScaledInvChi}\n\\text{ScaledInvChi}(x | \\sigma, k) \n&= \\frac{2 \\sqrt{2 \\sigma ^2} }{ \\Gamma(\\tfrac{k}{2})} { \\left(\\frac{1}{\\sqrt{2 \\sigma^2} x}\\right)}^{k+1} \\exp\\left\\{-\\left(\\frac{1}{2 \\sigma^2 x^2}  \\right)\\right\\}\n\\\\\n&=  \\text{Stacy}(x|\\tfrac{1}{\\sqrt{2 \\sigma^2}}, \\tfrac{k}{2}, -2)  \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\tfrac{1}{\\sqrt{2 \\sigma^2}}, \\tfrac{k}{2}, -2) \\notag \n\\end{align}\nUsed as a prior for the standard deviation of a normal distribution.\n\n\n\\dist{Inverse chi} distribution~\\cite{Lee2009}: \n\\begin{align}\n\\label{InvChi}\n\\text{InvChi}(x | k) \n&= \\frac{2\\sqrt{2} }{ \\Gamma(\\tfrac{k}{2})} { \\left(\\frac{1}{\\sqrt{2} x}\\right)}^{k+1} \\exp\\left\\{-\\left(\\frac{1}{2 x^2}  \\right)\\right\\}\n\\\\\n&=  \\text{Stacy}(x| \\tfrac{1}{\\sqrt{2}}, \\tfrac{k}{2}, -2)  \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\tfrac{1}{\\sqrt{2}} , \\tfrac{k}{2}, -2) \\notag \n\\end{align}\nThe standard inverse chi distribution.\n\n\n\\dist{Inverse Rayleigh} distribution~\\cite{Evans2000}:\n\\begin{align}\n\\label{InvRayleigh}\n\\text{InvRayleigh}(x | \\sigma) \n&= 2 \\sqrt{2 \\sigma ^2}    \\left(\\frac{1}{\\sqrt{2 \\sigma^2} x}\\right)^{3} \\exp\\left\\{-\\left(\\frac{1}{2 \\sigma^2 x^2}  \\right)\\right\\}\n\\\\\n&=  \\text{Stacy}(x|\\tfrac{1}{\\sqrt{2 \\sigma^2}}, 1, -2)  \\notag \\\\\n&=  \\text{Amoroso}(x| 0, \\tfrac{1}{\\sqrt{2 \\sigma^2}}, 1, -2) \\notag \n\\end{align}\nThe inverse Rayleigh distribution has been used to model  a failure time~\\cite{Voda1972}.\n\n\n\n\n\n\n\n\\subsection{Special cases: Extreme order statistics}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{pdfEVD}\n\\end{center}\n\\caption{Extreme value distributions}\n\\end{figure}\n\n\n\n\\dist{Generalized Fisher-Tippett} distribution~\\cite{Smirnov1949,Barndorff-Nielsen1963}:\n\\begin{align}\n\\label{GenFisherTippett}  \n \\text{GenFisherTippett}(x| a, \\omega, n, \\beta) \n\\notag\n&=\n\\frac{n^n}{\\Gamma(n)} \n\\left|\\frac{\\beta}{\\omega}\\right|\n\\left(\\frac{x-a}{\\omega}\\right)^{n \\beta -1}\n\\exp \\left\\{\n-  n \\left(\\frac{x-a}{\\omega}\\right)^{\\beta}\n\\right\\}\n\\\\\n& \\quad \\text{for positive integer } n\n\\\\ \\notag\n& = \\text{Amoroso}(x|a,{\\omega}\/{n^{\\frac{1}{\\beta} }},n,\\beta)\n\\end{align}\nIf we take $N$ samples from a probability distribution, then asymptotically for large $N$ and $n\\ll N$, the distribution of the $n$th largest (or smallest) sample follows a generalized Fisher-Tippett distribution. The parameter $\\beta$ depends on the tail behavior of the sampled distribution. Roughly speaking, if the tail is unbounded and decays exponentially then $\\beta$ limits to $\\infty$, if the tail scales as a power law then $\\beta<0$,  and if the tail is finite $\\beta>0$~\\cite{Gumbel1958}. In these three limits we obtain the Gumbel (\\ref{Gumbel}, \\ref{GenGumbel}), Fr\\'{e}chet (\\ref{Frechet}, \\ref{GenFrechet}) and Weibull (\\ref{Weibull},\\ref{GenWeibull}) families of extreme value distribution (Extreme value distributions types I, II and III) respectively. If $\\beta\/\\omega$ is negative we obtain distributions for the $n$th maxima, if positive then the $n$th minima.\n\n\n\n\\dist{Fisher-Tippett} (Generalized extreme value, GEV, von Mises-Jenkinson, von Mises extreme value) distribution~\\cite{Fisher1928, Mises1936, Gumbel1958,Johnson1995}: \n\\begin{align}\n\\label{FisherTippett}\n \\text{FisherTippett}(x| a , \\omega, \\beta) \n&=\n\\left|\\frac{\\beta}{\\omega}\\right|\n\\left(\\frac{x-a}{\\omega}\\right)^{ \\beta -1}\n\\exp \\left\\{\n-  \\left(\\frac{x-a}{\\omega}\\right)^{\\beta}\n\\right\\}\n\\\\ \\notag & = \\text{GenFisherTippett}(x|a, \\omega, 1, \\beta)\n\\\\ \\notag & = \\text{Amoroso}(x|a, \\omega, 1, \\beta)\n\\end{align}\nThe asymptotic distribution of the extreme value from a large sample. The superclass of type I, II and III (Gumbel, Fr\\'{e}chet, Weibull) extreme value distributions~\\cite{Mises1936}.  This is the distribution for maximum values with $\\beta\/\\omega<0$ and minimum values for $\\beta\/\\omega>0$.\n\n\nThe maximum of two Fisher-Tippett random variables (minimum if $\\beta\/\\omega>0$)  is again a Fisher-Tippett random variable. \n\\begin{align*}\n\\max\\Big[ \\text{FisherTippett}(a,\\omega_1,\\beta),  \\text{FisherTippett}(a, \\omega_2,\\beta)  \\Big]&\\\\  \\sim \n \\text{FisherTippett}(a, \\frac{(\\omega_1^{\\beta} + \\omega_2^{\\beta} )^{1\/\\beta}}{\\omega_1 \\omega_2},\\beta) \n\\end{align*}\nThis follows because taking the maximum of two random variables is equivalent to multiplying their cumulative distribution functions, and the Fisher-Tippett cumulative distribution function is $\\exp \\left\\{\n-  \\left(\\frac{x-a}{\\omega}\\right)^{\\beta}\n\\right\\}$.\n\n\n\\dist{Generalized Weibull} distribution~\\cite{Smirnov1949,Barndorff-Nielsen1963}:\n\\begin{align}\n\\label{GenWeibull}\n\\text{GenWeibull}(x | a , \\omega, n, \\beta) \n&=\t\\frac{n^n}{\\Gamma(n)}  \\frac{ \\beta}{| \\omega |} \\left(\\frac{x-a }{\\omega}\\right)^{n \\beta-1} \\exp\\left\\{-n \\left(\\frac{x-a }{\\omega}\\right)^{ \\beta}\\right\\} \n\\\\ \\notag &\\quad \\text{for } \\beta>0 \n\\\\ \\notag & = \\text{GenFisherTippett}(x|a, \\omega, n, \\beta)\n\\\\ \\notag\n&= \\text{Amoroso}(x| a , {\\omega}\/{n^{\\frac{1}{\\beta} }}, n, \\beta) \n\\end{align}\nThe limiting distribution of the $n$th smallest value of a large number of  identically distributed random variables that are at least~$a$. \nIf $\\omega$ is negative we obtain the distribution of the $n$th largest value.\n\n\n\\dist{Weibull} (Fisher-Tippett type III, Gumbel type III, Rosin-Rammler, Rosin-Rammler-Weibull, extreme value type III, Weibull-Gnedenko) distribution~\\cite{Weibull1951,Johnson1995}: \n\\begin{align}\n\\label{Weibull}\n\\text{Weibull}(x | a ,\\omega, \\beta) \n&=\t\\frac{\\beta}{| \\omega |} \\left(\\frac{x-a }{\\omega}\\right)^{\\beta-1} \\exp\\left\\{-\\left(\\frac{x-a }{\\omega}\\right)^{\\beta}\\right\\} \n\\\\ \\notag &\\quad \\text{for } \\beta>0 \n\\\\ \\notag\n& = \\text{FisherTippett}(x| a, \\omega, \\beta) \n\\\\ \\notag\n&= \\text{Amoroso}(x| a , \\omega, 1, \\beta) \n\\end{align}\nThis is the limiting distribution of the minimum of a large number of  identically distributed random variables that are at least~$a$.  If $\\omega$ is negative we obtain a {\\bf reversed Weibull} (extreme value type III) distribution for maxima.\nSpecial cases of the Weibull distribution include the exponential ($\\beta=1$) and Rayleigh ($\\beta=2$)  distributions.\n\n\n\\dist{Generalized Fr\\'{e}chet} distribution~\\cite{Smirnov1949,Barndorff-Nielsen1963}:\n\\begin{align}\n\\label{GenFrechet}\n\\text{GenFr\\'{e}chet}(x | a , \\omega, n, \\bar{\\beta}) \n \\notag \n&=\t\\frac{n^n}{\\Gamma(n)}  \\frac{\\bar{\\beta}}{| \\omega |} \\left(\\frac{x-a }{\\omega}\\right)^{-n\\bar{\\beta}-1} \n\\exp\\left\\{-n\\left(\\frac{x-a }{\\omega}\\right)^{-\\bar{\\beta}}\\right\\} \n\\\\ &\\quad \\text{for } \\bar{\\beta}>0 \n\\\\ \\notag\n& = \\text{GenFisherTippett}(x|a, \\omega, n, -\\bar{\\beta})\n\\\\ \\notag\n&= \\text{Amoroso}(x| a , {\\omega}\/{n^{\\frac{1}{\\beta} }},n,-\\bar{\\beta}),\n\\end{align}\nThe limiting distribution of the $n$th largest value of a large number identically distributed random variables whose moments are not all finite and are bounded from below by $a$.  (If the shape parameter $\\omega$ is negative then minimum rather than maxima.)\n\n\n\\dist{Fr\\'{e}chet} (extreme value type II , Fisher-Tippett type II, Gumbel type II, inverse Weibull) distribution~\\cite{Frechet1927,Gumbel1958}:\n\\begin{align}\n\\label{Frechet}\n\\text{Fr\\'{e}chet}(x | a , \\omega, \\bar{\\beta}) \n&=\t\\frac{\\bar{\\beta}}{| \\omega |} \\left(\\frac{x-a }{\\omega}\\right)^{-\\bar{\\beta}-1} \n\\exp\\left\\{-\\left(\\frac{x-a }{\\omega}\\right)^{-\\bar{\\beta}}\\right\\}\n\\\\ \\notag &\\quad \\text{for } \\bar{\\beta}>0 \n\\\\  \\notag\n& = \\text{FisherTippett}(x| a, \\omega, -\\bar{\\beta}) \n\\\\ \\notag \n&= \\text{Amoroso}(x| a , \\omega,1,-\\bar{\\beta})\\notag\n\\end{align}\nThe limiting distribution of the largest of a large number identically distributed random variables whose moments are not all finite and are bounded from below by $a$.  (If the shape parameter $\\omega$ is negative then minimum rather than maxima.)\nSpecial cases of the Fr\\'{e}chet  distribution include the inverse exponential ($\\bar{\\beta}=1$) and inverse Rayleigh ($\\bar{\\beta}=2$) distributions.\n \n\n\n\n\n\n\n\\subsection{Properties}\n\\begin{align*}\n\\text{support} \\quad & x \\geq a &  \\theta > 0\n\\\\\n&   x\\leq a  &  \\theta < 0 \n \\\\\n\\text{cdf} \\quad \n\n 1-Q(\\alpha, \\left(\\tfrac{x - a }{\\theta}\\right)^{\\beta}) \n& \\tfrac{\\beta}{\\theta}>0\n\\\\\n&\n Q(\\alpha, \\left(\\tfrac{x - a }{\\theta}\\right)^{\\beta}) \n& \\tfrac{\\beta}{\\theta}<0\n\\\\\n\\text{mode} \\quad&   a+ \\theta (\\alpha-\\tfrac{1}{\\beta})^{\\frac{1}{\\beta}} \n& \\alpha \\beta  \\geq 1\n\\\\ & a & \\alpha \\beta  \\le 1\n\\\\\n\\text{std. moments}\\quad &  \\frac{\\Gamma(\\alpha+\\frac{r}{\\beta})}{\\Gamma(\\alpha)}  \\quad ( a=0, \\theta=1) \n& \\alpha + \\tfrac{n}{\\beta} \\geq 0\n\\\\\n\\text{mean} \\quad& a  + \\theta \\frac{\\Gamma(\\alpha+\\frac{1}{\\beta})}{\\Gamma(\\alpha)} \n& \\alpha + \\tfrac{1}{\\beta} \\geq 0\n\\\\\n\\text{variance}  \\quad&   \\theta^2 \\left[  \\frac{\\Gamma(\\alpha+\\frac{2}{\\beta})}{\\Gamma(\\alpha)}  - \n\\frac{\\Gamma(\\alpha+\\frac{1}{\\beta})^2}{\\Gamma(\\alpha)^2}    \\right]\n& \\alpha + \\tfrac{2}{\\beta} \\geq 0\n \\\\\n\\text{entropy}  \\quad& \n\\ln \\frac{\\theta \\Gamma(\\alpha)}{|\\beta|} +\\alpha + \\left( \\frac{1}{\\beta} - \\alpha\\right) \\psi(\\alpha) &\n\\text{\\cite{Dadpay2007}}\n\\end{align*}\nHere, cdf is the cumulative distribution function, \n$Q(\\alpha,x)= \\Gamma(\\alpha,x) \/\\Gamma(\\alpha)$ is the regularized gamma function~\\cite{Abramowitz1965},\n$\\Gamma(\\alpha,x) = \\int_x^{\\infty} t^{\\alpha-1} e^{-t} dt$ is the incomplete gamma integral~\\cite{Abramowitz1965},\nand\n$\\psi(x)=\\frac{d}{dx} \\ln \\Gamma(x)$ is the digamma function~\\cite{Abramowitz1965}, the logarithmic derivative of the gamma function.\n Important special cases and limits include\n$\\Gamma(\\tfrac{1}{2})=\\sqrt{\\pi}$, $\\Gamma(\\tfrac{1}{2}, x) = \\sqrt{\\pi} \\text{erfc}(\\sqrt{x})$ and $\\Gamma(1,x) = \\exp(-x)$.\nThe derivative of the regularized gamma function is $\\tfrac{d}{dx} Q(\\alpha,x) = -\\tfrac{1}{\\Gamma(\\alpha)} x^{\\alpha-1} e^{-x}$. \n\n\nThe profile of the Amoroso distribution is bell shaped for $\\alpha\\beta\\geq1$, and otherwise L- or J- shaped with the  mode at the boundary. The moments are undefined if the side conditions are not satisfied. Expressions for skew and kurtosis are not simple, but can be deduced from the moments if necessary.\n\nThe Amoroso distribution can be obtained from the standard gamma distribution (\\ref{StdGamma}) with the change of variables,  $x \\mapsto \\left(\\tfrac{x-a}{\\theta}\\right)^\\beta$. Therefore, Amoroso random numbers can be obtained by sampling from the standard gamma distribution, for instance using the Marsaglia-Tsang fast gamma method~\\cite{Marsaglia2001} and applying the appropriate  transformation~\\cite{Knuth1997}.\n\n\n\n\n\n\n\\pagebreak[4]\n\\section{Log-gamma distributions}\n\\label{SecLogGamma}\n\n\n\n\n\n\nThe {\\bf log-gamma} (Coale-McNeil) distribution~\\cite{Bartlett1946,Prentice1974,Johnson1995} is  a three parameter, continuous, univariate, unimodal probability density with infinite range. The functional form in the most straightforward parameterization is\n\\begin{align}\n\\label{LogGamma}\n\\text{LogGamma}(x|\\nu, \\lambda, \\alpha) \n&=\n\\frac{1}{ \\Gamma(\\alpha) |\\lambda|}  \\exp\\left\\{ \\alpha \\left(\\frac{x-\\nu}{\\lambda}\\right) - \\exp\\left(\\frac{x-\\nu}{\\lambda}\\right)  \\right\\} \n\\\\ \\notag\n& \\qquad \\text{for } x,\\ \\nu,\\ \\lambda,\\ \\alpha,\\   \\text{in } \\mathbb{R}, \n\\ \\alpha>0, \\ \n\\\\ \\notag\n& \\qquad \\text{support } -\\infty \\leq x \\leq \\infty\n\\notag\n\\end{align}\nThe three real parameters consist of a location parameter $\\nu$, a scale parameter $\\lambda$, and a shape parameter $\\alpha$, which is inherited directly from the Amoroso distribution.\n\nThe name ``log-gamma'' arises because the standard log-gamma distribution is the logarithmic transform of the standard gamma distribution\n\\begin{align*}\n\\text{StdLogGamma}(\\alpha)  &\\sim \\ln\\Big( \\text{StdGamma}(\\alpha) \\Big)\n\\\\\n\\text{LogGamma}(\\nu, \\lambda, \\alpha)  &\\sim \\ln\\Big( \\text{Amoroso}(0, e^{\\nu},\\alpha, \\tfrac{1}{\\lambda}) \\Big)\n\\end{align*}\nNote that this naming convention is the opposite of that used for the log-normal distribution (\\ref{LogNormal}). The name  ``log-gamma''  has also been used for the antilog transform of the generalized gamma distribution, which leads to the unit-gamma distribution~\\cite{Gupta2004}.\n\n\nThe log-gamma distribution is a limit of the Amoroso distribution (\\ref{Amoroso}),  and itself  has a number of important limits and special cases (Table II), which we will discuss below. \n\\begin{align}\n&\\text{LogGamma}(x|\\nu, \\lambda, \\alpha) \n\\\\ \\notag & = \\lim_{\\beta\\rightarrow\\infty}   \n\\frac{1}{\\Gamma(\\alpha) |\\lambda|} \n\\left(1+ \\frac{1}{\\beta}\\frac{x-\\nu }{\\lambda}\\right)^{\\alpha\\beta-1}\n\\!\\!\\exp\\left\\{  - \\left(1+ \\frac{1}{\\beta}\\frac{x-\\nu }{\\lambda}\\right)^{\\beta} \\right\\}\n\\\\ \\notag &= \\lim_{\\beta\\rightarrow\\infty} \\text{Amoroso}(\\nu-\\beta \\lambda,\\beta \\lambda, \\alpha,\\beta)\n\\notag\n\\end{align}\nRecall that $\\lim_{\\beta\\rightarrow\\infty} (1+ \\tfrac{x}{\\beta})^{\\beta} = \\exp(x) $.\n\n\n\n\\begin{table}\n\\caption{The log-gamma family of distributions.}\n\\begin{align*}\n&\\text{LogGamma}(x|\\nu, \\lambda, \\alpha) \n\\\\&=\n\\frac{1}{ \\Gamma(\\alpha) | \\lambda |}  \\exp\\left\\{ \\alpha \\left(\\frac{x-\\nu}{\\lambda}\\right) - \\exp\\left(\\frac{x-\\nu}{\\lambda}\\right)  \\right\\} \\notag\n\\\\ \\notag\n& \\qquad \\text{for } x,\\ \\nu,\\ \\lambda,\\ \\alpha,\\   \\text{in } \\mathbb{R}, \n\\ \\alpha>0, \\ \n\\\\ \\notag\n& \\qquad \\text{support } -\\infty \\leq x \\leq \\infty\n\\notag\n\\end{align*}\n\n\\begin{tabular}{llcccl}\n(\\ref{LogGamma}) & log-gamma &  $\\nu$ & $\\lambda$ & $\\alpha$ \n\\\\ \\hline\n(\\ref{StdLogGamma}) & standard log-gamma &  $0$ & $1$ & $\\alpha$  \\\\\n(\\ref{LogChiSqr}) & log-chi-square &$\\ln 2 $ & $1$ & $\\tfrac{k}{2}$ \\\\ \n(\\ref{GenGumbel}) &generalized Gumbel    & . & . & $n$ &  \\\\\n(\\ref{Gumbel}) &Gumbel  &  . & . & 1 &  \\\\\n(\\ref{BHP}) &BHP   & . & . & $\\frac{\\pi}{2}$ &  \\\\\n(\\ref{StdGumbel}) &standard Gumbel &   0 & -1 & 1 & \\\\ \n\\\\\n & \\underline{Limits} \\\\\n(\\ref{Normal}) & normal  & . &  . & $\\alpha$& $\\lim_{\\alpha\\rightarrow\\infty}$  \\\\\n\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{pdfLogGamma}\n\\end{center}\n\\caption{$\\text{\\rm LogGamma}(x|0,1,\\alpha)$}\n\\end{figure}\n\n\n\\subsection{Special cases}\n\n\n\\dist{Standard log-gamma} distribution:\n\\begin{align}\n\\label{StdLogGamma}\n\\text{StdLogGamma}(x| \\alpha) \n=&\n\\frac{1}{\\Gamma(\\alpha) }  \\exp\\left\\{ \\alpha x - \\exp (x)  \\right\\} \n\\\\ \\notag =& \\text{LogGamma}(x|0,1,\\alpha)\n\\end{align} \nThe log-gamma distribution with zero location and unit scale.\n\n\\dist{Log-chi-square} distribution~\\cite{Lee2009}:\n\\begin{align}\n\\label{LogChiSqr}\n\\text{LogChiSqr}(x|k) \n &= \\notag\n\\frac{1}{2^{\\frac{k}{2}} \\Gamma(\\frac{k}{2})}  \\exp\\left\\{ \\frac{k}{2} x - \\frac{1}{2} \\exp(x)  \\right\\} \n\\\\ & \\qquad \\text{ for positive integer } k\n\\\\\n&= \\text{LogGamma}(x|\\ln 2, 1 ,\\tfrac{k}{2}) \n\\notag\n\\end{align}\nThe logarithmic transform of the chi-square distribution~(\\ref{ChiSqr}).\n\n\\dist{Generalized Gumbel} distribution~\\cite{Gumbel1958,Johnson1995}: \n\\begin{align}\n\\label{GenGumbel}\n\\text{GenGumbel}(x|u,\\bar{\\lambda},n) \n &= \\notag\n\\frac{n^n}{\\Gamma(n) |\\bar{\\lambda}|}  \\exp\\left\\{ - n \\left(\\frac{x-u}{\\bar{\\lambda}}\\right) - n \\exp\\left(- \\frac{x-u}{\\bar{\\lambda}}\\right)  \\right\\} \n\\\\ & \\qquad \\text{ for positive integer } n\n\\\\\n&= \\text{LogGamma}(x|u+\\bar{\\lambda} \\ln n,-\\bar{\\lambda},n) \n\\notag\n\\end{align}\nThe limiting distribution of the $n$th largest value of a large number of unbounded identically distributed random variables whose probability distribution has an exponentially decaying tail.\n\n\n\\dist{Gumbel}  (Fisher-Tippett type I, Fisher-Tippett-Gumbel, FTG, Gumbel-Fisher-Tippett, log-Weibull, extreme value (type I),  doubly exponential, double exponential) distribution~\\cite{Fisher1928,Gumbel1958, Johnson1995}:\n\\begin{align}\n\\label{Gumbel}\n\\text{Gumbel}(x|u,\\bar{\\lambda}) \n&=\n\\frac{1}{|\\bar{\\lambda}|}  \\exp\\left\\{ -\\left(\\frac{x-u}{\\bar{\\lambda}}\\right) - \\exp\\left(-\\frac{x-u}{\\bar{\\lambda}}\\right)  \\right\\} \n\\\\\n&= \\text{LogGamma}(x|u,-\\bar{\\lambda},1) \n\\notag\n\\end{align}\nThis is the asymptotic extreme value distribution for variables of ``exponential type'', unbounded with finite moments~\\cite{Gumbel1958}.\nWith positive scale $\\bar{\\lambda}>0$, this is an extreme value distribution of the maximum, with negative scale $\\bar{\\lambda}<0$ ($\\lambda>0$) an extreme value distribution of the minimum. Note that the Gumbel is sometimes defined with the negative of the scale used here.\n\nNote that the term ``double exponential distribution'' can refer to either the Gumbel or Laplace~\\cite{Johnson1995} distributions.\nThe Gompertz distribution is a truncated  Gumbel distribution~\\cite{Johnson1995}. \n\n\\dist{Standard Gumbel} (Gumbel) distribution~\\cite{Gumbel1958}:\n\\begin{align}\n\\label{StdGumbel}\n\\text{StdGumbel}(x) \n=&\n  \\exp\\left\\{- x - e^{-x} \\right\\}  \\\\\n=& \\text{LogGamma}(x|0,-1,1)  \\notag\n\\end{align}\nThe Gumbel distribution with zero location and a unit scale.\n\n\n\n\n\n\n\n\n\\dist{BHP} (Bramwell-Holdsworth-Pinton) distribution~\\cite{Bramwell1998}:\n\\begin{align}\n\\label{BHP}\n\\text{BHP}(x|\\nu,\\lambda) \n&=\n\\frac{1}{\\Gamma(\\tfrac{\\pi}{2}) |\\lambda|}  \\exp\\left\\{ \\frac{\\pi}{2}\\left(\\frac{x-\\nu}{\\lambda}\\right) -  \\exp\\left(\\frac{x-\\nu}{\\lambda}\\right)  \\right\\} \n\\\\\n&= \\text{LogGamma}(x|\\nu,\\lambda,\\frac{\\pi}{2})  \n\\notag\n\\end{align}\nProposed as a model of rare fluctuations in turbulence and other correlated systems.\n\n\n\\subsection{Properties}\n\n\\begin{align*}\n\\text{support}\\quad & -\\infty \\leq x \\leq +\\infty & \\\\\n\\text{cdf} \\quad &  1- Q(\\alpha, e^{\\frac{x-\\nu}{\\lambda} }) & \\text{ for } \\lambda>0 \n\\\\ & Q(\\alpha, e^{\\frac{x-\\nu}{\\lambda} }) & \\text{ for } \\lambda<0 \n\\\\\n\\text{cgf} \\quad & \\nu t + \\ln \\frac{\\Gamma(\\alpha+\\lambda t)}{\\Gamma(\\alpha)}  & \\text{\\cite{Johnson1995}}\n\\\\\n\\text{mode} \\quad &  \\nu -\\lambda \\ln \\alpha\n\\\\\n\\text{mean} \\quad & \\nu+ \\lambda \\psi(\\alpha) & \n\\\\\n\\text{variance}  \\quad &   \\lambda^2 \\psi_1(\\alpha) & \n\\\\\n\\text{skew}  \\quad &  \\text{sgn}(\\lambda) \\psi_2(\\alpha) \/  \\psi_1(\\alpha)^{3\/2} \n\\\\\n\\text{kurtosis}  \\quad &     \\psi_3(\\alpha) \/  \\psi_1(\\alpha)^{2}\n \\\\\n\\text{entropy}  \\quad & \\ln \\Gamma(\\alpha) |\\lambda|  - \\alpha \\psi(\\alpha) + \\alpha & \n\\end{align*}\nHere, cdf is the cumulative distribution function, cgf the cumulant generating function, $\\ln \\text{E}[ \\exp( t X)]$, $\\text{sgn}(z)$ is the sign function, $\\psi_n(z)= \\tfrac{d^n}{dz^n} \\ln \\Gamma(z)$ is the polygamma function and  $\\psi(z)\\equiv\\psi_0(z)$ is the digamma function.\n\n\n\\section{Miscellaneous limits}\n\n\\label{SecMisc}\n \n\\dist{Log-normal} (Galton, Galton-McAlister, antilog-normal,  logarithmic-normal, logarithmico-normal, Cobb-Douglas, $\\Lambda$) distribution~\\cite{Galton1879, McAlister1879, Johnson1994}: \n \\begin{align}\n \\label{LogNormal}\n \\text{LogNormal}(x|a, \\vartheta,\\sigma) \n&= \\frac{\\vartheta}{\\sqrt{2\\pi \\sigma^2}} \\left(\\frac{x-a}{\\vartheta}\\right)^{-1} \\exp\\left\\{-\\frac{1}{2\\sigma^2} \\left( \\ln \\frac{x-a}{\\vartheta} \\right)^2 \\right\\}\n\\\\ \\notag & \n=  \\lim_{\\beta\\rightarrow0} \\text{Amoroso}(x| a, \\vartheta (\\beta\\sigma)^{2\/\\beta} ,1\/(\\beta\\sigma)^2, \\beta)\n\\end{align}\nThe log-normal distribution is a limiting form of the gamma family.  To see this, make the requisite substitutions, \n\\begin{align*}\n& \\text{Amoroso}(x| a, \\vartheta (\\beta\\sigma)^{2\/\\beta} ,1\/(\\beta\\sigma)^2, \\beta)  \\\\\n&\\propto\n \\left(\\frac{x-a}{\\vartheta}\\right)^{-1} \n\\exp\\left\\{\n\\frac{1}{\\beta\\sigma^2} \\ln \\frac{x-a}{\\vartheta}  \n- \\frac{1}{\\beta^2\\sigma^2} \\exp \\left[ \\beta \\ln \\frac{x-a}{\\vartheta} \\right]\n\\right\\}\n\\notag\n\\end{align*}\nand in the limit ${\\beta\\rightarrow0}$  expand the second exponential to second order in $\\beta$.\n\nWith $a=0$, $\\vartheta=1$, $\\sigma=1$  we obtain the  {\\bf standard log-normal} (Gibrat) distribution~\\cite{Gibrat1931}.\nThe {\\bf two-parameter lognormal} distribution ($a=0$) arises from the multiplicative version of the central limits theorem: When the sum of independent random variables limits to normal, the product of those random variables limits to log-normal.\nThe log-normal distribution maps to the normal distribution with the transformation $x\\mapsto \\exp(x)$.\n\\begin{align*}\n\\text{LogNormal}(a, \\vartheta, \\sigma) &\\sim \\exp\\Big(\\text{Normal}(\\ln \\vartheta,\\sigma)\\Big) +a\n\\end{align*}\n\n \n \n \\dist{Normal} (Gauss, Gaussian, bell curve, Laplace-Gauss, de Moivre, error, Laplace's second law of error, law of error) distribution~\\cite{Moivre1733,Johnson1994}:\n\\begin{align}\n\\label{Normal}\n\\text{Normal}(x|\\mu,\\sigma) \n&=\n\\frac{1}{\\sqrt{2 \\pi \\sigma^2}}  \\exp\\left\\{ - \\frac{( x-\\mu)^2}{2\\sigma^2} \\right\\}\n \\\\ \\notag\n&= \\lim_{\\alpha\\rightarrow\\infty} \\text{Amoroso}(x| \\mu- \\sigma\\sqrt{\\alpha}, \\sigma\/\\sqrt{\\alpha}, \\alpha, 1)\n \\\\ \\notag\n&= \\lim_{\\alpha\\rightarrow\\infty} \\text{LogGamma}(x| \\mu-\\sigma\\sqrt{\\alpha}\\ln\\alpha, \\sigma\\sqrt{\\alpha}, \\alpha) \n\\end{align}\nWith $\\mu=0$ and $\\sigma = 1\/ \\sqrt{2} h$ we obtain the {\\bf error function} distribution, and\nwith $\\mu=0$ and $\\sigma=1$ we obtain the {\\bf standard normal} ($\\Phi$, $z$, unit normal)  distribution. In the limit that $\\sigma\\rightarrow\\infty$ we obtain an unbounded {\\bf uniform} (flat distribution, and in the limit $\\sigma\\rightarrow0$ we obtain a {\\bf delta} (degenerate) distribution. \n\nThe normal distribution is a limit of the Amoroso~\\cite{Johnson1994} and log-gamma distributions~\\cite{Prentice1974}. For Amoroso, make the requisite substitutions,\n\\begin{align*}\n& \\text{Amoroso}(x|  \\mu- \\sigma\\sqrt{\\alpha}, \\sigma\/\\sqrt{\\alpha}, \\alpha,1)  \\\\\n&\\propto \\\n\\exp\\left\\{ -\\sqrt{\\alpha} \\frac{x-\\mu}{\\sigma} + (\\alpha-1) \\ln \\left( 1+ \\frac{1}{\\sqrt{\\alpha}} \\frac{x-\\mu}{\\sigma}  \\right) \\right\\}\n\\end{align*}\nand expand the logarithm as $\\ln(1+x) = x - \\tfrac{x^2}{2} + O(x^3)$.\n\n\n\n\n\n\n\\dist{Power-law} (Pearson type XI, fractal) distribution~\\cite{Pearson1916}: \n\\begin{align}\n\\label{PowerLaw}\n\\text{PowerLaw}(x|p) \n&\\propto   \\frac{1}{(x-a)^p}\n\\\\&=  \\lim_{\\alpha\\rightarrow\\infty} \\text{Amoroso}(a,\\theta,\\alpha,(1-p)\/\\alpha) \\notag\n\\end{align}\nImproper (unnormalizable) power law distributions are obtained as a limit of the gamma distribution family.\n If $p=0$ we obtain the {\\bf half-uniform} distribution over the positive numbers; if $p=1$ we obtain {\\bf Jeffreys} distribution~\\cite{Jeffreys1948}, used as an uninformative  prior in Bayesian probability~\\cite{Jaynes2003}. \n\n~\n\n~\n\n~\n\n\n\\paragraph*{Acknowledgments:}\nI am grateful to  David Sivak, Edward E. Ayoub and Francis J. O'Brien for spotting various typos  and errors,  and, as always, to Avery Brooks for many insightful observations. In curating this collection of distributions, I have benefited mightily from Johnson, Kotz, and Balakrishnan's monumental compendiums~\\cite{Johnson1994, Johnson1995}, Eric Weisstein's MathWorld, and the myriad pseudo-anonymous contributors to Wikipedia.\nThis project was supported, in part, by Award Number R01GM092992 from the National Institute Of General Medical Sciences.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nRecently four groups \\cite{SDSS}, using data from the\nSloan Digital Sky Survey\\footnote{http:\/\/www.sdss.org\/}, published evidence\nfor features in the matter power spectrum on scales of $100\\,$Mpc.\nThese features, long predicted, hold the promise of another route to\nunderstanding the expansion history of the universe and the influence\nof dark energy \\cite{EisReview}.\n\nOscillations in the baryon-photon fluid at $z\\sim 10^3$ lead to a\nseries of almost harmonic peaks in the matter power spectrum, or a\nbump in the correlation function, arising from a preferred scale in\nthe universe: the sound horizon.\n(A description of the physics leading to the features can be found in\n\\cite{EHSS} or Appendix A of \\cite{MeiWhiPea}; a comparison of the Fourier\nand configuration space pictures is presented in \\cite{ESW06}.)\nIt was pointed out in Refs.~\\cite{CooHuHutJof,Eis03} that this scale could\nbe used as a standard ruler to constrain the distance-redshift relation, the\nexpansion of the universe and dark energy.  Numerous authors\n\\cite{Fisher} have now observed that a high-$z$ galaxy survey\\footnote{It\nis even possible that such oscillations could be seen in the Ly-$\\alpha$\nforest \\cite{Davis} or in very large cluster surveys \\cite{Ang}.} covering\nupwards of several hundred square degrees could place interesting\nconstraints on dark energy.\nKey to realizing this is the ability to accurately predict the physical scale\nat which the oscillations appear in the power spectrum plus the means to\nlocalize those primordial features in observations of galaxy spectra which\nnecessarily involve galaxy bias, non-linear evolution and redshift space\ndistortions.\nThe former problem seems well in hand \\cite{SSWZ,EisWhi}.\nPreliminary investigations of the latter problem were presented in\nRefs.~\\cite{Ang,Millenium,PMBaryon,SeoEis05,GuzBer}.\nWe continue these investigations in this paper using a large set of\nhigh resolution N-body simulations.\n\nThe outline of the paper is as follows: \\S\\ref{sec:sim} describes\nour N-body simulations and the construction of the mock galaxy catalogs\nusing halo model methods.  It also presents some basic properties of the\ngalaxy clustering.  \\S\\ref{sec:models} introduces the models for\nthe 2-point function that we consider in this paper.  \\S\\ref{sec:dxi}\nintroduces a new configuration space band-power statistic which we believe\nis useful for BAO work while \\S\\ref{sec:methodology} introduces our fitting\nmethodology.  Our primary results are described in \\S\\ref{sec:results}.\nSome preliminary investigations of the reconstruction method of \\cite{ESSS06}\nare described in \\S\\ref{sec:reconstruction} and our conclusions are\npresented in \\S\\ref{sec:conclusions}.\n\n\\section{Simulations} \\label{sec:sim}\n\nWe need an ``event generator'' which can be used to develop methods for\ngoing from observations of galaxies to cosmology.  Ideally this tool\nwould encode many of the complications we expect in real observations\nbut be based on a known cosmology.  To this end we use a series of\nsimulations of a $\\Lambda$CDM cosmology ($\\Omega_M=0.3=1-\\Omega_\\Lambda$,\n$\\Omega_B=0.046$, $h=0.7$, $n=1$ and $\\sigma_8=0.9$).  The linear theory\nmass spectrum was computed by evolution of the coupled Einstein, fluid\nand Boltzmann equations using the code described in \\cite{Boltz}.  (A\ncomparison of this code to CMBfast \\cite{CMBfast} is given in \\cite{SSWZ}.)\nFor this model the sound horizon\\footnote{We caution the reader that the\ndefinition of the sound horizon, like that of the epoch of last scattering,\nis one of convention.  Unfortunately several conventions exist in the\nliterature.  Along with fitting formulae of limited accuracy this makes it\ndifficult to compare quoted numbers at the percent level.  We define $s$ as\nthe integral of the sound speed up to the redshift where $\\tau=1$, excluding\nthe contribution from $z<z_{\\rm reion}$.}, $s=143\\,$Mpc or $100.2\\,h^{-1}$Mpc.\n\nFor tests in which we need large numbers of runs (i.e.~computing covariance\nmatrices) we use mock catalogs based on Gaussian density fields.\nWe first generate a Gaussian density field with the desired power spectrum\n(in our case the linear theory spectrum) on a $512^3$ grid in a box of side\n$1.1\\,h^{-1}$Gpc.  A `galaxy' is then placed at random in the cell with a\nprobability proportional to $\\exp[3\\nu-0.3\\nu^2]$ for $\\nu\\ge 0$ and\n$\\exp[3\\nu]$ for $\\nu<0$, where $\\nu=\\delta\/\\sigma$ is the peak height.\nSimilar techniques are used to construct mock galaxy redshift surveys in\n\\cite{CHWF}.  The non-linear mapping of the Gaussian density field mocks up\nthe action of gravity, inducing extra power on small scales and correlating\ndifferent scales in Fourier space.  The resulting `galaxy' field has a power\nspectrum with roughly constant bias on large scales and excess power on small\nscales, though the form does not match in detail the more realistic catalogs\nproduced with the halos found in N-body simulations.\n\nA more realistic catalog can be produced using N-body simulations which have\nenough spatial and force resolution to resolve the halos hosting the galaxies\nof interest for BAO surveys.\nThe basis for these calculations is a sequence of high force resolution\nN-body simulations in a periodic, cubical box of side $1.1\\,h^{-1}$Gpc.\nThese simulations were carried out with the HOT \\cite{HOT} and TreePM\n\\cite{TreePM} codes.\nIn all we ran 3 simulations, with different randomly generated Gaussian\ninitial conditions, which evolved $1024^3$ particles of mass\n$10^{11}\\,h^{-1}M_\\odot$ from $z=34$ to $z=1$.\nThe Plummer softening was $35\\,h^{-1}$kpc (comoving).\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{5.5in}{!}{\\includegraphics{massfn.ps}}\n\\end{center}\n\\caption{(Upper) The cumulative mass function of halos from the 3 simulations\nat $z=1$.  We define $M$ as $1.03\\times$ the sum of the masses of the particles\nin the FoF group.  The last point plotted in each case is the mass of the\nlargest halo in the box.  The solid line shows the fit of\nRef.~\\protect\\cite{ST} while the dotted line shows the results with $a=0.8$\nand $p=0.25$ as discussed in the text.  The horizontal dashed lines show the\nrange of number densities of most interest for this work.  (Lower) The ratio\nof the N-body results to the fit with $a=0.8$ and $p=0.25$.}\n\\label{fig:massfn}\n\\end{figure}\n\nThe simulations were chosen to be the largest we could practically run\nseveral realizations of, while retaining sufficient force resolution to\nresolve the halos likely to host the galaxies of interest.  This allowed\nus to simulate a volume comparable to that of proposed future surveys\n\\cite{WFMOS} at $z=1$.\nWhile observational campaigns could also study baryon oscillations at\nhigher redshift (e.g.~$z=3$) going to earlier times becomes increasingly\ndifficult for simulations.  The volume for a given survey area increases and\nthe characteristic mass scale of halos decreases to earlier times, making the\nrequired dynamic range infeasible for direct simulation at present.  Thus\nwe will focus our attention here on $z=1$.\n\nFor each output we generate a catalog of halos using the Friends-of-Friends\nalgorithm \\cite{FoF} with a linking length of $b=0.175$ in units of the\nmean inter-particle spacing.\nThis procedure partitions the particles into equivalence classes, by linking\ntogether all particle pairs separated by less than a distance $b$.\nThe groups correspond roughly to all particles above a density of about\n$3\/(2\\pi b^3)\\simeq 90$ times the background density and we keep all\ngroups with more than 16 particles.  Increasing the ``friends-of-friends''\nmass of the groups by a few percent gives a good match to the\nanalytic mass function of Ref.~\\cite{ST}.  However we find that even with\nthis increase the $z=1$ results are better fit if we change the parameters\nin the analytic mass function ($p$ which controls the low mass slope of the\nmass function and $a$ which controls the exponential suppression at high mass)\nfrom $a=0.707$ and $p=0.3$ to $a=0.8$ and $p=0.25$.\nOver the range $2\\times 10^{12}\\le M\\le 2\\times 10^{14}\\,h^{-1}M_\\odot$ the\nresulting fit is good to $5\\%$ in number density (see Fig.~\\ref{fig:massfn}).\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccc|cccc}\n\\multicolumn{4}{c|}{$\\bar{n}=10^{-3.0}\\,h^3\\,{\\rm Mpc}^{-3}$} &\n\\multicolumn{4}{c}{$\\bar{n}=10^{-3.5}\\,h^3\\,{\\rm Mpc}^{-3}$} \\\\\n$\\log_{10}M_{\\rm min}$ & $\\log_{10}M_1$ &\n  $\\langle b\\rangle$   & $\\log_{10}\\langle M\\rangle$ &\n$\\log_{10}M_{\\rm min}$ & $\\log_{10}M_1$ &\n  $\\langle b\\rangle$   & $\\log_{10}\\langle M\\rangle$ \\\\\n  12.8                 & 13.0           & 2.4 & 13.6 &\n  13.3                 & 13.0           & 3.1 & 13.9 \\\\\n  12.7                 & 13.5           & 1.9 & 13.3 &\n  13.1                 & 13.5           & 2.5 & 13.7 \\\\\n  12.6                 & 14.0           & 1.8 & 13.1 &\n  13.0                 & 14.0           & 2.3 & 13.5 \\\\\n  12.6                 & 14.5           & 1.7 & 13.1 &\n  13.0                 & 14.5           & 2.1 & 13.4\n\\end{tabular}\n\\end{center}\n\\caption{The HOD parameters of Eq.~\\protect\\ref{eqn:hod} for some of our\ncatalogs at $z=1$.  Masses are in units of $h^{-1}M_\\odot$.  Within each\nset the catalogs have a fixed space density, $\\bar{n}$.  The large-scale\nbias, $\\langle b\\rangle$, and the galaxy weighted mean halo mass,\n$\\langle M\\rangle$, are also listed.}\n\\label{tab:hod10}\n\\end{table}\n\nWe make mock catalogs using two different procedures.  First we apply\na simple threshold mass to the group catalogs, taking our tracers to be\nthe central galaxies of halos above the mass threshold.\nTo allow the inclusion of satellites we choose a mean occupancy of halos:\n$N(M)\\equiv\\left\\langle N_{\\rm gal}(M_{\\rm halo})\\right\\rangle$.\nEach halo either hosts a central galaxy or does not.\nFor each halo we define a galaxy to live at the minimum of the halo\npotential with probability $p={\\rm min}[1,N(M)]$.\nThe central galaxy inherits the velocity of the halo, which we take to\nbe the average velocity of the halo particles weighted by the square of the\npotential.  This weighting emphasizes particles near the halo center and\nallows the central galaxy to move with respect to the center-of-mass (com)\nof the halo, but the difference between the com velocity and the central\ngalaxy velocity is typically only a few tens of km$\\,{\\rm s}^{-1}$.\nFollowing Ref.~\\cite{KBWKGAP}, if $N(M)>1$ the mean number of satellites,\n$N_{\\rm sat}=N(M)-1$, is computed for the halo and a Poisson random number,\n$n_{\\rm sat}$, drawn.\nThen $n_{\\rm sat}$ dark matter particles, chosen at random, are anointed as\ngalaxies.  Our fiducial model thus has the satellite galaxies tracing the\ndark matter in the halo.\nThe galaxy velocity is taken to be the particle velocity, thus the satellites\nhave no velocity bias.  However since the central galaxy is nearly at rest\nwith respect to the halo, the population as a whole has a different velocity\nfield than the dark matter.\nThe characteristic mass, $M_\\star$, for our models at $z=1$ is\n$2\\times 10^{11}\\,h^{-1}M_\\odot$.  As all of our tracer galaxies live in\nhalos more massive than $M_\\star$ they have biases greater than 1.\n\nWe follow \\cite{PMBaryon} and choose a simple two-parameter form for $N(M)$:\n\\begin{equation}\n  N(M) = \\Theta(M-M_{\\rm min})\\ \\frac{\\left(M-M_{\\rm min}\\right)+M_1}{M_1}\n\\label{eqn:hod}\n\\end{equation}\nwhere $\\Theta(x)$ is the Heaviside step function.\nIf we take $M_1\\to\\infty$ our catalog reduces to the catalog of halos more\nmassive than $M_{\\rm min}$.  By holding $\\bar{n}$ fixed\nwe can specify a 1-parameter sequence of models with varying $M_{\\rm min}$,\nlarge-scale bias, $\\langle b\\rangle$, or galaxy weighted mean halo mass\n$\\langle M\\rangle$ (see Table \\ref{tab:hod10}).\nIn comparing the models with different HODs it is important to remember that\nwe hold $\\bar{n}$ fixed within each sequence, so variations in mean halo mass,\nsatellite fraction, bias etc are highly correlated.\nTo test the dependence on the slope of the satellite contribution we\nalso ran some models where $n_{\\rm sat}\\propto M^{1\/2}$.  The parameters of\nthese models are listed in Table \\ref{tab:hod05}.\nBoth theoretical \\cite{ConWecKra} and observational \\cite{YanMadWhi}\nresults suggest that at higher redshift $M_{\\rm min}\\approx M_1$.  These\nmodels have the larger biases, mean halo masses and satellite fractions.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccc|cccc}\n\\multicolumn{4}{c|}{$\\bar{n}=10^{-3.0}\\,h^3\\,{\\rm Mpc}^{-3}$} &\n\\multicolumn{4}{c}{$\\bar{n}=10^{-3.5}\\,h^3\\,{\\rm Mpc}^{-3}$} \\\\\n$\\log_{10}M_{\\rm min}$ & $\\log_{10}M_1$ &\n  $\\langle b\\rangle$   & $\\log_{10}\\langle M\\rangle$ &\n$\\log_{10}M_{\\rm min}$ & $\\log_{10}M_1$ &\n  $\\langle b\\rangle$   & $\\log_{10}\\langle M\\rangle$ \\\\\n  12.8                 & 13.0           & 2.0 & 13.3 &\n  13.2                 & 13.0           & 2.6 & 13.7 \\\\\n  12.7                 & 13.5           & 1.9 & 13.2 &\n  13.1                 & 13.5           & 2.4 & 13.6 \\\\\n  12.6                 & 14.0           & 1.8 & 13.2 &\n  13.0                 & 14.0           & 2.3 & 13.5 \\\\\n  12.6                 & 14.5           & 1.8 & 13.1 &\n  13.0                 & 14.5           & 2.2 & 13.4\n\\end{tabular}\n\\end{center}\n\\caption{The HOD parameters used for $z=1$ catalogs with\n$n_{\\rm sat}\\propto M^{1\/2}$.  Units and labels are as\nin Table \\protect\\ref{tab:hod10}.}\n\\label{tab:hod05}\n\\end{table}\n\nWhile the galaxy models above are not prescriptive, or likely even\nclose to ``right'', they are physically well motivated, easy to adjust and\nlead to galaxy catalogs with non-linear, scale-dependent, stochastic\nbiasing and redshift space distortions -- many of the complications we\nwill face in observations of the universe.\n\nThe statistics of counts in cubical cells allow us to infer the\nstochasticity of the bias and the degree to which the 1-point\ndistribution of the galaxies is Poisson.  We find that the\ngalaxy-mass cross-correlation coefficient \\cite{DekLah}, $r$, rises from\n$0.6-0.7$ (depending on sample) on scales of $1\\,h^{-1}$Mpc to $>95\\%$ on\nscales larger than $10\\,h^{-1}$Mpc.  The variance of the counts divided\nby their mean, which would be unity for a Poisson distribution, exceeds\none on scales $1-30\\,h^{-1}$Mpc with the largest value ($\\approx 20$)\non the largest scales.  This excess is easily understood as extra power\ncoming from large-scale clustering of the galaxies.  On Mpc scales the\nvalue is very close to Poisson for the less biased galaxies and close to\n$2$ for the more biased samples.\n\n\\section{BAO models} \\label{sec:models}\n\nThe cosmological signal that interests us is primarily contained in the\ntwo-point statistics of the galaxy density field, and we shall concentrate\non these statistics henceforth.  We begin by considering measurements in\nreal space, such as would be relevant to photometric or 2D surveys,\nand then include redshift space distortions which are relevant for\nspectroscopic surveys.  \n\n\\subsection{Real space}\n\nThere are now several models in the literature relating the (non-linear)\ngalaxy power spectrum to the (linear) dark matter power spectrum.\nWe have critically compared these to the power spectra and correlation\nfunctions measured in our mock surveys.\nTo our knowledge the performance of these models in matching the shape of\nthe power spectra and correlation function has not been compared to mock\ncatalogs produced with a wide range of HOD schemes.  In particular we find\nthat the correlation function is a very discriminating statistic, because it\nis sensitive to translinear scales.\nAs we discuss in \\S\\ref{sec:results} some of the models do not match the\nclustering of our mock galaxy samples, particularly for highly biased or rare\nsamples.\n\nWe now describe the five models we've investigated in this paper.\nThe simplest is the linear bias model, which is often motivated by\narguments like those presented in \\cite{SchWei}.\nThe linear bias model asserts that\n\\begin{equation}\n  \\Delta^2_{\\rm gal}(k)=b^2 \\Delta^2_{\\rm lin}(\\alpha k)\n\\label{eqn:linbias}\n\\end{equation} \nwhere $\\Delta^2(k)$ denotes the dimensionless power spectrum, or variance\nper $\\ln k$:\n\\begin{equation}\n  \\Delta^2(k)\\equiv \\frac{k^3P(k)}{2\\pi^2} \\qquad .\n\\end{equation}\nThe parameter $b$ in Eq.~\\ref{eqn:linbias} is the large scale galaxy bias\nand $\\Delta^2_{\\rm lin}(k)$ is the linear dark matter power spectrum.\nIn this study we have introduced the parameter $\\alpha$, which scales the\nwave number in Eqs.~\\ref{eqn:linbias}, \\ref{eqn:lrg}, \\ref{eqn:hmi}\nand \\ref{eqn:esw}, to parameterize small changes in the cosmology that\nresult in a stretch in the baryon signature.\nWe introduce this parameter in order to study potential degeneracies between\nthe other model parameters, which depend on the HOD, and the inferred\ncosmology.\nWhen $\\alpha\\ne 1$ the inferred length scale differs from the true length\nscale, leading to an incorrect estimate of the sound horizon and hence the\nconstraints on dark energy.  We will use biases and errors on $\\alpha$ as\nan indicator of how well the sound horizon can be measured.\nTo translate the error in $\\alpha$ into an error on dark energy parameters\nwe need to make further assumptions.  As a rough rule of thumb: if we assume\na constant equation of state, $w$, for the dark energy the fractional error\nin $w$ is five times that in $\\alpha$.\n \nIt has been observed several times in the literature \n\\cite{PMBaryon,SeoEis05,Coo04} that the large-scale bias may not be constant\nat the $1\\%$ level.  Halo bias \\cite{PeaksBias} will have a small scale\ndependence even for weakly non-linear scales.\nThe distribution of galaxies within dark matter halos and halo exclusion\neffects also lead to small changes in large-scale power.\nThe shifting of galaxy positions on $\\sim 10\\,h^{-1}$Mpc scales leads to\na smearing of the amplitude of the oscillations in the power spectrum.\nSeveral attempts have been made to model these effects.\n\nIn \\cite{BlaGla03} a method was introduced to empirically\nfit the scale dependence, using the form \n\\begin{equation}\n  \\Delta^2(k)= \\Delta^2_{\\rm ref}( k)\n  \\left[1+Ak\\,\\exp\\left\\{-\\left(\\frac{ k}{k_s}\\right)^{1.4}\\right\\}\n  \\sin\\left( \\frac{2 \\pi k}{k_A} \\right) \\right]\n\\label{eqn:blagla}\n\\end{equation}\nHere $A$ and $k_A$ are fit parameters in the decaying sinusoid used to\ncharacterize the baryon oscillations, $k_s\\equiv 0.1\\,h\\,{\\rm Mpc}^{-1}$\nis a constant and $\\Delta^2_{\\rm ref}(k)$ is a reference spectrum including\nthe effects of Silk damping but excluding the oscillations.  The reference\nspectrum is from the fitting formula in \\cite{EisHu99}.\nSince we are fitting non-linear power spectra we additionally allow a linear\nramp in power when doing our fits.  This approximates the broad-band power\nremoval suggested by \\cite{BlaGla03} without correlating the errors.\nThe fits do prefer a positive slope to this extra factor.\nIn Eq.~(\\ref{eqn:blagla}) a shift in $k_A$ corresponds to a shift in the\nsound horizon, so $k_A$ replaces the $\\alpha$ in our previous expression.\nWhile the true spectrum cannot be accurately fit by Eq.~(\\ref{eqn:blagla}),\nif we concentrate around the second peak $k_A\\simeq 0.058\\,h{\\rm Mpc}^{-1}$\nprovides a good fit to the peak position for our input model.\nAn alternative definition, used by \\cite{BlaGla03}, is\n$k_A=2\\pi\/s\\simeq 0.063\\,h{\\rm Mpc}^{-1}$ a difference of about 10\\%.\nIf we use the fitting function, Eq.~(26), of \\cite{EisHu98} for $s$ instead\nwe find $k_A\\simeq 0.060\\,h{\\rm Mpc}^{-1}$; midway between the former two\nvalues.\nThe latter approximation comes closest to our best fit $k_A$ (see below) so\nwe shall use that -- but we note again the uncertainty in quoted values of\n$s$ in the literature.\nIn our fitting we shall assume that $\\Delta_{\\rm ref}$ is known, and use the\ncorrect cosmological parameters for our runs.\n\nA recent analysis of the clustering of Luminous Red Galaxies (LRGs) in the\nSloan Digital Sky Survey (SDSS) instead used the fitting function\n\\cite{Pad06,Cole05}\n\\begin{equation}\n  \\Delta^2(k)=b^2\\,\\Delta_{\\rm lin}^2(k) \\frac{1+Qk^2}{1+ak}\n  \\qquad {\\rm for\\\/} \\ k<0.5\\,h\\,{\\rm Mpc}^{-1}\n\\label{eqn:lrg}\n\\end{equation}\nIn this description, the parameter $Q\\sim 10$ governs the scale dependence of\nthe bias and $a=1.7\\,h^{-1}$Mpc is a constant ($a$ becomes $1.4\\,h^{-1}$Mpc\nin redshift space).  To test for shifts in the sound\nhorizon we again replace $k$ with $\\alpha k$.  Note that for small $k$\nthis model looks like the quadratic, multiplicative bias model used in\n\\cite{PMBaryon}.\n\nIn \\cite{ToyModel}, motivated by analytic arguments using the halo model,\nthe scale dependence was modeled by adding a term to the expression in\nEq.~\\ref{eqn:linbias} proportional to $k^3$.  \nIn \\cite{SeoEis05} a similar treatment was used for slightly different\nreasons, involving $k^3$ times a quadratic in $k$.\nFor $k\\ll 1$ the leading order term will dominate and the two expressions\nare similar.  We consider\n\\begin{equation}\n  \\Delta_{\\rm gal}^2(k) = b^2 \\Delta_{\\rm lin}^2(\\alpha k)\n   e^{-(\\alpha k\/k_2)^2} + \\left(\\alpha k\/ k_1\\right)^3 \\qquad .\n\\label{eqn:hmi}\n\\end{equation}\nThe parameters $b$ and $k_1$ are the galaxy weighted large-scale halo bias\nand the amplitude of the 1-halo term, and the parameter $k_2$ governs a\nsuppression due to halo profiles and exclusion.\nWe introduce $k_2$ because very little clustering power on small scales\nis due to galaxies in separate halos.  It is most necessary for those\nmodels with low $M_1$, i.e.~many satellite galaxies, the strongest 1-halo\nterms, the largest mean halo mass, but the shape of the transition between\nthe 2- and 1-halo terms is not well constrained.  We will consider a\ndifferent transition below.\nSince the baryon oscillation signal is present in $\\Delta_{\\rm lin}^2(k)$, it\nis clear that the one halo term, which is a description of the impact of\nnon-linear physics, decreases the contrast of the oscillations.\nBecause the parameters $b$, $k_1$,and $k_2$ all depend on the HOD, the\ncontrast of the oscillations will depend somewhat on the type of galaxies\nbeing selected in the BAO survey, and on the mean redshift of the survey.\n\nAnother way of viewing the effects of non-linearities and \ngalaxy bias is found in \\cite{ESW06}.\nThat analysis seeks to describe the gradual erasure of the acoustic peak\nsignature by considering the\ndistribution of Lagrangian displacements of galaxies.\nThe authors of \\cite{ESW06} showed that to simultaneously model the smearing\ndue to galaxy displacement, and also the correct level of small scale power,\nit is useful to add back the broad-band power that is filtered out by the\nexponential suppression in Eq.~\\ref{eqn:hmi}.\nThe fitting form from \\cite{EisHu99} is again used for this purpose leading to\n\\begin{equation}\n  \\Delta_{\\rm gal}^2(k) = b^2 \\Delta_{\\rm lin}^2(\\alpha k)\n  e^{-(\\alpha k\/k_2)^2} + \\left(\\alpha k\/k_1\\right)^3 +\n  \\left( 1-e^{-(\\alpha k\/k_2)^2} \\right) b^2 \\Delta^2_{\\rm ref}(\\alpha k)\n\\label{eqn:esw}\n\\end{equation}\nThe role of $k_2$ in this form differs from its role in Eq. \\ref{eqn:hmi}\nin that here it controls the erosion of the baryon wiggles\ndue to galactic motions on $\\sim 10\\,h^{-1}$Mpc scales, while \npreserving the overall shape of the two halo portion of the\npower spectrum in the trans-linear regime.\n\n\\subsection{Redshift space}\n\nLittle work has been done on extending these models to redshift space.\nIn redshift space the power is enhanced, by a $z$-dependent factor, on\nlarge scales due to supercluster infall \\cite{Kaiser} and suppressed on\nsmall scales due to virial motions within halos\\footnote{This statement\nassumes a $k$-space picture.  In configuration space, on large scales,\noverdensities are the sites of convergent flows, so redshift space\ndistortions `sharpen' structure.  The correlation function thus falls\noff more quickly along the line-of-sight than transverse to it.}.\nTo include the dependence on the angle with respect to the line of sight,\n$\\mu\\equiv\\cos\\theta=\\hat{k}\\cdot\\hat{r}$, we decompose $\\Delta^2(\\vec{k})$\ninto Legendre polynomials, $P_\\ell(\\mu)$, as\n\\begin{equation}\n  \\Delta^2(k,\\mu) \\equiv \\sum_{\\ell=0}^{\\infty} \\Delta_\\ell^2(k)P_\\ell(\\mu)\n\\label{eqn:deltal}\n\\end{equation}\nso\n\\begin{equation}\n  \\Delta_\\ell^2(k) = \\frac{2\\ell+1}{2} \\int_{-1}^{+1}d\\mu\\ \n  \\Delta^2(k,\\mu)P_\\ell(\\mu)\n\\end{equation}\nSymmetry along the line-of-sight implies that the coefficients for odd $\\ell$\nmodes vanish.\n\nOn very large scales $(k\\to 0)$ supercluster infall modifies the power\nspectrum as \\cite{Kaiser,HamiltonReview}\n\\begin{equation}\n  \\Delta^2_{\\rm red}(k,\\mu) = \\Delta^2_{\\rm real}(k)\n  \\left(1+\\beta\\mu^2\\right)^2,\n\\end{equation}\nwhere $\\mu\\equiv\\hat{k}\\cdot\\hat{r}$ and\n$\\beta\\equiv f(\\Omega)\/b\\simeq \\Omega^{0.6}\/b$ assuming a scale-independent\nbias.  Here $f$ is the dimensionless linear growth rate of the growing mode\nin linear perturbation theory which can be approximated as $\\Omega^{0.6}$\n\\cite{Pee80}.\nThe coefficients of the first two multipole moments are thus\n\\begin{equation}\n  \\Delta^2_0(k) = \\left(1 + \\frac{2}{3}\\beta +\n  \\frac{1}{5}\\beta^2\\right)\\Delta_{\\rm real}^2(k)\n\\end{equation}\nand\n\\begin{equation}\n  \\Delta^2_2(k) = \\left( \\frac{4}{3}\\beta +\n  \\frac{4}{7}\\beta^2\\right)\\Delta_{\\rm real}^2(k).\n\\end{equation}\nTo capture the smaller-scale angular dependence one often adds a high-$k$\ncutoff; popular choices include Lorentzian, Gaussian or exponential, e.g.\n\\begin{equation}\n  \\Delta_{\\rm red}^2(k,\\mu)  = \\Delta_{\\rm real}^2(k)\n  \\left(1 + \\beta \\mu^2\\right)^2 e^{-k\\sigma|\\mu|}\n\\end{equation}\nWe will refer to these modifications collectively as ``streaming models''\n\\cite{HamiltonReview}.\nIn general $\\beta$ and $\\sigma$ are regarded simply as parameters to be fit\nto the data.\nThe analytic expressions for the resulting multipole moments are\nstraightforward but lengthy and will not be reproduced here.\n\nA more empirical model for the angular dependence is explored in \\cite{HatCol},\nwho found the quadrupole-to-monopole ratio in N-body simulations can be fit by\n\\begin{equation}\n  \\frac{\\Delta^2_2}{\\Delta^2_0} =\n  \\frac{\\frac{4}{3}\\beta+\\frac{4}{7}\\beta^2}\n       {1+\\frac{2}{3}\\beta+\\frac{1}{5}\\beta^2}\n  \\left[1-\\left(\\frac{k}{k_{\\rm rs}}\\right)^{1.22}\\right],\n\\label{eqn:hatcol}\n\\end{equation}\nwhere $k_{\\rm rs}$ is a free parameter analogous to the $\\sigma$ parameter\nin streaming models.  Though the authors of \\cite{HatCol} do not discuss\nthe shape of the small-scale downturn in $\\Delta_\\ell^2(k)$, an additional\nparameter needs to be introduced if we are to predict the run of\n$\\Delta_\\ell^2(k)$ with $k$.  Further discussion of the interplay between\nthe large-scale enhancement and small-scale suppression can be found in\n\\cite{SheDia01,HaloRed}.\n\nIn \\S\\ref{sec:redresult}, we will investigate the accuracy with which each\nthese forms reproduces the large- and intermediate-scale\n($k\\le 0.2\\,h\\,{\\rm Mpc}^{-1}$)\nangular dependence of the power spectrum in redshift space.\nOur approach also allows us to examine the effects of the HOD on the\nredshift-space distortions, with the hope of using this additional\ninformation to reduce degeneracies between cosmology and galaxy physics.\nWe do not consider in this paper how modeling of the anisotropic clustering\nallows us to constrain the line-of-sight and transverse distance scales\nseparately.\n\n\\section{A configuration space band-power estimator} \\label{sec:dxi}\n \nFor each of the forms in Eqs.~\\ref{eqn:linbias}-\\ref{eqn:esw}\nthere is a corresponding model for the correlation function $\\xi(r)$;\nthe probability, in excess of random, of finding a pair of tracers at\nseparation $r$.\nThe correlation function is related to the dimensionless power spectrum as\n\\begin{equation}\n  \\xi(r) =\\int \\frac{dk}{k}\\ \\Delta^2(k)\\ j_0(kr)\n  \\simeq  \\int \\frac{dk}{k}\\ \\Delta^2(k)\n  \\left[ 1 - \\frac{(kr)^2}{6} + \\cdots \\right]\n\\label{eqn:xi_wt}\n\\end{equation}\nwhere $j_0$ is the zeroth spherical Bessel function and the series expansion\nis valid for $kr\\ll 1$.\nMost of the scale dependence in the galaxy bias can be traced to the fact that\ngalaxies and dark matter transition from one to two halo dominance\nat disparate scales \\cite{ToyModel}.\nIn Fourier space this is spread over a range of $k$, but in configuration\nspace the 1-halo term is confined to scales much smaller than the scales of\ninterest to us.  Beyond $2-3\\,h^{-1}$Mpc more than 99.9\\% of the contribution\nto $\\xi(r)$ comes from the 2-halo term for all of our models.\nFor this reason we expect less scale dependence in the bias in configuration\nspace than in Fourier space (see also \\cite{GuzBer}), as shown in\nFig.~\\ref{fig:rbias}.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{5.5in}{!}{\\includegraphics{match.ps}}\n\\end{center}\n\\caption{An estimate of the scale-dependence of the bias in configuration\nspace.  The different symbols are $\\Delta\\xi$ (see text) for different HOD\nmodels (from Table~\\protect\\ref{tab:hod10}) each divided by a constant bias to\nmatch near $100\\,h^{-1}$Mpc.  The degree to which the shapes match indicates\nhow well each can be considered a constant times the dark matter correlation\nfunction.  The lower panel shows the residuals from one of the models, taken\nas a template.}\n\\label{fig:rbias}\n\\end{figure}\n\nFormally the correlation function, $\\xi(r)$, and the power spectrum, $P(k)$,\nare a Fourier transform pair.\nHowever the simulation volume is a periodic cube and our signal has support\nonly for $k$-modes which are integer multiples of the fundamental mode in\neach dimension.\nBecause of this restriction the correlation function computed in the box\ndiffers from the true continuum correlation function on scales approaching\nthe box size.  The modulation in power is non-trivial even on $100\\,$Mpc\nscales where we would like to work.\nFortunately the box is large enough to contain the modes of interest for\nbaryon oscillations and the difficulty is purely a technical issue.  We\nchoose to proceed by computing a quantity containing the same information\nas $\\xi(r)$ but which is less sensitive to the low-$k$ modes.\nSpecifically we compute\n\\begin{equation}\n  \\Delta\\xi(r) \\equiv \\bar{\\xi}(<r) - \\xi(r) =\n  \\frac{3}{r^3}\\int_0^r x^2\\,dx\\ \\xi(x) - \\xi(r)\n\\end{equation}\nfor which\n\\begin{equation}\n  \\Delta\\xi(r) = \\int \\frac{dk}{k}\\ \\Delta^2(k)\\ j_2(kr)\n  \\simeq  \\int \\frac{dk}{k}\\ \\Delta^2(k)\n  \\left[ \\frac{(kr)^2}{15} - \\frac{(kr)^4}{210} + \\cdots \\right]\n  \\label{eqn:dxi_wt}\n\\end{equation}\nwhere $j_2$ is the spherical Bessel function of order $2$.\nWe have effectively formed a band-pass filter in $k$-space; the slowly\nvarying, long-wavelength contribution is significantly down-weighted while\nretaining the sensitivity to $100\\,$Mpc scales.\nOnly 3\\% of the weight in the window function comes from scales below $kr=1$,\nso if we include the fact that $\\Delta^2(k)$ falls steeply as $k\\to 0$ we see\nthat modes near the fundamental contribute little to $\\Delta\\xi(r)$.\nWe believe that this configuration space band-power measurement is of\nconsiderable value in analyzing the acoustic oscillation signal.\nComputing $\\Delta\\xi(r)$ is straightforward once $\\xi(r)$ has been computed,\neither in simulations or from galaxy survey volumes.  In surveys of the sky,\nthe correlation function is computed by assuming a mean density given by the\nratio of total object number to the survey volume.\nThis necessarily drives the observed correlation function to zero at survey\nsized scales; a difficulty referred to as the integral constraint.\nThe quantity $\\Delta\\xi(r)$ is inherently less sensitive to the integral\nproblem, and is thus observationally useful apart from comparison to theory\nvia simulations.  A comparison of the relative robustness of $\\xi(r)$ and\n$\\Delta\\xi(r)$ is presented in \\S\\ref{sec:results}.\n\nAs an aside we note that further low-$k$ suppression can be obtained from a\nlinear combination of $\\xi(r)$, $\\bar{\\xi}(<r)$ and $\\bar{\\bar{\\xi}}(<r)$ with\n\\begin{equation}\n  \\bar{\\bar{\\xi}}(<r) \\equiv \\frac{5}{r^5}\\int_0^r x^4\\,dx\\ \\bar{\\xi}(<x)\n  = \\frac{5}{2}\\left[ \\bar{\\xi}(<r) - \\frac{3}{r^5}\\int_0^r x^4\\,dx\\ \\xi(x)\n  \\right]\n\\label{eqn:ddxi}\n\\end{equation}\nfor which the window function is $j_4(x)$, going as $x^4$ for $x\\ll 1$.\nThe generalization to higher orders is straightforward but will not be\nneeded here.\n\nWe do not expect any of the models of $\\Delta^2(k)$ to be accurate on small\nscales.  Since $\\bar{\\xi}(<r)$ is a cumulant, using the quantity\n$\\Delta\\xi(r)$ in place of $\\xi(r)$ has traded uncertainty at large scales for\nuncertainty at small scales.\nHowever, there is a distinct advantage because we know the functional\nform of the change in $\\Delta\\xi(r)$ at large separations due to any \nmodulation of small scale power.\nAssuming that the models of $\\xi(r)$ are accurate at large $r$ we can write\nthe change in $\\Delta\\xi(r)$ from inaccurately modeled (or measured) small\nscale power as an additive term\n\\begin{equation}\n  \\Delta\\xi(r) = \\Delta\\xi_{\\rm model}(r)+\\frac{{\\mathcal A}}{r^3}\n  \\quad {\\rm with\\\/}\\ {\\mathcal A}\\equiv\n  3\\int_0^r r'^2dr\\ \\left[\\xi(r')-\\xi_{\\rm model}(r')\\right] \\quad .\n\\end{equation}\nFor example, if the extra power were pure shotnoise, $(k\/k_1)^3$, it would\ngive ${\\mathcal A}\/r^3 = (3\\pi\/2)(k_1r)^{-3}$.  No matter what the small\nscale physics is doing to the true value of $\\xi(r)$, the integral quickly\nasymptotes to a constant value for all separations greater than\n${\\mathcal O}(10 h^{-1}$Mpc), for which the models are a good fit.\nIt is useful to know the functional form of the modification;\nit allows us to marginalize over the parameter \n${\\mathcal A}$ when connecting the observed galaxy correlation function to the \nHOD through $b$ and $k_2$, and to the cosmology, through $\\alpha$.  \n\nFinally we point out that the generalization of $\\Delta\\xi(r)$ to redshift\nspace is straightforward and gives the same kernel for the quadrupole as\nthe monopole allowing easy interpretation of their ratio.\nWriting the redshift space coordinate as $s$ with the cosine of the angle\nto the line of sight as $\\mu$ we can define\n\\begin{equation}\n  \\xi(s,\\mu) \\equiv \\sum_{\\ell=0}^{\\infty} \\xi_\\ell(s)P_\\ell(\\mu)\n\\end{equation}\nwhere $P_\\ell(\\mu)$ is a Legendre polynomials of order $\\ell$ and\n\\begin{equation}\n  \\xi_\\ell(s) = i^\\ell \\int\\frac{dk}{k}\\ \\Delta_\\ell(k)j_\\ell(ks)\n\\end{equation}\nNote that the sign of the quadrupole term in the correlation function\nis opposite that in the power spectrum.\nAgain we can reduce the dependence of $\\xi(s,\\mu)$ on low-$k$ modes\nby defining\n\\begin{equation}\n  \\Delta\\xi(s,\\mu) \\equiv \\bar{\\xi}(<s) - \\xi(s,\\mu)\n\\end{equation}\nwhich leads to a replacement of the $j_0$ in the monopole expression\nwith $j_2$ as in Eq.~(\\ref{eqn:dxi_wt}).  Both $\\Delta\\xi_0$ and\n$\\Delta\\xi_2$ then probe the same $k$-modes.\nThe hexadecapole can be isolated by using $\\xi$, $\\bar{\\xi}$ and\n$\\bar{\\bar{\\xi}}$ as in Eq.~(\\ref{eqn:ddxi}).\n\n\\section{Methodology} \\label{sec:methodology}\n\n\\subsection{Fourier space}\n\nTo compute the galaxy power spectrum for the periodic, constant time, slices\nwe first assign our tracers to the nearest grid point of a regular, periodic,\n$256^3$ Cartesian mesh and Fourier transform the resultant density\nfield\\footnote{We get the same results by using Cloud-in-Cell interpolation\nonto a $512^3$ grid.}.  The resulting $P(k)$ is corrected for the assignment\nto the grid using the appropriate window function and the Poisson shot noise\nis subtracted.  The power is averaged in bins spaced linearly in $k$.\nThe average $P(k)$ is plotted at the position of the average $k$ in each bin.\nWe obtain approximate error bars both by counting the number of modes in each\nbin and by dividing the data into 8 disjoint octants and using the\noctant-to-octant variance from the 3 different simulations -- for a total of\n24 sub-samples.\nThe latter method underestimates the errors on large scales, while the former\nneglects the mode coupling which occurs on small scales, the extent of which\ndepends on the galaxy sample under consideration.\nWhen computing the power spectrum of each octant we set the remaining 7\noctants to zero and correct for the windowing factor of 8 in power and the\nmodified shot noise when doing the FFT.\nFor HODs with $M_1\\gg M_{\\rm min}$ we find that mode counting and sub-sample\nvariance agree very well with\n\\begin{equation}\n  \\frac{\\delta P}{P} = \\sqrt{\\frac{2}{N_{\\rm mode}}}\n  \\left( 1 + \\frac{1}{\\bar{n}P(k)} \\right)\n  \\qquad .\n\\label{eqn:modecounting}\n\\end{equation}\nThe higher order shot-noise terms \\cite{MeiWhi} are subdominant for the\nrange of scales of interest to us.\nAs $M_1$ approaches $M_{\\rm min}$ there is excess variance on small scales\ncompared to the mode counting predictions.\nThis is not unexpected.  For models with $M_1\\simeq M_{\\rm min}$ the galaxy\npower spectrum goes non-linear at relatively low $k$, with the extra power\nbeing transferred from larger scales by gravitational collapse.  The number\nof modes driving the variance is thus smaller than one would estimate by\nmode counting at the measured $k$.\n\n{}From the 8 octants for each of 3 simulations we are able to compute the\ncovariance matrix of $P(k)$.  We find that for our chosen bin size the bins\nare almost independent at the scales of interest for the oscillations.\nOnce the data become significantly non-linear mode coupling induces a\ncorrelation.  For $k<0.2\\,h\\,{\\rm Mpc}^{-1}$ the correlation coefficient is\nbelow $5\\%$, reaching 10\\% by $k\\simeq 0.25\\,h\\,{\\rm Mpc}^{-1}$.\nThe degree of correlation between adjacent bins in Fourier space is, however,\na function of the HOD.  The more highly biased catalogs have more significant\ncorrelations between adjacent bins on larger scales.\nFor catalogs with a bias of $b=3.5$ for example, adjacent bins have\ncorrelations exceeding $0.2$ for $k>0.7\\,h{\\rm Mpc}^{-1}$.\nHowever, since the correlations are so small in the $k$-range of interest\nwe use Eq.~(\\ref{eqn:modecounting}) in the fits.\n\nWe compute our power spectra in redshift space assuming the distant observer\napproximation for all outputs and use the periodicity of the simulation to\nremap positions.  For the isotropically averaged spectra the power ratios\ndo not exactly recover the results of Ref.~\\cite{Kaiser} on large scales.\nWhether we should expect to reach \nthe those limits on the scales relevant to baryon\noscillations remains in doubt -- see Ref.~\\cite{Sco04} and references\ntherein for further discussion.\nOn small scales we are able to compute the $\\Delta^2_\\ell$ by direct\nsummation on the Cartesian $k$ grid, however on large scales we do not have\nenough modes.  For this reason we perform a least squares fit for the\n$\\Delta^2_\\ell$ up to $\\ell=6$ for each of the $k$ bins.\nThe line-of-sight angular dependence of the power spectrum on large scales\nis simple, as expected: the resulting Legendre coefficients above $\\ell=4$\nare small for $k\\le 0.3\\,h\\,{\\rm Mpc}^{-1}$.\n\n\n\\subsection{Configuration space}\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{2.7in}{!}{\\includegraphics{cmp_rxi.ps}}\n\\resizebox{2.7in}{!}{\\includegraphics{cmp_dxi.ps}}\n\\end{center}\n\\caption{A comparison of the different ways of computing $\\xi(r)$ and\n$\\Delta\\xi(r)$ discussed in the text.  In both panels open circles represent\ndirect pair counting in the periodic box, filled triangles the Landy-Szalay\nestimator and open squares the Fourier transform method.  For the\n$\\Delta\\xi(r)$ plot we also show the estimate of $\\Delta\\xi$ where $\\bar{\\xi}$\nis obtained from counts in spheres as the stars.  The different estimators\ndiffer in $\\xi(r)$ at small $r$, so for the $\\Delta\\xi$ plot we have added an\n$r^{-1}$ term to make $r^2\\Delta\\xi=100$ at $r=100\\,h^{-1}$Mpc for all\nof the lines.  For $\\xi(r)$ there is a noticeable shortfall in the power\nestimated by FFT methods at large $r$ which is largely absent for\n$\\Delta\\xi$.}\n\\label{fig:xi_cmp}\n\\end{figure}\n\nWe can compute $\\xi(r)$ either by directly counting pairs as a function of\nseparation in the periodic volume, \nusing FFT techniques in the periodic volume, or counting pairs and using the\nLandy \\& Szalay \\cite{LanSza} estimator\n\\begin{equation}\n  \\xi(r) = \\frac{\\langle DD\\rangle - 2\\langle DR\\rangle + \\langle RR\\rangle}\n                  {\\langle RR \\rangle}\n  \\qquad {\\rm pairs\\ in\\ }(r;dr)\n\\end{equation}\nin sub-volumes with vacuum boundary conditions -- here $D$ refers to the\ndata, $R$ refers to a random catalog with the same selection and the angled\nbrackets indicate the number of pairs in a given shell $(r;dr)$.\nA comparison of the different techniques is shown in Fig.~\\ref{fig:xi_cmp}.\nFor the FFT method we use a Fourier grid with $512^3$ points.\nBoth CIC and NGP charge assignment \\cite{HocEas} give the same answer for\n$\\xi(r)$ well above the grid scale.\nFor the Landy \\& Szalay method we used a random catalog with $10\\times$\nas many points as the data -- increasing the number did not alter the\nresults -- but in computing $\\langle RR\\rangle$ we only searched for pairs\naround the first $N_{\\rm data}$ random points.\nSince the $\\langle DR\\rangle$ point is limited to $N_{\\rm data}$ points\nthere is no loss in accuracy from limiting the $\\langle RR\\rangle$ term\nsimilarly but the cost scales as $N_{\\rm data}\\times N_{\\rm random}$ rather\nthan $N_{\\rm random}^2$.\nIn what follows we shall use the direct pair counting estimate of the\ncorrelation function, in bins of width $3\\,h^{-1}$Mpc.  We note in passing\nthat estimating $\\xi$ from future surveys will be non-trivial as we wish\nto work at very large scales with fine radial resolution.\n\nFormally $\\bar{\\xi}(<r)$ can be estimated in a similar manner to $\\xi(r)$\nexcept using counts within spheres of radius $r$ rather than shells at $r$\nof radius $dr$.  Unfortunately this method is prone to strong boundary effects\n(in the case of the sub-volumes with vacuum boundary conditions) because the\nsimulation volume available to each shell in the sphere is a strong function\nof radius.  Computing $\\xi$ in shells and then integrating up to find\n$\\bar{\\xi}$ is much more stable, and we do this for all of our estimators.\nWe tried several interpolation schemes and the results were insensitive to\nour choice.  We use linear interpolation between the measured $\\xi$ points\nin the results below.\n\nWe expect $\\Delta\\xi(r)$ to be less susceptible to finite volume effects than\n$\\xi(r)$.  To verify this we divided each simulation into $2^3$, $3^3$ or\n$4^3$ sub-cubes of side $L\/2$, $L\/3$ or $L\/4$ and computed $\\xi(r)$ and\n$\\Delta\\xi(r)$ using the estimator of \\cite{LanSza}.  We average the estimates\nover sub-cubes and simulations and compare them \nin Fig.~\\ref{fig:xi_vs_dxi} to the average result computed\nusing direct counting in \nthe full box with periodic boundary conditions for each simulation. \nAt larger $r$ the results become noisy, however\nit is clear that while both $\\xi(r)$ and $\\Delta\\xi(r)$ are insensitive to the\nsub-division at small $r$, $\\Delta\\xi(r)$ suffers far less than $\\xi(r)$ from\nsmall volume effects at larger $r$.  For reference the analytically predicted\nerror on $\\Delta\\xi$, averaged over all 3 simulations for this model, is 4-6\\%\nover the range $100-150\\,h^{-1}$Mpc, close to the difference seen in \nFig.~\\ref{fig:xi_vs_dxi} at higher $r$.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{5.5in}{!}{\\includegraphics{datares.ps}}\n\\end{center}\n\\caption{The fractional error in $\\xi(r)$ (thin, upper) and $\\Delta\\xi(r)$\n(thick, lower) computed in a smaller volume compared to the quantities in the\nfull simulation.  In each case we divided each simulation into $2^3$ (solid),\n$3^3$ (dashed) or $4^3$ (dotted) sub-cubes and computed $\\xi(r)$ or\n$\\Delta\\xi(r)$ using the estimator of \\protect\\cite{LanSza} (see text).\nThe curves display the deviation from the average computed with periodic\nboundary conditions in the full simulation. }\n\\label{fig:xi_vs_dxi}\n\\end{figure}\n\nThe different estimators of $\\xi(r)$ differ at small radius, and for\nthis reason our results for $\\Delta\\xi(r)$ can differ by an $r^{-3}$ term at\nlarge $r$.\nAs discussed earlier we also expect our models of $\\Delta^2(k)$ to misestimate\n$\\Delta\\xi(r)$ by a term ${\\mathcal A}\/r^3$.  For this reason we shall drop\nthe 1-halo contribution in $\\Delta^2(k)$ in favor of an additive $r^{-3}$ term\nin $\\Delta\\xi$ when doing the fits.  This term also soaks up any error in\ncomputing $\\Delta\\xi$ arising from uncertainties in our estimate of $\\xi(r)$\non small scales.  We shall marginalize over constant ${\\mathcal A}$ when\nfitting our theories to the observed $\\Delta\\xi(r)$ points.\n\nThe correlation function errors are highly correlated between different\nscales.  Although we can estimate\n${\\rm Cov}\\left[ \\Delta\\xi(r_1),\\Delta\\xi(r_2)\\right]$ from sub-divisions\nof the full volume we found that the covariances are so strong, and the\nnumerically estimated covariance matrix so noisy, that the results are\nunstable.  For this reason we compute the covariance analytically, making\nseveral approximations.\nFirst we assume that the small-scale fluctuations contributing to\n${\\mathcal A}$ are independent of the large-scale fluctuations contributing\nto $\\Delta\\xi(r)$ at large $r$.  For the large scale modes we assume Gaussian\nerrors on $P(k)$ with different $k$-modes being independent.\nThen it is straightforward to show that \\cite{Ber94,EisZal01,Coh06}\n\\begin{equation}\n  {\\rm Cov}\\left[ \\Delta\\xi(r_1),\\Delta\\xi(r_2)\\right] =\n  \\frac{(2\\pi)^2}{V}\\int \\frac{dk}{k^4} \\Delta^4(k) j_2(kr_1)j_2(kr_2)\n   + \\frac{\\sigma_{\\mathcal A}^2}{r_1^3r_2^3} + {\\rm s.n.}\n\\end{equation}\nwhere $V$ is the volume of the survey, in our case the simulation volume,\n$\\sigma_{\\mathcal A}^2$ is the variance of ${\\mathcal A}$ and ${\\rm s.n.}$\nindicates shot-noise terms.\nNote that the first term scales as $V^{-1}$, indicating that all of the\nerrors on $\\xi$ tend to zero as $V\\to\\infty$ but remain highly correlated.\nThis is to be contrasted with the errors on the power spectrum, which\nremain ${\\mathcal O}(1)$ for each $k$-mode but are uncorrelated between\nincreasingly finely spaced $k$-modes as $V\\to\\infty$.\nThe shot-noise term proportional to $1\/\\bar{n}$ is\n\\begin{equation}\n  \\frac{4}{V\\bar{n}}\\int \\frac{dk}{k} \\Delta^2(k) j_2(kr_1)j_2(kr_2)\n\\end{equation}\nFor $r_1>0$ and $r_2>0$ the $1\/\\bar{n}^2$ shot-noise terms are\n\\begin{equation}\n  \\frac{1}{2\\pi V\\bar{n}^2}\n  \\left[ \\frac{3}{r_1^3}\\Delta\\xi(r_2) + \\frac{3}{r_2^3}\\Delta\\xi(r_1) +\n  \\frac{3}{r_>^3}\\Delta\\xi(r_<) + \\frac{1+\\xi(r_1)}{r_1^2}\\delta(r_1-r_2)\\right]\n\\end{equation}\nwhere $r_<$ the lesser of $r_1$ and $r_2$ and $r_>$ is the greater.\nThe $\\delta$-function in the last term is rendered finite when we estimate\n$\\xi$ in bins of finite width.  For small bins we can replace $\\delta(r_1-r_2)$\nwith the inverse of the bin width.  Since $\\xi\\ll 1$ at large scales this\ngives $r_1^{-2}\\Delta r^{-1}$ along the diagonal.\nFinally the $1\/\\bar{n}^3$ term is simply\n\\begin{equation}\n  \\left(\\frac{1}{4\\pi}\\right)^2 \\frac{1}{V\\bar{n}^3}\n  \\ \\frac{3}{r_1^3}\\frac{3}{r_2^3}\n\\end{equation}\nWhile we show in \\S\\ref{sec:results} that making this Gaussian assumption\ndoes not bias the results for $\\alpha$, in future it would be better to use\na large set of mock catalogs to estimate the covariance.  From our limited\nnumbers of realizations, and using the non-linearly processed Gaussian fields,\nwe found that the shape and amplitude of the analytic expression were within\n${\\mathcal O}(25\\%)$ of the numerical results for scales near\n$100\\,h^{-1}$Mpc.  The agreement at smaller scales was considerably worse.\n\nIn configuration space there are no peculiar issues with estimating the $\\mu$\ndependence of $\\xi(s,\\mu)$ on the scales of interest.  We bin $\\xi(s,\\mu)$\nin 15 bins in $|\\mu|$ and then sum to get $\\xi_\\ell(s)$.  As \nwas found for $P(k)$, the\nmodes beyond the quadrupole are small at large $s$.  Performing a least\nsquares fit to $\\ell=0$, 2 and 4 yields the same result to better than a\npercent for $\\ell=0$, 2 and a few percent for the very small $\\ell=4$ mode.\nWe find in general that $\\Delta\\xi$ is much closer to spherical than $\\xi$\ndue to the large contribution from $\\bar{\\xi}(s)$.  For example from\n$75\\,h^{-1}$Mpc to $125\\,h^{-1}$Mpc, $\\Delta\\xi_2\/\\Delta\\xi_0$ falls from\napproximately $0.05$ to $\\simeq 0.02$ for our fiducial $b\\simeq 2$ catalog.\nThe contours of both $\\xi$ and $\\Delta\\xi$ are shown in\nFigure \\ref{fig:xicont}.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{2.7in}{!}{\\includegraphics{butterfly_xi.eps}}\n\\resizebox{2.7in}{!}{\\includegraphics{butterfly_dxi.eps}}\n\\end{center}\n\\caption{Contours of $\\xi(r_p,\\pi)$ (left) and $\\Delta\\xi(r_p,\\pi)$ (right)\nfor one of our $b\\simeq 2$ catalogs.  Contours are equally spaced in log,\nand dotted lines indicate negative values.  Here $r_p$ measures separations\nacross the line-of-sight and $\\pi$ along it.}\n\\label{fig:xicont}\n\\end{figure}\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{Dark matter}\n\nWe begin by presenting\\footnote{We thank Wayne Hu for suggesting we\nmake this point explicitly.} the power spectrum of the dark matter, in\nreal space, at the present epoch ($z=0$) from one of the simulations:\nFig.~\\ref{fig:pk_fit}.\nAlso on the plot we show the results of two ans\\\"{a}tze for non-linear\nspectra, that of Peacock \\& Dodds \\cite{PD96} based on an idea by\nHamilton et al.~\\cite{HKLM}, and another\nbased on halo model ideas \\cite{HaloFit}.\nThe former is seen to be a bad approximation as it implicitly assumes\nthat there exists a $1-1$ mapping between linear and non-linear power.\nWhile not an issue for smoothly varying spectra, this causes problems when\nthe spectrum contains features such as the baryon oscillations.  In reality\nmode coupling erases features, whereas the mapping procedure enhances them.\nWe could reduce some of the discrepancy by using a broad band measure of\nthe slope in the fitting function, but the underlying problem still remains.\nThe halo-model based methods perform better in this regard, as expected\n\\cite{Sel00}, since they model the non-linear power with an integral over\nthe linear theory power spectrum.  None of the fitting formulae approach\npercent level accuracy in the non-linear regime.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{5.5in}{!}{\\includegraphics{pk_fit.ps}}\n\\end{center}\n\\caption{The real space power spectrum of the dark matter at $z=0$ for\none of our simulations along with two ans\\\"{a}tze commonly used in the\nliterature (see text).  The lower panel shows the ratio of the fits and\nN-body points to the smooth spectrum of \\protect\\cite{EisHu99}.}\n\\label{fig:pk_fit}\n\\end{figure}\n\n\\subsection{Galaxies}\n\nNow we turn to the mock galaxy catalogs.\nWe show the results at $z=1$ for one of our HOD prescriptions, with\n$\\bar{n}=10^{-3}\\,h^3{\\rm Mpc}^{-3}$ and $b\\simeq 2$, in Fig.~\\ref{fig:pk_run7}\nalong with the predictions of linear theory multiplied by $b^2$.\nThe power is biased on large scales and shows a clear excess on small scales. \nWe shall now try to fit this behavior using the models described in\n\\S\\ref{sec:models}.  Our results for the sound horizon parameter $\\alpha$\nare reported in Table \\ref{tab:alpha}.\n\nWe have tried several methods for performing these fits.  We fit to both\n$P(k)$ and $\\Delta\\xi(r)$.  For the power spectrum we use errors from\nEq.~(\\ref{eqn:modecounting}), since they agree with the errors estimated from\nthe dispersion among the octants.\nFor the correlation function we use the analytic expression of\n\\S\\ref{sec:methodology}.\nThe multi-dimensional fitting was done using the Levenberg-Marquardt algorithm\n\\cite{NumRec}.  We experimented with several implementations and found good\nconvergence with both analytic and numerically computed derivatives.  From\nthese fits we also obtain an estimate of the parameter \ncovariance matrix from the\ncurvature of the likelihood around the best fit.\nTo test the Gaussianity of the likelihood surface we also ran Markov-Chain\nMonte-Carlo fits (see e.g.~\\cite{MCMC} for an introduction) for the power\nspectra for each model.\nWe provide comparisons of each of these methods for the different models below.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{5.5in}{!}{\\includegraphics{pk_run7.ps}}\n\\end{center}\n\\caption{The real space power spectrum of one of our mock galaxy catalogs\nwith $b\\simeq 2$ at $z=1$.  The solid line shows the predictions of linear\ntheory, multiplied by $b^2$.}\n\\label{fig:pk_run7}\n\\end{figure}\n\nWe begin with the linear bias model (Eq.~\\ref{eqn:linbias}) fit to\n$P(k)$ over the range $0.02<k<k_{\\rm max}\\simeq 0.15\\,h\\,{\\rm Mpc}^{-1}$.\nWe impose the lower $k$ cutoff so that there are enough independent modes\nin the bin that the Rayleigh distribution is close to symmetric.\nIn performing this and subsequent fits we convolve the known $k$-space\nsampling with the model at each bin to sample the theory in the same manner\nas the data, although this has only a small effect on our results.\nWe found that the strength of the small-scale excess is a strong function of\nthe HOD parameters\\footnote{In principle the form of the HOD for the sample\nbeing selected can be fit from the smaller scale clustering data that we are\nnot analyzing here.}.  The sense is as expected -- larger small-scale excess\nfor models with $M_1$ closer to $M_{\\rm min}$ or steeper power-law slopes\nfor the satellite term in $N(M)$.\nWhen $M_1\\simeq M_{\\rm min}$ multiple galaxies reside in a single halo,\nenhancing small-scale power.  Because the 1- and 2-halo terms shift by\ndifferent amounts in going from dark matter to galaxies, the bias is both\nlarger and more scale-dependent \\cite{ToyModel} with the galaxy 1-halo term\ndominating at larger characteristic scales than it does in the dark matter.\nWhen $M_1\\gg M_{\\rm min}$ the HOD has a long `plateau', with many halos\ncontaining only a single galaxy, and the galaxy bias is smaller and less\nscale-dependent (recall we hold $\\bar{n}$ fixed in our models).\nFor the former class of HODs, the 1-halo term can become significant at\na level of interest to BAO surveys at $k$ comparable to or even below the\nstandard choice for $k_{\\rm max}$.  This results in biases in the sound\nhorizon of up to $10\\%$, many times the formal fitting error.\nReducing $k_{\\rm max}$ further to compensate for these cases is problematic\nbecause -- as has been pointed out before \\cite{Fisher} -- the information\ncontent of a BAO experiment is very sensitive to this cutoff.\nThus safely fitting the power spectrum in this way requires throwing away\na significant amount of useful information.\nTo get around this one must accurately model the smaller scale or 1-halo\neffects.\n\nFor the catalogs with $\\bar{n}=10^{-3}\\,h^3{\\rm Mpc}^{-3}$ and the models of\nEqs.~(\\ref{eqn:lrg}-\\ref{eqn:esw}) we find we can safely fit up to\n$k_{\\rm max}\\simeq 0.3\\,h{\\rm Mpc}^{-1}$ before we notice a bias coming from\nincorrectly modeled small-scale physics.  We show the best fit $\\alpha$, and\nthe fit error, as a function of $k_{\\rm max}$ for several models in\nFig.~\\ref{fig:kmax}.  Note that the points are not independent because the\ninformation content is cumulative in $k_{\\rm max}$.  Beyond the range plotted\nthe bias in $\\alpha$ becomes significant.\nThe fitting form of Eq.~(\\ref{eqn:blagla}) does not fare as well.\nTo be conservative in what follows we choose\n$k_{\\rm max}=0.15\\,h{\\rm Mpc}^{-1}$ for the fits to Eq.~(\\ref{eqn:blagla})\nand $k_{\\rm max}=0.25\\,h{\\rm Mpc}^{-1}$ for the other models.  Beyond this\n$k_{\\rm max}$ our assumption of Gaussian uncorrelated errors becomes\nincreasingly suspect.\n\nFor the lower-density catalogs with $\\bar{n}=10^{-3.5}\\,h^3{\\rm Mpc}^{-3}$,\nwe find similar results when fitting the same models. Adopting a small-scale\ncutoff of $k_{\\rm max}=0.25\\,h{\\rm Mpc}^{-1}$ does not tend to bias the fitting\nresults. Significant differences between the fit results at differing\ndensities only begin to appear when dealing with highly biased populations.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{2.7in}{!}{\\includegraphics{fit_vs_kmax_05.ps}}\n\\resizebox{2.7in}{!}{\\includegraphics{fit_vs_kmax_07.ps}}\n\\end{center}\n\\caption{The sound horizon vs.~$k_{\\rm max}$ for two different HODs:\n(left)  $\\log_{10}M_{\\rm min}=12.8$ and $\\log_{10}M_1=13.0$ and\n(right) $\\log_{10}M_{\\rm min}=12.7$ and $\\log_{10}M_1=13.5$.\nOpen squares are Eq.~(\\protect\\ref{eqn:lrg}),\nsolid triangles Eq.~(\\protect\\ref{eqn:hmi})\nand open circles Eq.~(\\protect\\ref{eqn:esw}).\nPoints have been offset horizontally for clarity.}\n\\label{fig:kmax}\n\\end{figure}\n\nFitting the form of Eq.~(\\ref{eqn:blagla}) to the data in the range\n$0.02\\le k\\le 0.15\\,h\\,{\\rm Mpc}^{-1}$ and assuming the `correct'\n$\\Delta_{\\rm ref}$ we find the best fitting $k_A$.  Dividing the `true'\n$k_A\\simeq 0.06\\,h{\\rm Mpc}^{-1}$ by the best fit we find\n$\\alpha\\simeq 1.00$ for essentially all of our HOD models, with a mild\ntrend for higher $\\alpha$ (about $1\\sigma\\approx 1\\%$) for the less biased\nsamples with the smaller satellite fractions.  If we extend $k_{\\rm max}$\nbeyond $0.15\\,h\\,{\\rm Mpc}^{-1}$ we find $\\alpha$ increases and becomes\ninconsistent with unity.\nUsing all 3 of our runs, for a total survey volume of\n$3.8\\times 10^9\\,h^{-3}{\\rm Gpc}^3$, we are able to constrain $\\alpha$ to\n${\\mathcal O}(1\\%)$ for our catalogs -- consistent with the expectations of\nsimple error propagation.\n\nThe model of Eq.~(\\ref{eqn:lrg}) provides a reasonable fit to the data over\nthe relevant $k$ range: $0.02\\le k\\le 0.25\\,h\\,{\\rm Mpc}^{-1}$.\nThe Markov chains converge very rapidly for this model, and the best fits\nfor $b$ and $\\alpha$ were insensitive to the precise value of $a$ chosen.\nThis suggests that the first few elements of a Taylor series expansion in\nthe multiplicative bias would work as well.\nThe marginalized distributions for $b$, $Q$ and $\\alpha$ are reasonably well\nfit by Gaussians, with $\\alpha=0.996\\pm0.005$ for our $b\\simeq 2$ catalog for\nexample.  The best fit provided by the Levenberg-Marquardt algorithm for this\nmodel is $\\alpha=0.998\\pm0.004$.\nCompared to the halo model forms (Eqs.~\\ref{eqn:hmi}, \\ref{eqn:esw}) the best\nfitting models with Eq.~(\\ref{eqn:lrg}) have less intermediate scale power.\nTo compensate, the value for $b$ tends to be a few percent higher\n($b\\simeq 2.17$ for this model).\nThis also leads to a slightly higher $\\chi^2$ for the fit.\nComparing the galaxy $P(k)$ to that of the dark matter we find $b(k)$ rises\nfrom $2.05$ near the fundamental mode to $2.2$ at $k=0.2\\,h{\\rm Mpc}^{-1}$\nand $2.3$ at $k=0.4\\,h{\\rm Mpc}^{-1}$.  The best fit bias is thus\n${\\mathcal O}(10\\%)$ higher than the $k\\to 0$ value.\n\nFigure \\ref{fig:pk_real} shows the residuals around the fit to $P(k)$\nwhen using Eq.~(\\ref{eqn:hmi}) and the catalog with $b\\simeq 2$.\nAs is evident from the figure, Eq.~(\\ref{eqn:hmi}) is a good representation\nof the real-space power spectrum.\nComparisons between the fit results for different HODs show that $k_1$ (the\n1-halo parameter) and $b$ (the large-scale bias) vary as described in\n\\cite{ToyModel} with little scatter.\nThe Markov chains show the marginalized $\\alpha$ distribution is consistent\nwith unity, within $1\\sigma$, for all of the catalogs.\nWe also show in Fig.~\\ref{fig:pk_real} the best fit model using the\n$\\Delta\\xi(r)$ data for $80\\,h^{-1}{\\rm Mpc}<r<140\\,h^{-1}$Mpc.\nThe agreement between the $P(k)$ and $\\Delta\\xi(r)$ best fits is within\n$1\\sigma$ for the bias, $k_2$ and the sound horizon.\nIt is difficult to meaningfully compare the values of $k_1$, but the fit\nprefers a negligible 1-halo term for models with large $M_1$ as expected.\nFor the model of Eq.~(\\ref{eqn:hmi}) the predicted shape $\\Delta\\xi$ falls\nbelow the data for smaller $r$.  For this reason the values of the parameters\nreturned are sensitive to the range of $r$ chosen in the fit.\nIn general the fits to $\\Delta\\xi$ have slightly worse $\\chi^2$ values than\nfor $P(k)$, but the shape of the $\\chi^2$ surface for $\\alpha$ is similar.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{2.7in}{!}{\\includegraphics{pfit7.eps}}\n\\resizebox{2.7in}{!}{\\includegraphics{xfit7.eps}}\n\\end{center}\n\\caption{(Left) The real space power spectrum, $\\Delta^2(k)$, for one of our\nHOD models along with the halo model based fit of Eq.~(\\protect\\ref{eqn:hmi}).\nThe points are derived from the mock galaxy catalog, the dotted and dashed\nlines show the biased linear theory and shot-noise terms for the best fit\nwith the solid line being their sum.  The lower panel shows the ratio of\nthe theory and points to the smooth spectrum of \\protect\\cite{EisHu99} along\nwith the fit residuals.\n(Right) The correlation function, $\\Delta\\xi$, along with the best fit\n(solid) and the best fit to the power spectrum (dotted).  Note the errors\non $\\Delta\\xi$ have been computed analytically and so have been placed on\nthe best fit curve rather than the data (stars).  The errors are highly\ncovariant.}\n\\label{fig:pk_real}\n\\end{figure}\n\nWe show marginalized error contours (68 and 95\\% enclosed probability) for\nthe 4 parameters of our $P(k)$ fit to the same catalog in\nFig.~\\ref{fig:MargErr}.  These regions are derived from our Markov chains;\nthe error ellipses derived from the curvature of the likelihood at maximum\nare similar except for $k_1$ where the distribution is not well fit by a\nGaussian.\nExcept for the large-scale bias, $b$, the contours show little degeneracy\nbetween the cosmology, as parameterized by $\\alpha$, and the properties of\nthe galaxy sample.  The $b-\\alpha$ degeneracy is more pronounced for\ngalaxy populations with larger 1-halo terms.\nAs the small-scale power excess is increased, and the acoustic features are\nmore washed out, reducing the sound horizon (increasing $\\alpha$) becomes\na way of shifting power from large scales to small, and is thus difficult to\ndistinguish from a change in the amplitude of the 1-halo term.\nThe MCMC derived marginalized likelihoods are close to Gaussian for $b$,\n$k_2$ and $\\alpha$, but there is a significant tail to low $k_1$.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{5.5in}{!}{\\includegraphics{cont_multi.eps}}\n\\end{center}\n\\caption{The marginalized errors in each of the fit parameters of\nEq.~(\\protect\\ref{eqn:hmi}) for one of our catalogs with $b\\simeq 2$.\nContours represent $68\\%$ and $95\\%$ enclosed probability.}\n\\label{fig:MargErr}\n\\end{figure}\n\nThe form of Eq.~(\\ref{eqn:esw}) fits $\\Delta\\xi$ better to lower $r$ than\nEq.~(\\ref{eqn:hmi}).  As for Eq.~(\\ref{eqn:hmi}) there is a slight trend\nfor HOD models with higher $M_1$ to have lower $\\alpha$, but again the best\nfits are within $1\\%$ of unity for all of our catalogs.\nThe marginalized parameter distributions from the MCMC run on the $P(k)$\ndata are close to Gaussian except for $k_1$ and $k_2$, with the latter\npoorly constrained for this model.  For example, the marginalized distribution\nfor $\\alpha$ for the $b\\simeq 2$ data plotted above is well fit by\n$\\alpha=1.000\\pm0.007$.\n\nFinally we compare in Fig.~\\ref{fig:mcmc_all_ba} the marginalized errors in\n$b$ and $\\alpha$ from the Markov chain fits to the catalog in\nFig.~\\ref{fig:pk_run7}.  The value of $b$ preferred by Eq.~(\\ref{eqn:lrg})\nis slightly high, as discussed above.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{5.5in}{!}{\\includegraphics{mcmc_all_ba.eps}}\n\\end{center}\n\\caption{The marginalized errors in $b$ and $\\alpha$ computed from the\nMarkov chains fit to our $b\\simeq 2$ mock catalog for several of the\nmodels discussed above:\nEq.~(\\protect\\ref{eqn:blagla}) dot-dashed, Eq.~(\\protect\\ref{eqn:lrg}) dotted,\nEq.~(\\protect\\ref{eqn:hmi}) dashed and Eq.~(\\protect\\ref{eqn:esw}) solid.\nContours represent $68\\%$ and $95\\%$ enclosed probability.}\n\\label{fig:mcmc_all_ba}\n\\end{figure}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccccc}\n  $\\log_{10}M_{\\rm min}$ & $\\log_{10}M_1$  &\n  Eq.~(\\ref{eqn:lrg})    &\n  Eq.~(\\ref{eqn:hmi})    &\n  Eq.~(\\ref{eqn:esw})   \\\\\n  12.8 & 13.0 & $1.003\\pm0.005$ & $0.998\\pm0.007$ & $0.999\\pm0.008$ \\\\\n  12.7 & 13.5 & $0.996\\pm0.005$ & $0.998\\pm0.006$ & $1.000\\pm0.007$ \\\\\n  12.6 & 14.5 & $0.991\\pm0.005$ & $0.998\\pm0.006$ & $0.999\\pm0.006$\n\\end{tabular}\n\\end{center}\n\\caption{Results of fitting our catalogs to the various functional forms for\nthree typical choices of HOD parameters at\n$\\bar{n}=10^{-3.0}\\,h^3\\,{\\rm Mpc}^{-3}$.  We quote the parameters of a\nGaussian fit to the marginalized $\\alpha$ distribution from our Markov chains.}\n\\label{tab:alpha}\n\\end{table}\n\n\\subsection{Redshift space clustering} \\label{sec:redresult}\n\nThe results so far have been in real space, as appropriate for surveys\nmeasuring projected clustering statistics.  We now turn to measures in\nredshift space.\nWe begin by discussing the angle-averaged power spectrum\\footnote{We neglect\nthe contribution of $\\Delta_2^2$ to the covariance of $\\Delta_0^2$.  For\nour models, on the scales of relevance, this is a good approximation.}\nand correlation function.  The constraints on $\\alpha$ should thus be\ninterpreted as an average of the transverse and line-of-sight distances, or\napproximately a shift in $(D^2H^{-1})^{1\/3}$.\n\nSince the form of Eq.~(\\ref{eqn:blagla}) did not perform well for the\nreal space tests we begin by considering Eq.~(\\ref{eqn:lrg}).\nIn terms of the angle-averaged clustering pattern in redshift space we\nhave found that for $k<0.25\\,h{\\rm Mpc}^{-1}$ the exponential, Lorentzian\nor Gaussian streaming models fit the small-scale downturn to sufficient\naccuracy for our purposes, though they each prefer different values of\n$\\sigma$.  The goodness of fit is best for the Gaussian suppression.\nIf we multiply Eq.~(\\ref{eqn:lrg}) by a Gaussian suppression and a\n$1+2\\beta\/3+\\beta^2\/5$ prefactor (here $\\beta\\equiv\\Omega^{0.6}\/b$) we\nobtain good fits to most of the mock catalogs.\nTaking our $b\\simeq 2$ catalog for example, the marginalized real space\nconstraint is $\\alpha=0.996\\pm0.005$ while the redshift space constraint\nis $\\alpha=0.997\\pm0.008$.  For the bias we obtain $b=2.17\\pm0.01$\nvs.~$b=2.14\\pm0.02$ while the small-scale suppression is only poorly\nconstrained.\nThe difference in $b$ reflects the inapplicability of the large-scale\nenhancement factor.\nWith this $k_{\\rm max}$ there is a very slight bias in $\\alpha$ for the\nhighest $b$ HOD model.\nThe results are essentially unchanged if we use a Lorentzian suppression\nmodel, but if we use exponential suppression there is a 1\\% downward bias\non $\\alpha$ even for the $b\\simeq 2$ catalog and the goodness of fit is\nnoticeably worse.\n\nFor the forms Eqs.~(\\ref{eqn:hmi}, \\ref{eqn:esw}) we get good fits for either\na Gaussian or exponential suppression.  In each case the marginalized $\\alpha$\ndistribution is consistent with unity with an uncertainty of just under 1\\%.\nThe form of Eq.~(\\ref{eqn:esw}) does slightly better, as in the real space\ncase.\n\nNow let us consider the angle dependence of the clustering.\nFigure \\ref{fig:xicont} shows the anisotropy in configuration space,\ni.e.~for $\\xi$ and $\\Delta\\xi$ for one of our models.\nFigure \\ref{fig:q2m} shows the quadrupole-to-monopole ratio in Fourier\nspace for a number of our HOD models with\n$\\bar{n}=10^{-3}\\,h^3\\,{\\rm Mpc}^{-3}$.\nAs can be seen in the figure, we find a strong HOD dependence to the redshift\nspace clustering of galaxies.  In the limiting case of halos-only\n(i.e.~models with no satellite contribution) there is little small-scale\nsuppression even on scales as small as $1.5\\,h\\,{\\rm Mpc}^{-1}$.\n\nFurther insight into this behavior comes from considering the deviation\nbetween the true halo velocity in the simulation and that which would be\npredicted using linear theory from the (smoothed) density field in the\nsimulation at $z=1$ (see also \\cite{SheDia01,CroEfs94,Col00,PadBau} for\nsimilar calculations).\nThis is shown in Fig.~\\ref{fig:velcmp} for two smoothing scales.\nWhen the density field is estimated from the dark matter particles in the\nsimulation the rms deviation in each component of $v$ is between 50 and\n$80\\,$km\/s with smaller deviations for higher mass halos and smaller smoothing\nscales.  If the velocity is estimated from the halo catalog (weighting all\nhalos equally) then the rms rises to between 90 and $260\\,$km\/s.  The increase\nin the rms comes from the neglect of the mass contributed by the lower\nmass halos, the fact that more massive halos tend to live in denser\nenvironments and the assumption of equal weights per halo\n\\cite{SheDia01,Col00}.\nThe fact that the true velocities differ only slightly (the equivalent of a\nfew Mpc) from those predicted by linear theory suggests we should see little\nsmall-scale suppression in the redshift space halo power spectrum on the\nscale of interest.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{2.7in}{!}{\\includegraphics{qm_stream.eps}}\n\\resizebox{2.7in}{!}{\\includegraphics{qm_hc.eps}}\n\\end{center}\n\\caption{(Left) The quadrupole-to-monopole ratio vs.~scale for a number of\nour models at $z=1$ along with streaming model fits (see text).  The streaming\nmodel tends to overestimate $Q\/M$ at low-$k$ for the more biased models\n(lower panels).  (Right)  The same models with the fit of\nRef.~\\protect\\cite{HatCol}.  In our catalogs the high $k$ suppression closely\ntracks the mean galaxy weighted halo mass (see text).}\n\\label{fig:q2m}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{2.7in}{!}{\\includegraphics{velcmp_05.ps}}\n\\resizebox{2.7in}{!}{\\includegraphics{velcmp_10.ps}}\n\\end{center}\n\\caption{The true halo velocity in the simulations compared to the linear\ntheory prediction from a smoothed density field with Gaussian width\n$5\\,h^{-1}$Mpc (left) or $10\\,h^{-1}$Mpc (right).  The thick lines show\nthe histogram of velocity differences in each component when the density\nfield is estimated from the halo positions, the thin lines when the density\nfield is estimated directly from the dark matter particles in the simulation.}\n\\label{fig:velcmp}\n\\end{figure}\n\nThe other extreme case -- the strongest small-scale deviation from the\nKaiser approximation -- occurs where the mean galaxy-weighted halo mass\nand the satellite fraction are at their highest.  For these HOD models\nthe galaxies preferentially sample the highest peculiar velocities.\nThe other HODs occupy a continuum between these cases that is well explained\nby the fraction of satellite galaxies and the mean galaxy-weighted halo mass.\nOverplotted in Fig.~\\ref{fig:q2m} are the fits to the functional form\nderived from a streaming model with exponential small-scale\nsuppression.  We also attempted to fit streaming models with Gaussian\nand Lorentzian cutoffs, but these were generally poorer fits to the\ndata than the exponential cutoff. The streaming model fits shown are\nacceptable fits to the simulation quadrupole-to-monopole ratios only\nwhen the latter varies slowly with $k$; our more highly biased models\n(lower panels) are not well-represented.\nIn Fig.~\\ref{fig:q2m} we also compare our results with the fitting formula\nof Eq.~(\\ref{eqn:hatcol}).  This is quite a good fit to the simulations\nover the range of interesting scales.\n\nAdditional constraints on power spectrum model parameters can be gleaned from\nthe redshift-space distortions.  One option is to use the linear theory\nparameter $\\beta$ as a simultaneous constraint on the cosmology and the galaxy\nbias (recall the large-scale bias, $b$, was the parameter most degenerate\nwith $\\alpha$, see Fig.~\\ref{fig:MargErr}).\nWe examined this possibility, and found that even on the largest scales\nin the simulations, the best fits to the multipole moments did not approach\nthe linear theory value of $\\beta$ with enough precision for this to be useful.\n\nOne property of the angular dependence of the redshift-space distortions that\nis well constrained is the scale of the quadrupole zero-crossing -- represented\nby $k_{\\rm nl}$ in the empirical fit of \\cite{HatCol}, and by $\\sigma$ in the\nstreaming model.  This quantity is in fact very tightly correlated with the\nchoice of HOD parameters.  In this sense, at least one independent constraint\non the relevant galaxy physics derived from redshift-space distortions is\nfairly insensitive to the choice between these models.\n\nWe do not consider using the angular dependence of the redshift space\ndistortions to fit separately for the line-of-sight and transverse\ndistance scales.  Before further considering redshift space distortions\nwe want to include the (light-cone) evolution of clustering.\n\n\\section{Reconstruction} \\label{sec:reconstruction}\n\nThe common lore is that surveys targeting galaxy populations with a large\n1-halo term (small values of $k_1$), or surveys at low $z$ will have fewer\n`useful' modes than high redshift surveys or surveys of objects which singly\noccupy their halos.\nRecent work \\cite{ESSS06} has suggested that it may be possible to partially\nmitigate these effects and reconstruct the baryon oscillation signal despite\nthe corrosion due to non-linear collapse, even for surveys at low redshifts.\n\nThe original work of \\cite{ESSS06} did not make their measurements on `galaxy'\ncatalogs made by populating halos with a range of occupation distributions.\nWe have cross-checked their method using some of our catalogs and found very\nconsistent results.  We show the results from one of our simulations and\nmodels in Fig.~\\ref{fig:es3} in both configuration and Fourier space.  We\npresent the results here without redshift space distortions to better show\nthe degree to which reconstruction gains signal.\nIn agreement with \\cite{ESSS06} we find that the real space correlation\nfunction at low redshift is considerably sharpened by the reconstruction\nmethod.  The size of the effect is reduced as we go to higher $z$, and\nby $z=1$ the gains are significantly less pronounced.\n\n\\begin{figure}\n\\resizebox{2.7in}{!}{\\includegraphics{es3_pk.ps}}\n\\resizebox{2.7in}{!}{\\includegraphics{es3_xi.ps}}\n\\caption{The power spectrum (left) and correlation function (right) for one\nof our boxes at 3 different redshifts, $z=0.3$, 0.5 and 1.0, when using\nthe reconstruction method of Ref.~\\protect\\cite{ESSS06}.  For the correlation\nfunction we show both $\\xi(r)$ (lower curves) and $\\Delta\\xi(r)$ (upper curves).\nFor the power spectrum we have plotted the signal compared to the\nno-oscillation form of Ref.~\\cite{EisHu99}.\nThe squares are the original signal, the triangles show a reconstruction using\na smoothed field with $R_s=20\\,h^{-1}$Mpc and the circles with\n$R_s=10\\,h^{-1}$Mpc.}\n\\label{fig:es3}\n\\end{figure}\n\nSince the reconstruction procedure is inherently non-linear, we also tested\nwhether it induces correlations between otherwise uncorrelated modes.  To\ndo this we used the 100 non-linearly processed Gaussian density fields\ndescribed in \\S\\ref{sec:sim}.\nFor each we computed the power spectrum before and after reconstruction and\nhence the covariance matrix.  On the scales of relevance for the acoustic\noscillations the procedure does not seem to introduce significant correlations.\nFurther investigations of reconstruction will be deferred to a future\npublication.\n\n\\section{Conclusions} \\label{sec:conclusions}\n\nThe coupling of baryons and photons by Thomson scattering in the early\nuniverse leads to a rich structure in the power spectra of the CMB\nphotons and the matter.  The study of the former has revolutionized\ncosmology and allowed precise measurement of a host of important\ncosmological parameters.  The study of the latter is still in its\ninfancy, but holds the potential to constrain the nature of the dark\nenergy believed to be causing the accelerated expansion of the universe.\n\nFuture large redshift surveys offer the opportunity to measure a\ncharacteristic scale in the universe: the sound horizon at the time\nof photon-baryon decoupling.  This standard ruler, which we can calibrate\n{}from observations of the CMB, may allow us to tightly constrain the\nevolution of the scale factor and determine the nature of dark energy.\nTo ensure the success of these efforts we need to improve our understanding\nof the theoretical underpinnings of the method and generate simulated\nuniverses which can be used to refine and test our observational strategies.\n\nIn this paper we have made a first attempt to go all the way from (mock)\nobservations to constraints on the sound horizon for a number of galaxy\ncatalogs which display non-linear, scale dependent and stochastic bias.\nWe have performed fits in configuration and Fourier space for a number of\nmodels which have been proposed in the literature.  We investigate the\nshape of the likelihood function, parameter degeneracies and the range of\nvalidity of the fits.  We find that the forms of\nEqs.~(\\ref{eqn:lrg}-\\ref{eqn:esw}) fare quite well and lead to unbiased\nestimates of the sound horizon in both real and redshift space.\nIn agreement with earlier work \\cite{Fisher}, we find that a survey of\nseveral Gpc${}^3$ would constrain the sound horizon at $z\\sim 1$ to\nabout 1\\%.\n\nWe would like to thank Istvan Szapudi for helpful conversations on\ncorrelation function estimators and edge correction, Ravi Sheth for\nnumerous enlightening conversations on a number of issues and Joanne\nCohn and Eric Linder for conversations and comments on the manuscript.\nMJW also thanks the staff of the Aspen Center for Physics for their\nhospitality while part of this work was completed.\nThe analysis reported here was done on computers at NERSC and LANL.\nThis work was supported in part by NASA and the NSF.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:orgd6ac989}\n\nGravitational wave (GW) signals from the early Universe have the\npotential to provide a new observational perspective\non high energy physics phenomena. In this context, first order cosmological phase transitions\nprovide a compelling source of GWs. This was first proposed many years\nago (see\n\\Refs{Witten:1984rs,hogan,Turner:1990rc,Kosowsky:1991ua,Kosowsky:1992vn,Kamionkowski:1993fg}),\nbefore it became clear that the electroweak (EW) symmetry breaking\nproceeds as a crossover in the Standard Model\n\\cite{Kajantie:1996mn}. However, it has since emerged that many\nscenarios beyond the Standard Model (BSM) lead to first order phase\ntransitions at -- and beyond -- the EW scale, reopening the case for the\nstudy of GW production from first order phase transitions (for a GW-oriented review, see Ref.~\\cite{Caprini:2019egz}).\n\nThis is particularly interesting in the context of the Laser\nInterferometer Space Antenna (LISA), which is sensitive to a frequency window\nin the millihertz range~\\cite{Audley:2017drz}. In the context of primordial GW-sourcing processes\nthat are localized in time, such as a first order phase transition, the characteristic frequency of the GWs today can be connected to the characteristic time\/length scale of the source anisotropic stresses $L_*$, via\n\\begin{equation}\n\tf\\simeq 16.5 \\cdot 10^{-3} \\,\\mathrm{mHz} \\,\\frac{1}{L_*\\mathcal{H}_*} \\frac{T_*}{100\\,\\mathrm{GeV}} \\left(\\frac{g_*}{100}\\right)^{1\/6}.\n\\end{equation}\nHere a subscript $*$ denotes the epoch at which the phase transition occurs.\nThis shows that LISA can potentially detect the GW signal from\ntransitions in the window 100~GeV -- 1~TeV, if the\ncharacteristic time\/length scale of the anisotropic stresses is of the\norder $L_*\\sim 10^{-2}$ to $10^{-3}$ of the Hubble scale $\\mathcal{H}_*^{-1}$ at\nthe time of the phase transition. These can be considered as typical values for $L_*\\mathcal{H}_*$, given that $L_*$ is related to the mean bubble separation $R_*$ (see \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot~\\cite{Caprini:2015zlo,Caprini:2019egz} and references therein).\nLISA will therefore offer a new way to probe BSM physics, complementary to the Large Hadron Collider (see \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot~\\Refa{Caldwell:2022qsj}).\n\nThere are several processes possibly leading to sizable anisotropic\nstresses in connection with a first order phase transition. This\nrich phenomenology renders phase transitions particularly appealing as primordial GW sources. Bubble percolation, with the consequent breaking of spherical symmetry, is the most direct one~\\cite{Turner:1990rc,Kosowsky:1991ua,Kosowsky:1992vn}.\nThe GW generation by bubble collision has been analysed both with simulations~\\cite{Huber:2008hg,Weir:2016tov,Konstandin:2017sat,Cutting:2018tjt,Cutting:2020nla,Lewicki:2020jiv,Lewicki:2020azd,Gould:2021dpm} and analytical approaches~\\cite{Caprini:2007xq,Jinno:2016vai,Jinno:2017fby}.\n\nThe first 3-dimensional (3D) simulations of the coupled system of a scalar field and a relativistic fluid have shown that sound waves, produced in the fluid by expanding bubbles, are also a promising source of GWs \\cite{Hindmarsh:2013xza,Hindmarsh:2015qta,Hindmarsh:2017gnf}.\nThese  simulations have been performed using the code SCOTTS (Simulations of COsmic Thermal TransitionS) on which we will also rely in the present work.\nAnalytical insight into the  generation of GWs by sound waves has been developed in \\Refs{Hindmarsh:2016lnk,Hindmarsh:2019phv}, and a new simulation technique for computing the GW spectra in the sound shell limit has been proposed in \\Refa{Jinno:2020eqg}.\n\nRefs.~\\cite{Hindmarsh:2013xza,Hindmarsh:2015qta,Hindmarsh:2017gnf} performed simulations of first order phase transitions of weak to intermediate strength, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot~$\\alpha\\lesssim 0.1$, where $\\alpha$ is the ratio of the trace anomaly of the energy momentum tensor and the thermal energy. It was found that the bulk fluid motion consisted primarily of compressional modes, which correspond to sound waves. In this case, sound waves are expected to be the dominant GW source. However, shocks will develop in the fluid and are expected to convert the acoustic phase into a turbulent one \\cite{Pen:2015qta,Dahl:2021wyk}.\nThe characteristic time of shock formation can be estimated as $\\tau_\\mathrm{sh}\\sim R_*\/\\overline{v}_{\\parallel}$, where $\\overline{v}_{\\parallel}$ denotes the root-mean-squared (rms) velocity of the acoustic motion. For stronger transitions, in which the initial value of $\\overline{v}_{\\parallel}$ is typically larger, the onset of turbulence is expected to occur earlier and give a more substantial contribution to the total GW signal. Furthermore, recent simulations of transitions with $\\alpha\\sim 0.5$ found that, if the bubble wall velocity is subsonic, flows with substantial vorticity are already generated during the transition itself when fluid shells and bubble walls collide~\\cite{Cutting:2019zws}.\n\n\nIn the present work, we study the GW signal generated by a\nhypothetical turbulent phase in the aftermath of a first order\nphase transition. The first analyses of the GW signal from turbulence have relied on analytical modelling of the turbulent flow, and semi-analytical estimates of the GW signal~\\cite{Kamionkowski:1993fg,Caprini:2006jb,Gogoberidze:2007an,Kahniashvili:2008pe,Kahniashvili:2008er,Caprini:2009yp,Caprini:2009pr}.\nIn particular, Ref.~\\cite{Niksa:2018ofa}  evaluates the GW signal from all components (compressional, vortical, magnetic field) of both standard, and helical, freely-decaying magneto-hydrodynamic (MHD) turbulence. Various spectral shapes and scalings of the GW power spectrum have been found, depending on the relative amplitude of the compressional and vortical components, and on the presence of helicity.\nIn \\Refa{Niksa:2018ofa}, it was also argued that the correct auto-correlation time to be used in the equations describing GW production by turbulence is the Eulerian eddy turnover time (see Ref.~\\cite{kaneda_lagrangian_1993}).\nFar in the inertial range, the Eulerian eddy turnover time becomes  $\\tau_E(k)\\sim (k v_\\mathrm{sw})^{-1}$, where $v_\\mathrm{sw}\\simeq \\overline{v}\/3$ is a locally uniform velocity field sweeping the vortices, in accordance with the Kraichnan random sweeping approximation~\\cite{kraichnan:1964}, and $ \\overline{v}$ is the rms velocity of the turbulence.\nEarlier work used the Lagrangian eddy turnover time $\\tau_{\\ell}(k)\\sim (k v_{\\ell})^{-1}$, where $v_{\\ell}$ denotes the characteristic velocity on the scale $\\ell\\sim 1\/k$ \\cite{Gogoberidze:2007an,Kahniashvili:2008pe,Caprini:2009yp}.\nOur simulations confirm that the Eulerian eddy turnover time is the correct option.\n\n\nSimulations of both non-helical and helical MHD turbulence and the resulting GW generation have been carried out in \\Refs{Pol:2018pao,Pol:2019yex,Kahniashvili:2020jgm,Brandenburg:2021bvg}.\nIn these works there is an initial phase in which the MHD turbulence develops, starting from a nearly monochromatic electromotive force, or a kinetic forcing.\nThey find that the spectral shape of the GW signal varies, depending on whether a fully-developed turbulent magnetic and\/or kinetic field is present in the initial conditions, or whether it is sourced by the forcing.\n\nIn this paper, we develop a semi-analytical model of GW generation by freely-decaying kinetic, vortical, non-helical turbulence, which we support with the results of direct numerical simulations. We rely on the relativistic hydrodynamic code SCOTTS, developed by the authors of \\Refs{Hindmarsh:2013xza,Hindmarsh:2015qta,Hindmarsh:2017gnf,Cutting:2019zws}, but we use it to study the evolution of vortical fluid motion, excluding the dynamics of the scalar field undergoing the transition (see \\cref{sec:numerical}). We have chosen to over-simplify the turbulence model (no compressional modes, no helicity, no magnetic field, no initial forcing phase) in order to have full control of our analytical understanding of the GW production.\nInclusion of these additional features will be tackled in future works.\n\nThe layout of the paper is as follows. We begin with a short review of stochastic gravitational wave backgrounds in the context of vortical turbulence in \\cref{sec:sgwbreview}. We first introduce the equations of motion for the metric perturbations in an FLRW background, and write their solutions in terms of Green's functions and the anisotropic shear stress. The gravitational wave energy density power spectrum in the radiation era is then constructed, assuming a statistically homogeneous and isotropic signal. The solution for the gravitational wave energy density power spectrum is given in terms of an integral over the Green's function and the unequal time correlator (UETC) of the anisotropic stress. The section concludes by reviewing the computation of the anisotropic stress UETC in the context of vortical turbulence, decomposing the anisotropic stress UETC into a kernel which contains the fluid velocity UETC.\n\nWe continue by modelling freely decaying turbulence in \\cref{sec:free-decay-model}. In \\cref{sec:free-decay-velps}, we introduce the velocity power spectrum for a divergence free velocity field, along with important quantities such as the integral scale of the flow $\\xi$, and the rms velocity $\\overline{v}$. In our modelling, we assume that the turbulent flow is described by a von K\\'arm\\'an~spectrum, which has a Kolmogorov power law in the inertial range and a causal slope of $k^5$ at large scales, where $k$ refers to the power spectrum wavenumber. The evolution of the velocity field in decaying turbulence is considered in \\cref{sec:free-decay-evol}.\nHere we review the self-similarity properties of the turbulent power spectrum, starting from dimensional arguments and the assumption that there is a single length scale in the flow.\nOn further assuming that the eddy turnover time, $\\tau_\\xi$, is the only principal timescale in the flow, one arrives at exponential decay laws for the kinetic energy and integral scale, where the decay exponents of both quantities are in principle related. We conclude with \\cref{sec:UETCvelocity}, where we introduce a model for the velocity UETC. To model the decorrelation in time, we follow \\Refa{Niksa:2018ofa} and adopt the exponential decorrelation provided by the short-time analysis of  \\Refa{kaneda_lagrangian_1993}. This consistently reproduces the random sweeping model \\cite{kraichnan:1964} in the inertial range, which is supported by numerical works simulating decaying isotropic turbulence (see \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot Ref.~\\cite{he_computation_2004}).\n\nIn \\cref{sec:PosKer} we expand on the discussion of UETCs by considering them in the context of positive kernels and Mercer's theorem.\nEarlier works, such as \\Refa{Caprini:2009yp}, encountered the question of how to define a valid correlation function for a non-stationary process. Here, we consistently address this issue in the context of the theory of positive kernels, and propose to model the unequal time correlator (UETC) of the turbulent velocity field as a  Gibbs Kernel. This provides a way to symmetrize the unequal time correlator and, most importantly, it guarantees that the anisotropic stress two-point function is a valid correlation function. As a consequence, this leads to a GW energy density power spectrum which is always positive, as it should be.\n\nAfter discussing the theory, we move on to describe our direct numerical simulations in \\cref{sec:numerical}. We introduce the Minkowski space simulation code, SCOTTS, and discuss how it has been modified to study relativistic vortical turbulence. To prepare the initial conditions, we initialize the velocity field in momentum space, where each mode is randomly drawn using Gaussian statistics from a von K\\'arm\\'an~power spectrum. We also project the velocity field so that it is purely vortical. After outlining our suite of simulation runs, we specify the methodology for outputting the velocity power spectra and UETC, as well as the procedure for evolving the metric perturbations and obtaining the gravitational wave power spectrum in the simulations.\n\nIn the following two sections we present our results. First, we focus on the evolution of the velocity field in the direct numerical simulations, in \\cref{sec:results-vel}.\nWe study the evolution of the kinetic energy and of the integral scale of the flow in \\cref{sec:ResETC}, and link it to the model of \\cref{sec:free-decay-evol}. In the simulations, the velocity power spectrum displays self-similar scaling towards the end of the simulation, but\nthe theoretical relation between the decay law exponents and the velocity spectrum power law at large scales is not satisfied. We also test our model for the velocity UETC in \\cref{sec:results-uetc}. The combination of the Gibbs Kernel model of the UETC with the decorrelation of \\Refa{kaneda_lagrangian_1993} performs well for both the inertial range and for large scale modes. We compare our model to that of \\Refa{Niksa:2018ofa}, and find that our model has a closer agreement with the simulation data.\n\nOur results for the gravitational wave power spectrum are given in \\cref{sec:sgwbresults}.\nThere are three different cases. We begin by evaluating the gravitational wave power spectrum under the assumption that the turbulence is stationary, in \\cref{sec:stationary}.\nNext we cover the case of instantaneously generated turbulence in \\cref{sec:gw-instant}, which contains our main results regarding the SGWB spectrum.\nHere our initial conditions are a fully developed turbulent power spectrum.\nWe first show the SGWB extracted from the direct numerical simulations.\nWe then calculate the gravitational wave signal using our model of freely decaying turbulence outlined in \\cref{sec:free-decay-model,sec:PosKer}. In this case, to calculate the gravitational wave power spectrum, we use Monte Carlo integration with importance sampling to compute the integrals found in \\cref{sec:sgwbreview}. Using this technique, we confirm that the precise values of the decay law exponents are unimportant for the final spectral shape. We also describe the constant source approximation, in which the source of gravitational waves is taken to be constant in time, before it is turned off abruptly~\\cite{RoperPol:2022iel}.\nWe obtain in this context a ready-to-use formula \\cref{eq:constant_approx} for the SGWB spectrum from kinetic turbulence.\nWe compare the gravitational wave power spectrum under all of these methods, finding good agreement in their mutual regions of applicability. The final case we consider in \\cref{sec:GWcontinuous} is the inclusion of a simplified turbulence growth phase. Using our turbulence model and\nMonte Carlo integration, we calculate the gravitational wave power spectrum in the case of linear growth for both a $\\mathcal{C}^0$ and a $\\mathcal{C}^{1}$ {growth phase}. The inclusion of a growth phase substantially affects the shape of the gravitational wave power spectrum, underlining the importance of studying the creation mechanism of turbulence in the early Universe.\nFinally, we summarize our results and conclusions in \\cref{sec:discussion}.\n\nTo help the reader, in \\cref{tab:notation} we summarize some timescales and temporal parameters which appear in different sections of this paper.\n\\begin{table}\n  \\centering\n  \\begin{tabular}{ p{2cm}p{7cm}p{4,5cm}  }\n    \\multicolumn{3}{c}{\\emph{Direct numerical simulations (SCOTTS code) (\\cref{sec:numerical})}} \\\\\n    \\hline\n    time variable & description                            & properties                          \\\\\n    \\hline\n    $\\tdevel$     & initial time in the simulation         &                                     \\\\\n    $\\tend$       & final time in the simulation           & $k (\\tend-\\tdevel) \\gtrsim 10 $     \\\\\n    $\\tuetc$      & reference time at which UETCs measured & $\\tdevel < \\tuetc < \\tend$          \\\\\n  \\end{tabular}\n\n  \\vspace{1cm}\n\n  \\begin{tabular}{cll}\n    \\multicolumn{3}{c}{\\emph{Semi-analytical evaluation of GW spectrum (\\cref{sec:sgwbresults})}} \\\\\n    \\hline\n    time variable & description & properties \\\\\n    \\hline\n    $\\tini$       & initial integration time & turbulence starts \\\\\n    $\\tgro$       & \\parbox[t]{7cm}{timescale over which turbulence is sourced \\\\\n      (see Section \\ref{sec:GWcontinuous})} & \\\\\n    $\\tdevel$     & start of turbulence free decay & $\\tdevel=\\tini+\\tgro \\equiv \\mathcal{H}_*^{-1}$ \\\\\n    $\\tau_{\\xi_*}$    & initial eddy turnover time & $\\tau_{\\xi_*} = \\xi_* \/ \\overline{v}_*$ \\\\\n    $\\mathcal{N}_\\mathrm{e}$     & characteristic decay time of the source in units of  $\\tau_{\\xi_*}$ & dimensionless, $\\order{1}$ \\\\\n    $\\mathcal{N}_\\mathrm{cut}$       & \\parbox[t]{7cm}{ duration of the source in units of  $\\tau_{\\xi_*}$ \\\\\n    (used in the stationary source approximation of \\cref{sec:stationary}, and in the constant source approximation of \\cref{sec:const})} & dimensionless, $\\order{1}$ \\\\\n  \\end{tabular}\n\n  \\caption{\n    Important timescales and temporal parameters used in this article.\n    A $*$ subscript indicates quantities evaluated at $\\tdevel$.\n    In the above, $k$ refers to the magnitude of the non-zero wavenumbers on the discrete momentum space lattice in the simulations.}\n  \\label{tab:notation}\n\\end{table}\n\n\n\n\n\n\\section{Stochastic background of gravitational waves from vortical turbulence}\n\\label{sec:sgwbreview}\n\\subsection{Generation of gravitational waves}\n\\label{sec:gen-gw}\n\nIn the cosmological context, GWs are described by transverse and traceless tensor spatial perturbations $h_{ij}$ of the background FLRW metric for a homogeneous and isotropic Universe:\n\\begin{equation}\n\t\\dd{s}^2 = a^2(\\tau) \\qty[ - \\dd{\\tau}^2 + (\\delta_{ij} + 2 h_{ij}) \\dd{x}^i \\dd{x}^j ],\n\t\\label{eq:metric}\n\\end{equation}\nwith $\\partial_i h_{ij}=h_{ii}=0$.\nHere $a(\\tau)$ is the scale factor, and $\\tau$ the conformal time.\nIt follows from the linearized Einstein equations that $h_{ij}\\qty(\\vb{k}, \\tau)$ satisfies\n\\begin{equation}\n\t\\ddot{h}_{ij} + 2\\mathcal{H} \\dot{h}_{ij} + k^2 h_{ij} = 8 \\pi G a^2(\\tau) \\Pi^{(TT)}_{ij}\\qty(\\vb{k}, \\tau)\n\t\\label{eq:gw-propagation}\n\\end{equation}\nwhere $\\mathcal{H}=\\dot{a}\/{a}$ is the comoving Hubble parameter, $\\cdot = \\dv*{\\tau}$, $\\vb{k}$ denotes a comoving wave-vector, and $\\Pi_{ij}^{(TT)}$ is the transverse traceless part of the anisotropic stress tensor. The anisotropic stress is defined by\n\\begin{equation}\n\t\\Pi_{ij}=T_{ij} - p(\\delta_{ij}+2h_{ij})\\text,\n\t\\label{eq:Piij}\n\\end{equation}\nwith $T_{\\mu\\nu}$ the fluid energy momentum tensor, and $p$ its pressure.\nThe transverse traceless component of the anisotropic stress is extracted by means of the projector\n\\begin{align}\n\t\\Pi_{ij}^{(TT)}(\\vb{k},\\tau)            & =\\Lambda_{ij\\ell m}(\\vu{k})\\Pi_{\\ell m}(\\vb{k},\\tau),\n\t\\label{eq:Piij_proj}                                                                                                                            \\\\\n\t\\text{with}~~\\Lambda_{ij\\ell m}(\\vu{k}) & = \\bot_{i\\ell }(\\vu{k}) \\bot_{jm}(\\vu{k}) - \\frac{1}{2} \\bot_{ij}(\\vu{k}) \\bot_{\\ell m}(\\vu{k})\\text,\n\t\\label{eq:lambda_proj}                                                                                                                          \\\\\n\t\\text{and}~~\\bot_{ij}(\\vu{k})           & = \\delta_{ij} - \\hat{k}_i \\hat{k}_j\\text. \\label{eq:PSsim}\n\\end{align}\nWe work in the radiation era, and therefore we further define  $\\Pi_{ij}^{(TT)} = (4\\rho\/3) \\tilde{\\Pi}_{ij}$ with $\\rho$ the energy density in the early, radiation dominated Universe, and $p=\\rho \/3$.\nUsing Friedmann's equation, \\cref{eq:gw-propagation} can be rewritten as\n\\begin{equation}\n\t\\ddot{h}_{ij} + 2\\mathcal{H} \\dot{h}_{ij} + k^2 h_{ij} = 4 \\mathcal{H}^2 \\tilde{\\Pi}_{ij}\\qty(\\vb{k}, \\tau). \\label{eq:gw-propagation-rad}\n\\end{equation}\nAssuming that there were no changes in the relativistic degrees of freedom, $\\mathcal{H} = \\tau^{-1}$, and\nchanging variable from $h_{ij}(\\vb{k}, \\tau)$ to $\\tau h_{ij}(\\vb{k}, \\tau)$,  \\cref{eq:gw-propagation-rad} becomes the equation for a forced harmonic oscillator whose Green's function is known (see \\Refa{Caprini:2018mtu} for a review).\nThe solution and its time derivative are\n\\begin{align}\n\th_{ij}\\qty(\\vb{k}, \\tau)       & =\n\t4\\int_{\\tini}^\\tau\\frac{\\dd \\zeta}{\\zeta} \\frac{\\sin k(\\tau-\\zeta)}{k \\tau} \\tilde{\\Pi}_{ij}(\\vb{k}, \\zeta) ,\n\t\\label{eq:strain}                  \\\\\n\t\\dot{h}_{ij}\\qty(\\vb{k}, \\tau) & =\n\t4\\int_{\\tini}^{\\tau}\\frac{\\dd \\zeta}{\\zeta} \\qty[\\frac{\\cos k(\\tau-\\zeta)}{\\tau}-\\frac{\\sin k(\\tau-\\zeta)}{k\\tau^2}]\\tilde{\\Pi}_{ij}(\\vb{k}, \\zeta) .\n\t\\label{eq:hdot-short}\n\\end{align}\nwhere $\\tini$ is the conformal time at which the source turns on.\n\n\\subsection{GW energy density power spectrum}\n\\label{sec:gw-energy-ps}\n\nWe assume that the GWs are generated by a stochastic process (specifically vortical turbulence) and consist of signals coming from patches in the sky that were causally disconnected at the moment of their emission in the very early Universe.\nAssuming statistical homogeneity and isotropy, the two-point correlation function of $\\dot{h}_{ij}$ can be written as\n\\begin{equation}\n\t\\ev{\\dot{h}^*_{ij}(\\vb{k}, \\tau) \\dot{h}_{ij}(\\vb{q}, \\tau)} \\equiv\n\t(2\\pi)^3 \\delta(\\vb{q}-\\vb{k}) P_{\\dot{h}}(k, \\tau),\n\\end{equation}\nwhich defines the spectral density $P_{\\dot{h}}$.\nThe fractional GW energy density power spectrum at conformal time $\\tau$ in the radiation era is then\n\\begin{equation}\n\t\\eval{\\dv{\\Omega_\\mathrm{gw}}{\\ln k}}_\\tau =\n\t\\frac{k^3P_{\\dot{h}}(k, \\tau)}{2 (2\\pi)^3 G a^2(\\tau) \\rho(\\tau)}.\n\t\\label{eq:OmGW}\n\\end{equation}\nThe relevant statistical object of the GW source is the unequal time correlation function (UETC) of the anisotropic stress, defined by\n\\begin{equation}\n\t\\ev{\\tilde{\\Pi}_{ij}(\\vb{k}, \\tau) \\tilde{\\Pi}^*_{ij}(\\vb{q}, \\zeta)} \\equiv (2\\pi)^3\\delta(\\vb{k-q}) P_{\\tilde{\\Pi}}(k, \\tau,\\zeta).\n\t\\label{eq:Pi}\n\\end{equation}\nFrom \\cref{eq:hdot-short}, one then obtains\n\\begin{equation}\n\tP_{\\dot{h}}(k,\\tau) = \\frac{16}{\\tau^2} \\iint_{\\tini}^{\\tau}\\frac{\\dd{\\eta}}{\\eta} \\frac{\\dd{\\zeta}}{\\zeta} \\,\n\t\\mathcal{G}(k,\\tau, \\eta,\\zeta)\n\tP_{\\tilde{\\Pi}}(k, \\eta,\\zeta),\n\t\\label{eq:h-square}\n\\end{equation}\nwith\n\\begin{equation}\n\t\\mathcal{G}(k,\\tau, \\eta,\\zeta)=\n\t\\cos k(\\tau - \\eta)\\cos k(\\tau - \\zeta)-2\\frac{\\cos k(\\tau - \\eta)\\sin k(\\tau - \\zeta)}{k\\tau}+\\frac{\\sin k(\\tau - \\eta)\\sin k(\\tau - \\zeta)}{(k\\tau)^2}\\,.\n\t\\label{eq:green_with_all_terms}\n\\end{equation}\nThe GW energy density fraction \\cref{eq:OmGW} becomes then\n\\begin{equation}\n\t\\eval{\\dv{\\Omega_\\mathrm{gw}}{\\ln k}}_{\\tau} = \\frac{8}{3 \\pi^2}\n\tk^3\n\t\\iint_{\\tini}^{\\tau} \\mathcal{G}(k,\\tau, \\eta,\\zeta) P_{\\tilde{\\Pi}}(k, \\eta,\\zeta)\\frac{\\dd{\\eta}}{\\eta} \\frac{\\dd{\\zeta}}{\\zeta},\n\t\\label{eq:sbgw}\n\\end{equation}\nwhere we have used the Friedmann equation $3 \\mathcal{H}^2 = 8 a^2 \\pi G \\rho$.\nAs can be appreciated from \\cref{eq:sbgw}, the GW signal is  sourced by the UETC of the anisotropic stress energy of the turbulent fluid, $P_{\\tilde{\\Pi}}(k,\\tau, \\zeta)$.\n\n\n\\subsection{The unequal time anisotropic stress correlator}\n\\label{sec:uetc-stress}\n\nWe now review the computation of the UETC of the turbulent fluid anisotropic stress, defined in \\cref{eq:Pi}.\nThe starting point is the spatial, off-diagonal part of the energy momentum tensor of a relativistic fluid.\nIn order to simplify the computation, we neglect the spatial dependence of the fluid enthalpy density, and set the Lorentz factor $\\gamma = 1$.\nThis amounts to assuming non-relativistic turbulence.\nAs we shall see, we find an excellent agreement between the semi-analytical results based on the non-relativistic source model and the relativistic simulations, also in what concerns the GW signal.\nTherefore,\nthe assumption of non-relativistic turbulence does not significantly affect our final results, within the tested range of initial rms velocity.\nWe then approximate\n\\begin{equation}\n\tT_{ij} (\\vb{x}, \\tau) \\approx (\\rho + p)  v_i(\\vb{x}, \\tau) v_j(\\vb{x}, \\tau), \\quad \\text{for } i\\neq j.\n\t\\label{eq:Tlm}\n\\end{equation}\nOn using \\cref{eq:Piij,eq:Piij_proj},\nthe two point correlation of the normalized anisotropic stress becomes\n\\begin{multline}\n\t\\ev{\\tilde{\\Pi}_{ij}(\\vb{k}, \\tau) \\tilde{\\Pi}^*_{ij}(\\vb{k'}, \\zeta)} = \\\\\n\t\\Lambda_{ij\\ell m}(\\vu{k}) \\Lambda_{ijrs}(\\vu{k'})\\int \\frac{\\dd[3]{p}}{(2\\pi)^3} \\frac{\\dd[3]{h}}{(2\\pi)^3}\n\t\\ev{v_\\ell(\\vb{p}, \\tau) v_m (\\vb{k-p}, \\tau) v_r^*(\\vb{h}, \\zeta) v_s^* (\\vb{k'-h}, \\zeta) }.\n\\end{multline}\nWe assume Gaussianity of the velocity field to rewrite the four-point velocity correlator in terms of two-point correlators using Wick's theorem:\n\\begin{equation}\n\\begin{split}\n  \\ev{\\tilde{\\Pi}_{ij}(\\vb{k}, \\tau) \\tilde{\\Pi}^*_{ij}(\\vb{k'}, \\zeta)}  = &\n\t  \\Lambda_{ij\\ell m}(\\vu{k}) \\Lambda_{ijrs}(\\vu{k'})\n\t\\int \\frac{\\dd[3]{p}}{(2\\pi)^3} \\frac{\\dd[3]{h}}{(2\\pi)^3} \\\\\n\t& \\quad [ \\ev{v_\\ell(\\vb{p}, \\tau) v_m (\\vb{k-p}, \\tau)} \\ev{v_r^*(\\vb{h}, \\zeta) v_s^* (\\vb{k'-h}, \\zeta) } \\\\\n\t&  \\qquad + \\ev{v_\\ell(\\vb{p}, \\tau) v_r^*(\\vb{h}, \\zeta) } \\ev{v_m (\\vb{k-p}, \\tau)  v_s^* (\\vb{k'-h}, \\zeta) } \\\\\n\t & \\qquad + \\ev{v_\\ell(\\vb{p}, \\tau) v_s^* (\\vb{k'-h}, \\zeta} \\ev{v_m (\\vb{k-p}, \\tau) v_r^*(\\vb{h}, \\zeta)  }]. \\label{eq:Piquasinorm}\n\\end{split}\n\\end{equation}\nSimilarly to the anisotropic stress UETC of \\cref{eq:Pi}, we can define the turbulent velocity UETC.\nUnder the further assumption that the velocity field is divergence-free, statistically isotropic and homogeneous, the velocity UETC is\n\\begin{equation}\n\t\\ev{v_i(\\vb{k}, \\tau) v_j^*(\\vb{q}, \\zeta)} \\equiv (2\\pi)^3 \\delta\\qty(\\vb{k} - \\vb{q}) \\bot_{ij}(\\vu k) \\UETCv(k, \\tau, \\zeta).\n\t\\label{eq:velocity-ps}\n\\end{equation}\nThen on using $\\Lambda_{ij\\ell m}(\\vu{k}) \\Lambda_{ijrs}(\\vu{k}) = \\Lambda_{\\ell m r s}(\\vu{k})$, \\cref{eq:Piquasinorm} becomes\n\\begin{multline}\n\t\\ev{\\tilde{\\Pi}_{ij}(\\vb{k}, \\tau) \\tilde{\\Pi}^*_{ij}(\\vb{k'}, \\zeta)} = \\\\\n\t\\Lambda_{\\ell m r s}(\\vu{k}) \\delta(\\vb{k-k'})\n\t\\int \\dd[3]{p}\n\t\\UETCv(p, \\tau, \\zeta) \\UETCv(q, \\tau, \\zeta)\\qty[  \\bot_{\\ell r}(\\vu{p}) \\bot_{m s}(\\vu{q})\n\t\t+ \\bot_{\\ell s}(\\vu{p})\\bot_{m r}(\\vu{q})  ],\n\t\\label{eq:stress-full}\n\\end{multline}\nwhere $\\vb{q} = \\vb{k} - \\vb{p}$ (note that\nthe first term in \\cref{eq:Piquasinorm} does not contribute).\nThe contraction over indices can be carried out explicitly using \\cref{eq:Piij_proj,eq:lambda_proj,eq:PSsim},\n\\begin{align}\n\t2 \\Lambda_{\\ell m r s}(\\vu{k}) \\bot_{\\ell r}(\\vu{p}) \\bot_{m s}(\\vu{q}) & = 1 + 2 \\qty[(\\vu{k}\\vdot \\vu{p} )^2 + (\\vu{k}\\vdot \\vu{q})^2] + (\\vu{k}\\vdot \\vu{p} )^2  (\\vu{k}\\vdot \\vu{q})^2 \\label{eq:projection-niksa} \\\\\n\t2 \\Lambda_{\\ell m r s}(\\vu{k}) \\bot_{\\ell s}(\\vu{p}) \\bot_{m r}(\\vu{q}) & = 1 + (\\vu{k}\\vdot \\vu{p} )^2  (\\vu{k}\\vdot \\vu{q})^2 \\label{eq:projection-missing},\n\\end{align}\nso that \\cref{eq:velocity-ps} becomes\n\\begin{multline}\n\t\\ev{\\tilde{\\Pi}_{ij}(\\vb{k}, \\tau) \\tilde{\\Pi}^*_{ij}(\\vb{k'}, \\zeta)} = \\\\\n\t\\delta(\\vb{k-k'})\n\t\\int \\dd[3]{p} \\UETCv(p, \\tau, \\zeta) \\UETCv(q, \\tau, \\zeta)\n\t\\qty[1 + (\\vu{k}\\vdot \\vu{p} )^2 + (\\vu{k}\\vdot \\vu{q})^2 + (\\vu{k}\\vdot \\vu{p} )^2  (\\vu{k}\\vdot \\vb{q})^2].\n\\end{multline}\nWe can now extract the kernel of the anisotropic stress defined in \\cref{eq:Pi} and obtain\n\\begin{equation}\n\tP_{\\tilde{\\Pi}}(k, \\tau, \\zeta) =\n\t\\frac{1}{(2\\pi)^3}\n\t\\int \\dd[3]{p} \\UETCv(p, \\tau, \\zeta) \\UETCv(q, \\tau, \\zeta)\n\t\\qty[1 + (\\vu{k}\\vdot \\vu{p} )^2] \\qty[1 + (\\vu{k}\\vdot \\vu{q})^2].\n\t\\label{eq:unequal-stress-final}\n\\end{equation}\nThis result agrees with \\Refa{Caprini:2009yp}, but differs from \\Refa{Niksa:2018ofa}. The reason is that in \\Refa{Niksa:2018ofa}, the tensor indices on the two terms in square brackets on the right-hand side of \\cref{eq:stress-full} were assumed identical, but this is not the case as can be seen explicitly in \\cref{eq:projection-niksa}\nand \\cref{eq:projection-missing}.\n\nThe following \\cref{sec:free-decay-model,sec:PosKer} are devoted to the construction of a model for the velocity UETC $\\UETCv(p, \\tau, \\zeta)$ in \\cref{eq:unequal-stress-final}.\n\n\n\\section{Modelling freely decaying turbulence}\n\\label{sec:free-decay-model}\n\n\nIn this section, we provide definitions and review properties of freely decaying turbulence.\nThis will be complemented with new theoretical input regarding positive kernels in \\cref{sec:PosKer}, leading to the new velocity UETC model which we propose in this work.\nOur new model will be validated with relativistic, direct numerical simulations in \\cref{sec:results-vel}.\n\n\\subsection{Velocity power spectrum}\n\\label{sec:free-decay-velps}\n\nThe real-space two-point correlation function of a statistically homogeneous, isotropic and divergence-free velocity field reads~\\cite{Davidson:2004}\n\\begin{equation}\n\tb_{ij}(\\vb{r}, \\tau)=\\frac{\\ev{v_i(\\vb{x},\\tau)v_j(\\vb{x}+\\vb{r},\\tau)}}{\\overline{v}^2(\\tau)}\n\t\\equiv\\Sigma_t(r,\\tau)\\bot_{ij}(\\vu{r})+\\Sigma_\\ell(r,\\tau)\\hat r_i\\hat r_j,\\label{eq:SigmaGamma}\n\\end{equation}\nwhere $\\overline{v}^2(\\tau)\\equiv \\ev{v_i(\\vb{x},\\tau)v_i(\\vb{x},\\tau)}$ is the rms velocity squared, and the functions $\\Sigma_t$ and $\\Sigma_\\ell$ represent respectively the transverse and longitudinal correlation functions.\nSince the velocity field is divergence-free, we have that $\\pdv*{b_{ij}}{r_i} = 0$ and the correlation functions are related through\n\\begin{equation}\n\t\\Sigma_\\ell'(r)=\\frac{2}{r}[\\Sigma_t(r)-\\Sigma_\\ell(r)].\n\t\\label{eq:gamma-prime}\n\\end{equation}\nThe spectral density $\\Psdv(k,\\tau)$\n\\begin{equation}\n\t\\ev{v_i(\\vb{k}, \\tau) v_j^*(\\vb{q}, \\tau)} \\equiv (2\\pi)^3 \\delta\\qty(\\vb{k} - \\vb{q}) \\bot_{ij}(\\vu k) \\Psdv(k, \\tau)\n\t\\label{eq:velocity-sd}\n\\end{equation}\n(comparing with \\cref{eq:velocity-ps}, note that we denote quantities at equal time with one time variable only, for conciseness) can be expressed in terms of the correlation functions $\\Sigma_\\ell$ and $\\Sigma_t$ by matching the Fourier transform of \\cref{eq:velocity-sd} and the trace of \\cref{eq:SigmaGamma} \\cite{Caprini:2006jb}:\n\\begin{equation}\n\t\\Psdv(k,\\tau)=\\overline{v}^2(\\tau)\\int \\dd[3]{r}\\, \\e^{i\\vb{k}\\cdot\\vb{r}}\\qty[\\Sigma_t(r,\\tau)+\\frac{1}{2}\\Sigma_\\ell(r,\\tau)].\n\\end{equation}\nIn spherical coordinates and using \\cref{eq:gamma-prime}, this becomes\n\\begin{align}\n\t\\Psdv(k,\\tau) = &\n\t\\; 2 \\pi \\,\\overline{v}^2(\\tau) \\int \\dd{r}\\qty[ r^3 \\Sigma_\\ell'(r,\\tau) + 3 r^2 \\Sigma_\\ell(r,\\tau)] \\frac{\\sin kr}{kr}  \\label{eq:calP-1d}\n\t\\\\\n\t=               & \\; 2 \\pi\\, \\overline{v}^2(\\tau) \\int \\dd{r}\\qty[ r^3 \\Sigma_\\ell'(r,\\tau) + 3 r^2 \\Sigma_\\ell(r,\\tau)] \\qty[1 - \\frac{(kr)^2}{6} + \\order{(kr)^4}].\n\t\\label{eq:spectral-large-scale}\n\\end{align}\nWe assume that the correlation function of \\cref{eq:SigmaGamma} has compact support and vanishes outside the causal horizon.\nAs a consequence, the leading order term in \\cref{eq:spectral-large-scale} is $\\eval{r^3 \\Sigma_\\ell}_0^\\infty = 0$, and the spectral density on large scales is given by\n\\begin{equation}\n\t\\Psdv(k\\rightarrow 0,\\tau) = \\frac{2\\pi}{3}\\overline{v}^2(\\tau) k^2\\int_0^\\infty \\dd{r} r^4\\,\\Sigma_\\ell(r,\\tau) + \\order{(kr)^4} \\,.\\label{eq:Pvcausal}\n\\end{equation}\nTherefore, causality in the early Universe points to Batchelor turbulence, at least on super-horizon scales~\\cite{Caprini:2006jb}.\nThe coefficient of the amplitude of the large-scale part of the spectrum in \\cref{eq:Pvcausal} is the Loitsyansky integral $\\int \\dd{r} r^4 \\Sigma_\\ell(r)$.\nThis quantity was conjectured to be constant in time in the absence of long-range correlations   (see \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot \\Refa{Davidson:2004}), which would then be consistent with the causality requirement.\n\nThe power spectrum $\\Psv(k)$ encodes how\n$\\overline{v}^2$, or equivalently the non-relativistic kinetic energy per unit enthalpy,\nis distributed into the different length-scales:\n\\begin{equation}\n\t\\overline{v}^2(\\tau) \\equiv\n\t\\int \\frac{\\dd k}{k} \\Psv(k, \\tau).\n\t\\label{eq:kinetic-energy}\n\\end{equation}\nIn our notation, it is related to the spectral density through\n\\begin{equation}\n\t\\Psv(k,\\tau) = \\frac{k^3}{\\pi^2}\\Psdv(k,\\tau).\n\t\\label{eq:velocity_ps_mathcal}\n\\end{equation}\nNote that the power spectrum as defined here is related to the energy spectrum $E(k,\\tau)$ as defined in \\Refa{Davidson:2004} through\n\\begin{equation}\n\t\\Psv(k,\\tau) = 2kE(k,\\tau).\n\\end{equation}\nTo characterize the typical length-scale of the system, we also define the integral scale $\\xi$ in terms of the longitudinal correlation function, or equivalently in terms of the power spectrum\n\\begin{equation}\n\t\\xi(\\tau)\\equiv \\int_0^\\infty \\dd r\\,\\Sigma_\\ell(r,\\tau) = \\frac{\\pi}{4\\bar v^2} \\int k^{-1} \\Psv(k, \\tau) \\dd{\\ln k}.\n\t\\label{eq:integral-scale}\n\\end{equation}\nWe use the integral scale to define  dimensionless,  time-dependent wavenumbers\n\\begin{equation}\n    K(\\tau) \\equiv \\mathcal{A}\\, k\\, \\xi(\\tau)\\, ,\n\\end{equation} with $\\mathcal{A}$ a normalization constant given in \\cref{eq:constants1} below.\n\nConcerning the shape of the velocity power spectrum, we assume in our modelling\nthat\nin the inertial range it is determined by the well-known Kolmogorov $K^{-2\/3}$ law \\cite{1941DoSSR..30..301K}, and on large scales by the causal $K^5$ slope, motivated by \\cref{eq:Pvcausal}.\nThese properties are captured by the\nvon K\\'arm\\'an\\ spectrum \\cite{von_karman_progress_1948,Davidson:2004,Caprini:2009yp, Niksa:2018ofa}\n\\begin{equation}\n\t\\Psv\\qty(k, \\tau) = \\mathcal{B}\\, \\overline{v}^2(\\tau) \\frac{K^5(\\tau)}{\\qty[1 + K^2(\\tau)]^{17\/6}}.\n\t\\label{eq:power-spectrum}\n\\end{equation}\nThe coefficients $\\mathcal{A}$ and $\\mathcal{B}$ are chosen to ensure that the definitions of the kinetic energy and the integral scale of \\cref{eq:kinetic-energy,eq:integral-scale} are consistent\\footnote{Note that in \\Refa{Niksa:2018ofa}, the authors introduce a factor of $5\/12$ in the denominator of the velocity power spectrum: their motivation was to localize the peak of the spectrum around $K=1$. We do not follow this convention here. Other initial spectral shapes, derived from simulations, are also used for example in \\Refs{Brandenburg:2017neh,Pol:2019yex,RoperPol:2022iel}.}:\n\\begin{align}\n\t\\mathcal{A} &\\equiv \\frac{55 \\Gamma(1\/3)}{12 \\sqrt{\\pi} \\Gamma(17\/6)} \\approx 4.02\n\t\\label{eq:constants1}\\\\\n\t\\mathcal{B} &\\equiv \\frac{8  \\Gamma(17\/6)}{3 \\sqrt{\\pi} \\Gamma(1\/3)} \\approx 0.97.\n\t\\label{eq:constants2}\n\\end{align}\n\n\\cref{eq:power-spectrum} assumes that the von K\\'arm\\'an~spectral shape is preserved during the time evolution of the turbulent field.\nThis power spectrum will be used for the semi-analytical evaluation of the GW signal (see~\\cref{sec:numintegration}): self-similarity in time is therefore an underlying assumption of the resulting GW spectra.\nFurthermore, \\cref{eq:power-spectrum} will also be used as the initial condition for our hydrodynamical simulations.\nThe simulations will show, on the other hand, that the von K\\'arm\\'an~spectral shape is not fully preserved by time evolution: while for $K \\gtrsim 1$, a power law consistent with $K^{-2\/3}$ returns after $\\order{10}$ eddy turnover times, a shallower one than $K^5$ emerges for $K \\ll 1$ (see~\\cref{sec:ResETC}).\nNevertheless, the GW spectra output by the simulations, and those resulting from the semi-analytical integration, are in very good agreement in their common wavenumber $K$ region: we therefore anticipate that differences in the time evolution of the large-scale part of\nthe turbulent spectrum do not significantly affect the final results.\nThis is due to the fact that the production of gravitational waves is fast, as we shall see.\n\nAs a consequence of the fast GW sourcing, though, the particular choice of the initial\nvelocity power spectrum determines the final GW spectral shape.\nEquation~\\eqref{eq:power-spectrum} is the only input we use for exploring the SGWB signal from decaying turbulence.\nNevertheless, we are able to draw general lessons from our analysis.\n\n\\subsection{Evolution of the velocity field in decaying turbulence}\n\\label{sec:free-decay-evol}\n\nIn this section we provide a  very simplified review of the time evolution of the freely-decaying turbulent power spectrum, and of average quantities such as\nthe kinetic energy per unit enthalpy $\\overline{v}^2(\\tau)$ and the integral scale $\\xi(\\tau)$.\n\n\nIn line with the model of \\cref{sec:free-decay-velps}, we assume that there is only one principal length scale in the flow, $\\xi$ (see \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot \\Refa{george} for a turbulence description not satisfying this hypothesis).\nDimensional arguments then lead to (consistently with \\cref{eq:power-spectrum})\n\\begin{equation}\n\t\\Psv(k, \\tau) = \\qty(\\frac{\\overline{v}(\\tau)}{\\overline{v}_*})^2\\, \\varphi[k\\xi(\\tau)]\\,,\n\t\\label{eq:pv_scaled_one_scale}\n\\end{equation}\nwhere the function $\\varphi(k\\xi(\\tau))$ must satisfy integrability conditions following \\cref{eq:kinetic-energy,eq:integral-scale}, and $*$ indicates quantities evaluated at the start of turbulence decay.\nAt large scales, the absence of relevant dynamical processes implies a simple power law spectrum\n\\begin{equation}\n\t\\Psv[k\\xi(\\tau)\\ll 1, \\tau] \\simeq \\overline{v}^2(\\tau)\\,[k\\xi(\\tau)]^{\\beta+1}\\,,\n\\end{equation}\nwhere, for the moment, the spectral index $\\beta$ is left unspecified.\n\nFurther assuming that at large scales the power spectrum is constant in time\\footnote{This assumption excludes \\emph{a priori} the presence of a non-helical inverse cascade.\nWhile observed in simulations of MHD turbulence \\cite{Brandenburg:2014mwa,Reppin:2017uud,Christensson:2000sp} (see also \\Refs{Campanelli:2015ypt,Olesen:2015dga} for theoretical interpretations), this phenomenon seems to be absent in purely kinetic turbulence: see \\Refa{Brandenburg:2016odr} and the results presented in \\cref{sec:ResETC}.}, leads to\n\\begin{equation}\n\t\\overline{v}^2(\\tau)\\xi(\\tau)^{1+\\beta}=\\mathrm{const}\\,.\n\t\\label{eq:vxigeneral}\n\\end{equation}\nUnder this assumption, the self-similarity of the power spectrum takes the form \\cite{Christensson:2000sp,Brandenburg:2016odr}\n\\begin{equation}\n\t\\Psv(k, \\tau) = \\qty(\\frac{\\xi_*}{\\xi})^{ 1 + \\beta}\\varphi[k\\xi(\\tau)]\\,.\n\t\\label{eq:Pvxi}\n\\end{equation}\nThe individual time evolution of the rms velocity and the integral scale can be derived, if we further assume that there is only one principal timescale in the flow, the eddy turnover time $\\tau_\\xi=\\xi\/\\overline{v}$.\nWith this assumption, one can write\n\\begin{equation}\n\t\\dv{\\overline{v}^2}{\\tau} =  - c_v \\frac{\\overline{v}^2}{\\tau_\\xi},\n\t\\label{eq:kolmog}\n\\end{equation}\nwhere $c_v$ is a constant.\nTogether with \\cref{eq:vxigeneral}, one can solve \\cref{eq:kolmog} to derive\n\\begin{align}\n\t\\overline{v}^2(\\tau) = \\overline{v}_*^2 \\left(1 + \\frac{\\tau - \\tdevel}{\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}}  \\right)^{-p}\n\t\\label{eq:VelEvMod}\n\t\\\\\n\t\\xi(\\tau) = \\xi_* \\left(1 + \\frac{\\tau - \\tdevel}{\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}}  \\right)^{q}\\,,\n\t\\label{eq:XiEvMod}\n\\end{align}\nwhere the decay exponents $p$ and $q$ take the values\n\\begin{align}\n\tp& =\\frac{2(\\beta+1)}{\\beta+3}\n\t\\label{eq:pchialpha}\\\\\n\t& q=\\frac{2}{\\beta+3},\n\t\\label{eq:qchialpha}\n\\end{align}\nand $\\mathcal{N}_\\mathrm{e}$ is a constant representing the number of initial eddy turn-over times that the kinetic energy takes to decay, and the integral scale to grow.\nIn particular $p$ and $q$ satisfy\n\\begin{equation}\n\tp = (1 + \\beta)q = 2(1-q) \\,.\n\t\\label{eq:pVsq}\n\\end{equation}\nNote that this model does not predict the value of $\\beta$, although consistency with the arguments of  \\cref{sec:free-decay-velps} would indicate $\\beta=4$.\nIn \\cref{sec:ResETC} we will report on our numerical investigation of the evolution of the velocity power spectrum in the framework of the model described above.\n\n\nThe validity of the decay laws given in \\cref{eq:VelEvMod,eq:XiEvMod}, together with the actual value of the parameter $\\beta$ (and consequently of the decay law exponents through \\cref{eq:pVsq}), are currently open problems of the theory of turbulence.\nInvariance of the Navier-Stokes equations under appropriate rescaling leads to the self-similarity of the velocity power spectrum, and links the decay law exponents $p$ and $q$ to the scaling parameter \\cite{Olesen:1996ts,Ditlevsen:2002xs}.\nThe scaling parameter can then be related to the slope of the initial velocity power spectrum at large scales, if one assumes that the latter also satisfies the self-similarity conditions (see \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot \\Refa{Campanelli:2007tc}).\nIn this context, an initial power spectrum leading to $\\beta=4$, as motivated in \\cref{sec:free-decay-velps}, provides $p$ and $q$ values consistent with the classical Kolmogorov theory, namely $p=10\/7$ and $q=2\/7$ \\cite{landau2013fluid,Davidson:2004} (see also \\cite{2010PhST..142a4003T} for a numerical result).\nThe relation between the scaling parameter and the slope of the initial power spectrum at large scales has also been derived in the context of renormalization group analyses of the incompressible MHD equations, as holding at the fixed point \\cite{camargo_tasso,Shiromizu:1998he}.\nHowever, the initial velocity power spectrum does not need to necessarily satisfy either the self-similarity condition, nor does it need to exhibit power law behaviour.\nRelaxing these assumptions changes the prediction on the subsequent free decay model, disconnecting the value of $\\beta$ from the initial conditions \\cite{Campanelli:2015ypt,Brandenburg:2016odr}.\nRecently, new models of the turbulent free-decay have emerged, following which the turbulence decay is governed by the energy-containing scales, close to the integral one, rather than by the asymptotic behaviour at large scales \\cite{george,doi:10.1063\/1.4901448,schaefer,meldi:hal-01298925,Zhou:2021vah}.\nAccording to these findings there would be then no universal decay regime, since the details of the spectral shape near the peak are related to the turbulence production mechanism.\n\nNote that in both this Section and \\cref{sec:free-decay-velps} we have neglected the dissipative scales.\nThe turbulent kinetic energy dissipates at the viscous scale $\\lambda$, at which the Reynolds number is $\\mathrm{Re}(\\lambda)=v_\\lambda \\,\\lambda\\, a\/\\nu=1$, where $\\nu$ denotes the kinematic viscosity.\nThe viscous scale $\\lambda$ increases with time, and the turbulent velocity on a given scale $2\\pi\/k$ goes to zero once the dissipation scale has grown to $\\lambda(\\tau)=2\\pi\/k$.\nFurthermore, $\\lambda(\\tau)$ grows faster than the integral scale $\\xi(\\tau)$: the turbulent inertial range is therefore fully dissipated when the two scales become equal, so that $\\mathrm{Re}(\\xi)=1$.\nAs previously demonstrated \\cite{Caprini:2009pr,Caprini:2009yp}, turbulence is expected to free-decay for several Hubble times, before it is fully dissipated. For example, with the initial set of parameters $T_*= 100~\\mathrm{GeV}$, $\\overline{v}_*= 0.03$, $\\xi_*\\mathcal{H}_*= 0.01$, $q=2\/7$, $p=10\/7$, one obtains $\\mathrm{Re}(\\xi)=1$ at $T_\\mathrm{fin}\\sim 0.3~\\mathrm{GeV}$, corresponding to about $1000~\\tau_{\\xi_*}$.\nThe kinetic viscosity in the early Universe is indeed extremely small: for the same initial conditions, one finds $\\mathrm{Re}(\\xi_*)\\sim 10^{11}$, and the inertial range spans about $8$ orders of magnitude in wavenumber.\nIn principle, we should have inserted a cutoff at $k>2\\pi\/\\lambda(\\tau)$ in the turbulent power spectrum given in \\cref{eq:power-spectrum}, and we should account, in the GW production, for the viscous dissipation of the kinetic energy at each scale $k$ in addition to the overall free decay described by \\cref{eq:VelEvMod,eq:XiEvMod}.\nHowever, as we will discuss in \\cref{sec:gw-instant}, the bulk of the GW signal is generated faster than the turbulent decay: therefore, the dynamics at the viscous scale plays no role in practice in the present analysis, and it is justified to neglect it altogether.\n\n\n\\subsection{Unequal time correlator of the velocity field}\n\\label{sec:UETCvelocity}\n\nTogether with the overall free-decay, we also need to model the time decorrelation of the turbulent velocity field.\nThis is encapsulated by the normalized velocity UETC\n\\begin{equation}\n\t\\label{eq:NorVelUETC}\n\t\\tdec(k, \\tau, \\zeta)  \\equiv \\frac{\\UETCv(k,\\tau,\\zeta)}{\\sqrt{\\Psdv(k,\\tau)\\,\\Psdv(k,\\zeta)}}.\n\\end{equation}\nThe first model to have been developed was Kraichnan's random sweeping approximation \\cite{kraichnan:1964},\nin which the decorrelation function in the inertial range is a Gaussian in the variable $k\\overline{v}|\\tau - \\zeta| $,\nfor small time differences.\nThe thinking behind the model is that modes are decorrelated by being\n``swept'' by the large-scale flow, whose rms velocity is $\\overline{v}$.\nThis model has good numerical support in the inertial range for both stationary and decaying turbulence (see for example \\Refs{sanada_random_1992,he_computation_2004,Gorbunova:2021cpn}).\nWe review it in \\cref{sec:kraichnan}.\n\nHowever, for GW production calculations, we need to model the decorrelation around the peak of the velocity power spectrum.\nTo this end, we study --- for the first time with hydrodynamical simulations --- the decorrelation of the turbulent velocity field outside the inertial range, as far as possible in the low wavenumber part of the spectrum.\nThe simulation results are presented in \\cref{sec:results-uetc}, and show that a good model is obtained by replacing the rms velocity in the Gaussian with a time and scale dependent average decorrelation velocity, based on\nthe sweeping velocity proposed in Ref.~\\cite{kaneda_lagrangian_1993}, which we present in \\cref{sec:kaneda}.\n\nFurthermore, we also need to take into account the formal requirement that the UETCs (velocity and anisotropic stress) must be positive definite kernels, which is not manifest in the simple Gaussian model.\nThis will be presented in \\cref{sec:PosKer}.\n\n\\subsubsection{Kraichnan random sweeping approximation}\n\\label{sec:kraichnan}\n\nThe random sweeping approximation provides an explanation for the origin of the Gaussian functional form for the decorrelation.\nKraichnan's sweeping hypothesis is based on the assumption that\nFourier modes of the velocity field\nin the inertial range are advected without deformation by a locally uniform velocity field $\\vb{V}$, which may be time-dependent\n\\cite{kraichnan:1964}.\nThis amounts to neglecting the non-linear terms and writing\n\\begin{equation}\n\t\\pdv{\\vb{v}}{\\tau} + i \\qty[\\vb{k} \\vdot \\vb{V}(\\tau)] \\vb{v} \\simeq 0.\n\t\\label{eq:sweeping-ode}\n\\end{equation}\nSince $\\vb{V}$ is locally uniform, the evolution of modes with different wavenumbers \\(\\vb{k}\\) is decoupled.\nThe turbulent velocity field can then be explicitly integrated: from \\cref{eq:sweeping-ode} one finds\n\\begin{equation}\n\t\\vb{v}(\\vb{k}, \\tau) \\simeq \\vb{v}(\\vb{k},\\zeta) \\exp(-i\\int_{\\zeta}^{\\tau}\\vb{k}\\vdot \\vb{V}(s) \\dd{s}),\n\\end{equation}\nwhere $\\zeta < \\tau$.\nUnder the assumption that the locally uniform velocity field $\\vb{V}$ is statistically independent of the turbulent velocity field $\\vb{v}$ at the initial time, the unequal time correlator of $\\vb{v}$ can be expressed as\n\\begin{equation}\n\t\\UETCv(k,\\tau,\\zeta) \\simeq \\Psdv(k,\\zeta) \\ev{\\exp(-i\\int_{\\zeta}^{\\tau}\\vb{k}\\vdot \\vb{V}(s) \\dd{s})}.\n\\end{equation}\nThe average of the exponential can be calculated assuming that its argument is a Gaussian random variable.\nIt then follows that the velocity field decorrelates with a characteristic Gaussian law\\footnote{We recall that for a Gaussian random variable, \\begin{equation}\n\t\t\\ev{\\exp(X)} = \\ev{\\sum_{n=0}^{\\infty} \\frac{X^{n}}{n!}} = \\sum_{k=0}^{\\infty}\\frac{\\ev{X^{2}}^{k}}{2^{k} k!} = \\exp(\\frac{\\ev{X^{2}}}{2}).\n\t\\end{equation}},\n\\begin{equation}\n\t\\UETCv(k,\\tau,\\zeta) \\simeq \\Psdv(k,\\zeta) \\exp\\qty(- \\frac{1}{2} \\ev{X^2}),\n\\end{equation}where\n\\begin{equation}\n\tX^{2} \\equiv \\iint_{\\zeta}^{\\tau} (\\vb{k} \\vdot \\vb{V}(s))(\\vb{k} \\vdot \\vb{V}(s')) \\dd{s} \\dd{s'}.\n\\end{equation}\nThis expression can be further simplified assuming that on average  the locally uniform velocity has the same amplitude in the three spatial directions \\cite{kraichnan:1964}, leading to\n\\begin{equation}\n\t\\UETCv(k,\\tau,\\zeta) \\simeq \\Psdv(k,\\tau) \\exp\\qty[-\\frac{1}{2} k^2 v_\\mathrm{sw}^2(\\tau, \\zeta)(\\tau - \\zeta)^2],\n\t\\label{eq:expvsweep}\n\\end{equation}\nwhere $v_\\mathrm{sw}(\\tau,\\zeta)$ is the average sweeping velocity which takes the form \\cite{he_computation_2004,dong_study_2008}\n\\begin{equation}\n\tv_\\mathrm{sw}^{2}(\\tau, \\zeta) = \\frac{1}{3 (\\tau-\\zeta)^{2}}\\iint_{\\zeta}^{{\\tau}} \\ev{\\vb{V}(s)\\cdot \\vb{V}(s')} \\dd{s} \\dd{s'}.\n\t\\label{eq:sweep-equal-time}\n\\end{equation}\nKraichnan's model assumes stationary turbulence\nand sets the sweeping velocity to the rms velocity divided by $\\sqrt{3}$, under the hypothesis of statistical isotropy \\cite{kraichnan:1964,wilczek_wave-numberfrequency_2012}.\nSince this model has good numerical support in the inertial range \\cite{sanada_random_1992,he_computation_2004,Gorbunova:2021cpn}, any good model should reduce to\n\\begin{equation}\n\tv_\\mathrm{sw}^{2} (\\tau, \\tau) = C_v^2\\,\\frac{ \\overline{v}^2(\\tau)}{3}\n\t\\label{eq:Cv2}\n\\end{equation}\nat equal time and in the inertial range, with $C_v^2$ a numerical factor whose value will be inferred from  simulations.\nAs presented in \\cref{sec:results-uetc},  simulations indicate $C_v^2 \\simeq 1$, providing a good test of the random sweeping approximation and of statistical isotropy.\n\n\\subsubsection{Scale dependent sweeping velocity}\n\\label{sec:kaneda}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics{figures\/ir-sweeping.pdf}\n\t\\caption{Extension of the decorrelation velocity at scales larger than the integral scale, following the model given in \\cref{eq:Vlarge}. It interpolates smoothly between $\\overline{v}^2\/3$ in the inertial range as in \\cref{eq:Cv2} and $2\\overline{v}^2\/15$ on large scales. These limits are shown with the black dashed lines.}\n\t\\label{fig:IRsweep}\n\\end{figure}\n\n\n\n\nThe random sweeping approximation reviewed in \\cref{sec:kraichnan}\napplies in the inertial range.\nHowever, in order to evaluate the GW generation, we need to model the time decorrelation of the velocity field on all scales (see~for example \\cref{eq:unequal-stress-final}).\nThe decorrelation dynamics at scales larger than the integral scale in freely decaying turbulence have received little attention: numerical studies such as those performed in \\Refs{he_computation_2004,dong_study_2008}, for example, study decorrelation only in the inertial range.\nThe largest scale analysed in  \\Refa{sanada_random_1992} is $k=1$, corresponding to the scale of the forcing, namely the peak of the power spectrum: it appears from this analysis that even this scale decorrelates more slowly than those of the inertial range.\n\nIn the context of the literature dedicated to GW production by turbulence, various decorrelation models have been proposed, see  \\Refs{Gogoberidze:2007an,Caprini:2009yp,Niksa:2018ofa}.\n\\Refa{Caprini:2009yp} assumed that the large scales do not decorrelate. The only time dependence of the velocity spectrum outside the inertial range was therefore due to the free decay, and an exponential decorrelation was inserted for wavenumbers in the inertial range by means of a step function. This introduced a non-physical discontinuity (see~Eq.~(57) of \\cite{Caprini:2009yp}).\nFurthermore, as pointed out in \\Refa{Niksa:2018ofa}, \\Refs{Gogoberidze:2007an,Caprini:2009yp} used the Lagrangian eddy turnover time as typical decorrelation time,\nwhich is scale-dependent in the inertial range, and therefore\nin conflict with the numerical results of \\Refs{sanada_random_1992,he_computation_2004}.\n\n\\Refa{Niksa:2018ofa} recommended the use of the\nmodel of  Ref.~\\cite{kaneda_lagrangian_1993},\nwhich improves Kraichnan's random sweeping approximation by taking interactions between modes into account.\nThe result is an Eulerian eddy turnover time which is weakly scale-dependent for wavenumbers up to $\\order{10^2}$ times that corresponding to the integral scale.\nWe adopt the same form for the Eulerian eddy turnover time\nto construct our model of the velocity UETC, but\ncorrect it to ensure the mathematical consistency of\nthe correlation, as discussed in \\cref{sec:PosKer}.\n\n\nFollowing \\Refs{kaneda_lagrangian_1991,kaneda_lagrangian_1993},\nwe define an instantaneous scale-dependent decorrelation velocity\n\\begin{equation}\n\tv_\\mathrm{dc}^2(k, \\tau) \\equiv\n\t\\int_0^\\infty h\\qty(\\frac{q}{k}) \\Psv(q, \\tau) \\dd{\\ln q},\n\t\\label{eq:Vlarge}\n\\end{equation}\nwhere the function $h(x)$ is given by\\footnote{Note that the function $h(x)$ of \\Refs{kaneda_lagrangian_1991,kaneda_lagrangian_1993} has a factor of $2$ difference due to our definition of the power spectrum in \\cref{eq:kinetic-energy}. }\n\\begin{equation}\n\th(x) = \\frac{1}{48}(13-8x^2+3x^4) + \\frac{1}{32 x} (1-x^2)^3 \\ln \\frac{1+x}{\\abs{1-x}}.\n\t\\label{eq:weird-h}\n\\end{equation}\nNote that the calculations performed in \\Refa{kaneda_lagrangian_1993} in principle apply only to the inertial range.  We will see that extending it to lower wavenumbers is justified by the numerical data.\n\n\\cref{eq:Vlarge} reduces to $v_\\mathrm{dc}^2(\\tau)\\simeq \\overline{v}(\\tau)^2 \/ 3$ far in the inertial range, consistent with the result of the random sweeping hypothesis, and gives $C_v^2=1$ (see~\\cref{eq:Cv2}).\nHowever, it also has another term, relevant at scales closer to the energy containing scale $\\xi$, which is proportional to $- v_k^2\\sim - \\Psv(k)$.\nThis negative contribution comes from the analysis at small time intervals and appears to slow down the process of decorrelation.\nIn order to account for decorrelation at all scales, one can therefore attempt to use \\cref{eq:Vlarge} as an extension of the sweeping velocity \\cite{Niksa:2018ofa}.\nPushing \\cref{eq:Vlarge} to large scales, one can appreciate that $v_\\mathrm{dc}(k,\\tau)$ provides a continuous interpolation from $\\overline{v}^2 \/ 3$ in the inertial range to $2\\overline{v}^2 \/ 15$ at large scales, as shown in \\cref{fig:IRsweep}.\nIn the semi-analytical GW evaluation, we will solve the integral of \\cref{eq:Vlarge} numerically, although a good analytical fit to $v_\\mathrm{dc}(k,\\tau)$ is (we recall that $K=\\mathcal{A}k\\xi(\\tau)$)\n\\begin{equation}\n\tv_\\mathrm{dc}^2(k, \\tau) \\simeq \\frac{\\overline{v}^2}{3} \\qty(\\frac{1+0.2 K}{\\sqrt{5\/2} + 0.2 K})^2,\n\t\\label{eq:vdcapprox}\n\\end{equation}\nthat we compare to the numerical result in \\cref{fig:IRsweep} (note that this fit differs slightly from what was given in \\Refa{Niksa:2018ofa}).\n\nIn order to describe how the relevant scales in the velocity spectrum decorrelate, one might substitute, in the exponential of \\cref{eq:expvsweep}, the decorrelation velocity given in \\cref{eq:Vlarge}.\nThis is the approach adopted in \\Refa{Niksa:2018ofa}.\nHowever, this breaks the time symmetry of the UETC $\\UETCv(k,\\tau,\\zeta)$ (see~\\cref{eq:velocity-ps}). To restore this symmetry,\nthe decorrelation velocity entering the exponential in \\cref{eq:expvsweep} must depend on both times. \nThe next section is devoted to ensuring the symmetry and positivity requirements that are necessary to model a valid UETC.\n\n\n\n\\section{Unequal time correlators as positive kernels}\n\\label{sec:PosKer}\n\nThe two-point correlation function of any random variable must be a positive kernel \\cite{Genton:2002}, in other words it must satisfy certain properties --- most notably Mercer's theorem.\nThis applies to the turbulent velocity field UETC \\cref{eq:velocity-ps}, as well as to  $P_{\\tilde{\\Pi}}(k, \\tau,\\zeta)$ \\cref{eq:Pi},  describing the random anisotropic stresses arising from the turbulent field.\nThe model of the velocity field --- and consequently of the anisotropic stress --- UETCs that we construct in this section ensures that they are positive kernels.\nAccording to Mercer's theorem, this implies that the GW energy density power spectrum is indeed also positive, see~\\cref{eq:sbgw}.\nThis issue had already been raised in \\Refa{Caprini:2009yp}, but it is analysed here in more depth directly on the velocity field UETC.\n\n\n\\subsection{The unequal time correlation function as a non-stationary Gaussian kernel}\n\\label{sec:mercer}\n\nAs we have discussed in \\cref{sec:free-decay-model}, the turbulent rms velocity and  integral scale are evolving in time, while the turbulent source decorrelates.\nGWs are therefore generated by a non-stationary, random process, whose UETC we need to model.\nTo do so, we make use of some general properties of two-point correlators of stochastic processes, which we review in the following.\n\nThe two-point correlator of a complex valued\nstochastic process $\\phi(\\tau)$ is defined by\n\\begin{equation}\n\tK(\\tau_1, \\tau_2) \\equiv \\ev{\\phi^*(\\tau_1) \\phi(\\tau_2)}.\n\\end{equation}\nWe assume that the real and imaginary part of the stochastic process are uncorrelated, in which case $K(\\tau_1, \\tau_2)$ is real.\n$K(\\tau_1, \\tau_2)$ is a positive semi-definite kernel which satisfies several properties: symmetry $K(\\tau_1, \\tau_2)=K(\\tau_2, \\tau_1)$; the Cauchy-Schwartz inequality; and the Mercer condition (see \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot~\\Refa{Genton:2002})\n\\begin{equation}\n\t\\iint_{I\\times I} \\dd{\\tau_1} \\dd{\\tau_2} f^*(\\tau_1) f(\\tau_2) K(\\tau_1,\\tau_2)\n\t\\geq 0\\,,\n\t\\label{eq:mercer-condition}\n\\end{equation}\nfor any interval $I$.\nThis follows from the fact that, for any well-behaved function $f$, the stochastic variable\n$X = \\int \\dd{\\tau} f(\\tau) \\phi(\\tau)$ must have positive variance $\\langle \\abs{X}^2 \\rangle \\geq 0$.\nOne can show that the linear combination and\nmultiplication of two kernels yield another kernel~\\cite{1909RSPTA.209..415M}.\n\n\nThe velocity UETC $\\UETCv(k,\\tau,\\zeta)$ is the two-point correlation function of the stochastic process $v_i(\\vb{k}, \\tau)$ (see~\\cref{eq:velocity-ps}), and is therefore a positive semi-definite kernel.\nIn \\cref{sec:UETCvelocity} we analysed the time evolution and decorrelation properties of the velocity field,\nbased on our understanding of the turbulence physical process.\nIn this section, we explain how the fact that a UETC must be a positive semi-definite kernel leads to our choice of the decorrelation velocity  at unequal times $v_\\mathrm{dc}(k,\\tau,\\zeta)$, and how to symmetrize $\\UETCv(k,\\tau,\\zeta)$.\n\n\nThe simplest example of a positive semi-definite kernel is the separable kernel $K(\\tau_1, \\tau_2) = g^*(\\tau_1) g(\\tau_2)$, which\nfactorizes Mercer's condition, \\cref{eq:mercer-condition}, into a modulus.\nThis type of kernel was already proposed in the context of GW generation by turbulence and primordial magnetic fields in Refs~\\cite{Caprini:2009yp,Caprini:2009fx,Caprini:2009pr}, where it is referred to as the coherent assumption.\nAnother common form is the stationary kernel $K(\\tau_1, \\tau_2) =K(\\tau_1-\\tau_2)$.\nStationary kernels are positive if and only if there exist a positive finite function $F$ such that~$K(\\tau_1-\\tau_2) = \\int \\cos[\\omega (\\tau_1-\\tau_2)] F(\\omega) \\dd{\\omega}$.\nFrom this result one can easily see that a kernel of the form $K(\\tau_1, \\tau_2) \\propto \\delta(\\tau_1-\\tau_2)$ is positive definite: it is referred to as incoherent in Refs.~\\cite{Caprini:2009yp,Caprini:2009pr}.\n\n\nMost importantly for us, stationary kernels also include the Gaussian kernel $$K(\\tau_1, \\tau_2) = \\exp[-\\alpha(\\tau_1-\\tau_2)^2],$$ with $\\alpha>0$, which is the form provided by the Kraichnan sweeping scenario \\cref{eq:expvsweep}.\nHowever, here we would like to model the power spectrum of freely decaying turbulence, which is, by definition, non-stationary.\nA simple but well-defined departure from stationary kernels is provided by locally stationary kernels, of the form $K(\\tau_1, \\tau_2) = K_1\\qty({\\tau_1+\\tau_2}\/{2}) K_2\\qty(\\tau_1-\\tau_2)$,\nwhere $K_1$ is a non-negative function and $K_2$ a positive stationary kernel \\cite{Silverman:1957}.\nThe UETC of a freely-decaying turbulent velocity field, though, cannot assume this form.\nThis is because the sweeping velocity depends explicitly on time, as can be seen from \\cref{eq:expvsweep}.\nFor the problem at hand, we therefore also need to go beyond local stationarity.\n\nWe do so by introducing process-convolution kernels~\\cite{higdon:1999}.\nThis specific class of non-stationary kernels is defined through a two-valued\ncomplex function $g(\\tau,u)$ as follows:\n\\begin{equation}\n\tK(\\tau_1, \\tau_2) = \\int g^*(\\tau_1, u) g(\\tau_2, u) \\dd{u}.\n\t\\label{eq:pckernel}\n\\end{equation}\nIt is easy to show that process-convolution kernels satisfy Mercer's condition \\cref{eq:mercer-condition}.\nThe process-convolution kernel adapted to our case is the non-stationary Gaussian kernel, defined with the function\n\\begin{equation}\n\tg(\\tau,u) = \\qty(\\frac{2}{\\pi J(\\tau)})^ {1\/4}\\exp[-\\frac{(\\tau-u)^2}{J(\\tau)}],\n\t\\label{eq:gtwovalue}\n\\end{equation}\nwhere $J(\\tau)$ is a free function.\n\n\\subsection{Symmetrized unequal time  correlator of the velocity field}\n\\label{sec:unequal-time-velocity}\n\nTo define the kernel representing the velocity unequal time correlator, we choose for a given $k$\n\\begin{equation}\n\tJ_k(\\tau) =  \\frac{1}{[k\\,v_\\mathrm{dc}(k,\\tau)]^2},\n\\end{equation}\nwhere $v_\\mathrm{dc}(k,\\tau)$\nis given in \\cref{eq:Vlarge}.\nInserting this definition into \\cref{eq:gtwovalue} and further into \\cref{eq:pckernel}, one finds a family of non-stationary Gaussian kernels\n\\begin{equation}\n\tK_k(\\tau,\\zeta)  = \\sqrt{\\frac{2\\, v_\\mathrm{dc}(k, \\tau) v_\\mathrm{dc}(k, \\zeta) }{v_\\mathrm{dc}^2(k, \\tau) + v_\\mathrm{dc}^2(k, \\zeta)}}\n\t\\exp[-k^2 (\\tau-\\zeta)^2 \\frac{v_\\mathrm{dc}^2(k,\\tau) v_\\mathrm{dc}^2(k,\\zeta)}{v_\\mathrm{dc}^2(k,\\tau)+v_\\mathrm{dc}^2(k,\\zeta)}].\n\t\\label{eq:kernelvdc}\n\\end{equation}\nNote that these kernels satisfy $K_k(\\tau, \\tau) = 1$, and are therefore of a suitable form to be used as a normalized UETC $R(k,\\tau,\\zeta)$.\n\nWe can now use this result to construct a UETC $\\UETCv(k,\\tau,\\zeta)$ which satisfies the Mercer condition.\nFirst we note that the UETC evaluated at equal times is the power spectrum,\n$\\UETCv(k,\\tau,\\tau) = \\Psdv(k, \\tau)$.\nHence, multiplying the kernel \\cref{eq:kernelvdc} by\n$\\sqrt{\\Psdv(k, \\tau)\\Psdv(k, \\zeta)}$\nwill give a function of the two times which correctly\nreduces to the power spectrum at equal times.\nThis function is also a kernel,\nas $\\sqrt{\\Psdv(k, \\tau)\\Psdv(k, \\zeta)}$ is a separable kernel, and the product of two kernels is a kernel.\nOur model is then\n\\begin{equation}\n\t\\UETCv\\qty(k, \\tau, \\zeta) =\n\t\\sqrt{\\Psdv(k, \\tau)\\Psdv(k, \\zeta)} \\sqrt{\\frac{2 v_\\mathrm{dc}(k, \\tau) v_\\mathrm{dc}(k, \\zeta) }{v_\\mathrm{dc}^2(k, \\tau) + v_\\mathrm{dc}^2(k, \\zeta)}}\n\t\\exp[- \\frac{1}{2}k^2 (\\tau-\\zeta)^2 v_\\mathrm{dc}^2(k,\\tau, \\zeta)],\n\t\\label{eq:unequal-velocity-ps}\n\\end{equation}\nwhere the average decorrelation velocity $v_\\mathrm{dc}(k,\\tau,\\zeta)$ is given by the harmonic average of the equal-time decorrelation velocity:\n\\begin{equation}\n\tv_\\mathrm{dc}^{2}(k,\\tau,\\zeta)=2\\,\\frac{v_\\mathrm{dc}^{2}(k,\\tau)v_\\mathrm{dc}^{2}(k,\\zeta)}{v_\\mathrm{dc}^{2}(k,\\tau)+v_\\mathrm{dc}^{2}(k,\\zeta)}. \\label{eq:vsweepcomplete}\n\\end{equation}\nNote that this is symmetric in the two times $\\tau$ and  $\\zeta$.\n\nTo summarize, we have shown that the velocity UETC of \\cref{eq:unequal-velocity-ps} is a positive kernel, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot a well-defined two point correlator.\nThis guarantees that the anisotropic stress UETC $P_{\\tilde{\\Pi}}(k,\\tau,\\zeta)$ of \\cref{eq:unequal-stress-final} is also a positive kernel, being the product of two positive kernels.\nFinally, the integrand of \\cref{eq:sbgw} is also a positive kernel. Indeed, $\\mathcal{G}$ is a separable kernel since it results from the square of \\cref{eq:hdot-short}.\nAs a consequence, the GW energy density power spectrum defined in \\cref{eq:sbgw} is always positive, as it should be.\n\nHaving managed to model the decorrelation of the freely-decaying turbulence with \\cref{eq:unequal-velocity-ps}, we can go beyond the approach of \\Refa{Caprini:2009yp}.\nIn that paper the shear stress decorrelation was modelled directly in the anisotropic stress\nUETC as a top-hat function of the kind $\\Theta[k\\abs{\\tau-\\zeta} - x_c]$.\nWe remark that such a function is not a positive kernel, since it is the Fourier transform of a sinc function:\nin order to derive a positive definite GW power spectrum, one must indeed take $x_c <\\pi$\n\\cite{Caprini:2009yp,Caprini:2009pr}.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.49\\textwidth]{figures\/V-0000.png}\n\t\\includegraphics[width=0.49\\textwidth]{figures\/V-0001.png}\n\t\\caption{\\emph{Left panel}: Slice through simulation ($\\mathrm{A}'$) showing the velocity initial conditions in real space. \\emph{Right panel}: Same slice as the left panel but\n    after a time $\\Delta \\tau=20.6 \\tau_{\\xi_*}$ has elapsed.}\n\t\\label{fig:simslices}\n\\end{figure}\n\n\n\\section{Direct numerical simulations}\n\\label{sec:numerical}\n\nWe carry out a series of direct hydrodynamical simulations in order to\nstudy the freely-decaying turbulent flow, in particular the UETCs of\nthe velocity field and its overall time evolution, as well as the SGWB signal produced.  For our\nsimulations, we use a modified version of the relativistic\nhydrodynamics code SCOTTS, previously developed to study the coupled\nevolution of the scalar field and the relativistic fluid, and the\nproduction of GWs during a thermal phase\ntransition~\\cite{Hindmarsh:2013xza,Hindmarsh:2015qta,Hindmarsh:2017gnf,Cutting:2019zws}.\nHere we are mainly interested in the dynamics of the fluid after the\nphase transition has completed. We therefore\ninitialize the scalar field to lie in the broken phase at the start of\nthe simulation. Given the form of the effective potential and equation\nof state in Ref.~\\cite{Cutting:2019zws}, this leaves us\nsolely with the evolution of the\nrelativistic fluid.\n\nIdeally, one would want to simulate the coupled\nfield-fluid model~\\cite{Enqvist:1991xw}, in order to observe the\ngeneration of turbulence at all stages from the bubble collisions onwards~\\cite{Cutting:2019zws}.\nIn this work, however, we do not analyse the\ncomplete system up to fully developed turbulence.  We rather\nconcentrate on the\ndecay of turbulence,\ninserting the turbulence\nparameters (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot kinetic energy and integral scale) as input values,\ndisconnected from the rest of the phase transition dynamics.\nHere, we will initialize only the vortical component of the\nfluid velocity.\nThis permits a cleaner analysis of the results and comparison to the\nsemi-analytical modelling of \\cref{sec:free-decay-model,sec:PosKer}.\nWe defer studying the compressional component and\nsimulating the turbulence generation to future work.\n\nIn the rest of this section, we specify the equations of motion of the fluid that are solved by the code, and then explain how we fix the initial conditions and evaluate the UETCs. Finally, we discuss how we compute GW power spectra from the simulations.\n\n\n\\subsection{Hydrodynamic evolution}\n\\label{sec:hydroeq}\n\nGiven that we are discarding the scalar field dynamics, the energy momentum tensor of the system becomes that of a perfect fluid,\n\\begin{equation}\n\tT^{\\mu\\nu}= (\\epsilon + p) U^\\mu U^\\nu + g^{\\mu\\nu}p\\text,\n\\end{equation}\nwith $\\epsilon$ the internal energy density in the fluid, $p$ the\npressure, $U^\\mu=\\gamma(1,\\vb{v})$ the fluid four-velocity and with\nLorentz factor $\\gamma = 1\/\\sqrt{1-v^2}$.\nThe equation of state is\nthat of an ultrarelativistic gas with $p=\\epsilon\/3$.  It has been\nshown that the hydrodynamic equations of motion of an\nultrarelativistic fluid in an expanding universe expressed in comoving\ncoordinates and conformal time are the same as the hydrodynamic\nequations in Minkowski space-time, provided the dynamical quantities\nare replaced by appropriately scaled\nvariables~\\cite{Brandenburg:1996fc}.  However, the source term for the\nGWs does not scale in the same way. Therefore, the Minkowski\nspace-time code SCOTTS\ncan be used to study hydrodynamic turbulence in an expanding background, but the output GW power spectrum can be matched to the expanding background result only for sources which are  short-lived compared with the Hubble time \\cite{Hindmarsh:2015qta}.\n\nThe dynamical quantities that the code evolves are the fluid energy density $E=\\gamma \\epsilon$ with equation of motion~\\cite{Hindmarsh:2015qta}\n\\begin{equation}\n\t\\dot{E} + \\partial_{i}(Ev^i) + p\\left[\\dot{\\gamma} + \\partial_i(\\gamma v^i)\\right] = 0\\text,\n\\end{equation}\nand the fluid momentum density $Z_i = \\gamma^2(\\epsilon + p)v_i$, the components of which evolve according to\n\\begin{equation}\n\t\\dot{Z_i} + \\partial_j(Z_i v^j) + \\partial_i p = 0\\text.\n\\end{equation}\nThe evolution algorithm follows the approach taken in\nRef.~\\cite{WilsonMatthews} with a leapfrog and operator splitting\napproach to updating the dynamical quantities.\nFor this work we\nuse a van Leer scheme for the advection\nupdate~\\cite{VANLEER1977276,Anninos:2002gz}, whereas earlier uses of\nthe SCOTTS code used an upwind donor cell scheme. For all of our simulations we pick the lattice\nspacing and timestep such that their ratio is $\\dd x\/\\dd \\tau = 2.5$.  We denote\nthe starting time of our simulations as $\\tau = \\tdevel$.\nNote that while our simulations do not contain a physical viscosity,\nthe introduction of a lattice spacing $\\dd x$ effectively sets a\ndissipation scale. This is due to the numerical viscosity introduced\nby the discretization of the fluid equations. Therefore,\nsimulations with larger $\\xi_*\/\\dd x$ have a larger effective\nReynolds number.\n\n\n\\subsection{Initial conditions of the  simulation}\n\\label{sec:initial}\nIn our simulations the velocity field $\\vb{v}$ is not necessarily divergence-free. However, as outlined in the rest of this section, we initialize the velocity field to be divergence-free in our simulations. In \\cref{sec:appendix-sim-vrms-comp} we show that in this case the longitudinal component of the velocity field remains negligible throughout our simulations, and so we treat $\\vb{v}$ as divergence-free in subsequent sections.\n\nAt the start of our simulations, we initialize the fluid energy density $E$ everywhere. For our simulations we choose to set $E\\, \\dd x^4 = 1\/16$. Next we initialize the velocity field of the simulations in Fourier space. We do this indirectly by initializing the spatial components of the four-velocity $\\vb{U}=\\gamma \\vb{v}$, such that\n\\begin{equation}\n\t\\vb{U}_{\\vb{x}} = \\frac{1}{L^3}\\sum_{\\vb{k}} \\vb{U}_{\\vb{k}} \\e^{i \\vb{k} \\cdot \\vb{x}},\n\\end{equation}\nwhere $\\vb{x}$ and $\\vb{k}$ refer to the discrete real and momentum space coordinates on the lattice, and $L^3$ is the total number of lattice sites.\nEach mode $\\vb{U}_{\\vb{k}}$ is randomly distributed and follows Gaussian statistics with mean zero and variance determined by the desired power spectrum serving as initial condition.\nInitializing $\\vb{U}_{\\vb{k}}$ rather than $\\vb{v}_{\\vb{k}}$ prevents individual modes of $\\vb{v}_{\\vb{k}}$ being drawn from the tails of the Gaussian with unphysical velocities larger than one.\nThe $\\mathbf{U}_{\\vb{k}}$ field is then projected onto its vortical component with the projector $\\bot_{ij}(\\vu{k})$, see \\cref{eq:PSsim}.\nSince $\\vb{U}$ is real, we impose that $\\vb{U}_{\\vb{k}} = \\vb{U}^* _{-\\vb{k}}$.\nOnce the $\\vb{U}_{\\vb{k}}$ field is correctly initialized we perform a Fourier transform to find the real space configuration of the $\\vb{U}$ field, and from this we can calculate the velocity field $\\vb{v}$.\n\n\nThis method allows us to initialize the simulation with arbitrary choices for the $\\vb{U}$ field power spectrum $\\mathcal{P}_U(k,\\tau) = {k^3} P_U(k,\\tau)\/{\\pi^2}$.\nHere $P_U(k,\\tau)$ corresponds to the spectral density of the $\\vb{U}$ field, defined via the equal time correlator (see~also \\cref{eq:velocity-sd})\n\\begin{equation}\n\t\\langle U_{i}(\\vb{k},\\tau) U_{j}^*(\\vb{q},\\tau) \\rangle = (2\\pi)^3 \\bot_{ij}(\\vu{k}) \\delta(\\vb{k}-\\vb{q})P_{U}(k,\\tau)\\text,\n\\end{equation}\nwhere in our initial conditions we have set the longitudinal component of $\\mathbf{U}$ to zero.\nWe initialize the simulations with a relativistic version of the von K\\'arm\\'an~spectrum \\cref{eq:power-spectrum}, supplemented with a high-wavenumber cutoff:\n\\begin{equation} \\label{eq:sim-input}\n\t\\mathcal{P}_U(k,\\tdevel) =  C \\frac{(k\/\\tilde{k})^5}{[1 + (k\/\\tilde{k})^2]^{17\/6}} \\exp\\left[-\\left(\\frac{k}{k_\\mathrm{max}}\\right)^2\\right]\\text.\n\\end{equation}\nThe two constants $\\tilde{k}$ and $C$ determine respectively the peak location and amplitude.\nThe values of $C$ and $\\tilde{k}$ are chosen so that the initial rms velocity $\\overline{v}_*=\\sqrt{ \\overline{v}^2(\\tdevel)}$ (defined in \\cref{eq:kinetic-energy}) and integral scale $\\xi_* = \\xi(\\tdevel)$ (defined in \\cref{eq:integral-scale}) extracted from the simulation will be close to a desired value.\nA list of parameters for the simulations we perform is given in \\cref{tab:list}.\nIn \\cref{eq:sim-input} we insert an exponential cutoff to ensure that the power does not extend to the lattice scale, with $k_\\mathrm{max}\\, \\dd{x}=\\pi\/4$ used in our simulations.\n\n\n\\begin{table}\n\t\\centering\n\t\\resizebox{\\textwidth}{!}{%\n\t\t\\begin{tabular}{c D{.}{.}{-1} D{.}{.}{-1} D{.}{.}{-1} D{.}{.}{-1} D{.}{.}{-1} c c c c c c c}\n\t\t\t$L^3$                                           &\n\t\t\t\\multicolumn{1}{c}{$\\overline{v}_*$}                   &\n\t\t\t\\multicolumn{1}{c}{$\\xi_* \/ \\dd{x}$}            &\n\t\t\t\\multicolumn{1}{c}{$L \\dd{x}\/\\xi_*$}            &\n\t\t\t\\multicolumn{1}{c}{$(\\tend-\\tdevel)\/\\tau_{\\xi_*}$}  &\n\t\t\t\\multicolumn{1}{c}{$(\\tuetc-\\tdevel)\/\\tau_{\\xi_*}$} &\n\t\t\t\\multicolumn{1}{c}{$\\deltuetc\/\\tau_{\\xi_*}$}        &\n\t\t\tC                                               &\n\t\t\t$\\tilde{k} \\dd{x}$\n\t\t\t                                                & Label                                                                                            \\\\\n\t\t\t\\hline\n\t\t\t$4096^3$                                        & 0.0988 & 29.8 & 138  & 99.6 & 5.97  & $1.08\\,\\times\\, 10^{-2}$ & 1.99  & $9.19\\times10^{-3}$   &\n\t\t\t(A)                                                                                                                                                \\\\\n\t\t\t\\hline\n\t\t\t$2048^3$                                        & 0.0298 & 8.08 & 253  & 99.5 & 4.97  & $1.18\\,\\times\\, 10^{-3}$ & 0.995 & $4.06\\times10^{-2}$   &\n\t\t\t(B)                                                                                                                                                \\\\\n\t\t\t$2048^3$                                        & 0.0274 & 57.2 & 35.8 & 12.9 & 6.47  & $9.18\\,\\times\\, 10^{-4}$ & 0.129 & $3.28\\times10^{-3}$   &\n\t\t\t(C)                                                                                                                                                \\\\\n\t\t\t\\cline{2-9}\n\t\t\t$2048^3$                                        & 0.0984 & 8.08 & 253  & 97.4 & 4.87  & $1.32\\,\\times\\, 10^{-2}$ & 0.974 & $4.06\\times10^{-2}$   &\n\t\t\t(D)                                                                                                                                                \\\\\n\t\t\t$2048^3$                                        & 0.0955 & 64.4 & 31.8 & 11.9 & 0.594 & $1.02\\,\\times\\, 10^{-2}$ & 0.119 & $3.28\\times10^{-3}$   &\n\t\t\t(E)                                                                                                                                                \\\\\n\t\t\t\\cline{2-9}\n\t\t\t$2048^3$                                        & 0.279  & 8.11 & 253  & 103  & 4.13  & $1.18\\,\\times\\, 10^{-1}$ & 1.03  & $4.06\\times10^{-2}$   &\n\t\t\t(F)                                                                                                                                                \\\\\n\t\t\t$2048^3$                                        & 0.289  & 70.9 & 28.9 & 11.7 & 0.466 & $9.18\\,\\times\\, 10^{-2}$ & 0.117 & $3.28\\times10^{-3}$   &\n\t\t\t(G)                                                                                                                                                \\\\\n\t\t\t\\hline\n\t\t\t$2048^3$                                        & 0.0974 & 28.4 & 72.0 & 103  & 6.17  & $1.08\\,\\times\\, 10^{-2}$ & 2.06  & $9.19 \\times 10^{-3}$ &\n\t\t\t($\\mathrm{A}'$)                                                                                                                                    \\\\\n\t\t\t$1024^3$                                        & 0.0942 & 25.4 & 40.4 & 111  & 6.69  & $1.08\\,\\times\\, 10^{-2}$ & 2.23  & $9.19 \\times 10^{-3}$ &\n\t\t\t($\\mathrm{A}''$)                                                                                                                                   \\\\\n\t\t\\end{tabular}}\n\t\\caption{List of the simulations used in this work. We list\n\t\tthe number of sites in the lattice for each simulation\n\t\t$L^3$, the initial rms velocity $\\overline{v}_*=\\sqrt{\n\t\t\t\t\\overline{v}^2(\\tdevel)}$ (see \\cref{eq:kinetic-energy}) and\n\t\tintegral scale $\\xi_*=\\xi(\\tdevel)$ (see\n\t\t\\cref{eq:integral-scale}) extracted from the simulation\n\t\ton the initial timestep. The relative size of each\n\t\tsimulation to the integral scale $L \\dd{x}\/\\xi_*$ is\n\t\tgiven. We also list the final time of each simulation\n\t\t$\\tend$, the UETC reference time $\\tuetc$ and the UETC output interval $\\deltuetc$, all given in\n\t\tunits of the eddy turnover time at the integral scale\n\t\t$\\tau_{\\xi_*} = \\xi_* \/ \\overline{v}_*$. Finally we list the parameters\n\t\tfor the simulation input power spectrum as given in\n\t\t\\cref{eq:sim-input}. Simulations ($\\mathrm{A}'$) and ($\\mathrm{A}''$) are\n\t\tused to test the effects of finite volume and are discussed\n\t\tin \\cref{sec:appendix-finite-vol}.}\n\t\\label{tab:list}\n\\end{table}\n\n\\subsection{Velocity power spectra in the simulations}\n\\label{sec:sim-velps}\nDuring the simulation we regularly output the rotational velocity power spectra $\\Psv(k,\\tau)$.\nWe validate that the longitudinal piece of the velocity correlator $\\Psdv(k,\\tau)$ is negligible over the full duration of all our simulations, providing it is initialized to be zero, see \\cref{sec:appendix-sim-vrms-comp}.\nTo find the velocity power spectrum within our simulations, we first perform a discrete Fourier transform on the real space field to obtain the Fourier modes\n\\begin{equation}\n\t\\vb{v}_{\\vb{k}} = \\sum_{\\vb{x}} \\vb{v}_{\\vb{x}} \\e^{- i \\vb{x} \\cdot \\vb{k}}\\text.\n\\end{equation}\nThe rotational power spectrum is then approximated by projecting and binning these modes in momentum space.\nThe power spectrum for the $r$-th bin is constructed via\n\\begin{equation}\n\t\\Psv(k_r,\\tau) = \\frac{1}{L^6} \\frac{k_r}{\\Delta k} \\sum_{r \\leq \\tfrac{|\\vb{k}|}{\\Delta k}<r+1} \\bot_{ij}(\\mathbf{\\hat k}) v^i_{\\vb{k}}(\\tau) {v}^{j*}_{\\vb{k}}(\\tau)\\text,\n\\end{equation}\nwhere $k_r = 2\\pi(r+1\/2)\/L \\dd x$ is the central value of the bin in $k$, and $\\Delta k = 2\\pi\/L \\dd x$ is the bin width. Note that the power spectra from the simulation are formed from an average over a spherical shell in $\\mathbf{k}$ of Fourier modes.\n\n\\subsection{Unequal time correlators in the simulations}\n\\label{sec:sim-uetc}\n\nWe are also interested in studying the UETCs of the velocity field in these simulations as these enable us to model the GW source, see \\cref{eq:sbgw,eq:unequal-stress-final}.\nHowever, evaluating $\\UETCv(k,\\tau, \\zeta)$ for many values of $\\tau$ and $\\zeta$ is very costly within a  simulation, in part because each pair of $\\tau$ and $\\zeta$ represents snapshots of the field $\\vb{v}$ that must be stored concurrently in memory.\nInstead, we define a reference time $\\tuetc$ at which we store in memory the field $\\vb{v}(\\vb{k},\\tuetc)$, and then compute the UETC at regular intervals with the field on the current timestep, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot\n\\begin{equation}\n\t\\ev{ v_{i}(\\vb{k},\\tau) v_j^*(\\vb{q},\\tuetc) }= (2\\pi)^3 \\bot_{ij}(\\vu{k}) \\delta(\\vb{k}-\\vb{q})\\UETCv(k,\\tau, \\tuetc).\n\\end{equation}\nWe evaluate the unequal-time correlator at times $\\tau = \\tuetc + n\\deltuetc$, where $\\deltuetc$ is the interval between UETC outputs (see \\cref{tab:list}) and $n$ is some positive integer.\nUsing a single value of $\\tuetc$ per simulation is in principle a limitation of our analysis.\nNonetheless, we are confident that the latter still captures the general behaviour of the time decorrelation of the turbulent field, especially given the good agreement we find between the simulations results and the theoretical model, see~\\cref{sec:results-vel}.\n\n\\subsection{Gravitational waves}\n\\label{sec:gw-numerical}\n\nWe also output GW power spectra calculated from our simulations.\nAs SCOTTS is a Minkowski space code, we neglect cosmological expansion in the following.\nTo find the metric perturbations $h_{ij}$, we evolve an auxiliary tensor $u_{ij}$ according to \\Refa{Garcia-Bellido:2007fiu} (see~also \\cref{eq:gw-propagation})\n\\begin{equation} \\label{eq:eom-metric}\n\t\\ddot{u}_{ij} - \\nabla^2 u_{ij} = 8 \\pi T_{ij}\\text.\n\\end{equation}\nThe metric perturbations can then be recovered from our auxiliary tensor by performing a projection in momentum space,\n\\begin{equation}\n\th_{ij}(\\mathbf{k},\\tau) = G \\Lambda_{ij\\ell m}(\\vu{k}) u_{\\ell m}(\\mathbf{k},\\tau)\n\\end{equation}\nwhere $\\Lambda_{ij,\\ell m}(\\mathbf{k})$ is the projector onto transverse-traceless rank two tensors, \\cref{eq:lambda_proj}.\nWe can then define the power spectral density of the metric perturbations, $P_{\\dot{h}} (k,\\tau)$ via\n\\begin{equation}\n\t\\langle\\dot{h}_{ij}(\\mathbf{k},\\tau) \\dot{h}_{ij}(\\mathbf{q},\\tau) \\rangle = G^2 \\langle \\Lambda_{ijk\\ell }(\\vu{k}) \\Lambda_{ijmn}(\\vu{q}) \\dot{u}_{k\\ell}(\\mathbf{k}) \\dot{u}_{mn}(\\mathbf{q})\\rangle = (2\\pi)^3\\delta(\\mathbf{k}-\\mathbf{q}) P_{\\dot{h}}(k,\\tau)\\text,\n\\end{equation}\nand from this construct the power spectrum of the metric perturbations,\n\\begin{equation}\n\t\\mathcal{P}_{\\dot{h}}(k,\\tau) = \\frac{k^3}{2\\pi^2} P_{\\dot{h}} (k,\\tau)\\text.\n\\end{equation}\nFrom $\\mathcal{P}_{\\dot{h}}(k,\\tau)$ and a suitable scaling, one can then reconstruct the gravitational wave power spectrum (see~\\cref{eq:OmGW})\n\\begin{equation}\n\t\\left.\\frac{\\dd \\Omega_\\mathrm{gw}}{\\dd \\mathrm{ln}\\, k}\\right|_\\tau =  \\frac{\\mathcal{P}_{\\dot{h}}(k,\\tau)}{8\\pi G \\rho_*}\\text,\n\\end{equation}\nwhere $\\rho_*$ is the conserved average total energy density (note that we insert a $*$ subscript to make the connection with the expanding Universe quantity, see~\\cref{eq:Hstarsim}).\n\nDimensionful quantities in our simulations are always output normalized according to a corresponding length scale, which we have chosen to be the initial integral scale $\\xi_*$ (defined later in \\cref{eq:integral-scale}).\nNoting that the gravitational constant $G$ has been scaled out of our equation of motion in \\cref{eq:eom-metric}, the output from our code is\n\\begin{equation}\n\tO_\\mathrm{code}(k,\\tau) = \\frac{\\xi_*^2}{8\\pi \\rho_*}  \\frac{\\mathcal{P}_{\\dot{h}}(k,\\tau)}{G^2}\n\t=\\frac{\\xi_*^2}{G} \\left.\\frac{\\dd \\Omega_\\mathrm{gw}}{\\dd \\mathrm{ln} \\,k}\\right|_\\tau\n\t\\text.\n\\end{equation}\nIn this expression, the power spectra of the metric perturbations in the simulations are binned and constructed analogously to the velocity power spectra,\n\\begin{equation}\n\t\\frac{\\mathcal{P}_{\\dot{h}}(k_r,\\tau)}{G^2} = \\frac{1}{L^6} \\frac{k_r}{\\Delta k} \\sum_{r \\leq \\tfrac{|\\vb{k}|}{\\Delta k}<r+1} \\Lambda_{ij\\ell m}(\\mathbf{\\hat k}) \\dot{u}^{ij}_{\\vb{k}}(\\tau) \\dot{u}^{\\ell m*}_{\\vb{k}}(\\tau)\\text. \\label{eq:metric-pspec-dns}\n\\end{equation}\nThe GW power spectrum is thus recoverable from the simulations by fixing a value for $\\xi_*^2\/G$.\n\nWhile the GW evaluation is carried out in flat space-time,\nwe can still infer a value for the Hubble rate via the Friedmann equation\n\\begin{equation}\n\t\\mathcal{H}_*^2 = \\frac{8 \\pi G \\rho_*}{3}\\text.\\label{eq:Hstarsim}\n\\end{equation}\nThis allows us to rewrite the scaling for the gravitational wave power spectrum as (note that $\\rho_*$ is a conserved quantity throughout the simulation)\n\\begin{equation}\\label{eq:OmGWsimul}\n\t\\frac{1}{(\\mathcal{H}_*\\xi_*)^2} \\left.\\frac{\\dd \\Omega_\\mathrm{gw}}{\\dd \\mathrm{ln} \\,k}\\right|_\\tau = \\frac{3}{8\\pi \\rho_* \\xi_*^4}O_\\mathrm{code}(k,\\tau)\\text.\n\\end{equation}\nThe SGWB output of the simulations will be compared with the semi-analytic results in \\cref{sec:compamethods}.\nSince the simulations neglect the expansion of the Universe, one can only compare the spectral region $k>\\mathcal{H}_*$.\nThe wavenumbers accessible in the simulations satisfy $k\\xi_*>2\\pi\\xi_*\/(L \\dd{x}) > 0.03$ (see \\cref{tab:list}), and\nwe will therefore compare with semi-analytical results calculated setting  $\\mathcal{H}_*\\xi_*=0.001$ (see \\cref{sec:numintegration}).\n\n\n\n\\section{Results: evolution of the velocity field}\n\\label{sec:results-vel}\n\n\nIn this section, we verify the model of decaying turbulence presented in \\cref{sec:free-decay-model} with the hydrodynamic simulations.\nWe compare the time evolution laws of the turbulent kinetic energy and correlation scale  with the simulation results in \\cref{sec:ResETC}.\nThen in \\cref{sec:results-uetc}, we validate the  symmetrized velocity field decorrelation function constructed in  \\cref{sec:UETCvelocity,sec:PosKer}, and thereby show that it can also describe decorrelation at large scales, outside the inertial range.\nNote that the turbulence model we adopted has been developed in the context of non-relativistic turbulence.\nThe highest value of the initial rms velocity that we have tested in the simulations is $\\overline{v}_* \\simeq 0.3$ (see~\\cref{tab:list}), finding good agreement with the predictions of the non-relativistic turbulence modelling, as far as the UETC, free decay evolution and SGWB spectra are concerned.\n\n\\subsection{Velocity power spectrum and averaged quantities}\n\\label{sec:ResETC}\n\n\\begin{figure}\n\t\\centering \\subfloat[Simulation (A), unrescaled velocity power\n\t\tspectrum.\\label{fig:unrescaled}]{\n\t\t\\includegraphics[width=0.45\\textwidth]{figures\/power_spectra}}\n\t\\subfloat[Power spectrum rescaled with \\cref{eq:pv_scaled_one_scale}.\\label{fig:rescaledonescale}]{\n\t\t\\includegraphics[width=0.465\\textwidth]{figures\/pspec_scaled.pdf}}\n\t\\caption{Evolution of the velocity power spectrum from simulation\n\t\t(A). \\emph{Left panel}: unrescaled power\n\t\tspectrum. The solid black line is the\n\t\tinitial condition, \\cref{eq:power-spectrum} and coloured lines\n\t\tshow the time evolution from $\\tau=\\tdevel$ up to $\\tau = \\tend$\n\t\twith interval $\\Delta\\tau = 9.96\\, \\tau_{\\xi_*}$. Lighter colours refer to later times.\n\t\t\\emph{Right panel}:\n                power spectrum rescaled according to \\cref{eq:pv_scaled_one_scale}. The curves are plotted starting from $\\tau-\\tdevel\\simeq 40\\,\\tau_{\\xi_*}$, at fixed intervals $\\Delta \\tau = 1.99 \\tau_{\\xi_*}$.\n\t\t\\label{fig:Pv_and_first_rescale}}\n\\end{figure}\n\nHere we use the results from  simulation (A) (\\cref{tab:list}) to investigate the time evolution of the velocity power spectrum, and the values of the parameters $\\beta$, $p$ and $q$, introduced in \\cref{sec:free-decay-evol}.\n\nIn \\cref{fig:Pv_and_first_rescale} we show the time evolution of the velocity power spectrum.  \\cref{fig:unrescaled} shows the velocity power spectrum in its raw form, whilst \\cref{fig:rescaledonescale} shows the power spectrum rescaled with \\cref{eq:pv_scaled_one_scale}, indicating that there is indeed only one principal length scale in the flow, given our initial condition.\nThe self-similarity is established starting from about $40 \\tau_{\\xi_*}$, the time to which the first power spectrum line in \\cref{fig:rescaledonescale} corresponds.\nNote that the large-scale slope of the late-time power spectrum is significantly less steep than the $k^5$ corresponding to the Batchelor power spectrum imposed in the initial condition  \\cref{eq:power-spectrum}: therefore, the large-scale power spectrum is not constant in time in the simulation, before about $40 \\tau_{\\xi_*}$.\nWhile the large-scale slope could be subject to finite volume effects, its constancy in time seems to be well established at late times.\n\n\nTurning to the form of the  velocity power spectrum at wavenumbers higher than the peak,  we see from \\cref{fig:Pv_and_first_rescale}\nevidence for a power law in the range $1 \\lesssim k\\xi_* \\lesssim 5 $.\nThis power law is approximately consistent with a Kolmogorov power spectrum ($k^{-2\/3}$) at late times.\nAlthough this was also the power spectrum in the initial conditions, one can see that this power law is lost in the early evolution,\nbut later regained.\nWe do not have the inertial range\nto give a firm value for the late-time power law index.\nFor wavenumbers above $k\\xi_* \\gtrsim 5 $, we see a small build up in power followed by a sharp fall-off for $k\\xi_* \\gtrsim 10$.\nThe fall-off is set by the viscous damping scale of the numerical scheme, while the build up seems to be an associated feature of the van Leer advection scheme\\footnote{The upwind donor cell advection scheme previously used was associated with a fall-off in power at smaller $k\\xi_*$ consistent with it being a lower order advection scheme.}.\n\nIn \\cref{fig:vrms-xi-beta} we plot the\ncombination $\\overline{v}^2 \\xi^{1+\\beta}$, predicted to be constant when the large-scale power spectrum is constant as well, see \\cref{eq:vxigeneral}.\nIt can be seen that this combination indeed\nremains approximately constant after about $40\\tau_{\\xi_*}$, provided $\\beta=3$.\nConsequently, in \\cref{fig:rescaledbeta3} we analyse again the self similarity of the power spectrum, this time rescaled according to  \\cref{eq:Pvxi} setting $\\beta=3$. Again, we only plot the rescaled spectra for $\\tau-\\tdevel\\gtrsim 40\\,\\tau_{\\xi_*}$.\nIt appears that the self-similarity condition  \\cref{eq:Pvxi} is satisfied by the simulation with $\\beta=3$.\nThis result is consistent with the findings of \\Refa{Brandenburg:2016odr}.\nHowever, the large-scale power spectrum is less steep than $k^4$, which is the power law which would be inferred setting $\\beta=3$.\n\nSimulation (A), therefore, seems broadly consistent with the self-similar behaviour outlined in \\cref{eq:Pvxi}, at late times.\nWe have seen that $\\beta$ is not set by the slope of the power spectrum at low wavenumbers in the initial condition; rather, it seems to be part of the dynamics of self-similarity. We would need even larger lattices\nto study in detail the evolution of the power spectrum at low wavenumbers,\nwhere there appear to be correlations established at scales greater than the integral scale.\nLimitations of the size of the computational domain are known to possibly affect the interpretation of the turbulent decay features \\cite{doi:10.1063\/1.4901448}.\n\n\\begin{figure}\n\t\\centering\n\t\\subfloat[Evolution of $\\overline{v}^2 \\xi^{1+\\beta}$. \\label{fig:vrms-xi-beta}]{\\includegraphics[width=0.45\\textwidth]{figures\/globals}}\n\t\\subfloat[Power spectrum rescaled with \\cref{eq:Pvxi}, assuming\n\t\t$\\beta=3$.\\label{fig:rescaledbeta3}]{\n\t\t\\includegraphics[width=0.465\\textwidth]{figures\/collapse-beta3}}\n\t\\caption{\n\t\t\\emph{Left panel}: evolution of $\\overline{v}^2 \\xi^{1+\\beta}$ for different values of $\\beta$ in simulation (A). As shown in \\cref{eq:vxigeneral}, this quantity is expected to remain constant, thus indicating that $\\beta \\simeq 3$.\n\t\t\\emph{Right panel}:\n\t\tPower spectrum rescaled with\n\t\t\\cref{eq:Pvxi}, using $\\beta=3$.\n\t\tThe power spectra are plotted from $\\tau-\\tdevel\\simeq 40\\,\\tau_{\\xi_*}$ onward, at fixed interval $\\Delta \\tau = 1.99 \\tau_{\\xi_*}$.\n\t\t\\label{fig:collapse}}\n\t\\label{fig:beta3}\n\\end{figure}\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.47\\textwidth]{figures\/evolution}\n\t\\caption{Evolution of the velocity and integral scale in simulation (A).\n\t  The black-dashed line showcases \\cref{eq:VelEvMod,eq:XiEvMod} for $\\mathcal{N}_\\mathrm{e}=5$, $p=4\/3$ and $q=1\/3$, where the values of $p$ and $q$ correspond to $\\beta=3$; see the relationships in \\cref{eq:pchialpha,eq:qchialpha}.\n\t\t\\label{fig:xi-vrms-evol}}\n\\end{figure}\n\n\nIn \\cref{fig:xi-vrms-evol} we show the ratios\n$\\overline{v}^2(\\tau)\/\\overline{v}_*^2$ and $\\xi(\\tau)\/\\xi_*$  extracted from simulation (A), where $\\overline{v}^2(\\tau)$ and $\\xi(\\tau)$ are calculated by integrating the velocity power spectra of \\cref{fig:unrescaled}, according to \\cref{eq:kinetic-energy,eq:integral-scale}.\nWe also show the predicted evolution according to \\cref{eq:VelEvMod,eq:XiEvMod} with $\\mathcal{N}_\\mathrm{e} = 5$ (black, dashed lines), where this specific value has been chosen to match the simulation result.\nWe fix the decay exponents in \\cref{eq:VelEvMod,eq:XiEvMod} to $p=4\/3$ for the kinetic energy and $q=1\/3$ for the integral scale. These exponent values are provided by the condition $\\beta=3$ from \\cref{eq:pchialpha,eq:qchialpha} and provide a good fit to the simulation outcome.\nWe shall see later in \\cref{sec:numintegration} that the precise values of the decay exponents do not have a significant effect on the final SGWB spectrum.\nInstead, we will see in \\cref{sec:GWcontinuous} that the initial phase preceding the generation of the turbulence plays a much larger role in determining the spectral shape.\n\n\n\\begin{figure}\n\t\\centering\n\t\\subfloat[Instantaneous exponents $(p,q)$.\\label{fig:pq-diagram}]{\\includegraphics[width=0.49\\textwidth]{figures\/beta}}\n\t\\subfloat[Time evolution $p(\\tau)$, $q(\\tau)$.\\label{fig:p-q-evol}]{\\includegraphics[width=0.46\\textwidth]{figures\/scales}}\n\t\\caption{\\emph{Left panel}: Trajectory of the instantaneous exponents $(p,q)$ in simulation (A). Time is represented by the colour scheme: early times are shown with lighter shades and late times with darker shades, starting at $\\tau=\\tdevel$ and with interval $\\Delta \\tau \\simeq 2 \\tau_{\\xi_*}$ between markers.\n\t\tThe dark solid line represents the scale-invariance line $p=2(1-q)$, \\cref{eq:pVsq}. The coloured dashed, dotted and dash-dotted lines show the self-similarity relations $p=(1+\\beta)q$ for various choices of $\\beta$. \\emph{Right panel}: Evolution of the instantaneous kinetic energy and integral scale exponents $(p, q)$ as a function of time in simulation (A). The dash-dotted line shows the values expected for $p$ and long dashed line for $q$, with $\\beta = 3$.}\n\t\\label{fig:pq-explore}\n\\end{figure}\n\nWe further analyse the behaviour of the power law exponents $p$ and $q$ of the kinetic energy and integral scale in \\cref{fig:pq-explore}.\nIn \\cref{fig:pq-diagram}, we adopt the same approach as \\Refa{Brandenburg:2016odr} and plot the state of the system in $(p,q)$ space\nfor a range of times during the simulation.\nThe values of $p$ and $q$ are found by solving \\cref{eq:VelEvMod,eq:XiEvMod} for $p$ and $q$ in terms of the measured values of $\\overline{v}^2(\\tau)$, \\eqref{eq:kinetic-energy}, and $\\xi(\\tau)$, \\eqref{eq:integral-scale}.\nThe scale-invariance line $p = 2(1 - q)$, derived in \\cref{eq:pVsq}, is represented by the black line in \\cref{fig:pq-diagram}.\nThe self-similarity lines $p=(1+\\beta)q$ are shown for various values of $\\beta$ as dashed lines.\nAs in clear from the figure, the point $(p,q)$ is still evolving at the end of the simulation.\nHowever, the evolution is moving in the direction of the intersection of the self-similarity line $p = (1+\\beta)q$ with $\\beta=3$, and the scale invariance line, $p =2(1 -q)$.\nIn \\cref{fig:p-q-evol}, we display the evolution of $p$ and $q$\nagainst time, and also show the values that can be predicted from requiring scale-invariance and self-similarity for $\\beta = 3$, namely $p = 4\/3$ and $q =1\/3$.\n\n\n\n\n\n\n\\subsection{Unequal time velocity correlations}\n\\label{sec:results-uetc}\n\n\n\nIn this subsection we report on our results for the decorrelation of the velocity field at different times. We introduce a new model for the unequal time correlator and test it against the numerical simulations.\n\n\nOur model for the normalized UETC is\nobtained by combining \\cref{eq:expvsweep,eq:Vlarge,eq:vsweepcomplete}\ngiving\n\\begin{equation}\n\t\\tdec_\\mathrm{A+}(k,\\tau,\\zeta) =\n\t\\frac{v_\\mathrm{dc}(k,\\tau,\\zeta)}{\\sqrt{v_\\mathrm{dc}(k, \\tau) v_\\mathrm{dc}(k, \\zeta)}}\n\t\\exp\\qty[-\\frac{1}{2} k^2\n\t\tv_\\mathrm{dc}^{2}(k,\\tau,\\zeta)(\\tau-\\zeta)^2].\n\\end{equation}\nThe factor in front of the exponential guarantees that the shear stress UETC is a positive definite kernel and hence that the resulting gravitational wave power spectrum is positive definite, as explained in \\cref{sec:mercer}.\nThis is to be compared with the model of \\Refa{Niksa:2018ofa},\n\\begin{equation}\n\t\\label{eq:RNSSdef}\n\t\\tdec_\\mathrm{NSS}(k,\\tau,\\zeta) =\n\t\\exp\\qty[-\\frac{1}{2} k^2\n\t\tv_\\mathrm{dc}^{2}(k,\\zeta)(\\tau-\\zeta)^2], \\quad \\tau > \\zeta \\, .\n\\end{equation}\n\n\\begin{figure}\n\t\\centering\n\t\\subfloat[Model of \\Refa{kaneda_lagrangian_1993}. ]{\\includegraphics[width=0.48\\textwidth]{figures\/uetcs_A_kaneda.pdf}}\n\t\\subfloat[Our model. ]{\\includegraphics[width=0.48\\textwidth]{figures\/uetcs_A.pdf}}\n\t\\caption{Comparison of normalized unequal time correlator models with data from simulation (A): on the left, the model of \\Refa{kaneda_lagrangian_1993} as used in \\Refa{Niksa:2018ofa}, on the right our model. The $y$-axis displays the combination of data and model functions which should produce a Gaussian\n\tcurve in the argument of the $x$-axis;\n\tthis is discussed in further detail in connection with \\cref{eq:Rfigure2b}.\n\tThe solid dark line is the prediction of the model in each case.}\n\t\\label{fig:decorr}\n\\end{figure}\n\n\n\nIn \\cref{fig:decorr},  we compare the two models to data from simulation (A), by\nplotting combinations of data and model functions which should be of Gaussian form.\nResults from the other simulations are given in \\cref{sec:appD}.\nThe construction of the numerical UETC, $\\UETCv(k,\\tau,\\zeta)$\nis described in \\cref{sec:sim-uetc}; in our simulations we measure the correlation of the velocity field at times $\\zeta = \\tuetc$ and $\\tau$.\n\nFor our model, the data is plotted as the quantity\n\\begin{equation}\n\t\\tgauss(k, \\tau, \\tuetc) \\equiv \\frac{\\UETCv(k,\\tau,\\tuetc)}{\\sqrt{\\Psdv(k,\\tau)\\,\\Psdv(k,\\tuetc)}}  \\sqrt{\\frac{v_\\mathrm{dc}^2(k, \\tau) + v_\\mathrm{dc}^2(k, \\tuetc)}{2 v_\\mathrm{dc}(k, \\tau) v_\\mathrm{dc}(k, \\tuetc) }},\n\t\\label{eq:Rfigure2b}\n\\end{equation}\nagainst the combination $k\\,v_\\mathrm{dc}(k,\\tau,\\tuetc)(\\tau-\\tuetc)$,\nwhile for the model of \\Refa{Niksa:2018ofa}, we plot the normalized velocity UETC \\eqref{eq:NorVelUETC}\nas a function of $k\\,v_\\mathrm{dc}(k,\\tuetc)(\\tau-\\tuetc)$. We choose a range of values for the wavenumber $k$, both larger and smaller than the inverse integral scale $\\xi_*$.\nThe reference time is taken to be approximately one initial eddy turnover time, $\\tuetc \\simeq \\tau_{\\xi_*} = \\xi_*\/\\overline{v}_*$ (see \\cref{tab:list}).\nThe success of the models can be judged by how well all the data from simulations collapses onto a Gaussian in the $x$-axis argument,\nindicated by the black lines.\n\nFrom \\cref{fig:decorr}\nit appears that both models work well in the inertial range $k\\xi_* \\gg 1$.\nThis demonstrates that\nthe sweeping model, with sweeping velocity $\\overline{v}\/\\sqrt{3}$,\ncharacterizes very well the decorrelation of the system at high wavenumbers, and that we may take $C_v^2 = 1$ in \\cref{eq:Cv2}.\nThis also indicates that models using the scale-dependent Lagrangian eddy turn-over time\n$v^2_\\mathrm{dc}(k,\\tuetc) \\propto \\Psv(k, \\tuetc)$\nin the inertial range \\cite{Kamionkowski:1993fg,Gogoberidze:2007an}\ngive a poor fit in the inertial range.\n\nOn the other hand, the decorrelation model based on \\cref{eq:RNSSdef} degrades somewhat for $ 0.125\\leq k\\xi_*\\leq 1$.\nOur decorrelation model is a significant improvement on these scales, which we ascribe partly to the fact that the average decorrelation velocity takes into account\nthe slowing of the decorrelation with the decay of the kinetic energy.\n\nBecause of limitations in the dynamical range of the simulations, it is not possible to analyse very small values of $k\\xi_*$.\nHowever, we will see that the GW power spectrum decreases at small $k\\xi_*$, and so it is less important to model accurately the UETC at wavenumbers well below the peak.\n\nNote that according to recent work in \\Refa{Gorbunova:2021cpn}, the Gaussian form of the random sweeping hypothesis should change to an exponential form\nat large time differences.\nAgain, this effect should be most influential at low $k$, below the peak of the power spectrum. We leave further study of UETC models for future work.\n\n\n\\section{Results: the gravitational wave spectrum}\n\\label{sec:sgwbresults}\n\nThis section is organized as follows.\nIn \\cref{sec:stationary} we study a stationary source, meaning that we assume that $\\UETCv(p,\\tau,\\zeta)$ only depends on the time difference $\\tau-\\zeta$.\nThis is not the correct model for freely decaying turbulence, but we consider it nonetheless since it is useful to illustrate some properties of the GW power spectrum of a freely decaying source.\n\nWe present the case of freely decaying turbulence in the subsequent \\cref{sec:gw-instant,sec:GWcontinuous}.\n\\cref{sec:gw-instant} contains the main results of our work, and is dedicated to the case of instantaneously generated turbulence, meaning that we insert as initial conditions a fully developed turbulent spectrum.\nFinally, in \\cref{sec:GWcontinuous}, we go beyond the  instantaneous generation scenario, and include a growth phase for the turbulence kinetic energy.\n\n\nBefore continuing, we first set the basis for the calculation of the stochastic GW background that will follow.\nCombining \\cref{eq:sbgw,eq:unequal-stress-final}, one obtains (recall that $\\vb{q} = \\vb{k} - \\vb{p}$)\n\\begin{equation}\n\t\\eval{\\dv{{\\Omega}_\\mathrm{gw}}{\\ln k}}_{\\tau} =\n\t\\frac{k^3}{3 \\pi^5}\n\t\\int \\dd[3]{p}\n\t\\qty[1 + (\\vu{k}\\vdot \\vu{p} )^2] \\qty[1 + (\\vu{k}\\vdot \\vu{q})^2]\n\t\\iint_{\\tini}^{\\tau} \\mathcal{G}(k,\\tau,\\eta,\\zeta)\n\t\\UETCv(p, \\eta, \\zeta) \\UETCv(q, \\eta, \\zeta)\n\t\\frac{\\dd{\\eta}}{\\eta}\\frac{\\dd{\\zeta}}{\\zeta}.\n\t\\label{eq:OmPv}\n\\end{equation}\nFrom this expression, it appears that the computation of the GW background involves performing a five-dimensional integration for each mode $k$.\nIntegrating over the azimuthal angle of $\\vb{p}$ immediately gives a factor of $2\\pi$ and one is left with a four-dimensional integration.\nTechnical details on how to tackle this four-dimensional integration are presented in \\cref{sec:4dinteg}.\nIn particular, we advocate the change of variables $\\vb{p}=\\vb{k}\/2 + \\vb{h}$ (and hence $\\vb{q} = \\vb{k}\/2 - \\vb{h}$): this makes the symmetry around the zero of the cosine of the declination angle $\\alpha = \\hat{h}\\cdot\\hat{k}$ explicit.\n\n\n\nAfter the complete decay of turbulence, occurring at a time that we denote $\\tfin$, the GW power spectrum in the radiation era evolves as\n\\begin{equation}\n\t\\eval{\\dv{\\Omega_\\mathrm{gw}}{\\ln k}}_{\\tau\\geq \\tfin} = \\frac{8}{3 \\pi^2}\n\tk^3\n\t\\iint_{\\tini}^{\\tfin} \\mathcal{G}(k,\\tau, \\eta,\\zeta) P_{\\tilde{\\Pi}}(k, \\eta,\\zeta)\\frac{\\dd{\\eta}}{\\eta} \\frac{\\dd{\\zeta}}{\\zeta}\\label{eq:gwradera}.\n\\end{equation}\nThis can be obtained by matching the sourced and free solutions of \\cref{eq:gw-propagation} at time $\\tfin$.\nNote that the function $\\mathcal{G}$ of \\cref{eq:green_with_all_terms} depends explicitly on $\\tau$, meaning that the GW spectrum in principle still depends on time after the source has decayed.\nWe are interested in the GW spectrum at a time $\\tau$ in the radiation era long after the source has decayed, and such that all relevant wavenumbers are inside the horizon, $k\\tau>1$.\nThis way we can neglect the second and third terms in \\cref{eq:green_with_all_terms}, proportional to $1\/(k\\tau)$ and $1\/(k\\tau)^2$, since they become subdominant with respect to the first one.\nFurthermore, we can expand the first term of \\cref{eq:green_with_all_terms} to obtain\n\\begin{align}\n\t\\eval{\\dv{\\Omega_\\mathrm{gw}}{\\ln k}}_{\\tau\\geq \\tfin}\n\t=  \\, & \\frac{4}{3 \\pi^2} k^3 \\iint_{\\tini}^{\\tfin} \\cos k(\\eta - \\zeta) P_{\\tilde{\\Pi}}(k, \\eta,\\zeta)\\frac{\\dd{\\eta}}{\\eta} \\frac{\\dd{\\zeta}}{\\zeta}  \\nonumber                                                     \\\\\n\t      & + \\frac{4}{3 \\pi^2} k^3 \\cos(2k\\tau) \\iint_{\\tini}^{\\tfin} \\sin k(\\zeta + \\eta) P_{\\tilde{\\Pi}}(k, \\eta,\\zeta)\\frac{\\dd{\\eta}}{\\eta} \\frac{\\dd{\\zeta}}{\\zeta} \\nonumber                                       \\\\\n\t      & + \\frac{4}{3 \\pi^2} k^3 \\sin(2k\\tau) \\iint_{\\tini}^{\\tfin} \\cos k(\\zeta + \\eta ) P_{\\tilde{\\Pi}}(k, \\eta,\\zeta)\\frac{\\dd{\\eta}}{\\eta} \\frac{\\dd{\\zeta}}{\\zeta} + \\order{(k\\tau)^{-1}} \\label{eq:green-approx}\n\\end{align}\nThe residual time dependence consists of irrelevant, rapid oscillations in time.\nWe can therefore perform a time average of the free SGWB\nsolution over a long time interval around $\\tau$, satisfying $\\Delta\n\\tau \\gg 1\/k$ for every interesting $k$, to obtain\n\\begin{equation}\n\t\\dv{\\Omega_\\mathrm{gw}}{\\ln k}\n\t\\approx \\frac{4}{3 \\pi^2} k^3 \\iint_{\\tini}^{\\tfin} \\cos k(\\eta - \\zeta) P_{\\tilde{\\Pi}}(k, \\eta,\\zeta)\\frac{\\dd{\\eta}}{\\eta} \\frac{\\dd{\\zeta}}{\\zeta}\\,.\n\t\\label{eq:OmPvAVG}\n\\end{equation}\nThis is the effectively time-independent GW power spectrum that we aim to calculate.\nRecall that\n$\\Omega_\\mathrm{gw}$ is normalized to the radiation energy density, see \\cref{eq:sbgw}.\nTherefore, long after the source has decayed,\nthe averaged energy density of the GW signal would be expected to decrease as $a(\\tau)^{-4}$.\n\n\\subsection{The stationary assumption}\n\\label{sec:stationary}\n\nIn this section, we simplify the evaluation of the GW spectrum integral of \\cref{eq:OmPvAVG} by assuming that the turbulence is stationary.\nUnder this assumption, the rms velocity $\\overline{v}$ and the integral scale $\\xi$ remain constant in time.\nWe further set the decorrelation velocity to $v_\\mathrm{dc}\\equiv \\overline{v}_* \/ \\sqrt{3}$ on all scales.\nIn other words, we neglect the $k$-dependence in \\cref{eq:vdcapprox}.\nMotivated by what we see in the simulations, we also set $C_v^2=1$ in \\cref{eq:Cv2}.\nConsequently, the only time dependence left in the UETC in \\cref{eq:unequal-velocity-ps} is the time difference in the exponential.\nIn order to evaluate \\cref{eq:OmPvAVG} under the hypothesis that the source is stationary, it therefore becomes natural to change the time integration variables to  $\\tau_\\mathrm{mid}= (\\tau+\\zeta)\/2$ and $\\tau_\\mathrm{diff}=\\tau-\\zeta$, where the integration over the latter is symmetric around zero.\nWe give the form that \\cref{eq:OmPvAVG} takes under this stationary assumption as \\cref{eq:Omstationary}.\n\nIn order to proceed analytically, we rely on the exponential decorrelation, and extend the integration limits of\n$\\tau_\\mathrm{diff}$ to the interval $[0,\\infty[$.\nFurthermore, one can also assume that the time difference $\\tau_\\mathrm{diff}$ is small compared to the absolute time $\\tau_\\mathrm{mid}$.\nUnder these two further assumptions, which are not strictly speaking a direct consequence of stationarity, one can integrate \\cref{eq:Omstationary} analytically.\n\nA stationary source in principle lasts forever.\nHowever, this is not very realistic in the early Universe setting.\nWe will therefore fix the source duration to a finite value that we express in terms of $\\mathcal{N}_\\mathrm{cut}$, the number of eddy turnover times for which the turbulence lasts.\nIn other words, we integrate from $\\tdevel$ until $\\tdevel+\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*}$.\nThe result then depends explicitly on $\\mathcal{N}_\\mathrm{cut}$.\n\nIn \\cref{sec:analytical} we provide the details of the integration procedure.\nThe asymptotic behaviour of the resulting SGWB spectrum, and its scaling with the turbulence parameters are given by\n\\begin{align}\n\t{\\dv{{\\Omega}_\\mathrm{gw}}{\\ln k}}\n\t\\qty(K \\ll 1) & =\n\tc_1\\,\\frac{\\mathcal{N}_\\mathrm{cut}\\,(\\mathcal{H}_*\\xi_*)^2}{\\overline{v}_*+\\mathcal{N}_\\mathrm{cut}\\mathcal{H}_*\\xi_*}\\, \\overline{v}_*^6\\, \\qty(\\frac{K}{\\overline{v}_*})^3; \\label{eq:stat-ir} \\\\\n\t{\\dv{{\\Omega}_\\mathrm{gw}}{\\ln k}}\n\t\\qty(K \\gg 1) & = \\frac{\\mathcal{N}_\\mathrm{cut}\\,(\\mathcal{H}_*\\xi_*)^2}{\\overline{v}_*+\\mathcal{N}_\\mathrm{cut}\\mathcal{H}_*\\xi_*}\\,\n\t\\qty[c_2\\,\\overline{v}_*^6\\,\n\t\\qty(\\frac{K}{\\overline{v}_*})^{-\\frac{7}{3}}+\n\tc_3\\,\n\t\\overline{v}_*^{4\/3}\\,\n\t\\exp(-\\frac{3}{2 \\overline{v}_*^2})\n\t\\qty(\\frac{K}{\\overline{v}_*})^{-\\frac{5}{3}} ] \\label{eq:stat-uv}\n\\end{align}\n(note that $K=\\mathcal{A}\\xi k$, and the value of the numerical coefficients $c_i$ are given in \\cref{sec:analytical}).\n\n\\cref{eq:stat-ir,eq:stat-uv} show that the scaling of the GW signal with the parameter $\\mathcal{H}_*\\xi_*$ depends on the source duration.\nIf the sourcing lasts less than one Hubble time,\nmeaning $\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*}=\\mathcal{N}_\\mathrm{cut} \\xi_*\/\\overline{v}_*< \\mathcal{H}_*^{-1}$, one can neglect the second term in the denominator of \\cref{eq:stat-ir,eq:stat-uv}, and the SGWB scaling becomes quadratic in $\\mathcal{H}_*\\xi_*$.\nOn the other hand, if the source lasts longer than one Hubble time, the scaling is linear in  $\\mathcal{H}_*\\xi_*$\n(see for example \\Refs{Hindmarsh:2015qta,Hindmarsh:2017gnf,Caprini:2009yp,Caprini:2006jb,Caprini:2019egz}).\nIf we had set the duration of the source to infinity, as required in principle by the stationary assumption, we would have consistently\nfound a linear dependence on $\\mathcal{H}_*\\xi_*$.\n\nGravitational wave generation from sound waves also presents the same duration-dependent scaling with $\\mathcal{H}_*\\xi_*$.\nThis was previously derived within the sound shell model  \\cite{Hindmarsh:2015qta,Hindmarsh:2019phv,Hindmarsh:2020hop,Guo:2020grp}.\nIn the sound shell model one can also assume stationarity and set $\\mathcal{N}_\\mathrm{cut} \\sim 1$ as in \\Refa{Guo:2020grp}.\nNote that, in the acoustic case, the decorrelation time depends on the sound speed, instead of on the rms fluid velocity.\nThe typical lifetime of the source is connected to the kinetic energy.\nFor the turbulent case, low velocities correspond to longer eddy turnover times $\\tau_{\\xi_*}=\\xi_*\/\\overline{v}_*$, and therefore sources that last longer.\nThe kinetic energy is in turn expected to be connected to the PT strength (see for example \\Refa{Ellis:2020awk}).\nThe latter then ultimately influences the scaling of the SGWB amplitude \\cite{Caprini:2019egz}.\n\nThe turbulent kinetic energy also naturally enters the SGWB amplitude, through $\\overline{v}_*$.\nIn particular, the stationary assumption leads to a stronger scaling with $\\overline{v}_*$ than both the case of instantaneous generation (see \\cref{sec:gw-instant}) and the SGWB signal arising from sound waves \\cite{Hindmarsh:2019phv,Hindmarsh:2020hop}: at large scales and\/or for low initial turbulent velocity, \\cref{eq:stat-ir,eq:stat-uv} show that the SGWB amplitude from turbulence within the stationary assumption scales as $\\overline{v}_*^6$, as opposed to $\\overline{v}_*^4$ for both sound waves and an instantaneous turbulence generation (as we shall see in \\cref{sec:gw-instant}).\n\nWe remark that the value of $\\overline{v}_*$ also influences the high-frequency slope of the SGWB, see~\\cref{eq:stat-uv}. This is a consequence of the exponential decorrelation in the velocity power spectrum \\cref{eq:unequal-velocity-ps}. For large $\\overline{v}_* > 0.5$, the second term in \\cref{eq:stat-uv} dominates: consequently, the SGWB spectrum becomes shallower, and its amplitude scales as $\\overline{v}_*^{4\/3}$.\nThe slope of the SGWB at low wavenumber, on the other hand, is due to the steep (causal) increase of the velocity power spectrum \\cref{eq:power-spectrum}, causing the convolution in \\cref{eq:unequal-stress-final} to be flat for $k< 1\/\\xi_*$ (uncorrelated in space for scales larger than $\\xi_*$) \\cite{Caprini:2006jb}.\nThe complete spectral shape can be obtained by integrating numerically \\cref{eq:stationary}: it smoothly interpolates between the slopes given by \\cref{eq:stat-ir,eq:stat-uv}, peaking at the wavenumber $k \\simeq 2 \\overline{v}_* \/ (\\mathcal{A}\\xi_*)$.\nExamples of spectra are given in \\cref{fig:stationary}.\n\n\n\\subsection{Instantaneous turbulence generation}\n\\label{sec:gw-instant}\n\nIn this section, we consider turbulence generated instantaneously, and freely decaying afterwards.\nWe consider a situation where gravitational wave production begins when the fluid has\nthe velocity spectrum of fully developed turbulence, at time $\\tdevel$.\nWe use three different methods to compute the SGWB spectrum.\nFirst, we measure the SGWB in the direct numerical simulations.\nSecond, we numerically integrate \\cref{eq:OmPvAVG} inserting the model for freely-decaying turbulence derived in the first part of this paper, summarized in \\cref{eq:unequal-velocity-ps} (see also section \\cref{sec:results-vel}).\nThird, we assume that the GW source is constant in time, and analytically integrate  \\cref{eq:green-approx} under this assumption\\footnote{The reason why we do not perform the time average in this case will be clarified in \\cref{sec:const}.}.\nWe show that there is good agreement between the three methods for the resulting SGWB.\nIn particular, the analytical approximation provides an easy-to-use formula, \\cref{eq:constant_approx}, in which the scaling with the turbulence parameters (initial rms velocity, integral scale, duration) is apparent.\n\n\n\n\\subsubsection{Direct numerical simulations}\n\\label{sec:results-gw-sim}\n\nIn \\cref{fig:gwps-sim-vrms-0.1} we present GW power spectra from three simulations with $\\overline{v}_*\\approx 0.1$, corresponding to\nsimulations (A), (D) and (E) in \\cref{tab:list}. Equivalent\nplots for the other simulations are given in\n\\cref{sec:appendix-sim-gws}. We explore different regions of the power spectrum with each simulation.\nSimulation (D) has a large value\nof $L \\dd{x}\/\\xi_*$ and is therefore able\nto probe the GW power spectrum around the peak and towards smaller\n$k$, whereas simulation (E) is better able to probe the inertial range due to the smaller value of $\\dd\nx\/\\xi_*$.\nSimulation (A) has more lattice sites, allowing for a larger dynamic range. For this simulation we chose $L \\dd{x}\/\\xi_*$ and $\\dd x\/\\xi_*$ such that they lie in between the values of simulations (D) and (E). Simulation (A) then probes the peak of the spectrum and intermediate wavenumbers, confirming the power laws at low and high wavenumbers in these regions.\n\nThe SGWB power spectrum builds up very\nrapidly - this is further discussed in \\cref{sec:const}; its shape reflects the initial power spectrum, $\\mathcal{P}_U$ of \\cref{eq:sim-input}, representing fully developed turbulence.\n\\begin{figure}\n\t\\centering\n\t\\subfloat[Simulation (A), $\\Delta\\tau\/\\tau_{\\xi_*} = 3.98$.]{\\includegraphics[width=0.45\\textwidth]{figures\/A_gwspec.pdf}} \\\\\n\t\\subfloat[Simulation (D), $\\Delta\\tau\/\\tau_{\\xi_*} = 4.87$.]{\\includegraphics[width=0.45\\textwidth]{figures\/D_gwspec.pdf}}\n\t\\subfloat[Simulation (E), $\\Delta\\tau\/\\tau_{\\xi_*} = 0.594$.]{\\includegraphics[width=0.45\\textwidth]{figures\/E_gwspec.pdf}}\n\t\\caption{Gravitational wave power spectrum from our simulations with\n\t\t$\\overline{v}_* \\approx 0.1$.\n\t\tIn the y-axis, we divide the by $(\\mathcal{H}_*\\xi_*)^2$ since the simulations are in flat space-time, see~\\cref{eq:OmGWsimul} and the following discussion.\n\t\tThe coloured lines show the gravitational wave power spectrum\n\t\tat intervals $\\Delta \\tau$ starting from $\\tau - \\tdevel = \\Delta \\tau$ and finishing at $\\tau = \\tend$ .\n\t\tDarker shades correspond to later times. The black dashed line\n\t\tshows an average over the gravitational wave power spectrum in the\n\t\tlast\n\t\thalf of the elapsed simulation time.\n\t\tThe red dashed line shows a $k^1$ power-law, while the blue dashed line shows a $k^{-8\/3}$.\n\t\tWe have cut off the\n\t\tspectrum at high wavenumbers as it progressively gets polluted due to\n\t\tnumerical precision errors in projecting $\\dot{u}_{ij}(\\vb{k})$ to\n\t\t$\\dot{h}_{ij}(\\vb{k})$.\\label{fig:gwps-sim-vrms-0.1} }\n\\end{figure}\n\nBy averaging over the gravitational wave power spectrum in the latter\nhalf of each simulation, we are able to smooth out time-dependent\noscillations in the GW power spectrum. This averaged power spectrum is shown\nby the black dashed line in \\cref{fig:gwps-sim-vrms-0.1}. A clear $k^{-8\/3}$ power\nlaw is found for the inertial range in simulations (A) and (E), and an approximate $k^1$\npower law at small wavenumbers is seen in (A) and (D).\nTaking a time average over the second half of the simulation is justified since the simulation wavenumbers satisfy $k>2\/(\\tend-\\tdevel)$ for all simulations.\nThis can be verified by comparing the k-range in  \\cref{fig:gwps-sim-vrms-0.1,fig:gwps-sim-vrms-0.3,fig:gwps-sim-vrms-0.03} with the values given in \\cref{tab:list} (see also \\cref{tab:notation}).\n\nWhen we compute the power spectrum from our simulations, we bin modes which have a wavenumber within a given shell of $k$,\nas given by \\cref{eq:metric-pspec-dns}.\nAt late times, the frequency of the oscillations of each mode increases and the modes in each bin begin to oscillate out of phase. Consequently, our simulations average over these oscillations and do not capture them, even when we do not perform an average over time. This process can clearly be seen for high wavenumbers in \\cref{fig:gwps-sim-vrms-0.1}.\n\nTo show the variation of the GW power spectrum for different $\\overline{v}_*$\nand for different choices of $\\xi_*\/\\dd x$, we plot the averaged GW\npower spectrum from all simulations in \\cref{fig:gwps-sim-multi}. As can be seen,\nsimulations with different choices of $\\xi_*\/\\dd x$ explore different\nparts of the spectrum, with small $\\xi_*\/\\dd x$ mapping out low\nwavenumbers and the peak of the spectrum, and larger $\\xi_*\/\\dd x$\nexploring the inertial range.\nIndeed, for $\\overline{v}_*\\simeq0.1$ we find good agreement between simulations (A) and (D) from low wavenumbers up to the vicinity of the peak, and with (E) at high wavenumbers.\nThis gives us confidence\nthat the results\nobtained for other $\\overline{v}_*$ are also valid.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.7\\textwidth]{figures\/gw_multi_example.pdf}\n\t\\caption{Averaged GW power spectra for simulations (A)-(G) from \\cref{tab:list}. The coloured lines shown here correspond to  averaging the GW power spectra over the last half of the simulations. Simulations (B) and (C) have $\\overline{v}_*\\simeq0.03$, (A), (D) and (E) have $\\overline{v}_*\\simeq0.1$, and (F) and (G) have $\\overline{v}_*\\simeq0.3$. A cut off has been applied to the spectrum at high wavenumbers due to numerical precision noise.\n\t}\n\t\\label{fig:gwps-sim-multi}\n\\end{figure}\n\n\nIn subsequent sections we will show how these simulations can be compared with the results of the four dimensional numerical integration, and with the analytical result obtained under the assumption of a constant source in time.\nWe plot all three approaches in \\cref{fig:sim-analytic-comp} and find good agreement.\n\n\\subsubsection[GW spectrum via numerical integration of the\n\tanisotropic stress UETC]{Gravitational wave spectrum via numerical integration of the anisotropic stress UETC}\n\\label{sec:numintegration}\n\nWe have developed a numerical method allowing exact integration of the time-averaged SGWB spectrum, \\cref{eq:OmPvAVG}.\nThis includes integration of the full angular dependence as well, improving on previous attempts~\\cite{Caprini:2009yp,Niksa:2018ofa}.\nWe insert \\cref{eq:unequal-velocity-ps} into \\cref{eq:unequal-stress-final}, and in turn into \\cref{eq:OmPvAVG}.\nMaking use of \\cref{eq:velocity_ps_mathcal}, the initial spectral density $\\Psdv(k,\\tini)$ is set to the\nvon K\\'arm\\'an\\ spectrum  \\cref{eq:power-spectrum},\nrepresenting fully developed incompressible turbulence.\n\nWe also need to insert the time evolution laws for $\\overline{v}^2(\\tau)$ and $\\xi(\\tau)$.\nIn accordance with the initial conditions adopted in the simulations (see \\cref{sec:initial}), we assume instantaneous generation of turbulence.\nSpecifically, prior to $\\tdevel$ the velocity field is zero everywhere, then at time $\\tdevel$, turbulence appears fully developed with initial rms velocity $\\overline{v}_*$ and initial correlation scale $\\xi_*$.\nTurbulence then starts decaying on a timescale of order $\\tau_{\\xi_*}$, following \\cref{eq:VelEvMod,eq:XiEvMod}.\nTo summarize, for the instantaneous generation scenario:\n\\begin{align}\n\t & \\overline{v}^2(\\tau)=\n\t\\begin{cases}\n\t\t0                                                                              & \\text{if } \\tau < \\tdevel \\\\\n\t\t\\overline{v}_* ^2 \\qty(\\dfrac{\\tau - \\tdevel +\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}}{\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}})^{-p} & \\text{if } \\tau > \\tdevel\n\t\\end{cases}\n\t\\quad\\text{and} \\quad\n\t\\xi(\\tau) =\n\t\\begin{cases}\n\t\t\\text{Not defined}                                                      & \\text{if } \\tau < \\tdevel \\\\\n\t\t\\xi_*\\qty(\\dfrac{\\tau - \\tdevel +\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}}{\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}})^{q} & \\text{if } \\tau > \\tdevel\n\t\\end{cases}.\n\t\\label{eq:disc_evol}\n\\end{align}\n\nIn the following we set $\\mathcal{N}_\\mathrm{e}=5$, the value that best fits the simulations, see \\cref{fig:xi-vrms-evol}. For the decay exponents $p,~q$, we tested both sets of values that one infers from \\cref{eq:pchialpha,eq:qchialpha}, setting $\\beta=3$ and $\\beta=4$.\nThe unequal time decorrelation velocity appearing in the exponential of \\cref{eq:unequal-velocity-ps} is expressed in terms of the equal time one as in \\cref{eq:vsweepcomplete}, with $v_\\mathrm{dc}(k,\\tau)$ given in \\cref{eq:Vlarge}.\nThe four-dimensional integral of \\cref{eq:OmPvAVG} is calculated with Monte Carlo integration with importance sampling, using the VEGAS algorithm~\\cite{1978JCoPh..27..192L, Lepage:2020tgj}, an iterative and adaptive Monte Carlo scheme.\nMore details on the implementation are given in \\cref{sec:4dinteg}.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/figure_10.pdf}\n\t\\caption{\n\t\tExamples of GW power spectra computed via numerical integration of \\cref{eq:OmPvAVG} under the assumption of instantaneous turbulence generation (see \\eqref{eq:disc_evol}).\n\t\tThe spectra are computed using different values of the initial rms velocity, $\\overline{v}_*$, the initial integral scale relative to the Hubble scale, $\\mathcal{H}_*\\xi_*$.\n\t\tEach panel corresponds to a different value of $\\overline{v}_*$, as specified by the title, and each colour indicates a different value of $\\mathcal{H}_*\\xi_*$, as specified by the legend in the bottom right panel.\n\t\tThe solid lines are computed setting $\\beta = 3$, whereas the dashed lines setting $\\beta = 4$.\n\t\tWe recall that the evolution of $\\overline{v}(\\tau)$ and $\\xi(\\tau)$ are determined by $\\beta$ through \\cref{eq:pchialpha,eq:qchialpha}.}\n\t\\label{fig:varying-beta}\n\\end{figure}\n\n\nAs discussed at the end of \\cref{sec:free-decay-evol},  turbulence is fully dissipated after about $1000\\tau_{\\xi_*}$, for reasonable choices of the initial parameters $T_*,~\\overline{v}_*,~ \\xi_*\\mathcal{H}_*$.\nHowever, there is no need to time-integrate \\cref{eq:OmPvAVG} for so long: the bulk of the GW power spectrum is sourced on a much shorter timescale.\nThis is due to two reasons.\nFirst, the generation of GWs is very localized in time: GWs with wavenumber $k$ are produced by the source on a timescale  $\\order{1\/k}$, since the Green's function in \\cref{eq:OmPv}, given in \\cref{eq:green_with_all_terms}, has period $2\\pi\/k$.\nSecond, the decay of turbulence occurs over a timescale of the order of the initial eddy turnover time $\\tau_{\\xi_*}$, which effectively cuts off the source (more precisely, about five $\\tau_{\\xi_*}$, since $\\mathcal{N}_\\mathrm{e}\\simeq 5$ is the value that best fits the simulations -  see \\cref{sec:free-decay-evol}).\nAll GW wavenumbers with $k\\gtrsim 1\/(\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*})$ are therefore sourced before the turbulent kinetic energy has decayed appreciably. This includes the peak region of the SGWB spectrum.\nOn the other hand, GW wavenumbers with $k\\lesssim 1\/(\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*})$ are excited when the decay of the turbulent kinetic energy has already occurred: GW production on these wavenumbers is therefore negligible. This is the large scale spectral region, proportional to $k^3$.\nWe therefore expect the GW power spectrum to stop growing after a timescale of the order of $1\/(\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*})$.\nWe have numerically checked that the integration in \\cref{eq:OmPv} indeed converges to a fixed result after a few tens of $\\tau_{\\xi_*}$.\n\nIn the code, we actually integrate for $100 ~ \\tau_{\\xi_*}$: therefore, in practice we output the SGWB spectra long after the source has decayed, when the spectral amplitude does not grow in time any longer (the integral\nin \\cref{eq:OmPvAVG} has converged).\nFurthermore, the $k$-range of interest is such that $k>1\/(100\\,\\tau_{\\xi_*})$ for every chosen value of $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$, as can be appreciated from  \\cref{fig:varying-beta}.\nTherefore, we are allowed to integrate directly the time-averaged GW spectrum \\cref{eq:OmPvAVG}.\nWe have indeed checked that numerically integrating the full expression \\cref{eq:green-approx} provides a GW spectrum which is oscillating around the amplitude of the integral of the time-averaged GW spectrum, \\cref{eq:OmPvAVG}, shown in \\cref{fig:varying-beta}.\nWe do not plot these spectra, as they do not convey any additional information.\n\n\\cref{fig:varying-beta} shows the GW power spectrum computed from the numerical integration code, for different values of the parameters $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$ and for both $\\beta=3$ and $\\beta=4$.\nIt can be appreciated that the peak amplitude scales like $(\\mathcal{H}_*\\xi_*)^2$.\nThe spectral peak is independent of the value of $\\overline{v}_*$, and corresponds to the peak of the velocity power spectrum \\cref{eq:power-spectrum}, occurring at $k\\xi_*\\simeq 2.7\/\\mathcal{A}$.\nThis is consistent with the finding of Ref.~\\cite{Caprini:2009fx} for a source which is discontinuous in time.\n\nThe spectra which have $\\mathcal{H}_*\\xi_* = 10^{-3}$ in \\cref{fig:varying-beta} are\nminimally affected by the expansion of the Universe: the chosen values of $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$ satisfy $100\\,\\tau_{\\xi_*} \\leq \\mathcal{H}_*^{-1}$, and so the dynamics take place fully at  sub-horizon scales.\nIt is therefore appropriate to compare these spectra to the time-averaged spectra resulting from the simulations, in the region $k>\\mathcal{H}_*$ (see \\cref{eq:OmGWsimul} and discussion thereafter). We will do so in~sub-\\cref{sec:compamethods}.\nOn the other hand, for larger values of $\\mathcal{H}_*\\xi_*$ in \\cref{fig:varying-beta}, we integrate for longer than one Hubble time.\n\n\nWe find that the actual values of the exponents $p$ and $q$ do not substantially impact the SGWB spectrum, indicating that the precise details of the turbulent decay are unimportant in this respect.\nThis is due to the nature of the free decay of turbulence in combination with the typically short timescale associated with GW production.\n\\cref{fig:varying-beta}  shows the SGWB spectrum from the numerical integration for both $\\beta=3$ and $\\beta=4$.\nFor all values of $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$ the spectra only differ in the region $k< 1\/(\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*})$, as expected for reasons we have outlined above.\nFor the same reasons, we can safely ignore the role of the viscous scale $\\lambda$ in all our evaluations.\n\n\n\n\n\\subsubsection{Constant source approximation}\n\\label{sec:const}\n\nIn this section, we show that the GW source can be approximated as almost constant in time while the majority of the GW signal is being generated.\nThis has already been pointed out in \\Refa{RoperPol:2022iel}, in the case of MHD turbulence.\nSince the Green's function in \\cref{eq:OmPv} has period $2\\pi\/k$, the GW production for each wavenumber $k$ occurs on a timescale of the order of $1\/k$.\nTherefore, over the typical decay time for the turbulence source, namely $\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}$ -- as we assumed for \\cref{eq:disc_evol} --\nthere is sourcing of gravitational waves for wavenumbers $k>1\/(\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}) \\equiv \\overline{v}_*\/(\\mathcal{N}_\\mathrm{e}\\xi_*)$.\nThis constitutes a large fraction of the GW spectrum, including the region around the peak $k\\,\\xi_*\\simeq 2.7\/\\mathcal{A}$.\nConsequently, the SGWB signal on the scales of interest -- specifically, around the peak -- is fully established before the turbulence has decayed appreciably.\nOn scales larger than the peak, on the other hand, the GW production is less efficient due to the decay of the source.\nOn even larger scales, one expects the $k^3$ increase typical of uncorrelated sources in both time and space \\cite{Caprini:2009fx}.\n\nWe assume in what follows that the turbulent GW source is instantaneously generated, remains constant for a duration $\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*}$, and sharply turns off at a time $\\tdevel+\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*}$.\nThe number $\\mathcal{N}_\\mathrm{cut}$ is of the same order as $\\mathcal{N}_\\mathrm{e}$, and we set $\\mathcal{N}_\\mathrm{cut}$ to $7$ as it gives the best fit to the result of the numerical integration.\nWe neglect both the overall time dependence of the turbulent fluid velocity power spectrum and its time decorrelation in \\cref{eq:unequal-velocity-ps}\nThis means that the anisotropic stress spectral density \\cref{eq:unequal-stress-final} becomes time-independent, and the time and momentum integrals in \\cref{eq:OmPv} decouple.\nWe can then rewrite \\cref{eq:gwradera} as\n\\begin{equation}\n\t\\eval{\\dv{{\\Omega}_\\mathrm{gw}}{\\ln k}}_{ \\tdevel+\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*}}\n\t=\\frac{8}{3\\pi^2}\\,k^3\\,\\mathcal{T}_\\mathrm{gw}(k,\\tdevel,\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*})\\, P_{\\tilde\\Pi}(k,\\tdevel) \\,.\\label{eq:SGWBfinconst}\n\\end{equation}\nLet us first analyse  the part due to the double time integral $\\mathcal{T}_\\mathrm{gw}$.\nWe evaluate it at the source turn-off:\n\\begin{equation}\n\t\\mathcal{T}_\\mathrm{gw}(k,\\tdevel,\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*}) \\equiv \\iint_{\\tdevel}^{\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}} \\mathcal{G}(k, \\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}, \\eta, \\zeta)\\,\\frac{\\dd{\\eta}}{\\eta}\\frac{\\dd{\\zeta}}{\\zeta}.\n\\end{equation}\nRestricting our consideration of this integral to wavenumbers satisfying $k>1\/(\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*})$, we are again entitled to drop the second and third term from the Green's function \\cref{eq:green_with_all_terms}, to obtain~\\cite{RoperPol:2022iel}:\n\\begin{align}\n\t\\mathcal{T}_\\mathrm{gw}(k,\\tdevel,\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*}) & =\\iint_{\\tdevel}^{\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}} \\cos k(\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*} - \\eta)\\cos k(\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*} - \\zeta) \\frac{\\dd{\\eta}}{\\eta}\\frac{\\dd{\\zeta}}{\\zeta} \\label{eq:mathT} \\\\\n\t                                                  & = \\left\\{\\cos k(\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}) \\qty[\\cosi (k (\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*})) -\\cosi(k \\tdevel)] \\right. \\nonumber                                                                 \\\\\n\t                                                  & \\qquad + \\left.\\sin k(\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}) \\qty[\\sini (k (\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*} ))-\\sini(k \\tdevel)]\\right\\}^2.\n\\end{align}\nThe upper limits of both time integrals must be set to $\\tfin=\\tdevel+\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}$ in order to match the results of both the simulations and the numerical integration (see~\\cref{sec:compamethods}).\nAs pointed out in \\Refa{RoperPol:2022iel}, the abrupt switching off of the source means that the evolution after $\\tfin$ would give an enhancement of the SGWB power at high frequency, which would break this agreement.\nSince we evaluate the time integrals at $\\tfin$, we cannot perform a time average; this is why we do not use \\cref{eq:OmPvAVG} in this Section.\n\nIn \\Refa{RoperPol:2022iel}, a simplified expression for \\cref{eq:mathT} has been found, tuned to interpolate the envelope of the $k$-oscillations.\nIn this work, we want to compare with the results of the simulations (\\cref{sec:results-gw-sim}) and of the numerical integration (\\cref{sec:numintegration}), which both evaluate the time-averaged SGWB.\nWe therefore adapt the result of \\Refa{RoperPol:2022iel} to account for time averaging, as follows:\n\\begin{equation}\n\t\\mathcal{T}_\\mathrm{gw}(k,\\tdevel,\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*})\\approx\n\t\\begin{cases}\n\t\t\\ln^2\\qty(1 + \\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*} \/ \\tdevel) & \\text{ for}~\\sqrt{2}\\, k<1\/(\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*})\\,,  \\\\\n\t\t\\ln^2\\qty[1 + (\\sqrt{2}k\\tdevel)^{-1}]  & \\text{ for}~\\sqrt{2}\\, k>1\/(\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}) \\,.\n\t\\end{cases}\n\t\\label{eq:tgw-approx}\n\\end{equation}\nThe transition occurs at the wavenumber corresponding to the source lifetime, namely $\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}$ (see \\Refa{Caprini:2009fx}).\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics{figures\/pi_dimless_ps.pdf}\n\t\\caption{Anisotropic stress spectral density $P_{\\tilde{\\Pi}}(k,\\tdevel)$ at the initial time in the constant source approximation, from the exact integration of \\cref{eq:Piintegral} (black solid line) and the analytical approximation \\cref{eq:Piapprox} (red dashed line). We also show $P_{\\tilde{\\Pi}}(k,\\tau)$ extracted from simulation (A) (colored solid lines) between $\\tau=\\tdevel$ and $\\tau=\\tend$ with interval $\\Delta \\tau = 9.96 \\tau_{\\xi_*}$. Lighter colours indicate later times.}\n\t\\label{fig:plotPi}\n\\end{figure}\n\nWe now turn to the description of the anisotropic stress spectral density at initial time $P_{\\tilde\\Pi}(k,\\tdevel)$, also entering \\cref{eq:SGWBfinconst}.\nThis is given by\nthe convolution in momentum of \\cref{eq:unequal-stress-final} (without residual time dependence).\nIt takes the form (we use the notation of \\cref{sec:analytical}):\n\\begin{align}\n\tP_{\\tilde\\Pi}(k,\\tdevel) & =\\frac{\\pi^2}{2}\\mathcal{A}^3{\\xi_*^3}\n\t\\int_0^\\infty \\dd H\\,H^2\n\t\\int_0^1 \\dd\\alpha  \\,\n\t\\frac{\\mathrm{Proj}(H,\\alpha)}{\\mathrm{s}^3(H,\\alpha)\\mathrm{s}^3(H,-\\alpha)}\\Psv[\\mathrm{s}(H,\\alpha), \\tdevel]\\Psv[\\mathrm{s}(H,-\\alpha), \\tdevel]\\label{eq:Piintegral}\n\t\\\\\n\t                         & \\simeq\n\t\\frac{P_{\\tilde\\Pi}(0)}{1+\\qty(\\frac{\\mathcal{A} k\\xi_* }{3.0})^2+\\qty(\\frac{\\mathcal{A} k\\xi_* }{3.5})^{11\/3}}\\,,\\label{eq:Piapprox}\n\\end{align}\nwhere the factors of $3.0$ and $3.5$ come from an analytical fit, tuned to match  the exact numerical integration of \\cref{eq:Piintegral}, and $P_{\\tilde\\Pi}(0)$ is the value of the anisotropic stress spectral density at $k=0$: %\n\\begin{equation}\n\tP_{\\tilde\\Pi}(0)= \\mathcal{B}^2\\mathcal{A}^3\\,\\overline{v}_*^4\\,{\\xi_*^3} \\int_0^\\infty \\dd{H}  \\frac{1024\\,\\sqrt[3]{2}\\,\\pi^2 H^6}{(4 H^2+4)^{17\/3}} \\int_0^1 \\dd{\\alpha}  (\\alpha^2+1)^2= \\mathcal{B}^2\\mathcal{A}^3\\,\\overline{v}_*^4\\,{\\xi_*^3}\\,\\, \\frac{7 \\pi^{5\/2} \\,\\Gamma \\left(\\frac{13}{6}\\right)}{8\\, \\Gamma \\left(\\frac{17}{3}\\right)}\\,.\n\\end{equation}\nIn \\cref{fig:plotPi}, we  compare the anisotropic stress spectral density from simulation (A) with both the numerical result \\cref{eq:Piintegral}, and the analytical approximation\n\\cref{eq:Piapprox}.\nAs can be appreciated from the figure, at the initial time $\\tdevel$ there is excellent agreement between the simulation result and the anisotropic stress spectral density derived in the context of our turbulent model.\nFor illustration purposes, in \\cref{fig:plotPi} we also display the time evolution of the anisotropic stress spectral density from simulation (A): it is decaying, due in turn to the decay of the turbulent source itself.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/figure_12.pdf}\n\t\\caption{Reproduction of \\cref{fig:varying-beta}, where here we also show the constant source approximation given in \\cref{eq:constant_approx} for different values of $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$.  The solid lines match those of \\cref{fig:varying-beta} (with $\\beta=3$), whereas the dashed lines give the constant source approximation for an equivalent value of $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$. We fix $\\mathcal{N}_\\mathrm{cut}=7$ in the constant source approximation \\cref{eq:constant_approx} for all values of $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$.\n\t}\n\t\\label{fig:SGWB_num_int}\n\\end{figure}\n\nThe spectral shape of the SGWB from a constant source, \\cref{eq:SGWBfinconst}, is given by the combination of the $k$-dependence coming from both  $\\mathcal{T}_\\mathrm{gw}(k,\\tdevel,\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*})$ and  $P_{\\tilde{\\Pi}}(k,\\tdevel)$.\nRecalling that the values of $\\mathcal{A}$ and $\\mathcal{B}$ can be found in \\cref{eq:constants1,eq:constants2}, we find\n\\begin{multline}\n\t\\dv{{\\Omega}_\\mathrm{gw}}{\\ln k}\n\t= \\frac{7 \\sqrt{\\pi} \\,\\Gamma(13\/6)}{3 \\Gamma(17\/3)} \\frac{(\\mathcal{A}\\, k\\, \\xi_*)^3 \\mathcal{B}^2\\,\\overline{v}_*^4}{1+\\qty(\\mathcal{A} \\,k\\,\\xi_* \/ 3.0)^2+\\qty(\\mathcal{A}\\, k\\,\\xi_* \/ 3.5)^{11\/3}} \\\\\n\t\\times \\begin{cases}\n\t\t\\ln^2\\qty(1 + \\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*} \\mathcal{H}_*) & \\text{ for}~\\sqrt{2} k<1\/(\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*})   \\\\\n\t\t\\ln^2\\qty[1 +\\mathcal{H}_*\/ (\\sqrt{2}\\,k)]  & \\text{ for}~\\sqrt{2}\\, k>1\/(\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*})\n\t\\end{cases}\\text.\n\t\\label{eq:constant_approx}\n\\end{multline}\nThe $k$-dependence of $\\mathcal{T}_\\mathrm{gw}(k,\\tdevel,\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*})$ has been described in Ref.~\\cite{Caprini:2009fx} in the context of coherent sources,\nmeaning those for which the UETC can be factorized.\n\\cref{eq:mathT} is effectively a time-domain Fourier cosine transform, therefore the decay properties of $\\mathcal{T}_\\mathrm{gw}(k,\\tdevel,\\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*})$ at large $k$ depend on its time differentiability -- and in the case being studied here, it is discontinuous in time.\nThis gets combined with the behaviour of $P_{\\tilde{\\Pi}}(k,\\tdevel)$, which is uncorrelated for wavenumbers $k< 1\/(\\mathcal{A}\\xi_*)$, while at high $k$ it shares the same power-law decrease as $\\Psdv(k,\\tdevel)$ (hence, for an initially von K\\'arm\\'an spectrum, it decreases as $k^{-11\/3}$ at high $k$) \\cite{Caprini:2007xq}.\nAltogether, for instantaneous turbulence generation of a von K\\'arm\\'an spectrum, the expected slopes for low $\\overline{v}_*$ are (with notation $K=\\mathcal{A}\\xi_* k$):\n\\begin{align}\n\t\\text{if}~~\\mathcal{N}_\\mathrm{cut}\\mathcal{H}_*\\xi_*<\\overline{v}_* \\,, \\qquad &\n\t\\dv{{\\Omega}_\\mathrm{gw}}{\\ln k} \\propto \\label{eq:Omslopesprediction}\n\t\\begin{cases}\n\t\t\\mathcal{N}_\\mathrm{cut}^2\\, \\overline{v}_*^2\\, (\\mathcal{H}_*\\xi_*)^2\\, K^3 & \\text{ for}~\\mathcal{N}_\\mathrm{cut} K\\ll\\overline{v}_* \\,,          \\\\\n\t\t\\overline{v}_*^4\\, (\\mathcal{H}_*\\xi_*)^2\\, K^{1}         & \\text{ for}~ \\overline{v}_* \\ll \\mathcal{N}_\\mathrm{cut} K \\text{ and } K \\ll 1 \\,, \\\\\n\t\t\\overline{v}_*^4\\, (\\mathcal{H}_*\\xi_*)^2\\, K^{-8\/3}      & \\text{ for}~ 1 \\ll K \\,,\n\t\\end{cases} \\\\\n\t\\text{if}~~\\mathcal{N}_\\mathrm{cut}\\mathcal{H}_*\\xi_*>\\overline{v}_*\\,, \\qquad  &\n\t\\dv{{\\Omega}_\\mathrm{gw}}{\\ln k} \\propto \\label{eq:OmslopespredictionLONG}\n\t\\begin{cases}\n\t\t\\overline{v}_*^4 K^3 \\ln^2\\qty(1 + \\mathcal{N}_\\mathrm{cut} \\mathcal{H}_*\\xi_* \/ \\overline{v}_*) & \\text{ for}~\\mathcal{N}_\\mathrm{cut} K\\ll\\overline{v}_* \\,,                                    \\\\\n\t\t\\overline{v}_*^4 K^3 \\ln^2\\qty[1 + \\mathcal{A}\\mathcal{H}_*\\xi_*\/ (\\sqrt{2}\\,K)]\n\t\t                                                                & \\text{ for}~\\overline{v}_*\/\\mathcal{N}_\\mathrm{cut} \\ll K\\ll \\mathcal{A}\\mathcal{H}_*\\xi_* \\,, \\\\\n\t\t\\overline{v}_*^4\\, (\\mathcal{H}_*\\xi_*)^2\\, K^{1}                      & \\text{ for}~ \\mathcal{A}\\mathcal{H}_*\\xi_* \\ll \\mathcal{N}_\\mathrm{cut} K \\text{ and } K \\ll 1 \\,,     \\\\\n\t\t\\overline{v}_*^4\\, (\\mathcal{H}_*\\xi_*)^2\\, K^{-8\/3}                   & \\text{ for}~ 1 \\ll K \\,.\n\t\\end{cases}\n\\end{align}\nThis behaviour is in accordance with that found in \\Refs{Caprini:2009fx,RoperPol:2022iel}.\nNote that the $P_{\\tilde\\Pi}\/k^2$ slope observed in \\Refa{Brandenburg:2019uzj} at high $k$ can be interpreted in light of what has been discussed above as well as the results of \\Refa{Caprini:2009fx}.\n\nIn \\cref{fig:SGWB_num_int}, we plot the constant source approximation,  \\cref{eq:constant_approx}, and compare it to the result of the full numerical integration presented in \\cref{sec:numerical}.\nAs discussed above, we set $\\mathcal{N}_\\mathrm{cut}=7$ to match the SGWB spectra obtained from the numerical integration.\nThe spectral shapes of the SGWBs evaluated under the constant source approximation are in good agreement with those produced with the numerical integration,\nover the tested range of values for the input parameters $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$.\nThe discrepancy around the transition between regimes at $k = 1\/(\\sqrt{2} \\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*})$, see~\\cref{eq:constant_approx},\nis most notable for sources that last longer than one Hubble time (meaning $\\mathcal{N}_\\mathrm{cut}\\tau_{\\xi_*}\\mathcal{H}_*>1$), where the intermediate power law for wavenumbers smaller than the peak is less pronounced.\nNote that in \\Refa{RoperPol:2022iel} it was possible to adjust the source duration (i.e.~the value of $\\mathcal{N}_\\mathrm{cut}$) in the constant approximation, as a function of $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$. This was done by comparison with the SGWBs output by the simulations, which covered alsp the large scale part of the spectrum.\nHere, on the other hand, we fix  $\\mathcal{N}_\\mathrm{cut}=7$ for all initial $\\overline{v}_*$ and $\\mathcal{H}_*\\xi_*$.\n\nThe UETC of decaying turbulence \\cref{eq:unequal-velocity-ps} has both a factorizable (coherent) component and a decorrelating one.\nThe coherent component shapes the SGWB spectrum at large scales and around the peak.\nOn the other hand, the decorrelation influences the spectral shape at small scales, as discussed in \\cref{sec:stationary}.\nThe discrepancy that can be observed in \\cref{fig:SGWB_num_int} at high wavenumber for large $\\overline{v}_*$\ncan therefore be interpreted in light of  what found under the assumption of a purely stationary source, see~the asymptotic behaviour at high $k$ of \\cref{eq:stat-uv}.\nThe constant source approximation, in which the decorrelation is absent, predicts a steeper power law -- given in \\cref{eq:Omslopesprediction} --  than that found by the numerical integration, which contains the full time evolution of the source.\n\nAll in all, it emerges that both the overall decay of turbulence as well as its decorrelation play a relatively minor role in shaping the SGWB signal: the first influences the spectral shape only at large scales $k\\xi_*< \\overline{v}_*\/\\mathcal{N}_\\mathrm{e}$, while the second does so only for large initial velocity $\\overline{v}_*$ and at small scales $k\\xi_*\\gg 1$.\n\n\n\\subsubsection{Comparison of the SGWB spectra}\n\\label{sec:compamethods}\n\n\\begin{figure}\n\t\\centering\n\t{\\includegraphics[width=0.8\\textwidth]{figures\/sim-analytic-gw-comp.pdf}}\n\t\\caption{GW power spectrum for instantaneous turbulence generation. The gray lines show the analytical approximation of \\cref{eq:constant_approx} based on a constant source lasting for $\\mathcal{N}_\\mathrm{cut} = 7$ eddy turnover times. The black lines show the result of the 4d numerical integration of \\cref{sec:numintegration}. From top to bottom, these lines correspond to $\\overline{v}_*=0.3$, $\\overline{v}_*=0.1$ and $\\overline{v}_*=0.03$ respectively. In all cases we fix $\\mathcal{H}_* \\xi_* = 0.001$.\n\t\tWe also show the averaged GW power spectra for simulations (A)-(G) from \\cref{tab:list}, which are plotted using colored lines as specified in the legend.\n\t\tThe GW power spectra from simulations has been cut off at high wavenumbers due to numerical precision noise.\n\t}\n\t\\label{fig:sim-analytic-comp}\n\\end{figure}\n\n\nIn \\cref{sec:results-gw-sim,sec:numintegration,sec:const}, we have presented three different methods of estimating the GW power spectrum when turbulence is generated instantaneously.\nWe first presented the GW spectra from direct numerical simulations, then we performed an exact numerical integration and finally computed an analytical approximation assuming a constant source.\n\nIn this section, we compare the GW spectra obtained from the three different methods.\nThe comparison is performed (i) long after the source has dissipated and (ii) for the lowest value of $\\mathcal{H}_*\\xi_*$\nfor which we performed the numerical integration, namely $\\mathcal{H}_*\\xi_* = 10^{-3}$. Condition (i) is necessary to perform the time average, while\ncondition (ii) is necessary since the simulations neglect the effect of expansion. For more detail, we refer the reader to the discussions at the end of \\cref{sec:gw-numerical,sec:numintegration}.\n\nWe plot the late-time averaged GW power spectra from our simulations in \\cref{fig:sim-analytic-comp}, and compare them to those computed in the full numerical integration and in the constant source approximation.\nWe show the GW power spectra from the numerical integration and the constant source approximation for $\\overline{v}_*=0.03\\text,~0.1$ and $0.3$.\nThere is excellent agreement between the SGWBs obtained with the three methods, for all the values of $\\overline{v}_*$ we simulated.\nThey are all consistent with respect to the GW power spectrum amplitude, the breadth of the peak, the peak location at $k\\xi_* \\simeq 2.7\/\\mathcal{A}$ and the power laws at low and high wavenumbers.\nIn particular, they all agree in the regions we expect the simulations to be accurate, namely at low wavenumbers for simulations (B), (D) and (F), and at high wavenumbers for simulations (C), (E) and (G). Simulation (A), which is our largest simulation, has close agreement with the other methods for the full range of wavenumbers depicted.\nWe have therefore validated that the numerical integration and the constant approximation are\naccurate in the limit of $\\mathcal{H}_*\\xi_*\\ll1$. Furthermore, as they include the effect of expansion, we expect them to also be accurate outside this limit.\nAs discussed in the previous section and depicted in \\cref{fig:SGWB_num_int}, there is a discrepancy between the constant source approximation and the full numerical integration for wavenumbers close to the kink at $k = 1\/(\\sqrt{2} \\mathcal{N}_\\mathrm{cut} \\tau_{\\xi_*})$. This can also be seen in \\cref{fig:sim-analytic-comp}.\n\nAs the constant source approximation displays  excellent agreement around the peak and obtains the correct asymptotic power laws, \\cref{eq:constant_approx} can be used as a simple  approximation to the GW power spectrum which does not require simulations.\n\n\n\\subsection{Phase of turbulence growth: results from the numerical integration}\n\\label{sec:GWcontinuous}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tikzpicture}[scale=2,ultra thick, node distance=2mm]\n\n\t\t\\coordinate(begin) at (0,0);\n\t\t\\coordinate(onset) at (1,0);\n\t\t\\coordinate(onset2) at (0,1);\n\t\t\\coordinate(end) at (3,3);\n\t\t\\coordinate(tau) at (4,0);\n\t\t\\coordinate(zeta) at (0,3.5);\n\t\t\\coordinate(fin) at (3,0);\n\t\t\\coordinate(fin2) at (0,3);\n\t\t\\node (zone1) at (0.5, 0.5) {$1$};\n\t\t\\node (zone2) at (2, 2) {$2$};\n\t\t\\node (zone3) at (2, 0.5) {$3$};\n\t\t\\node (zone4) at (0.5, 2) {$4$};\n\n\n\t\t\\draw (begin) rectangle (end);\n\t\t\\draw[dashed] (onset) -- +(0, 3);\n\t\t\\draw[dashed] (onset2) -- +(3, 0);\n\t\t\\draw[-latex] (begin) -- +(tau);\n\t\t\\draw[-latex] (begin) -- +(zeta);\n\n\t\t\\node [below=of begin] {$\\tini$};\n\t\t\\node [below=of onset] {$\\tdevel=\\tini + \\tgro$};\n\t\t\\node [left=of onset2] {$\\tdevel=\\tini + \\tgro$};\n\t\t\\node [below=of tau] {$\\eta$};\n\t\t\\node [left=of zeta] {$\\zeta$};\n\t\t\\node [below=of fin] {$\\tau$};\n\t\t\\node [left=of fin2] {$\\tau$};\n\n\t\\end{tikzpicture}\n\n\t\\caption{Diagram for the evolution of turbulence in terms of $\\eta$ and $\\zeta$. The injection of kinetic energy starts at $\\tini$ and turbulence develops  in region $1$ on a timescale $\\tgro$, the value of which depends on the model: it might correspond to the PT duration $\\beta^{-1}$, or to the eddy turnover time $\\tau_{\\xi_*}$. In region $2$, the turbulence is freely decaying. The growth and free decay phases are correlated: regions $3$ and $4$ also contribute to the production of GWs.}\n\t\\label{fig:two-time}\n\\end{figure}\n\nIn this section we go beyond the scenario of instantaneous turbulence generation, considered so far, and include a growth phase for the turbulence kinetic energy.\nSince the Reynolds number in the early Universe is very large (of the order of $10^{13}$ at the  electroweak scale, see \\cref{sec:numintegration}), turbulence is expected to arise from vorticity generated during the PT \\cite{Cutting:2020nla},  or due to the interaction of shocks \\cite{Pen:2015qta,Dahl:2021wyk}.\nIn order to model the GW source properly, one would have to model the onset of turbulence by simulating the complete system of scalar field and fluid.  We leave this complicated problem  for future work.\n\nIn the present analysis we model the (purely vortical) turbulence growth phase heuristically.\nThis allows us to  gauge in a simple way the importance of this phase, as far as the SGWB spectral shape is concerned.\n\\Refs{Caprini:2009fx,Caprini:2009yp,Caprini:2009pr} have demonstrated analytically that time continuity of the GW sourcing process must be ensured, since it plays a role in shaping the SGWB signal, and Refs.~\\cite{Pol:2019yex,Brandenburg:2021aln,RoperPol:2021xnd} also obtained  different SGWB spectral shapes in the forced case.\nAs we shall see, we confirm this result in the present work, via the numerical integration of the SGWB source (see \\cref{sec:numintegration}).\n\nStarting from the initial time $\\tini$, we assume that turbulence is sourced on a timescale $\\tgro$.  At time $\\tdevel=\\tini + \\tgro$, turbulence is fully developed, and then starts decaying (see~\\cref{fig:two-time}).\nThe duration of the growth phase would depend on the particular mechanism sourcing the turbulence. If the sourcing process is related to the PT dynamics, one would expect the duration of the growth phase to be of the order of the duration of the PT.\nAlternatively, one can assume that it takes approximately one initial eddy turnover time $\\tau_{\\xi_*} = \\xi_* \/ \\overline{v}_*$ to build the turbulent cascade.\nThis is the assumption we make in the following, namely that $\\tgro = \\tau_{\\xi_*}$.\n\nConcerning the correlations of the velocity field, the Gaussian decorrelation scenario of \\cref{sec:kraichnan,sec:unequal-time-velocity} holds in principle only in the free-decay phase.\nHowever, in the absence of a better model, we assume that \\cref{eq:unequal-velocity-ps} is valid also during the growth phase, and in the mixed regions.\nTherefore, in the UETC of \\cref{eq:unequal-velocity-ps}, the times $\\tau$ and $\\zeta$ can pertain either both to the growth phase, or both to the free decay phase, or to the two phases mixed (see~\\cref{fig:two-time}).\n\nWe consider two scenarios for the generation of turbulence: a $\\mathcal{C}^0$ {growth phase} and a $\\mathcal{C}^{1}$ {growth phase}.\nFirst we adopt the model of Ref.~\\cite{Caprini:2009pr} and suppose that the vortical kinetic energy grows linearly in time before turbulence enters the free-decay phase. Furthermore, we assume that the integral scale remains constant during the growth phase, and starts growing during the phase of free decay in the $\\mathcal{C}^0$ scenario:\n\\begin{equation}\n\t\\begin{split}\n\t\t\\overline{v}^2(\\tau) & = \\overline{v}_* ^2 \\begin{cases}\n\t\t\t\\dfrac{\\tau - \\tini}{\\tgro}                                         & \\text{if } \\tau < \\tdevel \\\\\n\t\t\t\\qty(\\dfrac{\\tau - \\tdevel +\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}}{\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}})^{-p} & \\text{if } \\tau > \\tdevel\n\t\t\\end{cases} \\\\\n\t\t\\text{and} \\quad\n\t\t\\xi(\\tau) & = \\xi_*\\begin{cases}\n\t\t\t1                                                                  & \\text{if } \\tau < \\tdevel  \\\\\n\t\t\t\\qty(\\dfrac{\\tau - \\tdevel +\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}}{\\mathcal{N}_\\mathrm{e}\\tau_{\\xi_*}})^{q} & \\text{if } \\tau > \\tdevel.\n\t\t\\end{cases}\n\t\\end{split}\n\t\\label{eq:C0}\n\\end{equation}\n\nIn the second scenario the kinetic energy and the integral scale evolve in such a way that they are both  continuous and differentiable at time $\\tdevel$. The kinetic energy is also continuous and differentiable at initial time $\\tini$. This scenario requires defining two extra functions.\nFor the growth phase of the kinetic energy we use a smooth step function with the properties $\\smoothstep(1) = 1$,  $\\smoothstep'(1) = 0$ (and also, $\\smoothstep(0)=\\smoothstep'(0) = 0$):\n\\begin{equation}\n\t\\smoothstep(x) = \\begin{cases}\n\t\t0         & x< 0  \\\\\n\t\t3x^2-2x^3 & 0<x<1 \\\\\n\t\t1         & x>1.\n\t\\end{cases}\n\\end{equation}\nTo connect with the decay phase, we use a smooth power law with the property $\\smoothpl(1, p) = 1$ and $\\smoothpl'(1, p) = 0$:\n\\begin{equation}\n\t\\smoothpl(x, p) = (1 - p) x^p + p x^{p-1}\\, ,\n\\end{equation}\nwhere $p<1$.\nUsing the above functions, we set for the $\\mathcal{C}^1$ scenario:\n\\begin{equation}\n\t\\begin{split}\n\t\t\\overline{v}^2(\\tau) & = \\overline{v}_*^2 \\begin{cases}\n\t\t\t\\smoothstep\\qty(\\dfrac{\\tau - \\tini}{\\tgro})                                 & \\text{if } \\tau < \\tdevel \\\\\n\t\t\t\\smoothpl\\qty(\\dfrac{\\tau - \\tdevel+\\mathcal{N}_\\mathrm{e} \\tau_{\\xi_*}}{\\mathcal{N}_\\mathrm{e} \\tau_{\\xi_*}}, -p) & \\text{if } \\tau > \\tdevel\n\t\t\\end{cases} \\\\\n\t\t\\text{and} \\quad \\xi(\\tau) & = \\xi_* \\begin{cases}\n\t\t\t1                                                                           & \\text{if } \\tau < \\tdevel  \\\\\n\t\t\t\\smoothpl\\qty(\\dfrac{\\tau - \\tdevel+\\mathcal{N}_\\mathrm{e} \\tau_{\\xi_*}}{\\mathcal{N}_\\mathrm{e} \\tau_{\\xi_*}}, q) & \\text{if } \\tau > \\tdevel.\n\t\t\\end{cases}\n\t\\end{split}\n\t\\label{eq:C1}\n\\end{equation}\n\nDifferent choices for the growth phase lead to substantially different spectral shapes in the resulting SGWB.\nThis is demonstrated by \\cref{fig:c0vsc1}, showing the SGWB with both the $\\mathcal{C}^0$ and $\\mathcal{C}^1$ growth phases, as well as for the instantaneous generation scenario.\nWhile the spectra at large scales are comparable, inserting a continuous growth phase shifts the spectral peak position from the characteristic length scale of the source, $k_\\mathrm{peak}\\simeq \\xi_*^{-1}$, to the characteristic timescale of the source, $k_\\mathrm{peak}\\simeq \\tau_{\\xi_*}^{-1}$: the peak position becomes therefore velocity-dependent~\\cite{Caprini:2009fx}.\nThis behaviour is caused by the component in the velocity field UETC \\cref{eq:unequal-velocity-ps} which is factorizable in time,\ntermed the ``coherent'' case in \\Refa{Caprini:2009fx}; see the discussion below \\cref{eq:constant_approx}.\nFurthermore, the addition of a  growth phase which is continuous in time leads to a\nreduction in the GW power, and a steeper slope at high frequencies,\nas shown in \\cref{fig:gwps-smooth} (in this figure we only plot the results for the $\\mathcal{C}^1$ growth phase).\nSince the peak now corresponds to the inverse eddy turnover time, the $k^1$ region observed in the instantaneous growth scenario never develops, as illustrated by \\cref{fig:c0vsc1}.\nNote that the spectra in the $\\mathcal{C}^0$ and $\\mathcal{C}^1$ growth phase  are almost identical, contrary to what was observed in \\Refa{Caprini:2009fx}. Indeed, we would not expect \\Refa{Caprini:2009fx}\nto capture all the complexity of the spectral changes due to the insertion of a continuous growth phase, since it studied a very simplified case in which the anisotropic stress UETC was separable in time.\n\nAt first, it may seem counter-intuitive that the addition of a growth phase decreases the energy in GWs, given Mercer's condition (see \\cref{sec:mercer}).\nWe illustrate this using \\cref{fig:two-time}: Mercer's condition  \\cref{eq:mercer-condition} applies to intervals of the form $I\\times I$, hence the contributions from the regions $1$ and $2$ are positive.\nHowever, the contributions from the regions $3$ and $4$ can very well be negative, lowering the total GW energy.\nThis effect depends crucially on the correlations between the growth and the free decay phase. We leave the study of the growth of turbulence from more realistic initial conditions to a future work.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/figure_14.pdf}\n\t\\caption{Gravitational wave power spectrum in the  instantaneous generation scenario (solid lines), with a $\\mathcal{C}^0$ growth phase (dashed lines) and with a $\\mathcal{C}^1$ growth phase (dotted lines).\n\t\tFrom bottom to top, $\\overline{v}_* = 0.1, 0.3$ and $0.6$.\n\t\tThe left panel shows $\\mathcal{H}_* \\xi_* = 10^{-3}$, the middle panel $\\mathcal{H}_* \\xi_* = 10^{-2}$ and the right panel $\\mathcal{H}_* \\xi_* = 10^{-1}$.}\n\t\\label{fig:c0vsc1}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/figure_15.pdf}\n\t\\caption{Gravitational wave power spectrum in the scenario with $\\mathcal{C}^1$ growth phase.\n\t\tEach panel displays a different value for the initial integral scale $\\mathcal{H}_* \\xi_*$ (as specified in the panels titles), and each line corresponds to $\\overline{v}_* = 0.1, 0.2, 0.3, 0.4, 0.5,$ and $0.6$ from bottom to top.\n\t}\n\t\\label{fig:gwps-smooth}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\nIn this paper, we have studied the GW signal generated by a\nphase of freely decaying vortical turbulence in a relativistic fluid.\nWe were motivated by the fact that thermal first order phase transitions occurring in the early Universe are expected to lead to a turbulent flow, either directly from vorticity generation\nduring bubble collisions, or from\nthe interaction of shocks following the acoustic phase.\nSince the sourcing of the turbulent flow remains an active area of research, we have chosen to focus on GW production in the phase of turbulent free decay, inserting as our initial conditions a fully developed turbulent spectrum, as if the turbulence generation was instantaneous.\nWhile this is not the most realistic assumption, it\nsimplified\nour analysis,\nallowing us to build a thorough understanding of the temporal and spatial structure of both the velocity field and the subsequent GW signal.\nWe thereby derived a model for the UETC of the velocity field in freely decaying turbulence.\nThe GW source, built in the context of this model, was then integrated to compute the GW spectrum.\nThe validity of our model and of the integration technique was supported with a series of direct numerical simulations\nusing the Minkowski space relativistic hydrodynamics code SCOTTS. The simulations were prepared with equivalent initial conditions to our model, namely a fully developed turbulent spectrum.\n\n\nThe velocity of the turbulent fluid was assumed to follow the von K\\'arm\\'an\\ power spectrum, interpolating between a causal $k^5$ slope at low wavenumbers and a Kolmogorov $k^{-2\/3}$ one at high wavenumbers.\nThis basic choice influences the spectral shape of the anisotropic stress power spectrum and consequently of the GW power spectrum.\nConcerning the velocity UETC, motivated by the Kraichnan sweeping model,\nwe assumed that the velocity field  decorrelation is Gaussian in time, with a timescale set by the Eulerian eddy turnover time.\nWe implemented the decorrelation at larger scales, around the spectral peak, by extending the decorrelation velocity according to \\Refa{kaneda_lagrangian_1993}.\nWe further addressed the modelling of decorrelation in the context of the theory of positive kernels, and propose to model the UETC of the turbulent velocity field as a Gibbs kernel.\nThis both consistently enforces time symmetry in the UETC, and guarantees that the GW power spectrum is non-negative.\nWe have validated our model for the velocity field decorrelation\nwith a series of relativistic hydrodynamic simulations performed with SCOTTS, with the initial root-mean-squared velocity, $\\overline{v}_*$, ranging from $0.03$ to $0.3$.\nWe used our simulations also to study the time evolution of both the kinetic energy and\nthe integral scale, allowing us to fix the number of eddy turnover times that it takes for the turbulence to decay.\n\nWe have reviewed the analytical computation of the SGWB from a purely vortical fluid, arriving at an equation for the GW power spectrum in terms of an integral over the Green's function and the anisotropic stress UETC, which is provided by the turbulence model we have developed.\nThe computation of the GW spectrum could then be performed without any further approximation using Monte Carlo integration with importance sampling.\nVarying the exponents of the turbulence decay laws had limited impact on the resulting GW spectrum.\nWe therefore conclude that our predictions are robust with respect to this unknown.\nFor instantaneously generated turbulence, we found excellent agreement between  the GW signal evaluated with direct numerical integration and the one extracted from our direct numerical simulations.\n\nTo conclude our exploration of the GW signal in instantaneously generated turbulence, we studied the case of a turbulent source which is constant in time.\nAs the GW spectrum at high wavenumbers and around the peak is built up on short timescales in comparison to the decay of the source, the constant source approximation works well in this region.\nThe GW power spectrum derived within the constant source approximation is in very good agreement with both\nthe direct numerical simulation result and the power spectrum arising from direct numerical integration of the turbulence model we have built.\nWe provide an analytical form for the SGWB signal from turbulence, \\cref{eq:constant_approx}, which has the advantage that it can be readily used without the need to perform simulations or the challenging computation of 4-dimensional integrals.\n\nFinally, we explored how the GW spectrum was modified by inserting a growth phase for the turbulence, therefore removing the time discontinuity inherent in the instantaneous generation assumption.\nWe considered in particular a $\\mathcal{C}^0$ growth phase and a $\\mathcal{C}^{1}$ growth phase.\nThese models, although simplistic, allowed us to show how the GW signal might\nbe modified by the turbulence generation phase.\nIn contrast to the decay law exponents, the form of the initial growth phase has a large effect on the GW power spectrum: flows with equivalent $\\overline{v}_*$ can lead to GW signals with different spectral shapes and amplitudes, if the  initial condition differs.\nThis underlines the importance of understanding the turbulence formation mechanism in the context of first order phase transitions.\n\nFuture works will therefore focus on the generation of turbulence --- which we showed is crucial to make precise predictions for future GW experiments --- and on the inclusion of the longitudinal velocity field and a magnetic field.\n\n\\section*{Acknowledgements}\n\nThe work has been performed under the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the EC Research Innovation Action under the H2020 Programme; in particular, PA gratefully acknowledges the support of Mark Hindmarsh and the Department of Physics of the University of Helsinki and the computer resources and technical support provided by CSC.\nThis collaboration has been made possible thanks to the generosity of CNRS through the International Emerging Actions program 2020.\nPA and CC have completed most  of this project while affiliated at the APC laboratory, Paris.\nDJW (ORCID ID 0000-0001-6986-0517) was supported by Academy of\nFinland grant nos. 324882 and 328958; DC (ORCID ID 0000-0002-7395-7802) was supported by Academy of\nFinland grant nos. 328958 and 345070; KR (ORCID ID 0000-0003-2266-4716) was supported by Academy of Finland grants 319066, 320123 and 308791.\nThe work of PA (ORCID ID 0000-0002-4814-1406) was partially supported by the Wallonia-Brussels Federation Grant ARC \\textnumero 19\/24 - 103.\nWe acknowledge PRACE for awarding us access to HAWK at GCS@HLRS, Germany.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAn accurate, fast and robust estimation of 6D object pose is of great importance to many application fields such as robotic manipulation, augmented reality, scene interpretation, and autonomous driving. In robotics, for example, the 6D object pose facilitates spatial reasoning and allows an end-effector to act upon the object. In an augmented reality scenario, object pose can be used to enhance one's perception of reality by augmenting objects with extra information such as hints for assembly.\n\nThe introduction of consumer and industrial grade RGB-D sensors have allowed for substantial improvement in 6D object pose estimation as these sensors concurrently capture both the appearance and geometry of the scene. However, there still remain challenges to be addressed, including robustness against occlusion and clutter, scalability to multiple objects, and fast and reliable object modelling, including capturing of reflectance properties. Extending contemporary methods to work reliably and with sufficient execution speed in an industrial setting is still an open problem. Many recent methods focus on specific rigid objects, but pose estimation of deformable or articulated objects and of object categories is also an important research direction.\n\n\\begin{figure}[!t]\n\\begin{center}\n    \\includegraphics[width=1.0\\columnwidth]{teaser.jpg}\n    \\caption{\\label{fig:6d} Estimation of 6D pose, i.e. 3D translation and 3D rotation, of a specific rigid object. This task is considered in the BOP benchmark~\\cite{hodan2018bop}.}\n\\end{center}\n\\end{figure}\n\nThe field of 6D object pose estimation has gained more attention last years. A big achievement for the field is the best paper award of ECCV 2018 given to Martin Sundermeyer, Zoltan Marton, Maximilian Durner, Manuel Brucker, and Rudolph Triebel for their work titled \\emph{Implicit 3D Orientation Learning for 6D Object Detection from RGB Images}.\n\nThe 4th edition of the International Workshop on Recovering 6D Object Pose~\\cite{r6d2018}\\footnote{The previous editions were held at ICCV 2015~\\cite{r6d2015}, ECCV 2016~\\cite{r6d2016}, and ICCV 2017~\\cite{r6d2017}.} was organized in conjunction with ECCV 2018 and was attended by 100+ people from both academia and industry who shared up-to-date advances and discussed open problems. Four invited speakers talked about their current work (Section~\\ref{sec:invited_talks}), the BOP benchmark for 6D object pose estimation was introduced (Section~\\ref{sec:bop}), and the accepted workshop papers were presented.\n\nThe workshop covered the following topics: (a)~6D object pose estimation (a.k.a. 3D object detection) and tracking, (b)~3D object modeling and reconstruction, (c)~surface representation and registration, (d)~robustness to occlusion and background clutter, (e)~multiple object instance detection, (f)~pose estimation of non-rigid objects and object categories, (g)~robotic grasping and grasp affordances, and (h)~object manipulation and interaction.\n\nMany methods for 6D pose estimation of specific rigid objects (Fig.~\\ref{fig:6d}) have been published recently, but were usually compared with only a few competitors on a small subset of datasets. It had been therefore unclear which methods perform well and in which scenarios. To capture the \\emph{status quo} of the field, we organized the SIXD Challenge~\\cite{sixd2017} at the 3rd workshop edition held at ICCV 2017~\\cite{r6d2017}. The results submitted to the challenge were published in the BOP benchmark paper~\\cite{hodan2018bop} and presented at the 4th workshop edition.\n\nPapers submitted to the workshop were peer-reviewed. Out of 13 submissions, the program committee accepted 10 papers which were introduced at the workshop through oral and poster presentations. The best paper award was given to \\emph{Image-to-Voxel Model Translation with Conditional Adversarial Networks} by Vladimir Knyaz, Vladimir V. Kniaz, and Fabio Remondino.\n\n\n\\section{Invited talks} \\label{sec:invited_talks}\n\nThe invited talks were given by Federico Tombari from Technical University of Munich, Kostas Bekris from Rutgers University, Bertram Drost from MVTec, and Thibault Groueix from Ecole Nationale des Ponts et Chauss\u00e9es. The talks are summarized below and the slides are available on the workshop website~\\cite{r6d2018}.\n\n\\subsection{From 3D Descriptors to Monocular 6D Pose: What Have We Learned? -- \\textit{Federico Tombari}}\n\nWhile 6D rigid pose estimation has been an important research task for more than twenty years, recently the design of new algorithms is more and more focused on overcoming the limitations provided by real world applications, so to bridge the gap between lab research and products. This translates to the necessity to move on from the simplified scenario of estimating the pose of a single object on a clutter-less planar surface, towards scenarios with high clutter and occlusion~\\cite{tan2017looking}. At the same time, algorithms need to process input data at a very high frame rate, so to reduce the latency of the output and avoid lagging, provide accurate pose estimation (e.g. to reduce jitter during tracking) and cope with resource-limited hardware architectures such as mobile phones or embedded computers. This is particularly motivated by applications that already have a strong market interest such as augmented reality, personal\/industrial robotics and autonomous driving. A disruptive technology that strongly influenced the field of 6D rigid pose estimation in the past 5 years is deep learning. Hence, the talk presented an overview of the current trends and results regarding the use of deep learning for this task, in view of overcoming the aforementioned limitations. In particular, it highlighted two main research directions where deep learning has been leveraged to improve the state of the art: i) the definition of 3D descriptors for 3D data; ii) 6D object pose estimation from RGB (or monocular) data.  \n\nAs for the first aspect, we briefly went over the development of 3D descriptors for unorganized 3D representations, starting from the handcrafted ones \\cite{tombari10SHOT,rusu09FPFH} until the more recent \"learned\" ones. An important aspect regarding the influence of deep learning on this field is that certain 3D representations such as point clouds and 3D meshes, frequently utilized for 6D rigid pose estimation tasks, are not well suited to convolutions due to their intrinsic unorganized nature. Hence, an important step in the direction of learning 3D features was the introduction of methods such as PointNet \\cite{qi2017pointnet} and \\cite{wang18DGCNN}, that, conversely to approaches such as 3D Match \\cite{zeng173DMatch} and \\cite{khoury17CGN}, can directly operate on point clouds or meshes without the need to either voxelize or histogram the data. The recent global and fully-convolutional architecture for point cloud processing and scene understanding proposed in \\cite{rethage18} was also introduced. \n\nAs for the second research direction, the intuition of estimating the 6D pose of an object based on monocular information relies on the fact that humans can often have a rough idea of the pose of the objects in the surrounding 3D space simply from monocular cues, provided that they are familiar with the shape of the objects. Several works have recently explored this direction, which were briefly introduced and compared in terms of characteristics. One distinctive trait that differentiates such approaches is the type of output that the network is trained to infer: from the 8 corners of the projected bounding box \\cite{rad2017bb8,tekin18} to the regression of the 6D pose \\cite{xiang18posecnn,do18deep6dpose} or the classification of the viewpoint and in-plane rotations \\cite{kehl2017ssd}. The method in \\cite{manhardt18} was also introduced, that proposes to learn the 6D pose refinement from pairs of RGB patches using a CNN. Finally, the extension of monocular 6D pose estimation to the autonomous driving domain was also discussed, by referencing recent directions such as \\cite{chen16Mono3D,chen153DOP}.  \n\nIn conclusion, deep learning appears as a powerful tool for 6D rigid pose estimation, although seems still strongly limited by open issues such as generalizability, learning of geometric invariance and computational efficiency (especially in a field where most applications need to deal with resource-limited hardware). Monocular pose estimation can be promisingly carried out via deep learning, although not yet as accurately as with a depth sensor, as also showcased by a qualitative comparison on a real sequence between the two tracking approaches in \\cite{manhardt18} (monocular) and \\cite{tan2017looking} (RGB-D). Finally, deploying monocular 6D pose estimation jointly with monocular semantic SLAM such as CNN-SLAM \\cite{tateno17CNNSLAM}, which also leverages deep learning to obtain dense semantic reconstruction, appears as an interesting direction to explore towards full (i.e. semantic+geometry) scene understanding from monocular data. \n\n\n\\subsection{Towards Robust 6D Pose Estimation: Physics-based Reasoning and Data Efficiency -- \\textit{Kostas Bekris}}\n\nTowards the objective of robust 6 DoF pose estimation at an accuracy\nand speed level that allows robots to manipulate objects in their\nsurroundings, this talk focused on warehouses tasks, such as picking\nfrom bins, packing and sorting. Warehouses can be seen as a stepping\nstone between the success story of robotics in manufacturing and the\nvision of deploying robots in everyday human environments.  Warehouses\ninvolve a large variety of objects, which can appear in general,\nunpredictable configurations, as well as cluttered scenes and tight\nspaces, such as shelving units, which limit the observability of\non-board sensors and introduce occlusions. They allow, however, access\nto known object models, which frequently correspond to standard\ngeometric shapes, due to packaging.\n\nIn these setups, it is critical for a robot to both utilize\nphysics-based reasoning to achieve 3D scene-level understanding as\nwell as minimize the dependence of solutions to excessive human\nlabeling, which negatively impacts scalability. With these priorities\nin mind, the talk highlighted a pipeline for robust 6D pose\nestimation, which includes the following four steps: 1) semantic object segmentation given physically realistic synthetic data, 2) pose hypothesis generation through robust point registration, 3) pose improvement via consideration of physical constraints at the scene level, and 4) lifelong self-learning through active physical interaction with objects.\n\nRecently, deep learning methods, such as those employing Convolutional\nNeural Networks (CNN's), have become popular for object\ndetection \\cite{ren2015faster,redmon2016you} and pose estimation\n\\cite{kehl2017ssd,xiang2017posecnn}, outperforming alternatives in\nmost benchmarks. These results are typically\nobtained by training CNN's using datasets with a large number of labeled images. Such datasets, however, need to be\ncollected in a way that captures the intricacies of the environment\nthe robot is deployed in, such as lighting conditions, occlusion and clutter. This motivates the development of synthetic dataset that can capture the known parameters of the environment and generate data accordingly, while avoid overfitting to\nthe unknown parameters.\n\n\\emph{The first component of the proposed pipeline} is to use a\nphysics engine in the synthetic dataset generation pipeline\n\\cite{mitash2017self}. The physics engine defines environmental\nconstraints on object placement, which naturally capture in the\ntraining set, the distribution of object poses that can realistically\nappear during testing.  Furthermore, a physics engine is a convenient\ntool to parameterize the unknown scene features, such as illumination.\nA randomization over such parameters is very effective in avoiding overfitting to synthetic textures of objects.\n\nGiven semantic object segmentation, the problem of estimating the 6D object\nposes involves geometric reasoning regarding the position\nand orientation of the detected objects. Solutions that became popular\nin the context of the Amazon Picking Challenge (APC)\n\\cite{Correll:2016aa}, use a Convolutional Neural Network (CNN) for object segmentation \\cite{Princeton,hernandez2016team}\nfollowed by a 3D model alignment step using point cloud\nregistration techniques \\cite{mellado2014super,icp}. The quality of\nthe pose estimate, however, can still suffer due to over-reliance on\nthe learned models.\n\n\\emph{The second insight of the talk} is that CNN output can be\nseen as a probability for an object to be visible at each pixel. These\nsegmentation probabilities can then be used during the registration\nprocess to achieve robust and fast pose estimation. This requires\nsampling a base of points on a point cloud segment, such that all\npoints on the base belong to the target object with high\nprobability. The resulting approach, denoted as ``Stochastic Congruent\nSets'' (StoCS) \\cite{Mitash:2018aa}, builds a\nprobabilistic graphical model given the obtained soft segmentation and\ninformation from the pre-processed geometric object models. The\npre-processing corresponds to building a global model descriptor that\nexpresses oriented point pair features \\cite{drost2010model}.  This\ngeometric modeling, not only biases the base samples to lie within the\nobject bound, but is also used to constrain the search for finding the\ncongruent sets, which provides a substantial computational benefit.\n\n\\emph{The third key observation of the talk} is to treat\nindividual-object predictions with some level of uncertainty and\nperform a global, scene-level optimization process that takes object\ninteractions into account \\cite{Mitash:2018ab}. This information\narises from physical properties, such as respecting gravity and\nfriction as well as the requirement that objects do not penetrate one\nanother. In particular, a Monte Carlo Tree Search (MCTS) process\nutilizes local detections to achieve scene-level optimization. It\ngenerates multiple candidate poses per object and then searches over\nthe cartesian product of these individual object pose candidates to\nfind the optimal scene hypothesis. The scene is evaluated according to\na score defined in terms of similarity of the rendered hypothesized\nscenes against input data. The search performs constrained local\noptimization via physics correction and ICP \\cite{icp}.  Through\nthis physical reasoning, the resulting pose estimates for the objects\nare of improved accuracy and by default consistent. \n\nOnce the system has access to an object detector and a pose estimation\nprocess, it can already be deployed for the desired task.\nNevertheless, as the system performs its task, it also gets access to\ndata in the operation domain, which it did not have access to during\ntraining. This data could be very useful in further improving the\nperformance of the system but they are not labelled.\n\n\\emph{The fourth aspect of the talk} is a solution for automatically\nlabeling real images acquired from multiple viewpoints using a robotic\nmanipulator \\cite{mitash2017self}.  A robotic manipulator\nautonomously collects multi-view images of real scenes and labels them\nautomatically using the object detector trained with the above\nphysics-based simulation tool. The key insight is the fact that the\nrobot can often find a good viewing angle that allows the detector to\naccurately label the object and estimate its pose. The object's\npredicted pose is then used to label images of the same scene taken\nfrom more difficult and occluded views. The transformations between\ndifferent views are known because they are obtained by moving the\nrobotic manipulator. Overall, the data can be added to the existing\nsynthetic dataset to re-train the model for better performance as part\nof a lifelong, self-learning procedure.\n\n\n\\subsection{Detecting Geometric Primitives in 3D Data -- \\textit{Bertram Drost}}\n\nEven though methods based on (deep) learning lead to major advances in many areas of computer vision over the last years, the top-performing methods for 6D object detection are still hand-crafted, classic methods. This is evident from the recent results on the BOP benchmark (Section~\\ref{sec:bop}).\n\nMethods that currently perform best on BOP are based on a voting scheme that can be interpreted as a meet-in-the-middle between RANSAC and a generalized Hough transform.\nThe base method~\\cite{drost2010model} uses a local parametrization for the object pose, where an oriented scene point (reference point) is fixed and assumed to be on the target object. The remaining local parameters are then the point on the model surface that corresponds to the reference point and the rotation around the normal vector, which combined represent three degrees of freedom. The optimal local parameters are recovered using a voting scheme. For this, the reference point is paired with neighboring scene points, and similar point pairs on the model are searched using an efficient hashing scheme. Each such match then casts a vote.\nWhile the base method shows good results for 3D shapes with a distintive geometry, it has weaknesses for shapes that are symmetric of strongly self-similar. This is mostly because point pairs on such shapes are no longer very discriminative. The voting thus finds all symmetric poses simultaneously, which is both slower and less robust.\n\nThe talk presented a way of adapting the base method for geometric primitive shapes - spheres, planes, cylinders, and to some extend cones.\nFirst, the local parameter space was reduced by removing duplicate entries due to symmetries. For example, since a sphere is identical no matter the corresponding point on its surface or the rotation around the normal, the local parameter space becomes zero-dimensional, i.e. a single counter. Additionally, the local parameters can be extended by shape parameters such as the radius of a sphere or cylinder, or the opening angle of a cone. Second, instead of using a hashing scheme for matching point pairs between scene and model, an explicit point pair matching can be used thanks to the explicit nature of the geometric primitives.\nThose changes make the method both faster and more robust for such geometric primitives, while also allowing the recovery of shape parameters.\n\n\n\\subsection{Parameteric Estimation of 3D Surfaces and Correspondences -- \\textit{Thibault Groueix}}\n\nPose of a rigid object has 6 degrees of freedom and is well defined.\nA broader class of objects are non-rigid shapes such as articulated robots or humans. Articulated robots have an additional degree of freedom for each joint. Those additional intrinsic parameters also have to be estimated. In this case, it is easy to manually design a parametrization with N degrees of freedom, where each parameter naturally encodes the angle of a joint. Humans are much harder to parameterize and doing it manually is hard. We propose to learn the parameterization~\\cite{groueix20183d}.\n\nTo that end, we want to map each shape $S$ of category $C$, represented as a point cloud, to a parameter space. A relevant prior on deformable shapes is the ability to find a neutral template, already encoding lots of general information on humans.\nTo learn a parameterization, we use autoencoder neural networks. The encoder is a PointNet~\\cite{qi2017pointnet} and the decoder is a Shape Deformation Network~\\cite{groueix20183d} that deforms the template by iteratively transforming each point on the template to a point in 3D. Given correspondence annotation between a training example and the template, the L2 distance between the generated and the input point clouds is penalized. Annotation of correspondences is expensive and can be avoided through an unsupervised reconstruction loss, the chamfer distance, and a proper regularization. The parameterization is a mapping from the parameter space to the point cloud space, and is given by the learned decoder.\n\nAt test time, the parameters associated with this parametrization can be estimated using the encoder. We propose to use this estimate as an initialization for a local exploration of the parameter space through gradient descent. The learned parametrization is evaluated on the task of 3D correspondences on the FAUST dataset~\\cite{bogo2014faust} and achieves state-of-the-art-results. It also displays strong robustness to holes and noise in the data.\n\nThe method is not limited to humans, but applies to any category of shape for which a template can be found.  For broader category without natural templates such as chairs, or general furniture, the template can be replaced by a set a square patches~\\cite{groueix2018atlasnet}. The assumption is that any object can be reconstruct through the deformation of a set of patches. The proposed parametrization, Atlasnet, is learned on Shapenet and applied for the task of Single View Reconstruction, leading to state-of-the-art results. This confirms that learning a parametrization can be the key to extending 6D pose estimation beyond rigid objects.\n\n\n\\section{BOP: Benchmark for 6D Object Pose Estimation} \\label{sec:bop}\n\nThe BOP benchmark considers the task of 6D pose estimation of a rigid object from a single RGB-D input image, when the training data consists of a texture-mapped 3D object model or images of the object in known 6D poses. The benchmark comprises of: (i) eight datasets~\\cite{hinterstoisser2012accv,brachmann2014learning,tejani2014latent,doumanoglou2016recovering,hodan2017tless,rennie2016dataset} in a unified format that cover different practical scenarios, including two new datasets focusing on varying lighting conditions, (ii) an evaluation methodology with a pose-error function that deals with pose ambiguities, (iii) an evaluation of 15 diverse recent methods that captures the state of the art, and (iv) an online evaluation system at \\texttt{\\href{http:\/\/bop.felk.cvut.cz}{bop.felk.cvut.cz}} that is open for continuous submission of new results.\n\nThe evaluation shows that methods based on point-pair features~\\cite{vidal2018sixd,drost2010model}, introduced in 2010, currently perform best. They outperform template matching methods~\\cite{hodan2015detection}, learning-based methods~\\cite{brachmann2014learning,brachmann2016uncertainty,kehl2016deep,tejani2014latent} and methods based on 3D local features~\\cite{buch2016local,buch2017rotational}.\nOcclusion is a big challenge for current methods, as shown by scores dropping swiftly already at low levels of occlusion. As another important future research directions, our analysis identified robustness against object symmetries and similarities, varying lighting conditions, and noisy depth images.\n\n\\subsection{Future Plans}\n\nVarious possible extensions of the BOP benchmark regarding datasets, problem\nstatement and evaluation metrics were discussed at the workshop and are summarized in the following paragraphs.\n\nThe benchmark was started with the simple task of 6D localization of a single instance of a single object. This task allowed to evaluate most of the recent methods out of the box and is a common denominator of the other 6D localization variants -- a single instance of multiple objects, multiple instances of a single object, and multiple instances of multiple objects. The plan is to move step by step and add the more complicated variants to the online evaluation system in the near future, as well as the 6D detection task, where no prior information about the object presence is provided~\\cite{hodan2016evaluation}. Note that while it is easy to extend the evaluation to other tasks, it is non-trivial to extend the methods.\n\nThe plan is also to keep adding new datasets, e.g.~ \\cite{drost2017introducing}, and for every new dataset to reserve a set of test images for which the ground-truth annotation will be private. The datasets that are currently included in BOP are fully public.\n\nAnother topic of discussion were the pose-error functions. In the BOP benchmark, 6D pose estimates are evaluated using the Visible Surface Discrepancy (VSD) which measures misalignment of the visible part of the object surface.\nOne limitation of the current version of VSD is that it does not consider color. The same holds for the other pose-error functions commonly used in the literature~\\cite{hodan2016evaluation}. While the alignment of color texture is less relevant for a number of robotic applications, it may be important for augmented-reality applications. The plan is to fix this limitation in a new version of VSD with an extended pixel-wise test checking not only the distance of the object surface but also its color. \nWe are open to other feedback and discussion about the evaluation methodology.\n\n\\section{Conclusions}\n\nThis document summarized the 4th International Workshop on Recovering 6D Object Pose which was organized at ECCV 2018 and featured four invited talks, oral and poster presentations of accepted workshop papers, and an introduction of the BOP benchmark for 6D object pose estimation.\n\nAn accurate, fast and robust method for 6D object pose estimation is still in need, as is evident from the current scores on the BOP benchmark~\\cite{hodan2018bop}.\nWe would like to encourage authors of relevant methods to keep submitting results to the online evaluation system at \\texttt{\\href{http:\/\/bop.felk.cvut.cz}{bop.felk.cvut.cz}}. We will present snapshots of the leaderboards at the next workshop editions, which are planned for the upcoming major conferences.\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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Phys.} {{\\bf #1} {(#2)} {#3}}}\n\\newcommand{\\ibid}[3]{{\\sl ibid.} {{\\bf #1} {#2} {#3}}}\n\\newcommand{\\mbox{\\sl et al. }}{\\mbox{\\sl et al. }}\n\\newcommand{\\PE}{\\ \\\\ {\\centering {\\large Per Elmfors} \\\\ \\ \\\\\n   { NORDITA \\\\\n   Blegdamsvej 17 \\\\\n   DK-2100 K\\o benhavn \\O \\\\\n   Denmark \\\\\n   E-mail: elmfors@nordita.dk \\\\ }}}\n\n\n\\newcommand{{\\rm T}}{{\\rm T}}\n\\newcommand{\\cL^0_{eff}}{{\\cal L}^0_{eff}}\n\\newcommand{\\cL_{eff}^{\\beta,\\mu}}{{\\cal L}_{eff}^{\\beta,\\mu}}\n\\newcommand{\\cL_{0}^{\\beta,\\mu}}{{\\cal L}_{0}^{\\beta,\\mu}}\n\\newcommand{\\cL_{1}^{\\beta,\\mu}}{{\\cal L}_{1}^{\\beta,\\mu}}\n\\normalsize\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\newcommand{\\AUTHORS}{\\  {\\centering\n{\\large Per Elmfors}\\footnote{Email address: elmfors@nordita.dk.} \\\\\n{\\sl NORDITA \\\\\n   Blegdamsvej 17 \\\\\n DK-2100 Copenhagen \\O, Denmark \\\\ }\n{\\large\nDavid Persson\\footnote{Email address: tfedp@fy.chalmers.se.} and\nBo-Sture Skagerstam}\\footnote{Email address\naddress:tfebss@fy.chalmers.se. Research supported by the Swedish\nNational Research Council under contract no. 8244-103} \\\\\\\n\\vspace*{1mm}\n   {\\sl Institute of Theoretical Physics \\\\\n   Chalmers University of Technology and\\\\\nUniversity of G\\\"oteborg\\\\\n   S-412 96 G\\\"oteborg, Sweden\\\\ }}}\n\n\\begin{document}\n\\large\n\\thispagestyle{empty}\n\\begin{flushright} NORDITA--93\/35    P \\\\\n                   G\\3teborg ITP 92--22 \\\\\n                   April 1993  \\end{flushright}\n\\begin{center}\n\\normalsize\n{\\LARGE\\bf QED Effective Action at Finite Temperature\n      and Density\\\\}\n\\end{center}\n\\vspace*{0.5cm}\n\\AUTHORS\n\\begin{center}\n{\\bf Abstract} \\\\\n\\end{center}\n{\\normalsize\nThe QED effective action at finite temperature and density is\ncalculated to all orders in an external homogeneous and\ntime-independent magnetic\nfield in the weak coupling limit. The free energy, obtained\nexplicitly, exhibit the expected de\\ Haas -- van\\ Alphen oscillations.\nAn effective coupling at finite temperature and density is derived in\na closed form and is compared with renormalization group results.\n}\n\\newpage\n\\normalsize \\setcounter{page}{1}\n\\begin{center} \\section{Introduction}\n\\setcounter{equation}{0}\n\\label{intro}\n\\end{center}\nLarge magnetic fields are relevant in a number of physical\nsystems like supernovas\\cite{Ginzburg91}, where\n$B={\\cal O} (10^{10}){\\rm T}$, neutron stars \\cite{ShapiroT83}, where\n$B={\\cal O} (10^{8}){\\rm T}$, or white\nmagnetic dwarfs \\cite{Angel78} in which case\n$B={\\cal O}\n(10^{4}){\\rm T}$. (As a reference the electron mass in units of\ntesla is $m^2={\\cal O}(10^9){\\rm T}$.) The radiative corrections to the magnetic\nmoment of a Dirac fermion\nhas been estimated in the presence of such large magnetic fields and it was\nargued that they are extremely small \\cite{Skagerstam91,Studenikin90}.\nIt has recently been shown that a plasma at thermal equilibrium can sustain\nlarge fluctuations of the electromagnetic fields. For instance, in\nthe primordial Big-Bang\nplasma, the\namplitude of magnetic field (zero frequency) fluctuations at the time of\nthe\nprimordial nucleosynthesis can be as large as\n$B = {\\cal O}(10^{10}) {\\rm T}$\n\\cite{TajimaCSK92}. Other systems with large magnetic fields present are\nmergers of massive black holes \\cite{NatayanPP92},  where\n$B={\\cal O} (10^{13}){\\rm T}$ or superconducting strings \\cite{Witten85},  where\n$B={\\cal O} (10^{14}){\\rm T}$ or even larger. At the electroweak\nphase transition in\nthe\nvery early universe it has, furthermore, been argued that very large magnetic\nfields,\n$B={\\cal O} (10^{19}){\\rm T}$, can be generated due to\ngradients in the Higgs field\n\\cite{Vachaspati91}.\n\nIn many of these systems one has to consider the effects of thermal\nenvironments.\nCalculation of the QED effective potential, i.e. the free energy, has been\nattempted  before\neither at finite temperature \\cite{Dittrich79,Rojas92} or at\nfinite chemical potential \\cite{ChodosEO90}.  In\nthe latter case the effective action is unfortunately not complete but the\ncorrect form is\npresented here. At finite chemical potential and for sufficiently small\ntemperatures,\nthe QED effective action should exhibit a certain periodic dependence of\nthe the\nexternal field, i.e. the well-known de\\ Haas -- van\\ Alphen\noscillations in condensed\nmatter\nphysics. This was not obtained in Ref.\\cite{ChodosEO90}.\n\nLet ${\\cal L}_{eff}$ denote the QED effective action for a constant B-field\nat finite temperature $T=1\/\\beta$ and chemical potential $\\mu$. We\ncalculate\nthis effective action\nto all orders in $eB$ but with no virtual photons present, i.e. we\nconsider the weak coupling limit.\nA more detailed analysis will be\npresented elsewhere \\cite{ElmforsPS93}.\n\n\\begin{center}\\section{Derivation of the Effective Action ${\\cal L}_{eff}$}\n\\setcounter{equation}{0}\n\\label{derivLB}\n\\end{center}\nThe basic relation we need in order to derive the\none-loop correction to the effective Lagrangian\nis the identity\n\\begin{equation}\n\\frac{\\partial{\\cal L}_{eff}}{\\partial m}= i {\\rm Tr\\,} S_F(x;x)~~~,\n\\label{BI}\n\\end{equation}\nwhere $S_F(x;x')$ is the fermion propagator in the external magnetic\nfield\nand the trace is over spinor indices. It can be constructed from\nthe solutions of the Dirac equation\n$(i\\partial\\!\\!\\!\/-eA\\!\\!\\!\/-m)\\psi(x)=0$ in such a way that\n\\begin{equation}\n\\begin{array}{c}\nS_F(x;x')=\\langle 0|{\\bf T } [\\psi(x)\\overline{\\psi}(x')]|0\\rangle\\ .\n\\label{SF}\n\\end{array}\n\\end{equation}\nEquation (\\ref{BI}) determines the vacuum part,\n${\\cal L}_{1}={\\cal L}_{eff}(T=\\mu =0,B)$, of the effective action\nwhich should be added to the tree-level, ${\\cal L}_{0}=-B^2\/2$. At finite\ntemperature and\ndensity we simply replace the time-ordered vacuum expectation values\nin \\Eqref{SF} by a\nthermal average.\nIt can be shown that this replacement corresponds to the conventional\ncalculational rules of\nthermo field dynamics. The solutions of the Dirac equation with an external\nconstant\nmagnetic field parallel to the $z-$axis are the standard relativistic\nLandau levels with energy spectrum given by\n\\begin{equation}\nE_n(k_z) = \\sqrt{m^2+k^2_z+2eBn}\\ ,\n\\end{equation}\nwhere $n=0,1,2,...$ and $k_z$ is the momentum parallel to the magnetic\nfield. The construction of the propagator is similar to the zero\ntemperature case \\cite{KobayashiS83} except that the propagating\nparticles can now be exchanged with the heatbath. We find\n\\begin{equation}\n\\label{SF11}\n{\\rm Tr\\,} S_F(x;x)=  \\sum_{n=0}^\\infty \\int \\frac{d\\omega dk_y dk_z}{(2\\pi)^3}\n[\\Delta+(\\Delta^*-\\Delta)f_F(\\omega)]\\ 2m(I_n+I_{n-1})\\ ,\n\\end{equation}\nwhere we have introduced the scalar propagator\n\\begin{equation}\n\\Delta = \\inv{\\omega^2-k_z^2-m^2-2eBn+i\\epsilon}\\ .\n\\end{equation}\nThe thermal distribution $f_F(\\omega)$ is given by\n\\begin{equation}\nf_F(\\omega) = \\frac{\\theta(\\omega)}{e^{\\beta(\\omega-\\mu)}+1}\n+\\frac{\\theta(-\\omega)}{e^{\\beta(\\mu-\\omega)}+1}\\ ,\n\\end{equation}\nand we use the notation\n\\begin{equation}\nI_n = \\left( \\frac{eB}{\\pi} \\right)^{1\/2} \\exp \\left[\n  -  eB \\left( x \\!-\\! \\frac{k_{y}}{eB} \\right)^{2} \\right]\n \\frac{1}{n!} H_{n}^2 \\left[ \\sqrt{2eB} \\left( x \\!-\\! \\frac{k_{y}}\n {eB} \\right) \\right]\\ ,\n\\end{equation}\nwhere the functions $H_n$ are Hermite polynomials, and we define\n$I_{-1}=0$. From the propagator in \\Eqref{SF11} we get both\nthe vacuum correction ${\\cal L}_{1}$ and a thermal correction\n$\\cL_{eff}^{\\beta,\\mu}$. It is well-known that a real-time formalism at\nfinite temperature requires a doubling of the degrees of\nfreedom and it can be shown that \\Eqref{SF11} is the 11-component\nof the matrix propagator in thermo field dynamics \\cite{UmezawaMT82}.\nHere we only need the 11-component for the one-loop calculation.\nThe vacuum part  of \\Eqref{SF11} that survives when\n$f_F(\\omega)\\rightarrow 0$ reproduces the old result by Schwinger\n\\cite{Schwinger51}\n\\begin{equation}\n{\\cal L}_{1} = -\\frac{1}{8\\pi^{2}} \\int_{0}^{\\infty}\n \\frac{ds}{s^3}exp(-m^{2}s) \\left(esB\\coth(esB) -1 -\n\\frac{1}{3}(esB)^2\\right)~~.\n\\end{equation}\nHere ${\\cal L}_{1}$ has been renormalized  by adding\na second order polynomial in $eB$. We stress that the physics behind\nthis renormalization is related to the fact that the coefficient in\nfront of the quadratic term is proportional to the square of inverse\n(bare) coupling. This renormalization corresponds to a charge\nrenormalization as well as a wave function renormalization in such a way\nthat $eB$ is invariant. This charge renormalization also leads to the weak\ncoupling\nexpansion of the QED $\\beta$-function, i.e.\n\\begin{equation}\n\\lambda\\frac{d}{d\\lambda}\\alpha(\\lambda) = \\beta (\\alpha(\\lambda)) =\n\\frac{2}{3\\pi}\\alpha^{2}(\\lambda) + {\\cal O}(\\alpha^{3}(\\lambda))~~~,\n\\label{BETAF}\n\\end{equation}\nwhere $\\lambda$ is a momentum scale factor.\nIn order to calculate the thermal part $\\cL_{eff}^{\\beta,\\mu}$ of the effective action, we\nhave to be\ncareful with the convergence and the analytical structure. We therefore\nlet the sum over the quantum number $n$ only go to a finite $N$ and take\nthe limit $N\\rightarrow\\infty$ at the end. This gives\n\\begin{equation}\n\\label{SFbmu}\n{\\rm Tr\\,} S_F^{\\beta,\\mu}= \\lim_{N\\rightarrow\\infty} i\\,\\frac{mB}{\\pi^{3\/2}}\n\\,{\\rm Im}\\int_{-\\infty}^{\\infty}\\frac{d\\omega}{2\\pi}\n\\int_0^\\infty\\frac{ds}{s^{1\/2}}e^{i\\frac{3\\pi}{4}}\ne^{-is(\\omega^2-m^2-i\\epsilon)} \\left[\\frac{1+e^{i2sB}}{1-e^{i2sB}}-\n\\frac{2 e^{i2NsB}}{1-e^{i2sB}}\\right]\\ .\n\\end{equation}\nThe poles in the last factor cancel for finite\n$N$, and we cannot let $N\\rightarrow\\infty$ in a naive\nway before deforming the $s$ integration contour to the\nimaginary axis. After integrating \\Eqref{SFbmu} with\nrespect to $m$, to get $\\cL_{eff}^{\\beta,\\mu}$, and being careful\nwith the convergence when deforming the  contours of integration we\narrive at $\\cL_{eff}^{\\beta,\\mu} =\\cL_{0}^{\\beta,\\mu} +\\cL_{1}^{\\beta,\\mu}$, where\n\\begin{equation}\n\\cL_{0}^{\\beta,\\mu} =\n\\inv{3\\pi^2}\\int_{-\\infty}^\\infty d\\omega\n\\theta(\\omega^2-m^2)f_F(\\omega)(\\omega^2-m^2)^{3\/2} \\ ,\n\\end{equation}\nis the ideal gas contribution in absence of the external field $B$,\nand\n\\begin{eqnarray}\n\\label{Lbmueff}\n\\cL_{1}^{\\beta,\\mu} &=&\n\\int_{-\\infty}^\\infty d\\omega\n\\theta(\\omega^2-m^2)f_F(\\omega)\n\\Biggl[\\inv{4\\pi^{5\/2}}\\int_0^\\infty\\frac{ds}{s^{5\/2}}e^{-s(\\omega^2-m^2)}\n(seB\\coth (seB) -1)\\nonumber\\\\\n &&-\n\\inv{2\\pi^3}\\sum_{n=1}^\\infty \\left(\\frac{eB}{n}\\right)^{3\/2}\n\\sin\\left(\\frac{\\pi}{4}-\\frac{\\pi n}{eB}(\\omega^2-m^2)\\right) \\Biggr]\\ .\n\\end{eqnarray}\nThis is the main result of our paper. The term with the\nsum over $n$ was neglected in Ref.\\cite{ChodosEO90} and we\nshow in Section~\\ref{physical}  that it is essential to keep\nthis term in order to get the\ncorrect physical result.\n\nThe finite temperature part of the effective action is directly related to the\nfree energy of a gas of relativistic fermions in a constant $B$-field.\nIf $Z(B,T,\\mu)$ is the corresponding partition function, without the\ncontribution from the thermal photon gas, we can also write\n\\begin{eqnarray}\n\\label{F}\n\\cL_{eff}^{\\beta,\\mu}=\\frac{\\log Z(B,T,\\mu)}{\\beta V}\n&=&\\frac{eB}{\\beta(2\\pi)^2}\\sum_{\\lambda=1}^2\\sum_{n=0}^\\infty\n\\int_{-\\infty}^\\infty dk\\left\\{\\log(1+e^{-\\beta(E_{\\lambda,n}-\\mu)})\\right.\n\\nonumber \\\\\n&+&\n\\left. \\log(1+e^{-\\beta(E_{\\lambda,n}+\\mu)})\\right\\}\\, ,\n\\end{eqnarray}\nwhere $E_{\\lambda,n}=\\sqrt{m^2+k^2+2eB(n+\\lambda-1)}$,\nand $\\lambda$ labels the spin of the fermions.\nFor $\\Abs{\\mu} \\leq m$,  \\Eqref{F} can be rewritten in the\nphysically less transparent way\n\\begin{equation}\n\\label{Ditt}\n\\cL_{eff}^{\\beta,\\mu}=\\frac{1}{(2\\pi)^2}\n\\sum_{l=1}^\\infty (-1)^l\\int_0^\\infty\n\\frac{ds}{s^3}\\exp(-\\frac{\\beta^2l^2}{4s}-m^2s)\neBs\\coth(seB)\\frac{\\cosh(\\beta l\\mu)}{2}\\ ,\n\\end{equation}\nwhich for $\\mu = 0 $ also is an equation given in \\cite{Dittrich79}.\n However, it\nis not obvious, when written in this form, to see how\n to extract the physical\ncontents, and how to generalize $\\cL_{eff}^{\\beta,\\mu}$\nto $\\Abs{\\mu}\\geq m$, since then it appears to be divergent.\nIn particular we notice that the high $T$ behaviour given in\n\\cite{Dittrich79} is not correct. After a Poisson resummation in $l$,\nrewriting the sum over $l$ as a contour integral and and carefully\ndeforming the contours it is, however, possible to show that\n\\Eqref{Ditt} is equal to \\Eqref{Lbmueff} which, of course, is valid for all\n$T$ and $\\mu$.\n\\begin{center}\n\\section{The Physical Content of ${\\cal L}_{eff}$}\n\\setcounter{equation}{0}\n\\label{physical}\n\\end{center}\nThere are several dimensionful parameters related to ${\\cal L}_{eff}$, i.e.\n$T,\\ \\mu,\\ m$, and $B$, that can be large or small\ncompared to each other. We shall only focus on a few\nof these limits which we think are particularly\ninteresting.\n\nThe second term in \\Eqref{Lbmueff} has an oscillatory\nbehaviour that we can explore in the limit where\n$\\{T=0,eB\\ll \\mu^2-m^2\\ll m^2\\}$. This is a non-relativistic\nlimit (in the sense that the kinetic energy is much smaller\nthan $m$) with a degenerate Fermi sea and a weak external field.\nThe oscillating part ${\\cal L}_{osc}$ of $\\cL_{1}^{\\beta,\\mu}$ can be integrated in this\napproximation for which we obtain\n\\begin{equation}\n\\label{Losc}\n{\\cal L}_{osc}=-\\frac{(eB)^{5\/2}}{4\\pi^4 m}\\sum_{n=1}^\\infty\n\\inv{n^{5\/2}}\\cos\\left(\\frac{\\pi}{4}-n\\pi\\frac{\\mu^2-m^2}{eB}\\right)\\ .\n\\end{equation}\nThe oscillation frequency of this periodic function\nagrees with the one derived by Onsager \\cite{Onsager52}\nfor the de\\ Haas -- van\\ Alphen effect. Equation (\\ref{Lbmueff}) describes\nthe full relativistic generalization of this effect. The distance\nbetween the magnetic field of two\nadjacent minima of the magnetization is determined\nby\n\\begin{equation}\n\\Abs{\\inv{eB_i}-\\inv{eB_{i+1}}}= \\frac{2\\pi}{A}\\ ,\n\\end{equation}\nwhere $A$ is the area of an extremal cross section of the\nFermi sea.\n\nIn the limit of strong field, $\\{eB\\gg T^2,m^2,\\mu^2-m^2\\}$,\nwe can see from \\Eqref{F} that only the lowest Landau level\ncontribute and $\\cL_{eff}^{\\beta,\\mu}$ goes like a linear function of $eB$.\nWe shall now reproduce this result from \\Eqref{Lbmueff}\nand it turns out to be rather non--trivial. The leading $B$\ndependence in the first term in \\Eqref{Lbmueff} is obtained by\nscaling out $eB$ and taking $eB\\rightarrow \\infty $ in the remainder.\nThe total contribution is, apart from the thermal integration,\n\n\\begin{equation}\n\\frac{(eB)^{3\/2}}{4\\pi^{5\/2}}\\left[\n\\int_0^\\infty \\frac{dx}{x^{5\/2}}(x\\coth x -1)-\\sqrt{\\frac{2}{\\pi}}\n\\sum_{n=1}^\\infty\\inv{n^{3\/2}}\\right]\\ ,\n\\end{equation}\nbut this is actually identically zero. The next subleading\nterm can be shown to be\n\\begin{equation}\n\\cL_{1}^{\\beta,\\mu} = \\frac{eB}{2\\pi^2}\\int_{-\\infty}^\\infty\nd\\omega\\theta(\\omega^2-m^2)f_F(\\omega) \\sqrt{\\omega^2-m^2}~~~,\n\\end{equation}\nwhich is exactly the leading term from \\Eqref{F}. This\ncalculation shows that the oscillatory term in \\Eqref{Lbmueff}\nis absolutely necessary to cancel the $B^{3\/2}$ term and to give\nthe correct linear term.\n\nHaving shown that the thermal corrections in \\Eqref{Lbmueff}\nare correct and comprehensible in physical terms, we now\naddress the question of when they are important, i.e. when\nthey dominate over the vacuum correction. For\n$eB\\gg m^2,T^2,\\mu^2$, the vacuum correction goes like\n\\begin{equation}\n\\label{LARGEB}\n{\\cal L}_{1} \\approx \\frac{(eB)^2}{24\\pi^2}\\log\\left(\\frac{eB}{m^2}\\right)\\ ,\n\\end{equation}\nand it dominates over $\\cL_{1}^{\\beta,\\mu}$. However, when\n$\\{T=0,eB\\ll \\mu^2-m^2\\ll m^2\\}$, we have\n\\begin{equation}\n\\label{SMALLB}\n{\\cal L}_{1} \\approx \\frac{(eB)^2}{360\\pi^2}\n\\left(\\frac{eB}{m^2}\\right)^2\\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\cL_{1}^{\\beta,\\mu} \\approx \\frac{(eB)^2}{12\\pi^2}\\log \\left(\\frac{\\Abs{\\mu}}{m}\n      +\\sqrt{\\frac{\\mu^2}{m^2}-1}\\, \\right)\n      \\approx \\frac{(eB)^2}{12\\pi^2}\n\\left(\\frac{3\\pi^2 n}{m^3}\\right)^{1\/3}\\ ,\n\\end{equation}\nwhere $en$ is the charge density, and where we have neglected ${\\cal L}_{osc}$.\nThe density correction $\\cL_{1}^{\\beta,\\mu}$\ntherefore dominates over ${\\cal L}_{1}$ when\n\\begin{equation}\n\\left(\\frac{ n}{m^3}\\right)^{1\/3} \\gg\n\\inv{30(3\\pi^2)^{1\/3}}\\left(\\frac{eB}{m^2}\\right)^2\\ .\n\\end{equation}\nWhen $T^2\\gg m^2\\gg eB$, we have that\n\\begin{equation}\n\\cL_{1}^{\\beta,\\mu} \\approx \\frac{(eB)^2}{24\\pi^2}\\log\\left( \\frac{T^2}{m^2}\\right)\\ ,\n\\end{equation}\nand we do not agree with the high temperature and weak field\nlimit in \\cite{Dittrich79}.\n(We notice the similarity of our result with $\\cL^0_{eff}$ for\n$eB\\gg m^2$.) In this case the thermal contribution  $\\cL_{1}^{\\beta,\\mu}$\ndominates over ${\\cal L}_{1}$ as  given by Eq.(\\ref{SMALLB}) when\n\\begin{equation}\n\\frac{T}{m}\\gg \\exp\\left[\\inv{30}\\left(\\frac{eB}{m^2}\\right)^2\\right]\n\\approx 1\\ .\n\\end{equation}\n\nAnother useful way of extracting the physical information from\n${\\cal L}_{eff}$ is to define an effective coupling constant as\n\\cite{Schwinger51,ChodosOS88}\n\\begin{equation}\n\\label{effalp}\n\\frac{1}{\\alpha(T,\\mu,B)}=\\inv{\\alpha}\n-\\frac{1}{\\alpha B}\\frac{\\partial {\\cal L}_{eff}}{\\partial B}\\ ,\n\\end{equation}\nin analogy with the definition of the renormalized\ncoupling in the vacuum sector\nin connection with \\Eqref{BETAF}. Special care has\nto be taken when evaluating the\nderivative\nof the oscillating term in Eq.(\\ref{Lbmueff}).\nIn the limit when $eB=0$, we obtain the effective coupling\n$\\alpha(T,\\mu) = \\alpha(T,\\mu,B=0)$ given by\n\\begin{equation}\n\\label{AlphaTmu}\n\\frac{1}{\\alpha(T,\\mu)}= \\frac{1}{\\alpha } - \\frac{2}{3\\pi}\\int\n_{-\\infty}^{\\infty}d\\omega\n\\frac{\\theta(\\omega^{2}-m^{2})}{\n\\sqrt{\\omega^{2}-m^{2}}}f_{F}(\\omega)~~~.\n\\end{equation}\nWhen $T=0$, we therefore get an effective coupling\n$\\alpha(\\mu) = \\alpha(T=0,\\mu)$ such that\n\\begin{equation}\n\\label{Alphamu}\n\\frac{1}{\\alpha(\\mu)}= \\frac{1}{\\alpha }\n-\\frac{2}{3\\pi}\\log \\left( \\frac{|\\mu|}{m} + \\sqrt{\\frac{\\mu ^{2}}{m^{2}} -\n1}~\\right)\\ .\n\\end{equation}\nIn the limit $\\mu=0$, we find the following asymptotic behaviour\nof the corresponding effective coupling $\\alpha(T) = \\alpha(T,\\mu = 0)$:\n\\begin{equation}\n\\label{AlphaT}\n\\frac{1}{\\alpha(T)} = \\frac{1}{\\alpha } -\\frac{4}{3\\pi}\n\\int\n_{\\beta m}^{\\infty}\n\\frac{dx}{\n\\sqrt{x^{2}-(\\beta m)^{2}}}\\frac{1}{e^{x} + 1}\n\\approx \\frac{1}{\\alpha }\n-\\frac{2}{3\\pi}\\log \\left(\\frac{T}{m}\\right)~~~,\n\\end{equation}\nfor $T \\gg m $.\nIt is now clear that (only) for $\\mu \\gg m$ and $T \\gg m$ the\neffective couplings\n$\\alpha (\\mu)$ and $\\alpha (T)$ are solutions to the renormalization group\nequation (\\ref{BETAF}) when $\\lambda $ is identified with $\\mu$ and $T$\nrespectively\n(see in this context e.g. Refs.\\cite{Morley79,Rojas92}).\nWe also note that Eq.(\\ref{LARGEB}) leads to an effective\ncoupling $\\alpha(B) = \\alpha(T=0,\\mu = 0,B)$\nwith an asymptotic behaviour\n\\begin{equation}\n\\frac{1}{\\alpha(B)} \\approx \\frac{1}{\\alpha }\n-\\frac{1}{3\\pi}\\log \\left(\\frac{eB}{m^{2}}\\right)~~~.\n\\end{equation}\nThe effective coupling defined in \\Eqref{effalp}  can also be extracted\nfrom the residue of the thermal Debye-screened photon propagator\n(see Ref.\\cite{Morley79}).\n\nWe have only considered a few particular limits\nin this paper and there are many more to explore in\ndifferent physical situations. All information needed to\ndo that is contained\nin \\Eqref{Lbmueff}.\n\n\n\\begin{center}\\section{Conclusions}\n\\label{concl}\n\\setcounter{equation}{0}\n\\end{center}\nWe have established the correct form of the one-loop\nQED effective action at finite temperature and density\nto all orders in a constant external magnetic field, and the\nresult differs from earlier attempts. From the form of\n$\\cL_{eff}^{\\beta,\\mu}$ presented in \\Eqref{Lbmueff} we have checked several\nlimits that can be understood from a physical point of view.\nA great advantage  with our expression for $\\cL_{eff}^{\\beta,\\mu}$ is that\nthe thermal distribution function $f_F(\\omega)$ occurs explicitly.\nThis means that it is easy to study other thermal situations\nby simply replacing $f_F(\\omega)$ with some other distribution.\n\nThe importance of the thermal correction depends on the value\nof $B$, $T$ and $\\mu$. In many physically interesting cases\nthey are all large compared to $m$ and often of the same\norder of magnitude, which makes it difficult to obtain\nanalytical approximations. It is, however, straightforward\nto use \\Eqref{Lbmueff} for numerical calculations.\n\nEven though the correction to the free energy is small\ncompared to the value without the external field there\nare other quantities that are effected by the presence of\nthe heatbath. For instance, the magnetization of a degenerate\nFermi sea as was briefly discussed\nin Section~\\ref{physical}. One could also expect that\nQED radiative corrections at\nfinite\ntemperature and density and with the strong magnetic fields\ndiscussed in the Introduction could\neffect the\nelectroweak transition rates, relevant for the Big-Bang primordial\nnucleosynthesis. We will\nreturn to this issue elsewhere.\n\nWe have, furthermore, calculated an effective coupling constant\ndefined from the part of $\\cL_{eff}^{\\beta,\\mu}$ which is quadratic in $eB$.\nIt satisfies asymptotically a naive zero temperature\nrenormalization group\nequation where the renormalization scale is replaced by\n$T$, $\\mu$ or $\\sqrt{eB}$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nFeynman diagrams and the associated integrals are at the computational core of quantum field theory and their evaluation attracts considerable attention see e..g \\cite{SmirnovBooks}. Two important and well-known methods for their computation are Integration By Parts (IPB) \\cite{ChetyrkinTkachov1981} and Differential Equations (DE) \\cite{DE}. The more recent method called Symmetries of Feynman Integrals (SFI) \\cite{SFI} is closely related to them. It considers a Feynman integral $I$ associated with a diagram of fixed topology yet most general parameters, namely masses and kinematical invariants denoted collectively by $X$. Given a diagram SFI defines a set of partial differential equations in $X$, with the idea that the integral can be determined as a solution of this equation set (at least partially), rather than through direct integration.  \n\nThe method was applied to the two-loop vacuum diagram (or ``diameter diagram\") \\cite{SFI, locus}, the 1-loop propagator diagram (``bubble\") \\cite{bubble} and most recently to the vacuum seagull diagram, a certain 3 loop vacuum diagram, where novel evaluations were obtained for certain 3 mass scales. \n\nThe SFI method has aspects of geometrical symmetry as the equation set defines a continuous group $G$ which acts on parameter space $X$ and had a rather direct geometrical interpretation. More specifically, $G$ consists of all possible linear redefinitions of the loop currents and external momenta which preserve the propagator hyperspace in the space of quadratics in currents, see \\cite{locus,VacSeagull} for definitions.\n\nGenerically, on orbits of $G$ in $X$ SFI reduces the computation of the Feynman integral to a line integral over simpler diagrams (up to some boundary or initial condition such as a base point).  However, at some locus in $X$ the equations set is singular and produces algebraic rather than differential equations \\cite{locus}. On some components of this locus the algebraic equation allows to reduce $I$ into a linear combination of simpler diagrams (instead of a line integral).\n\nThe purpose of this note is to present algebraic methods for determining the singular locus and when relevant, the reduction to simpler diagrams. We start in section \\ref{sec:prelim} by recalling from linear algebra the role of maximal minors and their decomposition as well as the LU decomposition. In section \\ref{sec:reduction} we proceed to apply a generalization of these methods to the SFI equation set and to obtain our results. In section \\ref{sec:demon} the algebraic methods are demonstrated on the diameter diagram. We conclude with a discussion in section \\ref{sec:disc}.\n\n\n\\section{Linear algebra preliminaries}\n\\label{sec:prelim}\n\nIn this section we review maximal minors and the LU decomposition in linear algebra. This will serve as a basis for the next section.\n\n\\subsection*{Warm up}\n\nAs a first warm up consider an $n+1$ by $n$ matrix of rank $n$ (the maximal possible rank) \\begin{eqnarray}\nT^a_{~i}, && ~~ a=1 \\dots n+1,\\, i=1 \\dots n \\nonumber \\\\\n\\mbox{rk}(T) &=& n ~.\n\\end{eqnarray}\n The $n+1$ rows are linearly dependent, yet any $n$ of them are independent. Therefore there exists a single vector $l_a$ which is in the left hand side null subspace (kernel) of $T$, namely \\begin{equation}\n l_a\\,  T^a_{~i} = 0\n \\label{l_null1}\n \\end{equation}\nMoreover, the left null vector $l_a$ can be expressed in terms of $T$ as follows \\begin{equation}\nl_a =  \\eps_{a a_1 \\dots a_n}\\, \\eps^{i_1 \\dots i_n}\\,  T^{a_1}_{~i_1} \\dots T^{a_n}_{~i_n}\n\\label{def:null1}\n\\end{equation}\n This expression tells us that component $a$ of $l$ is gotten by erasing row  $a$ in $T$, computing the determinant (or minor) of the resulting square matrix, and finally multiplying by an appropriate sign. \n \nTo see why definition (\\ref{def:null1})  satisfies (\\ref{l_null1}) note that since each component of $l_a$ is given by an $n$ by $n$ determinant then for any column $i$ of $T$ $l_a\\,  T^a_{~i}$ represents an expansion of an $n+1$ by $n+1$ determinant gotten by joining with $T$ its column $i$. Since this matrix has a repeated column its determinant vanishes thereby proving the point. \n\nTo motivate  (\\ref{def:null1}) consider normalizing one of the component of $l_a$ to unity, say $l_{n+1}=1$. Now the remaining components are determined and can be expressed through Cramer's rule as a ratio of minors. Multiplying this vector by the common denominator (the minor gotten by erasing row $n+1$ of $T$) we arrive at (\\ref{def:null1}).\n\nSummarizing this first warm up example, it shows how maximal minors can be used to represent the null subspace of a matrix. \n\n\\vspace{.5cm} \\noindent {\\bf Example 2}. As a second warm up example consider an $n+1$ by $n+1$ matrix $T^a_{~i}$ of (sub-maximal) rank $n$, namely  \\begin{eqnarray}\nT^a_{~i}, && ~~ a=1 \\dots n+1,\\, i=1 \\dots n+1 \\nonumber \\\\\n\\mbox{rk}(T) &=& n ~.\n\\end{eqnarray}\nGiven its rank there should be a single left null vector $l_a$ and a single right null vector $r^i$ ($l_a,\\, r^i$ are unique up to rescaling), namely \\begin{equation}\n  l_a\\,  T^a_{~i} =0 \\qquad  T^a_{~i}\\, r^i =0 \n \\end{equation}\n\nIn order to find expressions for $l_a$ and $r^i$ one proceeds as follows. One defines the adjugate of $T$,  $M$, such that each of its components $M^i_{~a}$ is the minor gotten by erasing row $a$ and column $i$ of $T$ followed by multiplication by the sign $(-1)^{i+a}$. Equivalently \\begin{equation}\nM^i_{~a} =  \\eps^{i i_1 \\dots i_n}\\,  \\eps_{a a_1 \\dots a_n}\\, T^{a_1}_{~i_1} \\dots T^{a_n}_{~i_n} ~~.\n\\label{def:null2}\n\\end{equation}\n\nFor $T$ of any rank we have \\begin{eqnarray}\nT^a_{~i} \\, M^i_{~b} &=& \\det(T) \\delta^a_{~b} \\label{TM} \\\\\nM^i_{~a} \\, T^a_{~j} \\,  &=& \\det(T) \\delta^i_{~j} \\label{MT}\n\\end{eqnarray}\nWhen $\\det(T) \\neq 0$ the adjugate matrix leads to the inverse matrix through $T^{-1}=M \/ \\det(T)$. However, we assumed that $\\mbox{rk}(T)=n$ and hence $\\det(T)=0$. (\\ref{TM}) implies that each column $b$ of $M$ is in the right null subspace. Since all right null vectors are proportional to $r^i$ we must have $M_i^{~a} = r^i\\, \\tilde{l}_a$ where each component of $\\tilde{l}_a$ represents a possible rescaling factor and together they form the components of some row vector $\\tilde{l}$. Similarly, from (\\ref{MT}) one deduces that $M^i_{~a}$ must be of the form $M^i_{~a} = \\tilde{r}^i\\, l_a$ for some column vector $\\tilde{r}^i$. Together (\\ref{TM}, \\ref{MT}) imply that \\begin{equation}\nM^i_{~a} =c\\, r^i\\, l_a\n\\label{Factorization1}\n\\end{equation}\nwhere $c$ is some non-zero constant.  This tells us that the adjugate matrix, which is a matrix composed out of maximal minors, necessarily factorizes into a right null vector times a left null vector, and this is the purpose of this example. \n\n\n\\subsection{Factorization of maximal minors}\n\nAfter the warm up we proceed to the general case of a matrix of general size, $m$ by $n$, and a general rank $r$, namely \n\\begin{eqnarray}\nT^a_{~i}, && ~~ a=1 \\dots m,\\, i=1 \\dots n \\nonumber \\\\\n\\mbox{rk}(T) &=& r ~.\n\\label{def:T}\n\\end{eqnarray}\n \nMotivated by the previous examples we define \\begin{equation}\nM^I_{~A} :=  \\eps^{I i_1 \\dots i_r}\\,   \\eps_{A a_1 \\dots a_r}\\, T^{a_1}_{~i_1} \\dots T^{a_r}_{~i_r} ~~ \\\\\n\\label{def:M}\n\\end{equation}\nwhere $I=(i_{r+1} \\dots i_n)$ and $A=(a_{r+1} \\dots a_m)$ are multi-indices. The components of $M$ are the maximal minors of $T$. The following theorem describes the special properties of $M^I_{~A}$. It is the main point of the current section and will be useful in the next.\n\n\\vspace{.5cm} \\noindent {\\bf Maximal minor factorization theorem}. Given any matrix $T^a_{~i}$ as in (\\ref{def:T}) its associated maximal minors $M^I_{~A}$ defined in (\\ref{def:M}) satisfy \\begin{enumerate}\n\n\\item[(i)]  $M$ is null, namely  \\begin{eqnarray}\nT^a_{~i} \\, M^{i I'}_{~B} &=& 0 \\label{M_r_null} \\\\\nM^I_{~a A'} \\, T^a_{~j} \\,  &=& 0 \\label{M_l_null}\n\\end{eqnarray}\nwhere $I'$ is a multi index with $n-r-1$ indices and $A'$ has $m-r-1$ indices. Since $M^I_{~A}$ is anti-symmetric in both $I$ and $A$ the contracted index can be interchanged with any other.\n\n\\item[(ii)] $M$ factorizes as \\begin{equation}\nM^I_{~A} =c\\, r^I\\, l_A\n\\label{Factorization}\n\\end{equation}\nwhere $r^I$ and $l_A$ are antisymmetric and unique up to an overall multiplicative scalar.\n\n\\end{enumerate}\n\nThis theorem generalizes example 1 which demonstrated the minors to be null, and example 2 which demonstrated factorization. For completeness we include a proof.\n\n\\vspace{.5cm} \\noindent {\\bf Proof}. $M$ is seen to be null by an argument analogous to that in example 1, namely the product $M \\cdot T$ represents the expansion of the wedge product of $r+1$ columns of $T$, which must vanish given that $\\mbox{rk}(T)=r$ (and similarly for $T \\cdot M$).\n\nAn alternative proof for the null property would be to note that for minors that are not necessarily maximal multiplication by $T$ gives larger minors through the determinant expansion formula as follows \\begin{eqnarray}\nT^a_{~i} \\, M^{i I'}_{~b B'} &=& \\delta^a_{[b}\\,  M^{I'}_{~B']} \\nonumber \\\\\nM^{i I'}_{~a A'} \\, T^a_{~j} \\,  &=& \\delta^{[i}_{~j}\\, M^{I']}_{~A'} ~.\n\\end{eqnarray}\nNow, once we assume $M$ to be maximal the larger minors all vanish and we recover  (\\ref{M_r_null},\\ref{M_l_null}).\n\nIn order to see why $M$ factorizes, recall that $T$ defines a right null subspace of dimension $m-r$. It can be described as the span of a set of independent vectors $r_1^{i_1}, \\dots, r_{m-r}^{i_{m-r}}$. However, this description is considerably redundant as it can be replaced by any other spanning set. An alternative way to identify a subspace is through the wedge product \\begin{equation}\nr^I \\equiv r^{i_1 \\dots i_{m-r}} := \\bigwedge_{j=1}^{m-r} r_j^{i_j}   ~.\n\\end{equation}\n The wedge product defines an antisymmetric tensor (or multi-vector) $r^{i_1 \\dots i_{m-r}}$ which can be called the null right tensor. It satisfies \\begin{equation}\nT^a_{~i}\\, r^{i_1 \\dots i_{m-r-1} i} = 0\n\\end{equation}\nand it is the only such tensor which satisfies this equation up to an overall multiplicative factor. In this sense it characterizes the subspace in a unique and hence non-redundant way. \n\nThe uniqueness of of the right null tensor together with right null property of $M^I_{~B}$ for any value of the multi-index $B$, as described in (\\ref{M_r_null}), implies that $M^I_{~B} = r^I\\, \\tilde{l}_B$ for some tensor $\\tilde{l}_B$. Repeating the argument for the left null space, in analogy with example 2, implies the argued factorization, namely (\\ref{Factorization}), where $l_A$ is the left null tensor of $T$, and $c$ is a free constant whose value depends on the chosen normalizations for $r^I$ and $l_A$.\n\nThis completes the proof of the maximal minor factorization theorem. We now proceed to a discussion of several related comments.\n\n\\vspace{.5cm} \\noindent {\\bf Matrix of polynomials}. Below we shall be interested in a  matrix  $T=T(x)$ whose entries are polynomials in some variables denoted collectively by $x$. In this case each component of $M^I_{~A}$ is a polynomial in $x$ and one can be more specific about the choice of the scalar $c$:  it can be chosen to be the greatest common divisor of all the polynomials  $M^I_{~A}(x)$, namely \\begin{equation}\nc(x) = \\gcd \\( M^I_{~A} (x) \\) ~.\n\\label{def:cx}\n\\end{equation}\n $c(x)$  is unique up to multiplication by a number (a field element).\n\n\\vspace{.5cm} \\noindent {\\bf Gauss elimination}. The computation of each component of the maximal minor tensor $M^I_{~A}$ requires to evaluate a determinant. It is well known that evaluating determinants through their definition is computationally costly and a more efficient method is provided by the Gauss elimination method where through elementary operations on rows (or columns) the matrix $T^a_{~i}$ can be brought into an upper triangular form. This implies that $L^{-1}\\, T = U$ where $U$ is the upper triangular form, and $L^{-1}$ is a lower triangular matrix which records all the row operation carried on $T$. Since the row operations are invertible one also has \\begin{equation}\nT= L\\, U ~,\n\\label{LU_decomp}\n\\end{equation}\n which is known as an ``LU decomposition of $T$''. \n\nThe LU decomposition above is useful since $T$ and $U$ share the same right null subspace, yet the triangular form of $U$ makes it immediate to determine it.\n\nSimilarly Gauss elimination can be applied to columns to obtain (a possibly different) LU decomposition. Column operations produce a matrix $L$ of the same size as $T$ together with a square $U$, while row operations produce the opposite sizes, and hence if T is non-square the two decompositions are necessarily different.\n\nAltogether, minors and null subspaces can be computed either directly from the definitions (\\ref{def:M},\\ref{Factorization}) or by using an LU decomposition (which is essentially Gauss elimination). The choice of method depends on computational convenience. When applying an LU decomposition to a matrix of polynomials the $L,U$ factor generically would become rational (a ratio of polynomials0, yet in such a way that the minors remain polynomial.\n\n\\vspace{.5cm} \\noindent {\\bf Dualization}. The definition of the tensor of minors $M^I_{~A}$ in (\\ref{def:M}) can be thought to involve two steps -- first a wedge product of $T$ with several copies of itself, followed by a dualization on both the $a$ and $i$ indices. Both steps are performed by the $\\eps$ tensors  -- first assuring projection onto the antisymmetric sector and then performing dualization. Here we note that the wedge product is the more essential step, while dualization is convenient in the common case when the $r \\equiv \\mbox{rk} (T)$ is close to either $m$ or $n$.\n\n\\section{Reduction of a Feynman Integral}\n\\label{sec:reduction}\n\nAfter reviewing the factorization of maximal minors, we proceed to apply it to the Symmetries of Feynman Integrals (SFI) method \\cite{SFI}. \n\n\\vspace{.5cm} \\noindent {\\bf Set-up}.  SFI considers a Feynman diagram as a function of its most general possible parameters, namely the masses and the kinematical invariants, and a general spacetime dimension $d$. The parameter space is denoted by $X$ and each diagram is associated with set of partial differential equations in $X$. \n\nSchematically, the SFI equations are of the form \\begin{equation}\nc^a\\, I + (T^a)^j_{~i}\\, x_j\\, {\\partial}} \\def\\dag{\\dagger^i\\, I = J^a\n\\label{SFI_schematic}\n\\end{equation}\nwhere $a=1, \\dots , \\dim(G)$ labels each equation in the set, $c^a=c^a(d)$ are constants (namely, are independent of $X$),  the matrices $(T^a)^j_{~i}$ define a representation of a group $G$ on $X$, the range of $i,j$ is given by $i,j = 1, \\dots, \\dim(X)$, ${\\partial}} \\def\\dag{\\dagger^i={\\partial}} \\def\\dag{\\dagger\/{\\partial}} \\def\\dag{\\dagger x_i$ and finally  $J^a$ are terms composed of simpler diagrams. The group $G$ is called the SFI group and it is defined by the topology of the diagram in a natural way.\n\nHere we focus on the differential term $(T^a)^j_{~i}\\, x_j\\, {\\partial}} \\def\\dag{\\dagger_i\\, I$ which is fully defined by the representation of $G$ on $X$. We define the matrix $Tx$  by \\begin{equation}\n \\( Tx \\) ^a_{~i} := (T^a)^j_{~i}\\, x_j\n \\label{def:Tx}\n \\end{equation}\nThis matrix will be the object of our study and it will correspond to the general matrix $T^a_{~i}$ in the previous section. The size of the matrix $Tx$ is given by $m=\\dim(G)$ by $n=\\dim(X)$.\n\n\\vspace{.5cm} \\noindent {\\bf Factorization}. At any specific point in $x \\in X$ we may determine the rank of $Tx$ which equals the dimension of the tangent space to the G-orbit at $x$ and hence to the dimension of this G-orbit\n\\begin{equation}\n r(x) := {\\rm rk}(Tx) \\equiv \\dim \\( {\\rm G-orbit} (x) \\) ~.\n\\end{equation}\nThe we evaluate the maximal minors \\begin{equation}\nM^I_{~A}(x) := \\eps_{A a_1 \\dots a_r}\\, \\eps^{I i_1 \\dots i_r}\\,   Tx^{a_1}_{~i_1} \\dots Tx^{a_r}_{~i_r} ~~ \\\\\n\\label{def:Mx}\n\\end{equation}\n(gotten by substituting $T \\to Tx$ in the general definition  (\\ref{def:M}) ).\n\nFactorization of maximal minors (\\ref{Factorization}) in the presence of polynomials (\\ref{def:cx}) implies \\begin{equation}\nM^I_{~A}(x) =S(x)\\, Orb^I(x)\\, Stb_A(x)\n\\label{MxFactor}\n\\end{equation}\nwhere all the terms are defined up to a multiplicative $x$-independent constant. The notation reflects the interpretation of the various terms in the context of $Tx$  as we proceed to explain.\n\n$S(x)$ is the factor common to all minors, denoted by $c(x)$ in  (\\ref{Factorization},\\ref{def:cx}). At zeroes of $S(x)$ all the minors vanish reflecting a drop in the rank of $Tx$. We refer to this zero locus as the singularity locus, and accordingly the notation $S(x)$ was chosen to stand for singular.\n\n$Orb^I(x)$ is the right null tensor, denoted by $r^I$ in  (\\ref{Factorization}). This null subspace is composed of cotangent vectors at $x$ which annihilate all the rows in $Tx$ and hence annihilate (are perpendicular to) the orbit of $G$ through $x$.  The notation $Orb$ refers to this relation with the orbit. By its definition, the orbit tensor includes the differentials of all group invariants, namely $Orb \\wedge dInv=0$ where $Inv$ is any invariant of $G$.\n\n$Stb_A(x)$ is the left null tensor, denoted by $l^A$ in  (\\ref{Factorization}). Vectors in this subspace represent a combination of equations (rows of $Tx$) whose action  vanishes at $x$ and is hence known as the stabilizer of the group. Correspondingly, the notation $Stb$ stands for stabilizer.  Note that a multiplication of the SFI equation set from the left by a stabilizer vector generates by definition an equation with no derivatives, namely an algebraic rather than differential equation. For some $x$ the inner product of a stabilizing vector $Stb_a$ with $c^a$, the vector of constants in (\\ref{SFI_schematic}), is non-zero thereby generating a simple equation for $I$ which yields a reduction of $I$ to a linear combination of simpler integrals \\cite{locus}.\n \nThe preceding interpretation of the terms in (\\ref {MxFactor}) is summarized by the following list \\\\\n\\begin{tabular}{lcl}\n$c(x)$, the common factor for minors  & $\\to$ &  $S(x)$ whose zeroes are the singularity locus \\\\\nright null space & \t$\\to$ & co-orbit form subspace\t\\\\\nleft null space & \t$\\to$ & Stabilizer subspace. \t\n\\end{tabular} \n\nEquation (\\ref{MxFactor}) is the main result of this paper, providing a systematic way to compute the singularity locus, the orbit and its invariants and\/or the group stabilizer through minors of $Tx$.  Some stabilizers provide a reduction of the diagram under study into a linear combination of simpler ones (as discussed above).  Computationally, an LU decomposition (\\ref{LU_decomp}) might be performed to facilitate the evaluation of minors (see also a demonstration in the next section).\n\nSince all terms in (\\ref{MxFactor}) are polynomials in the parameters $x$ we obtain a useful relation between the degrees with respect to $x$ \\begin{equation}\n r = {\\rm deg}_x\\, S + {\\rm deg}_x\\, Orb + {\\rm deg}_x\\, Stb\n\\label{deg_balance}\n \\end{equation}\nwhere $r= {\\rm deg}_x\\, M^I_{~A}(x) ={\\rm rk}(Tx) \\equiv  \\dim \\( {\\rm G-orbit} (x) \\)$. \n\n\\vspace{.5cm} \\noindent {\\bf From maximal rank to lower ones}.\nLet us proceed to examine in more detail various loci in $X$ which correspond to a given rank of $Tx$, namely G-orbits of various dimensions.\n\n\\vspace{.5cm} \\noindent {Maximal rank}. Denote the highest possible rank by $Rk$ namely \\begin{equation}\nRk = \\max_x {\\rm rk} (Tx) ~.\n\\label{def:Rk}\n\\end{equation} \n$Rk$ is the generic rank, namely it is achieved in an open set in $X$.\n\nClearly $Rk$ is bounded by the size of $Tx$ namely $Rk \\le \\min \\{\\dim (G),\\dim(X)\\}$. If the inequality is exhausted such that $Rk = \\dim (G) \\le \\dim(X)$ then the stabilizer is trivial, namely $Stb =1$ in (\\ref{MxFactor}), and $Orb$ is non-trivial telling us about $G$ invariants. Similarly if $Rk=\\dim (X) \\le \\dim(G)$ then $Orb$ is trivial and hence the G-orbit is co-dimension zero, implying maximal effectiveness for SFI, while $Stb$ is non-trivial providing us with algebraic equations (probably an algebraic constraint on the sources).\n\n\\vspace{.5cm} \\noindent {Next to maximal rank}. Factorization at maximal rank defines the singularity function $S(x)$.  One proceeds to factorize $S(x) = S_1(x)\\, S_2(x) \\dots$. Each factor of $S(x)$ defines a component of the singular locus over which the rank is reduced, generically to $Rk-1$.  On any component $c=1,2,\\dots$ where $S_c(x)=0$ factorization of maximal minors (\\ref{MxFactor}) may be re-applied as follows \\begin{equation}\n  M_{c~~\\tilde{A}}^{~\\tilde{I}}(x) = \\hat{S}_c(x)\\, Orb_c^{~\\tilde{I}}(x)\\, Stb_{c~\\tilde{A}}(x)  \\mod S_c(x)\n\\label{Factor_at_sing}\n \\end{equation}\n where $\\tilde{I},\\tilde{A}$ are larger multi-indices. This time the minors are smaller, and accordingly the stabilizer and orbit subspaces strictly contain the respective maximal rank subspaces. In addition the factorization holds only at $S_c(x)=0$ or equivalently $\\mod S_c(x)$.\n\n\\vspace{.5cm} \\noindent {Second next to maximal rank}. It could be interesting to proceed to even lower rank at the locus of intersection of several singularity components and to determine the reduction of the Feynman diagram over there.\n\n\\vspace{.5cm} \\noindent {\\bf Original motivation and chronology}. The idea for this paper appeared while studying the 2-loop propagator diagram (sometimes called lizard eye or marshmallow) \\cite{InProgress}, yet notions closely related to maximal minors had appeared earlier in SFI papers and in a prominent way:  the wedge product in \\cite{locus}  and the determinant in \\cite{VacSeagull}.\n\n\\section{Demonstration}\n\\label{sec:demon}\n\nIn this section we demonstrate the algebraic method of the previous section on the 2-loop vacuum diagram shown in fig. \\ref{fig:diameter} which we call the diameter diagram. This provides a simple and non-trivial demonstration.\n\n\\begin{figure}\n\\centering \\noindent\n\\includegraphics[width=4cm]{diameter_fig}\n\\caption[]{The diameter diagram.}\n\\label{fig:diameter}\n\\end{figure}\n\nThe parameter space consists of the three possible masses-squared $X=\\(x_1,\\, x_2,\\, x_3 \\)$ where $x_i:=(m_i)^2, ~i=1,2,3$. The SFI equation set was found in \\cite{SFI}  eq. (6.9) from which we can read \\begin{equation}\nTx = \\[ \\begin{array}{ccc}\n -x_1\t\t& -x_2\t\t& -x_3 \t\\\\\n x_2-x_3\t& x_2\t\t& -x_3 \t\\\\\n -x_1 \t& x_3-x_1\t\t& x_3\t\\\\\n x_1 \t\t& -x_2\t\t& x_1- x_2 \\\\\n \\end{array} \\] \n \\label{def:Tx_diam}\n\\end{equation}\nThere are 4 equations and 3 parameters so $Tx$ is $4$ by $3$.\n\n\\vspace{.5cm} \\noindent {\\bf Maximal rank}. At a generic point the rank of $Tx$ is 3 and so the G-orbit is 3d. Computing the 3-minors according to (\\ref{def:Mx}) we find \\begin{equation}\nM_a = \\lambda} \\def\\hxi{{\\hat \\xi}  \\[  \\begin{array}{cccc} 0 & x_1 & x_2 & x_3 \\end{array} \\]\n\\label{diam_factor}\n\\end{equation}\nwhere \\begin{equation}\n \\lambda} \\def\\hxi{{\\hat \\xi} := x_1^2 + x_2^2 + x_3^2 - 2 x_1 x_2 - 2 x_1 x_3 - 2 x_2 x_3\n\\end{equation}\nis the Heron formula \/ K\\\"all\\'en invariant. \n\nComparing the expression for the minors with the general factorization (\\ref{MxFactor}) we identify \\begin{eqnarray}\n S &=& \\lambda} \\def\\hxi{{\\hat \\xi} \\label{s_diam} \\\\\n Stb &=& \\[  \\begin{array}{cccc} 0 & x_1 & x_2 & x_3 \\end{array} \\]\n \\label{stb_diam}\n \\end{eqnarray}\nso the singularities are the zeroes of $\\lambda} \\def\\hxi{{\\hat \\xi}$, thereby reproducing \\cite{SFI} eq. (6.16) and \\cite{locus} eq. (4.9), while the expression for the stabilizer reproduces \\cite{locus} eq. (4.12). Multiplying the equation set by this stabilizer produces a relation between sources which is valid everywhere in $X$. The degree balance (\\ref{deg_balance}) which corresponds to (\\ref{diam_factor})  is \\begin{equation}\n3 = 2 +1 ~.\n\\end{equation}\n\n\\vspace{.5cm} \\noindent {\\bf LU decomposition}. Alternatively, we may compute the minors and the associated factors through the LU decomposition. To obtain the stabilizer group (null left subspace) we may operate on the right, that is on columns. As a starting point it is convenient to reorder the rows of $Tx$ (\\ref{def:Tx_diam}) as \\begin{equation}\n\\widetilde{Tx}  = \\[ \\begin{array}{ccc}\n  x_1 \t& -x_2\t\t& x_1- x_2 \\\\\n   -x_1 \t& x_3-x_1\t\t& x_3\t\\\\\n x_2-x_3\t& x_2\t\t& -x_3 \t\\\\\n  -x_1\t\t& -x_2\t\t& -x_3 \t\\\\\n \\end{array} \\] \n \\label{Tx_diam_reorder}\n\\end{equation}\nAfter operating on columns one gets a lower triangular form \\begin{equation}\n \\[ \\begin{array}{ccc}\n  x_1 \t& 0\t\t& 0  \\\\\n   -x_1 \t& 1\t\t& 0\t\\\\\n x_2-x_3\t& -x_2\/x_1\t\t& 0\t\\\\\n  -x_1\t\t& x_2\/x_3\t\t& -\\lambda} \\def\\hxi{{\\hat \\xi} \t\\\\\n \\end{array} \\] \n\\end{equation}\nConsidering the third row we notice that all 3-minors are proportional to $\\lambda} \\def\\hxi{{\\hat \\xi}$ consistent with the singularity factor (\\ref{s_diam}). Moreover, from this form a left null vector can be read $\\[ \\begin{array}{cccc}  x_3\/x_1 &  x_2\/x_1 & 1  & 0 \\end{array} \\]$. After multiplying by $x_1$ and reordering to account for the ordering of $\\widetilde{Tx}$ we reproduce the stabilizer (\\ref{stb_diam}).\n\nFor completeness we present also the triangulation of  $\\widetilde{Tx}$ through operations on rows. In this case we reach the upper triangular form \\begin{equation}\n U = \\[ \\begin{array}{ccc}\n  x_1 \t& -x_2\t& x_1-x_2  \\\\\n   0\t \t& -2 s^3\t& 2 s^2\t\\\\\n  0\t\t& 0\t\t& 0\t\\\\\n  0\t\t& 0\t\t& -\\frac{2}{s^3}\\, \\lambda} \\def\\hxi{{\\hat \\xi} \t\\\\ \n \\end{array} \\]  ~.\n\\end{equation}\nwhere we have introduced the notation \\begin{equation}\n s^i := -\\frac{{\\partial}} \\def\\dag{\\dagger}{4\\, {\\partial}} \\def\\dag{\\dagger x_i}\\, \\lambda} \\def\\hxi{{\\hat \\xi} \n \\end{equation} \nOnly one minor is non-zero -- the one gotten by erasing row 3, and it is indeed proportional to $\\lambda} \\def\\hxi{{\\hat \\xi}$ (and the denominator cancels). To find the stabilizer we must record the row operations performed through the lower triangular matrix \\begin{equation}\n L^{-1} = \\[ \\begin{array}{ccc}\n  x_1 \t& -x_2\t& x_1-x_2  \\\\\n   0\t \t& -2 s^3\t& 2 s^2\t\\\\\n  0\t\t& 0\t\t& 0\t\\\\\n  0\t\t& 0\t\t& -\\frac{2}{s^3}\\, \\lambda} \\def\\hxi{{\\hat \\xi} \t\\\\ \n \\end{array} \\] \n\\end{equation}\nwhich satisfies $L^{-1}\\, \\widetilde{Tx} = U$. Now $\\tilde{s}=\\[ \\begin{array}{cccc}  0 &  0 & 1  & 0 \\end{array} \\]$ is a left null vector for $U$. Multiplication by $L^{-1}$ on its left reproduces the stabilizer (\\ref{stb_diam}), after some rescaling and reordering just as before.\n\n\n\\vspace{.5cm} \\noindent {\\bf Next to maximal rank}. At the singular\/algebraic locus $\\lambda} \\def\\hxi{{\\hat \\xi}=0$ the rank of $Tx$ and hence the dimension of the G-orbit reduces to 2 and one computes the maximal minors tensor \\begin{equation}\nN^i_{~ab} := \\( M_\\lambda} \\def\\hxi{{\\hat \\xi}\\)^i_{~ab}\n\\label{def:N}\n\\end{equation} by omitting column $i$ and rows $a,b$. \n\nAccording to the general procedure (\\ref{Factor_at_sing})  $N^i_{~ab}$ factorizes. We start by noticing that the co-orbit 1-form $Orb_\\lambda} \\def\\hxi{{\\hat \\xi} \\equiv Orb$ can be anticipated. By definition at $\\lambda} \\def\\hxi{{\\hat \\xi}=const$ the 1-form $d\\lambda} \\def\\hxi{{\\hat \\xi} \\equiv -4\\, s^i \\, dx_i$ annihilates all vectors tangent to the locus, and hence we recognize $Orb$ to be\n\\begin{equation}\nOrb^i  = \\[ \\begin{array}{c}\ns^1 \\\\\ns^2 \\\\\ns^3 \\\\\n\\end{array} \\]\n\\label{Orb_lam}\n\\end{equation}\n\nNext, $N^{i}_{~ab}$ should be divided by $Orb$ to yield $Stb_\\lambda} \\def\\hxi{{\\hat \\xi}$. Since this factorization holds for $\\lambda} \\def\\hxi{{\\hat \\xi}=0$, but not for all $x$, some more algebra is required.  Factoring, say, $M^3_{~ab}$ by $s^3 \\mod \\lambda} \\def\\hxi{{\\hat \\xi}$ can be done by eliminating one the coordinates, for instance $x_3$, in terms of $x_1,\\, x_2$ but this introduces square roots and makes the algebra awkward. Instead one can work $\\mod \\lambda} \\def\\hxi{{\\hat \\xi}$ and write $M^3_{~ab}(x)=s^3\\, Stb_{\\lambda} \\def\\hxi{{\\hat \\xi}~ab}(x) + \\lambda} \\def\\hxi{{\\hat \\xi}\\, k_{ab}(x)$ where $k_{ab}$ is some matrix. Now it is easier to impose $s^3=0$ by substituting $x_3 \\to x_1+x_2$. This allows to determine $k_{ab}$ which happens to be a matrix of constants, and now $ Stb_{\\lambda} \\def\\hxi{{\\hat \\xi}}$ can be determined to be\n\\begin{equation}\n Stb_{\\lambda} \\def\\hxi{{\\hat \\xi}~ab} = \\[ \\begin{array}{cccc}\n  0 \t\t& x_1\t& x_2\t& x_3 \t \\\\\n  -x_1\t&0\t \t& s^3\t& -s^2\t\\\\\n  -x_2\t& -s^3\t& 0\t\t& s^1 \t\\\\\n  -x_3\t& s^2\t& -s^1 \t& 0    \t\\\\ \n \\end{array} \\]  ~.\n\\label{Stb_lam}\n\\end{equation}\n\nSince at $\\lambda} \\def\\hxi{{\\hat \\xi}=0$ ${\\rm rk} \\( Tx \\) =2$ the stabilizer subgroup is 2d and hence the tensor $Stb_\\lambda} \\def\\hxi{{\\hat \\xi}$ tensor has rank 2, namely it is a bi-vector. Its first row confirms that it includes $Stb$ (\\ref{stb_diam}), the general stabilizer which is valid for all $x$ and leads to the algebraic constraint for the sources. \n\nAny of the remaining three rows can be used to obtain the algebraic solution, namely the reduction of the diameter to simpler diagrams. Their sum reproduces eq. (4.10) of \\cite{locus}. However, we notice that if we pick one of them, say the 1st, it becomes apparent that $\\mod \\lambda} \\def\\hxi{{\\hat \\xi}$ the expression for the algebraic solution  simplifies to \\begin{equation} \n\\left. I \\right|_\\lambda} \\def\\hxi{{\\hat \\xi} = \\frac{1}{d-3} \\[ s^3\\, j'(x_1)\\, j'(x_2) + cyc. \\]\n\\label{I_lam}\n\\end{equation}\nwhere the notation is the same as in \\cite{locus}. The simplification occurs due to a non-manifest cancellation of the denominator $\\mod \\lambda} \\def\\hxi{{\\hat \\xi}$ in eq. (4.11). \n\nSummarizing the factorization at $\\lambda} \\def\\hxi{{\\hat \\xi}=0$ we have \\begin{equation}\nN^i_{~ab}  = Orb^i\\,  Stb_{\\lambda} \\def\\hxi{{\\hat \\xi}~ab}  \\mod \\lambda} \\def\\hxi{{\\hat \\xi}\n\\label{N_factor}\n\\end{equation}\nwhere $Orb$ is given in (\\ref{Orb_lam}) and $\\left. Stb_{\\lambda} \\def\\hxi{{\\hat \\xi}~ab} \\right|_\\lambda} \\def\\hxi{{\\hat \\xi}$ in (\\ref{Stb_lam}). There is no non-trivial scalar common factor so $ \\hat{S}_\\lambda} \\def\\hxi{{\\hat \\xi}(x) =1$. The degree balance reads \\begin{equation}\n2 = 1 + 1 ~.\n\\end{equation}\n\n\n\\section{Summary and Discussion}\n\\label{sec:disc}\n\nIn this paper we analyzed certain algebraic aspects of the SFI equation set, and showed that factorization of maximal minors is useful to determine  the singular locus, the G-orbit together with the G-invariants and the stabilizer (\\ref{MxFactor}). On some orbits the latter provides a reduction of the diagram under study into a sum of simpler diagrams. Factorization can be performed over any G-orbit, including those whose dimension is lower than the generic value.\n\nThe method was illustrated through the diameter diagram. \n\nWe end with two comment. First, at the algebraic  locus we have an exact solution. It would be interesting to develop a perturbation theory in its vicinity. \n\nSecondly, the procedure described in this paper, depends only on the representation of $G$ on $X$ and it involves minors which are antisymmetric and hence suggest fermionic variables. In these respects it is similar to Group Cohomology, where the ghost and anti-ghosts are fermionic. It would be interesting to find out whether this similarity is more substantial.\n\n\\subsection*{Acknowledgments}\n\nIt is a pleasure to thank Subhajit Mazumdar, Lior Oppenheim, Amit Schiller and Ruth Shir and for collaboration on related projects and for comments on a presentation.\n\nThis research was supported by the ``Quantum Universe'' I-CORE program of the Israeli Planning and Budgeting Committee.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec1}\nThe influence of disorder on phase transitions and critical phenomena has\nattracted a considerable interest in the last decades. In the absence of\nfrustration, it is now well established that a first-order phase transition\nis smoothed by the introduction of randomness and can be made continuous\nat large enough disorder strength~\\cite{ImryWortis}. In 2D, an infinitesimal\ndisorder is sufficient to remove any discontinuity~\\cite{HuiBerker,%\nAizenman,Aizenman2}. When the pure system undergoes already a continuous\nphase transition, the Harris criterion predicts that the universality class\nof the pure model will be affected by disorder\nif the specific heat diverges, i.e. if the critical exponent $\\alpha$\nis positive~\\cite{Harris74}. In this context, the $q$-state Potts model\nhas been a useful toy model, because it displays a rich phase diagram\ninvolving two lines of respectively first and second-order phase transition.\nAlong the latter, the universality class depends on the number of states $q$.\nOn the practical side, efficient Monte Carlo and transfer matrix\nalgorithms exist for this model and Conformal Invariance can be used in\n2D in combination with Renormalisation Group (RG).\n \nIn comparison, correlated disorder was much less studied. Nevertheless\nin some experimental situations, impurities cannot be considered as\nuncorrelated. This is in particular the case when they carry an electric\ncharge or a magnetic moment and are coupled via an electromagnetic interaction.\nOn the theoretical side, Weinrib and Halperin studied the $\\phi^4$ model\nwith a random mass and showed that a new RG fixed point, distinct from the\nrandom and the pure ones, emerges in the phase diagram when the correlations\nof this mass decay algebraically~\\cite{Weinrib}.\nFor a sufficiently slow decay of these disorder correlations,\nthe new fixed point becomes stable. Denoting $a$ the exponent of the\nalgebraic decay of disorder correlations, the perturbation is relevant\nwhen $a<d$ if the correlation length exponent $\\nu$ of the pure model satisfies\nthe inequality\n$\\nu<2\/a$.\nAt the new fixed point, correlated disorder is marginally irrelevant,\nwhich implies that that $\\nu=2\/a$~\\cite{Honkonen}. The magnetic exponent\n$\\eta$ remains small compared to $\\epsilon=4-d$. Even though still\ncontroversial, these predictions were confirmed by Monte Carlo simulations\nof the 3D Ising model with $a=2$~\\cite{IMCorrele,IMCorrele2}.\n\nWe recently studied by large-scale Monte Carlo simulations the influence\nof correlated couplings on the 2D Potts model~\\cite{EPL,PRE}.\nLike in the absence of disorder correlation, the first-order phase transition,\noccurring for the  pure model when $q>4$, was shown to be smoothed and replaced\nby a continuous transition. However, the new universality class was shown\nto be $q$-independent, a feature shared by the strong-disorder fixed point\nof the $q$-state Potts model with a layered McCoy-Wu-like disorder. This\nresult is remarkably different from the continuous increase of the\nmagnetic scaling dimension $x_\\sigma$ observed for the Potts model with\nan uncorrelated disorder. More intriguing is the fact that the phase diagram\ndisplays a Griffiths phase, as in the McCoy-Wu model, where the magnetic\nsusceptibility diverges with the lattice size. Interestingly, such a phase\nhas been predicted by Weinrib and Halperin, but only above the upper\ncritical dimension $d_c=4$. Finally, the hyperscaling relation ${\\gamma\\over\\nu}\n=d-2{\\beta\\over\\nu}$ was observed to be broken in the Griffiths phase,\nas a result of large disorder fluctuations.\n\nHowever, these observations were made for finite systems so one cannot exclude\nthe possibility that, at much larger lattice sizes, the Griffiths phase\ncollapse into a single point, the critical point, where the hyperscaling\nrelation would be restored. Moreover, the estimate of the correlation length\nexponent $\\nu$ is incompatible with Weinrib-Halperin exact result $\\nu=2\/a$,\nwhich substantiates the idea that the lattice sizes considered could be too\nsmall and that a cross-over would be observed at much larger lattice sizes.\nOn the other hand, no significant evolution of the Griffiths phase could be\nobserved in the range of lattice sizes studied~\\cite{PRE}. Moreover, the\nconspiracy of two amplitudes that leads to the violation of the hyperscaling\nrelation is well verified and no sign of deviation at large lattice\nsizes is observed.\n\nSince larger lattice sizes are not accessible by Monte Carlo simulations,\nwe turn our attention in this work to larger exponents $a$ of the disorder\ncorrelations. The fact that Weinrib-Halperin predictions were confirmed by\nMonte Carlo simulations of the 3D Ising model with $a=2$ could indicate\nthat finite-size effects are weaker for larger values of $a$.\nIn refs~\\cite{EPL,PRE}, only small values of $a$ were considered\nbecause disorder configurations were generated by simulating an auxiliary\nAshkin-Teller model on a self-dual line where its critical exponents\nare known exactly. The polarisation density was then used to construct the\ncouplings $J_{ij}$ of the Potts model. Disorder correlations correspond\ntherefore to the polarisation-polarisation correlations of the auxiliary\nAshkin-Teller model. When moving along the self-dual line, only exponents\nin the range $a\\in [1\/4;3\/4]$ can be obtained.\n\nIn this work, we present new data for disorder correlation exponents\n$a=1\/3$ and $2\/3$ obtained by using an auxiliary Ashkin-Teller model.\nIn order to investigate the possible existence of a cross-over towards the\nWeinrib-Halperin fixed point, we considered also the values $a\\simeq 1.036$\nand $a=2$ obtained using the 3D and 4D Ising models as auxiliary models to\ngenerate the disorder configurations. In the first section, details about\nthe models and the Monte Carlo simulations are given. In the second section,\nthe behaviour of the magnetic susceptibility $\\bar\\chi$ is discussed.\nAs already observed in Ref.~\\cite{PRE}, $\\bar\\chi$ diverge\nalgebraically with the lattice size in a broad interval of temperatures,\nidentified as a Griffiths phase, when $a$ is sufficiently small.\nA simple explanation of this phenomena is to assume that disorder\nfluctuations induce a spreading of local transition temperatures. Because\nthese fluctuations vanish as $L^{-a\/2}$, this would imply that a single\npeak would be recovered at large lattice sizes. Moreover, the smaller the\nexponent $a$ and the larger the lattice sizes needed to observe a single peak.\nIn the second section,\nnumerical evidence is given that disorder fluctuations are not sufficient\nto explain the observed Griffiths phase, and therefore, that the latter\nphase cannot be expected to collapse as $L^{-a\/2}$. In the third section,\nthe possibility of a cross-over controlled by the amplitude of disorder\ncorrelations is considered. These amplitudes are compared for the\ndifferent disorder correlations considered and, then different disorder\nstrengths are studied. Finally, conclusions follow.\n\n\\section{Models and simulation}\n\\label{sec2}\nThe classical $q$-state Potts model is the lattice spin model defined by the\nHamiltonian~\\cite{Potts,Wu}\n   \\begin{equation}\n     H=-J\\sum_{(i,j)} \\delta_{\\sigma_i,\\sigma_j},\\hskip 1truecm\n     \\sigma_i=0,1,\\ldots,q-1\n     \\end{equation}\nwhere the spin $\\sigma_i$ takes $q$ possible values and is located on the\n$i$-th node of the lattice. The sum extends over all pairs $(i,j)$ of nearest\nneighbours on the lattice. In the following, the Potts model will be considered\non the square lattice. As mentioned in the introduction, the phase transition\nis continuous for $q\\le 4$ and of first-order when $q>4$. We will restrict\nourselves to the case $q=8$, i.e. a point in the regime of first-order\ntransition. Disorder is now introduced as bond-dependent random exchange\ncouplings $J_{ij}$. The Hamiltonian becomes\n   \\begin{equation}\n     H=-\\sum_{(i,j)} J_{ij}\\delta_{\\sigma_i,\\sigma_j}.\n     \\end{equation}\nThe spatial correlations between these couplings is assumed to decay\nalgebraically with an exponent $a$ at large distance:\n   \\begin{equation}\n     \\overline{J_{ij}J_{kl}}-\\overline{J_{ij}}\\ \\!\\overline{J_{kl}}\n     \\sim |\\vec r_i-\\vec r_k|^{-a}.\n     \\end{equation}\nFor convenience, we will restrict ourselves in the following to a binary\ncoupling distribution, i.e. $J_{ij}=J_1$ or $J_2$. The presence of disorder\ncorrelations does not affect the self-duality condition of the random Potts\nmodel. Imposing $J_1$ and $J_2$ to be equiprobable and self-dual of each other,\nthe self-dual line is given by the condition~\\cite{Kinzel}\n   \\begin{equation}\n     \\big(e^{\\beta J_1}-1\\big)\\big(e^{\\beta J_2}-1\\big)=q.\n   \\end{equation}\n\nThe coupling configurations are generated by independent Monte Carlo simulations\nof two auxiliary models: the Ising and Ashkin-Teller models. The former is\ndefined by the Hamiltonian\n   \\begin{equation}\n     H=-J_{IM}\\sum_{(i,j)} \\sigma_i\\sigma_j,\\hskip 1truecm\\sigma_i=\\pm 1\n     \\end{equation}\nand is equivalent to the $q=2$ Potts model. It is well known that this model\nundergoes a second-order phase transition in any dimension $d>1$. We considered\nhypercubic lattices of dimension $d=3$ and $d=4$. A few thousand spin\nconfigurations are generated at the critical point, corresponding to $\\beta\nJ_{IM}\\simeq 0.221655$ for $d=3$~\\cite{Pelissetto} and $\\beta J_{IM}\\simeq 0.149694$\nfor $d=4$~\\cite{Stauffer97}. For each spin configuration, a two-dimensional\nsection is cut and random couplings for the 2D Potts model are constructed as\n    \\begin{equation}\n      J_{ij}={J_1+J_2\\over 2}+{J_1-J_2\\over 2}\\sigma_i\n      \\end{equation}\nfor each pair $(i,j)$ of nearest neighbours in the 2D section. Note that,\nat any site $i$, two couplings, in two different directions, are identical.\nBy construction, disorder correlation functions $\\overline{J_{ij}J_{kl}}\n-\\overline{J_{ij}}\\ \\!\\overline{J_{kl}}$ decay as the spin-spin correlation\nfunctions of the auxiliary Ising model. Therefore, the decay is algebraic\nat large distances with an exponent $a=2\\beta\/\\nu\\simeq 1.036$ for the 3D\nIsing model~\\cite{Pelissetto} and $a=2$ for the 4D Ising model.\nNote that, in the second case, the exponent $a$ is equal to the dimension\n$d=2$ of the Potts model. Therefore, according to Weinrib and Halperin,\ndisorder correlations are expected to be irrelevant and the system falls\ninto the same universality class as the\nPotts model with independent random couplings.\n\nThe second auxiliary model is the 2D Ashkin-Teller model defined by the\nHamiltonian~\\cite{AshkinTeller,Fan}\n   \\begin{equation}\n     H=-\\sum_{(i,j)} \\big[J_{AT}\\sigma_i\\sigma_j+J_{AT}\\tau_i\\tau_j+\n       K_{\\rm AT}\\sigma_i\\sigma_j\\tau_i\\tau_j\\big],\\hskip 1truecm\n       \\sigma_i,\\tau_i=\\pm 1\n     \\end{equation}\nand corresponding to two Ising models coupled by their energy densities.\nOn the square lattice, the model is self-dual along the line of the phase\ndiagram given by $e^{-2K_{\\rm AT}}=\\sinh 2J_{\\rm AT}$. Thanks to a mapping onto\nthe eight-vertex model, the critical exponents are known exactly along this\nline. The random couplings for the Potts model are constructed from the\npolarisation density as\n    \\begin{equation}\n      J_{ij}={J_1+J_2\\over 2}+{J_1-J_2\\over 2}\\sigma_i\\tau_i.\n      \\end{equation}\nThe disorder correlations therefore decay as the polarisation-polarisation\ncorrelation functions of the auxiliary Ashkin-Teller model. In this work, we\nconsidered two points on the self-dual line of the Ashkin-Teller model ($y=0.50$\nand $y=1.25$ in the language of the eight-vertex model) corresponding to\nexponents $a=1\/3$ and $a=2\/3$.\n\nThe above-described spin models were simulated using Monte Carlo cluster\nalgorithms to reduce the critical slowing-down. For the Ising and Potts models,\nthe Swendsen-Wang algorithm was employed~\\cite{SW}. The Ashkin-Teller was\nsimulated using a cluster algorithm introduced by Wiseman and\nDomany~\\cite{Salas,Salas2}.\n\n\\section{Griffiths phase and disorder fluctuations}\n\\label{sec3}\nThe magnetic susceptibility $\\chi$ of a finite system undergoing a continuous\nphase transition in the thermodynamic limit is expected to display a peak whose\nmaximum diverges with the lattice size $L$ as $L^{\\gamma\/\\nu}$. The location of\nthis maximum goes towards the critical temperature $T_c$ in the limit of an\ninfinite system. A very different situation was observed in the 2D Potts\nmodel with strongly correlated disorder~\\cite{PRE}. As can be seen on\nfigure~\\ref{Fig2}, two peaks are present for $a=1\/3$ and $2\/3$. The data\nshow an algebraic increase of the average magnetic susceptibility for all\ntemperatures between these two peaks. For this reason, this region was\nconjectured to be a Griffiths phase, similar to the one observed in the\nMcCoy-Wu model. The absence of any evolution of the location of the two\npeaks was reported in the case $a=0.4$. In contrast, figure~\\ref{Fig2}\nshows a slow evolution in the case $a=2\/3$. Since only lattice sizes up\nto $L=128$ were studied, the possibility of a collapse of the Griffiths\nphase into a single point in the thermodynamic limit cannot be excluded.\nMoreover, such a collapse is even more clearly seen on figure~\\ref{Fig2}\nfor $a\\simeq 1.036$. Two peaks are still visible but they tend to come closer\nwhen the lattice size is increased. It seems natural in this case to assume\nthat the two peaks will merge into a single one at larger lattice sizes.\nFor disorder correlations with a faster decay $a=2$, only one peak is observed\n(Fig.~\\ref{Fig2}) and its location tends towards the critical value\n$\\beta_c=1$, expected from self-duality arguments.\n\n\n\\begin{figure}[ht]\n\\centering\n\\psfrag{chi}[Bc][Bc][1][1]{$\\overline{\\chi}$}\n\\psfrag{beta}[tc][tc][1][0]{$\\beta$}\n\\psfrag{L=24}[Bc][Bc][1][0]{\\tiny $L=24$}\n\\psfrag{L=32}[Bc][Bc][1][0]{\\tiny $L=32$}\n\\psfrag{L=64}[Bc][Bc][1][0]{\\tiny $L=64$}\n\\psfrag{L=48}[Bc][Bc][1][0]{\\tiny $L=48$}\n\\psfrag{L=96}[Bc][Bc][1][0]{\\tiny $L=96$}\n\\psfrag{L=128}[Bc][Bc][1][0]{\\tiny $L=128$}\n\\includegraphics[width=7.25cm]{Fig2a.eps}\\quad\n\\includegraphics[width=7.25cm]{Fig2b.eps}\\par\n\\includegraphics[width=7.25cm]{Fig2c.eps}\\quad\n\\includegraphics[width=7.25cm]{Fig2d.eps}\n\\caption{Average magnetic susceptibility of the 8-state Potts model\nwith different disorder correlation exponents ($a=1\/3$, $2\/3$, $1.036$ and $2$\nfrom left to right and top to bottom) for a disorder strength $r=J_1\/J_2=7.5$.\nThe different curves correspond to different lattice sizes.\nNote that the scale of the $y$-axis is logarithmic.}\n\\label{Fig2}\n\\end{figure}\n\nAs mentioned in the introduction, it may be assumed that the width of the\nGriffiths phase is due to large disorder fluctuations. It seems indeed natural\nto assume that the first peak is caused by the ferromagnetic ordering\nof large clusters with a high concentration of weak bonds $J_2$ while\nthe second one corresponds to clusters of strong bonds $J_1$. Such large\nclusters are more probable when disorder correlations decay slowly.\nIn the following, disorder fluctuations will be compared for the\ndifferent values of $a$ considered. To be more specific, consider the\ngeneral case of a lattice model with an energy density denoted\n$\\epsilon_{ij}=\\epsilon(\\sigma_i,\\sigma_j)$ on the edge between the\nspins on sites $i$ and $j$. The weak disorder limit of the partition\nfunction can be calculated using the replica trick:\n  \\begin{equation}\n  \\overline{\\ln{\\cal Z}}=\\lim_{n\\rightarrow 0} {1\\over n}\\big(\n  \\overline{{\\cal Z}^n}-1\\big).\n  \\end{equation}\nIntroducing the interaction energy $\\epsilon_{ij}^\\alpha=\\epsilon\n(\\sigma_i^\\alpha,\\sigma_j^\\alpha)$ between the two spins $\\sigma_i^\\alpha$\nand $\\sigma_j^\\alpha$ of the $\\alpha$-th replica, the partition function\nof $n$ replicas reads\n  \\begin{equation}\n  \\overline{{\\cal Z}^n}=\\sum_{\\{\\sigma_i^\\alpha\\}}\n  \\overline{e^{-\\beta \\sum_{(i,j),\\alpha}J_{ij}\\epsilon_{ij}^\\alpha}}\n    \\simeq \\sum_{\\{\\sigma_i^\\alpha\\}}\n    e^{-\\beta \\sum_{(i,j),\\alpha}\\overline{J_{ij}}\\epsilon_{ij}^\\alpha\n      +{\\beta^2\\over 2}\\sum_{(i,j),(k,l),\\atop\\alpha,\\beta}\n      \\big(\\overline{J_{ij}J_{kl}}-\\overline{J_{ij}}\\ \\!\\overline{J_{kl}}\\big)\n      \\epsilon_{ij}^\\alpha\\epsilon_{kl}^\\beta+\\ldots}.\n    \\label{DevPartF}\n  \\end{equation}\nThe first contribution of disorder to the partition function involves\nthe correlations $\\overline{J_{ij}J_{kl}}-\\overline{J_{ij}}\\ \\!\n\\overline{J_{kl}}$, and is obviously a function of $a$. In order to\ncharacterise the disorder strength by a scalar, we considered the sum\nof these correlations, which also corresponds to the fluctuations\nof the couplings:\n   \\begin{equation}\n     \\Delta J_2=\\overline{\\Big[{1\\over N}\n       \\sum_{(i,j)}(J_{ij}-\\bar J)\\Big]^2}^{1\/2}\n     \\label{DefSigma}\n   \\end{equation}\nwhere $N=2L^2$ is the number of bonds of the square lattice. Since the\ncouplings $J_{ij}$ are constructed from the polarisation density $\\sigma_i\n\\tau_i$ of the auxiliary Ashkin-Teller model (for $a=1\/3$ and $2\/3$), or\nfrom the magnetisation density $\\sigma_i$ of the auxiliary Ising model\n(for $a\\simeq 1.036$ and $2$), $\\Delta J_2$ is related, up to a prefactor\n${J_1-J_2\\over 2}$, to the fluctuations of the polarisation, or magnetisation,\ndensity. Therefore, $\\Delta J_2$ is expected to scale as\n   \\begin{equation}\n     \\Delta J_2\\sim L^{-a\/2}\n     \\label{ScaleSigma}\n   \\end{equation}\nfor both auxiliary models. This result is obtained by expanding the square\nin equation~(\\ref{DefSigma}) and integrating out the disorder correlations\nin the continuum limit. Up to a further factor $L^d$, $(\\Delta J_2)^2$ is\nalso proportional to the electric or magnetic susceptibility of the\nAshkin-Teller and Ising models. The hyperscaling relation for these\nauxiliary models leads to $\\Delta J_2\\sim L^{-\\beta\/\\nu}$ where the\nexponents $\\beta\/\\nu$ is equal to $a\/2$ by construction of the random\ncouplings. Equation~(\\ref{ScaleSigma}) shows that $\\Delta J_2$\nbehaves as a shift of the critical temperature $|T_c(L)-T_c(\\infty)|$\nin a finite system. Indeed, one expects $|T_c(L)-T_c(\\infty)|\n\\sim L^{-1\/\\nu}$ and, at the Weinrib-Halperin fixed point, $\\nu=2\/a$.\n\n\n\\begin{figure}[ht]\n\\centering\n\\psfrag{s1}[Bc][Bc][1][1]{$\\Delta J_1$}\n\\psfrag{s2}[Bc][Bc][1][1]{$\\Delta J_2$}\n\\psfrag{s4}[Bc][Bc][1][1]{$\\Delta J_4$}\n\\psfrag{L}[tc][tc][1][0]{$L$}\n\\psfrag{a=1\/3}[Bc][Bc][1][0]{\\tiny $a=1\/3$}\n\\psfrag{a=2\/3}[Bc][Bc][1][0]{\\tiny $a=2\/3$}\n\\psfrag{a=1.036}[Bc][Bc][1][0]{\\tiny $a\\simeq 1.036$}\n\\psfrag{a=2}[Bc][Bc][1][0]{\\tiny $a=2$}\n\\includegraphics[width=4.85cm]{Fig5a.eps}\n\\includegraphics[width=4.85cm]{Fig5b.eps}\n\\includegraphics[width=4.85cm]{Fig5c.eps}\n\\caption{Fluctuations $\\Delta J$ (up to a factor $(J_1-J_2)\/2$) of the\naverage random couplings versus the lattice size $L$. The different\ncurves correspond to different disorder correlations, i.e. to different\nexponents $a=1\/3$, $2\/3$, $1.036$ and $2$ (from top to bottom).}\n\\label{Fig5}\n\\end{figure}\n\nThe variance of the average coupling $\\Delta J_2$ is plotted on\nfigure~\\ref{Fig5} versus the lattice size in the four cases $a=1\/3$,\n$2\/3$, $1.036$ and $2$. Note that in the last two cases ($a\\simeq 1.036$\nand $a=2$), only the magnetisation in the two-dimensional section that\nwas used to construct the exchange couplings $J_{ij}$ is considered.\nAs expected, an algebraic decay with an exponent compatible with $a\/2$\nis observed. On figure~\\ref{Fig2}, the collapse of the two peaks of the\nmagnetic susceptibility is observed for $a\\simeq 1.036$ for lattice sizes\n$L\\sim {\\cal O}(10^2)$ when $a\\simeq 1.036$. According to figure~\\ref{Fig5},\nthis corresponds to disorder fluctuations of order $\\Delta J_2\\simeq 0.06$.\nFor $a=1\/3$ and $2\/3$, none of the lattice sizes that were considered\ncorrespond to so small disorder fluctuations. Indeed, $\\Delta J_2=0.159(4)$\nwhen $a=1\/3$ for the largest lattice size $L=128$ and $\\Delta J_2=0.090(3)$\nwhen $a=2\/3$. This strengthens the idea that the collapse will be\nobserved for larger lattice sizes for $a=1\/3$ and $2\/3$. Using the\nscaling law (\\ref{ScaleSigma}), one can even predict these sizes\nto be of the order of $L^*\\simeq 128(0.06\/0.16)^{-2\/a}\\simeq 44,000$\nfor $a=1\/3$ and $L^*\\simeq 128(0.06\/0.09)^{-2\/a}\\simeq 432$ for $a=2\/3$.\nOn the other hand, disorder fluctuations are small for $a=2$ ($\\Delta J_2\n=0.0618(4)$ already for $L=24$), smaller than for $a\\simeq 1.036$ with\nlattice sizes $L\\sim {\\cal O}(10^2)$. \n\nThese results do not depend on the quantity used to measure disorder\nfluctuations. The scaling law (\\ref{ScaleSigma}) suggests to use the\norder parameter, polarisation $\\overline{|\\sigma_i\\tau_i|}$\nor magnetisation $\\overline{|\\sigma_i|}$, of the auxiliary models as an\nalternative measure of the fluctuations of the couplings. This quantity will\nbe denoted $\\Delta J_1$ in the following. $\\Delta J_1$ and $\\Delta J_2$ give\nessentially the same information and, as can be seen on figure~\\ref{Fig5},\ntake sensibly the same value, but $\\Delta J_1$ presents the advantage of\nbeing more stable numerically. More surprising is the fact that the same\nconclusions can be drawn from the second contribution of disorder\nto the partition function (\\ref{DevPartF}). Expanding further, the next\nterm will involve the connected four-point correlation function\n$\\overline{J_iJ_jJ_kJ_l}_c$ of disorder. This quantity was assumed to\nbe irrelevant by Weinrib and Halperin. We considered the fourth-order\ncumulant\n   \\begin{equation}\n     \\Delta J_4=\\left(3\\overline{\\Big[{1\\over N}\\sum_{i,j} (J_{ij}-\\bar J)\n       \\Big]^2}^2-\\overline{\\Big[{1\\over N}\\sum_{i,j} (J_{ij}-\\bar J)\n       \\Big]^4}\\right)^{1\/4}.\n   \\end{equation}\nAs can be seen on figure~\\ref{Fig5}, no qualitative difference between\nthe three quantities $\\Delta J_1$, $\\Delta J_2$ and $\\Delta J_4$ is\nobserved.\n\n\nHowever, there are small differences between the four cases $a=1\/3,2\/3,1.036$\nand $a$ that cannot be explained only in terms of disorder fluctuations.\nThe value of $\\Delta J$ for the largest lattice size $L=128$ at $a=2\/3$ is\nclose to the one estimated for $L=48$ at $a\\simeq 1.036$. Therefore, the average\nsusceptibility should look qualitatively the same for $a=2\/3$ at $L=128$\nand for $a\\simeq 1.036$ at $L=48$. It is not clear that it is indeed the\ncase on figure~\\ref{Fig5}. Moreover, a nice collapse of the magnetic\nsusceptibility is observed at large $\\beta$ for $a=1\/3$ and $2\/3$ while\nit is not case for $a\\simeq 1.036$ and 2. Stronger statements might be\nformulated by comparing thermodynamic quantities displaying universal\nproperties. The natural candidate is the 4th-order Binder cumulant\n     \\begin{equation}\n       U_M=1-{\\overline{\\langle m^4\\rangle}\n       \\over 3\\overline{\\langle m^2\\rangle^2}}\n\\end{equation}\nwhose value at the intersection of two curves with respect to\ntemperature is expected to be universal in the thermodynamic limit.\nA notable difference between the different values of $a$ is that the\ncrossing points occur for inverse temperatures $\\beta$ well below\n$\\beta_c=1$ when $a=1\/3$ and $2\/3$ and very close to $\\beta_c=1$\nwhen $a\\simeq 1.036$ and $2$. Unfortunately, the error bars are large\nand do not allow to be conclusive.\n\n\\begin{figure}[ht]\n\\centering\n\\psfrag{Rm}[Bc][Bc][1][1]{$\\overline{\\langle m\\rangle^2}\n\/\\overline{\\langle m\\rangle}^2-1$}\n\\psfrag{beta}[tc][tc][1][0]{$\\beta$}\n\\psfrag{L=24}[Bc][Bc][1][0]{\\tiny $L=24$}\n\\psfrag{L=32}[Bc][Bc][1][0]{\\tiny $L=32$}\n\\psfrag{L=64}[Bc][Bc][1][0]{\\tiny $L=64$}\n\\psfrag{L=48}[Bc][Bc][1][0]{\\tiny $L=48$}\n\\psfrag{L=96}[Bc][Bc][1][0]{\\tiny $L=96$}\n\\psfrag{L=128}[Bc][Bc][1][0]{\\tiny $L=128$}\n\\includegraphics[width=7.25cm]{Fig3a.eps}\\quad\n\\includegraphics[width=7.25cm]{Fig3b.eps}\\par\n\\includegraphics[width=7.25cm]{Fig3c.eps}\\quad\n\\includegraphics[width=7.25cm]{Fig3d.eps}\n\\caption{Ratio $\\overline{\\langle m\\rangle^2}\/\\overline{\\langle m\\rangle}^2-1$\nof the 8-state Potts model with different disorder correlation exponents\n($a=1\/3$, $2\/3$, $1.036$ and $2$ from left to right and top to bottom)\nfor a disorder strength $r=J_1\/J_2=7.5$. The different curves correspond\nto different lattice sizes.}\\label{Fig3}\n\\end{figure}\n\nAnother quantity displaying universal properties is the ratio~\\cite{Wiseman}\n   \\begin{equation}\n     R_m={\\overline{\\langle m\\rangle^2}-\\overline{\\langle m\\rangle}^2\n     \\over \\overline{\\langle m\\rangle}^2}\n    \\end{equation}\nthat measures the sample-to-sample fluctuations of magnetisation.\nOutside of a critical point, all disorder realisations are expected\nto lead to the same average magnetisation $\\langle m\\rangle$ in the\nthermodynamic limit. Therefore, the ratio $R_m$ vanishes as $L\\rightarrow\n+\\infty$ and magnetisation is said to be self-averaging. This is no longer\ntrue at a fixed point where disorder is relevant. In this case, $R_m$\ngoes towards a finite value in the thermodynamic limit. This limit is\nexpected to be a universal quantity~\\cite{Aharony}. Numerical data\nfor this ratio $R_m$ are plotted on figure~\\ref{Fig3}. Two distinct\nbehaviours are observed. For $a=1\/3$ and $a=2\/3$, $R_m$ displays\na peak in the paramagnetic phase (small $\\beta=1\/k_BT$), followed by\na broad shouldering. The latter extends over a range of temperatures\nwhich roughly corresponds to the range between the two peaks of the\naverage magnetic susceptibility (see figure~\\ref{Fig2}). Interestingly,\nthe estimates of $R_m$ at any temperature in this shouldering are\ncompatible, within error bars, for all lattices sizes $L\\in[32;128]$.\nUnless a sudden decay of $R_m$ occurs at much larger lattices sizes,\nwe are led to the conclusion that magnetisation is a non-self-averaging\nquantity in the whole range of temperatures between the two peaks\nof the susceptibility. This conclusion is consistent with the assumption\nof the existence of a Griffiths phase. On the other hand, for $a\\simeq\n1.036$ and $a=2$, the peak in the paramagnetic phase is softer and is\nnot followed by a shouldering but by a monotonous decay. More interesting\nis the fact that the curves corresponding to different lattice\nsizes cross each other at a single point, close to the self-dual point\n$\\beta_c=1$. This is consistent with the existence of a unique critical\npoint at $\\beta_c=1$. Would it be possible that, in the case $a=2\/3$,\nthe shouldering disappears at large lattice sizes to be replaced by\na monotonous decay with a single crossing point for different lattice\nsizes? If the coupling fluctuations $\\Delta J$ provides a measure of\nthe width of the Griffiths phase as discussed above, it should also\ndetermine the range of temperatures around $\\beta_c$ for which $R_m$\nis finite and size-independent. Then the ratio $R_m$ should look\nsimilar for $a=2\/3$ at $L=128$ and for $a\\simeq 1.036$ at $L=48$.\nThis is definitely not the case on figure~\\ref{Fig3}. Therefore, the\nGriffiths phase is not solely the consequence of disorder fluctuations\nand there is no reason to expect the Griffiths phase to collapse into a\nsingle point as $L^{-a\/2}$.\n\n\\section{Griffiths phase and disorder strength}\nAll data presented in the previous section correspond to a disorder strength\n$r=J_1\/J_2=7.5$. Because the two peaks of the susceptibility were interpreted\nas the ordering of macroscopic clusters with a majority of strong, or\nweak, couplings, the disorder strength $r$ controls the width of the Griffiths\nphase. One can therefore wonder whether disorder is not too strong\nin the cases $a=1\/3$ and $2\/3$ which implies that a cross-over to the\nWeinrib-Halperin fixed point would be observed at larger lattice sizes.\nFor the Potts model with uncorrelated disorder, strong scaling corrections\ndepending on $r=J_1\/J_2$ were indeed observed. Accurate estimates of the critical\nexponents became accessible only after an appropriate disorder strength was\ndetermined. The by-far most\nefficient technique was, in this case, to compute an effective central charge\n$c_{\\rm eff}$ by transfer matrix techniques and search for the maximum of\n$c_{\\rm eff}$. The central charge is unfortunately difficult to measure by\nMonte Carlo simulations. Consequently, we will restrict ourselves to\nobserve the effect of a variation of the disorder strength $r$.\n\n\\begin{figure}[ht]\n\\centering\n\\psfrag{s2}[Bc][Bc][1][1]{$(2-a)(\\Delta J_2)^2 L^{a}$}\n\\psfrag{L}[tc][tc][1][0]{$L$}\n\\psfrag{a=1\/3}[Bc][Bc][1][0]{\\tiny $a=1\/3$}\n\\psfrag{a=2\/3}[Bc][Bc][1][0]{\\tiny $a=2\/3$}\n\\psfrag{a=1.036}[Bc][Bc][1][0]{\\tiny $a\\simeq 1.036$}\n\\psfrag{a=2}[Bc][Bc][1][0]{\\tiny $a=2$}\n\\includegraphics[width=7.25cm]{Fig8.eps}\n\\caption{Amplitude $w$ of disorder correlations (up to a factor $(J_1-J_2)^2\/4$)\nversus the lattice size $L$. The different curves correspond to different\ndisorder correlations, i.e. to different exponents $a=1\/3$, $2\/3$\nand $1.036$.}\n\\label{Fig8}\n\\end{figure}\n\nWeinrib and Halperin considered disorder correlations of the form\n   \\begin{equation}\n     C(r_{ik})=\\overline{J_{ij}J_{kl}}-\\overline{J_{ij}}\\ \\!\\overline{J_{kl}}\n     =v\\delta(\\vec r_i-\\vec r_k)+{w\\over |r_{ik}|^a}\n     \\label{CorrelationWH}\n   \\end{equation}\nwhere the two amplitudes $v$ and $w$ are irrelevant scaling fields at the\nlong-range random fixed point. When simulating a finite system with an\namplitude $w$ much larger (or much weaker) than the value $w^*$ taken at the\nfixed point, the critical behaviour may be affected by strong scaling\ncorrections. Indeed, in the neighbourhood of the Weinrib-Halperin random\nfixed point, the free energy density can be assumed to scale under\na dilatation with a scale factor $b$ as:\n  \\begin{equation}\n    f(t,h,1\/L,w)=b^{-d}f\\big(b^{y_t}t,b^{y_h}h,b\/L,b^{y_w}(w-w^*)\\big)\n  \\end{equation}\nwhere $t=|T-T_c|$ is the reduced temperature and $h$ the magnetic field.\nAt the critical point, i.e. $t=h=0$, and with $b=L$, the magnetic\nsusceptibility $\\chi=-{\\partial^2f\\over\\partial h^2}$ scales as\n  \\begin{equation}\n    \\chi(1\/L,w)=L^{\\gamma\/\\nu}{\\cal F}\\big(L^{y_w}(w-w^*)\\big)\n  \\end{equation}\nwhere $\\gamma\/\\nu=2y_h-d$. The scaling function ${\\cal F}$ involves a cross-over\nlength $\\ell\\sim (w-w^*)^{-1\/y_w}$ associated to disorder. The dominant\nfinite-size scaling behaviour $\\chi\\sim L^{\\gamma\/\\nu}$ will be hidden by\nscaling corrections if $L\\ll\\ell$.\n\nIn the previous section, the fluctuations of the average coupling have been\ncompared for different exponents $a$. To compare now the amplitudes $w$ of\ndisorder correlations, note that integrating out disorder correlations leads\non one hand to\n  \\begin{equation}\n    {1\\over N^2}\\sum_{ij,kl}\\big[\\overline{J_{ij}J_{kl}}\n    -\\overline{J_{ij}}\\ \\!\\overline{J_{kl}}\\big]\n    \\simeq {1\\over L^2}\\int_{L^2} C(\\vec r)d^2\\vec r\n    \\simeq {w\\over L^2}\\int_0^L {rdr\\over r^a}\n    =w{L^{-a}\\over 2-a}\n  \\end{equation}\nwhile on the other hand, the same quantity is equal to $(\\Delta J_2)^2$\naccording to equation (\\ref{DefSigma}).\nThe amplitude $w$ can therefore be recomputed as $w\\simeq (2-a)\n(\\Delta J_2)^2L^a$. This estimate is plotted on figure~\\ref{Fig8}.\nNote that the amplitude $w$ is not plotted for $a=2$ because the definition\nis inappropriate in this case (the integration of the correlations involves\na logarithm) and leads to $w=0$. As can be seen on figure~\\ref{Fig8},\nthe amplitude $w$ does not evolve monotonously with $a$. This should not\nbe a surprise because the couplings have been generated from different\nauxiliary models. The amplitude $w$ for $a\\simeq 1.036$ lies in between the\namplitudes for $a=1\/3$ and $a=2\/3$. Therefore, the Griffiths phase\nand the small $\\nu$ exponents reported in~\\cite{PRE} for $a=1\/3$ and\n$a=2\/3$ cannot be explained as the result of strong scaling corrections.\nIndeed, if one assume that the amplitude $w$ is close to $w^*$ in the case\n$a\\simeq 1.036$, which would explain why the collapse of the two peaks\nof $\\bar\\chi$ is observed for reachable lattice sizes, one can conceive that\nthe Griffiths phase is the result of too strong disorder correlations,\ni.e. $w>w^*$, in the case $a=1\/3$. However, it is hard to understand how\nweak correlations, i.e. $w<w^*$, would lead to a similar result\nfor $a=2\/3$. The explanation in terms of a cross-over is therefore not\nsupported by the numerical data.\n\n\\begin{figure}[ht]\n\\centering\n\\psfrag{chi}[Bc][Bc][1][1]{$\\overline{\\chi}$}\n\\psfrag{Rm}[Bc][Bc][1][1]{$\\overline{\\langle m\\rangle^2}\n\/\\overline{\\langle m\\rangle}^2-1$}\n\\psfrag{beta}[tc][tc][1][0]{$\\beta$}\n\\psfrag{L=32}[Bc][Bc][1][0]{\\tiny $L=32$}\n\\psfrag{L=64}[Bc][Bc][1][0]{\\tiny $L=64$}\n\\psfrag{L=48}[Bc][Bc][1][0]{\\tiny $L=48$}\n\\psfrag{L=96}[Bc][Bc][1][0]{\\tiny $L=96$}\n\\psfrag{L=128}[Bc][Bc][1][0]{\\tiny $L=128$}\n\\includegraphics[width=7.25cm]{Fig9a.eps}\\quad\n\\includegraphics[width=7.25cm]{Fig9b.eps}\\par\n\\includegraphics[width=7.25cm]{Fig9c.eps}\\quad\n\\includegraphics[width=7.25cm]{Fig9d.eps}\n\\caption{On the left, average magnetic susceptibility of the 8-state Potts\nmodel with a disorder correlation exponent $a=2\/3$ and a disorder strengths\n$r=J_1\/J_2=3.5$ (top) and $r=2$ (bottom). The different curves correspond\nto different lattice sizes. On the right, ratio $R_m=\\overline{\\langle m\n\\rangle^2}\/\\overline{\\langle m\\rangle}^2-1$ for the same models.\nNote that the horizontal scales are not the same.}\n\\label{Fig9}\n\\end{figure}\n\nHowever, as predicted by Weinrib and Halperin for the $\\phi^4$ model, the\namplitude $w^*$ at the fixed point can be a function of $a$. In the following,\nthe effect of a change of the disorder strength, and therefore of $w-w^*$,\nis studied in the case $a=2\/3$. Since ${\\cal F}$ depends on the scaling\nvariable $L^{y_w}(w-w^*)$, tuning the amplitude $w$ to come closer to $w^*$\nis expected to be equivalent to increasing the lattice size $L$.\nTherefore, the cross-over, if any, should be observed when $w$ is decreased.\nOn figure~\\ref{Fig9}, the magnetic susceptibility is plotted for two\ndifferent disorder strengths $r=J_1\/J_2=3.5$ and $r=2$. Comparing\nwith figure~\\ref{Fig2} where the case $r=7.5$ was plotted, it is\nclear that the width of the Griffiths phase is directly proportional\nto the disorder strength $r$, and therefore the amplitude $w$. The\ntwo peaks come closer as the disorder strength is reduced.\nHowever, even for $r=2$, the magnetic susceptibility\nis still very different from what is observed on figure~\\ref{Fig2} for\n$a\\simeq 1.036$ and $r=7.5$. One can doubt that the two curves will look\nsimilar for an even weaker disorder in the case $a=2\/3$. Very probably,\nthe two peaks of the magnetic susceptibility for $a=2\/3$ will collapse\nbut only when approaching $r=1$, i.e. the pure model.\n\nThe ratio $R_m=\\overline{\\langle m\\rangle^2}\/\\overline{\\langle m\\rangle}^2-1$\nprovides a stronger evidence that what is observed for $a\\simeq 1.036$\nis not what should be expected for $a=2\/3$ at weaker disorder.\nAs discussed when the figure~\\ref{Fig2} was commented, the signature\nof the Griffiths phase is a size-independent ratio $R_m$ over a finite\nrange of temperatures. In contrast, for $a\\simeq 1.036$, the ratios $R_m$\ncomputed at two different lattice sizes display a single crossing point at\na temperature evolving towards the self-dual point $\\beta_c=1$. As observed\non figure~\\ref{Fig9}, the diminution of the disorder strength induces\na reduction of the width of the Griffiths phase, and, as expected,\nof the range of temperatures where the ratio $R_m$ appears to be\nsize-independent. However, the ratio $R_m$ looks surprisingly similar,\nup to a temperature rescaling, at different disorder strengths.\nEven at a disorder strength $r=2$, the behaviour is still very different\nfrom what is observed for $a\\simeq 1.036$.\n\n\n\\section{Conclusions}\nNew Monte Carlo simulations of the 2D 8-state Potts model with a\ndisorder involving algebraically decaying correlations $C(r)\\sim r^{-a}$\nwith $a=1\/3,2\/3,1.036$ and $2$ are presented. While the analysis of the\nmagnetic susceptibility does not allow to exclude the possibility of a\ncollapse of the Griffiths phase into a single critical point, the study\nof the self-averaging ratio $R_m=\\overline{\\langle m\\rangle^2}\/\\overline{%\n\\langle m\\rangle}^2-1$ allows to be more conclusive. Two different behaviours\nare indeed observed for $a=1\/3$ and $2\/3$ on one hand and $a\\simeq 1.036$\nand $2$ on the other hand. The first case is compatible with the assumption\nof the existence of a Griffiths phase while in the second case, the signature\nof a single critical point is observed. These difference cannot be explained\nby larger disorder fluctuations in the first case. Moreover, if the width of\nthe Griffiths phase depends on the disorder strength for $a=2\/3$,\nno cross-over towards the Weinrib-Halperin critical behaviour is observed\nat weak disorder. These numerical results call for a theoretical understanding\nof the precise mechanism behind this Griffiths phase. We hope that a theoretician\nwill find them sufficiently surprising to get interested into this problem.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe technique of sparse domination was developed to provide a simpler approach, achieved by Lerner \\cite{Lerner:simple}, to the ``$A_2$ conjecture'' on sharp weighted norm inequalities for Calder\\'on--Zygmund operators, which was first proved with a different machinery by the author \\cite{Hytonen:A2}. However, beyond this original aim, sparse domination immediately led to significant further consequences and has by now been applied to a variety of new questions, of which \\cite{BHRT,BRS,BFP,CCDO:17,CKL}  is only a  sample. The method consists of two main steps that are largely independent of each other and essentially decouple the operator from the space or norm in which it should be estimated:\n\\begin{enumerate}\n  \\item\\label{it:domOp} Dominating an operator of interest by a suitable sparse operator\/form.\n  \\item\\label{it:domSp} Estimating the sparse form with respect to relevant norms of interest.\n\\end{enumerate}\n\nWhile sparse domination very efficiently captures the local size of an object under consideration, and this is precisely what is needed in many applications, it loses information about directions, which is sometimes relevant when dealing with vector-valued functions, and especially so, matrix-valued weights are involved. To extend the method to such questions, Nazarov et al. \\cite{NPTV:convex} developed the so-called convex body domination, where the numerical averages featuring in sparse domination are replaced by convex subsets of $\\mathbb{R}^n$, thus containing information about different behaviour in different directions. Since its introduction in the context of Calder\\'on--Zygmund operators and matrix $A_2$ weights by \\cite{NPTV:convex} (see also \\cite{CDPO:unif} for another approach but based on the same key idea), convex body domination has been applied to matrix $A_p$-weight and two-weight bounds by Cruz-Uribe et al.~\\cite{CUIM:18}, and extended to commutators of Calder\\'on--Zygmund operators by Isralowitz et al.~\\cite{IPR:21,IPT:commu} and rough singular integral operators by Di Plinio et al.~\\cite{DPHL} and Muller and Rivera-R\\'{\\i}os \\cite{MRR:22}. In a recent breakthrough, Bownik and Cruz-Uribe \\cite{BCU} extended the Rubio de Francia algorithm, and its key application to weighted extrapolation, to matrix-valued weights, by further development of the convex body philosophy.\n\nThe aim of this paper is to further explore this technique, providing extensions, new applications and---hopefully---some additional insight into the abstract underlying mechanisms. We begin by developing a somewhat general framework, but our claims for originality in this regard are relatively mild, as most of the ideas are at least implicit in the previous works in the existing literature. A certain justification for this framework comes from the observation that it applies almost verbatim to the case of Banach space -valued functions. To be precise, given a Banach space $E$, we consider functions taking values in $E^n$, and develop a version of convex body domination applicable to weighted norm inequalities involving matrix weights $W:\\mathbb{R}^d\\to\\mathbb{R}^{n\\times n}$, acting on $E^n$ in the natural way. That is, we make no attempt towards a fully operator-valued theory of weighted norm inequalities in infinite dimensions, yet the results that we obtain are still new even in this more modest generality. In particular, if $E$ is a Banach space with the UMD property, the classical Hilbert transform extends boundedly to the matrix-weighted space $L^2(W;E^n)$ of $E^n$-valued functions; see Corollary \\ref{cor:L2WE} for the result, and Section \\ref{sec:L2WE} for the relevant definitions and background. A key to this extension is the observation that the convex bodies arising from our framework are still $\\mathbb{R}^n$-valued in this generality---and not, for instance, $E^n$-valued, as one might have (and this author certainly had) initially expected. Thus the powerful Euclidean machinery, most notably the John ellipsoid theorem, is still available in this setting.\n\nAs for new applications of the theory, we build on a recent observation from Isralowitz et al. \\cite{IPR:21,IPT:commu} that convex body domination of an operator $T$ bootstraps to a domination of its commutators $[b,T]=bT-Tb$ with pointwise multipliers. As we will explore in Section \\ref{sec:commu}, this phenomenon is far more general, and can be used to estimate any operators of the form\n\\begin{equation*}\n  f\\mapsto \\sum_{i=1}^n a_i T(b_i f),\n\\end{equation*}\nwhere an operator $T$ satisfying convex body domination is pre- and post-composed with pointwise multipliers $a_i,b_i$. From this general principle, we can in particular recover and sharpen a recent sufficient condition \\cite{HLO:20} for the boundedness of iterated mixed commutators $[b^1,[b^2,T]]$ in terms of joint conditions on the pair of functions $(b^1,b^2)$, but also obtain new examples.\n\nIn contrast to the development of the abstract framework in the first part of the paper, we have not strived for the greatest generality in terms of the applications in the later sections. In many cases, it will be clear to an experienced reader that several variants and extensions could be obtained, and some of them will most likely be pursued in forthcoming works, by this author and others. Besides the concrete results contained in this paper, our aim is to hint at the many rich directions for the further development of the theory.\n\n\\section{Norms and convex bodies}\n\nLet $X$ be a real normed space. We denote by\n\\begin{equation*}\n  \\bar B_X:=\\{x\\in X:\\Norm{x}{X}\\leq 1\\}\n\\end{equation*}\nits closed unit ball, and by $X^*$ the normed dual, which is a Banach space.\nFor $x^*\\in X^*$, we define, as usual,\n\\begin{equation*}\n  \\Norm{x^*}{X^*}:=\\sup\\{\\abs{\\pair{x}{x^*}}:x\\in\\bar B_X\\}.\n\\end{equation*}\nAs a consequence of the Hahn--Banach theorem, we have\n\\begin{equation}\\label{eq:HB}\n  \\Norm{x}{X}=\\sup\\{\\abs{\\pair{x}{x^*}}:x^*\\in\\bar B_{X^*}\\}\n  =\\max\\{\\pair{x}{x^*}:x^*\\in\\bar B_{X^*}\\};\n\\end{equation}\nin particular, the supremum is reached as a maximum, and we have $\\Norm{x}{X}=\\pair{x}{x^*}$ for some $x^*\\in\\bar B_{X^*}$.\n\nFor $\\vec x=(x_i)_{i=1}^n\\in X^n$ and $x^*\\in X^*$, we define the $\\mathbb{R}^n$-valued pairing $\\pair{\\vec x}{x^*}:=(\\pair{x_i}{x^*})_{i=1}^n\\in\\mathbb{R}^n$ and the set-valued ``norm''\n\\begin{equation*}\n  \\sset{\\vec x}_X:=\\{\\pair{\\vec x}{x^*}:x^*\\in \\bar B_{X^*}\\}\\subset\\mathbb{R}^n.\n\\end{equation*}\n\n\\begin{remark}\nThe notation is adapted from Nazarov et al.~\\cite{NPTV:convex}, who introduced the version with $X=\\avL^1(Q)$, the space $L^1(Q)$ with the normalised norm $\\frac{1}{\\abs{Q}}\\Norm{\\ }{1})$. The extension to $X=\\avL^p(Q)$ (i.e., $L^p(Q)$ with the normalised norm $\\frac{1}{\\abs{Q}^{1\/p}}\\Norm{\\ }{p})$ is due to Di Plinio et al.~\\cite{DPHL}. Although our main applications will be concerned with spaces of functions (living on a cube $Q$), we find it illuminating to develop the basics of the theory on a completely abstract level. Among other things, this point of view will make it clear that there will be essentially no difference in treating a space $X=L^p(Q;E)$ of $E$-valued functions for an arbitrary Banach space $E$; for $\\vec{f}\\in X^n$, the corresponding $\\cave{\\vec f}_{X}$ will still be subsets of $\\mathbb{R}^n$ and not, say, of $E^n$. This will allow us to make effortless use of the powerful John ellipsoid theorem from Euclidean geometry, even when working with functions taking values in an infinite-dimensional Banach space! In other applications, a choice like $X=L\\log L(Q)$ might also be relevant.\n\\end{remark}\n\nFor $\\vec a\\in\\mathbb{R}^n$ and $\\vec x\\in X^n$, we define the $X$-valued dot product\n\\begin{equation*}\n \\vec a\\cdot\\vec x:=\\vec x\\cdot\\vec a:=\\sum_{i=1}^n a_i x_i. \n\\end{equation*}\nWe observe the easy identities\n\\begin{equation*}\n  \\vec a\\cdot\\pair{\\vec x}{x^*}=\\pair{\\vec a\\cdot\\vec x}{x^*},\\quad\\forall\\vec a\\in\\mathbb{R}^n,\\ \\vec x\\in X^n,\\ x^*\\in X^*,\n\\end{equation*}\nand\n\\begin{equation*}\n  \\lspan_X(\\vec{x}):=\\lspan\\{x_i\\}_{i=1}^n=\\{\\vec a\\cdot\\vec x:a\\in\\mathbb{R}^n\\}\\subset X.\n\\end{equation*}\n\n\n\\begin{lemma}\nFor each $\\vec x\\in X^n$, the set $\\sset{\\vec x}_X\\subset\\mathbb{R}^n$ is convex, compact, and symmetric about the origin.\n\\end{lemma}\n\n\\begin{proof}\nSymmetry, convexity and boundedness are immediate from the fact that $\\bar B_{X^*}$ has these properties. For compactness in $\\mathbb{R}^n$, it remains to show closedness, so suppose that $\\pair{\\vec x}{x_k^*}\\to\\vec e\\in\\mathbb{R}^n$ as $k\\to\\infty$, where each $x_k^*\\in\\bar B_{X^*}$; we need to show that $\\vec{e}\\in\\sset{x}_X$. For each $\\vec a\\in\\mathbb{R}^n$, it follows that\n\\begin{equation*}\n  \\abs{\\vec a\\cdot\\vec e}=\\lim_{k\\to\\infty}\\abs{\\vec{a}\\cdot\\pair{\\vec x}{x_k^*}}\n  =\\lim_{k\\to\\infty}\\abs{\\pair{\\vec{a}\\cdot\\vec x}{x_k^*}}\n  \\leq\\Norm{\\vec{a}\\cdot\\vec x}{X}.\n\\end{equation*}\nThis in turn implies that\n\\begin{equation*}\n  \\Lambda(\\vec a\\cdot\\vec x):=\\vec a\\cdot\\vec e,\\quad \\forall\\vec a\\cdot\\vec x\\in\\lspan_X(\\vec{x}),\n\\end{equation*}\ngives a well-defined linear functional of norm $1$ on the subspace $\\lspan_X(\\vec{x})\\subset X$. By the Hahn--Banach theorem, $\\Lambda$ is the restriction of some $x^*\\in\\bar B_{X^*}$. Hence\n\\begin{equation*}\n  \\vec{a}\\cdot\\vec{e}=\\Lambda(\\vec a\\cdot\\vec x)=\\pair{\\vec{a}\\cdot\\vec{x}}{x^*}=\\vec{a}\\cdot\\pair{\\vec{x}}{x^*}\\quad\\forall\\vec a\\in\\mathbb{R}^n,\n\\end{equation*}\nand thus $\\lim_{k\\to\\infty}\\pair{\\vec x}{x_k}=\\vec{e}=\\pair{\\vec{x}}{x^*}\\in\\sset{\\vec x}_X$, as we wanted to show.\n\\end{proof}\n\nFor $A,B\\subset\\mathbb{R}^n$, we define the Minkowski dot product\n\\begin{equation*}\n  A\\cdot B:=\\{\\vec a\\cdot\\vec b:\\vec a\\in A,\\vec b\\in B\\}\\subset\\mathbb{R}.\n\\end{equation*}\nIf $A,B\\subset\\mathbb{R}^n$ are convex, compact and symmetric, so is $A\\cdot B\\subset\\mathbb{R}$. On $\\mathbb{R}$, such sets are precisely intervals of the form $[-c,c]$. Hence we can, and sometimes will, identify $A\\cdot B=[-c,c]\\subset\\mathbb{R}$ with its right end-point $c\\in[0,\\infty)$.\nIn particular, for $\\vec x\\in X^n$ and $\\vec y\\in Y^n$, we will use this identification when dealing with\n\\begin{equation*}\n\\begin{split}\n  \\sset{\\vec x}_X \\cdot\\sset{\\vec y}_{Y}\n  &=\\{\\pair{\\vec x}{x^*}\\cdot\\pair{\\vec y}{y^*}:x^*\\in\\bar B_{X^*},y^*\\in\\bar B_{Y^*}\\} \\\\\n  &=\\Big\\{\\sum_{i=1}^n\\pair{x_i}{x^*}\\pair{y_i}{y^*}:x^*\\in\\bar B_{X^*},y^*\\in\\bar B_{Y^*}\\Big\\}.\n\\end{split}\n\\end{equation*}\n\n\n\\section{Bi-linear forms}\n\nLet $X,Y$ be real normed spaces, and suppose that we have a bilinear from $t:X\\times Y\\to\\mathbb{R}$. We define its extension acting on pairs of vectors $(\\vec x,\\vec y)\\in X^n\\times Y^n$ as follows. If $\\vec e\\in\\mathbb{R}^n$ and $x\\in X^n$, we have $\\vec x\\cdot\\vec e\\in F$ by our previous convention about the $X$-valued dot product. If $(e_i)_{i=1}^n$ is a fixed orthonormal basis of $\\mathbb{R}^n$, we then define\n\\begin{equation*}\n  t(\\vec f,\\vec g):=\\sum_{i=1}^n t(\\vec f\\cdot\\vec e_i,\\vec g\\cdot\\vec e_i),\n\\end{equation*}\nFor $x\\in X$, $y\\in Y$, and $\\vec e,\\vec u\\in\\mathbb{R}^n$, it follows that\n\\begin{equation*}\n  t(x \\vec e,y \\vec u):=\\sum_{i=1}^n t(x \\vec e\\cdot\\vec e_i,y \\vec u\\cdot\\vec e_i)\n  =t(x,y) \\sum_{i=1}^n (\\vec e\\cdot\\vec e_i)(\\vec u\\cdot\\vec e_i)\n  =t(x,y)\\vec e\\cdot\\vec u.\n\\end{equation*}\nIf $(\\vec u_i)_{i=1}^n$ is another orthonormal basis, then\n\\begin{equation*}\n\\begin{split}\n  \\sum_{i=1}^n t(\\vec x\\cdot\\vec  u_i,\\vec y\\cdot\\vec  u_i)\n  &=\\sum_{i,j,k=1}^n t(\\vec x\\cdot\\vec  e_j,\\vec y\\cdot\\vec  e_k)(\\vec e_j\\cdot \\vec u_i)(\\vec e_k\\cdot\\vec  u_i) \\\\\n  &=\\sum_{j,k=1}^n t(\\vec x\\cdot \\vec e_j,\\vec y\\cdot \\vec e_k)(\\vec e_j\\cdot \\vec e_k) \n  =\\sum_{j=1}^n t(\\vec x\\cdot \\vec e_j,\\vec y\\cdot \\vec e_j)=:t(\\vec x,\\vec y),\n\\end{split}\n\\end{equation*}\nso the definition of $t(\\vec f,\\vec g)$ is independent of the chosen orthonormal basis.\n\nIf $A\\in\\mathbb{R}^{n\\times n}$ is a linear transformation of $\\mathbb{R}^n$, acting in a natural way on $F^n$, then\n\\begin{equation*}\n\\begin{split}\n  t(A\\vec x,\\vec y)\n  &=\\sum_{i=1}^n t(A\\vec f\\cdot\\vec e_i,\\vec g\\cdot\\vec e_i)\n  =\\sum_{i=1}^n t(\\vec f\\cdot A^t\\vec e_i,\\vec g\\cdot\\vec e_i) \\\\\n  &=\\sum_{i,j=1}^n t(\\vec f\\cdot \\vec e_j,\\vec g\\cdot\\vec e_i)(\\vec e_j\\cdot A^t\\vec e_i) \\\\\n  &=\\sum_{i,j=1}^n t(\\vec f\\cdot \\vec e_j,\\vec g\\cdot\\vec e_i)(A\\vec e_j\\cdot \\vec e_i) \n  =\\sum_{j=1}^n t(\\vec f\\cdot \\vec e_j,\\vec g\\cdot A\\vec e_j)\n  =t(\\vec f,A^t\\vec g).\n\\end{split}\n\\end{equation*}\n\n\n\\section{From norm bounds to convex body bounds}\n\nThe idea of the following lemma lies behind many of the existing convex body domination results. To isolate the key point, we state it here in an operator-free version, involving functions and therir norms only.\n\n\\begin{lemma}\\label{lem:fi-vs-convex}\nLet $X,Y$ be normed spaces, and $\\vec f\\in X^n,\\vec g\\in Y^n$.\n\nLet $\\mathcal E_K$ be the John ellipsoid of $K:=\\cave{\\vec f}_{X}$ such that\n\\begin{equation*}\n  \\mathcal E_K\\subset K\\subset\\sqrt{n}\\mathcal E_K,\n\\end{equation*}\nand suppose that $\\mathcal E_K$ is non-degenerate (i.e., of full dimension).\nLet $R_K$ be a linear transformation such that $R_K\\mathcal E_K=\\bar B_{\\mathbb{R}^n}$, the closed unit ball of $\\mathbb{R}^n$, and let $(\\vec e_i)_{i=1}^n$ be an orthonormal basis of $\\mathbb{R}^n$. If\n\\begin{equation*}\n  f_i:=R_K\\vec f\\cdot \\vec e_i,\\quad g_i:=R_K^{-t}\\vec g\\cdot e_i,\\qquad i=1,\\ldots,n,\n\\end{equation*}\nthen\n\\begin{equation*}\n  \\sum_{i=1}^n\\Norm{f_i}{X}\\Norm{g_i}{Y}\\leq n^{3\/2}\\cave{\\vec f}_{X}\\cdot\\cave{\\vec g}_{Y}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nIf $\\phi\\in\\bar B_{X^*}$, then\n\\begin{equation*}\n\\begin{split}\n  \\pair{\\phi}{R_K\\vec f\\cdot\\vec e_i}\n  &=R_K\\pair{\\phi}{\\vec f}\\cdot\\vec e_i \\\\\n  &\\in R_K\\cave{\\vec f}_{X}\\cdot\\vec e_i\n  \\subset R_K \\sqrt{n}\\mathcal E_K\\cdot\\vec e_i\n  =\\sqrt{n}\\bar B_{\\mathbb{R}^n}\\cdot\\vec e_i=\\sqrt{n}[-1,1],\n\\end{split}\n\\end{equation*}\nand hence\n\\begin{equation*}\n  \\Norm{f_i}{X}=\\Norm{R_K\\vec f\\cdot\\vec e_i}{X}\\leq \\sqrt{n}.\n\\end{equation*}\n\nIf $\\psi\\in\\bar B_{Y^*}$, then\n\\begin{equation*}\n\\begin{split}\n  \\pair{\\psi}{R_K^{-t}\\vec g\\cdot\\vec e_i}\n  &=R_K^{-t}\\pair{\\psi}{\\vec g}\\cdot\\vec e_i \\\\\n  &\\in R_K^{-t} \\cave{\\vec g}_{Y}\\cdot\\vec e_i\n  \\subset [-M,M],\\quad M:=\\max\\{\\abs{\\vec y}:\\vec y\\in R_K^{-t} \\cave{\\vec g}_{Y}\\}.\n\\end{split}\n\\end{equation*}\nIt follows that $\\abs{\\pair{\\psi}{R_K^{-t}\\vec g\\cdot\\vec e_i}}\\leq M$, and hence\n\\begin{equation*}\n  \\Norm{g_i}{Y}=\\Norm{R_K^{-t}\\vec g\\cdot\\vec e_i}{Y}\\leq M.\n\\end{equation*}\n\nCombining the estimates, we have\n\\begin{equation*}\n  \\sum_{i=1}^n\\Norm{f_i}{X}\\Norm{g_i}{Y}\n  \\leq\\sum_{i=1}^n \\sqrt{n}M=n^{3\/2}M.\n\\end{equation*}\n\nOn the other hand,\n\\begin{equation*}\n  \\cave{\\vec f}_{X}\\cdot\\cave{\\vec g}_{Y}\n  \\supset \\mathcal E_K\\cdot\\cave{\\vec g}_{Y} \n  =R_K\\mathcal E_K\\cdot R_K^{-t}\\cave{\\vec g}_{Y} \n  =\\bar B_{\\mathbb{R}^n}\\cdot R_K^{-t}\\cave{\\vec g}_{Y} \n  =[-M,M],\n\\end{equation*}\nand hence\n\\begin{equation*}\n  M  \\leq \\cave{\\vec f}_{X}\\cdot\\cave{\\vec g}_{Y},\n\\end{equation*}\nusing the identification of the symmetric interval $\\cave{\\vec f}_{X}\\cdot\\cave{\\vec g}_{Y}$ with its right end-point in the last step.\nSubstituting back, this completes the proof.\n\\end{proof}\n\nThe following proposition contains the basic idea of bootstrapping norm bounds to convex body bounds. Unfortunately, it is a bit too simple for most actual applications, but we include it as an illustrative toy model for the more serious result to be presented after it.\n\n\\begin{proposition}\\label{prop:loc-sparse2convex-pre}\nLet $X,Y$ be normed spaces with subspaces $F\\subset X$ and $G\\subset Y$, and let $t:F\\times G\\to\\mathbb{R}$ be a bilinear form.\nConsider the following conditions:\n\\begin{enumerate}\n  \\item\\label{it:1scaleDomSca-pre} For all $(f,g)\\in F\\times G$, we have\n\\begin{equation*}\n  \\abs{t(f,g)}\\leq C\\Norm{f}{X}\\Norm{g}{Y}.\n\\end{equation*}\n  \\item\\label{it:1scaleDomVec-pre} For all $(\\vec f,\\vec g)\\in F^n\\times G^n $, we have\n\\begin{equation*}\n  \\abs{t(\\vec f,\\vec g)}\\leq C_n\\cave{\\vec f}_{X}\\cdot\\cave{\\vec g}_{Y}.\n\\end{equation*}\n\\end{enumerate}\nFor each $n\\in\\mathbb{Z}_+$, condition \\eqref{it:1scaleDomSca} implies condition \\eqref{it:1scaleDomVec} with $C_n=Cn^{3\/2}$.\n\\end{proposition}\n\n\\begin{proof}\nGiven $\\vec f$, consider the compact, convex, symmetric set \n\\begin{equation*}\n  K:=\\cave{\\vec f}_{X}, \n\\end{equation*}\nand denote by $\\mathcal E_K$ its John ellipsoid such that\n\\begin{equation*}\n  \\mathcal E_K\\subset K\\subset\\sqrt{n}\\mathcal E_K. \n\\end{equation*}\n\n\\subsubsection*{Case: $\\mathcal E_K$ is non-degenerate}\nLet $R_K$ be a linear transformation such that $R_K\\mathcal E_K=\\bar B_{\\mathbb{R}^n}$, the closed unit ball of $\\mathbb{R}^n$.\nLet $(\\vec e_i)_{i=1}^n$ be some orthonormal basis of $\\mathbb{R}^n$.\nWe then write\n\\begin{equation}\\label{eq:tvecfvecg-pre}\n\\begin{split}\n  t(\\vec f,\\vec g)\n  &=t(R_K^{-1}R_K\\vec f,\\vec g)\n  =t(R_K\\vec f,R_K^{-t}\\vec g) \\\\\n  &=\\sum_{i=1}^n t(R_K\\vec f\\cdot\\vec e_i,R_K^{-t}\\vec g\\cdot\\vec e_i)\n    =:\\sum_{i=1}^n t(f_i,g_i),\n\\end{split}\n\\end{equation}\nwhere $f_i$ and $g_i$ are as in Lemma \\ref{lem:fi-vs-convex}.\n\nBy assumption \\eqref{it:1scaleDomSca-pre} and Lemma \\ref{lem:fi-vs-convex}, it follows that\n\\begin{equation*}\n  \\abs{ t(\\vec f,\\vec g)}\n    \\leq\\sum_{i=1}^n \\abs{t(f_i,g_i)}\n    \\leq \\sum_{i=1}^n C\\Norm{f_i}{X}\\Norm{g_i}{Y}\n    \\leq Cn^{3\/2}\\cave{\\vec f}_X\\cdot\\cave{\\vec g}{Y},\n\\end{equation*}\nand this completes the proof in the case that $\\mathcal E_K$ is non-degenerate.\n\n\\subsubsection*{Case: $\\mathcal E_K$ is degenerate}\nSuppose then that $\\mathcal E_K$ is degenerate; hence $H:=\\lspan K$ is a strict subspace of $\\mathbb{R}^n$. Let $P$ denote the orthogonal projection of $\\mathbb{R}^n$ onto $H$. For each $x^*\\in\\bar B_{X^*}$, we have $\\pair{\\vec f}{x^*}\\in K\\subset H$, hence\n\\begin{equation*}\n  \\pair{\\vec f}{x^*}=P\\pair{\\vec f}{x^*}=\\pair{P\\vec f}{x^*},\n\\end{equation*}\nand thus $\\vec f=P\\vec f$. It follows that \n\\begin{equation}\\label{eq:tfg-vs-tfPg}\n  t(\\vec f,\\vec g)=  t(P\\vec f,\\vec g)=  t(\\vec f,P^t\\vec g)=  t(\\vec f,P\\vec g),\n\\end{equation}\nand similarly\n\\begin{equation}\\label{eq:caveg-vs-cavePg}\n\\begin{split}\n  \\cave{\\vec f}_{X}\\cdot\\cave{\\vec g}_{Y}\n  =P\\cave{\\vec f}_{X}\\cdot\\cave{\\vec g}_{Y} \n  =\\cave{\\vec f}_{X}\\cdot P^t\\cave{\\vec g}_{Y}\n =\\cave{\\vec f}_{X}\\cdot \\cave{P\\vec g}_{Y}.\n\\end{split}\n\\end{equation}\nSo it is enough to prove the claim with $P\\vec g$ in place of $\\vec g$, and hence we may assume without loss of generality that also $\\vec g=P\\vec g$. But then we can repeat the argument in the non-degenerate case, but with $\\mathbb{R}^n$ replaced by its subspace $H$ throughout; within this subspace, $\\mathcal E_K\\subset H$ is non-degenerate, and the previous case applies to give the desired result.\n\\end{proof}\n\nIn the following proposition, condition \\eqref{it:1scaleDomSca} is a typical intermediate step that is established in the course of proving a sparse domination result for an operator, while condition \\eqref{it:1scaleDomVec} is its convex body analogue. The proposition says that  \\eqref{it:1scaleDomSca} in fact implies \\eqref{it:1scaleDomVec}. It is essentially an abstraction (from $L^1$ averages to general dominating norms) of an idea already present in \\cite[Lemma 3.2]{NPTV:convex}.\n\n\\begin{proposition}\\label{prop:loc-sparse2convex}\nLet $X,Y$ be normed spaces with subspaces $F\\subset X$ and $G\\subset Y$.\nLet $Q_0\\in\\mathscr D$, and suppose that there are bilinear forms $t_Q:F\\times G\\to\\mathbb{R}$ indexed by all $Q\\in\\mathscr D(Q_0)$.\nConsider the following conditions:\n\\begin{enumerate}\n  \\item\\label{it:1scaleDomSca} For all $(f,g)\\in F\\times G$, there exist disjoint $\\hat Q_k\\subset Q_0$ with $\\sum_k\\abs{\\hat Q_k}\\leq\\eps\\abs{Q_0}$ and such that: whenever $Q_j\\subset Q_0$ are disjoint, not strictly contained in any $\\hat Q_k$, and cover all $\\hat Q_k$, then\n\\begin{equation*}\n  \\abs{t_{Q_0}(f,g)-\\sum_j t_{Q_j}(f,g)}\\leq C\\Norm{f}{X}\\Norm{g}{Y}.\n\\end{equation*}\n  \\item\\label{it:1scaleDomVec} For all $(\\vec f,\\vec g)\\in F^n\\times G^n $, there exist disjoint $Q_k\\subset Q_0$ with $\\sum_k\\abs{Q_k}\\leq\\eps_n\\abs{Q_0}$ and such that\n\\begin{equation*}\n  \\abs{t_{Q_0}(\\vec f,\\vec g)-\\sum_k t_{Q_k}(\\vec f,\\vec g)}\\leq C_n\\cave{\\vec f}_{X}\\cdot\\cave{\\vec g}_{Y}.\n\\end{equation*}\n\\end{enumerate}\nFor each $n\\in\\mathbb{Z}_+$, condition \\eqref{it:1scaleDomSca} implies condition \\eqref{it:1scaleDomVec} with $\\eps_n=n\\eps$ and $C_n=Cn^{3\/2}$.\n\\end{proposition}\n\nOf course, the condition $\\sum_k\\abs{Q_k}\\leq\\eps_n\\abs{Q_0}$ is only useful for $\\eps_n<1$. For a fixed $\\eps$ and $\\eps_n=n\\eps$, this would only allow us to conclude \\eqref{it:1scaleDomVec} for boundedly many values of $n$; so in order to obtain \\eqref{it:1scaleDomVec} for all $n\\in\\mathbb{N}$, we need \\eqref{it:1scaleDomSca} for arbitrarily small $\\eps>0$. This is seldom a problem in concrete situations.\n\n\\begin{proof}\nAs in the proof of Proposition \\ref{prop:loc-sparse2convex-pre}, given $\\vec f$, we consider the compact, convex, symmetric set \n\\begin{equation*}\n  K:=\\cave{\\vec f}_{X}, \n\\end{equation*}\nand denote by $\\mathcal E_K$ its John ellipsoid such that\n\\begin{equation*}\n  \\mathcal E_K\\subset K\\subset\\sqrt{n}\\mathcal E_K. \n\\end{equation*}\n\n\\subsubsection*{Case: $\\mathcal E_K$ is non-degenerate}\nLet $R_K$ be a linear transformation such that $R_K\\mathcal E_K=\\bar B_{\\mathbb{R}^n}$, the closed unit ball of $\\mathbb{R}^n$.\nLet $(\\vec e_i)_{i=1}^n$ be some orthonormal basis of $\\mathbb{R}^n$.\nAs in \\eqref{eq:tvecfvecg-pre}, we then write\n\\begin{equation}\\label{eq:tvecfvecg}\n\\begin{split}\n  t_{Q_0}(\\vec f,\\vec g)\n  &=t_{Q_0}(R_K^{-1}R_K\\vec f,\\vec g)\n  =t_{Q_0}(R_K\\vec f,R_K^{-t}\\vec g) \\\\\n  &=\\sum_{i=1}^n t_{Q_0}(R_K\\vec f\\cdot\\vec e_i,R_K^{-t}\\vec g\\cdot\\vec e_i)\n    =:\\sum_{i=1}^n t_{Q_0}(f_i,g_i),\n\\end{split}\n\\end{equation}\nwhere $f_i$ and $g_i$ are as in Lemma \\ref{lem:fi-vs-convex}.\n\nIt is from this point on that the present proof requires some elaboration compared to the proof of Proposition \\ref{prop:loc-sparse2convex-pre}. According to assumption \\eqref{it:1scaleDomSca}, for each of the pairs of functions $f_i:=R_K\\vec f\\cdot\\vec e_i$ and $g_i:=R_K^{-t}\\vec g\\cdot\\vec e_i$, we can find disjoint $\\hat Q_{i,k}\\subset Q_0$ with $\\sum_k\\abs{\\hat Q_{i,k}}\\leq\\eps\\abs{Q_0}$ and such that: whenever $Q_j\\subset Q_0$ are disjoint, not strictly contained in any $\\hat Q_{i,k}$, and cover all $\\hat Q_{i,k}$, then\n\\begin{equation}\\label{eq:tfigi}\n  \\abs{t_{Q_0}(f_i,g_i)-\\sum_j t_{Q_j}(f_i,g_i)}\\leq C\\Norm{f_i}{X}\\Norm{g_i}{Y}.\n\\end{equation}\nWe make the following specific choice of the cubes $Q_j$: Let $\\{Q_j\\}_{j=1}^\\infty$ be the maximal cubes among $\\{\\hat Q_{i,k}\\}_{1\\leq i\\leq n}^{1\\leq k<\\infty}$. Then\n\\begin{equation*}\n  \\sum_j\\abs{Q_j}\\leq\\sum_{i=1}^n\\sum_{k=1}^\\infty\\abs{\\hat Q_{i,k}}\n  \\leq\\sum_{k=1}^n\\eps\\abs{Q_0}\n  = n\\eps\\abs{Q_0},\n\\end{equation*}\nand \\eqref{eq:tfigi} holds with these $Q_j$ for each $i=1,\\ldots,n$. Using \\eqref{eq:tvecfvecg}, and observing that it also holds with $Q_0$ replaced by $Q_j$, it follows that\n\\begin{equation*}\n\\begin{split}\n   &\\abs{t_{Q_0}(\\vec f, \\vec g)-\\sum_j t_{Q_j}(\\vec f,\\vec g)}\n   \\leq\\sum_{i=1}^n \\abs{t_{Q_0}(f_i,g_i)-\\sum_j t_{Q_j}(f_i,g_i)} \\\\\n   &\\qquad\\leq C\\sum_{i=1}^n\\Norm{f_i}{X}\\Norm{g_i}{Y}\n   \\leq Cn^{3\/2}\\cave{\\vec f}_X\\cdot\\cave{\\vec g}_Y,\n\\end{split}\n\\end{equation*}\nusing Lemma \\ref{lem:fi-vs-convex} in the last step.\nThis completes the proof under the assumption that $\\mathcal E_K$ is non-degenerate.\n\n\\subsubsection*{Case: $\\mathcal E_K$ is degenerate}\nThis follows the corresponding case in the proof of Proposition \\ref{prop:loc-sparse2convex-pre} almost verbatim. Like there, let $H:=\\lspan K$, and let $P$ denote the orthogonal projection of $\\mathbb{R}^n$ onto $H$. We then have \\eqref{eq:tfg-vs-tfPg} for each $t=t_Q$, as well as \\eqref{eq:caveg-vs-cavePg}. So it is again enough to prove the claim with $P\\vec g$ in place of $\\vec g$, and hence we may assume without loss of generality that also $\\vec g=P\\vec g$. But then we can repeat the argument in the non-degenerate case, but with $\\mathbb{R}^n$ replaced by its subspace $H$ throughout; within this subspace, $\\mathcal E_K\\subset H$ is non-degenerate, and the previous case applies to give the desired result.\n\\end{proof}\n\n\n\\section{From single-scale bounds to global bounds}\n\nThis passage is by now a relatively routine part of the theory, but we include some details for completeness. The following lemma is again stated in an operator-free, and even function-free way, simply as a criterion for dominating a real number by sum over a sparse collection. A more concrete situation for applying this criterion is presented afterwards.\n\n\\begin{lemma}\\label{lem:loc2glob}\nConsider numbers $a\\in\\mathbb{R}$ and $a_Q,c_Q\\in\\mathbb{R}$ indexed by dyadic cubes $Q\\in\\mathscr D$, with the following properties:\n\\begin{enumerate}\n  \\item\\label{it:cover} There is a family $\\mathscr Q$ of disjoint dyadic cubes such that\n\\begin{equation*}\n   a=\\sum_{Q\\in\\mathscr Q}a_Q.\n\\end{equation*}\n  \\item\\label{it:local} For some $\\delta\\in(0,1)$ and each $Q\\in\\mathscr D$ that is contained in some $P\\in\\mathscr Q$, there is a family of disjoint $Q_k\\in\\mathscr D(Q)$ such that\n\\begin{equation*}\n   \\sum_k\\abs{Q_k}\\leq\\delta\\abs{Q},\\qquad\n   \\Babs{a_{Q}-\\sum_k a_{Q_k}}\\leq c_{Q}.\n\\end{equation*}\n  \\item\\label{it:limit} For some $\\alpha,C\\in[1,\\infty)$ and each $Q\\in\\mathscr D$ that is contained in some $P\\in\\mathscr Q$, we have $\\abs{a_Q}\\leq C\\abs{Q}^\\alpha$.\n\\end{enumerate}\nThen there is a $(1-\\delta)$-sparse family of dyadic cubes $\\mathscr S$ such that\n\\begin{equation*}\n   \\mathscr S\\subset\\bigcup_{Q\\in\\mathscr Q}\\mathscr D(Q),\\qquad  \\abs{a}\\leq \\sum_{S\\in\\mathscr S}c_S.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{remark}\\label{rem:loc2glob}\nIf $\\mathscr Q=\\{Q_0\\}$ consists of a single cube only, then condition \\eqref{it:cover} is automatic with $a=a_{Q_0}$.\n\\end{remark}\n\n\\begin{proof}\nLet $\\mathscr Q\\subset\\mathscr D$ be a disjoint collection provided by assumption \\eqref{it:cover}. For each $P\\in\\mathscr Q$, denote $\\mathscr S_0(P):=\\{P\\}$.\n Assuming that a disjoint $\\mathscr S_j(P)\\subset\\mathscr D(P)$ has already been constructed, for each $Q\\in\\mathscr S_j(P)$, let $\\mathscr S'(Q):=\\{Q_k\\}_{k=1}^\\infty$ be the collection provided by assumption \\eqref{it:local}, and let $\\mathscr S_{j+1}(P):=\\bigcup_{Q\\in\\mathscr S_j(P)}\\mathscr S'(Q)$. Let also \n  $\\mathscr S(P):=\\bigcup_{j=0}^\\infty\\mathscr S_j(P)$, and $\\mathscr S:=\\bigcup_{P\\in\\mathscr Q}\\mathscr S(P)$.\n\nFor $Q\\in\\mathscr S$, let $E(Q):=Q\\setminus\\bigcup_{R\\in\\mathscr S'(Q)}R$. From the construction it is clear that these sets $E(Q)$ are pairwise disjoint, and by assumption \\eqref{it:local} we have $\\abs{E(Q)}\\geq(1-\\delta)\\abs{Q}$.\n\nBy telescoping, for each $P\\in\\mathscr Q$, we have\n\\begin{equation*}\n  a_P=\\sum_{j=0}^{k-1}\\sum_{Q\\in\\mathscr S_j(P)}\\Big(a_Q-\\sum_{R\\in\\mathscr S'(Q)}a_R\\Big)+\\sum_{S\\in\\mathscr S_k(P)}a_S.\n\\end{equation*}\nand hence, using assumptions \\eqref{it:local} and \\eqref{it:limit},\n\\begin{equation*}\n  \\abs{a_P}\\leq\\sum_{j=1}^{k-1}\\sum_{Q\\in\\mathscr S_j(P)}c_Q+\\sum_{S\\in\\mathscr S_k(P)} C\\abs{S}^\\alpha\n\\end{equation*}\nBy an elementary inequality and induction, we have\n\\begin{equation*}\n  \\sum_{S\\in\\mathscr S_k(P)} \\abs{S}^\\alpha\n  \\leq\\Big(\\sum_{S\\in\\mathscr S_k(P)} \\abs{S}\\Big)^\\alpha\\leq(\\delta^k\\abs{P})^\\alpha,\n\\end{equation*}\nand hence\n\\begin{equation*}\n  \\abs{a_P}\\leq\\lim_{k\\to\\infty}\\sum_{j=1}^{k-1}\\sum_{Q\\in\\mathscr S_j(P)}c_Q=\\sum_{Q\\in\\mathscr S(P)}c_Q.\n\\end{equation*}\nSubstituting this into assumption \\eqref{it:cover}, we obtain the claim.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:easyConds}\nSuppose that $t$ is a bilinear form on $L^\\infty_c(\\mathbb{R}^d;E)\\times L^\\infty_c(\\mathbb{R}^d;H)$, and moreover bounded with respect to the norm of $L^p(\\mathbb{R}^d;E)\\times L^q(\\mathbb{R}^d;H)$ for some exponents with $1\/p+1\/q\\geq 1$. For $(\\vec f,\\vec g)\\in L^\\infty_c(\\mathbb{R}^d;E)^n\\times L^\\infty_c(\\mathbb{R}^d;H)^n$, the numbers\n\\begin{equation*}\n   a=t(\\vec f,\\vec g),\\quad a_Q=t(1_{3Q}\\vec f,1_Q \\vec g)\n\\end{equation*}\nsatisfy assumptions \\eqref{it:cover} and \\eqref{it:limit} of Lemma \\ref{lem:loc2glob}, provided that $\\mathscr D$ is a dyadic system without quadrants.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\mathscr D$ is without quadrants, each $Q\\in\\mathscr D$ is contained in some (large enough) $R\\in\\mathscr D$ that contains $\\supp \\vec f$. Thus the collection $\\mathscr Q$ of maximal cubes that do not contain $\\supp \\vec f$ form a cover of $\\mathbb{R}^d$. By maximality, it follows that $\\supp \\vec f\\subset 3Q$, and hence $\\vec f=1_{3Q}\\vec f$ for every $Q\\in\\mathscr Q$. On the other hand, any $Q$ with $\\ell(Q)<\\operatorname{diam}(\\supp \\vec f)$ cannot contain $\\supp\\vec f$; hence any $Q$  with $\\ell(Q)<\\frac12\\operatorname{diam}(\\supp \\vec f)$ cannot be among the maximal cubes $\\mathscr Q$, and thus every $Q\\in\\mathscr Q$ will have to satisfy $\\ell(Q)\\geq\\frac12\\operatorname{diam}\\vec f$. Since $\\vec g\\in L^\\infty_c(\\mathbb{R}^d;F)^n$, there are only finitely many $Q\\in\\mathscr Q$ with $1_Q \\vec g\\neq 0$. Hence, without any issues of convergence, we can write\n\\begin{equation*}\n  t(\\vec f,\\vec g)=t\\Big(\\vec f,\\sum_{Q\\in\\mathscr Q}1_Q \\vec g\\Big)=\\sum_{Q\\in\\mathscr Q}t(\\vec f,1_Q \\vec g)=\\sum_{Q\\in\\mathscr Q}t(1_{3Q}\\vec f,1_Q \\vec g),\n\\end{equation*}\nwhich is condition \\eqref{it:cover}.\n\nIf $n=1$, the assumed boundedness directly implies that\n\\begin{equation*}\n   \\abs{t(1_{3Q}f,1_Q g)}\\leq C\\Norm{1_{3Q}f}{L^p(\\mathbb{R}^d;E)}\\Norm{1_Q g}{L^{q}(\\mathbb{R}^d;F)}\n   \\leq C 3^{d\/p}\\Norm{f}{\\infty}\\Norm{g}{\\infty}\\abs{Q}^{1\/p+1\/q},\n\\end{equation*}\nwhere $\\alpha:=1\/p+1\/q\\geq 1$, as required for condition \\eqref{it:limit}. In general, if $(\\vec e_i)_{i=1}^n$ is an orthonormal basis of $\\mathbb{R}^n$ and $\\vec f=\\sum_{i=1}^n f_i\\vec e_i$ and similarly for $\\vec g$, we have\n\\begin{equation*}\n  \\abs{t(1_{3Q}\\vec f,1_Q \\vec g)}\\leq\\sum_{i=1}^n\\abs{t(1_{3Q}f_i,1_Q g_i)}\n   \\leq Cn 3^{d\/p}\\Norm{\\vec f}{\\infty}\\Norm{\\vec g}{\\infty}\\abs{Q}^{1\/p+1\/q},\n\\end{equation*}\nusing the previous bound in each component and trivial bounds like $\\Norm{f_i}{\\infty}\\leq\\Norm{\\vec f}{\\infty}$.\n\\end{proof}\n\nWe are finally ready to state a semi-generic convex body domination principle. Condition \\eqref{it:1scaleDomScaNew} below is a typical intermediate estimate in a number of sparse domination proofs for different operators. The conclusion is that it is already good enough to conclude convex body domination as well.\n\n\\begin{corollary}\\label{cor:1scale2cbd}\nLet $E$ and $H$ be Banach spaces, and suppose that $t$ is a bilinear form defined on $F\\times G:=L^\\infty_c(\\mathbb{R}^d;E)\\times L^\\infty_c(\\mathbb{R}^d;H)$ and bounded with respect to the norm of $L^p(\\mathbb{R}^d;E)\\times L^q(\\mathbb{R}^d;H)$ for some exponents with $1\/p+1\/q\\geq 1$, and suppose that \n\\begin{enumerate}\n  \\item\\label{it:1scaleDomScaNew} for all $(f,g)\\in F\\times G$ and all $Q\\in\\mathscr D$, there are disjoint $\\hat Q_k\\subset Q$ with $\\sum_k\\abs{\\hat Q_k}\\leq\\eps\\abs{Q}$ and such that: whenever $Q_j\\subset Q$ are disjoint, not strictly contained in any $\\hat Q_k$, and cover all $\\hat Q_k$, then\n\\begin{equation}\\label{eq:1scaleDomScaNew}\n  \\abs{t(1_{3Q}f,1_{Q}g)-\\sum_j t(1_{3Q_j}f,1_{Q_j}g)}\\leq c\\Norm{f}{X(Q)}\\Norm{g}{Y(Q)}\\abs{Q}\n\\end{equation}\nfor some norms $\\Norm{\\ }{X(Q)}$ on $L^\\infty_c(\\mathbb{R}^d;E)$ and $\\Norm{\\ }{Y(Q)}$ on $L^\\infty_c(\\mathbb{R}^d;H)$.\n\\end{enumerate}\nThen for all $(\\vec f,\\vec g)\\in F^n\\times G^n$, there is a $(1-\\eps_n)$-sparse collection $\\mathscr S\\subset\\mathscr D$ such that\n\\begin{equation*}\n   \\abs{t(\\vec f,\\vec g)}\\leq c_n\\sum_{S\\in\\mathscr S}\\cave{\\vec f}_{X(S)}\\cdot\\cave{\\vec g}_{Y(S)}\\abs{S},\n\\end{equation*}\nwhere $\\eps_n=n\\eps$ and $c_n=cn^{3\/2}$.\n\\end{corollary}\n\n\\begin{proof}\nLet us begin by considering a fixed cube $Q=Q_0\\in\\mathscr D$. We observe that assumption \\eqref{it:1scaleDomScaNew} of the present corollary coincides with condition \\eqref{it:1scaleDomSca} of  Proposition \\ref{prop:loc-sparse2convex} with\n\\begin{equation*}\n   t_Q(f,g):=t(1_{3Q}f,1_{Q}g),\\qquad C=c\\abs{Q},\\qquad X=X(Q),\\qquad Y=Y(Q).\n\\end{equation*}\nHence the said proposition, applied to each fixed $Q=Q_0\\in\\mathscr D$ at a time, implies:\n\\begin{enumerate}\\setcounter{enumi}{1}\n  \\item\\label{it:1scaleDomScaVecNew} For all $(\\vec f,\\vec g)\\in L^\\infty_c(\\mathbb{R}^d;E)^n\\times L^\\infty_c(\\mathbb{R}^d;H)^n$ and all $Q\\in\\mathscr D$, there are disjoint $Q_k\\subset Q$ with $\\sum_k\\abs{Q_k}\\leq\\eps_n\\abs{Q}$ and such that\n\\begin{equation*}\n  \\abs{t(1_{3Q}\\vec f,1_{Q}\\vec g)-\\sum_j t(1_{3Q_j}\\vec f,1_{Q_j}\\vec g)}\\leq c_n\\cave{\\vec f}_{X(Q)}\\cdot \\cave{g}_{Y(Q)}\\abs{Q},\n\\end{equation*}\nwhere $\\eps_n=n\\eps$ and $c_n=cn^{3\/2}$.\n\\end{enumerate}\n\nLet us then consider a fixed pair $(\\vec f,\\vec g)\\in L^\\infty_c(\\mathbb{R}^d;E)^n\\times L^\\infty_c(\\mathbb{R}^d;H)^n$. We observe that condition \\eqref{it:1scaleDomScaVecNew} above coincides with condition \\eqref{it:local} of Lemma \\ref{lem:loc2glob} with the choices\n\\begin{equation*}\n    a_Q=t(1_{3Q}\\vec f,1_{Q}\\vec g),\\qquad c_Q= c_n\\cave{\\vec f}_{X(Q)}\\cdot \\cave{g}_{Y(Q)}\\abs{Q},\\qquad \\delta=\\eps_n.\n\\end{equation*}\nOn the other hand, Lemma \\ref{lem:easyConds} shows that these same $a_Q$, together with $a:=t(\\vec f,\\vec g)$, also satisfy conditions \\eqref{it:cover} and \\eqref{it:limit} of Lemma \\ref{lem:loc2glob}. Thus, all assumptions, and hence the conclusions, of Lemma \\ref{lem:loc2glob} are valid for the said quantities, and these conclusions agree with the claims of the result that we are proving. The proof is thus complete.\n\\end{proof}\n\nTo facilitate the discussion of consequences of Corollary \\ref{cor:1scale2cbd}, we give\n\n\\begin{definition}\\label{def:cbd}\nSuppose that a pair of normed spaces $(X(Q),Y(Q))$ is associated to every dyadic cube $Q\\in\\mathscr D$.\nWe say that a bilinear form $t:F\\times G\\to\\mathbb{R}$ satisfies the $(X(Q),Y(Q))$ convex body domination of order $n\\in\\mathbb{N}$ if $F\\subseteq X(Q)$ and $G\\subseteq Y(Q)$ for every $Q\\in\\mathscr D$, and if for every $(f,g)\\in F^n\\times G^n$, there exists a $\\delta_n$-sparse collection $\\mathscr S\\subset\\mathscr D$ such that\n\\begin{equation*}\n    \\abs{t(\\vec f,\\vec g)}\\leq C_n\\sum_{Q\\in\\mathscr S}\\abs{Q}\\cave{\\vec f}_{X(Q)}\\cdot\\cave{\\vec g}_{Y(Q)}.\n\\end{equation*}\nWe say that $t:F\\times G\\to\\mathbb{R}$ satisfies the $(X(Q),Y(Q))$ convex body domination if it satisfies this for every $n\\in\\mathbb{N}$. We say that an operator $T:F\\to G^*$ satisfies these properties if its associated bilinear form $t(f,g):=\\pair{Tf}{g}$ does.\n\\end{definition}\n\n\nLet us now consider some examples:\n\n\\begin{example}[Calder\\'on--Zygmund operators]\\label{ex:CZO}\nLet $T$ be a Dini--Calder\\'on--Zygmund operator, i.e., $T$ is $L^2(\\mathbb{R}^d)$ bounded and has the representation\n\\begin{equation*}\n  Tf(x)=\\int_{\\mathbb{R}^d}K(x,y)f(y)\\ud y,\\qquad x\\notin\\supp f,\n\\end{equation*}\nwhere $\\abs{K(x,y)}\\leq c\\abs{x-y}^{-d}$ and, for $\\abs{x-x'}\\leq\\frac12\\abs{x-y}$,\n\\begin{equation}\\label{eq:CZomega}\n  \\abs{K(x,y)-K(x',y)}+\\abs{K(y,x)-K(y,x')}\\leq\\omega\\Big(\\frac{\\abs{x-x'}}{\\abs{x-y}}\\Big)\\frac{1}{\\abs{x-y}^d},\n\\end{equation}\nwhere $\\omega:[0,\\frac12]\\to[0,\\infty)$ is increasing, subadditive, and satisfies the Dini condition\n\\begin{equation*}\n  \\int_0^{1\/2}\\omega(t)\\frac{\\ud t}{t}<\\infty.\n\\end{equation*}\nThen \\eqref{it:1scaleDomScaNew} of Corollary \\ref{cor:1scale2cbd} holds for $t(f,g)=\\pair{Tf}{g}$ and $E=H=\\mathbb{R}$ and $X(Q)=\\avL^1(3Q)$, $Y(Q)=\\avL^1(Q)$, even in a stronger form. Namely, on the left oif \\eqref{eq:1scaleDomScaNew}, we have\n\\begin{equation}\\label{eq:Lerner}\n\\begin{split}\n     &\\Babs{\\Bpair{T(1_{3Q}f)}{1_{Q}g}-\\sum_j \\pair{T(1_{3Q_j}f)}{1_{Q_j}g}} \\\\\n     &\\leq\\BNorm{1_QT(1_{3Q}f)-\\sum_j 1_{Q_j}T(1_{3Q_j}f)}{L^\\infty(Q)}\\Norm{g}{L^1(Q)},\n\\end{split}\n\\end{equation}\nand even the $L^\\infty$ norm here is dominated by $\\Norm{f}{\\avL^1(3Q)}$, as essentially shown in \\cite[(3.4)]{Lerner:NYJM}. (Strictly speaking, \\cite[(3.4)]{Lerner:NYJM} is formally slightly weaker, but a straightforward modification of the argument gives the desired version, as observed in \\cite[Proof of Theorem 3.4]{NPTV:convex}.) Thus Corollary \\ref{cor:1scale2cbd} says that a Dini--Calder\\'on--Zygmund operator satisfies $(\\avL^1(3Q),\\avL^1(Q))$ convex body domination, but this was of course already known from \\cite{NPTV:convex} by essentially the same argument.\n\\end{example}\n\n\\begin{example}[Banach space -valued Calder\\'on--Zygmund operators]\\label{ex:CZO-Banach}\nLet $T$ be as in Example \\ref{ex:CZO} but now acting on the Bochner space $L^2(\\mathbb{R}^d;E)$ of Banach space $E$ -valued functions, and with an operator-valued kernel $K(x,y)\\in\\bddlin(E)$ satisfying the same estimates as above but for the operator norm in place of the absolute value, e.g., $\\Norm{K(x,y)}{\\bddlin(E)}\\leq c\\abs{x-y}^{-d}$. It is in general a difficult problem to check the $L^2(\\mathbb{R}^d;E)$-boundedness of such an operator, but we now take this as an assumption. For $g\\in L^2(\\mathbb{R}^d;E^*)$, we have \\eqref{eq:Lerner} with $L^\\infty(Q;E)$ and $L^1(Q;E^*)$ in place of $L^\\infty(Q)$ and $L^1(Q)$, and the same proof of \\cite[(3.4)]{Lerner:NYJM} (with same modifications pointed out in \\cite[Proof of Theorem 3.4]{NPTV:convex}) shows that the $L^\\infty(Q;E)$ norm is dominated by $\\Norm{f}{\\avL^1(3Q;E)}$. Thus we find that \\eqref{it:1scaleDomScaNew} of Corollary \\ref{cor:1scale2cbd} also holds with $X(Q)=\\avL^1(3Q;E)$ and $Y(Q)=\\avL^1(Q;E^*)$. The resulting sparse domination (i.e., case $n=1$ of the conclusion of Corollary \\ref{cor:1scale2cbd}) was known before, first in \\cite{HH:14} for a slightly smaller class of kernels, and since \\cite[discussion on page 193]{Lacey:elem} in the present generality. However, the convex body domination in this Banach space -valued setting is completely new.\n\\end{example}\n\n\n\\begin{example}[Operators with grand maximal function control]\\label{ex:grand}\nLet $1 \\leq q \\leq r$ and $s \\geq  1$. Suppose that $T$ is a linear operator\n\\begin{equation}\\label{eq:TLcL1loc}\n  T:L^\\infty_c(\\mathbb{R}^d)\\to L^1_{\\loc}(\\mathbb{R}^d),\n\\end{equation}\nthat $T$ has weak type $(q,q)$, and that the bi-sublinear maximal operator\n\\begin{equation*}\n  \\mathcal M_T(f,g)(x):=\\sup_{Q\\owns x} \\fint_Q \\abs{T(1_{(3Q)^c}f)}\\cdot\\abs{g}\n\\end{equation*}\nmaps boundedly $\\mathcal M_T:L^r\\times L^s\\to L^{\\nu,\\infty}$, where $1\/\\nu=1\/r+1\/s$. Then condition \\eqref{it:1scaleDomScaNew} of Corollary \\ref{cor:1scale2cbd} holds for $t(f,g)=\\pair{Tf}{g}$ and $E=H=\\mathbb{R}$ and $X(Q)=\\avL^r(3Q)$, $Y(Q)=\\avL^s(Q)$. This result is essentially contained in the proof of \\cite[Theorem 3.1]{Lerner:rough}, where it appears as an intermediate step towards the sparse domination (i.e., case $n=1$ of the conclusion of Corollary \\ref{cor:1scale2cbd}) for such operators. The extension to convex body domination was recently achieved in \\cite{MRR:22}, so Corollary \\ref{cor:1scale2cbd} only reproduces a known result here. A key example of concrete operators satisfying these assumptions consists of rough homogeneous singular integrals\n\\begin{equation*}\n  Tf(x)=\\int_{\\mathbb{R}^d}\\frac{\\Omega(y)}{\\abs{y}^d}f(x-y)\\ud y,\n\\end{equation*}\nwhere $\\Omega(y)=\\Omega(y\/\\abs{y})$ is a bounded function with vanishing average over the unit sphere.\n\nAs in Example \\ref{ex:CZO-Banach}, the abstract result above, involving a priori bounds of $T$ and $\\mathcal M_T$, extends straightforwardly to the Banach space -valued setting; however, verifying these bounds for concrete operators such as the rough homogeneous singular integrals may present a problem in this generality, since the scalar-valued versions depend on deep results of Seeger \\cite{Seeger:rough}, which so far lack a Banach space -valued extension.\n\\end{example}\n\n\n\\section{Matrix-weighted inequalities for Banach space -valued operators}\\label{sec:L2WE}\n\nA matrix weight is a locally integrable function $W:\\mathbb{R}^d\\to\\mathbb{R}^{n\\times n}$ that is a.e.\\ positive definite -valued. The space $L^p(W)$ consists of all measurable $\\vec f:\\mathbb{R}^d\\to\\mathbb{R}^n$ such that $W^{1\/p}\\vec f\\in L^p(\\mathbb{R}^d;\\mathbb{R}^n)$, and $\\Norm{\\vec f}{L^p(W)}:=\\Norm{W^{1\/p}\\vec f}{L^p(\\mathbb{R}^d;\\mathbb{R}^n)}$.\n\nFor a Banach space $E$, we extend this definition in a natural way: The space $L^p(W;E^n)$ consists of all measurable $\\vec f:\\mathbb{R}^d\\to E^n$ such that $W^{1\/p}\\vec f\\in L^p(\\mathbb{R}^d;E^n)$, and $\\Norm{\\vec f}{L^p(W;E^n)}:=\\Norm{W^{1\/p}\\vec f}{L^p(\\mathbb{R}^d;E^n)}$. Here, at each $x\\in\\mathbb{R}^d$, we define $(W^{1\/p}\\vec f)(x)\\in E^n$ as the vector with components $(W^{1\/p}\\vec f)_i(x):=\\sum_{j=1}^n (W^{1\/p}(x))_{ij}f_j(x)$, i.e., the matrix multiplication on $\\mathbb{R}^n$ is extended to $E^n$ in the natural way.\n\nWe now concentrate on $p=2$. For two matrix weights $W,V:\\mathbb{R}^d\\to\\mathbb{R}^{n\\times n}$, we define\n\\begin{equation*}\n  [W,V]_{A_2}:=\\sup_Q\\abs{\\ave{W}_Q^{1\/2}\\ave{V}_Q^{1\/2}}^2,\\qquad\n  [W]_{A_2}:=[W,W^{-1}]_{A_2},\n\\end{equation*}\nwhere we denote the operator norm in $\\mathbb{R}^{n\\times n}\\simeq\\bddlin(\\mathbb{R}^n)$ simply by $\\abs{\\ }$. We denote by $A_2(\\mathbb{R}^d;\\mathbb{R}^n)$ the class of matrix weights $W:\\mathbb{R}^d\\to\\mathbb{R}^{n\\times n}$ for which $[W]_{A_2}<\\infty$. We also define\n\\begin{equation*}\n  [W]_{A_\\infty}:=\\sup_{\\vec e\\in\\mathbb{R}^n}[x\\mapsto \\vec e\\cdot W(x)\\vec e]_{A_\\infty},\n\\end{equation*}\nwhere on the right we have $A_\\infty$ ``norms'' of some scalar weights, defined as usual by\n\\begin{equation*}\n  [w]_{A_\\infty}:=\\sup_Q\\frac{1}{w(Q)}\\int_Q M(1_Q w).\n\\end{equation*}\nAccording to \\cite[Remark 4.4]{NPTV:convex}, we have\n\\begin{equation}\\label{eq:Ainfty-vs-A2}\n  [W]_{A_\\infty}\\leq 4[W]_{A_2}.\n\\end{equation}\n\nAs a consequence of the Banach space -valued convex body domination from Example \\ref{ex:CZO-Banach}, we obtain:\n\n\\begin{theorem}\\label{thm:L2WE}\nLet $E$ be a Banach space, and $T\\in\\bddlin(L^2(\\mathbb{R}^d;E))$ be a Dini--Calder\\'on--Zygmund operator with $\\bddlin(E)$-valued kernel. For any $W\\in A_2(\\mathbb{R}^d;\\mathbb{R}^n)$, the operator $T$ extends boundedly to $L^2(W;E^n)$ and satisfies\n\\begin{equation*}\n  \\Norm{T}{\\bddlin(L^2(W;E^n))}\\leq c_{n,T}([W]_{A_2}[W]_{A_\\infty}[W^{-1}]_{A_\\infty})^{1\/2}\\leq c_{n,T}[W]_{A_2}^{3\/2}.\n\\end{equation*}\n\\end{theorem}\n\nNote that Theorem \\ref{thm:L2WE} applies to a general Banach space $E$, but contains the (difficult) a priori boundedness hypothesis that $T\\in\\bddlin(L^2(\\mathbb{R}^d;E))$. Concrete examples are available in the class of UMD spaces, treated in detail in \\cite{HNVW}.\n\n\\begin{corollary}\\label{cor:L2WE}\nLet $E$ be a UMD space, and $T\\in\\bddlin(L^2(\\mathbb{R}^d))$ be a scalar-valued Calder\\'on--Zygmund operator with a H\\\"older-type modulus of continuity $\\omega(t)=c t^\\delta$, $\\delta\\in(0,1]$ in \\eqref{eq:CZomega}. For any $W\\in A_2(\\mathbb{R}^d;\\mathbb{R}^n)$, the operator $T$ extends boundedly to $L^2(W;E^n)$ and satisfies\n\\begin{equation*}\n  \\Norm{T}{\\bddlin(L^2(W;E^n))}\\leq c_{n,E,T}([W]_{A_2}[W]_{A_\\infty}[W^{-1}]_{A_\\infty})^{1\/2}\\leq c_{n,E,T}[W]_{A_2}^{3\/2}.\n\\end{equation*}\nIn particular, this estimate holds when $T$ is the classical Hilbert transform.\n\\end{corollary}\n\n\\begin{proof}\nWe reduce Corollary \\ref{cor:L2WE} to Theorem \\ref{thm:L2WE} with the help of the $T(1)$ theorem of David and Journ\\'e \\cite{DJ:T1}, and its extension to UMD spaces by Figiel \\cite{Figiel:T1}. By the (easy half of) the David--Journ\\'e theorem, the assumptions on $T$ imply that that $T$ satisfies the so-called weak boundedness property as well as $T(1),T^*(1)\\in\\BMO(\\mathbb{R}^d)$. Then, by Figiel's theorem, an operator satisfying these conditions and the Calder\\'on--Zygmund kernel assumptions extends boundedly to $L^2(\\mathbb{R}^d;E)$, for any UMD space $E$. Thus $T$ satisfies the assumptions, and hence the conclusions, of Theorem \\ref{thm:L2WE}, and we are done.\n\\end{proof}\n\nThese results, even just for the Hilbert transform, and even in their qualitative form (i.e., just concluding the boundedness of $T$, without specifying any concrete bound for the norm), are completely new in the combined setting of matrix weights and Banach spaces. For $E=\\mathbb{R}$ and the Hilbert transform $T$, the qualitative form of Corollary \\ref{cor:L2WE} is due to Treil and Volberg \\cite{TV:angle}. The quantitative form for $E=\\mathbb{R}$ was obtained by Nazarov et al.\\ \\cite{NPTV:convex}, and this is the best that is known at the time of writing. For scalar-weights, the power $3\/2$ can be replaced by $1$ \\cite{Hytonen:A2}, and the product of $[W]_{A_\\infty}$ and $[W^{-1}]_{A_\\infty}$ by their sum \\cite{HytPer}, but extending these to the general matrix case consists of the outstanding open ``matrix $A_2$ conjecture''.\n\nTurning to the proof of Theorem \\ref{thm:L2WE}, we begin with:\n\n\\begin{remark}[Without loss of generality, we assume that $E$ is reflexive]\\label{rem:RNP}\nSince Theorem \\ref{thm:L2WE} is about the bounded extension of an operator, it suffices to prove an a priori estimate on a dense subspace of functions $\\vec f$. In particular, we can assume that each component $f_i$ takes its values in a finite-dimensional subspace of $E$. Since any finite-dimensional space is reflexive, {\\em we make the standing assumption, without loss of generality, that $E$ is reflexive}. (Note that this is automatic in Corollary \\ref{cor:L2WE} in any case, since UMD spaces are reflexive \\cite[Theorem 4.3.3]{HNVW}.) Under this assumption, we have $L^1(Q;E)^*=L^\\infty(Q;E^*)$ (see \\cite[Theorems 1.3.10 and 1.3.21]{HNVW}), which is convenient in view of calculations involving the convex bodies $\\cave{\\ }_{\\avL^1(Q;E)}$.\n\\end{remark}\n\n\\begin{lemma}\n\\begin{equation*}\n\\begin{split}\n  \\abs{Q} &\\cave{W^{1\/2}\\vec f}_{\\avL^1(3Q;E)}\\cdot \\cave{V^{1\/2}\\vec g}_{\\avL^1(Q;E^*)} \\\\\n  &\\leq \\int \\Big(1_Q(x)\\fint_{3Q}\\abs{V^{1\/2}(x)W^{1\/2}(y)}\\Norm{\\vec f(y)}{E^n}\\ud y\\Big)\\Norm{\\vec g(x)}{\\vec E^{*n}}\\ud x\n\\end{split}\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nUnder the standing assumption from Remark \\ref{rem:RNP}, we evaluate consider a generic element of the convex body on the left with $\\phi\\in\\bar B_{L^\\infty(Q;E^*)}$ and $\\psi\\in\\bar B_{L^\\infty(Q;E)}$:\n\\begin{equation*}\n\\begin{split}\n  &\\abs{Q}\\Babs{\\fint_{3Q} W^{1\/2}(y)\\pair{\\vec f(y)}{\\phi(y)}\\ud y\\cdot \\fint_Q V^{1\/2}(x)\\pair{\\vec g(x)}{\\psi(x)}\\ud x} \\\\\n  &=\\abs{Q}\\Babs{\\fint_Q\\fint_{3Q} V^{1\/2}(x)W^{1\/2}(y)\\pair{\\vec f(y)}{\\phi(y)}\\cdot\\pair{\\vec g(x)}{\\psi(x)}\\ud y\\ud x} \\\\\n  &\\leq\\int_Q\\fint_{3Q} \\abs{V^{1\/2}(x)W^{1\/2}(y)}\\Norm{\\vec f(y)}{E^n}\\Norm{\\vec g(x)}{\\vec E^{*n}}\\ud y\\ud x.\\qedhere\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\nSumming over a sparse collection, we obtain\n\\begin{equation}\\label{eq:cbd-vs-L}\n\\begin{split}\n   &\\sum_{Q\\in\\mathscr S} \\abs{Q} \\cave{W^{1\/2}\\vec f}_{\\avL^1(3Q;E)}\\cdot \\cave{V^{1\/2}\\vec g}_{\\avL^1(Q;E^*)}  \\\\\n   &\\leq \\int\\Big(\\sum_{Q\\in\\mathscr S}1_Q(x)\\fint_{3Q}\\abs{V^{1\/2}(x)W^{1\/2}(y)}\\Norm{\\vec f(y)}{E^n}\\ud y\\Big)\\Norm{\\vec g(x)}{\\vec E^{*n}}\\ud x \\\\\n   &=:\\int \\tilde L(\\Norm{\\vec f}{E^n})(x)\\Norm{\\vec g(x)}{\\vec E^{*n}}\\ud x,\n\\end{split}\n\\end{equation}\nwhere $\\tilde L$, here acting on the scalar-valued function $y\\mapsto\\Norm{\\vec f(y)}{E^n}$, is an operator denoted by the same symbol in \\cite[(5.8)]{NPTV:convex}. By \\cite[Lemma 5.6]{NPTV:convex}, we have\n\\begin{equation}\\label{eq:NPTV-L}\n  \\Norm{\\tilde L}{\\bddlin(L^2)}\\leq C([W,V]_{A_2}[W]_{A_\\infty}[V]_{A_\\infty})^{1\/2}.\n\\end{equation}\n\nBy duality and standard changes of variables, which present no essential difference in the Banach space -valued setting, an estimate of the form\n\\begin{equation*}\n  \\Norm{T\\vec f}{L^2(V;E^n)}\\leq N\\Norm{\\vec f}{L^2(V;E^n)}\n\\end{equation*}\nis equivalent to\n\\begin{equation}\\label{eq:dualBound2prove}\n  \\pair{T(W^{1\/2}\\vec f)}{V^{1\/2}\\vec g}\\leq N\\Norm{\\vec f}{L^2(\\mathbb{R}^d;E^n)}\\Norm{\\vec g}{L^2(\\mathbb{R}^d;E^{*n})}.\n\\end{equation}\nIf $T$ is an in Theorem \\ref{thm:L2WE}, it satisfies the $(\\avL^1(3Q;E),\\avL^1(Q;E^*))$ convex body domination by Example \\ref{ex:CZO-Banach}, which means that the left-hand side of \\eqref{eq:dualBound2prove} is dominated by the left-hand side of \\eqref{eq:cbd-vs-L}, and hence, by \\eqref{eq:cbd-vs-L} and \\eqref{eq:NPTV-L}, we have\n\\begin{equation*}\n  N\\leq c_{n,T}([W,V]_{A_2}[W]_{A_\\infty}[V]_{A_\\infty})^{1\/2}.\n\\end{equation*}\nThis is the desired bound, and concludes the proof of Theorem \\ref{thm:L2WE}.\n\n\\section{Convex domination and generalised commutators}\\label{sec:commu}\n\nFor an operator $T$ and two vector functions $\\vec a=(a_1,\\ldots,a_n)$ and $\\vec b=(b_1,\\ldots,b_n)$, let us consider the operator\n\\begin{equation*}\n  \\vec a\\cdot T\\vec b:f\\mapsto\\vec a\\cdot T(\\vec b f)=\\sum_{i=1}^n a_i T(b_i f).\n\\end{equation*}\nWe are mainly interested in the boundedness on $L^p(\\mathbb{R}^d)$, or a weighted $L^p(w)$, or between two such spaces, and the case when $T$ is a singular integral operator bounded on the space. However, we do not require that $a_i,b_i\\in L^\\infty(\\mathbb{R}^d)$, and hence the pointwise multipliers $f\\mapsto b_i f$ and $g\\mapsto a_i g$, and the compositions $f\\mapsto a_iT(b_i f)$, may be unbounded operators. Nevertheless, their sum $\\vec a\\cdot T\\vec b$ may still be bounded, thanks to cancellation between different terms.\n\nA case that has been much studied in the literature consists of $\\vec b=(1,b)$ and $\\vec a=(b,-1)$, in which case\n\\begin{equation*}\n  \\vec a\\cdot T(\\vec b f)= bTf-T(bf)=[b,T]f\n\\end{equation*}\nis the {\\em commutator} of $b$ and $T$, whose $L^p(\\mathbb{R}^d)$-boundedness is characterised by $b\\in\\BMO(\\mathbb{R}^d)$, the space of functions of bounded mean oscillation, which is strictly larger than $L^\\infty(\\mathbb{R}^d)$, and contains in particular functions like $b(x)=\\log\\abs{x}$.\n\nBy dualising with a function $g$, and denoting by $t(f,g)=\\pair{Tf}{g}$ the bilinear form of $T$, we arrive at\n\\begin{equation*}\n  \\pair{\\vec a\\cdot T(\\vec b f)}{g}=\\sum_{i=1}^n\\pair{T(b_i f)}{a_i g}\n  =t(\\vec b f,\\vec a g),\n\\end{equation*}\nwhere the action of the bilinear form is extended to vector-valued functions as before. To be precise, if $t$ in defined on $F\\times G$, we should now require that\n\\begin{equation*}\n  f\\in F_{\\vec b}:=\\{f\\in F: b_i f\\in F\\text{ for all }i=1,\\ldots,n\\},\n\\end{equation*}\nand $g\\in G_{\\vec a}$, defined similarly. If $F\\supseteq L^\\infty_c(\\mathbb{R}^d)$, then clearly $F_{\\vec b}$ contains in particular all $f\\in L^\\infty_c(\\mathbb{R}^d)$ with $\\supp f\\subseteq E_N:=\\{\\abs{\\vec b}\\leq N\\}$ for any $N\\in\\mathbb{N}$. For a.e.\\ finite-valued $b_i$, the union $\\bigcup_{N\\in\\mathbb{N}}E_N$ covers $\\mathbb{R}^d$ up to a null set, it is immediate that $F_{\\vec b}$ is dense in any $L^p(w)$ with finite $p$. \n\n\n\\begin{lemma}\\label{lem:cbd-implies}\nSuppose that $T$ satisfies the $(X(Q),Y(Q))$ convex body domination. Then for all relevant functions, we have\n\\begin{equation}\\label{eq:cbd-implies}\n  \\abs{\\pair{\\vec{a}\\cdot T(\\vec{b}f)}{g}}\n  \\leq C\\sum_{Q\\in\\mathscr S}\\abs{Q}\\cave{\\vec b f}_{X(Q)}\\cdot\\cave{\\vec a g}_{Y(Q)}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nThis is immediate by applying definition to $\\vec f=\\vec b f$ and $\\vec g=\\vec a g$.\n\\end{proof}\n\nWe take a closer look at the case when $X(Q)=Y(Q)=\\avL^1(\\gamma Q)$.\n\n\\begin{lemma}\\label{lem:cbdL1-further}\nFor all $s,t\\in(1,\\infty)$ and all functions in the relevant spaces, we have\n\\begin{equation*}\n  \\cave{\\vec b f}_{\\avL^1(Q)}\\cdot\\cave{\\vec a g}_{\\avL^1(Q)}\n  \\leq\\Norm{(x,y)\\mapsto\\vec a(x)\\cdot\\vec b(y)}{\\avL^{(s,t)}_{\\min}(Q\\times Q)}\n    \\Norm{f}{\\avL^{t'}(Q)}\\Norm{g}{\\avL^{s'}(Q)},\n\\end{equation*}\nwhere\n\\begin{equation*}\n  \\Norm{F}{\\avL^{(s,t)}_{\\min}(Q\\times Q)}\n  :=\\begin{cases} \\Big(\\fint_Q\\Big[\\fint_Q\\abs{F(x,y)}^s\\ud x\\Big]^{t\/s}\\ud y\\Big)^{1\/t}, & \\text{if }s\\leq t, \\\\\n   \\Big(\\fint_Q\\Big[\\fint_Q\\abs{F(x,y)}^t\\ud y\\Big]^{s\/t}\\ud x\\Big)^{1\/s}, & \\text{if }t\\leq s. \\end{cases} \n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nThe generic element of $\\cave{\\vec b f}_{X(Q)}\\cdot\\cave{\\vec a g}_{Y(Q)}$ has the following form, where $\\phi,\\psi\\in\\bar B_{L^\\infty(Q)}$:\n\\begin{equation*}\n\\begin{split}\n  &\\fint_Q \\vec b(y)f(y)\\phi(y)\\ud y\\cdot\\fint_Q \\vec a(x)g(x)\\psi(x)\\ud x \\\\\n  &=\\fint_Q\\fint_Q(\\vec a(x)\\cdot\\vec b(y))f(y)g(x)\\phi(y)\\psi(x)\\ud x\\ud y,\n\\end{split}\n\\end{equation*}\nand hence\n\\begin{equation*}\n\\begin{split}\n  &\\cave{\\vec b f}_{X(Q)}\\cdot\\cave{\\vec a g}_{Y(Q)}\n  \\leq\\fint_Q\\fint_Q\\abs{\\vec a(x)\\cdot\\vec b(y)} \\abs{f(y)}\\abs{g(x)}\\ud x\\ud y \\\\\n  &\\leq\\Norm{(x,y)\\mapsto a(x)\\cdot b(y)}{Z}\\Norm{(x,y)\\mapsto f(y)g(x)}{Z^*},\n\\end{split}\n\\end{equation*}\nfor either choice of\n\\begin{equation*}\n  (Z,Z^*)\\in\\{(\\avL^s_x(Q;\\avL^t_y(Q)),\\avL^{s'}_x(Q;\\avL^{t'}_y(Q))),\n  (\\avL^t_y(Q;\\avL^s_x(Q)),\\avL^{t'}_y(Q;\\avL^{s'}_x(Q)))\\},\n\\end{equation*}\nby H\\\"older's inequality for mixed-norm $L^p$ spaces. By Fubini's theorem, we have\n\\begin{equation*}\n   \\Norm{(x,y)\\mapsto f(x)g(y)}{Z^*}\n   =\\Norm{f}{\\avL^{t'}(Q)}\\Norm{g}{\\avL^{s'}(Q)}\n\\end{equation*}\nin either case, and hence, taking the minimum over the two choices of $Z$, we arrive at the factor\n\\begin{equation*}\n  \\min_Z\\Norm{(x,y)\\mapsto b(x)\\cdot a(y)}{Z}=\\Norm{(x,y)\\mapsto\\vec b(x)\\cdot\\vec a(y)}{\\avL^{(s,t)}_{\\min}(Q\\times Q)}.\n\\end{equation*}\n\\end{proof}\n\n\\begin{proposition}\\label{prop:gen-commu}\nLet $T$ be an operator that satisfies the $(\\avL^1(\\gamma Q),\\avL^1(\\gamma Q))$ convex body domination. Let $\\vec a,\\vec b\\in L^1_{\\loc}(\\mathbb{R}^d)^n$ be functions such that\n\\begin{equation*}\n  A_{s,t}:=\\sup_{Q}\\Norm{(x,y)\\mapsto \\vec a(x)\\cdot\\vec b(y)}{\\avL^{(s,t)}_{\\min}(Q\\times Q)}<\\infty.\n\\end{equation*}\nThen $\\vec a\\cdot T\\vec b$ extends to a bounded operator on $L^p(\\mathbb{R}^d)$ for all $p\\in(t',s)$. In particular, if $A_s:=A_{s,s}<\\infty$ for some $s\\in(2,\\infty)$, then $\\vec a\\cdot T\\vec b$ extends boundedly to $L^2(\\mathbb{R}^d)$.\n\\end{proposition}\n\n\\begin{proof}\nCombining Lemmas \\ref{lem:cbd-implies} and \\ref{lem:cbdL1-further}, we find that\n\\begin{equation*}\n\\begin{split}\n  \\abs{\\pair{\\vec aT(\\vec b f)}{g}}\n  &\\leq C\\sum_{Q\\in\\mathscr S}\\abs{Q}\\cave{\\vec b f}_{\\avL^1(\\gamma Q)}\\cdot\\cave{\\vec a g}_{\\avL^1(\\gamma Q)} \\\\\n  &\\leq C\\sum_{Q\\in\\mathscr S}\\abs{Q}\\Norm{(x,y)\\mapsto a(x)\\cdot b(y)}{\\avL^{(s,t)}_{\\min}(Q\\times Q)}\n     \\Norm{f}{\\avL^{t'}(Q)}\\Norm{g}{\\avL^{s'}(Q)} \\\\\n   &\\leq C\\sum_{Q\\in\\mathscr S}\\frac{\\abs{E(Q)}}{\\delta}A_{s,t}\\inf_Q M_{t'}f\\inf_Q M_{s'}g \\\\\n   &\\leq\\frac{C A_{s,t}}{\\delta}\\sum_{Q\\in\\mathscr S}\\int_{E(Q)}M_{t'}f M_{s'}g\n   \\leq\\frac{C A_{s,t}}{\\delta}\\int_{\\mathbb{R}^d}M_{t'}f M_{s'}g \\\\\n   &\\leq\\frac{C A_{s,t}}{\\delta}\\Norm{M_{t'}f}{L^p(\\mathbb{R}^d)}\\Norm{M_{s'}g}{L^{p'}(\\mathbb{R}^d)},\n\\end{split}\n\\end{equation*}\nwhere\n\\begin{equation*}\n  \\Norm{M_{t'}f}{L^p(\\mathbb{R}^d)}\\lesssim_{t,p}\\Norm{f}{L^p(\\mathbb{R}^d)},\\qquad\n  \\Norm{M_{s'}g}{L^{p'}(\\mathbb{R}^d)}\\lesssim_{s,p}\\Norm{g}{L^{p'}(\\mathbb{R}^d)}\n\\end{equation*}\nfor $p>t'$ and $p'>s'$, where the latter is equivalent to $p<s$.\n\\end{proof}\n\n\nLet us consider some examples:\n\n\\begin{example}[Classical commutators]\\label{ex:commu} \nAs we already observed, $\\vec a=(b,-1)$ and $\\vec b=(1,b)$ gives rise to the usual commutator $[b,T]$. In this case\n\\begin{equation*}\n  \\vec a(x)\\cdot\\vec b(y)= b(x)-b(y)\n\\end{equation*}\nand each $A_{s,t}$ is equivalent to $\\Norm{b}{\\BMO(\\mathbb{R}^d)}$ by elementary considerations and the John--Nirenberg inequality. Thus Proposition \\ref{prop:gen-commu} reproduces the well-known sufficient condition for the boundedness of commutators.\n\\end{example}\n\n\\begin{example}[Iterated commutators]\\label{ex:iter-commu} More generally, choosing $\\vec a,\\vec b$ so that\n\\begin{equation*}\n   \\vec a(x)\\cdot\\vec b(y)= (b(x)-b(y))^k=\\sum_{i=0}^k\\binom{k}{i}b(x)^{k-i}(-b(y))^i,\n\\end{equation*}\nthus e.g. $a_i(x)=\\binom{k}{i}b(x)^{k-i}$ and $b_i(y)=(-b(y))^i$, we reproduce the $k$th order commutator\n\\begin{equation*}\n  \\vec a\\cdot T\\vec b=T_{k,b}:=[b,T_{k-1,b}],\\qquad T_{0,b}:=T,\n\\end{equation*}\nand $A_{s,t}$ is equivalent to $\\Norm{b}{\\BMO(\\mathbb{R}^d)}^k$ by the John--Nirenberg inequality.\n\\end{example}\n\n\\begin{example}[Iterated commutators with different multipliers]\\label{ex:mixed-commu}\nLet us then choose $\\vec a,\\vec b$ so that\n\\begin{equation*}\n   \\vec a(x)\\cdot\\vec b(y)= (b^1(x)-b^1(y))(b^2(x)-b^2(y));\n\\end{equation*}\nwithout specifying the precise choice of $a_i(x)$ and $b_i(y)$, it is evident that such a choice can be easily written down, if desired. (We deliberately use superscript indices for $b^i$ above, since these not be the same as the components $b_i$ of $\\vec b$.) This reproduces the second order iterated commutator with two different functions,\n\\begin{equation*}\n   \\vec a\\cdot T\\vec b=[b^1,[b^2,T]].\n\\end{equation*}\nIt is well-known and classical that $b^i\\in\\BMO(\\mathbb{R}^d)$ for both $i=1,2$ is sufficient for the $L^2(\\mathbb{R}^d)$ boundedness of $[b^1,[b^2,T]]$; however, as recently observed in \\cite{HLO:20}, much weaker sufficient conditions can be given for the pair $(b^1,b^2)$. Namely, in \\cite[(1.1)]{HLO:20}, it shown that the pair of conditions\n\\begin{equation*}\n\\begin{split}\n  S_s &:=\\sup_Q\\Big(\\fint_Q\\abs{b^1(x)-\\ave{b^1}_Q}^s\\ud x\\Big)^{1\/s}\\Big(\\fint_Q\\abs{b^2(y)-\\ave{b^2}_Q}^s\\ud y\\Big)^{1\/s}<\\infty, \\\\\n  T_s &:=\\sup_Q\\Big(\\fint_Q\\abs{b^1(x)-\\ave{b^1}_Q}^s\\abs{b^2(x)-\\ave{b^2}_Q}^s\\ud x\\Big)^{1\/s}<\\infty,\n\\end{split}\n\\end{equation*}\nis sufficient for the $L^2(\\mathbb{R}^d)$ boundedness of $[b^1,[b^2,T]]$ for $s>2$. On the other hand, by Proposition \\ref{prop:gen-commu}, another sufficient condition for the same conclusion is $A_s<\\infty$.\n\nLet us compare the two. Adding and subtracting terms and multiplying out, we find that\n\\begin{equation*}\n\\begin{split}\n  &(b^1(x)-b^1(y))(b^2(x)-b^2(y)) \\\\\n  &=[(b^1(x)-\\ave{b^1}_Q)-(b^1(y)-\\ave{b^1}_Q)][(b^2(x)-\\ave{b^2}_Q)-(b^2(y)-\\ave{b^2}_Q)] \\\\\n  &=(b^1(x)-\\ave{b^1}_Q)(b^2(x)-\\ave{b^2}_Q)+(b^1(y)-\\ave{b^1}_Q)(b^2(y)-\\ave{b^2}_Q) \\\\\n  &\\qquad-(b^1(x)-\\ave{b^1}_Q)(b^2(y)-\\ave{b^2}_Q)-(b^1(y)-\\ave{b^1}_Q)(b^2(x)-\\ave{b^2}_Q).\n\\end{split}\n\\end{equation*}\nTaking $\\avL^s(Q\\times Q)$ and then supremum over $Q$ on both sides, we deduce that\n\\begin{equation*}\n  A_s\\leq 2(T_s+S_s),\n\\end{equation*}\nso that the new criterion provided by Proposition \\ref{prop:gen-commu} is at least as sharp as that of \\cite[(1.1)]{HLO:20}, and it seems less obvious to make any estimate in the other direction. Perhaps more importantly, the new condition $A_s<\\infty$ arises more ``naturally'' as an instance of a general principle.\n\n(Let us note that there is a more general criterion \\cite[Theorem 3.10]{HLO:20}, where the $\\avL^s$ norms in $S_s$ and $T_t$ are replaced by more general Orlicz norms. On the other hand, it is apparent that similar generalisations could be achieved in Proposition \\ref{prop:gen-commu}: what we used was the boundedness of the rescaled maximal operators $M_{t'}$ on $L^p(\\mathbb{R}^d)$ for $p>t'$, and this could be replaced having an Orlicz maximal operator $M_A$ with the same mapping property. A characterisation of this property in terms of the so-called $B_p$ condition on the Orlicz function $A$ is a classical result of P\\'erez \\cite{Perez:95}; this very result is used in \\cite{HLO:20}; see \\cite[Proposition 3.8]{HLO:20}.)\n\\end{example}\n\nLet us finally consider an ``exotic'' example with no obvious predecessor in the existing literature. We begin with a lemma:\n\n\\begin{lemma}\\label{lem:BMOpowers}\nSuppose that $0\\leq b\\in\\BMO(\\mathbb{R}^d)$. If $0\\leq\\alpha,\\beta$ and $\\alpha+\\beta\\leq 1$, then\n\\begin{equation*}\n  B(x,y):=b(x)^\\alpha b(y)^\\beta - b(x)^\\beta b(y)^\\alpha\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n  \\Big(\\fint_Q\\fint_Q\\abs{B(x,y)}^p\\ud x\\ud y\\Big)^{1\/p} \\leq (2\\Norm{b}{\\BMO^p(\\mathbb{R}^d)})^{\\alpha+\\beta}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\gamma:=\\min(\\alpha,\\beta)\\in[0,\\frac12]$ and $\\delta:=\\max(\\alpha,\\beta)-\\gamma\\in[0,1]$. Then\n\\begin{equation*}\n  \\abs{B(x,y)}=b(x)^\\gamma b(y)^\\gamma \\abs{b(x)^\\delta-b(y)^\\delta}.\n\\end{equation*}\nWe observe the following elementary inequality:\n\\begin{equation}\\label{eq:ud-vd}\n  \\abs{u^\\delta-v^\\delta}\\leq\\frac{\\abs{u-v}}{\\max(u,v)^{1-\\delta}},\\qquad\\forall u,v\\geq 0,\\ \\delta\\in[0,1].\n\\end{equation}\nIndeed, by symmetry and homogeneity, it is enough to consider $u=1$ and $v\\in[0,1]$, in which case we are reduced to proving that\n\\begin{equation*}\n  1-v^\\delta\\leq 1-v,\n\\end{equation*}\nwhich is immediate from the fact that $v\\leq v^\\delta$ for $v,\\delta\\in[0,1]$.\n\nUsing \\eqref{eq:ud-vd}, and noting that $\\delta+2\\gamma=\\alpha+\\beta\\in[0,1]$, it follows that\n\\begin{equation*}\n\\begin{split}\n  \\abs{B(x,y)} &\\leq b(x)^\\gamma b(y)^\\gamma\\frac{\\abs{b(x)-b(y)}}{\\max(b(x),b(y))^{1-\\delta}}  \n  \\leq \\frac{\\abs{b(x)-b(y)}}{\\max(b(x),b(y))^{1-\\delta-2\\gamma}} \\\\\n  &= \\Big(\\frac{\\abs{b(x)-b(y)}}{\\max(b(x),b(y))}\\Big)^{1-\\delta-2\\gamma}\\abs{b(x)-b(y)}^{\\delta+2\\gamma}\n  \\leq\\abs{b(x)-b(y)}^{\\alpha+\\beta},\n\\end{split}\n\\end{equation*}\nand hence\n\\begin{equation*}\n\\begin{split}\n  \\Big( &\\fint_Q\\fint_Q\\abs{B(x,y)}^p\\ud x\\ud y\\Big)^{1\/p}\n  \\leq \\Big(\\fint_Q\\fint_Q\\abs{b(x)-b(y)}^p\\ud x\\ud y\\Big)^{(\\alpha+\\beta)\/p} \\\\\n  &\\leq\\Big[\\Big(\\fint_Q\\abs{b(x)-c}^p\\ud x\\Big)^{1\/p}+\\Big(\\fint_Q\\abs{b(y)-c}^p\\ud y\\Big)^{1\/p}\\Big]^{\\alpha+\\beta}\n\\end{split}\n\\end{equation*}\nfor all constants $c$.\n\\end{proof}\n\n\\begin{corollary}\nLet $T$ be an operator satisfying $(\\avL^1(\\gamma Q),\\avL^1(\\gamma Q))$ convex body domination, let $0\\leq b\\in\\BMO(\\mathbb{R}^d)$ and $0\\leq\\alpha,\\beta$ with $\\alpha+\\beta\\leq 1$. Then\n\\begin{equation*}\n  \\Norm{b^\\alpha T(b^\\beta f)-b^\\beta T(b^\\alpha f)}{L^p(\\mathbb{R}^d)}\\lesssim_p \\Norm{b}{\\BMO(\\mathbb{R}^d)}^{\\alpha+\\beta}\\Norm{f}{L^p(\\mathbb{R}^d)}.\n\\end{equation*}\n\\end{corollary}\n\n\\begin{proof}\nBy Proposition \\ref{prop:gen-commu} with $s=t$, the $L^p(\\mathbb{R}^d)$ operator norm of $f\\mapsto b^\\alpha T(b^\\beta f)-b^\\beta T(b^\\alpha f)$ is dominated by\n\\begin{equation*}\n   A_s:= \\sup_Q\\Norm{(x.y)\\mapsto b(x)^\\alpha b(y)^\\beta-b(x)^\\beta b(y)^\\alpha}{\\avL^s(Q\\times Q)}\n\\end{equation*}\nif $p\\in(s',s)$, i.e., if $s>\\max(p,p')$. By Lemma \\ref{lem:BMOpowers} and the John--Nirenberg inequality, we have\n\\begin{equation*}\n   A_s\\leq(2\\Norm{b}{\\BMO^s(\\mathbb{R}^d)})^{\\alpha+\\beta}\\lesssim_s\\Norm{b}{\\BMO(\\mathbb{R}^d)}^{\\alpha+\\beta},\n\\end{equation*}\nand fixing (say) $s=2\\max(p,p')$, we obtain a dependence on $p$ only.\n\\end{proof}\n\n\n\\begin{remark}\nAside from the examples already discussed, the generalised commutators $\\vec a\\cdot T\\vec b$ also arise in the following question studied by Bloom \\cite{Bloom:85,Bloom:89}. Suppose that a matrix weight $W$ is given in the diagonalised form $W = U^*\\Lambda U$, where $U$ is unitary, $\\Lambda$ is diagonal, and the diagonal entries $\\lambda_k$ of $\\Lambda$ are scalar $A_2$ weights. What does one need to know about $U$ in order to conclude that $W\\in A_2$? (According to \\cite[Theorem 4.2]{Bloom:89}, the condition that $\\lambda_k\\in A_2$ is necessary for $W\\in A_2$, if in addition $U$ is assumed to be continuous.)\n\nLet $T$ be the Hilbert transform, or another operator whose boundedness on the matrix-weighted $L^2(W)$ characterises $W\\in A_2$. By connecting the $L^2(W)$ boundedness of $T$ to the boundedness of the classical commutators $[T,\\bar u_{ij}]$ between the weighted spaces $L^2(\\lambda_i)$ and $L^2(\\lambda_k)$ (sic: the condition involves triplets of indices $(i,j,k)$), \\cite[Theorem 5.1]{Bloom:85} shows that $u_{ij}\\in\\BMO_{\\sqrt{\\lambda_i\/\\lambda_k}}$ (a weighted BMO space, nowadays commonly referred to as Bloom-type BMO) is a sufficient condition. In the special case of $2\\times 2$ matrices, it is also necessary by \\cite[Theorem 4.3]{Bloom:89} but, over 30 years since these contributions, the general case seems to remain open. (The author is grateful to Amalia Culiuc for bringing this question to his attention \\cite{Culiuc:22}.)\n\nHere is a possible approach to the problem. As is well known, the $L^2(W)$ boundedness of $T$ is equivalent to the (unweighted) $L^2$ boundedness of\n\\begin{equation*}\n  W^{1\/2}TW^{-1\/2}=U^*\\Lambda^{1\/2}UTU^*\\Lambda^{-1\/2}U.\n\\end{equation*}\nMultiplication by $U$ and $U^*$ is isometric on $L^2$, and the $L^2$ boundedness of a matrix of operators is equivalent to the $L^2$ boundedness of each of the components\n\\begin{equation*}\n  (\\Lambda^{1\/2}UTU^*\\Lambda^{-1\/2})_{ij}\n  =\\sum_{k=1}^n\\lambda_i^{1\/2}u_{ik}T\\bar u_{jk}\\lambda_j^{-1\/2}\n  =\\lambda_i^{1\/2}\\vec u_i\\cdot T\\bar {\\vec u}_j\\lambda_j^{-1\/2},\n\\end{equation*}\nwhere $i,j=1,\\ldots,n$ and $\\vec u_i=(u_{ik})_{k=1}^n$. These are operators of the form $\\vec{a}\\cdot T\\vec{b}$ that we have studied here and, up to this point, we kept an exact equivalence with the original question; the question then would be, whether we can give useful conditions on the boundedness of these operators. A further equivalent condition is of course the two-weight boundedness\n\\begin{equation*}\n  \\vec u_i\\cdot T\\bar {\\vec u}_j:L^2(\\lambda_j)\\to L^2(\\lambda_i),\\qquad i,j=1,\\ldots,n,\n\\end{equation*}\nwhere the spaces are more complicated, but the multipliers are simply rows of the unitary matrix $U$.\n\\end{remark}\n\n\\begin{remark}\nWe have concentrated in this section on the application of convex body domination---an inherently vector-valued theory---to questions of generalised commutators acting on scalar-valued functions. We have made this choice for two reasons: to make the case that this vector-valued theory is useful even for such scalar-valued applications, and not to obscure the relatively simple basic philosophy behind too many technicalities of notation. This said, it is quite plain that the presented ideas can be immediately generalised to the case of vector-valued functions $\\vec f$ and $\\vec g$ (in place of scalar $f$ and $g$) and matrix-valued multipliers $A$ and $B$ (in place of the vectors $\\vec a$ and $\\vec b$). In the particular case of the classical-style commutator $[T,B]$ with a matrix-valued function, this idea has been developed in \\cite{IPT:commu}. \n\\end{remark}\n\n\n\\section{Stopping times and maximal functions involving convex bodies}\n\nThe aims of this final section are two-fold. Concretely, we establish a convex-body analogue of a result of Nieraeth \\cite{Nie:19}, which shows that the estimation of sums over sparse collection that arise in the usual sparse domination is equivalent to the estimation of certain maximal functions. On the way of achieving this, we develop some convex-body versions of the typical stopping time arguments involving averages of scalar-valued functions; these might have some independent interest elsewhere.\n\nWe begin with an estimate of a sum of convex-body ``norms'' over disjoint subsets.\n\n\\begin{lemma}\\label{lem:stoppingAux}\nLet $p,q\\in[1,\\infty)$ and $\\frac1r:=\\frac1p+\\frac1q$. Let $Q_i\\in\\mathscr D(Q_0)$ be disjoint cubes. Then\n\\begin{equation*}\n  \\sum_{i=1}^\\infty\\big(\\cave{\\vec f}_{L^p(Q_i)}\\cdot\\cave{\\vec g}_{L^q(Q_i)}\\big)^r\n  \\leq n^{\\max(r,1)+r\/2}\\big(\\cave{f}_{L^p(Q)}\\cdot\\cave{g}_{L^q(Q)}\\big)^r.\n\\end{equation*}\n\\end{lemma}\n\nNote that for $p,q\\in[1,\\infty)$, we have $\\frac1r=\\frac1p+\\frac1q\\leq 1+1=2$, and hence\n\\begin{equation*}\n  n^{\\max(r,1)+r\/2}\n  =\\big(n^{\\max(1,1\/r)+1\/2}\\big)^r\\leq \\big(n^{5\/2}\\big)^r.\n\\end{equation*}\n\n\\begin{proof}\nFor orientation, let us begin with the proof in the case $n=1$, i.e., with $\\Norm{\\ }{}$ in place of $\\cave{\\ }$ throughout. By H\\\"older's inequality with $1=\\frac{r}{p}+\\frac{r}{q}$, we have\n\\begin{equation*}\n\\begin{split}\n   \\sum_{i=1}^\\infty &\\big(\\Norm{f}{L^p(Q_i)}\\Norm{g}{L^q(Q_i)}\\big)^r\n   =\\sum_{i=1}^\\infty\\big(\\Norm{f}{L^p(Q_i)}^p\\big)^{r\/p}\\big(\\Norm{g}{L^q(Q_i)}^q\\big)^{r\/q} \\\\\n   &\\leq\\Big(\\sum_{i=1}^\\infty\\Norm{f}{L^p(Q_i)}^p\\Big)^{r\/p}\\Big(\\sum_{i=1}^\\infty\\Norm{g}{L^q(Q_i)}^q\\Big)^{r\/q} \\\\\n   &\\leq\\Big(\\Norm{f}{L^p(Q_0)}^p\\Big)^{r\/p}\\Big(\\Norm{g}{L^q(Q_0)}^q\\Big)^{r\/q}\n     =\\Big(\\Norm{f}{L^p(Q_0)}\\Norm{g}{L^q(Q_0)}\\Big)^{r}.\n\\end{split}\n\\end{equation*}\n\nIn the general case of the lemma, let\n\\begin{equation*}\n  A_i:=\\cave{\\vec f}_{L^p(Q_i)}\n  =\\Big\\{\\int_{Q_i}\\phi_i\\vec f:\\Norm{\\phi_i}{L^{p'}(Q_i)}\\leq 1\\Big\\},\\quad B_i:=\\cave{\\vec g}_{L^q(Q_i)}.\n\\end{equation*}\nThen we observe that\n\\begin{equation*}\n\\begin{split}\n  \\cave{\\vec f}_{L^p(Q)}\n  &=\\Big\\{\\int_{Q}\\phi\\vec f:\\Norm{\\phi}{L^{p'}(Q)}\\leq 1\\Big\\} \\\\\n  &\\supseteq\\Big\\{\\sum_{i=1}^\\infty a_i\\int_{Q_i}\\phi_i\\vec f:\\Norm{\\phi_i}{L^{p'}(Q_i)}\\leq 1, \\Norm{(a_i)}{\\ell^{p'}}\\leq 1\\Big\\} \\\\\n  &=\\Big\\{\\sum_{i=1}^\\infty a_i A_i: \\Norm{(a_i)}{\\ell^{p'}}\\leq 1\\Big\\}=:\\bigoplus_{\\ell^p}A_i=:A,\n\\end{split}\n\\end{equation*}\nand similarly\n\\begin{equation*}\n  \\cave{\\vec g}_{L^q(Q)}\\supseteq\\bigoplus_{\\ell^q}B_i=:B.\n\\end{equation*}\nHence, the lemma is reduced to proving that\n\\begin{equation*}\n  \\sum_{i=1}^\\infty\\big(A_i\\cdot B_i\\big)^r\n  \\leq n^{\\max(r,1)+r\/2}\\big(A\\cdot B\\big)^r,\\quad A:=\\bigoplus_{\\ell^p}A_i,\\quad B:=\\bigoplus_{\\ell^q}B_i.\n\\end{equation*}\n\nLet $\\mathcal E_A$ be the John ellipsoid of $A$, and let $R_A\\mathcal E_A=\\bar B_{\\mathbb{R}^n}$. Since $A_i\\cdot B_i=R_A A_i\\cdot R_A^{-t}B_i$, The claim above is equivalent to a version where each $A_i$ is replaced by $R_A A_i$ and each $B_i$ by $R_A^{-t}B_i$. Hence, without loss of generality, we assume that $\\mathcal E_A=\\bar B_{\\mathbb{R}^n}$ to begin with, hence $\\bar B_{\\mathbb{R}^n}\\subseteq A\\subseteq \\sqrt{n}\\bar B_{\\mathbb{R}^n}$. Thus\n\\begin{equation*}\n  A\\cdot B\\supset \\bar B_{\\mathbb{R}^n}\\cdot B=[-M,M],\\quad\\text{where}\\quad M:=\\max\\{\\abs{\\vec b}:\\vec b\\in B\\}.\n\\end{equation*}\nOn the other hand, if $(\\vec e_j)_{j=1}^n$ is some orthonormal basis of $\\mathbb{R}^n$, then\n\\begin{equation*}\n\\begin{split}\n  A_i\\cdot B_i &=\\{\\vec a\\cdot \\vec b:\\vec a\\in A_i,\\vec b\\in B_i\\} \\\\\n  &=\\Big\\{\\sum_{j=1}^n(\\vec a\\cdot\\vec e_j)(\\vec b\\cdot\\vec e_j):\\vec a\\in A_i,\\vec b\\in B_i\\} \n  \\subseteq\\sum_{j=1}^n (A_i\\cdot \\vec e_j)(B_i\\cdot\\vec e_j),\n\\end{split}\n\\end{equation*}\nor, using the identification of $[-s,s]$ with $s$,\n\\begin{equation*}\n  A_i\\cdot B_i \\leq \\sum_{j=1}^n (A_i\\cdot \\vec e_j)(B_i\\cdot\\vec e_j).\n\\end{equation*}\nThus\n\\begin{equation*}\n  (A_i\\cdot B_i)^r \\leq \\sum_{j=1}^n \\big((A_i\\cdot \\vec e_j)(B_i\\cdot\\vec e_j)\\big)^r,\\quad r\\in(0,1],\n\\end{equation*}\nand\n\\begin{equation*}\n  \\Big(\\sum_{i=1}^\\infty (A_i\\cdot B_i)^r\\Big)^{1\/r}\\leq\\sum_{j=1}^n \\Big(\\sum_{i=1}^\\infty \\big((A_i\\cdot \\vec e_j)(B_i\\cdot\\vec e_j)\\big)^r\\Big)^{1\/r},\\quad r\\in[1,\\infty).\n\\end{equation*}\nIn the sum over $i$, we use H\\\"older's inequality as in the toy model in the beginning:\n\\begin{equation*}\n\\begin{split}\n  \\sum_{i=1}^\\infty&\\big((A_i\\cdot \\vec e_j)(B_i\\cdot\\vec e_j)\\big)^r\n  =\\sum_{i=1}^\\infty\\big((A_i\\cdot \\vec e_j)^p\\big)^{r\/p}\\big((B_i\\cdot\\vec e_j)^q\\big)^{r\/q} \\\\\n  &\\leq\\Big(\\sum_{i=1}^\\infty(A_i\\cdot \\vec e_j)^p\\Big)^{r\/p}\\Big(\\sum_{i=1}^\\infty(B_i\\cdot\\vec e_j)^q\\Big)^{r\/q} \\\\\n  &=\\sup\\Big\\{\\Big(\\sum_{i=1}^\\infty a_i A_i\\cdot\\vec e_j\\Big)^{1\/r}\\Big(\\sum_{i=1}^\\infty b_i B_i\\cdot\\vec e_j\\Big)^{1\/r}:\n    \\Norm{(a_i)}{\\ell^{p'}}\\leq 1,\\Norm{(b_i)}{\\ell^{q'}}\\leq 1\\Big\\} \\\\\n  &=(A\\cdot \\vec e_j)^r(B\\cdot\\vec e_j)^r\n\\end{split}\n\\end{equation*}\nHere\n\\begin{equation*}\n  A\\cdot\\vec e_j\\subseteq \\sqrt{n}\\bar B_{\\mathbb{R}^n}\\cdot\\vec e_j=[-\\sqrt{n},\\sqrt{n}],\\quad A\\cdot\\vec e_j\\leq \\sqrt{n},\n\\end{equation*}\nand clearly\n\\begin{equation*}\n  B\\cdot\\vec e_j\\leq M.\n\\end{equation*}\nAltogether, writing $s:=\\max(r,1)$, we have\n\\begin{equation*}\n\\begin{split}\n  \\Big(\\sum_{i=1}^\\infty(A_i\\cdot B_i)^r\\Big)^{1\/s}\n  &\\leq \\sum_{j=1}^n   \\Big[\\sum_{i=1}^\\infty(A_i\\cdot \\vec e_j)^r(B_i\\cdot\\vec e_j)^r\\Big]^{1\/s} \\\\\n  &\\leq \\sum_{j=1}^n  \\Big[ (A\\cdot \\vec e_j)^r(B\\cdot\\vec e_j)^r\\Big]^{1\/s} \n  \\leq n [ n^{r\/2} M^r]^{1\/s},\n\\end{split}\n\\end{equation*}\nand hence\n\\begin{equation*}\n   \\sum_{i=1}^\\infty(A_i\\cdot B_i)^r\\leq n^s n^{r\/2} M^r= n^{\\max(1,r)+r\/2}(A\\cdot B)^r,\n\\end{equation*}\nwhich remained to be proved.\n\\end{proof}\n\nThe following lemma is a convex-body analogue of the basic principle underlying the simplest stopping time constructions: for a function on a cube $Q_0$, the total measure of the subcubes, where the average of a function is much bigger than on the whole $Q_0$, can be at most a fraction of the measure of $Q_0$.\n\n\\begin{lemma}\\label{lem:convexStopping}\nLet $A,p,q\\in[1,\\infty)$ and let $Q_i\\in\\mathscr D(Q_0)$ be disjoint cubes such that\n\\begin{equation*}\n  \\cave{\\vec f}_{\\avL^p(Q_i)}\\cdot\\cave{\\vec g}_{\\avL^q(Q_i)}\\geq A\\cave{\\vec f}_{\\avL^p(Q_0)}\\cdot\\cave{\\vec g}_{\\avL^q(Q_0)}.\n\\end{equation*}\nThen\n\\begin{equation*}\n  \\sum_{i=1}^\\infty\\abs{Q_i}\\leq \\frac{n^{\\max(r,1)+r\/2}}{A^{r}}\\abs{Q_0},\\qquad \\frac1r:=\\frac1p+\\frac1q.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nDirectly from the definition, it is easy to extend the basic identity $\\Norm{f}{\\avL^p(Q)}=\\abs{Q}^{-1\/p}\\Norm{f}{L^p(Q)}$ to convex bodies as\n\\begin{equation}\\label{eq:avLvsL}\n  \\cave{\\vec f}_{\\avL^p(Q)}=\\abs{Q}^{-1\/p}\\cave{\\vec f}_{L^p(Q)}.\n\\end{equation}\nFrom this, the assumption of the lemma can be rewritten as\n\\begin{equation*}\n  \\abs{Q_i}^{-1\/p-1\/q} \\cave{\\vec f}_{L^p(Q_i)}\\cdot\\cave{\\vec g}_{L^q(Q_i)}\\geq A\\abs{Q_0}^{-1\/p-1\/q}\\cave{\\vec f}_{\\L^p(Q_0)}\\cdot\\cave{\\vec g}_{\\L^q(Q_0)},\n\\end{equation*}\nor, rearranging,\n\\begin{equation*}\n  \\abs{Q_i}\\leq \\frac{A^{-r}\\abs{Q_0}}{\\big(\\cave{\\vec f}_{\\L^p(Q_0)}\\cdot\\cave{\\vec g}_{\\L^q(Q_0)}\\big)^r}\n      \\big(\\cave{\\vec f}_{L^p(Q_i)}\\cdot\\cave{\\vec g}_{L^q(Q_i)}\\big)^r.\n\\end{equation*}\nSumming over $i$ and using Lemma \\ref{lem:stoppingAux}, we obtain the claim.\n\\end{proof}\n\nWe now obtain the following proposition, which is a convex body analogue of a result of Nieraeth \\cite[Prop. 2.7; especially Eq. (2.7) for $m=1$]{Nie:19}. It says that estimating the sums over sparse collections, like those that arise from convex body domination, is equivalent to estimating related bi-sublinear maximal operators. In \\cite[Prop. 2.7]{Nie:19}, the result is formulated as a set of equivalent conditions for a tuple of weights. The formulation below has no reference to weights as such, but as soon as one starts asking questions about the boundedness of either side on spaces like $L^s(W)\\times L^{s'}(W')$, the proposition guarantees that one can equally well study this boundedness for the other side of the equivalence.\n\n\\begin{proposition}\nFor all $\\delta\\in(0,1)$, all dimensions $d,n\\geq 1$, exponents $p,q\\in[1,\\infty)$, and functions $\\vec f\\in L^p_{\\loc}(\\mathbb{R}^d)^n$, $\\vec g\\in L^q_{\\loc}(\\mathbb{R}^d)^n$, we have the two-sided estimate\n\\begin{equation*}\n  \\sup_{\\mathscr S}\\sum_{Q\\in\\mathscr S}\\cave{\\vec f}_{\\avL^p(Q)}\\cdot\\cave{\\vec g}_{\\avL^q(Q)}\\abs{Q}\n  \\eqsim\\BNorm{\\sup_{Q\\in\\mathscr D}1_Q\\cave{\\vec f}_{\\avL^p(Q)}\\cdot\\cave{\\vec g}_{\\avL^q(Q)}}{L^1(\\mathbb{R}^d)},\n\\end{equation*}\nwhere the supremum is taken over all $\\delta$-sparse collections of dyadic cubes in $\\mathbb{R}^n$, and the implied constants depend only on $n,p,q$, and $\\delta$.\n\\end{proposition}\n\n\\begin{proof}\nWith $\\vec f\\in L^p_{\\loc}(\\mathbb{R}^d)^n$ and $\\vec g\\in L^q_{\\loc}(\\mathbb{R}^d)^n$ fixed, let us denote\n\\begin{equation*}\n  a_Q:=\\cave{\\vec f}_{\\avL^p(Q)}\\cdot\\cave{\\vec g}_{\\avL^q(Q)}.\n\\end{equation*}\nThe estimate $\\lesssim$ is immediate: From $\\delta$-sparseness, we have $\\abs{Q}\\leq\\delta^{-1}\\abs{E(Q)}$ for some disjoint sets $E(Q)$, and hence\n\\begin{equation*}\n   \\sum_{Q\\in\\mathscr S}a_Q\\abs{Q}\n   \\leq\\frac{1}{\\delta}\\sum_{Q\\in\\mathscr S}a_Q\\abs{E(Q)}\n   =\\frac{1}{\\delta}\\int_{\\mathbb{R}^d}\\sum_{Q\\in\\mathscr S}a_Q 1_{E(Q)}\n   \\leq\\frac{1}{\\delta}\\int_{\\mathbb{R}^d}\\sup_{Q\\in\\mathscr D}a_Q 1_Q.\n\\end{equation*}\n\nThe estimate $\\gtrsim$ needs a bit more. By monotone convergence, it is enough to consider $\\mathscr D(Q_0)$ in place of $\\mathscr D$. Let $\\mathscr S_0:=\\{Q_0\\}$. For some $A>1$ to be chosen and $Q\\in\\mathscr D(Q_0)$, let $\\mathscr S'(Q)$ consist of all maximal $Q'\\in\\mathscr D(Q)$ such that $a_{Q'}>A a_Q$. By maximality, the cubes $Q'\\in\\mathscr S'(Q)$ are disjoint. By Lemma \\ref{lem:convexStopping}, we have\n\\begin{equation*}\n   \\sum_{Q'\\in\\mathscr S'(Q)} \\abs{Q'}\\leq\\frac{n^{\\max(1,r)+r\/2}}{A^r}\\abs{Q}\\leq(1-\\delta)\\abs{Q},\\qquad\\frac1r:=\\frac1p+\\frac1q,\n\\end{equation*}\nprovided that $A$ is chosen large enough, depending on $n,p,q$, and $\\delta$. Hence, defining inductively $\\mathscr S_{j+1}:=\\bigcup_{Q\\in\\mathscr S_j}\\mathscr S'(Q)$ and $\\mathscr S:=\\bigcup_{j=0}^\\infty\\mathscr S_j$, we find that $\\mathscr S$ is $\\delta$-sparse. If $Q\\in\\mathscr D(Q_0)$ and $S\\in\\mathscr S$ is the minimal stopping cube that contains $Q$, then $a_Q\\leq Aa_S$ by the way that the cubes $S\\in\\mathscr S$ were chosen, hence\n\\begin{equation*}\n  \\sup_{Q\\in\\mathscr D(Q_0)}1_Q a_Q\\leq  \\sup_{S\\in\\mathscr S}1_S Aa_S\\leq A\\sum_{S\\in\\mathscr S}1_S a_S,\n\\end{equation*}\nand thus\n\\begin{equation*}\n  \\BNorm{\\sup_{Q\\in\\mathscr D(Q_0)}1_Q a_Q}{L^1(\\mathbb{R}^d)}\\leq A\\BNorm{\\sum_{S\\in\\mathscr S}1_S a_S}{L^1(\\mathbb{R}^d)}\n  =A\\sum_{S\\in\\mathscr S}a_S\\abs{S}.\\qedhere\n\\end{equation*}\n\\end{proof}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe electronic transport properties of a molecular device depend much on the\nunderlying nuclear configuration and the electron-phonon coupling\\cite{galperin-review}.\nThe mutual interaction between them induces Joule heating in the molecule,\nwhich may have important consequences on the functionality and reliability of\na molecular device. Study of the heat transport by phonons in molecular\njunctions is crucial for a better understanding of their electronic\ntransport properties. Phonon thermal transport itself is also interesting.\nDetailed understanding of the underlying transport mechanism is especially\nuseful for the design of novel phononic\ndevices\\cite{li:184301,science-thermal}. Furthermore, combined study of the\nelectronic and phononic transport in molecular junctions is the first step\ntoward the design of molecular thermoelectric\ndevices\\cite{reddy07}.  Great experimental progress has been\nmade in these directions recently, which enables one to measure the thermal\nand thermoelectric transport properties of molecular\njunctions\\cite{wangzh07,reddy07}. Advances in experimental technique call for\nnew theoretical method to predict the thermal conductance of molecular\njunctions. Although semiempirical or minimal model calculation is helpful to\nunderstand the underlying physics\\cite{segal:6840,buldum99}.  For a detailed\nquantitative study, a parameter-free, first-principle method is highly\ndesirable\\cite{mingo08}. Furthermore, for many molecular structures there\nexists no empirical inter-atom potential.\n\nIn this paper we introduce such a method based on a standard quantum chemistry\nsoftware, the Gaussian03 parckage\\cite{gaussian03}.  Given the molecule\nstructure, we can obtain the force constant matrices after performing the\nenergy minimization. The thermal conductance can be calculated via available\ntheoretical methods. Among them are the nonequilibrium Green's function (NEGF)\nmethod\\cite{wang:033408,wang-2007,wang-review-epjb,mingo06,yamamoto06}, which\nhas been successfully used to  predict the electronic conductance of molecular\njunctions\\cite{taylor01,brandbyge02,damle01}. To study the transport of each\nphonon mode, we will also use the mode-matching method\\cite{ando91}, which is\nequivalent to the NEGF method in the ballistic limit\\cite{khomyakov05}.  Using\nthese methods, we first compare the thermal conductance of a benzene ring\namide-linked with two $(6,0)$ single-walled carbon nanotubes (SWCNTs) under\ndifferent compression or stretching conditions.  Then we study the effect of\nSWCNT terminating group at the cutting edges. Finally, we compare the phonon\ntransmission probability and the thermal conductance of benzene and alkane\nchains with the same leads. Although the electron contribution to the thermal\nconductance may be comparable with or larger than that of phonons, inclusion of\nthis effect is out of the scope of present study. \n\n\n\\section{Molecular structure and theoretical method}\n\\label{sec:theory}\nIn this section we first introduce the system Hamiltonian and the procedure to\nobtain it from the Gaussian03 package. Then we briefly outline the NEGF, the\nmode-matching method, and their relationship. For a detailed discussion, we\nrefer the reader to Ref.~\\onlinecite{wang-review-epjb}. The system we are\ninterested in is a molecular junction connected with two periodic semi-infinite\nleads at both sides. It is a standard treatment to divide the whole structure\ninto three parts: the center junction and two leads acting as thermal baths.\nThe boundaries between the center and the baths may be at arbitrary positions,\nand not correspond to any physical interface. But it is desirable to include\npart of the leads into the center region, since we need to make sure that there\nis no direct interaction between the two baths, which is required by the NEGF\nformalism. By doing this, we can also include the charge transfer effect\nbetween the leads and the molecule junction. In this setup, the system\nHamiltonian can be written as\n\\begin{equation}\n\\label{eq:h}\n{\\cal H} = \\!\\!\\!\\!\\!\\sum_{\\alpha=L,C,R}\\!\\!\\!\\!\\!H_\\alpha  + (u^L)^T V^{LC} u^C + (u^C)^TV^{CR} u^R + V_n,\n\\end{equation}\nwhere $H_{\\alpha} = \\frac{1}{2} {(\\dot{u}^\\alpha)}^T \\dot{u}^\\alpha +\n\\frac{1}{2} {(u^\\alpha)}^T K^\\alpha u^\\alpha$ represents harmonic\noscillators, $u^\\alpha$ is a column vector consisting of all the mass normalised\ndisplacement variables in region $\\alpha$, and $\\dot{u}^\\alpha$ is the\ncorresponding conjugate momentum.  $K^\\alpha$ is the spring constant matrix in the tight-binding form\nand $V^{LC}=(V^{CL})^T$ is the coupling matrix of the left lead to the\ncentral region; similarly for $V^{CR}$. $V_n$ is the nonlinear interaction\nin the center, which could be $V_n = \\frac{1}{3}\\sum_{ijk}t_{ijk}u_iu_ju_k$\nfor cubic nonlinearality. We ignore the nonlinear interaction in this paper and only briefly discuss its effect in Sec.~\\ref{sec:results}.\n\nWe study the phonon thermal conductance of benzene and alkane chains covalently\nbonded with two $(6,0)$ SWCNTs via the amide group. This is relevant to a\nrecent experimental setup\\cite{guo06}, where the SWCNT is oxidatively cutted,\nand the cutting edges are covalently bonded with molecular chains via the amide\ngroup.  To get the force constants of the system, we do two separate runs for\nthe center and the leads using Gaussian03. For the center, we include extra one\nand a half periods of SWCNT at each side, which is terminated by hydrogen atoms\nat the outer boundaries. The cutting edges may be terminated by H or COOH\ngroup. We optimize the center at the level of b3lyp density functional method\nusing the 6-31G basis set. We first relax the structure by constraining the\nleads to be coaxial to get an optimized distance between the two leads. Then we\nfix the outer half period of the SWCNTs and the H atoms at both sides and allow\nall other atoms to move freely. The optimized structure is used to get the\ndynamical matrix.  The inset of Fig.~\\ref{fig:coohh} shows the optimized\nbenzene structure terminated by the H and COOH groups at the cutting edges. For\nthe lead, it is desirable to use periodic boundary condition to compute the\ndynamical matrix, while this is out of the ability of Gaussian03. So we\noptimise $7$ SWCNT periods and extract the force constant from the central\nperiod to minimize the finite size effect. To connect the molecule with the two\nleads, we remove the outer fixed atoms in the molecule and connect the\nremaining period of SWCNT with one semi-infinite SWCNT at each side. For the\ncoupling matrix between the center and the leads, we only include coupling\nbetween one period of SWCNT atoms in the center and one period in the leads.\nThis is a good approximation so long as we include enough SWCNT atoms into the\ncenter molecule. \n\nIn the NEGF method as described in\nRefs.~\\onlinecite{wang:033408,wang-2007,wang-review-epjb,mingo06,yamamoto06}, thermal\nconductance of the molecular junction can be calculated from the \nLandauer formula ($\\hbar=1$ throughout the formula)\n\\begin{equation}\n\\label{eq:caroli}\n\\sigma = \n\\frac{1}{2\\pi} \\int_0^\\infty \\!\\!\\!d\\omega\\, \\omega\\, T[\\omega] \n\\frac{\\partial f(\\omega)}{\\partial T}\n\\end{equation}\nwith the transmission coefficient\n\\begin{eqnarray}\nT[\\omega] &=& \n{\\rm Tr} \\bigl\\{ G^r \\Gamma_L G^a \\Gamma_R \\bigr\\}. \n\\label{eq-eff-trans}\n\\end{eqnarray}\n$f(\\omega)$ is the Bose-Einstein distribution function.\nThe retarded Green's function $G^r$ is obtained from \n\\begin{equation}\n\tG^r[\\omega] = \\left( (\\omega+i0^+)^2-K^C-\\Sigma^r_L[\\omega]-\\Sigma^r_R[\\omega] \\right)^{-1},\n\t\\label{eq:dyson1}\n\\end{equation}\nwhere the retarded self-energy of lead $\\alpha$ is \n\\begin{equation}\n\t\\Sigma^r_\\alpha[\\omega] = V^{C\\alpha}g^r_\\alpha[\\omega]V^{\\alpha C},\n\t\\label{eq:sel}\n\\end{equation}\nand the lead surface Green's function $g^r_\\alpha[\\omega]$ can be calculated\nfrom the generalized eigenvalue method\\cite{wang-review-epjb}, e.g., \n\\begin{equation}\ng^r_R[\\omega] = \\left( (\\omega+i0^+)^2-k^R_{11}-k^R_{01}F^+_R(1) \\right)^{-1}.\n\t\\label{eq:surface}\n\\end{equation}\n$k^R_{11}$ and $k^R_{01}$ are the diagonal and off-diagonal parts of the\nright lead dynamical matrix. Their sizes are determined by the degrees of\nfreedom $M$ in each period of the lead. The matrix $F^+_R(s)$\ntranslates to the right the displacement in the\n$n$th period to the $(n+s)$th period $u^+_R(n+s)=F^+_R(s)u^+_R(n)$. \nIt is constructed from the eigen values and vectors of the generalized eigen value problem\n\\begin{equation}\n\\label{eq:eigen}\n\\left( \\begin{array}{cc} (\\omega+i0^+)^2I\\!\\!-\\!\\!k^R_{11} & -I \\\\\n             k^R_{10} & 0 \n           \\end{array}\\right) \n\\left( \\begin{array}{c} \\epsilon \\\\\n             \\zeta \n           \\end{array}\\right) = \\lambda\n\\left( \\begin{array}{cc} k^R_{01} & 0 \\\\\n             0 & I \n           \\end{array}\\right) \n\\left( \\begin{array}{c} \\epsilon \\\\\n             \\zeta \n           \\end{array}\\right)\n\\end{equation}\nas $F^+_R(s)=E^+_R\\Lambda_+^s(E^+_R)^{-1}$.  Here $I$ is an $M\\! \\times\\! M$\nidentity matrix. The diagonal matrix $\\Lambda_+^s$ consists of all the eigen\nvalues $|\\lambda_+^s| < 1$, and $E^+_R$ the corresponding eigen vectors $E^+_R\n= (\\varepsilon^+_1,\\varepsilon^+_2,\\cdots,\\varepsilon^+_{M'})$. Note that $M'$\nmay be less than $M$, in which case the matrix inverse becomes pseudo-inverse.\nA similar left-translation matrix $F^{-}_R(s)$ can be constructed from\n$\\Lambda_{-}^s$ and $E_R^-$, which include all the eigen values $|\\lambda_-^s|\n> 1$ (excluding infinity) and the corresponding eigen vectors.\n\nWhile the NEGF method is systematic and suitable to take into account the\nnonlinear interaction, the transmission coefficient from it is the sum of\nall the eigen modes from the leads. Thus it is difficult to analyse the\ncontribution from each mode. The mode-matching method provides another way\nto calculate the transmission coefficient in the ballistic\nlimit\\cite{ando91,khomyakov05,wang-review-epjb}. Single mode transmission and\nmode mixing effect can be studied by this method. The transmission matrix is\ngiven by\n\\begin{equation}\n\tt^{RL}_{mn} = \\sqrt{\\frac{v_{R,m}^+a_L}{v_{L,n}^+a_R}}\\tau^{RL}_{mn},\n\t\\label{eq:mm1}\n\\end{equation}\nand\n\\begin{equation}\n\t\\tau^{RL} = (E^+_R)^{-1}g^r_RV^{RC}G^rV^{CL}g^r_Lk_{10}^L(F^{+}_L(-1)-F^{-}_L(-1))E^+_L,\n\t\\label{eq:mm2}\n\\end{equation}\nwhere $v^+_{\\alpha,m}$ is the group velocity of the $m$th right propagating\nmode for the lead $\\alpha$. While the mode indices $m$ and $n$ are only for\npropagating modes, the matrices $E^{\\pm}_\\alpha$ include all the propagating\nand evanescent modes. The total transmission coefficient as in the Landauer\nformula is $T[\\omega] = {\\rm Tr} \\bigl\\{ (t^{RL})^\\dagger t^{RL} \\bigr\\}$.\nSimilar relations hold for waves incidented from the right lead and transmitted\nto the left $t^{LR}_{nm}$. \n\nThe NEGF and the mode-matching method are exactly equivalent in the ballistic\ncase as show in Ref.~\\onlinecite{khomyakov05}. All the matrices needed by\nEqs.~(\\ref{eq:mm1}--\\ref{eq:mm2}) can be obtained by solving the generalized\neigen value problem Eq.~(\\ref{eq:eigen}). It is interesting to note that by\ndoing a singular value decomposition on the transmission matrix $t$ we can get\nthe transmission eigenchannel information without any other efforts, for which\ndifferent methods have been developed in the electronic transport\nliterature\\cite{inglesfield:155120}.\n\n\\section{Numerical results and discussion}\n\\label{sec:results}\nWe now present our numerical results. We begin with the simplest case where\nthere is only one benzene ring at the center. We compare the thermal\nconductance of the molecular junction at different compression or stretching\nconfigurations by changing the distance between the two leads. The purpose of\nthis study is twofold. Firstly, we want to study the effect of the compression\nor stretching on the thermal conductance. Secondly, our optimization process\ndiscussed in Sec.~\\ref{sec:theory} can not ensure that we have reached the\nlowest energy configuration while keeping the two SWCNTs to be coaxial. This is because we have fixed the position of the outer C and H atoms at both sides. If the thermal conductance is not sensitive to the\ndistance between the two leads, our results make sense even if we do not find the\nlowest energy configuration. As we can see in Fig.~\\ref{fig:confs}, this is\nindeed the case. In Fig.~\\ref{fig:confs}, from $1$ to $5$ the distance between the two carbon\natoms connecting to the amide groups is $10.07$, $10.21$, $10.38$, $10.51$,\n$10.64$ \\AA, respectively. The inset shows the atom configuration of the five\ncases. In the compression states ($10.07$ and $10.21$ \\AA), the relative position between the benzene ring and the SWCNT leads changes compared to the full relaxed ($10.38$ \\AA) and stretching ($10.51$ and $10.64$ \\AA) states (inset of Fig.~\\ref{fig:confs}). Although the atom configurations change a lot, the ballistic thermal\nconductance of the molecular junction only changes slightly in all the\ntemperature range studied here. So we can conclude that the ballistic thermal\nconductance of the molecular junction is not sensitive to small compression or\nstretching of the molecule.\n\\begin{figure} \n\\includegraphics[scale=0.6]{fig1.eps}\n\\caption{\\label{fig:confs} Ballistic thermal conductance of the benzene junction terminated by H atoms at different configurations. $3$ is the full relaxed structure. $1-2$ are in the compression state, and $3-4$ are in the stretching state.}\n\\end{figure}\n\nIn a related experiment\\cite{guo06}, the cutting edge carbon atoms are\nexpected to be saturated by the COOH group, not the H group shown in\nFig.~\\ref{fig:confs}. In Ref.~\\onlinecite{renw07}, the authors show that the\nelectron transmission is largely affected by the terminating groups. It is also\ninteresting to know how the terminating group affects the quantum thermal\nconductance. We still use the single benzene structure to study this problem.\nFigure~\\ref{fig:coohh} shows the thermal conductance of the two configurations.\nThe full relaxed structures are shown in the inset. The electrical conductance\nis mainly determined by the energy channel near the chemical potential, while\nthe thermal conductance is jointly contributed by many phonon modes. The\nterminating group only has a large influence on the high energy (short\nwavelength) optical phonon modes.  The low energy (long wavelength) phonon\nmodes are not sensitive to the local environment at the cutting edges.  As a\nresult, the thermal conductance only shows quantitative difference, which\nbecomes larger at high temperatures.\n\\begin{figure}\n\\includegraphics[scale=0.6]{fig2.eps}\n\\caption{\\label{fig:coohh} Ballistic thermal conductance of the benzene junctions terminated by H and COOH group.}\n\\end{figure}\n\nWe now compare the conductance of the benzene rings and that of the alkane\nchains, with the SWCNTs terminated by H atoms. We expect that the COOH\nterminating structures show similar behavior. To ensure the two molecules have\ncomparable length, we include two benzene rings for the benzene structure and\n$8$ CH$_2$ groups for the alkane chain. The total transmission probability and\nthe thermal conductance are shown in Figs.~\\ref{fig:3} and \\ref{fig:4},\nrespectively. Since the nonlinear interaction is not considered here, phonon\nmodes with different energies are independent of each other. The transmission\nprobability is nonzero only in the overlapped energy range of the SWCNT leads\nand the center molecule.  Strong coupling with the leads gives rise to wide\nbroadening and strong shifting of the discrete molecule phonon energy. It is\nhard to make a one-to-one correspondence between the transmission peaks and the\nisolated molecule phonon eigen frequencies. This is especially true for low\nenergy phonon modes, which have relatively large spatial extent. The high\nenergy modes are highly localized and show sharp peaks in the transmission\nspectrum. Due to the highly localized nature and low Bose-Einstein weighting,\nthese modes can hardly transfer energy. There is a wide zero transmission gap\naround $0.07$ eV for the alkane chain, which is a characteristic of the alkane\nchain phonon spectrum\\cite{segal:6840}.\n\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{fig3.eps}\n\\caption{\\label{fig:3} Phonon transmission probability as a function of energy for the benzene and alkane chains.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{fig4.eps}\n\\caption{\\label{fig:4} Ballistic thermal conductance as a function of temperature for the benzene and alkane chains. The inset shows the crossing point of the thermal conductance at about $38$ K.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{fig5.eps}\n\\caption{\\label{fig:5} Transmission probability of different phonon branches: (a) TA modes, (b) LA mode, (c) twisting mode, (d) lowest optical modes.}\n\\end{figure}\n\nTo gain further insight into the transmission spectrum, we also study the\nsingle mode transmission using the mode-matching method. Figure~\\ref{fig:5}\nshows the transmission of several important branches in SWCNT phonon spectrum: Two transverse acoustic (TA) phonon\nmodes, one longitudinal acoustic (LA) mode, one twisting mode and the lowest two degenerate optical modes. These\nlow frequency modes are expected to contribute much to the thermal\nconductance. The degenerate phonon modes of SWCNTs show different transmission\nin both cases. This is consistent with the fact that the junction structure\ndestroys the degeneracy in transverse directions. At the energy range below\n$0.01$ eV, the main contribution comes from the TA\nmodes of the SWCNT. The alkane chain shows larger transmission than the\nbenzene chain in this energy range. Above $0.01$ eV, the benzene chain shows\nlarger transmission in most cases.  The twisting mode shows the largest\ndifference in two structures. While it can hardly be transmitted by the\nalkane chain, it has a large contribution in the benzene chain.\n\nThe thermal conductance of the molecular junction depends not only on the\ntransmission coefficient, but also on the phonon occupation number, which is\nreflected as the Bose-Einstein distribution in the Landauer formula. According\nthe analysis of the transmission spectrum, we may expect that the alkane chain\nhas larger thermal conductance at low temperatures, while at higher\ntemperatures the order reverses. This is confirmed in Fig.~\\ref{fig:4}. The\ncrossing temperature is about $38$ K. The room temperature thermal conductance\nis about $0.075$ nW\/K for alkane chains and $0.125$ nW\/K for benzene chains.\nIn a recent experiment\\cite{wangzh07}, the thermal conductance of alkane chain\nis found to be smaller than our theoretical value. This difference comes from\nthe effect of the leads. We are using SWCNTs here, while in the experiment it\nis the bulk gold connected with the alkane chain via the sufur atom. At least\ntwo factors from the leads may account for this decrease. The first is the\nsmaller phonon spectrum overlap between gold and alkane chain, and the second\nis the weaker coupling between them.\n\nSome comments are worthwhile. The nonlinear interaction may change the\ntransmission spectrum, and lead to a decrease of the thermal conductance.  A\nperturbative analysis of the single benzene structure shows that the room\ntemperature thermal conductance drops about $30\\%$ of the ballistic\nvalue if we include the cubic force constant calculated from Gaussion03.\nMean-filed approximation in the NEGF formalism only works well for simple\nstructures with relatively weak nonlinear interaction\\cite{wang-review-epjb}. It\nfails to converge in the self-consistent iteration in present case. So it is\nstill a challenge to find a good approximation for the nonlinear\nself-energies in NEGF method.\n\n\\section{Conclusions} In this paper, we introduce a straightforward method to\ncalculate the phononic thermal conductance of molecular junctions in the\nballistic regime from first-principles. The force constant matrices are\nobtained from Gaussian03 quantum chemistry software. The phonon transmission\nand thermal conductance are calculated using the NEGF or mode-matching method.\nFurther information can be obtained from the transmission spectrum of each\nsingle mode. Using this method we show that the ballistic thermal conductance\nof the benzene ring amide linked with SWCNTs is not sensitive to the distance\nbetween the two SWCNTs. The benzene rings show larger thermal conductance\nthan the alkane chains. This method is general, and can be easily applied to\nother material systems.\n\n\\begin{acknowledgments}\nThe authors thank Nan Zeng for discussions. JTL is grateful to the hospitality\nof Prof. J. C. Cao at Shanghai Institute of Microsystem and Information\nTechnology, where this paper was finished. This work was supported in part\nby a Faculty Research Grant (R-144-000-173-101\/112) of National University of\nSingapore.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\\subsection{Motivation}\n\\label{sec:bwp_intro}\nOur adaptive bandwidth allocation (BA) model is motivated by a new feature in 5G NR called bandwidth part (BWP)~\\cite[Section 6.10]{NR_BWP_2019_1},\n\\cite[Section 4.4.5]{NR_BWP_2019_2}. The BWP concept is based on the division of a wide bandwidth into multiple \\textit{contiguous} smaller chunks of bandwidth. The main aim of this division is to let the number of bandwidth chunks used by a wireless user depend on its type at a given time, namely on its current needs in terms of data rate or constraints in terms of hardware complexity and power consumption. This flexibility on what is allocated to users makes of BWP a new dimension of radio spectrum sharing. \n\n\n\n\\begin{table}[]\n\\caption{Types of wireless devices, wireless applications, and heterogeneous throughput demands. The numbers in parentheses refer to 2017 and 2022, respectively. (Sources: Cisco VNI Mobile, 2019~\\cite{Cisco_VNI} and MediaTek white paper~\\cite{mediatek}.)}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n\\multirow{3}{*}{} & \\multicolumn{2}{l|}{\\!\\!\\!\\! Wireless devices~\\cite{Cisco_VNI}} & \\multicolumn{2}{l|}{\\!\\!\\! Wireless applications~\\cite{mediatek}\\!\\!\\!} \\\\ \\cline{2-5} \n & \\multirow{2}{*}{\\begin{tabular}[c]{@{}l@{}}\\!\\!\\! \\% Share \\\\ \\!\\!\\!(2017, 2022)\\!\\!\\!\\end{tabular}} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}l@{}}\\!\\!\\! \\% Growth \\!\\!\\!\\\\ \\!\\!\\!(2017, 2022)\\!\\!\\!\\!\\!\\end{tabular}} & \\multicolumn{2}{l|}{\\multirow{2}{*}{\\!\\!\\!\\!\\! Throughput demand (Mbps)}\\!\\!\\!} \\\\\n &  &  & \\multicolumn{2}{l|}{} \\\\ \\hline\n\\multirow{2}{*}{\\begin{tabular}[c]{@{}l@{}}\\!\\!\\! \\!Smartphones \\\\ + phablets\\!\\!\\end{tabular}} & \\multirow{2}{*}{\\!\\!\\!(50, 44)} & \\multirow{2}{*}{\\!\\!\\!(88, 93)} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}l@{}}\\!\\!\\! \\!\\!Streaming\\!\\!\\!\\!\\!\\!\\!\\!\\! \\\\ \\!\\!\\!\\!\\! video\\end{tabular}} & \\!\\!\\!Video (1.5) \\\\ \\cline{5-5} \n &  &  &  & \\!\\!\\!HD video (5) \\\\ \\hline\n\\multirow{2}{*}{\\!\\!\\!M2M} & \\multirow{2}{*}{\\!\\!\\!(11, 31)} & \\multirow{2}{*}{\\!\\!\\!(1.8, 2.2)} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}l@{}}\\!\\!\\!\\!\\! Online \\\\ \\!\\!\\!\\!\\! gaming\\end{tabular}} & \\!\\!\\!Min (1) \\\\ \\cline{5-5} \n &  &  &  & \\!\\!\\!Full (25) \\\\ \\hline\n\\multirow{3}{*}{\\!\\!\\!Nonsmartphones \\!\\!\\!\\!\\!} & \\multirow{3}{*}{\\!\\!\\!(34, 10)} & \\multirow{3}{*}{\\!\\!\\!(1.3, 0.3)} & \\multirow{3}{*}{\\begin{tabular}[c]{@{}l@{}}\\!\\!\\!\\!\\! Video \\\\\\!\\!\\! \\!\\!\\! service\\\\  \\!\\!\\!\\!\\! \\& sharing \\!\\!\\!\\end{tabular}} & \\!\\!\\!Min (0.5) \\\\ \\cline{5-5} \n &  &  &  & \\!\\!\\!HD video (2) \\\\ \\cline{5-5} \n &  &  &  & \\begin{tabular}[c]{@{}l@{}}\\!\\!\\!HD video \\\\ \\!\\!\\!sharing\\! (10)\\end{tabular} \\\\ \\hline\n\\multirow{3}{*}{\\!\\!\\!Tablets} & \\multirow{3}{*}{\\!\\!\\!(2, 3)} & \\multirow{3}{*}{\\!\\!\\!(4.6, 2.9)} & \\multirow{3}{*}{\\begin{tabular}[c]{@{}l@{}}\\!\\!\\!\\!\\! VoIP \\end{tabular}} & \\!\\!\\!Voice (0.1) \\\\ \\cline{5-5} \n &  &  &  & \\!\\!\\!HD video (1.5)\\!\\!\\!\\!\\! \\\\ \\hline\n\\multirow{2}{*}{\\!\\!\\!PCs} & \\multirow{2}{*}{\\!\\!\\!(2, 1)} & \\multirow{2}{*}{\\!\\!\\!(4.3, 1.6)} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}l@{}}\\!\\!\\!\\!\\! Social \\\\ \\!\\!\\!\\!\\! media\\end{tabular}} & \\multirow{2}{*}{\\!\\!\\!Text (0.3)} \\\\\n &  &  &  &  \\\\ \\hline\n\\end{tabular}\n\\label{tab:device_types}\n\\end{table}\n\nThe BWP setting requires the allocation of a set of contiguous bandwidth chunks. But in general, e.g., in carrier aggregation in LTE, there is no restriction on a user to use contiguous bandwidth chunks~\\cite{CA}. In this paper, we propose a general adaptive BA model that allows one to analyze both the non-contiguous bandwidth chunk allocation of LTE and the contiguous case of BWP.\n\nSuch a bandwidth adaptation depending on the user type is particularly important in future wireless networks, e.g., $5$G networks, which need to accommodate a larger variety of wireless devices (see Table~\\ref{tab:device_types}), running in turn a larger variety of wireless applications with highly heterogeneous throughput demands (see Table~\\ref{tab:device_types}). More specifically, as shown in Table~\\ref{tab:device_types}, mobile video streaming constitutes the majority of wireless traffic and requires higher data rate and hence wider spectrum allocation\nin the adaptive BA setting. In contrast, text and e-mail applications have lower data rate requirements and would thus be allocated less bandwidth. For users of the latter type, the use of wide bandwidth leads to high costs, in particular, high idling power consumption by radio-frequency (RF) and baseband signal processing circuitry. Hence, the use of different bandwidth sizes allows a balance between data rate variations and power consumed by users.\n\nOne typical use case is that of web browsing, where the user is active for a short time on wide bandwidth to accommodate the bursty traffic (download of a web page with pictures), and then stays active on narrow bandwidth for the time until it again encounters bursty traffic situation. This power-saving feature of adaptive BA helps making wide bandwidth operations energy more efficient. \n\n\nThe benefits of the flexibility offered by adaptive BA are hence multiple. Not only this adaptation saves energy for those applications and devices with lower data rate requirement or bursty traffic, but this in turn diminishes the interference incurred by other nearby devices of all types. Indeed, due to the broadcast nature of the wireless medium, users interact with each other through mutual interference. The wider the bandwidth, the higher the interference. Hence the bandwidth adaption  has two competing effects. On the positive side, a wider bandwidth increases the signal power and hence the throughput. On the negative side, it increases the interference power, which has a detrimental collective effect. As a result, it is fair to say that there is no global understanding of the effect of adaptive BA on the per-type and the overall performance. \n\nTo the best of our knowledge, there is no analytical study of a resource allocation model that  captures the features of BWP. The existing studies on BWP (discussed in Section~\\ref{sec:rel_work}) are limited to the investigation of power savings and bandwidth switching using simulations. The study of the key wireless network performance metrics is an open area. In particular, there is currently no known way to predict the effect of the BA on the performance of a user of a given type.\n\nThe main motivation of the present paper can now be stated in simple terms: it is to provide a statistical model allowing one to analyze adaptive bandwidth allocation motivated by BWP in device-to-device (D2D) wireless networks, and more precisely to predict the key performance metrics of the typical user of a given type in this context. It is appropriate to stress the analogy with the theory of differentiated services in wireline networks (DiffServ~\\cite{DiffServ}), which was instrumental in classifying and managing different types (classes) of network traffic and in predicting their interactions. The aim of this paper is to make a first step in the direction of a quantitative theory for BWP-based bandwidth allocation and the management of different types of network traffic in this class of wireless networks.\n\n\n\\subsection{Contributions}\n\\textbf{An adaptive BA model.} The first contribution of this paper is a stochastic model for adaptive BA motivated by BWP in infrastructureless wireless networks,\\footnote{In 5G NR, although the BWP feature is proposed for cellular networks, we focus on an infrastructureless network (a simpler one compared to the cellular network) since this paper is the first attempt to analytically study a bandwidth allocation scheme motivated by BWP.}  where the type of the user is determined by the number of bandwidth chunks it uses which in turn adapts to the current user needs. As already explained, our model accommodates the following two BA approaches.\n\\begin{itemize}\n\\item Contiguous BA: A user that needs $k$ bandwidth chunks can select any set of $k$ \\textit{contiguous} chunks uniformly at random. Such a contiguous bandwidth allocation may be used in the BWP setting in 5G NR.\n\\item Random BA: A user that needs $k$ bandwidth chunks can select any set of $k$ chunks uniformly at random. Such a bandwidth allocation has applications in LTE resource allocation, e.g., carrier aggregation~\\cite{CA}.\n\\end{itemize} \nAlso, our adaptive BA model allows us to quantify service differentiation in wireless networks by capturing interaction among different types of wireless users. Specifically, the model allows one to calculate the performance achieved by each type of user.\n\n\\textbf{Performance analysis.} Using tools from stochastic geometry for D2D networks, we derive analytical expressions for key wireless network performance metrics, namely, success probability, meta distribution of the signal-to-interference ratio (SIR), Shannon throughput, and Shannon throughput per Joule. These expressions permit the evaluation of per-type and overall performance metrics.\n\n\\textbf{Different performance viewpoints.} Our model allows one to analyze the bandwidth allocation from the viewpoints of both users and operators. Per-type performance, although relevant to operators, is more important for users, while overall performance might be important too from the operator's viewpoint due to, e.g., the link between this and pricing. Thus, the machinery proposed here to predict both could be useful to help an operator make a choice between the following options: 1) allocate the entire bandwidth, 2) adaptively allocate bandwidth chunks.  \n\n\\textbf{The mean model.} The proposed BA model introduces additional randomness due to the probabilistic selection of the set of bandwidth chunks depending on the user type. We show that the increased variability in traffic due to adaptive BA may improve the performance for the same mean interference and the same mean signal powers. This is particularly useful when comparing two networks based on adaptive BA but with different mix of user types. \n\n\\textbf{Service differentiation.} We show that adaptive BA leads to a roughly egalitarian service in Shannon throughput per Hz and to a linear service differentiation in aggregated Shannon throughput.\n\n\n\n\\subsection{Related work}\n\\label{sec:rel_work}\nIn wireline networks, traffic-flow characterization has received significant attention for networks consisting of users running different types of applications. To understand the behavior of heterogeneous flows and their interactions, flows have been classified based on their features, e.g., traffic size (as \\textit{elephant} and \\textit{mouse})~\\cite{traffic_size_Thompson, traffic_size_Papagiannaki, traffic_size_Estan}.\nFor wireless networks as well, in the adaptive BA setting, one can make an analogy to elephants and mice: users needing wide bandwidth can be viewed as \\textit{elephants}, while users needing small bandwidth as \\textit{mice}.\n\n\nAs alluded to earlier, the $3$GPP has very recently considered the inclusion of BWP in $5$G NR to enable spectrum flexibility and power savings. The literature on how BWP affects power savings, throughput, and reliability is very limited. For instance, \\cite{Jeon_BWP_2018} discusses power savings due to BWP. Since a user need not transmit or receive outside the bandwidth allocated to it, the user consumes less power in some scenarios, for example, involving bursty traffic. The work in~\\cite{Fuad_BWP} shows that the bandwidth switching to save power results in increased latency and decreased throughput for low load and bursty traffic. The work in~\\cite{Arslan_reliability} studies the effect of BWP on reliability and fairness in wireless networks. But these works on BWP use simulations as a tool to evaluate the performance based on the dynamic BWP management. \n\nThis paper is focused on adaptive BA for D2D networks. There exists a large number of works on spectrum allocation including multi-channel scenarios in standalone D2D networks or D2D networks sharing spectrum with cellular networks. For instance, see~\\cite{Stefanatos_BWP,Kyasanur_BWP,Sun_BWP} and references therein. In relation to heterogeneity among devices in a D2D network based on allocated bandwidth, \\cite{Yuan_BWP} focuses on the dynamic allocation of bandwidth to unlicensed users based on data rate demand, provided the allocated spectrum is unoccupied by licensed users and other unlicensed users. This results in an orthogonal bandwidth allocation to avoid mutual interference. Using a simulation approach, the work in \\cite{Hamidouche_BWP} takes battery life into account and tries to maximize the number of completed transmissions as a function of already allocated bandwidth. \n\nStochastic geometry, which is the main tool used in this paper, has been extensively used to model and analyze both infrastructureless (e.g., D2D) and infrastructure (e.g., cellular) networks~\\cite{bb_book,mh_book,SG_tut,SG_cellular}. Especially, Poisson point process (PPP)-based models are very popular for the analysis of wireless networks. Heterogeneous PPP based cellular networks were in particular discussed in \\cite{SG_het}. These models feature several types of base stations and a single class of users associating e.g. to the closest base station. In contrast, the setting discussed here features different types of transmitters with dedicated receivers and adapting their bandwidths to their needs.\n\nIn \\cite{Jindal_BWP} the authors aim to maximize the density of successful transmissions given an outage constraint at the typical user. The user always selects one bandwidth chunk uniformly at random irrespective of its need. For the same bandwidth partitioning scheme as~\\cite{Jindal_BWP}, \\cite{Haenggi_BWP_2014} analyzes the mean local delay and \\cite{sanket_globecom} exhibits the tradeoff between the density of successful transmissions and the mean local delay. As in~\\cite{Jindal_BWP}, the work in~\\cite{Lu_BWP} maximizes the density of successful transmissions, but for frequency-selective channels, where again only one bandwidth chunk is selected for the transmission. To the best of our knowledge, no concrete adaptive BA models capturing BWP features and accommodating different types of users are available in the literature.\n\n\n\n\\section{System Model}\n\\label{sec:models}\n\n\\iffalse\n\\begin{figure}\n\\centering\n\\includegraphics[scale=.45]{PBN}\n\\caption{A snapshot of the Poisson bipolar network with $\\lambda = 1\/25$ and $R = 2$. A cross `$\n\\times$' represents a transmitter and a circle `$\\circ$' represents a receiver. Arrows represent the links between transmitters and their associated receivers.}\n\\label{fig:snapshot}\n\\end{figure} \n\\fi\n\\subsection{Network model}\nWe consider infrastructureless wireless networks such as ad hoc, D2D, and machine-to-machine (M2M) networks.\\footnote{As shown in Table~\\ref{tab:device_types}, in year 2022, $31\\%$ of mobile devices  are expected to belong to M2M networks.} The transmitters are randomly located according to a homogeneous PPP $\\Phi \\subset \\mathbb{R}^2$ of intensity $\\lambda$. Each transmitter has a receiver at fixed distance $R$ in a random direction~\\cite{bb_book}. Since the homogeneous PPP is stationary, one can just focus on the reference link between a receiver at the origin $o$ and its associated transmitter at $x_0 \\in \\Phi$ with $\\|x_0\\| = R$. Averaging over $\\Phi$, this representative link becomes the {\\em typical} link in that it has the same statistical properties as those obtained by averaging over all other links in the network.\n\nA transmission is subject to some path loss, where the path-loss function is given by $\\ell(r)$ for distance $r$. Furthermore, the transmissions experience Rayleigh fading, where the channel power gain is an exponential random variable with mean $1$. Let $h_x$ denote the channel power gain between the typical receiver at the origin and the transmitter at $x \\in \\Phi$ due to fading. Then $h_x \\sim \\exp(1)$. We focus on the interference-limited scenario, where the noise power is negligible compared to the interference power.\n\\subsection{Bandwidth allocation model}\n\\label{sec:bwp_model}\nIn this subsection, we describe our two BA models.\n\nLet $W$ be the total bandwidth available to users. Without loss of generality, we assume $W = 1$. \n\n\\textbf{Random BA model.} In the case without continuity requirement, our adaptive BA model is as follows:\n\\begin{itemize}\n\\item[(1)] The total bandwidth is divided into $K$ orthogonal chunks of equal bandwidth of $1\/K$.\n\\item[(2)] Depending on how many chunks a transmitter uses, the transmitters are categorized into $K$ types. A type-$i$ transmitter selects $i$ ($ 1\\leq i \\leq K$) chunks for its transmission. In other words, the type of the user is decided by the amount of bandwidth used by that user. Let $\\mathcal{T} = \\lbrace \\mathcal{T}_1, \\mathcal{T}_2, \\dotsc, \\mathcal{T}_K \\rbrace$, where $\\mathcal{T}_i$ is the set of all subsets $\\mathcal{T}_{i,q}$ of $[K] \\triangleq \\lbrace 1, 2, \\dotsc, K\\rbrace$ of cardinality $i$, i.e., $|\\mathcal{T}_{i,q}| = i$ with $q = 1, 2, \\dotsc, {K \\choose i}$. Here, ${K \\choose i}$ is the number of possible ways of selecting $i$ chunks from $K$ chunks, i.e., $|T_i| = {K \\choose i}$. For example, for $K = 3$, a type-$2$ transmitter, i.e., $i = 2$, can select two from three chunks $1$, $2$, and $3$, i.e., we have $\\mathcal{T}_2 = \\lbrace \\lbrace 1, 2\\rbrace, \\lbrace 2, 3\\rbrace, \\lbrace1, 3\\rbrace\\rbrace$.\n\\item[(3)] A type-$i$ transmitter further selects a set of chunks from $\\mathcal{T}_i$ uniformly at random. For the aforementioned example with $K = 3$, a type-$2$ transmitter selects $2$ chunks for transmission, and it does so by selecting one of the possible sets of chunks from $\\lbrace 1, 2\\rbrace, \\lbrace 2, 3\\rbrace$, and $\\lbrace 1, 3\\rbrace$ at random.\n\\item[(4)] Each transmitter independently decides to be of $i$th type with probability $p_i$ with $\\sum_{i = 1}^{K} p_i = 1$.\n\\end{itemize}\n\n\\textbf{Contiguous BA model.} The case of contiguous bandwidth chunks is a variant of the above random BA model because the set of contiguous chunks is a subset of $\\mathcal{T}_i$. For a type-$i$ user, there are $K-i+1$ sets of contiguous chunks. Again, a natural way for type-$i$ user to select one of $K-i+1$ sets of contiguous chunks is to do so uniformly at random.\\footnote{The contiguous BA model is also random in nature but with a restriction that only contiguous bandwidth chunks are allocated.}  For example, for $K = 3$ and $i = 2$, we have $\\mathcal{T}_2 = \\lbrace \\lbrace 1, 2\\rbrace, \\lbrace 2, 3\\rbrace\\rbrace$, and the type-$2$ transmitter can select one set from $\\lbrace 1, 2\\rbrace$ and  $\\lbrace 2, 3\\rbrace$ uniformly at random.\n\n\n\\textbf{User types.} For both BA models, the probabilities $p_i$, $1 \\leq i \\leq K$, quantify service requirements. Specifically, these probabilities can be obtained from the statistical analysis of the user traffic. For instance, $p_i$ could be set according to what proportion of user's data traffic consists of large data transfer such as video streaming and what proportion consists of small traffic such as text or e-mail transmissions. As Table~\\ref{tab:device_types} in Section~\\ref{sec:bwp_intro} shows, a user's traffic consists of traffic from different wireless applications with heterogeneous throughput demands. Thus, the probabilities $p_i$ could be set according to the user's throughput demands. Let us consider a use case: \\cite{Cisco_VNI} reports that, in 2018 approximately 63\\% of the traffic of a mobile user was video content, and it will grow to 79\\% in 2022. Due to higher throughput demand for video, a mobile user is likely to request a wider bandwidth (and hence a large number of bandwidth chunks).  \n\n\n\n\\subsection{Signal-to-interference ratio (SIR)}\n\\label{sec:sir}\nIn interference-limited wireless networks, many key performance metrics are based on SIR. The SIR at the typical receiver located at the origin $o$ with respect to its associated transmitter at $x_0 \\in \\Phi$ is given by $\\mathsf{SIR}_o \\triangleq  \\frac{S}{I}$, where $S$ and $I$ are the received signal power and the interference power at the origin, respectively. \n\nThe received SIR depends on the type of the typical user because the signal and interference powers are functions of the bandwidth allocated to the user. Without loss of generality, we condition on the fact that the typical transmitter is of type $k$, i.e., it uses $1 \\leq k \\leq K$ chunks for transmission. \n\n\\textbf{Signal power}: We assume that a transmitter spends power $P$ per chunk used for a transmission.\\footnote{This assumption is in line with the proposed BWP model for $5$G, where the transmit power is mentioned in terms of the power spectral density.} Hence, $P$ is expressed in Joule-s$^{-1}$-Hz$^{-1}$. The received signal power at the typical receiver is given by $S_k = kPh_{x_0}\\ell(x_0)$, since the typical transmitter selects $k$ chunks for transmission. Here, $h_{x_0}\\sim\\exp(1)$ denotes the channel power gain on the typical link.\n\n\\textbf{Interference power}: \n Let $t_x^{(k)}$ denote the number of such overlapping  chunks between an interferer at $x \\in \\Phi$ and the typical transmitter (of type $k$). Then, the interference power at the typical receiver is given by\n\\begin{align}\nI_k = \\sum_{x \\in \\Phi\\setminus\\lbrace x_0 \\rbrace} P t_x^{(k)} h_x  \\ell(x).\n\\label{eq:intf_pow_main}\n\\end{align} \nOur assumption in this paper is that a type-$i$ interfering transmitter selects $i$ chunks with probability $p_i$ independently of other transmitters. Hence, the original PPP $\\Phi\\setminus\\lbrace x_0\\rbrace$ of interferers can be split into $K$ independent PPPs of intensities $\\lambda p_i$, $i = 1, 2, \\dotsc, K$. Let $\\Phi_{i} \\setminus \\lbrace x_0\\rbrace$ denote the PPP of type-$i$ interferers. In random BA, the typical transmitter selects a set $T_{k, u}$ of $k$ chunks at random with $1 \\leq u \\leq {K \\choose k}$. In contiguous BA, the typical transmitter selects a set $T_{k, u}$ of $k$ contiguous chunks at random with $1 \\leq u \\leq K-k+1$. Also, in random BA, a type-$i$ interferer selects a set $T_{i,v}$  of $i$ chunks with $1 \\leq v \\leq {K \\choose i}$ at random and independently. Similarly, in contiguous BA, a type-$i$ interferer selects a set $T_{i,v}$ of $i$ contiguous chunks with $1 \\leq v \\leq K-i+1$ at random and independently. Then the interference power from the transmitter at $x \\in \\Phi\\setminus \\lbrace x_0\\rbrace$ to the typical receiver is $P|T_{k, u} \\cap T_{i, v}|h_x\\ell(x)$, where $|T_{k, u} \\cap T_{i, v}|$ is the number of overlapping chunks between a type-$i$ interferer and the typical transmitter. Let $0 \\vee (i+k-K) \\leq t \\leq k \\wedge i \\leq K$ denote the number of overlapping chunks between a type-$i$ interfering transmitter and the typical transmitter, where $0 \\vee (i+k-K)$ means $\\max(0, i+k-K)$ and $k \\wedge i$ means $\\min(k, i)$. Note that $t$ is a random variable since the typical and interfering transmitters select $k$ and $i$ chunks, respectively, uniformly at random. For notation simplicity, we do not always indicate the dependence of $t$ on $k$ and $i$. \n\nDepending on the number of overlapping chunks $t$, the PPP $\\Phi_i$ can further be partitioned into $t$ independent PPPs denoted by $\\Phi_{i, t}$. The PPP $\\Phi_{i, t}$ corresponds to transmitters located at $x \\in \\Phi_i$ that have $0 \\vee (i+k-K) \\leq t \\leq k \\wedge i$ chunks in common with the typical transmitter. Consequently, the interference power received at the typical receiver at the origin from type-$i$ interferers is given by\n\\begin{align}\nI_{k,i} =  \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\sum_{x \\in \\Phi_{i,t}\\setminus \\lbrace x_0 \\rbrace} tPh_x\\ell(x).\n\\label{eq:intf_pow_i}\n\\end{align}\nThe total interference power at the typical receiver of type $k$ follows as $I_k = \\sum_{i = 1}^{K} I_{k,i}$.\n\n\n\n\\textbf{SIR expression}: Following the expressions of signal and interference powers, the SIR experienced at the typical user of type $k$ can be expressed as\n\\begin{align}\n\\mathsf{SIR}_o^{(k)} = \\frac{kh_{x_0}\\ell(x_0)}{\\sum_{i = 1}^{K} \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\sum_{x \\in {\\Phi_{i,t}}\\setminus \\lbrace x_0 \\rbrace} th_x\\ell(x)}.\n\\label{eq:sir_main_ex}\n\\end{align}\nThe transmit power $P$ per chunk vanishes from \\eqref{eq:sir_main_ex}. \n\n\n\n\n\n\n\\section{Performance Metrics}\nIn this section, we define and discuss the three performance metrics that we consider in this work. These metrics are based on the SIR received at the typical receiver and shed light on different aspects of the performance of users. \n\nAlthough the definitions of the performance metrics are general, the calculation of these metrics depends on the type of the typical user. The overall performance can be evaluated by unconditioning with respect to the type of the typical transmitter. For instance, let $f_k$ be some performance function conditioning on the fact that the typical transmitter is of type $k$. Then, the overall performance $f$ of the typical user is given by $f = \\sum_{k = 1}^{K} p_k f_k$. For the notation simplicity, in this section, we do not explicitly show the dependence of the performance metrics on $k$ while defining them. However, we shall reintroduce the parameter `$k$' in the performance analysis done in Section~\\ref{sec:perf_ana}. \n\n\\subsection{Success probability}\n\\begin{definition}[Success probability]\nThe success probability of the typical user at the origin $o$ is the complementary cumulative distribution function (ccdf) of the SIR, which is\n\\begin{align}\np_{\\rm s}(\\theta) \\triangleq \\mathbb{P}^{!}_{o}(\\mathsf{SIR}_o > \\theta),\n\\label{eq:p_suc}\n\\end{align}\nwhere $\\theta \\in \\mathbb{R}^{+}$ is the target SIR threshold. \n\\end{definition}\n\\noindent Here, $\\mathbb{P}^{!}_{o}(\\cdot)$ denotes the reduced Palm probability of the receiver point process. $p_{\\rm s}$ is an outage-based performance metric: if the received SIR at the typical receiver is larger than $\\theta$, the transmission is considered successful. \n\nWhen the underlying point process is ergodic, $p_{\\rm s}$ can also be interpreted as the fraction of concurrent transmissions that achieve an SIR greater than $\\theta$ in each realization of the network. In other words, $p_{\\rm s}$ is nothing but a {\\em spatial average} in that it is evaluated by taking a certain expectation over the point process. This average is certainly very useful in wireless networks, but it does not provide information about individual user success probabilities. \nHence, to analyze the fine-grained performance, we need to quantify how individual user success probabilities are distributed around the average $p_{\\rm s}$. The meta distribution of the SIR defined below is one such fine-grained performance metric in wireless networks~\\cite{Haenggi_MD_2016}.\n\n\\subsection{Meta distribution of the SIR}\n\\label{sec:md_sir}\nWe are interested in the random variable $P_{\\rm s}$ defined as $P_{\\rm s}(\\theta, \\Phi) \\triangleq \\mathbb{P}(\\mathsf{SIR} > \\theta \\mid  \\Phi)$, where the conditional probability is taken over the fading and the random channel access scheme of interferers determined by the BA model. This conditional random variable is the probability that the fading and the random channel access scheme yielding an SIR at least $\\theta$ for the user under consideration for a given realization of $\\Phi$. Hence, $P_{\\rm s}$ is the success probability of that user conditioned on the point process $\\Phi$. The distribution of  $P_{\\rm s}$ obtained by taking an expectation over the point process is the meta distribution of the SIR. Formally, it is defined as follows.\n\\begin{definition}[Meta distribution]\nThe meta distribution of the SIR is the distribution function\n\\begin{align}\n\\bar{F}(\\theta, x) \\triangleq \\mathbb{P}^{!}_{o}(P_{\\rm s}(\\theta, \\Phi) > x), \\quad \\theta \\in \\mathbb{R}^{+}, x \\in [0, 1].\n\\label{eq:md_def}\n\\end{align}\n\\end{definition}\n\n\\noindent The meta distribution $\\bar{F}(\\theta, x)$ is the probability that the user under consideration has a reliability at least $x$ for the target SIR threshold of $\\theta$, where the reliability  is the conditional success probability $P_{\\rm s}(\\theta, \\Phi)$. When the underlying point process is ergodic (such as the PPP), $\\bar{F}(\\theta, x)$ can be interpreted as the fraction of users that achieve the target SIR of $\\theta$ with probability at least $x$. The parameter $x$ can be viewed as the target reliability. Note that the standard success probability $p_{\\rm s}(\\theta)$ given in \\eqref{eq:p_suc} is the mean of the conditional random variable $P_{\\rm s}(\\theta, \\Phi)$. Hence, the meta distribution of the SIR $\\bar{F}(\\theta, x)$ provides much sharper SIR performance compared to its mean $p_{\\rm s}(\\theta)$.\n\n\n\\subsection{Shannon throughput}\nThe success probability $p_{\\rm s}(\\theta)$ and the meta distribution $\\bar{F}(\\theta, x)$ correspond to the binary event whether the SIR is larger than some threshold $\\theta$ or not. Hence, they fail to use the SIR values larger than $\\theta$ and reduce the SIR threshold $\\theta$ to avoid outages if the SIR value is smaller than $\\theta$. Instead, a transmitter can (if possible) adapt to channel conditions and adjust the SIR threshold $\\theta$ to the maximum value such that $\\mathsf{SIR} \\geq \\theta$ (alternatively, adapt the coding rate). In this case, Shannon throughput is a more suitable performance metric to quantify the performance in wireless networks.\n\\begin{definition}[Shannon throughput]\nFor the bandwidth $B$ used by a user, the Shannon throughput is\n\\begin{align}\n\\mathcal{R} \\triangleq B\\mathbb{E}(\\log_2(1+\\mathsf{SIR})).\n\\label{eq:shannon_throughput}\n\\end{align} \n\\end{definition}\n\\noindent The Shannon throughput is the average of the instantaneous throughput $B\\log_2(1+\\mathsf{SIR})$ of a random user in the network with adaptive coding, where the SIR corresponds to that random user. The Shannon throughput is expressed in bits\/s.\n\n\\section{Performance Analysis}\n\\label{sec:perf_ana}\nIn this section, we calculate the expressions of the performance metrics.\n\n\\subsection{Success probability}\nWe obtain a closed-form expression of the success probability. Although our analysis can be generalized to arbitrary path loss models, we first focus on the standard power-law path loss function given as $\\ell(x) = \\|x\\|^{-\\alpha}$ with $\\alpha > 2$ being the path loss exponent. We later provide a simple closed-form expression of the success probability for the bounded path loss function $\\ell(x) = (c_0 +\\|x\\|^\\alpha)^{-1}$ with $c_0 > 0$.\n\\begin{lemma}\n\\label{lem:suc_prob_k}\nConditioned on the typical transmitter being of type $k$, i.e., when $x_0 \\in \\Phi_k$, the success probability $p_{\\rm s}^{(k)}$ of the typical receiver at the origin can be expressed as\n\\begin{align}\np_{\\rm s}^{(k)}(\\theta) & = \\prod_{i = 1}^{K} p_{\\rm s} ^{(k,i)}(\\theta),\n\\label{eq:cond_suc}\n\\end{align}\nwhere $p_{\\rm s} ^{(k,i)}$ is the success probability of the typical receiver due to the interference from type-$i$ interferers only conditioned on the typical transmitter being of type $k$.\n\\end{lemma}\n\\begin{IEEEproof} \nSee Appendix~\\ref{app:suc_prob_k}.\n\\end{IEEEproof}\nThe probabilities $p_{\\rm s} ^{(k,i)}$ quantify the performance of differentiated services. \nSpecifically, as shown in the following theorem, by just playing with the values of $k$ and $i$, one can investigate how elephants (users with wider bandwidth) affect the performance of mice (users with smaller bandwidth) and vice-versa. The following theorem gives a simple closed-form expression of $p_{\\rm s} ^{(k,i)}$ that allows one to quantify the effect of type-$i$ users on the success probability of a type-$k$ user.\n\\begin{theorem}\n\\label{thm:suc_ki}\nLet us condition on the typical transmitter being a type-$k$ user. For the power-law path loss model, the success probability $p_{\\rm s} ^{(k,i)}$ of the typical user is\n\\begin{align}\np_{\\rm s} ^{(k,i)}(\\theta) = \\exp\\left(-\\lambda p_i C \\theta^{\\delta} \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}p^{(t)}_{k,i}\\left(\\frac{t}{k}\\right)^\\delta\\right),\n\\end{align}\nwhere $C \\triangleq \\pi R^2 \\Gamma(1+\\delta)\\Gamma(1-\\delta)$ with $\\delta \\triangleq 2\/ \\alpha$ and $p^{(t)}_{k,i}$ is the probability that an interferer of type $i$ has $0 \\vee (i+k-K) \\leq t \\leq k\\wedge i$ chunks in common with the typical transmitter, conditioned on the fact that the latter is a type-$k$ user. For the random BA case, we have\n\\begin{align}\np^{(t)}_{k,i} = \\frac{{k \\choose t}{K-k \\choose i-t}}{{K \\choose i}},\n\\label{eq:pt}\n\\end{align}\nwhereas for the contiguous BA case,\n\n{{\\small\\begin{align}\np^{(t)}_{k,i} =\\begin{cases}\n\\frac{2(K+t-k-i+1)}{(K-k+1)(K-i+1)} & \\mathrm{if}~t < k \\wedge i~\\mathrm{and}~K \\geq k+i-t\\\\\n0 & \\mathrm{if}~t < k \\wedge i~\\mathrm{and}~K < k+i-t\\\\\n\\frac{i-k+1}{K-k+1} & \\mathrm{if}~t = k~\\mathrm{and}~k \\leq i \\\\\n\\frac{k-i+1}{K-i+1} & \\mathrm{if}~t = i~\\mathrm{and}~k > i \\\\\n\\frac{(K-k-i+1)(K-k-i+2)}{(K-k+1)(K-i+1)} & \\mathrm{if}~t = 0~\\mathrm{and}~K \\geq k+i.\n\\end{cases} \n\\label{eq:pt_cont}\n\\end{align}}}\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix~\\ref{app:suc_ki}.\n\\end{IEEEproof}\nFinally, unconditioning on the type of the typical transmitter yields the overall success probability $p_{\\rm s} = \\sum_{k = 1}^{K}p_k p_{\\rm s}^{(k)}(\\theta)$ as\n\\begin{align}\np_{\\rm s}(\\theta) = \\sum_{k = 1}^{K} p_k \\exp\\!\\left(\\!\\!-\\lambda C \\theta^{\\delta}\\sum_{i = 1}^{K}p_i  \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}p^{(t)}_{k,i} \\left(\\frac{t}{k}\\right)^\\delta\\right).\n\\label{eq:int_suc} \n\\end{align}\n\n\\textbf{Bounded path loss function}: For the bounded path loss function $\\ell(x) = (c_0 + \\|x\\|^{\\alpha})^{-1}$ with $c_0 > 0$, we have \n{{\\small\n\\begin{align}\np_{\\rm s}^{(k,i)}(\\theta) = \\exp\\!\\left(\\!\\!- p_i C_b\\theta \\!\\!\\!\\!\\!\\!\\!\\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\!\\!\\!\\!\\!\\!\\!\\!p^{(t)}_{k,i}\\!\\left(\\frac{t}{k}\\right)\\!\\! \\!\\left(\\frac{\\theta t}{k}(c_0 + R^{\\alpha}) + c_0\\!\\right)^{\\delta{-}1}\\! \\right)\\!\\!,\n\\end{align}}}\\vspace*{-4mm}\n\n\\noindent where $C_b \\triangleq \\lambda \\pi (c_0+R^{\\alpha})\\Gamma(1+\\delta)\\Gamma(1-\\delta)$. The proof follows the one of Theorem~\\ref{thm:suc_ki}. The standard path loss function $\\ell(x) = \\|x\\|^{-\\alpha} $ is a special case of the bounded path loss function $(c_0 + \\|x\\|^{\\alpha})^{-1}$ with $c_0 = 0$. The equations \\eqref{eq:cond_suc} and \\eqref{eq:int_suc} are also valid for the bounded path loss case.\n\n\\subsection{Meta distribution of the SIR}\n\nThe direct calculation of the SIR meta distribution, i.e., the ccdf in \\eqref{eq:md_def}, is impossible. Hence we take an indirect route, where we first calculate $b$th moments ($b \\in \\mathbb{C}$) of $P_{\\rm s}$ and use those moments to obtain the ccdf $\\bar{F}(\\theta, x)$ accurately using the Gil-Pelaez theorem~\\cite{Gil_Pelaez} or approximately by matching the first and second moments to those of the beta distribution~\\cite{Haenggi_MD_2016}. In other words, averaging over the point process $\\Phi$ and averaging over the fading and the random channel access scheme are done separately. This is unlike the calculation of the success probability $p_{\\rm s}$, where averaging over $\\Phi$, the fading, and the random channel access scheme are done simultaneously.\n\n\\begin{theorem}\n\\label{thm:Mb_k}\nConditioning on the typical transmitter being of type $k$, the $b$-th moment of $P_{\\rm s}^{(k)}$ is\n{{\\small\\begin{align*}\nM_{b}^{(k)} = \\exp\\!\\!\\left(\\!\\!-2\\pi\\lambda \\!\\!\\int_{0}^{\\infty}\\!\\!\\!\\left[\\!1-\\!\\!\\left(\\sum_{i = 1}^{K}p_i\\!\\!\\!\\!\\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\!\\frac{p^{(t)}_{k,i}}{1+\\theta \\frac{t}{k}\\frac{\\ell(r)}{\\ell(x_0)}}\\!\\!\\right)^{b}\\right]\\!\\mathrm{d}r\\!\\!\\right)\\!\\!,\n\\end{align*}}}\nwhere $p^{(t)}_{k,i}$ is given by \\eqref{eq:pt} for random BA and by \\eqref{eq:pt_cont} for contiguous BA.\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix~\\ref{app:Mb_k}.\n\\end{IEEEproof}\n\n\\subsection{Shannon throughput}\n\\begin{theorem}\nConditioning on the typical transmitter being of type $k$, the Shannon throughput $\\mathcal{R}^{(k)}$ is given by\n\\begin{align}\n\\mathcal{R}^{(k)} = \\frac{k}{K} \\int_{0}^{\\infty} p_{\\rm s}^{(k)}(2^y-1)~\\mathrm{d}y,\n\\end{align}\nwhere $p_{\\rm s}^{(k)}$ is given by~\\eqref{eq:cond_suc}.\n\\end{theorem}\n\\begin{IEEEproof}\nA transmitter of type $k$ has $k\/K$ Hz of bandwidth for the transmission. Hence, we have\n\\begin{align}\n\\mathcal{R}^{(k)} &= \\frac{k}{K}\\int_{0}^{\\infty} \\mathbb{P}(\\log_2(1 + \\mathsf{SIR}_o) > y \\mid x_0 \\in \\Phi_k)~\\mathrm{d}y \\nonumber \\\\\n&= \\frac{k}{K} \\int_{0}^{\\infty} p_{\\rm s}^{(k)}(2^y-1)~\\mathrm{d}y.\n\\label{eq:Shannon_throughput_no_joule}\n\\end{align}\n\\end{IEEEproof}\n\nThe overall Shannon throughput of the typical user is hence\n$\\mathcal{R} = \\sum_{k = 1}^{K} p_k \\mathcal{R}^{(k)}.$\n\n\\textbf{Shannon throughput per Joule}: In $5$G NR, a key motivation for proposing the inclusion of BWP is power savings through adaptive bandwidth allocation. Hence, we consider Shannon throughput per Joule of the energy spent, which is the Shannon throughput divided by the spent power $kP$. Conditioning on the typical transmitter being of type $k$, Shannon throughput per Joule $\\mathcal{R}^{(k)}_{J}$ is obtained by dividing \\eqref{eq:Shannon_throughput_no_joule} by $kP$ as\n\\begin{align}\n\\mathcal{R}^{(k)}_{J} = \\frac{1}{KP}\\int_{0}^{\\infty} p_{\\rm s}^{(k)}(2^y-1)~\\mathrm{d}y.\n\\end{align}\nHere, the Shannon throughput is expressed in bits\/Joule.\n\\begin{remark}\nNote that, in this paper, the Shannon throughput per Joule is equivalent to the Shannon throughput per Hertz in that they only differ in the multiplicative constant $P$. This is due to the assumption that a transmitter spends power $P$ per chunk. Thus, for a given type of the user, normalizing by the transmit power to obtain the Shannon throughput per Joule is equivalent to normalizing by the bandwidth used by the user. \n\\end{remark}\n\n\n\n\n\n\n\\section{The Mean Model}\n\\label{sec:mean_model}\n\n\\subsection{Comparison of two networks with adaptive BA}\nSuppose the network operator has decided to use adaptive BA. There are several BA configurations to choose from based on the values of the probabilities $p_k$. From the operator's viewpoint, it is natural to investigate how two networks with different $p_k$ fare in terms on the overall performance. To fairly compare two networks employing adaptive BA, it is natural to do so for the same mean signal and the same mean interference powers.\n\n\\textbf{Mean signal power}: For our BA model, since the typical transmitter is of type $k$ with probability $p_k$, the mean signal power at the typical receiver is given by\n\\begin{equation}\n\\bar{S}_{k} = P\\ell(x_0)\\sum_{k = 1}^{K}p_kk.\n\\label{eq:mean_sig_bwp}\n\\end{equation}\nFor another network with different $p_k$, say $p_k'$, we might have to adjust the transmit power per chunk $P'$ to have the mean signal power $\\bar{S}_{k}'$ same as $\\bar{S}_{k}$. From \\eqref{eq:mean_sig_bwp}, it follows that\n\\begin{equation}\nP' = \\frac{P\\sum_{k = 1}^{K}k p_k}{\\sum_{k = 1}^{K}k p_k'} \n\\label{eq:eq:_power}\n\\end{equation}\nleads to $\\bar{S}_{k} = \\bar{S}_{k}'$.\n\n\\textbf{Mean interference power}: For our BA model, conditioned on the fact that the typical user is of type $k$, the mean interference power is obtained by taking the expectation of $I_k$ given in \\eqref{eq:intf_pow_main} as\\vspace*{-3mm}\n\n{{\\small \\begin{align}\n\\mathbb{E}[I_k]  &= \\sum_{i = 1}^{K}\\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\mathbb{E}\\left[\\sum_{x \\in \\Phi_{i,t}\\setminus \\lbrace x_0 \\rbrace} tP\\ell(x)\\right] \\nonumber\\\\\n&\\overset{(a)}{=} \\lambda\\pi \\delta c_0^{\\delta - 1}\\Gamma(\\delta)\\Gamma(1-\\delta)P\\sum_{i = 1}^{K}p_i \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i} tp_{k,i}^{(t)},\n\\label{eq:cond_mean_intf}\n\\end{align}}}\\vspace{-3mm}\n\n\\noindent where $(a)$ is obtained by the straightforward application of the Campbell's theorem for the PPP $\\Phi_{i,t}$~\\cite{bb_book} and using the bounded path loss function $\\ell(x) = (c_0 +\\|x\\|^{\\alpha})^{-1}$ with $\\delta \\triangleq 2\/\\alpha$.\\footnote{The standard path loss function $\\ell(x) = \\|x\\|^{-\\alpha}$ follows from the bounded path loss function as $c_0 \\to 0$.} Unconditioning on the type of the typical user yields the mean interference power at the typical receiver as\n\\begin{align}\n\\bar{I} = \\sum_{k = 1}^{K} p_k \\mathbb{E}[I_k].\n\\label{eq:mean_intf}\n\\end{align}\n\n\\begin{figure*}[t]\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{suc_prob_N_3_uniform_alpha_4\n\t\t\t\\caption{\\footnotesize Per-type success probability $p_{\\rm s}^{(k)}$ versus the SIR threshold $\\theta$. $p_k = 1\/K = 1\/3$.}\n\t\t\t\\label{fig:suc_prob_N_3_uniform_alpha_4}\n\t\t\\end{center}\n\t\\end{minipage}\n\t\\hspace{0.01\\linewidth}\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{overall_suc_prob_K_3_alpha_4\n\t\t\t\\caption{\\footnotesize Overall success probability $p_{\\rm s}$ versus the SIR threshold $\\theta$.}\n\t\t\t\\label{fig:overall_suc_prob_K_3_alpha_4}\n\t\t\\end{center}\n\t\\end{minipage}\n\t\\hspace{0.01\\linewidth}\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{md_basic\n\t\t\t\\caption{\\footnotesize The SIR meta distribution $\\bar{F}(\\theta, x)$ versus the reliability threshold $x$. $p_k = 1\/K = 1\/3$.}\n\t\t\t\\label{fig:md_basic}\n\t\t\\end{center}\n\t\\end{minipage}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{shannon_rate_joule_no_joule_no_mean\n\t\t\t\\caption{\\footnotesize Per-type Shannon throughput versus intensity $\\lambda$. $P = 2$ and $p_k = 1\/K = 1\/3$. For solid curves: $k = 1, 2, 3$ (from bottom to top). For dashed curves: $k = 1, 2, 3$ (from top to bottom).}\n\t\t\t\\label{fig:shannon_rate_joule_no_joule_no_mean}\n\t\t\\end{center}\n\t\\end{minipage}\n\t\\hspace{0.01\\linewidth}\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{Overall_shannon_joule_no_joule\n\t\t\t\\caption{\\footnotesize Overall Shannon throughput versus intensity $\\lambda$. $P = 2$.}\n\t\t\t\\label{fig:Overall_shannon_joule_no_joule}\n\t\t\\end{center}\n\t\\end{minipage}\n\t\\hspace{0.01\\linewidth}\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{egalitarian\n\t\t\t\\caption{\\footnotesize Per-type Shannon throughput versus the type $k$ of the user for random and contiguous bandwidth allocation. $K = 10$, $p_i = 1\/K$.}\n\t\t\t\\label{fig:Mean_interference_contiguous}\n\t\t\\end{center}\n\t\\end{minipage}\n\\end{figure*}\n\n\n\nFor another adaptive BA-based network with probabilities $p_k'$, the mean interference power $\\bar{I}'$ can be calculated by replacing $p_k$ by $p_k'$ and $\\lambda$ by $\\lambda'$ in \\eqref{eq:cond_mean_intf} and \\eqref{eq:mean_intf}, where $\\lambda'$ is the intensity of the PPP corresponding to the network with $p_k'$. The intensity $\\lambda'$ is chosen such that the mean interference powers for two networks with $p_k$ and $p_k'$ are the same, i.e., $\\bar{I} = \\bar{I}'$. Hence, from \\eqref{eq:cond_mean_intf} and \\eqref{eq:mean_intf}, it follows that\n\\begin{align}\n\\lambda' = \\frac{\\lambda P\\sum_{k = 1}^{K}p_k \\sum_{i = 1}^{K}p_i \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i} tp_{k,i}^{(t)}}{P'\\sum_{k = 1}^{K}p_k' \\sum_{i = 1}^{K}p_i' \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i} tp_{k,i}^{(t)}},\n\\label{eq:eq_lambda}\n\\end{align}\nwhich results in $\\bar{I} = \\bar{I}_K$.\n\n\\subsection{Service differentiation}\nWe now quantify the service differentiation resulting from adaptive BA. We focus on mean values of signal and interference powers to estimate the Shannon throughput (data rate) a user gets depending on its type. As shown later in the paper in Section~\\ref{sec:var}, such a mean-based approach sheds light on the effect of traffic variability on the performance.\n\\begin{proposition}\nThe random BA is roughly egalitarian in Shannon throughput per Hertz and leads to a linear service differentiation in aggregate Shannon throughput.\n\\end{proposition}\nThis proposition is based on the following observation. For a user of type $k$, the mean signal power is $k$ times the mean signal power of a user of type $1$. Also, noting that \\eqref{eq:pt} is the probability mass function of the hypergeometric distribution, the term  $\\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i} tp_{k,i}^{(t)} = ik\/K$ in \\eqref{eq:cond_mean_intf} is the mean of the hypergeometric distribution. As a result, the mean interference at a user of type $k$ is $k$ times that at a user of type $1$. Hence, if we only think in terms of mean values, all users, regardless of their type, have the same mean signal-to-mean interference ratio (MSMIR), and hence roughly the same (egalitarian) Shannon throughput per Hertz.   Of course, since a user of type $k$ has a bandwidth $k$ times larger than that of a user of type 1, a user benefits linearly from its type in terms of aggregated Shannon throughput. This proposition is illustrated by Fig.~\\ref{fig:Mean_interference_contiguous}.\n\nSimilarly, for the contiguous BA as well, as Fig.~\\ref{fig:Mean_interference_contiguous} shows, the adaptive BA provides a roughly egalitarian service in Shannon throughput per Hertz and a linear service differentiation in aggregate Shannon throughput.\n\n\\section{Results and Discussions}\nIn this section, we discuss the results for the adaptive BA model. We divide this section in two parts. The first part provides results that focus on the per-type and the overall performance in the adaptive BA setting. The second part integrates the mean model discussed in Section~\\ref{sec:mean_model}.  \n\nWithout loss of generality, we assume the following model parameters. The number of bandwidth chunks is $K = 3$ unless otherwise mentioned. The intensity of the PPP is $\\lambda = 0.2$. For the bounded path loss model, the path loss exponent is $\\alpha = 4$ with $c_0 = 1$. The desired link distance is $R = 1$. Other parameters are given in the captions of the relevant figures. Unless otherwise mentioned, all plots correspond to the random BA model.\n\n\\subsection{Per-type and overall performance}\n\n\n\\textbf{Success probability.} Fig.~\\ref{fig:suc_prob_N_3_uniform_alpha_4} shows the per-type success probability. A type-$1$ user experiences a smaller interference compared to type-$2$ and type-$3$ users since it uses only one chunk selected randomly. Hence, a type-$1$ user achieves the highest success probability. As a consequence, when we uncondition on the type of the user and calculate the overall success probability, the network model where a user always selects only one chunk at random (the model with $p_1 = 1$) achieves the highest overall success probability, while a network model without adaptive BA (the case with $p_3 = 1$) performs the worst (see Fig.~\\ref{fig:overall_suc_prob_K_3_alpha_4}). Other cases of adaptive BA lie in between these two extreme cases of $p_1 = 1$ and $p_3 = 1$.\n\n\\textbf{Meta distribution.} Fig.~\\ref{fig:md_basic} plots the per-type SIR meta distribution against the reliability $x$ for different target SIR thresholds $\\theta$. The curves in~Fig.~\\ref{fig:md_basic} allow one to make precise statements about the fraction of users achieving an SIR of $\\theta$ with reliability $x$. Notice that the same trend as the per-type success probability in Fig.~\\ref{fig:suc_prob_N_3_uniform_alpha_4} holds for the SIR meta distribution, i.e., type-$1$ users outperform type-$2$ and type-$3$ users irrespective of the SIR threshold value. In other words, the fraction of type-$1$ users that achieve a reliability of $x$ for a given SIR threshold $\\theta$ is higher than the fractions of type-$2$ and type-$3$ users. For instance, the fraction of users achieving an SIR of $-5$ dB with reliability $60\\%$ is $0.78$ for type-$1$ users, $0.74$ for type-$2$ users, and $0.73$ for type-$3$ users. We skip the discussion on the overall SIR meta distribution since it follows a similar trend as that of the overall success probability in~Fig.~\\ref{fig:overall_suc_prob_K_3_alpha_4}.\n \n\\textbf{Shannon throughput.}\nFor the Shannon throughput defined in~\\eqref{eq:shannon_throughput}, as shown by solid curves in Fig.~\\ref{fig:shannon_rate_joule_no_joule_no_mean}, a new trend emerges for the per-type Shannon throughput, where a type-$3$ user outperforms type-$2$ and type-$1$ users. This trend occurs because a higher allocated bandwidth boosts the per-type Shannon throughput, which overcomes the increased interference due to higher bandwidth. Here, note that the users of higher types achieve a higher Shannon throughput at the expense of larger transmit power $kP$ (hence more power consumption), which grows linearly with $k$ (the type of the user). \n\n\\textbf{Shannon throughput per Joule.}\\footnote{Note that, as mentioned in Remark $1$, the trends observed here in the Shannon throughput per Joule hold true for the Shannon throughput per Hertz as well.}\nAs shown by dashed curves in Fig.~\\ref{fig:shannon_rate_joule_no_joule_no_mean}, the trend reverses when the per-type Shannon throughput is normalized by the transmit power, i.e., a type-$1$ user achieves a higher Shannon throughput than type-$2$ and type-$3$ users. This reveals the per-type throughput performance per Joule of energy spent, which is useful in understanding the effect of power consumption on the per-type throughput performance.\n\n\n\n\n\n\n\n\n\n\n\n\n\nFig.~\\ref{fig:Overall_shannon_joule_no_joule} shows that the intensity $\\lambda$ of the PPP plays a key role in determining the overall Shannon throughput. For small intensity $\\lambda$, as shown by solid curves in Fig.~\\ref{fig:Overall_shannon_joule_no_joule}, the network model with no adaptive BA (the model with $p_3 = 1$) achieves a higher overall Shannon throughput than two networks employing adaptive BA with $p_k = 1\/3$ and $p_1 = 1$. In contrast, for large $\\lambda$, the trend reverses in that the networks with adaptive BA outperform the network with no adaptive BA. The reason behind this behavior is that: for a small $\\lambda$, the interference power is relatively small. Hence, a wide bandwidth in the network without adaptive BA boosts the Shannon throughput and overcomes the negative impact of increased interference due to higher bandwidth. Whereas for large $\\lambda$, the effect of increased interference dominates and the network with adaptive BA achieves a higher Shannon throughput. \n\nAlso, again similar to the per-type Shannon throughput case, the network without adaptive BA achieves a higher Shannon throughput for small $\\lambda$ at the expense of larger transmit power $kP$ (hence more power consumption). Thus, when the Shannon throughput is normalized by the transmit power, the network with the least interference power (the network with $p_1 = 1$ in this case) achieves the highest Shannon throughput.\n\n\n\\begin{figure*}[t]\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{suc_prob_mean_model\n\t\t\t\\caption{\\footnotesize Overall success probability $p_{\\rm s}(\\theta)$ versus the SIR threshold $\\theta$. }\n\t\t\t\\label{fig:suc_prob_mean_model}\n\t\t\\end{center}\n\t\\end{minipage}\n\t\\hspace{0.01\\linewidth}\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{shannon_rate_3_10_7_10_N_3_mean_model\n\t\t\t\\caption{\\footnotesize Overall Shannon throughput versus intensity $\\lambda$. $P = 2$. }\n\t\t\t\\label{fig:shannon_rate_3_10_7_10_N_3_mean_model}\n\t\t\\end{center}\n\t\\end{minipage}\n\t\\hspace{0.01\\linewidth}\n\t\\begin{minipage}{0.32\\linewidth}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width = 2in,height=1.3in]{shannon_rate_3_10_7_10_N_3_Joule_mean_model\n\t\t\t\\caption{\\footnotesize Overall Shannon throughput per Joule versus intensity $\\lambda$. $P = 2$.}\n\t\t\t\\label{fig:shannon_rate_3_10_7_10_N_3_Joule_mean_model}\n\t\t\\end{center}\n\t\\end{minipage}\n\\end{figure*}\n\\textbf{Service differentiation.} Fig.~\\ref{fig:Mean_interference_contiguous} plots the per-type aggregated Shannon throughput and per-Hertz Shannon throughput against the user type $k$. The plots confirm Proposition $1$ that both random and contiguous BA result in a roughly equal Shannon throughput per Hertz to users of different types and a linear increase in aggregated throughput with user type. As shown by the dashed curves in Fig.~\\ref{fig:shannon_rate_joule_no_joule_no_mean}, this egalitarian property holds for different intensities $\\lambda$ of the PPP. Also, as $k$ increases, the Shannon throughput per Hertz decreases slightly. This can be attributed to the fact that a type-$k$ user experiences interference with a larger variance than a type-$j$ user if $k< j$, and such a higher variance is generally beneficial to most metrics as shown in~\\cite{Lee_F}. \nAnother interesting observation from Fig.~\\ref{fig:Mean_interference_contiguous} is that random BA results in a better Shannon throughput compared to contiguous BA since the former leads to higher traffic variability that benefits users in terms of data rates. We discuss this observation in more detail in Section~\\ref{sec:var}.\n\n\n\n\\subsection{Benefits of adaptive BA}\n\nWe now briefly discuss the benefits of adaptive BA from these numerical results. Let us start with the user performance viewpoint. It should be clear from Figs.~\\ref{fig:suc_prob_N_3_uniform_alpha_4} and \\ref{fig:md_basic} that smaller users (i.e., users of type 1) get a better success probability and a better meta distribution than bigger ones. Note that the improvement is more pronounced for higher SIR thresholds. Fig.~\\ref{fig:shannon_rate_joule_no_joule_no_mean} shows that the Shannon throughput per Joule is also better for smaller users than for\nbigger ones. We conclude that adaptive BA brings the expected service differentiation and protection of small users, both in terms of success probability and Shannon throughput per Joule. As for bigger users, we see in Fig.~\\ref{fig:shannon_rate_joule_no_joule_no_mean} that they nevertheless get a better Shannon throughput than smaller ones, to the\nexpense of a higher power consumption (proportional to the number of chunks they use). One also gets from first principles that the biggest users get less interference and hence a better Shannon throughput in the scenario with adaptive BA than in the scenario without adaptive BA. Hence, at least for this performance metric, adaptive BA is beneficial to bigger users as well.\n\nConsider now the point of view of operators, namely overall performance, which can be linked to revenue. When comparing the uniform case to the case with $p_3 = 1$ (no adaptive BA case) in Fig.~\\ref{fig:overall_suc_prob_K_3_alpha_4}, we see that the overall success probability is higher in the situation with adaptive BA than in the situation without.\nFig.~\\ref{fig:Overall_shannon_joule_no_joule} actually shows that the same conclusion holds for the overall Shannon throughput per Joule. We conclude that adaptive BA should be beneficial to operators as well. These conclusions are not limited to this special case with three chunks as the ordering of the curves is in fact the same when varying the number of chunks and\/or the other model parameters.\n\n\n\n\n\n\\subsection{Increasing traffic variability may improve performance}\n\\label{sec:var}\nSuppose the operator has decided to use adaptive BA and is interested in comparing the overall performance of two networks with different probabilities $p_k$. Consider a first network with uniformly distributed $p_k$, for $K = 3$, $p_k = 1\/K = 1\/3$. For the second network, also with $K = 3$, assume that $p_3 = 0.7$ and $p_1 = 0.3$. The choice of $p_k$ for the second network is inspired from the estimated proportion of the video traffic for year 2020~\\cite{Cisco_VNI}. It is expected that approximately $70\\%$ of mobile traffic will correspond to videos needing a wide bandwidth. Hence, $p_3 = 0.7$ corresponds to video traffic, while $p_1 = 0.3$ corresponds to non-video traffic due to applications requiring smaller bandwidths. For the second network, the transmit power per chunk $P'$ and the intensity $\\lambda'$ are calculated from \\eqref{eq:eq:_power} and \\eqref{eq:eq_lambda}, respectively, such that both networks have the same mean interference and the same mean signal powers.\n\nCompared to the first network, the second one has a more variable distribution of powers and hence a more variable interference. From Figs.~\\ref{fig:suc_prob_mean_model}, \\ref{fig:shannon_rate_3_10_7_10_N_3_mean_model}, and \\ref{fig:shannon_rate_3_10_7_10_N_3_Joule_mean_model}, it is clear that the second and more variable traffic network outperforms the first one in terms of all performance metrics: success probability, Shannon throughput, and Shannon throughput per Joule. \n\nFor the Shannon throughput case, the trends in both the Shannon throughput and the Shannon throughput per Joule with intensity $\\lambda$ are the same (see Figs.~\\ref{fig:shannon_rate_3_10_7_10_N_3_mean_model} and \\ref{fig:shannon_rate_3_10_7_10_N_3_Joule_mean_model}). This is different from the case without equality of the mean values as shown in Fig.~\\ref{fig:Overall_shannon_joule_no_joule}, where we observed that, for small $\\lambda$, the trend in Shannon throughput per Joule is opposite of that in Shannon throughput. Also, there is no a crossover in curves of Shannon throughput with $\\lambda$ in the model with balanced means.\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding Remarks and Future Directions}\nThis paper proposes a first analytic model for the prediction of the adaptive BA inspired by BWP. The proposed model allowed us to show that adaptive BA should benefit to both small and big users.\nWe showed that small users are well protected by adaptive BA in terms of success probability, meta distribution of the SIR, and Shannon throughput per Joule. On the other hand, big users achieve a better Shannon throughput than in the situation without adaptive BA. The analysis of overall performance allowed us to show that adaptive BA should also be beneficial to operators. Also, we observe that adaptive BA is roughly egalitarian per Hertz and leads to a linear service differentiation in aggregated Shannon throughput. \n\nThere are several future directions of research. A natural extension is to study adaptive BA in cellular settings. It would also be interesting to see how to strategically assign bandwidth chunks to users based on their local environment. Finally, this work is limited to a snapshot analysis of the network. The inclusion of dynamics will certainly be of great interest as well.\n\n\\section*{Acknowledgment}\nThe authors would like to thank Luis Guilherme Uzeda Garcia, Fuad Abinader, Andrea Marcano, and Dalia Popescu from Nokia Bell Labs, Nozay, France for the initial discussion on this problem and bringing to our attention the lack of analytical formulation of BWP.\n\n\n\n\\appendices\n\\section{Proof of Lemma~\\ref{lem:suc_prob_k}}\n\\label{app:suc_prob_k}\nConditioned on $x_0 \\in \\Phi_k$, the success probability $p_{\\rm s}^{(k)}$ is\\vspace*{-3mm}\n\n{{\\small \\begin{align*}\n&p_{\\rm s}^{(k)}(\\theta)  = \\mathbb{P}(\\mathsf{SIR}_{o}^{(k)} > \\theta \\mid x_0 \\in \\Phi_k) \\nonumber \\\\\n&= \\mathbb{P}\\!\\left(\\!h_{x_0} > \\theta \\frac{\\sum_{i = 1}^{K} \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\sum_{x \\in \\Phi_{\\Phi_{i,t}}\\setminus \\lbrace x_0 \\rbrace}th_x \\ell(x)}{k\\ell(x_0)}\\!\\right) \\nonumber \\\\\n&= \\mathbb{E}\\!\\left[\\!\\exp\\!\\left(\\!\\!-\\frac{\\theta \\sum_{i = 1}^{K} \\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\sum_{x \\in \\Phi_{\\Phi_{i,t}}\\setminus \\lbrace x_0 \\rbrace}t h_x\\ell(x)}{k\\ell(x_0)}\\!\\right)\\!\\!\\right]\\!\\!.\n\\end{align*}}}\nAveraging over interfering fading channels, it follows that\n\\begin{equation}\np_{\\rm s}^{(k)}(\\theta)  =\\prod_{i=1}^{K} \\underbrace{\\prod_{t = 0 \\vee (i+k-K)}^{k \\wedge i} \\mathbb{E}\\left[\\prod_{x \\in \\Phi_{i,t} \\setminus \\lbrace x_0 \\rbrace} \\frac{1}{1+\\frac{\\theta t \\ell(x)}{k\\ell(x_0)}}\\right]}_{p_{{\\rm s} \\mid k}^{(i)}(\\theta)}.\n\\label{eq:suc_ki_inter}\n\\end{equation}\nThe probability $p_{\\rm s}^{(k,i)}$ can be interpreted as the success probability of the typical receiver due to interference from type-$i$ interferers only conditioned on the typical transmitter being of type $k$. The reason behind this interpretation is that the expression of $p_{\\rm s}^{(k,i)}$ in \\eqref{eq:suc_ki_inter} can be obtained by calculating $\n\\mathbb{P}\\left(\\frac{S}{I_{k,i}} > \\theta\\right) = \\mathbb{P}\\left(\\mathsf{SIR}_o^{(k, i)} > \\theta\\right)$, where $S$ is signal power and $I_{k,i}$ given by \\eqref{eq:intf_pow_i} is the interference power received at the typical receiver only from type-$i$ interferers. Thus, $\\mathsf{SIR}_o^{(k,i)}$ is the SIR at the typical receiver when the interference from type-$i$ interferers only is taken into account.\n\n\\section{Proof of Theorem~\\ref{thm:suc_ki}}\n\\label{app:suc_ki}\nContinuing from \\eqref{eq:suc_ki_inter}, we have\n\\begin{align}\np_{\\rm s}^{(k,i)}(\\theta) = \\prod_{t = 0 \\vee (i+k-K)}^{k \\wedge i} \\mathbb{E}\\left[\\prod_{x \\in \\Phi_{i,t} \\setminus \\lbrace x_0 \\rbrace} \\frac{1}{1+\\frac{\\theta t \\ell(x)}{k\\ell(x_0)}}\\right].\n\\end{align}\nFor the power-law path loss model $\\ell(r) = r^{-\\alpha}$, by the probability generating functional (PGFL) of the PPP, it follows that\n\\begin{align}\np_{\\rm s}^{(k,i)}(\\theta) = \\hspace*{-6mm}\\prod_{t =0 \\vee (i+k-K)}^{k \\wedge i}\\hspace*{-6mm}\\exp\\left(-\\lambda_{i,t}\\int_{\\mathbb{R}^2}\\left(1- \\frac{1}{1+\\frac{t\\theta R^{\\alpha} \\|x\\|^{-\\alpha}}{k}}\\right)\\mathrm{d}x\\right)\\!\\!,\n\\label{eq:suc_ki_int}\n\\end{align}\nwhere $\\lambda_{i,t}$ is the intensity of the point process $\\Phi_{i,t}$. Here, $\\lambda_{i,t} = \\lambda_i p_{k,i}^{(t)}$ since the PPP $\\Phi_i$ of interferers of type $i$ is partitioned into $t$ independent PPPs of interferers of type $i$ having $t$ common chunks with the typical transmitter. Solving the integral in \\eqref{eq:suc_ki_int} and substituting $\\lambda_i = \\lambda p_i$, we have the desired expression of $p_{\\rm s}^{(k,i)}$.\n\n\n\\section{Proof of Theorem~\\ref{thm:Mb_k}}\n\\label{app:Mb_k}\nConditioning on the typical transmitter $x_0$ being of type $k$, the conditional success probability $P_{\\rm s}^{(k)}$ is given as\n\\begin{align*}\nP_{\\rm s}^{(k)}(\\theta) &= \\mathbb{P}(\\mathsf{SIR}_o^{(k)} > \\theta \\mid \\Phi) \\nonumber \\\\\n&=  \\mathbb{P}\\left(h_{x_0} > \\theta \\frac{\\sum_{x \\in \\Phi\\setminus\\lbrace x_0 \\rbrace} t_x h_x  \\ell(x)}{k\\ell(x_0)} \\mid \\Phi\\right)  \\\\ \n& = \\prod_{x \\in \\Phi\\setminus\\lbrace x_0\\rbrace} \\mathbb{E}\\left[\\exp\\left(-\\theta \\frac{ t_x h_x  \\ell(x)}{k\\ell(x_0)}\\right) \\mid  \\Phi\\right],\n\\end{align*}\nwhere the expectation is taken over the random channel access scheme of interferers determined by the BWP model and the fading. Recall that, for the BWP model, each interferer can be of type $i$ with probability $p_i$. Then, a type-$i$ interferer can have $ 0 \\vee (i+k-K)\\leq t \\leq k \\wedge i$ chunks common with the typical transmitter with probability $p^{(t)}_{k,i}$. Thus, by averaging over the channel access scheme, it follows that\\vspace*{-3mm} \n\n{{\\small \\begin{align*}\nP_{\\rm s}^{(k)}(\\theta) = \\hspace*{-3mm}\\prod_{x \\in \\Phi  \\setminus \\lbrace x_0 \\rbrace}\\!\\!\\!\\mathbb{E}\\!\\left[\\sum_{i = 1}^{K}p_i\\hspace*{-3mm}\\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\hspace*{-3mm}p^{(t)}_{k,i}\\exp\\!\\left(-\\theta \\frac{ t h_x  \\ell(x)}{k\\ell(x_0)}\\right)\\mid \\Phi\\right]\\!\\!.\n\\end{align*}}}\nNow, averaging over the fading on interfering channels yields\n\\begin{align*}\nP_{\\rm s}^{(k)}(\\theta) =  \\prod_{x \\in \\Phi  \\setminus \\lbrace x_0 \\rbrace}\\sum_{i = 1}^{K}p_i\\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\frac{p^{(t)}_{k,i}}{1+\\theta \\frac{t}{k}\\frac{\\ell(x)}{\\ell(x_0)}}.\n\\end{align*}\nThe $b$th ($b \\in \\mathbb{C}$) moment of $P_{\\rm s}^{(k)}$ can be expressed as\n{{\\small\\begin{align*}\nM_{b}^{(k)} = \\exp\\!\\!\\left(\\!\\!-2\\pi\\lambda \\!\\!\\int_{0}^{\\infty}\\!\\!\\left[\\!1-\\!\\left(\\sum_{i = 1}^{K}p_i\\hspace*{-3mm}\\sum_{t = 0 \\vee (i+k-K)}^{k \\wedge i}\\hspace*{-1mm}\\frac{p^{(t)}_{k,i}}{1+\\theta \\frac{t}{k}\\frac{\\ell(r)}{\\ell(x_0)}}\\!\\!\\right)^{b}\\right]\\!\\mathrm{d}r\\!\\!\\right)\\!\\!,\n\\end{align*}}}\nwhen making use of the PGFL of the PPP.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\nIn this article we present a new expression for certain elements in the Hecke algbra, related to the $q$-analogue of the Young symmetrizer, which generate its irreducible representations. We will obtain this new expression using Cherednik's fusion\nprocedure. This method originates from the work of Jucys\n\\cite{J}, and has already been used by Nazarov and Tarasov\n\\cite{N1, N2, NT}. However our approach differs by minimising the\nnumber of auxiliary parameters needed in the fusion procedure.\nThis is done by considering hooks of Young diagrams, rather than\ntheir rows or columns as in \\cite{N1, N2, NT}.\n\n\nLet $H_n$ be the finite \ndimensional Hecke algebra over the\nfield $\\mathbb{C}(q)$ of rational functions in $q$,\nwith the generators $T_1, \\ldots , T_{n-1}$ and the relations \n\\begin{equation}\\label{Hecke1}\n(T_i-q)(T_i+q^{-1})=0;\n\\end{equation}\n\\begin{equation}\\label{Hecke2}\nT_i T_{i+1} T_i=T_{i+1} T_i T_{i+1};\n\\end{equation}\n\\begin{equation}\\label{Hecke3}\nT_i T_j=T_j T_i,\n\\quad\nj\\neq i, i+1\n\\end{equation}\nfor all possible indices $i$ and $j$.\n\nThe generators $T_1, \\ldots , T_{n-1}$ are invertible since \n\\begin{equation}\\label{Tinverse}\nT^{-1}_i=T_i-q+q^{-1}\n\\end{equation}\ndue to (\\ref{Hecke1}). \n\nFor any index $i=1, \\ldots , n-1$ let $\\sigma_i=(i, i+1)$ be \nthe adjacent transposition in the symmetric group $S_n$. \nTake any element $\\sigma \\in S_n$\nand choose a reduced decomposition $\\sigma=\\sigma_{i_1}\\ldots \\sigma_{i_l}$.\nAs usual put $T_\\sigma=T_{i_1}\\ldots T_{i_l}$,\nthis element of the algebra $H_n$ does not depend on the\nchoice of reduced decomposition of $\\sigma$ due to (\\ref{Hecke2}) and\n(\\ref{Hecke3}).  The element of maximal length in\n$S_n$ will be denoted by $\\sigma_0$. We will write $T_0$ instead of \n$T_{\\sigma_0}$ for  short. The elements $T_\\sigma$ form a basis of $H_n$ \nas a vector space over the field $\\mathbb{C}(q)$. We will also use the \nbasis in $H_n$  formed by the elements~$T_\\sigma^{-1}$.\n\n\n\nA \\emph{partition} of $n$ is a sequence of weakly decreasing integers $\\lambda_1 \\geqslant \\lambda_2 \\geqslant \\cdots \\geqslant \\lambda_k$ whose sum is equal to $n$. The \\emph{Young diagram} of a\npartition $\\lambda$ is the set of boxes $(i, j) \\in \\mathbb{Z}^2$\nsuch that $1 \\leqslant j \\leqslant \\lambda_i$. In drawing such\ndiagrams we let the first coordinate $i$ increase as one goes\ndownwards, and the second coordinate $j$ increase from left to\nright. For example the partition $\\lambda = (3,3,2)$ gives the\ndiagram \\[ \\yng(3,3,2)\\]\n\nIf $(i,j)$ is a box in the diagram of $\\lambda$, then the\n$(i,j)$-\\emph{hook} is the set of boxes in $\\lambda$\n\\[  \\{ (i, j') : j' \\geqslant j \\} \\cup \\{ (i', j)\n: i' \\geqslant i \\}, \\] We call the $(i,i)$-hook the\n\\emph{$i^{\\mbox{\\small th}}$ principal hook}.\n\nA \\emph{standard tableau}, $\\Lambda$, is a filling of the diagram\n$\\lambda$ in which the entries are the numbers 1 to $n$, each\noccurring once. If the box $(i,j)$ contains $a$ we define the \\emph{content} of the box to be $c_a(\\Lambda) = j-i$.\n\n\nThe $\\mathbb{C}(q)$-algebra $H_n$ is semisimple; see \\cite[Section 4]{GU}\nfor a short proof of this well known fact. \nThe simple ideals of $H_n$ are\nlabeled by partitions $\\lambda$ of $n$, like the equivalence classes of \nirreducible representations of the symmetric group $S_n$.\n\nIn this article, for any standard tableau $\\Lambda$ of shape $\\lambda$ we \nwill construct\na certain non-zero element $F_\\Lambda\\in H_n$. Under left multiplication by\nthe elements of $H_n$, the left ideal $H_nF_\\Lambda\\subset H_n$ is an \nirreducible $H_n$-module. The $H_n$-modules $V_\\lambda$\nfor different partitions $\\lambda$ are pairwise non-equivalent; see\nCorollary \\ref{C3.6}. At $q=1$, the algebra $H_n(q)$ specializes\nto the group ring $\\mathbb{C}S_n$, where $T_\\sigma$ becomes the permutation $\\sigma \\in S_n$.  The $H_n(q)$-module\n$V_\\lambda$ then specializes to the irreducible representation of \n$S_n$, coresponding to the partition $\\lambda$, \\cite{Y1}.\n\nOur construction of $V_\\lambda$ employs a certain limiting process called \nthe \\emph{fusion procedure}. The idea of this construction goes \nback to \\cite[Section 3]{C2} were no proofs were given however.\nThe element $F_\\Lambda$ is related to the $q$-analogue\nof the Young symmetrizer in the group ring $\\mathbb{C} S_n$\nconstructed in \\cite{Gy}.\n\n\n\n\nFor each $i=1, \\ldots , n-1$ introduce the $H_n$-valued rational function in \ntwo variables $a \\neq 0$, $a \\neq b \\in\\mathbb{C}(q)$\n\\begin{equation}\\label{q-smallf}\nF_i(a, b)=T_i+\\frac{q-q^{-1}}{a^{-1}b - 1}.\n\\end{equation}\n\n\nNow introduce $n$ variables $z_1, \\ldots , z_n\\in\\mathbb{C}(q)$.\nEquip the set of all pairs $(i, j)$\nwhere $1\\leqslant i<j\\leqslant n$, with the following ordering.\nThe pair $(i, j)$ precedes another pair\n$(i',j')$\nif $j<j'$, or if $j=j'$ but $i<i'$. Call this the \\emph{reverse-lexicographic} ordering.\nTake the ordered product\n\\begin{equation}\\label{q-bigf}\nF_\\Lambda(z_1, \\ldots , z_n) = \\prod_{(i,j)}^{\\rightarrow}\\ F_{j-i}\n\\bigl( q^{2c_i(\\Lambda)}z_i, q^{2c_j(\\Lambda)}z_j\\bigr)\n\\end{equation}\nover this set. Consider the product (\\ref{q-bigf})\nas a rational function taking values in $H_n$,\nof the variables $z_1, \\ldots , z_n$.\nIf $i$ and $j$ sit on\nthe same diagonal in the tableau $\\Lambda$, then $F_{j-i}\n\\bigl( q^{2c_i(\\Lambda)}z_i, q^{2c_j(\\Lambda)}z_j\\bigr)$ has a pole at $z_i = z_j \\neq 0$.\n\nLet $\\mathcal{R}_\\Lambda$ be the vector subspace in $\\mathbb{C}(q)^n$\nconsisting of all tuples $(z_1, \\dots , z_n)$ such that $z_i =\nz_j$ whenever the numbers $i$ and $j$ appear in the same row of\nthe tableau $\\Lambda$.\n\nAs a direct calculation using (\\ref{Hecke1}) and (\\ref{Hecke2}) shows,\nthese functions satisfy \n\\begin{equation}\\label{q-triple}\nF_i(a, b)F_{i+1}(a, c)F_i(b, c)=\nF_{i+1}(b, c)F_i(a, c)F_{i+1}(a, b).\n\\end{equation}\nDue to (\\ref{Hecke3}) these rational functions also satisfy the relations\n\\begin{equation}\\label{q-commute}\nF_i(a, b) F_j(c, d)=F_j(c, d) F_i(a, b);\n\\qquad\nj\\neq i, i+1.\n\\end{equation}\n\nUsing (\\ref{q-triple}) and (\\ref{q-commute}) we may reorder the\nproduct $F_\\Lambda(z_1, \\dots , z_n)$ such that each singularity\nis contained in an expression known to be regular at $z_1 = z_2 =\n\\dots = z_n \\neq 0$, \\cite{N2}. It is by this method that it was shown\nthat the restriction of the rational function $F_\\Lambda(z_1,\n\\dots ,z_n)$ to the subspace $\\mathcal{R}_\\Lambda$ is regular at\n$z_1 = z_2 = \\dots = z_n \\neq 0$. Furthermore the following theorem was proved;\n\n\\begin{theorem} \\label{q-MNtheorem} Restriction to $\\mathcal{R}_\\Lambda$ of the\nrational function $F_\\Lambda(z_1, \\dots , z_n)$ is regular at $z_1\n= z_2 = \\dots = z_n \\neq 0$ and has value $F_\\Lambda \\in H_n$. The left ideal generated by this element is irreducible, and the $H_n$-modules for different partitions $\\lambda$ are pairwise non-equivalent.\n\\end{theorem}\n\n\nSimilarly, we may form another expression for $F_\\Lambda$ by\nconsidering the subspace in $\\mathbb{C}(q)^n$ consisting of all\ntuples $(z_1, \\dots , z_n)$ such that $z_i = z_j$ whenever the\nnumbers $i$ and $j$ appear in the same column of the tableau\n$\\Lambda$ \\cite{N1}.\n\n\nIn this article we present a new expression for the element $F_\\Lambda$ which minimises the number of auxiliary parameters\nneeded in the fusion procedure. We do this by considering hooks of\nstandard tableaux rather than their rows or columns.\n\nLet $\\mathcal{H}_\\Lambda$ be the vector subspace in $\\mathbb{C}(q)^n$\nconsisting of all tuples $(z_1, \\dots , z_n)$ such that $z_i =\nz_j$ whenever the numbers $i$ and $j$ appear in the same principal\nhook of the tableau $\\Lambda$. We will prove the following\ntheorem.\n\n\n\\begin{theorem}\\label{q-fulltheorem} Restriction to $\\mathcal{H}_\\Lambda$ of the\nrational function $F_\\Lambda(z_1, \\dots , z_n)$ is regular at $z_1\n= z_2 = \\dots = z_n \\neq 0$ and has value $F_\\Lambda \\in H_n$. The left ideal generated by this element is irreducible, and the $H_n$-modules for different partitions $\\lambda$ are pairwise non-equivalent. The element is the same as the element in Theorem \\ref{q-MNtheorem}.\n\\end{theorem}\n\n\nIn particular, this hook fusion procedure can be used to form\nirreducible representations of $H_n$ corresponding to Young\ndiagrams of hook shape using only one auxiliary parameter, $z$. By\ntaking this parameter to be 1 we find that no parameters are\nneeded for diagrams of hook shape. Therefore if $\\nu$ is a\npartition of hook shape, and $N$ a standard tableau of shape $\\nu$, we have\n\\begin{equation}\\label{q-bigfhook} F_N = F_N(z) =\n\\prod_{(p,q)}^\\rightarrow F_{j-i}(q^{2c_i(\\Lambda)}, q^{c_j(\\Lambda)}), \\end{equation} with\nthe pairs $(i, j)$ in the product ordered reverse-lexicographically.\n\n\nTo motivate the study of modules corresponding to partitions of\nhook shape first let us consider  the \\emph{Jacobi-Trudi\nidentities} \\cite[Chapter I3]{MD}. There is an isomorphism from the ring of symmetric functions to the Grothendieck ring of representations of the Hecke algebra. Therefore we can think of the following\nidentities as dual to the Jacobi-Trudi identities.\n\nIf $\\lambda = (\\lambda_1, \\dots , \\lambda_k)$ such that $n =\n\\lambda_1 + \\dots + \\lambda_k$ then we have the following\ndecomposition of the induced representation of the tensor product\nof modules corresponding to the rows of $\\lambda$;\n\\[ \\textrm{Ind}_{H_{\\lambda_1} \\times H_{\\lambda_2} \\times \\cdots\n\\times H_{\\lambda_k}}^{H_n } V_{(\\lambda_1)} \\otimes\nV_{(\\lambda_2)} \\otimes \\cdots \\otimes V_{(\\lambda_k)} \\cong\n\\bigoplus_{\\mu} (V_\\mu)^{\\oplus K_{\\mu \\lambda}},\\] where the sum\nis over all partitions of $n$. Note that $V_{(\\lambda_i)}$ is the\ntrivial representation of $H_{\\lambda_i}$, that sends generators $T_i$ to $q$. The coefficients\n$K_{\\mu \\lambda}$ are non-negative integers known as \\emph{Kostka\nnumbers}, \\cite{MD}. Importantly, we have $K_{\\lambda \\lambda} =\n1.$\n\nOn the subspace $\\mathcal{R}_\\Lambda$, if $z_i \/ z_j \\notin\nq^\\mathbb{Z}$ when $i$ and $j$ are in different rows of $\\Lambda$\nthen the above induced module may be realised as the left ideal in\n$H_n$ generated by $F_\\Lambda(z_1, \\dots, z_n)$.\\\\\nThe irreducible representation $V_\\lambda$ appears in the\ndecomposition of this induced module with coefficient 1, and is\nthe ideal of $H_n$ generated by $F_\\Lambda(z_1, \\dots ,\nz_n)$ when $z_1 = z_2 = \\dots = z_n \\neq 0$. The fusion procedure of\nTheorem \\ref{q-MNtheorem} provides a way of singling out this\nirreducible component.\n\nSimilarly we have the equivalent identity for columns, \\[\n\\textrm{Ind}_{H_{\\lambda'_1} \\times H_{\\lambda'_2} \\times \\cdots\n\\times H_{\\lambda'_l}}^{H_n } V_{(1^{\\lambda'_1})} \\otimes\nV_{(1^{\\lambda'_2})} \\otimes \\cdots \\otimes V_{(1^{\\lambda'_l})}\n\\cong \\bigoplus_{\\mu} (V_{\\mu})^{\\oplus K_{\\mu' \\lambda'}},\\]\nwhere $l$ is the number of columns of $\\lambda$. In this case\n$V_{(1^{\\lambda'_i})}$ is the alternating representation of\n$H_{\\lambda'_i}$. This induced module is isomorphic to the left\nideal of $H_n$ generated by $F_{\\Lambda}(z_1, \\dots,\nz_n)$ considered on the subspace $\\mathcal{R}_{\\Lambda'}$, with\n$z_i \/ z_j \\notin q^\\mathbb{Z}$ when $i, j$ are in different columns\nof $\\Lambda$. Again the irreducible representation $V_{\\lambda}$\nappears in the decomposition of this induced module with\ncoefficient 1, and is the ideal of $H_n$ generated by\n$F_{\\Lambda}(z_1, \\dots, z_n)$ when $z_1 = z_2 = \\dots = z_n \\neq 0$.\n\nThere is another expression known as the \\emph{Giambelli identity}\n\\cite{Gi}. Unlike the Jacobi-Trudi identities, this identity\ninvolves splitting $\\lambda$ into its principal hooks, rather than\nits rows or columns. A combinatorial proof of the Giambelli\nidentity can be found in \\cite{ER}.\n\nDivide a Young diagram $\\lambda$ into boxes with positive and\nnon-positive content. We may illustrate this on the Young diagram\nby drawing 'steps' above the main diagonal. Denote the boxes above\nthe steps by $\\alpha(\\lambda)$ and the rest by $\\beta(\\lambda)$.\nFor example, the following figure illustrates $\\lambda$,\n$\\alpha(\\lambda)$ and $\\beta(\\lambda)$ for $\\lambda = (3,3,2)$.\n\n\\begin{figure}[h]\n\\label{steps}\n\\begin{center}\n\\begin{picture}(150, 60)\n\\put(0,30){\\framebox(15,15)[r]{  }}\n\\put(15,30){\\framebox(15,15)[r]{ }}\n\\put(30,30){\\framebox(15,15)[r]{  }}\n\n\\put(0,15){\\framebox(15,15)[r]{  }}\n\\put(15,15){\\framebox(15,15)[r]{  }}\n\\put(30,15){\\framebox(15,15)[r]{  }}\n\n\\put(0,0){\\framebox(15,15)[r]{  }}\n\\put(15,0){\\framebox(15,15)[r]{}}\n\n\n\\put(72.5,30){\\framebox(15,15)[r]{  }}\n\\put(87.5,30){\\framebox(15,15)[r]{ }}\n\\put(125,30){\\framebox(15,15)[r]{ }}\n\n\\put(87.5,15){\\framebox(15,15)[r]{  }}\n\\put(125,15){\\framebox(15,15)[r]{  }}\n\\put(140,15){\\framebox(15,15)[r]{  }}\n\n\\put(125,0){\\framebox(15,15)[r]{ }}\n\\put(140,0){\\framebox(15,15)[r]{}}\n\n\n\\put(20,50){$\\lambda$} \\put(77.5,50){$\\alpha(\\lambda)$}\n\\put(130,50){$\\beta(\\lambda)$}\n\n\\linethickness{2pt} \\put(0,45){\\line(1,0){15}}\n\\put(15,30){\\line(1,0){15}} \\put(30,15){\\line(1,0){15}}\n\n\\put(15,30){\\line(0,1){15}} \\put(30,15){\\line(0,1){15}}\n\n\\end{picture}\n\\end{center}\n \\caption{The Young diagram $(3,3,2)$ divided into boxes with positive content and non-positive content}\n \\end{figure}\n\n\nIf we denote the rows of $\\alpha(\\lambda)$ by $\\alpha_1 > \\alpha_2\n> \\dots > \\alpha_d > 0$ and the columns of $\\beta(\\lambda)$\nby $\\beta_1 > \\beta_2 > \\dots > \\beta_d > 0$, then we have the\nfollowing alternative notation for $\\lambda$;\n\\[ \\lambda = ( \\alpha | \\beta ), \\]\nwhere $\\alpha = (\\alpha_1, \\dots , \\alpha_d)$ and $\\beta =\n(\\beta_1, \\dots , \\beta_d)$.\n\nHere $d$ denotes the length of the side of the \\emph{Durfee\nsquare} of shape $\\lambda$, which is the set of boxes\ncorresponding to the largest square that fits inside $\\lambda$,\nand is equal to the number of principal hooks in $\\lambda$. In our\nexample $d=2$ and $\\lambda = (2, 1 | 3, 2)$.\n\nWe may consider the following identity as a dual of the Giambelli\nidentity.\n\\[ \\textrm{Ind}_{H_{h_1} \\times H_{h_2} \\times \\cdots\n\\times H_{h_d}}^{H_n} V_{(\\alpha_1 | \\beta_1)} \\otimes\nV_{(\\alpha_2 | \\beta_2)} \\otimes \\cdots \\otimes V_{(\\alpha_d |\n\\beta_d)} \\cong \\bigoplus_{\\mu} (V_{\\mu})^{\\oplus D_{\\mu\n\\lambda}},\\] where $h_i$ is the length of the $i^{\\mbox{\\small th}}$\nprincipal hook, and the sum is over all partitions of $n$. This is\na decomposition of the induced representation of the tensor\nproduct of modules of hook shape. Further these hooks are the\nprincipal hooks of $\\lambda$. The coefficients, $D_{\\mu \\lambda}$,\nare non-negative integers, and in particular $D_{\\lambda \\lambda}\n=1$.\n\nOn the subspace $\\mathcal{H}_\\Lambda$, if $z_i \/ z_j \\notin\nq^\\mathbb{Z}$ when $i$ and $j$ are in different principal hooks of\n$\\Lambda$ then the above induced module may be realised as the\nleft ideal in\n$H_n$ generated by $F_\\Lambda(z_1, \\dots, z_n)$.\\\\\nThe irreducible representation $V_\\lambda$ appears in the\ndecomposition of this induced module with coefficient 1, and is\nthe ideal of $H_n$ generated by $F_\\Lambda(z_1, \\dots ,\nz_n)$ when $z_1 = z_2 = \\dots = z_n$.\n\nHence, in this way,  our hook fusion procedure relates to the\nGiambelli identity in the same way that Cherednik's original\nfusion procedure relates to the Jacobi-Trudi identity. Namely, it\nprovides a way of singling out the irreducible component\n$V_\\lambda$ from the above induced module.\n\n\nThe fusion procedure was originally developed in the study of\naffine Hecke algebras, \\cite{C1}. Our results may be regarded as an\napplication of the representation theory of these algebras,\n\\cite{OV}. Descriptions of the fusion procedure for the Symmetric group may be found in \\cite{NT} and \\cite{GP}. The hook fusion procedure for the Symmetric group was considered in \\cite{Gr}.\n\n\nAcknowledgements and thanks go to Maxim Nazarov for his\nsupervision, and for introducing me to this subject. I would also\nlike to thank EPSRC for funding my research.\n\n\\section{Fusion Procedure for a Young Diagram}\n\n\n\nWe fill a diagram $\\lambda$ by hooks to form a tableau $\\Lambda^\\circ$\nin the following way: For the first principal hook we fill the\ncolumn with entries $1$, $2$, \\dots , $r$ and then fill the row\nwith entries $r+1$, $r+2$, \\dots , $s$. We then fill the column of\nthe second principal hook with $s+1$, $s+2$, \\dots , $t$ and fill\nthe row with $t+1$, $t+2$, \\dots , $x$. Continuing in this way we\nform the hook tableau.\n\n\n{\\addtocounter{definition}{1} \\bf Example \\thedefinition .} On the\nleft is the hook tableau of the diagram $\\lambda = (3,3,2)$, and\non the right the same diagram with the content of each box.\n\n\\begin{normalsize}\n\n\\begin{center}\n\\begin{picture}(50,50)\n\\put(0,30){\\framebox(15,15)[r]{ 1 }}\n\\put(15,30){\\framebox(15,15)[r]{ 4 }}\n\\put(30,30){\\framebox(15,15)[r]{ 5 }}\n\\put(0,15){\\framebox(15,15)[r]{ 2 }}\n\\put(15,15){\\framebox(15,15)[r]{ 6 }}\n\\put(30,15){\\framebox(15,15)[r]{ 8 }}\n\\put(0,0){\\framebox(15,15)[r]{ 3 }}\n\\put(15,0){\\framebox(15,15)[r]{ 7 }}\n\\end{picture}\n\\qquad \\qquad \\qquad\n\\begin{picture}(50,50)\n\\put(0,30){\\framebox(15,15)[r]{ 0 }}\n\\put(15,30){\\framebox(15,15)[r]{ 1 }}\n\\put(30,30){\\framebox(15,15)[r]{ 2 }}\n\\put(0,15){\\framebox(15,15)[r]{ -1 }}\n\\put(15,15){\\framebox(15,15)[r]{ 0 }}\n\\put(30,15){\\framebox(15,15)[r]{ 1 }}\n\\put(0,0){\\framebox(15,15)[r]{ -2 }}\n\\put(15,0){\\framebox(15,15)[r]{ -1 }}\n\\end{picture}\n\\end{center}\n\\end{normalsize}\nTherefore the sequence $(c_1(\\Lambda^\\circ), c_2(\\Lambda^\\circ), \\dots , c_8(\\Lambda^\\circ))$ is given by $(0,\n-1, -2, 1, 2, 0 , -1, 1)$. {\\nolinebreak \\hfill \\rule{2mm}{2mm}\n\nConsider (\\ref{q-bigf}) as a rational function of the variables\n$z_1, \\dots , z_n$ with values in $H_n$. Using the substitution \\begin{equation}\\label{substitution} w_i = q^{c_i(\\Lambda)}z_i, \\end{equation} the factor\n$F_{i}(w_a, w_b)$ has a pole at $z_a = z_b$ if and\nonly if the numbers $a$ and $b$ stand on the same diagonal of a\ntableau $\\Lambda$. We then call the pair $(a, b)$ a\n\\emph{singularity}. And we call the corresponding term $F_{i}(w_a,w_b)$ a \\emph{singularity term}, or\nsimply a singularity.\n\nLet $a$ and $b$ be in the same principal hook of $\\Lambda$. If $a$ and $b$\nare next to one another in the column of the hook then, on\n$\\mathcal{H}_\\lambda$, $F_{i}(w_a,w_b) = T_{i} - q$. Since \\begin{equation}\\label{P-}(T_i - q)^2 = -(q+q^{-1})(T_i-q)\\end{equation}\nthen $\\frac{-1}{q+q^{-1}} F_{i}(w_a,w_b)$ is an\nidempotent. Denote this idempotent $P^-_i$.\n\nSimilarly, if $a$ and $b$ are next to one another in\nthe same row of the hook then $F_{i}(w_a,w_b)= T_{i} +\nq^{-1}$. And since \\begin{equation}\\label{P+}(T_i + q^{-1})^2 = (q+q^{-1})(T_i+q^{-1})\\end{equation} then\n$\\frac{1}{q+q^{-1}} F_{i}(w_a,w_b)$ is an idempotent. Denote this idempotent $P^+_i$.\\\\\n\nWe also have\n\\begin{equation}\\label{q-inverse} F_i(a, b) F_i(b, a)=1-\\frac{(q-q^{-1})^2 ab}{(a-b)^2}.\n\\end{equation}\nTherefore, if the contents $c_a(\\Lambda)$ and $c_b(\\Lambda)$ differ by a number\ngreater than one, then the factor $F_{i}(w_a,w_b)$\nis invertible in $H_n$ when $z_a = z_b \\neq 0$ for all values of $q$.\n\nThe presence of singularity terms in the product $F_\\Lambda(z_1,\n\\dots , z_n)$ mean this product may or may not be regular on the\nvector subspace of $\\mathcal{H}_\\lambda$ consisting of all tuples\n$(z_1, \\dots , z_n)$ such that $z_1 = z_2 = \\dots = z_n \\neq 0$. Using\nthe following lemma, we will be able to show that $F_\\Lambda(z_1,\n\\dots , z_n)$ is indeed regular on this subspace.\n\n\\begin{lemma}\\label{q-regular} Restriction of the rational function $F_i(a, b)F_{i+1}(a, c)F_i(b, c)$ to the set of\n$(a, b, c)$ such that $a=q^{\\pm 2}b$, \nis regular at $a = c \\neq 0$. \\end{lemma}\n\\begin{proof} Let us expand the product at the left hand side of (\\ref{q-triple}) in the\nfactor\n$F_{i+1}(a, c)$. By the definition (\\ref{q-smallf}) we will get the sum\n\\[\nF_i(a, b)T_{i+1}F_i(b, c)+\n\\frac{q-q^{-1}}{a^{-1}c-1}F_i(a, b)F_i(b, c).\n\\]\nHere the restriction to $a=q^{\\pm 2}b$ of the first summand is\nevidently regular at $a=c$. After the substitution \n$b=q^{\\mp2}a$, the second summand takes the form\n\\[\n\\frac{q-q^{-1}}{a^{-1}c-1}\n\\bigl( T_i\\mp q^{\\pm1}\\bigr)\n\\biggl(\nT_i+\\frac{q-q^{-1}}{q^{\\pm2}a^{-1} c-1}\n\\biggr)\n=\n\\frac{q-q^{-1}}{a^{-1} c-q^{\\mp2}}\n( q^{\\pm1}\\mp T_i).\n\\]\nThe rational function of $a, c$ at the right hand side\nof the last displayed equality is also evidently regular at $a=c$. \\end{proof}\n\nIn particular, if the middle term on the left hand side of (\\ref{q-triple}) is a singularity\nand the other two terms are an appropriate idempotent and\n\\emph{triple term}, then the three term product, or \\emph{triple} is regular at $z_1 = z_2 =\n\\dots = z_n \\neq 0$. we may now prove the first statement of Theorem\n\\ref{q-fulltheorem}.\n\n\\begin{proposition}\\label{q-jimtheorem1} The restriction of the rational function\n$F_\\Lambda (z_1, \\dots , z_n)$ to the subspace\n$\\mathcal{H}_\\lambda$ is regular at $z_1 = z_2 = \\dots = z_n \\neq 0$.\n\\end{proposition}\n\n\\begin{proof}\nConsider any standard tableau $\\Lambda'$ obtained from the tableau $\\Lambda$\nby an adjacent transposition of its entries, say by $\\sigma_k\\in S_n$.\nUsing the relations (\\ref{q-triple}) and (\\ref{q-commute}), we derive\nthe equality of rational functions in the variables $z_1, \\ldots , z_n$\n\\[\nF_\\Lambda(z_1, \\ldots , z_n)F_{n-k}\n\\bigl( \nq^{2c_{k+1}(\\Lambda)}z_{k+1}, q^{2c_k(\\Lambda)}z_k\n\\bigr)\n=\n\\]\n\\begin{equation}\\label{q-jimtheorem1equation}\nF_k\n\\bigl( \nq^{2c_k(\\Lambda)}z_k, q^{2c_{k+1}(\\Lambda)}z_{k+1}\n\\bigr)\nF_{\\Lambda^{\\prime}}(z'_1, \\ldots ,z'_n),\n\\end{equation}\nwhere the sequence of variables $(z'_1, \\ldots ,z'_n)$\\ is obtained from\nthe sequence $(z_1, \\ldots , z_n)$ by exchanging the terms $z_k$ and \n$z_{k+1}$. Observe that\n\\[\n(z'_1, \\ldots ,z'_n)\\in\\mathcal{H}_{\\Lambda'}\n\\quad\\Leftrightarrow\\quad\n(z_1, \\ldots , z_n)\\in\\mathcal{H}_\\Lambda.\n\\]\nAlso observe that here $| c_k(\\Lambda)-c_{k+1}(\\Lambda)|\\geqslant2$\nbecause the tableaux $\\Lambda$ and $\\Lambda'$ are standard.\nTherefore the functions \n\\[\nF_k\n\\bigl( \nq^{2c_k(\\Lambda)}z_k, q^{2c_{k+1}(\\Lambda)}z_{k+1}\n\\bigr)\n\\ \\quad\\textrm{and}\\ \\quad\nF_{n-k}\n\\bigl( \nq^{2c_{k+1}(\\Lambda)}z_{k+1}, q^{2c_k(\\Lambda)}z_k\n\\bigr)\n\\]\nappearing in the equality (\\ref{q-jimtheorem1equation}),\nare regular at $z_k=z_{k+1} \\neq 0$.\nMoreover, their values at $z_k=z_{k+1} \\neq 0$ are invertible\nin the algebra $H_n$, see the relation (\\ref{q-inverse}). \nDue to these two observations, the equality (\\ref{q-jimtheorem1equation})\nshows that Proposition \\ref{q-jimtheorem1} is equivalent to its counterpart for\nthe tableau $\\Lambda'$ instead of $\\Lambda$.\n\nLet us take the hook tableau $\\Lambda^\\circ$ of shape $\\lambda$.  \nThere is a chain $\\Lambda,\\Lambda', \\ldots ,\\Lambda^\\circ$ of standard tableaux\nof the same shape $\\lambda$, such that each subsequent tableau in the \nchain is\nobtained from the previous one by an adjacent transposition of the \nentries.\nDue to the above argument, it now suffices to prove Proposition \\ref{q-jimtheorem1} \nonly in the case $\\Lambda=\\Lambda^\\circ$.\n\n\nWe will prove the statement by reordering the factors of the\nproduct $F_{\\Lambda^\\circ} (z_1, \\dots , z_n)$, using relations\n(\\ref{q-triple}) and (\\ref{q-commute}), in such a way that each\nsingularity is part of a triple which is regular at $z_1 = z_2 =\n\\dots = z_n \\neq 0$, and hence the whole of $F_{\\Lambda^\\circ} (z_1, \\dots ,\nz_n)$ will be manifestly regular.\n\nDefine $g(a,b)$ to be the following; \\[ g(a,b) = \\left\\{\n\\begin{array}{ccc}\n  F_{b-a} (w_a, w_b) & \\textrm{if} & a<b \\\\\n  1 & \\textrm{if} & a \\geqslant b\n\\end{array} \\right. \\]\nwhere $w_i$ is the substitution (\\ref{substitution}).\n\nNow, let us divide the diagram $\\lambda$ into two parts,\nconsisting of those boxes with positive contents and those with\nnon-positive contents as in Figure \\ref{steps}. Consider the\nentries of the $i^{\\mbox{\\small th}}$ column of the hook tableau\n$\\Lambda^\\circ$ of shape $\\lambda$ that lie below the steps. If $u_1,\nu_2, \\dots , u_k$ are the entries of the $i^{\\mbox{\\small th}}$ column\nbelow the steps, we define\n\\begin{equation}\\label{q-cproduct} C_i = \\prod_{j=1}^n g(u_1 , j)\ng(u_2 , j) \\cdots g(u_k , j).\n\\end{equation} Now consider the entries of the $i^{\\mbox{\\small th}}$ row of $\\Lambda^\\circ$ that lie\nabove the steps. If $v_1, v_2, \\dots , v_l$ are the entries of the\n$i^{\\mbox{\\small th}}$ row above the steps, we define\n\\begin{equation}\\label{q-rproduct} R_i = \\prod_{j=1}^n g(v_1 , j)\ng(v_2 , j) \\cdots g(v_l , j).\n\\end{equation}\n\nOur choice of the hook tableau was such that the following is\ntrue; if $d$ is the number of principal hooks of $\\lambda$ then by\nrelations (\\ref{q-triple}) and (\\ref{q-commute}) we may reorder the\nfactors of $F_{\\Lambda^\\circ} (z_1, \\dots , z_n)$ such that \\[ F_{\\Lambda^\\circ}\n(z_1, \\dots , z_n) = \\prod_{i=1}^d C_i R_i . \\]\n\nNow, each singularity $(a,b)$ has its corresponding term $F_{b-a}\n(w_a,w_b)$ contain in some product $C_i$ or $R_i$.\nThis singularity term will be on the immediate left of the\n term $F_{b-a-1} (w_{a+1}, w_b)$. Also, this ordering has been chosen such\nthat the product of factors to the left of any such singularity in\n$C_i$ or $R_i$ is divisible on the right by $F_{b-a-1}(w_a,w_{b+1})$.\\\\\nTherefore we can replace the pair $F_{b-a}\n(w_a,w_b)F_{b-a-1} (w_{a+1}, w_b)$ in $C_i$ by the triple\n\\[ P_{b-a-1}^- F_{b-a}\n(w_a,w_b)F_{b-a-1} (w_{a+1}, w_b), \\] where\n$P_{b-a-1}^- = \\frac{-1}{q+q^{-1}} F_{b-a-1}(w_a,w_{a+1})$ is the\nidempotent (\\ref{P-}). Divisibility on the right by $F_{b-a-1}(w_a,w_{a+1})$ means the addition of the idempotent has no\neffect on the value of the product $C_i$. Similarly, in the product $R_i$ we can replace the pair by \\[ P_{b-a-1}^+ F_{b-a}\n(w_a,w_b)F_{b-a-1} (w_{a+1}, w_b). \\] \nBy Lemma \\ref{q-regular}, the above triples are regular at $z_1 =\nz_2 = \\dots = z_n \\neq 0$, and therefore, so too are the products $C_i$\nand $R_i$, for all $1 \\leqslant i \\leqslant d$. Moreover, this\nmeans $F_{\\Lambda^\\circ} (z_1, \\dots , z_n)$ is regular at $z_1 = z_2 =\n\\dots = z_n \\neq 0$.\n\\end{proof}\n\n{\\addtocounter{definition}{1} \\bf Example \\thedefinition .} As an\nexample consider the hook tableau of the Young diagram $\\lambda =\n(3,3,2)$, as shown in Example 2.1.\n\nIn the original reverse-lexicographic ordering the product $F_{\\Lambda^{\\circ}}(z_1,\n\\dots , z_n)$ is written as;\n\\[\n\\begin{array}{rl}\n   F_{\\Lambda^\\circ} (z_1, \\dots , z_n) = & F_1(w_1,w_2)F_2(w_1,w_3)F_1(w_2,w_3)F_3(w_1,w_4)F_2(w_2,w_4)F_1(w_3,w_4)\\\\&\n   F_4(w_1,w_5)F_3(w_2,w_5)F_2(w_3,w_5)F_1(w_4,w_5)F_5(w_1,w_6)F_4(w_2,w_6)\\\\&F_3(w_3,w_6)F_2(w_4,w_6)F_1(w_5,w_6)F_6(w_1,w_7)F_5(w_2,w_7)F_4(w_3,w_7)\\\\&F_3(w_4,w_7)F_3(w_4,w_7)F_2(w_5,w_7)F_1(w_6,w_7)F_7(w_1,w_8)F_6(w_2,w_8)\\\\&F_5(w_3,w_8)F_4(w_4,w_8)F_3(w_5,w_8)F_2(w_6,w_8)F_1(w_7,w_8)\\\\\n\\end{array}\n\\]\nwe may now reorder this product into the form below using\nrelations (\\ref{q-triple}) and (\\ref{q-commute}) as described in the\nabove proposition. The terms in square brackets are the singularity terms\nwith their appropriate triple terms.\n\\[\n\\begin{array}{rl}\n   F_{\\Lambda^\\circ} (z_1, \\dots , z_n) = &\n   F_1(w_1,w_2)F_2(w_1,w_3)F_1(w_2,w_3)F_3(w_1,w_4)F_2(w_2,w_4)F_1(w_3,w_4)\\\\&F_4(w_1,w_5)F_3(w_2,w_5)F_2(w_3,w_5)\\left[F_5(w_1,w_6)F_4(w_2,w_6)\\right]F_3(w_3,w_6)\\\\&F_6(w_1,w_7)\\left[F_5(w_2,w_7)F_4(w_3,w_7)\\right]F_7(w_1,w_8)F_6(w_2,w_8)F_5(w_3,w_8)\\\\&\n   \\cdot F_1(w_4,w_5)F_2(w_4,w_6)F_1(w_5,w_6)F_3(w_4,w_7)F_2(w_5,w_7)\\\\&\\left[F_4(w_4,w_8)F_3(w_5,w_8)\\right] \\cdot F_1(w_6,w_7)F_2(w_6,w_8)F_1(w_7,w_8)\\\\\n\\end{array}\n\\]\nWe may now add the appropriate idempotents to these singularity-triple term pairs to form triples. Since each of these triples are regular at $z_1 = z_2 = \\dots\n= z_n$ then so too is the whole of $F_{\\Lambda^\\circ} (z_1, \\dots , z_n)$.\n{\\nolinebreak \\hfill \\rule{2mm}{2mm}\n\n\\quad\n\nTherefore, due to the above proposition an element $F_\\Lambda \\in\nH_n$ can now be defined as the value of $F_\\Lambda (z_1,\n\\dots , z_n)$ at $z_1 = z_2 = \\dots = z_n \\neq 0$. Note that for $n=1$ we have $F_\\Lambda=1$. For any $n\\geqslant1$, \ntake the expansion of the element $F_\\Lambda\\in H_n$ in the basis of the \nelements $T_\\sigma$ where $\\sigma$ is ranging over $S_n$.\n\n\\begin{proposition}\\label{T_0coeff} The coefficient in $F_\\Lambda\\in H_n$ of the element $T_0$ is $1$.\n\\end{proposition}\n\\begin{proof} Expand the product (\\ref{q-bigf})\nas a sum of the elements $T_\\sigma$ with coefficients\nfrom the field of rational functions of $z_1, \\ldots , z_n$;\nthese functions take values in $\\mathbb{C}(q)$. \nThe decomposition in $S_n$\nwith ordering of the pairs $(i, j)$ as in (\\ref{q-bigf})\n\\[\n\\sigma_0=\\prod_{(i,j)}^{\\longrightarrow}\\ \\sigma_{j-i}\n\\]\nis reduced, hence the coefficient at $T_0=T_{\\sigma_0}$ in the\nexpansion of (\\ref{q-bigf}) is $1$. By the definition of $F_\\Lambda$,\nthen the coefficient of $T_0$ in $F_\\Lambda$  must be also $1$\n\\end{proof}\n\nIn particular this shows that $F_\\Lambda \\neq 0$ for any nonempty\ndiagram $\\lambda$. Let us now denote by $\\varphi$ the involutive\nantiautomorphism of the algebra $H_n$ over the field $\\mathbb{C}(q)$, defined by\n$\\varphi (T_i) = T_i$ for every $i \\in 1, \\dots , n-1$.\n\n\\begin{proposition}\\label{q-varphi} The element $F_\\Lambda T^{-1}_0$ is $\\varphi$-invariant. \\end{proposition}\n\\begin{proof} Any element of the algebra $H_n$ of the form\n$F_i(a, b)$ is $\\varphi$-invariant. \nHence applying the antiautomorphism $\\varphi$ to an element of \n$H_n$ the form (\\ref{q-bigf}) just reverses the ordering of\nthe factors corresponding to the pairs $(i, j)$.\nUsing the relations (\\ref{q-triple}) and (\\ref{q-commute}),\nwe can rewrite the reversed product as\n\\[\n\\prod_{(i,j)}^{\\longrightarrow}\\ F_{ n-j+i}\n\\bigl( q^{2c_i(\\Lambda)}z_i, q^{2c_j(\\Lambda)}z_j\\bigr)\n\\]\nwhere the pairs $(i, j)$ are again ordered as in (\\ref{q-bigf}).\nBut due to (\\ref{Hecke2}) and (\\ref{Hecke3}),\nwe also have the identity in the algebra $H_n$\n\\[\nF_{ n-i}(a,b)T_0=T_0 F_i(a, b).\n\\]\nThis identity along with the equality \n$\\varphi(T_0)=T_0$ implies that\nany value of the function $F_\\Lambda(z_1, \\ldots , z_n)T^{-1}_0$ is \n$\\varphi$-invariant. So is the element $F_\\Lambda T^{-1}_0 \\in H_n$\n\\end{proof}\n\n\\begin{proposition}\\label{q-stripping} If $\\lambda= (\\alpha_1,\n\\alpha_2, \\dots, \\alpha_d | \\beta_1, \\beta_2, \\dots , \\beta_d)$\nand $\\mu = (\\alpha_{k+1}, \\alpha_{k+2}, \\dots , \\alpha_d |\n\\beta_{k+1}, \\beta_{k+2}, \\dots , \\beta_d)$, then $F_{\\Lambda^\\circ} = P\n\\cdot F_{\\mathrm{M}^\\circ}$, for some element $P \\in H_n$.\n\\end{proposition}\n\\begin{proof}\nHere the shape $\\mu$ is obtained by removing the first $k$\nprincipal hooks of $\\lambda$. Let $x$ be last entry in the $k^{\\mbox{\\small th}}$ row\nof the hook tableau of shape $\\lambda$. By the ordering described in\nProposition \\ref{q-jimtheorem1},\n\\[ F_{\\Lambda^\\circ}(z_1, \\dots ,\nz_n) = \\prod_{i=1}^k C_iR_i \\cdot F_{\\mathrm{M}^\\circ}(z_{x+1}, \\dots,\nz_{n}),\\]\nwhere $C_i$ and $R_i$ are defined by (\\ref{q-cproduct}) and (\\ref{q-rproduct}).\\\\\nSince all products $C_i$ and $R_i$ are regular at $z_1 = z_2 =\n\\dots = z_n \\neq 0$, Proposition \\ref{q-jimtheorem1} then gives us the\nrequired statement. \\end{proof}\n\nIn any given ordering of $F_\\Lambda(z_1, \\dots , z_n)$, we want a\nsingularity term to be placed next to an appropriate triple term\nsuch that we may then form a regular triple. In that case we will\nsay these two terms are 'tied'. However, proving the divisibilities described in the next two propositions require some pairs\nto be 'untied', in which case we must form a new ordering. This is\nthe content of the following proofs. Some explicit examples will\nthen given in Example 2.10 below.\n\n\\begin{proposition}\\label{q-jimtheorem2} Suppose the numbers $u < v$ stand next to each\nother in the same column of the hook tableau $\\Lambda^\\circ$ of shape\n$\\lambda$. First, let $s$ be the last entry in the row containing\n$u$. If $c_v < 0$ then the element $F_{\\Lambda^\\circ} \\in H_n$\nis divisible on the left\nby $F_{u}(q^{2c_u(\\Lambda^\\circ)}, q^{2c_v(\\Lambda^\\circ)}) =\nT_{u} - q$. If $c_v \\geqslant 0$ then the element $F_{\\Lambda^\\circ} \\in\nH_n$ is divisible on the left by the product\n\\[ \\prod_{i = u, \\dots, s}^\\leftarrow \\left( \\prod_{j= s+1, \\dots,\nv}^\\rightarrow F_{i+j-s-1}(q^{2c_i(\\Lambda^\\circ)}, q^{2c_j(\\Lambda^\\circ)}) \\right) \\]\n\\end{proposition}\n\n\\begin{proof}\nLet $\\lambda$ and $\\mu$ be as in Proposition \\ref{q-stripping} with $\\lambda$ a partition of $n$ and $\\mu$ a partition of $m$. If $F_\\mathrm{M}(z_1, \\dots, z_m)$ is divisible on the left by $F_i(w_a,w_b)$ then $F_\\mathrm{M}(z_{x+1}, \\dots, z_n)$ is divisible on the left by $F_i(w_{a+x},w_{b+x})$. Then, by Proposition \\ref{q-varphi}, $F_\\mathrm{M}(z_{x+1}, \\dots, z_n)$ is divisible on the right by $F_{m-i}(w_{a+x},w_{b+x})$. Therefore, by Proposition \\ref{q-stripping}, $F_{\\Lambda^\\circ}(z_1, \\dots, z_n)$ is divisible on the right by $F_{m-i}(w_{a+x},w_{b+x})$ and so $F_{\\Lambda^\\circ}(z_1, \\dots, z_n)$ is divisible on the left by $F_{n-m+i}(w_{a+x}, w_{b+x}) = F_{i+x}(w_{a+x},w_{b+x})$. Hence we only need to prove the statement for $(u,v)$ such that $u$ is in the first row or first column of $\\Lambda^\\circ$.\n\n\nLet $r$ be the last entry in the first column of $\\Lambda^\\circ$, $s$\nthe last entry in the first row of $\\Lambda^\\circ$, and $t$ the last\nentry in the second column of $\\Lambda^\\circ$, as shown in Figure\n\\ref{jimtheorem2fig}.\n\n\\begin{figure}[h] \\label{jimtheorem2fig}\n\n\\begin{normalsize}\n\\begin{center}\n\\begin{picture}(275,200)\n\\begin{small}\n\\put(0,175){\\framebox(25,25)[c]{ 1 }}\n\\put(25,175){\\framebox(25,25)[c]{ $r + 1$ }}\n\\put(50,175){\\framebox(25,25)[c]{ $r+2$ }}\n\\put(0,150){\\framebox(25,25)[c]{ 2 }}\n\\put(25,150){\\framebox(25,25)[c]{ $s+1$ }}\n\\put(50,150){\\framebox(25,25)[c]{ $t+1$ }}\n\n\n\n\\put(0,25){\\line(0,1){125}} \\put(25,25){\\line(0,1){125}}\n\\put(0,0){\\framebox(25,25)[c]{ $r$ }}\n\n\\put(75,200){\\line(1,0){50}} \\put(75,175){\\line(1,0){50}}\n\\put(125,175){\\framebox(25,25)[c]{ $u$ }}\n\\put(150,200){\\line(1,0){100}} \\put(150,175){\\line(1,0){100}}\n\\put(250,175){\\framebox(25,25)[c]{ $s$ }}\n\n\\put(50,100){\\line(0,1){50}} \\put(25,75){\\framebox(25,25)[c]{ $t$\n}}\n\n\\put(75,150){\\line(1,0){50}} \\put(125,150){\\framebox(25,25)[c]{\n$v$ }} \\put(150,150){\\line(1,0){50}} \\put(200,150){\\line(0,1){25}}\n\n\\linethickness{1.5pt} \\put(0,200){\\line(1,0){25}}\n\\put(25,175){\\line(1,0){25}} \\put(50,150){\\line(1,0){25}}\n\\put(25,175){\\line(0,1){25}} \\put(50,150){\\line(0,1){25}}\n\n\\put(125,150){\\line(0,1){50}}\\put(150,150){\\line(0,1){50}}\n\\put(125,200){\\line(1,0){25}}\\put(125,150){\\line(1,0){25}}\n\n\\put(95,185){$\\dots$} \\put(170,185){$\\dots$}\\put(220,185){$\\dots$}\n\\put(95,160){$\\dots$} \\put(170,160){$\\dots$}\n\n\\put(10,120){$\\vdots$} \\put(35,\n120){$\\vdots$}\\put(10,80){$\\vdots$} \\put(10,40){$\\vdots$}\n\n\\end{small}\n\\end{picture}\n\\end{center}\n\\end{normalsize}\n \\caption{The first two principal hooks of the hook tableau $\\Lambda^\\circ$}\n \\end{figure}\n\n\n\n\nWe now continue this proof by considering three cases and showing\nthe appropriate divisibility in each.\n\n\\emph{(i)} \\quad  If $c_v < 0$ (i.e. $u$ and $v$ are in the first\ncolumn of $\\Lambda^\\circ$) then $v = u+1$ and $F_{\\Lambda^\\circ}(z_1, \\dots ,\nz_n)$ can be written as $F_{\\Lambda^\\circ}(z_1, \\dots, z_n) = F_{u}(w_u, w_v) \\cdot F$. \\\\\nStarting with $F_{\\Lambda^\\circ}(z_1, \\dots , z_n)$ written in the\nordering described in Proposition \\ref{q-jimtheorem1} and simply\nmoving the term $F_{v-u}(w_u, w_v)$ to the left\nresults in all the singularity terms in the product $F$ remaining\ntied to the same triple terms as originally described in that\nordering, and the index of $F_{v-u}(w_u, w_v)$ increases from $u-v$ to $u$. Therefore we may still form regular triples for\neach singularity in $F$, and hence $F$ is regular at $z_1 = z_2 = \\dots = z_n \\neq 0$.\\\\\nSo by considering this expression for $F_{\\Lambda^\\circ}(z_1, \\dots ,\nz_n)$ at $z_1 = z_2 = \\dots = z_n \\neq 0$ we see that $F_{\\Lambda^\\circ}$ will be\ndivisible on the left by $F_{u}(q^{2c_u(\\Lambda^\\circ)}, q^{2c_v(\\Lambda^\\circ)}) = T_{v-u} - q$.\n\n\\emph{(ii)} \\quad  If $c_v = 0$ then $v=s+1$, and $F_{\\Lambda^\\circ}(z_1,\n\\dots , z_n)$ can be written as \\[ F_{\\Lambda^\\circ}(z_1, \\dots , z_n) =\n\\prod_{i = u, \\dots , s}^\\leftarrow F_{i}(w_i, w_{s+1}) \\cdot F' . \\] Again, starting with the ordering\ndescribed in Proposition \\ref{q-jimtheorem1}, this results in  all\nthe singularity terms in the product $F'$ remaining tied to the\nsame triple terms as originally described in that ordering. Hence\n$F'$ is regular at $z_1 = z_2 = \\dots = z_n \\neq 0$. And so $F_{\\Lambda^\\circ}$\nis divisible on the left by\n\\[ \\prod_{i = u, \\dots , s}^\\leftarrow F_{i}(q^{2c_i(\\Lambda^\\circ)}, q^{2c_{s+1}(\\Lambda^\\circ)}).\n\\]\n\n\\emph{(iii)} \\quad  If $c_v > 0$ (i.e. $v$ is above the steps)\nthen $F_{v-u}(w_u, w_v)$ is tied to the singularity\n$F_{v-u+1}(w_{u-1}, w_v)$ as a triple term. To\nshow divisibility by $F_{v-u}(w_u, w_v)$ in this case\nwe need an alternative expression for $F_{\\Lambda^\\circ}(z_1, \\dots ,\nz_n)$ that is regular when $z_1 = z_2 = \\dots = z_n \\neq 0$. Define a\npermutation $\\tau$ as follows,\n\n\\[ \\tau = \\prod_{i = u, \\dots, s}^\\rightarrow \\left( \\prod_{j= s+1, \\dots,\nv}^\\leftarrow (i j) \\right)\n\\phantom{XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX}\n\\]\n\\[ = \\left(\n\\normalsize \\begin{array}{ccccccccccccccccc}\n                   1 & 2 & \\dots & u-1 & u & u+1 & \\dots &  & \\dots & \\dots  &  & \\dots & v-1 & v & v+1 & \\dots & n \\\\\n                   1 & 2 & \\dots & u-1 & s+1 & s+2 & \\dots & v-1 & v & u & u+1 & \\dots & s-1 & s & v+1 & \\dots & n \\\\\n                 \\end{array} \\right) \\]\n\n\nFrom the definition of $C_1$ in (\\ref{q-cproduct}) we now define\n$C'_1 = \\psi_\\tau C_1$, where $\\psi_\\tau$ is a homomorphism such that\n\\[ \\psi_\\tau F_{j-i}(w_i,w_j) = F_{j-i}(w_i,w_{\\tau j}). \\]\n\n\nDefine $R'_1$ as,\n\\begin{eqnarray*}\nR'_1 & = & \\prod_{i= r+2, \\dots, u-1}^\\leftarrow \\left(\n\\prod_{j=s+1, \\dots ,v}^\\rightarrow F_{i+j-s-r-1}(w_i,w_j) \\right) \\cdot \\prod_{i=\ns+1, \\dots, t-1}^\\rightarrow \\left( \\prod_{j=i+1, \\dots\n,t}^\\rightarrow f_{j-i+1}(w_i,w_j) \\right) \\\\\n && \\times  \\left( \\prod_{j=s+1, \\dots\n,t}^\\leftarrow F_{t+1-j}(w_{r+1},w_j) \\right) \\left( \\prod_{j=t+1, \\dots\n,v}^\\rightarrow F_{j-s}(w_{r+1},w_j) \\right) \\\\\n   && \\times \\prod_{i= r+1,\n\\dots, s-1}^\\rightarrow \\left( \\prod_{j=i+1, \\dots ,s}^\\rightarrow\nF_{j-i+v-s}(w_i,w_j) \\right) \\cdot \\prod_{j= v+1, \\dots, n}^\\rightarrow\n\\left( \\prod_{i=r+1, \\dots ,s}^\\rightarrow F_{j-i}(w_i,w_j) \\right). \\\\\n\\end{eqnarray*}\n\nFinally, define $C'_2$ as, \\[ C'_2 = \\prod_{j= t+1, \\dots,\nn}^\\rightarrow \\left( \\prod_{i=s+1, \\dots ,t}^\\rightarrow F_{j-i}(w_i,w_j)\n\\right). \\]\n\nThen, \\[ F_\\Lambda(z_1, \\dots , z_n) = \\prod_{i = u, \\dots ,\ns}^\\leftarrow \\left( \\prod_{j=s+1, \\dots, v}^\\rightarrow F_{i+j-s-1}(w_i,w_j)\n\\right) \\cdot C'_1 R'_1 C'_2 R_2 \\cdot \\prod_{i=3}^d C_iR_i, \\]\nwhere $d$ is the number of principal hooks of $\\lambda$.\n\nThe product $C'_1 R'_1 C'_2 R_2 \\cdot \\prod C_iR_i$ is regular at\n$z_1 = z_2 = \\dots z_n \\neq 0$ since, as before, for any singularity\n$(a,b)$ the terms $F_{i}(w_a,w_b)F_{i-1}(w_{a+1},w_b)$ can be replaced by the triple\n$P_{i-1}^\\pm F_{i}(w_a,w_b)F_{i-1}(w_{a+1},w_b)$ for some index $i$ -- except in the\nexpression $R'_1$ where the terms $F_{i-1}(w_a,w_l)F_{i}(w_a,w_b)$ are replaced by\n$F_{i-1}(w_a,w_l)F_{i}(w_a,w_b)P_{i-1}^+$, where $l$ is the entry to the\nimmediate left of $b$. Note that $l = b-1$ when $c_b(\\Lambda^\\circ) > 1$ and $l =\ns+1$ when $c_b(\\Lambda^\\circ) = 1$. \\\\\nAnd so by letting $z_1 = z_2 = \\dots =z_n \\neq 0$ we see that $F_{\\Lambda^\\circ}$\nis divisible on the left by \\[ \\prod_{i = u, \\dots , s}^\\leftarrow\n\\left( \\prod_{j=s+1, \\dots, v}^\\rightarrow F_{i+j-s-1}(q^{2c_i(\\Lambda^\\circ)}, q^{2c_j(\\Lambda)})\n\\right).\n\\]\n\\end{proof}\n\n\\begin{proposition}\\label{q-jimtheorem3} Suppose the numbers $u < v$ stand next to each\nother in the same row of the hook tableau $\\Lambda^\\circ$ of shape\n$\\lambda$. Let $r$ be the last entry in the column containing $u$.\nIf $c_u > 0$ then the element $F_{\\Lambda^\\circ} \\in H_n$ is\ndivisible on the left by $F_{u}(q^{2c_u(\\Lambda^\\circ)}, q^{c_v(\\Lambda^\\circ)}) = T_{u} +\nq^{-1}$. If $c_u \\leqslant 0$ then the element $F_{\\Lambda^\\circ} \\in\nH_n$ is divisible on the left by the product\n\\[ \\prod_{i = u, \\dots, r}^\\leftarrow \\left( \\prod_{j= r+1, \\dots,\nv}^\\rightarrow F_{i+j-r-1}(q^{c_i(\\Lambda^\\circ)}, q^{c_j(\\Lambda^\\circ)}) \\right) \\]\n\\end{proposition}\n\nWe omit the proof of this proposition as it is very similar to\nthat of Proposition \\ref{q-jimtheorem2}. \n\n\n\\begin{lemma}\\label{q-divisibilitybyadjacenttransposition} Let $\\Lambda$ and $\\tilde{\\Lambda}$ be tableaux of the same shape such that $k= \\Lambda(a,b) = \\Lambda(a+1,b)-1$ and $\\tilde{k}=\\tilde{\\Lambda}(a,b) = \\tilde{\\Lambda}(a+1,b)-1$. Then $F_\\Lambda \\in H_n$ is divisible on the left by $T_k - q$ if and only if $F_{\\tilde{\\Lambda}} \\in H_n$ is divisible on the left by $T_{\\tilde{k}} - q$. \\end{lemma}\n\n\\begin{proof}\nLet $\\sigma$ be the permutation such that $\\tilde{\\Lambda}=\\sigma\\cdot\\Lambda$.\nThere is a decomposition\n$\\sigma=\\sigma_{i_N}\\ldots\\sigma_{i_1}$ such that for each $M=1, \\ldots , N-1$\nthe tableau $\\Lambda_{ M}=\\sigma_{i_M}\\ldots\\sigma_{i_1}\\cdot\\Lambda$ is\nstandard. Note that this decomposition is not necessarily reduced.\n\nDenote $F_\\Lambda$ by $F_k(q^{2c_k(\\Lambda)},q^{2c_{k+1}(\\Lambda)}) \\cdot F$. Then, by using the relations\n(\\ref{q-triple}) and (\\ref{q-commute}) and  Proposition \\ref{q-jimtheorem1}, we have the following chain of equalities:\n\n\\[\nF_{\\tilde{k}}\n\\bigl( \nq^{2c_{\\tilde{k}}(\\tilde{\\Lambda})}, q^{2c_{\\tilde{k}+1}(\\tilde{\\Lambda})}\n\\bigr)\n\\ \\cdot\\ \n\\prod_{M=1, \\ldots , N}^{\\longleftarrow}\nF_{ n- i_M}\n\\bigl( \nq^{2 c_{i_M+1}(\\Lambda_M)}\n, \nq^{2 c_{i_M}(\\Lambda_M)}\n\\bigr) \\cdot F =  \\]\n\\[\\prod_{M=1, \\ldots , N}^{\\longleftarrow}\nF_{ i_M}\n\\bigl( \nq^{2 c_{i_M}(\\Lambda_M)}\n, \nq^{2 c_{i_M+1}(\\Lambda_M)}\n\\bigr)\n\\ \\cdot\\ \nF_{ k}\n\\bigl( q^{2c_k(\\Lambda)}, q^{2c_{k+1}(\\Lambda)}\\bigr) \\cdot F =\\]\n\\[\n\\prod_{M=1, \\ldots , N}^{\\longleftarrow}\nF_{ i_M}\n\\bigl( \nq^{2 c_{i_M}(\\Lambda_M)}\n, \nq^{2 c_{i_M+1}(\\Lambda_M)}\n\\bigr)\n\\ \\cdot\\ F_\\Lambda\\ =\n\\]\n\\[\nF_{\\tilde{\\Lambda}}\\ \\cdot\n\\prod_{M=1, \\ldots , N}^{\\longleftarrow}\nF_{ n- i_M}\n\\bigl( \nq^{2 c_{i_M+1}(\\Lambda_M)}\n, \nq^{2 c_{i_M}(\\Lambda_M)}\n\\bigr)\n\\]\n\nHence divisibility by $T_k - q$ for $F_\\Lambda$ implies its counterpart for the tableau\n$\\tilde{\\Lambda}$ and the index $\\tilde{k}$, and vice versa.\nHere we also use the equalities\n\\[\nF_{ k}\n\\bigl( q^{2c_k(\\Lambda)}, q^{2c_{k+1}(\\Lambda)}\\bigr)\n=T_k-q,\n\\]\\[\nF_{\\tilde{k}}\n\\bigl( \nq^{2c_{\\tilde{k}}(\\tilde{\\Lambda})}, q^{2c_{\\tilde{k}+1}(\\tilde{\\Lambda})}\n\\bigr)\n=T_{\\tilde{k}}-q.\n\\]\n\\end{proof}\n\n\\begin{corollary}\\label{q-divisibilitycorollary}\nIf $k=\\Lambda(a, b)$ and $k+1=\\Lambda(a+1, b)$\nthen the element $F_\\Lambda\\in H_n$ is divisible on the left by $T_k-q$. If $k=\\Lambda(a, b)$ and $k+1=\\Lambda(a, b+1)$\nthen the element $F_\\Lambda\\in H_n$ is divisible on the left by \n$T_k+q^{-1}$.\n\\end{corollary}\n\n\\begin{proof}\nDue to Lemma \\ref{q-divisibilitybyadjacenttransposition}\nit suffices to prove the first part of Corollary \\ref{q-divisibilitycorollary} for only one \nstandard tableau $\\Lambda$ of shape $\\lambda$. Therefore, using Proposition \\ref{q-jimtheorem2} and taking $\\tilde{\\Lambda}$ to be the hook tableau $\\Lambda^\\circ$ of shape $\\lambda$ we have shown the first part of Corollary \\ref{q-divisibilitycorollary} in the case $c_v(\\Lambda) < 0$.\n\nNext let $\\Lambda^{\\circ}(a,b)= u$, $\\Lambda^{\\circ}(a+1,b) = v$ and $s$ be the last entry in the row containing $u$. Then for $c_v({\\Lambda^\\circ}) \\geqslant 0$ Proposition \\ref{q-jimtheorem2} showed that $F_{\\Lambda^\\circ} \\in H_n$ is divisible on the left by \n\\begin{equation}\\label{q-divisibilitycorollaryequation2} \\prod_{i = u, \\dots, s}^\\leftarrow \\left( \\prod_{j= s+1, \\dots,\nv}^\\rightarrow F_{i+j-s-1}(q^{2c_i(\\Lambda^\\circ)}, q^{2c_j(\\Lambda^\\circ)}) \\right) \\end{equation}\nPut $k=u+v-s-1$,\nthis is the value of the index $i+j-s-1$ in (\\ref{q-divisibilitycorollaryequation2})\nwhen $i=u$ and $j=v$. Let $\\Lambda$ be the tableau\nsuch that $\\Lambda^\\circ$ is obtained from \nthe tableau $\\sigma_k\\cdot\\Lambda$ by the permutation\n\\[\n\\prod_{i =  u, \\ldots, s}^{\\longleftarrow}\\,\n\\biggl(\\ \n\\prod_{j = s+1, \\ldots,  v}^{\\longrightarrow}\n\\sigma_{ i+j-s-1} \\biggr).\n\\]\nThe tableau $\\Lambda$ is standard. Moreover, then\n$\\Lambda(a, b)=k$ and $\\Lambda(a+1, b)=k+1$.\nNote that the rightmost factor in the product (\\ref{q-divisibilitycorollaryequation2}),\ncorresponding to $i=u$ and $j=v$, is\n\\[\nF_{u+v-s-1}\n\\bigl( q^{2c_u(\\Lambda^\\circ)}, q^{2c_{v}(\\Lambda^\\circ)}\\bigr)\n=\nT_k - q.\n\\]\nDenote by $F$ the product of all factors in (\\ref{q-divisibilitycorollaryequation2})\nbut the rightmost one. Further, denote by $G$ the product obtained\nby replacing each factor in $F$\n\\[\nF_{i+j-s-1}\n\\bigl(q^{2c_i(\\Lambda^\\circ)}, q^{2c_j(\\Lambda^\\circ)}\\bigr)\n\\]\nrespectively by\n\\[\nF_{n-i-j+s+1}\n\\bigl(q^{2c_j(\\Lambda^\\circ)}, q^{2c_i(\\Lambda^\\circ)}\\bigr).\n\\]\nThe element $F \\in H_n$ is invertible, and we have\n\\[ F \\cdot F_\\Lambda = F_{\\Lambda^\\circ} \\cdot G = F \\cdot (T_k - q) \\cdot (C'_1R'_1C'_2R_2 \\prod C_iR_i) \\cdot G,\\] where the final equality is as described in Proposition \\ref{q-jimtheorem2}. Therefore the divisibility of \nthe element $F_{\\Lambda^\\circ}$ on the left by the product (\\ref{q-divisibilitycorollaryequation2})\nwill imply the divisibility of the element $F_\\Lambda$ on the left by \n$T_k - q$.\n\nThis shows the required divisibility for the tableau $\\Lambda = \\sigma \\sigma_k \\cdot \\Lambda^{\\circ}$. Using Lemma \\ref{q-divisibilitybyadjacenttransposition} again concludes the proof of the first part of Corollary \\ref{q-divisibilitycorollary}. \n\nThe second part of Corollary \\ref{q-divisibilitycorollary} may be shown similarly.\n\\end{proof}\n\n\n\n\n\\section{Generating irreducible representations of $H_n$}\n\nFor every standard tableau $\\Lambda$ of shape $\\lambda$ \nwe have defined an element $F_\\Lambda$ of the algebra $H_n$. \nLet us now assign to $\\Lambda$ another element of $H_n$,\nwhich will be denoted by $G_\\Lambda$. \n\nLet $\\rho\\in S_n$ be the permutation such \nthat $\\Lambda=\\rho\\cdot\\Lambda^\\circ$, that is\n$\\Lambda(a, b)=\\rho(\\Lambda^\\circ(a, b))$ for all possible $a$ and $b$. For any $j=1, \\ldots , n$ denote by $\\mathcal{B}_j$ the subsequence of the sequence\n$\\rho(1), \\ldots ,\\rho(n)$ consisting of all $i<j$ such that\n$\\rho^{-1}(i)<\\rho^{-1}(j)$. Let $|\\mathcal{B}_j|$ be the length of sequence $\\mathcal{B}_j$. \n\nConsider the rational\nfunction taking values in $H_n$, of the variables $z_1, \\ldots , z_n$\n\\[\n\\prod_{j=1,\\ldots, n}^{\\longrightarrow}\n\\biggl(\\ \n\\prod_{k=1,\\ldots,|\\mathcal{B}_j|}^{\\longrightarrow}\n\\ F_{j-k}\n\\bigl( q^{2c_i(\\Lambda)}z_i, q^{2c_j(\\Lambda)}z_j\\bigr)\n\\biggr)\n\\ \\quad\\textrm{where}\\ \\quad \ni=\\mathcal{B}_j(k).\n\\]\nDenote this rational function by $G_\\Lambda(z_1, \\ldots , z_n)$.\n\nFor any $j=1, \\ldots , n$ take the subsequence of the sequence\n$\\rho(1), \\ldots ,\\rho(n)$ consisting of all $i<j$ such that\n$\\rho^{-1}(i)>\\rho^{-1}(j)$. Denote by $\\mathcal{A}_j$ the result of reversing\nthis subsequence. Using induction on the length of the element $\\rho\\in S_n$, one can prove that\n\\[\nF_\\Lambda(z_1, \\ldots , z_n)\\ =\\  \nG_\\Lambda(z_1, \\ldots , z_n)\\ \\ \\times\n\\]\\[\n\\prod_{j=1,\\ldots, n}^{\\longleftarrow}\n\\biggl(\\ \n\\prod_{k=1,\\ldots,|\\mathcal{A}_j|}^{\\longleftarrow}\n\\ F_{ n-j+k}\n\\bigl( q^{2c_i(\\Lambda)}z_i, q^{2c_j(\\Lambda)}z_j\\bigr)\n\\biggr)\n\\ \\quad\\textrm{where}\\ \\quad \ni=\\mathcal{A}_j(k).\n\\]\nHence restriction of $G_\\Lambda(z_1, \\ldots , z_n)$ \nto the subspace $\\mathcal{H}_\\Lambda\\subset\\mathbb{C}(q)^{n}$\nis regular on the line $(z_1 = \\cdots = z_n \\neq 0)$ due to Proposition \\ref{q-jimtheorem1}.\nThe value of that restriction is our element $G_\\Lambda\\in H_n$ \nby definition. \n\nNote that $G_{\\Lambda^\\circ}=F_{\\Lambda^\\circ}$. Denote by $V_\\lambda$ the left ideal in the algebra $H_n$\ngenerated by the element $F_{\\Lambda^\\circ}$. The elements $G_\\Lambda\\in H_n$ for all\npairwise distinct standard tableaux $\\Lambda$ of shape $\\lambda$\nform a basis in the vector space $V_\\lambda$, \\cite{N1}. Let us now consider the left ideal $V_\\lambda\\subset H_n$ as \n$H_n$-module. Here the algebra $H_n$ acts via left multiplication.\n\nThe following theorems are stated without proofs. All proofs can in found in \\cite[Section 3]{N1}.\n\n\\begin{theorem}\\label{C3.6}\n{\\bf\\hskip-6pt.\\hskip1pt} \nThe $H_n$-modules $V_\\lambda$ for different partitions $\\lambda$ of $n$\nare irreducible and pairwise non-equivalent.\n\\end{theorem}\n\nFurthermore we have the following proposition about the elements $G_\\Lambda$;\n\n\\begin{proposition}\\label{Vkappa}\nThe vector $G_\\Lambda\\in V_\\lambda$ belongs to the $H_k$-invariant subspace\nin $V_\\lambda$, equivalent to the $H_k$-module $V_{\\kappa}$ where\nthe partition $\\kappa$ is the shape of the tableau obtained by\nremoving from $\\Lambda$ the entries $k+1, \\ldots , n$.\n\\end{proposition}\n\nThe properties of the vector $G_\\Lambda$ given \nby Proposition \\ref{Vkappa} for $k=1, \\ldots , n-1$,\ndetermine this vector \nin $V_\\lambda$ uniquely up to a non-zero factor from $\\mathbb{C}(q)$. \nThese properties can be restated for any irreducible\n$H_n$-module $V$ equivalent to $V_\\lambda$.\nExplicit formulas for the action of the generators $T_1, \\ldots , T_{n-1}$\nof $H_n$ on the vectors in $V$ determined by these properties,\nare known; cf.\\ \\cite[Theorem 6.4]{M}.\n\nSetting $q=1$, the algebra $H_n$ specializes to the symmetric group ring\n$\\mathbb{C}S_n$. The element $T_\\sigma\\in H_n$ then specializes to the\npermutation $\\sigma\\in S_n$ itself. \nThe proof of Proposition \\ref{q-jimtheorem1} demonstrates that the coefficients\nin the expansion of the element $F_\\Lambda\\in H_n$ relative to the basis\nof the elements $T_\\sigma$, are regular at $q=1$ as rational functions\nof the parameter $q$. Thus the specialization of the element\n$F_\\Lambda\\in H_n$ at $q=1$ is well defined. The same is true for the\nelement $G_\\Lambda\\in H_n$.\nThe specializations at $q=1$ of the basis vectors $G_\\Lambda\\in V_\\lambda$\nform the \\emph{Young seminormal basis} in the corresponding\nirreducible representation of the group $S_n$.\nThe action of the generators $\\sigma_1, \\ldots ,\\sigma_{n-1}$ of $S_n$ on the \nvectors of the latter basis was first given by \\cite[Theorem IV]{Y2}.\nFor the interpretation of the elements $F_\\Lambda$ and $G_\\Lambda$ using\nrepresentation theory of the affine Hecke algebra $\\widehat{H}_n$, \nsee \\cite[Section 3]{C2} and references therein.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\large Supplemental Material}\n\n\n\\subsection{Time Dependencies}\n\n\\subsubsection{Intermediate Times}\n\nIn our paper, we discussed the frequency dependence of the\noptical conductivity $\\text{Re} \\, \\sigma(\\omega)$ rather than\nthe time dependence of the spin-current autocorrelation function\n$C(t)$ as such. However, we determined $\\text{Re} \\, \\sigma(\\omega)$\nvia the finite-time Fourier transform of $C(t)$. Furthermore, it is\ninstructive to discuss the real-time decay of $C(t)$. Thus, we\nshow in Fig.\\ \\ref{FigS1} the time-dependent data underlying\nFig.\\ 3 in our paper.\n\nClearly, for all $0.5 \\leq W\/J \\leq 4.0$ and the two $\\Delta = 1.0$\nand $1.5$ depicted, the initial value $C(0)$ agrees well with the\nsum rule $J^2\/8$, which also shows the accuracy of our DQT approach. \nFor all $W$ depicted, the initial decay of $C(t)$ is fast with a\nrelaxation time $\\tau \\, J \\ll 10$. It is clearly visible that $C(t)$\ndevelops oscillatory behavior for large $W$, which is the origin\nof the maximum $\\sigma_\\text{max}$ located at the position\n$\\omega_\\text{max}$, as discussed in our paper. However, all\noscillations fully decay on a time scale $t \\, J \\leq 40$ and\nthere is no signature of a conserved Drude-weight contribution\nto $C(t)$ in the long-time limit. Therefore, $t \\, J \\leq\n40$ data is sufficient to precisely determine the $\\omega$\ndependence of $\\text{Re} \\, \\sigma(\\omega)$ in general and the\nvalue of $\\sigma_\\text{dc}$ in particular.\n\n\\subsubsection{Long Times}\n\nIn Fig.\\ \\ref{FigS2} (a) we show $C(t)$ at $\\Delta = 1$ and\n$W\/J = 3.5$ for even longer times $t J \\leq 400$ and as many\nas $N = 30000$ disorder realizations in a system of size $L=18$.\nClearly, $C(t)$ is practically zero for $t J \\gtrsim 50$. Thus, while\ntaking into account $t J \\leq 400$ in the Fourier transform certainly\nincreases frequency resolution, we find no change of the linear\nfrequency dependence down to a rather small scale of frequency, see\nFig.\\ \\ref{FigS2} (b). It is worth mentioning that the $W\/J = 3.5$\ncalculation depicted in Fig.\\ \\ref{FigS2} took about $20$ CPU years in\ntotal.\n\nIn Fig.\\ \\ref{FigS2} we also depict high-resolution data for the\nsame set of parameters except for $W\/J = 2.0$, $2.5$ and $N=5000$, cf.\\\nFig.\\ 4 in the main text. Apparently, this data is well described by power\nlaws with the exponent $\\alpha$ being close to 1 in both cases.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.95\\columnwidth]{FigS1.eps}\n\\caption{(Color online) Data underlying Fig.\\ 3 of the main text: Real-time decay of the\nhigh-temperature current autocorrelation function $C(t)$ of the spin-$1\/2$ Heisenberg\nchain at (a)-(d) $\\Delta = 1.0$ and (e)-(h) $\\Delta = 1.5$ in the transition from (a),\n(e) small disorder $W \/ J = 0.5$ over intermediate disorder (b), (f) $W \/J = 1.0$, (c),\n(g) $W \/ J = 2.0$ to strong disorder (d), (h) $W \/ J = 4.0$, as obtained numerically\nfor the grand-canonical ensemble $\\langle S^z \\rangle = 0$ and system sizes $L = 22$\nand $24$. The results shown are averaged over $N = 200$ different disorder realizations\nusing a uniform distribution $[-W,W]$. The sum rule is $C(0) \/ J^2 = 0.125$. In all\ncases (a)-(h), $C(t)$ decays fully on a time scale $t \\, J \\leq 40$. Damping of $C(t)$\nafter its first zero crossing causes the linear $\\omega$ dependence $\\text{Re} \\,\n\\sigma(\\omega) \\approx \\sigma_\\text{dc} + a |\\omega|$.}\n\\label{FigS1}\n\\end{figure*}\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{FigS2.eps}\n\\caption{(Color online) (a) Real-time decay of $C(t)$ for very long times $t J \\leq\n400$, averaged over as many as $N=30000$ disorder realizations in a system of size\n$L=18$. Remaining parameters: $\\Delta = 1$, $W\/J=3.5$, and $\\beta J \\to 0$. (b) Fourier\ntransform of (a) and, additionally, for $W\/J = 2.0, 2.5$ in a log-log plot. Remaining\nparameters: identical to (a) except for $N = 5000$. Note that $\\sigma_{\\mathrm dc}$ is\nsubtracted from the Fourier transform. Power-law fits are also indicated.}\n\\label{FigS2}\n\\end{figure}\n\n\\subsection{Binary Disorder}\n\nOur paper focused on local magnetic fields $B_r$ drawn at\nrandom from a uniform distribution in the interval $[-W,W]$. To\ndemonstrate that the results presented do not depend on the\nspecific distribution used, we repeat the $\\Delta=1.5$ calculations\nfor $W\/J = 1.0$ and $W\/J = 2.0$ in Fig.\\ 3 (f) and (g) for\na binary distribution with the same width, i.e., $B_r = \\pm \\sqrt{3}\n\\, W$. In Fig.\\ \\ref{FigS3} we compare the corresponding results.\nEvidently, the low-$\\omega$ behavior of the optical conductivity\n$\\text{Re} \\, \\sigma(\\omega)$  is the same for both distributions,\nwhile differences can only be seen in the high-$\\omega$ behavior,\nemerging for strong disorder $W$. These differences are not relevant\nfor the physics discussed in our paper.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{FigS3.eps}\n\\caption{(Color online) Optical conductivity $\\text{Re} \\, \\sigma(\\omega)$\nfor binary and uniform distribution and disorder strength (a)\n$W\/J = 1$ and (b) $W\/J = 2$. Remaining parameters: $\\Delta=1.5$,\n$\\beta \\, J \\to 0$, $L \\leq 24$, $t \\, J \\leq 40$, and $N=200$.\nClearly, the low-$\\omega$ behavior does not depend on the specific\nprobability distribution used, while high-$\\omega$ differences\nemerge for large $W$.}\n\\label{FigS3}\n\\end{figure}\n\n\\subsection{Finite-Size Effects:\\\\ Comparison to Clean Systems}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{FigS4.eps}\n\\caption{(Color online) Optical conductivity $\\text{Re} \\, \\sigma(\\omega)$ for\ncases without disorder but with (a) $\\Delta' = 0.5$ (non-integrable) and (b)\n$\\Delta' = 0.0$ (integrable). Remaining parameters: $\\Delta=1.5$, $\\beta\n\\, J \\to 0$, $L \\leq 34$, $t \\, J \\leq 20$, and $N=1$. The solid line in\n(b) is the perturbative result of \\cite{Ssteinigeweg2011}.\nInset: (a) for $t \\, J \\leq 100$, i.e., higher $\\omega$ resolution.}\n\\label{FigS4}\n\\end{figure}\n\nWe found in our paper that the optical conductivity $\\text{Re} \\,\n\\sigma(\\omega)$ has a maximum $\\sigma_\\text{max} > \\sigma_\\text{dc}$\nlocated at a position $\\omega_\\text{max} > 0$. Moreover, we observed\nlittle finite-size effects for large $L \\geq 22$. Even though not\nexpected, we cannot exclude a very slow convergence to the thermodynamic\nlimit $L \\to \\infty$. Note that estimating potential finite-size effects\non the basis of the non-interacting case $\\Delta = 0$ is not meaningful\nfor two reasons: First, for $\\Delta = 0$ and $W > 0$, the localization\nlength represents a natural scale for finite-size effects but is absent\nin the thermal phase of the $\\Delta > 0$ problem. Second, also the case\n$\\Delta = W = 0$ is well-known to feature huge finite-size effects\nbecause of the highly degenerated spectrum. Moreover, the mean free\npath is infinitely large.\n\n\\subsubsection{Non-Integrable Systems}\n\nTo provide further evidence for finite-size effects being negligibly small,\nwe compare to results for cases without disorder, i.e., $W = 0$. For such\ncases, and $\\Delta = 1.5$, the diffusion constant can be estimated\nperturbatively along the lines of \\cite{Ssteinigeweg2011},\nyielding $D\/J \\sim 0.6$. This value of $D$ corresponds to a mean free path\nof a few lattice sites, i.e., the mean free path is small compared to\ntypical system sizes considered. Thus, finite-size effects are most likely\nrelated to the Hilbert-space dimension being finite and not to a physical\nlength scale as such. Note that this line of reasoning is also meaningful\nfor disordered but thermal cases.\n\nWe again break the integrability of the XXZ spin-$1\/2$ chain but now by\nadding to Eq.\\ (1), where $W=0$, the next-to-nearest neighbor interaction\n\\begin{equation}\nH' = J \\, \\Delta' \\sum_{r=1}^L S_r^z S_{r+2}^z \\label{NNN}\n\\end{equation}\nwith the anisotropy $\\Delta'$. Adding Eq.\\ (\\ref{NNN}) does not break\ntranslation invariance and does not change the form of the spin-current\noperator.\n\nIn Fig.\\ \\ref{FigS4} (a) we depict the high-temperature optical conductivity\n$\\text{Re} \\, \\sigma(\\omega)$ for $\\Delta = 1.5$ and $\\Delta' = 0.5$, for\na large $L = 30$ and a small enough $L = 22$ to illustrate the role of\nfinite-size effects. It is clearly visible that, as long as $L \\ll 30$,\n$\\sigma_\\text{dc}$ decreases with $L$. Hence, together with the overall\nconvergence at frequencies $\\omega \/ J \\gtrsim 0.4$, Fig.\\ \\ref{FigS4}\n(a) proves $\\sigma_\\text{dc} < \\sigma_\\text{max}$ in another model. Note\nthat the largest-subspace dimension for $L = 30$ is comparable to the one\nof the $L=26$ disordered model.\n\nThis $\\omega$ dependence of $\\text{Re} \\, \\sigma(\\omega)$ has been found\nalso in \\cite{Smierzejewski2011} using Lanczos diagonalization\nand, for spin-$1\/2$ ladders, in \\cite{Skarrasch2015} using\ntime-dependent density-matrix renormalization group.\n\n\\subsubsection*{Integrable Systems}\n\nEventually, we contrast all our results presented so far against the\nlarge finite-size effects in the integrable cases $W = \\Delta' = 0$,\nas shown in Fig.\\ \\ref{FigS4} (b) for $\\Delta = 1.5$. Here, $\\text{Re}\n\\, \\sigma(\\omega)$ is governed by finite-size effects at $\\omega = 0$\nand $\\omega > 0$. Furthermore, these finite-size effects depend on\nthe $\\omega$ resolution used, see the inset of Fig.\\ \\ref{FigS4} (b).\nThus, a very careful analysis is needed to determine correctly the\nthermodynamic limit \\cite{Sprelovsek2004, Ssteinigeweg2011, Skarrasch2014}, \nyielding the dc value $\\sigma_\\text{dc} \/ \\beta \\, J\\sim 0.15$. While\nthis value is depicted in Fig.\\ 4 (a) of our paper, none of\nour actual results rely on any kind of extrapolation.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{FigS5.eps}\n\\caption{(Color online) \\label{FigS5} Optical conductivity $\\text{Re} \\,\n\\sigma(\\omega)$ at high temperatures $\\beta \\to 0$, as calculated by ED\n($L=14$), FTLM ($L=22$), and DQT ($L=22$), for the isotropic point $\\Delta\n=1.0$ and various disorder strengths (a) $W\/J = 0.5$ ,$1.0$ and (b) $W\/J =\n2.0$, $4.0$. ED data for $W\/J = 1.0$, $4.0$ is taken from\n\\cite{Sgopalakrishnan2015}.}\n\\end{figure}\n\n\\subsection{Comparison to Exact and Lanczos Diagonalization}\n\nTo additionally confirm the DQT results presented in our paper, we present\nresults from two other numerical techniques: standard exact diagonalization\n(ED) and the finite-temperature Lanczos method (FTLM) \\cite{Sprelovsek2013}.\nBoth numerical techniques have direct access to the frequency domain. While\nED data is binned in channels of width $\\delta \\omega \/J = 0.005$,\nthe resolution of FTLM depends on the number of Lanczos steps $M$ and\nthe energy span $\\Delta E$, i.e., $\\delta\\omega \\propto \\Delta E\/M$.\nHere, we use $M=400$. Furthermore, we use $10$ initial random vectors for\neach of the $N=100$ disorder realizations, to decease any remaining\nstatistical error associated with the initial state. \n\nIn Fig.\\ \\ref{FigS5} we show ED ($L=14$) and FTLM ($L=22$) data for\n$\\Delta = 1.0$, together with the DQT data ($L=22$) presented in the\nmain text. The overall agreement of all methods for $W\\ge1$ is\nremarkably good, despite the smaller system size $L=14$ accessible to ED.\nFor small $W$, significant finite-size effects are visible for $L=14$.\nNote that ED data for $W\/J = 1.0, 4.0$ is taken from \\cite{Sgopalakrishnan2015}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Secure Minimum Energy Routing}\n\\label{sec:algorithms}\n\nIn this section, we investigate the secure minimum energy routing\nproblem, where the cost of a path is given by~\\eqref{eq:pcost}. We\nbegin by establishing that it is NP-hard. Then, by exploiting the\nstructure of the optimal solution, we employ dynamic programming to\nobtain a pseudo-polynomial time algorithm that provides an exact\nsolution. This means that the problem is weakly NP-hard~\\cite{garey},\nthus fully polynomial time approximate schemes are possible.\nAccordingly, we conclude the section by presenting a fully polynomial\ntime $\\epsilon$-approximation algorithm for the problem,\nwhich takes an approximation parameter $\\epsilon > 0$ and after\nrunning for time polynomial in the size of the network and in\n$1\/\\epsilon$, it returns a path whose cost is at most $(1+\\epsilon)$\ntimes more than the optimal value.\n\n\n\\subsection{Computational Complexity}\nWe first show that our routing problem is NP-hard\nvia a reduction from the partition problem.\n\n\\begin{theorem}\nProblem SMER is NP-hard.\n\\end{theorem}\n\n\\begin{proof}\nWe describe a polynomial time reduction of the Partition\nproblem~\\cite{garey} to SMER. Given a set of integers $\\mc{S}=\n\\set{k_1, k_2, \\ldots, k_n}$, with $\\sum_{i=1}^{n} k_i = 2\\cdot K$,\nthe Partition problem is to decide whether there is a subset\n$\\mc{S'}$ of $\\mc{S}$ such that $\\sum_{i\\in \\mc{S'}} k_i = K$.\n\nGiven an instance $\\mc{S}= \\set{k_1, k_2, \\ldots, k_n}$ of the\nPartition problem, with $\\sum_{i=1}^{n} k_i  = 2\\cdot K$, we\nconstruct the following network. The set of nodes is identical to\n$\\mc{S}$. For $i=1$ to $n-1$, we interconnect node $k_i$ to node\n$k_{i+1}$ with two links, as follows: an ``upper'' link\n$\\ell_i^{(u)}$, to which we assign $\\mc{C}_1(\\ell_i^{(u)})= 2\\cdot K\n\\cdot k_i$ and $\\mc{C}_2(\\ell_i^{(u)})= 0$, and a ``lower'' link\n$\\ell_i^{(w)}$, to which we assign $\\mc{C}_1(\\ell_i^{(w)})= 0$ and\n$\\mc{C}_2(\\ell_i^{(w)})= k_i$.\n\n\\begin{lemma}\nThe answer to the Partition problem is affirmative {\\em iff} the\nsolution to SMER in the constructed network, \\ie\\ the minimum value\nof $(\\ref{eq:pcost2})$ of a path between nodes $k_1$ and $k_n$,\nequals $3 \\cdot K^2$.\n\\end{lemma}\n\n\\begin{proof}\nA path $\\Pi$ between nodes $k_1$ and $k_n$ consists of a (possibly\nempty) set of ``upper'' links $\\mc{U}$ and a (possibly  empty) set\nof ``lower'' links $\\mc{W}$. Let $\\mc{S}_u$ and $\\mc{S}_w$ be,\ncorrespondingly, the sets of indices of the links in $\\mc{U}$ and in\n$\\mc{W}$, \\ie\\ $i\\in \\mc{S}_u$ {\\em iff} $\\ell_i^{(u)}\\in \\mc{U}$\nand $i\\in S_w$ {\\em iff} $\\ell_i^{(w)}\\in \\mc{W}$. Clearly,\n$\\mc{S}_u \\cup \\mc{S}_w = \\mc{S}$. The cost of the path, per\n$(\\ref{eq:pcost2})$, is given by:\n\\begin{equation}\n\\label{eq:nph1}\n\\begin{split}\n    \\mc{C}(\\Pi)\n        &= \\sum_{\\ell_i^{(u)}\\in \\mc{U}} \\mc{C}_1(\\ell_i^{(u)}) + \\sum_{\\ell_i^{(w)}\\in \\mc{W}} \\mc{C}_1(\\ell_i^{(w)})\\\\\n          &\\quad+ \\Big( \\sum_{\\ell_i^{(u)}\\in \\mc{U}} \\mc{C}_2(\\ell_i^{(u)}) + \\sum_{\\ell_i^{(w)}\\in \\mc{W}} \\mc{C}_2(\\ell_i^{(w)}) \\Big)^2, \\\\\n    &= \\sum_{i\\in \\mc{S}_u} (2\\cdot K \\cdot k_i) + \\sum_{i\\in \\mc{S}_w} 0\n          + \\Big( \\sum_{i\\in \\mc{S}_u} 0 + \\sum_{i\\in \\mc{S}_w} k_i \\Big)^2\n          \\eqend\n\\end{split}\n\\end{equation}\nConsider first the case $\\sum_{i\\in \\mc{S}_u} k_i \\geq \\sum_{i\\in\n\\mc{S}_w} k_i$. Since $\\sum_{i\\in \\mc{S}_u} k_i + \\sum_{i\\in\n\\mc{S}_w} k_i = 2\\cdot K$, denote: $\\sum_{i\\in \\mc{S}_u} k_i =  K +\n\\delta$, $\\sum_{i\\in \\mc{S}_w} k_i =  K - \\delta$, for some $\\delta\n\\geq 0$. Then, from (\\ref{eq:nph1}), we have:\n\\begin{equation*}\n\\label{eq:nph2}\n\\begin{split}\n    \\mc{C}(\\Pi)\n        = 2\\cdot K \\cdot (K+\\delta) + \\Big( K-\\delta \\Big)^2 = 3 \\cdot K^2 + \\delta^2\n    \\eqend\n\\end{split}\n\\end{equation*}\nConsider now the case $\\sum_{i\\in \\mc{S}_u} k_i < \\sum_{i\\in\n\\mc{S}_w} k_i$. It follows similarly that\n\\begin{equation*}\n\\label{eq:nph3}\n    \\mc{C}(\\Pi) = 3 \\cdot K^2 + \\delta^2 \\eqend\n\\end{equation*}\nWe conclude that the length of a path between nodes $k_1$ and $k_n$\nis at least $3 \\cdot K^2$, and, furthermore, that value is attained\n{\\em iff} the set $\\mc{S}$ can be partitioned into two subsets\n$\\mc{S}_u$ and $\\mc{S}_w$, such that $\\sum_{i\\in \\mc{S}_u} k_i =\n\\sum_{i\\in \\mc{S}_w} k_i$, \\ie\\ {\\em iff} there is a subset $\\mc\n{S'}= \\mc{S}_u$ of $\\mc{S}$ such that  $\\sum_{i\\in \\mc{S'}} k_i =\nK$, and the lemma follows.\n\\end{proof}\nSince the Partition problem is NP-complete~\\cite{garey}, the\n\\mbox{theorem} follows.\n\\end{proof}\n\n\n\\subsection{Pseudo-Polynomial Time Exact Algorithm}\nFirst, scale the values of the $\\mc{C}_2(\\ell)$'s for any link $\\ell$\nin the network so that they are all integers.\\fnote{\\mbox{The value\nof ``$1$'' is determined by the precision at which we compute\n$\\mc{C}_2(\\ell)$'s}.} Let $B$ denote an upper-bound on the sum of the\n$\\mc{C}_2(\\ell)$'s on any simple path. A trivial bound is given by\n$B=(N-1)\\cdot\\mc{C}_2^{max}$, where $N$ is the number of nodes in the\nnetwork and $\\mc{C}_2^{max}$ is the maximum value of $\\mc{C}_2(\\ell)$\namong all network links.\nIn a network with $N$\nnodes, $\\mc{C}_2^{max}$ can be computed in $O(N^2)$ time via a\nbrute-force search.\n\nOur algorithm, termed DP-SMER, is listed below. \\mbox{DP-SMER} iterates\nover all values of $\\mc{C}_2(\\ell)$, \\ie\\\n$\\mc{C}_2(\\ell)=1,2,\\ldots,B$, and for each value of\n$\\mc{C}_2(\\ell)$, it minimizes $\\sum \\mc{C}_1(\\ell)$. Upon return,\nthe algorithm returns the cost of the optimal path from source $s$\nto destination $d$ along with the structure $\\Pi$ that contains the\nnetwork nodes that form the path.\n\n\\begin{algorithm}\n\\caption{DP-SMER (source $s$, dest. $d$, network $\\mc{N}$).}\n\\label{alg:dp-smer}\n\\renewcommand{\\algorithmiccomment}[1]{\/*~#1~*\/}\n\\begin{algorithmic}\n    \\STATE \\COMMENT{path cost from $s$ to itself is always $0$}\n        \\FOR {$b = 1 \\to B$}\n                \\STATE $C_s(b) = 0$\n            \\ENDFOR\n    \\STATE \\COMMENT{initial path cost from $s$ to any other node is infinite}\n        \\FORALL {$n_i \\in \\mc{N}$, $n_i \\neq s$}\n            \\FOR {$b = 1 \\to B$}\n                        \\STATE $C_i(b) = \\infty$\n            \\ENDFOR\n        \\ENDFOR\n\n\n    \\FOR {$b = 1 \\to B$}\n        \\STATE \\COMMENT{all node pairs can form a link and be neighbors}\n        \\FORALL {$n_i \\in \\mc{N}$}\n            \\FORALL {$n_j \\in \\mc{N}$}\n                \\STATE \\COMMENT{update path cost via the neighboring nodes}\n            \\IF {$b+\\mc{C}_2(\\ell_{ij}) \\leq B$}\n                \\STATE $ t = C_{i}(b) + \\mc{C}_1(\\ell_{ij})$\n                \\IF {$t < C_{j}(b+\\mc{C}_2(\\ell_{ij}))$}\n                            \\STATE $\\Pi_j(b+\\mc{C}_2(\\ell_{ij})) = i$ \\COMMENT {set $n_j$'s parent to $n_i$}\n                            \\STATE $C_{j}(b+\\mc{C}_2(\\ell_{ij})) = t$ \\COMMENT {update path cost}\n                \\ENDIF\n            \\ENDIF\n            \\ENDFOR\n            \\ENDFOR\n        \\ENDFOR\n\n    \\STATE \\COMMENT{include the ``$b$'' component, \\ie\\ $\\mc{C}_2$, in the path costs}\n    \\FOR {$b = 1 \\to B$}\n            \\STATE $\\hat{C}_d(b) = C_d(b) + b^2$\n    \\ENDFOR\n\n    \\STATE \\COMMENT{choose the best value for reaching the destination}\n    \\STATE $\\displaystyle b^* = \\arg \\min_{b} \\hat{C}_d(b)$\n        \\STATE {\\bf return} $[\\hat{C}_d(b^*), \\Pi(b^*)]$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{theorem}\n\\label{theo:DP} DP-SMER runs in time $O(N^2 \\cdot B)$, where $N$ is\nthe number of nodes in the network. Upon completion, the algorithm\nreturns an optimal solution to Problem SMER.\n\\end{theorem}\n\\begin{proof}\nThe first claim follows by noting that the computational complexity\nis dominated by an iteration on all values $1,2,\\ldots,B$, and for\neach such iteration, iterating on all pairs of nodes.\n\nWe turn to consider the second claim. First, it can be established,\nby induction on the values of $b$, that, upon completion of the\n$b$-th iteration of the main loop of the algorithm, for all nodes\n$n_i$, $C_i(b)$ is the length of a shortest path with respect to the\nmetric of the $\\mc{C}_1(\\ell_{ij})$ values, among all paths between\nthe source $s$ and node $n_i$, whose length with respect to the\nmetric of the $\\mc{C}_2(\\ell_{ij})$ values is precisely $b$.\\fnote{We\nnote that this shortest path may be non-simple, \\ie\\, include loops,\ndue to the potentially negative values of $\\mc{C}_1(\\ell_{ij})$'s;\nnonetheless, it is a finite path, and, furthermore, the optimal path\nreturned by the last step of DP-SMER is guaranteed to be simple, due\nto the monotonicity property explained at the end of Section\n\\ref{sec:pathcost}.} Furthermore, it is easy to verify that the\nvalues of $\\hat{C}_N (b)$, computed at the next step of the\nalgorithm, stand for the lengths of the above shortest paths with\nrespect to the metric considered by Problem SMER.\n\nNow, let $\\Pi^*$ be an optimal solution (\\ie\\ a path) to Problem\nSMER, and denote by $C^*$ its length with respect to the metric\nconsidered by SMER. Furthermore, denote $b^* =\n\\sum_{\\ell_{ij}\\in\\Pi^*} \\mc{C}_2(\\ell_{ij})$. It is easy to verify\nthat $\\Pi^*$ is a shortest path with respect to the metric of the\n$\\mc{C}_1(\\ell_{ij})$ values, among all paths between the source $s$\nand the destination $d$, whose length with respect to the metric of\nthe $\\mc{C}_2(\\ell_{ij})$ values is precisely $b^*$. Therefore, upon\ncompletion of the above steps of the algorithm, we will have\n$\\hat{C}_N (b^*) = C^*$; moreover, since $\\Pi^*$ is an optimal\nsolution to SMER, it must hold that $\\hat{C}_N (b^*) \\leq \\hat{C}_N\n(b)$ for all values of $b$. The theorem follows.\n\\end{proof}\n\n\n\\subsection{Fully Polynomial Time $\\epsilon$-Approximation}\nAs in the previous section, we scale the values of the\n$\\mc{C}_2(\\ell)$'s for any link $\\ell$ in the network so that they\nare all integers and denote by $B$ an upper-bound on the sum of the\n$\\mc{C}_2(\\ell)$'s on any simple path.\n\nThe above pseudo-polynomial solution indicates that SMER is only\nweakly NP-hard (see~\\cite{garey}), which enables us to apply\nefficient, $\\epsilon$-optimal approximation schemes of polynomial\ntime complexity, similar to the case of the widely investigated\nRestricted Shortest Path problem (RSP, see,\n\\eg~\\cite{Lorenz99asimple} and references therein). The RSP problem\nconsiders a network where each link has two metrics, say ``cost''\nand ``delay'', and some ``bound'' on the end-to-end delay. Then, for\na given source-destination pair, the problem is to find a path of\nminimum cost among those whose delay do not exceed the delay bound.\nThis weakly NP-hard problem admits efficient $\\epsilon$-optimal\napproximation schemes of polynomial\ncomplexity,~\\eg~\\cite{Lorenz99asimple}.\n\nWe turn to specify our approximation scheme for Problem SMER by a\nsimple employment of any solution to the RSP problem.\\fnote{Other\nsolutions, of reduced computational complexity, can be established,\nyet their structure is somewhat more complex.} First, a technical\ndifficulty arises in applying RSP approximation schemes to Problem\nSMER. Recall that while link costs as given by~\\eqref{eq:lcost} are\nnon-negative, $\\mc{C}_1(\\ell)$ can be negative for some links\n$\\ell$. In RSP, specifically in the approximation scheme\nof~\\cite{Lorenz99asimple}, it is assumed that link costs are\nnon-negative. Nevertheless, we show that the original network with\npossibly negative link weights can be safely transformed (\\ie\\\nwithout affecting the identity of the solution) to an expanded\nnetwork with non-negative link weights,  by employing the following\npre-processing step:\n\\begin{algorithm}\n\\caption{Expand\\_Network (source $s$, network $\\mc{N}$).}\n\\begin{enumerate}\n\\item Add the source node $s$ to the expanded network.\n\\item For each node $u$ $(u \\neq s)$ in the original network,\nadd $N-1$ replicas denoted by $u(1), u(2), \\ldots, u(N-1)$ to the\nexpanded network.\n\n\\item For each link $\\ell_{su}$ from node $s$ to node $u$ in the original network,\nadd a link from node $s$ to node $u(1)$ in the expanded network with\nthe same metrics as for the original link.\n\n\\item For each link $\\ell_{uv}$ in the original network, where \\mbox{$u \\neq s, u \\neq d,\nv \\neq s$}, and for each $h=1,\\ldots, N-2$, add a link between node\n$u(h)$ and node $v(h+1)$ in the expanded network with the same\nmetrics as for the original link.\n\n\\item For each link $\\ell$ in the expanded network, add some (identical to all links)\nbias $\\delta \\ge 0$\nto each link cost $\\mc{C}_1(\\ell)$ so that the new link costs would\nbe non-negative.\n\\end{enumerate}\n\\end{algorithm}\n\nThe following lemmas establish the relation between the shortest\npaths in the original network and the shortest paths in the expanded\nnetwork.\n\\begin{lemma}\n\\label{l:exp}\nA path that is shortest w.r.t. the biased metric $(\\mc{C}_1(\\ell) +\n\\delta)$ among those that obey a bound on the $\\sum \\mc{C}_2(\\ell)$\nand have precisely $h$ hops, is also shortest w.r.t. the unbiased\nmetric $\\mc{C}_1(\\ell)$ among those that obey the same bound on\n$\\sum \\mc{C}_2(\\ell)$ and have precisely $h$ hops.\n\\end{lemma}\n\\begin{proof}\nSuppose that this is not true. That is, there are paths $\\Pi$ and\n$\\Pi'$, both obeying the bound on $\\sum \\mc{C}_2(\\ell)$ and with $h$\nhops, in such a way that $\\Pi'$ is a shortest path with the bias yet\n$\\Pi$ is shorter without the bias. Therefore, $\\sum_{\\ell \\in \\Pi}\n\\mc{C}_1(\\ell) < \\sum_{\\ell \\in \\Pi'} \\mc{C}_1(\\ell)$, yet\n$\\sum_{\\ell \\in \\Pi} (\\mc{C}_1(\\ell) + \\delta) \\ge \\sum_{\\ell \\in\n\\Pi'} (\\mc{C}_1(\\ell) + \\delta)$. However, the second inequality can\nbe rewritten as:\\linebreak $\\sum_{\\ell \\in \\Pi} \\mc{C}_1(\\ell) + h \\cdot\n\\delta \\ge \\sum_{\\ell \\in \\Pi'} \\mc{C}_1(\\ell) + h \\cdot \\delta$,\nwhich contradicts the first inequality.\n\\end{proof}\n\n\\begin{lemma}\n\\label{l:hops} A shortest path from source $s$ to node $d(h)$ in the\nexpanded network has precisely $h$ hops.\n\\end{lemma}\n\\begin{proof}\nThe proof follows from the fact that the $i$-th hop on the shortest path\nfrom $s$ to $d(h)$ has to go from some node $v(i-1)$ to some node\n$u(i)$ (see the network expansion procedure).\n\\end{proof}\n\nThus, to compute an $\\epsilon$-optimal solution to Problem SMER, for\nevery bound on $\\sum \\mc{C}_2(\\ell)$, we find the shortest path with\n$h=1,\\ldots,N-1$ hops in the expanded network by repeatedly\nemploying an approximation solution to the RSP problem.\nFor a given\napproximation value $\\epsilon>0$, let \\mbox{$\\eta=\\epsilon\/3$}.\nFurthermore, let $L$ be the smallest integer for which $\\lceil\n(1+\\eta)^L \\rceil \\geq B$. Our algorithm, called $\\epsilon$-SMER, is\nlisted below. In this algorithm, $\\epsilon$-RSP refers to an\n$\\epsilon$-optimal approximation solution for the RSP problem.\n\n\\begin{algorithm}\n\\caption{$\\epsilon$-SMER (error $\\epsilon$, source $s$, dest. $d$,\nnet. $\\mc{N}$).}\n\\renewcommand{\\algorithmiccomment}[1]{\/* #1 *\/}\n\\begin{algorithmic}\n    \\STATE $\\mc{N}_x$ = Expand\\_Network($s$, $\\mc{N}$)\n\n    \\FORALL {$\\ell \\in \\mc{N}_x$}\n        \\STATE $cost(\\ell) = \\mc{C}_1(\\ell)$\n        \\STATE $delay(\\ell) = \\mc{C}_2(\\ell)$\n    \\ENDFOR\n\n    \\FOR {$l = 1 \\to L$}\n        \\STATE $delay\\_bound = \\lceil (1+\\eta)^l \\rceil$\n        \\STATE \\COMMENT{compute the approximate $h$-hop path}\n        \\FOR {$h = 1 \\to N-1$}\n            \\STATE $[C(l, h), \\Pi(l, h)] =$ $\\epsilon$-RSP$(\\epsilon, s, d(h), \\mc{N}_x)$\n         \\STATE \\COMMENT{compute the actual cost as per SMER metric}\n        \\STATE $\\hat{C}(l,h) = (C(l,h) - h \\cdot \\delta) + \\lceil (1+\\eta)^l \\rceil^2$\n        \\ENDFOR\n    \\ENDFOR\n\n    \\STATE \\COMMENT{choose the best $l$ and $h$ for reaching the destination}\n    \\STATE $\\displaystyle (l^*,h^*) = \\arg \\min_{l,h} \\hat{C}(l,h)$\n        \\STATE {\\bf return} $[\\hat{C}(l^*,h^*), \\Pi(l^*,h^*)]$\n\\end{algorithmic}\n\\end{algorithm}\n\nIn the $\\epsilon$-SMER algorithm, for each considered delay bound\n$\\lceil (1+\\eta)^l \\rceil$, $N-1$ instances of the approximation\nsolution to the RSP problem, for the same bound, are run on the\nexpanded network: in each instance $h$, we consider $s$ to be the\nsource and $d(h)$ to be the destination. Using Lemma~\\ref{l:hops},\nit is straightforward to verify that, in each instance $h$, the RSP\napproximation obtains a solution that satisfies the required delay\nbound with the restriction that the path has {\\em precisely} $h$\nhops (in both the expanded and the original network).\n\nTherefore, per considered bound on the $\\mc{C}_2(\\ell)$ metric and\nper possible number of hops up to $N-1$, we get an $\\epsilon$-optimal\npath with respect to the original metric $\\mc{C}_1(\\ell)$ (of\nprecisely that many hops). It follows from Lemmas~\\ref{l:exp} and\n~\\ref{l:hops}, that, by comparing all solutions (for all considered\nbounds on the $\\mc{C}_2(\\ell)$ metric and number of hops $h$), we\nwill find a shortest $\\epsilon$-optimal path that corresponds to an\n$\\epsilon$-optimal solution to SMER. This is established next\nthrough the following lemmas and theorem.\n\n\\begin{lemma}\n\\label{l:eps1} Let $\\Pi^*$ be an optimal solution (path) to SMER.\nDenote by $\\mc{C}(\\Pi^*)$ and $\\mc{C}(\\hat{\\Pi})$, the costs, per\nthe SMER metric, of the optimal solution and of the solution\nobtained by $\\epsilon$-SMER, correspondingly. Then:\n\\begin{equation}\n\\label{eq:eps1}\n    \\mc{C}(\\hat{\\Pi}) \\leq (1+\\epsilon) \\cdot \\mc{C}(\\Pi^*)\n    \\eqend\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet $\\bar{l}$ be the smallest integer such that\n\\begin{equation}\n\\label{eq:eps2}\n    \\Big( \\sum_{\\ell \\in \\Pi^*} \\mc{C}_2(\\ell)\n    \\Big)^2 \\leq \\Big( \\lceil {(1+\\eta)^{\\bar{l}}} \\rceil \\Big)^2\n    \\eqend\n\\end{equation}\nNote that this implies that:\n\\begin{equation}\n\\label{eq:eps3}\n    (1+\\eta)^2 \\Big( \\sum_{\\ell \\in \\Pi^*}\n    \\mc{C}_2(\\ell) \\Big)^2 \\geq \\Big( \\lceil (1+\\eta)^{\\bar{l}} \\rceil\n    \\Big)^2\n    \\eqend\n \\end{equation}\nLet $\\bar{h}$ be the number of hops of $\\Pi^*$. By construction,\n$\\Pi (\\bar{l},\\bar{h})$ is an $\\epsilon$-optimal approximation for\nRSP, for ``costs'' $\\mc{C}_1(\\ell)$, ``delays'' $\\mc{C}_2(\\ell)$,\n``delay bound'' $\\lceil (1+\\eta)^{\\bar{l}} \\rceil$ and precisely\n$\\bar{h}$ hops. Moreover, by (\\ref{eq:eps2}), the path $\\Pi^*$ obeys\nthis bound. Therefore:\n\\begin{equation}\n    \\sum_{\\ell \\in \\Pi (\\bar{l},\\bar{h})} \\mc{C}_1(\\ell) + \\bar{h} \\cdot \\delta \\leq\n    (1+\\epsilon) \\sum_{\\ell \\in \\Pi^*} \\mc{C}_1(\\ell) + \\bar{h} \\cdot \\delta,\n \\end{equation}\nor, equivalently,\n\\begin{equation}\n\\label{eq:eps4}\n    \\sum_{\\ell \\in \\Pi (\\bar{l},\\bar{h})} \\mc{C}_1(\\ell)\n    \\leq (1+\\epsilon) \\sum_{\\ell \\in \\Pi^*} \\mc{C}_1(\\ell)\n    \\eqend\n \\end{equation}\nSince $\\Pi (\\bar{l},\\bar{h})$ obeys the ``delay bound'' $\\lceil (1+\\eta)^{\\bar{l}}\n\\rceil$, we have:\n\\begin{equation}\n\\label{eq:eps5}\n    \\Big( \\sum_{\\ell \\in \\Pi (\\bar{l},\\bar{h})}\n    \\mc{C}_2(\\ell) \\Big)^2 \\leq \\Big( \\lceil (1+\\eta)^{\\bar{l}} \\rceil \\Big)^2\n    \\eqend\n \\end{equation}\nCombining (\\ref{eq:eps3}), (\\ref{eq:eps4}) and (\\ref{eq:eps5}), we\nhave:\n\\begin{equation}\n\\label{eq:eps6}\n\\begin{split}\n    &\\mc{C}(\\hat{\\Pi}) \\leq \\mc{C}( \\Pi (\\bar{l},\\bar{h}))\\\\\n    &\\quad\\leq (1+\\epsilon) \\sum_{\\ell \\in \\Pi^*} \\mc{C}_1(\\ell) +\n    (1+\\eta)^2 \\Big( \\sum_{\\ell \\in \\Pi^*} \\mc{C}_2(\\ell) \\Big)^2,\n\\end{split}\n\\end{equation}\nwhere the first transition is due to the way that $\\hat{\\Pi}$ is\nchosen. Since $\\eta={\\epsilon\\over 3}$, for small values of\n$\\epsilon$ (precisely, $\\epsilon < 3$), (\\ref{eq:eps6}) implies:\n\\begin{equation}\n\\label{eq:eps7} \\mc{C}(\\hat{\\Pi}) \\leq (1+\\epsilon) \\cdot\n\\mc{C}(\\Pi^*),\n\\end{equation}\nas required.\n\\end{proof}\n\n\\begin{lemma}\n\\label{l:eps2} The computational complexity of $\\epsilon$-SMER is\n\\mbox{$O(A\\cdot {1\\over \\epsilon} \\cdot \\log (B) \\cdot N^3)$}, where\n$O(A)$ is the computational complexity of the employed approximation\nscheme for RSP.\n\\end{lemma}\n\\begin{proof}\nLet $M$ be the number of links in the original network. Each time we\nemploy the RSP approximation scheme, we would incur a computational\ncomplexity of $O(A)$, where $A$ corresponds to a network with $N$\nnodes and $M$ links.\n\nFor each value of $l=1\\ldots,L$, we call the RSP approximation as\nfollows: once for a network with $N$ nodes and $O(N)$ links (\\ie\\\nfor the network that contains $s$ and all the $u(1)$'s), once for a\nnetwork with roughly $2N$ nodes and $M$ links (\\ie\\ for the network\nthat contains, in addition to the above, all the $u(2)$'s and links\nof the form $(u(1),v(2))$, once for a network with roughly $3N$\nnodes and $2M$ links (\\ie\\ for the network that contains, in\naddition to the above, all the $u(3)$'s and links of the form\n$(u(2),v(3))$, and so on up to, once (the $(N-1)$-th time) for a\nnetwork with roughly $(N-1)N$ nodes and $(N-2)M$ links. The above\n$N-1$ instances (more precisely, all but the first, which can be\nneglected due to smaller complexity) aggregate to:\n\n\n\\begin{equation}\n\\begin{split}\n    &O\\big(A\n      \\cdot (2\\cdot1+3\\cdot2+\\cdots+N\\cdot(N-1)) \\big) \\\\\n    &\\quad= O\\big( A \\cdot \\sum_{i=1}^{N-1} i(i+1) \\big)\n    = O\\big( A \\cdot N^3 \\big)\n    \\eqend\n\\end{split}\n\\end{equation}\n\nThe proof follows by noting that $L=O({1 \\over \\epsilon} \\cdot \\log\n(B))$.\n\\end{proof}\n\n\n\\begin{theorem}\n$\\epsilon$-SMER is an $\\epsilon$-optimal approximation scheme of\npolynomial complexity. In particular, when employing the\napproximation solution of~\\cite{Lorenz99asimple} to the RSP problem,\n$\\epsilon$-SMER runs in $O(N^6 \\cdot (\\log \\log N + {1 \\over\n\\epsilon}) \\cdot {1\\over \\epsilon} \\cdot \\log (B))$ time.\n\\end{theorem}\n\\begin{proof}\nThe RSP scheme of~\\cite{Lorenz99asimple} has computational\ncomplexity of $O\\big( (N \\cdot M \\cdot (\\log \\log N+1\/\\epsilon)\n\\big)$ for $N$ nodes and $M$ links. Depending on the limit on the\ntransmission power at each node, in worst-case we have $M=O(N^2)$,\n\\ie\\ all nodes may be neighbors\\fnote{Note that, typically, the network is sparse, \\ie\\ $M \\ll N^2$, hence\nthe dependency on $N$ is more like $N^5$.}.\nThe proof then follows from\nLemmas~\\ref{l:eps1}~and~\\ref{l:eps2}.\n\\end{proof}\n\nMore efficient versions of $\\epsilon$-SMER should be possible,\nyet our goal has been to show that fully polynomial time $\\epsilon$-approximation schemes\n(FPTAS) exist for the NP-hard problem SMER.\n\n\n\\subsection{Distributed Implementation}\nWhile it is not discussed in this paper, our routing algorithms can\nbe implemented in a distributed manner following standard techniques\nof distance-vector routing. Note that the power allocation at the\nphysical layer is a local operation performed by the transmitting\nnode of each link based on the information from the routing algorithm\nand topological information (collected, for instance, through\nneighbor discovery before running the routing algorithm).\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThis paper studied the problem of secure minimum energy routing in\nwireless networks. It was shown that while the problem is\n\\mbox{NP-hard}, it admits exact pseudo-polynomial and fully polynomial time\n$\\epsilon$-approximation algorithmic solutions.\nFurthermore, using simulations, we showed that our algorithms\nsignificantly outperform security-agnostic algorithms based on\nminimum energy routing. Finally, we note that our work can be\npotentially extended to incorporate other secrecy models. Such extensions are left for future work.\n\n\\section{Introduction}\n\\label{sec:intro}\n\nProtecting the secrecy of user messages has become a major concern\nin modern communication networks.  Due to the propagation properties\nof the wireless medium, wireless networks can potentially make the\nproblem more challenging by allowing an eavesdropper to have\nrelatively easy access to the transmitted message if countermeasures\nare not employed.  Our goal is to provide everlasting\nsecurity in this wireless environment; that is, we will consider\nmethods that will prevent an eavesdropper from ever decoding a\ntransmitted message - even if the eavesdropper has the capability\nto record the signal and attempt decryption over many years (or decades).\nThere are two different classes of security techniques of interest\nhere:  cryptographic approaches based on computational complexity,\nand information-theoretic approaches that attempt to obtain\nperfect secrecy.  Both have advantages and disadvantages\nfor the desired everlasting security in the wireless environment.\n\nThe traditional solution to providing security in a wireless\nenvironment is the cryptographic approach: assume that the\neavesdropper will get the transmitted signal without distortion,\nbut the desired recipient who shares a key with the transmitter\nis able to decode the message easily, while the eavesdropper\nlacking the key must solve a \\emph{hard} problem that is beyond\nher\/his computational capabilities~\\cite{stinson2006cryptography}.\nSince the eavesdropper is assumed to get the transmitted signal\nwithout distortion, cryptography addresses the key challenge in the\nwireless environment of thwarting an eavesdropper very near the\ntransmitter.  However, such an approach faces the concern that the\neavesdropper can store the signal, and, then, with later advances in\ncomputational capabilities or by breaking the encryption scheme, obtain\nthe message.  The desire for everlasting security then motivates\nadding countermeasures at the physical layer that inhibit even the\nrecording of the encrypted message by the eavesdropper that combine\nwith cryptography to facilitate a defense-in-depth approach~\\cite{NSA_defense}.\n\nIn the information-theoretic approach to obtain perfect\nsecrecy~\\cite{shannon1949communication}, the goal is to guarantee\nthat the eavesdroppers can never extract information from the\nmessage, regardless of their computational capability.\nWyner~\\cite{wyner1975wire} and succeeding\nauthors~\\cite{leung1978gaussian,csiszar1978broadcast} showed that\nperfect secrecy is possible if the channel conditions between the\ntransmitter and receiver were favorable relative to the channel\nconditions between the transmitter and eavesdropper. In this\nso-called \\emph{wiretap} channel, perfect secrecy at a positive rate\nwith no pre-shared key is possible~\\cite{wyner1975wire}.  This\nclearly satisfies the requirement for everlasting secrecy, but it\nrelies on favorable channel conditions that are difficult (if not\nimpossible) to guarantee in a wireless environment.  Hence,\ninformation-theoretic secrecy requires a network design which\ninhibits reception at the eavesdropper while supporting reception at\nthe desired recipient.\n\nOur work supports both a cryptographic (computational) approach or\ninformation-theoretic approach.  Per above, it is advantageous in\neither case to seek or create conditions so as to inconvenience\nreception at eavesdropper(s) while facilitating communication of the\nlegitimate system nodes. This has been actively considered in the\nliterature on the physical layer of wireless networks over the last\ndecade, with approaches based on both\nopportunism~\\cite{maurer93,bloch2008} and active channel\nmanipulation~\\cite{goel08,tekin2008general} being employed. Most of\nthese works have arisen in the information-theoretic community and\nconsidered small networks consisting of a source, destination,\neavesdropper, and perhaps a relay\nnode(s)~\\cite{bloch2008,goel08,tekin2008general,dong10,elgamal08,\nDennis2011JSAC}. More recently, there has been the active\nconsideration of large networks with the introduction of the secrecy\ngraph to consider secure\nconnectivity~\\cite{haenggi2008secrecy,pinto2008physical,pinto2009wireless}\nand a number of approaches to throughput scaling versus security\ntradeoffs~\\cite{liang2009secrecy, koyluoglu2010secrecy,\nvasudevan2010security}.  Hence, whereas there has been a significant\nconsideration of small single- and two-hop networks and\nasymptotically large multi-hop networks, there has been almost no\nconsideration of the practical multi-hop networks that lie between\nthose two extremes. It is this large and important gap that this\npaper fills.\n\nConsider a network where system nodes communicate with each other\nwirelessly, possibly over multiple hops, such as in wireless mesh\nnetworks and ad hoc networks. A set of {\\em eavesdroppers} try to passively listen\nto communications among legitimate network nodes. To prevent the\neavesdroppers from successfully capturing communications between\nlegitimate nodes, mechanisms to thwart such are employed at the\nphysical layer of the network. Two nodes that wish\nto communicate securely may need to do so over multiple hops in\norder to thwart eavesdroppers or simply because the nodes are\nnot within the reach of each other. While we make no argument about\nthe optimality or practicality of any specific physical layer\nsecurity mechanism, for the sake of concreteness, we focus on\n\\emph{cooperative jamming}, which has received considerable attention\n~\\cite{goel08,tekin2008general,dong10,elgamal08,Dennis2011JSAC,swindle11}.\nIn cooperative jamming, whenever a node transmits a message, a\nnumber of cooperative nodes, called {\\em jammers}, help the node conceal its\nmessage by transmitting a carefully chosen signal to raise the\nbackground \\emph{noise} level and degrade the eavesdropping channels.\nBecause our general philosophy applies to any physical layer\napproach, the framework can be extended to include other forms\nof physical layer security.  However, some of the attractive features\nof cooperative jamming that motivated us to study this technique include:\n\\begin{enumerate}\n\\item Opportunistic techniques~\\cite{maurer93,bloch2008} that exploit the time-varying wireless channel may suffer from excessive delays depending on the rate of channel fluctuations. For applications that require security without an excessive delay, active channel manipulation such as cooperative jamming should be adopted. The price to be paid, in this case, is the increased interference due to jamming.\n\n\\item Multi-antenna systems can also be used to jam eavesdroppers~\\cite{yates07,goel08}. However, the use of multiple antennas on every wireless device may not be feasible due to cost and size (\\eg\\ wireless sensors). Cooperative jamming is a distributed alternative to multi-antenna systems.\n\n\\item Node cooperation, while requiring a more complex physical layer, is incorporated in commercial wireless technologies such as LTE. Thus, we envision that cooperative jamming can be implemented in practice, as was demonstrated in a limited form (single jammer) in~\\cite{katabi11}.\n\n\\item Anonymous wireless communication is a challenging problem. Cooperative jamming can potentially be utilized for wireless anonymous communication, as it creates confusion for wireless localization techniques~\\cite{banerjee11}.\n\\end{enumerate}\n\nIn this general case, the main questions are: (1) how to choose the\nintermediate nodes that form a multi-hop path from the source node to\nthe destination node, and (2) how to configure each hop at the\nphysical layer with respect to the security and throughput\nconstraints of the path. Specifically, the problem we consider in\nthis paper is how to find a {\\em minimum cost} path between a source\nand destination node in the network, while guaranteeing a\npre-specified lower bound on the {\\em end-to-end secrecy} and {\\em\ngoodput} of the path. The cost of a path can be defined in terms of\nvarious system parameters. In a wireless network, transmission power\nis a critical factor affecting the throughput and lifetime of the\nnetwork. While increasing the transmission power results in increased\nlink throughput, excessive power actually results in high levels of\ninterference, hence reducing the network throughput due to\ninefficient spacial reuse. With cooperative jamming at the physical\nlayer, transmission power is even more important due to the\nadditional interference caused by jamming signals if they need to be\nemployed. Thus, in this work, we consider the amount of end-to-end\ntransmission power as the cost of a path with the objective of\nfinding secure paths that consume the least amount of energy. In\nturn, such paths, by minimizing interference in the network, result\nin \\emph{higher} throughput.  Note that solutions employing power\nonly at the nodes transmitting the messages (and no cooperative\njamming) are part of the space over which the optimization will be\nperformed; thus, if it is more efficient to not employ cooperative\njamming, such a solution will be revealed by our algorithms.\n\nWhile it might seem that physical layer security techniques can be\nextended to multi-hop networks by implementing them on a\nhop-by-hop basis, in general, such extensions sacrifice performance\nor are not feasible. The eavesdropping probability on a link\nis a function of the power allocation on that link.  A hop-by-hop\nimplementation is unable to determine the optimal eavesdropping\nprobability and consequently power allocation for each link in\norder to satisfy the end-to-end constraints (\\ie\\ the chicken-egg problem).\nMoreover, a hop-by-hop approach overlaid on a shortest path routing algorithm\nmight pay an enormous penalty to mitigate eavesdroppers on some links (\\eg\\\nby routing through a node with one or more links, that, because\nof system geometry, are very vulnerable to nearby eavesdroppers).\nA routing algorithm that is designed in conjunction with\nphysical layer security can selectively employ links that are\neasier to secure when it is power-efficient to do so and, in\nsuch a way, minimize the impact of the security constraint on\nend-to-end throughput.\n\n\n\nOur main contributions can be summarized as follows:\n\\begin{itemize}\n\\item We formulate the secure minimum energy routing problem with end-to-end security and goodput constraints as a constrained\noptimization of transmission power at the physical layer and link\nselection at the network layer.\n\n\\item We prove that the secure minimum energy routing problem is \\mbox{NP-hard}, and develop\nexact and $\\epsilon$-approximate solutions of, respectively, pseudo-polynomial and fully-polynomial time complexity for the problem.\n\n\\item We show how cooperative jamming can be used to establish a secure link between two nodes in the presence of multiple eavesdroppers or probabilistic information about potential eavesdropping locations by utilizing random linear coding at the network layer.\n\n\\item We provide simulation results that demonstrate the significant energy savings\nof our algorithms compared to the combination of security-agnostic minimum energy\nrouting and physical layer security.\n\\end{itemize}\n\nThe rest of the paper is organized as follows. Our system model is\ndescribed in Section~\\ref{sec:sysmodel}. The optimal link and path\ncost are analyzed in Sections~\\ref{sec:linkcost}\nand~\\ref{sec:pathcost}. Our routing algorithms are presented in\nSection~\\ref{sec:algorithms}. Simulation results are discussed in\nSection~\\ref{sec:simulation}. Section~\\ref{sec:related} presents an\noverview of some related work, while Section~\\ref{sec:conclusion}\nconcludes the paper.\n\n\\section{Secure Link Cost}\n\\label{sec:linkcost}\n\nThe link cost is composed of two components: (1) the source power, and (2)\nthe jammers' power. Let $\\mc{C}(\\ell_k)$ denote the cost of link\n$\\ell_k=\\enc{S_k, D_k, \\mc{E}_k, \\mc{J}_k}$ under the constraint of\neavesdropping probability $\\pi_k$. Then, $\\mc{C}(\\ell_k)$ is given by:\n\\begin{eqnarray}\n    \\mc{C}(\\ell_k) = P_S^{(k)} + P_J^{(k)},\n\\end{eqnarray}\nwhere $P_S^{(k)}$ and $P_J^{(k)}$ denote, respectively, the average\nsource and jammers power on link $\\ell_k$. In the following\nsubsections, we will compute the optimal values of $P_S^{(k)}$ and\n$P_J^{(k)}$ subject to a given $\\pi_k$.\n\n\n\\subsection{Source Transmission Power}\nAssume that the (complex) fading channel coefficient $h_{S_k,D_k}$ is\nknown at the source $S_k$ of the given link $\\ell_k$. Because we are\ntrying to maintain a fixed rate (and, hence, a fixed received power),\nthe source will attempt to invert the channel using power control.\nHowever, for a Rayleigh frequency-nonselective fading channel, as\nassumed here, the expected required power for such an inversion goes\nto infinity, and, hence {\\em truncated channel inversion} is employed\n\\cite[Pg.~112]{goldsmith2005}. In truncated channel inversion, the\nsource maintains the required link quality except for extremely bad\nfades, where the link goes into outage. When a link is in a bad fade,\nthe source will need to wait until the link improves before\ntransmitting the packet and delay will be incurred.  To limit the\ndelay, we maintain a given outage probability $\\rho$ per link. Then,\nfor a given packet, we need to transmit at rate $R = \\lambda \/ (1 -\n\\rho)$ to maintain the desired goodput $\\lambda$. Associated with\nthat rate $R$ is the SINR threshold $\\gamma_D = 2^{R}- 1$ required\nfor successful reception at the link\ndestination~\\cite{tse05wireless}.\n\nLet $P_S^{(k)}$ denote the average transmission power of $S_k$, and\nlet $P_S^{(k)}(|h_{S_k,D_k}|^2)$ denote the power used for a given\npacket as a function of the power $|h_{S_k,D_k}|^2$ in the fading\nchannel between $S_k$ and $D_k$.  Per above, below some threshold\n$\\tau$, the source will wait for a better channel.  From the\nRayleigh fading model employed,\n$|h_{S_k,D_k}|^2$ is exponential with parameter\n$1\/d_{S_k,D_k}^{\\alpha}$; hence, $\\tau = - \\ln(1 - \\rho) \\cdot d_{S_k,D_k}^{\\alpha}$ and truncated channel inversion yields:\n\\begin{equation}\n    P_S^{(k)}(|h_{S_k,D_k}|^2) = \\begin{cases}\n                                    \\frac{\\gamma_D}{|h_{S_k,D_k}|^2} \\cdot d_{S_k, D_k}^{\\alpha},\n                                             &       |h_{S_k,D_k}|^2 \\geq \\tau \\\\\n                                     0, &       |h_{S_k,D_k}|^2 < \\tau\n                                 \\end{cases}\n\\end{equation}\nThen, the average power employed on the link is given by:\n\\begin{eqnarray}\n\\label{eq:ps}\n    P_S^{(k)} & = & \\frac{1}{1 - \\rho} \\int_{\\tau}^{\\infty} \\frac{\\gamma_D}{x} \\cdot d_{S_k,D_k}^{\\alpha}\n            e^{-x} dx   \\nonumber\\\\\n          & = & \\gamma_D d_{S_k,D_k}^{\\alpha} \\frac{1}{1 - \\rho} \\int_{\\tau}^{\\infty} \\frac{e^{-x}}{x} dx \\nonumber\\\\\n          & = & \\gamma_D \\, k_{\\rho} \\, d_{S_k,D_k}^{\\alpha},\n\\end{eqnarray}\nwhere $k_{\\rho}$ is a constant that depends on the parameter\n${\\rho}$.  Hence, for a fixed network parameter $\\rho$ (which also\ndetermines $\\gamma_D$), the average power consumed on a given link\n$\\ell_k$ to achieve the secure goodput $\\lambda$ is proportional to\n$d_{S_k, D_k}^{\\alpha}$.\n\n\n\\subsection{Jammers' Transmission Power}\nOur physical layer security primitive described in\nSection~\\ref{sec:sysmodel} can provide security only against a single\neavesdropper at a fixed location. To achieve security in the presence\nof multiple eavesdroppers or uncertainty about the location of\neavesdroppers, we utilize \\emph{random linear coding}\\fnote{Other\nforms of coding, such as dividing each message to smaller\nchunks~\\cite{shamir79}, can be equally incorporated in our\nalgorithm.} on each link.\n\nConsider link $\\ell_k$ between transmitter $S_k$ and receiver $D_k$\nwith the associated set of potential eavesdropping locations\n$\\mc{E}_k=\\set{E_1, \\ldots, E_{|\\mc{E}_k|}}$. Transmitter $S_k$\nperforms coding over $|\\mc{E}_k|$ messages accumulated in its buffer\nfor transmission to $D_k$. To generate a coded message, $S_k$ selects\na random subset of the messages in its buffer and adds them together\n(module-$2$). To recover the original messages, the receiver needs to\ncollect $|\\mc{E}_k|$ linearly independent coded messages. In order to\ntransmit only linearly independent coded messages, $S_k$ keeps track\nof the coded messages it has transmitted so far. Let $m_i$ denote the\n$i$-th coded message that is being transmitted to $D_k$. To securely\ntransmit $m_i$, $S_k$ employs the cooperative jamming primitive of\nSection~\\ref{sec:sysmodel} assuming that there is an eavesdropper in\nlocation $E_i$. Since each coded message is hidden from at least one\neavesdropping location, it is guaranteed that an eavesdropper located\nat location $E_i$, for all $E_i \\in \\mc{E}_k$,  will not be able to\nobtain any information about the original messages.\n\nIn the following subsections, we compute the optimal jamming power\nper link. The derivation for the case of multiple eavesdroppers\nrelies on the jamming power computed for the single eavesdropper\ncase.\n\n\\subsubsection{Single Eavesdropper}\nBecause slow frequency non-selective fading is assumed and\nthe channel to the eavesdropper is unknown, there is some probability\nthat the eavesdropper will obtain the message by achieving a received\nSINR greater than a threshold $\\gamma_E$.\nLet $\\pi_k(|h_{S_k,D_k}|^2)$ denote the probability the eavesdropper\nachieves SINR greater than threshold $\\gamma_E$ for a given source to\ndestination channel $h_{S_k,D_k}$ (recall that the source power will\nfluctuate as $h_{S_k,D_k}$ fluctuates, and this will impact the\ninterception probability at the eavesdropper). Because we want to\navoid placing limitations on the capabilities of the eavesdropper,\nassume that the eavesdropper receiver is noiseless. Let $P_J^{(k)}$\nand $P_J^{(k)}(|h_{S_k,D_k}|^2)$ denote the average and instantaneous\ntransmission power allocated to jammers in $\\mc{J}_k$, respectively.\nThen, using~\\eqref{eq:success}, it is obtained that\n\\begin{equation*}\n    \\pi_k(|h_{S_k,D_k}|^2) =\n    \\quad \\frac{1}\n        {1 + \\frac{\\gamma_E d_{S_k, E_k}^{\\alpha}}\n    {P_S^{(k)}(|h_{S_k,D_k}|^2)}\n        \\big( \\sum_{J_i \\in \\mc{J}_k} \\frac{1}{d_{J_i, E_k}^{\\alpha}} \\big)\n    P_J^{(k)}(|h_{S_k,D_k}|^2)} \\eqend\n\\end{equation*}\nNow, to maintain a given $\\pi_k$, it is sufficient to\nmaintain $\\pi_k(|h_{S_k,D_k}|^2) = \\pi_k$ across all\n$|h_{S_k,D_k}|^2$. Under this condition, recognizing that both\n$P_S^{(k)}(|h_{S_k,D_k}|^2)$ and $P_J^{(k)}(|h_{S_k,D_k}|^2)$ are\nproportional to $|h_{S_k,D_k}|^2$, we have:\n\n\\begin{equation}\n\\begin{split}\n    P_J^{(k)}(|h_{S_k,D_k}|^2) =\n    \\frac{(1\/\\pi_k - 1) \\, P_S^{(k)}(|h_{S_k,D_k}|^2)}\n        {\\gamma_E \\, d_{S_k,E_k}^{\\alpha} (\\sum_{J_i \\in \\mc{J}_k} \\frac{1}{d_{J_i, E_k}^{\\alpha}})}\n         ,\n\\end{split}\n\\end{equation}\nand, taking expectations yields\n\\begin{equation}\n\\label{eq:P_J}\n    P_J^{(k)} =  \\frac{1\/\\pi_k - 1}\n        {\\gamma_E d_{S_k,E_k}^{\\alpha} (\\sum_{J_i \\in \\mc{J}_k} \\frac{1}{d_{J_i,E_k}^{\\alpha}})}\n        P_S^{(k)} ,\n\\end{equation}\nand,\n\\begin{equation}\n\\label{eq:pi1}\n    \\pi_k =  \\frac{1}\n        {1 + \\frac{\\gamma_E d_{S_k,E_k}^{\\alpha}}{P_S^{(k)}} \\big(\\sum_{J_i \\in \\mc{J}_k} \\frac{1}{d_{J_i, E_k}^{\\alpha}}\\big)P_J^{(k)} }\n        \\eqend\n\\end{equation}\n\n\n\\subsubsection{Multiple Eavesdroppers}\nRecall that our objective is to compute the minimum jamming power for\nthe link.\nLet $\\pi_k(i)$ denote the successful eavesdropping probability on\nlink $\\ell_k$ conditioned on having an eavesdropper at location\n$E_i$. The unconditional eavesdropping probability $\\pi_k$ on link\n$\\ell_k$ is then given by the approximate relation $\\pi_k =\n\\sum_{E_i \\in \\mc{E}_k} p_k(E_i) \\cdot \\pi_k(i)$, where $p_k(E_i)$ is\nthe probability of having an eavesdropper at location $E_i$. Since\njamming power depends on the location of the eavesdroppers, by\noptimally allocating jamming power to each potential eavesdropping\nlocation, we can minimize the total jamming power across all\neavesdropping locations for a given link.\n\nThe minimum jamming power for link $\\ell_k$ over all eavesdropping\nlocations $\\mc{E}_k$ is given by the solution of the following\noptimization problem:\n\\begin{equation}\n\\begin{split}\n     &\\min_{P_J^{(k)}(i)}  \\sum_{E_i \\in \\mc{E}_k} P_J^{(k)}(i) \\\\\n     &s.t. \\quad\n        \\displaystyle\\sum_{E_i \\in \\mc{E}_k} p_k(E_i) \\cdot \\pi_k(i) = \\pi_k,\n\\end{split}\n\\end{equation}\nwhere $P_J^{(k)}(i) = \\sum_{J_j \\in \\mc{J}_k} P_j^{(k)}(i)$ is the\njamming power conditioned on the eavesdropping location $E_i$, \\ie\\\nthe jamming power during the transmission of the coded message $m_i$.\nDefine $\\phi_k(i)$ as follows\n\\begin{equation}\n    \\phi_k(i) =  \\frac{\\gamma_E}{\\gamma_S k_{\\rho}} \\big(\\frac{d_{S_k, E_i}}{d_{S_k, D_k}}\\big)^{\\alpha} \\sum_{J_j \\in \\mc{J}_k} \\frac{1}{d_{J_j, E_i}^{\\alpha}}\n    \\eqend\n\\end{equation}\nAfter substituting for $\\pi_k(i)$ using~\\eqref{eq:pi1}, we obtain the\nfollowing optimization problem:\n\\begin{equation}\n\\label{e:multi}\n\\begin{split}\n     &\\min_{P_J^{(k)}(i)}  \\sum_{E_i \\in \\mc{E}_k} P_J^{(k)}(i) \\\\\n     &s.t. \\quad\n        \\displaystyle\\sum_{E_i \\in \\mc{E}_k}\n            \\frac{p_k(E_i)}\n            {1 + \\phi_k(i) P_J^{(k)}(i) }\n        = \\pi_k\n        \\eqend\n\\end{split}\n\\end{equation}\nThe optimization variables in this optimization problem are the\njamming powers $P_J^{(k)}(i)$. \nThe Lagrangian for the link cost\noptimization problem is expressed as follows\n\\begin{equation*}\n\\begin{split}\n    &L(P_J^{(k)}(1), \\ldots, P_J^{(k)}(|\\mc{E}_k|), \\nu)\\\\\n        & = \\sum_{E_i \\in \\mc{E}_k}\n                P_J^{(k)}(i)\n            + \\nu\n                \\Big(\n                    \\displaystyle\\sum_{E_i \\in \\mc{E}_k}\n                    \\frac{p_k(E_i)}\n                    {1 + \\phi_k(i) P_J^{(k)}(i) }   \n                    - \\pi\n                \\Big)\n                \\eqend\n\\end{split}\n\\end{equation*}\nUsing the Lagrange multipliers technique, it is obtained that\n\\begin{equation}\n\\label{eq:LL1}\n    \\frac{\\partial L}{\\partial P_J^{(k)}(i)}\n        = 1 - \\nu\n                \\frac{\\phi_k(i) p_k(E_i)}\n                    {(1 + \\phi_k(i) P_J^{(k)}(i))^2} ,\n\\end{equation}\nand,\n\\begin{equation}\n\\label{eq:LL2}\n    \\frac{\\partial L}{\\partial \\nu}\n        = \\sum_{E_i \\in \\mc{E}_k}\n                \\frac{p_k(E_i)} {1 + \\phi_k(i) P_J^{(k)}(i) }\n           - \\pi\n           \\eqend\n\\end{equation}\nUsing \\eqref{eq:LL1}, we have\n\\begin{equation}\n\\label{eq:LL1-s}\n    \\frac{p_k(E_i)} {1 + \\phi_k(i) P_J^{(k)}(i) }\n    =\n    \\frac{\\sqrt{p_k(E_i)}}{\\sqrt{\\nu \\phi_k(i)}}\n    \\eqend\n\\end{equation}\nBy substituting in~\\eqref{eq:LL2}, it follows that\n\\begin{equation}\n    \\sum_{E_i \\in \\mc{E}_k} \\sqrt{\\frac{p_k(i)}{\\nu \\phi_k(i)}} = \\pi ,\n\\end{equation}\nand, therefore,\n\\begin{equation}\n   \\frac{1}{\\sqrt{\\nu}}\n        = \\frac{\\pi}{\\sum_{E_i \\in \\mc{E}_k} \\sqrt{\\frac{p_k(i)}{\\phi_k(i)}} }\n        \\eqend\n\\end{equation}\nIt is then obtained that\n\\begin{equation}\n    \\pi_k(i) = \\frac{1}{\\phi_k(i)} \\frac{1\/\\sqrt{\\frac{p_k(E_i)}{\\phi_k(i)}}}\n                 { \\sum_{E_i \\in \\mc{E}_k} \\sqrt{\\frac{p_k(E_i)}{\\phi_k(i)}}} \\, \\pi_k,\n\\end{equation}\nand,\n\\begin{equation}\n\\label{eq:pji}\n   P_J^{(k)}(i) =\n            \\frac{1}{\\pi_k} \\sqrt{\\frac{p_k(E_i)}{\\phi_k(i)}} \\sum_{E_i \\in \\mc{E}_k} \\sqrt{\\frac{p_k(E_i)}{\\phi_k(i)}} - \\frac{1}{\\phi_k(i)}\n            \\eqend\n\\end{equation}\nFor a given link $\\ell_k$ and eavesdropping probability $\\pi_k$, we\ncan use~\\eqref{eq:pji} to compute the optimal jamming power\nallocation for each coded message $m_i$. Consequently, the average\njamming power per message on link $\\ell_k$ is given by:\n\\begin{equation}\n\\label{eq:pj1}\n\\begin{split}\n   P_J^{(k)} &= \\frac{1}{|\\mc{E}_k|} \\sum_{E_i \\in \\mc{E}_k} P_J^{(k)}(i)\\\\\n             &= \\frac{1}{\\pi_k} \\frac{1}{|\\mc{E}_k|} \\Bigg( \\sum_{E_i \\in \\mc{E}_k} \\sqrt{\\frac{p_k(E_i)}{\\phi_k(i)}} \\Bigg)^2\n                - \\frac{1}{|\\mc{E}_k|} \\sum_{E_i \\in \\mc{E}_k} \\frac{1}{\\phi_k(i)}\n        \\eqend\n\\end{split}\n\\end{equation}\n\n\n\\subsection{Discussion}\n\\subsubsection{Colluding Eavesdroppers}\nWhile we considered the case of non-colluding\neavesdroppers here, our model can be extended to handle\ncolluding eavesdroppers by requiring that at least of the coded messages be protected against all\neavesdroppers. Let $\\mc{E}_k=\\set{E_1, \\ldots, E_{|\\mc{E}_k|}}$ denote the set of colluding eavesdroppers. Assume that on link $\\ell_k$, $B_k$ messages are coded together for transmission, \\ie\\ $B_k$ is the length of the coding block. Then, the probability that a coded message $m$ is captured by all eavesdroppers is given by $\\prod_{E_i \\in \\mc{E}_k} \\pi_k(i)$. Thus, the probability that at least one message out of the $B_k$ coded messages is not received by all eavesdroppers is given by\n\\begin{equation}\n    1 - \\Big( \\prod_{E_i \\in \\mc{E}_k} \\pi_k(i) \\Big)^{B_k}\n        \\eqend\n\\end{equation}\nTo satisfy the link eavesdropping constraint $\\pi_k$, the following relation should be satisfied\n\\begin{equation}\n    1 - \\Big( \\prod_{E_i \\in \\mc{E}_k} \\pi_k(i) \\Big)^{B_k} = \\pi_k,\n\\end{equation}\nwhich yields\n\\begin{equation}\n    \\prod_{E_i \\in \\mc{E}_k} \\pi_k(i) = \\sqrt[B_k]{\\pi_k}\n    \\eqend\n\\end{equation}\nThis constraint can be used in the optimization problem~\\eqref{e:multi} to compute the optimal link cost for the case of colluding eavesdroppers.\n\nAn interesting observation is that\n\\begin{equation}\n    \\lim_{B_k \\rightarrow \\infty} \\pi_k(i) = 1, \\quad \\text{for all $E_i \\in \\mc{E}_k$}\n    \\eqend\n\\end{equation}\nThat is, by increasing the length of the coding block, the link cost can be significantly reduced. The cost to be paid is in terms of increased transmission delay.\n\n\n\\subsubsection{End-to-End Coding}\nRather than looking at individual links in isolation and then\nperforming hop-by-hop coding, we can perform coding on an end-to-end\nbasis only at the source node. Then by repeatedly finding paths that\nare secure against single eavesdropping per link, the source can\nsecurely communicate with the destination through multiple paths.\nThis approach is appropriate if there are only a few potential\neavesdropping locations in the network. If the maximum number of\neavesdropping locations per link is $m$, then the running time of\nthis approach is $m$ times that of the routing algorithm with single\neavesdropping location per link.\n\n\\section{Secure Path Cost}\n\\label{sec:pathcost} In this section, using the link cost formulation\nof the previous section, we formulate the optimal cost of a\n\\emph{given} path $\\Pi$ subject to an end-to-end eavesdropping\nprobability $\\pi$.\nThe problem essentially is to divide $\\pi$ across the links forming $\\Pi$ so that the path cost is minimized.\n\n\\subsection{Optimal Path Cost}\nConsider a given path $\\Pi$. We find the optimal cost of path $\\Pi$\nby solving the optimization problem~\\eqref{eq:smer-p}. Consider link\n$\\ell_k \\in \\Pi$, where $\\ell_k = \\enc{S_k, D_k, \\mc{E}_k,\n\\mc{J}_k}$. Define $x_k$ and $y_k$ as follows:\n\\begin{align*}\n    x_k = \\frac{1}{\\sqrt{|\\mc{E}_k|}} \\sum_{E_i \\in \\mc{E}_k} \\sqrt{\\frac{p_k(E_i)}{\\phi_k(i)}},\n\\end{align*}\nand,\n\\begin{align*}\n    y_k = \\frac{1}{|\\mc{E}_k|} \\sum_{E_i \\in \\mc{E}_k} \\frac{1}{\\phi_k(i)}\n    \\eqend\n\\end{align*}\nUsing the results obtained in the previous subsection, the following\nrelation holds:\n\\begin{align*}\n    \\pi_k =  \\frac{x^2_k}{y_k + P_J^{(k)}}\n    \\eqend\n\\end{align*}\nBy substituting the above expressions in the optimal routing\nformulation described in~\\eqref{eq:smer-p}, the following\noptimization problem is obtained for minimizing the cost\n$\\mc{C}(\\Pi)$ of route $\\Pi$:\n\\begin{equation}\n\\begin{split}\n     &\\displaystyle\\min_{P_J^{(k)}}  \\sum_{\\ell_k \\in \\Pi} P_S^{(k)} + P_J^{(k)} \\\\\n     &s.t. \\quad\n        \\displaystyle\\sum_{\\ell_k \\in \\Pi}\n                        \\Big(\n                         \\frac{x^2_k} {y_k + P_J^{(k)}}\n                        \\Big)\n                            = \\pi\n                            \\eqend\n\\end{split}\n\\end{equation}\nThe optimization variables in this optimization problem are jamming\npowers $P_J^{(k)}$. \nThe Lagrangian for the routing\noptimization problem is expressed as follows\n\\begin{equation*}\n\\begin{split}\n    &L(P_J^{(1)}, \\ldots, P_J^{(K)}, \\nu)\\\\\n        & = \\sum_{\\ell_k \\in \\Pi}\n            \\big( P_S^{(k)} + P_J^{(k)} \\big)\n            + \\nu\n                \\Big(\n                    \\sum_{\\ell_k \\in \\Pi}\n                        \\Big(\n                         \\frac{x^2_k} {y_k + P_J^{(k)}}\n                        \\Big)\n                    - \\pi\n                \\Big)\n                \\eqend\n\\end{split}\n\\end{equation*}\nUsing the Lagrange multipliers technique, it is obtained that\n\\begin{equation}\n\\label{eq:L1}\n    \\frac{\\partial L}{\\partial P_J^{(k)}}\n        = 1 - \\nu\n                \\frac{x^2_k} {(y_k + P_J^{(k)})^2} ,\n\\end{equation}\nand,\n\\begin{equation}\n\\label{eq:L2}\n    \\frac{\\partial L}{\\partial \\nu}\n        = \\sum_{\\ell_k \\in \\Pi}\n              \\Big(\n                 \\frac{x^2_k} {y_k + P_J^{(k)}}\n              \\Big)\n           - \\pi\n           \\eqend\n\\end{equation}\nUsing \\eqref{eq:L1}, we have\n\\begin{equation}\n\\label{eq:L1-s}\n    \\frac{x^2_k} {y_k + P_J^{(k)}}\n    =\n    \\frac{x_k}{\\sqrt{\\nu}}\n    \\eqend\n\\end{equation}\nBy substituting in~\\eqref{eq:L2}, it follows that\n\\begin{equation}\n    \\sum_{\\ell_k \\in \\ell} \\frac{x_k}{\\sqrt{\\nu}} = \\pi ,\n\\end{equation}\nand, therefore,\n\\begin{equation}\n   \\frac{1}{\\sqrt{\\nu}}\n        = \\frac{\\pi}{\\sum_{\\ell_k \\in \\Pi} x_k}\n        \\eqend\n\\end{equation}\nAfter substitution in~\\eqref{eq:L1-s}, the following relation for\nthe optimal eavesdropping probability $\\pi_k$ on link $\\ell_k$ is\nobtained\n\\begin{equation}\n\\label{eq:pi}\n    \\pi_k = \\frac{x_k}\n                {\\sum_{\\ell_i \\in \\Pi} x_i} \\, \\pi\n                \\eqend\n\\end{equation}\nFor a given route $\\Pi$ and end-to-end eavesdropping probability\n$\\pi$, we can use \\eqref{eq:pi} to divide $\\pi$ between links\n$\\ell_k \\in \\Pi$. Having computed $\\pi_k$, the optimal power allocated to\njammers on link $\\ell_k$ is given by the following expression:\n\\begin{equation}\n\\label{eq:pj}\n   P_J^{(k)} = \\frac{1}{\\pi} \\cdot x_k \\sum_{\\ell_i \\in \\Pi} x_i - y_k\n        \\eqend\n\\end{equation}\nUsing the above expression for $P_J^{(k)}$, the cost of link $\\ell_k\n\\in \\Pi$ is expressed as\n\\begin{equation}\n\\label{eq:lcost}\n    \\mc{C}(\\ell_k)\n        = \\big( (\\gamma_S k_{\\rho}) \\cdot d_{S_k, D_k}^{\\alpha} - y_k \\big)\n            + \\frac{1}{\\pi} \\big( x_k \\sum_{\\ell_i \\in \\Pi} x_i \\big)\n                \\eqend\n\\end{equation}\nConsequently, the cost of secure route $\\Pi$ is given by:\n\\begin{equation}\n\\label{eq:pcost}\n    \\mc{C}(\\Pi)\n        = \\sum_{\\ell_k \\in \\Pi} \\big( (\\gamma_S k_{\\rho}) \\cdot d_{S_k, D_k}^{\\alpha} - y_k \\big)\n            + \\frac{1}{\\pi} \\big( \\sum_{\\ell_k \\in \\Pi} x_k \\big)^2\n            \\eqend\n\\normalsize\n\\end{equation}\nTo this end, for a given route $\\Pi$ between the source and\ndestination, the optimal cost of $\\Pi$ subject to the end-to-end\neavesdropping constraint $\\pi$ is given by~\\eqref{eq:pcost}. The\noptimal cost is achieved by allocating $P_S^{(k)}$ and $P_J^{(k)}$ to\neach link $\\ell_k \\in \\Pi$ using~\\eqref{eq:ps} and \\eqref{eq:pj},\nrespectively. Such a power allocation scheme would result in minimum\ncost, while guaranteeing that the eavesdropping constraint would be\nsatisfied. Thus, SMER is reduced to finding a path, among all\npossible paths between the source and destination, that minimizes the\noptimal path cost~\\eqref{eq:pcost}. The following proposition\nformally states this result.\n\\begin{propose}\nSMER with end-to-end eavesdropping and goodput constraint $\\pi$ and\n$\\lambda$, respectively, is equivalent to finding a path that\nminimizes the optimal path cost $\\mc{C}(\\Pi)$ as given\nby~\\eqref{eq:pcost}.\n\\end{propose}\n\n\\subsection{Optimal Path Cost Structure}\nDefine $\\mc{C}_1(\\ell_k)$ and $\\mc{C}_2(\\ell_k)$ as follows:\n\\begin{equation}\n\\begin{split}\n    \\mc{C}_1(\\ell_k) &= (\\gamma_S k_{\\rho}) \\cdot d_{S_k, D_k}^{\\alpha} - y_k,\\\\\n    \\mc{C}_2(\\ell_k) &= \\frac{1}{\\sqrt{\\pi}} \\cdot \\sum_{\\ell_k \\in \\Pi} x_k \\eqend\n\\end{split}\n\\end{equation}\nThen the optimal path cost~\\eqref{eq:pcost} can be expressed as\n\\begin{equation}\n\\label{eq:pcost2}\n    \\mc{C}(\\Pi)\n        = \\sum_{\\ell_k \\in \\Pi} \\mc{C}_1(\\ell_k)\n          + \\Big( \\sum_{\\ell_k \\in \\Pi} \\mc{C}_2(\\ell_k) \\Big)^2\n            \\eqend\n\\end{equation}\n\n\nIt is important to note that, while the $\\mc{C}_1(\\ell_k)$'s may\nassume negative values, the path cost structure in (\\ref{eq:pcost2})\nis monotonous in the number of links, \\ie\\ if a path $\\hat{\\Pi}$ is a\nsubset of a path $\\Pi$, then $\\mc{C} (\\hat{\\Pi}) < \\mc{C} (\\Pi)$.\nThis is because $\\pi <1$, and it can be shown that\n$\\big( \\sum_{\\ell_k \\in \\Pi} x_k \\big)^2 > \\sum_{\\ell_k \\in \\Pi} y_k$.\nConsequently, (\\ref{eq:pcost2}) is minimized by a {\\em simple} path.\n\n\\section{Related Work}\n\\label{sec:related}\n\nA survey of prior work is presented in this section.\n\n\\spar{Secure Routing in Multi-hop Networks} While there are numerous\nworks on secure routing in wireless networks (see,\n\\eg~\\cite{guizani08} and references therein), their focus is on\npreventing malicious attacks that disrupt the operation of the\nrouting protocol using application level mechanisms such as\nauthentication and cryptography. The focus of this paper, on the\nother hand, is on secure transmission of messages via the most\ncost-effective paths in the network, which is orthogonal to the\nsecure routing problem considered in the existing literature.\n\n\n\\spar{Wireless Physical Layer Security}\nThe idea behind physical layer security is to exploit the\ncharacteristics of the wireless channel such as fading to provide\nsecure wireless communications.\nThe foundations of information\ntheoretic security, which is the theoretical basis for physical layer\nsecurity, were laid by Wyner and\nothers~\\cite{wyner1975wire,leung1978gaussian,csiszar1978broadcast}\nbased on Shannon's notion of perfect\nsecrecy~\\cite{shannon1949communication}.\nIn the classical wiretap\nmodel of Wyner, to achieve a strictly positive secrecy rate, the\nlegitimate user should have some advantage over the eavesdropper in\nterms of SNR. Later, Maurer~\\cite{maurer93} proved that even when a\nlegitimate user has a worse channel than an eavesdropper, it is\npossible to have secure communication.\nWhile some physical layer\nsecurity techniques allow for opportunistic exploitation of the\nspace\/time\/user diversity for secret\ncommunications~\\cite{maurer93,bloch2008}, others actively manipulate\nthe wireless channel to block eavesdroppers by employing techniques\nsuch as multiple antennas~\\cite{yates07} and\njamming~\\cite{tekin2008general,elgamal08}. While some of these\ntechniques have been successfully implemented in practical\nsystems~\\cite{katabi11}, physical layer security is focused on very\nspecial network topologies, \\eg\\ single-hop networks. In this work,\nwe have developed algorithms to extend these techniques to multi-hop\nnetworks.\n\n\n\\spar{Scaling Laws in Large Secure Networks} Motivated\nby~\\cite{gupta00}, recently, throughput scaling versus security\ntradeoffs have been investigated in the context of large wireless\nnetworks~\\cite{liang2009secrecy, koyluoglu2010secrecy,\nvasudevan2010security, Dennis2011JSAC}. Specifically, for cooperative\njamming when the eavesdroppers are uniformly distributed, it was\nshown that if the number of eavesdroppers grows sub-linearly with\nrespect to the number of legitimate nodes, a positive throughput for\nsecure communication is achievable~\\cite{Dennis2011JSAC}.\n\n\n\\spar{Security Based on Network Topology}\nWhen there is sufficient path diversity in a network, different\nmessages can be routed over different parts of the network in the\nhope that an eavesdropper would be incapable of capturing all\nmessages from across the network. To exploit network diversity for\nsecurity, various techniques based on multi-path\nrouting~\\cite{fang09,krunz10} and network\ncoding~\\cite{yeung02,jain04} have been investigated. While such techniques are suitable for wired networks,\ntheir application in wireless networks is challenging due to lack of\npath diversity at the source or destination of a communication\nsession. Moreover, there are considerable complications when splitting a flow among several paths, in particular,\nat the granularity of a single session.\nMoreover, network topology, in wireless networks, is a\nfunction of power allocation at the physical-layer and propagation\nenvironment, \\eg\\ fading.\nNevertheless, our approach is\ncomplimentary to these techniques, by providing a mechanism to find a\nminimum cost path that is information-theoretically secure,\nregardless of the network diversity.\n\n\n\\section{Simulation Results}\n\\label{sec:simulation}\n\n\\subsection{Simulation Environment}\nWe have implemented our routing algorithms in a custom-built\nsimulator to study their performance in a variety of network\nscenarios. We simulate a wireless network, in which nodes are\ndistributed uniformly at random in a square of area $5 \\times 5$ with\nnode density $\\sigma=3$. We also place a number of eavesdroppers in\nthe network with density $\\sigma_E$, as described later. We consider\none eavesdropper per link. We keep the number of eavesdroppers\nconsiderably less than that of the legitimate nodes in order to be\nable to establish secure routes as we put a limit on the maximum\ntransmission power of each node. Every node has a maximum\ntransmission power that is set in such a way that the resulting\nnetwork becomes connected (the absolute value of the maximum power\ndoes not affect the results). We choose two nodes $s$ and $d$ located\nat the lower left and the upper right corners of the network,\nrespectively, and find paths from $s$ to $d$. We then compute the\ntotal amount of energy consumed on each path using different routing\nalgorithms. The performance metric ``energy savings'' refers to the\npercentage difference between total energy used by different\nalgorithms with respect to the benchmark. For simulation purposes, we\nset $\\pi=0.1$, $\\sigma_E=1$, $N_0=1$, $\\gamma_D = 0.8$, and $\\gamma_E\n= 0.6$, unless otherwise specified. The numbers reported are obtained\nby averaging over $10$ simulation runs with different seeds.\n\n\n\\subsection{Simulated Algorithms}\nIn addition to\nDP-SMER and \\mbox{$\\epsilon$-SMER}, we have also implemented a\nsecurity-agnostic algorithm based on minimum energy routing as a\nbenchmark to measure energy savings achieved by our algorithms. The\nbenchmark algorithm, called {\\em security-agnostic shortest path\nrouting (SASP)}, is described below. Note that some of the\noptimizations described in Sections~\\ref{sec:linkcost}\nand~\\ref{sec:pathcost} have been incorporated in SASP, making it a\nconsiderably efficient benchmark (see Subsection~\\ref{sub:alloc}).\n\n\\begin{algorithm}\n\\caption{SASP (source $s$, dest. $d$, network $\\mc{N}$).}\n\\begin{enumerate}\n\\item Find a shortest path in terms of transmission power between\n$s$ and $d$ ignoring eavesdroppers. The standard Dijkstra's\nalgorithm can be used for this purpose.\n\n\\item Use~\\eqref{eq:pi} to allocate an optimal eavesdropping probability\nto each link of the computed path.\n\n\\item Use~\\eqref{eq:pj} to allocate sufficient power to jammers on each link\nwith respect to the allocated eavesdropping probabilities in step\n(2).\n\\end{enumerate}\n\\end{algorithm}\n\n\n\\subsection{Results and Discussion}\n\n\\spar{Effect of Eavesdropper Location on Link Cost}\nFor a fixed link between two nodes, the source transmission power is\nalso fixed as obtained in~\\eqref{eq:ps}. Thus, the cost of the link\ndepends only on the jamming power which is a function of the\neavesdropper location as given by~\\eqref{eq:pj1}.\nFig.~\\ref{fig:onelink} shows the cost of establishing a secure link\nbetween source $S$ (placed at the center) and destination $D$ for\ndifferent eavesdropper locations and $\\pi = 0.001$. In the figure,\nthe color intensity at each point is proportional to the amount of\nenergy required to establish the link if the eavesdropper is placed\nat that point. Clearly, by some maneuvering around an eavesdropper,\na significant reduction in energy cost can be achieved as the\neavesdropper becomes almost ineffective in some locations. This is\nthe main idea behind this work.\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfloat[Path-loss exponent $\\alpha=2$.]{\n        \\includegraphics[width=7cm]{link_a2}\n        \\label{fig:onelinka2}\n    }\n    \\hspace{1mm}\n    \\subfloat[Path-loss exponent $\\alpha=4$.]{\n        \\includegraphics[width=7cm]{link_a4}\n        \\label{fig:onelinka4}\n    }\n  \\caption{Effect of eavesdropper location on link cost.}\n  \\label{fig:onelink}\n\\end{figure}\n\n\n\\spar{Effect of Optimal Secrecy Allocation on Path Cost}\n\\label{sub:alloc} For a fixed path subject to an end-to-end secrecy\nrequirement $\\pi$, the optimal eavesdropping probability assigned to\neach link of the path is given by~\\eqref{eq:pi}, which in turn\ndetermines the optimal jamming power allocated to each link of the\npath using~\\eqref{eq:pj}. Specifically, this is how power allocation\nis performed in SASP in order to minimize power consumption.\nAlternatively, a simple heuristic is to divide $\\pi$ equally across\nthe links. That is, if the path contains $h$ links, then each link\n$\\ell_k$ is allocated sufficient jamming power to satisfy the\neavesdropping probability $\\pi_k = \\pi\/h$. In\nFig.~\\ref{fig:allocation}, we have depicted energy savings that can\nbe achieved ``solely'' by optimal secrecy allocation compared to\nequal allocation for a fixed path that is computed by SASP.\nInterestingly, as the number of eavesdroppers increases or the\nsignal propagation becomes more restricted, optimal secrecy\nallocation becomes even more important, achieving energy savings of\nup to $72\\%$ ($47\\%$) for $\\alpha=4$ ($\\alpha=2$) in the simulated\nnetwork.\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfloat[Effect of eavesdropper density.]{\n        \\includegraphics[width=8cm]{path_density}\n        \\label{fig:optdens}\n    }\n    \\subfloat[Effect of eavesdropping prob.]{\n        \\includegraphics[width=8cm]{path_prob}\n        \\label{fig:optprob}\n    }\n  \\caption{Energy savings achieved by optimal secrecy allocation.}\n  \\label{fig:allocation}\n\\end{figure}\n\n\n\\spar{Non-uniform Eavesdropper Placement}\nTo gain more insight about the behavior of different routing\nalgorithms, in this experiment, rather than randomly distributing\neavesdroppers in the network, we strategically place them close to\nthe line that connects the source and destination. Ideally, SMER and\n$\\epsilon$-SMER should avoid the shortest path that crosses the\nnetwork diagonally. This is indeed the behavior observed in the\nsimulations as depicted in Fig.~\\ref{fig:nonuniform} (`\\scalebox{0.6}{$\\bigstar$}'\ndenotes an eavesdropper). As expected, SASP blasts right through the\neavesdroppers, while SMER, $0.1$-SMER and $1.0$-SMER route around\nthem resulting in $88\\%$, $86\\%$ and $85\\%$ energy savings,\nrespectively.\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=8cm]{snapshot}\n  \\caption{Snapshot of paths computed by different algorithms.}\n  \\label{fig:nonuniform}\n\\end{figure}\n\n\n\\spar{Uniform Eavesdropper Placement} In this experiment,\neavesdroppers are placed in the network uniformly at random. As seen\nin Fig.~\\ref{fig:uniform}, our algorithms consistently outperform\nSASP for a wide range of eavesdropper densities and eavesdropping\nprobabilities. In particular, energy savings of up to $99\\%$ and\n$98\\%$ (for $\\alpha=4$) can be achieved by SMER and $0.1$-SMER,\nrespectively.\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfloat[Effect of eavesdropper density.]{\n        \\includegraphics[width=8cm]{uniform_density}\n        \\label{fig:uniform1}\n    }\n    \\subfloat[Effect of eavesdropping prob.]{\n        \\includegraphics[width=8cm]{uniform_prob}\n        \\label{fig:uniform2}\n    }\n  \\caption{Energy savings with uniform eavesdropper placement.}\n  \\label{fig:uniform}\n\\end{figure}\n\n\n\\spar{Effect of Network Size}\nFig.~\\ref{fig:size} shows the energy savings achieved by different\nalgorithms in networks with varying sizes. The ``network dimension''\nrefers to the length of one side of the square area that contains\nthe network nodes. As observed from the figure, the energy saving\nis an increasing function of the network size. Interestingly, as the\nnetwork size increases, the effect of the propagation environment\ndiminishes in such a way that energy savings for $\\alpha=2$ and\n$\\alpha=4$ converge to the same numbers as opposed to the previous\nscenarios. As the network size increases so does the average length\nof the path (in terms of the number of hops) between the source and\ndestination nodes. Those paths that are longer provide more\nopportunities for energy savings on each link of the path resulting\nin increased overall energy savings. This effect works in favor of\n$\\alpha=2$ as well as $\\alpha=4$. However, given the high values of\nenergy savings for $\\alpha=4$ (due to longer paths compared to\n$\\alpha=2$), the effect of longer paths is more prominent for\n$\\alpha=2$.\n\\begin{figure}[ht]\n    \\centering\n    \\subfloat[Non-uniform eavesdrop. placement.]{\n        \\includegraphics[width=8cm]{network_size_diag}\n        \\label{fig:size_diag}\n    }\n    \\subfloat[Uniform eavesdrop. placement.]{\n        \\includegraphics[width=8cm]{network_size}\n        \\label{fig:size_uniform}\n    }\n  \\caption{Effect of network size on energy savings.}\n  \\label{fig:size}\n\\end{figure}\n\n\n\\spar{Effect of Jamming Set} The cardinality of the jamming set\naffects the power allocation to jammers. In this experiment, we\nchange the number of jammers that participate in secure transmissions\non each link and compute the energy savings achieved by different\nalgorithms. Figs.~\\ref{fig:jammers}\\subref{fig:jammers_diag}\nand~\\ref{fig:jammers}\\subref{fig:jammers_uniform}, respectively, show\nthe energy savings achieved for non-unform and uniform placement of\neavesdroppers. Interestingly, in these scenarios, a small number of\njammers, namely $2$, is sufficient to obtain most of the benefits of\ncooperative jamming, which should greatly simplify any practical\nimplementation.\n\n\\begin{figure}[ht]\n    \\centering\n    \\subfloat[Non-uniform eavesdrop. placement.]{\n        \\includegraphics[width=8cm]{jammers_diag}\n        \\label{fig:jammers_diag}\n    }\n    \\subfloat[Uniform eavesdrop. placement.]{\n        \\includegraphics[width=8cm]{jammers_uniform}\n        \\label{fig:jammers_uniform}\n    }\n  \\caption{Effect of the jamming set on energy savings.}\n  \\label{fig:jammers}\n\\end{figure}\n\n\\section{System Model and Assumptions}\n\\label{sec:sysmodel}\n\nConsider a wireless network with arbitrarily distributed nodes.\nWe assume that each node (legitimate or eavesdropper) is equipped\nwith a single omni-directional antenna.\nA $K$-hop route $\\Pi$ between a source and a destination in the\nnetwork is a sequence of $K$ links connecting the source to the\ndestination\\fnote{Terms ``path'' and ``route'' are used\ninterchangeably throughout the paper.}. We use the notation $\\Pi =\n\\angles{\\ell_1, \\ldots, \\ell_K}$ to refer to a route that is formed\nby $K$ links $\\ell_1$ to $\\ell_K$. A link $\\ell_k \\in \\Pi$ is formed\nbetween two nodes $S_k$ and $D_k$ on route $\\Pi$. We assume that\nevery link $\\ell_k$ is exposed to a set of (potential) eavesdroppers\ndenoted by $\\mc{E}_k$. Whenever $S_k$ transmits a message to $D_k$,\na set of trusted nodes, called jammers, cooperate with $S_k$ to\nconceal its message from the eavesdroppers in $\\mc{E}_k$ by jamming\n$S_k$'s signal at the eavesdroppers. The set of the jammers\ncooperating with $S_k$ to conceal its transmissions from $\\mc{E}_k$\nis denoted by $\\mc{J}_k = \\set{J_1\\ldots, J_{|\\mc{J}_k|}}$, where $|\\mc{A}|$\ndenotes the cardinality of set $\\mc{A}$. The set of\njammers is potentially different for different links. Throughout the\npaper, we use the notation $\\enc{S_k, D_k, \\mc{E}_k,\n\\mc{J}_k}$ to identify link $\\ell_k$.\n\nIn the following subsections, we describe the models considered in\nthis paper for the wireless channel, eavesdroppers, physical-layer\nsecurity and end-to-end routing. For notational simplicity, we may\ndrop the link index $k$ whenever there is no ambiguity.\n\n\n\\subsection{Wireless Channel Model}\nConsider the discrete-time equivalent model for a transmission from node\n$S$ to node $D$. Let $x_S$ be the normalized (unit-power) symbol\nstream to be transmitted by $S$, and let $y_D$ be the received\nsignal at node $D$. We assume that transmitter $S$ is able to\ncontrol its power $P_S$ in arbitrarily small steps, up to some\nmaximum power $\\pmax$. Let $n_D$ denote the receiver noise at $D$, where\n$n_D$ is assumed to be a complex Gaussian random variable with\n$\\mathbb{E}\\left[|n_D|^2\\right] = N_0$. The received signal at $D$\nis expressed as\n\\begin{equation}\n    y_D = \\sqrt{P_S} \\, h_{S,D} \\, x_S + n_D,\n\\end{equation}\nwhere $h_{S,D}$ is the complex channel gain between $S$ and $D$. The\nchannel gain is modeled as $h_{S,D} = |h_{S,D}|e^{\\theta_{S,D}}$,\nwhere $|h_{S,D}|$ is the channel gain magnitude and $\\theta_{S,D}$\nis the uniform phase. We assume a non line-of-sight environment, implying\nthat $|h_{S,D}|$ has a Rayleigh distribution, and that\n$\\mathbb{E}[|h_{S,D}|^2] = 1\/d_{S,D}^\\alpha$, where  $d_{S,D}$ is the\ndistance between nodes $S$ and $D$,\nand $\\alpha$ is the path-loss exponent (typically between $2$ and\n$6$). This is the standard narrowband fading channel model\nemployed in the physical layer literature~\\cite{tse05wireless,goldsmith2005}.\n\n\n\\subsection{Adversary Model}\nWe limit our attention to passive eavesdroppers as in prior\nwork~\\cite{goel08,tekin2008general,dong10,elgamal08,Dennis2011JSAC,swindle11}.\nAlthough there are other forms of adversarial behavior, their\nconsideration is beyond the scope of this paper.\n\nWhile the literature on physical layer security often assumes not\nonly eavesdropper locations but also either perfect\n(\\eg~\\cite{tekin2008general}) or imperfect (\\eg~\\cite{swindle11})\nknowledge at the transmitters and jammers of the complex channel\ngains of the eavesdropping channels (\\ie\\ availability of\ninstantaneous eavesdropper channel state information (CSI)), we\nconsider the more realistic scenario, in which CSI for eavesdropping\nchannels is not available. In addition, our model requires only the\nknowledge of {\\em potential} eavesdropping locations in the network,\nyet we show that it provides guaranteed security by employing coding\nin conjunction with cooperative jamming.\n\nSpecifically, we assume that each link $\\ell_k$ is subject to\npotential eavesdropping from a set of locations denoted by $\\mc{E}_k\n= \\set{E_1, \\ldots, E_{|\\mc{E}_k|}}$, where the probability of\neavesdropping from location $E_i$ is given by $p(E_i)$ for $0 \\le\np(E_i) \\le 1$. This is a considerably general model that can be used\nto represent a wide range of eavesdropping scenarios\\fnote{Although\nour model cannot be applied to every possible scenario, it is more\ngeneral compared to the models in the literature on physical layer\nsecurity\n(see~\\cite{goel08,tekin2008general,dong10,elgamal08,Dennis2011JSAC,swindle11},\nand references therein).}. For example, setting all $p(E_i)$'s to $1$\nfor a link models multiple eavesdroppers for that link.  Other\nexamples include, for example, military scenarios where the locations\nof enemy installations are known, or wireless networks where a\nmalicious user(s) has been detected.  In general, for any given link,\nthere is only a limited region around the link that can be exploited\nfor eavesdropping. By dividing the effective eavesdropping region to\na few smaller areas~\\cite{tessellation}, one can compute the most\neffective eavesdropping location within each area, and consequently,\nconstruct the set of eavesdropping locations for that link.\n\n\\subsection{Physical Layer Security Model}\nConsider a secure link formed between source $S$ and receiver $D$\nwith the help of jammers $\\mc{J}$.  For the moment, we assume that\ncooperative jamming is implemented at the physical layer to deal with\na \\emph{single} eavesdropper $E$ located at a \\emph{fixed} position.\nLater, in Section~\\ref{sec:linkcost}, we show how this physical layer\nprimitive can be used to provide security against multiple\neavesdroppers or unknown eavesdropping locations.\n\nWhen node $S$ transmits a message, there are multiple ways in which\ncooperative jamming by system nodes can be exploited, ranging from\nrelatively simple noncoherent techniques to sophisticated beamforming\ntechniques~\\cite{mostafa_twireless}.  Since the {\\em implementation}\nof beamforming in other contexts, with the same challenges of\nsynchronization in the wireless environment, is advancing rapidly\n\\cite{beamform_overview,beamform_implementation}, we assume that the\njammers cooperatively \\emph{beamform} a common artificial noise\nsignal $z$ to the receiver in such a way that their signals cancel\nout at the receiver \\cite{brown_null}.  The noise signal $z$ is\ntransmitted in the \\emph{null~space} of the channel vector\n    $\\mx{h}_D = [h_{J_1,D}, h_{J_2,D}, \\ldots, h_{J_{|\\mc{J}|},D}]^\\mathrm{T}$\nwhere, $h_{J_i,D}$ denotes the channel gain between jammer $J_i$ and\ndestination $D$ and $\\mx{A}^\\mathrm{T}$ denotes the conjugate transpose of vector\n$\\mx{A}$. Thus, the signal transmitted by the jammers can be\nexpressed as\n    $\\mathbf{s}_J = \\mathbf{h}_D^\\perp \\, z$,\nwhere $\\mathbf{h}_D^\\perp$ is a vector chosen in the null\nspace of $\\mathbf{h}_D$. It follows that the total transmission\npower of the jammers is given by $P_J = \\parallel\\mathbf{h}_D^\\perp\\parallel^2$.\nAssuming that the source node transmits with power $P_S$, the\nsignals received at the destination and the eavesdropper are given\nby\n\\begin{equation}\n\\begin{split}\n\\label{eq:rsignals}\n    y_D &= \\sqrt{P_S} \\, h_{S,D} \\, x_S + n_D,\\\\\n    y_E &= \\sqrt{P_S} \\, h_{S,E} \\, x_S + \\mathbf{h}_E^\\mathrm{T} \\mathbf{h}_D^\\perp z + n_E,\n\\end{split}\n\\end{equation}\nwhere, $\\mathbf{h}_E = [h_{J_1,E}, h_{J_2,E}, \\ldots,\nh_{J_{|\\mc{J}|},E}]^\\mathrm{T}$ represents the channel gain vector between\nthe jammers and the eavesdropper, and $n_D$ and $n_E$ denote the\ncomplex Gaussian noise at the destination and eavesdropper,\nrespectively, with $\\E{|n_D|^2} = \\E{|n_E|^2} = N_0$.\n\nAlthough the jammers try to prevent the eavesdropper from\nsuccessfully receiving the message, there is still some probability\nthat the eavesdropper actually obtains the message due to the fact\nthat the channel to the eavesdropper is \\emph{unknown} in our model, \\ie\\ $\\mx{h}_E$ and\n$h_{S,E}$ are unknown. Recalling that the signal-to-interference-plus-noise\nratio ($\\sinr$) at the destination is controlled via power control, let\n$\\gamma_E$ denote the minimum required $\\sinr$ at the eavesdropper\nin order to violate the security constraints of the protocol\n(e.g. for the cryptographic case, the $\\sinr$ above which the eavesdropper\ncan record a meaningful version of the transmitted signal; in the\ninformation-theoretic case, the $\\sinr$ above which, for a given wire-tap code,\nthe equivocation does not equal the entropy of the message.)  Let $\\sinr_E$\ndenote the $\\sinr$ at the eavesdropper.  We have\n\\begin{equation}\n\\label{eq:success_beaming}\n\\begin{split}\n  &\\pr{ \\sinr_E \\ge \\gamma_E}\n    =\n      \\pr{\n          \\tfrac{ P_S|h_{S,E}|^2 }{ N_0 + \\mathbf{h}_E^\\mathrm{T}  \\mathbf{h}_D^\\perp {\\mathbf{h}_D^\\perp}^\\mathrm{T} \\mathbf{h}_E }\n          \\ge \\gamma_E }\\\\\n    &\\qquad= \\mathbb{E}_{\\mathbf{h}_E}\n        \\left[\n            \\pr{\n                \\frac{ P_S|h_{S,E}|^2 }{ N_0 + \\mathbf{h}_E  \\mathbf{h}_D^\\perp {\\mathbf{h}_D^\\perp}^\\mathrm{T} \\mathbf{h}_E^\\mathrm{T} }\n                \\ge \\gamma_E \\Big|\\, \\mathbf{h}_E  }\n        \\right] \\\\\n    &\\qquad= \\mathbb{E}_{\\mathbf{h}_E}\n        \\big[\n            e^{- \\frac{\\gamma_E d_{S,E}^\\alpha}{P_S} \\mathbf{h}_E^\\mathrm{T} \\mathbf{h}_D^\\perp {\\mathbf{h}_D^\\perp}^\\mathrm{T} \\mathbf{h}_E }\n        \\big]\n        e^{- \\frac{\\gamma_E N_0 d_{S,E}^\\alpha}{P_S}} ,\n \\end{split}\n\\end{equation}\nwhere $\\mathbb{E}_{\\mathbf{h}_E}$ means expectation with respect to\nchannel gain vector $\\mathbf{h}_E$. Using the results on quadratic forms~\\cite[Eq.~14]{turin} to calculate the expectation, it is obtained that ($\\mathbf{I}_N$ is\nthe identity matrix of size $N$)\n\\begin{equation}\n\\label{eq:success}\n\\begin{split}\n  \\pr{ \\sinr_E \\ge \\gamma_E}\n    &\\le \\frac{ e^{- \\frac{\\gamma_E N_0 d_{S,E}^\\alpha}{P_S}} }\n        { |\\mathbf{I}_N + \\frac{\\gamma_E d_{S,E}^\\alpha}{P_S}(\\sum_{J_i \\in \\mc{J}}{\\frac{1}{d_{J_i,E}^\\alpha}}) \\mathbf{\\mathbf{h}_D^\\perp {\\mathbf{h}_D^\\perp}^\\mathrm{T}}| } \\\\\n    &= \\frac{ e^{ -N_0 \\gamma_E \\frac{d_{S,E}^\\alpha}{P_S} } }\n            { 1 + \\frac{\\gamma_E d_{S,E}^\\alpha}{P_S}(\\sum_{J_i \\in \\mc{J}}{\\frac{1}{d_{J_i,E}^\\alpha}}) P_J }\n            \\eqend\n\\end{split}\n\\end{equation}\nwhere the final expression is derived from Sylvester's determinant\ntheorem:\n\\[\\det(\\mathbf{I}_m + \\mathbf{A}\\mathbf{B}) = \\det(\\mathbf{I}_n + \\mathbf{B}\\mathbf{A} ),\\]\nfor $\\mathbf{A}$ and $\\mathbf{B}$ being $m\\times n$ and $n\\times m$\nmatrices, respectively, and the fact that $P_J =\n{\\mathbf{h}_D^\\perp}^\\mathrm{T} \\mathbf{h}_D^\\perp$ (see\n\\eqref{eq:pj1}).\n\nIn the remainder of the paper, we use~\\eqref{eq:success} in equality form to compute the eavesdropping probability for a given jamming power $P_J$. While this results in a (slightly)\nconservative power allocation, it is sufficient to satisfy the security requirement of\neach link.\n\n\n\n\\subsection{Routing Model}\nConsider a $K$-hop route $\\Pi = \\angles{\\ell_1, \\ldots, \\ell_K}$ between a\nlegitimate source and destination in the network. Let $\\mc{L}$ denote the set\nof all possible routes between the source and destination. Let\n$\\mc{C}(\\Pi)$ denote the cost of route $\\Pi$, where the cost of a\nroute is defined as the summation of the costs of the links forming the\nroute. With slight abuse of the notation, we use $\\mc{C}(\\ell_k)$ to\ndenote the cost of link $\\ell_k$ as well. The secure routing problem\nis then defined as follows.\n\n\\bigskip\\noindent {\\bf SMER: Secure Minimum Energy Routing Problem}\\\\\n{\\em Consider a wireless network and a set of eavesdroppers\ndistributed in the network. Given a source and destination, find a\nminimum energy path $\\Pi^*$ between the source and destination\nsubject to constraints $\\pi$ and $\\lambda$ on the end-to-end successful eavesdropping\nprobability and goodput on the path respectively.}\n\\bigskip\n\nLet $\\lambda(\\Pi)$ and $\\lambda(\\ell_k)$ denote, respectively, the goodput\nof path $\\Pi$ and link $\\ell_k \\in \\Pi$. Then $\\lambda(\\Pi)$ can be expressed as\n    \\[ \\lambda(\\Pi) = \\min_{\\ell_k \\in \\Pi} \\lambda(\\ell_k) \\eqend\\]\nSince goodput of a link is an increasing function of the transmission\npower of the transmitter of that link, a necessary condition for minimizing power over the path $\\Pi$\nis given by $\\lambda(\\ell_k) = \\lambda$, for all $\\ell_k \\in \\Pi$, \\ie\\\nall links should just achieve the minimum goodput $\\lambda$. Thus, our power allocation scheme (see Section~\\ref{sec:linkcost}) establishes links that achieve exactly the minimum required goodput $\\lambda$. Consequently, the constraint on the end-to-end goodput is satisfied by any path in the network, and hence does not need to be explicitly considered when solving SMER.\nAs such, SMER can be formally described by the following optimization problem:\n\\begin{equation}\n\\begin{split}\n\\label{eq:smer}\n    \\Pi^* = \\, &\\displaystyle\\arg \\min_{\\Pi \\in \\mc{L}}  \\sum_{\\ell_k \\in \\Pi} \\mc{C}(\\ell_k) \\\\\n                &s.t. \\quad \\pr{\\text{eavesdropping on route $\\Pi$}} \\le \\pi,\n\\end{split}\n\\end{equation}\nfor some pre-specified $\\pi$ ($0 < \\pi < 1$). The constraint on\nthe route eavesdropping probability in the above optimization\nproblem can be expressed in terms of the eavesdropping probability\non individual links $\\ell_k$ that form the route $\\Pi$, as\n        $\\prod_{\\ell_k \\in \\Pi} (1 - \\pi_k) \\ge 1-\\pi$,\nwhere $\\pi_k$ ($ 0 < \\pi_k < 1$) denotes the successful\neavesdropping probability on link $\\ell_k$. We use the following\nresult to convert the above inequality constraint to an equality\nconstraint in the routing problem~\\eqref{eq:smer}.\n\n\\begin{lemma}\nThe cost of route $\\Pi$ is a monotonically\nincreasing function of\n$\\prod_{\\ell_k \\in \\Pi} (1 - \\pi_k)$.\n\\end{lemma}\n\\begin{proof}\nConsider a path $\\Pi$ between the source and destination nodes.\nDefine the end-to-end secrecy of path $\\Pi$, denoted by\n$\\omega(\\Pi)$, as follows:\n\\begin{equation}\n    \\omega(\\Pi) = \\prod_{\\ell_k \\in \\Pi} \\omega_k,\n\\end{equation}\nwhere $\\omega_k = (1 - \\pi_k)$.\n\nFirst, we show that the link cost $\\mc{C}(\\ell_k)$ is a\nmonotonically increasing function of the the link secrecy $\\omega_k$\nfor every link $\\ell_k \\in \\Pi$. Let $P_S^{(k)}$ and $P_J^{(k)}$\ndenote the source and jamming powers allocated to the link $\\ell_k$,\nrespectively. In Section~\\ref{sec:linkcost}, we show that: (i)\n$P_S^{(k)}$ is a constant for a given link independent of the link\nsecrecy, and (ii) $P_J^{(k)}$ is a function of the link secrecy and\nis given by\n\\begin{equation}\n    P_J^{(k)} = c_k \\cdot \\frac{\\omega_k}{1 - \\omega_k},\n\\end{equation}\nwhere $c_k$ is some constant independent of $\\omega_k$. Thus, for a\nfixed link $\\ell_k$, the link cost $\\mc{C}(\\ell_k) = P_S^{(k)} +\nP_J^{(k)}$ depends on $\\omega_k$ only through the jamming power\n$P_J^{(k)}$. Taking the derivative on the link cost with respect to\n$\\omega_k$ results in the following relation:\n\\begin{equation}\n    \\frac{d}{d \\omega_k} \\, \\mc{C}(\\ell_k) = c_k \\cdot \\frac{1}{(1 - \\omega_k)^2} > 0,\n\\end{equation}\nindicating that the link cost is an increasing function of the link\nsecrecy $\\omega_k$.\n\nLet $\\mc{C}^*(\\Pi)$ and $\\mc{C}(\\Pi)$ denote the optimal cost of the\npath $\\Pi$ computed by solving the optimization\nproblem~\\eqref{eq:smer-p}, with equality and inequality constraints,\nrespectively. Furthermore, let $\\omega^*(\\Pi)$ and $\\omega(\\Pi)$\ndenote the corresponding end-to-end path secrecies. We present a\nproof of the lemma based on contradiction by assuming that the\noptimal path cost with the inequality constraint is less than that\nwith the equality constraint. That is, we assume that\n\\begin{equation}\n    \\mc{C}(\\Pi) \\le \\mc{C}^*(\\Pi),\n\\end{equation}\nwhile,\n\\begin{equation}\n    \\omega(\\Pi) > \\omega^*(\\Pi) \\eqend\n\\end{equation}\n\nNext, by manipulating the link secrecy allocation vector $[\\omega_1,\n\\ldots, \\omega_k, \\ldots, \\omega_K]$, we construct a new link\nsecrecy allocation vector that satisfies the equality constraint,\nwhile having a cost smaller than $\\mc{C}^*(\\Pi)$. To this end,\nconsider some arbitrary link $\\ell_k$, and replace $\\omega_k$ by a\nnew $\\omega'_k$ as follows\n\\begin{equation}\n    \\omega'_k = \\frac{\\omega^*}{\\omega} \\cdot \\omega_k\n    \\eqend\n\\end{equation}\nSince $\\omega^* < \\omega$, it follows that $\\omega'_k < \\omega_k$.\nConsequently, the new cost of the link $\\ell_k$ with link secrecy\n$\\omega'_k$ is less than $\\mc{C}(\\ell_k)$, which in turn indicates\nthat the new path cost with secrecy allocation vector $[\\omega_1,\n\\ldots, \\omega'_k, \\ldots, \\omega_K]$ is less than $\\mc{C}(\\Pi)$.\nTherefore, we have\n\\begin{equation}\n    \\mc{C}'(\\Pi) < \\mc{C}(\\Pi) < \\mc{C}^*(\\Pi),\n\\end{equation}\nand,\n\\begin{equation}\n    \\omega(\\Pi) > \\omega'(\\Pi) = \\omega^*(\\Pi) \\eqend\n\\end{equation}\nThe proof follows by noting that this contradicts the assumption\nthat $\\mc{C}^*(\\Pi)$ is the minimum cost of path $\\Pi$ with the\nequality constraint.\n\\end{proof}\n\nThus, to minimize the cost of the optimal route, the inequality\nconstraint can be substituted by the equality constraint\n$\\prod_{\\ell_k \\in \\Pi} (1 - \\pi_k) = 1-\\pi$. On each link $\\ell_k$,\nit is desirable to keep the successful eavesdropping probability\n$\\pi_k$ close to $0$. In this case, the product $\\prod_{\\ell_k \\in\n\\Pi} (1 - \\pi_k)$ can be approximated by the expression $1 -\n\\sum_{\\ell_k \\in \\Pi} \\pi_k$. By substituting the approximate\nlinearized constraint in the routing problem, the following\noptimization problem is obtained\n\\begin{equation}\n\\begin{split}\n\\label{eq:smer-p}\n    \\Pi^* = \\, &\\displaystyle\\arg \\min_{\\Pi \\in \\mc{L}}  \\sum_{\\ell_k \\in \\Pi} \\mc{C}(\\ell_k) \\\\\n                     &s.t. \\quad \\displaystyle \\sum_{\\ell_k \\in \\Pi}\n                     \\pi_k = \\pi\n                     \\eqend\n\\end{split}\n\\end{equation}\nIn the rest of the paper, we focus on\nthis optimization \\mbox{problem}.\nWe first show that the problem is, in general,\nNP-hard and then develop exact and approximate algorithms to\nsolve it.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMolecules secreted and internalized by Eukaryotic cells follow well defined routes, the secretory or endocytic pathways, along which they are exposed to a succession of biochemical environments by sequentially visiting different membrane-bound organelles \\cite{kelly1985pathways}. Different organelles have different membrane compositions, as well as a distinct set of membrane-associated proteins, refered to henceforth as the membrane {\\em identity}. Interestingly, it has been shown that the identity of some organelles changes with time ; for example, the early endosome (a compartment digesting newly internalized content) has a different identity from the late endosome, which then becomes a lysosome \\cite{zerial:2001}. One fundamental issue underlying the organization of  intracellular transport  is whether progression along the various pathways occurs by exchange between organelles of fixed  biochemical identities (via the budding and scission of carrier vesicles), or by the biochemical maturation of the organelles themselves \\cite{kelly1985pathways,zerial:2001}. This question is particularly debated for the Golgi apparatus, where proteins undergo post-transcriptional maturation and sorting. The Golgi is divided into early ({\\em cis}), middle ({\\em medial}) and late ({\\em trans})  $\\unit{\\mu m}$-size compartments called cisternae.  In yeast, each  cisterna appears to undergo independent biochemical maturation from a {\\em cis} to a {\\em trans} identity in less than 1 min \\cite{matsuura:2006,losev:2006}. In higher eukaryotes, the cisternae form a tight and polarized stack with {\\em cis} and {\\em trans} ends, through which proteins travel in  about 20 min \\cite{emr:2009}. Whether transport through the stack occurs by inter-cisternal exchange  or by the maturation of entire cisternae remains controversial \\cite{emr:2009}.\n\nMaturation in an organelle membrane causes  different membrane identities to transiently coexist and may trigger the formation of transient membrane domains. Membrane components have indeed been seen to segregate into domains in both yeast \\cite{matsuura:2006,losev:2006} and  mammalian Golgi cisternae \\cite{patterson:2008}.\n This is the case of proteins of the Rab family, though to be essential identity labels of cellular organelles \\cite{zerial:2001}. The so-called {\\em Rab cascade}, in which the activation of one Rab inactivates the preceding Rab along the pathway, is thought to permit the  sequential maturation of the organelle identity \\cite{rivera:2009}. Domains could also emerge from the maturation of ceramids (present in cis-Golgi) into sphingomyelin (present in trans-Golgi), as these two species are known to lead to domain formation on vesicles \\cite{sot:2006}. Finally, there is a continuous gradient of membrane thickness from cis to trans Golgi compartments \\cite{mitra:2004} and thickness mismatch can lead to phase separation in model membranes \\cite{heberle:2013}.\n \n  It has been argued that membrane domains in organelles could undergo budding and scission, and hence control inter-organelle transport \\cite{pfeffer:2010b}. This raises the interesting possibility that the rate of domain formation could control the rate of transport. To quantitatively assess this possibility, we studied transient domain formation in an ideal two-component membrane. We consider an irreversible transformation (maturation) $A\\rightarrow B$ taking place between two components, with $A$ and $B$ representing distinct biochemical identities, and we investigate the phase behavior of a such membrane.\n\nThe kinetics of phase separation in binary mixtures have been abundantly studied \\cite{bray:1994}. In the context of fluid membranes, hydrodynamic flows in the membrane and the surrounding media make the problem quite complex. Several dynamical regimes have been reported, and a unified picture of has not yet emerged  \\cite{camley:2010,fan:2010}. For deformable fluid membranes such as cellular membranes, the budding of membrane domains \\cite{lipowsky:1992} makes the dynamics of phase separation even more complex \\cite{sunil:1998,sunil:2001,laradji:2004}. Here, we study transient phase separation on flat membranes, and we implement membrane deformability at a phenomenological level by introducing a critical domain size beyond which flat domains are unstable. If domains reach such a size, they undergo a budding transition and may serve as transport intermediates, provided a scission mechanism ({\\em e.g} the activity of specialized proteins such as dynamin \\cite{ferguson:2012}) separates budded domains from the rest of the membrane.\n\nThe budding of membrane domains may for instance be driven by the line energy associated with domain boundaries \\cite{lipowsky:1992}, expressed as the domain line tension $\\gamma$ times the boundary length. Budding is resisted by the membrane bending rigidity $\\kappa$ and surface tension $\\sigma$, and will occur for a finite range of domain size $R$ \\cite{pio_regul}:\\begin{equation}\n4 \\frac{\\kappa}{\\gamma} < R < 2 \\frac{\\gamma}{\\sigma}\\, .\n\\label{range}\n\\end{equation}\nFor typical values of the parameters: $\\kappa\\simeq10k_{{}_{\\rm B}}T$ and $\\gamma\\simeq1\\unit{pN}$, the lower bound is a fraction of the typical  size of Golgi cisternae ($\\sim\\unit{\\mu m}$), and the scenario of a line tension induced domain pinching appears realistic. The upper bound could be restrictive in a system with low area\/volume ratio, where pinching might increase membrane tension and lead to incomplete buds. This is often observed in artificial vesicles, but this constraint does not appear stringent in organelles such as the Golgi. We assume henceforth that the rate of membrane deformation is much faster than the rate of chemical maturation, so that only the lower bound of \\eq{range} is relevant. \n\n\nIn this article, we show that, in a membrane undergoing irreversible maturation, the maximal size of transient domains follows a power law with respect to the maturation rate. First, we illustrate two modes of domain growth in fluid membranes. We then show the influence of maturation on domain growth, and the predicted power laws are then confirmed numerically. Because budding depends on domain size, this means that organelle transport can be controlled by the maturation rate.\n\n\n\n\\section{Phase separation kinetics in fluid membranes}\n\n In order to elucidate the role of maturation, we only consider phase separation in a flat fluid membrane and we disregard the influence of the surrounding fluid. This approximation is valid for domains smaller than $\\sim\\eta_2\/\\eta_3$, where $\\eta_3$ is the three-dimensional viscosity of the surrounding fluid (the cytoplasm and the lumen of cellular organelles in the cell), and $\\eta_2$ is the two-dimensional viscosity of the membrane \\cite{saffman:1975}. For biological membranes, one expects $\\eta_2\/\\eta_3\\gtrsim\\unit{\\mu m}$.\n\nDomain growth is initially dominated by the so-called ``Ostwald ripening'', where large domains grow by adsorbing diffusing matter evaporated from smaller domains. At later time, hydrodynamic effects, leading in particular to domain coalescence, dominate the growth. In both cases, after an early nucleation stage, the domain size $R_c$ increases with time according to a power law \\cite{bray:1994}:\n\\begin{eqnarray}\nR_c^{1\/\\alpha-1} \\dot{R_c} \\propto \\tilde D(\\bar\\phi) \\quad:\\quad R_c\\sim t^\\alpha.\n\\label{powerlaw}\n\\end{eqnarray}\nThe  exponent $\\alpha$ and the transport coefficient $\\tilde D$ depend on the dominant growth process, and the latter also depends on composition, line tension, and component mobility. \n\n\\subsection{Thermodynamics}\n\nThe thermodynamics of phase separation in a two-component membrane containing distinct biochemical identities $A$ and $B$ can be studied using a local order parameter $\\phi$ varying continuously between  $\\phi=0$ for A-rich and $\\phi=1$ for B-rich membrane regions. We use the classical Landau free energy \\cite{bray:1994}:\n\\begin{eqnarray}\n&\\mathcal{F}=\\int d^2 \\mathbf{r} \\left[ V [\\phi(\\mathbf{r})] + \\frac{1}{2} \\zeta\\| \\boldsymbol{\\nabla} \\phi \\|^2 \\right] \\label{compofreeE} \\\\\n&V [\\phi] =\\frac{k_{{}_{\\rm B}}T}{a^2}\\left(\\phi \\log{\\phi} + (1-\\phi) \\log{(1-\\phi)}\\right)+ \\frac{K}{2a^2} \\phi ( 1- \\phi), \\nonumber\n\\end{eqnarray}\nwhere $a$ is a molecular size. This energy is the sum of an interfacial term (of parameter $\\zeta$) and a potential term $V$ that includes the translational entropy and the interaction between the two phases (repulsive if $K>0$). \n\n\nPhase separation occurs spontaneously inside the spinodal region of the phase diagram, defined by $K> k_{{}_{\\rm B}}T\/(\\bar{\\phi} (1-\\bar{\\phi}))$, where $\\bar{\\phi}$ is the mean  value of $\\phi$ in the system \\cite{chaikin:1995}. Within this region, the interface between A-rich and B-rich domains is sharp and the energy of interaction reduces to a line energy characterised by the line tension $\\gamma$ (see Supplementary Information - S.I. - for more details). \n\n\\begin{figure}[b]\n   \\includegraphics[width=7.5cm]{fig1} \n   \\caption{\\small (color online) Phase diagram of a binary mixture undergoing chemical maturation.  The blue line represents the spinodal line. The chemical reaction (maturation) increases the mean order parameter $\\bar{\\phi}$ from $0$ to $1$.}\n   \\label{spinomat}\n\\end{figure}\n\\subsection{Domain growth by Ostwald ripening}\n\n For a conserved order parameter, the dynamics of the order parameter is described by a  Cahn-Hilliard equation \\cite{bray:1994}:\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\partial_t \\phi = D\\boldsymbol{\\nabla}^2\\mu\/k_{{}_{\\rm B}}T \\qquad \\mu =a^2\\delta \\mathcal{F} \/\\delta \\phi\\ , \n\\label{cahnhilliard}\n\\ee\nwhere $\\mu$ is the chemical potential associated with $\\phi$, and $D$ is the monomer diffusion coefficient. Within the two-phases region of \\fig{spinomat}, \\eq{cahnhilliard} produces domains with sharp boundaries. Domains larger than a critical size $R_c$ grow at the expense of smaller domains, and inter-domain exchange occurs by monomer diffusion through the bulk phase. \n\nProvided diffusion is much faster than the evaporation of the smallest domains,  Lifschitz, Slyozov and Wagner (LSW) have shown that the characteristic domain size should obey the scaling law  $R_c \\sim t^{1\/3}$ \\cite{lifshitz:1961,wagner:1961}. This can be qualitatively explained:  if diffusion is fast, the order parameter profile outside the domain boundary satisfies the quasi-static approximation $\\boldsymbol{\\nabla}^2\\mu=0$, and the chemical potential inside domains is related to the line tension by $\\mu \\approx \\gamma \/ R_c$ \\cite{bray:1994}. If there is only one length scale $R_c$ in the system, mass conservation implies $\\dot R_c\\sim \\nabla\\mu\\sim \\mu\/R_c$. Identifying ${\\bf \\nabla.}$ with $1\/R_c$, one finds (see S.I.):\n\\begin{eqnarray}\nR_c^2 \\dot{R_c} \\propto D \\gamma a^2\/k_{{}_{\\rm B}}T \\label{rcsigma} \\, ,\n\\label{LS}\n\\end{eqnarray}\nleading to the classical Lifschitz-Slyozov-Wagner (LSW) dynamical scaling $R_c \\sim t^{1\/3}$ \\cite{lifshitz:1961,wagner:1961}. The size distribution of domains growing by Ostwald ripening is strongly peaked around the size $R_c$ \\cite{bray:1994}. Though this was strictly shown in dimension $3$ or higher, it has been confirmed numerically in two dimension \\cite{huse:1986}.\n\n  A similar scaling has been observed numerically {\\em at steady state} in the presence of reversible reaction $A \\rightleftarrows B$, time being replaced by the inverse of the reaction rate \\cite{glotzer:1995}. In the following we show analytically and numerically that a similar scaling  also exists for  irreversible reactions.\n\n\n\n\\subsection{Domain growth by coalescence}\n\n\n The role of hydrodynamics on phase separation in fluid membranes is still controversial, despite considerable recent attention \\cite{camley:2011,fan:2010}. For off-critical mixtures ($\\phi\\ne 1\/2$), hydrodynamic correlations result in the diffusion and coalescence of entire domains. At the scaling level \\cite{bray:1994}, the area of the largest domain can at most double at each coalescence event: $\\dot R_c \\le R_c\/\\tau_D$. The typical collision time $\\tau_D=L^2\/D_d$ depends on the domain diffusion coefficient  $D_d$ and the typical area per domain $L^2\\sim R_c^2\/\\bar\\phi$. Finally, one finds :\n\\begin{equation}}   \\def\\ee{\\end{equation}\nR_c\\dot R_c\\propto D_d\\bar\\phi\\ .\n\\label{hydro}\n\\ee\nIf viscous dissipation is mostly due to membrane hydrodynamics, the domain diffusion coefficient $D_d$ is only weakly dependent on domain size \\cite{saffman:1975}. One thus expects $R_c\\sim t^{1\/2}$ for constant composition, which dominates Ostwald ripening at long times.\n\nThe size distribution of domains can be studied using the Smoluchowski coagulation equation \\cite{smo:1916}:\n\\begin{eqnarray}\n\\partial_t C_n=J_n-kC_n N+\\frac{k}{2}\\sum_{m=1}^{n-1}C_{m}C_{n-m}\\cr {\\rm with} \\quad N=\\sum_{m=1}^\\infty C_m\n\\label{master}\n\\end{eqnarray}\nwhere $C_n$ is the concentration of domains containing $n$ monomers (if $R$ is the domain size: $n\\sim R^2$), $k$ is a typical diffusion rate, and where domain scission has been neglected. This model has been studied extensively for different forms of the diffusion rate \\cite{davies:1999}. Following the assumption that the diffusion coefficient of a domain is independent of its size \\cite{saffman:1975}, we choose a constant diffusion rate $k=D_d\/a^2$. $J_n$ is a source and sink term allowing for the creation or removal of domains \\cite{turner:2005}.\n\nIn the absence of maturation ($\\bar\\phi=$const., $J_n=0$), the size distribution is well approximated by an exponential with a characteristic domain size $\\bar n\\sim \\bar\\phi kt$, giving the domain radius $\\bar R(t)\\sim \\sqrt{D_d\\bar\\phi t}$, in agreement with \\eq{hydro}.\n\nThe domain size distribution is modified by the presence of sources and sinks. It has been shown in \\cite{turner:2005} that choosing a source and sink term that conserves the average concentration  $\\bar\\phi$ (i.e. $J_n=j_{in}\\delta_{n,1}-k_{\\rm off}C_n$) produces a steady-state power-law distribution $C_n\\sim n^{-3\/2}$, up to a characteristic domain size beyond which the distribution is exponential. The characteristic size obeys a scaling reminiscent of \\eq{hydro}: $R_c\\propto \\sqrt{D_d\\bar\\phi\/k_{\\rm off}}$.\n\n\\section{Transient phase separation under irreversible maturation}\n\nMaturation corresponds to the increase of $\\bar{\\phi}$ with time from $\\bar{\\phi}=0$ to $\\bar{\\phi}=1$ due to a chemical reaction. If $K>4 k_{{}_{\\rm B}}T$, the spinodal line (\\fig{spinomat}) is crossed twice, first at $\\bar{\\phi}=\\phi_a$ when phase separation starts, then at $\\bar{\\phi}=\\phi_b$ where the system tends to be homogenous once again . \nWe ask whether domains larger than a critical budding size, for instance given by \\eq{range}, can form during the time the system is prone to phase separation ({\\em i.e.} while $\\phi_a <\\bar\\phi <\\phi_b$). \n\n\\subsection{Dynamical scaling}\n \nFor Ostwald ripening, the quasi-static approximation above assumes that the order parameter profile outside domains adjusts quasi-statically to domain growth ($\\boldsymbol{\\nabla}^2\\phi=0$ in the bulk). In a maturing membrane,\nthe average membrane composition evolves according to $\\partial_t \\bar\\phi= k_r (1-\\bar\\phi)$, and  the quasi-static composition profile satisfies :\n\\begin{equation}}   \\def\\ee{\\end{equation}\nD\\boldsymbol{\\nabla}^2\\phi+k_r(1-\\phi)=0 \\, ,\\ee\n which defines a characteristic length scale $\\lambda_D=\\sqrt{D\/k_r}$.  \nFor small domains $R\\ll\\lambda_D$, \nmaturation does not modify the concentration profile, but merely changes the mean density outside the domains. Thus \\eq{LS} should hold, with a transport coefficient now depending on time through the mean concentration $\\bar\\phi$. This is shown with more details in the S.I.\n\nIn the regime dominated by coalescence, \\eq{hydro}, which assumes a single characteristic domain size, may be used with a time-dependent $\\bar\\phi$ under the assumption that the number of domains varies little between two coalescence events. This approximation is shown to be valid below.\n\nThe extent of phase separation can be characterized by the maximum size $R_{max}$ a domain can reach during the transient phase separation. Here, we are interested in membrane domains that may undergo budding, namely domains of the minority phase surrounded by the majority phase. Domains are thus of the mature species below $\\phi=1\/2$ and of the immature species for $\\phi>1\/2$ (see insets \\fig{spinomat}), and the maximum domain size  occurs for $\\phi=1\/2$. Integrating \\eq{powerlaw}, one finds:\n\\begin{equation}\nR_{max}^{1\/\\alpha} \\propto  \\int_{\\phi_a}^{1\/2}  \\tilde D (\\bar{\\phi}) \\frac{d\\bar\\phi }{\\dot{\\bar\\phi}}=  \\frac{1}{k_r} \\int_{\\phi_a}^{1\/2} \\frac{\\tilde D(\\bar\\phi)}{1-\\bar{\\phi}} d\\bar{\\phi} .\n\\label{scaling}\n\\end{equation}\nThe {\\em maximum}  size of transient domains in a maturing membrane is thus predicted to follow the dynamical scaling law observed for domain growth under fixed composition (\\eq{powerlaw}), where the maturation rate replaces $1\/t$: $R_{max}\\sim k_r^{-\\alpha}$. This dynamical scaling is reminiscent of the scaling observed {\\em at steady state}  in the presence of reversible reaction $A \\rightleftarrows B$ \\cite{glotzer:1995} or continuous recycling \\cite{turner:2005}. That it is  also applicable to irreversible reactions is remarkable, since the kinetics of domain growth in a membrane undergoing maturation do not follow the same scaling, as $\\bar\\phi$ changes with time. This kinetics is not easily obtained from scaling arguments for Ostwald ripening, as the $\\bar\\phi$ dependence of \\eq{LS} is not straightforward (see S.I.). For domain coalescence, a simple integration of \\eq{hydro} with $\\bar\\phi\\simeq k_r t$, valid at early time, shows that one expects $R_c(t)\\simeq\\sqrt{D_d k_r}t$. This linear growth contrasts with the $\\sim t^{1\/2}$ scaling in the absence of maturation.\n\n\n \\begin{figure}[t]\n   \\includegraphics[width=8.5cm]{fig2} \n   \\caption{\\small Growth by Ostwald ripening. Snapshots of Monte Carlo simulations of domain formation in an Ising model without maturation for different times (top row) and with maturation for different maturation rates (bottom row). The average concentration is $\\bar\\phi=1\/2$ in all cases, \n   and the interaction parameter  $J=0.75 k_{{}_{\\rm B}}T$ corresponds to a physiological line tension $\\gamma\\simeq4\\unit{pN}$ (see S.I.)}\n   \\label{num}\n\\end{figure}\n\n\\subsection{Numerical results - Ostwald ripening}\n\n\n\\begin{figure}[b]\n   \\includegraphics[width=8.5cm]{fig3} \n   \\caption{\\small Growth by Ostwald ripening. Domain size (in units of the molecule size $a$) as a function of time ($\\circ$) and maximum domain size as function of the inverse maturation rate $1\/k_r$ ($\\times$), from simulations with $J=0.75$.  Time is in units of the diffusion time $a^2 \/ D$ ($\\approx 10^{-5} s$). The dashed line corresponds to $k_r^{-1\/3}$.  The critical budding size is estimated based on the line tension driven budding scenario (\\eq{range})}\n   \\label{Lct}\n\\end{figure}\n\nThe prediction of \\eq{scaling} was tested numerically (simulation details are given in the S.I.). Without hydrodynamics, we performed Monte Carlo simulations of the Ising model with nearest-neighbor interaction (parameter $J$) and a discrete order parameter $s$ ($0$ or $1$). Monomer diffusion is implemented using Kawasaki dynamics (spin exchange between nearer neighbors), known to produce the LSW growth in a system without maturation \\cite{huse:1986,2005:krzakala}.  Maturation is implemented by letting each site with $s$=0 become a $s$=1 site with a probability $k_r dt$ at each time step (of duration $dt$). Snapshots of the simulations, shown in \\fig{num}, highlight the similarity between a system without maturation (fixed $\\bar\\phi$) after a time $t$, and a system which reaches the same concentration by maturation (after a time $t=-\\log(1-\\bar\\phi)\/k_r$, shown for $\\bar\\phi=1\/2$). \n\\fig{Lct}  compares the variation of the average domain size with time in a system without maturation (with $\\bar\\phi=1\/2$) and the variation of the maximum domain size with respect to the inverse maturation rate in a system undergoing maturation.\nBoth the LSW scaling without maturation ($R\\sim t^{1\/3}$), and our predicted scaling with maturation ($R_{max}\\sim k_r^{-1\/3}$) are apparent at late stages. Strikingly, prefactors of the power law appear very similar with or without maturation. \n \n\\subsection{Numerical results - Diffusion and coalescence}\n \n \nGrowth by coalescence was studied numerically by solving the Master equation \\eq{master} with the source term $J_n=\\delta_{n,1} k_r(1-\\bar\\phi)$ (with $\\bar\\phi=\\sum_n nC_n$), corresponding to monomers being constantly  ``created'' by maturation (see S.I. for details of the numerical scheme). The system contains no mature components at $t=0$ ($C_n(t=0)=0\\ \\forall\\ n$). We find that the domain size distribution crosses over from a power-law $C_n=A n^{-3\/2}$ for small $n<n^*(t)$ to an exponential decay over the size $n^*$ for large $n$. This result indicates that, in a system undergoing irreversible maturation, the domain size distribution is essentially stationary up to the cross-over size $n^*(t)$, computed analytically below. \n\n\\begin{figure}[b]\n\\includegraphics[width=3in]{fig4}   \n\\caption{Growth by coalescence.\tDistribution of domain size at $\\bar\\phi=1\/2$  for different maturation rates, obtained by numerically solving  \\eq{master}.  The log-log plot shows a power law behavior $\\sim n^{-3\/2}$ for small domains, as expected from scaling arguments. }\n   \\label{fig4}\n\\end{figure}\n\\begin{figure}[t]\n   \\includegraphics[width=3in]{fig5}\n   \\caption{\tGrowth by coalescence. Mean domain size ($\\bar{R}(t)$, grey circles)  and maximum domain size ($R_{max}(t)$, grey crosses) as a function of time in a system without maturation (constant $\\bar\\phi$), and mean domain size ($\\bar{R}(k_r)$, black circles) and maximum domain  size ($R_{max}(k_r)$, black crosses) as a function of the inverse maturation rate $1\/k_r$ in the presence of maturation. The dashed line is $\\sim t^{1\/4}$ and the solid line is $\\sim t^{1\/2}$.}\n   \\label{fig5}\n\\end{figure}\n\n\nWe first focus on the role of the maturation rate by analyzing domain size distribution for different values of $k_r$, when $\\bar\\phi$ reaches $1\/2$ ($t=\\log[2]\/k_r$), for which phase separation is most pronounced and one expects to observe the largest domains. \\fig{fig4} shows the domain size distribution for $\\bar\\phi=1\/2$ for different values of the maturation rate. The power-law $C_n=A n^{-3\/2}$ is confirmed up to a characteristic size that depends on $k_r$. This size may be computed as follows. Using \\eq{master} for $n=1$, stationarity of the monomer concentration $C_1$ imposes  $A\\simeq k_r(1-\\bar\\phi)\/(k N)$, with the total number of domains $N=\\sum_{n=1}^{\\infty}C_n\\simeq \\sum_{n=1}^{n^*}C_n=\\sqrt{(1-\\bar\\phi)k_r\/k}$. Using  the conservation relation $\\bar\\phi=\\sum_{n=1}^{\\infty}n C_n\\simeq\\sum_{n=1}^{n^*}n C_n\\simeq\\sqrt{(1-\\bar\\phi)k_r\/kn^*}$, the maximum domain size is found to be: $n^*\\sim k\/k_r\\times\\bar\\phi^2\/(1-\\bar\\phi)$, and the average domain size is $\\bar n\\equiv\\bar\\phi\/N=\\sqrt{n^*}$. The maximum and average domain size when $\\bar\\phi=1\/2$  are predicted to depend on the maturation rate according to\n\\begin{equation}}   \\def\\ee{\\end{equation}\nR_{max}\\sim\\sqrt{\\frac{D_d}{k_r}}\\quad{\\rm and}\\quad \\bar R \\sim\\left(\\frac{D_d a^2}{k_r}\\right)^{1\/4}\n\\ee\nThese predictions are confirmed numerically in \\fig{fig5} which shows the variation of these two characteristic length scales with the maturation rate.\n \n\n\n\n\\begin{figure}[b]\n\\includegraphics[width=3in]{fig6}\n   \\caption{Growth by coalescence. Variation of the maximum domain size as a function of time when maturation is present. The linear growth law $R_{max} \/ \\sqrt{k_r D} \\sim t$ predicted in the text is observed. The maximum domain size is defined as $R_{max}=\\sqrt{\\sum n^2 C_n\/\\bar\\phi}$.}\n  \n   \\label{fig6}\n\\end{figure}\n\n\nAs discussed above, the dynamical scaling for domain growth can be obtained for the entire maturation process (at least while the matured species is the minority, $\\bar{\\phi} \\in [0,1\/2]$) using  \\eq{hydro}: $R_c\\propto\\sqrt{D_d k_r}t$. The characteristic domain size is predicted to increase linearly with time due to the combined effect of domain coalescence and the increasing fraction of matured species, both accounting for $\\sqrt{t}$. This prediction was  verified numerically, as shown \\fig{fig6}.\n\n\\section{Discussion}\n\nThe dynamical scaling predicted by \\eq{scaling} is thus universally observed  whether domain growth proceeds  by Ostwald ripening or by domain coalescence.  On time scales consistent with biochemical maturation: $1\/k_r\\sim\\unit{min.}$, the maximum size of transient domains in a membrane undergoing irreversible maturation is $\\sim0.1\\unit{\\mu m}$ with Ostwald ripening (\\eq{LS} and \\fig{Lct}) and $\\sim 1 \\unit{\\mu m}$ by domain coalescence (\\eq{hydro} and S.I.), with $D=0.1\\unit{\\mu m^2\/s}$, $a=1\\unit{nm}$, and $\\gamma a\/k_{{}_{\\rm B}}T\\sim 1$.   \n \n On deformable membranes, domains within the size range of \\eq{range}  undergo line tension-driven budding. Membrane deformability does not modify the early stages of domain growth, but has a complex influence on late-stage growth. Numerical studies report the possible fusion of budded domains \\cite{sunil:1998,sunil:2001,laradji:2004}, but membrane-mediated repulsion between non-flat  domains may prevent their coalescence \\cite{yanagisawa:2007,ursell:2009}. In low membrane tension organelles, domains large enough to deform will form a complete bud \\cite{pio_regul} that may undergo scission, and the late-stage dynamics should be less relevant.\n \nWithin our framework, one may expect irreversible maturation of membrane components to lead to domain budding and irreversible morphological changes if transient domains can reach the critical budding size, a possibility that requires slow maturation rates. This prediction could be tested experimentally on artificial membrane systems (giant unilamellar vesicles). We predict that a large ($\\gtrsim\\unit{\\mu m}$) deformable vesicle undergoing chemical maturation would preserve its integrity if the reaction is fast, while it would split in two or more daughter vesicles if the reaction is slower ($\\gtrsim\\unit{min.}$) and the maximum transient domain size exceeds the budding size. One possibility would be to use the sphingomyelinase-induced maturation of ceramid into sphingomyelin in giant vesicles, as this reaction is of physiological interest since it occurs in the Golgi apparatus, and is known to produce lipid domains \\cite{fanani:2009}.\n\nExtending our results to multicomponent cellular membranes is not  straightforward, since many factors may influence domain growth and budding and  participate in domain size regulation. Specific membrane proteins  promote curvature and fission \\cite{mcmahon:2005} and may modify the critical budding size range compared to \\eq{range}. Interaction with the cytoskeleton may prevent domain diffusion and coalescence \\cite{fisher:2011,ehrig:2011}. However, transient sub-micron domains have been seen on yeast Golgi cisternae \\cite{matsuura:2006,losev:2006} and slightly larger domains in mammals \\cite{patterson:2008}. This suggests that domain formation is an important component impacting the dynamics of membrane-bound organelles. We thus venture the proposal that the  rate of maturation of membrane components could fundamentally affect the  morphology and dynamics of cellular organelles. \n\n\n\n\nOur study appears particularly interesting in the case of the Golgi apparatus. We argue that the two extreme Golgi organizations observed in nature can  be fitted within a single framework. Yeast Golgi (fast maturation, $k_r\\sim 1\/\\unit{min}$) could be made of dispersed cisternae undergoing independent maturation, because the maturation rate is too fast for the emergence of membrane domains that can reach the budding size. On the other hand, the fact that the Golgi of mammalian cells is a stack of interacting cisternae of different biochemical identities ({\\em cis,\\ medial,\\ trans}) could be made possible by a relatively slow maturation rate ($k_r\\sim 1\/20\\unit{min}$) allowing the formation of large mature domains. Although this simple picture is far from capturing the full complexity of the Golgi apparatus, and in particular the compositional complexity present in other models \\cite{heinrich:2005, gong:2008} or the recycling of resident Golgi enzymes by specific retrograde transport, our results suggest that an internal property of an organelle (the rate of chemical reaction in the Golgi apparatus) could control the structure and organization of this organelle. \n\n\n\\begin{acknowledgments} \nWe gratefully acknowledge B. Goud, N. Gov, R. Phillips and M. Rao for stimulating discussion.  \n\\end{acknowledgments}\n     \n\n\n\\bibliographystyle{apsrev4-1}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Question} Let $\\mathcal{GI}$ and $\\mathcal{GP}$ denote the classes of Gorenstein injective and Gorenstein projective modules, respectively. There is a ``Gorenstein version'' of the aforementioned recollements in \\cite{Gil16}, i.e. $\\xymatrix{\n\\mathbf{K}_{ex}(\\mathcal{GI})\\ar[r]^{} & \\mathbf{K}(\\mathcal{GI}) \\ar[r]^{}\\ar@<-0.6ex>[l]^{} \\ar@<0.6ex>[l]_{} &\\mathbf{D}(R)\\ar@<-0.6ex>[l]^{} \\ar@<0.6ex>[l]_{}}$ and $\\xymatrix{\n\\mathbf{K}_{ex}(\\mathcal{GP})\\ar[r]^{} & \\mathbf{K}(\\mathcal{GP}) \\ar[r]^{}\\ar@<-0.6ex>[l]^{} \\ar@<0.6ex>[l]_{} &\\mathbf{D}(R)\\ar@<-0.6ex>[l]^{} \\ar@<0.6ex>[l]_{}}$.\n\nIf the underlying ring is left-Gorenstein, it follows from \\cite{Chen10} that $\\mathbf{K}(\\mathcal{GP})\\simeq \\mathbf{K}(\\mathcal{GI})$; we also recovered this equivalence in \\cite{Ren19}. However, we do not know if it is true that $\\mathbf{K}_{ex}(\\mathcal{GP})\\simeq \\mathbf{K}_{ex}(\\mathcal{GI})$. We remark that one can not get an answer by simply restricting the equivalent functor $\\mathbf{K}(\\mathcal{GP})\\simeq \\mathbf{K}(\\mathcal{GI})$ in \\cite{Chen10}, or by the methods in \\cite{Ren19}.\n\n\n\n\\section { \\bf The proof of the theorem}\n\nThroughout the paper, let $R$ be a left-Gorenstein ring. All modules are left $R$-modules.\n\nLet $\\mathcal{A}$ be an abelian category with enough projectives and injectives. A pair of classes $(\\mathcal{X}, \\mathcal{Y})$ in $\\mathcal{A}$ is a cotorsion pair provided that $\\mathcal{X} =  {^\\perp}\\mathcal{Y}$ and $\\mathcal{Y} = \\mathcal{X}^{\\perp}$, where $^{\\perp}\\mathcal{Y} = \\{X \\mid \\mathrm{Ext}^{1}_{\\mathcal{A}}(X, Y) = 0,~~\\forall~~Y\\in \\mathcal{Y}\\}$ and\n$\\mathcal{X}^{\\perp} = \\{Y \\mid \\mathrm{Ext}^{1}_{\\mathcal{A}}(X, Y) = 0,~~\\forall~~X\\in \\mathcal{X}\\}$.\n\nThe cotorsion pair $(\\mathcal{X}, \\mathcal{Y})$ is complete provided that for any $M\\in \\mathcal{A}$,\nthere exist short exact sequences $0\\rightarrow Y\\rightarrow X \\stackrel{f}\\rightarrow M \\rightarrow 0$ and  $0\\rightarrow M\\stackrel{g}\\rightarrow Y^{'} \\rightarrow X^{'} \\rightarrow 0$ with $X, X^{'}\\in \\mathcal{X}$ and $Y, Y^{'}\\in \\mathcal{Y}$. In this case, for any $N\\in \\mathcal{X}$, $\\mathrm{Hom}_{\\mathcal{A}}(N, f): \\mathrm{Hom}_{\\mathcal{A}}(N, X)\\rightarrow \\mathrm{Hom}_{\\mathcal{A}}(N, M)$ is surjective since $\\mathrm{Ext}^{1}_{\\mathcal{A}}(N, Y) = 0$, and then $f: X\\rightarrow M$ is said to be a special $\\mathcal{X}$-precover of $M$. Dually, $g: M\\rightarrow Y^{'}$ is called a special $\\mathcal{Y}$-preenvelope of $M$.\n\nBy \\cite[Theorem 2.2]{Hov02}, an abelian model structure on $\\mathcal{A}$ is equivalent to a triple $(\\mathcal{A}_{c}, \\mathcal{A}_{tri}, \\mathcal{A}_{f})$ of subcategories, for which $\\mathcal{A}_{tri}$ is thick and both $(\\mathcal{A}_{c}, \\mathcal{A}_{f}\\cap \\mathcal{A}_{tri})$ and $(\\mathcal{A}_{c}\\cap \\mathcal{A}_{tri}, \\mathcal{A}_{f})$ are complete cotorsion pairs; see also \\cite[Chapter VIII]{BR07}. In this case, $\\mathcal{A}_{c}$ is the class of cofibrant objects, $\\mathcal{A}_{tri}$ is the class of trivial objects and $\\mathcal{A}_{f}$ is the class of fibrant objects. The model structure is called ``abelian'' since it is compatible with the abelian structure of the category in the following way: (trivial) cofibrations are monomorphisms with (trivially) cofibrant cokernel, (trivial) fibrations are epimorphisms with (trivially) fibrant kernel, and weak equivalences are morphisms which factor as a trivial cofibratin followed by a trivial fibration.\n\nFor convenience, we will use the triple $(\\mathcal{A}_{c}, \\mathcal{A}_{tri}, \\mathcal{A}_{f})$ to denote the corresponding model structure. The following is immediate from \\cite[Section 2]{Bec14} or \\cite[Theorem 4.7]{Gil08}.\n\n\\begin{lemma}\\label{lem 1}\nOn the category $\\mathrm{Ch}(R)$ of complexes, there is a singular contraderived model structure\n$\\mathcal{M}_{sing}^{ctr} = (ex\\widetilde{\\mathcal{P}}, (ex\\widetilde{\\mathcal{P}})^{\\perp}, \\mathrm{Ch}(R))$, and a singular coderived model structure $\\mathcal{M}_{sing}^{co} = (\\mathrm{Ch}(R), {^{\\perp}}(ex\\widetilde{\\mathcal{I}}), ex\\widetilde{\\mathcal{I}})$.\n\\end{lemma}\n\nFor a bicomplete abelian category $\\mathcal{A}$ with the model structure $\\mathcal{M} = (\\mathcal{A}_{c}, \\mathcal{A}_{tri}, \\mathcal{A}_{f})$, the associated homotopy category $\\mathrm{Ho}(\\mathcal{M})$ is constructed by localization with respect to weak equivalences. The homotopy category of an abelian model category is always a triangulated category. There is an equivalence of categories $\\mathrm{Ho}(i): \\mathcal{A}_{cf}\/\\omega = {\\mathcal{A}_{cf}\/\\sim}\\rightarrow \\mathrm{Ho}(\\mathcal{M})$ induced by the inclusion functor $i: \\mathcal{A}_{cf}\\rightarrow \\mathcal{A}$, where $\\mathcal{A}_{cf}=\\mathcal{A}_{c}\\cap \\mathcal{A}_{f}$, $f\\sim g: M\\rightarrow N$ if $g-f$ factors through an object in $\\omega = \\mathcal{A}_{c}\\cap \\mathcal{A}_{tri}\\cap \\mathcal{A}_{f}$; see e.g. \\cite[Section 1.2]{Hov99}.\n\n\\begin{corollary}\\label{cor 1}\nThere are equivalences $\\mathrm{Ho}(\\mathcal{M}_{sing}^{ctr})\\simeq \\mathbf{K}_{ex}(\\mathcal{P})$ and\n$\\mathrm{Ho}(\\mathcal{M}_{sing}^{co})\\simeq \\mathbf{K}_{ex}(\\mathcal{I})$.\n\\end{corollary}\n\n\\begin{proof}\nWe use $\\widetilde{\\mathcal{P}}$ (resp. $\\widetilde{\\mathcal{I}}$) to denote the subcategory of contractible complexes of projective (resp. injective) modules. It is well known that a complex $P\\in \\widetilde{\\mathcal{P}}$ if and only if $P$ is exact and each $\\mathrm{Ker}d_{i}^{P}$ is a projective module; similarly, complexes in $\\widetilde{\\mathcal{I}}$ are characterized. Note that for any chain maps $f$ and $g$, if $g-f$ factors through a complex in $\\widetilde{\\mathcal{P}}$ (or,  a complex in $\\widetilde{\\mathcal{I}}$), then $f$ is chain homotopic to $g$, denoted by $f\\sim g$. Since $ex\\widetilde{\\mathcal{P}}\\cap (ex\\widetilde{\\mathcal{P}})^{\\perp}=\\widetilde{\\mathcal{P}}$ and $ex\\widetilde{\\mathcal{I}}\\cap {^{\\perp}}(ex\\widetilde{\\mathcal{I}}) = \\widetilde{\\mathcal{I}}$, the equivalences hold directly.\n\\end{proof}\n\nLet $F=\\Lambda\\Omega$ and $G=\\Lambda\\Theta$ be functors on $\\mathrm{Ch}(R)$, where $\\Omega$ and  $\\Theta$ are functors from $\\mathrm{Ch}(R)$ to $\\mathrm{Mod}(R)$ such that for any $X\\in \\mathrm{Ch}(R)$, $\\Omega(X)= X_0\/\\mathrm{Im}d_{1}^{X}$ and $\\Theta(X)= \\mathrm{Ker}d_{0}^{X}$. Let $\\Lambda: \\mathrm{Mod}(R)\\rightarrow \\mathrm{Ch}(R)$ be a functor which sends every module to a stalk complex concentrated on degree zero.\n\n\n\\begin{lemma}\\label{lem 2}\nLet $X$, $Y$ be any $R$-complexes, and $f: X\\rightarrow Y$ a monomorphism of complexes. If $f$ is a quasi-isomorphism, then $\\Omega(f)$ is also a monomorphism of $R$-modules.\n\\end{lemma}\n\n\\begin{proof}\nWe consider the following commutative diagram\n$$\\xymatrix{\n0\\ar[r] & \\mathrm{Ker}d_{0}^{X} \/ \\mathrm{Im}d_{1}^{X} \\ar[r]\\ar[d] & X_{0} \/ \\mathrm{Im}d_{1}^{X} \\ar[r]\\ar[d]_{\\Omega(f)}\n& X_{0} \/ \\mathrm{Ker}d_{0}^{X}\\ar[r]\\ar[d] &0\\\\\n0\\ar[r] & \\mathrm{Ker}d_{0}^{Y} \/ \\mathrm{Im}d_{1}^{Y} \\ar[r] & Y_{0} \/ \\mathrm{Im}d_{1}^{Y} \\ar[r] & Y_{0} \/ \\mathrm{Ker}d_{0}^{Y}\\ar[r] &0 }$$\nSince $f$ is a quasi-isomorphism, we have an isomorphism induced by $f$:\n$$\\mathrm{H}_0(f): \\mathrm{H}_0(X)=\\mathrm{Ker}d_{0}^{X} \/ \\mathrm{Im}d_{1}^{X}\\longrightarrow \\mathrm{Ker}d_{0}^{Y} \/ \\mathrm{Im}d_{1}^{Y}=\\mathrm{H}_0(Y).$$\nThe chain map $f$ is monic, then the induced map of modules $X_{0} \/ \\mathrm{Ker}d_{0}^{X}\\cong \\mathrm{Im}d_{0}^{X}\\longrightarrow \\mathrm{Im}d_{0}^{Y}\\cong Y_{0} \/ \\mathrm{Ker}d_{0}^{Y}$ is also monic. Hence, by the ``Five Lemma'' for the above diagram, we get that $\\Omega(f): X_{0} \/ \\mathrm{Im}d_{1}^{X}\\longrightarrow Y_{0} \/ \\mathrm{Im}d_{1}^{Y}$ is a monomorphism. We mention that it is also direct to check injectivity of $\\Omega(f)$ by diagram chasing.\n\\end{proof}\n\nFor model categories $\\mathcal{C}$ and $\\mathcal{D}$, recall that an adjunction $(F, G): \\mathcal{C}\\rightarrow \\mathcal{D}$ is a Quillen adjunction if $F$ is a left Quillen functor, or equivalently $G$ is a right Quillen functor. That is, $F$ preserves cofibrations and trivial cofibrations, or $G$ preserves fibrations and trivial fibrations.\n\n\n\\begin{proposition}\\label{prop 1}\n$(F, G): (\\mathrm{Ch}(R), \\mathcal{M}_{sing}^{ctr})\\rightarrow (\\mathrm{Ch}(R), \\mathcal{M}_{sing}^{co})$ is a Quillen adjunction.\n\\end{proposition}\n\n\\begin{proof}\nLet $X$, $Y$ be any $R$-complexes. It follows from \\cite[Lemma 3.1]{Gil04} that $(\\Omega, \\Lambda): \\mathrm{Ch}(R)\\rightarrow \\mathrm{Mod}(R)$\nand  $(\\Lambda, \\Theta): \\mathrm{Mod}(R)\\rightarrow \\mathrm{Ch}(R)$ are adjunctions. Then we have the following natural isomorphisms:\n$\\mathrm{Hom}_{\\mathrm{Ch}(R)}(F(X), Y)\\cong \\mathrm{Hom}_{R}(\\Omega(X), \\Theta(Y))\\cong \\mathrm{Hom}_{\\mathrm{Ch}(R)}(X, G(Y))$.\nThis implies that $(F, G):  \\mathrm{Ch}(R)\\rightarrow \\mathrm{Ch}(R)$ is an adjunction.\n\nIt suffices to show that $F$ preserves cofibration and trivial cofibration. Let $f: X\\rightarrow Y$ be a cofibration in $\\mathcal{M}_{sing}^{ctr}$, i.e. $f$ is a monomorphism with $\\mathrm{Coker}f \\in ex\\widetilde{\\mathcal{P}}$. This yields that $f$ is a quasi-isomorphism, and by Lemma \\ref{lem 2}, $\\Omega(f)$ is monic. Then, we have an exact sequence $$0\\longrightarrow F(X)\\stackrel{F(f)}\\longrightarrow F(Y)\\longrightarrow F(\\mathrm{Coker}f)\\longrightarrow 0.$$ Since every complex is a cofibrant object in $\\mathcal{M}_{sing}^{co}$, this implies that $F(f)$ is a cofibration.\n\nNow suppose $f: X\\rightarrow Y$ is a trivial cofibration in $\\mathcal{M}_{sing}^{ctr}$, i.e. $f$ is a monomorphism with $\\mathrm{Coker}f \\in ex\\widetilde{\\mathcal{P}}\\cap (ex\\widetilde{\\mathcal{P}})^{\\perp}= \\widetilde{\\mathcal{P}}$. Then we have an exact sequence $$0\\longrightarrow F(X)\\stackrel{F(f)}\\longrightarrow F(Y)\\longrightarrow F(\\mathrm{Coker}f)\\longrightarrow 0.$$\nNote that $\\Omega(\\mathrm{Coker}f)$ is a projective module. For any complex $I\\in ex\\widetilde{\\mathcal{I}}$, it is easy to show that any chain map $F(\\mathrm{Coker}f) = \\Lambda\\Omega(\\mathrm{Coker}f)\\rightarrow I$ is null homotopic, and then $F(\\mathrm{Coker}f)\\in {^{\\perp}}(ex\\widetilde{\\mathcal{I}})$. Thus $F(f)$ is a trivial cofibration in $\\mathcal{M}_{sing}^{co}$. This completes the proof.\n\\end{proof}\n\n\nRecall that a module $M$ is Gorenstein projective if $M$ is a syzygy of a totally acyclic complex of projective modules; and dually, Gorenstein injective modules are defined; see \\cite{EJ00}. We use $\\mathcal{GP}$ and $\\mathcal{GI}$ to denote the classes of Gorenstein projective and Gorenstein injective modules, respectively. It is widely accepted that over a left-Gorenstein ring, $(\\mathcal{GP}, \\mathcal{W})$ and $(\\mathcal{W}, \\mathcal{GI})$ are complete cotorsion pairs, where $\\mathcal{W}$ is the class of modules with finite projective (injective) dimension. In \\cite[Theorem 2.7]{Ren18} we show that the cotorsion pair $(\\mathcal{GP}, \\mathcal{W})$ is cogenerated by a set, i.e. there exists a set $S$ such that $\\mathcal{W} =\\{S\\}^{\\perp}$. This also implies the completeness of $(\\mathcal{GP}, \\mathcal{W})$, and generalizes the Gorenstein projective model structure of $\\mathrm{Mod}(R)$ in \\cite[Theorem 8.3 and 8.6]{Hov02} from Iwanaga-Gorenstein rings to left-Gorenstein rings.\n\n\\begin{lemma}\\label{lem 3}\nLet $X$, $Y$ be complexes in $ex\\widetilde{\\mathcal{P}}$, and $f: X\\rightarrow Y$ a chain map. If $F(f)$ is a weak equivalence in $\\mathcal{M}_{sing}^{co}$, then $f$ is a weak equivalence in $\\mathcal{M}_{sing}^{ctr}$.\n\\end{lemma}\n\n\\begin{proof}\nIn the model category $(\\mathrm{Ch}(R), \\mathcal{M}_{sing}^{ctr})$, we can factor $f: X\\rightarrow Y$ as a trivial cofibration $i: X\\rightarrow Z$ followed by a fibration $p: Z\\rightarrow Y$. By Proposition \\ref{prop 1}, $F(i)$ is a trivial cofibration in $\\mathcal{M}_{sing}^{co}$, and then $F(i)$ is a weak equivalence. Then $F(f) = F(p)F(i)$ is a weak equivalence if and only if so is $F(p)$.\n\nLet $L = \\mathrm{Coker}i$ and $K =  \\mathrm{Ker}p$. It follows from the exact sequence $0\\longrightarrow X\\stackrel{i}\\longrightarrow Z\\longrightarrow L\\longrightarrow 0$ that $Z\\in ex\\widetilde{\\mathcal{P}}$, where  $X\\in ex\\widetilde{\\mathcal{P}}$ and $L\\in ex\\widetilde{\\mathcal{P}}\\cap (ex\\widetilde{\\mathcal{P}})^{\\perp} = \\widetilde{\\mathcal{P}}$. Moreover, it follows from the exact sequence $0\\longrightarrow K\\longrightarrow Z\\stackrel{p}\\longrightarrow Y\\longrightarrow 0$ that $K\\in ex\\widetilde{\\mathcal{P}}$.\n\nLet $M$ be any Gorenstein injective module. Then there exists a totally acyclic complex of injective module, saying $I$, such that $M\\cong \\Theta(I)$. It follows from \\cite[Lemma 4.2]{Gil08} that there is an isomorphism\n$$\\mathrm{Ext}^{1}_{\\mathrm{Ch}(R)}(F(K), I) = \\mathrm{Ext}^{1}_{\\mathrm{Ch}(R)}(\\Lambda\\Omega(K), I)\\cong \\mathrm{Ext}^{1}_{R}(\\Omega(K), \\Theta(I)).$$\nSince $F(p)$ is a weak equivalence, $F(K) = \\mathrm{Ker}(F(p)) \\in {^{\\perp}(ex\\widetilde{\\mathcal{I}})}$. For any Gorenstein injective module $M$, it yields that $\\mathrm{Ext}^{1}_{R}(\\Omega(K), M) = 0$. Since $(\\mathcal{W}, \\mathcal{GI})$ is a cotorsion pair, we get that $\\Omega(K)$ is a module of finite projective dimension. Moreover, $\\Omega(K)$ is a Gorenstein projective module since $R$ is a left-Gorenstein ring and $K\\in ex\\widetilde{\\mathcal{P}}$ is a totally acyclic complex of projective modules. By \\cite[Proposition 10.2.3]{EJ00}, the projective dimension of any Gorenstein projective module is either zero or infinity, so $\\Omega(K)$ is a projective module. Considering exact sequences $0\\rightarrow \\mathrm{Ker}d_{i}^{K}\\rightarrow K_{i}\\rightarrow \\mathrm{Ker}d_{i-1}^{K}\\rightarrow 0$ inductively, we can prove each syzygy of $K$ is projective, that is, $K$ is a complex in $\\widetilde{\\mathcal{P}}$. This implies that $p: Z\\rightarrow Y$ is a trivial fibration, and hence $f = pi$ is a weak equivalence, as desired.\n\\end{proof}\n\n\\begin{lemma}\\label{lem 4}\nLet $Y$ be an exact complex of injective $R$-modules. Then $\\varepsilon: FG(Y)\\rightarrow Y$ is a weak equivalence in $\\mathcal{M}_{sing}^{co}$, where $\\varepsilon$ is the counit of the adjoint pair $(F, G)$.\n\\end{lemma}\n\n\\begin{proof}\nFor $Y$, $G(Y)=\\Lambda\\Theta(Y)= \\cdots\\rightarrow 0 \\rightarrow \\mathrm{Ker}d_0^Y\\rightarrow 0\\rightarrow\\cdots$ is a stalk complex with $\\mathrm{Ker}d_0^Y$ concentrated in degree zero. It is easy to see that $FG(Y) = G(Y)$. Then the map $\\varepsilon: FG(Y)\\rightarrow Y$ is given by a natural embedding $\\varepsilon_0: \\mathrm{Ker}d_0^Y\\rightarrow Y_0$ and $\\varepsilon_i =0$ for any $i\\neq 0$.\nLet $C= \\mathrm{Coker}\\varepsilon$. Then $C=\\cdots\\longrightarrow Y_2\\stackrel{d_2^Y}\\longrightarrow Y_1\\stackrel{0}\\longrightarrow \\mathrm{Im}d_0^Y\\stackrel{\\iota}\\longrightarrow Y_{-1}\\stackrel{d_{-1}^Y}\\longrightarrow Y_{-2}\\longrightarrow\\cdots$, where $\\iota$ is an embedding. Let $Y_{\\sqsupset}=\\cdots\\rightarrow Y_2\\stackrel{d_2^Y}\\rightarrow Y_1\\rightarrow 0$ be a hard truncation, $D=0\\rightarrow \\mathrm{Im}d_0^Y\\stackrel{\\iota}\\rightarrow Y_{-1}\\stackrel{d_{-1}^Y}\\rightarrow Y_{-2}\\rightarrow\\cdots$. Then there is an exact sequence of complexes $0\\longrightarrow Y_{\\sqsupset}\\longrightarrow C\\longrightarrow D\\longrightarrow 0$.\n\nLet $E$ be any $R$-complex in $ex\\widetilde{\\mathcal{I}}$. Since $R$ is left-Gorenstein, then $E$ is totally acyclic, and for any $Y_i$, $\\mathrm{Hom}_{R}(Y_i, E)$ is an exact complex. By \\cite[Lemma 2.4]{CFH06}, the complex $\\mathrm{Hom}_{R}(Y_{\\sqsupset}, E)$ is exact.\nNote that $D$ is an exact sequence, and then $\\mathrm{Hom}_{R}(D, E_i)$ is an exact complex for any $i\\in \\mathbb{Z}$. By \\cite[Lemma 2.5]{CFH06}, the complex $\\mathrm{Hom}_{R}(D, E)$ is exact. Moreover, it follows from the short exact sequence\n$$0\\longrightarrow \\mathrm{Hom}_{R}(D, E)\\longrightarrow \\mathrm{Hom}_{R}(C, E)\\longrightarrow \\mathrm{Hom}_{R}(Y_{\\sqsupset}D, E)\\longrightarrow 0$$\nthat the complex $\\mathrm{Hom}_{R}(C, E)$ is exact. This implies that every map from $C$ to any complex in $ex\\widetilde{\\mathcal{I}}$ is null homotopic. Then $C\\in {^{\\perp}}ex\\widetilde{\\mathcal{I}}$. Hence, $\\varepsilon: FG(Y)\\rightarrow Y$ is a trivial cofibration in $\\mathcal{M}_{sing}^{co}$, and moreover, $\\varepsilon$ is a weak equivalence.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem 5}\nLet $Y$ be an exact complex of injective $R$-modules. Then $F(q): FQG(Y)\\rightarrow FG(Y)$ is a weak equivalence in $\\mathcal{M}_{sing}^{co}$, where $q: QG(Y)\\rightarrow G(Y)$ is a cofibrant replacement in the model category $(\\mathrm{Ch}(R), \\mathcal{M}_{sing}^{ctr})$.\n\\end{lemma}\n\n\\begin{proof}\nFor $Y$, $G(Y) = FG(Y) = \\cdots\\rightarrow 0 \\rightarrow \\mathrm{Ker}d_0^Y\\rightarrow 0\\rightarrow\\cdots$. By the completeness of the cotorsion pair $(\\mathcal{GP}, \\mathcal{W})$, there is an exact sequence of $R$-modules $0\\rightarrow W\\rightarrow M\\rightarrow \\mathrm{Ker}d_0^Y\\rightarrow0$ with $M\\in \\mathcal{GP}$ and $W\\in \\mathcal{W}$. Consider the totally acyclic complex $P$ of $M$, we have a short exact sequence $0\\rightarrow K\\rightarrow P\\stackrel{q}\\rightarrow G(Y)\\rightarrow 0$, see the following diagram\n$$\\xymatrix@C=20pt@R=10pt{\nK=\\cdots \\ar[r] &P_{1}\\ar[dd]_{=}\\ar[r]^{} &K_{0}\\ar[dd]\\ar@{-->}[rd]^{\\pi}\\ar[rr]^{} & &P_{-1}\\ar[r]\\ar[dd]_{=}&P_{-2}\\ar[r]\\ar[dd]_{=}&\\cdots \\\\\n & & & W\\ar@{-->}[ur]\\ar@{-->}[dd]^{}\\\\\nP= \\cdots \\ar[r] &P_{1}\\ar[dd]_{}\\ar[r]^{} &P_{0}\\ar[dd]\\ar@{-->}[rd]^{}\\ar[rr]^{} & &P_{-1}\\ar[r]\\ar[dd]_{}&P_{-2}\\ar[r]\\ar[dd]_{}&\\cdots\\\\\n & & & M\\ar@{-->}[ur]\\ar@{-->}[dd]^{}\\\\\nG(Y) = \\cdots \\ar[r] &0\\ar[r]^{} &\\mathrm{Ker}d_0^Y\\ar@{==}[rd]\\ar[rr]^{} & &0\\ar[r]_{} &0\\ar[r]&\\cdots\\\\\n & & & \\mathrm{Ker}d_0^Y \\ar@{-->}[ur]\n  }$$\n\nLet $K_{0\\supset}= \\cdots\\rightarrow P_2\\rightarrow P_1\\rightarrow \\mathrm{Ker}\\pi\\rightarrow 0$ and $K_{\\subset0}= 0\\rightarrow W\\rightarrow P_{-1}\\rightarrow P_{-2}\\rightarrow\\cdots$. Then there is a short exact sequence of complexes $0\\longrightarrow K_{0\\supset}\\longrightarrow K\\longrightarrow K_{\\subset0}\\longrightarrow 0.$\nLet $T$ be any complex in $ex\\widetilde{\\mathcal{P}}$. Note that $T$ is totally acyclic. Then it follows from\n\\cite[Lemma 2.5]{CFH06} that the complex $\\mathrm{Hom}_{R}(T, K_{\\subset0})$ is exact, and this implies that $K_{\\subset0}\\in (ex\\widetilde{\\mathcal{P}})^{\\perp}$. Note that $K_{0\\supset}$ is an exact complex. For any morphism\n$f: T\\rightarrow K_{0\\supset}$, we consider the following diagram:\n$$\\xymatrix@C=40pt{\n\\cdots \\ar[r] &T_{2}\\ar[d]_{f_2}\\ar[r]^{} &T_{1}\\ar[d]_{f_1}\\ar[r]^{}\\ar@{-->}[ld]_{s_1} &T_{0}\\ar[r]\\ar[d]_{f_{0}}\\ar@{-->}[ld]_{s_0}\n&T_{-1}\\ar[r]\\ar[d]^{}\\ar@{-->}[ld]_{s_{-1}} &\\cdots \\\\\n\\cdots \\ar[r] &P_{2}\\ar[r] &P_{1}\\ar[r]^{}  &\\mathrm{Ker}\\pi \\ar[r]&0\\ar[r]&\\cdots\n  }$$\nLet $s_i=0$ for any $i< 0$. Since $d_1^K: P_1\\rightarrow \\mathrm{Ker}\\pi$ is an epic and $T_0$ is a projective module, there is a map\n$s_0: T_0\\rightarrow P_1$ such that $f_0 = d_{1}^{K}s_0$. Since $d_{1}^{K}(f_{1} - s_{0}d_{1}^{T}) = d_{1}^{K}f_{1} - d_{1}^{K}s_{0}d_{1}^{T} = d_{1}^{K}f_{1} - f_{0}d_{1}^{T} = 0$, then $f_{1} - s_{0}d_{1}^{T}: T_1\\rightarrow \\mathrm{Ker}d_{1}^{K}$, and there exists a map $s_1: T_1\\rightarrow P_2$ such that $f_{1} - s_{0}d_{1}^{T} = d_{2}^{K}s_{1}$. Analogous to comparison theorem, we inductively get homotopy maps $\\{s_i\\}$ such that $f$ is null homotopic. Then $K_{0\\supset}\\in (ex\\widetilde{\\mathcal{P}})^{\\perp}$. Thus, we have $K\\in (ex\\widetilde{\\mathcal{P}})^{\\perp}$.\nNote that for any object in the model category $(\\mathrm{Ch}(R), \\mathcal{M}_{sing}^{ctr})$, its cofibrant replacement is precisely a special $ex\\widetilde{\\mathcal{P}}$-precover. Then it follows from the short exact sequence $0\\rightarrow K\\rightarrow P\\stackrel{q}\\rightarrow G(Y)\\rightarrow 0$ that $P$ is a cofibrant replacement of $G(Y)$, and we can set $QG(Y) = P$.\n\nNote that $F(K)= \\cdots \\rightarrow 0\\rightarrow W\\rightarrow 0\\rightarrow\\cdots$. Since $W$ is a module of finite projective dimension, for any complex $E\\in ex\\widetilde{\\mathcal{I}}$, $\\mathrm{Hom}_{R}(W, E)$ is exact. This implies that $F(K)\\in {^{\\perp}}(ex\\widetilde{\\mathcal{I}})$.\nFor $F(K)$, there is an exact sequence $0\\rightarrow F(K)\\rightarrow I\\rightarrow L\\rightarrow 0$ with $I\\in ex\\widetilde{\\mathcal{I}}$ and $L\\in {^{\\perp}}(ex\\widetilde{\\mathcal{I}})$. We consider the following push-out diagram:\n$$\\xymatrix@C=20pt@R=20pt{ & 0\\ar[d] & 0\\ar[d] \\\\\n0 \\ar[r]^{}  &F(K) \\ar[d] \\ar[r] & F(P) \\ar@{-->}[d]_{i}\n  \\ar[r]^{F(q)} &FG(Y) \\ar@{=}[d]  \\ar[r] &0 \\\\\n0 \\ar[r] & I \\ar@{-->}[r] \\ar[d] & J \\ar[r]^{p} \\ar[d] & GF(Y) \\ar[r] & 0\\\\\n  & L \\ar[d] \\ar@{=}[r] & L\\ar[d]\\\\\n  & 0 & 0\n  }$$\nIt is clear that $i$ is a trivial cofibration. By the left column, we have $I\\in {^{\\perp}}(ex\\widetilde{\\mathcal{I}})$. Then $I\\in ex\\widetilde{\\mathcal{I}}\\cap {^{\\perp}}(ex\\widetilde{\\mathcal{I}})$, and $p$ is a trivial fibration. Hence $F(q) = pi$ is a weak equivalence in $\\mathcal{M}_{sing}^{co}$.\n\\end{proof}\n\n\\subsection*{The proof of the theorem}\nIt follows from Proposition \\ref{prop 1} that $(F, G): (\\mathrm{Ch}(R), \\mathcal{M}_{sing}^{ctr})\\longrightarrow (\\mathrm{Ch}(R), \\mathcal{M}_{sing}^{co})$ is a Quillen adjunction.\nBy \\cite[Corollary 1.3.16]{Hov99}, there is a useful criterion for checking the given Quillen adjunction is a Quillen equivalence. Specifically, we need to show that $F$ reflects weak equivalences between cofibrant objects in $\\mathcal{M}_{sing}^{ctr}$ (i.e. complexes in $ex\\widetilde{\\mathcal{P}}$), see Lemma \\ref{lem 3}; moreover, for every fibrant object $Y$ in $\\mathcal{M}_{sing}^{co}$ (i.e. $Y\\in ex\\widetilde{\\mathcal{I}}$), we need to show that the composition $FQG(Y)\\stackrel{F(q)}\\rightarrow FG(Y)\\stackrel{\\varepsilon}\\rightarrow Y$ is a weak equivalence, where $\\varepsilon$ is the counit of the adjunction $(F, G)$, and $q: QG(Y)\\rightarrow G(Y)$ is a cofibrant replacement of $G(Y)$, see Lemma \\ref{lem 4} and \\ref{lem 5}.\n\n\n\\begin{ack*}\nThe author is supported by National Natural Science Foundation of China (11871125), Natural Science Foundation of Chongqing (cstc2018jcyjAX0541) and the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN201800509).\n\\end{ack*}\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\nIntermediate mass (IM) stars, namely stars whose mass is in the 2 to 8\nM$_\\odot$ range, are crucial in studies of star formation because they\nconstitute the link between low- and high-mass stars\n\\citep{DiF97,Man97,Man00}, and, therefore, can help to understand if\nand how much different are the processes at work in the two ends.  On\nthe one hand, low mass stars are can be formed isolated or in loose\ngroups of few objects per cubic parsec \\citep{Gom93}, while high-mass\nstars are usually found to form in tight clusters\n\\citep[e.g.][]{Hil98}. IM stars, on the other hand, are also found in\nclusters \\citep[e.g.][]{Tes98,Ner07,Fue07}, with a smooth transition\ntowards the low mass star, loose cluster regime for star masses around\n3.5 M$_\\odot$ \\citep{Tes99}. \\citet{Tes99} also concluded that IM\nstars mark the transition from low density aggregates of $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}}$ 10\nstars per cubic parsec of T Tauri stars to dense clusters of $\\gtrsim$\n10$^3$ stars per cubic parsec associated with early-type stars. In\nagreement with the different observed environments, several authors\nhave proposed that high mass stars are formed by coalescence of lower\nmass stars, whereas other authors favor the ``monolithic''\nformation (see for example the recent review by \\citet{Beu07}). In\nthis context, the IM stars study can greatly help the debate.  Indeed,\ndue to their intermediate position, the study of IM protostars will\nprovide crucial information on the transition between the two\nformation regimes as well as on the limits of the low mass and high\nmass formation scenarios.  Finally, IM stars are among the dominant\nsources of the Inter-Stellar FUV field \\citep[e.g.][]{Hab68,Gon75},\nwhich regulates the phases of the ISM in the Galaxy, and, in turns,\nthe overall Galaxy star formation process and history. Despite the\nfar-reaching importance of IM stars, very little is known about the\nformation and first evolutionary stages of these stars. The situation\nis so bad that to date we do not have a satisfying sample of Class 0\nIM objects, namely objects representing the first phases of stellar\nformation, where the protostar is embedded in its envelope and its\nluminosity is dominated by the accretion luminosity, nor a systematic\nstudy of their physical structure, as it is the case for low mass\nClass 0 sources \\citep[e.g.][]{Cec07,Dif07}.  This article is the\nfirst of a series that aims to fill this gap in our knowledge.\n\nIn this context, the Orion Molecular Cloud 2 (OMC2), the closest known\nregion where high to low mass star formation is going on, represents a\nprecious laboratory for these studies. Observed first by\n\\citet{Gat74}, OMC2 is located 15' ($\\sim$ 2 pc) North of the Orion\nnebula. It has a filamentary structure, elongated in the direction\nnorth-south, with active star formation concentrated in the central\nand densest region, shielded from the UV radiation from newly formed\nOB stars \\citep{Joh90}. The mass of the cloud amounts to about 1500\nM$_\\odot$ \\citep{Mez90}. Several extensive studies have shown that\nOMC2 is a rich star forming region, which harbors several young\nprotostars, including several Class 0 candidates\n\\citep{Ali95,Chi97,Lis98,Johnstone99,Rei99}.  Observations of\nmolecular lines have revealed several outflows emanating from the\nyoung protostars in the region. Many studies have focused on the\noutflows \\citep[e.g.][]{Wil03} and their impact on the cloud\n\\citep{Aso00,Wu05}. Only few of these studies, in contrast, have\naddressed the problem of the chemical structure of the forming stars\nin OMC2 \\citep{Johnstone03}.\n\nAmong the several protostars in OMC2, FIR4 stands out as the brightest\nsubmillimeter source \\citep{Mez90}. Located almost at the center of\nthe cloud, FIR4 is also a bright IRAS source and a VLA radio source\n\\citep{Rei99}. All these characteristics led \\citet{Rei99} to define\nFIR4 ``a bona fide Class 0 source''. The FIR4 integrated luminosity\nwas estimated to be about 400 L$_\\odot$ and the envelope mass is about\n35 M$_\\odot$. Such values led to identify FIR4 as an {\\it intermediate\n  mass protostar} \\citep{Johnstone03}. Because of its vicinity and its\nrelatively bright molecular lines, FIR4 is an ideal source for a\ndetailed study of the physical and chemical structure of an IM\nprotostar. Existing dust continuum and molecular line observations\npoint to an envelope with at least two components: a warm component\nwith an average temperature of about 40 K and a colder component at\nabout 15 K \\citep{Mez90,Johnstone03}.  \\citet{Jor06} modeled the 850\n$\\mu$m SCUBA map towards this source to reconstruct its temperature\nand density profiles. Based on the observed CO and H$_2$CO millimeter\nline emission, \\citet{Jor06} concluded that the FIR4 envelope is\nilluminated by an external FUV field amounting to $1\\times10^{4}$\ntimes the Interstellar FUV field. However, their interpretation\nsuffers of some contradictions emphasized by the same authors. For\nexample, such an intense FUV field would heat up the whole envelope to\na temperature larger than 25 K, the CO freezing temperature \\citep{Obe05}, in contradiction with the measured average CO\nabundance, ten times lower than the canonical value, which would\nrather testify for a large CO-frozen region \\citep{Jor06}. In\naddition, the maps of the fine structure lines of the O and C$^+$\natoms together with the CO 1-0 line led \\citet{Her97} to conclude that\nthe OMC2 region is illuminated by a FUV field 500 times the\nInterstellar field.\n\nGiven this puzzling situation, we decided to derive again the\ntemperature and density profiles of FIR4 by taking into account more\ndata than those considered by \\citet{Jor06} (\\S\n\\ref{sec:dust-dens-temp}). Using the derived dust temperature and\ndensity profiles, we then computed the gas temperature profile, by\nequating the heating and cooling terms across the envelope (\\S\n\\ref{sec:gas-temp-prof}). As shown by several authors\n\\citep[e.g.][]{Cec96,Dot97}, the gas cooling in protostellar envelopes\nis dominated by the emission from the rotational lines of CO and, more\nimportant, H$_2$O together with the fine structure lines of\nOI. Actually, water is a key molecule in the gas thermal balance for\ntwo reasons. First, in the warm regions where the grain mantles\nsublimate, it is the most abundant molecule; second, given its\nrelatively large dipole moment, water is a very powerful line emitter\nand, consequently, gas coolant.  Given its major role in the\nprediction of the gas temperature profile, we discuss the dependence\nof the derived gas temperature on the assumed water abundance profile,\nwhich is poorly known. Not surprising, FIR4 is in fact one of the few\nsources where the full spectrum between 50 and 2000 GHz is planned to\nbe observed at high spectral resolution with the Heterodyne Instrument\nfor the Far Infrared (HIFI) on board Herschel\n(http:\/\/herschel.esac.esa.int\/), to be launched in 2009. HSO, and\nspecifically the high resolution interferometer HIFI, will allow to\nobserve the water lines in the 500 to 2000 GHz range with\nunprecedented spectral and spatial resolution. Motivated by the\nHerschel mission, we report the predicted water line spectrum for the\ndifferent assumed water abundance profiles, and discuss the\nobservability by HIFI and PACS (\\S\n\\ref{sec:predicted-water-line}).  Section \\ref{sec:conclusion}\nconcludes the article.\n\n\\section{Dust density and temperature profiles}\\label{sec:dust-dens-temp}\nIn this section, we derive the dust density and temperature profiles\nby modelling the 350, 450 and 850 $\\mu$m maps of the region, plus the\nSpectral Energy Distribution (SED) from the millimeter to the\nMid-Infrared (MIR) wavelength range. We first describe the\nobservations we used in our analysis (\\S \\ref{sec:cont-emiss-data})\nand then the modeling (\\S \\ref{sec:cont-emiss-dusty}) and the result\nof the modeling (\\S \\ref{sec:dust-results}).\n\n\\subsection{Continuum emission: observational data}\\label{sec:cont-emiss-data}\nIn our analysis, we use the maps of the continuum emission at 850, 450\nand 350 $\\mu m$ obtained at JCMT and CSO respectively. In addition, we\ntake into account the Spectral Energy Distribution (SED) of FIR4 from\n24 to 850 $\\mu$m obtained considering also the IRAS and Spitzer\nobservations.\n\n\\noindent\n{\\it a) 850, 450 and 350 $\\mu m$ maps}\\\\\nWe retrieved the 450 and 850 $\\mu$m maps obtained by \\citet{Johnstone99}\n at the 15 m James Clerk Maxwell Telescope (JCMT) with the\nfocal-plane instrument SCUBA (Submillimeter Common-User Bolometer\nArray). The spatial resolution of the maps is 7.5$\"$ and 14.8$\"$ at\n450 and 850 $\\mu$m respectively. The calibration uncertainty and\nnoise levels are estimated by those authors $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}}$ 10\\% and 0.04\nJy beam$^{-1}$ at 850 $\\mu$m and $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}}$ 30\\% and 0.3 Jy\nbeam$^{-1}$ at 450 $\\mu$m, respectively. The 350 $\\mu$m map was\nobtained by \\citet{Lis98} at the 10.4 m telescope of the Caltech\nSubmillimeter Observatory (CSO). The instrument used was the bolometer\ncamera SHARC. The resolution of the map is 12$\"$. The calibration\nuncertainty has been evaluated $\\sim$ 25\\%-30\\%. The three maps are\nreported in Fig. \\ref{maps}.\n\\begin{figure*} \\centering\n\\rotatebox{270}{\\includegraphics[width=10cm]{maps.ps}}\n\\caption{Continuum emission maps around OMC2-FIR4 at 850 $\\mu$m (left\n  panel), 450 $\\mu$m (middle panel) and 350 $\\mu$m (right\n  panel). The contours mark the continuum flux from 5 \\% to 75 \\% of\n  the peak emission by steps of 5 \\%. The hatched regions have been\n  excluded when computing the brightness profile of the FIR4 envelope\n  (see text). The position of the three protostars in the regions,\n  FIR3, FIR4 and FIR 5 are marked in the central panel\n  figure. \\label{maps}}\n\\end{figure*}\nThey show the envelope surrounding\/forming FIR4 which extends for\nabout 20$\"$, but also the presence of two sources: FIR3, 25$\"$ North,\nand FIR5, 25$\"$ South. To evaluate the continuum brightness profile of\nthe FIR4 envelope, we averaged the continuum flux over ring at the\nsame distance from the FIR4 center, excluding the regions contaminated\nby the presence of FIR3 and FIR5 (dashed regions in\nFig. \\ref{maps}). The resulting brightness profiles are shown in\nFig. \\ref{fit}. Note that in the analysis of the envelope emission (\\S\n\\ref{sec:cont-emiss-dusty}) we subtracted the cloud contribution,\nestimated to be $\\sim$ 0.001, $\\sim$ 0.03 and $\\sim$ 0.05 Jy\narcsec$^{-2}$ at 850, 450 and 350 $\\mu$m respectively. Furthermore, in\norder to take into account that the SCUBA and SHARC maps were obtained\nwith the chop throw of 65$\"$ and 90-120$\"$ respectively, we only\nconsidered the inner 60$\"$ in our analysis.\n\n\\noindent\n{\\it b) SED}\\\\\nThe SED points at 850, 450 and 350 $\\mu$m, shown in Fig. \\ref{fit},\nwere obtained integrating the continuum emission over the envelope. We\nattributed an uncertainty of $\\sim$ 30 \\% to them to account for the\nuncertainty in the envelope size. We also considered the IRAS fluxes\nat 60 and 100 $\\mu$m, respectively, extracted from the IRAS maps at\nthese wavelengths.  The evaluation of the fluxes was done using the\nmethod previously employed for the maps at 850, 450 and 350 $\\mu$m,\nnamely excluding the same regions (dashed regions in Fig. \\ref{maps})\nto limit the contamination by FIR3 and FIR5 and integrating over the\nrings.  We also subtracted the cloud contribution, estimated to be\n$\\sim$ 0.06 and $\\sim$ 0.07 Jy arcsec$^{-2}$ at 60 and 100 $\\mu m$\nrespectively. To account for the possible contamination of FIR3 and\nFIR5 due to the large beam of IRAS and the non-sphericity of the\nsource, we took an uncertainty of 50 \\% on the fluxes.  Finally, we\nalso considered the integrated flux at 24 $\\mu$m extracted from the\nSpitzer Space Telescope's Multiband Imaging Photometer (MIPS) maps. To\nthis end, we retrieved the observations from the Spitzer Science\narchive ({\\it http:\/\/ssc.spitzer.caltech.edu\/archanaly\/}). The\nobservations were obtained the 6th October 2006 as part of the Program\n``Infrared Properties of Edge-on Young Stellar Object Disks'' (AOR:\n30765, PI: Karl Stapelfeldt).  The data reduction was performed using\nthe pipeline S16.0.1. The flux, ($5.0\\pm2.5$ Jy), in Fig. \\ref{fit}\nwas obtained by integration over a 15$\"$ radius.\n\n\\subsection{Continuum emission: modeling}\\label{sec:cont-emiss-dusty}\nTo derive the dust physical structure, namely the dust temperature and\ndensity profiles, we used the 1D radiative transfer code DUSTY\n\\citep{Ive97}.  Briefly, giving as input the temperature of the\ncentral object and a dust density profile, DUSTY computes\nself-consistently the dust temperature profile and the dust\nemission. The comparison between the computed 350, 450, 850 $\\mu$m\nbrightness profiles (namely the brightness versus the distance from\nthe center of the envelope) and SED with the observed profiles and SED\n(see previous paragraph) makes it possible to constrain the density\nprofile and, consequently, the temperature profile of the envelope.\n\nTo be compared with the observations, the theoretical emission is\nconvolved with the beam pattern of the telescope. Following the\nrecommendations for the relevant telescope, the beam is assumed to be\na combination of gaussian curves: at 850 $\\mu$m, we use HPBWs of\n14.5$\"$, 60$\"$, and 120$\"$, with amplitudes of 0.976, 0.022, and 0.002\nrespectively; at 450 $\\mu$m, the HPBWs are 8$\"$, 30$\"$, and 120$\"$\nwith amplitude ratios of 0.934, 0.06, and 0.006, respectively\n\\citep{San01}; at 350 $\\mu$m, we use HPBWs of 12$\"$ and 22$\"$, with\namplitude ratios of 0.7, 0.3, respectively \\citep{Hun96}.\n\nWe assumed that the envelope density follows a power law:\n\\begin{equation} \\label{density_power_law}\n  n(r)=n_0 \\times \\left( \\frac{r_0}{r} \\right)^\\alpha\n\\end{equation}\nwhere the power law index, $\\alpha$, is a free parameter of the model,\nas well as the density $n_0$, the density at $r_0$. Besides, the\nenvelope starts at a radius R$_{in}$ and extends up to R$_{out}$. Both\nR$_{in}$ and R$_{out}$ are free parameters of the model. The last\ninput to DUSTY is the temperature of the central source, T$_*$, here\nassumed to be 5000 K. We verified that the choice of this last\nparameter does not influence the results.  Finally, the opacity of the\ndust as function of the wavelength is another parameter of\nDUSTY. Following numerous previous studies\n\\citep{Van99,Eva01,Shi02,You03}, we adopted the dust opacity\ncalculated by \\citet{Oss94}, specifically their OH5 dust model, which\nrefers to grains coated by ice.\n\nIn summary, the output of DUSTY depends on $\\alpha$, $n_0$, R$_{in}$\nand R$_{out}$. In practice, the DUSTY input parameters are the power\nlaw index, $\\alpha$, the optical thickness at 100 $\\mu$m,\n$\\tau_{100}$, the ratio between the inner and outer radius, Y\n(=R$_{out}$\/R$_{in}$) and the temperature at the inner radius\nT$_{in}$. The optical thickness is, in turn, proportional to the dust\ncolumn density which depends on $n_0$ and the physical thickness of\nthe envelope.  Note that, since the beam of the available maps are\nrelatively large ($\\geq 7.5\"$ which corresponds to a linear length of\n$\\geq 3300$ AU), the inner region of the envelope is relatively\nunconstrained by the available observational data. In practice, we\nobtain a lower limit to T$_{in}$ of 300 K: any larger value would give\nsimilar results.  Finally, as explained in \\citet{Ive97}, DUSTY gives\nscaleless results (which makes it very powerful because the same grid\nof models can be applied to different sources). This means that to\ncompare the DUSTY output with actual observations, it is necessary to\nscale the output by the source bolometric luminosity L$_{bol}$ and the\ndistance. Note that the bolometric luminosity is in fact estimated by\nintegrating the emission over the full spectrum. By definition, this\ncan only be done when the entire SED is known. This is exactly one of\nthe outputs of the modeling. So we re-evaluated the luminosity of FIR4\niteratively from the best fit model, by minimizing the\n$\\chi^2$\\footnote{Note that, in the case of OMC2-FIR4, integrating the\n  model SED gives the same results than integrating under the observed\n  SED.}. We anticipate here that the new value is 1000 rather than 400\nL$_\\odot$, where we assumed the most recent estimation of the\ndistance, namely\n($437\\pm19$) pc \\citep{Hir07}.\\\\\n\nWe run a grid of models to cover the parameter space as reported in\nTable \\ref{DUSTY_input}. The same grid of models were run for four\nvalues of the illuminating FUV field : G$_0$ = 1, 10, 100 and 1000.\nIn all cases, we used the Inter-Stellar Radiation Field (ISRF)\nconstructed by \\citet{Eva01} : combination of the radiation field\nintroduced by \\citet{Bla94} with that of \\citet{Dra78}.  Note that,\nsince DUSTY makes the assumption of isotropic scattering, the computed\nMIR emission is largely overestimated in presence of strong\nexternal fields (Elitzur, private communication). To solve this\nproblem, we followed the suggestion by \\citet{You05} to neglect the\nscattering, artificially putting it to zero.\n\\begin{table}[h] \\centering\n\\begin{tabular}{|ll|} \\hline \\hline \nParameter   & Range    \\\\ \\hline\n$\\alpha$    & 0.2-3.9  \\\\ \nY           & 100-2200 \\\\ \n$\\tau_{100}$ & 0.1-4.6  \\\\\nT$_{in}$     & 300 K   \\\\\nT$_*$       & 5000 K   \\\\ \\hline \n\\end{tabular}\n\\caption{Range of the input parameters to DUSTY covered in the present\n  study. The range of the $\\alpha$, Y and $\\tau_{100}$ parameters\n  is covered by increasing by 20\\% their respective value at each\n  step of the grid. Note that T$_{in}$ and T$_{_*}$ are kept fixed\n  as they do not influence the results (see text). \\label{DUSTY_input}}\n\\end{table}\nThe best fit model has been found minimizing the $\\chi^2$ with an\niterated two-steps procedure.  First, we use the observed brightness\nprofiles at 350, 450 and 850 $\\mu m$ to constrain $\\alpha$ and Y,\nassuming a value for $\\tau_{100}$.  Second, we constrain the \noptical thickness $\\tau_{100}$ by comparing the computed and\nobserved SED, assuming the $\\alpha$ and Y of the previous step. The\nnew $\\tau_{100}$ is used for a new iteration and so on. In practice,\nthe iteration converges in two steps. This is because the normalized\nbrightness profiles very weakly depend on $\\tau_{100}$, while they\nvery much depend on the sizes of the envelope and on the slope of the\ndensity profile (see also \\citet{Jor02} and \\citet{Sch02}). On the\ncontrary, the optical thickness depends mostly on the\nabsolute column density of the envelope, constrained by the SED.\n\n\n\\subsection{Results}\\label{sec:dust-results} \n We run four grids of models, as discussed separately below: a)\n  with a standard illumination FUV field (G$_o$=1) and b) with a\n  10,100,1000 times enhanced field (G$_o$=10,100,1000) (see\n  Introduction). In paragraph c), we also discuss why larger G$_o$\n  were not considered, and in paragraph d) we summarize the results.\\\\\n\n\\noindent\n{\\it a) G$_o$=1}\\\\\nTable \\ref{best_fit_no_ISRF} presents the set of parameters $\\alpha$,\nY and $\\tau_{100}$, which better reproduce the observations \nassuming G$_o$=1. Figure \\ref{fit} shows the relevant derived\nbrightness profiles and SED against the observed ones. Figure\n\\ref{fig:chi2-contour} shows the $\\chi^2$ contours plots obtained by\nconsidering separately the brightness profiles at 350, 450 and 850\n$\\mu m$, and by combining the three profiles together. Figure\n\\ref{X2SED} shows the $\\chi^2$ dependence on the $\\tau_{100}$\nparameter.\n\\begin{table}[h] \\centering\n\\begin{tabular}{l|ccc|cc} \\hline \\hline \nObservation  & $\\alpha$ & Y & $\\tau_{100}$ & $\\chi _{red}^{2}$ & $\\nu$\\\\ \\hline\n850 $\\mu$m profile  & 1.4 & 160 & -    & 0.72 & 10 \\\\\n450 $\\mu$m profile  & 0.6 & 120 & -    & 0.63 & 10 \\\\\n350 $\\mu$m profile  & 0.5 & 170 & -    & 0.47 & 10 \\\\ \nAll profiles        & 0.6 & 120 & -    & 1.24 & 36 \\\\ \nSED                 & -   & -   & 0.6  & 0.55 &  3 \\\\ \\hline\n\\end{tabular}\n\\caption{Best fit parameters for the case G$_o$=1. Note that $\\chi\n  _{red}^{2} = \\chi ^{2}\/\\nu$ where $\\nu$ is the number of degrees of\n  freedom. The first line reports the best fit obtained using only the\n  850 $\\mu$m brightness profile; second line, using the 450 $\\mu$m\n  brightness profile; third line, using the 350 $\\mu$m brightness\n  profile; fourth line gives the best fit using the three profiles;\n  the last line gives the best fit using the\n  SED.\\label{best_fit_no_ISRF}}\n\\end{table}\n\\begin{figure*} \\centering\n\\rotatebox{90}{\\includegraphics[width=12cm]{FIR4_fit.ps}}\n\\caption{Observed brightness profiles at 350 (upper left panel), 450\n  (upper right panel), 850 $\\mu$m (lower left panel) and SED (lower\n  right panel).  The curves report the best fit obtained in the two\n  cases G$_o$=1 (solid line) and 1000 (dashed line). The dashed-dotted lines represent the beam pattern of the telescope adopted at 350 450 and 850 $\\mu$m. Note that\n    the SED plot reports the ISO-LWS spectrum between 45 and 200 $\\mu\n    m$ for completeness, although it has not been considered in the\n    $\\chi^2$ analysis, due to the relative larger calibration\n    uncertainty compared to the IRAS data.}\\label{fit}\n\\end{figure*}\n\\begin{figure*} \\centering\n\\centerline{\\rotatebox{90}{\\includegraphics[width=12cm]{X2_map_G1.ps}}}\n\\caption{$\\chi ^{2}_{red}$ contour plots (Y,$\\alpha$) for the models\n  with G$_{o}$=1. In these computations, $\\tau_{100}$ is equal to\n  0.6. The contours show the loci of the $\\chi ^{2}_{red}$ values\n    equal to 1.1, 1.5, 2.5 and 5 times the minimum $\\chi ^{2}_{red}$.\n The upper left panel is obtained by comparing the model predictions\n  with the 850 $\\mu$m brightness profile; the upper right panel refers\n  to the 450 $\\mu$m profile; the lower left panel refers to the 350\n  $\\mu$m profile; the lower right panel makes use of the three\n  profiles. \\label{fig:chi2-contour}}\n\\end{figure*}\n\\begin{figure} \\centering\n\\centerline{\\rotatebox{0}{\\includegraphics[width=10cm]{X2_SED_G1.ps}}}\n\\caption{$\\chi^2_{red}$ versus $\\tau_{100}$. In these computations, Y\n  is equal to 120 and $\\alpha$ is equal to 0.6. \\label{X2SED}}\n\\end{figure}\n\nThe three $\\chi _{350}^{2}$, $\\chi _{450}^{2}$ and $\\chi _{850}^{2}$\ncontour plots point to a value of Y around 100-200. Conversely, the\n$\\chi _{350}^{2}$ and $\\chi _{450}^{2}$ contour plots constraint\n$\\alpha$ to a lower value than 1, around 0.5-0.6, whereas the $\\chi\n_{850}^{2}$ would rather indicate a larger value for $\\alpha$,\nalthough the value 0.6 is still acceptable. Note that the solution\nfound by \\citet{Jor06} relies on the 850 $\\mu$m profile only, and,\ntherefore, gives a large $\\alpha$ value, consistent with our\n  $\\chi _{850}^{2}$ plot. The $\\chi _{SED}^{2}$ plot\n(Fig.\\ref{X2SED}) points to a value of $\\tau_{100}$ of 0.6. In\n  minimizing the $\\chi _{SED}^{2}$, we varied the source luminosity\n  from 400 to 1500 L$_\\odot$. The best fit is\n  obtained for a source luminosity equal to 1000 L$_\\odot$.\\\\\n\n\\noindent\n{\\it b) G$_o$=10,100,1000}\\\\\nThe best fit values of $\\alpha$ and Y for cases of an enhanced\nillumination UV field are presented in Figure \\ref{X2all_ISRF}.\n\\begin{figure*} \\centering\n\\rotatebox{90}{\\includegraphics[width=6cm]{cont_all_850um.ps}}\n\\rotatebox{90}{\\includegraphics[width=6cm]{cont_all_450um.ps}}\n\\rotatebox{90}{\\includegraphics[width=6cm]{cont_all_350um.ps}}\n\\rotatebox{90}{\\includegraphics[width=6cm]{cont_all_tot.ps}}\n\\caption{$\\chi ^{2}_{red}$ contour plots (Y,$\\alpha$) for the models\n  with G$_{o}$=1 (black lines), 10 (blue lines), 100 (green lines) and\n  1000 (red lines). In these computations, $\\tau_{100}$ is equal to\n  0.6. The contours show the loci of the $\\chi ^{2}_{red}$ values\n    equal to 2.5 times the minimum $\\chi ^{2}_{red}$. From top to\n  bottom: $\\chi^2$ contours of the 850 $\\mu$m, 450 $\\mu$m, 350 $\\mu$m\n  and the three together.\\label{X2all_ISRF}}\n\\end{figure*}\nThe first thing to notice is that the $\\chi _{all}^{2}$ does not\nchange appreciably for G$_o$ equal to 1,10,100 or 1000: the minimum\n$\\chi _{all_{red}}^{2}$ value is 1.24, 1.23, 1.19 and 1.20 for\nG$_o$=1, 10, 100 and 1000 respectively. In other words, the available\ncontinuum observations, both the profiles and the SED, cannot\ndistinguish which of the four models is better. Furthermore, \nFigure \\ref{X2all_ISRF} shows that the $\\chi _{all}^{2}$ contour\nplots point to the same Y and $\\alpha$ values. Similarly, the\n$\\tau_{100}$ value is 0.6 for the four cases G$_o$=1, 10, 100 and\n1000.  The situation is illustrated in Fig. \\ref{fit}, where the best\nfit predictions are compared to the observations for the two cases\nG$_o$=1 and 1000. Both models reproduce fairly well the observations,\nas implicit in the best-$\\chi^2$ similar values. Note, however, that\nthe G$_o$=1000 case predicts slightly larger fluxes, due to the\nenhanced temperature at the border of the envelope.\\\\\n\n\\noindent\n{\\it c) larger G$_o$ }\\\\\n We did not explore in detail the case of larger G$_0$ for three\n  reasons. The first one is that previous line observations showed\n  that the FUV field in the OMC2 region is ``only'' 500 times the\n  Interstellar field. Indeed, \\cite{Her97} mapped the OMC-2 cloud in\n  the CII-157 $\\mu$m, OI-63 and -146 $\\mu$m lines with the\n  spectrometer FIFI on board the Kuiper Airborne Observatory. They\n  detected extended emission associated with the Photo-Dissociation\n  Region (PDR) enveloping the whole OMC-2 molecular cloud. These\n  authors concluded that OMC-2 is illuminated by a FUV field whose\n  intensity is G$_0$ $\\sim$ 500. Note that this is the FUV field\n  impinging on the cloud and that the effective G$_0$ seen by the FIR4\n  envelope is probably lower than this. The second reason is that\nvarying G$_0$ from 1 to 10$^3$ does not improve the $\\chi^2$\nvalue. The third reason is that the G$_0$=10$^4$ case suffers of\nsevere convergence problems, and it was not possible to derive a large\nenough number of runs for a meaningful $\\chi^2$\nanalysis.\\\\\n\n\\noindent\n{\\it d) Summary of the adopted solution}\\\\\n Table \\ref{best_fit_phy_param} summarizes the value of the best\n  fit parameters, obtained by considering all the profiles and the SED\n  $\\chi^2$ contour plots and assuming the G$_0$=1 case. Some relevant\n  physical quantities are quoted in the same table. Fig. \\ref{T_n.ps}\n  shows the dust density and temperature profiles of the best fit\n  models with G$_0$=1 and 1000 respectively. Note that the dust\n  temperature in the skin of the envelope is larger by $\\sim$ 20-30 K\n  in the case G$_0$=1000 with respect to the G$_0$=1 case. This\n  increase concerns a relatively small region, of a few thousand\n  AU. \\citet{Jor06} found a larger warm region, of about\n  10$^4$ AU, because of the steeper adopted density distribution\n  ($\\alpha$=2): in this case, the FUV photons can penetrate deeper\n  into the envelope.\n\\begin{table}\n $$\n \\begin{array}{p{0.5\\linewidth}r}\n   \\hline\n   \\multicolumn{2}{c}{\\mathrm{Fixed\\ input\\ parameters}} \\\\\n   \\hline\n   \\noalign{\\smallskip}\n   Distance, $d$\t  & 437\\,\\mathrm{pc}\t  \\\\\n   Stellar temperature, $T_{\\star}$    & 5000\\,\\mathrm{K}\\\\\n   Dust temperature at $r_{\\mathrm{i}}$, $T_{\\mathrm{d}}(r_{\\mathrm{i}})$   & 300\\,\\mathrm{K}\\\\\n   Dust opacity (OH5) at 100\\,$\\mu$m, $\\kappa_{100}$ & 86.5\\,\\mathrm{cm}^2\\mathrm{g}^{-1} \\\\\n \\hline\n \\noalign{\\smallskip}\n \\multicolumn{2}{c}{\\mathrm{Best\\ fit\\ parameters}} \\\\\n \\hline\n \\noalign{\\smallskip}\n Luminosity, $L$   & 1000\\,\\mathrm{L}_{\\sun}\\\\\n Dust optical depth at 100\\,$\\mu$m, $\\tau_{100}$     & 0.6 \\\\\n Density power law index, $\\alpha$   & 0.6 \\\\\n Envelope thickness, $r_{\\mathrm{out}}$\/$r_{\\mathrm{i}}$  & 120 \\\\\n \\hline\n \\noalign{\\smallskip}\n \\multicolumn{2}{c}{\\mathrm{Physical\\ quantities}} \\\\\n \\hline\n \\noalign{\\smallskip}\n Inner envelope radius, $r_{\\mathrm{in}}$ \t & 100\\,\\mathrm{AU}\\\\\n Outer envelope radius, $r_{\\mathrm{out}}$ \t & 12000\\,\\mathrm{AU}\\\\\n Radius at T$_{dust}$ = 100 K,   $r_{\\mathrm{100K}}$ & 440\\,\\mathrm{AU}\\\\\n H$_2$ density at $r_{\\mathrm{100K}}$, $n_0$ \t & 4.3\\times 10^{6}\\,\\mathrm{cm}^{-3} \\\\\n Envelope mass, $M_{\\mathrm{env}}$\t\t & 30\\,\\mathrm{M}_{\\sun} \\\\\n \\hline\n \\noalign{\\smallskip}\n \\end{array}\n $$\n \\caption[]{ Summary of the dust radiative transfer analysis of\n     OMC2-FIR4. The first part lists the fixed input parameters, \n     the second part reports the best fit parameters, and some relevant \n     physical quantities corresponding to the\n     best fit model are reported in the third part.}\n \\label{best_fit_phy_param}\t \n\\end{table}\n\\begin{figure} \\centering\n\\rotatebox{0}{\\includegraphics[width=9cm]{T.ps}}\n\\rotatebox{0}{\\includegraphics[width=9cm]{n.ps}}\n\\caption{Dust temperature (upper panel) and H$_{2}$ density (lower panel) \nprofiles from the best fit obtained in the two cases G$_0$=1 and 1000. The \nplain line and the dotted line represent the cases G$_0$=1 and 1000 \nrespectively.}\\label{T_n.ps}\n\\end{figure}\n\n\n\\section{Gas temperature profile}\\label{sec:gas-temp-prof}\n\n\\subsection{Model description}\n\\citet{Cec96}, \\citet{Dot97} and \\citet{Mar02} showed that the gas is\nthermally decoupled from dust in the inner regions of low and high\nmass protostellar envelopes. The reason for that is the large water\nabundance in the gas phase caused by the sublimation of the grain\nmantles. The same phenomenon may occur in the envelopes of\nintermediate mass protostars, so we explicitly computed the gas\ntemperature profile of the envelope surrounding FIR4. For that we\nexplicitly computed the equilibrium temperature by equating the gas\ncooling and heating terms at each radius. Following the method\ndescribed in \\citet{Cec96}, we considered heating from the gas\ncompression (due to the collapse), dust-gas collisions and\nphoto-pumping of H$_2$O and CO molecules by the IR photons emitted by\nthe warm dust close to the center\\footnote{Cosmic rays ionization is a\n  minor heating term in the protostellar envelopes.}. The cooling is\nmainly due to rotational lines from H$_2$O and CO, plus the fine\nstructure lines from O. Therefore, the gas temperature depends on the\nabundance of these three species. In practice, though, only the water\nabundance is a real parameter of the model, because the CO and O lines\nare optically thick in the range of the CO and O abundances typical of\nprotostellar envelopes. For this reason, we computed various cases for\nthe water abundance, as it is generally poorly constrained in\nprotostellar envelopes, and totally unconstrained in FIR4 (see \\S\n\\ref{sec:predicted-water-line}).  We adopted a step function for the\nwater abundance profile to simulate the jump caused by the ices\nsublimation. The jump is assumed to occur at 100 K. We considered the\nH$_2$O abundance (with respect to H$_2$) X(H$_2$O)$_{out}$ in the\nouter envelope, where T$\\leq$ 100 K, equal to 10$^{-7}$, 10$^{-8}$ and\n10$^{-9}$.  We also considered three cases for the abundance in the\ninner region X(H$_2$O)$_{in}$, 10$^{-4}$, 10$^{-5}$ and\n10$^{-6}$. Finally, we studied the case with G$_0$ = 1000. The run\nparameters are summarized in Table \\ref{tab:abu-cool}.\n\\begin{table}[tbh]\n  \\centering\n  \\begin{tabular}{|cccccc|}\n    \\hline\n    Model & X(CO)            & X(O)             & X(H$_2$O)$_{out}$& X(H$_2$O)$_{in}$ & G$_0$\\\\ \\hline\n    1$^a$ & $1\\times10^{-4}$ & $5\\times10^{-4}$ & $1\\times10^{-8}$ & $1\\times10^{-5}$ & 1\\\\\n    2     & $1\\times10^{-4}$ & $5\\times10^{-4}$ & $1\\times10^{-8}$ & $1\\times10^{-6}$ & 1\\\\\n    3     & $1\\times10^{-4}$ & $5\\times10^{-4}$ & $1\\times10^{-8}$ & $1\\times10^{-4}$ & 1\\\\\n    4     & $1\\times10^{-4}$ & $5\\times10^{-4}$ & $1\\times10^{-9}$ & $1\\times10^{-5}$ & 1\\\\\n    5     & $1\\times10^{-4}$ & $5\\times10^{-4}$ & $1\\times10^{-7}$ & $1\\times10^{-5}$ & 1\\\\\n    6     & $1\\times10^{-4}$ & $5\\times10^{-4}$ & $1\\times10^{-7}$ & $1\\times10^{-4}$ & 1\\\\\n    7     & $1\\times10^{-4}$ & $5\\times10^{-4}$ & $1\\times10^{-8}$ & $1\\times10^{-5}$ & 1000\\\\ \\hline\n  \\end{tabular}\n  \\caption{The different run models (column 1). Column 2 to 5 report \n    the adopted abundances of the main gas coolants: X(CO), X(O) and the \n    H$_2$O abundance in the T$\\geq$100 K region X(H$_2$O)$_{in}$ and outer \n    region X(H$_2$O)$_{out}$. Column 5 reports the FUV illuminating \n    field G$_0$. Note: $^a$ Model 1 is the reference for the studies of\n    the water line spectrum presented in \\S \\ref{sec:predicted-water-line}. \n    \\label{tab:abu-cool}}\n\\end{table}\n\nTo compute the cooling from the lines we used the code described in\n\\citet{Cec96,Cec03} and \\citet{Par05}. The same code has been used in\nseveral past studies, whose results have been substantially confirmed\nby other groups (e.g. the analysis on IRAS16293-2422 by\n\\citep{Sch02}). Briefly, the code is based on the escape probability\nformalism in presence of warm dust (see \\citet{Tak83}), where the\nescape probability $\\beta$ is computed at each point by integrating\nthe line and dust absorption over the solid angle $\\Omega$ as follows:\n\\begin{equation} \n  \\beta = \\frac{k_\\mathrm{d}}{k_\\mathrm{L} + k_\\mathrm{d}} + \n  \\frac{k_\\mathrm{L}}{(k_\\mathrm{L} + k_\\mathrm{d})^2} \\int d\\mu \n  \\frac{1-\\exp \\left[ - \\left( k_\\mathrm{L} + k_\\mathrm{d} \\right) \n  \\Delta L_\\mathrm{th} \\right]} {\\Delta L_\\mathrm{th}} \n\\end{equation} \n\\noindent  \nwhere $k_\\mathrm{L}$ and $k_\\mathrm{d}$ are the line and dust\nabsorption coefficients respectively, and $\\Delta L_\\mathrm{th}$ is\nthe line trapping region, given by the following expressions:\n\\begin{equation} \n  \\Delta L_\\mathrm{th} = 2 \\Delta v_\\mathrm{th}  \n  \\left( \\frac{v}{r} \\left| 1-\\frac{3}{2} \\mu^2 \\right| \\right)^{-1}  \n\\end{equation} \n\\noindent\nin the infalling region of the envelope (where $\\mathrm{arcos} \\left(\n\\mu \\right)$ is the angle with the radial outward direction) and\n\\begin{equation} \n  \\Delta L_\\mathrm{th} = r \\left( 1 - \\frac{r}{R_\\mathrm{env}} \\right) \n\\end{equation} \n\\noindent \nin the static region (where $R_\\mathrm{env}$ is the envelope radius).\n In the present calculations, we assumed that the entire envelope is\ncollapsing in free-fall towards a central object of 2 M$_{\\sun}$.  In\npractice, the photons emitted by the dust can be absorbed by the gas\nand can pump the levels of the water molecules. This, indeed, is an\nimportant factor in the population of the water levels, and, for the\nhighest energy levels, even the dominant one (\\S\n\\ref{sec:predicted-water-line}). In addition, H$_2$O and CO\n  molecules can be pumped by absorption of the NIR photons emitted by\n  the innermost warm dust. Since the densities and temperatures of the\n  regions of the envelope targeted by this study are not enough to\n  populate the levels at the vibrational states, the effect of the NIR\n  photons is an extra heating of the gas, as described in the\n  Ceccarelli et al. (1996) article. Note that the code takes into\n  account the dust with temperatures up to 1500 K, by following the\n  algorithm described in Ceccarelli et al. (1996).\n\nFor the collisional coefficients of water with hydrogen molecules, we\nused the data by \\citet{Fau07} available for the temperature range\n20-2000K. This data set includes quasi-classical results for the\nhighest rates (those larger than 10$^{-12}$ cm$^3$s$^{-1}$) and quantum\nscaled H$_2$O-He results for the lowest rates. Recent quantum\ncalculations on ortho-H$_2$O by Dubernet and co-workers have shown\nthat the quasi-classical rates can be in error by as much as a factor\nof 100 but that, in general, they are accurate to within a factor of\n1-3 \\citep{Dub09}. It should be noted that\nthe rates of \\citet{Fau07} are currently the only complete and\nconsistent set of data for both ortho- and para-H$_2$O colliding with\nH$_2$. We also note that these rates have been recently extrapolated\nin order to cover energy levels and temperatures up to 5000K\n\\citep{Fau08}. Since the ortho to para conversion process of H$_2$ is\nchemical rather than radiative, the ortho-to-para ratio H$_2$ OPR is\nhighly uncertain in the interstellar medium. Here we assume that in\nwarmer gas it is in Local Thermal Equilibrium and, therefore, follows\nthe Boltzmann distribution:\n\\begin{equation}\nOPR =  \\frac{(2I_o+1) \\Sigma (2J+1) \\exp(-\\frac{E_o(J)}{kT})}\n            {(2I_p+1) \\Sigma (2J+1) \\exp(-\\frac{E_p(J)}{kT})}\n\\end{equation}\nwhere $I_o$ and $I_p$ are the total nuclear spin, corresponding to\nwhether the hydrogen nuclear spins are parallel ($I_o = 1$,\n$\\uparrow\\uparrow$) or anti-parallel ($I_p = 0$,\n$\\uparrow\\downarrow$). The sum in the numerator and denominator\nextends over all ortho and para levels J, respectively. Similarly to\nH$_2$, water comes in the ortho and para forms. In this case, since\nthe water is the dominant gas coolant only in the regions where the\ndust temperature exceeds 100 K, we assumed OPR equal to 3, strictly\nvalid for gas temperatures larger than 60 K.  Since the water lines\nare optically thick, the cooling depends on the velocity field,\nassumed to be that of an envelope collapsing in free-fall towards a\ncentral object of 2 M$_{\\sun}$ (see above). We checked the influence\nof our results against this assumption, running a case with a constant\nvelocity field of 0.5 km\/s. The difference in the gas temperature\nbetween the two cases never exceeds 10 K.\n\n\n\\subsection{Results}\\label{sec:results-gastemp}\nFigure \\ref{tg.ps} shows the computed gas temperature profile obtained\nwith different values of X(H$_2$O)$_{in}$ in the case G$_0$ = 1.\n\\begin{figure}\n\\rotatebox{0}{\\includegraphics[width=9cm]{Tg.ps}}\n\\caption{The gas temperature profile of the collapsing envelope of\n  OMC2-FIR4. The different curves refer to different values of the\n  inner envelope water abundance X(H$_2$O)$_{in}$: $1\\times10^{-6}$\n  (dotted), $1\\times10^{-5}$ (dashed) and $1\\times10^{-4}$\n  (dotted-dashed) respectively. In these computations,\n  X(H$_2$O)$_{out}$ is $1\\times10^{-8}$ and G$_0$=1. The solid line\n  refers to the dust temperature profile.\\label{tg.ps}}\n\\end{figure}\n\\begin{figure}\n\\rotatebox{0}{\\includegraphics[width=9cm]{heating.ps}}\n\\rotatebox{0}{\\includegraphics[width=9cm]{cooling.ps}}\n\\caption{Heating (top panel) and cooling (bottom panel) rates as\n  function of the radius, computed assuming that the inner H$_2$O\n  abundance is equal to $1\\times10^{-5}$ while the outer abundance is\n  $1\\times10^{-8}$. \\label{thermal.ps}}\n\\end{figure}\n Figure \\ref{thermal.ps} shows the different contributions to the\n  heating and cooling rates. Similarly to what had been found in low\nmass protostars \\citep{Cec96,Cec00,Mar02}, the gas temperature tracks\nthe dust temperature in the outer envelope while gas and dust are\ndecoupled in the inner part of envelope, where the icy grain mantles\nsublimate. The heating is dominated by compression of the\n  collapsing gas across the entire envelope, even though the dust-gas\n  collisions becomes comparable to the compression heating in the\n  inner envelope. Although important in the very inner regions, the\n  H$_2$O photo-pumping never dominates the heating contrarely to what\n  happens in the studied low mass protostars. The cooling, on the\n  other hand, is dominated by H$_2$O line emission in the inner\n  envelope, by the OI line emission in the intermediate region and by\n  the CO in the outermost regions of the envelope.\nNote that the increased water abundance causes an increased cooling\nof the gas, which brings the equilibrium gas temperature to lower\nvalues than the dust temperature. This phenomenon, already predicted\nin low mass protostars, is much more marked in the FIR4 case, leading\nto more than 100\\% of difference (with respect to the gas temperature)\nin the dust and gas temperatures for the case of the highest water\nabundance ($1\\times10^{-4}$). For example, at 100 AU the dust\ntemperature is 300 K, whereas the gas temperature varies from 200 to\n80 K depending on the assumed X(H$_2$O)$_{in}$, $1\\times10^{-6}$ and\n$1\\times10^{-4}$ respectively. The phenomenon is more marked in FIR4\nthan in the studied low mass protostars because of the relatively\nlower density of the region where the icy grain mantles sublimate in\nFIR 4 than in the low mass protostars, or, in other words, because the\nFIR4 envelope is warmer.  Note that we obtain similar results also for\nlarger illuminating FUV fields.\n\n We emphasize that this result is a consequence of the derived\n  shallow dependence of the density distribution, which is constrained\n  from the fit of the maps. The dependence is strictly valid only at\n  scales larger than the smaller telescope beam, namely 8$\"$\n  (equivalent to a radius of about 1700 AU) and the SED fit only gives\n  the total column density, which, coupled with the density dependence\n  on the radius (constrained by the maps), constrains the density at\n  these scales. While we cannot exclude the presence of a denser\n  compact object hidden by the envelope, it seems unlikely that the\n  envelope density gradient increases inwards, because this would be\n  unphysical.\n\nClearly, the water abundance in the inner region of FIR4 will have a\ngreat impact not only on the emerging water spectrum but also on the\nemerging line spectrum of any molecule (abundant in the inner region),\nand has to be correctly taken into account to give reliable molecular\nabundances. Conversely, given the large effect, in principle\nappropriate multiline observations of any molecule will be able to\nconstrain the inner region water abundance and the present model\npredictions. Note that varying the outer abundance X(H$_2$O)$_{out}$\ndoes not have effect on the gas temperature, as in the outer region\nthe cooling is dominated by the CO and O lines.\n\n\n\n\n\n \\section{Predicted water line spectrum}\\label{sec:predicted-water-line}\n\\subsection{Reference model}\nHere we report and discuss the predicted spectrum of our reference\nmodel. Next paragraph will discuss how it depends on the parameters of\nthe model. We adopted the Model 1 of Table \\ref{tab:abu-cool} as\nreference model . We first discuss the general water line spectrum by\nmeans of the synthetic rotational diagram, and then we discuss the\nspecific predictions for the two spectrometers on board Herschel: HIFI\nand PACS.\n\\begin{figure} \\centering\n\\rotatebox{0}{\\includegraphics[width=9cm]{rot_diag.ps}}\n\\caption{Synthetic rotational diagram derived from the water line emission \n using the reference model (Model 1, Table \\ref{tab:abu-cool})\n integrated over the whole envelope. \n Crosses and diamonds trace the ortho and para water, respectively.\n\\label{fig:rot-diagr}}\n\\end{figure}\n\nFigure \\ref{fig:rot-diagr} shows the synthetic rotational diagram\nderived from the line emission integrated over the whole envelope. As\nexpected, \n the theoretical points do not lie on a compact and straight\n  line, reflecting the different line optical depths, the gradients\nin density and temperature of the envelope and non-LTE effects.  An\nilluminating example is represented by the fundamental transitions of\nthe ortho and para water lines at 557 and 1113 GHz respectively. We\nwill discuss these two lines in detail because, first, they will\ncertainly be important observational diagnostics and, second, they\noffer a great pedagogic case. The situation is illustrated in\nFig.\\ref{fig:h2o-profile}, where we report the profile of the emission\nof the ortho and para H$_2^{16}$O and H$_2^{18}$O fundamental lines as\nfunction of the radius. Figure \\ref{fig:h2o-profile-2}, with the beta\nescape probability as function of the radius for the two fundamental\nH$_2^{16}$O lines, also greatly helps to interpret the emerging line\nfluxes for the two lines.  The ortho-H$_2^{16}$O fundamental line\nemission (Fig.\\ref{fig:h2o-profile}) peaks at the border of the\nenvelope and it decreases inwards because of the decreasing emitting\nvolume.  The para-H$_2$O fundamental line shows approximately the same\nbehavior.  If the lines were optically thin and LTE populated, the\nexpected flux ratio of the para over ortho fundamental line would be\nbetween 3 and 4 for a temperature between 50 and 200 K.  Any departure\nfrom this value originates from a combination of line opacity and\nnon-LTE effects.  In the outer region the ratio is lower than 1: the\npara-H$_2$O line is optically thin, whereas the ortho-H$_2$O lines is\nmoderately optically thick (Fig.\\ref{fig:h2o-profile-2}). Therefore,\nthe much lower emission of the para-H$_2$O line with respect to the\northo-line is due to the non-LTE population effect, more accentuated\nin the para-H$_2$O line.  The situation is reversed in the inner\nregion, where ices sublimate: the para-H$_2$O fundamental line becomes\nabout ten times brighter than the ortho-H$_2$O fundamental line\nbecause of the line opacity, which is much larger in the ortho-H$_2$O\nline than in the para-H$_2$O line (Fig.\\ref{fig:h2o-profile-2}). In\nfact, the increase in the water abundance by a factor 1000 gives rise to a jump in the\nline emission by a factor 3 in the ortho-H$_2$O line and 30 in the\npara-H$_2$O line, and this can only be due to the larger opacity of\nthe ortho-H$_2$O line as the excitation conditions do not change when\nices sublimate. In summary, the emission from the water lines is due,\nin principle, to a rather complex combination of line opacity, non-LTE\neffects and emitting volume (namely temperature and density\ngradient). Evidently, the intensity ratio of lines from the\nH$_2^{16}$O and H$_2^{18}$O isotopologues is far to give the\n``opacity'' of the line, as it is a combination of the penetration of\nthe line and the opacity itself.\n\\begin{figure} \\centering\n\\rotatebox{0}{\\includegraphics[width=9cm]{h2o_ground_prof.ps}}\n\\rotatebox{0}{\\includegraphics[width=9cm]{prof_h2_18o.ps}}\n\\caption[] {Emission profile, $R\\frac{dF}{dR}$ of the ortho (solid\n  line) and para (dashed line) water fundamental lines at 557 and 1113\n  GHz respectively as function of the radius, in the case of the\n  reference model (see Table \\ref{tab:abu-cool}). The plotted quantity\n  $R\\frac{dF}{dR}$ is the contribution of each shell to the flux\n  integrated over the whole envelope. H$_2^{16}$O and H$_2^{18}$O\n  emission profiles are represented on the top and bottom panel,\n  respectively.\\label{fig:h2o-profile}}\n\\end{figure}\n\n\\begin{figure} \\centering\n\\rotatebox{0}{\\includegraphics[width=9cm]{ratio_h2o_h2_18o.ps}}\n\\caption[] {Ratio of the H$_2^{16}$O escape probability over the\n  H$_2^{18}$O escape probability of the ortho (solid line) and para\n  (dashed line) water fundamental lines obtained with the reference\n  model (model 1 of Table \\ref{tab:abu-cool}) at 557 and 1113 GHz\n  respectively as function of the radius.\\label{fig:h2o-profile-2}}\n\\end{figure}\n\n Table \\ref{tab:spe-ref} lists the predicted water line fluxes\nfor the two spectrometers on board Herschel: HIFI and PACS. Note that,\nin both cases, we computed the signal after convolving the theoretical line intensity map \nwith the instrument beam which vary\n  from 39$\"$ to 13$\"$ with the frequency varying from 500 GHz to 2000\n  GHz (HIFI frequency range) and from 13$\"$ to 5$\"$ for wavelengths\n  from 210 $\\mu$m and 60 $\\mu$m (PACS wavelength range).\n\n\\begin{table}[h]\n  \\centering\n  \\begin{tabular}{|ccc|}\n    \\hline\n    PACS range & &                                                    \\\\ \\hline\n   Transition  & Wavelength & Flux                                    \\\\\n               & ($\\mu$m)   & (erg s$^{-1}$ cm$^-2$)                  \\\\ \\hline\n  H$_2^{16}$O  &            &                                        \\\\\n  2$_{ 2 1} \\rightarrow $ 3$_{ 3 0}$  &       66.44 &  2.99E-14     \\\\\n  2$_{ 2 0} \\rightarrow $ 3$_{ 3 1}$  &       67.09 &  1.13E-14     \\\\\n  3$_{ 0 3} \\rightarrow $ 3$_{ 3 0}$  &       67.27 &  2.28E-14     \\\\\n  2$_{ 1 2} \\rightarrow $ 3$_{ 2 1}$  &       75.38 &  1.55E-13     \\\\\n  3$_{ 1 2} \\rightarrow $ 4$_{ 2 3}$  &       78.74 &  4.51E-14     \\\\\n  2$_{ 1 1} \\rightarrow $ 3$_{ 2 2}$  &       89.99 &  5.09E-14     \\\\\n  4$_{ 0 4} \\rightarrow $ 5$_{ 1 5}$  &       95.63 &  1.05E-14     \\\\\n  4$_{ 1 4} \\rightarrow $ 5$_{ 0 5}$  &       99.49 &  4.21E-14     \\\\ \n  1$_{ 1 1} \\rightarrow $ 2$_{ 2 0}$  &      100.98 &  7.80E-14     \\\\ \n  1$_{ 1 0} \\rightarrow $ 2$_{ 2 1}$  &      108.07 &  1.43E-13     \\\\\n  3$_{ 0 3} \\rightarrow $ 4$_{ 1 4}$  &      113.54 &  8.45E-14     \\\\\n  4$_{ 2 3} \\rightarrow $ 4$_{ 3 2}$  &      121.72 &  1.14E-14     \\\\\n  3$_{ 1 3} \\rightarrow $ 4$_{ 0 4}$  &      125.36 &  5.34E-14     \\\\\n  4$_{ 1 4} \\rightarrow $ 4$_{ 2 3}$  &      132.41 &  2.92E-14     \\\\\n  3$_{ 2 1} \\rightarrow $ 3$_{ 3 0}$  &      136.49 &  1.35E-14     \\\\\n  2$_{ 0 2} \\rightarrow $ 3$_{ 1 3}$  &      138.53 &  7.84E-14     \\\\\n  3$_{ 1 3} \\rightarrow $ 3$_{ 2 2}$  &      156.20 &  2.25E-14     \\\\\n  2$_{ 1 2} \\rightarrow $ 3$_{ 0 3}$  &      174.62 &  7.42E-14     \\\\\n  1$_{ 0 1} \\rightarrow $ 2$_{ 1 2}$  &      179.53 &  1.17E-13     \\\\\n  2$_{ 1 2} \\rightarrow $ 2$_{ 2 1}$  &      180.49 &  4.97E-14     \\\\ \\hline\nHIFI range & & \\\\ \\hline\n   Transition  & Frequency  & Flux \\\\\n               & (GHz)      & (K Km s$^{-1}$) \\\\ \\hline\n H$_2^{16}$O  &            &                 \\\\\n  1$_{ 0 1} \\rightarrow $ 1$_{ 1 0}$  &      556.96 &  14.8 \\\\\n  2$_{ 0 2} \\rightarrow $ 2$_{ 1 1}$  &      752.04 &  1.06E+00 \\\\\n  1$_{ 1 1} \\rightarrow $ 2$_{ 0 2}$  &      987.95 &  1.31E+00 \\\\\n  3$_{ 0 3} \\rightarrow $ 3$_{ 1 2}$  &     1097.34 &  1.46E+00 \\\\\n  0$_{ 0 0} \\rightarrow $ 1$_{ 1 1}$  &     1113.35 &  4.32E+00 \\\\\n  2$_{ 2 1} \\rightarrow $ 3$_{ 1 2}$  &     1153.09 &  2.50E+00 \\\\\n  3$_{ 1 2} \\rightarrow $ 3$_{ 2 1}$  &     1162.93 &  6.97E-01 \\\\\n  2$_{ 1 1} \\rightarrow $ 2$_{ 2 0}$  &     1228.81 &  5.33E-01 \\\\ \n  4$_{ 1 3} \\rightarrow $ 4$_{ 2 2}$  &     1207.62 &  2.45E-01 \\\\ \n  5$_{ 1 4} \\rightarrow $ 5$_{ 2 3}$  &     1410.65 &  1.47E-01 \\\\\n  2$_{ 1 2} \\rightarrow $ 2$_{ 2 1}$  &     1660.99 &  2.33E+00 \\\\\n  4$_{ 0 4} \\rightarrow $ 4$_{ 1 3}$  &     1602.23 &  1.46E-01 \\\\\n  1$_{ 0 1} \\rightarrow $ 2$_{ 1 2}$  &     1669.87 &  5.46E+00 \\\\\n  2$_{ 1 2} \\rightarrow $ 3$_{ 0 3}$  &     1716.83 &  3.37E+00 \\\\\n  5$_{ 2 3} \\rightarrow $ 5$_{ 3 2}$  &     1867.75 &  3.13E-02 \\\\\n H$_2^{18}$O  &            &                 \\\\\n  1$_{ 0 1} \\rightarrow $ 1$_{ 1 0}$  &      556.96 &  1.88E-01 \\\\\n                 &            &                                 \\\\\n  0$_{ 0 0} \\rightarrow $ 1$_{ 1 1}$  &     1113.35 &  1.48E-01 \\\\\n &  &\\\\ \\hline\n  \\end{tabular}\n  \\caption{Predictions of the line fluxes (after subtraction of the continuum) \n    of the water lines observable with the Herschel spectrometers, HIFI and \n    PACS. The predictions refer to the reference model (model 1 of Table \n    \\ref{tab:abu-cool}). \\label{tab:spe-ref}}\n\\end{table}\n\n Based on the (preliminary) sensitivities reported on the Herschel\n  Observation Planning Tool HSpot: {\\it\n    http:\/\/herschel.esac.esa.int\/Tools.shtml}), several ortho and\npara lines are predicted to be detectable by the two Herschel\nspectrometers: about a dozen in the HIFI frequency range and twice\nmore in the PACS wavelength range. The H$_2^{18}$O ortho and para\nlines are also predicted to be detectable by HIFI, and 100 and 20\ntimes less bright than the respective lines of the H$_2^{16}$O\nrespectively. Note the counter-intuitive result: the para-H$_2^{16}$O\nline seems to be more optically thick than the ortho-H$_2^{16}$O line!\nAs explained above, this is not the case, of course: the line\nintensity ratio (from which the line optical depth is usually derived) is due to the combination of optical depth plus\nexcitation (non-LTE) effects, and the final result is not easily predictable. In our\nreference model, no observable line is predicted to be in\nabsorption.\n\n\\subsection{Other models}\nHere we explore the sensitivity of the results reported in the\nprevious paragraph against the variation of the three main parameters\nof the model: the water abundance in the inner (X(H$_2$O)$_{in}$) and\nouter (X(H$_2$O)$_{out}$) envelope, and the illuminating FUV field\nG$_o$. \n\nFigures \\ref{hifi_ratio_xout} and \\ref{hifi_ratio_xin} show the ratio\nbetween the line intensities of the reference model (Model 1 of Table\n\\ref{tab:abu-cool}) and the line intensities predicted by models with\ndifferent X(H$_2$O)$_{out}$ and X(H$_2$O)$_{in}$ respectively. As\nnoted by other authors \\citep{Cec00,Mar02}, lines with upper level\nenergies lower than about 200 cm$^{-1}$ are sensitive to\nX(H$_2$O)$_{out}$ and insensitive to X(H$_2$O)$_{in}$, because these\nlines mostly originate in the outer envelope for excitation and line\nopacity reasons. A variation of a factor 10 in X(H$_2$O)$_{out}$ leads\nto an almost similar variation in the line intensity of the lowest\nlying lines. The higher the upper level energy the smaller the\nvariation.  Conversely, lines with upper level energies larger than\nabout 200 cm$^{-1}$ are sensitive to X(H$_2$O)$_{in}$ and insensitive\nto X(H$_2$O)$_{out}$. In this case, variations by a factor 10 in\nX(H$_2$O)$_{in}$, going from $1\\times10^{-6}$ to $1\\times10^{-5}$, can\nlead to variations in the lines fluxes even 100 times larger. This\nextreme variation, 10 times larger than the difference in the\nX(H$_2$O)$_{in}$ variation, occurs to some lines in the 50-200 $\\mu$m\nwavelength range. This phenomenon is due to the fact that those lines\nare in absorption rather than in emission in the region just after the\nices sublimation, resulting in an additional decrease of the emerging\nline flux. The higher the X(H$_2$O)$_{in}$ the smaller the absorption\ndepth. When X(H$_2$O)$_{in}$ reaches $1\\times10^{-5}$ the absorption\nregion generally vanishes. In addition, many high lying lines are\n  prevalently populated by absorption of the photons emitted by the\n  dust, so that they are particularly sensitive to the dust\n  continuum.\n\\begin{figure} \\centering\n\\rotatebox{90}{\\includegraphics[width=6cm]{ratio_var_Xout_o.ps}}\n\\rotatebox{90}{\\includegraphics[width=6cm]{ratio_var_Xout_p.ps}}\n\\caption[] {Ratio between the line fluxes of the reference model\n  (model 1 of Table \\ref{tab:abu-cool}) and the line fluxes predicted\n  by models with same X(H$_2$O)$_{in}$ but different\n  X(H$_2$O)$_{out}$. Flux ratios between model 4 (model 5) and model 1\n  are represented by squares (triangles). Filled and empty symbols\n  refer to the lines emitted in the PACS and HIFI bands,\n  respectively. The upper (lower) panel reports ortho (para) water\n  line intensities ratios. \\label{hifi_ratio_xout}}\n\\end{figure}\n\\begin{figure} \\centering\n\\rotatebox{90}{\\includegraphics[width=6cm]{ratio_var_Xin_o.ps}}\n\\rotatebox{90}{\\includegraphics[width=6cm]{ratio_var_Xin_p.ps}}\n\\caption[] {Ratio between the line fluxes of the reference model\n  (model 1 of Table \\ref{tab:abu-cool}) and the line fluxes predicted\n  by models with same X(H$_2$O)$_{out}$ but different\n  X(H$_2$O)$_{in}$. Flux ratios between model 2 (model 3) and model 1\n  are represented by squares (triangles). Filled and empty symbols\n  refer to the lines emitted in the PACS and HIFI bands,\n  respectively. The upper (lower) panel reports ortho (para) water\n  line flux ratios. \\label{hifi_ratio_xin}}\n\\end{figure}\n\nNote that as mentioned in the Sect. \\ref{sec:cont-emiss-dusty}, the inner region of the envelope is relatively unconstrained by the available observational data. Therefore we derived water line spectrum predictions varying the power law index of the density profile $\\alpha$ of about $\\sim$ 30$\\%$ in the inner part. We observed a variation of the line intensity of a factor 5 - 10 for the transitions with upper level energy $\\gtrsim$ 300$-$400 cm$^{-1}$ and lesser than 2 for the lower lines.\nFinally, the predicted line intensities do not vary appreciably when\nthe illuminating FUV field changes from 1 to 1000. Therefore, observations of \nwater lines will be extremely helpful in constraining the water abundance across\nthe envelope, but will not be sensitive to the illuminating FUV field.\n\n\n\\subsection{Effect of gas-dust thermal decoupling}\n\nAs presented in \\S \\ref{sec:gas-temp-prof}, the large quantity of\nwater vapor injected into the gas in the inner part of the envelope\ncauses a dramatic decoupling between the dust and gas temperatures\n(see Fig. \\ref{tg.ps}). Obviously, this effect has a great impact in\nthe interpretation of the water line emission. This is illustrated in\nFigures \\ref{ratio_decoup}, where we report the ratio of the water\nline intensities obtained by considering the gas temperature\nself-consistently computed (model 1) over the case where T$_{gas}$ is\nassumed to be equal to T$_{dust}$.\n\\begin{figure} \\centering\n\\rotatebox{90}{\\includegraphics[width=6cm]{ratio_decoup_T_o-h2o_ref.ps}}\n\\rotatebox{90}{\\includegraphics[width=6cm]{ratio_decoup_T_p-h2o_ref.ps}}\n\\caption[]{Ratios between the line fluxes of the reference model\n  (model 1) and the reference model with gas-dust non-thermally\n  decoupled (namely T$_{gas}$=T$_{dust}$, as function of the upper\n  level energy of the transition. Upper and lower panels show ortho\n  and para H$_2$O lines, respectively. \\label{ratio_decoup}}\n\\end{figure}\nAssuming artificially T$_{gas}$ = T$_{dust}$ leads to differences in\nline fluxes up to two orders of magnitude.  Since the decoupling\noccurs in the inner part of the envelope, the larger the upper level\nenergy the larger the difference, for both ortho and para lines. Note\nthat the fluxes of the two fundamental ortho and para lines are not\naffected by the T$_{gas}$ = T$_{dust}$ choice. Finally, we did the\nsame study with the model 2 ( X(H$_2$O)$_{in}$ = 1O$^{-6}$). In this\ncase, due to the smaller X(H$_2$O)$_{in}$, the decoupling is less\nimportant than in the previous one, leading to changes in lines\nintensities up to one order of magnitude.\n\n\n\\subsection{Constraints from ISO data}\\label{sec:iso-data:water}\nObservations of FIR4 were obtained by the spectrometer ISO-LWS in the\ngrating mode (spectral resolution $\\sim$200) and Fabry-Perot (spectral\nresolution $\\sim 10^{4}$). We retrieved the data from the ISO Data\nArchive ({\\it http:\/\/iso.esac.esa.int\/ida\/}). No water lines are\ndetected. Table \\ref{ISO_upper} summarizes the upper limits obtained\nfor the predicted brightest lines together with the predictions of the\nreference model and model 6.\n\\begin{table*}\n \\centering\n\\begin{tabular}{|cc|c|c|cc|} \\hline \n&&Model 1 &Model 6          & ISO & \\\\\nWavelength    &  Transition        &   Intensity &   Intensity   & Instrument  & Upper Limit (3$\\sigma$)  \\\\\n ($\\mu$m)     & J$_{K_- K_+} \\rightarrow$ J'$_{K'_- K'_+}$ &   (erg.s$^{-1}$.cm$^{-2}$) &   (erg.s$^{-1}$.cm$^{-2}$) &&  (erg.s$^{-1}$.cm$^{-2}$) \\\\ \\hline\northo &&&&& \\\\\n75.4957       & 8$_{5 4} \\rightarrow$  8$_{4 5}$       &  2.6E-20        &      1.9E-19         & LWS04       & 9.78e-12 \\\\\n108.073       & 2$_{2 1} \\rightarrow$  1$_{1 0}$       &  1.3E-12        &      1.5E-12         & LWS04       & 2.21e-12 \\\\ \n179.527       & 2$_{1 2} \\rightarrow$  1$_{0 1}$       &  1.2E-12        &      7.0E-12         & LWS04       & 2.56e-11 \\\\  \n113.538       & 4$_{1 4} \\rightarrow$  3$_{0 3}$       &  8.3E-13        &      8.7E-13         & LWS01       & 1.2e-11 \\\\  \npara &&&&& \\\\\n100.983       & 2$_{2 0} \\rightarrow$  1$_{1 1}$       &  7.3E-13        &      9.1E-13         & LWS01       & 8.37e-12 \\\\\n138.527       & 3$_{1 3} \\rightarrow$  2$_{0 2}$       &  7.3E-13        &      7.3E-13         & LWS01       & 4.14e-12 \\\\\n156.197       & 3$_{2 2} \\rightarrow$  3$_{1 3}$       &  2.2E-13        &      3.4E-13         & LWS01       & 1.60e-12 \\\\ \\hline\n\\end{tabular}\n\\caption{The brightest lines predicted by models 1 and 6 (the model with the largest water abundance) compared with the upper limits derived by the ISO observations.)  \\label{ISO_upper}}\n\\end{table*}\nUnfortunately, the ISO sensitivity is not enough to put sensible\nconstraints to the water abundance across the FIR4 envelope.\n\n\n\n\\section{Concluding remarks}\\label{sec:conclusion}\nWe have analyzed in great detail the continuum emission from the\nIntermediate Mass protostar OMC2-FIR4, with the aim of deriving the\nphysical structure of its envelope, a mandatory first step for further\nstudies to understand the formation process. Our analysis led to a new\nestimate of the FIR4 luminosity, which is 1000 L$_\\odot$. The density\nof the envelope surrounding FIR4 has a shallow dependence on the\nradius, the density power law index being only 0.6. Since\n  systematic studies of the IM protostars envelopes have not been\n  published yet, we can tentatively compare the FIR4 envelope with low\n  and high mass protostellar envelopes, where similar studies have been\n  carried out. Specifically, \\citet{Jor02} analyzed 18 Class 0 and I\n  sources and found that the average power law index $\\alpha$ in Class\n  0 sources is $1.3\\pm0.4$ while in Class I sources it is $1.7\\pm0.1$,\n  significatively larger than the value we found in FIR4. Similarly,\n  \\citet{Van00} studied a sample of high mass protostars\n  and found $\\alpha=1.4\\pm0.4$. One has to notice that, however, there\n  are exceptions, with sources in both low and high mass showing a\n  smaller than unity $\\alpha$ value: L1527 ($\\alpha=0.6$) and L483\n  ($\\alpha=0.9$) in the Class 0 sources, GL7009S ($\\alpha=0.5$) in the\n  high mass protostars sample. It is not clear what makes these\n  sources ``anomalous'': the presence of strong asymmetries \\citep{Jor02} have been suggested as possible reason. The\n  case of FIR4 seems to fall in this ``anomalous sources'' category,\n  and further studies, possibly on chemistry, are needed to say more.\n\n  Giving the suggestion by \\citet{Jor06} that a strong FUV field\n  (G$_0$=$1\\times10^{4}$) illuminates the FIR4 envelope, we explored\n  the cases of different FUV fields.  As already noted by the same\n  authors, however, the dust continuum cannot really distinguish\n  whether a strong illuminating FUV field is impinging on the\n  envelope. In fact, \\citet{Jor06} adopted a steeper density distribution ($\\alpha$=2) which allows the FUV photons to penetrate deeper into the envelope. Their conclusions were based on\n  submillimeter lines from CO and H$_2$CO, which would be exceedingly\n  bright if they were emitted in the envelope. They attributed the\n  lines to the warm gas at the border of the envelope, heated up by\n  the hypothetical large FUV field. However, as discussed in \\S\n  \\ref{sec:dust-results}, OI and CII maps by \\citet{Her97} showed\n  that the entire OMC2 region is illuminated by a G$_0$=500 FUV field,\n  which would imply an even lower FUV field on the FIR4 envelope. One\n  has also to notice here that large scale maps by \\citet{sch82} show\n  that the CO (1$\\rightarrow$0) line is bright ($\\sim$40 K) over the\n  whole OMC2 region, a fact that lead \\citet{Her97} to attribute the\n  CO emission to the PDR associated with the cloud. In addition,\n  several outflows are known to \"pollute\" the CO emission in the\n  region, in particular the one originating from FIR3 (25$\"$\n  North of FIR4: Fig. \\ref{maps}) and reaching FIR4 and FIR5\n  \\citep{Wil03}.  \n All the above considerations together lead to\nconclude that the FUV field impinging FIR4 is not anomalously large\nand less than 500. Therefore, giving the presence of a bright PDR\n  and a ``polluting'' outflow from FIR3, caution is needed in\n  interpreting the low lying water lines, as much as lines from any\n  molecule, separately from the whole molecular cloud emission.\n\nOne major motivation of the present work is the prediction of the\nwater line spectrum from FIR4, as this source will be observed in the\n500-2000 GHz frequency range by the incoming Herschel Space\nObservatory (FIR4 is a target of the Key Program ``HIFI Spectral\nSurveys of Star Formation Regions'': {\\it\n  http:\/\/www-laog.obs.ujf-grenoble.fr\/heberges\/hs3f\/}). In the present\nstudy, we have shown that water is indeed a key molecular species,\nbecause of its great impact on the gas cooling in the region where the\ndust temperature exceeds 100 K, the sublimation temperature of the\ndust grain ices. The large quantity of water vapor injected into the\ngas by the sublimated ices very efficiently cools the gas, causing a\ndramatic decoupling between the dust and gas temperatures. Depending\non the abundance of the injected water vapor, the difference in the\ntemperature can be as high as 50 K at the sublimation radius (namely\n50\\%!) and even larger going inward. For example, at 100 AU the dust\ntemperature is predicted to be around 280 K whereas the gas\ntemperature is 80 K if the water abundance is\n$1\\times10^{-4}$. Obviously, this has a great impact in the\ninterpretation of the water line emission as much as the emission from\nany molecular species emitting in the inner region. In fact, the\ncomparison of the water line emission between the case where dust and\ngas are assumed to be thermally coupled and the case where the gas\ntemperature is self-consistently computed shows that the difference of\nthe line intensity can reach two orders of magnitudes for lines with\nlarge upper level energies (namely the lines excited in the innermost\nregion, where gas and dust decouple). Therefore, our important second\nconclusion is that caution has to be applied in interpreting the line\nemission from FIR4, as much as any source with a similar luminosity\nand envelope structure. Gas and dust temperature can be very different\nand in order to derive correct molecular abundances (including water\nabundance) they have to be both estimated, accounting for all terms of\nheating and cooling. Avoiding that may lead to very wrong conclusions.\n\n\\bigskip\n\\begin{acknowledgements}\n  We warmly thank Moshe Elitzur for his valuable help in using the\n  DUSTY code. We also wish to thank Neal Evans and Doug Johnstone for\n  helpful discussions, and Doug Johnstone and Darek Lis for providing\n  us with the JCMT and CSO continuum maps of the OMC2-FIR4 region. We thank an anonymous referee and Malcolm Walmsley for comments which helped improving the manuscriptOne of us (N.Crimier) is supported by a fellowship of the Minist\\`ere de\nl'Enseignement Sup\\'erieur et de la Recherche. \n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nLarge Language Models (LMs) \nachieve remarkable accuracy and generalization ability when fine tuned for NLP tasks~\\citep{peters2018deep,devlin2018bert,liu2019roberta,lan2020albert,raffel2020exploring}. They are also capable zero- and few-shot learners ~\\citep{brown2020gpt3}, with the ability to generalize to tasks not seen during training.\nA reliable way to improve LM accuracy in all of these settings is by scaling up: increasing the number of parameters and the amount of computation used during training and inference~\\citep{raffel2020exploring,brown2020gpt3,fedus2021switch}. In fact, some generalization properties only emerge in very large models, including much improved zero- and few-shot learning~\\citep{brown2020gpt3}.\n\nUnfortunately, the corresponding growth in computational resources required to train state-of-the-art language models is a barrier for many in the research community~\\cite{schwartz2019green}.\nThere is also a concern about the environmental costs associated with training and deploying such models~\\citep{strubell2019energy,gupta2021chasing,bender2021dangers,patterson2021carbon} motivating research into more efficient model designs~\\citep{lepikhin2021gshard,fedus2021switch,Lewis2021BASELS}.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/efficiency-gain.pdf}\n\\caption{\n\\textbf{Estimate of how much more efficient MoEs are relative to dense models.} A speedup factor of $y$ indicates that an MoE model can match the performance of the corresponding dense model---trained with $x$ ZFLOPs---using $y$ times less compute (i.e., $x\/y$ ZFLOPs). We estimate this factor according to validation perplexity for in-domain language modeling, the Pile perplexity for out-of-domain language modeling, and average accuracy across 6 tasks for zero-shot priming. See \\S\\ref{subsubsec:efficiency-gain} for more details.\n}\n\\label{fig:efficiency-gain}\n\\end{figure}\n\n\\emph{Sparse} models allow for increased number of learnable parameters without the associated computational costs. \nFor example, sparsely gated mixture of experts (\\emph{MoE})~\\cite{lepikhin2021gshard} have been successfully used for language modeling and machine translation~\\citep{lepikhin2021gshard,Lewis2021BASELS,roller2021hash}, but are yet to be shown effective for fine-tuning~\\cite{fedus2021switch} as well as zero- and few-shot learning. \nWe hypothesize that sparse models are comparatively accurate to dense models but at a much lower computational footprint. \nTo measure this claim, we train traditional dense and MoE language models ranging in size from several hundred million parameters to more than one trillion parameters and present a careful empirical comparison of these models on downstream tasks in zero-shot, few-shot and fully supervised settings.\n\nAs shown in Figure \\ref{fig:efficiency-gain}, we find that MoE models can indeed achieve similar downstream task performance as dense models at a fraction of the compute. For models with relatively modest compute budgets, a MoE model can perform on par with a dense model that requires almost four times as much compute.\nDownstream task performance improves with scale for both MoE models and dense models.\nWhile we observe that the performance gap narrows as we increase model size, even at larger compute budgets ($\\sim$ 5000 GPU days), our largest MoE model (1.1T parameters) outperforms a dense model with similar computational cost (6.7B parameters).\nWe further compare and contrast the performance of dense and sparse models with similar computational signatures and observe some performance variations across tasks and domains, suggesting this an interesting area for future research. In summary, our contributions are:\n\\begin{itemize}[leftmargin=*]\n    \\item We present a comprehensive study of sparse models for zero and few-shot learning at scale; \n    \\item We demonstrate that even at scale sparse MoE models can yield competitive zero and few-shot performance at a fraction of the computation for model training and inference;\n    \\item We observe some differences in how dense and sparse models generalize at scale suggesting complementary behaviour that could be an interesting future research direction.\n\\end{itemize}\n\n\n\n\\section{Background and Related Work}\n\n\\subsection{Large Language Models \/ GPT-3}\n\nProgress in the field of NLP has been driven by increasingly large Language Models (LMs) pretrained on large text datasets. While numerous variations have been proposed, such LMs are predominantly based on the transformer architecture~\\citep{vaswani2017attention}. Models are pretrained by hiding parts of the input: predicting the next word sequentially left-to-right, masking words in the text~\\citep{devlin2018bert,liu2019roberta}, or perturbing and\/or masking spans~\\citep{lewis-etal-2020-bart,raffel2020exploring}. The resulting models can be quickly adapted to perform new tasks at high accuracy by fine-tuning on supervised data~\\citep{devlin2018bert,liu2019roberta}.\n\nRecently, GPT-3~\\citep{brown2020gpt3} demonstrated that large LMs can perform zero- and few-shot learning without fine-tuning through in-context learning. Notably, many of these in-context zero- and few-shot learning behaviors emerge or amplify at scale. Concurrent to our work, \\citet{rae2022scaling} and \\citet{smith2022using} further explore scaling dense language models. %\n\n\n\\subsection{Sparse models}\nOne drawback of dense model scaling is that it grows increasingly computationally expensive. To more efficiently increase model capacity, conditional compute strategies have been developed \\citep{bengio2013estimating,davis2013low,cho2014exponentially,bengio2015conditional}, where each input  activates a subset of the model. Recent work \\citep{Lewis2021BASELS,lepikhin2021gshard,fedus2021switch,fan2021beyond} has studied different conditional compute strategies that work well with Transformer models for natural language tasks. In this work, we focus on Sparsely Gated Mixture of Expert (MoE) models \\citep{shazeer2017outrageously,lepikhin2021gshard}. Sparse MoE models replace the dense feed forward network block in every alternate Transformer layer with an MoE layer. The MoE layer has a routing gate that learns which tokens are to be mapped to which set of experts (we use top-2 experts). To ensure scalability and training efficiency, it is also common to include a weighted gate loss term as in \\citet{lepikhin2021gshard} to the cross entropy loss to encourage the tokens to be uniformly distributed to the experts. \nConcurrent to our work, \\citet{du2021glam}, \\citet{rajbhandari2022deepspeedmoe} and \\citet{clark2022unified} also study MoE scaling.\n\n\\begin{table*}[ht]\n\\begin{center}\n\\begin{small}\n\\addtolength{\\tabcolsep}{-2.5pt}\n\\begin{tabular}{crrrrrrrrrrrrrrrrrr}\n\\toprule\n& \\multicolumn{5}{c}{GPT-3 (dense)} && \\multicolumn{5}{c}{Ours (dense)} && \\multicolumn{5}{c}{Ours (MoE)} & \\\\\n\\cmidrule{2-6} \\cmidrule{8-12} \\cmidrule{14-18}\n& \\multicolumn{1}{c}{\\emph{size}} & \\multicolumn{1}{c}{\\emph{cost}} & \\multicolumn{1}{c}{$l$} & \\multicolumn{1}{c}{$h$} & \\multicolumn{1}{c}{$e$} &\n& \\multicolumn{1}{c}{\\emph{size}} & \\multicolumn{1}{c}{\\emph{cost}} & \\multicolumn{1}{c}{$l$} & \\multicolumn{1}{c}{$h$} & \\multicolumn{1}{c}{$e$} &\n& \\multicolumn{1}{c}{\\emph{size}} & \\multicolumn{1}{c}{\\emph{cost}} & \\multicolumn{1}{c}{$l$} & \\multicolumn{1}{c}{$h$} & \\multicolumn{1}{c}{$e$} &\n\\\\\n\\midrule\n& 125M & 0.36 & 12 & 768 & -- &\n& 125M & 0.36 & 12 & 768 & -- &\n& 15B & 0.43 & 12 & 768 & 512 &\n\\\\\n& 355M & 1.06 & 24 & 1024 & -- &\n& 355M & 1.06 & 24 & 1024 & -- &\n& 52B & 1.30 & 24 & 1024 & 512 &\n\\\\\n& 760M & 2.13 & 24 & 1536 & -- &\n& \\multicolumn{5}{c}{---} &\n& \\multicolumn{5}{c}{---} &\n\\\\\n& 1.3B & 3.57 & 24 & 2048 & -- &\n& 1.3B & 3.57 & 24 & 2048 & -- &\n& 207B & 4.53 & 24 & 2048 & 512 &\n\\\\\n& 2.7B & 7.08 & 32 & 2560 & -- &\n& 2.7B & 7.08 & 32 & 2560 & -- &\n& \\multicolumn{5}{c}{---} &\n\\\\\n& 6.7B & 17.12 & 32 & 4096 & -- &\n& 6.7B & 17.12 & 32 & 4096 & -- &\n& 1.1T & 22.27 & 32 & 4096 & 512 &\n\\\\\n& 13B & 32.67 & 40 & 5120 & -- &\n& 13B & 32.67 & 40 & 5120 & -- &\n& \\multicolumn{5}{c}{---} &\n\\\\\n& 175B & 430.17 & 96 & 12288 & -- &\n& \\multicolumn{5}{c}{---} &\n& \\multicolumn{5}{c}{---} &\n\\\\\n\\bottomrule\n\\end{tabular}%\n\\end{small}\n\\end{center}\n\\caption{\\textbf{Dense and mixture of expert (MoE) model details}. \\emph{size}: number of parameters, \\emph{cost}: training ZFLOPs, $l$: layers, $h$: hidden dimension, $e$: number of experts. All models are trained for 300B tokens with a sequence length of 2048 tokens. Models within the same row are roughly comparable. We estimate the training cost in ZFLOPs analytically (see Appendix~\\ref{app:flops}).}\n\\label{tab:models}\n\\end{table*}\n\n\\subsection{Zero-shot and Few-shot Learning} \\label{subsec:background_fewshot}\n\nRecent works \\citep{schick-schutze-2021-exploiting,radford2019language} have directly evaluated LMs on unseen tasks successfully (zero-shot learning), by recasting their task inputs as cloze-style prompt completion tasks. This is in contrast to the traditional approach of augmenting LMs with task-specific heads, followed by supervised fine-tuning \\citep{devlin2018bert,raffel2020exploring}. Subsequently, \\citet{brown2020gpt3} demonstrated that priming LMs with a few input-output examples (few-shot learning) before careful prompting can improve task performance, that grows with model scale without any fine-tuning, and this gave rise to new resources for prompt engineering \\citep{bach2022promptsource}. In this paper, we contrast the zero-shot, few-shot, and fully supervised fine-tuning performance of dense and MoE models. Finally, \\citet{schick2021smalllanguagemodels} perform few-shot learning by few-shot fine-tuning using pattern-exploiting training, whose efficiency can be improved by performing partial fine-tuning of a small number of additional task-specific parameters instead \\citep{lester2021power,li2021prefix,houlsby2019parameter}. %\n\n\n\n\\subsection{Large-scale training}\n\nMany of the models we consider in this work are too big to be trained using standard data parallel techniques, since parameter storage would exceed the usable memory of a single GPU.\nWe adopt several techniques to make these models feasible to train, including pure FP16 training, activation checkpointing and fully sharded data parallel training. These techniques are described in more depth in Appendix~\\ref{app:scaling}.\n\n\\section{Experimental Setup}\n\n\\subsection{Models}\n\nWe train autoregressive (decoder-only) transformer models that roughly match the sizes and architecture explored in~\\citet{brown2020gpt3}. Model sizes are summarized in Table~\\ref{tab:models}.\nWe use pre-normalization transformer blocks~\\citep{baevski2018adaptive,child2019generating} and GELU activations~\\citep{hendrycks2016gelu}.\nWe differ from \\citet{brown2020gpt3} in two ways: (1) we use only dense attention, while they alternate between dense and locally banded sparse attention; and (2) we train our models with sinusoidal positional embeddings, following Shortformer~\\cite{press2020shortformer}.\\footnote{Early experiments found this to produce comparable results with fewer learned parameters.}\n\nWe also train MoE models that mirror our dense model configurations (see the third set of columns in Table~\\ref{tab:models}), so that comparisons are approximately matched in terms of the number of floating point operations (\\emph{FLOP}s).\nOur MoE models follow the design proposed in \\citet{lepikhin2021gshard} with alternating dense and expert layers and top-2 expert selection.\nWe use 512 experts in each expert layer ($E=512$).\nEach expert has a \\emph{capacity} of $\\frac{C \\cdot B}{E}$ tokens, where $C$ is a \\emph{capacity factor} that we set to $2$ and $B$ is the total batch size in tokens. Capacity refers to the maximum number of tokens that are routed to each expert.\nOnce an expert is at capacity for a given batch, additional tokens are considered to be ``overflowed\" with their representations passed-through via the residual connection.\n\n\\citet{fedus2021switch} report instability training large MoE models and suggest rescaling the initial model weights, which we do not find necessary.\nWe instead observe that expert parameters have an $E$-times smaller batch size relative to dense (data parallel) parameters and accordingly rescale expert gradients by a factor $\\frac{1}{\\sqrt{E}}$.\nThis rescaling aligns with theory suggesting that an $E$-times increase in batch size should be accompanied by a $\\sqrt{E}$ increase in learning rate~\\citep{krizhevsky2014one}.\n\n\nFollowing \\citet{brown2020gpt3}, we train our models for 300B tokens\\footnote{While we control the total number of tokens to be the same as \\citet{brown2020gpt3}, our pretraining data is not the same. See \\S\\ref{subsec:pretraining_data} for further details.}\nwith a context size (sequence length) of 2048 tokens.\nThe batch size and learning rate are set according to the model size following \\citet{brown2020gpt3}.\nWe linearly warm-up the learning rate from $0$ over the first 375M tokens and linearly decay back to $0$ over the remaining tokens.\nWe use the Adam optimizer~\\cite{kingma2014adam} with $\\beta_1=0.9$, $\\beta_2=0.98$, $\\epsilon=10^{-8}$, weight decay of 0.01 and dropout of 0.1.\\footnote{We note that our 355M dense and 52B MoE models (FLOPs-matched) were trained without dropout, which we find slightly improves performance at smaller scale.}\n\nWe train our models in PyTorch~\\cite{paszke2017automatic} using \\textsc{fairseq}~\\citep{ott2019fairseq}.\n\n\\subsection{Pretraining data}\n\\label{subsec:pretraining_data}\n\nWe pretrain our models on a union of six English-language datasets, including the five datasets used to pretrain RoBERTa~\\citep{liu2019roberta} and the English subset of CC100, totalling 112B tokens corresponding to 453GB:\n\\begin{itemize}[leftmargin=*]\n\\item \\textbf{BookCorpus}~\\citep{zhu2015bookcorpus} consists of more than 10K unpublished books (4GB);\n\\item \\textbf{English Wikipedia}, excluding lists, tables and headers (12GB);\n\\item \\textbf{CC-News}~\\citep{nagel2016ccnews} contains 63 millions English news articles crawled between September 2016 and February 2019 (76GB);\n\\item \\textbf{OpenWebText}~\\citep{gokaslan2019openwebtext}, an open source recreation of the WebText dataset used to train GPT-2 (38GB);\n\\item \\textbf{CC-Stories}~\\citep{trinh2018simple} contains a subset of CommonCrawl data filtered to match the story-like style of Winograd schemas (31GB);\n\\item \\textbf{English CC100}~\\citep{wenzek-etal-2020-ccnet}, a dataset extracted from CommonCrawl snapshots between January 2018 and December 2018, filtered to match the style of Wikipedia (292GB).\n\\end{itemize}\nWe encode our data using the same Byte-Pair Encoding (BPE) as GPT-2~\\citep{radford2019language} and RoBERTa~\\citep{liu2019roberta} with a vocabulary of 50K subword units.\n\n\\subsection{Evaluation}\n\nWe evaluate models in terms of their in-domain and out-of-domain perplexity, as well as downstream task performance. \n\n\\subsubsection{Perplexity Evaluation}\n\nWe first evaluate our models on their ability to predict the next token in a sequence as measured by perplexity. Similar to training, we concatenate all documents in a given dataset using empty lines as separators, split the resulting sequence into non-overlapping blocks of 2048 tokens, and score each block independently.\\footnote{One limitation of this approach is that the first tokens in each block have limited context, as they do not condition on tokens from preceding blocks. Although more expensive, better results could be obtained using a sliding window approach. Nevertheless, this form of chunking the input is standard in language model evaluation.}\n\nWe evaluate and report perplexity in both \\textbf{in-domain} and \\textbf{out-of-domain} settings.\nIn-domain, we sample a held-out subset of the combined pretraining data (\\S\\ref{subsec:pretraining_data}).\nFor out-of-domain we use data from The Pile \\citep{gao2021thepile}, a public dataset that combines data from 22 diverse sources (e.g., ArXiv, Github, OpenSubtitles, etc.).\nWe report perplexities on the official test set of each individual subset, as well as the average across all subsets.\n\n\n\\subsubsection{Downstream Evaluation} \\label{subsec:downstream}\n\nWe target models that can perform downstream tasks well. Recent work shows that good perplexity performance does not always align with good performance on downstream tasks~\\cite{tay2021scale}. Hence, we evaluate our models accordingly.\n\n\\paragraph{Benchmarks.}\nWe evaluate our models on a subset of the tasks considered in \\citet{brown2020gpt3}.\nAs GPT-3 performance varies greatly across tasks and model sizes, we focus on tasks for which GPT-3 either demonstrated consistent gains from scaling, or consistent gains going from zero-shot to few-shot settings.\n\n\\textbf{Few-shot:} we use WinoGrande~\\citep{sakaguchi2020winogrande}, StoryCloze~\\citep{mostafazadeh-etal-2016-corpus} and OpenBookQA~\\citep{mihaylov-etal-2018-suit}, the only non-generation tasks for which \\citet{brown2020gpt3} reported meaningful gains over zero-shot at our scale.\\footnote{Defined as an improvement of at least 2 accuracy points over zero-shot learning and the majority class baseline for at least one GPT-3 model no bigger than 6.7B.}  We exclude SuperGLUE, since we were not able to reproduce results reported in \\citet{brown2020gpt3} using the public GPT-3 API.\\footnote{Different from other tasks, we were not able to reproduce GPT-3 results on SuperGLUE using the OpenAI API and our evaluation protocol. The authors confirmed that they used a different evaluation protocol for SuperGLUE through personal correspondence.}\n\n\\textbf{Zero-shot:} in addition to the 3 few-shot tasks, we evaluate on ReCoRD~\\citep{zhang2018record}, HellaSwag~\\citep{zellers-etal-2019-hellaswag} and PIQA~\\citep{bisk2020piqa}. \\citet{brown2020gpt3} reported strong results and monotonic improvements from scaling on these tasks.\n\n\\paragraph{Evaluation protocol.} Following \\citet{brown2020gpt3}, we report results on the development set for all tasks except OpenBookQA and StoryCloze, for which we use the test set. For few-shot learning, we report the average results across 25 runs, randomly sampling a different set of few-shot examples from the training set each time.\\footnote{StoryCloze does not have a training set, so we follow \\citet{brown2020gpt3} and sample few-shot examples from the development set instead.} For priming, we further shuffle the few-shot examples for each test instance. Following \\citet{brown2020gpt3}, we use k=50 few-shot examples for WinoGrande, k=70 for StoryCloze and k=100 for OpenBookQA.\nIn cases where this exceeds the maximum context length for the model, we truncate the prompt keeping the maximum number of full examples that fit.\n\n\\paragraph{Baselines.} We compare to the published GPT-3 numbers \\citep{brown2020gpt3} as our primary baseline. To validate our experimental framework, we also evaluate GPT-3 leveraging the OpenAI API using our own evaluation code and settings. Unfortunately, the correspondence between model sizes and model names in the OpenAI API is not published. We follow other published work \\citep{gao2021thepile} and guess the correspondence based on our results from the public API as compared to results in \\citet{brown2020gpt3} (see \\S\\ref{subsec:zero_shot_results}).\n\n\\paragraph{Methods.} We compare both priming and fine-tuning-based approaches.\n\\begin{itemize}[leftmargin=*]\n\\item \\textbf{Priming:} We use a language model to separately score each label choice using the same templates as \\citet{brown2020gpt3}, and pick the one with the highest score. For few-shot learning, we use a single newline to separate examples. Our scoring function follows the description in \\citet{brown2020gpt3}:\n\\begin{itemize}\n\\item{\\bf For WinoGrande}, we take the log-likelihood of the common suffix of the different candidates.\n\\item{\\bf For OpenBookQA}, we normalize by the unconditional probability of each candidate by taking $\\frac{p(\\mathtt{completion}|\\mathtt{context})}{p(\\mathtt{completion}|\\mathtt{answer\\_context})}$, where we use the string \\textit{``Answer: ''} as answer\\_context.\n\\item{\\bf For ReCoRD}, we take the sum of per-token log-probabilities.\\footnote{This is different from \\citet{brown2020gpt3}, who take the average per-token log-probability for this task. This worked worse in our preliminary experiments.}\n\\item{\\bf For all the other tasks}, we take the average of per-token log-probabilities, ignoring the common prefix of the different candidates.\n\\end{itemize}\n\\item \\textbf{Fine-tuning:} Although supervised fine-tuning of pre-trained LMs on task specific training data, $\\mathcal{D}$, requires updating and storage of all model parameters per task, the process typically produces significant task specific performance improvements. We contrast the fine-tuning performance of sparse models and their dense counterparts following \\cite{radford2018gpt}, which applies an additional task-specific linear layer $W_y$ on the representation from the final transformer block for each input candidate separately, followed by a softmax layer.\nWe fine-tune all model parameters using the entire training set (fully supervised learning). In addition to our zero-shot tasks, we also evaluate on 3 widely-used classification tasks: BoolQ~\\citep{clark-etal-2019-boolq}, MNLI~\\citep{williams-etal-2018-broad} and SST-2~\\citep{socher-etal-2013-recursive}. More details are in Appendix \\ref{sec:fine_tuning_settings}.\n\n\\end{itemize}\n\n\n\n\\begin{figure*}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.485\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figs\/ppl-valid.pdf}\n         \\caption{In-domain (validation)}\n         \\label{fig:ppl-valid}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.485\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figs\/ppl-thepile.pdf}\n         \\caption{Out-of-domain (the Pile)}\n         \\label{fig:ppl-thepile}\n     \\end{subfigure}\n     \\hfill\n    \\caption{\\textbf{Language modeling perplexity.} For the Pile, we report the average perplexity across the 22 subsets.}\n    \\label{fig:ppl}\n\\end{figure*}\n\n\\begin{figure*}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.485\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figs\/efficiency-gain-ppl.pdf}\n         \\caption{Language modeling (the Pile)}\n         \\label{fig:efficiency-gain-ppl}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.485\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figs\/efficiency-gain-zeroshot.pdf}\n         \\caption{Zero-shot priming}\n         \\label{fig:efficiency-gain-zeroshot}\n     \\end{subfigure}\n     \\hfill\n    \\caption{\n    \\textbf{Estimate of how much more efficient MoEs are relative to dense models in representative datasets.} A speedup factor of $y$ indicates that an MoE model can match the performance of the corresponding dense model using $y$ times less compute. Refer to \\S\\ref{subsubsec:efficiency-gain} for more details.\n    }\n    \\label{fig:efficiency-gain-datasets}\n\\end{figure*}\n\n\\subsubsection{MoE speedup factor}\n\\label{subsubsec:efficiency-gain}\n\nWe hypothesize that sparse models can achieve comparable performance at a smaller compute budget. As such, it is informative to measure how much more efficient MoEs are at achieving a specific performance level relative to dense models. We estimate how many FLOPs $\\flop (t)$ the model needs to achieve performance $t$ in a particular task (as measured by perplexity for language modeling and accuracy for downstream tasks) using either an MoE or a dense model. Given that we only have  discrete observations, we estimate exact missing values by interpolating on a logarithmic scale as follows:\n$$ \\flop(t) = \\exp \\left( \\log \\flop_{lo}(t) + r \\left(\\log \\flop_{hi}(t) - \\log \\flop_{lo}(t) \\right) \\right)$$\nwhere $r = \\frac{t - t_{lo}}{t_{hi} - t_{lo}}$, $t_{lo}$ and $t_{hi}$ are the closest performance to $t$ from the available models while being lower and higher than $t$, respectively, and $\\flop_{lo}(t)$ and $\\flop_{hi}$ are their corresponding training cost in ZFLOPs.\n\nThe interpolation gives us matching performance levels for dense and MoE models. We use them to compute the MoE speedup factor $\\flop_{dense}(t) \/ \\flop_{moe}(t)$. For example, if a dense model requiring 20 ZFLOPs achieves a performance of $90\\%$ on a given task and a MoE model requiring 5 ZFLOPs achieves the same performance, then the formula produces saving factor of 4. We visualize the savings curve using $\\flop_{dense}(t)$ in the $x$ axis, which allows us to contrast speedup in different tasks in a comparable scale.\n\n\n\n\n\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{small}\n\\addtolength{\\tabcolsep}{-2.5pt}\n\\resizebox{0.48\\textwidth}{!}{\n\\begin{tabular}{cr|cccccc|c}\n\\toprule\n&& RE & HS & PI & WG & SC & OB & avg \\\\\n\\midrule\n\\multirow{8}{*}{\\shortstack{GPT-3 \\\\ (paper)}}\n& 125M & 70.8 & 33.7 & 64.6 & 52.0 & 63.3 & 35.6 & 53.3 \\\\\n& 355M & 78.5 & 43.6 & 70.2 & 52.1 & 68.5 & 43.2 & 59.4 \\\\\n& 760M & 82.1 & 51.0 & 72.9 & 57.4 & 72.4 & 45.2 & 63.5 \\\\\n& 1.3B & 84.1 & 54.7 & 75.1 & 58.7 & 73.4 & 46.8 & 65.5 \\\\\n& 2.7B & 86.2 & 62.8 & 75.6 & 62.3 & 77.2 & 53.0 & 69.5 \\\\\n& 6.7B & 88.6 & 67.4 & 78.0 & 64.5 & 77.7 & 50.4 & 71.1 \\\\\n& 13B & 89.0 & 70.9 & 78.5 & 67.9 & 79.5 & 55.6 & 73.6 \\\\\n& 175B & 90.2 & 78.9 & 81.0 & 70.2 & 83.2 & 57.6 & 76.9 \\\\\n\\midrule\n\\multirow{4}{*}{\\shortstack{GPT-3 \\\\ (API)}}\n& ada & 77.4 & 42.9 & 70.3 & 52.9 & 68.6 & 41.0 & 58.9 \\\\\n& babb. & 83.1 & 55.1 & 74.5 & 59.4 & 73.3 & 45.6 & 65.2 \\\\\n& curie & 87.1 & 67.8 & 77.1 & 64.3 & 77.7 & 50.8 & 70.8 \\\\\n& davi. & -- & 78.8 & 80.0 & 70.0 & 83.1 & 58.8 & -- \\\\\n\\midrule\n\\multirow{6}{*}{\\shortstack{Ours \\\\ (dense)}}\n& 125M & 69.3 & 33.7 & 65.3 & 52.1 & 66.0 & 35.4 & 53.6 \\\\\n& 355M & 78.1 & 46.2 & 70.6 & 54.2 & 71.0 & 42.0 & 60.4 \\\\\n& 1.3B & 83.5 & 58.4 & 74.6 & 58.1 & 76.8 & 49.4 & 66.8 \\\\\n& 2.7B & 85.8 & 65.9 & 76.6 & 61.4 & 78.2 & 49.6 & 69.6 \\\\\n& 6.7B & 87.5 & 70.2 & 78.2 & 64.7 & 80.5 & 51.8 & 72.2 \\\\\n& 13B & 88.5 & 73.7 & 79.0 & 67.6 & 80.9 & 55.4 & 74.2 \\\\\n\\midrule\n\\multirow{4}{*}{\\shortstack{Ours \\\\ (MoE)}}\n& 15B & 77.8 & 53.2 & 74.3 & 53.4 & 73.6 & 42.0 & 62.4 \\\\\n& 52B & 83.4 & 64.9 & 76.8 & 57.4 & 75.9 & 51.0 & 68.2 \\\\\n& 207B & 86.0 & 70.5 & 78.2 & 60.9 & 78.1 & 50.8 & 70.7 \\\\\n& 1.1T & 88.0 & 78.6 & 80.3 & 66.4 & 81.8 & 55.2 & 75.0 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{small}\n\\end{center}\n\\caption{\n\\textbf{Zero-shot priming accuracy.}\n\\textit{GPT-3 (paper)} results taken from \\citet{brown2020gpt3}, all the other results were obtained by us as described in \\S\\ref{subsec:downstream}.\n\\texttt{RE}: ReCoRD, \\texttt{HS}: HellaSwag, \\texttt{PI}: PIQA, \\texttt{WG}: WinoGrande, \\texttt{SC}: StoryCloze, \\texttt{OB}: OpenBookQA.\nWe do not evaluate the largest GPT-3 model (davinci) on \\texttt{RE} given the high price.\n}\n\\label{tab:zeroshot}\n\\end{table}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/zeroshot-avg.pdf}\n\\caption{\n\\textbf{Zero-shot priming accuracy averaged across 6 tasks as a function of compute cost.} Each point corresponds to a different, fully-trained model (see Table \\ref{tab:models}).\n\\textit{GPT-3 (paper)} results taken from \\citet{brown2020gpt3}.\n}\n\\label{fig:zeroshot}\n\\end{figure}\n\n\n\n\\section{Results and Analysis}\\label{sec:results}\n\n\n\\subsection{Language modeling perplexity}\n\nWe report our perplexity results in Figure \\ref{fig:ppl}, and visualize the speedup curves in representative subsets of the Pile \\citep{gao2021thepile} in Figure \\ref{fig:efficiency-gain-ppl}. Refer to Appendix \\ref{app:full-results} for full results for all the 22 subsets of the Pile.\n\nWe observe that all MoE models outperform their dense counterparts in all datasets, but their advantage greatly varies across domains and models. MoEs are most efficient when evaluated in-domain, where they are able to match the performance of dense models trained with 8-16x more compute (see Figure \\ref{fig:efficiency-gain}). The improvement is more modest in out-of-domain settings, bringing a speedup of 2-4 on the Pile. This is reflected in Figure \\ref{fig:ppl}, where the gap between the MoE and dense curves is substantially smaller in out-of-domain settings. Moreover, the advantage of MoEs over dense models decreases at scale: MoEs need $\\sim$4 times less compute to match the performance of dense models trained with 2-6 ZFLOPs, but the speedup is $\\sim$2 for dense models trained with $\\sim$30 ZFLOPs. %\n\nWe also observe large difference across the subsets of the Pile, which correspond to different domains. As shown in Figure \\ref{fig:efficiency-gain-ppl}, MoEs obtain the largest speedups in subsets that are closest to the training corpus (e.g., CommonCrawl). The efficiency gains are more moderate but still remarkable for other domains like ArXiv and OpenSubtitles. Our largest MoE model barely outperforms its dense counterpart on DM Mathematics (7.63 vs. 7.66 perplexity), which is arguably very different from the training domain.\n\n\n\\subsection{Downstream task evaluation}\n\n\\subsubsection{Zero-shot learning}\\label{subsec:zero_shot_results}\n\nWe report zero-shot results in Table \\ref{tab:zeroshot}, and visualize how the different model families scale in Figure \\ref{fig:zeroshot}.\n\nOur dense models perform at par with their GPT-3 counterparts. This is consistent across different tasks, with our models doing marginally better on average. We are thus able to match \\citet{brown2020gpt3} despite some notable differences in our setup (e.g., different training corpus), establishing a solid baseline to evaluate MoE models on downstream tasks. Similarly, when using our own code to evaluate the strongest GPT-3 API backend (\\textit{davinci}), we obtain numbers that replicate those reported in the original paper for their largest model, which reinforces that our evaluation settings are comparable to \\citet{brown2020gpt3}.\\footnote{We assume that \\textit{ada} corresponds to the 355M model, \\textit{babbage} corresponds to the 1.3B model, and \\textit{curie} corresponds to the 6.7B model based on the API evaluation results.}\n\n\nAs with language modeling, MoEs outperform their dense counterparts for all datasets and model sizes. But, once again, we find the advantage narrows at scale as illustrated in Figure \\ref{fig:zeroshot}. Similar to the domain differences in language modeling, we observe differences across downstream tasks. As shown in Figure \\ref{fig:efficiency-gain-zeroshot}, MoEs obtain significant speedups in certain tasks like HellaSwag and PIQA, but this improvement is more modest in other tasks such as ReCoRD and Winogrande.\n\n\n\n\n\n\n\n\\subsubsection{Few-shot learning}\n\nWe report our few-shot results in Table \\ref{tab:fewshot} and plot the corresponding improvement over zero-shot in Figure  \\ref{fig:fewshot}.\n\nOur dense baselines perform at par or slightly better than GPT-3. We observe that the improvement over zero-shot is bigger for larger models, further supporting that certain capabilities in language models emerge at scale \\citep{brown2020gpt3}. Finally, we find that our larger MoE models also benefit from few-shot learning, outperforming their dense counterparts in all conditions. However, the improvements going from zero-shot to few-shot are smaller for MoE models compared to their dense counterparts. For example, the average for the 6.7B dense model improves by 3.6 points to 69.3 going from zero-shot to few-shot, whereas the corresponding 1.1T model improves by 2.3 points yielding 70.1.\n\n\\begin{table}[t!]\n\\begin{center}\n\\begin{small}\n\\addtolength{\\tabcolsep}{-2pt}\n\\resizebox{0.48\\textwidth}{!}{\n\\begin{tabular}{cr|ccc|c}\n\\toprule\n&& WG & SC & OB & avg \\\\\n\\midrule\n\\multirow{8}{*}{\\shortstack{GPT-3 \\\\ (paper)}}\n& {125M} & 51.3 \\scriptsize{--0.7} & 62.3 \\scriptsize{--1.0} & 37.0 \\scriptsize{+1.4} & 50.2 \\scriptsize{--0.1} \\\\\n& {355M} & 52.6 \\scriptsize{+0.5} & 70.2 \\scriptsize{+1.7} & 43.6 \\scriptsize{+0.4} & 55.5 \\scriptsize{+0.9} \\\\\n& {760M} & 57.5 \\scriptsize{+0.1} & 73.9 \\scriptsize{+1.5} & 48.0 \\scriptsize{+2.8} & 59.8 \\scriptsize{+1.5} \\\\\n& {1.3B} & 59.1 \\scriptsize{+0.4} & 76.1 \\scriptsize{+2.7} & 50.6 \\scriptsize{+3.8} & 61.9 \\scriptsize{+2.3} \\\\\n& {2.7B} & 62.6 \\scriptsize{+0.3} & 80.2 \\scriptsize{+3.0} & 55.6 \\scriptsize{+2.6} & 66.1 \\scriptsize{+2.0} \\\\\n& {6.7B} & 67.4 \\scriptsize{+2.9} & 81.2 \\scriptsize{+3.5} & 55.2 \\scriptsize{+4.8} & 67.9 \\scriptsize{+3.7} \\\\\n& {13B} & 70.0 \\scriptsize{+2.1} & 83.0 \\scriptsize{+3.5} & 60.8 \\scriptsize{+5.2} & 71.3 \\scriptsize{+3.6} \\\\\n& {175B} & 77.7 \\scriptsize{+7.5} & 87.7 \\scriptsize{+4.5} & 65.4 \\scriptsize{+7.8} & 76.9 \\scriptsize{+6.6} \\\\\n\\midrule\n\\multirow{6}{*}{\\shortstack{Ours \\\\ (dense)}}\n& {125M} & 52.2 \\scriptsize{+0.1} & 64.7 \\scriptsize{--1.3} & 35.0 \\scriptsize{--0.4} & 50.7 \\scriptsize{--0.5} \\\\\n& {355M} & 53.7 \\scriptsize{--0.5} & 72.2 \\scriptsize{+1.1} & 42.0 \\scriptsize{+0.0} & 56.0 \\scriptsize{+0.2} \\\\\n& {1.3B} & 60.1 \\scriptsize{+2.0} & 78.6 \\scriptsize{+1.9} & 49.4 \\scriptsize{+0.0} & 62.7 \\scriptsize{+1.3} \\\\\n& {2.7B} & 63.9 \\scriptsize{+2.5} & 82.1 \\scriptsize{+3.8} & 53.2 \\scriptsize{+3.6} & 66.4 \\scriptsize{+3.3} \\\\\n& {6.7B} & 67.6 \\scriptsize{+2.9} & 83.2 \\scriptsize{+2.7} & 57.0 \\scriptsize{+5.2} & 69.3 \\scriptsize{+3.6} \\\\\n& {13B} & 71.0 \\scriptsize{+3.5} & 85.0 \\scriptsize{+4.1} & 59.5 \\scriptsize{+4.1} & 71.8 \\scriptsize{+3.9} \\\\\n\\midrule\n\\multirow{4}{*}{\\shortstack{Ours \\\\ (MoE)}}\n& {15B} & 52.5 \\scriptsize{--0.9} & 71.4 \\scriptsize{--2.1} & 42.2 \\scriptsize{+0.2} & 55.4 \\scriptsize{--0.9} \\\\\n& {52B} & 58.1 \\scriptsize{+0.7} & 77.5 \\scriptsize{+1.6} & 48.9 \\scriptsize{--2.1} & 61.5 \\scriptsize{+0.1} \\\\\n& {207B} & 62.8 \\scriptsize{+1.9} & 81.1 \\scriptsize{+3.0} & 52.4 \\scriptsize{+1.6} & 65.4 \\scriptsize{+2.2} \\\\\n& {1.1T} & 68.6 \\scriptsize{+2.3} & 83.9 \\scriptsize{+2.1} & 57.7 \\scriptsize{+2.5} & 70.1 \\scriptsize{+2.3} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{small}\n\\end{center}\n\\caption{\n\\textbf{Few-shot priming accuracy and absolute improvement over zero-shot.}\n\\textit{GPT-3 (paper)} results taken from \\citet{brown2020gpt3}, all the other results were obtained by us as described in \\S\\ref{subsec:downstream}.\n\\texttt{WG}: WinoGrande, \\texttt{SC}: StoryCloze, \\texttt{OB}: OpenBookQA.}\n\\label{tab:fewshot}\n\\end{table}\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/fewshot-improvement.pdf}\n\\caption{\n\\textbf{Absolute accuracy improvement going from zero-shot to few-shot}, averaged across 3 tasks. Each point corresponds to a different, fully-trained model (see Table \\ref{tab:models}).\n\\textit{GPT-3 (paper)} results taken from \\citet{brown2020gpt3}.\n}\n\\label{fig:fewshot}\n\\end{figure}\n\n\n\n\n\n\\subsubsection{Supervised Fine-Tuning}\nTable \\ref{tab:finetuning} contrasts full fine-tuning performance of MoE models with their dense counterparts on 8 datasets, using zero-shot accuracy as a baseline for reference. We did not fine-tune the 6.7B and 13B dense models and the 1.1T MoE models, owing to their high resource needs. As expected, supervised fine-tuning yields substantial performance benefits for all dense models across all datasets, over zero-shot performance. In contrast, although fine-tuning of MoE models produces substantial benefits for Storycloze, BoolQ, SST-2, MNLI and some improvements on OpenBookQA, it results in worse performance for HellaSwag, PIQA, and Winogrande. For the cases where we see improvements, the accuracy of fine-tuned MoE models approach that of their corresponding dense models. For this comparison, we fine-tune MoE models exactly as we do the dense models. While MoE models may benefit from alternative fine-tuning approaches, for example, selective fine-tuning of the expert or non-expert parameters, we leave such exploration to future work. \n\n\\begin{table}[t!]\n\\begin{center}\n\\begin{small}\n\\addtolength{\\tabcolsep}{-2.5pt}\n\\resizebox{0.48\\textwidth}{!}{\n\\begin{tabular}{cr|cccc|ccc}\n\\toprule\n&& \\multicolumn{4}{c|}{Ours (Dense)} & \\multicolumn{3}{c}{Ours (MoE)} \\\\\n&& 125M & 355M & 1.3B & 2.7B & 15B & 52B & 207B \\\\ %\n\\midrule\n\\multirow{2}{*}{\\texttt{SC}}\n& zero-shot & 66.0 & 71.0 & 76.8 & 78.2 & 73.6 & 75.9 & 78.1 \\\\ %\n& fine-tune & 87.8  & 89.5 & 93.8 & 97.0 & 80.3 & 84.9 & 80.9 \\\\ %\n\\midrule\n\\multirow{2}{*}{\\texttt{OB}}\n& zero-shot & 35.4 & 42.0 & 49.4 & 49.6 & 42.0 & 51.0 & 50.8 \\\\ %\n& fine-tune & 50.6 & 59.0 & 67.4 & 70.8 & 51.2 & 51.4 & 51.0 \\\\ %\n\\midrule\n\\multirow{2}{*}{\\texttt{BQ}}\n& zero-shot & 56.1 & 58.6 & 58.7 & 60.3 & 60.9 & 56.0 & 54.2 \\\\ %\n& fine-tune & 73.2 & 75.2 & 79.6 & 84.6 & 71.6 & 75.3 & 77.5 \\\\ %\n\\midrule\n\\multirow{2}{*}{\\texttt{MN}}\n& zero-shot & 46.2 & 52.1 & 55.3 & 56.0 & 49.3 & 52.1 & 52.6 \\\\ %\n& fine-tune & 80.9 & 84.3 & 84.1 & 88.9 & 77.7 & 81.2 & 78.7 \\\\ %\n\\midrule\n\\multirow{2}{*}{\\texttt{SST-2}}\n& zero-shot & 50.9 & 50.9 & 51.6 & 51.9 & 51.6 & 50.9 & 50.9 \\\\ %\n& fine-tune & 92.9 & 92.9 & 94.8 & 93.4 & 89.3 & 90.1 & 90.3 \\\\ %\n\\midrule\n\\multirow{2}{*}{\\texttt{HS}}\n& zero-shot & 33.7 & 46.2 & 58.4 & 65.9 & 53.2 & 64.9 & 70.5 \\\\ %\n& fine-tune & 50.7 & 64.8 & 74.1 & 90.0 & 37.3 & 45.4 & 42.2 \\\\ %\n\\midrule\n\\multirow{2}{*}{\\texttt{PI}}\n& zero-shot & 65.3 & 70.6 & 74.6 & 76.6 & 74.3 & 76.8 & 78.2 \\\\ %\n& fine-tune & 68.2 & 71.7 & 71.2 & 80.3 & 66.3 & 66.1 & 68.3 \\\\ %\n\\midrule\n\\multirow{2}{*}{\\texttt{WG}}\n& zero-shot & 52.1 & 54.2 & 58.1 & 61.4 & 53.4 & 57.4 & 60.9 \\\\ %\n& fine-tune & 65.7 & 63.3 & 67.4 & 69.5 & 50.2 & 56.0 & 50.4 \\\\ %\n\\bottomrule\n\\end{tabular}}\n\\end{small}\n\\end{center}\n\\caption{\n\\textbf{Fully supervised fine-tuning accuracy compared with zero-shot accuracy.}  \\texttt{SC}: StoryCloze, \\texttt{OB}: OpenBookQA, \\texttt{BQ}: BoolQ, \\texttt{MN}: MNLI, \\texttt{HS}: HellaSwag, \\texttt{PI}: PIQA, \\texttt{WG}: WinoGrande. Largest models omitted owing to their high resource utilization.}\n\\label{tab:finetuning}\n\\end{table}\n\n\n\n\\section{Conclusion}\n\nWe present results for scaling sparse Language Models up to 1.1T parameters. We observe that up to this scale sparse models offer better performance vs. computation trade-off when compared to their dense counterparts for language modeling, zero- and few-shot learning. While the gap begins to close at scale our biggest sparse model outperforms its dense counterpart where the latter requires twice as much computation. These results confirm that sparse MoE models can provide an alternative to widely used dense architectures that saves computation and reduces model energy consumption.\n\n\n\n\\section*{Ethical considerations}\nPrevious work~\\cite{sheng2019woman,bordia2019identifying,nadeem2020stereoset,de2021stereotype} has observed that language models absorb bias and toxicity represented in the training data. So as to better understand the potential harms of our models in this front, we evaluated them on StereoSet~\\cite{nadeem2020stereoset} and CrowS-Pairs~\\cite{nangia-etal-2020-crows}, and report our results in Appendix \\ref{app:potential_harms}. Our results show that the percentage of bias and stereotype in dense and MoE models is comparable, especially at scale.\nMoreover, in general, we note worse performance (more bias\/stereotyping) at larger scales. This observation points to more research needed in order to mitigate such behavior. Intuitively however, we believe that sparse models may be inherently more controllable -- e.g.~designing specific experts -- than dense models. We leave this line of investigation for future research. \n\nAnother concern of scaling language models is the energy usage and the associated environmental impact required for training, which we discuss in detail in Appendix \\ref{app:co2}. Nevertheless, our work shows that MoEs can be a more compute-efficient alternative to traditional dense models, which could alleviate the environmental impact of future scaling efforts.\nMoreover, by releasing all of our pre-trained language models, we believe we have alleviated some exploration burden for the community and the environment, allowing for more efficient offsets for other researchers.\n\nIn the spirit of transparency and allowing for maximal replicability and accountability, we include data and model cards together with our code.  \n\n\n\\section*{Limitations}\n\nOur study is limited to one specific MoE configuration. In particular, our sparse and dense models use the same hyperparameters and model structure, closely following GPT-3. However, it is possible that this configuration is suboptimal for scaling MoE models. Similarly, we did not explore different MoE-specific hyperparameters, such as the number of experts. Finally, while our work reveals that the performance gap between MoE and dense models varies greatly across tasks and domains, the specific factors that make certain tasks more favorable for MoEs remain unclear.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEnergy levels, eigenvalues of a Hamiltonian, play the most primary role in\ndetermining properties of a quantum system. When energy levels cross\nor avoided cross as parameters of a Hamiltonian vary, various interesting\nphenomena happen. For example, if two instantaneous energy levels of a \ntime-dependent Hamiltonian are avoided crossing, the non-adiabatic tunneling \ncalled the Landau-Zener tunneling between them takes place~\\cite{Landau}. \nClosely related to this, the runtime of adiabatic quantum computation is \ninversely proportional to the square of the energy gap between the ground \nand first exited levels~\\cite{Farhi01}. An eigenstate encircling \nadiabatically degeneracy points accumulates a Berry phase in addition to \na dynamical phase~\\cite{Berry84,Shapere}. In quantum chemistry, a conical \nintersection of electronic energy surfaces of molecules plays a key role \nin understanding ultrafast radiationless reactions~\\cite{Yarkony96,Kuppermann}.\nA quantum phase transition, a dramatic change in a ground state as parameters \nof a system vary, is related with crossings or avoided crossings of\ntwo lowest energy levels~\\cite{Vojta03}. Kais {\\it et al}. have shown that \nthe finite size scaling method can be used for studying the critical behavior, \ni.e., the level degeneracy or absorption, of a few-body quantum Hamiltonian \n$H(\\lambda_1,\\cdots,\\lambda_k)$ as a function of a set of parameters \n$\\{\\lambda_i\\}$~\\cite{kais0,kais12}. These parameters could be the external \nfields, inter-atomic distances, nuclear charges for stability of negative \nions, cluster size, and optical lattice parameters such as the potential \ndepth~\\cite{kais34}. Thus, it is important to develop a way of finding level \ncrossings and to understand how eigenstates or relevant physical quantities \nchange at crossing or avoided crossings.\n\nRecently Bhattacharya and Raman presented a powerful algebraic method for \nfinding level crossings without solving an eigenvalue problem \ndirectly~\\cite{Bhattacharya06}. Along with this mathematical \nway, it is necessary to understand what physical quantities can be used to \ndetect or characterize crossings or avoided crossings. First of all, the \nmeasurement of a Berry phase could be a good way to detect level crossings \nbecause it due to level crossings. It is well known that avoided crossings \nor glancing intersections are not the source of Berry phases~\\cite{Yarkony96}.\nIs there any way that Berry phases can detect {\\it avoided level \ncrossings} ? Here we show that the marginal Berry phase of an entangled\nstate could be an indicator to avoided level crossings.\n\nThe entropy is an another indicator to level crossings. Since level crossing \nor avoided crossings are accompanied with a drastic change in eigenstates, \nany contents of information on relevant eigenstates may also vary. The Shannon \nentropy of the electron density measures the delocalization or the lack of \nstructure in the respective distribution. Thus the Shannon entropy is maximal \nfor uniform distribution, that is, for an unbound system, and is minimal when\nthe uncertainty about the structure of the distribution is minimal~\\cite{kais5}.\nGonz\\'alez-F\\'erez and Dehesa showed that the Shannon entropy could be \nused as an indicator of avoided crossings~\\cite{Gonzalez03}. The von Neumann \nentropy, the quantum version of the Shannon entropy, is a good entanglement \nmeasure for a bipartite pure state.  In quantum information, much attention \nhas been paid to the relation between entanglement and quantum phase \ntransitions~\\cite{Amico08,kais6}. Recently one of the authors investigated  \nthe relation between entanglement, Berry phases, and level crossings for two \nqubits with the XY-type interaction and found that the level crossing is not \nalway accompanied with the abrupt change in entanglement~\\cite{scoh08}. \n\nIn this paper, in order to study how entanglement and Berry phases vary at \nlevel crossings, we consider the Breit-Rabi Hamiltonian describing a hyperfine \ninteraction of electron and nuclear spins in a uniform magnetic \nfield~\\cite{Breit-Rabi}. It is shown that the von Neumann entropy of the\nelectron (or nuclear) spin is maximum at avoided crossings. \nIt is demonstrated that the significant changes in Berry phases and \nentanglement are closely related to level crossings. \nWe show that the marginal Berry phase of the electron (or nuclear) spin \ncould be a good indicator to avoided level crossings. \nThe marginal Berry phase has nodal points at the avoided crossing points. \n\nThe paper is organized as follows. In Sec.~\\ref{section_BR}, the Breit-Rabi\nHamiltonian is introduced. In Sec.~\\ref{section_hydrogen}, as a specific\napplication of the Breit-Rabi Hamiltonian, we consider a hyperfine interaction\nbetween a nuclear spin $1\/2$ and an electron spin $1\/2$ of a hydrogen atom \nin a magnetic field. We analyze the close relation between entanglement,\nBerry phases, and level crossings. In Sec.~\\ref{section_sodium}, we make \nan similar analysis for a sodium atom with a nuclear spin $3\/2$ and an \nelectron spin $1\/2$. In Sec.~\\ref{section_conclusion}, we summarize the main \nresults.\n\n\\section{Breit-Rabi Hamiltonian}\n\\label{section_BR}\n\nLet us consider an atom with a single valence electron in the ground state \nwith orbital angular momentum $L=0$. In the presence of a uniform magnetic \nfield $B$ in the $z$ direction, its atomic spectrum is described by \nthe Breit-Rabi Hamiltonian~\\cite{Breit-Rabi}, which is given by the sum of \nthe hyperfine interaction between a nuclear spin $\\bf I$ and an electron \nspin $\\bf S$ and their Zeeman couplings to the magnetic field \n\\begin{equation}\nH = A\\, {\\bf I}{\\bm \\cdot}{\\bf S} + (a S_z + b I_z)B \\,,\n\\label{Hamil_BR}\n\\end{equation}\nwhere $A$ is the hyperfine coupling constant, $a=\\gamma_e\\hbar$, and \n$b=\\gamma_n\\hbar$. Here $\\gamma_e$ and $\\gamma_n$ are the electron and \nnuclear gyromagnetic ratios, respectively. The electron spin operator \n${\\bf S}$ and the nuclear spin operator ${\\bf I}$ are measured in the unit \nof $\\hbar$. \n\nThe Breit-Rabi Hamiltonian~(\\ref{Hamil_BR}) is well studied to describe double\nresonance in nuclear magnetic resonance~\\cite{Slichter} and the muon spin \nrotation in semiconductors~\\cite{Patteson88}. Although simple and well\nunderstood, it still continues to provide new insights. Recently Bhattacharya \nand Raman found a new class of invariants of the Breit-Rabi\nHamiltonian~\\cite{Bhattacharya06}. As will be shown here, it is a prime example\nfor showing the close relation between level crossings, entanglement, and \ngeometric phases. Also it is related to a Hamiltonian of \nelectron spin qubits in quantum dots~\\cite{Loss98} where the Heisenberg \ninteraction between two electron spins can be turned on and off to \nimplement the controlled-not gate.\n\nBefore applying the Hamiltonian~(\\ref{Hamil_BR}) to specific systems, let us\nlook at its general properties. If $B =0$, then $H$ commutes with \nboth the square of the total spin operator ${\\bf J}^2$ and $J_z$, where the \ntotal spin operator is defined by ${\\bf J} \\equiv {\\bf I} + {\\bf S}$. However,\nfor $B\\ne 0$, due to the fact that $a \\ne b$, the Hamiltonian~(\\ref{Hamil_BR})\nno longer commutes with ${\\bf J}^2$, but still commutes with $J_z$. So the \neigenvalue $m$ of $J_z$ is a good quantum number for the Breit-Rabi \nHamiltonian. With ladder operators, $S_\\pm = S_x \\pm i S_y$ and \n$I_\\pm = I_x \\pm i I_y$, the Hamiltonian~(\\ref{Hamil_BR}) can be rewritten as\n\\begin{equation}\nH = A\\,I_zS_z + \\frac{A}{2}(S_{+}I_{-} + S_{-}I_{+}) + B (a S_z + b I_z)\\,.\n\\label{Hamil_BR2}\n\\end{equation}\nLet us use a simple notation $\\ket{m_S,m_I}$ to represent the product state \n$\\ket{S,m_S}\\otimes\\ket{I,m_I}$, where $\\ket{S,m_S}$ is an eigenstate of\n${\\bf S}^2$ and $S_z$, and $\\ket{I,m_I}$ is an eigenstate of ${\\bf I}^2$ and \n$I_z$. The first and third terms in Eq.~(\\ref{Hamil_BR2}) give the diagonal \nmatrix elements \n\\begin{subequations}\n\\begin{equation}\n\\bra{m_S\\,m_I}H\\ket{m_S\\,m_I} \n= f(m_S, m_I) = A\\,m_S\\,m_I  + m_S\\,aB + m_I\\,bB \\,.\n\\end{equation}\nThe second term in Eq.~(\\ref{Hamil_BR2}) corresponds to the off-diagonal \nmatrix elements\n\\begin{align}\n&\\bra{m_S'\\,m_I'}S_{+}I_{_-}\\ket{m_S\\,m_I} \\nonumber\\\\ \n&=\\sqrt{(S-m_S)(S+m_s+1)}\\,\\sqrt{(I+m_I)(I-m_I+1)}\\,\n  \\delta_{m_S',m_S+1}\\delta_{m_I',m_I-1}\\,.\n\\end{align}\n\\label{matrix_element}\n\\end{subequations}\nSince $m_S'-m_S =1$ and $m_I' - m_I= -1$ (or vice versa), one has the \nselection rule, $\\Delta m = (m_S'+m_I') -(m_S+m_I) =0$, that is, the magnetic\nquantum number $m=m_S + m_I$ is conserved. This implies that the \nHamiltonian~(\\ref{Hamil_BR}) is block diagonal in the basis set \n$\\{\\ket{m_S,m_I}\\}$ ordered by $m$. \n\n\\section{The Hydrogen Atom in a Uniform Magnetic Field}\n\\label{section_hydrogen}\n\n\\subsection{Eigenvalues and Eigenstates}\n\nAs a simple but real system described by the Hamiltonian~(\\ref{Hamil_BR}), \nlet us consider the interaction between the nuclear spin $I=1\/2$ and the \nelectron spin $S=1\/2$ of a hydrogen atom in a uniform magnetic field.\nSince $H$ commutes with the $z$-component of the total spin operator, \n$J_z = S_z + I_z$, it is convenient to arrange the product basis \n$\\{\\ket{m_S,m_I}\\}$ in the decreasing order of the magnetic quantum number \n$m$ of $J_z$ as \n$\\left\\{\\ket{\\tfrac{1}{2},\\tfrac{1}{2}}, \\ket{\\tfrac{1}{2},-\\tfrac{1}{2}},\n\\ket{-\\tfrac{1}{2},\\tfrac{1}{2}}, \\ket{-\\tfrac{1}{2},-\\tfrac{1}{2}}\\right\\}$. \nBy means of Eqs.~(\\ref{matrix_element}), the Breit-Rabi Hamiltonian for a\nhydrogen atom can be written in the ordered basis\n\\begin{align}\nH\n= \\frac{1}{4}\n\\begin{pmatrix}\nA + 2(a+b)B & 0     & 0           & 0 \\\\\n0      & -A+2(a-b)B & 2A          & 0 \\\\\n0      & 2A         & -A -2(a-b)B & 0 \\\\\n0      & 0          & 0           & A -2(a+b)B\n\\end{pmatrix}\\,.\n\\label{Hamil_Hydrogen}\n\\end{align}\nThe Hamiltonian~(\\ref{Hamil_Hydrogen}) is block diagonal, so it is \nstraightforward to obtain its eigenvalues and eigenvectors.\nThe subspace of $m=\\pm 1$ is spanned by \n$\\left\\{ \\ket{\\tfrac{1}{2},\\tfrac{1}{2}}, \n         \\ket{-\\tfrac{1}{2},-\\tfrac{1}{2}}\\right\\}$. \nThe block Hamiltonian on this subspace is already diagonal and has its \neigenvalues and eigenvectors\n\\begin{subequations}\n\\label{H_eigen1}\n\\begin{align}\nE_{\\pm 1}       &= \\frac{A}{4} \\pm \\frac{1}{2}(a + b)B \\,,\\\\\n\\ket{E_{\\pm 1}} &= \\ket{\\pm\\tfrac{1}{2},\\pm\\tfrac{1}{2}},\n\\label{H_eva}\n\\end{align}\n\\end{subequations}\nwhere the subscripts `$\\pm 1$' in $E_{\\pm 1}$ denote the magnetic quantum \nnumber $m=\\pm 1$. The block Hamiltonian with $m=0$ is defined on the subspace \nof $\\left\\{ \\ket{\\tfrac{1}{2}, -\\tfrac{1}{2}}, \n            \\ket{\\tfrac{1}{2}, -\\tfrac{1}{2}} \\right\\}$ and is written as \n\\begin{align}\nH_{m=0} \n&= \\frac{1}{4}\n   \\begin{pmatrix}\n   -A +2(a-b)B & 2A \\\\\n    2A         & -A - 2(a-b)B\n   \\end{pmatrix}\\,.\n\\end{align}\nOne can interpret the Hamiltonian $H_{m =0}$ as that of a spin in an \neffective magnetic field in the $x$-$z$ plane, ${\\bf B}_{\\rm eff} \n\\equiv (A\/2, 0, (a-b)B\/2)$. \nThe eigenvalues and eigenvectors of $H_{m=0}$ can be written easily as\n\\begin{subequations}\n\\label{H_eigen0}\n\\begin{align}\nE_{0}^{\\pm} \n&= -\\frac{A}{4} \\pm \\frac{1}{2}\\sqrt{(a-b)^2 B^2 + A^2}\\,, \\\\\n\\ket{E_{0}^{+}} \n&= \\phantom{+} \\cos\\frac{\\alpha}{2}\\, \\ket{ \\tfrac{1}{2},-\\tfrac{1}{2}} \n            +  \\sin\\frac{\\alpha}{2}\\, \\ket{-\\tfrac{1}{2}, \\tfrac{1}{2}}\\,,\n\\label{H_evb} \\\\\n\\ket{E_{0}^{-}} \n&= - \\sin\\frac{\\alpha}{2}\\, \\ket{ \\tfrac{1}{2},-\\tfrac{1}{2}} \n   + \\cos\\frac{\\alpha}{2}\\, \\ket{-\\tfrac{1}{2}, \\tfrac{1}{2}}\\,,\n\\label{H_evc}\n\\end{align}\n\\end{subequations}\nwhere $\\tan\\alpha\\equiv \\frac{A}{(a-b)B}$. In a weak magnetic field limit \n(so called the Zeeman region), the Zeeman energy is smaller\nthan the hyperfine coupling. At $B= 0$, i.e., $\\alpha = \\pi\/2$, \nthe ground eigenstate $\\ket{E_{0}^{-}}$ becomes the singlet state,\n$\\ket{E_{0}^{-}} = \\frac{1}{\\sqrt{2}}\\left(\\ket{-\\tfrac{1}{2},\\tfrac{1}{2}} -\n\\ket{\\tfrac{1}{2}, -\\tfrac{1}{2}} \\right)$. In a strong magnetic field called\nthe Paschen-Back region, the Zeeman couplings are dominant. That is, in   \nlimit of $B\\to \\infty$, one has $\\alpha \\to 0$ and $\\ket{E_{0}^{-}} \\to\n\\ket{-\\tfrac{1}{2},\\tfrac{1}{2}}$.\n\nThe eigenvalues and eigenstates of the Breit-Rabi Hamiltonian for a hydrogen \natom, Eqs.~(\\ref{H_eigen1}) and (\\ref{H_eigen0}) depend on two parameters: \nthe hyperfine constant $A$ and the magnetic field $B$. The hyperfine constant \n$A$ of the hydrogen atom in vacuum is positive. However, if a hydrogen atom\nis in an inert gas, the hyperfine constant $A$ could be \nnegative~\\cite{Foner60}, resembling the spin-spin coupling constant in \na Heisenberg model. We assume that $A$ as well as $B$ varies and can be \nnegative. To this end, $A$ in Eqs.~(\\ref{H_eigen1}) and (\\ref{H_eigen0}) is \nreplaced by $fA$ with $-1\\le f\\le 1$, so $A$ is still kept the positive \nconstant in vacuum. If $f$ is negative, so the hyperfine constant.\n\nDepending on $f$ and $B$, the ground state of the \nHamiltonian~(\\ref{Hamil_Hydrogen}) is given either by $\\ket{E_{\\pm 1}}$ or by \n$\\ket{E_{0}^{-}}$. It is convenient to plot the energy levels normalized by \n$A$.  Then, Eqs.~(\\ref{H_eigen1}) and (\\ref{H_eigen0}) become\n$E_{\\pm1}\/A = \\frac{f}{4} \\pm \\frac{1}{2}(a' + b') B$ and \n$E_{0}^{\\pm}\/A= -\\frac{f}{4} \\pm \\frac{1}{2}\\sqrt{(a'-b')^2 B^2 + f^2}$,\nwhere $a' \\equiv a\/A \\approx 19.767\\,\\,\\text{T}^{-1}$ and \n      $b'\\equiv b\/A \\approx -0.03\\,\\,\\text{T}^{-1}$ are taken \nform Ref.~\\cite{Arimondo77}, and $B$ is measured in the unit of tesla. \nThe energy levels $E_m\/A$ are plotted as functions of $B$ for $f=1$ in \nFig.~\\ref{Fig1} (a) and for $f=-0.5$ in Fig.~\\ref{Fig1} (b). For $f\\ge 0$, \nthe ground level is $E_{0}^{-}$. For $f<0$, two levels, $E_{0}^{\\pm}$ and\n$E_{\\pm1}$, with different magnetic quantum numbers cross at  \n$f =\\frac{2a'b'}{a'-b'} |B|$. Fig.~\\ref{Fig2} (a) shows the energy gap \n$\\Delta\/A$ between the ground and first exited states as a function of $f$ \nand $B$, where we take $a' =0.1\\,\\,\\text{T}^{-1}$ and \n$b' = -0.01\\,\\,\\text{ T}^{-1}$ to see clearly the phase diagram of \nthe ground state of the Hamiltonian~(\\ref{Hamil_Hydrogen}) determined by \nthe magnetic quantum number $m$. As shown in Fig.~\\ref{Fig2} (a), the energy\ngap $\\Delta\/A$ vanishes along the lines defined by \n$f = \\frac{2a'b'}{a'-b'} |B|$ \nand the negative $f$ axis.  In the region of $f <\\frac{2a'b'}{a'-b'} |B|$, \nthe ground state becomes either $\\ket{E_{+1}}$ or $\\ket{E_{-1}}$ with the \nmagnetic quantum number $m=1$ or $m=-1$, respectively. On the other hand, \nthe ground state in the region defined by $f >\\frac{2a'b'}{a'-b'} |B|$ is \ngiven by $\\ket{E_{0}^{-}}$ with the magnetic quantum number $m=0$.\nOne can see that the magnetic quantum number $m$ of the ground state \nchanges abruptly at the level crossing points.\n\n\\subsection{Entanglement}\n\nLet us discuss the relation between level crossings and entanglement. \nEntanglement refers to the quantum correlation between subsystems and has no \nclassical analog~\\cite{book-Gruska,kais7}. When level crossing happens as the \nparameter of the Hamiltonian varies, the ground state changes drastically. \nEntanglement as a physical quantity may also undergo a significant change. \nHowever, entanglement is not always a good indicator to level crossing as \nshown in Ref.~\\cite{scoh08}.\n\nFirst, let us examine the relation between entanglement and level crossings \nfor each eigenstate. The von Neumann entropy $S$ of a subsystem is a good \nentanglement measure for a pure bipartite system. If $\\ket{\\psi_{AB}}$ is \na quantum state of a system composed of two subsystems $A$ and $B$, the \nentanglement between $A$ and $B$ is measured by the von Neumann entropy of \nthe subsystem,\n$S(\\rho_A) = -{\\rm tr}(\\rho_A\\log\\rho_A)= S(\\rho_B) \n           = -{\\rm tr}(\\rho_B\\log\\rho_B)$, where the reduced density matrix \n$\\rho_A$ of the subsystem $A$ is obtained by tracing out the degrees of \nfreedom of $B$ as $\\rho_A ={\\rm tr}_B( \\ket{\\psi_{AB}} \\bra{\\psi_{AB}} )$.     \nIf the ground state is given by $\\ket{E_{\\pm 1}}$, i.e., a product state, \nthen the von Neumann entropy $S$ of the electron (or nuclear) spin \nis zero. On the other hand, for the quantum state $\\ket{E_0^{\\pm}}$ of \nthe electron and nuclear spins, the von Neumann entropy of the electron \n(or nuclear) spin can be written as\n\\begin{align}\nS(\\rho_A) = -\\frac{1+\\cos\\alpha}{2}\\,\\log_2\\frac{1+\\cos\\alpha}{2}\n            -\\frac{1-\\cos\\alpha}{2}\\,\\log_2\\frac{1-\\cos\\alpha}{2}\\,.\n\\end{align}\nFig.~\\ref{Fig1} (c) shows the von Neumann entropy of the electron (or nuclear) \nspin for each eigenstate as a function of $B$. For the eigenstates\n$\\ket{E_{0}^{\\pm}}$, it is maximum at $B=0$, i.e., at the avoided crossing\npoint. This is analogous to the sharp change in Shannon entropy at avoided \ncrossing in Ref.~\\cite{Gonzalez03}.\n\nNow, let us look at how entanglement changes at level crossings as the\nparameters of the Hamiltonian vary. Fig.~\\ref{Fig2} (b) plots the von Neumann \nentropy $S$ of the electron (or nuclear ) spin for the ground state as \na function of $f$ and $B$. Across the level crossing line, \n$f=\\frac{2a'b'}{a'-b'}|B|$, the von Neumann entropy changes abruptly. \nFor $f>0$, $S$ becomes 1 as $B$ goes to 0. Along the line of $f=0$, \nthe von Neumann entropy $S$ vanishes even though there is no level crossing. \n\n\\subsection{Berry Phase}\n\\label{hydrogen_berry}\n\nAn instantaneous eigenstate encircling the energy level crossing points \nacquires the Berry phase in addition to the dynamical phase. The information\non the level crossings is encoded in the Berry phase. At $B=0$ and $f=1$, the \ntwo levels $E_{\\pm1}$ cross and the other two levels $E_{0}^{\\pm}$ are avoided \ncrossing. Also $E_{\\pm1}$ and $E_{0}^{-}$ cross at $f=\\frac{2a'b'}{a'-b'}|B|$. \nHere we focus on the Berry phase due to the level crossing or avoided crossing \nat $B=0$.\n\nDue to the fact that $a\\gg |b|$, an electron spin rotates much faster than a \nnuclear spin. We assume the magnetic field ${\\bf B}$ is rotated slowly enough\nfor both the electron and nuclear spins to evolve adiabatically. The magnetic\nfield ${\\bf B} = B\\,{\\bf\\hat{n}}$ in the direction of ${\\bf\\hat{n}} = \n(\\sin\\theta\\cos\\phi, \\sin\\theta\\sin\\phi,\\cos\\theta)$ is constructed starting \nfrom ${\\bf B} = B\\,{\\bf \\hat{z}}$.  First, it is rotated about the $y$ axis by \nangle $\\theta$. And it is subsequently rotated about the $z$ axis by angle \n$\\phi$. By applying SU(2) rotations corresponding to the above SO(3) rotations \non the Hamiltonian~(\\ref{Hamil_BR}), one obtains the Breit-Rabi Hamiltonian in \nthe magnetic field ${\\bf B} = B\\, {\\bf\\hat{n}}$\n\\begin{align}\nH(\\theta,\\phi) \n= A\\,{\\bf I}{\\bm \\cdot}{\\bf S} \n+ a\\,{\\bf B}{\\bm \\cdot}{\\bf S} + b\\,{\\bf B}{\\bm \\cdot}{\\bf I}\\,.\n\\label{Hamil_Berry}\n\\end{align}\nThe hyperfine interaction $A\\,{\\bf I}{\\bm \\cdot}{\\bf S}$ is spherical\nsymmetric, so the eigenvalues and eigenvectors of the \nHamiltonian~(\\ref{Hamil_Berry}) are identical to those of the \nHamiltonian~(\\ref{Hamil_Hydrogen}) except replacing \n$\\ket{\\pm\\frac{1}{2}}$ by $\\ket{{\\bf \\hat{n}};\\pm\\frac{1}{2}}$.\nHere $\\ket{{\\bf \\hat{n}};\\pm\\frac{1}{2}}$ are eigenstates of \n${\\bf\\hat{n}}{\\bm \\cdot}{\\bf S}$ or ${\\bf\\hat{n}}{\\bm \\cdot}{\\bf I}$. \nIf the magnetic field ${\\bf B}$ is rotated slowly about the $z$ axis by $2\\pi$ \nto make a cone with a solid angle $\\Omega = 2\\pi(1-\\cos\\theta)$, then the\ninstantaneous eigenstate $\\ket{{\\bf \\hat{n}};\\pm\\frac{1}{2}}$ follows it and\naccumulates the Berry phase $\\beta_\\pm =\\mp\\frac{1}{2}\\Omega$. The total Berry \nphase $\\beta$ of electron and nuclear spins is the sum of two phases acquired \nby each one. It depends on the magnetic quantum number $m$ \n\\begin{align}\n\\beta = \\left\\{ \\begin{array}{cl}\n                \\mp\\Omega \\quad &\\text{for $m=\\pm 1$} \\,,\\\\\n                0         \\quad &\\text{for $m=0$} \\,.\n                \\end{array} \\right.\n\\end{align}\nAs expected, the Berry phase is nonzero only for real crossings, i.e., \n$m=\\pm1$. Fig.~\\ref{Fig2} (c) plots the total Berry phase as a function of \n$B$ and $f$ and shows that the total Berry phase jumps at the level crossings.\nThe zero Berry phase of the eigenstates $\\ket{E_{0}^{\\pm}(\\theta,\\phi)}$ can \nbe understood in two ways. First, two levels $E_{0}^{\\pm}$ are avoided \ncrossing at $B =0$, so it is zero. Another view is as follows.\nSince $\\ket{E_{0}^{\\pm}}$ is a superposition of \n$\\ket{\\tfrac{1}{2},-\\tfrac{1}{2}}$ and $\\ket{-\\tfrac{1}{2},\\tfrac{1}{2}}$, \nthe Berry phase of the electron spin is opposite to that of the nuclear spin \nand they cancel each other. \n\t  \nAlthough the entangled states $\\ket{E_{0}^{\\pm}}$ of electron and nuclear spins \naccumulates no Berry phase, each subsystem (electron spin or nuclear spin) can \nget nonzero marginal Berry phases of mixed states. Following the studies on\ngeometric phase of mixed states~\\cite{Sjoqvist00,Sjoqvist05} and the relation \nbetween entanglement and marginal Berry phases~\\cite{Yi04a,Yi04b,Sjoqvist05}, \nwe investigate the relation between avoided level crossings, marginal Berry \nphases, and entanglement. For an adiabatic cyclic evolution parameterized by\n$\\bf x$, an instantaneous eigenstate of a bipartite system $AB$ can be \nexpressed in a Schmidt decomposition $\\ket{\\psi({\\bf x})} = \\sum_{i=1}^{M}\n\\sqrt{p_i} \\ket{e_i({\\bf x})} \\otimes\\ket{f_i({\\bf x})}$, where \n$\\{\\ket{e_i({\\bf x})}\\}_{i=1}^{N_A}$ is an orthonormal basis for a subsystem\n$A$, $\\{\\ket{f_i({\\bf x})}\\}_{i=1}^{N_B}$ for a subsystem $B$, \n$M\\le\\min\\{N_A,N_B\\}$, and $\\sum_{i=1}^M p_i =1$. Here our attention is\nrestricted to the case that the Schmidt coefficients $\\sqrt{p_i}$ are\nindependent of ${\\bf x}$. After an adiabatic cyclic evolution implemented by\n${\\bf x}(0) = {\\bf x}(T)$, the total Berry phase of the bipartite system $AB$ \nis given by \n\\begin{align}\n\\beta=\\sum_{i=1}^{M} p_i \\left(\\beta^{A}_{i} + \\beta^{B}_{i}\\right)\\,,\n\\label{berry_sum}\n\\end{align}\nwhere $\\beta^{A}_{i} = i\\oint_C d{\\bf x}{\\cdot} \\bra{e_i({\\bf x})} \n                       \\nabla_{\\bf x} \\ket{e_i({\\bf x})}$. Then the marginal \nmixed state Berry phase $\\Gamma_A$ of a subsystem $A$ is defined by \n\\begin{align}\n\\Gamma_A =\\arg\\sum_{i}p_i \\exp\\left(i\\beta^A_i\\right)\\,.\n\\label{berry_mixed}\n\\end{align}\n\nWith Eqs.~(\\ref{berry_sum}) and~(\\ref{berry_mixed}), let us analyze how the \ntotal Berry phase and the marginal Berry phase of $\\ket{E_0^{-}}$ depend on\n$B$. The two Schmidt coefficients are given by $p_1 = \\sin^2\\tfrac{\\alpha}{2}$ \nand $p_2=\\cos^2\\tfrac{\\alpha}{2}$. It is easy to obtain the marginal \nBerry phase of the electron spin \n$\\Gamma_e = \\arctan\\left(\\cos\\alpha\\tan\\frac{\\Omega}{2}\\right)$\nand the average Berry phase of the electron spin \n$\\beta_e \\equiv p_1\\beta_1^e + p_2 \\beta_2^e =\\frac{\\Omega}{2}\\cos\\alpha$. \nIn the limit of $B\\gg1$, i.e., $\\alpha\\to 0$, one has $\\ket{E_0^{-}} \\to\n\\ket{-\\tfrac{1}{2},\\tfrac{1}{2}}$ and $\\beta_e={\\Omega}\/{2}$. Also the marginal\nBerry phase of the electron spin is given by $\\Gamma_e = {\\Omega}\/{2}$ for \n$0\\le \\theta <\\frac{\\pi}{2}$.  Fig.~\\ref{Fig3} plots $\\Gamma_e$\nas a function of $B$ and the azimuthal angle $\\theta$. The marginal Berry \nphase of the electron spin jumps at $\\theta=\\pi\/2$ and $B=0$.\nThe node at $B=0$ corresponds to the avoided crossing. \n\n\\section{The Sodium  Atom in a Uniform Magnetic Field}\n\\label{section_sodium}\n\n\\subsection{Energy Spectrum}\nNow we consider an $^{23}$Na atom in its $3S_{1\/2}$ ground state in the \npresence of a uniform magnetic field $B$ along the $z$ axis. The nuclear \nand electron spins of an $^{23}$Na atom are $I=2\/3$ and $S=1\/2$, respectively.\nAs in Sec.~\\ref{section_hydrogen}, it is convenient to arrange \nthe product basis $\\{\\ket{m_S,m_I}\\}$ in the decreasing order of \nthe magnetic quantum number $m$ of $J_z$ as follows.\n$\\left\\{\\ket{\\frac{1}{2},\\frac{3}{2}} \\right\\}$,\n$\\left\\{\\ket{\\frac{1}{2},\\frac{1}{2}}, \\ket{-\\frac{1}{2},\\frac{3}{2}}\\right\\}$,\n$\\left\\{\\ket{\\frac{1}{2},-\\frac{1}{2}},\\ket{-\\frac{1}{2},\\frac{1}{2}}\\right\\}$, \n$\\left\\{\\ket{\\frac{1}{2},-\\frac{3}{2}},\\ket{-\\frac{1}{2},-\\frac{1}{2}}\\right\\}$,\nand $\\left\\{\\ket{-\\frac{1}{2},-\\frac{3}{2}} \\right\\}$. For example, \n$\\left\\{ \\ket{\\frac{1}{2},\\frac{1}{2}}, \\ket{-\\frac{1}{2},\\frac{3}{2}}\\right\\}$ \nspans the subspace of $m=1$. In this ordered basis set, the \nHamiltonian~(\\ref{Hamil_BR}) for the sodium atom  can be represented by \na block-diagonal matrix\n\\begin{align}\nH = \n\\begin{pmatrix}\nf({\\frac{1}{2},\\frac{3}{2}}) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & f(\\frac{1}{2},\\frac{1}{2}) & \\frac{\\sqrt{3}}{2}A & 0 & 0 & 0 & 0 & 0 \\\\\n0 &\\frac{\\sqrt{3}}{2}A & f(\\frac{-1}{2},\\frac{3}{2}) & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & f(\\frac{1}{2},\\frac{-1}{2}) & \\frac{A}{2} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{A}{2} & f(\\frac{-1}{2},\\frac{1}{2}) & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & f(\\frac{1}{2},\\frac{-3}{2}) & \\frac{\\sqrt{3}}{2}A & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{\\sqrt{3}}{2}A & f(\\frac{-1}{2},\\frac{-1}{2}) & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & f(\\frac{-1}{2},\\frac{-3}{2})  \n\\end{pmatrix},\n\\label{Hamil_Sodium}\n\\end{align}\nwhere $f(m_S,m_I) \\equiv A\\,m_S\\,m_I + m_S\\,aB + m_I\\,bB$.\nEach block is at most a $2\\times 2$ matrix and  can be easily diagonalized.\nFirst, consider the subspace of $m = \\pm 2$. The corresponding eigenvalues \nand eigenvectors can be written as\n\\begin{subequations}\n\\label{Na_m2}\n\\begin{align}\nE_{\\pm 2} &= \\frac{3}{4}A \\pm \\frac{1}{2}(a+3b)B\\,,\\\\\n\\ket{E_{\\pm 2}} &= \\ket{\\pm\\tfrac{1}{2},\\pm\\tfrac{3}{2}} \\,. \n\\label{Na_ev2}\n\\end{align}\n\\end{subequations}\nNotice that Eqs.~(\\ref{Na_m2}) are comparable to Eqs.~(\\ref{H_eigen1}).\nSecond, in the subspace with $m= 1$ spanned by \n$\\left\\{\\ket{\\frac{1}{2},-\\frac{1}{2}}, \\ket{-\\frac{1}{2},\\frac{1}{2}}\\right\\}$, \none obtains the eigenvalues and eigenvectors,\n\\begin{subequations}\n\\label{Eq_plus}\n\\begin{align}\nE_{+1}^{\\pm} \n&= -\\frac{A}{4} + bB \\pm \\frac{1}{2}\\sqrt{\\bigl(A + (a-b)B\\bigr)^2 + 3A^2}\\,,\\\\\n\\ket{E_{+1}^+} \n&= \\phantom{+} \\cos\\frac{\\alpha_1}{2}\\, \\ket{ \\tfrac{1}{2},\\tfrac{1}{2}} \n            +  \\sin\\frac{\\alpha_1}{2}\\, \\ket{-\\tfrac{1}{2},\\tfrac{3}{2}}\\,,\n\\label{Na_p1p}\\\\\n\\ket{E_{+1}^-}&= -\\sin\\frac{\\alpha_1}{2}\\, \\ket{ \\tfrac{1}{2},\\tfrac{1}{2}} \n            +  \\cos\\frac{\\alpha_1}{2}\\, \\ket{-\\tfrac{1}{2},\\tfrac{3}{2}}\\,,\n\\label{Na_p1m}\n\\end{align}\n\\end{subequations}\nwhere $\\tan\\alpha_1\\equiv \\frac{\\sqrt{3} A}{A + (a-b)B}$. \nThird, the Hamiltonian of $m=-1$ is defined on the subspace \n spanned by $\\left\\{ \\ket{\\frac{1}{2},-\\frac{3}{2}}, \n\\ket{-\\frac{1}{2},-\\frac{1}{2}} \\right\\}$. \nIts eigenvalues and eigenstates are given by\n\\begin{subequations}\n\\label{Eq_minus}\n\\begin{align}\nE_{-1}^{\\pm} &= -\\frac{A}{4} -bB \n             \\pm \\frac{1}{2}\\sqrt{\\bigl(A - (a-b)B\\bigr)^2 + 3A^2}\\,,\\\\\n\\ket{E_{-1}^+} \n&= \\phantom{+} \\cos\\frac{\\alpha_2}{2}\\, \\ket{-\\tfrac{1}{2},-\\tfrac{1}{2}} \n            +  \\sin\\frac{\\alpha_2}{2}\\, \\ket{ \\tfrac{1}{2},-\\tfrac{3}{2}}\\,,\n\\label{Na_m1p} \\\\\n\\ket{E_{-1}^-} \n&= - \\sin\\frac{\\alpha_2}{2}\\, \\ket{-\\tfrac{1}{2},-\\tfrac{1}{2}} \n   + \\cos\\frac{\\alpha_2}{2}\\, \\ket{ \\tfrac{1}{2},-\\tfrac{3}{2}} \\,,\n\\label{Na_m1m}\n\\end{align}\n\\end{subequations}\nwhere $\\tan\\alpha_2\\equiv \\frac{\\sqrt{3}A}{A -(a-b)B}$. Note that\nEqs.~(\\ref{Eq_minus}) can be obtained from Eqs.~(\\ref{Eq_plus}) by replacing \n$B$ with $-B$. Finally, the subspace of $m=0$ is spanned by \n$\\left\\{\\ket{\\frac{1}{2},-\\frac{1}{2}},\\ket{\\frac{1}{2},-\\frac{1}{2}}\\right\\}$. \nThe corresponding eigenvalues and eigenvectors are given by\n\\begin{subequations}\n\\label{Na_ev0}\n\\begin{align}\nE_{0}^{\\pm} &= -\\frac{A}{4} \\pm \\frac{1}{2}\\sqrt{(a-b)^2B^2 + 4A^2} \\,,\\\\\n\\ket{E_0^+} &= \n\\phantom{+} \\cos\\frac{\\alpha_0}{2}\\, \\ket{ \\tfrac{1}{2},-\\tfrac{1}{2}} \n         +  \\sin\\frac{\\alpha_0}{2}\\, \\ket{-\\tfrac{1}{2}, \\tfrac{1}{2}} \\,, \n\\label{Na_0p} \\\\\n\\ket{E_0^-} &= \n        -   \\sin\\frac{\\alpha_0}{2}\\, \\ket{ \\tfrac{1}{2},-\\tfrac{1}{2}} \n\t+   \\cos\\frac{\\alpha_0}{2}\\, \\ket{-\\tfrac{1}{2}, \\tfrac{1}{2}}\\,,\n\\label{Na_0m}\n\\end{align}\n\\end{subequations}\nwhere $\\tan\\alpha_0\\equiv \\frac{A}{(a-b)B}$. As expected, Eqs.~(\\ref{Na_ev0})\nis very similar to Eqs.~(\\ref{H_eigen0}) in the case of a hydrogen atom.\n\n\n\\subsection{Entanglement}\n\nLet us examine the relation between entanglement and level crossings \nor avoided crossings for a sodium atom. \nWith the values of the parameters $A$, $a$, and $b$ of the $^{23}$Na \natom in Ref.~\\cite{Arimondo77}, energy levels $E_{m}^{\\pm}\/A$ are plotted\nin Fig.~\\ref{Fig4} (a). The von Neumann entropies of the electron \n(or nuclear) spin for each eigenstates are shown in Fig.~\\ref{Fig4} (b).\nThe ground state is given by $\\ket{E_{+1}^{-}}$ for $B>0$ and \n$\\ket{E_{-1}^{-}}$ for $B<0$. Two levels, $E_{+1}^{+}$ and \n$E_{+1}^{-}$, are avoided crossing and maximally entangled at \n$A -(a-b)B= \\sqrt{3}A$.  Another two levels, $E_{-1}^{+}$ and $E_{-1}^{-}$, \nare avoided crossing and maximally entangled at $A + (a-b)B= \\sqrt{3}A$.\nTwo levels with $m=0$, $E_{0}^{\\pm}$ are avoided crossing and maximally\nentangled at $B=0$. Two levels $E_{\\pm2}$ show real crossing at $B=0$ and have\nzero von Neumann entropies. Again one can see that the eigenstate is maximally\nentangled at the avoided crossing point. This is analogous to the results \nin Ref.~\\cite{Gonzalez03}, where Shannon entropy is used as an indicator of\navoided crossings.\n\n\n\\subsection{Berry phase}\n\nAs in Sec.~\\ref{hydrogen_berry}, let us consider an adiabatic cyclic evolution\nof nuclear and electron spins of a sodium atom by rotating the magnetic \nfield $B\\,{\\bf\\hat{n}}$ slowly. For a adiabatic rotation keeping the azimuthal \nangle $\\theta$ constant and varying the polar angle $\\phi$ from $0$ to $2\\pi$,\nthe instantaneous eigenstates accumulates the total Berry phases proportional \nto the magnetic quantum number $m$, $\\beta = \\mp m \\Omega$. In contrast to a\nhydrogen atom, the ground state is given either by $\\ket{E_{+1}^{-}}$ or by\n$\\ket{E_{-1}^{-}}$ with $m=\\pm 1$, so it acquires the total Berry phase $\\beta\n=\\mp\\Omega$.\n\nLet us analyze how the marginal Berry phase of the entangled state is related \nto the avoided crossings. We focus on the eigenstate, $\\ket{E_{+1}^{-}}$. It has \ntwo Schmidt coefficients, \n$p_1 = \\sin^2\\tfrac{\\alpha_1}{2}$ and $p_2=\\cos^2\\tfrac{\\alpha_1}{2}$.\nWith Eq.~(\\ref{berry_sum}), one obtains  the total phase as a sum of the Berry \nphases acquired by nuclear and electron spins with weights of the Schmidt \ncoefficients,\n\\begin{align}\n\\beta = \\sin^2\\tfrac{\\alpha_1}{2}\n        \\left( -\\tfrac{\\Omega}{2} - \\tfrac{\\Omega}{2}\\right)\n      + \\cos^2\\tfrac{\\alpha_1}{2}\n        \\left(+\\tfrac{\\Omega}{2}  - \\tfrac{3\\Omega}{2}\\right)\n      = -\\Omega\\,.\n\\end{align}\nFrom Eq.~(\\ref{berry_mixed}), one obtains the marginal Berry phases of an\nelectron spin $\\Gamma_e$ and of nuclear spin $\\Gamma_n$\n\\begin{subequations}\n\\begin{align}\n\\Gamma_n &= \\arg\\bigl[\\,\n             \\sin^2\\tfrac{\\alpha_1}{2}\\,e^{-i\\Omega\/2} \n\t   + \\cos^2\\tfrac{\\alpha_1}{2}\\,e^{-i3\\Omega\/2}\\, \\bigr] \\,,\\\\\n\\Gamma_e &=\\arctan\\left[\\,\\cos\\alpha_1\\tan\\tfrac{\\Omega}{2}\\,\\right]\\,.\n\\end{align}\n\\end{subequations}\nFig.~\\ref{Fig5} plots the marginal Berry phase of a nuclear spin $\\Gamma_n$ \nas a function of $B$ and the azimuthal angle $\\theta$. In the limit of \n$B\\gg 1$, i.e., $\\alpha_1\\to 0$, one has $\\ket{E_{+1}^{-}} \\to\n\\ket{-\\tfrac{1}{2}, \\tfrac{3}{2}}$, $\\Gamma_e=\\Omega\/2$, and\n$\\Gamma_n=-3\\Omega\/2$. It is clearly seen that \nthe node of the marginal Berry phase of a nuclear (or electron) spin \ncorresponds to the avoid crossing at $A + (a-b)B = \\sqrt{3}A$. Thus it could \nbe expected that the marginal Berry phase of a subsystem for an entangled \nstate has a node at avoided crossings.\n\n\n\\section{Conclusions}\n\\label{section_conclusion}\n\nWe have considered the Breit-Rabi Hamiltonians for hydrogen and sodium atoms, \ndescribing the hyperfine interaction between a nuclear spin and an electron \nspin in the presence of a magnetic field. We have examined the relation between\nlevel crossings, entanglement, and Berry phases. It is shown that entanglement \nbetween nuclear and electron spins is maximum at avoided crossing points. \nThe Berry phase and the von Neumann entropy change abruptly at level\ncrossings as the parameters of the Breit-Rabi Hamiltonian for a hydrogen atom \nvary. An entangled state encircling the avoided crossing acquires \nthe marginal Berry phase of an electron (or nuclear) spin like an eigenstate \nmoving around the real crossing accumulates a Berry phase.\nWe have shown that the nodal points of the marginal Berry phase of \nan entangled state corresponds to the avoided crossing points.\n\n\n\\begin{acknowledgments}\nThis work is in part supported by the visitor program of Max Planck Institute \nfor the Physics of Complex Systems. We would like also to thank the Binational \nIsrael-US Foundation (BSF) for financial support. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\\emph{Science} is commonly described as the ``discovery of natural laws through experimentation and observation''.\nResearchers in the natural sciences increasingly turn to machine learning (ML) to aid the discovery of natural laws from observational data alone, which is often abundantly available, hoping to bypass expensive and cumbersome targeted experimentation.\nWhile there may be fundamental limitations to what can be extracted from observations alone,\nrecent successes of ML in the entire range of natural sciences provide ample reason for excitement.\nIn this work, we focus on ordinary differential equations, a ubiquitous description of dynamical natural laws in physics, chemistry, and systems biology.\nFor a first order ODE $\\dot{y} := \\nicefrac{\\partial y}{\\partial t} = f(y, t)$, we call~$f$ (which uniquely defines the ODE) the underlying dynamical law.\nInformally, our goal is then to infer~$f$ in symbolic form given discrete time-series observations of a single solution $\\{y_i := y(t_i)\\}_{i=1}^n$ of the underlying ODE.\n\nContrary to ``black-box-techniques'' such as Neural Ordinary Differential Equations (NODE)~\\citep{chen2018neural} that aim at inferring a possible~$f$ as an arguably opaque neural network, we focus specifically on symbolic regression.\nFrom the perspective of the sciences, a law of nature is useful insofar as it is more broadly applicable than to merely describe a single observation.\nIn particular, the reason to learn a dynamical law in the first place is to dissect and understand it as well as to make predictions about situations that differ from the observed one.\nFrom this perspective, a symbolic representation of the law (in our case the function~$f$) has several advantages over block-box representations: they are compact and directly interpretable, they are amenable to analytic analysis, they allow for meaningful changes and thus enable assessment of interventions and counterfactuals.\n\nIn this work we present NSODE, a sequence-to-sequence transformer that maps observed trajectories, i.e., numeric sequences of the form $\\{(t_i, y_i)\\}_{i=1}^n$, directly to symbolic equations as strings, e.g., \\texttt{\"y**2+1.64*cos(y)\"}, which is the prediction for $f$.\nThis example directly highlights the benefit of symbolic representations in that the $y^2$ and $\\cos(y)$ terms tell us something about the fundamental dynamics of the observed system; the constant \\texttt{1.64} will have semantic meaning in a given context and we can, for example, make predictions about settings in which this constant takes a different value.\n\n\\begin{figure*}\n  \\centering\n  \\vspace{-7mm}\n  \\includegraphics[width=1.0\\textwidth]{figs\/pipeline_new.pdf}\n  \\caption{An overview illustration of the data generation (top) and training pipeline (bottom). Our dataset stores solutions in numerical (non-binarized) form on the entire regular solution time grid.}\n  \\label{fig:overview}\n  \\vspace{-2mm}\n\\end{figure*}\n\n\\section{Background and Related Work}\n\\label{sec:related_work}\nWhile NODE \\citep{chen2018neural} (with a large body of follow up work) is perhaps the most prominent method to learn ODEs from data in black-box form, we focus on various works that infer governing laws in symbolic form.\nClassically, symbolic regression aims at regular functional relationships (mapping $(x, f(x))$ pairs to~$f$ instead of mapping trajectories $(t, y(t))$ to the governing ODE $\\dot{y}=f(y, t)$) and has been approached by heuristics-based search, most prominently via genetic programming \\citep{koza}. Genetic programming randomly evolves a population of prospective mathematical expressions over many iterations and mimics natural selection by keeping only the best contenders across iterations, where superiority is measured by user-defined and problem-specific fitness functions.\nMore recently, symbolic regression has been approached with machine learning methods which exploit gradient information to optimize within the space of (finite) compositions of pre-defined basis functions.\n\\citet{sindy} use linear regression to identify a (sparse) linear combination of basis functions that yields the best fit for the observed data, while other approaches use neural networks with a diverse set of activation functions \\citep{eql2, pdenet2, rodenet}.\nAll these techniques deploy strong sparsity-promoting regularizers and fit a separate model for each observed trajectory.\n\nAlternatively, one can train a model to directly output the symbolic expressions.\nSupervised learning with gradient-based optimization for this approach is challenged by the formulation of a differentiable loss that measures the fit between the predicted symbolic expression and the observed data.\nThus, prior work resorted to reinforcement learning \\citep{deepsymres} or evolutionary algorithms \\citep{atkinson2019data, costa2021fast} for gradient-free optimization.\nFurthermore, inspired by common properties of known natural laws, \\citet{aifeynman2} devise a hybrid approach that combines a gradient-free heuristic search with neural network-based optimization. This approach has been extended to work with dynamical systems by \\citet{weilbach2021inferring}.\n\nThe closest works to ours use pre-trained, attention-based sequence-to-sequence models for symbolic regression \\emph{of functional relationships} \\citep{nesymres,SymbolicGPT2021,kamienny2022end, vastl2022symformer}.\nThey exploit the fact that symbolic expressions for (multi-variate) scalar functions can be both generated and evaluated on random inputs cheaply, resulting in essentially unlimited training data.\nLarge data including ground-truth expressions in symbolic form allow for a differentiable cross-entropy loss based directly on the symbols of the expression, instead of the numerical proximity between evaluations of the predicted and true expression.\nWhile the cross-entropy loss works well for operators and symbols (e.g. \\texttt{+,exp,sin,x,y}), a naive implementation is inefficient for numerical constants, e.g., \\texttt{1.452}. Previous works therefore resort to one of two strategies: \n1) represent all constants with a special \\texttt{<const>} token when training the sequence-to-sequence model and predict only the presence of a constant. Actual values are then inferred in a second, subsequent parameter estimation step where the structure of an expression is held fixed and only constants are optimized.\nThis second optimization procedure comes with substantial computational cost as constants have to be fit per inferred expression. In particular, we highlight that it does not transfer to inferring ODEs as it would require to first solve the predicted ODE $\\dot{y} = \\hat{f}(y)$ to obtain predicted $\\{\\hat{y}_i\\}_{i=1}^n$ values that can be compared to the set of observations $\\{y_i\\}_{i=1}^n$. While differentiable ODE solvers exist, optimizing constants this way is prohibitively expensive and typically highly unstable. \n2) A second popular strategy consists in rounding constants within the range of interest so that they can be represented with a finite number of tokens. This second strategy avoids a subsequent optimization step and enjoys clever encoding schemes with improved token efficiency yet represents values with an inherent loss in precision.\nAs an alternative, we develop a representation based on a ``two-hot'' encoding which avoids subsequent optimization steps as well as rounding.\n\n\\section{Method}\n\\label{sec:method}\n\n\\xhdr{Problem setting}\nGiven observations $\\{(t_i, y_i)\\}_{i=1}^n$ of a trajectory $y: [t_1, t_n] \\to \\ensuremath \\mathbb{R}$ that is a solution of the ODE $\\dot{y} = f(y)$, we aim to recover the function~$f$ in symbolic form---in our case as a string.\nIn this work, we focus on time-invariant (or autonomous) ODEs (i.e.,~$f(y, t) = f(y)$).\nSuch settings are a good starting point for investigation as they are commonly studied and can be thought of as ``evolving on their own'' without external driving forces or controls, i.e., once an initial condition is fixed the absolute time does not directly influence the evolution.\nWe explicitly assume that the observed system actually evolves according to an ODE in canonical form~$\\dot{y} = f(y)$ such that~$f$ can be expressed in closed form using the mathematical operators seen during training (see \\cref{sec:data}).\nIn this paper we restrict ourselves to the rich class of non-linear, scalar, first-order, autonomous ODEs but we discuss extensions of NSODE to higher-order systems of coupled non-autonomous ODEs in \\cref{app:extensions}.\n\n\\subsection{Data Generation}\n\\label{sec:data}\n\n\\xhdr{Sampling symbolic expressions}\n\\label{xhdr:sampling}\nTo exploit large-scale supervised pretraining we generate a dataset of $\\sim$63M ODEs in symbolic form along with numerical solutions for randomly sampled initial values.\nSince we assume ODEs to be in canonical form $\\dot{y} = f(y)$, generating an ODE is equivalent to generating a symbolic expression $f(y)$.\nWe follow \\citet{lample2019deep}, who sample such an expression $f(y)$ as a unary-binary tree, where each internal node corresponds to an operator and each leaf node corresponds to a constant or variable.\nThe algorithm consists of two phases: (1) A unary-binary tree is sampled uniformly from the distribution of unary-binary trees with up to $k\\in \\ensuremath \\mathbb{N}$ internal nodes, which crucially does not overrepresent small trees corresponding to short expressions. Here the maximum number of internal nodes $\\ensuremath K$ is a hyperparameter of the algorithm.\n(2) The sampled tree is ``decorated'', that is, each binary internal node is assigned a binary operator, each unary internal node is assigned a unary operator, and each leaf is assigned a variable or constant. \nHence, we have to specify a distribution over the $\\ensuremath N_{\\mathrm{bin}}$ binary operators, one over the $\\ensuremath N_{\\mathrm{una}}$ unary operators, a probability $\\ensuremath p_{\\mathrm{sym}} \\in (0,1)$ to decide between symbols and constants, as well as a distribution $\\ensuremath p_{\\mathrm{c}}$ over constants.\nFor constants we distinguish explicitly between sampling an integer or a real value.\nTogether with $\\ensuremath K$, these choices uniquely determine a distribution over equations $f$ and are described in detail in \\cref{app:modeltraining}.\n\\Cref{fig:overview} depicts an overview of the data generation procedure.\n\nThe pre-order traversal of a sampled tree results in the symbolic expression for~$f$ in prefix notation.\nAfter conversion to infix notation, we simplify each expression using the computer algebra system SymPy \\citep{sympy}, and filter out constant equations~$f(y) = c$ as well as expressions that contain operators or symbols that were not part of the original distribution. \nWe call the structure modulo the value of the constants of such an expression a \\textbf{skeleton}.\nAny skeleton containing at least one binary operator or constant can be represented by different unary-binary trees.\nVice versa many of the generated trees will be simplified to the same skeleton. To ensure diversity and to mitigate potential dataset bias towards particular expressions, we discard duplicates on the skeleton level. To further cheaply increase the variability of ODEs we sample $\\ensuremath N_{\\mathrm{const}}$ unique sets of constants per skeleton.\nWhen sampling constants we take care not to modify the canonical expression by adhering to the rules listed in \\cref{app:constant_rules}.\nWe provide summary statistics on operator frequencies and expression complexities for the generated dataset in \\cref{app:datastats}. Here, \\textbf{complexity} refers to overall count of symbols (e.g., $y$, or constants) as well as operators in an expression, a simple yet common measure in the symbolic regression literature.\n\n\\xhdr{Computing numerical solutions}\nWe obtain numerical solutions for all ODEs via SciPy's interface \\citep{scipy} to the LSODA software package \\citep{lsoda} with both relative and absolute tolerances set to $10^{-9}$.\nWe solve each equation on a fixed time interval $t \\in [0, \\ensuremath T]$ and store solutions on a regular grid of $\\ensuremath N_{\\mathrm{grid}}$ points.\nFor each ODE, we sample up to $\\ensuremath N_{\\mathrm{iv}}$ initial values $y(0) = y_0$ uniformly from $\\ensuremath (y_0^{\\min}, y_0^{\\max})$.\\footnote{Due to a timeout per ODE, fewer solutions may remain in cases when the numerical solver fails repeatedly.}\nWhile LSODA attempts to select an appropriate solver, numerical solutions still cannot be trusted in all cases.\nTherefore, we check the validity of solutions via the following quality control check: we use 9th order central finite differences to approximate the temporal derivative of the solution trajectory (on the same temporal grid as the proposed solution), denoted by $\\dot{y}_{\\mathrm{fd}}$, and filter out any solution for which $\\|\\dot{y}_{\\mathrm{fd}} - \\dot{y} \\|_{\\infty} > \\epsilon$, where we use $\\epsilon = 1$.\n\n\\subsection{Model Design Choices}\n\\label{sec:model}\n\nNSODE consists of an encoder-decoder transformer with architecture choices listed in \\cref{app:modeldesign}.\nWe provide a visual overview in \\cref{fig:overview}.\n\n\\xhdr{Representing input trajectories}\nA key difficulty in feeding numerical solutions $\\{y_i\\}_{i=1}^n$ as input is that their range may differ greatly both within a single solution as well as across ODEs.\nFor example, the linear ODE $\\dot{y} = c \\cdot y$ for a constant~$c$ is solved by an exponential $y(t) = y_0 \\exp(c t)$ for initial value $y(0) = y_0$, which may span many orders of magnitude on a fixed time interval.\nTo prevent numerical errors and vanishing or exploding gradients caused by the large range of values, we assume each representable 64-bit float value is a token and use its IEEE-754 encoding as the token representation \\citep{nesymres}.\nWe thus convert all pairs $(t_i, y_i)$ to their IEEE-754 64 bit representations, channel them through a linear embedding layer before feeding them to the encoder.\n\n\\xhdr{Representing symbolic expressions}\nThe target sequence (i.e., the string for the symbolic expression of~$f$) is tokenized on the symbol-level.\nWe distinguish two cases: (1) \\emph{Operators and variables:} for each operator and variable we include a unique token in the vocabulary. \n(2) \\emph{Numerical constants:} constants may come from both discrete (integers) as well as continuous distributions, as for example in \\texttt{y**2+1.64*cos(y)}. Hence, it is unfeasible to include individual tokens ``for each constant''.\nNaively tokenizing on the digit level, i.e., representing real values literally as the sequence of characters (e.g., \\texttt{\"1.64\"}), not only significantly expands the length of target sequences and thus the computational cost, but also requires a variable number of prediction steps for every single constant.\n\nInstead, we take inspiration from \\citet{schrittwieser2020mastering} and encode constants in a \\emph{two-hot} fashion.\nWe fix a finite homogeneous grid on the real numbers $x_1 < x_2 < \\ldots < x_m$ for some~$m \\in \\ensuremath \\mathbb{N}$, which we add as tokens to the vocabulary.\nThe grid range $(x_1, x_m)$ and number of grid points $m$ are hyperparameters that can be set in accordance to the problems of interest. Our choices are described in \\cref{app:modeldesign}.\n\nFor any constant $c$ in the target sequence we then find $i \\in \\{1, \\ldots, m-1\\}$ such that $x_i \\le c \\le x_{i+1}$ and encode~$c$ as a distribution supported on $x_i, x_{i+1}$ with weights $\\alpha, \\beta$ such that $\\alpha x_i + \\beta x_{i+1} = c$.\nThat is, the target in the cross-entropy loss for a constant token is not a strict one-hot encoding, but a distribution supported on two (neighboring) vocabulary tokens resulting in a lossless encoding of continuous values in $[x_1, x_m]$.\nThis two-hot representation can be used directly in the cross-entropy loss function.\n\n\\xhdr{Decoding constants}\nWhen decoding a predicted sequence, we check at each prediction step whether the argmax of the logits corresponds to one of the~$m$ constant tokens $\\{x_1, \\ldots, x_m\\}$.\nIf not, we proceed by conventional one-hot decoding to obtain predicted operators and variables.\nIf instead the argmax corresponds to, for example, $x_i$, we also pick its largest-logit neighbor ($x_{i-1}$ or $x_{i+1}$; suppose $x_{i+1}$), renormalize their probabilities by applying a softmax to all logits and use the resulting two probability estimates as weights $\\alpha, \\beta$.\nConstants are then ultimately decoded as $\\alpha x_i + \\beta x_{i+1}$.\n\n\\subsection{Evaluation and Metrics}\n\\label{sec:metrics}\n\n\\xhdr{Sampling solutions}\nTo infer a symbolic expression for the governing ODE of a new observed solution trajectory $\\{(t_i, y_i)\\}_{i=1}^n$, all the typical policies such as greedy, sampling, or beam search are available.\nIn our evaluation, we use beam search with 1536 beams and report top-$k$ results with $k$ ranging from 1 to 1536.\n\n\\xhdr{Metrics}\nWe evaluate model performance both numerically and symbolically.\nFor numerical evaluation we follow \\citet{nesymres}: suppose the ground truth ODE is given by $\\dot{y} = f(y)$ with (numerical) solution $y(t)$ and the predicted ODE is given by $\\hat{\\dot{y}} = \\hat{f}(y)$.\nTo compute numerical accuracy we first evaluate $f$ and $\\hat{f}$ on $\\ensuremath N_{\\mathrm{eval}}$ points in the interval $[\\min(y(t)), \\max(y(t))]$ (i.e., the interval traced out by the observed solution), which yields function evaluations $\\texttt{gt}=\\{\\dot{y}_i\\}_{i=1}^{\\ensuremath N_{\\mathrm{eval}}}$ and $\\texttt{pred}=\\{\\hat{\\dot{y}}_i\\}_{i=1}^{\\ensuremath N_{\\mathrm{eval}}}$.\nWe then assess whether \\texttt{numpy.allclose}\\footnote{\\texttt{numpy.allclose} returns True if \\texttt{abs(a - b) <= (atol + rtol * abs(b))} holds element-wise for elements $a$ and $b$ from the two input arrays.\nWe use \\texttt{atol=1e-10} and \\texttt{rtol=0.05}; $a$ corresponds to predictions, $b$ corresponds to ground truth.} returns \\texttt{True} as well as whether the coefficient of determination $\\mathrm{R}^2 \\geq 0.999$.\\footnote{For observations $y_i$ and predictions $\\hat{y}_i$ we have $\\mathrm{R}^2 = 1 - (\\sum_i (y_i - \\hat{y}_i)^2) \/ (\\sum_i (y_i - \\overline{y})^2 )$.}\nNumerical evaluations capture how closely the predicted function approximates the ground truth function within the interval $[\\min(y(t)), \\max(y(t))]$.\n\nHowever, a key motivation for symbolic regression is to uncover a \\emph{symbolic} mathematical expression that governs the observations.\nTesting for symbolic equivalence between ground truth expression $f(y)$ and a predicted expression $\\hat{f}(y)$ is unsuitable in the presence of real-valued constants as even minor deviations between true and predicted constants render the equivalence false.\nInstead, we regard the predicted expression $\\hat{f}(y)$ to be symbolically correct if $f(y)$ and $\\hat{f}(y)$ can be made equivalent by modifying only the values of constants in the predicted expression $\\hat{f}(y)$.\nThis is implemented using SymPy's \\texttt{match} function.\nIn order not to alter the structure of the predicted expression, we constrain modifications of constants such that all constants remain non-zero and retain their original sign.\nThis definition is thus primarily concerned with the structure of an expression, rather than precise numerical agreement.\nOnce the structure is known, the inference problem becomes conventional parameter estimation.\nWe report percentages of samples in a given test set that satisfies any individual metric (numerical and symbolic), as well as percentages satisfying symbolic and numerical metrics simultaneously.\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Benchmark Datasets}\\label{sec:datasets}\nWe construct several test sets to evaluate model performance and generalization in different settings.\n\\begin{itemize}[leftmargin=*,topsep=0pt,itemsep=0pt]\n    \\item \\textbf{testset-iv}: Our first test set assesses generalization within initial values not seen during training.\n    It consists of 5793 ODEs picked uniformly at random from our generated dataset but re-sampled initial values. We also employ the following constraints via rejection sampling: (a) All skeletons in testset-iv are unique. (b) As the number of unique skeletons increases with the number of operators, we allow at most 2000 examples per number of operators (with substantially fewer unique skeletons existing for few operators).\n    \n    \\item \\textbf{testset-constants}: Our second test set assesses generalization within unseen initial values and constants.\n    It consists of 2720 ODEs picked uniformly at random from our dataset (ensuring unique skeletons and at most 1000 examples per number of operators as above), but re-sampled intial values and constants.\n    \n    \\item \\textbf{testset-skeletons}: In principle, we can train NSODE on all possible expressions (using only the specified operators and number ranges) up to a specified number of operators.\n    However, even with the millions examples in our dataset, we have by far not exhausted the huge space of possible skeletons (especially for larger numbers of operators).\n    Hence, our third test set contains 100 novel random ODEs with skeletons that were never seen during training. \n    \n    \\item \\textbf{testset-iv-163}: This is a subset of testset-iv motivated by the fact that most symbolic regression models we want to compare to require a separate optimization for every individual example, which was computationally infeasible for our testset-iv.\n    For a fair comparison, we therefore subsampled up to 10 ODEs per complexity uniformly at random, yielding 163 examples.\n    \n    \\item \\textbf{testset-textbook}: To assess how NSODE performs on ``real problems'', we manually curated 12 scalar, non-linear ODEs from Wikipedia pages, physics textbooks, and lecture note from university courses on ODEs.\n    These equations are listed in \\cref{tab:textbook_equations} in \\cref{app:textbook_equations}.\n    We note that they are all extremely simple compared to the expressions in our generated dataset in that they are ultimately mostly low order polynomials, some of which with one fractional exponent.\n    \n    \\item \\textbf{testset-classic}:\n    To validate our approach on existing datasets we turn to benchmarks in the classic symbolic regression literature (inferring just the functional relationship between input-ouput pairs) and simply interpret functions as ODEs. In particular we include all scalar function listed in the overview in \\cite{mcdermott2012genetic} which includes equations from many different benchmarks \\cite{keijzer2003improving, koza, koza1994genetic, uy2011semantically, vladislavleva2008order}.\n    For example, we interpret the function $f(y) = y^3 + y^2 + y$ from \\citet{uy2011semantically} as an autonomous ODE $\\dot{y}(t) = f(y(t)) = y(t)^3 + y(t)^2 + y$, which we solve numerically for a randomly sampled initial value as described before.\n\\end{itemize}\n\n\\subsection{Baselines}\\label{sec:baselines}\nWe compare our method to recent popular baselines from the literature (see \\cref{sec:related_work}).\nWe briefly describe them including some limitations here and defer all details to \\cref{app:baselines}.\nFirst, no baseline is suited directly to infer dynamical laws, but only to infer functional relationships.\nTherefore, all baselines fit a separate regression function mapping $y(t) \\mapsto \\hat{\\dot{y}}(t)$ per individual ODE, using the coefficient of determination $\\mathrm{R}^2$ as optimization objective.\nSince derivatives $\\hat{\\dot{y}}(t)$ are typically not observed, we approximate them via finite differences using PySindy \\citep{pysindy}.\nHence, all these methods crucially rely on regularly sampled and noise-free observations, whereas our approach can easily be extended to take those into account (see \\cref{app:extensions}).\n\n\\begin{itemize}[leftmargin=*,topsep=0pt,itemsep=0pt]\n    \\item \\textbf{Sindy} \\citep{sindy}: Sindy builds a (sparse) linear combination of a fixed set of (non-linear) basis functions.\n    The resulting Lasso regression is efficient, but suffers from limited expressiveness.\n    In particular, Sindy cannot easly represent nested functions or non-integer powers as all non-linear expressions have to be added explicitly to the set of basis functions.\n    We cross-validate Sindy over a fairly extensive hyperparameter grid of 800 different combinations for each individual trajectory.\n    \n    \\item \\textbf{GPL}\\footnote{\\ttfamily \\url{gplearn.readthedocs.io\/}} (genetic programming):\n    GPL(earn) maintains a population of programs each representing a mathematical expression.\n    The programs are mutated for several generations to heuristically optimize a user defined fitness function.\n    While not originally developed for ODEs, we can apply GPLearn on our datasets by leveraging the finite difference approximation.\n    We use a population size of 1000 and report the best performance across all final programs.\n    Compared to sindy, GPLearn is more expressive yet substantially slower to fit.\n    \n    \\item \\textbf{AIFeynman} \\citep{aifeynman, aifeynman2}: AIFeynman is a physics-inspired approach to symbolic regression that exploits the insight that many famous equations in natural sciences exhibit well-understood functional properties such as symmetries, compositionality, or smoothness.\n    AIFeynman implements a neural network based heuristic search that tests for such properties in order to identify a symbolic expression that fits the data.\n    For every test sample AIFeynman computes a pareto front of solutions that trade off complexity versus accuracy.\n    We report the best performance across all functions on the pareto front. \n    Notably, AIFeynman performs quite an exhaustive search procedure such that running it even on a single equation took on the order of tens of minutes.\n\\end{itemize}\n\n\\subsection{Results}\\label{sec:results}\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}{1\\textwidth}\n    \\centering\n    \\includegraphics[width=0.65\\linewidth]{figs\/iv_const_skel\/legend_iv_const_skel.pdf}\n\\end{subfigure}%\n\\\\ \\vspace{0.25cm}\n\\begin{subfigure}{.25\\textwidth}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/iv_const_skel\/iv_const_skel_isclose.pdf}\n    \\caption{allclose}\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/iv_const_skel\/iv_const_skel_r2.pdf}\n    \\caption{R$^2 \\geq 0.999$}\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/iv_const_skel\/iv_const_skel_skelrec.pdf}\n    \\caption{skeleton recovery}\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/iv_const_skel\/iv_const_skel_skelrec_isclose.pdf}\n    \\caption{skel. recov. \\& allclose}\n\\end{subfigure}%\n\\caption{Numerical and symbolic performance evaluation on testset-iv.}\n\\label{fig:results}\n\\end{figure}\n\n\n\\xhdr{Model Performance}\n\\Cref{fig:results} shows NSODE's performance on our testset-iv, testset-constants, and testset-skeletons according to our numerical and symbolic metrics as well as combined skeleton recovery and allclose as we vary $k$ in the top-k considered candidates of the beam search.\nInvesting sufficient test-time-compute (i.e., considering many candidates) continuously improves performance.\nWhile we capped $k$ at 1536 due to memory limitations, we did not observe a stagnation of the roughly logarithmic scaling of all performance metrics with $k$.\nThis cannot be attributed to ``exhaustion effects'', where one may assume that all possible ODEs will eventually be among the candidates, because (a) the space of possible skeletons grows much faster than exponentially, and (b) the numerical metrics are extremely sensitive also to the predicted constant values in continuous domains.\n\nAs one may expect, performance decreases as we move from only new initial conditions, to also sampling new constants, to finally sampling entirely unseen skeletons.\nOn testset-iv with $k=1536$ we achieve about 50\\% skeleton recovery and still successfully recover more than a third skeletons of testset-skeletons with similar numbers for allclose.\nThe fact that the combined metric (symbolic + numerical) is only about half of that indicates that numerical and symbolic fit are indeed two separate measures, none of which need to imply the other.\nHence, a thorough evaluation of both is crucial to understand model performance in symbolic regression tasks.\n\n\\begin{table}[h!]\n\\centering\n\\caption{Comparing NSODE to the baselines. Results are average percentages across dataset. GPLearn often generates extremely long expressions which take SymPy up to half a minute to parse during evaluation. We denote this extra time in gray.}\\label{tab:resultsummary}\n\\begin{tabularx}{\\columnwidth}{l@{\\hspace{8pt}}l@{\\hspace{8pt}}Y@{\\hspace{6pt}}Y@{\\hspace{8pt}}Y@{\\hspace{0pt}}Y}\n\\toprule \\rowcolor{white}\nDataset                           & Metric                 & NSODE & Sindy & GPLearn & AIFeynman  \\\\ \n\\midrule\n                                   & skel-recov             & \\bf 37.4  & 3.7   & 2.5 &  14.1   \\\\ \n                                  & R$^2 \\geq 0.999$              & 24.5  &  31.9  & 3.7& \\bf 49.7     \\\\ \niv-163                    & allclose               & 42.3  & 25.8  & 14.7 & \\bf 55.8  \\\\ \n                                  & skel-recov \\& R$^2 \\geq 0.999$  & \\bf 15.3  & 3.1   & 1.8 &  13.5   \\\\ \n                                  & skel-recov \\& allclose & \\bf 15.3  & 3.1   & 1.8  &  13.5  \\\\ \n                                  & runtime [s]             & 5.4 & \\bf0.4 & 29 {\\color{gray}+22} & 1203.6 \\\\\n\\midrule\n                                    & skel-recov             & 41.7  & 33.3  & 8.3  & \\bf 91.7   \\\\ \n                                  & R$^2 \\geq 0.999$              &  16.7  & 50   & 0.0  & \\bf 75   \\\\ \ntextbook                            & allclose               & 25    &  58.3  & 8.3  & \\bf 75   \\\\ \n                                  & skel-recov  \\& R$^2 \\geq 0.999$ &  33.3   &  41.7  & 0 & \\bf 66.7      \\\\ \n                                  & skel-recov \\& allclose & 8.3  &  33.3  & 1.8 & \\bf 66.7   \\\\\n                                  & runtime [s]             & 6 & \\bf1 & 23 {\\color{gray}+22} & 1267.1 \\\\\n\\bottomrule\n\\end{tabularx}%\n\\end{table}\n\n\\xhdr{Comparison to baselines}\nIn \\cref{tab:resultsummary} we compare NSODE to all baselines using $k=1536$ in our beam search; full results on all datasets can be found in \\cref{app:detailedresults}.\nWe also include the average wallclock runtime per expression for each of the datasets.\n\nFirst, we note that on our subsampled testset-iv-163, NSODE outperforms competing approaches in terms of skeleton recovery by a wide margin and also performs best in terms of joint skeleton recovery and numerical measures, which is a strong indication of actually having recovered the governing ODE accurately.\nBy spending over 200x more time on its exhaustive heuristic search, AIFeynman manages to outperform NSODE in terms of numerical accuracy ($\\mathrm{R}^2$ and allclose).\n\\Cref{fig:distribution} shows the number of skeletons recovered by each method given the complexity of equations, results for other datasets can be found in \\cref{app:detailedresults}. \\footnote{Due to simplification, complexity is not upper bounded by the number of nodes in a unary-binary tree.}\nWhile AIFeynman and Sindy recover some of the low complexity expressions, NSODE is the only method to also recover some of the more complex skeletons. \n\nOn testset-textbook, AIFeynman outperforms all other methods on numerical and symbolic metrics. This can be understood with regard to the dataset where $8\/12$ expressions are polynomials with the remaining 4\/12 expressions having a polynomial skeleton with fractional or negative exponents. These expressions are particularly favorable for the heuristics implemented by AIFeynman which explicitly attempt to fit a polynomial to the data.\nHowever, even on these simple examples AIFeynman takes over 200x longer than our method, which in turn clearly outperforms Sindy and GPLearn in terms of skeleton recovery.\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}{1\\textwidth}\n    \\centering\n    \\includegraphics{figs\/complexity_vs_recovery\/legend2.pdf}\n\\end{subfigure}%\n\\\\\n\\begin{subfigure}{.25\\textwidth}\n    \\centering\n    \\includegraphics[width=.99\\linewidth]{figs\/complexity_vs_recovery\/complexity_vs_recovery_nsode_163.pdf}\n    \\caption{NSODE}\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n    \\centering\n    \\includegraphics[width=.99\\linewidth]{figs\/complexity_vs_recovery\/complexity_vs_recovery_aifeynman_163.pdf}\n    \\caption{AIFeynman}\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n    \\centering\n    \\includegraphics[width=.99\\linewidth]{figs\/complexity_vs_recovery\/complexity_vs_recovery_sindy_163.pdf}\n    \\caption{Sindy}\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n    \\centering\n    \\includegraphics[width=.99\\linewidth]{figs\/complexity_vs_recovery\/complexity_vs_recovery_gplearn_163.pdf}\n    \\caption{GPLearn}\n\\end{subfigure}%\n\\caption{Correctly recovered skeletons by each method on testset-iv-163 per complexity. AIFeynman and Sindy are mostly able to recover some of the low complexity skeletons, while NSODE performs much better also on higher complexities. GPLearn fails to recover most skeletons.}\\label{fig:distribution}\n\\vspace{-2mm}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe have developed a flexible and scalable method to infer ordinary differential equations $\\dot{y} = f(y)$ from a single observed solution trajectory.\nOur method follows the successful paradigm of large-scale pretraining of attention-based sequence-to-sequence models on essentially unlimited amounts of simulated data, where the inputs are the observed solution $\\{(t_i,y_i)\\}_{i=1}^n$ and the output is a symbolic expression for~$f$ as a string.\nOnce trained, our method is orders of magnitude more efficient than similarly expressive existing symbolic regression techniques that require a separate optimization for each instance and achieves strong performance in terms of skeleton recovery especially for complex expressions on various benchmarks.\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{secIntro}\n\nThe affine Kac-Moody superalgebra $\\AKMSA{gl}{1}{1}$ is an attractive candidate for study.  On the one hand, its highest weight theory is particularly easy to analyse.  On the other, one is naturally led to study indecomposable modules of the type that arise in logarithmic conformal field theory{}.  In \\cite{CR:GL11}, we reviewed and consolidated what was known about this superalgebra, drawing in particular upon the previous works \\cite{RozQua92,Rozansky:1992td,SalGL106,Creutzig:2007jy,CS09,Creutzig:2009zz,Creutzig:2010ne}.\n\nOne motivation for undertaking this work was to understand how one could reconcile the observation that conformal field theories{} with $\\AKMSA{gl}{1}{1}$ symmetry appeared to admit only continuous spectra, whereas one might expect that the Wess-Zumino-Witten{} model on the real form $\\SLSG{U}{1}{1}$ would have the same symmetry, but a discrete spectrum.  Another was to understand whether $\\AKMSA{gl}{1}{1}$ could be related to other infinite-dimensional algebras, thus providing relationships between certain (logarithmic) conformal field theories{}.  For the first question, we were able to show that certain discrete spectra seem to be consistent provided one \\emph{extends} the chiral algebra appropriately.  For the second, we identified a certain $\\AKMA{u}{1}$-coset of $\\AKMSA{gl}{1}{1}$ as the chiral algebra of the well-known $\\beta \\gamma$ ghost system.  Previous work \\cite{RidSL210} then links $\\AKMSA{gl}{1}{1}$ to the affine Kac-Moody algebra $\\AKMA{sl}{2}_{-1\/2}$ \\cite{RidSL208,RidFus10}, the triplet algebra $\\func{\\alg{W}}{1,2}$ of Gaberdiel and Kausch \\cite{GabRat96} and the symplectic fermions algebra \\cite{KauSym00} ($\\AKMSA{psl}{1}{1}$).\n\nThis article describes a certain family of \\emph{extended algebras} of $\\AKMSA{gl}{1}{1}$.  In \\cite{CR:GL11}, we noted that the fusion rules give rise to an infinite family of simple currents labelled by $n \\in \\mathbb{R}$ and $\\ell \\in \\mathbb{Z}$.  It follows that these algebra extensions may be computed algorithmically \\cite{RidSU206,RidMin07}.  Here, we perform the computations up to a certain order, using a well-known free field realisation \\cite{Guruswamy:1999hi}.  More precisely, we study the resulting W-algebras and show that, for certain infinite families of $n$ and $\\ell$, there is a bosonic subalgebra which we conjecture to be the $W^{\\brac{2}}_N$ algebra of Feigin and Semikhatov \\cite{Feigin:2004wb}.\n\n\\section{$\\SLSA{gl}{1}{1}$ and its representations} \\label{secFinAlg}\n\n\\subsection{Algebraic Structure}\n\nThe Lie superalgebra $\\SLSA{gl}{1}{1}$ consists of the endomorphisms of the super vector space $\\mathbb{C}^{1 \\mid 1}$ equipped with the standard graded commutator.  It is convenient to choose the following basis,\n\\begin{equation} \\label{eqngl11DefRep}\nN = \\frac{1}{2} \n\\begin{pmatrix}\n1 & 0 \\\\\n0 & -1\n\\end{pmatrix}\n, \\qquad E = \n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1\n\\end{pmatrix}\n, \\qquad \\psi^+ = \n\\begin{pmatrix}\n0 & 1 \\\\\n0 & 0\n\\end{pmatrix}\n, \\qquad \\psi^- = \n\\begin{pmatrix}\n0 & 0 \\\\\n1 & 0\n\\end{pmatrix}\n,\n\\end{equation}\nin which $N$ and $E$ are parity-preserving (bosonic) whereas $\\psi^+$ and $\\psi^-$ are parity-reversing (fermionic).  The non-vanishing brackets are then\n\\begin{equation} \\label{eqngl11Rels}\n\\comm{N}{\\psi^{\\pm}} = \\pm \\psi^{\\pm}, \\qquad \\acomm{\\psi^+}{\\psi^-} = E.\n\\end{equation}\nWe note that $E$ is central, so this superalgebra is not simple.  In fact, $\\SLSA{gl}{1}{1}$ does not decompose as a direct sum of ideals.  Equivalently, the adjoint representation of $\\SLSA{gl}{1}{1}$ is reducible, but indecomposable.\n\nThe standard non-degenerate bilinear form $\\killing{\\cdot}{\\cdot}$ on $\\SLSA{gl}{1}{1}$ is given by the supertrace of the product in the defining representation \\eqref{eqngl11DefRep}.  With respect to the basis elements \\eqref{eqngl11DefRep}, this form is\n\\begin{equation}\n\\killing{N}{E} = \\killing{E}{N} = 1, \\qquad \\killing{\\psi^+}{\\psi^-} = -\\killing{\\psi^-}{\\psi^+} = 1,\n\\end{equation}\nwith all other combinations vanishing.  From this, we compute the quadratic Casimir $Q \\in \\uealg{\\SLSA{gl}{1}{1}}$ (up to an arbitrary polynomial in the central element $E$).  We find it convenient to take\n\\begin{equation} \\label{eqnDefCasimir}\nQ = NE + \\psi^- \\psi^+.\n\\end{equation}\n\n\\subsection{Representation Theory} \\label{secFinRep}\n\nThe obvious triangular decomposition of $\\SLSA{gl}{1}{1}$ regards $\\psi^+$ as a raising (annihilation) operator, $\\psi^-$ as a lowering (creation) operator, and $N$ and $E$ as Cartan elements.  A highest weight state{} of a $\\SLSA{gl}{1}{1}$-representation is then defined to be an eigenstate of $N$ and $E$ which is annihilated by $\\psi^+$.  Such states generate Verma modules in the usual way and as $\\psi^-$ squares to zero in any representation, every Verma module has dimension $2$.  If $\\brac{n,e}$ denotes the weight (the $N$- and $E$-eigenvalues) of a highest weight state{} generating a Verma module, then its unique descendant will have weight $\\brac{n-1,e}$.  We will denote this Verma module by $\\VerMod{n-1\/2,e}$, remarking that the convention of characterising a highest weight module{} by the \\emph{average} $N$-eigenvalue of its states, rather than that of the highest weight state{} itself, turns out to symmetrise many of the formulae to follow.\n\nSuppose now that $\\ket{v}$ is a (generating) highest weight state{} of $\\VerMod{n,e}$.  It satisfies\n\\begin{equation}\n\\psi^+ \\psi^- \\ket{v} = \\acomm{\\psi^+}{\\psi^-} \\ket{v} = E \\ket{v} = e \\ket{v},\n\\end{equation}\nso the descendant $\\psi^- \\ket{v} \\neq 0$ is a singular vector if and only if $e=0$.  Verma modules are therefore irreducible for $e \\neq 0$, and have irreducible quotients of dimension $1$ when $e = 0$.  Modules with $e \\neq 0$ are called \\emph{typical} while those with $e = 0$ are \\emph{atypical}.  We will denote a typical irreducible by $\\TypMod{n,e} \\cong \\VerMod{n,e}$ and an atypical irreducible by $\\AtypMod{n}$.  Our convention of labelling modules by their average $N$-eigenvalue leads us to define the latter to be the irreducible quotient of $\\VerMod{n-1\/2,0}$.  This is summarised in the short exact sequence\n\\begin{equation} \\label{ESFinV}\n\\dses{\\AtypMod{n-1\/2}}{\\VerMod{n,0}}{\\AtypMod{n+1\/2}}\n\\end{equation}\nand structure diagram\n\\begin{equation}\n\\parbox[c]{0.4\\textwidth}{\n\\begin{tikzpicture}[auto,thick,\n\tnom\/.style={circle,draw=black!20,fill=black!20,inner sep=2pt}\n\t]\n\\node (q1) at (0,0) {$\\AtypMod{n+1\/2}$};\n\\node (s1) at (3,0) {$\\AtypMod{n-1\/2}$};\n\\node at (-2,0) [nom] {$\\VerMod{n,0}$};\n\\node at (-1.25,0) {$:$};\n\\draw [->] (q1) to node {$\\psi^-$} (s1);\n\\end{tikzpicture}\n} \\ .\n\\end{equation}\nSuch diagrams illustrate how the irreducible composition factors of a module are combined, with arrows indicating (schematically) the action of the algebra.\n\nAtypical modules also appear as submodules of larger indecomposable modules.  Of particular importance are the four-dimensional projectives\\footnote{We mention that the typical irreducibles are also projective in the category of finite-dimensional $\\SLSA{gl}{1}{1}$-modules.} $\\ProjMod{n}$ whose structure diagrams take the form\n\\begin{equation} \\label{picStaggered}\n\\parbox[c]{0.28\\textwidth}{\n\\begin{center}\n\\begin{tikzpicture}[auto,thick,\n\tnom\/.style={circle,draw=black!20,fill=black!20,inner sep=2pt}\n\t]\n\\node (top) at (0,1.5) [] {$\\AtypMod{n}$};\n\\node (left) at (-1.5,0) [] {$\\AtypMod{n+1}$};\n\\node (right) at (1.5,0) [] {$\\AtypMod{n-1}$};\n\\node (bot) at (0,-1.5) [] {$\\AtypMod{n}$};\n\\node at (0,0) [nom] {$\\ProjMod{n}$};\n\\draw [->] (top) to node [swap] {$\\psi^+$} (left);\n\\draw [->] (top) to node {$\\psi^-$} (right);\n\\draw [->] (left) to node [swap] {$\\psi^-$} (bot);\n\\draw [->] (right) to node {$-\\psi^+$} (bot);\n\\end{tikzpicture}\n\\end{center}\n}\n.\n\\end{equation}\nWe remark that these modules may be viewed as particularly simple examples of staggered modules \\cite{RidSta09}.  Indeed, they may be regarded as extensions of highest weight modules{} via the exact sequence\n\\begin{equation} \\label{ESFinP}\n\\dses{\\VerMod{n+1\/2,0}}{\\ProjMod{n}}{\\VerMod{n-1\/2,0}},\n\\end{equation}\nand one can verify that the Casimir $Q$ acts non-diagonalisably on $\\ProjMod{n}$, taking the generator associated with the top $\\AtypMod{n}$ factor to the generator of the bottom $\\AtypMod{n}$ factor, while annihilating the other states.\n\n\\subsection{The Representation Ring}\n\nThe relevance of the projectives $\\ProjMod{n}$ is that they appear in the representation ring generated by the irreducibles.\\footnote{It is perhaps also worth pointing out that the adjoint representation of $\\SLSA{gl}{1}{1}$ is isomorphic to $\\ProjMod{0}$.}  The tensor product rules governing this ring are \\cite{RozQua92}\n\\begin{equation} \\label{RepRing}\n\\begin{gathered}\n\\AtypMod{n} \\otimes \\AtypMod{n'} = \\AtypMod{n+n'}, \\qquad\n\\AtypMod{n} \\otimes \\TypMod{n',e'} = \\TypMod{n+n',e'}, \\qquad\n\\AtypMod{n} \\otimes \\ProjMod{n'} = \\ProjMod{n+n'}, \\\\\n\\TypMod{n,e} \\otimes \\TypMod{n',e'} = \n\\begin{cases}\n\\ProjMod{n+n'} & \\text{if $e+e'=0$,} \\\\\n\\TypMod{n+n'+1\/2,e+e'} \\oplus \\TypMod{n+n'-1\/2,e+e'} & \\text{otherwise,}\n\\end{cases}\n\\\\\n\\TypMod{n,e} \\otimes \\ProjMod{n'} = \\TypMod{n+n'+1,e} \\oplus 2 \\: \\TypMod{n+n',e} \\oplus \\TypMod{n+n'-1,e}, \\qquad\n\\ProjMod{n} \\otimes \\ProjMod{n'} = \\ProjMod{n+n'+1} \\oplus 2 \\: \\ProjMod{n+n'} \\oplus \\ProjMod{n+n'-1}.\n\\end{gathered}\n\\end{equation}\nThere are other indecomposables which may be constructed from submodules and quotients of the $\\ProjMod{n}$ by taking tensor products.  We will not need them and refer to \\cite{GotRep07} for further discussion.\n\n\\section{$\\AKMSA{gl}{1}{1}$ and its Representations} \\label{secAffine}\n\n\\subsection{Algebraic Structure} \\label{secAffAlg}\n\nOur conventions for $\\SLSA{gl}{1}{1}$ carry over to its affinisation $\\AKMSA{gl}{1}{1}$ in the usual way.  Explicitly, the non-vanishing brackets are\n\\begin{equation}\n\\comm{N_r}{E_s} = r k \\delta_{r+s,0}, \\qquad \\comm{N_r}{\\psi^{\\pm}_s} = \\pm \\psi^{\\pm}_{r+s}, \\qquad \\acomm{\\psi^+_r}{\\psi^-_s} = E_{r+s} + r k \\delta_{r+s,0},\n\\end{equation}\nwhere $k \\in \\mathbb{R}$ is called the level and $r,s \\in \\mathbb{Z}$.  We emphasise that when $k \\neq 0$, the generators can be rescaled so as to normalise $k$ to $1$:\n\\begin{equation} \\label{eqnLevelScaling}\nN_r \\longrightarrow N_r, \\qquad E_r \\longrightarrow \\frac{E_r}{k}, \\qquad \\psi^{\\pm}_r \\longrightarrow \\frac{\\psi^{\\pm}_r}{\\sqrt{k}}.\n\\end{equation}\nAs in the more familiar case of $\\AKMA{u}{1}$, we see that the actual value of $k \\neq 0$ is not physical.\n\nThe Virasoro generators are constructed using (a modification of) the Sugawara construction.  Because the quadratic Casimir of $\\SLSA{gl}{1}{1}$ is only defined modulo polynomials in $E$, one tries the ansatz \\cite{RozQua92}\n\\begin{equation} \\label{eqnDefT}\n\\func{T}{z} = \\mu \\func{\\normord{NE + EN - \\psi^+ \\psi^- + \\psi^- \\psi^+}}{z} + \\nu \\func{\\normord{EE}}{z},\n\\end{equation}\nfinding that this defines an energy-momentum tensor if and only if $\\mu = 1\/2k$ and $\\nu = 1\/2k^2$.  Moreover, the $\\AKMSA{gl}{1}{1}$ currents $\\func{N}{z}$, $\\func{E}{z}$ and $\\func{\\psi^{\\pm}}{z}$ are found to be Virasoro primaries of conformal dimension $1$ and the central charge is zero.\n\nThe structure theory of highest weight modules{} for $\\AKMSA{gl}{1}{1}$ turns out to be particularly accessible because of certain automorphisms.  These consist of the automorphism $\\mathsf{w}$ which defines the notion of conjugation and the family \\cite{SalGL106} of spectral flow automorphisms $\\sigma^{\\ell}$, $\\ell \\in \\mathbb{Z}$.  Explicitly,\n\\begin{equation}\n\\begin{aligned}\n\\func{\\mathsf{w}}{N_r} &= -N_{r}, \\\\\n\\func{\\sigma^{\\ell}}{N_r} &= N_r,\n\\end{aligned}\n\\qquad\n\\begin{aligned}\n\\func{\\mathsf{w}}{E_r} &= -E_{r}, \\\\\n\\func{\\sigma^{\\ell}}{E_r} &= E_r - \\ell k \\delta_{r,0},\n\\end{aligned}\n\\qquad\n\\begin{aligned}\n\\func{\\mathsf{w}}{\\psi^{\\pm}_r} &= \\pm \\psi^{\\mp}_{r}, \\\\\n\\func{\\sigma^{\\ell}}{\\psi^{\\pm}_r} &= \\psi^{\\pm}_{r \\mp \\ell},\n\\end{aligned}\n\\qquad\n\\begin{aligned}\n\\func{\\mathsf{w}}{L_0} &= L_0. \\\\\n\\func{\\sigma^{\\ell}}{L_0} &= L_0 - \\ell N_0.\n\\end{aligned}\n\\end{equation}\nThese automorphisms may be used to construct new modules $\\func{\\mathsf{w}^*}{\\mathcal{M}}$ and $\\func{\\sigma^*}{\\mathcal{M}}$ by twisting the action of the algebra on a module $\\mathcal{M}$:\n\\begin{equation} \\label{eqnInducedAction}\nJ \\cdot \\tfunc{\\mathsf{w}^*}{\\ket{v}} = \\func{\\mathsf{w}^*}{\\tfunc{\\mathsf{w}^{-1}}{J} \\ket{v}}, \\qquad J \\cdot \\tfunc{\\sigma^*}{\\ket{v}} = \\func{\\sigma^*}{\\tfunc{\\sigma^{-1}}{J} \\ket{v}} \\qquad \\text{($J \\in \\AKMSA{gl}{1}{1}$).}\n\\end{equation}\nNote that $\\func{\\mathsf{w}^*}{\\mathcal{M}}$ is precisely the module conjugate to $\\mathcal{M}$.\n\n\\subsection{Representation Theory} \\label{secAffRep}\n\nWe can now define affine highest weight states{}, affine Verma modules $\\AffVerMod{n,\\ell}$, and their irreducible quotients as before.  We remark only that \\eqref{eqnLevelScaling} suggests that we characterise modules by the invariant ratio $\\ell = e\/k$ rather than by the $E_0$-eigenvalue $e$.  The affine highest weight state{} $\\ket{v_{n,\\ell}}$ of $\\AffVerMod{n,\\ell}$, whose weight (its $N_0$- and $E_0\/k$-eigenvalues) is $\\brac{n + \\tfrac{1}{2}, \\ell}$, has conformal dimension\n\\begin{equation} \\label{eqnConfDim}\n\\Delta_{n,\\ell} = n \\ell + \\frac{1}{2} \\ell^2.\n\\end{equation}\nOf course, this formula also applies to singular vectors.  Again, the label $n$ refers to the average $N_0$-eigenvalue of the zero-grade subspace of $\\AffVerMod{n,\\ell}$, generalising the labelling convention of \\secref{secFinRep}.\n\nVerma modules for $\\AKMSA{gl}{1}{1}$ are infinite-dimensional and their characters have the form\n\\begin{equation} \\label{eqnCharVerma}\n\\ch{\\AffVerMod{n,\\ell}}{z;q} = \\traceover{\\AffVerMod{n,\\ell}} z^{N_0} q^{L_0} = z^{n+1\/2} q^{\\Delta_{n,\\ell}} \\prod_{i=1}^{\\infty} \\frac{\\brac{1 + z q^i} \\brac{1 + z^{-1} q^{i-1}}}{\\brac{1 - q^i}^2}.\n\\end{equation}\nFor the irreducible quotients, the case with $\\ell = 0$ is particularly easy.  As in \\secref{secFinRep}, we regard $\\brac{n,\\ell}$ (and modules so-labelled) as being \\emph{typical} if $\\AffVerMod{n,\\ell}$ is irreducible and \\emph{atypical} otherwise.\n\\begin{proposition} \\label{prop:ell=0}\nThe affine Verma module $\\AffVerMod{n,0}$ has an exact sequence\n\\begin{equation}\n\\dses{\\AffAtypMod{n-1\/2,0}}{\\AffVerMod{n,0}}{\\AffAtypMod{n+1\/2,0}}\n\\end{equation}\nin which the $\\AffAtypMod{n,0}$ are (atypical) irreducibles whose characters are given by\n\\begin{equation} \\label{eqnCharVac}\n\\ch{\\AffAtypMod{n,0}}{z;q} = z^n \\prod_{i=1}^{\\infty} \\frac{\\brac{1 + z q^i} \\brac{1 + z^{-1} q^i}}{\\brac{1 - q^i}^2}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSince $\\ell = 0$, every singular vector of $\\AffVerMod{n,0}$ has dimension $0$ by \\eqnref{eqnConfDim}.  The space of singular vectors is thus spanned by $\\ket{v_{n,0}}$ and $\\psi^-_0 \\ket{v_{n,0}}$.  Taking the quotient by the module generated by $\\psi^-_0 \\ket{v_{n,0}}$ gives a module with a one-dimensional zero-grade subspace.  The only singular vector is then the highest weight state{}, so this quotient is irreducible.  We denote it by $\\AffAtypMod{n+1\/2,0}$ as its zero-grade subspace has $N_0$-eigenvalue $n+\\tfrac{1}{2}$.  Its character follows trivially.  The submodule of $\\AffVerMod{n,0}$ generated by $\\psi^-_0 \\ket{v_{n,0}}$ is not a Verma module because $\\bigl( \\psi^-_0 \\bigr)^2 \\ket{v_{n,0}} = 0$.  It must therefore be a proper quotient of $\\AffVerMod{n-1,0}$ and, by the above argument, the only such quotient is the irreducible $\\AffAtypMod{n-1\/2,0}$.  The exact sequence follows.\n\\end{proof}\nFor $\\ell \\neq 0$, one proves by direct calculation \\cite{CR:GL11} that for $0 < \\abs{\\ell} < 1$, $\\AffVerMod{n,\\ell}$ is irreducible.  In other words, the corresponding irreducibles are typical, hence we denote them by $\\AffTypMod{n,\\ell}$.  For $\\abs{\\ell} \\geqslant 1$, the structure of the Verma modules now follows from considering the induced action of the spectral flow automorphisms.  More precisely, one proves \\cite{CR:GL11} that any Verma module is isomorphic to a twisted version of a Verma module with $-1 < \\abs{\\ell} < 1$ (or the conjugate of such a Verma module).  We summarise the result as follows.\n\\begin{proposition}\nWhen $\\ell \\notin \\mathbb{Z}$, the affine Verma module $\\AffVerMod{n,\\ell}$ is irreducible, $\\AffVerMod{n,\\ell} \\cong \\AffTypMod{n,\\ell}$, so its character is given by \\eqnref{eqnCharVerma}.  When $\\ell \\in \\mathbb{Z}$, the affine Verma module $\\AffVerMod{n,\\ell}$ has an exact sequence\n\\begin{equation}\n\\begin{gathered}\n\\dses{\\AffAtypMod{n+1,\\ell}}{\\AffVerMod{n,\\ell}}{\\AffAtypMod{n,\\ell}} \\qquad \\text{($\\ell = +1,+2,+3,\\ldots$),} \\\\\n\\dses{\\AffAtypMod{n-1,\\ell}}{\\AffVerMod{n,\\ell}}{\\AffAtypMod{n,\\ell}} \\qquad \\text{($\\ell = -1,-2,-3,\\ldots$),}\n\\end{gathered}\n\\end{equation}\nin which the $\\AffAtypMod{n,\\ell}$ are (atypical) irreducibles whose characters are given by\n\\begin{equation} \\label{eqnCharAtyp}\n\\ch{\\AffAtypMod{n,\\ell}}{z;q} = \n\\begin{cases}\n\\displaystyle \\frac{z^{n+1\/2} q^{\\Delta_{n,\\ell}}}{1 + zq^{\\ell}} \\prod_{i=1}^{\\infty} \\frac{\\brac{1 + z q^i} \\brac{1 + z^{-1} q^{i-1}}}{\\brac{1 - q^i}^2} & \\text{($\\ell = +1,+2,+3,\\ldots$),} \\\\\n\\displaystyle \\frac{z^{n+1\/2} q^{\\Delta_{n,\\ell}}}{1 + z^{-1} q^{-\\ell}} \\prod_{i=1}^{\\infty} \\frac{\\brac{1 + z q^i} \\brac{1 + z^{-1} q^{i-1}}}{\\brac{1 - q^i}^2} & \\text{($\\ell = -1,-2,-3,\\ldots$).}\n\\end{cases}\n\\end{equation}\n(The exact sequence and character for $\\ell = 0$ was given in \\propref{prop:ell=0}.)\n\\end{proposition}\n\\noindent Note that the $\\AffVerMod{n,\\ell}$ with $\\ell \\in \\mathbb{Z}$ have a non-trivial singular vector at grade $\\abs{\\ell}$.  We emphasise that the $\\AffAtypMod{n,\\ell}$ with $\\ell \\neq 0$ therefore possess a two-dimensional zero-grade subspace.\n\nThis description of the Verma modules, their irreducible quotients and characters relies upon being able to identify the result of applying the spectral flow automorphisms to modules.  For irreducibles, we have\n\\begin{equation}\n\\tfunc{\\bigl( \\sigma^{\\ell'} \\bigr)^*}{\\AffTypMod{n,\\ell}} = \\AffTypMod{n-\\ell',\\ell+\\ell'}, \\qquad \\tfunc{\\bigl( \\sigma^{\\ell'} \\bigr)^*}{\\AffAtypMod{n,\\ell}} = \\AffAtypMod{n-\\ell'+\\func{\\varepsilon}{\\ell+\\ell'}-\\func{\\varepsilon}{\\ell},\\ell+\\ell'},\n\\end{equation}\nwhere we introduce a convenient variant $\\varepsilon$ of the sign function on $\\mathbb{Z}$, defined by taking $\\func{\\varepsilon}{\\ell}$ to be $\\tfrac{1}{2}$, $0$ or $-\\tfrac{1}{2}$ according as to whether $\\ell \\in \\mathbb{Z}$ is positive, zero or negative, respectively.\n\n\\subsection{Fusion} \\label{secAffFus}\n\nThe fusion rules of the irreducible $\\AKMSA{gl}{1}{1}$-modules (among others) were first deduced in \\cite{Creutzig:2007jy} using three-point functions computed in a free field realisation and a conjectured completeness of the spectrum.  These rules and the spectrum conjecture were confirmed in \\cite{CR:GL11} through a direct argument involving the Nahm-Gaberdiel-Kausch fusion algorithm \\cite{NahQua94,GabInd96} and spectral flow.  The fusion ring generated by the irreducibles may be understood \\cite{Quella:2007hr} as a ``constrained lift'' of the representation ring \\eqref{RepRing} of $\\SLSA{gl}{1}{1}$ where the constraints are effectively implemented by spectral flow.  Explicitly, the rules are\n\\begin{equation} \\label{Fusion}\n\\begin{gathered}\n\\AffAtypMod{n,\\ell} \\mathbin{\\times} \\AffAtypMod{n',\\ell'} = \\AffAtypMod{n+n'-\\func{\\varepsilon}{\\ell,\\ell'},\\ell+\\ell'}, \\quad\n\\AffAtypMod{n,\\ell} \\mathbin{\\times} \\AffTypMod{n',\\ell'} = \\AffTypMod{n+n'-\\func{\\varepsilon}{\\ell},\\ell+\\ell'}, \\quad\n\\AffAtypMod{n,\\ell} \\mathbin{\\times} \\AffProjMod{n',\\ell'} = \\AffProjMod{n+n'-\\func{\\varepsilon}{\\ell,\\ell'},\\ell+\\ell'}, \\\\\n\\AffTypMod{n,\\ell} \\mathbin{\\times} \\AffTypMod{n',\\ell'} = \n\\begin{cases}\n\\AffProjMod{n+n'+\\func{\\varepsilon}{\\ell+\\ell'},\\ell+\\ell'} & \\text{if $\\ell+\\ell'=0$,} \\\\\n\\AffTypMod{n+n'+1\/2,\\ell+\\ell'} \\oplus \\AffTypMod{n+n'-1\/2,\\ell+\\ell'} & \\text{otherwise,}\n\\end{cases}\n\\\\\n\\AffTypMod{n,\\ell} \\mathbin{\\times} \\AffProjMod{n',\\ell'} = \\AffTypMod{n+n'+1-\\func{\\varepsilon}{\\ell'},\\ell+\\ell'} \\oplus 2 \\: \\AffTypMod{n+n'-\\func{\\varepsilon}{\\ell'},\\ell+\\ell'} \\oplus \\AffTypMod{n+n'-1-\\func{\\varepsilon}{\\ell'},\\ell+\\ell'}, \\\\\n\\AffProjMod{n,\\ell} \\mathbin{\\times} \\AffProjMod{n',\\ell'} = \\AffProjMod{n+n'+1-\\func{\\varepsilon}{\\ell,\\ell'},\\ell+\\ell'} \\oplus 2 \\: \\AffProjMod{n+n'-\\func{\\varepsilon}{\\ell,\\ell'},\\ell+\\ell'} \\oplus \\AffProjMod{n+n'-1-\\func{\\varepsilon}{\\ell,\\ell'},\\ell+\\ell'}.\n\\end{gathered}\n\\end{equation}\nHere, we have defined $\\func{\\varepsilon}{\\ell , \\ell'} = \\func{\\varepsilon}{\\ell} + \\func{\\varepsilon}{\\ell'} - \\func{\\varepsilon}{\\ell + \\ell'}$ for convenience.\n\nThese fusion rules also introduce the indecomposable modules $\\AffProjMod{n,\\ell}$ which are the counterparts of the projective $\\SLSA{gl}{1}{1}$-modules $\\ProjMod{n}$ discussed in \\secref{secFinRep}.\\footnote{More precisely, $\\AffProjMod{n,0}$ is the affine counterpart to $\\ProjMod{n}$ and the remaining $\\AffProjMod{n,\\ell}$ are obtained by spectral flow.}  The $\\AffProjMod{n,\\ell}$ are staggered with structure diagram\n\\begin{equation} \\label{picAffineStaggered}\n\\parbox[c]{0.28\\textwidth}{\n\\begin{center}\n\\begin{tikzpicture}[auto,thick,\n\tnom\/.style={circle,draw=black!20,fill=black!20,inner sep=2pt}\n\t]\n\\node (top) at (0,1.5) [] {$\\AffAtypMod{n,\\ell}$};\n\\node (left) at (-1.5,0) [] {$\\AffAtypMod{n+1,\\ell}$};\n\\node (right) at (1.5,0) [] {$\\AffAtypMod{n-1,\\ell}$};\n\\node (bot) at (0,-1.5) [] {$\\AffAtypMod{n,\\ell}$};\n\\node at (0,0) [nom] {$\\AffProjMod{n,\\ell}$};\n\\draw [->] (top) to (left);\n\\draw [->] (top) to (right);\n\\draw [->] (left) to (bot);\n\\draw [->] (right) to (bot);\n\\end{tikzpicture}\n\\end{center}\n}\n\\end{equation}\nand a non-diagonalisable action of the Virasoro mode $L_0$.  It follows that conformal field theories{} whose spectra contain typical modules will also contain such $\\AffProjMod{n,\\ell}$ (by fusion), and so will be \\emph{logarithmic}.\n\n\\section{W-Algebras extending $\\AKMSA{gl}{1}{1}$} \\label{secExtAlg}\n\n\\subsection{Chiral Algebra Extensions}\n\nOur search for extended algebras is guided by the following considerations:  First, note that if we choose to extend by a zero-grade field associated to any irreducible $\\AKMSA{gl}{1}{1}$-module, then we must include the rest of its zero-grade fields in the extension.  Second, the fields we extend by should be closed under conjugation.  Third, extending by fields from typical irreducibles will lead to logarithmic behaviour in the extended chiral algebra because fusing typicals with their conjugates yields the staggered indecomposable $\\AffProjMod{0,0}$.\n\nIt seems then that the most tractable extensions will involve zero-grade fields from atypical modules $\\AffAtypMod{n,\\ell}$ and their conjugates $\\AffAtypMod{-n,-\\ell}$.  The simplest extension we could hope for would involve a single atypical and its conjugate and have the further property that these extension fields generate no new fields at the level of the commutation relations.  This may be achieved for extension fields of integer or half-integer conformal dimension by requiring that the operator product expansions{} of the zero-grade fields of $\\AffAtypMod{n,\\ell}$ are regular.  From the fusion rules \\eqref{Fusion}, we obtain\n\\begin{equation}\n\\AffAtypMod{n,\\ell} \\mathbin{\\times} \\AffAtypMod{n,\\ell} = \\AffAtypMod{2n - \\func{\\varepsilon}{\\ell},2\\ell},\n\\end{equation}\nfrom which it follows that the zero-grade fields of $\\AffAtypMod{n,\\ell}$ will have regular operator product expansions{} with one another if $2 \\: \\Delta_{n,\\ell} \\leqslant \\Delta_{2n - \\func{\\varepsilon}{\\ell},2\\ell}$, that is, if\n\\begin{equation}\\label{eqdim}\n\\abs{\\ell} \\leqslant 2 \\: \\Delta_{n,\\ell}.\n\\end{equation}\nWe may take $\\ell$ positive without loss of generality.  Further, we require that the conformal dimension of the extension fields be a positive half-integer (so $2n\\ell \\in \\mathbb{Z}$).  \\eqnref{eqdim} then implies that there are $m$ distinct possibilities to extend by fields of dimension $m\/2$.\nWe denote by $\\alg{W}_{n,\\ell}$ the algebra obtained upon extending $\\AKMSA{gl}{1}{1}$ by the atypical module $\\AffAtypMod{n,\\ell}$ and its conjugate $\\AffAtypMod{-n,-\\ell}$.\n\n\\subsection{Characters of Extended Algebras}\n\nThe complete extended algebra also contains normally-ordered products of the extension fields and their descendants.  Indeed, the extended algebra $\\alg{W}_{n,\\ell}$ may be identified, at least at the level of graded vector spaces, with the orbit of the $\\AKMSA{gl}{1}{1}$ vacuum module under fusion by the simple current modules $\\AffAtypMod{n,\\ell}$ and $\\AffAtypMod{-n,-\\ell}$.  In other words,\n\\begin{equation}\n\\alg{W}_{n+1\/2,\\ell} = \\AffAtypMod{0,0} \\oplus \\bigoplus_{m=1}^{\\infty} \\bigl( \\AffAtypMod{mn+1\/2,m\\ell} \\oplus \\AffAtypMod{-mn-1\/2,-m\\ell} \\bigr).\n\\end{equation}\nThe character of the extended vacuum module is therefore\n\\begin{equation} \\label{eqnCharW}\n\\begin{split}\n\\ch{\\alg{W}_{n+1\/2,\\ell}}{y;z;q} &= \\ch{\\AffAtypMod{0,0}}{y,z;q} + \\sum_{m=1}^\\infty \\Bigl[ \\ch{\\AffAtypMod{mn+1\/2, m \\ell}}{y;z;q} + \\ch{\\AffAtypMod{-mn-1\/2, -m \\ell}}{y;z;q} \\Bigr] \\\\\n&= z \\sum_{m \\in \\mathbb{Z}} \\frac{y^{m \\ell} z^{mn} q^{\\brac{mn+1\/2} m \\ell + m^2 \\ell^2 \/ 2}}{1 + z q^{m \\ell}} \\cdot \\prod_{i=1}^{\\infty} \\frac{\\brac{1 + z q^i} \\brac{1 + z^{-1} q^{i-1}}}{\\brac{1 - q^i}^2}.\n\\end{split}\n\\end{equation}\nHere, we have introduced an additional formal variable $y$ in order to keep track of the eigenvalues of $E_0 \/ k$.  One can likewise identify the irreducible modules of the extended algebra with the other orbits of the extension modules.  We will not consider these modules, their characters, nor their interesting modular properties here, but will return to this in a future publication.\n\n\\subsection{Free Field Realisations} \\label{appFreeFields}\n\nThe affine Kac-Moody superalgebra $\\AKMSA{gl}{1}{1}$ has two well-known free field realizations, the standard Wakimoto realization \\cite{SalGL106} and one constructed from a pair of symplectic fermions, a euclidean boson, and a lorentzian boson \\cite{Guruswamy:1999hi}.  An explicit equivalence between the two realisations was established in \\cite{CR09}.  Here, we review the latter one.\n\nWe take the symplectic fermions $\\chi^{\\pm}$ and bosons $Y$, $Z$ to have the following operator product expansions{}:\n\\begin{equation}\n\\func{\\chi^+}{z} \\func{\\chi^-}{w} = \\frac{1}{\\brac{z-w}^2} + \\text{ regular terms}, \\qquad \n\\func{\\partial Y}{z} \\func{\\partial Z}{w} = \\frac{1}{\\brac{z-w}^2} + \\text{ regular terms}\n\\end{equation}\n(the others are regular).  The $\\AKMSA{gl}{1}{1}$ current fields are then given by\n\\begin{equation} \\label{eqnGL11FFR}\n\\func{E}{z} = k \\func{\\partial Y}{z}, \\qquad \\func{N}{z} = \\func{\\partial Z}{z}, \\qquad \\func{\\psi^{\\pm}}{z} = \\sqrt{k} \\vertop{\\pm \\func{Y}{z}} \\func{\\chi^{\\pm}}{z},\n\\end{equation}\nand a moderately tedious computation shows that the $\\AKMSA{gl}{1}{1}$ energy momentum tensor \\eqref{eqnDefT} indeed corresponds to the sum of those of the bosonic and symplectic fermion systems.\n\nIt remains to construct the $\\AKMSA{gl}{1}{1}$ primaries that generate our extended algebras.  As these correspond to atypical modules, this is relatively straight-forward.  First, we introduce some convenient notation:\n Let $X_{n,\\ell}$ be the bosonic linear combination $n Y + \\ell Z$ and define composite fields $F^{\\pm}_r$, with $r \\in \\mathbb{N}$, by $F^{\\pm}_0 = 1$ and $F^{\\pm}_r = \\normord{F^{\\pm}_{r-1} \\partial^{r-1} \\chi^{\\pm}}$ for $r \\geqslant 1$.  The conformal dimension of $F^{\\pm}_r$ is then $\\tfrac{1}{2} r \\brac{r+1}$.  The zero-grade fields of the atypicals $\\AffAtypMod{n,\\ell}$ for $\\ell > 0$ have conformal dimension $\\Delta_{n,\\ell} = \\ell \\brac{n + \\ell \/ 2}$ and are realised by\n\\begin{equation}\nV_{n,\\ell}^+ = \\vertop{X_{n+1\/2,\\ell}} F^-_{\\ell-1}, \\qquad V_{n,\\ell}^- = \\vertop{X_{n-1\/2,\\ell}} F^-_{\\ell}.\n\\end{equation}\nThis follows from their operator product expansions{} with the $\\AKMSA{gl}{1}{1}$ currents:\n\\begin{equation}\n\\begin{aligned}\n\\func{N}{z} \\func{V_{n,\\ell}^{\\pm}}{w} &= \\frac{\\brac{n \\pm 1\/2} \\: \\func{V_{n,\\ell}^{\\pm}}{w}}{z-w} + \\ldots , \\\\\n\\func{E}{z} \\func{V_{n,\\ell}^{\\pm}}{w} &= \\frac{\\ell k \\: \\func{V_{n,\\ell}^{\\pm}}{w}}{z-w} + \\ldots ,\n\\end{aligned}\n\\qquad\n\\begin{aligned}\n\\func{\\psi^+}{z} \\func{V_{n,\\ell}^-}{w} &= \\brac{-1}^{\\ell - 1} \\ell ! \\frac{\\sqrt{k} \\: \\func{V_{n,\\ell}^+}{w}}{z-w} + \\ldots , \\\\\n\\func{\\psi^-}{z} \\func{V_{n,\\ell}^+}{w} &= \\frac{(-1)^{\\ell-1}}{\\brac{\\ell-1}!} \\frac{\\sqrt{k} \\: \\func{V_{n,\\ell}^-}{w}}{z-w} + \\ldots ,\n\\end{aligned}\n\\end{equation}\nthe others being regular.  The zero-grade fields of the conjugate module $\\AffAtypMod{-n,-\\ell}$ are realised as\n\\begin{equation}\nV_{-n,-\\ell}^+ = \\vertop{X_{-n+1\/2,-\\ell}} F^+_{\\ell}, \\qquad V_{-n,-\\ell}^- = \\vertop{X_{-n-1\/2,-\\ell}} F^+_{\\ell-1}.\n\\end{equation}\nTheir operator product expansions{} with the current fields are similar.\n\n\\subsection{The Extended Operator Product Algebra}\n\nIn order to compute the leading contributions to the extended algebra operator product expansions{}, we need the expansion of the bosonic vertex operators.  To second order, this is\n\\begin{multline} \\label{eqnVertexOPE}\n\\vertop{\\func{X_{n,\\ell}}{z}} \\vertop{\\func{X_{n',\\ell'}}{w}} = \\brac{z-w}^{n\\ell'+n'\\ell} \\biggl[ \\vertop{\\func{X_{n+n',\\ell+\\ell'}}{w}} + \\normord{\\func{\\partial X_{n,\\ell}}{w} \\Vertop{\\func{X_{n+n',\\ell+\\ell'}}{w}}} \\brac{z-w} \\Biggr. \\\\\n\\Biggl. + \\frac{1}{2} \\normord{\\Bigl( \\func{\\partial X_{n,\\ell}}{w} \\func{\\partial X_{n,\\ell}}{w} + \\func{\\partial^2 X_{n,\\ell}}{w} \\Bigr) \\Vertop{\\func{X_{n+n',\\ell+\\ell'}}{w}}} \\brac{z-w}^2 + \\ldots \\biggr].\n\\end{multline}\nNote that it follows that $\\vertop{\\func{X_{n,\\ell}}{w}}$ and $\\vertop{\\func{X_{n',\\ell'}}{w}}$ will be mutually bosonic when $n\\ell' + n'\\ell$ is an even integer and mutually fermionic when $n\\ell' + n'\\ell$ is odd.  The implication of this for the statistics of the extended algebra generators $V_{n,\\ell}^{\\pm}$ and  $V_{-n,-\\ell}^{\\pm}$ is a little subtle.  It turns out that when $2n \\ell$ is even, these generators may be consistently assigned a bosonic or fermionic parity --- $\\alg{W}_{n,\\ell}$ is a superalgebra.  In fact, $V_{n,\\ell}^+$ and $V_{-n,-\\ell}^-$ will be fermions and $V_{n,\\ell}^-$ and $V_{-n,-\\ell}^+$ will be bosons in this case.  However, when $2n \\ell$ is odd, such an assignment is impossible --- $\\alg{W}_{n,\\ell}$ is \\emph{not} a superalgebra.  In this case, separately taking $V_{n,\\ell}^+$ and $V_{-n,-\\ell}^-$ to be bosons and $V_{n,\\ell}^-$ and $V_{-n,-\\ell}^+$ to be fermions is consistent, but the mutual locality of a boson and a fermion will now be $-1$ instead of $+1$.  We will remark further on this subtlety in \\secref{secExamples}.\n\nWe moreover need the leading terms of certain operator product expansions{} of the $F^{\\pm}_r$.  In particular,\n\\begin{equation} \\label{eqnCompositeSFOPEs}\n\\begin{split}\n\\func{F^+_r}{z} \\func{F^-_r}{w} &= \\brac{z-w}^{-r \\brac{r+1}} \\biggl[ \\mu_r^{\\brac{0}} + \\mu_{r-1}^{\\brac{2}} \\normord{\\func{\\chi^+}{w} \\func{\\chi^-}{w}} \\brac{z-w}^2 + \\ldots \\biggr] , \\\\\n\\func{F^-_{r-1}}{z} \\func{F^+_r}{w} &= \\brac{z-w}^{-\\brac{r-1} \\brac{r+1}} \\biggl[ \\mu_{r-1}^{\\brac{1}} \\: \\func{\\chi^+}{w} + \\ldots \\biggr] , \\\\\n\\func{F^-_r}{z} \\func{F^+_{r-1}}{w} &= \\brac{z-w}^{-\\brac{r-1} \\brac{r+1}} \\biggl[ \\mu_{r-1}^{\\brac{1}} \\: \\func{\\chi^-}{w} + \\ldots \\biggr] ,\n\\end{split}\n\\end{equation}\nwhere the coefficients $\\mu_r^{\\brac{a}}$, for $a = 0$, $1$, $2$, are given by\n\\begin{equation} \\label{eq:coeff}\n\\mu_r^{\\brac{a}} = \\sum_{\\sigma \\in \\group{S}_r} \\brac{-1}^{\\abs{\\sigma}} \\prod_{i=1}^r \\brac{i + \\func{\\sigma}{i} + a - 1}! = \\prod_{i=1}^r \\brac{i-1}! \\brac{i+a}!\n\\end{equation}\nThis last equality follows from recognising the $\\mu_r^{\\brac{a}}$ as determinants of Hankel matrices for which LU-decompositions are easily found.  In detail, consider the $r \\times r$ matrix $\\func{A_r}{a}$, for a non-negative integer $a$, with entries $\\brac{\\func{A_r}{a}}_{ij} = \\brac{i+j+a-1}!$  Defining $r \\times r$ matrices $\\func{L_r}{a}$ and $\\func{U_r}{a}$ by\n\\begin{equation}\n\\brac{\\func{L_r}{a}}_{ij} = \\frac{\\brac{i+a}!}{\\brac{j+a}!} \\binom{i-1}{j-1}, \\qquad \n\\brac{\\func{U_r}{a}}_{ij} = \\brac{i-1}! \\brac{j+a}! \\binom{j-1}{i-1},\n\\end{equation}\nand noting that $\\func{L_r}{a}$ is lower-triangular with diagonal entries equal to $1$ and $\\func{U_r}{a}$ is upper-triangular, we see that $\\func{L_r}{a} \\func{U_r}{a}$ is an LU-decomposition of $\\func{A_r}{a}$:\n\\begin{equation}\n\\begin{split}\n\\brac{\\func{L_r}{a} \\func{U_r}{a}}_{ij} &= \\sum_{k=1}^r \\frac{\\brac{i+a}! \\brac{i-1}! \\brac{j+a}! \\brac{j-1}!}{\\brac{k+a}! \\brac{k-1}! \\brac{i-k}! \\brac{j-k}!} = \\brac{j+a}! \\brac{i-1}! \\sum_{k=1}^r \\binom{i+a}{k+a} \\binom{j-1}{k-1} \\\\\n&= \\brac{j+a}! \\brac{i-1}! \\binom{i+j+a-1}{i-1} = \\brac{\\func{A_r}{a}}_{ij}.\n\\end{split}\n\\end{equation}\nSince $\\det \\: \\func{L_r}{a} = 1$, we obtain $\\det \\: \\func{A_r}{a} = \\det \\: \\func{U_r}{a} = \\prod_{i=1}^r \\brac{i-1}! \\brac{i+a}!$ and hence \\eqnref{eq:coeff}.\n\nWe are now in a position to obtain the leading contributions to the\noperator product expansions{} of the extension fields $V_{n,\\ell}^{\\pm}$ and their conjugates $V_{-n,-\\ell}^{\\mp}$.  Since we assume \\eqref{eqdim}, there are only four non-regular expansions and these take the form\n\\begin{equation} \\label{eqnGenExtAlgOPEs}\n\\begin{split}\n\\func{V_{n,\\ell}^+}{z} \\func{V_{-n,-\\ell}^+}{w} &= \\frac{\\mu_{\\ell-1}^{\\brac{1}} \\: \\func{\\psi^+}{w} \/ \\sqrt{k}}{\\brac{z-w}^{2 \\Delta_{n,\\ell} - 1}} + \\ldots , \\\\\n\\func{V_{-n,-\\ell}^-}{z} \\func{V_{n,\\ell}^+}{w} &= \\mu_{\\ell-1}^{\\brac{0}} \\Biggl[ \\frac{1}{\\brac{z-w}^{2 \\Delta_{n,\\ell}}} - \\frac{\\func{\\partial X_{n+1\/2,\\ell}}{w}}{\\brac{z-w}^{2 \\Delta_{n,\\ell} - 1}} + \\frac{\\ell \\brac{\\ell-1}}{2} \\frac{\\normord{\\func{\\chi^+}{w} \\func{\\chi^-}{w}}}{\\brac{z-w}^{2 \\Delta_{n,\\ell} - 2}} \\Biggr. \\\\\n& \\mspace{90mu} \\Biggl. + \\frac{1}{2} \\frac{\\normord{\\func{\\partial X_{n+1\/2,\\ell}}{w} \\func{\\partial X_{n+1\/2,\\ell}}{w}} - \\func{\\partial^2 X_{n+1\/2,\\ell}}{w}}{\\brac{z-w}^{2 \\Delta_{n,\\ell} - 2}} + \\ldots \\Biggr] , \\\\\n\\func{V_{-n,-\\ell}^+}{z} \\func{V_{n,\\ell}^-}{w} &= \\mu_{\\ell}^{\\brac{0}} \\Biggl[ \\frac{1}{\\brac{z-w}^{2 \\Delta_{n,\\ell}}} - \\frac{\\func{\\partial X_{n-1\/2,\\ell}}{w}}{\\brac{z-w}^{2 \\Delta_{n,\\ell} - 1}} + \\frac{\\ell \\brac{\\ell+1}}{2} \\frac{\\normord{\\func{\\chi^+}{w} \\func{\\chi^-}{w}}}{\\brac{z-w}^{2 \\Delta_{n,\\ell} - 2}} \\Biggr. \\\\\n& \\mspace{90mu} \\Biggl. + \\frac{1}{2} \\frac{\\normord{\\func{\\partial X_{n-1\/2,\\ell}}{w} \\func{\\partial X_{n-1\/2,\\ell}}{w}} - \\func{\\partial^2 X_{n-1\/2,\\ell}}{w}}{\\brac{z-w}^{2 \\Delta_{n,\\ell} - 2}} + \\ldots \\Biggr] , \\\\\n\\func{V_{n,\\ell}^-}{z} \\func{V_{-n,-\\ell}^-}{w} &= \\frac{\\mu_{\\ell-1}^{\\brac{1}} \\: \\func{\\psi^-}{w} \/ \\sqrt{k}}{\\brac{z-w}^{2 \\Delta_{n,\\ell} - 1}} + \\ldots\n\\end{split}\n\\end{equation}\nHere, we have used \\eqref{eq:coeff} to evaluate the ratios $\\mu_{r-1}^{\\brac{2}} \/ \\mu_r^{\\brac{0}} = \\tfrac{1}{2} r \\brac{r+1}$ appearing in these expansions.\n\n\\subsection{Examples} \\label{secExamples}\n\nLet us now illustrate the results of the above calculations with a few simple examples.  First, \\eqref{eqdim} tells us that the extended algebra $\\alg{W}_{n,\\ell}$ will be unique if we insist that the extension fields have conformal dimension $\\tfrac{1}{2}$.  Indeed, this requires $\\ell = 1$ and $n=0$.  We are therefore extending $\\AKMSA{gl}{1}{1}$ by the fields associated with the atypical modules $\\AffAtypMod{0,1}$ and $\\AffAtypMod{0,-1}$.  Since $2n \\ell = 0$ is even, the generators of the resulting extended algebra, $\\alg{W}_{0,1}$, may be assigned a definite parity:  $\\varkappa = V_{0,1}^+$ and $\\bar{\\varkappa} = V_{0,-1}^-$ are odd, $\\beta = V_{0,1}^-$ and $\\gamma = -V_{0,-1}^+$ are even.  The expansions \\eqref{eqnGenExtAlgOPEs} become\n\\begin{equation}\n\\begin{aligned}\n\\func{\\varkappa}{z} \\func{\\bar{\\varkappa}}{w} &= \\frac{1}{z-w} + \\func{N}{w} + \\frac{1}{2k} \\func{E}{w} + \\ldots , \\\\\n\\func{\\beta}{z} \\func{\\gamma}{w} &= \\frac{1}{z-w} + \\func{N}{w} - \\frac{1}{2k} \\func{E}{w} + \\ldots ,\n\\end{aligned}\n\\qquad\n\\begin{aligned}\n\\func{\\beta}{z} \\func{\\varkappa}{w} = +\\frac{\\func{\\psi^+}{w}}{\\sqrt{k}} + \\ldots , \\\\\n\\func{\\gamma}{z} \\func{\\bar{\\varkappa}}{w} = -\\frac{\\func{\\psi^-}{w}}{\\sqrt{k}} + \\ldots ,\n\\end{aligned}\n\\end{equation}\nwhich we recognise as a free complex fermion $\\brac{\\varkappa,\\bar{\\varkappa}}$ and a $\\beta \\gamma$ ghost system.  Because the mixed operator product expansions{} are regular, $\\alg{W}_{0,1}$ decomposes into the direct sum of the chiral algebras of these theories.\n\nIf we choose to extend by dimension $1$ fields, then there are two distinct choices:  $n = \\tfrac{1}{2}$ and $\\ell = 1$ or $n = \\tfrac{1}{2}$ and $\\ell = -2$.  We expect a current algebra symmetry in both cases.  Indeed, if we set $\\mathbf{H} = N + E \/ \\ell k$ and $\\mathbf{Z} = N - E \/ \\ell k$, then we discover that the $\\brac{\\mathbf{H} , \\mathbf{Z}}$-weights of the $\\AKMSA{gl}{1}{1}$ currents and the extension fields $V_{n,\\ell}^{\\pm}$, $V_{-n,-\\ell}^{\\pm}$ precisely match the $\\brac{\\mathbf{H} , \\mathbf{Z}}$-weights of the adjoint representation of $\\SLSA{sl}{2}{1}$.\\footnote{Here, $\\mathbf{H}$ and $\\mathbf{Z}$ should be associated with the matrices $\\diag \\set{1,-1,0}$ and $\\diag \\set{1,1,2}$ in the defining representation of $\\SLSA{sl}{2}{1}$.}  Moreover, we have\n\\begin{equation}\n\\func{\\mathbf{H}}{z} \\func{\\mathbf{H}}{w} = \\frac{2 \/ \\ell}{\\brac{z-w}^2} + \\ldots , \\qquad \\func{\\mathbf{Z}}{z} \\func{\\mathbf{Z}}{w} = \\frac{-2 \/ \\ell}{\\brac{z-w}^2} + \\ldots ,\n\\end{equation}\nand $\\func{\\mathbf{H}}{z} \\func{\\mathbf{Z}}{w}$ regular, which suggests that the extended algebra will be $\\AKMSA{sl}{2}{1}$ at level $1 \/ \\ell$.\n\nChecking this for the choice $\\ell = -2$ is easy.  As $2n \\ell = -2$ is even, $\\alg{W}_{1\/2,-2}$ admits a superalgebra structure.  Moreover, the fusion rules\n\\begin{equation}\n\\AffAtypMod{0,1} \\mathbin{\\times} \\AffAtypMod{0,1} = \\AffAtypMod{-1\/2,2}, \\qquad \\AffAtypMod{0,-1} \\mathbin{\\times} \\AffAtypMod{0,-1} = \\AffAtypMod{1\/2,-2}\n\\end{equation}\nimply that $\\alg{W}_{1\/2,-2}$ is a subalgebra of the extended algebra $\\alg{W}_{0,1}$ considered above.  One readily checks that by taking normally-ordered products, the $\\beta \\gamma$ ghost fields of $\\alg{W}_{0,1}$ generate the bosonic subalgebra $\\AKMA{sl}{2}_{-1\/2} \\subset \\AKMSA{sl}{2}{1}_{-1\/2}$, the complex fermion gives the $\\AKMA{u}{1}$-subalgebra, and the mixed products yield the remaining fermionic currents.  This establishes the superalgebra isomorphism $\\alg{W}_{1\/2,-2} \\cong \\AKMSA{sl}{2}{1}_{-1\/2}$.\n\nThe computation when $\\ell = 1$ is, however, more subtle because $2n \\ell = 1$ is odd, so $\\alg{W}_{1\/2,1}$ does not admit the structure of a superalgebra.  To impose the correct parities on the extended algebra currents, we must adjoin an operator-valued function $\\mu$ which is required to satisfy\n\\begin{equation} \\label{eqCocycle}\n\\mu_{a,b} \\mu_{c,d} = (-1)^{ad} \\mu_{a+b,c+d}, \\qquad \\text{($a,b,c,d \\in \\mathbb{Z}$).}\n\\end{equation}\nNote that the algebra generated by these operators has unit $\\mu_{0,0}$.  The currents are then given by\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E} &= +\\mu_{1,1} V_{1\/2,1}^+, \\\\\n\\mathbf{F} &= -\\mu_{-1,-1} V_{-1\/2,-1}^-,\n\\end{aligned}\n\\qquad\n\\begin{aligned}\n\\mathbf{H} &= N + E\/k, \\\\\n\\mathbf{Z} &= N - E\/k,\n\\end{aligned}\n\\qquad\n\\begin{aligned}\n\\mathbf{e}^+ &= -\\mu_{1,0} \\psi^+ \/ \\sqrt{k}, \\\\\n\\mathbf{f}^- &= +\\mu_{-1,0} \\psi^- \/ \\sqrt{k},\n\\end{aligned}\n\\qquad\n\\begin{aligned}\n\\mathbf{f}^+ &= \\mu_{0,-1} V_{-1\/2,-1}^+, \\\\\n\\mathbf{e}^- &= \\mu_{0,1} V_{1\/2,1}^-,\n\\end{aligned}\n\\end{equation}\nand routine computation now verifies that these currents indeed generate $\\AKMSA{sl}{2}{1}_1$.\n\nAs our final example, we briefly consider the case of extensions of conformal dimension $\\tfrac{3}{2}$.  There are now three distinct choices, corresponding to $n=1$, $\\ell=1$, or $n=-\\tfrac{1}{4}$, $\\ell=2$, or $n=-1$, $\\ell=3$.  The latter choice again results in an extended algebra which is a subalgebra of $\\alg{W}_{0,1}$ because\n\\begin{equation}\n\\AffAtypMod{0,1} \\mathbin{\\times} \\AffAtypMod{0,1} \\mathbin{\\times} \\AffAtypMod{0,1} = \\AffAtypMod{-1,3}.\n\\end{equation}\nBoth $\\alg{W}_{1,1}$ and $\\alg{W}_{-1,3}$ are superalgebras, while $\\alg{W}_{-1\/4,2}$ is not.  We expect, however, that a modification similar to \\eqref{eqCocycle} will restore the superalgebra parity requirements.  We will not analyse this in any detail as our interest in $\\Delta_{n,\\ell} = \\tfrac{3}{2}$ lies not with the full extended algebra, but rather with one of its subalgebras.\n\nWe start with the superalgebras $\\alg{W}_{1,1}$ and $\\alg{W}_{-1,3}$.  Both $V_{-n,-\\ell}^+$ and $V_{n,\\ell}^-$ are bosonic and upon defining\n\\begin{equation}\n\\begin{gathered}\n\\mathsf{g}^+ = \\sqrt{\\frac{3 \\alpha \\brac{3 \\alpha - 1}}{2 \\mu_{\\ell}^{\\brac{0}}}} \\: V_{-n,-\\ell}^+, \\qquad \n\\mathsf{g}^- = \\sqrt{\\frac{3 \\alpha \\brac{3 \\alpha - 1}}{2 \\mu_{\\ell}^{\\brac{0}}}} \\: V_{n,\\ell}^-, \\\\\n\\mathsf{j} = -\\alpha \\partial X_{n-1\/2,\\ell}, \\qquad \n\\mathsf{t} = \\frac{\\alpha}{2} \\normord{\\partial X_{n-1\/2,\\ell} \\partial X_{n-1\/2,\\ell}} - \\frac{\\ell \\brac{\\ell + 1}}{2} \\frac{\\alpha \\brac{3 \\alpha - 1}}{\\alpha + 1} \\frac{\\normord{\\psi^+ \\psi^-}}{k},\n\\end{gathered}\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha = \\frac{1}{\\brac{2n-1} \\ell},\n\\end{equation}\nwe obtain the defining relations of the \\emph{Bershadsky-Polyakov algebra} $W_3^{\\brac{2}}$ \\cite{PolGau90,BerCon91}:\n\\begin{equation}\n\\begin{gathered}\n\\func{\\mathsf{g}^+}{z} \\func{\\mathsf{g}^-}{w} = \\frac{\\brac{K+1} \\brac{2K+3}}{\\brac{z-w}^3} + \\frac{3 \\brac{K+1} \\func{\\mathsf{j}}{w}}{\\brac{z-w}^2} + \\frac{3 \\func{\\normord{\\mathsf{j} \\mathsf{j}}}{w} + \\tfrac{3}{2} \\brac{K+1} \\func{\\partial \\mathsf{j}}{w} - \\brac{K+3} \\func{\\mathsf{t}}{w}}{z-w} + \\ldots , \\\\\n\\func{\\mathsf{j}}{z} \\func{\\mathsf{g}^{\\pm}}{w} = \\frac{\\pm \\func{\\mathsf{g}^{\\pm}}{w}}{z-w} + \\ldots , \\qquad \n\\func{\\mathsf{j}}{z} \\func{\\mathsf{j}}{w} = \\frac{\\brac{2K+3}\/3}{\\brac{z-w}^2} + \\ldots , \\\\\n\\func{\\mathsf{t}}{z} \\func{\\mathsf{g}^{\\pm}}{w} = \\frac{3}{2} \\frac{\\func{\\mathsf{g}^{\\pm}}{w}}{\\brac{z-w}^2} + \\frac{\\func{\\partial \\mathsf{g}^{\\pm}}{w}}{z-w} + \\ldots , \\qquad \n\\func{\\mathsf{t}}{z} \\func{\\mathsf{j}}{w} = \\frac{\\func{\\mathsf{j}}{w}}{\\brac{z-w}^2} + \\frac{\\func{\\partial \\mathsf{j}}{w}}{z-w} + \\ldots , \\\\\n\\func{\\mathsf{t}}{z} \\func{\\mathsf{t}}{w} = \\frac{-\\brac{2K+3} \\brac{3K+1} \/ 2 \\brac{K+3}}{\\brac{z-w}^4} + \\frac{2 \\func{\\mathsf{t}}{w}}{\\brac{z-w}^2} + \\frac{\\func{\\partial \\mathsf{t}}{w}}{z-w} + \\ldots\n\\end{gathered}\n\\end{equation}\nHere, the $\\AKMA{sl}{3}$-level $K = \\tfrac{3}{2} \\brac{\\alpha - 1}$ is $0$ for $\\alg{W}_{1,1}$ and $-\\tfrac{5}{3}$ for $\\alg{W}_{-1,3}$.  The central charge of the $W_3^{\\brac{2}}$-subalgebra is in both cases $-1$.\n\nFor $\\alg{W}_{-1\/4,2}$, this procedure does not yield a Bershadsky-Polyakov algebra because $V_{-n,-\\ell}^+$ and $V_{n,\\ell}^-$ are, in this case, mutually fermionic.  Rather, these fields generate a copy of the $\\mathcal{N} = 2$ superconformal algebra of central charge $-1$.  Instead, we must consider the mutually bosonic fields $V_{n,\\ell}^+$ and $V_{-n,-\\ell}^-$.  Taking\n\\begin{equation}\n\\mathsf{g}^{+} = \\sqrt{3} \\: V_{1\/4,-2}^-, \\quad \\mathsf{g}^{-} = \\sqrt{3} \\: V_{-1\/4,2}^+, \\quad  \\mathsf{j} = -\\partial X_{1\/4,2}, \\quad \\mathsf{t} = \\frac{1}{2} \\normord{\\partial X_{1\/4,2} \\partial X_{1\/4,2}} - \\frac{1}{k} \\normord{\\psi^+ \\psi^-}\n\\end{equation}\nin particular, now leads to the Bershadsky-Polyakov algebra of level $0$ and central charge $-1$.  (In contrast, $V_{n,\\ell}^+$ and $V_{-n,-\\ell}^-$ are fermionic in both $\\alg{W}_{1,1}$ and $\\alg{W}_{-1,3}$, generating copies of the $\\mathcal{N} = 2$ superconformal algebra with central charges $1$ and $-1$, respectively.)\n\n\\subsection{$W^{\\brac{2}}_N$-subalgebras}\n\nIn the previous section, we found the Bershadsky-Polyakov algebra $W^{\\brac{2}}_3$, at certain levels, appearing as a subalgebra of the extended algebras $\\alg{W}_{1,1}$, $\\alg{W}_{-1\/4,2}$ and $\\alg{W}_{-1,3}$.  We now generalise this observation.  The algebra $W^{\\brac{2}}_3$ is defined \\cite{PolGau90,BerCon91} as the Drinfel'd-Sokolov reduction of $\\AKMA{sl}{3}$ corresponding to the non-principal embedding of $\\SLA{sl}{2}$ in $\\SLA{sl}{3}$.  Feigin and Semikhatov \\cite{Feigin:2004wb} found that it could also be realised as a subalgebra of $\\AKMSA{sl}{3}{1} \\oplus \\AKMA{u}{1}$ commuting with an $\\AKMA{sl}{3}$-subalgebra.  They then studied a generalisation $W^{\\brac{2}}_N \\subset \\AKMSA{sl}{N}{1} \\oplus \\AKMA{u}{1}$ which commutes with the obvious $\\AKMA{sl}{N}$-subalgebra.\n\nWhen $N=1$, these generalisations reduce to the chiral algebra of the $\\beta \\gamma$ ghost system.  For $N=2$, one gets $\\AKMA{sl}{2}$, and as mentioned above, $N=3$ recovers the Bershadsky-Polyakov algebra.  The examples studied in \\secref{secExamples} therefore lead us to the plausible conjecture that the $W^{\\brac{2}}_N$ algebras of Feigin and Semikhatov may be realised, at least for certain levels, as subalgebras of certain of our extended algebras $\\alg{W}_{n,\\ell}$.  We mention that there is a second construction of these $W^{\\brac{2}}_N$ algebras, but restricted to the critical level $K=-N$ (see \\eqref{eq:W_NC}), starting from the affine superalgebra $\\AKMSA{psl}{N}{N}$ at (critical) level $0$ \\cite{CGL}.\n\nFeigin and Semikhatov only computed the first few terms of the defining operator product expansions{} of $W^{\\brac{2}}_N$.  We will compare these terms with those obtained from our extended algebras, finding decidedly non-trivial agreement.  Our findings will, however, be stated as conjectures because the full operator product expansion{} of $W^{\\brac{2}}_N$ is not currently known.  $W^{\\brac{2}}_N$ is generated by two fields $\\mathcal{E}^{\\pm}_N$ of dimension $\\tfrac{1}{2} N$, a $\\AKMA{u}{1}$-current $\\mathcal{H}_N$ and an energy-momentum tensor $\\mathcal{T}_N$.  The defining expansions are:\n\\begin{equation}\\label{eq:W2nope}\n\\begin{split}\n\\func{\\mathcal{H}_N}{z} \\func{\\mathcal{H}_N}{w} &= \\frac{\\brac{N-1} K\/N + N-2}{\\brac{z-w}^2} + \\ldots , \\qquad\n\\func{\\mathcal{H}_N}{z} \\func{\\mathcal{E}^{\\pm}_N}{w} = \\pm \\frac{\\func{\\mathcal{E}^{\\pm}_N}{w}}{z-w} + \\ldots , \\\\\n\\func{\\mathcal{E}^+_N}{z} \\func{\\mathcal{E}^-_N}{w} &= \\frac{\\lambda_{N-1}}{\\brac{z-w}^N} +  \\frac{N \\lambda_{N-2} \\func{\\mathcal{H}_N}{w}}{\\brac{z-w}^{N-1}} - \\frac{\\brac{K+N} \\lambda_{N-3} \\func{\\mathcal{T}_N}{w}}{\\brac{z-w}^{N-2}} \\\\\n&\\mspace{-20mu} + \\frac{\\lambda_{N-3}}{\\brac{z-w}^{N-2}} \\sqbrac{\\frac{N \\brac{N-1}}{2} \\func{\\normord{\\mathcal{H}_N \\mathcal{H}_N}}{w} + \\frac{N \\bigl( \\brac{N-2} \\brac{K+N-1} - 1 \\bigr)}{2} \\func{\\partial \\mathcal{H}_N}{w}} + \\ldots\n\\end{split}\n\\end{equation}\nHere, $\\lambda_m = \\prod_{i=1}^m \\bigl( i \\brac{K+N-1} - 1 \\bigr)$, $K$ is the level of the $W^{\\brac{2}}_N$ algebra, and the central charge is given by\n\\begin{equation} \\label{eq:W_NC}\nC = -\\frac{\\bigl( \\brac{K+N} \\brac{N-1} - N \\bigr) \\bigl( \\brac{K+N} \\brac{N-2} N - N^2 + 1 \\bigr)}{K+N}.\n\\end{equation}\n\nSuppose first that $2n\\ell$ is even, so we can consider the bosonic subalgebra generated by the fields\n\\begin{equation}\n\\mathcal{E}^+_N = \\sqrt{\\frac{\\lambda_{N-1}}{\\mu_{\\ell}^{\\brac{0}}}} \\: V_{-n,-\\ell}^+, \\qquad \\mathcal{E}^-_N = \\sqrt{\\frac{\\lambda_{N-1}}{\\mu_{\\ell}^{\\brac{0}}}} \\: V_{n,\\ell}^-.\n\\end{equation}\nEvaluating the operator product expansion{} of these fields using \\eqref{eqnGenExtAlgOPEs} and comparing with \\eqref{eq:W2nope}, we find that the first two singular terms agree provided that $N = 2 \\Delta_{n,\\ell}$ and $\\mathcal{H}_N = -\\partial X_{n-1\/2,\\ell} \/ \\brac{2n-1} \\ell$.  This also fixes the $W^{\\brac{2}}_N$ level $K$.  Comparing the third terms fixes the form of the $W^{\\brac{2}}_N$ energy-momentum tensor $\\mathcal{T}_N$ and $\\mathcal{H}_N$ is then verified to have dimension $1$.  However, the $\\mathcal{E}^{\\pm}_N$ only have the required dimension $\\tfrac{1}{2} N = \\Delta_{n,\\ell}$ if $n=1$ or $2n+\\ell=1$.\\footnote{There is a third solution, $\\Delta_{n,\\ell} + \\ell + 1 = 0$, but this is invalid as we require $\\ell, \\Delta_{n,\\ell} > 0$.}  These constraints also let us check that $\\mathcal{T}_N$ is an energy-momentum tensor and the central charge turns out to be $C=-1$.  When $2n\\ell$ is odd, we instead consider the bosonic subalgebra generated by\n\\begin{equation}\n\\mathcal{E}^+_N = \\sqrt{\\frac{\\lambda_{N-1}}{\\mu_{\\ell-1}^{\\brac{0}}}} V_{-n,-\\ell}^-, \\qquad \\mathcal{E}^-_N = \\sqrt{\\frac{\\lambda_{N-1}}{\\mu_{\\ell-1}^{\\brac{0}}}} \\: V_{n,\\ell}^+.\n\\end{equation}\nA similar analysis reveals that this subalgebra agrees with $W^{\\brac{2}}_N$ up to the first three terms in the operator product expansions{} provided that $N = 2 \\Delta_{n,\\ell}$ and either $\\ell = 1$ or $\\ell = 2$.\\footnote{Taking $n = -\\tfrac{1}{2} \\brac{\\ell + 1}$ also satisfies these requirements, but then $2n\\ell$ is necessarily even.  Moreover, there is again a solution of the form $\\Delta_{n,\\ell} - \\ell + 1 = 0$, but it is easy to check that it leads to the wrong operator product expansion{} of $\\mathcal{T}_N$ with itself.}  In the first case, $C=1$; in the second, $C=-1$.\n\nWe summarise our findings as follows:\n\\begin{conjecture}\nThe extended algebra $\\alg{W}_{n,\\ell}$ has a subalgebra isomorphic to $W^{\\brac{2}}_N$ of level $K$ when:\n\\begin{itemize}\n\\item $\\ell = 1$ and $n = 0,1,2,\\ldots$  Then, $N = 2n + 1$ and $K = -2 \\brac{n-1} \\brac{2n+1} \/ \\brac{2n-1}$.\n\\item $\\ell = 1$ and $n = \\tfrac{1}{2}, \\tfrac{3}{2}, \\tfrac{5}{2}, \\ldots$  Then, $N = 2n + 1$ and $K = - \\brac{2n^2 - 1} \/ n$.\n\\item $\\ell = 2$ and $n = -\\tfrac{3}{4}, -\\tfrac{1}{4}, \\tfrac{1}{4}, \\ldots$  Then, $N = 4 \\brac{n+1}$ and $K = -2 \\brac{n+1} \\brac{4n+1} \/ \\brac{2n+1}$.\n\\item $n = -\\tfrac{1}{2} \\brac{\\ell - 1}$ and $\\ell = 1,2,3,\\ldots$  Then, $N = \\ell$ and $K = -\\brac{\\ell^2 - \\ell - 1} \/ \\ell$.\n\\end{itemize}\n\\end{conjecture}\n\\noindent Note that the examples considered in \\secref{secExamples} exhaust the $W^{\\brac{2}}_N$-subalgebras with $N \\leqslant 3$ except for $\\ell = 2$ and $n = -\\tfrac{3}{4}$.  This latter case is excluded if one insists, as we did with \\eqref{eqdim}, that the operator product expansion{} of $\\mathcal{E}^{\\pm}$ with itself is regular.  We mention that Feigin and Semikhatov actually computed the first \\emph{four} terms of the $W^{\\brac{2}}_N$ operator product expansions{}, finding in the fourth term a Virasoro primary field $\\mathcal{W}_N$ of dimension $3$ and $\\mathcal{H}_N$-weight $0$.  We have extended \\eqnTref{eqnVertexOPE}{eqnCompositeSFOPEs}{eqnGenExtAlgOPEs} to compute $\\mathcal{W}_N$ in our extended algebras and have checked that for each $\\ell$ and $n$ appearing in our conjecture, this field indeed has the required properties.  It follows that our conjecture has been verified for all $N \\leqslant 4$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\nA recent analysis of the Low Energy Neutrino Anomaly (LNA) \\cite{RNA11},\\cite{GiuLev10} led to a \nchallenging claim that this anomaly can be explained in terms of a new \nfourth neutrino with a much larger mass\nsquared difference. Assuming that the neutrino mass eigenstates are non \ndegenerate one finds\\cite{RNA11}\\cite{GiuLev10}:\n \\barr\n \\Delta m^2_{31}&\\approx& \\Delta m^2_{32}=|m_3^2-m_2^2|,\\nonumber\\\\ \\Delta m^2_{41}&\\approx& \\Delta m^2_{42}=|m_4^2-m_2^2|>1.5\\mbox{(eV)}^2\n \\label{deltam}\n \\earr\nwith a mixing angle:\n\\beq \n\\sin^2{2 \\theta_{14}}=0.14\\pm 0.08 (95\\%).\n\\eeq\n\nIt is obvious that this new neutrino should contribute to the oscillation \nphenomenon. In the present paper we will assume that the new neutrino is sterile, that is it does not participate in weak interaction. Even then, however, it has an effect on neutrino oscillations since it will tend to decrease the electron neutrino flux. This makes the analysis of oscillation experiments more \nsophisticated. In all the previous experiments the oscillation length is \nmuch larger than the size of the detector. So one is able to see the effect only \nif the detector is placed in the right distance from the source. It is, \nhowever, possible to design an experiment with an oscillation length of the \norder of the size of the detector, as it was proposed in \\cite{VERGIOM06},\\cite{VERNOV10}. This is \nequivalent to many standard experiments done simultaneously. \nThe main \nrequirements are as follows \\cite{VERNOV10}:\n\\begin{itemize}\n\\item The neutrinos should have as low as possible energy so that the oscillation \nlength can be minimized. At the same time it should not be too low, so that \nthe neutrino-electron cross section is sizable.\n\\item A monoenergetic neutrino source has the advantage that some of the features \nof the oscillation pattern are not washed out by the averaging over a \ncontinuous neutrino spectrum.\n\\item The life time of the source should be suitable for the experiment to be \nperformed. If it is too short, the time available will not be adequate for \nthe execution of the experiment. If it is too long, the number of counts \nduring the data taking will be too small. Then one will face formidable \nbackgrounds and\/or large experimental uncertainties.\n\\item The source should be cheaply available in large quantities. Clearly a \ncompromise has to be made in the selection of the source.\n\\end{itemize}\nAt low energies the only neutrino detector, which is sensitive to neutrino \noscillations, is one, which is capable of detecting recoiling electrons\\cite{VERGIOM06} or nuclei \\cite{VGN-NC11}:\n\nThe aim of this article is to show that the existence of a new fourth \nneutrino can be verified experimentally by the direct measurements\nof the oscillation curves for the monoenergetic neutrino-electron \nscattering. It can be done point-by-point within the dimensions of the detector, \nthus providing what we call  neutrino oscillometry \\cite{VERNOV10},\\cite{VERGIOMNOV}. \n\nThe electron neutrino, produced in weak interactions, can be expressed in \nterms of the standard mass eigenstates as follows:\n\\barr\n\\nu_e&=&\\cos_{\\theta_{14}}\\left[\\cos{\\theta_{12}} \\cos{\\theta_{13}}~\\nu_1+\\sin{\\theta_{12}} \\cos{\\theta_{13}}\\,\\nu_2+\\right .\\nonumber\\\\\n&&\\left . \\sin{\\theta_{13}}~ e^{i\\delta} \\nu_3 \\right]+\\sin{\\theta_{14}}e^{i\\delta_4} \\nu_4\n\\label{nue},\n\\earr\n where $\\sin{ \\theta_{13}}$ is a small quantity constrained by the\nCHOOZ experiment and  $\\sin{ \\theta_{14}}$ is the small mixing angle proposed for the resolution of  LNA\\cite{RNA11},\\cite{GiuLev10}.\n We can apply a four  neutrino oscillation analysis to write, under the approximations of Eq. \\ref{deltam},\n  the   $\\nu_e$ disappearance oscillation probability as follows:\n\\barr\n P(\\nu_e \\rightarrow \\nu_e )&\\approx&1- \n \\left [\n   \\sin ^2 {2\\theta_{12}}\n\\sin^2 {(\\pi \\frac{L}{L_{21}})}\\right .\\nonumber\\\\\n&+& \\left . \\sum_{n=3}^4  \\sin ^2{2\\theta _{1n}}\\sin^2{ (\\pi \\frac{L}{L_{n2}})} \\right]\n \\label{disap}\n\\earr\nwith\n\\beq \nL_{ij}=\\frac{4 \\pi E_{\\nu}}{m_i^2-m_j^2}.\n\\label{OscLength}\n\\eeq\nSince the oscillation lengths are very different, $L_{42}<<L_{32}<<L_{21}$, one may judiciously select the distance $L$ so that one observes  only one mode of oscillation, e.g. that to sterile neutrino. Thus\n\\beq\n P(\\nu_e \\rightarrow \\nu_e )\\approx1- \n \\left [\n \\sin ^2{2\\theta _{14}}\\sin^2{ (\\pi \\frac{L}{L_{42}})} \\right].\n \\label{sterdisap}\n\\eeq\nAs we have already mentioned in connection with the NOSTOS-project \\cite{VERGIOM06}, the \nexperiment involving electrons, unlike the hadronic target case, in addition to electron neutrino disappearance, is sensitive to \nthe other neutrino flavors, which  can also produce electrons. These flavors are generated via the appearance \noscillation. Since, however, the new neutrino is sterile, its presence will be manifested via  the mixing angle $\\theta_{14}$ due to the reduction of the electron neutrino flux only. \nThus the number of the scattered electrons, which bear this rather unusual\noscillation pattern, is proportional to\n the $(\\nu_e,e^-)$ scattering cross section, which can be cast in the form:\n  \\beq\n\\sigma(L, x, y_{th}) = \\sigma(0, x, y_{th}) (1-p(L, x))\n\\eeq\nwith $x = E_{\\nu }\/m_{e}$ and $y_{th} = (T_{e})_{th}\/m_{e}$ with \n($T_{e})_{th}$ the threshold electron energy imposed by the detector. Note that the function $\\chi(x)$, which appears when the neutrinos are not sterile \\cite{VERGIOM06},\ndoes not enter here.\nThe oscillation part due to the sterile neutrino  takes the form:\n\\beq\nP(L,x)=\\sin^{2}{\\left [ 2.48 \\left ((\\Delta m_{42}^{2}\/1\\mbox{eV}^{2})\\mbox{(L\/m)} \\right )\/x\\right ]}\\sin{^{2} (2\\theta_{14})}\n\\label{Eq5} \n\\eeq\nwith $L$ the source-detector distance (in meters) . \nThe total cross \nsection in the absence of oscillations can be written in the form :\n\\barr\n\\sigma(0,x,y_{th})&=&\\frac{G^2_F m^2_e}{2 \\pi} \\left (h(x)-h(y_{th})\\right ),\\nonumber\\\\\nh(x)&=&\\frac{x^2 \\left(17.7 x^2+15.3 x+3.36\\right)}{(2 x+1)^3}\n\\label{sigmatot2}\n\\earr\nwith $h(y_{th})\\rightarrow 0$, since the threshold effect is negligible in the case of the spherical TPC (STPC).\n\nWe will consider a spherical detector with the source at the origin and will \nassume that the volume of the source is much smaller than the volume of the \ndetector. \nThe number of events between $L$ and $L+dL$ is given by:\n\\beq\ndN=N_{\\nu} n_e \\frac{4 \\pi L^2dL}{4 \\pi L^2} \\sigma(L,x)=N_{\\nu} n_e dL \\sigma(L,x,y_{th})\n\\eeq\nor\n\\beq\n \\frac{dN}{dL}=N_{\\nu} n_e \\sigma(L,x,y_{th})\n\n \\label{eventsph}\n\\eeq\nor\n \\beq\n  R_0\\frac{dN}{dL}=\\Lambda\\tilde{\\sigma}(L,x,y_{th}),\n\\eeq\nwhere\n\\beq\n\\Lambda=\\frac{G^2_F m^2_e}{2 \\pi} R_0 N_{\\nu} n_e \n\\eeq\nwith $N_{\\nu}$ being the number of neutrinos emitted by the source, $n_e $ the density of electrons in the target, which is proportional to the atomic number Z, $R_0$ the radius of the target and $\\tilde{\\sigma}(L,x,y_{th})$ is the neutrino - electron cross section in units of \n${(G^2_F m^2_e)}\/{2 \\pi}$.\n\nOne can ask whether the relevant candidates for small length oscillation \nmeasurements exist in reality. A detailed analysis \nshows that there exist many cases of nuclei, which can  undergo  orbital electron capture yielding monochromatic neutrinos with low  energy.\n\nSince this process has the two-body mechanism, the total neutrino energy is \nequal to the difference of the total capture energy $Q_{EC}$ (which is the \natomic mass difference) and binding energy of captured electron $B_{i}$ and  the\nenergy of the final nuclear excited state $E^{\\ast } $, that is:\n\\beq\nE_{\\nu } = Q_{EC}  - B_{i }- E^{\\ast }. \n\\label{Eq12}\n\\eeq\nThis value can be easily determined because the capture energies are usually \nknown (or can be measured very precisely by the \nion-trap spectrometry \\cite{BNW10}) \nand the electron binding energies as well as the excited nuclear energies \nare tabulated \\cite{LARKINS},\\cite{AUDI03}.\n The main feature of the electron capture process is the \nmonochromaticity of neutrino. This paves the way for the neutrino \noscillometry \\cite{VERNOV10}. Since $\\Delta m_{42}^{2}  >  1.5\\mbox{(eV)}^{2} $\\cite{RNA11}, i.e. very \nlarge by neutrino mass standards, the oscillation length can be quite small \neven for quite energetic neutrinos. \n\n\\begin{table}[htbp]\n\\caption{\nProposed candidates for a new neutrino oscillometry at the \nspherical gaseous TPC. \nTabulated nuclear data have been taken from \\cite{AUDI03}, other data have been \ncalculated in this work (see the text for details. The mass of the source was assumed to be 0.1Kg).\n\\label{table1}}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n\\hline\n&   &  &  & & & &\\\\\nNucli-& \n$T_{1\/2}$ \\par & \n$Q_{EC}$ & \n$E_{\\nu }$ & \n$L_{32}$& \n$L_{42}$ &  \n$\\sigma(0,x) $& \n$N_{\\nu }$ \\\\\n& & & & && $10^{-45}$&\\\\\nde& \n(d)& \n(keV)& \n(keV)& \n(m)& \n(m)& \ncm$^2$& \n(s$^{-1})$ \\\\\n\\hline\n$^{37}$Ar& \n35 & \n814& \n811& \n842& \n1.35& \n5.69& \n$3.7\\times 10^{17}$ \\\\\n\\hline\n\n$^{51}$Cr& \n27.7 & \n753& \n747& \n742& \n1.23& \n5.12& \n$4.1\\times 10^{17}$ \\\\\n\\hline\n$^{65}$Zn& \n244 & \n1352& \n1343& \n1330& \n2.22&  \n10.5& \n$3.0\\times 10^{16}$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nIn other words, unlike the case involving $\\theta_{13}$ previously discussed \n\\cite{VERGIOM06},\\cite{VERNOV10},\\cite{VERGIOMNOV}, one can now choose much higher neutrino energy sources and thus \nachieve much higher cross sections. Thus our best candidates, see in Table \\ref{table1}, are \nnuclides, which emit monoenergetic neutrinos with energies higher than many \nhundreds of keV. Columns 2 and 3 show the decay characteristics of the \ncorresponding nuclides \\cite{NDSH}. The neutrino energies in column 4 have been \ncalculated by using equation (\\ref{Eq12}) taking  $Q_{EC}$ from \\cite{AUDI03} and $B_{i}$ \nfrom \\cite{LARKINS}. For these nuclides the capture is strongly predominant between the \nground states, thus $E^{\\ast }$ =0. Columns 5 and 6 give the oscillation \nlengths for the third and the fourth neutrino states. One can see that \n$L_{32}$ and $L_{42}$ are very different and that the two oscillation curves \ncan be disentangled.\nThe maximum energy of the recoiling electron can be \ncalculated by use of Eq. (2.4) in \\cite{VERNOV10}. Column 7 shows the neutrino-electron \ncross-sections calculated by the use of formula (\\ref{sigmatot2}). The last column presents \nthe neutrino source intensities which can be reasonably produced by \nirradiation of the corresponding targets of stable nuclides in the high flux \nnuclear reactors. \n\nThe goal of the experiment is to scan the monoenergetic neutrino electron \nscattering events by measuring the electron recoil counts in a function of \ndistance from the neutrino source prepared in advance at the reactor\/s. This \nscan means point-by-point determination of scattering events along the \ndetector dimensions within its position resolution.\n\nIn the best cases these events can be observed as a smooth curve, which \nreproduces the neutrino disappearance probability.\nIt is worthwhile to note again that the \noscillometry is suitable for monoenergetic neutrino, since it deals with a \nsingle oscillation length $L_{32}$ or $L_{42}$. This is obviously not a case \nfor antineutrino, since, in this instance, one extracts only an effective \noscillation length. Thus some information may be lost due to the folding \nwith the continuous neutrino energy spectrum.\n\nTable \\ref{table1} clearly shows that the oscillation lengths for a new neutrino \nproposed in \\cite{RNA11}, \\cite{GiuLev10} are much smaller compared to those previously considered \\cite{VERGIOMNOV} in connection with $\\theta_{13}$. They can thus  be  directly measured within \nthe dimensions of detector of reasonable sizes. One of the very promising \noptions could be the STPC  proposed \nin \\cite{VERGIOM06}. If necessary, a spherical Micromegas based on the micro-Bulk  \n technology \\cite{ADRIAM10},\n which will be developed in the near future, can be employed in the STPC. In fact a large detector 1.3 m in diameter has already been developed and it is under operation at the LSM (Laboratoire Souterrain de Modane) underground laboratory. The device provides sub-keV energy threshold and good energy resolution. \n A thin 50\n micron polyamide foil will be used as bulk material  to fabricate the\n detector structure. This detector provides an excellent energy  \n resolution, can\n reach high gains at high gas pressure (up to 10 bar) and has the advantage that its  \n radioactivity\n level \\cite{CEBRIAN10} should  fulfill the requirements of the proposed  \n experiment.\n \n In this spherical chamber with a modest radius of a few meters the shielded\nneutrino source can be situated in the center of the sphere.\nThe details of shielding, like the amount and the type of material surrounding the neutrino source, which is  required to reach an  \nappropriate background level, is under study. The electron \ndetector is also placed around the center of the smaller sphere with radius \n$r \\approx 1$m. The sphere volume out of the detector position is filled with a \ngas (a noble gas such as Ar or preferably Xe, which has a higher number of \nelectrons). The recoil electrons are guided by the strong electrostatic \nfield towards the Micromegas-detector \\cite{Giomataris},\\cite{GIOMVER08}. Such type of device has an \nadvantage in precise position determination (better than 0.1 m) and in \ndetection of very low electron recoils in 4$\\pi$-geometry (down to a few \nhundreds of eV, that well suits to  the nuclides of table \\ref{table1}).\n\nAssuming that we have a gas target under pressure $P$ and temperature $T_0$, the number of electrons in STPC can be determined by formula: \n\\beq\nn_e= Z\\frac{P}{kT_0}=4.4\\times 10^{27}m^{-3} \\frac{P}{10~{\\mbox{Atm}}}\\frac{Z}{18}\\frac{300}{T_0},\n\\eeq\nwhere $Z$ is the atomic number, while\n $P$ and $T_{0}$ stand for a gas pressure and \ntemperature.\n\n\nSince in the resolution of neutrino anomaly one can employ sources with quite \nhigh energy neutrinos of hundreds of keV, one expects large cross sections. \nTherefore a modest size source, so that it can easily fit inside the \ninner sphere of the detector, and a modest size detector say of radius of 4 \nm and pressure of 10 bar can be adequate. We will thus employ these \nparameters in this calculation and assume a running time equal to the life \ntime of the source. The result obtained for one of the candidates, nuclide \n$^{51}$Cr, is shown in Fig. \\ref{fig1}. This nuclide has previously been considered for oscillation measurements \\cite{VERNOV10}, \\cite{RNA11}, \\cite{GiuLev10}.\n\nAs can be seen from this figure the oscillometry curves are well \ndisentangled for different values of mixing angle $\\theta_{14 }$, which shows the \nfeasibility of this method for identification of the new neutrino existence \nas such. \n\nThe sensitivity for determination of $\\theta_{14 }$ can be deduced also from the \ntotal number of events in the fiducial volume of detector. After integration \nof equation (\\ref{eventsph}) over $L$ from 0 to 4 m it can be written in the form:\n\\barr\nN_{0} &=& A + B {\\sin^2{ (2\\theta_{14})}},\\quad A=N_{\\nu} n_e R_0 \\sigma(0,x),\\nonumber\\\\\n \\frac{B}{A}&=&- \\left [\\frac{1}{2}-\\frac{0.067}{R_0} x \\sin \\left(\\frac{7.45 R_0}{x}\\right) \\right ].\n\\label{Eq14}\n\\earr\nThus for 55 days of measurements with $^{51}$Cr we find:\n  $ A=1.59\\times 10^{4}$ and $B = -7.56 \\times 10^{3}$  . \n\nTaking these values we determined the sensitivity of $\\sin{^{2 }(2\\theta_{14})}$ = \n0.05 within 99{\\%} of confidence level reachable after two months of data handling in the STPC. This \nvalue is quite enough to access the validity of a new neutrino existence. \n\\begin{figure}[!ht]\n \\begin{center}\n \\includegraphics[width=3.3in,height=2.2in]{fig1.eps}\n\\hspace*{-0.0cm} { $L \\rightarrow$ meters}\\\\\n \\caption{ \n\n Oscillation spectra with different values of $\\sin{^{2}(2\\theta_{14})}$= \n0.07, 0.17 and 0.27 on the corresponding colored curves with the \nstatistical corridor of 1$\\sigma$. The values on the y-axis are obtained for 55 \ndays of measurement with a $^{51}$Cr source and an Ar target under a pressure of 10 bar. In all \ncases we have included distances up to $1.5\\times L_{42}$. The pattern is repeated \ntwo times up to the radius of the sphere $R_{0}$= 4 m.\n  } \n \\label{fig1}\n  \\end{center}\n  \\end{figure} \n\nThe results presented in Fig. \\ref{fig1}\ndid not take into consideration the electron energy threshold of 0.1 keV, \nwhich is too small in comparison with the neutrino energy and the average \nelectron recoil energy. We neglected also the Solar background of 2 counts \nper day derived from the measured Borexino results \\cite{BOREXINO8}, \\cite{BOREXINO9}. It is obvious that STPC  should be installed in an underground laboratory surrounded with appropriate shield against rock radioactivity.\n\n\nIn conclusion, we propose to use the oscillometry method for direct \nobservation of the fourth neutrino appearance. The calculations and analysis \nshows that neutrino oscillometry with the gaseous STPC is a powerful tool \nfor identification of a new neutrino in the neutrino-electron scattering. \nSince the expected mass-difference for this neutrino is rather high, the \ncorresponding oscillation length is going to be sufficiently  small for 1 MeV neutrino energy so that it can \nbe fitted into the dimensions of a spherical detector with the radius of a \nfew meters. The neutrino oscillometry can be implemented in this detector \nwith the use of the intense monochromatic neutrino sources which can be \nplaced at the origin of sphere and suitably shielded. The gaseous STPC with the Micromegas \ndetection has a big advantage in the 4$\\pi$-geometry and in very good position \nresolution (better than 0.1 m) with a very low energetic threshold ($\\approx \n$ 100 eV). The most promising candidates for oscillometry have been \nconsidered. The sensitivity for one of them, e.g. $^{51}$Cr, to the mixing angle \n$\\theta_{14}$ is estimated as $\\sin{^{2}(2\\theta_{14})}$ = 0.05 with the 99{\\%} of \nconfidence, which can be reached after two months of data handling. This value can be pushed further down by using  renewable sources.  The observation of the oscillometry curve suggested in this work will be a \ndefinite manifestation of the existence of a new type of neutrino, like the one\n recently proposed by the analysis of the low energy neutrino anomaly.\n\nA help of D. Nesterenko in preparation of this manuscript is very much \nacknowledged.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Historical Introduction}\n\nThe late 1500's and early 1600's were a remarkable period in the evolution of human thought. It might be reasonably argued that during this period Galileo Galilee put into practice the modern scientific method for describing and understanding natural processes. An equally important advancement in our way of thinking about the world was an emerging conviction of the universality of causes. This extraordinary new way of understanding the world around us is often associated with a slightly later period and with Isaac Newton. The notion that the laws of nature applied equally everywhere was indeed imagined in this earlier period by Johannes Kepler. In particular Kepler proposed that the principles that governed the movement of the planets was the same as on Earth. Kepler's thinking of a universal nature of physical properties both celestial and terrestrial is evident in his own words \\cite{Holton88} : ``I am occupied with the investigation of the physical causes. My aim in this is to show that the celestial machine is to be likened not to a divine organism but rather to a clockwork ..., insofar as nearly all the manifold movements are carried out by means of a single, quite simple magnetic force, as in the case of a clockwork all motion are caused by a simple weight. Moreover, I show how this physical conception is to be presented through calculation and geometry.\" Kepler's way of thinking about the motions of the planets and the universality of the laws of physics would be completely recognizable to every modern physicist.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=75mm]{Faraday.jpeg}\n\\caption{This figure is from Faraday's laboratory notebook describing the apparatus he constructed to conduct experiments on the relationship between gravity and electromagnetic induction \\cite{Faraday1885}. \\label{FaradayFig}}\n\\end{figure}\n\nWhile Kepler's conviction of the relationship between the motion of the planets and processes on Earth helped inspire our modern way of thinking, he was of course mistaken in making the association between gravity and a ``simple magnetic force''. However, even this mistake was an inspired effort to describe the world around us in terms of physical causes. The universality of physical principles quickly became a central theme in the development of physics and an inspiration for Newton and those that followed. This expectation of the universality of celestial and terrestrial processes and Kepler's expectation of  universality in a connection between magnetism and the motion of the planets is evident in Faraday's experimental investigations. Some time around the 1850's Faraday conducted experiments to demonstrate the possible connection between the gravitational field and the electromagnetic field. Faraday constructed an experimental apparatus in an effort to measure the magnitude of electromagnetic induction associated with a gravitational field as shown in Fig. \\ref{FaradayFig}. Faraday's results failed to demonstrate any relation between gravity and electricity but his commitment to this idea of universality was unwavering, \\cite{Faraday1885}, ``Here end my trials for the present. The results are negative. They do not shake my strong feeling of the existence of a relation between gravity and electricity, though they give no proof that such a relation exists.\"\n\nWhile Faraday's experiments were not successful, later theoretical research by Skobelev \\cite{Skobelev75} in 1975 supported Faraday's ``strong feeling\" by demonstrating a non-zero amplitude for the interaction of gravitons and photons in both scattering and annihilation. This association between gravity and electromagnetism was also described around the same time by Gibbons \\cite{gibbons} in noting that, ``Indeed since a `graviton' presumably in some sense carries light-like momentum the creation of one or more particles with time-like or light-like momentum would violate the conservation of momentum unless the created particles were massless and precisely aligned with the momentum of the graviton\". These kinematic restrictions for conversion of massless particles have also been studied more recently and in greater detail by Fiore and Modanese \\cite{Fiore96,Modanese95}. The processes of graviton and photon interaction described by Skobelev and Gibbons are exceedingly small \\cite{Skobelev75} but are non-zero. Our more recent research has expanded on this interaction of gravity\/gravitons and electromagnetism\/photons through annihilation and scattering processes by recognizing the contribution of the external gravitational field associated with a gravitational wave \\cite{Jones15,Jones16,Jones17,Jones18,Gretarsson18}. Our study of the vacuum production of light by a gravitational wave differs from Skobelev in that the amplitudes of the ``tree level diagrams\" would be dependent on the strength of the external gravitational field or equally the strain amplitude of the gravitational wave. This type of semi-classical conversion process between gravitational and electromagnetic fields was described more broadly by Davies \\cite{Davies01} ``One result is that rapidly changing gravitational fields can create particles from the vacuum, and in turn the back-reaction on the gravitational dynamics operates like a damping force.\" The back-reaction on the gravitational wave was shown to be small compared to the gravitational wave luminosity but sufficient to be detectable under the right circumstances \\cite{Jones17}. \n\nIn this brief review we will specifically outline the relationship between gravity and electricity for the special case of gravitational and electromagnetic radiation. While we take a historical perspective leading to current research no effort will be made to present the historical formalism. Instead we will present the ideas relating the association between gravitational and electromagnetic radiation and in particular the vacuum production of electromagnetic radiation by a gravitational wave using modern notation and mathematical formalism.\n\n\n\\section{Electromagnetism and light}\n\nA general review of the research on the relationship between gravity and electricity would completely preclude any possibility of being brief. We will instead focus our attention on the radiation regimes. The current understanding of the radiation regime for electricity began with James Clerk Maxwell's modification of Ampere's law \\cite{Maxwell2} to include the displacement current. This modification led to a wave equation solution to the equations of electromagnetism. Maxwell immediately recognized this wave equation as a description of the phenomena of light. In keeping with our intent to discuss the historical development of gravitational wave production of electromagnetic radiation using modern notation, the equations describing electromagnetic radiation will be presented in a form that is completely covariant. The Maxwell equations are written in terms of tensor relations and will have the same form in Minkowski space and curved space-time. The physical properties of electromagnetic radiation, such as luminosity, will be developed in terms of Newman-Penrose scalars  \\cite{Newman61,Teukolsky73}, in a form that is well suited for the comparison of electromagnetic and gravitational radiation \\cite{Jones17}.\n\nIn order to write the field equations for electrodynamics in a suitable \nform for curved space-time two tensors are defined in terms of the \nelectric and magnetic fields. The field strength tensor \n\\cite{Ellis73,Senego98,Hogan09,Palenzuela10,Lehner09,Lehner12_85,\nLehner12_86,Lehner16} is defined as  \\footnotemark \\footnotetext{Great \ncare is required with sign conventions in any covariant representation. \nThis is particularly true in the case of Maxwell's equations and here we \nare following Palenzuela {\\it et al.} \\cite{Palenzuela10}, which is consistent\nwith our metric. It is prudent to check the signs by confirming that the \ncovariant relations reduce correctly to the Maxwell equations in a \nLorentz inertial frame.}\n\n\\begin{equation}\nF^{\\mu \\nu }  = u^{\\mu} E^{\\nu}  - u^{\\nu} E^{\\mu} + e^{\\mu \\nu \\alpha \n\\beta} B_{\\alpha} u_{ \\beta},\n\\label{Faraday}\n\\end{equation}\n\n\\noindent and the dual to the field strength tensor as,\n\n\\begin{equation}\n^*F^{\\mu \\nu }  = u^{\\mu} B^{\\nu}  - u^{\\nu} B^{\\mu}  - e^{\\mu \\nu \n\\alpha \\beta} E_{\\alpha} u_{ \\beta},\n\\label{dFaraday}\n\\end{equation}\n\n\\noindent where $e^{\\mu \\nu \\alpha \\beta}$ is the ``Levi-Civita \npseudotensor of the space-time\" and $u_{\\nu}$ is the field frame 4-\nvelocity. The covariant expression for the field strength tensor \n\\eqref{Faraday} and the dual \\eqref{dFaraday} was originally developed \nby Ellis \\cite{Ellis73}. While these expressions are perhaps not widely \nknown, expanding \\eqref{Faraday} in a Lorentz inertial frame produces \nthe expected components for the field strength tensor. This covariant \nform of the field strength tensor has proven to be very useful in \nstudies of the relation between gravity and electromagnetism \n\\cite{Hogan09,Palenzuela10,Lehner09,Lehner12_85,Lehner12_86,Lehner16}. \nConversely, the electric and magnetic fields are found from the \ncontractions of the tensors with the 4-velocity,\n\n\\begin{equation}\nE^{\\mu} = F^{\\mu \\nu } u_{ \\nu},~ ~ B^{\\mu} = {^*}F^{\\mu \\nu } u_{ \\nu}.\n\\label{EBfieldsF}\n\\end{equation}\n\n\\noindent Using the field strength tensor and its dual the Maxwell \nequations can be written in a  covariant form that is the same in both Minkowski space and curved \nspace-time. The inhomogeneous Maxwell equations ({\\it i.e.} Gauss's law and Ampere's) \nlaw are,\n\n\\begin{equation}\n\\frac{1}{\\sqrt{-\\left| g_{\\mu \\nu} \\right|}} \\partial_{\\nu} \\left( \\sqrt{-\\left| g_{\\mu \\nu} \\right|} F^{\\mu \\nu } \n\\right)=  4 \\pi J^{\\mu}.\n\\label{Maxwell1}\n\\end{equation}\n\n\\noindent where $ \\left| g_{\\mu \\nu} \\right|= det [g_{\\mu \\nu}]$ is the determinant of the metric. The homogeneous \nGauss's law for magnetism and Faraday's law are \\cite{Greiner96, \nPalenzuela10},\n\n\\begin{equation}\n\\partial_{\\nu} \\left( \\sqrt{-\\left| g_{\\mu \\nu} \\right|} ~{^*}F^{\\mu \\nu } \\right)  = 0.\n\\label{Maxwell2}\n\\end{equation}\n\n\\noindent The covariant form of the conservation law is,\n\n\\begin{equation}\n\\partial_{\\mu} \\left( \\sqrt{-\\left| g_{\\mu \\nu} \\right|} J^\\mu \\right)= 0.\n\\label{Conservation}\n\\end{equation}\n\n\\noindent The field strength tensor can also be expressed in terms of \nthe electromagnetic 4-vector potential,\n\n\\begin{equation}\nF_{\\mu \\nu } = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu} ~.\n\\label{FourPotential}\n\\end{equation}\n\n\\noindent Maxwell came to the wave equation from the bottom up by recognizing that the displacement current term was missing from the traditional form of Ampere's law. In the modern notation the wave equation is a mathematical identity in the absence of source terms in the Maxwell equations \\cite{Tsaga05}. \n\nThe covariant form of the equations for the electromagnetic field appears naturally in the radiative expression for electrodynamics in the Newman-Penrose formalism \\cite{Newman61,Teukolsky73} through the introduction of the Newman-Penrose electromagnetic scalar, to be discussed shortly. In order to provide a means of comparison between electromagnetic and gravitational radiation using the Newman-Penrose formalism we will require the Lagrangian density for the electromagnetic field in curved space-time. Including the electric source terms the Lagrangian density is,\n\n\\begin{equation}\n \\mathcal{L}_\n{em}  =  - \\frac{1}{4}\\left( {\\partial _\\nu  A_\\mu   - \\partial _\\mu  A_\\nu  } \\right)\\left( {\\partial ^\\nu  A^\\mu   - \n\\partial ^\\mu  A^\\nu  } \\right) +J_\\mu A^\\mu.\n\\label{emLagrangian}\n\\end{equation}\n\n\\noindent The Lagrangian density can be simplified using the Lorenz gauge \\cite{Greiner96}, $\\partial_\\mu A^\\mu = 0$, and by restricting our attention to be source free ({\\it i.e.} $J_\\mu =0$) so that \\eqref{emLagrangian} becomes,\n\n\\begin{equation}\n \\mathcal{L}_{em}  =  - \\frac{1}{2}\\partial _\\mu  A_\\nu  \\partial ^\\mu\n A^\\nu .\n\\label{emLagrangianLG}\n\\end{equation}\n\n\\noindent Since we are only considering the radiation regime for the \nfield equations, we assume a plane wave solution for the \nelectromagnetic field and a massless vector field can then be \nexpressed in terms of a mode expansion \\cite{Greiner96} as follows,\n\n\\begin{equation}\nA_\\mu  \\left( {k,\\lambda ,x} \\right) = \\epsilon_{\\mu} \n^{(\\lambda)} \\phi ^{(\\lambda)} \\left( {k ,x} \\right).\n\\label{ModeExpN}\n\\end{equation}\n\n\\noindent with $k$ being the momentum, $x$ being the \nspace-time coordinates and $\\epsilon ^{(\\lambda)}$ being the \npolarization vectors with $\\lambda = 0, 1, 2, 3$. For example, the two \ntransverse modes are often labeled $\\lambda =1 , 2$ with $\\epsilon \n_\\mu ^{(1)} = (0,1,0,0)$ and $\\epsilon _\\mu ^{(2)} =  (0,0,1,0)$. The \n$\\lambda = 0, 3$ components are the time-like and longitudinal polarizations. The\nfour polarization vectors satisfy the orthogonality relationship\n$\\epsilon _\\mu ^{(\\lambda)} \\epsilon ^{\\mu ~(\\lambda')} = \n\\eta ^{\\lambda \\lambda'}$, with $\\eta ^{\\lambda \\lambda'}$ being the\nMinkowski metric.\n\nOne can define a complex scalar field using the $\\lambda = 1, 2$ \ncomponents of the real scalar fields {\\it i.e.} \n$\\phi ^{(1,2)} (k, x)$ as $\\varphi =\\frac{1}{\\sqrt{2}}\n(\\phi ^{(1)} + i \\phi ^{(2)})$. In terms of this complex scalar field \nthe Lagrange density of \\eqref{emLagrangianLG} can be written as\n\\begin{equation}\n\\label{emLagrangian2}\n{\\cal L}_{em} = - \\partial_\\mu \\varphi \\partial ^\\mu \\varphi ^* ~,\n\\end{equation}\nwhich is the Lagrange density for a massless complex scalar field. Below we will use a massless,\ncomplex scalar field as a stand-in for the massless photon. This substitution is justified \nby equation \\eqref{emLagrangian2}\n\nThe field equations following from the Lagrange density in \\eqref{emLagrangian2} are,   \n\n\\begin{equation}\n\\frac{1}{{\\sqrt{-\\left| g_{\\mu \\nu} \\right|}}} \\partial_{\\mu} \\sqrt{-\\left| g_{\\mu \\nu} \\right|} g^{\\mu \\nu} \\partial_{\\nu}\\varphi = 0.\n\\label{eomvarphi}\n\\end{equation}\n\n\\noindent This expression for the field equations is the same for both curved space-time and for Minkowski space-time. If we restricted our attention to plane waves propagating in Minkowski space-time the solution to \\eqref{eomvarphi} takes the form, \n\n\\begin{equation}\n\\varphi = B e^{i  \\text{k} u} + C,\n\\label{PlaneWave}\n\\end{equation}\n\n\\noindent where the $z$ axis is assumed to be along the direction of propagation, $B$ and $C$ are constants, and $k$ is the wave number. The solution \\eqref{PlaneWave} is written in the standard light cone coordinate $u = z - t$ and $\\partial_{z} - \\partial_{t} = 2 \\partial_{u}$ \\cite{Jones16}.\n\nThe previous discussion provides an outline of Maxwell's electromagnetic radiation in modern covariant form. The principle goal of this review is to realize Faraday's expectation of a relationship between gravity and electricity. We will demonstrate this relationship for gravitational and electromagnetic radiation by comparing the gravitational radiation luminosity to the luminosity of the corresponding electromagnetic radiation produced from the vacuum by the gravitational radiation. We have found that the Newman-Penrose formalism is a good method for calculating the luminosity of both gravitational radiation and the corresponding electromagnetic radiation. The electromagnetic luminosity is presented here in terms of the Newman-Penrose formalism and the gravitational radiation luminosity will be presented in the following section.\n\nThe radiated electromagnetic power per unit solid angle is found from the projection of the electromagnetic field strength onto the elements of a null tetrad ($l_\\mu, m^\\mu, n^\\mu , \\bar m^\\mu$) which gives us the electromagnetic Newman-Penrose scalar $\\Phi _2$. In the Newman-Penrose formalism the power per unit solid angle of emission for electromagnetic radiation is written as \\cite{Teukolsky73,Lehner09},\n\n\\begin{equation}\n\\frac{d E_{em} }{dt d\\Omega}  = \\mathop {lim}\\limits_{r \\to \\infty } \\frac{{r^2 }}{{4\\pi }}\\left| {\\Phi _2 } \\right|^2.\n\\label{emFlux0}\n\\end{equation}\n\n\\noindent The Newman-Penrose electromagnetic scalar in \\eqref{emFlux0}  \\cite{Newman61,Teukolsky73,Lehner09,Lehner12_86} is,\n\n\\begin{equation}\n\\Phi _2  = F_{\\mu \\nu } \\bar m^\\mu  n^\\nu.\n\\label{Phi2}\n\\end{equation}\n\n\\noindent The null tetrad of the Newman-Penrose formalism in \\eqref{Phi2} can be defined as \\cite{Lehner12_85},\n\n\\begin{equation}\n\\begin{array}{*{20}c}\n   {l^\\mu   = \\frac{1}{{\\sqrt 2 }}\\left( {1,0,0,1} \\right),} & {n^\\mu   = \\frac{1}{{\\sqrt 2 }}\\left( {1,0,0, - 1} \\right),}  \\\\\n   {m^\\mu   = \\frac{1}{{\\sqrt 2 }}\\left( {0,1,i,0} \\right),} & {\\bar m^\\mu   = \\frac{1}{{\\sqrt 2 }}\\left( {0,1, - i,0} \\right),}  \\\\\n\\end{array}\n\\label{null1}\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\n l \\cdot n =  - 1,~~m \\cdot \\bar m = 1 ,~~\nl \\cdot l = n \\cdot n = m \\cdot m = \\bar m \\cdot \\bar m = 0.\n\\label{null2}\n\\end{equation}\n\n\\noindent The electromagnetic field strength tensor in \\eqref{Phi2} is $F_{\\mu \\nu }  = \\partial _\\mu  A_\\nu  {\\kern 1pt}  - \\partial _\\nu  A_\\mu$, where $A_\\mu   = \\epsilon ^{(\\lambda)}  _\\mu \\phi ^{(\\lambda )} \\left( {t,z} \\right)$, and the plane polarization vectors are $\\epsilon ^{(1)} _\\mu   = \\left(0, 1, 0, 0 \\right), \\;\\epsilon ^{(2)} _\\mu   = \\left(\n0, 0, 1,0 \\right)$ \\cite{Jones17}. The electric and magnetic fields are determined by taking the derivatives of the scalar field \\eqref{PlaneWave}: $\\,\\partial _t \\varphi  =  - i  k B e^{i k \\left( {z - t} \\right)}$ and $\\partial_z \\varphi = i k B e^{i  k \\left( {z - t} \\right)}$. Collecting terms for the Newman-Penrose scalar of the ``out\" state of \\eqref{PlaneWave} \\cite{Lehner12_85,Jones17},\n\n\\begin{equation}\n\\Phi _2  = F_{\\mu \\nu } \\bar m^\\mu  n^\\nu =\n\\frac{1}{{\\sqrt 2 }}e^{ - i\\frac{\\pi }{4}} \\left( { \\partial _z \\varphi - \\partial _t \\varphi} \\right) =\ni e^{ - i\\frac{\\pi }{4}} {\\sqrt 2 }  k B e^{i k \\left( {z - t} \\right)}.\n\\label{emNPscalar0}\n\\end{equation}\n\n\\noindent The square of the electromagnetic scalar is then,\n\n\\begin{equation}\n\\left| {\\Phi _2 } \\right|^2  = 2  k^2 B^2.\n\\label{emNPscalar}\n\\end{equation}\n\nThe square of the Newman-Penrose scalar in  \\eqref{emNPscalar} is proportional to the electromagnetic flux, $F_{em} \\sim  \\left| {\\Phi _2 } \\right|^2 $. In Section \\ref{gRad} on gravitational radiation and in Section \\ref{production} on scalar field production we will show that by using the Newmam-Penrose scalars for the gravitational and the electromagnetic fields respectively, one can compare the fluxes of gravitational and electromagnetic radiation \\cite{Jones17}.\n\n\\section{Gravitational radiation}  \\label{gRad}\n\nA wave like solution to the vacuum equations for general relativity exist similar to that of electromagnetism \\cite{Schutz00}. This was recognized by Einstein soon after the development of general relativity and proposed even earlier by Poincar{\\' e} \\cite{Smoot16}. Initially there was doubt as to whether or not gravitational waves were physically real. Unlike the production of electromagnetic radiation there is no dipole source for gravitational radiation. This is because mass dipole production of radiation would violate conservation of 4-momentum \\cite{Schutz00,Smoot16}. However, there are also quadrupole source terms which lead to a wave solution and does not violate any conservation principles \\cite{Schutz00}.\n\n\nSince our interest here is in the relation between gravity and electromagnetism, in the radiation regime, we will restrict our attention to the plane wave solution of general relativity. The metric of a gravitational plane wave traveling in the $+z$ direction and with the $+$ polarization\ncan be written as \\cite{Schutz},\n\n\\begin{equation}\nds^2 = g_{\\mu \\nu} dx^{\\mu} dx^{\\nu} = -dt^2 + dz^2 + f^2 dx^2 + g^2 dy^2, \n\\label{GWmetric}\n\\end{equation}\n\n\\noindent were we set $c=1$. This metric is oscillatory with $f = 1 + \\varepsilon (u) $, $g = 1 - \\varepsilon(u)$ with $\\varepsilon(u) =h_+ e^{iku}$.  The coefficient $h_+$ is the gravitational wave strain amplitude and $k$ is the wave number. The coordinate variable in the metric is the standard light cone coordinate $u=z-t$. This metric only includes the ``plus\" polarization. Similar to electromagnetic radiation there are two degrees of freedom corresponding to two polarization states for gravitational radiation. The two polarization states for gravitational radiation are ``plus\" and ``cross\" polarization. They differ by an angle of $\\frac{\\pi}{4}$ in contrast to a phase angle difference of $\\frac{\\pi}{2}$ for electromagnetism \\cite{Schutz00, Schutz09}. Including the ``cross\" polarization would not change our discussion.\n\n\\begin{figure}[H]\n\\centering\n{\\caption{The ``discovery paper\" spectrogram of gravitational waves produced by a binary black hole in-spiral \\cite{LIGO}.} \\label{spectrogram}}\n{\\includegraphics[width=12cm]{SpectrogramBW.jpeg}}\n\\end{figure}\n\nThe luminosity of the gravitational radiation can be calculated using the Newman-Penrose formalism \\cite{Teukolsky73,Lehner09}. The relevant scalar for gravitational radiation is a projection of the Riemann tensor onto elements of a null tetrad. This projection is identified as the gravitational Newman-Penrose scalar $\\Psi _4$. The power per unit of solid angle for the gravitational radiation is written in terms of the gravitational scalar as,\n\n\\begin{equation}\n\\frac{d E_{gw} }{dt d\\Omega}  = \\mathop {lim}\\limits_{r \\to \\infty } \\frac{{r^2 }}{{16\\pi k^2 }}\\left| {\\Psi _4 } \\right|^2 .\n\\label{GWluminsity}\n\\end{equation}\n\n\\noindent Substituting the metric for the gravitational wave \\eqref{GWmetric} into the Riemann tensor for an outgoing gravitational plane wave in vacuum the gravitational Newman-Penrose scalar \\cite{Teukolsky73,Jones17} is,\n\n\\begin{equation}\n\\Psi _4  =  - R_{\\alpha \\beta \\gamma \\delta } n^\\alpha  \\bar m^\\beta  n^\\gamma  \\bar m^\\delta   = f\\partial_{u}^{2} f- g\\partial_{u}^{2} g .\n\\label{Psi}\n\\end{equation}\n\n\\noindent The partial derivatives, $\\partial_{u}$, are with respect to the light cone coordinate, $u$. For the plane wave metric, where $\\varepsilon = h_+ e^{iku}$, the gravitational Newman-Penrose scalar and the square are given by,\n\n\\begin{equation}\n \\Psi _4  =  - 2 h_+ k^2 e^{ik\\left( {z - t} \\right) } \\rightarrow |\\Psi _4 | ^2 = 4 h_+ ^2 k^4.\n \\label{Psi4}\n \\end{equation}\n\n\\noindent The Newman-Penrose scalars in \\eqref{emNPscalar} and \\eqref{Psi4} provide a convenient method for the comparison of the power of the gravitational radiation and the counterpart vacuum production of electromagnetic radiation which will follow. The flux of the gravitational wave can be calculated from \\eqref{GWluminsity} and  \\eqref{Psi} as \\cite{Schutz,Schutz09,Gretarsson18},\n\n\\begin{equation}\nF_{gw}= \\frac{c^3}{16 \\pi G} \\left| \\dot \\varepsilon \\right| =  \\frac{c^3 h_+ ^2 \\omega^2}{16 \\pi G},\n \\label{GWfluxh}\n \\end{equation}\n \n \\noindent which is a function of the strain amplitude $h$ and gravitational wave frequency $\\omega = 2 \\pi f$.\n \nThis review of the relationship between gravity and electromagnetism in the radiation regime might be considered of academic interest only except for the recent and remarkable discovery by the LIGO scientific collaboration of gravitational waves \\cite{LIGO}. The spectrogram from the discovery papers is provided in Fig. \\ref{spectrogram} for the binary in-spiral of two black holes. The detection of gravitational waves makes the discussion of the potential production of electromagnetic radiation by gravitational waves immediately relevant to current research in both the fundamental relation between gravity and electromagnetism as well as potential applications in astrophysics. If there were any lingering doubt about the certainty of the detection of gravitational waves the more recent detection of gravitational waves from a kilonova event \\cite{ligo2} has laid these doubts to rest. The kilonova event was first identified through the detection of gravitational waves by the LIGO scientific collaboration.  The kilonova was immediately verified across the electromagnetic spectra through the coordination of an international collaboration of observatories based around the World and in space. The kilonova observations have not only eliminated any reasonable doubt of the existence of gravitational waves but also ushered in a new era of ``multi-messenger\" astronomy and astrophysics.\n\n\n\\section{Electromagnetic fields in gravitational wave background}\n\n\\subsection{Gravitational waves and uniform magnetic field}\n\nThe first modern attempt to connect gravitational radiation and \nelectromagnetic fields was work by Gertsenshtein \\cite{Gertsenshtein60}\nwhich considered the linearized Einstein field equations coupled to an \nelectromagnetic plane wave.\n\n\\begin{equation}\n\\label{gert-1}\n\\Box {\\tilde h}^{\\mu \\nu} = -\\kappa t^{\\mu \\nu} ~,\n\\end{equation}\n\n\\noindent where $t^{\\mu \\nu} = \\frac{1}{4 \\pi} (F^{\\mu \\tau} F^\\nu _\\tau -\ng^{\\mu \\nu} F^{\\alpha \\beta} F_{\\beta \\alpha})$ is the energy-momentum\ntensor for the electromagnetic field, ${\\tilde h}^{\\mu \\nu} = h^{\\mu \\nu} \n- \\frac{1}{2} g^{\\mu \\nu} h$ is the trace reduced metric deviation \nof the metric tensor ({\\it i.e.} $g^{\\mu \\nu} = \\eta ^{\\mu \\nu} + h^{\\mu \n\\nu}$), and $\\kappa = 16 \\pi G$. Now the proposal in \n\\cite{Gertsenshtein60} was to generate gravitational waves by sending \nelectromagnetic waves through a constant magnetic field. This potential \neffect was compared to radio physics phenomenon of wave resonance. The \nidea being that despite the weak coupling of gravity one could \nnevertheless generate some significant amount of gravitational radiation \nby this method. \n\nNow if one takes the electromagnetic field to have a constant magnetic \nfield part (whose field strength tensor we denote by $F^{(0) \\mu \\nu}$) \nand a plane wave part (whose field strength we denote by $F ^{\\mu \\nu}$),\nand if we feed this into \\eqref{gert-1}, dropping the squared terms in\n$F^{(0) \\mu \\nu}$ and $F^{\\mu \\nu}$ and keeping only the cross terms \nwe arrive at\n\n\\begin{equation}\n\\label{gert-2}\n\\Box {\\tilde h}^{\\mu \\nu} = -\\frac{\\kappa}{2} \\left( F^{(0) \\mu \\tau} \nF^{\\nu} _\\tau - \\frac{1}{4} g^{\\mu \\nu} F^{(0) \\alpha \\beta} F_{\\alpha \\beta}    \\right) ~.\n\\end{equation}\n\n\\noindent One now assumes that the electromagnetic plane wave field and\ngravitational field propagate along the $z$ direction with wave number \n$k$ and have the form\n\n\\begin{equation}\n\\label{gert-3}\nF^{\\mu \\nu} = b(z) \\epsilon^{\\mu \\nu} e^{i(kz-\\omega t)}  ~~~;~~~\n{\\tilde h}^{\\mu \\nu} = a(z) \\zeta ^{\\mu \\nu} \\sqrt{\\frac{\\kappa}{k^2}}\ne^{i(kz - \\omega t)} ~,\n\\end{equation}\n\n\\noindent where $\\epsilon ^{\\mu \\nu} , \\zeta ^{\\mu \\nu}$ are the \nelectromagnetic and gravitational polarization tensors respectively. \nUsing \\eqref{gert-3} in \\eqref{gert-2} and assuming slowly varying\namplitudes $a(z) , b(z)$ one obtains the following relationship \nbetween the amplitudes\n\\begin{equation}\n\\label{gert-4}\ni \\frac{da (z)}{dz} = \\sqrt{\\frac{\\kappa}{16}} F^{(0) \\mu \\nu}\n\\epsilon_{\\beta \\nu} \\zeta^{\\beta} _\\mu b(z) ~.\n\\end{equation}\nUnder the assumption that $b(x) \\approx const.$ \\eqref{gert-4} can be \nintegrated to obtain $a(x)$ as\n\\begin{equation}\n\\label{gert-5}\n\\left| \\frac{a (z)}{b(0)} \\right|^2 = \\frac{\\kappa}{16 \\pi^2} \nB_0 ^2 T^2 ~,\n\\end{equation}\nwhere $B_0 \\simeq | F^{(0) \\mu \\nu} |$ is the constant magnetic field \nstrength, $T$ is the time that the electromagnetic wave traverses the\nuniform magnetic field, and $b(0)$ is the initial amplitude of the\nelectromagnetic wave. If one takes the cosmological sized magnetic fields\n($B_0 \\simeq 10 ^{-5}$ G) and assumes cosmological times for the \nelectromagnetic wave to travel through this constant magnetic field\n($T \\simeq 10^7$ years) one finds that the ratio of gravitational to \nelectromagnetic amplitude is of the order $|a\/b|^2 \\simeq 10^{-17}$. One \ncould increase this by having stronger magnetic fields and\/or longer \nperiods of travel. \n\nOne of the most interesting features of the above mechanism is that the\ngravitational wave frequency is the same as that of the electromagnetic \nwave. This gives the possibility of generating very high frequency \ngravitational waves compared to the ``natural'' sources of gravitational \nwaves from the first series of direct detections-- merging black hole, \nmerging neutron stars. These natural or astrophysical sources have \nfrequencies in the 100s to 1000s of Hertz, whereas electromagnetic \nradiation has a much broader range of frequencies which have been\nobserved -- from 1000s of Hertz to Gigahertz and beyond.\n\nIn the original work by Gertsenshtein \\cite{Gertsenshtein60} the focus \nwas on generating gravitational waves from electromagnetic waves. In this\nreview our focus is the exact opposite -- we are interested in \nelectromagnetic radiation generated from gravitational waves. This \nreversed possibility was pessimistically noted by Gertsenshtein with the\nconcluding comment ``From general relativity follows also the possibility\nof the inverse conversion of gravitational waves into light waves, \nbut this problem is hardly of interest.'' Nevertheless, several years \nafter Gertsenshtein's paper, Lupanov \\cite{lupanov67} did examine the \ninverse process of generating electromagnetic waves from gravitational\nwaves.  \n\nWe will follow the work in \\cite{lupanov67} by examining the reverse \nprocess of electromagnetic radiation generated from gravitational waves. \nThere are two  reasons for our focus on the reverse process: (i) \nelectromagnetic radiation, even weak radiation, is easier\nto detect, and (ii) we argue, beginning in the next subsection, that the\nconversion of gravitational waves to electromagnetic radiation occurs \neven in vacuum.\n\nBefore concluding this subsection we mention that there is more recent work\nin the spirit of Gertsenshtein's work \\cite{Gertsenshtein60}\nwhere the magnetic field is replaced by a Bose-Einstein condensate\n\\cite{fuentes}. In this case the interaction of the gravitational\nwave with the Bose-Einstein condensate is conjectured to lead to \nthe creation of phonons, just as in Gertsenshtein's work the \ninteraction of the gravitational wave with the magnetic field\nlead to the creations of photons. This creation of phonons\nwith a Bose-Einstein condensate has been put forward as a \npotential alternative mechanism to interferometers like LIGO\nto detect gravitational waves.  \n\n\n\\subsection{Feynman diagram approach to gravitational and electromagnetic\nradiation}\n\nTo check the assertion that electromagnetic radiation can be created in \nvacuum from gravitational radiation we turn to tree-level Feynman \ndiagrams for graviton-photon scattering. The four basic diagrams for this \nprocess are given in Fig. \\eqref{Skobelev} with curly lines \nrepresenting gravitons and wavy lines represent photons. The original \ncalculation was carried out by Skobelev \\cite{Skobelev75} with more \nrecent and more extensive calculations being found in references \n\\cite{Bohr14}. The diagrams in Fig. \\eqref{Skobelev} represent\n$graviton + photon \\to graviton + photon$ scattering. By rotating the \ndiagrams one can get $graviton + graviton \\to photon + photon$ or\n$photon + photon \\to graviton + graviton$ which can be viewed as \ncreation of photons (gravitons) from gravitons (photons). \n\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=110mm]{Skobelev.jpeg}\n\\caption{Tree level Feynman diagrams of graviton-photon transitions \n\\cite{Skobelev75}. Wavy lines represent photons and curly lines \nrepresent gravitons \\label{Skobelev}}\n\\end{figure}\n\nIn \\cite{Skobelev75} the process $graviton + photon \\to graviton + \nphoton$ is calculated first. After a long calculation the \ndifferential scattering cross section is found to be\n\\begin{equation}\n\\label{sko-1}\n\\frac{d \\sigma}{d \\cos \\Theta} =\\frac{\\kappa ^2 \\omega ^2}{64 \\pi}\n\\left( \\frac{1+ \\cos ^8 (\\Theta \/2) }{\\sin ^4 (\\Theta \/ 2)} \\right) ~,\n\\end{equation}\nwhere $\\kappa = 16 \\pi G$ as previously, $\\omega$ is the energy of \nthe system, and $\\Theta$ is the scattering angle.\n\nNow the process of interest in this review is where gravitational waves \ncreate electromagnetic waves or photons. In the Feynman diagram language \nthis means $graviton + graviton \\to photon + photon$. This process \ncan be obtained by rotating the diagrams in Fig. \\ref{Skobelev}  by $90^0$ degrees. Upon \ndoing this the differential cross section for $graviton + graviton \\to \nphoton + photon$ is found to be \\cite{Skobelev75}\n\\begin{equation}\n\\label{sko-3}\n\\frac{d \\sigma}{d \\cos \\Theta} =\\frac{\\kappa ^2 \\omega ^2}{64 \\pi}\n(\\cos ^8 (\\Theta \/2) + \\sin ^8 (\\Theta \/ 2) ) ~.\n\\end{equation}\nIntegrating \\eqref{sko-3} to obtain the total cross section \n\\begin{equation}\n\\label{sko-4}\n\\sigma =\\frac{\\kappa ^2 \\omega ^2}{160 \\pi} ~.\n\\end{equation}\nIf one takes the energy of the system to be the rest mass energy of \nthe electron, $\\omega = m_e c^2$, \\footnote{This energy is much larger\nthan the energy implied by the frequencies of the observed gravitational \nwave signals \\cite{LIGO}. For the energy associated with the frequencies\nimplies by the LIGO observations the cross section would be even smaller\n} one finds \\cite{Skobelev75} that  $\\sigma \\simeq 10^{-110} cm ^2$. \nThis is a very small number and indicates that at the level of individual\nphotons and gravitons this is not a large effect. However, given the \nenormous energy of the observed gravitational wave signals, which implies a\nlarge number of gravitons, we will argue that there are cases where the \nsmall cross section of \\eqref{sko-4} may be compensated for by the \nlarge number of gravitons\/strength of the gravitational wave. \n\n\\subsection{Massive Scalar field in Gravitational Plane Wave Background}\n\nThe idea of particle creation from a time dependent space-time was \nfirst considered in a series of papers by Parker \\cite{parker-1,parker-2,parker-3} which investigated particle production \nfrom the time dependent FRW cosmological space-time. The next major \ntime-dependent space-time to be studied in terms of particle production \nwas the gravitational plane wave space-time. The initial studies were \ncarried out by Gibbons \\cite{gibbons} and Deser \\cite{deser}, who \nconsidered the  production of a massive scalar field in a pulsed \ngravitational wave space-time. Reference \\cite{garriga} has a \nmore extensive discussion of particle creation from a gravitational \nwave background, again in the context of a massive scalar field. \nThe conclusion of all of these works was that massive scalar \nfields would not be created from such a plane wave  gravitational \nbackground. \n\nThis conclusion, of no particle creation from a plane gravitational \nwave background, appears at odds with recent work \\cite{Jones15,Jones16,Jones17,Jones18}. However these recent\nworks focus on the case of massless particles whereas references\n\\cite{gibbons,deser,garriga} focus on massive particles.\nAs noted by Gibbons one can already guess that the production of a \nmassive field from a gravitational plane wave would be forbidden\nby energy momentum conservation. A massless graviton can not \ndecay\/transform into massive particles since this would violate \nenergy-momentum conservation. It is the same reason that forbids \na photon from decaying into an electron-positron pair in free space.\n(A photon can decay\/transform into electron-positron pair in the\npresence of a heavy nuclei which acts to conserve energy and momentum). \n\nGeneral studies of when one type of massless field quanta can \ndecay\/transform into other massless field quanta can be found\nin \\cite{Modanese95} and \\cite{Fiore96}. Using energy-momentum \nconservation these two works show that some number of massless\nquanta can transform into some number of other massless field quanta\nso long as the incoming and outgoing particles lie along the same \ndirection. This is consistent with the Feynman diagram calculations\nof reference \\cite{Skobelev75} where the decay of gravitons to photons\n({\\it i.e.} $graviton + graviton \\to photon + photon$) is possible so long as the\nmomenta of all particles lie along the same direction. This is also the\ncondition under which the creation of massless fields\/field quanta \noccurs in references  \\cite{Jones15,Jones16,Jones17,Jones18}.\n\nIn the rest of this section we review the calculations of \\cite{gibbons,deser,garriga} which demonstrate the absence of\nparticle creation when the \nparticles are massive, since this will provide a nice segue to\nthe case of massless particles. We will follow the notation of\nreference \\cite{garriga}.\n\nGarriga and Verdaguer \\cite{garriga} begin by considering the plane\nwave metric of the form\n\\begin{equation}\n\\label{garr-1}\nds^2 = -dt^2 + dz^2 + g_{ab} (z, t) dx^a dx^b = \n-du dv + g_{ab} (u) dx^a dx^b ~,\n\\end{equation}\nwhere in the last expression the metric has been transformed to light\nfront coordinates defined as $u=z-t$ and $v=z+t$ with $c=1$. The \nindices $a, b =1, 2$ and run over the $x, y$ directions, which are \nperpendicular to the $+z$ direction of travel of the gravitational wave.\nFor a gravitational wave traveling in the $-z$ direction one would take\nthe metric components to be functions of the light front coordinate\n$v$ {\\it i.e.} $g_{ab} (v)$.\n\nNext a massive scalar field, $\\varphi$, is placed in the metric given by \n\\eqref{garr-1}. The equation for a massive scalar field in a curved\nbackground is given by\n\\begin{equation}\n\\label{garr-2}\n\\left [ \\frac{1}{{\\sqrt { -\\left| {g_{\\mu \\nu } } \\right|} }}\\left( \n{\\partial _\\mu  g^{\\mu \\nu } \\sqrt { - \\left| {g_{\\mu \\nu } } \\right|} \n\\partial _\\nu  } \\right) - m^2 \\right] \\varphi  = 0.\n\\end{equation}\nUsing  the metric \\eqref{garr-1} in equation \\eqref{garr-2}\nand applying separation of variables, one finds solutions for the \nscalar field of the form\n\\begin{equation}\n\\label{garr-3}\n\\varphi (u, v, x^a)  = \n\\frac{1}{(p_{-})^{1\/2} (det g_{ab} (u))^{1\/4} (2 \\pi)^{3\/2} }\n\\exp \\left[i p_a x^a - i  p_{-} v -\\frac{i}{4 p_{-}} \n\\int _0 ^u (g_{ab} p^a p^b +m^2) du\\right]~,\n\\end{equation}\nwith $p_{-}$ and $p_a$ being separation constants which physically\ncorrespond to momenta connected with the coordinates,\n$v$ and $x^a$ respectively. \n\nThe next step is to calculate the Bogoliubov  coefficients \\cite{davies} \nfor this scalar field for a sandwich space-time {\\it i.e.} one has a \nplane wave space-time for $u_1 < u < u_2 <0$, sandwiched between two \nMinkowski space-times for $u<u_1$ and $u >u_2$. The \n``in'' and ``out'' states for this sandwich space-time are \\cite{garriga}\n\\begin{eqnarray}\n\\label{garr-4}\n\\varphi ^{in} (u, v, x^a)  &=& \n\\frac{1}{(p_{-})^{1\/2} (2 \\pi)^{3\/2} }\n\\exp \\left[i p_a x^a - i  p_{-} v -\\frac{i}{4 p_{-}} \n(p_a p^a +m^2) u + i \\Delta \\right]~,\\\\\n\\varphi ^{out} (u,v, x^a) &=&\n\\frac{1}{(k_{-})^{1\/2} (2 \\pi)^{3\/2} }\n\\exp \\left[i k_a x^a - i  k_{-} v -\\frac{i}{4 k_{-}} \n(k_a k^a +m^2) u \\right]~,\n\\label{garr-5}\n\\end{eqnarray}\nwhere $\\Delta$ is a constant phase. For the exact expression for this\nphase as well as for the full details of the calculation and some \nsubtleties in the definition of the coordinates we refer the reader to\n\\cite{garriga}. The light front momentum $p_-$ and $k_-$ are given by \n\\begin{equation}\n\\label{garr-5a}\np_- = \\frac{\\omega_p - p_z}{2} ~~~;~~~ k_-=\\frac{\\omega_k - k_z}{2} ~,\n\\end{equation}\nwith $p_z , k_z$ being the three-momentum in the $z$ direction and \n$\\omega_p = \\sqrt{{\\bf p}^2 +m^2}$ and $\\omega_k = \\sqrt{{\\bf k}^2\n+m^2}$ are the energies associated with the wave solutions. \n\nFrom \\eqref{garr-4} and \\eqref{garr-5} one can calculate the\nBogoliubov beta coefficients to be\n\\begin{equation}\n\\label{garr-6}\n\\beta = - \\langle \\varphi ^{in} | \\varphi ^{out ~*}\\rangle \\propto \n\\delta (p_- + k_-) ~,\n\\end{equation}\nwith the Dirac delta being a function of $p_- , k_-$ coming from \nintegration over $dv$. Now if the scalar field is massive it is easy \nto see, using the expressions for $\\omega _p \\omega_k$ in equation \n\\eqref{garr-5a} that\n$p_- + k_- \\ne 0$ so that $\\beta =0$. However, if $m=0$ and the 3-\nmomentum are in the same direction ${\\bf p} = p_z = {\\bf k} = k_z$\nthen $p_- + k_- =0$ and $\\beta \\ne 0$ meaning that production of \nthe scalar field from the gravitational wave occurs. This is \nconsistent with the Feynman\ndiagram calculations of \\cite{Skobelev75,Bohr14} as well as the\ndiscussion in term of energy-momentum conservation of \nparticle decay\/production\/scattering of massless fields \n\\cite{Modanese95,Fiore96}. In the next section we investigate \nin more detail the possibility of producing massless fields\/particles \nfrom a gravitational wave background.   \n\n\\section{Particle Production from a Gravitational Wave Background}\n\nIn this section we review some recent work \\cite{Jones15,Jones16,Jones17,Jones18} on the production of massless fields from\ngravitational wave backgrounds. We use a massless scalar field\nto carry out the analysis, but our results also apply to the more\nrealistic case of a massless vector field from the results and\ndiscussion around equations \\eqref{emLagrangian},  \\eqref{emLagrangianLG}, \n\\eqref{ModeExpN}, and  \\eqref{emLagrangian2}.\nWe also look at the response of an Unruh-Dewitt \ndetector in a gravitational plane wave background which supports the\npicture of gravitons decaying\/transforming into photons. \n            \n\\subsection{Scalar field production} \\label{production}\n\nWe now repeat some of the calculations of the previous section but for a \nmassless scalar field. We will follow the work of \\cite{Jones16}. For\nthe gravitational plane wave background we take the metric of \n\\eqref{garr-1} to have the more specific form\n\n\\begin{equation}\n\\label{jones-1}\nds^2 = -dt^2 + dz^2 + f^2 (z,t) dx^2 + g^2 (z,t) dy^2 = \n-du dv +  f^2 (u) dx^2 + g^2 (u) dy^2  ~,\n\\end{equation}\nwhere we have again transformed to light front coordinates, $u, v$ and \ntaken $c=1$. The form of the metric in \\eqref{jones-1} assumes the\nplus-polarization for the gravitational plane wave, which we take \nwithout loss of generality. We further assume that the ansatz functions\nhave a oscillatory behavior of the form $f(u) = 1 + h_+ e^{iku}$ and\n$g(u) =1 - h_+ e^{iku}$, where $h_+$ is the dimensionless amplitude\nof the plus polarization and $k$ is the gravitational wave \nnumber. With $m=0$ the field equation for $\\varphi$ from \n\\eqref{garr-2} becomes\n\n\\begin{equation}\n\\label{jones-2}\n\\frac{1}{{\\sqrt { -\\left| {g_{\\mu \\nu } } \\right|} }}\\left( \n{\\partial _\\mu  g^{\\mu \\nu } \\sqrt { - \\left| {g_{\\mu \\nu } } \\right|} \n\\partial _\\nu  } \\right)  \\varphi  = 0.\n\\end{equation}\n\nUsing the plane wave metric from \\eqref{jones-1} along with the oscillatory form of the ansatz functions $f(u), g(u)$ equation \\eqref{jones-2}  becomes\n\n\\begin{equation}\n \\left( {4F (ku) \\partial _u \\partial _v  - 4ikG (ku) \\,\\partial _v  + \n H(ku) \\left( {\\partial _x^2  + \\partial _y^2 } \\right)} \\right)\\varphi  \n = 0,\n\\label{jones-3}\n\\end{equation}\n\n\\noindent where the functions $F(ku), G(ku), H(ku)$, are given by\n\n\\begin{equation}\n\\begin{array}{l}\n \\quad F\\left( {ku} \\right)  \\equiv  \\left( 1 - h_ + ^2 e^{2iku}  \\right) \n ^2, \\\\ \n \\quad G\\left( {ku} \\right)  \\equiv  h_ + ^2 e^{2iku} \\left( 1  -\n h_ + ^2 e^{2iku}  \\right), \\\\ \n \\quad H\\left( {ku} \\right)  \\equiv  \\left( {1 + h_ + ^2 e^{2iku} } \n \\right). \\\\ \n \\end{array}\n\\label{jones-4}\n\\end{equation}\n\n \\noindent In arriving at \\eqref{jones-3} we have assumed that the\n behavior of $\\varphi$ in the perpendicular $x, y$ directions are the \n same so that\n $\\partial _x \\varphi = \\partial _y \\varphi$ and \n $\\partial _x ^2 \\varphi = \\partial _y ^2 \\varphi$.\n\nTo solve \\eqref{jones-3} we employ separation of variables as\n$\\varphi (u, v, x, y) = U(u) V(v) X(x) Y(y)$. The ansatz functions\nin the $v, x, y$ directions are plane waves of the form\n\n\\begin{equation}\n X (x) = e^{ik_{xy} x} ~~~;~~~ Y(y) = e^{ik_{xy} y} ~~~;~~~~ \n V(v) =  e^{ik_v v} ~,\n\\label{jones-5}\n\\end{equation}\n\n\\noindent where we have enforced the equality of the $x$ and $y$ \ndirections by taking a common wave number, $k_{xy}$. With this \nset up and the solutions from \\eqref{jones-5} the solution for $U(u)$\nis \\cite{Jones16}\n\\begin{equation}\nU = B e^{\\frac{\\lambda }{k}} e^{  \\frac{- \\lambda }{{k\\left( {1 - h_ + \n^2 e^{2iku} } \\right)}}} \\left( {1 - h_ + ^2 e^{2iku} } \n\\right)^{\\frac{1}{2}\\left( {\\frac{\\lambda }{k} - 1} \\right)} e^{ - \ni\\lambda u} + C ~,\n \\label{jones-6}\n\\end{equation} \n\\noindent with $B, C$ being integration constants and \n$\\lambda = \\frac{k_{xy}^2}{2 k_v}$. Putting equations\n\\eqref{jones-5} \\eqref{jones-6} together, and taking $C=-B$ the \nscalar field in the plane wave background becomes\n\\begin{equation}\n\\varphi (u, v, x, y)   = B e^{\\frac{\\lambda }{k}} e^{ - \\frac{\\lambda }\n{{k\\left( {1 - \nh_ + ^2 e^{2iku} } \\right)}}} \\left( {1 - h_ + ^2 e^{2iku} } \n\\right)^{\\frac{1}{2}\\left( {\\frac{\\lambda }{k} - 1} \\right)} e^{ - \ni\\lambda u} e^{ik_v v} e^{ik_{xy} x} e^{ik_{xy} y} - B.\n\\label{jones-7}\n\\end{equation}\nIn the limit $h_+ \\to 0$ ({\\it i.e.} the gravitational background is  \nturned off) the scalar field in \\eqref{jones-7} becomes\n\\begin{equation}\n\\varphi _0  (t,x,y,z) = \nB e^{ - i\\lambda u} e^{ik_v v} e^{ik_{xy} x} e^{ik_{xy} y} \n-B \\rightarrow\n B e^{i\\left( {k_v  + \\lambda } \\right)t} e^{i\\left( {k_v  - \\lambda } \n \\right)z} e^{ik_{xy} x} e^{ik_{xy} y} -B ~,\n \\label{jones-8}\n\\end{equation}\n\\noindent where in the last step we have converted back to the original\n$t,x,y,z$ coordinates. One can see that $k_v + \\lambda$ plays the role\nof the wave energy and $k_v - \\lambda$ momentum in the $z$ direction. \nThe result in \\eqref{jones-8} is expected, since if the gravitational\nwave background is turned off one should recover a plane wave traveling\nin free space, which is what the solution in \\eqref{jones-8} represents. \n\nTaking the limit where all the wave numbers\/momenta go to zero \n({\\it i.e.} $k_v, \\lambda, k_{xy} \\to 0$) in equation \\eqref{jones-7}\none would expect the scalar field to vanish. However on taking this\nlimit one finds instead that \n\\begin{equation}\n\\varphi (u, v, x, y)   = B \\left[\\left( {1 - h_ + ^2 e^{2iku} } \n\\right)^{-\\frac{1}{2}} - 1 \\right] \\approx \\frac{B}{2} h_ + ^2 e^{2iku}\n+ \\frac{3B}{8} h_ + ^4 e^{4iku} ~.\n\\label{jones-9}\n\\end{equation}\nThe result in \\eqref{jones-9} shows that even when one tries to take\nthe wave to its vacuum state, namely $k_v , \\lambda, k_{xy} \\to 0$,\nthere is a non-vanishing and non-trivial scalar field. This \nnon-vanishing scalar field is the field\/field quanta created by the \ngravitational wave background. Note that if one takes $h_+ \\to 0$\nin \\eqref{jones-9} that one does get the expected value for the scalar\nfield $\\varphi \\to 0$ \\footnote{Setting the constant $C=-B$ in\n\\eqref{jones-6} is done to get $\\varphi \\to 0$ in this limit. If one takes \n$C=0$ the $h_+ \\to 0$ limit of \\eqref{jones-9} would be\n$\\varphi \\to B$ which is also a vacuum solution to the wave equation\nfor the massless $\\varphi$, but having $\\varphi \\to 0$ is more ``natural\".}\nThe four-current associated with $\\varphi$ is given by the standard\nexpression $j_\\mu = -i (\\varphi \\partial _\\mu \\varphi ^* - \\varphi^* \n\\partial_\\mu \\varphi)$. Inserting this solution from \\eqref{jones-7} in the expression for the four-current and time averaging gives \\cite{Jones16}\n\\begin{equation}\n\\langle j_\\mu \\rangle = -2 B^2 \\lambda - B^2 h_+ ^4 \\left(\n\\frac{9}{2} \\frac{\\lambda ^3}{k^2} -\\frac{12 \\lambda ^2}{k}\n+ \\frac{13}{2} \\lambda - k \\right) ~.\n\\label{jones-10}\n\\end{equation}\nThe constant $B$ is determined by choosing a normalization condition or\nconvention. Following references \\cite{stahl} and \\cite{Jones16} \nwe pick the normalization condition $B = \\frac{1}{\\sqrt{2 k V}}$. \nOther possible normalization conditions for $B$ are \ndiscussed in \\cite{halzen}. With this normalization the vacuum \nscalar field from \\eqref{jones-9} reads\n\\begin{equation}\n\\varphi (u, v, x, y)   =    \\frac{1}{\\sqrt{2 k V}}\n\\left[\\left( {1 - h_ + ^2 e^{2iku} } \n\\right)^{-\\frac{1}{2}} - 1 \\right] \\approx \n\\frac{1}{2 \\sqrt{2 k V}} h_ + ^2 e^{2iku}\n\\left( 1 + \\frac{3}{4} h_ + ^2 e^{2iku} \\right) ~.\n\\label{jones-11}\n\\end{equation}\nTime averaging this vacuum current from \\eqref{jones-10} gives\n\\begin{equation}\n\\langle j_\\mu \\rangle =   \\frac{\\rm{sign} (k) h_+ ^4 }{2 V}~.\n\\label{jones-12}\n\\end{equation}\n\n\\noindent In \\eqref{jones-11} we are using a normalization that assumes the \nscalar field is in a box of volume $V$. In the previous section we took  \n$B=\\frac{1}{(2 \\pi) ^{3\/2}}$ -- see \\eqref{garr-4} \\eqref{garr-5}.\n\nEquation \\eqref{jones-10} gives the effect, in terms of the \nfour-current, of passing a massless scalar field through a \ngravitational wave. On setting all the energy-momentum of the\nscalar field to zero one finds, from equations \\eqref{jones-11} and \n\\eqref{jones-12}, that the scalar field and scalar field current\ndo not vanish. This represents the production \nof scalar field\/scalar field quanta from the gravitational wave\nbackground. \n\nFollowing \\cite{Jones17} the ratio of the \nproduced electromagnetic radiation \\eqref{emFlux0} \nto the gravitational \\eqref{GWluminsity} radiation can be written \ndown in terms of electromagnetic and gravitational Newman-Penrose\nscalars from  \\eqref{Phi2} and \\eqref{Psi4},  \n\n\\begin{equation}\n\\frac{{dE_{em} }}{{dE_{gm} }} = \\frac{{\\left( {\\frac{1}{{4\\pi }}\\left| {\\Phi _2 } \\right|^2 } \\right)}}{{\\left( {\\frac{1}{{16\\pi k^2 }}\\left| {\\Psi_4 } \\right|^2 } \\right)}} = \\frac{F_{em}}{F_{gw}}.\n\\label{emgwRatioA}\n\\end{equation}\n\n\\noindent Switching to a normalization where $B=1$ in \\eqref{jones-9} \nthe amplitude of the leading term of the scalar field is $\\frac{h_+^2}{2}$.\nUsing this in the expression for $|\\Phi_2|^2$  calculated \nin \\eqref{emNPscalar}  we get $|\\Phi_2|^2 =  2 k^2 h_+^4$. \nNext from \\eqref{Psi4} we recall that \n$|\\Psi _4 |^2 = 4 h_+^2 k^4$ for a gravitational plane we. Using all this\nin \\eqref{emgwRatioA} yields a relationship between the electromagnetic wave \nflux and gravitational wave flux\n\n\\begin{equation}\nF_{em}  = 2 h_+^2 F_{gw}.\n\\label{emgwRatioB}\n\\end{equation}\n\n\\noindent Since the gravitational radiation is proportional to $h_+^2$ the electromagnetic production will be proportional to $h_+^4$. The result in equation \\eqref{emgwRatioB} is consistent with the result in equation\n\\eqref{jones-12}.\n\n\\subsection{Unruh-Dewitt detector approach}\n\nAnother approach to study the connection between gravitational and \nelectromagnetic radiation is through the use of an Unruh-DeWitt detector\n\\cite{unruh-det,dewitt-det}. An Unruh-DeWitt detector is a two-state, \nquantum system which is placed in a given space-time background. If the\nUnruh-DeWitt detector is excited from the low energy state to the \nhigh energy state, this is taken to indicate that the given space-time\nhas produced field quanta in order to excite this transition. \nTwo common examples of the use of an Unruh-DeWitt detector are placing \nit in the Schwarzschild space-time \\cite{davies,hawking} of a black hole or \nplacing it the Rindler space-time of an accelerating observer \\cite{unruh}. \nIn the first case the Unruh-DeWitt detector will detect the photons from \nHawking Radiation and in the second case the Unruh-DeWitt detector will \ndetect the photons from Unruh Radiation.\n\nIn this subsection we will summarize the work of reference \\cite{Jones15}\nwhich calculates the response of an Unruh-DeWitt detector interacting with \na plane gravitational wave. The expression for the spectrum, $S(E)$, \nof an Unruh-DeWitt detector is given by \n\\begin{equation}\n\\label{UD-1}\nS(E) = n_{general} - n_{inertial} = 2 \\pi \\rho (E) F(E) ~.\n\\end{equation}\nIn equation \\eqref{UD-1} $n_{general}$ and $n_{inertial}$ are the\nphoton density of a general space-time and inertial space-time\nrespectively. The difference between these two is a measure of the\nphotons created due to the general space-time. The terms $\\rho (E)$\nand $F(E)$ are, respectively, the density of states and response \nfunction both as a function of energy \\cite{davies,letaw,akhmedov,wilburn,rad}. \n\nThe detector response function is given by\n\\begin{equation}\n\\label{UD-2}\nF(E) = \\int _{-\\infty} ^{+\\infty} e^{-i \\Delta \\tau \\Delta E}\n(G^+ _{general} (\\Delta \\tau) - G^+ _{inertial} (\\Delta \\tau))\nd (\\Delta \\tau ) ~,\n\\end{equation}\nwhere we recall that $\\hbar =1$ and $c=1$ in the above\nformulas. $\\Delta E = E_{up} - E_{down}$ is the energy difference \nbetween the two states of the Unruh-DeWitt detector. For simplicity \nwe assume $E_{down} = 0$ so that $\\Delta E \\to E_{up} \\to E$ and \nthus the response function is written as just a function of $E$. \nThe terms $G^+ _{general} (\\Delta \\tau)$ and  $G^+ _{inertial} \n(\\Delta \\tau))$ are the  Wightman functions \\cite{davies,Jones15} for the detector path in a general space-time and \nthe detector path in the inertial space-time. The\nWightman function depends on the proper time difference \n$\\Delta \\tau$ \nfor the path through the given space-time. The space-time path for \nthe inertial  detector is $x^\\mu (\\Delta \\tau ) = (\\Delta \\tau , 0, \n0, 0)$. The Wightman function for this inertial detector is\n\\begin{equation}\n\\label{UD-3}\nG_{inertial} ^+ = \\frac{1}{4 \\pi ^2 x^\\mu x_\\mu} =\\frac{1}{4 \\pi ^2 \n\\Delta \\tau ^2} ~.\n\\end{equation}\nFor a gravitational wave traveling in the $+z$ direction and having\n$+$ polarization the space-time path is given by $x^\\mu (\\Delta \n\\tau ) = ( \\gamma \\Delta \\tau , \\Delta x, 0, 0)$, where $\\Delta x$ \nis the spatial displacement of the detector due to the gravitational \nwave and $\\gamma ^{-2} = 1 - \\Delta {\\dot x} ^2$. Without loss of \ngenerality the detector is taken to be aligned along the $x$ direction. \nThe Wightman function for the gravitational wave is \n\\begin{equation}\n\\label{UD-4}\nG_{wave} ^+ = \\frac{1}{4 \\pi ^2 x^\\mu x_\\mu} =\\frac{1}{4 \\pi ^2 \n(\\gamma ^2 \\Delta \\tau ^2 - \\Delta x^2)} ~.\n\\end{equation}\nUsing these two Wightman functions from \\eqref{UD-3} \\eqref{UD-4} in \n\\eqref{UD-2} we find\n\\begin{equation}\n\\label{UD-5}\nF(E) = \\frac{1}{2\\pi^2 }\\int _0 ^{+\\infty} \\cos (E \\Delta \\tau)\n\\left( \\frac{1}{ (\\gamma ^2 \\Delta \\tau ^2 - \\Delta x^2)}- \n\\frac{1}{\\Delta \\tau ^2} \\right)\nd (\\Delta \\tau ) ~.\n\\end{equation}\nTo evaluate \\eqref{UD-5} we need to give $\\Delta x$ as a function of \n$\\Delta \\tau$. This is done using the ${\\dot x} = (1 + \\frac{1}{2}h)$ \n\\cite{EFTaylor} which is the expression for the trajectory along a null \ngeodesic, to first order, for a gravitational wave background \ncharacterized by the amplitude $h (\\Delta \\tau, \\theta, \\psi) = h_0 \n\\sin^2 (\\theta) \\sin (2 \\psi ) \\sin (\\omega \\Delta \\tau )$. The angles\n$\\theta$ and $\\psi$ give the orientation of the axis of the detector \nwith respect to the incoming gravitational wave \\cite{hendry}. The \nseparation $\\Delta x$ between the particle undergoing geodesic \nmotion in the gravitational wave background characterized by \n$h (\\Delta \\tau, \\theta, \\psi)$ and an inertial observer is then\ngiven by $\\Delta x = (1 + \\frac{1}{2}h) \\Delta \\tau - \\Delta \\tau =\n\\frac{1}{2}h (\\Delta \\tau, \\theta, \\psi) \\Delta \\tau$. Using these\nresults in \\eqref{UD-5} and integrating over $\\Delta \\tau$ as well as \nintegrating over the orientation direction $\\theta$ and $\\psi$ gives\nthe detector response function as \\cite{Jones15}\n\\begin{equation}\n\\label{UD-6}\nF(E) = \\frac{3 \\pi}{256} h_0 ^2 (2 \\omega  - E)   \\Theta (2 \\omega - E),\n\\end{equation}\nwhere $\\Theta$ is the Heaviside step function. Thus $F(E) = 0 $ when\n$E > 2 \\omega$ which is a similar type of cut-off to that in \nmuon decay \\cite{halzen,griffiths}. This suggests a picture of\ngravitons ``decaying\" into photons -- $graviton + graviton \\to photon + \nphoton$ or $graviton \\to graviton + photon + photon$.\n\nUsing the response function from \\eqref{UD-6} and a density of\nstates $\\rho (E) = \\frac{E^2}{2 \\pi^2}$ \\cite{letaw} the spectrum can \nbe found from \\eqref{UD-1} as\n\\begin{equation}\n\\label{UD-7}\nS(E) = \\frac{3 }{256 \\pi \\hbar ^3 c^3} E^2 h_0 ^2 \n(2 \\hbar \\omega  - E)  \\Theta (2 \\hbar \\omega - E) ~.\n\\end{equation}\nWe have restored factors of $\\hbar$ and $c$ temporarily. The\nfunctional form of the spectrum from \\eqref{UD-7} is that of a\n$Beta (3,2)$ distribution which is reminiscent of particle decays. This\nagain supports the picture of gravitons decaying into photons. \n\nThe analysis of the present subsection is different from the proceeding\nsubsection in that here we place an Unruh-DeWitt detector\nin the presence of a gravitational plane wave background, whereas in\nthe previous subsection we focused on the response of the vacuum to \na gravitational wave. The Unruh-DeWitt calculation is closer in spirit to \nthe work of Gertsenshtein \\cite{Gertsenshtein60} where the gravitational \nwave interacts with a magnetic field. In both these cases there is some \nphysical object --  the Unruh-DeWitt detector or a magnetic field -- which \ninteracts with the gravitational wave. In the previous subsection the \ngravitational wave interacts with the quantum vacuum. Nevertheless all of \nthese calculations indicate that\na gravitational wave can create electromagnetic radiation, or in\nparticle language that gravitons can transform\/decay into photons. The work in \\cite{Calmet16}\nalso looks into this possibility of gravitons decaying\/transforming in to  photons and thus\nweakening the gravitational wave.\n\n\\section{Possible Observational Consequences\/Signatures}\n\nWhile gravitational waves have only been directly detected very recently, electromagnetic \nradiation has been observed for all of human existence. If gravitational waves produce \ncounterpart electromagnetic radiation as is outlined above, it is natural to ask what the \npossible observable consequence of this would be. In this section we address two possible observational\nconsequence: (i) the attenuation\/decay of the gravitational wave due to production electromagnetic radiation;\n(ii) the direction detection of the electromagnetic radiation produced by the gravitational wave. \n\n\\subsection{Decay\/attenuation of the gravitational wave}\n\nIf electromagnetic waves are produced from a gravitational wave, as suggested above, this\nshould weaken and attenuate the gravitational wave since this electromagnetic radiation must be\ncreated at the expense of the gravitational wave \\cite{Jones15,Jones16}. This is similar \nto how a black hole is conjectured to lose mass as a result of Hawking radiation -- the \nHawking radiation comes at the expense of the mass of the black hole. \n\nOne can connect particle\/field production rate, $\\Gamma$, with a current, $j_\\mu$, as in equation\n\\eqref{jones-12} via the relationship \\cite{stahl,nikolic,frob}\n\\begin{equation}\n    \\label{decay-1}\n    \\frac{\\Gamma}{V} \\Delta T \\approx | j_\\mu | ~,\n\\end{equation}\nwhere $\\Delta T$ is some characteristic time for the system and $V$ is the volume. Using $|j_\\mu| = \\frac{h_+ ^4}{2 V}$\nfrom \\eqref{jones-12} and taking $\\Delta T \\approx \\frac{1}{\\omega}$ (where $\\omega$ is the frequency of the gravitational\nwave) as the characteristic time we arrive at\n\\begin{equation}\n    \\label{decay-2}\n    \\Gamma \\approx \\frac{\\omega h_+ ^4}{2}~.\n\\end{equation}\nIf we denote the number of gravitons in the volume $V$ as, $N_g$ one can write out a rate of change of $N_g$ as\n\\begin{equation}\n    \\label{decay-3}\n    \\frac{d N_g}{dt} = - \\Gamma N_g \\to   c \\frac{d N_g}{dz} = - \\Gamma N_g ~.\n\\end{equation}\nIn the last step we have replaced $dt$ by $dz\/c$ since we want the decay as a function of distance rather than \ntime. Taking the number of gravitons to be proportional to the amplitude squared \\footnote{This is similar to QED where\nthe number of photons is proportional to the square of the vector potential -- $N_\\gamma \\propto A_\\mu A^\\mu$}\n({\\it i.e.} $N_g \\propto h_+ ^2$) and using the expression of $\\Gamma$ from \\eqref{decay-2} we arrive at an\nequation for how the amplitude, $h_+$, varies with distance, $z$,\n\\begin{equation}\n    \\label{decay-4}\nc \\frac{d {h_+ ^2}}{dz} = - \\frac{\\omega h_+ ^4}{2} (h_+ ^2) \\to \\frac{dh_+}{dz} = - \\frac{k h_+ ^5}{4}~.\n\\end{equation}\nOne can solve \\eqref{decay-4} for $h_+ (z)$ and find\n\\begin{equation}\n    \\label{decay-5}\nh_+ (z) = (k z + K_0) ^{-1\/4}~,\n\\end{equation}\nwhere $K_0 = (h_+ ^{(0)} ) ^{-4}$ and $h_+ ^{(0)}$ is the reference amplitude at $z=0$. From equation \n\\eqref{decay-5} one sees that the fall off of $h_+$ as a function of distance, $z$, is very slow. This is expected,\nsince this slow fall off tells us that the transformation of gravitational radiation (gravitons) into electromagnetic\nradiation (photons) is a very weak process. If a gravitational wave background did not produce electromagnetic radiation\nthen $h_+ (z)$ should remain constant (recall that in this plane wave approximation we do not take into account\nthe $\\frac{1}{r}$ fall of a real three dimensional wave).  \n\nTo get an idea of how weak the effect is we can calculate the ``half-distance\", $\\Lambda$,  which we define \nas the distance for the amplitude of the plane wave to fall to half of its initial value, $h_+ ^{(0)}$.  \nTaking $\\omega \\approx 300$ Hz, the approximate frequency of the signal from the first LIGO detection\n\\cite{LIGO}, gives $k = \\frac{\\omega}{c} = 10 ^{-6}$ m. Setting $h_+ (\\Lambda) = \\frac{1}{2} h_+ ^{(0)}$ \ngives the ``half-distance\" as\n\\begin{equation}\n    \\label{decay-6}\n\\Lambda = \\frac{15}{k (h_+ ^{(0)} )^4} = \\frac{1.5 \\times 10^7}{ (h_+ ^{(0)} )^4} \\rm{m}~,\n\\end{equation}\nIf one sets the ``half-distance\", $\\Lambda$,  equal to the size of \nthe observable Universe -- $\\Lambda = 10^{27}$ m -- then \nequation \\eqref{decay-6} gives an amplitude of $h_+ ^{(0)} \n\\approx 10^{-5}$, which is a very large amplitude. Equation\n\\eqref{decay-6} implies that as $h_+ ^{(0)}$ gets larger the \nhalf-distance, $\\Lambda$, gets smaller. Taking  $h_+ ^{(0)} \n\\approx 10^{-3}$ would give $\\Lambda \\approx 10 ^{19}$ m, which \nis 100 times smaller than the size of the Milky Way. We also want to stress again that\nthe above estimates based on equation \\eqref{decay-6} are for planes waves and do not take into account the\n$\\frac{1}{r}$ fall off of a more realistic three dimensional wave, but regardless they\nindicate that the decay\/attenuation of the gravitational wave due to vacuum production of\nelectromagnetic radiation is a small effect, except perhaps close to the source where \none might have amplitudes like $h_+ ^{(0)} \\approx 10^{-3}$ or larger.  \n\n\\subsection{Detection of electromagnetic radiation produced by gravitational waves}\n\nNext we look at the possibility of directly detecting the electromagnetic radiation that is\nproduced from the vacuum by the gravitational wave. Looking at \\eqref{jones-9} one can \nsee that the counterpart electromagnetic radiation production would have twice the\nfrequency of the gravitational wave. From \\eqref{jones-9} one can see there are also components that\nare at four times the frequency of the gravitational wave, but they are down by an extra factor of\n$h_+ ^2$ compared to the component at twice the frequency. \n\nThe first problem that occurs in potentially detecting the counterpart electromagnetic radiation is that\nit will have a very low frequency (VLF) and thus a very large wavelength. For example the discovery paper \n\\cite{LIGO} reported frequencies for the gravitational wave on the order of $100~\\rm{Hz}$. Even doubling this,\nthe electromagnetic radiation would have a frequency and wavelength of  $200~\\rm{Hz}$ and $1.5 \\times 10 ^6 \\rm{m}$\nrespectively. \n\nA second problem with detecting the VLF counterpart electromagnetic radiation is that there are various cutoff\nfrequencies due to the plasma in space. In the illustration and table below we give the plasma cutoffs for a detector\nlocated in one of three locations: on the Earth, in space but near Earth's orbit, and finally in interstellar space\nas shown in Fig. \\eqref{CutoffsFig}. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=130mm]{Interstellar.jpeg}\n\\begin{tabular}{|c|c|}\n\\hline  ~~ Region ~~  & \\  ~~ Observable frequency range ~~       \n\\\\   \n\\hline  On Earth & \\ $> 10 $ MHz          \n\\\\  \n\\hline  Interplanetary space  (near Earth's orbit) & \\ $>$ $20~\\rm{kHz} - 30~\\rm{kHz} $  \n\\\\\n\\hline  Interstellar space (outside the heliosphere) & $ > 2 $ kHz \n\\\\   \n\\hline\n\\end{tabular}\n\\caption{Log scale illustration of the regions of space within our solar system and galaxy \\cite{UCSB} \\label{MediumFig} and the associated plasma frequency cutoff in each region \\cite{Jones17,Lacki10}.}\n\\label{CutoffsFig}\n\\end{figure}\n\nFor a detector on Earth one can detect electromagnetic radiation with a frequency of $10~\\rm{MHz}$ or larger. \nAssuming this electromagnetic wave came from production by a gravitational wave, this would require a\ngravitational wave frequency of $5~\\rm{MHz}$. Since non-exotic gravitational waves sources are expected to have \nfrequencies that are orders of magnitude lower than this, this rules out an Earth based detector for such VLF electromagnetic \nradiation. \n\nFor a detector at the outer edge of the Solar System, near interstellar space, one has a plasma cutoff of \n$2~\\rm{kHz}$ which would require a gravitational wave frequency of $1~\\rm{kHz}$. The fundamental ({\\it i.e.} \nf-modes) of neutron star quakes have frequencies in the range $1 - 3~\\rm{kHz}$ \\cite{Kokkotas97} and thus upon \ndoubling this frequency could produce counterpart VLF electromagnetic radiation above the $2~\\rm{kHz}$ cutoff. \nIn fact \nthe Voyager probes did detect such VLF electromagnetic radiation \\cite{Kurth84} in the range of $2-3 ~\\rm{kHz}$\nshowing that detection of such VLF electromagnetic radiation is possible \\footnote{The source of the Voyager \ndetection of this VLF electromagnetic radiation was a mystery for some time, but the source of this\nVLF radiation is now thought to be due to the interaction of the solar wind with ions in the outer heliosphere \nduring times of intense solar activity \\cite{Kurth03, Webber09}.} Thus\none could detect counterpart VLF electromagnetic radiation from\nthe $f$-modes of neutron star quakes if the neutron star were\nclose enough. However, it requires sending a probes to the edge of\nthe Solar System or beyond. \n\nGiven that sending probes to the edge of the Solar System is costly both in terms of time and money \none could ask if there are gravitational wave sources which would give rise to VLF electromagnetic \nradiation, which could be detected near Earth's orbit. From Fig. \\eqref{CutoffsFig}  one can see that for\ndetection one needs the electromagnetic radiation to have a frequency greater than $20 - 30$ kHz. This implies that\nthe gravitational wave vacuum producing the electromagnetic radiation would need to have a frequency in the range\n$10-15$ kHz. Theoretical models show that of gravitational wave of this kHz frequency range  could be \nproduced from neutron star oscillations \\cite{Kokkotas97,Kokkotas01}. There are different types of \nneutron star oscillation modes. Three\nof the most common are: (i) {\\it p}-modes or ``pressure modes\" \\cite{Kokkotas97} with a frequency \nrange $5 - 9~ \\rm{kHz}$; (ii) {\\it f}-modes or  ``fundamental modes\" \\cite{Kokkotas97} with a\nfrequency range of $1 - 3~\\rm{kHz}$; (iii) {\\it w}-modes or ``space-time modes\" \\cite{Anderson96} with \na frequency range of $8 - 16~\\rm{kHz}$ or greater. From this list of oscillation modes \nthe {\\it w}-modes have the most promising frequency range in terms of detection of the \ncounterpart VLF electromagnetic radiation. \n\nWe now want to give a rough estimate of the strength of the electromagnetic flux produced by \ngravitational waves coming from a  {\\it w}-mode oscillation of a neutron star quake. First from\n\\cite{abadie} the gravitational wave amplitude at Earth for {\\it f}-mode generated gravitational waves \nfrom a neutron star that is $1$ kpc  or $3\\times 10^{19}$ m distant from Earth would be of order\n$h_+ \\sim 10^{-23}$. The associated {\\it w}-modes gravitational wave amplitude is expected to be\ndown from this by at least one order of magnitude $h_+ \\sim 10^{-24}$. Using this {\\it w}-mode \namplitude and the $1\/r$ fall off relation \n\\begin{equation}\n\\label{detect-1}\n    h_+ = 10^{-24} \\frac{1 ~ {\\rm kpc}}{r}\n\\end{equation}\nwe can determine the amplitude at some point close to the source. For this we take \n$r^{(0)}=3 \\times 10^4$ m \\cite{Jones17} -- this is far enough from the neutron star that the plane wave \napproximation we have used throughout should apply, at least roughly. Using \\eqref{detect-1} and \n$r^{(0)}=3 \\times 10^4$ m we find that the {\\it w}-mode amplitude near the source would be of \norder $h_+^{(0)} \\sim 10^{-9}$. Using this amplitude, a frequency of 10 kHz in the {\\it w}-mode\nrange $8-16$ kHz, we can determine the flux of the gravitational wave near the source \n({\\it i.e.} at $r^{(0)}=3 \\times 10^4$ m) using the formula \\cite{Schutz96}\n\\begin{equation}\n    \\label{detect-2}\n    F^{(0)} _{gw} = \\frac{c^3}{16 \\pi G} |{\\dot \\epsilon}|^2 = \n    \\left( 3 \\times 10 ^{35} \\frac{W s^2}{m^2} \\right) h_+^2 f^2 =\n    3 \\times 10^{25} \\frac{W}{m^2} ~,\n\\end{equation}\nwhere $\\epsilon = h_+ e^{iku}$ as defined below equation \\eqref{GWmetric}. We can now calculate\nthe flux of the counterpart VLF electromagnetic radiation using \\eqref{detect-2} in\n\\eqref{emgwRatioB} to give \n\\begin{equation}\n    \\label{detect-3}\n    F^{(0)} _{em} = 2 \\times (10^{-9})^2 \\times (3 \\times 10^{25} \\frac{W}{m^2})\n    = 6.0 \\times 10^7 \\frac{W}{m^2}~.\n\\end{equation}\nFrom \\eqref{detect-2} and \\eqref{detect-3} we find that $F^{(0)} _{gw} \\gg F^{(0)} _{em}$\nwhich is expected -- the electromagnetic radiation produced is much smaller than the\ngravitational wave which produced it. However, $F^{(0)} _{em}$ is nevertheless large enough\nthat even taking into account the $1\/r$ fall off one could potentially detect this\nelectromagnetic radiation at the location of the Earth's orbit. Taking\nthe flux $F^{(0)} _{em}$ from \\eqref{detect-3} one can determine the flux at the \nlocation of the Earth assuming that the neutron star is 1 kpc away.\n\\begin{equation}\n    \\label{detect-4}\n    F_{em}= F^{(0)} _{em} \\left( \\frac{r^{(0)}}{1 kpc}\\right) \\sim \n    6.0 \\times 10 ^{-23} \\frac{W}{m^2} ~,\n\\end{equation}\nwhere $r^{(0)} = 3 \\times 10 ^4$ m from before. A flux of the magnitude in\n\\eqref{detect-4} could be detected \\cite{Jones17} and given the frequency range\nof the {\\it w}-modes the associated VLF electromagnetic radiation would \nhave a\nfrequency that is above the plasma cutoff at the location of Earth's\norbit as given in Fig. \\eqref{CutoffsFig}. Thus the proposal to detect such\nthe hypothesized co-produced VLF electromagnetic radiation, coming {\\it w}-modes of\nneutron star quakes, would be to place a satellite capable of detecting such\nradiation near earth's orbit \\cite{Jones17, Gretarsson18}. The old Explorer 49\nsatellite was capable of detecting such VLF electromagnetic radiation. The Explorer\n49 satellite was a Lunar orbiting satellite which was periodically occulted\nby the Moon in order to block out interference from Solar emissions. The\noccultation allows one to detect weak signals like \\eqref{detect-4} above\nthe interference from the Sun. \n            \n\\section{Conclusion and Future prospects}\n\nFor over 100 years there was nothing to support Faraday's expectation of a relationship between \ngravity and electromagnetism. However, in the past 50 years we have seen the development of considerable \ntheoretical support for this relationship and in particular the relation between gravitational and \nelectromagnetic radiation. The earliest work was by Gertsenshtein \\cite{Gertsenshtein60} \ndemonstrating that electromagnetic radiation can produce gravitational radiation. This \nwas followed in 1975 by Skobelev \\cite{Skobelev75} who calculated the small but non-zero \namplitudes for the $graviton + graviton \\to photon + photon$ processes. Beginning around the same time as reference \\cite{Skobelev75}, there was work that \nexamined the production of electromagnetic fields\/multiple photons from a gravitational background \\cite{unruh-det,dewitt-det,hawking,unruh,Jones15}. Rrcently we have worked on calculations of the production of electromagnetic radiation \nby gravitational waves propagating in vacuum \\cite{Jones16,Jones17,Jones18}. Perhaps \nin the next 50 years we will see empirical evidence  of the relation between \ngravitational and electromagnetic radiation by either direct or indirect observation.\n\nThe most promising possibility for direct observation is the detection of VLF counterpart \nproduction by gravitational waves from neutron star quakes  \\cite{Jones17}. This would \nonly be possible using space based detectors such as the Voyager missions \n\\cite{Kurth84,Kurth03,Webber09} or with a lunar occulted detector similar to the \nExplorer 49 mission  \\cite{Gretarsson18}. However, detection of counterpart production \nlocally would be limited to the highest frequencies of the counterpart production \nfrom neutron star gravitational waves. Detectors in the outer heliosphere would be \nmuch more effective and rather remarkably the Voyager space craft are still making \nobservations in the $2-4~\\rm{kHz}$ range of expected counterpart production \\cite{Gurnett15}.\n\nEven without direct observation it is possible that the counterpart production of electromagnetic radiation would have important applications in astrophysical processes. One intriguing possibility is in the energetics of core collapse supernovae. The prompt production of gravitational waves from the core collapse would produce gravitational waves with quadruple amplitudes on the order of $1~\\rm{m}$ and strain amplitude of something like $10^{-5} - 10^{-4}$ in the star layers just outside the core. These strain amplitudes have the potential of producing counterpart radiation of sufficient energy to contribute to the energetics of the supernovae. Previous work on fully general relativistic magnetohydrodynamics \\cite{Font07} (MHD) have assumed the ``ideal\" MHD condition. This assumption suppresses any potential production of electromagnetic radiation from the strong gravitational wave background. More recent work \\cite{Obergaulinger14,Just18,Obergaulinger18} on the effects of magnetic fields and rotation on the energetics of core collapse supernovae have not been fully general relativistic and again could not include the energy from production of electromagnetic radiation by the outgoing gravitational wave.  Fully general relativistic MHD simulations have been implemented \\cite{Lehner12_86} for collapsing hyper-massive neutron stars but not for the study of core collapse supernovae. It is possible that the energy associated with electromagnetic  production by gravitational waves outside the iron core could contribute importantly to the supernova, but only fully general relativistic MHD simulations would account for this phenomena in the processes of core collapse and explosion.\n\nSince the production of electromagnetic radiation by gravitational waves is so fundamental it is likely that further study of this phenomena could illuminate our understanding of nature. One recent example of the potential importance of production of photons in a gravitational wave background is the investigation of graviton-photon oscillations in alternatives to general relativity \\cite{Cembranos18, ejlli}. This investigation did not directly study counterpart production and potential general relativity violations but does describe the significance of this phenomena in investigating theories of gravity. Counterpart production by gravitational waves \\cite{Ricciardone17} could also be important in studies of cosmology. Following the Planck epoch the Standard Model fields were still massless\nfor some time. It would be interesting to consider the production of the massless Standard Model particles by primordial gravitational waves during the grand unification epoch and prior to the Standard Model particles acquiring mass via the Higgs mechanism.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{To Do list}\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nIn the study of CP violation by CKM unitary triangle analysis,\nhadron matrix elements of four-fermion operators,\nsuch as $B_{\\rm K}$, play a vital role.\nAccurate calculations of this quantity from first principles\nare an important task for the lattice QCD community.\nIn such calculations, having chiral symmetry\nis crucial to avoid an operator mixing problem\nwhich causes uncontrollable systematic errors.\nAlthough lattice chiral fermions\n \\cite{Kaplan:1992bt,Shamir:1993zy,Neuberger:1997fp}\nare a clean formulation,\nthey require enormous computing power to perform dynamical simulations.\nIn comparison, ordinary fermion formulations, like Wilson type fermions\nand starggared fermions are relatively cheap.\nNowadays, however, thanks to the development of computer architecture\nand algorithms, dynamical simulations with\nlattice chiral fermions have become feasible\neven for three flavors \\cite{Noaki:2008gx}.\nIn particular,\nthe RBC\/UKQCD collaboration \\cite{Allton:2008pn} is currently using\ndomain-wall fermions (DWFs) to compute $B_{\\rm K}$.\nIn the course of their computation,\nthere are many sources of systematic errors which one has to control.\nAmong them, the non-perturbative renormalization (NPR)\ncould be serious.\nAt the moment, the collaboration has been using conventional schemes, such as,\nthe RI\/MOM scheme and its variants \\cite{Aoki:2007xm,Sturm:2009kb}.\nHowever, these schemes potentially contain ``large scale\nproblem\" which requires a quite large lattice volume.\nTo avoid such difficulties,\na new scheme was invented, known as the Schr\\\"odinger functional\n(SF) scheme \\cite{Luscher:1992an}.\nThis scheme provides a reliable way of estimating errors in the NPR.\nIf one wants to use this scheme for the renormalization\nof $B_K$ given by the RBC collaboration,\nfirst of all, one has to formulate DWF in the SF setup.\nThis is the purpose of this paper.\n\n\n\nWhile chiral fermions are useful\nfor computing the bare $B_{\\rm K}$\nto avoid the mixing problem, a formulation for such fermions\nin the SF setup was a non-trivial task\nbecause SF boundary conditions break chiral symmetry explicitly.\nWe will address this issue in the next section.\nHowever, Taniguchi \\cite{Taniguchi:2004gf} made the first attempt to formulate overlap fermions\nby using an orbifolding technique.\nSubsequently he provided a formulation for domain-wall fermions\nand then he and his collaborators \\cite{Nakamura:2008xz} calculated\na renormalized $B_{\\rm K}$ in quenched QCD.\nSint \\cite{Sint:2010eh} developed such\ntechniques by combining with a flavor twisting trick.\nHowever, these orbifolding formulations are constrained by\nthe requirement that the number of flavors be even.\nThus, apparently such formulations\nare incompatible with current trends toward dynamical three flavor simulations.\nTo overcome this difficulty, L\\\"uscher \\cite{Luscher:2006df} gave\na completely different approach\nrelying on a universality argument,\ndimensional power counting and symmetry considerations.\nSome perturbative calculations were performed in Ref.~\\cite{Takeda:2007ga}.\nA crucial property of this formulation is that\nthere is no restriction on the number of flavors.\nSince only overlap fermions were considered in Ref.~\\cite{Luscher:2006df},\nour main purpose here is to formulate\nthe other chiral fermions, namely, domain-wall fermions.\n\n\n\nThe rest of the paper is organized as follows.\nSection \\ref{sec:formulation} gives the formulation of domain-wall\nfermions in the SF setup, after a brief\nreview of the universality argument.\nWe present several pieces of numerical evidence\nin Section \\ref{sec:spectrum} and \\ref{sec:oneloop}\nto show that our formulation is working properly.\nWe also discuss the lattice artifacts\nfor the step scaling function in Section \\ref{sec:SSF}.\nIn the last section, we conclude by giving some remarks and outlook.\n\n\n\n\n\n\\section{Formulation}\n\\label{sec:formulation}\nIn the following, we assume that the reader\nis familiar with the SF in QCD \\cite{Luscher:1992an,Sint:1993un}.\nAfter giving a brief reminder of the universality argument,\nwe give a formulation for DWF\nand finally check the chiral symmetry breaking structure numerically.\n\n\\subsection{Universality argument}\n\nIn the massless continuum theory,\nthe Dirac operator $D$ satisfies the anti-commutation relation with\n$\\gamma_5$\n\\begin{equation}\n\\gamma_5D+D\\gamma_5=0.\n\\label{eqn:Dg5}\n\\end{equation}\nThe above is true even in the SF setup, although the boundary conditions,\n\\begin{eqnarray}\nP_+ \\psi(x)=0 &\\mbox{ at }& x_0=0,\n\\label{eqn:SFBC0}\n\\\\\nP_- \\psi(x)=0 &\\mbox{ at }& x_0=T,\n\\label{eqn:SFBCT}\n\\end{eqnarray}\nwith $P_\\pm=(1\\pm\\gamma_0)\/2$,\nbreak chiral symmetry explicitly.\nEq.(\\ref{eqn:Dg5}) means that the operator itself\ndoes not know about boundary conditions.\nIn the continuum theory, information such as boundary conditions\nis embedded in the Hilbert space.\nIn fact, the corresponding propagator,\nwhich is a solution of the inhomogeneous equation,\n\\begin{equation}\nDS(x,y)=\\delta(x-y),\n\\end{equation}\nfails to satisfy the anti-commutation relation.\nInstead, it follows\n\\begin{eqnarray}\n\\lefteqn{\\gamma_5S(x,y)+S(x,y)\\gamma_5=}\\nonumber\\\\\n&&\n\\int_{z_0=0}d^3{\\bf z} S(x,z)\\gamma_5 P_- S(z,y)\n+\n\\int_{z_0=T}d^3{\\bf z} S(x,z)\\gamma_5 P_+ S(z,y).\n\\label{eqn:Sg5}\n\\end{eqnarray}\nThis can be derived by using partial integration\non the SF manifold which has two boundaries\nat time slice $x_0=0$ and $T$.\nThe non-vanishing right-hand side in eq.(\\ref{eqn:Sg5})\nshows an explicit chiral symmetry breaking.\nSince such a breaking term is supported\nonly on the time boundaries,\nthe chiral symmetry is preserved in a bulk.\n\n\nIf someone naively tries to formulate\nchiral fermions on the lattice,\none may define an overlap operator, for example,\nwith the Wilson kernel in the SF setup \\cite{Sint:1993un}.\nHowever such an operator immediately satisfies\nthe Ginsberg-Wilson relation and thus cannot reproduce\neq.(\\ref{eqn:Sg5}) in the continuum limit.\nThis indicates that such naive formulation\ndoes not work and furthermore may\nbelong to another boundary universality class\nwhich is not what we want.\nIn this way, it is a non-trivial task to\nformulate chiral fermions in the SF setup.\n\n\nSome years ago, L\\\"uscher \\cite{Luscher:2006df} proposed a clever way\nto overcome this situation.\nFirst, consider the relation for the propagator\nin eq.(\\ref{eqn:Sg5}).\nThis indicates that the GW relation has to be modified\nby boundary effects.\nThus one has to find a modified overlap operator\nwhich breaks the GW relation near the time boundaries and correctly\nreproduces eq.(\\ref{eqn:Sg5}) in the continuum limit.\nActually, finding such a modified operator is not so hard.\nHowever, a new question naturally arising\nis how the SF boundary conditions emerge.\nFor the Wilson fermion case \\cite{Sint:1993un},\nbecause there is a transfer matrix,\nit is natural for fermion fields\nto follow the SF boundary conditions.\nHowever for chiral fermions,\nthere is no such transfer matrix\nwhich can be defined from nearest neighbor interaction\nin the time direction.\nTherefore it is not an easy task.\n\n\nL\\\"uscher \\cite{Luscher:2006df} gave another point of view\nto see how fields respect the boundary condition.\nIn the quantum field theory,\nthe correlation function can tell you\nwhat kinds of boundary conditions are imposed.\nAs an example, let us see\nhow the boundary conditions emerge for Wilson fermions\nwhose action is given by\n\\begin{eqnarray}\nS_{\\rm w}\n&=&\n\\sum_x\\bar\\psi(x)\nD_{\\rm w}(m)\n\\psi(x),\n\\\\\nD_{\\rm w}(m)\n&=&\n\\frac{1}{2}\n\\left[\n\\sum_\\mu (\\nabla_\\mu+\\nabla^{\\ast}_\\mu)\\gamma_\\mu\n-a\\sum_\\mu \\nabla_\\mu^{\\ast}\\nabla_\\mu\n\\right]+m,\n\\end{eqnarray}\nwhere $\\nabla_\\mu$ and $\\nabla_\\mu^{\\ast}$ are\nforward and backward covariant difference operators respectively,\n\\begin{eqnarray}\n\\nabla_\\mu\\psi(x)\n&=&\n\\frac{1}{a}\n\\left[\nU(x,\\mu)\\psi(x+a\\hat\\mu)-\\psi(x)\n\\right],\n\\\\\n\\nabla_\\mu^{\\ast}\\psi(x)\n&=&\n\\frac{1}{a}\n\\left[\n\\psi(x)-U(x-a\\hat\\mu,\\mu)^{-1}\\psi(x-a\\hat\\mu)\n\\right].\n\\end{eqnarray}\nIn the SF setup, the sum over $x$ in the action is a little bit subtle.\nWe assume that the dynamical fields are \n$\\psi(x)$ with $a\\le x_0 \\le T-a$ and\nthe fields $\\psi(x)$ with $x_0 \\le 0$ and $T\\le x_0$ are set to zero.\nFor this setup,\nthe propagator may be defined by\n\\begin{eqnarray}\n\\langle\\eta(x)\\bar\\psi(y)\\rangle&=&a^{-4}\\delta_{x,y},\n\\label{eqn:etapsi}\n\\\\\n\\eta(x)&=&\\frac{\\delta S_{\\rm w}}{\\delta\\bar\\psi(x)}.\n\\label{eqn:etaS}\n\\end{eqnarray}\nFor $2a \\le x_0 \\le T-2a$, eq.(\\ref{eqn:etaS}) turns out to be\n\\begin{equation}\n\\eta(x)=D_{\\rm w}(m)\\psi(x).\n\\end{equation}\nOn the other hand, at $x_0=a$, we obtain\n\\begin{eqnarray}\n\\eta(x)&=&\n\\frac{1}{a}P_+\\psi(x)-\\nabla_0P_-\\psi(x)\n\\nonumber\\\\\n&+&\n\\frac{1}{2}\n\\left[\n \\sum_k (\\nabla_k+\\nabla^{\\ast}_k)\\gamma_k\n-a\\sum_k \\nabla_k^{\\ast}\\nabla_k\n\\right]\n\\psi(x)\n+m\\psi(x).\n\\label{eqn:eta}\n\\end{eqnarray}\nBy substituting eq.(\\ref{eqn:eta}) into eq.(\\ref{eqn:etapsi}) with $x\\neq y$\nwe obtain\n\\begin{equation}\n\\frac{1}{a}P_+\\langle\\psi(x)\\bar\\psi(y)\\rangle|_{x_0=a}\n-\n\\nabla_0P_-\\langle\\psi(x)\\bar\\psi(y)\\rangle|_{x_0=a}\n+...\n=0.\n\\end{equation}\nIn the continuum limit, the first term is dominant\n\\begin{equation}\n\\frac{1}{a}P_+\\langle\\psi(x)\\bar\\psi(y)\\rangle|_{x_0=0}=0.\n\\end{equation}\nThis shows that in the naive continuum limit,\nthe Dirichlet type boundary condition ($P_+\\psi|_{x_0=0}=0$)\nis stable against the Neumann one ($\\nabla_0P_-\\psi|_{x_0=0}=0$), and in the end\nthe SF boundary conditions in eq.(\\ref{eqn:SFBC0}) emerge.\nIt is plausible that similar things happen also for the chiral fermions case,\nas long as the locality and symmetry\nare kept in a proper way,\nalthough we expect that the coefficient of the lowest dimensional operators\n($\\frac{1}{a}P_+\\psi$)\nmay be different from the above case,\nand more higher dimensional terms may appear in eq.(\\ref{eqn:eta}).\nThe important point here is that\ncontinuum SF boundary conditions emerge dynamically\nin the continuum limit of the correlation function.\nThis boundary condition is natural\nand automatically guaranteed to emerge\nfrom the dimensional order counting argument.\nTherefore, when we construct chiral fermions in the SF,\nwe only have to prepare a modified operator\nby introducing an additional term\nwhich breaks the chiral symmetry near the time boundaries.\nOnce this is fulfilled, then\nsuch an operator automatically\nturns out to be the desired one in the continuum limit without fine tuning.\nA final important note is that the form of the boundary term is irrelevant\nas long as it will go into a preferred boundary universality class.\nTherefore, there is a large amount of freedom when choosing boundary terms\nand one can use this freedom for practical purposes.\n\n\n\nFollowing these guiding principles,\nL\\\"uscher \\cite{Luscher:2006df} proposed the operator:\n\\begin{eqnarray}\n\\bar{a}D_{\\rm N}&=&1-\\frac{1}{2}(U+\\tilde U),\n\\\\\nU&=&A(A^{\\dag}A+caP)^{-1\/2},\\hspace{7mm}\\tilde U = \\gamma_5 U^{\\dag} \\gamma_5,\n\\\\\nA&=&1+s-aD_{\\rm w}(0),\\hspace{7mm}\\bar{a}=a\/(1+s),\n\\label{eqn:overlap}\n\\end{eqnarray}\nwith the parameter in the range $|s|<1$.\n$D_{\\rm w}(0)$ is the massless Wilson operator in the SF.\nThe key point here is the presence of the $P$\nterm in the inverse square root which is given by\n\\begin{equation}\naP(x,y)\n=\n\\delta_{{\\bf x},{\\bf y}}\\delta_{x_0,y_0}(\\delta_{x_0,a}P_-+\\delta_{x_0,T-a}P_+).\n\\end{equation}\nNote that this term is supported near the time boundaries\nand thus called a boundary operator.\nThe presence of this term breaks the GW relation explicitly\nand the breaking is given by\n\\begin{eqnarray}\n\\Delta_{\\rm B}=\\gamma_5D_{\\rm N}+D_{\\rm N}\\gamma_5-\\bar{a}D_{\\rm N}\\gamma_5D_{\\rm N}.\n\\end{eqnarray}\nIt was shown in ref.~\\cite{Luscher:2006df} that this term is local and\nsupported in the vicinity of the boundaries up to the\nexponentially small tails.\n\n\nAlthough this operator breaks chiral symmetry explicitly,\nother symmetries (the discrete rotational symmetries, $C$, $P$ and $T$,\nflavor symmetry and so on)\nhave to be maintained\nsince the boundary conditions in eq.(\\ref{eqn:SFBC0},\\ref{eqn:SFBCT}) are\ninvariant under these symmetries.\nIn addition, this operator has $\\gamma_5$-Hermiticity.\nIn this way, the universality formulation\ncan avoid breaking important symmetries, such as the flavor symmetry.\nThis is a distinctive feature of this formulation\ncompared with the orbifolding technique,\nwhere flavor symmetries cannot be maintained\nor, there is a constraint on the number of flavors.\n\n\n\nBefore leaving this subsection,\nlet us summarized the guiding principles\nof formulating chiral fermions in the SF setup.\nWhat we learned from this construction\nis that, for an original chiral fermion operator,\none has to introduce an additional term\nto break the chiral symmetry and\nthen demand that\nsuch breaking only appears near the time boundaries.\nFurthermore, one must\nmaintain important symmetries as well as $\\gamma_5$-Hermiticity.\nOnce these conditions are fulfilled,\nit is automatically guaranteed that the such a lattice operator\nwill correctly reproduce the continuum results\naccording to the universality argument.\n\n\n\\subsection{Formulation of domain-wall fermions }\nLet us apply the guiding principles given in the previous subsection\nto domain-wall fermions.\nWe propose a massless\\footnote{The mass term can be introduced\nin the usual way, namely\n$a^4m_{\\rm f}\\sum_{{\\bf x}}\\sum_{x_0=a}^{T-a}[\\bar\\psi(x,1)P_R\\psi(x,L_s)+\\bar\\psi(x,L_s)P_L\\psi(x,1)]$.}\ndomain-wall fermion action\n\\begin{equation}\nS=\na^4\\sum_{x,x^\\prime}\\sum^{L_s}_{s,s^\\prime=1}\n\\bar\\psi(x,s)(D_{\\rm DWF})_{xs,x^\\prime s^\\prime}\\psi(x^\\prime,s^\\prime),\n\\end{equation}\nwhere a massless operator with $L_s=6$\nfor example\\footnote{We restrict ourselves to an even number of $L_s$,\nwhich is the case usually implemented.}\nin four dimensional block form is given by\n\\begin{equation}\naD_{\\rm DWF}=\n\\left[\n\\begin{array}{cccccc}\na\\tilde{D}_{\\rm w}&-P_L&0&0&0&cB\\\\\n-P_R&a\\tilde{D}_{\\rm w}&-P_L&0&cB&0\\\\\n0&-P_R&a\\tilde{D}_{\\rm w}&-P_L+cB&0&0\\\\\n0&0&-P_R-cB&a\\tilde{D}_{\\rm w}&-P_L&0\\\\\n0&-cB&0&-P_R&a\\tilde{D}_{\\rm w}&-P_L\\\\\n-cB&0&0&0&-P_R&a\\tilde{D}_{\\rm w}\\\\\n\\end{array}\n\\right],\n\\label{eqn:DWF}\n\\end{equation}\nwith the chiral projections,\n\\begin{equation}\nP_{R\/L}=(1\\pm\\gamma_5)\/2.\n\\end{equation}\nWe also assume that the dynamical fields\nare $\\psi(x,s)$ with $a\\le x_0 \\le T-a$.\nThe block elements in eq.(\\ref{eqn:DWF}) are four dimensional operators\nand $a\\tildeD_{\\rm w}$ is given by\n\\begin{equation}\na\\tilde{D}_{\\rm w}=aD_{\\rm w}(-m_5)+1.\n\\end{equation}\nThe domain-wall height parameter usually takes a value in a range $0<am_5<2$.\n\n\nAn important ingredient here is the presence\nof $B$ in eq.(\\ref{eqn:DWF}) terms in the cross diagonal elements.\nThe reason for this $s$-dependence\nis to break the chiral symmetry\nin a similar way to the usual mass term \\cite{Christ:2004gc}.\nAs mentioned before, such chiral symmetry breaking should be\npresent only near time boundaries, therefore, we chose\nthe $B$ term as\n\\begin{equation}\nB(x,y)=\\delta_{{\\bf x},{\\bf y}}\\delta_{x_0,y_0}\\gamma_5\n(\\delta_{x_0,a}P_-+\\delta_{x_0,T-a}P_+),\n\\label{eqn:Bpm}\n\\end{equation}\nwhich is supported near the boundaries.\nIn this way, the time dependence is fixed.\nThe spinor structure ($\\gamma_5 P_\\pm$) is determined by imposing\nthe discrete symmetries C, P and T and $\\Gamma_5$-Hermiticity.\nThese requirements are not so strong to determine\nthe spinor structure completely and therefore there is some freedom.\nThe structure proposed here\nis only one of many solutions.\nActually, we examined several choices\nof the spinor structure in the boundary term\nand confirmed numerically the universal results\nin the continuum limit for the lowest eigenvalue.\nIn the following,\nwe take this boundary term in eq.(\\ref{eqn:Bpm}).\n\n\nThe boundary coefficient $c$ is supposed to be non-zero\nto correctly reproduce the continuum theory as we will see in\nSection \\ref{sec:spectrum}.\nIt also plays an important role to cancel\nboundary O($a$) cutoff effects and has\na perturbative expansion\n\\begin{equation}\nc=c^{(0)}+c^{(1)}g_0^2 + O(g_0^4).\n\\end{equation}\nIn the same way as in \\cite{Takeda:2007ga},\nwe tune the first coefficient $c^{(0)}$\nas a function of the domain-wall hight $am_5$,\n\\begin{eqnarray}\nc^{(0)}&=&0.5089-0.0067(am_5-1)+0.0488(am_5-1)^2\n\\nonumber\\\\\n&&\n-0.0216(am_5-1)^3+0.0673(am_5-1)^4.\n\\label{eqn:ctreeL}\n\\end{eqnarray}\nThis is valid in the region where\nbulk O($a$) can be neglected\n\\footnote{\nActually, we observe that\nO($a$) improvement program does not work for small $L_s$\nand values of $am_5$ which are far from $1$.\nFor example O($a$) terms in $f_A$ ad $f_P$\n(defined in appendix \\ref{subsec:operators})\nat tree level\ndo not vanish simultaneously with the same value of $c^{(0)}$.\n},that is, for sufficient large $L_s$.\nIn the process of this determination, one needs to define the operators\nof the axial vector current and pseudo scalar density.\nWe give their definition together\nwith that of the conserved axial current in appendix \\ref{subsec:operators}.\n\n\n\n\n\n\n\\subsection{Structure of chiral symmetry breaking at tree level}\n\\begin{figure}[!t]\n\\begin{center}\n  \\begin{tabular}{ccc}\n  \\hspace{-30mm}\n  \\scalebox{1.2}{\\includegraphics{Ls4_L20.eps}}&\n  \\hspace{-40mm}\n  \\scalebox{1.2}{\\includegraphics{Ls12_L20.eps}}&\n  \\hspace{-40mm}\n  \\scalebox{1.2}{\\includegraphics{Ls24_L20.eps}}\\\\\n  \\end{tabular}\n\\caption{\nThe color ranging $[-14,0]$ corresponds to the value of\n$\\ln||\\Delta^{(L_s)}(x_0,y_0)||_{\\rm spin}$\nfor the zero spatial momentum configuration with the parameters\n$T\/a=20$,\n$m_{\\rm f}=0$,\n$am_5=1$ and $c=1$.\n\\label{fig:GWbreaking}}\n\\end{center}\n\\end{figure}\n\nIn this subsection, \nlet us check an important properties of the operator defined\nin the previous section.\nA reader may worry that even though the additional\nboundary term is localized to time boundary,\nafter integrating over the fifth dimensional\ndegree of freedom such breaking effects may leak into the 4-dim bulk and\nruin the bulk chiral symmetry.\nTo settle this question, we numerically investigate the\nstructure of the chiral symmetry breaking\nby looking at the breaking of the GW relation\n\\begin{equation}\n\\Delta^{(L_s)}\n=\n\\gamma_5D_{\\rm eff}^{(L_s)}+D_{\\rm eff}^{(L_s)}\\gamma_5\n-2aD_{\\rm eff}^{(L_s)}\\gamma_5D_{\\rm eff}^{(L_s)},\n\\label{eqn:DeltaLs}\n\\end{equation}\nwith the effective four dimensional operator\n\\cite{Kikukawa:1999sy,Kikukawa:1999dk}\n\\begin{equation}\n\\detD_{\\rm eff}^{(L_s)}= \\det[D_{\\rm DWF}\/D_{\\rm PV}].\n\\end{equation}\nThe Pauli-Villars (PV) operator is defined as\nthe massive DWF operator with $am_{\\rm f}=-1$.\nTo obtain the effective operator,\nfirst of all, we have to define physical quark fields\n\\begin{eqnarray}\nq(x)&=&P_L\\psi(x,1)+P_R\\psi(x,L_s),\n\\label{eqn:q}\n\\\\\n\\bar{q}(x)&=&\\bar{\\psi}(x,1)P_R+\\bar{\\psi}(x,L_s)P_L.\n\\label{eqn:barq}\n\\end{eqnarray}\nIn terms of the propagator of domain-wall fermions defined from\n\\begin{equation}\nD_{\\rm DWF} S_{\\rm DWF}(x,y;s,t)=a^{-4}\\delta_{x,y}\\delta_{s,t},\n\\end{equation}\nthat of the physical field is expressed\n\\begin{eqnarray}\n\\left[q(x)\\bar{q}(y)\\right]_{\\rm F} \n&=&\nS_{\\rm q}(x,y) \n\\label{eqn:Sq}\n\\nonumber\\\\\n&=&\n P_LS_{\\rm DWF}(x,y;1,L_s)P_L\n+P_LS_{\\rm DWF}(x,y;1,  1)P_R\n\\nonumber\\\\\n&+&\n P_RS_{\\rm DWF}(x,y;L_s,L_s)P_L\n+P_RS_{\\rm DWF}(x,y;L_s,  1)P_R.\n\\end{eqnarray}\nIn terms of $S_{\\rm q}$\nthe effective operator is given by\n\\begin{equation}\naD_{\\rm eff}^{(L_s)}=(1+a^3S_{\\rm q})^{-1}.\n\\end{equation}\n\n\n\nIn the SF setup, $\\Delta^{(L_s)}$ in eq.(\\ref{eqn:DeltaLs})\ncontains not only the bulk chiral symmetry breaking\nbut also the boundary breaking.\nThe former is supposed to be removed by taking $L_s$ to infinity.\nIn such limit, boundary breaking effects remain\nand they are expected to be localized near time boundaries.\nTo see this situation, we numerically compute $\\Delta^{(L_s)}$\nfor a free operator.\nIn the free case, \nwe can perform the Fourier transformation for spatial directions.\nWe study the momentum configuration ${\\bf p}=(0,0,0)$ in the following.\nThe remaining dimensions are only the time direction and spinor space,\ntherefore for a given $L_s$ and the fixed spatial momentum configuration,\n$\\Delta^{(L_s)}$ is a matrix with dimension $4(T\/a-1)$.\nFigure \\ref{fig:GWbreaking} shows the magnitude of\n$\\ln(||\\Delta^{(L_s)}(x_0,y_0)||_{\\rm spin})$,\nwhere the norm is taken for the spinor space only.\nBy increasing $L_s$, the bulk symmetry breaking is reduced.\nFinally at $L_s=24$, only boundary breaking effects remain\nand they are localized exponentially near the time boundaries.\nThis is the expected behavior for overlap fermions \\cite{Luscher:2006df}.\nWe conclude that the presence of the boundary term\ncauses chiral symmetry breaking, which decays exponentially away from\nthe time boundaries\nfor the effective four dimensional operator.\n\n\n\n\n\n\\section{Spectrum of free operator}\n\\label{sec:spectrum}\nIn this section, we investigate\nthe free spectrum of the DWF operator\nto confirm universality at the tree level.\nWe set $T=L$ in this section.\n\n\\subsection{Spectrum of $D_{\\rm DWF}^{\\dag}D_{\\rm DWF}$}\n\\begin{figure}[!t]\n\\begin{center}\n  \\scalebox{1.1}{\\includegraphics{Lowest.eps}}\n\\caption{\nThe lowest eigenvalue of $L^2D_{\\rm DWF}^{\\dag}D_{\\rm DWF}$ with $a m_5=1$.\nSome combinations of parameters $c$ and $L_s$ are shown.\nThe continuum value is $\\pi^2\/4=2.467..$.\nFor $c=1$ case, since the $L_s$-dependence is so weak on on this scale,\nwe show only $L_s=4$ results as representative.\n\\label{fig:DWFEV}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\begin{center}\n  \\begin{tabular}{cc}\n  \\scalebox{1.1}{\\includegraphics{Lowest_c10.eps}}&\n  \\scalebox{1.1}{\\includegraphics{Lowest_c00.eps}}\\\\\n  \\end{tabular}\n\\caption{$||\\psi(x_0,s)||_{\\rm spin}$\nwith zero spatial momentum and the parameters\n$T\/a=L_s=40$, $\\theta=0$ and $am_5=1$.\nThe norm for the eigenvector is taken in the spinor space.\nThe left (right) panel is for $c=1$ ($c=0$).\n\\label{fig:WF}}\n\\end{center}\n\\end{figure}\n\nTo achieve better chiral symmetry,\nthe (physical) eigenmodes of the domain-wall operator should\nbe localized near the boundaries of the fifth direction\nand propagate in the space-time directions.\nThis should also be true in the SF setup,\nsince the chiral symmetry is supposed to be maintained in the bulk.\nTo observe such phenomena, we numerically compute the lowest eigenmode\nof the operator $D_{\\rm DWF}^{\\dag}D_{\\rm DWF}$ with the trivial gauge configuration $U(x,\\mu)=1$.\nIn the free case, \nwe can perform the Fourier transformation for spatial directions,\nand project out the momentum configuration ${\\bf p}=(0,0,0)$.\nThus, remaining indexes of the vector space\nare now the spinor, the time $x_0$ and the extra dimension $s$,\n\\begin{equation}\nD_{\\rm DWF}^{\\dag}D_{\\rm DWF}\\psi(x_0,s)=\\lambda\\psi(x_0,s),\n\\end{equation}\nwhere the spinor indexes are suppressed.\nWe set input parameters $am_5=1$ and $\\theta=0$.\n$\\theta$ is the parameter which\ncontrols the spatial boundary condition for fermion fields.\n(For more details, we refer to \\cite{Sint:1995ch}.)\n\nWe numerically compute the lowest eigenvalue and the corresponding\neigenfunction $||\\psi(x_0,s)||_{\\rm spin}$.\nWe examine not only for $c=1$ but also for $c=0$\nto investigate the importance of the presence of the boundary operator.\nThe scaling behavior of the eigenvalue is shown in Figure \\ref{fig:DWFEV}.\nFor $c=1$, $L_s$ dependence is too small to see on this scale.\nThe lowest eigenvalues converge to their continuum values properly,\ntherefore universality is confirmed.\nFurthermore, the associated eigenfunction shows\nnice localization behavior, namely\nbeing localizing for the fifth direction and propagating for the time direction,\nas shown in the left panel of Figure \\ref{fig:WF}.\nThis shows that this mode is a physical one.\n\nFor $c=0$ in Figure \\ref{fig:DWFEV},\nalthough all $L_s=4,16,32$ results tend to converge to the continuum limit,\nlarge $L_s$ results have a bending phenomenon\nin small $a\/L$ region and show no power decay in terms of $a\/L$. \nThis indicates that if one takes $L_s$ to infinite before taking $a\/L=0$ limit,\nthe eigenvalue will likely converge to zero.\nIf this is so, the theory with $L_s=\\infty$ does not\nbelong to a correct universality class.\nFurthermore, the eigenfunction in the right panel in Figure \\ref{fig:WF}\nis localized on edges in the time-$s$ plane.\nThis is a typical unphysical mode.\nOn the other hand, interestingly for small $L_s$,\nthe scaling behavior is rather mild.\nIn the small $L_s$ case, the chiral symmetry breaking\nof domain-wall fermions are rather similar\nto that of the ordinary Wilson fermions.\nAs in the Wilson fermions case,\nthe bulk chiral symmetry breaking for DWFs due to finite $L_s$\nplays some role in producing the correct continuum limit.\nThis is the reason why DWFs with smaller $L_s$ and no\nboundary term $B$ can produce the continuum results.\n\n\n\n\nThe results shown in this subsection show\nthat the boundary term with $c\\neq 0$ plays an important role\nfor the theory to be in the correct universality class.\n\n\n\n\n\\subsection{Spectrum of $D_{\\rm q}^{\\dag}D_{\\rm q}$}\n\n\\begin{figure}[!t]\n\\begin{center}\n  \\begin{tabular}{cc}\n  \\scalebox{1.1}{\\includegraphics{MIN10_theta00_BG1.eps}}&\n  \\scalebox{1.1}{\\includegraphics{MIN10_theta05_BG1.eps}}\\\\\n  \\end{tabular}\n\\caption{The $a\/L$ dependence of the lowest ten eigenvalues\nof $L^2D_{\\rm q}^{\\dag}D_{\\rm q}$ in the presence of the background gauge field.\nThe left (right) panel is for $\\theta=0$ ($\\theta=\\pi\/5$).\nThe parameters are set to $am_5=1$ and $c=1$.\nThe red points at $a\/L=0$ are continuum values \\cite{Sint:1995ch}.\n\\label{fig:lowest10}}\n\\end{center}\n\\end{figure}\n\n\nNot all eigenmodes of $D_{\\rm DWF}$\nare physical ones and elimination of unphysical mode is not clear.\nTo extract physical modes only, let us study the eigenmodes\nof $D_{\\rm q}$, where unphysical modes are excluded.\nThe operator $D_{\\rm q}$ is defined from $S_{\\rm q}$ in eq.(\\ref{eqn:Sq})\n\\begin{equation}\nD_{\\rm q} S_{\\rm q} (x,y)=a^{-4}\\delta_{x,y}.\n\\end{equation}\n\nWe numerically compute the lowest ten eigenvalues of\n$D_{\\rm q}^{\\dag}D_{\\rm q}$\nwith the parameter set $am_5=1$, $\\theta=0,\\pi\/5$ and $c=1$\nin the presence of the the background gauge field (choice A in\nRef.~\\cite{Luscher:1993gh}).\nThe values obtained for $L\/a=T\/a=6,12,24$ are summarized in Table \\ref{tab:MIN10}.\nAll tables are given in appendix \\ref{sec:tables}\nThe scaling behavior of the eigenvalues\nare shown in Figure \\ref{fig:lowest10}.\nAlthough we show two cases of $L_s$, namely $L_s=4$ and $L_s=32$,\nit is hard to see the difference on this scale.\nWe observe that they converge to the continuum values given\nin Ref~\\cite{Sint:1995ch}.\nThis behavior persists for a variety of values of $c$,\n$0.5 \\le c\\le 1.5$.\nThis confirms universality at the tree level.\n\n\n\n\n\\section{One-loop analysis of SF coupling}\n\\label{sec:oneloop}\nTo check further universality at the quantum level\nand renormalizability, we perform the one-loop order calculation\nof the SF coupling.\n\n\\subsection{Definition and results}\n\nWe compute the fermion contribution to\nthe SF coupling \\cite{Sint:1995ch}\n$p_{1,1}(L\/a,L_s)$\n(we set $L=T$ as usual)\nat one-loop order\nfor massless domain-wall fermions.\nThe one-loop coefficient is given as\n\\begin{equation}\np_{1,1}(L\/a,L_s)\n=\n\\left.\n\\frac{1}{k}\n\\frac{\\partial}{\\partial\\eta}\n\\ln \\det(D_{\\rm DWF}\/D_{\\rm PV})\n\\right|_{\\eta=\\nu=0},\n\\label{eqn:p11}\n\\end{equation}\nwith a normalization (See \\cite{Sint:1995ch} for details.)\n\\begin{equation}\nk=12(L\/a)^2[\\sin(\\gamma)+\\sin(2\\gamma)],\n\\hspace{5mm}\n\\gamma=\\frac{1}{3}\\pi(a\/L)^2.\n\\end{equation}\nThe parameters $\\eta$ and $\\nu$ parameterize\nthe background gauge field \\cite{Luscher:1993gh}.\nIn the actual calculation,\nwe expand the $\\eta$ derivative\nand use the fact that the determinant\nis factorized for individual spatial momentum ${\\bf p}$ and color sector $b$,\n\\begin{eqnarray}\np_{1,1}(L\/a,L_s)\n&=&\n\\frac{1}{k}\n{\\rm Tr}\n\\left[\nD_{\\rm DWF}^{-1}\\frac{\\partialD_{\\rm DWF}}{\\partial\\eta}\n-D_{\\rm PV}^{-1}\\frac{\\partialD_{\\rm PV}}{\\partial\\eta}\n\\right]\n\\nonumber\\\\\n&=&\n\\frac{1}{k}\n\\sum_{{\\bf p}}\n\\sum_{b=1}^{3}\n{\\rm tr}\n\\left[\n(D_{\\rm DWF}^b)^{-1}({\\bf p})\\frac{\\partialD_{\\rm DWF}^b({\\bf p})}{\\partial\\eta}\n\\right.\n\\nonumber\\\\&&\n\\left.\n-(D_{\\rm PV}^b)^{-1}({\\bf p})\\frac{\\partialD_{\\rm PV}^b({\\bf p})}{\\partial\\eta}\n\\right].\n\\label{eqn:p11expand}\n\\end{eqnarray}\nThe trace\n${\\rm tr}$\nconcerns the spinor, the time indices and fifth coordinate only.\nIt is maybe worthwhile to note that for our definition of DWF,\n\\begin{equation}\n\\frac{\\partialD_{\\rm DWF}}{\\partial\\eta}\n=\n\\frac{\\partialD_{\\rm PV}}{\\partial\\eta}\n\\end{equation}\nholds since the mass term does not involve the gauge field.\n\n\nWe compute $p_{1,1}$ on the lattices of size $L\/a=4,6,...,48$ and\n$L_s=6,8,10,12,16$ with parameters $0.7 \\le am_5 \\le 1.3$\nand $\\theta=\\pi\/5$.\nSubsets of the results are summarized in Table \\ref{tab:p11}\nfor $am_5=1$, $L_s=6$ and $L\/a=4,6,...,48$.\nSeparate contributions from DWF and PV are\nalso shown there.\n\n\n\n\\subsection{Coefficients of Symanzik's expansion}\n\\begin{figure}[!t]\n\\begin{center}\n  \\scalebox{1.3}{\\includegraphics{A0Ls.eps}}\n\\caption{\n$L_s$-dependence of $A_0$ for $0.7 \\le am_5 \\le 1.3$.\nThe horizontal lines show \nthe values of $A_0$ in the infinity $L_s$ limit,\nwhich are obtained by combining the results\nof previous literature\n\\cite{Sint:1995ch,Aoki:2003uf}.\n\\label{fig:A0Ls}}\n\\end{center}\n\\end{figure}\n\nFrom the Symanzik's analysis of the cutoff dependence\nof Feynman diagrams on the lattice,\none expects that the one-loop coefficient has\nan asymptotic expansion in terms of $a\/L$\n\\begin{equation}\np_{1,1}(L\/a,L_s)\n=\n\\sum^{\\infty}_{n=0}\n(a\/L)^n[A_n(L_s) + B_n(L_s) \\ln (L\/a)].\n\\label{eqn:p11symanzik}\n\\end{equation}\nNote that the coefficients $A_n$ and $B_n$ ($n=0,1,2,...$)\ndepend on $L_s$.\nWe can reliably extract the first few coefficients\nby making use of the method described in Ref.~\\cite{Bode:1999sm}.\n\n\nFor the usual renormalization of the coupling constant,\n$B_0$ at $L_s=\\infty$ should be $2b_{0,1}$ where $b_{0,1}$ is\nthe fermion part of the one-loop coefficient\nof the $\\beta$-function for $N_{\\rm f}$ flavors QCD,\n\\begin{eqnarray}\nb_0\n&=&\nb_{0,0}+N_{\\rm f}b_{0,1},\n\\\\\nb_{0,0}\n&=&\n\\frac{11}{(4\\pi)^2},\n\\\\\nb_{0,1}\n&=&\n-\\frac{2}{3}\n\\frac{1}{(4\\pi)^2}.\n\\end{eqnarray}\nWe confirmed that\n$B_0(L_s)$ for large $L_s$ (say $L_s=16$) converges to $2b_{0,1}=-0.008443...$\nup to three significant digits\nfor the values of $am_5$ which we investigated.\nWhen the tree-level O($a$) improvement is realized,\nwe expect that $B_1 = 0$ holds.\nWe check this to $10^{-3}$ for the same parameter region as before.\nThis shows that the formula\nfor the boundary coefficient in eq.(\\ref{eqn:ctreeL})\nworks well to achieve the tree-level O($a$) improvement\nto the precision considered here.\nIn the following analysis, we set exact values $B_0=2b_{0,1}=-1\/(12 \\pi^2)$\nand $B_1=0$.\n\n\n\n\n$A_0$ gives information about a ratio of $\\Lambda$-parameters.\nThe obtained values of $A_0(L_s)$\nas a function of $L_s$ are shown\nin Figure \\ref{fig:A0Ls}.\nBy combining the previous results\nfrom Ref.~\\cite{Sint:1995ch,Aoki:2003uf},\nthe values of $A_0$ at infinity $L_s$ can be obtained, and\nare shown in Figure \\ref{fig:A0Ls} as the horizontal lines.\nWe observe that our results at finite $L_s$\nproperly converge to the known results at infinity $L_s$.\n\n\nTo achieve one-loop O($a$) improvement,\nwe need to determine the coefficient\nof the fermion part of the boundary\ncounter-term, $c_{\\rm t}^{(1,1)}$ \\cite{Sint:1995ch}\nat one-loop order.\nIf one imposes an improvement condition\n\\cite{Sint:1995ch}, one finds that\n\\begin{equation}\nc_{\\rm t}^{(1,1)}=A_1\/2,\n\\label{eqn:ct11A1}\n\\end{equation}\ntherefore we need the value of $A_1$.\nThe obtained values of $A_1$ are given in Table \\ref{tab:A1}.\nFor future reference, we provide an interpolation formula for\n$c_{\\rm t}^{(1,1)}$ as a polynomial of $am_5$\nfor larger $L_s$, where value of $A_1$ is saturated,\n\\begin{equation}\nc_{\\rm t}^{(1,1)}\n=\n0.00434+0.01102(am_5-1)-0.00858(am_5-1)^2,\n\\label{eqn:ct11}\n\\end{equation}\nfor $0.7 \\le am_5 \\le 1.3$.\n\n\n\\section{Lattice artifacts of the step scaling function to one-loop order}\n\\label{sec:SSF}\n\\begin{figure}[!t]\n \\begin{center}\n  \\begin{tabular}{cc}\n  \\resizebox{70mm}{!}{\\includegraphics{deviation0.eps}}\n  &\n  \\resizebox{70mm}{!}{\\includegraphics{deviation1.eps}}\n  \\\\\n  \\end{tabular}\n  \\caption{We show the relative deviation with the various actions for tree level\n$O(a)$ improvement, $\\delta_{1,1}^{(0)}$ (Left),\nand one-loop $O(a)$ improvement, $\\delta_{1,1}^{(1)}$ (Right),\nas a function of $a\/L$ and $(a\/L)^2$ respectively.\nUpper part is for $\\theta=0$, and lower is for $\\theta=\\pi\/5$.\nFor comparison, those of the Wilson type fermion\nwith $c_{\\rm t}^{(1,1)}=0$ and the clover fermion with\n$c_{\\rm t}^{(1,1)}=0.019141$ \\cite{Sint:1995ch}\nare included in the plot of $\\delta_{1,1}^{(0)}$ and\n$\\delta_{1,1}^{(1)}$ respectively.\n  \\label{fig:deviation}}\n \\end{center}\n\\end{figure}\n\n\nIn this section, we investigate lattice artifacts of\nthe step scaling function (SSF) \\cite{Luscher:1991wu}\n$\\sigma(2,u)$, which describes the evolution\nof the running coupling $\\bar{g}^2(L)=u$\nunder changes of scale $L$ by a factor $2$,\n\\begin{equation}\n\\sigma(2,u)=\\bar{g}^2(2L),\n\\hspace{5mm}\nu=\\bar{g}^2(L).\n\\end{equation}\nThe lattice version of the step scaling function is\ndenoted by $\\Sigma(2,u,a\/L)$ which\ncontains lattice artifacts.\nSuch lattice artifacts\nare described by the relative deviation\n\\begin{equation}\n\\delta (u,a\/L) \n\\equiv\n\\frac{\\Sigma(2,u,a\/L) \n - \\sigma(2,u)}{\\sigma(2,u)}.\n\\end{equation}\nBy expanding the relative deviation\nin terms of the coupling constant $u$, one obtains\n\\begin{equation}\n\\delta (u,a\/L) \n= \n\\delta_1(a\/L) u  \n+ O(u^2),\n\\end{equation}\nwhere the one-loop deviation, $\\delta_1(s,a\/L)$,\nmay be decomposed into pure gauge and fermion part \\cite{Sint:1995ch},\n\\begin{equation}\n\\delta_1(a\/L)\n=\n\\delta_{1,0}(a\/L)\n+N_{\\rm f}\n\\delta_{1,1}(a\/L).\n\\end{equation}\nWe are currently only interested in the fermion part.\nWe consider domain-wall fermions,\nthus the fermion part of the one-loop deviation\n$\\delta_{1,1}(a\/L,L_s)$\ncontains $L_s$ dependence.\nIn terms of the\none-loop coefficient of the SF coupling $p_{1,1}$,\nthe one-loop deviation is given by\n\\begin{equation}\n\\delta_{1,1}(a\/L,L_s)\n= \n p_{1,1}(2L\/a,L_s)\n-p_{1,1}(L\/a,L_s)\n-2 b_{0,1} \\ln(2).\n \\label{eqn:deviation1}\n\\end{equation}\nDepending on the value of the\nboundary counter term $c_{\\rm t}^{(1,1)}$,\nwe denote with $\\delta^{(0)}_{1,1}$\nthe tree level O($a$) improved version\nwith $c_{\\rm t}^{(1,1)}=0$,\nand $\\delta^{(1)}_{1,1}$\nthe  one-loop O($a$) improved one\nfor $c_{\\rm t}^{(1,1)}$ in eq.(\\ref{eqn:ct11A1}).\n\n\nWe show numerical results for the one-loop deviation in\nTables \\ref{tab:deviation0} and \\ref{tab:deviation1}\nand the plots in Figure \\ref{fig:deviation},\nwhere we include those of the Wilson type fermions \\cite{Sint:1995ch}\nand overlap fermion \\cite{Takeda:2007ga}\nfor comparison.\n$L_s$-dependence of DWFs is small.\nIn the case of the clover action, $c_{\\rm t}^{(1,1)}$ is set\nto be the proper value to achieve one-loop $O(a)$ improvement,\nand for Wilson fermions\nit is set to $c_{\\rm t}^{(1,1)}=0$.\nWe observe that the lattice artifacts\nfor domain-wall fermions are small for\ntree level boundary O($a$) improvement case\ncompared with other fermions, while\nthey are large for one-loop boundary O($a$) improvement case.\n\n\n\n\n\n\\section{Conclusion and outlook}\n\\label{sec:conclusion}\n\nIn this paper, we provide a new formulation of domain-wall fermions\nin the SF setup by following the universality argument of L\\\"uscher.\nIn contrast to the previous formulation by Taniguchi,\nours can deal with the boundary O($a$) improvement properly,\nand there is no constraint on the number of flavors.\nTo check that our formulation works properly,\nwe investigate the spectrum and eigenmodes of the free operator,\nand perform a one-loop analysis of the SF coupling constant.\nThen we confirm universality at tree and the one-loop level\nand observe that all results investigated\nshow the desired behaviors.\n\n\nBefore starting simulations,\nthe boundary improvement coefficient $c$\nshould be determined to one-loop order.\nThis involves calculations of the SF\ncorrelators, $f_{\\rm A}$, $f_{\\rm P}$ etc. given in\nappendix \\ref{sec:operator}.\nThis could be done in a similar way to the case of the Wilson fermion.\n\n\nAs mentioned before, one of the most important\nproperties of the universality formulation\nis that there are no restriction of the number of flavors.\nBy taking advantage of this property,\nwe may compute the renormalization factor\nof $B_{\\rm K}$ for $N_{\\rm f}=3$ QCD.\n\n\n\n\\section*{Acknowledgments}\nWe would like to thank Sinya Aoki, Yasumichi Aoki,\nNorman Christ, Michael Endres,\nTaku Izubuchi, Changhoan Kim, Robert Mawhinney\nand the members of the RBC Collaboration\nfor helpful discussions.\nWe are grateful to Stefan Sint for his hospitality\nduring our stay at Trinity College in Dublin,\nwhere this work was initiated.\nThis work is supported by the U.S. Department of Energy\nunder Grant No. DE-FG02-92ER40699.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\thesectionumber. #1}}\n\n\\newcounter{subsectionumber}\n\\newcommand{\\ssubsection}[1]{\\refstepcounter{subsectionumber}\\subsection{\\thesectionumber.\\thesubsectionumber. #1}}\n\n\n\\renewcommand{\\figurename}{\\normalfont\\bfseries Extended Data Figure}\n\\renewcommand{\\thefigure}{\\arabic{figure}}\n\n\n\n\\newcommand{\\Schrader}[1]{\\ensuremath{_{\\text{Sch[#1]}}}}\n\\newcommand{*}{*}\n\\newcommand{\\inner}[2]{\\left(#1,\\,#2\\right)}\n\\newcommand{\\kket}[1]{| #1 \\!\\!\\succ}\n\n\\newcommand{\\refgordon}{28\\xspace}\n\\newcommand{\\refzakharov}{24\\xspace}\n\\newcommand{\\refKivsharAndMalomed}{23\\xspace}\n\n\n\\newcommand{1\\xspace}{1\\xspace}\n\\newcommand{5\\xspace}{5\\xspace}\n\n\\renewcommand\\theequation{S\\arabic{equation}}\n\n\\ssection{Energetically allowed fractional loss of particles for a single soliton}\n\\label{allowed_loss}\n\nConsider the stationary, single-soliton solution (normalized to 1) of Eq.~(1) with $V=0$ in the main text, $\\Psi_{\\text{1}}(z)=\\sqrt{c\/2}\\sech (c z)$, where $c=|g_{\\textrm{1D}}| N_{\\textrm{a}} m \/(2\\hbar^2)$.\n\nThe energy-per-particle of the soliton is given by\n\\begin{align*}\nE\/N_{\\textrm{a}}\n&=\\int_{-\\infty}^{\\infty}\\,dz\\,\\left(\\frac{\\hbar^{2}}{2m}|\\partial_{z}\\Psi_{\\text{1}}(z)|^2+\\frac{g_{\\textrm{1D}}N_{\\textrm{a}}}{2}|\\Psi_{\\text{1}}(z)|^{4}\\right)\n\\\\\n&=-d N_{\\textrm{a}}^{2},\\text{ with } d=\\frac{mg_{\\textrm{1D}}^{2}}{24\\hbar^2},\n\\end{align*}\nwhere $g_{\\textrm{1D}}<0$. A fracturing of a single soliton into two smaller ones, $N_{\\textrm{a}} \\to N_{1} + N_{2}$, will raise the energy-per-particle by $2d N_{1}N_{2}$. If there are multiple fragments, their energies are even less negative, so the increase in the energy-per-particle will be even greater; same if some of the particles are radiated away, as their energy will actually be positive. So the assumption of a fracture into exactly two smaller solitons will give the maximal possible particle loss. If the fracturing is to occur due to the soliton (with center-of-mass speed $v$) colliding with a barrier, this energy can come only from the kinetic energy per particle of the center of mass, $E_{\\text{K},1}=\\frac{1}{2}mv^{2}$. Let us write $N_{1}=N_{\\textrm{a}}-\\Delta N$ and $N_{2}=\\Delta N$, where, without loss of generality, we impose the condition that $0\\leqslant \\Delta N<N_{\\textrm{a}}\/2$. We therefore have $0\\leqslant 2d(N_{\\textrm{a}}-\\Delta N)\\Delta N\\leqslant E_{\\text{K},1}$; dividing through by $2d N_{\\textrm{a}}^{2}$, we get\n\\begin{equation}\n0\\leqslant \\frac{\\Delta N}{N_{\\textrm{a}}}-\\left(\\frac{\\Delta N}{N_{\\textrm{a}}}\\right)^{2}\\leqslant\\frac{E_{\\text{K},1}}{2d N_{\\textrm{a}}^{2}}\\,.\n\\label{ineqality}\n\\end{equation}\nNow we invoke the assumption that $E_{\\text{K},1} \\ll |E\/N_{\\textrm{a}}|$. Thus the right-hand side of the above relation is much less than one. The equation $x-x^{2}=\\epsilon$ has the solutions $x_{1,2}=\\frac{1}{2}\\pm\\sqrt{\\frac{1}{2}-\\epsilon}$, i.e. one close to zero, the other close to one. Because we imposed that $\\Delta N<N_{\\textrm{a}}\/2$, it follows that $\\Delta N\/N_{\\textrm{a}}$ must be much less than one; so we may neglect the quadratic term in Eq.~(\\ref{ineqality}), obtaining\n\\[0\\leqslant\\frac{\\Delta N}{N_{\\textrm{a}}}\\leqslant\\frac{E_{\\text{K},1}}{2d N_{\\textrm{a}}^{2}}\\,.\\]\n\nSee also Ref.~\\onlinecite{Kivshar1987_968} (the paragraph before Sec.~3).\n\n\\ssection{The natural units for the 1D nonlinear Schr{\\\"o}dinger equation}\n\\label{unit_conversions}\n\nConsider the 1D nonlinear Schr{\\\"o}dinger equation\n\\begin{equation}\ni\\hbar\\partial_{t}\\Psi=-\\frac{\\hbar^{2}}{2m}\\partial^{2}_{z}\\Psi+g_{\\textrm{1D}}N_{\\textrm{a}}|\\Psi|^{2}\\psi+\\tilde{V}(z)\\Psi\\,,\n\\label{SI_NLSE}\n\\end{equation}\nwhere $\\int_{-\\infty}^{\\infty}\\,|\\Psi(z,\\,t)|^{2}\\,dz=1$. By choosing to work in ``natural units,'' it may be written as\n\\[\ni\\partial_{t}\\psi=-\\frac{1}{2}\\partial^{2}_{z}\\psi\\pm\\,|\\psi|^{2}\\psi+V(z)\\psi\\,,\n\\]\nwhere the sign of the $|\\psi|^{2}\\psi$ term is the sign of $g_{\\textrm{1D}}$. In the present work, this is negative (i.e. attractive, or ``focusing''), so that it can support bright solitons. $\\int_{-\\infty}^{\\infty}\\,|\\psi(z,\\,t)|^{2}\\,dz=\\mathcal{N}$ is the new\nnormalization, which may be chosen at will. In the resulting system of units, the unit of length is $u_{L}=\\mathcal{N} \\hbar^{2} \/(m g_{\\textrm{1D}} N_{\\textrm{a}})$, of time, $u_{T}=\\mathcal{N}^{2} \\hbar^{3}\/(m g_{\\textrm{1D}}^{2} N_{\\textrm{a}}^{2})$, and the field is transformed as\n$\\Psi(z,\\,t)=\\psi(z\/u_{L},\\,t\/u_{T})\/\\sqrt{\\mathcal{N}u_{L}}$.\n\n\n\n\\ssection{The $N$-soliton solution of the nonlinear Schr{\\\"o}dinger equation}\n\\label{N-soliton_solution}\n\nThe multisoliton solutions of the equation\n\\begin{equation*}\ni\\partial_{t}\\psi(z,\\,t)=-\\frac{1}{2}\\partial^{2}_{z}\\psi(z,\\,t)-|\\psi(z,\\,t)|^{2}\\psi(z,\\,t)\n\\end{equation*}\nmay be constructed as follows \\cite{gordon1983_596}: given the $4N$ real\nparameters $A_{j}>0$, $v_{j}$, $z_{j,0}$, and $\\phi_{j,0}$, $j=1,\\,\\ldots,\\,N$,\nwe have that\n\\[\n\\psi(z,\\,t)=\\sum_{j=1}^{N}u_{j}(z,\\,t)\\,,\n\\]\nwhere the $u_{j}$'s are the solutions of the system of $N$ equations\n\\[\n\\sum_{k=1}^{N}\\,\\frac{1\/\\gamma_{j}+\\gamma_{k}^{*}}{\\lambda_{j}+\\lambda_{k}^{*}}u_{k}=1\\,,\\quad j=1,\\,\\ldots,\\,N,\n\\]\nwhere\n\\[\n\\lambda_{j}=A_{j}\/2+i v_{j}\n\\] and\n\\[\n\\gamma_{j}=\\exp\\left[\\lambda_{j}(z-z_{j,0})+i\\lambda_{j}^{2}t\/2+i\\phi_{j,0}\\right]\\,.\n\\]\nHere the parameters $z_{j,0}$, and $\\phi_{j,0}$ are \\emph{almost but not quite}\nthe position and phase, at $t=0$, of the $j$th soliton when it is spatially\nseparated from the others (see below), but the first two are precisely its norm\nand velocity, as we now explain. The norm of $\\psi$ is $\\sum_{j=1}^{N} A_{j}$;\nas $t\\to \\pm \\infty$, if the velocities $v_{j}$ are all distinct, $\\psi$\nbecomes a sum of $N$ 1-soliton solution, of norms $A_{j}$, traveling at\nvelocities $v_{j}$. More precisely, in the limit as the $j$th soliton becomes\nmore and more spatially separated from the others, its form converges to\n\\[\n\\frac{A_{j}}{2}\\sech\n\\left[\\frac{A_{j}}{2}(z-z_{j})+q_{j}\\right]\\exp[i(\\phi_{j}+\\Psi_{j})]\\,,\n\\]\nwhere\n\\begin{align*}\nz_{j}&=z_{j,0}+v_{j}t\\,,\n\\\\\n\\phi_{j}&=v_{j}(z-z_{j})+\\frac{1}{2}\\left(A_{j}^{2}\/4+v_{j}^{2}\\right)t+\\phi_{j,0}\\,,\n\\end{align*}\nand the real numbers $q_{j}$ and $\\Psi_{j}$ capture the interaction with the\nother solitons. They are given as\n\\[\nq_{j}+i\\Psi_{j}=\\sum_{\\substack{k=1 \\\\\nk\\neq j}}^{N}\\sign (z_{k}-z_{j})\\ln\n\\frac{A_{j}+A_{k}+2i(v_{j}-v_{k})}{A_{j}-A_{k}+2i(v_{j}-v_{k})}\\,,\n\\]\nwhere $\\sign{z}$ is -1, 0, or +1 if  $z<0$, $z=0$, or $z>0$, respectively.\n\nNow we see that the parameters $z_{j,0}$, and $\\phi_{j,0}$ \\emph{differ},\nthrough $q_{j}$ and $\\Psi_{j}$, respectively, from the actual position and\nphase at $t=0$ of the $j$th soliton when it is spatially separated from the\nothers. But the actual position and phase at $t=0$ is easily characterized:\nsince the displacements $q_{j}$ depend only on the ordering of the spatial\npositions of the solitons (and not on the magnitudes of the relative\ndistances), it follows that if one wants isolated solitons to sit at\n$\\bar{z}_{0,j}$ at $t=0$, one may proceed as follows: first set each $z_{j,0}$\nto $\\bar{z}_{0,j}$, and compute the $q_{j}$'s. Then set\n$z_{j,0}=\\bar{z}_{0,j}+2q_{j}\/A_{j}$, and proceed to compute the $N$-soliton\nsolution; a similar correction may be applied for the initial phases.\n\nThe case of degenerate $\\lambda_{j}$, when two or more constituent solitons\nhave both the same norm and the same velocity, may be treated by taking the\nappropriate limit of the system of $N$ equations above. In this case one\nobtains solutions that are qualitatively different from those discussed thus\nfar. For example, in the two-soliton case, as $t\\to \\pm \\infty$, one finds \\cite{zakharov1972_118} that\nthe distance between the solitons increases proportionally to $\\ln (A^{2} t)$\n(in natural units). Since here the solitons separate on\ntheir own on the time scale of $1\/A^{2}$, the collision experiment should last\nshorter than that. On the other hand, the breather needs to start sufficiently\nfar from the barrier so that it begins in an approximately integrable regime,\nand it needs to be sufficiently slow so that the kinetic energy per particle is\nmuch less than the chemical potential. It turns out that these constraints are\nimpossible to satisfy simultaneously, and thus the degenerate case is not of\ninterest for us.\n\n\n\n\\label{conjecture_argument}\n\n\n\n\n\\ssection{Derivation of the exact expression for $d\\lambda\/dt$, Eq.~(5\\xspace) in the main text}\n\\label{derivation_of_DlambdaDt}\n\nWe are dealing with the 1D nonlinear Schr{\\\"o}dinger equation in Eq.~(1\\xspace) in the main text,\n\\begin{multline}\ni\\hbar\\frac{\\partial}{\\partial t}\\Psi(z,\\,t)=-\\frac{\\hbar^{2}}{2m}\\frac{\\partial^{2}}{\\partial z^{2}}\\Psi(z,\\,t)\n\\\\\n+g_{\\textrm{1D}}N_{\\textrm{a}}|\\Psi(z,\\,t)|^{2}\\Psi(z,\\,t)+\\tilde{V}(z,\\,t)\\Psi(z,\\,t)\\,,\n\\label{td_NLSE}\n\\end{multline}\nwhere $g_{\\textrm{1D}}<0$, \\emph{in the presence of} the external integrability-breaking potential $\\tilde{V}(z,\\,t)$ (note that here we will allow this external potential to explicitly depend on time). In order to facilitate comparison with Ref.~\\onlinecite{Kivshar1989_763}, which treats the same kind of problem, we will work in the units in which $\\hbar=1$, $m=1\/2$, and $|g_{\\textrm{1D}}|N_{\\textrm{a}}=2$. Thus the NLSE becomes\n\\begin{multline}\ni\\frac{\\partial}{\\partial t} \\psi(z,\\,t)\n=\n\\left[- \\frac{\\partial^2}{\\partial z^2}  - 2 |\\psi(z,\\,t)|^2 \\right] \\psi(z,\\,t)\n\\\\\n + \\epsilon v_{ext}(z,t) \\psi(z,\\,t)\n\\label{td_NLSE_perturbed}\n\\,,\n\\end{multline}\nwhere the external potential in (\\ref{td_NLSE}) is factorized as\n\\begin{eqnarray*}\n&&\n\\tilde{V}(z,\\,t) \\equiv V_{0} v_{ext}(z,t)\n\\\\\n&&\n\\max_{z}[v_{ext}(z,\\,0)]=1\n\\,,\n\\end{eqnarray*}\nand the small parameter $\\epsilon$ is\n\\begin{eqnarray*}\n\\epsilon \\equiv \\frac{2 \\hbar^2 V_{0}}{(g_{\\textrm{1D}}N_{\\textrm{a}})^{2} m}\n\\,.\n\\end{eqnarray*}\n\nThe first Lax operator reads\n\\begin{eqnarray}\n\\hat{\\cal L}\n=\n\\left(\n\\begin{array}{cc}\n\\hat{L}            &   \\hat{M}\n\\\\\n-\\hat{M}^{\\dagger} & -\\hat{L}\n\\end{array}\n\\right)\n\\label{Lax}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&&\n\\hat{L} = -i \\frac{\\partial}{\\partial z}\n\\label{Lax_L}\n\\\\\n&&\n\\hat{M} = \\psi^{*}(z,\\,t)\n\\label{Lax_M}\n\\,\\,.\n\\end{eqnarray}\nFor each instance of time $t$, one can set up an eigenstate-eigenvalue problem, which is the central object of interest of this derivation:\n\\begin{eqnarray*}\n\\hat{\\cal L} | w \\!\\!\\succ = \\lambda | w \\!\\!\\succ\n\\,\\,,\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\n| w \\!\\!\\succ = \\left(\\begin{array}{c} u(z,\\,t) \\\\ v(z,\\,t) \\end{array}\\right)\n\\,.\n\\end{eqnarray*}\n\nTo convert expressions in this text to the ones appearing in Ref.~\\onlinecite{Kivshar1989_763}, one\nshould use the following replacement table:\n\\begin{center}\n\\begin{minipage}{.3\\textwidth}\n\\begin{eqnarray*}\n&&\nz \\to x\n\\\\\n&&\n\\psi(z,\\,t) \\to u(x,\\,t)\n\\\\\n&&\nu(z,\\,t) \\to \\psi^{(1)}(x,\\,t)\n\\\\\n&&\nv(z,\\,t) \\to \\psi^{(2)}(x,\\,t)\n\\\\\n&&\nv_{ext}(z,\\,t) \\psi(x,\\,t) \\to i (P[u])(x,\\,t)\\,.\n\\end{eqnarray*}\n$\\left.\\right.$ \\vspace{-.5\\baselineskip}\n\\end{minipage}\n\\end{center}\\vspace{\\baselineskip}\n\n\\ssubsection{Relevant functional analysis}\nFrom the fact that $\\hat{L}$ is Hermitian, $\\hat{L}^{\\dagger} = \\hat{L}$, it follows that the Lax operator (\\ref{Lax}) possesses the following property:\n\\begin{eqnarray*}\n\\hat{\\cal L}^{\\dagger} = \\hat{\\sigma}_{3} \\hat{\\cal L} \\hat{\\sigma}_{3}\n\\,.\n\\end{eqnarray*}\nThis property induces a particular Hermitian form $\\inner{\\cdot}{\\cdot}$, a pseudo-inner product:\n\\begin{align}\n\\inner{\\kket{w_{1}}}{\\kket{w_{2}}}&=\\prec \\!\\!\n  w_{1} | w_{2}\n\\!\\!\\succ\n\\equiv\n\\langle u_{1} | u_{2} \\rangle - \\langle v_{1} | v_{2} \\rangle\n\\notag\n\\\\\n&=\n\\int \\! dz \\,\n\\{\n  u_{1}^{*}(z) u_{2}(z) - v_{1}^{*}(z) v_{2}(z)\n\\}\n\\label{scalar_product}\n\\,\\,.\n\\end{align}\nThis Hermitian form lacks the property of being positive definite (i.e. lacks the property that, for all $| w \\rangle$, $\\langle w | w \\rangle \\ge 0$ and $\\langle w | w \\rangle = 0$ if and only if $| w \\rangle = | 0 \\rangle$). The rest of the inner product axioms, on the other hand, remain intact:\n\\begin{eqnarray*}\n&&\\prec \\!\\! w_{2} | w_{1} \\!\\!\\succ = \\prec \\!\\! w_{2} | w_{1} \\!\\!\\succ^{*}\n\\\\\n\\text{and}&&\n\\\\\n&&\\prec \\!\\! w_{1} | a w_{2} + b w_{3}\\!\\!\\succ = a\\! \\prec \\!\\! w_{1} | w_{2} \\!\\!\\succ + b\\! \\prec \\!\\! w_{1} | w_{3}\\!\\!\\succ\\,.\n\\end{eqnarray*}\nThe Lax operator $\\hat{\\cal L}$ from Eq.~(\\ref{Lax}) is symmetric with respect to this form:\n\\begin{eqnarray}\n\\inner{\\kket{w_{1}}}{ \\hat{\\cal L}\\kket{w_{2}}}=\\inner{ \\hat{\\cal L}\\kket{w_{1}}}{\\kket{w_{2}}}\n\\label{pseudo-Hermiticity}\n\\,\\,.\n\\end{eqnarray}\nThe property above justifies a standard notation\n$\n\\inner{\\kket{w_{1}}}{ \\hat{\\cal L}\\kket{w_{2}}}\n\\equiv\n\\prec \\!\\!\n  w_{1} | \\hat{\\cal L} | w_{2}\n\\!\\!\\succ\n$\nthat we are going to employ below.\n\nThe pseudo-Hermiticity property (\\ref{pseudo-Hermiticity}) implies the following properties of the eigenstates of $\\hat{\\cal L}$: Let\n$\\hat{\\cal L} | w_{1} \\!\\!\\succ = \\lambda_{1} | w_{1} \\!\\!\\succ$,\n$\\hat{\\cal L} | w_{2} \\!\\!\\succ = \\lambda_{2} | w_{2} \\!\\!\\succ$, and\n$\\hat{\\cal L} | w \\!\\!\\succ = \\lambda | w \\!\\!\\succ$.\nThen\n\\begin{itemize}\n\\item[1.]\neigenvectors whose eigenvalues are not complex conjugates of each other are mutually orthogonal,\n\\begin{eqnarray*}\n&&\n\\lambda_{1}^{*} \\not= \\lambda_{2}^{} \\,\\, \\Rightarrow \\,\\,\\, \\prec \\!\\! w_{1} | w_{2} \\!\\!\\succ = 0\\,;\n\\end{eqnarray*}\n\\item[2.]\nnon-zero norm eigenstates of $\\hat{\\cal L}$ correspond to real eigenvalues\n(a corollary of the above):\n\\begin{eqnarray*}\n&&\n\\lambda^{*} \\not= \\lambda^{} \\,\\, \\Rightarrow \\,\\,\\, \\prec \\!\\! w | w \\!\\!\\succ = 0\\,.\n\\end{eqnarray*}\n\\end{itemize}\nNote that the eigenspectrum of $\\hat{\\cal L}$ is not necessarily complete. In cases when the Lax operator (\\ref{Lax})\nrepresents a linear stability analysis equation of a nonlinear PDE, the missing states are associated with the continuous\nsymmetries of the PDE that is broken by the solution in question \\cite{castin2009_317}. (For a ``flat'' condensate, $\\psi(z) = \\mbox{const}$,\nwe found one missing state; there could be more. There seem to be none for a single soliton.)\n\nThe operator (\\ref{Lax}) also possesses properties specific to a particular form of the matrix elements, Eqs.~(\\ref{Lax_L}) and (\\ref{Lax_M}):\n\\begin{eqnarray*}\n&&\n\\hat{L}^{*} = - \\hat{L}\n\\,\\,.\n\\end{eqnarray*}\nThis property implies that:\n\\begin{itemize}\n\\item[1.]\nreal eigenvalues $\\lambda$ are doubly degenerate. The corresponding eigenstates,\n\\begin{eqnarray*}\n&&\n| w \\!\\!\\succ \\stackrel{\\cdot}{=} \\left(\\begin{array}{c}u(z) \\\\ v(z)\\end{array}\\right)\n\\\\\n&&\n| \\tilde{w} \\!\\!\\succ \\stackrel{\\cdot}{=} \\left(\\begin{array}{c} \\tilde{u}(z) \\\\ \\tilde{v}(z)\\end{array}\\right)\n\\,\\,,\n\\end{eqnarray*}\nare related\nby\n\\begin{eqnarray*}\n&&\n\\tilde{u}(z) = - v^{*}(z)\n\\\\\n&&\n\\tilde{v}(z) = + u^{*}(z)\n\\,\\,.\n\\end{eqnarray*}\nHere,\n\\begin{eqnarray*}\n&&\n\\hat{\\cal L}| w \\!\\!\\succ \\stackrel{\\cdot}{=} \\lambda | w \\!\\!\\succ\n\\\\\n&&\n\\hat{\\cal L}| \\tilde{w} \\!\\!\\succ \\stackrel{\\cdot}{=} \\lambda | \\tilde{w} \\!\\!\\succ\n\\,.\n\\end{eqnarray*}\n\n\\item[2.]\nComplex eigenvalues $\\lambda$ come in complex conjugate pairs, $\\lambda_{+}$, $\\lambda_{-}$ such that $\\lambda_{-} = (\\lambda_{+})^{*}$. The corresponding eigenstates,\n\\begin{eqnarray*}\n&&\n| w_{+} \\!\\!\\succ \\stackrel{\\cdot}{=} \\left(\\begin{array}{c} u_{+}(z) \\\\ v_{+}(z)\\end{array}\\right)\n\\\\\n&&\n| w_{-} \\!\\!\\succ \\stackrel{\\cdot}{=} \\left(\\begin{array}{c} u_{-}(z) \\\\ v_{-}(z)\\end{array}\\right)\n\\,\\,,\n\\end{eqnarray*}\nare related\nby\n\\begin{eqnarray}\n&&\nu_{-}(z) = - (v_{+})^{*}(z)\n\\nonumber\n\\\\\n&&\nv_{-}(z) = + (u_{+})^{*}(z)\n\\label{plus-minus_relation}\n\\,\\,.\n\\end{eqnarray}\nHere,\n\\begin{eqnarray*}\n&&\n\\hat{\\cal L}| w_{+} \\!\\!\\succ \\stackrel{\\cdot}{=} \\lambda_{+} | w_{+} \\!\\!\\succ\n\\\\\n&&\n\\hat{\\cal L}| w_{-} \\!\\!\\succ \\stackrel{\\cdot}{=} \\lambda_{-} | w_{-} \\!\\!\\succ\n\\\\\n&&\n\\lambda_{-} = (\\lambda_{+})^{*}\n\\,.\n\\end{eqnarray*}\n\\end{itemize}\n\nWithin the context of the Inverse Scattering Transform, the wavefuction $\\psi(x,\\,t)$ in the parent NLSE, Eq.~(\\ref{td_NLSE}), is assumed to be localized in space, while the\neigenstates of the Lax operator $\\hat{\\cal L}$ of Eq.~(\\ref{Lax}) are required to be finite at $x=\\pm\\infty$.\nIn this case, the real eigenvalues of $\\hat{\\cal L}$ form a continuum spectrum, while the complex eigenvalues are discrete.\nFinally, in the parent NLSE, the complex eigenvalues correspond to\nthe solitonic part of the scattering data, while the real eigenvalues correspond to the thermal noise.\n\nFrom now on, we will assume that the eigenstates of $\\hat{\\cal L}$ with substantially complex eigenvalues (i.e. the ``discrete spectrum'',  or ``bound states'') are normalized\nas\n\\begin{eqnarray}\n\\prec \\!\\! w_{-} | w_{+} \\!\\!\\succ = 1\\,.\n\\label{normalization}\n\\end{eqnarray}\n\nFor the case of a single soliton,\n\\begin{eqnarray*}\n&&\n\\psi(z) = -i \\, \\mbox{sech}(x)\n\\,,\n\\end{eqnarray*}\nthe corresponding eigenvalues and eigenstates are\n\\begin{eqnarray*}\n&&\n\\lambda_{+} = - \\frac{i}{2}\n\\\\\n&&\n| w_{+} \\!\\!\\succ \\stackrel{\\cdot}{=} \\frac{+i}{\n2\n} \\exp(x\/2) \\left(\\begin{array}{c} -1 + \\tanh(x) \\\\ \\mbox{sech}(x) \\end{array}\\right)\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n&&\n\\lambda_{-} = + \\frac{i}{2}\n\\\\\n&&\n| w_{-} \\!\\!\\succ \\stackrel{\\cdot}{=}  \\frac{-i}{\n2\n} \\exp(x\/2) \\left(\\begin{array}{c} -\\mbox{sech}(x)  \\\\ -1 + \\tanh(x) \\end{array}\\right)\n\\,.\n\\end{eqnarray*}\n\n\\ssubsection{The Hellmann-Feynman theorem}\nLet $\\hat{\\cal L}$ depend on a parameter $\\xi$: $\\hat{\\cal L} = \\hat{\\cal L}(\\xi)$. Its discrete eigenvalues $\\lambda_{\\pm}$ and the corresponding eigenstates, $| w_{\\pm} \\!\\!\\succ$ then also depend on $\\xi$:\n$\\lambda_{\\pm} = \\lambda_{\\pm}(\\xi)$, and\n$| w_{\\pm} \\!\\!\\succ = | w_{\\pm}(\\xi) \\!\\!\\succ$.\n\nLet us express the eigenvalue $\\lambda_{+}$ as\n\\begin{eqnarray*}\n\\lambda_{+}(\\xi) = \\prec \\!\\! w_{-}(\\xi) | \\hat{\\cal L}(\\xi) | w_{+}(\\xi) \\!\\!\\succ\\,.\n\\end{eqnarray*}\nThen, the derivative of $\\lambda_{+}$ with respect to $\\xi$ becomes\n\\begin{multline*}\n\\frac{d}{d\\xi}  \\lambda_{+}\n=\n\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\hat{\\cal L}\\kket{w_{+}}}\n\\\\\n+\n\\inner{\\kket{w_{-}}}{\\left(\\frac{d}{d\\xi}\\hat{\\cal L}\\right)\\kket{w_{+}}}+\n\\inner{\\kket{w_{-}}}{\\hat{\\cal L}\\frac{d}{d\\xi}\\kket{w_{+}}}\n\\end{multline*}\nAs in the proof of the usual Hellmann-Feynman theorem, the sum of the first and the last term will turn out to be proportional to the derivative of the norm; and since the norm is one, the derivative is zero. Indeed, the first term gives\n\\begin{align*}\n\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\hat{\\cal L}\\kket{w_{+}}}\n&\n=\n\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\lambda_{+}\\kket{w_{+}}}\n\\\\\n&\n=\n\\lambda_{+}\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\kket{w_{+}}}\\,.\n\\end{align*}\nSimilarly, the last term gives\n\\begin{align*}\n\\inner{\\kket{w_{-}}}{\\hat{\\cal L}\\frac{d}{d\\xi}\\kket{w_{+}}}\n&\n=\n\\inner{\\hat{\\cal L}\\kket{w_{-}}}{\\frac{d}{d\\xi}\\kket{w_{+}}}\n\\\\\n&\n=\n\\inner{\\lambda_{-}\\kket{w_{-}}}{\\frac{d}{d\\xi}\\kket{w_{+}}}\n\\\\\n&\n=\n(\\lambda_{-})^{*}\\inner{\\kket{w_{-}}}{\\frac{d}{d\\xi}\\kket{w_{+}}}\\,.\n\\end{align*}\nBut $(\\lambda_{-})^{*}= \\lambda_{+}$; thus, the sum of the two terms gives\n\\begin{multline*}\n\\lambda_{+}\\,\\left[\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\kket{w_{+}}}+\\inner{\\kket{w_{-}}}{\\frac{d}{d\\xi}\\kket{w_{+}}}\\right]\n\\\\\n=\\lambda_{+}\\,\\frac{d}{d\\xi}\\inner{\\kket{w_{-}}}{\\kket{w_{+}}}=\\lambda_{+}\\,\\frac{d}{d\\xi}\\,1=0\\,.\n\\end{multline*}\nThus, we get the following generalization of the Hellmann-Feynman theorem:\n\\begin{equation}\n\\frac{d}{d\\xi}  \\lambda_{+}  =  \\prec \\!\\! w_{-} | \\left(\\frac{d}{d\\xi} \\hat{\\cal L}\\right) | w_{+} \\!\\!\\succ\\,.\n\\label{HF}\n\\end{equation}\n\n\n\\ssubsection{The exact expression for $d\\lambda\/dt$ from the Hellmann-Feynman theorem}\nLet us set\n\\begin{eqnarray*}\n&&\n\\xi = t\n\\\\\n&&\n\\hat{\\cal L}(t)\n=\n\\left(\n\\begin{array}{cc}\n-i \\frac{\\partial}{\\partial z}           &   \\psi^{*}(z,\\,t)\n\\\\\n-\\psi^{}(z,\\,t) & +i \\frac{\\partial}{\\partial z}\n\\end{array}\n\\right)\n\\\\\n&&\n\\hat{\\cal L}(t)  | w(t)\\!\\!\\succ =  \\lambda(t) | w(t)\\!\\!\\succ\n\\\\\n&&\n\\frac{d}{dt}\\hat{\\cal L}(t) =\n\\\\\n&&\n\\left(\n\\begin{array}{cc}\n0           &   +(F[\\psi] + \\epsilon P[\\psi])^{*}(z,\\,t)\n\\\\\n-(F[\\psi] + \\epsilon P[\\psi])^{}(z,\\,t) & 0\n\\end{array}\n\\right)\n\\\\\n&&\n| w(t) \\!\\!\\succ =\n\\left(\n\\begin{array}{cc}\nu(z,\\,t)\n\\\\\nv(z,\\,t)\n\\end{array}\n\\right)\n\\\\\n&&\n2 \\int_{-\\infty}^{+\\infty} u(z,\\,t) v(z,\\,t) = 1\n\\,,\n\\end{eqnarray*}\nwhere $F[\\psi](z,\\,t) = -i \\left[- \\frac{\\partial^2}{\\partial z^2}  - 2 |\\psi(z,\\,t)|^2 \\right] \\psi(z,\\,t)$, and $P[\\psi](z,\\,t) = -i v_{ext}(z,\\,t)\\psi(z,\\,t)$.\nAccording to the Hellmann-Feynman theorem, Eq.~(\\ref{HF}), the time derivative of the Lax eigenvalue is\n\\begin{eqnarray*}\n\\frac{\\partial}{\\partial t}&& \\lambda\n=\n\\\\\n&&\n- \\int_{-\\infty}^{+\\infty} \\! dz \\,\n\\left\\{\n\tu_{+}^2(z,\\,t) \\epsilon P[\\psi](z,\\,t)\n\\right.\n\\\\\n&&\\hspace{10em}\\left.\n- v_{+}^2(z,\\,t) \\epsilon^{*} P^{*}[\\psi](z,\\,t)\n\\right\\}\n\\\\\n&&\n=\n(+i)\n\\int_{-\\infty}^{+\\infty} \\! dz \\,\n\\left\\{\n\t\\epsilon u^2(z,\\,t)\\psi(z,\\,t)\n\\right.\n\\\\\n&&\\hspace{10em}\\left.\n+ \\epsilon^{*} v^2(z,\\,t) \\psi^{*}(z,\\,t)\n\\right\\}v_{ext}(z,\\,t)\n\\,.\n\\end{eqnarray*}\nNotice that the contribution to $d\\lambda\/dt$ from $F[\\psi]$ disappears. Indeed this contribution\ndescribes the time derivative of the Lax eigenvalue in the time evolution according to the {\\it unperturbed} NLS; this derivative\nindeed vanishes as a consequence of integrability of the NLSE.\n\n\nA translation to the Kivshar-Malomed\nnotation system of Ref.~\\onlinecite{Kivshar1989_763} gives\n\\begin{multline*}\n\\frac{\\partial}{\\partial t} \\lambda_{n}\n=\n- \\frac{1}{2}\n\\frac{1}\n{\\int_{-\\infty}^{+\\infty} \\! dz \\, \\psi^{(1)}(x,\\,t) \\psi^{(2)}(x,\\,t) }\n\\\\\n\\times \\int_{-\\infty}^{+\\infty} \\! dz \\,\n\\left\\{\n\t(\\psi^{(1)})^2(x,\\,t,\\,\\lambda_{n}) \\epsilon P[\\psi](x,\\,t)\n\\right.\n \\\\\n\\left.\n - (\\psi^{(2)})^2(x,\\,t) \\epsilon^{*} P^{*}[\\psi](x,\\,t)\n\\right\\}\n\\,.\n\\end{multline*}\n\\mbox{}\\\\\n\\mbox{}\\\\\n\\mbox{}\\\\\n\\mbox{}\\\\\n\\mbox{}\\\\\n\\mbox{}\\\\\n\n\n\n\n\n\\newpage\n\n\\begin{widetext}\n\n\\begin{center}\n\\textbf{\\textsc{\\relsize{1}Extended Data}}\n\\end{center}\n\n\\vfill\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{Fig_4_01a_font}\n\\end{center}\n\\caption { \\label{Fig_4_01a}\n\\textbf{Integrable vs. nonintegrable case.} The initial state for the nonintegrable\ncase was prepared by time-propagating the breather at rest while the\nnonlinearity was slowly ramped from $|\\psi|^{2}$ to $|\\psi|^{2p}$ with $p=3\/2$.\nThe result was Galilei-boosted and scattered off of a Gaussian barrier whose\nwidth (for numerical reasons) was twice the reference value. All other\nparameters were at their reference values, including the number of particles. Also plotted is the integrable case, all of whose parameters are at their\nreference values. In particular, the number of particles is four times smaller than that for Fig.~3 in the main text, resulting in a degraded, but\nstill noticeable plateau at around 25\\% transmission. In the nonintegrable\ncase, no plateau can be discerned.\n          }\n\\end{figure}\n\n\\vfill\n\\mbox{}\n\\newpage\n\\mbox{}\n\\vfill\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{Fig_4_01b_font}\n\\end{center}\n\\caption { \\label{Fig_4_01b}\n\\textbf{Dependence on the phase of the\nbreathing cycle.} The $y$-axis: the value of $E_{\\text{kin}}\/V_{0}$ for which\ntransmission jumps from 0 to 1\/4; the $x$-axis: the time offset in the\nbreathing cycle for the initial state, relative to that in\nFig.~2 in the main text; all other parameters are as in that Figure.\n          }\n\\end{figure}\n\n\n\\vfill\n\\mbox{}\n\\newpage\n\\mbox{}\n\\vfill\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{Fig_4_01c_font}\n\\end{center}\n\\caption { \\label{Fig_4_01c}\n\\textbf{Transmission plot for a three-soliton breather solution.} The  constituent solitons have norms 1\/6,\\, 1\/3,\\, and 1\/2 (1:2:3 norm ratio, which does \\emph{not} belong to the sequence of odd number ratios). The dash-dotted horizontal lines are at transmissions values of 1\/6, 1\/2, and 1, corresponding, respectively, to only the smallest, norm-1\/6 soliton being transmitted, to the norm-1\/6 and norm-1\/3 solitons being transmitted, and to all three constituent solitons being transmitted.\n          }\n\\end{figure}\n\n\\vfill\n\\mbox{}\n\\newpage\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{solitons_and_breather_workv}\n\\end{center}\n\\caption { \\label{Fig_4_01d}\n\\textbf{Uncoupled solitons vs. breather.} How the ``staircase'' plot of Fig.~3 in the main text would look if the constituent solitons of the breather were completely uncoupled, as compared to now it is in reality. \\textbf{a,} The transmission plots for the scattering of single solitons, of norms $1\/4$ and $3\/4$, off a barrier. All parameters are as in Fig.~3 in the main text, with barrier width $w=w_{0}$. \\textbf{b,} dashed line: the weighted sum of the single-soliton transmission curves from panel a, with the norms used as weights. Solid line: the transmission curve for the breather for the same set of parameters. This is the same curve as the $w=w_{0}$ curve in Fig.~3 in the main text.\n          }\n\\end{figure}\n\n\\newpage\n\\mbox{}\n\\vfill\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{phase_imprinting_split_05_workv}\n\\end{center}\n\\caption { \\label{Fig_4_01e}\n\\textbf{Superheated integrability in the context of phase imprinting.} At $t=0$, the breather wavefunction is multiplied by the space-dependent pure phase $e^{i\\epsilon \\varphi(x)}$. Here \\mbox{$\\epsilon=0.01$} and $\\varphi(x) = \\sqrt{\\frac{2}{L}}\\sqrt{\\frac{3}{2M}}\\sum_{m=1}^{M}\\left[c_{m}\\cos (2\\pi m x\/L)+s_{m}\\sin (2\\pi m x\/L)\\right]$, with \\mbox{$L=16$} and \\mbox{$M=5$}; $c_{m}$ and $s_{m}$ were drawn from the uniform distribution on $[-1,\\,1]$, and in the realization shown here had the values $(c_{1},\\,\\ldots,\\,c_{5})=(0.307,\\, 0.622,\\, 0.648,\\, -0.738,\\, 0.304)$ and $(s_{1},\\,\\ldots,\\,s_{5})=(0.353,\\,  -0.0422,\\, -0.794 ,\\,  -0.746,\\,  0.721)$. The function $\\varphi(x)$ is plotted in the inset. The main plot shows the time evolution of the density $|\\psi|^{2}$, during which the constituent solitons separate. Once they are well-separated, one can verify that their norms are 0.25 and 0.75. In the context of Eq.~(5) in the main text: if one uses the approximation $e^{i\\epsilon \\varphi(x)}\\approx 1 + i\\epsilon \\varphi(x)$, then the process depicted in this Figure corresponds to $v_{\\text{ext.}}(x,\\,t)\\,\\psi(x,t)=i\\delta(t)\\,\\varphi(x)\\,\\lim_{\\tau\\to t^{-}}\\psi(x,\\tau)$. Just as in the case of collision with a barrier, the separation of scales between the real and imaginary parts of the Lax-operator eigenvalues $\\lambda$, which correspond to the constituent solitons, again imply that the soliton velocities ($\\sim \\text{Re}~\\lambda$) change but norms ($\\sim \\text{Im}~\\lambda$) do not.\n          }\n\\end{figure}\n\n\\vfill\n\\mbox{}\n\\newpage\n\n\\end{widetext}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Intro}\n The heavy ion collisions produce matter at extreme temperatures and densities where \nit is expected to be in the form of Quark Gluon Plasma \n(QGP), a phase in which the quarks and gluons can move far beyond  the size of a nucleon \nmaking color degrees of freedom dominant in the medium. \n  The experimental effort to produce such matter started with low energy CERN accelerator \nSPS and evolved through voluminous results \nfrom heavy ion collision at Relativistic Heavy Ion Collider (RHIC) \\cite{INTRO_Arsene, INTRO_Back, INTRO_Adams, INTRO_Adcox}.\nThe recent results from Large Hadron Collider (LHC) experiments \\cite{QGP_Tc} are \npointing towards formation of high temperature system in many ways similar to the matter\nproduced at RHIC. \n  One of the most important signal of QGP is the suppression of \nquarkonium states \\cite{SATZ}, both of the charmonium ($J\/\\psi$, $\\psi(2S)$, $\\chi_{c}$, etc) \nand the bottomonium ($\\Upsilon(1S)$ , $\\Upsilon(2S)$, $\\chi_{b}$, etc) families. This is thought to be a \ndirect effect of deconfinement, when the binding potential between the constituents of a quarkonium state, \na heavy quark and its antiquark, is screened by the colour charges of the surrounding light quarks and gluons. \n The ATLAS and CMS experiments have carried out detailed quarkonia measurements in PbPb collisions \nwith the higher energy and luminosity available at the LHC.\n The ATLAS measurements \\cite{ATLAS} show suppression of inclusive $J\/\\psi$ with high transverse momenta $p_T$  \nin central PbPb collisions compared to peripheral collisions at $\\sqrt s_{NN} = 2.76$ TeV. \n  Similarly, CMS measured a steady and smooth decrease of suppression \nof prompt $J\/\\psi$ as a function of centrality with nuclear modification factor $R_{\\rm AA}$ remaining $<$ 1 even \nin the peripheral bin \\cite{JCMS}. \n\n The melting temperature of the quarkonia states depends on their binding energy. The ground states, \n$J\/\\psi$ and $\\Upsilon(1S)$ are expected to dissolve at significantly higher temperatures than the \nmore loosely bound excited states. The difference in binding energies among different quarkonia indicate that\nthey melt in a hot QGP at different temperatures and the quarkonium spectrum can\nserve as plasma thermometer \\cite{SATZ2,Mocsy_Strik}.\n  The $\\Upsilon(2S)$ and $\\Upsilon(3S)$ have smaller binding energies as compared to ground\nstate $\\Upsilon(1S)$ and hence are expected to dissolve at a lower temperature. \n With the 2011 PbPb run the CMS published results on sequential suppression of \n$\\Upsilon(nS)$ states as a function of centrality \\cite{CMSU2} with enlarged statistics\nover their first measurement \\cite{UCMS}\nwhere a suppression of the excited $\\Upsilon$ states with respect to the ground state have been observed  \nin PbPb collisions compared to pp collisions at $\\sqrt s_{NN} = 2.76$ TeV.\n\n  The quarkonia yields in heavy ion collisions are also modified due to non-QGP effects such as\nshadowing, an effect due to change of the parton distribution functions inside the nucleus,\nand dissociation due to nuclear or comover interactions \\cite{Vogt}. Due to higher mass, the \nnuclear suppression is expected to be less for bottomonia over charmonia.\n  If large number of heavy quarks are produced in initial heavy ion collisions at LHC energy \nthis could even lead to enhancement of quarkonia via statistical recombination \\cite{Rapp1,Rapp2}. \n The effect of regeneration is expected to be less significant for bottomonia as compared \nto charmonia since bottom quarks are much smaller in number as compared to charm quarks. \n In addition, due to higher bottom mass the bound state properties obtained from \npotential models are more reliable. Thus recent years witness a shift in the \ninterest to bottomonia. \n The ratios of the yields of excited states to the ground states is considered \neven more robust QGP probe as the cold nuclear matter effects if any cancel out and can be \nneglected in the ratios. The calculation of ratios of $\\Upsilon$ states was also made in \nfew works e.g. \\cite{UPsi_Blaiz,UPsi_Guni} in past which showed that the $p_T$ dependence \nof such ratio would show large variations and this would be a direct probe of the QGP. \n \n In this paper, we calculate the bottomonia suppression due to color screening in an expanding\nQGP using the model by Chu and Matsui \\cite{CHU},\nwhich takes into account the finite QGP lifetime and spatial extent. \n We start by describing the properties of quarkonia obtained from potential models and then \ngive a brief description of the model which is extended to get the survival \nprobabilities of $\\Upsilon$ states as a function of centrality of the collisions. \n Finally we compare the model calculations with the experimental data recently \nmeasured by the CMS experiment.\n\n\\section{Properties of the $\\Upsilon$ states from potential models}\nInteraction between the heavy quark and its antiquark inside the quarkonium at zero temperature  \ncan be described by Cornell potential \\cite{QPOT1, QPOT2,UPsi_KarschMehr}.\nThe solution of the Schrodinger equation for such potential gives mass, bound state radius and\nthe formation time $\\tau_{F}$, the time needed to form a bound state after the production of heavy quark pairs.\n All parameters obtained with zero temperature \npotential using the parameter values given in \\cite{UPsi_KarschMehr,QPROP} are summarized in \nfirst three rows of Table I, which describe well the experimentally \nobserved quarkonia spectroscopy.          \n\nThe potential model can be extended to finite temperature with the main assumption that medium effects can be \naccounted for as a temperature-dependent potential. Instead of just looking at the individual\nbound states (at $T$ = 0 where quarkonium is well defined), one could rather obtain a unified treatment of bound states, \nthreshold and continuum by determining the spectral function. \n  Using a class of screened potentials based on lattice calculations of the static quark-antiquark free energy, \nspectral functions at finite temperature are calculated in a work \\cite{UPsi_Mocsy2,UPsi_Mocsy3} and it was found that all \nquarkonium states, except the 1S bottomonium, dissolve in the deconfined phase at temperatures smaller than 1.5$T_C$.\nAn upper limit on binding energy and the thermal width of different quarkonia states are then estimated using  \nspectral functions in the quark-gluon plasma. \n Corresponding upper bounds on their dissociation temperatures $T_{D}$ \\cite{UPsi_Mocsy3} are\ngiven in second last row of Table I. We used slightly lower values of $T_{D}$ given in the last row \nto obtain a good match with measured $R_{\\rm AA}$. \n\\renewcommand{\\arraystretch}{1.4}\n\\begin{table}[ph]\n\\tbl{Quarkonia properties from non-relativistic potential theory \\cite{UPsi_KarschMehr,UPsi_Mocsy3}.}\n{\\begin{tabular}{@{}cccccc@{}} \\toprule \n\\hline\\noalign{\\smallskip}\n  {\\rm Bottomonium properties}        &    $\\Upsilon(1S)$   & $\\chi_b(1P)$  &  $\\Upsilon(2S)$      & $\\Upsilon(3S)$  &  $\\chi_b(2P)$ \\\\\n\\hline\\noalign{\\smallskip}\n {\\rm Mass~[GeV\/$c^{2}$]}     &    9.46             &   9.99        &    10.02             & 10.36           &   10.26 \\\\\n\\hline\n{Radius \\rm [fm]}          &    0.28             &   0.44        &     0.56             & 0.78            &0.68    \\\\\n\\hline \n$\\tau_{F}$ \\rm [fm] \\cite{UPsi_KarschMehr}  &    0.76             &   2.60        &     1.9              & 2.4             &     \\\\\n\\hline \n$T_D$ \\rm [GeV] upper limit \\cite{UPsi_Mocsy3}    &    2~$T_C$          &   1.3~$T_C$   &     1.2~$T_C$        & 1~$T_C$       &     \\\\\n\\hline\n$T_D$ \\rm [GeV] used in the present work  &    1.8~$T_C$          &   1.15~$T_C$   &     1.1~$T_C$        & 1.0~$T_C$       &     \\\\\n\\hline\n\\end{tabular} }\n\\label{prop}\n\\end{table}\n\n\\section{Quarkonia suppression in finite size QGP}\n The bottomonia survival probabilities due to color screening in an expanding QGP\nare estimated using a dynamical model which takes into account \nthe finite lifetime and spatial extent of the system \\cite{CHU}. The competition between the resonance formation \ntime $\\tau_{F}$ and the plasma characteristics such as temperature, lifetime and spatial extent decide the \n$p_{T}$ dependence of the survival probabilities of $\\Upsilon$ sates.  We describe the essential \nsteps used to develop the model which is then extended to get \nthe survival probabilities as a function of centrality of the collision.\n\n  The model assumes that quark gluon plasma is formed at some initial entropy density \n$s_0$ corresponding to initial temperature $T_0$ at time $\\tau_{0}$ which undergoes an \nisentropic expansion by Bjorken's hydrodynamics~\\cite{UPsi_Bjork}. The plasma cools to an entropy density $s_D$ \ncorresponding to the dissociation temperature $T_D$ in time $\\tau_{D}$ which is given by \n\\begin{equation}\\label{bjork}\n \\tau_{D} = \\tau_0 \\left( { s_0 \\over s_D} \\right) = \\tau_0 \\left( \\frac{T_{0}}{T_{D}} \\right)^{3},\n\\end{equation}\n  As long as $\\tau_{D}$\/$\\tau_{F}$ $>$ 1, quarkonium formation will be suppressed. \n\n In the finite system produced in heavy ion collision, the suppression and entropy depend on the size \nof the system. The initial entropy density is assumed to be dependent on radius $R$ (decided by the\ncentrality of the collision) of the QGP \\cite{CHU} as \n\\begin{equation}\\label{eprofile}\n   s_0(r) = s_0 ~\\left(1 - \\left(\\frac{r}{R}\\right)^2\\right)^{1\/4},\n\\end{equation}\n  Using Eq.~(\\ref{bjork}) and Eq.~(\\ref{eprofile}) one can obtain the $r$ dependence of $\\tau_D$ as\n\\begin{eqnarray}\\label{tDr}\n   \\tau_{D}(r) & = &  \\tau_{D}(0)\\left(1 - \\left(\\frac{r}{R}\\right)^2\\right)^{1\/4}.\n\\end{eqnarray}\nwhere $\\tau_{D}(0)$ is the value of $\\tau_{D}$ for resonances produced in the center of the system.\n\n  Let a $Q\\overline{Q}$ pair is created at the position ${\\rm \\bf r}$ in the transverse plane with a \ntransverse momentum ${\\rm \\bf p_T}$ and transverse energy $E_{T}$ = $\\sqrt{M^2 + p_{T}^2}$.\n The $\\Upsilon$ formation time is $\\tau_{F}\\gamma$ which on equating with the screening duration $\\tau_{D}(r)$  given \nin Eq~(\\ref{tDr}) one obtains the critical radius $r_D$, which is the boundary of the suppression region as \n\\begin{equation}\n  r_D  = R\\left(1 - \\left(\\frac{\\gamma \\tau_{F}}{\\tau_{D}(0)}\\right)^{4}\\right)^{1\/2}.\n\\end{equation}\nwhere $\\gamma$ = $E_T\/M$ is the Lorentz factor associated with the transverse motion of the pair. \n A bottom-quark pair can escape the screening region $r_D$ and form $\\Upsilon$ if the position at \nwhich it is created satisfies\n\\begin{equation}\\label{taumax}\n| {\\rm \\bf r} + {\\tau_{F} {\\rm \\bf p_{T}} \\over M} | > r_D,\n\\end{equation}\nwhere the screening region $r$ $<$ $r_D$ is shrinking because of the cooling of the system.\n Defining $\\phi$ to be the angle between ${\\rm \\bf p_{T}}$  and ${\\rm \\bf r}$, the Eq.~(\\ref{taumax}) leads to a range \nof $\\phi$ for which the bottom-quark pair can escape:\n\\begin{equation}\\label{cosmax}\n   {\\rm cos} \\, \\phi \\ge z   ~~~~{\\rm where } ~~~~ \\nonumber \\\\ \n    z = \\frac{ r_D^2 - r^2 - (\\tau_{F}p_{T}\/M)^2}{2r \\,(\\tau_{F}p_{T}\/M)},\n\\end{equation}\n  With this we can then calculate probability for the pair created at ${\\rm \\bf r}$ with transverse momentum ${\\rm \\bf p_T}$\nto survive as\n\\begin{eqnarray}\n\\phi(r,p_{T})  &  =     1                 & \\,\\,\\, z\\le -1  \\nonumber \\\\\n               &  =  \\left( { {\\rm cos}^{-1}z \\over \\pi} \\right)    & \\,\\,\\, |z| < 1  \\nonumber \\\\\n               &  =    0                 & \\,\\,\\, z\\ge 1,   \\nonumber  \n\\end{eqnarray}\n  If the probability $\\rho(r)$ of a quark pair to be created at $r$ which is symmetric in transverse plane \nis parameterized as\n\\begin{equation}\n\\rho(r) = \\left(1 - \\left( {r \\over R} \\right)^2\\right)^{1\/2},\n\\end{equation}\nthe survival probability of quarkonia becomes \n\\begin{equation}\nS(p_{T}, R) = \\frac{\\int_0^Rdr~r~\\rho(r)~\\phi(r,p_{T})}{\\int_0^Rdr~r~\\rho(r)}.\n\\end{equation}\n\nThe survival probability as a function of centrality can be obtained by integrating over \n$p_T$ as follows \n\\begin{equation} \n  S(N_{\\rm part})  = \\int S(p_{T}, R(N_{\\rm part}) ) \\, Y(p_T) \\,dp_T.\n\\end{equation}\nHere Y($p_T$) is $p_T$ distribution (normalized to one) obtained from Pythia. \nThe size $R = R(N_{\\rm part})$ as a function of centrality is obtained in terms of the radius of the Pb \nnucleus given by $R_0 = r_0\\, A^{1\/3}$($r_0 = 1.2 \\,$ fm) and the total number of participants $N_{\\rm part0}=2A$ in head-on collisions as \n\\begin{equation}\\label{rnpart}\nR(N_{\\rm part}) = R_0 \\, \\sqrt{N_{\\rm part} \\over N_{\\rm part0} }.\n\\end{equation}\nWe assumed initial temperature $T_0$ is the temperature in  0-5\\% central collisions and calculated it\nfor a given initial time $\\tau_0$ by\n\\begin{equation}\\label{Int1}\nT_{0}^{3}\\tau_{0} = \\frac{3.6}{4a_{q}\\pi R_{0-5\\%}^{2}}\\left(\\frac{dN}{d\\eta}\\right)_{0-5\\%},\n\\end{equation}\nHere $(dN\/d\\eta)_{0-5\\%}$ = 1.5$\\times$1600 obtained from the charge particle multiplicity measured in \nPbPb collisions at 2.76 TeV \\cite{MULT} and $a_{q}$ = 37$\\pi^{2}$\/90 is the degrees of freedom we take in \nquark gluon phase. Using Eq.~(\\ref{rnpart}) we can obtain the transverse size of the system\nfor 0-5\\% centrality as $R_{0-5\\%}$ = 0.92$R_0$. \nFor $\\tau_{0}$ = 0.1 fm\/$c$, we obtain $T_{0}$ as 0.65 GeV using Eq.~(\\ref{Int1}). \nThe critical temperature is taken as $T_{C}$ = 0.170 GeV \\cite{QGP_Tc}. \nThe initial temperature as a function of centrality is calculated by \n\\begin{equation}\\label{Int2}\nT(N_{\\rm part})^3 = T_0^3 \\, \\left({dN\/d\\eta \\over N_{\\rm part}\/2}\\right) \/ \\left({dN\/d\\eta \\over N_{\\rm part}\/2}\\right)_{0-5\\%}.\n\\end{equation}\nwhere $(dN\/d\\eta)$ is the multiplicity as a function of number of participants measured by ALICE experiment \\cite{MULT}. \nBoth ALICE and CMS \\cite{CMSmult} measurements on multiplicity agree well with each other.\n  Equation~(\\ref{Int2}) giving the variation of initial temperature as a function of centrality \ndiffers from the approach taken in the work of Ref.~\\cite{STR1} where it is taken to vary\nas a third root of number of participants.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70\\textwidth]{Fig1_dNdEta.eps}\n\\caption{\na) Measured $(dN\/d\\eta)\/(N_{\\rm part}\/2)$ ~\\cite{MULT} as a function \nof $N_{\\rm part}$ along with the function $(dN\/d\\eta)\/(\\pi R^2)$. \n(b) The initial temperature obtained from measured multiplicity using Eq.~(\\ref{Int2})\n}\n\\label{fig:upsiRatio1}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n  \\includegraphics[width=0.48\\textwidth]{Fig2a_rD1s.eps}\n  \\includegraphics[width=0.48\\textwidth]{Fig2b_rD2s.eps} \\\\\n\\caption{ The screening radius $r_D$ (in fm) as a function of $p_T$ for $R=6.8$ fm \n(corresponding to head-on collisions) and $R=3.7$ fm (corresponding to minimum bias collisions)\nfor (a) $\\Upsilon(1S)$ and (b) $\\Upsilon(2S)$.\n The straight lines $|$ ${\\rm \\bf r} + {\\tau_{F} {\\rm \\bf p_{T}} \\over M}$ $|$\nmark the distance a bottom quark pair (created at $r=0$) will travel before forming a bound state.\n The mesh region in both the figures marks the escape region for \nbottom quark pair in case of head-on collisions and total shaded (mesh+lines) region marks\nthe escape region in case of minimum bias collisions. }\n\\label{fig:upsiRatio2}\n\\end{center}\n\\end{figure}\n\n  The nuclear modification factor, $R_{\\rm AA}$ is obtained from survival probability taking into account \nthe feed-down corrections as follows,\n\n\\begin{eqnarray}\n R_{\\rm AA}(3S) &=& S(3S)  \\nonumber \\\\\n\n R_{\\rm AA}(2S) &=& f_1~S(2S) + f_2~S(3S)  \\nonumber \\\\\n R_{\\rm AA}(1S) &=& g_1 ~S(1S) + g_2~S(\\chi_b(1P)) + g_3~S(2S) + g_4~S(3S)\n\\end{eqnarray}\nThe factors $f$'s and $g$'s are obtained from CDF measurement \\cite{CDF}. The values \nof $g_{1}$, $g_{2}$, $g_{3}$ and $g_{4}$ are 0.509, 0.27, 0.107 and 0.113 respectively.\nHere it is assumed that the survival probabilities of $\\Upsilon(3S)$ and $\\chi_{b}$(2P) \nare same and $g_4$ is their combined fraction. \nThe values of $f_{1}$ and $f_{2}$ are taken as 0.50 guided by the work from Ref.~\\cite{STR2}. \n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{Fig3a_Surv_MB.eps}\n\\includegraphics[width=0.48\\textwidth]{Fig3b_SurvFd_MB_3s.eps}\\\\\n\\caption{ (a) The survival probability as a function of \n$p_{T}$ for $\\Upsilon(1S)$, $\\Upsilon(2S)$, $\\Upsilon(3S)$ and $\\chi_{b}(1P)$ for $R=3.7$ fm \n(corresponding to average $N_{\\rm part}$ = 114 for minimum bias collisions). \n (b) The nuclear modification factor \nfor $\\Upsilon(1S)$, $\\Upsilon(2S)$ and $\\Upsilon(3S)$ which is obtained from survival probabilities including \nfeed down corrections.  The solid squares are $\\Upsilon(1S)$ $R_{\\rm AA}$ measured in the minimum \nbias PbPb collisions at $\\sqrt{s_{NN}} = 2.76$ TeV by CMS experiment \\cite{JCMS}. \n}\n\\label{fig:upsiRatio3}\n\\end{center}\n\\end{figure*}\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth]{Fig4_RaaALL.eps} \n\\caption{The nuclear modification factor, $R_{\\rm AA}$\nas a function of $N_{\\rm part}$ for $\\Upsilon(1S)$, $\\Upsilon(2S)$ and $\\Upsilon(3S)$. The solid squares and circles are \nmeasured $R_{\\rm AA}$ by CMS experiment in PbPb collisions at $\\sqrt{s_{\\rm NN}}$ = 2.76 TeV \\cite{CMSU2} for  \n$\\Upsilon(1S)$ and $\\Upsilon(2S)$ respectively and solid triangles are the minimum bias data points. \nThe boxes at unity are the common systematic uncertainties in pp luminosity \nmeasurement and the pp yield. The lines(solid for $\\Upsilon(1S)$ , dashed for $\\Upsilon(2S)$ and dotted for $\\Upsilon(3S)$) \nrepresent the present model calculations. }\n\\label{fig:upsiRatio4}\n\\end{center}\n\\end{figure*}\n\n\\section{Results and discussions}\n  Figure~\\ref{fig:upsiRatio1} (a) shows measured $(dN\/d\\eta)\/(N_{\\rm part}\/2)$ ~\\cite{MULT} as a function \nof $N_{\\rm part}$. The function $(dN\/d\\eta)\/(\\pi R^2)$ gives the multiplicity divided by transverse \narea obtained using Eq.(\\ref{rnpart}). Figure~\\ref{fig:upsiRatio1} (b) gives the initial temperature \nobtained from measured multiplicity using Eq.~(\\ref{Int2}). Except in peripheral collisions, the initial \ntemperature has weak dependence on centrality of collisions. \nFigure~\\ref{fig:upsiRatio2} demonstrates working of the model. It shows\nthe screening radius $r_D$ (in fm) as a function of $p_T$ for $R=6.8$ fm \n(corresponding to head-on collisions) and $R=3.7$ fm (corresponding to minimum bias collisions)\nfor (a) $\\Upsilon(1S)$ and (b) $\\Upsilon(2S)$.\n The straight lines $|$ ${\\rm \\bf r} + {\\tau_{F} {\\rm \\bf p_{T}} \\over M}$ $|$\nmark the distance a bottom quark pair (created at $r=0$) will travel before forming a bound state.\n The mesh region in both the figures marks the escape region for \nbottom quark pair in case of head-on collisions and total shaded (mesh+lines) region marks\nthe escape region in case of minimum bias collisions. \n If $r$ is non-zero, the region where a bottomonium can escape screening, enlarges.\n\n Figure~\\ref{fig:upsiRatio3} (a) shows the survival probability as a function of \n$p_{T}$ for $\\Upsilon(1S)$, $\\Upsilon(2S)$, $\\Upsilon(3S)$ and $\\chi_{b}(1P)$ for $R=3.7$ fm \n(corresponding to average $N_{\\rm part}$ = 114 for minimum bias collisions). \n  The survival probability $S(p_{T})$ has a unique $p_T$ dependence decided by the \n$T_D$ and $\\tau_{F}$ of each $\\Upsilon$ state. \n  In general, the survival probabilities of resonance states increase with increasing $p_T$ \nand become unity at different $p_T$ for different states corresponding to complete survival. \n Since $\\Upsilon(1S)$ is expected to dissolve at a higher temperature it has more probability to survive\nthe plasma region even at lower p$_{T}$ as compared to the cases of other bottomonia states.\n The model gives very similar survival probabilities for $\\Upsilon(2S)$ and $\\Upsilon(3S)$.\nThis is due to the fact that $\\Upsilon(3S)$ has large formation time even though its \ndissociation temperature is smaller in comparison to $\\Upsilon(2S)$.\n Figure~\\ref{fig:upsiRatio3} (b) shows the nuclear modification factor \nfor $\\Upsilon(1S)$, $\\Upsilon(2S)$ and $\\Upsilon(3S)$ which is obtained from survival probabilities \nincluding feed down corrections. \n The solid squares are $\\Upsilon(1S)$ $R_{\\rm AA}$ measured in the minimum bias PbPb collisions at \n$\\sqrt{s_{NN}} = 2.76$ TeV by CMS experiment \\cite{JCMS}. \n The model reproduces the trend of the $p_T$ dependence of low statistics \nmeasurements of $R_{\\rm AA}$ from 2010 PbPb collisions by CMS.  \n\n  Figure~\\ref{fig:upsiRatio4} shows the nuclear modification factor, $R_{\\rm AA}$\nas a function of $N_{\\rm part}$ for $\\Upsilon(1S)$, $\\Upsilon(2S)$ and $\\Upsilon(3S)$. The solid squares and circles are \nmeasured $R_{\\rm AA}$ by CMS experiment in PbPb collisions at $\\sqrt{s_{\\rm NN}}$ = 2.76 TeV \\cite{CMSU2} for  \n$\\Upsilon(1S)$ and $\\Upsilon(2S)$ respectively and solid triangles are the minimum bias data points. \nThe lines(solid for $\\Upsilon(1S)$, dashed for $\\Upsilon(2S)$ and dotted for $\\Upsilon(3S)$) represent \nthe present model calculations. \nThe common systematic uncertainties in pp luminosity measurement and the pp yield are \nrepresented by the boxes at unity. The model correctly reproduces the measured nuclear modification \nfactors of both $\\Upsilon(1S)$ and $\\Upsilon(2S)$ for \nall centralities using the parameters given in the Table I. The \nsurvival probabilities for $\\Upsilon(2S)$ and $\\Upsilon(3S)$ are very similar.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth]{Fig5_DRatio.eps}\n\\caption{Double ratio, $[\\Upsilon(2S)\/\\Upsilon(1S)]_{PbPb}$\/$[\\Upsilon(2S)\/\\Upsilon(1S)]_{pp}$ as a \nfunction of $N_{\\rm part}$ measured by CMS experiment \\cite{CMSU2} \nalong with the present calculation (solid line). The box at unity is\nthe common systematic uncertainty in the pp yield. \n}\n\\label{fig:upsiRatio5}\n\\end{center}\n\\end{figure}\n\n We also calculated the ratio of $R_{\\rm AA}$ of $\\Upsilon(2S)$ to that of $\\Upsilon(1S)$ \nwhich is equivalent to the so called double ratio $[\\Upsilon(2S)\/\\Upsilon(1S)]_{PbPb}$\/$[\\Upsilon(2S)\/\\Upsilon(1S)]_{pp}$.\nThe double ratio has the advantage that the effects such as initial-state nuclear effects and regeneration\nwhich we ignore in our calculations are supposedly canceled out. \n Figure~\\ref{fig:upsiRatio5} shows the double ratio measured by CMS experiment \\cite{CMSU2} \nalong with the present calculation. The calculations reproduce the measured double ratio \neven for the most peripheral data point.\n\n The most important parameters in above study are formation time \nand dissociation temperatures of bottomonia states. There are reliable calculations of formation time \nobtained from zero temperature potential models which reproduce the bottomonia spectroscopy very well. \nUpper limits are available for dissociation temperatures which are obtained from potential models \nat finite temperature. We used slightly lower values of the dissociation temperature to get a good\ndescription of the measured nuclear modification factors of $\\Upsilon(1S)$ and $\\Upsilon(2S)$. \n The dynamics of the system is affected by the initial conditions which in the present calculations are \nfixed using measured charged particle multiplicity at LHC.\nThere can be suppression due to initial nuclear effects which we assume to be \nmuch smaller than that due to colour screening and hence are ignored in the present work. \n  The calculations of shadowing in PbPb show that it will effect the bottomoina yields by \napproximately 20 \\% for most central collisions \\cite{Shadow}. Thus, the dissociation temperatures \nobtained by us are still considered to be the upper limits. Conversly there are other views which say that  \n$\\Upsilon$ ground state is not much affected by the color screening \nup to the temperatures of $\\sim 3-4T_C$ and regeneration of the states are not negligible at the LHC \\cite{Rapp}.\n  The bottom quark mass is 10 times higher than the temperature we \nare considering for the system and hence the regeneration effect can be safely ignored in calculating\nnuclear modification for bottomonia. The uncertainties in the measurements of feed-down fractions \nwould introduce uncertainties in the calculated nuclear modification factor. \n Finally we mention that the uncertainties arising from the effects other than colour screening are small and supposedly \nwill have little or no effect on the double ratio.\n\n\\section{Conclusions}\n  In summary, we calculate the survival probabilities of $\\Upsilon$ states and obtain the nuclear modification\nfactors due to colour screening in an expanding quark gluon plasma of finite lifetime and size produced \nduring PbPb collisions $\\sqrt{s_{NN}}=$ 2.76 TeV.\n  The formation time and dissociation temperatures of bottomonia states \nobtained from potential models are used as input parameters in the model. \nWe used slightly lower values of the dissociation temperatures to get a good\ndescription of the measured nuclear modification factors of $\\Upsilon(1S)$ and $\\Upsilon(2S)$. \n  The model reproduces the centrality dependence of measured nuclear modification \nfactors of $\\Upsilon(1S)$ and $\\Upsilon(2S)$ and the double ratio very well at $\\sqrt{s_{\\rm NN}}$ = 2.76 TeV.\n  The trend of $p_T$ dependence of low statistics measurements of nuclear modification factor \nfrom 2010 PbPb collisions of CMS is reproduced as well.  \n The uncertainties arising from effects other than colour screening are assumed to be \nsmall and supposedly will have little or no effect on the double ratio calculations.\n\\section{Acknowledgement}\nWe thank Vineet Kumar, Ramona Vogt and CMS heavy ion colleagues for many fruitful discussion.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the last fifteen years, large dimensional stationary factor models have achieved great success in the economic profession, especially in forecasting macroeconomic variables \\citep[see, e.g.,][]{Nowcasting}, and are now a common tool in several policy institutions. However, macroeconomic time series are typically non-stationary due to the presence of common and idiosyncratic stochastic trends, and the practice of differencing the data to achieve stationarity is a problem that not always has a clear-cut solution. Take for example the case of the unemployment rate, which is a highly-persistent time series, but at the same time economic theory forbids it to have a unit root; or, take as another example the case of inflation, which shows  periods of high-persistence in the late 70s early 80s, while more recently displays clear mean reversion. To avoid the risk of over- or under-differencing data, a Non-Stationary Dynamic Factor Model (NS-DFM) is then desirable, and it is studied in this paper. \n\nThe NS-DFM proposed in this paper captures several features of macroeconomic data as it takes into account the presence of common trends generating permanent fluctuations in the economy, as well as common transitory forces generating cyclical fluctuations. More technically, in our model, the common factors are a cointegrated vector process, thus containing both $I(1)$ trends and stationary components. Moreover, the NS-DFM addresses the possible presence of idiosyncratic trends, as well as the presence of secular (linear) trends, which can have either a constant slope (deterministic linear trends) or a time-varying slope (local linear trends). \n\nIn this paper, we study estimation of the NS-DFM by Quasi Maximum Likelihood (QML) implemented through the Expectation Maximization (EM) algorithm and the Kalman smoother (KS). Specifically, we extend the results in \\citet{BLqml} for the stationary case, to prove that when the common factors are the only source of non-stationarity, the common component estimated at a given point in time and for a given unit is $\\min(\\sqrt n,\\sqrt T)$-consistent. We also discuss extensions to the cases of (i) unit roots in the idiosyncratic components, and (ii) local levels and local linear trends. \n \nEstimation is implemented in two steps. First, given the observed data, by means of the KS we estimate the conditional mean of the latent factors,  which, together with its associated conditional covariance matrix, we use to compute the expected log-likelihood of the model (E-step).\\footnote{In a non-stationary setting the existence of the conditional mean of the factor as a minimizer of the mean-squared prediction error has been proved by \\citet[Theorem 1]{hannan67} and \\citet[Theorem 1]{sobel67}.} Second, we maximize the expected log-likelihood with respect to the loadings and the other parameters of the model (M-step). The use of an iterative procedure to extract unobserved components in the case of non-stationary data was proposed since the original work by \\citet{kalman60}. Although this is not the first paper using these techniques for non stationary data, this is the first paper to address consistency of factors. Moreover, QML estimation of autoregressive processes with unit roots is a classical problem studied at length by the literature \\citep{simsstockwatson,johansen91}.\\footnote{Solutions based on spectral analysis are also in \\citet{bell84} and \\citet{CT76}.}\n\nEstimation of NS-DFMs has also been studied by \\citet{baing04} and \\citet{BLL2} by PC analysis on differenced data. Both approaches are designed to account for non-stationary idiosyncratic components; however, only the latter is designed to deal with linear deterministic trends. \\citet{bai04} has used a factor model to estimate common trends via PC on data in levels. However, because of its nature, that approach is valid only if all idiosyncratic components are stationary, i.e., only if data are cointegrated.\n\nCompared to those PC based estimators, our approach has a number of practical advantages. First, it allows estimating the model even in the presence of missing values, which is crucial when using the model in real-time because macroeconomic data are published with delays and at non-synchronized dates. Second, it allows estimating jointly stochastic trends as well as (deterministic or local) linear trends, whereas \\citet{baing04} and \\citet{BLL2} are forced to remove the deterministic trends before running PC analysis. Third, it allows having time-varying parameters, such as, for example, the slope of linear trends. Fourth, it allows putting restrictions on the parameters, such as national accounts identities, or restrictions coming from economic theory.\n\nFrom a theoretical point of view, our estimator converges at a faster rate than those of \\citet{baing04} and \\citet{BLL2}. However, this faster convergence does not come for free. Indeed, our estimator is based on stronger assumptions than those of PC analysis: namely, it is derived under the assumption that we know which idiosyncratic components are $I(1)$ and which ones are stationary, and which series have a linear trend component. Under this assumption, we can model the $I(1)$ idiosyncratic components, and the time-varying slopes or means, as additional latent states in the KS, thus allowing to simultaneously estimate the entire model. This strategy is shown to work well in practice, provided the number of additional latent states is not too large.\n\nThe use of the EM in time series dates back to \\citet{SS77}, \\citet{shumwaystoffer82}, \\citet{watsonengle83}, \\citet{quahsargent93}, and \\citet{SAZ13}, among others. However, with the exception of the last two, none of the above works has considered the case of non-stationary data. Moreover, to the best of our knowledge, no asymptotic theory exists for the setting considered in this paper.\n\nThe rest of the paper proceeds as follows: in Section \\ref{sec:modelNS}, we present the NS-DFM and its assumptions. Estimation is outlined in Section \\ref{sec:estNS} where we also prove consistency. The extension to non-stationary idiosyncratic states is discussed in Section \\ref{sec:idioI1}. Numerical results are in Section \\ref{sec:mc2}. Section \\ref{sec:conclusion} concludes.\n\n\\section{Model and assumptions}\\label{sec:modelNS}\nWe define a NS-DFM driven by $q$ factors as \n\\begin{align}\nx_{it}&=\\alpha_{it}+\\beta_{it} t+\\bm b_i^\\prime(L) \\bm f_t +\\xi_{it},\\label{eq:NSDFM1}\\\\\n\\bm f_t&= \\bm{\\mathcal A}(L)\\bm f_{t-1}+\\bm u_t,\\label{eq:NSDFM2}\\\\\n\\xi_{it}&=\\rho_i\\xi_{it-1}+e_{it},\\label{eq:NSDFM3}\\\\\n\\alpha_{it}&=\\alpha_{it-1}+\\omega_{it},\\label{eq:NSDFM5}\\\\\n\\beta_{it}&=\\beta_{it-1}+\\eta_{it},\\label{eq:NSDFM4}\n\\end{align}\nfor  $i=1,\\ldots, n$, and $t=1,\\ldots, T$. We let $\\chi_{it}=\\bm b_i^\\prime(L) \\bm f_t$. Then, $\\bm\\chi_{nt}=(\\chi_{1t}\\cdots\\chi_{nt})^\\prime$ is the common component, $\\bm\\xi_{nt}=(\\xi_{1t}\\cdots\\xi_{nt})^\\prime$ the idiosyncratic component, $\\bm {\\mathcal B}_n(L)=(\\bm b_1(L)\\cdots\\bm b_n(L))^\\prime$ the $n\\times q$ polynomial matrix of factor loadings, $\\bm f_t=(f_{1t}\\cdots f_{qt})^\\prime$ the $q$ factors, $\\bm u_t=(u_{1t}\\cdots u_{qt})^\\prime$ the $q$ common shocks, $\\mathbf e_{nt}=(e_{1t}\\cdots e_{nt})^\\prime$ the idiosyncratic shocks, and we also define $\\bm\\omega_{nt}=(\\omega_{1t}\\cdots\\omega_{nt})^\\prime$ and $\\bm\\eta_{nt}=(\\eta_{1t}\\cdots\\eta_{nt})^\\prime$.\n\n\nWe make the following assumptions.\n\n\\begin{ass}\\label{ass:dynamic}\n\\begin{inparaenum}[(a)]\n\\item for all $i\\in\\mathbb N$ and $z\\in\\mathbb C$, $\\bm b_i(z)=\\sum_{k=0}^s \\bm b_{ik}z^k$, such that $\\bm b_{ik}$ are $q\\times 1$ and $s$ is a finite integer with $s\\ge 0$;\n\\item for all $n\\in\\mathbb N$, let $\\bm{\\mathcal B}_{kn}=(\\bm b_{1k}\\cdots \\bm b_{nk})^\\prime$, then $\\lim_{n\\to\\infty}\\Vert n^{-1}\\bm{\\mathcal B}_{kn}^\\prime\\bm{\\mathcal B}_{kn}-\\bm\\Sigma_{k}\\Vert=0$, with $\\bm\\Sigma_{k}$ being $q\\times q$, and $\\bm \\Sigma_0$ positive definite, while $\\mbox{rk}(\\bm\\Sigma_k)\\le q$ for $k=1,\\ldots,s$, \nmoreover, for all $i\\in\\mathbb N$ and $k=0,\\ldots, s$, $\\Vert\\bm b_{ik}\\Vert \\le M_B$ for some finite positive real $M_B$ independent of $i$ and $k$; \\item $\\bm\\Gamma^{\\Delta f}=\\E_{\\varphi_n}[\\Delta\\bm f_t\\Delta\\bm f_t^\\prime]$ is $q\\times q$ positive definite and there exists a finite positive real $M_f$, such that $\\Vert\\bm\\Gamma^{\\Delta f}\\Vert\\le M_f$;\n\\item $q$ is a finite positive integer, such that $q<n$ and is independent of $n$; \\item $\\bm{\\mathcal A}(z)=\\sum_{k=1}^{p} \\bm{\\mathcal A}_{k}z^{k-1}$, such that $\\bm{\\mathcal A}_{k}$ are $q\\times q$ and $p$ is a finite positive integer, and $\\det(\\mathbf I_q-\\bm{\\mathcal A}(z))\\ne 0$ for all $z\\in\\mathbb C$ such that $|z|< 1$; \\item $\\mbox{\\upshape rk}(\\bm{\\mathcal A}(1))=d$ with $0<d\\le q$; \\item $|\\rho_i|\\le 1$ for all $i\\in\\mathbb N$; \n\\item $\\alpha_{i0}$ and $\\beta_{i0}$ are finite reals.\n\\end{inparaenum}\n\\end{ass}\n\n\\begin{ass}\n\\begin{inparaenum}[(a)]\\label{ass:modelNS}\n \\item for all $t\\in\\mathbb Z$, $\\bm u_t\\sim \\mathcal N(\\mathbf 0_q,\\bm\\Gamma^u)$, such that $\\bm\\Gamma^u$ is $q\\times q$ and positive definite, and  $\\E_{\\varphi_n}[\\bm u_{t}\\bm u_{t-k}^\\prime]=\\mathbf 0_{q\\times q}$ for all $k\\ne 0$; \\item for all $t\\in\\mathbb Z$ and all $n\\in\\mathbb N$, $\\mathbf e_{nt}\\sim\\mathcal N(\\mathbf 0_n,\\bm\\Gamma_n^e)$, such that $\\bm\\Gamma_n^e$ is $n\\times n$ and positive definite, and $\\E_{\\varphi_n}[\\mathbf e_{nt}\\mathbf e_{nt-k}]=\\mathbf 0_{n\\times n}$ for all $k\\ne 0$; \n \\item for all $n\\in\\mathbb N$, $\\Vert\\bm\\Gamma_n^e \\Vert \\le M_e$, for some positive real $M_e$ independent of $n$; \\item $\\E_{\\varphi_n}[\\mathbf e_{nt}\\bm u_{s}^\\prime]=\\mathbf 0_{n\\times q}$ for all $n\\in\\mathbb N$ and $t,s\\in\\mathbb Z$; \\item for all $t\\in\\mathbb Z$ and all $n\\in\\mathbb N$, $\\bm \\omega_{nt}\\sim\\mathcal N(\\mathbf 0_n,\\bm\\Gamma_n^\\omega)$ and $\\bm \\eta_{nt}\\sim\\mathcal N(\\mathbf 0_n,\\bm\\Gamma_n^\\eta)$, such that $\\bm\\Gamma_n^\\omega$ and $\\bm\\Gamma_n^\\eta$ are diagonal, respectively with entries $0\\le \\sigma_{i\\omega}^{2} < M_\\omega$ and $0\\le \\sigma_{i\\eta}^{2} < M_\\eta$, for some positive reals $M_\\omega$ and $M_\\eta$ independent of $i$, and $\\E_{\\varphi_n}[\\bm \\omega_{nt}\\bm \\omega_{nt-k}]=\\mathbf 0_{n\\times n}$ and $\\E_{\\varphi_n}[\\bm \\eta_{nt}\\bm \\eta_{nt-k}]=\\mathbf 0_{n\\times n}$ for all $k\\ne 0$; \\item \n $\\E_{\\varphi_n}[\\bm\\omega_{nt}\\bm u_{s}^\\prime]=\\mathbf 0_{n\\times q}$,\n $\\E_{\\varphi_n}[\\bm\\eta_{nt}\\bm u_{s}^\\prime]=\\mathbf 0_{n\\times q}$, for all $n\\in\\mathbb N$ and $t,s\\in\\mathbb Z$;\n \\item  $\\E_{\\varphi_n}[\\bm\\omega_{nt}\\mathbf e_{ns}^\\prime]=\\mathbf 0_{n\\times n}$,\n $\\E_{\\varphi_n}[\\bm\\eta_{nt}\\mathbf e_{ns}^\\prime]=\\mathbf 0_{n\\times n}$, and\n $\\E_{\\varphi_n}[\\bm\\omega_{nt}\\bm \\eta_{ns}^\\prime]=\\mathbf 0_{n\\times n}$,\n for all $n\\in\\mathbb N$ and $t,s\\in\\mathbb Z$.\n\\end{inparaenum}\n\\end{ass}\n\n\\begin{ass}\\label{ass:I1}\nFor any given $n\\in\\mathbb N$, there exists sets $\\mathcal I_1\\in\\{1,\\ldots,n\\}$, $\\mathcal I_a\\in\\{1,\\ldots,n\\}$,  and $\\mathcal I_b\\in\\{1,\\ldots,n\\}$, such that:\n\\begin{inparaenum}[(a)]\n\\item if $i\\in\\mathcal I_1$ then $\\rho_i=1$, while $\\rho_i=0$ otherwise, moreover $\\# \\mathcal I_1=n_1$ such that $n_1n^{-1}\\to 0$, as $n\\to\\infty$;\n\\item if $i\\in\\mathcal I_a$ then $\\sigma_i^{2\\omega}\\ge C_\\omega$ for some positive real $C_\\omega$, while $\\sigma_i^{2\\omega}=0$ otherwise, moreover $\\# \\mathcal I_a=n_a$ such that $n_an^{-1}\\to 0$, as $n\\to\\infty$;\n\\item if $i\\in\\mathcal I_b$ then $\\sigma_i^{2\\eta}\\ge C_\\eta$ for some positive real $C_\\eta$, while $\\sigma_i^{2\\eta}=0$ otherwise, moreover $\\# \\mathcal I_b=n_b$ such that $n_bn^{-1}\\to 0$, as $n\\to\\infty$.\n\\end{inparaenum}\n\\end{ass} \n\n\nBy Assumption \\ref{ass:dynamic}(a) we are considering the case in which factors are loaded dynamically with a finite number of lags. We do not consider here the case of autoregressive filters, which has been studied in \\citet{FHLZ17} in the stationary case. By Assumption \\ref{ass:dynamic}(b) we are assuming for simplicity that all $q$ factors are pervasive at lag-zero, while at higher lags they might or might not have a pervasive effect depending on the rank of $\\bm\\Sigma_k$. In other words, \\eqref{eq:NSDFM1} can be seen as a factor model with $q(s+1)$ factors of which $q$ are strong factors, i.e., having an effect on all series, and the remaining $qs$ are weak factors, i.e., having an effect only on a subset of series. \n\nBy Assumptions \\ref{ass:dynamic}(d) and \\ref{ass:dynamic}(e), when $d<q$ we allow the dynamics of the factors to be driven by $(q-d)<q$ unit roots implying the presence of $(q-d)$ common trends \\citep{stockwatson88JASA}. Clearly, in this setting, the factors are cointegrated with cointegration rank $d$, thus representing the permanent and transitory aspects of macroeconomic dynamics. When $d=0$---i.e., the dynamics of the factors are driven by $q$ unit roots---the VAR for the common factors in levels in \\eqref{eq:NSDFM2} does not exist; instead, it exists a VAR for $\\Delta \\bm f_t$, or the factors can be modeled as $q$ independent random walks as in \\citet{bai04}. That said, the case $d>0$ is the relevant one, as there is full agreement in the economic profession that while some fluctuations in the economy are permanent (common trends), some others are only temporary.\n\nAssumption \\ref{ass:modelNS} characterizes the innovations of the model. In particular, by part (c)  the idiosyncratic innovations $e_{it}$ are allowed to be mildly cross-correlated, thus implying that $\\Delta x_{it}$ follows an approximate factor model. Moreover, by part (e) we allow some series to be driven by a time-varying intercept and\/or a trend with time-varying slope, modeled as in a local level and local linear trend model, respectively \\citep[Section 2.3.6, page 45]{harvey90}. Notice that, if we set $\\sigma_{i\\eta}^2=0$, then the trend becomes deterministic with slope $\\beta_{i0}$, which is fixed to a constant by Assumption \\ref{ass:modelNS}(h), and similarly if we set $\\sigma_{i\\omega}^2=0$, we have a deterministic, hence constant, intercept term equal to $\\alpha_{i0}$. Finally, by parts (d) and (f) all innovations are independent. Notice that gaussianity is not strictly needed, but it is a reasonable assumption in macroeconomics.\n\nUnder these assumptions, it can be shown that the covariance matrix of the differenced common component $\\Delta\\bm\\chi_{nt}$ has at least $q$ and at most $q(s+1)$ eigenvalues that diverge linearly as $n\\to\\infty$. In particular, letting the covariance matrix of $\\Delta \\bm \\chi_n$ be $\\bm\\Gamma_n^{\\Delta\\chi}$, and denoting as $\\mu_{jn}^{\\Delta\\chi}$ the $j$-th largest eigenvalue of $\\bm\\Gamma_n^{\\Delta\\chi}$, Assumptions \\ref{ass:dynamic}(b) and \\ref{ass:dynamic}(c) imply that, for $j=1,\\ldots, q$,\n\\begin{equation}\\label{eq:divevaldiff}\n\\underline K_j\\le \\lim\\inf_{n\\to\\infty} n^{-1} \\mu_{jn}^{\\Delta\\chi}\\le\\lim\\sup_{n\\to\\infty} n^{-1} \\mu_{jn}^{\\Delta\\chi}\\le \\overline K_j, \n\\end{equation}\nfor some positive reals $\\underline K_j$ and $\\overline K_j$. Moreover,  letting the covariance matrix of $\\Delta \\bm \\xi_n$ be $\\bm\\Gamma_n^{\\Delta\\xi}$, and denoting as $\\mu_{jn}^{\\Delta\\xi}$ the $j$-th largest eigenvalue of $\\bm\\Gamma_n^{\\Delta\\xi}$, Assumption \\ref{ass:modelNS}(c), implies that\n\\begin{equation}\n\\sup_{n\\in\\mathbb N} \\mu_{1n}^{\\Delta\\xi}\\le M_\\xi, \\label{eq:divevaldiff2}\n\\end{equation}\nfor some positive real $M_\\xi$. From \\eqref{eq:divevaldiff} and \\eqref{eq:divevaldiff2}, and Assumption \\ref{ass:modelNS}(e), by Weyl's inequality,  the $q$ largest eigenvalues of the covariance matrix of $\\Delta \\mathbf x_{nt}$ diverge linearly in $n$, while all other $(n-q)$ eigenvalues stay bounded for all $n\\in\\mathbb N$. \n\nMoreover, it can be shown that the $q$ largest eigenvalues of the spectral density of $\\Delta \\mathbf x_{nt}$ diverge with $n$ at all frequencies, but at zero-frequency, where, due to the presence of common trends, only $(q - d)$ eigenvalues diverge, all the others eigenvalues being bounded for all $n$ and all frequencies. Hence, by looking at the eigenvalues of the spectral density matrix of $\\Delta \\mathbf x_{nt}$ we can determine $q$ and $d$ (see \\citealp{hallinliska07}, and \\citealp{BLL2}, respectively). Moreover, notice that when all factors are pervasive at all lags, i.e., in Assumption \\ref{ass:dynamic}(b) we let $\\mbox{rk}(\\bm\\Sigma_k)=q$ for all $k=0,\\ldots,s$,  then \\eqref{eq:divevaldiff} holds for all $j=1,\\ldots,q(s+1)$.  Therefore, by looking at the eigenvalues of the covariance matrix of $\\Delta \\mathbf x_{nt}$, we can also determine $s$ \\citep{dagostinogiannone12}.\n\nThe model defined in \\eqref{eq:NSDFM1}-\\eqref{eq:NSDFM4} has $q$ latent states, given by the common factors $\\bm f_t$, and additional latent states given by those idiosyncratic components that are autocorrelated as in \\eqref{eq:NSDFM3}, and by the time-varying intercepts and trend slopes as in \\eqref{eq:NSDFM5} and \\eqref{eq:NSDFM4}. These additional latent states are such that they satisfy the following assumption. \n\nIn other words, we are assuming that some, but not all, idiosyncratic components are $I(1)$, and that some, but not all, series have a time-varying intercept and\/or a linear trend with time-varying slope. For simplicity, we are also assuming that stationary idiosyncratic components are serially uncorrelated. \n\nWe then  make the following identifying assumptions.\n\n\\begin{ass} \\label{ass:identNS}\nLet $\\mathbf M_n^{\\Delta \\chi}$ be the $q\\times q$ diagonal matrix with elements $\\mu_{1n}^{\\Delta \\chi},\\ldots,\\mu_{qn}^{\\Delta \\chi}$, and let \n$\\mathbf V_n^{\\Delta\\chi}$ be the $n\\times q$ matrix having as columns the corresponding normalized eigenvectors. Then: \\begin{inparaenum}[(a)]\n \\item $\\Delta \\bm f_t=(\\mathbf M_n^{\\Delta\\chi})^{-1\/2}\\mathbf V_n^{\\Delta\\chi\\prime} \\Delta\\bm\\chi_{nt}$;\n \\item the entries of $\\mathbf M_n^{\\Delta\\chi}$ are such that they satisfy \\eqref{eq:divevaldiff} and $\\overline K_{j+1}<\\underline K_j$ for $j=1,\\ldots, q-1$;\n \\item the entries of $\\mathbf V_n^{\\Delta\\chi}$ are such that $[\\mathbf V_n^{\\Delta\\chi}]_{1j}>0$ for all $j=1,\\ldots, q$.\n\n\\end{inparaenum}\n\\end{ass}\n \nParts (a) and (b) are standard in factor model literature for stationary processes and allow to identify the differenced factors up to a multiplication by a sign \\citep[see, e.g.,][]{FGLR09,FLM13}. We identify the first difference of the factors with the $q$ normalized principal components of $\\Delta\\bm\\chi_{nt}$ and this implies in Assumption \\ref{ass:dynamic}(b) that $\\bm\\Gamma^{\\Delta f}=\\mathbf I_q$. It can then be seen that the following must hold for the loadings\n\\begin{align}\n\\mathbf V_n^{\\Delta\\chi\\prime} \\bm{\\mathcal B}_{0n}= (\\mathbf M_n^{\\Delta\\chi})^{1\/2},\n\\end{align}\ntherefore we can choose $\\bm{\\mathcal B}_{0n}=\\mathbf V_n^{\\Delta\\chi}(\\mathbf M_n^{\\Delta\\chi})^{1\/2}$, and in Assumption \\ref{ass:dynamic}(a) we have that $\\bm\\Sigma_0$ is diagonal with entries given by $\\lim_{n\\to\\infty} (n^{-1}\\mu_{jn}^{\\Delta \\chi})$, which as requested are finite and positive because of \\eqref{eq:divevaldiff}. Part (c) is a way to fix the sign indeterminacy in the identification of the factors. Once $\\Delta\\bm f_t$ and $\\bm{\\mathcal B}_{0n}$ are identified, then the remaining loadings are obtained by projecting $\\Delta\\mathbf x_{nt}$ onto the lagged factors.\n\n\nThe identifying restrictions in Assumption \\ref{ass:identNS} are particularly useful for initializing the EM algorithm with the PC estimator (see the next section). However, it has to be stressed that this identification does not provide any economic meaning to the factors. In other words we are not interested here in giving any interpretation of the factors, but we are just interested in the common component, which is always identified.\n\n\n\\section{Estimation and asymptotic properties}\\label{sec:estNS}\nThroughout the rest of the section we assume to observe the $nT$-dimensional vector $\\bm X_{nT}=(\\mathbf x_{n1}^\\prime\\cdots\\mathbf x_{nT}^\\prime)^\\prime$ satisfying \\eqref{eq:NSDFM1}-\\eqref{eq:NSDFM4}. In order to derive an estimator of the common component, we need to estimate the factors vector $\\bm f_T=(\\bm f_{1}^\\prime\\cdots \\bm f_T^\\prime)^\\prime$ and the vector containing the true values of all parameters is \n \\[\n \\bm\\varphi_n=\\l(\\text{vec}(\\bm{\\mathcal B}_{0n}\\cdots \\bm{\\mathcal B}_{sn})^\\prime, \\text{vech}(\\bm\\Gamma_n^e)^\\prime, \\rho_1,\\ldots, \\rho_{n_1} ,\\text{vec}(\\bm{\\mathcal A}_1\\cdots \\bm{\\mathcal A}_p)^\\prime, \\text{vech}(\\bm\\Gamma^u)^\\prime,\\sigma_{1\\omega}^{2}\\cdots\\sigma^{2}_{n_a\\omega},\\sigma_{1\\eta}^{2}\\cdots\\sigma^{2}_{n_b\\eta}\\r)^\\prime, \n \\]\n where, without loss of generality, we assumed that $\\mathcal I_1=\\{1,\\ldots, n_1\\}$,  $\\mathcal I_a=\\{1,\\ldots, n_a\\}$, and  $\\mathcal I_b=\\{1,\\ldots, n_b\\}$.\n\nIn this Section, we provide asymptotic results when $n_1=0$, $n_a=0$, and $n_b=0$, thus assuming that all idiosyncratic component are stationary and that no time-varying term is present. At first sight this might seem as a strong requirement, but notice that in our framework introducing non-stationary idiosyncratic components and\/or local levels and\/or local linear trends implies just adding latent states. We discuss this extension in Section \\ref{sec:idioI1}. Moreover, in \\ref{app:prestNS}, we give all details of the EM algorithm together with explicit expressions for all estimators in the general case. \n\nWithout loss of generality, we fix $s=1,$ and we fix the VAR order in \\eqref{eq:NSDFM2} to $p=2$, thus $\\bm{\\mathcal A}(L)\\equiv(\\bm{\\mathcal A}_1 L+\\bm{\\mathcal A}_2L^2)$, and, in this way the stationary component of $\\bm f_t$ follows a non-trivial dynamics. For simplicity, we also assume that $\\alpha_{i0}=0$ and $\\beta_{i0}=0$. \n\nThe EM algorithm is an iterative procedure, which starts with an initial value of the parameters $\\widehat{\\bm\\varphi}_n^{(0)}$, and at each iteration $k\\ge 0$ produces estimates of the factors $\\bm f_{t|T}^{(k)}$ (KS and E-step) and of the parameters $\\widehat{\\bm\\varphi}_n^{(k+1)}$ (M-step).  When the EM algorithm converges, say at iteration $k^*$, it gives the estimated common component $\\widehat{\\chi}_{it}=\\widehat{\\bm b}_{i0}^{(k^*+1)\\prime}\\bm f_{t|T}^{(k^*+1)}+\\widehat{\\bm b}_{i1}^{(k^*+1)\\prime}\\bm f_{t-1|T}^{(k^*+1)}$. \n\nMore in detail, the NS-DFM in \\eqref{eq:NSDFM1}-\\eqref{eq:NSDFM2} can be written as\n\\begin{align}\n\\mathbf x_{nt}&=\\l(\\bm{\\mathcal B}_{0n}\\ \\bm{\\mathcal B}_{1n}\\r)\n\\l(\\begin{array}{c}\n\\bm f_t\\\\\n\\bm f_{t-1}\n\\end{array}\n\\r)+\\mathbf e_{nt},\\label{eq:SSNS001}\\\\\n\\l(\\begin{array}{c}\n\\bm f_t\\\\\n\\bm f_{t-1}\n\\end{array}\n\\r)&=\\l(\\begin{array}{cc}\n\\bm{\\mathcal A}_1&\\bm{\\mathcal A}_2\\\\\n\\mathbf I_q&\\mathbf 0_{q\\times q}\n\\end{array}\n\\r)\n\\l(\\begin{array}{c}\n\\bm f_{t-1}\\\\\n\\bm f_{t-2}\n\\end{array}\n\\r)\n+\\l(\\begin{array}{c}\n\\bm u_t\\\\\n\\mathbf 0_q\n\\end{array}\n\\r),\\label{eq:SSNS002}\n\\end{align} \nBy defining  $\\mathbf F_t=(\\bm f_t^\\prime\\; \\bm f_{t-1}^\\prime)^\\prime$ and $\\bm\\lambda_i=(\\bm{b}_{0i}^\\prime\\;\\bm{b}_{1i}^\\prime)^\\prime$, we see that, for given values of the parameters $\\widehat{\\bm\\varphi}_n^{(k)}$, we can easily estimate the factors via the KS applied to the state-space form in \\eqref{eq:SSNS001}-\\eqref{eq:SSNS002}. The estimated states are then $\\mathbf F_{t|T}^{(k)}=\\E_{\\widehat{\\varphi}_n^{(k)}}[\\mathbf F_t|\\bm X_{nT}]$, the first $q$-components of which give $\\bm f_{t|T}^{(k)}=\\E_{\\widehat{\\varphi}_n^{(k)}}[\\bm f_t|\\bm X_{nT}]$. Then, using the output of the KS, we can compute the expected log-likelihood, which is maximized by the loadings estimator $\\widehat{\\bm \\lambda}_{i}^{(k+1)}\\equiv(\\widehat{\\bm b}_{i0}^{(k+1)\\prime} \\; \\widehat{\\bm b}_{i1}^{(k+1)\\prime})^\\prime$, such that\n\\begin{align}\n\\widehat{\\bm \\lambda}_{i}^{(k+1)}\\!=\n\\l\\{\\sum_{t=1}^T\\E_{\\widehat{\\varphi}_n^{(k)}}\\!\\!\\l[\\l(\\begin{array}{ll}\n\\bm f_t\\bm f_t^\\prime&\\bm f_t\\bm f_{t-1}^\\prime\\\\\n\\bm f_{t-1}\\bm f_t^\\prime&\\bm f_{t-1}\\bm f_{t-1}^\\prime\n\\end{array}\\r)\\!\\!\\bigg\\vert \\bm X_{nT}\n\\r]\\r\\}^{\\!\\!-1}\\!\\!\\!\n\\l\\{\\sum_{t=1}^T\\E_{\\widehat{\\varphi}_n^{(k)}}\\!\\!\\l[\\l(\\begin{array}{l}\n\\bm f_tx_{it}\\\\\n\\bm f_{t-1}x_{it}\n\\end{array}\\r)\\!\\!\\bigg\\vert \\bm X_{nT}\n\\r]\\r\\}.\\label{eq:param1NS}\n\\end{align}\n\nThe initial value of the parameters $\\widehat{\\bm\\varphi}_n^{(0)}$ is determined as follows. For the loadings and the factors we use the approach proposed in \\citet{BLL2}, which makes use of the $q$ leading PCs of the model in first differences. Two comments are worth making. \nFirst, it important to stress that initializing the model in first differences (including when determining $q$ and $s$) is crucial, since it allows us to use PCs without incurring in spurious effects due to the presence of idiosyncratic unit roots (\\citealp{OW19}), or  linear trends (\\citealp{ngCG}). Second, in light of the previous comment, this approach provides consistent estimates of the loadings, even in the case in which Assumption \\ref{ass:I1} is satisfied with $n_1>0$ and $n_b>0$, but for constant intercepts and trend slopes \\citep[see also][in the case of no linear trends]{baing04}. In particular, our initialization delivers estimates of $\\alpha_{i0}$ and $\\beta_{i0}$, which, together with a given small initial value of the variances $\\widehat{\\sigma}_{i\\omega}^{2(0)}$ and $\\widehat{\\sigma}_{i\\eta}^{2(0)}$,  can be used to update the slope state in \\eqref{eq:NSDFM4}. Notice that the pre-estimators of those initial conditions do not need to be consistent for our results to hold.\nThe initialization is completed by estimating the parameters of  \\eqref{eq:NSDFM2} from an unrestricted VAR fitted on the estimated factors. This is a valid procedure when estimating an autoregressive model for cointegrated data (see \\citealp{simsstockwatson}). Consistency of the pre-estimators of the loadings and VAR coefficients is proved in \\citet[Lemma 3 and Proposition 2]{BLL2} (see also \\ref{app:consN}). \n\nFinally, notice also that we initialize the KF by setting the initial value of the covariance of the factors, ${\\mathbf P}_{0|0}$, to a very large value, as suggested by \\citet[Section 3.3.4, page 121]{harvey90}. \n\nConsistency of the estimated common component follows.\n\\newpage\n\\begin{prop}\\label{th:chiNS}\nUnder Assumptions \\ref{ass:dynamic}, \\ref{ass:modelNS}, \\ref{ass:identNS}, and if $\\mbox{rk}(\\bm\\Sigma_k)=q$ for all $k=0,\\ldots,s$, and $n_1=0$, $n_a=0$, and $n_b=0$,  as $n,T\\to\\infty$, for any given $i=1,\\ldots, n$ and $k=0,1$, $\\min(\\sqrt n,\\sqrt T)\\Vert \\widehat{\\bm b}_{ki}-\\bm b_{ki}\\Vert = O_p(1)$, \nand, for any given $t=\\bar t,\\ldots, T$,\n$\\min(\\sqrt n,\\sqrt T)\\Vert \\widehat{\\bm f}_{t|T}-\\bm f_{t}\\Vert= O_p(1)$. Moreover, $\\min(\\sqrt n,\\sqrt {T})\\,\\Vert \\widehat{\\chi}_{it}-{\\chi}_{it} \\Vert = O_p(1)$,\nfor any given $i=1,\\ldots,n$ and $t=\\bar t,\\ldots, T$, with $\\bar t\\ge 2$.\n\\end{prop}\n\n\n\nThe convergence rate depends on different ingredients. First, we show that the KS reaches a steady state within $\\bar t$ periods, where $\\bar t$ depends on the initial value $\\mathbf P_{0|0}$ and, as shown in Section \\ref{sec:mc2}, $\\bar t$ is typically very small.  Then, for the KS we show that,  given the true parameters, the factors are $\\sqrt N$-consistent. Third, given the true factors, the loadings estimator are consistent, with convergence rate $T$ for the loadings of the $I(1)$ components of the factors and convergence rate $\\sqrt T$ for the loadings of the stationary \ncomponent of the factors. As a result, for any given $i$, the whole loadings vector is $\\sqrt T$-consistent, unless $d=q$, in which case each all $q$ factors are random walks and then the loadings vector would be $T$-consistent. \n\nUnder the assumption $n_1=0$, $n_a=0$, and $n_b=0$, our model is equivalent to the model studied in \\citet{bai04}, who considers estimation by means of PCs in levels. In this respect, we notice that the rates in Proposition \\ref{th:chiNS} are very similar to those in  \\citet[Theorem 6 for the case $d<q$ and Theorem 4 for the case $d=q$]{bai04}. In other words, the QML estimator converges at the same rate than the estimator based on PC on the levels, which resembles the result in \\citet{BLqml} for the stationary case. However, in the simulation study in Section \\ref{sec:mc2} show that our estimator behave much better in finite samples.\n \n \n\n\n\n\n\\section{$I(1)$ idiosyncratic components, local levels, local linear trends}\\label{sec:idioI1}\n\nIf some idiosyncratic components are non-stationary, we can no longer use the EM algorithm described in the previous section. Indeed, when the residuals of \\eqref{eq:NSDFM1} are non-stationary, the M-step estimator of the loadings cannot be obtained by regressing $x_{it}$ is $I(1)$ onto  $\\bm f_t$ and $\\bm f_{t-1}$. However, the case in which some idiosyncratic components are $I(1)$ is the relevant one for large macroeconomic datasets, since otherwise all data would be cointegrated. This is, for example, shown by the empirical results in  \\citet{OGAP}, where the methodology proposed by \\citet{baing04} for testing for idiosyncratic unit roots is applied on a standard US macroeconomic dataset.\n\nIn this Section, we adapt the EM algorithm to model non-stationary idiosyncratic components as well as local levels and local linear trends. In particular, we borrow from the literature on nowcasting with stationary factor models which models autocorrelated idiosyncratic components by treating them as additional latent states (see, e.g., \\citealp{banburamodugno14}; and \\citealp{BGMR13}). \n\nLet us define $m=(n_1+n_a+n_b)$, as the number of additional latent states and recall that by Assumption \\ref{ass:I1}, $n^{-1}m\\to 0$, as $n\\to\\infty$. Define also the set $\\mathcal I_{m}=\\mathcal I_1\\cup\\mathcal I_a\\cup\\mathcal I_b$, and notice that $\\#\\mathcal I_m\\le m$, since it is possible that a variable has both a non-stationary idiosyncratic component as well as, for example, a linear trend. Then for all $i\\in\\mathcal I_{m}$,  we replace the measurement equation \\eqref{eq:NSDFM1} with\n\\begin{align}\nx_{it}&=\\alpha_{it}+\\beta_{it}t + \\bm b_i^\\prime(L) \\bm f_t + \\xi_{it} + \\nu_{it},\\label{eq:NSDFM1bis}\n\\end{align}\nsuch that Assumption \\ref{ass:dynamic} still hold, and, moreover, letting $\\bm\\nu_{mt}=(\\nu_{1t}\\cdots\\nu_{mt})^\\prime$, for all $t\\in\\mathbb Z$, we assume $\\bm\\nu_{mt}\\sim\\mathcal N(\\mathbf 0_m,\\phi\\mathbf I_m)$, with $\\phi>0$, and $\\E_{\\varphi_n}[\\bm\\nu_{mt}\\bm\\nu_{mt-k}] = 0_{m\\times m}$ for all $k\\ne 0$. If $i\\notin\\mathcal I_m$, then \\eqref{eq:NSDFM1} stays the same. Moreover, we leave the dynamics of the factors in \\eqref{eq:NSDFM2} unchanged, while we change \\eqref{eq:NSDFM3} to\n\\begin{align}\n\\xi_{it}&=\\xi_{it-1} + e_{it},\\;\\text{ if }\\; i\\in \\mathcal I_1,\\;\\text{ and }\\; \\xi_{it}= e_{it},\\;\\text{ if }\\; i\\notin \\mathcal I_1,\\label{eq:NSDFM3bis}\n\\end{align}\nwhere Assumptions \\ref{ass:modelNS}(b) and  \\ref{ass:modelNS}(c) still hold, and $\\E_{\\varphi_n}[\\nu_{it} e_{js}]=0$, for all $t,s\\in\\mathbb Z$, all $i\\in\\mathcal I_m$ and all $j=1,\\ldots, n$.\nFinally, according to \\eqref{eq:NSDFM5} and \\eqref{eq:NSDFM4}, we have the state equations\n\\begin{align}\n&\\alpha_{it}=\\alpha_{it-1} + \\omega_{it},\\;\\text{ if }\\; i\\in \\mathcal I_a,\\label{eq:NSDFM5bis}\\\\\n&\\beta_{it}=\\beta_{it-1} + \\eta_{it},\\;\\text{ if }\\; i\\in \\mathcal I_b,\\label{eq:NSDFM4bis}\n\\end{align}\nsuch that $\\E_{\\varphi_n}[\\nu_{it} \\omega_{js}]=0$, and $\\E_{\\varphi_n}[\\nu_{it} \\eta_{js}]=0$, for all $t,s\\in\\mathbb Z$, all $i\\in\\mathcal I_m$, and  all $j\\in\\mathcal I_a$ or $j\\in\\mathcal I_b$.  \n\nThe model, which has as measurement equation either \\eqref{eq:NSDFM1} or \\eqref{eq:NSDFM1bis} if $i\\in\\mathcal I_m$, and which has as state equations  \\eqref{eq:NSDFM2}, and, if needed, also equations \\eqref{eq:NSDFM3bis}, \\eqref{eq:NSDFM5bis} and \\eqref{eq:NSDFM4bis}, has a compact state space form which is given in  \\ref{app:prestNS}, together with the details on its estimation via the EM algorithm. In particular, letting $w_{it}=\\alpha_{it}+\\beta_{it}t+\\xi_{it}$, for all $i\\in\\mathcal I_m$ we show that, at a given iteration $k\\ge 0$ of the EM algorithm, the M-step gives the loadings estimators:\n\\begin{align}\n\\widehat{\\bm \\lambda}_{i}^{(k+1)}\\!=\n\\l\\{\\sum_{t=1}^T\\E_{\\widehat{\\varphi}_n^{(k)}}\\!\\!\\l[\\l(\\begin{array}{ll}\n\\bm f_t\\bm f_t^\\prime&\\bm f_t\\bm f_{t-1}^\\prime\\\\\n\\bm f_{t-1}\\bm f_t^\\prime&\\bm f_{t-1}\\bm f_{t-1}^\\prime\n\\end{array}\\r)\\!\\!\\bigg\\vert \\bm X_{nT}\n\\r]\\r\\}^{\\!\\!-1}\\!\\!\\!\n\\l\\{\\sum_{t=1}^T\\E_{\\widehat{\\varphi}_n^{(k)}}\\!\\!\\l[\\l(\\begin{array}{l}\n\\bm f_t(x_{it}-w_{it})\\\\\n\\bm f_{t-1}(x_{it}-w_{it})\n\\end{array}\\r)\\!\\!\\bigg\\vert \\bm X_{nT},\n\\r]\\r\\}.\\nonumber\n\\end{align}\nwhere $\\bm\\lambda_i=(\\bm{b}_{0i}^\\prime\\;\\bm{b}_{1i}^\\prime)^\\prime$, while for $i\\notin\\mathcal I_m$ the loadings estimator is the same as in \\eqref{eq:param1NS}. Formulas for all other estimators are given in \\ref{app:prestNS}. In order to be able to compute $\\widehat{\\bm \\lambda}_{i}^{(k+1)}$, we have to estimate the $m$ additional latent states $w_{it}$ and therefore we also need modify the KS accordingly (see \\ref{app:prestNS} for details). \n\nIn \\ref{app:misidioNS}, we provide an overview of the challenges involved by this task and we provide an informal derivation of the conditions necessary for consistent estimation, together with the related convergence rates.  Three main results emerge. First, the new latent states can be recovered only if they display also some degree of cross-sectional correlation, as if they were driven by some common factor which is weakly pervasive for the whole panel. The intuition is that, if the additional latent states are completely uncorrelated across the components of $\\mathbf x_{nt}$, then pooling many series does not help in recovering them, since their effect is always dominated by the factors. \n\nSecond, when the previous condition is verified, then we can still achieve $\\sqrt n$-consistency for the estimated factors (as in the proof of Proposition \\ref{th:chiNS}), regardless of $m$, but provided that the variance of the measurement error $\\nu_{it}$ in \\eqref{eq:NSDFM1bis} is fixed in such a way that $\\phi=o(n^{-1})$, that is, it is asymptotically negligible. Indeed, the presence of $\\nu_{it}$ represents a mis-specification of the original model in \\eqref{eq:NSDFM1}, which needs to be introduced only as a numerical device, since the KF is not be defined if $\\phi=0$. The smaller is $\\phi$, the smaller the effect of the mis-specification is, and, therefore, the estimation of the factors is unaffected by the additional states. \n\nAs a consequence of this result, our estimator converges at a faster rate than those proposed by \\citet{baing04} and \\citet{BLL2}, which are based on PC analysis on the differenced data. This faster convergence rate comes from the fact that we distinguish \\textit{a priori} between $I(1)$ and stationary idiosyncratic components. By contrast, due to differencing the estimator of \\citet{baing04} and \\citet{BLL2} essentially treat all idiosyncratic components as if they were $I(1)$. Of course, for the implementation of our estimator, it is crucial to be able to determine consistently which idiosyncratic component is $I(1)$---for example, using the test for idiosyncratic unit roots proposed by \\citet{baing04}.\n\nThird, to achieve consistency of the additional latent states a necessary condition is $mn^{-1}\\to 0$. This reflects the obvious intuition that the more latent states we need to estimate, the worse the performance of our estimator is going to be. Moreover, $\\sqrt n$-consistency for the new states can be obtained for any $m$, but only if we choose an even smaller value of $\\phi$, namely $\\phi=o((m\\sqrt n)^{-1})$. \n \nWe conclude with three remarks. First, the requirement that the new latent states display some degree of cross-sectional correlation is perfectly in line with Assumption \\ref{ass:modelNS}(c) according to which the idiosyncratic components can be cross-correlated. Moreover, we can relax Assumptions \\ref{ass:modelNS}(e) and \\ref{ass:modelNS}(g) to allow for some correlation across the innovations $e_{it}$, $\\omega_{it}$, and $\\eta_{it}$ in \\eqref{eq:NSDFM3bis}, \\eqref{eq:NSDFM5bis} and \\eqref{eq:NSDFM4bis}. Indeed, it is reasonable to assume that local linear trends are shared by real variables (e.g., GDP and GDI), or that local levels are more apt to capture time-varying mean of groups of variables belonging, for example, to the labor market. Nevertheless, as shown in the proof of Proposition \\ref{th:chiNS}, the fact that we estimate the $I(1)$ idiosyncratic components without modeling the cross-correlation between their innovations, will add miss-specification to our model, but will not affect the consistency of our estimates. \n\nSecond, as a far as estimation of the parameters given estimates of the states is concerned, we conjecture that nothing changes with respect to the results used in the proof of Proposition \\ref{th:chiNS}, provided the states estimators are $\\sqrt n$-consistent. Third, since the above are just  asymptotic arguments, the choice of $\\phi$ is not straightforward. A common way to proceed consists in initializing $\\phi$ to be very small for all $m$ additional states and then update its estimate at each iteration of the EM algorithm, thus adding $m$ additional parameters. This is the way we implement the EM algorithm in the next section (see also \\ref{app:prestNS}).\n\n\n\n\n\n\n\n\n\\section{MonteCarlo results}\\label{sec:mc2}\nThroughout, we let $n\\in\\{75,100,200,300\\}$, $T\\in\\{75,100,200,300\\}$, $q\\in\\{2,4\\}$, and $s\\in\\{0,1\\}$, and we simulate data according to \\eqref{eq:NSDFM1}, \\eqref{eq:NSDFM2}, \\eqref{eq:NSDFM3}, and \\eqref{eq:NSDFM4} as follows.\n\nFirst, the factor loadings are such that $[\\bm{\\mathcal B}_{kn}]_{ij} \\sim N(1,1)$ for $k=0,\\ldots, s$, and then if $s=1$, for all $j=1,\\ldots q$, we take $n\/2$ randomly selected elements of $[\\bm{\\mathcal B}_{1n}]_{\\cdot j}$ and we set them to zero. Second, for the common factors we set the VAR order $p=2$, and to generate $\\bm{\\mathcal A}(L)$ we use the Smith-McMillan factorization according to which $\\bm{\\mathcal A}(L)=\\mathbfcal{U}(L) \\mathbfcal{M}(L) \\mathbfcal{V}(L)$, where $\\mathbfcal{M}(L)= \\mbox{diag} \\left( (1-L)\\mathbf I_{q-d}, \\mathbf I_d\\right)$, $\\mathbfcal{V}(L)=\\mathbf I_q$, and $\\mathbfcal{U}(L)=(\\mathbf I_q-\\mathbfcal{U}_1 L)$, where $\\mathbfcal{U}_1=\\mu\\,\\widetilde{\\mathbfcal{U}}_1(\\nu^{(1)}(\\widetilde{\\mathbfcal{U}}_1))^{-1}$, where the diagonal elements of $\\widetilde{\\mathbfcal{U}}_1$ are drawn from a uniform distribution on $[0.5,0.8]$, while the off-diagonal elements from a uniform distribution on $[0,0.3]$, and $\\mu=0.5$. In this way, $\\bm f_t$ follows a VAR(2) with $q-d$ unit roots, or, equivalently, a VECM(1), where the number cointegration relations is set to $d=1$. The common innovations are such that $\\bm u_t\\stackrel{iid}{\\sim} \\mathcal{N}(\\mathbf 0_q,\\mathbf I_q)$, or $\\bm u_t\\stackrel{iid}{\\sim} t_4(\\mathbf 0_q,\\mathbf I_q)$.\n\nThird, each idiosyncratic component follows an AR(2) with roots $\\rho_{i1}$ and $\\rho_{i2}$, such that $\\rho_{i1}=1$ if $\\xi_{it}\\sim I(1)$, while $\\rho_{i1}=0$ otherwise, and $\\rho_{i2}$ is drawn from a uniform distribution on $[0.2,0.6]$. We randomly select $n_1$ idiosyncratic components to have a unit root, with $n_1\\in\\{0,25,50,75,100\\}$, provided $n_1<n$. The innovations are such that $\\mathbf e_t\\stackrel{iid}{\\sim} \\mathcal{N}(\\mathbf 0_n,\\bm\\Gamma^e_n)$, or $\\mathbf e_t\\stackrel{iid}{\\sim} t_4(\\mathbf 0_n,\\bm\\Gamma^e_n)$\n with $[\\bm\\Gamma^e_n]_{ij}=\\tau^{|i-j|}$ if $\\tau>0$, while, if $\\tau=0$, $\\bm\\Gamma^e_n$ is diagonal with entries drawn from a uniform distribution on $[0.5,1.5]$. We set $\\tau\\in\\{0,0.5\\}$.\n\nFourth, we randomly select $n_b$ variables to have a non-zero linear trend, with $n_b\\in\\{0,25,50,75,100\\}$, provided $n_b<n$. For those variables we draw $\\beta_{i0}$ from a uniform distribution on ${[0.3,0.5]}$, but we set $\\sigma_i^2=0$, thus considering only linear trends with constant slopes.\n\n\nLast, we rescale the first differences of each common and idiosyncratic component in such a way that the share of variance of the $i$-th variable explained by the common component is $\\theta(1+\\theta)^{-1}$. We set $\\theta=0.5$.\n\n\nWe consider $B=1000$ replications and we run the EM algorithm to estimate the NS-DFM by running the KS with $(q+n_1)$ latent states and estimating in the M-step only the diagonal terms of the idiosyncratic covariance matrix even when $\\tau>0$. Similarly we do not add idiosyncratic states even when $\\delta>0$. In other words, we always estimate a mis-specified model and, in this way, we are able to assess how robust our-estimators are with respect to mis-specifications. \n\nIn Table \\ref{tab:PttNS}, we report for different values of $n$ and for $t=1,\\ldots, 10$, the trace of the one-step-ahead, KF, and KS MSEs when $q=2$, $s=1$, $T=100$, $\\tau=0.5$, and $\\delta=0.2$ (serially and cross-correlated idiosyncratic components). The MSEs are computed using the true simulated value of the parameters in order to verify numerically convergence to the steady-state. First, as $n$ grows, the one-step-ahead MSE reaches a steady state within maximum five time periods and $\\mbox{tr}(\\mathbf P_{t|t-1})\/q\\simeq 1$. This is consistent with the fact that due to the presence of unit roots we inizialize the filter with a vary large value of $\\mathbf P_{0|0}$. Second, the KF and KS MSEs are very similar and both decrease to zero as $n$ grows and $\\mbox{tr}(\\mathbf P_{t|t})n\/q$ and $\\mbox{tr}(\\mathbf P_{t|T})n\/q$, computed when $t=10$, stabilize as $n$ grows thus showing that the rate of decrease is $n$.\n\n\\begin{table}[t!]\n\\setlength{\\tabcolsep}{0\\textwidth}\n\\caption{Simulation results NS-DFM}\\label{tab:PttNS} \\centering \\smallskip\n\n\\textsc{Kalman filter and Kalman smoother MSEs} \\smallskip\n\n\\scriptsize\n\\vskip .2cm\n\\begin{tabular}{L{.04\\textwidth} L{.08\\textwidth} | C{.11\\textwidth}C{.11\\textwidth}C{.11\\textwidth}C{.1\\textwidth}C{.11\\textwidth}C{.11\\textwidth}C{.11\\textwidth}C{.11\\textwidth}}\n\\multicolumn{10}{c}{Serially and cross-correlated idiosyncratic ($\\tau=0.5$, $\\delta=0.2$),  $n_1=0$, $n_b=0$, $q=2$, $s=1$}\\\\\n\\multicolumn{10}{c}{Gaussian innovations}\\\\[-4pt]\n\\\\\n\\hline\n\\hline\n&&\\\\[-4pt]\n \t&$n$&\t$5$\t&\t$10$\t&\t$25$\t&\t$50$\t&\t$75$\t&\t$100$\t&\t$200$\t&\t$300$\t\\\\[4pt]\n\\hline\n&&\\\\[-4pt]\n\\multicolumn{2}{l}{$\\mbox{tr}(\\mathbf P_{0|0})\/q$}\t\\vline \n\t&\t147.6161\t&\t154.1094\t&\t133.2495\t&\t152.7333\t&\t138.4666\t&\t109.5626\t&\t120.8897\t&\t134.7097\t\\\\[4pt]\n\\hline\n&&\\\\[-4pt]\n&$t=1$\t&\t2.874610\t&\t1.601115\t&\t1.132925\t&\t1.076572\t&\t0.981511\t&\t1.020192\t&\t0.947337\t&\t0.980840\t\\\\\n&$t=2$\t&\t2.768656\t&\t1.543904\t&\t1.110419\t&\t1.064744\t&\t0.971928\t&\t1.014110\t&\t0.945185\t&\t0.979096\t\\\\\n\\multirow{4}{*}{\\rotatebox{90}{$\\mbox{tr}(\\mathbf P_{t|t-1})\/q$}}\n&\t$t=3$\t&\t2.748894\t&\t1.537389\t&\t1.109305\t&\t1.064269\t&\t0.970874\t&\t1.013479\t&\t0.945034\t&\t0.978928\t\\\\\n&\t$t=4$\t&\t2.746597\t&\t1.536414\t&\t1.109239\t&\t1.064245\t&\t0.970735\t&\t1.013398\t&\t0.945022\t&\t0.978908\t\\\\\n&\t$t=5$\t&\t2.746357\t&\t1.536248\t&\t1.109235\t&\t1.064244\t&\t0.970716\t&\t1.013387\t&\t0.945021\t&\t0.978906\t\\\\\n&\t$t=6$\t&\t2.746329\t&\t1.536220\t&\t1.109235\t&\t1.064244\t&\t0.970714\t&\t1.013386\t&\t0.945020\t&\t0.978906\t\\\\\n&\t$t=7$\t&\t2.746325\t&\t1.536215\t&\t1.109235\t&\t1.064244\t&\t0.970713\t&\t1.013385\t&\t0.945020\t&\t0.978906\t\\\\\n&\t$t=8$\t&\t2.746324\t&\t1.536214\t&\t1.109235\t&\t1.064244\t&\t0.970713\t&\t1.013385\t&\t0.945020\t&\t0.978906\t\\\\\n&\t$t=9$\t&\t2.746324\t&\t1.536214\t&\t1.109235\t&\t1.064244\t&\t0.970713\t&\t1.013385\t&\t0.945020\t&\t0.978906\t\\\\\n&\t$t=10$\t&\t2.746324\t&\t1.536214\t&\t1.109235\t&\t1.064244\t&\t0.970713\t&\t1.013385\t&\t0.945020\t&\t0.978906\t\\\\\n[4pt]\n\\hline\n&&\\\\[-4pt]\n&\t$t=1$\t&\t1.665780\t&\t0.548360\t&\t0.161240\t&\t0.107121\t&\t0.049408\t&\t0.036570\t&\t0.015982\t&\t0.011897\t\\\\\n&\t$t=2$\t&\t1.386910\t&\t0.400762\t&\t0.116701\t&\t0.079362\t&\t0.037964\t&\t0.030533\t&\t0.011515\t&\t0.007687\t\\\\\n\\multirow{4}{*}{\\rotatebox{90}{$\\mbox{tr}(\\mathbf P_{t|t})\/q$}}\n&\t$t=3$\t&\t1.369813\t&\t0.393790\t&\t0.114352\t&\t0.078298\t&\t0.037113\t&\t0.030087\t&\t0.011295\t&\t0.007466\t\\\\\n&\t$t=4$\t&\t1.366557\t&\t0.392627\t&\t0.114174\t&\t0.078221\t&\t0.037001\t&\t0.030029\t&\t0.011276\t&\t0.007446\t\\\\\n&\t$t=5$\t&\t1.365967\t&\t0.392442\t&\t0.114163\t&\t0.078216\t&\t0.036985\t&\t0.030021\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=6$\t&\t1.365898\t&\t0.392411\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=7$\t&\t1.365890\t&\t0.392406\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=8$\t&\t1.365889\t&\t0.392405\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=9$\t&\t1.365889\t&\t0.392404\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=10$\t&\t1.365889\t&\t0.392404\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n[4pt]\n\\hline\n&&\\\\[-4pt]\n\\multicolumn{2}{l}{$\\mbox{tr}(\\mathbf P_{10|10})n\/q$}\t\\vline &\t3.414722\t&\t1.962022\t&\t1.427032\t&\t1.955403\t&\t1.386863\t&\t1.501011\t&\t1.127376\t&\t1.116435\t\\\\[4pt]\n\\hline\n&&\\\\[-4pt]\n&\t$t=1$\t&\t0.938577\t&\t0.368873\t&\t0.110011\t&\t0.074161\t&\t0.028644\t&\t0.017577\t&\t0.010943\t&\t0.008560\t\\\\\n&\t$t=2$\t&\t0.708279\t&\t0.253087\t&\t0.073988\t&\t0.051268\t&\t0.021393\t&\t0.014262\t&\t0.007372\t&\t0.005221\t\\\\\n\\multirow{4}{*}{\\rotatebox{90}{$\\mbox{tr}(\\mathbf P_{t|T})\/q$}}\n&\t$t=3$\t&\t0.697724\t&\t0.250786\t&\t0.072065\t&\t0.050450\t&\t0.020933\t&\t0.014096\t&\t0.007206\t&\t0.005061\t\\\\\n&\t$t=4$\t&\t0.696501\t&\t0.250387\t&\t0.071913\t&\t0.050388\t&\t0.020872\t&\t0.014075\t&\t0.007192\t&\t0.005046\t\\\\\n&\t$t=5$\t&\t0.696180\t&\t0.250328\t&\t0.071903\t&\t0.050384\t&\t0.020864\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=6$\t&\t0.696142\t&\t0.250318\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=7$\t&\t0.696138\t&\t0.250317\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=8$\t&\t0.696137\t&\t0.250316\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=9$\t&\t0.696137\t&\t0.250316\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=10$\t&\t0.696137\t&\t0.250316\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n[4pt]\n\\hline\n&&\\\\[-4pt]\t\t\t\t\t\t\t\t\t\t\t\n\\multicolumn{2}{l}{$\\mbox{tr}(\\mathbf P_{10|T})n\/q$}\t\\vline &\t1.740343\t&\t1.251582\t&\t0.898783\t&\t1.259595\t&\t0.782350\t&\t0.703592\t&\t0.719033\t&\t0.756698\t\\\\[4pt]\n\\hline\n\\hline\n\n\\end{tabular}\n\\end{table}\n\nIn Table \\ref{tab:mc1NS} and in Table \\ref{tab:mc1NSt}, we report the relative MSE of our estimator over the MSE of the common component estimators obtained by PC as in \\citet{bai04}, and by PC in first differences as in \\citet{baing04} and \\citet{BLL2}. Overall our estimator outperforms the others with the exception of the latter, which is show to perform better when $n_1$ becomes very large and about the same order of magnitude as $n$. This reflects the additional computational burden of our estimator which requires increasing the number of latent states when the idiosyncratic components are non-stationary and therefore we must include their dynamics in the model.\n\n\\begin{table}[ht!]\n\\setlength{\\tabcolsep}{0\\textwidth}\n\\caption{Simulation results - Common components}\\label{tab:mc1NS}\n\\centering\n\\small \\textsc{Relative Mean Squared Errors} \\smallskip\n\n\\footnotesize\n\\begin{tabular}{C{.085\\textwidth}C{.085\\textwidth}C{.085\\textwidth}C{.085\\textwidth} | C{.11\\textwidth} C{.11\\textwidth} C{.11\\textwidth} | C{.11\\textwidth} C{.11\\textwidth} C{.11\\textwidth} }\n\\multicolumn{10}{c}{Serially and cross correlated idiosyncratic components ($\\tau=0.5$, $\\delta =0.2$). Gaussian innovations}\\\\\n\\hline\n\\hline\n&&&&\\multicolumn{3}{c|}{$q=2$, $s=0$}&\\multicolumn{3}{c}{$q=2$, $s=1$}\\\\\\hline\n$n$ & $T$ & $n_1$& $n_b$& \\tiny{B}& \\tiny{BN}& \\tiny{BLL}& \\tiny{B}& \\tiny{BN}& \\tiny{BLL}\\\\\n\\hline\n75\t&\t75\t&\t0\t&\t0\t&\t0.58\t&\t0.00\t&\t0.28\t&\t0.54\t&\t0.01\t&\t0.53\t\\\\\n100\t&\t100\t&\t0\t&\t0\t&\t0.54\t&\t0.00\t&\t0.22\t&\t0.54\t&\t0.00\t&\t0.49\t\\\\\n200\t&\t200\t&\t0\t&\t0\t&\t0.45\t&\t0.00\t&\t0.12\t&\t0.59\t&\t0.00\t&\t0.39\t\\\\\n300\t&\t300\t&\t0\t&\t0\t&\t0.40\t&\t0.00\t&\t0.08\t&\t0.63\t&\t0.00\t&\t0.32\t\\\\\n\\hline\n75\t&\t75\t&\t25\t&\t25\t&\t0.01\t&\t0.02\t&\t0.44\t&\t0.04\t&\t0.04\t&\t0.83\t\\\\\n100\t&\t100\t&\t25\t&\t25\t&\t0.01\t&\t0.01\t&\t0.47\t&\t0.02\t&\t0.02\t&\t0.68\t\\\\\n200\t&\t200\t&\t25\t&\t25\t&\t0.00\t&\t0.00\t&\t0.53\t&\t0.01\t&\t0.01\t&\t0.66\t\\\\\n300\t&\t300\t&\t25\t&\t25\t&\t0.00\t&\t0.00\t&\t0.77\t&\t0.00\t&\t0.00\t&\t0.73\t\\\\\n\\hline\n75\t&\t75\t&\t50\t&\t50\t&\t0.05\t&\t0.21\t&\t1.55\t&\t0.11\t&\t0.23\t&\t1.47\t\\\\\n100\t&\t100\t&\t50\t&\t50\t&\t0.02\t&\t0.12\t&\t1.45\t&\t0.06\t&\t0.12\t&\t1.53\t\\\\\n200\t&\t200\t&\t50\t&\t50\t&\t0.00\t&\t0.02\t&\t0.82\t&\t0.01\t&\t0.02\t&\t0.75\t\\\\\n300\t&\t300\t&\t50\t&\t50\t&\t0.00\t&\t0.01\t&\t0.93\t&\t0.00\t&\t0.01\t&\t0.80\t\\\\\n\\hline\n100\t&\t100\t&\t75\t&\t75\t&\t0.03\t&\t0.23\t&\t1.44\t&\t0.08\t&\t0.23\t&\t1.48\t\\\\\n200\t&\t200\t&\t75\t&\t75\t&\t0.00\t&\t0.03\t&\t0.80\t&\t0.02\t&\t0.06\t&\t1.52\t\\\\\n300\t&\t300\t&\t75\t&\t75\t&\t0.00\t&\t0.02\t&\t1.11\t&\t0.01\t&\t0.02\t&\t0.92\t\\\\\n\\hline\n200\t&\t200\t&\t100\t&\t100\t&\t0.01\t&\t0.12\t&\t1.87\t&\t0.04\t&\t0.15\t&\t2.08\t\\\\\n300\t&\t300\t&\t100\t&\t100\t&\t0.00\t&\t0.03\t&\t1.08\t&\t0.01\t&\t0.04\t&\t1.45\t\\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\begin{tabular}{p{\\textwidth}}\\tiny\nThis table reports relative MSEs of the QML estimator proposed in this paper over the MSE of the common component estimators obtained by PC as in \\citet{bai04} (B), and by PC in first differences as in \\citet{baing04} (BN) and \\citet{BLL2} (BLL).\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}[t!]\n\\setlength{\\tabcolsep}{0\\textwidth}\n\\caption{Simulation results NS-DFM - Common components}\\label{tab:mc1NSt}\n\\centering\n\\small \\textsc{Relative Mean Squared Errors} \\smallskip\n\n\\footnotesize\n\\begin{tabular}{C{.085\\textwidth}C{.085\\textwidth}C{.085\\textwidth}C{.085\\textwidth} | C{.11\\textwidth} C{.11\\textwidth} C{.11\\textwidth} | C{.11\\textwidth} C{.11\\textwidth} C{.11\\textwidth} }\n\\multicolumn{10}{c}{Serially and cross correlated idiosyncratic components ($\\tau=0.5$, $\\delta =0.2$). Student $t_4$ innovations}\\\\\n\\hline\n\\hline\n&&&&\\multicolumn{3}{c|}{$q=2$, $s=0$}&\\multicolumn{3}{c}{$q=2$, $s=1$}\\\\\n$n$ & $T$ & $n_1$& $n_b$&Rel-MSE&Rel-MSE&Rel-MSE&Rel-MSE&Rel-MSE&Rel-MSE\\\\\n&&&&\\tiny B&\\tiny BN&\\tiny BLL&\\tiny B&\\tiny BN&\\tiny BLL\\\\\n\\hline\n75\t&\t75\t&\t0\t&\t0\t&\t0.58\t&\t0.00\t&\t0.30\t&\t0.58\t&\t0.01\t&\t0.57\t\\\\\n100\t&\t100\t&\t0\t&\t0\t&\t0.55\t&\t0.00\t&\t0.25\t&\t0.56\t&\t0.00\t&\t0.53\t\\\\\n200\t&\t200\t&\t0\t&\t0\t&\t0.45\t&\t0.00\t&\t0.12\t&\t0.60\t&\t0.00\t&\t0.40\t\\\\\n300\t&\t300\t&\t0\t&\t0\t&\t0.41\t&\t0.00\t&\t0.09\t&\t0.67\t&\t0.00\t&\t0.35\t\\\\\n\\hline\n75\t&\t75\t&\t25\t&\t25\t&\t0.01\t&\t0.02\t&\t0.43\t&\t0.06\t&\t0.04\t&\t0.83\t\\\\\n100\t&\t100\t&\t25\t&\t25\t&\t0.01\t&\t0.01\t&\t0.39\t&\t0.03\t&\t0.02\t&\t0.70\t\\\\\n200\t&\t200\t&\t25\t&\t25\t&\t0.00\t&\t0.00\t&\t0.40\t&\t0.01\t&\t0.00\t&\t0.64\t\\\\\n300\t&\t300\t&\t25\t&\t25\t&\t0.00\t&\t0.00\t&\t0.56\t&\t0.01\t&\t0.00\t&\t0.68\t\\\\\n\\hline\n75\t&\t75\t&\t50\t&\t50\t&\t0.06\t&\t0.17\t&\t1.51\t&\t0.14\t&\t0.17\t&\t1.45\t\\\\\n100\t&\t100\t&\t50\t&\t50\t&\t0.04\t&\t0.11\t&\t1.57\t&\t0.09\t&\t0.10\t&\t1.46\t\\\\\n200\t&\t200\t&\t50\t&\t50\t&\t0.00\t&\t0.01\t&\t0.57\t&\t0.01\t&\t0.01\t&\t0.69\t\\\\\n300\t&\t300\t&\t50\t&\t50\t&\t0.00\t&\t0.01\t&\t0.63\t&\t0.01\t&\t0.01\t&\t0.71\t\\\\\n\\hline\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n100\t&\t100\t&\t75\t&\t75\t&\t0.04\t&\t0.18\t&\t1.36\t&\t0.12\t&\t0.21\t&\t1.48\t\\\\\n200\t&\t200\t&\t75\t&\t75\t&\t0.01\t&\t0.02\t&\t0.65\t&\t0.03\t&\t0.05\t&\t1.41\t\\\\\n300\t&\t300\t&\t75\t&\t75\t&\t0.00\t&\t0.01\t&\t0.71\t&\t0.01\t&\t0.01\t&\t0.81\t\\\\\n\\hline\n200\t&\t200\t&\t100\t&\t100\t&\t0.02\t&\t0.10\t&\t1.68\t&\t0.06\t&\t0.13\t&\t1.99\t\\\\\n300\t&\t300\t&\t100\t&\t100\t&\t0.00\t&\t0.02\t&\t0.72\t&\t0.02\t&\t0.04\t&\t1.39\t\\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\begin{tabular}{p{\\textwidth}}\\tiny\nThis table reports relative MSEs of the QML estimator proposed in this paper over the MSE of the common component estimators obtained by PC as in \\citet{bai04} (B), and by PC in first differences as in \\citet{baing04} (BN) and \\citet{BLL2} (BLL).\n\\end{tabular}\n\\end{table}\n\nSome clarifications on the competing methods considered are necessary in order to interpret the results in Table \\ref{tab:mc1NS} and in Table \\ref{tab:mc1NSt} (we refer to the original papers for details). First, notice that all alternative approaches considered here do not allow for dynamic loadings, so here they are implemented by computing the first $q(s+1)$ PCs. \n\nSecond, despite the common practice in the literature, \\citet{bai04} did not propose its approach for factor model estimation, but rather to estimate common trends, and it is based on the crucial assumption of all idiosyncratic components being stationary. Indeed, we see from Table \\ref{tab:mc1NS} that when $n_1>0$ this approach fails completely. \n\nThird, the \\citet{baing04} approach delivers estimates of the common component which are obtained \n\\begin{inparaenum}[($i$)]\n\t\\item by  detrending the data by estimating the slope of the trend with the mean of the data in first difference; then\n\t\\item by estimating the factors in first differences; and, finally,\n\t\\item by cumulating the differenced estimator to obtain an estimate of the levels. \n\\end{inparaenum} \nAs such, this estimator is always subject to a location shift---it can be shown to converge to a Brownian bridge. Notice that this approach was introduced to test for the presence of unit roots rather than for factor model estimation, and, while the test is unaffected by location shifts, the use of the cumulated estimator for other scopes is not justified in general. As we see from Table \\ref{tab:mc1NS}, this approach fails to consistently reconstruct the common component in all cases considered. \n\nFourth, the approach in \\citet{BLL2} is based on the same ideas of \\citet{baing04}, but it takes care of the above mentioned issues related to detrending and cumulation, and, therefore, it is a valid alternative.\n\n\n\\section{Concluding remarks}\\label{sec:conclusion}\n\nThis paper considers estimation of large non-stationary approximate dynamic factor models by means of the Expectation Maximization algorithm, implemented jointly with the Kalman smoother. In our model the factors are a cointegrated vector process, thus containing both common $I(1)$ trends and stationary (cyclical) components.  We show that, as the cross-sectional dimension $n$ and the sample size $T$ diverge to infinity, the common factors, the factor loadings, and the common component estimated are $\\min(\\sqrt n,\\sqrt T)$-consistent at each $i$ and $t$.\n\nFurthermore, we show that the model can be extended to account for the possible presence of idiosyncratic trends, as well as the presence of secular (linear) trends, which can have either a constant slope (deterministic linear trends) or a time-varying slope (local linear trends). Consistent estimation of this case is also considered. \n\nFinally, the results in this paper provides the theoretical background for the application considered in \\citet{OGAP}, where the NS-DFM  is used to estimate the output gap in the US.\n\n\\singlespacing\n{\\small{\n\\setlength{\\bibsep}{.2cm}\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThroughout this paper $R$ is a commutative noetherian ring, $\\frak a$ is an ideal of $R$. Given a Serre subcategory $\\cS$  of $R$-modules, an $R$-module $M$ is said to be $\\cS$-$\\frak a$-{\\it cofinite} if $\\Supp M\\subseteq V(\\frak a)$ and $\\Ext_R^i(R\/\\frak a, M)\\in\\cS$ for all integers $i\\geq 0$. Let $\\cN$ be the subcategory of finitely generated $R$-modules.\n\n The extension subcategory induced by $\\cN$ and $\\cS$ is denoted by $\\cN\\cS$, consisting of those $R$-modules $M$ for which there exist an exact sequence $0\\To N\\To M\\To S\\To 0$ such that $N\\in\\cN$ and $S\\in\\cS$. It has been proved by [Y] that $\\cN\\cS$ is Serre. A well-known example for these subcategories is $\\cN\\cA$, the subcategory of {\\it minimax modules} studied by [Z] where $\\cA$ is the subcategory of artinian modules. Another example is $\\cN\\cF$, the subcategory of FSF modules introduced by [Q], where $F$ consists of all modules of finite support. When $\\cS=0$, an $\\cN\\cS$-$\\frak a$-cofinite module was known as an $\\frak a$-cofinite module which is defined for the first time by Hartshorne [H], giving a negative answer to a question of [G, Expos XIII, Conjecture 1.1]. Many people [BNS, M1, M2, NS] studied $\\frak a$-cofiniteness in various cases. \n\nThe main aim of this paper is to extend the fundamental results about $\\frak a$-cofinite modules at small dimensions to $\\cN\\cS$-$\\frak a$ cofinite modules.\n We recall that a Serre subcategory $\\cS$ satisfies the condition $C_{\\frak a}$ if for every $R$-module $M$, the following implication holds.\n\\begin{center}\n$C_{\\frak a}$: If $\\Gamma_{\\frak a}(M)=M$ and $(0:_M{\\frak a})$ is in $\\cS$, then\n$M$ is in $\\cS$.\n\\end{center}\n\nFor every $\\frak p\\in\\Spec R$, we denote by $\\cS({\\frak p})$ the smallest Serre subcategory of $R_{\\frak p}$-modules containing $\\cS_{\\frak p}=\\{M_{\\frak p}|\\hspace{0.1cm} M\\in\\cS\\}$. For every $R$-module $M$, $\\dim M$ means the dimension of $\\Supp_RM$ which is the length of the longest chain of prime ideals in $\\Supp_RM$. In Section 2, we prove the following result. \n\\begin{Theorem}\nLet $(R,\\frak m)$ be a local ring, let $S(\\frak p)$ satisfy the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. If $M$ is an $R$-module of dimension $d$ such that $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for each $i\\leq d$, then $\\Ext_R^i(N,M)\\in\\cN\\cS$ for each $i\\geq 0$ and each finitely generated $R$-module $N$ with $\\dim N\\leq 2$ and $\\Supp_RN\\subseteq V(\\frak a)$.\n\\end{Theorem}\n Assume that $M$ is an $R$-module such that $\\Supp_R M\\subseteq V(\\frak a)$. Melkersson [M1, Theorem 2.3] showed that if $\\dim R\/\\frak a=1$, then $M$ is $\\frak a$-cofinite if and only if $\\Hom_R(R\/\\frak a,M)$ and $\\Ext_R^1(R\/\\frak a,M)$ are finitely generated. The above theorem generalizes this result when $R$ is a local ring and $S(\\frak p)$ satisfies the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. Indeed we deduce that if $\\dim R\/\\frak a=1$, then  $M$ is $\\cN\\cS$-$\\frak a$-cofinite if and only if $\\Hom_R(R\/\\frak a,M)$ and $\\Ext_R^1(R\/\\frak a,M)\\in\\cN\\cS$. Moreover, if $R$ is a local ring of dimension $2$ such that $\\cS(\\frak p)$ satisfies the condition $C_{\\frak pR_{\\frak p}}$ for every prime ideal $\\frak p$ of $R$ with $\\dim R\/\\frak p\\leq 1$, then $M$ is $\\cN\\cS$-$\\frak a$-cofinite if and only if $\\Hom_R(R\/\\frak a,M)$ and $\\Ext_R^1(R\/\\frak a,M)\\in\\cN\\cS$. \n\nBahmanpour et all [BNS, Theorem 3.5] showed that if $R$ is a local ring such that $\\dim R\/\\frak a=2$, then $M$ is $\\frak a$-cofinite if and only if $\\Ext_R^i(R\/\\frak a,M)$ are finitely generated for $i=0,1,2$. As another conclusion, we generalize this result when $S(\\frak p)$ satisfies the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. To be more prcise, we deduce that if $\\dim R\/\\frak a=2$, then $M$ is $\\cN\\cS$-$\\frak a$-cofinite if and only if $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for $i=0,1,2$. Moreover, if $R$ is a local ring of dimension $3$ such that $\\cS(\\frak p)$ satisfies the condition $\\frak pR_{\\frak p}$ for every prime ideal $\\frak p$ with $\\dim R\/\\frak p\\leq 2$, then $M$ is $\\cN\\cS$-$\\frak a$-cofinite if and only if $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for $i=0,1,2$. We prove the following result about local cohomology which generalizes [NS, Theorem 3.7]. For the basic properties and unexplained terminology of local cohomology, we refer the reader to the textbook by Brodmann and Sharp [BS]. \n\n\\begin{Theorem}\nLet $(R,\\frak m)$ be a local ring, let $\\frak a$ be an ideal of $R$ such that $\\dim R\/\\frak a=2$ and let\n$S(\\frak p)$ satisfy the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. If $n$ is a non-negative integer such that $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for all $i\\leq n+1$, then the following conditions are equivalent.\n \n{\\rm (i)} $H_{\\frak a}^i(M)$ is $\\cN\\cS$-$\\frak a$-cofinite for all $i<n$.\n\n{\\rm (ii)} $\\Hom_R(R\/\\frak a,H_{\\frak a}^i(M))\\in\\cN\\cS$ for all $i\\leq n$. \n\\end{Theorem}\n \nIn Section 3, we define a new dimension of modules which is an upper bound of the dimension mentioned in Section 3. For every $R$-module $M$, we define $\\overline{\\dim}M=\\dim R\/\\Ann_RM$. For every non-negative integer $n$, we denote by $\\cD_{\\leq n}$, the subcategory of all $R$-modules $M$ satisfying $\\overline{\\dim} M\\leq n$. It is proved that $\\cD_{\\leq n}$ is Serre for each $n\\geq 0$. We show that the subcategory $\\frak a$-cofinite modules in $\\cD_{\\leq 1}$ is Serre, the subcategory of $\\frak a$-cofinite modules in $\\cD_{\\leq 2}$ is abelian. Finally we show that the kernel and cokernel of a homomorphism $f:M\\To N$ of $\\frak a$-cofinite modules in $\\cD_{\\leq 3}$ is $\\frak a$-cofinite if and only if $(0:_{\\Coker f}\\frak a)$ is finitely generated.\n\n  \n\\section{Cofiniteness with respect to Extension subcategories}\n\nLet $\\cC$ be an abelian category and $\\cS$ be a subcategory of $\\cC$. We denote by $\\cS_{\\rm sub}$ ($\\cS_{\\rm quot}$), the smallest subcategory of $\\cC$ containing $\\cS$ which is closed under subobjects (quotients). These subcategories can be specified as follows:    \n\n$$(\\cS)_{\\rm sub}=\\{N\\in\\cC|\\hspace{0.1cm} N \\hspace{0.1cm}{\\rm is\\hspace{0.1cm} a\\hspace{0.1cm} subobject\\hspace{0.1cm}\nobject\\hspace{0.1cm} of\\hspace{0.1cm} an \\hspace{0.1cm}object\\hspace{0.1cm} of}\\hspace{0.1cm} \\cS\\};$$\n\n\n$$(\\cS)_{\\rm quot}=\\{M\\in\\cC|\\hspace{0.1cm} M \\hspace{0.1cm}{\\rm is\\hspace{0.1cm} a\\hspace{0.1cm} quotient\\hspace{0.1cm}\nobject\\hspace{0.1cm} of\\hspace{0.1cm} an \\hspace{0.1cm}object\\hspace{0.1cm} of}\\hspace{0.1cm} \\cS.$$\n\nLet $\\cT$ be another subcategory of $\\cC$. We denote by $\\cS\\cT$, the extension subcategory of $\\cS$ and $\\cT$ which is:\n\n$$\\cS\\cT=\\{M\\in\\cC|\\hspace{0.1cm} {\\rm there \\hspace{0.1cm} exists \\hspace{0.1cm} an\\hspace{0.1cm}exact\\hspace{0.1cm}\n sequence}$$$$\n0\\To L\\To M\\To N\\To 0\\hspace{0.1cm} {\\rm with}\\hspace{0.1cm} L\\in\\cS\n\\hspace{0.1cm}{\\rm and}\\hspace{0.1cm} N\\in\\cT\\}.$$\n\nFor any $n\\in \\mathbb{N}_0$, we set $\\cS^0=\\{0\\}$ and\n$\\cS^n=\\cS\\cS^{n-1}$. In the case where $\\cS^2=\\cS$, we say\nthat $\\cS$ is {\\it closed under extension}. We also define\n$(\\cS)_{\\rm ext}=\\bigcup_{n\\geq 0}\\cS^n$ the smallest\nsubcategory of $\\cC$ containing $\\cS$ which is closed under\nextension. \n\n\nA full subcategory $\\cS$ of $\\cC$ is called {\\it Serre} if it is\nclosed under taking subobjects, quotients and extensions. For any subcategory $\\cX$ of $\\cC$, we denote by $(\\cX)_{\\rm Serre}$, the smallest Serre subcategory of $\\cA$ containing $\\cX$.\n\\medskip\n\\begin{Lemma}\\label{kan}\nLet $\\cS$ be a subcategory of $\\cC$. Then we have the following conditions.\n\n${\\rm (i)}$ $(\\cS)_{\\rm Serre}=(((\\cS)_{\\rm sub})_{\\rm quot})_{\\rm ext}$.\n\n${\\rm (ii)}$ $(\\cS)_{\\rm ext}=\\{N\\in\\cC|\\hspace{0.1cm}\\textnormal{there exists a finite filtration}\\hspace{0.1cm} 0\\subset N_1\\subset N_2\\subset\\dots \\subset N_n=N\\\\ \\hspace{0.1cm}\\textnormal{of subobjects of }\\hspace{0.1cm} N\\hspace{0.1cm}\\textnormal{such that}\\hspace{0.1cm} N_i\/N_{i-1}\\in\\cS\\hspace{0.1cm}\\textnormal{for each}\\hspace{0.1cm} 1\\leq i\\leq n\\}$\n\\end{Lemma}\n\\begin{proof}\n(i) See [K, Proposition 2.4]. (ii) Given $N\\in(\\cS)_{\\rm ext}$, there exists a positive integer $n$ such that $N\\in\\cS^n$. We prove by induction on $n$ that there exists a filtration \n $0\\subset N_1\\subset N_2\\subset\\dots \\subset N_n=N$ of subobjects of $N$ of length $n$ such that $N_i\/N_{i-1}\\in\\cS$ for each $1\\leq i\\leq n$. If $n=1$, there is nothing to prove. Since $N\\in\\cS^n$, there exists an exact sequence $0\\To N_1\\To N\\To N\/N_1\\To 0$ such that $N_1\\in\\cS$ and $N\/N_1\\in\\cS^{n-1}$. By the induction hypothesis, there exists a finite filtration  $0\\subset N_2\/N_1\\subset \\dots \\subset N_n\/N_1=N\/N_1$ such that $N_i\/N_{i-1}\\in\\cS$ for each $2\\leq i\\leq n$. Consequently, $0\\subset N_1\\subset N_2\\subset\\dots\\subset N_n=N$ is the desired filtration.  \n\\end{proof}\n\n\n\\medskip\n\n\n\n\n  In the rest of this paper, $\\frak a$ is an ideal of $R$, $\\cS$ is a Serre subcategory of $R$-modules and $\\cN$ is the subcategory of finitely generated modules in its corresponding module category (such as in the category of $R$-modules or in the category of $R_{\\frak p}$-modules for some prime ideal $\\frak p$ of $R$). We have the following lemma.\n\t\n\t\\begin{Lemma}\\label{yosh}\n\tThe extension subcategory $\\cN\\cS$ is Serre for every Serre subcategory $\\cS$ of $R$-modules. Moreover $\\cS\\cN\\subseteq \\cN\\cS$.\n\t\\end{Lemma}\t\n\t\\begin{proof}\n\tSee [Y, Corollary 3.3 and Theorem 3.2]. \n\t\\end{proof}\t\t\n\t\n\tAn $R$-module $M$ is said to be $\\cS$-$\\frak a$-{\\it cofinite} if $\\Supp M\\subseteq V(\\frak a)$ and $\\Ext_R^i(R\/\\frak a, M)\\in\\cS$ for all integers $i\\geq 0$. An $\\cN$-$\\frak a$-cofinite module is called $\\frak a$-{\\it cofinite}. We recall that $\\cS$ satisfies the condition $C_{\\frak a}$ if for every $R$-module $M$, the following implication holds.\n\\begin{center}\n$C_{\\frak a}$: If $\\Gamma_{\\frak a}(M)=M$ and $(0:_M{\\frak a})$ is in $\\cS$, then\n$M$ is in $\\cS$.\n\\end{center}\n\t\n\t\\medskip\n\t\\begin{Lemma}\\label{sr}\n\tThe Serre subcategory $\\cS$ of $R$-modules satisfies the condition $C_{\\frak a}$ if and only if it satisfies the condition $C{\\sqrt{\\frak a}}$.  \n\t\\end{Lemma}\n\t\\begin{proof}\n\tSee [SR, Proposition 2.4].\n\t\\end{proof}\n\t\\medskip\n\t\\begin{Lemma}\\label{ext}\nLet $N$ be a finitely generated $R$-module and $M$ be an arbitrary $R$-module such that for a non-negative integer $n$, we have $\\Ext_R^i(N,M)\\in\\cS$ for each $i\\leq n$. Then $\\Ext_R^i(L,M)\\in\\cS$ for each finitely generated $R$-module $L$ with $\\Supp_RL\\subseteq\\Supp_RN$ and each $i\\leq n$. \n\\end{Lemma}\n\\begin{proof}\nSee [AS, Lemma 2.1]\n\\end{proof}\n\n\tIn this section for every $R$-module $M$, $\\dim M$ means dimension of $\\Supp_RM$ which is the length of the longest chain of prime ideals in $\\Supp_RM$.\n\t\\medskip\n\t\n\\begin{Proposition}\\label{d1}\nLet $M$ be an $R$-module of dimension $d$ such that $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for all $i\\leq d$ and ${\\rm Max} M\\subseteq \\Supp\\cS$ (e.g. if $R$ is a local ring). Then \n\n${\\rm (i)}$ $H_{\\frak b}^i(M)$ is $\\cN\\cS$-$\\frak b$-cofinite for all $i\\geq 0$ and all ideals $\\frak b$ containing $\\frak a$ with $\\dim R\/\\frak b\\leq 1$ such that $\\cS$ satisfies the condition $C_{\\frak b}$. More precisely, $H_{\\frak b}^i(M)$ is $\\cN\\cS$-$\\frak b$-cofinite for all $i<d$ and $H_{\\frak b}^d(M)\\in\\cS$. \n\n${\\rm (ii)}$ $\\Ext_R^i(N,M)\\in\\cN\\cS$ for all $i\\geq 0$ and all finitely generated $R$-modules $N$ with $\\Supp_R N\\subseteq V(\\frak a)$ and $\\dim N\\leq 1$ such that $\\cS$ and satisfying the condition $C_{\\Ann_R N}$.\n\\end{Proposition}\n\\begin{proof}\n(i) Assume that $\\frak b$ an ideal of $R$ containing $\\frak a$ with $\\dim R\/\\frak b\\leq 1$. It follows from \\cref{ext} that $\\Ext_R^i(R\/\\frak b,M)\\in\\cN\\cS$ for all $i\\leq d$. If $d=0$, there exists an exact sequence $$0\\To N\\To \\Hom_R(R\/\\frak b,M)\\To S\\To 0 $$ such that $N$ has finite length and $S\\in\\cS$. Since ${\\rm Max} M\\subseteq \\Supp\\cS$, we deduce that $N\\in\\cS$ so that $\\Hom_R(R\/\\frak b,M)\\in\\cS$ and since $\\cS$ satisfies the condition $C_{\\frak b}$, we have $M\\in\\cS$. This implies that $\\Gamma_{\\frak b}(M)\\in\\cS$ and hence $\\Gamma_{\\frak b}(M)$ is $\\cN\\cS$-$\\frak a$-cofinite (we observe that $H_{\\frak b}^i(M)=0$ for all $i>0$). Assume that $d>0$ and we proceed by induction on $i$. If $\\dim R\/\\frak b=0$, since $\\Ext_R^i(R\/\\frak b,M)\\in\\cN\\cS$, using a similar argument as mentioned above, $\\Ext_R^i(R\/\\frak b,M)\\in\\cS$ for all $i\\leq d$; and hence it follows from [AM, Theorem 2.9] that $H_{\\frak b}^i(M)\\in\\cS$ for all $i\\geq 0$; and hence the assertion is clear in this case. If $\\dim R\/\\frak b=1$, it follows from [AS, Theorem 3.5] that $H_{\\frak b}^i(M)$ is $\\cN\\cS$-$\\frak b$-cofinite for all $i<d$ and $\\Hom_R(R\/\\frak a, H_{\\frak b}^d(M))\\in\\cN\\cS$. Since $\\Supp_R H_{\\frak b}^d(M)\\subseteq {\\rm Max}R$, using a similar argument as mentioned in the case $d=0$, we deduce that $\\Hom_R(R\/\\frak a, H_{\\frak b}^d(M))\\in\\cS$ and since $\\cS$ satisfies the condition $C_{\\frak b}$, we deduce that $H_{\\frak b}^d(M)\\in\\cS$. (ii) It follows from \\cref{ext} that $\\Ext_R^i(R\/\\Ann_R N,M)\\in\\cN\\cS$ for all $i\\geq 0$ if and only if $\\Ext_R^i(R\/\\sqrt{\\Ann_R N},M)\\in\\cN\\cS$ for all $i\\geq 0$; ad hence we may assume that $\\Ann_R N=\\sqrt{\\Ann_R N}$. Since $\\Supp_R N\\subseteq V(\\frak a)$, we have $\\frak a\\subseteq \\Ann N$; and hence (i) implies that $H_{\\Ann_R N}^i(M)$ is $\\cN\\cS$-$\\Ann_R N$-cofinite for all $i\\geq 0$. Then it follows from [AS, Theorem 3.5] that $\\Ext_R^i(R\/\\Ann_R N,M)\\in\\cN\\cS$ for all $i\\geq 0$. It thus follows from \\cref{yosh} and \\cref{ext} that $\\Ext_R^i(N,M)\\in\\cN\\cS$ for all $i\\geq 0$.\n\\end{proof}\n\n\\medskip\n\nFor every Serre subcategory $\\cS$ of $R$-modules and every $\\frak p\\in\\Spec R$, we denote by $\\cS({\\frak p})$ the smallest Serre subcategory of $R_{\\frak p}$-modules containing $\\cS_{\\frak p}=\\{M_{\\frak p}|\\hspace{0.1cm} M\\in\\cS\\}$. We have the following lemma.\n\n \n\\begin{Lemma}\\label{sqext}\nLet $\\frak p$ be an ideal of $R$. Then $\\cS_{\\frak p}$ is closed under subobjects and quotients. In particular, $\\cS(\\frak p)=(\\cS_{\\frak p})_{\\rm ext}$.\n\\end{Lemma}\n\\begin{proof}\nGiven $M_{\\frak p}\\in\\cS_{\\frak p}$ and a submodule $K$ of $M_{\\frak p}$, there exists a submodule $N$ of $M$ such that \n$K=N_{\\frak p}$. Since $M\\in\\cS$, we have $N, M\/N\\in\\cS$ and $M_{\\frak p}\/K=(M\/N)_{\\frak p}$ which yields $K,M_{\\frak p}\/K\\in\\cS_{\\frak p}$. The second assertion follows from \\cref{kan}.\n\\end{proof}\n\nBy virtue of \\cref{sqext}, if $(R,\\frak m)$ is a local ring, then we have $\\cS(\\frak m)=\\cS$.\n\n\n\n\\medskip\n\\begin{Lemma}\\label{ess}\nLet $\\frak p$ be a prime ideal of $R$ and $X\\in\\cS_{\\frak p}$. Then there exists an $R$-module $S\\in\\cS$ such that $S$ is an essential $R$-submodule of $S_{\\frak p}$ and $X=S_{\\frak p}$.\n\\end{Lemma}\n\\begin{proof}\nSince $X\\in\\cS_{\\frak p}$, there exists $T\\in\\cS$ such that $X=T_{\\frak p}$. If $\\phi: T\\To T_{\\frak p}$ is the canonical homomorphism, then $S=T\/\\ker \\phi$ is the desired module. \n\\end{proof}\n\n\n\\medskip \n\\begin{Theorem}\\label{pdim1}\nLet $(R,\\frak m)$ be a local ring, let $S(\\frak p)$ satisfy the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. If $M$ is an $R$-module of dimension $d$ such that $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for each $i\\leq d$, then $\\Ext_R^i(N,M)\\in\\cN\\cS$ for each $i\\geq 0$ and each finitely generated $R$-module $N$ with $\\dim N\\leq 1$ and $\\Supp_RN\\subseteq V(\\frak a)$.\n\\end{Theorem}\n\\begin{proof}\nIf $\\dim N=0$, then $\\Ann_RN$ is $\\frak m$-primary. It follows from the assumption and \\cref{sr} that $N$ satisfies the condition $C_{\\Ann_RN}$; and hence \\cref{d1} implies that $\\Ext_R^i(N,M)\\in\\cN\\cS$ for each $i\\geq 0$. Now, assume that $\\dim N=1$,  $\\Min\\Ann_RN=\\{\\frak p_1,\\dots,\\frak p_k\\}$ and $\\Ext_R^i(R\/\\frak p_j,M)\\in\\cN\\cS$ for each $i\\geq 0$ and each $1\\leq j\\leq k$. If we set $T=\\oplus_{j=1}^kR\/\\frak p_j$, then $\\Ext_R^i(T,M)\\in\\cN\\cS$ for each $i\\geq 0$ and we have $\\Supp_RT=\\Supp_RN$. It thus follows from \\cref{ext} that $\\Ext_R^i(N,M)\\in\\cN\\cS$ for each $i\\geq 0$; and hence we may assume that $N=R\/\\frak p$ for some $\\frak p\\in V(\\frak a)$ with $\\dim R\/\\frak p=1$.\nIt is clear that $\\Ext_{R_{\\frak p}}^i(R_{\\frak p}\/\\frak a R_{\\frak p},M_{\\frak p})\\in\\cN\\cS({\\frak p})$ for each $i\\leq d$; and hence it follows from \\cref{d1} that $\\Ext_{R_{\\frak p}}^i(R_{\\frak p}\/\\frak pR_{\\frak p},M_{\\frak p})\\in\\cN\\cS(\\frak p)$ for each $i\\geq 0$. Assume that $i\\geq 0$ and $L=\\Ext_R^i(R\/\\frak p,M)$. We show that $L\\in\\cN\\cS$. In view of the previous argument there exists a finitely generated submodule $K$ of $L$ such that $L_{\\frak p}\/K_{\\frak p}\\in\\cS(\\frak p)$. Consider the canonical morphism $\\phi:L\/K\\To (L\/K)_{\\frak p}$ with $\\Ker\\phi=L_1\/K$  where $L_1$ is a submodule of $L$ containing $K$. Clearly $(L_1\/K)_{\\frak p}=0$ and since $\\frak p\\subseteq \\Ann_R L_1\/K$ and $\\dim R\/\\frak p=1$, we have $\\Supp_RL_1\/K\\subseteq V(\\frak m)$. Considering $x\\in\\frak m\\setminus \\frak p$, there is an exact sequence $$0\\To R\/\\frak p\\stackrel{x.}\\To R\/\\frak p\\To R\/\\frak p+xR\\To 0$$ which yields the following exact sequence \n$$\\Ext_R^i(R\/\\frak p +xR, M)\\To \\Ext_R^i(R\/\\frak p,M)\\stackrel{x.}\\To\\Ext_R^i(R\/\\frak p,M).$$ We observe that $\\frak p +xR$ is $\\frak m$-primary and according to the assumption and \\cref{sr}, $\\cS$ satisfies the condition $C_{\\frak p+xR}$, and hence \\cref{d1} implies that $\\Ext_R^i(R\/\\frak p +xR,M)\\in\\cN\\cS$. Thus $(0:_Lx)\\in\\cN\\cS$.\nSince $(L\/K)_{\\frak p}\\in\\cS(\\frak p)$, using \\cref{kan}, there exists a positive integer $t$ such that $(L\/K)_{\\frak p}=\\cS^t_{\\frak p}$.  Without loss of generality, we may assume that $t=2$, the other cases are similar. Then there exists an exact sequence of $R_{\\frak p}$-modules $$0\\To S'_{\\frak p}\\To (L\/K)_{\\frak p}\\To S''_{\\frak p}\\To 0$$ such that $S',S''\\in\\cS$. Using \\cref{ess} we may assume that $S'$ and $S''$ are $R$-submodule of $S'_{\\frak p}$ and $S''_{\\frak p}$, respectively. Since $L\/L_1$ is an essential $R$-submodule of $(L\/L_1)_{\\frak p}=(L\/K)_{\\frak p}$, the $R$-submodule $S'\\cap L\/L_1$ is nonzero and $(S'\\cap L\/L_1)_{\\frak p}=S'_{\\frak p}$ and replacing $S'$ by $S'\\cap L\/L_1$, we may assume that $S'$ is a submodule of $L\/L_1$. Suppose that $S'=L_2\/L_1$ for submodule $L_2$ of $L$ containing $L_1$ and we have $(L\/L_2)_{\\frak p}=S''_{\\frak p}$. Since $(L_1\/K)_{\\frak p}=0$, we have $\\Supp_RL_1\/K\\subseteq V(\\frak m)$ and applying the functor $\\Hom_R(R\/xR,-)$ to the exact sequence $0\\To K\\To L_1\\To L_1\/K\\To 0$ and the previous argument, we deduce that $\\Hom_R(R\/xR,L_1\/K)\\in\\cN\\cS$. Then there is an exact sequence $$0\\To F\\To\\Hom_R(R\/xR,L_1\/K)\\To S\\To 0$$ such that $F$ is finitely generated and $S\\in\\cS$. Since $\\Supp_R\\Hom_R(R\/xR, L_1\/K)\\subseteq V(\\frak m)$, the module $F$ has finite length and hence $F\\in\\cS$ as $R\/\\frak m\\in\\cS$. This implies that $\\Hom_R(R\/xR,L_1\/K)\\in\\cS$ and hence $\\Hom_R(R\/\\frak m,L_1\/K)\\in\\cS$. Since $\\cS$ satisfies the condition $C_{\\frak m}$, we deduce that $L_1\/K\\in\\cS$ so that $L_1\\in\\cN\\cS$ and consequently $L_2\\in\\cN\\cS$. Consider the canonical homomorphism $\\phi_1:L\/L_2\\To (L\/L_2)_{\\frak p}$ with $L_3\/L_2=\\Ker \\phi_1$. Thus $(L_3\/L_2)_{\\frak p}=0$ so that $\\Supp_RL_3\/L_2\\subseteq V(\\frak m)$. From the induced essential monomorphism $L\/L_3\\To (L\/L_2)_{\\frak p}=(L\/L_3)_{\\frak p}$ and using a similar argument as mentioned above, we may assume that $S''$ is a submodule of $L\/L_3$ and so there exists a submodule $L_4$ of $L$ containing $L_3$ such that $S''=L_4\/L_3$ and $(L\/L_4)_{\\frak p}=0$. Therefore we have $\\Supp_RL\/L_4\\subseteq V(\\frak m)$.  The exact sequence $0\\To L_2\\To L_3\\To L_3\/L_2\\To 0$, \\cref{yosh} and the fact that $L_2\\in\\cN\\cS$ imply that $\\Hom_R(R\/xR,L_3\/L_2)\\in\\cN\\cS$ and so using a similar argument as mentioned above, we deduce that $L_3\/L_2\\in\\cS$ so that $L_3\\in\\cN\\cS$ and hence $L_4\\in\\cN\\cS$. Now applying $\\Hom_R(R\/xR,-)$ to the exact sequence $0\\To L_4\\To L\\To L\/L_4\\To 0$ and using again a similar argument as mentioned before, we deduce that $L\\in\\cN\\cS$.     \n\\end{proof}\n\\medskip\nThe following corollary generalizes a result due to Melkersson [M2, Theorem 2.3].\n\\begin{Corollary}\\label{t22}\nLet $(R,\\frak m)$ be a local ring, let $\\dim R\/\\frak a\\leq 1$ and let\n$S(\\frak p)$ satisfy the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. If $M$ is an $R$-module such that $\\Supp_RM\\subseteq V(\\frak a)$ and $\\Hom_R(R\/\\frak a,M), \\Ext_R^1(R\/\\frak a,M)\\in\\cN\\cS$, then $M$ is $\\cN\\cS$-$\\frak a$-cofinite.\n\\end{Corollary}\n\\begin{proof}\nThe assertion follows immediately from \\cref{pdim1}.\n\\end{proof}\n\n\n  \\medskip\n\\begin{Theorem}\\label{d2}\nLet $(R,\\frak m)$ be a local ring, let $S(\\frak p)$ satisfy the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. If $M$ is an $R$-module of dimension $d$ such that $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for each $i\\leq d$, then $\\Ext_R^i(N,M)\\in\\cN\\cS$ for each $i\\geq 0$ and each finitely generated $R$-module $N$ with $\\dim N\\leq 2$ and $\\Supp_RN\\subseteq V(\\frak a)$.\n\\end{Theorem}\n\\begin{proof}\nSimilar to the proof of \\cref{pdim1}, we may assume that $N=R\/\\frak p$ for some $\\frak p\\in V(\\frak a)$ with $\\dim R\/\\frak p=2$ and $\\Ext_{R_{\\frak p}}^i(R_{\\frak p}\/\\frak pR_{\\frak p},M_{\\frak p})\\in\\cN\\cS(\\frak p)$ for each $i\\geq 0$. Assume that $i\\geq 0$ and $L=\\Ext_R^i(R\/\\frak p,M)$ and we show that $L\\in\\cN\\cS$. There exists a finitely generated submodule $K$ of $L$ such that $L_{\\frak p}\/K_{\\frak p}\\in\\cS(\\frak p)$. Consider the canonical morphism $\\phi_{L\/K}:L\/K\\To (L\/K)_{\\frak p}$ with $\\Ker\\phi_{L\/K}=L_1\/K$ where $L_1$ is a submodule of $L$. Clearly $(L_1\/K)_{\\frak p}=0$ and since $\\frak p\\subseteq \\Ann_R L_1\/K$ and $\\dim R\/\\frak p=2$, every $\\frak q\\in\\Supp_RL_1\/K$ with $\\dim R\/\\frak q=1$ is a minimal prime ideal of $\\Ann_R L_1\/K$. Then the set $$\\cT=\\{\\frak q\\in\\Supp_RL_1\/K|\\hspace{0.1cm} \\dim R\/\\frak q=1\\}$$ is finite. Assume that $\\cT=\\{\\frak q_1,\\dots,\\frak q_n\\}$. Then $\\frak p\\subsetneq\\cap_{j=1}^n\\frak q_j$ and so there $x\\in\\cap_{i=1}^n\\frak q_i\\setminus \\frak p$. Then there is an exact sequence $$0\\To R\/\\frak p\\stackrel{x.}\\To R\/\\frak p\\To R\/\\frak p+xR\\To 0$$ which yields the following exact sequence \n$$\\Ext_R^i(R\/\\frak p +xR, M)\\To \\Ext_R^i(R\/\\frak p, M)\\stackrel{x.}\\To\\Ext_R^i(R\/\\frak p, M).$$ Since $\\dim R\/\\frak p+xR\\leq 1$, it follows from \\cref{pdim1} that $\\Ext_R^i(R\/\\frak p+xR,M)\\in\\cN\\cS$; and hence $(0:_Lx)\\in\\cN\\cS$. Now assume that $L_2\/K=\\Ker\\phi_{L_1\/K}$ where $\\phi_{L_1\/K}:L_1\/K\\To (L_1\/K)_{\\frak q_1}$. Then we have $(L_2\/K)_{\\frak q_1}=0$. Continuing this way, assume that $L_{i+1}\/K=\\Ker\\phi_{L_i\/K}$ where $\\phi_{L_i\/K}:L_i\/K\\To (L_i\/K)_{\\frak q_i}$ for $1\\leq i\\leq n$, we deduce that $\\Supp_RL_{i}\/K\\subseteq \\{\\frak q_i,\\dots \\frak q_n,\\frak m\\}$ for each $1\\leq i\\leq n$, $\\Supp_RL_{n+1}\/K\\subseteq \\{\\frak m\\}$ and we have a chain of submodules $$L_{n+1}\\subseteq L_n\\subseteq \\dots\\subseteq L_1\\subseteq L=L_0$$ such that the induced morphism $\\overline{\\phi_{L_i\/K}}:L_i\/ L_{i+1}\\To (L_i\/K)_{\\frak q_i}$ is an essential monomorphism for each $0\\leq i\\leq n$. We observe that $\\Supp_RL_i\/L_{i+1}\\subseteq \\{\\frak q_i,\\dots \\frak q_n,\\frak m\\}$ for each $1\\leq i\\leq n$. Since $(0:_{L}x)\\in\\cN\\cS$, we have $(0:_{L_{n+1}}x)\\in\\cN\\cS$. Now applying the functor $\\Hom_R(R\/xR,-)$ to the exact sequence $0\\To K\\To L_{n+1}\\to L_{n+1}\/K\\To 0$, we deduce that $\\Hom_R(R\/xR, L_{n+1}\/K)\\in\\cN\\cS$ and so there is an exact sequence of $R$-modules  $$0\\To F\\To\\Hom_R(R\/xR, L_{n+1}\/K)\\To S\\To 0$$ such that $F$ is finitely generated and $S\\in\\cS$. Using the same argument as mentioned in the proof of \\cref{pdim1}, we deduce that $L_{n+1}\/K\\in\\cS$ so that $L_{n+1}\\in\\cN\\cS$. Since $\\Hom_R(R\/xR,L_n)\\in\\cN\\cS$, $\\Hom_R(R\/xR,L_n)_{\\frak q_n}\\in\\cN\\cS(\\frak q_n)$ so that $\\Hom_R(R\/xR,L_n\/K)_{\\frak q_n}\\in\\cN\\cS(q)$ and a similar proof as mentioned above gives $(L_n\/K)_{\\frak q_n}=(L_n\/L_{n+1})_{\\frak q_n}\\in\\cS(\\frak \\frak q_n)$. To be convenient, set $\\frak q=\\frak q_n$. in view of \\cref{sqext}, we may assume that $(L_n\/L_{n+1})_{\\frak q}=\\cS_{\\frak q}^2$ and the other cases are similar. Then there exists an exact sequence \n$$0\\To S'_{\\frak q}\\To (L_n\/L_{n+1})_{\\frak q}\\To S''_{\\frak q}\\To .$$ Using \\cref{ess}, we may assume that $S'$ and $S''$ are $R$- submodules of $S'_{\\frak p}$ and $S''_{\\frak p}$, respectively. Since $L_n\/L_{n+1}$ is an essential submodule of $(L_n\/L_{n+1})_{\\frak q}$, the submodule $S'\\cap L_n\/L_{n+1}$ is nonzero and $(S'\\cap L_n\/L_{n+1})_{\\frak p}=S'_{\\frak p}$; and hence replacing $S'$ by $S'\\cap L_n\/L_{n+1}$, we may assume that $S'$ is a submodule of $L_n\/L_{n+1}$. Assume that $S'=L'_n\/L_{n+1}$ for some submodule $L'_n$ of $L_n$ and so $L'_n\\in\\cN\\cS$. Thus $(L_n\/L_n')_{\\frak q}=S''_{\\frak q}$. Consider the canonical exact sequence $0\\To L''_n\/L'_n\\To L_n\/L'_n\\To (L_n\/L'_n)_{\\frak q}$ which forces $\\Supp_R L''_n\/L'_n\\subseteq V(\\frak m)$. Since $\\Hom_R(R\/xR,L''_n)\\in\\cN\\cS$ and $L'_n\\in\\cN\\cS$, using a similar argument as above, we deduce that $L''_n\\in\\cN\\cS$. The essential monomorphism $L_n\/L''_n\\To (L_n\/L'_n)_{\\frak q}$ and a similar argument as above imply that $S''$ is a submodule of $L_n\/L''_n$ and so $S''=L''\/L''_n$ for some submodule $L''$ of $L_n$. The fact that $L''_n\\in\\cN\\cS$ implies that $L''\\in\\cN\\cS$. Since $(L_n\/L'')_{\\frak q}=0$, we deduce that $\\Supp_RL_n\/L''\\subseteq V(\\frak m)$. A similar argument as above implies that $L_n\/L''\\in\\cS$, and hence $L_n\\in\\cN\\cS$. Continuing this manner we deduce that $L=L_1\\in\\cN\\cS$. Therefore, we may assume that $\\phi_{L\/K}$ is an essential monomorphism. According to \\cref{sqext}, there exists a positive integer $t$ such that $(L\/K)_{\\frak p}\\in(\\cS_{\\frak p})^t$. Set $A=L\/K$. Without loss of generality, we may assume that $t=2$ and so by a similar argument as mentioned above, there exists an exact sequence of $R_{\\frak p}$-modules $0\\To S'_{\\frak p}\\To A_{\\frak p}\\To S''_{\\frak p}\\To 0$ such $S'$ is a submodule of $L\/K$ and $S''$ is an $R$-submodule of $A\/S'$ and so $((A\/S')\/S'')_{\\frak p}=0$; and hence $\\frak p\\notin\\Supp_R(A\/S')\/S''$. Since $\\frak p\\subseteq \\Ann_R (A\/S')\/S''$ and $\\dim R\/\\frak p=2$, every $\\frak q\\in\\Supp_R(A\/S')\/S''$ with $\\dim R\/\\frak q=1$ is a minimal prime ideal of $\\Ann_R (A\/S')\/S''$; and hence the set $$\\cU=\\{\\frak P\\in\\Supp_R(A\/S')\/S''|\\hspace{0.1cm} \\dim R\/\\frak P=1\\}$$ is finite. Assume that $\\cU=\\{\\frak p_1,\\dots\\frak p_m\\}$ and set $D=(A\/S')\/S''$. We notice that $\\Supp_RD=\\{\\frak p_1,\\dots,\\frak p_m,\\frak m\\}$. It follows from \\cref{d1} that $D_{\\frak p_j}\\in\\cN\\cS(\\frak p_j)$ for each $1\\leq j\\leq m$. Then for each $j$, there exists an exact sequence $$0\\To N_{j_{\\frak p_j}}\\To D_{\\frak p_j}\\To (D\/N)_{\\frak p_j}\\To 0$$ such that $N_j$ is a finitely generated $R$-module and $(D\/N)_{\\frak p_j}\\in (S_{\\frak p_j})_{\\rm ext}$. In view of the preceding arguments, we may assume that $S\\in\\cS_{\\frak p_j}$ so that there exists $S_j\\in\\cS$ such that $S=S_{j_{\\frak p_j}}$. Furthermore, we may assume that $S_j$ is a submodule of $D\/N_j$ and $((D\/N_j)\/S_j)_{\\frak p_j}=0$. Then there exists a submodule $X_j$ of $D$ such thst $S_j=X_j\/N_j$ and so this implies that $X_j\\in\\cN\\cS$ for each $1\\leq j\\leq m$. Now, set $X=X_1+\\dots+ X_m$ and so clearly $X\\in\\cN\\cS$ and $\\Supp D\/X\\subseteq V(\\frak m)$. We notice that there exists a submodule $A'$ of $A$ such that $S''=A'\/S'$  and so $A'\\in\\cS$. On the other hand, there exists a submodule $L'$ of $L$ such that $A'=L'\/K$ which implies that $L'\\in\\cN\\cS$ and $D=L\/L'$. Moreover, since $X$ is a submodule of $D$, there exists a submodule $L_1$ of $L$ containing $L'$ such that $X=L_1\/L'$ and hence $D\/X=L\/L_1$. We observe by \\cref{yosh} that $L_1\\in\\cN\\cS\\cN\\cS\\subseteq \\cN\\cS$. Considering the following exact sequence $$0\\To L_1\\To L\\To D\/X\\to 0,$$ we have $\\Hom_R(R\/xR,D\/X)\\in\\cN\\cS$ and using a similar proof as mentioned before, we deduce that $L\/L_1=D\/X\\in\\cS$. Now the fact that $L_1\\in\\cN\\cS$ forces $L\\in\\cN\\cS\\cS\\subseteq\\cN\\cS$.  \n\\end{proof}\n\n\nThe following corollary generalizes [BNS, Theorem 3.5].\n\\medskip\n\n\\begin{Corollary}\\label{t21}\nLet $(R,\\frak m)$ be a local ring, let $\\dim R\/\\frak a\\leq 2$ and let\n$S(\\frak p)$ satisfy the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. If $M$ is an $R$-module such that $\\Supp_RM\\subseteq V(\\frak a)$ and $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for $i=0,1,2$, then $M$ is $\\cN\\cS$-$\\frak a$-cofinite.\n\\end{Corollary}\n\\begin{proof}\nThe assertion follows immediately from \\cref{d2}. \n\\end{proof}\n\nThe subcategory of all $R$-modules of finite support is denoted by $\\cF$. It is clear that $\\cF$ is Serre and it satisfies the condtion $C_{\\frak a}$ for every ideal $\\frak a$ of $R$.\n\\medskip\n\\begin{Corollary}\\label{f}\nLet $(R,\\frak m)$ be a local ring, let $\\dim R\/\\frak a=2$ and let $M$ be an $R$-module such that $\\Supp_RM\\subseteq V(\\frak a)$. If $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cF$ for $i=0,1,2$, then $M$ is $\\cN\\cF$-$\\frak a$-cofinite.\n\\end{Corollary}\n\\begin{proof}\nIt is clear that $\\cF(\\frak p)$ satisfies the condition $C_{\\frak pR_{\\frak p}}$ for every prime ideal $\\frak p$ of $R$. Therefore, the assertion follows immediately from \\cref{d2}. \n\\end{proof}\n\\medskip\n\n\\begin{Theorem}\\label{loc}\nLet $(R,\\frak m)$ be a local ring, let $\\dim R\/\\frak a=2$ and let\n$S(\\frak p)$ satisfy the condition $\\frak pR_{\\frak p}$ for each $\\frak p\\in V(\\frak a)$. If $n$ is a non-negative integer such that $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for all $i\\leq n+1$, then the following conditions are equivalent.\n \n{\\rm (i)} $H_{\\frak a}^i(M)$ is $\\cN\\cS$-$\\frak a$-cofinite for all $i<n$.\n\n{\\rm (ii)} $\\Hom_R(R\/\\frak a,H_{\\frak a}^i(M))\\in\\cN\\cS$ for all $i\\leq n$. \n\\end{Theorem}\n\\begin{proof}\nWe proceed by induction on $n$. The case $n=0$ is clear. If we consider $\\overline{M}=M\/\\Gamma_{\\frak a}(M)$, there exists an exact sequence of $R$-modules $0\\To \\overline{M}\\To E\\To Q\\To 0$ such that $E$ is injective with $\\Gamma_{\\frak a}(E)=0$. This eaxct sequence implies that $H_{\\frak a}^i(Q)\\cong H_{\\frak a}^{i+1}(M)$ and $\\Ext_R^i(R\/\\frak a,Q)\\cong\\Ext_R^i(R\/\\frak a,\\overline{M})$ for each $i\\geq 0$; furthermore $\\Hom_R(R\/\\frak a,H_{\\frak a}^1(M))\\cong\\Ext_R^1(R\/\\frak a,\\overline{M})$. Moreover for each $i\\geq 1$ we have an exact sequence of $R$-modules,\n$$\\Ext_R^i(R\/\\frak a,\\Gamma_{\\frak a}(M))\\To\\Ext_R^i(R\/\\frak a,M)\\To\\Ext_R^i(R\/\\frak a,\\overline{M})$$$$\\To\\Ext_R^{i+1}(R\/\\frak a,\\Gamma_{\\frak a}(M))\\To\\Ext_R^{i+1}(R\/\\frak a,M)\\hspace{0.3cm}(\\dag_i).$$\nIf $n=1$, the exact sequence $(\\dag_1)$ and \\cref{t21} imply that (i) and (ii) are equivalent. For $n>1$, since $\\Gamma_{\\frak a}(M)\\in\\cN\\cS$, it follows from $(\\dag_i)$ and the previous isomorphisms that $\\Ext_R^j(R\/\\frak a,Q)\\in\\cN\\cS$ for all $i\\leq n$ so that (i) and (ii) are equivalent for $Q$ and non-negative integer $n-1$. Now, using again the previous isomorphisms, the conditions (i) and (ii) are equivalent for $M$ and non-negative integer $n$.\n\\end{proof}\n\n\n\\medskip\n\\begin{Corollary}\nLet $R$ be an local ring with $\\dim R\/\\frak a\\leq 2$, let $n$ be a non-negative integer such that $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cF$ for all $i\\leq n+1$. Then the following conditions are equivalent.\n \n{\\rm (i)} $H_{\\frak a}^i(M)$ is $\\cN\\cF$-$\\frak a$-cofinite for all $i<n$.\n\n{\\rm (ii)} $\\Hom_R(R\/\\frak a,H_{\\frak a}^i(M))\\in\\cN\\cF$ for all $i\\leq n$. \n\\end{Corollary}\n\\begin{proof}\nSince $\\cF(\\frak p)$ satisfies the condition $C_{\\frak pR_{\\frak p}}$ for every prime ideal $\\frak p$ of $R$, the results is obtained by \\cref{loc}.\n\\end{proof}\n\n\n\\medskip\n Given an arbitrary Serre subcategory $\\cS$ of $R$-modules, we say that $R$ admits the condition $P_n^{\\cS}(\\frak a)$ if for every $R$-module $M$, the following implication holds:\n\\begin{center}\n$P_n^{\\cS}(\\frak a):\\hspace{0.3cm}$ {\\it If $\\Ext_R^i(R\/\\frak a, M)\\in\\cS$ for all $i\\leq n$ and $\\Supp(M)\\subseteq V(\\frak a)$,\nthen $M$ is $\\cS$-$\\frak a$-cofinite.}\n \\end{center}\n\n\\medskip\n\n\n\n\\begin{Theorem}\\label{td}\nLet $R$ be a ring of dimension $d\\geq 1$ admitting the condition $P^{\\cS}_{d-1}(\\frak a)$ for all ideals $\\frak a$ of dimension $d-1$ (i.e. $ \\dim R\/\\frak a\\leq d-1$), then $R$ admits $P^{\\cS}_{d-1}(\\frak a)$ for all ideals $\\frak a$ of $R$. \n\\end{Theorem}\n\\begin{proof} \nLet $M$ be an $R$-module and $\\frak a$ be an arbitrary ideal of $R$ such that $\\Supp_RM\\subseteq V(\\frak a)$ and $\\Ext_R^i(R\/\\frak a,M)\\in\\cN\\cS$ for all $i\\leq d-1$. If there exists some positive integer $n$ such that $\\frak a^n=0$, then $M=(0:_M\\frak a^n)$. On the other hand, if $\\frak a=(a_1,\\dots,a_t)$, there is an exact sequence of $R$-modules $0\\To (0:_M\\frak a)\\To (0:_M\\frak a^n)\\stackrel{f}\\To a_1(0:_M\\frak a^n)\\oplus\\dots a_t(0:_M\\frak a^n)$ where $f(x)=(a_1x,\\dots,a_tx)$ . It is clear that $a_i(0:_M\\frak a^n)$ is a submodule of $(0:_M\\frak a^{n-1})$ for each $i$ and hence, since $\\cN\\cS$ is Serre, an easy induction on $n$ implies that $(0:_M\\frak a^n)\\in\\cN\\cS$. Since $V(\\frak a)=\\Spec R$, the module $M$ is $\\cN\\cS$-cofinite. Now, assume that $\\frak a$ is not nilpotent and so there is a positive integer $n$ such that $\\Gamma_{\\frak a}(R)=(0:_R\\frak a^n)$. Taking $\\overline{R}=R\/\\Gamma_{\\frak a}(R)$ and $\\overline{M}=M\/(0:_M\\frak a^n)$, it is clear that $\\overline{M}$ is an $\\overline{R}$-module and since $\\Gamma_{\\frak a}(\\overline{R})=0$, $\\frak a$ contains an $\\overline{R}$-regular element so that $\\dim R\/\\frak a+\\Gamma_{\\frak a}(R)\\leq d-1$. We observe that $\\Supp_R\\overline{M}\\subseteq V(\\frak a+\\Gamma_{\\frak a}(R))$ and it follows from \\cref{ext} that $\\Ext_R^i(R\/\\frak a+\\Gamma_{\\frak a}(R),\\overline{M})\\in\\cN\\cS$ for all $i\\leq d-1$. Thus the assumption implies that $\\overline{M}$ is $\\cN\\cS$-$\\frak a+\\Gamma_{\\frak a}(R)$-cofinite. We now show that $\\Ext_{\\overline{R}}^i(\\overline{R}\/\\frak a\\overline{R},\\overline{M})\\in\\cN\\cS$ for each $i\\geq 0$. Consider the Grothendieck spectral sequence $$E_2^{p,q}:=\\Ext_{\\overline{R}}^p(\\Tor_q^R(\\overline{R},R\/\\frak a+\\Gamma_{\\frak a}(R)),\\overline{M})\\Rightarrow H^{p+q}=\\Ext_R^{p+q}(R\/\\frak a+\\Gamma_{\\frak a}(R),\\overline{M}).$$\n For $i=0$, we have $\\Hom_{\\overline{R}}(\\overline{R}\/\\frak a\\overline{R},\\overline{M})=\\Hom_{\\overline{R}}(\\overline{R}\\otimes_RR\/\\frak a+\\Gamma_{\\frak a}(R),\\overline{M})\\cong \\Hom_R(R\/\\frak a+\\Gamma_{\\frak a}(R),\\overline{M})\\in\\cN\\cS$. Now, assume that $i>0$ and the result has been proved for all values smaller than $i$. Then $E_2^{p,0}=\\Ext_{\\overline{R}}^p(\\overline{R}\\otimes_RR\/\\frak a+\\Gamma_{\\frak a}(R),\\overline{M})\\in\\cN\\cS$ for all $0\\leq p<i$.  Since $\\Supp_{\\overline{R}}\\Tor_q^R(\\overline{R},R\/\\frak a+\\Gamma_{\\frak a}(R))\\subseteq V(\\frak a\\overline{R})$, It follows from \\cref{ext} that $E_2^{p,q}\\in\\cN\\cS$ for all $0\\leq p<i$ and all $q\\geq 0$. The exact sequence $E_2^{i-2,1}\\To E_2^{i,0}\\To E_3^{i,0}\\To 0$ and the induction hypothesis imply that $E_2^{i,0}\\in\\cN\\cS$ if $E_3^{i,0}\\in\\cN\\cS$. Continuing this manner, we deduce that $E_2^{i,0}\\in\\cN\\cS$ if $E_{\\infty}^{i,0}\\in\\cN\\cS$. But there are the following filtration \n$$0=\\Phi^{i+1}H^i\\subset\\dots\\subset\\Phi^1H^i\\subset \\Phi^0H^i\\subset H^i$$ such that $E_{\\infty}^{i,0}\\cong \\Phi^iH^i\/\\Phi^{i+1}H^i =\\Phi^iH^i$ is a submodule of $H^i=\\Ext_R^i(R\/\\frak a+\\Gamma_{\\frak a}(R),\\overline{M})$; and hence it is in $\\cN\\cS$. Therefore $\\Ext_{\\overline{R}}^i(\\overline{R}\/\\frak a\\overline{R},\\overline{M})\\in\\cN\\cS$ for all $i\\geq 0$. Now consider the Grothendieck spectral sequence $$E_2^{p,q}:=\\Ext_{\\overline{R}}^p(\\Tor_q^R(\\overline{R},R\/\\frak a),\\overline{M})\\Rightarrow H^{p+q}=\\Ext_R^{p+q}(R\/\\frak a,\\overline{M}).$$  \nUsing again \\cref{ext}, we deduce $E_2^{p,q}\\in\\cN\\cS$ for all $p,q\\geq 0$. For any $r>2$, the $\\overline{R}$-module $E_r^{p,q}$ is a subquotient of $E_{r-1}^{p,q}$ and so an easy induction yields that $E_r^{p,q}\\in\\cN\\cS$ for all $r\\geq 2$  so that $E_{\\infty}^{p,q}\\in\\cN\\cS$ for all $p,q\\geq 0$ . For any $0\\leq t\\leq n$, there is a finite filtration \n$$0=\\Phi^{t+1}H^t\\subset \\Phi^tH^t\\subset\\dots\\subset\\Phi^1H^t\\subset \\Phi^0H^t\\subset H^t$$ \nsuch that $\\Phi^pH^t\/\\Phi^{p+1}H^t\\cong E_{\\infty}^{p,t-p}$ where $0\\leq p\\leq t$. Since $E_{\\infty}^{p,t-p}\\in\\cN\\cS$ for all $0\\leq p\\leq t$ and $t\\geq 0$, we deduce that $H^t\\in\\cN\\cS$ for all $t\\geq 0$; and hence $\\overline{M}$ is $\\cN\\cS$-$\\frak a$-cofinite. On the other hand, since $(0:_M\\frak a^n)\\in\\cN\\cS$, we conclude that $M$ is $\\cN\\cS$-$\\frak a$-cofinite.\n\\end{proof}\n\\medskip\n\n\\begin{Corollary}\nLet $R$ be a local ring of dimension $2$ such that $\\cS$ satisfies the condition $C_{\\frak a}$ for every ideal $\\frak a$ of $R$ with $\\dim R\/\\frak a\\leq 1$. Then $R$ admits the condition $P_1^{\\cN\\cS}(\\frak a)$ for every ideal $\\frak a$ of $R$. \n\\end{Corollary}\n\\begin{proof}\nIt follows from [AS, Theorem 3.2] and \\cref{t22} that $R$ admits the condition $P_1^{\\cN\\cS}(\\frak a)$ for all ideals with $\\dim R\/\\frak a\\leq 1$. Now, the result follows from \\cref{td}.\n\\end{proof}\n\n\n\\medskip\n\\begin{Corollary}\nLet $R$ be a local ring of dimension $2$ such that $\\cS(\\frak p)$ satisfies the condition $\\frak pR_{\\frak p}$ for every prime ideal $\\frak p$ with $\\dim R\/\\frak p\\leq 1$. Then $R$ admits the condition $P_2^{\\cN\\cS}(\\frak a)$ for every ideal $\\frak a$ of $R$. \n\\end{Corollary}\n\\begin{proof}\nIt follows from \\cref{t22} that $R$ admits the condition $P_1^{\\cN\\cS}(\\frak a)$ for all ideals $\\frak a$ with $\\dim R\/\\frak a\\leq 1$. Now, the result follows from \\cref{td}.\n\\end{proof}\n\\medskip\n\n\\begin{Corollary}\nLet $R$ be a local ring of dimension $3$ such that $\\cS(\\frak p)$ satisfies the condition $\\frak pR_{\\frak p}$ for every prime ideal $\\frak p$ with $\\dim R\/\\frak p\\leq 2$. Then $R$ admits the condition $P_2^{\\cN\\cS}(\\frak a)$ for every ideal $\\frak a$ of $R$. \n\\end{Corollary}\n\\begin{proof}\nIt follows from \\cref{t21} that $R$ admits the condition $P_2^{\\cN\\cS}(\\frak a)$ for all ideals with $\\dim R\/\\frak a\\leq 2$. Now, the result follows from \\cref{td}.\n\\end{proof}\n\n\\medskip\n\\begin{Corollary}\nLet $R$ be a local ring of dimension $3$. Then $R$ admits the condition $P_2^{\\cN\\cF}(\\frak a)$. \n\\end{Corollary}\n\\begin{proof}\nIt follows from \\cref{f} that $R$ admits the condition $P_2^{\\cN\\cF}(\\frak a)$ for all ideals with $\\dim R\/\\frak a=2$. Now, the result follows from \\cref{td}.\n\\end{proof}\n\n\\section{Cofiniteness with respect to a new dimension}\n\nFor every $R$-module $M$, it is clear that $\\Supp_RM\\subseteq V(\\Ann_RM)$ and for the case where $M$ is finitely generated, they are equal. We define the {\\it upper dimension} of $M$ and we denote it by $\\overline{\\dim}M$ which is $\\overline{\\dim}M=\\dim R\/\\Ann_RM$. Clearly $\\dim M\\leq \\overline{\\dim}M$. We first recall some results which are needed in this section.  \n\n\\begin{Lemma}\\label{del}\nLet $S$ be a finitely generated $R$-algebra and let $M$ be an $S$-module. Then $M$ is $\\frak a$-cofinite if and only if $M$ is $\\frak a S$-cofinite (as an $S$-module).\n\\end{Lemma}\n\\begin{proof}\nSee [DM, Proposition 2].\n\\end{proof}\n\\medskip\n\\begin{Lemma}\nLet $S$ be a finitely generated $R$-algebra and let $M$ be an $S$-module. Then $M$ satisfies the condition $P^{\\cN}_n(\\frak a)$ if and only if $M$ satisfies the condition $P_n^{\\cN}(\\frak aS)$. \n\\end{Lemma}\n\\begin{proof}\nSee [KS, Proposition 2.15].\n\\end{proof}\n\n\n\n\\medskip\n\\begin{Lemma}\\label{mel}\nLet $M$ be an $R$-module such that $\\Supp_RM\\subseteq V(\\frak a)$. Then $M$ is Artinian and $\\frak a$-cofinite if and only if $(0:_M{\\frak a})$ has finite length.\n\\end{Lemma}\n\\begin{proof}\nSee [M1, Proposition 4.1].\n\\end{proof}\nFor every non-negative integer $n$, we denote by $\\cD_{\\leq n}$, the subcategory of all $R$-modules $M$  such that $\\overline{\\dim} M\\leq n$. We also denote by $\\cG$, the subcategory of all $R$-modules $F$ such that $V(\\Ann_RF)$ is a finite set. \n\n\\medskip\n\\begin{Lemma}\\label{ser}\nIf $0\\To N\\To M\\To M\/N\\To 0$ is an exact sequence of $R$-modules, Then $V(\\Ann_RM)=V(\\Ann_RN)\\cup V(\\Ann_RM\/N)$.\n\\end{Lemma}\n\\begin{proof}\nSince $\\Ann_RM\\subseteq \\Ann_RN\\cap\\Ann_R M\/N$, we conclude that $V(\\Ann_RN)\\cup V(\\Ann_RM\/N)\\subseteq V(\\Ann_RM)$. Now assume that $\\frak p\\in V(\\Ann_RM)$ and $\\frak p\\notin V(\\Ann_RN)$. Then there exists $r\\in\\Ann_RN\\setminus \\frak p$ and so for every $x\\in\\Ann_R M\/N$, we have $rx\\in\\Ann_RM\\subseteq \\frak p$ which implies that $x\\in\\frak p$. Consequently, $\\frak p\\in V(\\Ann_RM\/N)$. \n\\end{proof}\n\n\\medskip\n\n \n\n\\begin{Corollary}\nThe following conditions hold.\n\n{\\rm (i)} For every non-negative integer $n$, the suncategory $\\cD_{\\leq n}$ is Serre.\n\n{\\rm (ii)} The subcategory $\\cG$ is Serre.\n\\end{Corollary}\n\\begin{proof}\nThe proof is is straightforward by \\cref{ser}.\n\\end{proof}\n\n\n\\medskip \n\n\\begin{Proposition}\nLet $M$ be an $R$-module with $\\overline{\\dim}M=2$. Then $M$ is $\\frak a$-cofinite if and only if $\\Hom_R(R\/\\frak a,M)$ and $\\Ext_R^1(R\/\\frak a,M)$ are finitely generated.  \n\\end{Proposition}\n\\begin{proof}\nLet $\\overline {R}=R\/\\Ann_RM$. Using [KS, Proposition 2.15] we may assume that $\\dim R=2$. Now, the result follows from [NS, Corollary 2.4].\n\\end{proof}\n\n\\medskip \n\n\\begin{Proposition}\nLet $M$ be an $R$-module with $\\overline{\\dim}M=3$. Then $M$ is $\\frak a$-cofinite if and only if $\\Ext_R^i(R\/\\frak a,M)$ is finitely generated for $i=0,1,2$.  \n\\end{Proposition}\n\\begin{proof}\nLet $\\overline {R}=R\/\\Ann_RM$. Using [KS, Proposition 2.15] we may assume that $\\dim R=3$. Now, the result follows from [NS, Corollary 2.5].\n\\end{proof}\n\\medskip\n\n\\begin{Proposition}\\label{dim1}\nThe subcategory of $\\frak a$-cofinite modules in $\\cD_{\\leq 1}$ is Serre.\n\\end{Proposition}\n\\begin{proof}\n If $\\overline{\\dim} M=0$, then $\\dim M=0$ and so the module $(0:_M\\frak a)$ has finite length. Thus it follows from \\cref{mel} that $M$ is $\\frak a$-cofinite. Now, assume that $\\overline{\\dim} M=1$ and so $\\dim \\overline{R}=1$ where $\\overline{R}=R\/\\Ann_RM$. It is clear that $(0:_M\\frak a\\overline{R})=(0:_M\\frak a)$ is finitely generated and $\\Supp_{\\overline{R}}M\\subseteq V(\\overline{\\frak a})$. It follows from [M1, Proposition 4.5] that $M$ is $\\overline{\\frak a}$-cofinite. Finally, using \\cref{del}, $M$ is $\\frak a$-cofinite. The second assertion is straightforward. \n\\end{proof}\n\n\n\\medskip\n\\begin{Corollary}\nLet $M\\in\\cN\\cG$ with $\\Supp_RM\\subseteq V(\\frak a)$. Then $M$ is $\\frak a$-cofinite if and only if $(0:_M\\frak a)$ is finitely generated. \n\\end{Corollary}\n\\begin{proof}\nSince $M\\in\\cN\\cF$, there exists an exact sequence of $R$-modules $0\\To N\\To M\\To F\\To 0$ such that $N$ is finitely generated and $F$ has finite support. We notice that $(0:_F\\frak a)$ is finite, and it suffices to show that $F$ is $\\frak a$-cofinite and so we may assume that $V(\\Ann_RM)$ is a finite set so that $\\overline{\\dim}M\\leq 1$. It follows from [M1, Proposition 4.5] that $M$ is $\\overline{\\frak a}$-cofinite where $\\overline{R}=R\/\\Ann_RM$ and $\\overline{\\frak a}=\\frak a\\overline{R}$. Now \\cref{del} implies that $M$ is $\\frak a$-cofinite.  \n\\end{proof}\n\\medskip\n\n\\begin{Proposition}\n The subcategory of $\\frak a$-cofinite modules in $\\cD_{\\leq 2}$ is abelian.\n\\end{Proposition}\n\\begin{proof}\nAssume that $f:M\\To N$ be a morphism of $\\frak a$-cofinite modules and assume that $K=\\Ker f, I=\\Im f$ and $C=\\Coker f$. \n The assumption implies that $(0:_I\\frak a)=(0:_I\\frak a\\overline{R})$ is finitely generated $\\overline{R}$-module where $\\overline{R}=R\/\\Ann_RM$. If $\\dim \\overline{R}=0$, the module $(0:_I\\frak a)$ has finite length and so $I$ is Artinian. Now, \\cref{mel} implies that $I$ is $\\frak a$-cofinite. If $\\dim\\overline{R}=1$, it follows from \\cref{dim1} that $I$ is $\\frak a\\overline{R}$-cofinite as $M$ is $\\frak a\\overline{R}$-cofinite; and hence $I$ is $\\frak a$-cofinite using \\cref{del}. If $\\dim\\overline{R}=2$, it follows from [NS, Corollary 2.6] that $I$ is $\\frak a\\overline{R}$-cofinite and so is $\\frak a$-cofinite by using \\cref{del}. Now, using the exact sequences of $R$-modules $$0\\To K\\To M\\To I\\To 0;$$$$0\\To I\\To N\\To C\\To 0;$$ it is straightforward to show that $K$ and $C$ are $\\frak a$-cofinite modules.\n\\end{proof}\n\n\n\\medskip\n\n\\begin{Proposition}\nThe kernel and cokernel of a homomorphism $f:M\\To N$ of $\\frak a$-cofinite modules in $\\cD_{\\leq 3}$ is $\\frak a$-cofinite if and only if $(0:_{\\Coker f}\\frak a)$ is finitely generated.\n\\end{Proposition}\n\\begin{proof}\nBy the assumption we have $\\dim R\/\\Ann_RM\\leq 3$ and also $\\dim R\/\\Ann_RN\\leq 3$. If we put $\\frak b=\\Ann_RM\\cap \\Ann_RN$ and $\\overline{R}=R\/\\frak b$, we have $\\dim \\overline{R}\\leq 3$ and further $M$ and $N$ are $R\/\\frak b$-module. Clearly $(0:_{\\Coker f}\\frak a\\overline{R})$ is a finitely generated $\\overline{R}$-module. It follows from [NS, Theorem 2.8] that \n$\\ker f$ and $\\Coker f$ are $\\frak a\\overline{R}$-cofinite and so using \\cref{del}, they are $\\frak a$-cofinite.\n\\end{proof}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{}\n    \n\\noindent An exciting area where biology meets technology is evolutionary robotics \\cite{nolfi2000evolutionary,Vargas-2014,doncieux2015evolutionary}. The key idea behind the field is to optimize the design of robots through evolutionary computing \\cite{eiben2003introduction-to}. Using artificial evolution for robot design has a strong rationale.\n\n\\begingroup\n\\addtolength\\leftmargini{0.1cm}\n\\begin{quote}\n{\\bf As natural evolution has produced successful life forms for practically all possible environmental niches on Earth, it is plausible that artificial evolution can produce specialised robots for various environments and tasks.}\n\\end{quote}\n\\endgroup\n\n\n\\noindent The long-term vision foresees a technology where robots for a certain application can be `bred' through iterated selection and reproduction cycles until they satisfy the users' criteria. This approach is not meant to replace classic engineering when designing robots for structured environments with known conditions. However, for complex, unstructured environments with (partially) unknown and possibly changing conditions evolution offers great advantages \\cite{howard2019evolving}. A crucial difference between a system of evolving robots and a usual evolutionary algorithm is that the members of the population are not digital objects in a virtual space, but physical artefacts in the real-world. Such a system goes beyond evolutionary computing and implements the Evolution of Things with several new challenges rooted in the physical incarnation \\cite{Eiben2012Embodied-Artifi,eiben2015evolutionary}.\n\nAnother property distinguishing robot evolution from mainstream evolutionary optimization is that robots have agency, i.e., they are active artefacts that exhibit autonomous behavior. To this end, it is important to note that a robot is a combination of its body (morphology, hardware) and its brain (controller, software), and the behavior depends on both \\cite{Louise2011Beyond,weigmann2012does}. This implies that the evolution of robots should concern both the bodies and the brains \\cite{pfeifer2007body}. \n\nThus, in a full-fledged evolutionary robot system the morphologies as well as the controllers undergo evolution. This is in stark contrast with the current practice. Evolutionary robotics today is mainly concerned with evolving the controllers of simulated robots \\cite{radhakrishna2018survey}. Systems where morphologies and controllers of robots evolve simultaneously are rare and they work in simulation \\cite{Auerbach_Bongard2014Environmental}. Occasionally, an organism evolved in software is constructed in the real world, using hardware \\cite{lipson2000automatic} or `wetware' \\cite{Kriegman2020}, but the evolutionary process is simulated. This evolve-then-construct approach inevitably runs into the reality gap \\cite{jakobi1995noise}. On the other hand, there exist systems where physical objects are evolved in the real-world, but these are passive artefacts without agency \\cite{Rieffel-Sayles-2010,Kuehn-Rieffel-2012}. Complete systems where real robots undergo evolution are still ahead of us, but with the development of 3D-printing, rapid prototyping, and automated assembly such systems are becoming feasible, at least in an academic setting \\cite{brodbeck2015morphological,hale2019robot, jelisavcic2017real,Vujovic2017}. \n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=.33\\textwidth]{images\/Triangle-of-Life.pdf}\n    \\caption{Generic system architecture for robot evolution conceptualized by the Triangle of Life{}.\n    }\n\t\\label{fig:system}\n\\end{figure} \n\nA generic model to underpin ``evolving robots in real time and real space'' has been described recently \\cite{eiben2013triangle}. \nThis model, called the Triangle of Life{}, captures the universal life cycle of a robot, not from birth to death because that would not be a cycle, but from conception (being conceived) to conception (conceiving offspring). This cycle consists of three stages: morphogenesis (from conception to birth), infancy (from birth to becoming a fertile adult), and mature life (during which the robot can mate and conceive offspring multiple times), cf. Figure \\ref{fig:system}. \n\n\nA key insight behind this paper is that including a learning stage is not an arbitrary design choice, it is pivotal for mitigating a general problem. Namely, while it can be assumed that the parents had well-matching bodies and brains (otherwise they had not been fit enough to be selected for mating), in general it cannot be assumed that crossover preserves the good match. The mis-match in the offspring may be severe (e.g., having more sensors than there are inputs in the controller) or moderate (e.g., only requiring some parameter tuning in the brain), but in any case it is important to adjust and optimize the inherited brain quickly after `birth'. \n\nThe problem we highlight here is inherent to morphological robot evolution where a large number and variety of robots is produced through consecutive generations. All these robots with different and unpredictable morphologies have to learn the tasks required by the given application. Thus, a morphologically evolving robot system needs a learning method that works in any of the possible robots and finds a good controller efficiently. \n\n\n\n\n\nComparing with current studies, the main contributions of the paper are the following. \nFirstly, a new type of controller architecture based on coupled oscillators with sensory feedback that can be customized to any modular robot driven by joints. \nIn such a type of controller, a generic method is used to specify a frame of reference (coordinates for the robot modules) for any given body in our design space.\nA steering mechanism based on scaling the activation signals, depending on the coordinates of the joint and the angle between the direction to the target and the robot's heading, is used to drive the robot joints. \nThese controllers can be used in a closed-loop approach and steer a robot towards a target regardless of the specific shape of modular robots. \nThis makes it possible to generate a closed-loop controller for a modular robot with an arbitrary shape.\nSecondly, a generic learning method that allows modular robots with arbitrary shapes to learn approaching a target. \nWe validate our method with three different robots, rather than a fixed and special shape robot, in three different scenarios. To this end, we use three modular robots, a `spider', a `gecko', and their `baby' created by an evolutionary robotics project in our lab \\cite{jelisavcic2017real}. \n\n\n\n\n\n\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{images\/all_robots.pdf}\n    \\caption{The three robots in our test suite after \\cite{jelisavcic2017real}. The spider (b) and gecko (c) are the parents, the baby (a) is the offspring. (d) exhibits the overall network topologies of the controllers and the coordinates of the joints in the baby (left), spider (middle), and gecko (right) robot. The core block (head) is red, other blocks are black. Joints are represented by circles.\n   \n    (e) shows the inner components of the spider robot.}\n    \\label{fig:robots}\n\\end{figure}\n\n\n\n\n\\section{Good locomotion and how to learn it}\n\\label{sec:loco-learn}\n\nThe existing studies in learning locomotion on modular roobts can be divided into two categories in terms of the controllers, open-loop and closed-loop. The dominant approaches are using Central Pattern Generators (CPGs) \\cite{Ijspeert2008learning}.\nMost papers studied gait learning with real robots using an open-loop controller with no sensory feedback from the environment \\cite{Kamimura2005Automatic,Bongard2006Science,Kyrre2015real-world,Thakker2014ReBis}. \nFor closed-loop controllers, joint angle and foot contact are typically used to be the sensory feedback. \nThe studies \\cite{owaki2013simple,owaki2017quadruped,nordmoen2019evolved} used a CPG-based controller and foot contact from force sensors on each robot leg to produce coordinated gaits and increase its adaptability on various terrains. \nInertial measurement \\cite{wang2005motion,seo2010cpg,barasuol2013reactive,sartoretti2018central}, joint angles \\cite{kimura2007adaptive}, and touch sensing \\cite{righetti2008pattern,ajallooeian2013central} are used to be sensory feedback in CPG-based controllers to adjust robot behaviours for desired tasks. \nIn particular, the combination of a CPG-based controller and camera is used to achieve the closed-loop control for directed locomotion in a hexapod robot \\cite{shaw2019workspace}.\nSimilarly, there are other studies \\cite{wu2013neurally,Ijspeert2007From} that use various sensory feedback and controllers to achieve closed-loop control for different tasks.\nAlthough these studies proposed closed-loop controllers with various sensory feedback for locomotion, they focus on learning locomotion on a fixed shape robot such as a hexapod robot, a fish robot, a salamander robot. It is still generally lacks generic methodologies for integrating sensory feedback to adapt the locomotion for a modular robot with an arbitrary shape.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn this paper, we consider the problem of targeted locomotion on modular robots with arbitrary shapes. This is important, challenging, and novel. Targeted locomotion is important, because for many applications robots should be capable of going to a point of interest, be it a charging station, an object to fetch, or another robot to mate with. Targeted locomotion on evolvable modular robots is challenging because the number and the spatial arrangement of the joints, the length and branches of the limbs can vary and the overall shape can be irregular. This makes the simple adoption of the steering policy for wheeled robots --to turn left (right) apply more force to the wheels on the right (left)-- highly non-trivial. Finally, this problem is also novel; to our best knowledge there are no publications addressing this. \n\n\n\\subsection{Robots with a sense of direction}\n\\label{sec:loco}\n\nTo enable targeted locomotion of modular robots with arbitrary morphologies, we need a generic way of defining a \\textit{frame of reference}. To this end we define a coordinate system based on the fact that our robots have a core module with a camera, see Figure \\ref{fig:robots}. By definition, the `head', that is, the core module is the origin $(0,0)$ and the direction of the camera determines `North'. Now we can assign the coordinates $\\{(1,0), (2,0), ...\\}$ and $\\{(-1,0), (-2,0), ...\\}$ to the modules (passive bricks or active joints) as we go to East and West, respectively.\nSimilarly, if we go North and South we set the coordinates to $\\{(0,1), (0,2), ...\\}$ and $\\{(0,-1), (0,-2), ...\\}$, respectively. The coordinates of the three robots in this paper are shown in Figure \\ref{fig:robots}.\n\n\n\n\nTo make robots see the target we use a system proposed in \\cite{lan2018ICARCV}. When a target is detected its direction $\\alpha \\in [-\\beta, \\beta]$ w.r.t. the robot can be determined, where $[-\\beta, \\beta]$ is the cameras field of vision. This information can be combined with the frame of reference defined above: If $\\alpha < 0$, then the target is on the left; otherwise it is on the right.\n\nFollowing the literature we use coupled oscillators to control the joints of a robot and we arrange these oscillators in a network to form Central Pattern Generators (CPGs) \\cite{ijspeert2008central, steuer2019central}. Such oscillators generate tightly-coupled patterns of neural activity that drive rhythmic and stereotyped locomotion behaviors like walking, swimming, flying in vertebrate species and they have been proven to perform well in modular robots as well \\cite{hultborn2007spinal,ijspeert2008central,Ijspeert2007From}.\n\nTo provide useful information from the environment we introduce the notion of a sensory oscillator as shown in Figure \\ref{fig:framework} (a). \nSuch a sensory oscillator drives one joint in the robot by two coupled neurons $x$ and $y$, and an $out$-neuron that together generate a signal regulated by three weights as usual. \nSubsequently, the extra (square shaped) node generates the actual signal $sig$ for the joint by applying a function $f$ to the sensory information $sen$ and the signal of the $out$-neuron.\nThe novelty of this model lies in the extra node that considers the sensory input before generating the signal that actually drives the joint. The model is general, there is no restriction on the sensory input(s) and $f$ can be any function depending on the task at hand.\n\nThe subsystems described above can be combined into an adequate control system that enables the modular robots to move towards a target. To this end, we use the angle to the target $\\alpha$ as input and generate a scaling factor $d_p(\\alpha)$ and define a function $f$ to adjust the signals of the joints on the left and right side of the robots as necessary. The exact details are described in the Methods section, the overall effect is that if the target is on the left (right), then the joints on the left (right) apply less force and make the robot turn in the correct direction. The middle joints are never modified. The overall architecture is exhibited in Figure \\ref{fig:framework} (b).\n\\begin{figure}[!htbp]\n\t    \\centering\n\t    \\includegraphics[width=\\columnwidth]{images\/framework.pdf}\n\t\t\\caption{(a) A sensory oscillator with three neurons, $x$-neuron, $y$-neuron, $out$-neuron and an extra node $f$. The function $f$ combines the raw control signal of the $x$-neuron with the sensory information $sen$. (b) Overall scheme of steering by sending control commands to actuators depending on their lateral position and the direction of the target.}\n\t\t\\label{fig:framework}\n\\end{figure}\nThe overall control system of a robot is a CPG network where sensory oscillators of neighboring joints are connected as shown in Figure \\ref{fig:robots} (d).\n\n\n\\subsection{Learning method}\n\\label{sec:learn}\n\nLearning a task in our system amounts to finding proper weights for the controller. This boils down to optimizing a black-box objective function. Bayesian optimization (BO) is the state-of-the-art method in terms of data efficiency, but its computational complexity grows cubically with respect to the number of observations \\cite{jasper2012practical}. \nAlternatively, Evolutionary Algorithms (EA) take constant time for generating candidate solutions \\cite{eiben2015evolutionary}. This makes their overhead much less computation-intensive than that of the BO, at the expense of data efficiency. \n\nTo get the best of both worlds, here we use a combined algorithm, the Bayesian Evolutionary Algorithm (BEA), that starts with BO and runs this until the time efficiency becomes too low. At this point, a certain subset $S$ of the solutions generated so far is selected by considering their quality and diversity. This set $S$ is then used as the initial population for the EA that is run until a good solution is found or the given computing budget is exhausted. We refer the reader to the  Methods section for further details.\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nWe employed the BEA to learn adequate controllers in simulation. Then we tested the learned controllers in three test scenarios: 1) approaching a fixed target, 2) following a moving target, and 3) following another robot that is following a moving target. Let us note that the robots do not change or tune their controllers between scenarios. \n\nWe used the customized framework \\emph{Revolve} \\cite{hupkes_2018_revolve} to learn appropriate controllers for directed locomotion at a zero angle (thus: straight forward) on the virtual gecko, spider, and baby robots.\nThe fitness function for the BEA was a combination of the deviation from the required direction and the distance covered during the test period, see the Methods section for details. The duration of a test period was 60 seconds and the BEA was allowed to perform 1500 evaluations. \nDespite the heavy simulations, the computing times were acceptable, approximately one hour was enough to complete one run on a Linux PC with a 3.0GHz CPU, 64Gb RAM, and 32 cores with multi-threading. Conducting the whole learning process on the real robots would take several days, $1500 \\times 60$ seconds, plus overhead for re-positioning the robots between tests, charging the batteries, and handling breakdowns.\n\n\n\n\\subsection{Scenario 1: fixed target}\n\\label{sec:fixed}\nThe controllers learned in simulation are validated on the real robots, the spider, the gecko, and the baby. In the first series of experiments, we tested each robot with three different setups, one with a fixed target to the left, one with the target straight ahead of the robot, one with the target to the right. In all cases, the target was in the field of view of the robot camera (Raspberry Pi Camera Module v2) at the start of the test. We repeated each test five times and displayed the observed trajectories in Figure \\ref{fig:trajectories}.\nThe experiments were recorded with an overhead camera above the test arena, as shown in the supplementary videos.\nThe plots show a consistent behavior for all robots and test cases. The path of the spider shows that it is `wobbling', see the top left plot in Figure \\ref{fig:trajectories}. This is understandable because its middle-line lies over two of its limbs, hence straight forward movement is very hard without zigzagging. Comparing the blue curves with the other colors we can also see that the robots approach the target in the center faster. This result can be easily explained, because for a target on one of the sides, the robot must turn and that costs extra time.\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t    \\begin{adjustbox}{max width=0.49\\textwidth}\n\t\\begin{tabular}{c c c}\t\t\\includegraphics[width=0.4\\textwidth]{images\/spider9_trajectory_all.pdf}&\n\t\t\\includegraphics[width=0.4\\textwidth]{images\/gecko7_trajectory_all.pdf}&\n\t\t\\includegraphics[width=0.4\\textwidth]{images\/babyA_trajectory_all.pdf} \\\\\n\t\\end{tabular}\n    \\end{adjustbox}\n    \\begin{adjustbox}{max width=0.49\\textwidth}\n\t\\begin{tabular}{c c c}\n\t\t\\includegraphics[width=0.4\\textwidth]{images\/spider9_moving_tra_all.pdf}&\n\t\t\\includegraphics[width=0.4\\textwidth]{images\/gecko7_moving_tra_all.pdf}&\n\t\t\\includegraphics[width=0.4\\textwidth]{images\/babyA_moving_tra_all.pdf} \\\\\n\t\\end{tabular}\n    \\end{adjustbox}\n\t\\caption{Trajectories of the spider (left), the gecko (middle), and the baby (right). The solid lines show the average trajectories (five runs). The hexagonal bins show the number that robots located in the hexagonal area for all five repetitions, where the location of robots are collected per 0.5 second. Top row: moving towards a fixed target, the coloured circles. Bottom row: following a moving target. The red line shows the path of the target robot, the blue one belongs to the `chaser'.  \n\n\t}\n\t\\label{fig:trajectories}\n\\end{figure}\n\n\\subsection{Scenario 2: moving target}\n\\label{sec:moving}\nIn the second series of experiments, we tested each robot with a moving target. To this end, we used a wheeled Robobo robot that was pre-programmed to drive a given trajectory. In the initial position, the Robobo was approximately 30 cm ahead of the modular robot and started to drive to the right, then it turned and drove to the left. The bottom row of Figure \\ref{fig:trajectories} shows the trajectories after five repetitions with each modular robot. Additionally, we recorded the experiments with an overhead camera above the test arena (see Supplementary Video). These data indicate that the modular robots were able to follow the Robobo in all cases. This demonstrates that the controllers learned for a simple task (moving straight ahead) were applicable in a different and more difficult task. In turn, this proves the usefulness of our new controllers based on an internal frame of reference and sensory oscillators. \n\n\n\\begin{figure*}[!ht]\n\t\\centering\n    \\begin{adjustbox}{max width=0.95\\textwidth}\n\t\\begin{tabular}{c c c c c}\n\t\t\\includegraphics[width=0.385\\textwidth,angle=90]{images\/babyA2gecko1_1.png}& \\hspace{-4mm}\n        \\includegraphics[width=0.385\\textwidth,angle=90]{images\/babyA2gecko1_2.png}& \\hspace{-4mm}\n\t\t\\includegraphics[width=0.385\\textwidth,angle=90]{images\/babyA2gecko1_3.png}& \\hspace{-4mm}\n\t\t\\includegraphics[width=0.385\\textwidth,angle=90]{images\/babyA2gecko1_4.png}& \\hspace{-4mm}\n\t\t\\includegraphics[width=0.385\\textwidth,angle=90]{images\/babyA2gecko1_5.png} \\\\\n\t\t\\includegraphics[width=0.363\\textwidth,angle=90]{images\/gecko2spider1_1.png}& \\hspace{-4mm}\n\t\t\\includegraphics[width=0.363\\textwidth,angle=90]{images\/gecko2spider1_2.png}& \\hspace{-4mm}\n\t\t\\includegraphics[width=0.363\\textwidth,angle=90]{images\/gecko2spider1_3.png}& \\hspace{-4mm}\n\t\t\\includegraphics[width=0.363\\textwidth,angle=90]{images\/gecko2spider1_4.png}& \\hspace{-4mm}\n\t\t\\includegraphics[width=0.363\\textwidth,angle=90]{images\/gecko2spider1_5.png} \\\\\n\t\\end{tabular}\n    \\end{adjustbox}\n\t\\caption{Still images of the double moving target experiments. Top row: the gecko follows the hand-held target and the baby follows the gecko. Bottom row: the spider follows the hand-held target and the gecko follows the spider. The hand-held target is the red cylinder at the end of a white stick. The red and blue lines show the trajectories of the first and the second robot, respectively.}\n\t\\label{fig:double-overhead}\n\\end{figure*}\n\n\\subsection{Scenario 3: double moving target}\n\\label{sec:d-moving}\n\nIn the third series of real-world experiments, we challenged the modular robots even further. In this setup, we replaced the Robobo used in the previous experiments by one of the modular robots and hand-held a target in front of it. Another modular robot was placed on the regular starting position. Then the robots started at the same time, the first one following the target hand-held by the experimenter, the second one following the first one. In Figure \\ref{fig:double-overhead}, we show two of these tests, the case of the gecko following the hand-held target and the baby following the gecko and the case of the spider following the hand-held target and the gecko following the spider. \nThe figure captures the test by five still images taken by the overhead camera, more information is shown in Supplementary Video. These show that robots can follow a target even if it is an irregularly moving irregular shape (another modular robot). The trajectories also disclose the speed differences. In the first test, the distance between the gecko and the baby is gradually growing, which indicates that the baby is a little slower than the gecko. In particular, the baby encountered a little difficulty in turning right. This is not surprising, given that its morphology with the long right limb is not perfectly symmetrical. In the second test we see the opposite, the second robot is closing in on the first one, and before the end of the experiment the gecko hits the spider.\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\n\nThe wider context of this study is full-fledged robot evolution, where both morphologies and controllers evolve. We note that a morphologically evolving robot system needs a learning component that works in any of the possible robots and adjusts the inherited controller (or a randomly initialized controller) to the given morphology and task. \n\nOur experimental work is, in essence, a feasibility study addressing this general issue within our system of evolvable robots and a specific task. \nThe concrete problem we tackle is to enable targeted locomotion, that is, approaching and following an object, in modular robots without any assumption on their specific morphology. This forms a good test case because it is morphology dependent, challenging for robots with random shapes, and practically relevant. The solution we deliver is based on an internal frame of reference and the use of oscillators with sensory feedback to drive the joints. These are combined into a control system that modulates the force in the servo motors depending on the robots angle to the target and the location of the joint in the robots body. The overall control system is a CPG-based network where neighboring joints of the robot are connected. The other ingredient of the solution is a learning method that can find good parameter values for any given CPG-based network. \n\nTo validate our approach we designed nine test cases by three robots with different morphologies and three scenarios, one with a fixed target, one with a moving target, and one with a double moving target. The robots learned a good controller for moving straight forward and these controllers were evaluated in the test scenarios.\n\nIn the first scenario, all robots approached the target accurately, even though the spider exhibited a little offset to the right, cf. Figure \\ref{fig:trajectories} and Supplementary Video. In the second scenario, the target moved to the right first and turned to the left halfway the test. All three robots could adjust their trajectory and followed the target with a little delay as can be seen in Figure \\ref{fig:trajectories}. This proves that the sensory-motor feedback loop they learned for one basic skill (straight forward movement) was generalizable to a more dynamic and complex task.\n\nIn the third scenario, a modular robot had to follow another modular robot that followed a hand-held target. This test further confirmed the adequacy of the learned controllers and illuminated the differences between the locomotion abilities. For instance, the baby followed the gecko well in a left turn, but it fell behind when turning right because of the longer right-front limb as shown in the fifth frame of Figure \\ref{fig:double-overhead}.\nIn the other test, the gecko followed the spider easily because of the wiggly locomotion of the spider and the relatively high speed of the gecko.\n\nThe results of the experiments demonstrate that the new type of oscillators and CPGs with external feedback in combination with the internal frame of reference empower robots with different morphologies to perform object following. Our frame of reference is in essence a 2-dimensional coordinate system, where the origin and the `North' are defined by specific features in our robots, the head module and the direction of the camera, respectively. This concept can be easily extended to three dimensions and to different robots with other features. For instance, a designated head module is not required, the origin can be defined by any reasonable principle that applies to the given morphologies. Likewise, while using the direction of the camera to define `North' is a natural choice in our application, a coordinate system can be defined for robots without a camera as well. \n\nFurther extensions and generalizations are possible regarding the feedback from the environment. Although in this study we use a camera, our approach is applicable in a wide range of robot systems, because the sensory oscillators can handle inputs from different sensors, e.g., accelerometers, gyroscopes, IR sensors, sonars. Furthermore, in this paper all joints on the same side are treated the same way, cf. Equations \\ref{eq:l_joint_downscale} and \\ref{eq:r_joint_downscale}, but it is possible to define a variant where the exact coordinates, for instance the distance to the `spinal cord', are also taken into account. \n\nLast but not least, improvements are possible by employing another learning method. The BEA we use here is not application specific, in principle any derivative-free black-box optimization algorithm for learning the adequate parameter values for the controller is applicable. In our view, it is very important to consider both sample efficiency and time efficiency, that is, the number of trials or evaluations as well as the time needed to achieve a decent result. In this paper we used a budget of 1500 trials and the BEA spent about one hour on learning. This is certainly acceptable and delivers the proof of concept we aimed for, but we are convinced that these figures can be improved by more advanced methods. \n\nA particular aspect here is the dichotomy between the real-world and simulations. To this end, it is important to note that the use of simulations does not invalidate the concept of physical robot evolution. In fact, simulations can be used in both the evolutionary and the learning loop.\nAs outlined in \\cite{howard2019evolving} and \\cite{hale2019robot} there are great potential advantages in evolving real and simulated robots simultaneously. If the simulated and real robots share the same genetic language then we can `cross-breed' them, using a virtual `mother' and a physical `father'. In such a hybrid evolutionary system physical evolution is accelerated by the virtual component that can find good robot features with less time and resources than physical evolution, while simulated evolution benefits from the influx of genes that are tested favourably in the real world. Additionally, even if we have clean physical evolution and all robot `children' are produced in the real world, the learning process in the Infancy stage can use simulations to obtain a good controller for the given robot `child'. The learned controllers can suffer from the reality gap, which can be mitigated by a subsequent real-world learning process on the physical robot. The advantage of the combination is that virtual learning can be on a meso-scale, spending quite a few trials (e.g., hundreds, like in our case) and the real-world process on a micro-scale with much fewer trials (perhaps just a few dozens). For the sake of the argument, if 100 real-world trials were enough, then the learning process could be completed on the real robots in an afternoon. The current paper is on the simulated side. The controllers learned in simulation just worked for us here, thus we had no need to add a real-world learning process. \n\n\n\n\nReflecting from a broader perspective, this work provides an approach for learning tasks inside an evolutionary process that produces various morphologies. Our approach is generic, applicable to various types of evolvable robot systems and different tasks. For example, it can be applied in a system for evolving robots for exploring and decommissioning nuclear power plants \\cite{hale2019robot}, as well as in more fundamental studies regarding the evolution of embodied intelligence. \n\n\n\n\n\n\n\n\\section{Methods}\n\\label{sec:methods}\n\n\\subsection*{The general system}\n\\label{subsec:general_framework}\n\nThe robots in our system are based on RoboGen \\cite{auerbach2014robogen}, they consist of a core component that hosts the controller board, the battery and a camera, 3D-printed passive bricks, and joints driven by servo motors \\cite{jelisavcic2017real}. Figure \\ref{fig:robots} displays the three robots we use in the current experiments. The camera provides information about the environment that is processed by a new type of control system based on sensory oscillators that activate the servos. \nA schematic representation of the system is presented in Figure \\ref{fig:framework} (b). \nIn the next subsections, we discuss the details of each component of the system.\n\\subsection*{Robot Vision}\n\\label{subsec:robot_vision}\n\nThe closed-loop controllers are based on visual input delivered by a camera. The robot vision system must work accurately in real-time on our Raspberry Pi Camera Module v2, and ideally be power efficient. The two pivotal steps for this system are the recognition of objects of interest, and the calculation of the angle of a target object w.r.t. the orientation of the robot.\n\n\\paragraph{Object recognition} \nHere, we follow the approach proposed in \\cite{lan2018ICARCV,lan2019evolving} that allows to quickly recognize targets with low-performance hardware.\nThe method consists of two components:\n\\begin{itemize}[nolistsep]\n    \\item Detection of region of interests (ROIs) using \\textit{Fast ROIs Search} proposed in \\cite{lan2018ICARCV}.\n    \\item Object recognition using Histogram of Gradients (HOG) as a feature extractor and Support Vector Machines (SVM) with the linear kernel as a detection method.\n\\end{itemize}\nWe decided to use the combination of HOG and SVM instead of Deep Neural Networks since HOG and SVM perform much faster. In this project, we used the implementation provided by OpenCV.\n\n\\paragraph{Angle to target object}\nOnce a target is detected, its relative position from the robot's perspective can be expressed by the angle $\\alpha \\in [-\\beta, \\beta]$, where $\\beta$ is the angle determined by the field of vision of the given camera. The angle to the target $\\alpha$ is the angle between the orientation of the robot (i.e, the camera) and the target object. If the target is on the left-hand side of the robot's face, then $\\alpha$ is negative, whereas if the target is on the right-hand side of the robot's face, $\\alpha$ is positive. The value of $\\alpha$ is zero if the target is straight ahead of the robot. \nGiven an image registered by the Raspberry Pi Camera Module v2 with the parameters (the field of view ($2 \\times \\beta$) is $\\ang{62.2}$, the resolution is $3280 \\times 2464$ where 3280 is the number of pixel columns ($\\mathcal{N}_c$)), where the coordinate of the target can be recognized by the robot vision as $(x,y)$ in pixels, $\\alpha$ is calculated from this image by\n\\begin{equation}\n    \\alpha = \\frac{\\arctan(\\frac{x - \\mathcal{N}_c \/ 2} {\\mathcal{F}})} { \\pi \\times 180 }\n\\end{equation}\nwhere $\\mathcal{F}$ is a potential inherent factor that depends on the camera, can be calculated by \n\\begin{equation}\n    \\mathcal{F} = \\frac{ \\mathcal{N}_c \/2}{\\tan(\\frac{\\beta}{180 \\times \\pi})}\n\\end{equation}\n\n\n\n\n\n\nDue to the limited camera's field of view in the real world, the robots have to process the situation that the target is out of the camera's field of view. \nFor such a situation, we expect the robots to search the target until the target is in the camera's field of view.\nTo this end, we use two solutions: 1) If the target is out of the camera's field of view initially, the robots have to search the target until it is in the camera's field of view. For this purpose, the initial value of $\\alpha$ can be set as $\\beta\/2$ (or $-\\beta\/2$) for turning right (or left).\n2) If the target escapes from the camera's field of view during locomotion, the robots keep the previous behaviours to follow the target until the target comes back to the field of view. \nThat is, $\\alpha$ preserves the previous value if no target is in the field of view except the initial stage.\n\n\n\n\n\n\n\n\n\n\\subsection*{Controller}\n\\label{subsec:controller}\n\n\\paragraph{Modeling joints as oscillators}\nThe key element of the controller is the model of a single joint. Here, we propose a new oscillator with sensory feedback to properly represent oscillatory behavior often seen in nature \\cite{ijspeert2008central}.\nThe controller composed of new oscillators works in a closed-loop, in which the robot's action can be changed according to the sensory feedback about the target in the environment.\nIn general, the sensory input to the new oscillator can be generated by any sensors.\n\nA sensory oscillator has an $x$-neuron, a $y$-neuron, an $out$-neuron, and an extra node that implements a function $f$.\nFor each time step, neuron $x$ ($y$) feeds its activation value multiplied by the weight $w_{xy}$ ($w_{yx}$) to the neuron $y$ ($x$).\nAt a time step $t$, the changes of the activation value of an $x$-neuron and a $y$-neuron can be calculated as $\\Delta x_{(t)}  = w_{yx}y_{(t-1)}$ and $\\Delta y_{(t)}  = w_{xy}x_{(t-1)}$ respectively, \nwhere $t-1$ represents the last time step. \nThe $x$-neuron and the $y$-neuron generate the activation values $x_{(t)}$ and $y_{(t)}$ of oscillatory patterns over time according to the following expression:\n\\begin{align}\n    \\begin{split}\n        x_{(t)} = x_{(t-1)} + \\Delta x_{(t)} \\\\\n    \ty_{(t)} = y_{(t-1)} + \\Delta y_{(t)}  \n    \\end{split}\n    \\label{eq:activation}\n\\end{align}\n\n\\par The $x$-neuron feeds its activation value multiplied by the weight $w_{xo}$ to the $out$-neuron.\nThe $out$-neuron applies an activation function to generate the activation value.\nFor the oscillator in the CPG-based controller of modular robots with joints driven by servo motors, the activation values of the $out$-neurons have to meet two conditions due to the limited rotating angle of the joints. \nFirst, the activation value of $out$-neuron must be periodic that repeatedly returns to its initial condition.\nAccording to the stability criterion for linear dynamical systems \\cite{bubnicki2005modern}, it is beneficial to take $w_{yx} = - w_{xy}$. Such values of parameters lead to periodic signals that do not explode over time.\nIn this study, we use the predefined initial values $(x_{(0)}, y_{(0)}) = (-\\frac{1}{2}\\sqrt{2}, \\frac{1}{2}\\sqrt{2})$, and $(w_{xy}, w_{yx}) = (0.5, -0.5)$, but they can be randomly initialized except 0. \nSecond, the activation value of $out$-neuron should be bounded in an interval.\nTherefore, we use a variant of the sigmoid function, the hyperbolic tangent function ($tanh$), as the activation function of $out$-neurons to bound the output value in $[-1,1]$.\nAt a time step $t$, the $tanh$ activation value of $out$-neuron can be calculated as follows:\n\\begin{equation}\n    out_{(t)} = \\frac{e^{x_{(t)}} - e^{- x_{(t)}}}{e^{x_{(t)}} + e^{- x_{(t)}}} .\n    \\label{eq:output}\n\\end{equation}\n\n\nFinally, the new oscillator executes an extra operation $f$ that combines the activation value of $out$-neuron and the external sensory signal $sen$ to produce a signal $sig$, i.e., $sig = f(sen, out)$.\nIn general, $f$ could be any function, here we use multiplication of $out$ and $sen$.\nHence, at a time step $t$, the $sig$ value can be calculated by:\n\\begin{equation}\n    sig_{(t)} = sen_{(t)} \\times out_{(t)}\n   \n    \\label{eq:multiplication}\n\\end{equation}\nThe rationale behind our new oscillator model is to allow the inclusion of a sensory signal for a closed-loop control. We refer to this new model as the sensory oscillator (see Figure \\ref{fig:framework} (a)). \n\n\n\\paragraph{Network of oscillators}\n\nFor modular robots, e.g., the robots in Figure \\ref{fig:robots}, there are multiple joints affecting each other for achieving the actions.\nIn other words, we deal with a network of joints (oscillators) and we must take into account the influence of other oscillators.\nIn the CPG-based network controller, the neighboring oscillators are connected to each other as the blue arrow lines shown in Figure \\ref{fig:robots} (d).\nAs a result, we have a further expression for a single oscillator by including the neighborhood instead of Equation \\ref{eq:activation}.\nFor the $i$-th sensory oscillator, the activation values $x_{i(t)}$ of its $x$-neuron and $y_{i(t)}$ of its $y$-neuron can be calculated as:\n\\begin{align}\n    \\begin{split}\n        x_{i(t)} &= x_{i(t-1)} + \\Delta x_{i(t)} + \\sum_{j \\in \\mathcal{N}_i} x_{j(t-1)} w_{ji}\\\\\n    \ty_{i(t)} &= y_{i(t-1)} + \\Delta y_{i(t)}  \n    \\end{split}\n    \\label{eq:network}\n\\end{align}\nwhere \n$\\mathcal{N}_i$ is the set of indices of the oscillators connected to the $i$-th oscillator,\n$w_{ji}$ is the weight between $i$-th oscillator and $j$-th oscillator.\nThe connected oscillators in the CPG-based controller cooperate to achieve the desired task.\nThe number of weights that need to be optimized for the controllers of the robots, spider, gecko, and baby are 18, 13, and 16, respectively.\n\n\n\\paragraph{Steering}\nThe usual steering policy for wheeled robots is relatively easy, for left (right) turn the force needs to be reduced on the left (right) wheel. Here we generalize this idea to modular robots with no assumptions about the morphology. Our method allows for scaling the activation signals for the joints depending on the coordinates of the joint and the angle $\\alpha$ between the direction to the target and the robots heading. The key idea is to use a scaling factor \n\\begin{equation}\nd_p(\\alpha) = \\left(\\frac{\\beta - | \\alpha |}{\\beta}\\right)^p ,\n\\label{eq:cp}\n\\end{equation}\nwhere $p > 0$ is a user parameter that determines how strongly we penalize the deviation $\\alpha$. \nIn this study, we use the value of 7 for the parameter $p$ by the experiments of parameter tuning.\nRecall that robots field of vision is the region between $-\\beta$ and $\\beta$, hence $\\alpha \\in [-\\beta, \\beta]$, and $\\alpha < 0$ means that the target is on the left, $\\alpha > 0$ means the target is on the right.\n\nThis scaling factor is used to modify the signals to the joints on the left as follows: \n\\begin{equation}\n  sig = \\begin{cases}\n        d_p(\\alpha) \\cdot out & \\textit{if } \\alpha < 0 \\\\\n        out  & \\textit{if }\\alpha \\ge 0\n    \\end{cases}\n    \\label{eq:l_joint_downscale}\n\\end{equation}\nand, analogously, the signal for the joints on the right is modified as follows:\n\\begin{equation}\n  sig = \\begin{cases}\n        out & \\textit{if } \\alpha < 0 \\\\\n        d_p(\\alpha) \\cdot out  & \\textit{if }\\alpha \\ge 0 .\n    \\end{cases}\n    \\label{eq:r_joint_downscale}\n\\end{equation}\nThe signals for the middle joints are never modified.\n\n\nObserve that by these formulas we define a specific implementation of the square shaped extra node within a sensory oscillator in Figure \\ref{fig:framework}. We use $d_p(\\alpha)$ as the sensory information $sen$ and the function $f$ is simple multiplication. To our knowledge, this is a novel method. The only other existing work that is remotely similar is that of \\cite{Ijspeert2007From}, where an internal signal is used to modify the working of oscillators for two predefined locomotions, walking and swimming.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[!tbp]\n    \\centering\n    \\begin{adjustbox}{max width=0.99\\textwidth}\n\t\\begin{tabular}{c c c}\n\t    \\Large{spider} & \\Large{gecko} & \\Large{baby} \\\\\n\t    \\includegraphics[width=0.4\\textwidth]{images\/spider9_bo_ea_bea.pdf} &\n\t\t\\includegraphics[width=0.4\\textwidth]{images\/gecko7_bo_ea_bea.pdf} &\n\t\t\\includegraphics[width=0.4\\textwidth]{images\/babyA_bo_ea_bea.pdf} \n\t\\end{tabular}\n    \\end{adjustbox}\n    \\caption{The performance of Bayesian optimization (blue), an evolutionary algorithm (black), and the BEA (green) on the directed locomotion task in simulation. In each plot we provide the mean value (a solid line) with two standard deviations (a shadowed region). The purple dashed lines present the switch points. The plots on the left, in the middle and on the right represent results for spider, gecko, and baby, respectively.}\n    \\label{fig:bea_comparison}\n\\end{figure*}\n\n\\subsection*{Fitness function for directed locomotion}\n\\label{sec:function}\nThe fitness function used here is to evaluate the performance of controllers for the task of directed locomotion. This task is defined by a target direction $\\gamma$ that the robot has to follow. The good fitness function needs to combine two objectives: minimizing deviation with respect to the target direction $\\gamma$ and maximizing speed (i.e., displacement) over the evaluation period (60 seconds in our experiments). \nTo calculate the fitness value we need \n\\begin{itemize}[noitemsep,nolistsep]\n    \\item the robots starting position $p_0$ at the beginning of the evaluation period,\n    \\item the robots final position $p_1$ at the end of the evaluation period,\n    \\item the deviation angle $\\delta$ between the target direction $\\gamma$ and the line drawn between $p_0$ and $p_1$,\n    \\item the total length $L$ of the trajectory travelled during the evaluation period (which is not the distance between $p_0$ and $p_1$).\n\\end{itemize}  \n\nThe fitness value is then composed of several components. Component one is to maximize the distance travelled in the right direction. This distance can be expressed by the value $E_1 = d(p_0,p_1) \\times cot(\\delta)$, where $d(x,y)$ is the Euclidean distance between two points $x$ and $y$ and $cot$ is the cotangent function. Another component is to minimize the deviation w.r.t. the target angle. This can be expressed not only by $\\delta$ but also by the distance of the of the final position $p_1$ from the ideal trajectory starting at $p_0$ and following the target direction. This distance can be expressed as $E_2 = d(p_0,p_1) \\times tan(\\delta)$, where $tan$ is the tangent function. The third component is to reward locomotion in a straight line. This can be simply captured by the value $E_3 = d(p_0,p_1) \/ (L + \\epsilon$), where $\\epsilon$ is an infinitesimal constant. This value is maximized when the length $L$ of the travelled trajectory equals the distance between the starting point $p_0$ and the end point $p_1$. Combining these components into one formula we obtain the following fitness function:\n\n\\begin{equation}\n\\label{eq:fitness}\nF = E_3 \\cdot \\Big{(} \\frac{E_1}{\\delta + 1} - w \\cdot E_2 \\Big{)},\n\\end{equation}\nwhere $w > 0$ is a penalty factor. This function maximized when $d(p_0,p_1)$ is maximal, $\\delta$ (and hence $E_2$) is zero, and $L = d(p_0,p_1)$.\n\nThe fitness of a given controller in a robot is established by running the robot with that controller, measuring $p_0, p_1, \\delta$ and $L$ and calculating the value of $F$ as defined by Equation \\ref{eq:fitness}.\n\n\\subsection*{The Bayesian-Evolutionary Algorithm}\n\\label{subsec:BEA}\n\nThe Bayesian-Evolutionary Algorithm (BEA) consists of three stages: BO, switching, EA. In the first stage, i.e., the early iterations, Bayesian optimization is employed because its computation time is not yet large. In this paper we use standard Bayesian optimization from the flexible high-performance library Limbo \\cite{cully2018limbo}, using a Gaussian process with a Mat\\'ern 5\/2 kernel, the length scale $\\theta = 0.2$, and a GP-UCB acquisition function. This setting outperformed other hyperparameter settings in our preliminary experiments on parameter tuning.\n\nThe stage of switching starts when the time efficiency of BO drops to lower than that of EA. To determine the switch point we monitor the quality gain per time interval during the search process. This can be defined for any interval of $n$ consecutive iterations (objective function evaluations). If $t_1, \\dots, t_n$ denote the time instances of these iterations and $f_1, \\dots, f_n$ the resulting objective function values, then the gain over this time interval is $\\mathcal{G} = \\frac{f_n - f_1}{t_n - t_1}$. \nWe tested the gain of BO and EA on several well-known objective functions and found that a good moment to switch generally lies in the interval between 190 and 300 iterations.\nIn this study, the switch is triggered at 300 evaluations. \n\nTo seed the EA with a good initial population, we aim for quality (which can be exploited) as well as diversity (which assures appropriate exploration). Hence, if the intended population size of the EA is $K$, then we use K-means clustering in the top $50\\%$ of solutions found by BO and transfer the best solution in each cluster to the EA. \n\nIn the third stage, the search is continued by an evolutionary algorithm. In general, this can be any EAs, but here we use an evolution strategy where the mutation step-size is self-adaptive, but is also controlled by the quality gain per time interval. The idea is to use smaller mutation step size for exploitation when the gains are relatively large and larger mutation step sizes for exploration when gains are small over a period of iterations. \nFor parameters of the EA in BEA, the mutation rate is 0.8, the population size is 10, the tournament size is 2.\n\n\n\n\n\n\n\nWe performed additional experiments to compare the performance of the BEA against its components separately, namely, the BO and the EA. The goal of these additional experiments is to indicate the benefit of our approach compared to using either the BO or the EA alone. We report the performance of the three methods in \\autoref{fig:bea_comparison}. Additionally, we present a comparison of the BEA with the BO and the EA in terms of the achieved fitness value and the computational time in \\autoref{tab:comparison}. We notice that the BEA obtains around $20-30\\%$ better fitness value than the BO and around $45-70\\%$ better fitness value than the EA. Obviously, the BEA is slower than the EA (by around $10-20\\%$), but it is significantly faster than the BO (by about $70\\%$). Eventually, we see that the proposed optimization procedure allows to not only significantly reduce time complexity of the optimization as originally planned, but also leads to a better exploration\/exploitation, and, eventually, to better results than the standalone BO\n\nIn conclusion, we want to stress out the novelty of the proposed optimization strategy. First, the idea of combining BO and EAs is not widely-used. Typically, BO is applied to parameter tuning of EAs \\cite{karroblack, roman2016bayesian}. Here, we propose to optimize the initial population of the EA using BO. Second, running both algorithms one after the other is not necessarily beneficial. The crucial step is to decide about the moment to switch from a computationally heavy, but accurate procedure, to a lightweight generate-and-test method. Here, we discussed how to accomplish that by monitoring the time efficiency. Third, we propose heuristics to \\textit{transfer} solutions found by BO to the initial population of the EA. Last, we further propose a new self-adaptive mutation operation that takes into account information about the progress of the procedure, i.e., the gain in the fitness function value.\n\n\\begin{table}[!tbp]\n    \\centering\n    \\small\n    \\setlength{\\tabcolsep}{8pt}\n    \\renewcommand{\\arraystretch}{1.2}\n    \\begin{tabular}{l|l|ccc}\n    \\multicolumn{2}{c|}{}  &  \\multicolumn{1}{c}{spider9} & \\multicolumn{1}{c}{gecko7} & \\multicolumn{1}{c}{babyA}  \\\\ \n    \\hline\n    best fitness ($\\uparrow$) & BO & 122.9\\% & 135.6\\% & 133.6\\% \\\\\n    of BEA w.r.t. & EA  & 145.9\\% & 169.7\\% & 158.7\\% \\\\ \n   \n    \\hline\n    comp. time  ($\\downarrow$) & BO   & 31.0\\% &  33.8\\% & 32.4\\% \\\\\n    of BEA w.r.t. & EA  & 112.2\\% & 119.6\\% & 110.4\\% \\\\ \n    \\end{tabular}\n    \\caption{Simulation based comparison. Upper half: The performance of BEA in terms of fitness over a full run w.r.t. BO and EA defined by BEA\/BO and BEA\/EA respectively. Lower half: BEA vs. BO and the EA in terms of computation time defined as BEA\/BO and BEA\/EA respectively.}\n    \\label{tab:comparison}\n\\end{table}\n\n\\section*{Data and code availability}\nThe data and code in this study can be provided by the corresponding author \nupon reasonable request.\nA Supplementary Video is available for this paper at \\url{https:\/\/youtu.be\/U9n86ngVe-4}.\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nUnderstanding isotopic abundances on a large scale is a major  field of interest which has received a great deal of attention for its application to terrestrial environments (ocean, meteorites), the solar system (planets, comets) and in galactic interstellar space. The variation of isotopic ratios may give us some information about the link between solar system objects and galactic interstellar environments as discussed by \\cite{aleon:10}.\n We principally focus our study on interstellar environments where low temperature conditions may significantly impact the isotopic ratios of the molecular content. Isotopic molecules are detected in a variety of environments and offer an additional tool to determine physical conditions as they usually do not suffer from opacity problems.\n \n Early modeling studies on {\\tC} and {\\dO} isotopic enrichment were performed by \\cite{langer:84} who introduced  different isotopic exchange reactions, relying on previous theoretical and experimental studies by \\citealt{watson:76,smith:80}. Possible effects of selective photodissociation of CO have  subsequently been emphasized \\citep{glassgold:85,lebourlot:93,visser:09}, which tend to increase the {\\nCO\/\\tCO}  ratio. The use of the CN radical as a tracer of  {\\dC\/\\tC} isotopic ratio has been raised by \\cite{savage:02,milam:05}, who studied the corresponding gradient as a function of the galactic distance. The actual value of the {\\dC\/\\tC} isotopic ratio in the local InterStellar Medium (ISM) is assumed to be 68 \\citep{milam:05}.\n  \n The possibility of nitrogen isotopic fractionation in interstellar clouds has been investigated by \\cite{terzieva:00} who suggested various {\\fN} isotopic exchange reactions. \\cite{rodgers:04,rodgers:08b,rodgers:08a} used these suggested reaction rate constants to  predict nitrogen isotopic fractionation in chemical models of dense interstellar molecular cores. They specifically discussed the role of the atomic to molecular nitrogen ratio in the fractionation process and  the possible link between nitrogen hydrides and CN containing molecules (nitriles).\nThe corresponding observations are sparse however and difficult as the elemental {\\nN\/\\fN} ratio is high (441 $\\pm$ 5), as determined by the recent Genesis solar wind sampling measurement \\citep{marty:11} and assumed to hold in the local ISM. In addition, the zero point energy (ZPE) differences involved in nitrogen fractionation reactions are small and the predicted corresponding chemical enrichment is moderate. \n\\cite{lucas:98} reported {\\fHCN} absorption in diffuse clouds located in front of distant quasars with a {\\nHCN\/\\fHCN} ratio of 270$\\pm$ 27, close to value reported for the Earth. However, various new observations of isotopic nitrogen containing molecules have been reported, including {\\fN}  substituted ammonia and deuterated ammonia \\citep{gerin:09,lis:10} , the  diazenylium  ion (\\NNHp)  \\citep{bizzocchi:10,bizzocchi:13,daniel:13}, CN and HCN \\citep{ikeda:02,pillai:07,adande:12,hilyblant:13,daniel:13}, HCN and HNC \\citep{wampfler:14}.  The strong depletion found in  {\\fN}  variants of the {\\NNHp} isotopologue strongly contradicts model predictions \\citep{gerin:09}, which motivates a reinvestigation of the chemical processes at work.  With this in mind, the link between deuterated chemistry and the possible role of ortho\/para molecular hydrogen has been \nstudied by \\cite{wirstrom:12}. \n \nWe analyse in Section~\\ref{sec:reac} the various possible isotopic exchange reactions that are involved for carbon and nitrogen containing molecules. Indeed, most nitrogen fractionation observational results for CN containing molecules involve only {\\tC} and\n{\\fN} species so that the measure of the nitrogen isotopic ratio  assumes a  fixed {\\dC\/\\tC} fraction. We examine and extend the pioneering study of \\cite{terzieva:00}\nand check for the possible presence of barriers in the entrance channels of isotopic exchange reactions  through theoretical calculations. \nWe also update the zero point energy (ZPE) values involved   and derive the corresponding exothermicity values.\nWe present our new chemical model in Section~\\ref{sec:model} and compare with available observations and other models. Our conclusions are presented in Section~\\ref{sec:conclusion}\n\n\\section{Chemical reactions involving isotopic substitutes of {\\dC} and \\nN.}\n\\label{sec:reac}\n\\subsection{{\\tC} and {\\fN} exchange reactions}\nAt very low temperatures, isotopic exchange reactions may only occur  if no barrier is present between the interacting atoms, ions and molecules or if tunnelling plays an important role. Experimental information is crucial and we constrain the evaluation of rate constants using that information. If no experimental data are available, we apply theoretical  methods to determine the presence of a barrier: A first technique uses\n DFT (Density Functional Theory)  calculations (with the hybrid M06-2X functional developed by \\cite{zhao:08} which is well suited for thermochemical calculations, associated to the cc-pVTZ basis set using GAUSSIAN09 software). The alternative is provided by the MRCI+Q method (with the aug-cc-pVTZ basis set).\nFor barrierless cases, we derive the reaction rate constants  by using a simple capture theory \\citep{georgievskii:05}  for both ion-neutral and neutral-neutral reactions.\nWe consider four different families of isotopic exchange reactions:\n\\begin{itemize}\n\\item{{\\emph{A : direct reactions}}. The proton transfer in the $\\NNHp + \\NfiN \\rightarrow  \\NfNHp + \\NN$ reaction can serve as an example.  In this case and for reactions without a barrier, the reaction rate coefficient of the forward reaction is equal to the capture rate constant multiplied by a probability factor {\\it{f(B, M)}}, $f(B,M)$ depending on the rotational constant, mass and symmetry values of the reactants and products. In reactions involving{\\fN} and {\\tC} the mass ratio of reactants and products are very close and $f(B,M) \\cong \\sigma_{\\rm{entrance~channel}} \/ \\sigma _{\\rm{exit~channels} }$} (The symmetry number $\\sigma$ is equal to the number of pathways). The reverse reaction is calculated from the equilibrium constant $K$, as in \\cite{terzieva:00}: $K = k_f\/k_r=f(B,M) \\times exp( \\Delta E \/ kT)$.\n \n\\item{{\\emph{B : reactions involving adduct formation leading to direct products without isomerization}}. As an example, we refer to \n$\\fNp + \\NN \\rightarrow  \\NfiN  + \\Np$.  We first assume that the high pressure rate constant is equal to the capture rate constant (for reactions without a barrier).  We apply statistical theory  for the system at thermal equilibrium so that $ k_f + k_r = k_{capture} $. From the equilibrium constant expression, we then derive \n$ k_f=  k_{capture} \\times \\frac{f(B,M)}{[ f(B,M) + exp(- \\Delta E \/ kT)] }$ and  $ k_r=  k_{capture} \\times \\frac{exp(- \\Delta E \/ kT)}{[f(B,M) +exp(- \\Delta E \/ kT)]}$} \n\n\\item{{\\emph{C : reactions involving adduct formation with  isomerization pathways}}. Such a case holds for \n$\\tC + \\nHCN  \\rightarrow  \\dC  + \\tHCN$. We again assume that the high pressure rate constant is given by capture theory (for reactions without a barrier). The isotopic isomerization reaction competes with the dissociation of the adduct. The rate constant depends on the location of the transition state and statistical calculations are generally required to estimate the isomerization reaction rate constant.}\n \\item{{\\emph{D : other reactive exothermic channels exist}}}. The exchange reaction is   discarded generally (the possibility of  N atom exchange in the {\\fN} + CN and in {\\fN} + {\\CCN} reactions is discussed).\n\\end{itemize}\nThe knowledge of the exoergicity values $\\Delta E$ is  also a major concern. They are obtained from the differences of  the ZPEs between products and reactants. We recall in the Appendix  the corresponding expressions and derive their  values by using the most recent determinations of spectroscopic constants\n \n We summarize in Table \\ref{tab:exch}  the different isotopic exchange reactions considered and display the corresponding reaction rate constants. Detailed information is provided in the online material on the theoretical methods used for the different  systems. The reactions involving {\\fN} are displayed in the upper part of the Table. We also consider {\\tC} isotopic exchange reactions in the lower part of Table \\ref{tab:exch}.  \n %\n \\begin{table*}[h]\n\\caption{Isotopic exchange reactions.}             \n\\label{tab:exch}      \n\\begin{center}          \n\\begin{tabular}{l l l l l c  l}    \n\\hline\\hline       \n                     \nlabel \/  & \\multicolumn{3}{c}{Reaction}  &   k$_f$  $^*$ & $f(B,M)$ $^*$  & $\\Delta$E  $^*$  \\\\\ncomment  &  \\multicolumn{3}{c}{ }  &     (cm$^{3}$ s$^{-1}$) &   &   (K)\\\\\n\\hline                    \n(1)  A  & {\\bf{$\\NfiN + \\NNHp$}} & $\\rightleftharpoons$ &  {\\bf{$ \\NfNHp + \\NN$}} &  2.3 $\\times$ 10$^{-10}$ &  0.5  &  10.3   \\\\\n (2) A &{\\bf{$\\NfiN + \\NNHp$}} & $\\rightleftharpoons$ &  {\\bf{$ \\fNNHp + \\NN$}}  &       2.3 $\\times$ 10$^{-10}$   & 0.5   &   2.1   \\\\\n (3) A &{\\bf{$\\NfiN + \\fNNHp$}} & $\\rightleftharpoons$ &  {\\bf{$ \\NfNHp + \\NfiN$}}  &       4.6 $\\times$ 10$^{-10}$   & 1  &   8.1   \\\\\n(4)  B   &  {\\bf{$\\fNp + \\NN$}} & $\\rightleftharpoons$ &  {\\bf{$ \\Np + \\NfiN$}} & 4.8 $\\times  10 ^{-10} \\times   \\frac{2}{2+exp(-28.3\/T)}$ &2 & 28.3   \\\\\n(5)  C  & {\\bf{$\\fN  + \\CNCp$}} & $\\rightleftharpoons$ &      {\\bf{$ \\CfNCp  + \\nN$}} & 3.8 $\\times  10 ^{-12} \\times  (\\frac{T}{300})^{-1} $  &   1  & 38.1  \\\\ \n(6)  D &  {\\bf{$\\fNp  + \\nNO$}} & $\\rightleftharpoons$ &      {\\bf{$ \\Np + \\fNO$}} &   no react   &   -     &  24.3  \\\\\n(7) barrier   & {\\bf{$\\fN  + \\NNHp$}} & $\\rightleftharpoons$ &      {\\bf{$\\nN  +   \\NfNHp $}} &    no react &  -  &  38.5 \\\\ \n(8)  barrier &   {\\bf{$\\fN  + \\NNHp$}} & $\\rightleftharpoons$ &      {\\bf{$\\nN  +   \\fNNHp $}} &  no react  &  -  &  30.4 \\\\ \n(9)  barrier &   {\\bf{$ \\fNNHp + \\nH  $}} & $\\rightleftharpoons$ &      {\\bf{$\\nH  +   \\NfNHp $}} &  no react   &  -  &  8.1 \\\\ \n (10)   barrier  & {\\bf{$\\fN  + \\HCNHp$}} & $\\rightleftharpoons$ &      {\\bf{$\\nN  +   \\HCfNHp $}} &  no react  & -   & 37.1  \\\\ \n (11)  D & {\\bf{$\\fN  + \\nCN$}}  &  $\\rightleftharpoons$      &   {\\bf{$ \\nN  + \\CfiN$}} &  upper limit  : 2.0 $\\times$ 10$^{-10}$    $\\times$ &  1   & 22.9   \\\\ \n                                 &     &        &     &  (T\/300)$^{1\/6}$  $\\times$ $\\frac{1}{1+exp(-22.9\/T)}$   &   &  \\\\\n  (12)  B & {\\bf{$\\fN  + \\CCN$}} &  $\\rightleftharpoons$ &      {\\bf{$ \\nN  + \\CCfN $}} &     1.6 $\\times  10 ^{-10} \\times  (T\/300)^{1\/6} \\times$ &  1  &   26.7  \\\\ \n                                 &     &        &     &  $\\frac{1}{1+exp{\\bf{(-26.7\/T)}}}$   &   &  \\\\\n(13)  D &  {\\bf{$\\fN   + \\nNO$}} & $\\rightleftharpoons$ &      {\\bf{$\\nN  + \\fNO$}} &     -    &   -     &  24.3  \\\\ \n\\hline                    \n\\hline                    \n(14)   B &     {\\bf{$\\tCp  + \\nCO$}} & $\\rightleftharpoons$ &      {\\bf{$ \\Cp  + \\tCO$}} & 6.6 $\\times  10 ^{-10} \\times (T\/300)^{-0.45} $&  1   &   34.7   \\\\\n        &      &         &      &    $\\times$  exp(-6.5\/T)    $  \\times$  $ \\frac{1}{1+exp(-34.7\/T)}$  &     &      \\\\\n(15)  A  &  {\\bf{$\\tCO   + \\HCOp$}} & $\\rightleftharpoons$ &      {\\bf{$ \\nCO   + \\HtCOp$}} & 2.6 $\\times  10 ^{-10}  \\times (T\/300)^{-0.4} $  & 1 &  17.4 \\\\ \n(16)   B & {\\bf{$\\tCp  + \\nCN$}} & $\\rightleftharpoons$ &      {\\bf{$ \\Cp  + \\tCN$}} &   3.82 $\\times  10 ^{-9} \\times (T\/300)^{-0.4}$    &  1   &   31.1   \\\\ \n           &   &       &      &   $ \\times$   $ \\frac{1}{1+exp(-31.1\/T)}$  &      &      \\\\ \n (17)  B  &    {\\bf{$\\tC   + \\nCN$}} & $\\rightleftharpoons$ &      {\\bf{$ \\dC   + \\tCN$}} &  3.0 $\\times  10 ^{-10}   \\times \\frac{1}{1+exp(-31.1\/T)}$  &  1   &   31.1   \\\\ \n (18)  C & {\\bf{$\\tC   + \\nHCN$}} & $\\rightleftharpoons$ &      {\\bf{$ \\dC   + \\tHCN$}} & no react  & - &  48.4 \\\\ \n(19)  B &  {\\bf{$\\tC   + \\nCC$}} & $\\rightleftharpoons$ &      {\\bf{$ \\dC   + \\tCC$}} &   3.0 $\\times  10 ^{-10}  \\times  \\frac{2}{2+exp(-26.4\/T)}$  &  2   &   26.4   \\\\ \n  (19) barrier &{\\bf{$ \\tCH + \\nCO $}} & $\\rightleftharpoons$ &      {\\bf{$ \\tCO   + \\nCH$}} & no react & - &   28.6 \\\\ \n  \\hline                  \n\\end{tabular}\n\\end{center}\n$^*$    k$_f$ is the forward reaction rate coefficient. The reverse reaction rate coefficient,   k$_r$ , is obtained by   k$_r$ = $\\frac{k_f}{f(B,M)}$ exp(- $\\Delta$E\/T).\n\\end{table*}\nTable \\ref{tab:exch} shows two main discrepancies compared to previous studies by \\cite{terzieva:00} : The  exchange reactions between atomic {\\fN}  and {\\NNHp},  {\\HCNHp} are found to be unlikely to occur as significant barriers arise in the complex formation step. A similar result is obtained for {\\fNp} exchange with NO, whereas these reactions had been included in \\cite{terzieva:00}. The exchange reaction between atomic {\\fN} and CN, which was suggested by \\cite{rodgers:08b}, is found to be plausible.\nAdditional possibilities of exchange have also been considered such as the reaction between {\\fN}  and {\\CCN}. As far as {\\tC} possible fractionation is concerned, we find that CN could be enriched in  {\\tC} through the exchange reactions of CN with {\\tC} and {\\tCp}. However, such a mechanism does not hold for HNC as atomic carbon is found to react with HNC \\citep{loison:14}. $^{13}$C\nenrichment of HCN is also found to be unlikely as the calculated transition state lies above the entrance level in the hypothetical isomerization process (C mechanism). \n \n \\subsection{Ammonia synthesis}\n Ammonia synthesis proceeds mainly through a chain of reactions starting with {\\Np} and {\\HH}, as the reaction between N and {\\HHHp} has been shown to be inefficient \\citep{milligan:00}.\n \\label{sec:ammonia}\n \\subsubsection{The {\\Np}  +  {\\HH}   reaction and isotopic substitutions}\n\\label{sec:nph2}\nThis almost thermoneutral reaction deserves a special mention and has received  considerable attention. \\cite{lebourlot:91} first pointed out the possible role of ortho-{\\HH} in the interstellar chemistry of ammonia as the energy of its J=1 rotational level almost compensates the small endothermicity of the reaction $ \\Np   +   \\HH  \\rightarrow \\NHp + \\nH$. \\cite{dislaire:12}  subsequently  reanalysed the experimental data \\citep{marquette:88} and suggested new separate expressions for the reaction rate with {p-\\HH} and o-\\HH. Similar results were obtained by \\cite{zymak:13}, who also emphasized the possible role of the fine structure level of \\Np.  We follow the prescription derived by \\cite{dislaire:12} and  extend their analysis to deuterated forms and those including {\\fN} , as displayed in Table \\ref{tab:npHH}.   In the case of {\\fN}  substituted compounds, we have taken into account the (small) additional term due to the change in  ZPE. \n\\begin{table*}[h]\n\\caption{Reaction rate coefficients of   {\\mbox{N$^{+}$}}   +  {\\HH}  and isotopic variants.}             \n\\label{tab:npHH}      \n\\centering          \n\\begin{tabular}{l c l l c }    \n\\hline\\hline       \n                     \n\\multicolumn{3}{c}{Reaction}  &   k   (cm$^{3}$ s$^{-1}$)& Comment\\\\\n\\hline                    \n{\\bf{$\\Np  + p-\\HH$}} & $\\rightarrow$ &      {\\bf{$ \\NHp  + \\nH$}}& 8.35 $\\times$ 10$^{-10}$ $\\times$ exp(-168.5\/T)  & \\citealt{dislaire:12}\\\\  \n{\\bf{$\\Np  + o-\\HH$}} & $\\rightarrow$ &      {\\bf{$ \\NHp  + \\nH$}}& 4.2 $\\times$ 10$^{-10}$ $\\times$ (T\/300)$^{-0.17}$ $\\times$ exp(-44.5\/T) & \\citealt{dislaire:12} \\\\  \n{\\bf{$\\fNp  + p- \\HH$}} & $\\rightarrow$ &      {\\bf{$ \\fNHp  + \\nH$}}& 8.35 $\\times$ 10$^{-10}$$\\times$ exp(-164.3\/T)&  see text  \\\\  \n{\\bf{$\\fNp  + o- \\HH$}} & $\\rightarrow$ &      {\\bf{$ \\fNHp  + \\nH$}}& 4.2 $\\times$ 10$^{-10}$ $\\times$ (T\/300)$^{-0.17}$ $\\times$ exp(-39.7\/T)  & see text \\\\  \n{\\bf{$\\Np  + \\HD   $}} & $\\rightarrow$ &      {\\bf{$ \\NDp  + \\nH$}}& 3.17 $\\times$ 10$^{-10}$  $\\times$ exp(-16.3\/T) & \\citealt{marquette:88}  \\\\  \n{\\bf{$\\Np  + \\HD   $}} & $\\rightarrow$ &      {\\bf{$ \\NHp  + \\nD$}}& 3.17 $\\times$ 10$^{-10}$  $\\times$ exp(-594.3\/T) & see text\\\\  \n{\\bf{$\\fNp  + \\HD   $}} & $\\rightarrow$ &      {\\bf{$ \\fNDp  + \\nH$}}& 3.17 $\\times$ 10$^{-10}$  $\\times$ exp(-9.3\/T)  & see text \\\\  \n{\\bf{$\\fNp  + \\HD   $}} & $\\rightarrow$ &      {\\bf{$ \\fNHp  + \\nD$}}& 3.17 $\\times$ 10$^{-10}$  $\\times$ exp(-589.5\/T) & see text \\\\  \n{\\bf{$\\Np  + \\DD   $}} & $\\rightarrow$ &      {\\bf{$ \\NDp  + \\nD$}}& 2.37 $\\times$ 10$^{-10}$  $\\times$ exp(-197.9\/T)  &  \\citealt{marquette:88} \\\\  \n{\\bf{$\\fNp  + \\DD   $}} & $\\rightarrow$ &      {\\bf{$ \\fNDp  + \\nD$}}& 2.37 $\\times$ 10$^{-10}$  $\\times$ exp(-190.9\/T) &  \\citealt{marquette:88}  \\\\  \n \\hline                  \n\\end{tabular}\n\\end{table*}\n These expressions should be used with caution as in their kinetic expression, we consider that the exponential term represents the enthalpy difference between the products and reactants. The capture rate constant of  {\\fNp} + HD reaction is about (2\/3)$^{0.5}$\nsmaller than the rate of the {\\fNp} + {\\HH} reaction, due to the different mass dependences. The formation of {\\fNDp} is favored at low temperatures.\n\\subsubsection{ {\\NHHHp} + \\HH}\nThe final step of ion-molecule reactions leading to ammonia formation is the reaction between {\\NHHHp} and \\HH,  giving \\NHHHHp. Although exothermic, this reaction has a strong temperature dependence, displaying a minimum at T $\\sim$ 100K and a slow increase at lower  temperatures \\citep{barlow:87}, which is interpreted by the presence of a barrier to complex formation, as discussed in \\cite{herbst:91}. At temperatures close to 10K, the reaction is likely to proceed through tunneling which may take place or H atom  abstraction. However, the {\\NHHHp} reaction with D$_2$ is found to be slower when the temperature decreases as tunneling is not efficient  with deuterium. We thus reconsider the isotopic variants of this reaction, as shown in Table \\ref{tab:nhhhp}, where we give the present reaction rates compared to previous values which were derived from \\cite{anicich:86}.\nThese values are indeed different from  those used in our previous studies \\citep{roueff:05} where we assumed the same rate for  the channels resulting from the reactions of {\\NHHHp} and isotopologues with HD, based on pure statistical considerations. \n We discuss the resulting modifications in Section \\ref{sec:model}.\n\\begin{table*}[h]\n\\caption{Reaction rate coefficients of   {\\NHHHp}   +  {\\HH}  and isotopic variants at T = 10K.}             \n\\label{tab:nhhhp}      \n\\begin{center}         \n\\begin{tabular}{l c l c c}    \n\\hline\\hline       \n                     \n\\multicolumn{3}{c}{Reaction}  &   \\multicolumn{2}{c}{k   (cm$^{3}$ s$^{-1}$)} \\\\\n    &    &   &   present work & old value (*)  \\\\\n\\hline                    \n{\\bf{$\\NHHHp  +  \\HH$}} & $\\rightarrow$ &      {\\bf{$ \\NHHHHp  + \\nH$}}& 8.2 $\\times$ 10$^{-13}$  &  2.4 $\\times$ 10$^{-12}$ \\\\  \n {\\bf{$\\NHHHp  +  \\HD$}} & $\\rightarrow$ &      {\\bf{$ \\NHHHHp  + \\nD$}}& 8.2 $\\times$ 10$^{-13}$ &  1.2 $\\times$ 10$^{-12}$  \\\\  \n {\\bf{$\\NHHHp  +  \\HD$}} & $\\rightarrow$ &      {\\bf{$ \\NHHHDp  + \\nH$}}& 1.0 $\\times$ 10$^{-13}$  &  1.2 $\\times$ 10$^{-12}$ \\\\  \n{\\bf{$\\NHHHp  +  \\DD$}} & $\\rightarrow$ &      {\\bf{$ \\NHHHDp  + \\nD$}}& 1.0 $\\times$ 10$^{-13}$  &  2.4 $\\times$ 10$^{-12}$ \\\\  \n {\\bf{$ \\NHHDp  + \\HH$}} & $\\rightarrow$ &      {\\bf{$ \\NHHHDp  + \\nH$}}& 8.2 $\\times$ 10$^{-13}$  &  2.4 $\\times$ 10$^{-12}$ \\\\  \n {\\bf{$ \\NHHDp  + \\HD$}} & $\\rightarrow$ &      {\\bf{$ \\NHHHDp  + \\nD$}}& 8.2 $\\times$ 10$^{-13}$  & 1.2 $\\times$ 10$^{-12}$  \\\\  \n {\\bf{$ \\NHHDp  + \\HD$}} & $\\rightarrow$ &      {\\bf{$ \\NHHDDp  + \\nH$}}& 1.0 $\\times$ 10$^{-13}$  & 1.2 $\\times$ 10$^{-12}$  \\\\  \n{\\bf{$ \\NHHDp  + \\DD$}} & $\\rightarrow$ &      {\\bf{$ \\NHDDDp  + \\nH$}}& 1.0 $\\times$ 10$^{-13}$  &  2.4 $\\times$ 10$^{-12}$ \\\\  \n {\\bf{$ \\NHDDp  + \\HH$}} & $\\rightarrow$ &      {\\bf{$ \\NHHDDp  + \\nH$}}& 8.2 $\\times$ 10$^{-13}$   &  2.4 $\\times$ 10$^{-12}$ \\\\  \n {\\bf{$ \\NHDDp  + \\HD$}} & $\\rightarrow$ &      {\\bf{$ \\NHHDDp  + \\nD$}}& 8.2 $\\times$ 10$^{-13}$  &  1.2 $\\times$ 10$^{-12}$ \\\\  \n {\\bf{$ \\NHDDp  + \\HD$}} & $\\rightarrow$ &      {\\bf{$ \\NHDDDp  + \\nH$}}& 1.0 $\\times$ 10$^{-13}$  & 1.2 $\\times$ 10$^{-12}$  \\\\  \n{\\bf{$ \\NHDDp  + \\DD$}} & $\\rightarrow$ &      {\\bf{$ \\NHDDDp  + \\nD$}}& 1.0 $\\times$ 10$^{-13}$  & 2.4 $\\times$ 10$^{-12}$  \\\\  \n{\\bf{$\\NDDDp  +  \\HH$}} & $\\rightarrow$ &      {\\bf{$ \\NHDDDp  + \\nH$}}& 8.2 $\\times$ 10$^{-13}$  & 2.4 $\\times$ 10$^{-12}$  \\\\  \n {\\bf{$\\NDDDp  +  \\HD$}} & $\\rightarrow$ &      {\\bf{$ \\NHDDDp  + \\nD$}}& 8.2 $\\times$ 10$^{-13}$ &   1.2 $\\times$ 10$^{-12}$ \\\\  \n {\\bf{$\\NDDDp  +  \\HD$}} & $\\rightarrow$ &      {\\bf{$ \\NDDDDp  + \\nH$}}& 1.0 $\\times$ 10$^{-13}$   &  1.2  $\\times$ 10$^{-12}$ \\\\  \n{\\bf{$\\NDDDp  +  \\DD$}} & $\\rightarrow$ &      {\\bf{$ \\NDDDDp  + \\nD$}}& 1.0 $\\times$ 10$^{-13}$  &   2.4 $\\times$ 10$^{-12}$  \\\\  \n  \\hline                  \n\\end{tabular}\n\\end{center}\n(*) \\cite{roueff:05}\n\\end{table*}\nIdentical reaction rate coefficients  are used for the $^{15}$N isotopically substituted reactions. \n\n \\section{Models}\n\\label{sec:model}\n\\subsection{General features}\nChemical reactions involving nitrogen atoms and CH, CN and OH have  been   studied experimentally at low temperatures and shown to be less efficient than previously thought at low temperatures \\citep{daranlot:12,daranlot:13}, which was confirmed by theoretical studies.\nThe corresponding reaction rate constants have been implemented in the KIDA chemical data base \\citep{wakelam:13} and we have updated our chemical network accordingly. We also include the reactions discussed in \\cite{loison:14} in their study of HCN \/ HNC chemistry. We  explicitly include deuterium, {\\tC} and {\\fN} molecular compounds in our chemical network. The reactions displayed  in Table \\ref{tab:exch}, have been included,  which allows us to test the hypothesis of a constant {\\dC}\/{\\tC} isotopic ratio to derive the {\\nN\/\\fN} ratio in {\\CfiN} containing molecules. \nWe take into account the role of the ortho\/para ratio of  molecular {\\HH}  in the  {\\HHDp} + {\\HH} and {\\Np} ({\\fNp}) + {\\HH} chemical reactions   in an approximate way: we do not compute the full ortho\/para equilibrium  as in  the models of \\citealt{flower:06,pagani:11,faure:13} and rather introduce it as a model parameter, which can be varied.  \\cite{faure:13} have found a value of 10$^{-3}$    for temperatures below 15K\n\nApart from exchange reactions, reactions involving isotopic molecules are assumed to have the same total rate constant as those involving the main isotope except for the reaction of {\\fNp}  with H$_{2}$\/HD\/D$_{2}$. The various reaction channels  are obtained from statistical considerations, in the absence of experimental information. We restrict carbon containing molecules to 3 carbon atoms, nitrogen containing molecules to 2 nitrogen atoms and consider full deuteration as in our previous studies \\citep{roueff:05}. Within these constraints, the number of species considered in the model is 307 linked through more than 5400 chemical reactions. \n\\begin{table}[h]\n\\caption{Model definitions. The { C \/ \\tC}      and  {N \/ \\fN}  ratios are respectively taken as 68 \\citep{milam:05} and 440\n \\citep{marty:11}.  }   \n\\label{tab:model}      \n\\centering          \n\\begin{tabular}{lll }    \n\\hline\\hline       \n     & Model (a) &    Model (b)  \\\\\n     \\hline\ndensity {\\mbox{n$_H$}} (cm$^{-3}$) &    2 $\\times$ 10$^4$ &   2 $\\times$ 10$^5$  \\\\\nTemperature (K) &  10  &  10 \\\\\ncosmic ionization rate per H$_2$ (s$^{-1}$) &  1.3  $\\times$ 10$^{-17}$     &   1.3 $\\times$ 10$^{-17}$      \\\\\n\\hline\nHe {\/} H  & 0.1  &  0.1     \\\\ \nC {\/} H &   4.15 $\\times$ 10$^{-5}$   & 1.4 $\\times$ 10$^{-5}$     \\\\ \nN {\/} H &  6.4 $\\times$ 10$^{-5}$   &  2.1 $\\times$ 10$^{-5}$       \\\\ \nO {\/} H &   6  $\\times$ 10$^{-5}$   &   2.0 $\\times$ 10$^{-5}$    \\\\ \nS {\/} H  &  8.0  $\\times$ 10$^{-8}$  &  8.0   $\\times$ 10$^{-8}$     \\\\\nFe \/ H & 1.5  $\\times$ 10$^{-9}$  &  1.5   $\\times$ 10$^{-9}$     \\\\ \n  \\hline                    \n  \\hline                  \n\\end{tabular}\n\\end{table}\nWe consider two different models as displayed in Table \\ref{tab:model}. Model (a) may be considered as a template of TMC1,  and assumes a  density of hydrogen nuclei  n$_H$ = 2 $\\times$ 10$^4$ cm$^{-3}$.   The elemental abundance of carbon relative to hydrogen nuclei is taken as 4.15 $\\times$ 10$^{-5}$  to reproduce the derived relative abundance of CO \\citep{ohishi:92}. We derive the oxygen elemental abundance by imposing a C\/O ratio of 0.7 appropriate for TMC1 and take the nitrogen elemental abundance  used by \\cite{legal:14} in their work on nitrogen chemistry. The elemental abundance of sulfur is not well constrained and we have taken the low metal case value of 8.0   $\\times$ 10$^{-8}$.\nModel (b) is  more representative of  a pre stellar core with a density of 2 $\\times$ 10$^5$ cm$^{-3}$  similar to L134N or Barnard 1 (B1) where the elemental abundances of carbon, oxygen and nitrogen are reduced by a factor  of 3\nto account for depletion.\n T~=~10K  in both cases and the cosmic ionization rate $\\zeta$ per H$_2$ is 1.3  $\\times$ 10$^{-17}$ \\textbf{s$^{-1}$}  as in \\cite{legal:14}.\n\\subsection{Results}\n\nWe summarize  our results  obtained with a 10$^{-3}$ value of the o\/p ratio of  {\\HH}  in Table \\ref{tab:res} and give some observational values for comparison. Time dependent effects may be  visualized\n from the values reported at 10$^6$ years and at steady state for model (a). \n Steady state is reached after a few 10$^7$ and 10$^6$ years respectively for models (a) and (b).\n As there are  fewer $^{15}$N enrichment reactions  than previously assumed \\citep{terzieva:00}\n most nitrogen containing species are found to have isotopic abundance ratios close to the solar value ($^{14}$N\/$^{15}$N = 440) \ngiven by the Genesis mission \\citep{marty:11}. \n\\begin{table*}[h]\n\\caption{Model (a) and (b) results and observations. ss means stationary state.  The value of the o\/p ratio of {\\HH} is taken as 10$^{-3}$.  }   \n\\label{tab:res}      \n\\centering          \n\\begin{tabular}{l|cc|c|lll}    \n\\hline\\hline       \n     & \\multicolumn{2}{c|}{Model (a)} &    Model (b) & TMC1 & L1544 & B1\\\\\n     & t= 10$^6$ yrs& ss & ss &   &    &    \\\\  \n\n     \\hline\nelectronic fraction & 1.4 $\\times$ 10$^{-8}$   &   2.8 $\\times$ 10$^{-8}$   &   1.7 $\\times$ 10$^{-8}$ &   & &   \\\\\nN \/ \\fN  & 440 & 456  & 455 &     &  &    \\\\   \n2 $\\times$ N$_2$ \/ \\fN N &   438 &431& 437& &  & \\\\\nNH \/ \\fNH  & 429 &  428  &  421 &     &  &    \\\\  \nNH \/ ND  & 16  &  31  &   9 &     &  &    \\\\   \n\\NHHH \/ \\HH    &  6.7 10$^{-9}$  &      1.3 10$^{-9}$   & 6.0 10$^{-9}$& 2 10$^{-8}$   $^{(8)}$ &   &  \\\\\n\\NHHH \/ \\fNHHH  &  333 &      386       &      387  & &   & 300$^{+ 55}_{-40}$ $^{(1)}$\\\\\n\\NHHD \/ \\HH    &   3.8 10$^{-10}$ &     5.8 10$^{-11}$        &  3.3 10$^{-10}$  & 4 10$^{-10}$  $^{(9)}$&   &  \\\\\n\\NHHD \/ \\fNHHD  &  215 &      276  &      336  &  &  & 230$^{+ 105}_{-55}$ $^{(1)}$ \\\\\n\\NHHH \/ \\NHHD  &  18  &     22          &     18  & 50 $^{(9)}$&  &  \\\\\n\\NNHp \/ \\HH    &   4.8  10$^{-10}$ &     1.3  10$^{-10}$         & 2.1  10$^{-10}$ & 5 10$^{-10}$  $^{(8)}$ &   &  \\\\\n\\NNHp \/ \\NfNHp &  431 &  430   &   423  &  &1050$^{\\pm  220}$ $^{(2)}$  &   400$^{+ 100}_{-65}$ $^{(1)}$\\\\\n\\NNHp \/ \\fNNHp & 437 &   432  &   433   &  &   1110$^{\\pm  240}$ $^{(2)}$  &   $>$ 600 $^{(1)}$  \\\\\n\\NNHp \/ \\NNDp  &  16 &  29  &  8.6 & 12.5 $^{(9)}$   & & 2.9 $^{(1)}$  \\\\\nCN \/ \\HH    &   6.8 10$^{-9}$ &      5.5  10$^{-9}$  &  1.2  10$^{-9}$  & 3 10$^{-8}$ $^{(8)}$ &   &  \\\\\nCN \/ \\tCN   & 67 &   84  &    63    &   & &50$^{+19}_{-11}$ $^{(1)}$\\\\\nCN \/ \\CfiN  &  430 &  449  &   445 &   & & 240$^{+135}_{-65}$ $^{(1)}$\\\\\n\\tCN \/ \\CfiN  & 6.4 &   5.3  &  7.0 &   & 7.5$^{\\pm 1}$ $^{(3)}$  &\\\\\nHCN \/ \\HH    &   7.4  10$^{-9}$&   5.9  10$^{-10}$ &  5.4  10$^{-10}$ & 2 10$^{-8}$  $^{(8)}$ &   &  \\\\\nHCN \/ \\tHCN  & 93 &  168 &   114  &   &  & 30$^{+7}_{-4}$ $^{(1)}$\\\\\nHCN \/ \\fHCN  &  398 & 445  &    453   &  &   &  165$^{+30}_{-20}$ $^{(1)}$\\\\\n \\tHCN \/  \\fHCN & 4.3 & 2.6  &  4.0  &  2 - 4.5 $^{(5)}$  &   & 5.5 $\\pm$1 $^{(1)}$ \\\\\nHCN \/ DCN  &  43 &  96 &  22 &  &   &  20$^{+6}_{-10}$ $^{(1)}$\\\\\nHNC \/ \\HH    &   5.6 10$^{-9}$ &    7.4  10$^{-10}$  &   8.4  10$^{-10}$   & 2 10$^{-8}$  $^{(8)}$ &   &  \\\\\nHNC\/ \\tHNC &  93 & 180   & 121 & 54 - 72 $^{(4)}$ &   & 20$^{+5}_{-4}$ $^{(1)}$\\\\\nHNC\/ \\fHNC &  405 & 442   & 446 & 250 - 330  $^{(4)}$  &   &  75$^{+25}_{-15}$ $^{(1)}$ \\\\\n{\\tHNC} \/  {\\fHNC} &  2.5 & 1.75   & 3.7   &  4.6 $\\pm$ 0.6  $^{(4)}$ &   & 3.7 $\\pm$ 1 $^{(1)}$ \\\\\nHNC\/ DNC &  23 & 66   &  16 &  &   &  2.9$^{+1.1}_{-0.9}$ $^{(1)}$\\\\\nNO \/ \\HH    &  1. 10$^{-7}$  &  3.1  10$^{-8}$  & 4.1  10$^{-8}$ & 2.7 10$^{-8}$  $^{(10)}$ &   &  \\\\\nNO \/ {\\fN}O  & 438 & 451  &  446  &     &  &    \\\\ \nCO \/ \\HH    &  8.1 10$^{-5}$ &    8.0 10$^{-5}$ &  2.8  10$^{-5}$       & 8 10$^{-5}$  $^{(8)}$ &   &  \\\\\n  {CO} \/  {\\tC}O & 68 & 67.4 &  68 &    &   &     \\\\\nCH \/ \\HH    &   1.3 10$^{-9}$  & 1.7 10$^{-9}$&  1.7  10$^{-10}$  & 2 10$^{-8}$ $^{(8)}$  &   &  \\\\\n{CH} \/  {\\tC}H & 74 & 154  &  71 & $>$ 71 $^{(6)}$   &   &     \\\\\n\\HCOp \/ \\HH    &  2.5 10$^{-9}$  &    3.610$^{-10}$    & 5.7  10$^{-10}$ & 8 10$^{-9}$  $^{(8)}$ &   &  \\\\\n{\\HCOp} \/ {\\HtCOp} & 56 & 65 & 56 &    &   & 59 $^{(7)}$  \\\\\n{\\HCOp} \/ {\\DCOp} & 15 & 29  & 8.4  &  50 $^{(9)}$  &   &     \\\\\n\\hline                    \n  \\hline                  \n\\end{tabular}\n\\tablebib{(1)~\\citet{daniel:13};\n(2) \\citet{bizzocchi:13}; (3) \\citet{hilyblant:13}; (4) \\citet{liszt:12}\n; (5) \\cite{hilyblant:13b}; (6) \\citet{sakai:13b}, (7) \\citet{hirano:14} assuming CO\/ C$^{18}$O=500,\n(8) \\cite{ohishi:92}, (9) \\cite{tine:00}, (10) \\cite{gerin:93}.\n}\n\\end{table*}\n\\section{Discussion}\nWe  display the time dependence of various isotopic ratios  and  fractional abundances relative to {\\HH}  and discuss the chemical behavior involved \nin the fractionation processes for the two reported models. We first\n consider in Figure \\ref{fig:nn2}  the reservoirs of nitrogen, atomic and molecular nitrogen as well as {\\NNHp}  ions which are chemically linked to \\NN.\n    \\begin{figure*\n   \\centering\n  \\includegraphics[width=14cm]{Fig_1a.pdf}\n  \n   \\caption{ Upper panel : time dependence of N \/ {\\fN} isotopic ratios in atomic and molecular nitrogen  and {\\NNHp} ions. (a) and (b) correspond to the models defined in Table \\ref{tab:model}; the black heavy dotted line represents the elemental N \/ \\fN.  Lower panel :  time dependence of the fractional abundances relative to {\\HH} of N, {\\NN} and {\\NNHp} for models (a) and (b) . The value reported for {\\NNHp} towards TMC1 \\citep{ohishi:92} is displayed as a horizontal green dashed line.}\n              \\label{fig:nn2}%\n    \\end{figure*}\nAtomic nitrogen becomes depleted in {\\fN} after about 10$^5$ years whereas molecular nitrogen is slightly enriched. \n These evolution times are also required to build significant amounts of molecular compounds which compare satisfactorily with available observations. \n The overall dependence of {\\NNHp} follows closely that of {\\NN} as it is formed from {\\NN} + {\\HHHp} reaction, with a slight decoupling between {\\fNNHp}  and {\\NfNHp} at long evolution times. \nWe find that the isotopic ratio of the   {\\NNHp} ions displays an almost constant value close to the solar value after some 10$^5$ years. They are in good agreement with observations in B1 but disagree by a factor of 2 for L1544.\n The trend that {\\fNNHp} is less abundant than {\\NfNHp} is reproduced in our results, as a result of the differences of endothermicity in their reactions with \\NN.  We checked that introducing \\fN$_2$ and species containing two {\\fN} atoms had no effect on these ratios.\nWe could not find any gas-phase mechanism able to generate \n such a large ratio in pre-stellar core conditions. The high isotopic ratio found in L1544 implies an equivalently large ratio for molecular nitrogen, which is in strong contradiction with our findings.\nThese results are markedly different from those derived by \\cite{hilyblant:13} who found a moderate {\\fN} enrichment as these authors introduced the {\\fN} + {\\NNHp}  fractionation reaction  which we have found not to occur. \\footnote{These authors also interchanged the endothermicities of the {\\fNNHp} and {\\NfNHp} reactions with N and {\\NN}.\n\\subsection{Nitrogen hydrides}\n\\label{sec:NH}\nWe display in Figure \\ref{fig:nh3-fN} the time evolution of the isotopic ratios of  nitrogen hydrides   and of their fractional abundances relative to \\HH. {\\NHHH} and {\\NHH} have a very similar behavior as they both result from the reaction chain  starting with the {\\Np} + {\\HH} reactions. \n    \\begin{figure*\n   \\centering\n  \\includegraphics[width=14cm]{Fig_2.pdf}\n  \n   \\caption{ Upper panel :   Time dependence of N \/ {\\fN} isotopic ratios in nitrogen hydrides. (a) and (b) correspond to the models defined in Table \\ref{tab:model}; The black heavy dotted line represents the elemental N \/ \\fN.\n   Lower panel :  time dependence of the fractional abundances relative to {\\HH} of NH, {\\NHH}, {\\NHHH} and {\\NHHD} for models (a) and (b). The values reported for {\\NHHH} \\citep{ohishi:92} and {\\NHHD} \\citep{tine:00} towards TMC1 are displayed as  horizontal blue and red dashed lines respectively in the left panel.}              \\label{fig:nh3-fN}%\n    \\end{figure*}\n The species NH$_2$ and NH$_3$ are found to be enriched in $^{15}$N due to the $^{15}$N$^{+}$ + o-H$_{2}$ reaction which has a slightly smaller (weak) endothermicity, as reported in Table \\ref{tab:npHH},    than the \ncorresponding $^{14}$N$^{+}$ + o-H$_2$ reaction, a difference which  slightly favors  $^{15}$NH$^{+}$ formation. Nevertheless, despite the large uncertainties regarding \nthe rate of the N$^{+}$ + H$_2$ reaction, our results are in fair agreement with the results of \\cite{daniel:13} for prestellar core B1.\nAs the N$^{+}$ + HD $\\rightarrow$ ND$^{+}$ + H reaction has a smaller endothermicity than the N$^{+}$ + H$_{2}$ $\\rightarrow$ NH$^{+}$ + H reaction, \nthe modeled $^{14}$NH$_2$D \/ $^{15}$NH$_2$D ratio behaves somewhat differently than the $^{14}$NH$_3$ \/ $^{15}$NH$_3$  ratio.  The ratio exhibits large variations around 10$^5$ years and becomes smaller than that of  {\\NHHH}  at large times and at steady state.\nThe observed values are compatible with calculations at sufficiently large times. The values of this ratio reported in \\cite{gerin:09} have been found to be too large as a result of assuming a single excitation rotational temperature. The future availability  of collisional excitation rates of {\\NHHD} by {\\HH} \\citep{daniel:14} may give rise to additional changes.\n NH  does not follow the same trend as  NH$_2$ and NH$_3$ and is  only  slightly enriched in \\fN\n  As discussed by \\cite{hilyblant:13}, NH is mainly formed \n from the dissociative recombination (DR)  of \\NNHp,  a reaction which has been recently revisited by \\cite{vigren:12}\n who derive a branching ratio towards NH of 7\\%.\n We have checked that this analysis still holds for model (b) even if  the  reaction of  {\\NNHp} with CO becomes more efficient.\nThe formation route of NH through NH$_2^+$ recombination may take over for highly depleted CO. \n\n\\subsection{Nitriles and isonitriles}\nDeriving {\\fN} isotopic ratios of CN, HCN and HNC from observations is a difficult challenge as the transitions of the main isotopologues are optically thick.\nThen, most of the reported observational values of the {\\nN} \/ {\\fN} molecular ratios \n are obtained from the ratios of the minor isotopologues $^{13}$CN \/ C$^{15}$N, H$^{13}$CN \/ HC$^{15}$N  \n and HN$^{13}$C \/ H$^{15}$NC which is subsequently multiplied by an assumed C \/ {\\tC}  value, usually taken as 68 \\citep{milam:05}. \n\n\\subsubsection{ {\\tC}\/ {\\fN} ratios}\n %\n We test these hypotheses in our models by explicitly introducing fractionation reactions of \\tC, as discussed in Section \\ref{sec:reac}.\n Table \\ref{tab:res} shows that\nthe {\\tC} isotopic ratios of CN, HCN and HNC vary both with time and density. \n      \\begin{figure*\n   \\centering\n  \\includegraphics[width=14cm]{Fig_3.pdf}\n  \n\n   \\caption{Time evolution of \\tC \/ {\\fN} ratios in CN, HCN and HNC for models (a) and (b). The \\tC\/{\\fN} elemental ratio is displayed as heavy dotted line.}\n              \\label{fig:1315}%\n    \\end{figure*}\nFigure \\ref{fig:1315} displays the  $^{13}$CN \/ C$^{15}$N, H$^{13}$CN \/ HC$^{15}$N  \n and HN$^{13}$C \/ H$^{15}$NC ratios as a function of time for the two considered models.\n The deviation from the elemental ratio of 6.48 is\n significant  for HCN, HNC and CN.   \n\n %\n\\subsubsection{{\\tC} chemistry}\nWe now consider the time dependences  of   the $^{12}$C \/ $^{13}$C isotopic ratios in   CN, HCN and HNC  species as displayed in Figure \\ref{fig:tCN}.\nThe time dependent ratio displays large variations, which is\n due to the fact that there are various reactions incorporating $^{13}$C \nin the  molecules.\nThe elemental value of the ratio (68) is fulfilled in a narrow range around 1 Myr for model (a) and 2 $\\times$ 10$^5$ yrs for model (b)  but steady state values are significantly different  except for CN.  \n    \\begin{figure*\n   \\centering\n  \\includegraphics[width=14cm]{Fig_4.pdf}\n  \n   \\caption{time dependence of C \/ {\\tC} isotopic ratios in CN, HCN and HNC. (a) and (b) correspond to the models defined in Table \\ref{tab:model}}\n              \\label{fig:tCN}%\n    \\end{figure*}\n    %\nThis relatively complex behavior results from the many different reaction channels involved in {\\tC} chemistry. \nWe then also consider other {\\tC} containing species and display  the {\\dC}  \/ {\\tC}  ratio in C, CH, CO and HCO$^+$\nin Figure \\ref{fig:tC}  as well as their fractional abundances relative to \\HH.\n    \\begin{figure*\n   \\centering\n  \\includegraphics[width=14cm]{Fig_5.pdf}\n  \n   \\caption{ Upper panel : Time dependence of C \/ {\\tC} isotopic ratios in C, CH, CO and HCO$^+$. The black heavy dotted line represents the elemental {\\dC} \/ \\tC.\n   Lower panel :  Time dependence of the fractional abundances relative to {\\HH} of C,  \\Cp, CO, CH and {\\HCOp},   for models (a) and (b). The observational values  towards TMC1 \\citep{ohishi:92}  are displayed as  horizontal   dashed lines  with corresponding colors in the left panel. (a) and (b) correspond to the models defined in Table \\ref{tab:model}}\n              \\label{fig:tC}%\n    \\end{figure*}\nThe $^{12}$C\/$^{13}$C isotopic ratios of the various molecules are highly dependent\non the evolution time. \nThe transition from gas-phase atomic carbon  toward CO controls the $^{13}$C enrichment. As long as there is still a relatively \nhigh carbon atom concentration in the gas phase, there is enough free $^{13}$C  to allow strong enrichment of CN\nthrough the {\\tC} + CN reaction. When  CO molecules become the reservoir of carbon,  even if in that case the $^{13}$C concentration is low, the $^{13}$C$^{+}$ + $^{12}$CO  $\\rightarrow$ \n$^{12}$C$^{+}$ + $^{13}$CO reaction still leads to a small  $^{13}$CO enrichment \\citep{langer:90,milam:05}. Although this  small excess is not measurable in CO, significant amounts of $^{13}$C are locked up in CO and  most of the other carbon containing species become depleted in {\\tC}, as found for CH and other carbon chains. \n This effect is seen in Figure \\ref{fig:tC} for the isotopic ratio of CH in model (a) at steady state where CO is slightly enriched,\nleading to a significant depletion of {\\tC} in CH. \nOur results \nfor C and {\\HCOp} are similar to those of  \\cite{furuya:11} who studied  the {\\tC} fractionation of multiple carbon chains by \nexplicitly introducing the dependence of the {\\tC} position in the chain.\nWe see that\n{\\HCOp} is marginally enriched in {\\tC} at steady state whereas HCN and HNC are significantly depleted in \\tC. \n We also note that CN and {\\HCOp} may react via proton transfer  to give CO + HNC$^+$ and have checked\n, at the DFT level, the absence of any barrier. The corresponding reaction rate is 2.2 $\\times$ 10$^{-9}$ ($\\frac{T}{300}$)$^{-0.4}$\n cm$^{3}$ s$^{-1}$ when using the capture rate theory. This reaction has not been included in any chemical database up to now. \n \nHNC has been shown recently to react with atomic carbon \\citep{loison:14}, which leads to the different steady state isotopic ratios obtained for HCN and HNC. \n The CN chemistry is then  somewhat decoupled from that of   HCN and HNC.   HCN and HNC are formed at relatively long times via CN + H$_3$$^{+}$ $\\rightarrow$ HNC$^{+}$ \/HCN$^{+}$ + H$_2$, followed immediately by HNC$^{+}$ + H$_2$ $\\rightarrow$ HCNH$^{+}$ + H giving back HCN and HNC via DR \\citep{mendes:12}. With the adopted elemental abundances, the main CN destruction reactions are however O + CN and N + CN so that the HCNH$^{+}$\/HCN$^{+}$\/HNC$^{+}$\/HCN\/HNC\/CN \n network is not a closed system in contrast to models including coupled gas-grain chemistry \\citep{loison:14}.\n\n %\n \\subsubsection{ {\\fN} fractionation} \nWe now display also the N \/ {\\fN} fractionation in nitriles as well as that of NO in Figure~\\ref{fig:15CN}.\n    \\begin{figure*\n   \\centering\n  \\includegraphics[width=14cm]{Fig_6.pdf}\n  \n   \\caption{Time dependence of N \/ {\\fN} isotopic ratios in CN, HCN, HNC and NO. (a) and (b) correspond to the models defined in Table \\ref{tab:model}}\n     \\label{fig:15CN}%\n    \\end{figure*}\nThe time dependences of the isotopic ratios are markedly different in the  two models, except for NO. Whereas CN, HCN and HNC are somewhat enriched in {\\fN} for model (a) both at intermediates times and steady state, the opposite result is obtained in model (b)  after 10$^{5}$ years.  We display as well the time dependences of the fractional abundances of these molecules in Figure \\ref{fig:td} in order to better understand the previously described differences observed in the various fractionation ratios. \n    \\begin{figure*\n   \\centering\n  \\includegraphics[width=14cm]{Fig_7.pdf}\n  \n   \\caption{Time dependence of   CN, HCN, HNC and their isotopologues. (a) and (b) correspond to the models defined in Table \\ref{tab:model}}\n      \\label{fig:td}%\n    \\end{figure*}\n\nOur model (a) is intended to be representative of TMC1, a moderately dense cloud in an early evolutionary stage.  \nWe see that the various abundances and ratios  are very sensitive to the chosen \"age\", assumed to be the relevant chemical evolution time. Considering a  TMC1 age of 1Myr  leads to a reasonable agreement with the sparse  observations available, see Table \\ref{tab:res}.  \nModel (b) is more likely to be representative of a denser  evolved molecular cloud such as L134N, L1544, Barnard B1, ... where elemental C, O, N are partially depleted through sticking  on grains.  Steady state values obtained with significant depletion conditions \\citep{roueff:05} may be used as corresponding proxies. We see that ammonia isotopologues are satisfactorily accounted for in our model. The agreement between observed and calculated \\tC \/ {\\fN} ratios is somewhat misleading as the modeled value mainly results from a significant depletion in the {\\tC} species. The ratios involving N \/ {\\fN}  are found  close to the elemental value in our model (b) at steady state  (although {\\NHH} and {\\NHHH} are somewhat enriched in {\\fN} through the {\\fNp} + {\\HH} reaction as explained in subsection \\ref{sec:NH}), which is in agreement with the fact that  no significant \ngas phase fractionation mechanisms have been  found. The occurrence of small ratios in B1 observations \\citep{daniel:13},\nif real, implies other mechanisms at work. An obvious suggestion lies in the processes involved in adsorption\/{\\bf{desorption}}   on grains and possible surface reactions.\n\\subsection{Role of the {\\HH} ortho to para ratio}\nWe now test the role of the value of the ortho\/para ratio of molecular hydrogen (OPR) in the context of ammonia chemistry and fractionation determination. We run two additional models in the frame of models (a) and (b) by changing the OPR by a factor of 10 \nboth upwards and downwards. The abundances of p-{\\HH} and o-{\\HH} are expressed respectively as\nn(p-{\\HH}) = $\\frac{1}{1+\\rm{OPR}}$ n(\\HH) and \nn(o-{\\HH})~=$\\frac{\\rm{OPR}}{1+\\rm{OPR}}$ n(\\HH).\nThis ratio is introduced, in addition to the reaction of  {\\HH} with  {\\Np} (\\fNp), in the reverse\nreaction of the fractionation of \\HHHp, namely the \\HHDp + {\\HH} reaction which  plays a significant role in the deuterium \nfractionation of various molecules \\citep{pagani:11} as shown below:\n\\\\\n\\\\\n\\begin{tabular}{llll}\n{\\bf{$\\HHDp  + \\rm{p}-\\HH$}} &  $\\rightarrow$  & {\\bf{$ \\HHHp  + \\HD$}}& , k$_1$  (cm$^3$ s$^{-1}$)\\\\\n{\\bf{$\\HHDp  + \\rm{o}-\\HH$}}  & $\\rightarrow$ &   {\\bf{$ \\HHHp  + \\HD$}}& , k$_2$   (cm$^3$ s$^{-1}$)\\\\\\  \n\\end{tabular}\n\\\\\nwith k$_1$ = 2.0 $\\times$ 10$^{-9}$ $\\times$ exp(-232\/T)  and \nk$_2$ =  2.0 $\\times$ 10$^{-9}$ $\\times$ exp(-61.5\/T) . \n\\begin{table*}[h]\n\\caption{Dependence of the fractionation ratios on the o\/p ratio of {\\HH} for model (a). ss means stationary state.    }   \n\\label{tab:opa}      \n\\centering          \n\\begin{tabular}{|l|cc|cc|cc|}    \n\\hline\\hline       \n     & \\multicolumn{2}{c|}{OPR =  10$^{-4}$} &    \\multicolumn{2}{c|}{OPR =  10$^{-3}$} &\\multicolumn{2}{c|}{OPR =  10$^{-2}$}\\\\\n     & t= 10$^6$ yrs& ss &  t= 10$^6$ yrs& ss  &    t= 10$^6$ yrs& ss    \\\\  \n\n     \\hline\nelectronic fraction & 5.1 $\\times$ 10$^{-8}$  &  2.2 $\\times$ 10$^{-7} $   &4.8 $\\times$ 10$^{-8}$   &   2.1 $\\times$ 10$^{-7}$   &  3.7 $\\times$ 10$^{-8}$ &  4.2 $\\times$ 10$^{-8}$    \\\\\nN \/ \\fN  & 440 &   456  & 440 & 456 &  440   & 452   \\\\   \n2 $\\times$ N$_2$ \/ \\fN N &  438&   430  &  438 &431&  438  &  437  \\\\\nNH \/ \\fNH  &  431 &  429   & 429 &  428  &  426  &  418  \\\\  \nNH \/ ND  &  16  &   31  & 16  &  31  &   18   &  23 \\\\   \n\\NHHH \/ \\fNHHH  & 345 &  409   &   333 &       386          &    374   &  395   \\\\\n\\NHHD \/ \\fNHHD  &  206 &  267  &  215 &      276         &     265  &  303  \\\\\n\\NHHH \/ \\NHHD  & 8   &    14 & 17  &      22        &   43  &  63 \\\\\n\\NNHp \/ \\NfNHp &431  &  429   & 431 &  430  &    429 &  421\\\\\n\\NNHp \/ \\fNNHp & 437 &  432   & 437 &   432  &  436  &  433  \\\\\n\\NNHp \/ \\NNDp  & 16 &   30  & 16 &  29  & 17 &  21  \\\\\nCN \/ \\tCN   &  67 &  85   & 67 &   84  &  67   &  70 \\\\\nCN \/ \\CfiN  & 432 &   449  & 430 &  449  &  429 & 434    \\\\\nHCN \/ \\tHCN  & 92 & 168   &   93 &  168 &    93   &  165 \\\\\nHCN \/ \\fHCN  & 401  & 453    &  398 & 445  &   400    &  413 \\\\\nHCN \/ DCN  &  44 &   95  & 43 &  96 &  40  &  55  \\\\\nHNC\/ \\tHNC &  91 &   178  & 93 & 180   &  97   &  195   \\\\\nHNC\/ \\fHNC & 410 &   451   & 405 & 442   &  405 &   410   \\\\\nHNC\/ DNC &  23 &  66  & 23 & 66   &  22   &  32  \\\\\n\n{\\HCOp} \/ {\\HtCOp} & 57&  66   & 56 & 65 &   54  &  55   \\\\\n{\\HCOp} \/ {\\DCOp} &  15 &  30  & 15 & 29  &   16 &  19   \\\\\n\\hline                    \n  \\hline                  \n\\end{tabular}\n\\end{table*}\n %\n \\begin{table}[h]\n\\caption{Dependence of the fractionation ratios on the o\/p ratio of {\\HH} for model (b) at steady state.    }   \n\\label{tab:opb}      \n\\centering          \n\\begin{tabular}{l|c|c|c}    \n\\hline\\hline       \n     &  {OPR =  10$^{-4}$} &     {OPR =  10$^{-3}$} & {OPR =  10$^{-2}$}\\\\\n\n     \\hline\nelectronic fraction & 2.2$\\times$ 10$^{-8}$  &  1.7 $\\times$ 10$^{-8} $   &1.1 $\\times$ 10$^{-8}$      \\\\\nN \/ \\fN  & 457 &   455  & 445  \\\\   \n2 $\\times$ N$_2$ \/ \\fN N &  436&   437  & 438  \\\\\nNH \/ \\fNH  &  425 &  421   & 418  \\\\  \nNH \/ ND  &  9  &   9 &  11 \\\\   \n\\NHHH \/ \\fNHHH  &396 &  387   & 416   \\\\\n\\NHHD \/ \\fNHHD  &  322&  336  &  376  \\\\\n\\NHHH \/ \\NHHD  &11 & 18        &   33 \\\\\n\\NNHp \/ \\NfNHp &425  &  423 &  417 \\\\\n\\NNHp \/ \\fNNHp & 434 &   433  &  433 \\\\\n\\NNHp \/ \\NNDp  & 8.7 & 8.6  & 9.7  \\\\\nCN \/ \\tCN   &  65 &  63  &  60 \\\\\nCN \/ \\CfiN  & 449&  445  &   438    \\\\\nHCN \/ \\tHCN  & 115 &  114 &   107 \\\\\nHCN \/ \\fHCN  & 467  & 453   &   458 \\\\\nHCN \/ DCN  &  25 &   22 &  18 \\\\\nHNC\/ \\tHNC &  120 &   121  & 118  \\\\\nHNC\/ \\fHNC & 461 &   446 &   453   \\\\\nHNC\/ DNC &  19  & 16   &  12  \\\\\n\n{\\HCOp} \/ {\\HtCOp} &  58&  56 &51  \\\\\n{\\HCOp} \/ {\\DCOp} &  8.6&  8.4 & 9.4  \\\\\n\\hline                    \n  \\hline                  \n\\end{tabular}\n\\end{table}\n\nTables \\ref{tab:opa} and \\ref{tab:opb} display the fractionation values for the three different assumed values of the OPR for models (a) and (b)\nand Figures \\ref{fig:opa} and \\ref{fig:opb} display  the corresponding time evolutions. We see that the curves are \nalmost superposable for times less than 1 Myr for model (a) and less than several 10$^5$ yrs for model (b).\n The variations are significative at steady state. We find that the competition between the destruction channels of  {\\Np}\nthrough its reactions with o-{\\HH} and CO plays a major role.  If o-{\\HH} is the most efficient destruction channel, which occurs typically for \n$\\rm{OPR} >  200   \\times  x_{CO} $ at 10K, where x$_{CO}$ is the fractional abundance of CO relative to \\HH,\nformation of  {\\NHHHHp} proceeds efficiently. In the opposite case  {\\Np} is mainly destroyed through  reaction with CO to yield  {\\COp}, leading rapidly to {\\HCOp}, and {\\NOp} (which does not react with \\HH). As the DR rate coefficients of polyatomic ions increase significantly  for larger polyatomic ions (the {\\NHHHHp} + e$^-$ reaction is 3 times more rapid than {\\HCOp} + e$^-$  and 20 times more rapid than {\\HHHp} + e$^-$), these two different channels   impact the electron abundances and then affect many other species. \n Moreover, the reaction between  {\\NHHH} and {\\HHHp}  leads to  {\\NHHHHp}  + \\HH. {\\NHHHHp} then reacts mainly with electrons, recycling back to {\\NHHH} which subsequently converts to {\\NHHHHp}, amplifying the electron loss through DR. \n These effects occur when CO becomes the main carbon reservoir, for sufficiently large evolution times. \nAs a result, the electron fraction\nis found to decrease when  the OPR increases at large evolution times and at steady state.\nThe decreasing electron abundance with increasing OPR values acts to diminish the effect of all DR reactions\nand as a result to increase the abundances of {\\HHHp} and {\\HHDp} since DR is the main destruction channel in our conditions. The abundance of {\\HHDp} is even more enhanced through the {\\HHHp} + HD reaction. Then  the abundances of other deuterated molecular ions produced through deuteron transfer  reactions of abundant neutrals  with {\\HHDp} are increased as well, which impacts the subsequent formation of neutral deuterated compounds produced in the DR reactions. Eventually, the HCN\/DCN and HNC\/DNC ratios are shown to decrease for increasing OPR values.\nThis example demonstrates the complexity of the interplay between the different chemistries.\n \n    \\begin{figure*\n   \\centering\n  \\includegraphics[width=15cm]{Fig_8.pdf}\n  \n   \\caption{Time dependence of   fractionation ratios of CN, HCN, HNC in model (a) for 3 different OPR values. Black heavy dotted line: elemental ratio; green : OPR=10$^{-4}$, purple : OPR = 10$^{-3}$, red : OPR =10$^{-2}$}\n      \\label{fig:opa}%\n    \\end{figure*}\n    \\begin{figure*\n   \\centering\n  \\includegraphics[width=15cm]{Fig_9.pdf}\n  \n   \\caption{Time dependence of   fractionation ratios of CN, HCN, HNC in model (b) for 3 different OPR values. Black heavy dotted line: elemental ratio; green : OPR=10$^{-4}$, purple : OPR = 10$^{-3}$, red : OPR =10$^{-2}$}\n      \\label{fig:opb}%\n    \\end{figure*}\n\\section{Conclusions}\n\\label{sec:conclusion}\n We have built for the first time an isotopically substituted gas phase chemical network including D, {\\tC} and {\\fN} species and included it in a time dependent chemical model. Our model is based on a  careful  analysis of the possible gas phase mechanisms involved in carbon and nitrogen fractionation by scrutinizing the few available experimental studies and performing  DFT and ab-initio quantum calculations on  hypothetical reactions to check the possible presence of barriers in the reaction channels. One important result obtained is that the fractionation reaction of  {\\fN} with {\\NNHp} is unlikely, due to the presence of a barrier, in contrast to the previous hypothesis made by \\cite{terzieva:00}. As a result, the modeled isotopic ratios involved in the isotopologues of \\NNHp are  found to be very close to the elemental values and are similar to each other, in contradiction with observations towards L1544 \\citep{bizzocchi:13}. The availability of new collisional rate coefficients\n for the {\\NNHp - \\HH} system \\citep{lique:15} may however modify these conclusions. \n We also discarded the {\\fN} + {\\HCNHp} and  {\\fN}  + NO exchange reactions through similar arguments. Tentative reaction rate coefficients are also proposed for carbon fractionation reactions involving {\\tCp} and {\\tC} with CN and {\\nCC}.\nAdditionally, we have explicitly considered the various isotopologues  involved in  N$^{+}$ +  {\\HH} reaction, assuming that  the energy defect involved in the reactions of {\\Np} with para-{\\HH}  is a \"real\" endothermicity. This leads to a slight decline of the exponential term when {\\fNp} reacts with {\\HH} and with HD compared to {\\Np}. This explains satisfactorily that {\\fNHHD} is found to be more enriched in {\\fN} than {\\fNHHH} in the observations. \nComparison between observations of nitriles and isonitriles and simulated values is much more questionable, as carbon and nitrogen chemistries are interdependent. Observed isotopic ratios are usually large and suffer from large error bars due to opacity effects in the main isotopologue and difficulties linked to nuclear spin effects. Whereas the various isotopologues follow a similar evolution, the isotopic ratios display significant variations due to slight shifts in the position of the maximum fractional abundances. \nA reasonable agreement {\\bf{is}} obtained between the observed {\\tC} \/ {\\fN} ratios for most of the species in L134N and Barnard B1 and steady state model values.\nOur model results  show a strong depletion in {\\tC} and a near elemental ratio for  {\\nN} \/ {\\fN}, whereas  observations are usually  interpreted by assuming an elemental ratio for {\\tC} containing species which leads to the incorrect assumption of {\\fN} enrichment.  These considerations\nare undeniably dependent on the chosen elemental abundances, and in particular to the assumed C\/O ratio. \n We additionally point out a somewhat unexpected effect of the ortho to para ratio of \\HH, which affects significantly the fractional ionization, and consequently the level of deuterium fractionation  through the respective orders of magnitude of DR rate coefficients of polyatomic molecular ions. The importance of coupling C, O and N chemistries is emphasized. %\n\\section*{Acknowledgments} \n We acknowledge support of PCMI (Programme National de Physique et Chimie du Milieu Interstellaire). This work has been  partially funded by the Agence Nationale de la\nRecherche (ANR) research project IMOLABS (ANR-13-BS05-0008). \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA major outstanding puzzle in modern physics is the nature of dark matter (DM) \\cite{pdg2018}. Despite the ever-improving sensitivities of direct detection experiments, the simplest dark matter candidates have not been observed, motivating searches for a wider range of possible dark sectors. Moreover, challenges that simple cold DM candidates face on sub-galactic scales \\cite{Kuhlen:2012ft,Bullock:2017xww} might be relieved with more complex dark sectors. For instance, self-interacting dark matter has been investigated to address small-scale challenges such as the core-cusp problem \\cite{Spergel:1999mh,Loeb:2010gj}.\n\nNearly all current dark matter detection strategies, ranging from direct-detection efforts in the laboratory to indirect signals from DM annihilation (or decay), are based on the assumption that the dark matter is distributed around the universe as a gas of free particles with a large number density. This picture naturally emerges if self-interactions within the dark sector are weak, but is not strictly prescribed by existing observational constraints. The most stringent limits arise from observations of the Bullet Cluster, which restrict the self-interaction cross section per unit mass to be $\\sigma_{\\chi\\chi}\/m_{\\rm DM} \\lesssim 1 ~{\\rm cm}^2\/{\\rm g}$ \\cite{Spergel:1999mh}. It is apparent that the limit on $\\sigma_{\\chi\\chi}$ itself is significantly weakened if dark matter is clustered into composite states with large masses $m_{\\rm DM}$.\n\nIf the dark sector has strong self-interactions, it would undergo a nucleosynthesis process in the early universe much like the nuclei of the standard model (SM), whereby individual particles coalesce to form large composite states \\cite{Hardy:2014mqa,Wise:2014ola}. SM nucleosynthesis suffers from a number of theoretical accidents (such as the deuterium bottleneck) that render certain light elements unstable and thereby inhibit the production pathway of ultra-heavy elements. Still, the SM manages to produce large composite systems \\cite{ghiorso1955new}. It is thus not surprising that a completely unconstrained dark sector could also produce large composite objects. We will refer to these composite states as ultra heavy dark matter (UHDM) \\cite{Gresham:2017cvl,Grabowska:2018lnd}.\n\nDirect searches for canonical DM in the form of a gas of small particles leverage the large influx of these particles in a detector to compensate for their small cross section with SM particles. For example, about $10^{16}$ WIMP particles of mass 100 GeV would pass through a ${\\rm m}^3$ detector in a year, allowing for the direct detection of WIMP-nucleon scattering cross sections as low as $10^{-45}\\text{ cm}^2$ \\cite{Roszkowski_2018}. UHDMs, on the other hand, would arrive with significantly lower flux and  require a different detection strategy. Thus, an experimental strategy for UHDM detection should leverage generic signatures of large composite objects instead of focusing on the specifics of any one composite dark matter model, as the rich dynamics of an interacting dark sector can produce a plethora of models. Accordingly, we focus here on the fact that the many constituent particles of a UHDM will enhance its cross section with SM matter such that each rare UHDM incidence could result in a spectacular event with a distinct signature imitated by no SM effect. The detector sensitivity is therefore less important in such scenarios and can be traded for greater spacetime dimensions that maximize the probability of a rare UHDM transit. In this paper, our primary focus is to establish the detectability of such a signal. While we provide an example of a dark matter model that yields such a signal, it is straightforward to construct other examples of ultra-heavy particles that can cause similar damage (e.g., Q-balls \\cite{Kusenko:1997si}). \n\nDamage tracks left by particles passing through the Earth over geological times could be recorded by ancient rock samples buried underground. For this reason, such geological samples have been proposed and used as \"natural particle detectors\" in the past, including for magnetic monopoles \\cite{Fleischer1969,Eberhard1971,Kovalik1986,Jeon:1995rf,fleischer1969jan,fleischer1969aug,price1984,price1986}, WIMPs \\cite{Snowden-Ifft1995,Collar:1994mj,engel1995,SnowdenIfft:1997hd,Baum:2018tfw,Drukier:2018pdy,Edwards:2018hcf}, and neutrinos \\cite{Baum:2019fqm,Jordan:2020gxx}. The geological exposure times of ancient rock detectors range from the present back to the time they were last heated naturally to the point of annealing, which can be up to $\\sim 10^9$ years and is thus much longer than typical DM direct detection laboratory experiments (by factors of up to $10^9$). This advantage makes geological DM detectors ideal probes of sparser, higher-mass composite DM. As we discuss in Sec. \\ref{sec:feasibility}, searches with $10 ~{\\rm m}^2$ of billion-year old rock would probe DM masses up to $m_{\\rm DM} \\sim 10^{28}~{\\rm GeV}$. A major challenge for such a detection strategy is the ability to efficiently identify DM signatures in a large volume of rock and distinguish them from geological, radioactive, and cosmic ray backgrounds. Such discrimination is significantly simpler in searches for UHDMs, since the extremely long and continuous cylindrical damage patterns they generically leave are qualitatively different from the sporadic defects due to expected backgrounds.\n\nHere, we assess the use of geologically old quartz samples as solid-state particle detectors to search for damage tracks left by UHDMs. Quartz, a crystalline polymorph of silica SiO$_2$, is one of the most abundant and well-studied minerals in the lithosphere \\cite{gotze2012application}. Defects and damage tracks can be resolved down to the micron scale with SEM-CL: a scanning electron microscope (SEM) combined with a cathodoluminescence (CL) detector. Imaging provided by the SEM is supplemented with spectral information from CL, which reveals the nature of trace elements and point defects in the quartz \\cite{stevens2009cathodoluminescence}.  This modality has already proved successful at providing answers to key geological questions \\cite{vasyukova2013,macrae2013hyperspectral,leeman2012study,spear2009cathodoluminescence,ackerson2018low,hamers2017scanning,ackerson2015trace}. The technical advantages of SEM-CL mapping, as well as the considerable literature on its application in quartz, make it an appropriate choice for our readout method.\n\nNote that a similar search for long damage tracks was performed by Price and Salamon \\cite{price1986} in ancient mica crystals with null results. While they used this result to constrain the abundance of magnetic monopoles, the experiment is also sensitive to UHDMs with masses $m_{\\rm DM}\\lesssim 10^{26}\\,{\\rm GeV}$ \\cite{Bhoonah:2020dzs}. The readout method used in their experiment requires acid etching prior to microscopy in order to enlarge damage tracks and render them visible in an optical microscope. Such an etching process also enlarges background signals in the form of naturally-occurring lattice defects. Hence, the success of the experiment hinges on the low level of background in the samples and its scalability is limited by the availability of sufficiently pristine mica crystals. By contrast, the lack of etching in our proposed SEM-CL readout and the more readily available quartz mineral that meets our requirements allow us to scan over larger sample areas and extend the search pioneered by Price and Salamon to higher DM masses.\n\nThe rest of the discussion is organized as follows. In Sec.~\\ref{sec:feasibility} we present an experimental realization of the UHDM detection method sketched above, identify optimal samples, and assess its model-independent sensitivity. We then demonstrate its ability to probe a simple composite UHDM model, taking into account various existing constraints, in Sec.~\\ref{sec:model}. Finally, we conclude in Sec.~\\ref{sec:conclusion}.\n\n\\section{Detection Feasibility}\n\\label{sec:feasibility}\nIn this section, we investigate experimental issues for UHDM detection with quartz. Based on the considerations below, we establish the criteria for UHDM signatures to be robustly detectable with microscopy of about $\\SI{}{\\SI{}{\\micro}\\meter}$-resolution. We discuss the discovery reach of this approach in Sec. \\ref{sec:sensitivity} (see equations \\eqref{eqn:dragforcebound}-\\eqref{eqn:meltingenergypernucleus}).\n\n\\begin{figure}[htbp!]\n    \\centering\n    \\includegraphics[width=1\\columnwidth]{fig1.pdf}\n    \\caption{{Schematic of the proposed readout method.} {\\bf (a)} A quartz sample of size $\\sim{\\rm cm}^2 \\times {\\rm mm}$. The black straight line illustrates a damage track as a result of an ultra-heavy composite dark matter (UHDM) particle passing through the sample. The sample is sectioned into multiple sections of thickness $\\sim100~\\SI{}{\\SI{}{\\micro}\\meter}$. We show several sections where the top and bottom surfaces are highlighted, which would be scanned using SEM-CL.  {\\bf (b)} Correlated damage spots of micron-scale diameter over a macroscopic (mm-scale or longer) distance, between sections is the unique signature of the ultra-heavy DM particle interaction with quartz. Note that the probability of background features coincidently aligning reduces exponentially with the number of correlated layers. For a realistic feature density of $1000\/{\\rm cm}^2$, simulations show that correlations of 4 layers efficiently rejects false positive signals.}\n    \\label{fig:readout}\n\\end{figure}\n\n\\subsection{Damage Tracks}\nSolid-state systems \\cite{fleischer1964,fleischer1965,lannunziata2012} have been used as particle track detectors with applications ranging from nuclear science to geophysics, as such tracks yield information about the history of the sample and properties of the impinging particles \\cite{fleischer1965a}. For example, DM detection using crystal damage tracks has been proposed as a directional signal in semiconductors such as diamond, where a WIMP scattering event would give rise to damage tracks of $\\sim ~30-100 ~{\\rm nm}$ in length; the directional information in these tracks could enable such a detector to probe below the \"neutrino floor\" \\cite{Rajendran:2017ynw,Marshall:2020azl}. Paleodetection, which looks for damage tracks of a similar size in ancient rock samples, has also been investigated for WIMP detection \\cite{Baum:2018tfw,Drukier:2018pdy,Edwards:2018hcf}.\n \n\nWe propose using ancient quartz as a detector for ultra-heavy composite dark matter (UHDMs). As we discuss in Sec. \\ref{sec:sensitivity}, each UHDM could deposit enough energy to locally melt nearby quartz along its trajectory. Since quartz nucleation under ambient conditions is a very slow process \\cite{buckley2018nucleation}, the melted region would solidify into amorphous silica without recrystallizing. Detecting such amorphous micro-regions within quartz samples is feasible with SEM-CL, where defects in the tetrahedrally coordinated SiO$_2$ microstructure contribute to CL emission \\cite{stevens2013cathodoluminescence}. Even in the absence of melting, the same SEM-CL method would in principle be sensitive to linear tracks of lattice distortions left by UHDMs. However, quantitatively characterizing the sensitivity of this method to such tracks requires further study of the backgrounds in natural quartz, which we leave for future work. Thus, in what follows we focus on detecting UHDMs that can cause melting.\n\n\\subsection{Quartz Samples and Backgrounds}\n\\label{sec:background}\n\nThe signature of the proposed UHDM detection method is a long damage track, of micron-scale cross section, extending through the entire length of the quartz sample (see Figure \\ref{fig:readout}a). This signature has the distinct advantage that no known mechanism produces such a track, allowing for strong geometric rejection of background signals. A variety of effects may induce localized disruption of the crystal lattice on the micron scale, such as extended growth defects or radioactive decays, but these localized features cannot pass through the entire macroscopic crystal. And although particles with low interaction probability (e.g., neutrinos or relativistic cosmic rays) can pass through an entire sample, their low nuclear cross sections yield dispersed individual damage events rather than a continuous, micron-scale-diameter track. Fractures induced by historical geological stresses could similarly pass continuously through an entire sample, but would in general be two- or three- rather than one-dimensional, and would not leave behind amorphous quartz within the damage track.\n\nTo take advantage of the distinctive extended geometry of a UHDM signal, we propose a multi-phase scanning readout, where we search for correlated feature positions at multiple depths in the sample (see Figure \\ref{fig:readout}). This allows us to reject backgrounds, which have an exponentially suppressed likelihood of lying along a single line  (as shown by simulations discussed in Figure \\ref{fig:readout}). An expected background signal, given the imaging resolution of our method, is from  the presence of radioactive isotopes such as uranium, which lead to fission tracks and alpha recoil damage within the crystal lattice. These processes leave behind halos of size $\\sim~10~ \\SI{}{\\SI{}{\\micro}\\meter}$ \\cite{Bower2016}, which are readily detectable using our proposed SEM-CL protocol (see Figure \\ref{fig:CLimage}d). In a single two-dimensional SEM-CL scan, these could mimic a UHDM damage track, but would be disqualified as UHDM signals by lack of correlated damage in subsequent slices. The presence of some radioactivity-induced features is potentially beneficial to our analysis, as their preservation would indicate that recent annealing events would not have removed older, UHDM-induced features. As such, if the fission track age can be determined from the host quartz, the absence of UHDM-induced features implies the lack of UHDM interaction events since the time of occurrence of the fission track.\n\nQuartz samples with low impurity levels are essential for reducing background levels. High-resolution cathodoluminescence (CL) studies reveal both the microstructure of the samples and trace element inclusions. Titanium (Ti) and aluminum (Al) are the two of the most abundant impurities in quartz. Ti is the dominant CL activator while Al is not generally considered an activator \\cite{tailby2018,leeman2012study}. Trace element studies show that quartz samples of different geological origins have a wide range of Ti and Al concentrations. Low-temperature hydrothermal vein quartz (HVQ) has the lowest trace element concentrations: quartz typically has a Ti concentration of a few 100 ppm; but this number could be as low as 6 ppm for an HVQ sample, which simultaneously has a low Al concentration \\cite{rusk2008trace,ackerson2015trace}. Here, we characterize preliminary measurements to demonstrate that low-Ti vein quartz samples are a suitable choice for our proposed experiment (see Figure \\ref{fig:CLimage}).\n\n\\begin{figure*}\n\\includegraphics[width=2\\columnwidth]{fig2.png\n\\caption{\\label{fig:CLimage}Example quartz sample characterization. SEM-CL images of two samples, {\\bf (a)} magmatic quartz from Bishop Tuff with Ti concentration $51 \\pm 6$ ppm, and {\\bf (b)} vein quartz from Jack Hills with Ti concentration $5.2 \\pm 6.5$ ppm, measured on a mass spectrometer. The scan rate is $20~{\\rm s}\/{\\rm mm}^2$ with $1.5~\\SI{}{\\SI{}{\\micro}\\meter}$ resolution for magmatic quartz and $5~{\\rm s}\/{\\rm mm}^2$ with $3~\\SI{}{\\SI{}{\\micro}\\meter}$ resolution for vein quartz (we forecast the full-scale UHDM experiment time and resources using these values). In (b) we identify a few high-count pixels in the vein quartz image, which demonstrates the possibility of high-resolution detection of concentrated CL emission. The inset shows a zoomed-in image of the region of interest with high-count pixels. These pixels could be a melting track intersection, which needs to be investigated by correlating multiple sections as described in the text. {\\bf (c)} Normalized histogram of the pixel counts in arbitrary units for each of the two sample SEM-CL images. Vein quartz shows a lower CL noise level as well as smaller variation, making it a suitable target for our detection proposal. {\\bf (d)} SEM-CL signal from a uranium halo (measured in a different quartz sample from those shown in (a) and (b)).  Microscopic uranium inclusions have decayed over time; the fission products from these inclusions create crystal lattice damage, which emits cathodoluminescence (CL) upon excitation by the SEM. The CL signal from an ultra-heavy composite dark matter (UHDM) particle track would also result from crystal lattice defects at and around the track of melted quartz.  Any such uranium halos in a UHDM search would be disqualified as potential damage tracks by lack of correlated damage in other slices of the sample.}\n\\end{figure*}\n\nHVQ has yet other advantages. Hydrothermal fluid flow is commonly localized along fracture systems, fault systems, and shear zones that can produce vast arrays of quartz veins. When a fracture or fault remains open and under hydrothermal pressure for a sufficient period of time, hydrothermal vein quartz grows as large, euhedral, and high-purity crystals. These properties will enable us to analyze a large net exposure with the proposed protocol, using serial sectioning and SEM-CL scanning of large samples, with background rejection via correlation of damage spots across layers (see Figure \\ref{fig:readout}b).\n\nHVQ from the Jack Hills of Western Australia is an ideal source of quartz for the DM search. The siliciclastic units at Jack Hills contain numerous, large quartz veins that appear as prominent surface features (i.e., clusters of milky white outcrops that can form very localized topographic highs) observed throughout the range. The veins are generally either undeformed or very weakly deformed, and often show an abundance of high-purity, gem-quality quartz crystals. These HVQ systems can reach impressive sizes at several locations within the Jack Hills from cm-scale to 50 meters wide \\cite{spaggiarietal2007jack}. Several of the vein systems can be followed for several kilometers and appear to be associated with major episodes of brittle faulting. The combined work of Rasmussen et al. \\cite{rasmussen2010situ} and Spaggiari \\cite{spaggiari2007jack} provide strong evidence that units (including the hydrothermal quartz veins) at Jack Hills, particularly in the vicinity of the \"classic\" W74 location, have likely remained at temperature conditions less than 330-420 $^{\\circ}$C for $\\sim$ 1.7 Gyr. The fact that the tectonic environment can be evaluated in detail \\cite{trail2016li,cavosie2004internal,baxter1984jack,spaggiari2007jack} (including thermometry and age dating) and consistently demonstrates equilibria at such low temperature (i.e., at or below greenschist facies), is to the best of our knowledge unique to Jack Hills, making it an ideal source of HVQ for our proposed measurements.\n\n\\subsection{Experimental Protocol}\n\\label{sec:protocol}\n\nThe proposed experimental protocol is as follows:\n\\begin{enumerate}\n    \\item Identify quartz samples that are (i) old, having last annealed no less than 1 Gyr ago; and (ii) clean, with low CL noise level and less than a few thousand micron-scale resolved CL features per cm$^2$.\n    \\item Prepare about $10^4$ samples of size $\\sim {\\rm cm}^2 \\times {\\rm mm}$ (lateral area $\\times$ length) that satisfy the above conditions and prepare each of the samples into sections of thickness $\\sim 100~\\SI{}{\\SI{}{\\micro}\\meter}$ (see Figure \\ref{fig:readout}a). Polish the top and bottom surfaces of each section, then scan them with SEM-CL.\n    \\item Search for correlated damage spots, across the first few sections, that are aligned, section-to-section, along a straight line (see Figure \\ref{fig:readout}b).\n    \\item If such a damage track of interest is identified, perform a dedicated search in subsequent sections to reject false positives\n    \\item Repeat steps 3 and 4 for all the ${\\rm cm}^2 \\times {\\rm mm}$ samples.\n\\end{enumerate}\nThe scanning rate with SEM-CL depends on sample properties such as the concentration of CL activators. Given a typical data acquisition time of $\\sim$ 100 min per ${\\rm cm}^2$ with $\\SI{}{\\SI{}{\\micro}\\meter}$ resolution (for example see Figure \\ref{fig:CLimage}), we plan the experiment in three stages:\n\\begin{itemize}\n    \\item Quartz-$1 \\, {\\rm m}^2$: About two years of experiment time with four SEM-CL apparatuses will be required to scan samples with a total area of about 1 m$^2$. The total quartz exposure of $\\sim 1\\, {\\rm m}^2\\,{\\rm Gyr}$ for such a search would probe UHDMs of mass $m_{\\rm DM} \\lesssim 10^{27}~{\\rm GeV}$. This first stage search would probe a currently unconstrained mass range with a new technique; see Figure \\ref{fig:param_space}.\n    \\item Quartz-$10 \\, {\\rm m}^2$: 20 SEM apparatuses running for about four years would provide a total quartz exposure to UHDMs of $\\sim 10\\, {\\rm m}^2\\,{\\rm Gyr}$, yielding sensitivity $m_{\\rm DM} \\lesssim 10^{28}~{\\rm GeV}$.\n    \\item Quartz-$100 \\, {\\rm m}^2$: 100 SEM apparatuses running for about eight years would provide a total quartz exposure to UHDMs of $\\sim 100\\, {\\rm m}^2\\,{\\rm Gyr}$, yielding sensitivity $m_{\\rm DM} \\lesssim 10^{29}~{\\rm GeV}$.\n\\end{itemize}\n\n\\subsection{Model-Independent Sensitivity}\n\\label{sec:sensitivity}\nThe proposed\nexperiment would be sensitive to a wide range of ultra-heavy dark matter (UHDM) candidates, independent of the underlying dark sector microphysics, that (1) pass through the quartz sample with sufficiently high probability while (2) depositing enough energy in a sufficiently concentrated way to melt a micron-size lateral region. \n\nGiven a DM candidate of mass $m_{\\rm DM}$, we can estimate the expected number of DM transits in a sample of area $L\\times L$ over a duration $T$ to be\n\\begin{equation}\n    N \\sim 1 \\left(\\frac{10^{29}\\text{ GeV}}{m_{\\rm DM}}\\right) \\left(\\frac{L}{10\\,{\\rm m}}\\right)^2 \\left(\\frac{T}{10^9\\text{ year}}\\right)\\label{eqn:eventrate}\n\\end{equation}\nbased on the local DM density, $\\rho_{\\rm DM}\\approx 0.3\\,{\\rm GeV}\/\\text{cm}^3$. As described in the previous sections, the quartz samples under consideration are roughly $T\\sim 10^{9}\\text{ year}$ old, and a 100 ${\\rm m}^2$ sample area can be scanned in stage three. The requirement that $N\\gtrsim 1$ imposes an upper bound on the UHDM mass, $m_{\\rm DM}\\lesssim 10^{29}\\text{ GeV}\\sim 100\\text{ kg}$. The advantage afforded by the large spacetime volume of such a long-lived sample is manifest. \n\nAn UHDM moving through the Earth will collide with and deposit energy to SM particles along its path. The energy $E_1$ imparted to each SM nucleus can go as high as the kinematical limit of $10~{\\rm keV}$ (corresponding to nuclei acquiring twice the velocity of the UHDM in a collision) depending on how elastic these collisions are, while the stopping power $dE\/dx$ depends on $E_1$ as well as the UHDM radius. For simplicity, we assume in our estimates that the UHDM travels at least a few kilometers deep into the Earth's surface while maintaining its Milky Way virial velocity of $v_{\\rm DM}\\sim 10^{-3}c$. This amounts to an upper bound on the energy deposition rate\n\\begin{equation}\n    \\frac{dE}{dx}\\lesssim 10^{13}\\frac{{\\rm MeV}}{\\SI{}{\\angstrom}}\\left(\\frac{m_{\\rm DM}}{10^{29}\\text{ GeV}}\\right).\\label{eqn:dragforcebound}\n\\end{equation}\nMost of the deposited energy will likely go to SM nuclei. Only a tiny portion will go directly to electrons, whose low mass limits their kinetic energy gain (for kinematics reasons) and whose coupling to DM is severely limited by astrophysical and cosmological constraints \\cite{Green:2017ybv}. The nuclei and electrons will then thermalize, leading to a loosening of molecular bonds as the electrons acquire more energy, and eventually cause melting. Due to thermal diffusion, the melted region will enlarge and cool. What ultimately remains, in the case of quartz, is a long cylindrical trail of amorphous silica, precisely the kind of damage that is detectable with the method outlined above.\n\n\nIn order to leave a robustly detectable damage trail, the UHDM must deposit sufficient energy per unit length $dE\/dx$ exceeding the required latent heat to melt each unit-length segment of a micron-radius cylinder. This amounts to a $dE\/dx$ threshold for robust detection of\n\\begin{equation}\n    \\frac{dE}{dx}\\gtrsim \\frac{{\\rm MeV}}{\\SI{}{\\angstrom}}\\,.\\label{eqn:meltingenergydeposition}\n\\end{equation}\nSee Figure \\ref{fig:param_space}a for model-independent sensitivity projections. Further, since quartz has a melting point of $10^4~\\rm K \\sim 1~{\\rm eV}$ and energy tends to spread outward, the energy deposition must be sufficiently localized that the energy $E_1$ gained by each nucleus is greater than the melting temperature, i.e. it must lie in the range\n\\begin{equation}\n   1\\,{\\rm eV}\\lesssim E_1 \\lesssim 10\\, {\\rm keV} \\label{eqn:meltingenergypernucleus}\n\\end{equation}\nwhere the upper bound is the kinematical limit for energy transfer per nucleus (with mass number $A\\sim 10$).\n\n\\begin{figure}[htbp!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{fig3.png}\n    \\caption{{Sensitivity projections for the proposed ultra heavy dark matter (UHDM) search.} {\\bf (a)} Model-independent reach of the geological-quartz detector proposal expressed as stopping power $dE\/dx$ vs mass $m_{\\rm DM}$ of a passing UHDM particle, together with the existing constraints from MACRO for energy deposition per nucleus $E_1\\sim 1~{\\rm eV}$  \\cite{Ambrosio:2004ub, Scholz:2016} as well as from damage track searches in ancient mica \\cite{price1986}. The vertical and slanted boundaries of the quartz-detectable parameter space (for different effective detector areas) stem from the requirements of an $O(1)$ probability of transit, Eq.~\\eqref{eqn:eventrate}, and a negligible slowing of the UHDM up to a $1\\text{ km}$ depth, Eq.~\\eqref{eqn:dragforcebound}, respectively. The black horizontal line indicates the melting threshold for a micron-sized lateral region, Eq.~\\eqref{eqn:meltingenergydeposition}, above which robust detection is possible. {\\bf (b)} Parameter space of the UHDM model considered in Sec.~\\ref{sec:model}. \\textit{Left:} reach on coupling $g_{\\rm n}$ vs DM mass $m_{\\rm DM}$. \\textit{Right:} reach on coupling $g_{\\rm n}$ vs mediator mass $m_\\phi$. Also shown are existing constraints from ancient mica \\cite{price1986}, fifth force experiments \\cite{Green:2017ybv}, and stellar cooling of SN1987A \\cite{Green:2017ybv} and horizontal branch (HB) stars \\cite{Green:2017ybv} (note that the stellar cooling bounds are model-dependent \\cite{DeRocco:2020xdt}). In these $g_{\\rm n}$ plots, we set $g_\\chi$ to its upper bound $m_\\phi\/\\Lambda_\\chi$ from Eq.~\\eqref{stability}.}\n    \\label{fig:param_space}\n\\end{figure}\n\n\n\\section{Example UHDM Model}\n\\label{sec:model}\n\nIn this section, we consider an example of a simple, ultra-heavy composite dark matter (UHDM)  state \\cite{Grabowska:2018lnd} that can give rise to the desired damage tracks while being consistent with existing experimental and observational constraints. These composite objects consist of $N_\\chi$ dark fermions $\\chi$ whose mass, inverse size, and binding energy to form the UHDM are determined by a single scale $\\Lambda_\\chi$. It follows that they have mass $m_{\\rm DM} \\sim N_\\chi \\Lambda_\\chi$ and size $R \\sim N_\\chi^{1\/3} \\Lambda_\\chi^{-1}$. We assume that the fermions $\\chi$ interact with standard model nucleons $\\psi_{\\rm n}$ through a repulsive\\footnote{Attractive DM-nucleon interactions are just as compelling as the repulsive interaction considered here. We note that the attractive interactions might have more complicated dynamics as nuclei may get trapped and accumulate inside the UHDM.} Yukawa interaction mediated by a scalar $\\phi$ of mass $m_\\phi$:\n\\begin{equation}\n    \\mathcal{L}\\supset \\frac{1}{2}m_\\phi^2\\phi^2+g_{\\rm n}\\phi\\bar{\\psi}_{\\rm n}\\psi_{\\rm n}-g_{\\chi}\\phi\\bar{\\chi}\\chi\\,.\n\\end{equation}\nWe show that UHDMs with the following properties satisfy the robust detectability criteria detailed in Sec.~\\ref{sec:sensitivity} without running afoul of any existing constraints:\\footnote{Due to various constraints, this parameter space has a complicated geometry. Here we simply identified the lower and upper limits for each parameter.}\n\\begin{align}\n    10^{26} \\,{\\rm GeV} \\lesssim& m_{\\rm DM} \\lesssim 10^{29} \\,{\\rm GeV}\\label{eqn:massrange}\\\\\n    10 \\text{ nm}\\lesssim& R\\lesssim 1 \\text{ cm}\\label{eqn:radiusrange}\\\\\n    0.1\\,{\\rm eV}\\lesssim& m_\\phi\\lesssim {\\rm MeV} \\,\\leftrightarrow\\, \\,\n    10^2\\text{ fm}\\lesssim m_\\phi^{-1}\\lesssim 1\\,\\SI{}{\\SI{}{\\micro}\\meter}\n    \\label{eqn:mediatorrange}\\\\    \n    100 \\,{\\rm keV} \\lesssim& \\Lambda_\\chi \\lesssim 10 \\,{\\rm GeV}\\,.\\label{eqn:Lambdarange}\n\\end{align}\nThis allows us to probe wide ranges of the couplings $g_{\\rm n}$ and $g_{\\chi}$. Two slices of this parameter space are shown in Figure~\\ref{fig:param_space}b. Eq.~\\eqref{eqn:massrange} follows from Eq.~\\eqref{eqn:eventrate} and ancient mica constraints; Eq.~\\eqref{eqn:radiusrange} follows from\n\\eqref{eqn:meltingenergydeposition}, \\eqref{eqn:meltingenergypernucleus}, \\eqref{eqn:E1dEdx}, and the quartz sample size of 1 cm; Eq.~\\eqref{eqn:mediatorrange} follows from fifth force constraints and the requirement that the UHDM-nucleus interaction be treated classically; Eq.~\\eqref{eqn:Lambdarange} follows from Eqs.~\\eqref{eqn:massrange} and \\eqref{eqn:radiusrange}.\n\n\\subsection{Detectability with Quartz}\nThe optimal UHDM detection signature is expected for mediators with a range $m_\\phi^{-1}$ satisfying $\\Lambda_\\chi^{-1}\\ll m_\\phi^{-1} \\lesssim \\SI{}{\\SI{}{\\micro}\\meter}$, since this is the intermediate regime where the UHDM-nucleon coupling is enhanced by the number of constituents $\\chi$ of the UHDM within the range of the mediator $(m_\\phi^{-1}\/\\Lambda_\\chi^{-1})^3$ while simultaneously evading existing fifth force constraints. For simplicity of analysis we only consider part of the parameter space where $m_\\phi^{-1}\\ll R$. In doing so, we limit the UHDM's cross section to be at most geometrical.\n\nAn SM nucleus located inside the UHDM only sees the composite dark matter particle's constituents $\\chi$ within the range of the mediator $ m_\\phi^{-1}\\ll R$. Hence, to the SM nucleus each point in the bulk of the UHDM is just like any other, yielding a potential energy $V(r)$ as a function of the distance $r$ from the center of the UHDM with the following profile:\n\\begin{equation}\n    V(r)=\\begin{cases}\n    +V_0, &r<R\\\\\n    0, &r>R\n    \\end{cases}\n\\end{equation}\nwhere at the boundary $r\\approx R$ the potential drops to zero exponentially over a length scale of order $m_\\phi^{-1}$, and \n\\begin{align}\n    V_0\\sim  \\left(\\frac{\\Lambda_\\chi}{m_\\phi}\\right)^3\\frac{g_{\\chi}(10g_{\\rm n})}{m_\\phi^{-1}} \\label{eqn:V0}\n\\end{align}\nfor SM nuclei with mass number  $A\\sim 10$. As a result, from the perspective of a nucleus the UHDM is just a constant potential hill moving at a velocity $v_{\\rm DM}\\sim 10^{-3}c$.\n\nSince the de Broglie wavelengths $(10\\,{\\rm MeV})^{-1}$ of the SM nuclei are smaller than the mediator ranges $m_\\phi^{-1}$ of interest, we can treat the UHDM-nucleus interactions classically. When $V_0\\gtrsim 10 \\,{\\rm keV}$, the potential $V_0$ prevents any nucleus from entering the UHDM. The UHDM-nucleus collisions are thus elastic, and the energy $E_1$ transferred to a nucleus saturates the kinematical limit $E_1\\sim 10\\, {\\rm keV}$. If $V_0\\lesssim 10 \\,{\\rm keV}$, on the other hand, the nuclei can easily climb the potential hill, and the collisions between a nucleus and the UHDM's surface will be inelastic. When a nucleus encounters the surface of the UHDM, it receives a force $F\\sim V_0\/m_\\phi^{-1}$ due to the gradient of the Yukawa potential. This force is exerted throughout the duration $\\tau\\sim m_\\phi^{-1}\/v_{\\rm DM}$ of the collision, resulting in a nearly-instantaneous momentum kick $p_1\\sim F\\tau$ which translates to the kinetic energy $E_1\\sim 10\\,{\\rm keV}\\left(V_0\/10\\,{\\rm keV}\\right)^2$ per nucleus. To sum up, the energy imparted to a nucleus after the passage of a UHDM is\n\\begin{equation}\n    E_1\\sim 10\\,{\\rm keV} \\times \\text{min}\\left[1,\\left(\\frac{V_0}{10 \\,{\\rm keV}}\\right)^2\\right].\\label{eqn:E1V0}\n\\end{equation}\nUsing a lattice spacing of about $5\\SI{}{\\angstrom}$ for quartz, the energy deposition rate then follows:\n\\begin{equation}\n    \\frac{dE}{dx}\\sim \\frac{E_1}{5\\SI{}{\\angstrom}}\\left(\\frac{R}{5\\SI{}{\\angstrom}}\\right)^2. \\label{eqn:E1dEdx}\n\\end{equation}\nHaving linked the model parameters with the quantities characterizing quartz damage tracks, the detectable parameter space can be evaluated based on the considerations in Sec.~\\ref{sec:sensitivity} (see Figure \\ref{fig:param_space}b).\n\n\\subsection{Existing Constraints}\n\\label{sec:existing_constraints}\n\\subsubsection{The mediator}\nPast experiments and observations have placed limits on the coupling $g_{\\rm n}$ of the mediator $\\phi$ to standard model nucleons with varying severity for different masses $m_\\phi$ of the mediator.These include: collider constraints on the meson decay rate, laboratory {\\it fifth-force} searches, and stellar cooling bounds from observations of the SN1987A event and horizontal branch (HB) stars. Note, however, that the stellar cooling bounds are model-dependent \\cite{DeRocco:2020xdt}. The following parameter space is thus ruled out \\cite{Green:2017ybv}:\n\n\\begin{itemize}\n    \\item Meson decay: $g_{\\rm n}\\gtrsim 10^{-6}$, $m_\\phi\\lesssim 100\\,{\\rm MeV}$.\n    \\item Fifth force: $g_{\\rm n}\\gtrsim 10^{-12} (m_\\phi\/{\\rm eV})^3$, $m_\\phi\\lesssim 100\\,{\\rm eV}$.\n    \\item SN1987A: $3\\times10^{-10}\\lesssim g_{\\rm n}\\lesssim 3\\times10^{-7}$, $m_\\phi\\lesssim 30\\,{\\rm MeV}$.\n    \\item HB stars: $g_{\\rm n}\\gtrsim 10^{-13}$, $m_\\phi\\lesssim 100\\,{\\rm keV}$.\n\\end{itemize}\nFurthermore, the couplings of UHDM constituents $\\chi$ to the mediator $\\phi$ add extra self-interactions among $\\chi$ that may destabilize the UHDM. In order for the UHDM to be stable the mediated self-interaction potential $g_\\chi^2\\Lambda_\\chi^3m_\\phi^{-2}$ of a single $\\chi$ must not exceed the binding energy $\\Lambda_\\chi$. This puts an upper bound on the coupling $g_\\chi$ of $\\chi$ to the mediator $\\phi$:\n\\begin{equation}\n    g_{\\chi}\\lesssim \\frac{m_\\phi}{\\Lambda_\\chi}.\\label{stability}\n\\end{equation}\n\n\n\\subsubsection{Direct detection}\nOf the currently and previously running direct-detection DM experiments, MACRO puts a strong constraint on our scenario due to its large volume. MACRO is a scintillator experiment with spacetime dimensions of about $10^3\\text{ m}^2\\times 10\\text{ years}$ corresponding to $m_{\\rm DM} \\lesssim 10^{22}\\,{\\rm GeV}$ for 1 event over its decade-long lifespan. It is sensitive to energy depositions $\\gtrsim 10\\,{\\rm MeV}\/{\\rm cm}$ \\textit{to electrons} \\cite{Ambrosio:2004ub}. When a nucleus receives energy $E_1$ from interaction with a UHDM, only some fraction $Q(E_1)$, called the quenching factor, of that energy effectively goes to the electrons tied to the nucleus. It is this relatively small fraction of energy that is responsible for the processes of scintillation and ionization that may occur subsequently. We can translate MACRO's $10\\,{\\rm MeV}\/{\\rm cm}$ detection threshold to a sensitivity \\textit{to nuclear energy depositions} via effecting an increase by the quenching factor $Q(E_1)$ \\cite{Scholz:2016}. \n\nAn even more stringent bound on our model arises from direct searches for long damage tracks in muscovite mica crystals \\cite{price1986}. The non-observation of tracks extending beyond naturally occurring defects and radioactivity damage was originally used to constrain the abundance of magnetic monopoles, but also limits the UHDM parameter space. This past mica search involved total sample area $\\sim 1200 ~{\\rm cm}^2$ with sample ages $\\simeq 5 \\times 10^8 \\,{\\rm yr}$, corresponding to a dark matter reach of $m_{\\rm DM} \\lesssim 10^{26}~{\\rm GeV}$. The energy deposition threshold for detection in this experiment via etching and optical microscopy was identified as $dE\/dx \\gtrsim 6~{\\rm GeV}\/{\\rm cm}$.    \n\n\\subsubsection{Astrophysical and Cosmological limits}\nIndirect limits can also be placed on the couplings $g_{\\rm n}$ and $g_{\\chi}$ from the limits on DM-baryon and DM-DM cross sections. DM-baryon interactions in the early universe can affect baryon acoustic oscillations and is therefore constrained by CMB and LSS observations. This puts an upper bound on the DM-baryon momentum-transfer cross section that would be observed today: $\\sigma_{\\chi\\rm b}\/m_{\\rm DM}\\lesssim 10^{-3}\\text{ cm}^2\/\\text{g}$ \\cite{Dvorkin:2013cea}. Astronomical observations of the Bullet Cluster also place a limit on the DM-DM momentum-transfer cross section: $\\sigma_{\\chi\\chi}\/m_{\\rm DM}\\lesssim 1\\text{ cm}^2\/\\text{g}$ \\cite{Spergel:1999mh}. Since we are mainly interested in UHDMs with geometrical cross sections of order $\\SI{}{\\SI{}{\\micro}\\meter}^2$ and masses up to $10^{29}~{\\rm GeV} (100 ~{\\rm kg})$, these astrophysical and cosmological observations only impose significant constraints on the low mass side of our parameter space. Moreover, these constraints are alleviated if UHDMs constitute less than $10\\%$ of the total dark matter mass, in which case the maximum detectable mass would also be lowered by an order of magnitude.\n\n\\section{Conclusion and Outlook}\n\\label{sec:conclusion}\n\nGiven the diverse range of theoretically well-motivated dark sectors, it is critical to perform searches with techniques that are sensitive to a broad class of dark-sector phenomena. In this paper, we propose a detection method for ultra-heavy composite dark matter particles (UHDMs). Our proposed experiment is based on mapping damage tracks in ancient quartz samples with SEM-CL scanning. This method has two significant advantages: (1) the billion-year exposure time of such samples enables us to probe dark matter candidates with masses as high as $10^{29}~{\\rm GeV} (100 ~{\\rm kg})$, surpassing the reach of existing direct-detection experiments, and (2) the distinctly-long cylindrical damage trails left by such UHDMs are easily distinguished from other features at the relevant scales.\n\nIn this work, we focus on detecting long tracks of amorphous silica in quartz samples expected from passing UHDMs that impart enough energy to cause melting. In future work, we will consider the feasibility of extending the experimental sensitivity to energy deposition rates below the melting threshold. For that purpose, we intend to carry out a number of studies including: (i) signal calibration by artificially creating damage tracks in synthetic quartz samples with a high-power pulsed laser of variable intensity and comparing it with the resulting CL signal levels; and (ii) noise calibration by preparing a set of quartz samples, natural and synthetic, with different concentrations of CL activators and analysing their CL emission rates. These studies will provide us a better understanding of the signal-to-noise ratio as seen in SEM-CL imaging, which will allow us to better estimate the detection threshold.\n\nOur proposed experiment is largely agnostic to the detailed microphysics of the dark sector, as long as it results in long damage tracks in geological quartz. To demonstrate the projected reach of the proposed approach, we considered a QCD-like dark sector that interacts with the standard model repulsively via a light mediator. The particle spectrum of this theory includes heavy bound states, composed of a large number of elementary dark fermions, which could create interesting targets for detection. We identified experimentally-detectable regions of the parameter space that satisfy various limits derived from phenomenological considerations as well as past observations. In future work, it would be interesting to delineate a broader range of dark matter models than can lead to similar damage patterns in ancient rock.\n\n\n\\begin{acknowledgments}\nThis work was supported by the DOE QuANTISED program under Award No. DE-SC0019396; the Army Research Laboratory MAQP program under Contract No. W911NF-19-2-0181; and the University of Maryland Quantum Technology Center. SR is supported by the NSF under grant PHY-1818899, the SQMS Quantum Center and DOE support for MAGIS. \n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev4-1}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the study of nonnegative scalar curvature, one would like to formulate some notion of ``weak'' nonnegative scalar curvature for metrics that are not necessarily smooth.   The gold standard for a notion of weak nonnegative scalar curvature would be something like Alexandrov spaces as a weak notion of spaces with lower bounds on sectional curvature.   A good notion of weak nonnegative scalar curvature would be one that implies the same consequences as ``classical'' nonnegative scalar curvature---for example, the positive mass theorem in the asymptotically flat case, or topological restrictions in the compact case. \n\nAn important theorem in this direction was proved by P.~Miao \\cite{Miao:2002}, generalizing an earlier result of H.~Bray \\cite[Section 6]{Bray:2001}.  See also \\cite[Section~3]{Shi-Tam:2002} for the spin case, as well as a more recent proof by D.~McFeron and G.~Sz\\'{e}kelyhidi~\\cite{McFeron-Szekelyhidi}.  \n\n\\begin{thm}\\label{hypersurface}\nLet $M^n$ be a smooth manifold such that $n<8$ or $M$ is spin.\\footnote{The only point of this assumption is to make sure that the classical positive mass theorem is valid on $M$.  If the positive mass theorem is true in all dimensions, then this hypothesis can safely be eliminated}  Let $S$ be a smooth closed hypersurface in $M$, and let $g$ be a complete asymptotically flat metric on $M$ such such that $g$ is $C^2$ \\emph{up to} $S$ from each side of it (but not necessarily \\emph{across} it) and $C^{2,\\alpha}_\\mathrm{loc}$ away from $S$.  \n\nNear each point of $S$, $S$ divides $M$ into two sides, which we will call $A$ and $B$.  Let $H_A$ be the mean curvature vector of $S$ as computed by the metric on the $A$ side, and similarly define $H_B$.\n\nIf $g$ has nonnegative scalar curvature on the complement of $S$, and at each point of~$S$, $H_A-H_B$ either points toward side $A$ or is zero, then the mass of $g$ is nonnegative in each end.\n\\end{thm}\n\nNote that the hypotheses of the theorem above require $g$ to be Lipschitz everywhere.  One way to interpret this theorem is that when the singular set of $g$ is a hypersurface $S$ whose induced metric is well-defined regardless of which ``side'' of $S$ one uses to compute it, then the correct notion of weak nonnegative scalar curvature on $S$ is the pointwise mean curvature comparison condition that appears as a hypothesis of the theorem above.  \n\nIn this article we consider singular sets of lower dimension and ponder what conditions on $S$ correspond to weak nonnegative scalar curvature.  We find that if $S$ has low enough dimension, then no further conditions are needed.\n\\begin{thm}\\label{maintheorem}\nLet $M^n$ be a smooth manifold such that $n<8$ or $M$ is spin.  Let $g$ be a complete asymptotically flat Lipschitz metric on $M$, and let $S$ be a bounded subset whose $n\/2$-dimensional lower Minkowski content is zero.\nIf $g$ has bounded $C^2$-norm and nonnegative scalar curvature on the complement of $S$, then the mass of $g$ is nonnegative in each end.  \n\\end{thm}\nSee Section \\ref{definitions} for the definition of Minkowski content.  For now, recall that Minkowski content equals Hausdorff measure for well-behaved sets (\\textit{e.g.} submanifolds).\n\nThere is also a $W^{1,p}$ version of this theorem.\n\\begin{thm}\\label{w1p}\nLet $M^n$ be a smooth manifold such that $n<8$ or $M$ is spin.  Let $p>n$, let $g$ be a complete asymptotically flat $W^{1,p}_{\\mathrm{loc}}$ (and hence continuous) metric on $M$, and let $S$ be a bounded subset whose $\\frac{n}{2}(1-\\frac{n}{p})$-dimensional lower Minkowski content is zero.  If $g$ has bounded $C^2$-norm and nonnegative scalar curvature on the complement of $S$, then the mass of $g$ is nonnegative in each end.  \n\\end{thm}\n\nIt might seem surprising that one does not have to place any other conditions on the behavior of $g$ at singular set, but as we will see in the proof, the Lipschitz (or $W^{1,p}$) condition is very restrictive.  Essentially, $g$ is too regular for the scalar curvature to be truly singular on a small set.  We use the same technique as in \\cite{Miao:2002}.  Note that if $S$ is a closed submanifold, the proofs of Theorems \\ref{maintheorem} and \\ref{w1p} are much simpler.\n\nThe dimensional restriction of $n\/2$ seems to an unnecessary artifact of the conformal method used in the proof.  Also note that Theorems \\ref{maintheorem} and \\ref{w1p} do not include rigidity results for Euclidean space.\n\\begin{conj}\nLet $M^n$ be a smooth manifold such that $n<8$ or $M$ is spin.  Let $g$ be a complete asymptotically flat Lipschitz metric on $M$, and let $S$ be a bounded subset whose $(n-1)$-dimensional lower Minkowski content is zero.\nIf $g$ has bounded $C^2$-norm and nonnegative scalar curvature on the complement of $S$, then the mass of $g$ is nonnegative in each end.  Moreover, if the mass of any end is zero, then $(M,g)$ must be isometric to Euclidean space.\n\\end{conj}\nOne might try to prove the spin case of this conjecture using a spinor argument, following \\cite{Shi-Tam:2002}.  Or one might try to prove the conjecture using Ricci flow as in \\cite{McFeron-Szekelyhidi}.  One advantage of the Ricci flow method is that it is more likely to produce a rigidity result.\n\n\\section{Definitions}\\label{definitions}\n\\begin{defin}\nLet $g$ be a continuous Riemannian metric on a smooth manifold $M^n$ where $n\\geq3$.  Then $(M,g)$ is an \\emph{asymptotically flat manifold} if and only if there is a compact set $K\\subset M$ such that $M\\smallsetminus K$ is a disjoint union of ends, $E_\\ell$, such that\neach end is diffeomorphic to $\\rr^n$ minus a ball, and in each of\nthese coordinate charts, the metric $g_{ij}$ is $C^2$ and satisfies \n\\begin{align*}\ng_{ij}&=\\delta_{ij}+O(|x|^{-\\sigma})\\\\ \ng_{ij,k}&=O(|x|^{-\\sigma-1})\\\\\ng_{ij,kl}&=O(|x|^{-\\sigma-2})\\\\\n R_g&=O(|x|^{-\\tau}), \n\\end{align*} \nfor some $\\sigma>(n-2)\/2$ and $\\tau>n$, where the commas denote partial derivatives in the coordinate chart, and $R_g$ denotes the scalar curvature of~$g$.\n\nWe define the \\emph{mass} of each end $E_\\ell$ by the formula\n\\[m(E_\\ell,g)={1\\over\n2(n-1)\\omega_{n-1}}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho}\n\\sum_{i,j=1}^n(g_{ij,i}-g_{ii,j})\\nu_j d\\mu,\\]\n where $\\omega_{n-1}$ is the area of the standard unit\n$(n-1)$-sphere, $S_{\\rho}$ is the coordinate sphere in $E_\\ell$ of radius\n$\\rho$, $\\nu$ is its outward unit normal, and $d\\mu$ is the Euclidean area\nelement on $S_{\\rho}$.  The mass is well-defined on each end of an asymptotically flat manifold.\n\\end{defin}\n\n\\begin{defin}\nFor a subset $S$ of a Riemannian manifold$(M^n, g)$, the \\emph{$m$-dimensional lower Minkowski content} of $S$ is \n\\[ \\liminf_{\\epsilon\\to0} \\frac{\\mathcal{L}_g^n(S_\\epsilon)}{\\alpha_{n-m}\\epsilon^{n-m}}\\]\nwhere $\\mathcal{L}_g^n$ is Lebesgue measure with respect to $g$, $S_\\epsilon$ is the $\\epsilon$-neighborhood of $S$, and $\\alpha_{n-m}$ is the volume of the unit ball in $\\rr^{n-m}$.\n\\end{defin}\nThe $m$-dimensional lower Minkowski content provides an upper bound (up to constant) for $m$-dimensional Hausdorff measure, and they are the same for rectifiable sets (see \\cite[Chapter 3.2]{Federer-book} for details).  In particular, the condition of zero Minkowski content in Theorems \\ref{maintheorem} and \\ref{w1p} is only slightly stronger than the condition of zero Hausdorff measure.\n\n\n\\section{Proof of Theorem \\ref{maintheorem}}\\label{main}\n\nFirst, we briefly sketch out the proof, which is straightforward.  Choose $M^n$, $g$, and $S$ as in the statement of Theorem \\ref{maintheorem}.  We mollify the metric $g$ to get a smooth metric $g_\\epsilon$ in such a way that $g_\\epsilon=g$ outside of the $2\\epsilon$-neighborhood $S_{2\\epsilon}$.  The precise smoothing of $g$ does not matter much.  The only important property of the smoothing is that, using the hypotheses on~$g$, we have that $g_\\epsilon$, $g_\\epsilon^{-1}$, and $\\partial g_\\epsilon$ are bounded independently of $\\epsilon$, while $\\partial\\partial g_\\epsilon=O(\\epsilon^{-1})$, with respect to a particular atlas.  By the formula for the scalar curvature of $g_\\epsilon$, it follows that $R_{g_\\epsilon}=O(\\epsilon^{-1})$.  The hypothesis about Minkowski content tells us (roughly) that the volume of $S_\\epsilon$ is $o(\\epsilon^{n\/2})$.  Thus\n\\begin{equation}\\label{goal}\n\\int_{S_{2\\epsilon}} |R_{g_\\epsilon}|^{n\/2}\\,dg=o(1).\n\\end{equation}\nFrom there, a standard argument (as in \\cite{Miao:2002}) tells us that we can conformally deform $g_\\epsilon$ to have nonnegative scalar curvature, without changing the mass too much.  Applying the classical positive mass theorem to the new, smooth manifold of nonnegative scalar curvature, we find that the original manifold $(M,g)$ has mass greater than a small negative number that is $o(1)$ in $\\epsilon$.  Taking the limit as $\\epsilon$ approaches zero, the result follows.  In what follows, we describe an explicit smoothing that yields \\eqref{goal}.\n\nWe choose a finite atlas $U_1,\\ldots,U_N$ for $M$.  By asymptotic flatness and continuity of $g$, we can choose these $U_k$ so that $g$ is uniformly equivalent to the background Euclidean metric of each patch.  That is, on each coordinate patch, we have \n\\[ C^{-1}\\delta_{ij}\\leq g_{ij}\\leq C\\delta_{ij} \\]\nas positive definite symmetric bilinear forms.  \nWe choose a partition of unity  $\\psi_1,\\ldots, \\psi_N$ subordinate to this cover. \nOn each patch $U_k$, we will define a smoothing $g^k_\\epsilon$ of $g$ that is defined on the support of $\\psi_k$, which we denote $U'_k$.   We then obtain a smoothing $g_\\epsilon$ of $g$ by defining $g_\\epsilon=\\sum_{k=1}^N \\psi_k g^k_\\epsilon$.  \n\n\\begin{notation}In what follows, we will use a generic constant $C$ to mean some large number that may depend on $(M,g)$ and the choices of $U_k$ and $\\psi_k$.  The only thing that will be important to us is that $C$ is independent of $\\epsilon$.\n\\end{notation}\nGiven a coordinate patch $U_k$, we wish to define $g^k_\\epsilon$.  Let $\\varphi$ be a nonnegative smooth function supported on the unit ball in $\\rr^n$ whose integral is $1$.  The standard way to smooth $g$ is to convolve it with \n\\[\\varphi_\\epsilon(x) := \\epsilon^{-n}\\varphi(x\/\\epsilon).\\]\n  However, we want to smooth $g$ in such a way that it does not change $g$ away from a neighborhood of $S$.  In order to do that, we need the following simple lemma.  \n\n\\begin{lem}\\label{sigma}\nFor each $\\epsilon>0$, on each coordinate patch $U_k$, there exists a nonnegative smooth function $\\sigma$  such that $\\sigma=\\epsilon$ on the Euclidean neighborhood $S_\\epsilon$ of $S$ in $U_k$, and $\\sigma=0$ outside $S_{2\\epsilon}$, while $|\\partial\\sigma|\\le 3$ and $|\\partial\\partial\\sigma|\\le C\\epsilon^{-1}$ everywhere.\n\\end{lem}\n\\begin{proof}\nDefine a continuous function \n\\[s(x)=\\left\\{\\begin{array}{ll}\n\\epsilon & \\text{for }x\\in S_{4\\epsilon\/3}\\\\\n5\\epsilon - 3\\dist(x,S) & \\text{for }x\\in S_{5\\epsilon\/3}\\smallsetminus S_{4\\epsilon\/3}\\\\\n0 & \\text{for }x\\notin S_{5\\epsilon\/3}\n\\end{array}\\right.\\]\nThen we can define \n\\[\\sigma(x)=\\int_{\\rr^n} s(x-y)\\varphi_{\\epsilon\/6} (y)\\,dy.\\]\nClearly, $\\sigma$ is a nonnegative smooth function such that $\\sigma=\\epsilon$ on $S_\\epsilon$ and $\\sigma=0$ outside $S_{2\\epsilon}$.  We just need to check the bounds on derivatives.\n\\begin{align*}\n|\\sigma(x_1)-\\sigma(x_2)| & \\leq\\int_{\\rr^n} |s(x_1-y)-s(x_2-y)|\\varphi_{\\epsilon\/6} (y)\\,dy \\\\\n&\\leq \\int_{\\rr^n} 3 |x_1-x_2| \\varphi_{\\epsilon\/6} (y)\\,dy\\\\\n&=3|x_1-x_2|,\n\\end{align*}\nwhere we used the Lipschitz property of $s$ in the second line.  Thus $|\\partial\\sigma|\\le 3$.\n\nWe know that \n\\begin{align*}\n \\partial\\sigma(x)&= \\int_{\\rr^n} s(x-y) \\partial\\varphi_{\\epsilon\/6}(y)\\,dy \\\\ \n&=\\int_{\\rr^n} s(x-y)  \\left(\\frac{6}{\\epsilon}\\right)^{n+1}  \\partial\\varphi\\left(\\frac{6y}{\\epsilon}\\right)\\,dy \n\\end{align*}\nArguing as above, \n\\begin{align*}\n|\\partial\\sigma(x_1)-\\partial\\sigma(x_2)| \n&\\leq \\int_\\rr^n 3|x_1-x_2|    \\left(\\frac{6}{\\epsilon}\\right)^{n+1}    \\left|\\partial\\varphi \\left(\\frac{6y}{\\epsilon}\\right)\\right|\\,dy \\\\\n& = \\frac{18}{\\epsilon} |x_1-x_2| \\int_{\\rr^n} |\\partial\\varphi (y)|\\,dy \\\\\n&=\\frac{C}{\\epsilon}|x_1-x_2|.\n\\end{align*}\n Thus $|\\partial\\partial\\sigma|\\le C\\epsilon^{-1}$.\n\\end{proof}\n\nFor small enough $\\epsilon$, we define $g^k_\\epsilon$ on the patch $U'_k$ by the formula\n\\[ (g^k_\\epsilon)_{ij}(x)=\\int_{\\rr^n}g_{ij}(x-\\sigma(x)y)\\varphi(y)\\,dy \n=\\int_{\\rr^n}g_{ij}(y)\\varphi_{\\sigma(x)}(x-y)\\,dy.\\]\nKeep in mind that the function $\\sigma$ described by the lemma above depends on the patch $U_k$, the singular set $S$, and $\\epsilon$.  Clearly, each component of $g^k_\\epsilon$ is smooth.  We now claim that $|\\partial g^k_\\epsilon|\\le C$ and $|\\partial\\partial g^k_\\epsilon|\\le C\\epsilon^{-1}$.  For ease of notation, let us prove these inequalities for each component, individually.  The lemma below proves the claim.\n\n\\begin{lem}\\label{convolve}\nLet $\\sigma$ be the function described in Lemma \\ref{sigma}, and let $f$ be a Lipschitz function on $U_k$ that has bounded $C^2$-norm on the complement of $S$.  For small enough $\\epsilon$, if we define the function\n\\[ f_\\epsilon(x)=\\int_{\\rr^n}f(x-\\sigma(x)y)\\varphi(y)\\,dy \n=\\int_{\\rr^n}f(y)\\varphi_{\\sigma(x)}(x-y)\\,dy\\]\non the set $U'_k$, then $|\\partial f_\\epsilon|\\le C$ and $|\\partial\\partial f_\\epsilon|\\le C\\epsilon^{-1}$, where $C$ may depend on the supremum and Lipschitz constant of $f$.\n\\end{lem}\n\\begin{proof}\nOn the complement of $S_\\epsilon$, the result follows easily from differentiating the first formula for $f_\\epsilon$ above and using the $C^2$ bound on $f$ and the bounds on $|\\partial\\sigma|$ and $|\\partial\\partial\\sigma|$ from Lemma \\ref{sigma}.  So we need only consider the region $S_\\epsilon$.  But in this region we have $\\sigma(x)=\\epsilon$ by construction, and therefore\n\\[ f_\\epsilon(x)\n=\\int_{\\rr^n}f(y)\\varphi_\\epsilon(x-y)\\,dy \n\\]\nis just the usual mollification formula.  Then since $f$ is Lipschitz, a standard computation shows that $|\\partial f_\\epsilon|$ is bounded.  Moreover, for $x\\in S_\\epsilon$,\n\\begin{align*}\n\\partial f_\\epsilon (x)\n&=\\int_{\\rr^n} f(y) \\partial\\varphi_\\epsilon(x-y)\\,dy\\\\\n&=\\int_{\\rr^n} f(y) \\epsilon^{-n-1}\\partial\\varphi\\left(\\frac{x-y}{\\epsilon}\\right)\\,dy\\\\\n&=\\int_{\\rr^n} f(x-\\epsilon y) \\epsilon^{-1}\\partial\\varphi(y)\\,dy.\n\\end{align*}\nSo for any $x_1,x_2\\in S_\\epsilon$, \n\\begin{align*}\n|\\partial f_\\epsilon(x_1)-\\partial f_\\epsilon(x_2)|\n&=\\left| \\int_{\\rr^n} [f(x_1-\\epsilon y)-f(x_2-\\epsilon y)] \\epsilon^{-1}\\partial\\varphi(y)\\,dy\\right|\\\\\n&\\le  \\int_{\\rr^n} |f(x_1-\\epsilon y)-f(x_2-\\epsilon y)| \\epsilon^{-1}|\\partial\\varphi(y)|\\,dy\\\\\n&\\le  \\int_{\\rr^n}C|x_1-x_2| \\epsilon^{-1}|\\partial\\varphi(y)|\\,dy\\\\\n&\\le \\frac{C}{\\epsilon}|x_1-x_2|.\n\\end{align*}\nThe result follows.\n\\end{proof}\n\nSetting $g_\\epsilon=\\sum_{k=1}^N \\psi_k g^k_\\epsilon$, Lemma \\ref{convolve} implies that in each coordinate chart~$U_k$, \n$|\\partial (g_\\epsilon)_{ij}|\\le C$ and $|\\partial\\partial (g_\\epsilon)_{ij}|\\le C\\epsilon^{-1}$ for some $C$ independent of~$\\epsilon$.  From looking at how scalar curvature depends on the metric, it is clear that $|R_{g_\\epsilon}|\\le C\\epsilon^{-1}$ for some $C$.  Meanwhile, $g=g_\\epsilon$ outside $S_{2\\epsilon}$, and by our assumption on the Minkowski content of $S$, we have \n\\begin{align*}\n\\int_{S_{2\\epsilon}} |R_{g_\\epsilon}|^{n\/2}\\,dg &\\le \\mathcal{L}_g^n(S_{2\\epsilon})\\sup |R_{g_\\epsilon}|^{n\/2} \\\\\n&= o(\\epsilon^{n\/2}) O(\\epsilon^{-n\/2})\\\\\n&=o(1),\n\\end{align*}\nwhich is our desired estimate \\eqref{goal}.  The rest of the proof of Theorem \\ref{maintheorem} proceeds exactly as in \\cite{Miao:2002}.  (Technically, since we are using \\emph{lower} Minkowski content, it is inaccurate to say that $\\mathcal{L}_g^n(S_{2\\epsilon})= o(\\epsilon^{n\/2})$, but the argument still works since we only need to use a subsequence of $\\epsilon$'s approaching zero.  Also, we were careless about the distinction between defining $S_\\epsilon$ using the metric $g$ versus the Euclidean metric on each chart, but by uniform equivalent of metrics, this sloppiness is inconsequential.)\n\n\\section{Proof of Theorem \\ref{w1p}}\n\nThe proof of the $W^{1,p}$ version of Theorem \\ref{maintheorem} requires only slight modification.  Choose $M^n$, $g$, $S$, and $p$ as in the statement of Theorem \\ref{maintheorem}.  First, observe that because of the $C^2$ bounds on $g$, we can see that $|R_{g_\\epsilon}|=O(\\epsilon^{-1})$ on the complement of $S_\\epsilon$, just as in the Lipschitz case, and we now have even better bounds on $\\mathcal{L}^n_g(S_{2\\epsilon})$, so we have\n\\[\\int_{S_{2\\epsilon}\\smallsetminus S_\\epsilon} |R_{g_\\epsilon}|^{n\/2}\\,dg=o(1).\\] \n  Therefore, in order to establish \\eqref{goal}, it is sufficient to show that\n\\begin{equation}\\label{newgoal}\n\\int_{S_\\epsilon} |R_{g_\\epsilon}|^{n\/2}\\,dg=o(1).\n\\end{equation}\n\n\nNext we use a $W^{1,p}$ version of Lemma \\ref{convolve}.\n\\begin{lem}\\label{convolve-w1p}\nLet $\\sigma$ be the function described in Lemma \\ref{sigma}, and let $f\\in W^{1,p}_{\\mathrm{loc}}(U_k)$ such that $f$ has bounded $C^2$-norm on the complement of $S$.  For small enough $\\epsilon$, if we define the function\n\\[ f_\\epsilon(x)=\\int_{\\rr^n}f(x-\\sigma(x) y)\\varphi(y)\\,dy \n=\\int_{\\rr^n}f(y)\\varphi_{\\sigma(x)}(x-y)\\,dy\\]\non the set $U'_k$, then $\\|\\partial f_\\epsilon\\|_{L^p(S_\\epsilon\\cap U'_k)} \\le C$ and \n$|\\partial\\partial f_\\epsilon|\\le C\\epsilon^{-1-\\frac{n}{p}}$, where $C$ may depend on the $W^{1,p}$ norm of $f$.\n\\end{lem}\n\\begin{proof}\nRecall that for $x\\in S_\\epsilon$, $\\sigma(x)=\\epsilon$, so that the formula for\n$f_\\epsilon$ is the usual mollification formula.  A standard argument using H\\\"{o}lder's inequality shows that\n\\[ \\|\\partial f_\\epsilon\\|_{L^p(S_\\epsilon \\cap U'_k)} \\le \\|\\partial f\\|_{L^p(S_{2\\epsilon}\\cap U_k)}\\le C.\\]\nFor any $x\\in S_\\epsilon$, if $q$ is chosen so that $\\frac{1}{p}+\\frac{1}{q}=1$, then\n\\begin{align*}\n|\\partial\\partial f_\\epsilon(x)|\n&= \\left|\\partial\\partial \\int_{\\rr^n} f(x-y)\\varphi_\\epsilon(y)\\,dy\\right|\\\\\n&= \\left|\\partial\\int_{\\rr^n} \\partial f(x-y)\\varphi_\\epsilon(y)\\,dy\\right|\\\\\n&= \\left|\\partial\\int_{\\rr^n} \\partial f(y)\\varphi_\\epsilon(x-y)\\,dy\\right|\\\\\n&= \\left|\\int_{S_{2\\epsilon}\\cap U_k} \\partial f(y) \\partial\\varphi_\\epsilon(x-y)\\,dy\\right|\\\\\n&\\le \\|\\partial f\\|_{L^p(S_{2\\epsilon}\\cap U_k)}  \\left(\\int_{B_\\epsilon(x)} |\\partial\\varphi_\\epsilon(x-y)|^q\\,dy\\right)^{\\frac{1}{q}}\\\\\n&\\le C\\left(\\epsilon^{n} (\\epsilon^{-n-1})^q\\right)^{\\frac{1}{q}}\\\\\n&= C\\epsilon^{-1-\\frac{n}{p}}.\n\\end{align*}\n\\end{proof}\nSince the scalar curvature is a contraction of $\\partial\\partial g + g^{-1}*g^{-1}*\\partial g *\\partial g$, on any of the coordinate patches, we can use Lemma \\ref{convolve-w1p} to estimate\n\\begin{align*}\n \\int_{S_\\epsilon\\cap U_k} |R_{g_\\epsilon}|^{n\/2}\\,dg\n &\\le  C\\int_{S_\\epsilon\\cap U_k} |\\partial\\partial g_\\epsilon|^{n\/2}\\,dx +  C\\int_{S_\\epsilon\\cap U_k} |\\partial g_\\epsilon|^{n}\\,dx \\\\\n  &\\le  \\mathcal{L}^n(S_\\epsilon\\cap U_k) \\left(C\\epsilon^{-1-\\frac{n}{p}}\\right)^{n\/2} + o(1)\\\\\n  &= o\\left(\\epsilon^{\\frac{n}{2}\\left(1+\\frac{n}{p}\\right)}\\right) \\epsilon^{-\\frac{n}{2}\\left(1+\\frac{n}{p}\\right)}+o(1)\\\\\n  &=o(1).\n  \\end{align*}\n As in Section \\ref{main}, there is some justifiable carelessness in the computation above.  And once again, the result of the proof follows exactly as in \\cite{Miao:2002}.\n\n\\bibliographystyle{hamsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nMagnetic fields permeate the Universe and often play an important role in the dynamics of astrophysical processes \\citep{crutcher:12, vlemmings:13, crutcher:19}. It is difficult to directly observe magnetic fields; one typically has to use the polarization properties of the observed light \\citep[e.g.,][]{han:17}. For (sub)millimeter interferometers, such as ALMA, magnetic field detection is done mostly through dust \\citep[e.g.,][]{hull:17} and line polarization observations \\citep[e.g.,][]{vlemmings:17}. However, it has recently become increasingly clear that dust polarization does not always faithfully trace the magnetic field morphology, but instead it can be affected by processes such as self-scattering \\citep{kataoka:15, kataoka:17}. Line polarization observations are not affected by such processes, and therefore they likely trace the magnetic field structure of the observed region. However, to interpret line polarization observations, modelers have to defer to the theory of \\citet{goldreich:81}, which relies on the large velocity gradient (LVG) approximation, and therefore they cannot treat three-dimensional (3D, magnetic field) structures. \n\nIn this paper, we present POlarized Radiative Transfer Adapted to Lines (PORTAL)\\footnote{The source code of PORTAL is available on GitHub at \\url{https:\/\/github.com\/blankhaar\/PORTAL}.}, which is a 3D polarized radiative transfer code that simulates the emergence of polarization in the emission of atomic or molecular (sub)millimeter lines. PORTAL can be used in stand-alone mode or process the output of regular 3D radiative transfer codes. We are able to model the emergence of linear polarization in (sub)millimeter lines through two main approximations: (i) the strong magnetic field approximation and (ii) the anisotropic intensity approximation.\\ We show that both of them are valid in the majority of astrophysical regions. \n\nRegular radiative transfer models of astrophysical environments only take the total radiation intensity and its effect on the local isotropic populations into account \\citep{vandertak:07, brinch:10}. The local populations are determined by the balance of collisional and radiative events that both excite and de-excite the populations of the molecular and atomic species. Collisional events are isotropic and a function of the density and temperature of the environment. At the outset, for unaligned quantum states, spontaneous emission events are also isotropic, that is,~the direction of the next spontaneously emitted photon of a certain molecule is random. The probability of absorption of those randomly directed photons, however, need not be isotropic \\citep{goldreich:81}. Anisotropy in the local absorption of photons aligns the quantum states that are associated with the line-transition, which in turn leads to polarization in the emission \\citep{morris:85, landi:06}. By considering the directional dependance of the photon-escape probability in a medium with an anisotropic velocity gradient, \\citet{goldreich:81} showed that radiation emitted from such a system is partially polarized. This effect is known in the literature as the Goldreich-Kylafis (GK) effect. It is strongest for lines with optical depth around unity in regions where the collisional rates are not so high as to quench the molecular or atomic alignment. Provided the magnetic field precession rate is 10-100 times stronger than radiative and collisional rates, which we show to be the case in most astrophysical regions in Section \\ref{sec:comp_align}, the line polarization traces the magnetic field projected onto the plane of the sky with a $90^{\\circ}$ ambiguity. \n\nNumerical modeling of the GK effect has been based on the theory presented in \\citet{goldreich:82}. In such models, the perpendicular and parallel components (with respect to the projected magnetic field direction) of the radiation field are propagated through a medium with an anisotropic velocity gradient. The velocity-gradient is so strong that the large velocity gradient (LVG) approximation can be employed. The LVG escape probability is a function of the velocity-gradient and is therefore anisotropic. This leads to alignment in the molecular or atomic states associated with the transition under investigation. Because of this, the emitted radiation is partially polarized. \\citet{deguchi:84} later showed that in order to accurately model the GK effect, it is vital to perform comprehensive (polarized) excitation modeling of the molecular or atomic quantum states and also of the ones that are not associated with the transition under investigation. \\citet{cortes:05} showed that an external anisotropic radiation source, such as a nearby stellar object, can enhance the polarized emission significantly. These numerical models only considered the one-dimensional propagation of polarized radiation, and the representation of the radiation field in perpendicular and parallel components is only valid when the magnetic field direction is constant over the investigated path. Furthermore, because of its heavy reliance on the LVG approximation, numerical modeling based on \\citet{goldreich:82} can only consider the introduction of anisotropy in the escape probability through an anisotropic velocity gradient. In light of recently developed polarimetric capabilities of interferometers, such as ALMA, these types of approximations cannot be afforded anymore. Rather, one needs comprehensive modeling of the 3D radiative transfer and its anisotropy, taking both the spatial and velocity structure into account for the astrophysical region under investigation as well as the 3D structure of the magnetic field. \n\nIn this paper, we demonstrate how such modeling can be attained. By using two (main) approximations, we show that regular (nonpolarized) radiative transfer codes can be extended with polarization capabilities. In Section 2, we introduce these approximations and show their simplifying impact on the theory of line polarization. In Section 3, we show how our PORTAL code provides the option of computing the emerging polarization using the output from a regular 3D radiative transfer code, in particular LIME \\citep{brinch:10}. In Section 4, we present the capabilities of PORTAL through the simulation of the emergence of polarization in a protoplanetary disk and a collapsing sphere. We discuss our results in Section 5 and conclude in Section 6.\n\n\n\\section{Theory}\nWe describe the introduction of anisotropy in the molecular or atomic populations through an anisotropic radiation field using the formalism of \\citet{landi:06}. We make the following approximations: \n\nFirst, we assume the magnetic field precession rate is way higher than collisional and radiative rates. We call this the strong magnetic field approximation. The magnetic precession rate is in the order of s$^{-1}$\/mG for diamagnetic (i.e.,~weakly magnetizable) molecules. Typical collisional rates are on the order of $10^{-5} \\left(\\frac{n_{H_2}}{10^6 \\ \\mathrm{cm}^{-3}}\\right)\\ \\mathrm{s}^{-1}$ and radiative rates are on the order of $10^{-4} \\ \\mathrm{s}^{-1}$ for a transition at $100$ GHz with a dipole moment of $0.1$ Debye, which is shone upon isotropically by $400$ Kelvin black-body radiation. Therefore, for almost all molecules, magnetic field interactions already dominate at very weak magnetic fields ($\\mu$G). Under the assumption of a strong magnetic field, many terms in the polarized density-equations can be dropped \\citep{landi:06}. The strong magnetic field approximation is also invoked by \\citet{goldreich:81}. In Section \\ref{sec:comp_align} we discuss special cases where a dominant magnetic field cannot be assumed.\n\nSecond, we assume that only the total intensity of the radiation has an influence on the (polarized) populations of the molecular or atomic states. This is a reasonable assumption if the polarization fraction is low, which is corroborated by polarization observations of molecular emission lines. We refer to this approximation as the anisotropic intensity approximation. We discuss the validity of the anisotropic intensity approximation in more detail in Section \\ref{sec:anis_int}, where we also compare our modeling with that of \\citet{goldreich:81}, who take the influence of both the Stokes-I and -Q parameters on the alignment of the molecular states into account. \n \nThese assumptions lead to significant simplifications in the theory behind the alignment of molecular and atomic quantum states and the radiation with which they interact. They allow for the implementation of such a model as an extension to a regular line radiative transfer code. In the following, we introduce the formalism that we used to model the alignment to molecular or atomic quantum states. After this, we outline how aligned quantum states influence the propagation of polarized radiation. \n\\subsection{Polarized statistical equilibrium equations}\nThe polarizing mechanism we focus on is the anisotropic radiation field. Mathematically, anisotropy in the radiation field that affects the quantum state alignment is most easily described in terms of an irreducible tensor-element expansion. The irreducible tensor components of the radiation field, which are in direction $\\Omega$ and at the position $\\boldsymbol{r}$, are obtained as \\citep{landi:84}\n\\begin{align}\n\\mathcal{J}^K_Q (\\boldsymbol{r},\\nu , \\Omega) = \\sum_j \\mathcal{T}^K_Q(j,\\Omega) S_j(\\boldsymbol{r},\\nu,\\Omega),\n\\label{eq:irred_J}\n\\end{align}\nwhere $K$ represents the irreducible tensor rank and $Q$ is its projection, $S_j (\\nu,\\Omega)$ are the Stokes-parameters at frequency $\\nu,$ and $j$ runs over all four Stokes parameters. We define the Stokes parameters in relation to the complex electric field vector components as \n\\begin{subequations}\n\\begin{align}\nI = |E_x|^2 + |E_y|^2 , \\\\\nQ = |E_x|^2 - |E_y|^2 , \\\\ \nU = 2\\mathrm{Re}\\left[E_x E_y^* \\right], \\\\\nV = 2\\mathrm{Im}\\left[E_x E_y^* \\right], \n\\end{align} \n\\end{subequations}\nwhere $x$ and $y$ refer to the axes that are perpendicular to the propagation direction, $z$, and each other. In this work, we consistently chose the axis of $x$ along the rejection of the (local) magnetic field direction from the propagation direction. The transformation coefficients $\\mathcal{T}^K_Q(j,\\Omega)$ are defined in equation~(A6) from \\citet{landi:84}. If we only consider alignment by Stokes-I radiation and if we furthermore assume a dominant magnetic field, only the $K=0,2$ and $Q=0$ components are of interest \\citep{landi:06}. Under these conditions, the irreducible tensor components of the radiation field reduce to\n\\begin{subequations}\n\\begin{align}\n\\mathcal{J}^0_0(\\boldsymbol{r},\\nu,\\Omega) &= I(\\boldsymbol{r},\\nu,\\Omega), \\\\\n\\mathcal{J}^2_0(\\boldsymbol{r},\\nu,\\Omega) &= \\sqrt{\\frac{1}{2}}P_2 (\\mu ) I(\\boldsymbol{r},\\nu,\\Omega),\n\\end{align}\n\\label{eq:J_int}\n\\end{subequations}\nwhere $\\Omega = (\\theta,\\phi)$ is expressed in terms of the inclination and azimuth angles that are gauged with respect to the magnetic field direction. The quantity $P_2 (\\mu)$ is the second-order Legendre polynomial and $\\mu=\\cos \\theta$. The solid-angle integrated tensors at position $\\boldsymbol{r}$ are readily obtained as\n\\begin{subequations}\n\\begin{align}\nJ^0_0 (\\boldsymbol{r},\\nu) &= \\frac{1}{4\\pi}\\int_{-1}^1 d \\mu \\int_0^{2\\pi} d\\phi \\  I(\\boldsymbol{r},\\nu,\\mathrm{acos}(\\mu),\\phi), \\\\ \nJ^2_0 (\\boldsymbol{r},\\nu) &= \\frac{1}{4\\pi \\sqrt{2}} \\int_{-1}^1 d \\mu \\ P_2 (\\mu) \\int_0^{2\\pi} d\\phi \\  I(\\boldsymbol{r},\\nu,\\mathrm{acos}(\\mu),\\phi). \n\\end{align}\n\\label{eq:int_rad_tens} \n\\end{subequations}\nIn the following, we refer to the ratio $J_0^0 (\\boldsymbol{r},\\nu) \/ J_0^2 (\\boldsymbol{r},\\nu)$ as the relative alignment of the radiation field. For an isotropic radiation field ($I(\\boldsymbol{r},\\nu,\\Omega) = I(\\boldsymbol{r},\\nu)$), it should be noted that only the (isotropic) $J^0_0 (\\boldsymbol{r},\\nu)$-term survives.\n\nJust as for the radiation field, we represent the molecular or atomic quantum states as irreducible tensor elements in order to most optimally utilize their symmetry properties. Quantum states are denoted as $\\rho^K_Q (\\alpha J)$, where $K$ is the rank of the irreducible tensor element and $Q$ is its projection. The total angular momentum of the associated quantum state is $J$ and all other quantum numbers characterizing the quantum state are collected in $\\alpha$. The rank $K$ is positive and restricted to values of $\\leq 2J$. The elements $K\\geq 1$ of the population tensor relate to the alignment of the quantum state and the $K=0$ element relates to the population of the quantum state. Under the assumption of a strong magnetic field, we can neglect all but the $Q=0$ projection elements. Because of the symmetry of the radiation field, we only have to take elements into account where $K$ is even. \\citet{landi:06} presented the statistical equilibrium equations for the polarized quantum state $\\rho^K_0 (\\alpha, J)$ under the following conditions: \n\\begin{align}\n\\dot{\\rho}^K_0 (\\alpha J) &= \\sum_{\\alpha_l J_l K_l} \\rho^{K_l}_0 (\\alpha_l J_l) \\left[ [t_A]_{\\alpha J K}^{\\alpha_l J_l K_l} + \\sqrt{\\frac{[J_l]}{[J]}} \\delta_{K,K_l} [C_I^{(K)}]_{\\alpha J}^{\\alpha_l J_l} \\right] \\nonumber \\\\ \n&+  \\sum_{\\alpha_u J_u K_u} \\rho^{K_u}_0 (\\alpha_u J_u) \\left[ [t_S]_{\\alpha J K}^{\\alpha_u J_u K_u} + [t_E]_{\\alpha J K}^{\\alpha_u J_u K_u} \\right. \\nonumber \\\\ \n&+ \\left. \\sqrt{\\frac{[J_u]}{[J]}} \\delta_{K,K_u} [C_S^{(K)}]_{\\alpha J}^{\\alpha_u J_u} \\right] \\nonumber \\\\\n&-  \\sum_{K'} \\rho^{K'}_0 (\\alpha J) \\left[ [r_A]_{\\alpha J K K'} + [r_E]_{\\alpha J K K'} + [r_S]_{\\alpha J K K'} \\right. \\nonumber \\\\\n&+ \\left. \\delta_{KK'}\\left( \\sum_{\\alpha_u J_u} [C_I^{(0)}]_{\\alpha_u J_u}^{\\alpha J} + \\sum_{\\alpha_l J_l} [C_S^{(0)}]_{\\alpha_l J_l}^{\\alpha J} + D^{(K)} (\\alpha J)\\right) \\right].\n\\label{eq:stateq}\n\\end{align}\nIn Eq.~(\\ref{eq:stateq}), the rate of radiative absorption events toward the $\\rho^K_0 (\\alpha, J)$ from lower level $ \\rho^{K_l}_0 (\\alpha_l J_l)$ is given by $[t_A]_{\\alpha J K}^{\\alpha_l J_l K_l}$ \nand the collisional contribution is $[C_I^{(K)}]_{\\alpha J }^{\\alpha_l J_l }$. The rate of stimulated and spontaneous emission events toward the $\\rho^K_0 (\\alpha, J)$ from upper level $ \\rho^{K_u}_0 (\\alpha_u J_u)$ are given by $[t_S]_{\\alpha J K}^{\\alpha_u J_u K_u}$ and $[t_E]_{\\alpha J K}^{\\alpha_u J_u K_u}$,\nand the collisional contribution is $[C_S^{(K)}]_{\\alpha J}^{\\alpha_u J_u}$. The rates of absorption, stimulated emission, and spontaneous emission from the level $\\rho^K_0 (\\alpha, J)$ to all other levels is given by $[r_A]_{\\alpha J K K'}$, $[r_S]_{\\alpha J K K'}$, and $[r_E]_{\\alpha J K K'}$.\nFinally, the collisional depolarization rates are $D^{(K)}(\\alpha J)$. More detailed expressions for the radiative rates from Eq.~(\\ref{eq:stateq}) can be found in equations 7.20 from \\cite{landi:06}. By assuming a steady-state, $\\dot{\\rho}^K_0 (\\alpha J) = 0$, the statistical equilibrium equations can be solved as a linear set of equations. The solution yields the quantum state populations, including their relative alignment. \n\nWe should note that the statistical equilibrium equations of Eq.~(\\ref{eq:stateq}) are isomorphic to those presented in \\citet{deguchi:84}. While \\citet{deguchi:84} set up the statistical equilibrium equations in the standard angular momentum basis $\\ket{jm}$, where $j$ is the total angular momentum of the eigenstate and $m$ is its projection, we worked in a spherical tensor representation. We refer to \\citet{landi:06} for a detailed discussion on the relation between the two representations. We chose to work in a spherical tensor representation because of its symmetry properties. The properties of the spherical tensor expansion of both the molecular (or atomic) states and the radiation are such that truncation of higher-order $K$-terms in the $\\rho^K_0 (\\alpha J)$-expansion can be done with minimal loss of accuracy in the description of the statistical equilibrium equations for our system. Such truncation is not possible in the representation that \\citet{deguchi:84} used, and it results in a rapid and unmitigable increase in computational effort when high angular momentum states are considered. \n\\subsection{Polarized radiative transfer}\nAfter having obtained the (aligned) quantum state populations, we can evaluate their impact on the radiation propagation. Because of the strong magnetic field, (locally) only Stokes-Q radiation is produced.\nThe propagation of radiation around frequency, $\\nu_{\\alpha' J' , \\alpha J}$, associated with a transition $\\alpha' J' \\to \\alpha J$, can be described by\n\\begin{align}\n\\frac{d}{ds} \\boldsymbol{I}_{\\nu} = -\\boldsymbol{\\kappa}_{\\nu}^{\\alpha' J' , \\alpha J} \\boldsymbol{I}_{\\nu} + \\boldsymbol{\\epsilon}^{\\alpha' J' , \\alpha J},\n\\label{eq:polrad} \n\\end{align}  \nwhere $\\boldsymbol{I}_{\\nu}=[I_{\\nu},Q_{\\nu},U_{\\nu},V_{\\nu}]$ is the Stokes vector and the propagation matrix \n\\begin{align}\n\\boldsymbol{\\kappa}_{\\nu}^{\\alpha' J' , \\alpha J} = \\begin{bmatrix}\\eta_I^{\\alpha' J' , \\alpha J} (\\nu) & \\eta_Q^{\\alpha' J' , \\alpha J} (\\nu) & 0 & \\eta_V^{\\alpha' J' , \\alpha J} (\\nu) \\\\ \\eta_Q^{\\alpha' J' , \\alpha J} (\\nu) & \\eta_I^{\\alpha' J' , \\alpha J} (\\nu) & 0 & 0 \\\\ 0 & 0 & \\eta_I^{\\alpha' J' , \\alpha J} (\\nu) & 0 \\\\ \\eta_V^{\\alpha' J' , \\alpha J} (\\nu) & 0 & 0 & \\eta_I^{\\alpha' J' , \\alpha J} (\\nu) \\end{bmatrix} \n\\label{eq:kappa_mat}\n\\end{align}\nis significantly simplified if one assumes a dominant magnetic field. Because we only consider diamagnetic molecules with Zeeman splitting that are far weaker than the thermal broadening, the production of Stokes-V radiation through the Zeeman effect is negligible and we set $\\eta_V^{\\alpha' J' , \\alpha J} (\\nu) \\to 0$. Thus, in PORTAL, we only consider the propagation of linearly polarized radiation. The expressions for the $\\eta$-elements of Eq.~(\\ref{eq:kappa_mat}) are \\citep{landi:84} \n\\begin{subequations}\n\\begin{align}\n\\eta_I^{\\alpha' J' , \\alpha J} (\\nu) &= \\frac{h \\nu_{\\alpha' J',\\alpha J}}{4\\pi} B_{\\alpha' J' , \\alpha J} \\phi_{\\nu_{\\alpha' J' , \\alpha J}}(\\nu) \\left\\{  \\left( \\mathcal{N}_{\\alpha' J'}  - \\mathcal{N}_{\\alpha J} \\frac{[J']}{[J]}\\right) \\right. \\nonumber \\\\\n&+ \\left. \\left(\\mathcal{N}_{\\alpha' J'} w_{J'J}^{(2)} \\sigma^2_0 (\\alpha' J') - \\frac{[J']}{[J]} \\mathcal{N}_{\\alpha J} w_{JJ'}^{(2)} \\sigma^2_0 (\\alpha J) \\right) \\right. \\nonumber \\\\\n& \\times \\left. \\frac{3 \\cos^2 \\theta - 1}{2\\sqrt{2}} \\right\\}, \\\\\n\\eta_Q^{\\alpha' J' , \\alpha J} (\\nu) &= -\\frac{h \\nu_{\\alpha' J',\\alpha J}}{4\\pi} B_{\\alpha' J' , \\alpha J} \\phi_{\\nu_{\\alpha' J' , \\alpha J}} (\\nu) \\left( \\mathcal{N}_{\\alpha' J'} w_{J'J}^{(2)} \\sigma^2_0 (\\alpha' J') \\right. \\nonumber \\\\ \n&- \\left. \\frac{[J']}{[J]}\\mathcal{N}_{\\alpha J} w_{JJ'}^{(2)}  \\sigma^2_0 (\\alpha J) \\right) \\frac{3\\sin^2 \\theta }{2\\sqrt{2}},\n\\end{align}\n\\label{eq:eta}\n\\end{subequations}\nwhere $\\mathcal{N}_{\\alpha J} = \\mathcal{N} [J]^{1\/2} \\rho_0^0 (\\alpha J)$ is the number density of the quantum state $\\alpha J$, and $\\phi_{\\nu_{\\alpha' J' , \\alpha J}}$ denotes the normalized line-profile centered at $\\nu_{\\alpha' J' , \\alpha J}$ in frequency-space. The symbols \n\\begin{align}\nw_{J'J}^{(2)} = (-1)^{1+J+J'}\\sqrt{3[J']}\\begin{Bmatrix} 1 & 1 & 2 \\\\ J' & J' & J\\end{Bmatrix} \\nonumber\n\\end{align}\nwere introduced by \\citet{landi:84}. The quantity between curly brackets is a Wigner-6j symbol \\citep{biedenharn:81}. We use the short-hand notation\n$\n\\sigma_0^2 (\\alpha J) = \\rho_0^2 (\\alpha J) \/ \\rho_0^0 (\\alpha J)\n$ \nfor the relative alignment of the quantum state $\\alpha J$. The spontaneous emission events in the polarized radiative transfer equations of Eq.~(\\ref{eq:polrad}) are represented in the $\\boldsymbol{\\epsilon}$-vector. The spontaneous emission contribution to the Stokes-U is zero in the strong magnetic field limit we consider. The Zeeman effect for diamagnetic molecules is way smaller than the thermal broadening, so we can set $\\epsilon_V^{\\alpha' J', \\alpha J} \\to 0$. The contributions to the Stokes-I and -Q parameters are \\citep{landi:84}\n\\begin{subequations}\n\\begin{align}\n\\epsilon_I^{\\alpha' J' , \\alpha J} (\\nu) &= \\frac{h \\nu_{\\alpha' J',\\alpha J}}{4\\pi} A_{\\alpha' J' , \\alpha J} \\phi_{\\nu_{\\alpha' J' , \\alpha J}} (\\nu) \\nonumber \\\\\n& \\times \\mathcal{N}_{\\alpha' J'} \\left\\{  1  + w_{J'J}^{(2)} \\sigma^2_0 (\\alpha' J') \\frac{3 \\cos^2 \\theta - 1}{2\\sqrt{2}} \\right\\}, \\\\\n\\epsilon_Q^{\\alpha' J' , \\alpha J} (\\nu) &= -\\frac{h \\nu_{\\alpha' J',\\alpha J}}{4\\pi} A_{\\alpha' J' , \\alpha J} \\phi_{\\nu_{\\alpha' J' , \\alpha J}} (\\nu) \\nonumber \\\\\n&\\times\\mathcal{N}_{\\alpha' J'} w_{J'J}^{(2)} \\sigma^2_0 (\\alpha' J') \\frac{3\\sin^2 \\theta }{2\\sqrt{2}}.\n\\end{align}\n\\label{eq:eps}\n\\end{subequations}\n\\section{Methods}\n The formalism that we present in the previous section can be used to simulate the emergence of polarization in spectral lines using an (isotropic) excitation model as input. It can be directly used by assuming LTE excitation, or alternatively, the atomic and molecular excitation from any (3D) radiative transfer code can be input. We outline in the following how we used the LIME radiative transfer code \\citep{brinch:10}. \n\nLIME is a Monte Carlo 3D radiative transfer code that works with a (weighted) randomly chosen grid. A physical structure can be input, whereupon a random grid is chosen that is weighed over the molecular density and other parameters \\citep{ritzerveld:06}. After a number of Monte Carlo radiative transfer iterations, which are sped up by an accelerated lambda iteration \\citep{rybicki:91}, the simulation converges on a molecular and atomic excitation over all of the nodes in the simulation. Subsequently, this solution can be ray traced to simulate an image of the physical structure under investigation. \n\nRather than directly ray tracing the excitation solution, we used it to thoroughly map out the local anisotropy of the radiation field throughout the simulation. With the local anisotropy parameters of the radiation field, we modeled the polarized excitation of the molecular or atomic states under investigation. Having the polarized excitation mapped out throughout the simulation, we performed a polarized ray-tracing to obtain a polarized image of the physical structure under investigation. \n\nIn the following, we outline in more detail how we implemented PORTAL. In the first paragraph, we discuss setting up the polarized statistical equilibrium equations using the output of a line radiative transfer code. In order to do this, we dedicated most of our attention to the mapping of the local anisotropy of the radiation fields. In the second paragraph, we detail the polarized radiative transfer that was performed in the polarized ray-tracing. Especially for simulations with nonuniform magnetic fields, it is crucial to pay extra attention to the frame of reference of the polarized radiation and the proper way to relate different frames of reference.        \n\n In PORTAL, we used the anisotropic intensity approximation and formulated the polarized statistical equilibrium equations in terms of irreducible tensor elements. This approach differs from other efforts such as LinePol \\citep[][(submitted)]{kuiper:20}, which builds on LIME, is optimized for CO, and uses the formalism of \\citet{goldreich:82} to describe the propagation of polarized radiation and its interaction with the molecular medium. LinePol takes two polarization modes of the radiation into account and uses a polarized accelerated lambda iteration scheme to obtain the state-populations in the simulation. At minimal cost to the accuracy of our results (see Sections \\ref{sec:polsee} and \\ref{sec:anis_int}), the approximations in PORTAL speed up the simulation tremendously and lead to the possibility to treat more complex systems. PORTAL allows for complex geometries, magnetic field structures, and the treatment of molecules with extensive energy structures.\n\n\\subsection{Polarized statistical equilibrium equations}\n\\label{sec:polsee}\nThe quantum state alignment is dependent on the local anisotropy of the radiation field, so it is important to obtain a good angular sampling of the radiation field at the location of the simulation nodes. Different angular integrations of Eqs.~(\\ref{eq:int_rad_tens}) for the case of an internal source of radiation (e.g., a central stellar object) and the case of no internal radiation source were used. For the latter, the local angular integration was performed as\n\\begin{subequations}\n\\begin{align}\nJ^0_0 (\\boldsymbol{r},\\nu) &= \\frac{1}{4\\pi}\\sum_{i=1}^{N_{\\mu}} w_i^{\\mu} \\sum_{j=1}^{N_\\phi(\\mu_i)} w_j^{\\phi} I(\\boldsymbol{r},\\nu,\\mathrm{acos}( \\mu_i),\\phi_j),  \\\\ \nJ^2_0 (\\boldsymbol{r},\\nu) &= \\frac{1}{4\\pi\\sqrt{2}}  \\sum_{i=1}^{N_{\\mu}} w_i^{\\mu} P_2 (\\mu_i)  \\sum_{j=1}^{N_\\phi (\\mu_i)} w_j^{\\phi} I(\\boldsymbol{r},\\nu,\\mathrm{acos}(\\mu_i),\\phi_j) \n\\end{align}\n\\end{subequations}\nwhere $\\mu_i$ and $w_i^{\\mu}$ are the coordinates and the weights, which were taken from the $N_{\\mu}$-point Gaussian quadrature rule. The integration over $\\phi$ was performed over $N_{\\phi} (\\mu)\\propto \\sqrt{1-\\mu^2}$ equidistant points all with weight $2\\pi\/N_\\phi$. \n\nIn the case of an internal radiation source, it should be appreciated that the solid angle associated with the radiation coming from this internal source is well-defined. Therefore, the solid angle integration was divided up into rays coming from the internal radiation source; the number of rays is proportional to the solid-angle of the internal source $\\Delta \\Omega_* = \\pi(|\\boldsymbol{r}|\/R_*)^2$ and all of the other rays were distributed equally over the remaining sphere surface.\n\nThe local radiation field parameters of a node at the position $\\boldsymbol{r}$, summarized in $J^0_0 (\\boldsymbol{r},\\nu)$ and $J^2_0 (\\boldsymbol{r},\\nu)$, were obtained by ray tracing $N$ rays with direction $\\boldsymbol{k}_{\\mu,\\phi}$ to that node. The parameters $\\mu$ were chosen with respect to the magnetic field direction ($\\boldsymbol{b} \\cdot \\boldsymbol{k}_{\\mu,\\phi} = \\mu$). The angles $\\phi$ were gauged with respect to a canonical direction not parallel to the magnetic field. The choice of the canonical direction is free as the angle $\\phi$ is integrated out without weighing (see Eq.~\\ref{eq:J_int}). The ray-tracing was performed using the molecular populations that were output by LIME, while also using some of the relevant LIME-input parameters, such as (local) temperature, (local) velocity, and gridding. The ray-tracing yielded the local radiation field parameters that were subsequently used to obtain the quantum state populations and alignment.\n\nThe quantum state populations and alignment were obtained from the statistical equilibrium equations (SEE) given in Eq.~(\\ref{eq:stateq}). The SEE are a balance of the radiative and collisional transition events. The radiative transition events are dependent on local parameters for the (an)isotopic radiation field at frequencies of all of the allowed transitions and their associated Einstein coefficients. Collisional rates are dependent on the temperature-dependent collisional cross-sections and (local) number densities of the relevant collisional partners. \nThe relevant Wigner coupling symbols were calculated using the WIGXJPF package \\citep{johansson:16}. The SEE were formulated in terms of a set of linearly dependent equations and were subsequently solved via an LQ decomposition (using the LAPACK libraries, \\citet{LAPACK}) under the following physical constraint: $\\sum_i [j_i]^{1\/2} \\rho_0^0 (\\alpha_i j_i) = 1$. The solutions also included the isotropic populations that were compared to the LIME-output. We found that neglecting the quantum state alignment terms, $\\rho_0^k (\\alpha j)$ with $k>2$, introduces an error of $\\sim 1\\% $ in the state-alignment expressions, and for $k>4$ this error is already reduced to $\\sim 1\\permil$. In general, the quantum state alignment can be neglected for terms, $\\rho_0^k (\\alpha j)$ with $k>6$, with virtually no loss in precision and with great reduction of computational effort as a consequence\\footnote{For example, the dimensionality of the polarized SEE for the first 41 rotational levels of CO reduced from 861 to 151 by setting $k_{\\mathrm{max}}=6$.}.\n\n\\subsection{Polarized radiative transfer}\nThe quantum state populations and alignment obtained from the SEE were used to compute the (polarized) absorption and emission factors for each node in the simulation. The angle $\\theta$ in Eqs.~(\\ref{eq:eta}-\\ref{eq:eps}) was obtained from the local magnetic field direction and the ray-trace direction. The ray-trace direction was chosen by defining an inclination angle and azimuth angle. The polarized radiation was gauged with respect to a canonical axis, $\\boldsymbol{\\chi}_{\\mathrm{global}}$, perpendicular to the ray-tracing direction. The local and global Stokes parameters are related as \\citep{landi:06} \n\\begin{align}\n\\begin{pmatrix} Q_{\\mathrm{local}} \\\\ U_{\\mathrm{local}} \\end{pmatrix} = \\begin{pmatrix} \\cos 2\\alpha \\ & \\sin 2\\alpha \\\\ -\\sin 2\\alpha & \\cos 2\\alpha \\end{pmatrix} \\begin{pmatrix} Q_{\\mathrm{global}} \\\\ U_{\\mathrm{global}} \\end{pmatrix},\n\\label{eq:globtoloc}\n\\end{align}\nand $I_{\\mathrm{local}} = I_{\\mathrm{global}}$. In Eq.~(\\ref{eq:globtoloc}), $\\alpha$ is the angle between $\\boldsymbol{\\chi}_{\\mathrm{global}}$ and $\\boldsymbol{\\chi}_{\\mathrm{local}}$ and the local reference axis is the unit vector along the rejection of the local magnetic field direction from the ray-tracing direction. \n\nThe local Stokes-parameters were propagated using the polarized radiative transfer equations. Equations~(\\ref{eq:polrad}-\\ref{eq:kappa_mat}) show that only the Stokes-Q and -I coefficients are coupled in the polarized radiative transfer. That means that the propagation of the Stokes-U radiation is simply $U(s) = U(0) e^{-\\eta_I s}$. To evaluate the propagation of the other Stokes parameters, $\\boldsymbol{i} = [I,Q]$, the evolution operator formalism of \\citet{landi:06} was used, where the propagation is described by\n\\begin{align}\n\\boldsymbol{i}(s) = \\int_0^s ds' \\ \\boldsymbol{O}(s,s') \\boldsymbol{\\epsilon}(s') + \\boldsymbol{O}(s,0) \\boldsymbol{i}(0),\n\\end{align} \nand where\n\\[\n\\boldsymbol{O}(s,s') = e^{\\int_{s'}^s ds'' \\  \\eta_I} \\begin{bmatrix} \\cosh \\left( \\int_{s'}^s ds'' \\ \\eta_Q \\right) & -\\sinh \\left( \\int_{s'}^s ds'' \\ \\eta_Q \\right) \\\\ \n -\\sinh \\left( \\int_{s'}^s ds'' \\ \\eta_Q \\right) & \\cosh \\left( \\int_{s'}^s ds'' \\  \\eta_Q \\right) \\end{bmatrix}\n\\]\nis the evolution operator (see Chapter 8 of \\citet{landi:84}). The propagation for each crossed cell was considered, and within such a propagation, the coefficients $\\eta_I$ and $\\eta_Q$ as well as $\\epsilon_I$ and $\\epsilon_Q$ are constant. It is then straightforward to evaluate the integrals inside the evolution operator as well as the integral over the evolution operator: $\\int_0^s ds'\\ \\boldsymbol{O}(s,s')$. Having done so, the propagation of the Stokes-I and -Q within a single cell is given by \n\\begin{subequations}\n\\begin{align}\nI(s) &= o_{I} \\epsilon_I + o_Q \\epsilon_Q + \\left[ \\cosh (\\eta_Q s) I(0) - \\sinh(\\eta_Q s) Q(0) \\right]e^{-\\eta_I s},  \\\\\nQ(s) &= o_{Q} \\epsilon_I + o_I \\epsilon_Q + \\left[ \\cosh (\\eta_Q s) Q(0) - \\sinh(\\eta_Q s) I(0) \\right]e^{-\\eta_I s}, \n\\end{align} \n\\end{subequations}\nwhere\n\\begin{align}\no_I &= \\frac{\\eta_I}{\\eta_I^2 - \\eta_Q^2} \\left(1 - \\left[\\cosh(\\eta_Q s) + \\frac{\\eta_Q}{\\eta_I} \\sinh(\\eta_Q s)\\right] e^{-\\eta_I s}\\right), \\nonumber \\\\\no_Q &= -\\frac{\\eta_Q}{\\eta_I^2 - \\eta_Q^2} \\left(1 - \\left[\\cosh(\\eta_Q s) + \\frac{\\eta_I}{\\eta_Q} \\sinh(\\eta_Q s)\\right]e^{-\\eta_I s} \\right) \\nonumber \n\\end{align}\nare the factors that were obtained from integrating the elements of the evolution operator. \n\n\n\\section{Simulations}\nWe applied PORTAL to known astrophysical problems. We consider the standard problem of a spherically symmetric collapsing molecular cloud, and we investigate the emergence of polarization in molecular lines through an anisotropic radiation field in a standard protoplanetary disk system. It should be noted that neither of these problems illustrate the full 3D capabilities of PORTAL. We focus, however, on these models because of their more straightforward interpretation and we leave more complex modeling for further work.\n \n\\subsection{Collapsing spherical cloud}\nA benchmark problem in radiative transfer modeling, which furthermore allows for local anisotropy to establish itself in the radiation field, is the problem of a collapsing spherical cloud. We consider the emergence of polarization in HCO$^+$ lines. The density, velocity, and temperature distribution are taken from the \\citet{shu:77} collapse model, using the same parameters as \\citet{zadelhoff:02}. Only the ground vibrational state of HCO$^+$ is considered. We assume a uniform HCO$^+$ abundance of $10^{-9}$ and assume constant turbulent broadening of $200 \\ \\mathrm{m\/s}$. We assume that a strong radial magnetic field (origin: center of mass) permeates the cloud. \n\nFirst of all, an overview of the relevant isotropic and anisotropic interactions is instrumental to an eventual discussion of the quantum state alignment and radiation polarization characteristics. We report the cumulative radiative and collisional rates of the $J=2$ and $J=3$ level of HCO$^+$ in Figure \\ref{fig:sphere_rates}. Of the different interactions, only stimulated emission and absorption are anisotropic interactions. Using the spherical symmetry of the collapsing sphere-problem, we only plotted the rates as a function of the distance to the center. We observe that for the inner regions of the collapsing sphere, collisions become dominant as the density of this regions increases. Even though there is appreciable alignment of the radiation field, the quantum states do not align themselves because of the dominant isotropizing collisions. From about 400 AU, radiative interactions take over as the dominant interaction and the quantum states align themselves. We also give the magnetic precession rate for a magnetic field of $1$ mG and 1 $\\mathrm{\\mu G}$ and note that for a HCO$^+$-molecule in the collapsing sphere, the magnetic field can be taken to define the symmetry axis when it is $\\sim 10-100 \\times$ stronger than other interactions. From Figure \\ref{fig:sphere_rates}, we estimate this to be the case at magnetic field strengths of $\\sim 10-100 \\ \\mathrm{\\mu G}$.\n\nIn the same Figure \\ref{fig:sphere_rates}, we plotted the relative anisotropy of the radiation field and the relative alignment of the quantum states $J=2$ and $J=3$. We note that the radiation anisotropy increases, thus moving away from the collapsing-sphere center. The radiation anisotropy in the collapsing sphere is partly a result of the density structure and partly the result of the velocity structure. Both structures are spherically symmetric, but this spherical symmetry is only manifest when the center is taken as the origin. For any cell that is not located at the center of the collapsing sphere, the radiation field is therefore anisotropic. Higher anisotropy in the radiation is associated with a stronger alignment of the quantum states.\n\\begin{figure}[h!]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/sphere_rates_j2.png}\n    \\caption{}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/sphere_rates_j3.png}\n    \\caption{}\n  \\end{subfigure}\n  \\caption{Plot of the collapsing sphere's interaction rates (collisional, absorption, and stimulated emission as well as spontaneous emission) and relative alignment (radiative and quantum state) as a function of the radius for (a) the $J=2$ level and $J=2-1$ transition and (b) the $J=3$ level and $J=3-2$ transition. The interaction rates should be read from the left axis, the relative alignment from the right axis.}\n  \\label{fig:sphere_rates}\n\\end{figure}\n\nWe report the azimuthally averaged total intensity and polarization fraction of the HCO$^+$ $J=3-2$ and $J=2-1$ transitions in Figure \\ref{fig:sphere_frac}. Indeed, we note that close to the center of the collapsing sphere, the polarization fraction is the lowest and gradually increases when moving outward. Polarization fractions are above $1 \\%$ for a radial distance greater than $600$ AU for the $J=3-2$ transition and $900$ AU for the $J=2-1$ transition. We report the associated spectra at $R=1400$ AU in Figure \\ref{fig:sphere_spec}. We observe that the linear polarization spectra roughly follow the spectral shape of the total intensity. \n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{.\/figures\/sphere_frac.png}\n  \\caption{Total and polarized emission intensity (in Kelvin) of a collapsing sphere as a function of the radial distance.}\n  \\label{fig:sphere_frac}\n\\end{figure}\n\n\\begin{figure}[h!]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/sphere_spec_21.png}\n    \\caption{}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/sphere_spec_32.png}\n    \\caption{}\n  \\end{subfigure}\n  \\caption{Spectra of the (polarized) intensity (in Kelvin) of the (a) $J=2-1$ and (b) $J=3-2$ transitions from a collapsing sphere. The spectra are azimuthally averaged at $1400$ AU.}\n  \\label{fig:sphere_spec}\n\\end{figure}\n\nThe polarization angles are oriented in a radial fashion along the magnetic field lines. One should be aware that for a radial magnetic field, the angle between the magnetic field lines and the propagation direction toward the observer is a function of the propagation position. Accordingly, at the magic angle of $\\theta_{\\mathrm{magic}} \\approx 54.7$ or $z\/R = \\frac{1}{\\sqrt{3}}$, the propagation elements $\\eta_Q$ flip sign and some of the earlier produced polarization is negated.\n\\subsection{Protoplanetary disk}\nThe protoplanetary disk is a prime example of an anisotropic astrophysical structure. Both the anisotropy in the density and velocity structure produce a locally anisotropic radiation field. Magnetic fields in the protoplanetary disk have been conjectured through dust-polarization observations \\citep{stephens:17}, and recently, stringent limits have been put on the magnetic field strength through ALMA line circular polarization observations \\citep{vlemmings:19}. \n\nWe consider the polarization of $^{12}$CO in a general toy model of a protoplanetary disk having a number-density distribution of \n\\begin{align}\nn_{\\mathrm{H}_2}(r_c,h) = 4\\times 10^{14} \\left(\\frac{h}{\\mathrm{AU}}\\right)^{-2.25} e^{-50 \\frac{(h\/\\mathrm{AU})^{2}}{(r_c\/\\mathrm{AU})^{2.5}}} \\ \\mathrm{m}^{-3}, \n\\end{align}\nwhere $r_c$ is the radial distance and $h$ is the height. The disk is assumed to be rotating, resulting in a model velocity-field of\n\\begin{align}\n\\boldsymbol{v}(\\boldsymbol{r})= v(\\cos \\phi \\hat{\\boldsymbol{x}} - \\sin \\phi \\hat{\\boldsymbol{y}}), \n\\end{align}\nwhere\n\\[\nv = 2.11 \\times 10^4 \\left(\\frac{r_c}{\\mathrm{AU}}\\right)^{-1} \\ \\mathrm{m\/s},\n\\]\nand $\\tan \\phi = y\/x$. The temperature is given by\n\\begin{align}\nT(r_c) = 400 \\left(\\frac{r_c}{\\mathrm{AU}} \\right)^{-\\frac{1}{2}} \\ \\mathrm{K}.\n\\end{align}  \nFurthermore, we assume a constant CO abundance of $10^{-3}$ and a constant turbulent doppler broadening of $b_{\\mathrm{turb}}=200 \\ \\mathrm{m\/s}$. We only take the vibrational ground-state of $^{12}$CO into account. We neglect any line-overlap with transitions from other species. We explore the emergence of polarization in a protoplanetary disk for three types of (strong) magnetic fields: radial, toroidal, and poloidal.\n\nWe note that perhaps this toy model of the protoplanetary disk does not capture all features of the protoplanetary disk that are important in considering the polarization of thermal lines. For instance, we neglect to represent the inner midplane region by optically thick dust, so that the anisotropic radiation field resulting therefrom is not accounted for. Also, by not taking vibrationally excited levels and the transitions between different vibrational levels into account, we fail to include their significant aligning interactions (see Section \\ref{sec:comp_align}). We explore more detailed and thorough modeling of protoplanetary disk regions in future work. These results should be taken as a simplified, but generally indicative, model of the mechanisms involved in the polarization of thermal line radiation of radiation by a magnetic field in protoplanetary disk regions. \n                                 \nIt is important to map out the rates of isotropic and anisotropic interactions in order to understand the relative alignment of the molecules or atoms. Because of the cylindrical symmetry of the protoplanetary disk, we are able to analyze the interaction rates as a function of the radial distance and the height. In Figure \\ref{fig:pp_rates}, we report the cumulative radiative and collisional rates for the $J=3$ level of CO. The rates are plotted as a function of $r_c$ for different height-cross sections. We also report the magnetic precession rate of a $1\\ \\mathrm{\\mu G}$ and a $1 \\ \\mathrm{mG}$ magnetic field. It is apparent that magnetic interactions dominate other interactions and that we are justified in choosing the projection-axis along the magnetic field direction. Further, we observe a dominance of collisions over other interactions in a large region of the inner parts of the protoplanetary disk. In the disk midplane, isotropic collisions dominate the radiative interactions in the disk, but this dominance becomes weaker with the radial distance. In the outer parts of the disk, where the density drops, collisions become weaker and radiative events dominate.\n\nIn Figure \\ref{fig:pp_rates} we also plotted the relative anisotropy of the radiation field resonant with the $J=3-2$-transition and the relative alignment of the $J=3$ state. Both of these parameters are defined with respect to a toroidal magnetic field configuration. The radiation anisotropy is strongest in the outer parts of the disk and weakest in the bulk of the disk. The same dependence is seen for the relative alignment of the quantum states. The relative anisotropy of the radiation is almost constant as a function of the radial distance at a height of $1$ AU. This is because the disk is optically thick in the midplane. The local angular radiation profile is not isotropic because of the temperature gradient. Due to dominant collisions, the quantum state alignment in the midplane is not large enough to significantly polarize radiation that is coming through. \n\n\\begin{figure}[h!]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/pp_rates_1.png}\n    \\caption{}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/pp_rates_40.png}\n    \\caption{}\n  \\end{subfigure}\n\\caption{Plot of the protoplanetary disk's interaction rates (collisional, absorption, and stimulated emission as well as spontaneous emission) and relative alignment (radiative and quantum state, with respect to a toroidal magnetic field) as a function of the radial distance for (a) $1$ AU height and (b) $40$ AU height. The interaction rates should be read from the left axis, the relative alignment from the right axis.}\n  \\label{fig:pp_rates}\n\\end{figure}\n\n\\begin{figure*}[h!]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/rad_vec2.png}\n    \\caption{}\n    \\label{fig:pp_contour_rad}\n  \\end{subfigure}\n  ~\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/tor_vec2.png}\n    \\caption{}\n    \\label{fig:pp_contour_tor}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/rad_angle.png}\n    \\caption{}\n    \\label{fig:pp_angle_rad}\n  \\end{subfigure}\n  ~\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/tor_angle.png}\n    \\caption{}\n    \\label{fig:pp_angle_tor}\n  \\end{subfigure}\n  \\caption{Contour plots of (the logarithm of) the total intensity (in Kelvin) of a protoplanetary disk. The disk is viewed face on [(a) and (b)] and at an inclination of $45^o$ [(c) and (d)]. We overlayed the intensity plot with polarization vectors from PORTAL simulations that come from a radial magnetic field (a,c) and a toroidal magnetic field (b,d). Polarization vector lengths scale with the polarization fraction.}\n  \\label{fig:pp_contour}\n\\end{figure*}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{.\/figures\/pp_frac.png}\n  \\caption{(Polarized) emission intensity (in Kelvin) of a protoplanetary disk as a function of the radial distance. The polarized intensity is plotted for three different magnetic field configurations and the disk is seen face on.}\n  \\label{fig:pp_frac}\n\\end{figure}\n\nWe analyzed the emergence of polarization through two different magnetic field configurations: toroidal and radial. In Figure \\ref{fig:pp_contour_rad} we report the contour map of the $J=3-2$ CO-transition at $345.8$ GHz of the total intensity (in Kelvins) overlayed with polarization vectors resulting from the polarized emission of CO aligned with a radial magnetic field. The polarization vectors are scaled with respect to the polarization fraction and are parallel to the radial configuration of the magnetic field. Figure \\ref{fig:pp_contour_tor} gives the polarization map coming from a toroidal magnetic field. We note that the polarization fraction for the face-on view of the protoplanetary disk is cylindrically symmetric.\n\nIt is striking that the polarization vector maps viewed face on, for both the toroidal and radial magnetic field, have the same configurations. This similarity can be traced back to the anisotropy introduced in the molecular states via the anisotropic radiation field, $J^2_0 (\\boldsymbol{r})$ (Eq.~\\ref{eq:int_rad_tens}). When performing the integration to acquire $J^2_0 (\\boldsymbol{r})$, the $\\mu$-angle is gauged with respect to the magnetic field direction. The different gauges with respect to the toroidal and radial magnetic field configurations lead to the $J^2_0 (\\boldsymbol{r})$, which is associated with the toroidal magnetic field, to be negative, while the $J^2_0(\\boldsymbol{r})$ of the radial magnetic field is positive. Thus, in the region where polarization is produced, where furthermore the angle between propagation and the magnetic field $\\theta_{\\mathrm{prop}} > \\theta_{\\mathrm{magic}}$ for both magnetic field configurations, this gives rise to perpendicular and parallel orientations of the polarization vectors with respect to the toroidal and radial magnetic fields, that is,~polarization vectors that are identically oriented. Only when we view the disk at a significant inclination are we able to discern the orientation of the magnetic field from its polarization vectors, which can be seen in Figures \\ref{fig:pp_angle_tor} and \\ref{fig:pp_angle_rad}. \n\nThe polarization maps of a protoplanetary disk viewed at a $45^o$ inclination show large polarization fractions for the poloidal and toroidal magnetic field configurations. Lower but still significant polarization fractions are seen to emerge from the radial magnetic field configuration. The highest polarization fractions occur at the edges of the protoplanetary disk. In the disk midplane, almost no polarization arises. This effect can be ascribed to the high optical depth from this region; it should, however, also be noted that our method underestimates the polarization fraction coming from optically thick regions (see Section \\ref{sec:anis_int}). \n\nFor the face-on view of a protoplanetary disk that is permeated by a poloidal magnetic field, no significant polarization emerges even though the quantum states are aligned. This is because for a large part of the disk, the magnetic field is almost aligned along the propagation direction. When this is the case, the propagation coefficients are $\\eta_Q \\to 0$, and no polarization is produced. When the disk is viewed at a significant inclination, the poloidal magnetic field produces a large polarization fraction. \n\nFigure \\ref{fig:pp_frac} is a plot of the azimuthally averaged polarization fraction as a function of the radial distance. Near the center of the proto-planetary disk, the polarization fraction is low and increases as one moves outward. The maximum polarization fraction of the protoplanetary disk viewed face on is  $\\sim 0.5 \\%$, but polarization fractions up to $\\sim 9\\%$ are observed when the disk is viewed at an inclination of $45^o$. We analyze the azimuthally averaged ($r_c = 50$ AU) spectrum of the total (polarized) intensity in Figure \\ref{fig:pp_spec}. The polarization roughly follows the spectral shape of the total intensity.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{.\/figures\/pp_spec.png}\n  \\caption{Spectrum of the (polarized) intensity (in Kelvin) of the $J=3-2$ transition from a protoplanetary disk permeated by a toroidal magnetic field. The spectrum is azimuthally averaged at $60$ AU and the disk is seen face on.}\n  \\label{fig:pp_spec}\n\\end{figure}\nIt is a general trend that high-frequency transitions have a larger tendency to emit polarized radiation. This is because the radiative rates scale with the frequency. Radiative interactions of high-frequency transitions therefore tend to dominate over collisional interactions. At the same time, the transition optical depth falls (generally) with the transition frequency; for transitions that are too optically thin, radiation intensity is too low to align the quantum states. \n\n\\section{Discussion}\nThe anisotropic intensity approximation and the strong magnetic field approximation are central to the quality of the method we employed in PORTAL. We discuss these two approximations in the following two subsections. We discuss general remarks about the simulations of astrophysical regions using PORTAL in Section \\ref{sec:general_remarks}. \n\\subsection{The anisotropic intensity approximation}\n\\label{sec:anis_int}\nOur method heavily relies on the approximation that it is only the anisotropy in the total intensity that contributes to the alignment of the molecular or atomic states under investigation. We call this approximation the anisotropic intensity approximation. We were able to directly compare the anisotropic intensity approximation to the LVG problem of \\citet{goldreich:81}. \\citet{goldreich:81} accounted for the influence of the anisotropy of both the Stokes I and Stokes Q on the quantum state alignment. In the GK approach, the Stokes U component of the radiation field is neglected because the LVG method can only treat a constant magnetic field. The comparison is summarized in Figure \\ref{fig:GK_compare}.\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{.\/figures\/compare_GK.png}\n  \\caption{Comparison of the polarization fraction computed through the GK method (solid line) and the radiation anisotropy method we employ in this paper (dotted line). For more details on the simulation parameters, see \\citet{goldreich:81}. We consider a $J=1-0$ transition at $100$ GHz, with a strong magnetic field along the $\\hat{z}$-axis and a velocity gradient of $10^{-9}$ s$^{-1}$ in the $xy$-plane. We consider a temperature of $T=10$ K. Three ratios for the collision-radiative rates are considered and denoted inside the figure. The polarization fraction was computed for a ray traveling along the $\\hat{x}$-axis.}\n  \\label{fig:GK_compare}\n\\end{figure}\n\nWe note that below polarization fractions of $2\\%$, our method agrees with the GK effect for any optical depth. Furthermore, for low optical depth, $\\tau < 0.3$, our method reproduces the GK effect very well regardless of the polarization fraction. It is only for very high degrees of polarization and large optical depths that the polarization fraction obtained through the anisotropic intensity approximation starts to deviate from the GK polarization fraction. For strongly polarized lines ($p_L > 6\\%$), the polarization fraction can be underestimated by up to a factor $1.5$ for $\\tau > 1$ and this underestimation is sustained with increasing $\\tau$. We note that the polarization angle is identical for both methods.\n\nThe anisotropic intensity approximation loses its quality through the following: (i) the fact that a significant part of the radiation is polarized, which has an impact on the irreducible tensor representation of the radiation field (see Eq.~\\ref{eq:irred_J}), and (ii) that this simplification subsequently impacts the source function, resulting in a magnification of the error. The latter error is particularly manifest at high optical depths, and it is also a consequence of the local approximation of an LVG-like problem. We expect this error to be ameliorated when the local approximation is abandoned as in PORTAL. \n\nThe polarization of (sub)millimeter lines through the GK effect has been observed in a number of sources. For most line observations, the observed polarization fraction is lower than $2\\%$ \\citep{lai:03}. This can be taken as a direct indicator of the quality of the anisotropic intensity approximation. There is a fraction of emission lines for which high polarization fractions are observed; the most strongly polarized emission lines go up to $13 \\%$ \\citep{vlemmings:12, cortes:05}. The large polarization fractions are most probably due to large sources of external radiation in the vicinity. \n\nOne avenue to remedy the anisotropic intensity approximation is to iteratively perform the inward ray-tracing steps (see Section \\ref{sec:polsee}) for all radiative polarization modes and perform the irreducible tensor integration as Eq.~(\\ref{eq:irred_J}). After each iteration, the alignment of the quantum states for each cell is recomputed until convergence is attained. We plan to implement such a scheme in a later version of PORTAL, although this will significantly increase the calculation time.\n\\subsection{The strong magnetic field approximation}\n\\label{sec:comp_align}\nThe symmetry axis of the molecular and atomic states determines the (projected) direction of polarization. In our models, it is assumed that the symmetry axis is along the local magnetic field direction. This requires the magnetic precession rate to be $10$-$100$ times stronger than other directional interaction rates. If an alternative directional interaction is about as strong or stronger than the magnetic precession rate, then the symmetry axis of the quantum states is rotated. \n\nThe magnetic precession rate for a nonparamagnetic molecule is given by \n\\begin{align}\ng\\Omega = 4.8 g_{\\mathrm{mol}} \\left(\\frac{B}{\\mathrm{mG}} \\right) \\ \\mathrm{s}^{-1}, \n\\end{align} \nwhere $g_{\\mathrm{mol}}$ is the molecular g-factor: A dimensionless factor that determines the coupling of the molecule to the magnetic field. For linear molecules, $g_{\\mathrm{mol}}$ is the same for all rotational levels. The molecular g-factors of CO and HCO$^+$ that we consider in this work are $g^{\\mathrm{CO}} = -0.269$ \\citep{flygare:71} and $g^{\\mathrm{HCO}^+}=0.006$.\\footnote{\nWe computed the g-factor of HCO$^+$ using quantum chemical techniques since no experimental data are available. The quantum chemical calculations were performed at the CCSD(T) level of theory, using aug-cc-pVTZ basis sets, with the CFOUR program package \\citep{CFOUR}. We used a linear geometry of $r_{\\mathrm{CO}}=1.112$ \\AA and $r_{\\mathrm{CH}}=1.095$ \\AA. We note that the molecular g-factor of HCO$^+$ is anomalously low. Indeed, the only polarimetric observation of HCO$^+$ yielded no detection \\citep{glenn:97}. This could be an effect of weak Zeeman precession. However, one should not forget that HCO$^+$ has in fact a hyperfine structure, where each hyperfine-transition has its own g-factor \\citep[see, for instance,][]{lankhaar:18} that only averages to the rotational g-factor if all hyperfine-transitions have line-strengths proportional to the hyperfine-resolved Einstein A-coefficients.\n}\n\nWe compare the magnetic precession rate ($1$ mG and $1$ $\\mathrm{\\mu G}$) to the cumulative rate of stimulated emission in Figures \\ref{fig:sphere_rates} and \\ref{fig:pp_rates}. For the problems we considered, the magnetic precession rate is dominant over all other interactions and it is justified to assume that the quantum state symmetry axis is along the magnetic field direction.\n\nEarlier, we saw that HCO$^+$ had an exceptionally low magnetic moment. Conversely, the dipole moment of HCO$^+$ is very large. Thus radiative interactions for such a molecule are very strong, and therefore also a strong magnetic field is required to justify the dominant magnetic field approximation. Indeed, for a large region of the collapsing sphere, a $1 \\ \\mathrm{\\mu G}$ magnetic field would not determine the HCO$^+$ symmetry axis. We stress that for molecules that have strong radiative interactions, one should be extra vigilant and check the relevant interaction rates to verify that the magnetic field truly defines the symmetry axis of the quantum states and thus if the polarization vectors do indeed trace the magnetic field structure. \n\nIt is conceivable that a strong external radiation field that has a large angular size, such as a large stellar object, determines the quantum state symmetry axis. The directional rate of interaction of a general lower quantum state, $1$, by an external black-body radiation source at the solid angle $\\Delta \\Omega_*$ and with the temperature $T_*$ is \\citep{nedoluha:92,morris:85} \n\\begin{align}\nR_{12} = \\frac{g_2}{g_1} A_{21} \\left[e^{h\\nu_{21} \/ k_B T_*} - 1 \\right]^{-1} \\Delta \\Omega_*,\n\\end{align}\nwhere $g_i$ is the degeneracy of level $i$ and $A_{21}$ and $\\nu_{21}$ are the Einstein coefficient and frequency of the transition from upper level $2$ to lower level $1$. It is apparent from this expression that (sub)millimeter lines have relatively low interaction rates. Rather, vibrational transitions in the IR region have associated directional interaction rates that are far greater and are more likely to compete with magnetic interactions to determine the symmetry axis of the quantum states. For instance, the interaction rate of the $(v,J)$, $(0,0)\\to (1,1)$ transition of CO is $\\sim 7.8 \\ \\mathrm{s}^{-1}$ when it is excited by a $2000$ Kelvin black-body radiation source at $\\Delta \\Omega_* = 1$ sr. The rate drops quadratically with the distance to the external radiation source and it is not corrected for absorption. We implemented a module in PORTAL that can incorporate the interactions resulting from a bright external source of radiation through vibrational transitions. This is particularly important when investigating the circumstellar envelopes of evolved stars \\citep{morris:85, ramos:05}\n\nThe strong magnetic field approximation should be abandoned when multiple directional interactions have similar interaction rates. In that case, one must comprehensively model all anisotropies affecting the quantum state alignment. This can be done at the expense of a computational effort as it increases the dimensionality of the problem greatly. For example, in treating the first $41$ rotational levels of a linear rotor and by setting $k_{\\mathrm{max}}=6$, the dimensionality of the SEE would increase from $151$ to $1086$, provided that we neglect orientation elements of uneven $k$. The general theory of setting up the complete SEE can be found in Chapter 7 of \\citet{landi:06}. \n\\subsection{General remarks}\n\\label{sec:general_remarks}\n\\subsubsection{(Sub)millimeter line polarization in astrophysical regions}\nIt is clear from our calculations that the only requirement for the emergence of polarized emission is a source that has some form of anisotropy. This anisotropy may come from the velocity field, which has already been explored by \\citet{goldreich:81}, but it is not necessarily limited to this. To present the capabilities of PORTAL, we computed the emergence of polarized radiation in a protoplanetary disk and a collapsing sphere. In the protoplanetary disk, anisotropy mostly comes from the density structure. For the collapsing sphere, anisotropy comes from both the velocity-field and the density structure.\n\nFurthermore, we confirm the earlier observation of \\citet{goldreich:81}, which is that namely around optical depths of unity, the polarization of line emission is the strongest. The physical reason behind this is that for sources with some sort of anisotropy, around $\\tau \\sim 1,$ this anisotropy is most manifest in the local radiation field. Subsequently leading to the highest polarization degrees. \n\n\\subsubsection{Sampling} \nThe sampling of the space that we used is identical to the sampling used by LIME in which a random sampling, weighed by the density-structure, of the space is performed and neighboring cells are found through a Voronoi tessellation. We found that the extensive angular sampling that we performed to compute the local anisotropic radiation field generally requires a higher sampling of the space than would be necessary if one is generating a nonpolarized image. We found that for insufficient sampling of the space, strong local variation in the polarization fractions manifest themselves even though similar variations would not be visible in the total intensity. Also, local $90^o$ flips of the polarization vectors can be a product of sampling of the surrounding space that is too sparse. For a source with symmetry in both the magnetic field and the radiative transfer structure, it can be prudent to use symmetrical averages in the case of a sampling that is too sparse.\n\n\\subsubsection{Collisions} \nIn order for appreciable polarization in the emission from astrophysical regions to be produced, one requires the rate of (isotropic) collisions to be relatively low. When collisions occur more than $100$ times as frequent as the aligning absorption and stimulated emission events, no observable polarization is produced. Polarization is therefore not produced in regions of high number density and temperature. In general, regions that are in local thermal equilibrium show no appreciable polarization in their emission. \n\nIn the astrophysical problems that we analyzed, we represented collisions only by their rank-0 elements, that is, we assumed all magnetic substates to be equally pumped. At the same time, we assumed no depolarization through elastic collisions. The systematic errors of both assumptions are opposite. Such an approximation for the alignment characteristics of collisional rates is a common assumption in the modeling of alignment of quantum states \\citep{landi:06}. Indeed, collisional rates resolved at the level of magnetic substates are not readily available, even though it is possible to compute these using modern quantum-dynamical methods \\citep{alexander:79, faure:12, landi:06}. \n\n\n\\subsubsection{External radiation} \nWe found that an external source of directional radiation enhances the polarization appreciably. Similar conclusions have also been drawn in maser polarization theory \\citep{lankhaar:19} and also for the GK effect \\citep{deguchi:84, cortes:05}. In particular, \\citet{cortes:05} found that they could explain the $90^o$-flip in polarization angle between the CO $J=1-0$ and the $J=2-1$ transitions through the anisotropic radiation coming from an external source. We confirm that this is one possible explanation, but we stress that there are other avenues to attain such a polarization effect. According to our theory, this $90^o$-flip is most generally explained by the $\\eta_Q^{1-0}$ and the $\\eta_Q^{2-1}$ elements being of opposite signs. This does not necessarily require an external radiation source. \n\nIt should be emphasized that polarization enhancement through external radiation is most manifest when hot objects irradiate high-frequency transitions, such as vibrational lines. It is also the case for such transitions that are most likely to compromise the strong magnetic field approximation (see Section \\ref{sec:comp_align}). In this work, we have abstained from including higher vibrational states when computing the polarization maps, but we will further explore this when we use PORTAL in conjunction with more detailed models of astrophysical regions and the involved radiative processes. \n\n\\subsubsection{Alternative routes to polarization} \nDust emission is often observed to be partially polarized. This has been seen in protoplanetary disks \\citep{hull:17}, in circumstellar envelopes of evolved stars \\citep{vlemmings:17}, and molecular clouds \\citep{soler:13}. Polarized emission from dust follows from its alignment. Dust can get aligned to the magnetic field through the process of radiative torque alignment \\citep{draine:97}, but alignment to a strong external source of radiation \\citep{lazarian:07} or through self-scattering \\citep{kataoka:15} is also possible. \n\nThe dust polarization is indicative of the alignment and therefore does not always trace the (projected) magnetic field direction. Polarization fractions are observed to be up to a few percent. In PORTAL, we neglected the contribution of the dust polarization to the molecular state alignment because we used the anisotropic intensity approximation. In the ray-tracing step, we implemented the dust polarization module outlined in \\citet{padovani:12} and added it to the regular line polarization ray-tracing. We have found in the simulations we present here that the contribution of the dust polarization around the line-frequency is negligible because the line-opacity is some orders of magnitude greater than the dust opacity. This means that for strong enough magnetic fields (see Section \\ref{sec:comp_align}), line polarization faithfully traces the (projected) magnetic field direction with 90$^o$ ambiguity.\n\nRecently, it has been proposed that through forward scattering of radiation by a collective of molecules, a phase difference can be induced to the parallel and perpendicularly polarized components of the radiation field \\citep{houde:13}. The phase difference subsequently leads to a conversion of Stokes-U to Stokes-V radiation. This process, called anisotropic resonant scattering, would lead to the production of circular polarization at the cost of linear polarization, and it also changes the polarization angle. Observational evidence for this phenomenon is accruing \\citep{hezareh:13, chamma:18}. Anisotropic resonant scattering is typically thought to occur in a foreground cloud, between the observer and the source of polarized line emission \\citep{houde:13}, but it could also be a feature of the radiative transfer inside the source. A better estimate of the relative strength of anisotropic resonant scattering has to be developed before we can evaluate the importance of this effect on the emergence of linear polarization in thermal line emission. \n\n\\subsubsection{Ground state alignment.} \n\\citet{yan:06} showed that polarization can emerge in atomic (hyper)fine-structure lines through (i) a strong magnetic field that defines the symmetry axis and (ii) an external UV radiation field that induces directional transitions, aligning the quantum states. If the pumping rate is much lower than the spontaneous decay rates of the excited states, only the ground state of the atomic system is aligned. Collisions and stimulated emission events are neglected in the formalism of ground state alignment (GSA). Through neglecting collisions and stimulated emission events and adapting an idealized geometry, \\citet{yan:06} are able to formulate semianalytical expressions for the polarization fractions emerging from atomic lines. GSA has been proposed as a polarizing mechanism for atomic lines in the ISM \\citet{zhang:18}.\n\nPORTAL builds on the same theory as GSA, but it explicitly incorporates the effect of collisions and stimulated emission events. Furthermore, instead of assuming that a radiation field only comes from an external source, PORTAL maps out the full 3D radiation field structure of the medium in which the investigated species is embedded. In this work, we focus on the polarized radiative transfer of (sub)millimeter molecular and atomic lines because its radiative transfer does not involve any scattering \\citep{brinch:10}. We plan to extend our model to also incorporate the emergence of polarization in atomic fine-structure lines, where we will pay special attention to scattering in the radiative transfer of these systems.\n\n\\section{Conclusions}\nWe present PORTAL, a 3D polarized radiative transfer program that is adapted to lines. The program uses the strong magnetic field approximation and the anisotropic intensity approximation, both of which we show to hold for the majority of relevant astrophysical problems. PORTAL can be used in stand-alone mode using an LTE estimate of the molecular or atomic excitation. Alternatively, the output of existing 3D radiative transfer programs can be input in PORTAL. \n\nTo outline PORTAL's capabilities, we computed the polarization maps of a collapsing sphere and a simple protoplanetary disk model. The polarization spectrum of a collapsing sphere shows polarization in its spectral lines up to $2\\%$ with the associated polarization vectors aligned with the projected magnetic field direction. The protoplanetary disk when viewed face on shows polarization fractions up to $\\sim 0.5 \\%$, but the polarization fraction rises to $\\sim 9\\%$ at significant inclinations. The polarization vectors resulting from a radial and toroidal magnetic field configuration are identical for a face-on view of the protoplanetary disk, and they can only be distinguished when viewed at a significant inclination. In forthcoming papers, we plan to use PORTAL to analyze the emergence of polarization in spectral lines in more detailed models of protoplanetary disks, to a molecular outflow, and to the circumstellar envelopes of AGB stars.\n\n\\begin{acknowledgements} Support for this work was provided by the Swedish Research Council (VR). Simulations were performed on resources at the Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC). Ko-Yun Huang and Athol Kemball are acknowledged for sharing the results of their GK code. The authors thank Luis Velilla Prieto for helpful comments on a first draft of the manuscript. We thank the referee (Martin Houde) for comments that improved the paper.\n \\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{When a star cluster meets a cloud}\n\\label{sec:1}\nWe study the encounters between stars clusters and giant molecular\nclouds (GMCs) \\cite{gieles06b}. The effect of these encounters has\npreviously been studied analytically for two cases: 1) head-on\nencounters, for which the cluster moves through the centre of the GMC\n\\cite{1987gady.book.....B} and 2) distant encounters, where the\nencounter distance $p>3\\mbox{$R_{\\rm n}$}$, with $p$ the encounter parameter and \\mbox{$R_{\\rm n}$}\\ the\nradius of the GMC \\cite{1958ApJ...127...17S}. We introduce an\nexpression for the energy gain of the cluster due to GMC encounters\nvalid for all values of $p$ and \\mbox{$R_{\\rm n}$}\\ of the form\n\n\\begin{equation}\n\\Delta E  \\simeq \\frac{4.4\\,\\mbox{$r^2_{\\rm h}$}}{\\left(p^2+\\sqrt{\\mbox{$r_{\\rm h}$}\\,\\mbox{$R_{\\rm n}$}^{3}}\\right)^2}\\,\\left(\\frac{G\\mbox{$M_{\\rm n}$}}{\\mbox{$V_{\\rm max}$}}\\right)^2\\,M_c.\n\\label{eq:1}\n\\end{equation}\nHere \\mbox{$V_{\\rm max}$}\\ is the maximum relative velocity, $\\mbox{$M_{\\rm n}$}$ is the mass of the\nGMC, $G$ is the gravitational constant and \\mbox{$r_{\\rm h}$}\\ and \\mbox{$M_{\\rm c}$}\\ are the\nhalf-mass radius and mass of the cluster, respectively. We perform\n$N$-body simulations of encounters with different $p$ and compare the\nresulting ${\\rm \\Delta} E$ of the cluster to\nEq.~\\ref{eq:1}. Fig.~\\ref{fig:1} shows the very good agreement between\nsimulations and predictions of Eq.~1.  Snapshots of one simulation are\nshown in Fig.~2.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[height=3.8cm]{de_p.ps}\n\\caption{\\mbox{${\\rm \\Delta} E\/|E_0|$}\\ of a cluster as a function of $p$. The $N$-body results\n    are shown with diamonds. The result of \\cite{1987gady.book.....B}\n    and \\cite{1958ApJ...127...17S} for head-on and distant encounters\n    are shown as a filled circle and as a dashed line,\n    respectively. Eq.~\\ref{eq:1} is shown as a full line.}\n\\label{fig:1}      \n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[height=2.35cm]{snap_distant_5frames.ps}\n\\caption{Simulation of a close encounter between a GMC (grey shades)\nand a star cluster. The snapshots are viewed in the centre-of-mass\nframe of the cluster. \\mbox{$a_{\\rm c}$}\\ is the Plummer radius of the cluster.}\n\\label{fig:2}      \n\\end{figure*}\n\\vspace{-0.8cm}\n\\section{The cluster disruption time}\nFrom the simulations we find that the fractional mass loss ($\\mbox{${\\rm \\Delta} M\/|M_0|$}$) is only\n25\\% of \\mbox{${\\rm \\Delta} E\/|E_0|$}. This is because stars escape with velocities much higher\nthan the escape velocity. Defining the cluster disruption time as\n$\\mbox{$t_{\\rm dis}$}=\\mbox{$M_{\\rm c}$}\/\\dot{\\mbox{$M_{\\rm c}$}}$, we find a cluster disruption time of the form\n\n\\begin{equation}\n\\mbox{$t_{\\rm dis}$} = 2.0\\,S\\left(\\mbox{$M_{\\rm c}$}\/10^4\\,\\mbox{$M_{\\odot}$}\\right)^{\\gamma}{\\mbox{Gyr}},\n\\label{eq:2}\n\\end{equation}\nwith $S\\equiv1$ for the solar neighbourhood and scales with the global\n GMC density ($\\mbox{$\\rho_{\\rm n}$}$) as $S\\propto\\mbox{$\\rho_{\\rm n}$}^{-1}$. The index $\\gamma$ is\n defined as $\\gamma=1-3\\lambda$, with $\\lambda$ the index that relates\n the cluster half-mass radius to its mass ($\\mbox{$r_{\\rm h}$} \\propto\n \\mbox{$M_{\\rm c}$}^{\\lambda}$).  The observed shallow relation between cluster\n radius and mass (e.g. $\\lambda\\simeq0.1$), makes the index\n ($\\gamma=0.7$) similar to the index found both from observations\n \\cite{2005A&A...441..117L} and from simulations of clusters\n dissolving in tidal fields ($\\gamma\\simeq0.62$). The constant of 2.0\n Gyr, which is the disruption time of a $10^4\\,\\mbox{$M_{\\odot}$}$ cluster in the\n solar neighbourhood, is about a factor of 3.5 shorter than found from\n earlier simulations of clusters dissolving under the combined effect\n of the galactic tidal field and stellar evolution. It is only slightly\n higher than the observationally determined value of 1.3 Gyr\n \\cite{2005A&A...441..117L}, suggesting that the combined effect of\n tidal field and encounters with GMCs can explain the lack of old open\n clusters in the solar neighbourhood \\cite{1958RA......5..507O}. GMC\n encounters can also explain the (very) short disruption time that was\n observed for star clusters in the central region of M51\n \\cite{2005A&A...441..949G}, since there $\\mbox{$\\rho_{\\rm n}$}$ is an order of\n magnitude higher than in the solar neighbourhood.\n\n\t    \n\\vspace{-0.3cm}\n\\input{referenc}\n\n\n\\printindex\n\\end{document}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{The Ever-Increasing Threat of Cyber Crimes}\nIn recent years, the risk of cyber crimes has emerged as a top global concern. \nThis is made evident by the world economic forum's annual global risk report \\citep{wef2020}, which regularly places cyber attacks and theft of data in its ``Top 5 global risks in terms of likelihood''.\nOver the years, the frequency and severity of cyber attacks have increased significantly globally, and they are projected to continue to increase in the future.\nRecently, Cybersecurity Ventures estimated the cost of cyber crimes to rise to 10.5 trillion USD annually by 2025 \\citep{morgan2020cybercrimecost}, up from a world economic forum estimate of 3 trillion USD for 2015.  \nFurthermore, see a detailed descriptive analysis of cyber crimes in the business sector over the last 15 years in \\citep{shevchenko2022nature}.\n\nCyber crime can be initiated by various actors, including malicious actors within institutions and external adversaries such as rogue nation states, hackers, and cyber terrorists.\nThese attacks come in a variety of forms, ranging from denial-of-service (DoS) attacks \\citep{gupta2017taxonomy}, malware \\citep{tailor2017comprehensive}, ransomware \\citep{tailor2017comprehensive}, blackmail \\citep{rid2012cyber}, extortion \\citep{young1996cryptovirology}, and more \\citep{craigen2014defining, husak2018survey}.\nSuch criminal activities are being perpetrated on a massive scale, hitting individuals as well as organisations. A large array of organisations worldwide are targeted by cyber attacks, including government agencies, universities, financial sectors, and private corporations. \nIn particular, these attacks can affect important infrastructure units that play a key role in security and safety, such as emergency services and health care.\nSuch attacks have caused breaches and significant damages to those organisations, and often adversely affect downstream users of the compromised organisations and services.\nThe damages incurred include losses attributed to outcomes such as business interruption, loss of data, reduced reputation and trust of the organisation, legal liabilities, intellectual property theft, and potential for loss of life. \n\nThe seriousness of cyber attacks has been reflected in the {U.S.} President's executive order on \\textit{Strengthening the Cybersecurity of Federal Networks and Critical Infrastructure}, which calls for a cybersecurity framework that can ``support the cybersecurity risk management efforts of the owners and operators of the Nation's critical infrastructure\".\nCyber risk from a financial and insurance perspective has also been developed under international banking and insurance regulations. \nThe Basel~III banking Accords cover cyber risk as a key component of Operational Risk capital modelling and adequacy, and the Solvency~II insurance regulations discuss the significance of an emerging cyber insurance threat that affects insurers as well as reinsurers.\nFor example, see an overview of cyber risk from a financial and insurance perspective in \\cite{peters2018understanding, malavasi2022cyber}.\n\n \n\\subsection{Lack of Product Standardisation in the Cyber Risk Insurance Market} \nAlthough many security solutions have been developed and implemented in order to detect and prevent cyber attacks, achieving a complete security protection is not feasible \\citep{lu2018managing}.\nTo address this problem, there is an increasing demand to develop the market for cyber risk insurance and to understand the structuring of insurance products that will facilitate risk transfer strategies in the context of cyber risk; see discussions in \\cite{peters2018understanding, peters2017statistical, marotta2017cyber, bohme2010modeling} and the references therein. \n\n\nAs one can see from surveys such as \\cite{marotta2017cyber} and \\cite{shetty2010competitive}, the scope of such a marketplace is still very much in its infancy, despite the fact that financial institutions rank cyber losses in their top three loss events systematically when reporting Operational Risk loss events under Basel II\/III to national regulators. \nThe reason that the cyber risk insurance market has yet to emerge with standardised products is largely due to differences in opinion as how best to mitigate and reserve against these loss events. \nFrom a technology perspective, it is common to attempt to mitigate such events in contrast to insurance or capital reserving; see discussions in \\cite{bandyopadhyay2009managers}. \nFrom a financial risk perspective, practitioners often opt for Tier I capital reserving and avoid insurance products, as the capital reduction from Operational Risk insurance is capped under Basel regulations with a haircut of 20\\%; see discussions in \\cite{peters2011impact}. This disincentivises them to purchase insurance products that have excessive premiums. \nFrom the insurance industry's perspective, there is a lack of market standardisation on insurance contract specifications that would avoid excessive premiums to be charged when the bespoke insurance products are designed. \n\n\n \n\\subsection{Our Contributions}\n\nThe three aforementioned perspectives are beginning to change and we believe that it is a suitable time to revisit the perennial question of how best to set up an insurance marketplace for cyber loss events. \nIn this paper, we introduce for the first time a Bonus-Malus framework for cyber risk insurance that provides IT-specific incentive mechanisms for encouraging sound IT governance and technology developments. \nSpecifically, we explore a class of Bonus-Malus systems in which an insured enjoys a discount in the cost of risk transfer as a result of their upfront expenses in risk reduction in the form of self-mitigation measures; and like-wise an insurer benefits from encouraging risk reduction in their risk pools to provide competitive pricing of insurance premium. \nUnder this framework, we also demonstrate how to develop loss models and decision models under uncertainty.\n\nTo illustrate this framework and the associated decision problems, let us consider an organisation which provides a service that is exposed to the threat of distributed denial-of-service (DDoS) attacks.\nSuch an organisation may opt to use network traffic filtering as well as a content distribution network, and may use multiple servers to balance the network load. \nAs such, there are various countermeasures for mitigating the threats of DDoS attacks and ensuring the availability of the service, but each of such self-mitigation measures adds to the upfront costs of risk reduction. \nTherefore, the organisation needs to determine the quantum of its security (i.e., risk reduction) budget and distribute this across these self-mitigation measures. We argue that this budget for risk reduction needs to and can be determined in tandem with risk transfer decisions, i.e., the purchase of cyber risk insurance.\nMoreover, a cyber risk insurer can incorporate incentive mechanisms in their insurance policies to encourage such risk reduction provision through offsetting the expenses of risk reduction by a discount in the pricing of the insurance premium.\nThis highlights the need for a comprehensive framework in which losses incurred by cyber threats can be realistically modelled and the rational decisions of organisations in the face of cyber threats can be quantitatively analysed.\n\n\n\nOur main contributions are as follows: \n\\begin{enumerate}\n    \\item We introduce the Bonus-Malus system to cyber risk insurance as a mechanism to provide incentive for the insured to adopt self-mitigation measures against cyber risk. \n\\item We develop a mathematical model of cyber losses and cyber risk insurance, and subsequently analyse the optimal cybersecurity provisioning process of the insured under the stochastic optimal control framework. \n\\item We develop an efficient algorithm based on dynamic programming to accurately solve the stochastic optimal control problem, under the assumption that the loss severity follows a truncated version of the g-and-h distribution. We also formally prove the correctness of the proposed algorithm. \n\\item We demonstrate through a numerical experiment that a properly designed cyber risk insurance contract with a Bonus-Malus system can resolve the issue of moral hazard, and can provide benefits for the insurer.\n\\end{enumerate}\n\nThe rest of the paper is organised as follows. \nSection~\\ref{sec:relatedwork} discusses related studies in the literature.\nIn Section~\\ref{sec:model}, we introduce the mathematical model of cyber losses and cyber risk insurance with a Bonus-Malus system. In Section~\\ref{sec:stocoptctr}, we present the optimal cybersecurity provisioning process and the dynamic programming algorithm. In Section~\\ref{sec:g-and-h}, we introduce the g-and-h distribution and use it as the model for loss severity. We present results from the numerical experiment in Section~\\ref{sec:exp}. Finally, Section~\\ref{sec:conclusion} concludes the paper. \n\n\\section{Related Work}\n\\label{sec:relatedwork}\nRecently, many studies analysed cyber risk insurance from a technology perspective. \nCybersecurity frameworks involving cyber risk insurance have been developed for specific IT systems, \nincluding computer networks \\citep{fahrenwaldt2018pricing,xu2019cybersecurity}, \nheterogeneous wireless network \\citep{lu2018cyber}, wireless cellular network \\citep{lu2018managing}, plug-in electric vehicles \\citep{hoang2017charging}, cloud computing \\citep{chase2017scalable}, and fog computing \\citep{feng2018evolving}. \n\nSome studies considered the interplay between self-mitigation measures (i.e., risk reduction) and cyber risk insurance, e.g., \\cite{pal2010analyzing,pal2014will,pal2017security,khalili2018designing,dou2020insurance}. \nThese studies investigated two important challenges in cyber risk insurance: risk interdependence and moral hazard. They found that in order to incentivise the insured to invest in self-mitigation measures, some form of contract discrimination, i.e., adjusting the insurance premium based on the insured's security investment, is necessary. \n\\cite{yang2014security,schwartz2014cyber,zhang2017bi} investigated these challenges in a networked environment, where cyber attacks can spread between neighbouring nodes, further complicating these challenges.\n\nThere are also studies which took the insured's perspective and analysed the security provisioning process using dynamic models. \n\\cite{chase2017scalable} developed a framework based on stochastic optimisation to jointly provision cyber risk insurance and cloud-based security services across multiple time periods in cloud computing applications. \n\\cite{zhang2018optimal} modelled the decisions on self-protections of the insured by a Markov decision process and investigated the problem of insurance contract design.\n\nA critical drawback of many of the existing studies is that they neglected the highly uncertain nature of losses incurred by cyber incidents. These studies relied on over-simplified assumptions, e.g., by modelling cyber loss as: a fixed amount \\citep{pal2010analyzing,pal2014will,yang2014security,hoang2017charging,feng2018evolving,dou2020insurance}, random with finite support \\citep{chase2017scalable,zhang2018optimal,lu2018managing}, or random with a simple parametric distribution \\citep{zhang2017bi,khalili2018designing}. These assumptions limit the practicality of these studies, and their results remain conceptual and non-applicable to realistic insurance loss modelling under a classical Loss Distribution Approach (LDA) framework \\citep{moscadelli2004modelling, peters2006bayesian, maillart2010heavy, shevchenko2013loss, eling2019actual, zeller2022comprehensive}.\nLDA is one of the most common modelling methods in the Advanced Measurement Approach (AMA) under the Basel II Accords. \nUnder the LDA framework, the probability distributions of loss severity (i.e., the impact of a single loss event) and annual loss frequency are modelled and estimated separately. The aggregate annual loss is thus modelled by a compound distribution. \nFor detailed discussions about LDA and a comparison with the Internal Measurement Approach, see \\citep{frachot2001loss} and the references therein.\nIn addition, see \\citep{dutta2006tale} for an evaluation of specific distributional assumptions in LDA and their impacts on the estimation of Operational Risk capital. \nIn our study about the design of cyber risk insurance contracts, we have built LDA into our cyber loss model and we have also incorporated a quantitative model for the effects of self-mitigation measures on cyber losses (see Section~\\ref{ssec:lossmodel} and Section~\\ref{ssec:mitigation}).\n\nMoreover, many of the aforementioned studies do not take into account the interplay between the upfront costs of risk prevention, the consequent reduced risks, and the possibility to exploit this interplay to design practical cyber risk insurance products.\nA review of cyber insurance product prospectus by major insurers, such as AIG's CyberEgde\\footnote{\\url{https:\/\/www.aig.com\/business\/insurance\/cyber-insurance\/}, accessed on 2020-12-10}, Allianz's Cyber Protect\\footnote{\\url{https:\/\/www.agcs.allianz.com\/solutions\/financial-lines-insurance\/cyber-insurance.html}, accessed on 2020-12-10}, and Chubb's Cyber Enterprise Risk Management (Cyber ERM)\\footnote{\\url{https:\/\/www.chubb.com\/us-en\/business-insurance\/cyber-enterprise-risk-management-cyber-erm.html}, \\linebreak accessed on 2020-12-10} also indicates that the insurance products in the market have yet to explicitly factor in the benefits of upfront risk reduction, or to offset those costs against that of risk transfer.\nFor the first time, we introduce the Bonus-Malus system which is frequently used in vehicle insurance products to address this gap in cyber risk insurance product design. \nIn vehicle insurance, Bonus-Malus systems are experience rating systems in which an insured who had one or more accidents is penalised by premium surcharges or \\textit{maluses} and an insured who had a claim-free year is rewarded with premium discounts or \\textit{bonuses} \\citep{lemaire1995bonus, neuhaus1988bonus}; see, e.g., \\citep{baione2002development, ragulina2017bonus, gomez2018multivariate, tzougas2018bonus} for discussions about the design and analysis of Bonus-Malus systems.\nA key characteristic of Bonus-Malus systems is the bonus hunger mechanism \\citep{holtan2001optimal, charpentier2017optimal, tzougas2018bonus}, i.e., under a Bonus-Malus system, an insured is willing to carry small losses themself in order to avoid premium surcharges in the future.\nOur optimal cybersecurity provisioning model in Section~\\ref{sec:stocoptctr} captures this aspect of Bonus-Malus systems by allowing the insured to decide whether to make a claim at the end of each policy year.\n\n\\section{Cyber Risk Insurance Policy and Bonus-Malus System}\n\\label{sec:model}\n\n\nLet us first present an overview of our cyber risk insurance model.\nTo begin, let us specify the frequency and severity model under the Loss Distribution Approach (LDA) that defines the financial loss process resulted from cyber loss events.\nWe consider $T\\in\\N$ consecutive years, and we assume that throughout each year $t$, the insured may suffer a random number ($N_t \\in \\N$) of cyber loss events arising from cyber attacks. Their loss amounts are denoted by $X^{(t)}_1,\\ldots,X^{(t)}_{N_t}$. The aggregate annual cyber loss in year $t$ is therefore $\\sum_{k=1}^{N_t}X^{(t)}_{k}$. \nThe insured has several choices to attempt to mitigate these cyber loss events and reduce the risk, including enhancing the security and resilience of their IT infrastructure and reserving Tier~I capital to cover the incurred losses.\nIn regards to the internal IT infrastructure, it will be assumed that the insured has the option to adopt a self-mitigation measure that can reduce the severity of cyber loss events up to a fixed amount of loss. \nIn addition, the insured can choose to purchase a cyber risk insurance policy which gives the insured the right to claim the aggregate cyber loss incurred, up to a maximum cap imposed by the insurance contract, in an agreed interval of time (typically annually), minus a deductible. \nPlease refer to Table~\\ref{tab:model-notations} for notations used in the cyber risk insurance model.\n\n\\begin{table}\n\\begin{center}\n\\caption{Notations in the cyber risk insurance model}\n\\label{tab:model-notations}\n\\vspace{0.5em}\n\\begin{tabular}{ll}\n$T$ & number of policy years \\\\\n$N_t$ & number of cyber loss events in year $t$ \\\\\n$X^{(t)}_1,\\ldots,X^{(t)}_{N_t}$ & loss amounts of the cyber loss events in year $t$  \\\\\n$\\mathcal{W}$ & space representing annual loss frequency and severities \\\\ \n$(\\Omega,\\mathcal{F}_T,\\PROB,(\\mathcal{F}_t)_{t=0:T})$ & filtered probability space \\\\ \n$W_t$ & loss frequency and severities in year $t$ \\\\\n$\\psi_N(\\cdot)$ & probability generating function of the loss frequency \\\\ \n$F_X(\\cdot)$ & distribution function of the loss severity \\\\ \n$\\mathcal{D}$ & set representing self-mitigation measures \\\\\n$\\beta(d)$ & annual investment required by the self-mitigation measure $d\\in\\mathcal{D}$ \\\\\n$\\gamma(d)$ & loss reduction effect of the self-mitigation measure $d\\in\\mathcal{D}$ \\\\\n$L(d,w)$ & aggregate annual cyber loss with the self-mitigation measure $d\\in\\mathcal{D}$ \\\\\n$\\delta_{\\mathrm{i}\\mathrm{n}}(t)$ & initial sign-on fee of cyber risk insurance in year $t$ \\\\\n$\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)$ & withdrawal penalty of cyber risk insurance in year $t$ \\\\\n$\\delta_{\\mathrm{r}\\mathrm{e}}$ & re-activation penalty of cyber risk insurance \\\\\n$\\mathcal{B}$ & set representing Bonus-Malus levels \\\\\n$\\mathcal{I}$ & set representing states of the cyber risk insurance contract \\\\\n$\\mathcal{B}\\mathcal{M}(\\cdot,\\,\\cdot)$ & transition rules of the Bonus-Malus system \\\\\n$\\mathcal{B}\\mathcal{M}_0(\\cdot,\\,\\cdot)$ & transition rules of the Bonus-Malus system for inactive contract \\\\\n$p^{\\mathcal{B}\\mathcal{M}}(b,t)$ & insurance premium in Bonus-Malus level $b$ in year $t$ \\\\\n$l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}(b,t)$ & insurance deductible in Bonus-Malus level $b$ in year $t$ \\\\\n$l^{\\mathcal{B}\\mathcal{M}}_{\\max}(b,t)$ & maximum insurance compensation in Bonus-Malus level $b$ in year $t$ \\\\\n$\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,l)$ & insurance compensation in Bonus-Malus level $b$ in year $t$ with loss $l$\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Cyber Loss Model}\n\\label{ssec:lossmodel}\n\nLet us define $\\mathcal{W}:=\\bigcup_{n\\in\\Z_+}\\{n\\}\\times\\R^n_+$ to be the space representing all possible combinations of realisations of the number of cyber loss events per year (frequency) and individual loss amounts per event (severities).\nLet $\\mathfrak{B}(\\mathcal{W}):=\\sigma\\big(\\bigcup_{n\\in\\Z_+}\\{(n,B):B\\in\\mathfrak{B}(\\R^n_+)\\}\\big)$ be a $\\sigma$-algebra on $\\mathcal{W}$, where $\\mathfrak{B}(\\R^n_+)$ denotes the Borel subsets of $\\R^n_+$. \nLet us consider the space $\\Omega:=(\\mathcal{W})^T=\\underbrace{\\mathcal{W}\\times\\cdots\\times\\mathcal{W}}_{T \\text{ times}}$. For each $\\omega=(w_1,\\ldots,w_T)=\\big(\\big(n_1,x^{(1)}_1,\\ldots,x^{(1)}_{n_1}\\big),\\ldots,$ $\\big(n_T,x^{(T)}_1,\\ldots,x^{(T)}_{n_T}\\big)\\big)\\in\\Omega$, we define $W_t(\\omega):=w_t$ and $N_t(\\omega):=n_t$ for $t=1,\\ldots,T$. \nLet $\\PROB_1$ be a probability measure on $(\\mathcal{W},\\mathfrak{B}(\\mathcal{W}))$, where the subscript ``1'' indicates that it is a probability measure for the cyber loss events occurring in a single year.\nLet $\\PROB:=\\underbrace{\\PROB_1\\otimes\\cdots\\otimes\\PROB_1}_{T\\text{ times}}$, and let $(\\mathcal{F}_t)_{t=0:T}$ be a filtration on $\\Omega$, defined by \n$\\mathcal{F}_0:=\\{\\emptyset,\\Omega\\}$, $\\mathcal{F}_t:=\\sigma((W_s)_{s=1:t})$. Then, $(\\Omega,\\mathcal{F}_{T},\\PROB,(\\mathcal{F}_t)_{t=0:T})$ is a filtered probability space. Under these definitions, $W_1,\\ldots,W_T$ are independently and identically distributed ({i.i.d.})\\ random variables. \nLet $\\psi_{N}(s):=\\EXP[s^{N_1}]$ denote the probability generating function (pgf) of the loss frequency distribution. We assume that for $t=1,\\ldots,T$, \n\\begin{align}\n\\begin{split}\n&\\phantom{=}\\;\\;\\PROB\\Big[\\Big\\{\\omega=\\big(\\big(n_1,x^{(1)}_1,\\ldots,x^{(1)}_{n_1}\\big),\\ldots,\\big(n_T,x^{(T)}_1,\\ldots,x^{(T)}_{n_T}\\big)\\big):n_t=n,\\;x^{(t)}_k\\le z_k,\\forall 1\\le k\\le n\\Big\\}\\Big]\\\\\n&=\\PROB[N_t=n]\\prod_{k=1}^nF_X(z_k),\n\\end{split}\n\\end{align}\nwhere $F_X(\\cdot)$ is the distribution function of the severity distribution. This implies that given the loss frequency in a year, the individual loss amounts in that year are i.i.d. We assume that the severity distribution has finite expectation, i.e., $\\int_{\\R_+}|x|\\DIFFM{F_X}{\\DIFF x}<\\infty$. \nFor convenience, we write $W_t=\\big(N_t,X^{(t)}_1,\\ldots,X^{(t)}_{N_t}\\big)$. \nWe use $W$ to denote a random variable that has the same distribution as $W_1,\\ldots,W_T$. \nSimilarly, we use $N$ to denote a random variable that has the same distribution as $N_1,\\ldots,N_T$, and we use $X$ to denote a random variable that has the distribution function $F_X$. \n\n\\subsection{Self-Mitigation Measures}\n\\label{ssec:mitigation}\n\nLet us assume that there exists $D\\in\\N$ different self-mitigation measures, and the insured makes the decision to either adopt one of the self-mitigation measures or to not adopt any self-mitigation measure at the beginning of each year. The self-mitigation measure $d\\in\\mathcal{D}:=\\{0,1,\\ldots,D\\}$ requires an annual investment of $\\beta(d)\\in\\R_+$ per year, and reduces the severity of each cyber loss by up to $\\gamma(d)\\in\\R_+$, that is, the severity of a loss will be decreased from $X$ to\\footnote{Throughout the paper, we use the following notations: $(x)^+:=\\max\\{x,0\\}$, $x\\vee y:=\\max\\{x,y\\}$ and $x\\wedge y:=\\min\\{x,y\\}$.} $\\left(X-\\gamma(d)\\right)^+$ with the adoption of the self-mitigation measure $d$. We assume that $\\beta(0)=\\gamma(0)=0$. \nThus, if the insured decides to adopt the self-mitigation measure $d$ in a year, then the total loss suffered by the insured that year is given by\n\\begin{align}\nL(d,w):=\\sum_{k=1}^{n}\\left(x_k-\\gamma(d)\\right)^+,\n\\label{eqn:yearlyloss}\n\\end{align}\nwhen the corresponding loss frequency and severities that year are $w=(n,x_1,\\ldots,x_n)\\in\\mathcal{W}$.\n\n\n\\subsection{Cyber Risk Insurance Policy}\n\\label{ssec:cyberinsurance}\nLet us now consider a cyber risk insurance contract that lasts for $T$ years. \nAt the beginning of each year, the insured decides whether to activate the contract. In the case that the contract has been activated in a previous year, this corresponds to the insured deciding whether to continue the contract. If the contract is activated, the insured pays a premium to the insurer at the start of the year, in exchange for insurance coverage throughout the year. If the insured decides to withdraw from the contract, they no longer pays the premium and the contract is deactivated so that the insured receives no coverage. \nWe further assume that the insured pays the insurer an initial sign-on fee for fixed costs and contract origination the first time a contract is initiated, in addition to the premium, and that this amount varies deterministically over time and will be denoted by $\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\ge 0$ in year $t$. This can be used to incentivise the insured to activate the contract early. \nFurthermore, we also assume that the insured pays the insurer a deterministic and time-dependent penalty, denoted by $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\ge0$, when they withdraws from the contract in year $t$. Once withdrawn, the insured may re-activate the contract in a later year with a fixed penalty $\\delta_{\\mathrm{r}\\mathrm{e}}\\ge0$. \n\nSuppose that the aggregate cyber loss suffered by the insured in a year is $L$. At the end of the year, the insured decides whether to make a claim to the insurer. Once the claim is processed, the insured receives a payment of $(L-l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}})^+\\wedge l_{\\max}$ from the insurer as compensation, that is, the insured covers the loss up to the deductible $l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}\\ge 0$, and the insurer covers all of the remaining loss up to the maximum compensation $l_{\\max}\\ge0$. \n\n\\subsection{Bonus-Malus System}\n\\label{ssec:bonusmalus}\n\nWe now introduce the Bonus-Malus system to cyber risk insurance contracts. Let us assume that there are $B\\in\\N$ Bonus-Malus levels in the contract, denoted by $\\mathcal{B}:=\\{-\\underline{B},\\ldots,-1,0,1,\\ldots,\\overline{B}\\}$, where $\\underline{B}+\\overline{B}+1=B$. \nHere, the level $0$ is the initial Bonus-Malus level, and the lower the Bonus-Malus level, the higher the experience rating.\nConsequently, a negative Bonus-Malus level grants the insured premium discounts, while a positive Bonus-Malus level penalises the insured by premium surcharges.\nAt $t=0$, the insured starts in the initial Bonus-Malus level, denoted by $b_0=0$. At the end of the $t$-th year, given that the contract is still active, the insurer determines the Bonus-Malus level of the insured based on their previous level $b_{t-1}$ and the amount of insurance claim $C_t$ that was given out to the insured in the $t$-th year, that is, $b_{t}=\\mathcal{B}\\mathcal{M}(b_{t-1},C_t)$, where $\\mathcal{B}\\mathcal{M}:\\mathcal{B}\\times\\R_+\\to\\mathcal{B}$ denotes the deterministic rules that are transparent to the insured at the signing of the contract. We make the assumption that $\\mathcal{B}\\mathcal{M}(b,C)$ is non-decreasing in $C$ for each $b\\in\\mathcal{B}$. \nEven when the insured has withdrawn from the contract, we assume that their Bonus-Malus level is still updated annually. \nConcretely, let us define $\\mathcal{I}:=\\{\\mathrm{n}\\mathrm{o},\\mathrm{o}\\mathrm{n},\\mathrm{o}\\mathrm{f}\\Tf_1,\\ldots,\\mathrm{o}\\mathrm{f}\\Tf_{T}\\}$ as the set of all possible states of the cyber risk insurance contract. In $\\mathcal{I}$, ``$\\mathrm{n}\\mathrm{o}$'' denotes that the contract has not been signed yet, ``$\\mathrm{o}\\mathrm{n}$'' denotes that the contract is active, and ``$\\mathrm{o}\\mathrm{f}\\Tf_y$'' denotes that the contract is withdrawn where $y\\in\\Z_+$ is a counter variable that is updated annually as long as the insured does not re-activate the contract. \nLet $\\mathcal{B}\\mathcal{M}_{0}:\\mathcal{B}\\times\\mathcal{I}\\to\\mathcal{B}\\times\\mathcal{I}$ be a deterministic transition function that represents the update rules after the insured withdraws from the contract. At the end of the $t$-th year, given that the contract is inactive, the insurer determines the Bonus-Malus level $b_t$ and the insurance state $i_t$ of the insured based on their Bonus-Malus level and the insurance state in the previous year, that is, $(b_t,i_t)=\\mathcal{B}\\mathcal{M}_0(b_{t-1},i_{t-1})$. Since no such update is possible before the insured activates the contract, it is required that $\\mathcal{B}\\mathcal{M}_{0}(b,\\mathrm{n}\\mathrm{o})=(b,\\mathrm{n}\\mathrm{o})$ for all $b\\in\\mathcal{B}$. We will formally model the evolution of $(b_t,i_t)_{t=0:T}$ by a controlled stochastic process in Section~\\ref{ssec:provisionprocess}.\n\nWith the addition of the Bonus-Malus system to the cyber risk insurance contract, we assume that the premium depends on both time and the current Bonus-Malus level of the insured, and is given by $p^{\\mathcal{B}\\mathcal{M}}(b,t)$, where $p^{\\mathcal{B}\\mathcal{M}}:\\mathcal{B}\\times\\{1,\\ldots,T\\}\\to\\R_+$ is a deterministic function that is increasing in the first argument. The deductible and the maximum compensation are also assumed to be dependent on both time and the Bonus-Malus level, and are given by deterministic functions $l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}^{\\mathcal{B}\\mathcal{M}}:\\mathcal{B}\\times\\{1,\\ldots,T\\}\\to\\R_+$ and $l_{\\max}^{\\mathcal{B}\\mathcal{M}}:\\mathcal{B}\\times\\{1,\\ldots,T\\}\\to\\R_+$. We define the function $\\lambda^{\\mathcal{B}\\mathcal{M}}:\\mathcal{B}\\times\\{1,\\ldots,T\\}\\times\\R_+\\to\\R_+$ by\n\\begin{align}\n\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,l):=(l-l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}^{\\mathcal{B}\\mathcal{M}}(b,t))^+\\wedge l_{\\max}^{\\mathcal{B}\\mathcal{M}}(b,t)\n\\end{align}\nto simplify the notation for the claimable loss from the insurer.  \n\n\\begin{remark}\nOur loss model and insurance model are particularly suitable for analysing cyber risk insurance, due to the nature of cyber loss events.\nFirstly, the compound loss model in LDA is vital for capturing the low frequency and high impact nature of certain cyber loss events \\citep{maillart2010heavy, biener2015insurability, eling2017data, eling2015modelling, eling2019actual}. \nIn particular, this loss model allows us to use the highly flexible g-and-h distribution for the loss severity, in order to capture the right-skewness and the heavy-tail of cyber losses.\nThis provides adequate modelling flexibility so that one can tailor this loss model, particularly the skewness and the tail-index of the loss distribution, according to specific kinds of cyber threats and different business lines. \nWe will discuss the details of the g-and-h distribution in Section~\\ref{sec:g-and-h}.\nSecondly, as studies such as \\citep{oughton2019stochastic, armenia2021dynamic} have discussed, despite that complete mitigation of cyber risk is impractical, prevention of crippling damages from cyber attacks, which often include secondary damages to downstream customers, can be achieved with relatively low cybersecurity expenses. \nFor example, countermeasures such as the adoption of two-factor authentication and the regular update or reconfiguration of the software system \\citep{pate2018cyber} can often result in significant reduction of the risk of intrusion. \nMoreover, the adoption of self-mitigation measures generates positive externalities since it improves the overall security of the cyberspace. \nDue to these factors, there is a strong motive for cyber risk insurers to offer incentive mechanisms in their insurance products to encourage such upfront risk reduction effort and to alleviate the problem of moral hazard.\nOverall, the combination of this flexible loss model and the incentive mechanisms built into the Bonus-Malus system is novel in the context of cyber risk insurance. \nOn the other hand, the i.i.d.\\ assumption on the individual loss amounts in our model means that it does not account for the systemic aspect of cyber risk.\nWe would also like to remark that the general specifications of our cyber risk insurance model make it also applicable to other idiosyncratic risk types which can be partially prevented by self-mitigation measures. \nDespite that, our analyses and discussions about our model will be carried out in the context of cyber risk, and we will not generalise our model to other risk types in order not to obscure the main objective of this paper.\n\\label{rmk:cyber-specificity}\n\\end{remark}\n\n\n\\section{Optimal Cybersecurity Provisioning and Stochastic Optimal Control}\n\\label{sec:stocoptctr}\n\n\\subsection{Cybersecurity Provisioning Process}\n\\label{ssec:provisionprocess}\nNow, having introduced the model for cyber losses and cyber risk insurance, we consider the problem of optimal cybersecurity provisioning from the insured's point of view. \nIt is assumed that the cybersecurity provisioning process takes place for $T$ consecutive years (same as the length of the cyber risk insurance contract).\nBefore the first year, the state of the cyber risk insurance contract is initialised to ``$\\mathrm{n}\\mathrm{o}$'', i.e., $i_0=\\mathrm{n}\\mathrm{o}$, and the Bonus-Malus level is initialised to level~$0$, i.e., $b_0=0$.\nSubsequently, each year $t\\in\\{1,\\ldots,T\\}$ consists of the three following stages:\n\\begin{enumerate}\n\\item \\textbf{Provision Stage.} The insured decides: (i) the self-mitigation measure to adopt in this year, denoted by $d_t\\in\\mathcal{D}$ and (ii) whether to activate\/withdraw\/re-activate the cyber risk insurance contract, denoted by $\\iota_t\\in\\{0,1\\}$. Then, depending on the decision $\\iota_t$, a premium payment $p^{\\mathcal{B}\\mathcal{M}}(b_{t-1},t)$ and\/or a sign-on fee $\\delta_{\\mathrm{i}\\mathrm{n}}(t)$, a withdraw penalty $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)$, or a re-activation penalty $\\delta_{\\mathrm{r}\\mathrm{e}}$ will be incurred, as stated in Section~\\ref{ssec:cyberinsurance} and Section~\\ref{ssec:bonusmalus}.\n\n\\item \\textbf{Operation Stage.} The random cyber loss events and the corresponding cyber losses suffered by the insured in this year, denoted by $W_t$, is realised in this stage according to the model described in Section~\\ref{ssec:lossmodel} and Section~\\ref{ssec:mitigation}. The aggregate cyber loss incurred is given by $L(d_t,W_t)$.\n\n\\item \\textbf{Claim Stage.} If the insurance contract is active, i.e., $\\iota_t=1$, the insured decides whether or not to make a claim, denoted by $j_t\\in\\{0,1\\}$. In the case where a claim is made, i.e., $j_t=1$, the insured receives a compensation from the insurer according to the aggregate cyber loss that year, given by $\\lambda^{\\mathcal{B}\\mathcal{M}}(b_{t-1},t,L(d_t,W_t))$. Subsequently, the cyber risk insurance contract state $i_t$ and the Bonus-Malus level $b_t$ are updated based on the decisions $\\iota_t$ and $j_t$, as described in Section~\\ref{ssec:bonusmalus}. \nConcretely, if $\\iota_t=1$, then the insurance contract state will be ``$\\mathrm{o}\\mathrm{n}$'' and the Bonus-Malus level will be updated according to the function $\\mathcal{B}\\mathcal{M}(\\cdot,\\,\\cdot)$, i.e., $i_t=\\mathrm{o}\\mathrm{n}$,  $b_t=\\mathcal{B}\\mathcal{M}\\big(b_{t-1},j_t\\lambda^{\\mathcal{B}\\mathcal{M}}(b_{t-1},t,L(d_t,W_t))\\big)$.\nIf $\\iota_t=0$, then the insurance contract state and the Bonus-Malus level will be updated according to the function $\\mathcal{B}\\mathcal{M}_0(\\cdot,\\,\\cdot)$, i.e., $(b_t,i_t)=\\mathcal{B}\\mathcal{M}_0(b_{t-1},i_{t-1})$.\n\\end{enumerate}\n\n\\begin{table}\n\\begin{center}\n\\caption{Notations in the optimal cybersecurity provisioning process and stochastic optimal control}\n\\label{tab:dp-notations}\n\\vspace{0.5em}\n\\begin{tabular}{ll}\n$d_t$ & decision of self-mitigation measure adopted in year $t$ \\\\\n$\\iota_t$ & decision to activate\/withdraw\/re-activate the insurance contract in year $t$ \\\\\n$j_t$ & decision of whether to make a claim in year $t$  \\\\\n$\\Pi$ & set containing all admissible decision policies \\\\ \n$f_t(b,i,d,\\iota,j,w)$ & state transition function in year $t$ in stochastic optimal control \\\\ \n$g_t(b,i,d,\\iota,j,w)$ & cost function in year $t$ in stochastic optimal control \\\\\n$(b^\\pi_t,i^\\pi_i)_{t=0:T}$ & controlled stochastic process representing the Bonus-Malus and insurance states \\\\ \n$e^{-r}$ & discount factor in stochastic optimal control \\\\\n$V^\\pi_t$ & expected discounted future cost at year $t$ in stochastic optimal control \\\\ \n$\\mathbb{V}_0$ & the optimal value of the stochastic optimal control problem \\\\\n$\\mathcal{V}_t(b,i)$ & value function in dynamic programming \\\\\n$\\widehat{d}_t(b,i)$ & one-stage optimal decision of self-mitigation measure in year $t$ \\\\\n$\\widehat{\\iota}_t(b,i)$ & one-stage optimal decision of cyber risk insurance in year $t$ \\\\\n$\\widehat{j}_t(b,i,w)$ & one-stage optimal decision of insurance claim in year $t$ \\\\\n$P^\\star_t\\big[(b,i)\\rightarrow(b',i')\\big]$ & transition kernel of the optimally controlled process \\\\\n$\\overline{P}^\\star_t(b,i)$ & marginal state occupancy probability of the optimally controlled process \\\\\n$\\zeta^{(m)}_t(b,i,w)$ & a quantity of interest that depends on the state and the losses \\\\\n$\\overline{\\zeta}^{(m)}_t$ & expected value of a quantity of interest \\\\\n$\\overline{Z}_{\\zeta^{(m)}}$ & aggregate expected value of a quantity of interest\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\nNext, in order to formally define an optimal decision policy, let us introduce the stochastic optimal control formulation of the cybersecurity provisioning process. Please refer to Table~\\ref{tab:dp-notations} for notations used in the cybersecurity provisioning process and the stochastic optimal control formulation.\nLet\n\\begin{align}\n\\begin{split}\n\\Pi:=\\Big\\{\\pi=(d_t,\\iota_t,&j_t)_{t=1:T}:d_t:\\Omega\\to\\mathcal{D},\\;\\iota_t:\\Omega\\to\\{0,1\\} \\text{ are }\\mathcal{F}_{t-1}\\text{-measurable},\\\\\n&j_t:\\Omega\\to\\{0,1\\} \\text{ is }\\mathcal{F}_t\\text{-measurable},\\;\\{\\iota_t=0,j_t=1\\}=\\emptyset,\\text{ for }t=1,\\ldots,T\\Big\\}\n\\end{split}\n\\end{align}\ndenote the set of all admissible decision policies. The conditions in the above definition are explained as follows.\n\\begin{itemize}\n\\item The decisions $d_t$ and $\\iota_t$ are made before observing $W_t$, hence may depend on all available information up to year $t-1$.\n\\item The decision $j_t$ is made after observing $W_t$, hence may depend on all available information up to year $t$.\n\\item The condition $\\{\\iota_t=0,j_t=1\\}=\\emptyset$ requires that the insured may only make a claim (i.e., $j_t=1$) when the insurance contract is activated (i.e., $\\iota_t=1$) in year $t$. \n\\end{itemize}\nFor $t=1,\\ldots,T$, let $f_t:\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{D}\\times\\{0,1\\}\\times\\{0,1\\}\\times\\mathcal{W}\\to\\mathcal{B}\\times\\mathcal{I}$ be the state transition function for each year $t$, given by\n\\begin{align}\n\\begin{split}\nf_t(b,i,d,\\iota,j,w):=\\begin{cases}\n\\big(\\mathcal{B}\\mathcal{M}(b,j\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,w))),\\mathrm{o}\\mathrm{n}\\big) &\\text{if }\\iota=1,\\\\\n\\mathcal{B}\\mathcal{M}_0(b,i) &\\text{if }\\iota=0,\n\\end{cases}\n\\end{split}\n\\label{eqn:socstate}\n\\end{align}\nthat is, $f_t(b,i,d,\\iota,j,w)$ returns the Bonus-Malus level and the insurance state in year $t$ given that the Bonus-Malus level and the insurance state in year $t-1$ are $b$ and $i$, the decisions in year $t$ are $d$, $\\iota$, $j$, and the cyber loss events in year $t$ are $w$.\nFor $t=1,\\ldots,T$, let $g_t:\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{D}\\times\\{0,1\\}\\times\\{0,1\\}\\times\\mathcal{W}\\to\\R_+$ be the cost function for each year $t$, given by \n\\begin{align}\n\\begin{split}\ng_t(b,i,d,\\iota,j,w)&:=\\beta(d)+\\iota p^{\\mathcal{B}\\mathcal{M}}(b,t)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}\\\\\n&\\hspace{1.5cm}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}+L(d,w)-\\iota j\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,w)),\n\\end{split}\n\\label{eqn:soccost}\n\\end{align}\nwhere $\\beta(d)$ is the investment of adopting the self-mitigation measure $d$, $\\iota p^{\\mathcal{B}\\mathcal{M}}(b)$ corresponds to the cyber risk insurance premium, $\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}$ corresponds to the sign-on\/withdrawal\/re-activation costs, $L(d,w)$ is the aggregate cyber loss, and $\\iota j\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,w))$ corresponds to the compensation from the insurer. \n\nSubsequently, for any decision policy $\\pi=(d_t,\\iota_t,j_t)_{t=1:T}\\in\\Pi$, let us define the $(\\mathcal{F}_t)_{t=0:T}$-adapted controlled stochastic process $\\big(b^{\\pi}_t,i^{\\pi}_t\\big)_{t=0:T}$ as follows:\n\\begin{align}\n\\begin{split}\n(b^\\pi_0,i^\\pi_0)&:=(0,\\mathrm{n}\\mathrm{o}),\\\\\n(b^\\pi_t,i^\\pi_t)&:=f_t(b^\\pi_{t-1},i^\\pi_{t-1},d_t,\\iota_t,j_t,W_t)\\quad\\text{for }t=1,\\ldots,T.\n\\end{split}\n\\label{eqn:markovprocess}\n\\end{align}\nThen, $g_t(b^\\pi_{t-1},i^\\pi_{t-1},d_t,\\iota_t,j_t,W_t)$ corresponds to the cybersecurity cost in year $t$. For $\\pi\\in\\Pi$, and $t=1,\\ldots,T$, define $V^{\\pi}_t$ by\n\\begin{align}\nV^\\pi_t:=\\EXP\\Big[\\textstyle\\sum_{s=t+1}^Te^{-(s-t)r}g_s(b^\\pi_{s-1},i^\\pi_{s-1},d_s,\\iota_s,j_s,W_s)\\Big|\\mathcal{F}_{t}\\Big],\n\\end{align}\nwhere $0<e^{-r}\\le 1$ is the discount factor and $V^\\pi_T:=0$,\nthat is, $V^\\pi_t$ is the expected value of the discounted total future cybersecurity cost viewed at the end of year $t$.\n\nWe assume that in the optimal cybersecurity provisioning process, the objective of the insured is to minimise the expected value of the discounted total cybersecurity cost, i.e., $V^\\pi_0$, over all admissible decision policies $\\pi\\in\\Pi$. This is formulated as the following finite horizon stochastic optimal control problem:\n\\begin{align}\n\\mathbb{V}_0:=\\inf_{\\pi\\in\\Pi}V^\\pi_0=\\inf_{\\pi\\in\\Pi}\\EXP\\Big[\\textstyle\\sum_{t=1}^Te^{-tr}g_t(b^\\pi_{t-1},i^\\pi_{t-1},d_t,\\iota_t,j_t,W_t)\\Big]. \n\\label{eqn:stocoptctr}\n\\end{align}\n\n\n\\subsection{Dynamic Programming Algorithm}\n\\label{ssec:dpalgo}\nThe stochastic optimal control problem in (\\ref{eqn:stocoptctr}) can be solved efficiently via the dynamic programming algorithm. \nThe dynamic programming algorithm iteratively solves one-stage optimisation problems while going backward in time.\nIn the following, let us denote the optimal values of these one-stage optimisation problems by $(\\mathcal{V}_t)_{t=0:T}$, and denote their corresponding optimisers by $(\\widehat{d}_t,\\widehat{\\iota}_t,\\widehat{j}_t)_{t=1:T}$.\nFor $t=T,T-1,\\ldots,0$, let $\\mathcal{V}_{t}:\\mathcal{B}\\times\\mathcal{I}\\to\\R_+$ be recursively defined as follows: for every $b\\in\\mathcal{B},i\\in\\mathcal{I}$, let\n\\begin{align}\n\\begin{split}\n\\mathcal{V}_T(b,i)&:=0,\\\\\n\\mathcal{V}_{t-1}(b,i)&:=e^{-r}\\min_{d\\in\\mathcal{D},\\,\\iota\\in\\{0,1\\}}\\bigg\\{\\EXP\\bigg[\\min_{j\\in\\{0,1\\}}\\Big\\{g_{t}(b,i,d,\\iota,j,W)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,\\iota,j,W)\\big)\\Big\\}\\bigg]\\bigg\\}\\\\\n&\\hspace{280pt}\\text{for }t=T,T-1,\\ldots,1.\n\\end{split}\n\\label{eqn:dpvalue}\n\\end{align}\nMoreover, for $t=1,\\ldots,T$, let $\\widehat{d}_t:\\mathcal{B}\\times\\mathcal{I}\\to\\mathcal{D}$, $\\widehat{\\iota}_t:\\mathcal{B}\\times\\mathcal{I}\\to\\{0,1\\}$, and $\\widehat{j}_t:\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{W}\\to\\{0,1\\}$ be defined as follows: for every $b\\in\\mathcal{B},i\\in\\mathcal{I}$, let\n\\begin{align}\n\\big(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i)\\big)&\\in \\argmin_{d\\in\\mathcal{D},\\,\\iota\\in\\{0,1\\}}\\bigg\\{\\EXP\\bigg[\\min_{j\\in\\{0,1\\}}\\Big\\{g_{t}(b,i,d,\\iota,j,W)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,\\iota,j,W)\\big)\\Big\\}\\bigg]\\bigg\\},\\label{eqn:dpconcept-d-iota}\\\\\n\\widehat{j}_t(b,i,w)&:=\\begin{cases}\n1 & \\text{if }\\widehat{\\iota}_t(b,i)=1,\\;\\Big[g_{t}(b,i,\\widehat{d}_t(b,i),1,1,w)+\\mathcal{V}_{t}\\big(f_{t}(b,i,\\widehat{d}_t(b,i),1,1,w)\\big)\\Big] \\\\\n&\\hspace{2.5cm}< \\Big[g_{t}(b,i,\\widehat{d}_t(b,i),1,0,w)+\\mathcal{V}_{t}\\big(f_{t}(b,i,\\widehat{d}_t(b,i),1,0,w)\\big)\\Big],\\\\\n 0 & \\text{otherwise}.\n\\end{cases}\n\\label{eqn:dpconcept-j}\n\\end{align}\nThus, for $t=0,\\ldots,T$, $\\mathcal{V}_t(b,i)$ corresponds to the optimal total discounted future cybersecurity cost when viewing the optimal cybersecurity provisioning process at the end of year $t$ when the Bonus-Malus level and the insurance state are $b$ and $i$. \nObserve from (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}) that $\\big(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i)\\big)$ solves the minimisation outside the expectation in (\\ref{eqn:dpvalue}), while $\\widehat{j}_t(b,i,W)$ solves the minimisation inside the expectation in (\\ref{eqn:dpvalue}) with $(d,i)\\leftarrow\\big(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i)\\big)$.\nSince the minimisation over $j\\in\\{0,1\\}$ in (\\ref{eqn:dpvalue}) is inside the expectation, the decision $j$ is made after observing the realisation of the cyber loss events $W$, and thus the optimiser $\\widehat{j}_t$ is allowed to depend on $W$. On the other hand, since the minimisation over $d\\in\\mathcal{D}$, $\\iota\\in\\{0,1\\}$ is outside the expectation, these decisions need to be made before observing $W$, and thus $\\widehat{d}_t$ and $\\widehat{\\iota}_t$ cannot depend on $W$.\nIn the first iteration of the dynamic programming algorithm, one computes $\\big(\\widehat{d}_T(b,i),\\widehat{\\iota}_T(b,i),\\widehat{j}_T(b,i,w)\\big)$ from (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}) for every $(b,i,w)\\in\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{W}$. Moreover, one computes the values of $(\\mathcal{V}_{T-1}(b,i))_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}$ from (\\ref{eqn:dpvalue}).\nIn subsequent iterations of the dynamic programming algorithm, one iterates through $t=T-1,T-2,\\ldots,1$, where, in each iteration, one considers the one-stage optimisation problem in year $t$ given the values $(\\mathcal{V}_t(b,i))_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}$.\nSpecifically, one computes $\\big(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i),\\widehat{j}_t(b,i,w)\\big)$ from (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}) for every $(b,i,w)\\in\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{W}$, and computes the values of $(\\mathcal{V}_{t-1}(b,i))_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}$ from (\\ref{eqn:dpvalue}).\nThe following theorem shows how one can construct an optimal decision policy $\\pi^\\star$ for the stochastic optimal control problem (\\ref{eqn:stocoptctr}) by piecing together the one-stage optimisers $\\big(\\widehat{d}_t,\\widehat{\\iota}_t,\\widehat{j}_t\\big)_{t=1:T}$ obtained via dynamic programming.\n\n\n\\begin{theorem}\nLet the functions $(\\mathcal{V}_t)_{t=0:T}$ be defined as in (\\ref{eqn:dpvalue}), let the functions $(\\widehat{d}_t,\\widehat{\\iota}_t,\\widehat{j}_t)_{t=1:T}$ be defined as in (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}), and let $\\pi^\\star=(d^\\star_t,\\iota^\\star_t,j^\\star_t)_{t=1:T}\\in\\Pi$ be recursively defined as follows: \n\\begin{align}\n\\begin{split}\n&\\hspace{-0.75cm}\\big(b^{\\pi^\\star}_{0},i^{\\pi^\\star}_{0}\\big):=(0,\\mathrm{n}\\mathrm{o}),\\\\\n\\text{for }t&=1,\\ldots,T,\\text{ let:}\\\\\n&d^\\star_t:=\\widehat{d}_t\\big(b_{t-1}^{\\pi^\\star},i_{t-1}^{\\pi^\\star}\\big),\\quad\\quad \\iota^\\star_t:=\\widehat{\\iota}_t\\big(b_{t-1}^{\\pi^\\star},i_{t-1}^{\\pi^\\star}\\big),\\quad\\quad j^\\star_t:=\\widehat{j}_t\\big(b_{t-1}^{\\pi^\\star},i_{t-1}^{\\pi^\\star},W_t\\big).\n\\end{split}\n\\label{eqn:dpconcept-opt-process}\n\\end{align}\nThen, the following holds:\n\\begin{align}\n\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})=V_0^{\\pi^\\star}=\\mathbb{V}_0.\n\\end{align}\n\\label{thm:dp}\n\\end{theorem}\n\\begin{proof}\nSee Appendix~\\ref{apx:proofs}. \n\\end{proof}\n\nBesides an optimal decision policy $\\pi^\\star$ constructed in Theorem~\\ref{thm:dp}, we are often also interested in other quantities related to the optimal cybersecurity provisioning process, such as the transition kernels and the marginal state occupancy probabilities of the process $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ as well as other quantities of interest.\nSpecifically, given an optimal decision policy $\\pi^\\star$ constructed in Theorem~\\ref{thm:dp}, its associated optimally controlled process $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ is a discrete-time time-inhomogeneous Markov chain, and we are interested in its transition kernels $\\big(P^\\star_t\\big)_{t=1:T}$, where\n\\begin{align}\n\\begin{split}\nP^\\star_t\\big[(b,i)\\rightarrow(b',i')\\big]&:=\\PROB\\Big[\\big(b^{\\pi^\\star}_{t},i^{\\pi^\\star}_{t}\\big)=(b',i')\\Big|\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1}\\big)=(b,i)\\Big] \\\\\n&\\hspace{160pt}\\forall (b,i),(b',i')\\in\\mathcal{B}\\times\\mathcal{I},\\;\\text{for }t=1,\\ldots,T,\n\\end{split}\n\\label{eqn:dpalgo-kernel}\n\\end{align}\nas well as its marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$, where\n\\begin{align}\n\\overline{P}^\\star_t(b,i):=\\PROB\\Big[\\big(b^{\\pi^\\star}_{t},i^{\\pi^\\star}_{t}\\big)=(b,i)\\Big]\\qquad\\forall (b,i)\\in\\mathcal{B}\\times\\mathcal{I},\\;\\text{for }t=0,\\ldots,T.\n\\label{eqn:dpalgo-marg}\n\\end{align}\nUsing these probabilities, we are able to compute important quantities that will aid us in the design and evaluation of cyber risk insurance contracts, such as the expected number of years the insured will spend in each of the Bonus-Malus levels.\nMoreover, we are often interested in computing the expected values of other quantities of interest which can be used as evaluation metrics for the cyber risk insurance contract.\nFormally, suppose that we are interested in $M$ quantities, and let $\\zeta^{(m)}_t:\\mathcal{B}\\times\\mathcal{I}\\times\\R_+\\to\\R$ for $t=1,\\ldots,T$, $m=1,\\ldots,M$ satisfy\n\\begin{align}\n\\EXP\\big[\\big|\\zeta^{(m)}_t(b,i,W)\\big|\\big]<\\infty\\qquad \\forall (b,i)\\in\\mathcal{B}\\times\\mathcal{I},\\;\\text{for }t=1,\\ldots,T,\\;\\text{for }m=1,\\ldots,M.\n\\label{eqn:qoi}\n\\end{align}\nLet us define\n\\begin{align}\n\\overline{\\zeta}^{(m)}_t&:=\\EXP\\big[\\zeta^{(m)}_t\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1},W_t\\big)\\big]\\qquad\\text{for }t=1,\\ldots,T,\\;\\text{for }m=1,\\ldots,M,\\label{eqn:qoi-expval}\\\\\n\\overline{Z}_{\\zeta^{(m)}}&:=\\sum_{t=1}^T\\overline{\\zeta}_t^{(m)}\\hspace{139pt}\\qquad\\text{for }m=1,\\ldots,M.\\label{eqn:qoi-aggregate}\n\\end{align}\nThus, for $m=1,\\ldots,M$, $\\zeta^{(m)}_t\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1},W_t\\big)$ represents a quantity of interest in year $t$ that (possibly) depends on the Bonus-Malus level and the insurance state at the end of year $t-1$, i.e., $b^{\\pi^\\star}_{t-1}$ and $i^{\\pi^\\star}_{t-1}$, and the losses in year $t$, i.e., $W_t$. \nMoreover, for $m=1,\\ldots,M$, $\\overline{\\zeta}^{(m)}_t$ denotes the expected value of the quantity of interest $\\zeta^{(m)}_t\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1},W_t\\big)$ in year $t$, and $\\overline{Z}_{\\zeta^{(m)}}$ denotes the aggregate expected value of this quantity of interest over $T$ years. \nDue to the independence between $\\mathcal{F}_{t-1}$ and $\\sigma(W_t)$, one observes that $(\\overline{\\zeta}^{(m)}_t)_{t=1:T}$ and hence also $\\overline{Z}_{\\zeta^{(m)}}$ can be computed based on the marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ defined in (\\ref{eqn:dpalgo-marg}). Example~\\ref{exp:quantities} below provides concrete examples of quantities of interest that can aid the design and analysis of cyber risk insurance contracts.\n\n\\begin{example}[Quantities of interest]\nIn this example, we let $M=D+5$, and defined the following quantities of interest. \n\\begin{enumerate}[label=(\\roman*)]\n\\item \\label{sexp:quantities-mitiadoption}For every $d\\in\\mathcal{D}:=\\{0,1,\\ldots,D\\}$, let $\\zeta^{(d+1)}_{t}(b,i,w)=\\INDI_{\\{\\widehat{d}_t(b,i)=d\\}}$. Then, we have\n$\\overline{\\zeta}^{(d+1)}_{t}=$ \\linebreak$\\PROB\\big[\\widehat{d}_t\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1}\\big)=d\\big]$, which corresponds to the probability that the self-mitigation measure $d$ is adopted in year $t$ under the decision policy $\\pi^\\star$.\n\\item Let $\\zeta^{(D+2)}_{t}(b,i,w)=e^{-tr}\\beta(\\widehat{d}_t(b,i))$. Then, $\\overline{Z}_{\\zeta^{(D+2)}}$ corresponds to the expected value of the discounted total self-mitigation expenses. \n\\item \\label{sexp:quantities-premium}Let $\\zeta^{(D+3)}_{t}(b,i,w)=e^{-tr}\\big(\\widehat{\\iota}_t(b,i)\\big(p^{\\mathcal{B}\\mathcal{M}}(b)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o}\\}}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{n}\\mathrm{o},i\\ne\\mathrm{o}\\mathrm{n}\\}}\\big)+(1-\\widehat{\\iota}_t(b,i))\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n}\\}}\\big)$.\nThen, $\\overline{Z}_{\\zeta^{(D+3)}}$ corresponds to the expected value of the discounted total payment from the insured to the insurer. \n\\item \\label{sexp:quantities-mitieffect}Let $\\zeta^{(D+4)}_{t}\\big(b,i,w=(n,x_1,\\ldots,x_n)\\big)=e^{-tr}\\left(\\sum_{k=1}^{n}x_k-L(\\widehat{d}_t(b,i),w)\\right)$. Then, $\\overline{Z}_{\\zeta^{(D+4)}}$ corresponds to the expected value of the discounted total loss that is prevented by the self-mitigation measures. \n\\item \\label{sexp:quantities-benefit}Let $\\zeta^{(D+5)}_{t}(b,i,w)=e^{-tr}\\Big(\\widehat{j}_t(b,i,w)\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(\\widehat{d}_t(b,i),w))\\Big)$, then $\\overline{Z}_{\\zeta^{(D+5)}}$ corresponds to the expected value of the discounted total insurance compensation the insured receives. \n\\end{enumerate}\n\\label{exp:quantities} \n\\end{example}\n\nAs a consequence of Theorem~\\ref{thm:dp}, let us now introduce a concrete algorithm based on the dynamic programming principle to solve the stochastic optimal control problem (\\ref{eqn:stocoptctr}), which is presented in Algorithm~\\ref{alg:dpconcrete}. \nIn addition to an optimal decision policy $\\pi^\\star$, Algorithm~\\ref{alg:dpconcrete} also computes the state transition kernels $\\big(P^\\star_t\\big)_{t=1:T}$ and the marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ of the optimally controlled process $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$, defined in (\\ref{eqn:dpalgo-kernel}) and (\\ref{eqn:dpalgo-marg}).\nMoreover, given $M$ quantities of interest $\\big(\\zeta^{(m)}(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$ as input, Algorithm~\\ref{alg:dpconcrete} also computes their expected values defined in (\\ref{eqn:qoi-expval}) as well as their aggregate expected values defined in (\\ref{eqn:qoi-aggregate}). \nRemark~\\ref{rmk:dpalgo} provides detailed explanations of the inputs and outputs of Algorithm~\\ref{alg:dpconcrete} as well as explanations of some of the lines in Algorithm~\\ref{alg:dpconcrete}. \nTheorem~\\ref{thm:dpalgo} shows the correctness of Algorithm~\\ref{alg:dpconcrete}. \nMoreover, Remark~\\ref{rmk:comp} discusses the computational tractability of Algorithm~\\ref{alg:dpconcrete}.\n\n\n\n\n\n\\begin{algorithm}[p]\n\\KwIn{$\\mathcal{B}$, $\\mathcal{I}$, $\\mathcal{D}$, $\\mathcal{B}\\mathcal{M}$, $\\mathcal{B}\\mathcal{M}_{0}$, $p^{\\mathcal{B}\\mathcal{M}}$, $l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}$, $l^{\\mathcal{B}\\mathcal{M}}_{\\max}$, $\\beta(\\cdot)$, $\\gamma(\\cdot)$, $\\delta_{\\mathrm{i}\\mathrm{n}}(\\cdot)$, $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(\\cdot)$, $\\delta_{\\mathrm{r}\\mathrm{e}}$, $r$, $\\big(\\zeta^{(m)}_t(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$}\n\\KwOut{$\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})$, $\\pi^{\\star}$, $\\big(P^\\star_t\\big)_{t=1:T}$, $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$, $\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$, $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$}\n\\nl $\\mathcal{V}_T(b,i)\\leftarrow0$ for all $b\\in\\mathcal{B},i\\in\\mathcal{I}$. \\label{alglin:dp-value-init} \\\\\n\\nl \\For{$t=T,T-1,\\ldots,1$}{\n\\nl \\label{alglin:dp-forloop-b}\\For{$b\\in\\mathcal{B}$}{\n\\nl $\\underline{b}\\leftarrow\\mathcal{B}\\mathcal{M}(b,0)$, $\\overline{b}\\leftarrow\\max\\{\\mathcal{B}\\mathcal{M}(b,c):c\\in\\R_+\\}$. \\COMMENT{$\\underline{b}$ and $\\overline{b}$ are the lowest and highest possible Bonus-Malus levels reachable from level $b$} \\label{alglin:dp-b-b-def} \\\\\n\\nl \\For{$\\underline{b}\\le b'\\le\\overline{b}$}{\n\\nl $\\alpha_t(b,b')\\leftarrow \\mathcal{V}_{t}(b',\\mathrm{o}\\mathrm{n})-\\mathcal{V}_{t}(\\underline{b},\\mathrm{o}\\mathrm{n})$. \\COMMENT{temporary value to simplify computation} \\label{alglin:dp-alpha-def} \\\\\n\\nl $\\mathcal{L}_t(b,b')\\leftarrow\\left\\{c\\in\\R_+:\\mathcal{B}\\mathcal{M}(b,c)=b',c>\\alpha_t(b,b')\\right\\}$. \\COMMENT{the set of potential insurance compensation amounts such that the optimal decision is to make a claim and the updated Bonus-Malus level will be $b'$; see also Remark~\\ref{rmk:dpalgo}}\\label{alglin:dp-L-def} \\\\\n}\n\\nl \\For{$i\\in\\mathcal{I}$}{\n\\nl \\label{alglin:dp-forloop-d}\\For{$d\\in\\mathcal{D}$}{\n\\nl $H_t(b,i,d,1)\\leftarrow \\mathcal{V}_t(\\underline{b},\\mathrm{o}\\mathrm{n})-\\sum_{\\underline{b}\\le b'\\le\\overline{b}}\\EXP\\bigg[\\INDI_{\\{\\mathcal{B}\\mathcal{M}(b,\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W)))=b'\\}}\\Big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))-\\alpha_t(b,b')\\Big)^+\\bigg]$.\\label{alglin:dp-H-def1} \\\\\n\\nl $H_t(b,i,d,0)\\leftarrow \\mathcal{V}_t\\big(\\mathcal{B}\\mathcal{M}_0(b,i)\\big)$. \\COMMENT{$H_t(b,i,d,\\iota)$ for $\\iota\\in\\{0,1\\}$ are temporary values to simplify the one-stage optimisation problem in Line~\\ref{alglin:dp-d-iota-def}; see also Remark~\\ref{rmk:dpalgo}}\\label{alglin:dp-H-def2} \\\\\n}\n\\nl $(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i))\\leftarrow\\argmin_{d\\in\\mathcal{D},\\,\\iota\\in\\{0,1\\}} \\bigg\\{\\beta(d)+\\iota p^{\\mathcal{B}\\mathcal{M}}(b,t)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\EXP\\big[L(d,W)\\big]+H_t(b,i,d,\\iota)\\bigg\\}$.\\label{alglin:dp-d-iota-def} \\\\\n\\nl $\\widehat{j}_t(b,i,w)\\leftarrow\\INDI_{\\{\\widehat{\\iota}_t(b,i)=1\\}}\\INDI_{\\bigcup_{\\underline{b}\\le b'\\le\\overline{b}}\\mathcal{L}_t(b,b')}\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(\\widehat{d}_t(b,i),w))\\big)$.\\label{alglin:dp-j-def} \\\\\n\\nl $\\mathcal{V}_{t-1}(b,i)\\leftarrow e^{-r}\\min_{d\\in\\mathcal{D},\\,\\iota\\in\\{0,1\\}} \\bigg\\{\\beta(d)+\\iota p^{\\mathcal{B}\\mathcal{M}}(b,t)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\EXP\\big[L(d,W)\\big]+H_t(b,i,d,\\iota)\\bigg\\}$.\\label{alglin:dp-value-update} \\\\\n\\nl $P^\\star_t\\big[(b,i)\\rightarrow(b',i')\\big]\\leftarrow 0$ for all $(b',i')\\in\\mathcal{B}\\times\\mathcal{I}$. \\COMMENT{initialise all transition probabilities to 0} \\\\\n\\nl \\If{$\\widehat{\\iota}_t(b,i)=1$}{\n\\nl \\For{$\\underline{b}<b'\\le\\overline{b}$}{\n\\nl $P^\\star_t\\big[(b,i)\\rightarrow(b',\\mathrm{o}\\mathrm{n})\\big]\\leftarrow {\\PROB\\Big[\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(\\widehat{d}_t(b,i),W))\\in\\mathcal{L}_t(b,b')\\Big]}$. \\COMMENT{compute the transition probability to level $b'$}\\label{alglin:dp-kern1} \\\\\n}\n\\nl $P^\\star_t\\big[(b,i)\\rightarrow(\\underline{b},\\mathrm{o}\\mathrm{n})\\big]\\leftarrow 1-\\sum_{\\underline{b}<b'\\le\\overline{b}}P^\\star_t\\big[(b,i)\\rightarrow(b',\\mathrm{o}\\mathrm{n})\\big]$. \\label{alglin:dp-kern2} \\COMMENT{compute the transition probability to the lowest level}\\\\\n}\\nl \\Else{\n\\nl $P^\\star_t\\big[(b,i)\\rightarrow\\mathcal{B}\\mathcal{M}_0(b,i)\\big]\\leftarrow 1$. \\COMMENT{when the optimal decision is to not purchase insurance, the transition is deterministic} \\label{alglin:dp-kern3} \\\\\n}\n}\n}\n}\n\\nl $\\overline{P}_{0}^{\\star}(0,\\mathrm{n}\\mathrm{o})\\leftarrow 1$, $\\overline{P}_{0}^{\\star}(b,i)\\leftarrow 0$ for all $(b,i)\\ne(0,\\mathrm{n}\\mathrm{o})$. \\label{alglin:dp-marg-init} \\\\\n\\nl \\For{$t=1,2,\\ldots,T$}{\n\\nl For all $(b,i)\\in\\mathcal{B}\\times\\mathcal{I}$, $\\overline{P}_{t}^{\\star}(b,i)\\leftarrow\\sum_{(b',i')\\in\\mathcal{B}\\times\\mathcal{I}}P^\\star_t[(b',i')\\rightarrow(b,i)]\\overline{P}_{t-1}^{\\star}(b',i')$. \\label{alglin:dp-marg-iter} \\\\\n\\nl For $m=1,\\ldots,M$, $\\overline{\\zeta}^{(m)}_t\\leftarrow\\sum_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}\\EXP[\\zeta^{(m)}_t(b,i,W)]\\overline{P}_{t-1}^{\\star}(b,i)$. \\label{alglin:dp-qoi} \\\\\n}\n\\nl For $m=1,\\ldots,M$, $\\overline{Z}_{\\zeta^{(m)}}\\leftarrow\\sum_{t=1}^T\\overline{\\zeta}_t^{(m)}$. \\label{alglin:dp-qoi-aggregate} \\\\\n\\nl Define $\\pi^\\star=(d^\\star_t,\\iota^\\star_t,j^\\star_t)_{t=1:T}$ by (\\ref{eqn:dpconcept-opt-process}).\\\\\n\\nl \\Return $\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})$, $\\pi^{\\star}$, $\\big(P^\\star_t\\big)_{t=1:T}$, $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$, $\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$, $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$. \n    \\caption{{\\bf Dynamic Programming for Optimal Cybersecurity Provisioning}}\n    \\label{alg:dpconcrete}\n\\end{algorithm}\n\n\\begin{remark}[Details about Algorithm~\\ref{alg:dpconcrete}]\nThe inputs of Algorithm~\\ref{alg:dpconcrete} are explained as follows.\n\\begin{itemize}\n\\item $\\mathcal{B}$, $\\mathcal{I}$, $\\mathcal{D}$, $\\mathcal{B}\\mathcal{M}$, $\\mathcal{B}\\mathcal{M}_{0}$, $p^{\\mathcal{B}\\mathcal{M}}$, $l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}$, $l^{\\mathcal{B}\\mathcal{M}}_{\\max}$, $\\beta(\\cdot)$, $\\gamma(\\cdot)$, $\\delta_{\\mathrm{i}\\mathrm{n}}(\\cdot)$, $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(\\cdot)$, $\\delta_{\\mathrm{r}\\mathrm{e}}$ are specified in Section~\\ref{ssec:mitigation}, Section~\\ref{ssec:cyberinsurance}, and Section~\\ref{ssec:bonusmalus}.\n\\item $r\\ge0$ specifies the discount factor $e^{-r}$ in the objective of the optimal cybersecurity provisioning process.\n\\item $\\big(\\zeta^{(m)}_t(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$ specify $M$ quantities of interest that satisfy (\\ref{eqn:qoi}) whose expected values (\\ref{eqn:qoi-expval}) and aggregate expected values (\\ref{eqn:qoi-aggregate}) will be computed in Algorithm~\\ref{alg:dpconcrete}. Some examples of quantities of interest are shown in Example~\\ref{exp:quantities}. \n\\end{itemize}\nThe outputs of Algorithm~\\ref{alg:dpconcrete} are explained as follows.\n\\begin{itemize}\n\\item $\\pi^{\\star}$ is the optimal decision policy computed by the dynamic programming algorithm and $\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})$ is equal to the optimal value $\\mathbb{V}_0$ of the stochastic optimal control problem. The optimality of $\\pi^\\star$ is shown in Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo1}. \n\\item $\\big(P^\\star_t\\big)_{t=1:T}$ are the transition kernels of the discrete-time time-inhomogeneous Markov chain \\linebreak$\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ that are defined in (\\ref{eqn:dpalgo-kernel}). $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ are the marginal state occupancy probabilities of $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ defined in (\\ref{eqn:dpalgo-marg}). The correctness of these outputs is shown in Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo2}. \n\\item $\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$ are the expected values of the quantities of interest $\\big(\\zeta^{(m)}_t(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$ defined in (\\ref{eqn:qoi-expval}). $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$ are the aggregate expected values of these quantities of interest defined in (\\ref{eqn:qoi-aggregate}). The correctness of these outputs is shown in Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo3}. \n\\end{itemize}\nBelow is a list explaining some of the lines in Algorithm~\\ref{alg:dpconcrete}. \n\\begin{itemize}\n\\item Line~\\ref{alglin:dp-b-b-def} computes the lowest possible Bonus-Malus level $\\underline{b}$ and the highest possible Bonus-Malus level $\\overline{b}$ that the insured can transition to from level $b$. We only iterate through the Bonus-Malus levels between $\\underline{b}$ and $\\overline{b}$ in the remainder of the for-loop starting from Line~\\ref{alglin:dp-forloop-b}.\n\\item Line~\\ref{alglin:dp-L-def} computes a set $\\mathcal{L}_t(b,b')\\subseteq\\R_+$ which contains all potential insurance compensation amounts (i.e., $\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))$) such that the optimal decision of the insured is to make a claim, and that the updated Bonus-Malus level after making the claim is $b'$. \nIn particular, the following claim holds true.\n\\begin{enumerate}[label=\\underline{Claim~\\arabic*}:,leftmargin=3.8em]\n\\item for all $(b,i)\\in\\mathcal{B}\\times\\mathcal{I}$, $d\\in\\mathcal{D}$, the following equivalence holds:\n\\begin{align*}\n\\begin{split}\ng_{t}(b,i,d,1,1,w)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,1,1,w)\\big)&< g_{t}(b,i,d,1,0,w)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,1,0,w)\\big)\\\\\n&\\Updownarrow\\\\\n\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,w))&\\in\\bigcup_{\\underline{b}\\le b'\\le\\overline{b}}\\mathcal{L}_t(b,b').\n\\end{split}\n\\end{align*}\n\\end{enumerate}\nThis claim is justified in (\\ref{eqn:dpalgo-claim1-proof1}) and (\\ref{eqn:dpalgo-claim1-proof2}) in the proof of Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo1}. \nMoreover, notice that $\\mathcal{L}_t(b,b')$ is an interval since the set $\\left\\{c\\in\\R_+:\\mathcal{B}\\mathcal{M}(b,c)=b'\\right\\}$ is an interval in $\\R_+$ by the assumption that the function $\\mathcal{B}\\mathcal{M}(b,\\cdot)$ is non-decreasing.\n\\item In the for-loop starting from Line~\\ref{alglin:dp-forloop-d}, we compute temporary values $(H_t(b,i,d,\\iota))_{d\\in\\mathcal{D},\\iota\\in\\{0,1\\}}$ which are subsequently used in Line~\\ref{alglin:dp-d-iota-def} and Line~\\ref{alglin:dp-value-update}. In fact, the following claim holds true.\n\\begin{enumerate}[label=\\underline{Claim~\\arabic*}:,leftmargin=3.8em]\n\\setcounter{enumi}{1}\n\\item for all $(b,i)\\in\\mathcal{B}\\times\\mathcal{I}$, $d\\in\\mathcal{D}$, $\\iota\\in\\{0,1\\}$, it holds that\n\\begin{align*}\n\\begin{split}\n&\\phantom{=}\\;\\;\\EXP\\bigg[\\min_{j\\in\\{0,1\\}}\\Big\\{g_{t}(b,i,d,\\iota,j,W)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,\\iota,j,W)\\big)\\Big\\}\\bigg]\\\\\n&=\\beta(d)+\\iota p^{\\mathcal{B}\\mathcal{M}}(b,t)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}\\\\\n&\\qquad+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\EXP\\big[L(d,W)\\big]+H_t(b,i,d,\\iota).\n\\end{split}\n\\end{align*}\n\\end{enumerate}\nThis claim is justified in (\\ref{eqn:dpalgo-claim2-proof1}) and (\\ref{eqn:dpalgo-claim2-proof2}) in the proof of Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo1}. \n\\item The main step of the dynamic programming algorithm are in Line~\\ref{alglin:dp-value-init}, Line~\\ref{alglin:dp-d-iota-def}, Line~\\ref{alglin:dp-j-def}, and Line~\\ref{alglin:dp-value-update}. In Line~\\ref{alglin:dp-value-init}, the values of $(\\mathcal{V}_T(b,i))_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}$ are all initialised to 0 according to (\\ref{eqn:dpvalue}). \nIn Line~\\ref{alglin:dp-d-iota-def} and Line~\\ref{alglin:dp-j-def}, the one-stage optimisers $(\\widehat{d}_t,\\widehat{\\iota}_t,\\widehat{j}_t)$ are computed as in (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}), due to \\underline{Claim~1} and \\underline{Claim~2} above. \nIn Line~\\ref{alglin:dp-value-update}, the value of $\\mathcal{V}_t(b,i)$ is computed as in (\\ref{eqn:dpvalue}), due to \\underline{Claim~2} above.\n\\end{itemize}\n\\label{rmk:dpalgo}\n\\end{remark}\n\n\\begin{theorem}\nLet us assume that all inputs of Algorithm~\\ref{alg:dpconcrete} are set according to Remark~\\ref{rmk:dpalgo}, and let $\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})$, $\\pi^{\\star}$, $\\big(P^\\star_t\\big)_{t=1:T}$, $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$, $\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$, and $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$ be the outputs of Algorithm~\\ref{alg:dpconcrete}. \nThen, the following statements hold.\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\label{sthm:dpalgo1}$\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})=\\mathbb{V}_0$ and $\\pi^\\star$ is an optimal decision policy for (\\ref{eqn:stocoptctr}), i.e., $\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})=V^{\\pi^\\star}_0=\\mathbb{V}_0$. \n\n\\item \\label{sthm:dpalgo2}$\\big(P^\\star_t\\big)_{t=1:T}$ are equal to the transition kernels of $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ defined in (\\ref{eqn:dpalgo-kernel}), and $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ are equal to the marginal state occupancy probabilities of $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ defined in (\\ref{eqn:dpalgo-marg}).\n\n\\item \\label{sthm:dpalgo3}$\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$ are equal to the expected values of the quantities of interest $\\big(\\zeta^{(m)}_t(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$ defined in (\\ref{eqn:qoi-expval}), and $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$ are equal to the aggregate expected values of these quantities of interest defined in (\\ref{eqn:qoi-aggregate}).\n\n\\end{enumerate}\n\\label{thm:dpalgo}\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix~\\ref{apx:proofs}. \n\\end{proof}\n\n\\begin{remark}\nAssume that the following quantities either admit an analytically tractable expression, or can be efficiently approximated to high numerical precision:\n\\begin{enumerate}[label=(\\roman*)]\n\\item the expectation $\\EXP\\big[L(d,W)\\big]=\\EXP[N]\\EXP\\big[(X-d)^+\\big]$, where $d\\in\\mathcal{D}$; \\label{srmk:comp1}\n\\item the expectation $\\EXP\\!\\left[\\INDI_{I}(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W)))\\Big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))-\\alpha\\Big)^+\\right]$, where $b\\in\\mathcal{B}$, $d\\in\\mathcal{D}$, $t\\in\\{1,\\ldots,T\\}$, $\\alpha\\ge0$, and $I\\subset\\R_+$ is an interval; \\label{srmk:comp2}\n\\item the probability $\\PROB\\left[\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))\\in I\\right]$, where $b\\in\\mathcal{B}$, $d\\in\\mathcal{D}$, $t\\in\\{1,\\ldots,T\\}$, and $I\\subset\\R_+$ is an interval; \\label{srmk:comp3}\n\\item the expectation $\\EXP\\big[\\zeta^{(m)}_t(b,i,W)\\big]$, where $b\\in\\mathcal{B}$, $i\\in\\mathcal{I}$, $m\\in\\{1,\\ldots,M\\}$, $t\\in\\{1,\\ldots,T\\}$. \\label{srmk:comp4}\n\\end{enumerate}\nThen, Algorithm~\\ref{alg:dpconcrete} is computationally tractable, meaning that quantities in Algorithm~\\ref{alg:dpconcrete} can either be computed exactly or efficiently approximated to high numerical precision. \nIn addition, since $\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))$ is bounded above by $l^{\\mathcal{B}\\mathcal{M}}_{\\max}(b,t)$, we can assume without loss of generality that the interval $I$ in \\ref{srmk:comp2} and \\ref{srmk:comp3} above is bounded. A concrete model in which Algorithm~\\ref{alg:dpconcrete} is computationally tractable will be introduced in Section~\\ref{sec:g-and-h}. \n\\label{rmk:comp}\n\\end{remark}\n\n\n\n\\subsection{Discussion About the Pricing of the Insurance Premium}\n\\label{ssec:pricing}\nOne important consideration of the cyber risk insurer is the choice of the annual premium. Conventionally, for a fixed self-mitigation measure $d\\in\\mathcal{D}$, a fixed deductible $l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}\\ge 0$, and a fixed maximum compensation $l_{\\max}\\ge0$, one may consider the risk premium $\\EXP[(L(d,W)-l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}})^+\\wedge l_{\\max}]$. \nHowever, these quantities are variable in our dynamic model with the Bonus-Malus system.\nMoreover, the insured may choose to withdraw from the cyber risk insurance contract, thus further complicating the matter.\n\nEven though the optimal cybersecurity provisioning process minimises the total cybersecurity cost from the insured's point of view, the insurer can use the quantities of interest computed from this process to evaluate the insurance contract and compare different contract design options.\nOne of the evaluation criteria is how high the insurer can price the premium while retaining their customers, that is, what is the highest premium such that the cyber risk insurance policy remains attractive to a rational insured. \nTo answer this question, one can analyse the relationship between the marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ and the premium.\nWhen the premium is high, the insured may opt not to purchase the cyber risk insurance policy, or may withdraw from the insurance contract prematurely due to a transition into a higher Bonus-Malus level. \nThis evaluation criterion is crucial for the insurer, since a high retention rate would allow the insurer to build a large pool of customers with similar risk exposures.\n\nAnother criterion is the range of premium under which a rational insured would choose to adopt a self-mitigation measure. \nThe reasoning here is that if the insurance premium is set too low, then the insured will see cyber risk insurance as a low-cost alternative to self-mitigation measures and will thus choose not to adopt any self-mitigation measure.  \nThis range can be determined by studying the relationship between the values of\n$\\big(\\overline{\\zeta}^{(d+1)}_t\\big)_{t=1:T,d\\in\\mathcal{D}}$ in Example~\\ref{exp:quantities}\\ref{sexp:quantities-mitiadoption} and the premium.\nMoreover, the value of $\\overline{Z}_{\\zeta^{(D+4)}}$ in Example~\\ref{exp:quantities}\\ref{sexp:quantities-mitieffect} reflects the discounted total amount of loss prevented by the adoption of self-mitigation measures, and its relationship with the premium can also be studied.\nThese are important criteria due to the positive externalities generated by organisations adopting self-mitigation measures against cyber threats.\nCombining the criterion of loss prevention with the criterion of insurance customer retention, one can study whether the adoption of cyber risk insurance disincentivises the adoption of self-mitigation measures, which is an issue known as moral hazard.\n\n\nAn important question related to the design of the insurance contract is how much the insurer can financially gain from offering the cyber risk insurance policy. One simple criterion is to look at the discounted expectation of the difference between the total payment from the insured to the insurer and the total insurance compensation. Specifically, let $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}\\equiv\\overline{Z}_{\\zeta^{(D+3)}}$ in Example~\\ref{exp:quantities}\\ref{sexp:quantities-premium} and let $\\overline{Z}_{\\mathrm{c}\\mathrm{p}}\\equiv\\overline{Z}_{\\zeta^{(D+5)}}$ in Example~\\ref{exp:quantities}\\ref{sexp:quantities-benefit}. \nThen, one can see $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ as the insurer's expected profit from the cyber risk insurance policy.\nHowever, this approach implicitly makes the zero-sum assumption, i.e., the insurer's expected profit is the negative of the expected net payment from the insured to the insurer. \nSince we assume that the insured always has the option to not purchase cyber risk insurance, which would result in zero net payment to the insurer, the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ will always be non-positive. \nIn particular, in the case where the insurance policy is not purchased, the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ will be exactly 0.\nIn reality, there are a few important practical considerations that this zero-sum approach neglects.\nFirst, the insured and the insurer often perceive cyber risk differently. \nSpecifically, this is due to the insured having impartial or incomplete information as to the frequency and severity of the risk they face relative to an insurer who is privy to a portfolio of insured, giving them a competitive advantage in their assessments of such loss processes.\nSecond, the zero-sum approach assumes that the insured will act optimally according to the optimal cybersecurity provisioning process.\nIn reality, the insured will typically act sub-optimally due to the practical complexity of the required computation being beyond the scope and the available resources of the decision makers \\citep{targino2013optimal}.\nThird, regulatory transparency requirements will also play an important role in determining the profit margins that may arise if such products are issued.\nThe lack of these considerations indicates some of the limitations of this zero-sum approach. \nNevertheless, the relationship between the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ and the premium does shed some light on an important aspect of cyber risk insurance contract, that is, whether it is possible to make this quantity arbitrarily close to 0 by setting the premium appropriately. \nIf an insurance contract design allows such possibility, then there is a way to set the premium such that the insured becomes indifferent between purchasing and not purchasing the insurance policy up to an arbitrarily small tolerance, at which point the value of $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ would be approximately maximised. \nThen, the real profit of the insurer should become more preferable than the case where the value of $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ is far from zero.\nOn the contrary, if a design does not allow such possibility, then there is no way to set the premium such that the insured becomes approximately indifferent between purchasing and not purchasing the insurance policy, indicating that there is space for improving the contract design in order to hopefully further improve the real profit of the insurer.\nWe will discuss the aforementioned important criteria when evaluating the design of cyber risk insurance contracts with a concrete example in Section~\\ref{sec:exp}.\n\n\n\\section{Modelling Cyber Loss with Truncated g-and-h Distribution}\n\\label{sec:g-and-h}\n\nIn this section, we adopt specific distributional assumptions about the random variable $X$, which corresponds to the severity of a single cyber loss event. \nStudies such as \\cite{maillart2010heavy} and \\cite{wheatley2016extreme} have shown that the severity of cyber loss events have heavy-tailed distributions. \nIt is well-known that the g-and-h family introduced by \\cite{tukey1977exploratory} contains distributions with a wide range of skewness and kurtosis (see, e.g., Figure~3 of \\cite{dutta2006tale}), which makes it suitable for modelling Operational Risk \\citep{dutta2006tale,peters2006bayesian,cruz2015fundamental,peters2015advances}. \nIn \\cite{dutta2006tale}, the following advantages of the g-and-h distribution are discussed:\n\\begin{itemize}\n\\item it is flexible and it fits well to real data under many different circumstances, e.g., when considering losses of a company as a whole and when considering individual business lines or event types;\n\\item it produces realistic estimations of the Operational Risk capital;\n\\item it is easy to simulate random samples from.\n\\end{itemize}\nThe parameters in the g-and-h distribution can be robustly estimated based on quantiles~\\citep{xu2014robust} or L-moments~\\citep{peters2016estimating}. \nDue to these properties, we adopt the g-and-h distribution as a particular model for the severity of cyber loss events in this section. \n\nThe g-and-h distribution is a four-parameter family of distributions, given by the following definition:\n\\begin{align}\n\\begin{split}\n\\widetilde{X}\\text{ follows a g-and-h}&(\\alpha,\\varsigma,g,h)\\text{ distribution, if}\\\\\n\\widetilde{X}&=\\alpha+\\varsigma Y_{g,h}(Z),\\\\\n\\text{where }Z&\\sim\\text{Normal}(0,1),\\\\\nY_{g,h}(z)&:=\\begin{cases}\n\\frac{\\exp(gz)-1}{g}\\exp\\left(\\frac{hz^2}{2}\\right) & \\text{if }g\\ne 0,\\\\\nz\\exp\\left(\\frac{hz^2}{2}\\right) & \\text{if }g=0,\n\\end{cases}\n\\end{split}\n\\label{eqn:g-and-hdef}\n\\end{align}\nwhere $\\alpha\\in\\R$ is the location parameter, $\\varsigma>0$ is the scale parameter, $g\\in\\R$ is the skewness parameter, and $h\\ge0$ is the kurtosis parameter. \nIn this paper, we assume that the parameters $\\alpha$, $\\varsigma$, $g$, and $h$ are fixed and known.  \nBy (\\ref{eqn:g-and-hdef}), the distribution function of $\\widetilde{X}$ is given by\n\\begin{align}\n\\begin{split}\nF_{\\widetilde{X}}(x):=\\PROB[\\widetilde{X}\\le x]=\\Phi\\left(Y_{g,h}^{-1}\\left(\\tfrac{x-\\alpha}{\\varsigma}\\right)\\right),\n\\end{split}\n\\end{align}\nwhere $Y_{g,h}^{-1}$ denotes the inverse function of $Y_{g,h}$, and $\\Phi$ denotes the distribution function of the standard normal distribution. \nEven though $Y_{g,h}^{-1}$ cannot be expressed analytically, it can be efficiently evaluated using a standard root-finding procedure such as the bisection method and the Newton's method. Therefore, we treat $Y_{g,h}^{-1}$ as a tractable function. \nThe g-and-h distribution has the property that the $m$-th moment of $\\widetilde{X}$ exists when $h<\\frac{1}{m}$ (see, e.g., Appendix~D of \\cite{dutta2006tale}). Since we consider losses that are positively skewed and have finite expectation, from now on, we assume that $g>0$ and $0\\le h<1$. \n\nSince cyber losses are positive, we introduce a truncated version of the g-and-h distribution. \n\\begin{definition}[Truncated g-and-h distribution] For $\\alpha\\in\\R,\\varsigma>0,g>0,h\\in[0,1)$, the random variable $X$ has truncated g-and-h distribution with parameters $\\alpha,\\varsigma,g,h$, denoted by $X\\sim\\text{Tr-g-and-h}(\\alpha,\\varsigma,g,h)$, if $X$ has distribution function \n\\begin{align}\nF_X(x):=\\PROB[X\\le x]=\\PROB[\\widetilde{X}\\le x|\\widetilde{X}>0],\n\\label{eqn:tr-g-and-hdef}\n\\end{align}\nwhere $\\widetilde{X}\\sim\\text{g-and-h}(\\alpha,\\varsigma,g,h)$. \n\\label{def:tr-g-and-h}\n\\end{definition}\n\nThe next lemma shows some useful properties of the truncated g-and-h distribution.\n\\begin{lemma}\nSuppose that $X\\sim\\text{Tr-g-and-h}(\\alpha,\\varsigma,g,h)$ for $\\alpha\\in\\R,\\varsigma>0,g>0,h\\in[0,1)$. Then, the following statements hold.\n\\begin{enumerate}[label=(\\roman*)]\n\\item The distribution function of $X$ is given by\n\\begin{align}\n\\begin{split}\nF_{X}(x)=\\begin{cases}\n\\frac{F_{\\widetilde{X}}(x)-F_{\\widetilde{X}}(0)}{1-F_{\\widetilde{X}}(0)} & \\text{if }x>0,\\\\\n0 & \\text{if }x\\le0,\n\\end{cases}\n\\end{split}\n\\label{eqn:tr-g-and-h-df}\n\\end{align}\nwhere $F_{\\widetilde{X}}$ is defined in (\\ref{eqn:g-and-hdef}). \n\\label{slem:tr-g-and-h1}\n\\item Suppose that $U\\sim\\text{Uniform}[0,1]$, and let\n\\begin{align}\nX_U:=\\alpha+\\varsigma Y_{g,h}\\Big(\\Phi^{-1}\\Big(U+(1-U)F_{\\widetilde{X}}(0)\\Big)\\Big),\n\\label{eqn:tr-g-and-h-inv}\n\\end{align}\nthen $X_U\\sim\\text{Tr-g-and-h}(\\alpha,\\varsigma,g,h)$. \n\\label{slem:tr-g-and-h2}\n\\item For any $\\gamma\\ge0$, the expectation $\\EXP\\big[(X-\\gamma)^+\\big]$ is given by:\n\\begin{align}\n\\begin{split}\n\\EXP\\big[(X-\\gamma)^+\\big]&=\\frac{\\varsigma}{(1-F_{\\widetilde{X}}(0))g\\sqrt{1-h}}\\Bigg[\\exp\\left(\\frac{g^2}{2(1-h)}\\right)\\Phi\\left(\\left(\\frac{g}{1-h}-Y_{g,h}^{-1}\\left(\\tfrac{\\gamma-\\alpha}{\\varsigma}\\right)\\right)\\sqrt{1-h}\\right)\\\\\n&\\qquad-\\Phi\\left(-Y_{g,h}^{-1}\\left(\\tfrac{\\gamma-\\alpha}{\\varsigma}\\right)\\sqrt{1-h}\\right)\\Bigg]+\\frac{(\\alpha-\\gamma)(1-F_{\\widetilde{X}}(\\gamma))}{1-F_{\\widetilde{X}}(0)}.\n\\end{split}\n\\label{eqn:tr-g-and-h-mom}\n\\end{align}\n\\label{slem:tr-g-and-h3}\n\\end{enumerate}\n\\label{lem:tr-g-and-h}\n\\end{lemma}\n\\begin{proof}\nSee Appendix~\\ref{apx:proofs}. \n\\end{proof}\n\nLemma~\\ref{lem:tr-g-and-h}\\ref{slem:tr-g-and-h2} allows us to efficiently generate random samples from the severity distribution $F_X$, thus allowing us to approximate the distribution of quantities of interest in Example~\\ref{exp:quantities} via Monte Carlo.\nLemma~\\ref{lem:tr-g-and-h}\\ref{slem:tr-g-and-h3} shows that the assumption~\\ref{srmk:comp1} in Remark~\\ref{rmk:comp} is satisfied as long as the expected value of the frequency distribution, i.e., $\\EXP[N]$, is also tractable. \nLemma~\\ref{lem:tr-g-and-h}\\ref{slem:tr-g-and-h1} provides the distribution function of $X$ that can be used to approximate the distribution function of $L(d,W)$. Concretely, by adopting the fast Fourier transform (FFT) approach with exponential tilting (see, e.g., \\cite{embrechts2009panjer,cruz2015fundamental}), we approximate the distribution function of $L(d,W)$, denoted by $F_{L(d,W)}$, by a finitely supported discrete distribution $\\widehat{F}_{L(d,W)}(x)=\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\INDI_{\\big\\{a^{(d)}_j\\le x\\big\\}}$, where $\\big(a^{(d)}_j\\big)_{j\\in\\mathcal{A}}\\subset\\R_+$ is a finite set of atoms and $\\big(p^{(d)}_j\\big)_{j\\in\\mathcal{A}}$ are the corresponding probabilities. \nThe details of the FFT approach with exponential tilting are shown in Algorithm~\\ref{alg:fft}. \nAfter obtaining $(\\widehat{F}_{L(d,W)})_{d\\in\\mathcal{D}}$ from Algorithm~\\ref{alg:fft}, the quantities $\\EXP\\!\\left[\\INDI_{I}(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W)))\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))-\\alpha\\big)^+\\!\\right]$ and $\\PROB\\left[\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))\\in I\\right]$ in Remark~\\ref{rmk:comp} can be approximated by finite sums:\n\\begin{align*}\n\\EXP\\!\\left[\\INDI_{I}(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W)))\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))-\\alpha\\big)^+\\!\\right]&\\approx\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\!\\INDI_{I}\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}\\big(b,t,a^{(d)}_j\\big)\\big)\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}\\big(b,t,a^{(d)}_j\\big)-\\alpha\\big)^+,\\\\\n\\PROB\\!\\left[\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))\\in I\\right]&\\approx\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\!\\INDI_{I}\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}\\big(b,t,a^{(d)}_j\\big)\\big).\n\\end{align*}\nOne may increase the granularity parameter $K_{\\mathrm{g}\\mathrm{r}}$ in Algorithm~\\ref{alg:fft} to increase the precision of numerical approximation. \nConsequently, assumptions~\\ref{srmk:comp2} and \\ref{srmk:comp3} in Remark~\\ref{rmk:comp} are satisfied, and hence, Algorithm~\\ref{alg:dpconcrete} is tractable and efficient in this setting. In particular, Algorithm~\\ref{alg:fft} only needs to be executed once before executing Algorithm~\\ref{alg:dpconcrete}. \n\n\\begin{algorithm}[t]\n\\KwIn{$\\mathcal{D}$, $F_X(\\cdot)$, $\\psi_N(\\cdot)$, $\\gamma(\\cdot)$, $\\overline{l}$, $K_{\\mathrm{g}\\mathrm{r}}\\in\\N$, $\\theta>0$}\n\\KwOut{$(a^{(d)}_j,p_j^{(d)})_{j\\in\\mathcal{A},d\\in\\mathcal{D}}$, $\\widehat{F}_{L(d,W)}(x)=\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\INDI_{\\{a^{(d)}_j\\le x\\}}$ for each $d\\in\\mathcal{D}$}\n\\nl $\\varepsilon\\leftarrow(2^{K_{\\mathrm{g}\\mathrm{r}}}-1)^{-1}\\overline{l}$, $\\mathcal{A}\\leftarrow\\{0,1,\\ldots,2^{K_{\\mathrm{g}\\mathrm{r}}}-1\\}$. \\\\\n\\nl \\For{$d\\in\\mathcal{D}$}{\n\\nl $a^{(d)}_j\\leftarrow j\\epsilon$ for each $j\\in\\mathcal{A}$. \\\\\n\\nl $f^{(d)}_{j}\\leftarrow \\exp(-j\\theta)\\left[F_{X,d}(j\\varepsilon+\\frac{1}{2}\\varepsilon)-F_{X,d}(j\\varepsilon-\\frac{1}{2}\\varepsilon)\\right]$ for each $j\\in\\mathcal{A}$, where $F_{X,d}(y):=F_X\\big(y+\\gamma(d)\\big)\\INDI_{\\{y\\ge 0\\}}$. \\\\\n\\nl $\\varphi^{(d)}_j\\leftarrow\\sum_{k\\in\\mathcal{A}}\\exp(i\\pi2^{1-K_{\\mathrm{g}\\mathrm{r}}} jk)f^{(d)}_k$ for each $j\\in\\mathcal{A}$ via the FFT algorithm. \\\\\n\\nl $\\psi^{(d)}_j\\leftarrow\\psi_N(\\varphi^{(d)}_j)$ for each $j\\in\\mathcal{A}$. \\\\\n\\nl $p^{(d)}_j\\leftarrow\\exp(j\\theta)2^{-K_{\\mathrm{g}\\mathrm{r}}}\\sum_{k\\in\\mathcal{A}}\\exp(-i\\pi2^{1-K_{\\mathrm{g}\\mathrm{r}}} jk)\\psi^{(d)}_k$ for each $j\\in\\mathcal{A}$ via the inverse FFT algorithm. \\\\\n}\n\\nl \\Return $(a^{(d)}_j,p_j^{(d)})_{j\\in\\mathcal{A},d\\in\\mathcal{D}}$, $\\widehat{F}_{L(d,W)}(x)=\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\INDI_{\\{a^{(d)}_j\\le x\\}}$ for each $d\\in\\mathcal{D}$.\\\\\n    \\caption{{\\bf Fast Fourier Transform Approach with Exponential Tilting for Approximating $F_{L(d,W)}$} (see \\cite{embrechts2009panjer})}\n    \\label{alg:fft}\n\\end{algorithm}\n\n\\section{Numerical Experiments}\n\\label{sec:exp}\n\nIn Section~\\ref{sec:stocoptctr} and Section~\\ref{sec:g-and-h}, we formulated the optimal cybersecurity provisioning problem as a finite horizon stochastic optimal control problem, and developed a dynamic programming algorithm, i.e., Algorithm~\\ref{alg:dpconcrete}, to efficiently solve the problem under the assumption that the loss severity follows the truncated g-and-h distribution. Algorithm~\\ref{alg:dpconcrete} not only computes the optimal cybersecurity provisioning policy $\\pi^\\star$ for the insured, but also computes related quantities of interest, including the marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ of the optimally controlled process $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ as well as other quantities of interest such as those illustrated in Example~\\ref{exp:quantities}.\nAs discussed in Section~\\ref{ssec:pricing}, these quantities can guide the insurer when designing a suitable cyber risk insurance contract with a Bonus-Malus system. \nIn this section, we demonstrate how Algorithm~\\ref{alg:dpconcrete} aids the insurer when designing a cyber risk insurance contract and the benefits of the Bonus-Malus system by a numerical experiment.\\footnote{The code used in this work for the experiment is available on GitHub: \\url{https:\/\/github.com\/qikunxiang\/CyberInsuranceBonusMalus}} \nIn particular, we investigate two aspects of the cyber risk insurance contract with Bonus-Malus discussed in Section~\\ref{ssec:pricing}. \nThe first aspect is the issue of moral hazard, that is, whether the presence of the cyber risk insurance contract disincentivises the adoption of self-mitigation measures. \nThe second aspect is whether the Bonus-Malus system provides benefits to the insurer in terms of increased customer retention rates and expected profits.\n\n\\subsection{Experimental Settings}\n\nWe assume that all monetary quantities, including the severity of cyber loss events, the insurance premium, and the annual investment required by the self-mitigation measures are adjusted to the scale of the insured (e.g., their average annual revenue) and are unit-free. \nWe consider insurance policies that last for 20 years, that is, $T=20$. The discount factor $e^{-r}$ is fixed at 0.95. \nIn the cyber loss model, we let the frequency distribution be the Poisson distribution with rate 0.8. We set the severity distribution to be $\\text{Tr-}g\\text{-and-}h(\\alpha=0,\\varsigma=1,g=1.8,h=0.15)$, where the $g$ and $h$ parameters are set to be similar to those estimated in \\cite{dutta2006tale} from real Operational Risk data (see Table~8 of \\cite{dutta2006tale}). \nWe would like to remark that the heaviness of the tail of the loss severity distribution (i.e., the parameter $h$ in the truncated $g$-and-$h$ distribution) determines the probability of extreme risk events and is crucial in the computation of capital estimate \\citep{dutta2006tale}. Therefore, it is important that we specify a realistic value of the parameter $h$.\nIn Algorithm~\\ref{alg:fft}, we fix $\\overline{l}=10000$, $K_{\\mathrm{g}\\mathrm{r}}=20$, $\\theta=\\frac{20}{2^{K_{\\mathrm{g}\\mathrm{r}}}}=3.0518\\times10^{-4}$. \n\nFor simplicity, we consider the situation where only a single self-mitigation measure is available, that is, $D=1$. This self-mitigation measure requires an annual investment of 0.5, and has the effect of preventing 70\\% of the incoming cyber loss events and decreasing the severity of the remaining events by the 70th percentile of the severity distribution, that is, $\\beta(1)=0.5$, $\\gamma(1)=F^{-1}_X(0.7)$, where $X\\sim\\text{Tr-}g\\text{-and-}h(\\alpha=0,\\varsigma=1,g=1.8,h=0.15)$. \nWe consider the following simple cyber risk insurance policy with Bonus-Malus system. Let $\\mathcal{B}=\\{-2,-1,0,1\\}$, and let the functions $\\mathcal{B}\\mathcal{M}(b_{t-1},C_t)$ and $\\mathcal{B}\\mathcal{M}_0(b_{t-1},i_{t-1})$ be specified in Table~\\ref{tab:bm} below.\n\n\\begin{table}[h]\n\\label{tab:bm}\n\\caption{The $\\mathcal{B}\\mathcal{M}(\\cdot,\\cdot)$ and $\\mathcal{B}\\mathcal{M}_0(\\cdot,\\cdot)$ functions that represent the Bonus-Malus update rules}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n\\multicolumn{2}{|c|}{\\multirow{2}{*}{$\\mathcal{B}\\mathcal{M}(b_{t-1},C_t)$}} & \\multicolumn{2}{|c|}{$C_t$} \\\\ \n\\cline{3-4}\n\\multicolumn{2}{|c|}{} & $=0$ & $>0$ \\\\ \n\\cline{1-4}\n\\multirow{4}{*}{$b_{t-1}$} & $-2$ & $-2$ & $1$ \\\\ \n\\cline{2-4}\n & $-1$ & $-2$ & $1$ \\\\ \n\\cline{2-4}\n & $0$ & $-1$ & $1$ \\\\ \n\\cline{2-4}\n & $1$ & $0$ & $1$ \\\\ \n\\hline \n\\end{tabular}\n\\hspace{1cm}\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n\\multicolumn{2}{|c|}{\\multirow{2}{*}{$\\mathcal{B}\\mathcal{M}_0(b_{t-1},i_{t-1})$}} & \\multicolumn{2}{|c|}{$i_{t-1}$} \\\\ \n\\cline{3-4}\n\\multicolumn{2}{|c|}{} & $\\text{on}$ & $\\text{off}_1$ \\\\ \n\\cline{1-4}\n\\multirow{4}{*}{$b_{t-1}$} & $-2$ & $(-2,\\text{off}_1)$ & $(-1,\\text{off}_1)$ \\\\ \n\\cline{2-4}\n & $-1$ & $(-1,\\text{off}_1)$ & $(0,\\text{off}_1)$ \\\\ \n\\cline{2-4}\n & $0$ & $(0,\\text{off}_1)$ & $(0,\\text{off}_1)$ \\\\ \n\\cline{2-4}\n & $1$ & $(1,\\text{off}_1)$ & $(0,\\text{off}_1)$ \\\\ \n\\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\nThe above settings mean that when the contract is activated, the insured is migrated to level 1 in the subsequent policy year whenever a claim is made. When the insured does not make any claim in a policy year, their policy is migrated downwards by one level in the subsequent policy years until it reaches level $-2$. When the contract is deactivated, if the insured's policy is in level 1, it is migrated back to level 0 after one year. Otherwise, the policy is migrated upwards by one level each year until it reaches level 0. \nIn the experiment, we let the base premium $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$ be an adjustable parameter that is varied between 0 and 7 with an increment of $0.005$, and set the premium to be $60\\%,80\\%,100\\%,150\\%$ of the base premium for the Bonus-Malus levels $-2,-1,0,1$, respectively. That is, we let $p^{\\mathcal{B}\\mathcal{M}}(-2,t)=0.6p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$, $p^{\\mathcal{B}\\mathcal{M}}(-1,t)=0.8p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$, $p^{\\mathcal{B}\\mathcal{M}}(0,t)=p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$, $p^{\\mathcal{B}\\mathcal{M}}(1,t)=1.5p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$ for all $t\\in\\{1,\\ldots,T\\}$. We fix the maximum compensation to be 1000, that is, $l^{\\mathcal{B}\\mathcal{M}}_{\\max}(b,t)=1000$ for all $b\\in\\mathcal{B},t\\in\\{1,\\ldots,T\\}$.\nWe set the deductible to be 0.5 for all but the final policy year, and set the deductible to be 5 for the final policy year, that is, $l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}(b,t)=0.5$ for all $b\\in\\mathcal{B},t\\in\\{1,\\ldots,T-1\\}$ and $l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}(b,T)=5$ for all $b\\in\\mathcal{B}$. This is to prevent an issue caused by the finite horizon. Since after the final policy year there is no future benefit from the insurance policy and the insured is not incentivised to adopt the self-mitigation measure, a higher deductible is used as the incentive in the final policy year. \nIn addition, we let $\\delta_{\\mathrm{i}\\mathrm{n}}(t)=0.75(t-16)^+$, $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)=3+\\frac{5}{19}(t-1)$, and $\\delta_{\\mathrm{r}\\mathrm{e}}=3$. This setting has the effect of incentivising the insured to activate the insurance contract early on, and disincentivising withdrawal when close to the final policy year. \nAs a baseline for comparison, we also consider another cyber risk insurance policy without the Bonus-Malus system, which can be modelled by letting $\\mathcal{B}=\\{0\\}$. We fix the premium to be the base premium $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$, and leave everything else identical to the policy with the Bonus-Malus system. \n\n\\subsection{Results and Discussion}\n\nFigure~\\ref{fig:exp1policy} shows the expected number of years the insured's policy spends in each of the Bonus-Malus levels or being uninsured and the expected number of years the insured adopts the self-mitigation measure. The two panels compare the cyber risk insurance contract with the Bonus-Malus system with the one without.\nWith the contract that does not have the Bonus-Malus system, the decisions of the insured are completely deterministic, that is, they do not depend on the realisation of the cyber loss events. When $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\le4.410$, the optimal strategy of the insured is to purchase the cyber risk insurance policy every year and only adopt the self-mitigation measure in the final policy year (due to the higher deductible in the final policy year). When $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\ge4.415$, the optimal strategy of the insured is to never purchase the cyber risk insurance policy and always adopt the self-mitigation measure. Therefore, without the Bonus-Malus system, the issue of moral hazard is present and the insured will treat the cyber risk insurance policy and the self-mitigation measure as substitute goods. \nOn the other hand, when the Bonus-Malus system is introduced to the cyber risk insurance contract, the decisions of the insured depend on the realisation of the cyber loss events. When $4.495\\le p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\le4.930$, the optimal strategy of the insured is to always purchase the cyber risk insurance policy and adopt the self-mitigation measure. When $4.935\\le p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\le5.050$, the optimal strategy of the insured is to always adopt the self-mitigation measure but withdraw from the insurance contract when the expected future cost exceeds the expected future benefit of the insurance policy. As a result, the retention rate, i.e., the expected proportion of years the insured remains in the contract, drops when the base premium is increased. When $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\ge5.055$, the optimal strategy of the insured is to never purchase the cyber risk insurance policy and always adopt the self-mitigation measure. Hence, compared with the contract without Bonus-Malus, the contract with Bonus-Malus incentivises the insured to adopt the self-mitigation measure in addition to purchasing the cyber risk insurance policy. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{figures\/exp_ori_wo_BM1-eps-converted-to.pdf}\n~\n\\includegraphics[width=0.48\\linewidth]{figures\/exp_ori_w_BM1-eps-converted-to.pdf}\n\\caption{The retention rate of the cyber risk insurance policy and the expected years of adoption of the self-mitigation measure versus the base premium.}\n\\label{fig:exp1policy}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{figures\/exp_ori_wo_BM2-eps-converted-to.pdf}\n~\n\\includegraphics[width=0.48\\linewidth]{figures\/exp_ori_w_BM2-eps-converted-to.pdf}\n\\caption{The discounted total expected loss prevented by the self-mitigation measure and the discounted expected profit (measured by the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$) of the insurer versus the base premium. Left panel: the contract without the Bonus-Malus system. The dashed lines indicate the highest base premium before the insured chooses not to purchase the cyber risk insurance policy. Right panel: the contract with the Bonus-Malus system. The dashed lines indicate the highest base premium before the retention rate drops below 100\\%. The dotted lines indicate the highest base premium before the insured chooses not to purchase the cyber risk insurance policy.}\n\\label{fig:exp1profit}\n\\end{figure}\n\nFigure~\\ref{fig:exp1profit} compares both the expected value of the discounted total loss prevented by the self-mitigation measure and the expected value of the discounted profit of the insurer measured by the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ (defined in Section~\\ref{ssec:pricing}) in the two policies. The left panel of Figure~\\ref{fig:exp1profit} shows the case without Bonus-Malus. In that case, when $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\le4.410$, the insured will always purchase the cyber risk insurance policy but will only adopt the self-mitigation measure in the final policy year. Hence, the discounted total expected loss prevented stays at $0.505$, while the discounted expected profit of the insurer increases as the base premium increases. When $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\ge4.415$, the insured will not purchase the insurance policy but will always adopt the self-mitigation measure. As a result, the discounted total expected loss prevented will be $17.183$ but the insurer will earn no profit. The most the insurer can gain before losing the insured is $-10.510$, when the base premium is set to $4.410$. \nIn contrast, in the case with the Bonus-Malus system, as shown in the right panel of Figure~\\ref{fig:exp1profit}, the insurer can gain a discounted expected profit of at most $-0.860$ while always retaining the insured (i.e., the insured will never withdraw from the contract), when the base premium is set to $4.930$. The insurer can gain a discounted expected profit of at most $-0.006$ before losing the insured, when the base premium is set to $5.050$. With both of these base premiums, the insured will always adopt the self-mitigation measure, resulting in a discounted total expected loss prevention of~$17.183$. \n\nOverall, this experiment demonstrates two benefits of the Bonus-Malus system. First, the presence of the Bonus-Malus system in the cyber risk insurance contract incentivises the insured to adopt the self-mitigation measure. This results in a considerable increase in the prevention of cyber losses, which enhances the overall security of the cyberspace. Second, the Bonus-Malus system benefits the insurer, since it allows the insurer to gain more profit from the cyber risk insurance policy while remaining attractive to the insured. \nTo show that the observations from the results in this experiment and the conclusions drawn do not depend on the specific choice of the $h$ parameter in the loss severity distribution (which is the most impactful parameter in the truncated g-and-h distribution), and that they also do not depend on our distributional assumption, we have repeated the same experiment with slightly modified settings.\nIn the first modified setting, the $h$ parameter in the truncated g-and-h distribution is set to $0.10$, $0.20$, or $0.25$. \nIn the second modified setting, the loss severity distribution is replaced by a log-normal distribution where the parameters are determined by matching the first two moments to $\\text{Tr-}g\\text{-and-}h(\\alpha=0,\\varsigma=1,g=1.8,h=0.15)$. \nThe results obtained under these modified settings turned out to be very similar to the results in the original experiment.\\footnote{The results under these modified settings are available in the online appendix on GitHub: \\url{https:\/\/github.com\/qikunxiang\/CyberInsuranceBonusMalus}}\nThis shows that the benefits of the Bonus-Malus system in the experiment are in fact present across various loss distributional assumptions.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThis paper motivated the joint consideration of risk reduction and risk transfer decisions in the face of cyber threats. We introduced a cyber risk insurance contract with a Bonus-Malus system to provide incentive mechanisms to promote the adoption of cyber risk mitigation practices. We developed a model based on the stochastic optimal control framework to analyse how a rational insured allocates funds between self-mitigation measures and a cyber risk insurance policy. A dynamic programming-based algorithm was then developed to efficiently solve this decision problem. \nA numerical experiment demonstrated that this novel type of insurance contract can incentivise the adoption of self-mitigation measures and can allow the insurer to profit more from the policy while remaining attractive to the insured. \nFuture research could investigate the effects of the risk profile, i.e., the characteristics of the loss distribution such as the heaviness of its tail, on the effectiveness of the Bonus-Malus system and how one can tailor Bonus-Malus-based insurance contracts for different risk profiles.  \n\n\n\\section*{Acknowledgments}\n\\noindent\nAriel Neufeld gratefully acknowledges the financial support by his Nanyang Assistant Professorship Grant (NAP Grant) \\emph{Machine Learning based Algorithms in Finance and Insurance}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and related Research}\n\nDiscrete-time queueing networks are used to model a variety of scenarios, ranging from traffic control over parallel computing to wireless communication. They are closely related to the canonical control system\n\\begin{equation}\n\t\\label{eq::usual_control}\n\tx_{t+1} = A x_t + B_t v_t + D_t w_t\n\\end{equation}\nwith some significant differences: i) The controls $v_t$ are binary in nature and linearly constrained by $C v_t \\leq c$, e.g. due to the interference properties of wireless channels. ii) The state lives on the discrete set $x_t \\in \\mathbb{N}^n$ where it exhibits no inertia ($A = I$). iii) And crucially, the matrices $B_t$ and $D_t$ behave stochastically, implying that the effect of a control decision is not certain. Together with the class of back-pressure control policies, those systems form a well investigated subclass of control problems.\n\nThe prototype back-pressure policy, that we will call the max-weight policy ($\\mathsf{MW}$), was first introduced in \\cite{Tassiulas1992}, where the authors also proved its much praised property of being \\textit{throughput optimal}. This means, that $\\mathsf{MW}$ can manage any load of traffic, provided this load can somehow be supported by the network topology.\nOver time, many variations of $\\mathsf{MW}$ where developed, e.g. to allow for a generalized control objective \\cite{Meyn2009} \\cite{Kasparick2018}, or to increase its performance in special cases like networks with input-queued switches or time-varying channels \\cite{Mckeown1999} \\cite{Neely2005}. Specific shortcomings of $\\mathsf{MW}$, like e.g. high end-to-end delay, where investigated in \\cite{Khan2009} \\cite{Subramanian2007} \\cite{Ying2011} and later partially remedied by \\cite{Neely2005} \\cite{Ying2011} \\cite{Huang2013} \\cite{Xiong2011}, using e.g. shortest path algorithms to reduce delay especially in low traffic scenarios.\n\nIn this paper, we propose a novel control policy that is predictive in nature and that we will call predictive network control ($\\mathsf{PNC}$). It can be regarded as a generalization of $\\mathsf{MW}$, since it contains $\\mathsf{MW}$ as a special case. But while $\\mathsf{MW}$ and all its derivations are \\textit{myopic}, i.e. only aim to improve the system state for the \\textit{immediate} next time slot, $\\mathsf{PNC}$ aims to improve the system state for \\textit{multiple} time slots up until a prediction horizon. This leads to the calculation of an entire optimal \\textit{trajectory} of control vectors. However, instead of applying the entire trajectory for the next few time slots, only the first control vector is applied to the system and the process repeats in the consecutive time slot. This allows the controller to react to any unforeseen changes in the control system \\cite{Mayne2000}.\n\nSuch a control scheme is called model-predictive control (MPC), and therefore $\\mathsf{PNC}$ is a realization of MPC, tailored specifically towards queueing networks. MPC itself is a well established branch of control theory and can cope very easily with hard constraints and non-linearities, making it particularly suited for our control problem. However, its advantages are payed for by high requirements on computational resources.\nSo far, there has only been one attempt to bring MPC to queueing networks: In \\cite{VanLeeuwaarden2010a} the authors focus on a special case of the standard model, in which only the arrivals to the system are of stochastic nature. The investigation is limited to numerical simulations, which show better system performance (smoother time behavior) for a designed MPC controller compared to simple feedback control laws.\nSince our queueing network will include a much higher degree of stochastics, we will not follow up on their work.\n\nBecause a queueing network misses any inertia ($A = I$), a predictive control scheme can only tap into its full potential, if the stochastics for $B_t$ (or $D_t$) are complex enough. E.g. if both matrices behave according to white-noise, prediction over more than the next time slot yields close to no improvement over myopic strategies. Hence, the benefit of a predictive control scheme usually increases with complexity of the system model. Therefore, we let $B_t$ (the matrix which is responsible for the topology and the quality of the links between the nodes of the network) be governed by a discrete-time Markov chain (DTMC) and a Bernoulli trial. This gives the opportunity to model long-term and short-term effects, respectively. Take e.g. wireless relay networks with user mobility: here, short-term interference leading to packet loss can be modeled by the Bernoulli trial, while long-term change in channel quality due to the mobility can be expressed by the DTMC \\cite{Guzman2019}.\n\nControl systems, in which the model parameters change according to a DTMC are called Jump-Markov systems (JMS). (Since simple feedback controllers cannot detect this change, JMS are usually controlled with MPC controllers). There exist several control approaches towards JMS, covering cases with linear \\cite{Park2002} \\cite{Chitraganti2014} \\cite{Tonne2017a} and even nonlinear system dynamics \\cite{Liu2015} \\cite{Tonne2017}, where the referenced works mainly differ in the choice of considered constraints.\nHowever, all these works deal with conventional control systems, where the controller usually tries to follow a reference trajectory and noise ($w_t$) represents a stochastic disturbance with zero first moment. In contrast, from the perspective of queueing networks, the noise term represents the arrival of packets\/customers whose first moment is strictly positive, and the controller tries to maintain finite queues for any given arrival (hence, there is no need for a reference trajectory). For that reason, prior work on JMS is only partially applicable to our systems. To the best of our knowledge, we are the first to consider both JMS and MPC in the context of discrete-time queueing networks.\n\nOur \\textbf{contribution} is three-folded: i) We develop a JMS-adapted discrete-time queueing network and introduce a family of predictive control policies, based on the paradigms of MPC. ii) We proof throughput optimality (the equivalent of stability) for the most simple of our predictive control policies, thereby implying the same for the rest. And iii) we show the benefit of these policies over the conventional back-pressure control ($\\mathsf{MW}$), using numerical simulation. In particular, our policies seem to maintain their throughput optimality in networks with synchronized queues, making them unique.\n\n\\section{System Model \\& Prerequisites}\n\n\\subsection{System Model}\n\nWe begin by stating the constituting equation for our system model and clarify its components afterwards. Similar to the conventional control system, a discrete-time queueing network can be expressed by its one-step evolution and associated constraints\n\\def\\hspace{0.5mm}{\\hspace{0.5mm}}\n\\begin{gather}\n    \\label{eq::basic_queueing}\n    q_{t+1} = q_t + R M_t v_t + a_t\n    \\\\[1ex]\n    \\nonumber\n    \\text{subject to}\n    \\\\[1ex]\n    \\nonumber\n    \\left(\n\t\\begin{aligned}\n\t    C v_t & \\leq c \\\\\n\t    - R^- v_t & \\leq q_t\n    \\end{aligned}\n    \\right)\n    \\hspace{0.5mm} \\text{and} \\hspace{0.5mm}\n    \\left(\n    \\begin{gathered}\n\t    M_t \\sim \\mathcal{B}(W^{s_t})\n\t    \\\\\n\t    W^{s_t} \\in \\{W^1 ,\\dots W^{n_s} \\}\n\t    \\\\\n\t    (s_t) \\sim \\operatorname{DTMC}(\\{1,\\dots n_s\\},P,s_0)\n    \\end{gathered}\n    \\right)\n\\end{gather}\nThe \\textit{queue vector} $q_t = \\left( q_t^1 \\dots q^{n_q}_t \\right)^\\intercal \\in \\mathcal{Q} = \\mathbb{N}^{n_q}$ represents the system- (or queue-) state, where $q_t^i$ counts the number of packets, waiting in queue $i = 1,\\dots n_q$ in time slot $t$. Each queue itself is a node of the network.\n\nIn any time slot, packets can be transmitted from one queue to another if there exists a directed link between the two and the link is activated.\nThere are $n_v$ links, each of which can be represented by a vector $r^j \\in \\left\\{ -1,0,+1 \\right\\}^{n_q}$ ($j = 1,\\dots n_v$), that, by superpositioning with $q_t$, transfers a packet from one queue ($\\{-1\\}$) to another ($\\{+1\\}$). All links are collected as columns in the routing matrix $R \\in \\left\\{ -1,0,+1 \\right\\}^{n_q \\times n_v}$ which therefore holds the topology.\n\n[Remark:\nIn \\textit{conventional} networks, a link has exactly \\textit{one} $\\{-1\\}$ entry (origin) and \\textit{at most one} $\\{+1\\}$ entry (destination). This implicit constraint is a prerequisite for all back-pressure policies to develop their throughput optimality. Though we will also use this constraint throughout the paper, our novel control policies seem to maintain their throughput optimality even when it is violated (see section \\ref{subsec::synchronized_queues}), allowing us to control networks with synchronized queues.]\n\nThe controller may activate a link in a given time slot via the binary control vector $v_t \\in \\{ 0 , 1 \\}^{n_v}$.\nIf we could activate all links simultaneously ($v_t = \\mathbf{1}_{n_v}$), the control problem would become trivial. However, we are usually constrained in the activation (e.g. due to interference properties) by the \\textit{constituency constraint} $Cv_t \\leq c$. The dimensions of $C$ and $c$ are case dependent, their entries are from the set $\\mathbb{N}$. Furthermore, a packet can only be scheduled for transmission, if it is present at the corresponding queue, hence a packet may only traverse a single link per time slot. We will refer to this as the \\textit{positiveness constraint}, which is readily implemented by considering the maximum one-step efflux of the system, which is $R^- v_t$, where $R^-$ is equal to $R$ without its positive entries. Naturally, the maximum efflux cannot drain more packets than are actually present: $q_t + R^- v_t \\geq 0$. Note that this also guarantees that $q_t \\in \\mathbb{N}^{n_q}$.\n\nFor clarification, we refer to \\figref{fig::min_example}. Here, we stated topology and constituency matrices and derived the corresponding constraints. \nGiven only $C$ and $c$, both components of $v_t$ could be active simultaneously. However, if $q^2$ is empty ($q^2 = 0$), it is not possible to activate the second link $r^2$.\n\n\\def\\comR{\n\t$\n\tR = \\begin{pmatrix}\n\t-1 & 0 \\\\ +1 & -1\n\t\\end{pmatrix}\n\t$\n}\n\\def\\comC{\n\t$\n\tC = \\begin{pmatrix}\n\t0 & 0\n\t\\end{pmatrix}\n\t$\n}\n\\def\\comc{\n\t$ c = 1 $\n}\n\\def\\comRC{\n\t$ \\begin{pmatrix}\n\t1 & 0 \\\\ 0 & 1 \n\t\\end{pmatrix} v_t \\leq q_t$\n}\n\\def\\comCC{\n\t$ \\begin{pmatrix}\n\t\t0 & 0 \n\t\\end{pmatrix} v_t \\leq 1$\n}\n\\def\\comA{\n\t$\\Downarrow$\n}\n\n\\begin{figure}[htbp]\n\t\\centering\n\n\t\\includegraphics[]{network_example_f.pdf}\n\t\\caption{Minimal Example of a Queueing Network}\n\t\\label{fig::min_example}\n\\end{figure}\n\n\nStill, even an activated link $r^j$ might fail in its transmission, leaving source and destination queue unchanged. This is modeled by a stochastic variable $m^j_t \\in \\{0,1\\}$ which is Bernoulli distributed (coin-flip) with probability $\\e{m}^j_t \\in [0,1]$. I.e. $m^j_t \\sim \\mathcal{B}(\\e{m}^j_t)$. For a succinct notation, we collect all those quantities in the diagonal matrices $M_t = \\operatorname{diag}_{j = 1 ,\\dots n_v}\\{m^j_t\\}$ and $\\e{M}_t = \\operatorname{diag}_{j = 1 ,\\dots n_v}\\{\\e{m}^j_t\\}$ respectively such that $M_t \\sim \\mathcal{B}(\\e{M}_t)$ and of course $\\E{M_t} = \\e{M}_t$. \n\nThe Bernoulli trials on $\\e{M}_t$ are intended to model short-term stochastics. For long-term stochastics, we let $\\e{M}_t$ be picked from a predetermined set $\\mathcal{W} = \\{ W^1,W^2,\\dots \\}$ of weight matrices $W^i$ according to a DTMC $(s_t)$. If $\\mathcal{S} = \\{1 ,\\dots n_s\\}$ is the index set of $\\mathcal{W}$, we have $(s_t) \\sim \\operatorname{DTMC}(\\mathcal{S},P,s_0)$ where $P$ and $s_0$ are transition matrix and initial state, respectively. The entire selection process can therefore be expressed as $\\e{M}_t = W^{s_t}$. If $\\sigma_t$ describes a distribution for the DTMC, we have $\\lim_{t\\to\\infty} \\sigma_t = \\pi$, which we assume to be the only stable distribution, with $\\pi^{\\R{s}}$ being the average probability of $s_t = {\\R{s}}$.\n\nThe task for a controller is to steer the packets through the network to their destination nodes. Once reached, the packets leave the system, which can be modeled via links $r^j$ without the $\\{+1\\}$ entry. At the same time, new packets are created directly at the queues through an \\textit{arrival vector} $a_t \\in \\mathbb{N}^{n_q}$ of possibly stochastic nature. We call $\\e{a} = \\E{a_t}$ the \\textit{arrival rate} and make the usual assumption, that there is an upper bound, such that always $a_t \\leq \\hat{a}$.\n\nFinally, we remark that if different packets are destined for different final destination nodes, they are of different \\textit{class} (or belong to a different \\textit{flow}). Each class has to have its separate network of queues in order for the packets to be distinguishable. Thus, for each class a new copy of the system would have to be employed. While many authors model this by adding an additional dimension (the dimension of all classes) to all quantities, we will just assume, that the so far described system model already consists of those copies, stacked in a suitable way, thereby avoiding the introduction of another dimension to the system model.\n\n\\subsection{Control Policies and Throughput Optimality}\n\nWe already defined $q_t \\in \\mathcal{Q}$ and $s_t \\in \\mathcal{S}$. Note that $(s_t)$ is a DTMC and $q_{t-1}$ is not needed for a prediction of future states once $q_t$ is known. If we further assume the arrival vector to be independent of past realizations, there is no reason for a controller to use any but the last known realizations of $q_t$ and $s_t$ for its decision making.\nIf we also define the set of all control vectors by\n\\begin{equation}\n\t\\mathcal{V} = \\Set{ v \\in \\{0,1\\}^{n_v} | \n\t\t\\begin{aligned}\n\t    C v & \\leq c \\\\\n\t   \n    \t\\end{aligned}\n    }\n\\end{equation}\nthen we can express a control policy $\\phi$ as a mapping from the set of relevant, observed quantities onto the set of control vectors: $\\phi: \\mathcal{Q} \\times \\mathcal{S} \\to \\mathcal{V}$. In some cases, however, it makes sense to incorporate a stochastic process into the policy itself, in order to circumvent the discreteness of $\\mathcal{V}$. This way, given a fixed pair of observations $(q',s')$, we can not only access a fixed $v' = \\phi(q',s') \\in \\mathcal{V}$, but on average \\textit{any} predetermined element $\\sum_{v \\in \\mathcal{V}} \\lambda^v v$ of the convex hull $\\conv{\\mathcal{V}}$. Hence we define a \\textbf{control policy} as\n\\begin{equation}\n\t\\phi: \\mathcal{Q} \\times \\mathcal{S} \\times \\Omega \\to \\mathcal{V}\n\\end{equation}\nwhere $\\Omega$ is the sample space of the underlying stochastic process. (Remember that a control policy is only valid, if it complies with $R^- \\phi(q_t,s_t,\\omega_t) \\leq q_t$.)\n\nWe say that a control policy $\\phi$ \\textbf{stabilizes} a system for a given arrival rate $\\e{a}$ if it can compensate the arrival rate on average:\n\\begin{equation}\n\\label{eq::def_stability}\n\\begin{aligned}\n\t\\mathbf{0} &= \\lim_{\\tau \\to \\infty} \\frac{1}{\\tau} \\sum_{t=1}^\\tau \\left( \\vphantom{\\frac{1}{2}} a_t + R M_t \\phi(q_t,s_t,\\omega_t) \\right)\n\t\\\\\n\t&= \\hspace{10mm} \\e{a} \\hspace{10.7mm} + \\sum_{\\R{s} \\in \\mathcal{S}} \\pi^\\R{s} RW^\\R{s} \\phi(q_t,\\R{s},\\omega_t)\n\\end{aligned}\n\\end{equation}\nComparing with the system equation \\eqref{eq::basic_queueing}, this implies that the average queue state remains bounded.\n\nFinally, a control policy $\\phi$ is throughput optimal, if it stabilizes a given system for any arrival rate $\\e{a}$ for which at least one other (possibly unknown) policy stabilizes the system. This can readily be expressed by noting that a policy $v_t = \\phi(s_t,\\omega_t)$ can on average, for every state of $\\mathcal{S}$ separately, excess any predetermined element in the interior of the convex hull $\\conv{\\mathcal{V}}$. Note that this excludes the boundary of $\\conv{\\mathcal{V}}$, because due to the positiveness constraints, no policy can guarantee to never be forced to be idle (meaning $v_t = \\mathbf{0}$). Naturally, there are no other options for the average control vector than those in $\\conv{\\mathcal{V}}$.\nThus, $\\phi$ is \\textbf{throughput optimal}, if it stabilizes the system for all $\\e{a}$ with\n\\begin{equation}\n\t\\label{eq::def_to}\n\t\\e{a} + \\sum_{\\R{s} \\in \\mathcal{S}} \\pi^\\R{s} RW^\\R{s} \\sum_{v \\in \\mathcal{V}} \\lambda^{\\R{s},v}  v = -\\varepsilon \\mathbf{1}_{n_q}\n\t\\hspace{10mm}\n\t\\begin{aligned}\n\t\\lambda^{\\R{s},v} & \\geq 0 \\\\\n\t\\sum_v \\lambda^{\\R{s},v} & \\leq 1\n\t\\end{aligned}\n\\end{equation}\nwhere $\\varepsilon > 0$ is used to exclude the boundary of $\\conv{\\mathcal{V}}$.\n\n\\section{Predictive Network Control ($\\mathsf{PNC}$)}\n\nInspired by the common MPC paradigms, our novel control policy, $\\mathsf{PNC}$, works in 3 steps: i) An entire trajectory of optimal control decisions from $t$ up until $t+H-1$ is calculated as the result of a minimization of an objective function $J$. Here, $J$ is a function of the next $H$ future system states, which we can only \\textit{predict}. $H$ is called the prediction horizon. ii) Only the first (i.e. the immediate) control decision in this trajectory is actually applied to the system. iii) The process repeats (discarding the rest of the just calculated trajectory). W.l.o.g., for the rest of the paper, we assume the current time slot to be $t=0$.\n\nThe most often encountered objective is the sum of squares, which in our case translates to\n\\begin{equation}\n\t\\label{eq::pnc_objective}\n\tJ(q_0,s_0) = \\CE{ \\sum_{t=1}^{H} q_t^Tq_t }{q_0,s_0}\n\\end{equation}\nUsing this definition, minimizing $J$ means minimizing the amount of packets in the network, which can only be done by delivering the packets to their destinations.\nOver the system evolution \\eqref{eq::basic_queueing}, $J$ will be influenced by the choice of control vectors via\n\\begin{equation}\n\t\\label{eq::actual_expected_evolution}\n    \\CE{q_{T}}{q_0,\\sigma_0}\n    = q_0 + \\sum_{t = 0}^{T-1} \\sum_{\\R{s} \\in \\mathcal{S}} \\left( \\sigma_0 P^{(t)} \\right)^\\R{s} R W^\\R{s}\n    v_t\n    + T\\e{a}\n\\end{equation}\nwhere $\\sigma_0$ is the distribution corresponding to the initial state $s_0$ and $\\left( \\sigma_0 P^{(t)} \\right)^\\R{s}$ stands for the $\\R{s}$-th entry of the predicted distribution in time slot $t$. \n\nNote, that the prediction can be implemented in three different ways, varying in precision and required effort:\n\ni) The first one is the \\textit{true prediction}, which assigns a control vector to \\textit{every} time slot (up until $H$) and \\textit{every} possible set of realizations of the quantities in the system evolution. Since the ensemble of these realizations in time slot $t$ forms $q_{t+1}$ and $s_{t+1}$, this would mean making $v_t$ a function of $q_t$ and $s_t$ for the remainder of the prediction. The number of control vectors, required for such a prediction amounts to $H \\cdot n_s \\cdot k_q$, where $k_q$ is the number of all possible queue states, that can be realized in a single time slot (likely to depend itself on prior queue state, realization of arrival and Bernoulli trial). This obviously requires the maximum amount of computational resources but allows us to truly find the optimal control trajectory that minimizes $J$.\n\nii) In contrast, a \\textit{relaxed prediction} uses only a minimum of control vectors. I.e. in every time slot, a single control vector is chosen and thus $v_t$ is only a function of $t$ for the remainder of the prediction. (Using even less control vectors would not constitute a meaningful prediction for our purposes.) This amounts to only $H$ control vectors being required for the prediction, speeding up the calculation of an optimal control trajectory to minimize $J$ considerably. However, said trajectory might be sub-optimal compared to the true prediction from before and as a consequence the control performance might be worse.\n\niii) Finally, a mixture of both cases could be implemented, finding a balance between computational complexity and control performance. E.g. one could consider every future DTMC state $s_t$ (up until $s_H$) leading to $H \\cdot n_s$ control vectors that have to be determined in order to minimize $J$.\n\nWe will define or policy via case ii), i.e. the relaxed prediction and explain the reasoning for this in the end of the section. For what follows, we substitute the control vector $v_t$ with $u_t$ to emphasize, that this is not the actual control of the queueing network but rather the one used for the prediction. Hence, $u_t$ is a quantity that is used internally to define the $\\mathsf{PNC}$ policy according to\n\\begin{multline}\n\t\\label{eq::pnc_expected_evolution}\n    \\CE{q_{T}}{q_0,\\sigma_0}\n    = q_0 + \\sum_{t = 0}^{T-1} \\sum_{\\R{s} \\in \\mathcal{S}} \\left( \\sigma_0 P^{(t)} \\right)^\\R{s} R W^\\R{s} u_t + T\\e{a}\n\\end{multline}\nIf we define the trajectory\n\\begin{equation}\n    \\label{eq::def_w_tilde}\n    \\tr{u}_0^{H-1} = \\begin{pmatrix}\n        u_0 \\\\\n        \\vdots \\\\\n        u_{H-1}\n    \\end{pmatrix}\n\\end{equation}\nand substitute it together with \\eqref{eq::pnc_expected_evolution} into \\eqref{eq::pnc_objective}, the objective can be rewritten as\n\\begin{equation*}\n    J(q_0,s_0) = J_1(q_0) + J_2(q_0,s_0) \\tr{u}_0^{H-1} + \\left( {\\tr{u}_0^{H-1}} \\right)^\\intercal J_3(s_0) \\tr{u}_0^{H-1}\n\\end{equation*}\nAs can be seen, $J_1(q_0)$ will not be influenced by the minimization over $\\tr{u}_0^{H-1}$, and $J_3(s_0)$ will stay bounded, since it is not dependent on $q_0$. Therefore, for large enough $q_0$, the linear term $J_2(q_0,s_0) \\tr{u}_0^{H-1}$ will always dominate the minimization over $\\tr{u}_0^{H-1}$, making it prudent to define our actual objective only over this linear term. This simplifies the minimization from a quadratic to a linear one. (Note that a similar step is also taken in the definition of the $\\mathsf{MW}$ policy.) Expanding the remaining constraints from the original system in a straight forward way, we end up with the following definition of the $\\mathsf{PNC}$ policy:\n\\begin{equation}\n\t\\label{eq::pnc_policy_def}\n\\begin{gathered}\n    \\phi^{\\mathsf{PNC}}(q_0,s_0) = \\operatorname{first} \\argmin_{\\tr{u}_0^{H-1}} \\  J_2(q_0,s_0) \\tr{u}_0^{H-1}\n    \\\\[2ex]\n    \\text{subject to} \\qquad\n    \\begin{aligned}\n    \\tr{C} \\tr{u}_0^{H-1} &\\leq \\tr{c}\n    \\\\\n    \\tr{A} \\tr{u}_0^{H-1} &\\leq \\tr{b}(q_0)\n    \\\\\n    \\tr{u}_0^{H-1} &\\in \\{0,1\\}^{Hn_v}\n    \\end{aligned}\n\\end{gathered}\n\\end{equation}\nwhere $\\operatorname{first} \\argmin ()$ expresses that only the first argument of the trajectory is used as output. An overview of the utilized quantities can be found in Table~\\ref{table_01}.\n\nChoosing $H=1$, we end up with the common $\\mathsf{MW}$ policy, which is not surprising, since its definition also involves a quadratic objective. And indeed, $\\mathsf{PNC}$ would merely be the extension of $\\mathsf{MW}$ over multiple time slots, if the $\\mathsf{PNC}$ controller would follow a once calculated optimal trajectory to its end (i.e. for $H$ time slots). However, $\\mathsf{PNC}$ recalculates this trajectory each time slot, thereby discarding its entire tail. This results in a much improved behavior of $\\mathsf{PNC}$ (see section \\ref{subsec::synchronized_queues}) but also makes it impossible to infer any properties from $\\mathsf{MW}$ to $\\mathsf{PNC}$. For more comparisons between the two policies, we refer to \\cite{Schoeffauer2018a} and \\cite{Schoeffauer2019}.\n\nWe continue with the main theorem of this paper, which states throughput optimality of the $\\mathsf{PNC}$ policy. Note that this automatically implies throughput optimality for every other MPC controller, that uses a more precise prediction (under the same constraints and objective function).\n\\begin{theorem}\n\\label{theorem::to_of_pnc}\nThe $\\mathsf{PNC}$ policy \\eqref{eq::pnc_policy_def} is throughput optimal for the system \\eqref{eq::basic_queueing}.\n\\end{theorem}\n\n\\begin{table*}[ht!]\n\\caption{Extended Formulas for the Optimization Problem}\n\\centering\n\\normalsize\n\\begin{tabular}{|p{0.3\\linewidth}|p{0.64\\linewidth}|}\n\\hline\n\\hline\n\\begin{center}\nExpected value of weight matrix $W^{s_t}$\n\\end{center}\n&\n\\begin{center}\nLinear objective $J_2$\n\\end{center}\n\\\\\n\\vspace{-2.8mm}\n\\begin{equation}\n\\begin{gathered}\n    \\e{W}_t(s_0) = \n\t\\CE{W^{s_t}}{s_0} \n\t\\\\  =\n    \\left( \\sigma_0 P^{(t)} \\otimes I_{n_s} \\right)\n    \\begin{pmatrix}\n\t\tW^1 \\\\ W^2 \\\\ \\vdots \\\\ W^{n_s}\n    \\end{pmatrix}^\\intercal\n    \\end{gathered}\n\\end{equation}\n&\n\\begin{equation}\n    \\label{eq::J2_formula}\n    J_2 = \n    2q_0^\\intercal R\n\t\\begin{pmatrix}\n\t\tH \\e{W}_0(s_0)\n\t\t\\\\\n\t\t(H-1) \\e{W}_1(s_0)\n\t\t\\\\\n\t\t\\vdots\n\t\t\\\\\n\t\t \\e{W}_{H-1}(s_0)\n\t\\end{pmatrix}^\\intercal\n\t+ \\e{a}^\\intercal R\n\t\\begin{pmatrix}\n\t\t(H+1)(H-0) \\e{W}_0(s_0)\n\t\t\\\\\n\t\t(H+2)(H-1) \\e{W}_1(s_0)\n\t\t\\\\\n\t\t\\vdots\n\t\t\\\\\n\t\t2H \\e{W}_{H-1}(s_0)\n\t\\end{pmatrix}^\\intercal\n\\end{equation}\n\\\\\n\\hline\n\\begin{center}\nConstituency constraints\n\\end{center}\n&\n\\begin{center}\nPositiveness constraints\n\\end{center}\n\\\\\n\\begin{equation}\n\t\\label{eq::table_constituency}\n\\underbrace{\n    \t\t\\left( \\vphantom{\\frac{1}{2}} I_{H} \\otimes C \\right) \n\t}_{\\displaystyle \\tr{C} }  \t\n \t\\tr{u}_0^{H-1} \\leq\n\t\\underbrace{ \t\n \t\t\\mathbf{1}_{H} \\otimes c \\vphantom{\\left( \\vphantom{\\frac{1}{2}} I_{H} \\otimes C \\right) }\n \t}_{\\displaystyle \\tr{c} \\vphantom{\\tr{C}} }\n\\end{equation}\n&\n\\vspace{-2.6mm}\n\\begin{equation}\n\t\\label{eq::table_positiveness}\n    \\underbrace{\n    \\begin{pmatrix}\n        R^- &  & &  \\\\\n        R & R^- &   &  \\\\\n        \\vdots &  & \\ddots \\\\\n        R & \\dots & R & R^-\n    \\end{pmatrix}\n    }_{\\displaystyle \\tr{A}}\n    \\tr{u}_0^{H-1}\n    \\leq \n    \\underbrace{\n    \\begin{pmatrix}\n    \tq_0 \\\\ q_0 + \\e{a} \\\\ \\vdots \\\\ q_0 + (H-1) \\e{a}\n    \\end{pmatrix}\n    }_{\\displaystyle \\tr{b}(q_0) }\n\\end{equation}\n\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{table_01}\n\\end{table*}\n\n\\section{Proof of Theorem \\ref{theorem::to_of_pnc}}\n\\label{sec::proof}\n\nWe will now prove, that $\\phi^\\mathsf{PNC}$ is throughput optimal. And in contrast to the usual stability-related proofs employed for MPC controllers, we will not rely on a terminal set.\n\n\\subsection{Preliminaries}\n\nIt will often become necessary to upper and lower-bound certain expressions. We will use $K_i \\in \\mathbb{R}_+$, $i\\in \\mathbb{N}$ to denote those bounds or variables, whose values are of no further interest and are obvious to calculate. Crucially, any $K_i$ will be independent of the initial system state $q_0$!\n\nWe will use gothic letters to express realizations of random variables, such that e.g. $\\R{s_t}$ is a realization of $s_t$, hence $\\R{s_t} \\in \\mathcal{S}$. And because the corresponding set of realizations will always be very clear from the context, we will use the succinct notation $\\sum_{\\R{s_t}}$ instead of $\\sum_{\\R{s_t} \\in \\mathcal{S}}$ for the sum of all realizations (as is needed for e.g. expressing the expectation).\n\nGiven a trajectory $\\tr{x}$ of control vectors of certain length, we use $\\tr{x} \\in \\mathcal{P}(q_0)$, to express that $\\tr{x}$ abides to the positiveness constraints $\\tr{A} \\tr{x} \\leq \\tr{b}(q_0)$, where $\\tr{A}$ and $\\tr{b}(q_0)$ are defined as in \\eqref{eq::table_positiveness}, expect for a possibly different value of $H$ (depending on the length of $\\tr{x}$). Analogue, $\\tr{x} \\in \\mathcal{C}$ will express, that $\\tr{x}$ abides to the constituency constraints as in \\eqref{eq::table_constituency}.\n\nFinally, we make the definitions $\\Delta_0^T := q_T - q_0$. Keep in mind, that analogue to $q_t$ being a function of all prior stochastics and controls, $\\Delta_0^T$ is of course a function of all stochastics and controls in the time slot $0,\\dots T-1$. With the definition $\\norm{q_t} := q_t^\\intercal q_t$ (which is not meant to be a norm) this gives raise to\n\\begin{equation}\n\t\\label{eq::to_sub_to_q0}\n\t\\norm{q_T} = \\norm{\\Delta_0^T} + \\norm{q_0} + 2q_0^\\intercal \\Delta_0^T\n\\end{equation}\nNote, that this decomposition corresponds to the one for the objective function $J$ and we can now restate the objective of the $\\mathsf{PNC}$ policy, $J_2$, as\n\\begin{equation}\n\t\\label{eq::to_objective_better}\n\tJ_2(q_0,s_0) = \\CE{ \\sum_{t=1}^{H} 2q_0^\\intercal \\Delta_0^t }{q_0,s_0}\n\\end{equation}\nWith this notation we formulate the next lemmas, needed for the proof.\n\n\\begin{lemma}\nThe difference $\\Delta_0^T$ between two queue states is bounded (element-wise) by\n\\begin{equation}\n\t\\label{eq::to_gen_bounds}\n\t- T n_v \\mathbf{1}_{n_q}\n\t\\leq\n\t\\Delta_0^T\n\t\\leq\n\tT n_v \\mathbf{1}_{n_q}  + T \\hat{a}\n\\end{equation}\nleading to\n\\begin{equation}\n\t\\label{eq::to_quad_to_lin}\n\t\\norm{q_0} + 2q_0^\\intercal  \\Delta_0^T\n\t\\leq\n\t\\norm{q_T} \n\t\\leq \\norm{q_0} + 2q_0^\\intercal  \\Delta_0^T + K_1\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBetween time slots $0$ and $T$ we have at best a constant efflux of $n_v$ or at worst a constant influx of $n_v + \\hat{a}$ packets per queue per time slot (since there are at most $n_v$ links to fill or drain any given queue).\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma::diff}\nThe difference between the minimization that originates from the definition of the $\\mathsf{PNC}$ policy (using the formulation from \\eqref{eq::to_objective_better}), and the same minimization but without considering any positiveness constraints can be bounded by\n\\begin{gather}\n\\label{eq::to_lemma}\n\\begin{aligned}\n    \\min_{ \\tr{u}_0^{H-1} \\in \\mathcal{C} \\cap \\mathcal{P}(q_0) }\n    \t&\\CE{ \n    \t\t\\sum_{t=1}^{H}  2q_0^\\intercal \\Delta_0^t\n    \t}\n    \t{\n    \t\tq_0,s_0\n    \t}\n    \\\\\n    -\n    \\min_{ \\tr{u}_0^{H} \\in \\mathcal{C} } \\hspace{6mm}\n    \t&\\CE{ \n    \t\t\\sum_{t=1}^{H}  2q_0^\\intercal \\Delta_0^t\n    \t}\n    \t{\n    \t\tq_0,s_0\n    \t}    \n\\end{aligned}\n    \\\\ \\nonumber\n    \\leq\n    \\left( Hn_v^2-n_v \\right) \\left( H + 1 \\right)\n    = K_2\n\\end{gather}\n\\end{lemma}\n\\begin{proof}\nClearly, the maximum deficit, that $\\Delta_0^t$ can generate is (element-wise) less than $t n_v \\mathbf{1}_{n_q}$ (all links drain a queue over $t$ steps).\nFor a single queue, the \\textit{most} efflux in $H$ time slots is $Hn_v$ packets. \nHence, if $q_0 \\geq Hn_v \\mathbf{1}_{n_q}$, a control trajectory \\textit{cannot} violate the positiveness. Conversely, if a link cannot be activated due to the positiveness constraints, at least one entry of $q_0$ must be smaller than $H n_v$.\n\nDue to the linearity, the largest difference in the minimizations will be found, if $q_0 = (H n_v - 1) \\mathbf{1}_{n_q}$ (possibly denying any activation for the minimization with the positiveness constraints). This, together with the initial bound on $\\Delta_0^t$ leads directly to\n\\begin{equation}\n\t\\sum_{t=1}^H 2 (H n_v - 1) \\mathbf{1}_{n_q}^\\intercal  t n_v \\mathbf{1}_{n_q} = \\left( Hn_v^2-n_v \\right) \\left( H + 1 \\right)\n\\end{equation}\nwhich is an upper bound for the difference in question.\n\\end{proof}\n\nFinally, the following theorem will allow us to express our definition of stability by the means of a Ljapunov function.\n\\begin{lemma}\n\t\\label{lemma::foster}\n\tA policy $\\phi$ stabilizes the system \\eqref{eq::basic_queueing} under a certain arrival rate $\\e{a}$, if we can find a function $f: \\mathcal{Q} \\times \\mathcal{S} \\to \\mathbb{R}_+$ with the property\n\t\\begin{equation}\n\t\t\\label{eq::to_drift}\n    \t\\CE{ f(q_{1},s_{1}) - f(q_0,s_0) }{q_0} \\leq K_4 - K_5 \\mathbf{1}_{n_q}^\\intercal q_0\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWithout further ado, we take expectations and sum \\eqref{eq::to_drift} over multiple time slots to obtain the following sequence of arguments\n\\begin{gather}\n\t\\nonumber\n    \\E{f(q_T,s_T)} - \\E{f(q_0,s_0)}\n    \\leq T K_4 - K_5 \\sum_{t = 0}^{T - 1} \\E{ \\mathbf{1}_{n_q}^\\intercal q_t }\n\\\\\n\t\\nonumber\n    \\Longrightarrow \\qquad\n    -\\E{f(q_0,s_0)} \n    \\leq T K_4 - K_5 \\sum_{t = 0}^{T - 1} \\E{ \\mathbf{1}_{n_q}^\\intercal q_t }\n\t\\hphantom{\\Longrightarrow \\qquad}    \n\\\\\n    \\Longrightarrow \\qquad\n    \\frac{1}{T} \\sum_{t=0}^{T-1} \\E{ \\mathbf{1}_{n_q}^\\intercal q_t } \n    \\leq\n    \\frac{K_4}{K_5} + \\frac{\\E{f(q_0)}}{T K_5}\n    \\hphantom{\\Longrightarrow }\n\\\\\n\t\\nonumber\n    \\Longrightarrow \\qquad\n    \\lim_{T\\to\\infty}\n    \\frac{1}{T} \\sum_{t=0}^{T-1} \\E{ q_t } \\leq\n    \\frac{K_4}{K_5} \\mathbf{1}_{n_q} \n    \\hphantom{\\Longrightarrow \\qquad}\n\\end{gather}\nSince $\\E{q_t} \\geq \\mathbf{0}$ always, and the difference between consecutive states is bounded, all elements of the sequence $(\\E{q_t})$ must be bounded. From there, the stability condition \\eqref{eq::def_stability} follows immediately. (Note that $\\E{\\cdot}$ is the expectation of the stochastic in the system model and not over time.)\n\\end{proof}\n\n\\subsection{Main Proof}\n\nWe now start with the main part of the proof. We will define a Ljapunov function $f(q_t,s_t)$ and show that \\textit{if} the system is governed by the $\\mathsf{PNC}$ policy, $f$ fulfills Lemma \\ref{lemma::foster} for any possibly stabilizable arrival rate (see \\eqref{eq::def_to}).\n\nFor a $\\mathsf{PNC}$ policy with horizon $H+1$, we employ the following Ljapunov function:\n\\begin{equation}\n    \\label{eq::to_f_tilde}\n\\begin{gathered}    \n    f(q_0,s_0) =\n    \\min_{ \\tr{z}_0^{H-1} \\in \\mathcal{C}_z } \n    \\CE{ \n    \t\t\\sum_{t=1}^{H} \\norm{q_{t}}\n    }{q_0,s_0}\n\\end{gathered}\n\\end{equation}\nA few remarks are in order: i) The minimization in $f$ mimics the $\\mathsf{PNC}$ policy, but is in fact independent of it. ii) The horizon of the minimization of $f$ is chosen to be one step smaller than that of the $\\mathsf{PNC}$ policy. iii) The control vectors are now denoted by $z$ instead of $v$ or $u$, because they run independent of the actual control $v_t$ or the predicted control inside the $\\mathsf{PNC}$ controller $u_t$. Crucially, the control trajectory $\\tr{z}_0^{H-1}$ is state sensitive regarding the DTMC $(s_t)$ of the weight matrices and is \\textit{not} constraint by the positiveness constraints $\\mathcal{P}$. I.e.\n\\begin{multline}\n\t\\label{eq::to_expected_evolution}\n    \\CE{q_{T}}{q_0,\\sigma_0}\n    = q_0 + \\sum_{t = 0}^{T-1} \\sum_{\\R{s} \\in \\mathcal{S}} \\left( \\sigma_0 P^{(t)} \\right)^\\R{s} R W^\\R{s} z_t(\\R{s}) + T\\e{a}\n\\end{multline}\n\nThis last point is important: the minimization of the actual $\\mathsf{PNC}$ policy (with horizon $H+1$) uses the control trajectory $\\tr{u}_0^{H}$, which assigns to each time slot of the prediction \\textit{a single} control vector $u_t \\in \\mathcal{V}$. In contrast and per definition, the minimization in the Ljapunov function $f$ uses the control trajectory $\\tr{z}_0^{H-1}$, which assigns to each time slot \\textit{and} each possible realization of $s_t$ a control vector $z_t(s_t) \\in \\mathcal{V}$. For succinct notation we define $z_t^\\R{s} := z_t(s_t = \\R{s})$. The trajectory of the control vectors $z_t^\\R{s}$ is defined by first stacking over all realization of $(s_t)$ and then over all time slots:\n\\begin{equation}\n\t\\tr{z}_0^{H} \n\t=\n\t\\begin{pmatrix}\n\t\tz'_0 \\\\\n\t\tz'_1 \\\\\n\t\t\\vdots \\\\\n\t\tz'_H\n\t\\end{pmatrix}\n\t,\\qquad \\text{with} \\qquad\n\tz'_t =\n\t\\begin{pmatrix}\n\t\tz_t^1 \\\\\n\t\tz_t^2 \\\\\n\t\t\\vdots \\\\\n\t\tz_t^{n_s}\n\t\\end{pmatrix}\n\\end{equation}\n\nThe constraint $\\tr{z}_0^{H-1} \\in \\mathcal{C}_z$ in the definition of $f$ expresses, that each single control vector $z_t^\\R{s}$ is constraint by the constituency in the usual way ($C z_t^\\R{s} \\leq c$), and therefore $\\mathcal{C}_z$ is a straight forward expansion of $\\mathcal{C}$.\n\nWe can now start expressing the first term of \\eqref{eq::to_drift} (for now conditioning on $s_0$ as well) as\n\\begin{gather}\n\t\\label{eq::to_simple_first_term}\n\\CE{ f (q_1,s_1) }{ q_0,s_0 }\t =\n\t\\CE{\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C} }\n\t\t\\CE{\n\t\t\t\\sum_{t=2}^{H+1} \\norm{q_t}\n\t\t}{\n\t\t\tq_1 , s_1\n\t\t}\n\t}{\n\t\tq_0,s_0\n\t}\n\\end{gather}\nNote that this term is shifted in time. The control $v_0$, that leads from $q_0$ to $q_1$ is exactly the control, that is defined by the control policy $v_0 = \\phi^\\mathsf{PNC}$ and that actually affects the network. In contrast, the dummy controls $\\tr{z}_1^H$ are part of the function $f$, do not affect the actual network and therefore are independent of the chosen policy.\n\nUsing \\eqref{eq::to_sub_to_q0} the inner term of \\eqref{eq::to_simple_first_term} can become\n\\begin{align*}\n\t\\sum_{t=2}^{H+1} \\norm{q_t}\n\t&=\n\t\\sum_{t=2}^{H+1} \\Big(\n\t\t\\norm{q_0} + \\norm{\\Delta_0^t} + 2q_0^\\intercal \\Delta_0^t\n\t\\Big)\n\t\\\\\n\t&\\leq\n\tH \\norm{q_0} + K_6\n\t+\n\t\\sum_{t=2}^{H+1} \\Big(\n\t\t2q_0^\\intercal \\Delta_0^t\n\t\\Big)\n\\end{align*}\nThe individual sums of $\\Delta_0^t$ can be bounded according to \\eqref{eq::to_gen_bounds} by some constant $K_6$ which is unaffected by the minimization or the expectation from \\eqref{eq::to_simple_first_term}. The same holds for $\\norm{q_0}$ if we notice, that conditioning on $q_1$ is the same as conditioning on $q_0$ \\textit{and} $\\Delta_0^1$, since $q_1 = q_0 + \\Delta_0^1$. Hence, both terms can be pulled to the left-hand-side of \\eqref{eq::to_simple_first_term}, as seen in the first two lines of \\eqref{eq::to_long}. In what follows, we will step by step dissolve the outer expectation and expand the sum, which is possible, since the minimization is linear in $\\tr{z}_1^H$ and the constraints act on each control vector separately:\n\\begin{align}\n\t\\label{eq::to_long}\n\t& \\hspace{5mm}\n\t\\CE{ f (q_1,s_1) }{ q_0,s_0 } - H \\norm{q_0} - K_6 \n\\\\ \n\t\\nonumber\n\t& \\leq \n\t\\mathbb{E} \\Bigg[\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=2}^{H+1}\t\t\t\n\t\t\t\t2q_0^\\intercal \\Delta_1^t\n\t\t\\, \\Bigg| \\,\n\t\t\tq_1 , s_1\n\t\t\\Bigg]\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\\\ \n\t\\nonumber\n\t& = \n\t\\mathbb{E} \\Bigg[\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z}\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=2}^{H+1}\t\t\t\n\t\t\t\t2q_0^\\intercal \\Delta_1^t\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0 ,\\Delta_0^1, s_1\n\t\t\\Bigg]\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\\\ \n\t\\nonumber\n\t& = \n\t\\sum_{\\R{q_0},\\R{s_1}}\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=2}^{H+1}\t\t\t\n\t\t\t\t2q_0^\\intercal  \\Delta_1^t\n\t\t\\, \\Bigg| \\,\n\t\t\t\\R{q_0} , \\R{s_1}\n\t\t\\Bigg]\n\t\\CP{\n\t\t\\R{q_0} , \\R{s_1}\n\t}{\n\t\tq_0,s_0\n\t}\n\\\\\n\t\\nonumber\n\t& \\hspace{1mm}\n\t\\begin{aligned}\n\t=\n\t\\sum_{\\R{q_0},\\R{s_1}}\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\t\t\n\t\t\t\t2q_0^\\intercal  \\left(\n\t\t\t\tR M_t z_t^{s_t} + a_t \\right)\n\t\t\\, \\Bigg| \\,\n\t\t\t\\R{q_0} , \\R{s_1}\n\t\t\\Bigg] &\n\t\\\\\n\t\\cdot\t\n\t\\CP{\n\t\t\\R{q_0}\n\t}{\n\t\tq_0\n\t}\n\t\\CP{\n\t\t\\R{s_1}\n\t}{\n\t\ts_0\n\t} \\, &\n\t\\end{aligned}\n\\\\ \n\t\\nonumber\n\t& \\hspace{1mm}\n\t\\begin{aligned}\n\t=\n\t\\sum_{\\R{s_1}}\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\t\t\t\n\t\t\t\t2q_0^\\intercal  \\left(\n\t\t\t\tR M_t z_t^{s_t} + a_t \\right)\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0 , \\R{s_1}\n\t\t\\Bigg] &\n\t\\\\\n\t\\cdot\n\t\\CP{\n\t\t\\R{s_1}\n\t}{\n\t\ts_0\n\t} \\, &\n\t\\end{aligned}\n\\\\ \n\t\\nonumber\n\t& \\hspace{1mm}\n\t\\begin{aligned}\n\t=\n\t\\sum_{\\R{s_1}}\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\n\t\t\t\\sum_{\\R{s_t}}\t\t\t\n\t\t\t\t2 q_0^\\intercal  \\left(\n\t\t\t\tR W^\\R{s_t} z_t^\\R{s_t} + a_t \\right)\n\t\t\\CP{ \\R{s_t} }{\\R{s_1}} &\n\t\\\\[-2ex]\n\t\\cdot\n\t\\CP{\n\t\t\\R{s_1}\n\t}{\n\t\ts_0\n\t} &\n\t\\end{aligned}\n\\\\ \n\t\\nonumber\n\t& \\hspace{1mm}\n\t\\begin{aligned}\n\t=\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\sum_{\\R{s_1}}\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\n\t\t\t\\sum_{\\R{s_t}}\t\t\t\n\t\t\t\t2q_0^\\intercal  \\left(\n\t\t\t\tR W^\\R{s_t} z_t^\\R{s_t} + a_t \\right)\n\t\t\\CP{ \\R{s_t} }{\\R{s_1}} &\n\t\\\\[-2ex]\n\t\\cdot\n\t\\CP{\n\t\t\\R{s_1}\n\t}{\n\t\ts_0\n\t} &\n\t\\end{aligned}\n\\\\ \n\t\\nonumber\n\t& = \n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\n\t\t\t\\sum_{\\R{s_t}}\t\t\t\n\t\t\t\t2q_0^\\intercal   \\left(\n\t\t\t\tR W^\\R{s_t} z_t^\\R{s_t} + a_t \\right)\n\t\t\\CP{ \\R{s_t} }{s_0}\n\\\\ \n\t\\nonumber\n\t& = \n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\n\t\t\t\t2q_0^\\intercal   \\left(\n\t\t\t\tR M_{t} z_t^{s_t} + a_t \\right)\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0,s_0\n\t\t\\Bigg]\n\\\\ \n\t\\nonumber\n\t& = \n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=2}^{H+1}\n\t\t\t\t2q_0^\\intercal  \n\t\t\t\t\\Delta_1^t\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0,s_0\n\t\t\\Bigg]\n\\\\ \n\t\\nonumber\n\t& =\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H+1}\n\t\t\t\t2q_0^\\intercal  \n\t\t\t\t\\Delta_0^t\n\t\t\t\t- 2q_0^\\intercal  \\Delta_0^1\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0,s_0\n\t\t\\Bigg]\n\\\\ \n    \\label{eq::to_last_long}\n\t& \\leq\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z}\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H+1}\n\t\t\t\t2q_0^\\intercal  \n\t\t\t\t\\Delta_0^t\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0,s_0\n\t\t\\Bigg]\n\t-\n\t\\min_{ \\tr{z}_0^{0} \\in \\hphantom{\\mathcal{C}} \\mathclap{\\mathcal{C}_z} }\n\t\\mathbb{E} \\Bigg[\t\n\t2q_0^\\intercal  \\Delta_0^1\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\end{align}\nThough $\\Delta_0^1$ is steered by $v_0$ (the actual control of the system), every policy has to abide to the constituency, which allows for the last term to be formulated over $\\tr{z}_0^0 \\Leftrightarrow z_0^{s_0}$. Still, the first term of \\eqref{eq::to_last_long} depends on $v_0$ through $\\Delta_0^t$ so that we can rewrite it as\n\\begin{multline}\n\\label{eq::to_elim_a}\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t\\sum_{t=1}^{H+1}\n\t\t\t2q_0^\\intercal \n\t\t\t\\Delta_0^t\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\t=\n\t\\frac{ \\left( H+1 \\right) \\left( H+2 \\right) }{2} \\e{a}\n\\\\\n\t\t+\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t2q_0^\\intercal R \\left[\n\t\t    \\sum_0^H\n\t\t        W^{s_0} v_0\n\t\t    +\n\t\t    \\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} z_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\right]\n\\end{multline}\n\nTo interface the $\\mathsf{PNC}$ policy \\eqref{eq::pnc_policy_def}, we need to incorporate the positiveness constraints. To that end, we introduce a transformation in variables, centered around the idea, that each set $\\{ z_t^1,\\dots z_t^{n_s} \\}$ can be expressed by a common part $\\mu_t$, and $n_s$ differences $\\delta_t^i$:\n\\begin{equation}\n\\begin{gathered}\n\tz_t^i = \\mu_t + \\delta_t^i \\in \\mathcal{C}_z\n\t\\\\\n\t\\text{for} \\qquad  t = 1,\\dots H \\qquad \\text{and} \\qquad i = 1 ,\\dots n_s\n\t\\\\\n\t\\text{with}  \\qquad  \\mu_t,\\delta_t^i \\in \\{0,1\\}^{n_v} \\qquad \\text{and} \\qquad C \\mu_t \\leq c \n\\end{gathered}\n\\end{equation}\nWe define a suitable stacking of these new variables in such a way that\n$\\tr{z}_1^H = \\tr{\\mu}_1^H + \\tr{\\delta}_1^H$ and write $\\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H $, to express that $\\tr{\\delta}_1^H$ has to abide to usual constituency, if $\\tr{\\mu}_1^H$ has already been chosen (i.e. for each $\\delta_t^i$ separately it must hold that $C \\delta_t^i \\leq c - C \\mu_t$).\n\nNext, we substitute these variables into the last term of \\eqref{eq::to_elim_a}.\n\\def\\hspace{1cm}{\\hspace{1cm}}\n\\def\\hspace{3mm}{\\hspace{3mm}}\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align}\n    \\nonumber\n    &\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t2q_0^\\intercal  R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H}  W^{s_0} v_0\n\t\t +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} z_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\t\\min_{ \\tr{\\mu}_1^H + \\tr{\\delta}_1^H \\in \\mathcal{C}_z }\n\t\t2q_0^\\intercal  R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H} W^{s_0} v_0\n\t\t\\\\ \\nonumber\n\t\t& \\hspace{1cm} +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} \\left( \\mu_t + \\delta_t^\\R{s_t} \\right) \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\tr{\\mu}_1^H \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H} W^{s_0} v_0\n\t\t\\\\ \\nonumber\n\t\t& \\hspace{1cm} +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\left( \\sum_{\\R{s_t}} W^\\R{s_t} \\CP{\\R{s_t}}{s_0} \\right) \\mu_t \n\t\t\\\\ \\nonumber\n\t\t& \\hspace{1cm} +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} \\delta_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\tr{\\mu}_1^H \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H} W^{s_0} v_0\n\t\t\\\\ \\nonumber\n\t\t& \\hspace{1cm} +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\e{W}_t(s_0) \\mu_t\n\t\t\\\\ \\nonumber\t\n\t\t& \\hspace{1cm} +\t\t\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} \\delta_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t\\leq\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H}  \\left( \\mathbf{0}_{n_v}^\\intercal , \\left( \\tr{\\mu}_1^H\\right )^\\intercal \\right)^\\intercal \\in \\mathcal{C} \\cap \\mathcal{P}(q_0)}\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\cdot\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t\\overset{\\mathclap{\\mathsf{PNC}}}{=}\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\left( v_0^\\intercal , \\left( \\tr{\\mu}_1^H\\right )^\\intercal \\right)^\\intercal \\in \\mathcal{C} \\cap \\mathcal{P}(q_0) }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\cdot\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t\\overset{\\mathclap{\\text{Lemma \\ref{lemma::diff}}}}{\n\t\\leq\n\t}\n\tK_7 +\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\left( v_0^\\intercal , \\left( \\tr{\\mu}_1^H\\right )^\\intercal \\right)^\\intercal \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\cdot\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t\\leq\n\tK_7 +\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H}  v_0 \\in \\mathcal{C} }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\tr{\\mu}_1^H \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\cdot\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\tK_7 +\n\t\\min_{ \\tr{z}_1^H \\in \\mathcal{C}_z }\n\t\\hspace{3mm}\n\t\\min_{ v_0 \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H} W^{s_0} v_0\n\\\\ \\nonumber\n\t\t& \\quad +\t\t\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} z_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\tK_7 + \n\t\\min_{ \\tr{z}_0^H \\in \\mathcal{C}_z }\n\t2q_0^\\intercal R \n\t\t\\sum_{t=0}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} z_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\\\\\n    \\label{eq::to_intro_PNC}\n\t&\n\t=\n\tK_7 + \n\t\\min_{ \\tr{z}_0^H \\in \\mathcal{C}_z }\n\t\t\\CE{ \\sum_{t=1}^{H+1} 2q_0^\\intercal R \\Delta_0^t }{q_0,s_0}\n\\end{align}\nNote that we used the fact that $\\mu_t$, once separated from the $\\delta_t^i$ in a suitable manner, can be identified with the dummy control $u_t$ from the definition of the $\\mathsf{PNC}$ policy \\eqref{eq::pnc_objective}. Hence, the equality marked with the $\\mathsf{PNC}$ label \\textit{only} holds under the $\\mathsf{PNC}$ policy, since it chooses $v_0$ in such a way that the entire first term is minimized over the trajectory $\\tr{z}_0^H$ instead of only $\\tr{z}_1^H$.\n\nIf we now combine the results of \\eqref{eq::to_long}, \\eqref{eq::to_elim_a} and \\eqref{eq::to_intro_PNC} we get\n\\begin{gather}\n\t\\label{eq::to_first_drift_term}\n\t\\CE{ f (q_1,s_1) }{ q_0,s_0 } - H \\norm{q_0} - K_8 \n\t\\\\\n\t\\nonumber\n\t\\leq\n\t\\min_{ \\tr{z}_0^{H} \\in \\hphantom{\\mathcal{C}} \\mathclap{\\mathcal{C}_z} }\n\t\\mathbb{E} \\Bigg[\n\t\t\\sum_{t=1}^{H+1}   2q_0^\\intercal  \\Delta_0^t\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\t-\n\t\\min_{ \\tr{z}_0^{0} \\in \\hphantom{\\mathcal{C}} \\mathclap{\\mathcal{C}_z} }\n\t\\mathbb{E} \\Bigg[\t\n\t2q_0^\\intercal  \\Delta_0^1\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\end{gather}\nwhich completes the derivation for the first term of \\eqref{eq::to_drift}.\n\nIn a similar but much easier fashion, the second term of \\eqref{eq::to_drift} can be reshaped into\n\\begin{equation}\n\t\\label{eq::to_second_drift_term}\n\\begin{gathered}\n\t\\CE{ f (q_0,s_0) }{ q_0,s_0 } - H \\norm{q_0}\n\t\\\\\n\t\\geq\n\t\\min_{ \\tr{z}_0^{H-1} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t\\sum_{t=1}^{H}   2q_0^\\intercal  \\Delta_0^t\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\end{gathered}\n\\end{equation}\nCombining \\eqref{eq::to_first_drift_term} and \\eqref{eq::to_second_drift_term} results in\n\\begin{gather*}\n\t\\CE{ f (q_1,s_1) - f (q_0,s_0) }{ q_0,s_0 }\n\\\\\n\t\\leq\n\tK_{8} +\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t2q_0^\\intercal  \\Delta_1^{H+1}\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\end{gather*}\nTo alleviate the outer conditioning on $s_0$ we take the expectation $\\CE{\\cdot}{q_0}$ on both sides, conditioned only on $q_0$, and swap minimization and expectation operator:\n\\begin{gather*}\n\t\\CE{ f (q_1,s_1) - f (q_0,s_0) }{ q_0 }\n\\\\\n\t\\leq\n\tK_{9} +\n\t\\mathbb{E} \\Bigg[\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t2q_0^\\intercal  \\Delta_1^{H+1}\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\t\\, \\Bigg| \\,\n\t\tq_0\n\t\\Bigg]\n\\\\\n\t\\leq\n\tK_{9} +\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t2q_0^\\intercal  \\Delta_1^{H+1}\n\t\\, \\Bigg| \\,\n\t\tq_0\n\t\\Bigg]\n\\end{gather*}\nFinally, recall that for throughput optimality, this expression has to be negative for each $\\e{a}$ that can be expressed via \\eqref{eq::def_to}. Substituting this we obtain\n\\begin{gather*}\n\t\\CE{ f (q_1,s_1) - f (q_0,s_0) }{ q_0 }\n\\\\\n\\begin{aligned}\n\t& \\leq\n\tK_{9} +\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t2q_0^\\intercal \\left(\n\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\R{s}}  \\pi^\\R{s} R  W^\\R{s} z_t^\\R{s} + H \\e{a}\n\t\t\\right)\n\\\\ &\n\\begin{multlined}\n\t\\leq\n\tK_{9} +\n\t\\min_{ z' \\in \\mathcal{C}_z }\n\t\t2Hq_0^\\intercal \\Bigg(\n\t\t\\sum_{\\R{s}} \\pi^\\R{s} RW^\\R{s} z^\\R{s}\n\t\\\\\n\t\t-  \\varepsilon \\mathbf{1}_{n_q}\n\t\t-  \\sum_{\\R{s}} \\pi^\\R{s} R W^\\R{s} \\sum_{v \\in \\mathcal{V}} \\lambda^{\\R{s},v} v\n\t\t\\Bigg)\n\\end{multlined}\t\n\\end{aligned}\t\n\\end{gather*}\nBecause the minimization is linear, the optimum is found on the boundary and thus the first term in the bracket (which is subject to minimization) will at least cancel out the last term, leaving us with\n\\begin{align*}\n\t\\CE{ f (q_1,s_1) - f (q_0,s_0) }{ q_0 }\n\t& \\leq\n\tK_{9}\n\t-\n\t2 H \\varepsilon q_0^\\intercal  \\mathbf{1}_{n_q}\n\t\\\\\n\t& \\leq\n\tK_{9}\n\t-\n\tK_{10} \\mathbf{1}_{n_q}^\\intercal q_0 \n\\end{align*}\nwhich fulfills lemma \\ref{lemma::foster} and therefore proves throughput optimality of our $\\mathsf{PNC}$ policy.\n\\\\\n\\rightline{$\\Box$}\n\n\\section{Exemplary Applications of PNC}\n\\label{sec::examples}\n\n\\subsection{Dynamic Topology}\n\nWe employ a scenario as depicted in \\figref{fig::ext_1_scenario}, where a mobile user equipment (UE) crosses multiple sectors, each one designated to a specific access point (AP). In each sector, the UE can only communicate with the corresponding AP. The APs are connected to a global network from which they receive packets that they are supposed to transmit to the UE.\nThe derived queueing network is shown in \\figref{fig::ext_1_model}. We use a most simplified model to yield easily interpretable results: First, the DTMC is deterministic which allows us to fix the time behavior of the transmission success probabilities $\\e{m}^j_t$ of the links. Second, we model this deterministic time behavior as binary sequences which are depicted in \\figref{fig::ext_1_links}. This corresponds to the case, in which the UE travels with constant velocity along a known path and the sectors do not overlap. The UE remains in each sector for exactly 3 time slots, where it experiences perfect channel quality (guaranteed transmission success).\nFurther we assume that a single packet is created every second time slot at $q^1$, which represents the entire arrival to the system.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_DTMC_f.pdf}\n  \\caption{Extension 1 - Scenario}\n  \\label{fig::ext_1_scenario}\n\\end{figure}\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_DTMC_model_f.pdf}\n  \\caption{Example 1 - Queueing Network}\n  \\label{fig::ext_1_model}\n\\end{figure}\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_DTMC_links_f.pdf}\n  \\caption{Example 1 - Link Probabilities}\n  \\label{fig::ext_1_links}\n\\end{figure}\n\nThe simulation results in \\figref{fig::ext_1_simu_2} depict the accumulated amount of packets send (blue) and received by the UE (green and red). For visualization purposes, we averaged the resulting step functions, so that they are presented as lines. It can be observed that only around 33\\% of the packets reach the UE for the conventional back pressure policy, $\\mathsf{MW}$, (red). (Note that $\\mathsf{MW}$ can be expressed as a special case of the $\\mathsf{PNC}$ policy, in which the horizon is $H=1$.) The other 66\\% remain at already past base stations. This high packet loss is due to $\\mathsf{MW}$ requiring time to establish its throughput optimality. Indirectly, $\\mathsf{MW}$ functions by using misplaced packets as an indicator for later control decisions. The presented example, however, is based on a transient event where this indicator function of misplaced packets is only of limited use.\n\nAs can be seen, using the novel $\\mathsf{PNC}$ with horizon $H=2$, (the lowest green line) already nearly doubles the amount of packets that arrive at the UE to 60\\%. For $H=5$ we reach 80\\%, a significant performance boost.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_DTMC_simu_f.pdf}\n  \\caption{Example 1 - Simulation with Multiple Policies}\n  \\label{fig::ext_1_simu_2}\n\\end{figure}\n\n\\subsection{Networks with Synchronized Queues}\n\\label{subsec::synchronized_queues}\n\nThe following application is motivated by fact, that though $\\mathsf{MW}$ performs poorly in networks with dynamic topology, it is still able to achieve throughput optimality in the long run, if we assume $\\mathsf{MW}$ to be sensitive to the current state of the DTMC. And it is only fair to make this assumption, since we assume the same for our $\\mathsf{PNC}$ policy. Hence, throughput optimality seems to be shared by both policies, if we talk about conventional networks. However, in the next example, we forgo conventional networks and introduce \\textit{synchronized} queues. In networks with synchronized queues, only $\\mathsf{PNC}$ seems to maintain its throughput optimality while $\\mathsf{MW}$ fails, giving a strong incentive to employ the $\\mathsf{PNC}$ policy.\n\nQueues are \\textit{synchronized} (or \\textit{paired}), if they can only be served at the same time. This can be useful, if one wants to\nexploit constructive interference \\cite{Timotheou2016} or\nmodel parallel processing tasks in computing \\cite{Evdokimova2018} and social matchmaking \\cite{Buke2015}.\nWhile synchronized queues have been studied on their own \\cite{Harrison1973} \\cite{Fayolle1979} \\cite{Borst2008} \\cite{DeCuypere2014}, there has not been any research on how they behave in a network. Indeed, \\cite{Schoeffauer2018a} presents a simple example, proving that $\\mathsf{MW}$ loses its throughput optimality if the network contains only a single pair of synchronized queues. The reason for that can be found equation A.18 from the original proof in \\cite{Tassiulas1992}, which loses its generality. In layman's terms, the original proof is based on the fact, that the evolution of the queue vector constitutes a DTMC by itself. And for conventional networks, there exists a \\textit{finite} set of states (of that DTMC) which can be shown to be recurrent. This makes the entire DTMC recurrent which corresponds to throughput optimality. However, introducing synchronized queues, the finite set grows to infinite size, invalidating this correspondence. This leads to the questions, in how far back-pressure policies are suited for such networks and if there exists another policy, which guarantees throughput optimality.\n\nTo illustrate that $\\mathsf{PNC}$ might be that policy, we refer to the example, depicted in \\figref{fig::constructive_interference_set_up}. Set-up and thereof derived queueing network are shown on the left and right side, respectively. The example consists of an access point (AP) $q_1$ that can either transmit solitary (link $r^1$), or initiate synchronized transmission (link $r^3$) with a neighboring AP $q_2$. The synchronized transmission uses constructive interference and thus achieves higher throughput. However, before synchronized transmission can be initiated, the data packets have to be shared (link $r^2$), i.e. copied from $q_1$ to $q_2$.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_LT_f.pdf}\n  \\caption{Example 2 - Scenario and Queueing Network}\n  \\label{fig::constructive_interference_set_up}\n\\end{figure}\n\nFor comprehensiveness, we use a constant success probability matrix $\\e{M}$, i.e. we do not make use of an DTMC to select different matrices from $\\mathcal{W}$. The $\\e{m}^j$ (which are the diagonal elements of $\\e{M}$) are chosen in such a way, that it is beneficial to copy (share) the data and then transmit together, instead of broadcasting the data directly. \nSpecifically, we set $\\e{m}^1 = \\frac{1}{4}$ and $\\e{m}^2 = \\e{m}^3 = 1$ and assume all links to be disjunct. Note, that we can neglect $q^3$ in all further discussions, since it only symbolizes the destination queue.\n\nFor this simple example, it is prudent to forgo working in terms of the \\textit{control vector} $v_t$ and instead use the \\textit{control option} $u_t$, which we define to be $u_t = R \\e{M} v_t$ (this $u_t$ is not related to the one, that was used in the definition of the $\\mathsf{PNC}$ policy). We have $u_t \\in \\mathcal{U}$, which can be derived from the system description without further ado to be\n\\begin{equation}\n    \\mathcal{U} =\n    \\Set{\n    u^0 , u^1 , u^2 , u^3\n    }\n    = \n    \\Set{\n    \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix} ,\n    \\begin{pmatrix} -1 \\\\ 0 \\end{pmatrix} ,\n    \\begin{pmatrix} 0 \\\\ 4 \\end{pmatrix} ,\n    \\begin{pmatrix} -4 \\\\ -4 \\end{pmatrix}\n    }\n\\end{equation}\nwhere we scaled all elements of $\\mathcal{U}$ with the factor $4$ to simplify any calculations. We have $u^1, u^2, u^3$ represent single transmission, data sharing, and joint transmission, respectively and in each time slot, the controller may only choose one of these options to influence the expected queue state via $\\CE{q_{t+1}}{q_t} = q_t + u_t + a_t$.\n\nRegarding \\eqref{eq::def_to}, it is now very easy to express the set of all arrival rates $\\e{a}$, for which there exists a policy that stabilizes the system. We call this set the maximum stability region $\\mathcal{A}$ and have\n\\begin{equation}\n\t\\begin{aligned}\n\t\\mathcal{A} :&= \\Set{\n\t\t\\e{a} : \\quad\n\t\t\\e{a} + \\sum_{u \\in \\mathcal{U}} \\lambda^u u = -\\mathbf{1} \\varepsilon, \\quad\n\t\t\\begin{gathered}\n\t\t\t\\varepsilon > 0\n\t\t\t\\\\\n\t\t\t\\sum \\lambda^u \\leq 1\n\t\t\\end{gathered}\n\t}\n\t\\\\\n\t&=\n\t\\Set{\n\t\t\\e{a} : \\quad\n\t\t\\e{a} + \\sum_{u \\in \\mathcal{U}} \\lambda^u u =  \\mathbf{0}, \\quad \\hspace{4.5mm}\n\t\t\t\\sum \\lambda^u < 1\n\t}\n\t\\end{aligned}\n\\end{equation}\nRemember that throughput optimality is accomplished, if a policy can stabilize the system for all arrival rates in $\\mathcal{A}$.\nA graphical representation of $\\mathcal{A}$ is given in green on the left side of \\figref{fig::constructive_interference_stab_region}.\n\nNow, let us assume that there is no arrival at $q^2$, i.e. $\\e{a}^2 = 0$.\nUsing the control options $u^2$ and $u^3$ in alternating sequence (given that there are enough packets in $q^1$ to do so) would yield an efflux of $4$ packets every $2$ time slot, thus an efflux of $2$ packets per time slot. The corresponding point is shown on the right side of \\figref{fig::constructive_interference_stab_region}.\nIt is easy to check, that no other sequence of control options can match this efflux.\n\nHowever, conventional back-pressure policies like $\\mathsf{MW}$ are not able to access the control option $u^2$, resulting in the loss of its throughput optimality in this example. The only arrival rates, that $\\mathsf{MW}$ \\textit{can} stabilize are those in the red triangle on the right side of \\figref{fig::constructive_interference_stab_region}.\n\n\\begin{figure}[htbp]\n  \\centering\n \n  \\includegraphics[]{stab_region_simple_f.pdf}\n  \\vspace{0mm}\n    \\centering\n  \\caption{Example 2 - Stability Regions}\n  \\label{fig::constructive_interference_stab_region}\n\\end{figure}\n\nIn contrast, $\\mathsf{PNC}$ is able to select the missing control option $u^2$ and simulations suggest, that it stabilizes the example for any strictly positive arrival rate $\\e{a}$ from $\\mathcal{A}$. To substantiate this claim we refer to \\figref{fig::simu_comparison}. Here, we simulated the queue state $q^1$ over time $t$ for 3 different arrival rates $\\e{a}$ under 3 different policies. The respective positions of those $\\e{a}$ regarding $\\mathcal{A}$ are depicted in \\figref{fig::simu_points}. As for the policies, we chose $\\mathsf{MW}$ and $\\mathsf{PNC}$. Also, we added a third control policy, labeled $\\mathsf{fPNC}$ for \\textit{fixed} $\\mathsf{PNC}$. This policy mimics the $\\mathsf{PNC}$ policy, except that it uses the entire calculated control trajectory before repeating the optimization. In contrast, $\\mathsf{PNC}$ repeats the optimization every time slot again.\n\nAs predicted, we have $\\mathsf{MW}$ not stabilizing the blue and green arrival rates. Furthermore, it can be seen, that $\\mathsf{fPNC}$ loses some stabilizing properties with increasing horizon as the green arrival rate cannot be stabilized with $H=3$ (this is related to the horizon not being an even number). This proves, that the MPC paradigm of repeating the optimization in every step (and thereby discarding the rest of the trajectory) is an essential part in the $\\mathsf{PNC}$ policy.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{stab_region_points_f.pdf}\n  \\caption{Example 2 - Selected arrival rates for simulation; Corresponding stability regions: (red -- $\\mathsf{MW}$), (red+blue -- $\\mathsf{fPNC}_{H=3}$), (red+blue+green -- $\\mathsf{PNC}$, $\\mathsf{fPNC}_{H=2}$)}\n  \\label{fig::simu_points}\n\\end{figure}\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{horizon_dependent_stability_f.pdf}\n  \\caption{Example 2 - \n  Queue state (system state) $q^1$ as a function of time $t$ for various (color-coded) arrival rates}\n  \\label{fig::simu_comparison}\n\\end{figure}\n\n\\section{Conclusion}\nWe successfully modify a discrete-time queueing network with a JMS, i.e. with an additional DTMC that changes network parameters (even topology) on a mid- to long-term time scale. We then introduce a novel family of predictive control policies, $\\mathsf{PNC}$, based on the paradigms of MPC, and devise a special implementation of the underlying prediction, that allows the policy to be executed in the fastest way possible. The policy is especially well suited to control the mentioned systems and outperforms conventional control approaches as is illustrated in numerical simulations. In our main contribution, we prove throughput optimality of $\\mathsf{PNC}$. Looking ahead, we see an intriguing application in networks that consist of synchronized queues (e.g. found in parallel computing or manufacturing chains). Those networks still elude known control strategies but seem to be stabilizable under $\\mathsf{PNC}$ policies with suitably chosen prediction horizon.\n\n\\section*{Acknowledgment}\nThis work is part of and thereby funded by the DFG Priority Program 1914\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Motivation} \n Information on hyperon resonances is generally not as extensive as for nucleon resonances. The study of $\\overline K N\\rightarrow \\overline K N$, $\\overline K N\\rightarrow \\pi \\Lambda$, and $\\overline K N\\rightarrow \\pi \\Sigma$ could lead to the better understanding of $\\Lambda^*$s and $\\Sigma^*$s.\n\n     Most previous partial-wave analyses (PWAs) of $\\overline K N\\rightarrow \\overline K N$, $\\overline K N\\rightarrow \\pi \\Lambda$, and $\\overline K N\\rightarrow \\pi \\Sigma$ {\\cite{Armenteros1969, Conforto1971, Horn1975_1, Hemingway1975, Baillon1975,  Gopal1977}}, were based on the assumption that  partial-wave amplitudes could be represented by a simple sum of resonant and  background terms. Such an assumption violates  unitarity of the partial-wave $S$-matrix. In this work, we report on our investigation of the reactions $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$, $K^-p\\rightarrow \\pi^0\\Lambda$, and $K^-p\\rightarrow\\pi^+\\Sigma^-$, $K^-p\\rightarrow\\pi^0\\Sigma^0$, and $K^-p\\rightarrow\\pi^-\\Sigma^+$ via single-energy analyses and a subsequent energy-dependent analysis.  All available differential cross section, polarization, polarized cross section, and cross-section data up to a maximum c.m.\\ energy of about 2.1 GeV were fitted. In order to ensure that our amplitudes had a relatively smooth variation with energy, we introduced several constraints that will be described in detail below. The determination of resonance parameters from our subsequent energy-dependent analysis is discussed in Ref. \\cite{manoj2013}.\n\n\n         \n \n\n   \n   \\section{Formalism and Fitting Procedures}\n        Here, we summarize the formalism for the single-energy partial-wave analyses.\n        The differential cross section $\\rm d\\sigma \/\\rm d\\Omega$ and polarization $P$ for unpolarized scattering of spin-0 mesons off spin-$\\frac12$ nucleons are given by \\cite{bransden73}\n               \\begin{equation}\n               \\frac{{\\rm d}\\sigma}{{\\rm d}\\Omega} = {\\lambdabar}^2(|f|^2+|g|^2)~,\n                \\end{equation}\n                \\begin{equation}\n                P\\frac{{\\rm d}\\sigma}{{\\rm d}\\Omega} =2{\\lambdabar}^2\\rm Im(fg^\\ast)~,\n                \\end{equation}\n        where $\\lambdabar = {\\hbar}\/{k}$,\n         with $k$ the magnitude of c.m.\\ momentum of the incoming meson.\n         Here, $f = f(W,\\theta)$ and $g = g(W,\\theta)$ are the usual spin-non-flip and spin-flip amplitudes at c.m.\\ energy $W $ and meson c.m.\\ scattering angle $\\theta$. In terms of partial waves, $f$ and $g$ can be expanded as\n       \n       \\begin{equation}\n        f(W,\\theta) = \\sum_{l=0}^{\\infty} [(l+1)T_{l+} + lT_{l-}]P_l(\\cos\\theta)~,\n        \\end{equation}\n        \\begin{equation}\n        g(W,\\theta) = \\sum_{l=1}^{\\infty} [T_{l+} - T_{l-}]P_l^1(\\cos\\theta)~,\n        \\end{equation}\n        where $l$ is the initial orbital angular momentum, $P_l(\\cos\\theta)$ is a Legendre polynomial and $P_l^1(\\cos\\theta) = \\sin\\theta \\cdot {\\rm d} P_l(\\cos\\theta)\/{\\rm d}(\\cos\\theta$). The total angular momentum for the amplitude $T_{l+}$ is $J=l+\\frac12$, while that for the amplitude $T_{l-}$ is $J=l-\\frac12$.\n              For the initial $\\overline K N$ system, we have $I = 0$ or $I =1$ so that the amplitudes $T_{l\\pm}$ can be expanded in terms of  isospin amplitudes as \n         \\begin{equation}\n               T_{l\\pm} = C_{0}T^{0}_{l\\pm} + C_{1}T^{1}_{l\\pm}~,\n       \\end{equation}\n       \\newline\n       where $T^I_{l\\pm}$ are partial-wave amplitudes\n        with isospin $I$  and total angular momentum $J = l\\pm\\frac12$ with $C_I$ the appropriate isospin Clebsch-Gordon coefficients for a given reaction.\n       For $K^- p \\rightarrow K^- p$, for example, we have $C_{0} = {\\frac12}$ and $C_{1}={\\frac12}$.\n\\newline\n\n        The total $K^- p$ cross section is given by $\\sigma_{\\rm total} = 4\\pi {\\lambdabar}^2 {\\rm Im} f(W, 0)$, or\n\\begin{equation}\\label{eq:Sigma_total}\n\\sigma_{\\rm total} = 4\\pi \\lambdabar^2 \\sum_{l=0}^{\\infty}[(l+1) {\\rm Im}T_{l^+} + l{ \\rm Im}T_{l^-}],\n\\end{equation}\nwhere here the $T_{l\\pm}$ are partial-wave amplitudes for elastic $\\overline KN$ scattering.\n\n\n  The integrated cross section  for a particular two-body reaction is\n\\begin{equation}\n\\sigma=4\\pi {\\lambdabar}^2 \\sum_{l=0}^{\\infty} [(l+1) |T_{l^+}|^2+l |T_{l^-}|^2] .\n\\end{equation}\n\n  Tables I, II, and III summarize  the available quantity and types of data in each energy bin for the three reactions $\\overline K N\\rightarrow \\overline K N$, $\\overline K N\\rightarrow \\pi \\Lambda$, and $\\overline K N\\rightarrow \\pi \\Sigma$, respectively.   \n   Single-energy fits were performed separately for (i)  $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$, for (ii) $K^-p\\rightarrow \\pi^0\\Lambda$, and for (iii) $K^-p\\rightarrow\\pi^+\\Sigma^-$, $K^-p\\rightarrow\\pi^0\\Sigma^0$, and $K^-p\\rightarrow\\pi^-\\Sigma^+$. In each case the available data were analyzed in  c.m.\\ energy bins of width 20 MeV. This choice of bin width was appropriate because the data for smaller widths had unacceptably low  statistics and for larger  widths, some amplitudes varied too much over the energy spread of the bin. \n \n\n\n\n\\begin{table*}[htbp]\n\\caption{Summary of database for $\\overline{K} N \\rightarrow \\overline{K} N$. Column 1 lists the central energy $W_0$ of each energy bin, columns 2 and 3 list the number of differential cross-section data points in each bin for $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$, respectively, column 4 lists the number of polarization data points for $K^-p\\rightarrow K^-p$ in each bin, column 5 lists the number of polarized cross-section data points for $K^-p\\rightarrow K^-p$ in each bin, column 6 lists the number of integrated cross-section data points for $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$ in each bin, column 7 lists the number of $K^-p$ total cross-section data points in each bin, column 8 lists the total number of data points for all kinds of data in each bin, and column 9 lists the references for the measurements referred to in columns 2-5.}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccc}\n\\multicolumn{1}{c}{$W_{0}$} & \\multicolumn{1}{c}{{${d\\sigma}\/{d\\Omega}$ } } & \\multicolumn{1}{c}{${d\\sigma}\/{d\\Omega}$ { } }& \\multicolumn{1}{c}{{\\mbox{$P$}}}& \\multicolumn{1}{c}{{\\mbox{$P$}${d\\sigma}\/{d\\Omega}$ }} & \\multicolumn{1}{c}{\\multirow{2}{*}{$\\sigma$}}& \\multicolumn{1}{c}{\\multirow{2}{*}{$\\sigma_{\\rm total}$}} & \\multicolumn{1}{c}{ Total} & \\multicolumn{1}{c}{\\multirow{2}{*}{ References}} \\\\\n\\multicolumn{1}{c}{\\rm (MeV)} & ($K^- p$) &($\\overline{K}^0 n$) & ($K^- p$)& ($K^- p$)& & & \\multicolumn{1}{c}{ No.} &  \\multicolumn{1}{c}{} \\\\\n\n\\hline\n1480\t&\t 57&\t 24&\t&\t\t&15\t&\t5&   96\t&\t \\cite{Mast1976}\\\\\n1500\t&\t114&\t120&\t&&\t\t23\t&\t8&  257\t\t& \\cite{Mast1976}\\\\\n1520\t&\t100&\t100&\t&&\t\t28\t&  10&  228\t\t& \\cite{Mast1976}\\\\\n1540\t&\t178&\t120&\t&&\t\t25\t&\t8&  323\t\t& \\cite{Armenteros1970, Mast1976}\\\\\n1560&\t117&\t100&\t&&\t\t14\t&\t5&  231\t\t& \\cite{Armenteros1970, Alston1978_1, CrystalBall2005}\\\\\n1580&\t78&\t112&\t&\t&\t10\t&\t3&   200\t\t\t& \\cite{Armenteros1970, Alston1978_1, CrystalBall2005}\\\\\n1600&\t79&\t116&\t&\t&\t14\t&\t6&   209\t\t\t& \\cite{Armenteros1970, Alston1978_1, CrystalBall2005}\\\\\n1620&\t147&\t120&\t&&\t\t14\t&\t6&  281\t\t& \\cite{Armenteros1970, Adams1975, Alston1978_1, CrystalBall2005}\\\\\n1640\t&\t113&\t148&\t&&\t\t14\t&\t6&  275\t\t& \\cite{Armenteros1970, Adams1975, Alston1978_1, CrystalBall2005}\\\\\n1660\t&\t149&\t132&\t&&\t\t19\t&\t6&  300\t\t& \\cite{Armenteros1970, Adams1975, Alston1978_1, CrystalBall2005}\\\\\n1680\t&\t194&\t210&\t&&\t\t30\t&\t9&  434\t\t& \\cite{Armenteros1968, Armenteros1970, Conforto1971, Adams1975, Alston1978_1, CrystalBall2005}\\\\\n1700\t&\t150&\t112&\t&&\t\t11\t&\t6&  283\t\t& \\cite{Armenteros1968, Armenteros1970, Conforto1971, Adams1975, Alston1978_1}\\\\\n1720\t&\t150&\t150&\t&&\t\t20\t&\t6&  320\t\t& \\cite{Armenteros1968, Conforto1971, Adams1975, Alston1978_1}\\\\\n1740\t&\t216&\t241&\t&27&\t\t29\t&\t6&    513\t\t & \\cite{Armenteros1968, Albrow1971, Conforto1971, Jones1975, Adams1975, Alston1978_1}\\\\\n1760&\t176&\t193&\t&26&\t\t25\t&\t4&    420\t\t & \\cite{Armenteros1968, Albrow1971, Conforto1971, Jones1975, Adams1975, Alston1978_1}\\\\\n1780\t&\t185&\t196&\t&27&\t\t34\t&\t9&    442\t\t & \\cite{Armenteros1968, Andersson1970, Conforto1971, Jones1975, Conforto1976, Alston1978_1}\\\\\n1800&\t132&\t79&\t&52&\t\t18\t&\t4& 281\t\t\t& \\cite{Armenteros1968, Conforto1971, Jones1975, Conforto1976}\\\\\n1820&\t146&\t60&\t&27&\t\t18\t&\t5& 251\t\t\t& \\cite{Armenteros1968, Albrow1971, Conforto1971, Conforto1976}\\\\\n1840&\t185&\t80&\t&27&\t\t23\t&\t9& 315\t\t\t& \\cite{Armenteros1968, Andersson1970, Conforto1971, Conforto1976}\\\\\n1860&\t227&\t100&\t&30&\t\t22\t&\t4&    379\t\t & \\cite{Armenteros1968, Andersson1970, Conforto1971, Griselin1975, Conforto1976}\\\\\n1880&\t266&\t120&\t&28&\t\t21\t&\t6&    435\t\t & \\cite{Armenteros1968, Albrow1971, Conforto1971, Griselin1975, Conforto1976}\\\\\n1900\t&\t175&\t60&\t&56&\t\t18\t&\t6& 309\t\t\t& \\cite{Armenteros1968, Andersson1970, Albrow1971, Conforto1971, Griselin1975, Conforto1976}\\\\\n1920\t&\t146&\t60&\t&27&\t\t17\t&\t3& 250\t\t\t& \\cite{Andersson1970, Griselin1975, Conforto1976}\\\\\n1940&\t110&\t80&\t&30&\t\t18\t&\t5& 238\t\t\t& \\cite{Albrow1971, Griselin1975, Conforto1976}\\\\\n1960&\t64&\t40&\t23&&\t\t14\t&\t4&  141\t\t& \\cite{Daum1968, Griselin1975, Conforto1976}\\\\\n1980&\t34&\t20&\t&&\t\t11\t&\t3&    65\t\t& \\cite{Griselin1975, Abe1975}\\\\\n2000&\t23&\t20&\t23&&\t\t9\t&\t3&   75\t\t& \\cite{Daum1968, Abe1975}\\\\\n2020\t&\t23&\t&\t23&&\t\t12\t&\t5&    58\t\t& \\cite{Daum1968}\\\\\n2040\t&\t54&\t&\t22&&\t\t10\t&\t4&    86\t\t& \\cite{Daum1968, Abe1975}\\\\\n2060&\t23&\t&\t23&&\t\t10\t&\t3&    56\t\t& \\cite{Daum1968}\\\\\n2080\t&\t22&\t&\t22&&\t\t9\t&\t3& 53\t\t& \\cite{Daum1968}\\\\\n2100&\t53&\t&\t22&&\t\t7\t&\t2& 83\t\t& \\cite{Daum1968, Abe1975}\\\\\n2120\t&\t46&\t&\t23&46&\t\t12\t&\t5&    104\t\t& \\cite{Daum1968, Andersson1970}\\\\\n2140&\t23&\t&\t23&&\t\t5\t&\t2& 51\t\t& \\cite{Daum1968}\\\\\n2160&\t32&\t&\t&&\t\t14\t&\t5&  46\t\t& \\cite{Abe1975}\\\\\n\\end{tabular}\n\\label{Table:KN}\n\\end{ruledtabular}\n\\end{center}\n\\end{table*}\n\n\\begin{table*}[htbp]\n\\caption{Summary of database for $\\overline{K} N \\rightarrow \\pi \\Lambda$. Column 1 lists the central energy $W_0$ of each energy bin, column 2 lists the number of differential cross-section data points in each bin for $K^-p\\rightarrow \\pi^0\\Lambda$, column 3 lists the number of polarization data points in each bin, column 4 lists the number of polarized cross-section data points in each bin, column 5 lists the number of integrated cross-section data points in each bin, column 6 lists the total number of data points for all kinds of data in each bin, and column 7 lists the references for the measurements referred to in columns 2-4.}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccc\n\\multicolumn{1}{c}{$W_{0}$} & \\multicolumn{1}{c}{\\multirow{2}{*}{${d\\sigma}\/{d\\Omega}$}} & \\multicolumn{1}{c}{\\multirow{2}{*}{\\mbox{$P$}}} & \\multicolumn{1}{c}{\\multirow{2}{*}{\\mbox{{$P$}${d\\sigma}\/{d\\Omega}$}}} & \\multicolumn{1}{c}{\\multirow{2}{*}{$\\sigma$}} & \\multicolumn{1}{c}{ Total}  & \\multicolumn{1}{c}{\\multirow{2}{*}{ References}} \\\\\n\\multicolumn{1}{c}{\\rm (MeV)} & & & & & \\multicolumn{1}{c}{ No.} & \\multicolumn{1}{c}{} \\\\\n\\hline\n1480\t&\t\t&  &\t\t&\t4\t&\t4\t\t\t&\t\t \\cite{Baldini1988}\\\\\n1500&\t\t&  &\t\t&\t8\t&\t8\t\t\t&\t\t \\cite{Baldini1988}\\\\\n1520\t&\t\t&  &\t\t&\t9\t&\t9\t\t\t&\t\t \\cite{Baldini1988}\\\\\n1540&\t40\t&  &\t16\t&\t7\t&\t63\t\t\t&\t\t \\cite{Armenteros1970}\\\\\n1560&\t76\t&16&\t26\t&\t4\t&\t122\t\t\t&\t \\cite{Armenteros1970, CrystalBall2005}\\\\\n1580&\t56\t&16&\t19\t&\t2\t&\t93\t\t\t&\t \\cite{Armenteros1970, CrystalBall2005}\\\\\n1600\t&\t56\t&16&\t20\t&\t5\t&\t97\t\t\t&\t \\cite{Armenteros1970, CrystalBall2005}\\\\\n1620\t&\t56\t&16&\t20\t&\t4\t&\t96\t\t\t&\t \\cite{Armenteros1970, CrystalBall2005}\\\\\n1640&\t81\t&32&\t20\t&\t6\t&\t139\t\t\t&\t \\cite{Armenteros1970, Baxter1973, CrystalBall2005}\\\\\n1660\t&\t74\t&16&\t19\t&\t9\t&\t118\t\t\t&\t \\cite{Armenteros1970, Baxter1973, CrystalBall2005}\\\\\n1680\t&\t114\t&22&\t28\t&\t13\t&\t177\t\t\t&\t \\cite{Armenteros1968, Armenteros1970, Baxter1973, CrystalBall2005}\\\\\n1700\t&\t58\t&7 &\t\t8\t&\t13\t&\t86\t\t\t\t&\\cite{Armenteros1968, Armenteros1970, Baxter1973}\\\\\n1720\t&\t101\t&27&\t\t&\t9\t&\t137\t\t\t&\t \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1740&\t185\t&65&\t\t&\t12\t&\t262\t\t\t&\t \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1760\t&\t138\t&54&\t\t&\t11\t&\t203\t\t\t&\t \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1780&\t160\t&88&\t\t&\t10\t&\t258\t\t\t&\t \\cite{Armenteros1968, Jones1975, Conforto1976}\\\\\n1800\t&\t80\t&46&\t\t&\t4\t&\t130\t\t\t&\t \\cite{Armenteros1968, Jones1975, Conforto1976}\\\\\n1820\t&\t60\t&35&\t\t&\t4\t&\t99\t\t\t&\t \\cite{Armenteros1968, Conforto1976}\\\\\n1840&\t80\t&36&\t\t&\t5\t&\t121\t\t\t&\t \\cite{Armenteros1968, Conforto1976}\\\\\n1860&\t100\t&26&\t\t&\t7\t&\t133\t\t\t&\t \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1880\t&\t120\t&44&\t\t&\t7\t&\t171\t\t\t&\t \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1900\t&\t60\t&18&\t\t&\t5\t&\t83\t\t\t&\t \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1920\t&\t80\t&21&\t\t&\t7\t&\t108\t\t\t&\t \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1940\t&\t100\t&23&\t\t&\t8\t&\t131\t\t\t&\t \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1960\t&\t60\t&24&\t\t&\t5\t&\t89\t\t\t&\t \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1980&\t40\t&6&\t\t&\t5\t&\t51\t&\t\t\t \\cite{Berthon1970, Griselin1975}\\\\\n2000\t&\t40\t&13&\t&\t4\t&\t57\t&\t\t\t \\cite{Berthon1970, Griselin1975}\\\\\n2020\t&\t20\t&9&\t\t&\t4\t&\t33\t\t&\t \\cite{Berthon1970}\\\\\n2040\t&\t20\t&8&\t\t&\t2\t&\t30\t\t&\t \\cite{Berthon1970}\\\\\n2060&\t30\t&6&\t\t&\t3\t&\t39\t\t&\t \\cite{Berthon1970, London1975}\\\\\n2080&\t40\t&8&\t\t&\t3\t&\t51\t\t&\t \\cite{Berthon1970, London1975}\\\\\n2100&\t30\t&&\t\t&\t2\t&\t32\t\t&\t \\cite{London1975}\\\\\n2120\t&\t70\t&11&\t&\t4\t&\t85\t\t&\t \\cite{Berthon1970, London1975}\\\\\n2140&\t\t& &\t\t&\t\t&\t0\t   \t &\tNA\\\\\n2160\t&\t40\t&9&\t\t&\t2\t&\t51\t\t&\t \\cite{Berthon1970}\\\\\n\\end{tabular}\n\\label{Table:Lambda}\n\\end{ruledtabular}\n\\end{center}\n\\end{table*}\n\n\\begin{table*}[htbp]\n\\caption{Summary of database for $\\overline{K} N \\rightarrow \\pi \\Sigma$. Column 1 lists the central energy $W_0$ of each energy bin, columns 2, 3, and 4 list the number of differential cross-section data points in each bin for $K^-p\\rightarrow \\pi^+\\Sigma^-$, $K^-p\\rightarrow \\pi^0\\Sigma^0$, and $K^-p\\rightarrow \\pi^-\\Sigma^+$, respectively, column 5 and 6 list the number of polarization data points for $K^-p\\rightarrow \\pi^0\\Sigma^0$ and $K^-p\\rightarrow \\pi^-\\Sigma^+$, respectively, in each bin, column 7 and 8 list the number of polarized cross-section data points for $K^-p\\rightarrow \\pi^0\\Sigma^0$ and $K^-p\\rightarrow \\pi^-\\Sigma^+$, respectively, in each bin, column 9 lists the number of integrated cross-section data points for $K^-p\\rightarrow \\pi^+\\Sigma^-$, $K^-p\\rightarrow \\pi^0\\Sigma^0$, and $K^-p\\rightarrow \\pi^-\\Sigma^+$ in each bin, column 10 lists the total number of data points for all kinds of data in each bin, and column 11 lists the references for the measurements referred to in columns 2-8.}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccccc\n\\multicolumn{1}{c}{$W_{0}$} & \\multicolumn{1}{c}{{${d\\sigma}\/{d\\Omega}$}} & \\multicolumn{1}{c}{{${d\\sigma}\/{d\\Omega}$}} & \\multicolumn{1}{c}{{${d\\sigma}\/{d\\Omega}$}} & \\multicolumn{1}{c}{{\\mbox{$P$}}}& \\multicolumn{1}{c}{{\\mbox{$P$}}} & \\multicolumn{1}{c}{{\\mbox{{$P$}${d\\sigma}\/{d\\Omega}$}}}& \\multicolumn{1}{c}{{\\mbox{{$P$}${d\\sigma}\/{d\\Omega}$}}} & \\multicolumn{1}{c}{{$\\sigma$}} & \\multicolumn{1}{c}{ Total}  & \\multicolumn{1}{c}{{ References}} \\\\\n\\multicolumn{1}{c}{\\rm (MeV)} &{($\\pi^+ \\Sigma^-$})  &{($\\pi^0 \\Sigma^0$}) & {($\\pi^- \\Sigma^+$}) & {($\\pi^0\\Sigma^0$})& {($\\pi^- \\Sigma^+$})& {($\\pi^0\\Sigma^0$})& {($\\pi^- \\Sigma^+$})& & \\multicolumn{1}{c}{ No.} &  \\multicolumn{1}{c}{} \\\\\n\\hline\n1480&\t\t&\t\t&\t\t&&&\t\t& &\t13\t&\t13\t\t&\t \\cite{Baldini1988}\\\\\n1500\t&\t\t&\t\t&\t\t&&&\t\t&\t&21\t&\t21\t\t&\t \\cite{Baldini1988}\\\\\n1520\t&\t\t&\t\t&\t\t&&&\t\t&\t&23\t&\t23\t\t&\t \\cite{Baldini1988}\\\\\n1540\t&\t40\t&\t19\t&\t40\t&&&\t19\t&18\t&22\t&\t158\t\t&\t  \\cite{Armenteros1970}\\\\\n1560\t&\t60\t&\t39\t&\t60\t&9&&\t30\t&30&\t12\t&\t240\t\t&\t  \\cite{Armenteros1970, CrystalBall2005, CrystalBall2008_1}\\\\\n1580&\t40\t&\t29\t&\t40\t&9&\t&20\t&20\t&6\t&\t164\t\t&\t  \\cite{Armenteros1970, CrystalBall2005, CrystalBall2008_1}\\\\\n1600&\t40\t&\t29\t&\t40\t&9&\t&20\t&20\t&10\t&\t168\t\t&\t  \\cite{Armenteros1970, CrystalBall2005, CrystalBall2008_1}\\\\\n1620\t&\t40\t&\t29\t&\t40\t&9&\t&20\t&20\t&13\t&\t171\t\t&\t  \\cite{Armenteros1970,CrystalBall2005, CrystalBall2008_1}\\\\\n1640\t&\t40\t&\t48\t&\t40\t&18&&\t20\t&20&\t11\t&\t\t&\t  \\cite{Armenteros1970, Baxter1973, CrystalBall2005, CrystalBall2008_1}\\\\\n1660\t&\t40\t&\t49\t&\t40\t&9&\t&20\t&\t19&17\t&\t194\t\t&\t  \\cite{Armenteros1970, Baxter1973, CrystalBall2005, CrystalBall2008_1}\\\\\n1680\t&\t80\t&\t49\t&\t80\t&9&\t&30\t&29\t&25\t&\t302\t\t&\t  \\cite{Armenteros1968, Armenteros1970, Baxter1973, CrystalBall2005, CrystalBall2008_1}\\\\\n1700\t&\t40\t&\t30\t&\t40\t&&\t&10\t&9\t&16\t&\t145\t\t&\t  \\cite{Armenteros1968, Armenteros1970, Baxter1973}\\\\\n1720&\t76\t&\t30\t&\t75\t&&10&\t\t&&\t15\t&\t206\t&\t  \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1740&\t154\t&\t20\t&\t157\t&&24&\t\t&&\t24\t&\t379\t\t&\t  \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1760\t&\t114\t&\t20\t&\t114\t&&25&\t\t&&\t20\t&\t293\t\t&\t  \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1780\t&\t148\t&\t\t&\t147\t&&34&\t\t&&\t24\t&\t353\t\t&\t  \\cite{Armenteros1968, Jones1975, Conforto1976}\\\\\n1800\t&\t72\t&\t\t&\t74\t&&22&\t\t&&\t10\t&\t178\t\t&\t  \\cite{Armenteros1968, Jones1975, Conforto1976}\\\\\n1820&\t60\t&\t\t&\t60\t&&15&\t\t&&\t11\t&\t146\t\t&\t  \\cite{Armenteros1968, Conforto1976}\\\\\n1840\t&\t80\t&\t\t&\t80\t&&13&\t\t&&\t14\t&\t187\t\t&\t  \\cite{Armenteros1968, Conforto1976}\\\\\n1860&\t100\t&\t\t&\t100\t&&11&\t\t&&\t18\t&\t229\t\t&\t  \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1880&\t120\t&\t\t&\t120\t&&24&\t\t&&\t18\t&\t282\t\t&\t  \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1900&\t60\t&\t\t&\t60\t&&14&\t\t&&\t8\t&\t142\t\t&\t  \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1920\t&\t80\t&\t\t&\t79\t&&15&\t\t&&\t12\t&\t186\t\t&\t  \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1940\t&\t99\t&\t\t&\t100\t&&14&\t\t&&\t17\t&\t230\t\t&\t  \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1960\t&\t60\t&\t\t&\t59\t&&17&\t\t&&\t10\t&\t146\t\t&\t  \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1980\t&\t40\t&\t\t&\t38\t&&\t&\t\t&\t&7\t&\t85\t\t&\t  \\cite{Berthon1970, Griselin1975}\\\\\n2000\t&\t40\t&\t\t&\t40\t&&\t&\t&\t&9\t&\t89\t&\t\t  \\cite{Berthon1970, Griselin1975}\\\\\n2020\t&\t19\t&\t\t&\t19\t&&\t&\t&\t&5\t&\t43\t\t&\t  \\cite{Berthon1970}\\\\\n2040&\t20\t&\t\t&\t16\t&&\t&\t&\t&2\t&\t38\t\t&\t  \\cite{Berthon1970}\\\\\n2060\t&\t19\t&\t10\t&\t18\t&&\t&\t&\t&5\t&\t52\t\t&\t  \\cite{Berthon1970, London1975}\\\\\n2080\t&\t20\t&\t10\t&\t19\t&&\t&\t&\t&5\t&\t54\t\t&\t  \\cite{Berthon1970, London1975}\\\\\n2100\t&\t\t&\t26\t&\t\t&&\t&\t&\t&3\t&\t29\t\t&\t  \\cite{London1975}\\\\\n2120\t&\t39\t&\t8\t&\t37\t&&\t&\t&\t&4\t&\t88\t\t&\t  \\cite{Berthon1970, London1975}\\\\\n2140\t&\t\t&\t\t&\t\t&&\t&\t&\t\t&\t0       \t\t    &\t  NA\\\\\n2160&\t36\t&\t\t&\t35\t&&\t&\t&\t&5\t&\t76\t\t&\t  \\cite{Berthon1970}\\\\\n\\end{tabular}\n\\label{Table:Sigma}\n\\end{ruledtabular}\n\\end{center}\n\\end{table*}\n \n \n The general qualitative behavior of the partial-wave amplitudes that we wanted to determine is known from earlier partial-wave analyses. Therefore it was convenient to make use of this information in our single-energy fits. In 2007, one of us (J. Tulpan) completed a multichannel fit {\\cite{Tulpan2007}} of published partial-wave amplitudes for $\\overline{K}N$ scattering to several final states, including $\\overline{K}N$, $\\overline{K}^*N$, $\\overline{K}\\Delta$, $\\pi \\Lambda$, $\\pi \\Lambda(1520)$, $\\pi \\Sigma$, and $\\pi \\Sigma(1385)$. The channels $\\sigma\\Lambda$, $\\sigma\\Sigma$, and $\\eta\\Sigma$ (for $S_{11}$) were included as ``dummy channels\",  where $\\sigma$ denotes the broad isoscalar S-wave $\\pi \\pi$ interaction. Also, $\\eta\\Lambda$ was included for $S_{01}$. Our fit of $S_{01}$ amplitudes included data for $\\sigma(K^-p\\rightarrow\\eta\\Lambda)$ up to a c.m.\\ energy of 1685 MeV (see Fig.\\ \\ref{etalambda}). The dummy channels were channels without data and were included to satisfy unitarity in some partial waves.\nTulpan's work resulted in an energy-dependent solution that is consistent with { $S$}-matrix unitarity.\nWe refer to his solution as the {\\it initial global fit}. Within each energy bin, each partial-wave amplitude with a given isospin amplitude was approximated by a first-order Taylor series expansion:\n\\begin{equation}\n       T(W) = T(W_0) + T'(W_0)(W - W_0)\n\\end{equation}       \n        where $W$ is the c.m.\\ energy of the data point in the bin and $W_0$ is the central energy of the bin. Here, for simplicity $T(W)$ denotes an isospin amplitude $T_{l\\pm}^I$. The complex $T$-matrix amplitude $T(W_0)$ belongs to the parameter set to be optimized at c.m.\\ energy $W_0$, and $T'(W_0)$ is called the slope parameter. During fits, the slope parameter was held fixed so that the real and imaginary parts of $T(W_0)$ were our fitting parameters.\nDuring our initial single-energy partial-wave analyses, we calculated the slope parameters $T'(W_0)$ from the initial global fit and kept these parameters constant in our fits.\n\n\\begin{figure}[htpb]\n\\scalebox{0.35}{\\includegraphics{.\/etalambda.pdf}}\n\\caption{Integrated cross section for $K^-p\\rightarrow\\eta\\Lambda$ compared with the results of our energy-dependent fit. Data are from Starostin 2001 \\cite{starostin2001}.}\n\\label{etalambda}\n\\end{figure}\n\nBecause the database is somewhat sparse, additional constraints were introduced in order to determine partial-wave amplitudes with a reasonably smooth variation with energy.\nTo decrease the number of free parameters to be searched, we also held fixed the very small {$ T$}-matrix amplitudes (those with $|{ T}(W_0)|<0.05$).\nThis constraint is expected to introduce only a small bias to our final energy-dependent partial-wave solution.\n\nAs an additional constraint, we held fixed the ${ D}_{03}$ amplitudes for $\\overline{K}N \\rightarrow \\overline{K}N$ and $\\overline{K}N \\rightarrow \\pi \\Sigma$ at the values from the initial global fit in the bin with $W_0=1520$ MeV. This constraint was introduced because of the well-known narrow $\\Lambda (1520)$ resonance, which has a width of only about 16 MeV.\nEven with this constraint, we ultimately concluded that we could not determine reliable amplitudes in this bin for the  reactions $\\overline{K}N \\rightarrow \\overline{K}N$ and $\\overline{K}N \\rightarrow \\pi \\Sigma$.\n\nFinally, in our single-energy fits, we introduced a {\\it penalty term} to the $\\chi^2$ function that we minimized. This penalty term constrained our fitted amplitudes from differing greatly from the values of the initial global fit. \nFor calculating the uncertainties in our single-energy amplitudes, we carried out a {\\it zero-iteration} fit in which the initial values of all amplitudes were replaced by the values determined by our $\\chi^2$ minimization procedure.\nIn the zero-iteration fit, all partial-wave amplitudes except ${ G}_{17}$ were treated as free parameters.\nThe ${ G}_{17}$ amplitude was held fixed in this procedure to remove the ambiguity in determining the global overall phase of our amplitudes. Spin-9\/2 waves were not needed in our solution.\n\nOnce we had obtained a complete set of amplitudes for $\\overline{K}N$, $\\pi \\Lambda$, and $\\pi \\Sigma$ reactions from our single-energy analyses, we carried out  global multichannel energy-dependent fits using a procedure similar to that of Tulpan \\cite{Tulpan2007}.\nThe key new ingredient is that our global fit (details of how the partial-wave $S$-matrix was constructed can be found in Ref. \\cite{manoj12}) used our own single-energy amplitudes for the $\\overline{K}N$, $\\pi \\Lambda$, and $\\pi \\Sigma$ channels.\nFor other final states,\nwe used the same input information as Tulpan did from Refs. \\cite{Gopal1977, Cameron1978, Cameron1977, Cameron1978_1, Litch1974}. We assumed the same uncertainties used by Tulpan {\\cite{Tulpan2007}} for obtaining the inital global fit ($\\pm 0.025$ for $\\overline{K}N$, $\\pm 0.035$ for $\\pi \\Lambda$ and $\\pi \\Sigma$, and $\\pm 0.050$ otherwise). These uncertainties were necessary because previous published partial-wave amplitudes were without error bars. These uncertainties were estimated by comparing like partial-wave amplitudes from different energy-dependent analyses and estimating the average differences for the real and imaginary parts. The smaller error bars implied the analyses agreed well with each other and the larger error bars implied the analyses agreed less well with each other.\n\nWe reduced the number of free amplitudes for a new set of single-energy solutions. At this stage, our free amplitudes included only $ S_{01}, S_{11}, P_{01}, P_{11}, P_{13},$ and $ D_{03}$.\nAll other amplitudes were held fixed at the values determined from our first new global fit.\nIn addition, the slope parameters were recalculated based on the new global fit and kept constant during this stage of the single-energy analyses.\nWe were able to obtain excellent agreement with the observables.\nNext, we repeated our global energy-dependent analysis to refit the new set of single-energy amplitudes for $ S_{01}, S_{11}, P_{01}, P_{11}, P_{13},$ and $ D_{03}$.\nWe then compared our new predictions with the observables in our single-energy fits.\nStill the agreement was less than satisfactory, so we carried out yet another round of single-energy analyses.\nAt this stage, we successfully reduced our free amplitudes to include only $ S_{01}, S_{11}$, and $ P_{01}$.\nAll other amplitudes were unchanged at the values from our last global fit, and slope parameters were again recalculated from the last global fit, and then held fixed in the single-energy fits.\n\n\\section{\\emph{\\bf  RESULTS AND DISCUSSION }}\nThe final single-energy fits resulted in an excellent agreement with all observables ($\\rm d\\sigma\/{\\rm d\\Omega}$, $P$, $P\\rm d\\sigma\/{\\rm d\\Omega}$, and $\\sigma$) yielding a fairly smooth set of partial-wave  amplitudes within the energy range of our analysis. The energy-dependent solutions were finally used to compare with the observable data.\nFigures 2 - 7 show representative energy-dependent results for the differential cross section of each $\\overline KN$ reaction included in our single-energy fits. \nThe cross sections are shown as a function of $\\cos\\theta$, where $\\theta$ is the c.m.\\ scattering angle of the meson.\nFigure 2 shows the comparison of differential cross section data for $K^-p\\rightarrow K^-p$ with our energy-dependent solution at four lab momenta of 514, 935, 1165, and 1483 MeV. Although  the data are from the 1960s and 1070s \\cite{Armenteros1970, Conforto1971, Conforto1976, Daum1968} they are in excellent agreement with our solution. For $K^- p \\rightarrow \\overline K^0 n$ (Fig.\\ 3) the Crystal Ball data with smaller error bars at $ P_{\\rm Lab}$ = 514 MeV and 714 MeV are well described by our solution in the forward and backward angles with a slight under fitting in the intermediate angles. The other data \\cite{Jones1975, Griselin1975} at $P_{\\rm Lab}$ = 936 MeV and 1434 MeV with larger error bars are in good agreement with our energy-dependent solution. Similarly, Fig.\\ 4 shows the excellent agreement between our solution and differential cross section data at $P_{\\rm Lab}$ = 514, 750, 1153, and 1465 MeV for $K^-p\\rightarrow \\pi^0\\Lambda$. Figure 5 shows a comparison of data from Refs. \\cite{Armenteros1970, Armenteros1968, Conforto1976, Berthon1970} with our solution for $K^-p\\rightarrow \\pi^+\\Sigma^-$. Except for some under representation of data at $P_{\\rm Lab}$ = 1245 MeV we have an excellent agreement with the data. Figures 6 and 7 show an excellent agreement of our energy-dependent solution with the differential cross section data at various lab momenta of kaons for $K^-p\\rightarrow \\pi^0\\Sigma^0$ and $K^-p\\rightarrow \\pi^-\\Sigma^+$, respectively.\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_11_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_11_935.pdf}}\n\\vspace{-25.5mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_11_1165.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_11_1483.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow K^- p$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Conforto 1971 \\cite{Conforto1971}, and Duam 1968 \\cite{Daum1968}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_12_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_12_750.pdf}}\n\\vspace{-25.5mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_12_1165.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_12_1434.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\overline K^0 n$ differential cross section. Data are from Prakhov 2009 \\cite{CrystalBall2005}, Conforto 1976 \\cite{Conforto1976}, and Griselin 1975 \\cite{Griselin1975}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_2_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_2_750.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_2_1153.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_2_1465.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Lambda$ differential cross section. Data are from Prakhov 2009 \\cite{CrystalBall2005}, Armenteros 1968 \\cite{Armenteros1968}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_31_495.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_31_935.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_31_1245.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_31_1462.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^+\\Sigma^-$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_32_495.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_32_560.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_32_659.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_32_714.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Sigma^0$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970} and Prakhov 2009 \\cite{CrystalBall2005}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_33_495.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_33_935.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_33_1245.pdf}} \n\\vspace{-25mm}\n\\vspace{3mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_33_1462.pdf}}\n\\vspace{-2mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^-\\Sigma^+$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\nFigures 8, 9, 10, and 11 show representative energy-dependent fit results for the polarization in reactions $K^- p \\rightarrow K^- p$, $K^- p \\rightarrow \\pi\\Lambda$, $K^- p \\rightarrow \\pi^0\\Sigma^0 $, and $K^- p \\rightarrow \\pi^- \\Sigma^+$, respectively. The polarizations are shown as a function of $\\cos\\theta$, where $\\theta$ is the c.m.\\ scattering angle of the meson. Figure 8 shows the excellent agreement of our energy-dependent solution with the $K^-p\\rightarrow K^-p$ polarization at $P_{\\rm Lab}$ = 1383, 1483, 1584, 1684 MeV from Ref. \\cite{Daum1968}. For $K^-p\\rightarrow\\pi^0\\Lambda$ (Fig.\\ 9) our solution is in good agreement with the polarization data at $P_{\\rm Lab}$ = 514, 936, and 1165 MeV. Our solution also agrees well the Crystal Ball data at $P_{\\rm Lab}$ = 714 MeV at forward angles but there is a slight under representation of data at backward angles. Similarly, Fig.\\ 10 shows a very good agreement between our solution and the $K^-p\\rightarrow \\pi^0\\Sigma^0$ polarization data within the given uncertainties at $P_{\\rm Lab}$ = 514, 581, 687, and 750 MeV, all from Crystal Ball Collaboration. Finally, Fig.\\ 11 shows a comparison of our solution with the $K^-p\\rightarrow \\pi^-\\Sigma^+$ polarization data at  $P_{\\rm Lab}$ = 862, 936, 1001, and 1125 MeV. Except for small forward angles at $P_{\\rm Lab}$ = 1001 MeV, we have good agreement with the data.\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_11_1383.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_11_1483.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_11_1584.pdf}} \n\\vspace{-25mm}\n\\vspace{3mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_11_1684.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow K^-p$ differential cross section. Data are from Daum 1968 \\cite{Daum1968}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_2_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_2_714.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_2_936.pdf}} \n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_2_1165.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Lambda$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_32_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_32_581.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_32_687.pdf}} \n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_32_750.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^-\\Sigma^+$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_33_862.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_33_936.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_33_1001.pdf}} \n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_33_1125.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^-\\Sigma^+$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\nFigures 12, 13, 14, and 15 show representative energy-dependent fit results for the polarized cross section in reactions $K^- p \\rightarrow K^- p$, $K^- p \\rightarrow \\pi\\Lambda$, $K^- p \\rightarrow \\pi^0\\Sigma^0 $, and $K^- p \\rightarrow \\pi^- \\Sigma^+$, respectively. The polarized cross sections are shown as a function of $\\cos\\theta$, where $\\theta$ is the c.m.\\ scattering angle of the meson.\nWithin the uncertainties associated with the polarized cross section data our results are in good agreement with the data for  these reactions.\n\n\\begin{comment}\n\\begin{figure}[htpb]\n\\vspace{-10mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_11_865.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_11_1082.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_11_1330.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_11_1732.pdf}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow K^- p$ polarized cross section. Data are from Albrow 1971 \\cite{Albrow1971}, Andersson 1970 \\cite{Andersson1970}, and Armenteros 1970 \\cite{Armenteros1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/PdSigma_11_865.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/PdSigma_11_1082.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/PdSigma_11_1330.pdf}} \n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/PdSigma_11_1732.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow K^- p$ polarized cross section. Data are from Albrow 1971 \\cite{Albrow1971}, Andersson 1970 \\cite{Andersson1970}, and Armenteros 1970 \\cite{Armenteros1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\\begin{figure}[htpb]\n\\vspace{-10mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_2_514.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_2_554.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_2_637.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_2_719.pdf}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Lambda$ polarized cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\vspace{-10mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_32_495.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_32_597.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_32_699.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_32_793.pdf}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Sigma^0 $ polarized cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\\begin{figure}[H]\n\\vspace{-10mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_33_495.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_33_597.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_33_699.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_33_793.pdf}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^- \\Sigma^+$ polarized cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n Figure 16 shows our prediction for the total $K^-p\\rightarrow$ cross section.\n Figure 17 shows our prediction for the $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$ integrated cross sections, Fig.\\ 18 shows our prediction for the $K^-p\\rightarrow \\pi^0\\Lambda$ integrated cross section, and Fig. 19 shows our prediction for the $K^-p\\rightarrow \\pi^+\\Sigma^-$, $K^-p\\rightarrow \\pi^0\\Sigma^0$, and $K^-p\\rightarrow\\pi^-\\Sigma^+$ integrated cross sections. Figures 17 and 18 do not show predictions for c.m.\\ energies below 1540 MeV because it was not possible to obtain single-energy amplitudes in this region where only integrated cross-section data are available.\n\n\\begin{figure}[H]\n\\begin{center}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/All1.pdf}}\n\\caption{(Color online) Total $K^-p$ cross section compared with the results of our energy-dependent fit. Data are from Baldini 1988 \\cite{Baldini1988}}\n\\label{fig:dSigma_11_New}\n\\end{center}\n\\end{figure}\n \n\\begin{figure}[htpb]\n\\begin{center}\n\\vspace{-9.5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma1_11.pdf}}\n\\vspace{-10mm}\n\\vspace{-5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma11_12.pdf}}\n\\vspace{-10mm}\n\\caption{(Color online) Integrated cross sections for $K^- p \\rightarrow K^-p$ and $K^- p \\rightarrow \\overline K^0n$ compared with the results of our energy-dependent fit. Data are from Baldini 1988 \\cite{Baldini1988}}\n\\label{fig:dSigma_11_New}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\vspace{-9.08mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma21_2.pdf}}\n\\caption{(Color online) Integrated cross section for $K^- p \\rightarrow \\pi^0\\Lambda$ compared with the results of our energy-dependent fit. Data are from Baldini 1988 \\cite{Baldini1988}.}\n\\label{fig:dSigma_11_New}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\vspace{-9.5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma1_31.pdf}}\n\\vspace{-10mm}\n\\vspace{-5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma1_32.pdf}}\n\\vspace{-10mm}\n\\vspace{-5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma31_33.pdf}}\n\\vspace{-10mm}\n\\caption{(Color online) Integrated cross sections for $K^- p \\rightarrow \\pi^+\\Sigma^-$, $K^- p \\rightarrow \\pi^0\\Sigma^0$, and $K^- p \\rightarrow \\pi^-\\Sigma^+$ compared with the results of our energy-dependent fit. Data are from Baldini 1988 \\cite{Baldini1988}.}\n\\label{fig:dSigma_11_New}\n\\end{center}\n\\end{figure}\n\n\\clearpage\n\\section{Summary and Conclusions}\nWe have investigated $\\overline KN\\rightarrow \\overline KN$, $\\overline K N\\rightarrow \\pi\\Lambda$, and $\\overline KN\\rightarrow \\pi\\Sigma$ reactions through single-energy analyses constrained by a global unitary energy-dependent fit from threshold to a c.m.\\ energy of 2.1 GeV. We found partial waves up to G-waves necessary to describe the available data for the reactions. This work was motivated, in part, by the relatively recent measurements for $K^-p\\rightarrow \\overline K^0n$, $K^-p\\rightarrow \\pi^0\\Lambda$, $K^-p\\rightarrow\\pi^0\\Sigma^0$, and $K^-p\\rightarrow \\eta\\Lambda$ by the Crystal Ball Collaboration. We were successful in describing these data in addition to older data from constrained single-energy analyses. The partial-wave amplitudes thus extracted were used in our global multichannel fit. A discussion of the resonance parameters from this global fit, which is the most comprehensive multichannel fit to date for $\\overline K N$ scattering reactions, is presented in a separate publication \\cite{manoj2013}. \n\n\n\\acknowledgements{This work was supported by the U.S. Department of Energy Grant No. DE-FG02-01ER41194. \n       \n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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@@ -0,0 +1,5 @@
+{"text":"\n\n\\section{Overview}\n\n\nThe Smooth 4-Dimensional Poincar\\'e Conjecture (SPC4)\nproposes that every\nclosed smooth 4-manifold $\\Sigma$ that is homotopy equivalent to $\\sphere{4}$\nis diffeomorphic to the standard $\\sphere{4}$.\nBy work of Freedman \\cite{freedman1982topology},\nit is known that if $\\Sigma$ is homotopy equivalent to $\\sphere{4}$,\nthen $\\Sigma$ is in fact homeomorphic to $\\sphere{4}$.\nIn stark contrast to the SPC4,\nit might be the case that every compact smooth 4-manifold\nadmits infinitely many distinct smooth structures.\nThe existence of an exotic homotopy 4-sphere is equivalent\nto the existence of an exotic\ncontractible compact manifold with\n$\\sphere{3}$ boundary \\cite[p.\\ 113]{MR0190942},\nhenceforth called an \\emph{exotic homotopy 4-ball}.\n\nOne possible approach to proving that a proposed exotic homotopy 4-ball $\\mathcal{B}$\nis in fact exotic is to find a knot $K \\subset \\sphere{3} = \\partial \\mathcal{B}$,\nsuch that there is a smooth properly embedded disk\n$\\disk{2} \\hookrightarrow \\mathcal{B}$,\nwith $\\partial \\disk{2}$ mapped to $K$,\nwhere $K$ is not smoothly slice in the usual sense\nin the standard 4-ball $\\ball{4}$.\nA knot is (topologically\/smoothly) slice in $\\ball{4}$ \nif and only if it is null-concordant in\n$\\sphere{3} \\times {I} = \\sphere{3} \\times [0, 1]$,\ni.e.\\ there is a properly embedded (locally flat\/smooth) cylinder\n$\\sphere{1} \\times {I} \\hookrightarrow \\sphere{3} \\times {I}$\nwhose oriented boundary is\n$K \\subset \\sphere{3} \\times \\{ 0 \\}$ together with\nthe unknot $U \\subset \\sphere{3} \\times \\{ 1 \\}$.\nAnother way of thinking about this strategy is\nthat we want to find a knot $K$ in\n$\\sphere{3} = \\partial \\mathcal{B}$\nthat bounds a properly embedded smooth disk in $\\mathcal{B}$\nbut does not bound any such disk that is contained in a \ncollar $\\sphere{3} \\times {I}$ of the boundary of $\\mathcal{B}$.\nIn this case, to verify the sliceness of $K$, we have to go ``deep'' into $\\mathcal{B}$.  \n\nAn easier task might be to find a homology 4-ball $X$\nwith $\\sphere{3}$ boundary such that there is a knot in the boundary\nthat bounds a smooth properly embedded disk in $X$ but not in $\\ball{4}$,\nhowever, this is also an open problem.\nIn \\cite{Freedman_2010}, the authors investigate the possibility of proving\nthat a homotopy 4-ball $\\mathcal{B}$ with $\\sphere{3}$ boundary is exotic by taking a\nknot in the boundary that bounds a smooth properly embedded disk in $\\mathcal{B}$\nand computing the $s$-invariant of $K$, in the hopes that $s(K) \\neq 0$,\nwhereby they could then conclude that $\\mathcal{B}$ is exotic.\nUnfortunately for this approach as noted in the paper,\nit turns out that the homotopy 4-ball that they were studying\nwas in fact diffeomorphic to $\\ball{4}$, see \\cite{akbulut2010cappell}.\nIt is still open whether the $s$-invariant can obstruct the\nsliceness of knots in $\\ball{4}$ that are slice in some homotopy 4-ball,\nas is noted in the corrigendum to \\cite{kronheimer2013gauge}.\n\nMotivated by this, we make the following definitions:\nFor a 3-manifold $M^3$ containing a knot\n$K \\colon \\sphere{1} \\hookrightarrow M$, we say\nthat \\emph{$K$ is null-concordant in $M \\times {I}$}\nif there is a smoothly properly embedded annulus\n$\\sphere{1} \\times {I} \\hookrightarrow M \\times I$\ncobounding $K \\subset M \\times \\{ 0 \\}$ on one end\nand an unknot contained in a 3-ball $U \\subset B^3 \\subset M \\times \\{ 1 \\}$\non the other.\nEquivalently, $K \\subset M \\times \\{ 0 \\}$ bounds a smoothly properly\nembedded disk in $M \\times {I}$.\n\n\\begin{definition}[Deep slice\/Shallow slice]\n    Let $X^{4}$ be a smooth compact 4-manifold with nonempty boundary $\\partial X$.\n    We call a knot $K \\subset \\partial X$ \\emph{slice} in $X$ if there is a smooth properly embedded disk in $X$\n    with boundary $K$.\n    We call a knot $K \\subset \\partial X$ \\textit{shallow slice} in $X$\n    if there is a smooth properly embedded disk in $\\partial X \\times I$\n    with boundary $K$ -- this is equivalent\n    to $K$ being null-concordant in the collar $\\partial X \\times {I}$.\n    If $K$ is slice in $X$ but not shallow slice,\n    we will call it \\textit{deep slice} in $X$.\n    See \\autoref{fig:deep_shallow_schematic} for a schematic\n    illustration of these definitions.\n\\end{definition}\n\n\\begin{figure}\n    \\centering\n    \\begin{overpic}[width=0.35\\textwidth]{pictures\/deep_slice_schematic}\n        \\definecolor{blue-green}{rgb}{0.0, 0.87, 0.87}\n        \\put(34, 96){\\color{blue} $K$}\n        \\put(6, 87){$\\partial X$}\n        \\put(35, 74){\\Large \\color{blue-green} ?}\n        \\put(70, 80.5){\\color{gray} collar}\n        \\put(5, 40){$X^4$}\n        \\put(50, 20){\\color{blue} $\\Delta^2$}\n    \\end{overpic}\n    \\caption{\n        Schematic of a deep slice disk $\\Delta^2$ (blue)\n        in a 4-manifold $X^4$,\n        with boundary the knot $K \\subset \\partial X$.\n        The knot $K$ is called \\emph{deep slice}\n        if it does not bound a properly embedded disk in a collar\n        of the boundary, indicated by the (light blue) dashed lines.\n    }\n    \\label{fig:deep_shallow_schematic}\n\\end{figure}\n \nIn this language,\nProblem 1.95 on Kirby's list \\cite{kirby1995problems}\n(attributed to Akbulut)\ncan be reformulated as follows:\nAre there contractible smooth 4-manifolds\nwith boundary an integral homology 3-sphere\nwhich contain deep slice knots that are null-homotopic in the boundary?\nNote that any knot that is not nullhomotopic in the boundary\nwill not be shallow slice and\nthus if it is slice, it will be deep slice.\nFor this reason we will be looking for deep slice knots\nthat are null-homotopic in the boundary.\nWe will often consider our knots\nto be contained in 3-balls in the boundary,\nwhich we call \\emph{local knots},\nso we can freely consider them in the boundary of any 4-manifold and\ndiscuss if they are slice there.\nTo avoid confusion\nwhen we say that a (local) knot in a 3-manifold $M^3$ is slice\nwe will usually qualify it with ``in $X^4$''.\n\n\n\n\n\\subsection{Outline}\n\nIn the first part of this paper\nwe will restrict ourselves to the smooth category,\nstarting in \\autoref{sec:nonexistence_deep_slice}, \nwhere we discuss a condition that guarantees that some 4-manifolds\nhave no deep slice knots and related results.\nIn \\autoref{sec:existence_deep_slice}, we prove that every 2-handlebody\nhas a deep slice knot in its boundary.\nTo do this we employ the Wall self-intersection number\nand a result of Rohlin\nwhich we discuss briefly.\n\nIn \\autoref{sec:universal_slicing}, we recall the Norman-Suzuki trick\nand observe that every 3-manifold bounds a 4-manifold\nwhere every knot in the boundary bounds a properly embedded disk.\nIn contrast, if we restrict to slice disks trivial in\nrelative second homology, \nwe will see that every compact topological $4$-manifold\nwith boundary $\\sphere{3}$\ncontains a knot which\ndoes not bound a null-homologous topological slice disk.\nWe finish with some questions and suggestions for\nfurther directions in \\autoref{sec:questions}.\n\n\n\n\n\\subsection{Conventions}\n\nIn the literature, \nproperly embedded slice disks in a $4$-manifold $X$ are \noften assumed to be null-homologous\nin $H_{2}(X, \\partial X)$.\nWe will make this extra assumption on homology only\nin \\autoref{sec:universal_slicing} when discussing the ``universal slicings''.\nFor the first part \\emph{deep slice} and \\emph{shallow slice}\nwill describe the existence of\na embedded disks with the relevant properties\nwithout conditions on the homology class.\n\nStarting from\nan $n$-manifold $M^{n}$ without boundary,\nwe obtain a \\emph{punctured $M$}\n(more precisely a \\emph{bounded punctured $M$})\nby removing\na small open $n$-ball $M^{\\circ} \\coloneqq M \\setminus \\interior \\disk{n}$,\nwhich yields a\nmanifold with boundary\n$\\partial M^{\\circ} = \\sphere{n-1}$.\nObserve that a punctured $M$ is the same\nas a connected sum\n$M^{\\circ} \\cong M \\# \\disk{n}$\nwith a $n$-ball.\n\n\\subsection{Acknowledgments}\n\nThe authors would like to thank\nAnthony Conway, Rob Kirby, Mark Powell, \nArunima Ray and Peter Teichner for helpful conversations,\ntheir encouragement and guidance.\nBR would like to thank Thorben Kastenholz\nfor asking about the decidability of the embedded genus problem\nin 4-manifolds, which motivated \\autoref{sec:universal_slicing}.\nWe are especially grateful to\nAkira Yasuhara for pointing us to related literature\nand the work of Suzuki.\nWe are also especially grateful to the anonymous referee for numerous helpful\nsuggestions that improved the exposition and a suggestion that\nlead to our subsequent work relating the techniques in this paper\nto the embeddings of traces of surgeries on links.  \nThe Max Planck Institute for Mathematics in Bonn\nsupported us financially and with a welcoming research environment.\n\n\\section{Nonexistence of deep slice knots}\n\\label{sec:nonexistence_deep_slice}\n\n\n\\noindent For starters, we have:\n\\begin{proposition} \\label{1-handles}\nThere are no deep slice knots in\n$\\natural^k \\sphere{1} \\times \\ball{3}$.\n\\end{proposition}\n\n\\begin{proof}\nLet\n$K \\subset \\#^k \\sphere{1} \\times \\sphere{2} \n= \\partial(\\natural^k \\sphere{1} \\times \\ball{3})$\nsuch that $K$ is slice in $\\natural^k \\sphere{1} \\times \\ball{3}$.\nThen, thinking of $\\natural^k \\sphere{1} \\times \\ball{3}$\nas a wedge of $k$ copies of $\\sphere{1}$ thickened to be 4-dimensional,\nif $D$ is any slice disk for $K$ we can isotope $D$ such that\nit does not intersect a one-dimensional wedge of circles that\n$\\natural^k \\sphere{1} \\times \\ball{3}$ deformation retracts onto.\nTherefore, $D$ can be isotoped to be contained in\na collar neighborhood of the boundary $\\#^k \\sphere{1} \\times \\sphere{2}$\nand thus $K$ is shallow slice.   \n\\end{proof}\n\n\nThe following might be a surprise,\nas one could expect that additional topology\nin a 3-manifold $M^3$\ncreates more room for concordances:\n\\begin{proposition}[{Special case of \\cite[Prop. 2.9]{nagel2019smooth}}]\n    \\label{prop:local_knot_slice}\n    If a local knot $K \\subset \\ball{3} \\subset M^{3}$\n    is null concordant in $M^{3} \\times {I}$,\n    then $K$ is null concordant in $\\sphere{3} \\times {I}$. \n\\end{proposition}\n\n\\begin{proof}[Proof sketch]\n    Let $D$ be a properly embedded disk in\n    $M \\times {I}$ with boundary $K$\n    and let $\\widetilde{M}$ be the universal cover of $M$.\n    Then $D$ lifts to a properly\n    embedded disk $\\widetilde{D} \\subset \\widetilde{M} \\times {I}$.\n    Further, since $K$ is contained in a 3-ball $B$,\n    all of the lifts of $K$ to $\\widetilde{M}$ are just copies of $K$,\n    and therefore, the boundary of $\\widetilde{D}$\n    is a copy of $K$, considered inside of $\\widetilde{M}$.\n    As a consequence of geometrization \\cite{perelman2003finite},\n    we know that every universal cover\n    of a punctured compact 3-manifold smoothly embeds into $\\sphere{3}$,\n    as was observed in \\cite[Lem.\\ 2.11]{boden2017concordance}.\n    It follows then that there is an embedding\n    $\\widetilde{M} \\times {I} \\hookrightarrow \\sphere{3} \\times {I}$.\n    But then the image of $\\widetilde{D}$\n    under this embedding shows that $K$ bounds a disk in $\\sphere{3} \\times {I}$. \n\\end{proof}\n\nWe have added a proof of this proposition here to\nhighlight that this lifting argument \nbreaks down in the case of higher genus\nsurfaces if their inclusion\ninduces a nontrivial map on fundamental groups.\nIf $K$ bounds a genus $g$ surface \nwith one boundary component $\\Sigma_{g, 1}$\nin $M \\times {I}$, we can only\nlift this to the universal cover (and subsequently find\na genus $g$ surface for $K$ in $\\sphere{3}$ via this method)\nunder the condition that the inclusion\nof $\\Sigma_{g, 1}$ in $M \\times {I}$ is $\\pi_{1}$-trivial.\nSo this argument does not work if the surface\nreally ``uses the extra topology of $M$''.\n\n\\begin{example}\n    Take a non-orientable\n    3-manifold $M$ containing the connected\n    sum $K \\# K$ of two copies of\n    a local invertible knot $K$ with\n    smooth 4-ball genus $g^{4}(K \\# K) \\ge 2$.\n    As an explicit example,\n    $K$ a left-handed trefoil will work,\n    and we illustrate the following\n    in \\autoref{fig:torus_movie}.\n    Describe an embedded torus in\n    $M \\times {I}$ with the motion picture method:\n    Use the connected sum band to split the sum with a saddle.\n    Then let one of the summands travel around an orientation\n    reversing loop in $M$\n    while leaving the other one fixed.\n    The summand traveling around the loop\n    was reflected in the process and since it is invertible\n    it is isotopic to\n    $-K = r \\overline{K}$\n    in a 3-ball neighborhood in $M$.\n    Fusing the summands back together\n    along a connected sum band\n    we now obtain\n    $K \\# -K$ as a local knot.\n    Finally cap this off with the usual ribbon disk\n    for the connected sum of a knot with\n    its concordance inverse.  Therefore $g^{M \\times I}(K \\# K) \\leq 1$ and by\n    \\autoref{prop:local_knot_slice} in fact $g^{M \\times I}(K \\# K) = 1$.\n    The 4-ball genus $g^{4}(K \\# K) \\ge 2$\n    of this example is strictly\n    larger than its\n    \\emph{$(M^{3} \\times {I})$-4-genus},\n    which we define as\n    \\[\n        g^{M \\times {I}}(J)\n        \\coloneqq\n        \\min \n        \\{ \n            g \\mid\n            \\exists \\text{ smooth proper embedding }\n            \\Sigma_{g, 1} \\hookrightarrow M \\times I\n            \\text{ with }\n            \\partial \\Sigma_{g, 1} = J \\subset M \\times \\{ 0 \\}\n        \\}\n    \\]\n    Observe that in this notation\n    the usual 4-ball genus is\n    $g^{4} = g^{\\sphere{3} \\times {I}}$\n    and we can rephrase \\autoref{prop:local_knot_slice} as\n    $g^{M \\times {I}}(K) = 0$ implies\n    $g^4(K) = 0$\n    for local knots $K$.\n    Similar notions of 4-genera\n    were introduced in Celoria's investigation\n    of almost-concordance \\cite[Def.\\ 12]{celoria2018concordance}.\n\\end{example}\n\n\\begin{figure}[ht]\n    \\begin{subfigure}{.495\\textwidth}\n      \\centering\n     \n      \\begin{overpic}[width=\\textwidth]{pictures\/torus_movie_frame_1}\n        \\label{fig:torus_movie_frame_1}\n      \\end{overpic}\n      \\caption{\n            Saddle move to separate the summands\n            of $K \\# K$.\n        }\n    \\end{subfigure}\n   \n    \\begin{subfigure}{.495\\textwidth}\n      \\centering\n     \n      \\begin{overpic}[width=\\textwidth]{pictures\/torus_movie_frame_2}\n        \\label{fig:torus_movie_frame_2}\n      \\end{overpic}\n      \\caption{\n            One of the summands travels around an orientation\n            reversing loop in $M$.\n        }\n    \\end{subfigure}\n   \n    \\begin{subfigure}{.495\\textwidth}\n      \\centering\n     \n      \\begin{overpic}[width=\\textwidth]{pictures\/torus_movie_frame_3}\n        \\label{fig:torus_movie_frame_3}\n      \\end{overpic}\n      \\caption{\n            It returns mirrored, now add a fusion band.\n        }\n    \\end{subfigure}\n   \n    \\begin{subfigure}{.495\\textwidth}\n      \\centering\n     \n      \\begin{overpic}[width=\\textwidth]{pictures\/torus_movie_frame_4}\n        \\label{fig:fig:torus_movie_frame_4}\n      \\end{overpic}\n      \\caption{\n            Finish off the movie with the standard\n            ribbon disk for $K \\# -K$.\n        }\n    \\end{subfigure}\n   \n   \n    \\caption{\n        Four frames of the movie of a properly embedded\n        punctured torus in $M^3 \\times {I}$ with boundary\n        $K \\# K \\subset M \\times \\{ 0 \\}$,\n        where $M$ is a non-orientable 3-manifold.\n        }\n    \\label{fig:torus_movie}\n\\end{figure}\n\n\nIt would be interesting to find an example \nof an orientable $3$-manifold $M^3$ where the\n$g^{M \\times {I}}(K)$ genus of\nsome local knot $K \\subset \\disk{3} \\subset M$\nis strictly smaller than\nthe $4$-ball genus $g^{4}(K)$, or prove that no such $M$ exists.\nLocal $K$ satisfy $g^{M \\times {I}}(K) \\le g^4(K)$\nas cobordisms in $\\sphere{3} \\times {I}$\ncan be embedded into $M \\times {I}$.\nBecause of \\autoref{prop:local_knot_slice}\nan example where these values differ can only appear\nfor $g^{4}(K) \\ge 2$.\nMoreover, as we will see in \\autoref{thm:embedding_4-sphere_shallow}\nsuch an $M$ would necessarily not embed in $S^4$.  \nAnother special case is treated in\n\\cite[Thm.\\ 2.5]{Davis_2018} where \na handle cancellation argument shows that\nthere is no difference\nfor local knots in $M = \\sphere{1} \\times \\sphere{2}$,\nthat is the equality \n$g^{\\sphere{1} \\times \\sphere{2} \\times {I}} = g^{4}$ holds\n(and also analogous statements for $\\#^{k} \\sphere{1} \\times \\sphere{2}$).\nTopological concordance in $\\sphere{1} \\times \\sphere{2} \\times {I}$\nis investigated in \\cite{friedl2019satellites}.\n\n\nWe now give a criterion that shows that certain 4-manifolds\nhave no local deep slice knots in the boundary.\nThis idea is also contained in \\cite[Thm.\\ 0]{suzuki1969localknots}\nand its variants.\n\n\\begin{proposition} \n    \\label{thm:embedding_4-sphere_shallow}\n    Let $X^4$ be a compact 4-manifold with a \n    local knot $\\gamma \\subset \\ball{3} \\subset \\partial X$\n    that is slice in $X$.\n    If there is a cover of $X$ which can be smoothly embedded into $\\sphere{4}$,\n    then $\\gamma \\subset \\ball{3} \\hookrightarrow \\sphere{3} = \\partial \\ball{4}$\n    is slice in $\\ball{4}$.\n    Hence, $\\gamma$ is shallow slice in $X$.    \n\\end{proposition}\n\n\\begin{proof}\n    Let $\\widetilde{X}$ be a cover of $X$\n    with an embedding $\\widetilde{X} \\subset \\sphere{4}$ into $\\sphere{4}$\n    and let $\\widetilde{D}$ be a lift of a slice disk for $\\gamma$ to\n    $\\widetilde{X}$\n    with $\\widetilde{\\gamma} = \\partial \\widetilde{D}$.\n    Note that the knot $\\widetilde{\\gamma}$ is the same as $\\gamma$,\n    since $\\gamma$ is contained in a 3-ball and the only covers of a 3-ball\n    are disjoint unions of 3-balls. Puncture $S^4$ by removing a small ball $B$ close to $\\widetilde{\\gamma}$ and such that $\\widetilde{\\gamma}$ can be connected by an annulus  disjoint from $\\tilde{X}$ to $\\partial B$ and such that the other end of the annulus is (the mirror image of) $K \\subset \\partial B$.\n    Then since $S^4 - \\interior{B} \\cong B^4$, \n    the annulus together with $\\widetilde{D}$ show that $K$ is slice in\n    the $B^4$ which is the complement of the small ball.    \n    Therefore $\\gamma$ is shallow slice in $X$.  \n\\end{proof}\n\nAs an example, \\autoref{thm:embedding_4-sphere_shallow}\nimplies that $\\natural^k \\sphere{2} \\times \\disk{2}$\ncontains no deep slice local knots,\nsince these manifolds can all be embedded in $\\sphere{4}$.\nHowever, these manifolds all contain deep slice knots, necessarily non-local, \nas will be seen shortly.\nAdditionally, we have:\n\\begin{corollary}\n    \\label{cor:universal_cover_S4_no_deep_slice}\n    Suppose that $X$ is a closed smooth 4-manifold with universal cover $\\mathbb{R}^4$ or $\\sphere{4}$,\n    and let $X^{\\circ}$ denote the punctured version.\n    Then $X^{\\circ}$ has no deep slice knots.  \n\\end{corollary}\n\n\n\\section{Existence of deep slice knots}\n\\label{sec:existence_deep_slice}\n\nA \\emph{2-handlebody} is a 4-manifold whose handle decomposition\ncontains one 0-handle, some nonzero number of 2-handles\nand no handles of any other index.\nExamples of this are knot traces,\nwhere a single $2$-handle is attached along a framed knot\nto the 4-ball.\nIn this section, we prove:\n\\begin{theorem} \n    \\label{existence}\n    Every 2-handlebody $X$ contains a null-homotopic\n    deep slice knot in its boundary.  \n\\end{theorem}\n\n\\begin{remark}\n    For the special case of the $2$-handlebody\n    $\\disk{2} \\times \\sphere{2}$\n    the existence of such knots was\n    already observed in\n    \\cite[Thm.\\ B]{Davis_2018}\n    (here only winding number $w = 0$ gives null homotopic knots).\n    Furthermore the authors construct an infinite family of slice knots\n    which are pairwise different in topological concordance in\n    a collar of the boundary.\n\\end{remark}\n\n\\autoref{existence} breaks up naturally into two cases depending on whether\nthe boundary has nontrivial $\\pi_1$ or not (i.e. if it is or is not $S^3$).\nIn the case where $\\pi_1(\\partial X) \\neq 1$,\nthere is a concordance invariant for knots in arbitrary 3-manifolds,\nclosely related to the Wall self-intersection number\n(see \\cite{wall1999surgery}, \\cite{freedman1990topology},\nand \\cite{schneiderman2003algebraic}),\nthat will allow us to show that some obviously slice knots are not shallow slice.\nIn the case where $\\pi_1 (\\partial X)$ is trivial,\nand therefore by the \n3-dimensional Poincar\\'e conjecture \\cite{perelman2003finite}\n$\\partial X = \\sphere{3}$, the Wall self-intersection number is of no use.\nHowever, in this case, the consideration of whether a knot\nthat is slice in $X$ is deep slice in $X$ is related\nto the existence of spheres representing various homology\nclasses in the manifold obtained by closing $X$ off with a 4-handle.   \n\n\\begin{remark}\n    If there was a direct proof\n    that every closed homotopy 3-sphere\n    smoothly bounds a contractible 4-manifold,\n    then we would not need to\n    invoke the $3$-dimensional Poincar\\'e conjecture.\n\\end{remark}\n\nFollowing \\cite{yildiz2018note} and \\cite{schneiderman2003algebraic},\nwe briefly introduce the Wall self-intersection number\nin the setting that we will be working in, and state some of its basic properties.\nLet $Y^3$ be a closed oriented 3-manifold and let\n$\\gamma \\colon \\sphere{1} \\hookrightarrow Y$ be a knot in $Y$.\nLet $\\mathcal{C}_\\gamma(Y)$ denote the set of concordance classes\nof oriented knots in $Y$ that are freely-homotopic to $\\gamma$.\nIn particular $\\mathcal{C}_{U}(Y)$\ndenotes the set of concordance classes of\noriented null-homotopic knots in $Y$,\nwhere we write $U$ for the local unknot in $Y$.\nGiven an  oriented null-homotopic knot $K \\subset Y$,\nby transversality there exists an oriented immersed disk\n$D$ in $Y \\times {I}$ with boundary\n$K \\subset Y \\times \\{ 0 \\} = Y$\nthat has only double points of self-intersection.\nLet $\\star \\in Y$ denote a basepoint which we implicitly use for\n$\\pi_1(Y) = \\pi_1(Y \\times {I})$ throughout.\nChoose an arc, which we will call a whisker, from $\\star$ to $D$.\nFor each double point of self-intersection $p \\in D$\nchoose a numbering of the two sheets of $D$ that intersect at $p$.\nThen let $g_p \\in \\pi_1(Y)$ be the homotopy class of the loop in $Y \\times {I}$\nobtained by starting at $\\star$, taking the whisker to $D$,\ntaking a path to $p$ going in on the first sheet,\ntaking a path back to where the whisker meets $D$ that leaves $p$ on the second sheet,\nand then returning to $\\star$ using the whisker.\nNote that changing the order of the two sheets would transform $g_p$ to $g_p^{-1}$.\nAlso, since $K$ and $Y$ are oriented,\n$D$ and $Y \\times {I}$ obtain orientations\nwith the convention that $K \\subset Y \\times \\{ 0 \\} = Y$,\nand therefore, for every self-intersection point $p \\in D$,\nthere is an associated sign which we will denote by $\\operatorname{sign}(p)$.  \n\n\\noindent Let\n\\[\n    \\widetilde{\\Lambda}\n    \\coloneqq\n    \\frac{   \\mathbb{Z}[\\pi_1(Y)]  }\n\t{  \\langle \\{g - g^{-1} \\mid g \\in \\pi_1(Y)\\} \\rangle \\oplus \\mathbb{Z}[1]  }\n\\]\nwere the quotient is a quotient as abelian groups.\nThe \\emph{Wall self-intersection number} of $K$ is defined to be \n\\[\n    \\mu(K) = \\sum_p \\operatorname{sign}(p) \\cdot g_p \\in \\widetilde{\\Lambda}\n\\]\nSee \\cite{schneiderman2003algebraic}\nfor a proof that it is independent of the choice of $D$, the choice of whisker,\nand the choice of orderings of the sheets of $D$ around the double points.\nFurther, $\\mu$ is a concordance invariant in $Y \\times {I}$,\nand therefore defines a map:\n\\[\n    \\mu \\colon \\mathcal{C}_{U}(Y) \\to \\widetilde{\\Lambda}\n\\]\nNotice that if $g \\in \\pi_1(Y)$ and $g \\neq 1$ then $g$ is also nonzero in $\\widetilde{\\Lambda}$.  \n\n\\begin{proof}[{Proof of \\autoref{existence}, Case 1}]\nWe are now in position to handle \\autoref{existence}\nin the case where $\\pi_1(\\partial X) \\neq 1$.\nNow $X$ is described by attaching 2-handles to $\\disk{4}$\nalong some framed link $L \\subset \\partial \\disk{4}$.\nSince $\\pi_1(\\partial X)$ is \n(normally) generated by the meridians of $L$\nand $\\pi_1(\\partial X) \\neq 1$, there is some meridian $\\gamma$ of\n$L$ that is nontrivial in $\\pi_1(\\partial X)$.\nNotice that if we are given a 2-handlebody described by a framed link $L$\nand $K$ is a knot in the boundary of the 2-handlebody that is shown in the\nframed link diagram as an unknot (possibly linked with $L$),\nthen $K$ is slice in the 2-handlebody -- just forget all of the other\n2-handles and take an unknotting disk\nwhose interior has been pushed into the 0-handle. \nNow, take $K$ in $\\partial X$ to be a\nWhitehead double of $\\gamma$ as in \\autoref{fig:meridian_Whitehead_double},\nwhich is a null-homotopic knot in the boundary. \nBy the previous observation, since $K$ is unknotted in the boundary of the 0-handle,\n$K$ is slice in $X$.\nAdditionally, one computes that $\\mu(K) = \\gamma \\neq 1 \\in \\widetilde{\\Lambda}$,\nfor example using the null-homotopy in \\autoref{fig:Whitehead_double_nullhomotopy}.\nTherefore, $K$ is not null-concordant in $\\partial X$, so $K$ is deep slice in $X$.\n\\end{proof}\n\n\\begin{figure}\n    \\centering\n    \\begin{overpic}[width=0.9\\textwidth]{pictures\/meridian_Whitehead_double}\n        \\put(5, 18){$L_{i}$}\n        \\put(15, 10){\\color{blue} $\\gamma$}\n        \\put(36.5, 25){$\\star$}\n        \\put(63, 10){\\color{blue} $\\operatorname{Wh}(\\gamma) = K$}\n        \\put(63, 18){$L_{i}$}\n        \\put(94, 25){$\\star$}\n    \\end{overpic}\n    \\caption{The Whitehead double of a nontrivial meridian\n    $\\gamma$ to one of the surgery link components is deeply slice in $X$.}\n    \\label{fig:meridian_Whitehead_double}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\begin{overpic}[width=0.9\\textwidth]{pictures\/Whitehead_double_nullhomotopy}\n    \\end{overpic}\n    \\caption{\n        Track of a homotopy from the Whitehead double of $\\gamma$\n        to the unknot giving an immersed disk with a\n        single double point (red in the middle frame).\n        The red double point loop based at the green basepoint\n        calculates that $\\mu(K) = \\gamma$.\n    }\n    \\label{fig:Whitehead_double_nullhomotopy}\n\\end{figure}\n\nNotice that if $\\pi_1(\\partial X) = 1$,\nthen $\\mu$ is of no use since $\\widetilde{\\Lambda} = 0$.\nNow assume that $\\pi_1(\\partial X) = 1$ so that $\\partial X = \\sphere{3}$.\nAgain $X$ is obtained by attaching 2-handles to some framed link $L$.\nLet $\\widehat{X}$ denote the closed 4-manifold\nobtained by closing off $X$ with a 4-handle.\nWe will need a lemma on surfaces in 2-handlebodies,\nwhose statement is\nstandard and could alternatively be concluded from\nthe KSS-normal form for surfaces\nas in \\cite[{Thm.\\ 3.2.7}]{kamada2017surface} and \\cite{MR672939}.\n\\begin{lemma}\n    \\label{lem:disk_in_the_bottom}\n    Let $X$ be a closed smooth 4-manifold with a handle decomposition consisting\n    of only 0-, 2-, and 4-handles,\n    with exactly one 0-handle and one 4-handle.\n    Every element of $H_2(X;\\mathbb{Z})$ can be represented\n    by a smooth closed orientable surface whose\n    intersection with the union of the 0- and 2-handles of $X$\n    is a single disk.\n\\end{lemma}\n\n\\begin{proof}\t\nLet $X_{\\le 2}$ denote the union of the 0- and 2-handles of $X$,\nso that $X = X_{\\le 2} \\cup \\ball{4}$.\nFor every 2-handle $h_i$, there is an element $H_2(X; \\mathbb{Z})$\nobtained by taking the co-core disk $D_i$ for $h_i$\nand capping it off\nwith an orientable surface in the 4-handle. \nLet $\\{F_i\\}$ denote a choice of these surfaces, one for each 2-handle.\nThese surfaces form a basis for $H_2(X;\\mathbb{Z})$ and note\nthat each has the desired property that $F_i \\cap X_{\\le 2} = D_i$ is a disk.\n\nGiven an arbitrary element $x \\in H_2(X; \\mathbb{Z})$, we have\n$x = a_1 [F_1] + \\cdots + a_n[F_n]$ for some $a_i \\in \\mathbb{Z}$.\nTherefore, by taking parallel copies of the $F_i$ for each summand,\nwe can find an immersed (possibly disconnected)\norientable surface $F'$ representing $x$,\nwith $F' \\cap X_{\\le 2}$ a union of $\\sum \\abs{a_{i}}$ disjoint disks.\nBy taking arcs in $\\partial X_{\\le 2}$ that connect\nthe different boundaries of the disks all together,\nand attaching tubes to $F'$ along these arcs,\nwe obtain a connected orientable immersed surface $F''$ representing $x$\nwhose intersection with $X_{\\le 2}$ is now a disk.\nIn particular, the tubing is done so that half of the tube is contained in $X_{\\le 2}$ and the other half is in the 4-handle, and therefore $F'' \\cap X_{\\le 2}$ is the result of boundary summing together the disks in $F' \\cap X_{\\leq 2}$.\n\nTo make $F''$ into an embedded surface,\nwe can resolve the double points in the 4-handle,\nby increasing the genus,\nand arrive at a surface\nrepresenting $x$ with the desired property.  \n\\end{proof}\n\nThe main ingredient for the proof of the second case of\n\\autoref{existence} is the following theorem of Rohlin,\nand in particular the corollary that follows.\nRohlin's theorem has been used in a similar way\nto study slice knots in punctured connected sums\nof projective spaces, for example in \\cite{yasuhara1991torusknot} \nand \\cite{yasuhara1992complexplane}.\n\\begin{theorem}[{Rohlin, \\cite{rokhlin1971two}}]\n\t\\label{Rohlin}\n\tLet $X$ be an oriented closed smooth 4-manifold with $H_1(X; \\mathbb{Z}) = 0$.\n\tLet $\\psi \\in H_2(X; \\mathbb{Z})$ be an element that is divisible by 2,\n\tand let $F$ be a closed oriented surface of genus $g$\n\tsmoothly embedded in $X$ that represents $\\psi$. Then\n\t$$\n\t\t4g \\geq | \\psi \\cdot \\psi - 2\\sigma(X) | - 2b_2(X)\n\t$$\n\\end{theorem}\n\n\\begin{corollary}\n    \\label{cor:no_sphere}\n    Let $X$ be a closed smooth 4-manifold with $H_1(X; \\mathbb{Z}) = 0$, and $H_2(X; \\mathbb{Z}) \\neq 0$.  Then there exists a homology class $\\psi \\in H_2(X; \\mathbb{Z})$ that cannot be represented by a smoothly embedded sphere.  \n\\end{corollary}\n\n\\begin{proof}[Proof of \\autoref{cor:no_sphere}]\nTo apply \\autoref{Rohlin}, we must find a homology class $\\psi$\nthat is divisible by 2 where the right hand side\n$| \\psi \\cdot \\psi - 2\\sigma(X) | - 2b_2(X) > 0$.\nSince the intersection form on $X$ is unimodular,\nthere exists some element $\\alpha$ with $\\alpha \\cdot \\alpha \\neq 0$.\nFrom Poincar{\\'e} duality together with the universal coefficient\ntheorem and our hypothesis that $H_{1}(X; \\mathbb{Z}) =0$, we know that $H_{2}(X; \\mathbb{Z})$ \nis torsion free.\nThen by taking $k$ to be a sufficiently large integer,\nwe can make $|(2k \\alpha) \\cdot (2k \\alpha) - 2 \\sigma(X) |$ arbitrarily large.\nBy taking $\\psi = 2k \\alpha$, the result follows.  \n\\end{proof}\n\n\\begin{proof}[{Proof of \\autoref{existence}, Case 2}]\nBy \\autoref{cor:no_sphere}, let $\\psi \\in H_2(\\widehat{X}; \\mathbb{Z})$\nbe a homology class that can not be represented by an embedded sphere.\nUsing \\autoref{lem:disk_in_the_bottom}, let $F$ be a smooth closed orientable surface\nrepresenting $\\psi$ whose intersection with $X = \\widehat{X}_{\\le 2}$ is a disk $D$,\nas illustrated schematically in \\autoref{fig:surface_schematic}.\n\\begin{figure}\n    \\centering\n    \\begin{overpic}[width=0.45\\textwidth]{pictures\/surface_in_handlebody_schematic}\n        \\definecolor{blue-green}{rgb}{0.0, 0.87, 0.87}\n        \\put(73, 88){\\color{blue} $\\partial D$}\n        \\put(-11.5, 60.5){\\color{gray} $\\sphere{3} \\times {I}$}\n        \\put(60, 69){4-h.}\n        \\put(81, 33){$X = \\widehat{X}_{\\le 2}$}\n        \\put(60, 36){2-h.}\n        \\put(60, 28){0-h.}\n        \\put(36, 20){\\color{blue} $D$}\n    \\end{overpic}\n    \\caption{\n        Schematic of the blue surface $F$\n        in the 2-handlebody, intersecting $X=$ the\n        union of the 0- and 2-handles\n        in a disk $D$. If $\\partial D$\n        was shallow slice (dashed light blue) in $X$,\n        disk $D$ union the shallow slice disk\n        flipped into the 4-handle (solid light blue)\n        would be an impossible sphere representative\n        of the homology class of $F$.\n    }\n    \\label{fig:surface_schematic}\n\\end{figure}\nThen $\\partial D \\subset \\partial X$ is deep slice in $X$,\nsince otherwise the surface obtained by intersecting $F$\nwith the 4-handle could be replaced with a disk\nwithout altering the homology class,\nviolating the assumption that $\\psi$ cannot be represented\nby an embedded sphere.\nTo see that the homology class is not altered,\nobserve that in any 2-handlebody the homology class of a surface\nis determined by how it intersects the 0- and 2-handles. \nAlso observe that in this case the deep slice knot\n$\\partial D \\subset \\partial X = \\sphere{3}$\nis local.\nThis concludes the proof of \\autoref{existence}.\n\\end{proof}\n\n\n\\section{Universal slicing manifolds do not exist}\n\\label{sec:universal_slicing}\n\nThe Norman-Suzuki trick\n\\cite[Cor.\\ 3]{MR246309}, \\cite[Thm.\\ 1]{suzuki1969localknots}\ncan be used to show\nthat any knot $K \\subset \\sphere{3}$ bounds\na properly embedded disk in a punctured\n$\\sphere{2} \\times \\sphere{2}$:\nThe track of a null-homotopy of $K$ in $\\disk{4}$\ncan be placed in the punctured $\\sphere{2} \\times \\sphere{2}$\nwhich gives a disk\nthat we can assume to be a generic immersion,\nmissing $\\sphere{2} \\vee \\sphere{2} \\subset (\\sphere{2} \\times \\sphere{2})^{\\circ}$,\nand with a finite number\nof double points.\nBy tubing into the spheres\n$\\sphere{2} \\times \\{ \\textrm{pt} \\}, \\{ \\textrm{pt} \\} \\times \\sphere{2}$\nwe can remove all the intersections\n-- but observe that\nthis changes the homology class of the disk.\n\n\\begin{proposition}\n    \\label{prop:all_slice}\n\tLet $M^3$ be a closed orientable 3-manifold.\n\tThere exists a compact orientable 4-manifold $X^4$ constructed\n\twith only a 0-handle and 2-handles, with $\\partial X = M$\n\tsuch that every knot in $M$ is slice in $X$.\n\\end{proposition}\n\n\\begin{proof}\n\tStart by taking any compact 4-manifold $X'$\n\twith only 0-, 2-handles\n\tand boundary $M$\n\tand let $X = X' \\# (\\sphere{2} \\times \\sphere{2})$.\n\tLet $K \\subset M = \\partial X$ be a knot.\n\tSince $X'$ and $X$ are simply connected,\n\t$K$ bounds an immersed disk which we can assume\n\tlives completely in the $X'$-summand of the connected sum.\n\tNow the Norman-Suzuki trick works to remove intersection points\n\tof the immersion\n\tby tubing into the coordinate spheres\n\tof the $\\sphere{2} \\times \\sphere{2}$-summand.\n\\end{proof}\n\n\\begin{remark}\n    In contrast to the homologically nontrivial\n    disks constructed in the Norman-Suzuki trick,\n    a knot is slice via a null-homologous disk in\n    some connected sum $\\#^{n} \\sphere{2} \\times \\sphere{2}$\n    if and only if its Arf-invariant is zero.\n    $\\operatorname{Arf} K = 0$ implies that the knot is band-pass\n    equivalent to the unknot, and a band pass\n    can be realized by sliding the (oppositely oriented) strands\n    of a pair of bands over the coordinate spheres\n    in a $\\sphere{2} \\times \\sphere{2}$ factor.\n    Conway-Nagel \\cite{conway2020stably}\n    defined and studied the minimal number of summands\n    needed \n    to find a disk \n    in a punctured $\\#^{n} \\sphere{2} \\times \\sphere{2}$.\n\\end{remark}\n\n\\noindent \\textbf{Convention:} From now until the\nend of this section, properly embedded\nslice disks $\\Delta^{2} \\subset X^{4}$ in a $4$-manifold\nare always required\nto be \\textbf{null-homologous}.\nWe will still add the qualifier ``null-homologous''\nin the statements to emphasize this.\nSince our obstructions work in the topologically locally\nflat category, we will formulate everything in this more general setting.\n\n\\begin{definition}\n    A knot $K \\subset \\sphere{3}$ is\n    \\emph{(topologically\/smoothly)\n    null-homologous slice in the (topological\/smooth)\n    $4$-manifold $X^{4}$} with\n    $\\partial X = \\sphere{3}$,\n    if $K = \\partial \\Delta$, where\n    $\\Delta^{2} \\subset X$ is a (locally flat\/smooth) properly embedded\n    disk such that\n    $[\\Delta, \\partial \\Delta] = 0 \\in H_{2}(X, \\partial X)$.\n\\end{definition}\n\nOne way of studying if a knot $K$ is slice in $\\disk{4}$\nis to approximate $D^4$ by varying the $4$-manifold $X$.\nBy restricting the intersection form\nand looking at simply-connected 4-manifolds $X$\nthis gives rise to various filtrations of the\nknot concordance group\n(notably the $(n)$-solvable filtration $\\mathcal{F}_{n}$\nof Cochran-Orr-Teichner \\cite{COT2003}\nand the positive and negative variants\n$\\mathcal{P}_{n}, \\mathcal{N}_{n}$ \\cite{MR3109864}).\n\nWe say that the properly embedded disk\n$\\Delta$ is \\emph{null-homologous}\nif its fundamental class\n$[\\Delta, \\partial \\Delta] \\in H_{2}(X, \\partial X)$\nis zero.\nSince by Poincar{\\'e} duality the intersection pairing \n$H_{2}(X) \\otimes_{\\mathbb{Z}} H_{2}(X, \\partial X) \\xrightarrow{\\pitchfork} \\mathbb{Z}$\nis non-degenerate, \na null-homologous disk is characterized\nby the property that it intersects all closed second homology classes\nalgebraically zero times.\nFor slicing in arbitrary $4$-manifolds,\nwe here restrict to null-homologous disks\nto exclude constructions\nas in the Norman-Suzuki trick.\n\nFor every fixed knot $K \\subset \\sphere{3}$,\nthere is a $4$-manifold in which $K$ is null-homologically slice.\nNorman \\cite[Thm.\\ 4]{MR246309} already\nobserves that it is possible\nto take as the 4-manifold a punctured connected sum\nof the twisted 2-sphere bundles $\\sphere{2} \\widetilde{\\times} \\sphere{2}$.\nSimilarly, \\cite[Lem.\\ 3.4]{cochran1986unknotting} discuss that\nfor any knot $K \\subset \\sphere{3}$\nthere are numbers $p, q \\in \\mathbb{N}$\nsuch that $K$ is null-homologous slice\nin the punctured connected sum \n$\\#^{p} \\CP{2} \\#^{q} \\overline{\\CP{2}}$ of\ncomplex projective planes.\nThe argument starts with a sequence of positive and negative\ncrossing changes leading from $K$ to the unknot, \nand then realizes say a positive crossing change\nby sliding a pair of oppositely\noriented strands over the $\\CP{1}$\nin a projective plane summand. The track of this\nisotopy, together with a disk bounding the final unknot\ngives a motion picture of a null-homologous slice disk.\nSince both positive and negative crossing changes might be\nnecessary, it is important that both orientations\n$\\CP{2}, \\overline{\\CP{2}}$ are allowed to appear in the connected sum.\n\nIn view of $(\\sphere{2} \\times \\sphere{2})^{\\circ}$\nwhere every knot in the boundary bounds a disk\n(which is rarely null-homologous)\nand $(\\#^{p} \\CP{2} \\#^{q} \\overline{\\CP{2}})^{\\circ}$,\nin which we find plenty of null-homologous disks\n(but only know how many summands $p, q$ we need\nafter fixing a knot on the boundary)\na natural question concerns the\nexistence of a\n\\emph{universal slicing}\nmanifold.\nIs there a fixed compact, smooth, oriented\n$4$-manifold $V^{4}$ with $\\partial V = \\sphere{3}$\nsuch that any knot $K \\subset \\sphere{3}$\nis slice in $V$ via a null-homologous disk?\nIt turns out that a signature estimate shows such\na universal solution cannot exist.\n\\begin{theorem}\n\t\\label{thm:no_universal_slicing}\n\tAny compact oriented $4$-manifold $V^{4}$ \n\twith $\\partial V = \\sphere{3}$\n\tcontains a knot\n\tin its boundary\n\tthat is not \n\ttopologically null-homologous slice in $V$.\n\\end{theorem}\n\n\\begin{remark}\n    If we drop the assumption that $V$ should be compact,\n    a punctured infinite connected sum of projective planes\n    does the job:\n    \\begin{equation*}\n       \\disk{4} \\#^{\\infty} (\\CP{2} \\# \\overline{\\CP{2}})\n    \\end{equation*}\n    For any fixed knot on the\n    boundary there\n    is a compact slice disk in a finite stage\n    \\begin{equation*}\n        \\disk{4} \\#^{k} \\CP{2} \\#^{l} \\overline{\\CP{2}} \\# \\disk{4}\n        \\subset\n        \\disk{4} \\#^{\\infty} (\\CP{2} \\# \\overline{\\CP{2}}).\n    \\end{equation*}\n\\end{remark}\n\nThe remainder of this section is concerned with\na proof of \\autoref{thm:no_universal_slicing}.\nAs preparation, let us specialize a result\n\\cite[Thm.\\ 3.8]{conway2020stably},\nwhich is a generalization of the Murasugi-Tristram inequality\nfor links bounding surfaces in $4$-manifolds,\nto the case of knots.\nHere $\\sigma_{\\omega}(K)$ is the\n\\emph{Levine-Tristram signature} of the knot $K$,\ndefined as the signature\nof the hermitian matrix\n$(1-\\omega)V + (1 - \\overline{\\omega})V^T$,\nwhere $V$ is a Seifert matrix of $K$\nand $\\omega$ a unit complex number not equal to $1$.\nReferences for this signature include\n\\cite{MR253348}, \\cite{tristram1969some}\nand the recent survey \\cite{conway2019levine}.\nThe following inequality only holds for specific values of $\\omega$,\nand will adopt the notation\n$\\sphere{1}_{!}$ for unit complex numbers\n$\\omega \\in \\sphere{1} - \\{ 1 \\}$\nwhich do not appear as a zero of an integral Laurent polynomial\n$p \\in \\mathbb{Z}[t, t^{-1}]$ with $p(1) = \\pm 1$.\n\\begin{theorem}[{\\cite[Special case of Thm.\\ 3.8]{conway2020stably}}]\n    Let $X$ be a closed oriented topological $4$-manifold\n    with $H_{1}(X; \\mathbb{Z}) = 0$.\n    If $\\Sigma \\subset (\\sphere{3} \\times {I}) \\# X$\n    is a null-homologous (topological)\n    cobordism between two knots\n    $K \\subset \\sphere{3} \\times \\{ 0 \\}$ and\n    $-K' \\subset - (\\sphere{3} \\times \\{ 1 \\})$,\n    each contained\n    in one of the two boundary component $\\sphere{3}$'s\n    of $(\\sphere{3} \\times {I}) \\# X$, then\n    \\begin{equation*}\n         \\abs{ \\sigma_{K'}(\\omega) - \\sigma_{K}(\\omega) + \\sign(X) }\n        - \\chi(X) + 2\n        \\le\n        - \\chi(\\Sigma)\n    \\end{equation*}\n    for all $\\omega \\in \\sphere{1}_{!}$.\n\\end{theorem}\n\n\n\\noindent For $K \\subset \\partial X^{\\circ}$ which is\nnull-homologous slice in $X$ and $\\Sigma$ an annulus,\nwe can further simplify:\n\\begin{corollary}\n    \\label{cor:MT_general_manifold}\n    Let $X$ be a closed topological $4$-manifold with\n    $H_{1}(X; \\mathbb{Z}) = 0$.\n    If the knot $K \\subset \\sphere{3}$ is topologically\n    null-homologous slice in $X^{\\circ}$ then\n    for $\\omega \\in \\sphere{1}_{!}$ we have\n    \\begin{equation*}\n        \\abs{ \\sigma_{K}(\\omega) + \\sign(X) }\n        - \\chi(X) + 2\n        \\le\n        0\n    \\end{equation*}\n\\end{corollary}\n\nTo prove \\autoref{thm:no_universal_slicing}\nit will be enough to obstruct the sliceness of a single knot\nin the boundary.\nThe strategy is to use surgery to trivialize $H_{1}$,\nthen pick the knot $K$ in the original manifold boundary\nand arrive at a contradiction to \\autoref{cor:MT_general_manifold}\nin the surgered manifold if $K$ was null-homologous slice.\n\n\\begin{proof}[Proof of \\autoref{thm:no_universal_slicing}]\n\tLet $V$ be a compact topological $4$-manifold\n\twith boundary $\\sphere{3}$, we want to\n\tfind a knot in its boundary which is not slice.\n\tPick a set of disjointly embedded loops\n\t$\\gamma_{1}, \\ldots, \\gamma_{l}$ in $V$ whose homology classes\n\tgenerate $H_{1}(V)$.\n\tIf $V$ already satisfies $H_{1}(V) = 0$,\n\tset $l=0$ for the remainder of the proof and\n\tomit the surgery altogether.\n\tLet $K$ be a knot in $\\sphere{3}$ whose signature\n\t(at the unit complex number $\\omega = -1$)\n\tsatisfies\n\t\\begin{equation*}\n\t\t\\abs{\\sigma_{K}(-1)} \\ge \\abs{\\sign(V)} + \\abs{\\chi(V)} + 2l.\n\t\\end{equation*}\n\tNote that\n\tthe constant on the right hand side only depends\n\ton the signature, Euler characteristic,\n\tand number of generators of $H_{1}(V)$,\n\tand not on the knot $K$.\n    For example, since signature is additive\n    under connected sum,\n    the self-sum $K_{n} = \\#^{n} K$ with $n$ large enough\n    has arbitrarily high signature at $\\omega = -1$\n    if we start with a $K$\n    that has positive signature $\\sigma_{K}(-1)$\n    (for example, taking $K$ to be the left-handed trefoil knot).\n\t\n\tSuppose that $K$ is slice in $V$ via a null-homologous disk $\\Delta$.\n\tBeing null-homologous in\n\tthe relative second homology group\n\tmeans geometrically that there is a locally flat embedded 3-manifold\n\t$M^3$ with boundary the slice disk $\\Delta$ union\n\ta Seifert surface for $K$ in the boundary $\\sphere{3}$,\n\tsee \\cite[Lem.\\ 8.14]{lickorish1997introduction}.\n\tWe can remove the closed components from $M$,\n\twhat remains is a 3-manifold with nonempty boundary in $V$.\n\tGenerically the embedded circles $\\gamma_1, \\ldots, \\gamma_l$\n\twill intersect the\n\t$3$-manifold $M$ in points, but we can push these intersection\n\tpoints off the boundary of $M$\n\tvia an isotopy of the curves in $V$.\n\tWe will still keep the notation\n\t$\\gamma_1, \\ldots, \\gamma_l$ for the isotoped curves\n\twhich are now disjoint from $M$.\n\tEssentially, this finger move supported in a\n\tneighborhood of $M$ is guided by pairwise disjoint arcs in $M$\n\tconnecting the intersections points to the boundary.\n\t\n\tPerform surgery on the loops\n\t$\\gamma_{1}, \\ldots, \\gamma_{l}$,\n\ti.e.\\ for each $\\gamma_{i}$ remove an open\n\ttubular neighborhood $\\nu(\\gamma_{i}) \\cong \\sphere{1} \\times \\interior \\disk{3}$\n\tand glue\n\tcopies of $\\disk{2} \\times \\sphere{2}$ to the\n\tnew $\\sphere{1} \\times \\sphere{2}$ boundary components\n\tvia the identity map\n\t$\\sphere{1} \\times \\sphere{2} \\rightarrow \\sphere{1} \\times \\sphere{2}$.\n\tAfter this surgery we have a compact $4$-manifold $V'$ with $H_{1}(V')=0$,\n\tand the original disk $\\Delta$ survives into $V'$\n\tin which we will call it $\\Delta'$.\n\tObserve that this ``new'' disk $\\Delta'$ is still null-homologous\n\tin $V'$, since the $3$-manifold is still present after\n\tthe surgery.\n    Each circle surgery in a $4$-manifold increases\n    the Euler characteristic by 2, thus $\\chi(V') = \\chi(V) + 2l$.\n    By construction, the 4-manifolds $V$ and $V'$\n    are cobordant, and so their signatures $\\sign(V') = \\sign(V)$ agree.\n\t\n\tStarting with a knot $K$\n\twith large enough signature, if there existed\n\ta null-homologous $\\Delta'$,\n\tsince $H_{1}(V') = 0$:\n\t\\begin{equation*}\n\t    \\abs{ \\sigma_{K}(-1) + \\sign(V') } - \\chi(V') + 2\n\t    =\n\t    \\abs{ \\sigma_{K}(-1) + \\sign(V) } - (\\abs{\\chi(V)} + 2l) + 2\n\t    >\n\t    0\n\t\\end{equation*}\n\twhich contradicts the inequality in\n\t\\autoref{cor:MT_general_manifold}.\n\tTherefore $\\Delta$ for $K$ cannot exist.\n\\end{proof}\n\n\\begin{remark}\n    Earlier sources for results\n    in the smooth category\n    include Gilmer and Viro's\n    \\cite{MR603768} version\n    of the Murasugi-Tristram inequality for the classical signature\n    as stated in\n    \\cite[Thm.\\ 3.1]{yasuhara1996homologyclasses}.\n    Our preference for using\n    \\cite{conway2020stably} in the proof of\n    \\autoref{thm:no_universal_slicing} comes from\n    the result being stated in the topological\n    locally flat category.\n\\end{remark}\n\n\n\\section{Speculation and Questions}\n\\label{sec:questions}\n\n\\subsection{Connection to other conjectures}\n\nAn alternative approach to the SPC4  is to find a compact 3-manifold $M$\nthat embeds smoothly in some homotopy 4-sphere $\\Sigma^{4}$, but not in $\\sphere{4}$.\nNotice that if a smooth integral\nhomology sphere $M$ smoothly embeds in $\\Sigma$,\nthen $M$ is the boundary of a smooth\nhomology 4-ball \\cite[Prop.\\ 2.4]{MR3653313}.\nHowever, there is no known example of a 3-manifold $M$ that is the boundary of\na smooth homology 4-ball but that does not embed into $\\sphere{4}$.\nBoth this and the approach in the introduction\nare hung up at the homological level.\nFurther discussion of knots in homology spheres\nand concordance in homology cylinders\ncan be found in, for example, \\cite{hom2018knot}, \\cite{davis2019concordance}.\n\n\\autoref{cor:universal_cover_S4_no_deep_slice} has some relevance to\nthis which we now discuss (a similar discussion also appears in a comment by\nIan Agol on Danny Calegari's blogpost \\cite{scharlemannonschoenflies}).\nThe unsolved Schoenflies conjecture proposes that if $\\mathcal{S} \\subset \\sphere{4}$\nis a smoothly embedded submanifold with $\\mathcal{S}$ homeomorphic to $\\sphere{3}$,\nthen $\\mathcal{S}$ bounds a submanifold $B \\subset \\sphere{4}$\nthat is diffeomorphic to $\\disk{4}$.\nThe SPC4  implies the Schoenflies conjecture.  \n\n\\begin{question} \\label{all embed}\n\tDoes every exotic homotopy 4-ball $\\mathcal{B}$ smoothly embed into $\\sphere{4}$?\n\\end{question}\n\nNote that if the answer to \\autoref{all embed} is yes,\nthen the Schoenflies conjecture implies the SPC4  and hence the\ntwo conjectures are equivalent: If any homotopy 4-ball\nwould embed into $\\sphere{4}$ and thus,\nby the Schoenflies conjecture, would be diffeomorphic to $\\disk{4}$,\nhence all homotopy 4-balls would be standard,\nso all homotopy 4-spheres would be standard.\nWe have:\n\\begin{observation}\n    If the answer to \\autoref{all embed} is yes,\n    then no homotopy 4-ball can have deep slice knots.  \n\\end{observation}\n\nThus by \\autoref{cor:universal_cover_S4_no_deep_slice},\nif the answer to \\autoref{all embed} is yes,\nthe approach towards SPC4  mentioned in this section would never succeed.\nSimilarly, there would be no 3-manifold that would smoothly embed into a homotopy 4-sphere\nbut not into $\\sphere{4}$. This is because any such embedding into a homotopy sphere avoids a standard 4-ball and after removing this ball the complement is a homotopy 4-ball which we assume embeds into $S^4$, so this approach to SPC4  would also be a dead end.  \n\n\\subsection{More questions}\n\n\n\n\\begin{question}\n\tAre there any 2-handlebodies $X$ other that\n\t$\\natural^k (\\sphere{2} \\times \\ball{2}), k \\ge 0$,\n\twith the property that all $K \\subset \\ball{3} \\subset X$ that are slice in $X$\n\tare also slice in $\\ball{4}$?\n\tIn other words, are there always deep slice local knots\n\twhen $X \\neq \\natural^k (\\sphere{2} \\times \\ball{2})$? \n\\end{question}\n\nOne strategy for answering this question would be to start with a framed link $L$\ndescribing a 2-handlebody other than $\\natural^k (\\sphere{2} \\times \\ball{2})$\nand to handle-slide $L$ to a new framed link $L'$ that \ncontains a knot $K$ that is not slice in $\\ball{4}$.\nThen this knot $K$ when considered in a 3-ball\n$K \\subset \\ball{3} \\subset \\partial X$ is an example of such a deep slice knot in $X$.\nThis strategy fails to find any non-slice knots $K$ (as it must)\nfor $\\natural^k (\\sphere{2} \\times \\disk{2})$ when we start with $L$ being\nthe 0-framed unlink -- since then all resulting knots $K$\nwill be ribbon hence slice in $\\ball{4}$.  \n\nIn view of the $2$-handlebodies constructed in\n\\autoref{prop:all_slice}, \none could ask whether this extension of the Norman-Suzuki trick is\nthe only way to make any knot in the boundary of a manifold\nbound an embedded disk:\n\\begin{question}\n\tIf $X$ is a 2-handlebody with the property that every knot in the boundary\n\tof $X$ is slice in $X$\n\t(no assumption on the relative homology class of the disk), \n\tdoes it follow that $X$ decomposes\n\tas $X = X_0 \\# (S^2 \\times S^2)$ or $X = X_0 \\# (S^2 \\widetilde{\\times} S^2)$?\n\tMore generally, what about the same question\n\twithout the hypothesis that $X$ be a 2-handlebody?\n\\end{question}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\nThe model of Lipschitz percolation was introduced in \\cite{DiDoGrHoSc-10}. \nSince its introduction it has been the subject of numerous articles \nand has shown various connections and applications to other topics such as lattice embeddings, plaquette, entanglement and comb percolation or the pinning of interfaces in random media (see e.g.\n\\cite{GrHo-10}, \\cite{DiDoSc-11}, \\cite{GrHo-12}, \\cite{GrHo-12b}, \\cite{HoMa-14}).\nIn the present article we investigate the critical probability for the existence of a Lipschitz surface of open sites\nthat lies above a hyperplane which is tilted (along one or several\ncoordinate axes) by an angle $\\gamma$. We are particularly interested in the asymptotics of this critical probability as\n$d\\to \\infty$ and $\\gamma \\uparrow \\pi\/4$. An immediate consequence of our results is the existence of non-negative stationary supersolutions to the problem\n$$\nu_t(x,t) = \\Delta u(x,t) + f(x, \\bar a\\cdot x + u(x,t),\\omega) + F\n$$\nfor $\\bar a\\in(-\\alpha,\\alpha)^d$ and $F>0$ independent of $\\bar a$ for some $\\alpha>0$ in the sense of \\cite{DiDoSc-11}, i.e., where $f$ describes randomly placed local obstacles. This setting is related to the study of singular homogenization problems, since -- as a cell problem -- it determines the effective velocity $H(\\bar a)$ of an interface with slope $\\bar a$.\n\nOur context is that of site percolation in $\\ensuremath{\\mathbb{Z}}^{d+1}$ with parameter $p \\in [0,1]$. That is, $\\Omega:= \\{0,1\\}^{\\mathbb Z^{d+1}}$ is the set of configurations and the corresponding probability distribution $\\P_p$ is the product measure of Bernoulli distributions with parameter $p$. A site $x \\in \\mathbb Z^{d+1}$ is called \\emph{open} (with respect to $\\omega$) if $\\omega(x)=1$, and \\emph{closed} \nif $\\omega(x)=0$.\n \nOur main object of study are Lipschitz functions and surfaces defined as follows. A function $F: \\ensuremath{\\mathbb{Z}}^d\\rightarrow \\ensuremath{\\mathbb{Z}}$ is called \\emph{Lipschitz} if for any $\\bar x,\\bar y \\in \\ensuremath{\\mathbb{Z}}^d$ the implication \n\\begin{align*}\n \\Vert \\bar x-\\bar y \\Vert_1 =1 \\Rightarrow \\vert F(\\bar x) - F(\\bar y)\\vert \\leq 1\n\\end{align*}\nholds true. We use the term \\emph{Lipschitz surface} to refer to a subset of $\\ensuremath{\\mathbb{Z}}^{d+1}$ that is the graph of a Lipschitz function. Furthermore, given a realization $\\omega \\in \\Omega$, we\n call the Lipschitz surface \\emph{open} if all sites \n in the Lipschitz surface are open in the sense of site percolation, i.e., if $\\omega(\\bar x, F(\\bar x))= 1$ for all ${\\bar x \\in \\ensuremath{\\mathbb{Z}}^d}$.\n\nIt was proven in \\cite{DiDoGrHoSc-10} that the event of existence of an open Lipschitz surface completely contained in the upper half-plane $\\ensuremath{\\mathbb{Z}}^d\\times\\ensuremath{\\mathbb{N}}$ undergoes a phase transition. That is, for any dimension $d \\ge 1$ there exists a critical probability $p_L(d) \\in (0,1)$ such that the following holds: For $p < p_L(d)$ one has that $\\P_p$-a.s. there exists no open Lipschitz\nsurface in $\\ensuremath{\\mathbb{Z}}^d\\times\\ensuremath{\\mathbb{N}}$, whereas for $p > p_L(d)$ one has that $\\P_p$-a.s. there exists an open Lipschitz surface in $\\ensuremath{\\mathbb{Z}}^d\\times\\ensuremath{\\mathbb{N}}$.\n  Furthermore, an upper bound for $p_L(d)$ and tail estimates for the height of the minimal surface  were established\n  for $p$ sufficiently large. These results\n  were improved in \\cite{GrHo-12}, where in particular exponential tails for the height of the minimal Lipschitz surface \n  have been established for all $p>p_L(d)$. The results were complemented with an asymptotic lower bound yielding $1\/d$ as the correct order of magnitude for $1-p_L(d)$. \n  Applications and related results can be found in \\cite{DiDoSc-11}, \\cite{\nGrHo-12b}, \\cite{GrHo-10}.\n\nWhile the investigation of Lipschitz percolation up to now has been focused on  Lipschitz surfaces that stay above the plane\n$L:=\\ensuremath{\\mathbb{Z}}^d\\times\\{0\\}$, we are interested in the effect of \\lq tilting\\rq\\, this plane. To make this more precise let us define  \n for any $d \\in \\ensuremath{\\mathbb{N}}, \\, \\alpha \\in [0,1)$ and $\\eta \\in \\{-1,0,+1\\}^d$ the \\emph{tilted planes}\n \\begin{align*}\n  L^{\\alpha,d}_{\\eta} := \\Big\\{ (x_1, \\ldots, x_{d+1}) \\in \\ensuremath{\\mathbb{Z}}^{d+1} \\mid x_{d+1} = \n  \\Big \\lfloor \\alpha \\sum_{i=1}^d \\eta_ix_i \\Big \\rfloor\\Big\\}.\n \\end{align*}\nFor computational convenience we introduce the parameter $\\alpha$ as in the above definition, instead of directly working with the angle $\\gamma $ by which a plane is tilted along all the coordinate axes in direction $e_i$ for which $\\eta_i = 1$, $1 \\le i \\le d,$\nin the above choice of $\\eta$ (and $-\\gamma$ in the case that $\\eta = -1$). However, given $\\eta,$ there is a natural\n one-to-one correspondence between $\\alpha$ and \nthe angle $\\gamma$. Also, note that the case of $\\alpha =0$ as well as the case $\\eta = 0$ correspond to $\\gamma = 0$ and thus to standard Lipschitz percolation.\nThe restriction to $\\alpha \\in [0,1)$, resp. $\\gamma < \\pi\/4$, is natural, once one realizes that for \n$\\eta \\ne 0$, $\\alpha \\geq 1$ (resp. $\\gamma \\geq \\pi\/4$), and any $p<1$, $\\P_p$-a.s. there exists no open Lipschitz surface above the plane $L^{\\alpha,d}_{\\eta}$. \n\nIn the study of Lipschitz percolation above tilted planes, the related concept of Lipschitz percolation above \\lq inverted pyramids\\rq\\ turns out to be helpful. Thus, we introduce for any $d \\in \\ensuremath{\\mathbb{N}}, \\, \\alpha \\in [0,1)$ and $\\eta \\in \\{-1,0,+1\\}^d$ the \n\\emph{inverted pyramid} $\\nabla_\\eta^{\\alpha, d}$ as\n\\begin{align*}\n& \\nabla^{\\alpha, d}_\\eta \n:=\n\t\\Big \\{(x_1, \\dots, x_{d+1}) \\in \\mathbb Z^{d+1} \\mid x_{d+1} = \n\t\\max_{\\substack{\\eta' \\in \\{-1,0,+1\\}^d\\\\ \\Vert \\eta' \\Vert_1 = \\Vert \\eta \\Vert_1}} \n\t\\Big \\{ \\Big \\lfloor \\alpha \\sum_{i=1}^d \\eta_i'  x_i \\Big \\rfloor \\Big \\} \\Big \\}.\n\\end{align*}\nWe can now formulate our main result:\n\n\\begin{theorem}\n \\label{thm:mainthm_intro}\n There exists a phase transition for both Lipschitz percolation above planes and\n Lipschitz percolation above inverted pyramids, and their critical probabilities coincide. This critical probability $p_L(\\alpha,d,\\eta)$ is nontrivial and depends on $\\eta$ only via\n $\\Vert \\eta \\Vert_1$. Furthermore,\n \\begin{equation}\\label{eq:mr_d}\n 1-p_L(\\alpha, d, \\eta) \\asymp d^{-\\frac{1}{1-\\alpha}}, \\qquad \\text{ as } d \\rightarrow \\infty, \n \\end{equation}\n and\n\\begin{equation}\\label{mr_alpha}\n 1-p_L(\\alpha, d, \\eta) \\asymp (1-\\alpha)^d, \\qquad \\text{ as }\\alpha \\rightarrow 1. \n \\end{equation}\n\\end{theorem}\nHere we write $f(s) \\asymp g(s)$ as $ s\\rightarrow \\bar s$ for two functions $f$ and $g$\n if there exist positive and finite constants $c,C$ such that $\\liminf_{s \\rightarrow \\bar s} f(s)\/g(s) \\geq c$ and $\\limsup_{s \\rightarrow \\bar s} f(s)\/g(s) \\leq C$.\n\nFor the reader's convenience, Theorem \\ref{thm:mainthm_intro} is a concise\nsummary of the principal asymptotics for $p_L(\\alpha, d, \\eta)$ obtained in this article. The actual asymptotics\n we obtain are more precise and will be given as individual results below.\n\nThe article is structured as follows.\nSection \\ref{sec:Lipschitz} is concerned with general results on Lipschitz percolation in the set-up of tilted planes.\n Proposition \\ref{prop:equiv} establishes the non-trivial phase transition for $p_L(\\alpha, d, \\eta)$,\n  whereas Lemma \\ref{lem:mon} exposes the monotonicity relations for the individual parameters.\n\nSection \\ref{sec:bounds} outlines all bounds on the critical probabilities separated into two subsections, one for lower and one for upper bounds. Using the notation of \\eqref{eq:qDef}, the asymptotics\n\\eqref{eq:mr_d} and \\eqref{mr_alpha} follow by combining Propositions \\ref{prop:allAlphaUBp} and \\ref{prop:UBqAsympD}, \nas well as Propositions \\ref{prop:lbq_alpha} and \\ref{prop:UBqAsympAlpha}, respectively. As explained in Proposition \\ref{prop:D1}, for $d=1$ we obtain the exact asymptic behavior for $\\alpha \\rightarrow 1$. In addition, Proposition \\ref{prop:qLBkDepOnd} provides lower bounds for the critical probabilities, depending on how the number of tilted axes behaves asymptotically with the dimension. \n\nThe corresponding proofs and further auxiliary results are contained in Section \\ref{sec:proofs}.\n\n\n\\section{Further notation and auxiliary results} \\label{sec:Lipschitz}\n\nWe begin by defining the events to be considered and to this end denote by $L^{\\alpha, d}_{\\eta, \\geq}$ the upper half space above $L^{\\alpha, d}_{\\eta}$.\nFor this purpose, denote the set of all Lipschitz functions by $\\Lambda$.\n\n\\begin{definition}\n Let $\\mathsf{LIP}^{\\alpha, d}_{\\eta}$ denote the event that there exists an open Lipschitz surface \ncontained in  $L^{\\alpha, d}_{\\eta,\\ge}$, i.e.,\n\t\\[\\mathsf{LIP}^{\\alpha, d}_{\\eta} := \n\t\\Big \\{ \\omega \\in \\Omega \\mid \\exists F \\in \\Lambda : \\forall \\bar{x} \\in \\mathbb Z^{d}: \\omega((\\bar{x}, F(\\bar{x})))=1 \\text{ and } F(\\bar{x}) > \\Big \\lfloor \\alpha \\sum_{i=1}^d \\eta_i \\bar{x}_i \\Big \\rfloor \\Big \\}.\\]\nSimilarly to the case of planes we use $\\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta})$ to denote the event of existence of a Lipschitz surface above the inverted pyramid $\\nabla_\\eta^{\\alpha, d}$, i.e.,\n\\begin{align*}\n&\\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta}):= \\\\\n& \\Big \\{\\omega \\in \\Omega \\mid \\exists F \\in \\Lambda  :  \\forall \\bar{x} \\in \\mathbb Z^{d} :  \\omega((\\bar{x}, F(\\bar{x})))=1 \n\t\\text{ and } F(\\bar{x}) >  \n\t\\max_{\\substack{\\eta' \\in \\{-1,0,+1\\}^d\\\\ \\Vert \\eta' \\Vert_1 = \\Vert \\eta \\Vert_1}} \n\t \\Big \\{ \\Big\\lfloor \\alpha \\sum_{i=1}^d \\eta_i'  \\bar x_i \\Big \\rfloor \\Big \\}   \\Big\\}.\n\t\\end{align*}\n\\end{definition}\n\n\\begin{proposition}\n \\label{prop:equiv}\nFor any $d\\ge 1,$   $\\alpha \\in [0,1)$ and $\\eta \\in \\{-1,0,+1\\}^d$, there exists a critical probability \n $p_L(\\alpha,d, \\eta) \\in (0,1)$ \nsuch that\n\\begin{align}\\label{eq:planePhaseTrans}\n&  \\P_p(\\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta})) = \\mathbb P_p(\\mathsf{LIP}^{\\alpha, d}_{\\eta}) = \\begin{cases}\n\t0, \\;   p\\in [0,p_L(\\alpha,d, \\eta)),\\\\\n        1, \\;  p \\in (p_L(\\alpha,d,\\eta ),1].\n        \\end{cases}\n\\end{align}\nIn fact, for any $\\eta' \\in \\{-1,0,1\\}$ with $\\Vert \\eta \\Vert_1 = \\Vert \\eta'\\Vert_1$,\n\\begin{align} \\label{eq:pNormDep}\np_L(\\alpha,d, \\eta) = p_L(\\alpha,d, \\eta' ).\n\\end{align}\nTherefore, $p_L(\\alpha,d, \\eta)$ depends on $\\eta$ only through the number of nonzero entries.\n\\end{proposition}\n\nThis means that there exists a phase transition for both Lipschitz percolation above tilted planes and above inverted pyramids, and their critical probabilities coincide.\nDue to \\eqref{eq:pNormDep} it is convenient to define $p_L(\\alpha, d,k) := p_L(\\alpha,d,\\eta)$ for any $\\eta \\in \\{-1,0,+1\\}$ such that $\\Vert \\eta \\Vert_1 = k\\in\\{0,\\ldots,d\\}$. Furthermore, we set\n\\begin{equation} \\label{eq:qDef}\nq_L(\\alpha, d,k) := 1-p_L(\\alpha, d,k).\n\\end{equation}\nFor notational convenience we will formulate most of our results for $q_L$ instead of $p_L$ since the latter usually tends to $1$ and hence the former to $0$.\n\n\\begin{proof}[Proof of Proposition \\ref{prop:equiv}]\nFirst observe that due to the symmetries of $\\ensuremath{\\mathbb{Z}}^d$ and the i.i.d.-product structure of $\\P_p$,  the quantity $\\P_p(\\mathsf{LIP}^{\\alpha, d}_{\\eta})$ depends on $\\eta$ only through  $\\Vert \\eta \\Vert_1$. Thus, if the postulated critical probabilities exist,\nthen they must fulfill \\eqref{eq:pNormDep}.\n\nWe now start with showing the second equality in  \\eqref{eq:planePhaseTrans} for some $p_L(\\alpha,d, \\eta) \\in [0,1]$.\nSince $\\mathsf{LIP}^{\\alpha, d}_{\\eta}$  is an \nincreasing event,  it is immediate that $\\mathbb P_p(\\mathsf{LIP}^{\\alpha, d}_{\\eta})$ is nondecreasing in $p.$\nTherefore, it is sufficient to show \n that it takes values in $\\{0,1\\}$ only.\n\nDefine the shift  $\\theta: \\omega \\mapsto \\omega(\\cdot, \\ldots, \\cdot +1)$ \nin the $(d+1)$-st coordinate. Then $\\theta$ is measure preserving for $\\P_p$ and ergodic with respect to $\\P_p.$ As a consequence, since $\\theta^{-1}(\\mathsf{LIP}^{\\alpha, d}_{\\eta}) \\subset \\mathsf{LIP}^{\\alpha, d}_{\\eta}$ and $\\P_p(\\theta^{-1}(\\mathsf{LIP}^{\\alpha, d}_{\\eta}))=\\P_p(\\mathsf{LIP}^{\\alpha, d}_{\\eta})$, the event $\\mathsf{LIP}^{\\alpha, d}_{\\eta}$ is $\\P_p$-a.s. invariant with respect to $\\theta$, i.e. $\\P_p(\\mathsf{LIP}^{\\alpha, d}_{\\eta} \\triangle \\theta^{-1}(\\mathsf{LIP}^{\\alpha, d}_{\\eta}))=0,$ and by Proposition 6.15 in \\cite{Breiman} this already implies \n\\begin{align*}\n \\P_p (  \\mathsf{LIP}^{\\alpha, d}_{\\eta} )\\in \\{0,1\\}.\n\\end{align*}\nThis establishes the second equality in  \\eqref{eq:planePhaseTrans} for some $p_L(\\alpha,d, \\eta) \\in [0,1]$.\n\nIn order to obtain the first equality of \\eqref{eq:planePhaseTrans}, due to the \nsecond\nequality in \\eqref{eq:planePhaseTrans} and\n$\\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta}) \\subseteq  \\mathsf{LIP}^{\\alpha, d}_{\\eta},\n$\nit remains to show\nthat\n$\n \\mathbb P_p(\\mathsf{LIP}^{\\alpha, d}_{\\eta}) = 1$ implies\n$\\P_p(\\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta})) = 1.$\nBy symmetries, $ \\mathbb P_p(\\mathsf{LIP}^{\\alpha, d}_{\\eta}) = 1$\n already yields\n\\begin{equation*}\n\\P_p \\Big( \\bigcap_{\\substack{\\eta' \\in \\{-1,0,+1\\}^d\\\\ \\Vert \\eta' \\Vert_1 = \\Vert \\eta \\Vert_1}} \\mathsf{LIP}^{\\alpha, d}_{\\eta'} \\Big) = 1.\n\\end{equation*}\nNote that the pointwise maximum of Lipschitz functions is a Lipschitz function again and thus \n \\begin{align*}\n  \\bigcap_{\\substack{\\eta' \\in \\{-1,0,+1\\}^d\\\\ \\Vert \\eta' \\Vert_1 = \\Vert \\eta \\Vert_1}} \\mathsf{LIP}^{\\alpha, d}_{\\eta'}  \\subseteq \\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta}).\n \\end{align*}\n Thus \\eqref{eq:planePhaseTrans} holds true.\n \n It remains to show the nontriviality of the phase transition, i.e., that $p_L(\\alpha,d, \\eta) \\in (0,1).$\nProposition \\ref{prop:allAlphaUBp} below in particular shows that $p_L(\\alpha, d, d) < 1$ for all $\\alpha \\in [0,1)$ and $d \\ge 1$;  hence, using \\eqref{eq:etaMon} below, we deduce $p_L(\\alpha, d, k) < 1$ for all $0 \\le k \\le d.$ On the other hand, $p_L(\\alpha, d, k) > 0$ for all $0 \\le k \\le d$ follows from the fact that the critical probability for the existence of an infinite connected component in the $1$-norm\nin $(d+1)$-dimensional Bernoulli site-percolation (which is a lower bound for $p_L(\\alpha, d, k)$) is strictly positive.\n\\end{proof}\n\nUsing the above result one can obtain  some simple but helpful monotonicity results for the critical probabilities. \n\n\\begin{lemma} \\label{lem:mon}\n For all $d \\in \\mathbb N,$ and $\\alpha, \\alpha' \\in [0,1)$ such that $\\alpha \\leq \\alpha'$, we have \n\\begin{equation} \n\\forall\\, k = 0, \\ldots, d: \\quad p_L({\\alpha},d, k) \\leq p_L({ \\alpha'}, d, k), \\label{eq:alphaMon}\n\\end{equation}\n\\begin{equation}\n \\forall\\, k = 0, \\ldots, d:\\quad p_L({\\alpha}, d, k)\n \\leq p_L({\\alpha}, d+1, k), \\label{eq:dMon}\n \\end{equation}\n and\n \\begin{equation}\n\\forall\\, k = 0, \\ldots, d-1:\\quad p_L({\\alpha},d, k) \\leq p_L({\\alpha}, d, k+1).  \\label{eq:etaMon} \n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe start by proving the monotonicity in $\\alpha$, which is best seen considering Lipschitz surfaces above inverted pyramids. Note that for $\\alpha' \\geq \\alpha$, one has $ \\nabla^{\\alpha',d}_\\eta \\geq \\nabla^{\\alpha, d}_\\eta,$ in the sense that for any $(\\bar y, y_{d+1}^{\\alpha'}) \\in \\nabla^{ \\alpha',d}_\\eta$ and $(\\bar y, y_{d+1}^{\\alpha}) \\in \\nabla^{\\alpha,d}_\\eta$ we have $y^{ \\alpha'}_{d+1} \\geq y^{\\alpha}_{d+1}$. Hence $\\mathsf{LIP}(\\nabla^{\\alpha', d}_{\\eta}) \\subseteq\n\\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta}),$ which implies \\eqref{eq:alphaMon}.\n\nOn the other hand, to prove \\eqref{eq:dMon} choose $\\eta \\in \\{-1,0,+1\\}^{d+1}$ with $\\Vert \\eta \\Vert_1 = k,$ and let $1 \\le j \\le\nd+1$ be such that $\\eta_j = 0.$\nThen \\eqref{eq:dMon}\n follows directly from the fact that the cross section of a Lipschitz surface in $L^{\\alpha, d+1}_{\\eta, \\ge}$ with $\\ensuremath{\\mathbb{Z}}^{j-1} \\times\n\\{0\\}\\times\\mathbb Z^{d-j+1}$ mapped to $\\ensuremath{\\mathbb{Z}}^d$ by eliminating the $j$-th coordinate\nis again a Lipschitz surface contained in  $L^{\\alpha, d}_{\\eta^{(j)}, \\ge}$, for $\\eta^{(j)}:=(\\eta_1, \\ldots, \\eta_{j-1},\\eta_{j+1}, \\ldots, \\eta_{d+1})$, combined with the fact that $\\Vert \\eta^{(j)} \\Vert_1 = k$ and \\eqref{eq:pNormDep}.\n\nLastly, \\eqref{eq:etaMon} follows from the fact that for any $1\\le j \\le d,$ $ \\nabla^{\\alpha, d}_{\\eta_{j \\to 0}} \\geq \\nabla^{\\alpha, d}_\\eta$ in the above sense and thus $\\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta_{j \\to 0}}) \\supset \\mathsf{LIP}(\\nabla^{\\alpha, d}_{\\eta})$,\nwhere $\\eta_{j \\to 0}$ is obtained from $\\eta$ by replacing the $j$-th coordinate by $0.$\n\\end{proof}\n\n\n\\section{Bounds on the Critical Probabilities} \\label{sec:bounds}\n\nFor functions $f,g$ we write $f(s) \\lesssim g(s)$ as $s\\rightarrow \\bar s$, if $\\limsup_{s \\rightarrow \\bar s} f(s)\/g(s) \\leq 1$, \nwe write $f(s) \\gtrsim g(s)$ as $s\\rightarrow \\bar s$, if $\\liminf_{s \\rightarrow \\bar s} f(s)\/g(s) \\geq 1$, and\nasymptotic equivalence is denoted by\n $f(s) \\sim g(s), \\,s\\rightarrow \\bar s$ (i.e., if $f(s) \\lesssim g(s)$ and $f(s) \\gtrsim g(s)$ as $s\\rightarrow \\bar s$). With this notation we can write the results on the bounds in \\cite{GrHo-12} as\n\\begin{align} \\label{eq:GH12}\n\\begin{split}\n q_L(0,d,0) &\\geq (8d)^{-1}, \\quad \\text{ for all } d \\in \\ensuremath{\\mathbb{N}},\\\\\n q_L(0,d,0) &\\lesssim (2d)^{-1}, \\quad \\text{ as } d\\rightarrow \\infty.\n \\end{split}\n\\end{align}\n\n\\subsection{Lower Bounds for $q_L(\\alpha, d, k)$ }\n\n\n\\begin{proposition}[General bound]\\label{prop:allAlphaUBp}\n For any $d \\geq 1$ and $\\alpha \\in [0,1)$ one has\n \\begin{align*\n  q_L(\\alpha, d, d) \\geq \\frac{1}{2} (4d)^{-\\frac{1}{1-\\alpha}}.\n \\end{align*}\n\\end{proposition}\n\n\nNote that for $\\alpha =0$ this is exactly the lower bound of \\eqref{eq:GH12}.\nIn a similar way one can find bounds for the critical probability in the case that the number $k$ of axes along which the plane is tilted depends on the dimension $d$:\n\\begin{proposition} \\label{prop:qLBkDepOnd}\n Consider a function $\\varphi : \\ensuremath{\\mathbb{N}} \\to \\ensuremath{\\mathbb{N}}_0$ with $\\varphi(d) \\le d$ for all $d \\in \\ensuremath{\\mathbb{N}}$. \n\\begin{enumerate}\n \\item If for some $\\alpha \\in [0,1)$ one has that $\\varphi (d) \\in o(d^{1-\\alpha})$ as $d\\rightarrow \\infty$, then\n\\begin{align*}\n q_L(\\alpha,d,\\varphi(d))  \\gtrsim\\frac{1}{8} d^{-1}, \\quad \\text{ as }d\\rightarrow \\infty.\n\\end{align*}\n \\item If for some\n $\\alpha \\in [0,1)$ and $c \\in [0,1]$ one has $\\varphi(d) \\sim cd^{1-\\alpha}$ as $d\\rightarrow \\infty$, \n then there exists a constant $C(c,\\alpha)>0$ such that \n\\begin{align*}\n  q_L(\\alpha,d,\\varphi(d))  \\gtrsim C(c,\\alpha) d^{-1}, \\quad \\text{ as }d\\rightarrow \\infty.\n\\end{align*}\n \\item If for some\n  $c \\in (0,1]$ one has\n $\\varphi(d) \\sim cd$ as $d \\rightarrow \\infty$, then for $\\alpha \\in (0,1)$,\n\\begin{align*}\n q_L(\\alpha, d, \\varphi(d)) \\gtrsim \\frac{1}{4} (1-\\alpha)(cd)^{-\\frac{1}{1-\\alpha}}, \\quad \\text{ as }d\\rightarrow \\infty.\n\\end{align*}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{remark}\n The constant in Proposition \\ref{prop:qLBkDepOnd}, (b), satisfies $C(c,0) = C(0,\\alpha)= 1\/8$ for any $c \\in [0,1]$, $\\alpha \\in [0,1)$;\n this is what one would hope for, given that these cases correspond to standard Lipschitz percolation.\n \n The bound in Proposition \\ref{prop:qLBkDepOnd}, (c), is an improvement compared to Proposition \\ref{prop:allAlphaUBp} at the expense of being of asymptotic nature only. \n\\end{remark}\n\n\n\\begin{proposition}[]\\label{prop:lbq_alpha}\n For each $d \\geq 1$ and each $k = 1,\\ldots,d$ there exists a constant $C(k,d)>0$ such that for all $\\alpha \\in [0,1)$ one has\n \\begin{align*\n  q_L(\\alpha,d,k) \\geq C(k,d)(1-\\alpha)^k.\n \\end{align*}\n \\end{proposition}\n \n \\begin{proposition}\\label{prop:D1}\n For $d=1$ one has $q_L(\\alpha,1,1) \\gtrsim (1-\\alpha)$ as $\\alpha \\rightarrow 1$, which together with Proposition \\ref{prop:UBqAsympAlpha}  below yields\n \\begin{align*}\n  q_L(\\alpha,1,1) \\sim (1-\\alpha), \\qquad \\text{ as } \\alpha \\rightarrow 1.\n \\end{align*}\n \\end{proposition}\n \n\\subsection{Upper Bounds for $q_L(\\alpha, d, k)$}\n \n\n \\begin{proposition}[Asymptotic behavior for $d \\rightarrow \\infty$]\\label{prop:UBqAsympD}\n  For every $\\alpha \\in [0,1)$ there exists a constant $C(\\alpha)$ such that\n  \\begin{align*}\n   q_L(\\alpha,d,d) \\lesssim C(\\alpha) d^{-\\frac{1}{1-\\alpha}}, \\qquad \\text{ as } d\\rightarrow \\infty.\n  \\end{align*}\n  More precisely, $C(\\alpha) = \\theta^{\\frac{1}{1-\\alpha}}\/({\\rm e}^{\\theta} -1)$, where $\\theta$ is the unique solution to \n  $\\theta {\\rm e}^{\\theta}\/({\\rm e}^{\\theta} -1) = 1\/(1-\\alpha)$ and $C(0) = 1$.\n \\end{proposition}\n\n\n \\begin{proposition}[General bound]\\label{prop:UBqAsympAlpha}\n  For any $\\alpha \\in [0,1)$ and $d \\in \\ensuremath{\\mathbb{N}}$ \n  \\begin{align*}\n  q_L(\\alpha,d,d) \\leq \\frac{d!(1-\\alpha)^d}{1+d!(1-\\alpha)^d} \\leq d!(1-\\alpha)^d.\n  \\end{align*}\n \\end{proposition}\n \\begin{remark}\nSince $q_L(\\alpha, d,k) \\leq q_L(\\alpha, k,k)$ by Lemma \\ref{lem:mon}, Proposition \\ref{prop:UBqAsympAlpha}\nimmediately implies upper bounds for $q_L(\\alpha, d, k)$ for any $k=1,\\ldots,d$ also.\n \\end{remark}\n \n \n\n\\section{Proofs} \\label{sec:proofs}\n\nAs explained in \\cite{DiDoGrHoSc-10} and \\cite{GrHo-12} for standard Lipschitz percolation, the lowest open Lipschitz surface (above $L^{\\alpha,d}_{\\eta}$) may be constructed as a blocking surface to a certain type of paths called (admissible) $\\lambda$-paths. This characterization is the core of the proofs in this section.\n\nDenote by $e_1, \\ldots, e_{d+1} \\in \\ensuremath{\\mathbb{Z}}^{d+1}$ the standard basis vectors of $\\ensuremath{\\mathbb{Z}}^{d+1}$.\n\n\\begin{definition} \\label{def:adLamPa}\n For $x,y \\in \\ensuremath{\\mathbb{Z}}^{d+1}$ a \\emph{$\\lambda$-path} from $x$ to $y$ is any finite sequence $x=u_0, \\ldots, u_n=y$ \n of distinct sites in $\\ensuremath{\\mathbb{Z}}^{d+1}$ such that for all $ i = 1, \\ldots,n$\n \\begin{align*}\n    u_i - u_{i-1} \\in \\{e_{d+1}\\} \\cup \\{-e_{d+1} \\pm e_{j} \\mid j = 1, \\ldots, d\\}.\n \\end{align*}\n Such a path will be called \\emph{admissible (with respect to $\\omega$)}, if for all $i = 1, \\ldots, n$ the following implication holds:\n \\begin{align*}\n  \\text{ If } u_i-u_{i-1} = e_{d+1}, \\text{ then } u_i \\text{ is closed (with respect to $\\omega$).}\n \\end{align*}\n\\end{definition}\nFor any $x,y \\in \\ensuremath{\\mathbb{Z}}^{d+1}$ denote by $x \\rightarrowtail y$ the event that there exists an admissible $\\lambda$-path from $x$ to $y$. \nWe then define for all $ x \\in \\ensuremath{\\mathbb{Z}}^d$,  $\\alpha \\in [0,1)$, $d \\in \\ensuremath{\\mathbb{N}}$ and $\\eta \\in \\{-1,0,+1\\}$  the function\n\\begin{align} \\label{eq:LSfromlambda}\n\\; F^{\\alpha, d}_{\\eta}(\\bar x):= \\sup\\{n \\in \\ensuremath{\\mathbb{Z}} \\mid \\exists y \\in L^{\\alpha, d}_{\\eta}:\\; y \\rightarrowtail (\\bar x, n) \\} + 1.\n\\end{align}\nNote that the graph of $F$ is contained in $L^{\\alpha,d}_{\\eta}$. As in \\cite{DiDoGrHoSc-10} and  \\cite{GrHo-12}, it is easy to see that  \nthe function defined in \\eqref{eq:LSfromlambda} \ndescribes a Lipschitz function whose graph consists of open sites,\nif and only if it is finite  for all $\\bar x \\in \\ensuremath{\\mathbb{Z}}^d$. This in turn holds true if and only if it is finite at $\\bar x = 0$. Thus, in the analysis of the existence of an open Lipschitz surface we can focus on the behavior of $F^{\\alpha, d}_{\\eta}(0)$ as defined above.\n\nIt will be useful to define $L^{\\alpha, d}_{\\eta}(h) := L^{\\alpha,d}_{\\eta} + he_{d+1}$ and denote by $\\mathcal L^{\\alpha,d}_{\\eta}(h)$ the random set of sites in $L^{\\alpha, d}_{\\eta}(h)$ reachable by an admissible $\\lambda$-path started in the origin.\nWe have taken the practice of marking elements of $\\ensuremath{\\mathbb{Z}}^d$ with a bar as in $\\bar x \\in \\ensuremath{\\mathbb{Z}}^d$ in order to distinguish them from  canonical elements $x \\in \\ensuremath{\\mathbb{Z}}^{d+1}$. In the same vein, for $x = (x_1, \\ldots, x_{d+1})\\in \\ensuremath{\\mathbb{Z}}^{d+1}$, we use $\\bar x$ to refer to $(x_1, \\ldots, x_d)$ as well as $(\\bar x,x_{d+1})$ to denote $x$. In addition, by a slight abuse of notation we use $0$ to denote the origin of $\\ensuremath{\\mathbb{Z}}, \\ensuremath{\\mathbb{Z}}^d$ and $\\ensuremath{\\mathbb{Z}}^{d+1}$. As we have tacitly done above already,\n it will be necessary to distinguish between $\\ensuremath{\\mathbb{N}}$ and $\\ensuremath{\\mathbb{N}}_0$. For a set $A$ we will use $\\vert A \\vert$ to denote its cardinality. \n\nIn addition, due to the symmetries of $\\ensuremath{\\mathbb{Z}}^d$ and the product structure of $\\P_p$, we will w.l.o.g.\n from now on assume that for any $k = 1, \\ldots, d$, the vector $\\eta$ is of the form\n\\begin{align*}\n \\eta = (\\underbrace{1, \\ldots,1}_{ k \\text{ times}},\\underbrace{0,\\ldots,0}_{d-k \\text{ times}}).\n\\end{align*}\n\n\\subsection{Lower Bounds for $q_L(\\alpha, d, k)$}\n\n\nWe begin with a criterion ensuring the existence of an open Lipschitz surface by providing suitable conditions for the $\\P_p$-a.s. finiteness of $F^{\\alpha, d}_{\\eta}$ as defined in \\eqref{eq:LSfromlambda}.\n\\begin{lemma}[Criterion for existence of an open Lipschitz surface]\\label{lem:surfaceExistence}\nLet $F^{\\alpha, d}_{\\eta}$ be defined as in \\eqref{eq:LSfromlambda}. Then, for any $\\bar x \\in \\ensuremath{\\mathbb{Z}}^d$ and $h \\in \\ensuremath{\\mathbb{N}}$,\n\\begin{equation} \\label{eq:tailEstCond}\n\\P_p \\Big ( F^{\\alpha,d}_{\\eta}(\\bar{x}) - \\Big \\lfloor \\alpha \\sum_{i=1}^d \\eta_i \\bar{x}_i \\Big \\rfloor \\ge h \\Big) \\le  \\ensuremath{\\mathbb{E}}_p [\\vert \\mathcal L_{\\eta}^{\\alpha, d}(h-2) \\vert ].\n\\end{equation} \n In particular, if\n \\begin{align} \\label{eq:summability}\n  \\lim_{h \\rightarrow \\infty} \\ensuremath{\\mathbb{E}}_p[\\vert \\mathcal L^{\\alpha,d}_{\\eta}(h) \\vert ] = 0,\n \\end{align}\nthen \n\\begin{equation} \\label{eq:surfaceEx}\n\\P_p(\\mathsf{LIP}^{\\alpha, d}_{\\eta})  = 1.\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:surfaceExistence}]\nIn order to prove \\eqref{eq:tailEstCond} we start by observing that \n for every $\\bar{x} \\in \\mathbb Z^{d},$ \n\\begin{align} \\label{eq:stochDom}\n\\text{the random variable }F^{\\alpha,d}_{\\eta}(0)+1 \\text{ stochastically dominates }\nF^{\\alpha,d}_{\\eta}(\\bar{x}) - \\Big \\lfloor \\alpha \\sum_{i=1}^d \\eta_i \\bar{x}_i \\Big \\rfloor,\n\\end{align}\nwhere the $+1$ stems from lattice effects.\nNow we estimate \n\\begin{align*}\n \\P_p(F^{\\alpha,d}_{\\eta}(0)  \\ge h + 1) & =  \\mathbb P_p \t\\Big( \\exists z \\in L^{\\alpha, d}_{\\eta}\\,: \\, z \\rightarrowtail (0, h) \\Big)  \n\t\t\\leq \\sum_{z \\in L^{\\alpha, d}_{\\eta}} \\mathbb P_p ( z \\rightarrowtail (0,h) )\\\\\n\t\t & \\le\t\\sum_{z \\in L^{\\alpha, d}_{\\eta}(h)} \\mathbb P_p  ( 0 \\rightarrowtail z ) = \\ensuremath{\\mathbb{E}}_p[\\vert \\mathcal L_{\\eta}^{\\alpha, d}(h) \\vert ]. \n\\end{align*}\nIn combination with \\eqref{eq:stochDom}, this supplies us with \n\\eqref{eq:tailEstCond} which finishes the proof. Note that we used the fact that if a site \n $x = (\\bar x, h)$ with $h \\ge \\Big \\lfloor \\alpha \\sum_{i=1}^d \\eta_i \\bar{x}_i \\Big \\rfloor$ is reachable from $L^{\\alpha, d}_{\\eta}$ by an admissible $\\lambda$-path, then so is any site \n$x = (\\bar x, i)$ with $\\Big \\lfloor \\alpha \\sum_{i=1}^d \\eta_i \\bar{x}_i \\Big \\rfloor \\le i \\le h$. This stems from the observation that if we remove the last step the admissible $\\lambda$-path took in the upward direction and then trace it, we obtain again an admissible $\\lambda$-path reaching the site right below $x$.\n\nThe fact that \\eqref{eq:summability} implies \\eqref{eq:surfaceEx} follows immediately from \\eqref{eq:tailEstCond} in combination with\nthe\nobservation below \\eqref{eq:LSfromlambda}.\n\\end{proof}\n\n\nThe common core of the proofs of Propositions \\ref{prop:allAlphaUBp} and  \\ref{prop:qLBkDepOnd} can be summarized in the following, somewhat technical lemma.\n\n\\begin{lemma}[A general lower bound]\\label{lem:LBwithPi}\n Let $\\alpha \\in [0,1)$, $d \\in \\ensuremath{\\mathbb{N}}$ and $k = 0, \\ldots, d$. Then for any choice of\n \\begin{align} \\label{eq:pchoice}\n   p_1, p_2, p_3, p_4 \\in (0,1) \\text{ such that } \\sum_{i=1}^4 p_i = 1\n \\end{align}\n we obtain \n\\begin{align}\\label{eq:LBqWithp_i}\n q_L(\\alpha, d,k) \\geq \\min\\left\\{\\frac{1}{k} p_1 \\sqrt{p_2 p_3}\\, ,\\, p_1 \\Big( \\frac{p_3}{k}\\Big)^{\\frac 1{1-\\alpha}}\\, ,\\, \\frac{p_1p_4}{2(d-k)}\\right\\}.\n\\end{align}\n\\end{lemma}\n Note that the above holds true for all possible choices of our parameters -- in particular for $ k \\in \\{0,d\\}$ -- if we use the convention of $1\/0=\\infty$. This somewhat unelegant agreement may be justified in this case as it avoids  the need of repeating analogous \n computations without the respective terms.\n\n\n \n \\begin{proof}[Proof of Lemma \\ref{lem:LBwithPi}]\n  In order to obtain the existence of an open Lipschitz surface and thus the lower bound through Lemma \\ref{lem:surfaceExistence}, we will show the following estimate under appropriate  assumptions on $q = 1-p$:\n \nFor $d \\ge 1,$ $\\alpha \\in [0,1)$, $k = 1, \\ldots, d$ and  $q$\nsmaller than the right-hand side of \\eqref{eq:LBqWithp_i},\n there exist constants $\\delta \\in (0,1)$ and $C>0$ such that for all $h\\in \\ensuremath{\\mathbb{N}},$\n\\begin{equation} \\label{eq:tailEst}\n   \\ensuremath{\\mathbb{E}}_p [\\vert \\mathcal L_{\\eta}^{\\alpha, d}(h) \\vert ] \\leq C\\delta^{ h-1}.\n \\end{equation}\n \nWe will say that the $j$-th step of a $\\lambda$-path $(u_n)$ is \\emph{positive downward}, if $u_j-u_{j-1} \\in \\{-e_{d+1} + e_l \\mid l = 1, \\ldots, k\\}$ and \\emph{negative downward} if $u_j-u_{j-1} \\in \\{-e_{d+1} - e_l \\mid l = 1, \\ldots, k\\}$ and use $D^+=D^+(u)$, resp $D^-=D^-(u)$ to denote the number of these steps. In analogy, $D=D(u)$ will denote the number of \\emph{downward} steps such that $u_j-u_{j+1} \\in \\{-e_{d+1} \\pm e_l \\mid l = k+1, \\ldots, d\\}$ and $U = U(u)$ will be the number of \\emph{upward} steps, i.e., those for which $u_j -u_{j-1} = e_{d+1}$.\n\nNow for any natural numbers $U,$ $D^+,$ $D^-$ and $D$, the number of  $\\lambda$-paths \nstarting in the origin with $U$ upward steps as well as $D^+$ positive, $D^-$ negative and $D$ neutral downward steps, respectively, can be estimated from above by\n\\[\\binom{U+D^++D^-+D}{U,D^+,D^-,D}k^{D^++D^-}(2(d-k))^D.\\]\nThus the expected number of such paths which are admissible can  be upper bounded by\n\\begin{equation} \\label{eq:expNrPathUB}\n\\binom{U+D^++D^-+D}{U,D^+,D^-,D}k^{D^++D^-}(2(d-k))^Dq^U.\n\\end{equation}\nIn addition, due to the multinomial theorem, for any $p_1,\\ p_2,\\ p_3,\\ p_4$ chosen as in \\eqref{eq:pchoice} we have\n\\begin{align*}\n{\\binom{U+D^++D^-+D}{U,D^+,D^-,D} p_1^U p_2^{D^+} p_3^{D^-} p_4^D \\le 1,}\n\\end{align*}\nand hence\n\\begin{align} \\label{eq:multinomEst}\n\\binom{U+D^++D^-+D}{U,D^+,D^-,D} \\le \\Big(\\frac{1}{p_1}\\Big)^U \\Big(\\frac{1}{p_2}\\Big)^{D^+} \\Big(\\frac{1}{p_3}\\Big)^{D^-}\\Big(\\frac{1}{p_4}\\Big)^D.\n\\end{align}\nIn order to simplify notation, note that the \\lq best strategy\\rq \\ for admissible $\\lambda$-paths is to go for the  negative orthant in the first $d$ coordinate axes, in the sense that\n\\begin{equation*}\n\\sum_{y \\in L^{\\alpha, d}_{\\eta}(h)}\t\\mathbb P_p(0 \\rightarrowtail y) \t\n \\le 2^d \\sum_{y \\in L^{\\alpha, d}_{\\eta}(h) \\cap ((-\\ensuremath{\\mathbb{N}}_0)^d \\times \\ensuremath{\\mathbb{Z}})}\t\\mathbb P_p(0 \\rightarrowtail y) \t.\n\\end{equation*}\n Since at each downward step of a $\\lambda$-path the $(d+1)$-st coordinate of the path is decreased by one, the total number $U(u)$ \n of upward steps of a $\\lambda$-path $(u_n)$  starting in 0  and ending in $L^{\\alpha, d}_{\\eta}(h) \\cap ((-\\ensuremath{\\mathbb{N}}_0)^d \\times \\ensuremath{\\mathbb{Z}})$ fulfills\n\\begin{align*}  \nU(u) & = D^+(u) + D^-(u) + D(u) +  \\floor{\\alpha (D^+(u)-D^-(u))} + h\n\\end{align*}\nand\n$$\nD^+(u)-D^-(u)  \\leq 0.\n$$\nUsing \\eqref{eq:expNrPathUB} and \\eqref{eq:multinomEst} and choosing $q <p_1$ \nwe can thus estimate\n\\begin{align}\n \\sum_{y \\in L^{\\alpha, d}_{\\eta}(h) \\cap ((-\\ensuremath{\\mathbb{N}}_0)^d \\times \\ensuremath{\\mathbb{Z}})}  \\mathbb P_p(0 \\rightarrowtail y)\\notag\\\\\n &\t\\hspace{-65pt} \\leq \\sum_{\\substack{D^+,D^-,D\\geq 0\\,:\\,D^+-D^-\\leq 0\\\\U =  h+D^++D^-+D +\\floor{\\alpha(D^+-D^-)} }}\\left(\\frac{q}{p_1}\\right)^U\\left(\\frac{k}{p_2}\\right)^{D^+}\\left(\\frac{k}{p_3}\\right)^{D^-}\\left(\\frac{2(d-k)}{p_4}\\right)^{D}\\notag\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t& \\hspace{-65pt} \\leq \\sum_{\\substack{D^+, D^-, D \\geq 0,\\\\ D^- - D^+ \\geq 0}} \\left(\\frac{q}{p_1}\\right)^{ D^++D^-+D+\\floor{\\alpha(D^+-D^-)}+h}\\left(\\frac{k}{p_2}\\right)^{D^+}\\left(\\frac{k}{p_3}\\right)^{D^-}\\left(\\frac{2(d-k)}{p_4}\\right)^{D}\\notag\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t& \\hspace{-65pt} =  \\sum_{n \\geq 0}\\sum_{\\Delta \\geq 0}  \\sum_{m \\geq 0}\\left(\\frac{q}{p_1}\\right)^{ n+\\Delta + n+ m +\\floor{-\\alpha\\Delta} + h}\\left(\\frac{k}{p_2}\\right)^{n}\\left(\\frac{k}{p_3}\\right)^{\\Delta + n}\\left(\\frac{2(d-k)}{p_4}\\right)^{m}\\notag\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t&\\hspace{-65pt}  = \\left(\\frac{q}{p_1}\\right)^h \\sum_{n \\geq 0} \\left(\\frac{q^2k^2}{p_1^2p_2p_3}\\right)^n\\sum_{\\Delta \\geq 0}\\left(\\frac{q}{p_1}\\right)^{\\Delta + \\floor{-\\alpha\\Delta}}\\left(\\frac{k}{p_3}\\right)^{\\Delta}\\sum_{m \\geq 0}\\left(\\frac{2(d-k)q}{p_1p_4}\\right)^{m}\\notag\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t& \\hspace{-65pt} \\leq \\left(\\frac{q}{p_1}\\right)^h \\sum_{n \\geq 0} \\left(\\frac{q^2k^2}{p_1^2p_2p_3}\\right)^n\\sum_{\\Delta \\geq 0}\\left(\\frac{q}{p_1}\\right)^{\\Delta(1-\\alpha)-1}\\left(\\frac{k}{p_3}\\right)^{\\Delta}\\sum_{m \\geq 0}\\left(\\frac{2(d-k)q}{p_1p_4}\\right)^{m} \\notag\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t& \\hspace{-65pt}  = \\left(\\frac{q}{p_1}\\right)^{h-1} \\sum_{n \\geq 0} \\left(\\frac{q^2k^2}{p_1^2p_2p_3}\\right)^n\\sum_{\\Delta \\geq 0}\\left(\\frac{q^{1-\\alpha}k}{p_1^{1-\\alpha}p_3}\\right)^{\\Delta}\\sum_{m \\geq 0}\\left(\\frac{2(d-k)q}{p_1p_4}\\right)^{m}. \\label{eq:EstReachability}\n\\end{align}\nNow note that if \n\\begin{align}\\label{eq:boundq}\n q  < \\min\\left\\{\\frac{1}{k} p_1 \\sqrt{p_2 p_3}\\, ,\\, p_1 \\Big( \\frac{p_3}{k}\\Big)^{\\frac 1{1-\\alpha}}\\, ,\\, \\frac{p_1p_4}{2(d-k)}\\right\\}\n\\end{align}\nthen all sums in \\eqref{eq:EstReachability} converge and $q\/p_1<1$. Thus \n\\begin{align*}\n \\ensuremath{\\mathbb{E}}_p [\\vert \\mathcal L_{\\eta}^{\\alpha, d}(h) \\vert ]\t& = \\sum_{y \\in L^{\\alpha, d}_{\\eta}(h)}\t\\mathbb P_p(0 \\rightarrowtail y) \\\\\n\t\t\t\t\t\t\t& \\leq 2^d\\left(\\frac{q}{p_1}\\right)^{h-1} \\frac{1}{1-\\frac{q^2k^2}{p_1^2p_2p_3}} \\frac{1}{1-\\frac{q^{1-\\alpha}k}{p_1^{1-\\alpha}p_3}} \\frac{1}{1-\\frac{2(d-k)}{p_1p_4}}\n\\end{align*}\nand with \n\\begin{align*}\n \\delta=\\delta(q,p_1) & := \\frac{q}{p_1} \\qquad \\text{and} \\\\\n  \\qquad C=C(\\alpha,d,k,q,p_1,p_2,p_3,p_4) & := 2^d\\frac{p_1^2p_2p_3}{p_1^2p_2p_3-q^2k^2} \\frac{p_1^{1-\\alpha}p_3}{p_1^{1-\\alpha}p_3-q^{1-\\alpha}k}\\frac{p_1p_4}{p_1p_4-2(d-k)q}\n\\end{align*}\nwe obtain the claim in \\eqref{eq:tailEst}. Lemma \\ref{lem:surfaceExistence} then guarantees the existence of an open Lipschitz surface for $q$ as in \\eqref{eq:boundq} which completes the proof.\n \\end{proof}\n\n Depending on our choice of the parameters $p_1, p_2, p_3, p_4$ we now obtain different bounds for the critical probability leading to the results of Propositions \\ref{prop:allAlphaUBp} and  \\ref{prop:qLBkDepOnd}.\n \n\n\\begin{proof}[Proof of Proposition \\ref{prop:allAlphaUBp}]\n\nIn order to obtain Proposition \\ref{prop:allAlphaUBp} set\n\\begin{align*}\n p_1 = \\frac{1}{2} \\qquad \\text{and} \\qquad p_2=p_3=\\frac{1}{4} -\\frac{1}{2}p_4.\n\\end{align*}\nNote that since we consider the case of $k=d,$  the last term on the right-hand side of \\eqref{eq:LBqWithp_i} \nis infinite and hence irrelevant. Comparing the first two terms on the right-hand side of \\eqref{eq:LBqWithp_i}, one can easily see that the second is the dominating one. \nThus, taking $p_4 \\downarrow 0$,  from \\eqref{eq:LBqWithp_i} we  can  deduce the validity of Proposition \\ref{prop:allAlphaUBp}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:qLBkDepOnd}]\n\n\\textit{(a)} Assume $\\varphi(d) \\in o(d^{1-\\alpha})$ as $ d\\rightarrow \\infty$. Then for any fixed choice of $p_1, \\ldots, p_4$,\nas $d \\rightarrow \\infty$  the last term on the right-hand side of \\eqref{eq:LBqWithp_i} is the minimal one\n and thus determines the lower bound for the critical probability given in \\eqref{eq:LBqWithp_i}. For every $\\varepsilon >0$, choosing $p_1 = 1\/2, p_2=p_3=\\varepsilon\/2$ and $p_4 = 1\/2-\\varepsilon$, we get\n\\begin{align*}\n \\liminf_{d \\rightarrow \\infty} q_L(\\alpha, d, \\varphi(d)) d\\geq \\frac{1}{2^2}\\left(\\frac{1}{2}-\\varepsilon\\right).\n\\end{align*}\nSince this is true for any $\\varepsilon >0$, the claim follows.\n\n\\textit{(b)} Now consider the case that\nfor some $c \\in [0,1]$ and $\\alpha >0$\none has $\\varphi(d) \\sim c d^{1-\\alpha}$ as $d\\rightarrow \\infty$. Then the second and third term on the right-hand side of \\eqref{eq:LBqWithp_i} are asymptotically equivalent and smaller than \nthe first term. Hence, they dictate the bound. The claim then holds for any feasible choice of $p_1,\\ldots, p_4$ and\n\\begin{align*}\n C(\\alpha,c):= \\min\\left\\{p_1 \\left(\\frac{p_3}{c}\\right)^{\\frac{1}{1-\\alpha}}, \\frac{1}{2}\\frac{p_1 p_4}{1-c}\\right\\}.\n\\end{align*}\nFor $\\alpha = 0$ we have to take into consideration all three terms of the right-hand side of \\eqref{eq:LBqWithp_i}, and thus obtain the claim with \n\\begin{align*}\n C(0,c):= \\min\\left\\{\\frac{p_1 \\sqrt{p_2p_3}}{c} , p_1 \\frac{p_3}{c}, \\frac{1}{2}\\frac{p_1 p_4}{1-c}\\right\\}.\n\\end{align*}\n\n\\textit{(c)} Now assume that for some $c \\in (0,1]$ one has\n $\\varphi(d) \\sim c d$ as $d\\rightarrow \\infty$. In this case, the second term on the right-hand side of\n\\eqref{eq:LBqWithp_i} is  the asymptotically decisive  contribution. Again, for any $\\varepsilon >0$, choosing\n\\begin{align*}\n p_1 = \\frac{1-\\alpha}{2-\\alpha} - 2\\varepsilon, \\quad p_2 = p_4 = \\varepsilon, \\quad \\text{and} \\quad  p_3 = 1-\\frac{1-\\alpha}{2-\\alpha} = \\frac{1}{2-\\alpha}\n\\end{align*}\nyields\n\\begin{align*}\n \\liminf_{d \\rightarrow \\infty} q_L(\\alpha, d, \\varphi(d)) d^{\\frac{1}{1-\\alpha}} \\geq \\left(\\frac{1-\\alpha}{2-\\alpha} - 2\\varepsilon\\right)\\left(\\frac{1}{2-\\alpha}\\frac{1}{c}\\right)^{\\frac{1}{1-\\alpha}}.\n\\end{align*}\nSince $\\varepsilon$ was arbitrary, \n\\begin{align*}\n \\liminf_{d \\rightarrow \\infty} q_L(\\alpha, d, \\varphi(d)) d^{\\frac{1}{1-\\alpha}} &  \\geq \\frac{1-\\alpha}{2-\\alpha}\\left(\\frac{1}{2-\\alpha}\\frac{1}{c}\\right)^{\\frac{1}{1-\\alpha}}\\\\\n\t\t\t\t\t\t\t\t\t\t  & = (1-\\alpha)\\left(1-\\frac{1-\\alpha}{2-\\alpha} \\right)^{\\frac{2-\\alpha}{1-\\alpha}}\\left(\\frac{1}{c}\\right)^{\\frac{1}{1-\\alpha}} \\geq (1-\\alpha)\\frac{1}{4}\\left(\\frac{1}{c}\\right)^{\\frac{1}{1-\\alpha}}.\n\\end{align*}\n\\end{proof}\n\n\nThe next step is to prove Proposition \\ref{prop:lbq_alpha}.\n\\begin{proof}[Proof of Proposition \\ref{prop:lbq_alpha}]\n We will again want to apply Lemma \\ref{lem:surfaceExistence}. In order to derive an upper bound for the expectation in \\eqref{eq:summability}, instead of \n directly looking at $\\lambda$-paths, we will consider a coarse-grained version of them and estimate the probability of these paths reaching a certain height. \n  The reason for coarse-graining is the following: if $q$ is approximately equal to $q_L(\\alpha,d,k)$, then an admissible $\\lambda$-path starting in 0 (say) will on average pick up at most $1-\\alpha$ closed sites per horizontal step and if $q$ is slightly above  $q_L(\\alpha,d,k)$, then such a path will certainly exist. When $\\alpha$ is very close to one, then  the average number of sites which such a path visits between two successive visits of closed sites\nwill be of the order $(1-\\alpha)^{-1}$  (which is large). If $d \\ge 2$, then there will automatically be lots of admissible $\\lambda$-paths visiting exactly the same closed sites (in the same order) but taking different routes in between\nsuccessive visits to closed sites, the factor increasing to infinity as $\\alpha$ approaches 1. This means that estimating the probability that there exists an admissible $\\lambda$-path (with a certain property) by the expected number of\nsuch paths (via Markov's inequality) becomes very poor when $\\alpha$ is close to 1. Therefore, we will define larger boxes in $\\ensuremath{\\mathbb{Z}}^{d+1}$ and define equivalence classes of paths by just observing the sequence of larger boxes they visit. The boxes will then be tuned such that the number of closed sites inside a box is of order one. \n\n Recall that w.l.o.g. we assume $\\eta_i \\in \\{0,1\\}$, $i = 1, \\ldots, d$. To facilitate\n reading, we have structured the proof into three steps.\n \n \\noindent \\textit{\\uline{Step 1:} Coarse-grained $\\lambda$-paths.} In order to define the abovementioned paths we partition $\\ensuremath{\\mathbb{Z}}^{d+1}$ by dividing $\\ensuremath{\\mathbb{R}}^{d+1}$ into boxes as illustrated in Figure \\ref{fig:only}:\nDefine \n\\begin{align*}\n B^{\\alpha,d,\\eta}_0:= \\Big\\{ r \\in \\ensuremath{\\mathbb{R}}^{d+1} \\, \\mid \\,& \\forall i =1, \\ldots, d:\\; \\big ( \n \\eta_i = 0 \\Rightarrow r_i \\in [0,1) \\big) \\land \\big(\\eta_i \\neq 0 \\Rightarrow  r_i \\in [0, ({1-\\alpha})^{-1} ) \\big),\\\\\n\t\t\t\t\t      &  \\qquad \\qquad r_{d+1} \\in \\Big( \\alpha \\sum_{i=1}^d \\eta_ir_i -1,\\alpha \\sum_{i=1}^d \\eta_ir_i\\Big]\\Big\\}\n\\end{align*}\nand likewise for $a \\in \\ensuremath{\\mathbb{Z}}^{d+1}$ set $B^{\\alpha,d,\\eta}_a:= B^{\\alpha,d,\\eta}_0 + v(a)$, where\n\\begin{align*}\n v(a):\t& = \\sum_{i\\,:\\, \\eta_i=0} a_ie_i + \\sum_{i\\,:\\, \\eta_i\\neq0} a_i\\frac{1}{1-\\alpha}(e_i + \\alpha\\eta_ie_{d+1}) + a_{d+1}e_{d+1}\\\\\n\t& = \\sum_{i\\,:\\, \\eta_i=0} a_ie_i + \\sum_{i\\,:\\, \\eta_i\\neq0} a_i\\frac{1}{1-\\alpha}e_i + \\left(\\sum_{i \\, : \\, \\eta_i\\neq0}a_i\\frac{\\alpha}{1-\\alpha}\\eta_i + a_{d+1}\\right)e_{d+1}.\n\\end{align*}\n\\begin{SCfigure} \\label{fig:only}\n\\includegraphics[width=.7\\textwidth]{coarse}\n\\caption{$L^{\\alpha,d}_{\\eta}$ is marked by the black dots and the corresponding coarse-grained boxes are hatched. $B^{\\alpha,d,\\eta}_0$ is double hatched.}\n\\end{SCfigure}\nNote that these boxes are translations of $B^{\\alpha,d,\\eta}_0$ shifted either in the direction of $e_{d+1}$ or parallel to the inclination of $L^{\\alpha,d}_{\\eta}$ and are such that $\\ensuremath{\\mathbb{Z}}^{d+1}=\\bigcup_{a \\in \\ensuremath{\\mathbb{Z}}^{d+1}} \\big( B^{\\alpha,d,\\eta}_a \\cap \\ensuremath{\\mathbb{Z}}^{d+1}\\big),$ where the union is over disjoint sets. For any $y\\in \\ensuremath{\\mathbb{Z}}^d$ the coordinates of the box it is contained in are given by $a(y)\\in \\ensuremath{\\mathbb{Z}}^{d+1}$ as\n\\begin{align*}\n a_i(y):=\\begin{cases}\n          y_i,\t& i=1,\\ldots,d, \\eta_i =0,\\\\\n          \\lfloor (1-\\alpha)y_i\\rfloor,\t& i=1,\\ldots,d, \\eta_i \\neq0,\\\\\n          y_{d+1}-\\lfloor \\alpha \\sum_{i=1}^d\\eta_iy_i\\rfloor,\t& i=d+1.\n         \\end{cases}\n\\end{align*}\nWe will refer to these as the \\emph{coarse-grained coordinates}. Note that they describe the position of the boxes relative to $L^{\\alpha,d}_{\\eta}$. Note that for $y \\in \\ensuremath{\\mathbb{Z}}^d$ the $(d+1)$-st coordinate of its coarse-grained coordinates $a(y)$ gives its height (or distance in the \n$(d+1)$-st coordinate) relative to $L^{\\alpha, d}_\\eta$.\nSince $\\alpha,d$ and $\\eta$ are fixed for this proof, we will often drop the superscripts for the sake\nof better readability. With the above partition\nof $\\ensuremath{\\mathbb{Z}}^{d+1}$ at hand,\n we can now define coarse-grained $\\lambda$-paths.\n A \\emph{coarse-grained $\\lambda$-path} is any path that takes values in $\\bigcup_{a \\in \\ensuremath{\\mathbb{Z}}^{d+1}} \\{B_a\\},$\nsuch that it can go from $B_a$ to  $B_{a'}$ in one  time step if and only if \n\\begin{align}\n a'-a \\in \t&\\; \\{ e_{d+1}\\} \\cup \\{-\\eta_ie_i\\mid i=1,\\ldots,d, \\eta_i\\neq0 \\} \\label{eq:cgsteps}\\\\\n\t\t&  \\qquad\\cup \\{ - e_{d+1}\\}\\cup \\{\\pm e_i - e_{d+1}\\mid i=1,\\ldots,d\\} \\cup \\{\\eta_ie_i -2e_{d+1} \\mid i=1,\\ldots,d, \\eta_i\\neq0 \\}. \\notag\n\\end{align}\nIn particular, if we sample a standard $\\lambda$-path only on the boxes $\\{B_a\\},$ $a \\in \\ensuremath{\\mathbb{Z}}^{d+1}$, it visits,\nthen this supplies us with a coarse-grained $\\lambda$-path (however, there might be coarse-grained $\\lambda$-paths that\ncannot be obtained by this sampling procedure).\nWe call a box $B_{a}$  \\emph{closed} (with respect to $\\omega$) if and only if $\\omega(x) = 0$ for at least one $x \\in B_a.$\nSimilarly to the case of $\\lambda$-paths, we will call a coarse-grained $\\lambda$-path \\emph{admissible} if for each of its \n\\emph{upward steps}, i.e., those steps for which $a'-a = e_{d+1}$, the box $B_{a'}$ \nis closed. Now since the above sampling procedure maps admissible $\\lambda$-paths to\nadmissible coarse-grained $\\lambda$-paths, the existence of an admissible $\\lambda$-path from some $x \\in \\ensuremath{\\mathbb{Z}}^{d+1}$ to $y\\in \\ensuremath{\\mathbb{Z}}^{d+1}$ implies the existence of an admissible coarse-grained $\\lambda$-path from $B_{a(x)}$ to $B_{a(y)}$.  We\ntherefore investigate the behavior of these coarse-grained $\\lambda$-paths more closely.\n\n\\noindent \\textit{ \\uline{Step 2:} An estimate for coarse-grained $\\lambda$-paths.}\n Recalling \\eqref{eq:cgsteps}, note that there is only one kind of step in a coarse-grained $\\lambda$-path that will not \n change its height relative to $L_\\eta^{\\alpha, d}$, i.e., its coarse-grained coordinate in the $(d+1)$-st dimension, namely those of the form $ -\\eta_ie_i$ with $i$ such that $\\eta_i\\neq0$. Use $\\mathsf{CG}(M)$ to denote the set of all coarse-grained $\\lambda$-paths starting with $B_0$ of length $M\\in \\ensuremath{\\mathbb{N}}$ whose endpoint, i.e. its last box, is above or intersects\n $L^{\\alpha,d}_{\\eta}$. For  $\\pi \\in \\mathsf{CG}(M)$, use $U=U(\\pi)$ to denote the number of its \\lq up\\rq\\,-steps, i.e., those steps that increase the $(d+1)$-st coarse-grained coordinate. Similarly, use $D=D(\\pi)$ to denote the number of steps that decrease the $(d+1)$-st\n  coarse-grained coordinate (possibly by more than 1) and $D^i_0=D^i_0(\\pi)$ the number of steps in each dimension $i=1,\\ldots,d$, that do \\emph{not} alter the \n  $(d+1)$-st coarse-grained coordinate. Due to the natural restrictions on the movements, $D^i_0 = 0$ for any $i$ such that $\\eta_i =0$. We can now make the following observation: In order for $\\pi$ to end in a box above or intersecting\n  $L^{\\alpha,d}_{\\eta}$, we necessarily have \n$$\nU \\geq D.\n$$\n In addition, observe that due to the length of the boxes in the corresponding directions being $1\/(1-\\alpha)$, between two steps of type $D^i_0$ (for the same $i$) there needs to be at least one step of type $D$ or $U$ (not $D^j_0, j\\neq i$). This implies that \n $$\n D^i_0 \\leq D+U+1.\n $$\n  Therefore, for a coarse-grained $\\lambda$-path $\\pi \\in \\mathsf{CG}(M)$, recalling that it ends above \n  or intersecting $L^{\\alpha,d}_{\\eta}$,\n\\begin{align} \\label{eq:estU}\n\\begin{split}\n M \t& = U + D + \\sum_{i=1}^{d} D^i_0  \\leq 2U + \\Vert \\eta\\Vert_1 (2 U + 1) = 2U(k+1) + k \\\\ \n \\Longleftrightarrow \\qquad U\t& \\geq \\frac{M-k}{2(k+1)}.\n  \\end{split}\n\\end{align}\nThus, we will now  estimate the probability of the event on the right-hand side in the above display. Write $m(\\pi)$ for the number of \\emph{distinct} boxes visited by a path $\\pi \\in \\mathsf{CG}(M)$. Then the exponential Chebychev inequality yields for any $\\beta >0$ and \n $\\gamma \\in (0,1)$ that\n \n\\begin{align} \\label{eq:EstClosedSitesInPath}\n\\P_p(& \\text{there exists }  \\pi \\in \\mathsf{CG}(M) \n\\text{ whose boxes contain at least } \\gamma M \\text{ closed sites})\\notag\\\\\n\t& \\leq \\sum_{ \\pi \\in \\mathsf{CG}(M)} \\P_p(\\text{boxes of $\\pi$ contain at least } \\gamma M \\text{ closed sites})\\notag\\\\\n\t& \\leq \\sum_{ \\pi \\in \\mathsf{CG}(M)} \\frac{1}{\\exp(\\beta \\gamma M)} \\ensuremath{\\mathbb{E}}_p[\\exp(\\beta ( \\# \\text{ of closed sites in boxes of } \\pi))] \\notag\\\\\n\t& = \\sum_{ \\pi \\in \\mathsf{CG}(M)} \\frac{1}{\\exp(\\beta \\gamma M)} \\ensuremath{\\mathbb{E}}_p[\\exp(\\beta ( \\# \\text{ of closed sites in $m(\\pi)$ distinct boxes}))] \\notag\\\\ \n\t& = \\sum_{ \\pi \\in \\mathsf{CG}(M)} \\frac{1}{\\exp(\\beta \\gamma M)} (\\ensuremath{\\mathbb{E}}_p[\\exp(\\beta ( \\# \\text{ of closed sites in } B_0))])^{m(\\pi)} \\notag\\\\\n\t& = \\sum_{ \\pi \\in \\mathsf{CG}(M)} \\frac{1}{\\exp(\\beta \\gamma M)} (\\exp(\\beta)q + (1-q))^{\\lceil \\frac{1}{1-\\alpha}\\rceil^km(\\pi)} \\notag\\\\\n\t& \\leq \\sum_{ \\pi \\in \\mathsf{CG}(M)} \\frac{1}{\\exp(\\beta \\gamma M)} (\\exp(\\beta)q + (1-q))^{\\lceil \\frac{1}{1-\\alpha}\\rceil^kM} \\notag\\\\\n\t& \\leq (2(2d+1))^M \\frac{1}{\\exp(\\beta \\gamma M)} (\\underbrace{\\exp(\\beta)q + (1-q)}_{ \\leq \\exp(q (\\exp(\\beta)-1))})^{\\lceil \\frac{1}{1-\\alpha}\\rceil^kM} \\notag\\\\\n\t& \\leq \\exp \\Big (M \\Big (\\log(4d+2)- \\beta\\gamma + q(\\exp(\\beta)-1)\\left(\\frac{2-\\alpha}{1-\\alpha}\\right)^k \\Big ) \\Big),\n\\end{align}\nwhere in the penultimate inequality we estimated the total number of coarse-grained $\\lambda$-paths of length $M$ by $(2(2d+1))^M.$\nObserve that, choosing $\\beta = \\frac{1+\\epsilon}{\\gamma}\\log(4d+2)$ for some $\\epsilon >0$ the expression inside the exponential is negative if, and only if, \n\\begin{align}\\label{eq:uglyboundonq}\n  -\\epsilon\\log(4d+2) +& q(\\exp(\\frac{1+\\epsilon}{\\gamma}\\log(4d+2))-1)  \\left(\\frac{2-\\alpha}{1-\\alpha}\\right)^k  < 0 \\notag\\\\\n    \\Leftrightarrow \\qquad q & < \\frac{\\epsilon \\log(4d+2)}{\\exp((1+\\epsilon)\\gamma^{-1}\\log(4d+2))-1}\\left(\\frac{1-\\alpha}{2-\\alpha}\\right)^k.\n\\end{align}\n\n\\noindent\\textit{\\uline{Step 3:} Returning to $\\lambda$-paths.}\nIn order to apply Lemma \\ref{lem:surfaceExistence} we need to estimate the probability of reaching a site $y \\in L^{\\alpha,d}_{\\eta}(h)$ with an admissible $\\lambda$-path. Recall that coarse-grained  $\\lambda$-paths were defined in such a way that\n the existence of an admissible $\\lambda$-path from $0 \\in \\ensuremath{\\mathbb{Z}}^{d+1}$ to $y \\in \\ensuremath{\\mathbb{Z}}^{d+1}$ implies the existence of an admissible coarse-grained $\\lambda$-path from $B_0$ to $B_{a(y)}$. This path then has length $M$ at least $\\Vert a(y) \\Vert_1$ and thus\n\\begin{align*}\n M \t& \\geq \\Vert a(y) \\Vert_1 \\\\\n\t& \\geq \\sum_{i=1}^d \\vert a_i(y)\\vert + h\\\\\n\t& = \\sum_{i\\,:\\, \\eta_i=0} \\vert y_i \\vert + \\sum_{i\\,:\\, \\eta_i \\neq 0} \\vert \\lfloor(1-\\alpha)y_i\\rfloor \\vert +h\\\\\n\t& \\geq \\sum_{i\\,:\\, \\eta_i=0} \\vert y_i \\vert + \\sum_{i\\,:\\, \\eta_i \\neq 0} ((1-\\alpha)\\vert y_i \\vert - 1) +h\\\\ \n\t& \\geq (1-\\alpha)\\Vert \\bar y \\Vert_1 - k + h.\n\\end{align*}\nTherefore, for any $h \\in \\ensuremath{\\mathbb{N}}$ and $y \\in L^{\\alpha,d}_{\\eta}(h)$ using \\eqref{eq:estU} in the third step,\n\\begin{align*}\n \\P_p(&0 \\rightarrowtail y)\t \\leq \\P_p(\\text{there exists an admissible coarse-grained $\\lambda$-path from $B_0$ to $B_{a(y)}$})\\\\\n\t\t\t\t& \\leq \\P_p(\\text{there exists } \\pi \\in \\mathsf{CG}( (1-\\alpha)\\Vert \\bar y \\Vert_1 - k + h ) \\text{ admissible})\\\\\n\t\t\t\t& \\leq \\P_p(\\text{there exists } \\pi \\in \\mathsf{CG}( (1-\\alpha)\\Vert \\bar y \\Vert_1 - k + h ) \\\\\n\t\t\t\t& \\qquad\\quad \\text{ whose boxes contain at least } \\frac{(1-\\alpha)\\Vert \\bar y \\Vert_1 - k + h-k}{2(k+1)} \\text{ closed sites})\\\\\n\t\t\t\t& \\leq  \\exp\\Big(\\big((1-\\alpha)\\Vert \\bar y \\Vert_1 - k + h\\big)\\\\\n\t\t\t\t& \\qquad \\qquad \\quad \\times \\big(-\\epsilon\\log(4d+2) + q(\\exp((1+\\epsilon)4(k+1)\\log(4d+2))-1) \\left(\\frac{2-\\alpha}{1-\\alpha}\\right)^k\\big)\\Big),\n\\end{align*}\nwhere\n we choose $h \\geq 3k$ and set $\\gamma := \\frac{1}{4(k+1)}$ to apply \\eqref{eq:EstClosedSitesInPath} for the last inequality. Assuming \n\\begin{align*}\n q  < \\underbrace{\\frac{\\epsilon \\log(4d+2)}{\\exp((1+\\epsilon)4(k+1)\\log(4d+2))-1}\\frac{1}{2^k}}_{=:C(k,d,\\epsilon)}(1-\\alpha)^k\n\\end{align*}\n\\eqref{eq:uglyboundonq} holds and combining the observations above we can estimate \\eqref{eq:summability} by \n \\begin{align*}\n \\sum_{y \\in L^{\\alpha,d}_{\\eta}(h)}  \\P_p(0 \\rightarrowtail y)\t& \\leq \\sum_{y \\in L^{\\alpha,d}_{\\eta}(h)} \\exp\n \\Big( ((1-\\alpha)\\Vert \\bar y \\Vert_1 - k + h)\\\\\n\t\t\t\t\t\t\t\t& \\qquad \\qquad \\times (\\underbrace{-\\epsilon\\log(4d+2) + q(\\exp(\\frac{1+\\epsilon}{\\gamma}\\log(4d+2)-1) \\lceil\\frac{1}{1-\\alpha}\\rceil^k}_{=:\\bar c(k,d,\\epsilon,\\alpha, q)=\\bar c < 0 }) \\Big) \\\\\n\t\t\t\t\t\t\t\t& = \\exp((-k+h)\\bar c)\\sum_{y \\in L^{\\alpha,d}_{\\eta}(h)}\\exp((1-\\alpha)\\Vert \\bar y \\Vert_1\\bar c)\\\\\n\t\t\t\t\t\t\t\t& \\leq \\exp((-k+h)\\bar c)\\underbrace{\\sum_{i=1}^{\\infty}\\exp((1-\\alpha)i\\bar c)(2d+1)^i}_{<\\infty}.\n \\end{align*}\nThus\n\\begin{align*}\n  \\lim_{h\\rightarrow \\infty} \\ensuremath{\\mathbb{E}}_p[\\vert \\mathcal L^{\\alpha,d}_{\\eta}\\vert ] = \\lim_{h\\rightarrow \\infty} \\sum_{y \\in L^{\\alpha,d}_{\\eta}(h)}  \\P_p(0 \\rightarrowtail y) = 0.\n\\end{align*}\nTherefore, the assumptions of Lemma \\ref{lem:surfaceExistence} hold which implies the existence of an open Lipschitz surface. Hence,\n\\begin{align*}\n q_L(\\alpha,k,d) \\geq C(k,d,\\epsilon)(1-\\alpha)^k.\n\\end{align*}\nNote that for our result, any $\\varepsilon >0$ is sufficient. However, the optimal $\\varepsilon$ is given by $\\varepsilon = \\frac{1 + h}{4(k+1)\\log(4d+2)}$, where $h$ is such that $-\\exp(-1-4(k+1)\\log(4d+2))=h\\exp(h)$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:D1}]\n\nIn order to prove the lower bound for $q_L(\\alpha,1,1)$ we show the existence of an open Lipschitz surface for sufficiently small $q$ by analyzing the existence of an admissible $\\lambda$-path starting in $L^{\\alpha,1}_{(1)}$ reaching the site $(0,h)$ for large $h \\in \\ensuremath{\\mathbb{N}}_0$. Writing $x \\overset{A}{\\rightarrowtail} y$ for the event of existence of an admissible $\\lambda$-path from $x \\in \\ensuremath{\\mathbb{Z}}^2$ to $y \\in \\ensuremath{\\mathbb{Z}}^2$ that only uses sites in the set $A \\subseteq \\ensuremath{\\mathbb{Z}}^2$, we observe that\n\\begin{align}\\label{eq:est1}\n \\P_p(L^{\\alpha,1}_{(1)} \\rightarrowtail (0,h)) & = \\P_p(L^{\\alpha,1}_{(1)} \\overset{L^{\\alpha,1}_{(1), \\geq}}{\\rightarrowtail} (0,h)) \\leq 2 \\P_p\\big(\\bigcup_{n \\in \\ensuremath{\\mathbb{N}}_0}\\big\\{(n, \\floor{\\alpha n}) \\overset{L^{\\alpha,1}_{(1), \\geq}}{\\rightarrowtail} (0,h)\\big\\}\\big) \\notag \\\\\n\t\t\t\t\t\t& \\leq 2 \\sum_{n =0}^{\\infty}\\P_p((n, \\floor{\\alpha n}) \\overset{L^{\\alpha,1}_{(1), \\geq}}{\\rightarrowtail} (0,h)),\n\\end{align}\nfor any $h \\in \\ensuremath{\\mathbb{N}}_0$. Therefore we need to find suitable upper bounds for the summands. \n\nA first helpful bound, albeit without the restriction on the space, can be obtained similarly to \\eqref{eq:expNrPathUB}. Observe that any $\\lambda$-path from $(n, \\floor{\\alpha n})$ to $(0,h)$ must have made a total of $4k + \\ceil{(2-\\alpha)n} + h$ steps for some $k \\in \\ensuremath{\\mathbb{N}}_0$: $n+k$ to the downward left, $k$ to the downward right and $n-\\floor{\\alpha n} + h + 2k$ upwards. Then, counting the number of admissible $\\lambda$-paths under consideration\n\\begin{align}\\label{eq:genBound}\n \\P_p((n, \\floor{\\alpha n}) \\overset{L^{\\alpha,1}_{(1), \\geq}}{\\rightarrowtail} (0,h)) \t& \\leq  \\P_p((n, \\floor{\\alpha n}) \\rightarrowtail (0,h))\\\\\n\t\t\t\t\t\t\t\t\t\t\t&\\leq \\sum_{k \\in \\ensuremath{\\mathbb{N}}_0} \\binom{2n + h - \\floor{\\alpha n} + 4k}{n+k, k, n-\\floor{\\alpha n} + h + 2k} q^{n-\\floor{\\alpha n} + h + 2k}.\n\\end{align} \nThis upper bounds the terms for small $n$ in \\eqref{eq:est1}, but can also be used to obtain an adequate estimate for large $n$. This is, however, more elaborate: For $n \\in \\ensuremath{\\mathbb{N}}_0$ define\n\\begin{align*}\n A_n & := \\{-n, -(n-1), \\ldots, -1, 0, 1, \\ldots \\}\\times \\ensuremath{\\mathbb{Z}},\\\\\n Y_n & := \\max\\{ r \\in \\ensuremath{\\mathbb{Z}} \\mid (0,0) \\overset{A_n}{\\rightarrowtail }(-n,r) \\}.\n\\end{align*}\n$Y_n$ is the height of the highest site above $-n$ reachable by an admissible $\\lambda$ path started in $0$ under the restriction of using only the sites in $A_n$. Now note that denoting by $\\bar Y_0$ a copy of $Y_0$, independent of $(Y_n)_{n \\in \\ensuremath{\\mathbb{N}}_0}$,\n\\begin{align} \\label{eq:StDom}\n \\bar Y_0 \\text{ stochastically dominates } Y_{n+1} - (Y_n -1) \\text{ under } \\P_p(\\cdot\\mid Y_i, i \\leq n)\n\\end{align}\nsince the conditioning can be seen as discarding those paths in the construction using any site below the $Y_i, i \\leq n$. Therefore,\n a closer study of the distribution of $\\bar Y_0$ seems advisable. Using \\eqref{eq:genBound},\n\\begin{align} \\label{eq:YdistrUB}\n\\begin{split}\n \\P_p(\\bar Y_0 \\geq m) \t& \\leq \\P_p((0,0) \\rightarrowtail (0,m)) \\\\\n\t\t\t& \\leq q^m + \\sum_{k \\in \\ensuremath{\\mathbb{N}}}3^{ m + 4k}q^{m + 2k} \\\\\n\t\t\t& \\leq q^m + (3q)^m \\frac{(9q)^2}{1-(9q)^2},\n\\end{split}\n\\end{align}\nfor $q < 1\/9$. Hence, we can upper bound the expectation\n\\begin{align*}\n \\ensuremath{\\mathbb{E}}_p[\\bar Y_0]\t& \\leq q + \\sum_{m = 2}^{\\infty} q^m + \\frac{(9q)^2}{1-(9q)^2}\\sum_{m =1}^{\\infty}(3q)^m \\\\\n\t\t& \\leq q + Cq^2\n\\end{align*}\nfor a suitable $C>0$ and small $q$. As a consequence, assuming $q$ sufficiently small for \n\\begin{align}\\label{eq:condq}\nq + Cq^2 - 1< -\\alpha \n\\end{align}\n to hold, \\eqref{eq:StDom} and a large deviation principle (the required exponential moments exist due to \\eqref{eq:YdistrUB})\n yield the existence of $c_1, c_2>0$ such that\n\\begin{align*}\n \\P_p( Y_n \\geq -\\alpha n)\t& \\leq c_1\\exp( - n c_2).\n\\end{align*}\nObserve that an admissible $\\lambda$ path started in some $(n, \\floor{\\alpha n})$ and reaching $\\{0\\}\\times\\ensuremath{\\mathbb{N}}_0$ going only through $L^{\\alpha,1}_{(1), \\geq}$ has only used sites to right of $\\{0\\}\\times\\ensuremath{\\mathbb{Z}}$ until the first time it hits $\\{0\\}\\times\\ensuremath{\\mathbb{N}}_0$. Hence,\n\\begin{align*}\n \\P_p((n, \\floor{\\alpha n}) \\overset{L^{\\alpha,1}_{(1), \\geq}}{\\rightarrowtail} (0,h)) & \\leq \\P_p((n, \\floor{\\alpha n}) \\overset{L^{\\alpha,1}_{(1), \\geq}}{\\rightarrowtail} \\{0\\}\\times \\ensuremath{\\mathbb{N}}_0)\\\\\n\t\t      & \\leq \\P_p(Y_n \\geq -n \\alpha) \\leq c_1\\exp( - n c_2).\n\\end{align*}\nThis is the last component needed to estimate \\eqref{eq:est1} as it allows us to choose $N \\in \\ensuremath{\\mathbb{N}}$ such that for any $h \\in \\ensuremath{\\mathbb{N}}$\n\\begin{align*}\n \\sum_{n = N}^{\\infty} \\P_p((n, \\floor{\\alpha n}) \\overset{L^{\\alpha,1}_{(1),\\geq}}{\\rightarrowtail} (0,h) )\\leq \\frac{1}{8}.\n \\end{align*}\nOn the other hand, using \\eqref{eq:genBound} again, we may now choose $H$ sufficiently large such that for all $h \\geq H,$ \n\\begin{align*}\n \\sum_{n = 0}^{N-1} \\P_p((n, \\floor{\\alpha n}) \\overset{L^{\\alpha,1}_{(1),\\geq}}{\\rightarrowtail} (0,h) )\\leq \\frac{1}{8}.\n\\end{align*}\nHence, by \\eqref{eq:est1} choosing $q$ as in \\refeq{eq:condq} implies\n\\begin{align*}\n \\P_p(L^{\\alpha,1}_{(1)} \\rightarrowtail (0,h)) \\leq \\frac{1}{2}\n\\end{align*}\nfor all $h \\geq H$ and thus $q < q_L(\\alpha, 1, 1)$. \n\nThe corresponding upper bound is already given by Proposition \\ref{prop:UBqAsympAlpha}.\n\\end{proof}\n\n\\subsection{Upper Bounds for $q_L(\\alpha, d, k)$}\n\nIt will be useful in this section to consider what we call reversed $\\lambda$-paths. A sequence of sites $x_0, x_1, \\ldots, x_n \\in \\ensuremath{\\mathbb{Z}}^{d+1}$ is called an (admissible) \\emph{reversed $\\lambda$-path}, if $x_n, x_{n-1},x_{n-2}, \\ldots, x_0$ is an (admissible) $\\lambda$-path in the sense of Definition \\ref{def:adLamPa}. \n\nFurthermore, the proof of Proposition \\ref{prop:UBqAsympD} will take advantage of a comparison to so-called $\\rho$-percolation, see e.g. \\cite{MeZu-93} and \\cite{KeSu-00}. Here the setting is that of oriented site-percolation in $\\ensuremath{\\mathbb{Z}}^d$, i.e., where in addition to our standard setting\nof Bernoulli site percolation we assume the nearest neighbor edges of $\\ensuremath{\\mathbb{Z}}^d$ to be oriented in the direction of the positive coordinate vectors\n(which is the sense of orientation for the rest of this section). We say that \\emph{$\\rho$-percolation occurs} for $\\omega \\in \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^d}$ if there exists an oriented nearest neighbor path $0=\\bar x_0, \\bar x_1, \\ldots$ in $\\ensuremath{\\mathbb{Z}}^d$ starting in the origin, such that \n\\begin{align*}\n \\liminf_{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{i=1}^n (1-\\omega(\\bar x_i)) \\geq \\rho.\n\\end{align*}\nAny such path is called a \\emph{$\\rho$-path}.\nThe probability of the existence of such a path exhibits a phase transition in the parameter $q$ and the corresponding critical probability is denoted by $q_c(\\rho, d)$. Theorem 2 in \\cite{KeSu-00} states that for every $\\rho \\in (0, 1]$,\n\\begin{align} \\label{eq:RhoConst}\n \\lim_{d \\rightarrow \\infty} d^{\\frac{1}{\\rho}}q_c(\\rho, d) = \\frac{\\theta^\\frac{1}{\\rho}}{e^{\\theta} -1} =:R(\\rho),\n\\end{align}\nwhere $\\theta$ is the unique solution to $\\theta e^{\\theta}\/(e^{\\theta} -1) = 1\/\\rho$, and $R(1)=1$. Note that we have interchanged the role of `open' and `closed' (and thus $p$ and $q$) with respect to \\cite{KeSu-00} in order to adapt the result to its application in our proof. \n\nBefore turning to the proof of Proposition \\ref{prop:UBqAsympD}, we observe a useful property of the critical probability of $\\rho$-percolation.\n\n\n\\begin{lemma}[Continuity of $q_c$]\\label{lem:rho}\n The critical probability of $\\rho$-percolation is continuous in $\\rho$, i.e. for any $d \\in \\ensuremath{\\mathbb{N}}$ the map \n \\begin{align} \\label{eq:qcCont}\n[0,1) \\ni  \\rho \\mapsto q_c(\\rho, d)\n \\end{align}\nis continuous.\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:rho}]\n Since $d$ is fixed and we only consider $\\ensuremath{\\mathbb{Z}}^d$ in this proof, the index is dropped for better readability. It is easy to see that the event of $\\rho$-percolation also undergoes a phase-transition in $\\rho$ (for fixed $q$) and thus we define\n\\begin{align*}\n \\rho_c(q):= \\sup\\{\\rho \\mid \\P_{1-q}(\\rho\\text{-percolation occurs}) = 1\\}.\n\\end{align*}\nNote that strict monotonicity of $\\rho_c(q)$ for $q \\in [0,\\bar q]$, where $\\bar q := \\sup\\{ q\\mid \\rho_c(q) <1\\}$, \nwould imply the desired continuity of $q_c(\\rho)$ on $[0,1)$. In order to prove this strict monotonicity, we will, however, first consider a different quantity: Still in the setting of oriented percolation in $\\ensuremath{\\mathbb{Z}}^d$, for any $\\omega \\in \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^d}$ \nlet\n\\begin{align*\n Y_{0,n}(\\omega) := \\max \\Big\\{\\, r \\in \\ensuremath{\\mathbb{N}}_0 \\mid \\exists  \\text{ directed nearest neighbor path }& 0 = x_0, x_1, \\ldots, x_{n}:\\\\\n\t\t\t\t\t\t      & \\qquad \\qquad \\, \\sum_{i=1}^{n} (1- \\omega(x_i)) = r\n\t\t\t\t\t\t      \\Big\\},\n\\end{align*}\nand denote by $\\hat X_n$ the site with the lowest lexicographical order that is the endpoint of such a directed nearest neighbor path on which the value of $Y_{0,n}$ is attained. Then, for $m\\geq n$  define\n\\begin{align*\n Y_{n,m}(\\omega) := \\max \\Big\\{\\, r \\in \\ensuremath{\\mathbb{N}}_0 \\mid \\exists  \\text{ directed nearest neighbor path }& \\hat X_n = x_0, x_1, \\ldots, x_{m-n}:\\\\\n\t\t\t\t\t\t      & \\qquad \\qquad \\, \\sum_{i=1}^{m-n} (1- \\omega(x_i)) = r\n\t\t\t\t\t\t      \\Big\\}.\n\\end{align*}\n\nBy the Subadditive Ergodic Theorem (see e.g. \\cite{Durrett}, Theorem 6.6.1)\n the sequence $(Y_{0,n}\/n)_{n \\in \\ensuremath{\\mathbb{N}}}$ converges $\\P_{1-q}$-a.s. and in $L^1(\\P_{1-q})$ to a (deterministic) limit that we denote by $\\gamma(q)$. In fact, \n \\begin{equation} \\label{eq:limitEquality}\n\\gamma (q) = \\rho_c(q).\n \\end{equation}\n To see this, fix $q \\in (0,1)$ and choose $\\rho < \\rho_c(q)$. \nThen for $\\P_{1-q}$-almost any $\\omega \\in \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^d}$ there exists an oriented nearest neighbor path $X_1(\\omega), X_2(\\omega), \\ldots$ such that \n\\begin{align*}\n \\rho \\leq \\liminf_{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{i=1}^n (1-\\omega(X_i(\\omega))).\n\\end{align*}\nSince by definition $ \\sum_{i=1}^n (1-\\omega(X_i(\\omega))) \\leq Y_{0,n}(\\omega)$\nfor $\\P_{1-q}$-almost all $\\omega\\in \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^d}$ and $n \\in \\ensuremath{\\mathbb{N}}$, taking the limes inferior on both sites gives $\\rho \\leq \\gamma(q)$, which implies $\\rho_c(q) \\leq \\gamma(q)$. To prove the converse inequality, choose, for any $\\varepsilon >0$ an $N \\in \\ensuremath{\\mathbb{N}}$ such that $\\frac{1}{N}\\ensuremath{\\mathbb{E}}_{1-q}[Y_{0,N}] \\geq \\gamma(q)-\\varepsilon$. For any $\\omega \\in \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^d}$ let $X_1(\\omega),X_2(\\omega),\\ldots, X_N(\\omega)$ be an (oriented nearest neighbor) path, such that $Y_{0,N} = \\sum_{i=1}^{N} (1- \\omega(X_i(\\omega)))$. Using i.i.d. copies of $(X_1, \\ldots, X_N)$, one can construct an infinite oriented nearest neighbor path $(\\widetilde X_i)_{i \\in \\ensuremath{\\mathbb{N}}_0}$ with the property that by the law of large numbers\n\\begin{align*}\n \\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{i=1}^n \\widetilde X_i = \\frac{1}{N}\\ensuremath{\\mathbb{E}}_{1-q}[Y_{0,N}] \\geq \\gamma(q)-\\varepsilon \\qquad \\P_{1-q}\\text{-a.s.}.\n\\end{align*}\nThus $\\gamma(q) - \\varepsilon \\leq \\rho_c(q)$ and since $\\varepsilon$ was arbitrary, $\\gamma(q) \\leq \\rho_c(q)$, which in combination with the above establishes \\eqref{eq:limitEquality}.\n\nThe strict monotonicity of $\\gamma(\\cdot)$ (and thus $q_c(\\cdot)$) can now be proven through a suitable coupling argument. Denote by $\\mathcal U_{[0,1]}$ the uniform measure on the interval $[0,1]$ and define $\\mu:= \\mathcal U_{[0,1]}^{\\otimes \\ensuremath{\\mathbb{Z}}^{d+1}}$ as the product measure on the space $\\mathcal W := [0,1]^{\\ensuremath{\\mathbb{Z}}_d}$. For any $w \\in \\mathcal W$, $q \\in (0,1)$ and $n \\in \\ensuremath{\\mathbb{N}}_0$ define\n\\begin{align*}\n Y^q_n(w): = \\max \\Big \\{r \\in \\ensuremath{\\mathbb{N}}_0 \\mid \\exists  \\text{ directed nearest neighbor path }& 0 = x_0, x_1, \\ldots, x_{n}:\\\\\n\t\t\t\t\t\t      & \\qquad \\qquad \\, \\sum_{i=}^{n} \\mathds{1}_{[0,q]}(w(x_i)) = r\n\t\t\t\t\t\t      \\Big \\}.\n\\end{align*}\nObserve that $\\mathcal L_{\\mu} ((Y^q_n)_{n \\in \\ensuremath{\\mathbb{N}}_0}) = \\mathcal L_{\\P_{1-q}}((Y_{0,n})_{n \\in \\ensuremath{\\mathbb{N}}_0})$, where $\\mathcal L_{\\nu}$ denotes the law with respect to the measure $\\nu$. Therefore \n\\begin{align*}\n \\lim_{n \\rightarrow \\infty} \\frac{1}{n} Y^q_n = \\gamma(q) \\qquad \\mu\\text{-a.s. and in } L^1(\\mu).\n\\end{align*}\nAs before, for any $q \\in (0,1)$, $w \\in \\mathcal W$ and $n \\in \\ensuremath{\\mathbb{N}}_0$, let $X^{q,n}_1(w), \\ldots, X^{q,n}_n(w)$ be an oriented nearest neighbor path such that $Y^q_n = \\sum_{i=1}^n \\mathds{1}_{[0,q]}(w(X^{q,n}_i(w)))$. Choose $0\\leq q < q' \\leq \\bar q$, then\n\\begin{align} \\label{eq:DiffInY}\n Y^{q'}_n = \\sum_{i=1}^n \\mathds{1}_{[0,q']}(w(X^{q',n}_i(w))) & \\geq \\sum_{i=1}^n \\mathds{1}_{[0,q']}(w(X^{q,n}_i(w))) \\notag\\\\\n\t\t\t\t\t\t\t    & = Y^q_n + \\sum_{i=1}^n \\mathds{1}_{[q,q']}(w(X^{q,n}_i(w))).\n\\end{align}\nSet $\\mathcal F_q := \\sigma(w \\mapsto \\mathds{1}_{[0,q]} (w(x)) \\mid x \\in \\ensuremath{\\mathbb{Z}}^d)$. Then, obviously, the $Y^q_n$ are $\\mathcal F_q$-measurable and the $\\mathds{1}_{[0,q]}(w(X^{q,n}_i(w)))$, $1 \\le i \\le n,$ are independent given $\\mathcal F_q$. In addition,\n\\begin{align*}\n \\mu(\\mathds{1}_{[q,q']}(w(X^{q,n}_i(w))) = 1 \\mid \\mathcal F_q) = \\frac{q'-q}{1-q} \\mathds{1}_{\\{w(X^{q,n}_i(w))>q\\}}.\n\\end{align*}\nThus using \\eqref{eq:DiffInY} we obtain\n\\begin{align*}\n \\ensuremath{\\mathbb{E}}_{\\mu}[Y^{q'}_n-Y^q_n \\mid \\mathcal F_q] \\geq \\ensuremath{\\mathbb{E}}_{\\mu}\n \\Big [ \\sum_{i=1}^n \\mathds{1}_{[q,q']}(w(X^{q,n}_i(w)))\\mid \\mathcal F_q \\Big] = \\left( n - Y^q_n\\right)\\frac{q'-q}{1-q},\n\\end{align*}\nwhere $\\ensuremath{\\mathbb{E}}_{\\mu}$ denotes the expectation with respect to $\\mu$. Using the $L_1(\\mu)$ convergence\n\\begin{align*}\n \\gamma(q') - \\gamma(q) & = \\lim_{n \\rightarrow \\infty} \\ensuremath{\\mathbb{E}}_{\\mu}\n \\Big[ \\ensuremath{\\mathbb{E}}_{\\mu} \\Big[\\frac{1}{n}\\left(Y^{q'}_n - Y^q_n\\right)\\mid \\mathcal F_q \\Big ] \\Big] \\\\\n\t\t\t& \\geq \\lim_{n \\rightarrow \\infty} \\ensuremath{\\mathbb{E}}_{\\mu} \\Big[ \\frac{1}{n} \\left(n-Y^q_n\\right) \\frac{q'-q}{1-q} \\Big]\\\\\n\t\t\t& = (1-\\gamma(q))\\frac{q'-q}{1-q}\n\\end{align*}\nand the right-hand side is positive, since $\\gamma(q)=\\rho_c(q) < 1$ for $q < \\bar q$. This shows the strict monotonicity\nof the function $\\rho_c$  on  $[0,\\overline q]$ and hence implies \\eqref{eq:qcCont}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:UBqAsympD}]\n\nWe will compare $\\rho$-paths in $\\ensuremath{\\mathbb{Z}}^d$ with reversed admissible $\\lambda$-paths in $\\ensuremath{\\mathbb{Z}}^{d+1}$.\n To this end define for any $\\omega \\in \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^{d+1}}$ and $\\bar x \\in \\ensuremath{\\mathbb{Z}}^d$ the quantity\n\\begin{align*}\n H_{\\omega}(\\bar x): = \\min\\Big\\{h \\in \\ensuremath{\\mathbb{N}}_0 \\mid \\exists & \\text{ an oriented \n nearest neighbor path } 0=\\bar x_0, \\ldots, \\bar x_m=\\bar x \\in \\ensuremath{\\mathbb{Z}}^d, \\\\\n\t\t\t\t\t\t     & \\text{ and a sequence } 0=h_0, \\ldots, h_m = h \\in \\ensuremath{\\mathbb{N}}_0 \\text{ s.t. } \\\\\n\t\t\t\t\t\t     & \\qquad \\qquad h_{i+1} = \\begin{cases}\n\t\t\t\t\t\t\t\t\t    h_i, & \\text{ if }\\omega(\\bar x_i, h_i) = 0,\\\\\n\t\t\t\t\t\t\t\t\t    h_i+1, & \\text{ otherwise}.\n\t\t\t\t\t\t\t\t\t\\end{cases}  \\quad \\Big\\}.\n\\end{align*}\nA second's thought reveals that this map is defined in such\na way that there is an admissible $\\lambda$-path from $(\\bar x, H_{\\omega}(\\bar x))$ to the origin,\n which takes advantage of many closed sites in the\n configuration $\\omega$. (It is, however, not optimal, as it does not make use of consecutive \\lq piled up\\rq\\ closed sites in one step.) With this we can then define a map $T : \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^{d+1}} \\rightarrow \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^d}$ as\n\\begin{align*}\n (T(\\omega)) (\\bar x) := \\begin{cases}\n                     \\omega(\\bar x, H_{\\omega}(\\bar x)), & \\text{ if } \\bar x \\in \\ensuremath{\\mathbb{N}}_0^d,\\\\\n                     \\omega(\\bar x, 0), & \\text{ otherwise.}\n                    \\end{cases}\n\\end{align*}\nThe purpose of $T$ is to map a configuration $\\omega \\in \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^{d+1}}$ to a configuration $\\bar \\omega \\in \\{0,1\\}^{\\ensuremath{\\mathbb{Z}}^d}$, \nfor which there exists an oriented path picking up almost as many closed sites as the oriented reversed admissible $\\lambda$-path in $\\omega$ with lowest $(d+1)$-st coordinate. In order to be more precise, we add an index to the probability measure used to indicate the space it is defined on. \nI.e., $\\P_{p,d}$ will denote the Bernoulli product-measure on $\\ensuremath{\\mathbb{Z}}^d$ with parameter $p$. Since the value of $H(\\bar x)$ only depends on the state of the sites $\\bar y \\in \\ensuremath{\\mathbb{N}}_0^d$ with $\\Vert \\bar y \\Vert < \\Vert \\bar x \\Vert$, $\\P_{p,d+1} \\circ T^{-1} = \\P_{p,d}$. Thus, if $q > q_c(\\rho,d)$, we have that\n\\begin{align}\\label{eq:rhopercas}\n 1 \t& = \\P_{p,d}(\\text{$\\rho$-percolation occurs}) \\notag\\\\\n\t& = \\P_{p,d+1} \\Big(\\text{there exists an admissible reversed $\\lambda$-path } 0 = (\\bar x_0, h_0), (\\bar x_1, h_1), \\dots \\\\\n\t& \\hspace{300pt} \\text{ s.t. } \\limsup_{n \\rightarrow \\infty} \\frac{1}{n} h_n \\leq 1-\\rho \\Big). \\notag\n\\end{align}\nNow choose $\\rho > 1-\\alpha$ and set $\\delta := 1- \\rho + (\\alpha - (1-\\rho))\/2 \\in ( 1-\\rho, \\alpha)$. Then \\eqref{eq:rhopercas} implies the existence of a (deterministic) $N \\in \\ensuremath{\\mathbb{N}}$ such that for all $n \\geq N,$\n\\begin{align*}\n\\P_{p,d+1}\\Big(& \\text{there exists an admissible reversed $\\lambda$-path } \\\\ \n\t    & \\qquad 0 = (\\bar x_0, h_0), (\\bar x_1, h_1), \\dots, (\\bar x_n, h_n)  \\text{ s.t. }  h_n \\leq \\delta n\\Big) \\geq \\frac{1}{2}.\n\\end{align*}\nNote that if there exists an admissible reversed $\\lambda$-path from the origin to some $(\\bar x_n, h_n)$ with $h_n \\leq \\delta n$, then there actually exists an admissible $\\lambda$-path from $L^{\\alpha,d}_{\\eta} - \\floor{(\\alpha - \\delta)n}e_{d+1} $ to the origin. Thus, by translation invariance of $\\P_{p,d+1}$, we obtain that\n\\begin{align*}\n \\forall n \\geq N:\\; \\P_{p,d+1} \\big( L^{\\alpha,d}_{\\eta} \\rightarrowtail (0, \\floor{(\\alpha - \\delta)n}) \\big) \\geq \\frac{1}{2}\n\\end{align*}\nwhich, since $\\alpha - \\delta >0$, implies\n\\begin{align*}\n \\P_{p,d+1}\\left( (\\mathsf{LIP}^{\\alpha,d}_{\\eta})^c\\right) = \\lim_{n \\rightarrow \\infty} \\P_{p,d+1} \\big( L^{\\alpha,d}_{\\eta} \\rightarrowtail (0, (\\alpha - \\delta)n ) \\big) \\geq \\frac{1}{2}.\n\\end{align*}\nBy Proposition \\ref{prop:equiv} we deduce that $\\P_p\\left(\\mathsf{LIP}^{\\alpha,d}_{\\eta}\\right) = 0$ and\nhence $q \\geq q_L(\\alpha, d,d)$. We have thus shown that for any $\\rho > 1-\\alpha$ one has $ \\;q_c(\\rho,d) \\geq q_L(\\alpha,d,d)$. Since $q_c(\\rho,d)$ is continuous in $\\rho$ by Lemma \\ref{lem:rho},\nthen the claim follows from \\eqref{eq:RhoConst}. \n\\end{proof}\n\n\n\\begin{lemma}[Criterion for non-existence of an open Lipschitz surface] \\label{lem:CritLSNonExist}\n For any $\\alpha >0$, and $d \\in \\ensuremath{\\mathbb{N}}$ define \n \\begin{align*}\n  T := \\inf \\{ m \\in \\ensuremath{\\mathbb{N}}_0 \\mid \\exists \\bar x \\in \\ensuremath{\\mathbb{N}}_0^d:\\; \\Vert \\bar x \\Vert_1 = m \\text{ and } (\\bar x, \\Vert \\bar x \\Vert_1) \\text{ is closed}\\}.\n \\end{align*}\nIf for $p \\in (0,1)$ one has \n\\begin{align}\\label{eq:CritLSNonExist}\n \\ensuremath{\\mathbb{E}}_p[T] < \\frac{1}{1-\\alpha},\n\\end{align}\nthen $\\P$-a.s. there exists no open Lipschitz surface and $q= 1-p \\geq q_L(\\alpha,d,d)$. \n\\end{lemma}\nCondition \\eqref{eq:CritLSNonExist} has an intuitive interpretation: $1\/(1-\\alpha)$ is the number of \\lq downward-diagonal\\rq\\ steps a $\\lambda$-path can take before decreasing its distance to the plane with inclination $\\alpha$ by one. $\\ensuremath{\\mathbb{E}}_p[T]$ on the other hand is the expected number of such steps an admissible $\\lambda$ path must take before encountering a closed site and thus being able to take an upwards step. \\eqref{eq:CritLSNonExist} therefore means that this path will -- on average -- encounter a closed site strictly before decreasing its distance to the plane by one, thus increasing the distance in the long run and preventing the existence of an open Lipschitz surface above it.\n\n\n\\begin{proof}\nAs in the proof of Proposition \\ref{prop:UBqAsympD}, the idea is to construct admissible reversed $\\lambda$-paths starting in $0$ such that their endpoints (i.e. the starting points of the respective $\\lambda$-paths) are arbitrarily far below $L^{\\alpha,d}_{\\eta}$. With a simple shifting argument we can then see that the Lipschitz surface would, with probability bounded away from $0$, have to have arbitrarily large height in $0$ and can therefore almost surely not exist.\n \n We begin with the construction of the \\emph{reversed} $\\lambda$-paths. To this end, set $ X_0:=Y_0:=0$. Let $(\\bar z_i)_{i \\in \\ensuremath{\\mathbb{N}}_0}$ be an ordering of $\\ensuremath{\\mathbb{N}}_0^d$ compatible with $\\Vert \\cdot \\Vert_1$ in the sense that $\\Vert z_{i+1} \\Vert_1 \\geq \\Vert z_i \\Vert_1$, for all $i \\in \\ensuremath{\\mathbb{N}}_0$. Then define for any $n \\in \\ensuremath{\\mathbb{N}}_0$,\n \\begin{align*}\n  \\iota_{n+1} \t& := \\inf\\{i \\in \\ensuremath{\\mathbb{N}}_0 \\mid (\\bar z_i, \\Vert\\bar z_i \\Vert_1) + Y_n \\text{ is closed}\\},\\\\\n  X_{n+1} \t& := (\\bar z_{\\iota_n}, \\Vert \\bar z_{\\iota_n}\\Vert_1), \\\\\n  Y_{n+1}\t& := Y_n + X_{n+1} - e_{d+1}.\n \\end{align*}\nBy construction, there always exists an admissible $\\lambda$-path from any $Y_n$ to 0. Note also that $(\\iota_n)_{n \\in \\ensuremath{\\mathbb{N}}}$ and $(X_n)_{n \\in \\ensuremath{\\mathbb{N}}}$ are i.i.d. sequences where $\\iota_1$ is geometric on $\\ensuremath{\\mathbb{N}}_0$ with parameter $q$ and $\\Vert \\bar X_1 \\Vert_1 = X_1 \\cdot e_{d+1}$\nis distributed as $T$. \n\nWe are now interested in the height of the starting points of these $\\lambda$-paths relative to $L^{\\alpha,d}_{\\eta}$. This is given by \n\\begin{align*}\n H(n)\t& := \\floor{\\alpha \\Vert \\bar Y_n \\Vert_1} - Y_n\\cdot e_{d+1}\\\\\n\t& = \\Big \\lfloor \\alpha \\sum_{j = 1}^n \\Vert \\bar X_j \\Vert_1 \\Big \\rfloor - \\sum_{j=1}^n ( X_n - e_{d+1})\\cdot e_{d+1}\\\\\n\t& =  \\Big \\lfloor \\alpha \\sum_{j = 1}^n \\Vert \\bar X_j \\Vert_1 \\Big \\rfloor - \\sum_{j = 1}^n \\Vert \\bar X_j \\Vert_1 + n.\n\\end{align*}\n The law of large numbers then yields\n \\begin{align*}\n  \\lim_{n \\rightarrow \\infty} \\frac{1}{n}H(n) = (\\alpha - 1)\\ensuremath{\\mathbb{E}}_p[T] +1 \\qquad \\P_p\\text{-a.s.}\n \\end{align*}\nand the right-hand side is strictly negative by assumption. Thus with $\\Delta := -\\left( (\\alpha - 1)\\ensuremath{\\mathbb{E}}_p[T] +1 \\right)\/2 >0$ we have in particular the existence of a deterministic $N \\in \\ensuremath{\\mathbb{N}}$ such that\n\\begin{align*}\n \\forall n \\geq N: \\; \\P_p( H(n) \\leq - \\Delta n) \\geq \\frac{1}{2}.\n\\end{align*}\nNow note that on the event $\\{H(n) \\leq - \\Delta n\\}$ there exists an admissible $\\lambda$-path starting in $L^{\\alpha,d}_{\\eta} - \\Delta n e_{d+1}$ and reaching 0, since $Y_n$ is below the plane $L^{\\alpha,d}_{\\eta} - \\Delta n e_{d+1}$. Hence, by translation invariance of $\\P_p$ we have that\n\\begin{align*}\n \\forall n \\geq N: \\; \\P_p(L^{\\alpha,d}_{\\eta} \\rightarrowtail (0, \\Delta n) \\geq \\frac{1}{2}\n\\end{align*}\nwhich implies\n\\begin{align*}\n \\P_p\\left( (\\mathsf{LIP}^{\\alpha,d}_{\\eta})^c\\right) = \\lim_{n \\rightarrow \\infty} \\P_p(L^{\\alpha,d}_{\\eta} \\rightarrowtail (0, \\Delta n) \\geq \\frac{1}{2}.\n\\end{align*}\nBy Proposition \\ref{prop:equiv}, $\\P_p\\left(\\mathsf{LIP}^{\\alpha,d}_{\\eta}\\right) = 0$ and $p \\leq p_L(\\alpha,d,d)$, i.e., $q \\geq q_L(\\alpha, d,d)$. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:UBqAsympAlpha}]\n Recall the ordering $(\\bar z_i)_{i \\in \\ensuremath{\\mathbb{N}}_0}$ of $\\ensuremath{\\mathbb{N}}_0^d$ compatible with $\\Vert \\cdot \\Vert_1$ from the proof of Lemma \\ref{lem:CritLSNonExist} and define the random variable\n \\begin{align*}\n \\iota_1 :=  \\inf\\{i \\in \\ensuremath{\\mathbb{N}}_0 \\mid (\\bar z_i, \\Vert\\bar z_i \\Vert_1) \\text{ is closed}\\}, \n \\end{align*}\n which has a geometric distribution on $\\ensuremath{\\mathbb{N}}_0$ with parameter $q$. With $B(j):= \\{ \\bar x \\in \\ensuremath{\\mathbb{N}}_0^d \\mid \\Vert \\bar x \\Vert_1 \\leq j\\}$ denoting the ball with radius $j \\in \\ensuremath{\\mathbb{N}}_0$, define the function \n \\begin{align*}\n  r(i) := \\inf\\{j \\in \\ensuremath{\\mathbb{N}}_0 \\mid \\vert B(j) \\vert - 1 \\geq i\\}\n \\end{align*}\nthat gives the radius of the smallest ball such that its cardinality (without the origin) is larger than\nor equal to a given $i \\in \\ensuremath{\\mathbb{N}}_0$. Note that $r(\\iota_1)$ is distributed as $T$,\n for $T$ defined in Lemma \\ref{lem:CritLSNonExist}.\nUsing\n\\begin{align*}\n \\vert B(j) \\vert = \\binom{j+d}{d} \\geq  \\frac{(j+1)^d}{d!}\n\\end{align*}\nwe obtain\n\\begin{align*}\n i \\geq \\vert B(r(i)-1) \\vert \\geq  \\frac{r(i)^d}{d!}\n\\end{align*}\n and can thus upper bound the expectation\n \\begin{align*}\n  \\ensuremath{\\mathbb{E}}_p[T] = \\ensuremath{\\mathbb{E}}_p[r(\\iota_1)] \t& \\leq \\left( d!\\ensuremath{\\mathbb{E}}_p[\\iota_1] \\right)^{\\frac{1}{d}}  \\leq \\left( d!\\left(\\frac{1}{q}-1\\right)\\right)^{\\frac{1}{d}},\n \\end{align*}\n where we used Jensen's inequality in the first inequality.\nThe right-hand side is strictly smaller than $1\/(1-\\alpha)$ if and only if \n\\begin{align*}\n q > \\frac{d!(1-\\alpha)^d}{1+d!(1-\\alpha)^d}.\n\\end{align*}\nThus Lemma \\ref{lem:CritLSNonExist} then implies that for such values of $q$ no open Lipschitz surface can exist, i.e. $q \\geq q_L(\\alpha,d,d)$, and the claim follows.\n\\end{proof}\n\n\n\\subsection*{Acknowledgement}\nWe thank Patrick W. Dondl for helpful suggestions and valuable discussions.\n\n\\bibliographystyle{alpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMartin-L\u00f6f type theory can be characterized in a syntax independent way as the initial category with families (cwf)  with extra structure for the type formers \\cite{castellan:tlca2015,castellan:lmcs}. The main contribution of this note is a similar syntax independent characterization of the notion of finitely presented generalized algebraic theory as the initial cwf with extra structure.\n\nGeneralized algebraic theories (gats) were introduced by Cartmell in his PhD thesis \\cite{cartmell:phd} as a dependently typed generalization of many sorted algebraic theories. Each gat is specified by a signature with (possibly infinite) sets of sort symbols, operator symbols, and equations. Cartmell's definition of gats \\cite{cartmell:phd,cartmell:apal} is based on a notion of {\\em derived rule} expressed in terms of a traditional syntactic system for dependent type theory. He also defines a notion of model whereby sort symbols are interpreted as families of sets.\n\nCategories with families (cwfs) \\cite{dybjer:torino} were introduced as a new notion of model of dependent type theory. Cwfs arise by reformulating the notion of category with attributes in Martin Hofmann's sense \\cite{hofmann:csl}. The key point is that cwfs arise as models of a certain generalized algebraic theory closely related to Martin-L\u00f6f's substitution calculus \\cite{martinlof:gbg92}. As such the notion of cwf becomes a useful intermediary between traditional syntactic systems for dependent type theory and a variety of categorical notions of model.\n\nThe gat of cwfs is thus a kind of idealized formal system of dependent type theory. In contrast to Martin-L\u00f6f's substitution calculus, and other syntactic systems for dependent type theory, it is {\\em not} formulated in terms of grammars and inference rules for the forms of judgment of type theory. Instead it is formulated in terms of the sort symbols (corresponding to the judgment forms), operator symbols (corresponding to the formation, introduction, and elimination rules), and equations (corresponding to the equality rules for the type formers) of the gat. Some of the general reasoning (about equality, substitution, and assumptions) is taken care of by the underlying infrastructure of dependent types. This makes it possible to abstract away from details in the formulation of grammars and inference rules. In contrast to the various syntactic systems, the gat of cwfs has a canonical flavour. \n\nIn this note we explore the interdependence between gats and cwfs. We already explained that cwfs can be defined as models of a gat. \nIn the other direction, the notion of gat relies on the notion of cwf, in the sense that the latter models the underlying infrastructure of dependent types. \n\n\n\n\\subsection*{Plan of the paper}\n\nIn Section 2 we recall the definition of the category $\\mathbf{CwF}$ of categories with families and morphisms preserving cwf-structure on the nose. Section 3 contains our main definition of a syntax independent notion of valid signature $\\Sigma$ for a gat and the category $\\mathbf{CwF}_\\Sigma$ of cwfs with a $\\Sigma$-structure. In Section 4 we construct an initial object ${\\mathcal T}_\\Sigma$ in $\\mathbf{CwF}_\\Sigma$. In Section 5 we show several examples of gats: for monoids, categories, cwfs, and cwfs with extra structure for one universe. We point out that cwfs with extra structure for gats of monoids, categories, cwfs are cwfs with an internal monoid, category, and cwf, respectively. We also sketch how to extend our approach to some countably presented gats, and show the example of contextual cwfs, a variant of Cartmell's contextual categories \\cite{cartmell:phd,cartmell:apal}. Finally, in Section 6 we discuss related work, for example relating to Voevodsky's initiality conjecture \\cite{voevodsky:initiality} and Altenkirch and Kaposi's quotient inductive-inductive types \\cite{altenkirch:qiits}.\n\nOur development can be formulated in a constructive set theory,\nas described for instance by Aczel \\cite{MR519801}, although the set theory\nwe use for formulating the notion of cwf with a $\\Sigma$-structure is probably\nmuch weaker. As emphasized by Voevodsky~\\cite{voevodsky:initiality}, we study structures invariant\nunder {\\em isomorphisms} and not under {\\em equivalences}, and it is actually misleading\nto call them ``category'' (and this is why Voevodsky used the term ``$C$-system''\nfor what Cartmell called ``contextual category'').\nAs he also noticed, this\nimportant distinction between categories and notions invariant under isomorphisms becomes\nprecise in the setting of univalent foundations where not all collections of objects\nare constructed from sets.\n\n\n\\subsection*{Remarks on terminology and notation}\nLike Cartmell, we have chosen to use the term {\\em sort symbol} from many-sorted universal algebra. However, in our semantic notion of signature sort symbols are interpreted as {\\em type families} in a cwf. A cwf consists of a base category where the objects of the base category are (semantic) contexts and the morphisms are (semantic) substitutions. Moreover, we have a family-valued presheaf mapping contexts to families of (semantic) terms indexed by (semantic) types. Thus the reader should be aware of the mismatch between the word {\\em sort} from universal algebra and the word {\\em type} in the cwf semantics.\n\nAnother possible source of confusion is that cwfs appear on two different levels. In Section \\ref{sec:def_cwf} we recall the definition of cwf in set-theoretic metalanguage, where we use $\\mathrm{Ty}$ to denote the family of types indexed by contexts and $\\mathrm{Tm}$ to denote the family of terms indexed by contexts and types. This notion of cwf is then used to define the semantic notions of signature and category of models of a gat. Then in Section \\ref{gat-cwf} we define the gat of {\\em internal cwfs}. This gat has sort symbols $\\mathrm{ty}$ for {\\em internal types} and $\\mathrm{tm}$ for {\\em internal terms} using lower case to highlight the difference from $\\mathrm{Ty}$ and $\\mathrm{Tm}$ in the model cwf. \n\nFurthermore, we often use the same notation both on the semantic and the syntactic level. For example, in Section \\ref{gat-sig-mod},  where we are syntax independent, the letter $S$ denotes a semantic sort symbol, whereas in Section \\ref{initial-gat}, where we construct the initial model, it denotes a syntactic sort symbol.\n\n\n\\section{Categories with families}\\label{sec:def_cwf}\n\n\\subsection{The category of cwfs and strict cwf-morphisms}\n\n\n\\begin{definition}\\label{def:catFam}\n$\\textbf{Fam}$ is a category whose objects are\nset-indexed families of sets, denoted as $(U_x)_{x\\in X}$.\nA morphism of $\\textbf{Fam}$ with source $(U_x)_{x\\in X}$ and target $(V_y)_{y\\in Y}$\nconsists of a re-indexing function $f: X\\to Y$ together with a family\n$(g_x)_{x\\in X}$ of functions $g_x : U_x \\to V_{f(x)}$.\n\\end{definition}\n\nThe next step is to define the category $\\mathbf{CwF}$.\nWe split this definition in two: first the objects,\nwhich are called \\emph{categories with families}, in Definition~\\ref{def:Cwfobj},\nand then the morphisms in Definition~\\ref{def:Cwfmor}.\nSince $\\mathbf{CwF}$ has been developed as a categorical framework for the semantics of\ntype theory, much of the terminology (contexts, substitutions,\ntypes, terms) refers to the syntax of type theory,\nsuggesting the intended interpretation of this syntax in the\nso-called $\\mathbf{CwF}$-semantics.\n\nThe main novelty of this note is to use $\\mathbf{CwF}$ as a framework\nto define a new notion of a generalized algebraic theory.\nContexts, substitutions, types, and terms also make\nsense in relation to gats.\n\n\\begin{definition}\\label{def:Cwfobj}\nA category with families (cwf) consists of the following data:\n\n\\begin{itemize}\n\\item A category ${\\mathcal C}$;\n\n\\item A $\\textbf{Fam}$-valued presheaf on ${\\mathcal C}$, that is, a functor\n$T : \\C^\\op \\to \\textbf{Fam}$;\n\n\\item A terminal object $1\\in {\\mathcal C}$, and unique maps\n$\\tuple{}_\\Gamma \\in {\\mathcal C}(\\Gamma, 1)$ for all objects $\\Gamma$ of ${\\mathcal C}$;\n\n\\item Operations ${.\\,},~\\tuple{\\_,\\_},~\\mathrm{p}$ and $\\mathrm{q}$\nexplained in the following paragraphs.\nThese four operations and their associated equations\nare referred to as \\emph{context comprehension}.\n\\end{itemize}\n\nWe let $\\Gamma, \\Delta,\\ldots$ range over objects of ${\\mathcal C}$,\nand refer to them as \\emph{contexts}.\nWe let $\\delta, \\gamma,\\ldots$ range over morphisms,\nand refer to them as \\emph{substitutions}.\nWe refer to $1$ as the \\emph{empty} context; the terminal maps\n$\\tuple{}_\\Gamma$ represent the \\emph{empty} substitutions.\n\nIf $T(\\Gamma) = (U_x)_{x\\in X}$, we write $\\mathrm{Ty}(\\Gamma)$ for the set $X$.\nWe call the elements of $\\mathrm{Ty}(\\Gamma)$ \\emph{types in context $\\Gamma$},\nand let $A, B, C$ range over such types.\nFurthermore, for $A \\in \\mathrm{Ty}(\\Gamma)$, we write $\\mathrm{Tm}(\\Gamma, A)$ for the set $U_A$\nand call the elements of $\\mathrm{Tm}(\\Gamma, A)$\n\\emph{terms of type $A$ in context $\\Gamma$}.\n\nFor $\\gamma : \\Delta \\to \\Gamma$,\nthe functorial action of $T$ yields a morphism\n\\[\nT(\\gamma) \\in  \\textbf{Fam}\\left((\\mathrm{Tm}(\\Gamma, A))_{A\\in \\mathrm{Ty}(\\Gamma)},\n                (\\mathrm{Tm}(\\Delta, B))_{B\\in \\mathrm{Ty}(\\Delta)}\\right)\n\\]\nconsisting of a reindexing function $\\_\\,[\\gamma] : \\mathrm{Ty}(\\Gamma) \\to\n\\mathrm{Ty}(\\Delta)$ referred to as \\emph{substitution in types}, and for each $A\\in\n\\mathrm{Ty}(\\Gamma)$ a function $\\_\\,[\\gamma] : \\mathrm{Tm}(\\Gamma, A) \\to \\mathrm{Tm}(\\Delta,\nA[\\gamma])$, referred to as \\emph{substitution in terms}.\n\nNow we turn to the explanation of the operations\n${.\\,},~\\tuple{\\_,\\_},~\\mathrm{p},~\\mathrm{q}$.\nGiven $\\Gamma \\in {\\mathcal C}$, $A \\in \\mathrm{Ty}(\\Gamma)$, $\\gamma : \\Delta \\to \\Gamma$,\nand $a\\in \\mathrm{Tm}(\\Delta, A[\\gamma])$, we have\n\\[\n\\Gamma . A \\in {\\mathcal C}\n\\quad\\qquad\n\\mathrm{p}_{\\Gamma, A} : \\Gamma . A \\to \\Gamma\n\\quad\\qquad\n\\mathrm{q}_{\\Gamma, A} \\in \\mathrm{Tm}(\\Gamma. A, A[\\mathrm{p}_{\\Gamma,A}])\n\\quad\\qquad\n\\tuple{\\gamma, a}_A : \\Delta \\to \\Gamma . A.\n\\]\nWe call $\\Gamma . A$ the \\emph{extended} context\nand $\\tuple{\\gamma, a}_A$ the \\emph{extended} substitution.\n\nThe operations  ${.\\,},~\\tuple{\\_,\\_},~\\mathrm{p},~\\mathrm{q}$\nsatisfy the following universal property:\n$\\tuple{\\gamma, a}_A$ is the unique substitution satisfying\n\\[\n\\mathrm{p}_{\\Gamma, A} \\circ \\tuple{\\gamma, a}_A = \\gamma\n\\qquad \\text{and}\\qquad\n\\mathrm{q}_{\\Gamma, A} [\\tuple{\\gamma, a}_A] = a\\,.\n\\]\nWe refer (colloquially) to $\\mathrm{p}$ as the \\emph{first projection},\nand to $\\mathrm{q}$ as the \\emph{second projection}.\n{Note that the first equation implies that\n$\\mathrm{Tm}(\\Delta,A[\\mathrm{p}_{\\Gamma,A}][\\tuple{\\gamma, a}]) = \\mathrm{Tm}(\\Delta,A[\\gamma])$\nso that $\\mathrm{q}_{\\Gamma, A} [\\tuple{\\gamma, a}]$ and $a$ are elements of the same set.}\nHere and below, subscripts are omitted from ${.\\,},~\\tuple{\\_,\\_},~\\mathrm{p},~\\mathrm{q}$\nwhen they can be reconstructed from the context (no pun intended).\n(End Definition~\\ref{def:Cwfobj}.)\n\\end{definition}\n\nA cwf is thus a structure $({\\mathcal C},1,\\tuple{},T,.\\, , \\tuple{\\_,\\_},\\mathrm{p}, \\mathrm{q})$,\nsubject to equations, for the category and the presheaf, and universal\nproperties, formulated purely equationally, for the terminal object and for context comprehension.\nThe morphisms to be defined next preserve this structure,\neven in a strict way, `on the nose'.\nWe often shorten the notation of a cwf to $({\\mathcal C},T)$, or even just ${\\mathcal C}$,\nleaving the remaining structure implicit.\n\n\\begin{definition}\\label{def:Cwfmor}\nA \\emph{(strict) cwf-morphism $F$ between cwfs $({\\mathcal C},T_{\\mathcal C})$ and $(\\mathcal{D},T_\\mathcal{D})$}\nconsists of\n\n\\begin{itemize}\n\n\\item A functor $F_\\mathrm{fun} : {\\mathcal C} \\to \\mathcal{D}$;\n\\item A natural transformation $F_\\mathrm{nat} : T_{\\mathcal C} \\Rightarrow (T_\\mathcal{D} \\circ F_\\mathrm{fun}^\\text{op})$;\n\\item The terminal object is preserved on the nose: $F_\\mathrm{fun}(1_{{\\mathcal C}}) = 1_{\\mathcal{D}}$;\n\\item Context comprehension is preserved on the nose, see below.\n\\end{itemize}\n\nSince $F_\\mathrm{nat}$ is a natural transformation between $\\textbf{Fam}$-valued presheaves,\n$F_\\mathrm{nat}$ has a component for any object $\\Gamma$ of ${\\mathcal C}$, and\nthese components are morphisms in $\\textbf{Fam}(T_C(\\Gamma),T_\\mathcal{D}(F_\\mathrm{fun}(\\Gamma)))$.\nRecall that morphisms in $\\textbf{Fam}$ consist of a reindexing function\nand a family of functions. It is convenient to denote $F_\\mathrm{fun}$,\nall reindexing functions, as well as all members of the families of functions,\nsimply by $F$. Thus we have $F(A) \\in \\mathrm{Ty}_\\mathcal{D}(F(\\Gamma))$\nand $F(a) \\in \\mathrm{Tm}_\\mathcal{D}(F(\\Gamma), F(A))$, for all $\\Gamma$\nand $A\\in\\mathrm{Ty}_{\\mathcal C}(\\Gamma)$ and $a\\in \\mathrm{Tm}_{\\mathcal C}(\\Gamma, A)$.\n\nNaturality of $F_\\mathrm{nat}$\namounts to preservation of substitution, {i.e.}, for all\n$\\gamma : \\Delta \\to \\Gamma$ in ${\\mathcal C}$, we have\n\\[\nF(A[\\gamma]) = F(A)[F(\\gamma)] \\qquad \\qquad\nF(a[\\gamma]) = F(a)[F(\\gamma)]\\,.\n\\]\n\nLast but not least, we turn to the preservation of context comprehension\non the nose, and require\n\\[\nF(\\Gamma. A) = F(\\Gamma). F(A) \\qquad\nF(\\mathrm{p}_{\\Gamma, A}) = \\mathrm{p}_{F(\\Gamma), F(A)} \\qquad\nF(\\mathrm{q}_{\\Gamma, A}) = \\mathrm{q}_{F(\\Gamma), F(A)}\\,.\n\\]\n\nNote that the universal property implies that\n$F(\\tuple{\\gamma,a}) = \\tuple{F(\\gamma),F(a)}$.\nThe same is true for the terminal maps:\n$F(\\tuple{}_\\Gamma) = \\tuple{}_{F(\\Gamma)}$.\n(End Definition~\\ref{def:Cwfmor}.)\n\\end{definition}\n\nSmall cwfs with strict cwfs-morphisms form a category, written $\\mathbf{CwF}$.\n\n\\section{Signatures and models of generalized algebraic theories}\\label{gat-sig-mod}\n\nWe now come to the main point of this note.\nWe define how to build a valid gat signature $\\Sigma$ and the associated\ncategory $\\mathbf{CwF}_{\\Sigma}$ of cwfs with a $\\Sigma$-structure.\nEach object of $\\mathbf{CwF}_{\\Sigma}$ is a cwf with extra structure and\neach morphism is a cwf-morphism preserving $\\Sigma$-structure.\nFor this definition, we will need the following auxiliary notions.\n\nA {\\em uniform family of contexts} is a family $\\Gamma = (\\Gamma_{{\\mathcal C}})$ with $\\Gamma_{\\mathcal C}$ a context in \n${\\mathcal C}$ for each ${\\mathcal C} \\in \\mathbf{CwF}_{\\Sigma}$, such that\n$F(\\Gamma_{\\mathcal C}) = \\Gamma_\\mathcal{D}$ for all morphisms $F \\in \\mathbf{CwF}_{\\Sigma}({\\mathcal C},\\mathcal{D})$.\nIf $\\Gamma$ is such a family, a {\\em uniform family of types} over $\\Gamma$ is a\nfamily of types $A = (A_{{\\mathcal C}})$ with $A_{{\\mathcal C}}$ a type over $\\Gamma_{{\\mathcal C}}$ and\n$F(A_{{\\mathcal C}}) = A_{\\mathcal{D}}$ for all morphisms $F \\in \\mathbf{CwF}_{\\Sigma}({\\mathcal C},\\mathcal{D})$.\nFinally, given $\\Gamma$ and $A$, a {\\em uniform family of terms} is a family\nof terms $a = (a_{{\\mathcal C}})$ with $a_{\\mathcal C} \\in \\mathrm{Tm}_{{\\mathcal C}}(\\Gamma_{{\\mathcal C}},A_{{\\mathcal C}})$ such that\n$F(a_{{\\mathcal C}}) = a_{\\mathcal{D}}$ for all morphisms $F \\in \\mathbf{CwF}_{\\Sigma}({\\mathcal C},\\mathcal{D})$. \n\n\\begin{remark}\nUniform families appear in Freyd's proof of the adjoint functor theorem \\cite{freyd:abelian}, in Reynolds' \\cite{reynolds:impredicative} and Reynolds and Plotkin's construction \\cite{plotkin-reynolds} of an initial algebra for an endofunctor from an impredicative encoding of an inductive type, and in Awodey, Frey, and Speight's  \\cite{awodey:impredicative} construction of an impredicative encoding of a higher inductive type. The common idea in these works is to first construct a weakly initial object and then the initial object is obtained by taking uniform families.\n\\end{remark}\n\n\\begin{definition}\\label{def-sig-mod}\nWe define inductively (actually inductive-recursively) how to build a valid signature $\\Sigma$ and the category $\\mathbf{CwF}_\\Sigma$ of cwfs with a $\\Sigma$-structure and cwf-morphisms that preserve $\\Sigma$-structure. First, the base case:\n\\begin{description}\n\\item[The empty signature] The empty signature $\\emptyset$ is valid and $\\mathbf{CwF}_\\emptyset = \\mathbf{CwF}$.\n\\end{description}\nAssume now that we have defined $\\Sigma$ as a valid signature and the associated\ncategory $\\mathbf{CwF}_{\\Sigma}$.\nThen we can add a new sort symbol, or a new operator symbol, or a new equation, to get a new valid signature,\nas follows:\n\\begin{description}\n\\item[Adding a sort symbol]\n  Let $\\Gamma = (\\Gamma_{\\mathcal C})$ be a uniform family of contexts indexed by ${\\mathcal C} \\in \\mathbf{CwF}_{\\Sigma}$.\n  Then we can extend $\\Sigma$ with a new sort symbol $S$ relative to $\\Gamma$, to obtain\n  the gat $\\Sigma' = (\\Sigma,(\\Gamma,S))$.\n  The objects of $\\mathbf{CwF}_{\\Sigma'}$ are pairs $({\\mathcal C},S_{{\\mathcal C}})$, where ${\\mathcal C}$ is an object of $\\mathbf{CwF}_{\\Sigma}$\n  and $S_{{\\mathcal C}} \\in \\mathrm{Ty}_{\\mathcal C}(\\Gamma_{{\\mathcal C}})$.\n  A morphism in $\\mathbf{CwF}_{\\Sigma'}(({\\mathcal C},S_{{\\mathcal C}}), (\\mathcal{D},S_{\\mathcal{D}}))$\n  is a morphism $F \\in \\mathbf{CwF}_{\\Sigma}({\\mathcal C},\\mathcal{D})$ such that $F(S_{\\mathcal C}) = S_\\mathcal{D}$.\n\\item[Adding an operator symbol]\n  If $\\Gamma$ is a uniform family of contexts and $A$ a uniform family of\n  types over $\\Gamma$, \n \n  then we can extend $\\Sigma$ with a new operator\n  symbol $f$ relative to $\\Gamma$ and $A$, to obtain\n  the gat $\\Sigma' = (\\Sigma,(\\Gamma,A,f))$.\n  An object of $\\mathbf{CwF}_{\\Sigma'}$\n  is a pair $({\\mathcal C},f_{{\\mathcal C}})$ where ${\\mathcal C}$ is an object in $\\mathbf{CwF}_{\\Sigma}$ and $f_{{\\mathcal C}} \\in \\mathrm{Tm}_{\\mathcal C}(\\Gamma_{\\mathcal C},A_{\\mathcal C})$.\n  A morphism in $\\mathbf{CwF}_{\\Sigma'}(({\\mathcal C},f_{{\\mathcal C}}),(\\mathcal{D},f_{\\mathcal{D}}))$ is a morphism $F \\in \\mathbf{CwF}_{\\Sigma}({\\mathcal C},\\mathcal{D})$ such that $F(f_{{\\mathcal C}}) = f_{\\mathcal{D}}$\n\\item[Adding an equation]\n  If $\\Gamma$ is a uniform family of contexts,\n  $A$ is a uniform family of types over $\\Gamma$\n  and $a,a'$ are uniform families of terms in $A$, \n \n  then we can extend $\\Sigma$ with a new equation $a = a'$ relative to $\\Gamma$ and $A$, to obtain\n  the gat $\\Sigma' = (\\Sigma,(\\Gamma,A,a,a'))$.\n  In this case,\n  $\\mathbf{CwF}_{\\Sigma'}$ is a full subcategory of $\\mathbf{CwF}_{\\Sigma}$. An object ${\\mathcal C}$ in\n  $\\mathbf{CwF}_{\\Sigma'}$ is an object ${\\mathcal C}$ of $\\mathbf{CwF}_\\Sigma$ such that $a_{{\\mathcal C}} = a'_{{\\mathcal C}}$.\n\\end{description}\n\n\\end{definition}\n\nThis definition is {\\em syntax independent}. In the next subsection we then show the purely {\\em syntactic} construction of an {\\em initial object} ${\\mathcal T}_{\\Sigma}$ in $\\mathbf{CwF}_{\\Sigma}$ (for an arbitrary valid signature $\\Sigma$) in terms of grammars and inference rules. A context in ${\\mathcal T}_{\\Sigma}$ will be an equivalence class $[ \\Gamma ]$ of raw contexts, and similarly for substitutions, types, and terms. To give a uniform\n  family of contexts $\\Gamma_{\\mathcal C}$ is then equivalent to giving a context $[ \\Gamma ] \\in {\\mathcal T}_{\\Sigma}$, since $\\Gamma_{\\mathcal C} = \\inte{ [ \\Gamma ] }_{\\mathcal C}$ where $\\inte{-}_{\\mathcal C}$ is the interpretation morphism from ${\\mathcal T}_\\Sigma$ to ${\\mathcal C}$.\nUniform families of types and terms arise from types and terms in ${\\mathcal T}_\\Sigma$ in a similar way.\n\nWe refer to Section \\ref{monoids} where we show a simple example: the construction of a signature $\\Sigma$ for internal monoids and its associated category of models $\\mathbf{CwF}_\\Sigma$ of cwfs with an internal monoid. We also show how to construct the initial cwf with an internal monoid ${\\mathcal T}_\\Sigma$. \n\n\\begin{remark}\nCartmell's notion of gat \\cite{cartmell:phd,cartmell:apal} also makes it possible to stipulate equations between type expressions. However, neither of our examples makes use of this extra generality. In particular,  in Section \\ref{sec:examples} we present the gat of cwfs with extra structure for $\\mathsf{N}, \\Pi$ and a first universe $\\mathsf{U}_0$ without needing type equations. The reason is that\nequations between types become equations between terms in our rendering\nof dependent type theory as a gat. See Remark \\ref{remark:typeequations} in Section \\ref{sec:u-example} for more explanation.\n\nLike Cartmell, we could consider gats with type equations, but we prefer not to make such equations part of our notion.\n\n\n\\end{remark}\n\n\n\n\n\\section{The construction of an initial object in $\\mathbf{CwF}_\\Sigma$}\\label{initial-gat}\n\nIn Section 3 we gave a {\\em syntax independent specification} of a generalized algebraic theory as the initial object of the category $\\mathbf{CwF}_\\Sigma$ of models of a (semantic) signature $\\Sigma$. Now we show our main theorem: the {\\em syntactic construction} of such an initial object ${\\mathcal T}_\\Sigma$. This construction is done in several steps. We first define the ``raw\" syntactic expressions. Then we define four families of partial equivalence relations (pers) over those raw expressions, corresponding to the four equality judgments. The term model ${\\mathcal T}_\\Sigma$ is obtained by quotienting with these pers.\n\nThis theorem can be viewed as a generalization of Birkhoff's completeness theorem for equational logic \\cite{birkhoff}:\n\\begin{theorem}\nThe category $\\mathbf{CwF}_\\Sigma$ has an initial object ${\\mathcal T}_\\Sigma$,\nfor every valid signature $\\Sigma$.\n\\end{theorem}\n\nThe construction of ${\\mathcal T}_\\Sigma$ will be by induction on the construction of $\\Sigma$. It is based on construction of initial cwfs in \\cite{castellan:tlca2015,castellan:lmcs} and we refer the reader to those papers for more details. Here we only provide a sketch and focus on how to extend the construction to ${\\mathcal T}_\\Sigma$.\n\nFor each $\\Sigma$ we will define the following.\n\\begin{itemize}\n\\item\nA grammar for the {\\em raw syntax}, that is, raw contexts in ${\\tt Ctx}_\\Sigma$, raw substitutions in ${\\tt Sub}_\\Sigma$, raw types in ${\\tt Ty}_\\Sigma$, and raw terms in ${\\tt Tm}_\\Sigma$.\n\\item\nA system of inference rules that generate four families of partial equivalence relations (pers) by a mutual inductive definition:\n$$\n\\Gamma = \\Gamma' \\vdash_\\Sigma\n\\qquad\n\\Gamma \\vdash_\\Sigma A = A'\n\\qquad\n\\Delta \\vdash_\\Sigma \\gamma = \\gamma' : \\Gamma\n\\qquad\n\\Gamma \\vdash_\\Sigma a = a' : A\n$$\nwhere $\\Gamma, \\Gamma' \\in {\\tt Ctx}_\\Sigma, \\gamma, \\gamma' \\in {\\tt Sub}_\\Sigma, A, A' \\in {\\tt Ty}_\\Sigma,$ and $a,a' \\in {\\tt Tm}_\\Sigma$. These pers correspond to valid equality judgments of a variable-free version of dependent type theory with explicit substitutions based on the cwf-combinators. The ordinary judgments will be defined as the reflexive instances of these equality judgments. For example $\\Gamma \\vdash_\\Sigma$ (meaning ``$\\Gamma$ is a valid context\") is defined as the reflexive instance $\\Gamma = \\Gamma \\vdash_\\Sigma$.\n\\item\nA cwf ${\\mathcal T}_\\Sigma$ is then constructed from the equivalence classes of derivable judgments. For example, the contexts in ${\\mathcal T}_\\Sigma$ are equivalence classes $[\\Gamma]$, such that $\\Gamma \\vdash_\\Sigma$. We will show that ${\\mathcal T}_\\Sigma$ is a cwf with a $\\Sigma$-structure, that is, an object of $\\mathbf{CwF}_\\Sigma$.\n\\item\nA $\\mathbf{CwF}_\\Sigma$-morphism $\\inte{-} : {\\mathcal T}_\\Sigma \\to {\\mathcal C}$ for every ${\\mathcal C} \\in \\mathbf{CwF}_\\Sigma$. This is the {\\em interpretation morphism}. This morphism is a partial function defined by induction on the raw syntax, that (whenever it is defined) maps raw contexts to contexts in ${\\mathcal C}$, raw substitutions to substitutions in ${\\mathcal C}$, raw types to types in ${\\mathcal C}$, and raw terms to terms in~${\\mathcal C}$. We show that these partial functions preserve the partial equivalence relations so that we can define the interpretation morphism on the equivalence classes. Finally we show that it indeed is a $\\mathbf{CwF}_\\Sigma$-morphism and the unique such into ${\\mathcal C}$.\n\\end{itemize}\n\nWe begin with the construction for the base case: the {\\bf empty signature} $\\emptyset$.\n\\begin{itemize}\n\\item\nWe start with the {\\em raw syntax}. This following grammar for raw contexts, raw substitutions, raw types, and raw terms.\n\\begin{eqnarray*}\n\\Gamma \\in {\\tt Ctx}_\\emptyset &::=& 1  \\ |\\ \\Gamma. A\\\\\n\\gamma \\in {\\tt Sub}_\\emptyset  \\ &::=& \\gamma \\circ \\gamma \\ |\\ \\mathrm{id}_\\Gamma \\ |\\ \\langle\\rangle_\\Gamma \\ |\\ \\mathrm{p}_{A} \\ |\\ \\langle \\gamma, a \\rangle_A\\\\\nA \\in {\\tt Ty}_\\emptyset  &::=& A[\\gamma]\\\\\na \\in {\\tt Tm}_\\emptyset  &::=& a[\\gamma] \\ |\\ \\qI_A\n\\end{eqnarray*}\nThese grammars generate a language of {\\em cwf-combinators}.\n\\item\nThe system of inference rules is displayed in \\cite{castellan:tlca2015,castellan:lmcs}. It is a system of {\\em general rules}, rules for dependent type theory which come before we introduce any sort symbols and operator symbols and equations (or any rules for the type formers of intuitionistic type theory). We do not have room here to display them, but note that they can be divided into four groups:\n\\begin{itemize}\n\\item the per rules, amounting to symmetry and transitivity for the four forms of equality judgments;\n\\item preservation rules for judgments, amounting to substitution of equals for equals (an example of such a rule is the {\\em type equality rule});\n\\item congruence rules for operators expressing that the cwf-combinators preserve equality;\n\\item conversion rules for the cwf-combinators.\n\\end{itemize}\n\\item\nNote that the initial cwf ${\\mathcal T}_\\emptyset$ is rather uninteresting: its category of contexts contains only a terminal object (the empty context), and there are no types and terms. Nevertheless, the grammar and inference rules used in its definition form a starting point. The grammar for raw types and raw terms will be extended each time we add a new sort symbol or operator symbol, respectively. For each such new symbol and each new equation we will add a new inference rule. As a consequence we will generate a non-trivial ${\\mathcal T}_\\Sigma$.\n\\end{itemize}\n\nAssume now for the induction step that we have defined the grammar, the inference rules, ${\\mathcal T}_\\Sigma$ and the interpretation morphism $\\inte{-} : {\\mathcal T}_{\\Sigma} \\to {\\mathcal C}$ in $\\mathbf{CwF}_\\Sigma$.\nLet $\\Sigma'$ be $\\Sigma$ extended by a new sort symbol, a new operator symbol, or a new equation. We shall now explain how to define ${\\mathcal T}_{\\Sigma'}$.\n\\begin{description}\n\\item[Adding a sort symbol] If $\\Gamma \\vdash_\\Sigma$, then we can introduce a new sort symbol $S$ in the context $\\Gamma$ representing the sequence of types of the arguments of $S$.\n\\begin{itemize}\n\\item\nWe add a new production for raw types\n$$\nA ::= S\n$$\nto the productions for ${\\mathcal T}_\\Sigma$.\n\\item\nWe add  the inference rule\n\\begin{mathpar}\n    \\inferrule\n    {}\n    {\\Gamma \\vdash_{\\Sigma'} S}\n  \\end{mathpar}\nto the inference rules for ${\\mathcal T}_\\Sigma$.\n\\item\nWe define $S_{{\\mathcal T}_{\\Sigma'}} = [S]$, so that $ {\\mathcal T}_{\\Sigma'}$ has a $\\Sigma'$-structure $({\\mathcal T}_\\Sigma,[S])$.\n\\item\nWe extend the definition of the interpretation morphism $\\inte{-}$  to an interpretation morphism $\\inte{-}' : {\\mathcal T}_{\\Sigma'} \\to {\\mathcal C}$ by\n$$\n\\inte{[S]}' = S_{\\mathcal C}\n$$\nIt follows  that this is a morphism in $\\mathbf{CwF}_{\\Sigma'}$ and that it is unique.\n\\end{itemize}\n\n\\item[Adding an operator symbol] If $\\Gamma \\vdash_\\Sigma A$, then we can introduce a new operator symbol $f$, where the context $\\Gamma$ represents the sequence of types of the arguments and $A$ is the type of the result.\n\\begin{itemize}\n\\item\nWe add a new production for raw terms\n$$\na ::= f\n$$\nto the productions for ${\\mathcal T}_\\Sigma$.\n\\item\nWe add  the inference rule\n\\begin{mathpar}\n    \\inferrule\n    {}\n    {\\Gamma \\vdash_{\\Sigma'} f : A}\n  \\end{mathpar}\nto the inference rules for ${\\mathcal T}_\\Sigma$.\n\\item\nWe define $f_{{\\mathcal T}_{\\Sigma'}} = [f]$, so that $ {\\mathcal T}_{\\Sigma'}$ has a $\\Sigma'$-structure $({\\mathcal T}_\\Sigma,[f])$.\\item\nWe extend the definition of the interpretation morphism $\\inte{-}$  to an interpretation morphism $\\inte{-}' : {\\mathcal T}_{\\Sigma'} \\to {\\mathcal C}$ by\n$$\n\\inte{[f]}' = f_{\\mathcal C}\n$$\nIt follows  that this is a morphism in $\\mathbf{CwF}_{\\Sigma'}$ and that it is unique.\n\\end{itemize}\n\n\\item[Adding an equation] If $\\Gamma \\vdash_\\Sigma a : A$ and $\\Gamma \\vdash_\\Sigma a' : A$ we can introduce a new equation $a = a'$.\n\\begin{itemize}\n\\item\n${\\mathcal T}_\\Sigma'$ has the same productions as ${\\mathcal T}_\\Sigma$.\n\\item\nWe add  the inference rule\n\\begin{mathpar}\n    \\inferrule\n    {}\n    {\\Gamma \\vdash_{\\Sigma'} a = a' : A}\n  \\end{mathpar}\nto the inference rules for ${\\mathcal T}_\\Sigma$.\n\\item\n${\\mathcal T}_{\\Sigma'}$ is based on the same raw syntax as ${\\mathcal T}_\\Sigma$ but the equivalence relation has changed. To show that ${\\mathcal T}_{\\Sigma'} \\in \\mathbf{CwF}_{\\Sigma'}$ we just need to show that $[ a ] = [ a' ]$ but this follows from the inference rule $\\Gamma \\vdash_{\\Sigma'} a = a' : A$.\n\\item\nIn order to define $\\inte{-}'$ we first define the partial function on the raw syntax to be identical to the partial function on the raw syntax for $\\inte{-}$. We then prove that this partial function preserves the extended partial equivalence relation and define $\\inte{-}'$ on the new equivalence classes. It follows  that $\\inte{-}'$ is unique.\n\\end{itemize}\n\\end{description}\nThis concludes the proof of the theorem. \\qed\n\n\n\n\\section{Examples of generalized algebraic theories}\\label{sec:examples}\n\nWe will now display the sort symbols, operator symbols, and equations for the generalized algebraic of internal monoids, internal categories and of internal cwfs. We will then show how to add operator symbols and equations when adding internal notions of $\\Pi, \\mathsf{N}$ and a universe closed under $\\Pi$ and $\\mathsf{N}$ to the gat of internal cwfs. The reason for prefacing these notions by the word ``internal\" is that the models of the theories are internal monoids, categories, and cwfs in $\\mathbf{CwF}_\\Sigma$ for the respective signatures for these theories. Moreover, internal monoids, categories, and cwfs in the cwf~$\\Set$ are small monoids, categories, and cwfs, respectively. Note that the cwfs defined in Section 2 need not be small, and hence not internal cwfs in the cwf~$\\Set$.\n\n\nWe begin by using the recipe in Definition \\ref{def:Cwfmor} to construct the semantic signature for internal monoids and its associated category of models, that is, of cwfs with an internal monoid. We then follow the recipe in Definition \\ref{initial-gat} and construct the initial cwf with an internal monoid. \n\nFor ease of readability, we will only present the sort symbols, operator symbols, and equations in the remaining examples by using an informal notation with named variables, rather than the formal notations using cwf-combinators employed in Definitions \\ref{def:Cwfmor} and \\ref{initial-gat}.\n\nOur final example is the gat of internal contextual cwfs, a variant of Cartmell's contextual categories. The contexts in such contextual cwfs come with a length $n$. We sketch how this can be axiomatized as a gat with countably many sort symbols $\\mathrm{ctx}_n, \\mathrm{sub}_n, \\mathrm{ty}_n, \\mathrm{tm}_n$ for an external natural number $n$ (and similarly for the operator symbols and equations). We also indicate how our framework can be extended to cover such gats.\n\n\\subsection{Internal monoids}\\label{monoids}\n The one-sorted algebraic theory of monoids has two operator symbols,\n$\\mathrm{e}$ for identity and $\\mathrm{*}$ for composition, and associativity and identity laws as equations.\nAs any other one-sorted algebraic theory, the theory of monoids yields a\ngeneralized algebraic theory. In ordinary notation with variables it might be rendered as follows, where $\\mathrm{M}$ is the only sort:\n\\begin{eqnarray*}\n&\\vdash& \\mathrm{M}\\\\\n&\\vdash& e : \\mathrm{M}\\\\\nx, y : \\mathrm{M} &\\vdash& \\mathrm{*}(x,y) : \\mathrm{M}\\\\\ny : \\mathrm{M} &\\vdash& \\mathrm{*}(\\mathrm{e},y) = y : \\mathrm{M}\\\\\nx : \\mathrm{M} &\\vdash& \\mathrm{*}(x,\\mathrm{e}) = x : \\mathrm{M}\\\\\nx, y, z : \\mathrm{M} &\\vdash& \\mathrm{*}(\\mathrm{*}(x,y),z) = \\mathrm{*}(x,\\mathrm{*}(y,z)) : \\mathrm{M}\n\\end{eqnarray*}\n\nWe now show how the corresponding semantic signature for monoids $\\Sigma$ and its associated category of models $\\mathbf{CwF}_\\Sigma$ are constructed step-wise in the sense of Definition \\ref{def-sig-mod}.\n\nAs always, we begin with the empty signature $\\emptyset$ and its category of models $\\mathbf{CwF}_\\emptyset = \\mathbf{CwF}$.\n\\begin{description}\n\\item[Adding the sort symbol $\\mathrm{M}$] Each cwf ${\\mathcal C}$ has a chosen empty context (terminal object) $1_{\\mathcal C}$. Since cwf-morphisms preserve empty contexts on the nose, $1 = (1_{\\mathcal C})$ is a uniform family of contexts in $\\mathbf{CwF}_\\emptyset$. Hence we can introduce a new constant sort symbol $\\mathrm{M}$ in the empty context. The resulting signature  is\n$$\n\\Sigma_1 = (\\emptyset, (1,\\mathrm{M}))\n$$\nThe objects of $\\mathbf{CwF}_{\\Sigma_1}$  are pairs $({\\mathcal C},\\mathrm{M}_{\\mathcal C})$, where ${\\mathcal C}$ is a cwf and $\\mathrm{M}_{\\mathcal C} \\in \\mathrm{Ty}_{\\mathcal C}(1_{\\mathcal C})$.\n\\item[Adding the operator symbol for the identity]\nSince, morphisms in $\\mathbf{CwF}_{\\Sigma_1}$ preserve both empty contexts $1_{\\mathcal C}$ and types $\\mathrm{M}_{\\mathcal C}$ on the nose, we have a uniform family of contexts $1 = (1_{\\mathcal C})$ and a uniform family of types $\\mathrm{M} = (\\mathrm{M}_{\\mathcal C})$ in $\\mathbf{CwF}_{\\Sigma_1}$. Hence we can introduce a new constant operator symbol $e$ (the identity of the monoid).  The resulting signature  is\n$$\n\\Sigma_2 = (\\Sigma_1, (1,\\mathrm{M},e))\n$$\nThe objects of $\\mathbf{CwF}_{\\Sigma_2}$  are triples $({\\mathcal C},\\mathrm{M}_{\\mathcal C},e_{\\mathcal C})$, where ${\\mathcal C}$ is a cwf, $\\mathrm{M}_{\\mathcal C} \\in \\mathrm{Ty}_{\\mathcal C}(1_{\\mathcal C})$ and $e_{\\mathcal C} \\in \\mathrm{Tm}_{\\mathcal C}(1_{\\mathcal C},\\mathrm{M}_{\\mathcal C})$.\n\\item[Adding the operator symbol for composition]\nAgain using that cwf-morphisms preserve all cwf-structure and $\\mathrm{M}_{\\mathcal C}$, we deduce that we have a uniform family of contexts $1.\\mathrm{M}.\\mathrm{M}[\\mathrm{p}]$ and a uniform family of types $\\mathrm{M}[\\mathrm{p}][\\mathrm{p}]$ in $\\mathbf{CwF}_{\\Sigma_2}$. Thus we can add a binary operator symbol $\\mathrm{*}$. The resulting signature is\n$$\n\\Sigma_3 = (\\Sigma_2, (1.\\mathrm{M}.\\mathrm{M}[\\mathrm{p}],\\mathrm{M}[\\mathrm{p}][\\mathrm{p}],\\mathrm{*}))\n$$\nThe objects of $\\mathbf{CwF}_{\\Sigma_3}$  are quadruples $({\\mathcal C},\\mathrm{M}_{\\mathcal C},e_{\\mathcal C},*_{\\mathcal C})$, where ${\\mathcal C}$ is a cwf, $\\mathrm{M}_{\\mathcal C} \\in \\mathrm{Ty}_{\\mathcal C}(1_{\\mathcal C})$, $e_{\\mathcal C} \\in \\mathrm{Tm}_{\\mathcal C}(1_{\\mathcal C},\\mathrm{M}_{\\mathcal C})$, and $*_{\\mathcal C} \\in \\mathrm{Tm}_{\\mathcal C}((1.\\mathrm{M}.\\mathrm{M}[\\mathrm{p}])_{\\mathcal C},(\\mathrm{M}[\\mathrm{p}][\\mathrm{p}])_{\\mathcal C})$.\n\\item[Adding the left identity law]\nFurthermore, we extend the signature with the equations stating that $\\mathrm{e}$ is a left identity as follows:\n$$\n\\Sigma_4 = (\\Sigma_3, (1.\\mathrm{M}, \\mathrm{M}[\\mathrm{p}], \\mathrm{*}[\\tuple{\\tuple{\\tuple{},\\mathrm{e}[\\tuple{}]},\\mathrm{q}}], \\mathrm{q}))\n$$\nThe uniform family of context $1.\\mathrm{M}$ expresses that the equation has one variable of type $\\mathrm{M}$, the uniform family of types $\\mathrm{M}[\\mathrm{p}]$ expresses that the two sides of the equation have type $\\mathrm{M}$, and the uniform families of terms $\\mathrm{*}[\\tuple{\\tuple{\\tuple{},\\mathrm{e}[\\tuple{}]},\\mathrm{q}}]$ and $\\mathrm{q}$ express the two sides of the equation.\n$\\mathbf{CwF}_{\\Sigma_4}$ is the full subcategory of $\\mathbf{CwF}_{\\Sigma_3}$ with objects ${\\mathcal C}$\nsuch that $(\\mathrm{*}[\\tuple{\\tuple{\\tuple{},\\mathrm{e}[\\tuple{}]},\\mathrm{q}}])_{\\mathcal C} = \\mathrm{q}_{\\mathcal C}$.\n\\item[Adding the right identity and the associativity laws]\nFinally we add the right identity equation and the associativity equation to get the signatures $\\Sigma_5$ and $\\Sigma_6$. We omit the details. \n\\end{description}\nWe call the resulting signature for internal monoids $\\Sigma = \\Sigma_6$. The category $\\mathbf{CwF}_\\Sigma$ is the category of cwfs with an {\\em internal monoid}. This is a cwf-version of the notion of internal monoid which can be defined in any category with finite products. Ordinary (small) monoids come out as internal monoids in $\\Set$, the cwf of small sets.\n\nWe also sketch the construction of the initial object ${\\mathcal T}_\\Sigma$ of $\\mathbf{CwF}_\\Sigma$ following the recipe for introducing sort symbols, operators symbols, and equations in Section \\ref{initial-gat}. (We omit the index $\\Sigma$ in $\\vdash_\\Sigma$.)\n\\begin{description}\n\\item[Adding the sort symbol $\\mathrm{M}$]\nFirst, we have $1 \\vdash$ for the empty signature, so we can\nadd a production for the constant sort symbol $\\mathrm{M}$ and the inference rule:\n\\begin{eqnarray*}\n1 &\\vdash& \\mathrm{M}\n\\end{eqnarray*}\nFor later use we infer $1.\\mathrm{M} \\vdash$ and, using $\\mathrm{p}:1.\\mathrm{M} \\to 1$, $1.\\mathrm{M}\\vdash\\mathrm{M}[\\mathrm{p}]$,\nso $1.\\mathrm{M}.\\mathrm{M}[\\mathrm{p}] \\vdash$.\n\\item[Adding the operator symbol for identity]\nWe then add a production for the constant operator symbol $\\mathrm{e}$ and the inference rule:\n\\begin{eqnarray*}\n1 &\\vdash& \\mathrm{e} : \\mathrm{M}\n\\end{eqnarray*}\nAgain for later use we infer $1.\\mathrm{M}\\vdash \\mathrm{e}[\\mathrm{p}] :\\mathrm{M}[\\mathrm{p}]$.\nNote that here $\\mathrm{p} = \\tuple{}$, the empty substitution $1.\\mathrm{M} \\to 1$,\nsince there is only one substitution $1.\\mathrm{M} \\to 1$.\n\\item[Adding the operator symbol for composition]\nWe then add a production for the binary operator symbol $\\mathrm{*}$.\nUsing another $\\mathrm{p}: 1.\\mathrm{M}.\\mathrm{M}[\\mathrm{p}] \\to 1.\\mathrm{M}$ (note the different type),\nwe can derive $1.\\mathrm{M}.\\mathrm{M}[\\mathrm{p}] \\vdash \\mathrm{M}[\\mathrm{p}][\\mathrm{p}]$, so we can add the inference rule\n\\begin{eqnarray*}\n1.\\mathrm{M}.\\mathrm{M}[\\mathrm{p}] &\\vdash& \\mathrm{*} : \\mathrm{M}[\\mathrm{p}][\\mathrm{p}]\n\\end{eqnarray*}\nNote that we project $\\mathrm{M}$ on the right twice, reflecting that $\\mathrm{*}$ is binary.\n\\item[Adding the left identity law]\nWe can derive $1.\\mathrm{M} \\vdash \\mathrm{q} : \\mathrm{M}[\\mathrm{p}]$. With some effort,\nusing previous inferences, we can  derive\n$1.\\mathrm{M} \\vdash \\mathrm{*}[\\tuple{\\tuple{\\tuple{},\\mathrm{e}[\\tuple{}]},\\mathrm{q}}] : \\mathrm{M}[\\mathrm{p}]$.\nHence we can add the inference rule for the equation ($\\mathrm{e}$ is a left identity):\n\\begin{eqnarray*}\n1.\\mathrm{M} &\\vdash& \\mathrm{*}[\\tuple{\\tuple{\\tuple{},\\mathrm{e}[\\tuple{}]},\\mathrm{q}}] = \\mathrm{q} : \\mathrm{M}[\\mathrm{p}]\n\\end{eqnarray*}\n\\item[Adding the right identity and the associativity laws]\nWe omit the details.\n\\end{description}\nThe resulting initial object ${\\mathcal T}_\\Sigma = {\\mathcal T}_{\\Sigma_6}$ is generated by a system of grammar and inference rules for dependent type theory with an internal monoid. In this theory we can prove statements such as \n$$\n\\Gamma \\vdash a : \\mathrm{M}\n$$\nstating that $a$ is a well-formed monoid expression in the context $\\Gamma$ and\n$$\n\\Gamma \\vdash a = a': \\mathrm{M}\n$$\nstating that $a = a'$ is a derivable equation between monoid expressions in the context $\\Gamma$. Note that both contexts and monoid expressions use cwf-combinators and are variable-free. \n\nThis example illustrates that gats indeed generalise ordinary algebraic theories.\nThe remaining examples will use dependent types in an essential way. However, for reasons of readability we will from now on only use ordinary notation with named variables. Hopefully, it is clear from the above how to formally construct the corresponding semantic signatures, categories of models, and initial models using cwf-combinators. For example, these constructions for the theory of internal categories are similar to the constructions for the theory of internal monoids.\n\n\n\\subsection{Internal categories} The gat of categories was one of Cartmell's motivating examples. It has the following sort symbols, operator symbols, and equations. Again, note that in our case the models are internal categories in a cwf. To emphasize the difference between the internal notions of category and cwf and the external notions (introduced in Section 2), our notation for sort symbols in the gat of internal cwfs use lower case letters ($\\mathrm{obj}, \\mathrm{hom}, \\mathrm{ty}, \\mathrm{tm}$). This is in contrast to the upper case letters for the external versions ($\\mathrm{Ty}, \\mathrm{Tm}$). We will however overload notation for operator symbols, and for example use $\\circ$ both for the cwf-combinator and for the operator symbol in the gat of internal categories.\n\nSort symbols:\n\\begin{eqnarray*}\n&\\vdash& \\mathrm{obj}\\\\\n\\Delta, \\Gamma : \\mathrm{obj} &\\vdash& \\mathrm{hom}(\\Delta,\\Gamma)\\\\\n\\end{eqnarray*}\n\nOperator symbols:\n\\begin{eqnarray*}\n\\Gamma : \\mathrm{obj} &\\vdash& \\mathrm{id}_\\Gamma : \\mathrm{hom}(\\Gamma,\\Gamma)\\\\\n\\Xi,\\Delta,\\Gamma : \\mathrm{obj}, \\gamma : \\mathrm{hom}(\\Delta,\\Gamma), \\delta : \\mathrm{hom}(\\Xi,\\Delta) &\\vdash&\n\\gamma \\circ \\delta : \\mathrm{hom}(\\Xi,\\Gamma)\n\\end{eqnarray*}\n\nEquations:\n\\begin{eqnarray*}\n\\Delta, \\Gamma : \\mathrm{obj}, \\gamma : \\mathrm{hom}(\\Delta,\\Gamma) &\\vdash& \\mathrm{id}_\\Gamma \\circ \\gamma = \\gamma : \\mathrm{hom}(\\Delta,\\Gamma)\\\\\n\\Delta, \\Gamma : \\mathrm{obj}, \\gamma : \\mathrm{hom}(\\Delta,\\Gamma) &\\vdash& \\gamma \\circ \\mathrm{id}_\\Delta = \\gamma : \\mathrm{hom}(\\Delta,\\Gamma)\\\\\n\\Theta, \\Xi,\\Delta,\\Gamma : \\mathrm{obj}, \\gamma : \\mathrm{hom}(\\Delta,\\Gamma), \\delta : \\mathrm{hom}(\\Xi,\\Delta), \\xi : \\mathrm{hom}(\\Theta,\\Xi) &\\vdash&\n(\\gamma \\circ \\delta) \\circ \\xi = \\gamma \\circ (\\delta \\circ \\xi): \\mathrm{hom}(\\Theta,\\Gamma)\n\\end{eqnarray*}\nNote that composition is officially an operator symbol with five arguments. In the official notation we should write $\\gamma \\circ_{\\Xi,\\Delta,\\Gamma} \\delta$, but we suppress the context arguments $\\Xi,\\Delta,\\Gamma$. We will do so for some other operations too.\n\nThe rendering of the gat of categories in cwf-combinator language and the proof that it indeed yields a valid signature are similar to what they were for the gat of monoids. The inference rules for the two sort symbols in cwf-combinator language are\n\\begin{eqnarray*}\n1 &\\vdash& \\mathrm{obj}\\\\\n1.\\mathrm{obj}.\\mathrm{obj}[\\mathrm{p}] &\\vdash& \\mathrm{hom}\n\\end{eqnarray*}\nand the operator symbols for identity\n\\begin{eqnarray*}\n1.\\mathrm{obj} &\\vdash& \\mathrm{e} : \\mathrm{hom}[\\tuple{\\mathrm{id}_{1.\\mathrm{obj}},\\mathrm{q}_{1,\\mathrm{obj}}}]\n\\end{eqnarray*}\nWe omit the verbose cwf-renderings of the operator symbol for composition and the equations.\n\nA cwf with extra structure for the generalized algebraic theory of categories is a cwf with an {\\em internal category}. This is a cwf-based analogue of the usual notion of internal category in a category with finite limits. As shown by Martin Hofmann \\cite{hofmann:csl,hofmann:cambridge}, every category with finite limits yields a category with attributes, and hence a cwf. However, not every cwf has finite limits. To achieve this we need more structure. As shown by Clairambault and Dybjer \\cite{ClairambaultD11,ClairambaultD14} the 2-category of categories with finite limits is biequivalent to the 2-category of democratic cwfs that support $\\Sigma$-types and extensional identity types.\n\nAn internal category in the cwf $\\Set$ of small sets is a small category.\n\n\n\\subsection{Internal cwfs}\\label{gat-cwf}\n\nThe gat of internal cwfs is obtained by extending the gat of internal categories with new sort symbols, operator symbols, and equations for a family valued functor, and then new operator symbols and equations for a terminal object, and context comprehension. We here rename the sort $\\mathrm{obj}$ of objects of the category of contexts to $\\mathrm{ctx}$.\n\n\\subsubsection{The extension with a family valued functor}\n\\mbox{ }\n\nSort symbols:\n\\begin{eqnarray*}\n\\Gamma : \\mathrm{ctx} &\\vdash& \\mathrm{ty}(\\Gamma)\\\\\n\\Gamma : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma) &\\vdash& \\mathrm{tm}(\\Gamma,A)\n\\end{eqnarray*}\n\nOperator symbols:\n\\begin{eqnarray*}\n\\Gamma,\\Delta : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma), \\gamma : \\mathrm{hom}(\\Delta,\\Gamma) &\\vdash&\nA[\\gamma] : \\mathrm{ty}(\\Delta)\\\\\n\\Gamma,\\Delta : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma), \\gamma : \\mathrm{hom}(\\Delta,\\Gamma), a:\\mathrm{tm}(\\Gamma,A) &\\vdash&  a[\\gamma] : \\mathrm{tm}(\\Delta,A[\\gamma])\n\\end{eqnarray*}\n\nEquations:\n\\begin{eqnarray*}\n\\Gamma : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma) &\\vdash& A[\\mathrm{id}_\\Gamma] = A : \\mathrm{ty}(\\Gamma)\\\\\n\\Gamma : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma), a:\\mathrm{tm}(\\Gamma,A) &\\vdash& a[\\mathrm{id}_\\Gamma] = a : \\mathrm{tm}(\\Gamma,A)\\\\\n\\Xi,\\Delta,\\Gamma : \\mathrm{ctx}, \\delta : \\mathrm{hom}(\\Xi,\\Delta), \\gamma : \\mathrm{hom}(\\Delta,\\Gamma),\nA:\\mathrm{ty}(\\Gamma) &\\vdash& A[\\gamma\\circ\\delta] = A[\\gamma][\\delta]: \\mathrm{ty}(\\Xi)\\\\\n\\Xi,\\Delta,\\Gamma : \\mathrm{ctx}, \\delta : \\mathrm{hom}(\\Xi,\\Delta), \\gamma : \\mathrm{hom}(\\Delta,\\Gamma),\nA:\\mathrm{ty}(\\Gamma), a:\\mathrm{tm}(\\Gamma,A) &\\vdash&\na[\\gamma\\circ\\delta] = a[\\gamma][\\delta]: \\mathrm{tm}(\\Xi,A[\\gamma\\circ\\delta])\n\\end{eqnarray*}\n\n\\subsubsection{The extension with a terminal object}\nNo new sorts are required.\n\nOperator symbols:\n\\begin{eqnarray*}\n&\\vdash& 1 : \\mathrm{ctx}\\\\\n\\Gamma : \\mathrm{ctx} &\\vdash& \\tuple{}_\\Gamma : \\mathrm{hom}(\\Gamma,1)\n\\end{eqnarray*}\n\nEquations:\n\\begin{eqnarray*}\n &\\vdash& \\mathrm{id}_1 = \\tuple{}_1 : \\mathrm{hom}(1,1)\\\\\n\\Gamma,\\Delta : \\mathrm{ctx}, \\gamma : \\mathrm{hom}(\\Delta,\\Gamma) &\\vdash&\n\\tuple{}_\\Gamma\\circ\\gamma = \\tuple{}_\\Delta : \\mathrm{hom}(\\Delta,1)\n\\end{eqnarray*}\n(The latter two equations are better for term rewriting than the\nobvious single one expressing the uniqueness of $\\tuple{}_\\Gamma$.)\n\n\\subsubsection{The extension with context comprehension}\n\nNo new sorts are required.\n\nOperator symbols:\n\\begin{eqnarray*}\n\\Gamma : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma) &\\vdash& \\Gamma. A : \\mathrm{ctx}\\\\\n\\Gamma,\\Delta : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma), \\gamma : \\mathrm{hom}(\\Delta,\\Gamma), a:\\mathrm{tm}(\\Delta,A[\\gamma]) &\\vdash& \\tuple{\\gamma,a} : \\mathrm{hom}(\\Delta,\\Gamma. A)\\\\\n\\Gamma : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma) &\\vdash& \\mathrm{p}: \\mathrm{hom}(\\Gamma. A,\\Gamma)\\\\\n\\Gamma : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma) &\\vdash& \\mathrm{q}: \\mathrm{tm}(\\Gamma. A,A[\\mathrm{p}])\n\\end{eqnarray*}\n\nEquations:\n\\begin{eqnarray*}\n\\Gamma,\\Delta : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma), \\gamma : \\mathrm{hom}(\\Delta,\\Gamma), a:\\mathrm{tm}(\\Delta,A[\\gamma]) &\\vdash& \\mathrm{p}\\circ\\tuple{\\gamma,a} = \\gamma : \\mathrm{hom}(\\Delta,\\Gamma)\\\\\n\\Gamma,\\Delta : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma), \\gamma : \\mathrm{hom}(\\Delta,\\Gamma), a:\\mathrm{tm}(\\Delta,A[\\gamma]) &\\vdash& \\mathrm{q}[\\tuple{\\gamma,a}] = a : \\mathrm{tm}(\\Delta,A[\\gamma]) \\\\\n\\Gamma,\\Delta,\\Xi : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma), \\gamma : \\mathrm{hom}(\\Delta,\\Gamma), a:\\mathrm{tm}(\\Delta,A[\\gamma]), \\delta : \\mathrm{hom}(\\Xi,\\Delta) &\\vdash&\n\\tuple{\\gamma,a} \\circ \\delta = \\tuple{\\gamma\\circ\\delta,a[\\delta]} :\n\\mathrm{hom}(\\Xi,\\Gamma. A) \\\\\n\\Gamma : \\mathrm{ctx}, A:\\mathrm{ty}(\\Gamma) &\\vdash&\n\\mathrm{id}_{\\Gamma. A} = \\tuple{\\mathrm{p},\\mathrm{q}} : \\mathrm{hom}(\\Gamma. A,\\Gamma. A)\n\\end{eqnarray*}\n(If $\\mathrm{p}\\circ\\delta = \\gamma$ and $\\mathrm{q}[\\delta]=a$, we get\n$\\tuple{\\gamma,a}=\\tuple{\\mathrm{p}\\circ\\delta, \\mathrm{q}[\\delta]} = \\tuple{\\mathrm{p},\\mathrm{q}}\\circ\\delta =\n\\delta$, the uniqueness requirement of the universal property.\nHowever, the equation for surjective pairing is not left-linear and with\na variable on one side, which is not good for rewriting.)\n\nA cwf with extra structure supporting the generalized algebraic theory of cwfs is a cwf with an\n\\emph{internal cwf}. An internal cwf in the cwf $\\Set$ of small sets is a {\\em small cwf},\nthat is, it is a cwf in the ordinary sense (see Definition~\\ref{def:Cwfobj})\nexcept that it has small sets of objects, morphisms, types, and terms.\n\nAn example of a cwf with an internal cwf is provided by the cwf $\\Set$ of small sets with an internal category of very small sets. We can make this precise if we work in set theory with two Grothendieck universes $\\mathrm{V}_0 \\in \\mathrm{V}_1$. We call the members of $\\mathrm{V}_1$ ``small sets\" and the members of $\\mathrm{V}_0$ ``very small sets\". The category of contexts of the cwf $\\Set$ is the usual category of small sets, by which we here mean that its objects are in $\\mathrm{V}_1$. Moreover, the types are also the small sets in $\\mathrm{V}_1$. To get an internal cwf, we interpret its sort of objects $\\mathrm{ctx}$ as the small set $\\mathrm{V}_0$ of very small sets, and the sorts of types $\\mathrm{ty}(\\Gamma)$ also as $\\mathrm{V}_0$.\n\n\n\\subsection{Internal cwfs with $\\Pi$-types}\nWe add three operator symbols in addition to the operator symbols for cwfs in Section 5.2 and 5.3:\n\\begin{eqnarray*}\n\\Gamma : \\mathrm{ctx}, A : \\mathrm{ty}(\\Gamma), B : \\mathrm{ty}(\\Gamma.A)&\\vdash& \\Pi(A,B) : \\mathrm{ty}(\\Gamma)\\\\\n\\Gamma : \\mathrm{ctx}, A : \\mathrm{ty}(\\Gamma), B : \\mathrm{ty}(\\Gamma.A), b : \\mathrm{tm}(\\Gamma.A, B) &\\vdash& \\lambda(b) : \\mathrm{tm}(\\Gamma,\\Pi(A,B))\\\\\n\\Gamma : \\mathrm{ctx}, A : \\mathrm{ty}(\\Gamma), B : \\mathrm{ty}(\\Gamma.A), c :  \\mathrm{tm}(\\Gamma,\\Pi(A,B)), a : \\mathrm{tm}(\\Gamma, A) &\\vdash& \\mathsf{app}(c,a) : \\mathrm{tm}(\\Gamma, B[\\tuple{\\mathrm{id},a}])\n\\end{eqnarray*}\n(again omitting some of the official arguments)\nand equations for $\\beta, \\eta$ (also omitting the context and type of the equality judgment)\n \\begin{eqnarray*}\n \\mathsf{app}(\\lambda(b),a) &=& b[\\tuple{\\mathrm{id},a}]\\\\\n \\lambda(\\mathsf{app}(c[\\mathrm{p}],\\mathrm{q})) &=& c\n\\end{eqnarray*}\nand commutation with respect to substitution.\n\\begin{eqnarray*}\n\\Pi(A,B)[ \\gamma ] &=& \\Pi(A [ \\gamma ], B[ \\gamma^+ ])\\\\\n\\lambda(b) [ \\gamma ] &=& \\lambda(b[\\gamma^+ ])\\\\\n\\mathsf{app}(c,a) [ \\gamma ] &=& \\mathsf{app}(c[ \\gamma ], a[ \\gamma ] )\n\\end{eqnarray*}\nwhere $\\gamma^+ = \\tuple{\\gamma \\circ \\mathrm{p}, \\mathrm{q}}$.\n\n\\subsection{Internal cwfs with $\\Pi$ and $\\mathsf{N}$}\nFurthermore, we add the operator symbol\n\\begin{eqnarray*}\n\\Gamma : \\mathrm{ctx} &\\vdash& \\mathsf{N}_\\Gamma : \\mathrm{ty}(\\Gamma) \n\\end{eqnarray*}\nWe also add operator symbols for $0, \\mathrm{s}, \\mathrm{R}$ and the equations for $\\mathrm{R}$ and for commutativity with substitution, but omit the details. Note that the type of the primitive recursion operator $\\mathrm{R}$ relies on the signature for $\\Pi$-types.\n\n\\subsection{Internal cwfs with $\\mathsf{U}_0$ closed under $\\Pi$ and $\\mathsf{N}$}\\label{sec:u-example}\nWe add four more operator symbols\n\\begin{eqnarray*}\n\\Gamma : \\mathrm{ctx} &\\vdash& (\\mathsf{U}_0)_\\Gamma : \\mathrm{ty}(\\Gamma)\\\\\n\\Gamma : \\mathrm{ctx}, a : \\mathrm{tm}(\\Gamma,(\\mathsf{U}_0)_\\Gamma) &\\vdash& {\\mathrm{T}_0}(a) : \\mathrm{ty}(\\Gamma)\\\\\n\\Gamma : \\mathrm{ctx} &\\vdash& (\\mathsf{N}^0)_\\Gamma : \\mathrm{tm}(\\Gamma,(\\mathsf{U}_0)_\\Gamma) \\\\\n\\Gamma : \\mathrm{ctx},\na : \\mathrm{tm}(\\Gamma,(\\mathsf{U}_0)_\\Gamma),\nb :  \\mathrm{tm}(\\Gamma \\cdot \\mathrm{T}_0(a), (\\mathsf{U}_0)_\\Gamma))\n&\\vdash&\n \\Pi^0(a,b) : \\mathrm{tm}(\\Gamma,(\\mathsf{U}_0)_\\Gamma)\n\\end{eqnarray*}\n$(\\mathsf{U}_0)_\\Gamma$ is the universe (a type) relative to the context $\\Gamma$; $\\mathrm{T}_0$ is the decoding operation mapping a term in the universe to the corresponding type; $\\mathsf{N}^0$ is the code for $\\mathsf{N}$ in the universe, and $\\Pi^0$ forms codes for $\\Pi$-types in the\n universe. (Note that we have dropped the context argument of $\\mathrm{T}_0$ and $\\Pi^0$.)\n\nWe add the decoding equations:\n\\begin{eqnarray*}\n\\mathrm{T}_0(\\mathsf{N}^0_\\Gamma) &=& \\mathsf{N}_\\Gamma\\\\\n\\mathrm{T}_0(\\Pi^0(a,b)) &=& \\Pi(\\mathrm{T}_0(a),\\mathrm{T}_0(b))\n\\end{eqnarray*}\nand the equations for preservation of substitution:\n\\begin{eqnarray*}\n{(\\mathsf{U}_0)}_\\Gamma [ \\gamma ] &=& {(\\mathsf{U}_0)}_\\Delta\\\\\n\\mathrm{T}_0(a) [ \\gamma ] &=& \\mathrm{T}_0(a[ \\gamma ] )\\\\\n\\mathsf{N}^0_\\Gamma [ \\gamma ] &=&\\mathsf{N}^0_\\Delta\\\\\n\\Pi^0(a,b)[ \\gamma ] &=& \\Pi^0(a [ \\gamma ], b[ \\gamma^+ ])\n\\end{eqnarray*}\n\n\\begin{remark}\\label{remark:typeequations}\nNote that all equations are between {\\em terms} in the gat of cwfs with extra structure for $\\mathsf{N}, \\Pi,$ and $\\mathsf{U}_0$; we do not need the extra generality of stipulating type equations as discussed in the introduction. For example, $\\mathrm{T}_0(\\mathsf{N}^0_\\Gamma) = \\mathsf{N}_\\Gamma$ is an equation between {\\em internal} types, that is, terms of type $\\mathrm{ty}(\\Gamma)$.\n\\end{remark}\n\n\\begin{remark}\nAlso note that the gat for the universe is inevitably {\\em \\`a la Tarski} in the sense that we distinguish between types and terms in a cwf and we must have an operation decoding a term into a type. However, Martin-L\u00f6f's distinction between {\\em \\`a la Russell} and {\\em \\`a la Tarski} \\cite{martinlof:padova} is a distinction between two different formulation of the raw syntax and inference rules of type theory.\n\\end{remark}\n\n\\subsection{A possible refinement to internal contextual cwfs}\n\nOur treatment can be adapted to some non finitely presented gats.\nIf we have an increasing sequence of signatures $\\Sigma_n$ we can consider their\nunion.\nFor instance, we can describe a gat of contextual cwfs \\cite{castellan:lambek} (similar to Cartmell's contextual categories and Voevodsky's $C$-systems) by\nthe following stratification of the theory of cwfs. We replace the sort $\\mathrm{ctx}$\nby a sequence of sorts $\\mathrm{ctx}_0,\\,\\mathrm{ctx}_1,\\,\\dots ,$ where $\\mathrm{ctx}_n$ represents the sort\nof contexts of length $n$ and a corresponding sequence of sorts\n$\\mathrm{ty}_n(\\Gamma)$ for $\\Gamma$ in $\\mathrm{ctx}_n$\nand $\\mathrm{tm}_n(\\Gamma,A)$ for $A$ in $\\mathrm{ty}_n(\\Gamma)$. Context extension $\\Gamma.A$ is now in $\\mathrm{ctx}_{n+1}$\nif $A$ is in $\\mathrm{ty}_n(\\Gamma)$ and so on.\nWe also add {\\em destructors}: we have\n$\\mathrm{ft}(\\Gamma)$ in $\\mathrm{ctx}_n$\nand $\\mathrm{st}(\\Gamma)$ in $\\mathrm{ty}_n(\\mathrm{ft}(\\Gamma))$\nwith $\\Gamma = \\mathrm{ft}(\\Gamma).\\mathrm{st}(\\Gamma)$.\nSimilarly we have a stratification of the sort of substitutions\n$\\mathrm{hom}_{n,m}(\\Delta,\\Gamma)$ for $\\Delta$ in $\\mathrm{ctx}_n$ and $\\Gamma$ in $\\mathrm{ctx}_m$.\nThe resulting models are {\\em internal contextual cwfs} in a cwf.\n\n\\begin{remark}\nGeneralized algebraic presentations of contextual categories (C-systems) have been given by Voevodsky \\cite{voevodsky:c-systems} and Cartmell \\cite{cartmell:gat-contextual}.\n\\end{remark}\n\n\n\n\\section{Related work}\n\nStreicher \\cite{streicher:semtt} defined {\\em doctrines of constructions} (contextual categories with suitable extra structure) as a notion of model of the Calculus of Constructions. He also constructed a term model and remarked that it is an initial object in a category of doctrines of constructions.\nRecently, Brunerie et al \\cite{brunerie:initiality} presented a formalized proof in the Agda system that a formal system for Martin-L\u00f6f type theory forms an initial object in a category of contextual categories with extra structure for\nthe type formers.\n\nMore generally, Voevodsky \\cite{voevodsky:initiality} outlined a new vision of the theory of syntax and semantics of dependent type theories. In this vision formal systems for dependent type theory are proved to be initial in suitable categories of models ({\\em the initiality conjecture}). The above-mentioned contributions by Streicher and Brunerie et al are two examples of such characterizations. However, Voevodsky's aim was to go further and characterize a whole class of type theories and prove a general initiality result for them with the aim to form the basis for a general metatheory of dependent type theory. Our work can be viewed as a contribution to Voevodsky's programme, since we prove an initiality theorem for the whole class of finitely presented generalized algebraic theories. A characterization of a more general class of dependent type theories and their initial models has been proposed by Uemura \\cite{uemura:general-framework}. Another related contribution is Palmgren and Vickers' \\cite{palmgrenvickers} construction of initial models of essentially algebraic theories.\n\nAltenkirch and Kaposi \\cite{altenkirch:qiits} gave several examples of {\\em quotient inductive-inductive types (qiits)}. Their main example is a definition of dependent type theory with $\\Pi$-types and a universe, as a simultaneous definition in the Agda system \\cite{agda-wiki} of the data types $\\mathrm{Ctx}$ of contexts, $\\mathrm{Sub}(\\Delta,\\Gamma)$ of substitutions, $\\mathrm{Ty}(\\Gamma)$, and $\\mathrm{Tm}(\\Gamma,A)$ of terms. Their definition is {\\em inductive-inductive} \\cite{nordvallforsberg:iids}, since the index sets of $\\mathrm{Sub}, \\mathrm{Ty},$ and $\\mathrm{Tm}$ are generated simultaneously, and as a consequence are not indexed inductive definitions in the usual sense where the index sets are fixed in advance. Furthermore, it is a quotient inductive-inductive type since they also have constructors for identity types, as in a {\\em higher inductive type}.\nThere is a close relationship between this qiit and our initial internal cwf with $\\Pi$-types and a universe. Like our definition, their qiit-definition uses cwf-combinators. Moreover, our sort symbols correspond to their formation rules (data type constructors), our operator symbols correspond to their introduction rules (constructors), and our equations correspond to their propositional identities. However, the fact that our equations are judgmental equalities while theirs are propositional identities is an important difference. As a consequence they need to use transport maps when moving between identical types. \n\n\nKaposi, Kov{\\'{a}}cs, and Altenkirch \\cite{kaposi:qiits} developed a general theory of qiits. This includes a notion of signature for a qiit, the notion of an algebra of such a signature, and a construction of initial algebras. For these constructions they work in cwfs with $\\Pi$-types, identity types, and a universe. This is in contrast to our work which is based on plain cwfs without extra structure for type formers. Although gats and qiits are related notions, neither is a generalization of the other. Gat is a basic notion independent of Martin-L\u00f6f type theory, whereas qiit is the latest in a series of generalizations of inductive type (inductive family, inductive-recursive type and family, inductive-inductive type, higher inductive type) extending intensional Martin-L\u00f6f type theory.\n\n\n\n\n\\subsection*{Acknowledgements} \nWe are grateful to the anonymous referees for constructive criticism and pointers to related work. We would also like to thank Andrej Bauer and John Cartmell for further feedback.\n\n\\input{BCDE-ArXiV.bbl}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{rf-spectroscopy protocol : Theoretical link between the spectral function $A(\\mc{E},\\bk)$ and the transfer rate $\\Gamma$.}\n\n\nThe aim of this section is to give the explicit derivation of Eq.(1) of the main text. We start with the general expression of the spectral function for non-interacting disordered systems. Second, we derive the expression of transfer rate following the Fermi golden rule and discuss the validity of the approach. Last, we discuss the effect of the inter-atomic interactions in the mean-field regime.\n\n\\subsection{Theoretical expression for $A(\\mc{E},\\bk)$.}\n\n\nThe spectral function for the energy $\\mc{E}_\\delta$ can be generally expressed as \\cite{bruus}:\n\n\\begin{equation}\n     A(\\mc{E}_\\delta,\\bk) = - \\frac{1}{\\pi}\\,\\mathrm{Im}\\, \\overline{G}(\\mc{E}_\\delta,\\mathbf{k})=  \\overline{\\bra{\\mathbf{k}}  \\delta ( \\mc{E}_\\delta - H )  \\ket{\\mathbf{k}}}\n \\label{Ak_def_general}\n\\end{equation}\nwhere $\\overline{\\dotsb}$ denotes the averaging over disorder realizations, ${G}$ is the retarded Green's function and $H=\\bs{p}^2\/2m +V(\\bs{r})$ is the single particle Hamiltonian of the system. If one introduces the eigenstates $\\ket{\\psi_\\alpha}$ of the Hamiltonian $H$ at energy $\\mathcal{E}_\\alpha$, Eq.~(\\ref{Ak_def_general}) can be rewritten as:  \n\\begin{equation}\n      A(\\mc{E}_\\delta,\\bk) =   \\overline{\\sum_\\alpha |\\langle \\mathbf{k} | \\psi_\\alpha \\rangle |^2\\ \\delta(\\mc{E}_\\delta-\\mc{E}_\\alpha)} \n \\label{Ak_sum_eigenstates}\n\\end{equation}\nFor a single realization of the disorder $V(\\bs{r})$, the overlap function $| \\langle \\mathbf{k} | \\psi_\\alpha \\rangle |^2$ exhibits large fluctuations with the energy  $\\mc{E}_\\alpha$. However, once disorder averaging is performed, the fluctuations vanish and the overlap function becomes a smooth function. One can then factor the averaged coupling $\\overline{ |\\langle \\mathbf{k} | \\psi_\\delta \\rangle |^2}$ at the energy $\\mc{E}_\\delta$ from the sum, yielding:\n\\begin{eqnarray}\n A(\\mc{E}_\\delta,\\bk) & \\sim & \\overline{ |\\langle \\mathbf{k} | \\psi_\\delta  \\rangle|^2} \\cdot \\overline{\\sum_\\alpha \\delta(\\mc{E}_\\delta-\\mc{E}_\\alpha)} \\nonumber \\\\\n& \\sim &  \\overline{ |\\langle \\mathbf{k} | \\psi_\\delta  \\rangle |^2}  \\cdot \\rho (\\mathcal{E}_\\delta)\\; , \\label{Eq_Ak_rho}\n\\end{eqnarray}\nwhere $  \\rho(\\mc{E})$ is the disorder averaged density of states.\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{figS1.pdf}\n\\caption{\\textbf{Measurement scheme of the spectral function}. The initial state is a BEC created in the disorder insensitive internal state $\\ket{1}$. Its spatial wavefunction is approximately $\\ket{\\mathbf{k}=0}$. A weak rf field drives the transition from the initial state to the internal final state $\\ket{2}$ that is sensitive to the disordered potential $V(\\bs{r})$, with a transfer rate $\\Gamma$. The spatial wavefunctions of the final states are denoted $\\psi_\\alpha$. These targeted states have an energy centered around the resonant condition $\\mathcal{E}_\\delta=\\hbar \\delta$, where $\\delta$ is the rf-detuning with respect to the bare transition frequency between states $\\ket{1}$ and $\\ket{2}$. The energy spread around $\\mathcal{E}_\\delta$ is Fourier limited by the duration $t$ of the coupling: $\\Delta \\mathcal{E}=\\hbar\/t$.   }\n\\label{figSupp_scheme}\n\\end{figure}\n\n\\subsection{Expression of the transfer rate $\\Gamma$}\n\n\nFor clarity, the rf-transfer scheme presented in the letter is reproduced here in figure~\\ref{figSupp_scheme}, with some additional details. We consider the simple case where the two internal states $\\ket{1}$ and $\\ket{2}$ are directly coupled by an oscillating rf magnetic field at frequency $\\omega_\\mathrm{rf} $. The two-photon rf transition used in the experiment (see Fig.~\\ref{fig:freeRes}) is described in the specific section ``Rf-spectroscopy protocol: Experimental details\" below. The magnetic dipolar interaction is $\\hat{W}= \\hat{W}_0 \\,e^{-i\\omega_\\mathrm{rf} t}\\, +\\, \\mathrm{c.c}$, where $\\hat{W}_0= \\hbar \\Omega \\, |2\\rangle\\langle1| $ is the static coupling and $\\Omega $ the Rabi frequency.\n\nA key feature is that  the energy spectrum of the targeted states in the disordered potential can be considered as a continuum, due to the finite energy resolution $\\Delta \\mc{E}=\\hbar\/t$, where $t$ is the duration of the rf pulse (see Fig.~\\ref{figSupp_scheme}). If the Rabi frequency $\\Omega$ is weak enough, the transfer rate can be expressed by the Fermi golden rule (see e.g.~Ref.~\\cite{grynberg2010introduction}): \n\\begin{equation}\n\\Gamma=\\frac{2\\pi}{\\hbar} \\sum_{f}|\\braket{f |  \\hat{W}_0 | i  }|^2\\;\\delta_\\mathrm{t}(E_f- E_i -\\hbar \\omega_\\mathrm{rf}).\\label{Eq_rof}\n\\end{equation}\nHere $\\ket{i}=\\ket{1}\\ket{\\psi_\\mathrm{BEC}}$ is the initial state of \\emph{ total} energy $E_i$ and $\\ket{f}=\\ket{2}\\ket{\\psi_\\alpha}$ refers to the final states of \\emph{ total }energy $E_f$. \nThe function $\\delta_t (E)=2\\hbar\\sin^2(Et\/2\\hbar)\/\\pi t E^2$ is a narrow function of width $\\Delta \\mc{E}=\\hbar\/t$ [see Fig.~(\\ref{figSupp_scheme})]. \nFor long durations, it behaves essentially as a Dirac function, with $\\lim_{t\\rightarrow+\\infty} \\delta_t(E) = \\delta(E)$. Since the coupling between the initial and final states writes $\\braket{f |  \\hat{W}_0 | i  }=\\hbar \\Omega \\cdot \\braket{\\psi_\\alpha | \\psi_\\mathrm{BEC}}$, Eq.~(\\ref{Eq_rof}) can be directly simplified as:\n\\begin{equation}\n\\Gamma(\\delta)=2\\pi \\hbar \\Omega^2 \\sum_{\\alpha}|\\braket{\\psi_\\alpha |\\psi_\\mathrm{BEC}  }|^2\\;\\delta_\\mathrm{t}(\\mc{E}_\\alpha-\\mc{E}_\\delta),\\label{Eq_rof_overlapp}\n\\end{equation}\nwhere we used the relation $E_f- E_i -\\hbar \\omega_\\mathrm{rf}=\\mc{E}_\\alpha-\\mc{E}_\\delta$, with $\\mc{E}_\\delta=\\hbar \\delta$ is the final \\emph{external} energy targeted by the rf frequency (see fig.~\\ref{figSupp_scheme}).\n\nIn order to make the link with the expression~(\\ref{Eq_Ak_rho}) of the spectral functions, we use two assumptions. First, we identify the BEC's wavefunction with a purely uniform one, i.e. $\\ket{\\psi_\\mathrm{BEC}}\\sim\\ket{\\mathbf{k}=0}$. This assumption is meaningful since the BEC coherence length is much longer than the correlation length $\\sigma$ of the disordered potential. \nSecond, we use an ``ergodic'' hypothesis, i.e. replace the averaging over states within the band of width $\\Delta \\mc{E}$ by an averaging over different realizations of the disordered potential.  \nAs for the ensemble averaging, the energy averaging smoothes the overlap function $|\\braket{\\psi_\\alpha | \\mathbf{k}=0 }|^2$, which becomes a slowly varying function at the energy scale given by the resolution. One can thus replace the discrete sum by an integral involving the averaged density of states $\\rho(\\mc{E})$~\\footnote{Note that additional disorder averaging could be performed in the experiment. However, the energy averaging for a single disorder realization already reduces fluctuations considerably, and no residual fluctuations were observed in the experiment.}. Altogether we obtain:\n \\begin{equation}\n\\Gamma(\\delta)=2\\pi \\hbar \\Omega^2   \\; \\overline{|\\braket{\\psi_\\delta | \\mathbf{k}=0  }|^2}\\; \\rho(\\mc{E}_\\delta) \\propto A(\\mc{E}_\\delta,\\bk=0) \\; .\\label{Eq_Gamma_Aek}\n\\end{equation}\n\n\n\n\n\nThe derivation of Eq.~(\\ref{Eq_Gamma_Aek}) relies on the assumption that the overlap function $\\overline{|\\braket{\\psi_\\delta |\\mathbf{k}=0 }|^2}$ is a slowly varying function of the energy. The characteristic energy scale  $\\hbar \\Delta_\\mathrm{w}$ of variation of this overlap function is the energy width of the continuum that is coupled to the initial state. In our specific case, it is nothing but the width of the spectral function $A(\\mc{E},\\bk=0).$\nThe generic case of an initial state coupled to a continuum of finite width has been studied in great details in the literature, see e.g.  Ref.~\\cite{grynberg2010introduction}. \nThe Fermi golden rule is valid provided that the Rabi frequency $ \\Omega $ remains much smaller than the continuum width, i.e.:\n\\begin{equation} \n\\Omega \\ll \\Delta_\\mathrm{w}\\; .\n\\label{Eq_crit_Rabi}\n\\end{equation} \nOtherwise the system cannot differentiate the continuum from a genuine discrete two level system, and Rabi oscillations take place. In all cases, the condition $\\Omega\\ll \\Delta_\\mathrm{w}$ has been carefully checked in the experiment. In particular we verified that the shape of the measured spectral function is not modified when $\\Omega$ is varied.\n \n Another parameter is the duration $t$ of the rf pulse. Here we choose $t$ long enough for the energy resolution $\\Delta \\mc{E}$ to be much smaller that the width of the spectral function, and short enough to deplete only a very small fraction of the initial state (typically a few percent). Altogether, we operate in the regime:\n \\begin{equation} \n \\hbar\/\\Delta_\\mathrm{w} \\ll t \\ll 1\/\\Gamma\n\\label{Eq_crit_duration}\n\\end{equation} \n\n\\subsection*{Raman \\emph{v}s rf coupling scheme}\n\nIn the protocol described in figure~\\ref{figSupp_scheme}, we measure the spectral functions at $\\mathbf{k}=0$ because the rf field carries a negligible momentum. The scheme can be adapted to measure the spectral function with a finite momentum $\\mathbf{k}$ by using a two-photon Raman transition instead (see e.g.~\\cite{Dao07,Dao09}). In that case, a net momentum of $\\mathbf{\\Delta k}=\\mathbf{k}_2-\\mathbf{k}_1$, where $\\mathbf{k}_{1,2}$ are the respective wave numbers of the Raman beams, is transferred to the atoms. By varying the angle between the two lasers, the momentum transfer can be tuned between $0$ and $2 k_\\mathrm{L}$, where $k_1\\sim k_2\\equiv k_\\mathrm{L}$.\nFormally, the coupling matrix element writes now as $\\braket{f | \\hat{W}_0 | i  }=\\hbar \\Omega_\\mathrm{eff} \\cdot \\braket{\\psi_\\alpha| e^{i \\mathbf{\\Delta k}\\cdot \\bs{r}} | \\psi_\\mathrm{BEC}}$, where $\\Omega_\\mathrm{eff}$ is the effective Rabi coupling. Taking once again $\\ket{\\psi_\\mathrm{BEC}}\\sim \\ket{\\mathbf{k}=0}$, one has $\\braket{f |  \\hat{W}_0 | i  }=\\hbar \\Omega_\\mathrm{eff} \\cdot \\braket{\\psi_\\alpha |\\mathbf{\\Delta k}}$ and $\\Gamma \\propto A(\\mathbf{\\Delta k},\\mc{E})$. \n\n\n\\subsection*{Effect of mean-field interactions and the harmonic trap}\n\n\n\nThe above derivation of Eq.~(\\ref{Eq_Gamma_Aek}) can be extended to interacting particles, provided that the interactions can be treated at the mean-field level (i.e. for dilute atomic samples). In that case, the interactions play the role of an effective potential that should be taken into account in the energy resonance condition $E_f=E_i+\\hbar \\omega_\\mathrm{rf}$.\n\nThe short range interactions depend on the atomic internal states $(i,j)$. They are fully characterized by the scattering lengths $a_{ij}$, or equivalently by the interaction parameters $g_{ij}=4\\pi\\hbar^2 a_{ij}\/m$. More precisely, the mean-field interaction energy for one particle immersed in the BEC writes $\\mc{E}_{i,\\mathrm{mf}}=g_{11} \\, n_\\mathrm{BEC}(\\bs{r})$ where $n_\\mathrm{BEC}(\\bs{r})$ is the BEC density. Once the particle has been transferred in state $\\ket{2}$, it experiences the mean-field $1-2$ interaction with the remaining atoms in the BEC, together with the $2-2$  with the atoms already transferred. For weak rf coupling, the atomic density in state $\\ket{2}$ is very low and the latter interaction can be neglected. Since $a_{11}\\simeq a_{12}$ for $^{87}$Rb atoms~\\cite{Egorov13}, the mean-field interaction energy is the same in states  $\\ket{1}$ and  $\\ket{2}$ and cancels out in the energy resonance condition. Similarly, the shallow harmonic trapping potential is equal in both states and also cancels out.\n\n\n \n\\section*{Rf-spectroscopy protocol: Experimental details}\n\nThe following subsections give experimental details on the BEC parameters, the implementation of the two-photon rf transition used in the experiments and the transfer energy resolution. \n\n\\subsection*{Preparation of the BEC}\n\n\nThe BEC is created starting from $^{87}$Rb atoms in the hyperfine state $\\ket{1}\\equiv\\ket{F=1,m_F=-1}$ that are confined in a crossed optical dipole trap formed by two orthogonal laser beams at a wavelength of $1064\\,\\mathrm{nm}$ (as described in Ref.~\\onlinecite{jendrzejewski2012b}). In addition, a magnetic field gradient $B'=30.46\\,\\mathrm{G}.\\mathrm{cm}^{-1}$ is applied in the vertical direction in order to counteract gravity. We use forced evaporation to obtain an almost pure and dilute BEC with around $2\\times 10^5$ atoms, the trapping frequencies being around $\\omega_\\mathrm{T}\/2\\pi\\sim10\\,\\textrm{Hz}$ in all directions. The BEC's chemical potential is $\\mu_\\mathrm{BEC}=g_{11}\\, n_\\mathrm{BEC}(\\bs{r}=0)\\sim h \\times 140$~Hz (see section ``Effect of mean-field atomic interaction\" above), and the Thomas-Fermi (TF) radius is $R_\\mathrm{TF}=\\sqrt{2\\mu_\\mathrm{BEC}\/ m \\omega_\\mathrm{T}^2}\\;\\sim18\\,\\mu\\textrm{m}$. \n\nThe BEC's size $R_\\mathrm{TF}$ is much larger than the typical correlation length $\\sigma$ of the laser speckle potential (see section `` Statistical properties : measurement of the spatial auto-correlation function\" below).\n\n\\begin{figure}\n\\includegraphics[width=0.99\\columnwidth]{figS2.pdf}\n\\caption{\\textbf{Two-photon transition scheme.} (a) Two-photon transition between the ``clock states'' $\\ket{1}$ and $\\ket{2}$ (via the intermediate state $\\ket{0}$) of the ground state hyperfine structure of $^{87}$Rb. The two-photon transition uses a microwave magnetic field at frequency $\\omega_\\mathrm{MW}$ and a rf magnetic field at frequency $\\omega_\\mathrm{rf}$.  (b) Two-photon resonance in the absence of disordered potential for a coupling duration $t=100$~ms. The resonance curve has a Fourier limited width $\\Delta \\mc{E}\/h\\sim 10$~Hz. }\n\\label{fig:freeRes}\n\\end{figure}\n\n\n\\subsection*{Two-photon rf transition}\n\n\nThe hyperfine ``clock states'' $\\ket{1}\\equiv\\ket{F=1,m_F=-1}$ and $\\ket{2}\\equiv\\ket{F=2,m_F=1}$ used in the experiment are shown in Fig.~\\ref{fig:freeRes}(a). We suppressed the broadening effect of magnetic field fluctuations by working at a bias field $B=3.23\\,\\textrm{G}$, for which the magnetic moments of $\\ket{1}$ and $\\ket{2}$ are equal \\cite{Harber02}.\n\n\nDue to selection rules (the angular momentum difference is $\\Delta m_F=2$), these two states can only be coupled using a two-photon rf transition via the intermediate state $\\ket{0}\\equiv\\ket{F=2,m_F=0}$. This ``two-photon'' transition is realized with (i) a microwave magnetic field at frequency $\\omega_{\\textrm{MW}}\/2\\pi \\sim 6831.9\\,\\textrm{MHz}$ that drives the $\\ket{1}\\rightarrow\\ket{0}$ transition and (ii)  a rf field at frequency $\\omega_{\\textrm{rf}}\/2\\pi\\sim 2.8\\,\\textrm{MHz}$ that drives the $\\ket{0}\\rightarrow \\ket{2} $ transition.\n\nThe single photon transition to the intermediate state $\\ket{0}$ was detuned by $\\delta_{\\mathrm{1ph}} \/ 2\\pi\\sim 0.5$~MHz. In that case, the entire system can be seen as an effective 2-level system, with an effective Rabi frequency $\\Omega= \\Omega_{\\mathrm{MW}}\\Omega_{\\mathrm{rf}}\/\\delta_{\\mathrm{1ph}}$, where $\\Omega_{\\mathrm{MW}}$ and $\\Omega_{\\mathrm{rf}}$ are the Rabi frequencies associated to each transitions.  $\\Omega$ was measured by observing two-photon Rabi-oscillations from $\\ket{1}$ to $\\ket{2}$ in the absence of disordered potential. In practice, we adjusted the coupling $\\Omega$ and the duration $t$ of the transfer for each disorder amplitude $V_0$ in order to fulfill the criteria~(\\ref{Eq_crit_Rabi}) and~(\\ref{Eq_crit_duration}). Consistently, the maximum number of transferred atoms stayed fairly small in these conditions (on the order of $5\\times10^3$ atoms, i.e.~a few percent of the atoms in the initial BEC). Note that the effective Rabi frequency ranged up at maximum to $\\Omega\/ 2\\pi\\sim 80$~Hz, i.e.~much lower than the single photon detuning $\\delta_{\\mathrm{1ph}}$.\n\n\n\\subsection*{Transfer resolution}\n\nThe transfer resolution of the two-photon rf transition results from the convolution of the resolution associated with the transfer time, i.e.  $  \\Delta \\mathcal{E}=\\hbar \/ t$. \nAdditional sources of broadening are estimated to be negligible:\n(i) The broadening due to the magnetic field gradient across the size of the condensate is estimated of the order of $1\\,\\textrm{Hz}$ (ii) The residual broadening due to difference of mean-field interaction between the excited $\\ket{2}$ and initial $\\ket{1}$ state is \nof the order of $(g_{12}-g_{11})n_\\mathrm{BEC}(\\bs{r})\\sim (1-a_{12}\/a_{11})\\;\\mu_\\mathrm{BEC}\/h$ (see section ``Effect of mean-field atomic interaction\" above). Since $a_{12}\/a_{11}\\approx0.976$ \\cite{Egorov13}, an upper bound is $\\sim 3$~Hz.\n\n Altogether, the transfer resolution is then Fourier limited for our experimental conditions. In practice, we adapted the resolution to the typical energy span of the spectral function for each disorder amplitude $V_0$, so that it does not affect the observed profile. More precisely, the duration was varied from $t=5$~ms ($\\Delta \\mc{E}\/h =200$~Hz for the strongest disorder amplitude $V_0\/h\\sim4$~kHz) to $t=100$~ms at maximum ($\\Delta \\mc{E}\/h =10$~Hz for the lowest disorder $V_0\/h\\sim60$~Hz). As an example, Fig.~\\ref{fig:freeRes} shows the two-photon resonance in the absence of disordered potential for the largest duration time, i.e. $t=100$~ms.\n\n\n\n\\vspace{3ex}\n\\section*{Laser speckle disordered potential: experimental characterization and numerical simulation}\n\n\\subsection*{State dependent disordered potential}\n\nIn order to obtain a state dependent disordered potential we used a laser speckle tuned close to the optical transition   $F=2 \\rightarrow F'=3$ (see Figs.~2 and 1(a) in the main article), with a detuning $\\Delta_\\mathrm{L}$ much smaller than the hyperfine splitting of the ground state separating the ``clock states'' $|\\Delta|\\ll\\Delta_{\\textrm{HFS}}$ (similar to the ``tune-in'' scheme as described by \\cite{leblanc07}).  \nWith this configuration, the selectiveness $V_{\\ket{2}}\/V_{\\ket{1}}$  of the disorder potential for $\\ket{2}$ is roughly given by $\\Delta_{\\textrm{HFS}}\/|\\Delta|\\approx 100$.\n\n\\begin{figure}\n\\includegraphics[width=0.99\\columnwidth]{figS3.pdf}\n\\caption{\\textbf{Amplitude of the disordered potential.} (a) Hyperfine structure of the excited state and detunings of the laser speckle ($\\Delta_\\mathrm{red}$ and $\\Delta_\\mathrm{blue}$ for the red-detuned and blue-detuned case respectively). (b) Disorder amplitude $V_0$ in state $\\ket{2}$ as a function of the speckle laser frequency around the $F=2 \\rightarrow F'=3$ resonance.}\n\\label{fig:excState}\n\\end{figure}\n\n\nBeing close to the resonance, the precise hyperfine structure of the excited state has to be taken into account, as shown in panel (a) of Fig.~\\ref{fig:excState} of the present supplemental material. Furthermore, the speckle laser was frequency offset-locked to a saturation spectroscopy stabilized laser, allowing us to control its detuning with an accuracy of $1\\,\\mathrm{MHz}$. The exact detuning for the attractive and repulsive disordered potentials was calibrated by performing two-photon rf-spectroscopy of atoms in homogeneous light field (instead of a speckle field) produced by the same laser. The absolute light shifts created by the red and blue detuned light were adjusted to be equal.\nThe experimentally determined values for $\\Delta_{\\textrm{blue}}=2\\pi\\times81\\,\\textrm{MHz}$ and $\\Delta_{\\textrm{red}}=-2\\pi\\times 73\\,\\textrm{MHz}$ are in agreement with an optical dipole potential model taking all excited hyperfine states into account [see Fig.~ \\ref{fig:excState}(b)].\n\nIn order to calibrate precisely the disorder amplitude $V_0$, we took benefit of the excellent agreement between experiments and numerics. The calibration was done by adjusting our measured spectral function with the numerically-computed spectral function at $V_0\/h=-121\\,\\textrm{Hz}$ (attractive laser speckle disorder). It leads to a calibration of the disorder amplitude with a 4$\\%$ precision. Note that this calibration corresponds to a global correction of about 14$\\%$ compared to an independent calibration based on photometric measurements. Such deviation is not surprising (it is comparable to the one found in~\\cite{semeghini2014}) since the light intensity field cannot be measured at the location of the atoms (experiments take place inside a vacuum chamber). Here we estimate that the uncertainty mainly originates from a slight mis-estimation of the spatial extension of the laser speckle field.\n\n\nIn practice, the laser speckle power was varied between $0.1$ and $10\\,\\mu\\textrm{W}$ to vary the disorder amplitude from $V_0\/h\\sim60$~Hz to $V_0\/h\\sim4$~kHz (see Fig.~2 of the main text). Despite the proximity to the resonance, no heating or losses due to inelastic scattering were observed at the short time scales considered in the experiments. \n\n\n\\subsection*{Statistical properties of the disordered potential: measurement of the spatial auto-correlation function}\n\nThe laser speckle is created by passing a laser beam through a diffusive plate.  As illustrated in Fig.~\\ref{fig:corr}(a), the incoming wave that illuminates the diffusive plate is converging around the position $d$ of the atoms, such that they experience the Fraunhofer's diffraction pattern of ground plate (i.e.~the so-called Fourier speckle configuration~\\cite{goodman2007}). The intensity profile of the illumination on the diffusive plate is a Gaussian shape, of waist $w$  (radius at $1\/e^2$), truncated by a circular diaphragm of diameter $D$. The estimated geometrical parameters are: $D=20.3(1)$~mm, $w=9(1)$~mm and $d=15.2(5) $~mm. The diaphragm sets the maximal numerical aperture to $\\mathrm{NA}=\\sin(\\theta_\\mathrm{max})=0.55(2)$.\n\n\n\n\n\n\n In order to characterize the laser speckle field, the random intensity pattern was recorded at the position of the atoms with an high-resolution optical microscope, see Fig.~\\ref{fig:corr}(b). The measurement was done \\emph{ex-situ}, i.e. outside the vacuum chamber, but with an exact replica of the geometrical configuration. The extracted two-point correlation functions in the transverse ($z$ direction, same along $y$) and longitudinal directions ($x$ direction) are shown as blue squares in Fig.~\\ref{fig:corr}(c), together with the half-width-at-half-maximum (HWHM) lengths. They correspond to $\\mathrm{HWHM}_\\perp \\approx 0.42(1)\\, \\mu\\mathrm{m}$ and $\\mathrm{HWHM}_\\parallel \\approx 2.05(5)\\,\\mu\\mathrm{m}$. More details about the experimental configuration and the laser speckle characterization can be found in Ref.~\\onlinecite{richard15}.\n\nFor consistency with previous theoretical work~\\cite{pasek2016anderson}, we define the correlation lengths of the speckle potential as $\\sigma_{\\parallel,\\perp} = \\mathrm{HWHM}_{\\parallel,\\perp} \/ 1.39156$,\nwhich yields $\\sigma_\\perp \\approx 0.306\\, \\mu\\mathrm{m}$ and $\\sigma_\\parallel \\approx 1.45\\, \\mu\\mathrm{m}$. These values are used for the calculation of the correlation energy $\\mathcal{E}_\\sigma = \\hbar^2\/m(\\sigma_\\perp^2 \\sigma_\\parallel)^{2\/3}\\sim441~\\mathrm{Hz} $ (see main text).\n\n\\bigskip\n\n\n\n\n\n\\subsection*{Numerical simulation of the laser speckle}\n\nWe designed the numerical disorder to reproduce the experimental geometry represented in Fig.~\\ref{fig:corr}(a). In particular, the numerical speckle generation takes into account the presence of a diaphragm, and hence deviates from the usual pure Gaussian and Lorentzian character (see e.g.~\\cite{goodman2007}) for the two-point correlation function of the intensity in the transverse and longitudinal directions [see Fig.~\\ref{fig:corr}(c)]. \n\\begin{widetext}\nTo do so, the generation of the numerical disorder was done through the use of a phase mask:\n\\begin{equation}\\label{supsingle}\n\\mathcal{P} (\\mathbf k) =\\delta(|\\mathbf{k}|-k_L) \\exp\\left[-\\frac{(\\tan\\theta)^2}{(w\/d)^2}\\right]  \\Theta \\left[  \\frac{D}{2 d} - \\tan|\\theta|   \\right],\n\\end{equation}\nwhere $k_\\mathrm{L}$ is the wavenumber of the monochromatic laser beam, $\\theta \\in [-\\pi\/2,\\pi\/2]$ the angle between the wavevector $\\mathbf{k}$ and the beam axis, $w$ the beam waist, $D$ the diaphragm diameter and $d$ the position where the illumination field converges.\n\\end{widetext}\nThe numerical potential was then generated starting from a local random complex field $E_{u}(\\mathbf r)$, whose real and imaginary parts are uncorrelated Gaussian random variables.\nThe convolution of this random field with the phase mask in Eq.~(\\ref{supsingle}) yields the proper statistical distribution of the amplitude of the electric field~\\cite{goodman2007}\n\\begin{equation}\\label{sup2}\nE(\\mathbf r)= \\sum_{\\mathbf k}E_u(\\mathbf k) \\mathcal{P} (\\mathbf k) e^{i \\mathbf k \\cdot \\mathbf r},\n\\end{equation}\nwhich reproduces the coherent superposition of the various plane waves scattered by the diffusive plate. \nParameters of the numerical phase mask, Eq.~(\\ref{supsingle}), that yield the best agreement with measured two-point correlation functions of the speckle intensity, see Fig.~\\ref{fig:corr}(b), are $D = 20.4\\,\\mathrm{mm}$, $d = 15.2\\,\\mathrm{mm}$ and $w = 9.9\\,\\mathrm{mm}$, in excellent agreement with the experimentally-inferred values. \n\n\\begin{figure}[b!]\n\\includegraphics[width=1\\columnwidth]{figS4.pdf}\n\\caption{\\textbf{Laser speckle characterization.} (a) Schematic representation of the experimental configuration used for the speckle generation, with $\\mathrm{NA}=\\sin(\\theta_\\mathrm{max})=0.55(2)$ (see text). (b) 3D representation of the experimental laser speckle field recorded with an high aperture microscope at the position of the atoms.  (c) Transverse  and longitudinal correlation functions of the laser speckle, respectively along the $z$ (same along $y$) and $x$ directions. Blue squares: experimental autocorrelation functions extracted from the measurement shown in (b). Red solid lines: autocorrelation function for numerically simulated laser speckle [following Eqs.~(\\ref{supsingle}) and~(\\ref{sup2})]. Horizontal black arrows indicates the half-maxima $\\mathrm{HWHM}_\\perp=0.42(1)$~$\\mu$m and $\\mathrm{HWHM}_\\parallel=2.05(5)$~$\\mu$m of the correlation function.}\n\\label{fig:corr}\n\\end{figure}\n\n\n\n\\section*{Numerical computation of the spectral function} \n\n\n\\subsection*{Method}\n\nWe start with the definition, Eq.~(\\ref{Ak_def_general}), of the spectral function $A(\\mathcal{E},\\mathbf{k})  =  \\overline{\\bra{\\mathbf{k}}  \\delta ( \\mathcal{E} - H ) |\\mathbf k \\rangle} =  - \\frac{1}{\\pi}\\,\\mathrm{Im}\\, \\overline{G}(\\mc{E}_\\delta,\\mathbf{k})$. The spectral function can be evaluated numerically in many ways. The most common in the literature starts from the spectral, so-called Lehmann, representation of the Green's function (as used in, e.g., Ref.~\\onlinecite{semeghini2014}). We choose to use instead the temporal representation of the Green's function:\n\\begin{equation}\n\\label{Ak_temp}\nA(\\mathcal{E},\\mathbf{k}) = \\frac{1}{\\pi\\hbar}\\,\\mathrm{Re}\\,\\int_0^{\\infty} \\overline{\\langle \\mathbf{k}|\\mathrm{e}^{-\\mathrm{i} H t\/\\hbar}|\\mathbf k \\rangle}  \\mathrm{e}^{\\mathrm{i} \\mathcal{E}t\/\\hbar}\\ \\mathrm{d} t.\n\\end{equation}\nThe numerical computation of the spectral function $A(\\mathcal{E},\\mathbf{k})$ thus amounts to: (1) Computing the overlap between an initial plane-wave excitation $\\ket{\\mathbf{k}}$ \nand its time-evolution under the (disordered) Hamiltonian $H$; (2) Averaging this overlap over multiple disorder configurations (typically between 2000 and 4000 realizations depending on the disorder amplitude); (3) Using the Fourier transform to go from time- to energy-representation.\n\n\nFor the time-propagation algorithm, we use an iterative scheme based on the expansion of the evolution operator $\\exp (-i H\\Delta t \/\\hbar)$ over a time-step $\\Delta t$ in a series of Chebyshev polynomials of the Hamiltonian \\cite{roche1997,fehske2009,trappe2015}.\nIn order to check the numerical accuracy of our results, we verify that they satisfy the first three sum rules of the spectral function, namely \\cite{trappe2015}:\n\\begin{align}\n\\int \\mathrm{d} \\mathcal{E}\\, A_\\mathbf{k} (\\mathcal{E}) &= 1,\\\\\n\\int \\mathrm{d} \\mathcal{E}\\, \\mathcal{E} A_\\mathbf{k} (\\mathcal{E}) &= \\frac{\\hbar^2\\mathbf{k}^2}{2m} + \\overline{V},\\\\\n\\int \\mathrm{d} \\mathcal{E}\\, \\mathcal{E}^2 A_\\mathbf{k} (\\mathcal{E}) &= \\left( \\frac{\\hbar^2\\mathbf{k}^2}{2m} + \\overline{V} \\right)^2 + \\overline{\\delta V^2},\n\\end{align}\nwhere $\\overline{\\delta V^2}=V_0^2$ is the variance of the speckle potential distribution.\n\n\n\n\n\n\n\n\\subsection*{Effect of the residual disorder in state $\\ket{1}$}\n\nIn the experiment, the residual disorder on state $\\ket{1}$ is of the order of $0.01|V_0|$ (see section ``State dependent disordered potential'' above). \nFor the largest value of $|V_0|$ ($\\approx h\\times 4\\,\\mathrm{kHz}$), it results in a potential around 40 Hz on the state $\\ket{1}$ that perturbs the initial BEC (of chemical potential $\\mu_\\mathrm{BEC}\/h\\sim 140 $~Hz), and consequently the transfer $|1\\rangle \\to |2\\rangle$. This effect can be approximately taken into account in the numerical calculation of the spectral function.\nTo do so, we performed a preliminary propagation of the initial $\\ket{\\mathbf{k}=0}$ state in the presence of the same disorder realization but with residual amplitude $V_{\\ket{1}}=-0.01|V_0|$ (which is always blue-detuned since $\\Delta_\\mathrm{HFS}\\gg\\Delta_\\mathrm{L}$, see Fig.~1 of the main text), before performing the time-evolution under the disordered Hamiltonian for atoms in state $\\ket{2}$ [i.e. step (1) in the above-described numerical procedure]. \nThe duration of this preliminary propagation was taken to half the experimental transfer time, though different durations had little influence on the resulting spectral function. This is because the product $|V_{\\ket{1}}|t\/h$ is at maximum of the order of $0.2,$ so that the state $\\ket{1}$ is only slightly distorted by the residual disorder over the duration of the experiment.  The resulting spectral functions are displayed as solid brown lines in Panels (I.f) and (II.f) of Fig.~2 of the main text.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect:intro}\n\nIn \\citet{paper1}, henceforth Paper~I, we described the design and\nmaintenance of the Virtual Observatory (VO) Registry as a distributed\ninformation system.  Conceptually, it is a collection of, by now, about\n15000 registry records.  To give the Registry's users --\nastronomers, the library\ncommunity, or even the general public -- access to this collection,\nfacilities have to be provided that allow focused queries against it.\nThis includes common bibliographic constraints (by author,\ntitle or abstract term, year, etc), but also constraints specific to a\nregistry mainly concerned with\ndata services (e.g., supported protocols or query parameters,\nmetadata of published tables).  In the design of such facilities,\nseveral challenges have to be addressed:\n\n\\begin{enumerate}\n\\item different users have very different expectations and requirements\n\\item the underlying data collection (i.e., the set of registry records)\nis changing over time\n\\item the underlying data structure is fairly complex, and evolves\nitself as new standards and techniques are introduced in the VO\n\\item as many uses require only a small subset of the types of metadata\ncontained, partial resource descriptions should be retrievable\n\\item the total data set cannot efficiently be transferred to clients as\na whole\n\\item registry records are frequently authored by persons not entirely\nfamiliar with the data model, resulting in inconsistent quality\n\\end{enumerate}\n\nIn consequence, no single \\emph{user} interface to the Registry can be\nsufficient.  Instead, the VO community designed \\emph{client}\ninterfaces,\ni.e., network endpoints with rigorously defined behavior and\nsemantics, designed for use by\nprograms that then present the actual user interfaces to Registry data.\n\nWe will begin this paper with a brief review of the various client\ninterfaces that are or were used in the VO (section~\\ref{sect:history}).\nIn section~\\ref{sect:clients}, we proceed to describe the use some\nselected clients make of these facilities and the ways they apply and\nexpose information obtained from the registry.  A major part of the\npaper, section~\\ref{sect:regtap}, is devoted to a thorough discussion of\nthe Registry Relational Model (RegTAP for short), one of the two\nregistry interfaces currently being developed and deployed in response to the\ndeficiencies of previous standards.  In section~\\ref{sect:fulltext}, the\nother new-generation interface is described. \n\nWhile laying out some common use cases of Registry data\nin section~\\ref{sect:commonqueries} we also point out\ncommon query patterns.\nSection~\\ref{sect:openissues} concludes with some speculation about\nprobable future developments.\n\nIn the following, we refer to common Registry standard texts by their\nabbreviated names as introduced in Paper~I, and again the capitalized\nword ``Registry'' refers to the abstract concept, while concrete\nservices are written in lower case (e.g., a ``publishing registry'').\nConcepts from VOResource and its extensions are written in\n\\vorent{small caps}.\n\n\\section{History}\n\\label{sect:history}\n\nAlthough only explicitly written down in 2011, \nthe use cases collected on the IVOA\nwiki \\citep{wiki:regusecase} outline some of the challenges faced by the\ndesigners of the first client interfaces to the registry in the\nmid-2000s -- finding tables containing columns with certain physics,\nlocating services implementing certain protocols, and the like.\n\nWhile on the maintenance side of the registry the ecosystem\naround OAI-PMH \\citep{std:OAIPMH} \nprovided guidance for many technology choices,  \nin developing the client interfaces much more new ground had to be broken.\nFor instance, the OPACs (Online Public Access Catalogs; see\n\\citet{kani2008userperc} for a treatment from about the time of RI1\ndesign)\nestablished in the library community, while comparable for the purpose of\nlocating information resources,\ncould not efficiently address the use cases, and no\nbroadly accepted standard for client, rather than user, interfaces to\nOPACs, lent itself to adoption by the VO community.\n\nGiven that the interface to be designed was expected to be expressive\nenough for requests of the type ``find all TAP services exposing a table\nhaving some word in the description and a column with a given \nUCD\\footnote{Unified Content Descriptors or\nUCDs in the VO denote phyiscal concepts like ``angular distance'' or\n``radio flux'' in a simple formal language \\citep{std:UCD}}'',\nit was determined fairly early on that an interface based on\nsimple, atomic parameters would not be sufficent, and\nRegistry information \ncrucial to certain\ndiscovery tasks would not be queryable through it. \nClient interfaces making explicit too much of the\nunderlying data model would also unduly restrict future developments of\nthat data model. Thus, at least one\ninterface to the Registry would have to support a full query\nlanguage.  Since the Registry data model was defined in XML Schema, \nan obvious choice for the query language was XQuery\n\\citep{std:XQUERY}, a language that essentially extends SQL concepts to\nquerying XML trees.\n\nHowever, factors against the adoption of XQuery included:\n\\begin{itemize}\n\\item the heavy use VOResource makes of XML\nnamespaces, which tended to make queries hard to write by hand; \n\\item the much\nlarger installed base of relational databases compared to XQuery-capable\nengines (compounded by the fact that translating XQuery to a given\nrelational schema is hard);\n\\item the desire to open up the full registry data\nmodel to queries written by end users, i.e., astronomers.\nAs it was expected that many\nof these would familiarize themselves with the VO's SQL dialect ADQL\n(Astronomical Data Query Language; \\citet{std:ADQL}), \nrequiring yet another query language\nfor Registry access appeared undesirable.\n\\end{itemize}\n\nWith these considerations, it was decided to base the primary\nRegistry interface on conventional relational technology.\n\nWhile the complex queries XQuery and ADQL allow were needed for\nidentified use cases, it was also acknowledged that ``Google-like''\nsearches -- more or less loose matching of words in documents modelled\nas bags of words -- was the dominant mode of searching for resources\noutside of the VO in the targeted\nuser base.  At least if common ``comfort'' features like stemming or\nphrase searches are desired, this type of search is hard or\nimpossible to simulate through plain ADQL given its very basic set of\ntext search capabilities.  Therefore, a keyword search\noperation with significant freedom for implementors was also defined.\n\nThe result of these considerations was section 2 of RI1\n\\citep{std:RI1}.\nIt defines two required search operations\n\\textit{Search} (with constraints in ADQL) and \\textit{KeywordSearch}\n(with operator-defined matching of keywords against an\noperator-ex\\-ten\\-sible minimal set of fields) as well as an optional\n\\textit{XQuerySearch} operation.  All search operations return\neither identifier lists or sequences of full resource records in OAI-PMH\nstyle.  In addition, two OAI-PMH-like operations were defined,\n\\textit{GetResource} to obtain a resource record from an identifier, and\n\\textit{GetIdentity} to discover metadata about the registry service\nitself.\n\nSeveral implementations of the standards are available; services are\nprovided by STScI, ESA, and AstroGrid.\n\nAs the RI1 design significantly predates the final standardizations of both\nADQL \\citep{std:ADQL} and the transport protocol for queries and results\n-- that was eventually defined in the TAP standard \\citep{std:TAP} --, \nRI1 further defined an ad-hoc transport based on the RPC\nmechanism SOAP, and it adopted ADQL at a time when experiments were\nunderway with passing ADQL statements to client interfaces\nin parsed (XML) form.  In\nconsequence, modern TAP clients cannot use registry endpoints, and\nwriting queries in the aging XML serialization of ADQL became at least\ndifficult as software components translating SQL expressions into the\nXML forms went unmaintained.\n\nFurther critique came from implementor feedback\n\\citep[e.g.,][]{talk:topcatri1} and was collected together with the use\ncases \\citep{wiki:regusecase}.  For instance, in practice the\nuse of a restricted set of XPath to specify constraints instead of\ndefining an actual relational schema\nlead to severe interoperability problems between\ndifferent registries, which were further exacerbated by not specifying\nrules for case folding.  The apparent flexibility towards registry\nextensions provided by the XPath-based column references also did not\npay off as originally expected since registries still needed to do\ninternal mapping as registry extensions were developed.  In contrast\nto the (optional) XQuery interfaces, the (mandatory) ADQL interfaces\nfrequently lagged behind standards deployment.\n\nIn this situation, the most advanced Registry clients relied on the\noptional XQuery\ninterface or even used entirely proprietary interfaces.\n\nAs TAP services entered the registry in the early 2010s, RI1's response\nformat also became a liability.  Registry records contain table\nmetadata, and with TAP services exposing many tables, resource records \nof several megabytes are not exceptional.  \nThis made relatively common\nqueries like ``Retrieve basic metadata on all TAP services'' expensive\nin terms of transfer time and processing required.\n\nTherefore, starting in 2011, it was decided to design a new\nRegistry interface, dubbed ``RESTful'' to contrast it from the RI1\nSOAP-based protocol.  \nWith TAP and ADQL now available, a replacement of the RI1\n\\textit{Search} operation was mainly a matter of designing a schema and\na mapping to this schema from VOResource. This can be seen as creating a\nsecond serialization of an abstract\ndata model implicit in VOResource's XML schema files.\n\nThe combination of a defined\nschema and a TAP service had a model in ObsCore \\citep{std:OBSCORE}.\nThe resulting new standard (``RegTAP''), \ndiscussed in section~\\ref{sect:regtap},\nis in the last phases of IVOA peer review as this article\nis written.\n\nA replacement for the \\textit{KeywordSearch} operation is also being\ndeveloped.  Here, the wide availability of feature-rich fulltext engines\nsuch as Apache Lucene\noffers the possibility of enriching the bag-of-words model\nand allows some advanced operators as well.  We will revisit\nthis development in section~\\ref{sect:fulltext}.\n\n\n\\section{Registry Use in Clients}\n\\label{sect:clients}\n\nMany VO clients integrate Registry access, frequently without \nadvertising the actual source of the data.  Depending on the scope of\nthe application, different parts of Registry metadata are used, and\ndifferent presentations of this information appear appropriate.  In the\nfollowing, we look at Registry usage in a number of, we believe,\nrepresentative applications, concluding with an in-depth look at\nTOPCAT's use of the registry.\n\n\\begin{figure}[thm]\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{taphandle.png}\n\\end{center}\n\\caption{TAPHandle uses registry information to provide \ninput completion for TAP service access URLs, where the completion items\nare complemented by additional metadata.}\n\\label{fig:TAPHandle}\n\\end{figure}\n\nTAPHandle\\footnote{Online at \\texttt{http:\/\/saada.unistra.fr\/taphandle}.}\nis a TAP client operated through web browsers\n\\citep{2014ASPC..485...15M}.  It uses the Registry\nto discover all registered TAP services.  With this information, it can\nprovide input completion in the selector for the TAP service queried\n(Fig.~\\ref{fig:TAPHandle}), thus facilitating simple discovery tasks\n(``I want to query the CADC TAP server'').  As it is a TAP client, \nis is natural for TAPHandle \nto use RegTAP as its Registry interface. Indeed, its use case is one\nof the standard\ntasks identified in the collection of requirements for a\nrevised Registry interface \\citep{wiki:regusecase}.\nIt uses a hard-wired RegTAP endpoint, performing essentially a\nsingle query per session within its\nserver component. Thus, TAPHandle users are isolated from technical\ndetails of registry access and are also not exposed to visible registry\nqueries.\n\nSimilarly, the spectral analysis tool VOSpec \\citep{2005ASPC..347..198O}\nqueries the registry for all services implementing specific standards\n(spectral and line access, in this case),\nbut since in contrast to TAPHandle\nit has no server-side component, it does so directly\nfrom the user's client, using one of two built-in registry endpoints\nimplementing the RI1 \\textit{Search} operation.  Data extraction from\nthe registry records retrieved is\nperformed with an\nXSLT stylesheet. The discovered resources are\npresented to the user in a tree view for individual selection or\nde-selection.  This UI is employed both in the selection of the spectral\nservices and in the selection of servers providing information on the\nlocation of spectral lines.\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{splat.png}\n\\end{center}\n\\caption{SPLAT's rendering of the metadata of spectral services in the\nVO: Various metadata obtained from the registry records are made\nselectable through checkboxes.}\n\\label{fig:SPLAT}\n\\end{figure}\n\nAnother VO-enabled spectral tool, SPLAT \\citep{2014ascl.soft02008C}, takes this\napproach somewhat further by exposing \\vorent{dataSource} (from\nSimpleDALRegExt, allowing, for instance, the separation of theoretical and\nobservational services) and \\vorent{waveband} from VODataService via\ncheckboxes in its UI (Fig.~\\ref{fig:SPLAT}).  The UI shown is built from\na simple query for all services implementing SSAP. SPLAT furthermore\nallows users to add, as it were, private registry records (e.g., for\nunpublished services) that are then integrated into this interface.  \n\nA drawback of hiding actual registry queries in this way is that\nmetadata quality of the resource records directly influences the user\nexperience for the application itself.  For instance, when a resource\nrecord author neglected to give correct waveband metadata, users knowing\na certain resource serves optical spectra were frequently confused when\nthe service was deselected after restricting queries to optical data.\n\nThe VO client Aladin \\citep{soft:Aladin} supporting the major VO\nprotocols could build upon a registry-like system called GLU that\npredates the definition of the VO Registry \\citep{2003ASPC..295...43F}.\nGLU, with automatic mirror selection and an Aladin-customized\nmetadata format, to this day distributes Registry\ninformation to Aladin.  Registry records enter GLU not through a client\ninterface but rather by harvesting an OAI-PMH endpoint.  The operators\nof the GLU system at CDS perform additional curation, e.g., by removing invalid\nrecords or records for known-defective services.  By removing all\nresource metadata not immediately relevant to the client,\nAladin can keep, in effect, a local cache of the entire\nGLU content, which is impractical for\nthe actual Registry content, as that would currently\nentail managing and updating\nseveral hundreds of megabytes.\nResponsiveness is further enhanced by persisting this data\nbetween executions of Aladin.\n\nThere are also clients specifically built around the Registry.  One of the most\nadvanced to date is VOExplorer \\citep{2008ASPC..394..159T}, developed by\nthe UK's VO project\nAstroGrid in the late 2000s as part of its VODesktop suite.\nIn its user interface it guides users\nin the construction of constraints, mixing menu-based selection with\nfree-text queries as appropriate.\nVOExplorer communicates with registries via the XQuery client interface. \nBy thus retrieving only parts of the full VOResource record it\nsignificantly reduces network traffic compared to a RI1 \\emph{Search}\nclient. When a full VOResource record is required, it is cached \nfor use in future query results.\n\nThough the major discovery protocols defined while VODesktop was still\nbeing developed are supported, the application's focus is clearly the\nintegration of Registry data into a workflow, and it offloads\nvisualization to specialized clients by use of the SAMP\ninter-application communication protocol \\citep{std:SAMP}.  Regrettably,\nVODesktop's development ceased in 2009, with the demise of\nthe AstroGrid project.\n\nTo replace the comprehensive graphical Registry UI provided by\nVODesktop,\nWIRR\\footnote{online at \\texttt{http:\/\/dc.g-vo.org\/WIRR}.} \nwas developed.  It is essentially a\nbrowser-based query builder for RegTAP, where, much like in VODesktop,\nthe user can successively\nadd constraints on the search results.  Notable constraint types include\nqueries for\nresources containing columns with specific UCDs, ``inverted queries''\nto obtain registry information from a service's access URL -- which\nis useful for finding contact information when services fail --, or query\nwith regular expressions on IVORNs.  Even more than VODesktop,\nWIRR\nrelies on external applications to use the resources found, employing\nSAMP messages for transmitting resource lists.  TOPCAT is one application\nthat already supports these.\n\nTo support Registry use from within custom user programs, libraries have\nbeen written that encapsulate details of registry access.  Given the\nwidespread adoption of Astropy \\citep{2013A&A...558A..33A}, we note here\nthe registry functions within the Astropy affiliated\npackage PyVO \\citep{2014ascl.soft02004G}.  For Registry access, it\ncontains a single function \\texttt{regsearch} that supports constraints\nby keywords (essentially, a full-text search\nwithin resource record text fields), service types (e.g., image or\nspectral service) and wavebands.  The function also has a parameter to\npass in custom SQL fragments executed within the VAO's registry.  Due to\nthe limitations of RI1 standard client protocols, a custom,\nVOTable-based interface is employed at the moment, with a change\nto RegTAP in the back-end planned.\n\nFinally, there are uses of Registry client interfaces not directly\nconnected to actual VO clients.  As an example we mention VO\nFresh\\footnote{Online at http:\/\/dc.g-vo.org\/regrss},\nan RSS feed of metadata for services newly\npublished or updated\nin the Registry; new resources are also announced through microblogging\nservices.  VO Fresh\ninitially obtained registry information from a full registry's OAI-PMH\nendpoint but moved to obtaining registry information through RegTAP as\nthat became available.\n\n\\subsection{Case Study: TOPCAT}\n\nTOPCAT \\citep{2005ASPC..347...29T}\nis a tool for analysis of astronomical tables.\nPart of its function is to provide a user-friendly GUI for\nacquiring tabular data from Virtual Observatory services,\nmost importantly\nTAP \\citep{std:TAP} and Cone Search \\citep{std:SCS},\nbut also SIA \\citep{std:SIAP} and SSA \\citep{std:SSAP}.\nTo achieve this, it needs the Registry to locate\nservices with the relevant capabilities and to allow the user\nto assess their suitability for the science job at hand.\n\nFrom a user point of view, TOPCAT's registry interaction consists\nof selecting a particular type of data service,\noptionally supplying some keywords to match against one or\nmore of a handful of fixed resource metadata fields,\nand dispatching a search which results in\npresentation of basic metadata for each matching service.\nThe user then peruses this list and selects one of the returned\nservices for subsequent use in the application.\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{topcat.png}\n\\end{center}\n\n\\caption{TOPCAT's Registry interface: \nthe user specifies a query using\na selector for registry service endpoint\nand interface protocol (RegTAP or RI1),\na keyword text field,\nand a set of checkboxes for what resource fields to match against.\nAn additional constraint is the service type,\nwhch depends on the context these widgets are shown in.\nBelow the input widgets is a listing of matched resources\nfrom which one may be selected.\n``Accept Resource List'' allows filling the\nresource selector from SAMP messages.}\n\\label{fig:topcat}\n\\end{figure}\n\nTOPCAT makes only a single type of registry query to support\nthis functionality, the user interface to which is illustra\\-ted in\nFig.~\\ref{fig:topcat}: \nlocate all registry resources which offer\na fixed standard capability (e.g.\\ TAP) and which\nsatisfy zero or more additional user search constraints\n(e.g.\\ ``Title contains the term UKIDSS''),\nand for each one return a small fixed amount of metadata\n(ID, Title, Publisher, Access URL and a few others).\nThere is other information stored in the Registry records\nthat TOPCAT may require, such as {\\tt vs:CatalogService}\nrecords describing table and column metadata.\nHowever, for newer VO protocols such information is also available from the\nregistered data services themselves, and TOPCAT prefers to acquire\nit from the latter source, since it may be more reliable and\nis also available for unregistered services.\n\nImplementing these queries in RI1 presented some difficulties.\nThe {\\em KeywordSearch\\\/} operation is unsuitable since keyword\nsearches cannot be combined with restrictions on service type.\nThe {\\em XQuerySearch\\\/} operation offers suitable functionality,\nbut being an optional part of the standard it is only available\nfrom a subset of registry services\n(in fact, only the AstroGrid implementation),\nand so would\nhave restricted the choice of registries with which the tool could interact.\nThe only remaining option is the {\\em Search\\\/} operation.\nSyntax and semantics of the fields to match in the required\nADQL queries were somewhat under-documented, and\nthere is a problem with the way case sensitivity is defined,\nbut the most serious issue is that {\\em Search\\\/}\nalways returns the whole, perhaps large, record\nfor each matched resource, the bulk of which is not needed.\nPatchy service implementation quality also contributed to make\nRI1-based registry interaction generally slow and unreliable.\n\nWith the introduction of RegTAP,\nregistry interaction is much improved.\nThe user interface is almost unchanged, but queries are\nmore precise,\nthanks to more careful mapping of the RM data model\ninto its relational counterpart,\nand much faster,\nsince it is possible to restrict the query response to\nitems of interest only.\nThis latter point can lead to a reduction of two orders of magnitude\nin the required data transfer.\nTo give an admittedly drastic -- but in practice not uncommon --\nexample, the response size for a query\nfor all TAP services registered in May 2014\nwent down to about $150\\,\\rm kB$ from previously roughly\n$25\\,\\rm MB$.\n\nNote that although for both RI1 and RegTAP the client uses an essentially\nSQL-like language to select resources,\nwhat the user sees is a keyword-based or ``Google-like'' interface.\nMapping from the latter to the former can result in verbose\nquery text, but this text is not difficult for client code to generate.\nTherefore, for this purpose there has been no requirement for an\nessentially keyword-like client interface to the registry.\n\nThere is scope for richer interaction with the registry from TOPCAT,\nfor instance queries on fine-grained metadata (column UCDs)\nor more detailed display of descriptive or curation metadata\nfrom selected records.\nThese options may be explored in the future.\n\n\\section{The Registry Relational Model}\n\\label{sect:regtap}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{schema.pdffig}\n\\end{center}\n\\caption{A sketch of the database schema of the relational registry,\nadapted from \\citet{adasspaper}.\nThe arrows indicate foreign key relationships, the ``attributes''\nenumerate the fields most likely interesting to clients or scientists\nwriting queries.  When joining through\n\\rtent{relationship} (red) in the discovery of data collection access services\n(cf.~section~\\ref{sect:relationship}), the green tables would be\n``resource-bound'', the blue ones ``capability-bound'', whereas the\nyellow ones might be either resource- or capability-bound depending on\nquery semantics.}\n\\label{fig:schema}\n\\end{figure*}\n\nThe Registry Relational Model -- briefly called RegTAP for mainly\ntechnical reasons -- is the successor to the \\textit{Search} method in\nRI1.  It essentially defines a relational schema and rules to map\nVOResource records into this schema.  Using TAP as an access\nprotocol and ADQL as the query language, this is enough to completely\ndefine a client interface to the registry.\n\nA sketch of this relational schema is given in Fig.~\\ref{fig:schema}.\nAlthough the authors first experimented with alternative structures that would\nhave been derived from VOResource algorithmically \\citep{wiki:vormap},\nit turned out design considerations did not lend themselves to\nformalization, as discussed in the next subsection.\n\n\\subsection{Design Goals and their consequences}\n\nIn the following discussion of RegTAP's design, the model is derived from\nseveral partly conflicting design goals, which are written\n\\designgoal{slanted} in the following.  Additionally, RegTAP names are \nmarked up in\n\\rtent{slanted typewriter}, while we continue to write\nVOResource concepts\nin \\vorent{small caps}.\n\nWhile RegTAP attempts to \\designgoal{represent all concepts} of\nVOResource that could plausibly be of use in locating resources, it is\nnot a full relational mapping of VOResource.  An overarching design goal\nwas to \\designgoal{keep the model compact}.  In version~1, the model\ndefines 13 tables, a number that would have been significantly higher\nfor a full mapping without proportionally adding discovery capabilities.\nFrom VOResource and its current extensions, the model primarily left out:\n\n\\begin{itemize}\n\\item From TAPRegExt \\citep{std:TAPREGEXT} the descriptions of user defined \nfunctions;  these say what extra functions are available in ADQL\nqueries, how to call them, and what they do.  Representing this would\nhave required an extra table, which appeared hard to justify given that\nno major discovery scenarios were found for this metadata (it is there\nfor TAP client use, and the TAP clients get the information directly\nfrom the service's capabilities endpoint).\n\\item Also from TAPRegExt the declaration of how clients can upload\ntables into TAP services, with a similar rationale.\n\\item From StandardsRegExt \\citep{std:STDREGEXT} the \nenumerations of the input\nparameters defined by the standard itself. They do not appear\nvaluable for discovery given the moderate number of existing standards.\nRepresenting them would, however, break the simple foreign \nkey relationship between interface\nand capability if they were kept in the interface table, and an\nadditional relatively complex table otherwise.\n\\item Also from StandardRegExt the detailed information on\nthe versions of documents issued.  This would have required an extra\ntable, and again, given the moderate number of standards, no credible\ndiscovery scenario is apparent.\n\\item VOResource's ability to have multiple access URLs for a single\ninterface; this feature has essentially not been used in practice, and\nkeeping it would have introduced another join in all queries for access\nURLs and hence the vast majority of current Registry queries.\nIt is planned to drop the feature in future VOResource\nversions; resources that actually need multiple access URLs for a single\ninterface would then have to represent each endpoint as an interface of\nits own.\n\\end{itemize}\n\nTo leverage existing VOResource expertise, RegTAP tries to\n\\designgoal{follow VOResource names}.  However, again compromises had to\nbe made\nto meet some other design goals.  First, as the subject domain of\nVOResource partly coincides with the data definition language of SQL,\nmany of the terms in VOResource are reserved words in database engines.\nAs RegTAP also tries to \\designgoal{avoid requiring delimited\nidentifiers}\\footnote{Delimited identifiers in SQL, syntactically marked\nby enclosing the identifier name with double quotes, allow using\narbitrary strings as column names and also suspend SQL's case folding.}\nfor\nusability reasons (e.g., difficult to understand parse errors resulting\nfrom forgotten quotes), conflicting names were amended with tags\nindicating entities' roles.  In this way, VOResource's \\vorent{table} becomes\n\\rtent{res\\_table}, and \\vorent{column} \\rtent{ta\\-ble\\_column}.\n\nAnother important design goal was to \\designgoal{hide foreign key\nrelationships}.  This is again a usability concern -- having to\nwrite explicit join conditions would necessitate a more intimate\nfamiliarity with the data model than can be expected from a\npossibly casual user.  Instead, query writers should need\nonly to identify\ncolumns of interest and then use \\texttt{NATURAL JOIN} to build their\nquery's \\texttt{FROM} clause.\n\nThis implies more name mangling, as in VOResource many elements can be\nchildren of different parents, for instance \\vorent{type},\n\\vorent{name}, \\vorent{description}.  Again,\ndisambiguation is effected\nusing tags prepended with an underscore, indicating the\nsource table, abbreviated when names would attain excessive length.\nThus, \\vorent{description} in \\vorent{resource} becomes\n\\rtent{res\\_description}, whereas in \\vorent{capability} it becomes\n\\rtent{cap\\_description}.  Only the two co\\-lumn-like tables\n(\\rtent{table\\_column} and \\rtent{intf\\_param}) are an exception.  \nThis implies that \\rtent{intf\\_param} and\n\\rtent{table\\_co\\-lumn} are the only tables that cannot be naturally\njoined.\n\nThe key used for joining is obvious for all tables directly referencing\n\\rtent{resource}, as Registry semantics ensure \\rtent{ivoid} -- the\nrecord's IVORN -- is a\nsuitable primary key for that table.  However, RegTAP also has foreign\nkeys into the tables \\rtent{capability}, \\rtent{interface}, and\n\\rtent{res\\_schema}, for which VOResource does not provide suitable\nprimary keys, as the respective relationships are represented by lexical\ninclusion in XML.\nRegTAP instead introduces surrogate keys, the nature of which is\nimplementation-defined. Hence queries should never explicitly use them,\nand since the tables are naturally joinable, they have no reason to do\nso.  In general, as the declaration of primary and foreign keys has no\nimpact on service behavior, RegTAP makes no requirements in this area\nbut restricts itself to recommendations.\n\nA further design goal requiring changes to VOResource names is that\n\\designgoal{quoting must not hurt}.  It is not uncommon that SQL authors\nand query generators\nemploy delimited identifiers when they do not need to.\nIn these cases, mixed-case column names\neasily lead to execution errors that again may not be easy to understand.\nTherefore, all identifiers in the standard are completely lowercase.  Internal\ncapitalization to indicate compound words is not uncommon in VOResource,\nhowever.  In RegTAP compound words are concatenated with underscores,\nsuch that, for instance, \\vorent{relationshipType} becomes\n\\rtent{relationship\\_type}.\n\nIf only for reasons of ease of implementation across different back-end\ndatabase engines, it was important for us to \\designgoal{not grossly\nviolate the relational model}.  However, an analogue of the\nobject-relational impedance mismatch impacts RegTAP as well: for VOResource,\nbeing an XML application, hierarchy and sequences are natural and easy.  In\na relational model, these translate into foreign keys and extra tables\nand thus complicate the schema.  In order to avoid an inflation of\ntables, RegTAP supports what in effect are arrays of simple strings.\n\nThese are only used where \nvalues are taken from controlled vocabularies, specifically for\n\\vorent{level} and \\vorent{type} from \\vorent{content},\n\\vorent{waveband} and \\vorent{rights} from \\vorent{Resource},\n\\vorent{flag} from \\vorent{column}, and \\vorent{queryType} from\n\\vorent{interface}.  Here, multiple values from VOResource are\nconcatenated with hash characters (\\#).  To allow reliable querying in\nthese columns,\nRegTAP services must implement an ADQL user defined function called\n\\texttt{ivo\\_hashlist\\_has}.  We specifically did not use that pattern\nfor \\vorent{subject}, as its vocabulary, while governed by a\nrecommendation to use the IVOA Thesaurus, is deliberately open -- indeed, it\nis well conceivable that, for instance, hashtags might at some point be\nused here -- and it stands to reason that\ncomplex queries over \\rtent{res\\_subject} will be performed\nwhen clients make use of, say,\nontologies that may themselves be represented in database tables.\n\nIn a model so heavily dealing with natural language, another violation\nof strict relationality is almost unavoidable: Treating text as, at\nleast, bags of words.  RegTAP therefore requires conforming services to\noffer a user-defined function\n\n\\begin{verbatim}\nivo_hasword(txt VARCHAR(*), pat VARCHAR(*)) \n  -> INTEGER\n\\end{verbatim}\n\n\\noindent that returns true at least when \\texttt{pat} is present in\n\\texttt{txt}.  Operators are urged to match \\texttt{pat} to \\texttt{txt}\nin an information-retrieval (IR) sense (i.e., ``Google-style'' as\ndocument vectors).  This is the main violation of RegTAP's design goal that\n\\designgoal{different registries yield identical results} for identical\nqueries.  This\nviolation is regrettable, as experience shows that users are at least\nconfused if their familiar result lists change after\na change in the registry endpoint used by their client.  However,\ngiven that IR facilities in back-end databases are inconsistent with each\nother and an independent implementation of them is nontrivial, the\ndesign goal that the standard \\designgoal{does not exclude a \nmajor database back-end}\noverrode the consistency concern.\n\nA final salient design goal is that\n\\designgoal{Registry extensions are possible without schema updates}.\nRegistry extensions change the XML schema, and hence RegTAP would have \nto represent arbitrary XML trees within a fixed relational schema if\nthis design goal were to be fully achieved.  The\nresult would have been very hard to query indeed.  RegTAP's designers\ntherefore identified a\nsubset of extensions that is relatively straightforward\nin queries,  powerful enough to satisfy foreseeable use cases, and\nreasonably compact: atomic values in 1:n relationships over either\nresource or capability. The result is RegTAP's \\rtent{res\\_detail} table.\n\nThis table on the one hand references resources or capabilities by their\n\\rtent{ivoid} and, as appropriate, the surrogate key on\n\\rtent{capability} (which is NULL\nfor items pertaining to the entire resource).   On the other hand it\ncontains keys (\\rtent{detail\\_xpath}) and values\n(\\rtent{detail\\_value}).  The keys in this table are essentially XPath\nexpressions within the resource record, much like the references in RI1\nquery constraints.  The values are always strings, even when the VOResource\nelements represented have other types.  \n\nThus, a data collection's \\vorent{accessURL} child is accessible through\nthe key \\texttt{\/accessURL}, the maximum size of files returned from an\nimage service (defined in SimpleDALRegExt) is retrieved as its decimal\nserialization under the key \\texttt{\/capability\/maxFileSize}, and the\nauthorities managed by a registry are in, if necessary multiple, rows\nwith the key \\texttt{\/managedAuthority}.\n\nWhen mapping existing Registry extensions, it was found this was\nsufficient to express the concepts contained with the exceptions\noutlined above.  A full list of the keys from the registry extensions\npublished before RegTAP is given in \\citet{std:RegTAP}, and future registry\nextensions should specify which additional keys they define.\n\n\n\\subsection{Addressing Particular Issues}\n\nIn going from VOResource to a relational schema, properties of either\nthe relational or the XML model or restrictions of the query language\nforced us to introduce additional rules for several entities.  We\nmention some major special cases in this subsection.\n\n\\paragraph{Case issues} A particular challenge in the mapping rules\nfrom VOResource to RegTAP were case-insensitive values.  For instance,\nIVORN \\citep{std:VOID}, UCDs, and utypes\nin current VO usage \\citep{note:utypeusage} all have to be compared\nignoring case.  Even if ADQL had an operator for case-insensitive string\ncomparison, having to consider case issues in comparisons would\ninvite bugs in queries that are hard to detect -- when a query author\nforgets that a column must be compared ignoring case, the queries\nmight still return some records and thus appear to work. RegTAP\ntherefore mandates that all such values must be lowercased during ingestion.  In\nthis way, queries not taking into account case insensitivity will at\nleast reliably produce an empty result list.  RegTAP further\ncase-normalizes other columns filled from controlled vocabularies to\nbe as consistent as possible.  Only columns intended\nfor presentation (essentially the descriptions and titles, role name,\nand subject) and those where case normalization might lead to\nambiguities (mainly \\rtent{detail\\_xpath} and \\rtent{detail\\_value}) are\nexempt from normalization. Where case normalized comparisons are desired\nfor such mixed-case columns, RegTAP offers a UDF \\rtent{ivo\\_nocase\\_match} in\naddition to \\rtent{ivo\\_hasword} (that ignores case as well).\n\n\\paragraph{Order} While in most parts of VOResource, the order implied\nin XML trees is irrelevant and thus no particular attention is necessary\nin the translation to the sets of the relational model, \\vorent{creator}\nis an exception.  Typically used to convey authorship information, order\nthere matters to many data providers.  Rather than add sequencing\ncapabilities to \\rtent{res\\_role}, RegTAP adds a \ncolumn \\rtent{creator\\_seq} to \\rtent{resource} that contains a\npre-for\\-mat\\-ted author list.  This has the additional benefit that clients\ndo not need worry about reconciling the (correct) practice of having one\nauthor per \\vorent{creator} element with the (widespread) practice of\nincluding multiple names in one element\nin order to produce a flat author list at least for display purposes; \nany necessary special\nhandling happens at the registry.\n\n\\paragraph{QNames} Several VOResource values are really XML qualified\nnames (QNames).  This concerns some fairly fundamental VOResource\nconcepts, in particular the types of resources, interfaces, and\ncapabilities.  For instance, a query might be interested in locating all\nresources that are VODataService \\vorent{DataCollection}s.  These are\nidentified by having an XML schema type of\n\\texttt{\\{http:\/\/www.ivoa.net\/xml\/VO\\-Data\\-Ser\\-vice\/v1.1\\}Da\\-taCollection},\nusing the conventional notation that prepends the namespace part of\na QName in curly brackets.  As this notation is cumbersome for input,\nserialized XML maps the namespace URIs to namespace prefixes. \nVOResource and extensions strongly\nrecommend the use of canonical prefixes that would, for instance, bind\nthe prefix ``\\texttt{vs}'' to\nthe VODataService namespace.  Hence, the above name becomes a much more\nmanageable \\texttt{vs:Data\\-Collection}.  Unfortunately, the canonical\nprefixes are not mandatory in VOResource, which means that registries\nmight use entirely different prefixes, and indeed, in registry practice,\nseveral do.\n\nAs long as the RegTAP ingestor knows which attributes contain QNames --\nand that is defined in VOResource XML schema files --, it can, however,\nunify prefixes by turning the namespace prefixes of the instance\ndocument into namespace URIs and then translating them back into the\ncanonical prefixes.  To ensure consistent results over registries,\nRegTAP requires this prefix normalization.  \nEssentially, the recommendation to use its\ncanonical prefix contained in all VOResource standards becomes a hard\nrequirement for RegTAP.\n\n\\section{Full-text Based Registry Interface}\n\\label{sect:fulltext}\n\nIn parallel with the relatively complex RegTAP interface, a successor to\nRI1's \\textit{KeywordSearch} is also being developed.\nThis full-text based registry addresses the difficulty of\nextracting information from the previous registry interface. As field\nvalues describing resources are mostly text, a full-text search engine\nsuch as the Apache Lucene library -- as used in popular server components\nlike ElasticSearch or Apache Solr --\nis suitable to index and search\nthe contents of\nthe registry. \nIt was therefore decided to develop a RESTful\nAPI using ElasticSearch as a client interface to the Registry. The\nrequirements were to fulfill the registry use cases defined by the IVOA\nRegistry Working Group \\citep{wiki:regusecase} and to support\nthe web clients developed at VO-Paris Data\nCentre, among them several Registry curation tools. \nThe full specification of the RESTful interface is currently maintained\nat \\texttt{http:\/\/api.vo.obspm.fr\/registry\/}. That page also provides\nseveral examples.  \n\n\n\\subsection{Query Interface}\n\nThe \\texttt{\/search} method is derived from the \\textit{Search} and \n\\textit{KeywordSearch}\noperations of the RI1 searching interface. It allows querying common\nRegistry items individually or all together. \nFor requests specifying multiple constraints\nlogical AND is used by default, but a logical OR is available by adding\na parameter ``orValues'' in the query string\n\nThe initial set of fields to filter a request has been extended to\nfulfill the requirements of all clients using it.  One of the VO client\nuse cases is to find services by capability type (Spectrum, Image, TAP,\netc),\nand specific words or expressions from its\n\\vorent{description}, \\vorent{content.subject}, \\vorent{title}, or \n\\vorent{shortName} attributes.  Other useful selection criteria come\nfrom \\vorent{curation}'s \\vorent{publisher}, \\vorent{creator.name}, \nand \\vorent{contributor}.\nThe \\vorent{coverage} is also used to filter\nservices in spectral, time, and soon in spatial domain. The spatial\ncoverage actually is an optional field in the resource description and\nnot well described in the service declaration. However this information\nwill soon be available in a standard way, using the HEALPix Multi-Order\nCoverage maps (cf.~\\ref{sect:coverage}).\n\n\nAs the developers of the full-text search based interfaces are also data\ncurators in the VO Registry, some specific information was kept and is\navailable for selected resources:\n\n\\begin{itemize}\n\\item the IVORN of the resource record\n\\item the dates of the publication and last update of the resource\nrecord\n\\item the registry where the resource has been declared and which is responsible for this record\n\\item information from \\vorent{validation}.\n\\end{itemize}\n\n\\subsection{Query Examples}\n\nA prototype service implementing the full-text query is being maintained\nat VO Paris\\footnote{Access URL\n\\texttt{http:\/\/voparis-registry.obspm.fr\/vo\/ivoa\/1\/voresources\/search}.}.\nThe following example queries can be executed there by passing them as\nURL query strings; for reasons of readability, the query strings are\nshown here not URL-encoded.\n\n\\begin{itemize}\n\\item Search all resources containing the keyword ``infrared'':\\\\\n{\\small\n\\hspace*{1em}\\texttt{keywords=infrared}\n}\n \n\\item Ditto, but only return services implementing the Simple Image\nAccess protocol:\\\\\n{\\small\n\\hspace*{1em}\\texttt{keywords=infrared}\\\\\n\\hspace*{2em}\\texttt{+\"ivo:\/\/ivoa.net\/std\/SIA\"}\n}\n \n\\item  Search for all\nresources published by the Centre de donn\u00e9es de Strasbourg (CDS)\nimplementing the Simple Cone Search protocol, with a\n\\vorent{contentLevel} of Research,\nand return the 100 resources starting from match number 200:\\\\\n{\\small\n\\hspace*{1em}\\texttt{keywords=publisher:cds}\\\\\n\\hspace*{2em}\\texttt{+standardid:\"ivo:\/\/ivoa.net\/std\/ConeSearch\"}\\\\\n\\hspace*{2em}\\texttt{+contentlevel:Research}\\\\\n\\hspace*{1em}\\texttt{\\&max=100}\\\\\n\\hspace*{1em}\\texttt{\\&from=200}\n}\n \n\\item Return the full resource record for some IVORN:\\\\\n{\\small\n\\hspace*{1em}\\texttt{identifier=ivo:\/\/vopdc.obspm\/luth\/exoplanet}\n}\n \n\\end{itemize}\n\n\\subsection{Service Response}\n\nAs the aim of the full-text query interface is to provide the simplest\nsystem for VO application developers, and most of the new clients are\nJavaScript based, the API returns query results formatted in JSON\nresponses.  Those responses give back the most useful fields as a subset\nof the whole service declaration plus a link to the original VOResource\nXML file in the registry. The subset of information returned is \nrelatively compact and has proven sufficient for the clients already\nusing the API.\n\n\n\n\\section{Common Registry Queries}\n\\label{sect:commonqueries}\n\nVOResource is a complex data model that sometimes\noffers multiple ways of expressing apparently very\nsimilar concepts.  Driven by both registry record authoring practices and\nthe queries employed by popular clients, some usage patterns have evolved that\nshould be followed for successful Registry use.  Other patterns are\nrecommended for ease of use. We describe the\npatterns in RegTAP terms, but most would be equally\napplicable to endpoints speaking XQuery, and partly even to\nkeyword-based services.\n\nWhat is special to RegTAP is the query construction technique.  The way\nthe schema is designed, one looks for the fields to be constrained and\nfor the fields to be retrieved in a schema description such as\nthe RegTAP specification, an implementing service's\n\\texttt{TAP\\_SCHEMA}, or its VOSI table metadata.  The query can then be\nwritten by collecting all source tables, concatenating\nthem by \\texttt{NATURAL JOIN} and treating the result as a single table.\n\n\\subsection{Locating Standard Services}\n\\label{sect:capquery}\n\nA very common type of query is finding \\vorent{service}s implementing a\ncertain standard.  In VOResource terms, it is actually not the service\nbut one of its capabilities that complies with a standard.  For instance,\na service could at the same time implement a cone search for telescope\npointings, and two image services each conforming to a specific version\nof SIAP.  Each facility is then represented as a different\n\\vorent{capability}.\n\nThe \\rtent{capability} table offers two ways to identify the kind of\ninterface -- one could constrain \\rtent{cap\\_type} or\n\\rtent{standard\\_id}.  The correct constraint is on \\rtent{standard\\_id},\nas it would be perfectly legal to register an SSA service, say, with a\n\\rtent{cap\\_type} of \\texttt{vr:capability} (i.e., the minimal capability\ndescription only consisting of a standard identifier and the\ninterfaces).  While the record would miss essential metadata, clients\nshould have no trouble operating a service registred in this way, and\nhence the Registry query should find it.  All known clients' queries by\nservice type follow the pattern of matching against the standard\nidentifier.\n\nAs the standard identifer is an IVORN, it needs to be lowercased in\nqueries for RegTAP.  So, to locate all services (say, by their IVORN and\ntitles) having SSA interfaces, the query would look like this:\n\n\\begin{lstlisting}\nSELECT ivoid, res_title\nFROM rr.resource\n  NATURAL JOIN rr.capability\nWHERE\n  standard_id='ivo:\/\/ivoa.net\/std\/ssa'\n\\end{lstlisting}\n\nOther relevant standard identifiers are given in the respective\nspecifications or in one of the examples in RegTAP.\n\n\\subsection{Locating Standard Interfaces}\n\\label{sect:q-standardintf}\n\nLocating the capability is not enough to operate a service. In\naddition, the endpoint -- which in VO practice is identified by an\naccess URL -- needs to be located.  A single capability can have\nmultiple interfaces, and while this practice is not recommended,\nthere are resource records that have capabilities\ndeclaring adherence to a standard with interfaces for web browsers in\naddition to the standard interface.\n\nVOResource's \\vorent{interface} element has a \\vorent{role} attribute to\ndistinguish the standard interfaces from custom ones.  For the former,\n\\vorent{role} would contain a special string formed according to certain\nrules.  In practice, many resource record authors have neglected to set\n\\vorent{role}, and therefore actual clients started to ignore it.\nCurrent VO practice therefore is to regard the (hopefully unique)\ninterface of type \\vorent{vs:ParamHTTP} as the interface exposing the\nstandard.\n\n\nHence, the pattern to locate interfaces complying to standards right now\nis, in RegTAP (this time looking for TAP interfaces):\n\n\\begin{lstlisting}\nSELECT ivoid, access_url \nFROM rr.capability\n  NATURAL JOIN rr.interface\nWHERE standard_id='ivo:\/\/ivoa.net\/std\/tap'\n  AND intf_type='vs:paramhttp'\n\\end{lstlisting}\n\nWe expect this pattern to be stable, mainly because the development of\nStandardsRegExt now very strongly suggests that services supporting\nmultiple versions of a single standard will have to treat each such\ninterface in a single capability.  Hence, distinguishing different\nversions by the \\vorent{role} attribute (or the \\rtent{intf\\_role}\ncolumn in RegTAP) appears dispensable, and assuming the\n\\vorent{vs:ParamHTTP} interface  within a standard capability must be\nthe standard service endpoint is straightforward and robust.\n\n\\subsection{Query by Physics}\n\nA type of discovery query not yet widely supported in Registry UIs\nis the query by\nphysics.  The fact that resource records can and in many cases do\ncontain table metadata giving \nUCDs helps locating resources exposing a certain type of data.\nAs UCDs follow a grammar that ADQL does not understand, it is frequently\nadvisable to use wildcards in such queries.  For instance,\ncolumns containing infrared magnitudes could be found like this:\n\n\\begin{lstlisting}\nSELECT name, ucd, column_description\nFROM rr.table_column\nWHERE ucd LIKE 'phot.mag;em.i\n\\end{lstlisting}\n\nTo illustrate again RegTAP's principle of natural joins, let us\nshow how to add a constraint on the embedding table here:\n\n\\begin{lstlisting}\nSELECT name, ucd, column_description,\n  table_description\nFROM rr.table_column\n  NATURAL JOIN rr.res_table\nWHERE 1=ivo_hasword(\n    table_description, 'quasar')\n  AND ucd LIKE 'phot.mag;em.i\n\\end{lstlisting}\n\n\\noindent\n-- then constraining on tables accessible via TAP is simply a matter of\ncombining select list and the FROM and WHERE clauses\nfrom subsection~\\ref{sect:q-standardintf} with this query.\n\n\\subsection{A Sketch of TOPCAT's Query}\n\nBy way of example, we present a query submitted by TOPCAT to\nacquire service metadata for presentation to the user,\nincorporating some user-supplied constraints.\nThe following ADQL would locate TAP services concerning galaxies:\n\n\\begin{lstlisting}\nSELECT ivoid, short_name, res_title, \n  reference_url, base_role, role_name, \n  email, intf_index, access_url, \n  standard_id, cap_type, cap_description, \n  std_version, res_subjects\nFROM rr.resource AS res\n  NATURAL JOIN rr.interface\n  NATURAL JOIN rr.capability\n  NATURAL LEFT OUTER JOIN rr.res_role\n  NATURAL LEFT OUTER JOIN (\n    SELECT \n      ivoid, \n      ivo_string_agg(res_subject, ', ')\n        AS res_subjects\n     FROM rr.res_subject GROUP BY ivoid\n  ) AS sbj\nWHERE \n  standard_id='ivo:\/\/ivoa.net\/std\/tap'\n  AND intf_type='vs:paramhttp'            \n  AND (\n    1=ivo_hasword(res_title, 'galaxy')\n    OR 1=ivo_hasword(res_subjects, 'galaxy')))\n\\end{lstlisting}\n\nNote how in this query outer joins are used to make sure rows are\nreturned even for records that, for instance, do not give roles.  In the\ncase of \\rtent{res\\_subject}, VOResource guarantees that at least one\nsubject must always be present, so doing an outer join here should\nnot be necessary.  On the other hand, in particular in queries executed\non behalf of a UI, it is good practice to assume minor violations of\nVOResource will be present in the Registry.\n\nThe sub-query for \\rtent{res\\_subjects} also shows an example for how to\nreduce the number of rows transferred by server-side aggregation.\nAnother application for this pattern could be, using suitable strings\nas separators, retrieving pairs of capability identifiers and their\naccess URLs.\n\n\\section{Open Issues}\n\\label{sect:openissues}\n\nEven after the introduction of RegTAP, Registry development is not\ncompleted.  In addition to Registry extensions as new service and\nresource types are defined within the VO, several fields of work are\ncurrently actively being explored.  We discuss them here as they delineate\nwhat the Registry should be doing but does not do so far, as well as\nto document approaches tried in the VO to problems that\nmay similarly arise in other communities.\n\n\n\\subsection{Data Collection and Relationships}\n\\label{sect:relationship}\n\nSome TAP services today expose dozens or hundreds of\ntables\\footnote{Examples for such services include\n\\texttt{ivo:\/\/org.gavo.dc\/tap},\n\\texttt{ivo:\/\/nasa.heasarc\/services\/xamin}, as well as the\nTAP interface to VizieR.}.  In ObsCore\nservices\\footnote{The CADC TAP service \\texttt{ivo:\/\/cadc.nrc.ca\/tap}\nbelongs in this category.} \\citep{std:OBSCORE}, data from many\nindividual data collections are queryable through a single endpoint.  In\nthe same way, some SIAP services make data from several individual\nobservatories accessible.\n\nIn all these cases, the contributing data collections should all be\npresent with their full metadata in the Registry. Using GAVO's Lens\nImage Archive\\footnote{\\texttt{ivo:\/\/org.gavo.dc\/lensunion\/q\/im}} as an\nexample, a title query for one of the contributing data collections,\nMiNDSTEp, say, should yield the full metadata for the data\ncollection\\footnote{In this case,\n\\texttt{ivo:\/\/org.gavo.dc\/danish\/red\/data}.}, and clients should, from\nthere, be able to infer the access URL of the service exposing the data.\n\nThe \\vorent{DataCollection} type of VODataService provides a type for\nsuch cases, and through \\vorent{relationship} -- in this case, with a\n\\vorent{relationshipType} of \\texttt{servedBy} -- the associate data\nservice can be successfully located.\n\nHowever, client support for querying through \\vorent{relationship}\nhas been lacking, even in the most\nadvanced registry clients.  WIRR at least shows the presence of\nrelated resources\nexplicitly, but an additional query is required to retrieve them.\nAlso, a query for ``Image services exposing data from\nMiNDSTEp'' would fail unless the registry record of the embedded service\nwere carefully crafted.  In RegTAP, writing ADQL for queries that would\nsimultaneously find ``direct'' (i.e., services exposing exactly one data\ncollection) and ``indirect'' (i.e., data collection metadata managed\nseparately from service metadata) services is at least highly nontrivial\n\\citep{talk:uneasy}.  To understand why joining tables through\nrelationship requires great care, consider again\nFig.~\\ref{fig:schema}.  The colors there distinguish between\n``capability-bound'' metadata that in such queries would have to be\nqueried from the service (e.g., access URL, capability ids,\naccepted parameters) and ``resource-bound'' metadata that needs to come\nfrom the data collection itself (e.g., description, title, or UCDs from\na published table).  The two tables that can reference both\n\\rtent{resource} and \\rtent{capability}, shown in yellow in the figure,\nadditionally complicate query construction.  In any case,\nnatural joins of tables from different groups will not produce\nmeaningful results, thus requiring\nquery authors to add explicit join conditions.\n\nThese difficulties have spurred activity to consider changing VOResource\nsuch that clients do not need to follow relationships to locate access URLs.\nA reasonable solution explored was adding\n(some of) the capabilities of the data service to the resource\nrecords of the data collections themselves.  That, however, leads to an\ninflation of such capabilities that will make the very common queries\nfor all resources implementing a certain standard as laid out in\nsubsection \\ref{sect:capquery} much harder to handle for clients.  For\ninstance, to prepare for an all-VO query for images, clients would have\nfilter out duplicate access URLs in order to avoid querying a service\nexposing $n$ resources $n$ times (instead of once).\n\nThe least burdensome solution is still to be found.  Discussions within the\nRegistry working group are currently investigating the use of ``auxiliary''\nstandard identifiers for the capabilities on the data collections,\nwhich would, by lexical convention, facilitate the discovery of either\nunique services (by using the standard identifiers already in use) or\nall endpoints exposing data constrained by further metadata (by using an\nappropriate and index-friendly regular expression).\n\n\\subsection{Education and Internationalization}\n\nIn the context of work done within the IVOA working group on Education\n\\citep{note:edumatters}, the issue of multilinguality arose.  While in\nprofessional astronomy, all-English metadata seems sufficient and,\nindeed, preferable, the situation is not as clear when certain resources\n-- in this case, educational material -- should be made discoverable for\neducators or even the general public.  For instance, if a worked-out use-case\non open clusters is available in Italian, should it not be\ndiscoverable by querying for ``Ammasso Aperto''?  This would entail\nallowing the relevant text fields (\\vorent{title}, \\vorent{description},\npossibly \\vorent{subject}) to be present multiple times in resource\nrecords, each element containing text in a different language, and it\nwould probably also entail allowing language constraints in client\ninterfaces to avoid losing precision due to homographs in different\nlanguages.  Alternatively, different registries might be set up for\ndifferent languages.\n\nSo far, the Education WG only plans to allow discovering which language specific\nresources are available in rather than supporting queries in non-English\nlanguages.  If, however, takeup of Registry technologies outside of the\nresearch community were to increase, the issue would have to be\nrevisited, presumably from both the client and the data model side.\n\n\\subsection{Coverage in Space and Time}\n\\label{sect:coverage}\n\nVODataService allows the specification of resource coverage,\ni.e., the spatial area covered on the sky as well as the ranges in time\nand spectrum, in resource records.  Apart from the controlled vocabulary\nin \\vorent{waveband}, this is done through embedding STC-X\n\\citep{note:STCX} within registry records.  No standard way of querying\nthis information exists to date.  An attempt to include coverage\ninformation through four tables giving sets of coordinate intervals per\nresource as proposed in \\citet{talk:ri2} did not gain much traction,\npartly because of the flexibility and complexity of the underlying STC\ndata model, partly because it was felt that for spatial coverage,\ncoordinate ranges in the equatorial system were too inflexible to be\ngenerally useful even for discovery purposes.  An obvious example\nillustrating the shortcomings would be a survey along the galactic\nequator: either many ICRS ranges would have to be given, or the coverage\nwould be dramatically overrepresented.\n\nIn the meantime, multi-order coverage maps \\citep[][MOCs]{std:MOC}\nwere developed as a standard way of representing spatial coverages.  Work is\nongoing on how these could be integrated into VOResource on the data\nmodel side and exposed to the clients; if these were to be included in\nan extension to RegTAP, \nan ideally indexable way of representing MOCs in databases would\nbe required, and no technically feasible solution has been proposed so\nfar.\n\n\\section{Conclusions}\n\nThe VO Registry is an essential source of metadata about the services\nand data that can be used within the VO, and no non-trivial interaction\nwith the VO can take place without using its discovery capabilities.\nMany VO clients embed Registry information and protocols in various\nforms.\n\nBy necessity, standardization of the Registry protocols occurred\nrelatively early in the history of the VO.  While standards on the\nserver side have held out very well, the early standards on the client\nside have, in the meantime, proved insufficient for today's advanced\nRegistry use.\n\nThis resulted in the creation of second-generation client interfaces.\nIn this article, we have discussed principles and design goals of the\ntwo currently developed interfaces.\nThe keyword search interface provides a simple language to\nconstrain results that accomodates users' habits in taking up patterns\nfrom general search engines, with a response format designed for easy\nintegration into browser-based applications.\nRegTAP, on the other hand, is a relatively faithful mapping of\nessentially the entire data model to a relational database schema,\ntargeted towards \n``thick'' clients and expert users writing ADQL queries by hand.\n\nWhile RegTAP goes much further than RI1 in defining the mapping between\nthe XML schema that defines the Registry data model and the relational\nmodel that is in practice used to represent the data set in queryable\nform, there are still some small areas where the relational schema and\nits XML counterpart do not precisely match each other's expressiveness,\nprecluding, for instance, roundtrip ingestion and recreation of registry\nrecords through a RegTAP tableset.  We propose as a lesson to be learned\nfrom this that future data modelling efforts should be done in an\nimplementation-neutral language with well-defined and well-understood\nmappings to the common implementation\nlanguages.  Within the VO, an effort is underway to enable this\n\\citep{std:VODML}.\n\nThis is not to say that the Registry model needs fundamental work or a\ntechnology switch any time soon. The new interfaces to the Registry\nexpose its functionality fairly completely and interoperably\nbetween their implementations, and building on\nproven technologies like TAP and JSON, they also lower the cost of\nintegrating VO registry information into client programs.\n\nSome open issues remain in the registry's client interface; the most\nurgent ones are probably the formulation of contraints on spatial\ncoverage and the handling of capabilities associated with data\ncollections.\n\n\\section*{Acknowledgements}\n\nWe thank Laurent Michel, Pierre Fernique, and Pedro Osuna for providing\ninformation on the Registry interfaces of TAPHandle, Aladin, and VOSpec.\n\nThis work was in part supported by the German Astrophysical Virtual\nObservatory GAVO, BMBF grant 05A11VH3.\n\n\\section*{References}\n\n\\bibliographystyle{elsarticle-harv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nThe study of isometric group actions on Riemannian manifolds has seen a number of important applications in Riemannian geometry.\n\nMany of them fall under the umbrella of the so-called \\emph{Grove's program}, whose goal is to study the properties of Riemannian manifolds with non-negative (or even almost non-negative) sectional curvature in the presence of symmetry. This program has been extremely fruitful both in producing new examples of manifolds with non-negative sectional curvature, and in proving important conjectures in the area when some symmetry is added (cf. \\cite{KWW21}, \\cite{GKS20}, \\cite{FGT17}, \\cite{GW14}, \\cite{GZ00}, \\cite{GVZ11}, \\cite{Dea11}, etc.)\n\nThe concept of an isometric group action can be generalized by a \\emph{singular Riemannian foliation}, \nwhich roughly speaking is the partition of a Riemannian manifold into smooth and equidistant submanifolds of possibly varying dimensions, called leaves (and the leaves can be thought as a generalization of the orbits of an isometric group action). \nIt turns out that, while being more flexible than group actions (cf. for example \\cite{Rad14}), singular Riemannian foliations\nstill retain a lot of the same structure of isometric group actions (cf. \\cite{MR19}, \\cite{GGR15}, \\cite{GR15}, \n\\cite{CM20}, \\cite{Mor19}, etc.).\n\nGiven the action of a compact Lie group, the orbits are homogeneous spaces and thus have a very restricted topology, \nwhich can be employed to extrapolate topological properties of the ambient manifold (e.g. \\cite{GZ12} and \\cite{GYW19}). \nIn \\cite{GYW19}, the authors ask to what extent the leaves of a singular Riemannian foliation on a non-negatively curved space \nare also topologically restricted. In \\cite{GGR15}, Galaz-Garcia and the first author proved that if $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$, then the fundamental group of a generic leaf is a product $A\\times K_2$ \nof an abelian group $A$ and a 2-step nilpotent 2-group $K_2$ - in particular, it is nilpotent. In the present paper, \nwe continue exploring the topology of the leaves of singular Riemannian foliations $(M,\\mathcal{F})$.\n\nThe first result states that if $M$ is simply connected, then a generic leaf $L_0$ of $\\mathcal{F}$ is a \\emph{nilpotent space}, i.e. $\\pi_1(L_0)$ acts nilpotently on $\\pi_n(L_0)$ for all $n>1$:\n\n\\begin{maintheorem}\\label{main-thm:leaves-nilpotent}\nIf $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$,  \nthen the principal leaves of $\\mathcal{F}$ are nilpotent spaces. Furthermore, all leaves are finitely covered by a nilpotent space.\n\\end{maintheorem}\n\nThis answers the first part of Problem 4.8 in \\cite{GYW19}:\n\\begin{nonumberquestion}\nLet $\\mathcal{F}$ be a closed singular Riemannian foliation on a closed (simply connected) Riemannian manifold M of almost non-negative curvature. Are the leaves of $\\mathcal{F}$ finitely covered by a nilpotent space, which moreover is rationally elliptic?\n\\end{nonumberquestion}\n\nOur result does not in fact use the curvature assumption. On the rationally elliptic part of the question, we make the following remarks:\n\\begin{enumerate}\n\\item The very question of whether the leaves are rationally elliptic, only makes sense the moment we know that the leaves are (virtually) nilpotent spaces: these are in fact the spaces on which rational homotopy theory applies, and the rational dichotomy of rationally elliptic vs. rationally hyperbolic spaces holds.\n\\item Assuming the question above to be true, and applying it to the product foliation $(M\\times \\mathbb{S}^n,M\\times \\{pts.\\})$ with $M$ simply connected and almost non-negatively curved, would imply that every simply connected, almost non-negatively curved Riemannian manifold is rationally elliptic, which is the statement of the celebrated (and out of reach) Bott-Halperin-Grove Conjecture. \nIn particular, the rationally elliptic part of the question is so far out of reach.\n\\end{enumerate}\n\nThe second result analyzes more in detail the structure of the fundamental group of a generic leaf $L_0$ \nof a singular Riemannian foliation $(M,\\mathcal{F})$ with $M$ simply connected:\n \n\\begin{maintheorem}\\label{main-thm:non-abelian part}\nLet $(M,\\mathcal{F})$ be a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$. \nIf $L_0$ is a principal leaf of $\\mathcal{F}$, then the non-abelian part $K_2$ of the fundamental group of $L_0$ is of the form \n$$K_2\\cong (\\prod_{j=1}^s \\Z_{2^{a_j}}\\times \\Z_2^b\\times \\prod_{i=1}^k G_i)\/({\\Z_2^{c}\\times\\Z_4^{d}}),$$\nwhere each $G_{i}$ is isomorphic to a central product of copies of $Q_8$, with possibly one copy of $D_8$ or $\\Z_4$.\n\\end{maintheorem}\n\nThe groups $G_i$ in the theorem are called \\emph{generalized extraspecial}. These 2-groups already occur as fundamental groups of orbits of orthogonal representations and hence are impossible to avoid (e.g. $\\mathrm{SO}(3)$ acting on $\\mathbb{S}^4$), \nsee also a family of examples from Section \\ref{SS:examples}.\n\nFinally, we extend Theorem A from \\cite{GGR15} by showing that when $M$ has virtually nilpotent fundamental group, the leaves of any closed singular Riemannian foliation $(M,\\mathcal{F})$ have virtually nilpotent fundamental group as well:\n\n\\begin{maintheorem}\\label{main-thm:virtually nilpotent}\nSuppose $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on compact Riemannian manifold $M$ with virtually nilpotent fundamental group. Then the leaves of $\\mathcal{F}$ have virtually nilpotent fundamental group as well.\n\\end{maintheorem}\n\nIn the fundamental paper \\cite{KPT10}, the authors show that every Riemannian manifold with almost non-negative \nsectional curvature is finitely covered by a nilpotent space. With this in mind, Theorem \\ref{main-thm:virtually nilpotent} gives the following steaightforward corollary:\n\n\\begin{maincorollary}\nGiven a closed singular Riemannian foliation $(M, \\mathcal{F})$ on an almost non-negatively curved manifold $M$, the leaves have virtually nilpotent fundamental group.\n\\end{maincorollary}\n\n\nThis paper is organized as follows. In Section \\ref{S:preliminaries}, we collect some preliminaries about topological results \nfor singular Riemannian foliations, and the main notation for bilinear and quadratic forms we need in the proof of Theorem \\ref{main-thm:non-abelian part}. In Section \\ref{S:topology of leaves}, we prove Theorem \\ref{main-thm:leaves-nilpotent}. \nIn Section \\ref{S:fundamental group}, we prove Theorem \\ref{main-thm:non-abelian part} and provide a family of examples showing that the generalized extraspecial groups can indeed appear in the fundamental group of principal orbits of orthogonal representations. Finally, in Section \\ref{S:nilpotent fundamental group}, we prove Theorem \\ref{main-thm:virtually nilpotent}.\n \n\\section{Preliminaries}\\label{S:preliminaries}\n\n\\subsection{Singular Riemannian foliations}\n\nLet $M$ be a Riemannian manifold. A singular Riemannian foliation on $M$ is a partition $\\mathcal{F}$\nof $M$ into connected, injectively immersed submanifolds called leaves such that every geodesic that starts perpendicular \nto a leaf remains perpendicular to all the leaves it meets, and moreover, M admits a family of smooth vector fields \nthat spans the leaves at all points.\n\nA singular Riemannian foliation is called closed if all of its leaves are closed in $M$. \nGiven a singular Riemannian foliation $(M,\\mathcal{F})$ on a complete manifold $M$ we define the \\emph{dimension \nof $\\mathcal{F}$}, denoted $\\dim\\mathcal{F}$, as the maximal dimension of its leaves. \nThe codimension of $\\mathcal{F}$ is defined by $\\dim M-\\dim\\mathcal{F}$.\n\nA leaf $L$ of the foliation $\\mathcal{F}$ is called regular if its dimension is maximal, or equivalently, $\\dim L=\\dim\\mathcal{F}$.\nThe union of all regular leaves is an open, dense and connected submanifold, which is called the principal stratum of $M$ \nand is denoted by $M_0$. The union of all other leaves is called the singular stratum of $(M,\\mathcal{F})$ \nand the connected components of the singular stratum are called singular strata. \n\nFor a closed singular Riemannian foliation $(M,\\mathcal{F})$, the canonical projection $\\pi:M\\to M\/{\\mathcal{F}}$\ninduces a metric space structure on the leaf space $M\/{\\mathcal{F}}$, where the metric \nis given by $d_{M\/{\\mathcal{F}}}(\\pi(p),\\pi(q))=d_M(L_p,L_q)$. If in addition all the leaves of $\\mathcal{F}$ are regular, \nthen the leaf space is a Riemannian orbifold. In particular, given a closed singular Riemannian foliation $(M,\\mathcal{F})$, \nthe quotient space ${M_0}\/{\\mathcal{F}}$ is an orbifold.\n\nWe then call a leaf $L\\subset M_0$ \\emph{principal} if it projects to a manifold point of $M_0\/\\mathcal{F}$. Clearly, the set of principal leaves is open and dense in $M_0$.\n\n\n\\subsection{Slice Theorem}\\label{SS:Slice Theorem}\nIn this section we describe the structure of a singular Riemannian foliation around a leaf. For more details, we refer the interested reader to \\cite{MR19}.\n\nLet $(M, \\mathcal{F})$ be a closed singular Riemannian foliation, let $p\\in M$, and let $L_p$ denote the leaf through $p$. Define the \\emph{horizontal space to $\\mathcal{F}$ at $p$}, $\\nu_pL_p\\subseteq T_pM$, as the subspace perpendicular to $T_pL_p$. Then there exists a singular Riemannian foliation $(\\nu_pL_p,\\mathcal{F}_p)$ called the \\emph{infinitesimal foliation of $\\mathcal{F}$ at $p$}, with two important properties:\n\\begin{enumerate}\n\\item $\\mathcal{F}_p$ is  invariant under rescalings,\n\\item In an $\\epsilon$-neighbourhood $\\nu_p^{\\epsilon}L_p$ of the origin in $\\nu_pL_p$, the exponential map $\\exp_p:\\nu_p^\\epsilon L_p\\to M$ takes the leaves of $\\mathcal{F}_p$ onto the connected components of the intersections $L\\cap \\exp \\nu_p^\\epsilon L_p$, with $L\\in \\mathcal{F}$. \n\\end{enumerate}\nFurthermore, there is a group of isometries $K\\subseteq O(\\nu_pL_p)$, sending leaves of $L_p$ to (possibly different) leaves of $\\mathcal{F}_p$, with the property that for any $v\\in \\nu_p^\\epsilon L_p$, the leaf $L_v\\in \\mathcal{F}_p$ satisfies the following:\n\\[\n\\exp_p(K\\cdot L_v)=L_{\\exp_p(v)}\\cap \\exp_p\\nu_p^\\epsilon L_p\n\\]\nIn other words, two leaves of $\\mathcal{F}_p$ are in the same $K$-orbit if and only if they exponentiate to different connected components of an intersection $L \\cap \\exp_p\\nu_p^\\epsilon L_p$, for some $L\\in \\mathcal{F}$.\n\nIn \\cite{MR19}, the following Slice Theorem establishes a model for a singular Riemannian foliation around a leaf:\n\\begin{nonumbertheorem}[Foliated Slice Theorem]\nGiven a closed singular Riemannian foliation $(M, \\mathcal{F})$ and a point $p\\in M$, let $(\\nu_pL_p, \\mathcal{F}_p)$ be the infinitesimal foliation of $\\mathcal{F}$ at $p$. Then there exists a compact Lie group $K\\subset O(\\nu_pL_p)$ and a principal $K$-bundle $P\\to L_p$ such that the foliation $\\mathcal{F}$ in an $\\epsilon$-neighbourhood of $L_p$ is foliated diffeomorphic to\n\\[\n(P\\times_K\\nu_pL,P\\times_K\\mathcal{F}_p)\n\\]\n\\end{nonumbertheorem}\n\nIt follows directly from the Slice Theorem that all principal leaves are diffeomorphic to each other, and for any leaf $L_p$, there is a locally trivial fiber bundle $L_0\\to L_p$ from a principal leaf $L_0$, whose fiber is an orbit $K\\cdot L_v$ for some principal point $v\\in (\\nu_pL_p, \\mathcal{F}_p)$, and it consists of a finite disjoint union of principal leaves of $\\mathcal{F}_p$.\n\n\n\n\\subsection {The Molino bundle}\\label{SS:molino}\n\nLet $(M,\\mathcal{F})$ be a closed singular Riemannian foliation of codimension $q$ on a compact Riemannian manifold $M$. \nThe principal $\\mathrm{O}(q)$-bundle $\\hat{M}\\to M_0$, where $\\hat{M}$ is the collection of orthonormal frames \nof ${TM_0}\/{T\\mathcal{F}}$, is called the Molino bundle. The foliation $\\mathcal{F}$ lifts to a foliation \n$\\hat{\\mathcal{F}}$ on $\\hat{M}$ whose leaves are diffeomorphic to the leaves of $\\mathcal{F}$ \non an open dense set. Moreover, the leaves of $\\hat{\\mathcal{F}}$ are given by fibers of a submersion \n$\\theta:\\hat{M}\\to W$, where $W$ is the frame bundle of the orbifold ${M_0}\/{\\mathcal{F}}$. \n\nConsider the fibration $\\hat{\\theta}:{\\hat{M}}_{\\mathrm{O}(q)}\\to W_{\\mathrm{O}(q)}$ induced by $\\theta$,\nwhere ${\\hat{M}}_{\\mathrm{O}(q)}={\\hat{M}}\\times_{\\mathrm{O}(q)}\\mathrm{EO}(q)$ and $W_{\\mathrm{O}(q)}=W\\times_{\\mathrm{O}(q)}\\mathrm{EO}(q)$ denote the Borel constructions of $\\hat{M}$ and $W$, respectively. \nNote that $\\hat{\\theta}:{\\hat{M}}_{\\mathrm{O}(q)}\\to W_{\\mathrm{O}(q)}$ and $\\theta:\\hat{M}\\to W$ \nhave the same fibers and hence the fiber of $\\hat{\\theta}$ is diffeomorphic to $L_0$, where $L_0$ is a principal leaf \nof $\\mathcal{F}$. In addition, ${\\hat{M}}_{\\mathrm{O}(q)}$ is homotopy equivalent to ${\\hat{M}}\/{\\mathrm{O}(q)}=M_0$ \nand $W_{\\mathrm{O}(q)}$ coincides with the Haefliger's classifying space $B$ of ${M_0}\/{\\mathcal{F}}$. \nTherefore, we get  the following fibration (up to homotopy):\n$$L_0\\overset{\\iota_0}{\\rightarrow}M_0\\overset{\\hat{\\theta}}{\\rightarrow}B.$$\n\n\\subsection{Bilinear and quadratic forms over $\\Z_2$}\\label{SS:quadratic}\n\nLet $V$ be a finite dimensional vector space over a field $F$. A quadratic form on $V$ is a map $Q:V\\to F$ \nsuch that $Q(\\lambda v)=\\lambda^2 Q(v)$ for all $\\lambda\\in F$ and $v\\in V$, and moreover, the map \n$B_Q:V\\times V\\to F$ defined by $B_Q(u,v)=Q(u+v)-Q(u)-Q(v)$ is a bilinear form. \nGiven a basis $\\{v_1,\\ldots,v_{\\ell}\\}$ of $V$, it follows that\n\\begin{equation}\\label{eq:quadratic form}\nQ(x_1v_1+\\ldots+x_{\\ell}v_{\\ell})=\\sum_{i=1}^{\\ell} Q(v_i)x_i^2+\\sum_{1\\leq i<j\\leq\\ell} B_Q(v_i,v_j)x_ix_j,\n\\end{equation}\nTwo quadratic forms $Q:V\\to F$ and $Q':V\\to F$ are called \\emph{isometric} (or equivalent) \nif there exists an invertible linear map $f:V\\to V$ such that $Q(v)=Q'(f(v))$ for all $v\\in V$. \n\nFinally, given quadratic forms $Q:V\\to F$ and $Q':V'\\to F$, one defines the \\emph{orthogonal sum} $Q\\oplus Q':V\\oplus V'\\to F$ \nby $(Q\\oplus Q')(v,v'):=Q(v)+Q'(v')$.\n\n\\section{The topology of leaves}\\label{S:topology of leaves}\n\nLet $(M,\\mathcal{F})$ be a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$. \nThe goal is to prove Theorem \\ref{main-thm:leaves-nilpotent}, that the principal leaves are nilpotent manifolds.\n\nWe begin by collecting some of the results proved in \\cite{GGR15} about the fundamental group of the principal leaves of $\\mathcal{F}$.\n\n\\subsection{Known results on the topology of leaves}\\label{SS:known-results}\n\nSince the fundamental group of $M$ does not change if we delete the strata of $\\mathcal{F}$ with codimension $>2$,\nwe can assume that we only have singular strata of codimension $\\leq 2$. Furthermore, it is known that there are no strata \nof codimension one, which reduces $\\mathcal{F}$ to only having strata of codimension two. \n\nLet $\\Sigma_1,\\ldots,\\Sigma_m$ denote the singular strata of $\\mathcal{F}$ of codimension two.\nFor $i=1,\\ldots, m$, choose a singular leaf $L'_i$ in $\\Sigma_i$, and let $L_i$ be a principal leaf at some distance $\\epsilon_i$ \nfrom $L'_i$. For $\\epsilon_i$ small enough, the foot-point projection $\\pi_i:L_i\\to L'_i$ is a circle bundle.\nFix a point $p_i\\in L_i$, and let $[c_i]\\in \\pi_1(L_i,p_i)$ be the element represented by the fiber $c_i$ \nof $\\pi_i$ through $p_i$.\n\nFixing a principal leaf $L_0$ and $p_0\\in L_0$, we can choose, for each $i=1,\\ldots, m$,  a diffeomorphism $h_i:L_i\\to L_0$, \nand define $k_i=(h_i)_*([c_i])\\in\\pi_1(L_0,p_0)$. The group $K$ generated by the elements $k_i$ is then a normal subgroup \nof $\\pi_1(L_0,p_0)$. Furthermore, there exists a homotopy fibration\n\\[\nL_0\\overset{\\iota_0}{\\rightarrow}M_0\\overset{\\hat{\\theta}}{\\rightarrow}B,\n\\]\nas described in Section \\ref{SS:molino}. One has the following (see the proof of Theorem A in \\cite{GGR15}):\n\\begin{enumerate}\n\\item $\\pi_1(L_0,p_0)$ is generated by the subgroup $K$ and the image of the boundary map \n$\\partial:\\pi_2(B,b_0)\\to\\pi_1(L_0,p_0)$.\n\\item $H:=\\im(\\partial)$ is central in $\\pi_1(L_0,p_0)$\n\\item Any two non-commuting generators $k_i$ and $k_j$ of $K$ satisfy $k_ik_j=k_j^{-1}k_i$.\n\\item Let $N\\subseteq K$ be the subgroup generated by the non-central $k_i$'s, and let $Z_{(2)}$ \ndenote the Sylow $2$-subgroup of $Z(K)$. Then $\\pi_1(L_0,p_0)$ is nilpotent, and equal to $A\\times K_2$, \nwhere $A$ is abelian and $K_2=N\\cdot Z_{(2)}$.\n\\end{enumerate}\n\n\\subsection{Proof of Theorem \\ref{main-thm:leaves-nilpotent}}\n\nAs discussed in Section \\ref{SS:known-results}, the principal leaves of $\\mathcal{F}$ have nilpotent fundamental groups. \nAs a first step towards the proof of Theorem \\ref{main-thm:leaves-nilpotent}, we prove that the principal leaves \nare nilpotent spaces:\n\n\\begin{proposition}\\label{P:princ-leaves-nilp}\nSuppose $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$. \nLet $L_0$ denote a principal leaf of $\\mathcal{F}$ and let $p_0\\in L_0$. Then $\\pi_1(L_0,p_0)$ acts trivially on $\\pi_n(L_0,p_0)$ \nfor $n\\geq 2$.\n\\end{proposition}\n\n\\begin{proof}\nLet $[\\gamma]\\in\\pi_1(L_0,p_0)$ and $[\\omega]\\in\\pi_n(L_0,p_0)$. The goal is to prove that $[\\gamma]$\nacts trivially on $[\\omega]$. By the discussion in Section \\ref{SS:known-results}, we may assume that either $[\\gamma]\\in H$ \nor $[\\gamma]=k_i$ for some $i$. \n\\par     \nFirst, consider the case in which $[\\gamma]=k_i$ for some $i$. Note that ${\\bf p}_i:=\\pi_i\\circ h_i^{-1}:L_0\\to L'_i$ \nis a circle bundle whose fiber is represented by $k_i$. This means that $k_i\\in\\ker(({\\bf p}_i)_*)$, where $({\\bf p}_i)_*$\nis the induced map on $\\pi_n$. Hence we have:\n$$({\\bf p}_i)_*([\\gamma]\\cdot[\\omega])=({\\bf p}_i)_*(k_i\\cdot[\\omega])=(({\\bf p}_i)_*(k_i))\\cdot(({\\bf p}_i)_*([\\omega]))=({\\bf p}_i)_*([\\omega]).$$\nBy the long exact sequence of homotopy groups associated to the fibration \n$\\mathbb{S}}\\newcommand{\\Ca}{\\mathrm{Ca}}\\newcommand{\\pp}{\\mathbb{P}^1\\to L_0\\overset{{\\bf p}_i}{\\rightarrow} L'_i$, it follows that the homomorphism \n$({\\bf p}_i)_*$ is injective in $\\pi_n$ for $n\\geq 2$. \nThis, together with $({\\bf p}_i)_*([\\gamma]\\cdot[\\omega])=({\\bf p}_i)_*([\\omega])$, implies that $[\\gamma]$ \nacts trivially on $[\\omega]$.\n\\par\nSuppose now that $[\\gamma]\\in H=\\im(\\partial)$ and choose $[\\beta]\\in\\pi_2(B,b_0)$ such that $[\\gamma]=\\partial([\\beta])$.\nConsider the fibration \n$$L_0\\overset{\\iota_0}{\\rightarrow}M_0\\overset{\\hat{\\theta}}{\\rightarrow}B.$$\nNote that the action of $\\pi_1(L_0,p_0)$ on $\\pi_n(L_0,p_0)$ satisfies $[\\gamma]\\cdot[\\omega]=(\\iota_0)_*([\\gamma])\\cdot[\\omega]$ (see \\cite[Exercise 4.3.10]{Hat02}). Therefore, \n$$[\\gamma]\\cdot[\\omega]=(\\iota_0)_*([\\gamma])\\cdot[\\omega]=(\\iota_0)_*(\\partial([\\beta]))\\cdot[\\omega]=e\\cdot[\\omega]=[\\omega].$$\nThis completes the proof.\n\\end{proof}\n\nMoving to the non-principal leaves, we first prove that every leaf has a virtually nilpotent fundamental group.\n\n\\begin{lemma}\\label{L:other-leaves}\nSuppose $(M,\\mathcal{F})$ is a closed singular Riemannian foliation with principal leaf $L_0$. If $\\pi_1(L_0)$ is virtually nilpotent, \nthen so is the fundamental group $\\pi_1(L)$ of every leaf $L$ of $\\mathcal{F}$.\n\\end{lemma}\n\n\\begin{proof}\nFor any leaf $L$ of $\\mathcal{F}$, the foliated Slice Theorem (cf. Section \\ref{SS:Slice Theorem}) implies that there is a fibration $L_0\\to L$ whose fiber $F$ \nhas finitely many connected components. From the long exact sequence in homotopy one then has\n\\[\n\\pi_1(L_0)\\to \\pi_1(L)\\to \\pi_0(F)\n\\]\nfrom which it follows that $\\pi_1(L)$ is a finite extension of a quotient of $\\pi_1(L_0)$, therefore it is virtually nilpotent as well.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{main-thm:leaves-nilpotent}]\nThe statement about principal leaves has been proved in Proposition \\ref{P:princ-leaves-nilp}, so we now have to only consider non-principal leaves.\n\nGiven a leaf $L$, choose $p\\in L$. Recall that, by the Foliated Slice Theorem (cf. \\ref{SS:Slice Theorem}), there is a locally trivial fibration $\\phi:L_0\\to L$ whose fiber $F$ has finitely many connected components, all diffeomorphic to a principal leaf of the infinitesimal foliation $(\\nu_p L_p,\\mathcal{F}_p)$. Furthermore, the action $\\pi_1(L)\\to \\operatorname{Diff}(F)$ induces an action $\\pi_1(L)\\to \\operatorname{Aut}(\\pi_*(F))$, which factors as $\\pi_1(L)\\stackrel{\\psi}{\\to} \\pi_0(K)\\to \\operatorname{Aut}(\\pi_*(F))$. In particular:\n\\begin{enumerate}\n\\item The subgroup $G_1:=\\ker\\psi \\subseteq \\pi_1(L)$ has finite index in $\\pi_1(L)$ and it acts trivially on $\\pi_*(F)$.\n\\item The fibration induces a map $\\pi_1(L_0)\\stackrel{\\phi_*}{\\to} \\pi_1(L)\\to \\pi_0(F)$. Thus $G_2:=\\phi_*(\\pi_1(L_0))$ is a nilpotent subgroup of $\\pi_1(L)$ with finite index.\n\\end{enumerate}\nConsider $G:=G_1\\cap G_2\\subseteq \\pi_1(L)$, which is by the points above a nilpotent subgroup with finite index. We will now show that $G$ acts nilpotently on each $\\pi_n(L)$, i.e. the \\emph{lower central series} $\\Gamma^m_G(\\pi_n(L))\\subseteq\\pi_n(L)$ defined iteratively by\n\\[\n\\Gamma_G^1(\\pi_n(L))=\\pi_n(L), \\qquad \\Gamma_G^{m+1}(\\pi_n(L))=\\{\\gamma\\cdot \\alpha-\\alpha\\mid \\gamma\\in G,\\,\\alpha\\in  \\Gamma_G^{m}(\\pi_n(L))\\}\n\\]\neventually becomes trivial.\n\nConsider the long exact sequence\n\\[\n\\cdots\\to\\pi_n(F)\\to\\pi_n(L_0)\\stackrel{\\phi_*}{\\to}\\pi_n(L)\\stackrel{\\partial}{\\to}\\pi_{n-1}(F)\\to\\cdots\n\\]\nLet $\\alpha\\in \\pi_n(L)$, and $\\gamma=\\phi_*(\\gamma_0)\\in G$, where $\\gamma_0\\in \\pi_1(L_0)$. Recall that $\\partial(\\gamma\\cdot \\alpha)=\\gamma\\cdot \\partial(\\alpha)$, where the action on the left is $\\pi_1(L)$ acting on $\\pi_*(L)$, while on the right we have the $\\pi_1(L)$-action on $\\pi_*(F)$. Since $G\\subseteq G_1$, we have\n\\[\n\\partial(\\gamma\\cdot\\alpha)=\\partial (\\alpha)\\quad\\Rightarrow \\quad \\partial(\\gamma\\cdot \\alpha-\\alpha)=0\n\\]\nand therefore\n\\[\n\\Gamma_G^2(\\pi_n(L))\\subseteq \\ker (\\partial)=\\phi_*(\\pi_n(L_0))=\\phi_*(\\Gamma_{\\pi_1(L_0)}^1(\\pi_n(L_0))).\n\\]\nFinally, we notice that if $\\alpha=\\phi_*(\\alpha_0)$ with $\\alpha_0\\in \\pi_n(L_0)$ then\n\\[\n\\gamma\\cdot \\alpha=(\\phi_*(\\gamma_0))\\cdot(\\phi_*(\\alpha_0))=\\phi_*(\\alpha_0)\\Rightarrow \\gamma\\cdot \\alpha-\\alpha=\\phi_*(\\gamma_0\\cdot \\alpha_0-\\alpha_0).\n\\]\n\nBy induction on $m$, one then has\n\\[\n\\Gamma_G^{m+1}(\\pi_n(L))\\subseteq \\phi_*\\big(\\Gamma_{\\pi_1(L_0)}^{m}(\\pi_n(L_0))\\big).\n\\]\nSince by Proposition \\ref{P:princ-leaves-nilp}, $\\Gamma_{\\pi_1(L_0)}^{2}(\\pi_n(L_0))=0$, we have $\\Gamma_G^{3}(\\pi_n(L))=0$ which proves that $G$ acts nilpotently on $\\pi_n(L)$, hence finishing the proof.\n\\end{proof}\n\n\\section{Fundamental groups of the principal leaves}\\label{S:fundamental group}\n\nThis section consists of two parts. The first part is devoted to the proof of Theorem \\ref{main-thm:non-abelian part}.\nIn the second part, we provide examples of singular Riemannian foliations whose principal leaves have fundamental groups \nof the form discussed in Theorem \\ref{main-thm:non-abelian part}.\n\nSuppose that $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$. \nFix a principal leaf $L_0$ of $\\mathcal{F}$ and $p_0\\in L_0$. Let $N$ and $K_2$ be the subgroups of $\\pi_1(L_0,p_0)$ \ndiscussed in Section \\ref{SS:known-results}.\n\nConsider the graph $\\Gamma$ with vertices the generators of $N$ and an edge between $k_i$ and $k_j$ \nif and only if $k_ik_jk_i^{-1}=k_j^{-1}$. Note that for every generator $k_i$ of $N$, there exists another generator \nwhich does not commute with $k_i$. Therefore, $\\Gamma$ does not contain any isolated vertices. \nNote moreover that for every connected component $\\Gamma_i$ of $\\Gamma$, all vertices of $\\Gamma_i$ \nsquare to the same element $c_i$. In addition, by proof of Theorem A in \\cite{GGR15}, for any generator $k_i$ of $N$,\nwe have $k_i^4=1$ and $k_i^2$ is central in $K$. Therefore, $c_i$ is a central element of $N$ of order two.\nAltogether, we get that there is a map $C:\\pi_0(\\Gamma)\\to Z(N)$ defined by $C(\\Gamma_i)=c_i$.\n\n\\begin{notation}\\label{N:N_c}\nFrom now on, we fix an element $c$ of $Z(N)$ which is of the form $k_i^2$ for some generator $k_i$ of $N$.\nMoreover, $N_c$ denotes the subgroup of $N$ that is generated by all the vertices in $\\Gamma_c:=C^{-1}(c)$.\n\\end{notation}\n\nRecall that given a group $G$, the \\emph{Frattini subgroup} $\\Phi(G)$ is the intersection of all the maximal subroups of $G$. Furthermore, we recall the following:\n\n\\begin{definition}\\label{def:Generalized ES}\nA $2$-group $G$ is called \\emph{generalized extraspecial} if $\\Phi(G)$ is central, and $\\Phi(G)=[G,G]=\\Z_2$.\n\\end{definition}\n\nWe prove two important properties of the groups $N_c$. \n\n\\begin{lemma}\\label{L:abt-N_c}\nLet $\\{N_c\\}_{c\\in \\textrm{Im}(C)}$ be the collection of groups defined above. Then:\n\\begin{enumerate}\n\\item For $c\\neq c'$, the groups $N_{c}$ and $N_{c'}$ commute.\n\\item Each $N_c$ is a generalized extraspecial $2$-group.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFirst, we prove Statement (1). Let $k_1,\\ldots, k_{\\ell}$ be the generators of $N_c$, and $k'_1,\\ldots, k'_r$ be the generators\nof $N_{c'}$. As vertices of $\\Gamma$, there is no edge between any $k_i$ and any $k'_j$, which means that each $k_i$ commutes with any $k'_j$ in $K$. Hence the result follows.\n\\par\nAs for Statement (2), if $k_1,\\ldots, k_{\\ell}$ denote the generators of $N_c$, then $V={N_c}\/{\\langle c\\rangle}$ \nis isomorphic to $\\Z_2^{\\ell}$ and is generated by $[k_1],\\ldots,[k_{\\ell}]$. \nIt follows that $N_c$ fits into a short exact sequence\n\\[\n1\n\\to \\langle c\\rangle \\to N_c\\to V\\to 1\n\\]\nand in particular one has that both $N_c^2:=\\langle g^2\\mid g\\in N_c\\rangle$ and the commutator subgroup $[N_c,N_c]$ \ncoincide with $\\langle c \\rangle\\simeq \\Z_2$. Therefore, the same is true for the Frattini subgroup $\\Phi(N_c)$\nsince for a $2$-group $G$, one has $\\Phi(G)=G^2\\cdot [G,G]$.\n\\end{proof}\n\nGiven generalized extraspecial groups $G_1$ and $G_2$, with Frattini subgroups generated by $c_1$ and $c_2$, respectively, define the \\emph{central product} $G_1*G_2$ by $G_1*G_2:={(G_1\\times G_2)}\/\\langle (c_1,c_2)\\rangle$. \nThis is again a generalized extraspecial group, since\n\\[\\Phi(G_1*G_2)=\\Phi(G_1)\\times_{\\Z_2}\\Phi(G_2) \\cong \\Z_2.\\]\n\nThe $*$ operation is furthermore associative, and thus it makes sense to define, for a generalized extraspecial group $G$, \nthe central product powers\n\\[\n(G)^{*m}:=\\underbrace{G*G*\\ldots*G}_\\text{m\\textrm{ times}}\n\\]\n\nGeneralized extraspecial 2-groups are, as the name suggests, a generalization of \\emph{extraspecial $2$-groups}, \nthat is 2-groups such that $\\Phi(G)=Z(G)=[G,G]\\cong \\Z_2$. These groups have been thoroughly studied at least since the 60's \\cite{Hal56}. They are extremely simple: an extraspecial group has the form $(Q_8)^{*m}$ or $(Q_8)^{*(m-1)}*D_8$ \nfor some $m\\geq 1$, where $Q_8$ is the quaternion group and $D_8$ is the dihedral group of order $8$\n(cf. Theorem 2.2.11 of \\cite{LGM05}). It then follows from Lemma 3.2 in \\cite{Sta02} that\n\n\\begin{theorem}\nA generalized extraspecial $2$-group is of the form $G\\times \\Z_2^n$, where $G$ is one of\n\\[\nQ_8^{*m},\\qquad Q_8^{*(m-1)}*D_8,\\qquad Q_8^{*(m-1)}*\\Z_4.\n\\]\n\\end{theorem}\n\n\\subsection{The associated quadratic form}\\label{SS:associated quadratic form}\n\nLet $G$ be a generalized extraspecial $2$-group with $\\Phi(G)=G^2=\\langle c\\rangle$,\nand let $V:=G\/{\\langle c\\rangle}$. It is easy to check that $V$ is a vector space over $\\Z_2$. \n\nDefine the function $Q_G:V\\to\\Z_2$ by $Q_G([g])=k$, where $g^2=c^k$.  Since $c$ is central in $G$ and has order two, \nfor any $g\\in G$, we have $(cg)^2=cgcg=c^2g^2=g^2$ and thus $Q_G([cg])=Q_G([g])$. \nTherefore, $Q:=Q_G$ is well-defined and in fact a quadratic form as defined in Section \\ref{SS:quadratic}. \nFurthermore, the bilinear form $B_Q$ associated to $Q$ (cf. Section  \\ref{SS:quadratic}) satisfies \n$$ghg^{-1}h^{-1}=c^{B_Q([g],[h])},~{\\text{for}}~g,h\\in G.$$ \nIn order to see this, note that both $g^2$ and $h^2$ are central elements of $G$. Therefore,\n$$c^{B_Q([g],[h])}=c^{Q([g]+[h])}c^{-Q([g])}c^{-Q([h])}=(gh)^2g^{-2}h^{-2}=ghg^{-1}h^{-1}.$$\n\n\nThe quadratic form of each generalized extraspecial group can be explicitly computed. For this, consider the quadratic forms:\n\\begin{alignat*}{3}\n & H_+: \\Z_2^2\\to \\Z_2 && \\qquad H_-:\\Z_2^2\\to\\Z_2 && \\qquad Q_1:\\Z_2\\to \\Z_2\\\\\n & H_+(x,y)=xy && \\qquad H_-(x,y)=x^2+y^2+xy && \\qquad Q_1(x)=x^2.\n\\end{alignat*}\nWe have the following:\n\n\\begin{proposition}\\label{prop:quad-forms}\nSuppose that $G$ is a generalized extraspecial $2$-group and let $V:= G\/\\Phi(G)$. \n\\begin{enumerate}\n\\item If $G=(Q_8)^{*m}$, then $V\\simeq\\Z_2^{2m}$ and\n\\[\nQ_G=H_-^{\\oplus m}=\\begin{cases}\nH_+^{\\oplus m}& m~{\\rm{even}}\\\\\nH_-\\oplus H_+^{\\oplus (m-1)} & m~{\\rm{odd}}\n\\end{cases}\n\\]\n\\item If $G=(Q_8)^{*(m-1)}*D_8$, then $V\\simeq\\Z_2^{2m}$ and\n\\[\nQ_G=H_-^{\\oplus (m-1)}\\oplus H_+=\\begin{cases}\nH_+^{\\oplus m} & m~{\\rm{odd}}\\\\\nH_-\\oplus H_+^{\\oplus (m-1)}& m~{\\rm{even}}\n\\end{cases}\n\\]\n\\item If $G=(Q_8)^{m}*\\Z_4$, then $V\\simeq\\Z_2^{2m+1}$ and\n\\[\nQ_G=H_+^{\\oplus m}\\oplus Q_1=H_-^{\\oplus m}\\oplus Q_1\\]\n\\item If $G=G'\\times \\Z_2^n$ with $G'$ as in the previous points, then $V\\simeq V'\\oplus \\Z_2^n$ and $Q_G=Q_{G'}\\oplus0^{\\oplus n}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThis proposition follows easily from the following straightforward facts:\n\\begin{enumerate}\n\\item For $G=Q_8$, $G\/\\Phi(G)\\simeq \\Z_2^2$ and $Q_G=H_-$.\n\\item For $G=D_8$, $G\/\\Phi(G)\\simeq \\Z_2^2$ and $Q_G=H_+$.\n\\item For $G=\\Z_4$, $G\/\\Phi(G)\\simeq \\Z_2$ and $Q_G=Q_1$.\n\\item Given $G_1$ and $G_2$ with quotients $V_i=G_i\/\\Phi(G_i)$, one has\n\\[\n(G_1*G_2)\/\\Phi(G_1*G_2)=V_1\\oplus V_2\\quad\\textrm{and}\\quad Q_{G_1*G_2}=Q_{G_1}\\oplus Q_{G_2}.\n\\]\n\\item Given $G$ with quotient $V=G\/\\Phi(G)$, one has\n\\[\n(G\\times \\Z_2^n)\/\\Phi(G\\times \\Z_2^n)\\simeq V\\oplus \\Z_2^n\\quad\\textrm{and}\\quad Q_{G\\times \\Z_2^n}=Q_G\\oplus 0^{\\oplus n}.\n\\qedhere\\]\n\\end{enumerate}\n\\end{proof}\n\n\\begin{remark}\\label{remark:N_c}\nThe group $N_c$ discussed above is generated by elements of order four, that is the $k_i$'s. \nMoreover, for each $k_i$, there exists $k_j$ such that $k_ik_jk_i^{-1}k_j^{-1}=c$.\nThis is reflected in the corresponding quadratic form $Q:V\\to\\Z_2=\\{0,1\\}$ as follows.\nThere exists a basis $\\{v_1,\\ldots,v_{\\ell}\\}$ of $V\\cong\\Z_2^\\ell$ with the property that $Q(v_i)=1$ for all $i$, \nand for each $v_i$, there exists $v_j$ such that $B_Q(v_i,v_j)=1$. We call such quadratic forms \\emph{admissible}. \n\\end{remark}\n\nThe next step consists of understanding which of the quadratic forms in Proposition \\ref{prop:quad-forms} is admissible. \nWe start by reducing the problem to quadratic forms without trivial summands:\n\\begin{lemma}\\label{lem:splitting}\nLet $Q:V\\to\\Z_2$ be a quadratic form. If there exists a splitting $V=V_1\\oplus V_2$ such that $Q$ splits as $Q=q\\oplus 0^{\\oplus n}$ with $Q|_{V_1}=q$ and $Q|_{V_2}=0^{\\oplus n}$, then $Q$ is admissible if and only if $q$ is admissible.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $Q$ is admissible and choose a basis\n$$\\{(v_1,w_1),\\ldots, (v_{m+n},w_{m+n})\\}$$\nof $V_1\\oplus V_2$ with the property that $Q(v_i,w_i)=1$, and for every $(v_i,w_i)$ there exists $(v_j,w_j)$\nwith $B_Q((v_i,w_i),(v_j,w_j))=1$. After possibly rearranging basis elements of $V_1\\oplus V_2$, \nwe may assume that $\\{v_1,\\ldots, v_m\\}$ forms a basis for $V_1$. \nSince $Q(v_i,w_i)=q(v_i)$ and $B_Q((v_i,w_i),(v_j,w_j))=B_q(v_i,v_j)$, the basis $\\{v_1,\\ldots, v_m\\}$ of $V_1$ \nis admissible for $q$. On the other hand, if $\\{v_1,\\ldots, v_m\\}$ is admissible for $q$ and $\\{w_1,\\ldots, w_n\\}$ \nis any basis of $V_2$, then \n$$\\{(v_i,{\\bf{0}})\\mid  i=1,\\ldots, m\\}\\cup\\{(v_1,w_j)\\mid j=1,\\ldots, n\\},$$\nforms an admissible basis for $Q$.\\end{proof}\n\nWe now apply Lemma \\ref{lem:splitting} to classify the admissible quadratic forms.\n\n\\begin{theorem}\\label{thm:admissible}\nAny admissible quadratic form $Q:\\Z_2^{\\ell}\\to\\Z_2$ is isometric to one of the following, up to orthogonal sum \nwith $0^{\\oplus n}$:\n\\begin{equation}\\label{eq:admissible}\nH_+^{\\oplus m}~(m\\geq 2),\\qquad H_-\\oplus H_+^{\\oplus m-1},\\qquad H_+^{\\oplus m}\\oplus Q_1~(m\\geq 2).\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nSince the quadratic forms over $\\Z_2$ are classified (see Proposition \\ref{prop:quadratic form}), \nwe only need to check the admissibility condition. By Lemma \\ref{lem:splitting}, we may assume that $Q$ does not split \nas $q\\oplus 0^{\\oplus m}$. We break the proof into cases.\n\n\\smallskip\n\n{\\textbf{Case 1:}} $Q=H_-\\oplus H_+^{\\oplus m-1}$, where $2m=\\ell$. The quadratic form $Q$ is given by\n$$Q(x,y,z_1,z_2,\\ldots,z_{2m-2})=x^2+xy+y^2+z_1z_2+\\ldots+z_{2m-3}z_{2m-2}.$$\nLet $e_1,\\ldots,e_{\\ell}$ denote the standard basis elements of $\\Z_2^{\\ell}$ and consider the following basis:\n\\begin{align*}\n & v_1=e_1+e_2, \\quad v_2=e_3+e_4,\\quad\\ldots\\quad  v_{m}=e_{2m-1}+e_{2m},\\\\\n & v_{m+1}=e_1,\\\\\n & v_{m+2}=e_1+e_3, \\quad v_{m+3}=e_1+e_5,\\quad \\ldots\\quad v_{2m}=e_1+e_{2m-1}.\n\\end{align*}\nThen $Q(v_i)=1$ for all $i$, and for every $v_i$, there exists $v_j$ such that $B_Q(v_i,v_j)=1$. Hence $Q$ is admissible. \n\n\\smallskip\n\n{\\textbf{Case 2:}} $Q=H_+^{\\oplus m}$, where $2m=\\ell$. Note that the only element of $\\Z_2^2$ \nthat is mapped to $1$ by $H_+$ is $(1,1)$. Therefore, $H_+$ is not admissible. However, if $m\\geq 2$, \nthen the following basis of $\\Z_2^{\\ell}$ is admissible for $Q$:\n\\begin{align*}\n  &v_1=e_1+e_2, \\quad v_2=e_3+e_4\\quad \\ldots\\quad v_{m}=e_{2m-1}+e_{2m},\\\\\n &v_{m+1}=e_1+e_{2m-1}+e_{2m},\\\\\n & v_{m+2}=e_1+e_2+e_4, \\quad v_{m+3}=e_3+e_4+e_6,\\ldots \\\\\n &v_{2m}=e_{2m-3}+e_{2m-2}+e_{2m}.\n\\end{align*}\n\n\\smallskip\n\n{\\textbf{Case 3:}} $Q=H_+^{\\oplus m}\\oplus Q_1$, where $m\\geq 2$ and $2m+1=\\ell$. \nLet $\\{v_1,\\ldots, v_{2m}\\}$ denote the basis constructed for $H_+^{\\oplus m}$ in Case 2, \nand let $v_{2m+1}=e_1+e_{2m+1}$. Then $\\{v_1,\\ldots, v_{2m+1}\\}$ forms an admissible basis for $Q$.\n\nFor $Q=H_+\\oplus Q_1$, the elements with non-zero quadratic form are $(1,1,0)$, $(1,0,1)$, $(0,1,1)$, $(0,0,1)$. \nAmong these, the only vectors with non-zero bilinear form are the first three, which are linearly dependent \nand thus do not form a basis. Hence $H_+\\oplus Q_1$ is not admissible.\n\\end{proof}\n\nRecall that the group $N_c$ (cf. Notation \\ref{N:N_c}) is a generalized extraspecial group with an admissible basis. \nFrom the previous theorem, we then get:\n\n\\begin{corollary}\\label{C:N_c}\nIf $N_c$ is a generalized extraspecial group whose corresponding quadratic form is admissible,\nthen, up to a direct product with copies of $\\Z_2$, the group $N_c$ is isomorphic to one of the following:\n\\begin{equation}\\label{eq:N_c}\n(Q_{8})^{*m_1},\\qquad (Q_{8})^{*(m_1-1)}*D_8~(m_1\\geq 2),\\qquad (Q_{8})^{*m_1}*\\Z_4~(m_1\\geq 2).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nThis follows trivially by comparing the quadratic forms in Proposition \\ref{prop:quad-forms} with the classification of admissible quadratic forms in Theorem \\ref{thm:admissible}.\n\\end{proof}\n\nFinally, we prove Theorem \\ref{main-thm:non-abelian part}. \n\n\n\n\\begin{proof}[Proof of Theorem \\ref{main-thm:non-abelian part}]\nFix $p_0\\in L_0$. As discussed in Section \\ref{SS:known-results}, the non-abelian part $K_2$ of $\\pi_1(L_0,p_0)$ \nis a $2$-group of the form $K_2=N\\cdot Z_{(2)}$, where $N$ is generated by the non-central generators of $K$\nand $Z_{(2)}$ denotes the Sylow $2$-subgroup of $Z(K)$. Furthermore, by the discussion in Section \\ref{S:fundamental group}, $N=N_{c_1}\\cdot \\ldots\\cdot N_{c_k}$, where the elements $c_i\\in Z(K)$ have order two.\nBy Corollary \\ref{C:N_c}, each $N_{c_i}$ is of the form $G_i\\times \\Z_2^{a_i}$, where $G_i$ is one of the groups \nlisted in Equation \\eqref{eq:N_c}. Let $a=\\sum_i a_i$. Finally, since all the groups $N_{c_i}$ commute \nwith one another by Lemma \\ref{L:abt-N_c}, one has $N_{c_i}\\cap N_{c_j}\\subseteq Z(N_{c_i})\\cap Z(N_{c_j})$, \nand $Z(N_{c_i})\\subseteq Z(K_2)$. Therefore\n\\[\nK_2\\cong (Z_{(2)}\\times \\prod_{i=1}^kN_{c_i})\/Z'=(Z_{(2)}\\times \\Z_2^a\\times \\prod G_i)\/Z',\n\\]\nwhere $Z'\\subseteq Z_{(2)}\\times \\prod_i Z(N_{c_i})$ is the subgroup of $K_2$ generated by the intersections $H_{ij}=N_{c_i}\\cap N_{c_j}$ and $H_{0j}=Z_{(2)}\\cap N_{c_j}$. Since the groups $H_{ij}$, $H_{0j}$ are all abelian \nand central, commute with one another, and have elements of order $2$ or $4$ (because $Z(N_{c_i})=\\Z_2^{a_i}\\times \\Z_2$ \nor $\\Z_2^{a_i}\\times \\Z_4$) it follows that $Z'=\\Z_2^\\alpha\\times Z_4^\\beta$ for some $\\alpha$ and $\\beta$.\n\\end{proof}\n\n\n\\subsection{Examples of fundamental groups of principal leaves}\\label{SS:examples}\n\nThe family of examples below shows that the non-abelian groups $G_i$ discussed in Theorem \\ref{main-thm:non-abelian part} \nactually arise as fundamental groups of principal leaves of homogeneous singular Riemannain foliations.\n\nLet $\\{e_1,\\ldots, e_n\\}$ be the standard basis of $\\mathbb{R}}\\newcommand{\\C}{\\mathbb{C}}\\newcommand{\\HH}{\\mathbb{H}^n$. The Clifford algebra $Cl(0,n)$ on $\\mathbb{R}}\\newcommand{\\C}{\\mathbb{C}}\\newcommand{\\HH}{\\mathbb{H}^n$\nis defined as the associative algebra generated by $e_1,\\ldots, e_n$, where multiplication of the elements $e_i$ \nis given by:\n$$e_i^2=-1,\\hspace{0.3cm}e_ie_j=-e_je_i.$$\nConsider the subset $E(n)=\\{\\pm e_{i_1}\\ldots e_{i_{2k}}\\}\\subseteq Cl(0,n)$ containing products of even numbers of the $e_i$'s. This is easily seen to be a group under the product of $Cl(0,n)$. In \\cite{CHM09}, Czarnecki, Howe, and McTavish prove that for the action of $G=\\mathrm{SO}(n)\\times\\mathrm{SO}(n)$ \non $M_{n\\times n}(\\mathbb{R}}\\newcommand{\\C}{\\mathbb{C}}\\newcommand{\\HH}{\\mathbb{H})$ defined by $(g,h)\\cdot A=g^TAh$, the fundamental group of a principal orbit is of the form $E(n)\\times\\Z_2$. In this section, we investigate the structure of $E(n)$.\n\n\\begin{lemma}\nLet $G_{0,n-1}$ be the group defined by generators $-1,e_1,\\ldots,e_{n-1}$ and relations\n$$(-1)^2=1,\\qquad (e_i)^2=-1,\\hspace{0.5cm}[e_i,e_j]=-1~(i\\neq j),\\qquad [e_i,-1]=1.$$\nThen the groups $E(n)$ and $G_{0,n-1}$ are isomorphic.\n\\end{lemma}\n\n\\begin{proof}\nWe have: \n$$G_{0,n-1}=\\{\\pm e_{i_1}\\ldots e_{i_{\\ell}}\\mid 1\\leq i_j\\leq n-1, e_i^2=-1, e_ie_j=-e_je_i\\}.$$\nGiven an ordered set $I=(i_1,\\ldots, i_m)$ with indices $i_j$ in $\\{1, \\ldots, n-1\\}$, let $e_I=e_{i_1}\\ldots e_{i_m}$.\nNotice that if $I=(i_1,\\ldots, i_m)$ and $J=(j_1,\\ldots, j_p)$, then $e_Ie_J=e_{I\\cup J}$,\nwhere $I\\cup J=(i_1,\\ldots, i_m,j_1,\\ldots, j_p)$. Now, define the map $\\psi:G_{0,n-1}\\to E(n)$ by\n\\begin{equation*}\n\\psi(e_I)=\n\\begin{cases}\ne_I & |I|~{\\text{even}}\\\\\ne_{I\\cup\\{n\\}} & |I|~{\\text{odd}}\n\\end{cases}\n\\end{equation*}\nFirst, we claim that $\\psi(e_Ie_J)=\\psi(e_I)\\psi(e_J)$ for multi-indices $I$ and $J$.\n\n\\smallskip \n\n{\\textbf{Case 1.}} $|I|$ and $|J|$ are both even. In this case, we have:\n$$\\psi(e_Ie_J)=\\psi(e_{I\\cup J})=e_{I\\cup J}=e_Ie_J=\\psi(e_I)\\psi(e_J).$$\n\n\\smallskip \n\n{\\textbf{Case 2.}} $|I|$ and $|J|$ are both odd. In this case, we have:\n$$\\psi(e_Ie_J)=\\psi(e_{I\\cup J})=e_{I\\cup J}=e_Ie_J=e_Ie_J(-e_ne_n)=e_{I\\cup\\{n\\}}e_{J\\cup\\{n\\}}=\\psi(e_I)\\psi(e_J).$$\n\n\\smallskip \n\n{\\textbf{Case 3.}} If $|I|$ is even and $|J|$ is odd, then\n$$\\psi(e_Ie_J)=\\psi(e_{I\\cup J})=e_{I\\cup J\\cup\\{n\\}}=e_Ie_{J\\cup\\{n\\}}=\\psi(e_I)\\psi(e_J).$$\n\n\\smallskip \n\n{\\textbf{Case 4.}} If $|I|$ is odd and $|J|$ is even, then\n$$\\psi(e_Ie_J)=\\psi(e_{I\\cup J})=e_{I\\cup J\\cup\\{n\\}}=e_{I\\cup\\{n\\}}e_J=\\psi(e_I)\\psi(e_J).$$\n\n\\smallskip\n\nTherefore, $\\psi$ is a homomorphism. It is easy to see that $\\psi$ is injective, and hence an isomorphism\nsince the groups $G_{0,n-1}$ and $E(n)$ have the same order.\n\\end{proof}\n\nThe groups $G_{0,n-1}$ have been classified by Salingaros \\cite{Sal81,Sal82,Sal84} (cf. \\cite{AVW18}).\nWe use this classification to write the group $E(n)\\cong G_{0,n-1}$ as a central product.\nThis gives rise to the following list for fundamental groups of the principal orbits of the $G$-action on $M_{n\\times n}(\\mathbb{R}}\\newcommand{\\C}{\\mathbb{C}}\\newcommand{\\HH}{\\mathbb{H})$:\\\\\n\\begin{equation*}\nE(n)\\times\\Z_2\\cong\n\\begin{cases}\n((Q_8)^{*\\frac{n-4}{2}}*D_8)\\times \\Z_2^2 & n\\equiv 0~({\\text{mod}}~8)\\vspace{0.1cm}\\\\\n(Q_8)^{*\\frac{n-1}{2}}\\times\\Z_2 & n\\equiv 1, 3~({\\text{mod}}~8)\\vspace{0.1cm}\\\\\n((Q_8)^{*\\frac{n-2}{2}}*\\Z_4)\\times\\Z_2 & n\\equiv 2, 6~({\\text{mod}}~8)\\vspace{0.1cm}\\\\\n(Q_8)^{*\\frac{n-2}{2}}\\times\\Z_2^2 & n\\equiv 4~({\\text{mod}}~8)\\vspace{0.1cm}\\\\\n((Q_8)^{*\\frac{n-3}{2}}*D_8)\\times\\Z_2 & n\\equiv 5, 7~({\\text{mod}}~8)\n\\end{cases}\n\\end{equation*}\n\nWe do not know, however, whether \\emph{all} groups in Theorem \\ref{main-thm:non-abelian part} do in fact arise \nas fundamental groups of principal leaves in a simply connected manifold.\n\\smallskip\n\n\\section{Virtually nilpotent fundamental group}\\label{S:nilpotent fundamental group}\n\nIn this section, we consider singular Riemannian foliations $(M,\\mathcal{F})$, where the fundamental group of $M$ is virtually nilpotent.\nAs the following example shows, the fundamental group of a principal leaf is not necessarily nilpotent in this case.\n\n\\begin{example}\nLet $\\hat{M}={\\mathbb{C}}^2\\times{\\mathbb{S}}^1$ and consider the homogeneous foliation $\\hat{\\mathcal{F}}$ \non $\\hat{M}$ induced by the linear action of $T^3=T^2\\times S^1$. Let $M={\\hat{M}}\/{\\Z_2}$, where \nthe non-trivial element $g$ of $\\Z_2$ acts by $g\\cdot(z_1,z_2,t)=({\\bar{z}}_1,{\\bar{z}}_2,t+\\frac{1}{2})$.\nNote that $M$ inherits a singular Riemannian foliation $\\mathcal{F}={\\hat{\\mathcal{F}}}\/{\\Z_2}$.\n\\par\nThe manifold $M$ is orientable, and is homotopy equivalent to ${\\mathbb{S}}^1$. In particular, $M$ is nilpotent.\nHowever, the principal leaf of $\\mathcal{F}$ is $T^3\/{\\Z_2}$ which has fundamental group\n$$G=\\Z^2\\rtimes\\Z=\\langle a, b, c: cac^{-1}=a^{-1}, cbc^{-1}=b^{-1}, ab=ba\\rangle.$$\nSince $G_{\\ell}=\\langle a^{2^{\\ell}}, b^{2^{\\ell}}\\rangle$ for any $\\ell$, $G$ is not nilpotent.\n\\end{example}\n\nNevertheless, in what follows, we prove that the fundamental groups of the leaves contain a nilpotent subgroup of finite index. \n\n\\begin{notation}\nThroughout the rest of this section, $L_0$ denotes a principal leaf of $\\mathcal{F}$. Furthermore, we fix $p_0\\in L_0$, \nand $K=\\langle k_1,\\ldots, k_m\\rangle$ denotes the normal subgroup of $\\pi_1(L_0,p_0)$ discussed at the beginning \nof Section \\ref{S:fundamental group}. Recall that there is a homotopy fibration\n$$L_0\\overset{\\iota_0}{\\rightarrow}M_0\\overset{\\hat{\\theta}}{\\rightarrow}B.$$\nwhich induces a long exact sequence\n\\[\n0\\to H\\to \\pi_1(L_0,p_0)\\stackrel{(\\iota_0)_*}{\\to} \\pi_1(M_0,p_0)\\stackrel{\\hat{\\theta}_*}{\\to} \\pi_1(B,b)\\to 1,\n\\]\nwhere $H=\\partial(\\pi_2(B))$, as well as an action of $\\pi_1(B,b)$ on $L_0$. Denote by $\\hat{K}$ the group generated by $H$ \nand $c\\cdot K$, for $c\\in \\pi_1(B,b)$. Notice that for every $\\gamma\\in \\pi_1(M_0,p_0)$ with $c=\\hat{\\theta}_*(\\gamma)$, \nand every $g\\in\\pi_1(L_0,p_0)$, $(\\iota_0)_*(c\\cdot g)=\\gamma (\\iota_0)_*(g)\\gamma^{-1}$.\n\\end{notation}\n\\begin{lemma}\\label{lemma:central}\nLet $(M,\\mathcal{F})$ be a closed singular Riemannian foliation on a compact Riemannian manifold $M$.\nIf $\\pi_1(M)$ is $n$-step nilpotent, then $(\\pi_1(L_0,p_0))_{n+1}\\subseteq \\hat{K}$, where $(\\pi_1(L_0,p_0))_{n+1}$ \ndenotes the $(n+1)$-th group in the lower central series of $\\pi_1(L_0,p_0)$.\n\\end{lemma}\n\n\\begin{proof}\nSince removing strata of codimension $> 2$ does not change the fundamental group of $M$, we can assume that $M$ \nonly contains singular strata of codimension $\\leq 2$. In particular, we use the notation and results in Section \n\\ref{SS:known-results}.\n\nLetting $\\iota:L_0\\to M$ denote the inclusion, one then has\n\\[\n\\iota_*((\\pi_1(L_0,p_0))_{n+1})\\subseteq (\\pi_1(M,p_0))_{n+1}=1.\n\\]\nTherefore, given any curve $\\alpha$ representing an element of $(\\pi_1(L_0,p_0))_{n+1}$, there exists a disk \n$\\bar{\\iota}:\\mathbb{D}^2\\to M$ extending $\\iota(\\alpha)$. By transversality, this can be deformed to only intersect, \ntransversely, the singular strata $\\Sigma_1,\\ldots,\\Sigma_m$ of codimension 2, and the intersection consists of finitely many points $\\{q_1,\\ldots q_r\\}$ with $q_j\\in \\Sigma_{i_j}$. For each $j=1,\\ldots, r$, let $q'_j$ be a point in $\\bar\\iota(\\mathbb{D}^2)$ \nclose to $q_j$, let $u_j$ be a curve in $\\bar\\iota(\\mathbb{D}^2)$ connecting $p_0$ to $q'_j$, and let $\\psi_j$ a small loop \nin $\\bar\\iota(\\mathbb{D}^2)$ based at $q'_j$, turning once around $q_j$. Finally, let $\\gamma_j=u_j\\star \\psi_j\\star u_j^{-1}$. Then:\n\\begin{enumerate}\n\\item For each $i=1,\\ldots, r$, $[\\gamma_j]\\in \\pi_1(M_0,p_0)$ is conjugate to $(\\iota_0)_*(k_{i_j})$ \nwith $k_{i_j}\\in K\\subseteq \\pi_1(L_0,p_0)$. By the discussion before the proposition, it follows that $[\\gamma_j]=(\\iota_0)_*(c_j\\cdot k_{i_j})$ for some $c_j\\in \\pi_1(B,b)$.\n\\item $(\\iota_0)_*[\\alpha]=[\\gamma_1]\\star\\cdots\\star[\\gamma_r]=(\\iota_0)_*((c_1\\cdot k_{i_1})\\star\\cdots\\star (c_r\\cdot k_{i_r}))$ in $\\pi_1(M_0,p_0)$.\n\\end{enumerate}\n\nSince $H=\\ker((\\iota_0)_*)$, it follows that $[\\alpha]=h((c_1\\cdot k_{i_1})\\star\\cdots\\star (c_r\\cdot k_{i_r}))$ \nfor some $h\\in H$. In particular, $[\\alpha]\\in \\hat{K}$, and therefore $(\\pi_1(L_0,p_0))_{n+1}\\subseteq \\hat{K}$.\n\\end{proof}\n\nWe are finally ready to prove that if $(M, \\mathcal{F})$ is a closed singular Riemannian foliation with $\\pi_1(M)$ virtually nilpotent, then the fundamental group of every leaf is virtually nilpotent as well.\n\n\\begin{proof}[Proof of Theorem \\ref{main-thm:virtually nilpotent}]\nNotice that if $\\pi:\\hat{M}\\to M$ is a finite cover, and $(\\hat{M},\\hat{\\mathcal{F}})$ is the lifted singular Riemannian foliation, one has that a leaf $\\hat{L}\\in \\hat{\\mathcal{F}}$ has virtually nilpotent fundamental group if and only the corresponding leaf $\\pi(\\hat{L})\\in \\mathcal{F}$ does. Therefore, up to replacing $M$ with a finite cover $\\hat{M}$, we can assume that $\\pi_1(M)$ is nilpotent.\n\nLet $L_0$ be a principal leaf, and consider the Hurewicz homomorphism $h:\\pi_1(L_0,p_0)\\to H_1(L_0;\\Z)$ and let $G=h^{-1}(2\\cdot H_1(L_0;\\Z))$. \nClearly, $G$ has finite index in $\\pi_1(L_0,p_0)$, Since $\\pi_1(L_0,p_0)\/G\\cong H_1(L_0;\\Z)\/2\\cdot H_1(L_0;\\Z)$ \nis finite. We claim that if $\\pi_1(M)$ is $n$-step nilpotent, then $G$ is $(n+1)$-step nilpotent.\n\nBy Lemma \\ref{lemma:central}, $G_{n+1}\\subseteq G\\cap \\hat{K}$. The proof is complete once we prove that $G$ \ncommutes with $\\hat{K}$. Notice that $\\hat{K}$ is generated by $H$, and elements of the form $c\\cdot k_i$ for $c\\in \\pi_1(B,b)$ \nand $k_i$ one of the generators of $K$. Recall that $H$ is central in $\\pi_1(L_0,p_0)$ (in particular, $G$ commutes with $H$),\nand for each $g\\in \\pi_1(L_0,p_0)$, $gk_ig^{-1}=k_i^{\\pm 1}$. Since $\\pi_1(B,b)$ acts on $\\pi_1(L_0,p_0)$ \nby group automorphisms, it also follows that for every $g\\in \\pi_1(L_0,p_0)$, $g(c\\cdot k_i)g^{-1}=(c\\cdot k_i)^{\\pm 1}$.\n\nNotice that if $g(c\\cdot k_i)g^{-1}=(c\\cdot k_i)^\\epsilon$ (for $\\epsilon=\\pm1$), then $g^{-1}(c\\cdot k_i)g=(c\\cdot k_i)^{\\epsilon}$ as well. In particular, for every $g_1,g_2\\in \\pi_1(L_0,p_0)$, and every $(c\\cdot k_i)\\in \\hat{K}$, one has:\n\\[\n[g_1,g_2]\\cdot (c\\cdot k_i)[g_1,g_2]^{-1}=(c\\cdot k_i).\n\\]\nThe main observation is that, by definition, any element $g\\in G$ can be written as $g=g_3^2[g_1,g_2]\\cdots [g_{2k-1},g_{2k}]$ \nfor some $g_1,\\ldots g_{2k}\\in \\pi_1(L_0,p_0)$ and therefore, for any generator $(c\\cdot k_i)$ of $\\hat{K}$, one has:\n\\begin{align*}\ng(c\\cdot k_i)g^{-1}=&g_3^2[g_1,g_2]\\cdots [g_{2k-1},g_{2k}](c\\cdot k_i)[g_{2k-1},g_{2k}]^{-1}\\cdots [g_1,g_2]^{-1}g_3^{-2}\\\\\n=&g_3^2(c\\cdot k_i)g_3^{-2}=g_3(c\\cdot k_i)^\\epsilon g_3^{-1}\\\\\n=&(c\\cdot k_i)^{\\epsilon^2}=(c\\cdot k_i).\n\\end{align*}\nTherefore, $G$ commutes with $\\hat{K}$ and hence $G_{n+2}=[G,G_{n+1}]\\subseteq [G,\\hat{K}]=\\{1\\}$.\n\nThis proves that the principal leaves of $\\mathcal{F}$ have virtually nilpotent fundamental group. The corresponding statement for the non-principal leaves then follows from Lemma \\ref{L:other-leaves}.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDirect photon production is of special importance in relativistic heavy-ion collisions (for reviews, see \\cite{Gale:2009gc,Chatterjee:2009rs}). Since photons couple with the quark-gluon plasma (QGP) only through the electromagnetic interaction, their mean free path is much longer than the dimension of the reaction zone and they can escape from it carrying the information of QGP at the instance of their emission. These photons are called thermal photons, whose observation is regarded as an indirect evidence for the formation of hot thermalized states of quarks and gluons in relativistic heavy-ion collisions \\cite{Shuryak:1978ij,Kapusta:1991qp}. There are still other various photon sources over the time evolution in a heavy-ion collision event in addition to thermal photons from the QGP phase. For example, at the very initial stage, partonic hard collisions (such as the gluon Compton scattering $gq\\to \\gamma  q$ and quark annihilation $q\\bar q \\to \\gamma g$) produce the photons, which are called prompt photons \\cite{Owens:1986mp,Turbide:2007mi,Klasen:2013mga}. At the late stage after hadronization, scattering processes between hadrons also produce photons \\cite{Kapusta:1991qp,Holt:2015cda}. Thus, identifying and quantifying photon sources become a vital issue for extracting direct information on QGP from the photon observation.\n\n\nMeasurements of the photons in relativistic heavy-ion collisions have been performed both at RHIC and LHC, and the results of transverse momentum ($p_T$) distributions $dN^\\gamma\/dp_\\perp^2 dy$ and elliptic flow coefficients ($v_2^\\gamma$) at mid-rapidity $y\\sim 0$ are reported \\cite{Adare:2014fwh,Adare:2015lcd,Adam:2015lda,Lohner:2012ct} (see also \\cite{David:2019wpt} for recent experimental review). From the exponential slopes of the $p_T$ distributions, ``{\\it effective temperatures}'' of produced photons at RHIC ($\\sqrt{s_{NN}}=200~$GeV) and LHC ($\\sqrt{s_{NN}}=2.76~$TeV) are found to be $T_{\\rm eff}\\sim 230~$MeV and $T_{\\rm eff}\\sim 300$~MeV, respectively. Very interestingly, the elliptic flow $v_2^\\gamma$ of the direct photon distribution is found be as large as that of hadrons. Notice that although the ``decay photons'' produced in hadron decays (mainly due to $\\pi^0\\to 2 \\gamma$) have been subtracted from the total yield to obtain the direct photon yield, it is still a mixture of photons from various sources. Despite many theoretical attempts to reproduce these experimental results, any theoretical model so far seems to be incapable of explaining the photon data adequately. For example, hydrodynamic models which explain the hadronic spectra and anisotropic flow very well, tend to underpredict the amplitude of photon elliptic flow (e.g., see \\cite{Chatterjee:2021gwa}). In particular, even an up-to-date hydrodynamic model calculation with various possible effects included still underestimates the photon yield \\cite{Paquet:2017wji,Gale:2021emg}. Other model calculations such as blast-wave type fireball model and ideal hydrodynamic model also give rise to smaller thermal photon yield and elliptic flow than the experimental data by PHENIX an ALICE \\cite{vanHees:2014ida}. Furthermore, the parton-hadron-string dynamics (PHSD) model shares the same tendency in direct photon yield and elliptic flow \\cite{Linnyk:2015tha}. Meanwhile, the pre-equilibrium Glasma stage is discussed in Refs.~\\cite{Berges:2017eom,Monnai:2019vup} as a possible photon production source, but its contribution is also likely to suppress the elliptic flow.\n\nThis situation is referred to as the \"direct photon puzzle\".\nThe difficulty comes from two seemingly contradictive aspects of the photon data, large yields and strong collective flow. The large yield could be attributed to early stage of the evolution with higher temperatures, while the strong collective flow prefers large photon emission at later stage when momentum anisotropy of QGP is well developed. Therefore, it is not easy to explain these two points within a single theoretical model. However we should note that there is a reservation about the experimental results because the large yield of photons measured by PHENIX has not been confirmed by STAR.\n\nGiven this situation, it should be very worthwhile to examine another source of photons which has been overlooked so far and is inherent to late stages of the QGP time evolution. This brought us to think of a possibility of photon radiations at hadronization of QGP, which is, in fact, natural from the viewpoint of ordinary electromagnetic plasmas. It is well-known that an electromagnetic plasma radiates numerous photons when it goes back to an atomic gas. This process is called the ``radiative recombination\" and is seen in various astrophysical situations (for an overview see \\cite{graham2012recombination, hahn1997electron}. This is natural because photon emission is advantageous for satisfying energy conservation in the formation of bound states (electrically neutral atoms). We can expect similar phenomena occur in the hadronization processes, where valence quarks and antiquarks form bound states. In the present paper, we formulate photon production at the hadronization stage in analogy with the radiative recombination in electromagnetic plasmas and investigate properties of the produced photons. Photon production during QCD phase transition and hadronization has been attracting much attention indeed, and been discussed in different frameworks, for example, in \\cite{vanHees:2014ida,Campbell:2015jga,Young:2015adw}. \n\n\nThe present paper is organized as follows. In the next section, we explain the radiative recombination in ordinary electromagnetic plasma and comment on similarities to and differences from the QGP. Then in Sec.~III, we provide the basic theoretical framework for radiative recombination applied to hadronization.\nNumerical results are presented in Sec.~IV, where we include the thermal photon contributions obtained by a hydrodynamic simulation to compare the results to the observed data.\nDiscussions and summary are given in Sec.~V.  \n\nIn Appendix A, we present photon production formulas in radiative recombination model.\nIn Appendix B, we give a brief description of thermal photon calculation in a hydrodynamic model \\cite{Miyachi-Nonaka}.\n\n\n\n\\section{Radiative recombination in electromagnetic plasmas}\n\nIn this section, we explain what is known about radiative recombination in ordinary plasmas and in other relevant processes, which are helpful to understand how the radiative recombination in QGP could be formulated. The primary and simplest example reads an electromagnetic plasma made of electrons and protons. When the temperature is decreased, the plasma decays and neutralizes into a hydrogen gas through the microscopic mechanism of the radiative recombination $e^-+p^+\\to {\\rm H}^0 +\\gamma$. Emission of a photon compensates the energy difference between the initial continuum state and the final bound state. The ordinary plasma flashes when it decays. This process is well-known in plasma physics and is also called ``free-bound transition\" \\cite{fujimoto2004plasma}. The secondary example is the glow discharge which is seen as a typical picture of a plasma. The third example is radiative recombination in the early universe: There was a ``recombination era\" and without treating the radiative recombination we would not be able to accurately evaluate the ``recombination temperature'' which is essential for understanding the cosmic microwave background (CMB) data \\cite{weinberg2008cosmology}. The fourth example is gas nebulae, such as Orion Nebula, which have beautiful radiations containing continuum spectra due to the radiative recombination \\cite{osterbrock2006astrophysics}. Lastly, in addition to these examples induced by the electromagnetic interaction, we also have examples in nuclear reactions. We know that in the sun there are two important processes called the ``pp chain\" and the ``CNO cycle,\" both of which include formation of bound states accompanied by a photon emission (such as $D + p \\to {}^{3}{\\rm He}+\\gamma$ in the pp chain and $p+{}^{12}{\\rm C}\\to {}^{13}{\\rm N}+\\gamma$ in the CNO cycle) \\cite{clayton1968principles}. All these examples indicate that radiation is naturally inherent to the formation of bound states, which suggests the possibility of photon emissions at hadronization, provided that hadronization can be modeled as a coalescence process of valence partons.\n\n\n\nThere are two key equations for the description of radiative recombination in electromagnetic plasmas \\cite{rybicki2008radiative,weinberg2008cosmology}. One is the Kramers-Milne relation which relates recombination cross section to that of its inverse process (photo-ionization $\\gamma + A \\to e^- + A^+$) and corresponds to the detailed balance relation in thermal equilibrium states. The other is the Saha equation which gives ionization ratio $X=n_{\\rm ion}\/(n_{\\rm ion}+n_{\\rm atom})$ between the atom and ion numbers, $n_{\\rm atom}, n_{\\rm ion}$, as a function of a temperature when the electrons, ions, and atoms are in thermal equilibrium. The Kramers-Milne relation is useful because the photo-ionization rate is easily measured by experiments, and the Saha equation applies to an ordinary plasma because one can control the lifetime of a plasma much longer than the microscopic time scale of radiation reaction. These relations are, unfortunately, not suitable to the hadronization from QGP in heavy-ion collisions since it should be treated as a nonequilibrium process. Hadronization occurs in a small lump of QGP at the time scale of the strong interaction, and therefore the photons are just emitted without re-absorption. To this situation we cannot use the relations which assume a system in bulk equilibrium. \n\n\nWhen the density of an electromagnetic plasma is relatively high, another type of recombination will be possible. It is the ``three-body recombination\" $2 e^- + A^+\\to e^- + A$, where the energy conservation is satisfied by the spectator electron. \nIf one defines the effective photon emission rate at some density, it will be reduced by the presence of the three-body, or in general, multi-body recombination. We may expect similar phenomena in the hadronization process. Formation of bound states without photon emission will be possible if the valence partons interact with other particles in the medium. We will be able to absorb this kind of effects into an effective recombination rate including density dependence. In fact, as we will discuss later,  since we treat the recombination rate as a parameter, we expect that such kind of effects are included in the overall normalization parameter which is determined to fit the experimental data. Alternatively, effects of multi-body recombination without a photon emission will be described by ``off-shell'' valence partons. However, within our framework in the present analysis, it is not easy to define off-shell partons in QGP. \n\nLastly, we comment on a very similar process in $e^+e^-$ collisions.  Recall that the $e^+e^-$ collisions have been used to discover new particles by changing the invariant mass. Pronounced resonances such as $\\rho, \\omega, \\phi$ and $J\/\\psi$ mesons are measured with energies {\\it below} the invariant mass of the $e^+e^-$ system. This is called the {\\it radiative return}, and is quite important for the analysis of $R$-ratio around threshold and $(g-2)$ of leptons (for reviews, see \\cite{Actis:2010gg, Druzhinin:2011qd}). Notice that the formation of a resonance below the $e^+ e^-$ invariant mass is accompanied by photon emissions and the typical radiative return is represented as \n$\ne^+ + e^- \\to \\text{hadrons} + n\\gamma\\, ,\n$\nwhere the number $n$ of emitted photons is not necessarily one, $n\\ge 1$. In particular, a clean process with a single meson and a single photon \n$\ne^++e^-\\to \\text{meson} + \\gamma\n$ \nis experimentally measured. For example, $J\/\\psi$ production was observed at BaBar \\cite{Aubert:2003sv}, and more recently $\\chi_c$ and $\\eta_c$ at BESIII \\cite{Ablikim:2014hwn, Ablikim:2017ove}. The counterpart in purely QED case such as $e^+e^-\\to \\mu^+\\mu^-\\gamma$ can be perturbatively calculated (though quite tedious), and one can study interplay between the initial state radiation and the final state radiation. On the other hand, hadron production suffers from ambiguity related to the coupling between the virtual photon and a composite hadron. For example, $e^++e^- \\to \\gamma^* \\to \\text{meson}+\\gamma$ includes the transition of a virtual photon into a meson which may be phenomenologically described by the vector meson dominance.  Heavy quarkonium production will have less ambiguity, but the NRQCD formalism developed for the calculation of heavy-quarkonium production involves nonperturbative matrix elements (see for example, \\cite{Chung:2008km, Sang:2009jc}). \n\n\nThere are Monte-Carlo generators for the radiative return called PHOKHARA  (for real photon emission) and EKHARA (for virtual photon emission) \\cite{Czyz:2017veo}. These generators treat radiation by the vertices like $meson^*\\to meson + \\gamma\\, ({\\rm or}\\ \\gamma^*)$ which are given by effective lagrangian \\cite{Czyz:2012nq}. Here, $meson^*$ could be a virtual (or off-shell) meson. This kind of picture will be useful in our problem. Another lesson from the radiative return is that the final state could involve several hadrons, typically light mesons such as pions and kaons. For example, final states with four mesons like $\\pi^+ \\pi^- K^+ K^-$ were extensively studied \\cite{Actis:2010gg}. We expect similar multiple hadron production in the radiative hadronization. As we will discuss in the next section, we will adopt the ``Recombination model\" and modify it so that it allows photon emission. This recombination model provides the number of produced mesons by the overlapping between a $q\\bar q$ state and a meson state, and implicitly assumes a single meson production from a $q\\bar q$ state. However, the multiple hadron production seen in the $e^+e^-$ collisions suggests that the one-to-one correspondence between a $q\\bar q$ state and a meson should not exactly hold. Still, we may be able to effectively absorb such effects into the recombination rate. We should be aware that the overall recombination rate could contain many different physical effects. Having said all these suggestions and caveats, we are now ready for the problem how to formulate the radiative recombination at hadronization. \n\n\n\n\n\\section{Radiative hadronization: formulation}\n\nHadronization is a nonperturbative process because it takes place around the critical temperature $T\\sim T_c$ and the typical strong coupling $\\alpha_s(T\\sim T_c)$ is not small. It is also a nonequilibrium phenomenon in the sense mentioned in the previous section. One of the possible frameworks to describe the hadronization will be to work in an effective theory that includes both hadronic and constituent-quark degrees of freedom (see \\cite{Young:2015adw} for an analysis in the quark-meson coupling theory). Within this framework, one can compute the hadron production cross sections similarly to the radiation return. We will, however, take an alternative approach. In fact, as already commented before, we know a simple and phenomenologically successful model for hadronization which is based on the coalescence of constituent quark degrees of freedom. It is the recombination  model  and we utilize it to describe the radiative hadronization. Below we first explain briefly the basic strategy of the recombination model, and then discuss how to modify it to include the photon emission. \n\n\n\n\n\\subsection{Recombination model}\n\nThe recombination\/coalescence models provide a phenomenological description of hadronization for hadron production in the intermediate $p_T$ region ($2 \\simle p_T \\simle 5$~GeV\/c) and give a natural explanation for intriguing phenomena such as the anomalous baryon\/meson ratio and constituent quark number (CQN) scaling of the elliptic flow \\cite{Fries:2003vb,Greco:2003xt,Molnar:2003ff,Hwa:2003bn} (for a review, see \\cite{Fries:2008hs}). There are several recombination\/coalescence models with some differences \\cite{Fries:2003kq, Greco:2003mm,Hwa:2004ng}, and we will adopt the ReCo model that was developed by Duke group \\cite{Fries:2003kq}. \n\n\nThe ReCo model starts by defining the number of hadrons that one can find in the quark\/antiquark distributions. For example, the total number of mesons is defined as an overlap between the meson state $|M;\\bm{P}\\rangle$ and the reduced two-body density matrix $\\hat \\rho_{ab}$ which represents the partonic system with partons, $a$ and $b$, undergoing hadronization:\n\\begin{eqnarray}\nN_M=\\sum_{ab}\\int \\frac{d^3\\bm{P}}{(2\\pi)^3}\\, \\langle M; \\bm{P}|\\, \n\\hat \\rho_{ab} \\, |M;\\bm{P}\\rangle    \\, ,\n\\end{eqnarray}\nwhere summation is taken over all the possible combinations of partons $a, b$ that have nonzero overlap with a mesonic state.\nIn this sense, the ReCo model simply projects the partonic picture of the QGP onto the hadron picture, and does not describe dynamical processes of hadron formation. However, since the formula includes  matrix elements like $\\langle M;\\bm{P}|\\bm{r}_1,\\bm{r}_2\\rangle$ with $|\\bm{r}_1,\\bm{r}_2\\rangle$ being a state having a quark at $\\bm{r}_1$ and an antiquark at $\\bm{r}_2$, it allows for an intuitive understanding of coalescence processes\\footnote{Note however that this matrix element will be interpreted as a quark-antiquark component of a meson wavefunction.} like  $q\\bar q\\to M(\\text{meson})$ and $qqq\\to B(\\text{baryon})$. \n\n\nAfter some manipulations (see \\cite{Fries:2003kq} for details), one obtains the momentum distribution of mesons made of partons $a$ and $b$ as \n\\begin{equation}\nE\\frac{dN_M}{d^3\\bm{P}}=C_M \\int_\\Sigma \\frac{P\\cdot u(R)}{(2\\pi)^3}\\int_0^1 dx\\,  w_a(R;x\\bm{P})\\, \\left|\\phi^{}_M(x)\\right|^2\\, w_b(R;(1-x)\\bm{P}) \\, , \n\\label{dN\/dP}\n\\end{equation}\nwhere $P^\\mu=(E,\\bm{P})$ is the four-momentum of the meson $M$, and $R$ is a four-vector specifying a point on the hypersurface $\\Sigma$ where the hadronization takes place, and $u^\\mu (R)$ is a unit vector orthogonal to the hypersurface $\\Sigma$ at $R$. The function $\\phi_M(x)$ is the light-cone wavefunction of a meson with $x$ being the momentum fraction of one of the two quarks, and $w_a(R;x\\bm{P})$ is a one-particle phase space distribution of parton $a$,\n\n\nWe assume the longitudinal boost-invariant expansion (Bjorken expansion) of the QGP so that the hadronization hypersurface $\\Sigma$ has a constant longitudinal proper time $\\tau=\\sqrt{t^2-z^2}=\\, $const and a point $R^\\mu$ on it is specified as\n$$\nR^\\mu = (t,x,y,z)=(\\tau \\cosh \\eta,\\, \\rho \\cos \\phi,\\, \\rho \\sin \\phi ,\\,  \\tau \\sinh \\eta)\\, \n$$\nwith the space-time rapidity $\\eta$, the transverse radial coordinate $\\rho$, and the azimuthal angle $\\phi$. Then the forward normal vector $u^\\mu(R)$ orthogonal to the hypersurface $u\\cdot dR|_{\\tau= \\rm const.}  =0$, is given by $u^\\mu(R)=(\\cosh \\eta,0,0,\\sinh \\eta)$.\n\n\nRegarding $\\phi_M(x)$, we expect that our results are insensitive to its details and take $|\\phi_M(x)|^2=\\delta(x-1\/2)$ for analytic evaluation and $\\phi_M(x) =\\sqrt{30}\\, x(1-x)$ for numerical evaluation. Notice that both examples of the wavefunction have a peak at $x=1\/2$, and therefore $w_a(R,\\tfrac{1}{2}\\bm{P})w_b(R,\\tfrac{1}{2}\\bm{P})\\sim {\\rm e}^{-P\/T}$ gives rise to the dominant configuration in the $x$ integration. \nThe overall factor $C_M$ counts state degeneracy. The corresponding formula for baryon production has the factor $C_B$ and the wavefunctions of the three-quark state should have a peak around $x=1\/3$, and therefore $w_a(R,\\tfrac{1}{3}\\bm{P})w_b(R,\\tfrac{1}{3}\\bm{P})w_c(R,\\tfrac{1}{3}\\bm{P})\\sim {\\rm e}^{-P\/T}$. Thus, this model predicts that the ratio of the proton to the pion yield at the common $P_T$ at mid-rapidity, $R_{p\/\\pi}\\equiv \\frac{dN_p}{d^2P_Tdy}\\big{\/}\\frac{dN_{\\pi^0}}{d^2P_Tdy}$, is essentially given by a ratio $C_B\/C_M$, which amounts to $\\sim 2$. This is indeed consistent with the experimental result known as the anomalous baryon\/meson ratio, which is in contrast to the expectation from parton fragmentation processes in perturbative QCD, $R_{p\/\\pi}\\sim 0.2$.\n\n\nFor $w_a(R;x\\bm{P})$, \nwe assume that at the onset of hadronization the quarks\/antiquarks are in local thermal equilibrium with a fluid flow velocity $\\bar v^\\mu(R)$, which we parameterize as \n\\begin{equation}\n\\bar v^\\mu(R)=(\\cosh \\eta^{}_L \\cosh \\eta^{}_T,\\, \\sinh \\eta^{}_T \\cos \\phi, \\, \\sinh \\eta^{}_T \\sin \\phi,\\, \\sinh \\eta^{}_L \\cosh \\eta^{}_T)\\, , \\label{normalized_flow_vector}\n\\end{equation}\nwhere $\\eta^{}_L$ and $\\eta^{}_T$ are longitudinal and transverse flow rapidities, respectively. This four-velocity is normalized as $\\bar v_\\mu \\bar v^\\mu=1$.\nIn the Bjorken expansion, which we assume in this work, the longitudinal flow rapidity $\\eta_L$ is identified with the space-time rapidity $\\eta$, i.e., $\\eta^{}_L=\\eta$, and the longitudinal flow velocity is $v_L=\\tanh \\eta = z\/t$. On the other hand, the transverse rapidity $\\eta_T$ is related to the transverse flow velocity $v^{}_T=\\sqrt{v_x^2+v_y^2}$ by\n\\begin{equation}\n  v^{}_T=\\tanh \\eta^{}_T \\, ,\n  \\label{transverse_velocity}\n\\end{equation}\nat mid-rapidity ($\\cosh \\eta_L = 1$).\nWe assume that $v_T$ is independent of the transverse radius $\\rho$ for computational simplicity.\n\n\nBy using this flow velocity $\\bar v^\\mu(R)$, we take the one-body phase space distribution of parton $a$ as \n\\begin{equation}\nw_a(R;p)=\\gamma_a \\, {\\rm e}^{-p\\cdot \\bar v(R)\/T} {\\rm e}^{-\\eta^2 \/2\\Delta^2}\nf(\\rho, \\phi)\\, ,\n\\label{phase_space_dist}\n\\end{equation}\nwhere $\\gamma_a$ is the fugacity factor of parton $a$.\nThe factor ${\\rm e}^{-\\eta^2 \/2\\Delta^2} f(\\rho, \\phi)$ describes the spatial profile of the hot medium.\nFor central collisions, one may simply assume a constant profile $f(\\rho,\\phi)=\\theta(\\rho_0-\\rho)$\nwithin the transverse radius $\\rho_0$ of the fireball at the recombination time $\\tau$.\nIn more general cases, the transverse profile will be adjusted to reproduce the collision-centrality dependence of observed meson yields.\nMeanwhile, the $\\eta$-dependence may be ignored as far as the mid rapidity region is concerned. These approximations will be used in analytic evaluations of the ReCo model below.\n\n\n\nWe have assumed that the quark momentum distribution in $w_a$ has the thermal profile\n${\\rm e}^{-p \\cdot \\bar v(R)\/T}$\nof temperature $T$, boosted by the collective flow $\\bar v(R)$.\nWe introduce elliptic anisotropy in this momentum distribution by a weak modulation of the transverse flow rapidity $\\eta_T$\naround the mean $\\overline{\\eta}_T$:\n\\begin{align}\n   \\eta_T(\\phi;p_T)=\\overline{\\eta}_T (1 - h(p_T)\\cos 2\\phi) \\, .\n\\label{etaT_phi}\n\\end{align}\nThe modulation amplitude $h(p_T)$ is assumed to be $p_T$-dependent:\n\\begin{align}\n    h(p_T)=\\frac{\\alpha}{1+(p_T\/p_0)^a}\n\\label{hpt-profile}\n\\end{align}\nwith $\\alpha=(1-r)\/(1+r)$ fixed by the transverse aspect ratio $r$ of the almond-shape collision zone,\nand with $a$ and $p_0$ the constant parameters controlling the momentum dependence.\nWe then calculate the parton distribution eq.~(\\ref{phase_space_dist}),\nto find the elliptic flow coefficient $v_2^a(p^{}_T)$ of parton $a$ defined by \n\\begin{eqnarray}\nw_a(R;p)=\\overline w_a(R;p)\\Big(1+2v_2^a(p^{}_T)\\cos 2\\phi \\Big)\\, ,\n\\end{eqnarray}\nwhere $\\overline w_a(R;p)$ is the part independent of the azimuthal angle $\\phi$.\nInserting this distribution into the formula (\\ref{dN\/dP}), we can compute the elliptic flow coefficient for\na meson transverse momentum ($P_T$) distribution at mid-rapidity:\n\\begin{equation}\n  v_2^M(P_T)\\equiv \\langle \\cos 2\\Phi \\rangle^{}_{P_L=0}\n  =\\frac{{\\displaystyle \\int d\\Phi \\cos 2\\Phi \\left(\\frac{dN_M}{d^2P_TdP_L}\\Big|_{P_L=0}\\right)} }\n        {{\\displaystyle \\int d\\Phi  \\left(\\frac{dN_M}{d^2P_TdP_L}\\Big|_{P_L=0}\\right)}}\\, ,\n\\end{equation}\nwhere $\\Phi$ is the azimuthal angle of the produced meson momentum.\nIf we adopt the $\\delta$-function approximation for the light-cone wavefunction $|\\phi_M(x)|^2=\\delta(x-1\/2)$\nand assume a universal elliptic flow coefficient for all quark flavors,\n$v_2^q(p_T)\\equiv v^a_2(p_T)$,\nthen the elliptic flow of the meson momentum distribution is analytically evaluated as \n\\begin{eqnarray}\nv_2^M(P_T)=\\frac{2\\, v^q_2(\\frac12 P_T)}{1+2\\, v^q_2(\\frac12 P_T)^2}\n\\, ,\n\\end{eqnarray}\nwhich simplifies further for $v^q_2(p_T)\\ll 1$ to\n\\begin{eqnarray}\nv_2^M(P_T)\\simeq 2v^q_2(P_T\/2)\\, .\n\\end{eqnarray}\nSimilarly, for baryons one finds \n\\begin{eqnarray}\nv_2^B(P_T)\\simeq 3v^q_2(P_T\/3)\\, .\n\\end{eqnarray}\nTherefore, the elliptic flow coefficient of a hadron with $n$ constituents satisfies\nthe following scaling:\n\\begin{eqnarray}\nv_2^h(P_T)\\simeq nv^q_2(P_T\/n)\\, .\n\\label{eq:CQNscaling}\n\\end{eqnarray}\nIf we plot $v_2^h(P_T)\/n$ as a function of $P_T\/n$ for various hadrons, the results will collapse into a single curve which is determined by the quark  elliptic flow coefficient $v_2^q$. This ``CQN scaling\" is indeed observed in experimental data, and is regarded as one of the evidences for the quark recombination in hadronization and also for formation of a thermalized QGP in relativistic heavy-ion collision experiments. Nevertheless, as is obvious from the above derivation of the scaling, it appears only in an idealized situation and a certain deviation is expected from the scaling limit even when the recombination mechanism dominates in hadronization. Such deviations will have different sources, from which we can extract physical information on hadronization.\n\n\n\nAs we emphasized before, the recombination model describes the meson (baryon) formation as a 2-to-1 (3-to-1) process. If we take this literally, it implies that the model violates the energy conservation law because the invariant mass of the initial state with two (three) partons cannot be the same as the energy of a bound state. In Ref.~\\cite{Fries:2003kq}, it is argued that the energy conservation would be preserved by the effects of interactions with the medium (which generates a width in parton dispersion) and that omission of explicit treatment of such effects would not significantly affect the bulk features of hadron production.\nThe medium effect on the recombination reminds us of the three-body recombination in electromagnetic plasmas, where one of the three is just a spectator to keep the energy conservation.\n\n\nBut, at the same time, we notice another process which satisfies the energy conservation law in electromagnetic plasmas. It is the radiative recombination. The counterpart process in hadronization should be also possible.\\footnote{While it is possible to make energy conserved within a dynamical model including the effects of resonances as developed in \\cite{Ravagli:2007xx}, we will take a different picture. } In the next subsection we will discuss how to modify the ReCo model so that it describes the radiative hadronization. \n\n\n\\subsection{Radiative hadronization model}\n\nWe modify the ReCo model so as to allow for a photon emission, which we call radiative ReCo model.\nThen, the meson formation process becomes a 2-to-2 process,\n\\begin{equation}\n  q+\\bar q \\to M+\\gamma\n  \\, ,\n  \\label{radiative_hadronization}\n\\end{equation}\nwhich fulfills the energy conservation law. A similar modification for baryon formation with a photon emission: $qqq\\to B + \\gamma$ should be also possible.\n\n\n\\begin{figure}[t] \n\\begin{center}\n\\vspace*{-1mm}\n   \\includegraphics[width=0.7\\hsize]{NewReCo.pdf}\n\\end{center}\n\\caption{Radiative ReCo model with a photon emission.}\n\\label{fig:ReCo}\n\\end{figure}\n\n\nIn our radiative ReCo model, we re-interpret the original ReCo model \\cite{Fries:2003kq} as a tool for picking up a ``preformed state'' consisting of two partons and assume the preformed state emits a photon to form a bound state (see Fig.~\\ref{fig:ReCo}). Notice that we do not consider this preformed state as any physical resonance but just as an intermediate state in radiative meson production.\n\nThe preformed state consisting of two partons with momenta $p_1^\\mu=(E_1, \\bm{p}_1)$ and $p_2^\\mu=(E_2, \\bm{p}_2)$,\nhas the invariant mass $M_*$ and momentum $\\bm{P}$ determined by\n\\begin{eqnarray}\nE \\equiv \\sqrt{M_*^2+\\bm{P}_{}^2}&=&\\sqrt{m_1^2+\\bm{p}_1^2}\n+\\sqrt{m_2^2+\\bm{p}_2^2}\\, ,\\\\\n\\bm{P}&=&\\bm{p}_1 + \\bm{p}_2\\, ,\n\\end{eqnarray}\nwhere $m_1$ and $m_2$ are the constituent quark masses.\nThe invariant mass $M_*$ is a function of $\\bm{p}_1$ and $\\bm{p}_2$ and is evidently larger than $m_1+m_2$:\n$$\nM_*\\ge m_1+m_2\\, .\n$$\nOn the other hand, in the constituent quark picture, the mass of the bound state $M$ should be smaller than $m_1+m_2$ by the binding energy: $M < m_1+m_2$. The surplus energy of $M_* - M>0$ should be carried away by a photon emission here.\nThen, the number of the photons emitted in the formation of a meson $M$ reads\n\\begin{equation}\nE_\\gamma \\frac{d N_\\gamma}{d^3k}=\\kappa \\int dM_*\\, \\varrho(M_*)\\int d^3P \n\\left(\\frac{dN_{M_*}}{d^3P}\\right)\\left(E_\\gamma \\frac{dn_\\gamma(k; M_*,P)}{d^3k }\\right) \\, ,\n\\label{photon_distribution}\n\\end{equation}\nwhich means that it is given by the product of the number of preformed states and the photon distribution emitted from a preformed state. We explain each ingredient below.\n\nFirst of all, ${dN_{M_*}}\/{d^3P}$ is the number of the preformed states which is characterized by momentum ${\\bm P}$ and invariant mass $M_*$. We evaluate this part by the original ReCo model. Although we do not know the light-cone wavefunction of the preformed state, we expect that the results are insensitive to its details as we already commented concerning the original ReCo model. We use the same light-cone wavefunctions as in the original ReCo model. \n\nNext, $E_\\gamma {dn_\\gamma(k; M_*,P)}\/{d^3k}$ corresponds to the photon distribution emitted from the preformed state moving with the momentum $\\bm{P}$. We adopt a tree picture $M_*\\to M+\\gamma$ which is the leading order with respect to QED coupling $\\alpha$. We assume no specific polarization of the preformed states, and the photon distribution is treated as isotropic in the rest frame of the preformed state. More explicitly, we use the following photon distribution:\n\\begin{equation}\n  E_\\gamma \\frac{dn_\\gamma}{d^3 k}\n  = \\frac{1}{4\\pi k_0}\\, \\delta (E_{\\gamma _{\\rm CM}} - k_0)\n  = \\frac{M_*}{4\\pi k_0}\\, \\delta (k \\cdot P - k_0 M_*), \n\\label{photon_CM_dist}\n\\end{equation}\nwhere $E_{\\gamma _{\\rm CM}}$ is the photon energy\nand $k_0\\equiv (M^2_*-M^2)\/(2M_*)$ is the momentum magnitude of the photon and the meson in the center-of-mass (CM) frame of the preformed state ($M_*$-rest frame) with $M$ being the mass of the accompanying meson. The photon distribution is normalized as $\\int dn_\\gamma =1$.\n\n\nWe also introduced $\\varrho(M_*)$ to represent an invariant mass  distribution of the preformed states of two partons. It should have a support for $M_*\\ge m_1+m_2$. Considering the thermal distributions of quarks and antiquarks, we expect that the number of preformed states will rapidly decrease with increasing $M_*$. Therefore, we will replace it with the threshold contribution, {\\it i.e.}, $\\varrho(M_*)= \\delta(M_*-(m_1+m_2))$ in this paper.\n\n\nFinally and importantly, we comment on the overall factor $\\kappa$ which is introduced to reflect other possible effects on radiative hadronization. Consider the recombination process mediated by a preformed state $q+\\bar q \\to M_* \\to X$ where $X$ stands for any physical final states, including the radiative hadronization $X=M+\\gamma$.\nIn general, however, $X$ can be multiple hadron states (with photons), as we discussed previously in relation to the radiation return in $e^+e^-$ collisions.\nOnce we compute all possible diagrams for the decay of $M_*$, we are able to define the ``branching ratio\" for the radiative hadronization which corresponds to the factor $\\kappa$. Since the transition probability for $M_*\\to M+\\gamma$ would be proportional to the QED coupling $\\alpha=1\/137$, one may naively expect that $\\kappa$ would be of the order of $\\alpha$. On the other hand, we also know empirically that the CQN scaling works very well up to several GeV of meson momentum $p_T$. This fact suggests that a single meson formation would be the dominant process over multiple meson production. If so, the single photon emission attached to this dominant process may have different value of $\\kappa$ from the naive expectation.\nTherefore, we leave the overall factor $\\kappa$ as a parameter to be determined\nby the experimental data.\\footnote{One can consider gluon radiation,\n  $M+g$, but the gluons are strongly interacting and will be re-absorbed into medium.}\n\n\nWe remark here that the number of mesons is given by the same formula as eq.~\\eqref{photon_distribution},\nwith the last factor replaced by the meson distribution emitted from the preformed state:\n\\begin{equation}\nE_M \\frac{d N_M^{\\rm radReCo}}{d^3K}= \\kappa \\int dM_*\\, \\varrho(M_*)\\int d^3P \n\\left(\\frac{dN_{M_*}}{d^3P}\\right)\n\\left(E_M \\frac{dn_M(K; M_*,P)}{d^3K}\\right) \\, ,\n\\label{meson_distribution}\n\\end{equation}\nwhere $K^\\mu$ is a momentum of the produced meson, satisfying $P_\\mu=K_\\mu + k_\\mu^\\gamma$. The distribution $dn_M\/d^3K$ can be defined in a similar way to the photon case. \n\n\nRecall that the original ReCo model naturally explains the CQN scaling.\nIn our modified ReCo model, on the other hand, the scaling should appear at the level of the distribution of preformed states $M_*$ in the integrand of eq.~(\\ref{meson_distribution}) and\nthe meson production accompanying a photon emission may violate the CQN scaling to some extent.\nWe check this point both analytically and numerically in this paper.\nBut we emphasize here that the main contribution to hadron yield at around 2 GeV is given by the original ReCo model\nand the radiative hadronization \\eqref{meson_distribution} is a subdominant process of order $\\kappa$ at most.\nMoreover, since the photon carries away a fraction of the preformed-state momentum, the meson spectrum of the radiative hadronization is shifted to the lower momentum region and therefore at a given momentum $p_T$ its yield is more suppressed than the value simply expected by the factor $\\kappa$.\nHowever, we stress here that to photon production the radiative hadronization can give a significant contribution.\n\n\n\n\\subsection{Characteristics of the radiative hadronization}\n\n\n\nIn order to understand characteristics of the radiative hadronization, let us evaluate the momentum distributions of photons (\\ref{photon_distribution}) and mesons (\\ref{meson_distribution}) under certain approximations.\n\n\n\\subsubsection{Distribution of the preformed states}\n\n\nWe compute the number of preformed states by using the formula (\\ref{dN\/dP}) of the original ReCo model \\cite{Fries:2003kq} and we recap the formulas of\nReCo model here. The transverse momentum spectrum of the preformed state at mid rapidity $\\eta= 0$ is given by  \\cite{Fries:2003kq}\n\\begin{equation}\nE_{M_*}\\left.\\frac{dN_{M_*}}{d^3P}\\right|_{\\eta=0}\\sim C_{M_*} M_{*T} \\frac{\\tau A_T}{(2\\pi)^3} \\; 2\\gamma_a\\gamma_b I_0\\left(\\frac{P_T\\sinh \\eta_T}{T_{\\rm reco}}\\right)\n\\int_0^1 dx|\\phi_{M_*}(x)|^2 k_{M_*}(x,P_T)\n\\end{equation}\nwith \n\\begin{equation}\nk_{M_*}(x,P_T) \\equiv K_1\\left(\\frac{\\cosh \\eta_T}{T_{\\rm reco}}\\left\\{\\sqrt{m_a^2+x^2P_T^2}+\\sqrt{m_b^2+(1-x)^2P_T^2}\\right\\}\\right)\\, ,\n\\end{equation}\nwhere $M_{*T}=\\sqrt{P_T^2+M_*^2}$ is the transverse mass,\n$A_T$ is the transverse area of the parton system at hadronization, representing the collision geometry, and $I_0(x)$ and $K_1(x)$ are the modified Bessel functions.\nThe parameter $T_{\\rm reco}$ is the recombination temperature at which quark recombination $q+\\bar q\\to M_*$ occurs.\nFor analytic evaluation, we simply take $|\\phi_{M_*}(x)|^2=\\delta(x-1\/2)$\nand perform the integration over $x$ for pion production:\n$$\n\\int_0^1 dx|\\phi_{M_*}(x)|^2 k_{M_*}(x,P_T)\\simeq K_1\\left(\\frac{\\cosh \\eta_T}{T_{\\rm reco}}M_{*T}\\right)\\, ,\n$$\nwhere  $M_{*T}$ appears in the argument because we have taken $m_a=m_b=m$\nand $M_*= 2m$ (recall that we adopt $\\varrho(M_*)=\\delta (M_*-(m_1+m_2))$). By using the asymptotic forms of the modified Bessel functions $I_0(z)\\sim {\\rm e}^z\/\\sqrt{2\\pi z}$ and $K_1(z)\\sim \\sqrt{\\pi\/2z}\\ {\\rm e}^{-z}$ for large $z\\gg 1$ and large $P_T$ approximation $M_{*T}\\simeq P_T$,\nwe find the following approximate form for the $P_T$ distribution:\n\\begin{equation}\nE_{M_*}\\left.\\frac{dN_{M_*}}{d^3P}\\right|_{\\eta=0}\\sim \n{\\rm e}^{-P_T \/ T_{\\rm eff}^*}\\, ,\n\\label{M*pt_distribution}\n\\end{equation}\nwhere $T_{\\rm eff}^*$ is the effective temperature for the preformed state defined by\n\\begin{equation}\nT_{\\rm eff}^*=T_{\\rm reco}\\ {\\rm e}^{\\eta^{}_T}=T_{\\rm reco} \\, \\sqrt{\\frac{1+v^{}_T}{1-v^{}_T}}\\label{eff_temp}\n\\; .\n\\end{equation} \nThe recombination temperature $T_{\\rm reco}$ is identified with\nthe hadronization temperature in the original ReCo model.\nThe multiplicative factor $e^{\\eta_T}>1$ reflects the effect of the nonzero transverse flow of the quarks,\nwhich blue-shifts the inverse slope parameter of the preformed state distribution\nfrom $T_{\\rm reco}$ to $T_{\\rm eff}^*$.\nFor example, this factor $T_{\\rm eff}^*\/T_{\\rm reco}$ amounts to  $\\sim $1.7 for $v_T=0.5$ and $2$ for $v_T=0.6$. \n\n\n\\subsubsection{Transverse momentum distributions of photons and mesons}\n\nGiven the distribution of the preformed states, let us discuss the\nphoton distribution eq.~(\\ref{photon_distribution}) at mid-rapidity $k_L=0$.\nWe can perform the integration over $\\Phi$ in eq.~(\\ref{photon_distribution}) with the $\\delta$-function\nin the photon distribution (\\ref{photon_CM_dist}) in the laboratory frame,\n\\begin{align}\n  E_\\gamma  \\frac{dn^\\gamma}{d^2k_T dk_L}\n  &=  \\frac{M_*}{4\\pi k_0}\\delta(\n  k E_* - k_T P_T \\cos(\\Phi-\\phi) - k_0 M_*)\n  ,\n\\end{align}\nwhere $\\Phi$ ($\\phi$) is the azimuthal angle of the preformed state (photon) from the reaction plane,\nas shown in Fig.~{\\ref{fig:decay}.\nThen we obtain\n\\begin{align}\n  \\left .\n  E_\\gamma  \\frac{dN_\\gamma}{d^2 k_{T} dk_L}\n  \\right |_{k_L=0}  &=\n\\int_{-P_{L\\rm max}}^{P_{L\\rm max}} dP_L \\int_{P_{T\\rm min}}^{P_{T\\rm max}} d P_T \\, \n\\frac{dN_*}{d^2P_T dP_L}\\,\n\\frac{1}{4\\pi k_0}\n \\frac{2M_*}{k_T |\\sin \\theta|}\n \\, ,\n\\label{PT_integral}\n\\end{align}\nwhere \n\\begin{align}\n  \\cos \\theta =\\cos(\\Phi-\\phi)=\n  \\frac{k E_{M*}  -k_0 M_*}{k_T P_T} \n  \\, .\n  \\label{eq:theta}\n\\end{align}\nThe integration ranges of the longitudinal and transverse momenta,\n$\\pm P_{L\\rm max}$ and $P_{T\\rm min,max}$, of the preformed state are determined by decay kinematics (See Appendix A).\n\n\n\\begin{figure}[t] \n\\begin{center}\n  \\includegraphics[width=0.4\\textwidth,bb=0 0 600 600]{Decay-v3.png}\n\\hspace{0.05\\textwidth}\n\\includegraphics[width=0.35\\textwidth,bb=0 0 450 300]{Decay-align-v4.png}\n\\end{center}\n\\caption{Left: Kinematics of the photon emission from the preformed state, ${\\boldsymbol P} \\to {\\boldsymbol k} + {\\boldsymbol K}$.\n  Right: Momentum boost from the CM to the laboratory frame\n  for the photon momentum parallel (top) and anti-parallel (bottom) to ${\\boldsymbol P}$.\n}\\vspace{-2mm}\n \\label{fig:decay}\n\\end{figure}\n\n\n\n\nWe derive here approximate formulas valid for large photon momentum $k_T \\gg M_*$.\nThe distribution of the preformed states (\\ref{M*pt_distribution}) is a steeply-falling\nfunction of $P_T$ for large $P_T \\gg T_{\\rm eff}^*$,\nand therefore dominant contribution to the integral comes from the lower end of the $P_T$ integration\nwith $P_L \\simeq 0$.\nIn other words,\nit comes from the configurations in which ${\\boldsymbol P}$ is almost parallel to ${\\bm k}$, \n{\\it i.e.}, $\\cos \\theta \\sim 1$ and $P_L \\simeq 0$.\nIn this configuration,\nthe momentum $k_T$ is simply related with the CM-frame momentum $k_0$ by the transverse boost along $P_T = M_* \\sinh y_T^*$ via\n\\begin{align}\n  k_T=k_0 \\cosh y_T^* + k_0 \\sinh y_T^*  \\sim 2k_0\\frac{P_T}{M_*}\n  =(1-\\frac{M^2}{M_*^2}) P_T\n\\label{mom-shift}\n\\end{align}\nas $\\cosh y_T^* \\sim \\sinh y_T^*=P_T\/M_*$.\nThe momentum deficit is carried by the accompanying meson emitted in the opposite direction in the\n$M_*$-rest frame: $K_T= -k_0 \\cosh y_T^* + \\sqrt{M^2+k_0^2} \\sinh y_T^*\n\\sim\n(M^2\/M_*^2)P_T ,\n$\nand $k_T + K_T = P_T$ holds as it should (Fig.~\\ref{fig:decay} top-right). \n\nSince the number of the radiated photons is proportional to that\nof preformed states, the photon distribution should behave as\n(See Appendix A for more details)\n\\begin{align}\n\\left .  E_\\gamma \\frac{dN_\\gamma}{d^2 k_T dk_L}\n\\right |_{k_L=0} \\sim\n\\exp \\left ( -\\frac{P_{T}}{T^*_{\\rm eff}(M_*)} \\right ) \n\\sim\n       \\exp \\left ( -\\frac{k_{T}}{T^\\gamma_{\\rm eff}(M_*)} \\right )\n\\end{align}\nwith\n\\begin{align}\n         T^\\gamma_{\\rm eff}(M_*)  = \\left ( 1-\\frac{M^2}{M_*^2} \\right )\n         T^*_{\\rm eff}      \\,  .\n         \\label{gamma_T}\n\\end{align}\nThus we find that the effective temperature of the photon $T^\\gamma_{\\rm eff}$ becomes lower than $T_{\\rm eff}^*$.\n\n\n\nOn the other hand, the main contributions for the meson production by the radiative ReCo model come\nfrom the configuration in which the meson momentum is parallel to $P_T$ in the $M_*$-rest frame\n(See the bottom-right panel in Fig.~\\ref{fig:decay}), \nthe same calculation yields\n\\begin{align}\n  K_T = k_0 \\cosh y_T^* + \\sqrt{M^2+k_0^2} \\sinh y_T^* \\sim P_T,\n\\end{align}\nand the accompanied photon has nearly zero energy and momentum $k_T \\sim 0$ due to its massless nature.\nAccordingly, we find\n\\begin{align}\n   \\left . E_M \\frac{dN_M}{d^2K_TdK_L}\\right\\vert_{K_L=0}\n   \\sim\n   \\exp \\left(-\\frac{K_T}{T_{\\rm eff}^{\\rm meson}(M_*)}\\right)\n\\end{align}\nwith\n\\begin{align}\n   T_{\\rm eff}^{\\rm meson}(M_*)=T_{\\rm eff}^*\n   \\,   .\n\\label{eq:meson_T}\n\\end{align}\nThe meson has the same effective temperature as the preformed state, up to some corrections.\n\n\n\nThe radiative hadronization predicts that there is an ordering in the effective temperatures of\nphotons, mesons, and preformed states: \n\\begin{equation}\n  T_{\\rm eff}^\\gamma(M_*) < T_{\\rm eff}^{\\rm meson}(M_*)\n  \\sim T_{\\rm  eff}^* \\, .\n  \\label{eq:ordering}\n\\end{equation}\nNote that the thermal exponential form of the photon and meson distributions reflects the shape of the parton distributions\nand thus the origin of higher effective temperatures of photons and mesons than the hadronization temperature $T_{\\rm reco}$\ncan be attributed to the partonic collective flow built up during the QGP evolution.\n\n\n\n\n\\subsubsection{$v_2$ of photons and mesons}\n\nWithin the same approximations, we can evaluate the elliptic flow\ncoefficient for the photons, $v_2^\\gamma(k_T)$, defined by \n\\begin{equation}\n  v_2^\\gamma(k_T)\\equiv \\frac\n  {{\\displaystyle\n      \\int d\\phi \\cos 2\\phi\n      \\left(\\left. k \\frac{dN_\\gamma}{d^2k_Tdk_L }\\right|_{k_L=0} \\right)}}\n  {{\\displaystyle\n      \\int d\\phi\n      \\left(\\left. k \\frac{dN_\\gamma}{d^2k_T dk_L }\\right|_{k_L=0}\\right)}}\\, .\n  \\label{v2_gamma}\n\\end{equation}\nThe distribution of the preformed state is computed with the original ReCo model,\n\\begin{equation}\n\\frac{dN_*}{d^2P_T dP_L}=\n   \\overline{ \\frac{dN_*}{d^2P_T dP_L}}\\,\n   \\Big(1+2v_2^*(P_T)\\cos 2\\Phi \\Big)\\, ,\n\\label{v2_preformed}\n\\end{equation}\nwhere \n$\\overline{dN_*\/d^2 P_T dP_L}$ is $\\Phi$-independent part of the spectrum and the nonzero elliptic flow $v_2^*(P_T)$ is inherited from the quark\/anti-quark elliptic flow.\nThen, in place of eq.~(\\ref{PT_integral}), the integration over $\\Phi$ with the $\\delta$-function\nyields\n\\begin{align}\n  \\left .\n  k \\frac{dN_\\gamma}{d^2 k_{T} dk_L}\n  \\right |_{k_L=0}  &=\n\\int_{-P_{L\\rm max}}^{P_{L\\rm max}} dP_L \\int_{P_{T\\rm min}}^{P_{T\\rm max}} d P_T \\,\n \\overline{\\frac{dN_*}{d^2P_T dP_L}}\\,\n (1+2v_2^*(P_T) \\cos 2 \\phi \\cos 2 \\theta)\\frac{1}{4\\pi k_0}\n \\frac{2M_*}{k_T |\\sin \\theta|}\n,\n\\label{gamma_dist_2}\n\\end{align}\nwhere we have used $\\sum_{i=\\pm}\\cos 2\\Phi_i = 2 \\cos 2\\phi \\cos 2\\theta$\nwith $\\Phi_\\pm = \\phi \\pm \\theta$.\nWe insert eq.~(\\ref{gamma_dist_2}) into the definition (\\ref{v2_gamma}),\nand evaluate the momentum integral approximately with its threshold value near $P_T \\sim P_{T\\rm min}$ and $P_L \\sim 0$,\nwhere we can put $\\cos 2\\theta \\sim 1$.\nThus we find that, after the $\\phi$ integration, the elliptic flow coefficient is unchanged\nbut its momentum argument is replaced with $k_T = (1-M^2\/M_*^2)P_T$~:\n\\begin{equation}\n  v_2^\\gamma(k_T) \\sim v_2^*\\left(\\frac{k_T}{1-M^2\/M_*^2}\\right)\n  \\, .\n\\label{photon_v2_scaling}\n\\end{equation}\nIn the same manner, we find the coefficient for the meson distribution in radiative ReCo model as \n\\begin{equation}\nv_2^{\\rm meson}(K_T)\\sim v_2^*\\left(K_T \\right)\\, .\n\\label{meson_v2_scaling}\n\\end{equation} \nWe emphasize here that the elliptic flow coefficients $v_2$ of the photons and mesons are both given by $v_2^*$ of the preformed states, \nwhich approximately satisfies the CQN scaling, $v_2^*(P_T) \\sim 2 v_2^q(P_T\/2)$~(see eq.~\\eqref{eq:CQNscaling}). \n\n\nWe have shown that the effective temperature and the elliptic flow coefficient of \nthe meson distribution are the same as those of the preformed-state distribution, \nwhile for the photon distribution these parameters are estimated in the similar manner but with a simple momentum shift \\eqref{mom-shift}.\nThis means that even if the contributions of the radiative hadronization are added,\nnot only the meson elliptic flow still follows the CQN scaling,\nbut also the photon elliptic flow does so approximately. \n\n\n\n\\section{Numerical results}\n\nFor our numerical study we employ a 2-dimensional (2D) model, neglecting the longitudinal momentum  of the preformed state $M_*$ ($P_L=0$),\nsince we have seen that the $P_L$-integration plays only a minor role in the modification of the photon and meson distributions from that of the preformed state.\n\n\nThe photon distribution of 2D radiative ReCo model reads \n\\begin{align}\n  \\left .\n  k \\frac{d N_\\gamma}{d^2k_T dk_L} \\right |_{k_L=0}\n  &=\\kappa \\int_{P_{T\\rm min}}^{P_{T\\rm max}} dP_T  \\,\n  \\sum_{i=\\pm} \\frac{dN_{M_*}}{d^2P_TdP_L}(P_T, \\Phi_i,0)\n  \\frac{1}{2\\pi} \\frac{M_*}{k_T  |\\sin \\theta |}\n  \\, ,\n  \\label{2d_photon_distribution}\n\\end{align}\nwhere \n$\\Phi_\\pm=\\phi\\pm \\theta$ with $\\theta$ defined in eq.~\\eqref{eq:theta}.\nWe generate the distribution of the preformed states of mass $M_*$,\nusing the original ReCo model \\cite{Fries:2003kq,Fries:2003vb}.\nBut we change the recombination temperature $T_{\\rm reco}$ to 155 MeV, which is within the range of the pseudo-critical temperature obtained\nin lattice QCD calculations \\cite{Borsanyi:2010cj,Bazavov:2011nk}.\nAccordingly the transverse flow at hadronization is set to $v_T=0.6$ to reproduce the $p_T$ distribution of the mesons,\nand the freeze-out time $\\tau$ and the fireball radius $\\rho_0$ are also adjusted adequately (see Table \\ref{tab:ParamSet}).\nWe set $M_* = 2M_{ud}$ for the preformed state based on the constituent quark model picture. \n\n\n\n\\begin{table}\n  \\begin{tabular}{cccccccccc}\n  \\hline\n&  $T_{\\rm reco}$~\/MeV & ~ $v_T$ ~  & ~$\\tau$~\/fm ~ & ~ $\\rho_0$~\/fm ~ &~ $\\gamma_{u,d}$~ & ~$\\gamma_{\\bar u,\\bar d}$~ &  $M_{ud}$~\/MeV  & $p_0$~\/GeV & $a$ \\\\\n  \\hline\n  RHIC &   155 & 0.6  &  8.0  & 12.5 & 1  & 0.9 &  260  &  1.0 & 2.5\\\\\n  LHC  &   155 & 0.65 &  15.0 & 20.0 & 1  & 1   &  260  &  1.1 & 2.5\\\\ \n\\hline\n\\end{tabular}\n  \\caption{\n    Model parameters.\n  }\n\\label{tab:ParamSet}\n\\end{table} \n\n\n\\subsection{Model characteristics}\nFirst let us numerically test the characteristics of particle distributions of our radiative ReCo model,\nwhich was discussed in the previous section.\nNote that in this subsection we set the normalization factor $\\kappa=1$ of the radiative ReCo model to study the model characteristics,\nwhile in the next subsections we will adjust the parameter $\\kappa$ so that the model reproduces the observed photon distributions.\n\n\n\\subsubsection{Transverse momentum spectrum}\nWe show $p_T$ distributions of the pions and photons produced by the radiative ReCo model at $\\sqrt{s_{NN}}=200$ GeV\nin Fig.~\\ref{fig:model_char_pion} (left), along with the distribution of the preformed states.\nAt a given momentum, yields of the pions and photons are much lower than that of the preformed state because\neach of them shares a fraction of the momentum of the preformed state,\nand therefore their distributions are shifted to the lower $p_T$ region.\nBut their $p_T$ slopes are similar to each other in the relevant momentum region.\n\n\nWe define the effective temperature $T_{\\rm eff}$ here\nby fitting each of the $p_T$ distributions with an exponential function $\\exp(-p_T\/T_{\\rm eff})$\nin the momentum range $2 < p_T < 5$ GeV.\\footnote{%\n  %\n The slope parameter $T_{\\rm eff}$ is centrality-independent in this model\n since the quark distribution \\eqref{phase_space_dist}\n depends on collision centrality only by the weak modulation of the transverse flow \\eqref{etaT_phi}\n and by the overall factor of the hot-zone size $f(r,\\phi)$.\n}~\nIn Fig.~\\ref{fig:model_char_pion} (right) we check the $M_*$ dependence of these effective temperatures,\n$T_{\\rm eff}^{\\pi}$ of pions and $T_{\\rm eff}^\\gamma$ of photons produced in the radiative ReCo model.\nThey follow the ordering eq.~\\eqref{eq:ordering} derived in the previous section.\n$T_{\\rm eff}^\\pi = T_{\\rm eff}^*$ is predicted by eq.~\\eqref{eq:meson_T}, and \nthe difference between $T_{\\rm eff}^{\\pi}$ and $T^*_{\\rm eff}$ may be understood\nas a correction due to subleading $p_T$-dependence of the distributions.\nOn the other hand, $T_{\\rm eff}^\\gamma$ of photons is lower than $T^*_{\\rm eff}$ and\napproaching it with increasing $M_*$, as predicted by eq.~\\eqref{gamma_T}. \n\nOur model restricts the invariant mass of the preformed state to $M^*= 2 M_{ud}=520$~MeV for pion production,\nbut in more general treatments, the preformed states with $M^* \\ge 2M_{ud}$ may well contribute to the pion production.\nHowever, from Fig.~\\ref{fig:model_char_pion} (right), the slope parameter $T_{\\rm eff}$ of the photons and pions from the radiative ReCo model\nseem rather insensitive to the $M_*$ value, as far as $M_*$ is much larger than the meson mass $M$.\n\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=7.5cm]{radReCo_Npt_MstMeGam.pdf}\n \\hfil\n\\includegraphics[width=7.5cm]{radReCo_Teff_vs_Mst.pdf}\n \\caption{%\n   Left: Comparison of transverse momentum distributions $d^2N\/(2\\pi p_T dp_T dy)$\n   of the photons (green bold solid), pions (blue thin solid), and preformed state (black dashed) in radiative ReCo model\n   ($M_* = 2M_{ud}$, $\\kappa=1$ with $b=5.5$ fm at $\\sqrt{s_{NN}}=200$ GeV).\n   Right: \n$M^*$-dependence of the slope parameters of the photons $T_{\\rm eff}^\\gamma$ (green circle),\n   of pions $T_{\\rm eff}^\\pi$ (blue triangle), and of preformed states $T_{\\rm eff}^*$ (black square).\n   The parameter $T$ is determined by fitting the function $\\propto e^{-p_T\/T}$ \n   to the $p_T$ distributions in the momentum range $2 < p_T < 5$ GeV.\n   \\label{fig:model_char_pion} } \n\\end{figure}\n\n\n\nWe show the results for kaon production with $M_* = M_s + M_{ud}$ and $M_s = 460$ MeV in Fig.~\\ref{fig:model_char_kaon},\nwhere the particle yields become smaller at $p_T \\lesssim 2$ GeV than the pion case due to the mass effect. \nMoreover, the $p_T$ slope of the photon distribution is much steeper than those of the kaon and preformed state distributions, \nbecause, unlike the pion mass, the kaon mass $M_K=495$~MeV is comparable to $M_*=720$~MeV here (see eq.~\\eqref{gamma_T}).\nWe see that the photon yield associated with radiative hadronization of the kaons is small compared with that of the pions,\nand we neglect it in this work.\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=7.5cm]{radReCo_Npt_MstMeGam_kaon.pdf}\n \\caption{%\n    The same plot as in Fig.~\\ref{fig:model_char_pion} (right),\n  but associated with kaon production ($M_s =460$ MeV).\n  \\label{fig:model_char_kaon}\n }\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=7.5cm]{radReCo_v2_scaling.pdf}\n \\caption{%\n   Rescaled elliptic flow coefficients $v_2(p_T\/n_q)\/n_q$\n   of the pions (blue thin solid), photons (green thick solid),\n   and preformed state (black dashed) at $b=5.5$ fm with $n_q=2$.\n   The flow coefficient $v_2(p_T)$ of the quarks (purple dotted)\n   is shown for comparison.\n   Parameters are the same as in Fig.~\\ref{fig:model_char_pion}.  \n   \\label{fig:check_QNS_v2_rhic}\n }\n\\end{figure}\n\n\n\\subsubsection{Elliptic flow $v_2$}\n\nWe introduce an azimuthal angle dependence of the quark\/antiquark flow $v_T(p_T)$\nby the modulation amplitude $h(p_T)$ of the transverse flow rapidity $\\eta_T(\\phi; p_T)$ as in eq.~(\\ref{etaT_phi}).\nThe magnitude of $h(p_T)$ is determined by the aspect ratio of the initial collision zone.\nIn Fig.~\\ref{fig:check_QNS_v2_rhic} shown is the result of the elliptic flow coefficients $v_2$\nof the pions (cyan thin) and photons (green bold) as well as that of the preformed states (black dashed),\nafter divided by the constituent quark number $n_q=2$.\nThe quark\/antiquark elliptic flow coefficient $v_2^q$ is also shown (purple dotted) for comparison.\n\n\nWe find that all these $v_2$ have the same magnitude,\nwhich is taken over from the quark\/antiquark flow through the radiative recombination.\nWe also note that the flow coefficient $v_2^\\pi$ of the pions exactly follows\nthat of the preformed state $v_2^*(p_T)$ at $p_T \\gtrsim 1$ GeV,\nwhile the photon $v_2^\\gamma(p_T)$ lies slightly below them.\nIndeed, we have confirmed that,\nwhen plotted in the shifted momentum $\\bar p_T=p_T\/(1-M^2\/M_*^2)$,\nthe photon $v_2^\\gamma(p_T)$ curve overlaps with that of the preformed states at larger $p_T$,\nas discussed in eq.~\\eqref{photon_v2_scaling}.\n\n\n\n\n\n\n\n\\subsection{RHIC}\n\n\nNext we study the contributions of radiative hadronization to the photon production in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV.\nIn order to make a comparison of the photon yield to the direct photon data at RHIC \\cite{Adler:2003qi}, \nwe include the thermal photon contributions evaluated with a 3-dimensional viscous hydrodynamic model with the kinetic freezeout temperature $T_{\\rm fo}=116$ MeV \\cite{Miyachi-Nonaka}\n(see Appendix B).\nWe assign the impact parameter $b=3.0, 5.5, 7.5$ and $9.0$ fm in our model for the centrality classes, 0--10, 10--20, 20--30 and 30--40 $\\%$ of the collision events, respectively, based on the Glauber model estimate \\cite{PHENIX:2003iij}. \n\n\n\nIn Fig.~\\ref{fig:centrality_rhic} we compare the $p_T$ distributions of $\\pi^0$ obtained\nby the ReCo (black solid) and radiative ReCo (cyan dashed) models to PHENIX data at different centralities \\cite{Adler:2003qi}.\nThe parameter $\\kappa=0.2$ of the radiative ReCo model is determined so that the sum of the photons from radiative hadronization and\nthe thermal photons reproduces the observed photon yield (see Fig.~\\ref{fig:photon_centrality_rhic} below).\nWe are reassured here that the original ReCo model reproduces the pion $p_T$ distributions for different centrality classes, in the $p_T$ range from 2 to 4 GeV,\nwhere the quark recombination is regarded as the dominant hadronization mechanism.\nOutside this region, other production mechanisms, hydrodynamic process at the lower $p_T$ and parton fragmentation at the higher $p_T$, are important.\nIn contrast, the contribution from the radiative ReCo model takes only a small fraction of the pion yield (less than 10 \\% of the original ReCo model) at a given momentum $p_T$,\nand it brings no obstruction on the success of the original ReCo model in describing meson production in this momentum region \\cite{Fries:2003kq,Fries:2003vb}. \n\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{radReCo_Npt_wData-NoGam.pdf}\n \\caption{Transverse momentum distributions of $\\pi^0$\n   computed with ReCo model (black solid) and radiative ReCo model (cyan dashed)\n   for impact parameter $b=3$, 5.5, 7.5 and 9 fm in Au+Au  collisions\n   at $\\sqrt{s_{NN}}=200$ GeV.\n   The $\\pi^0$ data set of 0--10 \\%, 10--20 \\%, 20--30 \\% and 30--40 \\%  centrality classes\n   is adopted from \\cite{Adler:2003qi}.\n   \\label{fig:centrality_rhic} } \n\\end{figure}\n\n\n\\begin{figure}[th]\n  \\centering\n  \\includegraphics[width=10cm]{Prompt+Thermal+radReCo_photons_Npt.pdf}\n \\caption{\n Transverse momentum distributions of direct photons \n computed with radiative ReCo model (green dashed) for impact parameter $b=5.5$ (left) and $9$ fm (right)\n in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV.\n Thermal photon distribution obtained by a viscous hydrodynamic model (purple dotted),\n rescaled prompt photons (black dot-dashed),  and their sum (red solid) are also shown.\n The data is adopted from \\cite{Adare:2014fwh}, and the parameter $\\kappa=0.2$ is determined to fit the data. \n\\label{fig:photon_centrality_rhic}\n}\n\\end{figure}\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{radReCo_v2pt_pi.pdf}\n \\caption{\n   Pion elliptic flow coefficient $v_2$ from ReCo model (black solid)\n   and radiative ReCo model (blue dashed) as a function of $P_T$\n   for $b=5.5$ fm (left) and $b=9.0$ fm (right) in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV. \n   Data $\\pi^0$ (blue circles) are taken from PHENIX \\cite{Adare:2013wop}.\n   \\label{fig:v2_rhic}\n}\n\\end{figure}\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{Prompt+Thermal+radReCo_photons_v2pt.pdf}\n \\caption{\n   Elliptic flow coefficient $v_2^\\gamma$ of the direct photons (red solid) for impact parameter\n   $b=5.5$ (left) and 9 fm (right) in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV.\n The $v_2^\\gamma$ of the photons from a viscous hydrodynamic model \n and $v_2^\\gamma$ of the photons from radiative ReCo model are shown in purple dotted and green dashed curves, respectively.\nThe normalization $\\kappa=0.2$ for the radiative ReCo model is adopted.\n   Data for direct photons (blue solid stars) is adopted from \\cite{Adare:2015lcd}.\n   \\label{fig:v2_photon_rhic}\n}\n\\end{figure}\n\n\n\nNext in Fig.~\\ref{fig:photon_centrality_rhic}\nwe show the $p_T$ distributions of the photons emitted in radiative pion production (green dashed)\nfor $b=5.5$ fm (left panel) and $b=9.0$ fm (right panel),\ntogether with those of the thermal photons (purple dotted) and the prompt photons (black dot-dashed),\nand the total (red solid).\nFor thermal photon production, we adopt the thermal photon rates of QGP in \\cite{Arnold:2001ms} and\nthat of the hadronic phase in \\cite{Holt:2015cda,Turbide:2003si,Heffernan:2014mla},\nand integrate these rates over the evolution profile obtained by a 3D viscous hydrodynamic simulation (See Appendix B for details).\nOur estimate of the thermal photon contribution is consistent with other model studies \\cite{Paquet:2015lta}.\nRegarding prompt photon production in AA collisions, which dominates the total photon distribution at higher $p_T$,\nwe use the empirical fit of the photon distribution in pp collisions, $a_1 (1+p_T^2\/a_2)^{a_3}$ ($a_{1,2,3}$ are constants),\nscaled with the number of nucleon collisions for AA collisions, \nas is done by PHENIX \\cite{Adare:2014fwh}.\nThe experimental data of the direct photons in the 0--20 \\% (for $b=5.5$ fm) and 20--40 \\% (for $b=9.0$ fm) centrality classes\nare taken from PHENIX \\cite{Adare:2014fwh}\\footnote{%\n  A new data analysis is published in Ref.~\\cite{PHENIX:2022rsx} and\n  consistent with that in Ref.~\\cite{Adare:2014fwh}.\n}. \n\n\nWe set the normalization of the radiative ReCo model to $\\kappa=0.2$ so that the sum of the two photon contributions, thermal radiation and radiative hadronization, reproduces the observed photon yield for $p_T < 3$ GeV.\nIndeed, the photon $p_T$ distributions for two centrality classes $0-20$ \\% and $20-40$ \\% are reproduced fairly well with the same normalization $\\kappa=0.2$.\nWe notice that the photon yield from the radiative ReCo model is estimated to be several times larger than that from the thermal radiation and that the $p_T$ slope of the resultant photon distribution is mostly determined by the contribution from the radiative ReCo model for $2<p_T<4 $ GeV and is consistent with the data.\nNotice that the prompt photon contributions become non-negligible only for the higher $p_T \\gtrsim 3$ GeV.\n\n\nNext we study the elliptic flows of the pion and photon distributions.\nWe present in Fig.~\\ref{fig:v2_rhic}\nthe elliptic flow coefficient $v_2(p_T)$ of the pion distribution obtained by the ReCo model (solid line) as well as that of the radiative ReCo model (dashed line), separately,\ndespite we know that the latter contribution to pion production is very small.\nWe find that the two curves overlap on top of each other for $p_T \\gtrsim 1$ GeV.\n\n\nConcerning the photon $v_2^\\gamma$ in our model, we recall that there are three different sources of the photons, {\\it i.e.}, the thermal radiation, the radiative hadronization, and the prompt processes.\nIn Fig.~\\ref{fig:v2_photon_rhic} we show the flow coefficient $v_2^\\gamma(p_T)$ (red solid) of the total photon distribution as well as those of the thermal photon contribution (purple dotted) and of the radiative ReCo model contribution (green dashed), separately.\nIn addition, we presumed the prompt photons have no collective flow (black dot-dashed).\nThe thermal photons have a nonzero $v_2^\\gamma$ but its value is systematically below the observed values \\cite{Adare:2014fwh}.  \nOn the other hand, the photons from the radiative ReCo model have $v_2^\\gamma$ as large as the pion $v_2$, and its $p_T$ dependence is almost the same as that of the pions.\nSince the photon yield of the radiative ReCo model is estimated to be several times larger than the thermal photon yield, the resultant $v_2^\\gamma(p_T)$ of the total photons is close to that of the radiative ReCo model component and is consistent with the data for $p_T \\lesssim  2$ GeV, albeit a large uncertainty. At larger $p_T$, the prompt photons dominate and the flow is suppressed.\n\n\n\n\n\n\n\\subsection{LHC}\n\nWe perform the same analysis for photon production in heavy-ion collisions at the LHC energy, $\\sqrt{s_{NN}}=2.76$ TeV.\nIn this calculation we use the same recombination temperature $T_{\\rm reco}=155$ MeV,\nbut with a stronger transverse flow $v_T=0.65$ of the quarks at the hadronization.\nWe also assume full equilibration of $\\bar{u}$ and $\\bar{d}$ quarks and set the fugacity factors $\\gamma_{\\bar{u}}=\\gamma_{\\bar{d}}=1$.\nSee Tab.~\\ref{tab:ParamSet} for other parameters.\nWe adopt here the following correspondence between the impact parameter $b$ and the centrality classes,\n$b=4.0$ fm for 0--10~\\%, $6.0$ fm for 10--20~\\%,\n$7.8$ fm for 20--30~\\% and $9.2$ fm for 30--40~\\% centrality class \\cite{ALICE:2013hur}. \n\n\n\nIn Fig.~\\ref{fig:central-pt-lhc} shown are the $p_T$ distributions of the pions\nfor several centrality classes.\nThe original ReCo model (black solid) describes the $p_T$ spectra of the pions in the region $2<p_T<5$ GeV\nat different centralities.\nWith the normalization $\\kappa=0.05$, which we determine so as to reproduce the photon yield below,\nthe radiative ReCo model (cyan dashed) gives only a few \\% of the pion yields in the relevant momentum region.\nHence the success of the original ReCo model is unaltered by the contribution of the radiative ReCo model.\n\n\n\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[width=10cm]{radReCo_Npt_lhc.pdf}\n \\caption{\n Centrality dependence of transverse momentum distributions of $\\pi$ \n in $\\sqrt{s_{NN}}=2.76$ TeV Pb+Pb collisions at the LHC.\n Data points for $\\pi^0$ in the\n 0--10 \\%, 10--20 \\% and 20--40 \\%  centrality classes  \\cite{Abelev:2014ypa} \n and  $\\pi^+$ in the 20--30 \\% centrality class \\cite{Abelev:2013vea}\n are adopted from ALICE.  \n \\label{fig:central-pt-lhc}\n} \n\\end{figure}\n\n\n\nIn Fig.~\\ref{fig:pt_lhc} \nwe compare the photon $p_T$ distribution of our model at $b=6.0$ (9.2) fm\nwith the experimental data \\cite{Abelev:2013vea} at 0--20 (20--40) \\% centrality.\nUnlike in the RHIC case, the thermal photon yield (purple dotted) is not far off the observed data in the momentum region $1 < p_T < 2$ GeV,\nand we can reasonably fit the data in the region $p_T \\lesssim 2$ GeV\nby adding the photons from the radiative ReCo model (green dashed) with the normalization factor $\\kappa = 0.05$, both in central and mid-central collisions.\nSince prompt photon data in pp collisions is unavailable in this low $p_T$ region, and theory calculations for $p_T \\lesssim 2$ GeV have a large ambiguity,\nwe decided here to use the same model of photon distribution $a_1 (1+p_T^2\/a_2)^{a_3}$ as adopted by PHENIX \\cite{Adare:2014fwh},\nsetting the infrared cutoff $a_2=4$ GeV$^2$ and\ntuning the parameter $a_1=1.2\\times 10^{-2}$ GeV$^{-2}$ and $a_3 =-2.7$ to fit the available pp-collision data around $p_T \\sim 10$ GeV at $\\sqrt{s}=8$~TeV \\cite{ALICE:2018mjj}.\nWe estimate the prompt photons in AA collisions by rescaling this model \\cite{Adare:2014fwh}. \nThe parameter $a_2$ may be regarded as an infra-red cutoff of the prompt photon model.\nWe note here that the normalization parameter $\\kappa$ is not much sensitive to the choice of $a_2$ as far as it is of order $O(1~\\text{GeV}^2)$.\nAlthough we obtained a reasonable fit of the data in this low $p_T$ region with $\\kappa=0.05$,  we don't have a clear explanation for the decrease of the $\\kappa$ value\nfrom the RHIC case.\n(We will come back to this point in the next section.)\n\n\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{Thermal+radReCo_photons_Npt_lhc_b4.pdf}\n \\caption{\n   Transverse momentum distributions of direct photons \n (red solid) for $b=6$ fm (left) and  $b=9.2$ fm (right). \n The photon yields from the hydrodynamic model and radiative ReCo model are shown\n in purple dotted and green dashed curves, respectively.\n The data in the corresponding centrality classes (gray circles) is adopted from \\cite{Adam:2015lda}.\n \\label{fig:pt_lhc} \n} \n\\end{figure}\n\n\nNext we study the elliptic flow coefficient $v_2$ of the pions and photons.\nWe use the parameters $p_0 = 1.1$ GeV and $a=2.5$ for the quark elliptic flow $v_2(p_T)$ (see eq.~\\eqref{hpt-profile}).\nFigure \\ref{fig:pion_v2_lhc} shows the pion elliptic flow $v_2$ for $b=6.0$ (left) and 9.2 fm (right)\nalong with ALICE data \\cite{Adam:2015lda}.\nWe find that the pion $v_2$ is well described by the original ReCo model\nand the situation is unaltered by the addition of the radiative hadronization contribution. \n\n\nFigure \\ref{fig:v2_lhc} shows the photon elliptic flow coefficient $v_2^\\gamma$ for $b=6.0$ (left) and 9.2 fm (right).\nSince in $1 < p_T < 2$ GeV the thermal radiation and radiative hadronization contribute almost equally to the photon yield,\nthe elliptic flow coefficient $v_2^\\gamma$ of the total photon yield becomes an average of the two sources and\nlies just in between $v_2^\\gamma$ of the thermal photon (purple dotted)  and $v_2^\\gamma$ of radiative ReCo photons (green dashed) at lower $p_T$.\nIndeed, this is consistent with the fact that the photon $v_2^\\gamma$ is somewhat smaller than the pion $v_2$ in the observed data at the LHC energy.\nAt higher $p_T$, it is dominated by prompt photon contribution, which we assume has the zero elliptic flow.\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{radReCo_v2pt_lhc2.pdf}\n \\caption{\n   Elliptic flow $v_2$ of the pions of ReCo model (black solid) and of radiative ReCo model (cyan dashed) for $b=6.0$ (left) and 9.2 fm (right), \ncorresponding to centralities, 0--20 \\% and 20--40 \\%, respectively.\nNote that centralities of data (blue circles) are 10--20 \\% (left) and 30--40 \\% (right) \\cite{ALICE:2014wao}, respectively.\n   \\label{fig:pion_v2_lhc}} \n\\end{figure}\n\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{Thermal+radReCo_photons_v2pt_lhc_b4.pdf}\n \\caption{\nElliptic flow $v_2$ of direct photons (red solid) for $b=6.0$ (left) and 9.2 fm (right).\nData (gray circles) is adopted from \\cite{Lohner:2012ct}. \nOther notations are the same as in Fig.~\\ref{fig:v2_photon_rhic}.\n   \\label{fig:v2_lhc}} \n\\end{figure}\n\n\n\n\n\n\\section{Summary and discussions}\n\n\nIn QED plasma physics, radiative recombination is one of the fundamental processes, which occur when a plasma is de-ionized.\nIn analogy with it,\nwe have proposed here ``radiative hadronization'' \\cite{Fujii:2017nbv}\nas an additional photon source which had not been taken seriously so far in heavy-ion collision physics.\n\n\nIn this paper, we have embodied the radiative hadronization process by a two-step model, that is, picking up a preformed state of quarks,\nwhich we describe by the original ReCo model \\cite{Fries:2003kq,Fries:2003vb}, and its radiative decay to a photon and a hadron.\nWe call this model ``radiative ReCo'' model.\nThose particles produced by radiative ReCo model inherit collective flows from the quark fluid.\nWe can thus expect that those photons fill up the deficit of the direct photon sources for reproducing the observed yield\nand they give rise to the strong collective flow of photons.\nWe derived approximate formulas for the effective temperatures, $T_{\\rm eff}^\\pi$ and~$T_{\\rm eff}^\\gamma$, and elliptic flows of photons and pions in this model,\nand found that their elliptic flow coefficients, $v_2(p_T)$, still satisfy the CQN scaling relation approximately~(Fig.~\\ref{fig:check_QNS_v2_rhic}).\n\n\nFor numerical study of direct photons, we considered three kinds of photon sources, prompt photons, thermal photons and photons from the radiative hadronization.\nPhoton production from the radiative hadronization was evaluated with the 2-dimensional radiative ReCo model, while thermal photon yield was\nobtained by integrating  the rate formulas over hydrodynamic-model event profiles and a simple parametrization was used for prompt photon yield.\n\n\nWe have shown that the radiative ReCo model can account for a significant fraction of the total photon yield, while it produces only a small fraction of the total pion yield,\nto which the original ReCo gives the dominant contribution at around $ p_T \\sim  2$~GeV.\nIndeed, by adding the photon contribution from the radiative ReCo model with the overall factor $\\kappa=0.2$ to the thermal and prompt photon contributions,\nwe can fit simultaneously the large direct photon yield and the strong elliptic flow observed at RHIC.\nFor the LHC data, the same procedure resulted in a reasonable fit of the $p_T$ distribution and the elliptic flow $v_2^\\gamma(p_T)$ of photons in $ 1 < p_T < 4$~GeV,\nbut with the smaller value of $\\kappa=0.05$.\nThe apparent reason for this decrease of $\\kappa$ is the fact that the estimated thermal photon yield is not far off the observed direct photon yield compared to the RHIC case.\nThe origin of this change is unclear and open for future study.\n\n\n\n\nHadronization from QGP is a non-perturbative and non-equilibrium process in high-energy heavy-ion collisions.\nThe empirical success of the recombination models seems to suggest that complexity of hadronization is involved to some extent in these models.\nThe overall factor $\\kappa$ of the radiative ReCo model may be interpreted as a ratio of the radiative recombination to the original recombination.\nThe obtained values $\\kappa =0.2$ for RHIC and 0.05 for the LHC data are indeed small, but still larger than the naive expectation of $O(\\alpha_{\\rm em})=1\/137$.\nThese $\\kappa$ values should reflect various dynamical effects in hadronization.\nFor example, there would be quark recombination process with gluon radiation which makes an octet quark pair to a singlet one, which is\nparametrically of order $O(\\alpha_s)$. But, of course, the colored gluons cannot come out from the collision zone.\nOn the other hand, the photons are penetrating from the zone.\n\n\n\n\n\nConcerning the preformed states in our model,\nit is admittedly unclear whether we can assign any resonance interpretation to them and we intentionally avoid such an interpretation\nunder the lack of our understanding of hadronization mechanism.\nIn this work, we have assumed that the invariant mass distribution of the preformed state, or a quark pair in a simple term, is peaked at $M_*=2m$,\nand replaced it with a delta-function. If we relax this restriction, we will have\n\\begin{equation}\n  k \\left.\\frac{dN_\\gamma}{d^2k_T^\\gamma dk_L}\\right\\vert_{k_L=0}\n  \\sim\n  \\int_{2m}^\\infty dM_*\\, \\varrho(M_*)\\, \\# \\, \\exp \\left(-\\frac{k_T^\\gamma}{T_{\\rm eff}^\\gamma(M_*)}\\right)\n\\end{equation}\nwith $T_{\\rm eff}^\\gamma(M_*)$ being the same as in eq.~(\\ref{gamma_T}).\nThe $M_*$ distribution $\\varrho(M_*)$ should modify our estimate of the photon yield,\nbut part of modification may be effectively absorbed in the adjustable overall factor $\\kappa$.\n\nIncidentally, the pion production in radiative hadronization implies that the preformed state with spin one decays to the pseudo-scalar state with a photon emission,\nbut our radiative ReCo model does not specify the spin quantum number of the preformed state.\nThe parameter $\\kappa$ may include this kind of uncertainties, too.\n\n\n\n\nWe have studied direct photon production in an ad-hoc way by simply adding three sources, prompt photons, thermal photons and photons from the radiative hadronization of pions,\nevaluated with different models.\nThe radiative hadronization of heavier hadrons give much smaller contribution to the photon yield at $p_T \\sim 2$ GeV (Fig.~\\ref{fig:model_char_kaon}).\nBut there are other possible photon sources; jet fragmentation produces photons at relatively higher $p_T$, and initial pre-equilibrium evolution called Glasma stage also contributes to the photon yield but its elliptic flow will be small \\cite{Berges:2017eom,Monnai:2019vup}.\nIn this sense, our estimate for $\\kappa$ should be regarded as  an upper bound for the radiative hadronization contribution.\n\n\n\nPhoton production is a unique penetrating probe, along with the dilepton production. \nThe so-called ``direct photon puzzle'' has been an important challenge in understanding the evolution of heavy-ion collisions, QGP formation and its decay.\nA unified model for describing the collision events is strongly desired\nand we believe that the radiative hadronization occupies a crucial position in understanding the photon production in heavy ion collision physics.\n\n\n\n \n  \\begin{acknowledgments}\n\n    This work was supported in part by JSPS KAKENHI Grant Numbers, JP16K05343 and JP21K03568 (HF), and\n    JP20H00156, JP20H00581 and JP17K05438 (CN), and JP19K03836 (KI).\n\n  \\end{acknowledgments}\n  \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec-intro}\n\nThe total stellar mass of a galaxy is probably one of the most important properties that determine its structure and evolution. It is a footprint of its formation history, and a driver for future evolution. Galaxies accumulate gas through both secular processes as well as merging events \\citep[e.g.][]{cold-gas-accretion, galaxy-evolution-review}. If this gas condenses, it can form molecular clouds, in which star formation occurs. Hence, more evolved systems have a higher stellar mass. Moreover, the stellar mass also correlates with other galaxy properties, such as metallicity, star formation rate and halo mass \\citep{galaxy-physical-properties, mass-metallicity, gsmf-mass-metallicity, lara-lopez2010, mannucci2010, stellar-halo-mass}. The galaxy stellar mass function (GSMF), together with the scaling relations involving stellar mass, poses the most important constraint for a model of galaxy formation and evolution \\citep{eagle, illustris-stellar-mass}.\n\nUnfortunately, the stellar mass of a galaxy, or equivalently the stellar mass-to-light ratio $M\/L$ in a given broadband, is not a property that can be measured directly from observations. There are, however, a few ways to estimate $M\/L$. All of these estimates rely on a stellar population synthesis (SPS), and hence there can be systematic uncertainties. For example, different simple stellar population (SSP) models have different prescriptions for thermally pulsing asymptotic giant branch (TP-AGB) stars \\citep{bc03, maraston}. This is a variable phase of stellar evolution with many uncertainties, but its emission can dominate NIR wavelengths for intermediate age galaxies \\citep{conroy-sedfit-review}. An initial mass function (IMF) has to be assumed \\citep[e.g.][]{salpeter, chabrier2003}, as well as a star formation history (SFH). Depending on the band, a dust attenuation law also has to be present. We are interested in estimating $M\/L$ (rather than $M_*$), since it is intrinsic and captures these different physical processes in one number.\n\nTo first order, we can expect the stellar mass to scale with the total stellar luminosity, and hence also with the luminosity in a particular band. In other words, we can approximate the stellar mass by assuming a single, average $M\/L$. This works best for a NIR band, which mostly traces the old stellar population \\citep{nir-ml-1, nir-ml-2}. Since the NIR is less affected by the current star formation rate (SFR), the $M\/L$ varies over a much smaller range than in the optical. Moreover, the NIR is less affected by dust compared to the optical regime. It also benefits from the age-metallicity relation: young, metal-rich stars have a similar $M\/L$ in the $3.6 \\, \\mu \\textrm{m}$ band as old, metal-poor stars \\citep{age-metallicity, meidt2014}. From the assumption of constant $M\/L$, the $3.6 \\, \\mu \\textrm{m}$ image can be converted into a stellar mass map. The drawback of using the NIR to estimate stellar mass is that there are higher systematic uncertainties, in particular regarding TP-AGB stars \\citep{mcgaugh-ml-uncertainty}. This method also assumes an age-metallicity relation, but galaxies can deviate from this relation due to an atypical evolution \\citep{age-metallicity, meidt2014}.  \n\nInstead of using a constant $M\/L$, we can estimate the $M\/L$ from one or two colours \\citep{bell2001,zibetti2009,meidt2014}. In particular, this allows us to use optical fluxes, which are easier to obtain than NIR fluxes. This estimation relies on the fact that older stellar populations have a higher optical $M\/L$ than young stellar populations. Since the spectrum of older populations is redder than for younger populations, redder colours imply a higher $M\/L$. This is complicated by metallicity and dust attenuation, which both also have a reddening effect. Fortunately, dust attenuation also implies a higher $M\/L$, since we receive a lower luminosity. This roughly (but not exactly) preserves the colour - $M\/L$ relation \\citep{bell2001}. The effect of metallicity on $M\/L$ depends on the used colour and reference band \\citep{meidt2014}. \n\nFinally, if the optical and NIR are well sampled, we can estimate the total stellar mass from spectral energy distribution (SED) fitting. Model SEDs are created from several building blocks: star formation history, simple stellar population (for a given IMF), and an attenuation law. These are parametrized, and through a Bayesian approach it is possible to estimate various parameters (and their errors), including the stellar mass \\citep{cigale,magphys, galmc, fast}. Comprehensive reviews regarding SED fitting are provided by \\citet{walcher-sedfit-review} and \\citet{conroy-sedfit-review}. This method can make use of all available input fluxes and their uncertainties, which leads to more accurate predictions. The spectral range allows for a better estimation of metallicity and attenuation, compared to a single colour method. The estimation is resistant against one of the bands having a large error, which is not the case for methods which use one or two bands. However, the method is more computationally expensive, since a large number of SED models need to be built and fitted. The previously discussed methods (one band, one or more colours) make use of a precomputed library of SED models, bin these by the required bands or colours, and store the median $M\/L$ of that bin. In other words, they marginalize over the SED model grid in order to save computing time. \n\nAlthough SED fitting is one of the more accurate methods for estimating stellar masses, it is still prone to systematic uncertainties. Like other SPS methods, we have to assume a parametric form for the IMF, SFH, SSP, and dust attenuation curve. Different parametric forms can result in very similar SEDs, but different intrinsic parameters \\citep{cigale-2}. Moreover, since the method makes use of global fluxes, it is prone to \\textit{outshining}. Even though the old stars are usually responsible for most of the mass, the young stars can dominate the global flux at UV and optical wavelengths. The SED fitting then results in a lower $M\/L$, possibly missing a large part of an older stellar population. This bias strongly depends on specific star formation rate (sSFR = SFR \/ M$^*$), and can be avoided by using pixel-by-pixel SED fitting \\citep{unresolved-sed1,unresolved-sed2}.\n   \nIn SED fitting, each observed band helps constrain the derived properties. We investigate whether morphology can serve as an additional constraint on the stellar $M\/L$. The Hubble sequence--by definition a sequence of morphological types--correlates with many properties  \\citep{review-hubble-sequence}. One of the strongest correlations is with colour: early-type galaxies usually contain a much redder stellar population than late-type galaxies \\citep{hubble-colour}. Since the $M\/L$ depends primarily on the age of the stars, $M\/L$ also varies over the Hubble sequence. This explains why morphology can be a valuable constraint on $M\/L$. Late type galaxies also tend to have more dust and a lower metallicity than early type galaxies \\citep{review-hubble-sequence, cortese-2012}, influencing both colour and $M\/L$.\n\nThis work has the following goals. First, we want to investigate how strong the correlation between morphology and $M\/L$ is. This is done by predicting $M\/L$ from a single g-band image. While the g-band image suffers from dust extinction, we use this as an advantage instead of a drawback: the g-band image emphasizes morphological features such as spiral arms and rings, and gives a good view of both the younger and the older stars. Next, we want to investigate whether the information provided by morphology is still helpful in the presence of one or more global colours. One possible benefit of morphology is the distinction between an older stellar population and dust attenuation. \n  \nSince morphology is not easily analytically quantified, we use machine learning techniques, which is now the state-of-the-art in most image related tasks. Given a large enough dataset, convolutional neural networks (CNNs) excel at image recognition \\citep{alexnet, inception, resnet}. Maybe the most remarkable feature is that the method directly takes an image as input. This avoids the need for complex preprocessing pipelines that were used to detect and measure simple shapes in the image. In astronomy, CNNs have been used for photometric redshift estimation \\citep{ml-photoz, ml-photoz-2, ml-photoz-3}, point source detection \\citep{ml-pointsource}, host galaxy determination \\citep{ml-host}, morphology detection \\citep{dieleman, sanchez, morpho-dai}, and more. Past work on morphology detection has shown that machine learning is able to reproduce morphological information from catalogues such as Galaxy Zoo 2 \\citep{gz2}. Typically, these models output the probability of a particular feature being present (such as the probability of having a bar, a given number of spiral arms, or a ``just noticeable bulge''). Machine learning benefits from the large amount of data that is available through surveys, and this will only improve with future surveys.\n   \nMachine learning methods require a training target -- a ground truth. For this, we use global UV-NIR SED fitted $M\/L$. This implies that we can not improve on current state-of-the-art $M\/L$ estimators, and that is not the goal of this work. We also have to keep in mind that morphology is not accessible for high redshift galaxies. The main question we are trying to answer is: when estimating $M\/L$, can morphology be a valuable replacement for, or added value to, colour information?  \n\nThe paper is structured as follows. In the next section, we discuss the selection of the sample, the preprocessing, and the optimization goal. In Sect. \\ref{sec-ml}, we go more in depth about the machine learning pipeline which was used to predict the $M\/L$. In Sect. \\ref{sec-results}, we show the results and interpret our machine learning pipeline. Finally, we conclude in Sect. \\ref{sec-conclusions}. Appendix~\\ref{app-terminology} explains some of the machine learning terminology more in depth. In Appendix~\\ref{app-loss-functions}, we show that our conclusions do not depend on the choice of loss function. \n   \n\\section{Methods}\n\\label{sec-methods}\n\n\\subsection{Sample and data}\nTraining a machine to recognize morphology requires a sufficiently large dataset. For example, \\citet{dieleman} used about 55\\,000 gri images to train their morphology classifier. For this work, we need a large sample of well-resolved g-band images, as well as the target $M\/L$. Since we have access to the g-band, all our $M\/L$ are defined in the g-band ($M\/L_g$). These were acquired from the GALEX-SDSS-WISE Legacy Catalog, version 2 \\citep[GSWLC;][]{gswlc, gswlc2}. This catalogue used GALEX, SDSS and WISE global fluxes to estimate the star formation rate, stellar mass, and dust attenuation parameters for 700\\,000 low-redshift ($0.01 < z < 0.3$) galaxies with $r_{\\rm{petro}} < 18.0$. For this, they used the Bayesian SED fitting tool CIGALE \\citep{cigale, cigale-2}. We cross-correlated GSWLC with $D_{25}$ estimates from the HyperLeda database\\footnote{http:\/\/leda.univ-lyon1.fr\/} \\citep{hyperleda}, where $D_{25}$ is the projected major axis at $25\\, \\textrm{mag}\\, \\textrm{arcsec}^{-2}$ in the B-band. We limited the parent sample to galaxies with $D_{25} > 0.4 \\, \\textrm{arcmin}$. This makes sure that the galaxies are large enough in order for the morphology to be determined. The g-band images for each of these 84\\,723 objects were downloaded from the SDSS Science Archive \\citep[DR12;][]{sdss-dr12}. We used the SDSS mosaicking service\\footnote{https:\/\/dr12.sdss.org\/mosaics\/} which combines the maximum number of scans possible for the final image. The mosaicking service employs Swarp \\citep{swarp}, which aligns and combines the background offsets in the separate images. The result is a deep, background-subtracted image of each galaxy in the g-band. We limited ourselves to a field of view of $1.125\\cdot D_{25}$ since even the deep SDSS mosaics rarely reach the $25\\, \\textrm{mag}\\,\\textrm{arcsec}^{-2}$ surface brightness level. \n   \n\\subsection{Data preprocessing}\n   \nThe images were star subtracted using PTS\\footnote{http:\/\/www.skirt.ugent.be\/pts}, the python toolkit for SKIRT (\\citealt{skirt}; Verstocken et al., in prep.). This makes use of the SDSS point source catalogue as a prior for star positions, and then tries to find a peak around the positions which resembles a true point source. These point sources are then replaced by the local background using bicubic interpolation.\n\nThe star-subtracted images were used to calculate a few features (such as the total g-band luminosity), further discussed in Sect. \\ref{sec-ml}. After the extraction of these features, we log-scaled the images in order to emphasize lower brightnesses, especially at the outskirts of galaxies. First, the image flux was linearly rescaled to the interval [0, 1]. We then log-scaled the pixels in the following way:\n   \n\\begin{equation}\n   F' = \\frac{\\log \\left(1 + a F \\right)}{\\log \\left( 1 + a \\right)}.\n\\label{eq-scauto}\n\\end{equation}\n   \nHere, $F$ is the original pixel value (between 0 and 1). The log-scaled pixel value $F'$ also ranges from 0 to 1, and these then serve as input for the machine learning (Sect.~\\ref{sec-ml}). The 1 inside the log prevents very small values from dominating the output scale. The \\textit{scaling value a} determines how much lower brightnesses are emphasized. Small values of $a$ ($a <  1$) result in a nearly linear scaling. Large values of $a$ result in a pure log scaling, which boosts fainter regions. The scaling value was determined independently for each of the objects. First, the noise level of the input $F$ was determined by sigma clipping the image, and then defining the noise level as two standard deviations above the mean. The scaling value was then fitted so the output (i.e. log-scaled) noise level equals 0.2. The result is that images with a high signal to noise ($S\/N$) have a larger scaling value, which allows for their fainter features to stand out. Images where the noise is more prevalent get a smaller scaling value, which in turn prevents the noise from being mistaken with a feature of the galaxy. This is demonstrated for a low, median, and high scaling value galaxy in Fig.~\\ref{fig-scauto}, where the bottom row shows the automatic scaling value procedure. The result is a more consistent background, boosting features without blowing up the noise. The background noise value of 0.2 was picked visually to distinguish faint features from noise (see bottom panel of Fig.~\\ref{fig-scauto}). We argue that if humans can distinguish the two, deep neural networks should also be able to learn the difference.\n   \n\\begin{figure}\n   \\centering\n   \\includegraphics[width=\\hsize]{figures\/scauto_examples.pdf}\n      \\caption{Demonstration of the automatic scaling value. The rows present different scalings, with the top row being a linear scale (equivalent to a scaling value $a \\ll 1$), the middle row using a constant scaling value of 78 (the median of the automatic scaling values), and the bottom row using the automatic scaling value (which fixes the output noise level to 0.2). For the bottom row, the determined scaling values are given as an inset for the different galaxies. The columns show three different galaxies, which from left to right have an increasing $S\/N$ (and thus an increasing scaling value).}\n         \\label{fig-scauto}\n\\end{figure}\n   \nSo far, we discussed how the input of the machine learning (the g-band image) was processed. Using GSWLC 2, we combined the Bayesian estimate of the stellar mass with a Bayesian estimate of $L_g$ to produce our target $M\/L$. The Bayesian luminosities were taken directly from the GSWLC SED models. A flat prior over the parameter range of the model grid is used, so the Bayesian values are likelihood-weighted averages. Contrary to a least $\\chi_r^2$ method, this allows us to get an uncertainty on $M\/L$ for each galaxy. It should be noted that there is little difference between best-model (i.e. least $\\chi_r^2$) and Bayesian estimates of the $M\/L$, since stellar mass is one of the parameters that can be derived most accurately from SED fitting \\citep{conroy-sedfit-review}. To further improve the $M\/L$ estimate, GSWLC uses a two-component star-formation history (SFH), which allows for a larger old stellar component. The current SFR then only fixes the young component, without constraining the older stellar population (which happens for a single component SFH). This greatly reduces the outshining bias \\citep{gswlc}.\n   \n  \\begin{figure}\n   \t\\centering\n   \t\\includegraphics[width=\\hsize]{figures\/ml_hist.pdf}\n   \t\\caption{Histogram of the $M\/L$ values from GSWLC 2, after applying the threshold $D_{25} > 0.4$ arcmin. The galaxies with $\\chi_r^2 >= 5$ were not used for the machine learning. From the remaining galaxies, $59\\,637$ were used for training, $6\\,627$ for validation and $7\\,363$ galaxies for testing, as described in Sect.~\\ref{ssec-optimization-setup}. These samples were randomly drawn, and hence their distribution is similar.}\n   \t\\label{fig-ml-hist}\n\\end{figure}\n   \nOur sample so far is only limited by the minimum angular size ($D_{25} > 0.4$ arcmin), which results in 84\\,723 galaxies. We found that the distribution on $M\/L$ was quite broad, with some galaxies having $M\/L < 0.1$ and others having $M\/L > 10$ (all $M\/L$ are given in solar units). Most of these outliers can however be removed by setting an upper limit on the fitting $\\chi_r^2$ (i.e. the goodness of the CIGALE fit). A large $\\chi_r^2$ means that even the best model did not fit the observed fluxes well, and hence the resulting properties can be inaccurate. These high $\\chi_r^2$ objects are possible mismatches between optical and UV sources, or sources where the UV was compromised by a lower resolution. We decided to use only galaxies for which the $\\chi_r^2$ was below 5. This significantly reduced the number of outliers: the number of galaxies with a $M\/L$ below 0.1 is now 2 (from 36), while 20 galaxies have a $M\/L$ above 10 (from 50). These two criteria ($\\chi_r^2$ and angular resolution) result in our final sample, which contains 73\\,627 galaxies. This sample has a minimum, median, and maximum $M\/L$ of 0.09, 2.7 and 16.6, while without the $\\chi_r^2$ cut-off we had a minimum and maximum $M\/L$ of 0.04 and 30.8 respectively. The distribution of $M\/L$ can be seen in Fig.~\\ref{fig-ml-hist}, for the different subsamples (see Sect.~\\ref{ssec-optimization-setup}). We clearly see a bimodality, which (after inspecting the individual images) roughly correspond to elliptical galaxies for high $M\/L$ and disk galaxies for low $M\/L$. This distribution is specific for our sample: the low $M\/L$ spirals tend to be less luminous and hence they more often fall outside our selection criteria (both $D_{25}$ and the brightness cut from GSWLC). Our sample has a minimum, median, and maximum pixelsize of 0.08 kpc, 0.44 kpc, and 2.44 kpc respectively. The $D_{25}$ criterion selects mostly the more nearby galaxies, so the median redshift is now 0.05. The median seeing for the SDSS g-band is 1.4 arcsec.\n   \n\\subsection{Optimization setup}\n\\label{ssec-optimization-setup}\n\nIn order to learn from the data, the machine learning algorithm minimizes an optimization objective (also called a \\textit{loss function}). Since we have access to a Bayesian $M\/L$ as well as its uncertainty, we decided to use a L1 loss that takes into account the uncertainty on $M\/L$. We denote it with $\\mathcal{L}_1$ to distinguish it from the standard L1 without uncertainty. It is defined as follows:\n\n\\begin{equation}\n\t\\mathcal{L}_1 = \\frac{1}{N}\\sum_{i = 1}^N \\left|\\frac{\\Upsilon_{\\textrm{pred}, i} - \\Upsilon_{\\textrm{true}, i}}{\\Delta \\Upsilon_{\\textrm{true}, i}}\\right|.\n\\label{eq-l1loss}\n\\end{equation}\n\n$\\Upsilon_{\\textrm{true}, i}$ denotes the `true' $M\/L$ for the $i^{\\textrm{th}}$ galaxy, which is the Bayesian estimate from GSWLC. $\\Delta\\Upsilon_{\\textrm{true}, i}$ is the corresponding Bayesian error, and $\\Upsilon_{\\textrm{pred}, i}$ is the value predicted by our machine learning method. $N$ is the number of galaxies in the considered set. We usually define a separate $\\mathcal{L}_1$ for the training, validation, and test set (see below). Optimizing $\\mathcal{L}_1$ is equivalent to optimizing a weighted mean absolute error (MAE), with the weights defined as $w_i = 1 \/ \\left( \\Delta \\Upsilon_{\\textrm{true}, i} \\right)$. Since $\\Delta \\Upsilon_{\\textrm{true}, i}$ is derived from the likelihood over the model grid in CIGALE, and does not take into account systematic uncertainties, some galaxies can have a very low error. In order to prevent a few galaxies from dominating the weights, we used a minimum relative error on the Bayesian $M\/L$ of 5\\% (this affects about a quarter of our sample). Galaxies with a high $M\/L$ typically have a larger $\\Delta \\Upsilon_{\\textrm{true}, i}$. We first experimented with a squared loss ($\\mathcal{L}_2$), but found that this was dominated by a few outliers (mostly samples with a low $M\/L$ and hence a low error on $M\/L$). The $\\mathcal{L}_1$ loss ensures that we focus more on the general trend \\citep{mae}. In Appendix~\\ref{app-loss-functions}, we experiment with different loss functions, and show that our results still apply for other common loss functions (such as the L2 variant $\\mathcal{L}_2$). The $\\mathcal{L}_1$ loss performs well on a range of metrics. We will use the term `loss' to describe the optimization criterion on the training set, while `metric' is used for how well the predictor performs on the test set.\n\nWe split our sample randomly in three parts, thus creating subsets that are representative of the whole $M\/L$ distribution (see Fig.~2). First, 10\\% is kept apart as a test set (7\\,363 galaxies). From the remaining set, another 10\\% is split off as a validation set (6\\,627 galaxies). The other 59\\,637 make up the training set. The goal of the algorithm is to minimize $\\mathcal{L}_1$ (the loss) on unseen examples. The main way this is accomplished is by minimizing $\\mathcal{L}_1$ on the training set. However, care has to be taken that the algorithm does not \\textit{overfit}. Overfitting happens when the model is too complex, and the behaviour of individual training samples is learned (instead of the general trend). The result is a low training set loss but a high test and validation set loss. The purpose of the validation set is to prevent overfitting. We optimize the algorithm's hyperparameters (which determine the model complexity), including the duration of training, by using the ones that produce the lowest validation set loss. This means that the validation set is no longer an unbiased estimate of how our algorithm performs on unseen examples, which is why we set apart a test set. This is a typical setup for machine learning \\citep{ml-goodfellow, ml-python}.\n   \n   \n\\section{Machine learning}\n\\label{sec-ml}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=17cm]{figures\/pipeline.pdf}\n\t\\caption{Schematic overview of the machine learning pipeline. The black arrows denote the order in which properties are derived from each other. The pink boxes show the optimization objective: three CNNs are optimizing the Galaxy Zoo 2 probabilities, using a mean squared error (L2) loss. The LightGBM and the fourth CNN optimize $M\/L$ according to a $\\mathcal{L}_1$ loss. The initial weights of this last CNN were set to the final values of the first CNN (pretraining). The boxes with the bright green outline were used as features for the LightGBM, after disregarding the ones that always were zero (dead neurons). For each of the neural layers or blocks, the output dimension is provided on the left.}\n\t\\label{fig-pipeline}\n\\end{figure*}\n\n\nOur algorithm can be subdivided in two parts. The first part consists of four CNNs \\citep{lecuncnn} which are trained to detect morphology information. The second part is a gradient boosting machine \\citep[GBM;][]{gradient-boosting}, more specifically Microsoft's LightGBM \\citep{lightgbm}. The GBM combines the morphology information from the CNNs with other information, such as redshift and total image luminosity, in order to predict the $M\/L$. A schematic overview of the complete pipeline can be found in Fig.~\\ref{fig-pipeline}. For more information on the machine learning terminology used here, we refer to Appendix~\\ref{app-terminology}.\n\nThe benefit of using this two-part algorithm is that the task of predicting $M\/L$ from the images is split in two easier tasks. The first part detects what features are present in the image (spiral structure, a bar, a possible merger, etc.). The second part then determines how this morphological information correlates with $M\/L$. We have tried using a single CNN trained on $M\/L$, but this often got stuck in local minima, predicting an average $M\/L$ for all samples. Using this two part algorithm also allows us to better interpret the results, since we can directly correlate the $M\/L$ with the morphology features.\n\n\n\\subsection{CNN - detecting morphology features}\n\\label{ssec-cnn}\n\nCNNs are a type of neural network that make use of the 2D image structure. It consists primarily of convolutional layers, each having multiple convolutional kernels (also called filters). These kernels are trained through gradient based optimization, in order to minimize the training loss. In our networks, most kernels are of size $3\\times 3$; the number of trainable parameters is drastically reduced compared to fully connected layers. The kernels in early layers detect simple features such as edges. The implicit assumption in CNNs is translational invariance: a kernel that detects a feature in one part will detect the same feature in other parts of the image. Throughout the architecture, the image typically gets downscaled, giving rise to higher level features (which in our case can learn to detect spiral structure, bulges, bars, etc.). The final layers are often fully connected (also referred to as dense), combining all features into the final prediction. \n\nWhile the final goal is to learn $M\/L$, we started with training our networks on the morphology information from Galaxy Zoo 2 \\citep[GZ2;][]{gz2,gz2-hart}. Since this made use of the SDSS DR7, we crossmatched GZ2 with our catalogue and used the 58\\,966 galaxies ($\\sim 80$\\% of our sample) for which the sky separation was less than 3.6 arcsec. This sky separation was chosen to cleanly separate our matches (>~99\\% of which are closer than 1 arcsec) from possible mismatches (>~99\\% separated by more than 10 arcsec). While we could probably find a one-to-one relation between each of their DR7 and our DR12 galaxies, it is beneficial to train the morphology on a subsample, in order for the GBM to also learn on training samples for which the morphology is known less precisely (the GBM trains on the full training set, but the CNNs are only trained on the subset that has a GZ2 match). Unlike past endeavors to predict morphology information from GZ2 \\citep{dieleman,sanchez}, our CNN only uses the g-band image. \n\nGZ2 contains 11 questions, with 37 answers in total. We decided to use the weighted vote fractions as probabilities for each answer. We did not use the distance debiased vote fractions, since the GBM has access to redshift and can apply any necessary corrections. Since some questions are only answered after particular answers of previous questions, we converted the weighted vote fractions to unconditional probabilities (i.e. multiplying by the probability of the question being asked). For example, answer four gives the probability of being an edge-on disk, which has to be smaller than or equal to the probability of the galaxy having a disk or feature (answer two, the parent question). For a list of all GZ2 questions and answers, see Fig.~2 of \\citet{gz2-hart}. The last dense layer of all networks have a ReLU activation, making sure the output for an answer is larger than or equal to zero (this is a regression approach also taken in \\citealt{dieleman}). A post-processing layer then takes care of the normalization. First, all answers for a particular question have to sum to one. Then, all answers for that question are multiplied by the estimated probability of that question being asked, determined by the features from higher up answers. This way, the network automatically produces valid unconditional probabilities. All steps of the post-processing layer are differentiable.\n\nInstead of using a single CNN, we used an ensemble of CNNs. Different network architectures will make different errors, and combining the extracted features leads to more robust results \\citep{ensembling}. Since the purpose of the CNNs is to detect the morphology, we used these different CNNs as input to our GBM. The GBM can learn in which scenarios a particular CNN is more accurate than another, and can make use of the combined information. Our final model is based on five extracted feature layers from four different networks. Different setups can lead to similar results, and it might be possible to significantly simplify the setup without too much degradation of the test $\\mathcal{L}_1$. The first architecture is the ResNet50, part of the residual learning framework which won the ILSVRC2015 competition \\citep{resnet}. The residual blocks ensure that only residuals from the previous layer have to be learned, making it possible to build much deeper networks. The second architecture is Xception \\citep{xception}, which was based on an inception architecture \\citep{inception}. The idea is to separate spatial features from depth (channel) features by doing multiple convolutions in parallel, starting from a pointwise convolution. These two networks produce state of the art results on many imaging datasets. We applied the transfer learning technique, starting the network weights from their Imagenet values \\citep{imagenet}. We used the keras python library\\footnote{https:\/\/keras.io\/}, in which these models are already implemented. We only kept the convolutional part, after which we applied global average pooling, a dense-256 layer (i.e. a fully connected layer with 256 neurons) with ReLU activation, followed by a dense-37 layer (matching the 37 answers in GZ2). This was then followed by the probability normalization layer, described above. The optimization objective was to minimize the L2 loss (regular RMSE) of the predicted and ground truth probabilities. The ResNet50 architecture used $197\\times 197$ images as input (the minimum required), while the Xception architecture used $128\\times 128$ images. Prior to the training of these networks, all training samples are scaled to the corresponding resolution (pixel area interpolation for shrinking, bicubic interpolation for zooming), with all networks keeping the same field of view per galaxy. Since these networks are pretrained on ImageNet, which has three input colour channels, we duplicated each image across the three channels to avoid changing the architecture.\n\nA third CNN architecture is a more traditional, shallow network (further called the \\textit{custom} network). It consists of 4 convolutional layers followed by two fully connected layers. The number of channels (depth) in the consecutive layers is: 32, 64, 128, 128, 512, 37. The first three convolutional layers are followed by $2\\times2 $ max pooling, after which dropout is applied \\citep{dropout}. The last convolutional layer is followed by a global max pooling but no further dropout. The first and second convolutional kernels are $5\\times 5$ and $4\\times 4$, respectively, and the last two convolutional layers are $3\\times 3$. No zero padding is applied. This architecture is inspired by \\citet{dieleman}, the main differences being that we only have one input channel and that we do not use their view preprocessing pipeline. The input dimensions are $69\\times 69$. Since there is no pretrained variant of this network, we used Glorot uniform random initialization \\citep{glorot-initialization}.\n\nFor these first three networks, we used the 37 estimated GZ2 answer probabilities as features for the GBM. We then used the ResNet50 architecture to extract more features. First, we extract the 2048 features that followed the convolutional part (before the fully connected layers). Furthermore, we took the whole architecture and replaced the probability normalization layer by a dense-1 layer. This network was then further trained to predict $M\/L$ (minimizing $\\mathcal{L}_1$). This again is a form of transfer learning: we pretrain the network on morphology, and then train on $M\/L$. This makes training easier, and we experienced fewer problems with local minima. From this retrained network, we extract the 37 features from the next to last layer, which no longer directly correspond to the 37 answer probabilities (although they are primed on them). Just like the other networks, these features only make use of the log-scaled (Equation \\ref{eq-scauto}) image, without any extra input such as luminosity or redshift. This means that they are still purely morphological features (i.e. depending only on galaxy structure), even though they do not have a clear interpretation like the GZ2 probabilities do. We will refer to the four CNNs as CNN 1, 2, 3 and 4, where we use the order in which they appear in Fig.~\\ref{fig-pipeline} (from left to right).\n\nIn order to make the networks generalize better, we applied data augmentation at training time. The images were randomly rotated (between 0 and 360 degrees), zoomed (between 0.7 and 1.3), and flipped (horizontal and vertical). This means that for every pass through the training set (epoch), the networks see slightly different images. We used the Adam optimizer \\citep{adam} with Nesterov's momentum. We applied a factor of 0.3 learning rate decay when the validation loss did not improve for 4 epochs, and stopped training after 30 epochs (since the validation loss did not seem to improve further). \n\n\\subsection{GBM - combining all information}\n\\label{ssec-gbm}\n\nSo far, the different CNN architectures produced the following morphological features: the custom CNN, ResNet50 and Xception each produce 37 GZ2 features, the ResNet50 provides 2048 features from the last convolutional layer, and a retrained (on $M\/L$) ResNet50 gives 37 features which are primed on GZ2. In total, these account for 2196 features. In addition, the GBM uses features extracted from the images. The following luminosity features are used, where we used the corresponding flux and multiplied by $4\\pi D^2$ (where $D$ is the distance in Mpc): sum (over all pixels), mean, maximum, minimum, standard deviation, central pixel, mean around central pixel ($5\\times 5$ pixels around the centre), the original image size, and the scaling value (as described in Sect.~\\ref{sec-methods}). After the log-scaling (which also scales the images between 0 and 1), we also extract some image statistics: the sum, mean, standard deviation, central pixel value, and mean around the central pixel. We also added two features which required extra metadata: the redshift and pixel size (in kpc). These allow the network to distinguish between a faint but nearby galaxy and a bright, distant galaxy. After removing 17 features that were always zero (dead neurons, 10 from CNN 3 and 7 from the inner part of CNN 1), we are left with 2195 features. The units of all the features are of no importance at this point: decision trees -- on which the GBM is based -- are scale invariant.\n\nGradient boosting traditionally makes use of decision trees as a base classifier. The trees are built sequentially, where each tree learns from the mistakes made from previous ones (as with ResNets, we learn residuals). LightGBM makes use of several optimizations compared to traditional gradient boosting. Since the morphology features were already extracted via the CNN, training was fast (around five minutes on a dual-core CPU). This allowed us to do 5-fold cross-validation in order to optimize the hyperparameters. We ended up using trees with 40 leaves, with a minimum of 150 samples per leaf. Each tree only had access to a (random) subset of 40\\% of the features, and 80\\% of the data (bagging). The validation set was used for early stopping, in order to prevent overfitting. \n   \n   \n\\section{Results and discussion}\n\\label{sec-results}\n\n\\subsection{Single band predictions}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\hsize]{figures\/g_predictors.pdf}\n\t\\caption{ \\textit{Left:} Comparing the predicted $M\/L$ to the true $M\/L$ for a gradient boosting machine without morphology (top) and with morphology (bottom). Both predictors only make use of the g-band. The colour of the points is a 2D gaussian kernel density estimate. \\textit{Right:} histogram of dex error (top) and $\\mathcal{L}_1$ (bottom) for each galaxy. Purple is used for the predictor without morphology, while green is used for the predictor that includes morphology. For both quantities, closer to zero is better, positive numbers denote overpredictions, and negative values are underpredictions. The figure only includes galaxies from the test set. The dashed lines show 0.15 dex errors.}\n\t\\label{fig-g-predictors}\n\\end{figure}\n\nOur first goal is to investigate how good a single g-band image can constrain $M\/L$. As described in Sect.~\\ref{ssec-gbm}, our GBM combines the morphology information from the CNNs with other information (luminosity statistics, distance, and pixelsize). To evaluate the benefit of morphology, we compare this pipeline to a similar predictor that does not make use of morphology, nor any other resolved data (such as the luminosity features and the pixel size). Instead, we use two features: the g-band luminosity $L_g$ (calculated from the SDSS modelmag flux and distance), and the redshift. This reference method is shown in the top left panel of Fig.~\\ref{fig-g-predictors}, while our method (including morphology) is shown in the bottom left panel. These panels plot the predicted $M\/L$ against the ground truth (the Bayesian $M\/L$ from GSWLC). Even though the reference method only uses $L_g$ and redshift, it does not perform all that bad. We note that this works differently than using a single 3.4~$\\mu$m band, constant $M\/L$ assumption. This reference estimator can use the trend that low $M\/L$ spirals tend to have fewer stars (and hence they are less luminous) than a typical high $M\/L$ elliptical \\citep{review-hubble-sequence}. As discussed in Sect.~\\ref{ssec-interpretation}, it can also use the redshift feature to make use of Malmquist bias.\n\nIt is however clear that including morphology features clearly improves the results. The test set $\\mathcal{L}_1$ (Eq.~\\ref{eq-l1loss}) improves from 4.52 to 2.29. If we disregard the Bayesian uncertainty on the ground truth, we can use the root mean square logarithmic error (RMSLE), which improves from 0.227 dex to 0.124 dex. Including morphology also leads to less biased estimates, as seen on the right panels of Fig.~\\ref{fig-g-predictors}. For our estimator, 85.4\\% of the test set falls within 0.15 dex, while this is only 55.1\\% for the reference method without morphology. The reference method seems to be biased towards underpredictions (although there is a long tail towards overpredictions extending outside the histogram). This is mainly caused by the lack of predictions above 3.3: it seems like these high $M\/L$ cases are not easily found by $L_g$ and redshift alone. We will see in Sect.~\\ref{ssec-interpretation} that these galaxies mainly correspond to edge-on disks.\n\nInterestingly, the outliers in $\\Upsilon_{\\textrm{pred}, i} \/ \\Upsilon_{\\textrm{true}, i}$ do not necessarily match the outliers regarding $\\mathcal{L}_1$. For example, most underpredictions have a large error on $\\Upsilon_{\\textrm{true}, i}$. The largest outliers in $\\mathcal{L}_1$ (when including morphology) are overestimations. These have a low actual $M\/L$, which often results in a lower error on $M\/L$, and hence these datapoints are punished harder by our loss. A different loss function will weight galaxies differently, but we found that this does not change our conclusions (see Appendix~\\ref{app-loss-functions}). \n  \nIncluding morphology allows us to detect galaxies with $M\/L > 3.3$. However, the highest $M\/L$ that is predicted is 5.6 (while the ground truth values run up to 16.6). For one, there are only a limited number of these extreme cases, and it is safer to predict a lower $M\/L$. Moreover, this suggests that there is no easy way to detect these samples from the g-band images alone. These samples also have a larger error on $M\/L$, and hence a more conservative estimate is not punished as hard for these samples.\n  \n\\subsection{Interpretation}\n\\label{ssec-interpretation}\n\n\\begin{figure}\n \t\\centering\n \t\\includegraphics[width=\\hsize]{figures\/permutation_importances.pdf}\n \t\\caption{Feature importance ranking, by the amount the validation $\\mathcal{L}_1$ increases after permuting that feature. The CNN 4 features come from the ResNet50 that was retrained on $M\/L$ (the rightmost network in Fig.~\\ref{fig-pipeline}). All luminosity features are derived from the raw (star subtracted but not scaled) images, as described in Sect.~\\ref{ssec-gbm}. \\textit{Top:} standard predictor as described in Sect.~\\ref{sec-ml}, using all features. \\textit{Bottom:} a (freshly trained) predictor which does not make use of CNN 4.}\n \t\\label{fig-feat-importances}\n\\end{figure}\n\nOne of the useful properties of our pipeline is that it decouples morphology extraction and $M\/L$ prediction. The morphology detection by a CNN can be understood by inspecting the different layers. The first layers learn simple features such as edges, while deeper layers can learn to detect spiral arms, bars, or other features \\citep{dieleman}. The LightGBM can be interpreted by looking at the feature importances. These are presented in Fig.~\\ref{fig-feat-importances}. We have used \\textit{permutation importances}, which proved to be a robust feature importance measure in the study of \\citet{permutation-importance}. A certain feature's importance is calculated by permuting all observations of that feature, calculating the validation set $\\mathcal{L}_1$, and subtracting the non-permuted $\\mathcal{L}_1$ from this. The permuting leads to randomizing that feature without losing the distribution's properties. If the GBM heavily relies on a particular important feature, the permutation should increase $\\mathcal{L}_1$ considerably, leading to a larger importance.\n\nFrom the top panel of Fig.~\\ref{fig-feat-importances}, we can see that the ten most important features contain a mix of luminosity features, distance related features, and morphology features. The morphology features which are used the most are the ones from CNN 4, which was retrained to optimize $M\/L$. As discussed further below, these tend to correlate directly with $M\/L$. They no longer directly correspond to the GZ2 probabilities, but since CNN 4 only uses the log-normalized image, its features only depend on the galaxy's morphology. CNN 4 essentially eases the work for the GBM by moving part of the $M\/L$ prediction to that CNN. Since this hinders the interpretability of the model, we also show the feature importance for a GBM that does not make use of CNN 4 (bottom panel of Fig.~\\ref{fig-feat-importances}). This predictor is slightly worse, with a test $\\mathcal{L}_1$ of 2.41 (instead of 2.29). Although simple GZ2 questions such as the presence of galaxy features (e.g. spiral arms) correlate well with $M\/L$ (see below), they are not part of the most important features. Instead, the luminosity statistics seem to be more robust features. Since ellipticals and spirals have different brightness profiles, the luminosity statistics (such as the ratio of the mean centre luminosity and total luminosity) do provide morphological information. The GZ2 features that are most important (if CNN 4 is not present) look for a lack of bulge, and for irregular galaxies. Apparently, these two features can not be easily substituted by luminosity statistics.\n\nDue to the large number of features, the model can be resistant against the removal of some features. For example, if we remove the total luminosity feature (sum over all pixels), there is still the mean around the central pixel luminosity which can serve as a proxy. So if we train the LightGBM after leaving out the total luminosity feature, the test set $\\mathcal{L}_1$ only increases by 0.02. This shows an important difference between the permutation importance and so-called drop-out importance (increase in $\\mathcal{L}_1$ after retraining the model without that feature). If we have highly correlated features, retraining the model without one of those features will allow the similar features to make up for its lack. Our permutation importance measures something different: how important is that feature in the current model. We found that by only using the top 50 features (and redoing the cross-validation), the results stay the same. The computational time, however, decreases dramatically, with training only taking about 40 CPU seconds (from 16.7 CPU minutes) and evaluating on the training set taking only 4.2 CPU seconds (instead of 20 CPU seconds), using two threads on a Intel i5 processor.\n\n\\newcommand\\Tstrut{\\rule{0pt}{2.6ex}}      \n\\newcommand\\Bstrut{\\rule[-0.9ex]{0pt}{0pt}}\n\\newcommand{\\TBstrut}{\\Tstrut\\Bstrut}\n\\begin{table}\n\t\\caption{Test set $\\mathcal{L}_1$ for a LightGBM model that uses only the features which are checked. CNN 1-3 refers to the first three CNNs (from the left) in Fig~\\ref{fig-pipeline}, all of which are only trained on the Galaxy Zoo probabilities. CNN 4 is the ResNet50 which was retrained on $M\/L$. When leaving out the distance feature (redshift and pixelsize), we also replace all luminosity features by the corresponding flux features. The baseline $\\mathcal{L}_1$ (i.e. minimizing $\\mathcal{L}_1$ when predicting a single value) is 6.52.}    \n\t\\label{tab-feature-removing}  \n\t\\centering                         \n\t\\begin{tabular}{c c c c c}   \n\t\t\\hline\\hline         \n\t\tCNN 1-3 & CNN 4 & Luminosity & Distance & Test $\\mathcal{L}_1$ \\TBstrut \\\\    \n\t\t\\hline                    \n\t\t\\checkmark & \\checkmark & \\checkmark & \\checkmark & 2.29 \\Tstrut \\\\\n\t\t& \\checkmark & \\checkmark & \\checkmark& 2.37 \\\\\n\t\t\\checkmark & & \\checkmark & \\checkmark & 2.41 \\\\\n\t\t\\checkmark & \\checkmark & & \\checkmark& 2.32 \\\\\n\t\t\\checkmark & \\checkmark & \\checkmark & & 2.67 \\\\\n\t\t&& \\checkmark& \\checkmark & 3.38 \\\\\n\t\t\\hline                            \n\t\\end{tabular}\n\\end{table}\n\n\\begin{figure*} \n\t\\centering\n\t\\includegraphics[width=17cm]{figures\/scaling_relations.pdf}\n\t\\caption{Influence of a few features on $M\/L$, for the training set. Due to the large number of datapoints, we use (hexagonal) bins. The opacity of each bin corresponds to the number of galaxies in the bin, in a non-linear way (ensuring that lower densities are still visible). The total luminosity denotes the total g-band luminosity in $L_{\\odot, g}$, and is shown in log space. For the top left panel, the $P_{\\rm{feature}}$ feature from CNN 1 estimates the GZ2 probability of the galaxy having a feature or disk. For the bottom left panel, the morphology is determined from the last layer of CNN 1. A galaxy is defined as irregular if the predicted probability of being irregular is larger than 20\\%, it is considered edge-on if it is not irregular but has a probability of being edge-on larger than 40\\%, and it is a feature or disk if the corresponding probability is larger than 40\\% (but it is not in the previous two categories). The ellipticals are the remaining datapoints. }\n\t\\label{fig-scaling-relations}\n\\end{figure*}\n\nMaybe even more important is what happens when we leave out a group of features. The results of such an ablation study can be seen in Table~\\ref{tab-feature-removing}. We see that a reliable distance estimate (in our case the redshift from SDSS) is quite important. It should be noted that we need a distance estimate to go from $M\/L$ to stellar mass anyway. The luminosity is less important in order to estimate $M\/L$. This does not contradict with the total luminosity being an important feature: as discussed further below, the luminosity features allow the machine to roughly distinguish between high and low $M\/L$. However, in the absence of luminosity features the morphology features can take their place. We also see that CNN 4 and CNN 1-3 complement each other well: we see a clear improvement when all CNNs are combined. The results are clearly worse when no CNN is present (bottom row, $\\mathcal{L}_1 = 3.38$), although this predictor still has access to the resolved luminosity features, allowing it to outperform the reference method (top panel of Fig.~\\ref{fig-g-predictors}; $\\mathcal{L}_1 = 4.52$).\n   \nOf course, our model does not use the actual GZ2 probabilities as input: this ensures that no human interaction is required when making new predictions. CNN 1 to 3 exist to estimate the GZ2 probabilities. These estimations are not perfect, and we might wonder what the effect of these errors might be on the final prediction. To determine this, we replaced the custom network (CNN 3) by the actual GZ2 cumulative probabilities. The resulting $\\mathcal{L}_1$ of this cheating model is 2.25 (compared to 2.29 for the standard estimator). This shows that only minor improvements can be made by further improving the GZ2 predictions. It also shows that we can trust our CNNs to make good morphology detections, and hence that decisions made regarding the CNN pipeline are not negatively impacting our further analysis.\n\nWe can see the effect of the different features by looking at their influence on $M\/L$. In Fig.~\\ref{fig-scaling-relations}, we show how the target $M\/L$ correlates with some of the features. These correlations are the driving force behind the machine learning. In the top left panel, we can see that the luminosity feature can distinguish roughly between high $M\/L$ and low $M\/L$ galaxies: galaxies with $L_g < 10^{6.5} L_{\\odot, g}$ tend to have a low $M\/L$, while galaxies with $L_g > 10^{7.5} L_{\\odot, g}$ tend to have a high $M\/L$. With the help of the Galaxy Zoo probabilities estimated by the ResNet50 architecture (CNN 1), we can further distinguish between the two groups even in the case of intermediate luminosities. In this case, the GZ2 probability $P_{\\textrm{feature}}$ is used as a colour scale, where $P_{\\textrm{feature}}$ gives the probability of a morphological feature or disk being present (in contrast to being smooth, or a ``star or artifact''). For constant luminosity, galaxies with features tend to have a lower $M\/L$. One exception is the cloud of feature galaxies with $M\/L > 4$, which is explained in the next paragraph. Looking at lower $M\/L$ (< 2), we see that the probability of the galaxy having a feature increases with luminosity. This can be attributed to a distance-dependent classification bias \\citep{gz2-hart}. Essentially, the spiral structure is hard (or impossible) to see for fainter, more distant galaxies (with a lower $S\/N$). The algorithm can detect these low $S\/N$ galaxies (through luminosity features, redshift and scaling value), and react by predicting a lower $M\/L$ than is typical when $P_{\\rm{feature}}$ is low. \n\nThe bottom left panel is similar to the top left, but has combined three Galaxy Zoo features ($p_{\\textrm{feature}}$, $p_{\\textrm{edge-on}}$ and $p_{\\textrm{irregular}}$) to create four categories. We notice that irregulars have a very low $M\/L$, probably because a merger-triggered star formation burst leads to a young stellar population \\citep{merger-starburst}. As expected, there is the bimodality between disk galaxies and ellipticals, where ellipticals are believed to be more evolved objects with an older stellar population (and hence a higher $M\/L$). The big exception here is edge-on disks, which seem to have a very high $M\/L$. This is the result of our definition of the stellar luminosity $L$, where we directly multiplied the flux by $4\\pi D^2$. Edge-on disks are more attenuated and hence we receive less light, resulting in a higher $M\/L$. While our definition ignores anisotropy (the luminosity now depends on the viewing angle), the $M\/L$ only serves as a bridge to estimate the total stellar mass. The CIGALE models assume no particular geometry. We have inspected the influence of $p_{\\textrm{edge-on}}$ on the total stellar mass $M$, and found no clear trend: these two variables have a Spearman correlation coefficient of only 0.03. This suggests that we can still estimate the stellar mass, even when attenuation in edge-on disks causes the observed luminosity to be lower than the intrinsic luminosity (averaged over all directions).\n\nThe top right panel of Fig.~\\ref{fig-scaling-relations} clearly shows the effects of Malmquist bias. Our main selection criterion is $D_{25} > 0.4$ arcmin. At higher redshift, we only include very large (and thus often luminous) objects. These tend to have a high $M\/L$. The GBM makes use of this bias by predicting a high $M\/L$ for higher redshift galaxies. The result is that for our sample, the predictions are actually more accurate for further away galaxies. This stresses the importance that the test set (or any set on which the machine learning is evaluated) should have the same selection criteria as the training set. We learn by example, and so the assumption is that new samples are similar to the training set. \n\nThe CNN 1 features from the left two panels are actually not often present in the trees of the GBM, due to the presence of CNN 4. CNN 4 was trained to correlate more directly with $M\/L$, as seen in the bottom right panel. The result is that the CNN 4 features are no longer directly interpretable. Leaving out CNN 4 increases the test $\\mathcal{L}_1$ by only 0.12, as seen from Table~\\ref{tab-feature-removing}, so the relations from the left two panels do give us some insight in the behaviour of the machine learning. \n\n\\subsection{Using colour and morphology}\n\\label{ssec-gi-morph}\n\n\\begin{figure*}  \n\t\\centering\n\t\\includegraphics[width=17cm]{figures\/gi_predictors.pdf}\n\t\\caption{\\textit{Top left, top right and bottom left}: predicted $M\/L$ vs ground truth for $g-i$ power law, global luminosity GBM and the morphology GBM respectively. Only the galaxies in the test set are shown. \\textit{Bottom left}: histogram of dex errors. }\n\t\\label{fig-gi-predictors}\n\\end{figure*}\n \nSo far, we have shown that it is possible to make reasonable $M\/L$ (and hence stellar mass) predictions with observations in only one band (and ideally a distance estimate). This of course does not replace traditional stellar mass methods, but shows that the morphology of a galaxy does provide valuable information. Now we can wonder: does morphology give the same information as colour, or is there a benefit in using morphology in addition to global g and i luminosities? To investigate this, we added the g-band luminosity $L_g$, i-band luminosity $L_i$ and g - i colour $L_g\/L_i$ as features to the LightGBM. These are derived from the SDSS modelmags, which were also used for the SED fitting \\citep{gswlc, gswlc2}. After training has completed, we selected the 50 features that were used the most in the GBM, and retrained using only those. The result is shown in the bottom left panel of Fig.~\\ref{fig-gi-predictors}. The resulting test set $\\mathcal{L}_1$ is 1.12. We compare this against a standard method to estimate the $M\/L$ from a single colour: a power law between $M\/L$ and $g-i$ colour \\citep{zibetti2009}. The two power law parameters were fit on the training set, minimizing $\\mathcal{L}_1$ (just like the machine learning). The result is shown in the top left panel of Fig.~\\ref{fig-gi-predictors}, although the test metrics exclude two datapoints with extreme mispredictions. This already shows one of the drawbacks of this method: it is not applicable if the two fluxes are `incompatible' (e.g. due to large uncertainties or observational artifacts). To make a fairer comparison, we also compare against a more sophisticated single colour method. Instead of assuming a power law, we used a LightGBM regressor (which was also used for the morphology method). This method made use of four features: redshift, $L_i$, $L_g$ and $L_g \/ L_i$. The last feature is beneficial since the individual decision trees only split based on one feature. This method, which does not make use of morphology, is shown in the top right panel of Fig.~\\ref{fig-gi-predictors} and achieves a $\\mathcal{L}_1$ of 1.26. \n \nThere is a clear improvement when going from a power law to a GBM. The power law is unable to make a good fit for both low and high $M\/L$ (low and high $M\/L$ refer to the bimodality also seen in Fig.~\\ref{fig-ml-hist}). There's also a large number of outliers, and these influence the fit to keep the $\\mathcal{L}_1$ under control. A GBM can easily improve on this: every point in the feature space is assigned a $M\/L$ which minimizes the corresponding $\\mathcal{L}_1$, which hence avoids the bias that can be seen in the power law. This can also be verified by looking at the distribution of dex errors, in the bottom right panel of Fig.~\\ref{fig-gi-predictors}. The $g-i$ power law has a strong tail towards overpredictions: 5.1\\% of galaxies have a logarithmic error larger than 0.15 dex (overpredictions), while only 1.8\\% have a logarithmic error smaller than -0.15 dex (underpredictions). For the GBM method (without morphology), only 2.3\\% have a logarithmic error outside 0.15 dex (over- and underpredictions).\n \nAdding morphology to the GBM (bottom left panel of Fig.~\\ref{fig-gi-predictors}) further reduces the test $\\mathcal{L}_1$ to 1.12, resulting in a better estimator. An important question to investigate is whether a test $\\mathcal{L}_1$ of 1.12 (with morphology) is \\textrm{significantly} better than a test $\\mathcal{L}_1$ of 1.26 (without morphology). The reported $\\mathcal{L}_1$ are the mean $\\mathcal{L}_1$ over the $7\\,363$ test samples. Hence, we can look at the distribution of the individual $\\mathcal{L}_1$. First, we did a Kolmogorov-Smirnov two-sample test, for a one-sided comparison. The null hypothesis, which states that the model with morphology does not have a significantly lower $\\mathcal{L}_1$ than the model without morphology, can be rejected with very high confidence (p-value $7.6\\times 10^{-9}$). Next, we took 100\\,000 bootstrap samples of the $\\mathcal{L}_1$ distribution of both models. The mean $\\mathcal{L}_1$ of each of these bootstrap samples has no overlap for the two models. The bootstrap estimate for the mean $\\mathcal{L}_1$ of the method without morphology is $1.26 \\pm 0.01$, while the estimate for the method with morphology is $1.12 \\pm 0.01$. Hence, we can conclude that adding the morphology features gives a significant improvement.\n \nWe notice most improvement at $M\/L > 4$, which are dominated by edge-on galaxies, as can be seen from the bottom left panel of Fig.~\\ref{fig-scaling-relations}. We can quantify the improvement for each of the morphology classes from that panel. By including morphology features, $\\mathcal{L}_1$ improves from 1.15 to 1.08 for irregulars, from 1.31 to 1.18 for feature (disk) galaxies, from 1.14 to 1.07 for ellipticals, and from 1.52 to 1.14 for edge-on galaxies (the biggest improvement). A single colour often underestimates the stellar mass for these edge-on galaxies, and morphology information can be used to prevent this. \n \nWhen comparing these single colour predictors to the g-band image predictor (bottom left panel of Fig.~\\ref{fig-g-predictors}, $\\mathcal{L}_1 = 2.29$), we see that morphology can not replace colour. It is, however, impressive that a g-band $M\/L$ predictor gets close to a $g-i$ power law, even though the power law is also fitted to this dataset. This refitting was necessary, since the original \\citet{zibetti2009} power law has a test set $\\mathcal{L}_1$ of 7.47 (mainly due to the different SPS model grid than GSWLC). Due to its easy application, a single colour power law is commonly used \\citep[e.g.][]{scpl-2, scpl-1}, but as shown here it should be used with caution. Machine learning techniques can help improve these estimates without needing to do SED fitting at evaluation time. The morphology can help assist predictions of $M\/L$. Since it is available for more nearby galaxies anyway, it can be a valuable improvement over techniques that only use global flux information (and hence dismiss all information from the resolved image).\n\n\\subsection{Morphology assisted $M\/L$ with multiple colours: limitations of the ground truth}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\hsize]{figures\/ugriz_buildup_morphology.pdf}\n\t\\caption{Comparison of the performance (test set $\\mathcal{L}_1$) of different GBM predictors. The predictors only make use of the denoted broadbands. The yellow bars only make use of global information (luminosity, distance and colours), while the green bars also make use of morphology.}\n\t\\label{fig-ugriz-buildup}\n\\end{figure}\n\nAfter looking at single band and single colour predictors, we might wonder what happens when multiple colours are available. Similar to Sect.~\\ref{ssec-gi-morph}, we train gradient boosting machines with and without morphology. For a sequence of available SDSS broadbands, the results are shown in Fig.~\\ref{fig-ugriz-buildup}. We see that the performance stagnates at an $\\mathcal{L}_1$ of 1, when the predictions are as accurate as the uncertainty on the ground truth. Including morphology improves the results for all cases, although the stagnation at $\\mathcal{L}_1 = 1$ limits the benefit when multiple broadbands are available.\n\nThe problem is that we are limited by our ground truth (i.e. the prediction target): the GSWLC $M\/L$ come from Bayesian SED fitting to \\textrm{global} fluxes. This means that the morphology can only make up for missing broadbands, but not for the uncertainties that come from using only global fluxes. It is possible to apply SED fitting pixel-by-pixel, and then sum the stellar masses of the individual pixels. The assumption that a spectrum is the sum of SSPs with a simple attenuation law applies better to individual pixels than to complete galaxies. So while pixel-by-pixel SED fitting is believed to be more accurate \\citep{unresolved-sed1, unresolved-sed2}, it is also more expensive. Pixel-matched panchromatic datasets are required, where the band with the worst resolution effectively sets the working resolution. A high $S\/N$ is required for all relevant pixels. This method is also much more computationally intensive, and hence the number of models that can be fit is limited. The result is that at the time of writing, no large pixel-by-pixel SED fitted catalogues exist. Should they become available in the future, our method can be retrained which can make it possible to beat global flux methods. \n\nAnother way to improve the ground truth is by using more information. Currently, GSWLC uses the WISE observations to estimate the total infrared luminosity $L_{IR}$ \\citep{gswlc2}. Assuming energy balance, this then constrains the total energy absorbed by the dust, allowing us to make better estimates of the unattenuated stellar spectrum. Although the uncertainty on this $L_{IR}$ estimation is only 0.08 dex, it uses just a single WISE band. This can make it troublesome for galaxies with large uncertainties on that WISE band, or for galaxies where the correction for AGN contribution leads to additional uncertainties. The best way to constrain $L_{IR}$ is still to measure it with FIR observations, and hence galaxies with FIR data will have a slightly better ground truth $M\/L$. In addition to using UV-FIR broadbands, spectroscopy can be used as an additional constraint for the SED fitting \\citep{beagle, beagle-spectroscopy, bagpipes, prospector}. Limiting the training to samples were this additional information (more broadbands and\/or spectroscopy) is available unfortunately implies that the size of the training set will be smaller.\n\nOf course, the best case scenario would be that our ground truth were the actual stellar $M\/L$. Unfortunately, there is no way to directly measure stellar mass, instead of estimating it through SPS. There is however a situation in which we know the stellar mass: cosmological simulations. With radiative transfer, it is possible to create mock observations of these simulated galaxies \\citep[e.g.][]{camps2018}. These then could serve as a good training target, since we no longer have to deal with the limitations of SED fitting. The radiative transfer treats the effects of dust rigorously (in contrast to an empirical attenuation law), and star forming regions can be treated with subgrid prescriptions. Recently, some successes have been achieved with training CNNs on cosmological simulations, while testing them on real galaxies \\citep[e.g.][]{simulation-cnn-1,simulation-cnn-2}. The main limitation of this approach is that there are still discrepancies between the observed and the simulated universes.\n\n \n\\subsection{Applications and discussion}\n\nThe success of using morphology information to predict stellar mass depends on the quality of the images. In this work, we limited ourselves to $D_{25} > 0.4$ arcmin to make sure that we have enough pixels for each galaxy. Upcoming surveys will allow for deeper and higher resolution observations, drastically increasing the number of galaxies that are well resolved. In particular, Euclid will have a very broad optical band ($r+i+z$) which is useful to get deeper images. These will be combined with ground-based photometry ($griz$) and Euclid photometry ($YJH$) \\citep{euclid}. The goal is to have 1.5 billion galaxies with very accurate morphometric information. These will be an excellent target for training and testing morphological stellar mass estimates. Besides Euclid, LSST \\citep{lsst} and WFIRST \\citep{wfirst} will also provide wide-field optical\/NIR imaging which could benefit from our method. As discussed in the previous section, the hardest but most rewarding challenge to solve will be to acquire more accurate ground truth $M\/L$, such as from pixel-by-pixel SED fitting. The morphology can then use the resolved information to improve on a global colour estimate. \n\nGSWLC 1 contains about $700~000$ galaxies and is one of the largest SED fitted catalogues to date. This already shows that even global SED fitting will be computationally challenging for Euclid's 1.5 billion galaxies, without significantly reducing the number of fitted models. A machine learning approach (with or without morphology) can be a good alternative. With the use of a single GPU, CNN evaluation is more than an order of magnitude faster than (global) SED fitting on a 100 core CPU cluster. If we train on pixel-by-pixel SED fits, we further avoid the outshining bias. So training on a small but accurate $M\/L$ subset of Euclid, and evaluating on the remaining $> 1$ billion galaxies seems promising. We found that if we train the GBM with only half of the data, the test $\\mathcal{L}_1$ degrades only slightly to 2.31 (from 2.29), confirming that the quality of the training data is more important than the quantity.\n \nOur pipeline can also be used to predict other galaxy properties, such as SFR or metallicity. For both of these properties, spectroscopy can be a valuable constraint on the ground truth. We hope that this two-step process (CNN + GBM) can further improve our understanding of which morphological properties best correlate with the physical properties of a galaxy. This can then further constrain galaxy evolution models. \n \n\n\\section{Summary and conclusions}\n\\label{sec-conclusions}\n\nWe made use of a machine learning framework to make morphology assisted $M\/L$ predictions. First, we predicted $M\/L$ from a single g-band image. The pipeline can be split in two parts: a first part estimates morphology features such as the probability of the galaxy being featureless, edge-on, merging, etc. This information is then combined with redshift, pixel size, and a few g-band luminosity features in order to predict $M\/L$. We optimized a $\\mathcal{L}_1$ loss that weights down samples with a large uncertainty on $M\/L$. Our best model has a test set $\\mathcal{L}_1$ of 2.29, and a RMSLE of 0.124 dex. The morphology from the g-band can partially make up for a lack of observed colour. These predictions are made possible because featureless ellipticals tend to have a higher $M\/L$ than galaxies with features such as spirals (left two panels of Fig.~\\ref{fig-scaling-relations}). Irregular galaxies tend to have a low small $M\/L$, while highly inclined disk galaxies tend to have a very high $M\/L$. Even though the spiral features can not be detected for more distant, dimmer galaxies, the algorithm is trained to produce unbiased results. \n\nObserving multiple bands does lead to a better constrained $M\/L$.  A $g-i$ power law, recalibrated on our dataset achieves a $\\mathcal{L}_1$ of 1.90 (compared to 2.29 for our g-band only method). The $g-i$ power law has trouble fitting both small and large $M\/L$. This can be avoided by using a GBM (or other machine learning method). We find that a GBM that makes use of global $g$ and $i$ fluxes and a distance estimate achieves a $\\mathcal{L}_1$ of 1.26. Including the g-band morphology features further improves the $\\mathcal{L}_1$ to 1.12, showing that morphology information does have an added benefit over only global colours. Even though this improvement is small, we have shown that it is significant. Most of the improvement happens for edge-on disk galaxies. With global fluxes only, it is hard to distinguish a more inclined and hence attenuated galaxy from an older one (both effects lead to redder colours), and we find that the $M\/L$ tends to be underpredicted in those cases.\n\nIn future work, we hope that this machine learning framework can be trained on better target estimates for $M\/L$. Currently, our target values are derived from unresolved fluxes, limiting the benefit of our method over global colour methods. Our method can be fit to reproduce pixel-by-pixel SED fitted $M\/L$, but has less strict requirements on pixel $S\/N$, and is faster at evaluation time. \n\n\n \\begin{acknowledgements}\n       We thank the anonymous referee for helpful comments which improved this paper. W.D., S.V. and M.B. gratefully acknowledge support from the Flemish Fund for Scientific Research (FWO-Vlaanderen). W.D. is a pre-doctoral researcher of the FWO-Vlaanderen. Special thanks to the Flemish Supercomputer Centre (VSC) for providing computational resources and support. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http:\/\/www.sdss3.org\/. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.\n \\end{acknowledgements}\n\n   \n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Bethe--Sommerfeld conjecture is the following statement: for any $d \\geq 2$ and any periodic function $V:{\\mathbb R}^d \\to {\\mathbb R}$, the spectrum of the Schr\\\"odinger operator \n\\[\nL_V := -\\nabla^2 + V\\]\nhas only finitely many gaps. This was studied by many people with important advances in \\cite{HelMoh98,Karp97,PopSkr81,Skr79,Skr84,Skr85,Vel88}, and culminating in the paper of Parnovskii \\cite{Parn2008AHP}. One way to think about the Bethe--Sommerfeld conjecture is that any energy $E$ that is very large relative to the potential $V$ lies in the spectrum of $L_V$. Since discrete Schr\\\"odinger operators are bounded, the high-energy region is absent, so the appropriate discrete version of the Bethe--Sommerfeld conjecture lies in the region of small $V$. Discrete versions of the conjecture were proved on square lattices by Embree--Fillman in dimension $d=2$ \\cite{EmbFil2017} and by Han--Jitomirskaya in arbitrary dimensions $d \\geq 2$ \\cite{HanJit2017}. In those works, the spectrum of a discrete periodic Schr\\\"odinger operator on the square lattice $\\ell^2({\\mathbb Z}^d)$ with a small potential was shown to consist of at most two intervals. Moreover, they showed that as soon as at least one period of the potential is odd, then the spectrum is an interval, and, in the event that a gap opens perturbatively, it must happen at the exceptional energy $E=0$.\n\nMany interesting physical models occur with different underlying lattice geometries beyond the standard square lattice. One of the most prominent such models is supplied by graphene, a two-dimensional material that consists of carbon atoms at the vertices of a hexagonal lattice. The fascinating properties of graphene have led to a substantial amount of attention in mathematics and physics, see e.g.\\ \\cite{BZ2018,BHJ2018,CGPNG,DelMon2010,FW12,HKR2016,KP07,N11} and references therein. \nIn view of this, we are motivated to study the Bethe--Sommerfeld conjecture for the hexagonal lattice and for the corresponding dual lattice (the triangular lattce).\n\nIn addition to the hexagonal and triangular lattices, we also study the square lattice with next-nearest neighbor interactions, which is motivated by the extended Harper model (EHM). The EHM was proposed by Thouless \\cite{Thouless83} and has also led to a lot of study in mathematics and physics \\cite{AJM17,H17,H18,HJ17,Thouless94,JM15}; it corresponds to an electron in a square lattice that interacts not only with its nearest neighbors but also its next-nearest neighbors.\nIn the following, we will refer to square lattice with next-nearest neighbor interactions as the {\\it EHM lattice}, in order to distinguish it from the standard square lattice.\n\n\n\n\n\n\n{Let us mention in particular the closely related work \\cite{HKR2016}.} In \\cite{HKR2016}, Helffer, Kerdelhu\\'e and Royo-Letelier developed a Chambers analysis for magnetic Laplacians on the hexagonal lattice (and its dual lattice: triangular lattice) with rational flux.\nThey showed that for a non-trivial rational flux $p\/q\\notin{\\mathbb Z}$, the magnetic Laplacians on hexagonal and triangular lattices have non-overlapping (possibly touching) bands.\nThis recovers a similar feature of the square lattice \\cite{BelSim82}.\nHowever, unlike the square lattice that has no touching bands except at the center for $q$ even \\cite{VMou89}, \nthey were able to give an explicit example of non-trivial touching bands for hexagonal and triangular lattices. \nIndeed they showed that the triangular Laplacian has touching bands at energy $E=-\\sqrt{3}$ for $p\/q=1\/6$, and the hexagonal Laplacian has touching bands at energies $E=\\pm \\sqrt{3}$ and $0$ for $p\/q=1\/2$.\nTherefore, the underlying geometry is greatly responsible for the formation of touching bands.\nBut it {has remained} unclear that whether there will be other touching bands for different fluxes (and if any, what are the locations).\nIn our work we are able to give a sharp criterion of the formation of touching bands for the free Laplacians on these lattices and the EHM lattice, see Theorems \\ref{t:bsc:tri}, \\ref{t:bsc:hex} and \\ref{t:bsc:nnn}.\n\nMotivated by these models, we prove the Bethe--Sommerfeld conjecture for the triangular, hexagonal, and EHM lattices. \nSimilar to the square lattice case, we show that small perturbations of the free Laplacian may only open gaps at certain {\\it exceptional energies}.\nOur proof uses the perturb-and-count technique developed in \\cite{HanJit2017}.\nThe overall  strategy is to argue by contradiction. \nNamely, we assume two adjacent spectral bands of the free Laplacian have a trivial overlap containing a single energy $E$.\nThen, we carefully choose a Floquet parameter and perturb all the Floquet eigenvalues along two different directions. \nIt is then argued that different directions lead to different counting of eigenvalues that move above\/below $E$, hence a contradiction.\nAt the exceptional energies, we are able to develop a {\\it sharp} criterion, in terms of the periods, of whether the gaps could possibly open {under an infinitesimal perturbation}.\nWe also construct potentials that do open (the theoretically existing) gaps at these exceptional energies.\n\nAlthough the general strategy follows that of \\cite{HanJit2017}, {there are several challenges to overcome in the present work.}:\n\\begin{itemize}\n\\item\nThe Floquet parameters and perturbation directions that we choose in the perturb-and-count technique are strongly model-dependent in a subtle fashion. For example, at non-exceptional energies, we locate Floquet parameters and a perturbation direction in a way such that the Floquet eigenvalues with vanishing linear terms have quadratic terms of the same sign along this direction. \nAt the exceptional energy of the triangular lattice, we choose two directions such that the eigenvalues with vanishing gradients have quadratic terms of different signs along the two directions; for a more detailed discussion, see Remark~\\ref{rem:tri}.\nThis is similar to what was done in \\cite{HanJit2017} for the square lattice case.\nHowever, for the EHM lattice, any direction will lead to the same number of positive and negative quadratic terms; see Remark \\ref{rem:sqn}.\nThis issue is resolved by a new construction: we find a direction that moves approximately $2\/3$ of the degenerate eigenvalues  up while the other $1\/3$ move down.\nAll these constructions depend heavily on the Floquet representation of the eigenvalues, and thus get more difficult as the underlying geometry gets more complicated.\n\\item Applying the perturb-and-count ideas directly to the hexagonal lattice  is quite difficult,\ndue to the fact that the Floquet eigenvalues do not have simple expressions; compare \\eqref{eq:hexTriBandRel}.\nHowever, one can relate Laplacians and Floquet matrices for the triangular and hexagonal lattices in a fairly elegant fashion ({see \\cite{HKR2016} and our \\eqref{eq:hexSquareTri}}). Thus, we prove the Bethe--Sommerfeld conjecture directly for the triangular lattice and then derive the corresponding statement for the hexagonal lattice via a somewhat soft argument.\n\\item Because of the more complicated structure of the lattices involved, constructing potentials that open gaps at the exceptional energies is substantially more difficult than in the square lattice. \nIn particular, we need to construct (2,2)-periodic potentials that live on eight vertices for the hexagonal lattice, and (3,3)-periodic potential for the EHM lattice.\nIn this paper we develop an {robust} technique to study these finite volume problems in a sharp way.\nIndeed, we can not only prove that a gap exists, but also estimate its size up to a constant factor (see Theorems~\\ref{thm:triExampleGapLength}, \\ref{thm:hexQ}, and \\ref{thm:nnnExGapLength}). \nIn the case of the triangular lattice, we are even able to use our technique \\emph{exactly} compute the gap, not only estimate its size (Theorem~\\ref{thm:triExampleGapLength}).\n\\end{itemize}\n\n\\bigskip\n\n\\subsection{Main Results}\n\nLet us now describe more precisely the setting in which we work and the results that we prove. By a \\textit{graph}, we shall mean a pair $\\Gamma = ({\\mathcal V}, {\\mathcal E})$ where ${\\mathcal V}$ is a nonempty set and ${\\mathcal E}$ is a nonempty subset of ${\\mathcal V} \\times {\\mathcal V}$ with the following properties:\n\\begin{itemize}\n\\item For no $v \\in {\\mathcal V}$ does one have $(v,v) \\in {\\mathcal E}$;\n\\item If $(u,v) \\in {\\mathcal E}$, then $(v,u) \\in {\\mathcal E}$.\n\\end{itemize}\nIf $(u,v) \\in \\mathcal{E}$, we write $u\\sim v$ and we say that $u$ and $v$ are neighbors or neighboring vertices. We think of ${\\mathcal E}$ as the set of \\emph{directed edges}; $(u,v)$ represents the edge that originates at $u$ and terminates at $v$.\n\nGiven such a graph, we consider $\\mathcal{H}_\\Gamma = \\ell^2(\\mathcal{V})$ and the associated \\emph{graph Laplacian} $\\Delta_\\Gamma: \\mathcal{H}_\\Gamma \\to \\mathcal{H}_\\Gamma$, which acts via\n\\[\n[\\Delta_\\Gamma \\psi]_u\n=\n\\sum_{v \\sim u} \\psi_v,\n\\quad\nu \\in \\mathcal{V}, \\; \\psi \\in \\mathcal{H}_\\Gamma.\n\\]\nTechnically, this is the adjacency operator of the graph. Other authors use $\\psi_v - \\psi_u$ where we have only $\\psi_v$. Our convention is slightly more natural for the setting in which we wish to work. Concretely, all of the graphs that we consider in the present work have uniform degree (all vertices in a given graph have the same number of incident edges), and hence leaving off the $-\\psi_u$ term merely costs us a multiple of the identity operator, and it simplifies the appearance of a few calculations.\n\nBy a \\emph{Schr\\\"odinger operator} on $\\Gamma$, we mean an operator of the form $H_Q = H_{\\Gamma,Q} = \\Delta_\\Gamma + Q$, where $Q:{\\mathcal V} \\to {\\mathbb R}$ is a bounded function that acts on ${\\mathcal H}_\\Gamma$ by multiplication:\n\\[\n[Q\\psi]_u\n=\nQ(u) \\psi_u,\n\\quad\nu \\in \\mathcal{V}, \\; \\psi \\in \\mathcal{H}_\\Gamma.\n\\]\n\\begin{figure*}[t]\n\\begin{tikzpicture}[yscale=.6,xscale=.6]\n\\draw [-,line width = .06cm] (0,0) -- (6,0);\n\\draw [-,line width = .06cm] (0,0) -- (0,6);\n\\draw [-,line width=.06cm] (0,2) -- (6,2);\n\\draw [-,line width = .06cm] (2,0) -- (2,6);\n\\draw [-,line width=.06cm] (0,4) -- (6,4);\n\\draw [-,line width = .06cm] (4,0) -- (4,6);\n\\draw [-,line width=.06cm] (0,6) -- (6,6);\n\\draw [-,line width = .06cm] (6,0) -- (6,6);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](2,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](0,2) circle (.2);\n\\filldraw[color=black, fill=black](2,2) circle (.2);\n\\filldraw[color=black, fill=black](4,2) circle (.2);\n\\filldraw[color=black, fill=black](6,2) circle (.2);\n\\filldraw[color=black, fill=black](0,4) circle (.2);\n\\filldraw[color=black, fill=black](2,4) circle (.2);\n\\filldraw[color=black, fill=black](4,4) circle (.2);\n\\filldraw[color=black, fill=black](6,4) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=black, fill=black](2,6) circle (.2);\n\\filldraw[color=black, fill=black](4,6) circle (.2);\n\\filldraw[color=black, fill=black](6,6) circle (.2);\n\\end{tikzpicture}\n\\caption{The square lattice.}\n\\end{figure*}\nIn the present work, we study ${\\mathbb Z}^2$-periodic graphs. That is, we consider graphs whose vertices $\\mathcal{V}$ comprise a subset of ${\\mathbb R}^2$ and for which there exist linearly independent translations $\\bm{a}_1, \\bm{a}_2 \\in {\\mathbb R}^2$ which leave $\\Gamma$ invariant. That is to say:\n\\begin{itemize}\n\\item For any vertex $v \\in {\\mathcal V}$, $v + \\bm{a}_j \\in {\\mathcal V}$ for $j=1,2$;\n\\item For any edge $(u,v) \\in {\\mathcal E}$, $(u + \\bm{a}_j,v + \\bm{a}_j) \\in {\\mathcal E}$  for $j=1,2$.\n\\end{itemize}\nWe will then be most interested in studying the case when the potential $Q$ is itself periodic. In general, we will say that $Q:{\\mathcal V} \\to {\\mathbb R}$ is $\\bm{p} = (p_1,p_2)$-periodic for some $p_1,p_2 \\in {\\mathbb Z}_+$ if and only if \n\\[\nQ(u+p_1 \\bm{a}_1) = Q(u + p_2 \\bm{a}_2) = Q(u),\\quad\n\\text{for all } u \\in {\\mathcal V}.\n\\]\n\nThe square lattice is the graph with vertices ${\\mathcal V}_{\\mathrm{sq}} = {\\mathbb Z}^2$ and where \n\\[\n\\bm{n} \\sim \\bm{n}'\n\\iff\n\\| \\bm{n} - \\bm{n'} \\|\n=\n1.\n\\]\nHere and throughout the paper, $\\|\\cdot\\|$ denotes the Euclidean norm on ${\\mathbb R}^2$. It is easy to see that the associated Laplacian acts on $\\ell^2({\\mathbb Z}^2)$ via\n\\[\n[\\Delta_{\\mathrm{sq}} \\psi]_{n,m}\n=\n\\psi_{n-1,m} + \\psi_{n+1,m} + \\psi_{n,m-1} + \\psi_{n,m+1}.\n\\]\n\n\n\n\nPart of the motivation for the present work comes from \\cite{EmbFil2017, HanJit2017, KrugPreprint}. In \\cite{EmbFil2017}, Embree and Fillman showed that if $Q: {\\mathbb Z}^2 \\to {\\mathbb R}$ is $(p_1,p_2)$-periodic and sufficiently small, then $\\sigma(\\Delta_{\\mathrm{sq}}+Q)$ consists of one or two intervals and that the spectrum consists of exactly one interval whenever at least one of $p_1$ or $p_2$ is odd, which generalized the work of Kr\\\"uger, who proved a similar result under the stricter condition that the periods were coprime \\cite{KrugPreprint}. In \\cite{HanJit2017}, Han and Jitomirskaya showed that if $Q:{\\mathbb Z}^d \\to {\\mathbb R}$ is $(p_1,\\ldots,p_d)$-periodic and small, then the same results hold true: the spectrum has no more than one gap and has no gaps as long as at least one period is odd.\n\n\n\n\\subsection{The Triangular Lattice} \n\\begin{figure*}[b]\n \\begin{minipage}{0.45\\textwidth}\n \\centering\n\\begin{tikzpicture}[yscale=.85,xscale=.85]\n\\draw [-,line width = .06cm] (0,0) -- (6,0);\n\\draw [-,line width = .06cm] (0,{sqrt(3)}) -- (6,{sqrt(3)});\n\\draw [-,line width = .06cm] (0,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .06cm] (0,{3*sqrt(3)}) -- (6,{3*sqrt(3)});\n\\draw [-,line width=.06cm] (0,{2*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width=.06cm] (0,0) -- (3,{3*sqrt(3)});\n\\draw [-,line width=.06cm] (2,0) -- (5,{3*sqrt(3)});\n\\draw [-,line width=.06cm] (4,0) -- (6,{2*sqrt(3)});\n\\draw [-,line width=.06cm] (0,{2*sqrt(3)}) -- (2,0);\n\\draw [-,line width=.06cm] (1,{3*sqrt(3)}) -- (4,0);\n\\draw [-,line width=.06cm] (3,{3*sqrt(3)}) -- (6,0);\n\\draw [-,line width=.06cm] (5,{3*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [->,line width=.06cm,color=blue] (1,{sqrt(3)}) -- (2,{2*sqrt(3)});\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](2,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](1,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](3,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](5,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](0,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](2,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](4,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](6,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](1,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](3,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](5,{3*sqrt(3)}) circle (.2);\n\\draw [->,line width=.06cm,color=blue] (1,{sqrt(3)}) -- (2,{2*sqrt(3)});\n\\draw [->,line width=.06cm,color=blue] (1,{sqrt(3)}) -- (3,{sqrt(3)});\n\\node [above] at (1,{sqrt(3)+.3}) {\\cold{$\\bm{a}_2$}};\n\\node [below] at (1.7,{sqrt(3)-.1}) {\\cold{$\\bm{a}_1$}};\n\\end{tikzpicture}\n\\caption{A portion of the triangular lattice}\\label{fig:trilat}\n\\end{minipage}\n\\hfill\n \\begin{minipage}{0.45\\textwidth}\n \\centering\n\\begin{tikzpicture}[yscale=.75,xscale=.75]\n\\draw [-,line width = .06cm] (0,0) -- (6,0);\n\\draw [-,line width = .06cm] (0,0) -- (0,6);\n\\draw [-,line width=.06cm] (0,2) -- (6,2);\n\\draw [-,line width = .06cm] (2,0) -- (2,6);\n\\draw [-,line width=.06cm] (0,4) -- (6,4);\n\\draw [-,line width = .06cm] (4,0) -- (4,6);\n\\draw [-,line width=.06cm] (0,6) -- (6,6);\n\\draw [-,line width = .06cm] (6,0) -- (6,6);\n\\draw [-,line width = .06cm] (0,6) -- (6,0);\n\\draw [-,line width = .06cm] (2,0) -- (0,2);\n\\draw [-,line width = .06cm] (4,0) -- (0,4);\n\\draw [-,line width = .06cm] (2,6) -- (6,2);\n\\draw [-,line width = .06cm] (6,4) -- (4,6);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](2,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](0,2) circle (.2);\n\\filldraw[color=black, fill=black](2,2) circle (.2);\n\\filldraw[color=black, fill=black](4,2) circle (.2);\n\\filldraw[color=black, fill=black](6,2) circle (.2);\n\\filldraw[color=black, fill=black](0,4) circle (.2);\n\\filldraw[color=black, fill=black](2,4) circle (.2);\n\\filldraw[color=black, fill=black](4,4) circle (.2);\n\\filldraw[color=black, fill=black](6,4) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=black, fill=black](2,6) circle (.2);\n\\filldraw[color=black, fill=black](4,6) circle (.2);\n\\filldraw[color=black, fill=black](6,6) circle (.2);\n\\end{tikzpicture}\n\\caption{The triangular lattice after shearing.}\\label{fig:trishear}\n\\end{minipage}\n\\end{figure*}\n\nThe first graph that we consider is the \\emph{triangular lattice}. The graph has vertices\n\\[\n{\\mathcal V}_{\\mathrm{tri}}\n=\n\\set{n \\bm{a}_1 + m \\bm{a}_2 : n,m \\in {\\mathbb Z}},\n\\]\nwhere the generating vectors are\n\\[\n\\bm{a}_1\n=\n\\begin{bmatrix}\n1 \\\\ 0\n\\end{bmatrix},\n\\quad\n\\bm{a}_2\n=\n\\frac{1}{2} \\begin{bmatrix}\n1 \\\\ \\sqrt{3}\n\\end{bmatrix}.\n\\]\nOne then declares $v \\sim w$ for $v,w \\in \\mathcal{V}$ if $\\|v - w\\| = 1$. Thus, every $v \\in \\mathcal{V}$ has 6 neighbors; more specifically, if $v = n\\bm{a}_1 + m \\bm{a}_2$, then $v$ has neighbors\n\\[\n(n \\pm1) \\bm{a}_1 + m \\bm{a}_2, \\quad\nn\\bm{a}_1 + (m\\pm 1) \\bm{a}_2,\\quad\n(n \\pm 1) \\bm{a}_1 + (m\\mp 1) \\bm{a}_2.\n\\]\nConsequently, after identifying $n \\bm{a}_1 + m \\bm{a}_2$ with the point $(n,m) \\in {\\mathbb Z}^2$, we may view the Laplacian on the triangular lattice as an operator on $\\ell^2({\\mathbb Z}^2)$ via\n\\begin{equation} \\label{eq:triLaplacianSqVersion}\n[\\Delta_{\\rm tri} \\psi]_{n,m}\n=\n[\\Delta_{\\rm sq} \\psi]_{n,m}\n+\n\\psi_{n-1,m+1} + \\psi_{n+1,m-1}.\n\\end{equation}\nThis correspondence amounts to shearing and stretching the the triangular lattice, and essentially maps the triangular lattice to the square lattice with skewed next-nearest-neighbor interactions added. See Figures~\\ref{fig:trilat} and \\ref{fig:trishear}.\n\n\n\\begin{theorem}[Bethe--Sommerfeld for the triangular lattice] \\label{t:bsc:tri}\nFor all $\\bm{p} = (p_1,p_2 )\\in {\\mathbb Z}_+^2$, there is a constant $c = c_{\\bm{p}} > 0$ such that, if $Q:{\\mathcal V}_{\\mathrm{tri}} \\to {\\mathbb R}$ is $\\bm{p}$-periodic and $\\|Q\\|_\\infty \\leq c$, the following hold true for $H_Q = \\Delta_{\\mathrm{tri}} + Q$:\n\\begin{enumerate}\n\\item[{\\rm(1)}] $\\sigma(H_Q)$ consists of no more than two intervals.\n\\item[{\\rm(2)}] If at least one of $p_1$ or $p_2$ is odd, then $\\sigma(H_Q)$ consists of a single interval.\n\\end{enumerate}\nMoreover, the gap in the first setting may only open at the energy $E = -2$.\n\\end{theorem}\n\nThis theorem is sharp vis-\\`a-vis the number of intervals in the spectrum and the arithmetic restrictions on the periods. Concretely, we exhibit a $(2,2)$-periodic potential that perturbatively opens a gap at $-2$.\n\n\\begin{theorem} \\label{t:triExamples}\nThere exists $Q:{\\mathcal V}_{\\mathrm{tri}} \\to {\\mathbb R}$ which is $(2,2)$-periodic, such that $\\sigma(H_{\\lambda Q})$ has exactly two connected components for any sufficiently small $\\lambda > 0$.\n\\end{theorem}\n\n\\subsection{The Hexagonal Lattice} The set of vertices of the hexagonal lattice is closely related to that of the triangular lattice. Concretely, define $\\bm{b}_\\pm$ by\n\\[\n\\bm{b}_\\pm\n=\n\\frac{1}{2}\n\\begin{bmatrix}\n 3 \\\\ \\pm \\sqrt{3}\n\\end{bmatrix}.\n\\]\nThen, we obtain the hexagonal lattice by deleting the centers of some of the hexagons formed by the triangular lattice; more precisely,\n\\[\n{\\mathcal V}_{\\mathrm{hex}}\n=\n\\set{n \\bm{a}_1 + m \\bm{a}_2 \\in {\\mathcal V}_{\\mathrm{tri}}:n,m \\in {\\mathbb Z}} \\setminus \\{- \\bm{a}_1 + k \\bm{b}_+ + \\ell \\bm{b}_- : k,\\ell \\in {\\mathbb Z}\\}.\n\\]\nEquivalently, it is not hard to check that $\\{0, \\bm{a}_1\\}$ is a fundamental set of vertices and hence every $v \\in {\\mathcal V}_{\\mathrm{hex}}$ may be written uniquely as either $n \\bm{b}_+ + m \\bm{b}_-$ or $\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-$ for integers $n,m$, so we have\n\\begin{align*}\n{\\mathcal V}_{\\mathrm{hex}}\n& =\n\\set{n \\bm{b}_+ + m \\bm{b}_- : n,m \\in {\\mathbb Z}} \\cup \\set{\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_- : n,m \\in {\\mathbb Z}}.\n\\end{align*}\nWe define ${\\mathcal E}_{\\mathrm{hex}}$ by declaring $u \\sim v$ for $u, v \\in {\\mathcal V}_{\\mathrm{hex}}$ if $\\|u - v\\|_2 = 1$. After some calculations, we see that\n\\begin{align*}\n[\\Delta_{\\mathrm{hex}} \\psi]_{n \\bm{b}_+ + m \\bm{b}_-}\n& =\n\\psi_{\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-} + \\psi_{\\bm{a}_1 + n \\bm{b}_+ + (m-1) \\bm{b}_-} + \\psi_{\\bm{a}_1 + (n-1) \\bm{b}_+ + m \\bm{b}_-} \\\\\n[\\Delta_{\\mathrm{hex}} \\psi]_{\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-}\n& =\n\\psi_{n \\bm{b}_+ + m \\bm{b}_-} + \\psi_{n \\bm{b}_+ + (m+1) \\bm{b}_-} + \\psi_{(n+1) \\bm{b}_+ + m \\bm{b}_-}\n\\end{align*}\n\n\n\\begin{figure*}[t]\n\\begin{tikzpicture}[yscale=.75,xscale=.75]\n\\draw [-,line width = .06cm] (0,0) -- (1,{sqrt(3)});\n\\draw [-,line width = .06cm] (0,{2*sqrt(3)}) -- (1,{sqrt(3)});\n\\draw [-,line width = .06cm] (0,{2*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (0,{4*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width = .06cm,color=red] (1,{sqrt(3)}) -- (3,{sqrt(3)});\n\\draw [-,line width = .06cm] (1,{3*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (4,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .06cm] (4,0) -- (3,{sqrt(3)});\n\\draw [-,line width = .06cm] (4,{2*sqrt(3)}) -- (3,{sqrt(3)});\n\\draw [-,line width = .06cm] (4,{2*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (4,{4*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (4,0) -- (6,0);\n\\draw [-,line width = .06cm] (4,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .06cm] (4,{4*sqrt(3)}) -- (6,{4*sqrt(3)});\n\\draw [-,line width = .06cm] (6,0) -- (7,{sqrt(3)});\n\\draw [-,line width = .06cm] (6,{2*sqrt(3)}) -- (7,{sqrt(3)});\n\\draw [-,line width = .06cm] (6,{2*sqrt(3)}) -- (7,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (6,{4*sqrt(3)}) -- (7,{3*sqrt(3)});\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=red, fill=red](1,{sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](3,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](7,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](0,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](4,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](6,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](1,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](3,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](7,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](0,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](4,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](6,{4*sqrt(3)}) circle (.2);\n\\draw [->,line width = .06cm,color=blue] (1.2,{sqrt(3)-.1}) -- (3.8,{.1});\n\\draw [->,line width = .06cm,color=blue] (1.2,{sqrt(3)+.1}) -- (3.8,{2*sqrt(3)-.1});\n\\node at (2,{1.8*sqrt(3)}) {\\cold{$\\bm{b}_+$}};\n\\node at (2,{.2*sqrt(3)}) {\\cold{$\\bm{b}_-$}};\n\\end{tikzpicture}\n\\caption{A portion of the hexagonal lattice. A fundamental domain is highlighted in red.}\\label{fig:hexlat}\n\\end{figure*}\nSee Figure~\\ref{fig:hexlat}. The formula for $\\Delta_{\\mathrm{hex}}$ can be made more compact if we view the associated Hilbert space as \n\\[\n\\ell^2({\\mathbb Z}^2,{\\mathbb C}^2)\n=\n\\set{\\Psi:{\\mathbb Z}^2 \\to {\\mathbb C}^2 : \\sum_{n,m} \\|\\Psi_{n,m}\\|^2 < \\infty},\n\\]\nwhere the standard basis of ${\\mathbb C}^2$ corresponds to the left and right vertices of the fundamental domain, respectively. More precisely, given $\\psi \\in \\ell^2({\\mathcal V}_{\\mathrm{hex}})$, define $\\Psi \\in \\ell^2({\\mathbb Z}^2,{\\mathbb C}^2)$ by \n\\[\n\\Psi_{n,m}\n=\n\\begin{bmatrix}\n\\psi_{n \\bm{b}_+ + m \\bm{b}_-} \\\\ \\psi_{\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-}\n\\end{bmatrix}.\n\\]\nIdentifying $\\ell^2({\\mathcal V}_{\\mathrm{hex}})$  and $\\ell^2({\\mathbb Z}^2,{\\mathbb C}^2)$ in this fashion, the Laplacian for the hexagonal lattice is given by\n\\begin{align*}\n[\\Delta_{\\mathrm{hex}} \\Psi]_{n,m}\n& =\nU(\\Psi_{n,m-1}+ \\Psi_{n-1,m}) + L(\\Psi_{n,m+1}+  \\Psi_{n+1,m}) + J \\Psi_{n,m},\n\\end{align*}\nwhere\n\\[\nU\n=\n\\begin{bmatrix}\n0 & 1 \\\\ 0 & 0\n\\end{bmatrix},\n\\quad\nL \n=\nU^\\top\n=\n\\begin{bmatrix}\n0 & 0 \\\\ 1 & 0\n\\end{bmatrix},\n\\quad\nJ\n=\nU+L\n=\n\\begin{bmatrix}\n0 & 1 \\\\ 1 & 0\n\\end{bmatrix}.\n\\]\n{Equivalently, if we denote by $S_1, S_2 : \\ell^2({\\mathbb Z}^2) \\to \\ell^2({\\mathbb Z}^2)$ the shift operators\n\\[\n[S_1 \\psi]_{n,m}\n=\n\\psi_{n+1,m},\n\\quad\n[S_2\\psi]_{n,m}\n=\n\\psi_{n,m+1},\n\\]\nwe have\n\\[\n\\Delta_{\\mathrm{hex}} \\Psi\n=\n\\begin{bmatrix}\n(S_1^* + S_2^* + {\\mathbb I})\\psi^- \\\\  (S_1 + S_2 + {\\mathbb I})\\psi^+\n\\end{bmatrix}\n\\quad\n\\text{for any} \\quad \n\\Psi \n=\n\\begin{bmatrix} \\psi^+ \\\\ \\psi^- \\end{bmatrix}\n\\in \\ell^2({\\mathbb Z}^2,{\\mathbb C}^2).\n\\]\nAbbreviating somewhat, we write:\n\\begin{equation} \\label{eq:hexDecomp}\n\\Delta_{\\mathrm{hex}}\n=\\begin{bmatrix}\n0 & S_1^* + S_2^* + {\\mathbb I} \\\\\nS_1 + S_2 + {\\mathbb I} & 0\n\\end{bmatrix}.\n\\end{equation}}\n\n\\begin{theorem}[Bethe--Sommerfeld for the hexagonal lattice] \\label{t:bsc:hex}\nFor all $\\bm{p} = (p_1,p_2 )\\in {\\mathbb Z}_+^2$, there is a constant $c = c_{\\bm{p}} > 0$ such that, if $Q:{\\mathcal V}_{\\mathrm{hex}} \\to {\\mathbb R}$ is $\\bm{p}$-periodic and $\\|Q\\|_\\infty \\leq c$, the following statements hold true for $H_Q = \\Delta_{\\mathrm{hex}} + Q$:\n\\begin{enumerate}\n\\item[{\\rm(1)}]  $\\sigma(H_Q)$ consists of no more than four intervals.\n\\item[{\\rm(2)}] If at least one of $p_1$ or $p_2$ is odd, then $\\sigma(H_Q)$ consists of no more than two intervals.\n\\end{enumerate}\nMoreover, gaps may only open at $0$ and $\\pm1$ in the first case, and only at zero in the second case.\n\\end{theorem}\n\nMoreover, this theorem is sharp in the following sense: there exists a $(1,1)$-periodic potential $Q_1$ which infinitesimally opens a gap at zero, and there is a $(2,2)$-periodic potential $Q_2$ which infinitesimally opens gaps at $-1$, $0$, and $1$ in the following sense:\n\n\\begin{theorem} \\label{t:hexExamples}\n\\begin{enumerate}\n\\item[{\\rm(1)}] There exists $Q_1: {\\mathcal V}_{\\mathrm{hex}} \\to {\\mathbb R}^2$ which is $(1,1)$-periodic such that $\\sigma(H_{\\lambda Q_1})$ has exactly two connected components for all $\\lambda>0$.\n\\item[{\\rm(2)}] There exists $Q_2: {\\mathcal V}_{\\mathrm{hex}} \\to {\\mathbb R}^2$  which is $(2,2)$ periodic such that $\\sigma(H_{\\lambda Q_2})$ has exactly four connected components for any sufficiently small $\\lambda > 0$.\n\\end{enumerate}\n\\end{theorem}\n\nLet us remark that Theorem~\\ref{t:hexExamples}.(1) is well-known; we merely list it for completeness. The example in Theorem~\\ref{t:hexExamples}.(2) is novel.\n\n\\subsection{The EHM Lattice}\n\nThe EHM lattice also has vertex set ${\\mathcal V}_{\\mathrm{sqn}} = {\\mathcal V}_{\\mathrm{sq}} = {\\mathbb Z}^2$. However, now, one connects $\\bm{n}$ and $\\bm{n}'$ if and only if they are nearest neighbors or next-nearest-neighbors in the square lattice. Equivalently, one declares\n\\[\n\\bm{n} \\sim \\bm{n}'\n\\iff\n\\|\\bm{n} - \\bm{n}' \\|_\\infty\n=\n1.\n\\] \nThe associated Laplacian acts on $\\ell^2({\\mathbb Z}^2)$ via\n\\[\n[\\Delta_{\\mathrm{sqn}} \\psi]_{n,m}\n=\n[\\Delta_{\\mathrm{sq}}]_{n,m}+\\psi_{n-1,m-1}+\\psi_{n-1,m+1}+\\psi_{n+1,m-1}+\\psi_{n+1,m+1}.\n\\]\nSee Figure~\\ref{fig:ehmlat}.\n\\begin{figure*}[t]\n\n\\begin{tikzpicture}[yscale=.6,xscale=.6]\n\\draw [-,line width = .06cm] (0,0) -- (6,0);\n\\draw [-,line width = .06cm] (0,0) -- (0,6);\n\\draw [-,line width=.06cm] (0,2) -- (6,2);\n\\draw [-,line width = .06cm] (2,0) -- (2,6);\n\\draw [-,line width=.06cm] (0,4) -- (6,4);\n\\draw [-,line width = .06cm] (4,0) -- (4,6);\n\\draw [-,line width=.06cm] (0,6) -- (6,6);\n\\draw [-,line width = .06cm] (6,0) -- (6,6);\n\\draw [-,line width = .06cm] (0,0) -- (6,6);\n\\draw [-,line width = .06cm] (2,0) -- (6,4);\n\\draw [-,line width = .06cm] (4,0) -- (6,2);\n\\draw [-,line width = .06cm] (0,2) -- (4,6);\n\\draw [-,line width = .06cm] (0,4) -- (2,6);\n\\draw [-,line width = .06cm] (6,0) -- (6,6);\n\\draw [-,line width = .06cm] (0,6) -- (6,0);\n\\draw [-,line width = .06cm] (2,0) -- (0,2);\n\\draw [-,line width = .06cm] (4,0) -- (0,4);\n\\draw [-,line width = .06cm] (2,6) -- (6,2);\n\\draw [-,line width = .06cm] (6,4) -- (4,6);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](2,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](0,2) circle (.2);\n\\filldraw[color=black, fill=black](2,2) circle (.2);\n\\filldraw[color=black, fill=black](4,2) circle (.2);\n\\filldraw[color=black, fill=black](6,2) circle (.2);\n\\filldraw[color=black, fill=black](0,4) circle (.2);\n\\filldraw[color=black, fill=black](2,4) circle (.2);\n\\filldraw[color=black, fill=black](4,4) circle (.2);\n\\filldraw[color=black, fill=black](6,4) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=black, fill=black](2,6) circle (.2);\n\\filldraw[color=black, fill=black](4,6) circle (.2);\n\\filldraw[color=black, fill=black](6,6) circle (.2);\n\\end{tikzpicture}\n\\caption{A portion of the EHM lattice.}\\label{fig:ehmlat}\n\\end{figure*}\n\n\\begin{theorem}[Bethe--Sommerfeld for the EHM lattice] \\label{t:bsc:nnn}\nFor all $\\bm{p} = (p_1,p_2 )\\in {\\mathbb Z}_+^2$, there is a constant $c = c_{\\bm{p}} > 0$ such that, if $Q:{\\mathcal V}_{\\mathrm{sqn}} \\to {\\mathbb R}$ is $\\bm{p}$-periodic and $\\|Q\\|_\\infty \\leq c$, the following hold true for $H_Q = \\Delta_{\\mathrm{sqn}} + Q$:\n\\begin{enumerate}\n\\item[{\\rm(1)}] $\\sigma(H_Q)$ consists of no more than two intervals.\n\\item[{\\rm(2)}] If at least one of $p_1$ or $p_2$ is not divisible by three, then $\\sigma(H_Q)$ consists of a single interval.\n\\end{enumerate}\nMoreover, the gap in the first setting may only open at the energy $E = -1$.\n\\end{theorem}\nThis theorem is also sharp:\n\\begin{theorem} \\label{t:nnnExamples}\nThere exists $Q:{\\mathbb Z}^2 \\to {\\mathbb R}$ which is $(3,3)$-periodic such that $\\sigma(H_{\\lambda Q})$ has exactly two connected components for any sufficiently small $\\lambda>0$.\n\\end{theorem}\n\n\\bigskip\n\nThe remainder of the paper is organized as follows. Section~\\ref{sec:floquet} recalls Floquet theory for ${\\mathbb Z}^2$-periodic graphs. We work with the triangular lattice in Section~\\ref{sec:tri}, proving Theorems~\\ref{t:bsc:tri} and \\ref{t:triExamples}. We then work with the hexagonal lattice in Section~\\ref{sec:hex}, proving Theorems~\\ref{t:bsc:hex} and \\ref{t:hexExamples}. Finally, we conclude with the EHM lattice in Section~\\ref{sec:nnn}, proving Theorems~\\ref{t:bsc:nnn} and \\ref{t:nnnExamples}.\n\n\\section{Floquet Theory for Periodic Schr\\\"odinger Operators on Periodic Graphs}\n\\label{sec:floquet}\n\nLet $\\Gamma= ({\\mathcal V},{\\mathcal E})$ be a ${\\mathbb Z}^2$-periodic graph with translation symmetries $\\bm{a}_1, \\bm{a}_2 \\in {\\mathbb R}^2$, and suppose $Q:{\\mathcal V} \\to {\\mathbb R}$ is $\\bm{p} = (p_1,p_2)$-periodic, that is,\n\\[\nQ(u + p_j \\bm{a}_j)\n=\nQ(u),\\quad\nu \\in {\\mathcal V},\\; j = 1,2.\n\\]\nWe will briefly describe Floquet theory for $H_Q = \\Delta_\\Gamma + Q$, following \\cite{KorSab2014}. The main purpose of this section is to establish notation, so we do not give any proofs. One may write $H_Q$ as a constant-fiber direct integral over the fundamental domain. Concretely, let\n\\[\n{\\mathcal V}_{\\rm f}\n=\n{\\mathcal V} \\cap \\set{s \\bm{a}_1 + t \\bm{a}_2 : 0 \\le s < p_1, \\; 0 \\le t < p_2}.\n\\]\nBy periodicity, $|{\\mathcal V}_{\\rm f}| = P := p_0 p_1 p_2$, where\n\\[\np_0\n=\n|{\\mathcal V} \\cap \\set{s\\bm{a}_1 + t \\bm{a}_2 : 0 \\le s,t < 1}|.\n\\]\nHere, and throughout the paper, we use $|S|$ to denote the cardinality of the set $S$. For each edge $(u,v) \\in {\\mathcal E}$ there exist unique vertices $u_{\\rm f}, v_{\\rm f} \\in {\\mathcal V}_{\\rm f}$ and unique integers $n,m,n',m' \\in {\\mathbb Z}$ with\n\\[\nu = u_{\\rm f} + np_1 \\bm{a}_1 + mp_2 \\bm{a}_2,\n\\quad\nv = v_{\\rm f} + n'p_1\\bm{a}_1 + m'p_2 \\bm{a}_2,\n\\]\nWe then define the \\emph{index} of $(u,v)$ by $\\tau(u,v) = (n'-n,m'-m)$. Finally, for $u,v \\in {\\mathcal V}_{\\rm f}$, we define $B(u,v)$ to be the set of all translates of $v$ that connect to $u$ via an edge of $\\Gamma$:\n\\[\nB(u,v)\n=\n\\set{w \\in {\\mathcal V} : w \\sim u \\text{ and } w = v + np_1 \\bm{a}_1 + mp_2 \\bm{a}_2 \\text{ for some }n,m \\in {\\mathbb Z}}.\n\\]\n\nThen, for each $\\bm{\\theta} =(\\theta_1,\\theta_2) \\in {\\mathbb R}^2$, the corresponding Floquet matrix is a self-adjoint operator on ${\\mathcal H}_{\\rm f}:= \\ell^2({\\mathcal V}_{\\rm f}) = {\\mathbb C}^{{\\mathcal V}_{\\rm f}}$ defined by\n\\begin{equation} \\label{eq:floqMatDef}\n\\langle \\delta_u, H_Q(\\bm{\\theta}) \\delta_v \\rangle\n=\n\\sum_{w \\in B(u,v)} \\exp\\Big(i \\big\\langle \\tau(u,w), \\bm{\\theta} \\big\\rangle\\Big).\n\\end{equation}\nIn the event that the sum in \\eqref{eq:floqMatDef} is empty, $\\langle \\delta_u, H_Q(\\bm{\\theta}) \\delta_v \\rangle = 0$. Clearly, if $\\theta_j' - \\theta_j \\in 2\\pi{\\mathbb Z}$ for $j=1,2$, then $H_Q(\\bm{\\theta}) = H_Q(\\bm{\\theta}')$, so $H_Q(\\bm{\\theta})$ descends to a well-defined function of $\\bm{\\theta} \\in {\\mathbb T}^2 := {\\mathbb R}^2 \/ (2\\pi{\\mathbb Z})^2 \\cong [0,2\\pi)^2$. We will freely use $\\bm{\\theta} \\in {\\mathbb R}^2$ or $\\bm{\\theta} \\in {\\mathbb T}^2$ depending on which is more convenient in a given setting.\n\nInformally, \\eqref{eq:floqMatDef} represents the restriction of $H_Q$ to the discrete torus \n\\[\n({\\mathbb Z}\\bm{a}_1 \\oplus {\\mathbb Z}\\bm{a}_2) \/ (p_1 {\\mathbb Z} \\bm{a}_1 \\oplus p_2 {\\mathbb Z} \\bm{a}_2)\n\\cong\n{\\mathbb Z}_{p_1} \\oplus {\\mathbb Z}_{p_2}.\n\\]\nwith the following boundary conditions: wrapping once around the torus in the positive $\\bm{a}_1$ direction accrues a phase $e^{i\\theta_1}$ and wrapping around once in the positive $\\bm{a}_2$ direction accrues a phase $e^{i\\theta_2}$. More precisely, we may view $H_Q(\\bm{\\theta})$ in the following manner. The operator $H_Q$ acts on the space ${\\mathbb C}^{\\mathcal V}$ of arbitrary (not necessarily square-summable) functions ${\\mathcal V} \\to {\\mathbb C}$. When $Q$ is $(p_1,p_2)$-periodic, then for each $\\bm{\\theta} \\in {\\mathbb T}^2$, $H_Q$ preserves the subspace\n\\[\n\\mathcal{H}(\\bm{\\theta})\n=\n\\set{\\psi \\in {\\mathbb C}^{\\mathcal V} : \\psi(u+p_j \\bm{a}_j) = e^{i\\theta_j} \\psi(u)}.\n\\]\nThen, $H_Q(\\bm{\\theta})$ is equivalent to the restriction of $H_Q$ to ${\\mathcal H}(\\bm{\\theta})$.\n\nFor each $\\bm{\\theta}$, order the eigenvalues of $H_Q(\\bm{\\theta})$ as\n\\[\nE_1(\\bm{\\theta})\n\\leq\n\\cdots\n\\leq\nE_P(\\bm{\\theta})\n\\]\nwith each eigenvalue listed according to its multiplicity. Then, for $1 \\le j \\le P$, the $j$th spectral \\emph{band} of $H_Q$ is defined by\n\\[\nF_j\n=\nF_j(Q)\n:=\n\\mathrm{ran}(E_j)\n=\n\\set{E_j(\\bm{\\theta}) : \\bm{\\theta} \\in {\\mathbb T}^2}\n=\n\\set{E_j(\\bm{\\theta}) : \\bm{\\theta} \\in {\\mathbb R}^2}.\n\\]\n\n\\begin{theorem} \\label{t:floquet}\nWith notation as above,\n\\[\n\\sigma(H_Q)\n=\n\\bigcup_{\\bm{\\theta} \\in {\\mathbb T}^2} H_Q(\\bm{\\theta})\n=\n\\bigcup_{j=1}^P F_j.\n\\]\n\\end{theorem}\n\nWe will use Theorem~\\ref{t:floquet} in the following way. Making the dependence on the potential $Q$ explicit, one may write\n\\[\nF_j = F_j(Q)\n=\n[E_j^-(Q),E_j^+(Q)].\n\\]\nThe key fact is the following: by standard perturbation theory for self-adjoint operators, $E_j^\\pm(Q)$ are 1-Lipschitz functions of $Q$. Here, one views $Q$ as an element of ${\\mathbb R}^P$ and the perturbation is with respect to the uniform metric thereupon. In particular, if an energy $E$ satisfies $E \\in \\mathrm{int}(F_j(Q))$, then $(E-\\delta,E+\\delta) \\subseteq F_j(Q)$ for some positive $\\delta$, and it follows that $E \\in F_j(Q') \\subseteq \\sigma(H_{Q'})$ for any $(p_1,p_2)$-periodic $Q'$ with $\\|Q-Q'\\|_\\infty < \\delta$. Note that here it is very important that one views the periods as fixed: one may only perturb within ${\\mathbb R}^P$ for a fixed $P$. Thus, our analysis revolves around determining for a given energy $E$, whether $E$ belongs to the interior of some band of the Laplacian, where the Laplacian is viewed as a degenerate $(p_1,p_2)$-periodic operator. \n\n\n\n\n\\section{Triangular Laplacian}\\label{sec:tri}\n\nWe view the triangular Laplacian as acting on the square lattice $\\ell^2({\\mathbb Z}^2)$, but with extra connections as in \\eqref{eq:triLaplacianSqVersion}:\n\\begin{align*}\n[\\Delta_{\\rm tri} u]_{n,m}\n& =\nu_{n-1,m} + u_{n+1,m} + u_{n,m-1} + u_{n,m+1} + u_{n-1,m+1} + u_{n+1,m-1} \\\\\n& =\n[\\Delta_{\\rm sq}u]_{n,m} + u_{n-1,m+1} + u_{n+1,m-1}.\n\\end{align*}\nNow, given $p_1,p_2 \\in {\\mathbb Z}_+$, we view $\\Delta_{\\mathrm{tri}}$ as a $\\bm{p}$-periodic operator and perform the Floquet decomposition. Define $P:=p_1p_2$ as in Section~\\ref{sec:floquet}, and put\n\\[\n\\Lambda :=\n{\\mathbb Z}^2 \\cap \\Big( [0,p_1) \\times [0,p_2) \\Big).\n\\]\nFor $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb R}^2$, it is straightforward to check that\n\\[\n\\sigma(H(\\bm{\\theta}))\n=\n\\set{e_{\\bm{\\ell}}^{\\Lambda}(\\bm{\\theta}) : \\bm{\\ell}\\in \\Lambda },\n\\]\nwhere $\\bm{\\ell}=(\\ell_1,\\ell_2)$ and\n\\[\ne^{\\Lambda}_{\\bm{\\ell}}(\\bm{\\theta})\n=\n2\\cos\\left( \\frac{\\theta_1+2\\pi \\ell_1}{p_1}\\right) \n+ 2\\cos\\left(\\frac{\\theta_2+2\\pi \\ell_2}{p_2}\\right) \n+ 2\\cos\\left(\\frac{\\theta_1 + 2\\pi \\ell_1}{p_1}  - \\frac{\\theta_2 + 2\\pi \\ell_2}{p_2}\\right).\n\\]\nLet us point out that one needs to be somewhat careful at this point; namely, $e^\\Lambda_{\\bm{\\ell}}(\\bm{\\theta})$ is not a well-defined function of $\\bm{\\theta} \\in {\\mathbb T}^2$. However, the error incurred in using a different coset representative of $\\bm{\\theta} \\in {\\mathbb T}^2$ is simply a change in the index $\\bm{\\ell}$, and one can check that the \\emph{family} $\\set{e^\\Lambda_{\\bm{\\ell}}(\\bm{\\theta}) : \\bm{\\ell} \\in \\Lambda}$ is a well-defined function on ${\\mathbb T}^2$ (as well it should, since the \\emph{operator} $H(\\bm{\\theta})$ is itself a well-defined function of $\\bm{\\theta} \\in {\\mathbb T}^2$). In any case, the ambiguity disappears when one considers the covering space ${\\mathbb R}^2$, which we do for most of the paper. One could also use the minimal covering space ${\\mathbb R}^2 \/ (p_1{\\mathbb Z} \\oplus p_2 {\\mathbb Z})$ on which the $e_{\\bm{\\ell}}^\\Lambda$ are well-defined, but this does not accrue any benefits vis-\\`a-vis the present work, so we simply use ${\\mathbb R}^2$.\n\nAs in Section~\\ref{sec:floquet}, we label these eigenvalues in increasing order according to multiplicity by\n\\[\nE_1^{\\Lambda}(\\bm{\\theta})\n\\le \nE_2^{\\Lambda}(\\bm{\\theta})\\le \\cdots E_P^{\\Lambda}(\\bm{\\theta})\n\\]\nand denote the $P$ spectral bands by\n\\[\nF_k^{\\Lambda}\n=\n\\set{E_k^{\\Lambda}(\\bm{\\theta}) : \\bm{\\theta} \\in {\\mathbb R}^2},\n\\quad\n1 \\le k \\le P.\n\\]\nStraightforward computations shows that $\\sigma(\\Delta_{\\rm tri})=[-3,6]$, and thus\n\\[\n\\bigcup_{k=1}^P F_k^{\\Lambda}\n=\n[-3,6].\n\\]\nHenceforth, we view $p_1$ and $p_2$ as fixed and so we drop $\\Lambda$ from the superscripts.\nOur main theorem of this section is the following.\n\\begin{theorem}\\label{thm:trimain}\nLet $p_1,p_2 \\in {\\mathbb Z}_+$ be given.\n\\begin{enumerate}\n\\item[{\\rm 1.}]\nEach $E\\in (-3, 6)\\setminus \\{-2\\}$ belongs to $\\mathrm{int}(F_k)$ for some $1\\leq k\\leq P$.\n\\item[{\\rm 2.}] If one of the periods $p_1, p_2$ is odd, then $E=-2$ belongs to $\\mathrm{int}(F_k)$ for some $1\\leq k\\leq P$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}[Proof of Theorem~\\ref{t:bsc:tri}]\nAs already discussed, this follows immediately from Theorem~\\ref{thm:trimain}.\n\\end{proof}\n\\subsection{Proof of Theorem \\ref{thm:trimain}} \nWe will divide the proof into two different cases: $E\\neq -2$ and $E=-2$.\nOur general strategy is to argue by contradiction.\nMore specifically, we assume $E=\\min F_{k+1}=\\max F_k$ for some $1\\leq k\\leq P-1$, and show that this leads to a contradiction. We will use the following two lemmas, whose proofs we provide at the end of the present section.\n\\begin{lemma}\\label{lem:constructiontri}\nFor any $E \\in {[-3,6]}$, there exist $x, y \\in [0,2\\pi)$ such that\n\\begin{align}\n\\label{eq:xyCondA} \\cos(x) + \\cos(y) + \\cos(x-y) & = \\frac{E}{2} \\\\\n\\label{eq:xyCondB} \\sin(x) + \\sin(y) & = 0.\n\\end{align}\nFurthermore, {if $E \\neq -2$}, we have\n\\begin{align}\n\\label{eq:xyCondC}\\cos(x) + \\cos(y)=-1+\\sqrt{E+3}  \\neq 0\n\\end{align}\nfor any $x,y$ that satisfy conditions \\eqref{eq:xyCondA} and \\eqref{eq:xyCondB}.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lem:triJ0empty}\nConsider the following system:\n\\begin{equation} \\label{eq:triJ0syst}\n\\begin{cases}\n\\cos(x) + \\cos(y) + \\cos(x-y)  = \\frac{E}{2},\\\\\n\\sin(x)+\\sin(x-y)=0,\\\\\n\\sin(y)-\\sin(x-y)=0.\n\\end{cases}\n\\end{equation}\nFor any $E \\in (-3,6) \\setminus \\{-2\\}$, the solution set of \\eqref{eq:triJ0syst} is empty. For $E = -2$, the solutions of \\eqref{eq:triJ0syst} in $[0,2\\pi)^2$ are $(0,\\pi)$, $(\\pi,0)$ and $(\\pi, \\pi)$.\n\\end{lemma}\n\nWe will use Lemma \\ref{lem:constructiontri} in the $E\\neq -2$ case, and Lemma \\ref{lem:triJ0empty} in the $E=-2$ case.\n\n\\subsubsection{$E\\neq -2$}\\\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:trimain}.1]\nLet $E \\in (-3,6) \\setminus\\{-2\\}$ be given and suppose for the purpose of establishing a contradiction that $E = \\max F_k = \\min F_{k+1}$ for some $1 \\le k < P$. Let $(x,y)$ denote a solution to \\eqref{eq:xyCondA} and \\eqref{eq:xyCondB} from Lemma \\ref{lem:constructiontri}, and take $\\widetilde{\\bm{\\theta}}=(\\widetilde{\\theta}_1,\\widetilde{\\theta}_2)\\in [0,2\\pi)^2$ and  $\\bm{\\ell}^{(1)}=(\\ell_1^{(1)},\\ell_2^{(1)})\\in \\Lambda$ such that \n\\[p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(1)})=x,\\, \\ \\text{and }\\ \\ p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2^{(1)})=y.\n\\]\nIt is clear that $\\widetilde{\\bm{\\theta}}$ and $\\bm{\\ell}^{(1)}$ are uniquely determined by $x$ and $y$.\nLet us also note that \\eqref{eq:xyCondA} is equivalent to \n\\[e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=E.\\]\nDefine $\\Lambda_E(\\widetilde{\\bm{\\theta}}) \\subseteq \\Lambda$ to be the set of all $\\bm{\\ell} \\in \\Lambda$ such that $e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = E$. Then $r := |\\Lambda_E(\\widetilde{\\bm{\\theta}})|$ is the multiplicity of $E \\in \\sigma(H(\\widetilde{\\bm{\\theta}}))$ and clearly $\\bm{\\ell}^{(1)} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}})$.\n\n\nSince $E \\in F_k$ by assumption, let $s\\in {\\mathbb Z}\\cap [1,r]$ be chosen so that\n\\[E_{k-s}(\\widetilde{\\bm{\\theta}})<E_{k-s+1}(\\widetilde{\\bm{\\theta}})=\\cdots=E_k(\\widetilde{\\bm{\\theta}})=\\cdots=E_{k+r-s}(\\widetilde{\\bm{\\theta}})<E_{k+r-s+1}(\\widetilde{\\bm{\\theta}}).\\]\nSince all the eigenvalues are continuous in $\\bm{\\theta}$, we can take $\\varepsilon>0$ small enough such that \n\\[E_{k-s}(\\bm{\\theta})<E_{k-s+1}(\\bm{\\theta})\\, \\text{ and }\\, E_{k+r-s}(\\bm{\\theta})<E_{k+r-s+1}(\\bm{\\theta})\\]\nhold whenever $\\|\\bm{\\theta}-\\widetilde{\\bm{\\theta}}\\|_{{\\mathbb R}^2}<\\varepsilon$. Our goal is to perturb about the point $\\widetilde{\\bm{\\theta}}$ in two directions, one of which is ``generic'' and one of which is carefully chosen. The generic perturbation moves half of the eigenvalues to the right and half to the left, which we shall use to conclude that $r = 2s$. The non-generic perturbation is carefully chosen to contradict this.\n\nGiven {$\\bm{\\ell}\\in \\Lambda$ and} a unit vector $\\bm{\\beta}=(\\beta_1, \\beta_2)$, we have\n\\begin{align}\ne_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}+t \\bm{\\beta}) & =e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})+t\\bm{\\beta} \\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})+O(t^2) \\label{eq:pertgeneralbetatri1order} \\\\\n& = e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})+t\\bm{\\beta} \\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) \\label{eq:pertgeneralbetatri2order}\\\\\n&\\qquad -\\frac{t^2}{2}\\Bigg[\\frac{\\beta_1^2}{p_1^2}\\cos\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}\\Big)\n+\\frac{\\beta_2^2}{p_2^2}\\cos\\Big(\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big) \\notag\\\\ \n&\\qquad +\\Big(\\frac{\\beta_1}{p_1}-\\frac{\\beta_2}{p_2}\\Big)^2 \\cos\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}-\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\Bigg] +O(t^3).  \\notag\n\\end{align}\nIn particular, we will use \\eqref{eq:pertgeneralbetatri1order} if $\\bm{\\beta}\\cdot  \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\neq 0$, and \\eqref{eq:pertgeneralbetatri2order} otherwise.\n\n\n\nFor any vector $\\bm{\\beta}\\in {\\mathbb R}^2\\setminus \\{\\vec{0}\\}$, let \n\\begin{equation}\\label{def:Jbetatri}\n\\begin{aligned}\n\\mathcal{J}_{\\bm{\\beta}}^0\n& =\n{\\mathcal J}_{\\bm{\\beta}}^0(\\widetilde{\\bm{\\theta}})\n:=\n\\{\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}):\\ \\bm{\\beta}\\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})=0\\},\\\\\n\\mathcal{J}_{\\bm{\\beta}}^{\\pm}\n& =\n{\\mathcal J}_{\\bm{\\beta}}^\\pm(\\widetilde{\\bm{\\theta}})\n:=\n\\{ \\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) :\\ \\pm \\bm{\\beta} \\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})>0\\}.\n\\end{aligned}\n\\end{equation}\nConsequently, we always have \n\\begin{align}\\label{eq:sumJbetatri}\n|\\mathcal{J}_{\\bm{\\beta}}^0|+|\\mathcal{J}_{\\bm{\\beta}}^+|+|\\mathcal{J}_{\\bm{\\beta}}^-|=r.\n\\end{align}\nWe also define $\\mathcal{J}_0$ as follows\n\\begin{align}\\label{def:J0tri}\n\\mathcal{J}_0\n=\n{\\mathcal J}_0(\\widetilde{\\bm{\\theta}})\n:=\\{ \\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) :\\ \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = \\bm{0}\\}.\n\\end{align}\nSince $E \\neq -2$, Lemma~\\ref{lem:triJ0empty} clearly implies $\\mathcal{J}_0=\\emptyset$.\n\nWe choose $\\bm{\\beta}_1=(\\beta_{1,1},\\beta_{1,2})=(p_1,p_2)\/\\sqrt{p_1^2+p_2^2}$. Then \\eqref{eq:xyCondB} is equivalent to \n\\[\\bm{\\beta}_1 \\cdot \\nabla e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=0,\\]\nhence $\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset$. \n\n\nNext we are going to perturb the point $\\widetilde{\\bm{\\theta}}$ and count the eigenvalues.\nSince $\\mathcal{J}_0 = \\emptyset$, we can choose a unit vector $\\bm{\\beta}_2$ such that \n\\begin{align}\\label{eq:beta2trinon-empty}\n\\bm{\\beta}_2\\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\neq 0,\n\\end{align}\nholds for any $\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}})$.\nThus, $\\mathcal{J}_{\\bm{\\beta}_2}^0=\\emptyset$, so one concludes\n\\begin{align}\\label{eq:beta2trinon-empty'}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|+|\\mathcal{J}_{\\bm{\\beta}_2}^-|=r.\n\\end{align}\n\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_2$}\nWe first perturb the eigenvalues along the $\\bm{\\beta}_2$ direction.\nSince $\\mathcal{J}_{\\bm{\\beta}_2}^0 = \\emptyset$, we will always employ \\eqref{eq:pertgeneralbetatri1order}.\n\nFor $t > 0$ small enough, we have the following.\n\\begin{itemize}\n\\item  If ${\\bm{\\ell}}  \\in \\mathcal{J}_{\\bm{\\beta}_2}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta2+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|\\leq r-s.\n\\end{align}\n\n\\item If ${\\bm{\\ell} } \\in \\mathcal{J}_{\\bm{\\beta}_2}^-$, we have \n\\[\nE_{k-s}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\\leq s.\n\\end{align}\n\\end{itemize}\n{In view of \\eqref{eq:beta2trinon-empty'}, Equations~\\eqref{eq1:Jbeta2+tri} and \\eqref{eq2:Jbeta2+tri} imply\n\\begin{equation} \\label{eq:Jbeta2MinusCard+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\n=\ns.\n\\end{equation}\nUpon realizing that $\\mathcal{J}_{-\\bm{\\beta}_2}^0 = \\emptyset$ and $\\mathcal{J}_{-\\bm{\\beta}_2}^\\pm = \\mathcal{J}_{\\bm{\\beta}_2}^\\mp$, we may apply the analysis above with $\\bm{\\beta}_2$ replaced by $-\\bm{\\beta}_2$ and conclude that\n\\begin{equation} \\label{eq:JMinusBeta2+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+| = |\\mathcal{J}_{-\\bm{\\beta}_2}^-| = s.\n\\end{equation}\nIn particular, \\eqref{eq:Jbeta2MinusCard+tri} and \\eqref{eq:JMinusBeta2+tri} imply\n\\begin{align}\\label{eq4:Jbeta2tri}\nr=2s.\n\\end{align}} \n\n\\begin{comment}\n\\begin{align}\\label{eq3:Jbeta2+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|=r-s.\n\\end{align}\n\n\n\nWhen $t$ is small enough and $t < 0$, we have the following.\n\\begin{itemize}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_2}^+$, we have\n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1+t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta2-tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|\\leq s.\n\\end{align}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_2}^-$, we have \n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2-tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\\leq r-s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:beta2trinon-empty'}, \\eqref{eq1:Jbeta2-tri} and \\eqref{eq2:Jbeta2-tri} into account, we have\n\\begin{align}\\label{eq3:Jbeta2-tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|=s.\n\\end{align}\nCombining \\eqref{eq3:Jbeta2+tri} with \\eqref{eq3:Jbeta2-tri}, we arrive at\n\\begin{align}\\label{eq4:Jbeta2tri}\nr=2s.\n\\end{align}\n\\end{comment}\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_1$}\nNow we perturb the eigenvalues along $\\bm{\\beta}_1$.\nThe case when ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^{\\pm}$ is similar to that of $\\bm{\\beta}_2$.\nThe difference here is $\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset$.\n\nBy Lemma~\\ref{lem:constructiontri}, we have\n\\begin{align}\n\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}\\Big) + \\cos\\Big(\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)\n=\n-1+\\sqrt{E+3}\n\\neq\n0\n\\end{align}\nfor ${\\bm{\\ell} = (\\ell_1,\\ell_2)} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0$.\nThus, by employing \\eqref{eq:pertgeneralbetatri2order}, we obtain\n\\begin{align}\n&e_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\\notag\\\\\n&=\nE - {\\frac{ t^2}{2(p_1^2+p_2^2)}} \\Big(\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}\\Big) + \\cos\\Big(\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big) \\Big) + O(t^3)\\notag\\\\\n& = E - {\\frac{ t^2}{2(p_1^2+p_2^2)}} \\Big(-1+\\sqrt{E+3} \\Big)+O(t^3).\\label{eq:Jbeta10tri}\n\\end{align}\nNotice that the choice of $\\bm{\\beta}_1$ causes the {third} $t^2$ term of \\eqref{eq:pertgeneralbetatri2order} to drop out.\n\nWithout loss of generality, we assume $E\\in (-2, 6)$. The other case can be handled similarly.\nFor $E\\in (-2,6)$, \\eqref{eq:Jbeta10tri} implies that \n\\begin{align}\\label{eq:Jbeta10tri'}\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1) < E = \\min F_{k+1},\n\\end{align}\nholds for $|t|>0$ small enough and for any $\\bm{\\ell} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0$. \n\nCombining \\eqref{eq:Jbeta10tri'} with \\eqref{eq:pertgeneralbetatri1order}, we have the following.\n\nFor $t>0$ small enough,\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1+tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\\leq r-s\n=\ns,\n\\end{align}\n{where the equality follows from \\eqref{eq4:Jbeta2tri}.}\n\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0 \\bigcup \\mathcal{J}_{\\bm{\\beta}_1}^-$, we have \n\\[\nE_{k-s-1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1+tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^-|\\leq s.\n\\end{align}\n\\end{itemize}\n{In view of \\eqref{eq:sumJbetatri} and \\eqref{eq4:Jbeta2tri}, Equations~\\eqref{eq1:Jbeta1+tri} and \\eqref{eq2:Jbeta1+tri} yield\n\\begin{equation} \\label{eq3:Jbeta1+tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\n=\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\ns.\n\\end{equation}\nAs before, we may observe that ${\\mathcal J}_{-\\bm{\\beta}_1}^0 = {\\mathcal J}_{\\bm{\\beta}_1}^0$ and ${\\mathcal J}_{-\\bm{\\beta}_1}^\\pm = {\\mathcal J}_{\\bm{\\beta}_1}^\\mp$. Then, the analysis above applied with $\\bm{\\beta}_1$ replaced by $-\\bm{\\beta}_1$ forces\n\\begin{equation} \\label{eq3:Jbeta1-tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^+|\n=\ns.\n\\end{equation}\nTaken together, \\eqref{eq3:Jbeta1+tri} and \\eqref{eq3:Jbeta1-tri} imply $|{\\mathcal J}_{\\bm{\\beta}_1}^0| = 0$, which contradicts ${\\mathcal J}_{\\bm{\\beta}_1}^0 \\neq \\emptyset$.\n}\n\n\\begin{comment}\nTaking \\eqref{eq:sumJbetatri}, \\eqref{eq1:Jbeta1+tri} and \\eqref{eq2:Jbeta1+tri} into account, we have\n\\begin{align}\\label{eq3:Jbeta1+tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|=r-s.\n\\end{align}\n\n\n\nFor $t<0$ small enough, \n\\begin{itemize}\n\\item If $(\\ell_m,k_m)\\in  \\mathcal{J}_{\\bm{\\beta}_1}^0 \\bigcup \\mathcal{J}_{\\bm{\\beta}_1}^+ $, we have\n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n<\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1-tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^+|\\leq s.\n\\end{align}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_1}^-$, we have \n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n>\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1-tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-|\\leq r-s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:sumJbetatri}, \\eqref{eq1:Jbeta1-tri} and \\eqref{eq2:Jbeta1-tri} into account, we have\n\\begin{align}\\label{eq3:Jbeta1-tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^+|=s.\n\\end{align}\nCombining \\eqref{eq3:Jbeta1+tri} with \\eqref{eq3:Jbeta1-tri}, we arrive at\n\\begin{align}\\label{eq4:Jbeta1tri}\nr=2s-|\\mathcal{J}_{\\bm{\\beta}_1}^0|.\n\\end{align}\nSince $\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset$, \\eqref{eq4:Jbeta1tri} is in contradiction with \\eqref{eq4:Jbeta2tri}.\n\\end{comment}\n\n\\end{proof}\n\n\\subsubsection{$E=-2$}\\\n\nFirst, we would like to make a remark on our strategy of the proof of the $E=-2$ case, and on the importance of one of the periods being odd.\n\\begin{remark}\\label{rem:tri}\nWe will choose $\\widetilde{\\bm{\\theta}}=(\\widetilde{\\theta}_1, \\widetilde{\\theta}_2)$ and $\\bm{\\ell}^{(1)}=(\\ell_1^{(1)}, \\ell_2^{(1)})$ such that $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=-2$ and \n$\\nabla e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=\\bm{0}$. \nLemma \\ref{lem:triJ0empty} yields three possibilities $(p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(1)}), p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2^{(1)}))=(0,\\pi)$, $(\\pi, 0)$ or $(\\pi, \\pi)$.\nDepending on which one of $p_1, p_2$ is odd, we will choose $(0,\\pi)$ (if $p_1$ is odd), or $(\\pi, 0)$ (if $p_2$ is odd).\nThis choice guarantees that the only eigenvalue located at $-2$ with vanishing gradient is $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})$.\nConsequently, it suffices to control the second order perturbation of (a single eigenvalue) $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})$ along a given direction $(\\beta_1, \\beta_2)$.\nWhen $p_1$ is odd, this is equivalent to controlling the sign of the following expression (compare \\eqref{eq:remtri}):\n$$-\\beta_{2}\\Big(\\frac{\\beta_{1}}{p_1}-\\frac{\\beta_{2}}{p_2}\\Big).$$\nWe can easily choose two directions such that the expression above has different signs, which leads to un-even {eigenvalue counts and hence to the desired contradiction}.\n\n{{\\it A posteriori}, the existence of a $(2,2)$-periodic potential satisfying the conclusion of Theorem~\\ref{t:triExamples} implies that this argument must fail if both $p_1$ and $p_2$ are even; let us briefly describe why this must be the case.} If both $p_1, p_2$ are even, there will be three eigenvalues at $-2$ with vanishing gradients, corresponding to all three solutions $(0,\\pi)$, $(\\pi, 0)$, $(\\pi, \\pi)$.\nTrying to control the second order perturbations of all these three eigenvalues along $(\\beta_1, \\beta_2)$ is equivalent to controlling the signs of the following three expressions simultaneously\n\\begin{align*}\n-\\beta_{2}\\Big(\\frac{\\beta_{1}}{p_1}-\\frac{\\beta_{2}}{p_2}\\Big),\\ \\ \n\\beta_{1}\\Big(\\frac{\\beta_{1}}{p_1}-\\frac{\\beta_{2}}{p_2}\\Big),\\ \\ \\text{and}\\ \\ \n\\beta_1\\beta_2.\n\\end{align*}\nA simple inspection of these three expressions yields that two of them are always non-negative with the other one being non-positive. \nTherefore we can never choose two different directions that lead to un-even {eigenvalue counts}.\nThis explains why at least one of the periods must be odd for our argument to work. \n\\end{remark}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:trimain}.2]\nNow let us give a detailed proof. Without loss of generality, assume $p_1$ is odd, let $E = -2$, and assume for the sake of contradiction that $E = \\max F_k = \\min F_{k+1}$ for some $k$.\nWe choose $\\widetilde\\bm{\\theta}$ and $\\bm{\\ell}^{(1)}$ via\n\\begin{align*}\n\\widetilde{\\theta}_1=0,\\ {\\ell_1^{(1)}} = 0,\\ \\ \\ \n(\\widetilde{\\theta}_2, {\\ell_2^{(1)}})=\n\\begin{cases}\n\\Big(0, \\frac{p_2}{2}\\Big),\\ \\ \\text{if } p_2 \\text{ is even},\\\\\n\\Big(\\pi, \\frac{p_2-1}{2}\\Big),\\ \\ \\text{if } p_2 \\text{ is odd}.\n\\end{cases}\n\\end{align*}\n\nWith these choices of $\\bm{\\ell}^{(1)}$ and $\\widetilde{\\bm{\\theta}}$, one can check that $e_{{\\bm{\\ell}^{(1)}}}(\\widetilde\\bm{\\theta}) = -2 = E$. As before, let $r$ denote the multiplicity of $E$ and let $\\Lambda_E(\\widetilde{\\bm{\\theta}})$ denote the set of $\\bm{\\ell} \\in \\Lambda$ with $e_{\\bm{\\ell}}(\\widetilde\\bm{\\theta}) = -2$.\nNote that we also have $\\nabla e_{{\\bm{\\ell}^{(1)}}}(\\widetilde\\bm{\\theta}) = \\bm{0}$, and thus $\\mathcal{J}_0\\neq \\emptyset$. Moreover, we claim that ${\\mathcal J}_0 = \\{\\bm{\\ell}^{(1)}\\}$. To see this, suppose there exists $\\bm{\\ell} \\neq \\bm{\\ell}^{(1)}$ in ${\\mathcal J}_0$. In view of Lemma~\\ref{lem:triJ0empty}, we must have\n\\[\\frac{\\widetilde{\\theta}_1+2\\pi\\ell_1}{p_1}=\\pi,\\]\nwhich implies $p_1 = 2\\ell_1$, which is impossible, since $p_1$ is odd.\nConsequently,\n\\[\\mathcal{J}_0=\n{\\{ \\bm{\\ell}^{(1)} \\}}.\\]\n\nLet us choose $\\bm{\\beta}_1=(\\beta_{1,1},\\beta_{1,2})=(0,1)$ and a unit vector\n\\[\\bm{\\beta}_2=(\\beta_{2,1},\\beta_{2,2}) \\sim (2p_1, p_2)\/\\sqrt{4p_1^2+p_2^2}\\] \nsuch that \n\\begin{align}\\label{eq:beta2triodd1}\n\\beta_{2,2}\\Big(\\frac{\\beta_{2,1}}{p_1}-\\frac{\\beta_{2,2}}{p_2}\\Big)>0,\n\\end{align}\nand\n\\begin{align}\\label{eq:beta2triodd2}\n\\bm{\\beta}_2\\cdot \\nabla e_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}})\\neq 0\\ \\ \\text{holds for any }\\ \\ell \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) \\setminus \\{\\bm{\\ell}^{(1)}\\}.\n\\end{align}\n\nWe will use \\eqref{eq:beta2triodd1} to control the perturbation of $e_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}})$ along the $\\bm{\\beta}_2$ direction. \nWe also note that \\eqref{eq:beta2triodd2} simply says \n\\begin{align}\\label{eq:beta20triodd}\n\\mathcal{J}_{\\bm{\\beta}_2}^0=\\mathcal{J}_{0}\n=\n\\{{\\bm{\\ell}^{(1)}}\\}.\n\\end{align}\n\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_2$}\nWe first perturb the eigenvalues along the $\\bm{\\beta}_2$ direction.\\\n\nBy \\eqref{eq:beta20triodd}, we need only consider first-order perturbation theory as in \\eqref{eq:pertgeneralbetatri1order} for $\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) \\setminus \\{\\bm{\\ell}^{(1)}\\}$.\nSince ${\\bm{\\ell}^{(1)}} \\in \\mathcal{J}_{0}$, we need to employ \\eqref{eq:pertgeneralbetatri2order} for $e_{{\\bm{\\ell}^{(1)}}}$.\nIndeed, by \\eqref{eq:pertgeneralbetatri2order}, we have for $|t|>0$ small enough,\n\\begin{align}\\label{eq:remtri}\ne_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n& =\ne_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}})\n-\\frac{t^2}{2}\n\\Bigg[\\frac{\\beta_{2,1}^2}{p_1^2}-\\frac{\\beta_{2,2}^2}{p_2^2}-\\Big(\\frac{\\beta_{2,1}}{p_1}-\\frac{\\beta_{2,2}}{p_2}\\Big)^2\\Bigg]+O(t^3) \\notag\\\\\n& =-2-\\frac{\\beta_{2,2}}{p_2}\\Big(\\frac{\\beta_{2,1}}{p_1}-\\frac{\\beta_{2,2}}{p_2}\\Big)t^2+O(t^3)\\\\\n& <-2 \\notag\\\\\n& = \\min F_{k+1}, \\notag\n\\end{align}\nwhere we used \\eqref{eq:beta2triodd1} in the last inequality.\n\nFor $t > 0$ small enough, we then have the following.\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_2}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta2+triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|\\leq r-s.\n\\end{align}\n\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_2}^-\\bigcup \\mathcal{J}_0$, we have \n\\[\nE_{k-s}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2+triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|+|\\mathcal{J}_0|\\leq s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:sumJbetatri}, {\\eqref{eq:beta20triodd},} \\eqref{eq1:Jbeta2+triodd}, and \\eqref{eq2:Jbeta2+triodd} into account, we have\n\\begin{align}\\label{eq3:Jbeta2+triodd}\n{|\\mathcal{J}_{\\bm{\\beta}_2}^-|=s-1}.\n\\end{align}\n{Replacing $\\bm{\\beta}_2$ by $-\\bm{\\beta}_2$ as in previous phases of the argument, we arrive at\n\\begin{equation} \\label{eq3:Jbeta2-triodd}\n|{\\mathcal J}_{\\bm{\\beta}_2}^+|\n=\ns-1.\n\\end{equation}}\n\n\\begin{comment}\nWhen $t$ is small enough and $t < 0$, we have the following.\n\\begin{itemize}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_2}^+\\bigcup \\mathcal{J}_0$, we have\n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1+t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta2-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|+|\\mathcal{J}_0|\\leq s.\n\\end{align}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_2}^-$, we have \n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\\leq r-s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:sumJbetatri}, \\eqref{eq1:Jbeta2-triodd} and \\eqref{eq2:Jbeta2-triodd} into account, we have\n\\begin{align}\\label{eq3:Jbeta2-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|+|\\mathcal{J}_0|=s.\n\\end{align}\n\\end{comment}\nCombining \\eqref{eq3:Jbeta2+triodd} with \\eqref{eq3:Jbeta2-triodd}, we arrive at\n\\begin{align}\\label{eq4:Jbeta2triodd}\n{r\n= |{\\mathcal J}_{\\bm{\\beta}_2}^+| + |{\\mathcal J}_{\\bm{\\beta}_2}^-| + |{\\mathcal J}_0|\n=\n2s-1.}\n\\end{align}\n\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_1$}\nNow we perturb the eigenvalues along $\\bm{\\beta}_1 = (0,1)$.\nThe case when ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^{\\pm}$ is similar to that of $\\bm{\\beta}_2$.\nThe difference here is the {behavior of} perturbations of $e_{{\\bm{\\ell}^{(1)}}}$ {in the direction $\\bm{\\beta}_1$.}\nIndeed, by \\eqref{eq:pertgeneralbetatri2order}, we have\n\\begin{align*}\ne_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n&=e_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}})\n-\\frac{t^2}{2}\n\\Bigg[\\frac{\\beta_{1,1}^2}{p_1^2}-\\frac{\\beta_{1,2}^2}{p_2^2}-\\Big(\\frac{\\beta_{1,1}}{p_1}-\\frac{\\beta_{1,2}}{p_2}\\Big)^2\\Bigg]+O(t^3)\\\\\n&=-2+\\frac{t^2}{p_2^2}+O(t^3)\\\\\n&>-2=\\max F_{k}.\n\\end{align*}\nThus, the perturbations of $e_{{\\bm{\\ell}^{(1)}}}$ {in the direction $\\bm{\\beta}_1$} always move up.\n\nFor $t>0$ small enough,\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^+\\bigcup \\mathcal{J}_0$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1+triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|+|\\mathcal{J}_0| \\leq r-s.\n\\end{align}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^-$, we have \n\\[\nE_{k-s-1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1+triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-| \\leq s.\n\\end{align}\n\\end{itemize}\nIn view of \\eqref{eq:sumJbetatri}, Equations~\\eqref{eq1:Jbeta1+triodd} and \\eqref{eq2:Jbeta1+triodd} yield\n\\begin{align}\\label{eq3:Jbeta1+triodd}\n{|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\ns.}\n\\end{align}\n{Applying the usual symmetry argument, we also arrive at $|{\\mathcal J}_{\\bm{\\beta}_1}^+| = s$, which leads to\n\\[\nr\n=\n|{\\mathcal J}_{\\bm{\\beta}_1}^+| + |{\\mathcal J}_{\\bm{\\beta}_1}^-| + |{\\mathcal J}_0|\n=\n2s+1,\n\\]\nwhich in turn contradicts \\eqref{eq4:Jbeta2triodd}.}\n\\end{proof}\n\n\\begin{comment}\nFor $t<0$ small enough, \n\\begin{itemize}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_1}^+ $, we have\n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n<\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\\leq s.\n\\end{align}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_1}^-\\bigcup \\mathcal{J}_0$, we have \n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n>\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-|+|\\mathcal{J}_0|\\leq r-s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:sumJbetatri}, \\eqref{eq1:Jbeta1-triodd} and \\eqref{eq2:Jbeta1-triodd} into account, we have\n\\begin{align}\\label{eq3:Jbeta1-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|=s.\n\\end{align}\nCombining \\eqref{eq3:Jbeta1+triodd} with \\eqref{eq3:Jbeta1-triodd}, we arrive at\n\\begin{align}\\label{eq4:Jbeta1triodd}\nr=2s+|\\mathcal{J}_0|=2s+1.\n\\end{align}\nThis contradicts \\eqref{eq4:Jbeta2triodd}.\n\\end{comment}\n\n\n\n\\subsection{Proof of Lemmas \\ref{lem:constructiontri} and \\ref{lem:triJ0empty}}\\label{sec:constructionlemmaproof}\n\\begin{proof}[Proof of Lemma~\\ref{lem:constructiontri}]\nLet {$E \\in [-3,6]$ be given}, let $x$ be as-yet-unspecified, set $y = 2\\pi - x$, and note that \\eqref{eq:xyCondB} holds. Then, using $y = 2\\pi - x$, we note that\n\\begin{align*}\n\\cos(x) + \\cos(y) + \\cos(x-y)\n& =\n2\\cos(x) + \\cos(2x) \\\\\n& =\n2\\cos(x) + 2\\cos^2(x) - 1.\n\\end{align*}\nSetting $z = \\cos(x)$, we seek to solve $2z+2z^2-1 = E\/2$, which gives\n\\[\nz^2 + z - \\frac{1}{2} - \\frac{E}{4} = 0\n\\implies\nz = \n\\frac{-1 \\pm \\sqrt{3+E}}{2}.\n\\]\nThus, we may take $x$ so that\n\\[\n\\cos(x)\n=\n\\frac{-1+\\sqrt{3+E}}{2}.\n\\]\nIn fact, since $-3 \\le E \\le 6$, we may take $0 \\le x \\le 2\\pi\/3$. Thus, with this choice of $x$ (and $y = 2\\pi - x$), we get \\eqref{eq:xyCondA}. \n\nFinally, suppose $x$ and $y$ solve \\eqref{eq:xyCondA} and \\eqref{eq:xyCondB} for $E \\neq -2$. From \\eqref{eq:xyCondB}, we deduce that either $x+y = 2\\pi$ or $|x-y|=\\pi$. The second option leads to $E = -2$, so we must have $y=2\\pi - x$. Then, $E \\neq -2$ guarantees\n\\[\n\\cos(x) + \\cos(2\\pi - x)\n=\n2\\cos(x)\n=\n-1 + \\sqrt{3+E}\n\\neq 0,\n\\]\nwhich proves \\eqref{eq:xyCondC}.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:triJ0empty}]\nSuppose that $x$ and $y$ solve\n\\begin{align}\n\\label{eq:triJ0empty:cossum}\n\\cos(x) + \\cos(y) + \\cos(x-y)\n& =\n\\lambda \\\\\n\\label{eq:triJ0empty:sin1}\n\\sin(x) + \\sin(x-y) & = 0 \\\\\n\\label{eq:triJ0empty:sin2}\n\\sin(y) - \\sin(x-y) & = 0\n\\end{align}\nfor some $\\lambda \\in (-3\/2,3)$. Adding \\eqref{eq:triJ0empty:sin1} and \\eqref{eq:triJ0empty:sin2}, we arrive at \n\\[\n\\sin(x) = - \\sin(y).\n\\]\nFor $(x,y) \\in [0,2\\pi)^2$, this forces either $|x-y| = \\pi$ or  $x+y = 2\\pi$. In the case $|x-y| = \\pi$, substituting in to \\eqref{eq:triJ0empty:sin1} and \\eqref{eq:triJ0empty:sin2} gives $\\sin(x) = \\sin(y) = 0$, forcing $x,y \\in \\{0,\\pi\\}$. Plugging the various possibilities into \\eqref{eq:triJ0empty:cossum}, one either gets $\\lambda = 3 \\notin (-3\/2,3)$ (when $x=y=0$) or $\\lambda = -1$ (when at least one of $x$ or $y$ is $\\pi$).\n\nAlternatively, if $x = 2\\pi - y$, \\eqref{eq:triJ0empty:sin1} yields $\\sin(x) + \\sin(2x) = 0$,  which leads to\n\\[\n\\sin(x)(1 + 2\\cos(x))\n=\n0.\n\\]\nSetting $\\sin(x) = 0$ yields $x \\in \\{0,\\pi\\}$ which leads to the same solutions as before. Setting $1 +2\\cos(x) = 0$ yields $(x,y) = (2\\pi\/3,4\\pi\/3)$ or $(x,y) = (4\\pi\/3, 2\\pi\/3)$. Plugging in either possibility into \\eqref{eq:triJ0empty:cossum} yields\n\\[\n\\cos(x) + \\cos(y) + \\cos(x-y)\n=\n-\\frac{3}{2} \\notin (-3\/2,3),\n\\]\nas claimed.\n\\end{proof}\n\n\n\n\n\n\\subsection{\\boldmath Opening a Gap at $-2$}\nLet us exhibit a $(2,2)$-periodic potential that perturbatively opens a gap at energy $E = -2$ for the triangular lattice.\n\n\\begin{figure*}[t]\n\n\\begin{tikzpicture}[yscale=.8,xscale=.8]\n\\draw [-,line width = .06cm] (0,0) -- (9,0);\n\\draw [-,line width = .06cm] (0,0) -- (0,9);\n\\draw [-,line width=.06cm] (0,9) -- (9,9);\n\\draw [-,line width = .06cm] (9,0) -- (9,9);\n\\draw [-,line width = .06cm] (0,9) -- (9,0);\n\\draw [-,line width = .06cm] (3,0) -- (3,9);\n\\draw [-,line width = .06cm] (6,0) -- (6,9);\n\\draw [-,line width = .06cm] (0,3) -- (9,3);\n\\draw [-,line width = .06cm] (0,6) -- (9,6);\n\\draw [-,line width = .06cm] (0,3) -- (3,0);\n\\draw [-,line width = .06cm] (0,6) -- (6,0);\n\\draw [-,line width = .06cm] (3,9) -- (9,3);\n\\draw [-,line width = .06cm] (6,9) -- (9,6);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](0,3) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=black, fill=black](0,9) circle (.2);\n\\filldraw[color=black, fill=black](3,0) circle (.2);\n\\filldraw[color=red, fill=red](3,3) circle (.2);\n\\filldraw[color=red, fill=red](3,6) circle (.2);\n\\filldraw[color=black, fill=black](3,9) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=red, fill=red](6,3) circle (.2);\n\\filldraw[color=red, fill=red](6,6) circle (.2);\n\\filldraw[color=black, fill=black](6,9) circle (.2);\n\\filldraw[color=black, fill=black](9,0) circle (.2);\n\\filldraw[color=black, fill=black](9,3) circle (.2);\n\\filldraw[color=black, fill=black](9,6) circle (.2);\n\\filldraw[color=black, fill=black](9,9) circle (.2);\n\\draw [-,line width = .06cm,color=red] (3,3) -- (3,6);\n\\draw [-,line width = .06cm,color=red] (3,3) -- (6,3);\n\\draw [-,line width = .06cm,color=red] (6,3) -- (6,6);\n\\draw [-,line width = .06cm,color=red] (3,6) -- (6,6);\n\\draw [-,line width = .06cm,color=red] (6,3) -- (3,6);\n\\node  at (3.8,3.5) {$\\hot{q_1=1}$};\n\\node  at (6.8,3.5) {$\\hot{q_2=1}$};\n\\node  at (3.8,6.5) {$\\hot{q_3=1}$};\n\\node  at (7,6.5) {$\\hot{q_4=-1}$};\n\\end{tikzpicture}\n\\caption{A $(2,2)$ periodic potential on the triangular lattice with a gap at $E = -2$ for all positive coupling constants.}\\label{fig:tri2x2fundDomain}\n\\end{figure*}\n\n\\begin{theorem} \\label{thm:triExampleGapLength}\nDefine\n\\[\nQ_{n,m} \n= \n(-1)^{mn}\n=\n\\begin{cases}\n1 & \\text{ if } m \\text{ or } n \\text{ is even}, \\\\\n-1 & \\text{ if both } m \\text{ and } n \\text{ are odd,}\n\\end{cases}\n\\]\nand denote $H_\\lambda = \\Delta_{\\mathrm{tri}} + \\lambda Q$. For all $\\lambda > 0$, $\\sigma(H_\\lambda)$ has two connected components. Moreover, for all $\\lambda > 0$ sufficiently small, the gap that opens about $E=-2$ is precisely equal to\n\\[\n\\mathfrak{g}_\\lambda\n=\n\\left(-\\sqrt{4+\\lambda^2},-2+\\lambda \\right).\n\\]\nIn particular,\n\\[\n|\\mathfrak{g}_\\lambda|\n=\n\\lambda + \\left( \\sqrt{4+\\lambda^2}-2 \\right)\n\\sim\n\\lambda + \\frac{\\lambda^2}{2},\n\\]\nso the gap opens linearly as $\\lambda \\downarrow 0$.\n\\end{theorem}\n\nThe following lemma will be used:\n\n\\begin{lemma} \\label{lem:triGapLength:trigPolyEst}\nFor all $\\bm{\\theta} \\in {\\mathbb T}^2$ and all $0 \\leq a \\leq 54$,\n\\[\n4(\\sin\\theta_1 + \\sin\\theta_2 - \\sin(\\theta_1+\\theta_2))^2\n+ a(1+\\cos\\theta_1+\\cos\\theta_2+\\cos(\\theta_1+\\theta_2))\n\\geq 0.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nDefine\n\\[\ng(\\theta_1,\\theta_2,a)\n=\n4(\\sin\\theta_1 + \\sin\\theta_2 - \\sin(\\theta_1+\\theta_2))^2\n+ a(1+\\cos\\theta_1+\\cos\\theta_2+\\cos(\\theta_1+\\theta_2)).\n\\]\nWe begin by checking the boundary of ${\\mathbb T}^2 \\times [0,54]$. It is easy to see that $g \\geq 0$ if $a = 0$. For $a = 54$, define $h(\\bm{\\theta}) = g(\\bm{\\theta},54)$. Using the identities \n\\begin{align*}\n\\sin x+\\sin y-\\sin(x+y)&=4\\sin\\left(\\frac{x}{2}\\right) \\sin\\left(\\frac{y}{2}\\right) \\sin\\left(\\frac{x+y}{2}\\right),\\\\\n\\cos x-\\cos (x+y)&=2\\sin\\left(\\frac{y}{2}\\right) \\sin\\left(x+\\frac{y}{2}\\right)\\\\\n\\sin x+\\sin (x+y)&=2\\cos\\left(\\frac{y}{2}\\right) \\sin\\left(x+\\frac{y}{2}\\right),\n\\end{align*}\nwe may simplify $\\nabla h$ to get\n\\begin{align}\\label{eq:A3}\n\\frac{\\partial h}{\\partial \\theta_1}\n& =\n4\\sin\\left(\\theta_1+\\frac{\\theta_2}{2}\\right)\n\\Bigg[\n16\\sin\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_2}{2}\\right)\n\\Bigg],\\\\\n\\label{eq:A4}\n\\frac{\\partial h}{\\partial \\theta_2}\n& =\n4\\sin\\left(\\theta_2+\\frac{\\theta_1}{2}\\right)\n\\Bigg[\n16\\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_1}{2}\\right)\n\\Bigg].\n\\end{align}\nConsequently, setting $\\nabla h = 0$ leads to four cases. For notational convenience, define\n\\[\n\\alpha\n=\n\\arcsin\\sqrt[4]{\\frac{27}{32}} .\n\\]\n\\noindent \\textbf{Case 1.}\n\\[\\sin\\left(\\theta_1+\\frac{\\theta_2}{2}\\right)=\\sin\\left(\\theta_2+\\frac{\\theta_1}{2}\\right)=0.\\]\nThis implies $\\theta_1 + \\frac{1}{2}\\theta_2 \\in \\pi {\\mathbb Z}$ and $\\theta_2 + \\frac{1}{2}\\theta_1 \\in \\pi {\\mathbb Z}$. Solving the resulting systems for solutions in $[0,2\\pi)$ yields three points:\n\\[\\bm{\\theta}=(0,0),\\ \\left( \\frac{2\\pi}{3},\\frac{2\\pi}{3}\\right),\\ \\left( \\frac{4\\pi}{3},\\frac{4\\pi}{3}\\right).\\]\n\\noindent \\textbf{Case 2.}\n\\[\\sin\\left(\\theta_1+\\frac{\\theta_2}{2}\\right)=16\\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_1}{2}\\right)=0.\\]\nAs before, the first condition forces $\\theta_1 + \\frac{1}{2} \\theta_2 \\in \\pi{\\mathbb Z}$. Plugging the various possibilities that this yields into the second condition gives three solutions:\n\\begin{align*}\n\\bm{\\theta}=(\\pi, 0),\\ \\ (2\\alpha,2\\pi - 4\\alpha),\\ \\ (2\\pi-2\\alpha,4\\alpha).\n\\end{align*}\n\\noindent \\textbf{Case 3.}\n\\[\\sin\\left(\\theta_2+\\frac{\\theta_1}{2}\\right)=16\\sin\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_2}{2}\\right)=0.\\]\nArguing as in Case~2, there are three solutions:\n\\begin{align*}\n\\bm{\\theta}=(0,\\pi),\\ \\ (2\\pi - 4\\alpha,2\\alpha),\\ \\ (4\\alpha,2\\pi-2\\alpha).\n\\end{align*}\n\\noindent \\textbf{Case 4.}\n\\begin{align}\n\\label{eq:triGapLength:trigPolyEst:Case4a}\n16\\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_1}{2}\\right)&=0\\\\\n\\label{eq:triGapLength:trigPolyEst:Case4b}\n16\\sin\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_2}{2}\\right)&=0.\n\\end{align}\nMultiply \\eqref{eq:triGapLength:trigPolyEst:Case4a} by $\\sin(\\theta_2\/2)$, multiply \\eqref{eq:triGapLength:trigPolyEst:Case4b} by $\\sin(\\theta_1\/2)$, and subtract the results to obtain\n\\[\n\\sin\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)\n=\n0.\n\\]\nUsing this, we see that the solutions are \n\\begin{align*}\n\\bm{\\theta}=(\\pi, \\pi),\\ \\ (2\\alpha,2\\alpha),\\ \\ (2\\pi - 2\\alpha,2\\pi - 2\\alpha)\n\\end{align*}\nEvaluating $g$ at these points, we find out $\\max h(\\bm{\\theta})=216$ attained at $(0,0)$, \n$\\min h(\\bm{\\theta})= 0$, attained at\n\\[\n\\bm{\\theta}\n=\n\\left( \\frac{2\\pi}{3},\\frac{2\\pi}{3}\\right),\n\\;\n\\left( \\frac{4\\pi}{3},\\frac{4\\pi}{3}\\right),\n\\; (\\pi,0), \\; (0,\\pi), \\; (\\pi,\\pi).\n\\]\n\n\n\nFinally, we need to look at critical points of $g$  in the interior of ${\\mathbb T}^2 \\times [0,54]$. However, this is easy. Any zero of $\\nabla g$ must in particular satisfy $\\frac{\\partial g}{\\partial a} = 0$, which forces\n\\[\n1+\\cos\\theta_1 + \\cos\\theta_2 + \\cos(\\theta_1+\\theta_2)=0,\n\\]\nwhich clearly implies $g \\geq 0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:triExampleGapLength}]\nFor $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb T}^2$, denote by $H_\\lambda(\\bm{\\theta})$ the Floquet matrix corresponding to $H_\\lambda$. Ordering the vertices of the $2\\times 2$ fundamental domain as shown in Figure~\\ref{fig:tri2x2fundDomain}, we obtain\n\\[\nH_\\lambda(\\bm{\\theta}) -(-2+\\varepsilon) {\\mathbb I}\n=\n\\begin{bmatrix}\n2+\\lambda-\\varepsilon & 1+ e^{-i\\theta_1} & 1 + e^{-i\\theta_2}& 1 + e^{-i(\\theta_1+\\theta_2)} \\\\\n1 + e^{i\\theta_1}& 2+\\lambda -\\varepsilon & e^{i\\theta_1} + e^{-i\\theta_2} & 1 + e^{-i\\theta_2} \\\\\n1 + e^{i\\theta_2} & e^{-i\\theta_1} + e^{i\\theta_2} & 2+\\lambda -\\varepsilon & 1 + e^{-i\\theta_1} \\\\\n1 + e^{i(\\theta_1+\\theta_2)}& 1 + e^{i\\theta_2} & 1 + e^{i\\theta_1} & 2 - \\lambda -\\varepsilon\n\\end{bmatrix}\n\\]\n\n\n\n\nFor $\\bm{\\theta} \\in {\\mathbb T}^2$, $\\lambda>0$, and $\\varepsilon \\in {\\mathbb R}$, define\n\\[\np(\\bm{\\theta},\\lambda,\\varepsilon)\n=\n\\det(H_\\lambda(\\bm{\\theta}) - (-2+\\varepsilon) {\\mathbb I}).\n\\]\nAfter some calculations, one observes that\n\\begin{align*}\np(\\bm{\\theta},\\lambda,\\varepsilon)\n=\n-\\lambda^4 & - 4\\lambda^3 + X(\\bm{\\theta}) - 4\\varepsilon \\lambda\\left(3 - \\frac{\\lambda^2}{2}- \\cos\\theta_1 - \\cos\\theta_2 - \\cos(\\theta_1+\\theta_2) \\right) \\\\\n & + 4\\varepsilon^2 (3+3\\lambda - \\cos\\theta_1 - \\cos\\theta_2 - \\cos(\\theta_1+\\theta_2)) \\\\\n & - 2\\varepsilon^3(4 + \\lambda) + \\varepsilon^4,\n\\end{align*}\nwhere \n\\[\nX(\\bm{\\theta}) = \n-4\\big(\\sin\\theta_1+\\sin\\theta_2 - \\sin(\\theta_1+\\theta_2)\\big)^2\n\\]\nClearly $X(\\bm{\\theta}) \\leq 0$ for all $\\bm{\\theta}$, so we have\n\\[\n\\det(H_\\lambda(\\bm{\\theta}) + 2{\\mathbb I})\n=\np(\\bm{\\theta},\\lambda,0)\n\\leq\n-\\lambda^4 - 4\\lambda^3 <0\n\\]\nfor all $\\lambda > 0$; consequently $-2 \\notin \\sigma(H_\\lambda)$ for all $\\lambda > 0$, which proves the first claim of the theorem. Introducing $W_1(\\lambda,\\varepsilon) := -\\lambda^4 -4\\lambda^3 + 2 \\varepsilon\\lambda^3 + 12\\varepsilon^2\\lambda - 2\\varepsilon^3(4+\\lambda) + \\varepsilon^4$, we may rewrite $p$ as\n\\begin{align} \\label{eq:triPrewrittenW1}\np(\\bm{\\theta},\\lambda,\\varepsilon)\n=\n X(\\bm{\\theta}) - 4\\varepsilon(\\lambda-\\varepsilon)(3-\\cos\\theta_1-\\cos\\theta_2-\\cos(\\theta_1+\\theta_2)) +W_1(\\lambda,\\varepsilon).\n \\end{align}\nBy standard eigenvalue perturbation theory, we know that $|g_\\lambda^\\pm+2| \\leq \\lambda$, so we need only concern ourselves with $|\\varepsilon| \\leq \\lambda$. Since $X(\\bm{\\theta}) \\leq 0$ for all $\\bm{\\theta}$ and the second term of \\eqref{eq:triPrewrittenW1} is nonpositive whenever $0 \\leq \\varepsilon \\leq \\lambda$, we arrive at\n\\[\np(\\bm{\\theta},\\lambda,\\varepsilon)\n\\leq\n-\\lambda^4 -4\\lambda^3 + 2 \\varepsilon\\lambda^3 + 12\\varepsilon^2\\lambda - 2\\varepsilon^3(4+\\lambda) + \\varepsilon^4 \n= \n W_1(\\lambda,\\varepsilon)\n\\]\nfor all $\\bm{\\theta} \\in {\\mathbb T}^2$, all $\\lambda > 0$, and all $0 \\leq \\varepsilon \\leq \\lambda$. Moreover, we observe that $p(\\bm{0},\\lambda,\\varepsilon) = W_1(\\lambda,\\varepsilon)$, so this bound is sharp. Factoring $W_1$, we arrive at\n\\[\nW_1(\\lambda,\\varepsilon)\n=\n(\\lambda - \\varepsilon)^2(\\varepsilon^2 - 8\\varepsilon - \\lambda^2 - 4\\lambda).\n\\]\nConsequently, we see that $W_1(\\lambda,\\varepsilon) < 0$ for $\\varepsilon \\in [0,\\lambda)$, which implies that $p(\\bm{\\theta},\\lambda,\\varepsilon) < 0$ for all $\\bm{\\theta} \\in {\\mathbb T}^2$, all $\\lambda > 0$, and all $0 \\le \\varepsilon < \\lambda$; consequently, $[-2,-2+\\lambda) \\cap \\sigma(H_\\lambda) = \\emptyset$, which is to say:\n\\begin{equation} \\label{eq:triGapRightSide}\n[-2,-2+\\lambda) \\subseteq \\mathfrak{g}_\\lambda.\n\\end{equation}\nOn the other hand, $p(\\bm{0},\\lambda,\\lambda) = 0$, so \n\\begin{equation} \\label{eq:triGapRightSide-2+lambda}\n-2+\\lambda \\in \\sigma(H_\\lambda(\\bm{0})) \\subseteq \\sigma(H_\\lambda)\n\\end{equation} \nAlternatively, $-2+\\lambda \\in \\sigma(H_\\lambda)$ is clear from eigenvalue perturbation theory as soon as one has $[-2,2+\\lambda)\\cap \\sigma(H_\\lambda) = \\emptyset$.\n\n\nNow, for $- \\lambda \\leq \\varepsilon \\leq 0$, we have to be more careful with the term\n\\[\nq(\\bm{\\theta},\\lambda,\\varepsilon)\n:=\n-4\\varepsilon(\\lambda - \\varepsilon)(3- \\cos\\theta_1 - \\cos\\theta_2 - \\cos(\\theta_1+\\theta_2)),\n\\]\nas $q$ can be positive when $-\\lambda < \\varepsilon < 0$. Naively, one can bound\n\\[\n3-\\cos\\theta_1-\\cos\\theta_2 - \\cos(\\theta_1+\\theta_2)\n\\leq\n\\frac{9}{2},\n\\]\nwhich leads to the upper bound of $X(\\bm{\\theta}) + q(\\bm{\\theta},\\lambda,\\varepsilon) \\leq -18\\varepsilon(\\lambda - \\varepsilon)$. However, the maximum of $q$ occurs at the global \\emph{minimum} of $X$, so we can do better. Indeed, for $\\lambda > 0$ small and $-\\lambda \\leq \\varepsilon \\leq 0$, we have\n\\begin{equation} \\label{eq:triGapRefinedXQbound}\nX(\\bm{\\theta}) + q(\\bm{\\theta},\\lambda,\\varepsilon) \\leq - 16\\varepsilon(\\lambda-\\varepsilon).\n\\end{equation}\nIn particular, by Lemma~\\ref{lem:triGapLength:trigPolyEst}, the bound in \\eqref{eq:triGapRefinedXQbound} holds for all $\\varepsilon$ such that $-\\lambda \\leq \\varepsilon\\leq 0$ as long as $8 \\lambda^2 < 54$, i.e.\\ $0 < \\lambda < \\frac{3\\sqrt{3}}{2}$. This then leads us to\n\\begin{align*}\np(\\bm{\\theta},\\lambda,\\varepsilon)\n\\leq\nW_2(\\lambda,\\varepsilon)\n:&=\n-\\lambda^4 -4\\lambda^3 + 2 \\varepsilon\\lambda^3 + 12\\varepsilon^2\\lambda - 2\\varepsilon^3(4+\\lambda) + \\varepsilon^4 - 16\\varepsilon(\\lambda-\\varepsilon) \\\\\n& =\nW_1(\\lambda,\\varepsilon) - 16\\varepsilon(\\lambda-\\varepsilon)\n\\end{align*}\nfor $\\lambda > 0$ small and $-\\lambda \\leq \\varepsilon \\leq 0$. Factoring $W_2$ yields\n\\[\np(\\bm{\\theta},\\lambda,\\varepsilon)\n\\leq\nW_2(\\lambda,\\varepsilon)\n=\n(\\varepsilon - \\lambda)(\\varepsilon - \\lambda- 4)(\\varepsilon^2 - 4\\varepsilon - \\lambda^2)\n\\]\nfor $\\lambda > 0$ small and $-\\lambda \\leq \\varepsilon \\leq 0$. It is straightforward to find the roots of $W_2$ and to observe that $W_2(\\lambda,\\varepsilon) < 0$ when\n\\[\n2 - \\sqrt{4+\\lambda^2}\n<\n\\varepsilon\n\\leq \n0.\\]\nAs a result, this implies $p(\\bm{\\theta},\\lambda,\\varepsilon) < 0$ for all $\\bm{\\theta}$, all $\\lambda > 0$ small, and all $\\varepsilon \\in (2-\\sqrt{4+\\lambda^2},0]$, which in turn yields\n\\begin{equation} \\label{eq:triGapLeftSide}\n(-\\sqrt{4+\\lambda^2},-2] \\subseteq \\mathfrak{g}_\\lambda.\n\\end{equation}\nOn the other hand, \n\\[\np\\left((\\pi,\\pi),\\lambda,2 - \\sqrt{4+\\lambda^2}\\right) \n=\n W_2\\left(\\lambda,2 - \\sqrt{4+\\lambda^2}\\right)\n=\n0,\n\\] \nwhich leads us to conclude \n\\begin{equation} \\label{eq:triGapLeftSide-sqrt4+lambda2}\n-\\sqrt{4+\\lambda^2} \\in \\sigma(H_\\lambda(\\pi,\\pi)) \\subseteq \\sigma(H_\\lambda).\n\\end{equation}\nPutting together \\eqref{eq:triGapRightSide}, \\eqref{eq:triGapRightSide-2+lambda}, \\eqref{eq:triGapLeftSide}, and \\eqref{eq:triGapLeftSide-sqrt4+lambda2}, we obtain\n\\[\n\\mathfrak{g}_\\lambda\n=\n\\left(-\\sqrt{4+\\lambda^2},-2+\\lambda \\right)\n\\]\nfor small $\\lambda$, as promised.\n\n\\end{proof}\n\n\n\n\nThe effort involved in proving Lemma~\\ref{lem:triGapLength:trigPolyEst} in order to improve the constant ``18'' to ``16'' is nontrivial, but worthwhile. In particular, this is exactly what enables the exact factorization of $W_2$ and hence the ability to exactly compute the gap edges.\n\n\n\\section{Hexagonal Laplacian} \\label{sec:hex}\n\n\nWe now continue with the Laplacian on the hexagonal lattice. Let $\\Gamma_{\\mathrm{hex}} = ({\\mathcal V}_{\\mathrm{hex}},{\\mathcal E}_{\\mathrm{hex}})$ and\n\\[\n\\bm{b}_\\pm\n=\n\\frac{1}{2} \\begin{bmatrix}  3 \\\\ \\pm \\sqrt{3} \\end{bmatrix}\n\\]\nbe as in the introduction, let periods $p_1,p_2 \\in {\\mathbb Z}_+$ be given, and view $H = \\Delta_{\\mathrm{hex}}$ as a $(p_1,p_2)$-periodic operator.\nFor this setting, there are two vertices of ${\\mathcal V}_{\\mathrm{hex}}$ in $\\set{s\\bm{b}_+ + t \\bm{b}_- : 0 \\le s, t < 1}$, so our Floquet operator $H(\\bm{\\theta})$ will be $P \\times P$ with $P = 2p_1p_2$. As usual, define $\\Lambda = \\big( [0,p_1)\\times[0,p_2)\\big) \\cap {\\mathbb Z}^2$, denote the eigenvalues of $H(\\bm{\\theta})$ by\n\\[\nE_1^\\Lambda(\\bm{\\theta})\n\\leq\n\\cdots\n\\leq E_P^\\Lambda(\\bm{\\theta}),\n\\]\nand let $F_k^\\Lambda$ for $1 \\le k \\le P$ denote the bands of the spectrum. Our main theorem in this section is the following result.\n\n\n\\begin{theorem} \\label{thm:hexmain}\nLet $p_1,p_2 \\in {\\mathbb Z}_+$ be given.\n\\begin{enumerate}\n\\item Every $E \\in (-3,3) \\setminus \\set{-1,0,1}$ belongs to $\\mathrm{int}(F_j)$ for some $1 \\le j \\le P$.\n\\item If at least one of $p_1$ or $p_2$ is odd, then $-1 \\in \\mathrm{int}(F_k)$ and $+1 \\in \\mathrm{int}(F_\\ell)$ for some $1 \\le k \\le \\ell \\le P$\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{t:bsc:hex}]\nThis follows immediately from Theorem~\\ref{thm:hexmain}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:hexmain}] \nFrom \\eqref{eq:hexDecomp}, we have\n\\[\n\\Delta_{\\mathrm{hex}}\n=\n\\begin{bmatrix}\n0 & S_1^* + S_2^* + {\\mathbb I} \\\\ S_1 + S_2 + {\\mathbb I} & 0 \\end{bmatrix},\n\\]\nwhere $S_j: \\ell^2({\\mathbb Z}^2) \\to \\ell^2({\\mathbb Z}^2)$ denote the shifts\n\\[\n[S_1 \\psi]_{n,m} = \\psi_{n+1,m},\n\\quad\n[S_2 \\psi]_{n,m} = \\psi_{n,m+1}.\n\\]\nIt is easy to see that\n\\[\nS_1 + S_1^* + S_2 + S_2^* + S_1S_2^* + S_1^*S_2 = \\Delta_{\\mathrm{tri}},\n\\]\nthe triangular Laplacian. Thus, a simple calculation shows that\n\\begin{equation} \\label{eq:hexSquareTri}\n[\\Delta_{\\mathrm{hex}}^2 \\Psi]_{\\bm{n}}\n\\begin{bmatrix} [\\Delta_{\\mathrm{tri}} \\psi^+]_{\\bm{n}} + 3\\psi_{\\bm{n}}^+ \\\\ [\\Delta_{\\mathrm{tri}} \\psi^-]_{\\bm{n}} + 3\\psi_{\\bm{n}}^- \\end{bmatrix}\n\\quad\n\\text{for } \\Psi = \\begin{bmatrix} \\psi^+ \\\\ \\psi^- \\end{bmatrix} \\in \\ell^2({\\mathbb Z}^2,{\\mathbb C}^2).\n\\end{equation}\nThis calculation extends to the Floquet matrices, so we see that for each $1 \\le k \\le P$, the bands of $H = \\Delta_{\\mathrm{hex}}$ obey\n\\[\nF^\\Lambda_{k,{\\mathrm{hex}}}\n=\n-F^\\Lambda_{P+1-k,{\\mathrm{hex}}}\n\\]\nand \n\\begin{equation} \\label{eq:hexTriBandRel}\nF_{k,{\\mathrm{hex}}}^\\Lambda\n=\n\\begin{cases}\n\\sqrt{F^\\Lambda_{k - \\frac{P}{2},{\\mathrm{tri}}}+3} & \\frac{P}{2} < k \\leq P \\\\[3mm]\n-\\sqrt{F^\\Lambda_{\\frac{P}{2}+1-k,{\\mathrm{tri}}}+3} & 1 \\leq k \\leq \\frac{P}{2}\n\\end{cases}\n\\end{equation}\nFrom this, we deduce that $E \\in (-3,3)$ lies in the interior of some $F_{k,{\\mathrm{hex}}}$ if and only if $E^2-3$ lies in the interior of some $F_{\\ell,{\\mathrm{tri}}}$. For $E \\in (-3,3) \\setminus \\set{-1,0,1}$, $E^2-3 \\in (-3,6) \\setminus \\{-2\\}$, while $(\\pm1)^2-3 = -2$. Thus, the conclusions of the theorem follow from Theorem~\\ref{thm:trimain}.\n\n\\end{proof}\n\n\\subsection{\\boldmath Opening gaps at $0$ and $\\pm 1$}\n\nDefine the $(1,1)$-periodic potential $Q_1$ on ${\\mathcal V}_{\\mathrm{hex}}$ by $Q_1(\\bm{0}) = 1$ and $Q_1(\\bm{a}_1) = -1$, that is,\n\\[\nQ_1(n \\bm{b}_+ + m \\bm{b}_-) = 1,\n\\quad\nQ_1(\\bm{a}_1  + n \\bm{b}_+ + m \\bm{b}_-) = -1,\n\\quad n,m \\in {\\mathbb Z}.\n\\]\nAfter identifying $\\ell^2({\\mathcal V}_{\\mathrm{hex}})$ with $\\ell^2({\\mathbb Z}^2,{\\mathbb C}^2)$ in the usual way, we get (as an operator) $[Q_1 \\Psi]_{\\bm{n}}= Z\\Psi_{\\bm{n}}$, where\n\\[\nZ=\\begin{bmatrix}\n1 & 0 \\\\ 0 & -1\n\\end{bmatrix}.\n\\]\n\nFrom the calculations $Z U = U = -UZ$ and $Z  L= -L = -L Z$, we deduce that $Q_1 \\Delta_{\\mathrm{hex}} + \\Delta_{\\mathrm{hex}} Q_1 = 0$, and hence\n\\[\n(\\Delta_{\\mathrm{hex}} + \\lambda Q_1)^2\n=\n\\Delta_{\\mathrm{hex}}^2 + \\lambda^2 \n\\geq\n\\lambda^2.\n\\]\nConsequently, $(-\\lambda,\\lambda) \\cap \\sigma(\\Delta_{\\mathrm{hex}} + \\lambda Q_1) = \\emptyset$ and there is a gap at zero. In particular, the gap is precisely $(-\\lambda,\\lambda)$, and so opens linearly at the maximal possible rate.\n\\medskip\n\nLet us consider the $(2,2)$-periodic case. We parameterize our potential as $(q_1,\\ldots,q_8) \\in {\\mathbb R}^8$ as shown in Figure~\\ref{fig:hex22period}.\n\\begin{figure*}[t]\n\n\\begin{tikzpicture}[yscale=1]\n\\draw [-,line width = .1cm,color=black] (0,0) -- (1,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (0,{2*sqrt(3)}) -- (1,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (0,{2*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (0,{4*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (1,{sqrt(3)}) -- (3,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (1,{3*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (4,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .1cm] (4,0) -- (3,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (4,{2*sqrt(3)}) -- (3,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (4,{2*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (4,{4*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (4,0) -- (6,0);\n\\draw [-,line width = .1cm,color=red] (4,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .1cm] (4,{4*sqrt(3)}) -- (6,{4*sqrt(3)});\n\\draw [-,line width = .1cm] (6,0) -- (7,{sqrt(3)});\n\\draw [-,line width = .1cm] (6,{2*sqrt(3)}) -- (7,{sqrt(3)});\n\\draw [-,line width = .1cm] (6,{2*sqrt(3)}) -- (7,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (6,{4*sqrt(3)}) -- (7,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (-2,0) -- (0,0);\n\\draw [-,line width = .1cm,color=red] (-2,{2*sqrt(3)}) -- (0,{2*sqrt(3)});\n\\draw [-,line width = .1cm] (-2,{4*sqrt(3)}) -- (0,{4*sqrt(3)});\n\\draw [-,line width = .1cm] (-2,0) -- (-3,{sqrt(3)});\n\\draw [-,line width = .1cm] (-2,{2*sqrt(3)}) -- (-3,{sqrt(3)});\n\\draw [-,line width = .1cm] (-2,{2*sqrt(3)}) -- (-3,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (-2,{4*sqrt(3)}) -- (-3,{3*sqrt(3)});\n\n\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=red, fill=red](1,{sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](3,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](7,{sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](0,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](4,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](6,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](1,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](3,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](7,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](0,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](4,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](6,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](-2,{0*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](-2,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](-2,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](-3,{1*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](-3,{3*sqrt(3)}) circle (.2);\n\\node at (-2.5,{2*sqrt(3)}){\\hot{$q_1$}};\n\\node at (.5,{2*sqrt(3)}){\\hot{$q_2$}};\n\\node at (.5,{3*sqrt(3)}){\\hot{$q_3$}};\n\\node at (3.5,{3*sqrt(3)}){\\hot{$q_4$}};\n\\node at (.5,{sqrt(3)}){\\hot{$q_5$}};\n\\node at (3.5,{sqrt(3)}){\\hot{$q_6$}};\n\\node at (3.5,{2*sqrt(3)}){\\hot{$q_7$}};\n\\node at (6.5,{2*sqrt(3)}){\\hot{$q_8$}};\n\\end{tikzpicture}\n\\caption{A portion of the hexagonal lattice. A fundamental domain for a $(2,2)$-periodic potential is highlighted in red.}\\label{fig:hex22period}\n\\end{figure*}\n\nWe now turn to the construction of a potential that opens gaps at $0$, $1$, and $-1$ simultaneously. We show that it opens gaps linearly at zero, quadratically at $\\pm1$. Later on, we will show that one cannot open gaps linearly at $\\pm 1$ on both sides.\n\n\\begin{theorem}\\label{thm:hexQ}\nOrder the vertices of a $2\\times 2$ fundamental cell of the hexagonal lattice as shown in Fig.~\\ref{fig:hex22period}, define a $(2,2)$-periodic potential $Q$ by\n\\[\n(q_1,\\ldots,q_8)\n=\n(1,-1,1,2,-2,-1,1,-1),\n\\]\nand denote $H_\\lambda = \\Delta_{\\mathrm{hex}} + \\lambda Q$. Then, for $ |\\lambda| > 0$ sufficiently small, $\\sigma(H_\\lambda)$ consists of four connected components. Moreover, if $\\mathfrak{g}_{E,\\lambda} = (g_{E,\\lambda}^{-}, g_{E,\\lambda}^{+})$ denote the gaps of $\\sigma(H_{\\lambda})$ that open at $E=0,\\pm 1$, one has\n\\[\n \\Big(\\pm 1-\\frac{\\lambda^2}{20}, \\pm 1+\\frac{\\lambda^2}{20}\\Big)\\subset  \\mathfrak{g}_{\\pm 1,\\lambda} {\\subseteq \\Big(\\pm 1 - \\frac{1}{2}\\lambda^2,\\pm1 + \\frac{1}{2}\\lambda^2 \\Big)},\\]\nand\n\\[ \\ \\ \\ \\ \\Big(-\\frac{\\lambda}{{5}}, \\frac{\\lambda}{{5}}\\Big)\\subset  \\mathfrak{g}_{0,\\lambda} \\subset \\Big(-\\frac{\\lambda}{4}, \\frac{\\lambda}{4}\\Big)\\]\nfor all $|\\lambda| > 0$ sufficiently small.\n\\end{theorem}\n\nWe point that we do not carefully optimize the constants; it is possible to get better constants than $1\/20$, ${1\/2}$, ${1\/5}$, and $1\/4$.\n\n\\begin{proof}\nFor $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb T}^2$, let $H_\\lambda(\\bm{\\theta})$ denote the Floquet matrix corresponding to $H_\\lambda$. Ordering the vertices of the fundamental domain as in Figure~\\ref{fig:hex22period}, we obtain:\n\\begin{align}\\label{eq:hexH}\nH_\\lambda(\\bm{\\theta})\n=\n\\begin{bmatrix}\n\\lambda & 1 & 0 & e^{-i\\theta_1} & 0 & e^{-i\\theta_2} & 0 & 0 \\\\\n1 & -\\lambda & 1 & 0 & 1 & 0 &0&0 \\\\\n0 & 1 & \\lambda & 1 & 0 & 0 & 0 & e^{-i\\theta_2} \\\\\ne^{i\\theta_1} & 0 & 1 & 2\\lambda & 0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 & - 2\\lambda & 1 & 0 & e^{-i\\theta_1} \\\\\ne^{i\\theta_2} & 0 & 0 & 0 & 1 & - \\lambda & 1 & 0  \\\\\n0 & 0 &0 & 1 & 0 & 1 & \\lambda & 1 \\\\\n0 & 0 & e^{i\\theta_2} & 0 & e^{i\\theta_1} & 0 & 1 & -\\lambda\n\\end{bmatrix}\n\\end{align}\n\n\nFirst, let us consider the gaps at $E=\\pm 1$.\nCalculations yield\n\\begin{align}\\label{eq:hexsumXZ}\n\\det\\big(H_{\\lambda}(\\bm{\\theta})- (\\pm 1+ s\\lambda^2) {\\mathbb I} \\big)\n= \nX_0^\\pm(\\bm{\\theta}) + X_4^\\pm(\\bm{\\theta},s) \\lambda^4 + X_6^\\pm(\\bm{\\theta},s) \\lambda^6 + O(\\lambda^8)\n\\end{align}\nin which\n\\begin{align*}\nX_0^\\pm(\\bm{\\theta})&=\n-4 (-\\sin(\\theta_1) + \\sin(\\theta_1-\\theta_2) + \\sin(\\theta_2))^2\\\\\nX_4^\\pm(\\bm{\\theta},s)\n&=\n8(s\\pm1)(2s\\mp 1) (3 -\\cos(\\theta_1)-\\cos(\\theta_1-\\theta_2)-\\cos(\\theta_2))\\\\\nX_6^\\pm(\\bm{\\theta},s)\n&=\n-1 \\mp 12 s + 72 s^2 \\mp 16 s^3-4 s^2 (\\pm 4 s+1)(\\cos(\\theta_1) + \\cos(\\theta_1-\\theta_2) + \\cos(\\theta_2)\n\\end{align*}\nIt is clear that \n\\begin{align}\\label{eq:hexX0Z0}\nX_0^\\pm(\\bm{\\theta})\\leq 0 \\quad \\text{for all }\\bm{\\theta} \\in {\\mathbb T}^2.\n\\end{align}\nSince $\\cos(\\theta_1) + \\cos(\\theta_1-\\theta_2) + \\cos(\\theta_2)\\leq 3$, we also have \n\\begin{align}\\label{eq:hexX4Z4}\nX_4^\\pm(\\bm{\\theta},s)\\leq 0 \\quad \\text{for all } \\bm{\\theta} \\in {\\mathbb T}^2, \\; |s|\\leq 1\/2.\n\\end{align}\nWe also have for $|s|\\leq 1\/4$,\n\\[\nX_6^+(\\bm{\\theta},s)\\leq -1-12 s+72 s^2-16 s^3+12 s^2 (4s+1)=:T(s),\n\\]\nand \n\\[\nX_6^-(\\bm{\\theta},s)= X_6^+(\\bm{\\theta},-s) \\leq T(-s).\n\\]\nOne easily checks that $T(s)$ is decreasing on $[-0.05, 0.05]$, and\n\\[T(-0.05)=-0.194.\\]\nHence for $|s|\\leq 0.05$,\n\\begin{align}\\label{eq:hexX6Z6}\nX_6^\\pm(\\bm{\\theta},s)\\leq -0.194.\n\\end{align}\nCombining \\eqref{eq:hexX0Z0}, \\eqref{eq:hexX4Z4}, and \\eqref{eq:hexX6Z6}, we obtain that for $|\\lambda|>0$ sufficiently small, and $|s| \\leq 1\/20$,\n\\[\\det(H_{\\lambda}(\\bm{\\theta})- (\\pm 1+ s\\lambda^2) {\\mathbb I})\\leq -0.1\\lambda^6<0.\\]\nThis proves the claimed lower bound on the gaps at $\\pm 1$.\n\nOn the other hand, let us note that $X_0^\\pm(0,0) =X_0^\\pm(\\pi,\\pi) = 0$, while\n\\begin{align*}\n\\begin{cases}\nX_4^+(\\bm{\\theta}, 0.5)= 0 \\quad \\text{and}\\quad X_6^+((\\pi,\\pi),0.5)=12,\\\\\nX_4^+((0,0),s)=0 \\quad \\text{and}\\quad X_6^+((0,0),-0.5)=28,\\\\\nX_4^-((0,0),s)=0\\quad \\text{and} \\quad X_6^-((0,0),0.5)=28,\\\\\nX_4^-(\\bm{\\theta}, -0.5)=0 \\quad \\text{and}\\quad X_6^-((\\pi, \\pi), -0.5)=12.\n\\end{cases}\n\\end{align*}\nThus for small $\\lambda>0$, we have\n\\begin{align*}\n\\begin{cases}\n\\det(H_\\lambda(\\pi,\\pi)-(1+0.5\\lambda^2){\\mathbb I}) > 0,\\\\\n\\det(H_\\lambda(0,0)-(1-0.5\\lambda^2){\\mathbb I})>0,\\\\\n\\det(H_\\lambda(0,0)-(-1+0.5\\lambda^2){\\mathbb I})>0,\\\\\n\\det(H_\\lambda(\\pi,\\pi)-(-1-0.5\\lambda^2){\\mathbb I})>0.\n\\end{cases}\n\\end{align*}\nWe also easily check that \n\\[X_0^\\pm\\left(\\frac{\\pi}{2},\\pi\\right)=-16,\\]\nwhich implies that for small $\\lambda>0$, we have\n\\[\\det\\left(H_\\lambda\\left(\\frac{\\pi}{2},\\pi \\right)-(\\pm 1\\pm 0.5 \\lambda^2 {\\mathbb I}) \\right)<0.\\]\nWe therefore conclude that \n\\[\\pm 1 + 0.5 \\lambda^2 \\in \\sigma(H_\\lambda) \\text{ and } \\pm 1 - 0.5 \\lambda^2 \\in \\sigma(H_\\lambda),\\]\nwhich proves the upper bounds on the gaps at $\\pm 1$.\n\nNow let us consider the gap at $E=0$. After calculations, we have\n\\begin{align}\\label{eq:hexsumY}\n\\det(H_{\\lambda}(\\bm{\\theta})- s\\lambda {\\mathbb I} )=Y_0(\\bm{\\theta})+Y_2(\\bm{\\theta},s)\\lambda^2+Y_4(\\bm{\\theta},s)+O(\\lambda^6),\n\\end{align}\nwhere \n\\begin{align*}\nY_0(\\bm{\\theta})\n& =\n15 + 2\\cos(2\\theta_1) - 4 \\cos(\\theta_1 - 2\\theta_2) + 2 \\cos(2\\theta_1-2\\theta_2) \n- 4 \\cos(2\\theta_1 - \\theta_2) \\\\ & \\qquad + 2\\cos(2\\theta_2) - 4 \\cos(\\theta_1 + \\theta_2),\n\\end{align*}\n\\[\nY_2(\\bm{\\theta}, s)=2[5-26 s^2+(2+4 s^2)(\\cos(\\theta_1)+\\cos(\\theta_1-\\theta_2)+\\cos(\\theta_2))],\n\\]\nand \n\\[\nY_4(\\bm{\\theta},s)=(1-s^2)[-3 - 42 s^2 + 4 (2 + s^2) (\\cos(\\theta_1)+\\cos(\\theta_1-\\theta_2)+\\cos(\\theta_2))]\n\\]\nWe claim that\n\\begin{equation}\n\\label{eq:hexY0geq0}\nY_0(\\bm{\\theta})\\geq 0\n\\quad\n\\text{for all } \\bm{\\theta} \\in {\\mathbb T}^2.\n\\end{equation} \nLet us see how to use \\eqref{eq:hexY0geq0} to prove the claimed gap at zero and defer the proof of \\eqref{eq:hexY0geq0} for a moment. Using\n\\[\\cos(\\theta_1)+\\cos(\\theta_1-\\theta_2)+\\cos(\\theta_2)\\in \\left[-\\frac{3}{2}, 3\\right],\\]\nwe obtain that for $|s|<1\/5$\n\\begin{align}\\label{eq:hexY2}\nY_2(\\bm{\\theta}, s)\\geq 2(5-26s^2-3(1+2s^2))=4(1-16 s^2)>\\frac{36}{25}.\n\\end{align}\nCombining \\eqref{eq:hexsumY} with \\eqref{eq:hexY2}, we obtain that for $|\\lambda|>0$ sufficiently small\n\\[\\det(H_{\\lambda}(\\bm{\\theta})- s\\lambda\\, {\\mathbb I} )>\\lambda^2.\\]\nThis proves the claimed lower bound of the gap at $0$, modulo the claim that $Y_0(\\bm{\\theta}) \\geq 0$ for all $\\bm{\\theta} \\in {\\mathbb T}^2$. \n\nTo prove the upper bound, we compute\n\\begin{align*}\n\\begin{cases}\nY_0\\left(\\frac{2\\pi}{3}, \\frac{4\\pi}{3}\\right)=0,\\\\\nY_2\\left(\\left(\\frac{2\\pi}{3}, \\frac{4\\pi}{3}\\right), s\\right)=4(1-16 s^2),\\\\\nY_4\\left(\\left(\\frac{2\\pi}{3}, \\frac{4\\pi}{3}\\right), s\\right)=3(s^2-1)(16s^2+5),\n\\end{cases}\n\\end{align*}\nwhich implies that for small $\\lambda>0$,\n\\[\\det\\left(H_{\\lambda}\\left(\\frac{2\\pi}{3}, \\frac{4\\pi}{3}\\right)\\pm 0.25 \\lambda\\, {\\mathbb I} \\right)<0.\\]\nWe also compute that $Y_0(0,0)=9$, which shows for small $\\lambda>0$, \n\\[\\det(H_{\\lambda}(0,0)\\pm 0.25 \\lambda\\, {\\mathbb I})>0.\\]\nThus we conclude that\n\\[\\pm 0.25 \\lambda \\in \\sigma(H_\\lambda),\\]\nwhich proves the claimed upper bound of the gap at $0$.\n\nTo complete the argument, all that remains is to show $Y_0(\\bm{\\theta})\\geq 0$ for all $\\bm{\\theta}\\in {\\mathbb T}^2$.\nTo that end, introduce two auxiliary variables\n\\[\nz := \\cos\\left(\\frac{\\theta_1 - \\theta_2}{2} \\right),\n\\quad\nw := \\cos\\left(\\frac{\\theta_1 + \\theta_2}{2} \\right),\n\\]\nand write $g(z,w)$ to mean $Y_0(\\bm{\\theta})$ in the variables $z$ and $w$. Thus, to optimize $Y_0(\\bm{\\theta})$ on ${\\mathbb T}^2$, it suffices to optimize $g(z,w)$ on the square $[-1,1]^2$. To execute this change of variables, first note the following simple consequences of standard identities:\n\\begin{align*}\n\\cos(2\\theta_1) +  \\cos(2\\theta_2)\n& =\n2(2z^2-1)(2w^2-1),\\\\\n\\cos(2\\theta_1 - 2\\theta_2)\n& =\n2(2z^2-1)^2-1,\\\\\n\\cos(\\theta_1 + \\theta_2)\n& =\n2w^2 -1,\\\\\n\\cos(\\theta_1 - 2\\theta_2) + \\cos(2\\theta_1 - \\theta_2)\n& =\n2 zw(4z^2 - 3).\n\\end{align*}\nPutting all this together,\n\\begin{align*}\ng(z,w)\n& =\n15+ 4(2z^2-1)(2w^2-1) - 8zw(4z^2-3)+ 2(2(2z^2-1)^2-1) - 4(2w^2-1).\n\\end{align*}\nIt is easy to check that $g \\geq 0$ holds on the boundary; concretely,\n\\begin{align*}\ng(\\pm 1,w)\n& =\n15 + 4(2w^2-1) \\mp 8w + 2 - 4(2w^2-1) \\\\\n& =\n17 \\mp 8w \\\\\n& \\geq\n17 - 9 \\\\\n& > 0.\n\\end{align*}\nand\n\\begin{align*}\ng(z,\\pm 1)\n& =\n15 + 4(2z^2-1) \\mp 8z(4z^2-3) + (16z^4-16z^2+2) - 4 \\\\\n& =\n16 z^4 \\mp 32 z^3 - 8z^2 \\pm 24z + 9 \\\\\n& =\n(3 \\pm 4z - 4z^2)^2 \\\\\n& \\geq 0.\n\\end{align*}\nSo, we now seek zeros of $\\nabla g$ for $|z| < 1$ and $|w| < 1$.\nOne easily computes $\\partial_z g$ and $\\partial_w g$:\n\\begin{align*}\n\\partial_z g\n& = \n8(w-2z)(3+4z(w-z)) \\\\\n\\partial_w g\n& =\n8(3z - 4z^3 + 4w(z^2-1)).\n\\end{align*}\nSetting $\\partial_w g = 0$ yields\n\\begin{equation} \\label{eq:Y0critical:wval}\nw\n=\n\\frac{4z^3 - 3z}{4(z^2-1)}.\n\\end{equation}\nSince we are working on the interior of $[-1,1]^2$, $z \\neq \\pm 1$ and the denominator does not vanish. Substituting this expression for $w$ into $\\partial_z g$ and simplifying, we get\n\\[\n\\partial_zg \\left(z,\\frac{4z^3 - 3z}{z^2-1} \\right)\n=\n2z \\left( \\frac{1}{(z^2-1)^2} - 16 \\right).\n\\]\nSetting this equal to zero, we obtain three values of $z$ with $|z| <1$: $0$ and $\\pm \\sqrt{3}\/2$. Inserting these $z$ values into \\eqref{eq:Y0critical:wval}, the corresponding $w$ values are all readily seen to be zero. Plugging in the three critical points $(0,0)$ and $(\\pm \\sqrt{3}\/2,0)$ into $g$ yields 25 and 16, respectively, which concludes the proof that $g \\geq 0$ and hence \n\\[\nY_0(\\bm{\\theta}) \\geq 0\n\\]\nfor all $\\bm{\\theta} \\in {\\mathbb T}^2$, proving \\eqref{eq:hexY0geq0}.\n\\end{proof}\n\n\nNext, we show that for any $(2,2)$-periodic potential, it is impossible that it opens linear order gaps on both sides of $E=\\pm 1$ simultaneously.\n\\begin{theorem}\\label{thm:E=pm1linear}\nFor any $(2,2)$-periodic potential $Q$ and any constant $c>0$, the following holds for all sufficiently small $\\lambda>0$:\n\\[ \\left( (-1-c\\lambda, -1+c\\lambda) \\cup (1-c\\lambda, 1+c\\lambda) \\right) \\cap \\sigma(H_{\\lambda})\\neq \\emptyset\\]\n\\end{theorem}\n\\begin{proof}\nLet $(q_1,q_2,\\ldots,q_8)$ be the potential on a $2\\times 2$ fundamental cell, as shown in Fig.~\\ref{fig:hex22period}. \nThe corresponding Floquet matrix $H_{\\lambda}(\\bm{\\theta})$ is \n\\[\nH_\\lambda(\\bm{\\theta})\n=\n\\begin{bmatrix}\n\\lambda q_1 & 1 & 0 & e^{-i\\theta_1} & 0 & e^{-i\\theta_2} & 0 & 0 \\\\\n1 & \\lambda q_2& 1 & 0 & 1 & 0 &0&0 \\\\\n0 & 1 & \\lambda q_3& 1 & 0 & 0 & 0 & e^{-i\\theta_2} \\\\\ne^{i\\theta_1} & 0 & 1 & \\lambda q_4& 0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 & \\lambda q_5& 1 & 0 & e^{-i\\theta_1} \\\\\ne^{i\\theta_2} & 0 & 0 & 0 & 1 & \\lambda q_6& 1 & 0  \\\\\n0 & 0 &0 & 1 & 0 & 1 & \\lambda q_7& 1 \\\\\n0 & 0 & e^{i\\theta_2} & 0 & e^{i\\theta_1} & 0 & 1 & \\lambda q_8\n\\end{bmatrix}\n\\]\nFor $0<|s|<c$, let us consider \n\\[\\det\\big(H_\\lambda(\\bm{\\theta})-(\\pm 1+s\\lambda){\\mathbb I} \\big)\n=\n\\sum_{k=0}^8 X_k^\\pm (\\bm{\\theta}, s) \\lambda^k.\\]\nAfter a calculation, we obtain\n\\begin{align}\\label{eq:X0Y00}\nX_0^\\pm(\\bm{0},s)\n=\nX_1^\\pm(\\bm{0},s)\n=\nX_2^\\pm(\\bm{0},s)\n=0\\quad\n\\text{ for all }s\n\\end{align}\nand\n\\begin{align}\\label{eq:X3Y30}\n&X_3^+(\\bm{0}, s)=-X_3^-(\\bm{0}, s) = a_0 + a_2 s^2 + 64s^3,\n\\end{align}\nwhere\n\\begin{align*}\na_0 & =\n-2[(q_1 + q_2 + q_7 + q_8) (q_4 + q_5) (q_3 + q_6) + (q_1 + q_8) (q_2 + q_7) (q_3 + q_4 + q_5 + q_6)] \\\\\n  & \\quad +8[(q_1+q_8)(q_2+q_7)+(q_3+q_6)(q_4+q_5)+(q_1+q_2+q_7+q_8)(q_3+q_4+q_5+q_6)]\\\\\na_2  &= -24 \\sum_{k=1}^8 q_k .\n\\end{align*}\nBy \\eqref{eq:X3Y30}, we have \n\\[\nX_3^+(\\bm{0}, s_0) = -X_3^-(\\bm{0}, s_0)\\neq 0\n\\]\nfor some $s_0$ such that $0<|s_0|<c$.\nWithout loss of generality, we assume\n\\[X_3^+(\\bm{0}, s_0)>0>X_3^-(\\bm{0},s_0).\\]\nCombining this with \\eqref{eq:X0Y00}, we obtain\n\\begin{equation} \\label{eq:det00>0}\n\\det(H_\\lambda(\\bm{0})-(1+s_0\\lambda) {\\mathbb I})>0\n\\end{equation}\nfor small $\\lambda > 0$. We also have\n\\begin{align}\\label{eq:X0Y0Pi\/4}\nX_0^\\pm((\\pi\/4, 3\\pi\/4), s_0)=-4.\n\\end{align}\nIn particular, \\eqref{eq:X0Y0Pi\/4} implies that \n\\begin{align}\\label{eq:detPi\/4<0}\n\\det(H_\\lambda(\\pi\/4,3\\pi\/4))-(1+s_0\\lambda) {\\mathbb I})<0\n\\end{align}\nfor all $\\lambda \\geq 0$ small.\n\nCombining \\eqref{eq:detPi\/4<0} with \\eqref{eq:det00>0}, for any sufficiently small $\\lambda > 0$, there exists $\\bm{\\theta}$ such that\n\\[\\det(H_{\\lambda}(\\bm{\\theta})-(1+s_0\\lambda) {\\mathbb I})=0.\\]\nHence \n\\[(1-c\\lambda, 1+c\\lambda)\\cap \\sigma(H_{\\lambda})\\neq \\emptyset\\]\nas claimed.\n\\end{proof}\n\n\\section{Square Laplacian with Next-Nearest Neighbor Interactions}\\label{sec:nnn}\nWe now turn our attention to the EHM lattice, whose Laplacian is given by\n\\begin{align*}\n[\\Delta_{\\mathrm{sqn}} u]_{n,m}\n& =\nu_{n-1,m} + u_{n+1,m} + u_{n,m-1} + u_{n,m+1} + u_{n-1,m+1} + u_{n-1,m+1} + u_{n+1,m-1} + u_{n+1,m+1}\\\\\n& =\n[\\Delta_{\\rm sq}u]_{n,m} + u_{n-1,m-1} + u_{n-1,m+1} + u_{n+1,m-1} + u_{n+1,m+1} \\\\\n& =\n[\\Delta_{{\\mathrm{tri}}}u]_{n,m} + u_{n-1,m-1} + u_{n+1,m+1}.\n\\end{align*}\nNow, given $p_1,p_2 \\in {\\mathbb Z}_+$, we define $P = p_1p_2$ and $\\Lambda = {\\mathbb Z}^2 \\cap \\big([0,p_1) \\times [0,p_2)\\big)$ as before and view $\\Delta_{\\mathrm{sqn}}$ as a $(p_1,p_2)$-periodic operator and perform the Floquet decomposition. For $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb R}^2$, it is straightforward to check that\n\\[\n\\sigma(H(\\bm{\\theta}))\n=\n\\set{e_{\\bm{\\ell}}(\\bm{\\theta}) : \\bm{\\ell}\\in \\Lambda },\n\\]\nwhere $\\bm{\\ell}=(\\ell_1,\\ell_2)$ and\n\\begin{align*}\ne_{\\bm{\\ell}}(\\bm{\\theta})\n=\n2\\cos\\left( \\frac{\\theta_1+2\\pi \\ell_1}{p_1}\\right) \n+ 2\\cos\\left(\\frac{\\theta_2+2\\pi \\ell_2}{p_2}\\right) \n+ &2\\cos\\left(\\frac{\\theta_1 + 2\\pi \\ell_1}{p_1}  - \\frac{\\theta_2 + 2\\pi \\ell_2}{p_2}\\right)\\\\\n+ &2\\cos\\left(\\frac{\\theta_1 + 2\\pi \\ell_1}{p_1}  + \\frac{\\theta_2 + 2\\pi \\ell_2}{p_2}\\right).\n\\end{align*}\n\\begin{comment}\nWe now consider this as a $(p_1,p_2)$-periodic operator, and hence we seek (generalized) eigenfunctions $u$ with $u_{n+p_1,m} \\equiv e^{2\\pi i \\theta_1} u_{n,m}$ and $u_{n,m+p_2} \\equiv e^{2\\pi i y} u_{n,m}$ with $x,y \\in [0,1]$. Take\n\\[\nu_{n,m}^{(\\ell_1,\\ell_2)}\n=\n\\exp\\left(2\\pi i\\left(n \\frac{x+\\ell}{p_1} + m\\frac{y+k}{p_2} \\right) \\right)\n\\]\nwith $\\ell,k \\in {\\mathbb Z}$ and $0 \\le \\ell < p_1$, $0 \\le k < p_2$. It is straightforward to check that this leads to eigenvalues\n\nFor later use, we compute the derivative of $e_{\\ell,k}$:\n\\[-\\frac{1}{4 \\pi}\n\\nabla e_{\\ell,k}(x,y)\n=\n\\left( \\frac{1}{p_1} \\sin\\left(\\hat{x}\\right) + \\frac{1}{p_1}\\sin\\left(\\hat{x} - \\hat{y}\\right), \\frac{1}{p_2}\\sin(\\hat{y}) - \\frac{1}{p_2}\\sin(\\hat{x} - \\hat{y})   \\right)\n\\]\nwith $\\hat{x} = 2\\pi p_1^{-1} (x+\\ell)$ and $\\hat{y} = 2\\pi p_2^{-1} (y+k)$.\n\\end{comment}\nAs in Section~\\ref{sec:floquet}, we label these eigenvalues in increasing order according to multiplicity by\n\\[\nE_1(\\bm{\\theta})\n\\le \nE_2(\\bm{\\theta})\\le \\cdots E_P(\\bm{\\theta})\n\\]\nand denote the $P$ spectral bands by\n\\[\nF_k\n=\n\\set{E_k(\\bm{\\theta}) : \\bm{\\theta} \\in {\\mathbb R}^2},\n\\quad\n1 \\le k \\le P.\n\\]\nStraightforward computations shows that $\\sigma(\\Delta_{\\mathrm{sqn}})=[-4,8]$, hence \n\\[\n\\bigcup_{k=1}^P F_k\n=\n[-4,8].\n\\]\nOur main theorem of this section is\n\\begin{theorem}\\label{thm:sqnmain}\nLet $p_1,p_2 \\in {\\mathbb Z}_+$ be given.\n\\begin{enumerate}\n\\item[{\\rm 1.}]\nEach $E\\in (-4, 8)\\setminus \\{-1\\}$ belongs to $\\mathrm{int}(F_k)$ for some $1\\leq k\\leq P$.\n\\item[{\\rm 2.}] If one of the periods $p_1, p_2$ is not divisible by three, then $E=-1$ belongs to $\\mathrm{int}(F_k)$ for some $1\\leq k\\leq P$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{t:bsc:nnn}]\nThis follows immediately from Theorem~\\ref{thm:sqnmain}.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:sqnmain}} \n\nAs with the proof of Theorem~\\ref{thm:trimain}, we will divide the proof into two different cases: $E\\neq -1$ and $E=-1$ and argue by contradiction.\nTo that end, assume for the sake of establishing a contradiction that $E=\\min F_{k+1}=\\max F_k$ for some $1\\leq k\\leq P-1$.\n\nWe will use the following lemmas, whose proofs we provide at the end of the present section.\n\\begin{lemma}\\label{lem:constructionsqn}\nLet us consider the following system:\n\\begin{align}\\label{eq:xyCondABsqn}\n\\cos(x) + \\cos(y) + \\cos(x-y)+\\cos(x+y) & = \\frac{E}{2} \\\\\n\\sin(x) + \\sin(x-y)+\\sin(x+y) & = 0. \\notag\n\\end{align}\nFor any $E \\in (-4,8) \\setminus \\{-1\\}$, the solution set of \\eqref{eq:xyCondABsqn} in $[0,2\\pi)^2$ satisfies\n\\begin{align}\\label{eq:solutionx=0sqn}\nx=0,\\ \\  1+2\\cos(y)=\\frac{E+1}{3},\n\\end{align}\nor \n\\begin{align}\\label{eq:solutionx=pisqn}\nx=\\pi,\\ \\ 1+2\\cos(y)=-(E+1).\n\\end{align}\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:sqnJ0empty}\nConsider the following system:\n\\begin{equation} \\label{eq:sqnJ0syst}\n\\begin{cases}\n\\cos(x) + \\cos(y) + \\cos(x+y) +\\cos(x-y) = \\frac{E}{2},\\\\\n\\sin(x)+\\sin(x-y)+\\sin(x+y)=0,\\\\\n\\sin(y)-\\sin(x-y)+\\sin(x+y)=0.\n\\end{cases}\n\\end{equation}\nFor any $E \\in (-4,8) \\setminus \\{0, -1\\}$, the solution set of \\eqref{eq:sqnJ0syst} is empty. \nFor $E=0$, the unique solution of \\eqref{eq:sqnJ0syst} in $[0,2\\pi)^2$ is $(\\pi,\\pi)$.\nFor $E =-1$, the solutions of \\eqref{eq:sqnJ0syst} in $[0,2\\pi)^2$ are $({2\\pi}\/{3},{2\\pi}\/{3})$, $({2\\pi}\/{3},{4\\pi}\/{3})$, $({4\\pi}\/{3},{2\\pi}\/{3})$ and $({4\\pi}\/{3},{4\\pi}\/{3})$.\n\\end{lemma}\n\nWe will use Lemma \\ref{lem:constructionsqn} in the $E\\neq -1$ case, and Lemma \\ref{lem:sqnJ0empty} in the $E=-1$ case.\n\n\\subsubsection{$E\\neq -1$}\\\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:sqnmain}.1]\nLet $E \\in (-4,8)\\setminus\\set{-1}$ be given, and suppose towards a contradiction that $E = \\max F_k = \\min F_{k+1}$ for some $k$. Define $\\widetilde{\\bm{\\theta}}=(\\widetilde{\\theta}_1,\\widetilde{\\theta}_2)\\in [0,2\\pi)^2$ and  $\\bm{\\ell}^{(1)}=(\\ell_1^{(1)},\\ell_2^{(1)})\\in \\Lambda$ via\n\\begin{align}\\label{def:ttheellsqnE}\n\\widetilde{\\theta}_1=0,\\ \\ell_1^{(1)}=0,\\ \\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2^{(1)}}{p_2}=\\arccos\\Big(\\frac{E-2}{6}\\Big)\\in  (0,\\pi).\n\\end{align}\nNote that since $E\\in (-4, 8)$, we have $\\frac{E-2}{6}\\in (-1,1)$, hence $\\arccos\\Big(\\frac{E-2}{6}\\Big)$ is always well-defined.\nNote also that $\\widetilde{\\theta}_2$ and $\\ell_2^{(1)}$ are uniquely determined.\nUsing \\eqref{def:ttheellsqnE}, one easily checks that \n\\[e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=E,\\]\nand\n\\begin{align}\\label{eq:sqnJbeta10}\n(1,0)\\cdot \\nabla e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=0.\n\\end{align}\nAs in the proof of Theorem~\\ref{thm:trimain}, denote $\\Lambda_E(\\widetilde{\\bm{\\theta}}) = \\{\\bm{\\ell} \\in \\Lambda : e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = E\\}$, let $r := |\\Lambda_E(\\widetilde{\\bm{\\theta}})|$ be the multiplicity of $E$ as an eigenvalue of $H(\\widetilde{\\bm{\\theta}})$, and choose $s\\in {\\mathbb Z}\\cap [1,r]$ such that \n\\[E_{k-s}(\\widetilde{\\bm{\\theta}})<E_{k-s+1}(\\widetilde{\\bm{\\theta}})=\\cdots=E_k(\\widetilde{\\bm{\\theta}})=\\cdots=E_{k+r-s}(\\widetilde{\\bm{\\theta}})<E_{k+r-s+1}(\\widetilde{\\bm{\\theta}}).\\]\nSince all the eigenvalues are continuous in $\\bm{\\theta}$, we can take $\\varepsilon>0$ small enough such that \n\\[E_{k-s}(\\bm{\\theta})<E_{k-s+1}(\\bm{\\theta}),\\, \\text{ and }\\, E_{k+r-s}(\\bm{\\theta})<E_{k+r-s+1}(\\bm{\\theta}),\\]\nholds whenever $\\|\\bm{\\theta}-\\widetilde{\\bm{\\theta}}\\|_{{\\mathbb R}^2}<\\varepsilon$. Given $\\bm{\\ell}\\in \\Lambda$ and a unit vector $\\bm{\\beta}=(\\beta_1, \\beta_2)$, we have\n\\begin{align}\ne_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}+ t \\bm{\\beta})\n& = e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})+t\\bm{\\beta} \\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})+O(t^2) \\label{eq:pertgeneralbetasqn1order} \\\\\n& =e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})+t\\bm{\\beta} \\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) \\label{eq:pertgeneralbetasqn2order}\\\\\n&\\qquad -\\frac{t^2}{2}\\Bigg[\\frac{\\beta_1^2}{p_1^2}\\cos\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}\\Big)\n+\\frac{\\beta_2^2}{p_2^2}\\cos\\Big(\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big) \\notag\\\\ \n&\\qquad +\\Big(\\frac{\\beta_1}{p_1}-\\frac{\\beta_2}{p_2}\\Big)^2 \\cos\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}-\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\notag\\\\\n&\\qquad +\\Big(\\frac{\\beta_1}{p_1}+\\frac{\\beta_2}{p_2}\\Big)^2 \\cos\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}+\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\Bigg] +O(t^3).  \\notag\n\\end{align}\nIn particular, we will use \\eqref{eq:pertgeneralbetasqn1order} if $\\bm{\\beta}\\cdot  \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\neq 0$, and \\eqref{eq:pertgeneralbetasqn2order} otherwise.\n\n\n\nFor any vector $\\bm{\\beta}\\in {\\mathbb R}^2\\setminus \\{\\bm{0}\\}$, let \n\\begin{equation}\\label{def:Jbetasqn}\n\\begin{aligned}\n\\mathcal{J}_{\\bm{\\beta}}^0\n=\n{\\mathcal J}_{\\bm{\\beta}}^0(\\widetilde{\\bm{\\theta}}):\n&=\n\\{\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) :\\ \\bm{\\beta}\\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})=0\\},\\\\\n\\mathcal{J}_{\\bm{\\beta}}^{\\pm} = {\\mathcal J}_{\\bm{\\beta}}^\\pm(\\widetilde{\\bm{\\theta}}):\n&=\n\\{\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) :\\ \\pm \\bm{\\beta} \\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})>0\\}.\n\\end{aligned}\n\\end{equation}\nBy definition, we must have\n\\begin{align}\\label{eq:sumJbetasqn}\n|\\mathcal{J}_{\\bm{\\beta}}^0|+|\\mathcal{J}_{\\bm{\\beta}}^+|+|\\mathcal{J}_{\\bm{\\beta}}^-|=r\n\\end{align}\nfor any $\\bm{\\beta}$. We also define $\\mathcal{J}_0$ as follows\n\\begin{align}\\label{def:J0sqn}\n\\mathcal{J}_0\n={\\mathcal J}_0(\\widetilde{\\bm{\\theta}})\n:=\n\\{\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}):\\ \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = \\bm{0}\\}.\n\\end{align}\nIf $E\\neq 0$, Lemma~\\ref{lem:sqnJ0empty} directly implies $\\mathcal{J}_0=\\emptyset$.\nIf $E=0$, $\\mathcal{J}_0$ is also empty. \nTo see this, suppose on the contrary that $\\bm{\\ell} = (\\ell_1,\\ell_2) \\in {\\mathcal J}_0$.\nLemma~\\ref{lem:sqnJ0empty} implies that\n\\begin{equation} \\label{eq:sqn:J0emptyE=0eq1}\n\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}=\\pi,\n\\end{equation}\nand \\eqref{def:ttheellsqnE} forces\n\\begin{equation} \\label{eq:sqn:J0emptyE=0eq2}\n\\frac{\\widetilde{\\theta}_2+2\\pi \\ell^{(1)}_2}{p_2}=\\arccos\\left(-\\frac{1}{3}\\right).\n\\end{equation}\nSubtracting \\eqref{eq:sqn:J0emptyE=0eq1} from \\eqref{eq:sqn:J0emptyE=0eq2} yields\n\\[\\frac{\\ell^{(1)}_2-\\ell_2}{p_2}=\\frac{1}{2\\pi}\\arccos\\left(-\\frac{1}{3}\\right)-\\frac{1}{2}.\\]\nHowever, this implies that $(2\\pi)^{-1}\\arccos\\left(-1\/3\\right)$ is a rational number, which contradicts the following well-known fact, whose proof we supply at the end of the present section.\n\\begin{lemma}\\label{lem:arccos1\/3}\n\\[\\frac{1}{2\\pi}\\arccos\\left(-\\frac{1}{3}\\right)\\in {\\mathbb R}\\setminus {\\mathbb Q}.\\]\n\\end{lemma}\nTherefore $\\mathcal{J}_0=\\emptyset$ for any $E\\neq -1$.\n\nWe choose $\\bm{\\beta}_1=(1,0)$. Then \\eqref{eq:sqnJbeta10} implies $\\bm{\\ell}^{(1)} \\in {\\mathcal J}_{\\bm{\\beta}_1}^0$, and hence\n\\begin{align}\\label{eq:beta1sqnnon-empty}\n\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset.\n\\end{align} \n\n\nNext we are going to perturb the point $\\widetilde{\\bm{\\theta}}$ and count the eigenvalues.\nSince $\\mathcal{J}_0=\\emptyset$, we can choose a unit vector $\\bm{\\beta}_2$ such that \n\\begin{align}\\label{eq:beta2sqnnon-empty}\n\\bm{\\beta}_2\\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\neq 0,\n\\end{align}\nholds for any $\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}})$.\nThus $\\mathcal{J}_{\\bm{\\beta}_2}^0=\\emptyset$ and\n\\begin{align}\\label{eq:beta2sqnnon-empty'}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|+|\\mathcal{J}_{\\bm{\\beta}_2}^-|=r.\n\\end{align}\nArguing as in the proof of Theorem~\\ref{thm:trimain}.1, we deduce \\begin{comment}\nWe first perturb the eigenvalues along the $\\bm{\\beta}_2$ direction.\nSince $\\mathcal{J}_{\\bm{\\beta}_2}^0 = \\emptyset$, we will always employ \\eqref{eq:pertgeneralbetasqn1order}.\n\nFor $t > 0$ small enough, we have the following.\n\\begin{itemize}\n\\item  If ${\\bm{\\ell}}  \\in \\mathcal{J}_{\\bm{\\beta}_2}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq1:Jbeta2+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|\\leq r-s.\n\\end{align}\n\n\\item If ${\\bm{\\ell} } \\in \\mathcal{J}_{\\bm{\\beta}_2}^-$, we have \n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\\leq s.\n\\end{align}\n\\end{itemize}\n{In view of \\eqref{eq:beta2sqnnon-empty'}, Equations~\\eqref{eq1:Jbeta2+sqn} and \\eqref{eq2:Jbeta2+sqn} imply\n\\begin{equation} \\label{eq:Jbeta2MinusCard+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\n=\ns.\n\\end{equation}\nUpon realizing that $\\mathcal{J}_{-\\bm{\\beta}_2}^0 = \\emptyset$ and $\\mathcal{J}_{-\\bm{\\beta}_2}^\\pm = \\mathcal{J}_{\\bm{\\beta}_2}^\\mp$, we may apply the analysis above with $\\bm{\\beta}_2$ replaced by $-\\bm{\\beta}_2$ and conclude that\n\\begin{equation} \\label{eq:JMinusBeta2+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+| = |\\mathcal{J}_{-\\bm{\\beta}_2}^-| = s.\n\\end{equation}\nIn particular, \\eqref{eq:Jbeta2MinusCard+sqn} and \\eqref{eq:JMinusBeta2+sqn} imply \\end{comment}\n\\begin{align}\\label{eq4:Jbeta2sqn}\nr=2s.\n\\end{align}\n\n\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_1$}\nNow we perturb the eigenvalues along $\\bm{\\beta}_1=(1,0)$.\nThe case when ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^{\\pm}$ is similar to that of $\\bm{\\beta}_2$.\nThe difference here is that, according to \\eqref{eq:beta1sqnnon-empty}, $\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset$.\n\nBy Lemma~\\ref{lem:constructionsqn}, we have that for $(\\ell_1,\\ell_2)\\in \\mathcal{J}_{\\bm{\\beta}_1}^0$,\n\\begin{align}\\label{eq:beta1sqnt2sign}\n(E+1)\\Bigg[\n\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}\\Big)\n+&\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}-\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)\\\\\n&\\qquad +\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}+\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big) \n\\Bigg]\n>0. \\notag\n\\end{align}\nIndeed, if $(\\ell_1,\\ell_2)\\in \\mathcal{J}_{\\bm{\\beta}_1}^0$, \n$(x,y) = (p_1^{-1} (\\widetilde{\\theta}_1 + 2\\pi \\ell_1), p_2^{-1}(\\widetilde{\\theta}_2 + 2\\pi \\ell_2))$ is a solution to \\eqref{eq:xyCondABsqn}.\nHence Lemma~\\ref{lem:constructionsqn} implies that \nwe have \neither \n\\[\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}=0,\\ \\ 1+2\\cos\\Big(\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)=\\frac{E+1}{3},\\]\nor \n\\[\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}=\\pi,\\ \\ 1+2\\cos\\Big(\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)=-(E+1).\\]\nClearly, both cases lead to \\eqref{eq:beta1sqnt2sign}.\n\nBy employing \\eqref{eq:pertgeneralbetasqn2order}, we obtain\n\\begin{align}\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n=\nE & - {\\frac{ t^2}{2 p_1^2}} \\Bigg[\n\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}\\Big)\n+\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}-\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)\\\\\n& \\qquad\\qquad\\qquad\n+\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}+\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big) \n\\Bigg] + O(t^3)\\notag\n\\end{align}\nfor $\\bm{\\ell} \\in {\\mathcal J}_{\\bm{\\beta}_1}^0$. Combining this with \\eqref{eq:beta1sqnt2sign}, we obtain that for $|t|>0$ small enough\n\\begin{align}\\label{eq:Jbeta10sqn}\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n\\begin{cases}\n<E,\\ \\ \\text{if } E+1>0,\\\\\n>E,\\ \\ \\text{if } E+1<0.\n\\end{cases}\n\\end{align}\nNotice that the choice of $\\bm{\\beta}_1$ causes the second $t^2$ term of \\eqref{eq:pertgeneralbetasqn2order} to drop out.\n\nWithout loss of generality, we assume $E\\in (-1, 8)$. The complementary case when $E\\in (-4, -1)$ can be handled similarly.\nFor $E\\in (-1, 8)$, \\eqref{eq:Jbeta10sqn} implies that \n\\begin{align}\\label{eq:Jbeta10sqn'}\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n<\nE\n=\n\\min F_{k+1},\n\\end{align}\nholds for $|t|>0$ small enough and for any $\\bm{\\ell} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0$. \n\nCombining \\eqref{eq:Jbeta10sqn'} with \\eqref{eq:pertgeneralbetasqn1order}, we have the following.\n\nFor $t>0$ small enough,\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\\leq r-s\n=\ns,\n\\end{align}\n{where the equality follows from \\eqref{eq4:Jbeta2sqn}.}\n\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0 \\bigcup \\mathcal{J}_{\\bm{\\beta}_1}^-$, we have \n\\[\nE_{k-s-1}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^-|\\leq s.\n\\end{align}\n\\end{itemize}\n{In view of \\eqref{eq:sumJbetasqn} and \\eqref{eq4:Jbeta2sqn}, Equations~\\eqref{eq1:Jbeta1+sqn} and \\eqref{eq2:Jbeta1+sqn} yield\n\\begin{equation} \\label{eq3:Jbeta1+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\n=\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\ns.\n\\end{equation}\nAs before, we may observe that ${\\mathcal J}_{-\\bm{\\beta}_1}^0 = {\\mathcal J}_{\\bm{\\beta}_1}^0$ and ${\\mathcal J}_{-\\bm{\\beta}_1}^\\pm = {\\mathcal J}_{\\bm{\\beta}_1}^\\mp$. Then, the analysis above applied with $\\bm{\\beta}_1$ replaced by $-\\bm{\\beta}_1$ forces\n\\begin{equation} \\label{eq3:Jbeta1-sqn}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^+|\n=\ns.\n\\end{equation}\nTaken together, \\eqref{eq3:Jbeta1+sqn} and \\eqref{eq3:Jbeta1-sqn} imply $|{\\mathcal J}_{\\bm{\\beta}_1}^0| = 0$, which contradicts \\eqref{eq:beta1sqnnon-empty}.\n}\n\\end{proof}\n\n\\subsubsection{$E=-1$}\\\n\nFirst, we would like to make a remark on our strategy of the proof of the $E=-1$ case, and on the importance of one of the period being not divisible by $3$.\n\\begin{remark}\\label{rem:sqn}\nFor the exceptional energy {$E = -1$} of the EHM lattice, we can not use eigenvalues with vanishing gradients to create un-even numbers of counting unless neither $p_1$ nor $p_2$ is divisible by $3$.\nThe reason is the following: suppose {\\it only} $p_1$ is not divisible by $3$ and we choose $\\widetilde{\\bm{\\theta}}=(\\widetilde{\\theta}_1, \\widetilde{\\theta}_2)$ and $\\bm{\\ell}^{(1)}=(\\ell_1^{(1)}, \\ell_2^{(1)})$ such that $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=-1$ and $\\nabla e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=\\bm{0}$. \nLemma \\ref{lem:sqnJ0empty} yields four possibilities $(p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(1)}), p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2^{(1)}))=(2\\pi\/3,2\\pi\/3)$, $(2\\pi\/3,4\\pi\/3)$, $(4\\pi\/3,2\\pi\/3)$ or $(4\\pi\/3,4\\pi\/3)$.\nWithout loss of generality, we choose $(2\\pi\/3, 2\\pi\/3)$, the other three choices are essentially the same.\nSince $p_2$ is divisible by $3$, there exists $\\bm{\\ell}^{(2)}$, such that \n$(p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(2)}), p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2^{(2)}))=(2\\pi\/3,4\\pi\/3)$.\nHence $e_{\\bm{\\ell}^{(2)}}(\\widetilde{\\bm{\\theta}})$ is also located at $-1$ with vanishing gradient.\nPerturbing $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})$ and $e_{\\bm{\\ell}^{(2)}}(\\widetilde{\\bm{\\theta}})$ along a given direction $(\\beta_1, \\beta_2)$ is equivalent to controlling the signs of the following two expressions:\n$$\\beta_1\\beta_2\\ \\ \\text{and}\\ \\ -\\beta_1\\beta_2.$$\nThis means we can never choose two different directions that lead to un-even {counts}.\nTherefore we need to develop a new argument for this case.\n\nIndeed, when $p_1$ is not divisible by $3$, we choose $p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(1)})=2\\pi\/3$ and $\\widetilde{\\theta}_2$ such that \n$p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2)\\notin \\{2\\pi\/3, 4\\pi\/3\\}$ {{\\it regardless} of the choice of $\\ell_2$}. Such choices guarantee that there are in total $p_2$ eigenvalues located at $-1$, which are \n$\\{e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}),\\ \\ \\ell_1=\\ell_1^{(1)}\\}$. It then suffices to control the movements of these eigenvalues along any given direction.\nA key observation is that along any direction, approximately $2p_2\/3$ eigenvalues will move up (down) while the other $p_2\/3$ eigenvalues move down (up), see \\eqref{eq:E=-1Jbbe}.\nThis leads to un-even counting that we need.\nLet us point out that if both $p_1, p_2$ are divisible by $3$, this argument does not work {(as it must, given the example constructed in Theorem~\\ref{t:nnnExamples})}: there will be $2p_2$ eigenvalues located at $-1$, and $p_2$ of them move up while the other $p_2$ of them move down along any given direction.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:sqnmain}.2]\nWithout loss of generality, we assume $p_1$ is not divisible by $3$.\nLet $p_j=3p_j'+k_j$, where $p_j',k_j \\in {\\mathbb Z}$ with $0 \\le k_j < 3$ and then define $\\widetilde\\bm{\\theta}$ by\n\\begin{align*}\n\\widetilde{\\theta}_1= \\frac{2\\pi k_1}{3},\\ \\ \n\\widetilde{\\theta}_2={\\frac{k_2+1}{4}\\pi}.\n\\end{align*}\n{As usual, denote $\\Lambda_E(\\widetilde{\\bm{\\theta}}) = \\Lambda_{-1}(\\widetilde{\\bm{\\theta}}) = \\{\\bm{\\ell} \\in \\Lambda : e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = -1\\}$. We first claim that\n\\begin{equation} \\label{eq:sqnE=-1:multiplicity}\n\\Lambda_{-1}(\\widetilde{\\bm{\\theta}})\n=\n\\set{(p_1', \\ell_2) : 0 \\leq \\ell_2 < p_2 \\text{ and } \\ell_2 \\in {\\mathbb Z}}.\n\\end{equation}}\nLet us consider the trigonometric equation\n\\begin{align}\\label{eq:E=-1}\n\\cos(x)+\\cos(y)+\\cos(x-y)+\\cos(x+y)=-\\frac{1}{2}=\\frac{E}{2}.\n\\end{align}\nUsing the identity $\\cos(x-y)+\\cos(x+y)=2\\cos(x) \\cos(y)$, we see that \\eqref{eq:E=-1} is equivalent to \n\\[(2\\cos(x)+1) (2\\cos(y)+1)=0,\\]\nwhose solutions are $\\cos(x)=-1\/2$ or $\\cos(y)=-1\/2$. {With our choice of $\\widetilde{\\bm{\\theta}}$, it is clear that\n\\begin{equation} \\label{eq:E=-1the1ell1}\n\\frac{\\widetilde{\\theta}_1 + 2\\pi p_1'}{p_1}\n=\n\\frac{2\\pi}{3},\n\\quad\n\\cos\\left(\\frac{\\widetilde{\\theta}_1 + 2\\pi p_1'}{p_1}\\right)\n=\n-\\frac{1}{2}.\n\\end{equation}\nConsequently, \n\\begin{equation} \\label{eq:sqn:E=-1:LambdaEeq1}\ne_{(p_1',\\ell_2)}(\\widetilde{\\bm{\\theta}}) = -1 \\quad \\text{for every } 0 \\le \\ell_2 < p_2.\n\\end{equation}}\nDue to our choice of $\\widetilde{\\theta}_2$, we get\n\\begin{align}\\label{eq:E=-1J0}\n\\cos(p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2))\\neq -\\frac{1}{2}\\quad\n\\text{for any } \\ell_2 \\in [0,p_2) \\cap {\\mathbb Z}.\n\\end{align}\nIndeed, since $p_2^{-1}(\\widetilde{\\theta}_2+2\\pi\\ell_2) \\in [0,2\\pi)$, $\\cos(p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2))=-1\/2$ would force \n\\[\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2} \\in \\set{\\frac{2\\pi}{3}, \\frac{4\\pi}{3}},\\]\nwhich, after doing some algebra, leads to {\n\\[\n3(8\\ell_2+k_2+1) \\in \\{8p_2,16p_2\\},\n\\]\nwhich is plainly impossible, since $\\ell_2,p_2 \\in {\\mathbb Z}$ and $k_2 \\in \\{0,1,2\\}$.}\nAdditionally, due to our choice of $\\widetilde{\\theta}_1$, we also have\n\\begin{align}\\label{eq:E=-1Jbeta10}\n{\\cos(p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1))\\neq -1\/2 \\quad\n\\text{for any } \\ell_1 \\in \\big([0,p_1)\\cap{\\mathbb Z}\\big)\\setminus\\{p_1'\\}}.\n\\end{align}\nTo see this, suppose on the contrary that \\eqref{eq:E=-1Jbeta10} fails. This forces\n\\[\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}=\\frac{4\\pi}{3}\\]\nfor some $0 \\leq \\ell_1 < p_1$ with $\\ell_1\\neq p_1'$. Since\n\\[\\frac{\\widetilde{\\theta}_1+2\\pi p_1'}{p_1}=\\frac{2\\pi}{3},\\]\nthis implies\n\\[\\frac{2\\pi (\\ell_1-p_1')}{p_1}=\\frac{2\\pi}{3},\\]\nwhich is impossible since $p_1$ is not divisible by $3$. Combining \\eqref{eq:E=-1J0} and \\eqref{eq:E=-1Jbeta10} yields\n\\begin{equation} \\label{eq:sqn:E=-1:LambdaEeq2} \n{e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\neq - 1\n\\quad\n\\text{for any } \\bm{\\ell} =(\\ell_1,\\ell_2)\\in\\Lambda \\text{ such that } \\ell_1 \\neq p_1'.}\n\\end{equation}\nTaken together, \\eqref{eq:sqn:E=-1:LambdaEeq1} and \\eqref{eq:sqn:E=-1:LambdaEeq2} imply \\eqref{eq:sqnE=-1:multiplicity}.\n\nLet us choose $\\bm{\\beta}=(\\beta_1,\\beta_2)=(1,0)$. \nWe have that for any $\\bm{\\ell}\\in \\Lambda$:\n\\begin{align*}\n&\\bm{\\beta}\\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\\\\n=&-\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}\\Big)\n-\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}-\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\n-\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}+\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\\\\n=&-\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}\\Big)\\Bigg[1+2\\cos\\Big(\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\Bigg].\n\\end{align*}\nBy \\eqref{eq:sqnE=-1:multiplicity}, \\eqref{eq:E=-1the1ell1}, and \\eqref{eq:E=-1J0}, we have the following {for any $\\bm{\\ell} = (\\ell_1, \\ell_2) \\in \\Lambda_{-1}(\\widetilde{\\bm{\\theta}})$}:\n\\[\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}\\Big)=\\frac{\\sqrt{3}}{2},\\ \\ \\cos\\Big(\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\neq -\\frac{1}{2}.\\]\nThis implies\n\\begin{align}\\label{eq:E=-1Jbbe}\n\\mathcal{J}_{\\bm{\\beta}}^0=\\emptyset,\\ \\ \\text{and }\\ \\ \n\\mathcal{J}_{\\bm{\\beta}}^{\\pm}\n=\n\\Bigg\\lbrace \\bm{\\ell} \\in \\Lambda_{-1}(\\widetilde{\\bm{\\theta}}) :\\ \\ \\mp \\frac{1}{2}  \\mp \\cos\\Big(\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)>0 \\Bigg\\rbrace.\n\\end{align}\nHence we expect that $|{\\mathcal J}_{\\bm{\\beta}}^+|\\sim p_2\/3$, and $|{\\mathcal J}_{\\bm{\\beta}}^-|\\sim 2p_2\/3$.\nMore precisely, we note that\n\\[\n{{\\mathcal J}_{\\bm{\\beta}}^+\n=\n\\Big\\lbrace (p_1', \\ell_2):\\ \\ \\frac{2\\pi}{3} < \\frac{(k_2+1)\\pi\/4 + 2\\pi \\ell_2}{p_2} < \\frac{4\\pi}{3} \\Big\\rbrace}.\\]\nUsing $p_2=3p_2'+k_2$, we obtain\n\\[\n{{\\mathcal J}_{\\bm{\\beta}}^+\n=\n\\Big\\lbrace (p_1',\\ell_2):\\ \\ p_2'+\\frac{5k_2-3}{24} <  \\ell_2 < 2p_2'+\\frac{13k_2-3}{24} \\Big\\rbrace}.\n\\]\nConsequently,\n\\begin{align*}\n{\\mathcal J}_{\\bm{\\beta}}^+=\n\\begin{cases}\n\\{(p_1', \\ell_2):\\ \\ p_2'\\leq \\ell_2 \\leq 2p_2'-1\\},\\ \\ \\text{if } k=0,\\\\\n\\{(p_1', \\ell_2):\\ \\ p_2'+1\\leq \\ell_2 \\leq 2p_2'\\},\\ \\ \\text{if } k=1, 2.\n\\end{cases}.\n\\end{align*}\nTherefore \n\\begin{align}\\label{eq:Jbetapm}\n(|{\\mathcal J}_{\\bm{\\beta}}^+|, |{\\mathcal J}_{\\bm{\\beta}}^-|)=\n\\begin{cases}\n(p_2', 2p_2'),\\ \\ \\text{if } k=0,\\\\\n(p_2', 2p_2'+1),\\ \\ \\text{if } k=1,\\\\\n(p_2', 2p_2'+2),\\ \\ \\text{if } k=2.\n\\end{cases}\n\\end{align}\nNote that $p_2'\\geq 1$ whenever $k_2 = 0$.\nThus, a direct consequence of \\eqref{eq:Jbetapm} is\n\\begin{align}\\label{eq:Jbetapmneq}\n|{\\mathcal J}_{\\bm{\\beta}}^+|\\neq |{\\mathcal J}_{\\bm{\\beta}}^-|.\n\\end{align}\nOn the other hand, since ${\\mathcal J}_{\\bm{\\beta}}^0 = \\emptyset$, following the same argument as in the proof of Theorems~\\ref{thm:trimain}.1 yields $|{\\mathcal J}_{\\bm{\\beta}}^+|=|{\\mathcal J}_{\\bm{\\beta}}^-|$, which contradicts \\eqref{eq:Jbetapmneq}.\n\\end{proof}\n\n\\subsection{Proofs of Lemmas \\ref{lem:constructionsqn}, \\ref{lem:sqnJ0empty}, and \\ref{lem:arccos1\/3}}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:constructionsqn}]\nLet $x$ and $y$ solve \\eqref{eq:xyCondABsqn} with $E \\neq -1$. The second condition therein yields\n\\[\n\\sin(x) + 2\\sin(x)\\cos(y) = 0,\n\\]\nleading to two possibilities: $\\sin(x) = 0$ or $\\cos(y) = -1\/2$. If $\\sin(x) = 0$, we get $x = 0$ or $x = \\pi$, which yields \\eqref{eq:solutionx=0sqn} and \\eqref{eq:solutionx=pisqn} upon plugging in to the first condition in \\eqref{eq:xyCondABsqn}. In the event that $\\cos(y) = -1\/2$, we arrive at\n\\begin{align*}\n\\cos(x) + \\cos(y) + \\cos(x-y) + \\cos(x+y)\n& =\n\\cos(x) + \\cos(y) + 2\\cos(x)\\cos(y) \\\\\n& =\n\\cos(x) - \\frac{1}{2} - \\cos(x) \\\\\n& =\n-\\frac{1}{2},\n\\end{align*}\nin contradiction with $E \\neq -1$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:sqnJ0empty}]\nSuppose $x$ and $y$ satisfy \\eqref{eq:sqnJ0syst}. From the proof of Lemma~\\ref{lem:constructionsqn}, the second condition of \\eqref{eq:sqnJ0syst} implies $\\sin(x) = 0$ or $\\cos(y) = -1\/2$. Thus, $x = 0$, $x = \\pi$, $y=2\\pi\/3$, or $y = 4\\pi\/3$. \nWhen $\\sin(x) = 0$, the third condition of \\eqref{eq:sqnJ0syst} forces $\\sin(y) = 0$. The four points so obtained yield $E = 8$ when $(x,y) = (0,0)$, $E=-4$ when $(x,y) = (0,\\pi),(\\pi,0)$ and $E = 0$ when $(x,y) = (\\pi,\\pi)$. \nAlternatively, when $\\cos(y) = -1\/2$, the third condition of \\eqref{eq:sqnJ0syst} yields $\\cos(x) = -1\/2$, which impies $x = 2\\pi\/3$ or $x = 4\\pi\/3$. As in the Proof of Lemma~\\ref{lem:constructionsqn}, the four points corresponding to\n\\[\nx,y \\in \\set{\\frac{2\\pi}{3}, \\frac{4\\pi}{3}}\n\\]\nall yield $E = -1$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:arccos1\/3}]\nSuppose \n\\begin{align}\\label{eq1:cos1\/3}\n\\cos\\left(\\frac{2\\pi m}{n}\\right)=-\\frac{1}{3},\n\\end{align}\nfor $m\/n\\in {\\mathbb Q}$.\nLet $T_n(\\cdot)$ denote the $n$-th degree Cheybeshev polynomial so that\n\\begin{align}\\label{eq2:cos1\/3}\nT_n\\left(\\cos\\left(\\frac{2\\pi m}{n}\\right)\\right)=\\cos(2\\pi m)=1.\n\\end{align}\nIt is well-known that $T_n(x)=\\sum_{k=0}^n a_k x^k$, where $a_n=2^{n-1}$ and $a_k\\in {\\mathbb Z}$ for any $k$.\nHence \\eqref{eq1:cos1\/3} and \\eqref{eq2:cos1\/3} imply\n\\[2^{n-1}\\left(-\\frac{1}{3}\\right)^n+\\sum_{k=0}^{n-1} a_k \\left(-\\frac{1}{3}\\right)^{k} =1.\\]\nMultiplying by $(-3)^n$ on both sides of the equation, we obtain\n\\[2^{n-1}-3\\sum_{k=0}^{n-1} a_k (-3)^{n-k-1}=(-3)^n,\\]\nwhich implies $2^{n-1}$ is divisible by $3$.\nContradiction.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{\\boldmath Opening a gap at $-1$}\n\n\\begin{figure*}[b]\n\\begin{tikzpicture}[yscale=0.8, xscale=0.8]\n\\draw [-,line width = .1cm] (0,0) -- (12,0);\n\\draw [-,line width = .1cm] (0,0) -- (0,12);\n\\draw [-,line width=  .1cm] (0,3) -- (12,3);\n\\draw [-,line width=  .1cm] (3,0) -- (3,12);\n\\draw [-,line width=  .1cm] (0,6) -- (12,6);\n\\draw [-,line width=  .1cm] (6,0) -- (6,12);\n\\draw [-,line width=  .1cm] (0,9) -- (12,9);\n\\draw [-,line width=  .1cm] (9,0) -- (9,12);\n\\draw [-,line width=  .1cm] (0,12) -- (12,12);\n\\draw [-,line width=  .1cm] (12,0) -- (12,12);\n\\draw [-,line width=  .1cm] (0,0) -- (12,12);\n\\draw [-,line width=  .1cm] (3,0) -- (12,9);\n\\draw [-,line width=  .1cm] (6,0) -- (12,6);\n\\draw [-,line width=  .1cm] (9,0) -- (12,3);\n\\draw [-,line width=  .1cm] (0,3) -- (9,12);\n\\draw [-,line width=  .1cm] (0,6) -- (6,12);\n\\draw [-,line width=  .1cm] (0,9) -- (3,12);\n\\draw [-,line width=  .1cm] (3,0) -- (0,3);\n\\draw [-,line width=  .1cm] (6,0) -- (0,6);\n\\draw [-,line width=  .1cm] (9,0) -- (0,9);\n\\draw [-,line width=  .1cm] (12,0) -- (0,12);\n\\draw [-,line width=  .1cm] (12,3) -- (3,12);\n\\draw [-,line width=  .1cm] (12,6) -- (6,12);\n\\draw [-,line width=  .1cm] (12,9) -- (9,12);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](3,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](9,0) circle (.2);\n\\filldraw[color=black, fill=black](0,3) circle (.2);\n\\filldraw[color=red, fill=red](3,3) circle (.2);\n\\filldraw[color=red, fill=red](6,3) circle (.2);\n\\filldraw[color=red, fill=red](9,3) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=red, fill=red](3,6) circle (.2);\n\\filldraw[color=red, fill=red](6,6) circle (.2);\n\\filldraw[color=red, fill=red](9,6) circle (.2);\n\\filldraw[color=black, fill=black](0,9) circle (.2);\n\\filldraw[color=red, fill=red](3,9) circle (.2);\n\\filldraw[color=red, fill=red](6,9) circle (.2);\n\\filldraw[color=red, fill=red](9,9) circle (.2);\n\\filldraw[color=black, fill=black](12,0) circle (.2);\n\\filldraw[color=black, fill=black](12,3) circle (.2);\n\\filldraw[color=black, fill=black](12,6) circle (.2);\n\\filldraw[color=black, fill=black](12,9) circle (.2);\n\\filldraw[color=black, fill=black](12,12) circle (.2);\n\\filldraw[color=black, fill=black](0,12) circle (.2);\n\\filldraw[color=black, fill=black](3,12) circle (.2);\n\\filldraw[color=black, fill=black](6,12) circle (.2);\n\\filldraw[color=black, fill=black](9,12) circle (.2);\n\\draw [-,line width=  .1cm,color=red] (3,3) -- (3,9);\n\\draw [-,line width=  .1cm,color=red] (6,3) -- (6,9);\n\\draw [-,line width=  .1cm,color=red] (9,3) -- (9,9);\n\\draw [-,line width=  .1cm,color=red] (3,3) -- (9,3);\n\\draw [-,line width=  .1cm,color=red] (3,6) -- (9,6);\n\\draw [-,line width=  .1cm,color=red] (3,9) -- (9,9);\n\\draw [-,line width=  .1cm,color=red] (3,6) -- (6,9);\n\\draw [-,line width=  .1cm,color=red] (3,3) -- (9,9);\n\\draw [-,line width=  .1cm,color=red] (6,3) -- (9,6);\n\\draw [-,line width=  .1cm,color=red] (3,6) -- (6,3);\n\\draw [-,line width=  .1cm,color=red] (3,9) -- (9,3);\n\\draw [-,line width=  .1cm,color=red] (6,9) -- (9,6);\n\\node at (3.9,3.3){\\hot{$q_1$}};\n\\node at (6.9,3.3){\\hot{$q_2$}};\n\\node at (9.9,3.3){\\hot{$q_3$}};\n\\node at (3.9,6.3){\\hot{$q_4$}};\n\\node at (6.9,6.3){\\hot{$q_5$}};\n\\node at (9.9,6.3){\\hot{$q_6$}};\n\\node at (3.9,9.3){\\hot{$q_7$}};\n\\node at (6.9,9.3){\\hot{$q_8$}};\n\\node at (9.9,9.3){\\hot{$q_9$}};\n\\end{tikzpicture}\n\\caption{A $3\\times 3$ potential on the square lattice that opens a gap at $E=-1$ with small positive positive coupling.}\\label{fig:sqn33period}\n\\end{figure*}\n\n\\begin{theorem} \\label{thm:nnnExGapLength}\nEnumerate the vertices of a $3\\times 3$ fundamental cell of the square lattice as in Figure~\\ref{fig:sqn33period}, denote $r=\\sqrt{4-\\sqrt{15}}$, define a $(3,3)$-periodic potential $Q$ on ${\\mathbb Z}^2$ via\n\\[\n(q_1,\\ldots,q_9)\n=\n\\Big(-r-\\frac{1}{r}+2,\\ -r,\\ -r+\\frac{1}{r}-2,\\ -\\frac{1}{r},\\ 0,\\ +\\frac{1}{r},\\ r-\\frac{1}{r}-2,\\ r,\\ r+\\frac{1}{r}+ 2\\Big),\n\\]\nand denote $H_\\lambda = \\Delta_{\\mathrm{sqn}} + \\lambda Q$. Then, for all $ \\lambda > 0$ sufficiently small, $\\sigma(H_\\lambda)$ consists of two connected components. Moreover, if $\\mathfrak{g}_\\lambda$ denotes the gap that opens at energy $-1$, one has\n\\[\n\\left(-1 - \\frac{\\lambda}{10}, -1 + \\frac{\\lambda}{10} \\right)\n\\subseteq\n\\mathfrak{g}_\\lambda\n\\subseteq\n\\left(-1 - \\frac{\\lambda}{4},-1+\\frac{\\lambda}{4}\\right).\n\\]\nIn particular, the gap opens linearly.\n\\end{theorem}\n\nLet us observe that the proof below can be refined a bit to yield sharper constants than $1\/10$ and $1\/4$.\n\n\\begin{proof}\nFor $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb T}^2$, let $H_\\lambda(\\bm{\\theta})$ denote the Floquet matrix corresponding to $H_\\lambda$. Ordering the vertices of the fundamental domain as in Figure~\\ref{fig:sqn33period}, we obtain:\n\\[\nH_\\lambda(\\bm{\\theta})\n=\n\\begin{bmatrix}\n\\lambda q_1 & 1 & e^{-i\\theta_1} & 1 & 1 & e^{-i\\theta_1} & e^{-i \\theta_2} & e^{-i\\theta_2} &e^{-i(\\theta_1+\\theta_2)} \\\\\n1 & \\lambda q_2 & 1 & 1 & 1 & 1 & e^{-i \\theta_2}& e^{-i\\theta_2} & e^{-i\\theta_2} \\\\\ne^{i \\theta_1} & 1 & \\lambda q_3& e^{i\\theta_1} & 1 & 1 & e^{i(\\theta_1-\\theta_2)} & e^{-i\\theta_2}  & e^{-i\\theta_2}\\\\\n1 & 1 & e^{-i\\theta_1} & \\lambda q_4 & 1 & e^{-i\\theta_1} & 1 & 1 &e^{-i\\theta_1} \\\\\n1 & 1 & 1 & 1 & \\lambda q_5 & 1 & 1 & 1 &1 \\\\\ne^{i\\theta_1} & 1 & 1 & e^{i\\theta_1} & 1 &  \\lambda q_6 & e^{i\\theta_1} & 1 & 1  \\\\\ne^{i\\theta_2} & e^{i\\theta_2} & e^{-i(\\theta_1-\\theta_2)} & 1 & 1 &e^{-i\\theta_1} & \\lambda q_7& 1 & e^{-i\\theta_1} \\\\\ne^{i\\theta_2} & e^{i\\theta_2} & e^{i\\theta_2} & 1 & 1 & 1 & 1 & \\lambda q_8 & 1 \\\\\ne^{i(\\theta_1+\\theta_2)} & e^{i\\theta_2} & e^{i\\theta_2} & e^{i\\theta_1} &1 &1 & e^{i\\theta_1} & 1 &\\lambda q_9  \n\\end{bmatrix}.\n\\]\n\\begin{comment}\nStraightforward (Mathematica!) calculations reveal\n\\begin{align*}\n\\det(H_{\\lambda}(\\bm{\\theta}) + {\\mathbb I})\n& =\nQ_0(\\bm{\\theta})+\\sum_{k=1}^9 Q_k(\\bm{\\theta}) \\lambda^k.\n\\end{align*}\nwhere\n\\begin{align*}\nQ_0(\\bm{\\theta})& =\n4096 \\sin^6\\left(\\frac{\\theta_1}{2}\\right) \\sin^6\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_1(\\bm{\\theta})& =0\\\\\nQ_2(\\bm{\\theta})&=10240 \\sin^4\\left(\\frac{\\theta_1}{2}\\right) \\sin^4\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_3(\\bm{\\theta})&=1024 \\sin^4\\left(\\frac{\\theta_1}{2}\\right) \\sin^4\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_4(\\bm{\\theta})&=5824 \\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_5(\\bm{\\theta})&=1024 \\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_6(\\bm{\\theta})&=176 - 80 \\cos(\\theta_1)-80 \\cos(\\theta_2) - 16 \\cos(\\theta_1) \\cos(\\theta_2) - 8 \\sin(\\theta_1) \\sin(\\theta_2)\\\\\nQ_8(\\bm{\\theta})&=12\\\\\nQ_9(\\bm{\\theta})&=0.\n\\end{align*}\nNext we will show $Q_6\\geq 0$, which will lead to $\\det(H_{\\lambda}(\\bm{\\theta}) + {\\mathbb I})\\geq 12\\lambda^8>0$.\n\nTo this end, we compute\n\\begin{align*}\n\\nabla Q_6(\\bm{\\theta})=(&80\\sin(\\theta_1)-8\\cos(\\theta_1) \\sin(\\theta_2)+16\\cos(\\theta_2)\\sin(\\theta_1),\\\\\n &\\qquad 80 \\sin(\\theta_2) -8\\cos(\\theta_2) \\sin(\\theta_1)+16 \\cos(\\theta_1) \\sin(\\theta_2) ).\n\\end{align*}\nSetting $\\nabla Q_6(\\bm{\\theta})=(0,0)$, we arrive tat\n\\begin{align}\n10\\sin(\\theta_1)-\\cos(\\theta_1)\\sin(\\theta_2)+2\\sin(\\theta_1)\\cos(\\theta_2)=0,\\label{eq1}\\\\\n10\\sin(\\theta_2)-\\cos(\\theta_2)\\sin(\\theta_1)+2\\sin(\\theta_2)\\cos(\\theta_1)=0.\\label{eq2}\n\\end{align}\nAdding \\eqref{eq1} and \\eqref{eq2}, we obtain\n\\begin{align}\\label{eq3}\n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)\\left(10\\cos\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)+\\cos\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)\\right)=0.\n\\end{align}\nSubtracting \\eqref{eq2} from \\eqref{eq1}, we obtain\n\\begin{align}\\label{eq4}\n\\sin\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)\\left(10\\cos\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)+3\\cos\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)\\right)=0.\n\\end{align}\nSolving \\eqref{eq3} and \\eqref{eq4} yields\n\\[\n\\cos\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)=\\cos\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)=0,\n\\]\nor \n\\[\n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)=\\sin\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)=0.\n\\]\nHence all the four solutions are $\\bm{\\theta}=(0,0), (0, \\pi), (\\pi, 0), (\\pi, \\pi)$.\n\nEvaluating $Q_6(\\bm{\\theta})$ at these points yields\n\\[Q_6(0,0)=0,\\ \\ Q_6(0,\\pi)=192,\\ \\ Q_6(\\pi,0)=192,\\ \\ Q_6(\\pi,\\pi)=320.\\]\nIn conclusion, we have shown $Q_6(\\bm{\\theta})\\geq 0$ for all $\\bm{\\theta}$.\nHence $\\det(H_{\\lambda,\\theta_1,\\theta_2} + {\\mathbb I})\\geq 12\\lambda^8>0$, thus $E=-1$ is not in the spectrum.\n\\end{comment}\nFor $s\\in (-1,1)$, let us consider \n\\[\n\\det(H_{\\lambda}(\\bm{\\theta}) + (1+ s\\lambda) {\\mathbb I})\n=\n\\sum_{k=0}^9 X_k(\\bm{\\theta}, s) \\lambda^k.\n\\]\nOur goal is to show $\\det(H_{\\lambda}(\\bm{\\theta}) + (1+ s\\lambda) {\\mathbb I})$ never vanishes for sufficiently small $\\lambda>0$ and for $|s| < 0.1$.\nDirect computations yield\n\\begin{align*}\nX_0(\\bm{\\theta}, s)&=4096 \\sin^6\\left(\\frac{\\theta_1}{2}\\right) \\sin^6\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_1(\\bm{\\theta}, s)&=0\\\\\nX_2(\\bm{\\theta}, s)&=Y_2(s) \\sin^4\\left(\\frac{\\theta_1}{2}\\right) \\sin^4\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_3(\\bm{\\theta}, s)&=Y_3(s) \\sin^4\\left(\\frac{\\theta_1}{2}\\right) \\sin^4\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_4(\\bm{\\theta}, s)&=Y_4(s) \\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_5(\\bm{\\theta}, s)&=Y_5(s) \\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_6(\\bm{\\theta}, s)&=Y_{6,1}(s)+Y_{6,2}(s)\\cos(\\theta_1)+Y_{6,3}(s)\\cos(\\theta_2)\\\\ & \\qquad +Y_{6,4}(s)\\cos(\\theta_1)\\cos(\\theta_2)+Y_{6,5}(s) \\sin(\\theta_1)\\sin(\\theta_2)\\\\\nX_7(\\bm{\\theta}, s)&=0\\\\\nX_8(\\bm{\\theta}, s)&=Y_8(s)\\\\\nX_9(\\bm{\\theta}, s)&=Y_9(s),\n\\end{align*}\nin which\n\\begin{align*}\nY_2(s)&=512 (20-9 s^2)\\\\\nY_3(s)&=256 (4 - 20 s + 3 s^3)\\\\\nY_4(s)&=16(364 + 144 s - 504 s^2 + 81 s^4)\\\\\nY_5(s)&=16 (64 - 196 s - 48 s^2 + 104 s^3 - 9 s^5)\\\\\nY_{6,1}(s)&=176 + 704 s - 3132 s^2 - 496 s^3 + 1376 s^4 - 96 s^6\\\\\nY_{6,2}(s)&=-80 + (96 \\sqrt{15}-320) s + (1380 +144 \\sqrt{15}) s^2 + 208 s^3 - (584+54 \\sqrt{15}) s^4 + 42 s^6\\\\\nY_{6,3}(s)&=-80 - (320 +96 \\sqrt{15}) s + (1380 - 144 \\sqrt{15}) s^2 + 208 s^3 - (584- 54 \\sqrt{15}) s^4 + 42 s^6\\\\\nY_{6,4}(s)&=-16 - 64 s + 372 s^2 + 80 s^3 - 208 s^4 + 12 s^6\\\\\nY_{6,5}(s)&=8(2s-1)^3\\\\\nY_8(s)&=12 + 32 s - 360 s^2 - 512 s^3 + 1025 s^4 + 96 s^5 - 224 s^6 + 9 s^8\\\\\nY_9(s)&=12 s + 16 s^2 - 120 s^3 - 128 s^4 + 205 s^5 + 16 s^6 - 32 s^7 + s^9.\\\\\n\\end{align*}\nOne simple observation is that \n\\begin{align}\\label{eq:sumY6}\nY_{6,1}(s)+Y_{6,2}(s)+Y_{6,3}(s)+Y_{6,4}(s)=0.\n\\end{align}\nIt is easy to see that for $|s|<0.1$, \n\\[Y_2(s), Y_3(s),Y_5(s)>0.\\]\nIt is easy to compute that\n\\[\nY'_9(s)=12 + 32 s - 360 s^2 - 512 s^3 + 1025 s^4 + 96 s^5 - 224 s^6 + 9 s^8\n=\nY_8(s).\n\\]\nThus, \n\\begin{equation} \\label{eq:Y9prime(s)}\nY_9'(s)>12 - 32 \\times 0.1 - 360 \\times (0.1)^2  - 512 \\times (0.1)^3 - 96 \\times (0.1)^5 - 224\\times (0.1)^6{>4.5}>0\\end{equation}\nfor $|s|<0.1$, which implies\n\\begin{align}\\label{eq:Y9s}\nY_9(s)\\geq Y_9(-0.1)>-1\n\\end{align}\nfor all $|s| < 0.1$. Carefully estimating $Y_4(s)$ and $Y_8(s)$ will help us bound the $\\lambda^6$ order term from below using the AM-GM inequality.\n\\begin{align}\\label{eq:Y4Y8s}\nY_4(s)&\\geq 16(364-144\\times 0.1-504\\times (0.1)^2 {-81 \\times (0.1)^4})>5500,\\\\\nY_8(s)&\\geq 12  - 32 \\times 0.1 - 360 \\times (0.1)^2 - 512 \\times (0.1)^3 - 96 \\times (0.1)^5 - 224 \\times (0.1)^6>4.5. \\notag\n\\end{align}\n{In fact, since $Y_8 = Y_9'$, the second inequality already follows from \\eqref{eq:Y9prime(s)}.}\nFor the $Y_{6,j}$ terms, we have\n\\begin{equation}\\label{eq:Y6s}\n\\begin{aligned}\nY_{6,1}(s)&\\geq 176 - 704 \\times 0.1 - 3132 \\times (0.1)^2 - 496 \\times (0.1)^3 - 96 \\times (0.1)^6>0,\\\\\n  Y_{6,2}(s)&\\leq -80 + (96 \\sqrt{15}-320) \\times 0.1 + (1380 +144 \\sqrt{15}) \\times (0.1)^2 \\\\   & \\qquad\\qquad + 208 \\times (0.1)^3  + 42\\times (0.1)^6<0 \\\\\n  Y_{6,3}(s)&\\leq -80 + (320 + 96 \\sqrt{15}) \\times 0.1 + (1380 - 144 \\sqrt{15}) \\times (0.1)^2 \\\\   & \\qquad\\qquad+ 208 \\times (0.1)^3  + 42 \\times (0.1)^6<0, \\\\\nY_{6,4}(s)&\\leq -16 + 64 \\times 0.1 + 372 \\times (0.1)^2 + 80 \\times (0.1)^3  + 12 \\times (0.1)^6<0, \\\\\n-14 & \\leq Y_{6,5}(s)< 0.\n\\end{aligned}\n\\end{equation}\nUsing \\eqref{eq:sumY6} and \\eqref{eq:Y6s}, we obtain\n\\begin{align} \\label{eq:X6LB}\n\\notag\nX_6(\\bm{\\theta}) & \\geq Y_{6,1}(s)+Y_{6,2}(s)+Y_{6,3}(s)+Y_{6,4}(s) + Y_{6,5}(s)\\sin(\\theta_1)\\sin(\\theta_2) \\\\\n\\notag\n& =\nY_{6,5}(s)\\sin(\\theta_1)\\sin(\\theta_2) \\\\\n& \\geq\n-14 |\\sin(\\theta_1)\\sin(\\theta_2)|.\n\\end{align}\nIn particular, the first line uses $Y_{6,2}, Y_{6,3}, Y_{6,4} < 0$, the second line uses \\eqref{eq:sumY6}, and the final line uses $-14 \\leq Y_{6,5} < 0$.\n\nNow we combine our estimates together. \nNote that \n\\begin{align}\\label{eq:sumX0235}\nX_0(\\bm{\\theta},s)+X_2(\\bm{\\theta},s)\\lambda^2+X_3(\\bm{\\theta},s)\\lambda^3+X_5(\\bm{\\theta},s)\\lambda^5\\geq 0.\n\\end{align}\nUsing $a^2+b^2\\geq 2|ab|$, we obtain the following from \\eqref{eq:Y4Y8s}\n\\[\nX_4(\\bm{\\theta},s)\\lambda^4+\\frac{1}{2}X_8(\\bm{\\theta},s)\\lambda^8 \\geq 2\\sqrt{2.25\\times 5500} \\left|\\sin\\left(\\frac{\\theta_1}{2}\\right) \\sin\\left(\\frac{\\theta_2}{2}\\right)\\right| \\lambda^6.\n\\]\nUsing $2|\\sin(x\/2)|\\geq 2|\\sin(x\/2)\\cos(x\/2)|=|\\sin(x)|$, we obtain from above that\n\\[X_4(\\bm{\\theta},s) {\\lambda^4} + \\frac{1}{2}X_8(\\bm{\\theta},s) {\\lambda^8} \n\\geq \n55 |\\sin(\\theta_1)\\sin(\\theta_2)|{\\lambda^6}.\\]\nCombining this with \\eqref{eq:X6LB}, we have\n\\begin{equation} \\label{eq:sumX486}\nX_4(\\bm{\\theta},s)\\lambda^4+\\frac{1}{2}X_8(\\bm{\\theta},s)\\lambda^8+X_6(\\bm{\\theta},s)\\lambda^6 \n\\geq\n41 |\\sin(\\theta_1)\\sin(\\theta_2)|\\lambda^6\n\\geq 0.\n\\end{equation}\n\nFinally using \\eqref{eq:Y9s} and \\eqref{eq:Y4Y8s}, we have\n\\begin{align}\\label{eq:sumY89}\n\\frac{1}{2}X_8(\\bm{\\theta},s)\\lambda^8+X_9(\\bm{\\theta},s)\\lambda^9=\\frac{1}{2}Y_8(s)\\lambda^8+Y_9(s)\\lambda^9\\geq 2.25\\lambda^8-\\lambda^9>0.25 \\lambda^8,\n\\end{align}\nprovided that $\\lambda<2$.\nCombining \\eqref{eq:sumX0235}-\\eqref{eq:sumY89}, we have\n\\[\\det(H_{\\lambda}(\\bm{\\theta}) + (1+ s\\lambda) {\\mathbb I}) \\geq 0.25\\lambda^8>0,\\]\nfor any $\\bm{\\theta}\\in {\\mathbb T}^2$ and $|s|<0.1$. \nThis proves the lower bound on the gap.\n\n{For the upper bound, observe that $X_j\\big((\\pi,0),s\\big) = 0$ for all $s$ and for every $0 \\le j \\le 5$ and\n\\[\nX_6\\big((\\pi,0),\\pm 1\/4\\big)\n<-85.\n\\]\nThus, for small $\\lambda > 0$, \n\\[\n\\det\\big(H_\\lambda(\\pi, 0) + (1\\pm\\lambda\/4){\\mathbb I} \\big) < -85\\lambda^6 + O(\\lambda^8)<0.\n\\]\nIt is also clear that $X_0((0,0),s)=4096$,\nwhich implies \n\\[\\det \\big(H_\\lambda(0, 0) +(1\\pm\\lambda\/4){\\mathbb I} \\big) =4096+O(\\lambda)>0.\\]\nThus we conclude that \n\\[1\\pm \\frac{\\lambda}{4}\\in \\sigma(H_\\lambda),\\]\nwhich concludes the proof of the upper bound on the length of the gap.}\n\\end{proof}\n\n\\section*{Acknowledgement}\nWe would like to thank Svetlana Jitomirskaya for comments on an earlier version of the manuscript, and Tom Spencer for useful discussions.\nR.H. would like to thank IAS, Princeton, for its hospitality during the 2017-18 academic year,\nand Virginia Tech for its hospitality during which part of the work was done.\nR.H. is supported in part by the National Science Foundation under Grant No. DMS-1638352. \nJ.F.\\ was supported in part by an AMS-Simons Travel Grant 2016--2018.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNowadays, the axion mechanism represents our best solution to the long standing strong CP puzzle, that is, the non-observation of CP violation in the strong interactions that should have manifested itself as an electric dipole moment for the neutron~\\cite{Abel:2020pzs}.\n\nThe axion mechanism relies on the spontaneous breaking of a new symmetry, the PQ symmetry~\\cite{PQ}, and on the subsequent realignment of the associated Goldstone boson, the axion~\\cite{Weinberg:1977ma,Wilczek:1977pj}, by strong interaction effects that kills off any CP violation in the QCD Lagrangian. This solution is thus tailored to the problem it is intended to solve and, as such, may appear a bit ad-hoc. In addition, unsuccessful experimental searches for the axion have ruled out its simplest incarnation, leaving us with essentially two classes of scenarios in which the axion is extremely light (well below the eV scale) and very weakly coupled to normal matter: the KSVZ~\\cite{KSVZ} framework in which new very heavy colored fermions are introduced, and the DFSZ~\\cite{DFSZ} scenario in which at least two Higgs doublets are required. Though the strong CP puzzle is extremely serious, additional motivations appear desirable to motivate such departures from the Standard Model (SM) matter content. To that avail, knowing that the axion could also make up for the observed dark matter (DM) offers a strong incentive to pursue this route~\\cite{DMaxion}. \n\nYet, current axion models cannot explain why the DM relic density is so close to that of baryonic matter. Though this may be totally coincidental, it nevertheless suggests a link between DM and baryogenesis~\\cite{Kaplan:1991ah}, another prominent cosmological enigma. Actually, it suggests DM is not foreign to baryon $\\mathcal{B}$ or lepton $\\mathcal{L}$ number (see Ref.~\\cite{Alonso-Alvarez:2021oaj} and references therein for a recent analysis), or that DM is somehow related to $\\mathcal{B}$ being spontaneously broken~\\cite{Dulaney:2010dj}. In parallel, there have been many attempts at involving axions in the baryogenesis mechanism, see e.g. Refs.~\\cite{Craig:2010au,Servant:2014bla,Jeong:2018ucz,Co:2019wyp,Krauss:2022usd,Domcke:2020kcp}, though in general still relying on the SM anomalous $\\mathcal{B}+\\mathcal{L}$ effects. \n\nOur goal here is to go one step further and entangle the PQ symmetry with $\\mathcal{B}$ and $\\mathcal{L}$ from the start. As a matter of principle, accidental symmetries are not particularly attractive, but while we can live with the PQ symmetry, assuming some dynamics hide behind it, $\\mathcal{B}$ and $\\mathcal{L}$ cannot be viable since, as said before, the electroweak non-perturbative dynamics break them, and baryogenesis asks for their violation. By unifying the PQ symmetry with $\\mathcal{B}$ and $\\mathcal{L}$, all three are broken spontaneously, but a single Goldstone field remains, the axion (for some recent works along this line, see Refs.~\\cite{Reig:2018yfd,Ohata:2021rkh}). In this way, the complex scalar field whose pseudoscalar component is the axion becomes charged under $\\mathcal{B}$ and $\\mathcal{L}$ and, at the high scale, protects the model from additional $\\mathcal{B}$ and\/or $\\mathcal{L}$ violation. At the same time, though the axion has no charge, it inherits a $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating phenomenology. Whether this is sufficient to relate the DM and baryonic relic densities remains to be seen, and is beyond the scope of the present paper, but we think these constructions may direct us in the right direction.\n\nIn this paper, we will use scalar and vector leptoquarks and diquarks to entangle the PQ, $\\mathcal{B}$, and $\\mathcal{L}$ symmetries. Such states are well motivated in various theoretical settings (see Ref.~\\cite{Dorsner:2016wpm} for a review) and, furthermore, supported by a number of anomalies like the $W$ boson mass, B decays~\\cite{LQreview} or $(g-2)_{\\mu}$~\\cite{Dorsner:2019itg}, or even combinations of them~\\cite{Athron:2022qpo,Bhaskar:2022vgk}. Our goal is to systematically analyze the $\\mathcal{B}$ and\/or $\\mathcal{L}$ symmetry breaking patterns that can arise combining the DFSZ and KSVZ scenarios with leptoquarks and diquarks and, in each case, to analyze the impact on the axion phenomenology.\n\nThe paper is organized as follows. In section~\\ref{Sec2a}, we briefly introduce the KSVZ and DFSZ axion models and, in Section~\\ref{Sec2b}, discuss in some details the ambiguities arising from the $\\mathcal{B}$ and $\\mathcal{L}$ fermionic currents~\\cite{Quevillon:2020hmx,Quevillon:2020aij}. Then in section~\\ref{Sec2c}, we set up the leptoquark and diquark sector, describing all the possible $\\mathcal{B}$ and $\\mathcal{L}$ explicit breaking patterns achievable with these states. This forms the basis for combining the axion and leptoquark\/diquark sectors in Sec.~\\ref{SecAxLQDQ}. We analyze first the KSVZ setting in Sec.~\\ref{SecKSVZ} and describe the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,\\pm1), (2,0), (1,\\pm3)$ spontaneous breaking patterns, further adding to them a spontaneously generated $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ seesaw mechanism for neutrino masses. These scenarios are then trivially adapted to the DFSZ setting in Sec.~\\ref{SecDFSZ}. In the final Sec.~\\ref{SecSpont}, we show how to force $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ effects to involve one or more axion fields. The phenomenology is then quite different, and we briefly describe some possible consequences for the neutron lifetime anomaly or neutron-antineutron oscillation experiments. Finally, our results are summarized in Sec.~\\ref{Ccl}.\n\n\\section{Axion and leptoquark models}\n\nIn this section, the KSVZ~\\cite{KSVZ} and DFSZ~\\cite{DFSZ} axion models are introduced, and their connection to baryon and lepton numbers, $\\mathcal{B}$ and $\\mathcal{L}$, are detailed. Then, we introduce separately the leptoquarks and diquarks that can be coupled to SM fermions, and discuss how their couplings drive specific $\\mathcal{B}$ and $\\mathcal{L}$ violating patterns. This sets the stage for the next section, where both axion models and leptoquarks\/diquarks will be put together.\n\n\\subsection{Introducing the KSVZ and DFSZ models\\label{Sec2a}}\n\nIn both the KSVZ and DFSZ constructions, the axion emerges as the pseudoscalar component of a complex scalar field. This state is neutral under all the SM gauge interactions, $\\phi=(\\mathbf{1},\\mathbf{1},0)$ under $SU(3)_{C}\\otimes SU(2)_{L}\\otimes U(1)_{Y}$, but its kinetic term is invariant under the rephasing $\\phi\\rightarrow e^{i\\alpha}\\phi$. This invariance is promoted to a\nspontaneously broken symmetry $U(1)_{\\phi}$ by postulating a rephasing invariant scalar potential with the usual Mexican hat shape, $V(\\phi^{\\dagger}\\phi)=\\mu^{2}\\phi^{\\dagger}\\phi+\\lambda(\\phi^{\\dagger}\\phi)^{4}$, $\\mu^{2}<0$ and $\\lambda>0$. In that case, the components of $\\phi$ can be written\n\\begin{equation}\n\\phi=\\frac{1}{\\sqrt{2}}(v_{\\phi}+\\rho)\\exp(i\\eta_{\\phi}\/v_{\\phi})\\ ,\n\\label{PhiPolar}\n\\end{equation}\nwith $\\eta_{\\phi}$ the associated Goldstone boson and $v_{\\phi}^{2}=-\\mu^{2}\/\\lambda$ the vacuum expectation value (VEV). As the breaking scale $v_{\\phi}$ naturally tunes all the $\\eta_{\\phi}$ couplings, it is assumed much higher than the electroweak scale to avoid exclusion bounds.\n\nTo solve the strong CP puzzle, $\\eta_{\\phi}$ must interact with SM particles~\\cite{Weinberg:1977ma,Wilczek:1977pj}, in particular with gluons via a $\\eta_{\\phi}G^{\\alpha,\\mu\\nu}\\tilde{G}_{\\mu\\nu}^{a}$ coupling~\\cite{PQ}. What differentiates the KSVZ and DFSZ models is how these couplings are introduced. The former~\\cite{KSVZ} adds a vector-like colored fermion $\\Psi_{L,R}\\sim(\\mathbf{R},\\mathbf{T},Y)$ for some complex representation $\\mathbf{R}$ of $SU(3)_{C}$, but otherwise arbitrary weak representation $\\mathbf{T}$ and hypercharge $Y$, and postulates the Lagrangian (the rest of the SM couplings are understood)\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ}}  &  =\\partial_{\\mu}\\phi^{\\dagger}\\partial^{\\mu\n}\\phi-V(\\phi)+\\bar{\\Psi}_{L,R}(i\\slashed D)\\Psi_{L,R}+(y\\phi\\bar{\\Psi}_{L}\\Psi_{R}+h.c.)\\nonumber\\\\\n&  -\\bar{u}_{R}\\mathbf{Y}_{u}q_{L}H-\\bar{d}_{R}\\mathbf{Y}_{d}q_{L}H^{\\dagger\n}-\\bar{e}_{R}\\mathbf{Y}_{e}\\ell_{L}H^{\\dagger}-\\bar{\\nu}_{R}\\mathbf{Y}_{\\nu\n}\\ell_{L}H+h.c.\\ . \\label{KSVZ0}%\n\\end{align}\nThe covariant derivative acting on $\\Psi_{L,R}$ is as appropriate to its chosen gauge quantum numbers. What characterizes this model is first that the Goldstone boson of the PQ symmetry does not mix with that of the $SU(2)_{L}\\otimes U(1)_{Y}$ breaking (the phase of the Higgs doublet $H$). Thus, the axion is simply $a^{0}=\\eta_{\\phi}$, and it has no direct coupling to any of the SM particles. It only couples to $\\Psi_{L}$ and $\\Psi_{R}$, which necessarily have different charges under $U(1)_{\\phi}$. Then, axion to SM gauge boson couplings first arise at one-loop, via anomalous $\\Psi_{L,R}$ triangle loops, while those to SM fermions require a further gauge boson loop. Since $\\Psi_{L,R}$ can be massive in the electroweak unbroken phase, its loops do not break $SU(2)_{L}\\otimes U(1)_{Y}$ and the couplings to gauge bosons have the $SU(2)_{L}\\otimes U(1)_{Y}$ invariant form~\\cite{Georgi:1986df}\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ}}^{eff}=-\\frac{1}{16\\pi^{2}v_{\\phi}}a^{0}(g_{s}%\n^{2}d_{L}C_{C}G_{\\mu\\nu}^{a}\\tilde{G}^{a,\\mu\\nu}+g^{2}d_{C}C_{L}W_{\\mu\\nu}%\n^{i}\\tilde{W}^{i,\\mu\\nu}+g^{\\prime2}d_{L}d_{C}C_{Y}B_{\\mu\\nu}\\tilde{B}^{\\mu\n\\nu})\\ , \\label{GaugeCouplAno}\n\\end{equation}\nwith the quadratic invariants and dimensions of the $\\mathbf{R}$ and $\\mathbf{T}$ representations denoted $C_{C,L}$ and $d_{C,L}$, and $C_{Y}=Y^{2}\/4$.\n\nThe DFSZ model~\\cite{DFSZ} does not introduce new fermions, but requires two Higgs doublets. The important couplings are%\n\\begin{align}\n\\mathcal{L}_{\\mathrm{DFSZ}}  &  =\\partial_{\\mu}\\phi^{\\dagger}\\partial^{\\mu\n}\\phi-V(\\phi^{\\dagger}\\phi)+\\phi^{2}H_{u}^{\\dagger}H_{d}+V(H_{u}^{\\dagger\n}H_{u},H_{d}^{\\dagger}H_{d})\\nonumber\\\\\n&  -\\bar{u}_{R}\\mathbf{Y}_{u}q_{L}H_{u}-\\bar{d}_{R}\\mathbf{Y}_{d}q_{L}%\nH_{d}^{\\dagger}-\\bar{e}_{R}\\mathbf{Y}_{e}\\ell_{L}H_{d}^{\\dagger}-\\bar{\\nu}%\n_{R}\\mathbf{Y}_{\\nu}\\ell_{L}H_{u}+h.c.\\ . \\label{DFSZ0}%\n\\end{align}\nThe potentials and Yukawa couplings are invariant under three independent $U(1)$s, corresponding to the rephasing of $\\phi$, $H_{u}$, and $H_{d}$. A combination of these is explicitly removed by the mixing term $\\phi^{2}H_{u}^{\\dagger}H_{d}$ (we could equally take $\\phi H_{u}^{\\dagger}H_{d}$, but at the cost of introducing a new mass scale), so that only two Goldstone bosons arise. Explicitly, if we adopt for $H_{u,d}$ a polar representation similar as in Eq.~(\\ref{PhiPolar}), with their pseudoscalar components denoted as $\\eta_{u,d}$ and their VEVs as $v_{u,d}$, the $\\phi^{2}H_{u}^{\\dagger}H_{d}$ coupling translates as a mass term for the combination $\\pi^{0}\\sim2\\eta_{\\phi}\/v_{\\phi}-\\eta_{u}\/v_{u}+\\eta_{d}\/v_{d}$. One of the two remaining Goldstone bosons is eaten by the $Z$ boson. Since $H_{u,d}$ have the same hypercharge, the would-be Goldstone state $G^{0}$ must be $G^{0}\\sim v_{u}\\eta_{u}+v_{d}\\eta_{d}$. The last remaining Goldstone mode, orthogonal to both $\\pi^{0}$ and $G^{0}$, stays massless and is the axion:\n\\begin{equation}\na^{0}\\sim\\eta_{\\phi}+\\frac{v_{EW}}{v_{\\phi}}\\sin2\\beta(\\cos\\beta\\eta_{u}\n-\\sin\\beta\\eta_{d})+\\mathcal{O}(v_{EW}^{2}\/v_{\\phi s}^{2})\\ , \\label{DFSZA0}%\n\\end{equation}\nwith $\\tan\\beta=v_{u}\/v_{d}$ and $v_{EW}^{2}=v_{u}^{2}+v_{d}^{2}\\approx (246\\,$GeV)$^{2}$. The net result of all this is that the axion components in $H_{u,d}$ are suppressed by $v_{u,d}\/v_{\\phi}$. The leading couplings of the axion to SM particles come from the Yukawa couplings, with\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ}}^{eff}=-i\\frac{v_{EW}}{v_{\\phi}}\\sin2\\beta\n\\sum_{f=u,d,e}\\frac{m_{f}}{v_{EW}}\\chi_{P}^{f}\\,a^{0}\\bar{\\psi}_{f}\\gamma\n_{5}\\psi_{f}\\ ,\\ \\ \\chi_{P}^{u}=\\frac{1}{\\tan\\beta}\\ ,\\ \\chi_{P}^{d}=\\chi\n_{P}^{e}=\\tan\\beta\\ . \\label{PseudoCoupl}%\n\\end{equation}\nTo reach this form, the mass terms are identified as $\\sin\\beta v_{EW}\\mathbf{Y}_{u}\\equiv\\sqrt{2}\\mathbf{m}_{u}$ and $\\cos\\beta v_{EW}\\mathbf{Y}_{d,e}\\equiv\\sqrt{2}\\mathbf{m}_{d,e}$ and the fermions are rotated to their mass basis. In the DFSZ setting, the axion couplings to gauge bosons only arise through SM fermion loops. As shown in Ref.~\\cite{Quevillon:2019zrd} (see also Refs.~\\cite{Bonnefoy:2020gyh,Quevillon:2021sfz}), starting from the pseudoscalar couplings in Eq.~(\\ref{PseudoCoupl}), the final couplings to gauge boson do not have the form shown in Eq.~(\\ref{GaugeCouplAno}), but instead explicitly break $SU(2)_{L}\\otimes U(1)_{Y}$ invariance. Naively, this is easily understood since SM fermions only acquire masses after the $SU(2)_{L}\\otimes U(1)_{Y}$ breaking.\n\n\\subsection{Introducing baryon and lepton numbers\\label{Sec2b}}\n\nIn the following, when introducing leptoquark states, baryon and lepton numbers $\\mathcal{B}$ and $\\mathcal{L}$ will play a central role. The purpose in this section is to gather a few important facts about the interplay of these accidental symmetries with the PQ symmetry. Additional information on this topic can be found in Ref.~\\cite{Quevillon:2020hmx}.\n\nBy definition, the $U(1)$ symmetry associated to the axion state is called the PQ symmetry. Given the scalar couplings described in the previous section, the PQ charges of all the scalar states is well-defined in the KSVZ and DFSZ models. Explicitly, we have in the KSVZ setting\n\\begin{equation}\n\\begin{tabular}[c]{ccc}\\hline\nKSVZ & $\\phi$ & $H$\\\\\\hline\n$U(1)_{\\phi}$ & $1$ & $0$\\\\\n$U(1)_{H}$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\ \\ \\ \\ \\ \\Longrightarrow%\n\\begin{tabular}[c]{ccc}\\hline\nKSVZ & $\\phi$ & $H$\\\\\\hline\n$U(1)_{PQ}$ & $1$ & $0$\\\\\n$U(1)_{Y}$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nand in the DFSZ, choosing the two independent $U(1)$ symmetries as those associated to Higgs doublet rephasings\\footnote{The PQ charges of $\\phi$, $H_{u}$ and $H_{d}$ are simply the coefficients of $\\eta_{\\phi,u,d}$ in Eq.~(\\ref{DFSZA0}), up to a choice of normalization.},\n\\begin{equation}\n\\begin{tabular}[c]{cccc}\\hline\nDFSZ & $\\phi$ & $H_{u}$ & $H_{d}$\\\\\\hline\n$U(1)_{Hu}$ & $1\/2$ & $1$ & $0$\\\\\n$U(1)_{Hd}$ & $-1\/2$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\ \\ \\ \\ \\ \\Longrightarrow\n\\begin{tabular}[c]{cccc}\\hline\nDFSZ & $\\phi$ & $H_{u}$ & $H_{d}$\\\\\\hline\n$U(1)_{PQ}$ & $(x+1\/x)\/2$ & $x$ & $-1\/x$\\\\\n$U(1)_{Y}$ & $0$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\ \\ \\label{DFSZScalars}\n\\end{equation}\nwith the conventional notation $\\tan\\beta\\equiv1\/x$. Note that the $U(1)_{Y}$ and $U(1)_{PQ}$ charges of the two Higgs doublets are not `orthogonal', reflecting the fact that the original $U(1)_{Hu}$ and $U(1)_{Hd}$ charges for the three states $(\\phi,H_{u},H_{d})$ were not. Also, it is important to keep in mind that though well-defined, these PQ charges are only defined in the electroweak broken phase, since they are function of $x\\equiv v_{d}\/v_{u}$.\n\nFor fermions, identifying the PQ charge is less trivial because the Yukawa couplings allow for two accidental symmetries, $\\mathcal{B}$ and $\\mathcal{L}$ (no particular structure is assumed for $\\mathbf{Y}_{u,d,e,\\nu}$, so individual flavors are not conserved a priori). Looking at the Lagrangian, the KSVZ model prescribes\n\\begin{equation}\n\\begin{tabular}[c]{ccccccccc}\\hline\nKSVZ & $\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ &\n$e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\alpha$ & $\\alpha-1$ & $\\beta$ & $\\beta$ & $\\beta$ & $\\gamma$ &\n$\\gamma$ & $\\gamma$\\\\\n$U(1)_{Y}$ & $Y$ & $Y$ & $1\/3$ & $4\/3$ & $-2\/3$ & $-1$ & $-2$ & $0$\\\\\\hline\n\\end{tabular}\n\\label{KSVZfermions}\n\\end{equation}\nwhere $\\alpha$, $\\beta$, and $\\gamma$ are arbitrary, and correspond to conserved $\\Psi$ number, baryon number, and lepton number, respectively. Similarly, for the DFSZ model,\n\\begin{equation}\n\\begin{tabular}[c]{ccccccc}\\hline\nDFSZ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\beta$ & $\\beta+x$ & $\\beta-1\/x$ & $\\gamma$ & $\\gamma-1\/x$ &\n$\\gamma+x$\\\\\n$U(1)_{Y}$ & $1\/3$ & $4\/3$ & $-2\/3$ & $-1$ & $-2$ & $0$\\\\\\hline\n\\end{tabular}\n\\label{DFSZfermions}\n\\end{equation}\nSince $\\beta$ and $\\gamma$ are aligned with baryon and lepton numbers, it is tempting to set $\\beta=\\gamma=0$. This is not acceptable. For the DFSZ scenario, all the SM fermions do couple to the axion, but these couplings are not $SU(2)_{L}\\otimes U(1)_{Y}$ invariant. Looking at Eq.~(\\ref{PseudoCoupl}), no value of $\\beta$ or $\\gamma$ make perfect sense since the PQ charge of the Dirac $u$ and $d$ states are different, so that of $q_{L}$ cannot be defined. The situation appears simpler in the KSVZ case, where it seems rather natural to set $\\beta=\\gamma=0$ since the SM fermions are not directly coupled to the scalar field $\\phi$. Yet, even that is not tenable.\n\nTo see this, let us set off a seesaw mechanism~\\cite{TypeI}. Given the quantum numbers of the $\\nu_{R}$ field, we can either allow for a Majorana mass term $M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, a coupling $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, or a coupling $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$. These three cases are mutually exclusive since they impose different PQ charges to $\\nu_{R}$. Let us consider the $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ case, which in effect identifies the PQ symmetry with lepton number symmetry, and the axion with the Majoron~\\cite{Langacker:1986rj,Shin:1987xc,Clarke:2015bea} (see also \\cite{Heeck:2019guh}). It imposes non-zero values for $\\gamma$~\\cite{Quevillon:2020hmx}\n\\begin{align}\n\\text{KSVZ}  &  :\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}\\rightarrow\\gamma=\\frac{1}{2}\\ ,\\\\\n\\text{DFSZ}  &  :\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}\\rightarrow\\gamma=\\frac{1-3x^{2}}{4x}\\ .\n\\end{align}\nIn both cases, the PQ current acquires a component aligned with the lepton number current, $J_{\\mathcal{L}}^{\\mu}=\\bar{\\ell}_{L}\\gamma^{\\mu}\\ell_{L}+\\bar{e}_{R}\\gamma^{\\mu}e_{R}+\\bar{\\nu}_{R}\\gamma^{\\mu}\\nu_{R}$. In other words, $\\ell_{L}$ and\/or $e_{R}$ do end up PQ charged also. Yet, in the KSVZ case, a look at the Lagrangian shows that neither are directly coupled to $\\phi$. Because of $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, the axion does end up coupled to right-handed neutrinos, with a $a^{0}\\rightarrow\\nu_{R}\\nu_{R}$ vertex, but no such $\\Delta\\mathcal{L}=2$ coupling exists with the other leptons since it is forbidden by hypercharge. Only at the cost of extra Higgs doublet insertions could a $a^{0}\\rightarrow\\nu_{L}\\nu_{L}$ exist, as arising from an effective PQ- and hypercharge-neutral operator $\\phi^{\\dagger}H\\ell_{L}H\\ell_{L}$ (or $\\phi^{\\dagger}H_{u}\\ell_{L}H_{u}\\ell_{L}$ in the DFSZ model), while obviously, any $\\Delta\\mathcal{L}=2$ coupling to charged lepton would require either extra gauge fields, or charged Higgs bosons.\n\nThe ambiguous nature of the PQ charges of fermions is not purely academic. In most phenomenological studies of the axion, the starting point is the effective Lagrangian that is obtained by reparametrizing fermion fields to make them PQ neutral (even if that is usually not explicitly stated):\n\\begin{equation}\n\\psi\\rightarrow\\exp(-iPQ(\\psi)a^{0}\/v_{\\phi})\\psi\\ , \\label{ReparamG}%\n\\end{equation}\nwhere $\\psi$ denotes generically the PQ-charged fermions. Since the underlying physics is PQ neutral, this looks innocuous. Yet, it modifies the Lagrangian of the model in two important ways. First, it removes the axion field from Yukawa interactions (both for the SM and heavy fermions, if present), and replace them by shift-symmetric derivative couplings of the axion to the\nfermionic PQ current, as adequate for a Goldstone boson\n\\begin{equation}\n\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}=\\frac{\\partial_{\\mu}a^{0}}{v_{\\phi}%\n}J_{PQ}^{\\mu}\\ ,\\ J_{PQ}^{\\mu}=\\sum_{\\psi}PQ(\\psi)\\bar{\\psi}\\gamma^{\\mu}\\psi\\ .\n\\end{equation}\nSecond, the PQ symmetry being anomalous, the fermion reparametrizations in Eq.~(\\ref{ReparamG}) change the fermionic measure. To account for this, one must introduce anomalous couplings to the gauge bosons,\n\\begin{equation}\n\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}=\\frac{a^{0}}{16\\pi^{2}v_{\\phi}}\\left(\ng_{s}^{2}\\mathcal{N}_{C}G_{\\mu\\nu}^{a}\\tilde{G}^{a,\\mu\\nu}+g^{2}%\n\\mathcal{N}_{L}W_{\\mu\\nu}^{i}\\tilde{W}^{i,\\mu\\nu}+g^{\\prime2}\\mathcal{N}%\n_{Y}B_{\\mu\\nu}\\tilde{B}^{\\mu\\nu}\\frac{{}}{{}}\\right)  \\;,\n\\end{equation}\nwhere the coefficients $\\mathcal{N}_{C,L,Y}$ are functions of the PQ charges of all the fermions, and generically given by\n\\begin{equation}\n\\mathcal{N}_{X}=\\sum_{\\psi}PQ(\\psi)C_{X}(\\psi)\\ ,\n\\end{equation}\nwith $C_{C,L,Y}(\\psi)$ the quadratic invariant of the field $\\psi$ under $SU(3)_{C}$, $SU(2)_{L}$ or $U(1)_{Y}$. The effective Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\text{\\textrm{Eff}}}=\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}%\n}+\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}\\ , \\label{AxionEL}%\n\\end{equation}\nis in general the basis in which the axion phenomenology is studied, with the common further assumption that $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ is model-dependent and subleading compared to the model independent $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$. Yet, since the PQ charge of the fermions are ambiguous, both $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ and $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$ are also ambiguous. This is most striking in the DFSZ case, where $\\mathcal{N}_{L}\\sim3\\beta+\\gamma$. This conundrum was analyzed in Ref.~~\\cite{Quevillon:2019zrd}, where in particular it was shown that $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ and $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$ do in fact contribute at the same order to physical observables, and that this ensures all the ambiguities in $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ and $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$ cancel each other systematically. This means that the couplings to (chiral) gauge bosons cannot be read off $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$, and that $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ cannot be neglected.\n\nFor our purpose, it is important to emphasize how this translates for the baryon and lepton numbers. Thus, consider the KSVZ scenario with the fermion charges in Eq.~(\\ref{KSVZfermions}), keeping $\\alpha$, $\\beta$, and $\\gamma$ arbitrary, and let us perform the reparametrization of Eq.~(\\ref{ReparamG}) for all the fermions. The PQ current is then identified as\n\\begin{equation}\nJ_{PQ}^{\\mu}=\\bar{\\Psi}_{R}\\gamma^{\\mu}\\Psi_{R}+\\alpha J_{\\Psi}^{\\mu}+3\\beta\nJ_{\\mathcal{B}}^{\\mu}+\\gamma J_{\\mathcal{L}}^{\\mu}\\ ,\n\\end{equation}\nwhere%\n\\begin{align}\nJ_{\\Psi}^{\\mu}  &  =\\bar{\\Psi}_{L}\\gamma^{\\mu}\\Psi_{L}+\\bar{\\Psi}_{R}%\n\\gamma^{\\mu}\\Psi_{R}=\\bar{\\Psi}\\gamma^{\\mu}\\Psi\\ ,\\\\\nJ_{\\mathcal{B}}^{\\mu}  &  =\\frac{1}{3}\\bar{q}_{L}\\gamma^{\\mu}q_{L}+\\frac{1}%\n{3}\\bar{u}_{R}\\gamma^{\\mu}u_{R}+\\frac{1}{3}\\bar{d}_{R}\\gamma^{\\mu}d_{R}%\n=\\frac{1}{3}\\bar{u}\\gamma^{\\mu}u+\\frac{1}{3}\\bar{d}\\gamma^{\\mu}d\\ ,\\ \\\\\nJ_{\\mathcal{L}}^{\\mu}  &  =\\bar{\\ell}_{L}\\gamma^{\\mu}\\ell_{L}+\\bar{e}%\n_{R}\\gamma^{\\mu}e_{R}+\\bar{\\nu}_{R}\\gamma^{\\mu}\\nu_{R}=\\bar{e}\\gamma^{\\mu\n}e+\\bar{\\nu}\\gamma^{\\mu}\\nu\\ .\n\\end{align}\nAt first sight, one may think to discard the vector currents $J_{\\Psi}^{\\mu}$, $J_{\\mathcal{B}}^{\\mu}$, and $J_{\\mathcal{L}}^{\\mu}$ from the derivative interactions since upon integration by part, $\\partial_{\\mu}a^{0}\\bar{\\psi}\\gamma^{\\mu}\\psi=-a^{0}\\partial_{\\mu}\\bar{\\psi}\\gamma^{\\mu}\\psi=-a^{0}\\bar{\\psi}(m-m)\\psi=0$. This is incorrect though. The vector Ward identity does not survive to the presence of chiral gauge interactions. While $J_{\\Psi}^{\\mu}$ can indeed safely be discarded since $\\Psi$ is vector-like, the baryon and lepton currents are anomalous in the presence of chiral gauge fields:\n\\begin{equation}\n\\partial_{\\mu}J_{\\mathcal{B}}^{\\mu}=\\partial_{\\mu}J_{\\mathcal{L}}^{\\mu}%\n=-\\frac{N_{f}}{16\\pi^{2}}\\left(  \\frac{1}{2}g^{2}W_{\\mu\\nu}^{i}\\tilde\n{W}^{i,\\mu\\nu}-\\frac{1}{2}g^{\\prime2}B_{\\mu\\nu}\\tilde{B}^{\\mu\\nu}\\right)  \\ .\n\\end{equation}\nObviously, these contributions trivially cancel the $\\beta$ and $\\gamma$-dependent Jacobian terms generated by the fermion reparametrization, which have precisely the same form and origin. Thus, in the KSVZ setting, it seems\nthat the sole role of the SM fermions derivative interactions aligned with the $\\mathcal{B}$ and $\\mathcal{L}$ current is to kill the correspondingly spurious anomalous gauge interactions.\n\nThere is a problem in this reasoning though. This cancellation occurs whether a $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling is assumed initially present or not, since the value of $\\gamma$ is irrelevant. This is puzzling since in the presence of $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, the axion should retain some couplings to $\\nu_{R}$. In the above argument, the step at which we lost the $a\\nu_{R}\\nu_{R}$ coupling is in the Ward identity. After the spontaneous symmetry breaking (SSB), $\\mathcal{L}$, as part of the PQ symmetry, is no longer conserved and the equation of motion (EoM) of $\\nu_{R}$ breaks explicitly the anomalous vector Ward identity. In practice, $(\\partial_{\\mu}a^{0}\/v)\\bar{\\nu}_{R}\\gamma^{\\mu}\\nu_{R}$ does generate the $(M_{R}\/v)a^{0}\\nu_{R}\\nu_{R}$ coupling. This means that whether the axion is coupled to $\\nu_{R}$ or not is not apparent at the level of the effective axion Lagrangian, but hides in the EoM of $\\nu_{R}$.\\ Further, these EoM spoil the $1\/v_{\\phi}$ scaling of the effective Lagrangian operators, since they contain terms of $\\mathcal{O}(v_{\\phi})$. Phenomenologically, this failure of the effective interactions to manifestly exhibit all the possible axion interactions is clearly an important point to keep in mind.\n\nTo conclude, let us stress again:\n\n\\begin{itemize}\n\\item The PQ symmetry has some room for $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating effects. In the presence of such violation, the PQ symmetry eats part of the $\\mathcal{B}$ and $\\mathcal{L}$ accidental $U(1)$s, and the PQ current inherits some $J_{\\mathcal{B}}^{\\mu}$ and\/or $J_{\\mathcal{L}}^{\\mu}$ components.\n\n\\item Incorporating a $\\mathcal{B}$ and\/or $\\mathcal{L}$ component in the PQ current does not modify the leading order axion to gauge boson couplings.\n\n\\item The $\\mathcal{B}$ and\/or $\\mathcal{L}$ components of PQ current do not tell us much about the couplings of the axion to SM fermions. Most of the $\\partial_{\\mu}a^{0}J_{\\mathcal{B}}^{\\mu}$ and $\\partial_{\\mu}a^{0}J_{\\mathcal{L}}^{\\mu}$ couplings are just there to cancel spurious local anomalous terms.\n\n\\item Any $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating couplings must break explicitly the (already anomalous) $\\mathcal{B}$ and\/or $\\mathcal{L}$ vector Ward identities. In their presence, the EoM of the SM fermions will ensure the derivative interactions $\\partial_{\\mu}a^{0}J_{\\mathcal{B}}^{\\mu}$ and $\\partial_{\\mu}a^{0}J_{\\mathcal{L}}^{\\mu}$ do include the expected $\\Delta\\mathcal{B}$ and\/or $\\Delta\\mathcal{L}$ couplings of the axion.\n\\end{itemize}\n\nAs we will see in the following, introducing leptoquark states often forces us to entangle $\\mathcal{B}$ and\/or $\\mathcal{L}$ with the PQ symmetry. These points are thus crucial to understand the phenomenological consequences.\n\n\\subsection{Introducing leptoquarks and diquarks\\label{Sec2c}}\n\nLeptoquarks (LQ) are scalars or vectors that couple simultaneously to a quark-lepton pair, while diquarks (DQ) couple to quark pairs (for a review, see e.g. Ref.~\\cite{Dorsner:2016wpm}). Given the quantum numbers of the SM fermions, only a finite number of LQ and DQ can couple to normal matter, and only a few of them can have both LQ and DQ couplings. Though the full list of possible LQ and DQ states is well-known, let us nevertheless go through this construction as it will play an important role in the following, and permits to conveniently introduce our notations.\n\nAll the LQ are color triplets, while DQ are triplets (using $1\\supset 3\\otimes3\\otimes3$) or sexplets (using $1\\supset3\\otimes3\\otimes\\bar{6}$). From the point of view of $SU(2)_{L}$, these states can be either triplet, doublets, or singlets, depending on the involved SM fermions. Once $SU(2)_{L}\\otimes SU(3)_{C}$ contractions are set, the hypercharge is then fixed to accommodate specific couplings to SM fermions. In this regard, one should remember that scalars couple to $\\bar{\\psi}_{L}\\psi_{R}$ or $\\bar{\\psi}_{R}\\psi_{L}$, vectors to $\\bar{\\psi}_{R}\\gamma_\\mu\\psi_{R}$ or $\\bar{\\psi}_{L}\\gamma_\\mu\\psi_{L}$, and that charge conjugation $\\mathrm{C}$ flips the chirality. This means that a scalar can couple to $\\bar{\\psi}_{R}^{\\mathrm{C}}\\psi_{R}$ for example. By constructing all possible pairs of SM leptons, including conjugate fields, the standard list of possible states are recovered, with the scalars LQ states\n\\begin{align}\n(\\mathbf{3},\\mathbf{2},+1\/3)  &  :S_{2}^{1\/3}\\times(\\bar{d}_{R}\\ell_{L}\\ ,\\ \\bar{q}_{L}\\nu_{R})\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{2},+7\/3)  &  :S_{2}^{7\/3}\\times(\\bar{u}_{R}\\ell_{L}\\ ,\\ \\bar{q}_{L}e_{R})\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-2\/3)  &  :S_{1}^{2\/3}\\times(\\bar{d}_{R}\\nu_{R}^{\\mathrm{C}}\\ \\ ,\\ \\bar{u}_{R}e_{R}^{\\mathrm{C}}\\ ,\\ \\bar{q}_{L}\\ell_{L}^{\\mathrm{C}})\\ ,\\ \\ (\\mathbf{3},\\mathbf{3},-2\/3):S_{3}^{2\/3}\\times\\bar\n{q}_{L}\\ell_{L}^{\\mathrm{C}}\\ ,\\label{LQ1}\\\\\n(\\mathbf{3},\\mathbf{1},+4\/3)  &  :S_{1}^{4\/3}\\times\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-8\/3)  &  :S_{1}^{8\/3}\\times\\bar{d}_{R}e_{R}^{\\mathrm{C}}\\ ,\\nonumber\n\\end{align}\nand the vector LQ states$\\ $%\n\\begin{align}\n(\\mathbf{3},\\mathbf{2},+1\/3)  &  :V_{2,\\mu}^{1\/3}\\times(\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}\\ ,\\ \\bar{q}_{L}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}})\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{2},-5\/3)  &  :V_{2,\\mu}^{5\/3}\\times(\\bar{d}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}\\ ,\\ \\bar{q}_{L}\\gamma^{\\mu}e_{R}^{\\mathrm{C}})\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},+4\/3)  &  :V_{1,\\mu}^{4\/3}\\times(\\bar{u}_{R}\\gamma^{\\mu}\\nu_{R}\\ ,\\ \\bar{d}_{R}\\gamma^{\\mu}e_{R}\\ ,\\ \\bar{q}_{L}\\gamma^{\\mu}\\ell_{L})\\ ,\\ \\ (\\mathbf{3},\\mathbf{3},+4\/3):V_{3,\\mu}^{4\/3}\\times\\bar{q}%\n_{L}\\gamma^{\\mu}\\ell_{L}\\ ,\\label{LQ2}\\\\\n(\\mathbf{3},\\mathbf{1},10\/3)  &  :V_{1,\\mu}^{10\/3}\\times\\bar{u}_{R}\\gamma^{\\mu}e_{R}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-2\/3)  &  :V_{1,\\mu}^{2\/3}\\times\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}\\ .\\nonumber\n\\end{align}\nMany notations exist for these states, in particular $S_{i}$, $\\tilde{S}_{i}$, $\\bar{S}_{i}$ when several states occur with the same $SU(3)_{C}\\otimes SU(2)_{L}$ quantum numbers~\\cite{Dorsner:2016wpm}. Here, we denote all states as color triplets $S_{t}^{y}$ or $V_{t}^{y}$, with $t$ the $SU(2)_{L}$ dimensionality and $y$ the absolute value of the $U(1)_{Y}$ hypercharge. Note also that $V_{1,\\mu}^{2\/3}$ and $S_{1}^{4\/3}$ exist only in the presence of $\\nu_{R}$, and are thus often discarded. Concerning diquarks, there are only six possible combinations of quark fields, leading to%\n\\begin{align}\n(\\mathbf{3},\\mathbf{2},+1\/3)  &  :V_{2,\\mu}^{1\/3}\\times\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{2},-5\/3)  &  :V_{2,\\mu}^{5\/3}\\times\\bar{u}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-2\/3)  &  :S_{1}^{2\/3}\\times(\\bar{q}_{L}^{\\mathrm{C}}q_{L}\\ ,\\ \\bar{d}_{R}^{\\mathrm{C}}u_{R})\\ ,\\ (\\mathbf{3},\\mathbf{3},-2\/3):S_{3}^{2\/3}\\times\\bar{q}_{L}^{\\mathrm{C}}q_{L}\\ ,\\label{LQ3}\\\\\n(\\mathbf{3},\\mathbf{1},+4\/3)  &  :S_{1}^{4\/3}\\times\\bar{d}_{R}^{\\mathrm{C}}d_{R}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-8\/3)  &  :S_{1}^{8\/3}\\times\\bar{u}_{R}^{\\mathrm{C}}u_{R}\\ .\\nonumber\n\\end{align}\nAll these states are already present in the LQ list. Note that each of the above quark state can also couple to a DQ transforming like $\\mathbf{\\bar{6}}$ under $SU(3)_{C}$, with the same $SU(2)_{L}\\otimes U(1)_{Y}$ quantum numbers. In that case, they do not have LQ couplings. We will adopt the same notation for these states, relying on the context to make clear whether they transform as $\\mathbf{3}$ or $\\mathbf{\\bar{6}}$.\n\n\n\\begin{table}[t] \\centering\n\\begin{tabular}[c]{l}\n\\begin{tabular}[c]{lllllll}\\hline\n$\\Delta\\mathcal{B}$ & $\\Delta\\mathcal{L}$ & Dim. &\n\\multicolumn{2}{l}{Operators (no $\\nu_{R}$)} &  & \\\\\\hline\n$+0$ & $+2$ & $5$ & $H^{\\dagger2}\\ell_{L}^{2}$ &  &  & \\\\\n$+1$ & $+1$ & $6$ & $q_{L}^{3}\\ell_{L}$ & $u_{R}^{2}d_{R}e_{R}$ & $q_{L}%\nu_{R}d_{R}\\ell_{L}$ & $q_{L}^{2}u_{R}e_{R}$\\\\\n$+1$ & $-1$ & $7$ & $H^{\\dagger}d_{R}^{3}\\ell_{L}^{\\mathrm{C}}$ & $Hd_{R}%\n^{2}q_{L}e_{R}^{\\mathrm{C}}$ & $Hd_{R}^{2}u_{R}\\ell_{L}^{\\mathrm{C}}$ &\n$Hq_{L}^{2}d_{R}\\ell_{L}^{\\mathrm{C}}$\\\\\n$+2$ & $+0$ & $9$ & $d_{R}^{4}u_{R}$ & $d_{R}^{3}u_{R}q_{L}^{2}$ & $d_{R}%\n^{2}q_{L}^{4}$ & \\\\\n$+1$ & $+3$ & $9$ & $u_{R}^{2}q_{L}\\ell_{L}^{3}$ & $u_{R}^{3}\\ell_{L}^{2}%\ne_{R}$ &  & \\\\\n$+1$ & $-3$ & $10$ & $Hd_{R}^{3}\\ell_{L}^{\\mathrm{C},3}$ &  &  &\n\\end{tabular}\n\\\\\n\\begin{tabular}[c]{llllllll}\\hline\n$\\Delta\\mathcal{B}$ & $\\Delta\\mathcal{L}$ & Dim. &\n\\multicolumn{2}{l}{Operators (one $\\nu_{R}$)} &  &  & \\\\\\hline\n$+0$ & $+2$ & $5$ & $H^{\\dagger2}e_{R}\\nu_{R}$ &  &  &  & \\\\\n$+1$ & $+1$ & $6$ & $q_{L}^{2}d_{R}\\nu_{R}$ & $d_{R}^{2}u_{R}\\nu_{R}$ &  &  &\n\\\\\n$+1$ & $-1$ & $7$ & $H^{\\dagger}d_{R}^{2}q_{L}\\nu_{R}^{\\mathrm{C}}$ &\n$Hd_{R}q_{L}u_{R}\\nu_{R}^{\\mathrm{C}}$ & $Hq_{L}^{3}\\nu_{R}^{\\mathrm{C}}$ &  &\n\\\\\n$+2$ & $+0$ & $9$ & -- &  &  &  & \\\\\n$+1$ & $+3$ & $9$ & $d_{R}u_{R}^{2}\\ell_{L}^{2}\\nu_{R}$ & $d_{R}q_{L}u_{R}%\n\\ell_{L}^{2}\\nu_{R}$ & $u_{R}^{3}e_{R}^{2}\\nu_{R}$ & $u_{R}^{2}q_{L}\\ell\n_{L}e_{R}\\nu_{R}$ & $q_{L}^{2}u_{R}\\ell_{L}^{2}\\nu_{R}$\\\\\n$+1$ & $-3$ & $10$ & $Hd_{R}^{3}\\ell_{L}^{\\mathrm{C}}e_{R}^{\\mathrm{C}}\\nu\n_{R}^{\\mathrm{C}}$ & $Hd_{R}^{2}q_{L}\\ell_{L}^{\\mathrm{C},2}\\nu_{R}%\n^{\\mathrm{C}}$ &  &  & \\\\\\hline\n\\end{tabular}\n$\\ $\n\\end{tabular}\n$\\ $\n\\caption{Leading effective operators with non-trivial $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ charges in the SM, involving no or one $\\nu_R$ field.\nWe do not include redundant patterns, e.g. all the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=n\\times(0,2),n\\times\n(1,1),...$ with $n=2,3,...$ operators, or operators of higher dimensions within each $(\\Delta\\mathcal{B},\\Delta\\mathcal\n{L})$ class. With even more fields, the next unique patterns involve eight fermions, and induce $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,5)$ transitions at dimension 12, and $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-5)$ transitions at dimension 13 (with an extra Higgs field). All these processes involve at least one $\\nu\n_{R}$ field at these orders. Still higher in dimensionality, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(3,1)$ and $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,7)$ come at the ten-fermion level, via dimension-15 operators. Only the SM Higgs doublet $H$ is used in the Table together with SM fermions, but the extension to the THDM is trivial.}%\n\\label{TableLQBL}\n\\end{table}\n\nIntroducing scalar or vector states that couple to quarks and leptons can impact the accidental $\\mathcal{B}$ and $\\mathcal{L}$ symmetries (for a recent review, see e.g. Ref.~\\cite{Assad:2017iib}). Depending on which states are introduced and, if several of them are present, depending also on how they are coupled, the symmetry pattern can be quite different. Actually, these symmetry patterns are reminiscent of those of the possible effective operators involving SM fields but carrying non-trivial $\\mathcal{B}$ and\/or $\\mathcal{L}$ charges~\\cite{Weinberg,WeinbergPRD22,Weldon:1980gi}. Those are listed in Table~\\ref{TableLQBL}. This connection is easily understood from tree diagrams with the external fermions linked together by virtual LQ\/DQ exchanges\\footnote{The notation LQ\/DQ generically refers to any of the pure LQ, pure DQ, or mixed LQ\/DQ state introduced in Eqs.~(\\ref{LQ1}),~(\\ref{LQ2}), and~(\\ref{LQ3}).}. Obviously, these external fermion states must be $SU(3)_{C}\\otimes SU(2)_{L}\\otimes U(1)_{Y}$ invariant since the LQ\/DQ are. Further, operators with six or less fermions are the most relevant when only renormalizable interactions among the LQ\/DQ are present. Being colored, these states can at most have quadratic or cubic interactions, hence induce four or six fermion interactions. More complicated fermion interactions can arise, but they would require multiple cubic interactions, and would not open additional phenomenologically interesting channels. Indeed, the above set contains already the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ operators for neutrino masses, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators for neutron-antineutron oscillations, and all the others for proton decay. Note, finally, that one can understand why some states have both LQ and DQ couplings while others do not from the fact that dimension-six operators are necessarily $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$, see Table~\\ref{TableLQBL}. As tree-level exchanges of states with both LQ and DQ couplings (Fig.~\\ref{Fig1}$a$) must match onto these operators, only $V_{2}^{y}$ and $S_{1}^{y}$ can occur since they couple to a quark-lepton (or antiquark-antilepton) pair\\footnote{This condition is sometimes quantified using $\\mathcal{F}=3\\mathcal{B}+\\mathcal{L}$ as a quantum numbers~\\cite{Dorsner:2016wpm}, so that those states with both LQ and DQ couplings have $\\mathcal{F}=\\pm2$, and the others $\\mathcal{F}=0$. We prefer here to use $\\mathcal{B}\\pm\\mathcal{L}$.}.\n\nWith the above picture in mind, let us see in more details how the various\n$(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ patterns of Table~\\ref{TableLQBL} can arise:\n\n\\begin{enumerate}\n\\item[A.] Exact $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}:$ Whenever a given $S$ or $V$ state with only LQ or DQ coupling is present, $\\mathcal{B}$ and $\\mathcal{L}$ can still be unambiguously defined. The LQ or DQ state simply carries some specific $\\mathcal{B}$ and $\\mathcal{L}$ quantum numbers, but overall, $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ is still exact. This remains true even in the presence of several different states, so long as they do not couple together.\n\n\\item[B.] Exact $U(1)_{\\mathcal{B}-\\mathcal{L}}:$ When a state with both LQ and DQ couplings is present, the symmetry gets reduced to $U(1)_{\\mathcal{B}-\\mathcal{L}}$, with the $\\mathcal{B}-\\mathcal{L}$ quantum numbers $-2\/3$ for $S_{1}^{y}$ and $V_{2}^{y}$, $+1\/3$ and $-1$ for quarks and leptons, respectively. This remains true if more than one DQ\/LQ state is present provided any couplings among them is compatible with these charge assignments, which further requires the $\\mathcal{B}-\\mathcal{L}$ quantum numbers of $S_{2}^{y}$ and $V_{1}^{y}$ to be $+4\/3$. For example, a scenario with $S_{1}^{2\/3}$ and $S_{1}^{4\/3}$ but without an $S_{1}^{2\/3}S_{1}^{2\/3}S_{1}^{4\/3}$ interaction, or with $S_{2}^{7\/3}$, $S_{1}^{2\/3}$ and a coupling $H^{\\dagger}S_{2}^{7\/3}S_{1}^{2\/3}S_{1}^{2\/3}$, or with $S_{2}^{1\/3}$, $S_{1}^{2\/3}$ and a coupling $HS_{2}^{1\/3}S_{1}^{2\/3}S_{1}^{2\/3}$ all preserve $U(1)_{\\mathcal{B}-\\mathcal{L}}$ (note that the antisymmetric color contraction requires at least two different $S_{1}^{2\/3}$). For all these scenarios, the $S$ and\/or $V$ mass has to be pushed at the GUT scale since $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators induce proton decay (Fig.~\\ref{Fig1}$a$).\n\n\\item[C.] Exact $U(1)_{3\\mathcal{B}+\\mathcal{L}}:$ A peculiar situation arises for the $S_{2}^{1\/3}$ state, because the $H^{\\dagger}S_{2}^{1\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ is allowed by all the SM gauge symmetries. In its presence, the active symmetry gets reduced to $U(1)_{3\\mathcal{B}+\\mathcal{L}}$ since $S_{2}^{1\/3}$ is neutral for that specific combination. This situation is again problematic because $H^{\\dagger}S_{2}^{1\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ collapses to $Hd_{R}^{\\dagger3}\\ell_{L}^{3}$ and can induce a $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ proton decay. No other combination of scalar leptoquarks contributes to this $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ channel.\n\n\\item[D.] No exact $U(1):$ In the presence of two states having different $\\mathcal{B}-\\mathcal{L}$ quantum numbers, there is no remaining symmetry whenever those states have all their gauge-allowed couplings to SM fermions turned on, and when they are coupled together. For example, introducing both $S_{2}^{1\/3}$ and $S_{1}^{2\/3}$ with a $\\mu HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ coupling, $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ are entirely broken. As seen earlier, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ proton decay is induced by $S_{1}^{2\/3}$, pushing its mass to the GUT range. But the total absence of accidental $U(1)$s means the other classes of $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ operators are also generated. The simplest is the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ operator, generating neutrino masses via the diagram of Fig.~\\ref{Fig1}$b$.\n\n\\item[E.] Exact $U(1)_{\\mathcal{B}}:$ Adding to the scenarios A a seesaw mechanism for neutrino masses, i.e., a $\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ term, then $U(1)_{\\mathcal{L}}$ is explicitly broken but $U(1)_{\\mathcal{B}}$ remains exact, preventing proton decay. The same pattern can be obtained using mixing terms among some carefully chosen LQ\/DQ states, such that an effective neutrino mass term is generated but proton decay cannot occur. For example, introducing $S_{2}^{1\/3}$, $S_{1}^{2\/3}$, the mixing term $\\mu HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ but turning off the DQ couplings of $S_{1}^{2\/3}$ (or alternatively, with the mixing term $\\mu S_{2}^{1\/3}S_{2}^{1\/3}S_{1}^{2\/3}$ but turning off the LQ couplings of $S_{1}^{2\/3}$), the dimension-five $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ operator arises, see Fig.~\\ref{Fig1}$b$. In these scenarios, $S_{2}^{1\/3}$, $S_{1}^{2\/3}$ acquire well defined $\\mathcal{B}$ numbers, $U(1)_{\\mathcal{B}}$ is conserved, and proton decay is forbidden.\n\n\\item[F.] Exact $U(1)_{\\mathcal{B}+\\mathcal{L}}:$ Another possible symmetry pattern corresponds to taking again $S_{2}^{1\/3}$, $S_{1}^{2\/3}$, and the $\\mu HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ coupling but turning off the LQ couplings of $S_{1}^{2\/3}$ (or with $\\mu S_{2}^{1\/3}S_{2}^{1\/3}S_{1}^{2\/3}$ but turning off the DQ couplings of $S_{1}^{2\/3}$). In this case, no neutrino masses can be generated, but proton decay is back. Yet, the proton decay channels do not match those induced by the dimension-six Weinberg operators. With the $\\mu HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ coupling, the simplest processes lead to the dimension-seven $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ effective operators, see Fig.~\\ref{Fig1}$c$, while the $\\mu S_{2}^{1\/3}S_{2}^{1\/3}S_{1}^{2\/3}$ coupling generates $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ transitions but with an extra lepton-antilepton pair.\n\n\\item[G.] Exact $U(1)_{\\mathcal{L}}:$ Another pattern is obtained by introducing several states but now allowing only for DQ couplings, and turning on some mixing terms (this kind of construction was considered recently e.g. in Refs.~\\cite{Arnold:2012sd,FileviezPerez:2015mlm}). These latter mixings are necessary since otherwise, $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ remains exact. The simplest scenarios are those with $S_{1}^{2\/3}$, $S_{1}^{4\/3}$, and the cubic coupling $\\mu S_{1}^{2\/3}S_{1}^{2\/3}S_{1}^{4\/3}$, or $S_{1}^{4\/3}$, $S_{1}^{8\/3}$, and the cubic coupling $\\mu S_{1}^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}$. In both cases, only the DQ couplings are allowed, and $S_{1}^{4\/3}$ ($S_{1}^{8\/3}$) must transform as $\\mathbf{\\bar{6}}$ in the first (second) case, respectively. As a result, neither neutrino masses nor proton decay are induced, but the dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators do arise, and contribute to neutron-antineutron oscillations, see Fig.~\\ref{Fig1}$d$.\n\n\\item[H.] Exact $U(1)_{3\\mathcal{B}-\\mathcal{L}}:$ As for the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ case, dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ operators are attainable by taking $S_{1}^{2\/3}$, $S_{1}^{4\/3}$, and the cubic coupling $\\mu S_{1}^{2\/3}S_{1}^{2\/3}S_{1}^{4\/3}$, or $S_{1}^{4\/3}$, $S_{1}^{8\/3}$, and the cubic coupling $\\mu S_{1}^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}$, but turning on only the LQ couplings (since all LQ transform as $\\mathbf{3}$, the color contraction requires three different LQ to be present). Yet, only interactions involving $\\nu_{R}$ can occur because of the LQ coupling of $S_{1}^{4\/3}$ to $\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}$, so proton decay is suppressed. The dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ operators not involving $\\nu_{R}$ require a combination of scalar and vector LQ, for example $S_{1}^{2\/3}V_{2}^{1\/3}V_{2}^{1\/3}$ can induce both $\\bar{q}_{L}\\ell_{L}^{\\mathrm{C}}\\bar{u}_{R}\\gamma_{\\mu}\\ell_{L}^{\\mathrm{C}}\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}$ and $\\bar{u}_{R}e_{R}^{\\mathrm{C}}\\bar{u}_{R}\\gamma_{\\mu}\\ell_{L}^{\\mathrm{C}}\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}$.\n\\end{enumerate}\n\n\\begin{figure}[ptb]\n\\begin{center}\n\\includegraphics[height=2.3134in,width=5.3195in]{Fig1.jpg}\n\\caption{LQ\/DQ processes inducing proton decay via $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators ($a.$), a neutrino Majorana mass term ($b.$), proton decay via $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators ($c.$), and neutron-antineutron oscillations via a $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operator ($d.$).}%\n\\label{Fig1}\n\\end{center}\n\\end{figure}\n\nThis concludes our list of symmetry patterns. It is quite remarkable that a relatively simple scenario exists for all the possible patterns of Table~\\ref{TableLQBL}, with in each case the `orthogonal' $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ pattern remaining as an exact accidental $U(1)$ symmetry. What this list does not show is that actually, not so many other scenarios do exist to generate most of these symmetry-breaking patterns. Indeed, in most cases, allowing for several LQ\/DQ states, both scalar and vector, and some couplings among them, one simply ends up with no accidental symmetries. The interesting situations in which some accidental symmetries do remain are quite constrained, and those can be classified once and for all.\n\nFirst, notice that at the renormalizable level, there are only two classes of couplings among the LQ\/DQ: those with bilinear color contractions, typically $\\mathbf{3}\\otimes\\mathbf{\\bar{3}}$ or $\\mathbf{6}\\otimes\\mathbf{\\bar{6}}$, and those with cubic contractions, typically $\\mathbf{3}\\otimes\\mathbf{3}\\otimes\\mathbf{3}$ or $\\mathbf{3}\\otimes\\mathbf{3}\\otimes\\mathbf{\\bar{6}}$. For the former, barring partial derivatives acting on the LQ\/DQ fields, the only non-trivial LQ\/DQ bilinear couplings compatible with the SM gauge symmetries are%\n\\begin{equation}\nHS_{2}^{1\/3\\dagger}S_{1}^{2\/3}\\ ,\\ HS_{1}^{4\/3\\dagger}S_{2}^{1\/3}%\n\\ ,\\ HS_{2}^{7\/3\\dagger}S_{1}^{4\/3}\\ ,\\ HV_{2,\\mu}^{1\/3\\dagger}V_{1}^{2\/3,\\mu\n}\\ ,\\ HV_{1,\\mu}^{4\/3\\dagger}V_{2}^{1\/3,\\mu}\\ ,\\ HV_{1,\\mu}^{2\/3\\dagger}%\nV_{2}^{5\/3,\\mu}\\ . \\label{LQOpsBmL}\n\\end{equation}\nThe $HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ coupling was used to illustrate the symmetry patterns, but all the others are completely similar: $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ are entirely broken when all the LQ\/DQ couplings are present (case D), $U(1)_{\\mathcal{B}}$ stays exact with only LQ couplings (case E), or $U(1)_{\\mathcal{B}+\\mathcal{L}}$ remains if $S_{1}^{y}$ or $V_{2}^{y}$ have only DQ couplings (case F). This last situation is probably the most interesting phenomenologically since each coupling in Eq.~(\\ref{LQOpsBmL}) produces a specific subset of the dimension-seven $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators in Table~\\ref{TableLQBL}.\n\nFor cubic interactions, though there are a total of $37$ such couplings, most of them involve LQ\/DQ of different $\\mathcal{B}-\\mathcal{L}$ charges and conserve either $U(1)_{\\mathcal{B}-\\mathcal{L}}$ (case B) or $U(1)_{\\mathcal{B}+\\mathcal{L}}$ (case F). Yet, compared to the dimension 6 and 7 operators in Table~\\ref{TableLQBL}, they necessarily produce an extra lepton-antilepton pair. The symmetry patterns typical of six-fermion states, i.e., leading to the dimension 9 or 10 operators in Table~\\ref{TableLQBL}, are obtained with three LQ\/DQ with the same $\\mathcal{B}-\\mathcal{L}$ charge, and this leaves only eight possibilities:\n\\begin{align}\n&  S_{1}^{2\/3}V_{2,\\mu}^{1\/3}V_{2}^{1\/3,\\mu}\\ ,\\ S_{1}^{4\/3}V_{2,\\mu}%\n^{1\/3}V_{2}^{5\/3,\\mu}\\ ,\\ S_{1}^{2\/3}S_{1}^{2\/3}S_{1}^{4\/3}\\ ,\\ S_{1}%\n^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}\\ ,\\label{LQOpsB2}\\\\\n&  H^{\\dagger}S_{2}^{1\/3}S_{2}^{1\/3}S_{2}^{1\/3}\\ ,\\ H^{\\dagger}S_{2}%\n^{1\/3}V_{1,\\mu}^{2\/3}V_{1}^{4\/3,\\mu}\\ ,\\ H^{\\dagger}S_{2}^{7\/3}V_{1,\\mu\n}^{\\prime2\/3}V_{1}^{2\/3,\\mu}\\ ,\\ HS_{2}^{1\/3}V_{1,\\mu}^{\\prime2\/3}%\nV_{1}^{2\/3,\\mu}\\ . \\label{LQOpsBL3}%\n\\end{align}\nThe scenarios in the first line lead to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ or $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ operators (case G and H), and those in the second line to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ operators (case C). Note that for $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,\\pm3)$ transitions, the LQ must transform as $\\mathbf{3}$, and the color contraction is necessarily antisymmetric. When this is not compensated by an antisymmetric $SU(2)_L$ contraction, the three LQs must be different (hence one of the two $V_{1,\\mu}^{2\/3}$ fields is primed in the last two operators of Eq.~(\\ref{LQOpsBL3})). This does not apply to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators, for which it is always possible to take one of the DQ to transform as a symmetric $\\mathbf{\\bar{6}}$. As a final remark, it should be noted that scalar or vector color-singlet dileptons could also be introduced, opening the door to quartic couplings among the new states, and correspondingly, to eight-fermion $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,\\pm2)$ operators~\\cite{Helset:2021plg}. This will not be considered here.\n\nThroughout this paper, when estimating bounds on LQ\/DQ masses from proton decay or neutron-antineutron oscillations, the LQ\/DQ couplings to SM fermions is assumed flavor universal, or at the very least non-hierarchical in flavor space. As was shown in Ref.~\\cite{MFVBandL}, this is a strong assumption for $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating operators. The $SU(3)^{5}$ flavor\nsymmetry would ask instead for a strong hierarchy because of the systematic presence of the three quark generations in all the operators in Table~\\ref{TableLQBL}. In the present context, such hierarchies would first require LQ\/DQ to carry flavor quantum numbers, and then to extend the minimal flavor violating formalism to the LQ\/DQ sector~\\cite{Davidson:2010uu}. This will not be analyzed here, but such kind of flavor suppression should be kept in mind, especially given the context in B physics. There, a number of puzzles in leptonic and semileptonic decays can be explained by introducing new LQ states with particular flavor hierarchies (for a recent review, see e.g. Ref.~\\cite{LQreview}). Typically, the favored LQ is $V_{1,\\mu}^{4\/3}\\sim(\\mathbf{3},\\mathbf{1},+4\/3)$ thanks to its $q_{L}\\gamma^{\\mu}\\ell_{L}$ couplings, but other states could also occur in principle. The connection of some of these models with axions has been investigated e.g. in Ref.~\\cite{Fuentes-Martin:2019bue} (for some considerations of axions in the context of the B physics anomalies see e.g. \\cite{Baek:2020ovw}, whereas axions in a more broad flavor context have also been studied in Refs.~\\cite{Ema:2016ops,Calibbi:2016hwq,Arias-Aragon:2017eww,Bonnefoy:2020llz}, but to our knowledge, no systematic studies has been performed yet. In the present paper, our goal is mainly to analyze symmetry breaking patterns involving both LQ\/DQ and axions, so the LQ\/DQ couplings to SM fermions will simply be assumed $\\mathcal{O}(1)$ for all flavors whenever deriving bounds on their masses. Turning on non-trivial flavor structures is left for future studies.\n\n\\section{Coupling axions to leptoquarks and diquarks\\label{SecAxLQDQ}}\n\nIn the previous section, we have established the possible global accidental symmetries in the presence of LQ and DQ states. Here, we want to add to these scenarios a KSVZ or DFSZ sector. The consequences are rather different for both models, since the SM fermions can be PQ neutral in the former case, but not in the latter. Yet, so long as the $\\phi$ (and the heavy KSVZ fermions $\\Psi_{L,R}$) are not directly coupled to the LQ\/DQ states, the axion stays rather insensitive to the possible $\\mathcal{B}$ and\/or $\\mathcal{L}$ violation.\n\nTo illustrate this, consider the KSVZ scenario. Without direct couplings of $\\phi$ or $\\Psi_{L,R}$ to the LQ\/DQ states, the $U(1)_{\\phi}$ symmetry stays separate from the accidental symmetries $U(1)_{\\mathcal{B},\\mathcal{L}}$, so the PQ breaking proceeds trivially as\n\\begin{equation}\nU(1)_{\\phi}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}\\overset\n{\\text{Explicit}}{\\rightarrow}U(1)_{\\phi}\\otimes U(1)_{X}\\simeq U(1)_{PQ}%\n\\otimes U(1)_{X}\\overset{\\text{Spontaneous}}{\\rightarrow}U(1)_{X}\\ ,\\ \\\n\\end{equation}\nThe specific LQ\/DQ scenario fixes which accidental symmetry\n\\begin{equation}\nU(1)_{X}=U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}},\\ U(1)_{\\mathcal{B}%\n\\pm\\mathcal{L}}\\ ,U(1)_{\\mathcal{B}}\\ ,\\ U(1)_{\\mathcal{L}}%\n,\\ U(1)_{3\\mathcal{B}\\pm\\mathcal{L}}\\ ,...\\ , \\label{SurvivingU1}%\n\\end{equation}\nsurvives, by introducing couplings that explicitly break $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}\\backslash U(1)_{X}$. Yet, the axion does not break $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ or $U(1)_{X}$, only the dynamics of the SM and LQ\/DQ fields does. Of course, the axion being coupled to SM gauge fields and SM fermions, it does end up coupled to leptoquarks and possibly acquires some $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating decay channels, but this is indirect. A good example for this situation is the KSVZ model with a Majorana mass $M_{R}\\nu_{R}\\nu_{R}$. The Majorana mass term explicitly breaks $U(1)_{\\mathcal{L}}$ at all scale, but such that $U(1)_{X}=U(1)_{\\mathcal{B}}$ stays exact at all scales. Clearly, the axion dynamics does not break $U(1)_{\\mathcal{L}}$, only neutrino masses do. Thus, any $\\Delta\\mathcal{L}=2$ effect would come indirectly, e.g. as in $a^{0}\\rightarrow\\bar{\\nu}_{R}\\nu_{L}\\rightarrow\\nu_{R}\\nu_{L}$. The situation in the DFSZ scenario is similar, though the $U(1)_{PQ}$ arises from a specific combination of $U(1)_{\\phi}$ and $U(1)_{Y}$, see Eq.~(\\ref{DFSZScalars}). This situation also corresponds to that often found in simple GUT models. For example, in $SU(5)$, gauge interactions break $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ down to $U(1)_{\\mathcal{B}-\\mathcal{L}}$ independently of the axion field (for a detailed account of how the PQ, $\\mathcal{B}$, and $\\mathcal{L}$ symmetries are entangled in the $SU(5)$ setting, see Ref.~\\cite{Quevillon:2020aij}).\n\nOur goal is to consider situations in which the symmetry above the PQ scale entangles $U(1)_{\\phi}$ within $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$. Breaking $U(1)_{\\phi}$ spontaneously then means breaking a linear combination of $\\mathcal{B}$ and $\\mathcal{L}$ (or both) spontaneously. Taking again the KSVZ scenario for illustration, this is accomplished by introducing some set of couplings that are only invariant under a subgroup of $U(1)_{\\phi}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$. In most cases of interests, $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ stays active at the high scale, but $\\phi$ carries some definite $\\mathcal{B}$ and\/or $\\mathcal{L}$ quantum numbers, so that the breaking chain becomes\n\\begin{equation}\nU(1)_{\\phi}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}\\overset\n{\\text{Explicit}}{\\rightarrow}U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}%\n}\\simeq U(1)_{PQ}\\otimes U(1)_{X}\\overset{\\text{Spontaneous}}{\\rightarrow\n}U(1)_{X}\\ .\n\\end{equation}\nThe simplest example illustrating this situation is the KSVZ model with the $\\phi^{\\dagger}\\bar{\\nu}^\\mathrm{C}_{R}\\nu_{R}$ couplings, so that $\\phi$ becomes a $(\\mathcal{B},\\mathcal{L})=(0,2)$ state, $U(1)_{PQ}=U(1)_{\\mathcal{L}}$ is spontaneously broken, but $U(1)_{X}=U(1)_{\\mathcal{B}}$ stays exact. Compared to the previous case, the main difference is that the axion has a $\\Delta\\mathcal{L}=2$ coupling $a^{0}\\rightarrow\\nu_{R}\\nu_{R}$ of $\\mathcal{O}(1)$. Of course, phenomenologically, whether one adds $M_{R}\\bar{\\nu}^\\mathrm{C}_{R}\\nu_{R}$ or $\\phi^{\\dagger}\\bar{\\nu}^\\mathrm{C}_{R}\\nu_{R}$ is irrelevant, but this may not be the case for scenarios in which $U(1)_{\\mathcal{B}}$ is spontaneously broken. Our goal here is to systematically study these scenarios, taking advantage of the fact that LQ\/DQ open many routes to entangle $U(1)_{\\phi}$ within $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ at the renormalizable level (with only SM fields, the $\\phi^\\dagger\\bar{\\nu}^\\mathrm{C}_{R}\\nu_{R}$ coupling is the only possibility). Note, finally, that in the KSVZ context, there is actually an extra accidental symmetry corresponding to $\\Psi$ number, $U(1)_{\\Psi}$, that will either survive or be incorporated within $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ via explicit breaking terms independent of $\\phi$. In this way, the final surviving $U(1)_{X}$ accidental symmetry is independent of $U(1)_{\\Psi}$, and still given by Eq.~(\\ref{SurvivingU1}).\n\nIn practice, to entangle the $U(1)_{\\phi}$ symmetry with the accidental symmetries, the strategy is to turn on some direct couplings between $\\phi$ and the LQ\/DQ, and for these latter, to turn on some or all of their couplings to SM fields such that no direct $\\mathcal{B}$ and\/or $\\mathcal{L}$ violation occurs. It is important to stress that we do not assign $U(1)$ charges to the fields. Instead, we let the theory tell us what are the exact accidental $U(1)$ symmetries, and what are the charges of the fields. Indeed, it is well-known that symmetries and charges are entirely fixed given a set of couplings in the Lagrangian, but often one identifies them by inspection, or starts from the charges to infer the allowed couplings. In the present case, as we will see, the set of couplings can be quite large, and the surviving $U(1)$s assign quite intricate charges to the fields. Typically, a naive inspection of the Lagrangian couplings would most likely miss some of the surviving $U(1)$s, or outright fail to identify possible scenarios. In practice, starting from the Lagrangian also provides a very systematic procedure: to find the surviving $U(1)$ symmetries, it suffices to express the charge constraint corresponding to each coupling, and solve this system of equations. When this system is under-determined, each parametric under-determination corresponds to a surviving $U(1)$. The charges of $\\phi$ under these $U(1)$ then tell us which combination is spontaneously broken.\n\n\\subsection{KSVZ scenarios with leptoquarks and diquarks\\label{SecKSVZ}}\n\nOur requirements for the KSVZ scenarios are first that there should be only one Higgs doublet, neutral under the PQ and all accidental symmetries, and no direct mixing of the heavy fermions $\\Psi_{L,R}$ with SM quarks to avoid FCNC or CKM unitarity constraints. Also, our goal is to force proton decay, neutron-antineutron oscillations, or a Majorana mass terms for $\\nu_{R}$ (or more generally, neutrino-less double beta decays~\\cite{Deppisch:2012nb}) to only arise through the spontaneous symmetry breaking of $U(1)_{\\phi}$. Thus, none of these observables should be immediately allowed by LQ\/DQ transitions. Typically, the strategy to achieve this is, starting from some Lagrangian with a specific set of couplings among $\\phi$, some chosen LQ\/DQs, and the SM fermions, to identify the accidental symmetries, and then make sure these accidental symmetries forbid any other renormalizable Lagrangian couplings. This will be made clear going through specific examples. But, before that, let us describe some generic features of the scenarios and their consequence for the axion effective Lagrangian.\n\nIn all scenarios, there will be some $\\phi^{2}S_{i}^{\\dagger}S_{j}$, $\\phi HS_{i}^{\\dagger}S_{j}$, and\/or $\\phi S_{i}S_{j}S_{k}$ couplings. In this representation, the axion ends up coupled to the LQ\/DQ, as can be seen plugging in Eq.~(\\ref{PhiPolar}) in these couplings (remember $\\eta_{\\phi}=a^{0}$ in the KSVZ setting). Importantly, these couplings are never suppressed by the PQ breaking scale, since for example\n\\begin{equation}\n\\phi S_{i}S_{j}S_{k}\\rightarrow\\frac{1}{\\sqrt{2}}(v_{\\phi}+i\\eta_{\\phi}+...)S_{i}S_{j}S_{k}\\ . \n\\label{ExampleSSS}\n\\end{equation}\nThough as a matter of principle, the axion $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating couplings are not suppressed by $v_{\\phi}$, this scale nevertheless indirectly limits them. Indeed, the leading $v_{\\phi}$ term produces a direct coupling among the LQ\/DQ such that one falls into any one of the situations described in Sec.~\\ref{Sec2c}, with some $U(1)_{X}$ smaller than $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ remaining exact. At low energy, these LQ\/DQ couplings can induce $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating processes, hence set rather strong bounds on the LQ\/DQ masses. Now, the largest $v_{\\phi}$ is, the tightest these bounds are, so indirectly, the $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating axion couplings to SM fermions decrease for increasing $v_{\\phi}$.\n\nComing back to the point of principle, one may wonder how is it that the axion couplings is not suppressed by $v_{\\phi}$ in the effective axion Lagrangian language of Eq.~(\\ref{AxionEL}). Indeed, as a result of the $\\phi^{2}S_{i}^{\\dagger}S_{j}$, $\\phi HS_{i}^{\\dagger}S_{j}$, and\/or $\\phi S_{i}S_{j}S_{k}$ couplings, some or all of the LQ\/DQ become charged under $U(1)_{PQ}$. This means that if, along with Eq.~(\\ref{ReparamG}) for the fermions, we reparametrize them as\n\\begin{equation}\nS_{i}\\rightarrow\\exp(-iPQ(S_{i})a^{0}\/v_{\\phi})S_{i}\\ , \n\\label{ReparamS}\n\\end{equation}\nthe axion field is entirely removed from all the Lagrangian couplings. Indeed, the Lagrangian is PQ-symmetric, so the $\\exp(ia^{0}\/v_{\\phi})$ factors always compensate exactly. Their kinetic terms $D_{\\mu}S_{i}^{\\dagger}D^{\\mu}S_{i}$ are not invariant under the reparametrization though, and as for fermions, this is embodied in dimension-five interactions\n\\begin{equation}\n\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}=\\frac{1}{v_{\\phi}}\\partial_{\\mu}%\na^{0}J_{PQ}^{\\mu}\\text{ ,\\ \\ \\ }J_{PQ}^{\\mu}=\\sum_{i}PQ(S_{i})(S_{i}^{\\dagger\n}(D^{\\mu}S_{i})-(D^{\\mu}S_{i}^{\\dagger})S_{i})+...\n\\end{equation}\nThis representation is deceptive because the axion couplings to LQ\/DQ appear suppressed by $v_{\\phi}$. Yet, they are not suppressed because the EoM of the $S_{i}$ have $\\mathcal{O}(v_{\\phi})$ terms, like that coming from a $v_{\\phi}S_{i}S_{j}S_{k}$ coupling in the example of Eq.~(\\ref{ExampleSSS}). The same happens if LQ\/DQ are integrated out before the reparametrization Eq.~(\\ref{ReparamS}). They then do not occur in $J_{PQ}^{\\mu}$, but SM fermions do, and their EoM now have inherited $\\mathcal{O}(v_{\\phi}\/M^{n}$)\nterms for some $n$, with $M$ the LQ\/DQ mass scale. In all cases, the axion keeps its $\\mathcal{O}(v_{\\phi}^{0})$ $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating couplings, as it should.\n\nThis shows explicitly that the shift-symmetric $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ is not well-suited to these scenarios, at least for what concerns couplings to matter fields. For gauge boson, the situation is a bit different. The fermion reparametrization Eq.~(\\ref{ReparamG}) generates spurious anomalous interactions to chiral gauge fields that are cancelled by the anomalies in the $J_{PQ}^{\\mu}$ current, exactly as before, but the LQ\/DQ obviously do not. Thus, for them, the axion effective Lagrangian after the reparametrization of Eq.~(\\ref{ReparamS}) correctly captures the fact that triangle graphs with LQ\/DQ running in the loop are not anomalous, and vanish at the dimension-five level for a massless axion. Thus, none of the axion to gauge boson couplings is affected by the LQ\/DQ at leading order.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}+\\mathcal{L}$}\n\nWe have seen that $\\mathcal{B}+\\mathcal{L}$ is immediately broken whenever a given $S_{i}$ or $V_{i}$ has both LQ and DQ couplings. For example, $S_{1}^{8\/3}$ with its couplings to $\\bar{d}_{R}e_{R}^{\\mathrm{C}}$ and $\\bar{u}_{R}^{\\mathrm{C}}u_{R}$ can induce $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators and proton decay. A possible strategy to adapt this scenario and force these operators to appear only through the SSB of $\\phi$ is to consider two such states, one LQ and one DQ, with a $\\phi$-dependent mixing term:\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{8\/3}\\bar\n{d}_{R}e_{R}^{\\mathrm{C}}+\\tilde{S}_{1}^{8\/3}\\bar{u}_{R}^{\\mathrm{C}}%\nu_{R}+\\phi^{2}S_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}+h.c.\\ , \\label{LagrKSVZ1a}%\n\\end{equation}\nwith $\\mathcal{L}_{\\mathrm{KSVZ}}$ given in Eq.~(\\ref{KSVZ0}), and LQ\/DQ kinetic terms are understood. We also do not write explicitly the LQ\/DQ scalar potential terms made of bilinears like $S_{1}^{8\/3\\dagger}S_{1}^{8\/3}$ or $\\tilde{S}_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}$ since those are neutral under any $U(1)$ symmetry. Solving for the $U(1)$ charges of all the fields under the requirement that the Higgs doublet is neutral (to avoid mixing with $U(1)_{Y}$), a triple under-determination remains, which we can identify as\\footnote{Evidently, the normalization of each line is free, but that for $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ is chosen to reproduce conventional quark and lepton $\\mathcal{B}$ and $\\mathcal{L}$ of $1\/3$ and $1$, respectively.}\n\\begin{equation}\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{8\/3}$ & $\\tilde{S}_{1}^{8\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ &\n$q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\Psi}$ & $0$ & $0$ & $0$ & $1$ & $1$ & $0$ & $0$ & $0$ & $0$ & $0$ &\n$0$\\\\\n$U(1)_{\\mathcal{B}}$ & $1\/2$ & $1\/3$ & $-2\/3$ & $-1\/2$ & $0$ & $1\/3$ & $1\/3$ &\n$1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $1\/2$ & $1$ & $0$ & $-1\/2$ & $0$ & $0$ & $0$ & $0$ &\n$1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\label{ChargKSVZ1a}\n\\end{equation}\n\n\nWhat this table shows is that $\\phi$ carries a $U(1)_{\\mathcal{B}+\\mathcal{L}}$ charge, which thus gets spontaneously broken, while $U(1)_{\\mathcal{B}-\\mathcal{L}}$ stays exact. This model is essentially identical to that introduced long ago in Ref.~\\cite{WeinbergPRD22}, except that the Goldstone boson is here identified with the axion. This pattern of symmetry breaking is easily understood from the Lagrangian couplings and the diagram in Fig.~\\ref{Fig2}. Plugging in the polar representation of $\\phi$, Eq.~(\\ref{PhiPolar}), the effective operator at the low-scale is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)}^{eff}=\\exp\n(2ia^{0}\/v_{\\phi})\\frac{v_{\\phi}^{2}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\bar{u}%\n_{R}^{\\mathrm{C}}u_{R}\\bar{d}_{R}^{\\mathrm{C}}e_{R}+h.c.\\ , \\label{EffHKSVZ1}%\n\\end{equation}\nwhere we have identified $\\eta_{\\phi}$ as the axion $a^{0}$, and denoted the $S_{1}^{8\/3}$ and $\\tilde{S}_{1}^{8\/3}$ masses as $m_{S}$ and $m_{\\tilde{S}}$, respectively. Note well that this operator arises entirely through the SSB: the charges in Eq.~(\\ref{ChargKSVZ1a}) explicitly prevent a DQ coupling for $S_{1}^{8\/3}$, and a LQ coupling for $\\tilde{S}_{1}^{8\/3}$. Expanding the exponential, the leading term involves only SM particles, and contributes to proton decay. Thus, $m_{S}$ and $m_{\\tilde{S}}$ have to be pushed quite high, though a bit lower that in the usual GUT scenarios. For instance, while the scale of the dim-6 operators is typically pushed above $10^{14}$~GeV, we only need $m_{S}\\approx m_{\\tilde{S}}>10^{11}$~GeV when $v_{\\phi}=10^{9}$~GeV. With these parameters, the proton decay modes involving the axion are thus totally negligible. Finally, notice that the axion totally disappears from $\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)}^{eff}$ under the reparametrization Eq.~(\\ref{ReparamG}), with the PQ charges identified as $(\\mathcal{B}+\\mathcal{L})\/2$. As stated earlier, the $v_{\\phi}a^{0}\\bar{u}_{R}^{\\mathrm{C}}u_{R}\\bar{d}_{R}^{\\mathrm{C}}e_{R}$ effective coupling would then hide in the $\\partial_{\\mu}a^{0}J_{PQ}^{\\mu}\/v_{\\phi}$ terms since the quarks and leptons inherit from $v_{\\phi}^{2}\\bar{u}_{R}^{\\mathrm{C}}u_{R}\\bar{d}_{R}^{\\mathrm{C}}e_{R}$ some $\\mathcal{O}(v_{\\phi}^{2})$ terms in their EoM.%\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.3586in,width=1.5826in]{Fig2.jpg}\n\\caption{Proton decay operator generated by the spontaneous breaking of $U(1)_{\\mathcal{B}+\\mathcal{L}}$.}%\n\\label{Fig2}\n\\end{center}\n\\end{figure}\n\nAs shown in Eq.~(\\ref{ChargKSVZ1a}), $U(1)_{\\Psi}$ remains as an exact accidental symmetry, which means that the $\\mathcal{B}+\\mathcal{L}$ charges of $\\Psi_{L,R}$ are not unambiguously defined. To fix them requires $\\Psi_{L,R}$\nto couple to SM fermions, and this is possible only for some specific gauge quantum numbers. If we further ask that $S_{1}^{8\/3}$ ($\\tilde{S}_{1}^{8\/3}$) should always (never) couple to leptons, the only possibilities are%\n\\begin{equation}%\n\\begin{tabular}\n[c]{c|cccccc}\\hline\n$Y,\\mathcal{B},\\mathcal{L}$ & $S_{1}^{8\/3}\\bar{\\Psi}_{L}\\ell_{L}^{\\mathrm{C}}$\n& $S_{1}^{8\/3}\\bar{\\Psi}_{R}e_{R}^{\\mathrm{C}}$ & $S_{1}^{8\/3}\\bar{\\Psi}%\n_{R}\\nu_{R}^{\\mathrm{C}}$ & $\\tilde{S}_{1}^{8\/3}\\bar{\\Psi}_{L}^{\\mathrm{C}%\n}q_{L}$ & $\\tilde{S}_{1}^{8\/3}\\bar{\\Psi}_{R}^{\\mathrm{C}}u_{R}$ & $\\tilde\n{S}_{1}^{8\/3}\\bar{\\Psi}_{R}^{\\mathrm{C}}d_{R}$\\\\\\hline\n$\\Psi_{L}$ & $-\\dfrac{5}{3},\\dfrac{1}{3},0\\rule[-0.14in]{0in}{0.36in}$ &\n$-\\dfrac{2}{3},\\dfrac{5}{6},\\dfrac{1}{2}$ & $-\\dfrac{8}{3},\\dfrac{5}{6}%\n,\\dfrac{1}{2}$ & $\\dfrac{7}{3},\\dfrac{1}{3},0$ & $\\dfrac{4}{3},\\dfrac{5}%\n{6},\\dfrac{1}{2}$ & $\\dfrac{10}{3},\\dfrac{5}{6},\\dfrac{1}{2}$\\\\\n$\\Psi_{R}$ & $-\\dfrac{5}{3},-\\dfrac{1}{6},-\\dfrac{1}{2}\\rule[-0.14in]%\n{0in}{0.36in}$ & $-\\dfrac{2}{3},\\dfrac{1}{3},0$ & $-\\dfrac{8}{3},\\dfrac{1}%\n{3},0$ & $\\dfrac{7}{3},-\\dfrac{1}{6},-\\dfrac{1}{2}$ & $\\dfrac{4}{3},\\dfrac\n{1}{3},0$ & $\\dfrac{10}{3},\\dfrac{1}{3},0$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThese couplings are mutually exclusive since they impose different hypercharges for $\\Psi_{L,R}$. Also, $S_{1}^{8\/3}\\bar{\\Psi}_{R}e_{R}^{\\mathrm{C}}$ and $\\tilde{S}_{1}^{8\/3}\\bar{\\Psi}_{R}^{\\mathrm{C}}u_{R}$ would allow for direct $\\bar{\\Psi}_{R}d_{R}$ and $\\bar{\\Psi}_{R}u_{R}$ couplings, respectively, hence must be discarded. Note how the peculiar choice of couplings completely twists the $\\mathcal{B},\\mathcal{L}$ charges, in the sense that they do not correspond to the naive assignments of $\\mathcal{B}=1\/3$ and $\\mathcal{L}=0$ one may have expected for the \"heavy quarks\" of the KSVZ mechanism. As said before, the charges of the fields have to be deduced from the set of couplings of the Lagrangian, and not the other way around.\n\nSimilar scenarios can be constructed using $S_{1}^{2\/3}$, $S_{1}^{4\/3}$, $V_{2}^{1\/3}$ or $V_{2}^{5\/3}$. Actually, $S_{1}^{2\/3}$ was considered in Ref.~\\cite{Reig:2018yfd}, though the model built there is more complicated (here the PQ symmetry is directly identified with $\\mathcal{B}+\\mathcal{L}$ and only a single Higgs doublet is introduced instead of four). Each time, two such states are taken, with one having LQ couplings, and the other DQ couplings, and a $\\phi$-driven mixing term is introduced. The only difference in each case is the specific $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operator(s) that can be spontaneously generated, see Table~\\ref{TableLQBL}, and thereby, the induced pattern of proton decay modes. In this respect, it is worth to look at the $S_{1}^{4\/3}$ scenario, since it has only LQ couplings to $\\nu_{R}$:\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{4\/3}\\bar\n{u}_{R}\\nu_{R}^{\\mathrm{C}}+\\tilde{S}_{1}^{4\/3}\\bar{d}_{R}^{\\mathrm{C}}%\nd_{R}+\\phi^{2}S_{1}^{4\/3\\dagger}\\tilde{S}_{1}^{4\/3}+\\phi\\bar{\\nu}%\n_{R}^{\\mathrm{C}}\\nu_{R}+h.c.\\ . \\label{LagrKSVZ1b}%\n\\end{equation}\nLet us also turn on a coupling to $\\Psi_{L,R}$, to fix its charges, and for definiteness, let us take $S_{1}^{4\/3}\\bar{\\Psi}_{L}\\ell_{L}^{\\mathrm{C}}$. Because of the $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling, solving for the $U(1)$ charges of all the fields now leaves a single under-determination:\n\\begin{equation}%\n\\begin{tabular}\n[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{4\/3}$ & $\\tilde{S}_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ &\n$q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $2$ & $2\/3$ & $-10\/3$ & $5\/3$ & $-1\/3$ & $5\/3$ & $5\/3$ & $5\/3$ &\n$-1$ & $-1$ & $-1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThis time, neither $\\mathcal{B}$ nor $\\mathcal{L}$ survives. Starting from $U(1)_{\\phi}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$, two $U(1)$s are explicitly broken by $\\mathcal{L}_{\\mathrm{KSVZ+LQ}}$, while the remaining exact $U(1)$ is identified with $U(1)_{PQ}$ and spontaneously broken by $\\phi$. The interest in this scenario is that $S_{1}^{4\/3}$ couples only to $\\nu_{R}$, whose mass is pushed at the PQ breaking scale by the $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling. At the low-energy scale, the leading proton decay operator will scale as%\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)}^{eff}=\\exp\n(2ia^{0}\/v_{\\phi})\\frac{v_{\\phi}^{2}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\bar{d}%\n_{R}d_{R}^{\\mathrm{C}}\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}+h.c.\\rightarrow\n\\exp(2ia^{0}\/v_{\\phi})\\frac{v_{\\phi}v_{EW}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\bar{d}%\n_{R}d_{R}^{\\mathrm{C}}\\bar{u}_{R}\\nu_{L}^{\\mathrm{C}}+h.c.\\ .\n\\label{FinalScale11}%\n\\end{equation}\nThanks to this extra suppression, the PQ breaking scale, which is also the neutrino seesaw scale, and the LQ\/DQ mass scale, can all sit at around $10^{9}$ GeV. They could thus naturally have a common UV origin.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}-\\mathcal{L}$}\n\nWith only LQ\/DQ, scenarios in which $\\mathcal{B}-\\mathcal{L}$ is explicitly broken typically arise from any one of the $HS_{i}^{\\dagger}S_{j}$ or $HV_{i}^{\\dagger}V_{j}$ couplings in Eq.~(\\ref{LQOpsBmL}). Those couplings always involve a pure LQ state together with a mixed LQ\/DQ state. The $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ pattern arises when the latter has only DQ couplings. All these scenarios can be adapted to force $\\mathcal{B}-\\mathcal{L}$ to be broken spontaneously instead of explicitly. Let us take the $HS_{2}^{7\/3\\dagger}S_{1}^{4\/3}$ case as an example, the others being totally similar. To entangle the KSVZ symmetry with $\\mathcal{B}-\\mathcal{L}$, we start from the Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{4\/3}\\bar\n{d}_{R}^{\\mathrm{C}}d_{R}+S_{2}^{7\/3}(\\bar{u}_{R}\\ell_{L}+\\bar{q}_{L}%\ne_{R})+\\phi HS_{2}^{7\/3\\dagger}S_{1}^{4\/3}+h.c.\\ , \\label{LagrKSVZ2}%\n\\end{equation}\nwhere again kinetic terms and LQ\/DQ potential terms are understood. For definiteness, we also include the $S_{1}^{4\/3}\\bar{\\Psi}_{L}\\ell_{L}^{\\mathrm{C}}$ coupling to get rid of $U(1)_{\\Psi}$ and fix the quantum numbers of $\\Psi_{L,R}$. Then, there remain only a $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ symmetry with charges\n\\begin{equation}\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{2}^{7\/3}$ & $S_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $1$ & $1\/3$ & $-2\/3$ & $-2\/3$ & $-5\/3$ & $1\/3$ & $1\/3$\n& $1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-1$ & $-1$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0$ & $1$\n& $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nand $U(1)_{\\mathcal{B}-\\mathcal{L}}$ is spontaneously broken when $\\phi$ acquires its vacuum expectation value. Note that these charges prevent the LQ couplings of $S_{1}^{4\/3}$ (taking $\\phi HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ instead, they would further forbid the $HS_{2}^{1\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ coupling). The final operators are part of the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ dimension-seven ones in Table~\\ref{TableLQBL} because of the Higgs doublet appearing in the $\\phi HS_{2}^{7\/3\\dagger}S_{1}^{4\/3}$ mixing term (see Fig.~\\ref{Fig3}):\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\exp\n(ia^{0}\/v_{\\phi})\\frac{v_{\\phi}}{m_{S}^{2}m_{\\tilde{S}}^{2}}H\\bar{d}_{R}%\nd_{R}^{\\mathrm{C}}(\\bar{u}_{R}\\ell_{L}+\\bar{q}_{L}e_{R})+h.c.\\ .\n\\end{equation}\nThe situation is thus similar to that in Eq.$~$(\\ref{FinalScale11}). Further lowering the LQ\/DQ scale by about an order of magnitude is possible starting from the $HV_{1,\\mu}^{2\/3\\dagger}V_{2}^{5\/3,\\mu}$ coupling, as $V_{1,\\mu}^{2\/3\\dagger}$ couples only to $\\nu_{R}$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.3984in,width=1.9009in]{Fig3.jpg}\n\\caption{Proton decay operator generated by the spontaneous breaking of $U(1)_{\\mathcal{B}-\\mathcal{L}}$.}\n\\label{Fig3}\n\\end{center}\n\\end{figure}\n\nIn this regard, note that all these scenarios are again compatible with a seesaw mechanism. Adding either a $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, or $M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling to $\\mathcal{L}_{\\mathrm{KSVZ+LQ}}$ in Eq.~(\\ref{LagrKSVZ2}), a single exact $U(1)$ remains at the PQ scale, with charges\n\\begin{equation}\n\\begin{tabular}[c]{ccccccccccccc}\\hline\n&  & $\\phi$ & $S_{2}^{7\/3}$ & $\\tilde{S}_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$\n& $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2$ & $4\/3$ & $-2\/3$\n& $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $-1$ & $-1$ & $-1$\\\\\n$\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2\/3$ & $0$\n& $-2\/3$ & $-1$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$\\\\\n$M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $1$ & $1\/3$ & $-2\/3$\n& $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $0$ & $0$ & $0$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nFor all these cases, the axion still emerges as a massless Goldstone boson, and is associated to both $U(1)_{\\mathcal{B}-\\mathcal{L}}$ and $U(1)_{\\mathcal{L}}$ spontaneous breakings.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}$}\n\nThe spontaneous breaking of $\\mathcal{B}$ first arose at the dimension-9 level in Table~\\ref{TableLQBL} since it necessarily involves six fermions. As seen in Sec.~\\ref{Sec2c}, typical scenarios thus require a cubic coupling among DQ states. Let us start with\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{4\/3}\\bar\n{d}_{R}^{\\mathrm{C}}d_{R}+S_{1}^{8\/3}\\bar{u}_{R}^{\\mathrm{C}}u_{R}+\\phi\nS_{1}^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}+h.c.\\ , \\label{LagrKSVZ3}%\n\\end{equation}\nwhere $S_{1}^{4\/3}\\sim(\\mathbf{3},\\mathbf{1},+4\/3)$ and $S_{1}^{8\/3}\\sim(\\mathbf{\\bar{6}},\\mathbf{1},-8\/3)$. Though not compulsory, we add the coupling $S_{1}^{8\/3}\\bar{\\Psi}_{L}^{\\mathrm{C}}q_{L}$ to break $U(1)_{\\Psi}$ and fix the charges of $\\Psi_{L,R}$. With this Lagrangian, only two $U(1)$s are exact:\n\\begin{equation}\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{2}^{8\/3}$ & $S_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $2$ & $-2\/3$ & $-2\/3$ & $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$\n& $1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $1$ &\n$1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThus, $U(1)_{PQ}=U(1)_{\\mathcal{B}}$ is broken spontaneously by two units, but $U(1)_{\\mathcal{L}}$ stays exact. This model is actually very similar to that of Ref.~\\cite{Barbieri:1981yr} (see also Ref.~\\cite{Berezhiani:2015afa}), except that the Goldstone boson associated to the $U(1)_{\\mathcal{B}}$ breaking is identified with the axion. In turn, the axion ends up coupled to neutron pairs, via the diagram shown in Fig.~\\ref{Fig4a}. The corresponding operator is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)}^{eff}=\\exp\n(ia^{0}\/v_{\\phi})\\frac{v_{\\phi}}{m_{S^{4\/3}}^{4}m_{S^{8\/3}}^{2}}\\bar{d}%\n_{R}^{\\mathrm{C}}d_{R}\\bar{d}_{R}^{\\mathrm{C}}d_{R}\\bar{u}_{R}^{\\mathrm{C}%\n}u_{R}+h.c.\\ . \\label{ScaleNN}%\n\\end{equation}\nTypical bounds on the scale of the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators are at around $100$~TeV~\\cite{Mohapatra:2009wp,Phillips:2014fgb} if the couplings implicit in Eq.~(\\ref{LagrKSVZ3}) are all $\\mathcal{O}(1)$. The PQ scale of $10^{9}$~GeV pushes the DQ scale slightly higher than those $100\\ \\text{TeV}$, but given that their masses appear to the sixth power, this is marginal (less than an order of magnitude). The presence of the axion also leads to an effective operator%\n\\begin{equation}\n\\frac{1}{m_{S^{4\/3}}^{4}m_{S^{8\/3}}^{2}}a^{0}\\bar{d}_{R}^{\\mathrm{C}}d_{R}%\n\\bar{d}_{R}^{\\mathrm{C}}d_{R}\\bar{u}_{R}^{\\mathrm{C}}u_{R}+h.c.\\ \\rightarrow\n\\frac{\\Lambda_{QCD}^{6}}{m_{S^{4\/3}}^{4}m_{S^{8\/3}}^{2}}a^{0}\\bar\n{n}^{\\mathrm{C}}\\gamma_{5}n+h.c.\\ ,\n\\end{equation}\nwith the QCD confinement scale $\\Lambda_{QCD}$ of the order of $300$~MeV. Because the DQ mass scale is pushed rather high by the dimension-nine operator in Eq.~(\\ref{ScaleNN}), this direct coupling is very suppressed. Note, though, that it could have consequences in a cosmological context~\\cite{Mohapatra:2009wp}.%\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.8844in,width=1.7167in]{Fig4a.jpg}\n\\caption{Neutron-antineutron oscillation operators generated by the spontaneous breaking of $U(1)_{\\mathcal{B}}$.}\n\\label{Fig4a}\n\\end{center}\n\\end{figure}\n\nAs for the previous two scenarios, a seesaw mechanism can be implemented by adding a $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, or $M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling to the Lagrangian in Eq.~(\\ref{LagrKSVZ3}). For the former two, this identifies the axion as the Majoron~\\cite{Barbieri:1981yr}. The only change is, in some sense, to assign a $\\mathcal{B}$ number to $\\nu_{R}$, hence by extension, to the leptons:\n\\begin{equation}\n\\begin{tabular}[c]{ccccccccccccc}\\hline\n&  & $\\phi$ & $S_{2}^{8\/3}$ & $S_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ &\n$q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2$ & $-2\/3$ & $-2\/3$\n& $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $-1$ & $-1$ & $-1$\\\\\n$\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2$ &\n$-2\/3$ & $-2\/3$ & $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1$ & $1$ & $1$\\\\\n$M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2$ & $-2\/3$ &\n$-2\/3$ & $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $0$ & $0$ & $0$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nNote that the charges imposed by the presence of $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ open the door to the $S_{1}^{4\/3}\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}$ coupling also, and thus to direct proton decay via an $S_{1}^{4\/3}$ Fermi interaction. For the other two scenarios, proton decay remains forbidden since all its decay modes include an odd number of leptons, but only $\\Delta\\mathcal{L}=2n$ transitions are made possible by the Lagrangian couplings.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}\\pm3\\mathcal{L}$}\n\nFrom Eq.~(\\ref{LQOpsB2}), it is clear that the scenarios leading to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ can be adapted to generate $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ effects. All that is needed is to replace all DQ couplings by LQ couplings. The only difficulty is to account for the antisymmetric color contraction, since LQ necessarily transform as $\\mathbf{3}$ under $SU(3)_{C}$. If we insist on introducing at most two different LQ, the only available scenario is%\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{2\/3}(\\bar\n{d}_{R}\\nu_{R}^{\\mathrm{C}}+\\bar{u}_{R}e_{R}^{\\mathrm{C}}+\\bar{q}_{L}\\ell\n_{L}^{\\mathrm{C}})+V_{2,\\mu}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}%\n^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}})+\\phi S_{1}%\n^{2\/3}V_{2,\\mu}^{1\/3}V_{2}^{1\/3,\\mu}+h.c.\\ . \\label{LagrKSVZ4}%\n\\end{equation}\nAs usual, the $U(1)_{\\Psi}$ is broken explicitly, this time by adding $V_{2,\\mu}^{1\/3}\\bar{\\Psi}_{L}\\gamma^{\\mu}e_{R}^{\\mathrm{C}}$ to force the hypercharge of $\\Psi_{L,R}$ to be different from that of SM quarks. If instead of the LQ couplings, DQ couplings were allowed, this scenario produces the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ symmetry pattern discussed in the previous section. Now, with these LQ couplings and no DQ couplings, the charges are\n\\begin{equation}%\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{2}^{1\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $1$ & $1\/3$ & $1\/3$ & $1\/3$ & $-2\/3$ & $1\/3$ & $1\/3$ &\n$1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $3$ & $1$ & $1$ & $0$ & $-3$ & $0$ & $0$ & $0$ & $1$ &\n$1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThe PQ symmetry is identified with $U(1)_{\\mathcal{B}+3\\mathcal{L}}$, and dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ proton decay operators appear at the low scale, see Fig.~\\ref{Fig4b}$a$ (a similar LQ model was proposed in Ref.~\\cite{WeinbergPRD22} to break $\\mathcal{B}+3\\mathcal{L}$ spontaneously). The fact that these operators are dimension-nine allows to lower the LQ scale, but qualitatively, this scenario is not very different from the $\\mathcal{B}\\pm\\mathcal{L}$ ones. Also, a seesaw mechanism can be added with either $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ or $M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, but not with $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ as this would allow back the DQ couplings of both $S_{1}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$. It should be noted that these charges allow for the $D^{\\mu}HS_{1}^{2\/3}V_{2,\\mu}^{1\/3\\dagger}$ and $HD^{\\mu}S_{1}^{2\/3}V_{2,\\mu}^{1\/3\\dagger}$couplings. If not initially present, they are immediately generated via a fermion loop. Yet, these operators carry $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,0)$ and cannot help create simpler proton decay processes. They could turn on some new four-fermion semileptonic FCNC operators though, but these effects are beyond our scope.%\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=2.2329in,width=4.9536in]{Fig4b.jpg}\n\\caption{Proton decay operators generated by the spontaneous breaking of $U(1)_{\\mathcal{B}+3\\mathcal{L}}$ ($a.$) and $U(1)_{\\mathcal{B}-3\\mathcal{L}}$ ($b.$).}\n\\label{Fig4b}\n\\end{center}\n\\end{figure}\n\nThe final pattern is $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$, and this one is quite difficult to induce spontaneously. The operators in Eq.~(\\ref{LQOpsBL3}) being already of dimension four, we cannot proceed as for the other cases and simply multiply them by $\\phi$. One way to proceed is to start with an operator from the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$\nclass in Eq.~(\\ref{LQOpsB2}), and then switch $\\mathcal{L}$ by six units using $\\Delta\\mathcal{L}=2$ operators of Eq.~(\\ref{LQOpsBmL}). For instance, the Lagrangian\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}  &  =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{2}%\n^{1\/3}(\\bar{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})+V_{1,\\mu}^{2\/3}\\bar{d}%\n_{R}\\gamma^{\\mu}\\nu_{R}\\nonumber\\\\\n&  \\ \\ \\ \\ +\\phi(HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}+H^{\\dagger}V_{1}%\n^{2\/3\\dagger,\\mu}V_{2,\\mu}^{1\/3}+S_{1}^{2\/3\\dagger}V_{2,\\mu}^{1\/3\\dagger}%\nV_{2}^{1\/3,\\mu\\dagger})+h.c.\\ , \\label{LagrKSVZ5}%\n\\end{align}\ndoes lead to the desired $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ pattern, as shown in Fig.~\\ref{Fig4b}$b$. With four LQ states, it is certainly more complex than the other scenarios, though it should be noted that there is a certain symmetric flavor to the presence of $S_{2}^{1\/3}$, $S_{1}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$, $V_{1,\\mu}^{2\/3}$. Also, it is not compulsory for $\\phi$ to appear in all of the last three couplings, but when it does, only some combinations do give a $\\mathcal{B}-3\\mathcal{L}$ charge to $\\phi$. Further adding $V_{1,\\mu}^{2\/3}\\bar{\\Psi}_{R}\\gamma^{\\mu}e_{R}$ to fix the $\\Psi_{L,R}$ quantum numbers, we find%\n\\begin{equation}\n\\begin{tabular}[c]{cccccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{2}^{1\/3}$ & $S_{2}^{1\/3}$ & $V_{1}^{2\/3}$ &\n$\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ &\n$\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $1\/4$ & $1\/12$ & $1\/12$ & $1\/3$ & $1\/3$ & $7\/12$ &\n$1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-3\/4$ & $-1\/4$ & $-1\/4$ & $-1$ & $-1$ & $-3\/4$ & $0$ &\n$0$ & $0$ & $0$ & $1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThe PQ symmetry is thus indeed $U(1)_{\\mathcal{B}-3\\mathcal{L}}$. Note that these charges forbid all the SM fermion couplings of $S_{1}^{2\/3}$ and $V_{2}^{1\/3}$, as well as all other possible cubic interactions among the LQ and DQ states\\footnote{Some derivative interactions are possible though, but those necessarily involve the LQs whose SM fermion couplings are forbidden, hence they do not alter the symmetry breaking pattern, and would lead to more suppressed proton decay operators.}. However, given the complicated structure shown in Fig.~\\ref{Fig4b}$b$, the final operators are of dimension 16 instead of dimension 10:\n\\begin{align}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)}^{eff}  &\n=\\frac{\\phi^{4}(H^{\\dagger}H)}{m_{S1\/3}^{2}m_{V2\/3}^{4}m_{S2\/3}^{2}%\nm_{V1\/3}^{4}}H^{\\dagger}(\\bar{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})\\bar{d}%\n_{R}\\gamma_{\\mu}\\nu_{R}\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}\\nonumber\\\\\n&  \\rightarrow\\frac{v_{\\phi}^{4}v_{EW}^{3}}{m_{S1\/3}^{2}m_{V2\/3}^{4}m_{S2\/3}%\n^{2}m_{V1\/3}^{4}}(\\bar{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})\\bar{d}_{R}%\n\\gamma_{\\mu}\\nu_{R}\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}\\ .\n\\end{align}\nBesides, turning on a seesaw mechanism with $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ (as $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ would allow back some $S_{1}^{2\/3}$ and $V_{2}^{1\/3}$ couplings to SM fermions), a further suppression of $(v_{EW}\/v_{\\phi})^{2}$ to connect two $\\nu_{R}$ to light fermions arises. Altogether, assuming a common scale for all the LQs, their mass can be as low as around 100 TeV when $v_{\\phi}\\approx10^{9}$ GeV. This is much lower than in GUT scenarios, and actually falls within the ballpark of the scale required by neutron-antineutron oscillation from Eq.~(\\ref{ScaleNN}).\n\n\\subsection{DFSZ scenarios with leptoquarks and diquarks\\label{SecDFSZ}}\n\nAll the scenarios discussed in the KSVZ case can readily be adapted to the DFSZ model. Basically, one removes the $\\Psi_{L,R}$ state but introduces a $\\phi^{2}H_{u}^{\\dagger}H_{d}$ coupling, while the $\\phi$ couples to various combinations of LQ\/DQ states exactly as in the KSVZ scenarios. A number of peculiarities are worth mentioning though:\n\n\\begin{enumerate}\n\\item The symmetry patterns are more difficult to analyze in the DFSZ case because the PQ and hypercharge symmetries are entangled, see Eq.~(\\ref{DFSZScalars}). Thus, further entangling $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ with the $U(1)$s associated to $H_{u}$ and $H_{d}$ rephasing blurs the picture completely. In practice, the PQ charges of $\\phi$, $H_{u}$, and $H_{d}$ are always fixed to $PQ(H_{u})=x$, $PQ(H_{d})=-1\/x$, $PQ(\\phi)=\\left(  x+1\/x\\right)  \/2$, see Eq.~(\\ref{DFSZScalars}), no matter the amount of $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ that is entangled within $U(1)_{PQ}$.\n\n\\item Because $H_{u}$, and $H_{d}$ carry PQ charges, so does the SM fermions, even without the presence of LQ\/DQ states. As shown in Eq.~(\\ref{DFSZfermions}), these charges have ambiguities reflecting the exact accidental symmetries. Thus, any entanglement of $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ with $U(1)_{PQ}$ will be reflected in that arbitrariness. Typically, only one free parameter will remain instead of the $\\beta$ and $\\gamma$ parameters of Eq.~(\\ref{DFSZfermions}). Thus, looking at this remaining arbitrariness permits to identify the combination of $\\beta$ and $\\gamma$, i.e., $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$, that has been spontaneously broken.\n\n\\item Because $U(1)_{PQ}$ has always a component within $U(1)_{Hu}\\otimes U(1)_{Hd}$, the PQ charge of the SM fermions are never fully aligned with some combinations of $\\mathcal{B}$ and $\\mathcal{L}$. As a result, LQ states are often restricted to couple to only a single SM fermion LQ or DQ pair. For example, the gauge symmetries allow both $S_{2}^{1\/3}\\bar{d}_{R}\\ell_{L}$ and $S_{2}^{1\/3}\\bar{q}_{L}\\nu_{R}$, but the PQ charge do not since $PQ(\\bar{d}_{R}\\ell_{L})=\\gamma-\\beta+1\/x$ and $PQ(\\bar{q}_{L}\\nu_{R})=\\gamma-\\beta+x$, and this is true independently of $\\beta$ and $\\gamma$. In some cases, this actually makes the choice of LQ\/DQ couplings more natural than in the KSVZ case, since once some of them are selected, the others are immediately forbidden.\n\n\\item With $H_{u,d}$ at hand, many new couplings to LQ\/DQ states are a priori possible already in the scalar potential. For instance, replacing $H$ by $H_{u}$ or $H_{d}$ in any of the couplings in Eqs.~(\\ref{LQOpsB2}) or~(\\ref{LQOpsBL3}) would couple the axion to $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating operators. However, these situations correspond to breaking $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ at the electroweak scale, by entangling them with $U(1)_{Y}$. Indeed, $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating operators would involve $\\eta_{u}$ or $\\eta_{d}$ (the pseudoscalar components of $H_{u}$ and $H_{d}$), and thus also the Would-be Goldstone associated to $U(1)_{Y}$ since $G^{0}\\sim v_{u}\\eta_{u}+v_{d}\\eta_{d}$. The axion has only tiny $\\eta_{u}$ and $\\eta_{d}$ components, see Eq.~(\\ref{DFSZA0}). Turning on some $H_{u}^{\\dagger}H_{d}S_{j}^{\\dagger}S_{i}$ couplings would prevent any $G^{0}$ coupling, but would similarly lead to tiny axion couplings via its $\\cos\\beta\\eta_{u}-\\sin\\beta\\eta_{d}$ component. For these reasons, all the scenarios discussed below start from coupling $\\phi$ to the LQ\/DQ states, so that $\\mathcal{B}$ and\/or $\\mathcal{L}$ are broken at the PQ scale and the axion inherits some large $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating couplings. These scenarios are thus constructed exactly as in the KSVZ case.\n\\end{enumerate}\n\nAfter these general comments, let us briefly go through each of the $\\mathcal{B}$ and\/or $\\mathcal{L}$ spontaneous breaking scenario.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}+\\mathcal{L}$}\n\nBy analogy with the KSVZ scenario, Eq.~(\\ref{LagrKSVZ1a}), let us take%\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}=\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}^{8\/3}\\bar\n{d}_{R}e_{R}^{\\mathrm{C}}+\\tilde{S}_{1}^{8\/3}\\bar{u}_{R}^{\\mathrm{C}}%\nu_{R}+\\phi^{2}S_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}+h.c.\\ , \\label{LagrDFSZ1}%\n\\end{equation}\nwith $\\mathcal{L}_{\\mathrm{DFSZ}}$ given in Eq.~(\\ref{DFSZ0}). Solving for the $U(1)$ charges under the constraint that $PQ(H_{u})=x$, $PQ(H_{d})=-1\/x$, which fixes $PQ(\\phi)=\\left(  x+1\/x\\right)  \/2$, leaves a single under-determination. In this way, we identify the remaining symmetry as $U(1)_{\\mathcal{B}-\\mathcal{L}}$, with%\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{8\/3}$ & $\\tilde{S}_{1}^{8\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ &\n$\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\dfrac{1}{x}-x\\rule[-0.14in]{0in}{0.36in}$ & $-2x$ & $0$ & $x$\n& $\\dfrac{1}{x}$ & $-\\dfrac{1}{x}-x$ & $-x$ & $-\\dfrac{1}{x}$\\\\\n$U(1)_{\\mathcal{B}-\\mathcal{L}}$ & $-2\/3$ & $-2\/3$ & $1\/3$ & $1\/3$ & $1\/3$ &\n$-1$ & $-1$ & $-1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nwith $\\phi$, $H_{u}$, and $H_{d}$ neutral under $U(1)_{\\mathcal{B}-\\mathcal{L}}$. This shows that $U(1)_{\\mathcal{B}+\\mathcal{L}}\\subset U(1)_{PQ}\\subset U(1)_{Hu}\\otimes U(1)_{Hd}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ is spontaneously broken. Note well that the quoted $U(1)_{PQ}$ charges are just one possible choice, since $U(1)_{\\mathcal{B}-\\mathcal{L}}$ remains as an ambiguity. We could also have written the charges as\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{8\/3}$ & $\\tilde{S}_{1}^{8\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ &\n$\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\dfrac{1}{x}-x-2\\xi\\rule[-0.14in]{0in}{0.36in}$ & $-2x-2\\xi$ &\n$\\xi$ & $x+\\xi$ & $\\dfrac{1}{x}+\\xi$ & $-\\dfrac{1}{x}-x-3\\xi$ & $-x-3\\xi$ &\n$-\\dfrac{1}{x}-3\\xi$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nwith $\\xi$ the free parameter corresponding to $U(1)_{\\mathcal{B}-\\mathcal{L}}$. We can also see that this corresponds to Eq.~(\\ref{DFSZfermions}) with $\\beta=\\xi$ and $\\gamma=-\\dfrac{1}{x}-x-3\\xi$. This shows that the dimension-five axion to gauge boson couplings are unaffected by the LQ\/DQ, since they are independent of $\\beta$ and $\\gamma$. Also, one should not conclude that the axion does not couple to $q_{L}$, even though that coupling is absent from the axion effective Lagrangian since $PQ(q_{L})$ is set to zero.\n\nConcerning the axion $\\mathcal{B}+\\mathcal{L}$ violating operator, the same effective interactions arises as in the KSVZ scenario, see Eq.~(\\ref{EffHKSVZ1}). This is evident from Fig.~\\ref{Fig2}, which is independent of how the axion emerges. The only difference is that the pseudoscalar component of $\\phi$ is not purely the axion, but this is only a totally negligible $\\mathcal{O}(v_{EW}\/vs)$ effect, see Eq.~(\\ref{DFSZA0}).\n\nFinally, exactly as in the KSVZ scenario, the remaining $U(1)_{\\mathcal{B}-\\mathcal{L}}$ freedom permits to set up a PQ-induced seesaw mechanism by adding $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ or $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$. In both cases, this simply fixes the parameter $\\xi$ and removes the remaining $U(1)_{\\mathcal{B}-\\mathcal{L}}$ ambiguity. Yet, the final PQ charges do not reflect at all the peculiar symmetry breaking pattern, with $U(1)_{\\mathcal{B}+\\mathcal{L}}$ and $U(1)_{\\mathcal{L}}$ being separately, but concurrently, spontaneously broken at the PQ scale. By the way, exactly the same PQ charges arise if $\\phi^{2}S_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}$ is replaced by $H_{d}^{\\dagger}H_{u}S_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}$, though as discussed before, the symmetry breaking chain is very different, as are the axion couplings.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}-\\mathcal{L}$}\n\nPursuing our adaptation of the KSVZ scenario, let us consider now%\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}=\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}^{4\/3}\\bar\n{d}_{R}^{\\mathrm{C}}d_{R}+S_{2}^{7\/3}\\bar{u}_{R}\\ell_{L}+\\phi H_{u}%\nS_{2}^{7\/3\\dagger}S_{1}^{4\/3}+h.c.\\ . \\label{LagrDFSZ2}%\n\\end{equation}\nBoth fermionic couplings of $S_{2}^{7\/3}$ cannot be present at the same time for the PQ symmetry to exist, so we choose to keep $S_{2}^{7\/3}\\bar{u}_{R}\\ell_{L}$ and discard $S_{2}^{7\/3}\\bar{q}_{L}e_{R}$. Also, we introduced $H_{u}$ in the quartic scalar coupling, but could equally well have used $H_{d}$. From this Lagrangian, the PQ charges are found to be\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{4\/3}$ & $S_{2}^{7\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ &\n$e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $-\\dfrac{2}{x}\\rule[-0.14in]{0in}{0.36in}$ & $\\dfrac{3x}%\n{2}-\\dfrac{3}{2x}$ & $0$ & $x$ & $\\dfrac{1}{x}$ & $\\dfrac{3}{2x}-\\dfrac{x}{2}$\n& $\\dfrac{5}{2x}-\\dfrac{x}{2}$ & $\\dfrac{3}{2x}+\\dfrac{x}{2}$\\\\\n$U(1)_{\\mathcal{B}+\\mathcal{L}}$ & $-2\/3$ & $-2\/3$ & $1\/3$ & $1\/3$ & $1\/3$ &\n$1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nSo, $U(1)_{\\mathcal{B}-\\mathcal{L}}$ is spontaneously broken at the PQ scale, but $U(1)_{\\mathcal{B}+\\mathcal{L}}$ remains. As before, we could rewrite the PQ charge introducing a free parameter to reflect the exact $U(1)_{\\mathcal{B}+\\mathcal{L}}$ symmetry, hence one should not interpret $PQ(q_{L})=0$ as meaning it has no coupling to the axion. The $U(1)_{\\mathcal{B}+\\mathcal{L}}$ ambiguity can then be used to allow for a $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ or $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling, and set up the seesaw mechanism.\n\nThe final $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operator is again one of the dimension-seven operators listed in Table~\\ref{TableLQBL}. Note, though, that because the PQ symmetry restricts the LQ couplings to SM fermions, only a single operator is induced. This is a generic characteristic of the DFSZ implementation: compared to the KSVZ case, it is more restrictive. Phenomenologically, this could show up as definite decay patterns for the proton (if ever observed). Starting from Eq.~(\\ref{LagrDFSZ2}), the operator arising at tree level is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac{1}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\phi H_{u}\\bar{d}_{R}d_{R}^{\\mathrm{C}}\\bar\n{u}_{R}\\ell_{L}+h.c.\\ ,\n\\end{equation}\nNote that some other gauge and PQ invariant operators may arise at higher loops via Yukawa insertions, but those are more suppressed. The leading proton decay operator is thus proportional to $v_{\\phi}v_{u}\/m_{S}^{4}$, and the constraints are similar as in the KSVZ scenario. Concerning the axion coupling, notice that\n\\begin{equation}\n\\phi H_{u}\\ell_{L}\\rightarrow\\frac{1}{2}v_{u}v_{\\phi}\\exp i\\left(  \\frac{\\eta\n_{u}}{v_{u}}+\\frac{v_{\\phi}}{v_{\\phi}}\\right)  \\nu_{L}\\ ,\n\\end{equation}\nso the combination that occurs in the effective operator is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{v_{\\phi}v_{u}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\left(  1+i\\frac{G^{0}}{v}%\n+i\\frac{a^{0}}{v_{\\phi}}\\frac{3x^{2}+1}{x^{2}+1}+i\\frac{\\pi^{0}}{v}x\\right)\n\\bar{d}_{R}d_{R}^{\\mathrm{C}}\\bar{u}_{R}\\nu_{L}\\ .\n\\end{equation}\nFor comparison, the $\\mu H_{u}S_{2}^{7\/3\\dagger}S_{1}^{4\/3}$ coupling would lead to\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac{\\mu\nv_{u}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\left(  1+i\\frac{G^{0}}{v}+i\\frac{a^{0}%\n}{v_{\\phi}}\\frac{2x^{2}}{x^{2}+1}+i\\frac{\\pi^{0}}{v}x\\right)  \\bar{d}_{R}%\nd_{R}^{\\mathrm{C}}\\bar{u}_{R}\\nu_{L}\\ ,\n\\end{equation}\nwith $\\mu$ some mass scale. The $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operator arises at the $v_{\\phi}$ scale from $\\phi H_{u}S_{2}^{7\/3\\dagger}S_{1}^{4\/3}$, but at a lower scale from $H_{u}S_{2}^{7\/3\\dagger}S_{1}^{4\/3}$ since we would expect $\\mu$ to be at the LQ\/DQ scale, $\\mu\\sim m_{S}$, or even at the electroweak scale, $\\mu\\sim v$. Note that in both cases, the $G^{0}$ enters as expected for a would-be Goldstone, and would disappear in the unitary gauge. The axion coupling is $\\mathcal{O}(v_{EW}\/v_{\\phi})$ compared to the four-fermion operator, exactly like in the KSVZ scenario.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}$}\n\nNeutron-antineutron oscillations can be induced in the same way in the DFSZ and KSVZ models, see Fig.~\\ref{Fig4a}. Starting with\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}=\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}^{4\/3}\\bar\n{d}_{R}^{\\mathrm{C}}d_{R}+S_{1}^{8\/3}\\bar{u}_{R}^{\\mathrm{C}}u_{R}+\\phi\nS_{1}^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}+h.c.\\ , \\label{LagrDFSZ3}%\n\\end{equation}\nwhere $S_{1}^{4\/3}\\sim(\\mathbf{3},\\mathbf{1},+4\/3)$ and $S_{1}^{8\/3}\\sim(\\mathbf{\\bar{6}},\\mathbf{1},-8\/3)$, we get the PQ charges%\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{4\/3}$ & $S_{1}^{8\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ &\n$e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\dfrac{x}{2}-\\dfrac{5}{6x}\\rule[-0.14in]{0in}{0.36in}$ &\n$\\dfrac{7}{6x}-\\dfrac{3x}{2}$ & $-\\dfrac{x}{4}-\\dfrac{7}{12x}$ & $\\dfrac\n{3x}{4}-\\dfrac{7}{12x}$ & $\\dfrac{5}{12x}-\\dfrac{x}{4}$ & $0$ & $\\dfrac{1}{x}$\n& $x$\\\\\n$U(1)_{\\mathcal{L}}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThus, $U(1)_{\\mathcal{B}}$ is broken spontaneously, but $U(1)_{\\mathcal{L}}$ stays exact. The phenomenology is the same as in the KSVZ model, see Fig.~\\ref{Fig4a} and Eq.~(\\ref{ScaleNN}). Majorana neutrino masses can be generated spontaneously with $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, but not with $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ as the PQ charges of the leptons would then allow for the $S_{1}^{4\/3}\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}$ coupling, and thereby to tree-level proton decay.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}\\pm3\\mathcal{L}$}\n\nThe last two scenarios are those producing exotic $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,\\pm3)$ proton decay operators. Let us start with the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ case, and the Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}=\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}^{2\/3}\\bar\n{q}_{L}\\ell_{L}^{\\mathrm{C}}+V_{2,\\mu}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell\n_{L}^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}})+\\phi^{\\dagger}\nS_{1}^{2\/3}V_{2,\\mu}^{1\/3}V_{2}^{1\/3,\\mu}+h.c.\\ . \\label{LagrDFSZ4}%\n\\end{equation}\nThe $S_{1}^{2\/3}(\\bar{d}_{R}\\nu_{R}^{\\mathrm{C}}+\\bar{u}_{R}e_{R}^{\\mathrm{C}})$ and $S_{1}^{2\/3}\\bar{q}_{L}\\ell_{L}^{\\mathrm{C}}$ couplings cannot both be present, and we take only the latter, while the $V_{2,\\mu}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu}\\nu _{R}^{\\mathrm{C}})$ couplings are compatible with each other. The $U(1)$ charges are then\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{2\/3}$ & $V_{2,\\mu}^{1\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$\n& $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\dfrac{1}{6x}-\\dfrac{x}{2}\\rule[-0.14in]{0in}{0.36in}$ &\n$\\dfrac{x}{2}+\\dfrac{1}{6x}$ & $0$ & $x$ & $\\dfrac{1}{x}$ & $\\dfrac{1}%\n{6x}-\\dfrac{x}{2}$ & $\\dfrac{7}{6x}-\\dfrac{x}{2}$ & $\\dfrac{1}{6x}+\\dfrac\n{x}{2}$\\\\\n$U(1)_{3\\mathcal{B}-\\mathcal{L}}$ & $0$ & $0$ & $1$ & $1$ & $1$ & $-1$ & $-1$\n& $-1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThe $U(1)_{3\\mathcal{B}-\\mathcal{L}}$ symmetry remains, and its orthogonal combination $U(1)_{\\mathcal{B}+3\\mathcal{L}}$ is spontaneously broken. Dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ proton decay operators thus appear at the low scale (as well as semileptonic $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,0)$ operators since these charges allow for the $D^{\\mu}HS_{1}^{2\/3}V_{2,\\mu}^{1\/3\\dagger}$ couplings). Once more, there is enough room for a seesaw mechanism with $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ and\/or $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$. Depending on the LQ couplings of $S_{1}^{2\/3}$, it is always possible to choose the seesaw operator that sets PQ charges forbidding the DQ couplings of both $S_{1}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$, and thus proton decay via dimension-six operators.\n\nConcerning the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ operators, we start from\n\\begin{align}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}  &  =\\mathcal{L}_{\\mathrm{DFSZ}}+S_{2}%\n^{1\/3}\\bar{d}_{R}\\ell_{L}+V_{1,\\mu}^{2\/3}\\bar{d}_{R}\\gamma^{\\mu}\\nu\n_{R}\\nonumber\\\\\n&  +\\phi(H_{u}S_{2}^{1\/3\\dagger}S_{1}^{2\/3}+H_{d}^{\\dagger}V_{1}%\n^{2\/3\\dagger,\\mu}V_{2,\\mu}^{1\/3}+S_{1}^{2\/3\\dagger}V_{2,\\mu}^{1\/3\\dagger}\nV_{2}^{1\/3,\\mu\\dagger})+h.c.\\ , \\label{LagrDFSZ5}%\n\\end{align}\nwith the $S_{2}^{1\/3}\\bar{q}_{L}\\nu_{R}$ removed. Several choices are possible for introducing the doublets $H_{u}$ and $H_{d}$ in these couplings, and we opt for the one most symmetrical with the Yukawa couplings, see Eq.~(\\ref{DFSZ0}). Only one of the fermionic couplings of $S_{2}^{1\/3}$ can be turned on, and we choose $S_{2}^{1\/3}\\bar{d}_{R}\\ell_{L}$. Then, the $U(1)$\ncharges are found to be\n\\begin{equation}%\n\\begin{tabular}\n[c]{ccccccccccc}\\hline\n& $S_{1}^{2\/3}$ & $V_{2,\\mu}^{1\/3}$ & $S_{2}^{1\/3}$ & $V_{1,\\mu}^{2\/3}$ &\n$q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}\\rule[-0.14in]{0in}{0.39in}$ & $\\dfrac{x^{2}+5}{6x}$ & $\\dfrac\n{x^{2}-1}{6x}$ & $\\dfrac{5x^{2}+4}{3x}$ & $\\dfrac{2x^{2}+4}{3x}$ & $0$ & $x$ &\n$\\dfrac{1}{x}$ & $\\dfrac{5x^{2}+1}{-3x}$ & $\\dfrac{5x^{2}-2}{-3x}$ &\n$\\dfrac{2x^{2}+1}{-3x}$\\\\\n$U(1)_{3\\mathcal{B}+\\mathcal{L}}$ & $0$ & $0$ & $0$ & $0$ & $1$ & $1$ & $1$ &\n$1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThis time, $U(1)_{3\\mathcal{B}+\\mathcal{L}}$ remains and $U(1)_{\\mathcal{B}-3\\mathcal{L}}$ is spontaneously broken. The induced operator, from a process easily adapted from that of Fig.~\\ref{Fig4b}, is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)}^{eff}=\\frac\n{\\phi^{4}(H_{d}^{\\dagger}H_{d}^{\\dagger}H_{u})}{m_{S1\/3}^{2}m_{V2\/3}^{4}m_{S2\/3}^{2}%\nm_{V1\/3}^{4}}\\bar{d}_{R}\\ell_{L}\\bar{d}_{R}\\gamma_{\\mu}\\nu\n_{R}\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}\\ .\n\\end{equation}\nAgain, phenomenologically, there is not much difference between the DFSZ and KSVZ implementation.\n\n\\subsection{Axion-induced proton decay and neutron-antineutron oscillations\\label{SecSpont}}\n\nIn both the KSVZ and DFSZ cases, we can induce spontaneously proton decay or neutron-antineutron oscillations. But, in all the scenarios discussed up to now, the processes involving the axion were $\\mathcal{O}(v_{EW}\/v_{\\phi})$ with respect to that without it. The reason is of course that in all cases, some coupling of $\\phi$ to the LQ\/DQ states was introduced, and $\\phi=(v_{\\phi}+\\rho)\\exp(i\\eta_{\\phi}\/v_{\\phi})\\approx v_{\\phi}+\\rho+ia^{0}$ (see Eq.~(\\ref{ExampleSSS})). The purpose here is to kill off the leading term, leaving only axion-induced $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating processes. The only way to achieve this is to consider derivative couplings of $\\phi$ to pairs of LQ\/DQs, and there are only three renormalizable options\n\\begin{equation}\n\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}\\ ,\\ \\partial_{\\mu}\\phi\nS_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}\\ ,\\ \\partial_{\\mu}\\phi S_{1}^{4\/3\\dagger\n}V_{1}^{4\/3,\\mu}.\n\\label{DerScenars}\n\\end{equation}\nIn these cases, the axion enters as $\\partial_{\\mu}\\phi\\approx\\partial_{\\mu}\\rho+i\\partial_{\\mu}a^{0}$, without a leading term tuned by $v_{\\phi}$. Though we have not attempted at constructing UV complete models generating such interactions, their structure is evocative of that which could arise if both $\\phi$ and scalar LQ\/DQ were somehow related to the fields giving masses to the vector LQ\/DQ. Such a situation can happen in simple GUT models: In Ref.~\\cite{Wise:1981ry} for example, $\\phi$ is identified with the phase of the complex $H_{24}$ field breaking $SU(5)$ down to the SM gauge group. Note, though, that the PQ breaking scale and the LQ\/DQ mass scale would be related in such models. In the present section, the two will be kept independent, with the latter usually much lower than the former.\n\nLet us see which symmetry breaking patterns can be achieved with these building blocks. We will use the KSVZ setting throughout as the alignments of the PQ with some combination of $\\mathcal{B}$ and $\\mathcal{L}$ are manifest, but the adaptation to the DFSZ scenario is immediate. Also, we will discard $\\Psi_{L,R}$ from the discussion. As in Sec.~\\ref{SecKSVZ}, their charge can always be set separately by introducing some couplings to the LQ\/DQ.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}-\\mathcal{L}$}\n\nThe scenarios with $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators are immediately obtained using any one of the three couplings in Eq.~(\\ref{DerScenars}). For example, we can take\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{2\/3}(\\bar\n{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{1,\\mu}^{2\/3}%\n\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}+\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}%\nV_{1}^{2\/3,\\mu}+h.c.\\ , \\label{LagrSSB1}%\n\\end{equation}\nand get two active $U(1)$s, with charges\n\\begin{equation}%\n\\begin{tabular}[c]{cccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{1,\\mu}^{2\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ &\n$\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $-1$ & $-2\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $0$ &\n$0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $1$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $1$ & $1$ &\n$1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThus, $\\phi$ spontaneously breaks $U(1)_{\\mathcal{B}-\\mathcal{L}}$, leaving $\\mathcal{B}+\\mathcal{L}$ as an exact accidental symmetry. As before, we can add a $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling to generate neutrino masses and forbid the LQ couplings of $S_{1}^{2\/3}$.\n\nThe situation starting from the other derivative interaction is similar, hence we can generate:\n\\begin{align}\n\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}  &  \\rightarrow\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{1}{m_{S}^{2}m_{V}^{2}}\\partial_{\\mu}\\phi(\\bar{q}_{L}^{\\mathrm{C}}q_{L}%\n+\\bar{d}_{R}^{\\mathrm{C}}u_{R})\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}\\nu\n_{R}^{\\mathrm{C}}\\ ,\\\\\n\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}  &  \\rightarrow\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{1}{m_{S}^{2}m_{V}^{2}}\\partial_{\\mu}\\phi\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu\n}q_{L}(\\bar{d}_{R}^{\\mathrm{C}}\\ell_{L}^{\\mathrm{C}}+\\bar{q}_{L}^{\\mathrm{C}%\n}\\nu_{R}^{\\mathrm{C}})\\ ,\\\\\n\\partial_{\\mu}\\phi S_{1}^{4\/3\\dagger}V_{1}^{4\/3,\\mu}  &  \\rightarrow\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{1}{m_{S}^{2}m_{V}^{2}}\\partial_{\\mu}\\phi\\bar{d}_{R}^{\\mathrm{C}}d_{R}(\\bar\n{u}_{R}^{\\mathrm{C}}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}}+\\bar{d}_{R}^{\\mathrm{C}%\n}\\gamma^{\\mu}e_{R}^{\\mathrm{C}}+\\bar{q}_{L}^{\\mathrm{C}}\\gamma^{\\mu}\\ell\n_{L}^{\\mathrm{C}})\\ .\n\\end{align}\nAll these situations induce $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators, see Fig.~\\ref{Fig5}$a$, but necessarily involving the axion, with for example\n\\begin{align}\n\\frac{\\partial_{\\mu}\\phi}{m_{S}^{2}m_{V}^{2}}\\bar{d}_{R}\\gamma^{\\mu}%\nq_{L}^{\\mathrm{C}}\\bar{d}_{R}\\ell_{L}+h.c. & \\rightarrow\\frac{1}{m_{S}^{2}%\nm_{V}^{2}}\\partial_{\\mu}a^{0}\\bar{d}_{R}\\gamma^{\\mu}q_{L}^{\\mathrm{C}}\\bar\n{d}_{R}\\ell_{L}+h.c. \\nonumber \\\\ \n & \\rightarrow\\frac{\\Lambda_{QCD}^{3}}{m_{S}^{2}m_{V}^{2}%\n}(\\partial_{\\mu}a^{0}\\bar{p}\\gamma^{\\mu}\\left(  1-\\gamma^{5}\\right)\n\\ell + \\partial_{\\mu}a^{0}\\bar{n}\\gamma^{\\mu}\\left(  1-\\gamma^{5}\\right)\n\\nu + ... + h.c.)\\ ,\n\\label{OpsAPL}\n\\end{align}\nwhere (...) denotes operators with additional light mesons. Given the proton decay bounds, and taking $\\Lambda_{QCD}\\approx 300$~MeV, this imposes a quite high bound $m_{S}\\approx m_{V}>10^{4}$~TeV, close to the PQ breaking scale and quite lower than the GUT scale. With those values, such $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators cannot affect significantly the phenomenology of the axion, as its coupling to photons or gluons remain much larger.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.5766in,width=4.7547in]{Fig5.jpg}%\n\\caption{The $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators involving one ($a.$) and two axions ($b.$).}%\n\\label{Fig5}%\n\\end{center}\n\\end{figure}\n\nThe situation can be different for an axion-like particle (ALP) with a mass above that of the proton. If the mass is just slightly above but below that of the neutron, $m_{p}-m_{e}<m_{a}<m_{n}$, it could explain the neutron lifetime anomaly by opening up $n\\rightarrow a^{0}+\\nu$, realizing the mechanism proposed in Ref.~\\cite{NeutronTau}. The branching ratio for that latter process can reach $1\\%$ for $m_{S}^{2}\\approx m_{V}^{2}$ around the TeV, i.e., very close to the bounds from direct LHC searches~\\cite{CMS:2020gru,ATLAS:2020dsk,ATLAS:2020xov,CMS:2020wzx,ATLAS:2021oiz}. Such a large rate may be surprising for TeV scale LQ\/DQ, but can be understood from the fact that most of the $m_S$ powers in the operator of Eq.~(\\ref{OpsAPL}) are compensated by $\\Lambda_{QCD}$. For comparison, the electroweak neutron beta decay rate is proportional to $G_F m_e^5$, with $m_e$ the electron mass and $G_F \\sim 1\/v_{EW}^2$, bringing a much harsher scale suppression.\n\nFor this scenario to be viable, the ALP should not decay too quickly to light particles, since that would allow $p\\rightarrow e(a^{0\\ast}\\rightarrow X)$ at an unacceptable rate. Usually, ALPs have two gluon and\/or two photon couplings. To analyze this situation, let us turn on the usual ALP photon coupling tuned by some very high $f_{a}$ scale:\n\\begin{equation}\n\\frac{e^{2}}{f_{a}}a^{0}F_{\\mu\\nu}\\tilde{F}^{\\mu\\nu}\\rightarrow\\Gamma\n(a^{0}\\rightarrow\\gamma\\gamma)=\\frac{4\\pi\\alpha^{2}m_{a}^{3}}{f_{a}^{2}}\\ .\n\\end{equation}\nBy comparison, the decay rate into proton plus antilepton (or antiproton plus lepton) is\n\\begin{equation}\n\\Gamma(a^{0}\\rightarrow p\\bar{\\ell})=\\frac{m_{a}}{4\\pi}\\left(  1-\\frac{m_{p}^{2}%\n}{m_{a}^{2}}\\right)  ^{2}\\left(  \\frac{m_{p}\\Lambda_{QCD}^{3}}{m_{S}^{2}%\nm_{V}^{2}}\\right)  ^{2}\\ ,\n\\end{equation}\nwhile the neutron decay rate into ALP is\n\\begin{equation}\n\\Gamma(n\\rightarrow a^{0}\\nu)=\\frac{m_{n}}{4\\pi}\\left(  1-\\frac{m_{a}^{2}%\n}{m_{n}^{2}}\\right)  ^{2}\\left(  \\frac{m_{n}\\Lambda_{QCD}^{3}}{m_{S}^{2}%\nm_{V}^{2}}\\right)  ^{2}\\ .\n\\label{neutdec}\n\\end{equation}\nNow, kinematically, proton decay can occur via the process $p\\rightarrow\\ell(a^0\\rightarrow\\gamma\\gamma)$, whose decay rate is ($r_{a}\\equiv m_{a}\/m_{p}$, and the lepton mass is set to zero)\n\\begin{equation}\n\\Gamma\\left(  p\\rightarrow\\ell(a^{0\\ast}\\rightarrow\\gamma\\gamma)\\right)\n=\\frac{2\\alpha^{2}m_{p}^{3}}{\\pi f_{a}^{2}}\\left(  \\frac{m_{p}\\Lambda_{QCD}%\n^{3}}{m_{S}^{2}m_{V}^{2}}\\right)  ^{2}\\int_{0}^{1}dz\\frac{z^{2}\\left(\n1-z\\right)  ^{2}}{(r_{a}^{2}-z)^{2}}\\overset{r_{a}\\rightarrow1}{\\approx}%\n\\frac{2\\alpha^{2}m_{p}^{3}}{3\\pi f_{a}^{2}}\\left(  \\frac{m_{p}\\Lambda_{QCD}%\n^{3}}{m_{S}^{2}m_{V}^{2}}\\right)  ^{2}\\;.\n\\end{equation}\nFor $m_{p}<m_{a}<m_{n}$, if we impose that the $p\\rightarrow\\ell(a^{\\ast}\\rightarrow\\gamma\\gamma)$ lifetime is greater than $10^{32}$ years, \n\\begin{equation}\nm_S \\approx m_V > 900\\ \\text{GeV} \\left( \\frac{10^{16}\\ \\text{GeV}}{f_a} \\right)^{1\/4}\\;.\n\\end{equation}\nPlugging this in Eq.~(\\ref{neutdec}), the branching ratio for $n\\rightarrow a^{0}+\\nu$ is at around $1\\%$ provided $f_{a}$ is pushed at the GUT scale, $f_a \\approx 10^{16}\\ \\text{GeV}$, see Fig.~\\ref{FigNeut}. Note that for that value of $f_{a}$, the ALP still decays mostly into $\\gamma\\gamma$ as $\\Gamma(a^{0}\\rightarrow p\\bar{\\ell})>\\Gamma(a^{0}\\rightarrow\\gamma\\gamma)$ requires $f_{a}$ about an order of magnitude larger. The $p\\rightarrow\\ell(a^{0\\ast}\\rightarrow\\gamma\\gamma)$ decay can happen only for $\\ell=e,\\mu$, but the underlying LQ couplings could actually exhibit non-trivial flavor hierarchies. If they couple preferentially to the $\\tau$, then proton decay would be forced to occur via more suppressed channels, e.g. via $p\\rightarrow\\pi\\nu_{\\tau}(a^{0\\ast}\\rightarrow\\gamma\\gamma)$, and $f_{a}$ could be brought down by a few orders of magnitude. Thus, an ALP could realize the scenario proposed in Ref.~\\cite{NeutronTau} to solve the neutron lifetime puzzle, though it does not alleviate its inherent fine tuning of the dark particle mass. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=11cm]{Neutron.jpg}%\n\\caption{Evolution of the $a^{0}\\rightarrow\\gamma\\gamma$, $a^{0}\\rightarrow p\\bar{\\ell}$, $n\\rightarrow a^{0}+\\nu$ and $p\\rightarrow\\ell(a^{0\\ast}\\rightarrow\\gamma\\gamma)$ widths as functions of $f_a$. The dashed line indicates the observed neutron lifetime discrepancy, $\\Gamma_{bottle}-\\Gamma_{beam} \\approx 7.1\\times 10^{-30}$~GeV~\\cite{NeutronTau}. The ALP mass is kept fixed at $m_a =0.9384$~GeV. The LQ\/DQ mass is adjusted so that $\\tau \\left( p\\rightarrow\\ell(a^{0\\ast}\\rightarrow\\gamma\\gamma) \\right) = 10^{32}$~yr. For the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operator, Eq.~(\\ref{OpsAPL}), $m_{S}\\approx m_{V}$ then follows the indicated line (Case I), and must be a bit below the TeV to reproduce the observed $\\Gamma_{bottle}-\\Gamma_{beam}$ discrepancy. Concerning the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ scenario, for which leptons and antileptons should be interchanged, the extra factor of $f_a$ in Eq.~(\\ref{OpsAPantiL}) sets $m_{S}\\approx m_{V}\\approx 90$~TeV independently of $f_a$ (Case II).}\n\\label{FigNeut}%\n\\end{center}\n\\end{figure}\n\n\nFor heavier ALPs, neutron decay is kinematically closed, and the $\\Gamma(a^{0}\\rightarrow p\\bar{\\ell},n\\bar{\\nu})>\\Gamma(a^{0}\\rightarrow\\gamma\\gamma)$ pattern can arise for lower $f_{a}$ values (though still very large from the axion point of view), with for example $\\Gamma(a^{0}\\rightarrow p\\bar{\\ell},n\\bar{\\nu})>\\Gamma(a^{0}\\rightarrow\\gamma\\gamma)$ for $f_{a}>10^{13}$ GeV if $m_{a}=100~$GeV. This, however, requires also to boost the $a\\rightarrow p\\bar{\\ell},n\\bar{\\nu}$ rate by allowing light LQ\/DQ at around the TeV scale. Even if $\\Gamma(a^{0}\\rightarrow p\\bar{\\ell},n\\bar{\\nu})$ does not dominate, such decay channels could have some cosmological implications. From a baryogenesis point of view, it is interesting to remark that the present scenario has all the necessary ingredients. Provided the LQ\/DQ couple to more than one SM fermion states, several operators will simultaneously contribute to the $a^{0}\\rightarrow p\\bar{\\ell},n\\bar{\\nu}$ and $a^{0}\\rightarrow\\bar{p}\\ell,\\bar{n}\\nu$ decay processes, and since the LQ\/DQ couplings to SM fermions are a priori complex, their rates would be different (slightly, as rescattering is required). In this picture, note that if there are several LQ\/DQ states with a non-trivial mass spectrum, their decay chains may first generate an asymmetry when $m_{S,V}>m_{a}$~\\cite{Arnold:2012sd}, but it would be washed out and regenerated at a lower scale by the ALP decays. Whether this mechanism is sufficient to generate the observed baryon asymmetry is left for a future study.\n\nTo close this section, let us mention another scenario in which $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators require two derivative couplings, and proton decay is associated to axion pair production. Specifically, if we start from\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}  &  =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R}%\n)+S_{2}^{1\/3}(\\bar{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})\\nonumber\\\\\n&  \\ \\ \\ \\ +\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}+\\partial\n_{\\mu}\\phi V_{2}^{1\/3,\\mu\\dagger}S_{2}^{1\/3}+\\phi H^{\\dagger}V_{1,\\mu\n}^{2\/3\\dagger}V_{2}^{1\/3,\\mu}+h.c.\\ ,\n\\end{align}\nthe mixing between $S_{1}^{2\/3}$ and $S_{2}^{1\/3}$ can only occur through that of $V_{1,\\mu}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$. Specifically, with this specific choice of couplings,\n\\begin{equation}%\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{1,\\mu}^{2\/3}$ & $S_{2}^{1\/3}$ & $V_{2,\\mu\n}^{1\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}%\n$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $-1\/3$ & $-2\/3$ & $-1\/3$ & $1\/3$ & $0$ & $1\/3$ & $1\/3$\n& $1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $1\/3$ & $0$ & $-1\/3$ & $-1$ & $-2\/3$ & $0$ & $0$ & $0$\n& $1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThanks to these charges, which crucially follow from whether $\\phi$ or $\\phi^{\\dagger}$ are introduced in the couplings, no other SM fermion couplings of the LQ\/DQ, nor any other renormalizable couplings among the LQ\/DQ, is allowed. Turning on $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ does not change this picture, except for the operator $\\phi^{\\dagger}S_{1}^{2\/3}S_{2}^{1\/3}S_{2}^{1\/3}$. This is quite natural looking at the Lagrangian, since $\\phi^{\\dagger}S_{1}^{2\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ followed by $S_{1}^{2\/3}\\rightarrow\\bar{q}_{L}^{\\mathrm{C}}q_{L}$ and $S_{2}^{1\/3}\\rightarrow\\bar{q}_{L}\\nu_{R}$ permits to recover $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ by closing the $q_{L}$ loops. As the $\\phi^{\\dagger}S_{1}^{2\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ and $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ have the same quantum numbers, both carrying $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$, they are both able to generate neutrino masses only, and do not affect the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ pattern.\n\nThe leading operator for proton decay is now (Fig.~\\ref{Fig5}$b$):%\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{1}{m_{S}^{4}m_{V}^{4}}\\phi\\partial_{\\mu}\\phi\\partial^{\\mu}\\phi H^{\\dagger\n}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})(\\bar{d}%\n_{R}^{\\mathrm{C}}\\ell_{L}^{\\mathrm{C}}+\\bar{q}_{L}^{\\mathrm{C}}\\nu\n_{R}^{\\mathrm{C}})\\ ,\n\\end{equation}\nand it contains in particular\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{v_{\\phi}v\\Lambda_{QCD}^{3}}{m_{S}^{4}m_{V}^{4}}\\partial_{\\mu}a^{0}%\n\\partial^{\\mu}a^{0}\\bar{p}\\left(  1-\\gamma^{5}\\right)  \\ell+...+h.c.\\ .\n\\end{equation}\nPhenomenologically, proton decay is suppressed, even for relatively low $m_{S}\\approx m_{V}$ of $\\mathcal{O}(10~$TeV$)$. On the other hand, if $a^{0}$ is an ALP with twice its mass above the proton but below the neutron mass, this setting is less interesting because the LQ\/DQ masses need to be too low to reach $B(n\\rightarrow a^{0}+a^{0}+\\bar{\\nu})\\approx1\\%$. Whether ALP or axions, the cosmological implications of this scenario would be worth further study though, as the consequences of opening up (possibly CP-violating) $a^{0}+p\\leftrightarrow a^{0}+\\ell$ and $a^{0}+\\bar{p}\\leftrightarrow a^{0}+\\bar{\\ell}$ scattering processes could provide a new baryogenesis mechanism.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}+\\mathcal{L}$}\n\nTo attain $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators, the trick is to start from the previous scenario, but use some additional LQ couplings to switch $\\mathcal{L}$ by two units. Specifically, we can consider%\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}  &  =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{2,\\mu\n}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu\n}\\nu_{R}^{\\mathrm{C}})\\nonumber\\\\\n&  \\ \\ \\ \\ +\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}+\\phi\nH^{\\dagger}V_{1,\\mu}^{2\/3\\dagger}V_{2}^{1\/3,\\mu}+h.c.\\ . \\label{LagrSSB2}%\n\\end{align}\nProvided $V_{1,\\mu}^{2\/3}$ has no couplings to SM fermions, and only those two interactions among $\\phi$ and the LQ\/DQ are present, two $U(1)$s are present in the Lagrangian, with charges\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{1,\\mu}^{2\/3}$ & $V_{2}^{1\/3,\\mu}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $-1\/2$ & $-2\/3$ & $-1\/6$ & $1\/3$ & $1\/3$ & $1\/3$ &\n$1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-1\/2$ & $0$ & $1\/2$ & $1$ & $0$ & $0$ & $0$ & $1$ &\n$1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nSo, the $U(1)_{\\mathcal{B}+\\mathcal{L}}$ symmetry is spontaneously broken, while $\\mathcal{B}-\\mathcal{L}$ remains. If neutrino masses are generated by adding the $\\phi\\bar{\\nu}_{R}\\nu_{R}^{\\mathrm{C}}$ coupling, the remaining exact $U(1)$ symmetry suffices to keep off all other interactions among $\\phi$ and the LQ\/DQ, as well as the LQ\/DQ couplings to SM fermions not already present in the Lagrangian, except for a $\\phi^{\\dagger}S_{1}^{2\/3}V_{2}^{1\/3,\\mu}V_{2,\\mu}^{1\/3}$ which carries the same quantum number as $\\phi^{\\dagger}{\\bar{\\nu}_{R}}^{\\mathrm{C}}\\nu_{R}$. Neither is able to open new $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ patterns for proton decay.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.3517in,width=2.0254in]{Fig5b.jpg}\n\\caption{Axion-induced $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators.}\n\\label{Fig5b}\n\\end{center}\n\\end{figure}\n\nThe leading $\\mathcal{B}+\\mathcal{L}$ violating operator is (see Fig.~\\ref{Fig5b})\n\\begin{align}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)}^{eff}  &  =\\frac\n{1}{m_{S}^{2}m_{V}^{4}}\\phi\\partial_{\\mu}\\phi H^{\\dagger}(\\bar{q}%\n_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})(\\bar{u}_{R}^{\\mathrm{C}%\n}\\gamma^{\\mu}\\ell_{L}+\\bar{q}_{L}^{\\mathrm{C}}\\gamma^{\\mu}\\nu_{R})\\nonumber\\\\\n&  \\rightarrow\\frac{v_{EW}v_{\\phi}}{m_{S}^{2}m_{V}^{4}}\\partial_{\\mu}a^{0}(\\bar\n{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})(\\bar{u}%\n_{R}^{\\mathrm{C}}\\gamma^{\\mu}e_{L}+\\bar{d}_{L}^{\\mathrm{C}}\\gamma^{\\mu}\\nu\n_{R})\\ .\n\\label{OpsAPantiL}\n\\end{align}\nPhenomenologically, thanks to the $v_{EW}v_{\\phi}$ from the $\\phi HV_{2,\\mu}^{1\/3\\dagger}V_{1}^{2\/3,\\mu}$ coupling, the LQ\/DQ scale can be lower by about an order of magnitude without violating proton decay bounds. For ALPs, the main difference with the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ scenario is that $f_{a}=v_{\\phi}$ occurs in the $n\\rightarrow a^{0}\\bar{\\nu}$ and $a^0\\rightarrow p\\ell$ amplitudes, but cancels out from the $p\\rightarrow\\bar{\\ell}(a^0\\rightarrow\\gamma\\gamma)$ rate. This means that $m_{S}\\approx m_{V}$ cannot be as low as before, but must above $90$~TeV. Yet, this high scale is compensated in the $n\\rightarrow a^{0}\\bar{\\nu}$ rate by the $v_{\\phi}$ factor, so its branching ratio can still reach $\\mathcal{O}(1\\%)$. Actually, the dependencies of the various rates on $f_a$ stays exactly as depicted in Fig.~\\ref{FigNeut}, but for $m_{S}\\approx m_{V}\\approx 90$~TeV.\n\nNote, finally, that $\\mathcal{B}+\\mathcal{L}$ violating operators are not easily forced to involve pairs of axions. The pattern of LQ\/DQ couplings to SM fermions, and their hypercharge, does not leave many options if only renormalizable operators are allowed. The simplest we could find would require two different Higgs doublets, so would be suitable for the DFSZ model\n\\begin{align}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}  &  =\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}%\n^{4\/3}\\bar{d}_{R}^{\\mathrm{C}}d_{R}+S_{1}^{2\/3}(\\bar{d}_{R}\\nu_{R}%\n^{\\mathrm{C}}+\\bar{u}_{R}e_{R}^{\\mathrm{C}}+\\bar{q}_{L}\\ell_{L}^{\\mathrm{C}%\n})\\nonumber\\\\\n&  \\ \\ \\ \\ +\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}+\\partial\n_{\\mu}\\phi S_{1}^{4\/3\\dagger}V_{1}^{4\/3,\\mu}+H_{u}^{\\dagger}H_{d}^{\\dagger\n}V_{1,\\mu}^{2\/3\\dagger}V_{1}^{4\/3,\\mu}+h.c.\\ .\n\\end{align}\nAs the phenomenology is similar as that for $\\mathcal{B}-\\mathcal{L}$ violating operators, we do not detail this situation further.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}$}\n\nGiven that we want to start from the derivative couplings, which are all quadratic in the LQ\/DQ, we will need to add at least some cubic interactions. This quickly increases the number of new state needed, and phenomenologically, the longest the chain, the smallest the predicted rate given that LQ\/DQ masses are at least of a few TeV.\n\nThe simplest processes correspond to the skeleton graph $\\partial_{\\mu}\\phi\\rightarrow X_{i}(X_{l}\\rightarrow X_{j}X_{k})$, with the final\n$X_{i}X_{j}X_{k}$ set allowing for $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ transitions, so $X_{i,j,k}=S_{1}^{y}$ or $V_{2}^{y}$ for some $y$. If $X_{l}$ is integrated out, the effective operator involves $\\partial_{\\mu}\\phi X_{i}X_{j}X_{k}$ plus some Higgs fields. The simplest such operators are of dimension six, and only seven of them are compatible with the gauge symmetries,\n\\begin{align}\n&  \\partial_{\\mu}\\phi~H^{\\dagger}~(S_{1}^{2\/3}S_{1}^{4\/3}V_{2}^{1\/3,\\mu\n},\\ S_{1}^{4\/3}S_{1}^{4\/3}V_{2}^{5\/3,\\mu},\\ V_{2}^{1\/3,\\mu}V_{2,\\nu}%\n^{1\/3}V_{2}^{1\/3,\\nu})\\ ,\\\\\n&  \\partial_{\\mu}\\phi~H~(V_{2}^{5\/3,\\mu}V_{2,\\nu}^{1\/3}V_{2}^{1\/3,\\nu}%\n,\\ S_{1}^{2\/3}S_{1}^{2\/3}V_{2}^{1\/3,\\mu},\\ S_{1}^{2\/3}S_{1}^{4\/3}%\nV_{2}^{5\/3,\\mu},\\ S_{1}^{4\/3}S_{1}^{8\/3}V_{2}^{1\/3,\\mu})\\ ,\n\\end{align}\nwhere $\\partial_{\\mu}\\phi$ could be replaced by $\\partial_{\\mu}\\phi^{\\dagger}$ wherever required. Starting from the three derivative interactions of Eq.~(\\ref{DerScenars}), there are several ways to reach these operators using a $HX_{l}X_{j}X_{k}$ or $H^{\\dagger}X_{l}X_{j}X_{k}$ coupling. Since $X_{l}=S_{2}^{y}$ or $V_{1}^{y}$, these operators alone cannot induce $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ processes. Further, if $X_{l}$ transforms as a $6$, it does not couple to SM fermions hence these operators cannot lead to proton decay either. If $X_{l}$ transforms as a $3$, one must make sure the PQ charges forbid $X_{l}\\rightarrow\\ell q$. All this nevertheless leaves many possible mechanisms, though many of them turn out to be essentially equivalent phenomenologically, so let us take an example.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.9787in,width=4.7158in]{Fig6.jpg}\n\\caption{One and two axion induced neutron-antineutron oscillation $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators.}\n\\label{Fig6}\n\\end{center}\n\\end{figure}\n\nConsider the Lagrangian\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}  &  =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{2,\\mu\n}^{1\/3}\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\nonumber\\\\\n&  +\\partial_{\\mu}\\phi V_{1}^{2\/3,\\mu\\dagger}S_{1}^{2\/3}+HV_{1,\\mu}^{2\/3}%\nS_{1}^{2\/3}V_{2}^{1\/3,\\mu}+h.c.\\ , \\label{LagrSSB3a}%\n\\end{align}\nwhere $S_{1}^{2\/3}$ and $V_{1}^{2\/3,\\mu}$ transform as $\\mathbf{3}$, but $V_{2}^{1\/3,\\mu}\\sim\\mathbf{\\bar{6}}$ since the final operator $\\partial_{\\mu}\\phi HS_{1}^{2\/3}S_{1}^{2\/3}V_{2}^{1\/3,\\mu}$ would cancel for $V_{2}^{1\/3,\\mu}\\sim\\mathbf{3}$. Dropping the $\\Psi_{L,R}$, as their charge can independently be fixed by turning on some couplings to the LQ, the active $U(1)$s are then\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{2,\\mu}^{1\/3}$ & $V_{1,\\mu}^{2\/3}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $2$ & $-2\/3$ & $-2\/3$ & $4\/3$ & $1\/3$ & $1\/3$ & $1\/3$\n& $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $1$ & $1$ &\n$1$\\\\\\hline\n\\end{tabular}\n\\ \\label{SponB}%\n\\end{equation}\nTurning on any of the LQ couplings of $S_{1}^{2\/3}$ or $V_{1,\\mu}^{2\/3}$ would break $U(1)_{\\mathcal{B}+\\mathcal{L}}$, and induce proton decay (compare Eq.~(\\ref{LagrSSB3a}) with Eq.~(\\ref{LagrSSB2})). At this level, their presence is thus forbidden by the still active $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ symmetries. For an even stricter protection, the PQ symmetry can be extended to prevent these couplings. It suffices to add a seesaw mechanism with the $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling, something we should do anyway (the $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ would instead allow all the LQ couplings). Note that $S_{1}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$ can mix via a $D^{\\mu}HS_{1}^{2\/3\\dagger}V_{2,\\mu}^{1\/3}$ term, but this is inessential since they have the same $\\mathcal{B}$ and $\\mathcal{L}$ quantum numbers. This scenario lead to neutron-antineutron oscillation operators, with the diagram of Fig.~\\ref{Fig6}$a$, corresponding to\n\\begin{align}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)}^{eff}  &  =\\frac\n{1}{m_{S}^{8}}\\partial_{\\mu}\\phi H(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}%\n_{R}^{\\mathrm{C}}u_{R})(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}%\n}u_{R})\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}+h.c.\\ \\nonumber\\\\\n&  \\rightarrow\\frac{v\\Lambda_{QCD}^{6}}{m_{S}^{8}}\\partial_{\\mu}a^{0}\\bar\n{n}^{\\mathrm{C}}\\gamma^{\\mu}\\gamma_{5}n + ... + h.c.\\ , \\label{SponB2}%\n\\end{align}\nwhere we have set all the DQ masses to a common $m_{S}$ value. Since there is no associated $n-\\bar{n}$ operator, this scale can in principle be quite low. The best low energy limits come from nuclear transitions, as this operator also contributes to $nn\\rightarrow a$, but those do not push $m_{S}$ well above the TeV scale~\\cite{Heeck:2020nbq}. The main constraint thus come from LHC searches~\\cite{CMS:2020gru,ATLAS:2020dsk,ATLAS:2020xov,CMS:2020wzx,ATLAS:2021oiz}. Note, though, that the generic leptoquark searches may not apply to this case: all these states decay to diquark pairs and, furthermore, $V_{1}^{2\/3,\\mu}$ could end up quite long lived if it is lighter than $V_{2}^{1\/3,\\mu}$ and $S_{1}^{2\/3,\\mu}$, and would show up in channels with at least four jets.\n\nEven if $m_{S}$ can be quite low, at around the TeV say, the $a^{0}nn$ coupling is significantly smaller than the other couplings, including to $n\\bar{n}$, as can be estimated setting $f_{a}\\equiv v_{\\phi}$:\n\\begin{equation}\n\\frac{1}{f_{a}}\\approx\\frac{v\\Lambda_{QCD}^{6}}{m_{S}^{8}}\\Leftrightarrow\nm_{S}\\approx10~\\text{GeV}\\times\\left(  \\frac{f_{a}}{10^{9}\\ \\text{GeV}%\n}\\right)  ^{1\/8}\\ ,\n\\end{equation}\nfor $\\Lambda_{QCD}\\approx300$ MeV. Even with $f_{a}$ close to the Planck scale, the LQ mass would need to be well below the TeV scale, which would again be ruled out by direct searches. For $m_{S}$ around the TeV, the $a^{0}nn$ coupling is at best $10^{-16}$ smaller than that to $a^{0}n\\bar{n}$. Thus, $a^{0}nn$ does not represent a competitive signature for direct axion searches.\n\nIndirectly, the $a^{0}nn$ coupling may nevertheless open new routes by relying instead on neutron-antineutron oscillation phenomena. Indeed, while $a^{0}nn$ cannot generate $n\\rightarrow\\bar{n}$ in vacuum, oscillations could now be catalyzed by an axion dark matter background. While the typical high frequency of the coherent axion field precludes any observation using standard beam searches for $n-\\bar{n}$ oscillations (the induced $\\delta m_{n-\\bar{n}}$ would average to zero), transient variations of the axion field may be observable in this way. Another possibility would be to exploit the magnetic splitting between $n$ and $\\bar{n}$ states, which in a $1~$T magnetic field would be of about $10^{-7}$~eV~\\cite{Phillips:2014fgb}, larger than the axion mass if $f_{a}>10^{14}$~GeV. Note that the neutron beam go through a 4.6~T magnet in neutron lifetime experiments, Ref.~\\cite{Nico:2004ie,Yue:2013qrc}, and that axion-induced mixing effects, if they occur, would not have been excluded by the recent mirror neutron search of Ref.~\\cite{Broussard:2021eyr}, which relies on hypothesized mirror neutrons capabilities to pass through normal matter.\n\nTwo other features compared to the usual neutron oscillations are worth mentioning: the coupling is axial, $\\bar{n}^{\\mathrm{C}}\\gamma^{\\mu}\\gamma_{5}n$, instead of the usual scalar $\\bar{n}^{\\mathrm{C}}n$ oscillation operator, so the spin dependencies are different~\\cite{Gardner:2014cma}, and the $\\partial_{\\mu}a^{0}\\bar{n}^{\\mathrm{C}}\\gamma^{\\mu}\\gamma_{5}n$ coupling can be CP violating~\\cite{Berezhiani:2015uya,McKeen:2015cuz,Berezhiani:2018xsx} since the DQ couplings are a priori complex, so $n$ and $\\bar{n}$ may react differently to an axionic background. Also, compared to neutron-mirror neutron oscillations, like those invoked to explain the neutron lifetime anomaly~\\cite{Berezhiani:2018eds}, the antineutron would not be invisible but would either decay to antiproton, or annihilate with the surrounding matter. A quantitative analysis of these signatures is clearly called for but would require a detailed study, which go beyond our scope. Also, other manifestations of the $a^{0}nn$ coupling in an astrophysical and cosmological context are left for a future study.\n\nWith only three LQ, another rather simple scenario can lead to the $\\partial_{\\mu}\\phi H^{\\dagger}V_{2}^{1\/3,\\mu}V_{2,\\nu}^{1\/3}V_{2}^{1\/3,\\nu}$ operator by virtual $S_{2}^{1\/3}$ exchanges, and involves only states transforming as $\\mathbf{3}$:\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}  &  =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{2,\\mu\n}^{1\/3}\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\nonumber\\\\\n&  +\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}+H^{\\dagger}%\nS_{2}^{1\/3}V_{2,\\nu}^{1\/3}V_{2}^{1\/3,\\nu}+h.c.\\ . \\label{LagrSSB3b}%\n\\end{align}\nThe same $U(1)_{\\mathcal{B}}$ charges are found as in Eq.~(\\ref{SponB}), with $V_{1,\\mu}^{2\/3}\\rightarrow S_{2}^{1\/3}$. Also, as before, adding the $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling prevents all the LQ couplings of $V_{2,\\mu}^{1\/3}$, $S_{1}^{2\/3}$, and $S_{2}^{1\/3}$. Proton decay is now forbidden by the existence of the PQ symmetry at the high scale, and does not arise at the low scale thanks to the specific $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ and $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ symmetry breaking pattern. The final operator is phenomenologically similar to that in Eq.~(\\ref{SponB2}).\n\nMany other choices of DQ states are possible, but they lead to similar patterns. We will not investigate more complicated processes, except for the following that leads to a different phenomenology:\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} &  =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{2,\\mu\n}^{1\/3}\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\nonumber\\\\\n&  \\ \\ \\ \\ +\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}+\\partial\n_{\\mu}\\phi V_{1}^{2\/3,\\mu\\dagger}S_{1}^{2\/3}+\\phi^{\\dagger}S_{2}^{1\/3}V_{2,\\mu}%\n^{1\/3}V_{1}^{2\/3,\\mu}+h.c.\\ .\\label{LagrSSB3c}%\n\\end{align}\nIn some senses, it combines the previous two scenarios, and gives the same charges as in Eq.~(\\ref{SponB}), with $V_{1,\\mu}^{2\/3}$ and $S_{2}^{1\/3}$ having $\\mathcal{B}=4\/3$. Also, the $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling now suffices to prevent the LQ couplings of the four states, $V_{2,\\mu}^{1\/3}$, $V_{1,\\mu}^{2\/3}$, $S_{1}^{2\/3}$, $S_{2}^{1\/3}$. What differs however is how the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ effects are induced at the low-energy scale. The two derivative couplings are needed, and $\\phi$ further occurs in the cubic DQ coupling, so the leading operator is (see Fig.~\\ref{Fig6}$b$)\n\\begin{align}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)}^{eff} &  =\\frac\n{1}{m_{S}^{10}}\\phi\\partial_{\\mu}\\phi^\\dagger\\partial_{\\nu}\\phi^\\dagger(\\bar{q}_{L}%\n^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})\\bar{d}_{R}^{\\mathrm{C}%\n}\\gamma^{\\mu}q_{L}\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\nu}q_{L}+h.c.\\ \\nonumber\\\\\n&  \\rightarrow\\frac{v_{\\phi}\\Lambda_{QCD}^{6}}{m_{S}^{10}}\\partial_{\\mu}%\na^{0}\\partial^{\\mu}a^{0}\\bar{n}^{\\mathrm{C}}\\gamma_{5}n+...+h.c.\\ .\n\\end{align}\nThough this operator is now of dimension 14 instead of that of dimension 12 in Eq.~(\\ref{SponB2}), the extra suppression is compensated by the $v_{\\phi}$ factor since $v_{\\phi}\\Lambda_{QCD}\/m_{S}^{2}$ is of $\\mathcal{O}(1)$ for $m_{S}$ around the tens of TeV scale and $v_{\\phi}$ at around $10^{6}$~TeV. The nuclear transition bounds are thus similar as in the single axion case, and in any case not competitive with direct collider searches for new colored states. Phenomenologically, neutron-antineutron conversion now requires pairs of axions, and would occur through scattering processes like $a^{0}+n\\leftrightarrow a^{0}+\\bar{n}$ or $n+n\\leftrightarrow a^0+a^0$ and $\\bar{n}+\\bar{n}\\leftrightarrow a^0+a^0$. Though unlikely to be ever observed, these processes could play a cosmological role.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}\\pm3\\mathcal{L}$}\n\nThe $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ scenarios are trivially obtained from any of the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ Lagrangians of the previous section by switching all DQ couplings to LQ couplings. For example, starting from Eq.~(\\ref{LagrSSB3b}),\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} &  =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{d}_{R}\\nu_{R}^{\\mathrm{C}}+\\bar{u}_{R}e_{R}^{\\mathrm{C}}+\\bar\n{q}_{L}\\ell_{L}^{\\mathrm{C}})+V_{2,\\mu}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell\n_{L}^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}})\\\\\n&  +\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}+H^{\\dagger}%\nS_{2}^{1\/3}V_{2,\\nu}^{1\/3}V_{2}^{1\/3,\\nu}+h.c.\\ ,\n\\end{align}\nleads to the charges\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{2,\\mu}^{1\/3}$ & $S_{1}^{2\/3}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $-1$ & $-2\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ &\n$0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-3$ & $-2$ & $1$ & $1$ & $0$ & $0$ & $0$ & $1$ & $1$ &\n$1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nBy analogy, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ transitions can be induced by taking the Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{2}^{1\/3}(\\bar\n{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})+V_{1,\\mu}^{2\/3}\\bar{d}_{R}\\gamma^{\\mu}%\n\\nu_{R}+\\partial_{\\mu}\\phi V_{1}^{2\/3,\\mu\\dagger}S_{1}^{2\/3}+\\phi S_{1}%\n^{2\/3}S_{2}^{1\/3}S_{2}^{1\/3}+h.c.\\ ,\n\\end{equation}\nwith the charges are\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $S_{2}^{1\/3}$ & $V_{1,\\mu}^{2\/3}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $1\/2$ & $-1\/6$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$\n& $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-3\/2$ & $1\/2$ & $-1$ & $-1$ & $0$ & $0$ & $0$ & $1$ &\n$1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nNote that for each case, additional $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,0)$ couplings involving pairs of LQs are possible, like $\\phi HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ or $D^{\\mu}HS_{2}^{1\/3\\dagger}V_{1,\\mu}^{2\/3}$ for the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ scenario. Those can neither affect the symmetry pattern, nor open new routes for proton decay.\n\nPhenomenologically, these scenarios are very similar to the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,\\pm1)$ ones described before, so we will not detail them further. The main difference is the extra suppression of proton and neutron decays due to the higher dimensionality of the operators, and of the many particles in the final states. These scenarios thus have essentially the same phenomenology whenever these suppressions can be compensated by lowering the LQ mass scale without violating LHC bounds.\n\n\\section{Conclusions\\label{Ccl}}\n\nIn this paper, the opportunities arising from combining leptoquarks and diquarks with axions have been systematically analyzed. From a phenomenological standpoint, our main results are:\n\n\\begin{enumerate}\n\\item The PQ symmetry of which the axion is the Goldstone boson can be identified with any combination of baryon $\\mathcal{B}$ and lepton $\\mathcal{L}$ numbers. In this way, $\\mathcal{B}$ and $\\mathcal{L}$ appear less accidental, in the sense that one combination of $\\mathcal{B}$ and\/or $\\mathcal{L}$ is spontaneously broken, while the orthogonal combination remains exact and is actually protected by the PQ symmetry. Reminiscent of the possible $\\Delta\\mathcal{B}$ and\/or $\\Delta\\mathcal{L}$ operators made of SM fields (see Table~\\ref{TableLQBL}), the simplest scenarios identify $U(1)_{PQ}$ with $U(1)_{\\mathcal{B}\\pm\\mathcal{L}}$, $U(1)_{\\mathcal{B}\\pm3\\mathcal{L}}$, $U(1)_{\\mathcal{B}}$, or $U(1)_{\\mathcal{L}}$, and induce spontaneously either proton decay, neutron-antineutron oscillations, or a Majorana mass terms for $\\nu_{R}$ (or more generally, neutrinoless double beta decays).\n\n\\item All scenarios can be supplemented with a seesaw mechanism. The axion is then not only the Goldstone boson associated to $U(1)_{\\mathcal{B}\\pm\\mathcal{L}}$, $U(1)_{\\mathcal{B}\\pm3\\mathcal{L}}$, or $U(1)_{\\mathcal{B}}$ breaking, but becomes also the Majoron associated to the $U(1)_{\\mathcal{L}}$ breaking. Though no accidental symmetry (besides of course $U(1)_{PQ}$ itself) remains, each scenario retains a specific phenomenology. For example, when $U(1)_{PQ}$ is identified both with $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2n,0)$ and $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2n)$ transitions are possible, but proton decay cannot occur.\n\n\\item For each pattern of symmetry breaking, it is also possible to prevent axion-free proton decay, neutron-antineutron oscillations, or neutrinoless double beta decays. In other words, one can make sure $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ effects always involve at least one axion field. Phenomenologically, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ scenarios open the door to $p\\rightarrow a^{0}+\\ell,$ $n\\rightarrow a^{0}+\\nu$, $p\\rightarrow2a^{0}+\\ell,$ $n\\rightarrow2a^{0}+\\nu$, and scattering processes like $a^{0}+(p,n)\\leftrightarrow a^{0}+(\\ell,\\nu)$. Scenarios with $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ or $(1,\\pm3)$ are similar. Following the strategy proposed in Ref.~\\cite{NeutronTau}, if $a^{0}$ is an ALP of just the right mass, such that proton decay is forbidden but neutron decay is not, these scenarios are able to solve the neutron lifetime puzzle, see Fig.~\\ref{FigNeut}.\n\n\\item When applied to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators, being forced to include an axion field could lead to very peculiar effects. The phenomenology of the $\\partial_{\\mu}a^{0}\\bar{n}^{\\mathrm{C}}\\gamma^{\\mu}\\gamma_{5}n$ and $\\partial_{\\mu}a^{0}\\partial^{\\mu}a^{0}\\bar{n}^{\\mathrm{C}}\\gamma_{5}n$ interactions have, to our knowledge, not been investigated in detail yet. Though a dedicated analysis is called for, we do not expect these interactions to be phenomenologically relevant in vacuum, but they could open interesting channels in an axionic dark matter background, or transitions like $n\\rightarrow\\bar{n}+a^{0}$ or $n\\rightarrow\\bar{n}+a^{0}+a^0$ in an intense magnetic field.\n\\end{enumerate}\n\nBesides these phenomenological aspects, we have also analyzed the consequences on the foundations of axion effective Lagrangians. Whenever the axion is associated to some patterns of $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ breaking, the SM fermions become charged under the PQ symmetry. Typically, they thus occur in the usual $f_{a}$-suppressed derivative interactions, but through vector current interactions, $\\partial_{\\mu}a^{0}\\bar{\\psi}\\gamma^{\\mu}\\psi$ (since $\\mathcal{B}$ and $\\mathcal{L}$ are vectorial). Often, these interactions are discarded owing to the naive vector Ward identity, but this is incorrect for two reasons:\n\n\\begin{enumerate}\n\\item[5.] Axion-gauge field interaction are usually expected to be $(g_{X}^{2}\/f_{a})\\mathcal{N}_{X}a^{0}X_{\\mu\\nu}\\tilde{X}^{\\mu\\nu}$, $X=G^{a}$, $W^{i}$, $B$, with $\\mathcal{N}_{X}$ summing up the contribution of all the fields charged under both the PQ symmetry and the $X$ gauge interactions of strength $g_{X}$.\\ Thus, $\\mathcal{N}_{X}$ depend on the SM fermion charges, with in particular $\\mathcal{N}_{W}$ and $\\mathcal{N}_{B}$ depending on how $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ are embedded in $U(1)_{PQ}$. Yet, as shown in Ref.~\\cite{Quevillon:2019zrd}, the SM fermion contributions to $\\mathcal{N}_{W}$ and $\\mathcal{N}_{B}$ arising from $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ systematically cancel with that coming from triangle graphs built on the corresponding $\\partial_{\\mu}a^{0}\\bar{\\psi}\\gamma^{\\mu}\\psi$ interactions.\\ At the end of the day, the $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ components of $U(1)_{PQ}$ do not alter the axion to gauge boson couplings, even though this is not apparent at the level of the effective Lagrangian.\n\n\\item[6.] The counting rule in powers of $1\/f_{a}$, central in constructing the axion effective Lagrangian (see e.g. Ref.~\\cite{Georgi:1986df}), is invalid when $\\mathcal{B}$ and\/or $\\mathcal{L}$ are broken spontaneously along with the PQ symmetry. Indeed, the equations of motion of the SM fermions (or that of the leptoquarks if they have not been integrated out) inherit $\\mathcal{O}((f^{\\alpha})^{n}),$ $n\\geq1$ terms, so that $\\mathcal{O}(f_{a}^{n-1})$ interactions are hidden inside $f_{a}^{-1}\\partial_{\\mu}a^{0}\\bar{\\psi}\\gamma^{\\mu}\\psi$. In practice, in the present paper, all these interactions were suppressed by some relatively high power of the leptoquark masses, which are pushed above the TeV by direct collider searches. Thus, in all the scenarios considered here, the $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating interactions are not expected to be dominant compared to e.g. the two photon or two gluon modes for $f_{a}$ below the Planck scale. Still, as this relative suppression has nothing to do with $f_{a}$, there is no guarantee it always happens.\n\\end{enumerate}\n\nIn conclusion, even if entangling the PQ symmetry with the accidental symmetries of the SM requires new leptoquarks states, and often several of them, these scenarios end up being more economical from a $U(1)$ global symmetry point of view. The axion becomes a central piece, not only solving the strong CP puzzle, and maybe making up for the observed dark matter, but also setting off the seesaw mechanism and introducing potentially CP violating baryon number violation. With all its capabilities, the axion could hold the keys to many of the standing cosmological enigmas.\n\n\\subsubsection*{Acknowledgements} This work is supported by the labex \\textit{Enigmass}, and by the\nCNRS\/IN2P3 Master project \\textit{Axions from Particle Physics to Cosmology}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and main results}\nWe start with some notation and terminology. Let\n$(\\R^d,\\<\\cdot,\\cdot\\>,|\\cdot|)$ be the $d$-dimensional Euclidean\nspace, and $\\R^d\\otimes\\R^m$ the collection of all $d\\times m$\nmatrices endowed with the Hilbert-Schmidt norm $\\|\\cdot\\|$. For\nfixed $r_0>0$, $\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r:=C([-r_0,0];\\R^d)$ stands for the family of all\ncontinuous functions $f:[-r_0,0]\\rightarrow\\R^d$ which is a Banach space\nwith the uniform norm $\\|f\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle:=\\sup_{-r_0\\le v\\le 0}|f(v)|$. Given any\ninteger $ p\\ge1$, we use $\\Theta$ to denote a bounded, open and convex\nsubset of $\\R^p$ whose closure is written as $\\bar\\Theta$. Let $\\mathcal\n{P}(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$ be the totality of all probability measures on $\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$. Set\n$\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r):=\\{\\mu\\in\\mathcal\n{P}(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r):\\mu(\\|\\cdot\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2):=\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\\|\\xi\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\xi)<\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\}$.\n$(\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r),\\W_2)$ is a Polish space under the Warsserstein\ndistance $\\mathbb{W}_2$ on $\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$ defined by\n$$\\W_2(\\mu,\\nu):= \\inf_{\\pi\\in \\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r(\\mu,\\nu)} \\bigg(\\int_{\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\\times\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r} \\|\\xi-\\eta\\|^2_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\pi(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D \\xi,\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D \\eta)\\bigg)^{\\ff 1 2},\\ \\ \\mu,\\nu\\in \\mathcal\n{P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r),$$  where $\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r(\\mu,\\nu)$ is the set of couplings for $\\mu$\nand $\\nu$. As usual, we use $\\lfloor a\\rfloor$ to denote the integer part of $a\\ge0.$\n\n\n\n\n\n\n\n\n\n\n\nThe time evolution for most of stochastic dynamical systems depends\nnot only on the present state but also on the past path. So,\npath-dependent (i.e., functional) SDEs are much more desirable; see,\ne.g., the monograph \\cite{M84}. Since the pioneer work \\cite{IN} due\nto It\\^o and Nisio, path-dependent SDEs have been investigated\nconsiderably owing to their theoretical and practical importance;\nsee, e.g., Hairer et al. \\cite{HMS11}, Wang \\cite{W18} and the\nreferences within.\n\n McKean-Vlasov SDEs, which are SDEs with coefficients dependent on the\n law, were initiated by \\cite{Mc} inspired by Kac's programme in Kinetic\n theory. An excellent and thorough account of the general theory of McKean-Vlasov\n SDEs and their particle approximations can be found in \\cite{sz}. McKean-Vlasov\n SDEs\nare alternatively referred to as mean-field SDEs in the literature, which have\nwide applications in interacting particle systems, optimal control problems,\ndifferential games, just to mention but a few. Recently, McKean-Vlasov SDEs have been\nextensively investigated on, e.g., wellposedness of strong\/weak solutions (cf.\n\\cite{DST,HSS,LMb,MV,W16}), Freidlin-Wentzell large deviation principles (cf. \\cite{DST}), ergodicity (cf. \\cite{B14,EGZ,Ve}), links with nonlinear partial differential\nequations (cf. \\cite{BLPR,HRW,HRW}), and distribution properties (cf.\\cite{Huang,W18}).\n\nOn the other hand, from stochastic and\/or statistical aspects, there exist\nunknown parameters in various type SDEs arising in mathematical modeling (cf.\n\\cite{B08}). Hence, there are vast of investigations paying attention to\nparameter estimations for SDEs via maximum likelihood estimator,\nleast squares estimator (LSE for short), trajectory-fitting estimator, among others.\nSee, for instance, \\cite{Ku04,LS01,M05,P199,SY06}. In the same vein, the parameter estimations for SDEs (without path-dependence) with small noises have been developed very\nwell; see, e.g., \\cite{GS,HL,Long10,L09,LMS,LSS,M10,SM03,U04,U08}, and references\ntherein.\n\nFrom above discussion, it is very natural to consider SDEs together with all four features of path dependence, distribution dependence, small noises and unknown parameter. So,\nin the present work, we focus on the following path-distribution SDE\n\\begin{equation}\\label{eq1}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D X^\\vv(t)=b(X_t^\\vv,\\mathscr{L}_{X_t^\\vv},\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt+\\vv\\,\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^\\vv,\\mathscr{L}_{X_t^\\vv})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t),\n~~~t>0,~~~~X_0^\\vv=\\xi\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r.\n\\end{equation}\nHerein, $\\vv\\in(0,1)$ is the scale parameter; for fixed $t$,\n$X_t^\\vv(v):=X^\\vv(t+v), v\\in[-r_0,0],$ is called the segment (or\nwindow) process generated by $X^\\vv(t)$; $\\mathscr{L}_{X_t^\\vv}$\nstands for the distribution of $X_t^\\vv$; $b:\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\\times\\mathcal\n{P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)\\times\\Theta\\rightarrow\\R^d$ and $\\sigma} \\def\\ess{\\text{\\rm{ess}}:\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\\times\\mathcal\n{P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)\\rightarrow\\R^d\\otimes\\R^m$ are continuous w.r.t.  the\nfirst variable and the second variable;\n  $\\Theta\\ni\\theta $ is an unknown parameter whose true value is\n  written as\n$\\theta_0\\in\\Theta$; and $(B(t))_{t\\ge0}$ is an $m$-dimensional\nBrownian motion on a filtered probability space\n$(\\OO,\\F,(\\F_t)_{t\\ge0},\\P)$ satisfying the usual conditions, that\nis, $\\F_t$ is non-decreasing (i.e., $\\F_s\\subseteq\\F_t, s\\le t),$\n$\\F_0$ contains all $\\P$-null sets and $\\F_t$ is right continuous\n(i.e., $\\F_t=\\F_{t+}:=\\bigcap_{s\\uparrow t}\\F_s$).\n\n\nTo guarantee the existence and uniqueness of solutions to \\eqref{eq1},\nwe assume that, for any $\\zeta_1,\\zeta_2\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$,  $\\mu,\\nu\\in\\mathcal\n{P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$, and $\\theta\\in\\Theta,$\n\\begin{enumerate}\n\\item[({\\bf A1})]  There exist\n$\\aa_1,\\aa_2>0$ such that\n\\begin{equation*}\n\\<\\zeta_1(0)-\\zeta_2(0),b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\nu,\\theta)\\>\\le\\aa_1\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\aa_2\\mathbb{W}_2(\\mu,\\nu)^2;\n\\end{equation*}\n\n\n\\item[({\\bf A2})]  There exist $\\bb_1,\\bb_2>0$ such that\n\\begin{equation*}\n\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_1,\\mu)-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_2,\\nu)\\|^2\\le\n\\bb_1\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\bb_2\\mathbb{W}_2(\\mu,\\nu)^2.\n\\end{equation*}\n\\end{enumerate}\n\nFrom \\cite[Theorem 3.1]{HRW}, \\eqref{eq1} has a unique\nstrong solution $(X^\\vv(t))_{t\\ge-r_0}$ under the assumptions ({\\bf\nA1}) and ({\\bf A2}). For any $\\zeta_1,\\zeta_2\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$,\n$\\mu,\\nu\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$, and $\\theta\\in\\Theta,$ if there\nexist $\\aa,\\bb>0$ such that\n\\begin{equation*}\n\\<\\zeta_1(0)-\\zeta_2(0),b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\mu,\\theta)\\>\\le\\aa\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\n\\end{equation*}\nand\n\\begin{equation*}\n|b(\\zeta_2,\\mu,\\theta)-b(\\zeta_2,\\nu,\\theta)|\\le\n\\bb\\mathbb{W}_2(\\mu,\\nu),\n\\end{equation*}\nthen  ({\\bf A1}) holds.\n\n\nWithout loss of generality, we arbitrarily fix the time horizontal $T>0$ and assume that there exist positive integers $n,M$ sufficiently large such that\n$\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho:=\\frac{T}{n}=\\ff{r_0}{M}$. Now we define the continuous-time\ntamed Euler-Maruyama (EM) scheme (see, e.g., \\cite{HJK}) associated\nwith \\eqref{eq1}\n\\begin{equation}\\label{q3}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D Y^\\vv(t)=b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t+ \\vv\\,\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t),~~~~t>0\n\\end{equation}\nwith the initial value $Y^\\vv(t)=X^\\vv(t)=\\xi(t)$ for any $\nt\\in[-r_0,0]$, where\n\\begin{itemize}\n\\item $t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho:=\\lfloor t\/\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rfloor\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho$ for $t\\ge0;$\n\n\\item For  any $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ and $\\mu\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$,\n\\begin{equation}\\label{e4}\nb^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mu,\\theta):=\\ff{b(\\zeta,\\mu,\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|},~~~~\\aa\\in(0,1\/2];\n\\end{equation}\n\n\\item For $k=0,1,\\cdots,n,$\n$\\bar Y_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv=\\{ \\bar Y_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv(s):-r_0\\le s\\le0\\}$, a\n$\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$-valued random variable, is  defined by\n\\begin{equation}\\label{w2}\n\\bar\nY_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv(s)=Y^\\vv((k-i)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\ff{s+i\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\{Y^\\vv((k-i)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\n-Y^\\vv((k-i-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\}\n\\end{equation}\nfor any $s\\in[-(i+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho,-i\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho]$, $i=0,1,\\cdots,M-1$, that is, $\\bar\nY_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv$ is the linear interpolation of the points\n$(Y^\\vv(l\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho))_{l=k-M,\\cdots,k}$.\n\\end{itemize}\n\nWe denote $(Y_t^\\vv)_{t\\ge0}$ by the segment process generated by\n$(Y^\\vv(t))_{t\\ge-r_0}$. It is worthy to point out that $\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ is defined by \\eqref{w2} rather than by $\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}(s)=\\bar Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho+s)$ for any $s\\in[-r_0,0]$\n Based on the\ncontinuous-time tamed EM algorithm \\eqref{q3}, we design the\nfollowing contrast function\n\\begin{equation}\\label{eq2}\n\\Psi_{n,\\vv}(\\theta)=\\vv^{-2}\\delta^{-1}\\sum_{k=1}^nP_k^*(\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta),\n\\end{equation}\nin which, for  $k=1,\\cdots,n$,\n\\begin{equation}\\label{w1}\nP_k(\\theta):=Y^\\vv(k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-Y^\\vv((k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho,~\n\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv):=(\\sigma} \\def\\ess{\\text{\\rm{ess}}\\si^*)^{-1}(\\bar\nY_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}).\n\\end{equation}\nFor more motivations on the construction of constrast function\nabove, we refer to Ren-Wu \\cite{RW}. To obtain the LSE of\n$\\theta\\in\\Theta$, it   is sufficient  to choose an element\n$\\hat\\theta_{n,\\vv}\\in\\Theta$ satisfying\n\\begin{equation*}\n\\Psi_{n,\\vv}(\\hat\\theta_{n,\\vv})=\\min_{\\theta\\in\\Theta}\\Psi_{n,\\vv}(\\theta).\n\\end{equation*}\nWhence, for\n\\begin{equation*}\n\\Phi_{n,\\vv}(\\theta):=\\vv^2(\\Psi_{n,\\vv}(\\theta)-\\Psi_{n,\\vv}(\\theta_0)),\n\\end{equation*}\none has\n\\begin{equation}\\label{eq4}\n\\Phi_{n,\\vv}(\\hat\\theta_{n,\\vv})=\\min_{\\theta\\in\\Theta}\\Phi_{n,\\vv}(\\theta).\n\\end{equation}\nWe shall rewrite $\\hat\\theta_{n,\\vv}\\in\\Theta$ such that \\eqref{eq4}\nholds true as\n\\begin{equation*}\n\\hat\\theta_{n,\\vv}=\\arg\\min_{\\theta\\in\\Theta}\\Phi_{n,\\vv}(\\theta),\n\\end{equation*}\nwhich   is called the LSE of the unknown parameter\n$\\theta\\in\\Theta$.\n\nTo discuss the consistency of LSE (see Theorem \\ref{th1} below), we\nfurther suppose that, for any $\\zeta,\\zeta_1,\\zeta_2\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$,\n$\\mu,\\nu\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$, and $\\theta\\in\\Theta,$\n\\begin{enumerate}\n\n\\item[({\\bf B1})] There exist $q_1,L_1>0$ such that\n\\begin{equation*}\n|b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\nu,\\theta)|\\le\nL_1\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\nu)\\Big\\};\n\\end{equation*}\n\\item[({\\bf B2})] There exist $q_2,L_2>0$ such that\n\\begin{equation*}\n\\sup_{\\theta\\in\\bar\\Theta}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(\\zeta_1,\\mu,\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)(\\zeta_2,\\nu,\\theta)\\|\\le\nL_2\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_2}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_2})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\nu)\\Big\\},\n\\end{equation*}\nwhere $(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)$ is the gradient operator w.r.t. the third\nspatial variable;\n\n\\item[({\\bf B3})] $(\\sigma} \\def\\ess{\\text{\\rm{ess}}\\si^*)(\\zeta,\\mu)$ is invertible, and\nthere exist $q_3,L_3>0$ such that\n\\begin{equation*}\n\\|(\\sigma} \\def\\ess{\\text{\\rm{ess}}\\si^*)^{-1}(\\zeta_1,\\mu)-(\\sigma} \\def\\ess{\\text{\\rm{ess}}\\si^*)^{-1}(\\zeta_2,\\nu)\\|\\le\nL_3\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_3}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_3})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\nu)\\Big\\};\n\\end{equation*}\n\\item[({\\bf B4})] There exists a constant $K>0$ such that\n\\begin{equation*}\n|\\xi(t)-\\xi(s)|\\le K|t-s|,~~~t,s\\in[-r_0,0],\n\\end{equation*}\nwhere $\\xi(\\cdot)$ stands for the initial value of \\eqref{eq1}.\n\\end{enumerate}\n\nIn order to reveal the asymptotic distribution of LSE (see Theorem\n\\ref{th2} below), we   in addition assume that\n\\begin{enumerate}\n\\item[({\\bf C})] There exist $q_4,L_4>0$ such that\n\\begin{equation*}\n\\begin{split}\n&\\sup_{\\theta\\in\\bar\\Theta}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*))(\\zeta_1,\\mu,\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*))(\\zeta_2,\\nu,\\theta)\\|\\\\\n&\\le\nL_4\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_4}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_4})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\nu)\\Big\\},\n\\end{split}\n\\end{equation*}\nwhere $b^*$ means the transpose of $b.$\n\\end{enumerate}\n\n\n\nNext we consider the following deterministic path-dependent ordinary\nequation\n\\begin{equation}\\label{k1}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D X^0(t)=b(X_t^0,\\mathscr{L}_{X_t^0},\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt,~~~t>0,~~~X_0^0=\\xi\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r.\n\\end{equation}\nUnder the assumption ({\\bf A1}), \\eqref{k1} is wellposed. In\n\\eqref{k1}, $\\mathscr{L}_{X_t^0}$ is indeed a Dirac's delta measure\nat the point $X_t^0$ as $X_t^0$ is deterministic. To unify the\nnotation, we keep the notation $\\mathscr{L}_{X_t^0}$ in lieu of\n$\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{X_t^0}$. We remark that ({\\bf B4}) is imposed to guarantee\nthat the linear interpolation $\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}$ tends to $X_t^0$\nin the moment sense, see Lemma \\ref{le1} below.\n\n\n\n\n For any random variable $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ with\n$\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$,  set\n\\begin{equation}\\label{p1}\n\\Gamma(\\zeta,\\theta,\\theta_0):=b(\\zeta,\\mathscr{L}_\\zeta,\\theta_0)\n-b(\\zeta,\\mathscr{L}_\\zeta,\\theta),~~\n~~\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\theta,\\theta_0):=b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mathscr{L}_\\zeta,\\theta_0)\n-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mathscr{L}_\\zeta,\\theta),\n\\end{equation}\nand, for any $\\theta\\in\\Theta,$\n\\begin{equation*}\\label{e0}\n\\Xi(\\theta)=\\int_0^T\\Gamma^*(X_t^0,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Gamma(X_t^0,\\theta,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt,\n\\end{equation*}\nwhere $(X_t^0)_{t\\ge0}$ is the functional solution  to \\eqref{k1}.\n\nThe  theorem  below is concerned with the consistency of the LSE for\nthe parameter $\\theta\\in\\Theta$, which is the first contribution of\nour work.\n\n\n\\begin{thm}\\label{th1}\n  Let $({\\bf A1})-({\\bf A2})$ and $({\\bf B1})-({\\bf B4})$ hold   and\nassume further that $\\Xi(\\theta)>0$ for $\\theta\\neq\\theta_0$. Then\n\\begin{equation*}\n\\hat\\theta_{n,\\vv}\\rightarrow\\theta_0~~~~\\mbox{ in probability as }\n \\vv\\rightarrow0 ~~\\mbox{ and }~~n\\rightarrow}\\def\\l{\\ell\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle.\n\\end{equation*}\n\n\\end{thm}\n\nFor $A:=(A_1,A_2,\\cdots,A_p)\\in\\R^p\\otimes\\R^{pd}$ with  $A_{k}\\in\n\\R^p\\otimes\\R^d$, $k=1,\\cdots,p,$ and $B\\in\\R^d$, define $A\\circ\nB\\in\\R^p\\otimes\\R^p$ by\n\\begin{equation*}\nA\\circ B=(A_1B,A_2B,\\cdots,A_pB).\n\\end{equation*}\nFor any $\\theta\\in\\Theta$, set\n\\begin{equation}\\label{0z3}\nI(\\theta):=\\int_0^T(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t,\n\\end{equation}\n\\begin{equation}\\label{0z2}\n\\begin{split}\nK(\\theta):&=-2\\int_0^T(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\nb^*)(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\n\\Gamma(X_t^0,\\theta,\\theta_0)\\Big)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t,\n\\end{split}\n\\end{equation}\nwhere $(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b^*):=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*))$, and,\nfor any random variable $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ with\n$\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$,\n\\begin{equation}\\label{0s0}\n\\Upsilon(\\zeta,\\theta_0)=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta,\\mathscr{L}_{\\zeta},\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta,\\mathscr{L}_{\\zeta}).\n\\end{equation}\n\nAnother main result in this paper is presented as below, which\nreveals the asymptotic distribution of $\\hat\\theta_{n,\\vv}.$\n\n\n\\begin{thm}\\label{th2}\n  Let the assumptions of Theorem \\ref{th1} hold and suppose\nfurther that $({\\bf C})$ holds and that $I(\\cdot)$ and $K(\\cdot)$\ndefined in \\eqref{0z3} and \\eqref{0z2}, respectively, are\ncontinuous. Then,\n\\begin{equation*}\n\\vv^{-1}(\\hat\\theta_{n,\\vv}-\\theta_0)\\rightarrow\nI^{-1}(\\theta_0)\\int_0^T\\Upsilon(X_t^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t)~~~~\\mbox{ in\nprobability }\n\\end{equation*}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$, where  $\\Upsilon(\\cdot)$\nis given in \\eqref{0s0}.\n\\end{thm}\n\n\nWith contrast to the existing literature, the innovations of this\npaper lie  in:\n\\begin{itemize}\n\\item[(i)] The classical contrast function for LSE is based on EM\nalgorithm. Whereas, under the monotone condition, the EM scheme   no\nlonger  works. Hence in the present work we adopt a tamed EM method\nto establish the corresponding contrast function. The above is our\nfirst innovation.\n\n\\item[(ii)]For the classical setup,   the discrete-time\nobservations at the gridpoints are sufficient to construct the\ncontrast function. Nevertheless, for our present model, the\ndiscrete-time observations are insufficient to establish the\ncontrast function since the SDEs involved are path-dependent. In\nthis paper, we overcome the difficulty mentioned by linear\ninterpolation w.r.t. the discrete-time observations. The above is\nour second innovation.\n\n\\item[(iii)] Our model is much more applicable, which allow the\ncoefficients to be distribution-dependent and weakly monotone. In\nparticular, the drift terms are allowed to be singular (e.g.,\nH\\\"older continuous). The above is our third innovation.\n\n\n\\end{itemize}\n\n\n\nNow, we provide a concrete  example to demonstrate Theorems\n\\ref{th1} and \\ref{th2}.\n\\begin{exa}\\label{exa}\nFor any $\\vv\\in(0,1)$, consider the following scalar\npath-distribution dependent SDE\n\\begin{equation}\n\\begin{split}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D X^\\vv(t)&=\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\\Big(-\n(X^\\vv(t))^3+X^\\vv(t)+\\int_{-r_0}^0X^\\vv(t+\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\theta+\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\n \\zeta(\\theta) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\theta\\Big)\\mathscr{L}_{X^\\vv_t}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\\\\n&\\quad+\\vv\\,\\Big(1+ \\int_{-r_0}^0X^\\vv(t+\\theta)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\theta\\Big)\\,\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t),~~~t\\ge0\n\\end{split}\n\\end{equation}\nwith the initial value $X_0^\\vv=\\xi\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$  which is Lipschitz,\nwhere, for some $c_1<c_2$ and $c_3<c_4 $,\n$\\theta=(\\theta^{(1)},\\theta^{(2)})^*\\in\\Theta_0:=(c_1,c_2)\\times(c_3,c_4)\\subset\\R^2_+$\n is an unknown parameter with the\ntrue value $\\theta_0=(\\theta^{(1)}_0,\\theta^{(2)}_0)^*\\in\\Theta_0$.\nLet $\\hat\\theta_{n,\\vv}$ be the LSE for the unknown parameter\n$\\theta=(\\theta^{(1)},\\theta^{(2)})^*\\in\\Theta_0$. Then,\n\\begin{equation*}\n\\hat\\theta_{n,\\vv}\\rightarrow\\theta_0~~~~\\mbox{ in probability as }\n \\vv\\rightarrow0 ~~\\mbox{ and }~~n\\rightarrow}\\def\\l{\\ell\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle,\n\\end{equation*}\nand\n\\begin{equation*}\n\\vv^{-1}(\\hat\\theta_{n,\\vv}-\\theta_0)\\rightarrow\nI^{-1}(\\theta_0)\\int_0^T\\Upsilon(X_t^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t)~~~~\\mbox{ in\nprobability }\n\\end{equation*}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$, where\n\\begin{equation*}\nI(\\theta_0)=\\left(\\begin{array}{ccc}\n \\int_0^T\\ff{1}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s  &  \\int_0^T\\ff{b_0(X_s^0,X_s^0)}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s \\\\\n  \\int_0^T\\ff{b_0(X_s^0,X_s^0)}{(1+|X_s^0|)^2} \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s &  \\int_0^T\\ff{b_0^2(X_s^0,X_s^0)}{(1+|X_s^0|)^2} \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n  \\end{array}\n  \\right),\n\\end{equation*}\nand, for $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r,$\n\\begin{equation*}\n \\Upsilon(\\zeta,\\theta_0) = \\ff{1}{1+|\\zeta|}\\left(\\begin{array}{c}\n  1 \\\\\n  b_0(\\zeta,\\zeta)\\\\\n  \\end{array}\n  \\right).\n\\end{equation*}\n\\end{exa}\n\n\n\n\nThe remaining contents are organized as below: In Section\n\\ref{sec2}, we intend to complete the proof of Theorem \\ref{th1} on\nthe basis of numerous auxiliary lemmas; In Section \\ref{sec3}, we\naim to implement the proof  of Theorem \\ref{th2}; In the final\nsection, we shall finish the proof of\n Example\n\\ref{exa}. Throughout this paper, we emphasize that  $c>0$ is a\ngeneric constant whose value may change from line to line.\n\n\n\\section{Proof of Theorem \\ref{th1}}\\label{sec2}\nTo complete the proof of Theorem \\ref{th1}, we provide some\ntechnical lemmas. The lemma below expounds that the path associated\nwith \\eqref{q3} is uniformly bounded in the $p$-th moment sense.\n\\begin{lem}\n  Let $({\\bf A1})$ and $({\\bf A2})$ hold. Then,  for any $p>0$  there\nis a constant $C_{p,T}>0$ such that\n\\begin{equation}\\label{r0}\n\\sup_{0\\le t\\le T}\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p\\le C_{p,T}(1+\\|\\xi\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p),\n\\end{equation}\nand\n\\begin{equation}\\label{r6}\n\\sup_{0\\le t\\le T}\\E\\Big(\\sup_{-r_0\\le s\\le t}|Y^\\vv(s)|^p\\Big)\\le\nC_{p,T}(1+\\|\\xi\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p).\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nWith the assumption ({\\bf A1}) in hand, the proof of \\eqref{r0} can\nbe achieved by the chain rule and the Gronwall inequality.  We\nherein omit the details since it is\n standard. Now we turn to show the argument of\n\\eqref{r6}. By H\\\"older's inequality, it suffices to verify that\n\\eqref{r6} holds for any $p>4.$ By It\\^o's formula,   we deduce that\n\\begin{equation*}\n\\begin{split}\n|Y^\\vv(t)|^p&=|Y^\\vv(0)|^p+\\int_0^t\\Big\\{p|Y^\\vv(s)|^{p-2}\\<Y^\\vv(s),b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)\\>+\n\\ff{p}{2}|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}^*(\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\\\\\n&\\quad +\\ff{p(p-2)}{2}|Y^\\vv(s)|^{p-4}|\\sigma} \\def\\ess{\\text{\\rm{ess}}^*(\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})Y^\\vv(s)|^2\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+p\\int_0^t|Y^\\vv(s)|^{p-2}\\<Y^\\vv(s),\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(s)\\>\\\\\n&\\le p\\int_0^t|Y^\\vv(s)|^{p-2}\\<Y^\\vv(s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho),b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)\\>\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+p\\int_0^t|Y^\\vv(s)|^{p-2}\\<Y^\\vv(s)-Y^\\vv(s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho),b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)\\>\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+\\ff{p(p-1)}{2}\\int_0^t|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+p\\int_0^t|Y^\\vv(s)|^{p-2}\\<Y^\\vv(s),\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(s)\\>\\\\\n&=:\\sum_{i=1}^4\\Pi_i(t),~~~~~~t\\in[0,T].\n\\end{split}\n\\end{equation*}\nWhence, for any $t\\ge0$ one has\n\\begin{equation}\\label{w6}\n\\begin{split}\n\\Upsilon(t):=\\E\\Big(\\sup_{-r_0\\le s\\le\nt}|Y^\\vv(s)|^p\\Big)\\le\\|\\xi\\|^p_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\sum_{i=1}^4\\E\\Big(\\sup_{0\\le\ns\\le t}\\Pi_i(s)\\Big).\n\\end{split}\n\\end{equation}\nIn the sequel, we are going to claim that\n\\begin{equation}\\label{w7}\n\\Upsilon(t) \\le 2\\|\\xi\\|^p_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+ c\\,t+c \\int_0^t \\Upsilon(s) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation}\nIf \\eqref{w7} was true, thus \\eqref{r6} follows directly from\nGronwall's inequality. So, it remains to verify that \\eqref{w7}\nholds true.\n\n\nLet $\\zeta_0(s)\\equiv  {\\bf 0}\\in\\R^d$\nfor any $s\\in[-r_0,0].$ For  $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ and $\\mu\\in\\mathcal\n{P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$,   we deduce from ({\\bf A1}) that\n\\begin{equation}\\label{r1}\n\\begin{split}\n\\<\\zeta(0),b(\\zeta,\\mu,\\theta)\\>&=\\<\\zeta(0)-\\zeta_0,b(\\zeta,\\mu,\\theta)-b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta)\\>\n+\\<\\zeta(0),b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta)\\>\\\\\n&\\le\\aa_1\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\aa_2\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2+|\\zeta(0)|^2+|b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta)|^2\\\\\n&\\le c\\,(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2),\n\\end{split}\n\\end{equation}\nand from ({\\bf A2}) that\n\\begin{equation}\\label{r2}\n\\begin{split}\n\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta,\\mu)\\|^\n&\\le\n2\\bb_1\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+2\\bb_2\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2+2\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\|^2\\\\\n&\\le c\\,(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2).\n\\end{split}\n\\end{equation}\nAccording to \\eqref{w2}, we obtain that\n\\begin{equation}\\label{r5}\n\\begin{split}\n&\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\\\\n&=\\max_{k=0,\\cdots,M-1}\\sup_{-(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le s\\le-k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv(s)|\\\\\n&=\\max_{k=0,\\cdots,M-1}\\sup_{-(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\ns\\le-k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big|\\ff{s+(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho-k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-\\ff{s+k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho-(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\Big|\\\\\n&\\le 2\\sup_{-r_0\\le s\\le t}|Y^\\vv(s)|.\n\\end{split}\n\\end{equation}\nFurthermore, recall the Young inequality:\n\\begin{equation}\\label{r3}\na^\\aa b^{1-\\aa}\\le \\aa a+(1-\\aa)b,~~~~~a,b\\ge0,~~\\aa\\in[0,1],\n\\end{equation}\nand the fundamental fact that: for any $q>0$,\n\\begin{equation}\\label{w8}\n\\E|B(t)|^q\\le c\\, t^{q\/2}.\n\\end{equation}\nBy virtue of \\eqref{w2}, we notice that\n\\begin{equation}\\label{r8}\n\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv(0)=Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).\n\\end{equation}\nThen, by exploiting \\eqref{r1}, \\eqref{r5} as well as \\eqref{r8}, it\nfollows from \\eqref{r3} and H\\\"older's inequality that\n\\begin{equation}\\label{w3}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le t}\\Pi_1(s)\\Big) \\\\&= p\\,\\E\\Big(\\sup_{0\\le\ns\\le t}\\int_0^s\\ff{|Y^\\vv(u)|^{p-2}}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa |b(\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)|}\\<Y^\\vv(u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho),b(\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)\\>\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nu\\Big)\\\\\n&= p\\,\\E\\Big(\\sup_{0\\le s\\le\nt}\\int_0^s\\ff{|Y^\\vv(u)|^{p-2}}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa |b(\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)|}\\<\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}(0),b(\\bar Y^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)\\>\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nu\\Big)\\\\\n&\\le c\\,\\E\\Big(\\sup_{0\\le s\\le\nt}\\int_0^s\\ff{|Y^\\vv(u)|^{p-2}}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa| b(\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)|}\\Big\\{1+\\|\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\nc\\int_0^t\\Big\\{1+\\E|Y^\\vv(s)|^p+\\E\\|\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le c\\int_0^t\\{1+\\Upsilon(s)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nIt is straightforward to see that, for any $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$,\n$\\mu\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$, and $\\theta\\in\\Theta$,\n\\begin{equation}\\label{r4}\n|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mu,\\theta)|=\\ff{|b(\\zeta,\\mu,\\theta)|}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|}\\le\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{-\\aa}.\n\\end{equation}\nTaking \\eqref{r2}  and \\eqref{r4} into consideration and making use\nof \\eqref{w8} and $\\aa\\in(0,1\/2]$, for any $q\\ge2$, we derive that\n\\begin{equation}\\label{r7}\n\\begin{split}\n\\E|Y^\\vv(t)-Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^q&\\le\nc\\,\\Big\\{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q(1-\\aa)}+\\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^q\\E|B(t)-B(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^q\\Big\\}\\\\\n&\\le\nc\\,\\Big\\{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q(1-\\aa)}+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q\/2}\\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^q\\Big\\}\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q\/2}\\Big\\{1+\n\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|^q+\\mathbb{W}_2(\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^q\\Big\\}\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q\/2}\\Big\\{1+ \\E\\Big(\\sup_{-r_0\\le s\\le\nt}|Y^\\vv(s)|^q\\Big)\\Big\\},\n\\end{split}\n\\end{equation}\nwhere in the last procedure we have used H\\\"older's inequality and\n\\eqref{r5}. Thus, taking advantage of \\eqref{r4} and \\eqref{r7} and\nemploying H\\\"older's inequality yields that\n\\begin{equation}\\label{w4}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le t}|\\Pi_2(s)|\\Big)&\\le\np\\E\\int_0^t|Y^\\vv(s)|^{p-2}|Y^\\vv(s)-Y^\\vv(s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|\\cdot|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\np\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{-\\aa}\\int_0^t\\E(|Y^\\vv(s)|^{p-2}|Y^\\vv(s)-Y^\\vv(s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le\np\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{-\\aa}\\int_0^t\\Big(\\E(|Y^\\vv(s)|^p)\\Big)^{\\ff{p-2}{p}}\\Big(\\E|Y^\\vv(s)-Y^\\vv(s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{p}{2}}\\Big)^{\\ff{2}{p}}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\np\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{1}{2}-\\aa}\\int_0^t\\Big(\\E(|Y^\\vv(s)|^p)\\Big)^{\\ff{p-2}{p}}\\Big\\{1+\n\\E\\Big(\\sup_{-r_0\\le s\\le\nt}|Y^\\vv(s)|^{\\ff{p}{2}}\\Big)\\Big\\}^{\\ff{2}{p}}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le c \\int_0^t\\{1+\\Upsilon(s)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s,\n\\end{split}\n\\end{equation}\nwhere  in the last display we  used\n $\\aa\\in(0,1\/2]$ and  \\eqref{r3}. Next, we observe\nthat\n\\begin{equation}\\label{q1}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le t}\\Pi_3(s)\\Big)&\\le\n\\ff{p(p-1)}{2}\\int_0^t\\E(|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nUsing Burkhold-Davis-Gundy's (BDG's for short) inequality and\n\\eqref{r3}, we infer that\n\\begin{equation}\\label{q2}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le t}\\Pi_4(s)\\Big)&\\le p\\,\\E\\Big(\\sup_{0\\le s\\le\nt}\\Big|\\int_0^s|Y^\\vv(u)|^{p-2}\\<Y^\\vv(u),\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(u)\\>\\Big|\\Big)\\\\\n&\\le 4\\ss2\\,p\\,\\E\\Big(\\int_0^t|Y^\\vv(s)|^{2(p-2)}|\\sigma} \\def\\ess{\\text{\\rm{ess}}^*(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})Y^\\vv(s)|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le 4\\ss2\\,p\\,\\E\\Big(\\sup_{0\\le s\\le\nt}|Y^\\vv(s)|^p\\int_0^t|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le\\ff{1}{2}\\Upsilon(t)+16p^2\\int_0^t\\E(|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nSubsequently, one gets from \\eqref{q1} and \\eqref{q2} that\n\\begin{equation}\\label{w5}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le t}\\Pi_3(s)\\Big)+\\E\\Big(\\sup_{0\\le s\\le\nt}\\Pi_4(s)\\Big)\\\\&\\le\n\\ff{1}{2}\\Upsilon(t)+c\\int_0^t\\E(|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le \\ff{1}{2}\\Upsilon(t)+ c\\int_0^t\\{\\E|Y^\\vv(s)|^p+\\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^p\\}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le \\ff{1}{2}\\Upsilon(t)+ c\\int_0^t\\Big\\{1+\\E|Y^\\vv(s)|^p+\\E\\|\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^p\\Big\\}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le \\ff{1}{2}\\Upsilon(t)+ c \\int_0^t\\{1+\\Upsilon(s)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s,\n\\end{split}\n\\end{equation}\nwhere  we have adopted \\eqref{r3} in the second inequality, used\n\\eqref{r2} in the last two step, and utilized H\\\"older's inequality,\nin addition to \\eqref{r5}, in the last procedure. Substituting\n\\eqref{w3}, \\eqref{w4}, and \\eqref{w5} into \\eqref{w6} gives that\n\\begin{equation*}\n\\Upsilon(t) \\le \\|\\xi\\|^p_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+ \\ff{1}{2}\\Upsilon(t)+ c\n\\int_0^t\\{1+\\Upsilon(s)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation*}\nAs a consequence, \\eqref{w7} is now available.\n\\end{proof}\n\n\nThe following lemma shows that the linear interpolation $\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}$ approaches $X_t^0$ in the mean square sense as\n $\\vv$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho$ go to zero.\n\\begin{lem}\\label{le1}\n  Assume $({\\bf A1}), ({\\bf A2}), ({\\bf B1})$ and $({\\bf B4})$. Then,\nfor any $\\bb\\in(0,1)$, there exists $c_\\bb>0$\n\n\\begin{equation}\\label{a9}\n \\sup_{0\\le t\\le T}\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\le\n c_\\bb(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\vv^2+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}),\n\\end{equation}\nwhere $\\aa\\in(0,1\/2]$ is introduced in \\eqref{e4}.\n\\end{lem}\n\n\\begin{proof}\nFor any  $\\bb\\in(0,1)$ and $t\\in[0,T]$, by H\\\"older's inequality and\n$Y_0^\\vv=X_0^0=\\xi$, we find that\n\\begin{equation}\\label{a7}\n\\begin{split}\n\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2&\\le3\\,\\E\\|Y_t^\\vv-\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+3\\,\\E\\|Y^\\vv_t-X_t^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+3\\,\\E\\|X^\\vv_t-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\\\\n&\\le3\\,\\E\\Big(\\sup_{-r_0\\le v\\le 0}|Y^\\vv(t+v)-\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}(v)|^2\\Big)+3\\,\\E\\Big(\\sup_{0\\le s\\le\nt}|Y^\\vv(s)-X^\\vv(s)|^2\\Big)\\\\\n&\\quad+3\\,\\E\\Big(\\sup_{0\\le s\\le t}|X^\\vv(s)-X^0(s)|^2\\Big)\\\\\n&\\le3\\,\nM^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E\\Big(\\sup_{-(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\nv\\le-k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}| Y^\\vv(t+v)-\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}(v)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+3\\,\\E\\Big(\\sup_{0\\le s\\le t}|Y^\\vv(s)-X^\\vv(s)|^2\\Big)\n+3\\,\\E\\Big(\\sup_{0\\le s\\le\nt}|X^\\vv(s)-X^0(s)|^2\\Big)\\\\\n&=:\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho),\n\\end{split}\n\\end{equation}\nwhere $M>0$ such that $M\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho=r_0.$ Hereinafter, we intend to estimate\n$\\LL_i(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)$, $i=1,2,3,$ respectively. In the first place, we\nshall show that\n\\begin{equation}\\label{e3}\n\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb,~~~~t\\in[0,T].\n\\end{equation}\nFor   $t\\in[0,T)$, there is an integer $k_0\\ge0$ such that\n$t\\in[k_0\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho,(k_0+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).$ From \\eqref{w2}, it follows that\n\\begin{equation}\\label{a00}\n\\begin{split}\n&\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\\\&\\le c\\,\nM^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E\\Big(\\sup_{(k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\ns\\le(k_0+1-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\nY^\\vv(s)-Y^\\vv((k_0-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+\n c\\,\nM^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E\\Big(\\sup_{(k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\ns\\le(k_0+1-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\nY^\\vv(s)-Y^\\vv((k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\le\n c\\,\nM^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E\\Big(\\sup_{(k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\ns\\le(k_0+1-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\nY^\\vv(s)-Y^\\vv((k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+c\\, M^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E|\nY^\\vv((k_0-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-Y^\\vv((k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)^{1-\\bb}.\n\\end{split}\n\\end{equation}\nIn case of  $k\\ge k_0+1$, by virtue of ({\\bf B4}),  one has\n\\begin{equation*}\n\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\,M^{1-\\bb}\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^2\\le c\\,r_0^{1-\\bb}\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb.\n\\end{equation*}\nIn terms of ({\\bf B1}), for any $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ and $\\mu\\in\\mathcal\n{P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$,\n\\begin{equation}\\label{q4}\n\\begin{split}\n|b(\\zeta,\\mu,\\theta_0)|&\\le|b(\\zeta,\\mu,\\theta_0)-b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta_0)|+|b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta_0)|\\\\\n&\\le\nL_1\\Big\\{(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1})\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}+|b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta_0)|.\n\\end{split}\n\\end{equation}\nLet  $k'\\ge0$ be an arbitrary integer. For any\n$t\\in[k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho,(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho]$, note from BDG's inequality followed by\nH\\\"older's inequality that\n\\begin{equation*}\n\\begin{split}\n&\\E\\Big(\\sup_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le t\\le\n(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|Y^\\vv(t)-Y^\\vv(k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\\\&\\le\nc\\,\\E\\Big(\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta_0)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{\\ff{2}{1-\\bb}}+c\\,\\E\\Big(\\sup_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le t\\le\n(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big|\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^t\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(s)\\Big|^{\\ff{2}{1-\\bb}}\\Big)\\\\\n&\\le c\\,\\E\\Big(\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta_0)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{\\ff{2}{1-\\bb}}+c\\,\\E\\Big(\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{\\ff{1}{1-\\bb}}\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{\\bb}{1-\\bb}}\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big\\{\\E|b(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta_0)|^{\\ff{2}{1-\\bb}}+ \\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^{\\ff{2}{1-\\bb}}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s,\n\\end{split}\n\\end{equation*}\nwhere in the last display we have used the fact that\n\\begin{equation}\\label{t7}\n|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mu,\\theta_0)|\\le|b(\\zeta,\\mu,\\theta_0)|,~~~~\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r,~~~\\mu\\in\\mathcal\n{P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r).\n\\end{equation}\nSubsequently, taking  \\eqref{r6}, \\eqref{r2} and \\eqref{q4} into\naccount and making use of H\\\"older's inequality  yields that\n\\begin{equation}\\label{d1}\n\\begin{split}\n&\\E\\Big(\\sup_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le t\\le\n(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|Y^\\vv(t)-Y^\\vv(k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\\\\n&\\le\nc\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{\\bb}{1-\\bb}}\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big\\{1+\\E\\|\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{\\ff{2(1+q_1)}{1-\\bb}}+\n\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^{\\ff{2}{1-\\bb}}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\nc\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{\\bb}{1-\\bb}}\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big\\{1+\\E\\|\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{\\ff{2(1+q_1)}{1-\\bb}}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{1}{1-\\bb}}.\n\\end{split}\n\\end{equation}\nHence,    it follows from  \\eqref{a00} and \\eqref{d1} with\n$k^\\prime=k_0-k-1$ that\n\\begin{equation*}\n\\begin{split}\n\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\, M^{1-\\bb}\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb\n\\end{split}\n\\end{equation*}\nprovided that $k\\le k_0-1$. Whenever $k=k_0$, we deduce from\n\\eqref{a00}, \\eqref{d1} with $k'=0$ as well as ({\\bf B4}) that\n\\begin{equation*}\n\\begin{split}\n\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)&\\le c\\, M^{1-\\bb}\\Big(\\E\\Big(\\sup_{0\\le s\\le\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\nY^\\vv(s)-Y^\\vv(0)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+ c\\, M^{1-\\bb}\\Big(\\E\\Big(\\sup_{-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le s\\le0}|\nY^\\vv(s)-Y^\\vv(-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+c\\, M^{1-\\bb}|\nY^\\vv(0)-Y^\\vv(-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^2\\\\\n&\\le c\\, M^{1-\\bb}\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb.\n\\end{split}\n\\end{equation*}\nNext, we are going to claim that\n\\begin{equation}\\label{a8}\n\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho) \\le c\\,\\vv^2,~~~~t\\in[0,T].\n\\end{equation}\nFollowing the argument to derive \\eqref{r6}, we deduce that, for\nsome constant $C_{p,T}>0,$\n\\begin{equation}\\label{0r6}\n\\sup_{0\\le t\\le T}\\E\\|X^\\vv_t\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p\\le C_{p,T}(1+\\|\\xi\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p).\n\\end{equation}\nBy the It\\^o formula  and $X^\\vv_0=X_0^0=\\xi$, we observe that\n\\begin{equation*}\n\\begin{split}\n&|X^\\vv(t)-X^0(t)|^2\\\\\n&=\\int_0^t\\{2\\<X^\\vv(s)-X^0(s),b(X_s^\\vv,\\mathscr{L}_{X_s^\\vv},\\theta_0)-b(X_s^0,\\mathscr{L}_{X_s^0},\\theta_0)\\>+\\vv^2\\|\n\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})\\|^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+2\\,\\vv\\int_0^t\\<X^\\vv(s)-X^0(s),\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(s)\\>.\n\\end{split}\n\\end{equation*}\nThus, by using BDG's inequality and  \\eqref{r3} and noting that\n$X^\\vv_0=X_0^0=\\xi$, we infer from ({\\bf A1}) and \\eqref{r2} that\n\\begin{equation*}\n\\begin{split}\n\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)&\\le2\\int_0^t\\{\\aa_1\\E\\|X_s^\\vv-X_s^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\aa_2\\mathbb{W}_2(\\mathscr{L}_{X_s^\\vv},\\mathscr{L}_{X_s^0})^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+c\\,\\vv^2\\int_0^t\\{1+\\E\\|X_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{X_s^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+8\\ss2\\,\\vv\\E\\Big(\\int_0^t|\\sigma} \\def\\ess{\\text{\\rm{ess}}^*(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})(X^\\vv(s)-X^0(s))|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le c\\int_0^t\\LL_3(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s+c\\,\\vv^2\\int_0^t\\{1+\\E\\|X_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+8\\ss2\\,\\vv\\E\\Big(\\sup_{0\\le s\\le\nt}|X^\\vv(s)-X^0(s)|^2\\int_0^t\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le\\ff{1}{2}\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+c\\int_0^t\\LL_3(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns+c\\,\\vv^2\\int_0^t\\{1+\\E\\|X_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation*}\nSo, one has\n\\begin{equation*}\n\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho) \\le c\\int_0^t\\LL_3(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns+c\\,\\vv^2\\int_0^t\\{1+\\E\\|X_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation*}\nThus, \\eqref{a8} follows from \\eqref{0r6} and Gronwall's inequality.\nFinally, we intend to verify that\n\\begin{equation}\\label{s1}\n\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\,(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}),~~~~t\\in[0,T].\n\\end{equation}\nAlso, by It\\^o's formula, we derive from $X_0^\\vv=Y_0^\\vv=\\xi$ that\n\\begin{align*}\n|X^\\vv(t)-Y^\\vv(t)|^2&=2\\int_0^t\\<X^\\vv(s)-Y^\\vv(s),\nb(X_s^\\vv,\\mathscr{L}_{X_s^\\vv},\\theta_0)-b(Y_s^\\vv,\\mathscr{L}_{Y_s^\\vv},\\theta_0)\\>\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+2\\int_0^t\\<X^\\vv(s)-Y^\\vv(s),\nb(Y_s^\\vv,\\mathscr{L}_{Y_s^\\vv},\\theta_0)-b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)\\>\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+2\\int_0^t\\<X^\\vv(s)-Y^\\vv(s),\nb(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)\\>\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+\\vv^2\\int_0^t\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+2\\,\\vv\\int_0^t\\<X^\\vv(s)-Y^\\vv(s),(\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}))\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(s)\\>\\\\\n&=:\\Xi_1(t)+\\Xi_2(t)+\\Xi_3(t)+\\Xi_4(t)+\\Xi_5(t).\n\\end{align*}\nIn view of ({\\bf A1}), we deduce   that\n\\begin{equation}\\label{a22}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le\nt}\\Xi_1(s)\\Big)&\\le2\\int_0^t\\{\\aa_1\\E\\|X_s^\\vv-Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\aa_2\\mathbb{W}_2(\\mathscr{L}_{X_s^\\vv},\\mathscr{L}_{Y_s^\\vv})^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\,\\int_0^t\\E\\|X_s^\\vv-Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nCarrying out a similar argument to derive \\eqref{e3}, for any\n$\\kk>2$, we have\n\\begin{equation}\\label{w0}\n\\sup_{0\\le t\\le T}\\E\\|Y^\\vv_t-\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^\\kk\\le\nc\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{\\kk}{2}-1}.\n\\end{equation}\nTaking  ({\\bf A1}),  \\eqref{r6} and \\eqref{w0} into consideration\nand applying H\\\"older's inequality that\n\\begin{equation}\\label{a2}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le\nt}|\\Xi_2(s)|\\Big)\\\\&\\le\\int_0^t\\{\\E|X^\\vv(s)-Y^\\vv(s)|^2+\n\\E|b(Y_s^\\vv,\\mathscr{L}_{Y_s^\\vv},\\theta_0)-b(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\int_0^t\\E|X^\\vv(s)-Y^\\vv(s)|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+c\\int_0^t\\E\\{(1+\\| Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q_1}+\\|\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q_1})\\|Y^\\vv_s-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{Y_s^\\vv},\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})^2\\} \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s+c\\int_0^t\\Big(\\E\\|Y^\\vv_s-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{\\ff{2}{1-\\bb}}\\Big)^{1-\\bb}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+c\\int_0^t\\Big(\\E\\Big(1+\\| Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q_1}+\\|\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q_1}\\Big)^{\\ff{1}{\\bb}}\\Big)^{\\bb}\\Big(\\E\\|Y^\\vv_s-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{\\ff{2}{1-\\bb}}\\Big)^{1-\\bb}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nAccording to \\eqref{e4} and in view of \\eqref{r6} and \\eqref{q4}, it\nfollows from H\\\"older's inequality that\n\\begin{equation}\\label{a3}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le\nt}|\\Xi_3(s)|\\Big)\\\\&\\le2\\int_0^t\\E\\{|X^\\vv(s)-Y^\\vv(s)|\\cdot\n|b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le c\\int_0^t\\E\\Big\\{|X^\\vv(s)-Y^\\vv(s)|^2+\n \\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}|b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|^4}{(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|\n b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|)^2}\n \\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n &\\le c\\int_0^t\\Big\\{\\E|X^\\vv(s)-Y^\\vv(s)|^2+\n\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}\\E|b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|^4\n \\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n &\\le c\\int_0^t\\Big\\{\\E|X^\\vv(s)-Y^\\vv(s)|^2+\n\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}\\{1+\\E\\|\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q_1)}+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^4\\}\n \\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n &\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\n Next, owing to $\\vv\\in(0,1)$, $({\\bf A2})$, and\n\\eqref{e3}, one gets that\n\\begin{equation}\\label{a1}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le t}\\Xi_4(s)\\Big)&\\le\nc\\int_0^t\\{\\E\\|X_s^\\vv-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{X_s^\\vv},\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\int_0^t\\{\\E\\|X_s^\\vv-Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\E\\|Y_s^\\vv-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\int_0^t\\{\\LL_1(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nNext, for $\\vv\\in(0,1)$, BDG's inequality and Young's inequality\n\\eqref{r3}, besides \\eqref{a1},  give that\n\\begin{equation}\\label{a5}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le\nt}|\\Xi_5(s)|\\Big)\\\\&\\le8\\ss2\\,\\E\\Big(\\int_0^t\n|(\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}))^*(X^\\vv(s)-Y^\\vv(s))|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le8\\ss2\\,\\E\\Big(\\sup_{0\\le\\le\nt}|X^\\vv(s)-Y^\\vv(s)|^2\\int_0^t\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le\n\\ff{1}{2}\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+c\\int_0^t\\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\n\\ff{1}{2}\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nThus, \\eqref{a22}, \\eqref{a2}-\\eqref{a5} yield that\n\\begin{equation*}\n\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le\\ff{1}{2}\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\nc\\,(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa})+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation*}\nNamely,\n\\begin{equation*}\n\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le\nc\\,(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa})+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation*}\nAs a result, we obtain from Gronwall's inequality that\n\\begin{equation}\\label{a6}\n\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\,(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}).\n\\end{equation}\nInserting \\eqref{e3}, \\eqref{a8}, and \\eqref{a6} back into\n\\eqref{a7} leads to the desired assertion \\eqref{a9}.\n\\end{proof}\n\n\n\\begin{rem}\n{\\rm The convergence rate of EM scheme for path-independent SDEs\nunder the global Lipschitz condition is one half. Taking $\\aa=1\/2$\nin \\eqref{s1}, we conclude that the convergence rate of  the tamed\nEM scheme constructed in \\eqref{q3} is close sufficiently to one\nhalf. This demonstrate the distinct features between path-dependent\nSDEs and path-independent SDEs.  }\n\\end{rem}\n\n\nThe lemma below plays a crucial role in revealing the asymptotic\nbehavior of the LSE of the unknown parameter $\\theta\\in\\Theta$.\n\n\n\n\n\n\n\n\n\\begin{lem}\\label{lem}\n  Let $({\\bf A1})-({\\bf A2})$ and $({\\bf B1})-({\\bf B4})$ hold. Then,\n\\begin{equation}\\label{t4}\n\\begin{split}\n&\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\\\\n&\\rightarrow\\Xi(\\theta):=\\int_0^T\\Gamma(X_t^0,\\theta,\\theta_0)^*\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Gamma(X_s^0,\\theta,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\n\\end{split}\n\\end{equation}\nin $L^1$ as $\\vv\\rightarrow0$ and $\\delta\\rightarrow0 $ $($i.e.,\n$n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$$)$. Moreover,\n\\begin{equation}\\label{q6}\n \\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0) \\rightarrow0\n\\end{equation}\nin $L^2$ as $\\vv\\rightarrow0$.\n\\end{lem}\n\n\n\\begin{proof}\nObserve that\n\\begin{equation*}\n\\begin{split}\n&\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)-\n\\int_0^T\\Gamma^*(X_t^0,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Gamma(X_t^0,\\theta,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\\\\n&=\\int_0^T\\Big\\{(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\n-\\Gamma^*(X_t^0,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Gamma(X_t^0,\\theta,\\theta_0)\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\\\\n&=\\int_0^T \\Big(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)-\\Gamma(X_t^0,\\theta,\\theta_0)\\Big)^*\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\quad+\\int_0^T\\Gamma(X_t^0,\\theta,\\theta_0)^*\n \\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)-\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Big)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\\\\n&\\quad+\\int_0^T\\Gamma(X_t^0,\\theta,\\theta_0)^*\n\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Big(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)-\\Gamma(X_t^0,\\theta,\\theta_0)\\Big) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\\\\n&=:J_1(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+J_2(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+J_3(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).\n\\end{split}\n\\end{equation*}\nFrom ({\\bf B1}) and \\eqref{q4}, a direct calculation shows  that,\nfor any random variables $\\zeta_1,\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ with\n$\\mathscr{L}_{\\zeta_1},\\mathscr{L}_{\\zeta_2}\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$,\n\\begin{equation}\\label{t1}\n\\begin{split}\n&|\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\theta,\\theta_0)-\\Gamma(\\zeta_2,\\theta,\\theta_0)|\\\\\n&=|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta_0)+\nb(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\n|\\\\\n&\\le|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta_0)|\n+|b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)-b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\\\\n&\\quad+|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)|\n+|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\\\\n&=|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta_0)|\n+|b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)-b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Big|\\ff{|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)|}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)|}b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)\\Big|\n+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Big|\\ff{|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}\nb(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\Big|\\\\\n&\\le|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta_0)|\n+|b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)-b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\{|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)|^2+|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|^2\\}\\\\\n&\\le\nc\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mathscr{L}_{\\zeta_1},\\mathscr{L}_{\\zeta_2})\\Big\\}\\\\\n&\\quad+c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\Big\\{1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q_1)}+\\mathbb{W}_2(\\mathscr{L}_{\\zeta_1},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big\\}.\n\\end{split}\n\\end{equation}\nNext, for a  random variable  $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ with\n$\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$, by  \\eqref{q4} and\n\\eqref{t7}, it follows that\n\\begin{equation}\\label{t3}\n\\begin{split}\n|\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\theta,\\theta_0)|+|\\Gamma(\\zeta,\\theta,\\theta_0)\n\\le c\\,\\Big\\{1+\n \\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1} +\n \\mathbb{W}_2(\\mathscr{L}_\\zeta,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}\n \\end{split}\n\\end{equation}\nand, due to ({\\bf B3}), that\n\\begin{equation}\\label{t2}\n\\|\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)\\|\\le\\|\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)-\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(0)\\|+\\|\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(0)\\|\n\\le c\\,\\Big\\{1+\n \\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_3} +\n \\mathbb{W}_2(\\mathscr{L}_\\zeta,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}.\n\\end{equation}\nConsequently, combining   \\eqref{t1} with \\eqref{t3} and\\eqref{t2},\nfor $q:=q_1\\vee q_3$, we deduce from \\eqref{r0} that\n\\begin{align*}\n&|J_1(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|+|J_3(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|\\\\&\\le c\\int_0^T\\Big\\{(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1}+\\|X_s^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\mathscr{L}_{X_t^0})\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\Big(1+\\|\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q_1)}+\\mathbb{W}_2(\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big)\\Big\\}\\\\\n&\\quad\\times\\Big\\{1 +\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1}+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1} + \\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_0)\\Big\\}\\\\\n&\\quad\\times\\Big\\{1 +\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_3}+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_3} + \\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\le c\\int_0^T\\Big\\{(1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^q)\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\ss{\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big\\}\\\\\n&\\quad\\times\\Big\\{1+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+ q)} + \\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\int_0^T\\Big\\{1+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+ q)} + \\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^4\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t.\n\\end{align*}\nThis, by exploiting \\eqref{r6} and \\eqref{a7} and using H\\\"older's\ninequality, gives that\n\\begin{equation}\\label{t5}\n\\begin{split}\n&\\E|J_1(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|+\\E|J_3(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|\\\\&\\le c\\,\\int_0^T\\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big\\{1 +\\E\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+ q)}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\quad+c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\int_0^T\\Big\\{1+\\E\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+ q)} \\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\rightarrow0\n\\end{split}\n\\end{equation}\n  as  $ \\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$.\nNext,  making use of ({\\bf B3})   and \\eqref{t3}, we derive that\n\\begin{equation*}\n\\begin{split}\n |J_2(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|&\\le c\\int_0^T(1+\n \\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1})\\Big(1+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1} + \\ss{\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big)\\\\\n&\\quad\\times\\Big((1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_3}+\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_3})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t.\n\\end{split}\n\\end{equation*}\nAgain, using \\eqref{r0}, \\eqref{r6} and \\eqref{a9}  and utilizing\nH\\\"older's inequality gives that\n\\begin{equation}\\label{t6}\n\\begin{split}\n\\E|J_2(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|&\\le c\\,\\int_0^T\\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big\\{1 +\\E\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+ q)}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\rightarrow0\n\\end{split}\n\\end{equation}\n as  $ \\vv\\rightarrow0$ and  $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$.\nHence,   \\eqref{t4} follows immediately from \\eqref{t5} and\n\\eqref{t6}.\n\n\nIn the sequel, we are going to show that \\eqref{q6} holds. In terms\nof \\eqref{q3}, we obtain that\n\\begin{equation}\\label{q5}\n\\begin{split}\n&\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\\\\n&=\\vv\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\n\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y^\\vv_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})(B(k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-B((k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho))\\\\\n&=\\vv\\int_0^T(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv) \\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t).\n\\end{split}\n\\end{equation}\nBy the It\\^o isometry and the H\\\"older  inequality, we derive from\n\\eqref{r2}, \\eqref{t3}, and \\eqref{t2} that\n\\begin{equation*}\n\\begin{split}\n&\\E\\Big|\\int_0^T(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv) \\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(t)\\Big|^2\\\\\n&=\\int_0^T\\E|(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv) \\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\n t\\\\\n &\\le\\int_0^T\\E\\{|\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)|^2\\cdot\\|\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\|^2\\cdot\\| \\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\le c\\,\\int_0^T\\E\\Big\\{\\Big(1+\\|\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big)\\\\\n&\\quad\\times\\Big(1+\n \\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q_3)} +\n \\mathbb{W}_2(\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big)\\\\\n&\\quad\\times\\Big(1+\n \\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q_1)} +\n \\mathbb{W}_2(\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big)\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\le c\\,\\int_0^T\\{1+\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+q)}\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t.\n\\end{split}\n\\end{equation*}\nThis, together with \\eqref{r6}, leads to\n\\begin{equation*}\n\\begin{split}\n&\\E\\Big|\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\Big|^2\\\\\n&\\le c\\,\\vv^2\\int_0^T\\{1+\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+q)}\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\le c\\,\\vv^2.\n\\end{split}\n\\end{equation*}\nAs a consequence, we obtain \\eqref{q6} immediately.\n\\end{proof}\n\n\n\nSo far, with Lemma \\ref{lem} in hand, we are in the position to\ncomplete the\n\\begin{proof}[ Proof of Theorem \\ref{th1}]\nA direction calculation shows that\n\\begin{equation}\\label{h1}\n\\begin{split}\n&\\Phi_{n,\\vv}(\\theta)\\\\\n&=\\delta^{-1}\\sum_{k=1}^n\\Big\\{P_k^*(\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)-P_k^*(\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\Big\\}\\\\\n&=\\delta^{-1}\\sum_{k=1}^n\\Big\\{\\Big(P_k(\\theta_0)+(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big)^*\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv) \\Big(P_k(\\theta_0)\n+\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_{k-1}}^\\vv,\\theta,\\theta_0)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big)\\\\\n&\\quad -P_k^*(\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\Big\\}\\\\\n&=2\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0).\n\\end{split}\n\\end{equation}\nBy virtue of  Lemma \\ref{lem}, we therefore infer from Chebyshev's\ninequality that\n\\begin{equation*}\n\\sup_{\\theta\\in\\Theta}|-\\Phi_{n,\\vv}(\\theta)-(-\\Xi(\\theta))|\\rightarrow0~~~~\\mbox{\nin probability.}\n\\end{equation*}\n Next, for any\n$\\kk>0,$ due to $\\Xi(\\cdot)>0$,\n\\begin{equation*}\n \\sup_{|\\theta-\\theta_0|\\ge\\kk}(-\\Xi(\\theta))<-\\Xi(\\theta_0)=0.\n\\end{equation*}\nFurthermore,  one has $-\\Phi_{n,\\vv}(\n\\hat\\theta_{n,\\vv})\\ge-\\Phi_{n,\\vv}(\\theta_0)=0$.  Consequently, we\ndeduce  from \\cite[Theorem 5.9]{V98}  with\n$M_n(\\cdot)=-\\Phi_{n,\\vv}(\\cdot)$ and $M(\\cdot)=-\\Xi(\\cdot)$ therein\nthat $\\hat\\theta_{n,\\vv}\\rightarrow\\theta_0$ in probability as\n$\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$. We henceforth complete the\nproof.\n\\end{proof}\n\n\n\n\n\\section{Proof of Theorem \\ref{th2}}\\label{sec3}\nBefore we start to finish the argument of Theorem \\ref{th2}, we also\nneed to  prepare some auxiliary lemmas below. For any random\nvariable $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ with $\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$,\nset\n\\begin{equation*}\n\\Upsilon^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\theta):=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*\n(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta,\\mathscr{L}_{\\zeta}).\n\\end{equation*}\n\n\n\n\n\n\\begin{lem}\\label{le2}\n Let $({\\bf A1})-({\\bf A2})$ and $({\\bf B1})-({\\bf B4})$ hold. Then,\n\\begin{equation}\\label{v1}\n\\vv^{-1}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta)\\rightarrow-2\\int_0^T\\Upsilon(X_t^0,\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(t)~~~~\\mbox{ in probability }\n\\end{equation}\nwhenever $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$, where\n$\\Upsilon(\\cdot,\\cdot)$ is introduced in \\eqref{0s0}.\n\\end{lem}\n\n\\begin{proof}\nBy the chain rule,   one infers from \\eqref{q3} and \\eqref{h1} that\n\\begin{equation}\\label{c1}\n\\begin{split}\n&\\vv^{-1}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta)\\\\%&=2\\vv^{-1}\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\\\\n&=2\\,\\vv^{-1}\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\n\\Big\\{P_k(\\theta_0)+\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big\\}\\\\\n&=2\\,\\vv^{-1}\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)\\\\\n&=-2\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\\\\n&\\quad\\times(B(k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-B((k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho))\\\\\n&=-2\\int_0^T\\Upsilon^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t),\n\\end{split}\n\\end{equation}\nwhere in the last two display we used the fact that\n\\begin{equation}\\label{c2}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)=-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}) (\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta).\n\\end{equation}\nTo achieve \\eqref{v1}, in terms of \\cite[Theorem 2.6, P.63]{F98}, it\nis sufficient to claim that\n\\begin{equation}\\label{v2}\n\\int_0^T\\|\\Upsilon^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta)-\\Upsilon(X_t^0,\\theta)\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\rightarrow0~~~~\\mbox{ in probability }\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ Observe that\n\\begin{equation*}\n\\begin{split}\n&\\Upsilon^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta)-\\Upsilon(X_t^0,\\theta)\\\\\n&=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^* (\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0,\\mathscr{L}_{X_t^0})\\\\\n&=\\{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^* (\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\}\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\\\\n&\\quad+(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\{\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)-\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\}\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\n\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\\\\n&\\quad+(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^* (X_t^0,\\mathscr{L}_{X_t^0},\\theta)\n\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\{\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0,\\mathscr{L}_{X_t^0})\\}\\\\\n&=:\\Sigma_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\Sigma_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\Sigma_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).\n\\end{split}\n\\end{equation*}\n\n\n\n\n\n\n\n\n\n\n\nBy a straightforward calculation, for any random variable\n$\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ with $\\mathscr{L}_{\\zeta}\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$, one\nhas\n\\begin{equation}\\label{v3}\n\\begin{split}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})(\\zeta,\\mathscr{L}_\\zeta,\\theta)&=\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Big(\\ff{b(\\zeta,\\mu,\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|}\\Big)\\\\\n&=\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(\\zeta,\\mu,\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|}-\\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa( b\nb^*)(\\zeta,\\mu,\\theta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(\\zeta,\\mu,\\theta)}{|b(\\zeta,\\mu,\\theta)|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|)^2}.\n\\end{split}\n\\end{equation}\nNext, for   any random variables $\\zeta_1,\\zeta_2\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ with\n$\\mathscr{L}_{\\zeta_1},\\mathscr{L}_{\\zeta_2}\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$,\nit follows from \\eqref{v3} that\n\\begin{equation}\\label{v5}\n\\begin{split}\n&\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*\n(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)\\|\\\\\n&=\\Big\\|\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^* (\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta) -\\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)}{(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|)^2|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}\\Big\\|\\\\\n&=\\Big\\|\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}-\\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}\\\\\n&\\quad-\\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)}{(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|)^2|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}\\Big\\|\\\\\n&\\le\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\n(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)\\|\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\cdot\\{\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb) (\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)\\|+\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\n(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\|\\},\n\\end{split}\n\\end{equation}\nwhere in the last step we utilized the facts that $\\|A\\|=\\|A^*\\|$\nfor a matrix $A$ and that\n\\begin{equation*}\n\\begin{split}\n&\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\|^2\\\\\n&=\\mbox{trace}\\Big(((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta))^*(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\Big)\\\\\n&=\\mbox{trace}\\Big((b\nb^*)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta))((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\Big)\\\\\n&=\\mbox{trace}\\Big(((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b) (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)(b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)(b b^*)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)) \\Big)\\\\\n&=|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|^4\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\|^2.\n\\end{split}\n\\end{equation*}\nMoreover, from  ({\\bf B2}), one has\n\\begin{equation}\\label{v4}\n\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b) (\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)\\|\\le\nc\\Big\\{1+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_2}+\\mathbb{W}_2(\\mathscr{L}_{\\zeta_2},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}.\n\\end{equation}\nNow, taking ({\\bf B2}), \\eqref{v5}, and \\eqref{v4}, in addition to\n\\eqref{r2} and \\eqref{t2},  into account yields that\n\\begin{equation*}\n\\begin{split}\n\\|\\Sigma_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|&\\le c\\Big\\{(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_2}+\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_2})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\mathscr{L}_{X_t^0})\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Big(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1}+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big)\\Big(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_2}+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big)\\Big\\}\\\\\n&\\quad\\times\\Big\\{1+\n \\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_3} +\n \\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}\\times\\Big\\{1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}.\n\\end{split}\n\\end{equation*}\nFor $q:=q_1\\vee q_2\\vee q_3,$ simple calculations and \\eqref{r6}\n give  that\n\\begin{equation*}\n\\begin{split}\n\\|\\Sigma_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|&\\le c\\Big\\{(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\ss{\\E\\|\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big\\}\\\\\n&\\quad\\times\\Big\\{1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)}+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\Big\\}\\\\\n&\\quad+c\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Big\\{1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)}+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\Big\\}^2\\\\\n&\\le c (1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\\\\n&\\quad+c (1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)})\\ss{\\E\\|\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}+c\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)})\\\\\n&=:\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).\n\\end{split}\n\\end{equation*}\nFor any $\\rho>0$, by virtue of H\\\"older's inequality, together with\n\\eqref{r6} and \\eqref{a9}, it follows that\n\\begin{equation}\\label{v6}\n\\begin{split}\n&\\P\\Big(\\int_0^T\\|\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\ge\\rho\\Big)\\\\\n&\\le\\P\\Big(c\\int_0^T (1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+q)})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\ge\\rho\\Big)\\\\\n&\\le\\P\\Big(c\\int_0^T (1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{9(1+q)})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\ge\\rho\\Big)\\\\\n&\\le \\ff{c}{\\rho}\\int_0^T  (1+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{18(1+q)})\\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2} \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\rightarrow0\n\\end{split}\n\\end{equation}\nwhenever $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ On the other hand,\nby means of \\eqref{r6}, and \\eqref{a9}, it follows that\n\\begin{equation}\\label{v7}\n\\begin{split}\n\\E\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_2^2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\E\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_3^2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)&\\le c (1+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)})\\E\\|\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+c\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa(1+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+q)})\\\\\n&\\le c(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\vv^2+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa})\\\\\n&\\rightarrow0\n\\end{split}\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ As a consequence, we\ninfer from \\eqref{v6} and \\eqref{v7} that\n\\begin{equation}\\label{b1}\n \\int_0^T \\|\\Sigma_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho) \\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\rightarrow0~~~~\\mbox{ in\n probability }\n\\end{equation}\nwhen $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ Next, taking advantage\nof ({\\bf A2}), ({\\bf B3}), \\eqref{r2}, and \\eqref{v4} leads to\n\\begin{equation*}\n\\begin{split}\n&\\|\\Sigma_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2+\\|\\Sigma_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2\\\\&\\le c\\Big\\{1+\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)}\\Big\\}\\\\\n&\\quad\\times\\Big\\{(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q}+\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\mathscr{L}_{X_t^0})^2\\Big\\}\\\\\n&\\quad\\times (1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2)\\\\\n&\\le  c\\Big\\{(1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)}) \\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|^2_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\Big\\}\\\\\n&\\quad\\times  \\Big\\{1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|^2_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\Big\\}\\\\\n&\\le c (1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(2+q)}) \\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle +c(1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2)\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|^2_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\\\\n&=:\\Xi_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\Xi_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho),\n\\end{split}\n\\end{equation*}\nin which we adopted \\eqref{r6} in the last procedure. Via H\\\"older's\ninequality, we obtain from \\eqref{r6} and \\eqref{a9} that\n\\begin{equation}\\label{v8}\n \\E \\Xi_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c(1+\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(2+q)}) \\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\rightarrow0\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ Also, by \\eqref{r6} and\n\\eqref{a9}, one has\n\\begin{equation}\\label{v9}\n \\E \\Xi_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c(1+\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2)  \\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\rightarrow0\n\\end{equation}\nprovided that  $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ Therefore,\n\\eqref{v8} and \\eqref{v9} lead to\n\\begin{equation}\\label{b2}\n \\E \\|\\Sigma_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2+  \\E \\|\\Sigma_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2\\rightarrow0\n\\end{equation}\nif  $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ At last, the desired\nassertion \\eqref{v1} holds from \\eqref{b1} and \\eqref{b2}.\n\\end{proof}\n\n\n\n\\begin{lem}\\label{le3}\n  Let $({\\bf A1})-({\\bf A3}), ({\\bf B1})-({\\bf B4})$, and $({\\bf\n  C})$\nhold. Then\n\\begin{equation}\\label{c3}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)\\rightarrow\nK_0(\\theta):=K(\\theta)+I(\\theta)~~~~\\mbox{ in probability }\n\\end{equation}\nas $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$ and $\\vv\\rightarrow0$, where $I(\\cdot)$ and\n$K(\\cdot)$ are introduced in \\eqref{0z3} and \\eqref{0z2},\nrespectively.\n\\end{lem}\n\n\n\n\\begin{proof}\n From \\eqref{c1} and \\eqref{c2}, we deduce that\n\\begin{equation*}\n\\begin{split}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)& =2\n\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)\\Big)\\\\\n&\\quad+2 \\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta P_k)(\\theta)\\\\\n& =-2 \\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}(b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)\\Big)\\\\\n&\\quad+2 \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^* (\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta).\n\\end{split}\n\\end{equation*}\nFor any random variable $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ with\n$\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$, by the chain rule, we\ninfer from \\eqref{v3} that\n\\begin{equation}\\label{e6}\n\\begin{split}\n\\Big(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}(b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\n&=\\bigg(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\bigg(\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|}\\bigg)\\bigg)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\\\\n&\\quad-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\bigg(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\bigg(\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*)( b\nb^*)}{|b|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^2}\\bigg)\\bigg)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\\\\n&=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b^*)\n(\\zeta,\\mathscr{L}_\\zeta,\\theta)-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Theta_1(\\zeta,\\mathscr{L}_\\zeta,\\theta).\n\\end{split}\n\\end{equation}\nNext, the chain rule shows that\n\\begin{equation*}\n\\begin{split}\n& \\Theta_1(\\zeta,\\mathscr{L}_\\zeta,\\theta)\n:= \\bigg(\\ff{|b|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b^*)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|}+\\ff{\\Big(b^*(\\ff{\\partial}{\\partial\\theta_1}b)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*,\\cdots,b^*(\\ff{\\partial}{\\partial\\theta_p}b)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*\\Big)_{p\\times pd}}{|b|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^2}\\\\\n&\\qquad\\qquad\\quad+\\ff{\\Big((\\ff{\\partial}{\\partial\\theta_1}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*))( b\nb^*),\\cdots,(\\ff{\\partial}{\\partial\\theta_p}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*)( b\nb^*)\\Big)_{p\\times pd}}{|b|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^2}\\\\\n&\\qquad\\qquad\\quad+\\ff{\\Big((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*( (\\ff{\\partial}{\\partial\\theta_1}b)\nb^*+b\\ff{\\partial}{\\partial\\theta_1}b^*),\\cdots,(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*(\n(\\ff{\\partial}{\\partial\\theta_p}b)\nb^*+b\\ff{\\partial}{\\partial\\theta_p}b^*)\\Big)_{p\\times pd}}{|b|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^2}\\\\\n&\\quad-\\ff{1+3\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|}{|b|^3(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^3}\\Big(\n\\Big(b^*\\Big(\\ff{\\partial}{\\partial\\theta_1}b\\Big)\\Big)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*( b\nb^*),\\cdots,\\Big(b^*\\Big(\\ff{\\partial}{\\partial\\theta_p}b\\Big)\\Big)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*( b b^*)\\Big)\\bigg)_{p\\times\npd}(\\zeta,\\mathscr{L}_\\zeta,\\theta).\n\\end{split}\n\\end{equation*}\n Thanks to \\eqref{v3}, it follows\nthat\n\\begin{equation}\\label{e5}\n\\begin{split}\n&\\Big((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta) = \\Big((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta)-  \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\n\\aa}\\Theta_2(\\zeta,\\mathscr{L}_\\zeta,\\theta),\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation*}\n\\begin{split}\n\\Theta_2(\\zeta,\\mathscr{L}_\\zeta,\\theta):&=\\bigg( \\ff{ (2|\nb|+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}| b |^2)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*) \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\n}{(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa| b |)^2}  +\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*) \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(   b\n\\,\\, b^*) (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\n  b) }{|\nb |(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|\nb |)^3} \\\\\n&\\quad+ \\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*) (  b \\, b^*) \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb) }{| b |(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa| b |)^3} - \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{ \\aa} \\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*) (  b\n\\, b^*) \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)( b \\, b^*) (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b) }{| b\n|^2(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa| b |)^4}\\bigg)(\\zeta,\\mathscr{L}_\\zeta,\\theta).\n\\end{split}\n\\end{equation*}\nThus, taking \\eqref{e5} and \\eqref{e6} into consideration yields\nthat\n\\begin{equation*}\n\\begin{split}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)& =  -2\n\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}b^*)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\Big)\\\\\n&\\quad+2 \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n\\Big((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\\Big)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\\\\n&\\quad-2 \\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}b^*)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\Big)\\\\\n&\\quad-2\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa \\sum_{k=1}^n\\Theta_1(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)\\Big)\\\\\n&\\quad- \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{ 1+\\aa}\\sum_{k=1}^n\\Theta_2(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\\\\n&=:\\sum_{i=1}^5I_i(n,\\vv).\n\\end{split}\n\\end{equation*}\nBy following the argument to derive \\eqref{t4},   we deduce from\n({\\bf A3}) that\n\\begin{equation}\\label{c6}\nI_1(n,\\vv)\\rightarrow K(\\theta) ~~\\mbox{ and\n}~~I_2(n,\\vv)\\rightarrow I(\\theta),~~~~\\mbox{ in probability }\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$. Notice from ({\\bf A3})\nand \\eqref{q4} that\n\\begin{equation}\\label{c4}\n\\begin{split}\n\\|\\Theta_1\\|(\\zeta,\\mathscr{L}_\\zeta,\\theta)&\\le c\\Big((|b|+\n\\|\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b\\|+(1+3 |b|)\\|\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\nb\\|)\\|\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b\\|\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\\\\n&\\le c(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)}+\\mathcal\n{W}_2(\\mathscr{L}_\\zeta,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^4).\n\\end{split}\n\\end{equation}\nOn the other hand, owing to \\eqref{v4}, \\eqref{t2}, and \\eqref{q4},\none has\n\\begin{equation}\\label{c5}\n\\begin{split}\n\\|\\Theta_2\\|(\\zeta,\\mathscr{L}_\\zeta,\\theta)&\\le 2\\Big(|\nb|~\\|\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b\\|^2\\| \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)\\| (1+ 2| b | )\n\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\\\\n&\\le c(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)}+\\mathcal\n{W}_2(\\mathscr{L}_\\zeta,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^4).\n\\end{split}\n\\end{equation}\nThus, by mimicking the argument of \\eqref{q6}, we obtain from\n\\eqref{c4} that\n\\begin{equation}\\label{c7}\nI_3(n,\\vv)\\rightarrow0~~\\mbox{ and\n}~~I_4(n,\\vv)\\rightarrow0~~~\\mbox{ in probability }\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$. Furthermore, \\eqref{r6}\nand \\eqref{c5} enable us to get that\n\\begin{equation}\\label{c8}\nI_5(n,\\vv)\\rightarrow0 ~~~\\mbox{ in probability }\n\\end{equation}\nwhenever  $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$.\n Thus, the desired\nassertion \\eqref{c3} follows from \\eqref{c6}, \\eqref{c7}, as well as\n\\eqref{c8}.\n\\end{proof}\n\n\n\n\nNow, we move forward to complete the\n\\begin{proof}[ Proof of Theorem \\ref{th2}]\nWith Lemmas \\ref{le2} and \\ref{le3} at hand, the proof of Theorem\n\\ref{th2} is parallel to that of \\cite[Theorem 4.1]{RW}. Whereas, to\nmake the content self-contained, we give an outline of the proof. In\nterms of Theorem \\ref{th1}, there exists a sequence\n$\\eta_{n,\\vv}\\rightarrow0$ as $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$\nsuch that $\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)\\subset\\Theta$, $\\P$-a.s.  By the Taylor\nexpansion, one has\n\\begin{equation}\\label{a4}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\hat\\theta_{n,\\vv})=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta_0)+D_{n,\\vv}(\\hat\\theta_{n,\\vv}-\\theta_0),~~~\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)\n\\end{equation}\nwith\n\\begin{equation*}\nD_{n,\\vv}:=\\int_0^1(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0\n+u(\\hat\\theta_{n,\\vv}-\\theta_0))\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D u,~~~~\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0).\n\\end{equation*}\n Observe that, for $\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)$,\n\\begin{equation*}\n\\begin{split}\n\\|D_{n,\\vv}-K_0(\\theta_0)\\|&\\le\\|D_{n,\\vv}-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)\\|+\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)-K_0(\\theta_0)\\|\\\\\n&\\le\\int_0^1\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0\n+u(\\hat\\theta_{n,\\vv}-\\theta_0))-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)\\|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nu\\\\\n&\\quad+ \\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)-K_0(\\theta_0)\\|\\\\\n&\\le \\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)\\|+\n\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)-K_0(\\theta_0)\\|\\\\\n&\\le\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-K_0(\\theta)\\|+\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|K_0(\\theta)-K_0(\\theta_0)\\|\\\\\n&\\quad+2\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)-K_0(\\theta_0)\\|.\n\\end{split}\n\\end{equation*}\n This, together\nwith Lemma \\ref{le3} and  continuity of $K_0(\\cdot)$, gives that\n\\begin{equation}\\label{a0}\nD_{n,\\vv}\\rightarrow K_0(\\theta_0)~~~~\\mbox{ in probability }\n\\end{equation}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle.$ By following the exact\nline of \\cite[Theorem 2.2]{LSS}, we can deduce that $D_{n,\\vv}$ is\ninvertible on the set\n\\begin{equation*}\n\\Gamma_{n,\\vv}:=\\Big\\{\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-K_0(\\theta_0)\\|\\le\\ff{\\aa}{2},~~\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0) \\Big\\}\n\\end{equation*}\nfor some constant $\\aa>0.$  Let\n\\begin{equation*}\n\\mathscr{D}_{n,\\vv}=\\{D_{n,\\vv} \\mbox{ is invertible },\n\\hat\\theta_{n,\\vv}\\in B_{\\eta_{n,\\vv}}(\\theta_0) \\}.\n\\end{equation*}\nBy virtue of Lemma \\ref{le3}, one has\n\\begin{equation}\\label{n1}\n\\lim_{\\vv\\rightarrow0,n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle}\\P\\Big(\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-K_0(\\theta_0)\\|\\le\\ff{\\aa}{2}\\Big)=1.\n\\end{equation}\nOn the other hand, recall that\n\\begin{equation}\\label{n2}\n\\lim_{\\vv\\rightarrow0,n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle}\\P\\Big(\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)\\Big)=1.\n\\end{equation}\nBy the fundamental fact: for any events $A,B$,\n$\\P(AB)=\\P(A)+\\P(B)-\\P(A\\cup B)$, we observe that\n\\begin{equation}\\label{n3}\n\\begin{split}\n1\\ge\\P(\\Gamma_{n,\\vv})&\\ge\\P\\Big(\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-K_0(\\theta_0)\\|\\le\\ff{\\aa}{2}\\Big)\\\\\n&\\quad+\\P\\Big(\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)\\Big)-1.\n\\end{split}\n\\end{equation}\nThus, taking advantage of \\eqref{n1}, \\eqref{n2} as well as\n\\eqref{n3}, we deduce from Sandwich theorem that\n\\begin{equation}\\label{n4}\n\\P(\\mathscr{D}_{n,\\vv})\\ge \\P(\\Gamma_{n,\\vv})\\rightarrow1\n\\end{equation}\n as $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$. Set\n\\begin{equation*}\nU_{n,\\vv}:=D_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}}+I_{p\\times\np}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c},\n\\end{equation*}\nwhere $I_{p\\times p}$ is a $p\\times p$ identity matrix. For\n$S_{n,\\vv}:=\\vv^{-1}(\\hat\\theta_{n,\\vv}-\\theta_0)$, we deduce from\n\\eqref{a4}  that\n\\begin{align*}\nS_{n,\\vv}&=S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}}+ S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c}\\\\\n&=U_{n,\\vv}^{-1}D_{n,\\vv}S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}}+ S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c}\\\\\n&=\\vv^{-1}U_{n,\\vv}^{-1}\\{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\hat\\theta_{n,\\vv})-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta_0)\\}{\\bf1}_{\\mathscr{D}_{n,\\vv}}\n+ S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c}\\\\\n&=-\\vv^{-1}U_{n,\\vv}^{-1}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta_0){\\bf1}_{\\mathscr{D}_{n,\\vv}}\n+ S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c}\\\\\n&\\rightarrow I^{-1}(\\theta_0)\\int_0^T\\Upsilon(X_s^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(s),\n\\end{align*}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty$, where in the forth\nidentity we dropped the term\n$(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\hat\\theta_{n,\\vv})$ according to the\nnotion of LSE and Fermat's lemma, and the last display follows from\nLemma \\ref{le2}, \\eqref{a0} as well as \\eqref{n4} and by noting\n$K_0(\\theta_0)=I(\\theta_0)$. We therefore complete the proof.\n\\end{proof}\n\n\n\n\\section{Proof of Example \\ref{exa}}\n\n\\begin{proof}[Proof of Example \\ref{exa}]\nIt is sufficient to check all of assumptions in Theorems \\ref{th1}\nand Theorem \\ref{th2} are fulfilled.\n\nFor any $\\zeta,\\zeta'\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$, $\\mu\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$ and\n$\\theta=(\\theta^{(1)},\\theta^{(2)})^*\\in\\Theta_0$, set\n\\begin{equation}\\label{ex1}\nb_0(\\zeta,\\zeta'):=-\\zeta^3(0)+\\zeta(0)+\\int_{-r_0}^0\\zeta(v)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nv+\\int_{-r_0}^0\\zeta'(v)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D v,\n\\end{equation}\n\\begin{equation*}\nb(\\zeta,\\mu,\\theta):=\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\zeta,\\zeta')\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta') ~~\\mbox{ and\n}~~\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta):=\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta,\\mu):=1+\\int_{-r_0}^0|\\zeta(v)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D v.\n\\end{equation*}\nThen, \\eqref{d1} can be reformulated as \\eqref{eq1}. By  \\eqref{ex1}\nand H\\\"older's inequality, we find out some constants $c_1,c_2>0$\nsuch that\n\\begin{equation}\\label{d3}\n\\begin{split}\n&\\<\\zeta_1(0)-\\zeta_2(0),b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\mu,\\theta)\\>\\\\\n&= \\theta^{(2)} \\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r \\<\\zeta_1(0)-\\zeta_2(0),b_0(\\zeta_1,\\zeta)-\nb_0(\\zeta_2,\\zeta)\\>\\mu_t(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta) \\\\\n&\\le c_1\\Big\\{|\\zeta_1(0)-\\zeta_2(0)|^2+\\int_{-r_0}^0|\\zeta_1(v)-\\zeta_2(v)|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D v\\Big\\}\\\\\n&\\le c_2 \\,\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2,~~~~~\\mu\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r),\n~~\\zeta_1,\\zeta_2 \\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\n\\end{split}\n\\end{equation}\nNext, we deduce from \\eqref{ex1} that for some constant $c_3>0,$\n\\begin{equation*}\n\\begin{split}\n|b(\\zeta,\\mu,\\theta)-b(\\zeta,\\nu,\\theta)|&\\le\\theta^{(2)}\\Big|\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\zeta,\\zeta_1)\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta_1)-\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\zeta,\\zeta_2)\\nu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta_2)\\Big|\\\\\n&\\le\\theta^{(2)}\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r| b_0(\\zeta,\\zeta_1)-\nb_0(\\zeta,\\zeta_2)|\\pi(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta_1,\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta_2)\\\\\n&\\le c_3\\,\\mathbb{W}_2(\\mu,\\nu),~~~~\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r,~~~\\mu,\\nu\\in\\mathcal\n{P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r),\n\\end{split}\n\\end{equation*}\nin which $\\pi\\in\\mathcal {C}(\\mu,\\nu)$. Therefore, ({\\bf A1}) holds\ntrue. Next,\n for any\n$\\zeta_1,\\zeta_2\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$ and $\\mu,\\nu\\in\\mathcal {P}_2(\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r)$, we obtain\nthat\n\\begin{equation*}\n |\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_1,\\mu)-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_2,\\nu) |\\le\\int_{-r_0}^0|\\zeta_1(\\theta)-\\zeta_2(\\theta)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\theta\\le r_0\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle.\n\\end{equation*}\nSo ({\\bf A2}) is satisfied. For any\n$\\zeta_1,\\zeta_2,\\zeta^{(1)},\\zeta^{(2)}\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r$, note that\n\\begin{equation}\\label{ex2}\n\\begin{split}\n&|b_0(\\zeta_1,\\zeta^{(1)})-b_0(\\zeta_2,\\zeta^{(2)})|\\\\&\\le\n|\\zeta_1^3(0)-\\zeta_2^3(0)|+|\\zeta_1(0)-\\zeta_2(0)|+\\int_{-r_0}^0|\\zeta_1(v)-\\zeta_2(v)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nv+\\int_{-r_0}^0|\\zeta^{(1)}(v)-\\zeta^{(2)}(v)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D v\\\\\n&\\le\nc_4(1+\\zeta_1^2(0)+\\zeta_2^2(0))|\\zeta_1(0)-\\zeta_2(0)|+r_0\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+r_0\\|\n\\zeta^{(1)}-\\zeta^{(2)}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\\\\n&\\le\nc_5(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2)\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+r_0\\|\n\\zeta^{(1)}-\\zeta^{(2)} \\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\n\\end{split}\n\\end{equation}\nfor some constants $c_4,c_5>0.$ Next, we have\n\\begin{equation}\\label{d4}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)(\\zeta,\\mu,\\theta)=\\Big(1,\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\zeta,\\zeta')\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta')\\Big)^*~~~\\mbox{ and\n}~~~(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b))(\\zeta,\\mu,\\theta)={\\bf 0}_{2\\times2},\n\\end{equation}\nwhere ${\\bf 0}_{2\\times2}$ stands for the $2\\times 2$-zero matrix.\nThus, \\eqref{ex2} and \\eqref{d4} enable us to deduce that ({\\bf B2})\nand ({\\bf C}) hold, respectively. Furthermore, due to \\eqref{ex2},\nwe find that\n\\begin{equation*}\n\\begin{split}\n|b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\nu,\\theta)|&\\le\\theta^{(2)}\\Big|\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\zeta_1,\\zeta^{(1)})\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta^{(1)})-\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\zeta_2,\\zeta^{(2)})\\nu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta^{(2)})\\Big|\\\\\n&\\le\\theta^{(2)}\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r| b_0(\\zeta_1,\\zeta^{(1)})-\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\zeta_2,\\zeta^{(2)})|\\pi(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta^{(1)},\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta^{(2)})\\\\\n&\\le\nc_6(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2)\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+c_6\\mathbb{W}_2(\\mu,\\nu).\n\\end{split}\n\\end{equation*}\nTherefore, we infer that ({\\bf B1})  holds. Next, observe that\n\\begin{equation*}\n|\\sigma} \\def\\ess{\\text{\\rm{ess}}^{-2}(\\zeta_1,\\mu)-\\sigma} \\def\\ess{\\text{\\rm{ess}}^{-2}(\\zeta_2,\\nu)| \\le\nc_7\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\n\\end{equation*}\nfor some $c_7>0.$ Consequently, ({\\bf B3}) is true.\n\n\n\n\n\n\n\n\nThe discrete-time EM scheme associated with \\eqref{d1} is given by\n\\begin{equation}\nY^\\vv(t_k)=Y^\\vv(t_{k-1})+\\Big(\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\hat Y^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho+\\vv\\,\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\hat\nY^\\vv_{t_{k-1}})\\triangle B_k,~~~k\\ge1,\n\\end{equation}\nwith $Y^\\vv(t)=X^\\vv(t)=\\xi(t), t\\in[-r_0,0],$ where $(\\hat\nY^\\vv_{t_k})$ is defined as in \\eqref{w2}. According to \\eqref{eq2},\nthe contrast function admits the form below\n\\begin{eqnarray*}\n\\Psi_{n,\\vv}(\\theta)&=&\\vv^{-2}\\delta^{-1}\\sum_{k=1}^n\\ff{1}{(1+|Y^\\vv(t_{k-1})|)^2}\\Big|Y^\\vv(t_k)\n-Y^\\vv(t_{k-1})\\\\\n&&\\qquad\\qquad-\\Big(\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\hat Y^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big|^2.\n\\end{eqnarray*}\nObserve that\n\\begin{equation*}\n\\begin{split}\n\\ff{\\partial}{\\partial\\theta^{(1)}}\\Psi_{n,\\vv}(\\theta)&=-2\\,\\vv^{-2}\\sum_{k=1}^n\\ff{1}{(1+|Y^\\vv(t_{k-1})|)^2}\\Big\\{Y^\\vv(t_k)-Y^\\vv(t_{k-1})\\\\\n&\\quad-\\Big(\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big\\},\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n\\ff{\\partial}{\\partial\\theta^{(2)}}\\Psi_{n,\\vv}(\\theta)&=-2\\,\\vv^{-2}\\sum_{k=1}^n\\ff{1}{(1+|Y^\\vv(t_{k-1})|)^2}\\Big\\{Y^\\vv(t_k)-Y^\\vv(t_{k-1})\\\\\n&\\quad-\\Big(\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big\\} \\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat Y^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta).\n\\end{split}\n\\end{equation*}\nSubsequently, solving the equation below\n\\begin{equation*}\n\\ff{\\partial}{\\partial\\theta^{(1)}}\\Psi_{n,\\vv}(\\theta)=\\ff{\\partial}{\\partial\\theta^{(2)}}\\Psi_{n,\\vv}(\\theta)=0,\n\\end{equation*}\nwe then obtain the LSE\n$\\hat\\theta_{n,\\vv}=(\\hat\\theta_{n,\\vv}^{(1)},\\hat\\theta_{n,\\vv}^{(2)})^*$\nof the unknown parameter\n$\\theta=(\\theta^{(1)},\\theta^{(2)})^*\\in\\Theta_0$ with the following \n\\begin{equation*}\n\\hat\\theta_{n,\\vv}^{(1)}=\\ff{A_2A_5-A_3A_4}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho(A_1A_5-A_4^2)}~~~~~\\mbox{\nand }~~~~~\n\\hat\\theta_{n,\\vv}^{(2)}=\\ff{A_1A_3-A_2A_4}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho(A_1A_5-A_4^2)},\n\\end{equation*}\nwhere\n\\begin{equation*}\nA_1:=\\sum_{k=1}^n\\ff{1}{(1+|Y^\\vv(t_{k-1})|)^2},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n~~A_2:=\\sum_{k=1}^n\\ff{Y^\\vv(t_k)-Y^\\vv(t_{k-1})}{(1+|Y^\\vv(t_{k-1})|)^2},\n\\end{equation*}\n\\begin{equation*}\nA_3:=\\sum_{k=1}^n\\ff{(Y^\\vv(t_k)-Y^\\vv(t_{k-1}))\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)}{(1+|Y^\\vv(t_{k-1})|)^2},~~~A_4:=\\sum_{k=1}^n\\ff{\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r\nb_0(\\hat Y^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)}{(1+|Y^\\vv(t_{k-1})|)^2},\n\\end{equation*}\nand\n\\begin{equation*}\nA_5:=\\sum_{k=1}^n\\ff{\\Big(\\int_\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)^2}{(1+|Y^\\vv(t_{k-1})|)^2}.\n\\end{equation*}\nIn terms of Theorem \\ref{th1}, $\\hat\\theta_{n,\\vv}\\rightarrow\\theta$\nin probability as $\\vv\\rightarrow0$ and $n\\rightarrow\\infty$. Next,\nfrom \\eqref{d4}, it follows that\n\\begin{equation*}\nI(\\theta_0)=\\left(\\begin{array}{ccc}\n  \\int_0^T\\ff{1}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s & \\int_0^T\\ff{b_0(X_s^0,X_s^0)}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n  \\int_0^T\\ff{b_0(X_s^0,X_s^0)}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s & \\int_0^T\\ff{b_0^2(X_s^0,X_s^0)}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n  \\end{array}\n  \\right),\n\\end{equation*}\nand, for $\\zeta\\in\\scr C}      \\def\\aaa{\\mathbf{r}}     \\def\\r{r,$\n\\begin{equation*}\n\\int_0^T\\Upsilon(X_s^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(s)=\\left(\\begin{array}{c}\n  \\int_0^T\\ff{1}{1+|X^0(s)|}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(s) \\\\\n \\int_0^T\\ff{ b_0(X_s^0,X_s^0)}{1+|X^0(s)|}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(s)\\\\\n  \\end{array}\n  \\right).\n\\end{equation*}\nAt last, according to Theorem \\ref{th2}, we conclude that\n\\begin{equation*}\n\\vv^{-1}(\\hat\\theta_{n,\\vv}-\\theta_0)\\rightarrow\nI^{-1}(\\theta_0)\\int_0^T\\Upsilon(X_s^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(s)~~~~\\mbox{ in\nprobability }\n\\end{equation*}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$ provided that $I(\\cdot)$\nis positive definite.\n\n\n\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nIt is widely believed that the cosmic microwave background (CMB) will\nbecome the premier laboratory for the study of the early universe and\nclassical cosmology.  This belief relies on the high precision of the\nupcoming MAP\\footnote{{\\tt http:\/\/map.gsfc.nasa.gov}}\nand Planck Surveyor\\footnote{{\\tt http:\/\/astro.estec.eas.nl\/SA-general\/Projects\/Planck}} satellite missions and the high accuracy of theoretical\npredictions of CMB anisotropies given a definite model for\nstructure formation (\\cite{Hu95}\\ 1995).  To realize the potential of the CMB,\naspects of structure formation affecting anisotropies at only the\npercent level in power must be taken into account. \n\nA great uncertainty in models for structure formation is the extent\nand nature of reionization.  \nFortunately, this uncertainty is largely not reflected in\nthe CMB anisotropies due to the low optical depth to Thomson\nscattering at the low redshifts in question.\nReionization is known to be essentially\ncomplete by $z \\sim 5$ from the absence of the Gunn-Peterson effect in\nquasar absorption spectra (\\cite{Gun65}\\ 1965). Significant\nreionization before $z\\sim 50$ will be ruled out once the tentative\ndetections of the CMB acoustic peaks at present are confirmed \n(\\cite{Sco95} 1995). It should be possible to\ndeduce the reionization redshift $z_i$ through CMB polarization\nmeasurements (\\cite{Zal97}\\ 1997). \n\nNevertheless, the\nduration of time spent in a partially ionized state will remain\nuncertain. \nMoreover as emphasized by\n\\cite{Kai84} (1984) secondary anisotropies generated by the Doppler\neffect in linear perturbation theory are suppressed on small scales\nfor geometric reasons (gravitational instability generates potential flows, \nleading to cancellations between positive and negative Doppler shifts).  Higher\norder effects which are not generally included in the theoretical\nmodeling of CMB anisotropies are likely to be the main source of\nsecondary anisotropies from reionization below the degree scale.  Such\neffects rely on modulating the Doppler effect with spatial variations\nin the optical depth.  Incarnations of this general mechanism include\nthe Vishniac effect from linear density variations (\\cite{Vis87}\\\n1987), the kinetic Sunyaev-Zel'dovich effect from clusters (\\cite{SZ}\\\n1970), and the effect considered here: the spatial variation of the\nionization fraction.\n\nReionization commences when the first baryonic objects form stars or\nquasars that convert a part of the nuclear or gravitational energy\ninto UV photons.  Each such source then blows out an ionization sphere around it.  Before these\nregions overlap is a period when the universe is ionized in patches.\nThe extent of this period and the time evolution of the size and\nnumber density of these patches depend on the nature of the ionizing\nengines in the first baryonic objects. Theories of reionization do not\ngive robust constraints \n({\\it cf.} \\cite{Teg94}\\ 1994; \\cite{Ree96}\\ 1996;\n\\cite{Agh96}\\ 1996; \\cite{Loe97}\\ 1997; \\cite{Hai97}\\ 1997; \\cite{Sil98}\\ 1998;\n\\cite{Hai98}\\ 1998). \n\nWe therefore take a phenomenological approach to\nstudying the effects of patchy reionization on the CMB.  We introduce\na simple but illustrative three parameter model for the reionization\nprocess based on the redshift of its onset $z_i$, the duration before\ncompletion $\\delta z$, and the typical comoving size of the patches\n$R$. It is then straightforward to calculate the CMB anisotropies\ngenerated by the patchiness of the ionization degree of the\nintergalactic medium.\n\nWe find that only the most extreme models of reionization can produce\ndegree scale anisotropies that are observable in the power spectrum\n given the cosmic\nvariance limitations. A large signal on degree scales requires early\nionization, $z_i\\gtrsim 30$, long duration, $\\delta z\\sim z_i$, and\nionization in very large patches, $R\\gtrsim 30$Mpc. Thus the\npatchiness of reionization is unlikely to affect cosmological\nparameter estimation from the acoustic peaks in the CMB (\\cite{Jun96}\\\n1996;\n\\cite{Zal97}\\ 1997; \\cite{Bon97}\\ 1997).\n   \nOn the other hand, the patchy reionization signal on the sub-arcminute\nscale can, in principle, surpass both the primary and the secondary Vishniac\nsignals.  These may be detectable by the Planck Surveyor and upcoming \nradio interferometry measurements (\\cite{Par97}\\ 1997) \nif point sources can be removed\nat the $\\Delta T\/T \\sim 10^{-6}$ level. \n\nAn explicit expression for the CMB anisotropies power spectrum\ngenerated in a universe reionized in patches is given \\S \\ref{sec:explicit}. \nSimple\norder of magnitude estimate of the anisotropy from patches is given in\n\\S \\ref{sec:order}. In \\S \\ref{sec:power} we give a rigorous definition of our three-parameter\nreionization model, and calculate the patchy part of the power\nspectrum. We discuss illustrative examples in \\S \\ref{sec:discussion}.\n\n\n\\section{CMB power spectrum}\n\\subsection{Explicit expression}\n\\label{sec:explicit}\nTemperature perturbations $\\Delta \\equiv \\delta T\/T$ are generated by\nDoppler shifts from Thomson scattering. For small optical depths\n\\begin{equation}\n\\Delta =-\\int d{\\bf l}\\cdot {\\bf v}\\sigma _Tnx_e.\n\\label{eqn:fundamental}\n\\end{equation}\nAll quantities here are in physical units. The integral is along the line\nof site, ${\\bf v}$ is the peculiar velocity of matter, $c=1$,\n$\\sigma_T$ is the Thomson cross section, $n$ is the number density of\nfree and bound electrons, $x_e$ is the local ionization fraction. \n\nTo evaluate equation~(\\ref{eqn:fundamental}) explicitly one must specify\nthe cosmological model.  For simplicity, we take a universe\nwith critical density in matter throughout; we describe the generalization\nto an arbitrary FRW universe in \\S \\ref{sec:power}. \nWe furthermore use comoving coordinates ${\\bf x}$ and conformal time\n$\\eta\\equiv \\int (1+z)dt= (1+z)^{-1\/2} \\eta_0$, \nwhere $\\eta_0=2\/H_0$ is the present particle horizon. \nWe observe at ${\\bf x}=0$ and conformal time $\\eta_0$\nalong the direction of a unit vector $\\hat{\\gamma}$; light propagation\nis given by ${\\bf x}=\\hat{\\gamma}(\\eta _0-\\eta )$. \nEquation~(\\ref{eqn:fundamental})\ncan be\nwritten as\n\\begin{equation}\n\\Delta (\\hat{\\gamma})=-\\tau_0 \\eta_0^3\n\\int {d\\eta \\over \\eta^4} \\hat{\\gamma}\\cdot {\\bf v}[ \\eta ,\\hat{\\gamma}(\\eta _0-\\eta )] x_e[ \\eta ,\\hat{\\gamma}(\\eta _0-\\eta )] .\n\\end{equation}\nHere $\\tau _0\\equiv \\sigma _Tn_0 \\eta_0$ is\nthe optical depth to Thomson scattering across the present particle horizon.\n\nThe\nscales contributing to the peculiar velocity are still in the linear\nregime, therefore\n\\begin{equation}\n{\\bf v}(\\eta ,{\\bf x})={\\eta \\over \\eta _0}{\\bf v}({\\bf x}),\n\\end{equation}\nwhere ${\\bf v}({\\bf x})$ is the peculiar velocity today. The final\nexplicit expression for the CMB temperature perturbation generated\nduring reionization is\n\\begin{equation}\n\\Delta (\\hat{\\gamma})=-\\tau_0 \\eta_0^2\n\\int {d\\eta \\over \\eta ^3} \\hat{\\gamma}\\cdot {\\bf v}[ \\hat{\\gamma}(\\eta _0-\\eta )] x_e[ \\eta ,\\hat{\\gamma}(\\eta _0-\\eta )] .\n\\label{eqn:temperature}\n\\end{equation}\nThe correlation function of the temperature perturbations is defined\nas\n\\begin{equation}\nC(\\theta )=\\left<\\Delta (\\hat{\\gamma}_1)\\Delta\n(\\hat{\\gamma}_2)\\right>|_{\\hat{\\gamma}_1\\cdot \\hat{\\gamma}_2=\\cos \\theta}.\n\\end{equation}\nWith temperature perturbations given by equation~(\\ref{eqn:temperature}), \nthis becomes\n\\begin{eqnarray}\nC(\\theta )&=&\\tau_0^2 \\eta _0^4 \\int {d\\eta _1\\over \\eta _1^3}\n\\int {d\\eta _2\\over \\eta _2^3} \\big<\n\\hat{\\gamma}_1\\cdot {\\bf v}({\\bf x}_1)\n\\hat{\\gamma}_2\\cdot\n{\\bf v}({\\bf x}_2)\n\\nonumber\\\\ &&\\quad \\times \\, x_e(\\eta _1,{\\bf x}_1)x_e(\\eta _2,{\\bf x}_2)\\big>,\n\\label{eqn:correlationgeneral}\n\\end{eqnarray}\nwhere we denote ${\\bf x}_1\\equiv \\hat{\\gamma}_1(\\eta _0-\\eta _1)$ and\n${\\bf x}_2\\equiv \\hat{\\gamma}_2(\\eta _0-\\eta _2)$.\n\n\\subsection{Order of magnitude estimates}\n\\label{sec:order}\n\nConsider the following patchy reionization scenario. The universe was\nreionized in randomly distributed patches with a characteristic\ncomoving size $R$. The patches appeared at random in space and\ntime. Once a reionized patch appears, it moves with matter. The\naverage ionization fraction, that is the filling fraction of fully\nionized patches, grows monotonically from $X_e=0$ at high redshifts to\n$X_e=1$ at low redshifts. We consider late reionization (optical depth\nto Thomson scattering is small) and small patches (smaller than the\ncharacteristic length scale of the peculiar velocity field). We assume\nthat reionization occurred at redshift $z_i$, and the patchy phase\nduration is given by $\\delta z$.\n\nThe angular scale of the patchy signal is given by the ratio of the\nsize of patches to the distance to them in comoving coordinates, \n$\\theta \\sim R\/(\\eta_0-\\eta_i)$. \nAssuming that the patches are uncorrelated, the spectrum of \nfluctuations should be white noise above this scale which \nagrees with the exact result as we shall see (eq.~[\\ref{eqn:Cl}]). \n\nThe rms CMB temperature fluctuation $\\Delta$ on scales $\\theta$\ndue to the patchiness can be estimated as follows. Since, by\nassumption, different patches are independent, $\\Delta \\sim\nN^{1\/2}\\Delta _p$. Here $N$ is the number of patches on a line of\nsite, $\\Delta _p$ is a temperature fluctuation from one patch, $\\Delta\n_p\\sim \\tau _pv(z_i)$. Here $v(z)=(1+z)^{-1\/2}v(0)$ is the rms peculiar\nvelocity at redshift $z$, and $\\tau _p$ is the optical depth for one\npatch, $\\tau _p \\sim (1+z_i)^2\\sigma _Tn_0R$. The number of patches\n$N\\sim \\delta \\eta\/R\\sim (1+z_i)^{-3\/2}\\delta z\\eta _0\/R$. Collecting\nall the factors, we get the following estimate for the rms\nanisotropies from patches\n\\begin{equation}\n\\Delta \\sim \\tau_0\\left<v^2\\right>^{1\/2}({R\/\\eta _0})^{1\/2}(1+z_i)^{3\/4} (\\delta z)^{1\/2},\n\\end{equation}\nwhich again \nagrees with the exact result (eq.~[\\ref{eqn:amplitude}], up to a dimensionless\nmultiplier).  \n\n\n\\subsection{Power spectrum}\n\\label{sec:power}\n\nWe can factor the general expression for the temperature correlation\n(\\ref{eqn:correlationgeneral}) as\n\\begin{eqnarray}\nC(\\theta )&=&\\tau_0^2 \\eta_0^4 \\int {d\\eta _1\\over \\eta _1^3}\n\\int {d\\eta _2\\over \\eta _2^3}\n\\left<\\hat{\\gamma}_1\\cdot {\\bf v}({\\bf x}_1)\\hat{\\gamma}_2\\cdot {\\bf\nv}({\\bf x}_2)\\right>\n\\nonumber \\\\\n&&\\quad\\times\\,\n\\left<x_e(\\eta _1,{\\bf x}_1)x_e(\\eta _2,{\\bf x}_2)\\right>.\n\\end{eqnarray}\nThis assumes that $x_e$ and ${\\bf v}$ are independent random\nfields. This is not strictly correct. The ionization fraction $x_e$\nmust be determined by the density perturbation $\\delta$, and the\ndensity perturbation is not independent of the peculiar velocity (for\nexample in the linear regime $\\delta =-{1\\over 2} \\eta \\nabla \\cdot\n{\\bf v}$). However, the ionizing radiation is presumably coming from\nstrongly nonlinear objects, where first stars or quasars are\nlightening up. At high $z$, the length scales where the density is\nnonlinear are $\\ll 10$Mpc comoving, which is much smaller than the\nlength scales contributing to the peculiar velocity. Under the\nassumption of scale separation, velocity and density (and hence $x_e$)\nare indeed independent.\n\nThe correlation function for the local ionization fraction $\\left<x_e x_e\\right>$\nis not known. Our model parameterizes the correlation function\nthrough the the patch size $R$ and \na mean (cosmic time - dependent) ionization\nfraction $X_e(\\eta )$,\n\\begin{eqnarray}\n&&\\left<x_e(\\eta _1,{\\bf x}_1)x_e(\\eta _2,{\\bf x}_2)\\right> = X_e(\\eta _1)X_e(\\eta\n_2)\\nonumber\\\\ \n&& \\qquad + [ X_e(\\eta _{\\rm\nmin})-X_e(\\eta _1)X_e(\\eta _2)] e^{-{({\\bf x}_1-{\\bf x}_2)^2\\over\n2R^2}}\\qquad.\n\\label{eqn:xexemodel}\n\\end{eqnarray} \nHere $\\eta _{\\rm min}={\\rm min}(\\eta _1,\\eta _2)$. The Gaussian\nfunction is chosen for simplicity; it could have been any function of\nthe separation $x$ which equals 1 at $x=0$ and gradually turns to zero\nat $x>R$. For the mean ionization fraction $X_e$ we assume a change\nfrom 0 to 1 at a redshift $z_i$, with the transition occurring in a\nredshift interval $\\delta z$. We also assume $\\delta z\\ll\nz_i$ (this is true in all of the models of reionization that we are aware of).\n\n\nCMB anisotropies generated (and erased) due to the spatially constant\npart of the correlation function (\\ref{eqn:xexemodel}) are obviously the same as in the\nmodel with a uniform time-dependent reionization. The spatially\nvarying part is responsible for generating new anisotropies; its\ncontribution to erasing the primary anisotropies is negligible. The\nanisotropy suppression is mainly determined by the total optical depth\nto Thomson scattering and is insensitive to the small-scale structure\nof the ionization fraction $x_e(\\eta ,{\\bf x})$.\n\nThe CMB correlation function due to the patchy part only is\n\\begin{eqnarray}\n\\label{eqn:cthetaintegral}\nC^{\\rm (p)}(\\theta ) &=& \\tau_0^2 \\eta_0^4\\int_{0}^{\\eta _0} {d\\eta\n_1\\over \\eta _1^3} \\int_{0}^{\\eta _0} {d\\eta _2\\over \\eta\n_2^3}I_{12} \\\\ && \\quad\\times \\left<\\hat{\\gamma}_1\\cdot {\\bf\nv}({\\bf x}_1)\\hat{\\gamma}_2\\cdot {\\bf v}({\\bf x}_2)\\right>e^{-{({\\bf\nx}_1-{\\bf x}_2)^2\\over 2R^2}} \\nonumber,\n\\end{eqnarray}\nwhere we denote $I_{12}\\equiv X_e(\\eta _{\\rm min})-X_e(\\eta\n_1)X_e(\\eta _2)$.  The correlation function is non-negligible only for\n$|{\\bf x}_1-{\\bf x}_2|\\lesssim R$. By assumption, $R$ is much smaller\nthen the characteristic scale of the peculiar velocity field. Also\n$|{\\bf x}_1-{\\bf x}_2|\\lesssim R$ requires that the lines of sight\n$\\hat{\\gamma}_1$ and $\\hat{\\gamma}_2$ be nearly parallel. Then\n\\begin{equation}\n\\left<\\hat{\\gamma}_1\\cdot {\\bf v}({\\bf x}_1)\\hat{\\gamma}_2\\cdot {\\bf\nv}({\\bf x}_2)\\right>\\approx {1\\over 3}\\left<v^2 \\right>,\n\\end{equation}\nwhere $\\left<v^2 \\right>$ is the mean squared peculiar velocity today. For $z_i\\gg\n1$, the integral (\\ref{eqn:cthetaintegral}) is dominated by small conformal times, and we\nhave $|{\\bf x}_1-{\\bf x}_2|^2\\approx \\theta ^2(\\eta _0-\\eta_i)^2+(\\eta _1-\\eta\n_2)^2$. Then\n\\begin{eqnarray}\nC^{\\rm (p)}(\\theta )&\\approx& {1\\over 3}\\tau_0^2\\eta_0^4\\left<v^2\\right>e^{-{(\\eta\n_0 - \\eta_i)^2\\theta ^2\\over 2R^2}}\\nonumber \\\\\n&&\\times \\int_{0}^{\\infty }{d\\eta _1\\over \\eta _1^3}\n\\int_{0}^{\\infty } {d\\eta _2\\over \\eta _2^3}I_{12}e^{-{(\\eta _2-\\eta\n_1)^2\\over 2R^2}}.\n\\end{eqnarray}\nWe assume that $\\eta _i\\delta z\/(1+z_i)\\gg R$ (with $\\eta\n_i\\equiv \\eta (z_i)$) and that during the patchy phase,\n$z_i>z>z_i-\\delta z$, the ionization fraction $X_e$ grows linearly from\n0 to 1 such that eventually both hydrogen and helium are fully ionized. \nThen\n\\begin{equation}\nC^{\\rm (p)}(\\theta )=Ae^{-{\\theta ^2\\over 2\\theta _0^2}},\n\\label{eqn:cthetagen}\n\\end{equation}\nwhere the characteristic angular scale is\n\\begin{equation}\n\\theta _0={R\\over \\eta _0 - \\eta_i} = {R \\over \\eta_0} {(1+z_i)^{1\/2} \\over\n(1+z_i)^{1\/2} - 1},\n\\label{eqn:angularscale}\n\\end{equation}\nand the amplitude is\n\\begin{equation}\nA={\\sqrt{2\\pi }\\over 36} \\tau _0^2\\left<v^2\\right>{R\\over \\eta _0}\\delta\nz(1+z_i)^{3\/2}.\n\\label{eqn:amplitude}\n\\end{equation}\nNote that a critical matter-dominated universe is assumed in this\nexpression.  To generalize this result replace $\\eta_0-\\eta_i$ by\nthe comoving angular diameter distance in \nequation~(\\ref{eqn:angularscale}) and a factor of $(1+z_i)$ in\nequation~(\\ref{eqn:amplitude}) with the appropriate velocity growth\nfactor.\n\nThe power spectrum is given by the spherical harmonics decomposition\n\\begin{equation}\nC_l^{\\rm (p)}=2\\pi \\int d\\cos \\theta P_l(\\cos \\theta )w_p(\\theta ).\n\\end{equation}\nFor $l\\gg 1$,\n\\begin{equation}\nC_l^{\\rm (p)}\\approx 2\\pi \\int_0^{\\infty} \\theta d\\theta J_0(l\\theta )w_p(\\theta\n)=2\\pi A\\theta _0^2e^{-{\\theta _0^2l^2\\over 2}}.\n\\end{equation}\nThe power per octave is\n\\begin{equation}\n{l^2C_l^{\\rm (p)}\\over 2\\pi }=Al^2\\theta _0^2e^{-{\\theta _0^2l^2\\over 2}}.\n\\label{eqn:Cl}\n\\end{equation}\n\nThe anisotropy power reaches the maximal value\n\\begin{equation}\n({l^2C_l^{\\rm (p)}\\over 2\\pi })_{\\rm max}={\\sqrt{2\\pi }\\over 18{\\rm e}} \\tau\n_0^2\\left<v^2\\right>{R\\over \\eta _0}\\delta z(1+z_i)^{3\/2},\n\\end{equation}\nat\n\\begin{equation}\nl_{\\rm max}={\\sqrt{2} \\eta _0\\over R} [1 - (1+z_i)^{-1\/2}].\n\\end{equation}\n\n\n\\subsection{Discussion}\n\\label{sec:discussion}\n\nThe signal from patchy reionization in our model depends on four\nquantities: the rms peculiar velocity \n$\\left< v^2 \\right>^{1\/2}$ today, the redshift of reionization $z_i$,\nits duration $\\delta z$ and the characteristic comoving size of the patches $R$.\nThe structure formation model specifies\nthe power spectrum of fluctuations which in turn tells us the \nrms peculiar velocity. Let us now consider the patchy reionized signal in \nthe context of a specific model for structure formation.  \n\nFor illustrative purposes, let us consider\na cold dark matter model with $h=0.5$, $\\Omega_b  =0.1$, and a scale-invariant $n=1$ \nspectrum of initial fluctuations.  Normalizing the spectrum to the\nCOBE detection via the fitting formulae of \\cite{Bun97} (1997) (their\nequations [17]-[20]) and employing the analytic fit to the transfer function \nof \\cite{Eis98} (1998) (their equations [15]-[24])\nwe find an rms velocity  \nof $\\left<v^2\\right>^{1\/2}=3.9\\times 10^{-3}$.  \nWith the present optical depth of $\\tau_0 = 0.122 \\Omega_b h = 0.0061$,\nwe have a maximal anisotropy of\n\\begin{equation}\n({l^2C_l\\over 2\\pi })_{\\rm max}=2.41\\times 10^{-15}{R\\over {\\rm\nMpc}}\\delta z(1+z_i)^{3\/2},\n\\end{equation}\nat\n\\begin{equation}\nl_{\\rm max}={16958\\over R\/{\\rm Mpc}} [ 1 - (1+z_i)^{-1\/2}].\n\\end{equation}\n\nThe power spectrum of the model in principle also tells us the\nremaining parameters of the ionization: its redshift $z_i$,\nduration $\\delta z$ and typical patch size $R$.\nUnfortunately, these quantities depend on details of the cooling and\nfragmenting of the first baryonic objects to form the ionizing \nengines.  We therefore consider $5 \\lesssim z_i \\lesssim 50$ which spans \nthe range of estimates in the literature\n(\\cite{Teg94}\\ 1994; \\cite{Ree96}\\ 1996;\n\\cite{Agh96}\\ 1996; \\cite{Loe97}\\ 1997; \\cite{Hai97}\\ 1997; \\cite{Sil98}\\ 1998;\n\\cite{Hai98}\\ 1998).  Reionization, once it commences, is generally completed in\na time short compared with the expansion time at that epoch $\\delta z \/(1 +z_i) \n< 1$ by the coalescence of patches that are small compared with the \nhorizon at the time $R\/\\eta_i\n\\ll 1$ at the time.  Again the exact relations depend on the efficiency with\nwhich the first objects form and create ionizing radiation \n(see e.g. \\cite{Teg94}\\ 1994). \n\n\\begin{figure}[htb]\n\\psfig{figure=patchf1.eps,width=3.3in}\n\\caption{CMB anisotropy power spectra in a CDM model with extreme patchiness. \nShown here are the primary anisotropy \nand the patchy reionization anisotropy, eq.~(\\protect\\ref{eqn:Cl}) with\n$z_i=10$, $\\delta z=3$, $R=20$Mpc. These signals are compared with\nthe cosmic variance of the primary anisotropy and the noise of\nthe MAP satellite (in logarithmic bins).}\n\\label{fig:patch10}\n\\end{figure}\n\n\nLet us consider an extreme example of $z_i=10$, $\\delta z=3$, $R=20{\\rm Mpc}$. \nThen the maximal\npower is $\\approx 5.3\\times 10^{-12}$ at $l\\approx 590$, the primary\nsignal at these scales is $\\approx 3\\times 10^{-10}$ -- the\ncontribution of the patchy reionization is small in comparison \n(see Fig.~\\ref{fig:patch10}).  However, in light of \nthe high precision measurements expected from the MAP and Planck satellites \nsuch a signal is not necessarily negligible.  \nThe ultimate limit of detectability through\npower spectrum measurements is provided by so-called cosmic variance.\nThis arises since we can only measure $2\\ell+1$ realizations\nof any given multipole such that power spectrum estimates will vary by\n\\begin{equation}\n\\delta C_\\ell = \\sqrt{2 \\over 2\\ell+1} C_\\ell^{(\\rm primary)}.\n\\end{equation}\nDetection of a broad feature such as that from patchy reionization is assisted\nin that we may reduce the cosmic variance by averaging over many $\\ell$'s.  \nWe show an example of this averaging in Fig.~\\ref{fig:patch10} (lower left boxes).\nIn this model, the patchy reionization signal can be detected at the several\n$\\sigma$ level if cosmic variance were the main source of uncertainty.  \nOf course a realistic experiment also has noise and systematic errors.\nWe also show the noise error contributions expected from the MAP experiment\nin Fig.~\\ref{fig:patch10}.     \n\nAn important additional source of uncertainty is provided by other\nunknown aspects of the model.  Indeed it is hoped that the CMB power spectrum\ncan be used to measure fundamental cosmological parameters to high precision.\nExcess variance from patchy reionization can in principle cause problems for\ncosmological parameter estimation from the CMB if not included in the model.\nIt would remain undetected and produce parameter misestimates if\nits signal can be accurately mimicked by variations in the other parameters.\nFortunately, the angular signature we find here -- $\\ell^2$ white noise until some\ncut off due to the patch size -- does not resemble the signature of other cosmological\nparameters which alter the positions and amplitudes of the acoustic peaks \n(see \\cite{Bon97}\\ 1997; \\cite{Zal97}\\ 1997).  Coupled with the small amplitude of the effect on the 10 arcminute to degree scale for even this extreme model, it is unlikely that patchy reionization will significantly affect parameter\nestimation through the CMB. \n\nWe have called the ($z_i=10,\\delta z=3,R=20$) model extreme, because of the size of patches; the reionization redshift and duration would be considered reasonable by a number of theories. For example the early quasar model of Haiman \\& Loeb (1998) does predict $z_i\\sim 10$ and $\\delta z \\sim 3$. However, their ``medium quasar'' emits only $\\sim 10^{67}$ ionizing photons during its life time. These photons cannot ionize a bubble larger than $R\\sim 1$Mpc comoving.\n\nPerhaps more interesting is the case where reionization takes \nplace at a higher redshift with\nfor example \n$z_i=30$, $\\delta z=5$, $R=3{\\rm Mpc}$.  The reduction in the patch size\ncauses the signature to move to smaller angles where the primary signal\nis negligible due to dissipational effects at recombination.  \nThe increase in the optical depth at this higher redshift is counterbalanced\nby the reduction in the rms fluctuation due to the number of patches along\nthe line of site such that the amplitude of the signal increases\nonly moderately.  Here the maximal \npower is $\\approx 6.2\\times10^{-12}$ at $l\\approx 4650$ \n(see Fig.~\\ref{fig:patch30}).  Patchy reionization effects exceed \nthe Vishniac signal at these\nscales ($\\approx 3\\times 10^{-12}$) which is believed to be\nthe leading other source of secondary anisotropies \n(\\cite{Hu96}\\ 1996).  \n\n\\begin{figure}[htb]\n\\psfig{figure=patchf2.eps,width=3.3in}\n\\caption{CMB anisotropy power spectra in a CDM model with early reionization. \nShown here are the primary anisotropy suppressed by rescattering \nand the patchy reionization anisotropy, eq.~(\\protect\\ref{eqn:Cl}) with\n$z_i=30$, $\\delta z=5$, $R=3$Mpc. These signals are compared with\nthe cosmic variance of the primary anisotropy achievable by an ideal\nexperiment in the absence of galactic and extragalactic foregrounds.}\n\\label{fig:patch30}\n\\end{figure}\n\nAlthough the morphology and amplitude \nof the patchy reionization and Vishniac signals are similar, the Vishniac\neffect is fully specified by the ionization redshift and the spectrum\nof initial fluctuations and hence may be removed once these are determined\nfrom parameter estimation at larger angular scales.  Likewise, \nsince the rms peculiar velocity $\\left< v \\right>$ and the ionization\nredshift $z_i$ will be specified by the large scale observations, \nthe amplitude of the signal can be used to estimate\nthe duration of reionization $\\delta z$ and its angular location \nthe typical comoving size of the bubbles $R$. \n\nIn summary, the patchiness of reionization leaves a potentially observable \nimprint on the CMB power spectrum, but one that is unlikely to affect cosmological\nparameter estimation from the acoustic peaks in the CMB.   We show how the signature\nscales with the gross properties of reionization -- its redshift, duration, and\ntypical patch size.   Observational detection of this signature would provide \nuseful constraints on the presently highly uncertain reionization scenarios but\nwill likely require experiments with angular resolution of an \narcminute or better and foreground subtraction at better than the \n$\\delta T\/T \\sim 10^{-6}$ level.\n\n\\acknowledgements\nWe thank \nR. Juszkiewicz, A. Liddle, M. Tegmark and M. White\nfor discussions.  This work\nwas supported by NSF PHY-9513835.  WH was also supported by the\nW. M. Keck Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe fascinating beauty of the  theory of the functions of complex variable reveals\nitself, for example, in the harmony of the algebraic fractals on the Euclidian\nplane. It makes many researches look for the analogous number systems, the elements\nof which could be correlated not to the points on the plane but to the points of the\n4-dimensional space-time. In case of the success of such a search, we could really\ntrust the famous Pythagoras saying 'all the existing is number'. On this way, the\ninteresting results were obtained for quaternions [1], biquaternions [2-4], octaves\n[5] and so forth. Nevertheless, none of these number system theories can be compared\neven to the theory of the relatively simple 2-component complex numbers. The main\nreason for this seems to be the lack of the commutativity (and sometimes even of the\nassociativity) of the multiplication in these algebras. Although the authors of this\npaper realize the conceptual bases of all the variety of algebras, the commutativity\nof the multiplication is the integral property of all the principal number systems\nthat contain natural, integer, rational, real and complex numbers. Finally, the\ncommutativity and the associativity of the multiplication are among the axioms of\narithmetic which presents the foundation of mathematics, and it would be strange if\nthe algebraic system which is the most natural for our real world does not\ncorrespond to the rules of regular counting.\n\nOne of the systems free from this drawback is the algebra of the commutative and\nassociative hyper complex numbers, related to the direct sum of the four real\nalgebras, which will be denoted as $H_4$. The algebra of these numbers is isomorphic\nto the algebra of the 4-dimensional square real diagonal matrices, and the\ncorresponding space is a linear Finsler space with the Berwald-Moor metric (the last\nfact was proved by the authors in [6]). It should be mentioned that Finsler space\nwith the Berwald-Moor metric has been known and partially investigated for a long\ntime [7--8].\n\nOne of the main properties of this space is the existence of such a range of the\nparameters that the 3-dimensional distances (from the point of view of the observer\nwho uses the radar method to measure them [9]) correspond to the positively defined\nmetric function the limit of which is the quadratic form [10]. In other words, the\n3-dimensional world observed by an \"$H_4$ inhabitant\" is Euclidian within certain\naccuracy. Moreover, when one passes to the relativistic velocities, the\n4-dimensional intervals between the $H_4$ events present the Minkowski space\ncorrelations [11]. All this makes possible to suggest that the $H_4$ space and the\ncorresponding Finsler geometry can be used as a mathematical model of the real\nspace-time, and maybe this model would be even more productive than the pseudo\nRiemannian constructions prevailing in Physics now.\n\n\\mes\n\nAny hyper complex algebra is completely defined by the multiplication rule for the\nelements of a certain fixed basis. In the H4 number system there is a special --\nisotropic -- basis $e_1, e_2, e_3, e_4$, such that\n \\begin{equation}\\label{gp1}\n e_ie_j=p^k_{ij}e_k \\, \\qquad  p^k_{ij} = \\left\\{  \\begin{array}{l}\n                                                    1\\, , \\hbox{\\q if~} \\; i=j=k \\, , \\\\\n                                                    0\\, , \\hbox{\\q else} \\, .\n                                                  \\end{array}\n   \\right.\n \\end{equation}\n Any analytical function in this basis can be given as\n \\begin{equation}\\label{gp2}\n F(X) = f^1(\\xi^1)e_1 + f^2(\\xi^2)e_2 + f^3(\\xi^3)e_3 + f^4(\\xi^4)e_4 \\, ,\n \\end{equation}\nwhere\n \\begin{equation}\\label{gp3}\n H_4 \\ni X = \\xi^1e_1 + \\xi^2e_2 + \\xi^3e_3 + \\xi^4e_4 \\, ,\n \\end{equation}\nand $f^i$ are four arbitrary smooth functions of a single real variable.\n\nIn $H_4$ there is one more -- orthogonal -- selected basis $1, \\, j, \\, k, \\, jk$,\nwhich is related to the isotropic basis by the following formulas\n \\begin{equation}\\label{gp4}\n \\left.  \\begin{array}{l}\n          1 = e_1 + e_2 + e_3 + e_4 \\, , \\\\[2pt]\n          j = e_1 + e_2 - e_3 - e_4 \\, , \\\\[2pt]\n          k = e_1 - e_2 + e_3 - e_4 \\, , \\\\[2pt]\n          jk = e_1 - e_2 - e_3 + e_4 \\, ,\n        \\end{array}\n  \\right\\}\n \\end{equation}\nwhere {\\it 1} is the unity of algebra, and the corresponding component of the\nanalytical function of the $H_4$ variable is defined by the formula\n \\begin{equation}\\label{gp5}\n u = \\frac{1}{4} \\left[  f^1(\\xi^1) + f^2(\\xi^2) + f^3(\\xi^3) + f^4(\\xi^4)  \\right] \\, .\n \\end{equation}\n\nIf $X$ is a radius vector, then the coordinate space  $\\xi^1, \\, \\xi^2, \\, \\xi^3, \\,\n\\xi^4$  is a Berwald-Moor space with the length element\n \\begin{equation}\\label{gp6}\n ds = \\sqrt[4]{d\\xi^1d\\xi^2d\\xi^3d\\xi^4} \\equiv\n \\sqrt[4]{g_{ijkl}d\\xi^id\\xi^jd\\xi^kd\\xi^l} \\, ,\n \\end{equation}\nwhere\n \\begin{equation} \\label{gp7}\n g_{ijkl} = \\left\\{\n \\begin{array}{l}\n  \\frac{1}{4!} \\, ,  {~~(i\\ne j\\ne k\\ne l)}   , \\\\[9pt]\n \\; 0\\, \\, ,  {~~(else)}  .\n \\end{array} \\right.\n \\end{equation}\nFor this geometry the tangent indicatrix equation is\n \\begin{equation}\\label{gp8}\n g^{ijkl}p_ip_jp_kp_l - 1 = 0 \\, ,\n \\end{equation}\nwhere\n \\begin{equation}\\label{gp9}\n g^{ijkl} = \\left\\{  \\begin{array}{l}\n \\displaystyle\\frac{4^4}{4!} \\, ,  {~~(i\\ne j\\ne k\\ne l)}  , \\\\[9pt]\n \\; 0\\, \\, , \\hbox{~~(else)} ,\n \\end{array} \\right.\n \\end{equation}\n \\begin{equation}\\label{gp10}\n p_i = \\frac{g_{ijkl}d\\xi^jd\\xi^kd\\xi^l}{\\left(g_{mrst}d\\xi^md\\xi^rd\\xi^sd\\xi^t\n \\right)^{3\/4}} \\,\n \\end{equation}\nare the components of the generalized momentum or generalized momenta.\n\nIf we have tensors  $p^k_{ij}$, $g_{ijkl}$, $g^{ijkl}$  and vector fields of the\nanalytical functions $F_{(A)}(X)$ of the $H_4$ variables, we could construct the\nmetric tensors in the 4-dimensional space-time in many ways. For example,\n \\begin{equation}\\label{gp11}\n g_{ij}(\\xi) = g_{ijkl}f^k_{(1)}f^l_{(2)} \\, ,\n \\end{equation}\nNow one can investigate the obtained Riemannian geometry. The main drawback of this\napproach is the variety of the ways to construct it.\n\n It is known [12] that if the tangent indicatrix equation is defined as\n \\begin{equation}\\label{gp12}\n \\Phi(p;\\xi) = 0 \\, ,\n \\end{equation}\nthen the geodesics will be the solutions of the canonical system of differential\nequations\n \\begin{equation}\\label{gp13}\n \\dot{\\xi}^i = \\frac{\\partial \\Phi}{\\partial p_i} \\cdot \\lambda(p;\\xi) \\, , \\qquad \\dot{p}_i\n = - \\frac{\\partial \\Phi}{\\partial \\xi^i} \\cdot \\lambda(p;\\xi) \\, ,\n \\end{equation}\n $\\lambda (p;\\xi )\\neq 0$  is an arbitrary smooth function, and a dot above $\\xi^i$ and $p_i$ means the derivation by the evolution parameter, $\\tau$.\n\n\\section{Construction of the metric function\\\\ of the pseudo Riemannian space}\n\nLet us regard a space which is conformally connected to the $H_4$ space, that is to\nthe space with the length element\n \\begin{equation}\\label{gp14}\n ds' = \\kappa(\\xi)\\cdot \\sqrt[4]{g_{ijkl}d\\xi^id\\xi^jd\\xi^kd\\xi^l} \\, ,\n \\end{equation}\nwhere $\\kappa(\\xi) > 0$ is a scalar function which is a contraction-extension\ncoefficient depending on the point.\n\nLet there be a normal congruence of geodesics (world lines). Then there is a scalar\nfunction $S(\\xi)$ (see, e.g. [12]) such that its level hyper surfaces are\ntransversal to this normal congruence of the world lines and this function is a\nsolution of the equation\n \\begin{equation}\\label{gp15}\n g^{ijkl}\\frac{\\partial S}{\\partial \\xi^i}\\frac{\\partial S}{\\partial\n \\xi^j}\\frac{\\partial S}{\\partial \\xi^k}\\frac{\\partial S}{\\partial\n \\xi^l} = \\kappa(\\xi)^4  \\, ,\n \\end{equation}\nwhile the generalized momenta along this congruence of the world lines are related\nto $S(\\xi)$ by\n \\begin{equation}\\label{gp16}\n p_i = \\frac{\\partial S}{\\partial \\xi^i} \\, ,\n \\end{equation}\nThe equations for the world lines obtain the form\n \\begin{equation}\\label{gp17}\n \\dot{\\xi}^i = g^{ijkl}\\frac{\\partial S}{\\partial\n \\xi^j}\\frac{\\partial S}{\\partial \\xi^k}\\frac{\\partial S}{\\partial\n \\xi^l} \\cdot \\lambda(\\xi) \\, ,\n \\end{equation}\nwere  $\\lambda (\\xi )\\neq 0$.\n\nIn Physics the function $S(\\xi)$ is called \"action as a function of coordinates\"\\,\nand (\\ref{gp15}) is known as the Hamilton-Jacoby equation. In \\cite{10} the function\n$S(\\xi)$ was called the \\emph{World function}.\n\nIf there is a congruence of the world lines, then the evolution of every point in\nspace is known, particularly, the velocity field is known, but the energy\ncharacteristics of the material objects (observers) corresponding to a given world\nline are not known. The knowledge of the World function $S(\\xi)$ makes it possible\nto calculate the generalized momenta $p_i$, corresponding to the energy\ncharacteristics, and the invariant energy characteristic, $\\kappa(\\xi)$, which has\nalso the meaning of the local contraction-extension coefficient of the plane $H_4$\nspace.\n\nSo, if our world view is the classical mechanics, then any pair out of the three:\nWorld function, congruence of the world lines, Finsler geometry - gives us the\ncomplete knowledge of the World.\n\nLet us construct a twice contravariant tensor {\\it g}{\\it ij}{\\it (?)} in the\nfollowing way:\n \\begin{equation}\\label{gp18}\n g^{ij}(\\xi) = \\frac{1}{\\kappa(\\xi)^4} \\cdot g^{ijkl}\\frac{\\partial\n S}{\\partial \\xi^k}\\frac{\\partial S}{\\partial \\xi^l} \\, .\n \\end{equation}\nSince\n \\begin{equation}\\label{gp19}\n det(g^{ij}(\\xi))= - \\frac{4^4}{3^3 \\kappa(\\xi)^8} \\neq 0 \\, ,\n \\end{equation}\nthen everywhere where the geometry (\\ref{gp14}) is defined, one can construct a\ntensor $g_{ij}(\\xi)$ such that\n \\begin{equation}\\label{gp20}\n g^{ik}(\\xi)g_{kj}(\\xi)=\\delta^i_j  \\, ,\n \\end{equation}\n \\begin{equation}\\label{gp21}\n  g_{ij}(\\xi)   = 4\\cdot \\left(\n \\begin{array}{cccc}\n  -2\\left(\\frac{\\partial S}{\\partial \\xi^1}\\right)^2 & \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^2} & \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^3} & \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^4}\n  \\\\[9pt]\n  \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^2} & -2\\left(\\frac{\\partial S}{\\partial \\xi^2}\\right)^2  & \\frac{\\partial S}{\\partial \\xi^2}\\frac{\\partial S}{\\partial \\xi^3} & \\frac{\\partial S}{\\partial \\xi^2}\\frac{\\partial S}{\\partial \\xi^4} \\\\[9pt]\n  \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^3} & \\frac{\\partial S}{\\partial \\xi^2}\\frac{\\partial S}{\\partial \\xi^3} & -2\\left(\\frac{\\partial S}{\\partial \\xi^3}\\right)^2  & \\frac{\\partial S}{\\partial \\xi^3}\\frac{\\partial S}{\\partial \\xi^4} \\\\[9pt]\n  \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^4} & \\frac{\\partial S}{\\partial \\xi^2}\\frac{\\partial S}{\\partial \\xi^4} & \\frac{\\partial S}{\\partial \\xi^3}\\frac{\\partial S}{\\partial \\xi^4} & -2\\left(\\frac{\\partial S}{\\partial \\xi^4}\\right)^2\n \\end{array}\n \\right) .\n \\end{equation}\n\nNo doubt that in the same coordinate space  $\\xi ^{1} ,\\xi ^{2} ,\\xi ^{3} ,\\xi ^{4}$\nsuch tensor $g_{ij}(\\xi)$ defines a Riemannian or pseudo Riemannian geometry with\nthe length element\n \\begin{equation}\\label{gp22}\n ds'' = \\sqrt{g_{ij}(\\xi)d\\xi^id\\xi^j} \\, .\n \\end{equation}\n\nThe construction of tensor $g_{ij}(\\xi)$ leads directly to the conclusion: the\nchange of geometry (\\ref{gp14}) to the geometry (\\ref{gp22}) does not lead to the\nchange of the initial congruence of the world lines and corresponding World function\n$S(\\xi)$.\n\nTherefore, in our concept one and the same World, i.e. the pair \\{World function;\ncongruence of the world lines\\}, corresponds to a whole class of related but\nqualitatively different Finsler geometries.\n\n\\section{Analyticity condition and the Minkowski space}\n\nLet the World function $S(\\xi)$ be the (unity) component of an analytical function\nof the $H_4$ variable in the orthogonal basis (\\ref{gp4}), that is\n \\begin{equation}\\label{gp23}\n S(\\xi) = \\frac{1}{4} \\left[  f^1(\\xi^1) + f^2(\\xi^2) + f^3(\\xi^3) + f^4(\\xi^4)  \\right] \\, .\n \\end{equation}\nThen\n \\begin{equation}\\label{gp24}\n g^{ijkl}\\frac{\\partial S}{\\partial \\xi^i}\\frac{\\partial S}{\\partial\n \\xi^j}\\frac{\\partial S}{\\partial \\xi^k}\\frac{\\partial S}{\\partial\n \\xi^l} = \\frac{\\partial f^1(\\xi^1)}{\\partial \\xi^1}\\frac{\\partial\n f^2(\\xi^2)}{\\partial \\xi^2}\\frac{\\partial f^3(\\xi^3)}{\\partial\n \\xi^3}\\frac{\\partial f^4(\\xi^4)}{\\partial \\xi^4} = \\kappa(\\xi)^4 > 0  \\, ,\n \\end{equation}\nand this leads to the limitation on the functions, {\\it f}{\\it i}:\n \\begin{equation}\\label{gp25}\n \\frac{\\partial f^1(\\xi^1)}{\\partial \\xi^1}\\frac{\\partial\n f^2(\\xi^2)}{\\partial \\xi^2}\\frac{\\partial f^3(\\xi^3)}{\\partial\n \\xi^3}\\frac{\\partial f^4(\\xi^4)}{\\partial \\xi^4} > 0 \\, .\n \\end{equation}\n\nIt follows from (\\ref{gp24}) that the space with the length element (\\ref{gp14}) can\nbe obtained from the space with the length element (\\ref{gp6}) with the help of the\nconformal transformation, which means that the condition of the analyticity of the\nWorld function can be treated in a sense as the condition of the conformal symmetry.\n\nLet us construct tensor $g_{ij}(\\xi)$ following the algorithm developed in the\nprevious section. It turns out that in a region where functions $f^i$  have no\nsingularities there will always be such a coordinate system $x^0, \\, x^1, \\, x^2, \\,\nx^3$ in which the length element $ds''$ has a form\n \\begin{equation}\\label{gp26}\n ds'' = \\sqrt{(x^0)^2 - (x^1)^2 - (x^3)^2 - (x^3)^2} \\, .\n \\end{equation}\n\nLet us express the coordinates $x^0,\\, x^1,\\, x^2,\\, x^3$ in terms of the initial\ncoordinates  $\\xi ^{1} ,\\xi ^{2} ,\\xi ^{3} ,\\xi ^{4} $:\n\\begin{equation}\\label{gp27}\n\\left.\n\\begin{array}{l}\n  x^0 = \\displaystyle\\frac{\\;\\, 1 \\;}{4}       \\left(  f^1(\\xi^1) + f^2(\\xi^2) + f^3(\\xi^3) +\nf^4(\\xi^4)  \\right) \\, ,\\\\[12pt]\n  x^1 = \\displaystyle\\frac{\\sqrt{3}}{4}\\left(  f^1(\\xi^1) + f^2(\\xi^2) - f^3(\\xi^3) -\nf^4(\\xi^4)  \\right)  \\, ,\\\\[12pt]\n  x^2 = \\displaystyle\\frac{\\sqrt{3}}{4}\\left(  f^1(\\xi^1) - f^2(\\xi^2) + f^3(\\xi^3) -\nf^4(\\xi^4)  \\right) \\, ,\\\\[12pt]\n  x^3 =\\displaystyle \\frac{\\sqrt{3}}{4}\\left(  f^1(\\xi^1) - f^2(\\xi^2) - f^3(\\xi^3) +\nf^4(\\xi^4)  \\right) \\, .\n\\end{array}\n\\right\\}\n\\end{equation}\n\nTherefore, to obtain the non-trivial curving of the space-time one should use the World functions with the broken conformal symmetry.\n\n\\section{Newtonian  potential}\n\nLet us show that there are World functions that lead to the non-trivial pseudo\nRiemannian 4-dimensional spaces. Let us regard a function\n \\begin{equation}\\label{gp28}\n S(\\xi) = \\frac{1}{4}\\left( \\xi^1 + \\xi^2 + \\xi^3 + \\xi^4 \\right) +\n \\alpha\\cdot\\psi(\\varrho) \\, ,\n \\end{equation}\nwhere $\\alpha$ is the parameter characterizing the break of the analyticity of the\nWorld function (the break of the conformal symmetry in the $H_4$ space),  $\\psi$ is\nan arbitrary function of a single argument\n \\begin{equation}\\label{gp29}\n \\varrho = \\sqrt{(y^1)^2+(y^2)^2+(y^3)^2} \\, ,\n \\end{equation}\nand $y^0,\\, y^1,\\, y^2,\\, y^3$ are the coordinates in the orthogonal basis $1, j, k,\njk$:\n \\begin{equation}\\label{gp30}\n \\left.\n \\begin{array}{c}\n y^0 = \\displaystyle \\frac{1}{4}(\\xi^1+\\xi^2+\\xi^3+\\xi^4) \\, , \\\\[12pt]\n y^1 = \\displaystyle \\frac{1}{4}(\\xi^1+\\xi^2-\\xi^3-\\xi^4) \\, , \\\\[12pt]\n y^2 = \\displaystyle \\frac{1}{4}(\\xi^1-\\xi^2+\\xi^3-\\xi^4) \\, , \\\\[12pt]\n y^3 = \\displaystyle \\frac{1}{4}(\\xi^1-\\xi^2-\\xi^3+\\xi^4) \\, .\n \\end{array}\n \\right\\} \\,\n \\end{equation}\nThen the derivatives of the World functions over the coordinates $\\xi^i$  can be\nexpressed in the following way:\n\\begin{equation}\\label{gp31}\n\\left.\n\\begin{array}{c}\n\\displaystyle\\frac{\\partial S}{\\partial \\xi^1} =\n\\displaystyle \\frac{1}{4}\\left[1+\\frac{\\alpha}{\\varrho}\\frac{d\\psi}{d\\varrho}\\left( y^1+y^2+y^3  \\right)\\right] \\, , \\\\[15pt]\n\\displaystyle\\frac{\\partial S}{\\partial \\xi^2} =\n\\displaystyle \\frac{1}{4}\\left[1+\\frac{\\alpha}{\\varrho}\\frac{d\\psi}{d\\varrho}\\left( y^1-y^2-y^3  \\right)\\right] \\, , \\\\[15pt]\n\\displaystyle\\frac{\\partial S}{\\partial \\xi^3} =\n\\displaystyle \\frac{1}{4}\\left[1+\\frac{\\alpha}{\\varrho}\\frac{d\\psi}{d\\varrho}\\left( -y^1+y^2-y^3  \\right)\\right] \\, , \\\\[15pt]\n\\displaystyle\\frac{\\partial S}{\\partial \\xi^4} =\n\\displaystyle\\frac{1}{4}\\left[1+\\frac{\\alpha}{\\varrho}\\frac{d\\psi}{d\\varrho}\\left(\n-y^1-y^2+y^3  \\right)\\right] \\, .\n\\end{array}\n\\right\\} \\,\n\\end{equation}\n\nLet us calculate the components of the metric tensor in coordinates $y^0,\\, y^1,\\,\ny^2,\\, y^3$ using the invariance of the square of the length element\n \\begin{equation}\\label{gp32}\n g_{ij}(\\xi) d\\xi^id\\xi^j = \\tilde{g}_{ij}(y)dy^idy^j\n \\end{equation}\nGrouping the terms, one gets\n \\begin{equation}\\label{gp33}\n \\tilde{g}_{00}= 1 - 3\\alpha^2\\left( \\frac{d\\psi}{d\\varrho} \\right)^2\n \\, , \\qquad \\tilde{g}_{\\beta\\beta_-}=-3\\left\\{1 + \\alpha^2\\left(\n \\frac{d\\psi}{d\\varrho} \\right)^2\\left[ 1 -\n \\frac{4(y^\\alpha)^2}{3\\rho^2}\\right]\\right\\} \\, ,\n \\end{equation}\n \\begin{equation}\\label{gp34}\n 2\\tilde{g_{0\\beta}} = - 4 \\left[\n \\alpha\\frac{d\\psi}{d\\varrho}\\frac{\\;\\; y^\\beta}{\\varrho} +\n 3\\alpha^2\\left(\\frac{d\\psi}{d\\varrho}\\right)^2\\cdot\\frac{y^1y^2y^3}{y^\\beta\\varrho^2}\n \\right] \\,  ,\n \\end{equation}\n \\begin{equation}\\label{gp35}\n 2\\tilde{g}_{\\beta\\gamma} = - 4 \\left[ 3\n \\alpha\\frac{d\\psi}{d\\varrho}\\frac{\\;\\, y^\\delta}{\\varrho} +\n \\alpha^2\\left(\\frac{d\\psi}{d\\varrho}\\right)^2\\cdot\\frac{y^\\beta\n y^\\gamma}{\\varrho^2} \\right] \\, ,\n \\end{equation}\nwhere $\\beta,\\, \\gamma,\\, \\delta,\\,  = 1, 2, 3$; $\\beta\\equiv\\beta_-$ but no\nsummation is performed here; in the last formula all the indices $\\beta,\\, \\gamma,\\,\n\\delta\\,$ are different.\n\n If $\\alpha = 0$, then\n \\begin{equation}\\label{gp36}\n (\\tilde{g}_{ij}) = diag(1,-3,-3,-3) \\, .\n \\end{equation}\nThis means that the real physical coordinates $x^0,\\, x^1,\\, x^2,\\, x^3$ of the\nspace-time are expressed by the coordinates $y^0,\\, y^1,\\, y^2,\\, y^3$ in the\nfollowing way\n \\begin{equation}\\label{gp37}\n x^0 = y^0 \\, , \\qquad x^\\beta = \\sqrt{3}\\cdot y^\\beta \\, .\n \\end{equation}\n\nLet us pass to the physical coordinates $x^0,\\, x^1,\\, x^2,\\, x^3$:\n \\begin{equation}\\label{gp38}\n \\tilde{g}_{ij}(y)dy^idy^j = \\bar{g}_{ij}(x)dx^idx^j \\, ,\n \\end{equation}\nwhere\n \\begin{equation}\\label{gp39}\n \\bar{g}_{00} =   \\tilde{g}_{00} \\, , \\qquad \\bar{g}_{0\\beta} =\n \\frac{1}{\\sqrt{3}}\\cdot \\tilde{g}_{0\\beta} \\, , \\qquad \\bar{g}_{\\beta\\gamma} =\n \\frac{1}{3}\\cdot \\tilde{g}_{\\beta\\gamma} \\,.\n \\end{equation}\n Let us denote\n \\begin{equation}\\label{gp40}\n r = \\sqrt{(x^1)^2+(x^2)^2+(x^3)^2} \\equiv \\sqrt{3}\\cdot\\varrho \\, ,\n \\end{equation}\nThen\n \\begin{equation}\\label{gp41}\n \\bar{g}_{00}= 1 - 9\\alpha^2\\left( \\frac{d\\psi}{dr} \\right)^2 \\, ,\n \\qquad \\bar{g}_{\\beta\\beta_-}=-\\left\\{1 + 3\\alpha^2\\left(\n \\frac{d\\psi}{dr} \\right)^2\\left[ 1 -\n \\frac{4(x^\\alpha)^2}{3r^2}\\right]\\right\\} \\, ,\n \\end{equation}\n \\begin{equation}\\label{gp42}\n 2\\bar{g_{0\\beta}} = - 4 \\left[ \\alpha\\frac{d\\psi}{dr}\\frac{\\;\\;\n x^\\beta}{r} +\n 3\\sqrt{3}\\alpha^2\\left(\\frac{d\\psi}{dr}\\right)^2\\cdot\\frac{x^1x^2x^3}{x^\\beta\n r^2} \\right] \\,  ,\n \\end{equation}\n \\begin{equation}\\label{gp43}\n 2\\bar{g}_{\\beta\\gamma} = - 4 \\left[ \\sqrt{3}\n \\alpha\\frac{d\\psi}{dr}\\frac{\\;\\, x^\\delta}{r} +\n \\alpha^2\\left(\\frac{d\\psi}{dr}\\right)^2\\cdot\\frac{x^\\beta x^\\gamma}{r^2} \\right] \\, .\n \\end{equation}\n\nThe metric tensor  $\\bar{g}_{ij} (x)=\\bar{g}_{ij} (x^{1} ,x^{2} ,x^{3} )$  depends\nonly on the space coordinates $x^1,x^2,x^3$, and this corresponds to the stationary\ngravitational field, stationary Universe. The probe particle of mass $m$ moves along\nthe geodesic of the pseudo Riemannian space with metric tensor  $\\bar{g}_{ij} (x^{1}\n,x^{2} ,x^{3} )$.\n\nLet a particle move in a fixed frame and have velocity much less than the light\nvelocity, $c$:\n \\begin{equation}\\label{gp44}\n \\frac{dx^\\beta}{dt} = v^\\beta \\, , \\qquad   |v^\\beta| \\ll c \\, ,\n \\end{equation}\nThe gravitational fields are weak, that is the condition  $|v^{\\beta } |<<1$ remains\nvalid for all the time of the particle motion. Let us obtain the Lagrange function,\n{\\it L}, to describe such non-relativistic motion of the probe particle in the weak\ngravity field. To do this, develop the right hand side of the expression\n \\begin{equation}\\label{gp45}\n L = -mc\\cdot \\frac{\\sqrt{\\bar{g}_{ij}(x^1,x^2,x^3)dx^idx^j}}{dt}\n \\end{equation}\nWithin the accuracy of $\\left(\\frac{v}{c}\\right)^2$\n\\begin{equation}\\label{gp46}\nL = -mc^2 \\sqrt{\\bar{g}_{00}} \\cdot \\sqrt{1+ \\frac{1}{\\bar{g}_{00}}\\left(\n2\\bar{g}_{0\\beta}\\frac{v^\\beta}{c} + \\bar{g}_{\\beta\\gamma}\\frac{v^\\beta\nv^\\gamma}{c^2} \\right)} \\, ,\n\\end{equation}\n\\begin{equation}\\label{gp47}\nL \\simeq -mc^2 \\sqrt{\\bar{g}_{00}} \\cdot \\left\\{1+ \\frac{1}{2\\bar{g}_{00}}\\left(\n2\\bar{g}_{0\\beta}\\frac{v^\\beta}{c} + \\bar{g}_{\\beta\\gamma}\\frac{v^\\beta\nv^\\gamma}{c^2} \\right) - \\frac{1}{8\\bar{g}^2_{00}}\\left(\n2\\bar{g}_{0\\beta}\\frac{v^\\beta}{c} \\right)^2\\right\\} \\, .\n\\end{equation}\nOpening the brackets in the right hand side, we get an additive term which is the\nfull time derivative of a certain function {\\it f(r)}, it depends linearly on the\nvelocity components and, thus, it can be omitted. Leaving the same designation for\nthe Lagrange function, we get\n\\begin{equation}\\label{gp48}\nL \\simeq -mc^2 \\sqrt{\\bar{g}_{00}} \\cdot \\left\\{1+ \\frac{1}{2\\bar{g}_{00}}\\cdot\n\\bar{g}_{\\beta\\gamma}\\frac{v^\\beta v^\\gamma}{c^2} - \\frac{1}{8\\bar{g}^2_{00}}\\left(\n2\\bar{g}_{0\\beta}\\frac{v^\\beta}{c} \\right)^2\\right\\} \\, .\n\\end{equation}\n\nOur goal is the Lagrange function of the form\n \\begin{equation}\\label{gp49}\n L = \\frac{m\\vec{v}^2}{2} - U(\\vec{x}) \\, ,\n \\end{equation}\nwhere  $U(\\vec{x})$  is the potential energy of the probe particle,  $\\vec{x}\\equiv\n(x^{1} ,x^{2} ,x^{3} ),~ \\vec{v}\\equiv (v^{1} ,v^{2} ,v^{3} ),~ r^{2} =\n\\vec{x}^{\\,2}$,  $\\vec{v}^{\\,2} =(v^{1} )^{2} +(v^{2} )^{2} +(v^{3} )^{2} \\equiv\nv^{2} $. To reach it we have to make some assumptions about the correlation between\nthe parameter, $\\alpha$ and light velocity:\n \\begin{equation}\\label{gp50}\n \\alpha = \\frac{\\nu}{c} \\, , \\quad \\hbox{when} \\quad c\\rightarrow\n \\infty \\quad \\alpha\\rightarrow 0 \\, .\n \\end{equation}\nBesides, let $\\alpha$ be of the same order (or smaller) with the relation\n$\\left|\\displaystyle\\frac{v}{c}\\right|$. Then leaving only the terms that don't\ndisappear at  $c\\to \\infty $  in the (\\ref{gp48}), one gets\n \\begin{equation}\\label{gp51}\n L \\simeq -mc^2 + mc^2 \\frac{9}{2}\\frac{\\nu^2}{c^2} \\left(\n \\frac{d\\psi}{dr} \\right)^2 + m\\cdot \\frac{v^1v^1+v^2v^2+v^3v^3}{2}\\, .\n \\end{equation}\nSince $(-mc^2)$ is a full time derivative of function $(-mc^2\\cdot t)$, we omit it\nand get\n \\begin{equation}\\label{gp52}\n L \\simeq  \\frac{m\\vec{v}^2}{2} + \\frac{9m\\nu^2}{2} \\left(\n \\frac{d\\psi}{dr} \\right)^2 \\, .\n \\end{equation}\n\nLet a mass $M$ be motionless in the frame origin, and then the potential energy of\nthe probe particle with mass $m$ located at $x^1,x^2,x^3$ is equal to\n \\begin{equation}\\label{gp53}\n U(r) = - \\gamma \\frac{mM}{r} \\, ,\n \\end{equation}\nwhere $\\gamma$ is the gravitational constant. Comparing (\\ref{gp49}) and\n(\\ref{gp52}), we get the equation for $\\psi(r)$:\n\\begin{equation}\\label{gp54}\n\\frac{9m\\nu^2}{2} \\left(\\frac{d\\psi}{dr} \\right)^2 = \\gamma \\frac{mM}{r} \\qquad\n\\Rightarrow \\qquad  \\frac{d\\psi}{dr} = \\pm \\frac{\\sqrt{2\\gamma\nM}}{3\\nu}\\frac{1}{r^{1\/2}}   \\, .\n\\end{equation}\nTherefore,\n\\begin{equation}\\label{gp55}\n\\psi(r) = \\pm \\frac{2\\sqrt{2\\gamma M}}{3\\nu}\\cdot r^{1\/2} + \\psi_0  \\qq (\\psi_0 =\nconst).\n\\end{equation}\n\nFinally, the World function is equal to\n\\begin{equation}\\label{gp56}\nS = x^0 \\pm \\frac{2\\sqrt{2\\gamma M}}{3c}\\cdot r^{1\/2} + C_0 \\qq (C_0 = const),\n\\end{equation}\nWhen it performs a conformal transformation of the length element of the plane\nBerwald-Moor space, it induces a pseudo Riemannian geometry in the Minkowski space.\nFor a non-relativistic probe particle of mass {\\it m}, this geometry gives the\nmotion equations for the Kepler problem for the point mass {\\it M} located in the\norigin of the space frame.\n\n The more complicated World function, maybe also leading to the stationary Universe, has the form\n\\begin{equation}\\label{gp57}\nS(\\xi) = \\frac{1}{4}\\left( \\xi^1 + \\xi^2 + \\xi^3 + \\xi^4 \\right)\\left[\n1+\\alpha_1\\cdot\\psi_1(\\varrho) \\right] + \\alpha_2\\cdot\\psi_2(\\varrho) \\, ,\n\\end{equation}\nwhere $\\alpha_A$ are the parameters of the analyticity break of the World function\n(parameters of the conformal symmetry break in the $H_4$ space), $\\psi_A$ are the\narbitrary functions of single argument $\\varrho$ (\\ref{gp29}), (\\ref{gp30}).\n\n \\section*{Conclusion}\n\nThe results obtained in this paper point at the deep correlation between the\nEinstein geometries and Finsler spaces with Berwald-Moor metric. We managed to find\nthe concrete Finsler space with the Berwald-Moor metric which in the limit appeared\nto be related to the curved pseudo Riemannian space with the Newtonian gravitational\npotential. This fact points at the principal possibility to built more interesting\nconstructions, particularly, such Finsler spaces whose limit cases would be the\nknown relativistic solutions.\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nI was asked to talk about thermal radiation from\nisolated neutron stars (NSs). In this meeting, George Pavlov\nreviewed the X-ray properties of pulsars and thermally emitting NSs\n(see Kaplan et al.~2011), and Wynn Ho discussed central compact\nobjects and their magnetic fields (see Halpern \\& Gotthelf 2010;\nShabaltas \\& Lai 2012; Vigano \\& Pons 2012). Recent works on\ntheoretical modelling of NS surface emission can be found in Potekhin\net al.~(2012) (see also Pavlov et al.~1995; Harding \\& Lai 2006 and\nvan Adelsberg \\& Lai 2006 for reviews). Since these subjects were\nadequately covered in the meeting, I decided to focus on a different\ntopic that did not receive much attention in this meeting but is\nlikely to become increasingly important in the coming decade.\n\n\nMerging NS binaries have been studied since 1970s, with major\nactivities in the relativity community since the early 1990s because\nof their importance as a source of gravitational waves (GWs)\n(e.g. Cutler et al.~1993).  They are of great current interest for two\nreasons: (i) Merging NS\/NS or NS\/Black-Hole (BH) binaries have been\nidentified as the leading candidate for the central engine of short\nGRBs\n(Berger 2011).  They are also expected to produce optical and radio\ntransients that may be detected by wide-field, fast imaging telescopes\nthat are coming online (e.g. PTF, LSST) in the next few years\n(Nissanke et al.~2012).  (ii) After several decades of developmemt and\npromise,\ngravitational wave astronomy in the Hz-kHz band may finally take off\nin the next decade. The initial LIGO reached the design sensitivity\n($h_c\\simeq 10^{-21}$) in 2006, and the enhanced LIGO (with a factor of 2\nreduction in $h_c$) is taking or analysing data. The Advanced LIGO and\nVIRGO are expected to begin observations in 2015 and reach full\nsensitivity (a factor of 10 reduction in $h_c$) in 2018-19 --- at which\ntime the detection of GWs from many merging NS binaries seems\nguaranteed.\n\nThe last three minutes of a NS binary's life may be divided into two\nphases: the inspiral phase, producing quasi-periodic GWs, and the\ncoalescence phase, where physical collision results in ``messy'' GWs.\nThe recent years, 3D simulations of the final merger in full general\nrelativity (GR) have become possible (see Shibata \\& Taniguchi 2006;\nFoucart et at.~2012; Sekiguchi et al.~2012). It has long been\nrecognized that the final merger waveforms can provide a useful probe of NS\nequation of state (EOS; e.g., Cutler et al.~1993; Bildsten \\& Cutler\n1992; Lai \\& Wiseman 1996; Wiggins \\& Lai 2000). The idea is \nsimple: By measuring the ``cut-off'' frequency $\\propto\n(GM_t\/R^3)^{1\/2}$ associated with binary contact or tidal disruption,\ncombined with the precise mass measurement from the inspiral waveform,\none can obtain the NS radius (see Bauswein et al.~2012 for recent \nsimulations which put such a idea into concrete footing; see\nalso Sekiguchi et al.~2012; Faber \\& Rasio 2012 for reviews). \n\nIn the following sections I will focus on the pre-merger phase.\n\n\n\\section{Hydrodynamics of merging NS binaries}\n\nPrior to binary merger, tidal effects may affect the orbital decay and\nthe GWs. There are two types of tides: {\\it equilibrium tides} and\n{\\it dynamical tides}. The equilibrium tides correspond to global deformation \nof the NS, which leads to the interaction potential between the two stars\n(with the NS mass $M$ and radius $R$, the companion mass $M'$ -- treated as a \npoint mass, and the binary separation $a$) \n\\begin{equation}\nV(r)=-{MM'\/a}-{\\cal O}\\left(k_2{{M'}^2R^5\/a^6}\\right),\n\\end{equation}\nwhere $k_2$ is the so-called\nLove number. This would lead to a correction to the number of GW cycles,\n$dN=dN^{(0)}\\left[1-{\\cal O}(k_2M'R^5\/Ma^5)\\right]$.  For a Newtonian\npolytropic NS model, simple analytic expressions can be found in Lai et\nal.~(1994). Recent semi-analytic GR calculations of such equilibrium\ntidal effects (including the more precise determination of the Love number)\ncan be found in numerous papers (e.g., Flanagan \\& Hinderer 2008;\nBinnington \\& Poisson 2009; Damour \\& Nagar 2009; Penner et al.~2012, \nFerrari et al.~2012). Obviously\nthis effect is only important at small orbital separations (just prior\nto merger) -- there is some prospect of measuring this, thereby\nconstraining the EOS, but it may be challenging (Damour et al.~2012).\nAt small orbital separations, the quadrupole approximation is not valid; \ntherefore one\nmust use the numerically computed GR quasi-equilibrium binary sequences\nto characterize the tidal effect -- such sequences have been\nconstructed by several groups since the 1990s (e.g., Baumgarte et al.~1998;\nUryu et al.~2009) or use fully numerical simulations.\n\nAnother aspect of the equilibrium tide concerns tidal dissipation,\nwhich leads to a lag of the tidal bulge with respect to the binary\naxis.  It was shown already in the 1990s (Bildsten \\& Cutler 1992;\nKochanek 1992) that because of the rapid GW-driven orbital decay,\nviscous tidal lag cannot synchronize the NS spin. Thus the NS will\nbe close to irrotational (approximated as a Riemann-S ellipsoid;\nLai et al.~1994; Wiggins \\& Lai 2000; Ferrari et al.~2012). \nNear the final phase\nof the inspiral, the rapid orbital decay gives rise to a finite lag\nangle (even with zero viscosity), but this cannot synchronize the NS\n(Lai \\& Shapiro 1995; Dall'Osso \\& Rossi 2012).\n\n\nThe situation is more complicated for {\\bf dynamical tides}, which\nmanifest as resonant excitations of internal oscillations of the NS: As\ntwo NSs spiral in, the orbit can momentarily come into resonance with\nthe normal modes (frequency $\\omega_\\alpha$) of the NS:\n\\begin{equation}\n\\omega_\\alpha=m\\Omega_{\\rm orb},\\qquad m=2,3,\\cdots\n\\end{equation}\nBy drawing energy from the orbital\nmotion and resonantly exciting the modes, the rate of inspiral is\nmodified, giving rise to a phase shift in the gravitational\nwaveform. This problem was studied by Reisenegger \\& Goldreich (1994),\nLai (1994) and Shibata (1994) in the case of non-rotating NSs, where\nthe only modes that can be resonantly excited are g-modes (with\ntypical mode frequencies$\\lo 100$~Hz). It was found that the effect is\nsmall for typical NS parameters (mass $M=1.4M_\\odot$ and radius\n$R=10$~km) because the coupling between the g-mode and the tidal\npotential is weak. Ho \\& Lai (1999) studied the effect of NS rotation,\nand found that the g-mode resonance can be strongly enhanced even by a\nmodest rotation (e.g., the phase shift in the waveform $\\Delta\\Phi$\nreaches up to 0.1~radian for a spin frequency $\\nu_s\\lo 100$~Hz).\nThey also found that for a rapidly rotating NS ($\\nu_s\\go 500$~Hz),\nf-mode resonance becomes possible (since the inertial-frame f-mode\nfrequency can be significantly reduced by rotation) and produces a\nlarge phase shift. In addition, NS rotation gives rise to r-mode\nresonance whose effect is appreciable only for very rapid (near\nbreakup) rotations. Lai \\& Wu (2006) further studied resonant\nexcitations of other inertial modes (of which r-mode is a member) and\nfound similar effects.  Flanagan \\& Racine (2006) studied the\ngravitomagnetic resonant excitation of r-modes and and found that the\npost-Newtonian effect is more important than the Newtonian tidal\neffect (and that the phase shift reaches 0.1~radian for $\\nu_s\\sim\n100$~Hz). Tsang et al.~(2012) examined crustal modes and found that \nthe GW phase correction is small\/modest and suggested that tidal resonance\ncould shatter the NS crust, giving rise to the pre-cursor of short GRBs.\nTaken together, these studies suggest that for canonical NS\nparameters ($R \\simeq 10$~km, $\\nu_s\\lo 100$~Hz), tidal resonances\nhave a small effect on the gravitational waveform during binary\ninspiral.  However, it is important to keep in mind that the effect is \na strong function of $R$ (e.g., $\\Delta\\Phi\\propto R^4$ for g-modes\nand $\\propto R^{3.5}$ for inertial modes). A larger radius ($R\\simeq \n15$~km), appropriate for stiff EOS, \nwould make the effect important. In the case of g-modes, the magnitude\nof the effect depends on the symmetry energy of nuclear matter \nand could be non-negligible (W. Newton \\& D. Lai 2013, in prep).\n\n\n\\section{Electrodynamics of merging NS binaries}\n\nFor magnetic NSs, magnetic interactions may play a role.  If the\nbinary is embedded in a vacuum, then the interaction potential is\n$V(r)=-MM'\/a-{\\cal O}(\\mu\\mu'\/a^3)$ (where $\\mu,\\mu'$ are the magnetic\ndipole moments of the two stars). It is easy to check that such\nmagnetic interaction would lead to negligible effect on the GWs unless\nboth NSs have superstrong fields ($\\gg 10^{15}$~G) -- this is unlikely\n(e.g., the double pulsars PSR J0737-3039 has $10^{10}$~G for pulsar A\nand $2\\times 10^{13}$ for pulsar B). \n\n\\begin{figure}[b]\n\\vspace*{-0.8 cm}                                                             \n\\begin{center}\n \\includegraphics[width=3.4in]{f1.eps}\n\\vspace*{-1.0 cm}                                                             \n \\caption{DC circuit (unipolar induction) \nmodel of magnetic interactions in binary \nsystems {\\it a la} Goldreich \\& Lynden-Bell (1969).}\n   \\label{fig1}\n\\end{center}\n\\end{figure}\n\n\nOf course, as in the case of isolated pulsars,\nthe circumbinary environment cannot be vacuum.\nThe following discussion is based on Lai (2012).\nConsider a binary system consisting of a magnetic NS (the\n``primary'', with mass $M$, radius $R$, spin $\\Omega_s$, and magnetic\ndipole moment $\\mu$) and a non-magnetic companion (mass $M_c$, radius\n$R_c$). The orbital angular\nfrequency is $\\Omega$.  The magnetic field strength at the surface of\nthe primary is $B_\\star=\\mu\/R^3$.  The whole binary system is embedded\nin a tenuous plasma (magnetosphere).  We assume for simplicity that\n${\\bf\\Omega}$, ${\\bf\\Omega_s}$ and ${\\mbox{\\boldmath $\\mu$}}$ are all aligned.\nThe motion of the non-magnetic companion relative to the magnetic field of the\nprimary produces an EMF ${\\cal E} \\simeq 2R_c |E|$, where ${\\bf E}=                         \n{\\bf v}_{\\rm rel}\\times {\\bf B}\/c$, with\n${\\bf v}_{\\rm rel}=(\\Omega-\\Omega_s)a\\,{\\hat{\\mbox{\\boldmath $\\phi$}}}$ and\n${\\bf B}=(-\\mu\/a^3){\\hat{\\bf z}}$.\nThis gives\n${\\cal E}\\simeq {(2\\mu R_c\/ca^2)}\\Delta\\Omega$,\nwhere $\\Delta\\Omega=\\Omega-\\Omega_s$\\footnote{If the magnetic field does\nhave time to fully penetrate the companion star due to the rapid orbital decay,\nthen the EMF will be reduced by a factor of order the ratio of the skin depth\nand the stellar radius. I thank Anatoly Spitkovsky for\npointing this out to me at the IAU conference.}. \nThe EMF drives a current along the magnetic\nfield lines in the magnetosphere, connecting the primary and the companion\nthrough two flux tubes. The current in the circuit is given by\n${\\cal I}={{\\cal E}\/({\\cal R}_{\\rm tot})}$,\nwhere the total resistance of the circuit is\n${\\cal R}_{\\rm tot}={\\cal R}+{\\cal R}_c+2{\\cal R}_{\\rm mag}$,\nwith ${\\cal R},\\,{\\cal R}_c,\\,{\\cal R}_{\\rm mag}$ the resistances of the magnetic\nstar, the companion and the magnetosphere, respectively.\nThese resistances depend on the properties of the binary components and the\nmagnetosphere, and can vary widely for different types of systems.\nThe energy dissipation rate of the system is then\n$\\dot E_{\\rm diss}=2{\\cal I}^2R_{\\rm tot}={2{\\cal E}^2\/{\\cal R}_{\\rm tot}}$,\nwhere the factor of 2 accounts for both the upper and lower sides of the circuit.\n\nThe total magnetic force (in the azimuthal direction) on the companion is\n$F_\\phi\\simeq (2R_c) (2{\\cal I}B_z\/c)$, with $B_z=-\\mu\/a^3$.\nThus the torque acting on the binary's orbital angular momentum is\n$T=\\dot J_{\\rm orb}\\simeq ({4\/c})a\\,R_c{\\cal I}B_z\\simeq\n-({4\\mu R_c\/ca^2})({{\\cal E}\/{\\cal R}_{\\rm tot}})$.\nThe torque on the primary's spin is $I\\dot\\Omega_s=-T$ (where $I$ is\nthe moment of inertia).\nThe orbital energy loss rate associated with $T$ is then\n$\\dot E_{\\rm orb}=T\\Omega$.\n\nThe equations above show that the binary\ninteraction torque and energy dissipation associated with the DC\ncircuit increase with decreasing total resistance ${\\cal R}_{\\rm tot}$.\nIs there a problem for the DC model when ${\\cal R}_{\\rm tot}$ is too small?\nThe answer is yes.\nThe current in the circuit produces a toroidal magnetic field, which\nhas the same magnitude but opposite direction above and below\nthe equatorial plane. The toroidal field just above the\ncompanion star (in the upper flux tube) is $B_{\\phi+}\\simeq -(2\\pi\/c){\\cal J}_r$,\nwhere ${\\cal J}_r\\simeq -4{\\cal I}\/(\\pi R_c)$ is the (height-integrated) surface current.\nThus the azimuthal twist of the flux tube is\n$\\zeta_\\phi=-{B_{\\phi+}\/B_z}=\n={16 v_{\\rm rel}\/(c^2{\\cal R}_{\\rm tot})}$,\nwhere $v_{\\rm rel}=a\\Delta\\Omega=a(\\Omega-\\Omega_s)$.\nClearly, when ${\\cal R}_{\\rm tot}$ is less than $16v_{\\rm rel}\/c^2$,\nthe flux tube will be highly twisted.\n\nGoldreich \\& Lynden-Bell (1969) speculated that the DC circuit would break\ndown when the twist is too large. (For the Jupiter-Io system\nparameters,  the twist $|\\zeta_\\phi|\\ll 1$.) \nNumerous works have since confirmed that this is indeed the case.\nTheoretical studies and numerical simulations, usually carried out\nin the contexts of solar flares\nand accretion disks, have shown that as a flux tube is twisted beyond\n$\\zeta_\\phi\\go 1$, the magnetic pressure associated with $B_\\phi$\nmakes the flux tube expand outward and the magnetic fields open up,\nallowing the system to reach a lower energy state (e.g., Aly 1985; \nLynden-Bell \\& Boily 1994;\nLovelace et al.~1995; Uzdensky et al.~2002).\nThus, a DC circuit with $\\zeta_\\phi\\go 1$ cannot be realized: The\nflux tube will break up, disconnecting the linkage between the two\nbinary components.\nA binary system with ${\\cal R}_{\\rm tot}\\lo 16v_{\\rm rel}\/c^2$\ncannot establish a steady-state DC circuit.\nThe electrodynamics is likely rather complex, only\na quasi-cyclic circuit may be possible (Lai 2012; see Aly \\& Kuijpers 1990):\n(a) The magnetic field from the primary penetrates\npart of the companion, establishing magnetic linkage between the two\nstars; (b) The linked fields are twisted by differential rotation, generating\ntoroidal field from the linked poloidal field; (c) As the toroidal magnetic field\nbecomes comparable to the poloidal field, the fields inflate and\nthe flux tube breaks, disrupting the magnetic linkage;\n(d) Reconnection between the inflated field lines relaxes the shear and restore\nthe linkage. The whole cycle repeats.\n\nIn general, we can use the dimensionless azimuthal twist $\\zeta_\\phi$\nto parameterize the magnetic torque and energy dissipation rate:\n\\begin{equation}\nT= {1\\over 2}aR_c^2B_zB_{\\phi+}\n= -\\zeta_\\phi{\\mu^2 R_c^2\\over 2a^5},\\quad\n\\dot E_{\\rm diss} = -T \\Delta\\Omega\n=\\zeta_\\phi\\Delta\\Omega {\\mu^2 R_c^2\\over 2a^5}.\n\\label{eq:emax}\\end{equation}\nThe maximum torque and dissipation are obtained by setting $\\zeta_\\phi\\sim 1$.\nIf the quasi-cyclic circuit discussed in the last paragraph is\nestablished, we would expect $\\zeta_\\phi$ to vary between $0$ and $\\sim 1$.\nNote that in the above, $T$ is negative since we\nare assuming $\\Omega>\\Omega_s$. \n\nGravitational wave (GW) emission drives the orbital decay of the \nNS binary, with timescale\n$t_{\\rm GW}={a\/|\\dot a|}=\n0.012\\left({a\/30\\,{\\rm km}}\\right)^{4}{\\rm s}$,\nwhere we have adopted $M=1.4M_\\odot$ and mass ratio\n$q=M_c\/M=1$.\nThe magnetic torque tends to spin up the primary when $\\Omega$$>$$\\Omega_s$.\nSpin-orbit synchronization is possible only if\nthe synchronization time $t_{\\rm syn}=I\\Omega\/|T|$ is less than\n$t_{\\rm GW}$ at some orbital radii.  With\n$I=\\kappa M R^2$, we find\n\\begin{equation}\nt_{\\rm syn}={2\\kappa(1+q)\\over\\zeta_\\phi\\Omega}\n\\left(\\!{GM^2\\over B_\\star^2R^4}\\!\\right)\n\\!\\left(\\!{a\\over R_c}\\!\\right)^2\n\\simeq 2\\times 10^7\\zeta_\\phi^{-1}\\!\\left(\\!{B_\\star\\over 10^{13}\\,{\\rm G}}\n\\!\\right)^{\\!-2}\\!\\left({a\\over 30\\,{\\rm km}}\\right)^{7\/2}{\\rm s},\n\\label{eq:tsyn}\\end{equation}\nwhere on the right we have adopted $\\kappa=0.4$ and $R=R_c=10$~km.\nClearly, even with magnetar-like field strength ($B_\\star\\sim 10^{15}$~G) and\nmaximum efficiency ($\\zeta_\\phi\\sim 1$), spin-orbit synchronization cannot be\nachieved by magnetic torque. For the same reason, the effect of magnetic torque on the\nnumber of GW cycles during binary inspiral is small. \n\nThe energy dissipation rate is\n\\begin{equation}\n\\dot E_{\\rm diss}=\\zeta_\\phi\\left(\\!{v_{\\rm rel}\\over c}\\!\\right){B_\\star^2R^6\nR_c^2c\\over 2a^6}\n= 7.4\\times 10^{44}\\zeta_\\phi\\left(\\!{B_\\star\\over\n10^{13}\\,{\\rm G}}\\!\\right)^{\\!2}\\!\\left(\\!{a\\over 30\\,{\\rm km}}\\!\\right)^{\\!\\!-13\/2}\n\\!{\\rm erg\\,s}^{-1},\n\\end{equation}\nwhere on the right we have used $v_{\\rm rel}\\simeq a\\Omega$ (for\n$\\Omega_s\\ll \\Omega$) and adopted canonical parameters\n($M=M_c=1.4M_\\odot$, $R=R_c=10$~km).\nThe total energy dissipation per $\\ln a$ is\n\\begin{equation}\n{dE_{\\rm diss}\\over d\\ln a}=\\dot E_{\\rm diss}t_{\\rm GW}\n\\simeq 8.9\\times 10^{42}\\zeta_\\phi\\!\\left(\\!{B_\\star\\over\n10^{13}\\,{\\rm G}}\\!\\right)^{\\!2}\\!\\left({a\\over 30\\,{\\rm km}}\\right)^{\\!\\!-5\/2}\n\\!{\\rm erg}.\n\\end{equation}\nSome fraction of this dissipation will emerge as electromagnetic\nradiation counterpart of binary inspiral. It is possible that \nthis radiation is detectable at extragalactic distance. But this will depend \non the microphysics in the magnetosphere, including particle acceleration and \nradiation mechanism (e.g., Vietri 1996; Hansen \\& Lyutikov 2001).\n\nIf one assumes that the magnetosphere resistance is given by the\nimpedance of free space, ${\\cal R}_{\\rm mag}=4\\pi\/c$, then the corresponding twist\nis $\\zeta_\\phi=2v_{\\rm rel}\/(\\pi c)$, which satisfies our upper limit.\nWe then have\n\\begin{equation}\n\\dot E_{\\rm diss}=\\left(\\!{v_{\\rm rel}\\over c}\\!\\right)^2\\!\n{B_\\star^2R^6 R_c^2c\\over \\pi a^6}\n= 1.7\\times 10^{44}\\left(\\!{B_\\star\\over 10^{13}\\,{\\rm G}}\\!\\right)^{\\!2}\n\\!\\left({a\\over 30\\,{\\rm km}}\\right)^{\\!\\!-7}{\\rm erg\/s}.\n\\end{equation}\nThis is in agreement with the estimate of Lyutikov (2011).\n\nThe situation is similar for NS\/BH binaries.\nIn the membrane paradigm (Thorne et al.~1986), a BH\nof mass $M_H$ resembles a sphere of radius $R_c=R_H=2GM_H\/c^2$\n(neglecting BH spin)\nand impedance ${\\cal R}_H=4\\pi\/c$. Neglecting the resistances of the\nmagnetosphere and the NS, the azimuthal twist of the flux tube in the DC\ncircuit is\n$\\zeta_\\phi={4v_{\\rm rel}\/(\\pi c)}$,\nwhich satisfies our upper limit.\nThe energy dissipation rate is (cf. Lyutikov 2011; McWilliams \\& Levin 2011)\n\\begin{equation}\n\\dot E_{\\rm diss}=\\left(\\!{v_{\\rm rel}\\over c}\\!\\right)^2\\!\n{2B_\\star^2R^6 R_H^2c\\over \\pi a^6}\n\\simeq 5.7\\!\\times\\! 10^{42}\\!\\left(\\!{B_\\star\\over 10^{13}\\,{\\rm G}}\\!\\right)^{\\!2}\n\\!\\!\\left(\\!{M_H\\over 10M_\\odot}\\!\\right)^{\\!\\!\\!-4}\n\\!\\!\\!\\left(\\!{a\\over 3R_H}\\!\\right)^{\\!\\!-7}\\!\\!\\!{\\rm erg\\,s}^{-1},\n\\end{equation}\nwhere we have assumed $M_{BH}\/M\\gg 1$.\nAgain, it is uncertain whether this radiation is detectable for binaries\nat extragalactic distances.\n\n\\vspace{1ex}\n{\\bf Acknowledgements}:\nThis work has been supported in part by NSF grants AST-1008245 and\nAST-1211061, and NASA grants NNX12AF85G and NNX10AP19G.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStrongly interacting systems in the high energy (or small $x$) limit\nare very nonlinear systems in spite of the smallness of\nthe coupling constant $\\alpha_{\\mathrm{s}}$. This is due to the large phase\nspace available for semihard gluon radiation that increases \nthe occupation numbers of gluonic modes in the hadron or nucleus \nwavefunction. Thus high energy scattering has to be understood in\nterms of gluon recombination and saturation that enforce the unitarity\nrequirements of the $S$-matrix. This happens naturally in\nthe Color Glass Condensate (CGC) effective theory of the\nhigh energy wavefunction. In the context of \ndeep inelastic scattering (DIS) the CGC \nleads to the dipole picture that naturally\ngives a consistent description of both inclusive\nand diffractive scattering.\nThe nonlinearities in high energy scattering are enhanced when the target\nis changed from a proton to a heavy nucleus. Thus there is a great\nopportunity to understand them by studying nuclear DIS in \nnew collider experiments, such as the EIC~\\cite{Deshpande:2005wd} \nor the LHeC~\\cite{Dainton:2006wd}. The particular process we discuss in this\npaper is diffractive DIS on nuclei.\n\n\n\nIn the Good-Walker~\\cite{Good:1960ba} picture of diffraction one needs to identify \nthe states\nthat diagonalize the imaginary part of the $T$-matrix. In the case of \nnuclear DIS at high energy these states are the ones with the virtual photon \nfluctuating into a dipole of a fixed size $r$ and with the nucleons in\nthe nucleus at fixed transverse positions $b_i$. In coherent diffraction\nthe nucleus is required to stay intact, which corresponds to performing\n the average over the nuclear wavefunction at the level of the\nscattering amplitude. \nAveraging the cross section, instead of the amplitude,  \nover the nucleon positions allows for the nucleus\nto break up,  giving the sum of incoherent and coherent cross sections,\ni.e. the quasielastic cross section. \nFor a  more formal discussion of this we point the reader e.g. to \nRef.~\\cite{Caldwell:2009ke}.\nThe $t$-dependence of the incoherent cross section therefore directly \nprobes the fluctuations and correlations in the nuclear wavefunction, which\nhave turned out to be a crucial ingredient in understanding the initial \nconditions in heavy ion collisions \\cite{Miller:2007ri,*Alver:2010gr}.\n\nThe average gluon density probed in the \ncoherent process is very smooth, meaning that the cross section is dominated\nby small values of momentum transfer to the nucleus, $t \\sim - 1\/R_A^2$.\nMeasuring such a small momentum transfer accurately is \nvery challenging.\n At momentum scales corresponding to the nucleon \nsize $t \\sim - 1\/R_p^2$ the diffractive cross section is almost purely incoherent.\nThe larger momentum transfer should\nalso be easier to reconstruct experimentally even without measuring\nthe transverse momentum of the \nnuclear remnants, by accurately reconstructing the outgoing electron\nand $J\/\\Psi$ momenta and using momentum conservation.\nBy taking these processes into account in the detector design\none should be capable of measuring diffractive events at a higher accuracy\nthan was done at HERA.\nIn the dilute limit (for small dipoles) there is no multiple scattering, and the\nincoherent cross section is given by $A$ times the corresponding one for\nprotons.  The\ndeviation of the $t$-slope from the proton measures the transverse size of the \nfluctuating areas in the nucleus.  \n\n\nIn the black disc limit \nthe nucleus is smooth not only on average, but event-by-event, \nleading to a strong suppression of the incoherent cross section. Incoherent\ndiffraction gets contributions from the edge of the nucleus, making the\ncross section asymptotically behave as $\\sim A^{1\/3}$ in contrast to\n$\\sim A$ in the dilute limit.\nThe suppression in the\nnormalization relative to the proton is a measure of the approach to the unitarity\nlimit in the dipole cross section. It is a clear signal of how individual nucleons\nhave lost their identity in the sense that they cannot be resolved by the virtual photon.\nIt is precisely this suppression that we are proposing to \nuse to quantitatively access saturation effects in the nuclear  wavefunction.\nThe purpose of this paper  is to provide a realistic estimate of \nthe nuclear suppression in diffractive\ncross sections in a  regime that could be measured \nin future nuclear DIS experiments. \n\n\nNuclear DIS data from fixed target experiments, in particular \nE665~\\cite{Adams:1994bw} and NMC~\\cite{Arneodo:1994qb,*Arneodo:1994id} \nhave already been much discussed in the literature as\ndemonstrations of \\emph{color transparency} (see e.g. \nRefs.~\\cite{Frankfurt:1991nx,Frankfurt:1993it,Brodsky:1994kf,Kopeliovich:2001xj,Frankfurt:2005mc}). The form of nuclear modification to the incoherent\ndiffraction in terms of the dipole cross section \nthat we have rederived is not new (see \ne.g.~\\cite{Kopeliovich:1991pu,Kopeliovich:2001xj}).\nSo far, however, less  attention\nhas been paid to inelastic diffraction in future DIS experiments.\nThe production \ncross sections have not been calculated using the same\nCGC inspired cross sections that have been used\nsuccessfully to confront HERA data, as we intend to do here.\nIn this work we concentrate on the $J\/\\Psi$ \nbecause its small size means that the interaction\nof the dipole with the target is calculable in weak coupling even at\nsmall $Q^2$.\n\nThe importance of diffraction in understanding gluon saturation has been discussed\nand our basic setup motivated in Ref.~\\cite{Kowalski:2007rw}.\nNuclear modifications \nto the diffractive structure functions, integrated over the momentum transfer\n$t$, were computed in Ref.~\\cite{Kowalski:2008sa}. Vector meson production \nat future DIS experiments was recently \ndiscussed from a more experimental point of view in\nRef.~\\cite{Caldwell:2009ke}, and coherent production cross sections (integrated\nover $t$) calculated in Ref.~\\cite{Goncalves:2009za}.\nAn interesting discussion on coherent and incoherent diffraction\nand gluon saturation in the nucleus can be found in Ref.~\\cite{Tuchin:2008np}.\nIn this study we want to take a step beyond the discussion \nof inclusive diffraction in Refs.~\\cite{Kowalski:2007rw,Kowalski:2008sa}\n to understand the $t$ dependence in more detail.\n\n  \n\\section{Dipole cross sections}\n\\label{sec:dipxs}\n\nThere are many dipole cross section parametrizations available \nin the literature, and\nwe have taken for this study two representative samples. One is the \nIIM~\\cite{Iancu:2003ge} dipole cross section, which is a \nparametrization including the most important features \nof BK~\\cite{Balitsky:1995ub,*Kovchegov:1999yj,*Kovchegov:1999ua}\nevolution. The detailed expression for the\ndipole cross section can be found in Ref.~\\cite{Iancu:2003ge};\nwe use here the values of the parameters from the newer\nfit to HERA data including charm~\\cite{Soyez:2007kg}\nthat was also used to compute diffractive structure functions in\nRef.~\\cite{Marquet:2007nf}. We also want to compare\nour results to a parametrization with an eikonalized DGLAP-evolved gluon distribution.\nFor this purpose we will use an approximation of the IPsat dipole \ncross section~\\cite{Kowalski:2003hm,Kowalski:2006hc}. \n\n\nTo extend the dipole cross section from protons to nuclei\n we will take the independent\nscattering approximation that is usually used in Glauber theory \nand write the $S$-matrix as\n\\begin{equation}\\label{eq:sfact}\nS_A({\\mathbf{r}_T},{\\mathbf{b}_T},x) = \\prod_{i=1}^A S_p({\\mathbf{r}_T},{\\mathbf{b}_T}-{\\mathbf{b}_T}_i,x).\n\\end{equation}\nHere we conventionally parametrize the energy dependence of the scattering\namplitude with $x$, the Bjorken variable of the DIS event\\footnote{\nNote that strictly speaking the relation between $x$ and the \nenergy of the dipole-target scattering depends \non $Q^2$, not only $r$. Using $x$ here is justified\nin a high energy approximation\nwhere the energy of the dipole in the target rest frame\nis approximately the same as that of the virtual photon.}.\nThe variables ${\\mathbf{b}_T}_i$  in Eq.~\\nr{eq:sfact} \nare the nucleon coordinates that we will discuss in \nSec.~\\ref{sec:comp}.\nThis independent scattering assumption is\nnatural in IPsat-like parametrizations or the MV~\\cite{McLerran:1994ni} model, \nwhere, denoting $r = |{\\mathbf{r}_T}|,$ $S({\\mathbf{r}_T}) \\sim e^{-r^2 Q_\\mathrm{s}^2\/4}$ with a  saturation scale \n$Q_\\mathrm{s}^2$ proportional to the nuclear thickness $T_A(b)$.\nHigh energy evolution, however, introduces an anomalous dimension that leads,\nin the nuclear case, to what could be called leading twist shadowing.\nWith an anomalous dimension\n$S\\sim e^{-(Q_\\mathrm{s} r)^{2\\gamma}}$ with $\\gamma \\neq 1$, a proportionality\n$Q_\\mathrm{s}^2 \\sim T_A(b)$ is not equivalent to Eq.~\\nr{eq:sfact}. A solution to this \nproblem (see also the more detailed discussion in~\\cite{Kowalski:2008sa}) \nwould require a realistic impact parameter dependent solution to the \nBK equation which, we feel fair to say, is not yet available.\nWe point the reader e.g. to Ref.~\\cite{GolecBiernat:2003ym} for a \ndiscussion of the  difficulties. These are related to the long distance \nCoulomb tails that, physically, are regulated at the confinement length\nscale that is not enforced in a first principles weak coupling calculation.\nThe effect of BK evolution is important for the CGC description\nof the forward suppression of particle production\nin dAu-collisions at RHIC (for a review see~\\cite{Jalilian-Marian:2005jf}).\nIn our case the difficulty is greater since we are interested\nnot only in the relatively smooth average gluon density, but\nits variations at smaller length scales of the order of the\nproton radius.\nWe thus leave the modifications of Eq.~\\nr{eq:sfact} due to\nthe effects of evolution to a future study.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{amplitude.pdf}\n\\caption{\nThe $r$-dependence of the different proton dipole cross  sections used,\nat  $x=0.0001$ and $b=0$.\nAs discussed in Sec.~\\ref{sec:res}, the ``IPnonsat''-curve is\nEq.~\\nr{eq:BEKWfact} linearized in $r^2F(x,r)$.\n} \\label{fig:sigmap}\n\\end{figure}\n\nThe IIM parametrization assumes, either explicitly or implicitly,\na factorizable ${\\mathbf{b}_T}$ dependence \n\\begin{eqnarray}\\label{eq:factbt}\n{\\frac{\\ud \\sigma^\\textrm{p}_\\textrm{dip}}{\\ud^2 \\bt}}({\\mathbf{b}_T},{\\mathbf{r}_T},x) &=&  2 \\left( 1 - S_p({\\mathbf{r}_T},{\\mathbf{b}_T},x)\\right)\n\\\\ \\nonumber\n&=& 2 \\,T_p({\\mathbf{b}_T}) {\\mathcal{N}}(r,x),\n\\end{eqnarray}\nWe take, following Ref.~\\cite{Marquet:2007nf}, a Gaussian profile\n$T_p({\\mathbf{b}_T}) = \\exp\\left(-b^2\/2 B_p\\right)$ with \n$B_p=5.59\\ \\textrm{GeV}^{-2}$ (see Sec.~\\ref{sec:res} for a discussion of \nthis largish numerical value). \n\nIn the IPsat model the impact parameter dependence is\nincluded in the saturation scale as\n\\begin{equation}\\label{eq:unfactbt}\n{\\frac{\\ud \\sigma^\\textrm{p}_\\textrm{dip}}{\\ud^2 \\bt}}({\\mathbf{b}_T},{\\mathbf{r}_T},x)\n = 2\\,\\left[ 1 - \\exp\\left(- r^2  F(x,r) T_p({\\mathbf{b}_T})\\right) \n\\right].\n\\end{equation}\nHere $T_p({\\mathbf{b}_T}) = \\exp\\left(-b^2\/2 B_p\\right)$ \nis the impact parameter profile function in the proton \nwith $B_p=4.0\\ \\textrm{GeV}^2$ and $F$ is proportional to the \nDGLAP evolved gluon distribution~\\cite{Bartels:2002cj}\n\\begin{equation}\nF(x,r^2) = \n\\frac{1}{2 \\pi B_p}\n\\frac{ \\pi^2 }{2 {N_\\mathrm{c}}} \\alpha_{\\mathrm{s}} \\left(\\mu_0^2 + \\frac{C}{r^2} \\right) \nx g\\left(x,\\mu_0^2 + \\frac{C}{r^2} \\right),  \n\\label{eq:BEKW_F}\n\\end{equation}\nwith $C$ chosen as 4 and $\\mu_0^2=1.17\\ \\textrm{GeV}^2$ resulting from the \nfit~\\cite{Kowalski:2006hc}. The proton dipole cross sections used are\nplotted in Fig.~\\ref{fig:sigmap}  for $x=0.0001$.\n\nWe would generally prefer the unfactorized $b$-dependence\nof Eq.~\\nr{eq:unfactbt} to the factorized one in Eq.~\\nr{eq:factbt}\nbecause it allows for the correct unitarity\nlimit of the scattering amplitude at all impact parameters \n(see the discussion in Ref.~\\cite{Kowalski:2008sa}). \nHowever, there seems to be no clear difference between the two in\nterms of the quality of the description of HERA data, and for the sake \nof computational simplicity we will in this work limit ourselves to\nthe factorized dependence and approximate the IPsat dipole cross section\nby\n\\begin{equation}\\label{eq:BEKWfact}\n{\\frac{\\ud \\sigma^\\textrm{p}_\\textrm{dip}}{\\ud^2 \\bt}}({\\mathbf{b}_T},{\\mathbf{r}_T},x)\n \\approx  2 T_p({\\mathbf{b}_T}) \\,\\left[ 1 - \\exp\\left(- r^2  F(x,r)\\right)\n\\right]\n\\end{equation}\nusing the same $F(x,r)$ defined in Eq.~\\nr{eq:BEKW_F}. This approximation\nbrings the IPsat parametrization to the form Eq.~\\nr{eq:factbt}\nwith  ${\\mathcal{N}}(r,x)=\\left[ 1 - \\exp\\left(- r^2  F(x,r)\\right)\\right]$;\nin fact this is the form used already in Ref.~\\cite{Bartels:2002cj}; we however\nuse the gluon distribution from the IPsat fit~\\cite{Kowalski:2006hc} \nfor convenience. Improving this\ndescription goes hand in hand with giving up the approximation\nof independent scatterings off the nucleons, Eq.~\\nr{eq:sfact},\nand is left for future work. As we shall see in the following, these \napproximations enable us to write the cross section for incoherent\ndiffraction in a form which is much simpler to evaluate numerically \nthan one with a general $b$-dependence.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{totxs-boosted.pdf}\n\\caption{Comparison of the used dipole cross sections to HERA \ndata~\\cite{Chekanov:2004mw,*Aktas:2005xu} on diffractive vector meson production.\n} \\label{fig:hera}\n\\end{figure}\n\n\n\n\\section{Computing diffractive cross sections}\n\\label{sec:comp}\n\n\n\nThe cross section for quasielastic vector meson \nproduction in nuclear DIS is\n\\begin{equation} \\label{eq:xsec}\n\\frac{\\, \\mathrm{d} \\sigma^{\\gamma^* A \\to V A }}{\\, \\mathrm{d} t} \n= \\frac{R_g^2(1+\\beta^2)}{16\\pi} \\Aavg{|{\\mathcal{A}}({x_\\mathbb{P}},Q^2,{\\boldsymbol{\\Delta}_T})|^2}.\n\\end{equation}\nwith $t=-{\\boldsymbol{\\Delta}_T}^2$. \nThe dipole cross section is evaluated at the \nenergy scale corresponding to the rapidity gap between the \nvector meson and the target ${x_\\mathbb{P}}$.\nTo translate this into the photon-target center of mass energy $W$  \nthat is often used to present experimental results note that \n${x_\\mathbb{P}} = (M_{J\/\\Psi}^2 + Q^2)\/(W^2 + Q^2)$.\nThe factor $1+\\beta^2$ accounts for the\nreal part of the scattering amplitude and the factor $R_g^2$ corrects\nfor the skewedness effect, i.e. that the gluons in the target are probed at \nslightly different $x$~\\cite{Shuvaev:1999ce,*Martin:1999wb}. \nFor these corrections\nwe follow the prescription of Ref.~\\cite{Watt:2007nr}, taking them \nas\n\\begin{eqnarray}\n\\beta &=& \\tan \\frac{\\pi \\lambda}{2}\n\\\\\nR_g &=& \\frac{2^{2 \\lambda+3}}{\\sqrt{\\pi}}\\frac{\\Gamma(\\lambda + 5\/2)}{\\lambda+4} \n\\quad \\textrm{ with}\n\\\\\n\\lambda &=& \\frac{\\partial \\ln {\\mathcal{A}}_{T,L}^{\\gamma^*p\\to J\/\\Psi p}}{\\partial \\ln 1\/{x_\\mathbb{P}}}.\n\\end{eqnarray}\nThese corrections depend, in general, on $t$, which we take into account in our calculation. \nFor the full IPsat model $\\lambda$ changes by about 5\\% between $t=0$ and \n$-t=0.5\\ \\textrm{GeV}^2$. For the factorized impact parameter dependence in \nEqs.~\\nr{eq:factbt} and~\\nr{eq:BEKWfact} $\\lambda$ is \nindependent of $t$.\nWe calculate the correction terms from the energy dependence of the nucleon\nscattering amplitudes and use the same values for the nucleus at the \nsame $Q^2,{x_\\mathbb{P}}$. Since the difference in $\\lambda$ extracted from the nucleus \nand the nucleon cross sections is small (compared to the value of $\\lambda$) and \n$R_g$ and $\\beta$ are in themselves corrections to the cross section, this \napproximation is justified. In addition this approximation has the advantage that \nthese corrections cancel  on the nucleus\/nucleon cross section ratio.\nThe real part and skewedness corrections, especially $R_g$ are, however, a significant \nfactor in the absolute normalization of the cross section and are \nnecessary for the agreement with HERA data.\n\nThe imaginary part of the scattering amplitude is the Fourier-transform of the\ndipole cross section from ${\\mathbf{b}_T}$ to ${\\boldsymbol{\\Delta}_T}$ contracted with the \noverlap between the vector meson and virtual photon wave functions:\n\\begin{multline}\\label{eq:ampli}\n{\\mathcal{A}}({x_\\mathbb{P}},Q^2,{\\boldsymbol{\\Delta}_T}) \n= \\int \\, \\mathrm{d}^2 {\\mathbf{r}_T} \\int \\frac{\\, \\mathrm{d} z}{4\\pi} \\int \\, \\mathrm{d}^2 {\\mathbf{b}_T} \n\\\\\n\\times [\\Psi_V^* \\Psi](r,Q^2,z)\ne^{-i {\\mathbf{b}_T} \\cdot  {\\boldsymbol{\\Delta}_T}}  \n{\\frac{\\ud \\sigma_\\textrm{dip}}{\\ud^2 \\bt}}({\\mathbf{b}_T},{\\mathbf{r}_T},{x_\\mathbb{P}}),\n\\end{multline}\nwhere we have followed the normalization convention of~\\cite{Kowalski:2006hc}.\nFor the virtual photon--vector meson wavefunction overlap we\nuse the ``boosted Gaussian'' parametrization from Ref.~\\cite{Kowalski:2006hc}.\nWe have also tested the ``gaus-LC'' wavefunction also used in \nRef.~\\cite{Kowalski:2006hc}. Although the ``boosted Gaussian'' seems preferred\nby HERA data, also the ``gaus-LC'' parametrization is compatible with \nthe data within the experimental errors. The cross sections for the\nproton differ by factors of the order of 10\\%.\nThe interaction of the gluon target with the dipole can in general depend also on \n${\\boldsymbol{\\Delta}_T}$, which introduces terms that couple ${\\mathbf{r}_T},$ ${\\boldsymbol{\\Delta}_T}$  and $z$ in \nEq.~\\nr{eq:ampli}. For the $J\/\\Psi$ and the range in $t$ considered in this paper\n${\\boldsymbol{\\Delta}_T}$ is sufficiently small compared to the relevant values of $1\/r$ that \nwe can neglect this coupling, which simplifies the structure  considerably.\nLighter vector mesons would require a more general treatment.\n\nThe average over the positions of the nucleon in the nucleus is denoted here by\n\\begin{equation} \\label{eq:aavg}\n\\Aavg{\\mathcal{O}(\\{ {\\mathbf{b}_T}_i \\})} \n\\equiv \\int \\prod_{i=1}^{A}\\left[ \\, \\mathrm{d}^2 {\\mathbf{b}_T}_i T_A({\\mathbf{b}_T}_i) \\right] \n\\mathcal{O}(\\{ {\\mathbf{b}_T}_i \\}).\n\\end{equation}\nHere $T_A$ is the Woods-Saxon distribution with nuclear radius \n$R_A = (1.12 A^{1\/3}-0.86 A^{-1\/3})\\ \\textrm{fm} $ and surface thickness \n$d=0.54\\ \\textrm{fm}$.\nThis expectation value is equivalent to the average\nover nucleon configurations in a Monte Carlo Glauber \ncalculation.\nWe are assuming that the positions ${\\mathbf{b}_T}_i$ are independent, i.e. \nneglecting nuclear correlations that would be a subject of\ninterest in their own right (see e.g.~\\cite{Alvioli:2009ab}).\nThe coherent cross section is obtained by averaging the amplitude\nbefore squaring it, $|\\Aavg{{\\mathcal{A}}}|^2$, and the incoherent one is\nthe variance $\\Aavg{|{\\mathcal{A}}|^2} -  |\\Aavg{{\\mathcal{A}}}|^2$ that measures the fluctuations\nof the gluon density inside the nucleus. Because\n$\\Aavg{{\\mathcal{A}}}$ is a very smooth function of ${\\mathbf{b}_T}$, its Fourier transform \nvanishes rapidly for $\\Delta \\gtrsim 1\/R_A$. Therefore at large $\\Delta$\nthe quasielastic cross section \\nr{eq:xsec} is almost purely incoherent.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{coherent_q0.pdf}\n\\caption{The quasielastic and coherent diffractive $J\/\\Psi$ cross sections in gold\nnuclei  at  $Q^2= 0$ and  ${x_\\mathbb{P}} = 0.001$.\nShown are \nthe IPsat and IIM parametrizations. We also show the \nresult for the linearized  ``IPnonsat'' version \n(used e.g. in Ref.~\\cite{Caldwell:2009ke}) where the incoherent\ncross section is explicitly $A$ times that of the proton. \nOur approximation \\nr{eq:amplisq} \nis not valid for small  $|t|$; the corresponding part of the distribution\nhas been left out.\n} \\label{fig:dsigmavst}\n\\end{figure}\n\n\nThe cross section for quasielastic vector meson production is now expressed\nin terms of the dipole scattering amplitude as\n\\begin{multline} \\label{eq:incxsec}\n\\frac{\\, \\mathrm{d} \\sigma^{\\gamma^* A \\to V A^* }}{\\, \\mathrm{d} t} \n= \n\\frac{R_g^2(1+\\beta^2)}{16\\pi}  \n\\int \n\\frac{\\, \\mathrm{d} z}{4 \\pi}  \\frac{\\, \\mathrm{d} z'}{4 \\pi}\n\\, \\mathrm{d}^2 {\\mathbf{r}_T} \\, \\mathrm{d}^2 {\\mathbf{r}_T}'\n\\\\ \\times\n\\left[ \\Psi^*_V \\Psi \\right] (r,z,Q)\n\\, \\left[ \\Psi^*_V  \\Psi \\right](r',z',Q)\n\\\\ \\times\n\\Aavg{ \\left| \\mathcal{A}_{q\\bar{q}}\\right|^2({x_\\mathbb{P}},r,r',{\\boldsymbol{\\Delta}_T}) } \\, .\n\\end{multline}\nWe now average the square of the dipole scattering amplitude over the \nnucleon coordinates, using the assumptions of\nEqs.~\\nr{eq:sfact} and~\\nr{eq:factbt} and taking the large $A$ limit.\nWe are additionally assuming that $T_A$ is a smooth function on the \ndiscance scale defined by $B_p$.\nAveraging the square of the amplitude gives the total quasielastic \ncontribution, but we only keep the terms \nthat contribute at large $|t| \\gg 1\/R_A^2$, which leaves us \nwith the expression \n\\begin{multline}\\label{eq:amplisq}\n\\left| \\mathcal{A}_{q\\bar{q}}\\right|^2({x_\\mathbb{P}},r,r',{\\boldsymbol{\\Delta}_T})  \n=\n16 \\pi B_p \\int \\, \\mathrm{d}^2 {\\mathbf{b}_T} \\sum_{n=1}^A\n\\frac{1}{n} \\binom{A}{n} \n\\\\ \\times \ne^{-B_p {\\boldsymbol{\\Delta}_T}^2\/n}\ne^{-2 \\pi B_p A T_A(b)\n\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \n\\\\ \\times\n\\left( \\frac{\\pi B_p {\\mathcal{N}}(r){\\mathcal{N}}(r') T_A(b) }\n  {1 - 2 \\pi B_p T_A(b)\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \\right)^n\n.\n\\end{multline}\nNote that  Eqs.~\\nr{eq:sfact} and~\\nr{eq:factbt} have enabled us to\nwrite the leading contributions as proportional to the\n(Gaussian) proton impact parameter profile, which can then be \nFourier-transformed analytically. Giving up either of these approximations\nwould force us to numerically Fourier-transform the ``lumpy'' \n$b$-dependence corresponding to a fixed configuration\nof the nucleon positions. This would make the numerical calculation \nmuch more demanding and is left for future work. \n\nThe terms with $n \\geq 2$ correspond to scattering off a system of several \noverlapping nucleons simultaneously, leading to slower suppresion with $|t|$.\nIn practice we have verified numerically that they do not\ncontribute to our results at the values of  $t$ we are interested in \n(the $n=2$ contribution is typically $\\lesssim 2\\%$ of the $n=1$-one,\nonly reaching $5\\%$ at $-t\\gtrsim 0.5 \\ \\textrm{GeV}^2$ ) and \nwill neglect them in the following. This leaves us with the expression\n\\begin{multline}\\label{eq:amplisqn1}\n\\left| \\mathcal{A}_{q\\bar{q}}\\right|^2({x_\\mathbb{P}},r,r',{\\boldsymbol{\\Delta}_T})  \n=\n16 \\pi B_p A \\int \\, \\mathrm{d}^2 {\\mathbf{b}_T} \n\\\\ \\times \ne^{-B_p {\\boldsymbol{\\Delta}_T}^2}\ne^{-2 \\pi B_p A T_A(b)\n\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \n\\\\ \\times\n\\left( \\frac{\\pi B_p {\\mathcal{N}}(r){\\mathcal{N}}(r') T_A(b) }\n  {1 - 2 \\pi B_p T_A(b)\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \\right).\n\\end{multline}\nEquation \\nr{eq:amplisqn1} has a very clear interpretation. The \nsquared amplitude is proportional to $A$ times the squared amplitude\nfor scattering off a proton, corresponding to the dipole scattering \nindependently off the nucleons in a nucleus. This sum of independent\nscatterings is then multiplied by a nuclear attenuation factor\n\\begin{multline}\n\\frac{ e^{-2 \\pi B_p A T_A(b) \\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } } \n {1 - 2 \\pi B_p T_A(b)\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \n\\approx\n\\\\\n e^{-2 \\pi (A-1) B_p T_A(b) \\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } ,\n\\end{multline}\nwhich accounts for the requirement that the dipole must \\emph{not}\nscatter inelastically off the other $A-1$ nucleons in the target (otherwise the\ninteraction would not be diffractive). \nNote that the factor \n$4 \\pi B_p {\\mathcal{N}}(r,{x_\\mathbb{P}})={ \\sigma^\\textrm{p}_\\textrm{dip} }(r,{x_\\mathbb{P}})$ is the proton-dipole cross section for  a\ndipole of size $r$. Thus this attenuation corresponds to the\nprobability of a dipole with a cross section which is the average \nof dipoles with $r$ and $r'$ to pass though the nucleus.\nA similar expression \ncan be found e.g. in Ref.~\\cite{Kopeliovich:2001xj}.\n\n\n\nFor comparison, the coherent cross section in our approximation is given by\n\\begin{equation} \\label{eq:coh}\n\\frac{\\, \\mathrm{d} \\sigma^{\\gamma^* A \\to V A }}{\\, \\mathrm{d} t} \n=\\frac{R_g^2(1+\\beta^2)}{16\\pi} \\left| \\Aavg{{\\mathcal{A}}({x_\\mathbb{P}},Q^2,{\\boldsymbol{\\Delta}_T})} \\right|^2,\n\\end{equation}\nwhere in the large $A$ and smooth nucleus limit the amplitude is\n\\begin{multline}\\label{eq:cohampli}\n\\Aavg{{\\mathcal{A}}({x_\\mathbb{P}},Q^2,{\\boldsymbol{\\Delta}_T}) }\n= \\int \\frac{\\, \\mathrm{d} z}{4\\pi} \\, \\mathrm{d}^2 {\\mathbf{r}_T}  \\, \\mathrm{d}^2 {\\mathbf{b}_T} e^{-i {\\mathbf{b}_T} \\cdot  {\\boldsymbol{\\Delta}_T}}  \n\\\\\n \\times [\\Psi_V^*\\Psi](r,Q^2,z)\n\\,  2 \\left[ 1-\\exp\\left\\{ - 2 \\pi B_p A T_A(b) {\\mathcal{N}}(r,{x_\\mathbb{P}}) \\right\\} \\right].\n\\end{multline}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{plot_Q.pdf}\n\\caption{The ``nuclear transparency'' \nratio of cross sections vs. $Q^2$ for IPsat, IIM \nparametrizations at ${x_\\mathbb{P}}= 10^{-2}$ (the upper three curves, blue)\nand $10^{-4}$ (the lower 3 curves, black).\nFor comparison we also include\nwe also include the result if unitarization effects are included \nat the nucleus but not at the nucleon level in the IPsat-parametrization. \n(See text for discussion).\n}\\label{fig:ratiovsq}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{plot_Q_gaus-lc.pdf}\n\\caption{The ``nuclear transparency'' \nratio of cross sections vs. $Q^2$ using the ``Gaus-LC'' vector meson \nwavefunctions. The labeling is the same as in Fig.~\\ref{fig:ratiovsq}.\n}\\label{fig:ratiovsq_gauslc}\n\\end{figure}\n\n\n\n\n\\section{Results and discussion}\n\\label{sec:res}\n\nWe first test our dipole cross section parametrizations and vector meson wave\nfunctions by comparing them to HERA results~\\cite{Chekanov:2004mw,*Aktas:2005xu} \non diffractive $J\/\\Psi$ production\nthat is known to be well described by dipole model \nfits~\\cite{Kowalski:2006hc,Marquet:2007qa}. The comparison is quite satisfactory, as\ncan be seen from Fig.~\\ref{fig:hera}.  In addition to the factorized \napproximation (Eq.~\\nr{eq:BEKWfact}, ``factorized IPsat'' in the figure) that \nwe are using in the rest of this paper, also shown is the result with the original\nIPsat parametrization (Eq.~\\nr{eq:unfactbt}, denoted ``IPsat'' in the figure). The \nfactorized approximation differs from the original one slightly at small \n$Q^2$, but the difference is not significant for our purpose of establishing a \nreasonable baseline for computing nuclear effects.\n\nWe note here that the diffractive slope parameters\nin the parametrizations are different, $B_p=4.0\\ \\textrm{GeV}^{-2}$ for IPsat and \n$B_p= 5.59\\ \\textrm{GeV}^{-2}$\nfor IIM; since these are correlated with the other parameters in the fits leading to\nthe parameter values used we do not wish to alter them here.\nOur approximation of a factorized $b$-dependence with a constant $B$ \ndoes not allow us to describe the observed weak energy and $Q^2$ dependence of the\ndiffractive slope. \nThe larger $B$ that we use for IIM comes from the $\\sigma_0$ normalization\nin a fit to inclusive $F_2$ data, and also agrees with the observed slopes in \ninclusive diffraction at large $\\beta$ and small \n${x_\\mathbb{P}}$~\\cite{Chekanov:2004hy,*Aktas:2006hx} and exclusive $\\rho$ and $\\phi$\ndata~\\cite{Adloff:1999kg,*Chekanov:2005cqa}. The HERA $J\/\\Psi$-data,\non the other hand, has a smaller slope \n$\\sim 4\\ \\textrm{GeV}^{-2}$~\\cite{Chekanov:2004mw,*Aktas:2005xu}. \nThe $t$-slope in\nthe IPsat parametrization is mostly determined by this $J\/\\Psi$-measurement,\nand an agreement with the larger measured slopes for $\\rho$ and $\\phi$\nis obtained by taking into account the larger size of the wavefunctions\nof these lighter mesons.\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{plot_x.pdf}\n\\caption{The ``nuclear transparency''  ratio of cross sections vs. ${x_\\mathbb{P}}$ \nusing the IPsat and IIM  parametrizations for $Q^2=0$ and $Q^2=10\\ \\textrm{GeV}^2$.\n}\\label{fig:ratiovsx}\n\\end{figure}\n\n\n\nThe differential cross section $\\, \\mathrm{d} \\sigma^{\\gamma^* A \\to J\/\\Psi A}\/\\, \\mathrm{d} t$\n for $A=197$ (gold) as a function of $t$ is presented in Fig.~\\ref{fig:dsigmavst}. \nWe show the cross \nsection at ${x_\\mathbb{P}} = 0.001$ \nfor photoproduction.  As we performed\nthe nuclear wavefunction average leading to  Eq.~\\nr{eq:amplisq} in the \napproximation where $|t|$ is large, neglecting the coherent contribution,\nwe cannot extend our incoherent curves to small $|t|$. \nFor comparison we show the corresponding\n``IPnonsat'' result where the IPsat model is linearized in $r^2 F(x,r)$. \nThis curve corresponds to the calculation done in Ref.~\\cite{Caldwell:2009ke},\nincluding both the coherent and incoherent contributions, but without the effect\nof multiple scattering off different nucleons (i.e. the incoherent cross section is\nexplicitly $A$ times the one for a proton). \nAs one can see, the nuclear modification\ndue to multiple scattering (resulting mostly from the\nfactor $e^{-2 \\pi B_p  A T_A(b) \\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } $  in \nEq.~\\nr{eq:amplisq}) is very large. \n In the full black disk limit \nof ${\\mathcal{N}}(r)=1$ this factor becomes $\\approx e^{-0.5 A^{1\/3}}$ and completely\nsupresses the contribution from the center of a large nucleus, leaving only an area\nof $\\approx 2 \\pi d R_A \\sim A^{1\/3}$ contributing to the integral over ${\\mathbf{b}_T}$.\nThus the cross section in \nthe black disc limit behaves as $\\sim A^{1\/3}$ compared to $\\sim A$ in the dilute\nlimit, so a large suppression is to be expected.\n\n\nWe also show in Fig.~\\ref{fig:dsigmavst} the coherent cross sections \n(using Eq.~\\nr{eq:cohampli}). They are also suppressed compared to the linearized \nversion (IPnonsat), but not by as much as the incoherent one. In the linearized\nversion (as can be seen explicitly in Ref.~\\cite{Caldwell:2009ke} where this case\nwas considered) the ratio between the coherent cross section at $t=0$ and the incoherent\none extrapolated to $t=0$ is $A$. In the IPsat model we get\n$270$ ($250$) and in the IIM model $300$ ($270$) at $Q^2=0$ ($Q^2=10\\ \\textrm{GeV}^2$).\nThis would make it slightly easier to  measure the first \ndiffractive dip in the coherent cross section, since the background from the incoherent\nprocess is smaller by a factor of 2 than the linearized \nestimate~\\cite{Caldwell:2009ke}.\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{ratio_Q.pdf}\n\\caption{\nThe incoherent cross section integrated over the interval $0.1\\ \\textrm{GeV}^2< -t < 0.3 \\ \\textrm{GeV}^2$\ndivided by the coherent cross section integrated over $0 < -t < 0.1 \\ \\textrm{GeV}^2$\nas a function of $Q^2 + M_{J\/\\Psi}^2$.\n} \\label{fig:incohovercohQ}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{ratio_x.pdf}\n\\caption{\nThe incoherent cross section integrated over the interval $0.1\\ \\textrm{GeV}^2< -t < 0.3 \\ \\textrm{GeV}^2$\ndivided by the coherent cross section integrated over $0 < -t < 0.1 \\ \\textrm{GeV}^2$\nas a function of ${x_\\mathbb{P}}$.\n} \\label{fig:incohovercohx}\n\\end{figure}\n\n\nTo demonstrate the nuclear dependence further we show in Fig.~\\ref{fig:ratiovsq}\nthe ratio of the cross section in a gold  nucleus to that in a nucleon as a \nfunction of $Q^2$. Historically this ratio is known as the ``nuclear transparency''.\nIts smallness at low energy, similarly to coresponding quantities in \nhadron-nucleus scattering, is due to the interactions of the $J\/\\Psi$ as it propagates\nthrough the nucleus. The growth of the transparency towards $1$ for increasing\n$Q^2$~\\cite{Adams:1994bw,Arneodo:1994qb,*Arneodo:1994id} is \na demonstration of \\emph{color transparency} (see e.g. \nRef.~\\cite{Frankfurt:1991nx,Frankfurt:1993it,Brodsky:1994kf,Kopeliovich:2001xj,Frankfurt:2005mc,Miller:2010eh}),\nnamely that at large\n$Q^2$ the interacting components of the photon wavefunction are of smaller\nsize $r$ and interact weakly. In our framework color transparency is\nautomatically present in the fact that the dipole cross section approaches zero\nfor $r\\to0$. In Fig.~\\ref{fig:ratiovsq} we also show the result (labeled\n``IPsat, nonsatp'') of using a nonsaturated dipole-nucleon cross section\nin Eq.~\\nr{eq:amplisq}. This corresponds to including \nunitarity effects at the nucleus level but not \nfor a single nucleon. The observed nuclear suppression in this unphysical \nscenario is significantly larger than for the saturated full IPsat\nparametrization, showing the sensitivity of the nuclear transparency\nto saturation effects already at the proton level.\n\nThe IIM parametrization has a much larger nuclear \nsuppression in incoherent diffraction, with the nuclear transparency\nratio close that of an unsaturated dipole-proton cross section.\nTo put this in perspective recall that both parametrizations\ngave an equally good description of the elastic cross section\nmeasured at HERA (Fig.~\\ref{fig:hera}). Since IIM does this with a\nlarger $B_p$ than IPsat, we can infer that the typical ${\\mathcal{N}}$ is smaller, \nso that the elastic cross section  $\\sigma^\\textrm{el}\n\\sim B_p {\\mathcal{N}}^2$ is of the same order. \nThe nuclear transparency ratio, on the other hand, depends on the\ntotal dipole-nucleon cross section \n$\\sim B_p {\\mathcal{N}} \\sim \\sigma^\\textrm{el}\/{\\mathcal{N}}$ which is thus \nlarger for IIM. Thus we have a situation where both parametrizations\nhave been fitted to inclusive $F_2$ data\\footnote{Although we have here \napproximated the original IPsat parametrization by factorizing the $b$-dependence.},\nreproduce well the HERA $J\/\\Psi$ cross section, but differ in their\nresult for incoherent diffraction in nuclei. This stresses the importance of\nperforming a global analysis of both inclusive and diffractive data\nto constrain the dipole cross sections, and demonstrates\nthe utility of eventual incoherent diffractive measurements in such \nan analysis.\n\nFigure \\ref{fig:ratiovsq_gauslc} shows the same $Q^2$-dependence\nusing the ``gaus-LC'' wavefunction. It puts more\nweight on large dipole sizes, leading to a stronger nuclear suppression. \nThe cross section ratio typically decreases by $\\sim 0.04$ from \nthe ``boosted Gaussian'' wavefunction, but the\nrelative structure between the different dipole cross sections stays the same.\nThe difference between the cross sections themselves is larger,\nbut much of the it cancels in the ratio. The existing HERA data \nis not precise enough to fully discriminate between different models for\nthe vector meson wavefunction, a situation which should also improve\nwith planned new DIS experiments.\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{cohincoh.pdf}\n\\caption{\nThe ratio of the coherent (at $t=0$) and incoherent (at $t=-0.5\\ \\textrm{GeV}^2$,\nbut in our approximation this does not depend on $t$) \ncross sections to the corresponding ones for a proton; normalized with \n$A^2$ and $A$ respectively. Plotted as a function of $A$,\nfor ${x_\\mathbb{P}} = 0.001$ and $Q^2=10\\ \\textrm{GeV}^2$.\n} \\label{fig:cohincoh}\n\\end{figure}\n\n\n\nThe energy dependence of the nuclear suppression (again for $A=197$)\nis shown in Fig.~\\ref{fig:ratiovsx} for both IPsat and IIM parametrizations \nat $Q^2=0$ and $Q^2=10\\ \\textrm{GeV}^2$. Again \nwe see the larger nuclear suppression in the IIM model than in IPsat.\nThe differences in the energy (i.e. ${x_\\mathbb{P}}$) dependence of\nthe two dipole cross sections are more clearly visible in the \nphotoproduction result. This is natural, since in the IPsat model \nthe energy dependence at the initial scale of the DGLAP evolution\n(probed at smaller $Q^2$) is almost flat, in stark contrast to the\ntypical behavior resulting from BK evolution. At higher $Q^2$\nthe difference in the $x$-dependence is smaller, although \nthere the IPsat-model, driven by the DGLAP evolution, \nturns over to a \\emph{faster}\nenergy dependence. We have not extrapolated our curves to higher \nenergies, since there is no prospect of experimental measurements. One does \nhowever see from Fig.~\\ref{fig:ratiovsx} that the curves\ncontinue to go down when extrapolated to smaller ${x_\\mathbb{P}}$. This is to be \nexpected since, as discussed previously, one has not yet reached the  \nblack disk limit. \n\nIn a realistic experimental setup it might be possible to detect \nor veto the nuclear breakup even when the momentum transfer\n$t$ is not measured very accurately. In this case it will be interesting \nto understand how the relative magnitudes of the incoherent and coherent\ncross sections behave as a function of $Q^2$ and ${x_\\mathbb{P}}$. Generally when approaching \nthe black disk limit the coherent cross section increases and the incoherent one\ndecreases. The relative change shows, however, a smaller dependence on $Q^2$ and \n${x_\\mathbb{P}}$ than the nucleus\/nucleon cross section ratio. This is shown in our \nparametrization in  Figs.~\\ref{fig:incohovercohQ} and~\\ref{fig:incohovercohx}, where\nwe plot the \nthe incoherent cross section integrated over the interval $0.1\\ \\textrm{GeV}^2< -t < 0.3 \\ \\textrm{GeV}^2$\ndivided by the coherent cross section integrated over $0 < -t < 0.1 \\ \\textrm{GeV}^2$\nas a function of $Q^2 + M_{J\/\\Psi}^2$ and ${x_\\mathbb{P}}$.\nFigure \\ref{fig:cohincoh} further demonstrates the relative similarity of the \nnuclear suppression in the coherent and incoherent cross sections. \nShown is the $A$ dependence of the ratios \n$ (\\, \\mathrm{d}\\sigma^A_\\mathrm{incoh}\/\\, \\mathrm{d} t)\/(A \\, \\mathrm{d} \\sigma^p\/\\, \\mathrm{d} t)$\n(which, in our approximation, is independent of $t$)\nand \n$\\left. (\\, \\mathrm{d}\\sigma^A_\\mathrm{coh}\/\\, \\mathrm{d} t)\/(A^2 \\, \\mathrm{d} \\sigma^p\/\\, \\mathrm{d} t)\\right|_{t=0}$\nfor $Q^2=10\\ \\textrm{GeV}^2$ and ${x_\\mathbb{P}}=0.001$. Note that the coherent and the incoherent \ncross sections are normalized by different powers of $A$ and that\nwidth of the coherent peak at small $t$ also depends on $A$.\n\nFigures \\ref{fig:dsigmavst} and \\ref{fig:ratiovsq} are our main result.\nOur calculation uses as input only well tested parametrizations that have been fit\nto existing HERA data and nuclear geometry. We work strictly in the\nsmall $x$-limit which makes our formalism simple and transparent.\nThis paper provides realistic estimates for the absolute cross sections \nthat could be measured in future nuclear DIS experiments. \nWe have, however, made several simplifying assumptions in our calculation, the most \nimportant being a) the factorized impact parameter dependence Eq.~\\nr{eq:factbt},\nb) the assumption of independent scattering off different nucleons\nEq.~\\nr{eq:sfact} and c) neglecting nucleon-nucleon correlations. Including\nthese effects in a physically correct manner and discussing how they could be \nstudied experimentally is left for future work. As can be seen from \nthe values of the nuclear suppression in Figs.~\\ref{fig:ratiovsq} and~\\ref{fig:ratiovsx},\nthe effects of high  densities, gluon saturation and unitarity on the\nincoherent cross section are large.\nThus incoherent diffraction in future nuclear DIS experiments will\nbe a sensitive probe of small-$x$ physics.\n\n\n\n\\section*{Acknowledgements}\nWe thank E. Aschenauer, H. Kowalski and W. Horowitz for discussions and K.~J.~Eskola\nfor a careful reading of the manuscript.\nThe work of T.L. has been supported by the Academy of Finland, project \n126604. T.L. wishes to thank the INT  at the \nUniversity of Washington for \nits hospitality during the completion of this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nLocalising the source of an unknown or uncertain scalar field has attracted significant attention in recent years. Extremum seeking can then be understood as driving the state of an agent or network of agents to the source, and maintaining a steady state in the neighbourhood of this optimal state in the unknown field. The widespread applications include internal combustion engine calibration~\\cite{killingsworth2009hcci}, locating RF leakage\\cite{al2012position}, optimising energy distribution~\\cite{ye2016distributed}, and maximising the output of bioreactors~\\cite{wang1999optimizing}. The main challenge in general is the approximation of the field, or a valid descent direction, with the additional challenge in the multi-agent case of coordinating the agents to improve the estimation. In this work, we consider discrete time extremum seeking, for the more classical continuous time extremum seeking problem see~\\cite{tan2010extremum} and the references therein.\n\nExtremum seeking with a single agent primarily uses ``dither'' or other motion patterns to estimate a descent direction. In~\\cite{cochran2009nonholonomic, zhang2007extremum}, extremum seeking with a single agent is investigated relying only on the measurements of the scalar field, without usage of the agent's position. Both approaches use a sinusoidal dither signal to estimate the gradient of the unknown field. Using finite difference with previous measurements, tracking and estimation error bounds for the minima of a time-varying scalar field are derived in~\\cite{shames2019online}, along with extensive numerical studies using a single agent. A hybrid controller is defined in~\\cite{mayhew2007robust}, conducting a series of line minimisations to construct the descent direction, with stability and convergence results.\n\nUsing a network of agents allows for a more robust estimate of the gradient, as the measurements are typically assumed to be simultaneous and thus unaffected by a time-varying source. In\\cite{biyik2008gradient} a network is used with a single leader determining the estimated gradient, employing a zero mean dither signal, with the followers only keeping formation. The authors show that with a fast dither and slow formation keeping, the followers only track the gradient descent movement of the leader. Using multiple ``leader'' agents and only inter-agent bearing measurements, the authors in~\\cite{zhao2015bearing,zhao2015translational} stabilise a formation in arbitrary dimension with leaders following reference velocities or trajectories. In addition, using only bearing measurements allows for formation scaling and rotation. In multi-agent approaches, the set of measurements from each agent can be used to compute an estimated gradient, assuming a single sensor aboard each agent\\cite{khong2014multi,ogren2004cooperative,skobeleva2018planar,vandermeulen2017discrete,vweza2015gradient}. All of these publications use some form of the simplex gradient\\cite{regis2015calculus}, as do we in this paper. The controller design derived in~\\cite{khong2014multi} uses a centralised extremum seeking controller, with access to all of the agents' measurements, which provides reference velocities to each of the agents. Convergence guarantees are provided for a variety of formation and extremum seeking methods satisfying their assumptions. A centralised controller is implemented in~\\cite{ogren2004cooperative} to track the estimated gradient using least squares estimation and refined by Kalman filtering. The agents are tasked with formation keeping around a virtual leader, which climbs the gradient of the unknown field. However, the problem formulation only considers finite manoeuvres, and the formation may move extremely slowly. For networks of $3$ agents in $2$ dimensions a distributed control law with exponential convergence guarantees is investigated in\\cite{skobeleva2018planar}. The agents in\\cite{vandermeulen2017discrete} use a dynamic consensus algorithm to coordinate the gradient estimation, combined with a zero mean dither to construct a local gradient estimation. Finally, in a series of papers\\cite{circular2015distributed,circular2013consensus,circular2011collaborative,circular2010source}, a group of unicycle agents performing distributed extremum seeking in circular formations is examined. The agents stabilise their formation and gradient estimate using a consensus algorithm, and performs well even with lossy communication and time-varying communication networks. The algorithm described in~\\cite{circular2015distributed} is implemented in Section~\\ref{sec:simul} to compare to the results derived in this paper.\n\n\nThis paper provides a novel analysis of multiagent extremum seeking focused on a time-varying source without using a centralised coordinator or dither motion. This differs from the majority of the literature, which assumes a static or slowly drifting scalar field. We show that given an appropriate choice of the agents' formation, a simple gradient descent algorithm enables the network of agents to converge to a bounded neighbourhood of the time-varying extremum sets. At each iteration, we only assume that the time varying field is represented by a function which has Lipschitz continuous gradient (bounded second derivative), and satisfies the Polyak-\\L{}ojasiewicz inequality. The Polyak-\\L{}ojasiewicz inequality assumption is also weaker than many which are used to provide the linear convergence of gradient descent algorithms, such as convexity or quadratic growth\\cite{plstuff}. The authors' previous investigation into this problem \\cite{michael2020optimisation} included a more complicated control law than is presented here, with results restricted to $2$ dimensions. In this analysis, we simplify the control law, derive stronger convergence guarantees, and broaden the method to arbitrary dimension.\n\nThe paper is organised as follows. Section~\\ref{sec:probForm} is devoted to basic assumptions on the time-varying field and agent dynamics. Section~\\ref{sec:coopGradDesc} discusses the distributed control law for extremum seeking and formation keeping, along with convergence guarantees. Section~\\ref{sec:gradEstimation} provides an example of cooperative gradient estimation, an improvement and generalisation of the results from~\\cite{michael2020optimisation}. We provide numerical studies and comparison in Section~\\ref{sec:simul}, and conclude in Section~\\ref{sec:conclude}.\n\n\\section{Problem Formulation}\\label{sec:probForm}\n\nConsider a network of $n$ agents where $x_k^{(i)} \\in \\mathbb{R}^d$ denotes the position of the $i$-th agent for $i\\in \\{1,...,n\\}$ at iteration $k$. We use bold variables throughout the paper to describe the stacked vector for all agents, i.e. ${\\bf x}_k$ to denote the vector of all agents' states stacked vertically. Let $\\mathcal{G}=(\\mathcal{V},\\mathcal{E})$ be the underlying graph of the network with the vertex set $\\mathcal{V}=\\{1,...,n\\}$ representing the agents and the edge set $\\mathcal{E}\\subseteq \\mathcal{V}\\times\\mathcal{V}$ representing the communication topology. For each agent $i$, we define a set of neighbours $\\mathcal{N}^{(i)}:= \\{ j \\mid (j,i)\\in\\mathcal{E} \\}$ from which agent $i$ receives information at each iteration step.\n\n\\begin{assum}\\label{ass:networkProp}\nAssume that the agent communication graph $\\mathcal{G}=(\\mathcal{V},\\mathcal{E})$ is connected and time invariant.\n\\end{assum}\nThe agents are modeled as single integrators, with dynamics\n\\begin{align}\nx_{k+1}^{(i)} = x_k^{(i)} + \\alpha p_k^{(i)}. \\label{eq:dyn}\n\\end{align}\nThe constant term $\\alpha$ is determined by the time-varying field and is time invariant and uniform across the network. At each iteration $k$, the time-varying field is represented by the function $f_k:\\mathbb{R}^{d}\\rightarrow\\mathbb{R}$ with the non-empty minimiser set $\\mathcal{X}^*_{f_k} := \\textrm{argmin}_{x\\in\\mathbb{R}^{d}} f_k(x)$. The agents can only measure the function value at their location at each iteration, i.e. the value $f_k(x_k^{(i)})$. For any dimension $m\\in\\mathbb{Z}^+$ we define the distance between a point $x\\in\\mathbb{R}^{m}$ and a set $\\mathcal{S}\\subseteq\\mathbb{R}^{m}$ as\n\\begin{align}\n    d(x,\\mathcal{S}) = \\inf_{y\\in\\mathcal{S}} ||y-x||, \\label{eq:pointSetDist}\n\\end{align}\nwhere $||\\cdot||$ is the Euclidean norm. Additionally, for a function $h:\\mathcal{D}\\rightarrow\\mathbb{R}$ we will use the shorthand\n\\begin{align}\n    h^* := \\inf_{x\\in\\mathcal{D}} h(x), \\label{eq:funcMinVal}\n\\end{align}\nto represent the minimum value of that function.\n\n\\begin{assum}(Differentiability and Lipschitz Gradient):\\label{ass:Lipschitz}\nFor all $k\\geq0$, the functions $f_k:\\mathbb{R}^d \\rightarrow \\mathbb{R}$ are at least once continuously differentiable. The gradients are $L_{f}-$Lipschitz continuous, i.e. there exists a positive scalar $L_{f}$ such that, for all $k \\geq 0, x\\in\\mathbb{R}^d,\\; y\\in\\mathbb{R}^d$, $$|| \\nabla f_k(x) - \\nabla f_k(y) || \\leq L_{f}||x-y||, $$ or equivalently $$ f_k(y) \\leq f_k(x) + \\nabla f_k(x)^T(y-x) + \\frac{L_{f}}{2}||y-x||^2.$$\n\\end{assum}\n\n\\begin{assum}(Polyak-\\L{}ojasiewicz Condition):\\label{ass:polyak}\nFor all $k\\geq0$, there exists a positive scalar $\\mu_{f}$ such that $$\\frac{1}{2}||\\nabla f_k(x)||^2\\geq \\mu_{f}(f_k(x) - f^*_k),$$ for $f^*_k$ defined in~\\eqref{eq:funcMinVal}.\n\\end{assum}\nThe assumption that a function has an $L-$Lipschitz continuous gradient is equivalent to assuming the second derivative has bounded norm, if it is twice differentiable. The Polyak-\\L{}ojasiewicz condition simply requires that the gradient grows faster than a quadratic as we move away from the optimal function value. The Polyak-\\L{}ojasiewicz condition does not require the minima to be unique, although it does guarantee that every stationary point is a global minimum~\\cite{plstuff}. In addition to Assumptions~\\ref{ass:Lipschitz}-\\ref{ass:polyak} on each $f_k$, we quantify the ``speed'' with which the field may vary in the following assumption.\n\\begin{assum}(Bounded Drift in Time):\\label{ass:drift}\nThere exist positive scalars $\\eta_0$ and $\\eta^*$ such that $|f_{k+1}(x)-f_k(x)|\\leq \\eta_0$ for all $x\\in\\mathbb{R}^d$ and $|f^*_k-f^*_{k+1}|\\leq \\eta^*$.\n\\end{assum}\nThe problem of interest is given below.\n\\begin{prob}\\label{prob:onlyProb}\nFor a network of $n$ agents with dynamics~\\eqref{eq:dyn} and communication topology satisfying Assumption~\\ref{ass:networkProp}, let $\\{f_k\\}$ be a sequence of functions with a corresponding sequence of minimiser sets $\\{\\mathcal{X}^*_{f_k}\\}$ satisfying Assumptions~\\ref{ass:Lipschitz}-\\ref{ass:drift}. Given the measurements $\\mathcal{Y}_k^{(i)}=\\{f_k(x_k^{(j)}) \\mid j\\in\\mathcal{N}^{(i)}\\cup\\{i\\}\\}$, find $\\alpha,p_k^{(i)}$ and a constant $M$ for all agents $i\\in\\mathcal{V}$ and for all $k\\geq0$ such that $\\lim\\limits_{k\\rightarrow\\infty}d(x_k^{(i)},\\mathcal{X}^*_{f_k})\\leq M$.\n\\end{prob}\nPrevious research in extremum seeking of stationary sources have resulted in stronger convergence guarantees, such as semi-global practical asymptotic stability~\\cite{mayhew2007robust,khong2014multi,kvaternik2012analytic}. The authors of~\\cite{zhang2007extremum} are able to show local exponential stabilty of their extremum seeking algorithm for slowly moving source, i.e. the source velocity is significantly less than the dither speed. Further, the stability results are only extended for moving sources with periodic or ``almost-periodic'' trajectories. In this paper we have a more general definition of a ``moving source'' than found in the literature, and show that the control law described in the following sections is able to stabilise the steady state tracking error $d(x_k^{(i)},\\mathcal{X}^*_{f_k})$.\n\n\\section{Cooperative Gradient Descent}\\label{sec:coopGradDesc}\n\nIn this section we discuss our primary result, showing that a network of agents cooperating can reach a bounded neighbourhood of the minimiser set. In this section, for simplicity, we assume each agent uses an $\\varepsilon-$\\emph{gradient oracle} at each iteration to construct a step direction.\n\n\\begin{defn}\\label{def:epsOracle}\n\\emph{($\\varepsilon$-gradient oracle):} Given the function $f_k:\\mathbb{R}^{d}\\rightarrow\\mathbb{R}$ and the state of the agents in the network ${\\bf x}_k\\in\\mathbb{R}^{d}$, the oracle returns $O(f_k,{\\bf x}_k,\\mathcal{N}^{(i)}) = \\nabla f_k(x^{(i)}_k) + \\varepsilon_k$.\n\\end{defn}\n\nIn order to motivate the incorporation of formation control, we first analyse the erroneous gradient descent dynamics with input $p^{(i)}_k = -O(f_k,{\\bf x}_k,\\mathcal{N}^{(i)})$,\n\\begin{align}\n\\begin{split}\nx^{(i)}_{k+1} :=& x^{(i)}_{k} + \\alpha (-O(f_k,{\\bf x}_k,\\mathcal{N}^{(i)})) \\\\\n=& x^{(i)}_{k} - \\alpha (\\nabla f_k(x^{(i)}_k) + \\varepsilon_k)),\n\\end{split}\\label{eq:naiveDyn}\n\\end{align}\nand provide the following lemma on the convergence properties of the system.\n\n\\begin{lem}\\label{lem:noisyGradDesc}\nFor a sequence of functions $\\{f_k\\}$ with minimiser sets $\\{\\mathcal{X}^*_{f_k}\\}$ satisfying Assumptions~\\ref{ass:Lipschitz}-\\ref{ass:drift}, the system with dynamics~\\eqref{eq:naiveDyn} satisfies\n\\begin{align}\n\\begin{split}\n\\frac{1}{2}d(x^{(i)}_k,\\mathcal{X}^*_{f_k})^2 &\\leq \\beta(d(x^{(i)}_0,\\mathcal{X}^*_{f_0})^2,k) \\\\\n&\\hspace{-1cm} + \\frac{\\alpha}{2\\mu_{f}}\\sum_{t=0}^{k}(1 - \\alpha\\mu_{f})^{k-t}||\\varepsilon_t||^2  + \\frac{\\eta_{0}+\\eta^*}{\\alpha\\mu_{f}^2}, \\label{eq:noisyGradNeighbourhood}\n\\end{split}\n\\end{align}\nfor $\\beta\\in\\mathcal{KL}$, $\\alpha\\in(0,\\frac{1}{L_{f}}]$ with $L_{f},\\mu_{f}$ from Assumptions~\\ref{ass:Lipschitz}-\\ref{ass:polyak}, and $d(x^{(i)}_k,\\mathcal{X}^*_{f_k})$ defined in~\\eqref{eq:pointSetDist}.\n\\end{lem}\n\n\\begin{pf}\nSee Appendix~\\ref{app:noisyGradDesc}.\n\\end{pf}\n\n\\begin{rem}\n  Lemma~\\ref{lem:noisyGradDesc} seems to imply that if $\\alpha$ is chosen to be $\\frac{1}{\\mu_f}$, the impact of the gradient error from steps before $k$ is zero. To understand why this is so, note that the Lipschitz constant $L_{f}$ and Polyak-\\L{}ojasiewicz constant $\\mu_{f}$ satisfy the following\n  \\begin{align}\n  \\frac{\\mu_{f}}{2}d(x^{(i)}_k,\\mathcal{X}^*_{f_k})^2 \\leq f_k(x) - f^*_k \\leq \\frac{L_{f}}{2}d(x^{(i)}_k,\\mathcal{X}^*_{f_k})^2, \\label{eq:quadraticSqueeze}\n  \\end{align}\n  see~\\cite{plstuff} for in depth discussion regarding the Polyak-\\L{}ojasiewicz inequality. Requiring that $\\alpha \\leq \\frac{1}{L_{f}}$ then implies $\\alpha \\leq \\frac{1}{\\mu_{f}}$. Therefore, if $\\alpha \\approx \\frac{1}{\\mu_f}$, then we must have that $\\mu_f\\approx L_{f}$ and $f_k$ is approximately a scaled norm as a consequence of~\\eqref{eq:quadraticSqueeze}. For the scaled norm function, the gradient dynamics~\\eqref{eq:naiveDyn} would take the agent directly to the minimiser, except for the error term from the most recent gradient estimate in~\\eqref{eq:noisyGradNeighbourhood} and the drift error term $\\frac{\\eta_0+\\eta^*}{\\mu_f}$.\n\\end{rem}\n\nWe see from Lemma~\\ref{lem:noisyGradDesc} that the system with dynamics~\\eqref{eq:naiveDyn} converges to a neighbourhood dependent on the magnitude of the gradient error terms $||\\varepsilon_k||^2$ and a constant term due to drift. In using function samples to estimate the gradient, the error in estimation is generally a function of the geometry of the samples taken. By incorporating formation control into the dynamics, we are able to bound the error terms $||\\varepsilon_k||^2$. We show a specific example of this in Section~\\ref{sec:gradEstimation}, but make minimal assumptions in this section on the specifics of how to construct a gradient estimate from sample points.\n\nTo expand the focus onto the entire network's behavior, we define the collective time-varying function $F_k:\\mathbb{R}^{nd}\\rightarrow\\mathbb{R}$ as\n\\begin{align*}\nF_k({\\bf x}_k) := \\sum_{i\\in\\mathcal{V}} f_k(x^{(i)}_k),\n\\end{align*}\nand note that $F_k$ satisfies Assumptions~\\ref{ass:Lipschitz}-\\ref{ass:polyak} with the same constants $L_{f},\\mu_{f}$. The time-varying minimiser set of $F_k({\\bf x}_k)$ is\n\\begin{align*}\n\\mathcal{X}^*_{F_k} = \\overbrace{\\mathcal{X}^*_{f_k}\\times\\mathcal{X}^*_{f_k}...\\times\\mathcal{X}^*_{f_k}}^{n}.\n\\end{align*}\nTo incorporate formation control, we use a formation potential function $\\phi({\\bf x}_k):\\mathbb{R}^{nd}\\rightarrow\\mathbb{R}^+$ which takes the full state vector of all agents and returns a scalar which is minimised when the agents are in formation. Let the minimum be $\\phi^* := \\min\\limits_{{\\bf x}\\in\\mathbb{R}^{nd}}\\phi({\\bf x})$.\n\\begin{defn}\\label{def:formFuncs}\nWe define $\\phi:\\mathbb{R}^{nd} \\rightarrow \\mathbb{R}^+$ to be the \\emph{formation potential function} for the network, with minimisers $\\mathcal{X}^*_{\\phi}$, and assume the following properties. The function $\\phi({\\bf x}_k)$\n\\begin{enumerate}\n    \\item is continuously differentiable on $\\mathbb{R}^{nd}$ with gradient which is Lipschitz continuous with constant $L_{\\phi}$;\n    \\item satisfies the Polyak-\\L{}ojasiewicz inequality (Assumption~\\ref{ass:polyak}), with constant $\\mu_{\\phi} \\geq \\mu_{f}$;\n    \\item has gradient component $\\nabla_{x^{(i)}_k} \\phi({\\bf x}_k)$ which is computable using only the state of agent $i$ and neighbours $j\\in\\mathcal{N}_{i}$;\n    \\item satisfies $\\phi({\\bf x}_k) \\geq \\frac{c}{2}\\sum_{i\\in\\mathcal{V}}||\\varepsilon^{(i)}_k||^2$ for $c > \\frac{1}{\\mu_{f}}$.\n\\end{enumerate}\n\\end{defn}\nIn the definition of the formation potential functions, the first two conditions ensure that $\\phi({\\bf x}_k)$ shares the minimal properties that make $f_k$ amenable to analysis. The third property ensures that the local information each agent has is sufficient for computation of the descent direction. Finally, the fourth property formalises the connection between the formation and the gradient estimation error. This final property formalises the relationship between the gradient estimation error and the formation. In Section~\\ref{sec:simul} we provide the example $\\phi({\\bf x}_k) = \\phi^* + L_{f} \\sum_{i\\in\\mathcal{V}} ||x^{(i)}-x^{(j)}-\\hat{x}^{(ij)}||^2$, where the terms $\\hat{x}^{(ij)}$ define the optimal formation and the constant $\\phi^*$ ensures condition $4$ from Definition~\\ref{def:formFuncs} is satisfied when the agents are in perfect formation. The constant offset does not change the dynamics, it simply allows $\\phi({\\bf x}_k)$ to bound the gradient error in the convergence analysis, see the proof of Theorem~\\ref{thm:compositeConvergence}.\n\nWith the formation potential function defined, we define the ``composite'' function $\\hat{f}_k:\\mathbb{R}^{nd} \\rightarrow \\mathbb{R}$ as\n\\begin{align}\n\\hat{f}_k({\\bf x}_k) := F_k({\\bf x}_k) + \\phi({\\bf x}_k),\\label{eq:compositeFunc}\n\\end{align}\nwith corresponding minimisers in the set $\\mathcal{X}^*_{\\hat{f}_k}$, and the new system dynamics\n\\begin{align}\nx^{(i)}_{k+1} := x^{(i)}_k - \\alpha(\\nabla_{x^{(i)}_k} \\hat{f}_k + \\varepsilon_k). \\label{eq:compositeDyn}\n\\end{align}\nEach agent can compute the gradient $\\nabla_{x^{(i)}_k} \\phi({\\bf x}_k)$ with only local information, so the gradient of the composite function, being the sum of $f_k$ and $\\phi$, can be estimated by using the same $\\varepsilon-$gradient oracle for $f_k$. Both $F_k$ and $\\phi$ satisfy Assumption~\\ref{ass:Lipschitz} with constants $L_{f},L_{\\phi}$ respectively, and Assumption~\\ref{ass:polyak} with constants $\\mu_{f},\\mu_{\\phi}$. Therefore, the composite function satisfies both Assumptions~\\ref{ass:Lipschitz}-\\ref{ass:polyak} with constants $L_{\\hat{f}} := L_{f}+L_{\\phi}$ and $\\mu_{\\hat{f}} \\geq \\min(\\mu_{f},\\mu_{\\phi}) = \\mu_{f}$.\n\n\\begin{lem}\\label{lem:minimiserRelation}\nFor the composite function $\\hat{f}_k$, as defined in~\\eqref{eq:compositeFunc}, we have\n\\begin{align*}\n\\hat{f}^*_k := \\min_{{\\bf x}\\in\\mathbb{R}^{nd}}\\hat{f}_k({\\bf x}) \\leq \\phi^* + \\frac{\\min(L_{f},L_{\\phi})}{2} d(\\mathcal{X}^*_{F_k},\\mathcal{X}^*_{\\phi})^2,\n\\end{align*}\nwhere we define the distance between the minimiser sets as\n\\begin{align*}\nd(\\mathcal{X}^*_{F_k},\\mathcal{X}^*_{\\phi}) := \\min \\{ ||x^*_{\\phi} - x^*_{F_k}|| \\mid  x^*_{\\phi}\\in\\mathcal{X}^*_{\\phi} \\; , \\;x^*_{F_k}\\in \\mathcal{X}^*_{F_k} \\}.\n\\end{align*}\n\\end{lem}\n\n\\begin{pf}\nSee Appendix~\\ref{app:minimiserRelation}.\n\\end{pf}\nThe primary result is given in Theorem~\\ref{thm:compositeConvergence}.\n\n\\begin{thm}\\label{thm:compositeConvergence}\nFor a sequence of functions $\\{\\hat{f}_k\\}$ as defined in~\\eqref{eq:compositeFunc} with minimisers $\\{\\mathcal{X}^*_{\\hat{f}_k}\\}$, the system with dynamics~\\eqref{eq:naiveDyn} satisfies\n\\begin{align*}\n\\begin{split}\n\\frac{1}{2}d(x^{(i)}_{k+1},\\mathcal{X}^*_{\\hat{f}_{k+1}})^2 &\\leq \\beta(d(x^{(i)}_{0},\\mathcal{X}^*_{\\hat{f}_0})^2,k) \\\\\n&\\hspace{-1cm} + \\frac{\\alpha}{c\\mu}\\sum_{t=0}^{k}(1 - \\alpha\\mu')^{k-t}\\hat{f}^*_t + \\frac{\\eta_{0}+\\eta^*}{\\alpha\\mu\\mu'},\n\\end{split}\n\\end{align*}\nfor $\\beta\\in\\mathcal{KL}$, $\\alpha\\in(0,\\frac{1}{L_{\\hat{f}}}]$, and $\\mu' = \\mu_{f}-\\frac{1}{c}$. Therefore, we have\n\\begin{align}\n\\lim_{k\\rightarrow\\infty} \\frac{1}{2}d(x^{(i)}_{k+1},\\mathcal{X}^*_{\\hat{f}_{k+1}})^2 \\leq  \\frac{\\underset{k\\rightarrow\\infty}{\\lim} \\sup \\hat{f}^*_k}{\\mu'} + \\frac{\\eta_{0}+\\eta^*}{\\alpha\\mu\\mu'}. \\label{eq:limBound}\n\\end{align}\n\\end{thm}\n\n\\begin{pf}\nSee Appendix~\\ref{app:compositeConvergence}.\n\\end{pf}\nIn a typical gradient descent scenario, we may expect a convergence bound similar to~\\eqref{eq:limBound}, without the $\\limsup \\hat{f}^*_k$ term. However, the gradient being used is only an estimation, and $\\limsup \\hat{f}^*_k$ represent the limit of the ``quality'' of the gradient estimate. Even on a stalimsuptionary function, i.e. $\\eta_0 = \\eta^* = 0$, we would not expect asymptotic convergence with erroneous gradient information.\n\nIn Theorem~\\ref{thm:compositeConvergence}, we demonstrate that by incorporating a formation potential function, which bounds the gradient estimation error, we are able to converge to a bounded neighbourhood of the time varying minimiser set $\\mathcal{X}^*_{\\hat{f}_{k}}$. Further, the system does not require the delineation of leaders and followers, a separate time-scale for the formation-keeping, or any centralised computing to guide the formation.\n\\section{Gradient Estimation and Error}\\label{sec:gradEstimation}\nIn Section~\\ref{sec:coopGradDesc}, we assume that each agent has access to an estimate local gradient $\\nabla f(x_k^{(i)}) + \\epsilon^{(i)}$. In this section, we provide a method by which agent $i$ can estimate $\\nabla f(x_k^{(i)})$ as well as compute an error bound for the estimate. The error bound and gradient estimation method apply to \\emph{any} function which satisfies Assumption~\\ref{ass:Lipschitz}. This method is a significant improvement of our previous work\\cite{michael2020optimisation}, which generalises to any dimension with any number of neighbours. We make the following assumption on the neighbour set.\n\n\\begin{assum}\\label{ass:fullRankNeighbours}\nFor each agent $i\\in\\mathcal{V}$ with state $x^{(i)}_k\\in\\mathbb{R}^d$, the neighbour set cardinality satisfies $|\\mathcal{N}^{(i)}| \\geq d$. Further, the vectors $\\{x^{(l)}_k - x^{(i)}_k\\}_{l\\in\\mathcal{N}^{(i)}}$ span $\\mathbb{R}^d$.\n\\end{assum}\n\nThe requirement that the agents do not arrange on a low dimensional subspace is the primary motivation for incorporating formation control. We define three variables before proceeding, as they will figure heavily in the analysis,\n\\begin{align}\n\\begin{split}\ns^{(ij)}_k &:= \\frac{f_k(x^{(j)}_k) - f_k(x^{(i)}_k)}{||x^{(j)}_k - x^{(i)}_k||}, \\\\\nv^{(ij)}_{k} &:= \\frac{x^{(j)}_k - x^{(i)}_k}{||x^{(j)}_k - x^{(i)}_k||}, \\\\\na^{(ij)}_k &:= \\frac{L_{f}}{2}||x^{(j)}_k - x^{(i)}_k||.\n\\end{split}\n\\label{eq:usefulConstants}\n\\end{align}\nFor an agent $i$, $s^{(ij)}_k$ is the average slope between themselves and agent $j$, $v^{(ij)}_k$ the unit vector pointing from agent $i$ to $j$, and $a^{(ij)}_k$ is the distance between the agents, scaled by the Lipschitz constant $L_{f}$. We will use ${\\bf s}^{(i)}_k,{\\bf a}^{(i)}_k$ to denote the vertically stacked vectors of $s^{(ij)}_k,a^{(ij)}_k$ for all neighbours $j\\in\\mathcal{N}^{(i)}$.\n\n\\begin{lem}\\label{lem:gradPolyhedron}\nFor a function $f_k$ satisfying Assumption~\\ref{ass:Lipschitz} and an agent $i$ with neighbour set $\\mathcal{N}^{(i)}$ satisfying Assumption~\\ref{ass:fullRankNeighbours}, there exists a \\textbf{bounded} polyhedron\n\\begin{align}\n\\mathcal{P}^{(i)}_k := \\{ x\\in\\mathbb{R}^d \\mid \\begin{bmatrix} A^{(i)}_k \\\\ -A^{(i)}_k \\end{bmatrix} x \\leq b^{(i)}_k  \\} \\label{eq:polyhedron}\n\\end{align}\nsuch that $\\nabla f(x^{(i)}_k) \\in \\mathcal{P}^{(i)}_k$, for $A^{(i)}_k\\in\\mathbb{R}^{|\\mathcal{N}^{(i)}|\\times d}$ and $b_k\\in\\mathbb{R}^{2|\\mathcal{N}^{(i)}|\\times d}$.\n\\end{lem}\n\n\\begin{pf}\nSee Appendix~\\ref{app:gradPolyhedron}.\n\\end{pf}\nFrom Lemma~\\ref{lem:gradPolyhedron}, there exists a bounded space $\\mathcal{P}^{(i)}_k$ within which the gradient $\\nabla f(x^{(i)}_k)$ must exist. In~\\cite{michael2020optimisation}, we restricted the error bound analysis to $2$ dimensions with $2$ neighbours. The same method is not computationally feasible in higher dimension, as it requires computation of the largest diagonal in the $d$-parallelotope, which has $2^{d-1}$ diagonals. Instead, we compute a ``rounding'' of the polytope by defining an ellipse\n\\begin{align}\nm^{(i)}_{k} &:= \\sqrt{\\sum_{j\\in\\mathcal{N}^{(i)}} (|s^{(ij)}_k-(g^{(i)}_k)^Tv^{(ij)}_k| + a^{(ij)}_k)^2} \\label{eq:ellipseScaling}\\\\\n\\mathcal{E}^{(i)}_k &:= \\left\\{ x\\in\\mathbb{R}^{d} \\mid \\left\\|\\frac{A^{(i)}_k(x-g^{(i)}_k)}{m^{(i)}_k}\\right\\|^2 \\leq 1 \\right\\}, \\label{eq:ellipseDef}\n\\end{align}\nwith $g^{(i)}_k$ the center of $\\mathcal{E}^{(i)}_k$, $A^{(i)}_k$ the matrix defined in Lemma~\\ref{lem:gradPolyhedron}, and $s^{(ij)}_k,v^{(ij)}_k,a^{(ij)}_k$ defined in~\\eqref{eq:usefulConstants}. We define $g^{(i)}_k$ to be\n\\begin{align}\ng^{(i)}_k := ((A^{(i)}_k)^TA^{(i)}_k)^{-1}(A^{(i)}_k)^T{\\bf s}^{(i)}_k. \\label{eq:centerDef}\n\\end{align}\nNote that the center of the ellipse will serve as the gradient estimate for agent $i$, and is equivalent to the simplex gradient\\cite{regis2015calculus} of agent $i$ and its neighbours. In the following theorem we present the main result for this section, an error bounding process which works in arbitrary dimension for any number of neighbours, given Assumption~\\ref{ass:fullRankNeighbours}.\n\n\\begin{thm}\\label{thm:boundingEllipse}\nFor a function $f_k$ satisfying Assumption~\\ref{ass:Lipschitz} and an agent $i$ with neighbour set $\\mathcal{N}^{(i)}$ satisfying Assumption~\\ref{ass:fullRankNeighbours}, let $\\mathcal{P}^{(i)}_k$ be the polytope defined in Lemma~\\ref{lem:gradPolyhedron}. Then $\\mathcal{P}^{(i)}_k\\subseteq \\mathcal{E}^{(i)}_k$, for $\\mathcal{E}^{(i)}_k$ the ellipse defined in~\\eqref{eq:ellipseDef} with center $g^{(i)}_k$ defined in~\\eqref{eq:centerDef}. Further, if $|\\mathcal{N}^{(i)}| = d$, and we assume $B(r,c) = \\{ x\\in\\mathbb{R}^{d} \\mid ||x-c||_2 \\leq r\\}$ is the smallest bounding ball such that $\\mathcal{P}^{(i)}_k \\subseteq B(r,c)$, then\n\\begin{align}\n\\frac{||{\\bf a}_k^{(i)}||}{\\sigma_{\\textrm{max}}(A^{(i)}_k)} \\leq r \\leq \\frac{||{\\bf a}_k^{(i)}||}{\\sigma_{\\textrm{min}}(A^{(i)}_k)} \\label{eq:radiusBounds}\n\\end{align}\nfor $\\sigma_{\\textrm{max\/min}}$ the largest\/smallest singular values of $A^{(i)}_k$ and ${\\bf a}_k^{(i)}$ the vector of $a_k^{(ij)}$ for all $j\\in\\mathcal{N}^{(i)}$.\n\\end{thm}\n\n\\begin{pf}\nSee Appendix~\\ref{app:boundingEllipse}.\n\\end{pf}\nThe result in~\\eqref{eq:radiusBounds} may be interpreted as ``the radius of the smallest bounding ball lies between the largest and smallest radii of $\\mathcal{E}^{(i)}_k$.'' A simple example of the ellipse~\\eqref{eq:ellipseDef} with $2$ neighbours labelled \\emph{uniform scaling} (due to the uniform scaling of the shape matrix) compared the smallest bounding ball is shown in Figure~\\ref{fig:2neigh}.\n\\begin{figure}[thpb]\n \\centering\n \\includegraphics[scale=0.45]{2neighFont9797.pdf}\n \\caption{Ellipse bounding demonstration of Theorem~\\ref{thm:boundingEllipse}. \\label{fig:2neigh}}\n\\end{figure}\n\nGiven that finding the smallest bounding ball which contains a polytope is an NP hard problem, even for the relatively simple centrally symmetric parallelotopes\\cite{bodlaender1990computational}, this approximation is sufficient for the primary goal of gradient estimation. Further, this approximation method gives the smallest $2$-norm bound on the error in the simplest case, with $d$ neighbours distributed in a lattice around agent $i$, as demonstrated in Corollary~\\ref{cor:boundingBall}.\n\n\\begin{cor}\\label{cor:boundingBall}\nIf agent $i$ has neighbour set with cardinality $|\\mathcal{N}^{(i)}|=d$, and $(v^{(ij)}_k)^Tv^{(il)}_k=0$ for all $j,l\\in\\mathcal{N}^{(i)}$ with $j\\neq l$, then $\\mathcal{E}^{(i)}_k$ as defined in~\\eqref{eq:ellipseDef} is the smallest bounding ball such that $\\mathcal{P}^{(i)}_k\\in\\mathcal{E}^{(i)}_k$.\n\\end{cor}\n\n\\begin{pf}\nIf all neighbours are orthogonal, then $A^{(i)}_k$ as defined in Lemma~\\ref{lem:gradPolyhedron} is an orthogonal matrix, i.e. $(A^{(i)}_k)^TA^{(i)}_k = I$. Therefore, $\\mathcal{E}^{(i)}$ is a ball. Further, from Theorem~\\ref{thm:boundingEllipse}, the smallest bounding ball radius lies between the largest and smallest radii of $\\mathcal{E}^{(i)}_k$, which in this case are the same radius. Therefore, $\\mathcal{E}^{(i)}$ is the smallest bounding ball containing $\\mathcal{P}^{(i)}_k$.\n\\end{pf}\nFor any number of neighbours satisfying Assumption~\\ref{ass:fullRankNeighbours}, Theorem~\\ref{thm:boundingEllipse} guarantees a gradient estimation error bound of the form\n\\begin{align}\n||g^{(i)}_k - \\nabla f_k(x^{(i)}_k)|| \\leq \\frac{m^{(i)}_k}{\\sigma_{\\min}(A_k^{(i)})}, \\label{eq:gradErrorBound}\n\\end{align}\nfor $g^{(i)}_k$ the estimated gradient~\\eqref{eq:centerDef} and $m^{(i)}_k$ as defined in~\\eqref{eq:ellipseScaling}.\n\n\\subsection{Bounding Ellipse for large Neighbour Sets}\n\nThe ellipse definition from~\\eqref{eq:ellipseDef} performs well for smaller sets of neighbours, but tends to be conservative when the neighbour set is larger than $d$. We provide an additional bounding ellipse here, which shares many of the useful properties of the ellipse defined in~\\eqref{eq:ellipseDef}, but tends to be significantly less conservative in larger problems. The potentially large scaling factor in the denominator of ellipse definition~\\eqref{eq:ellipseDef} is distributed to each row, rather than applied uniformly, which mitigates some of the inflation from redundant neighbours. We define a matrix $B^{(i)}_{k}\\in\\mathbb{R}^{|\\mathcal{N}^{(i)}|\\times d}$ with the $j$-th row $B^{(i)}_{k}[j]$ defined\n\\begin{align}\nB^{(i)}_{k}[j] := \\frac{(v^{(ij)}_{k})^T}{\\sqrt{|\\mathcal{N}^{(i)}|}(|s^{(ij)}_k-(g^{(i)}_k)^Tv^{(ij)}_{k}| + a^{(ij)}_k)} \\label{eq:otherEllipseRow}\n\\end{align}\nfor $g^{(i)}_k\\in\\mathbb{R}^{d}$ the center of the ellipse. The second ellipsoidal approximation of $\\mathcal{P}^{(i)}_k$ can then be defined as\n\\begin{align}\n\\bar{\\mathcal{E}}^{(i)}_k := \\left\\{ x\\in\\mathbb{R}^{d} \\mid \\left\\|B^{(i)}_k(x-g^{(i)}_k)\\right\\|^2 \\leq 1 \\right\\}, \\label{eq:otherEllipseDef}\n\\end{align}\nIt can be verified that $\\bar{\\mathcal{E}}^{(i)}_k$ defined in~\\eqref{eq:otherEllipseDef} also contains $\\mathcal{P}^{(i)}_k$. However, the radius of the smallest bounding ball is not guaranteed to lie between the largest and smallest eigenvalues, and thus $\\bar{\\mathcal{E}}^{(i)}_k$ does not satisfy the claims of Corollary~\\ref{cor:boundingBall}. For problems with larger sets of neighbours however, the authors note that $\\bar{\\mathcal{E}}^{(i)}_k$ seems to be a tighter approximation of $\\mathcal{P}^{(i)}_k$, based on a large number of trials generating random neighbour sets and comparing the ellipses. An example comparing the ``uniform scaling ellipse'' from~\\eqref{eq:ellipseDef} to the ``row scaling ellipse'' from~\\eqref{eq:otherEllipseDef} is included in Figure~\\ref{fig:4neigh}.\n\\begin{figure}[thpb]\n \\centering\n \\includegraphics[scale=0.5]{4neighFont8838.pdf}\n \\caption{Comparing the bounds~\\eqref{eq:ellipseDef} and~\\eqref{eq:otherEllipseDef}. \\label{fig:4neigh}}\n\\end{figure}\n\n\\section{Simulations}\\label{sec:simul}\n\nIn this section we provide numerical studies to illustrate the results from the previous sections, as well as comparison to another distributed extremum seeking algorithm. For the time varying scalar field, we use convex quadratic functions $f_k(x) = \\frac{1}{2}(x-c(k))^TQ(x-c(k)) + \\zeta^T(x-c(k)) + p$, for positive semi-definite $Q$. The values used in the following plots are\n\\begin{align*}\nQ = \\begin{bmatrix} 2.66 &-0.36 \\\\ -0.35 &1.74 \\end{bmatrix} \\; , \\; &\\zeta = [-1.28,4.66]^T \\;,\\; p = 6.26,\\\\\nc(k) = 10\\sin(\\frac{\\sqrt{2}k}{100}) &+ 10\\sin(\\frac{\\sqrt{3}k}{100}) +\\frac{k}{100},\n\\end{align*}\nwith $L_f,\\mu_f$ the largest and smallest eigenvalues of $Q$ respectively. For the formation control function, we designate a set of neighbours for each agent $\\mathcal{N}^{(i)}$ along with a corresponding set of ideal displacements $\\hat{x}^{(ij)}$. The formation potential function is then\n\\begin{align}\n\\phi({\\bf x}_k) = \\phi^* + L_{f}\\sum_{i\\in\\mathcal{V}}\\sum_{j\\in\\mathcal{N}^{(i)}} ||x^{(i)} - x^{(j)} - \\hat{x}^{(ij)}||^2_2.  \\label{eq:formationPotential}\n\\end{align}\nIn~\\cite{michael2020optimisation} we derive the error bound on the gradient estimation in two dimensions, and show that the estimation error is proportional to the distance between the agents, with proportionality constant $L_{f}$, so the the Lipschitz constant $L_{f}$ and the minimum value $\\phi^*$ in~\\eqref{eq:formationPotential} ensure that $\\phi({\\bf x}_k) $ satisfies the assumptions in Definition~\\ref{def:formFuncs}. The minimum value $\\phi^*$ is chosen as an upper bound on the gradient approximation error when the agents are in perfect formation, derived from the gradient estimation error bounds in Theorem~\\ref{thm:boundingEllipse}.\n\nThe simulated methods include the composite method derived in Section~\\ref{sec:coopGradDesc} using two different formations, as well as the consensus for circular formations from~\\cite{circular2015distributed} for comparison. For the composite method, as described in Section~\\ref{sec:coopGradDesc}, we use the simplex gradient as the local gradient estimation method at each iteration, as presented in Algorithm~\\ref{alg:compDyn}.\n\n\\begin{algorithm}\n\\caption{Distributed Composite Dynamics\\label{alg:compDyn}}\n\\begin{algorithmic}\n\\For{$k=1,2,...$}\n    \\For{$i\\in\\{1,2,...,n\\}$}\n        \\State $g^{(i)}_k = ((A^{(i)}_k)^TA^{(i)}_k)^{-1}(A^{(i)}_k)^T{\\bf s}^{(i)}_k$ \n    \\EndFor\n    \\For{$i\\in\\{1,2,...,n\\}$}\n        \\State $x^{(i)}_{k+1} = x^{(i)}_{k} - \\frac{1}{L}(g^{(i)}_k+\\nabla_{x^{(i)}_k} \\phi({\\bf x}_k))$\n    \\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\nThe circular formation controller is presented in Algorithm~\\ref{alg:circDyn}, and is written exactly as in~\\cite{circular2015distributed} accounting for the notation of this paper. The parameters used within Algorithm~\\ref{alg:circDyn} are the same as used in the original paper~\\cite{circular2015distributed}, in the example provided therein without noise. The radius of the formation $D=3$, the rotation velocity $\\omega=1$, $\\epsilon=0.5$ and $\\alpha=1$. The consensus matrix used is also the same as shown in~\\cite{circular2015distributed}, for $6$ agents we have used\n\\begin{align*}\n P = \\begin{bmatrix}\n 0.5 & 0.25 & 0 & 0 & 0 & 0.25 \\\\\n 0.25 & 0.5 & 0.25 & 0 & 0 & 0 \\\\\n 0 & 0.25 & 0.5 & 0.25 & 0 & 0 \\\\\n 0 & 0 & 0.25 & 0.5 & 0.25 & 0 \\\\\n 0 & 0 & 0 & 0.25 & 0.5 & 0.25 \\\\\n 0.25 & 0 & 0 & 0 & 0.25 & 0.5\n \\end{bmatrix}.\n\\end{align*}\n\\begin{algorithm}\n\\caption{Circular Source Seeking\\label{alg:circDyn}}\n\\begin{algorithmic}\n\\For{$i=1,...,n$}\n    \\State $h^{(i)}_0 = \\tilde{g}^{(i)}_0=h^{(i)}_{-1}=c^{(i)}_{0}+f_{0}(x^{(i)}_0)(x^{(i)}_0-c^{(i)}_0)$\n    \\State $\\phi^{(i)} = i\\frac{2\\pi}{n}$\\;\n\\EndFor\n\\For{$k=1,2,...$}\n    \\For{$i=1,...,n$}\n        \\State $g^{(i)}_{k}=c^{(i)}_{k}+\\frac{2}{D^2}f(x^{(i)}_{k})(x^{(i)}_{k}-c^{(i)}_{k})$\n        \\State $\\tilde{g}^{(i)}_{k}=(1-{\\bf \\alpha})\\tilde{g}^{(i)}_{k-1}+\\alpha\\tilde{g}^{(i)}_{k}$\n        \\State $\\tilde{h}^{(i)}_{k}=h^{(i)}_{k-1}+\\tilde{g}^{(i)}_{k-1}-\\tilde{g}^{(i)}_{k-2}$\n    \\EndFor\n    \\State ${\\bf h_{k}} = (P\\otimes I_{2})(\\bf \\tilde{h}_{k})$\n    \\For{$i=1,...,n$}\n        \\State $c^{(i)}_{k} = (1-\\varepsilon)c^{(i)}_{k-1}+\\varepsilon h^{(i)}_{k}$\n        \\State $x^{(i)}_{k} = c^{(i)}_{k} + D R(\\phi^{(i)}+\\omega k)$\n    \\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\nChoosing six agents forces the use of a regular hexagon for~\\cite{circular2015distributed}. We therefore included the composite method using a regular hexagon formation for comparison. The neighbours are chosen to be the adjacent vertices as in Figure~\\ref{fig:formation2d}.\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[t]{0.2\\textwidth}\n        \\begin{tikzpicture}\n       \n        \\node[draw,minimum size=3cm,regular polygon,regular polygon sides=6] (a) {};\n\n       \n        \\foreach \\x in {1,2,...,6}\n          \\fill (a.corner \\x) circle[radius=2pt];\n\n        \\end{tikzpicture}\n        \\caption{Hexagonal\\label{fig:formation2d}}\n    \\end{subfigure}%\n    ~\n    \\begin{subfigure}[t]{0.2\\textwidth}\n      \\begin{tikzpicture}\n     \n      \\node[draw,minimum size=3cm,regular polygon,regular polygon sides=4] (a) {};\n     \n      \\node[draw,minimum size=3cm,regular polygon,regular polygon sides=4,right of= a,xshift=1.12cm] (b) {};\n\n     \n      \\foreach \\x in {1,2,...,4}\n        \\fill (a.corner \\x) circle[radius=2pt];\n\n     \n      \\foreach \\x in {1,4}\n        \\fill (b.corner \\x) circle[radius=2pt];\n\n      \\end{tikzpicture}\n      \\caption{Rectangular\\label{fig:rectangleForm}}\n    \\end{subfigure}\n    \\caption{Neighbour topology for six agents in two dimensions.}\n\\end{figure}\n\nWhile the circular motion controller in~\\cite{circular2015distributed} requires this hexagonal arrangement for six agents, the framework proposed in this paper is flexible in the choice of formation by changing the ideal displacements $\\hat{x}^{(ij)}$. To this end we also include a rectangular formation, illustrated in Figure~\\ref{fig:rectangleForm}. As shown in~\\cite{michael2020optimisation}, the gradient estimation error bound is a function of the orthogonality of the neighbours as well as the distance between them, so the rectangular formation will have lower gradient estimation error than the hexagonal formation with the same neighbour distances.\n\nFigures~\\ref{fig:trajComp} shows the resulting trajectories from the composite method. We exclude the trajectories from other methods, as they are visually identical. Instead, we include the comparison of the tracking error $\\frac{1}{2}d(x_{k+1},\\mathcal{X}^*_{\\hat{f}_{k+1}})^2$ in Figure~\\ref{fig:minErrorComp} for each method, including the theoretical bounds from Theorem~\\ref{thm:compositeConvergence}.\n\n\\begin{figure}[thpb]\n \\centering\n \\includegraphics[scale=0.45]{traj.pdf}\n \\caption{Agent Trajectories using the composite method from Section~\\ref{sec:coopGradDesc}. \\label{fig:trajComp}}\n\\end{figure}\n\n\\begin{figure*}[thpb]\n \\centering\n \\includegraphics[scale=0.33]{errorComp.pdf}\n \\caption{Comparison of formation distance from the signal source. \\label{fig:minErrorComp}}\n\\end{figure*}\n\nWe can see from Figure~\\ref{fig:minErrorComp} that the theoretical minimiser error bound derived in Theorem~\\ref{thm:compositeConvergence} holds in simulation. All methods exhibit similar performance, including the periodic increases in tracking error, i.e. the five ``bumps'' in Figure~\\ref{fig:minErrorComp}. These coincide with the source accelerating around the curves of the path. The circular formation has higher tracking error, but the method in~\\cite{circular2015distributed} is not explicitly designed to operate on time-varying scalar fields. The rectangular and hexagonal formations using the composite method track nearly identically, although the rectangular formation converges slightly closer to the optimal value set due to the lower gradient error.\n\nAlthough all methods achieve similar performance in tracking the time varying minimiser, the composite method derived in this paper can work in any formation, with any number of agents, in any dimension, without the need to delineate leaders and followers, and is guaranteed to converge to a bounded neighbourhood of the minimiser. Further, we can see from Figure~\\ref{fig:minErrorComp} that formations with orthogonal neighbours improve gradient information, and lead the better source tracking, as the rectangular formation of agents has a lower tracking error by the end of the simulation.\n\nIn Figure~\\ref{fig:gradError}, we show the error of the estimated gradient, as well as the error bound for each agent derived from the results of Theorem~\\ref{thm:boundingEllipse}, defined in~\\eqref{eq:gradErrorBound}.\n\n\\begin{figure}[thpb]\n \\centering\n \\includegraphics[scale=0.37]{gradError.pdf}\n \\caption{Gradient estimation error (black) and estimation error bound (red) for each agent from Figure~\\ref{fig:trajComp} \\label{fig:gradError}}\n\\end{figure}\n\nAs the results from Section~\\ref{sec:coopGradDesc} generalise to any dimension, we provide an example in three dimensions, as well as an implementation of the extremum seeking algorithm from Section~\\ref{sec:coopGradDesc}, at the provided link.\\footnote{\\url{https:\/\/tinyurl.com\/yc4fzpv2}}\n\n\\section{Conclusion}\\label{sec:conclude}\n\nIn this paper we consider a formation of agents tracking the optimum of a time varying scalar field with no gradient information, in arbitrary dimension. At each iteration, the agents take measurements, communicate with their neighbours to estimate a descent direction, and converge to a neighbourhood of the optimum. We derive distributed control laws which drive the agents to a bounded neighbourhood of the optimiser set, without the delineation of leaders\/followers or the use of communication intensive consensus protocols. The method is flexible to the choice of formation and gradient estimation method, and we provide examples using two formations and gradient estimation using the simplex gradient. By blending formation control with extremum seeking, the agents are able to minimise the gradient estimation error, improving the neighbourhood of convergence. We concluded with numerical studies showing that the proposed method is comparable with other extremum seeking methods, converging to a tighter neighbourhood while being more flexible in the choice of formation. Further research will focus on the relaxing of the assumptions on the formation potential functions, allowing for potential functions with non unique minima which do not satisfy the Polyak-\\L{}ojasiewicz inequality, and incorporating time-varying neighbour sets.\n\n\n\n\\bibliographystyle{plain}       \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\n\\subsection{Overview and Motivation}\n\n\n\n\nModern technologies enable Internet resources such as routers, computing servers and cables to be abstracted from the physical layer to a \\emph{virtual} layer, facilitating a quick response to demands for setting up communication networks or processing computing jobs. Virtual servers comprising different sets of physical resources are assigned to arriving customers who use these resources for a period of time and then return them to a pool when they depart. \n\n\n\n\nSuch networks are just particular examples of more general systems where users of different types arrive with a desire to be allocated resources of various kinds, to use these resources and then return them. Users are often indifferent to the precise set of resources that they are allocated, they just require allocation of some resources that will enable them to accomplish the task at hand. In such circumstances a network manager has the task of deciding whether an arriving customer should be admitted into the system and, if so, which set of resources should be assigned to satisfying their requirements.\n\nIn this paper we describe and analyze a very general model for such systems. Specifically, we study a system in which $J$ \\emph{resource pools}, each made up of finite numbers of \\emph{resource units} (RUs), await allocation to incoming requests of $L$ different types. We refer to the number of RUs in a resource pool as its \\emph{capacity}. Each resource\npool is potentially shared and \\emph{competed} for by many requests, but \\emph{reservation} of RUs for still-to-arrive requests is also allowed. When a request has been accommodated by a resource pool, an appropriate number of RUs of this type are occupied by the request until it leaves the system. The released RUs can be reused by other requests. A request is permitted to occupy RUs from more than one resource pool simultaneously.  In this context, the number of requests of the same type that are accommodated by a group of resource pools varies according to a stochastic process, where the transition rates are affected by the resource allocation policy employed. Several such processes associated with the same resource pool are coupled by its capacity limitations.\n\n\nBy strategically assigning requests to appropriate combinations of RUs, we aim to maximize the long-run average revenue, defined as the difference between the long-run average reward earned by serving the requests and the long-run average cost incurred by using the resource pools. \nSuch a resource allocation problem can be easily applied to a rich collection of classical models, such as loss networks in telecommunications, resource allocation for logistic systems, and job assignment in parallel computing. \n\n\\cite{kelly1991loss} published a comprehensive analysis of \\emph{loss network} models with and without \\emph{alternative routing}. In the latter case, network traffic can be re-routed onto alternative paths when the original path fails or is full. \nIn \\cite{kelly1991loss}, a list of alternative paths as choices of resource pools is given for each call\/request. The alternative paths are selected in turn after if preceding offered paths are unavailable. In contrast, the manager of a typical resource allocation problem described above is potentially able to change the priorities of paths dynamically. How this should be done is a key focus of this paper.\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{.45\\textwidth}\n\\centering\n\\includegraphics[width=0.4\\linewidth]{loss_network.eps}\n\\caption{A simple loss network.}\\label{fig:loss_network}\n\\end{minipage}\n\\begin{minipage}{.45\\textwidth}\n\\centering\n\\includegraphics[width=0.4\\linewidth]{queue_model.eps}\n\\caption{A simple parallel queueing model.}\\label{fig:queue_model}\n\\end{minipage}\n\\vspace{-0.5cm}\n\\end{figure}\nTo illustrate the kind of problem of interest here, consider the simple loss network model shown in Figure~\\ref{fig:loss_network}. Links $a$, $b$ and $c$ are abstracted as resource pools with capacities equal to 1, 3 and 3, respectively: link $a$ consists of one channel as an RU, and links $b$ and $c$ each have 3 channels. Requests asking for a connection from $A$ to $B$ occupying one channel can be served by either path $\\{a\\}$ or $\\{b,c\\}$, but requests requiring two channels for each connection from $A$ to $B$ are able to be accommodated only by path $\\{b,c\\}$. We refer to the former and the latter as type-I and type-II requests, respectively.  An arrival of a type-I request results in one of the paths $\\{a\\}$ and $\\{b,c\\}$ being chosen by the optimizer depending on current traffic loads on the three links, where links $b$ and $c$ might be shared with existing type-II requests. Occupied channels or RUs are released immediately and simultaneously when relevant requests are completed.  \n\nResource allocation problems with small values of $L$ and $J$, such as the example above, can be modeled by a Markov Decision Processes (MDP), and solved through dynamic programming. However, in real-world applications, where $L$ and $J$ are large, resulting in high dimensionality of the state and action spaces, such an approach is often intractable.\n\n\n\nIn this paper we use an analysis inspired by techniques applied to Restless Multi-Armed Bandit Problems (RMABPs).\nThe standard RMABP consists of parallel MDPs with binary actions (they can either be ``pulled', that is activated, or not), which are competing for a limited possibility of being selected at each decision epoch. Each of the MDPs, referred to as a \\emph{bandit process}, has its own individual state-dependent reward rates and transition probabilities when it is activated and when it is not. \n\nAttempts to solve the problem are faced with exponential growth in the size of the state space as the number of parallel bandit processes increases.  \nThis class of problems was described  by \\cite{whittle1988restless}, who proposed a heuristic management policy that was shown to be  asymptotically optimal under non-trivial extra conditions  by~\\cite{weber1990index};\nthis policy approaches optimality as the number of bandit processes tends to infinity.\nThe policy, subsequently referred to as the \\emph{Whittle index policy}, always prioritizes bandit processes with higher state-dependent \\emph{indices} that intuitively represent marginal rewards earned by processes if they are selected.\nThe Whittle indices can be computed independently for each bandit process - a process that imposes significantly reduced computational complexity. The Whittle index policy is scalable to a RMABP with a large number of bandit processes.\nAlso, the asymptotic optimality property, if it is satisfied, guarantees a bounded performance degradation in a large-scale system and is appropriate for large problems where optimal solutions are intractable.\nThe non-trivial extra conditions required by the asymptotic optimality proof in \\cite{weber1990index} are related to proving the existence of a global attractor of a stochastic process.\n\nRMABPs have been widely used in scheduling problems, such as channel detecting (see \\cite{liu2012learning,wang2019whittle} ), job assignments in data centers (see \\cite{fu2016asymptotic}), web crawling (see \\cite{avrachenkov2016whittle}), target tracking (see \\cite{krishnamurthy2007structured,le2010scheduling}) and job admission control (see \\cite{nino2012admission,nino2019resource}).\nHere we treat the resource allocation problem described above as a set of RMABPs coupled by linear inequalities involving random state and action variables.\n\n\n\\subsection{Main Contributions}\n\n\n\n\nWe propose a modified \\emph{index policy} that takes into account the capacity constraints of the problem. The index policy prioritizes combinations of RUs with the highest indices, each of which is a real number representing the marginal revenue of using its associated RUs. The policy is simple, scalable and appropriate for a large scale resource allocation problem.\n\n\n \nOur analysis of asymptotic optimality of the index policy proceeds through a relaxed version of the problem and study of a global attractor of a stochastic process defined in \\eqref{eqn:z_process} below.\nWe prove that the process \\eqref{eqn:z_process}  will almost surely converge to a global attractor in the asymptotic regime regardless of its initial point, and hence the index policy is asymptotically optimal if and only if this global attractor coincides with an optimal solution of the resource allocation problem. \nFollowing ideas similar to those of \\cite{weber1990index}, optimality of the global attractor for the resource allocation problem can be deduced from its optimality for the relaxed problem, which can be analyzed with  remarkably reduced computational complexity. \n\nA sufficient condition for the global attractor and optimal solution to coincide is that the offered traffic for the entire system is \\emph{heavy} and the resource pools in our system are \\emph{weakly coupled}.\nWe rigorously define these concepts in Section~\\ref{subsec:sufficient_condition}.\nThese results are enunciated in Theorems~\\ref{theorem:main_second}, \\ref{theorem:main} and Corollary~\\ref{coro:main} in Section~\\ref{sec:asym_opt}.\n\nWhen the above-mentioned sufficient conditions are not satisfied, an asymptotically optimal index policy can still exist. In this case, we propose a method that can derive the parameters required by the asymptotically optimal policy.\nAlthough asymptotic optimality is not guaranteed, \nTheorem~\\ref{theorem:main} provides a verifiable sufficient condition, less stringent than the one mentioned above, to check asymptotic optimality of the index policy with adapted parameters. \nWe numerically demonstrate the effectiveness of this method in Section~\\ref{sec:example}.\n\n\nThe index policy exhibits remarkably reduced computational complexity, compared to conventional optimizers, and its potential asymptotic optimality is appropriate for large-scale systems where computational power is a scarce commodity. Furthermore, simulation studies indicate that an index policy can still be good in the pre-limit regime.\nAs mentioned earlier, our problem can be seen as a set of RMABPs coupled by the capacity constraints. When the capacities of all resource pools tend to infinity,  the index policy reduces to the Whittle index policy because the links between RMABPs no longer exist.\n\n\nTo the best of our knowledge, no existing work has proved asymptotic optimality in resource allocation problems, where resource competition and reservation are potentially permitted, nor has there been a previous analysis of such a combination of multiple, different RMABPs, resulting in a much higher dimensionality of the state space.\n\n\nThe remainder of the paper is organized as follows.\nIn Section~\\ref{sec:model}, we describe the resource allocation problem.\nIn Section~\\ref{sec:relaxation}, we apply the Whittle relaxation technique. \nIn Section~\\ref{sec:index_policy}, we propose an algorithm to implement an index policy.\nIn Section~\\ref{sec:asym}, we define the asymptotic regime and we prove the asymptotic optimality of the index policy under some conditions.\nTo demonstrate the effectiveness of the proposed policies, numerical results are provided in Section~\\ref{sec:example}.\nIn Section~\\ref{sec:conclusions}, we present conclusions.\n\n\n\n\n\n\\subsection{Relation to the Literature}\nThe classical Multi-Armed Bandit Problem (MABP) is a optimization problem in which only one bandit process (BP) among $K$ BPs can be activated at any one time, while all the other $K-1$ BPs are \\emph{frozen}: an active BP randomly changes its state, while state transitions will not happen to the frozen BPs.\nIn 1974, Gittins and Jones published the well-known \\emph{index theorem} for the MABP \\cite{gittins1974dynamic}, and in 1979, \\cite{gittins1979bandit} proved the optimality of a simple \\emph{index policy}, subsequently referred to as the \\emph{Gittins index policy}.\nUnder the Gittins index policy, an index value, referred to as the \\emph{Gittins index}, is associated with each state of each BP, and the BP with the largest index value is activated, while all the other BPs are frozen.\nMore details about Gittins indices can be found in \\cite[Chapter 2.12]{gittins2011multiarmed} (and the references therein).\n\nThe optimality of the Gittins index policy for the conventional MABP fails for the general case where the $K-1$ BPs that are not selected can also change their states randomly; such a process is known as a Restless Multi-Armed Bandit Process (RMABP).\nThe RMABP was proposed by \\cite{whittle1988restless}.\nThe RMABP allows $M=1,2,\\ldots,K$ BPs to be active simultaneously.\nIn a similar vein to the Gittins index policy, Whittle assigned a state-dependent index value, referred to as the \\emph{Whittle index}, to each BP and always activated the $M$ BPs with the highest indices. \nThe Whittle indices are calculated from a \\emph{relaxed} version of the original RMABP obtained by randomizing the action variables. \n\\cite{whittle1988restless} defined a property of a RMABP, referred to as \\emph{indexability}, under which the \\emph{Whittle index policy} exists.\nWhittle conjectured in \\cite{whittle1988restless} that the Whittle index policy, if it exists, is \\emph{asymptotically optimal}.\n\\cite{papadimitriou1999complexity} proved that the optimization of RMABPs is PSPACE-hard in general;\nnonetheless, \\cite{weber1990index} were able to establish asymptotic optimality of Whittle index policy under mild conditions. \n\n\n\n\\cite{nino2001restless} proposed a Partial Conservation Law\n(PCL) for the optimality of RMABP;  this is an\nextension of  the General Conservation Law (GCL) published in\n\\cite{bertsimas1996conservation}.  \nLater, \\cite{nino2002dynamic} defined a group of problems that satisfies\nPCL-indexibility and proposed a new index policy that improved the\nWhittle index. \nThe new index policy was proved to be optimal for problems with PCL-indexibility.  \nPCL-indexibility implies (and is stronger than) Whittle indexibility.\nA detailed survey about the optimality of bandit problems can be found in \\cite{nino2007dynamic}. \n\n\n\\cite{verloop2016asymptotically} proved the asymptotic optimality of the Whittle index policy in an extended version of an RMABP, where BPs randomly arrive and depart the system. She proposed an index policy that was not restricted to Whittle indexable models and numerically demonstrated its near-optimality. \n\\cite{larranaga2015asymptotically} applied this extended RMABP to a queueing problem assuming convex, non-decreasing functions for both holding costs and measured values of people's impatience.\nMore results on  asymptotic optimality of index-like polices can be\nfound in \\cite[Chapter IV]{fu2016thesis}. \n\n\n\n\n\n\nAsymptotically optimal policies for cost-minimization problems in network systems using a fluid approximation technique have been considered in \\cite{bauerle2000asymptotic,bauerle2002optimal,stolyar2004maxweight,\nnazarathy2009near} and \\cite{bertsimas2015robust}.\nThe fluid approximation to the stochastic optimization problem can be much simpler than the original.\nA key problem here is to establish an appropriate fluid problem and translate its optimal solution to a policy amenable to the stochastic problem.\nAsymptotic optimality of the translated stochastic policy can be established if the fluid solution provides an upper\/lower bound of the stochastic problem and the policy coincides with this bound asymptotically.\nThe reader is referred to \\cite{meyn2008control} for a detailed description of fluid approximation across various models. \n\n\n\nAlthough the fluid approximation technique helps with asymptotic analysis in a wide range of (cost-minimization) network problems, existing results cannot be directly applied to our problem, where the arrival and departure rates of request queues are state-dependent and capacity violation over resource pools is strictly forbidden.\nOur system is always stable for any offered traffic because of the strict capacity constraints.\nIn our case, the form of the corresponding fluid model remains unclear for generic policies. \nEven given the optimal solution of a well-established fluid model, the synthesis of an explicit policy in the stochastic model remains a challenge.\n\nWe adopt another approach, following   the ideas of \\cite{whittle1988restless} and \\cite{weber1990index}.\nOur asymptotic optimality is derived from an optimal solution of a relaxed version of the stochastic optimization problem. The relaxed problem is still a stochastic optimization problem with a discrete state space. We propose a policy based on intuition captured by the relaxed problem, of which the optimal solution provides a performance upper bound of the original problem.\nThen, we prove, under certain conditions, that this policy coincides with the upper bound asymptotically. \nThe detailed analysis comprises the main content of the paper.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A Resource Allocation Problem}\\label{sec:model}\n\n\n\n\nWe use $\\mathbb{N}_{+}$ and $\\mathbb{N}_{0}$ to denote the sets of positive and non-negative integers, respectively, and for any $N\\in\\mathbb{N}_{+}$, let $[N]$ represent the set $\\{1,2,\\ldots,N\\}$ with $[0]=\\emptyset$.\nLet $\\mathbb{R}$, $\\mathbb{R}_{+}$ and $\\mathbb{R}_{0}$ be the set of all, positive and non-negative reals, respectively.\n\n\n\n\\subsection{System Model}\\label{subsec:model}\n\nRecall that there are $L$ types of requests and $J$ pools of RUs, all\npotentially different, with  resource pool $j\\in[J]$ having capacity $C_j$\nRUs that can be dynamically allocated to and released by the $L$ types\nof requests.\n\n\nEach request comes with an associated list of candidate resource combinations. \nSpecifically, requests from \\emph{request type} $\\ell \\in [L]$ can be accommodated by one of a set $\\mathscr{P}_{\\ell}$ of candidate \\emph{patterns}.\nOne of these candidate patterns will be selected by a policy.\nPatterns are  indexed by $i\\in\\mathbb{N}_+$. \nIf a request is accommodated by pattern $i$, $w_{j,i}$\nRUs of pool $j\\in [J]$ are occupied until the request is completed and\ndeparts. \nWe can thus identify pattern $i$ with the \\emph{weight vector} $\\bm{w}_i = (w_{j,i})$ that defines its requirement.\nPreemption or re-allocation of requests\nare not allowed.\nA request is blocked if there is not enough capacity on any\nof its corresponding patterns. We might also want to block a request in\nother circumstances, if accepting it would be detrimental to future\nperformance.  In either case, we model the situation where a request\nis blocked by assigning it to the dummy pattern $d(\\ell)$ with the weight vector set to $\\bm{0}$.\n\n\n\n\n\n\n\nIt is possible for different RTs to be satisfied by the same pattern (this occurs, in particular with the dummy pattern). In such cases, we consider there to be multiple copies of each pattern, one for each RT that it can satisfy. \nThis enables us to consider the sets $\\mathscr{P}_\\ell$ \nto be mutually\nexclusive; that is,\n$\\mathscr{P}_{\\ell_1}\\cap \\mathscr{P}_{\\ell_2} = \\emptyset$ for any\n$\\ell_1\\neq \\ell_2$.  Given\n$|\\mathscr{P}_{\\ell}|$ patterns for each RT $\\ell$, we have in total\n$I = \\sum_{\\ell\\in[L]}|\\mathscr{P}_{\\ell}|$ patterns \nassociated with weight vectors $\\bm{w}_i\\in\\mathbb{N}_0^{J}$,\n$i\\in[I]$.\nFor any pattern $i$, let $\\ell(i)$ be the unique RT that is satisfied by that pattern.\n\nLet $\\mathcal{W}$ be a $J\\times I$ matrix with entries $w_{j,i}$.\nWe assume that there is  no row and exactly $L$ columns in $\\mathcal{W}$ with all zero entries.\nEach of these zero columns corresponds to one of the dummy patterns $d(\\ell)$ where requests of type $\\ell\\in[L]$ are blocked.\n\nRequests of RT $\\ell$ arrive at the system sequentially, following a Poisson process, with rates $ \\lambda_{\\ell}$ \nand the occupation times of the requests accommodated by pattern $i\\in\\mathscr{P}_{\\ell}$ are exponentially distributed with parameter $\\mu_{i}$. \nAlthough there might be situations when it is reasonable to assume that the occupation time depends only on the request type $\\ell$, there might also be cases where the lifetime of a request depends on the resources accommodating it, which is why we allow the occupation time distribution to depend on $i$.\nThe RUs used to accommodate a request are  occupied and released at the same time.\nNeither the request nor the system knows the lifespan of a request until it is accomplished and departs the system.\n\n\n\nSince there are similarities between our problem and a parallel\nqueueing model, we present a second example to clarify the similarities and differences. \nConsider two\nresource pools corresponding to two queues as illustrated in\nFigure~\\ref{fig:queue_model}, where both capacities are set to three; that is, $J=2$ and $C_1=C_2=3$.  There are two types\nof requests: if a type-one request is accommodated in the system, it\nwill simultaneously occupy one RU of both pools;\nand a type-two request can be accommodated by two RUs of either\npool. In other words, $L=2$, $\\mathscr{P}_{1}=\\{1,2\\}$,\n$\\mathscr{P}_2=\\{3,4,5\\}$, patterns $2$ and $5$ are dummy patterns\nwith $\\bm{w}_2=\\bm{w}_5=\\bm{0}$, $\\bm{w}_1=(1,1)$, $\\bm{w}_3=(2,0)$,\n$\\bm{w}_4=(0,2)$ and $I=5$.  \n\n\nIn this case, the number of occupied RUs in both resource pools may decrease or increase\nby one simultaneously, or by two exclusively for  an arrival or\ndeparture event.\nThe transition rates are affected by the system\ncontroller: if the capacity constraints are not violated, there are\ntwo choices, resource pool one or two, for accommodating a type-two request. The task of a system manager is to find a policy for deciding which of these choices to take in order to maximize some long-term objective.\nEach choice will result in a parallel queueing model with dependencies\nbetween the sizes of queues,  between the policy employed and\nqueue transition rates.   As mentioned in Section~\\ref{sec:introduction},\nconventional optimization methods cannot be  applied directly when $L$ and $J$ are large.\n\n\n\n\n\n\n\\subsection{A Stochastic Optimization Problem}\\label{subsec:RMABPs:general_case}\n\nWe focus here an explanation of\nthe stochastic mechanism of the resource allocation problem.\n\n\nAn \\emph{instantiation} is \ngenerated in the memory of the system controller\nwhen a request of RT $\\ell\\in[L]$ is accommodated by a pattern $i\\in\\mathscr{P}_{\\ell}$. \nOnce the request departs the system, the associated instantiation will\nbe removed from the controller's memory.  As requests are accommodated\nand completed, the number of instantiations associated with each\npattern forms a birth-and-death process,  indicating the\nnumber of requests being served by this pattern.  As mentioned\nin the second example, the birth-and-death processes for all patterns\n$i\\in[I]$ are coupled by capacity constraints and affected by control\ndecisions.\n\n\nLet $N_i(t)$, $t\\geq 0$,  represent the number of instantiations\nfor pattern $i$ at time $t$.  The\nprocess $N_i(t)$ has state space $\\mathscr{N}_i $ that  is a discrete,\nfinite set of possible values. The finiteness of $\\mathscr{N}_i$ derives from the finite\ncapacities $C_j$.  \nIf  $N_i (t)$ is known for all $i \\in [I]$, \nthe number of occupied RUs in pool $j \\in [J]$ at time $t$ is given by\n$S_j(t)=\\sum_{i \\in [I]} w_{j,i}N_i(t)$, which must be less than $C_j$.\nThe  vector $\\bm{N}(t)=(N_i(t):\\ i\\in[I])$ is  the state variable of the entire system taking values in\n$\\mathscr{N}\\coloneqq\\prod_{i\\in[I]}\\mathscr{N}_i$,\nwhere $\\prod$ represents Cartesian product.\nSince the state variables are further subject to capacity constraints to be discussed in Section~\\ref{subsubsec:capacity_constraints}, $\\mathscr{N}$ is larger than necessary. With slightly abused notation, we still refer to $\\mathscr{N}$ as the state space of the system. \n\n\\subsubsection{Action Constraints}\\label{subsubsec:action_constraints}\nWe associate an action variable $a_i(\\bm{n})\\in\\{0,1\\}$ with\nprocess $i\\in[I]$ when the system is\nin state $\\bm{n}\\in\\mathscr{N}$, and $\\bm{a}(\\bm{n})=(a_i(\\bm{n}):\\ i\\in[I])$.  The action variable $a_i(\\bm{n})$ tells us what to do with a potential new request of type $\\ell(i)$. If $a_i(\\bm{n}) =1$, then such a pattern will be allocated to pattern $i$.\nThe \\emph{action constraint},\n\\begin{equation}\\label{eqn:constraint:action}\n\\sum\\limits_{i\\in\\mathscr{P}_{\\ell}} a_i(\\bm{n}) = 1,~\n\\forall \\ell\\in[L],~\n\\forall \\bm{n}\\in\\mathscr{N},\n\\end{equation}\nensures that exactly one pattern (which may be the dummy pattern $d(\\ell)$) is selected for each RT $\\ell$ and current state $\\bm{n}$.\n\n\nAt any time $t$, we say that the arrival process for pattern $i$ is \\emph{active} or \\emph{passive} according to whether $a_i(\\bm{N}(t))$ is $1$ or $0$ respectively.\nThe birth rate of process $i\\in\\mathscr{P}_{\\ell}$,\n$\\ell\\in[L]$, is $\\lambda_{\\ell}$ if $a_i(\\bm{N}(t))=1$;\nand zero otherwise. The death rate of process $i$ is\n$\\mu_{i} N_{i}(t)$.\nThe time proportion that $a_{d(\\ell)}(\\bm{N}(t))=1$ is the \\emph{blocking probability} for requests of type $\\ell$.\n \n \n\n\\subsubsection{Capacity Constraints}\\label{subsubsec:capacity_constraints}\n\nTo ensure feasibility of an allocation of a request of type $\\ell(i)$ to pattern $i$ when the state is $\\bm{n}$, we need \n\\begin{equation}\\label{eqn:constraint:resourcesi}\n\\mathcal{W}\\left(\\bm{n}+ \\bm{e}_i\\right) \\leq \\bm{C},~\n\\end{equation}\nwhere $\\bm{e}_i$ is a vector with a one in the $i$th position and zeros everywhere else and $\\bm{C}\\in\\mathbb{N}_{+}^{J}$ is a vector with entries $C_j$. In view of the action constraint (\\ref{eqn:constraint:action}), a neat way to collect together the constraints (\\ref{eqn:constraint:resourcesi}) for all $i \\in \\mathscr{P}_{\\ell}$ is to write them in the form\n\\begin{equation}\\label{eqn:resources1}\n\\mathcal{W}\\left(\\bm{n}+\\mathcal{E}_{\\ell}\\bm{a}(\\bm{n})\\right) \\leq \\bm{C},~\\forall \\bm{n}\\in\\mathscr{N},\n\\end{equation}\nwhere $\\mathcal{E}_{\\ell}$ is a diagonal matrix of size $I$ with entries $e_{\\ell,i,i}=1$ if $i\\in\\mathscr{P}_{\\ell}$ and $e_{\\ell,i,i}=0$ if $i\\in[I]\\backslash \\mathscr{P}_{\\ell}$.\n\n\n\nFor two different request types $\\ell_1$ and $\\ell_2$, a constraint of the form \n\\begin{equation}\\label{eqn:resources12}\n\\mathcal{W}\\left(\\bm{n}+\\mathcal{E}_{\\ell_1}\\bm{a}(\\bm{n})+\\mathcal{E}_{\\ell_2}\\bm{a}(\\bm{n})\\right) \\leq \\bm{C},~\\forall \\bm{n}\\in\\mathscr{N},\n\\end{equation} \ncaptures the idea that the action vector $\\bm{a}(\\bm{n})$ must be such that the allocation decisions for $\\ell_1$ and  \n$\\ell_2$ ensure enough capacity to implement both of them when both requests arrive simultaneously while the state is\n$\\bm{n}$. Another way to think about this is that, if a request of type $\\ell_1$ is allocated to a non-dummy pattern $i_1$ when the state is $\\bm{n}$, the decision for a request of type $\\ell_2$ when the state is $\\bm{n}$ must satisfy constraint \\eqref{eqn:resources1} when the state is $\\bm{n}+\\bm{e}_{i_1}$. In particular, if there is not enough capacity to accommodate a request of type $\\ell_2$ when the state is $\\bm{n}+\\bm{e}_{i_1}$, then a request of type $\\ell_2$ must be allocated to the dummy pattern $d(\\ell_2)$, when the state is $\\bm{n}$.\nThis can be viewed as giving priority to \\emph{reserving} resources for a type $\\ell_1$ request over a type $\\ell_2$ request\nwhen the state is $\\bm{n}$. \nAs we shall see below, the decision to do this will be made in order to optimize a long-term reward function.\n\n\n\nObserving that $\\sum_{\\ell \\in [L]} \\mathcal{E}_{\\ell} = I$, we see that the constraint \n\\begin{equation}\\label{eqn:constraint:resources}\n\\mathcal{W}\\left(\\bm{n}+ \\bm{a}(\\bm{n})\\right) = \\mathcal{W}\\Bigl(\\bm{n}+ \\bigl(\\sum_{\\ell \\in [L]} \\mathcal{E}_{\\ell}\\bigr) \\bm{a}(\\bm{n})\\Bigr) \\leq \\bm{C},~\n\\forall \\bm{n}\\in\\mathscr{N},\n\\end{equation}\ncan be thought of as an extended version of \\eqref{eqn:resources12}.\nIn \\eqref{eqn:constraint:resources}, requests of all types are taken into account when the state is $\\bm{n}$  and allocation decisions for some types are made in order to reserve resources for other types that turn out to be more profitable in the long run. In particular, resources are reserved for those request types $\\ell$ which are allocated to non-dummy patterns $i$ at the expense of those types that are allocated to less profitable patterns or the corresponding dummy patterns.\nIn this paper, all the results presented are based on capacity constraint~\\eqref{eqn:constraint:resources}.\n\n\nFrom \\eqref{eqn:constraint:resources}, there is an upper bound, $\\min_{j\\in[J]}\\lceil C_j\/w_{j,i} \\rceil$, on  the number of instantiations of pattern $i$, and this serves as a bounding state at which no further instantiation of this pattern can be added; that is, $\\mathscr{N}_i=\\{0,1,\\ldots,\\min_{j\\in[J]}\\lceil C_j\/w_{j,i}\\rceil\\}$ and $|\\mathscr{N}_i|=\\min_{j\\in[J]}\\lceil C_j\/w_{j,i}\\rceil +1 <+\\infty$.\nIn this context, Equation~\\eqref{eqn:constraint:resources} implies the condition\n\\begin{equation}\\label{eqn:constraint:resources:zero}\na_i(\\bm{n}) = 0, ~\\text{if}~i\\notin \\{d(\\ell):\\ \\ell\\in[L]\\} \\text{ and } n_i = |\\mathscr{N}_i|-1.\n\\end{equation}\n\n\n\n\\subsubsection{Objective}\nA \\emph{policy} $\\phi$ is defined as a mapping $\\mathscr{N}\\rightarrow \\mathscr{A}$ where\n$\\mathscr{A}\\coloneqq\\prod_{\\ell\\in[L]}\\{0,1\\}^{|\\mathscr{P}_{\\ell}|}$,\ndetermined by the action variables $\\bm{a}(\\bm{n})$ defined above.\nWhen we are discussing a system operating with a given policy $\\phi$, we rewrite the action and state variables as $\\bm{a}^{\\phi}(\\cdot)$ and $\\bm{N}^{\\phi}(t)$, respectively.\n\nBy serving a request of type $\\ell\\in[L]$ and occupying an RU of pool\n$j$ for one unit of time, we gain expected reward $r_{\\ell}$ and pay\nexpected cost $\\varepsilon_j$.  \nThe expected reward for a whole service is gained at the moment the service is completed. \nIt corresponds to the situation where a request allocated to pattern $i$ earns reward at rate $r_{\\ell(i)} \\mu_i$ for as long as it is in the system (so that the expected revenue per customer is $(r_{\\ell(i)} \\mu_i) . (1\/\\mu_i) = r_{\\ell(i)}$).\nThe value of $\\epsilon_j$ is the cost per unit time of using a unit of capacity from resource pool $j$ in which case the expected cost of accommodating the request in pool $j$ as part of pattern $i$ is $\\epsilon_j\/\\mu_i$.\nWe seek a policy that\nmaximizes the \\emph{revenue}: the difference between expected reward\nand cost, by efficiently utilizing the limited amount of\nresources. \n\nThe objective is to maximize the long-run average rate of earning revenue, which exists because, for any policy $\\phi$, the process can be modeled by a finite-state Markov chain.\nLet $\\bm{r}=(r_{\\ell}:\\ \\ell\\in[L])$ and $\\bm{\\varepsilon}=(\\varepsilon_{j}:\\ j\\in[J])$. \nFor all $\\ell\\in[L]$ and $i\\in[I]$, define a $L\\times I$ matrix $\\mathcal{U}$ with  entries\n$u_{\\ell,i}\\coloneqq \\mu_{i}\\mathds{1}_{i\\in\\mathscr{P}_{\\ell}}$.\nBy the Strong Law of Large Numbers for Continuous Time Markov Chains, see for example \\cite{serfozo2009basics} Theorem 45 in Chapter 4, noting the subsequent discussion of the case where rewards are earned at jump times, the long-run average rate of earning revenue when the policy is $\\phi$ is given by\n\\begin{equation}\nR^{\\phi} \\coloneqq \\mathbb{E}_{\\pi^{\\phi}} \\Bigl[\\bm{r}\\mathcal{U} - \\bm{\\varepsilon}\\mathcal{W}\\Bigr] =\\sum\\limits_{i\\in[I]} \\sum_{n_i\\in\\mathscr{N}_i} \\pi_i^{\\phi}(n_i) \\Bigl(r_{\\ell(i)} \\mu_i  - \\sum_{j\\in\\mathscr{J}} w_{j,i} \\varepsilon_j\\Bigr)n_i,\n\\end{equation}\nwhere $\\pi_i^{\\phi}(n_i)$ is the stationary probability that the state of process $i$ is $n_i$ when the policy is $\\phi$. Then we wish to find the policy $\\phi$ that maximizes $R^\\phi$, that is we wish to find\n\\begin{equation}\\label{eqn:objective}\nR \\coloneqq \\max\\limits_{\\phi}R^{\\phi}.\n\\end{equation}\nDefine\n$\\Phi$ to be the set of all policies\nwith  the constraints in~\\eqref{eqn:constraint:action} and \\eqref{eqn:constraint:resources} satisfied.\nEach policy in $\\Phi$ is then a \\emph{feasible policy} for our resource allocation problem.\n\n\n\\section{Whittle Relaxation}\\label{sec:relaxation}\nOur resource\nallocation problem with objective function defined by \\eqref{eqn:objective} and constraints given by \\eqref{eqn:constraint:action} and \\eqref{eqn:constraint:resources} can be modeled as a set of RMABPs coupled by\ncapacity constraints.\nWe leave the\nspecification of the RMABPs to Appendix~\\ref{app:RMABPs}.\n\n\n\nIn this section, we provide a theoretical analysis of the resource allocation problem,\nfollowing the idea of Whittle relaxation \\cite{whittle1988restless}.\nIn the vein of a RMABP, \nwe randomize the action variable $\\bm{a}^{\\phi}(\\bm{n})$ so that its elements take values from $\\{0,1\\}$ with probabilities determined by the policy $\\phi$ and relax constraint~\\eqref{eqn:constraint:action} to require that\n\\begin{equation}\\label{eqn:constraint:relax:action}\n\\lim\\limits_{t\\rightarrow +\\infty}\\mathbb{E}\\biggl[\\sum\\limits_{i\\in\\mathscr{P}_{\\ell}} a^{\\phi}_i(\\bm{N}^{\\phi}(t)) \\biggr]= 1,~\n\\forall \\ell\\in[L].\n\\end{equation}\nFollowing similar ideas, we relax \\eqref{eqn:constraint:resources} into two equations:\n\\begin{equation}\\label{eqn:constraint:relax:resources}\n\\lim\\limits_{t\\rightarrow +\\infty}\\mathbb{E}\\biggl[\\mathcal{W}\\Bigl(\\bm{N}^{\\phi}(t)+ \\bm{a}^{\\phi}(\\bm{N}^{\\phi}(t))\\Bigr)\\biggr] \\leq \\bm{C},\n\\end{equation}\nand  \n\\begin{equation}\\label{eqn:constraint:dummy}\n\\lim\\limits_{t\\rightarrow+\\infty} \\mathbb{E}\\Bigl[a^{\\phi}_i(\\bm{N}^{\\phi}(t))\\ \\mathds{1}_{N^{\\phi}_i(t)=|\\mathscr{N}_{i}|-1}\\Bigr] = 0,~\\forall i\\in[I]\\backslash\\{d(\\ell):\\ \\ell\\in[L]\\}.\n\\end{equation}\n{\\bf Remark} \nEquation~\\eqref{eqn:constraint:relax:resources} \nis derived by taking expectations for both sides of Equation~\\eqref{eqn:constraint:resources},\nand \\eqref{eqn:constraint:dummy} is a consequence of  \\eqref{eqn:constraint:resources:zero}, \nso constraints described by~\\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy} are relaxed versions of the constraints described by~\\eqref{eqn:constraint:resources}.\nThe justification for Equation~\\eqref{eqn:constraint:dummy} will be discussed in Section~\\ref{subsec:asym_regime}, in conjunction with the physical meanings of all variables, when we increase the scale of the entire system.\nWe refer to the problem with objective~\\eqref{eqn:objective}, constraints~\\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy} and randomized control variables $\\bm{a}^{\\phi}(\\bm{n})$, for all $\\bm{n}\\in\\mathscr{N}$, as the \\emph{relaxed problem}.\n\n\nA value $a$ in $(0,1)$ can be interpreted as a randomisation between taking $a_i^{\\phi}(\\bm{n}) = 0$ and $a_i^{\\phi}(\\bm{n}) = 1$. Specifically we take $a_i^\\phi(n) = 1$ with probability $a$.\nWe represent the set of policies that correspond to assigning such values $a\\in(0,1)$ as $\\tilde{\\Phi}$.\nFor $n_i\\in \\mathscr{N}_i$, $\\phi\\in\\tilde{\\Phi}$, $i\\in[I]$, define\n\\begin{itemize}\n\\item $\\alpha^{\\phi}_i(n_i) \\coloneqq \\lim\\limits_{t\\rightarrow+\\infty} \\mathbb{E}\\left[a^{\\phi}_i(\\bm{N}^{\\phi}(t))\\ |\\ N^{\\phi}_i(t)=n_i\\right]$, \nwhich is the expectation with respect to the stationary distribution when policy $\\phi$ is used,\nand the vector $\\bm{\\alpha}^{\\phi}_i \\coloneqq (\\alpha^{\\phi}_i(n_i) :\\ n_i\\in\\mathscr{N}_i)$;\n\\item the stationary probability that $N^{\\phi}_i(t)=n_i$ under policy $\\phi$ to be $\\pi_{i,n_{i}}^{\\phi}$, and the vector $\\bm{\\pi}^{\\phi}_i \\coloneqq (\\pi_{i,n_i}^{\\phi}:\\ n_i\\in\\mathscr{N}_i)$.\n\\end{itemize}\nLet $\\bm{\\Pi}^{\\phi}_n \\coloneqq \\left( \\bm{\\pi}^{\\phi}_{i}\\cdot (\\mathscr{N}_i):\\ i\\in[I]\\right)^{T}$ and $\\bm{\\Pi}^{\\phi}_a \\coloneqq \\left(\\bm{\\pi}^{\\phi}_{i}\\cdot \\bm{\\alpha}^{\\phi}_i:\\ i\\in[I]\\right)^{T}$, where \n$(\\mathscr{N}_i)$ represents the vector $(0,1,\\ldots,|\\mathscr{N}_i|-1)$.\nThe Lagrangian function for the optimization problem with objective function (\\ref{eqn:objective}) and constraints \\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy} is\n\\begin{multline}\\label{eqn:dual_func}\ng(\\pmb{\\gamma},\\bm{\\nu},\\bm{\\eta})\\coloneqq\n\\max\\limits_{\\phi\\in\\tilde{\\Phi}}\n(\\bm{r}\\mathcal{U}-\\bm{\\varepsilon}\\mathcal{W})\\bm{\\Pi}^{\\phi}_n-\\sum\\limits_{\\ell=1}^{L}\\nu_{\\ell}\\Bigl(\\sum\\limits_{i\\in\\mathscr{P}_{\\ell}}\\bm{\\pi}^{\\phi}_i\\cdot \\bm{\\alpha}^{\\phi}_i -1 \\Bigr)\\\\\n- \\pmb{\\gamma}\\cdot \\Bigl(\\mathcal{W}(\\bm{\\Pi}^{\\phi}_n+\\bm{\\Pi}^{\\phi}_a)-\\bm{C}\\Bigr)\n-\\sum\\limits_{i\\in[I]\\backslash\\{d(\\ell):\\ \\ell\\in[L]\\}}\\eta_i \\pi^{\\phi}_{i,|\\mathscr{N}_i|-1} \\alpha^{\\phi}_{i}(|\\mathscr{N}_i|-1),\n\\end{multline}\nwhere\n$\\bm{\\nu}\\in\\mathbb{R}^{L}$, $\\pmb{\\gamma}\\in\\mathbb{R}_{0}^{J}$ and $\\bm{\\eta}\\in\\mathbb{R}^{I-L}$ are Lagrange multiplier vectors for constraints~\\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy}, respectively. \nIn~\\eqref{eqn:dual_func}, the constraints no longer apply to variables $\\bm{\\alpha}^{\\phi}_i$ ($i\\in[I]$) but appear in the maximization as cost items weighted by their Lagrange multipliers.\nFor $i\\in[I]\\backslash\\{d(\\ell):\\ \\ell\\in[L]\\}$,\ndefine functions \n\\begin{multline}\\label{eqn:dual_func_i}\n\\Lambda_{i}(\\phi,\\pmb{\\gamma},\\nu_{\\ell(i)},\\eta_i)\n\\coloneqq (r_{\\ell(i)}\\mu_i-\\bm{\\varepsilon}\\cdot\\bm{w}_i)\\bm{\\pi}^{\\phi}_i\\cdot (\\mathscr{N}_i)-\\nu_{\\ell(i)}\\bm{\\pi}^{\\phi}_i\\cdot \\bm{\\alpha}^{\\phi}_i \n- \\pmb{\\gamma}\\cdot \\bigl(\\bm{w}_i(\\bm{\\pi}^{\\phi}_i\\cdot (\\mathscr{N}_i)+\\bm{\\pi}^{\\phi}_i\\cdot \\bm{\\alpha}^{\\phi}_i)\\bigr)\\\\\n-\\eta_i \\pi^{\\phi}_{i,|\\mathscr{N}_i|-1}\\alpha^{\\phi}_{i}(|\\mathscr{N}_i|-1),\n\\end{multline}\nwhere we recall that $\\bm{w}_i$ is the weight vector of pattern $i$ given by the $i$th column vector of $\\mathcal{W}$; similarly, for $\\ell\\in[L]$, $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$ and $\\eta\\in\\mathbb{R}$, define\n$\\Lambda_{d(\\ell)}(\\phi,\\pmb{\\gamma},\\nu_{\\ell},\\eta) \\coloneqq -\\nu_{\\ell}\\alpha^{\\phi}_{d(\\ell)}(n)$, where $n$ is the only state in $\\mathscr{N}_{d(\\ell)}$.\nFrom Equation \\eqref{eqn:dual_func}, for $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$, $\\bm{\\nu}\\in\\mathbb{R}^L$ and $\\bm{\\eta}\\in\\mathbb{R}^{I-L}$,\n\\begin{equation}\\label{eqn:dual_func2}\ng(\\pmb{\\gamma},\\bm{\\nu},\\bm{\\eta})=\\max\\limits_{\\phi\\in\\tilde{\\Phi}}\\sum\\limits_{i\\in[I]}\\Lambda_{i}(\\phi,\\pmb{\\gamma},\\nu_{\\ell(i)},\\eta_i)\n+\\sum\\limits_{\\ell\\in[L]}\\nu_{\\ell}\n+ \\pmb{\\gamma}\\cdot \\bm{C}.\n\\end{equation}\nwhere $\\eta_{d(\\ell)}$ $(\\ell\\in[L])$ are unconstrained real numbers that are used for notational convenience. \n\n\n\n\n\nIn the maximization problem on the right hand side of~\\eqref{eqn:dual_func2}, there is no constraint that restricts the value of one $\\Lambda_i(\\phi,\\pmb{\\gamma},\\nu_{\\ell(i)},\\eta_i)$ once the others are known. \nAs a result, we can maximize the sum in \\eqref{eqn:dual_func2} by maximizing each of the summands independently. \nWe can thus write \\eqref{eqn:dual_func2} as\n\\begin{equation}\ng(\\pmb{\\gamma},\\bm{\\nu},\\bm{\\eta})=\\sum\\limits_{i\\in[I]}\\max\\limits_{\\phi\\in\\tilde{\\Phi}}\\Lambda_{i}(\\phi,\\pmb{\\gamma},\\nu_{\\ell(i)},\\eta_i)\n+\\sum\\limits_{\\ell\\in[L]}\\nu_{\\ell}\n+ \\pmb{\\gamma}\\cdot \\bm{C},\n\\end{equation}\nbut with the maximum over $\\phi\\in\\tilde{\\Phi}$. \nObserve now that maximizing $\\Lambda_i$ over $\\phi$ is equivalent to choosing $\\bm{\\alpha}^{\\phi}_i(n_i)$ from $[0,1]^{|\\mathscr{N}_i|}$, by interpreting $\\alpha^{\\phi}_{i,n}\\in[0,1]$ as the probability that process $i$ is activated under policy $\\phi$ when it is in state $n$. Thus,\n\\begin{equation}\\label{eqn:dual_problem}\ng(\\pmb{\\gamma},\\bm{\\nu},\\bm{\\eta})\n=\\sum\\limits_{i\\in[I]}\\max\\limits_{\\bm{\\alpha}^{\\phi}_i\\in[0,1]^{|\\mathscr{N}_i|}}\\Lambda_{i}(\\phi,\\pmb{\\gamma},\\nu_{\\ell},\\eta_i)\n+\\sum\\limits_{\\ell\\in[L]}\\nu_{\\ell} + \\pmb{\\gamma}\\cdot \\bm{C}.\n\\end{equation}\n\n\n\nBy slightly abusing notation, we refer to the policy $\\phi$ determined by an action vector $\\bm{\\alpha}^{\\phi}_i$ as the policy for pattern $i$,\nand define $\\Phi_{i}$ as the set of all policies for pattern $i$.\n\n\n\n\n\n\n\\begin{definition}\\label{def:sub-problem}\nThe maximization of $\\Lambda_{i}(\\phi,\\pmb{\\gamma},\\nu_{\\ell},\\eta_i)$ over $\\bm{\\alpha}^{\\phi}_{i}\\in[0,1]^{|\\mathscr{N}_{i}|}$ is the \\emph{sub-problem} for pattern $i\\in[I]$. \n\\end{definition}\n\nFor given $\\pmb{\\gamma}$, $\\bm{\\nu}$ and $\\bm{\\eta}$, the sub-problem for any pattern is an MDP, so that it can be numerically solved by dynamic programming. By solving the sub-problems for all patterns $i\\in[I]$, we obtain $g(\\pmb{\\gamma},\\bm{\\nu},\\bm{\\eta})$.\nFor any $\\pmb{\\gamma}$, $\\bm{\\nu}$ and $\\bm{\\eta}$, the Lagrangian function $g(\\pmb{\\gamma},\\bm{\\nu},\\bm{\\eta})$ is a performance upper bound for the primal problem described in \\eqref{eqn:objective}, \\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy}, which is a relaxed version of the original resource allocation problem. \nThus there will be a non-negative gap between this upper bound and the maximized performance of the original problem.\n\n\\subsection{Analytical Solutions}\\label{sec:indexability}\n\\begin{proposition}\\label{prop:a_opt}\nFor given $\\bm{\\nu}$ and  $\\pmb{\\gamma}$, there exists $\\bm{E}\\in\\mathbb{R}^{I-L}$ such that, for any $\\bm{\\eta}>\\bm{E}$, a policy of the sub-problem  for pattern $i$, referred to as $\\bar{\\varphi}\\in\\Phi_i$, determined by action vector $\\bm{\\alpha}^{\\bar{\\varphi}}_i\\in[0,1]^{|\\mathscr{N}_i|}$ is optimal for this sub-problem, if, for $n\\in\\mathscr{N}_i$,\n\\begin{numcases}{\\alpha^{\\bar{\\varphi}}_{i}(n)}\n=1 &\\text{if }$0<\\lambda_{\\ell}(r_{\\ell}-\\frac{1}{\\mu_{i}}\\sum\\limits_{j\\in \\mathscr{J}_i}\\varepsilon_{j}w_{j,i})\n- (1+\\frac{\\lambda_{\\ell}}{\\mu_{i}})\\sum\\limits_{j\\in \\mathscr{J}_i}w_{j,i}\\gamma_{j}-\\nu_{\\ell} \\text{ and } n< |\\mathscr{N}_i|-1$,\\label{eqn:a_opt:a}\\\\\n\\in [0,1]\n&\\text{if }$0=\\lambda_{\\ell}(r_{\\ell}-\\frac{1}{\\mu_{i}}\\sum\\limits_{j\\in \\mathscr{J}_i}\\varepsilon_{j}w_{j,i})\n- (1+\\frac{\\lambda_{\\ell}}{\\mu_{i}})\\sum\\limits_{j\\in \\mathscr{J}_i}w_{j,i}\\gamma_{j}-\\nu_{\\ell} \\text{ and } n< |\\mathscr{N}_i|-1$,\\label{eqn:a_opt:b}\\\\\n=0&\\text{otherwise},\\label{eqn:a_opt:c}\n\\end{numcases}\nwhere $\\ell=\\ell(i)$.\n\\end{proposition}\nThe proof will be given in Appendix~\\ref{app:prop:a_opt} in the e-companion to this paper.\nIn the maximization of $\\Lambda_i(\\phi,\\pmb{\\gamma},\\nu_{\\ell(i)},\\eta_i)$), the only term of $\\Lambda_i$ dependent on $\\bm{\\eta}$ is $-\\eta_i\\pi^{\\phi}_{i,|\\mathscr{N}_i|-1}\\alpha^{\\phi}_i(|\\mathscr{N}_i|-1)$.\nThe choice of a sufficiently large $\\eta_i$ guarantees that $\\alpha^{\\phi}_i(|\\mathscr{N}_i|-1)$ is $0$ for an optimal policy of the sub-problem, so that constraints~\\eqref{eqn:constraint:dummy} of the relaxed problem are satisfied. \nFor convenience, in what follows we fix $\\bm{\\eta}$ to be one of these large values \nso that $\\alpha^{\\phi}_i(|\\mathscr{N}_i|-1)$ is also fixed to be $0$ for any optimal policy $\\phi$ of the sub-problem for pattern $i$.\nBy slightly abusing notation, in all subsequent equations and discussions, we directly require $\\alpha^{\\phi}_i(|\\mathscr{N}_i|-1)=0$ ($i\\in[I]\\backslash\\{d(\\ell):\\ell\\in[L]\\}$) unless specified otherwise.\n\n\n{\\bf Remark}\nRecall that the action variables $\\bm{\\alpha}^{\\phi}_i$ for any pattern $i\\in[I]$ and policy $\\phi\\in\\Phi_i$ are potentially state-dependent. However, the right hand sides of equations~\\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c} are independent of the state variable $n$ which appears on their left hand side, provided that this is less than $|\\mathscr{N}_i|-1$. This state-independence phenomenon is a consequence of the linearity of the reward and cost rates in the state variable, $N^{\\phi}_i(t)$, for pattern $i\\in[I]\\backslash\\{d(\\ell):\\ \\ell\\in[L]\\}$: from our definition in Section~\\ref{sec:model}, the reward and cost rates of process $i$ in state $N^{\\phi}_i(t)$ are $r_{\\ell(i)}\\mu_{i}N^{\\phi}_i(t)$ and $\\sum_{j\\in\\mathscr{J}_i}\\varepsilon_j w_{j,i}N^{\\phi}_i(t)$, respectively.\nA detailed analysis is provided in the proof of Proposition~\\ref{prop:a_opt}.\n\n\n\n\n\n\nUsing an argument similar to that in \\cite{whittle1988restless}, we can derive from \\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c} an abstracted \\emph{priority} for each \\emph{pattern-state pair} (PS pair) $(i,n)$ with $n\\in\\mathscr{N}_i\\backslash\\{|\\mathscr{N}_i|-1\\}$ and $i\\in[I]$; unlike in \\cite{whittle1988restless}, here, this priority is $(\\pmb{\\gamma},\\bm{\\nu})$-dependent. \nThe priority of a PS pair $(i,n)$ with $n\\in\\mathscr{N}_i\\backslash\\{|\\mathscr{N}_i|-1\\}$\nis determined by the \\emph{index}\n\\begin{equation}\\label{eqn:index_value}\n\\Xi_{i}(\\pmb{\\gamma},\\bm{\\nu})\\coloneqq\\lambda_{\\ell(i)}\\Bigl(r_{\\ell(i)}-\\frac{1}{\\mu_{i}}\\sum\\limits_{j\\in \\mathscr{J}_i}\\varepsilon_{j}w_{j,i}\\Bigr)\n- \\Bigl(1+\\frac{\\lambda_{\\ell(i)}}{\\mu_{i}}\\Bigr)\\sum\\limits_{j\\in \\mathscr{J}_i}w_{j,i}\\gamma_{j}-\\nu_{\\ell(i)},\n\\end{equation}\nand \\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c} can be characterized as comparing $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$ with $0$. \nWhen there is strict inequality in the comparison (that is, the cases described in \\eqref{eqn:a_opt:a} and \\eqref{eqn:a_opt:c}), the value of $\\alpha^\\phi_i(n)$ is specified, since PS pairs $(i,n)$ for all $n\\in\\mathscr{N}_i\\backslash\\{|\\mathscr{N}_i|-1\\}$ correspond to the same $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$ value. \nHowever, there is still freedom to decide different values of $\\alpha^\\phi_i(n)$,\nwhen $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})=0$ (the case described in \\eqref{eqn:a_opt:b}).\nA detailed discussion about priorities of PS pairs corresponding to the same $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$ will be provided in Section~\\ref{subsec:decomposable}.\nBy solving the sub-problem of dummy pattern $d(\\ell)$ $(\\ell\\in[L])$ which involves only one state $n\\in\\mathscr{N}_{d(\\ell)}$, we obtain an optimal policy $\\varphi$ determined by\n\\vspace{-0.1cm}\n\\begin{equation}\\label{eqn:a_opt:dummy}\n\\alpha^{\\varphi}_{d(\\ell)}(n) \\left\\{\\begin{array}{ll}\n=1, &\\text{if } 0 < -\\nu_{\\ell}, \\vspace{-0.1cm}\\\\\n\\in[0,1], &\\text{if }0 = -\\nu_{\\ell},\\vspace{-0.1cm}\\\\\n=0, &\\text{otherwise}. \n\\end{array}\\right.\n\\end{equation}\n\\vspace{-0.1cm}\nThe priority of the state of a dummy pattern is then $\\Xi_{d(\\ell)}(\\pmb{\\gamma},\\bm{\\nu}) \\equiv -\\bm{\\nu}$ for any $\\pmb{\\gamma}$. \n\nIn addition, from Equation~\\eqref{eqn:a_opt:c} in Proposition~\\ref{prop:a_opt}, for any given $\\bm{\\nu}\\in\\mathbb{R}^I$ and $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$, there exists $\\bm{\\eta}\\in\\mathbb{R}^{I-L}$ such that it is optimal to make states $|\\mathscr{N}_i|-1$ passive (that is, $\\alpha^{\\bar{\\varphi}}_i (|\\mathscr{N}_i| - 1) = 0$) for all $i\\in[I]\\backslash\\{d(\\ell):\\ \\ell\\in[L]\\}$. \nAmong all PS pairs $(n,i)$ ($n\\in\\mathscr{N}_i$, $i\\in[I]$), we assign, without loss of generality, the least  priority to  those PS pairs $(i,|\\mathscr{N}_i|-1)$ for which $i\\in[I]\\backslash\\{d(\\ell):\\ \\ell\\in[L]\\}$. \n\n\nThe policy $\\bar{\\varphi}$ determined by \\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c} and \\eqref{eqn:a_opt:dummy} is optimal for the relaxed problem described by \\eqref{eqn:objective}, \\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy}, if the given multipliers $\\bm{\\nu}$ and $\\pmb{\\gamma}$ that appear in \\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c} and \\eqref{eqn:a_opt:dummy}  satisfy the \\emph{complementary slackness conditions} of this relaxed problem, defined by \\vspace{-0.3cm}\n\\begin{condition}{Complementary Slackness}\\label{cond:complementary_slackness}\n\\vspace{-0.3cm}\n\\begin{equation}\n\\label{eqn:relaxaction:slack}\n\\nu_{\\ell}\\Bigl(\\sum\\limits_{i\\in\\mathscr{P}_l} \\bm{\\pi}^{\\phi}_i\\cdot\\bm{\\alpha}^{\\phi}_i -1\\Bigr)=0,~\n\\forall l\\in[L],\\vspace{-0.3cm}\n\\end{equation}\nand \\vspace{-0.3cm}\n\\begin{equation}\n\\label{eqn:relaxconstraint:slack}\n\\gamma_{j} \\Bigl(\\bm{\\omega}_j\\cdot \\left(\\bm{\\Pi}^{\\phi}_n+\\bm{\\Pi}^{\\phi}_a\\right)-C_j\\Bigr) = 0,~\\forall j\\in[J], \\vspace{-0.3cm}\n\\end{equation}\nwhere $\\bm{\\omega}_j=(w_{j,i}:\\ i\\in[I])$ is the $j$th row of matrix $\\mathcal{W}$.\\vspace{-0.3cm}\n\\end{condition}\nIn this context, if resource pool $j\\in[J]$ is very popular so that the capacity constraint corresponding to the $j$th row in  \\eqref{eqn:constraint:relax:resources} achieves equality, then $\\gamma_j$ is allowed to be positive, leading to a lower value of $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$ than for $\\gamma_j=0$.\nOn the other hand, if resource pool $j\\in[J]$ cannot be fully occupied and the $j$th capacity constraint in \\eqref{eqn:constraint:relax:resources} is satisfied with a strict inequality, then the \ncomplementary slackness condition described in \\eqref{eqn:relaxconstraint:slack} forces $\\gamma_j$ to be zero.\nFollowing this mechanism, when resource pool $j\\in[J]$ is overloaded and its priority is reduced by increasing $\\gamma_j$, the offered traffic to this resource pool will be reduced in line with its priority.\n\n\nIf\nthere exist multipliers $\\bm{\\nu}$, $\\pmb{\\gamma}$ and a policy $\\bar{\\varphi}$ determined by \\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c}, such that  the complementary slackness conditions~\\eqref{eqn:relaxaction:slack} and \\eqref{eqn:relaxconstraint:slack} are satisfied by taking $\\phi=\\bar{\\varphi}$, then, by the strong duality theorem, this policy $\\bar{\\varphi}$ is optimal for the relaxed problem;\nthis observation, together with Theorem~\\ref{theorem:main_second} in Section~\\ref{sec:asym_opt}, leads to the existence of an asymptotically optimal policy feasible for the original problem, derived with priorities of patterns induced by the descending order of $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$. More details about the analysis in the asymptotic regime will be provided in Section~\\ref{sec:asym}.\nHere we focus on the non-asymptotic regime, and specifically on the choice and computation of $\\pmb{\\gamma}$ and $\\bm{\\nu}$.  \n\n\n\\subsection{Decomposable Capacity Constraints}\\label{subsec:decomposable}\n\n\n\n\n\nIn the general case, it is not clear whether the \nthe complementary slackness conditions~\\eqref{eqn:relaxaction:slack} and \\eqref{eqn:relaxconstraint:slack}\ncan be satisfied and, \neven if they are, what the values of $\\pmb{\\gamma}$ and the corresponding $\\bm{\\nu}$ are.\nMore important is the question of how the multipliers help with proposing the asymptotically optimal policy applicable to the original problem.\n\n\nIn Sections~\\ref{subsec:decomposable} and \\ref{subsec:sufficient_condition}, we concentrate on the complementary slackness conditions and the existence of an optimal policy (for the relaxed problem) satisfying \\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c}.\nRecall that \\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c} intuitively suggest priorities of patterns induced by $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$. Later in Section~\\ref{sec:index_policy}, a policy feasible for the original problem will be proposed based on given priorities of patterns, and its asymptotic optimality will be discussed in Section~\\ref{sec:asym_opt}.\n\n\\subsubsection{Priorities of PS Pairs}\\label{subsubsec:priorities_of_pairs}\n\nAs described in Section~\\ref{sec:indexability}, \nthe priorities of PS pairs are determined by the descending order of $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$, with higher priorities given by higher values of $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$. \nIt may happen that, because of different tie-breaking rules, the same $\\pmb{\\gamma}$ and $\\bm{\\nu}$ lead to different priorities. For given $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$ and $\\bm{\\nu}\\in\\mathbb{R}^L$, let $\\mathscr{O}(\\pmb{\\gamma},\\bm{\\nu})$ represent the set of all rankings of PS pairs according to the descending order of $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$ ($i\\in[I]$). \nAlso, for notational convenience, let $\\mathscr{O}$ represent the set of all PS pair rankings.\n\n\nTo emphasize the priorities of these PS pairs, according to a given ranking $o\\in\\mathscr{O}$,  we label all these pairs by their order $\\iota^{o}\\in[N]$ with $N\\coloneqq \\sum_{i\\in[I]} |\\mathscr{N}_i|$ and $(i_{\\iota^o},n_{\\iota^o})$ giving the pattern and the state of the $\\iota^{o}$th PS pair.\nWe will omit the superscript $o$ and use $\\iota$ for notational simplicity unless it is necessary to specify the underlying ranking.\nThere exists one and only one $\\ell\\in[L]$ satisfying $i_{\\iota} \\in \\mathscr{P}_{\\ell}$ for any PS pair labeled by $\\iota$. Such an $\\ell$ is denoted by $\\ell_{\\iota}$.\n\n\n\\IncMargin{1em}\n\\begin{algorithm}\n\\small \n\\linespread{0.4}\\selectfont\n\n\\SetKwFunction{FPriorityPolicy}{PriorityPolicy}\n\\SetKwProg{Fn}{Function}{:}{\\KwRet}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\SetAlgoLined\n\\DontPrintSemicolon\n\\Input{a vector of non-negative reals $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$ and a ranking of PS pairs $o\\in\\mathscr{O}$.}\n\\Output{a policy $\\bar{\\varphi}(o)\\in\\tilde{\\Phi}$ determined by action variables $\\bm{\\alpha}^{\\bar{\\varphi}(o)}_i\\in[0,1]^{|\\mathscr{N}_i|}$ for all $i\\in[I]$ and a vector of reals $\\bm{\\nu}(o,\\pmb{\\gamma})$.}\n\n\n\n\\Fn{\\FPriorityPolicy{$o,\\pmb{\\gamma}$}}{\n\n\n    $\\bm{\\alpha}^{\\bar{\\varphi}}_i\\gets \\bm{0}$ for all $i\\in[I]$ \\tcc*{Variables $\\bm{\\alpha}^{\\bar{\\varphi}}_i$ determine a policy $\\bar{\\varphi}$}\n    Initializing the list of candidate PS pairs as the list of all PS pairs\\;\n    $\\iota \\gets 0$ \\tcc*{Iteration variable} \n    \\While {$\\iota<N $ {\\bf and} the list of candidate PS pairs is not empty}{\t\n\t \t    $\\iota\\gets \\iota+1$\\;\n\t\t\t\\If {PS pair $\\iota$ is not in the list of candidate PS pairs}{\n\t\t\t    {\\bf continue}\\;\n\t\t\t}\n\t    $a_1 \\gets \\inf \\left\\{\\left\\{\\alpha^{\\bar{\\varphi}}_{i_{\\iota}}(n_{\\iota}) \\in [0,1] \\left|~\\sum_{i\\in\\mathscr{P}_{\\ell_{\\iota}}}\\bm{\\pi}^{\\bar{\\varphi}}_i\\cdot \\bm{\\alpha}^{\\bar{\\varphi}}_i = 1\\right.\\right\\}\\cup\\{1\\}\\right\\}$\\;\n\n\t    \\tcc*{The maximal probability of activating PS pair $\\iota$ such that }\n\t\t\\tcc*{the action constraint for RT $\\ell_{\\iota}$ is not violated.}\n\n  \t    $a_2 \\gets \\inf \\left\\{\\left\\{\\alpha^{\\bar{\\varphi}}_{i_{\\iota}}(n_{\\iota}) \\in [0,1] \\left|~\\exists j\\in[J], \\bm{\\omega}_j\\cdot\\left(\\bm{\\Pi}^{\\bar{\\varphi}}_n+\\bm{\\Pi}^{\\bar{\\varphi}}_a\\right)=C_j\\right.\\right\\}\\cup\\{1\\}\\right\\}$\\;\n\n\t\t\\tcc*{The maximal probability of activating PS pair $\\iota$ such that}\n\t \t\\tcc*{the capacity constraints are not violated.}\n\n\t\t$\\alpha^{\\bar{\\varphi}}_{i_{\\iota}}(n_{\\iota})\\gets \\min\\{a_1,a_2\\}$ \\;\n\t\t\\tcc*{Update $\\alpha^{\\bar{\\varphi}}_{i_{\\iota}}(n_{\\iota})$ with the maximal activating probability}\n\t\t\\tcc*{without violating any constraint.}\n\n\t\t\\uIf{$\\sum_{i\\in\\mathscr{P}_{\\ell_{\\iota}}}\\bm{\\pi}^{\\bar{\\varphi}}_i\\cdot \\bm{\\alpha}^{\\bar{\\varphi}}_i = 1$}{\n\t\t\t\\tcc*{If the action constraint achieves equality under policy $\\bar{\\varphi}$}\n\t\t\t\\tcc*{determined by updated $\\bm{\\alpha}^{\\bar{\\varphi}}_i$, $i\\in[I]$.}\n\t\t    $\\nu_{\\ell_{\\iota}}(o,\\pmb{\\gamma})\\gets\\Xi_{i_{\\iota}}(\\pmb{\\gamma},\\bm{0})$\\;\n\n\t       remove all PS pairs $\\iota'>\\iota$ with $\\ell_{\\iota'}=\\ell_{\\iota}$ from the list of candidate PS pairs\\;\n\t   \t}\n   \t\t\\ElseIf{$\\exists j\\in[J],\\ \\bm{\\omega}_j\\cdot\\left(\\bm{\\Pi}^{\\bar{\\varphi}}_n+\\bm{\\Pi}^{\\bar{\\varphi}}_a\\right)=C_j$}{\n\t\t\t\\tcc*{If a capacity constraint achieves equality under policy $\\bar{\\varphi}$}\n\t\t\t\\tcc*{determined by updated $\\bm{\\alpha}^{\\bar{\\varphi}}_i$, $i\\in[I]$.}\n\n\t\t    remove all PS pairs $\\iota'>\\iota$ with $w_{j,i_{\\iota'}}>0$ from the list of candidate PS pairs\\;\n\t\t}\n\n}\n\n$\\bm{\\alpha}^{\\bar{\\varphi}(o)}_i\\gets \\bm{\\alpha}^{\\bar{\\varphi}}_{i}$ for all $i\\in[I]$\\;\n\n}\n\\caption{Priority-style policy for the relaxed problem}\\label{algo:varphi_gamma}\n\\end{algorithm}\n \\DecMargin{1em}\n\n\nFor any given ranking of PS pairs $o\\in\\mathscr{O}$, we can generate a policy $\\bar{\\varphi}(o)$ with priorities of PS pairs defined by $o$, such that \n\\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy} are satisfied:\nthe policy $\\bar{\\varphi}(o)$ is feasible for the relaxed problem but not necessarily feasible for the original problem.\nThe pseudo-code for generating $\\bar{\\varphi}(o)$ is presented in Algorithm~\\ref{algo:varphi_gamma}.\nThe key idea for generating such a $\\bar{\\varphi}(o)$ is to initialize $\\bm{\\alpha}_i^{\\bar{\\varphi}(o)}$ to $\\bm{0}$ for all $i\\in[I]$, and sequentially activate the PS pairs according to their priorities defined by $o$ until either a relaxed action or capacity constraint described in \\eqref{eqn:constraint:relax:action} and \\eqref{eqn:constraint:relax:resources}, respectively, achieves equality. In particular, \n\\begin{enumerate}[label=(\\Roman*)]\n\\item\nif a relaxed action constraint described in \\eqref{eqn:constraint:relax:action} achieves equality by activating PS pairs less than or equal to $\\iota$, then the multiplier $\\nu_{\\ell_{\\iota}}$ is set to $\\Xi_{i_{\\iota}}(\\pmb{\\gamma},\\bm{0})$, and all later PS pairs $\\iota'>\\iota$ with $\\ell_{\\iota'}=\\ell_{\\iota}$ are \\emph{disabled} from being activated and are removed from the \\emph{list of candidate pairs} awaiting later activation; \\label{case:equality:action}\n\\item similarly, if a relaxed capacity constraint described in \\eqref{eqn:constraint:relax:resources} associated with resource pool $j\\in[J]$ achieves equality by activating PS pairs less than or equal to $\\iota$, then all later PS pairs $\\iota'>\\iota$ with $w_{j,i_{\\iota'}}>0$ are disabled and removed from the list of candidate states.\\label{case:equality:capacity}\n\\end{enumerate}\nMaintaining an iteratively updated list of candidate pairs in this way continues until all \naction constraints in \\eqref{eqn:constraint:relax:action} achieve equality: the policy $\\bar{\\varphi}(o)$ is determined by the resulting $\\bm{\\alpha}^{\\bar{\\varphi}(o)}_i$ ($i\\in[I]$), and the multipliers $\\bm{\\nu}$ are updated in \\ref{case:equality:action}.\nThe vector of these multipliers is denoted by $\\bm{\\nu}(o,\\pmb{\\gamma})$.\nThe PS pair labeled by $\\iota$ satisfying the condition described in \\ref{case:equality:capacity} is called the \\emph{critical pair}, with the corresponding resource pool $j$ referred to as the \\emph{critical pool} of PS pair $\\iota$, denoted by $j_{\\iota}(o)$. Note that, from the description in \\ref{case:equality:capacity}, there might be more than one resource pool for which the capacity constraints achieve equality simultaneously while activating PS pair $\\iota$; we choose one of them to be $j_{\\iota}(o)$ and refer to this resource pool as the critical pool of $\\iota$.\nLet $\\mathscr{I}(o)$ represent the set of all critical pairs with respect to the policy $\\bar{\\varphi}(o)$. \n\\begin{lemma}\\label{lemma:critical_pattern}\nFor any $o\\in\\mathscr{O}$ and $\\iota,\\iota'\\in\\mathscr{I}(o)$, if $\\iota \\neq \\iota'$ then $i_{\\iota}\\neq i_{\\iota'}$.\n\\end{lemma}\n\\proof{Proof.}\nConsider critical pairs $\\iota,\\iota'\\in\\mathscr{I}(o)$ with $\\iota\\neq \\iota'$, and assume $\\iota < \\iota'$ without loss of generality. Since $\\iota$ is a critical pair, there is a critical resource pool $j_{\\iota}$ which is fully occupied. In this case, if $i_{\\iota} = i_{\\iota'}$, then pair $\\iota'$ must require some resource units from pool $j_{\\iota}$ and so $\\alpha^{\\bar{\\varphi}(o)}_{\\iota'}=0$.\nPS pair $\\iota'$ cannot be critical, which violates the condition $\\iota'\\in\\mathscr{I}(o)$. \nHence, $i_{\\iota} \\neq i_{\\iota'}$. This proves the lemma.\n\\endproof\n\n\n\n\nRecall, for any ranking $o$, the policy $\\bar{\\varphi}(o)$ must satisfy the action and capacity constraints~\\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy}. Also, since \\eqref{eqn:constraint:relax:action} holds, the \ncomplementary slackness conditions \ncorresponding to the action constraints~\\eqref{eqn:relaxaction:slack} are satisfied by taking $\\phi=\\bar{\\varphi}(o)$. However, the complementary slackness conditions \ncorresponding to the capacity constraints~\\eqref{eqn:relaxconstraint:slack} and equations~\\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c} are not necessarily satisfied if we plug in $\\phi=\\bar{\\varphi}(o)$ and $\\pmb{\\gamma}$: the policy $\\bar{\\varphi}(o)$ is a heuristic policy  applicable for the relaxed problem defined by \\eqref{eqn:objective}, \\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy} derived by intuitively prioritizing PS pairs according to their ranking $o\\in\\mathscr{O}$.\n\nIn Section~\\ref{subsec:sufficient_condition} we shall define a particular class of resource allocation models, for which we can show the complementary slackness conditions are indeed satisfied.\n\n\n\\begin{definition}\\label{define:decomposable}\nThe system said to be \\emph{decomposable} if there exist multipliers $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$, $\\bm{\\nu}\\in\\mathbb{R}^L$ and a ranking $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{\\nu})$ such that $\\bm{\\nu}=\\bm{\\nu}(o,\\pmb{\\gamma})$ and the complementary slackness conditions~\\eqref{eqn:relaxaction:slack} and \\eqref{eqn:relaxconstraint:slack} are achieved by taking $\\phi=\\bar{\\varphi}(o)$. In this case the optimal values of the dual variables are called \\emph{decomposable values}. \n\\end{definition}\n\n\nRecall that, in the general case, for $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$ and $\\bm{\\nu}\\in\\mathbb{R}^L$, even if $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{\\nu})$, the policy $\\bar{\\varphi}(o)$ is not necessarily optimal (because it does not necessarily satisfy \\eqref{eqn:a_opt:a}-\\eqref{eqn:a_opt:c}). \nWhen the policy $\\bar{\\varphi}(o)$ is optimal for the relaxed problem, the ranking $o$ can be used to construct an index policy applicable to the original problem (detailed steps are provided in Section~\\ref{sec:index_policy}).  Theorem~\\ref{theorem:main_second} (in Section~\\ref{sec:asym_opt}) then ensures that such an index policy is asymptotically optimal.\n\n\\subsubsection{Derivation of the Pair Ranking} \\label{subsubsec:derive_ranking}\n\nWe start with a proposition that shows how the values of the Lagrange multipliers $\\bm{\\nu}$ and $\\pmb{\\gamma}$ can be derived from a knowledge of the critical pair and critical resource pool corresponding to a given order $o\\in \\mathscr{O}$.\n\n\\begin{proposition}\\label{prop:solution_existence}\nFor any given $\\pmb{\\gamma}_0\\in\\mathbb{R}_0^J$ and $o\\in\\mathscr{O}$, the linear equations\n\\begin{equation}\\label{eqn:necessary_gamma}\n\\nu_{\\ell_{\\iota}}(o,\\pmb{\\gamma}_0)=\\Xi_{i_{\\iota}}(\\pmb{\\gamma},\\bm{0}),~\\forall \\iota\\in\\mathscr{I}(o)\n\\vspace{-0.3cm}\n\\end{equation}\nand \\vspace{-0.3cm}\n\\begin{equation}\\label{eqn:necessary_gamma:zero}\n\\gamma_{j} = 0,~\\forall j \\notin \\{j_{\\iota}(o)\\in[J] ~|~\\iota\\in\\mathscr{I}(o)\\}\n\\end{equation}\nhave a unique solution $\\pmb{\\gamma}\\in\\mathbb{R}^J$.\n\\end{proposition}\nThe proof of Proposition~\\ref{prop:solution_existence} will be given in Appendix~\\ref{app:prop:solution_existence} in the e-companion. \nFor a ranking $o\\in\\mathscr{O}$, define an function $\\mathcal{T}^o$ of $\\pmb{\\gamma}_0\\in\\mathbb{R}_0^J$ with respect to $o\\in\\mathscr{O}$: $\\mathcal{T}^o(\\pmb{\\gamma}_0)\\coloneqq \\pmb{\\gamma}$ where $\\pmb{\\gamma}$ is the unique solution of \\eqref{eqn:necessary_gamma} and \\eqref{eqn:necessary_gamma:zero}. \nLet $\\mathcal{T}^o_j(\\pmb{\\gamma}_0)$ represent the $j$th element of $\\mathcal{T}^o(\\pmb{\\gamma}_0)$.\n\n\n\\begin{proposition}\\label{prop:converge_gamma}\nIf there exist $\\pmb{\\gamma}_0\\in\\mathbb{R}_0^J$ and $o\\in\\mathscr{O}(\\pmb{\\gamma}_0,\\bm{0})$ such that $\\mathcal{T}^o(\\pmb{\\gamma}_0)=\\pmb{\\gamma}_0$, then  \n$\\pmb{\\gamma}_0$ is a vector of decomposable multipliers and the policy $\\bar{\\varphi}(o)$ based on  ranking $o$ is optimal for the relaxed problem defined by \\eqref{eqn:objective}, \\eqref{eqn:constraint:relax:action}, \\eqref{eqn:constraint:relax:resources} and \\eqref{eqn:constraint:dummy}.\n\\end{proposition}\nThe proof of Proposition~\\ref{prop:converge_gamma} will be given in Appendix~\\ref{app:prop:converge_gamma} in the e-companion.  Recall that $\\mathscr{I}(o)$ is the set of critical pairs with respect to the policy $\\bar{\\varphi}(o)$, $j_{\\iota}(o)$ is the critical resource pool corresponding to critical pair $\\iota\\in\\mathscr{I}(o)$ according to ranking $o$, and $\\nu_{\\ell_{\\iota}}(o,\\pmb{\\gamma}_0)$ is an output of Algorithm~\\ref{algo:varphi_gamma} when the inputs are $o$ and $\\pmb{\\gamma}=\\pmb{\\gamma}_0$.\n\n{\\bf Remark} Proposition~\\ref{prop:converge_gamma} provides a way of checking decomposability of $\\pmb{\\gamma}_0$ and optimality of $\\bar{\\varphi}(o)$.\nBy Proposition~\\ref{prop:converge_gamma}, any fixed point $\\pmb{\\gamma}_0\\in\\mathbb{R}_0^J$ of the function $\\mathcal{T}^o$ with respect to a ranking $o\\in\\mathscr{O}(\\pmb{\\gamma}_0,\\bm{0})$ is a decomposable vector. \nThe decomposability of $\\pmb{\\gamma}_0$ can be checked without requiring knowledge of any $\\bm{\\nu}\\in\\mathbb{R}^L$. \nAlso, we present the following corollary of Proposition~\\ref{prop:converge_gamma}.\n\\begin{corollary}\\label{coro:converge_gamma}\nFor $\\pmb{\\gamma}_0\\in\\mathbb{R}_0^J$ and $o\\in\\mathscr{O}(\\pmb{\\gamma}_0,\\bm{0})$, if $\\mathcal{T}^o(\\pmb{\\gamma}_0)\\neq \\pmb{\\gamma}_0$, $\\mathcal{T}^o(\\pmb{\\gamma}_0)\\in \\mathbb{R}_0^J$ and $o\\in\\mathscr{O}(\\mathcal{T}^o(\\pmb{\\gamma}_0),\\bm{0})$, then $\\mathcal{T}^o(\\mathcal{T}^o(\\pmb{\\gamma}_0))=\\mathcal{T}^o(\\pmb{\\gamma}_0)$. \n\\end{corollary}\nNote that the hypothesis of Corollary~\\ref{coro:converge_gamma} requires all components of $\\mathcal{T}^o(\\pmb{\\gamma}_0)$ to be nonnegative. This is not such an easy condition to satisfy. \nThe proof of Corollary~\\ref{coro:converge_gamma} will be given in Appendix~\\ref{app:coro:converge_gamma} in the e-companion.\n\n\n\nIn this context, consider a given $\\pmb{\\gamma}_0\\in\\mathbb{R}_0^J$ and a ranking $o\\in\\mathscr{O}(\\pmb{\\gamma}_0,\\bm{0})$. If $\\pmb{\\gamma}_0$ is a fixed point of $\\mathcal{T}^o$, then it is the vector of decomposable multipliers; if it is not but $\\mathcal{T}^o(\\pmb{\\gamma}_0)$ is a nonnegative fixed point of $\\mathcal{T}^o$, then $\\mathcal{T}^o(\\pmb{\\gamma}_0)$ represents the decomposable multipliers. \nHowever, in both cases we need to propose a specific $\\pmb{\\gamma}_0$; it requires prior knowledge of the multipliers, which is, in general, not available.\nSection~\\ref{subsec:sufficient_condition} will discuss a special case where the decomposability is provable and we have a method of deriving the decomposable multipliers.\nHere,  to make a reasonably good choice of the Lagrangian multipliers in a general system, we embark on a \\emph{fixed point iteration method}.\n\n\nSince Proposition~\\ref{prop:converge_gamma} requires a fixed point $\\pmb{\\gamma}$ of the function $\\mathcal{T}^o$ with $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{0})$, we need to iterate not only the value of $\\pmb{\\gamma}$ but also the corresponding ranking $o$ which affects the function $\\mathcal{T}^o$ and should be an element of $\\mathscr{O}(\\pmb{\\gamma},\\bm{0})$.\nFollowing the idea of conventional\nfixed point interation methods, for $k\\in\\mathbb{N}_0$, \nlet $\\pmb{\\gamma}_{k+1}= \\bigl(\\mathcal{T}^{o_k}(\\pmb{\\gamma}_k)\\bigr)^+$ with initial $\\pmb{\\gamma}_0$ and $o_0\\in\\mathscr{O}(\\pmb{\\gamma}_0,\\bm{0})$, \nwhere $(\\bm{v})^+\\coloneqq (\\max\\{0,v_i\\}: i\\in[N])$ for a vector $\\bm{v}\\in \\mathbb{R}^N$ ($N\\in\\mathbb{N}_+$). \nConstruct a ranking $o_{k+1}\\in\\mathscr{O}(\\pmb{\\gamma}_{k+1},\\bm{0})$ according to $o_k$:\nfor any two different PS pairs $(i,n)$ and $(i',n')$ with $\\Xi_i(\\pmb{\\gamma}_{k+1},\\bm{0})=\\Xi_{i'}(\\pmb{\\gamma}_{k+1},\\bm{0})$, $(i,n)$ precedes $(i',n')$ in the ranking $o_{k+1}$ if and only if $(i,n)$ precedes $(i',n')$ in the ranking $o_k$.\nHere, the operation $(\\cdot)^+$ is used to make all the elements of $\\pmb{\\gamma}_{k+1}$ non-negative, so that $\\pmb{\\gamma}_{k+1}$ is feasible for the function $\\mathcal{T}^{o_{k+1}}$.\nThus the ranking $o_{k+1}$ inherits the tie-breaking rule used for $o_k$ so that the difference between $o_k$ and $o_{k+1}$, which must satisfy $o_k\\in\\mathscr{O}(\\pmb{\\gamma}_k,\\bm{0})$ and $o_{k+1}\\in\\mathscr{O}(\\pmb{\\gamma}_{k+1},\\bm{0})$, is minimized.\nCorollary~\\ref{coro:converge_gamma} can be used to check whether the $\\pmb{\\gamma}_{k+1}$ is a fixed point of the function $\\mathcal{T}^{o_k}$.\nAlso, $\\pmb{\\gamma}_{k+1}$ and $o_{k+1}$ are uniquely determined by $\\pmb{\\gamma}_k$ and $o_k$. We can consider $(\\pmb{\\gamma}_k,o_k)$ as an entity which is an argument delivered to the function $\\mathcal{T}^{o_k}(\\pmb{\\gamma}_k)$, and wish to find a fixed point in this sense.\n\n\nIn the general case, the function $\\mathcal{T}^{o_k}(\\pmb{\\gamma}_k)$ is discontinuous in $\\pmb{\\gamma}_k$ and the sequence $\\{\\pmb{\\gamma}_k\\}_{k=0}^{\\infty}$ is heuristically generated with no proof of convergence to a fixed point. \nIn fact, the choice of $\\pmb{\\gamma}_{k+1}= \\bigl(\\mathcal{T}^{o_k}(\\pmb{\\gamma}_k)\\bigr)^+$ may result in the sequence $\\{\\pmb{\\gamma}_k\\}_{k=0}^{\\infty}$ being trapped in oscillations. \nTo avoid this,  with slight abuse of notation, we modify the iteration as $\\pmb{\\gamma}_{k+1}= \\bigl(c\\mathcal{T}^{o_k}(\\pmb{\\gamma}_k)+(1-c)\\pmb{\\gamma}_k\\bigr)^+$ with a parameter $c\\in [0,1]$, which captures the effects of exploring the new point $\\mathcal{T}^{o_k}(\\pmb{\\gamma}_k)$.\nNumerical examples of iterating $\\pmb{\\gamma}_k$ will be provided in Section~\\ref{sec:example}.\n\n\n\nWith an upper bound, $U\\in\\mathbb{N}_+$, we take $k^*\\coloneqq \\arg\\min_{k=1,2,\\ldots,U} \\lVert \\pmb{\\gamma}_{k-1}-\\pmb{\\gamma}_k\\rVert$ and consider $o_{k^*}$ as a reasonably good ranking of PS pairs. \nSuch $o_{k^*}$ is pre-computable with computational complexity no worse than $O(U(N^2+J^2))$, where $N^2$ and $J^2$ result from ordering the $N$ pairs and solving the $J$ linear equations, respectively.\nIn Section~\\ref{sec:index_policy}, we show that an index policy feasible for the original problem can always be generated with such an $o_{k^*}$, and the implementation complexity is $O(I)$  in terms of computation and storage. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{-0.3cm}\n\\subsection{Weakly Coupled Constraints}\\label{subsec:sufficient_condition}\n\\vspace{-0.3cm}\nHere, we discuss a sufficient condition under which the sequence $\\{\\pmb{\\gamma}_k\\}_{k=0}^{\\infty}$ is provably convergent; and, in Section~\\ref{sec:example}, when this condition fails, we show via numerical examples that the sequence might still converge.  \n\n\n\\begin{definition}\nRecall the matrix $\\mathcal{W}=(w_{j,i})$ defined in Section~\\ref{subsec:model}.\nWe say that row $j\\in[J]$ is \n\\begin{enumerate}\n\\item a \\emph{type-1} row if there is at most one $i\\in[I]$ with $w_{j,i}>0$; \\vspace{-0.15cm}\n\\item a \\emph{type-2} row if there is more than one $i\\in[I]$ with $w_{j,i}>0$.\n\\end{enumerate} \\vspace{-0.3cm}\n\\end{definition}\nThat is, row $j$ is a type-1 row if resource pool $j$ is not shared by patterns of different types; and is a type-2 row, otherwise.\nDenote by $\\mathscr{J}_i=\\{j\\in[J]\\ |\\ w_{j,i}>0\\}$ the set of resource pools used by pattern $i$. \nWe then define a condition. \\vspace{-0.3cm}\n\\begin{condition}{Weak Coupling}\\label{cond:hypothesis}\nA system is weakly coupled if, for any pattern $i$, there is at most one $j\\in\\mathscr{J}_i$ with row $j$ of $\\mathcal{W}$ being a type-2 row. \\vspace{-0.3cm}\n\\end{condition}\nThis condition implies that there is at most one shared resource pool associated with each pattern. \nIn a weakly coupled system, if pattern $i_1$ shares a resource pool $j_{12}$ with pattern $i_2$ and pattern $i_1$ shares a resource pool $j_{13}$ with pattern $i_3$ then $j_{12}=j_{13}$. \nA system where each of the patterns requires only one resource pool is clearly weakly coupled.\nNote that, in a  weakly coupled system, dependencies between state variables of different patterns still exist, because \neach resource pool can be shared by requests of multiple RTs.\n\n\\begin{definition}\\label{define:w_star}\nFor a weakly coupled system define, for each $i\\in[I]\\backslash\\{d(\\ell):\\ell\\in[L]\\}$, $w^*_i=w_{j,i}$ where $j$ is the only resource pool in $\\mathscr{J}_i$ shared with other patterns, if there is one; or any member of the set $\\arg\\min\\limits_{j'\\in\\mathscr{J}_i}\\frac{C_{j'}}{w_{j',i}}$, otherwise.\n\\end{definition}\n\\begin{definition}\\label{define:o_star}\n\nFor a weakly coupled system define, for $\\bm{\\nu}\\in\\mathbb{R}^L$, a set of PS rankings $\\mathscr{O}^*(\\bm{\\nu})\\subset \\mathscr{O}$ such that, for any $o\\in\\mathscr{O}^*(\\bm{\\nu})$, PS pairs $\\iota\\in[N]$ are ranked according to the descending order of \n\\begin{equation}\\label{eqn:main:index}\n\\Xi_{\\iota}^*= \\left\\{\n\\begin{array}{ll}\n\\frac{\\Xi_{i_{\\iota}}(\\bm{0},\\bm{0})-\\nu_{\\ell_{\\iota}}}{w^*_{i_{\\iota}}(1+\\lambda_{\\ell_{\\iota}}\/\\mu_{i_{\\iota}})}, &\\text{if } \\nexists \\ell\\in[L],~i_{\\iota}=d(\\ell),\\\\\n0, & \\text{otherwise},\n\\end{array}\\right.\n\\end{equation} \n\\end{definition}\n\n\n\\begin{proposition}\\label{prop:equal_opt}\nIf the system is weakly coupled and there exists a ranking $o\\in\\mathscr{O}^*(\\bm{0})$ satisfying $\\nu(o,\\bm{0})=\\bm{0}$, then the capacity constraints described in \\eqref{eqn:constraint:relax:resources} are decomposable and the policy $\\bar{\\varphi}(o)$ is optimal for the relaxed problem defined by \\eqref{eqn:objective} and \\eqref{eqn:constraint:relax:action}-\\eqref{eqn:constraint:dummy}.\nIn particular, there exist decomposable multipliers $\\pmb{\\gamma}\\in\\mathbb{R}_0^{J}$ satisfying, for $j\\in[J]$,\n\\begin{enumerate}[label=\\roman*)]\n\\item if there is a critical PS pair $\\iota\\in\\mathscr{I}(o)$ with critical resource pool $j=j_{\\iota}(o)$, and no \n$j'\\neq j$ with $j'\\in\\mathscr{J}_{i_{\\iota}}$ is critical for any other PS pair $\\iota' \\in\\mathscr{I}(o)$, then\n\\vspace{-0.3cm}\n\\begin{equation}\\label{eqn:equal_opt:a}\n\\gamma_j = \\frac{\\Xi_{i_{\\iota}}(\\bm{0},\\bm{0})-\\nu_{\\ell_i}}{w_{j,i_{\\iota}}\\left(1+\\lambda_{\\ell_{\\iota}}\/\\mu_{i_{\\iota}}\\right)};\n\\vspace{-0.3cm}\n\\end{equation}\n\\item if there are critical PS pairs $\\iota$ and $\\iota'$ in $\\mathscr{I}(o)$ with critical resource pools $j=j_{\\iota}(o)\\neq j_{\\iota'}(o)$ and $ j_{\\iota'}(o)\\in\\mathscr{J}_{i_{\\iota}}$, then  \\vspace{-0.3cm}\n\\begin{equation}\\label{eqn:equal_opt:b}\n\\gamma_j=\\frac{w_{j_{\\iota'}(o),i_{\\iota}}}{w_{j,i_{\\iota}}}\\left(\\frac{\\Xi_{i_{\\iota}}(\\bm{0},\\bm{0})-\\nu_{\\ell_{\\iota}}}{w_{j_{\\iota'}(o),i_{\\iota}}\\left(1+\\lambda_{\\ell_{\\iota}}\/\\mu_{i_{\\iota}}\\right)}\n-\\frac{\\Xi_{i_{\\iota'}}(\\bm{0},\\bm{0})-\\nu_{\\ell_{\\iota'}}}{w_{j_{\\iota'}(o),i_{\\iota'}}\\left(1+\\lambda_{\\ell_{\\iota'}}\/\\mu_{i_{\\iota'}}\\right)}\\right);\\vspace{-0.3cm}\n\\end{equation}\n\\item otherwise, \\vspace{-0.3cm}\n\\begin{equation}\\label{eqn:equal_opt:c}\n\\gamma_j = 0. \\vspace{-0.3cm}\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\nThe proof is given in Appendix~\\ref{app:prop:equal_opt} in the e-companion. \nNote that, from Lemma~\\ref{lemma:critical_pattern}, for any critical PS pairs $\\iota,\\iota'\\in\\mathscr{I}(o)$ with $\\iota\\neq \\iota'$, it follows that $i_{\\iota}\\neq i_{\\iota'}$.\nIf the system is weakly coupled, for any $j\\in[J]$, there exist at most two different critical pairs $\\iota\\in\\mathscr{I}(o)$ satisfying $j\\in\\mathscr{J}_{i_{\\iota}}$. Also, in a weakly coupled system, for the second case stated in Proposition~\\ref{prop:equal_opt},  if there are critical PS pairs $\\iota$ and $\\iota'$ in $\\mathscr{I}(o)$ with critical resource pools $j=j_{\\iota}(o)\\neq j_{\\iota'}(o)$ and $ j_{\\iota'}(o)\\in\\mathscr{J}_{i_{\\iota}}$, then $ j_{\\iota}(o)\\notin \\mathscr{J}_{i_{\\iota'}}$ because there is at most one resource pool in $\\mathscr{J}_{i_{\\iota}}$ shared with other patterns.\n\n\n\nIn Proposition~\\ref{prop:equal_opt}, \nthe assumption that the system is weakly coupled constrains the way in which resource pools are shared by different requests. \nThe case where there is an $o\\in\\mathscr{O}^*(\\bm{0})$ with $\\bm{\\nu}(o,\\bm{0})=\\bm{0}$ will occur when the relaxed action constraint~\\eqref{eqn:constraint:relax:action} is satisfied with $\\alpha^{\\bar{\\varphi}(o)}_{d(\\ell)}(n)>0$ for the only $n\\in\\mathscr{N}_{d(\\ell)}$ and for all $\\ell\\in[L]$. \nTo see this, note that\nthe construction of the policy $\\bar{\\varphi}(o)$ guarantees that the resulting multipliers $\\bm{\\nu}(o,\\bm{0})$ will be non-negative, and so\nit follows from~\\eqref{eqn:a_opt:dummy} that $\\alpha^{\\bar{\\varphi}(o)}_{d(\\ell)}(n) > 0$ only if $\\nu_{\\ell}(o,\\bm{0}) = 0$. That is, having $\\nu_{\\ell}(o,\\bm{0}) = 0$ is associated with there being a positive probability that the dummy pattern $d(\\ell)$ is selected in the relaxed system.\nFurthermore, if there is a PS pair $\\iota$ (for a non-dummy pattern $i_{\\iota}\\in\\mathscr{P}_{\\ell}$) which satisfies the condition described in \\ref{case:equality:action}, that is PS pair $\\iota$ causes the relaxed action constraint~\\eqref{eqn:constraint:relax:action} to bite, Algorithm~\\ref{algo:varphi_gamma} will ensure that $\\alpha^{\\bar{\\varphi}(o)}_{i_{\\iota'}}(n_{\\iota'}) = 0$ for all PS pairs $\\iota'$ \nranked lower than $\\iota$ according to the order $o$. In particular, this will cause  $\\alpha^{\\bar{\\varphi}(o)}_{d(\\ell)}(n) = 0$ for the only $n\\in\\mathscr{N}_{d(\\ell)}$.\n\nSo if $\\alpha^{\\bar{\\varphi}(o)}_{d(\\ell)}(n) > 0$, it is because the relaxed capacity constraints~\\eqref{eqn:constraint:relax:resources} bite before the relaxed action constraints~\\eqref{eqn:constraint:relax:action}.\nIf this is true for all $\\ell$, then the capacity constraints are biting for every request type, and so we refer to the case where there is an $o\\in\\mathscr{O}^*(\\bm{0})$ with $\\bm{\\nu}(o,\\bm{0})=\\bm{0}$ as a \\emph{heavy traffic} condition.\n\n\n\n\n\n\\begin{condition}{Heavy Traffic}\\label{cond:heavy_traffic}\nThe system is in heavy traffic if there is a ranking $o\\in\\mathscr{O}^*(\\bm{0})$ such that $\\bm{\\nu}(o,\\bm{0})=\\bm{0}$. \\vspace{-0.5cm}\n\\end{condition}\n\n{\\bf Remark} \nThe property of being weakly coupled and in heavy traffic simplifies the analysis of the complementary slackness condition of the relaxed problem. In particular, the index related to a pattern, described in Equation~\\eqref{eqn:index_value}, is affected only by the multipliers of resource pools $j \\in[J]$ with $w_{j,i}>0$. Weak coupling helps reduce the number of such multipliers $\\gamma_j$, so that the index of a pattern is affected by at most one $\\gamma_ j$, which in turn affects other pattern indices. When the system is weakly coupled and in heavy traffic, we can analytically solve the $I$ linear equations~\\eqref{eqn:necessary_gamma} and \\eqref{eqn:necessary_gamma:zero} and derive the $\\phi$ and $\\pmb{\\gamma}$ that satisfy the complementary slackness condition described in equaitons~\\eqref{eqn:relaxaction:slack} and \\eqref{eqn:relaxconstraint:slack}.  A detailed discussion is provided in the proof of Proposition~\\ref{prop:equal_opt}.\n\n\nProposition~\\ref{prop:equal_opt} guarantees the decomposability of a system when it is weakly coupled and in heavy traffic.\nThe property of being weakly coupled and in heavy traffic is stronger than necessary for decomposability, but it is simple to check and is satisfied in a number of common resource allocation problems. We consider examples about how to easily define such a system.\n\n\nAs explained above, the heavy traffic property is usually satisfied when the service capacity is not enough (or just enough) to address its high traffic load.\nOn the other hand, the weak coupling specifies the structure of the weight matrix $\\mathcal{W}$. \nFor instance, if each pattern involves only one resource pool (that is, for all $i\\in[I]\\backslash\\{d(\\ell)|\\ell\\in[L]\\}$, $|\\mathscr{J}_i|=1$), then the system is weakly coupled as each resource pool is still potentially shared by requests of different types. \n\n\nWithin the above framework, we can model skill-based resource pooling in call centers (see \\cite{wallace2004resource,cezik2008staffing}) as a weakly coupled resource allocation problem; and when its traffic load is also heavy, the system is decomposable.\nIn each call center, agents are trained for several skills, such as two or three languages, and are able to handle some but not all of the incoming calls. We classify these agents into multiple call centers according to their trained skills; that is, all agents in the same call center have the same skills and are able to serve the same types of calls. In this context, a call corresponds to a request, an agent corresponds to a resource unit, a call center is a resource pool and a call type is a request type.\n\nSince each call is served by an agent with appropriate skills, each pattern consists of only one call center (resource pool) and selecting a pattern means selecting an agent (a resource unit) from the corresponding call center: this problem is weakly coupled. Note that agents of each call center are potentially serving different types of calls simultaneously, and the capacity constraints \\eqref{eqn:constraint:resources} still restrict the system because of the limited number of agents in each call center. \n\n\nIn particular, the trained skills are used to establish the availability of an agent to serve a call, and do not relate to any concept defined in the resource allocation problem. When an agent of call center $j\\in[J]$  is able to serve calls of type $\\ell\\in[L]$, regardless of the skills needed for this service, there is a pattern $i$ in $\\mathscr{P}_{\\ell}$ with $w_{j,i}=1$ and $w_{j',i}=0$ for all $j'\\in[J]\\backslash\\{j\\}$. \nFor instance, a call center has agents who can speak English and Chinese, and there are two types of calls: one requires English or French-speaking agents and the other Chinese or Japanese speakers. A call of either type can be served by an agent of this call center, and many calls of both types can be served by this call center simultaneously. \n\n \n\nOther problems with similar features, such as health-care task scheduling for agents with different  qualifications (see \\cite{lieder2015task}) and home health-care scheduling (see \\cite{fikar2017home}), can also be modeled as weakly coupled systems. And, of course, when the systems are also in heavy traffic, they are decomposable.\n\n\nA virtual machine (VM) replacement problem can be modeled as a resource allocation problem (see \\cite{stolyar2013infinite,stolyar2017large}). VM replacement is about consolidating multiple VMs onto a set of physical machines\/servers, where each physical server can usually accommodate more than one VM simultaneously. To consolidate a VM, certain numbers of physical units, such as  CPU cycles, memory, disk, or I\/O ports, located on a server will be occupied by this VM until it is completed. The VMs and servers are potentially different, and, because of different server profiles or user preference, a server is not necessarily able to accommodate every VM. \nConsider a simple version, for which the capacity of a server is determined by the total amount of only one type of physical unit: this server has a plentiful supply of the others or is not aware of other physical units.\nIn this case, regarding a VM as a request, a server as a resource pool and a physical unit of the shortage type as a resource unit, we obtain a resource allocation problem that is weakly coupled. \nSimilar problems in computer networks, such as the virtual node embedding (see \\cite{esposito2016distributed}), server provisioning in distributed cloud environments (see \\cite{wei2017data}), and wireless resource scheduling (see \\cite{chen2017wireless}), can potentially be modeled as weakly coupled resource allocation problems. And as before, when the weakly coupled systems are in heavy traffic, the decomposability property holds.\n\n\nAs in \\cite{stolyar2013infinite,stolyar2017large}, for general VM replacement problems,  each server capacity is not necessarily constrained by physical units of just one type. As above, we model a VM as a request, a physical unit as a resource unit, and the set of all physical units of the same type located on the same server as a resource pool. In this context, the capacity of each resource pool is determined by the total number of its associated physical units of a given type on a server and the weak coupling property cannot hold in general. \nIt follows that, unlike the preceding examples, the system is not necessarily decomposable.\nHowever, as discussed in Section~\\ref{subsubsec:derive_ranking}, a decomposable system that is not weakly coupled or in heavy traffic can be found by finding a fixed point $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$ of the function $\\mathcal{T}^o$ ($o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{0})$). Numerical examples of such systems will be provided in Section~\\ref{sec:example}.\n\n\n\\section{The Index Policy: Its Implementation in the Non-Asymptotic Regime}\\label{sec:index_policy}\nIn Section~\\ref{sec:relaxation}, we considered the relaxed problem with constraints \\eqref{eqn:constraint:relax:action}-\\eqref{eqn:constraint:dummy}. Here, we return to the original problem with constraints \\eqref{eqn:constraint:action} and \\eqref{eqn:constraint:resources}.\n\n\nFor each RT $\\ell\\in[L]$, we refer to a policy $\\varphi\\in\\Phi$ as an \\emph{index policy} according to PS-pair ranking $o\\in\\mathscr{O}$, if it always prioritizes a candidate process in \na PS pair with a ranking equal or higher than those of all the other candidate processes.\nThis  policy $\\varphi$ is applicable to the original problem while, the policy $\\bar{\\varphi}(o)$ proposed in Section~\\ref{subsubsec:priorities_of_pairs} is not in general.\nThe method of implementing such a $\\varphi$ is not unique; for instance, the computation of the ranking of the PS pairs can vary. Here we propose one possible implementation.\n\n\nFor $t>0$, we maintain a sequence of $I$ ordered PS pairs $(i,N^{\\varphi}_i(t))$ ($i\\in[I]$) that are associated with the $I$ patterns, according to the given ranking $o$ and the state vector $\\bm{N}^{\\varphi}(t)$: PS pair $(i,N^{\\varphi}_i(t))$ is placed ahead of $(i',N^{\\varphi}_{i'}(t))$ if and only if the former precedes the latter in the ranking $o$.\nLet $i^o_{\\sigma}(\\bm{N}^{\\varphi}(t))$ ($\\sigma\\in[I]$) represent the pattern associated with the $\\sigma$th PS pair in this ordered sequence.\n\nFor a general ranking $o\\in\\mathscr{O}$, the variables $i^o_{\\sigma}(\\bm{N}^{\\varphi}(t))$ are potentially updated at each state transition. Nonetheless, for the purpose of this paper, we mainly focus on the rankings $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{\\nu})$ (for some $\\pmb{\\gamma}\\in\\mathbb{R}^J_0$ and $\\bm{\\nu}\\in\\mathbb{R}^L$) that follow the descending order of $\\Xi_i(\\pmb{\\gamma},\\bm{\\nu})$.\nIn this case, the variables $i^o_{\\sigma}(\\bm{N}^{\\varphi}(t))$ are updated only if a pattern $i\\in[I]\\backslash\\{d(\\ell)|\\ell\\in[L]\\}$ transitions into or out of its boundary state $|\\mathscr{N}_i|-1$.\n\n\n\nConsider the capacity constraints\n\\begin{equation}\\label{eqn:constraint:resources:epsilon:1}\n\\sum\\limits_{i'\\in[I]}w_{j,i'}N^{\\varphi}_{i'}(t) + \\sum\\limits_{i'\\in[I],i'\\neq i}w_{j,i'}a^{\\varphi}_{i'}(\\bm{N}^{\\varphi}(t))+ a^{\\varphi}_i(\\bm{N}^{\\varphi}(t)) \\leq \\Bigl\\lceil C_j\\bigl(1-\\bar{\\epsilon}_{j,\\iota(i,N^{\\varphi}_i(t))}\\bigr)\\Bigr\\rceil,~\\forall j\\in[J], i\\in[I],\n\\end{equation}\nwhere $\\iota(i,N^{\\varphi}_i(t))\\in[N]$ represents the order of PS pair $(i,N^{\\varphi}_i(t))$ in the ranking $o$ and $\\bar{\\bm{\\epsilon}}\\in [0,1]^{J\\times [N]}$ is a given matrix of parameters.\nApart from this matrix of parameters, constraints~\\eqref{eqn:constraint:resources:epsilon:1}  are the same as constraints~\\eqref{eqn:constraint:resources}.\nAs we shall discuss in Section~\\ref{subsec:policies}, we choose the $\\bar{\\epsilon}_{j,\\iota}$ such that $\\bar{\\epsilon}_{j,\\iota}C_j \\geq w_{j,i_{\\iota}}-1$ and, for any $j\\in[J]$ and PS pairs $\\iota<\\iota'$ with respect to the given ranking $o$, if $w_{j,i_{\\iota}},w_{j,i_{\\iota'}}>0$, then $\\bar{\\epsilon}_{j,\\iota} < \\bar{\\epsilon}_{j,\\iota'}$.\nIn this context, if $\\bar{\\epsilon}_{j,\\iota}C_j \\in [ w_{j,i_{\\iota}}-1,w_{j,i_{\\iota}})$ for all $\\iota\\in[N]$ and $j\\in[J]$, then constraints~\\eqref{eqn:constraint:resources:epsilon:1} reduce to \\eqref{eqn:constraint:resources}; otherwise, they are more stringent than \\eqref{eqn:constraint:resources}.\nThe parameter $\\bar{\\bm{\\epsilon}}$ is used to specify the trajectory of the underlying process $\\bm{N}^{\\varphi}(t)$ when the system is scaled to the asymptotic regime. \nThis specification is required \nfor proving the asymptotic optimality of the index policy $\\varphi$. \n\n\nIn the interests of notational consistency, we shall use the form \\eqref{eqn:constraint:resources:epsilon:1} throughout\nbut, here, since we do not worry about the asymptotic behavior, we consider the case with $\\bar{\\epsilon}_{j,\\iota}C_j \\in [ w_{j,i_{\\iota}}-1,w_{j,i_{\\iota}})$ for all $\\iota\\in[N]$ and $j\\in[J]$ so that \\eqref{eqn:constraint:resources:epsilon:1} reduces to \\eqref{eqn:constraint:resources}.\nA detailed discussion about the scaling procedure and the role of $\\bar{\\bm{\\epsilon}}$ in the asymptotic case will be provided in Section~\\ref{sec:asym}.\n\nUnder the index policy $\\varphi$, we select $L$ patterns to accept new arrivals of $L$ types according to their orders in sequence $i^o_{\\sigma}(\\bm{N}^{\\varphi}(t))$ ($\\sigma\\in[I]$).\nIn particular, at a decision making epoch $t>0$, we initialize $a^{\\varphi}_i(\\bm{N}^{\\varphi}(t))=0$ for all $i\\in[I]$ and a set of \\emph{available patterns} to be $[I]$. \nIf, for $i=i^o_1(\\bm{N}^{\\varphi}(t))$, constraints~\\eqref{eqn:constraint:resources:epsilon:1}\nwill not be violated by setting $a^{\\varphi}_i(\\bm{N}^{\\varphi}(t))=1$, then set $a^{\\varphi}_i(\\bm{N}^{\\varphi}(t))=1$ and remove all patterns associated with request type $\\ell(i)$ from the set of available patterns.\n\n\n\nThe other $L-1$ patterns are selected similarly and iteratively.\nThat is, we look for the smallest $\\sigma\\in\\{2,3,\\ldots,I\\}$ such that \n\\begin{itemize}\n\\item $i^o_{\\sigma}(\\bm{N}^{\\varphi}(t))$ remains in the set of available patterns; and\n\\item the capacity constraints~\\eqref{eqn:constraint:resources:epsilon:1} will not be violated by setting $a^{\\varphi}_i(\\bm{N}^{\\varphi}(t))=1$ where $i=i^o_{\\sigma}(\\bm{N}^{\\varphi}(t))$.\n\\end{itemize}\nIf there is such a $\\sigma$, set $a^{\\varphi}_i(\\bm{N}^{\\varphi}(t))=1$ for $i=i^o_{\\sigma}(\\bm{N}^{\\varphi}(t))$, remove all patterns associated with request type $\\ell(i)$ from the set of available patterns and continue selecting the remaining $L-2$ patterns in the same manner.\nWhen all of the $L$ patterns have been selected we can stop. \nDetailed steps are provided in Algorithm~\\ref{algo:index_policy}, \nwhich has a computational complexity that is linear in $I$.\n\n\n\\IncMargin{1em}\n\\begin{algorithm}\n\\small \n\\linespread{0.4}\\selectfont\n\n\\SetKwFunction{FIndexPolicy}{IndexPolicy}\n\\SetKwProg{Fn}{Function}{:}{\\KwRet $\\bm{a}^{\\varphi}(\\bm{n})$}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\SetAlgoLined\n\\DontPrintSemicolon\n\\Input{a ranking of PS pairs $o\\in\\mathscr{O}$ and a given state $\\bm{n}\\in\\mathscr{N}$.}\n\\Output{the action variables $\\bm{a}^{\\varphi}(\\bm{n})$ under the index policy $\\varphi\\in\\Phi$ with respect to ranking $o$ when the system is in state $\\bm{n}$.}\n\\Fn{\\FIndexPolicy{$o,\\bm{n}$}}{\n\t$\\bm{a}^{\\varphi}(\\bm{n}) \\gets \\bm{0}$ \\tcc*{Initializing the action variables}\n\t$\\mathscr{P}\\gets [I]$ \\tcc*{Initializing the set of available patterns}\n\t$\\sigma \\gets 1$ \\tcc*{Starting with the pattern with the highest priority}\n    \\While {$\\mathscr{P}\\neq \\emptyset $}{\t\n\t \t    $i\\gets i^o_{\\sigma}(\\bm{n})$\\;\n\t\t\t\\If {$i\\in\\mathscr{P}$ {\\bf and} Constraints~\\eqref{eqn:constraint:resources:epsilon:1} are not violated by setting $a^{\\varphi}_i(\\bm{n}) =1$ and $\\bm{N}^{\\varphi}(t)=\\bm{n}$}{\n\t\t\t    $a^{\\varphi}_i(\\bm{n})\\gets 1$\\;\n\t\t\t\tRemove all patterns $i'\\in\\mathscr{P}$ with $\\ell(i')=\\ell(i)$ from $\\mathscr{P}$\\;\n\t\t\t}\n\t    \t$\\sigma\\gets \\sigma +1$\\;\n\t}\n}\n\\caption{Implementing the index policy $\\varphi$ with respect to ranking $o$.}\\label{algo:index_policy}\n\\end{algorithm}\n \\DecMargin{1em}\n\n\n\nThe performance of $\\varphi$ is mainly determined by the given order $o\\in\\mathscr{O}$. \nBased on later discussion of the asymptotic regime, if the policy $\\bar{\\varphi}(o)$ is optimal for the relaxed problem in the asymptotic regime, then $\\varphi$ is asymptotically optimal for the original problem.\nEven without the proved asymptotic optimality, the ranking $o$ should ensure good performance of $\\varphi$ as it is always rational to prioritize patterns according to their potential profits.\nAs long as there are reasonably good $\\pmb{\\gamma}$ and $\\bm{\\nu}$ leading to a $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{\\nu})$, which correctly reflects the potential profits of patterns, the performance degradation of $\\bar{\\varphi}(o)$ is likely to be limited for the relaxed problem and close to the optimal solution of the original problem; and the index policy $\\varphi$ derived from $o$ is a promising choice for managing resources.\n\n\nThe selection of $\\pmb{\\gamma}$, $\\bm{\\nu}$  and $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{\\nu})$ is discussed in Section~\\ref{sec:relaxation}. The key point is to guarantee good performance of $\\bar{\\varphi}(o)$: the policy that is guaranteed to be optimal for the relaxed problem when the system is decomposable.\n\n\n\n\\section{Stochastic Optimization in a Scaled System} \\label{sec:asym}\n\n\nIn this section, we establish asymptotic optimality of $\\varphi$.\n\n\\vspace{-0.3cm}\n\\subsection{Scaling Parameter}\\label{subsec:asym_regime}\n\\vspace{-0.3cm}\nWith a parameter $h\\in\\mathbb{N}_+$, let $\\bm{C}\\coloneqq h\\bm{C}^{0}$, $\\bm{C}^{0}\\in\\mathbb{N}_{+}^{J}$, and the arrival rates scale as $\\bm{\\lambda}\\coloneqq h\\bm{\\lambda}^{0}$, $\\bm{\\lambda}^{0}\\in\\mathbb{R}_{+}^{L}$.\nWe refer to the parameter $h$ as the \\emph{scaling parameter}, and the \\emph{asymptotic regime} as the limiting case with $h\\rightarrow +\\infty$.\n\n\n\nWe split the process associated with pattern $i$ into $h$ identical \\emph{sub-processes}\n$(i,k)$, $k\\in[h]$, and divide $N^{\\phi}_{i}(t)$, the number of instantiations for pattern $i$ under policy $\\phi$ at time $t$, into $h$ pieces.\nThe number of instantiations of the $k$th piece is $N^{\\phi}_{i,k}(t)$, so that $N^{\\phi}_{i}(t) = \\sum_{k=1}^{h}N^{\\phi}_{i,k}(t)$.\nWe refer to $N^{\\phi}_{i,k}(t)$ as the number of instantiations for \\emph{sub-pattern} $(i,k)$.\nThe counting process given by $N^{\\phi}_{i,k}(t)$ ($k\\in [h], i\\in [I]$) has state space \n$\\mathscr{N}_{i}^{0} \\coloneqq \\{0,1,\\ldots,\\min_{j\\in \\mathscr{J}_i}\\lceil C_{j}^{0}\/w_{j,i}\\rceil\\}$.\nFor any dummy  pattern $d(\\ell)$, we take $\\mathscr{N}^0_{d(\\ell)}= \\mathscr{N}_{d(\\ell)} = \\{0\\}$.\n\n\n\nThe objective and constraints defined by \\eqref{eqn:objective}, \\eqref{eqn:constraint:action} and \\eqref{eqn:constraint:resources} still apply to the sums of variables $ \\sum_{k=1}^{h}N^{\\phi}_{i,k}(t)\\coloneqq  N^{\\phi}_i(t)$, $i\\in[I]$.\nWe say the process associated with pattern $i$ is \\emph{replaced} by the $h$ sub-processes associated with sub-patterns $(i,k)$, $k\\in[h]$.\nEach sub-pattern $(i,k)$ earns reward $r_{\\ell(i)}$ per each served request and the cost rate that a request accommodated by this sub-pattern imposes on resource pool $j\\in[J]$ is $\\varepsilon_jw_{j,i}$; that is,\nthe sub-process $(i,k)$ maintains the same reward and cost rates in states $n\\in\\mathscr{N}_i^0$ as process $i$.\nLet $\\bm{N}_{h}^{\\phi}(t) = (N^{\\phi}_{i,k}(t):\\ i\\in[I], k\\in[h])$ be the state variable after this replacement,\nand $a^{\\phi}_{i,k}(\\bm{N}^{\\phi}_h(t))\\in\\{0,1\\}$ ($i\\in[I],k\\in[h]$) be the action variables  with respect to the process $\\bm{N}^{\\phi}_h(t)$.\nTo clarify, we rewrite the objective described in \\eqref{eqn:objective} as\n\\begin{equation}\\label{eqn:objective:h}\n\\max\\limits_{\\phi}  \\frac{1}{h}\\sum\\limits_{i\\in[I]} \\sum\\limits_{k\\in[h]}\\sum_{n_i\\in\\mathscr{N}_i} \\pi_{i,k}^{\\phi,h}(n_i) \\Bigl(r_{\\ell(i)} \\mu_i  - \\sum_{j\\in\\mathscr{J}} w_{j,i} \\varepsilon_j\\Bigr)n_i,\n\\end{equation}\nwhere $\\pi^{\\phi,h}_{i,k}(n_i)$ represents the stationary probability that the state of sub-process $(i,k)$ is $n_i$ under policy $\\phi$ with scaling parameter $h$.\nWe divide the total revenue earned by all sub-patterns by $h\\in\\mathbb{N}_+$ so that the objective function is always bounded when $h\\rightarrow +\\infty$. \nThe policy $\\phi$ in \\eqref{eqn:objective:h}  is determined by the action variables $a^{\\phi}_{i,k}(\\bm{N}^{\\phi}_h(t))$ ($i\\in[I],k\\in[h]$)\nsubject to \\vspace{-0.3cm}\n\\begin{equation}\\label{eqn:constraint:action:h}\n\\sum\\limits_{i\\in\\mathscr{P}_{\\ell}}\\sum\\limits_{k\\in[h]}a^{\\phi}_{i,k}(\\bm{N}^{\\phi}_h(t))=1,~\\forall \\ell\\in[L],~\\forall t \\geq 0, \\vspace{-0.3cm}\n\\end{equation}\nand  \\vspace{-0.3cm}\n\\begin{equation}\\label{eqn:constraint:resources:h}\n\\sum\\limits_{i\\in[I]}\\frac{w_{j,i}}{h}\\sum\\limits_{k\\in[h]}\\Bigl(N^{\\phi}_{i,k}(t)+a^{\\phi}_{i,k}(\\bm{N}^{\\phi}_h(t))\\Bigr)\\leq C_j^0,~\\forall j\\in[J],~\\forall  t \\geq 0.\n\\end{equation}\nThe constraints in \\eqref{eqn:constraint:resources:h} are obtained by substituting $C_j^0h$ for $C_j$ in the constraints stated in \\eqref{eqn:constraint:resources}, and thus \\eqref{eqn:constraint:resources:h} is equivalent to \\eqref{eqn:constraint:resources}.\nAlso, to guarantee that the maximal value of each $N^{\\phi}_{i,k}(t)$ ($k\\in[h]$), $\\min_{j\\in\\mathscr{J}_i}\\lceil C_j^0\/w_{j,i}\\rceil$, is not exceeded, define, for $k\\in[h]$ and $i\\in[I]\\backslash\\{d(\\ell):\\ \\ell\\in[L]\\}$,\n\\vspace{-0.4cm}\n\\begin{equation}\\label{eqn:constraint:resources:zero:h}\na^{\\phi}_{i,k}(\\bm{N}^{\\phi}_h(t))=0,~\\text{if }  N^{\\phi}_{i,k}(t)=|\\mathscr{N}_i^0|-1,\n\\vspace{-0.4cm}\n\\end{equation}\nwhich corresponds to the redundant constraints described in \\eqref{eqn:constraint:resources:zero}.\n\n\n\n\n{\\bf Remark}\nAs in Section~\\ref{sec:introduction},  we activate exactly one sub-process $(i,k)$ ($i\\in\\mathscr{P}_{\\ell}$, $k\\in[h]$) for RT $\\ell\\in[L]$ regardless of the scaling parameter $h\\in\\mathbb{N}_+$.\nThe birth and death rates of this active sub-process are $h\\lambda_{\\ell}^0$ and $N^{\\phi}_{i,k}(t)\\mu_{i}^0$, respectively, so that if $h\\lambda^0_{\\ell} \\gg (|\\mathscr{N}^0_i|-1)\\mu_{i}^0$, the number of instantiations of pattern $i$ will increase rapidly until it is restricted by the capacity constraints.\n\n\nA model with a single active sub-process at any time has different stochastic properties compared to the case where the number of active sub-processes is proportional to $h$ (which was discussed in \\cite{weber1990index}). \nTo illustrate the difference, we present an example in Appendix~\\ref{app:example3} of the e-companion.\n\n\n\n\nThe optimization problem consisting of the $hI$ sub-processes associated with $hI$ sub-patterns, coupled through constraints~\\eqref{eqn:constraint:action:h}-\\eqref{eqn:constraint:resources:zero:h} can be analyzed and relaxed along the same lines as in Section~\\ref{sec:relaxation}. \nLet $\\alpha^{\\phi}_{i,k}(n)\\coloneqq \\lim_{t\\rightarrow +\\infty}\\mathbb{E}\\{a^{\\phi}_{i,k}(\\bm{N}^{\\phi}_h(t))| N^{\\phi}_{i,k}(t)=n\\}$ ($n\\in\\mathscr{N}^0_{i}$, $i\\in[I]$, $k\\in[h]$) represent the action variables of the $hI$ sub-problems for the relaxed problem scaled by $h$. \n\n\nAll the sub-processes corresponding to a given pattern $i\\in[I]$ in the same state $n\\in\\mathscr{N}_i^0$ are equivalent.\nThe controller then is concerned only  with the total number of active sub-processes of a given pattern in a given state. \nDefine the random variable $Z^{\\phi,h}_{\\iota}(t)$ to be the proportion of sub-processes in PS pair $\\iota$ at time $t$ under policy $\\phi$ where $h$ is the scaling parameter; that is, \\vspace{-0.3cm}\n\\begin{equation}\\label{eqn:z_process}\nZ^{\\phi,h}_{\\iota}(t)=\\frac{1}{hI}\\Bigl|\\bigl\\{(i,k)\\in[I]\\times[h]\\ \\left|\\ N^{\\phi}_{i,k}(t)=n_{\\iota},\\ i_{\\iota}=i\\right.\\bigr\\}\\Bigr|. \\vspace{-0.3cm}\n\\end{equation} \n\nLet $\\bm{Z}^{\\phi,h}(t)=(Z^{\\phi,h}_{\\iota}(t):\\ \\iota\\in[N])$ and $\\mathscr{Z}$ be the probability simplex $\\{\\bm{z}\\in[0,1]^{N}\\ |\\ \\sum_{\\iota\\in[N]}z_{\\iota}=1\\}$.\nIn this model, the process $\\bm{Z}^{\\phi,h}(t)$ is analogous to the counting process $\\bm{N}^{\\phi}_h(t)$ in the original process.\nWhen the process $\\bm{Z}^{\\phi,h}(t)$ takes value $\\bm{z}\\in\\mathscr{Z}$, it can transition only  to a state of the form $\\bm{z}+\\bm{e}_{\\iota,\\iota'}\\in\\mathscr{Z}$ with $i_{\\iota}=i_{\\iota'}$, where $\\bm{e}_{\\iota,\\iota'}\\in \\mathbb{R}^{N}$ is a vector with $\\iota$th element $+1\/hI$, $\\iota'$th element $-1\/hI$ and all the other elements set to zero. \nFor our birth-and-death process, a transition will happen only with $n_{\\iota'} = n_{\\iota}\\pm 1$ corresponding to the arrival and departure of a request, respectively.\nFor any given $h\\in\\mathbb{N}_+$, \nthe state space of the process $\\bm{Z}^{\\phi,h}(t)$ is a subset of $\\mathscr{Z}$ and thus the system is always stable.\nWe refer to the system with $h\\rightarrow +\\infty$ as the \\emph{asymptotic regime}.\n\n\n\nNote that any resource allocation problem in the non-asymptotic regime coincides with a scaled problem described in \\eqref{eqn:objective:h}-\\eqref{eqn:constraint:resources:zero:h} with given $h<+\\infty$. The scaling parameter $h$ is introduced to rigorously specify the trajectory of the entire system going from a non-asymptotic regime to an asymptotic regime.\n\n\\vspace{-0.5cm}\n\n\\subsection{Index Policies in a Scaled System}\\label{subsec:policies}\n\\vspace{-0.3cm}\n\nIn Section~\\ref{sec:index_policy}, for a ranking $o\\in\\mathscr{O}$, we proposed an index policy $\\varphi\\in\\Phi$ for the resource allocation problem in the non-asymptotic regime; this coincides with the problem described in \\eqref{eqn:objective:h}-\\eqref{eqn:constraint:resources:zero:h} with given $h<+\\infty$. \nFor clarity, we translate the description of $\\varphi$ to a policy used for a scaled system with the notation described in Section~\\ref{subsec:asym_regime}.\n\n\nFor a ranking $o\\in\\mathscr{O}$, the index policy $\\varphi$ \\emph{activates} a sub-process in the first PS pair $\\iota\\in[N]$ in ranking $o$ with $Z^{\\varphi,h}_{\\iota}(t)>0$ and the action and capacity constraints holding; that is, $\\varphi$ selects a sub-process $(i_{\\iota},k)$ ($k\\in[h]$) satisfying $N^{\\varphi}_{i_{\\iota},k}(t)=n_{\\iota}$ and sets $a^{\\varphi}_{i_{\\iota},k }(\\bm{N}^{\\varphi}_h(t))$ to $1$. The condition $Z^{\\varphi,h}_{\\iota}(t)>0$ is required because there has to be some sub-processes in PS pair $\\iota$ for us  to be able to activate.\nOnce a sub-process in PS pair $\\iota$ is selected for activation, the action constraint~\\eqref{eqn:constraint:action:h} for RT $\\ell_{\\iota}$ achieves equality: there is exactly one active sub-process for a specific RT $\\ell\\in[L]$.\nResource units in associated resource pools are reserved for this activated sub-process in PS pair $\\iota$. \nIn this way, $L$ sub-processes in $L$ different PS pairs will be activated iteratively, according to the ranking $o$, for the $L$ RTs.\n\nUnder the index policy $\\varphi$,  the transition matrix of process $\\bm{Z}^{\\varphi,h}(t)$ is determined by the value of $\\sum_{k\\in[h],N^{\\varphi}_{i_{\\iota},k}(t)=n_{\\iota}}a^{\\varphi}_{i_{\\iota},k }(\\bm{N}^{\\varphi}_h(t))$ for each PS pair $\\iota\\in[N]$, which is either $0$ or $1$  and is dependent on $\\bm{N}^{\\varphi}_h(t)$ through only $\\bm{Z}^{\\varphi,h}(t)$.\nDefine $\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z})$, $\\iota\\in[N]$, $\\bm{z}\\in \\mathscr{Z}$, to be the ratio of the number of active sub-processes in PS pair $\\iota$, for which the corresponding sub-patterns are prepared to accept a request,  to the total number of sub-processes in this PS pair under $\\varphi$, when the proportions of sub-processes in all PS pairs are currently specified by $\\bm{z}$. \nThat is, at time $t$, for $\\iota\\in[N]$,\n\\begin{equation}\n\\upsilon^{\\varphi,h}_{\\iota}\\bigl(\\bm{Z}^{\\varphi,h}(t)\\bigr)= \\frac{\\sum_{k\\in[h],N^{\\phi}_{i_{\\iota},k}(t)=n_{\\iota}}a^{\\phi}_{i_{\\iota},k}(\\bm{N}^{\\varphi}_h(t))}{IhZ^{\\varphi,h}_{\\iota}(t)},\n\\end{equation}\nwhere we recall that \nthe numerator on the right hand side relies on $\\bm{N}^{\\varphi}_h(t)$ through $\\bm{Z}^{\\varphi,h}(t)\\in\\mathscr{Z}$.\nNote that, for arbitrarily large $h$, the value of $\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z})Ihz_{\\iota}$ ($\\bm{z}\\in\\mathscr{Z}$), representing the number of active sub-processes in PS pair $\\iota$, is never more than $1$ because the policy $\\varphi$ must satisfy the action constraints \\eqref{eqn:constraint:action:h}.\nLet $\\bm{\\upsilon}^{\\varphi,h}(\\bm{z})=(\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z}):\\ \\iota\\in[N])$.\nAlthough different tie-breaking rules lead to the same process $\\bm{Z}^{\\varphi,h}(t)$, \nwe shall stipulate that, when there is more than one sub-process $(i,k)$ ($i\\in [I], k\\in[h]$)  in the same PS pair available for activation, we prioritize the one with the smaller value of $k$.\nIn this context, the variables $\\bm{\\upsilon}^{\\varphi,h}(\\bm{z})$, $\\bm{z}\\in\\mathscr{Z}$, provide sufficient information for the index policy $\\varphi$ to make decisions on the counting process $\\bm{N}^{\\varphi}_h(t)$.\n\n\n\n\nLet $\\zeta_{\\iota}^{\\varphi,h}(\\bm{z})$ represent the maximal proportion of sub-processes in PS pair $\\iota$ that can be active if we consider only the capacity constraints defined by \\eqref{eqn:constraint:resources:h} (neglecting the action constraints defined by \\eqref{eqn:constraint:action:h}) with proportions of sub-processes in all PS pairs specified by $\\bm{z}$ under policy $\\varphi$.\nWe obtain that, for $\\iota\\in[N]\\backslash\\{d(\\ell):\\ \\ell\\in[L]\\}$,\n\\begin{equation}\\label{eqn:tilde_z}\n\\zeta^{\\varphi,h}_{\\iota}(\\bm{z}) =\n\\min\\Biggl\\{z_{\\iota},\\max\\biggl\\{0,\\min\\limits_{j\\in\\mathscr{J}_i} \\frac{1}{w_{j,i}Ih}\\Bigl\\lceil hC_{j}^{0}(1-\\epsilon^h_{j,\\iota})-\\sum\\limits_{\\iota'=1}^{N}w_{j,i_{\\iota'}}n_{\\iota'}z_{\\iota'}Ih-\\sum\\limits_{\\iota'\\in \\mathscr{N}_{\\iota}^{+}}w_{j,i_{\\iota'}}\\upsilon^{\\varphi,h}_{\\iota'}(\\bm{z})z_{\\iota'}Ih\\Bigr\\rceil\\biggr\\}\\Biggr\\},\n\\end{equation}\nwhere $\\mathscr{N}_{\\iota}^{+}$, $\\iota\\in[N]$, is the set of PS pairs $\\iota'\\in[N]$ with $\\iota' <\\iota$ (with higher priorities than pair $\\iota$) with respect to ranking $o$, \nand $\\epsilon^h_{j,\\iota}\\in [0,1]$ corresponds to $\\bar{\\epsilon}_{j,\\iota}$ in \\eqref{eqn:constraint:resources:epsilon:1}.\n\nHere, the parameter $\\epsilon^h_{j,\\iota}$ is defined so that\n\\begin{equation}\\label{eqn:epsilon_condition}\n0<\\lim\\limits_{h\\rightarrow +\\infty}\\epsilon^h_{j,\\iota} < \\lim\\limits_{h\\rightarrow +\\infty}\\epsilon^h_{j,\\iota'} \\leq 1\n\\end{equation}\nfor any $\\iota < \\iota'$, $w_{j,i_{\\iota}}>0$ and $w_{j,i_{\\iota'}}>0$.\nWe need $\\epsilon^h_{j,\\iota}$ to indicate the priorities of PS pairs in resource pool $j$, because\nthe last term in \\eqref{eqn:tilde_z},\n$\\sum_{\\iota'\\in \\mathscr{N}_{\\iota}^{+}}w_{j,i_{\\iota'}}\\upsilon^{\\varphi,h}_{\\iota'}(\\bm{z})z_{\\iota'}Ih$,\nis $o(h)$.\nIn particular, \nin order to follow the strict capacity constraints described in \\eqref{eqn:constraint:resources:h}, we need to define the $\\epsilon^h_{j,\\iota}$ so that $\\epsilon^h_{j,\\iota}C_{j}^0 h \\geq w_{j,i_{\\iota}}-1$\nfor all $j\\in[J]$, $\\iota\\in[N]$ and $h\\in\\mathbb{N}_+$ and $\\lim\\limits_{h\\rightarrow+\\infty}\\epsilon^h_{j,\\iota}$ exists.\nLet $\\bm{\\epsilon}^h \\coloneqq (\\epsilon^h_{j,\\iota}:\\ j\\in[J],\\iota\\in[N])$ and $\\bm{\\epsilon} \\coloneqq \\lim\\limits_{h\\rightarrow +\\infty}\\bm{\\epsilon}^h$. \nDefine $\\mathscr{E}^h$, $h\\in\\mathbb{N}_+ \\cup \\{+\\infty\\}$, and $\\Psi$ as the sets of all such vectors $\\bm{\\epsilon}^h$ and sequences of such vectors $\\psi\\coloneqq (\\bm{\\epsilon}^1,\\bm{\\epsilon}^2,\\ldots,)$, respectively.\n\n\n\nEquation \\eqref{eqn:epsilon_condition} specifies possible trajectories of $\\bm{\\epsilon}^h$ as $h\\rightarrow +\\infty$, and is required for subsequent proofs of asymptotic optimality.\nNote that, although the asymptotic regime is a limiting situation,  using an asymptotically-optimal policy is likely to be appropriate for systems with finite but large $h$. \n\n\nIn \\eqref{eqn:tilde_z}, the value of $\\zeta^{\\varphi,h}_{\\iota}(\\bm{z})$ is constrained by the remaining capacities of relevant resource pools, the proportion of sub-processes currently in  PS pair $\\iota$ and the proportions of active sub-processes in PS pairs with higher priorities.\nRecall that  $\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z})$ represents  the proportion of active sub-processes in PS pair $\\iota$, for which the corresponding sub-patterns are prepared to accept a request, when the proportions of sub-processes in all PS pairs are currently specified by $\\bm{z}$.\nTogether with the action constraints described in \\eqref{eqn:constraint:action:h}, under an index policy $\\varphi$, \nfor $z_{\\iota}>0$,\\vspace{-0.15cm}\n\\begin{equation}\\label{eqn:probability_active}\n\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z}) = \\frac{1}{z_{\\iota}hI}\n\\min\\biggl\\{\\zeta^{\\varphi,h}_{\\iota}(\\bm{z})hI,\\  \\max\\Bigl\\{0,1- \\sum\\limits_{\\begin{subarray}\\ \\iota'\\in \\mathscr{N}^{+}_{\\iota}\\\\ l_{\\iota'}=l_{\\iota}\\end{subarray}}\\zeta^{\\varphi,h}_{\\iota'}(\\bm{z})hI\\Bigr\\}\\biggr\\}.\n\\end{equation}\nIf $z_\\iota = 0$, then there are no sub-processes in PS pair $\\iota$ and so $\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z})$ can take any value in $[0,1]$ without making a difference to the evolution of the process.\nFor completeness, define, for $\\bm{z}$ with $z_{\\iota}=0$ and $\\bm{z}_{\\iota}^{x} \n\\coloneqq (z_{1},z_{2},\\ldots,z_{\\iota-1},x,z_{\\iota+1},\\ldots,z_{N})$, \n$\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z})=\\lim\\limits_{x\\downarrow 0}\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z}^{x}_{\\iota})$.\nFor any given $\\bm{z}\\in\\mathscr{Z}$, $\\zeta^{\\varphi,h}_{\\iota}(\\bm{z})$ and $\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z})$ can be obtained iteratively using equations \\eqref{eqn:tilde_z} and \\eqref{eqn:probability_active} from $\\iota=1$ to $N$.\n\n\n{\\bf Remark}\nAlthough capacity constraints were not considered in \\cite{weber1990index}, the construction of $\\bm{\\upsilon}^{\\varphi,h}(\\bm{z})$ ($\\bm{z}\\in\\mathscr{Z}$) follows ideas similar to those used in that paper. \nRecall that, for a given ranking, $o\\in\\mathscr{O}$,  the policy $\\bar{\\varphi}(o)$ is generated by Algorithm~\\ref{algo:varphi_gamma} and is infeasible for the original problem. \nThis gives rise to the interesting property that values of $\\upsilon^{\\varphi,h}_{\\iota}(\\bm{z})$ and $\\alpha^{\\bar{\\varphi}(o)}_{i_{\\iota},k}(n_{\\iota})$ ($k\\in[h]$) for all $h\\in\\mathbb{N}_+\\cup\\{+\\infty\\}$, are always independent of those of PS pairs $\\iota'$ with $\\iota' > \\iota$: the PS pairs with lower priorities than $\\iota$.\nThe property is important \nfor the proofs of Theorems~\\ref{theorem:main_second} and \\ref{theorem:main}.\n\n\\vspace{-0.3cm}\n\n\\subsection{Asymptotic Optimality}\\label{sec:asym_opt}\n\\vspace{-0.3cm}\n\nFor given $h\\in\\mathbb{N}_+$, define the long-run average revenue normalized by $h$ of the resource allocation problem under policy $\\phi$ to be $R^{\\phi,h}$; that is, \\vspace{-0.2cm}\n\\begin{equation}\nR^{\\phi,h} \\coloneqq \\frac{1}{h}\\sum\\limits_{i\\in[I]} \\sum\\limits_{k\\in[h]}\\sum_{n_i\\in\\mathscr{N}_i} \\pi_{i,k}^{\\phi,h}(n_i) \\Bigl(r_{\\ell(i)} \\mu_i  - \\sum_{j\\in\\mathscr{J}} w_{j,i} \\varepsilon_j\\Bigr)n_i, \\vspace{-0.2cm}\n\\end{equation}\nthe objective function described in \\eqref{eqn:objective:h}.\n\\vspace{-0.3cm}\n\\begin{definition}\nWe say that the index policy $\\varphi$ derived from PS ranking $o$ by iterating \\eqref{eqn:tilde_z} and \\eqref{eqn:probability_active} is \\emph{asymptotically optimal} if \n$\n\\lim\\limits_{\\lVert \\bm{\\epsilon}\\rVert \\to \\bm{0}}\\lim\\limits_{h\\rightarrow+\\infty }|R^{\\varphi,h}-\\max\\limits_{\\phi\\in\\Phi}R^{\\phi,h}| = 0$.\n\\end{definition}\nRecall that the index policy $\\varphi$ described in Section~\\ref{subsec:policies} is dependent on the parameter $\\bm{\\epsilon}^h$ with $\\bm{\\epsilon} \\coloneqq \\lim\\limits_{h\\rightarrow +\\infty}\\bm{\\epsilon}^h$. The $\\bm{\\epsilon}$ is used to guarantee strict priorities of the sub-processes in the asymptotic regime as discussed in Section~\\ref{subsec:policies}. \nThe policy $\\bar{\\varphi}(o)$, proposed in Section~\\ref{subsubsec:priorities_of_pairs} for the relaxed problem, is generally not applicable to the original resource allocation problem.\nAlthough the policies $\\bar{\\varphi}(o)$ and $\\varphi$ both rely on the same ranking $o\\in\\mathscr{O}$, they are different policies. \n\\begin{theorem}\\label{theorem:main_second}\nFor given $o\\in\\mathscr{O}$, \n$\\varphi$ derived from $o$ by iterating \\eqref{eqn:tilde_z} and \\eqref{eqn:probability_active} is asymptotically optimal if and only if\n\\vspace{-0.5cm}\n\\begin{equation}\\label{eqn:main_second}\n\\lim\\limits_{h\\rightarrow +\\infty}|R^{\\bar{\\varphi}(o),h} -\\max\\limits_{\\phi\\in\\Phi}R^{\\phi,h}|=0.\n\\end{equation}\n\\end{theorem}\nThe proof is given  in Appendix~\\ref{app:theorem:main_second}.\nTheorem~\\ref{theorem:main_second} indicates that asymptotic optimality of $\\varphi$ is equivalent to the convergence between $R^{\\bar{\\varphi}(o),h}$ and the maximized long-run average revenue of the original problem as $h\\rightarrow +\\infty$.\nIt is proved by showing the existence of $\\lim_{h\\rightarrow +\\infty} \\lim_{t \\rightarrow +\\infty}\\mathbb{E}[\\bm{Z}^{\\bar{\\varphi}(o),h}(t)]$ and\na global attractor of the process $\\bm{Z}^{\\varphi,h}(t)$ as $t,h\\rightarrow +\\infty$ and $\\lVert\\bm{\\epsilon}\\rVert\\rightarrow 0$, and specifically that they coincide with each other.\nThe long-run average revenue $R^{\\varphi,h}$ then coincides with $R^{\\bar{\\varphi}(o),h}$ as $h\\rightarrow +\\infty$ and $\\lVert \\bm{\\epsilon}\\rVert \\rightarrow 0$. \n\n\nA similar condition relevant to the global attractor was required in \\cite{weber1990index} for asymptotic optimality of the Whittle index policy in a general RMABP. It does not necessarily hold. However, in our problem, each sub-process is a queueing process with the departure rate increasing in its queue size.\nSuch a sub-process is a special case of a general bandit process.\nWe prove in general that the underlying process $\\bm{Z}^{\\varphi,h}(t)$, regardless of its initial point, will converge to any specified neighborhood of a fixed point almost surely as $t,h\\rightarrow +\\infty$ and $\\lVert \\bm{\\epsilon}\\rVert \\rightarrow 0$.\nDetailed proofs are provided in Appendix~\\ref{app:theorem:main_second}.\n\n\nTheorem~\\ref{theorem:main_second}, in itself,  does not provide a verifiable condition.\nThis is given in our next theorem. If there exists $H\\in\\mathbb{R}$ such that, for all $h>H$, the system is decomposable, we say the system is decomposable in the asymptotic regime.\n\\begin{theorem}\\label{theorem:main}\nIf the capacity constraints described in \\eqref{eqn:constraint:resources:h} (or equivalently \\eqref{eqn:constraint:resources}) are decomposable with decomposable multipliers $\\pmb{\\gamma}\\in\\mathbb{R}^J_0$ in the asymptotic regime, then there exist $\\bm{\\nu}\\in\\mathbb{R}^L$ and a PS pair ranking $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{\\nu})$ such that the index policy $\\varphi$ derived from $o$ is asymptotically optimal.\n\\end{theorem}\nThe proof, based on Theorem~\\ref{theorem:main_second}, is given in Appendix~\\ref{app:theorem:main}. \nRecall that we discussed decomposability of multipliers in Section~\\ref{sec:relaxation}\nand provided examples of provably decomposable systems.\n\n\n{\\bf Remark}\nTheorem~\\ref{theorem:main} binds asymptotic optimality of $\\varphi$ to the decomposability of the relaxed problem. \nFor a decomposable system (see Definition~\\ref{define:decomposable}), there always exists a ranking $o\\in\\mathscr{O}$ such that $\\bar{\\varphi}(o)$ is optimal for the relaxed problem.\nIf a system is decomposable in the asymptotic regime \nthen \\eqref{eqn:main_second} is satisfied.\nThis follows because, for any $h\\in\\mathbb{N}_+\\cup\\{+\\infty\\}$, $R^{\\varphi,h} \\leq \\max_{\\phi\\in\\Phi}R^{\\phi,h} \\leq \\max_{\\phi\\in\\tilde{\\Phi}}R^{\\phi,h}$ where $\\Phi$ and $\\tilde{\\Phi}$ are the sets of feasible policies for the original and relaxed problem, and $R^{\\varphi,h}$ coincides with $R^{\\bar{\\varphi}(o),h}$ as $h\\rightarrow +\\infty$ and $\\lVert\\bm{\\epsilon}\\rVert \\rightarrow 0$.\n\n\n\n\nSimilarly, we say the system is in heavy traffic in the asymptotic regime if there exists $H\\in\\mathbb{R}$ such that, for all $h>H$, the system is in heavy traffic.\n\\vspace{-0.3cm} \n\\begin{corollary}\\label{coro:main}\nIf the system is in heavy traffic in the asymptotic regime and is weakly coupled,\nthen there exist decomposable multipliers $\\pmb{\\gamma}\\in\\mathbb{R}^J_0$, satisfying \\eqref{eqn:equal_opt:a}-\\eqref{eqn:equal_opt:c}, and a PS pair  ranking $o\\in\\mathscr{O}^*(\\bm{0})$,\nso that  the index policy $\\varphi$ derived from $o$ is asymptotically optimal. \\vspace{-0.3cm}\n\\end{corollary}\nThe proof, invoking Theorem~\\ref{theorem:main} and Proposition~\\ref{prop:equal_opt}, is given in Appendix~\\ref{app:coro:main}. \nIn particular, the PS pair ranking $o$, described in Corollary~\\ref{coro:main}, exists in closed form: its PS pairs are ranked according to the descending order of $\\Xi_{\\iota}^*$ with $\\bm{\\nu}=\\bm{0}$. \n\n\n\n\\section{Numerical Results}\\label{sec:example}\n\n\nWe demonstrate by simulation the performance of the index policy $\\varphi$, defined in Section~\\ref{subsec:policies} (or equivalently, defined in Section~\\ref{sec:index_policy} for a given $h<+\\infty$), in systems that are not weakly coupled or in heavy traffic in comparison with baseline policies.\n\n\n\nIn this section, the confidence intervals of all the simulated average revenues at the $95\\%$ level based on the Student's t-distribution are maintained within $\\pm 3\\%$ of the observed mean.\nWe recall that the capacities $\\bm{C}$ and arrival rates $\\bm{\\lambda}$ are scaled by the scaling parameter $h$.\n\n\n\n\n\n\n\n\n\n\nAlong with the fixed point iteration method proposed in Section~\\ref{subsubsec:derive_ranking},\nwe have been able to find systems which are not weakly coupled or in heavy traffic but are decomposable.\nHere, we provide two examples, where $L$ and $J$ are sampled uniformly from the sets $\\{2,3,4,5\\}$ and $\\{10,11,\\ldots,20\\}$, respectively.\nLet\n$\\epsilon_M = \\max_{j\\in[J],\\iota\\in[N]}\\epsilon_{j,\\iota}$.\nWe refer to an index policy $\\varphi$ with specific $\\epsilon_M\\in [0,1]$ as INDEX($\\epsilon_M$).\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.45\\linewidth]{OR-OPT-seed457-v3.eps}\\label{fig1:opt_v1}}\n\\subfigure[]{\\includegraphics[width=0.45\\linewidth]{OR-OPT-seed406-v3.eps}\\label{fig1:opt_v3}}\n\\caption{Relative difference of a specific policy to $R(o_{k^*})$ against the scaling parameter of the system: (a) diverse performance and non-zero decomposable multipliers; (b) similar performance and non-zero decomposable multipliers; and (c) zero decomposable multipliers.}\\label{fig:fig1}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\nWe consider three baseline policies: two greedy policies that prioritize patterns with maximal reward rates and minimal cost rates, and one policy randomly uniformly selecting an available pattern for each request type. We refer to the three policies as \\emph{Max-Reward}, \\emph{Min-Cost} and \\emph{Random}.\nThe Max-Reward and Min-Cost policies are in fact index policies with PS pairs ranked according to the descending order of their reward rates and the ascending order of their cost rates, respectively. \nThe Random policy was proposed by \\cite{stolyar2017large} for a VM replacement problem, aiming to minimize the system blocking probabilities in the case with finite capacities.\nIt is not a feasible policy of the original problem with capacity constraints \\eqref{eqn:constraint:resources} because it does not reserve resource units for a specific pattern that is more profitable than the others.\nWhen there are not enough resource units in a pool to accommodate multiple request types that have chosen their patterns involving this pool, the Random policy will always assign the resource units to the request that arrives first. \n\n\nIn Figure~\\ref{fig:fig1}, we compare the performance of INDEX(0), INDEX($0.01$), the baseline policies and $\\bar{\\varphi}(o_{k^*})$, where $o_{k^*}$ is the ranking of the multipliers $\\pmb{\\gamma}_{k^*}$ resulting from the fixed point iteration method (described in Section~\\ref{subsubsec:derive_ranking}) with parameter $c=0.5$ and initial point $\\pmb{\\gamma}_0=\\bm{0}$. \nThe system parameters are listed in Appendix~\\ref{app:simulation:opt} and are generated by pseudo-random functions.  \nThe discovered multipliers $\\pmb{\\gamma}_{k^*}$ for simulations in Figures~\\ref{fig1:opt_v1} and \\ref{fig1:opt_v3} are \n$(269.555,0,0,0,0,273.11,0,$ $347.995,0,0,0,8.323\\times 10^{-7},9.726\\times 10^{-5},0)$  and $\\bm{0}$,\nrespectively, satisfying $\\mathcal{T}^{o_{k^*}}(\\pmb{\\gamma}_{k^*})=\\pmb{\\gamma}_{k^*}$ in the asymptotic regime.\nBy Proposition~\\ref{prop:converge_gamma}, these $\\pmb{\\gamma}_{k^*}$ are decomposable multipliers and, by Theorem~\\ref{theorem:main}, the index policies derived from the rankings $o_{k^*}$ are asymptotically optimal.\nLet $R(o)\\coloneqq \\lim\\limits_{h\\rightarrow +\\infty}R^{\\bar{\\varphi}(o),h}$ ($o\\in\\mathscr{O}$) of which the existence is guaranteed in the proof of Theorem~\\ref{theorem:main_second}.\nFor the decomposable systems with $h<+\\infty$ and $\\bar{\\varphi}(o_{k^*})$ optimal for the relaxed problem in the asymptotic regime, the asymptotic long-run average revenue, $R(o_{k^*})$, is no less than the optimum of the original problem:  $R(o_{k^*})$ is an upper bound of $R^{\\phi,h}$ for any $\\phi\\in\\Phi$.\n\nFigure~\\ref{fig:fig1} illustrates the relative difference of average revenues,\n$\\bigl(R(o_{k^*}) - R^{\\phi,h}\\bigr)\/R(o_{k^*})$ for $\\phi=\\text{INDEX}(0),\\text{INDEX}(0.01)$, Max-Reward, Min-Cost and Random,\nagainst the scaling parameter $h$. \n\n\nIn this context, there are two aspects of performance evaluation presented in Figure~\\ref{fig:fig1}.\nFirst, we see the performance of the index policies in the non-asymptotic regime by comparing their long-run average revenues with an upper bound on the optimum. \nIn particular, Figures~\\ref{fig1:opt_v1} and \\ref{fig1:opt_v3} show that INDEX($0.01$) significantly outperforms INDEX(0) for large $h$: the small but positive $\\bm{\\epsilon}$ does affects the performance of $\\varphi$.\nThe performance of INDEX($0.01$) is close to the upper bound of the optimal solution with  relative difference less than $5\\%$ for $h$ greater than $50$ in all three examples: its performance degradation against the optimal solution is limited in the non-asymptotic regime.\n\n\nOn the other hand, by comparing to $R(o_{k^*})$, a trend of coincidence between $R^{\\text{INDEX}(0.01),h}$ and $R(o_{k^*})$ is observed in Figure~\\ref{fig:fig1} as $h$ increases from $1$ to $100$, consistent with the proved asymptotic optimality of $\\varphi$.\nRecall that the examples presented in Figure~\\ref{fig:fig1} are not for systems with weak coupling or heavy traffic but the index policy $\\varphi$ is still proved to be asymptotically optimal here.\nAlso, the performance of $\\varphi$ is close to the optimum without requiring extremely large $h$.\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.45\\linewidth]{OR-nonOPT-seed10-max50-v3.eps}\\label{fig2:nonopt_v1}}\n\\subfigure[]{\\includegraphics[width=0.45\\linewidth]{OR-nonOPT-seed10-scale50-v3.eps}\\label{fig2:nonopt_v2}}\n\\caption{(a) Relative difference of a specific policy to $R(o_{k^*})$ against scaling parameter of the system; (b) Relative difference of a specific policy to $R(o_k)$  against $k$.}\\label{fig:fig2}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\nIn Figure~\\ref{fig:fig2}, we consider another example with multipliers that are not decomposable (that is, $\\mathcal{T}^{o_{k^*}}(\\pmb{\\gamma}_{k^*})\\neq \\pmb{\\gamma}_{k^*}$).\nSimilar to Figure~\\ref{fig:fig1}, in Figure~\\ref{fig2:nonopt_v1}, we plot the relative difference of revenue of INDEX(0),  INDEX($0.01$) and the baseline policies to $R(o_{k^*})$ against the scaling parameter; while, in Figure~\\ref{fig2:nonopt_v2}, fixing the scaling parameter $h=50$, we illustrate curves of the relative differences, $\\bigl(R(o_k)-R^{\\phi,h}\\bigr)\/R(o_k)$ ($\\phi=\\text{INDEX}(0),\\text{INDEX}(0.01), \\text{Max-Reward}, \\text{Min-Cost}, \\text{Random}$), against the number of iterations $k$ for the fixed point iteration method. \nNote that the rankings $o_k$ are potentially different as $k$ varies, so as $R(o_k)$.\nIn Figure~\\ref{fig2:nonopt_v1}, the INDEX($0$) and INDEX($0.01$) are proposed based on the ranking $o_{k^*}$, while, with slightly abused notation, in Figure~\\ref{fig2:nonopt_v2},  INDEX($0$) and INDEX($0.01$) represent the index policies $\\varphi$, which are derived from the rankings $o_k$ associated with the varying $k$, with $\\epsilon_M=0$ and $0.01$, respectively. \nThe system parameters for the simulations in Figure~\\ref{fig:fig2} are listed in Appendix~\\ref{app:simulation:opt}.  \n\n\nFigure~\\ref{fig2:nonopt_v1} can be read in a similar way to Figure~\\ref{fig:fig1} except that $R(o_{k^*})$ is not a proved upper bound for the average revenue for the original problem.\nHere, INDEX($0$) and INDEX($0.01$) perform similarly and numerically converge to $R(o_{k^*})$ as $h$ increases although the system is not necessarily decomposable. \nThe convergence is consistent with Theorem~\\ref{theorem:main_second} which generally holds without requiring decomposability.\nOn the other hand, for each finite $h$ (which corresponds to the non-asymptotic regime), \nINDEX($0$) and INDEX($0.01$) significantly outperform all the other baseline policies, although the system is not proved to be decomposable, and their performance advantages are likely to maintain as $h$ continues increasing.\n\nFigure~\\ref{fig2:nonopt_v2} illustrates the performance trajectory as the iteration number $k$ (the x-axis) for the fixed point iteration method increases for a system with $h=50$ (in the non-asymptotic regime).\nRecall that, for the simulations presented here, the average revenues of INDEX($0$) and INDEX($0.01$) and $R(o_k)$ are varying with $k$ while all of the baseline policies are independent of $k$. \nWe observe a shape jump on the curves between $k=1$ and $5$.\nThis is caused by the initial setting, $\\pmb{\\gamma}_0=\\bm{0}$, which is not a good choice of multipliers.\nAfter several steps of the iteration method, the curves in Figure~\\ref{fig2:nonopt_v2} become almost flat; \nthat is, the values of $R(o_k)$, $R^{\\text{INDEX}(0)}$ and $R^{\\text{INDEX}(0.01)}$ become relatively stable for $k=5$ to $50$. \nAlso, in Figure~\\ref{fig2:nonopt_v2}, after the performance becomes stable, INDEX($0$) and INDEX($0.01$) achieve clearly higher long-run average revenues than those of the baseline policies: given the poor setting at the beginning, the fixed point iteration method can still lead to a reasonably good ranking $o_{k^*}$ and its associated multipliers $\\pmb{\\gamma}_{k^*}$.\n\n\n\n\\vspace{-0.3cm}\n\\section{Conclusions}\\label{sec:conclusions}\n\\vspace{-0.3cm}\n\n\nWe have modeled a resource allocation problem, described by \\eqref{eqn:objective}, \\eqref{eqn:constraint:action} and \\eqref{eqn:constraint:resources}, as a combination of various RMABPs coupled by limited capacities\nof the shared resource pools, which are shared, competed for, and reserved by requests.  \nThis presents us with an optimization problem for a stochastic system, aimed at maximizing the long-run average revenue by dynamically accommodating requests into appropriate resource pools.\n\n\n\n\nUsing the ideas of Whittle relaxation \\cite{whittle1988restless} and the asymptotic optimality proof of \\cite{weber1990index},\nwe have proved the asymptotic optimality of an index policy (referred to as $\\varphi$) if the capacity constraints are decomposable with multipliers $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$ (Theorem~\\ref{theorem:main}).\nThe asymptotic optimality is proved based on the existence of a global attractor $\\bm{z}\\in\\mathscr{Z}$ for the underlying process $\\bm{Z}^{\\varphi,h}(t)$ as $h\\rightarrow +\\infty$ and $\\bm{\\epsilon}\\rightarrow \\bm{0}$. \nWe have proved in general that such a global attractor exists, and then proposed a necessary and sufficient condition for asymptotic optimality in Theorem~\\ref{theorem:main_second}: the performance of the attractor $\\bm{z}$ approaches the optimum of the original problem in the asymptotic regime.\nThis condition holds if the system is decomposable. \n\n\nWe have proved a sufficient condition, described as the property of being weakly coupled and in heavy traffic, \nfor the existence of such decomposable multipliers as well as the asymptotic optimality of policy $\\varphi$ (Corollary~\\ref{coro:main}).\nThe property is not necessary, but is easy to verify and covers a significant class of resource allocation problems. We have listed examples of systems with the property satisfied in Section~\\ref{subsec:sufficient_condition}.\n\n\n\n\nIn a general system, we have proposed a fixed point method to fine tune the multipliers $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$ and a ranking $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{0})$. \nWe have proved that, if there exists a fixed point $\\pmb{\\gamma}\\in\\mathbb{R}_0^J$ of the function $\\mathcal{T}^o$ satisfying $o\\in\\mathscr{O}(\\pmb{\\gamma},\\bm{0})$, then this $\\pmb{\\gamma}$ is a vector of decomposable multipliers.\nWe have successfully discovered the decomposable multipliers in some situations without assuming weak coupling or heavy traffic by applying the fixed point method.\nAlso, in Section~\\ref{sec:example}, we have compared the index policy $\\varphi$ with different parameter $\\bm{\\epsilon}$ to baseline policies through simulations for systems that are not weakly coupled or in heavy traffic in the non-asymptotic regime. The index policy achieves clearly higher performance than the baseline policies.\nTo the best of our knowledge, no existing work provides rigorous asymptotic optimality for a resource allocation problem where dynamic allocation, competition and reservation are permitted.\n\n\\vspace{-0.3cm}\n\n\n\n\\section*{Acknowledgment}\nJing Fu's and Peter Taylor's research is supported by the Australian Research Council (ARC) Centre of Excellence for the Mathematical and Statistical Frontiers (ACEMS) and ARC Laureate Fellowship FL130100039.\n\\vspace{-0.1cm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\nThe necessity to generalize the Dirac equations and spinors in momentum space\nto free unstable spin-$1\/2$ particles has recently been recognized in\nconnection with the wave-function renormalizations of mixed systems of Dirac\n\\cite{Kniehl:2008cj,Kniehl:2014dra} and Majorana fermions\n\\cite{Kniehl:2014gfa}.\nIn this report, we discuss their construction when the fundamental requirement\nof Lorentz covariance is taken into account.\nWe also derive the generalized adjoint Dirac equations and spinors, and explain\nthe very simple relation that exists, in our formulation, between the\ngeneralized Dirac equations and spinors and the corresponding expressions for\nstable fermions.\\footnote{%\nFor brevity, spin-$1\/2$ particles are henceforth called fermions.}\nWe illustrate the application of the generalized spinors by evaluating the\nprobability density.   \n\nDefining the complex mass $M$ of the unstable fermion as the zero of the\ninverse propagator, a frequently used parametrization is \\cite{Smith:1996xz}\n\\begin{equation}\nM=m-i\\frac{\\Gamma}{2},\n\\label{eq:pole}\n\\end{equation}\nwhere $m$ and $\\Gamma$ are its mass and width, respectively.\n\nWe define the four-momentum of the unstable fermion according to\n\\begin{equation}\np^0=M\\gamma c,\\qquad\n\\vec{p}=M\\gamma\\vec{v},\n\\label{eq:p}\n\\end{equation}\nwhere $\\gamma=(1-\\vec{v}^{\\,2}\/c^2)^{-1\/2}$ and $\\vec{v}$ is the particle's\nvelocity in the chosen inertial frame.\nSince Eq.~(\\ref{eq:p}) differs from the expressions of special relativity\nfor stable fermions by only the constant factor $M\/m$, $p^\\mu=(p^0,\\vec{p}\\,)$\ntransforms as a four-vector.\nEquation~(\\ref{eq:p}) can also be written as\n\\begin{equation}\np^\\mu=Mu^\\mu,\n\\label{eq:u}\n\\end{equation}\nwhere $u^\\mu=\\gamma(c,\\vec{v}\\,)$ is the four-velocity.\nFrom Eqs.~(\\ref{eq:p}) and (\\ref{eq:u}) one finds the basic relation\\footnote{%\nIn this paper, we adopt the notations and conventions of\nRefs.~\\cite{bjorken,mandl}.}\n\\begin{equation}\np_\\mu p^\\mu={p^0}^2-\\vec{p}^{\\,2}=M^2c^2.\n\\label{eq:ps}\n\\end{equation}\nEvaluating the real and imaginary parts of $p^0$ from Eq.~(\\ref{eq:ps}) and\nusing the relation $\\mathop{\\mathrm{Im}}\\nolimits\\vec{p}^{\\,2}=-[m\\Gamma\/(m^2-\\Gamma^2\/4)]\\mathop{\\mathrm{Re}}\\nolimits\\vec{p}^{\\,2}$\nthat follows from Eqs.~(\\ref{eq:pole}) and (\\ref{eq:p}), one can express $p^0$\nin terms of $\\mathop{\\mathrm{Re}}\\nolimits\\vec{p}^{\\,2}$, $m$, and $\\Gamma$ as\n\\begin{equation}\np^0=\nM\\left[\\frac{\\mathop{\\mathrm{Re}}\\nolimits\\vec{p}^{\\,2}+(m^2-\\Gamma^2\/4)c^2}{m^2-\\Gamma^2\/4}\\right]^{1\/2}.\n\\label{eq:p0}\n\\end{equation}\nIn the limit $\\Gamma\\to0$, $\\vec{p}^{\\,2}$ is real, $p^0=E\/c$, and\nEq.~(\\ref{eq:p0}) becomes\n\\begin{equation}\nE=(\\vec{p}^{\\,2}c^2+m^2c^4)^{1\/2},\n\\end{equation}\nthe well-known energy-momentum relation for stable particles.\nIn the rest frame, $\\vec{p}=\\vec{0}$ and Eq.~(\\ref{eq:p0}) reduces to $p^0=Mc$,\nin agreement with Eq.~(\\ref{eq:p}) when $\\vec{v}=\\vec{0}$.\n\nIn momentum space, the generalizations of the Dirac equations to free unstable\nfermions are\n\\begin{equation}\n(\\slashed{p}-Mc)u_r(\\vec{p}\\,)=0,\\qquad\n(\\slashed{p}+Mc)v_r(\\vec{p}\\,)=0,\n\\label{eq:dirac}\n\\end{equation}\nwhere $\\slashed{p}=p_\\mu\\gamma^\\mu$ and $r=1,2$ labels the two independent\nsolutions.\nRecalling Eq.~(\\ref{eq:p}), we note that Eq.~(\\ref{eq:dirac}) can be derived by\nmultiplying the corresponding Dirac equations for stable fermions by $M\/m$.\nThe four independent solutions can be written explicitly in the form\n\\begin{equation}\nu_r(\\vec{p}\\,)=\\left(\\frac{p^0+Mc}{2Mc}\\right)^{1\/2}\\left(\n\\begin{array}{c}\n\\chi_r \\\\\n\\frac{\\vec{\\sigma}\\cdot\\vec{p}}{p^0+Mc}\\chi_r\n\\end{array}\\right),\n\\qquad\nv_r(\\vec{p}\\,)=\\left(\\frac{p^0+Mc}{2Mc}\\right)^{1\/2}\\left(\n\\begin{array}{c}\n\\frac{\\vec{\\sigma}\\cdot\\vec{p}}{p^0+Mc}\\chi_r^\\prime \\\\\n\\chi_r^\\prime\n\\end{array}\\right),\n\\label{eq:spinor}\n\\end{equation}\nwhere $\\sigma^i$ are the Pauli matrices and $\\chi_r$ and $\\chi_r^\\prime$ are\ntwo-dimensional constant and orthogonal spinors frequently chosen as\n\\begin{equation}\n\\chi_1=\\chi_2^\\prime=\\left(\n\\begin{array}{c}\n1 \\\\\n0\n\\end{array}\\right),\n\\qquad\n\\chi_2=\\chi_1^\\prime=\\left(\n\\begin{array}{c}\n0 \\\\\n1\n\\end{array}\\right).\n\\label{eq:chi}\n\\end{equation}\nWith this choice, $u_r(\\vec{p}\\,)$ and $v_r(\\vec{p}\\,)$ are eigenstates of the\n$z$ component of spin in the rest frame of the fermion, with eigenvalue\n$+\\hbar\/2$ (spin up) for $u_1(\\vec{p}\\,)$ and $v_2(\\vec{p}\\,)$ and $-\\hbar\/2$\n(spin down) for $u_2(\\vec{p}\\,)$ and $v_1(\\vec{p}\\,)$.\n\nIncluding the space-time dependencies, the plane-wave solutions associated with\nthe spinors $u_r(\\vec{p}\\,)$ and $v_r(\\vec{p}\\,)$ are\n$u_r(\\vec{p}\\,)\\exp(-ip\\cdot x)$ and $v_r(\\vec{p}\\,)\\exp(ip\\cdot x)$,\nrespectively.\nUsing Eqs.~(\\ref{eq:pole})--(\\ref{eq:u}), we have\n\\begin{equation}\ne^{-ip\\cdot x}=e^{-imu\\cdot x}e^{-(\\Gamma\/2)u\\cdot x}.\n\\label{eq:exp}\n\\end{equation}\nThe first factor on the right-hand side of Eq.~(\\ref{eq:exp}) is the space-time\ndependence in the stable case, while the second factor reflects the fact that\nthe fermion is unstable.\nThe amplitude $u_r(\\vec{p}\\,)\\exp(-imu\\cdot x)$ is a solution of the usual\nDirac equation for stable fermions and is, therefore, time-reversal invariant.\nBy contrast, the second factor,\n$\\exp[-\\Gamma\\gamma(c^2t-\\vec{v}\\cdot\\vec{x})\/2]$, is not invariant under the\ntime-reversal transformation $t\\to-t$ and $\\vec{v}\\to-\\vec{v}$.\n\nAnother simple way to show that the generalized Dirac equations are not\ninvariant under time reversal is the following:\nwe recall that the operator that relates the wave functions at times $t$ and\n$t^\\prime=-t$ is antiunitary, namely of the form $KU$, where $U$ is a unitary\nmatrix and $K$ means complex conjugation.\nIf the Hamiltonian $H(t)$ at time $t$ involves the complex mass $M$, as is the\ncase in the formulation of the generalized Dirac equations, when $K$ acts on\n$H(t)$ it transforms $M\\to M^*$.\nAs a consequence, $H(t^\\prime)$ differs from $H(t)$ by the same change\n$M\\to M^*$, and the proof of time-reversal invariance, explained for instance\nin Ref.~\\cite{bjorken}, breaks down.\n\nThe Hermitian adjoints of Eq.~(\\ref{eq:dirac}) are\n\\begin{equation}\n\\bar{u}_r(\\vec{p}\\,)(\\slashed{p}^*-M^*c)=0,\\qquad\n\\bar{v}_r(\\vec{p}\\,)(\\slashed{p}^*+M^*c)=0,\n\\label{eq:diracc}\n\\end{equation}\nwhere\n\\begin{equation}\n\\bar{u}_r(\\vec{p}\\,)=u_r^\\dagger(\\vec{p}\\,)\\gamma^0,\\qquad\n\\bar{v}_r(\\vec{p}\\,)=v_r^\\dagger(\\vec{p}\\,)\\gamma^0\n\\end{equation}\nare the usual adjoint spinors and $\\slashed{p}^*=p_\\mu^*\\gamma^\\mu$.\nAt first sight, the presence of the complex conjugates $p_\\mu^*$ and $M^*$ seems\nto complicate matters.\nHowever, we note from Eq.~(\\ref{eq:u}) that $p^\\mu\/M=u^\\mu$ is real.\nTherefore, we have the important relation\n\\begin{equation}\n\\left(\\frac{p^\\mu}{M}\\right)^*=\\frac{p^\\mu}{M}.\n\\label{eq:ratio}\n\\end{equation}\nInserting $p_\\mu^*=(M^*\/M)p_\\mu$ and multiplying by $M\/M^*$,\nEq.~(\\ref{eq:diracc}) becomes\n\\begin{equation}\n\\bar{u}_r(\\vec{p}\\,)(\\slashed{p}-Mc)=0,\\qquad\n\\bar{v}_r(\\vec{p}\\,)(\\slashed{p}+Mc)=0.\n\\label{eq:diraca}\n\\end{equation}\nThe four generalized Dirac equations shown in Eqs.~(\\ref{eq:dirac}) and\n(\\ref{eq:diraca}) were postulated in Ref.~\\cite{Kniehl:2014dra} without\napplying Eq.~(\\ref{eq:ratio}) and with a different interpretation of the\nadjoint spinors $\\bar{u}_r(\\vec{p}\\,)$ and $\\bar{v}_r(\\vec{p}\\,)$.\nIn the case of unstable fermions, Eqs.~(\\ref{eq:dirac}) and (\\ref{eq:diraca})\nplay an important role in the implementation of the\nAoki-Hioki-Kawabe-Konuma-Muta (AHKKM) \\cite{Aoki:1982ed} renormalization\nconditions in general theories with intergeneration mixing, as pointed out\nin Refs.~\\cite{Kniehl:2008cj,Kniehl:2014dra,Kniehl:2014gfa}.\n\nSince $p^\\mu$ transforms as a four-vector, the proof of Lorentz covariance of\nEq.~(\\ref{eq:dirac}) follows the same steps as the proof of the Lorentz\ncovariance of the Dirac equation in coordinate space (see, for example,\nchapter~2 in Ref.~\\cite{bjorken}).\nSpecifically, if $p^\\mu$ and $u_r(\\vec{p}\\,)$ are the four-momentum and the\nspinor in the Lorentz frame $O$ and $p^{\\prime\\mu}$ and\n$u_r^\\prime(\\vec{p}^{\\,\\prime})$ are those in the Lorenz frame $O^\\prime$, one\nexpresses, for example, the first equality in Eq.~(\\ref{eq:dirac}) in terms of\nthe $O^\\prime$ variables by means of the relations\n$p_\\mu=a_{\\phantom{\\nu}\\mu}^\\nu p_\\nu^\\prime$, where $a_{\\phantom{\\nu}\\mu}^\\nu$ are the\ncoefficients of the Lorentz transformation between the four-vectors $p_\\mu$ and\n$p_\\mu^\\prime$, and $u_r(\\vec{p}\\,)=S^{-1}u_r^\\prime(\\vec{p}^{\\,\\prime})$, where $S$\nis a matrix that satisfies the relations\n$a_{\\phantom{\\nu}\\mu}^\\nu S\\gamma^\\mu S^{-1}=\\gamma^\\nu$ and\n$S^{-1}=\\gamma^0S^\\dagger\\gamma^0$.\nThen the first equality in Eq.~(\\ref{eq:dirac}) becomes\n$(\\slashed{p}^\\prime-Mc)u_r^\\prime(\\vec{p}^{\\,\\prime})=0$, which demonstrates its\nLorentz covariance.\nCarrying out the Hermitian conjugation of the $O^\\prime$ Dirac equation and\nusing Eq.~(\\ref{eq:ratio}), one finds\n$\\bar{u}_r^\\prime(\\vec{p}^{\\,\\prime})(\\slashed{p}^\\prime-Mc)=0$, which shows the\nLorentz covariance of the corresponding adjoint Dirac equation,\nEq.~(\\ref{eq:diraca}).\n\nThe adjoint spinors with respect to $u_r(\\vec{p}\\,)$ and $v_r(\\vec{p}\\,)$ in\nEq.~(\\ref{eq:spinor}) are\n\\begin{equation}\n\\bar{u}_r(\\vec{p}\\,)=\\left(\\frac{p^0+Mc}{2Mc}\\right)^{1\/2}\\left(\n\\chi_r^\\dagger,-\\chi_r^\\dagger\\frac{\\vec{\\sigma}\\cdot\\vec{p}}{p^0+Mc}\\right),\n\\quad\n\\bar{v}_r(\\vec{p}\\,)=\\left(\\frac{p^0+Mc}{2Mc}\\right)^{1\/2}\\left(\n\\chi_r^{\\prime\\dagger}\\frac{\\vec{\\sigma}\\cdot\\vec{p}}{p^0+Mc},-\\chi_r^{\\prime\\dagger}\n\\right),\n\\label{eq:spinora}\n\\end{equation}\nwhere we have again applied Eq.~(\\ref{eq:ratio}) to eliminate $p_\\mu^*$ and\n$M^*$.\nIn particular, Eq.~(\\ref{eq:ratio}) implies that $[(p^0+Mc)\/(2Mc)]^{1\/2}$ and\n$\\vec{p}\/(p^0+Mc)$ are real.\n\nIt is interesting to note that the four generalized Dirac equations shown in\nEqs.~(\\ref{eq:dirac}) and (\\ref{eq:diraca}) as well as their spinor solutions\npresented in Eqs.~(\\ref{eq:spinor}) and (\\ref{eq:spinora}) can be obtained from\nthe corresponding ones for stable fermions, for $\\Gamma=0$, by simply\nsubstituting $m\\to M$ in their explicit mass dependencies and in the definition\nof $p^\\mu$ in Eqs.~(\\ref{eq:p}) and (\\ref{eq:u}).\n\nSince $M$ cancels in the ratios $(p^0+Mc)\/(2Mc)$ and $\\vec{p}\/(p^0+Mc)$, these\nfactors are the same as in the $\\Gamma=0$ case.\nIt hence follows that the spinor solutions in Eqs.~(\\ref{eq:spinor}) and\n(\\ref{eq:spinora}) satisfy the same normalization and completeness relations as\nin the case of stable fermions, which are given, for example, by Eqs.~(A.29)\nand (A.30) in Ref.~\\cite{mandl}.\nIn particular,\n\\begin{eqnarray}\n\\bar{u}_r(\\vec{p}\\,)u_s(\\vec{p}\\,)&=&-\\bar{v}_r(\\vec{p}\\,)v_s(\\vec{p}\\,)\n=\\delta_{rs},\n\\nonumber\\\\\n\\bar{u}_r(\\vec{p}\\,)v_s(\\vec{p}\\,)&=&-\\bar{v}_r(\\vec{p}\\,)u_s(\\vec{p}\\,)=0.\n\\end{eqnarray}\n\nIn some applications, the two-component spinors $\\chi_r$ and $\\chi_r^\\prime$ in\nEq.~(\\ref{eq:chi}) are replaced by helicity eigenstates $\\phi_s$ and\n$\\phi_s^\\prime$, respectively.\nBecause, in the case of unstable particles, $\\vec{p}$ is complex [cf.\\\nEqs.~(\\ref{eq:p}) and (\\ref{eq:u})], the usual helicity projection,\n\\begin{equation}\n\\frac{\\vec{p}}{|\\vec{p}\\,|}\\cdot\\frac{\\vec{\\sigma}}{2}\\phi_s=s\\phi_s,\n\\qquad\n\\frac{\\vec{p}}{|\\vec{p}\\,|}\\cdot\\frac{\\vec{\\sigma}}{2}\\phi_s^\\prime\n=-s\\phi_s^\\prime,\n\\label{eq:hel}\n\\end{equation}\nwhere $|\\vec{p}\\,|=(\\vec{p}^{\\,*}\\cdot\\vec{p}\\,)^{1\/2}$, leads to complex\neigenvalues $s=\\pm(\\vec{p}^{\\,2})^{1\/2}\/(2|\\vec{p}\\,|)$, as can be checked by\napplying the helicity projection operator twice.\nA consistent alternative is to define the helicity projection operator in\nEq.~(\\ref{eq:hel}) according to $\\vec{u}\/|\\vec{u}|\\cdot\\vec{\\sigma}\/2$, where\n$\\vec{u}=\\vec{p}\/M$ are the spatial components of the four-velocity $u$ given\nbelow Eq.~(\\ref{eq:u}), or, equivalently,\n$\\vec{v}\/|\\vec{v}|\\cdot\\vec{\\sigma}\/2$.\n\nIt is also instructive to calculate the probability current using the spinor\nsolutions in Eqs.~(\\ref{eq:spinor}) and (\\ref{eq:spinora}).\nWe find\n\\begin{equation}\nc\\bar{u}_r(\\vec{p}\\,)\\gamma^\\mu u_r(\\vec{p}\\,)=\nc\\bar{v}_r(\\vec{p}\\,)\\gamma^\\mu v_r(\\vec{p}\\,)=\\frac{p^\\mu}{M}=u^\\mu.\n\\end{equation}\nAs expected, these currents transform as four-vectors.\nIn particular, for $\\mu=0$ we have\n\\begin{equation}\ncu_r^\\dagger(\\vec{p}\\,)u_r(\\vec{p}\\,)=\ncv_r^\\dagger(\\vec{p}\\,)v_r(\\vec{p}\\,)=u^0=\\gamma c.\n\\label{eq:pro}\n\\end{equation}\nSince $cu_r^\\dagger(\\vec{p}\\,)u_r(\\vec{p}\\,)$ and\n$cv_r^\\dagger(\\vec{p}\\,)v_r(\\vec{p}\\,)$ are the probability densities, they\nshould be real and positive, consistent with Eq.~(\\ref{eq:pro}).\n\nAnother interesting application is to examine the space-time dependence of the\nprobability density.\nIncorporating the space-time factor $\\exp(-ip\\cdot x)$ of Eq.~(\\ref{eq:exp}) in\n$cu_r^\\dagger(\\vec{p}\\,)u_r(\\vec{p}\\,)$, we find that the latter is multiplied by\nan overall factor $\\exp[-\\Gamma\\gamma(c^2t-\\vec{v}\\cdot\\vec{x})]$, which\nimplies that the probability density for the positive-energy states decreases\nexponentially with time, reflecting the fermion's instability.\n\nIn summary, (i) we have proposed a simple definition of the (complex)\nfour-momentum of a free unstable spin-$1\/2$ particle [cf.\\ Eqs.~(\\ref{eq:p})\nand (\\ref{eq:u})] and shown that it indeed transforms as a four-vector,\n(ii) we have derived the generalized Dirac equations in momentum space [cf.\\\nEq.~(\\ref{eq:dirac})] and found explicit spinor solutions [cf.\\\nEq.~(\\ref{eq:spinor})],\n(iii) we have derived the generalized adjoint Dirac equations [cf.\\\nEq.~(\\ref{eq:diraca})] and spinors [cf.\\ Eq.~(\\ref{eq:spinora})] by Hermitian\nconjugation of Eqs.~(\\ref{eq:dirac}) and (\\ref{eq:spinor}), respectively,\ntaking into account the basic relation in Eq.~(\\ref{eq:ratio}),\n(iv) using the important fact that $p^\\mu$ transforms as a four-vector, we have\nshown how the proof of Lorentz covariance carries over to the generalized Dirac\nequations and their adjoints,\n(v) we have pointed out the very simple relation that exists, in our\nformulation, between the generalized Dirac equations and spinors and the\ncorresponding expressions for stable fermions,\n(vi) in particular, we have shown that our spinors and adjoint spinors satisfy\nthe same normalization and completeness relations as in the case of stable\nfermions,\n(vii) we have proposed a modified definition of the helicity projection\noperator for unstable fermions that leads to real eigenvalues,\n(viii) as an illustration, we have applied our spinors and adjoint spinors to\ncalculate the probability density and found that it satisfies the expected\ntheoretical properties,\nand (ix) we have discussed the behavior of the generalized Dirac equations\nunder time reversal.\nAs mentioned after Eq.~(\\ref{eq:diraca}), the four generalized Dirac equations\nin Eqs.~(\\ref{eq:dirac}) and (\\ref{eq:diraca}) play an important role in the\nimplementation of the AHKKM \\cite{Aoki:1982ed} renormalization conditions for\nunstable fermions in general theories with intergeneration mixing.\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank the Werner-Heisenberg-Institut for the hospitality extended to us\nduring a visit when this paper was prepared.\nThis research was supported in part by the German Research Foundation through\nthe  Collaborative Research Center No.~676 {\\it Particles, Strings and the\nEarly Universe---The Structure of Matter and Space Time}.\nThe work of A.S. was supported in part by the National Science Foundation \nthrough Grant No.\\ PHY--1316452. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMotion of a symmetric top can be studied by using either a cubic function or effective potential.\nThe cubic function is mostly used in works that utilize geometric techniques \\cite{Routh, Scarborough, MacMillan, ArnoldMaunder, Groesberg, JoseSaletan},\nand effective potential is mostly used in works considering physical parameters \\cite{Symon, McCauley, LandauLifshitz, MarionThornton, Taylor}.\nIn some other works, both the cubic function and effective potential are used \\cite{Goldstein, Arnold, Corinaldesi, MatznerShepley, Arya, Greiner, FowlesCassiday}.\n\nEffective potential shows different characteristics when one of the conserved angular momenta greater than the other one or equal to.\nOne can find different aspects of effective potential in the literature when magnitudes of the conserved angular momenta are equal to each other \\cite{Symon, Tanriverdi_abeql}.\nHowever, it is not studied when magnitudes of the conserved angular momenta are not equal to each other except in Greiner's work, \nand his study does not cover different possibilities related to the conserved angular momenta and the minimum of effective potential \\cite{Greiner}. \nStudying this topic helps understand the motion of a spinning heavy symmetric top,\nand in this study, we will study this case together with the relation between the minimum of effective potential and a constant derived from parameters of gyroscope and conserved angular momenta.\n\nIn section \\ref{frst}, we will give a quick overview of constants of motion and effective potential.\nIn section \\ref{scnd}, we will study effective potential when magnitudes of the conserved angular momenta are not equal to each other.\nThen, we will give a conclusion.\nIn the appendix, we will compare the cubic function with effective potential.\n\n\n\\section{Constants of motion and effective potential}\n\\label{frst}\n\nFor a spinning heavy symmetric top, Lagrangian is \\cite{Goldstein}\n\\begin{eqnarray}\n\tL&=&T-U \\nonumber \\\\ \n\t&=&\\frac{I_x}{2}(\\dot \\theta ^2 + \\dot \\phi ^2 \\sin^2 \\theta)+\\frac{I_z}{2}(\\dot \\psi+\\dot \\phi \\cos \\theta)^2-M g l \\cos \\theta, \n\t\\label{lagrngn}\n\\end{eqnarray}\nwhere $M$ is the mass of the symmetric top, $l$ is the distance from the center of mass to the fixed point, $I_x=I_y$ and $I_z$ are moments of inertia, $g$ is the gravitational acceleration, $\\theta$ is the angle between the stationary $z'$-axis and the body $z$-axis, $\\dot \\psi$ is the spin angular velocity, $\\dot \\phi$ is the precession angular velocity and $\\dot \\theta$ is the nutation angular velocity. \nThe domain of $\\theta$ is $[0,\\pi]$.\nFor a spinning symmetric top on the ground $\\theta$ should be smaller than $\\pi\/2$, and if $\\theta>\\pi\/2$, then the spinning top is suspended from the fixed point. \n\nThere are two conserved angular momenta which can be obtained from Lagrangian,\nand one can define two constants $a$ and $b$ by using these conserved angular momenta as \\cite{Goldstein}\n\\begin{eqnarray}\n\ta&=&\\frac{I_z}{I_x}(\\dot \\psi+\\dot \\phi \\cos \\theta), \\\\\n\tb&=&\\dot \\phi \\sin^2 \\theta + a \\cos \\theta,\n\\end{eqnarray}\nwhere $a=L_z\/I_x$ and $b=L_{z'}\/I_x$.\nHere, $L_z$ and $L_{z'}$ are conserved angular momenta in the body $z$ direction and stationary $z'$ direction, respectively.\n\nOne can define a constant from energy as \n\\begin{equation}\n\tE'=\\frac{I_x}{2} \\dot \\theta ^2 +\\frac{I_x}{2} \\dot \\phi^2 \\sin^2 \\theta + Mgl \\cos \\theta \\label{eprime},\n\\end{equation}\nand its relation with the energy is $E'=E-I_x^2 a^2\/(2 I_z)$.\n\nBy using change of variable $u=\\cos \\theta$, one can obtain the cubic function from \\eqref{eprime} as\\cite{Goldstein}\n\\begin{equation}\n\tf(u)=(\\alpha - \\beta u)(1-u^2)-(b-a u^2)\t\n\t\\label{cubicf}\n\\end{equation}\nwhich is equal to $\\dot u^2$,\nwhere $\\alpha=2 E'\/I_x$ and $\\beta=2 Mgl\/I_x$.\nThis cubic function can be used to find turning angles.\n\nFrom $E'=I_x \\dot \\theta ^2\/2+U_{eff}$ \\cite{LandauLifshitz}, it is possible to define an effective potential\n\\begin{equation}\n\tU_{eff}(\\theta)= \\frac{I_x}{2}\\frac{(b-a \\cos \\theta)^2}{\\sin^2 \\theta}+Mgl \\cos \\theta.\n\t\\label{ueff}\n\\end{equation}\n\nBy using the derivative of $U_{eff}$ with respect to $\\theta$\n\\begin{equation}\n\t\\frac{d U_{eff}(\\theta)}{d \\theta}= \\frac{I_x}{\\sin^3 \\theta} \\left[ (b-a \\cos \\theta)(a-b \\cos \\theta)- \\frac{Mgl}{I_x} \\sin^4 \\theta \\right],\n\t\\label{dueff}\n\\end{equation}\nit is possible to find the minimum of $U_{eff}$.\nThe factor $\\sin \\theta$ is equal to zero when $\\theta$ is equal to $0$ or $\\pi$, and effective potential goes to infinity at these angles.\nThe root of equation \\eqref{dueff} is between $0$ and $\\pi$, and it will be designated by $\\theta_r$ giving the minimum of effective potential,\nand it can be found numerically.\nThen, the form of effective potential is like a well. \nThe general structure of $U_{eff}$ together with $E'$ can be seen in figure \\ref{fig:ueffg}.\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\includegraphics[width=7cm]{ueff_gn2_d.pdf}\n\t\t\\caption{General structure of $U_{eff}(\\theta)$ and $E'$. $\\theta_{min}$ and $\\theta_{max}$ show turning angles, and $\\theta_r$ represents the angle where minimum of $U_{eff}$ occurs.\n\t\tCurve (red) shows $U_{eff}$, dashed (blue) line shows $E'$ and horizontal continious (black) line shows the minimum of $U_{eff}$.\n\t\t}\n\t\t\\label{fig:ueffg}\n\t\\end{center}\n\\end{figure}\n\nBy using equation \\eqref{dueff}, one can write \\cite{Goldstein}\n\\begin{equation}\n\t\\dot \\phi^2 \\cos \\theta- \\dot \\phi a+\\frac{Mgl}{I_x}=0.\n\\end{equation}\nThe root of this equation can also be used to obtain the minimum of $U_{eff}$.\nBy using the discriminant of this equation, \none can define a parameter $\\tilde a=\\sqrt{4 Mgl \/I_x}$ to make a disrimination between \"strong top\" (or fast top) where $a> \\tilde a$ and \"weak top\" (or slow top) where $a< \\tilde a$ \\cite{KleinSommerfeld, Tanriverdi_abdffrnt}. \n\nThe position of the minimum and the shape of $U_{eff}$ can be helpful in understanding the motion.\nIf $E'$ is equal to the minimum of $U_{eff}$ then the regular precession is observed.\nIf $E'$ is greater than the minimum of $U_{eff}$, like figure \\ref{fig:ueffg}, the intersection points of $E'$ and $U_{eff}$ give turning angles.\nAnd, symmetric top nutates between these two angles periodically.\nThere can be different types of motion, and some of these motions can be determined by using relations between $E'$ \\& $Mglb\/a$ and $a$ \\& $b$ when $|a|\\ne|b|$ \\cite{Tanriverdi_abdffrnt}.  \n\n\n\\section{Effective potential}\n\\label{scnd}\n\nThe relation between $a$ and $b$ can affect effective potential.\nThere are three possible relation between $a$ and $b$: $|a|>|b|$, $|a|<|b|$ and $|a|=|b|$. \nWe will consider two different possibilities, $|a|>|b|$ and $|a|<|b|$, to study effective potential since the third one is studied previously, i.e. $|a|=|b|$ \\cite{Symon, Tanriverdi_abeql}.\nWe will give examples to studied cases, and for examples, the following constants will be used: $Mgl=0.068 \\,J$, $I_x=0.000228 \\,kg \\,m^{2}$ and $I_z=0.0000572 \\,kg \\,m^{2}$.\n\n\\subsection{Effective potential when $|a|>|b|$}\n\nIn this section, we will study the case when $|a|>|b|$.\nAfter factoring equation \\eqref{dueff}, it can be written as\n\\begin{equation}\n\t\\frac{d U_{eff}(\\theta)}{d \\theta}= \\frac{a^2 I_x}{ \\sin^3 \\theta} \\left[ (\\frac{b}{a}- \\cos \\theta)(1- \\frac{b}{a} \\cos \\theta)- \\frac{Mgl}{I_x a^2} \\sin^4 \\theta \\right].\n\t\\label{dueff3}\n\\end{equation}\nThe angle, making the terms in the parentheses zero, gives the minimum of effective potential.\nIf $|a|>|b|$, the second term in the parentheses is always negative, and then $b\/a-\\cos \\theta$ should also be positive for the root.\nTherefore, the inclination angle should satisfy $\\pi>\\theta>\\arccos b\/a$.\nIn the limit where $a$ goes to infinity, $\\theta_r$ goes to $\\arccos b\/a$.\nIn $a$ goes to zero limit, $b$ should also go to zero since $|a|>|b|$, then the first term goes to zero (see equation \\eqref{dueff}) and the second term should also go to zero for the root which is possible when $\\theta_r$ goes to $\\pi$.\nIf both $a$ and $b$ are negative or positive, $\\theta_r$ is between $\\pi\/2$ and $\\pi$ when $|a|$ is close to zero, and it is between $0$ and $\\pi\/2$ when $|a|$ and $|b|$ are great enough.\nIf only one of them is negative, then $\\theta_r$ is always greater than $\\pi\/2$.\n\nWhen $b=0$, in $|a|$ goes to infinity limit $\\theta_r$ goes to $\\pi\/2$, and $a$ goes to zero limit does not change and remains as $\\pi$.\n\nThese shows that $\\theta_r \\in (\\arccos b\/a,\\pi)$. \nIf $b\/a$ goes to $1$, then $\\arccos b\/a$ goes to $0$.\nTherefore, $\\theta_r$ can take values between $0$ and $\\pi$ depending on signs of $a$ and $b$, the ratio $b\/a$ and greatness of $a$ and $b$.\n\nNow, we will consider the change of $U_{eff_{min}}$ when $|a|>|b|$.\nWe have seen that as $|a|$ goes to zero, $\\theta_r$ goes to $\\pi$ .\nThen, it can be seen from equation \\eqref{ueff} that $U_{eff_{min}}$ goes to $-Mgl$ as $|a|$ goes to zero.\nAs $|a|$ goes to infinity $\\theta_r$ goes to $\\arccos b\/a$, then $U_{eff_{min}}$ goes to $Mgl b\/a$ from below.\nThen, $Mglb\/a$ is always grater than $U_{eff_{min}}$ when $|a|>|b|$.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\subfigure[$U_{eff}$]{\n\t\t\t\\includegraphics[width=4.0cm]{ueff_a3.pdf}\n\t\t\n\t\t\t}\n\t\t\t\\subfigure[$\\theta_r$]{\n\t\t\t\t\\includegraphics[width=4.0cm]{loop_a_theta.pdf}\n\t\t\t\n\t\t\t\t}\n\t\t\t\t\\subfigure[$U_{eff_{min}}$]{\n\t\t\t\t\t\\includegraphics[width=4.0cm]{loop_a_umin.pdf}\n\t\t\t\t\n\t\t\t\t\t}\n\t\t\t\t\t\\caption{ $U_{eff}$, change of $\\theta_r$ with respect to $a$ and change of $U_{eff_{min}}$ with respect to $a$.\n\t\t\t\t\ta) Three different effective potential: $a=10 \\, rad \\,s^{-1}$ (green dashed-dotted curve), $a=30 \\, rad \\,s^{-1}$ (blue dashed curve) and $a=60 \\, rad \\,s^{-1}$ (red continious curve), and all of them satisfy $b\/a=0.5$. Black line shows $Mglb\/a$.\n\t\t\t\t\tb) Change of $\\theta_r$ with respect to $a$ for constant $b\/a=0.5$ ratio (red curve). Black line shows $\\arccos(b\/a)=1.05$. Vertical dotted line shows position of $\\tilde a$.\n\t\t\t\t\tc) Change of $U_{eff_{min}}$ with respect to $a$ for constant $b\/a=0.5$ ratio (red curve). Black line shows $Mglb\/a$. Vertical dotted line shows position of $\\tilde a$.\n\t\t\t\t\t}\n\t\t\t\t\t\\label{fig:ueffg5}\n\t\\end{center}\n\\end{figure}\n\nAs an example, we will consider that there is a constant ratio between $a$ and $b$: $b\/a=0.5$.\nIn figure \\ref{fig:ueffg5}(a), three different effective potentials for three different $a$ values are shown together with $Mglb\/a$.\nIn this figure, it can be seen that the form and magnitude of the minimum of $U_{eff}$ are changing as $a$ changes, and it can also be seen that $\\theta_r$ is also changing.\nIn figure \\ref{fig:ueffg5}(b), it can be seen that $\\theta_r$ takes very close values to $\\pi$ for very small values of $a$ and goes to $\\arccos 0.5=1.05 \\,rad$ as $a$ increases.\nIn figure \\ref{fig:ueffg5}(c), it can be seen that the minimum of $U_{eff}$ takes very close values to $-Mgl$ when $a$ is small, and it goes to $Mglb\/a$ as $a$ goes to infinity.\nThese are consistent with previous considerations.\n\nIt can be considered that there is a shift in the behaviour of $\\theta_r$ and $U_{eff_{min}}$ near $a=\\tilde a$.             \nBut this shift is not sudden, and one can say that the usage $\\tilde a$ gives an approximate separation when $|a|>|b|$.\n\nIn some cases, $Mgl$ can be negative and there are some differences in effective potential in these cases.\nWhen $Mgl$ is negative, the second term in equation \\eqref{dueff3} becomes positive, and then $\\arccos b\/a> \\theta > 0$ for the root.\nIn the limit where $a$ goes to infinity, again $\\theta_r$ goes to $\\arccos b\/a$.\nIn $a$ goes to zero limit, $\\theta_r$ goes to $0$.\nThese show that the interval for the minimum of effective potential changed from $(\\arccos b\/a,\\pi)$ to $(0,\\arccos b\/a)$ when $Mgl$ changed sign from positive to negative.\nIf both $a$ and $b$ are negative or positive, $\\theta_r$ is between $0$ and $\\pi\/2$.\nIf only one of them is negative, then $\\theta_r$ can be greater than $\\pi\/2$ when $|a|$ is great enough.\nThe minimum of $U_{eff}$ goes to $-|Mgl|$ when $a$ goes to $0$, and it goes to $-|Mgl| b\/a$ when $a$ goes to infinity when $Mgl$ is negative.\n\n\\subsection{Effective potential when $|b|>|a|$}\n\nIn this section, we will study the case when $|b|>|a|$.\nAfter factoring equation \\eqref{dueff} in another way, it can be written as \n\\begin{equation}\n\t\\frac{d U_{eff}(\\theta)}{d \\theta}= \\frac{b^2 I_x}{\\sin^3 \\theta}  \\left[ (1-\\frac{a}{b} \\cos \\theta)(\\frac{a}{b} - \\cos \\theta)- \\frac{Mgl}{I_x b^2} \\sin^4 \\theta \\right].\n\t\\label{dueff2}\n\\end{equation} \nSimilar to the previous case, the first term should be positive, and $a\/b- \\cos \\theta$ should be positive when $|b|>|a|$ for the root, and then $\\pi>\\theta>\\arccos a\/b$.\nIn $b$ goes to infinity limit, the second term in the parentheses goes to zero.\nThen, as $|b|$ goes to infinity, $\\theta_r$ should go to $\\arccos a\/b$.\nIn $b$ goes to zero limit, $\\theta_r$ goes to $\\pi$ which can be seen from equation \\eqref{dueff} similar to the previous section. \nThen, $\\theta_r$ goes to $\\pi$ when $b$ goes to zero, and it goes to $\\arccos a\/b$ when $|b|$ goes to infinity.\n\nWhen $a$ and $b$ are both positive or negative, as $|b|$ increases from zero to infinity, $\\theta_r$ decreases from $\\pi$ to $\\arccos a\/b<\\pi\/2$.\nIf only one of them is positive, then $\\theta_r$ is always greater than $\\pi\/2$ and shows a similar decrease to both positive or negative cases.\n\nWhen $a=0$, as $|b|$ goes to infinity $\\theta_r$ goes to $\\pi\/2$ and it goes to $\\pi$ as $|b|$ goes to $0$.\n\nSimilar to the previous case, $\\theta_r$ can take values between $0$ and $\\pi$ depending on signs of $a$ and $b$, the ratio $a\/b$ and greatness of $a$ and $b$.\n\nThe magnitude of the minimum of $U_{eff}$ changes with respect to $b$.\nIn $b$ goes to zero limit, $U_{eff_{min}}$ goes to $-Mgl$ since $\\theta_r$ goes to $\\pi$.\nIn $b$ goes to infinity limit, $\\theta_r$ goes to $\\arccos a\/b$, and then the minimum of $U_{eff}$ goes to infinity with $I_x b^2(1-(a\/b)^2)\/2$.\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\subfigure[$U_{eff}$]{\n\t\t\t\\includegraphics[width=4.2cm]{ueff_b3.pdf}\n\t\t\t\\label{fig:ueffg4a}\n\t\t\t}\n\t\t\t\\subfigure[$\\theta_r$]{\n\t\t\t\t\\includegraphics[width=4.2cm]{loop_b_theta.pdf}\n\t\t\t\t\\label{fig:ueffg4b}\n\t\t\t\t}\n\t\t\t\t\\subfigure[$U_{eff_{min}}$]{\n\t\t\t\t\t\\includegraphics[width=4.2cm]{loop_b_umin.pdf}\n\t\t\t\t\t\\label{fig:ueffg4c}\n\t\t\t\t\t}\n\t\t\t\t\t\\caption{ $U_{eff}$, change of $\\theta_r$ with respect to $b$ and change of $U_{eff_{min}}$ with respect to $b$.\n\t\t\t\t\ta) Three different effective potential: $b=10 \\, rad \\,s^{-1}$ (green dashed-dotted curve), $b=30 \\, rad \\,s^{-1}$ (blue dashed curve) and $b=60 \\, rad \\,s^{-1}$ (red continious curve) with $a\/b=0.5$. Black line shows $Mglb\/a$.\n\t\t\t\t\tb) Change of $\\theta_r$ with respect to $b$ for constant $a\/b=0.5$ ratio (red curve). Black line shows $\\arccos(a\/b)=1.05 \\,rad$. Vertical dotted line shows the position of $b=2 \\tilde a$.\n\t\t\t\t\tc) Change of $U_{eff_{min}}$ with respect to $b$ for constant $a\/b=0.5$ ratio (red curve). Black line shows $Mglb\/a$. Vertical dotted line shows position of $b=2 \\tilde a$. Dotted curve shows $I_x b^2(1-(a\/b)^2)\/2$.\n\t\t\t\t\t}\n\t\t\t\t\t\\label{fig:ueffg6}\n\t\\end{center}\n\\end{figure}\n\nFor examples, similar to the previous case, a constant ratio between $a$ and $b$ is considered: This time $a\/b=0.5$.\nIn figure \\ref{fig:ueffg6}(a), three different effective potentials for three different $b$ values are shown similar to the previous section.\nIn this figure, there are some similarities and differences from figure \\ref{fig:ueffg5}(a).\nOne can see that $\\theta_r$ is also different for different $b$ values similar to the previous section.\nIt can be seen that as $b$ takes different values, the form and magnitude of the minimum of $U_{eff}$ becomes different similar to previous case, and it can be greater than $Mglb\/a$, unlike the previous case.\nIn figure \\ref{fig:ueffg6}(b), it can be seen that for very small values of $b$, $\\theta_r$ is close to $\\pi$ and it goes to $\\arccos 0.5=1.05 \\,rad$ as $b$ increases.\nIn figure \\ref{fig:ueffg6}(c), it can be seen that the minimum of $U_{eff}$ is close to $-Mgl$ if $b$ is small, and it goes to infinity with $I_x b^2(1-(a\/b)^2)\/2$ as $b$ goes to infinity.\nThese are the expected results from the explanations given above.\n\nBy considering these results, it can be said that $Mglb\/a$ is not important differently from $|a|>|b|$ case.\nFrom figures \\ref{fig:ueffg6}(b) and \\ref{fig:ueffg6}(c), one can say that the shift in the behaviour of $\\theta_r$ and $U_{eff_{min}}$ does not take place around $a=\\tilde a$, and the usage of $\\tilde a$ for seperation is not suitable when $|b|>|a|$.\n\nWhen $Mgl$ is negative, the second term in equation \\eqref{dueff3} becomes positive, and then in this case, $a\/b-\\cos \\theta$ should be negative which is possible when $\\arccos a\/b> \\theta > 0$.\nIn the limit where $b$ goes to infinity, again $\\theta_r$ goes to $\\arccos a\/b$.\nIn $b$ goes to zero limit, $\\theta_r$ goes to $0$.\nSimilar to the previous case, the interval for the minimum of effective potential changed from $(\\arccos b\/a,\\pi)$ to $(0,\\arccos b\/a)$.\nIf both $a$ and $b$ are negative or positive, $\\theta_r$ is between $0$ and $\\pi\/2$.\nIf only one of them is negative, then $\\theta_r$ can be greater than $\\pi\/2$ when $|b|$ goes to infinity, and $\\theta_r$ goes to $0$ as $b$ goes to zero.\nWhen $a=0$, in $|b|$ goes to infinity limit $\\theta_r$ goes to $\\pi\/2$, and $|b|$ goes to zero limit does not change and remains as $0$.\nIf $Mgl$ is negative, the minimum of $U_{eff}$ goes to $-|Mgl|$ when $b$ goes to $0$, and it goes to infinity as $|b|$ goes to infinity.\n\n\n\\section{Conclusion}\n\nEffective potential can be helpful in understanding the motion of a symmetric top in different ways.\n$E'$ should be equal to or greater than the minimum of $U_{eff}$ for physical motions.\nBy using the limits given in section \\ref{scnd}, one can say that the regular precession takes place at greater angles when $a$ and $b$ are small, \nand as $a$ and $b$ increase, it takes place at smaller angles.\nTo observe regular precession smaller than $\\pi\/2$, $a$ and $b$ should have the same sign and have greater magnitudes.\nThe limiting angle when $|a|$ or $|b|$ goes to infinity can be found by using inverse cosine of $b\/a$ and $a\/b$ when $|a|>|b|$ and $|b|>|a|$, respectively.\nIf $E'$ is greater than the minimum of $U_{eff}$, then different types of motions can be seen \\cite{Tanriverdi_abdffrnt}.\nThese motion will take place closer angles to $\\theta_r$ when $E'$ is close to the minimum of $U_{eff}$,\nand by considering signs and magnitudes of $a$ and $b$ one can have an opinion on the angles where the motion takes place.\n\nIf $a$ and\/or $b$ are small, then there can be a high asymmetry in the form of $U_{eff}$.\nFrom the definitions of $U_{eff}$ and $E'$, one can say that $\\dot \\theta$ is propotional to the difference $E'-U_{eff}(\\theta)$ for a specific $\\theta$ value.\nTherefore, one can say that as $\\theta$ increases from $\\theta_{min}$ to $\\theta_r$, the change in $\\dot \\theta$ is gradual, and as $\\theta$ increases from $\\theta_{r}$ to $\\theta_{max}$, the change in $\\dot \\theta$ is more rapid when $a$ and\/or $b$ are small.\nAs $\\theta$ changes from $\\theta_{max}$ to $\\theta_{min}$, this change in $\\dot \\theta$ is firstly rapid and then gradual.\n\nIf $a$ and $b$ are great enough and the difference $E'-U_{eff_{min}}$ is small enough, then the asymmetry in $U_{eff}$ can be ignored.\nIn these cases, one can make an approximation and find an exact solution for this approximation \\cite{Goldstein, Arnold}.\nThis approximation works better when the asymmetry in $U_{eff}$ is least.\n\nWe have seen that comparison of $|a|$ with $\\tilde a$ can be used when $|a|>|b|$ for an approximate seperation, and it is not suitable when $|b|>|a|$.\nBut comparison between $|b|$ and $\\tilde a$ can be used when $|b|>|a|$, and if it is used, one should use a naming other than \"strong top\" or \"weak top\". \nWe should note that comparison of $|a|$ with $\\tilde a$ is very useful when $|a|=|b|$  \\cite{Tanriverdi_abeql}.\n\nAnother thing that should be taken into account is the relation between $Mglb\/a$ and $E'$ \\cite{Tanriverdi_abdffrnt}.\nThis study has shown that the minimum of $U_{eff}$ is always smaller than $Mglb\/a$ when $|a|>|b|$, which shows that one can always observe all possible motions when $|a|>|b|$.\nOn the other hand, $Mglb\/a$ can be greater than or smaller than the minimum of $U_{eff}$ when $|b|>|a|$.\n\nThese results show that effective potential has different advantages over the cubic function in understanding the motion of a spinning heavy symmetric top.\nHowever, the cubic function is still important since it is better for proofs.\n\n\n\\section{Appendix}\n\nThere is an alternative to effective potential: the cubic function given in equation \\eqref{cubicf}.\n\nHere, we will compare the cubic function with effective potential.\nThe cubic function is equal to $\\dot u^2$, and its roots give the points where $\\dot u=0$.\n$\\dot \\theta$ is equal to zero at two of these three points, and the third root is irrelevant to turning angles.\nThen, one can use the cubic function to obtain turning angles. \nIf these two roots are the same, i.e. double root, then one can also say that this case gives regular precession.\nThese turning angles can also be obtained from effective potential by using $E'=U_{eff}(\\theta)$.\nAnd, if $E'=U_{eff_{min}}$ then the regular precession is observed as explained above.\n\nOn the other hand, there is not any correspondence between the minimum of $U_{eff}$ and the maximum of $f(u)$.\nThe reason for this is the multiplication with $1-u^2$ during the change of variable.\nThen, $f(u)$ can not be used to make further analyses similar to $U_{eff}$, given above.\n\nWe will consider a case satisfying $\\alpha=575.1 \\, s^{-2}$, $a=10 \\, rad \\, s^{-1}$, $b=2 \\, rad \\, s^{-1}$ as an example.\nFor the symmetric top with previously given parameters, $\\beta$ becomes $ 596.5 \\, s^{-2}$.\n$U_{eff}$ and $f(u)$ can be seen in figure \\ref{fig:ueff_fu}.\nOne can see that $\\theta_{min}=1.83 \\, rad$ and $\\theta_{max}=2.57 \\, rad$ can be obtained from $\\arccos(u2)=1.83 \\, rad$ and $\\arccos(u1)=2.57 \\, rad$, respectively.\nOn the other hand, $\\theta_r=2.28 \\, rad$ can not be obtained from $\\arccos(u_m)=2.18$.\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\subfigure[$U_{eff}$]{\n\t\t\t\\includegraphics[width=5.5cm]{ueff_comp.pdf}\n\t\t\t}\n\t\t\t\\subfigure[$f(u)$]{\n\t\t\t\t\\includegraphics[width=5.5cm]{fu_comp.pdf}\n\t\t\t\t}\n\t\t\t\t\\caption{$U_{eff}$ and $f(u)$ when $\\alpha=575.1 \\, s^{-2}$, $\\beta=596.5 \\, s^{-2}$, $a=10 \\, rad \\, s^{-1}$ and $b=2 \\, rad \\, s^{-1}$.\n\t\t\t\ta) $U_{eff}$ continious (red) curve, $E'=-0.0150 \\, J$ dashed (blue) line, $\\theta_{min}=1.83 \\, rad$, $\\theta_{max}=2.57 \\, rad$, $\\theta_r=2.28 \\, rad$ and $U_{eff_{min}}=-0.0299 J$.   \n\t\t\t\tb) $f(u)$ continious (red) curve, $u_1=-0.841$, $u_2=-0.258$, $u_3=1.05$, $u_m=-0.575$ and $f_{max}=81.6 \\, s^{-2}$.\n\t\t\t\t}\n\t\t\t\t\\label{fig:ueff_fu}\n\t\\end{center}\n\\end{figure}\n\nThese show that $f(u)$ can be used to obtain turning angles, however, it can not be used to obtain $\\theta_r$ where the minimum of $U_{eff}$ occurs.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\mathbb{T}=\\mathbb{R}\/\\mathbb{Z}$ and identify it with $[0,1]$ in the usual way. For integers $d\\ge 2$, let $F$ be the $d$-adic Bernoulli map \n$$F: \\mathbb T\\rightarrow \\mathbb T,\\ t\\mapsto d\\cdot t\\mod 1.$$\nLet $f$ be a function (weight) on $\\mathbb{T}$. Consider the weighted transfer operator $L$ associated to $F$:\n\\begin{align*}\n(Lu)(t)\n&=\\frac1 d\\sum_{s\\in F^{-1}(t)} f(s)u(s)\\\\\n&=\\frac1 d\\sum_{i=0}^{d-1} f\\left(\\frac{t+i}{d}\\right)u\\left(\\frac{t+i}{d}\\right),\\ t\\in\\mathbb{T}\n\\end{align*}\nwhere $u$ is a function on $\\mathbb{T}$. Such operators are also called Ruelle (or Ruelle-Perron-Frobenius) operators. They can also be defined associated to more general maps and on more general spaces (cf. Hennion \\cite{Hennion1993} and Baladi \\cite{Baladi2000} for more background).\n\nIn this paper, we study spectral properties of $L$ as an operator acting on $C(\\mathbb{T})$, the space of continuous functions on $\\mathbb{T}$. When $f\\equiv 1$, this question has been extensively studied,\nespecially in the case $d=2$ (cf. Vep\\v{s}tas \\cite{V} and references therein). For more general weights $f$, there are Perron-Frobenius type theorems that describe spectral properties of $L$ (cf. \\cite[Theorem~1.5]{Baladi2000}). However, such theorems often require $f$ to be strictly positive, which is not met by the main examples we are interested in:\n$$(c)\\ f(t)=|\\cos(\\pi t)|^q \\hspace{1cm} (s)\\ f(t)=|\\sin(\\pi t)|^q$$\nwhere $q>0$. In Section \\ref{sec:quasicompact}, we develop Perron-Frobenius type theorems for transfer operators $L$ with such `degenerate' weights (more precisely, weights that have exactly one zero on $\\mathbb{T}$). The theorems are derived using notions of quasicompactness and Krein property, which we verify by exploiting the specific structure of the Bernoulli map $F$; see also Fan and Lau \\cite{FanLau1998} for similar treatments. As a corollary, we conclude that the operator $L$ satisfies classical Perron-Frobenius theorems in all cases of $d$ and $(c)\/(s)$, except for the case $d=2$ and $(c)$. \n\nIn Section \\ref{sec:special-weights}, we study in more detail the spectral properties of $L$ in the non-exceptional cases. When $q$ is an even integer, we obtain explicit computations of $\\rho(L)$ (the spectral radius of $L$) by reducing to a finite-dimensional problem. When $q$ is not an even integer, evaluating $\\rho(L)$ is more difficult. We derive in this case estimations of $\\rho(L)$, particularly for $d=3$ (note that when $d$ is odd, $(c)$ and $(s)$ are equivalent). As an application, we obtain asymptotic behavior of some integrals of the form\n$$I_n=\\int_{\\mathbb{T}} \\prod_{j=0}^{n-1} f(d^j t)dt,\\quad \\text{as } n\\rightarrow\\infty.$$\nIn particular, we extend a result of Strichartz \\cite{Strichartz1990} concerning the Fourier transform of the middle-third Cantor set. We also study geometric properties of the function $L1$, \nas well as asymptotic behavior of $\\rho(L)$ as $q\\rightarrow\\infty$. For the latter question it turns out that one needs to distinguish the case when $d$ is even and $f$ is given by $(s)$.\n\nIn Section \\ref{sec:binary}, we give a detailed account of the exceptional case $d=2$ and $(c)$. Using an explicit formula for the iterates $L^n 1$, we find the spectral radius and eigenfunctions of $L$ explicitly (see also Fan and Lau \\cite{FanLau1998} for related results), and obtain geometric properties of $L^n1$ for $n\\ge 1$ (especially for $q\\le 1$ and even $q$'s). The spectral problem in this case is closely related to the case $f\\equiv 1$ mentioned above, and has to do with the Hurwitz zeta functions.\n\nIn Section \\ref{sec:Lp}, we study the spectral problem on Lebesgue spaces. In Section \\ref{sec:application}, we give an application to Fourier multipliers.\n\n\\section{Quasicompact transfer operators}\\label{sec:quasicompact}\n\nLet $f:\\mathbb{R}\\to\\mathbb{R}$ be a continuous nonnegative 1-periodic function, and let $d\\ge 2$ be an integer.\nWe consider the transfer operator\n\\begin{equation}\\label{L:L}\n (Lu)(t)=\\frac1 d\\sum_{i=0}^{d-1} f\\left(\\frac{t+i}{d}\\right)u\\left(\\frac{t+i}{d}\\right) .\n\\end{equation}\nLet $\\mathbb{T}=[0,1]$ with $0$ and $1$ identified (circle). Let $C(\\mathbb{T})$ be the Banach space of continuous complex-valued functions on $\\mathbb{T}$\nendowed with the maximum norm $\\|\\cdot\\|_\\infty$.\nThen $L:C(\\mathbb{T})\\to C(\\mathbb{T})$ is a bounded linear operator. Moreover, $L$ is positive in the sense that $u\\ge 0$ implies $Lu\\ge 0$.\n\nDefine a map $F:\\mathbb{T}\\to \\mathbb{T}$ by $F(t)=d\\cdot t \\mod 1$. Then we can write\n\\[ (Lu)(t)=\\frac1d \\sum_{s\\in F^{-1}(t)} f(s)u(s), \\quad t\\in\\mathbb{T} .\\]\nFor each $n\\in\\mathbb{N}$, set\n\\[ f_n(t)=\\prod_{j=0}^{n-1} f\\big(F^j(t)\\big) .\\]\nThen\n\\begin{equation}\\label{L:Ln}\n(L^n u)(t)=\\frac{1}{d^n} \\sum_{s\\in F^{-n}(t)} f_n(s)u(s) .\n\\end{equation}\n\nLet $0<\\alpha\\le 1$. \nConsider the Banach space $\\H$ \nof H\\\"older continuous functions $u:\\mathbb{T}\\to \\mathbb{C}$ with the norm\n\\[ \\|u\\|_\\alpha=\\sup_{s\\neq t} \\frac{|u(s)-u(t)|}{|s-t|^\\alpha} + \\|u\\|_\\infty .\\]\nIf $f\\in \\H$ then $L_\\alpha u=Lu$ defines a bounded linear operator $L_\\alpha:\\H\\to \\H$.\n\nLet $T$ be a bounded linear operator on a Banach space $X$. We denote its spectral radius by $\\rho(T)$.\n$T$ is called quasicompact if there exists a compact operator $K$ on $X$ such that $\\rho(T-K)<\\rho(T)$.\nIf $T$ is quasicompact and $\\lambda\\in\\mathbb{C}$ is in the spectrum of $T$ with $|\\lambda|>\\rho(T-K)$, then $\\lambda$ is an eigenvalue of\n$T$.\n\nThe following theorem is proved in \\cite[pages 3--4]{Sm}.  In \\cite[Proposition 1]{Sm} it is assumed that $f$ is positive while we assume here that $f$ is nonnegative. However positivity of $f$ is not used on pages 3--4 of \\cite{Sm}. See also \\cite{Hennion1993}.\n\n\\begin{thm}\\label{L:t1}\nLet $0\\le f\\in \\H$ for some $0<\\alpha\\le 1$.\nThen $\\rho(L)=\\rho(L_\\alpha)$. Furthermore, if  $\\rho(L)>0$, then $L_\\alpha$ is quasicompact.\n\\end{thm}\n\nFor each $n\\in\\mathbb{N}$, set\n\\begin{equation}\\label{L:hn}\n h_n=L^n 1.\n \\end{equation}\nDefine\n\\begin{equation}\\label{L:rnRn}\n r_n=\\min_{t\\in\\mathbb{T}}  h_n(t),\\quad R_n=\\max_{t\\in\\mathbb{T}} h_n(t) .\n\\end{equation}\nIt is easy to show that\n\\[ r_{m+n}\\ge r_mr_n,\\quad R_{m+n}\\le R_mR_n .\\]\nTherefore, the limits\n\\begin{equation}\\label{L:rR}\n r=\\lim_{n\\to\\infty} (r_n)^{1\/n} =\\sup_n\\ (r_n)^{1\/n},\\quad R=\\lim_{n\\to\\infty} (R_n)^{1\/n}=\\inf_n\\ (R_n)^{1\/n}\n\\end{equation}\nexist. In particular, for every $n\\in\\mathbb{N}$ we have\n\\begin{equation}\\label{L:est1}\n r_n^{1\/n}\\le r\\le R\\le R_n^{1\/n}.\n\\end{equation}\nMoreover, since $R_n=\\|L^n\\|_{C(\\mathbb{T})\\rightarrow C(\\mathbb{T})}$, by Gelfand's formula, $R=\\rho(L)$.\n\n\\begin{thm}\\label{L:t2}\nLet $w\\in C(\\mathbb{T})$ be a unit, that is, $w(t)>0$ for all $t\\in\\mathbb{T}$.\nThen\n\\begin{equation}\\label{L:est2}\n\\min_{t\\in\\mathbb{T}} \\frac{(Lw)(t)}{w(t)}\\le r\\le R\\le \\max_{t\\in\\mathbb{T}} \\frac{(Lw)(t)}{w(t)} .\n\\end{equation}\n\\end{thm}\n\\begin{proof}\nWe define a bounded linear operator $S$ on $C(\\mathbb{T})$ by\n\\[ (Su)(t)= \\frac{1}{w(t)} L(wu)(t),\\]\na sequence of functions\n\\[ \\tilde h_n= S^n 1,\\]\nand sequences of numbers\n\\[ \\tilde r_n =\\min_{t\\in\\mathbb{T}} \\tilde h_n(t),\\quad \\tilde R_n=\\max_{t\\in\\mathbb{T}} \\tilde h_n(t) .\\]\nSince $S$ is positive, we obtain for every $n\\in\\mathbb{N}$\n\\[\n (\\tilde r_n)^{1\/n}\\le \\lim_{n\\to\\infty}  (\\tilde r_n)^{1\/n}=: \\tilde r\\le \\tilde R:= \\lim_{n\\to\\infty}  (\\tilde R_n)^{1\/n}\\le (\\tilde R_n)^{1\/n} .\n\\]\nSince $w$ is a unite, there are constants $a,b>0$ such that\n\\[ a\\le w(t)\\le b\\]\nfor all $t\\in\\mathbb{T}$. This implies\n\\[ a h_n(t)=(L^na)(t) \\le w(t) \\tilde h_n(t) \\le (L^nb)(t)=b h_n(t) .\\]\nFrom this we obtain\n\\[  r_n\\le \\frac{b}{a} \\tilde r_n,\\quad \\tilde r_n\\le \\frac{b}{a} r_n,\\quad  R_n\\le \\frac{b}{a} \\tilde R_n,\\quad \\tilde R_n\\le \\frac{b}{a} R_n .\\]\nThus\n\\[ r=\\tilde r,\\quad R=\\tilde R .\\]\nNow \\eqref{L:est2} follows from\n\\[ \\tilde r_1\\le \\tilde r=r\\le \\tilde R=R\\le \\tilde R_1.\\]\n\\end{proof}\n\nWe say that $L$ is a Krein operator if, for all $u\\in C(\\mathbb{T})$ such that $u(t)\\ge 0$ for all $t\\in\\mathbb{T}$ but $u(t_0)>0$ for at least one $t_0\\in\\mathbb{T}$, there is $n\\in\\mathbb{N}$ such that $L^nu$ is a unit. Note that $n$ may depend on $u$.\nIt is easy to show that a Krein operator carries units to units (cf. \\cite[Lemma 5.2]{A}).\nIt follows from \\eqref{L:Ln} that if $f(t)>0$ for all $t$ then $L$ is a Krein operator,\nAlso, if $f$ vanishes on an interval of positive length, then $L$ cannot be a Krein operator.\n\n\\begin{lemma}\\label{L:Krein1}\nSuppose $f$ has exactly one zero in $[0,1)$.\nIf $f_n$ has four zeros that form an arithmetic progression with step size $d^{-n}$, then $d=2$ and $f(\\frac12)=0$.\n\\end{lemma}\n\\begin{proof}\nLet $s_0+\\mathbb{Z}$ be the set of zeros of $f$.\nSuppose that $t_i=t+id^{-n}$ is a zero of $f_n$ for $i=0,1,2,3$.\nThen there exist integers $0\\le k_i\\le n-1$, and integers $j_i$ such that\n\\begin{equation}\\label{eq}\n t_i=d^{-k_i}(s_0+j_i),\\quad i=0,1,2,3.\n\\end{equation}\nWe will assume that $k_0=\\max\\{k_0,k_1,k_2,k_3\\}$ (the other cases are mentioned at the end of the proof.)\nClearly, $k_0>k_1$.\nSince we can replace $s_0$ by $s_0+j$ with any integer $j$, we will assume that $j_1=0$ in order to simplify the notation.\nFrom $t_1-t_0=t_2-t_1=d^{-n}$, we obtain\n\\[ d^{-k_1}s_0-d^{-k_0}(s_0+j_0)=d^{-n},\\quad d^{-k_2}(s_0+j_2) -d^{-k_1}s_0=d^{-n} .\\]\nEliminating $s_0$ from these equations, we find\n\\[ (1-j_0d^{n-k_0})(d^{k_0-k_1}-d^{k_0-k_2})= (1-j_2d^{n-k_2})(1-d^{k_0-k_1}) .\\]\nThis is an equation involving only integers. Since $n>k_2$, $k_0>k_1$, the right-hand side is not divisible by $d$.\nTherefore, we must have $k_0=k_2$. But this is impossible when $d>2$. So we must have $d=2$ and $k_0=k_2=n-1$.\n\nNow suppose that $d=2$ and $k_0=k_2=n-1$.\nWithout loss of generality we take $j_0=0$, $j_2=1$.\nSince $t_1-t_0=t_3-t_2=2^{-n}$ we obtain\n\\[ 2^{n-k_1}(s_0+j_1)=2s_0+1,\\quad 2^{n-k_3}(s_0+j_3)=2s_0+3 .\\]\nEliminating $s_0$ this gives\n\\[ (3-j_32^{n-k_3})(2^{n-k_1-1}-1)=(1-j_12^{n-k_1})(2^{n-k_3-1}-1) .\\]\nIt is clear that $k_1<n-1$, $k_3<n-1$. Therefore, the equation yields that\n$2^{n-k_3-2}-3\\cdot 2^{n-k_1-2}+1$ is an even integer. This implies that $k_1=n-2$ or $k_3=n-2$.\nIf $k_1=n-2$ then $2s_0+1=4(s_0+j_1)$, which implies that $s_0-\\frac12$ is an integer.\nIf $k_3=n-2$, we have the same conclusion.\n\nIf $k_2=\\max\\{k_0,k_1,k_2,k_3\\}$, the proof is almost the same. We again obtain $k_0=k_2=n-1$ and $d=2$ after the first part of the proof.\nThe other two cases can be reduced to the treated ones by replacing $s_0$ by $-s_0$.\n\\end{proof}\n\n\\begin{lemma}\\label{L:Krein2}\nSuppose $f$ has exactly one zero $s_0\\in[0,1)$.\nThen $L$ is a Krein operator unless $s_0=\\frac12$ and $d=2$.\n\\end{lemma}\n\\begin{proof}\nLet $u\\in C(\\mathbb{T})$ be nonnegative but not identically zero,\nChoose $n$ so large that $u(t)>0$ for $m d^{-n}\\le t\\le (m+4) d^{-n}$ for some integer $m$, $0\\le m\\le d^n-4$.\nWe claim that $(L^n u)(t)>0$ for all $t\\in\\mathbb{T}$. In fact, by Lemma \\ref{L:Krein1}, if $t\\in\\mathbb{T}$ then among the four points $t_i=d^{-n}(t+m+i)$, $i=0,1,2,3$,\nat least one satisfies $f_n(t_i)>0$. Then \\eqref{L:Ln} implies $(L^n u)(t)\\ge f_n(t_i)u(t_i)>0$.\n\\end{proof}\n\nIf $d=2$ and $f(\\frac12)=0$ then $L$ is not a Krein operator: If $u(0)=0$ then $(L^nu)(0)=0$ for all $n\\in\\mathbb{N}$.\n\n\\begin{thm}[See also \\cite{FanLau1998}]\\label{L:t3}\nSuppose that $L$ is a Krein operator. Then the following statements hold.\\\\\n(a) $R>0$.\\\\\n(b) If $Lv=\\lambda v$ with $v\\ne 0$ and $|\\lambda|=R$,\nthen there is a constant $\\theta\\in\\mathbb{R}$ such that $e^{-i\\theta} v$ is a unit. \\\\\n(c) $L$ has no eigenvalue $\\lambda$ on the circle $|\\lambda|=R$ except possibly $\\lambda=R$.\\\\\n(d) If $R$ is an eigenvalue of $L$, then its algebraic multiplicity is $1$.\\\\\n(e) If $0\\le f\\in \\H$, then  $L_\\alpha$ is quasicompact, $r=R$, and $R$ is an eigenvalue of $L_\\alpha$ of algebraic multiplicity $1$ with a unit eigenfunction. \\end{thm}\n\\begin{proof}\n(a) Since $L$ is a Krein operator, $h_1=L1$ is a unit, so $0<r_1\\le R$.\n\n\\noindent (b)\n$L v =\\lambda v$ implies $z:=L|v|-R|v|\\ge 0$.\nSuppose that $z$ is not identically zero. Then there is $n\\in\\mathbb{N}$ such that $L^nz$ and $w:=L^n |v|$ are units.\nIt follows that there is $\\delta>0$ such that\n\\[ (L^nz)(t)=(Lw)(t)-R w(t)\\ge\\delta w(t),\\quad t\\in \\mathbb{T}.\\]\nApplying Theorem \\ref{L:t2}, we obtain the contradiction\n\\[ R+\\delta\\le \\min_{t\\in\\mathbb{T}} \\frac{(Lw)(t)}{w(t)}\\le R .\\]\nTherefore, $z=0$ and\n\\begin{equation}\\label{L:eq}\nL|v|=R|v|.\n\\end{equation}\nThen\n\\[ |L^n v|=R^n |v|=L^n|v|=w.\\]\nTherefore, $|v|$ is a unit.\nWe claim that there is a constant $\\theta\\in\\mathbb{R}$ such that $e^{-i\\theta} v(t)>0$ for all $t$.\nSuppose this is not true. Since $L|v|=|Lv|$, there is $n\\in\\mathbb{N}$ and $1\\le i<j\\le d^n$ such that $v(s)\/v(t)\\not\\in(0,\\infty)$ for all $s\\in I_{n,i}$, $t\\in I_{n,j}$.\nSince $L$ is a Krein operator, $f$ does no vanish on an interval of positive length. Then also $f_n$ does not vanish on an interval of positive length.\nTherefore, there is $s\\in I_{n,i}$, $t\\in I_{n,j}$ with $(t-s)d^{-n}\\in\\mathbb{Z}$ such that $f_n(s)\\ne 0$, $f_n(t)\\ne 0$.\nHence, by \\eqref{L:Ln}, $|(L^n v)(s)|<(L^n |v|)(s)$, which is a contradiction.\nTherefore, the claim is proved.\n\n\\noindent (c) follows from (b).\n\n\\noindent (d)\nSuppose that $R$ is an eigenvalue of $L$. By (b), each corresponding eigenfunction is a constant multiple of a unit. It follows that\nthe geometric multiplicity of the eigenvalue $R$ is $1$.\nNow assume that there are $u,w\\in C(\\mathbb{T})$ such that $L u-Ru = w$, $L w=Rw$, where $w$ is a unit. We may assume that $u$ is a unit. There is $\\delta>0$ such that $\\delta u\\le w$.\nThen Theorem \\ref{L:t2} leads to the contradiction\n\\[ R+\\delta\\le \\min_{t\\in\\mathbb{T}} \\frac{(Lu)(t)}{u(t)}\\le R .\\]\nThis shows that the algebraic multiplicity of the eigenvalue $R$ is $1$.\n\n\\noindent (e)\nSince $\\rho(L)=R>0$, it follows from Theorem \\ref{L:t1} that $L_\\alpha$ is quasicompact. Then $L_\\alpha$ has an eigenvalue $\\lambda$ on the circle $|\\lambda|=R$.\nBy (c), $R$ is an eigenvalue of $L_\\alpha$. There is a corresponding unit eigenfunction.\nNow $r=R$ follows from Theorem \\ref{L:t2}.\n\\end{proof}\n\n\\begin{thm}[See also \\cite{FanLau1998}]\\label{L:t4}\nSuppose that $0\\le f\\in \\H$ and that $L$ is a Krein operator.\nLet $P$ be the spectral projection onto the eigenspace of $L_\\alpha$ corresponding to the eigenvalue $R$.\\\\\n(a) The sequence $R^{-n}L_\\alpha^n$ converges to $P$ as $n\\to\\infty$ with respect to the operator norm.\\\\\n(b) The sequence $R^{-n} h_n$ converges in $\\H$ to an eigenfunction of $L_\\alpha$ corresponding to the eigenvalue $R$.\n\\end{thm}\n\\begin{proof}\n(a)\nBy Theorem \\ref{L:t3}, $L_\\alpha$ is quasicompact and the eigenvalue $R$ of $L_\\alpha$ is an isolated point of its spectrum.\nTherefore, there exists the spectral projection $P$ onto the one-dimensional root subspace belonging to the eigenvalue $R$.\nThe Banach space $\\H$ is a direct sum of the subspaces $P\\H$ and $(1-P)\\H$. Both subspaces are invariant under $L_\\alpha$.\nOn $P\\H$, $L_\\alpha$ acts as $R$ times the identity.\nSet $S:=R^{-1}(1-P)L_\\alpha$. By Theorem \\ref{L:t3}, the spectral radius of $S$ is less than $1$ so $S^n$ converges to $0$ as $n\\to\\infty$ in the operator norm.\nWe have $R^{-n} L_\\alpha^n =S^n+ P$ which implies statement (a).\n\n\\noindent (b)\nWe have $R^{-n}h_n=R^{-n}L^n_\\alpha1\\to P1$ as $n\\to\\infty$. Since\n\\[1\\le R^{-n}R_n\\le \\|R^{-n} h_n\\|_\\alpha\\to \\|P1\\|_\\alpha,\\]\nwe have $P1\\ne0$.\n\\end{proof}\n\nWe consider now the following problem that was the original motivation for this paper.\nLet $f:\\mathbb{R}\\to\\mathbb{C}$ be a bounded measurable and 1-periodic function, and let $d\\ge 2$ be an integer. For $n\\in\\mathbb{N}$, define as before\n\\[ f_n(t)=\\prod_{j=0}^{n-1} f(d^j t) .\\]\nThe problem is to find the behavior of the sequence of integrals\n\\[ I_n(f)=\\int_0^1 f_n(t)\\,dt \\]\nas $n\\to\\infty$. In particular, we want to find $c(f)$ defined by\n\\begin{equation}\\label{L:c}\n c(f)=\\limsup_{n\\to\\infty} |I_n(f)|^{1\/n}.\n \\end{equation}\n\nThe sequence $I_n$ is related to the bounded linear operator\n\\begin{equation}\\label{T}\n (Tx)(t)=f(t)x(d\\cdot t)\n\\end{equation}\nwhich maps $L^2(\\mathbb{T})$ to itself. Note that\n\\[ f_n=T^n 1 \\]\nand\n\\begin{equation}\\label{Im}\n I_n(f) =\\langle T^n1,1\\rangle\n\\end{equation}\nwith the inner product $\\langle\\cdot,\\cdot\\rangle$ in $L^2(\\mathbb{T})$.\nIn particular, \n\\[ |I_n|\\le \\|T^n\\|\\]\nand\n\\begin{equation}\\label{L:eq1}\nc(f)\\le \\lim_{n\\to\\infty} \\|T^n\\|^{1\/n}=\\rho(T).\n\\end{equation}\n\nWe show that $c(f)$ is equal to the spectral radius of a transfer operator under suitable assumptions on $f$. See also \\cite{FanLau1998} for related results.\n\n\\begin{thm}\\label{A:t1}\nSuppose that $0\\le f\\in \\H$ for some $0<\\alpha\\le 1$, and that the transfer operator $L$ defined by \\eqref{L:L} is a Krein operator. Then $r=c(f)=R=\\rho(L)$, where\n$r,R$ are defined in \\eqref{L:rR}. Moreover, we can replace $\\limsup$ by $\\lim$ in definition \\eqref{L:c}.\n\\end{thm}\n\\begin{proof}\nThe adjoint $T^\\ast$ of $T$ agrees with the operator $L$ when considered as an operator on $L^2(\\mathbb{T})$.\nLet $h_n, r_n, R_n$ be defined by \\eqref{L:hn}, \\eqref{L:rnRn}.\nIt follows from \\eqref{Im} that\n\\[ I_n=\\langle 1, h_n\\rangle=\\int_0^1 h_n(t)\\,dt .\\]\nThus\n\\[r_n^{1\/n}\\le I_n^{1\/n}\\le R_n^{1\/n}.\\]\nBy Theorem \\ref{L:t3}(e), the sequences $r_n^{1\/n}$ and $R_n^{1\/n}$ converge to the same limit $r=R$.\nTherefore, the sequence $I_n^{1\/n}$ converges and we obtain $r=c(f)=R$.\n\\end{proof}\n\nUsing Theorem \\ref{A:t1} in connection with \\eqref{L:est1} or Theorem \\ref{L:t2} we can estimate $c(f)$. We will look at some examples in the next section.\n\nWe mention two special classes of functions $f$ for which $c(f)$ can be calculated explicitly.\n\n1)\nSuppose that $f$ is a step function such that $f(t)=f_i={\\rm const}$ for $\\frac{i-1}{d}\\le t< \\frac{i}{d}$, $i=1,\\cdots,d$.\nThen it is easy to show that\n\\[ I_n(f)= \\left(\\int_0^1 f(t)\\,dt\\right)^n.\\]\nTherefore,\n\\[ c(f)=\\left|\\int_0^1 f(t)\\,dt\\right| .\\]\nIf $f$ is any nonnegative bounded measurable $1$-periodic function, we may introduce\ntwo step function $g,h$ defined by\n\\begin{eqnarray*}\n g(t)&=&\\inf\\left \\{ f(s): \\frac{i-1}{d}\\le s<\\frac{i}{d}\\right \\}\n \\quad\\text{for $\\frac{i-1}{d}\\le t<\\frac{i}{d}$},\\\\\nh(t)&=&\\sup\\left \\{ f(s): \\frac{i-1}{d}\\le s<\\frac{i}{d}\\right \\}\n\\quad\\text{for $\\frac{i-1}{d}\\le t<\\frac{i}{d}$}.\n \\end{eqnarray*}\nThen we obtain the estimate\n\\begin{equation}\\label{ineq}\nI_n(g)\\le I_n(f)\\le I_n(h)\n\\end{equation}\nand so\n\\begin{equation}\\label{ineq1}\n\\int_0^1 g(t)\\,dt\\le c(f)\\le \\int_0^1 h(t)\\,dt .\n\\end{equation}\n\n2) Let $f$ be any bounded measurable $1$-periodic function with\nFourier expansion\n\\[ f(t)=\\sum_{k\\in\\mathbb{Z}} a_k e^{2\\pi i k t} .\\]\nWe represent the operator $T$ by an infinite matrix in the orthonormal basis $\\{e^{2\\pi ikt}\\}_{k\\in\\mathbb{Z}}$.\nThe matrix of $T$ is\n\\begin{equation}\\label{T:matrix}\n \\left( a_{k-d\\ell} \\right)_{k,\\ell \\in\\mathbb{Z}} .\n\\end{equation}\nIn this notation $k$ is the row index and $\\ell$ is the column index.\nIf we write\n\\[ f_n(t)=\\sum_{k\\in\\mathbb{Z}} a_{k,n} e^{2\\pi i kt},\\]\nthen we obtain the coefficients $a_{k,n+1}$ from $a_{k,n}$ by application of $T$, so\n\\[ a_{k,n+1}=\\sum_{\\ell\\in\\mathbb{Z}} a_{k-d\\ell}\\, a_{\\ell,n} .\\]\nNote that $I_n=a_{0,n}$.\n\nIn particular, suppose that $f(t)$ is a trigonometric polynomial of degree $N$, so\n\\[ f(t)=\\sum_{k=-N}^N a_k e^{2\\pi i k t} .\\]\nWe set\n\\[ K:=\\left\\lfloor \\frac{N-1}{d-1}\\right\\rfloor.\\]\nConsider the central $2K+1$ by $2K+1$ submatrix $B$ of $T$ consisting of rows $-K\\le k\\le K$ and columns $-K\\le \\ell\\le K$.\nNotice that all entries in the rows $-N\\le k\\le N$ outside the central submatrix vanish. Therefore, we obtain the recursion\n\\[ a_{k,n+1} =\\sum_{\\ell=-K}^K a_{k-d\\ell}\\,a_{\\ell,n} \\quad\\text{if $-K\\le k\\le K$.}\\]\nHence we can calculate $I_n=a_{0,n}$ by computing the powers of the matrix $B$.\nIt is clear that\n\\begin{equation}\\label{eq2}\n c(f)\\le \\rho(B)\n\\end{equation}\nbut it is not immediately clear whether we have equality in \\eqref{eq2}.\nIt depends on how the constant function $1$ is represented in a Jordan basis of $B$ (whether the basis vectors associated with largest eigenvalue of $B$ contribute to the expansion of $1$.)\n\nThe situation is clear if the matrix $B$ is nonnegative and primitive ($B^p$ is a positive matrix for some $p\\in\\mathbb{N}$.)\nThen the spectral radius of $B$ is a simple positive eigenvalue and we can use Theorem 8.5.1 in \\cite{HJ} to show that there is equality in \\eqref{eq2}.\nSuppose that $a_k>0$ for all $k=-N,-N+1,\\cdots,N$. Then all entries in the main diagonal, the subdiagonal and superdiagonal of $B$ are positive. Therefore, $B$ is primitive.\n\nIf we have symmetry $a_{-k}=a_k$ then we can replace the matrix $B$ by a $K+1$ by $K+1$ matrix $C$ whose entries are\n\\[ c_{i,0}=a_i, \\quad c_{i,j}=a_{i-dj}+a_{i+dj} \\quad\\text{if $0\\le i\\le K$, $1\\le j\\le K$.}\\]\nSee the next section for examples.\n\n\\section{The special cases $f(t)=|\\cos(\\pi t)|^q$ and $f(t)=|\\sin(\\pi t)|^q$}\\label{sec:special-weights}\n\nIn this section we consider the functions\n\\[f(t)=|\\cos(\\pi t)|^q,\\quad \\tilde f(t)=|\\sin(\\pi t)|^q\\quad \\text{where $q>0$}.\n\\]\nWe set\n\\[ c(q):=c(f),\\quad \\tilde c(q):=c(\\tilde f).\\]\nObviously, $0\\le c(q), \\tilde c(q)\\le 1$. Note that $f, \\tilde f\\in \\H$ with $\\alpha=\\min(q,1)$. By Lemma \\ref{L:Krein2}, $L$ is a Krein operator except when $f(t)=|\\cos(\\pi t)|^q$ and $d=2$.\nThis is an exceptional case that will be considered in the next section.\nExcept for this case we can apply Theorems \\ref{L:t3} and \\ref{A:t1}.\n\nIf $d$ is odd, then $I_n(f)=I_n(\\tilde f)$, and consequently $c(f)=c(\\tilde f)$. In fact, in this case the transfer operators weighted by $f$ and $\\tilde f$ are conjugate to each other.\n\n\\begin{thm}\\label{S:t1}\nThe functions $c(q)$ and $\\tilde c(q)$ are convex and nonincreasing in $q>0$. Moreover, \n\\[ \\lim_{q\\to\\infty} c(q)=\\frac1 d, \\quad\n\\lim_{q\\to\\infty} \\tilde c(q)=\\begin{cases} \\frac1d & \\text{if $d$ is odd,}\\\\ 0 & \\text{if $d$ is even,}\\end{cases}\n\\]\nand\n\\[ \\lim_{q\\to 0^+} c(q)= \\lim_{q\\to 0^+} \\tilde c(q)=1.\\]\n\\end{thm}\n\\begin{figure}[h]\n\\includegraphics[scale=0.4]{cq.png}\n\\caption{A graph of $c(q)$ for $0<q<6$, when $d=3$.}\n\\vspace{2em}\n\\includegraphics[scale=0.4]{tildecq.png}\n\\caption{A graph of $\\tilde c(q)$ for $0<q<20$, when $d=2$.}\n\\end{figure}\n\\begin{proof}\nMonotonicity of $c(q)$ and $\\tilde c(q)$ are clear. Convexity follows from H\\\"older's inequality and Young's inequality. By looking at the function $h_1$ associated with $f$, we see that $R_1=\\max_{t\\in[0,1]} h_1(t)$ converges to $d^{-1}$ as $q\\to\\infty$. So\n\\[\\limsup_{q\\to\\infty} c(q)\\le d^{-1}.\\]\nIf we use an estimate of the form $f(t)\\ge 1-\\frac{t}{a}$ for $0\\le t\\le a$ for some $a\\in(0,1)$ depending on $q$, then we can estimate\n\\[ f_n(t)\\ge \\prod_{j=0}^{n-1} \\left(1-\\frac{d^j t}{a}\\right)\\ge \\exp\\left(-(\\ln 4)\\, \\frac{d^n-1}{d-1} \\frac{t}{a}\\right)\\quad\\text{for $0\\le t\\le \\frac{a}{2 d^{n-1}}$} .\\]\nIt then follows from \\eqref{L:c} that $c(q) \\ge d^{-1}$ for all $q>0$. Hence $$\\lim_{q\\to\\infty} c(q)=d^{-1}.$$\n\nWhen treating $\\tilde c(q)$, we may assume that $d$ is even. The proof is similar to the preceding one. To determine the limit of $\\tilde c(q)$ as $q\\to\\infty$, we use $c(\\tilde f)\\le R_2^{1\/2}$.\n\nTo show the limits as $q\\to 0^+$, it suffices to show that\n\\[c(q)^{1\/q}\\ge 1\/2,\\quad \\tilde c(q)^{1\/q}\\ge 1\/2\\]\nfor all $q>0$. To this end,\nlet $g(t)=|\\cos(\\pi t)|$ (or $|\\sin(\\pi t)|$). By Jensen's inequality, we have\n$$I_n^{1\/q}=\\left (\\int_0^1 |g_n(t)|^q dt\\right )^{1\/q} \n\\ge \\exp\\left (\\int_0^1 \\ln |g_n(t)|dt\\right )$$\nfor all $q>0$. \nOn the other hand, since\n$$\\int_0^1 \\ln |\\cos(\\pi t)|dt=\\int_0^1 \\ln |\\sin(\\pi t)|dt=\\ln (1\/2),$$\nwe have\n$$\\int_0^1 \\ln |g_n(t)|dt=\\sum_{j=0}^{n-1} \\int_0^1 \\ln |g(d^j t)|dt=n\\ln (1\/2).$$\nThus, for any $q>0$,\n$$c(q)^{1\/q}\\ (\\text{or }\\tilde c(q)^{1\/q})=\\lim_{n\\to\\infty} I_n^{1\/(nq)}\\ge 1\/2.$$\nThis completes the proof.\n\\end{proof}\n\n\\subsection{The case $f(t)=\\cos^{2N}(\\pi t)$, $d=3$}\nFor $N\\in\\mathbb{N}$, consider the trigonometric polynomial\n\\[ f(t)=\\cos^{2N}(\\pi t). \\]\nThe degree of $f$ is $N$ and, for $-N\\le k\\le N$,\n\\[ a_k=2^{-2N} \\binom{2N}{k+N}>0 .\\]\nWe use the method 2) from Section \\ref{sec:quasicompact}.\nIf $N=1,2$, then $K=0$ and so $c(2)=\\frac12$ and $c(4)=\\frac38$ (note that this recovers a result of Strichartz \\cite{Strichartz1990}).\nIf $N=3$, then $K=1$ and\n\\[ C=\\frac1{64} \\left[\\begin{array}{rr} 20 & 2\\\\ 15 & 6 \\end{array}\\right]. \\]\nSo\n\\[ c(6)=\\rho(C)=\\frac1{64}(13+\\sqrt{79}) =0.342003\\cdots.\\]\nIf $N=4$, then $K=1$ and\n\\[ C=\\frac1{256} \\left[\\begin{array}{rr} 70 & 16\\\\ 56 & 29 \\end{array}\\right]. \\]\nIt follows that\n\\[ c(8)=\\rho(C)= \\frac1{512}(99+9\\sqrt{65})=0.335078\\cdots.\\]\nWhen $N\\ge 5$, the formulas for $c(2N)$ become more complicated, but $c(2N)$ is easy to compute numerically.\nFor example, we obtain\n\\[ c(10)=0.333691\\cdots.\\]\n\nThe same method can be used to determine $c(2N)$ for other values of $d$.\n\n\\subsection{The case $f(t)=|\\sin(\\pi t)|$, $d=3$}\\label{subsec:d=3}\nEven if $f$ is not a trigonometric polynomial, we can still use matrix methods to estimate $c(f)$.\nAs an example, consider\n\\begin{equation}\\label{S:fs}\n f(t)=|\\sin(\\pi t)|.\n\\end{equation}\nThe Fourier coefficients of $f$ are\n\\[ a_k=-\\frac{2}{\\pi} \\frac{1}{(2k-1)(2k+1)},\\ k\\in\\mathbb{Z} .\\]\nNote that $a_0>0$ but all other $a_k$ are negative.\nLet $N\\in\\mathbb{N}$. We estimate\n\\[ f(t)\\le  \\sum_{k=-N}^N a_k e^{2\\pi i k t} +\\frac2\\pi \\sum_{k=N+1}^\\infty \\frac{2}{(2k-1)(2k+1)} \\]\nso we have\n\\[ f(t)\\le h(t) ,\\]\nwhere\n\\[ h(t)=\\frac2\\pi \\frac1{2N+1} +  \\sum_{k=-N}^N a_k e^{2\\pi i k t} .\\]\nUsing\n\\[ c(f)\\le c(h)\\le \\rho(C) \\]\nwe get upper bounds for $c(f)=c(1)$ (when $d=3$):\n\n\\begin{center}\n\\begin{tabular}{r|c}\n$N$ & $\\rho(C)$ \\\\\n\\hline\n1 & 0.848826\\\\\n2 & 0.763943\\\\\n3 & 0.737463\\\\\n4 & 0.717381\\\\\n5 & 0.704696\\\\\n10& 0.678384\\\\\n20& 0.663593\\\\\n30& 0.658613\\\\\n50 & 0.654552\\\\\n100 & 0.651436\n\\end{tabular}\n\\end{center}\nAs far as we know the exact value of $c(1)$ is not known. We conjecture that\n$c(1)=0.648314\\cdots$.\n\nWe also obtain\n\\[ g(t)\\le f(t) \\]\nwhere\n\\[ g(t)= -\\frac2\\pi \\frac1{2N+1} + \\sum_{k=-N}^N a_k e^{2\\pi i k t} .\\]\nSince $a_0>0$ and all other $a_k<0$, we can easily show that $g(t)\\ge 0$.\nTherefore, we have\n\\[ c(g)\\le c(f) .\\]\nHowever, the trigonometric polynomial $g$ does not have positive coefficients, so we do not know whether $c(g)=\\rho(C)$. Therefore, we do not obtain lower bounds by this method.\nFor $N=100$, one would get $\\rho(C)=0.645194\\cdots $.\n\nSomewhat surprisingly, the functions $h_n$ associated with \\eqref{S:fs}\ncan be represented in a fairly explicit way.\nIf\n\\[ u(t)=\\cos\\left(\\pi as\\right),\\quad s=t-\\tfrac12, \\]\nthen\n\\[ (Lu)(t)= \\tfrac16 \\big(1+2\\cos\\tfrac\\pi3(1+a)\\big)\\cos\\tfrac\\pi3(1+a)s+\\tfrac16\\big(1+2\\cos\\tfrac\\pi3(1-a)\\big) \\cos\\tfrac\\pi3(1-a) s .\\]\nIterating this formula, we see that $h_n$ is a sum of $2^{n-1}$ many terms of the form\n\\[ A\\cos(\\pi a s) \\]\nwhere $A>0$ and $a=\\frac13\\pm\\frac19\\pm\\frac1{27}\\pm\\cdots\\pm \\frac{1}{3^n}\\in (0,\\tfrac12)$.\nIt follows that\n\\[ r_n=h_n(0),\\quad R_n=h_n(\\tfrac12).\\]\nBy \\eqref{L:rR},\n\\[ \\left(h_n(0)\\right)^{1\/n}\\le c(f)\\le \\left(h_n(\\tfrac12)\\right)^{1\/n} .\\]\nFor example, if $n=1$ we get the bounds\n\\[ \\frac13\\sqrt3\\le c(f)\\le \\frac23.\\]\n\nSince $0<a<\\frac12$, we have\n\\[ R_n\\le \\sqrt{2} r_n. \\]\nTherefore,\n\\[ (R_n)^{1\/n} \\le 2^{1\/(2n)} (r_n)^{1\/n} .\\]\nIn agreement with Theorem \\ref{A:t1}, we get\n\\[ r=R=c(f) .\\]\nWe also find that\n\\[ (R_n)^{1\/n}-(r_n)^{1\/n}\\le (R_n)^{1\/n}(1-2^{-1\/(2n)})\\le \\tfrac23 (1-2^{-1\/(2n)}) .\\]\nFor example, if $n=10$, then  $c(f)$ is enclosed in the interval $[(r_n)^{1\/n},(R_n)^{1\/n}]$ of length at most $\\frac23(1-2^{-1\/20})=0.0277\\cdots$ (the actual length is\n$0.008390\\cdots$.)\n\nBy numerical computation, we get the following bounds:\n\\begin{center}\n\\begin{tabular}{r|r|r}\n$n$ & $(h_n(0))^{1\/n}$ & $(h_n(\\tfrac12))^{1\/n}$ \\\\\n\\hline\n1 & 0.577350 & 0.666666  \\\\\n2 & 0.615672 & 0.656538  \\\\\n3 & 0.626102 & 0.653844  \\\\\n4 & 0.631603 & 0.652453  \\\\\n5 & 0.634908 & 0.651623   \\\\\n10& 0.641576 & 0.649967  \\\\\n15& 0.643815 & 0.649415\n\\end{tabular}\n\\end{center}\n\n\\subsection{Properties of $h_n(t)$ when $f(t)=|\\sin(\\pi t)|^q$}\nIf we use $r_n^{1\/n}$ and $R_n^{1\/n}$ to bound $c(f)$, we are faced with the problem to compute the maximum and minimum values of the function $h_n$.\nTherefore, it is of interest to discuss the behavior of the function $h_n$.\nConsider\n\\[ f(t)=|\\sin(\\pi t)|^q, \\quad d\\ge 2.\\]\nThen we have\n\\[ h_1(t)=\\frac1d \\sum_{i=0}^{d-1} \\left|\\sin\\Big(\\pi \\frac{t+i}{d}\\Big)\\right|^q,\\quad 0\\le t\\le 1. \\]\nNote that $h_1(t)=h_1(1-t)$.\nFor this function we have the following result.\n\n\\begin{lemma}\\label{S:l1}\n(a) If $q=2,4,6,\\cdots,2(d-1)$, then \n$$h_1(t)\\equiv \\frac{1}{2^q}\\binom{q}{q\/2}.$$\\\\\n(b) Define the intervals \n\\[ Q_k=\\begin{cases} \\big(2(k-1),2k\\big) & \\text{if $k=1,\\cdots,d-1$,}\\\\ \n\\big(2(d-1),\\infty\\big) & \\text{if $k=d$.}\\end{cases}\n\\]\nThen for $q\\in Q_k$ we have\n$$(-1)^{k-1}h_1'(t)>0,\\quad 0<t<1\/2.$$\n\\end{lemma}\n\n\\begin{proof}\n(a) follows from the matrix representation of $T^\\ast$, the adjoint of the matrix \\eqref{T:matrix}.\\\\\n(b) We differentiate $h_1(t)$ to get\n\\begin{equation}\\label{diff}\n \\frac{d^2}{\\pi q} h_1'(t)=\\sum_{j=1}^d s_j^q c_j,\\quad 0<t<1,\n \\end{equation}\nwhere\n\\[ s_j=\\sin\\left(\\pi\\frac{t+j-1}{d}\\right),\\quad c_j=\\cot\\left(\\pi\\frac{t+j-1}{d}\\right) .\\]\nBy (a), the left-hand side of \\eqref{diff} is zero for $q=2,4,\\cdots,2(d-1)$.\nTherefore, we obtain a linear system $V c= b$, where $V$ is a $d\\times d$ generalized Vandermonde matrix with entries $s_j^{q_i}$ in the $i$th row and $j$th column, where $q_1=q$ and $q_i=2(i-1)$ for $i=2,3,\\cdots,d$. The column vector $c$ has components $c_i$, and the column vector $b$\nhas first component $\\frac{d^2}{\\pi q} h_1'(t)$ and all other components equal to $0$.\nSuppose that $0<t<\\frac12$ and $q\\in Q_k$ for some $k=1,2,\\cdots,d$.\nThen both $s_1,s_2,\\cdots,s_d$ and $q_1,q_2,\\cdots,q_d$ are mutually distinct. It is known (cf.\\cite[page~76]{GK}) that\nthis implies that $\\det V\\ne 0$, and  $\\det V>0$ if $s_1<s_2<\\cdots<s_d$ and $q_1<q_2<\\cdots<q_d$.\nIf we solve the linear system $Vc=b$ for $c$ by Cramer's rule we find that $c_1$ has the same sign as $(-1)^{k-1} h_1'(t)$.\nSince $c_1>0$, we have $(-1)^{k-1}h_1'(t)>0$ when $0<t<\\frac12$ and $q\\in Q_k$.\n\\end{proof}\n\nUsing the bounds $r_1\\le c(f)\\le R_1$ together with Lemma \\ref{S:l1}, one can deduce for $d=3$\n$$\\begin{cases}\n\\frac23\\left(\\frac{\\sqrt3}{2}\\right)^q \\le  c(q) \\le  \\frac13+\\frac23\\left(\\frac12\\right)^q&\\quad \\text{if $0<q\\le 2$ or $q\\ge 4$},\\\\\n\\frac13+\\frac23\\left(\\frac12\\right)^q\\le  c(q) \\le \\frac23\\left(\\frac{\\sqrt3}{2}\\right)^q&\\quad \\text{if $2\\le q\\le 4$}.\n\\end{cases}$$\nNote also that Lemma \\ref{S:l1}(a) implies\n$$c(q)=\\tilde c(q)= \\frac{1}{2^q}\\binom{q}{q\/2}\\quad \\text{when } q=2,4,6,\\cdots,2(d-1).$$\n\nWe would like to extend Lemma \\ref{S:l1} to $h_2, h_3,\\cdots$.\nBased on computer experiments we conjecture the following.\n\n\\begin{conj}\\label{S:conj1}\nLemma \\ref{S:l1}(b) is true for every $h_n$, $n\\in\\mathbb{N}$.\n\\end{conj}\n\nWe obtain sharper lower and upper bounds for $c(f)$ when we\nchoose $w(t)=h_n(t)$ in \\eqref{L:est2}. More precisely, we get\n\\begin{equation}\\label{S:q3}\n\\min_{t\\in[0,1]} \\frac{h_{n+1}(t)}{h_n(t)}\\le c(f)\\le \\max_{t\\in[0,1]} \\frac{h_{n+1}(t)}{h_n(t)} .\n\\end{equation}\nHere we are faced with the problem\nto determine the extrema of the quotients $\\frac{h_{n+1}(t)}{h_n(t)}$.\nComputer calculations suggest the following.\n\n\\begin{conj}\\label{S:conj2}\nLet $d=3$. If $0<q<2$ and $n$ is odd, then $\\frac{h_{n+1}(t)}{h_n(t)}$ attains its maximum at $t=0$ and its minimum at $t=\\frac12$.\nIf $0<q<2$ and $n$ is even, then $\\frac{h_{n+1}(t)}{h_n(t)}$ attains its maximum at $t=\\frac12$ and its minimum at $t=0$.\nIf $2<q<4$, then $\\frac{h_{n+1}(t)}{h_n(t)}$ attains its maximum at $t=0$  and its minimum at $t=\\frac12$.\nIf $q>4$, then $\\frac{h_{n+1}(t)}{h_n(t)}$ attains its maximum at $t=\\frac12$  and its minimum at $t=0$.\n\\end{conj}\n\nIf we believe these conjectures then $c(f)$ would lie between $\\frac{r_{n+1}}{r_n}$ and $\\frac{R_{n+1}}{R_n}$ for every $n$.\nIn the case $d=3$ we get the following estimates for $c(1)$:\n\n\\begin{center}\n\\begin{tabular}{r|r|r}\n$n$ & lower bound & upper bound \\\\\n\\hline\n1 & 0.577350 & 0.666666  \\\\\n2 & 0.646564 & 0.656538  \\\\\n3 & 0.648297 & 0.648396\n\\end{tabular}\n\\end{center}\n\nWe see that these bounds are much better than those from Section \\ref{subsec:d=3}. Unfortunately, we used conjecture \\ref{S:conj2} but for small $n$ it can be proved by direct computation.\n\n\\section{The case $f(t)=|\\cos(\\pi t)|^q$, $d=2$}\\label{sec:binary}\n\nIn the exceptional case\n\\[ f(t)=|\\cos(\\pi t)|^q, \\quad d=2\\]\nwe can obtain more explicit computations. See also \\cite{FanLau1998} for related results.\n\n\\subsection{The integrals $I_n$}\nUsing the identity\n\\[\n\\prod_{j=0}^{n-1} \\cos(2^j t)=\\frac{\\sin(2^n t)}{2^n \\sin(t)} ,\n\\]\nwe can write\n\\begin{equation}\\label{1:fm}\nf_n(t)=\\frac{1}{2^{q n}} \\frac{|\\sin(\\pi 2^n t)|^q}{|\\sin(\\pi t)|^q}.\n\\end{equation}\nTherefore,\n\\begin{equation}\\label{1:fm-integral}\nI_n=\\frac{1}{2^{q n}} \\int_0^1 \\frac{|\\sin(\\pi 2^n t)|^q}{|\\sin(\\pi t)|^q}\\, dt=\\frac{1}{2^{q n-1}} \\int_0^{1\/2} \\frac{|\\sin(\\pi 2^n t)|^q}{|\\sin(\\pi t)|^q}\\, dt.\n\\end{equation}\n\n\\begin{thm}\\label{1:t}\nFor $d=2$, we have\n\\[ c(q)=\\lim_{n\\to\\infty} I_n^{1\/n}=\n\\begin{cases} \n2^{-q} & \\text{if $0<q\\le 1$,}\\\\\n\\frac12 & \\text{if $q>1$.}\n\\end{cases}\n\\]\n\\end{thm}\n\\begin{proof}\nSubstituting $u=2^n t$ in \\eqref{1:fm-integral}, we get\n\\[ I_n=\\frac{1}{2^{n-1}} \\int_0^{2^{n-1}} \\frac{|\\sin(\\pi u)|^q}{2^{q n}|\\sin(\\pi 2^{-n} u)|^q}\\,du .\\]\nUsing\n\\[ \\frac{2}{\\pi} t\\le \\sin t\\le t,\\quad 0\\le t\\le \\frac{\\pi}{2},\\]\nwe find\n\\begin{equation}\\label{1:ineq}\n\\frac{\\pi^{-q}}{2^{n-1}} \\int_0^{2^{n-1}} \\frac{|\\sin(\\pi  u)|^q}{u^q}\\, du\\le I_n \\le \\frac{2^{-q}}{2^{n-1}} \\int_0^{2^{n-1}} \\frac{|\\sin(\\pi  u)|^q}{u^q}\\, du  .\n\\end{equation}\nIf $q>1$, the integral\n\\begin{equation}\\label{1:int}\n\\int_0^\\infty \\frac{|\\sin(\\pi  u)|^q}{u^q}\\,du\n\\end{equation}\nconverges. Therefore, the statement of the theorem follows for $q>1$.\nIf $q=1$, the integral \\eqref{1:int} diverges and the integrals in \\eqref{1:ineq} behave like $\\ln(2^n)$. Since $n^{1\/n}$ converges to $1$ as $n\\to\\infty$,\nwe obtain the statement of the theorem when $q=1$. If $0<q<1$, the integrals in \\eqref{1:ineq}  behave like $2^{n(1-q)}$ which implies the statement of the theorem\nfor $0<q<1$.\n\\end{proof}\n\n\\subsection{Spectral radius} \nBy \\eqref{L:Ln}, we have\n\\[ h_n(t)=\\frac{1}{2^{n}}\\sum_{k=0}^{2^n-1} f_n\\left (\\frac{t+k}{2^{n}}\\right ) .\\]\nCombining with \\eqref{1:fm}, we get\n\\begin{equation}\\label{2:hm}\nh_n(t)=\\frac{|\\sin(\\pi t)|^q}{2^n 2^{q n}} \\sum_{k=0}^{2^n-1} \\frac{1}{|\\sin(\\pi2^{-n}(t+k))|^q}.\n\\end{equation}\nBy estimating the sum in \\eqref{2:hm} we obtain the following.\n\n\\begin{thm}\\label{2:t}\nFor $f(t)=|\\cos(\\pi t)|^q$, $d=2$, we have\n\\[ r=\\frac12 \\quad  \\text{for all $q>0$}\\]\nand\n\\[ R=\\begin{cases}\n2^{-q} & \\text{if $0<q\\le 1$,}\\\\\n\\frac12 & \\text{if $q>1$.}\n\\end{cases}\n\\]\n\\end{thm}\n\\begin{proof}\nUsing only the term with $k=0$ in \\eqref{2:hm}, we obtain, for $0\\le t\\le \\frac12$,\n\\[ h_n(t)\\ge \\frac{1}{2^n}\\frac{1}{2^{q n}}\\frac{|\\sin(\\pi t)|^q}{|\\sin(\\pi 2^{-n}t)|^q}\\ge \\frac{1}{2^n}\\frac{2^q}{\\pi ^q}.\\]\nThis inequality together with $h_n(0)=2^{-n}$ proves $r=\\frac12$.\nThe proof of the formula for $R$ is elaborated in Section \\ref{subsec:convergence}.\n\\end{proof}\n\n\\subsection{Eigenfunctions}\\label{subsec:eigenfunctions}\nLet $\\alpha=\\min\\{1,q\\}$.\nBy Theorem \\ref{L:t1}, the spectral radii of $L$ and $L_\\alpha$ agree, and $L_\\alpha$ is quasicompact.\nSince $L$ is also a positive operator, $\\lambda=R$ must be an eigenvalue, so there must exist a corresponding eigenfunction.\nBut $L$ is not a Krein operator (cf. \\cite{A}), so we do not know whether the eigenfunction is unique (up to a constant factor) \nor whether it is positive on $\\mathbb{T}$.\n\nWe want to find nontrivial solutions $u\\in C(\\mathbb{T})$ to the equation  $Lu=\\lambda u$, particularly for $\\lambda=R$.\nInterestingly, we can find these eigenfunctions fairly explicitly.\nIn fact, if we substitute\n\\[ u(t)=|\\sin(\\pi t)|^q g(t) \\]\nin $Lu=\\lambda u$, we find\n\\begin{equation}\\label{3:ber}\n\\tfrac12 g\\left(\\tfrac12 t\\right)+\\tfrac12 g\\left(\\tfrac12 (t+1)\\right) =\\mu g(t)\\quad \\text{where }\\mu=2^q \\lambda .\n\\end{equation}\nNote that $g(t)$ will usually be continuous only on the open interval $(0,1)$.\nMuch is known about equation \\eqref{3:ber} (cf. \\cite{V}).\nClearly, $g(t)=1$ is a solution to \\eqref{3:ber} with $\\mu=1$. Therefore, $u(t)=|\\sin(\\pi t)|^q$ is an eigenfunction of $L$ corresponding\nto the eigenvalue $\\lambda=2^{-q}$. If $0<q\\le 1$, then this is an eigenfunction corresponding to the spectral radius eigenvalue $R$.\nFurthermore, $g(t)=B_n(t)$ with $B_n(t)$ denoting a Bernoulli polynomial, is also a solution to \\eqref{3:ber} corresponding to $\\mu=2^{-n}$.\nThis gives us many more eigenfunctions of $L$, but they do not give us eigenfunctions corresponding to the eigenvalue $\\lambda=R$ if $q>1$.\n\nUsing an idea from \\cite{V}, we find eigenfunctions corresponding to $\\lambda=R$ when $q>1$.\nFor $s>1$, consider the Hurwitz zeta function\n\\begin{equation}\\label{3:g}\n\\zeta(s,t)=\\sum_{k=0}^\\infty \\frac{1}{(t+k)^s},\\quad 0<t<1.\n\\end{equation}\nIt is easy to check that $g(t)=\\zeta(s,t)$ is a solution to \\eqref{3:ber} with $\\mu =2^{s-1}$.\nIf we let \n\\begin{equation}\\label{3:Gst}\nG(t)=G(s,t):=g(t)+g(1-t),\\quad 0<t<1.\n\\end{equation}\nThen $G(t)$ is also a solution to \\eqref{3:ber} and has symmetry $G(t)=G(1-t)$. If $s\\le q$, \nthen\n\\[u(t)=|\\sin(\\pi t)|^q G(t)\\]\nis a continuous eigenfunction of $L$ corresponding to the eigenvalue $\\lambda=2^{s-q-1}$.\nIn particular, choosing $s=q$, we obtain an eigenfunction corresponding to the eigenvalue $R=\\frac12$.\n\nSuppose $q\\ge 2$ is an even integer. Let $s=2,3,\\cdots,q$.\nConsider \n\\[ \\widetilde G(t)=\\zeta(s,t)+(-1)^s\\zeta(s,1-t)=\\sum_{k=-\\infty}^\\infty \\frac{1}{(t+k)^s}. \\]\nSince\n\\[ \\pi\\cot(\\pi t)=\\lim_{N\\to\\infty} \\sum_{k=-N}^N \\frac1{t+k}, \\]\nwe obtain\n\\begin{equation}\\label{3:G}\n\\widetilde G(t)=\\frac{(-1)^{s-1}\\pi}{(s-1)!}\\left(\\frac{d}{dt}\\right)^{s-1} \\cot(\\pi t) ,\n\\end{equation}\nand correspondingly the eigenfunctions \n\\[ u(t)=|\\sin(\\pi t)|^q \\widetilde G(t) .\\]\nObviously, these eigenfunctions are trigonometric polynomials. For example, if $q=s=2$, we obtain the eigenfunction\n\\[ u(t)=|\\sin(\\pi t)|^2 \\widetilde G(t)=|\\sin(\\pi t)|^2 G(t)= \\pi^2 ,\\]\nand, if $q=s=4$, then\n\\begin{equation}\\label{3:q4}\nu(t)=\\frac{\\pi^4}{3}\\big(\\cos(2\\pi t)+2\\big) .\n\\end{equation}\n\nWe can also find these eigenfunctions in a different way.\nThe space of trigonometric polynomials $\\sum_{k=-K}^K c_k e^{2\\pi i kt}$ with $K=(q-2)\/2$ is an invariant subspace of $L$.\nThe matrix representation of the restriction of $L$ to this invariant subspace with respect to the basis $\\{e^{2\\pi i kt}\\}$ is\n\\[ B=(a_{2\\ell-k})_{-K\\le k, \\ell\\le K},\\]\nwhere the $a_k$ denote the Fourier coefficients in\n\\[ |\\cos(\\pi t)|^q= \\sum_{k=-K}^K a_k e^{2\\pi ikt}.\\]\nFor example, if $q=4$ then\n\\[ B=\\left[\\begin{array}{rrr} \n\\frac14 & \\frac{1}{16} & 0 \\\\[0.3em]\n\\frac14 & \\frac38 & \\frac14 \\\\[0.3em]\n0 & \\frac1{16} & \\frac14 \\\\\n\\end{array}\\right]\n\\]\nThis matrix is nonnegative and primitive. Its largest eigenvalue is $\\frac12$, which follows from the fact that the column sums are all equal to $\\frac12$.\nAn eigenvector corresponding to the eigenvalue $\\lambda=\\frac12$ is $[1,4,1]^T$.\nTherefore,\n\\[ u(t)=\\cos(2\\pi t)+2\\]\nis an eigenfunction of $L$ for $q=4$ corresponding to the eigenvalue $\\lambda=\\frac12$.\nApart from a constant factor, this is the same eigenfunction we found in \\eqref{3:q4}.\nThis is true in general. The eigenfunctions of the form $u(t)=|\\sin(\\pi t)|^q \\widetilde G(t)$ with $\\widetilde G$ from \\eqref{3:G} with $s=2,3,\\cdots,q$ match the eigenfunctions obtained from the matrix $B$. In particular, we see that the matrix $B$ has eigenvalues $\\frac12, \\frac14,\\cdots,2^{-q+1}$.\n\nIn Section \\ref{subsec:h_n-properties}, we study in more detail the trigonometric polynomials obtained from \\eqref{3:G}.\n\n\\subsection{Convergence of $h_n(t)$}\\label{subsec:convergence}\nThe following proposition shows that, after appropriately normalized, the function $h_n$ converges to the eigenfunctions we found in Section \\ref{subsec:eigenfunctions} corresponding to $\\lambda=R$.\n\n\\begin{prop}\\label{prop:convergence} The following limits hold in $C^q(\\mathbb{T})$.\\\\\n(a) If $0<q<1$, then\n$$\\lim_{n\\rightarrow\\infty} 2^{q n} h_n(t)=\\frac{\\Gamma(\\frac{1}{2}-\\frac{q}{2})}{\\sqrt{\\pi}\\Gamma(1-\\frac{q}{2})}|\\sin(\\pi t)|^q.$$\n(b) If $q=1$, then\n$$\\lim_{n\\rightarrow\\infty} \\frac{2^{n}}{n} h_n(t)=\\frac{2\\ln 2}{\\pi}|\\sin(\\pi t)|.$$\n(c) If $1<q<\\infty$, then\n$$\\lim_{n\\rightarrow\\infty} 2^{n} h_n(t)=\\pi^{-q}|\\sin(\\pi t)|^q G(q,t).$$\nwhere $G(q,t)$ is given by \\eqref{3:Gst}.\n\\end{prop}\n\n\\begin{proof}\nWe show here the pointwise convergence of $h_n$. The norm convergence can be shown by slight refinements of the argument.\\\\\n(a) By \\eqref{2:hm}, we have\n$$h_n(t)=\\frac{|\\sin(\\pi t)|^q}{2^n 2^{q n}} \\sum_{k=0}^{2^n-1} \\frac{1}{|\\sin(\\pi2^{-n}(t+k))|^q}.$$\nSince \n$$\\int_0^1 \\frac{1}{|\\sin(\\pi x)|^q}dx=\\frac{\\Gamma(\\frac{1}{2}-\\frac{q}{2})}{\\sqrt{\\pi}\\Gamma(1-\\frac{q}{2})},$$\nto prove the statement it suffices to show that\n$$\\lim_{n\\rightarrow\\infty}\\frac{1}{2^{n}}\\sum_{k=0}^{2^n-1} \\frac{1}{|\\sin(\\pi 2^{-n}(t+k))|^q}=\\int_0^1 \\frac{1}{|\\sin(\\pi x)|^q}dx.$$\nHowever, this follows easily by treating the left-hand side as a Riemann sum, using the monotonicity of the integrand and the assumption that $q<1$.\n\n\\noindent(c) Similar as in the proof of (a), we only need to show that\n$$\\lim_{n\\rightarrow\\infty}\\frac{1}{2^{q n}}\\sum_{k=0}^{2^n-1} \\frac{1}{|\\sin(\\pi 2^{-n}(t+k))|^q}=\\pi^{-q} G(q,t).$$\nBy symmetry, this reduces to showing\n$$\\lim_{n\\rightarrow\\infty}\\frac{1}{2^{q n}}\\sum_{k=0}^{2^{n-1}-1} \\frac{1}{|\\sin(\\pi 2^{-n}(t+k))|^q}=\\pi^{-q}\\sum_{k=0}^{\\infty} \\frac{1}{(t+k)^q}.$$\nHowever, this follows easily from the basic limit\n$$\\lim_{x\\rightarrow 0}\\frac{\\sin x}{x}=1$$\nand the fact that the series on the right-hand side is convergent.\n\n\\noindent(b) By symmetry, it suffices to show that\n$$\\lim_{n\\rightarrow\\infty}\\frac{1}{n 2^{n}}\\sum_{k=0}^{2^{n-1}-1} \\frac{1}{\\sin(\\pi 2^{-n}(t+k))}=\\frac{\\ln 2}{\\pi}.$$\nTo this end, for any given $\\varepsilon>0$ we fix $\\delta>0$ such that\n$$\\left |\\frac{x}{\\sin x}-1\\right |<\\varepsilon,\\quad 0<x<\\delta.$$\nWe can then write\n\\begin{align*}\n&\\sum_{k=0}^{2^{n-1}-1} \\frac{1}{\\sin(\\pi 2^{-n}(t+k))}\\\\\n=&\\sum_{k:\\,\\pi 2^{-n}(t+k)<\\delta}\\frac{\\pi 2^{-n}(t+k)}{\\sin(\\pi 2^{-n}(t+k))}\\frac{1}{\\pi 2^{-n}(t+k)}\\\\\n&+\\sum_{k:\\,\\pi 2^{-n}(t+k)\\ge\\delta}\\frac{1}{\\sin(\\pi 2^{-n}(t+k))}\\\\\n=&I+II.\n\\end{align*}\nBy our choice of $\\delta$,\n\\begin{align*}\nI\n&=(1+O(\\varepsilon))\\sum_{k:\\,\\pi 2^{-n}(t+k)<\\delta} \\frac{1}{\\pi 2^{-n}(t+k)}\\\\\n&=\\frac{2^n}{\\pi}(1+O(\\varepsilon))\\sum_{k:\\,\\pi 2^{-n}(t+k)<\\delta} \\frac{1}{t+k}\\\\\n&=\\frac{2^n}{\\pi}(1+O(\\varepsilon))(1+o(1))\\ln (2^n)\\\\\n&=n 2^n(1+O(\\varepsilon))\\frac{\\ln 2}{\\pi},\\quad \\text{as } n\\to\\infty.\n\\end{align*}\nOn the other hand, using\n$$\\sin x\\ge \\frac{2}{\\pi}x,\\quad 0\\le x\\le\\pi\/2,$$\nwe have\n\\begin{align*}\nII\n&=O(2^n) \\sum_{k:\\,\\pi 2^{-n}(t+k)\\ge\\delta}\\frac{1}{t+k}\\\\\n&=O(2^n) \\ln\\Big(\\frac{1}{\\delta}\\Big)\\\\\n&=o(n 2^n),\\quad \\text{as } n\\to\\infty.\n\\end{align*}\nCombining these, we get\n$$\\frac{1}{n 2^n}(I+II)=(1+O(\\varepsilon))\\frac{\\ln 2}{\\pi},\\quad \\text{as } n\\to\\infty.$$\nSince $\\varepsilon$ is arbitrary, this completes the proof.\n\\end{proof}\n\n\\subsection{Properties of $h_n(t)$}\\label{subsec:h_n-properties} It turns out that the functions $h_n, n\\in\\mathbb{N}$ share some common geometric properties. We were able to prove some of them.\n\n\\begin{prop}\\label{prop:convexity}\n(a) If $0<q\\le 1$, then $h_n''(t)<0$ for all $n\\in\\mathbb{N}$ and $t\\in (0,1)$.\\\\\n(b) If $1<q<2$, then $h_n''(1\/2)<0$ for $n=1,\\cdots,N(q)$, where $N(q)$ satisfies $\\lim_{q\\rightarrow 1^+}N(q)=\\infty$.\\\\\n(c) If $2<q<\\infty$, then $h_n''(1\/2)>0$ for $n=1,\\cdots,N(q)$, where $N(q)$ satisfies $\\lim_{q\\rightarrow\\infty}N(q)=\\infty$.\n\\end{prop}\n\\begin{proof}\n\\noindent(a) In the case $n=0$, $h_0(t)\\equiv 1$, so the statement obviously holds with strict inequality replaced by equality. Assume that $h_{n-1}''(t)\\le 0$ for all $t\\in (0,1)$. We now show that $h_n''(t)<0$ for all $t\\in (0,1)$.\n\nBy definition, we have\n$$h_n(t)=\\frac{1}{2}\\Big|\\cos\\Big(\\pi \\frac{t}{2}\\Big)\\Big|^q h_{n-1}\\Big(\\frac{t}{2}\\Big)+\\frac{1}{2}\\Big|\\cos\\Big(\\pi \\frac{t+1}{2}\\Big)\\Big|^q h_{n-1}\\Big(\\frac{t+1}{2}\\Big).$$\nSince the second term equals the first term after the change of variable $t\\rightarrow 1-t$, it suffices to show $(fg)''(t)<0$ for all $t\\in (0,1\/2)$, where\n$$f(t)=|\\cos(\\pi t)|^q,\\quad g(t)=h_{n-1}(t).$$\nHowever, by the product rule,\n$$(fg)''=f''g + 2 f'g' + fg''.$$\nSince $q\\le 1$, we have $f''(t)<0$ for all $t\\in (0,1\/2)$. Also, by symmetry we have $g'(1\/2)=0$, and so the induction hypothesis implies  $g'(t)\\ge 0$ for all  $t\\in (0,1\/2)$. Combining these we get $f''g<0$, $f'g'\\le 0$, and $fg''\\le 0$, which gives $(fg)''(t)<0$ for all $t\\in (0,1\/2)$. This completes the proof by induction.\n\n\\noindent(b) The proof is similar to that of (a). Using the same notation, we observe that\n$$f''(t)=\\pi^2 q |\\cos(\\pi t)|^q \\Big(q|\\sin(\\pi t)|^2-1\\Big),\\quad 0<t<1\/2$$\nnow changes sign at\n$$t=t_q:=\\pi^{-1} \\arcsin(\\sqrt{1\/q}).$$\nNotice that $t_q>1\/4$ if $q<2$, and $t_q<1\/4$ if $q>2$; moreover,\n$$\\lim_{q\\rightarrow 1^+} t_q = 1\/2,\\quad\n\\lim_{q\\rightarrow \\infty} t_q = 0.$$\nIn order to determine the sign of\n\\begin{equation}\\label{eqn:2nd-derivative}\n(fg)''=f''g + 2 f'g' + fg'',\n\\end{equation}\nas before we want all the three terms to have the same sign.\n\nIn the case $n=1$, since $g\\equiv 1$, we have $(fg)''(t)=f''(t)<0$ for all $t\\in(0,2t_q)$, where $2t_q>1\/2$ and $2t_q\\rightarrow 1$ as $q\\rightarrow 1^+$. This implies $h_1''(t)<0$ for all $t\\in (1-2t_q,2t_q)$. By symmetry we have $h_1'(t)>0$ for all $t\\in (1-2t_q,1\/2)$. Now proceeding by induction, we see that, using \\eqref{eqn:2nd-derivative},\n$$h_n''(t)<0,\\quad t\\in (2^{n-1}(1-2t_q),1-2^{n-1}(1-2t_q))$$\nand\n$$h_n'(t)>0,\\quad t\\in (2^{n-1}(1-2t_q),1\/2).$$\nIn particular, if $q>1$ is sufficiently close to 1, we have $2^{n-1}(1-2t_q)<1\/2$ and thus $h_n''(1\/2)<0$, as desired.\n\nThe proof for (c) is similar.\n\\end{proof}\n\nWhen $q$ is an even integer, we can have more information.\n\n\\begin{prop}\nIf $q\\ge 4$ is an even integer, then\n$$h_\\infty(t):=\\pi^{-q}|\\sin(\\pi t)|^q G(q,t)$$\nsatisfies $h_\\infty''(1\/2)>0$; moreover, $h_\\infty'(t)<0$ for all $t\\in(0,1\/2)$.\n\\end{prop}\n\n\\begin{proof}\nBy \\eqref{3:G}, we have\n$$G(q,t)=\\frac{(-1)^{q-1}\\pi}{(q-1)!}\n\\left (\\frac{d}{dt}\\right )^{q-1}\\cot (\\pi t).$$\nLemma \\ref{lem:polynomial} below shows that, after simplification,\n$$h_\\infty(t)=\\frac{2}{(q-1)!} P_{q-1}(\\cos(\\pi t))$$\nwhere $P_{q-1}(x)$ is a polynomial consisting of the even powers $1, x^2, \\cdots, x^{q-2}$ and has positive coefficients. By direct computation, we then have\n$$h_\\infty'(t)=-\\frac{2\\pi}{(q-1)!}P_{q-1}'(\\cos(\\pi t)) \\sin(\\pi t)$$\nand\n$$h_\\infty''(1\/2)=\\frac{2\\pi^2}{(q-1)!} P_{q-1}''(0).$$\nThe desired conclusions now follow immediately from the properties of $P_{q-1}(x)$ mentioned above.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:polynomial}\nFor all $n\\in\\mathbb{N}$, we have\n$$\\left (\\frac{d}{dt}\\right )^{n}\\cot t=(-1)^n\\frac{P_n(\\cos t)}{(\\sin t)^{n+1}}$$\nwhere $P_n(x)$ is a polynomial of degree $n-1$ whose coefficients are nonnegative integers. Moreover, when $n$ is odd, $P_n(x)$ consists of the even powers $1,x^2,\\cdots,x^{n-1}$; when $n$ is $even$, $P_n(x)$ consists of the odd powers $x,x^3,\\cdots,x^{n-1}$.\n\\end{lemma}\n\n\\begin{proof}\nIt is easy to see that $P_0(x)=x$ and $P_1(x)=1$. Moreover, by direct computation we have\n$$P_{n+1}(x)=(n+1)xP_{n}(x)+(1-x^2)P_{n}'(x).$$\nSuppose the statement holds for $P_n(x)$, i.e.\n$$P_n(x)=a_{n-1}x^{n-1}+\\sum_{j=0}^{n-2} a_j x^j$$\nwhere $a_{n-1}$ is a positive integer and the $a_j$'s ($j\\le n-2$) are nonnegative integers. Then\n\\begin{equation}\\label{eqn:coefficients}\nP_{n+1}(x)=2 a_{n-1} x^{n} + \\sum_{j=0}^{n-2} (n-j+1) a_j x^{j+1}+\\sum_{j=0}^{n-1} j a_j x^{j-1}.\n\\end{equation}\nTherefore $P_{n+1}(x)$ is a polynomial of degree $n$ whose coefficients are nonnegative integers. By induction, this completes the proof of the first part of the lemma.\n\nThe fact that $P_n(x)$ consists of either the even powers $1,x^2,\\cdots,x^{n-1}$ or the odd powers powers $x,x^3,\\cdots,x^{n-1}$ (depending on whether $n$ is odd or even) follows easily from the recursion formula \\eqref{eqn:coefficients} and induction.\n\\end{proof}\n\nWe believe that the $N(q)$'s the Proposition \\ref{prop:convexity} should not be present, but we have not been able to remove them. By examining $h_\\infty''(1\/2)$ in its dependence on $q$, we make the following conjecture, where $\\zeta(s)$ denotes the Riemann zeta function.\n\n\\begin{conj}\nThe function\n$$F(s)=2(s+1)(2^{s+2}-1)\\zeta(s+2)-2\\pi^2 (2^s-1)\\zeta(s),\\ 1<s<\\infty.$$\nis strictly increasing. In particular $s=2$ is the unique zero of $F(s)$.\n\\end{conj}\n\n\\begin{figure}[h]\n\\includegraphics[scale=0.4]{Fs.png}\n\\caption{A graph of $F(s)$ for $1<s<4$.}\n\\end{figure}\n\n\\newpage\n\n\\section{$L^p$ space}\\label{sec:Lp}\nLet $1\\le p\\le\\infty$. We can also consider the transfer operator $L$ on the Lebesgue space $L^p(\\mathbb T)$. Here we consider only the case $f(t)=|\\cos(\\pi t)|^q$. Other cases can be treated similarly.\n\nLet $d\\ge 2$ be an integer. Let $L_q=L: L^p(\\mathbb T)\\rightarrow L^p(\\mathbb T)$ be the operator given by \n\\begin{align*}\n(L_q u)(t)\n=\\frac{1}{d}\\sum_{i=0}^{d-1}\n\\left |\\cos\\left (\\pi\\frac{t+i}{d}\\right )\\right |^q\nu\\left (\\frac{t+i}{d}\\right ).\n\\end{align*}\nThen $L_q$ defines a bounded linear operator on $L^p(\\mathbb T)$.\nThe adjoint of $L_q$ is given by \n$T_q=L_q^*: L^{p'}(\\mathbb T)\\rightarrow L^{p'}(\\mathbb T)$ (where $p'=p\/(1-p)$),\n\\begin{align*}\n(T_q x)(t)\n=|\\cos(\\pi t)|^q x(d\\cdot t).\n\\end{align*}\nNotice that, for $p'<\\infty$,\n\\begin{align*}\n\\|T_q x\\|_{p'}^{p'}\n&=\\int_{\\mathbb T} |\\cos(\\pi t)|^{q p'} |x(d\\cdot t)|^{p'}dt\\\\\n&=\\int_{\\mathbb T} (L_{q p'}1)(t) |x(t)|^{p'}dt.\n\\end{align*}\nThus\n\\begin{align*}\n\\|L_q\\|_{p\\rightarrow p}\n&=\\|T_q\\|_{p'\\rightarrow p'}\\\\\n&=\\left (\\sup_{\\|x\\|_{p'}\\le 1} \\|T_q x\\|_{p'}^{p'} \\right )^{1\/p'}\\\\\n&=\\left (\\sup_{\\||x|^{p'}\\|_{1}\\le 1}\n\\int_{\\mathbb T} (L_{q p'}1)(t) |x(t)|^{p'}dt \\right )^{1\/p'}\\\\\n&=\\|L_{q p'}1\\|_\\infty^{1\/p'}.\n\\end{align*}\nBy Lemma \\ref{S:l1}, the function $h_1(t)=(L_{q p'}1)(t)$ attains its maximum at either $t=0$ or $t=1\/2$ depending on the value of $q p'$. (Note that the function $h_1$ for $f=|\\cos(\\pi t)|^q$ and that for  $f=|\\sin (\\pi t)|^q$ differ only by a translation of $d\/2$.) In particular, we obtain an explicit formula for the operator norm\n$$\\|L_q\\|_{p\\rightarrow p}=\\max\\left \\{(L_{q p'}1)(0),(L_{q p'}1)(1\/2)\\right \\}.$$\nMore generally, for any $n\\in\\mathbb{N}$, the same argument as above gives\n$$\\|L_q^{n}\\|_{p\\rightarrow p}=\\|L_{q p'}^{n}1\\|_\\infty^{1\/p'}.$$\nTherefore, to compute the spectral radius of $L_q$ on $L^p(\\mathbb T)$, it suffices to find\n\\begin{align*}\n\\rho_p(L_q):=\\lim_{n\\rightarrow\\infty} \\|L_q^n\\|^{1\/n}_{p\\rightarrow p}\n&=\\lim_{n\\rightarrow\\infty}\\|L_{q p'}^{n}1\\|_\\infty^{1\/(np')}\\\\\n&=\\lim_{n\\rightarrow\\infty}\\|h_n\\|_\\infty^{1\/(np')}\n\\end{align*}\nwhere $h_n(t)=(L_{q p'}^{n}1)(t)$. However, the last expression is the $p'$th root of the spectral radius of $L_{q p'}$ on $C(\\mathbb T)$. So we obtain the following.\n\n\\begin{prop} For $p>1$, we have\n$$\\rho_p (L_q)=\\big[\\rho (L_{q p'})\\big]^{1\/p'}.$$\n\\end{prop}\n\nSimilar as in Section \\ref{sec:binary}, in the special case $d=2$, we can find eigenfunctions of $L_q$ in $L^p(\\mathbb T)$ explicitly. We consider two different cases.\n\nCase 1: $q p'\\le 1$.  In this case we have, by Theorem \\ref{2:t},\n$$\\rho (L_{q p'})=2^{-q p'},$$\nand so\n$$\\rho_p (L_q)=2^{-q}.$$\nSince $q\\le 1$, the spectral radius of $L_q$ on $L^p(\\mathbb T)$ coincides that on $C(\\mathbb T)$. In particular, we have the same eigenfunction\n$$u(t)=|\\sin(\\pi t)|^q\\in L^p(\\mathbb{T})$$\ncorresponding to the eigenvalue $\\lambda=2^{-q}$.\n\nCase 2: $q p'>1$.  In this case we have\n$$\\rho (L_{q p'})=\\frac12,$$\nand so\n$$\\rho_p (L_q)=2^{-1\/p'}.$$\nNote that $1\/p'<q$. Following the same idea as in Section \\ref{sec:binary}, we consider functions of the form\n$$u_s(t)=|\\sin(\\pi t)|^q G(s,t)$$\nwhere $s>1$ and $G(s,t)=\\zeta(s,t)+\\zeta(s,1-t)$ \n\\footnote{More generally, one can take $G(s,t)$ to be linear combinations of $\\zeta(s,t)$ and $\\zeta(s,1-t)$.}\nis as in \\eqref{3:Gst}. Since\n$$\\zeta(s,t)\\sim t^{-s},\\quad \\text{as }t\\rightarrow 0^+,$$\nwe have that $u_s\\in L^p(\\mathbb T)$ if and only if $(s-q)p<1$, i.e.\n$$s<q+\\frac{1}{p}.$$\nSince $q p'>1$ exactly when $q+\\frac{1}{p}>1$,\nwe can take\n$$s=q+\\frac{1}{p}-\\varepsilon$$\nfor sufficiently small $\\varepsilon>0$ to obtain an eigenfunction in $L^p(\\mathbb T)$ corresponding to the eigenvalue\n$$2^{-q+(s-1)}=2^{-1\/p'-\\varepsilon}.$$\nTherefore, as $\\varepsilon\\rightarrow 0$, $u_s(t)$ gives an `approximate' eigenfunction corresponding to $\\rho_p (L_q)=2^{-1\/p'}$.\nNote that when $\\varepsilon=0$, $u_s(t)$ gives an eigenfunction in the Lorentz space $L^{p,\\infty}(\\mathbb T)$.\n \n\\section{An application to Fourier multipliers}\\label{sec:application}\nIn this section, we present an application to some Bochner-Riesz type multipliers introduced by Mockenhaupt in \\cite[Section~4.3]{Mockenhaupt}. Let $E\\subset\\mathbb{R}$ be the middle-third Cantor set obtained from dissecting the interval $[-1\/2,1\/2]$, and let $\\mu$ be the Cantor measure on $E$. It is well known that\n$$\\dim E=\\alpha:=\\frac{\\log 2}{\\log 3}$$\nand that the Fourier transform of $\\mu$ is given by\n\\begin{align}\\label{fourier-cantor}\n\\hat{\\mu}(x)=\\int_{\\mathbb R} e^{-\\pi i x \\xi}d\\mu(\\xi)=\\prod_{j=1}^\\infty {\\cos(\\pi 3^{-j} x)}.\n\\end{align}\nLet $\\chi\\in C_c^\\infty(\\mathbb{R})$ be a bump function with $\\hat{\\chi}\\ge 0$. For $\\delta>0$, let\n\\begin{align*}\nm_\\delta(\\xi)\n=\\frac{\\chi(\\cdot)}{|\\cdot|^{\\alpha-\\delta}}*\\mu(\\xi)\n=\\int_\\mathbb{R} \\frac{\\chi(\\xi-\\eta)}{|\\xi-\\eta|^{\\alpha-\\delta}}d\\mu(\\eta).\n\\end{align*}\nNote that $m_\\delta$ defines a bounded function only when $\\delta>0$. In particular, $m_\\delta$ is an $L^2$-Fourier multiplier if and only if $\\delta>0$. \n\n\\begin{thm}\\label{multiplier}\n$m_\\delta$ is an $L^1$-Fourier multiplier if and only if\n$$\\delta>\\frac{\\log 2}{\\log 3}+\\frac{\\log c(1)}{\\log 3}=0.236\\cdots$$\nwhere $c(1)$ is as in Section \\ref{sec:special-weights} (with $d=3$).\n\\end{thm}\n\n\\begin{proof}\nRecall that an $L^p$-Fourier multiplier is a function $m(\\xi)$ such that\n\\begin{equation}\\label{eq:multiplier}\n\\|\\mathcal{F}^{-1}\\big(m(\\xi)\\hat f(\\xi)\\big)\\|_{L^p(\\mathbb{R})}\\le C \\|f\\|_{L^p(\\mathbb{R})}\n\\end{equation}\nholds for a constant $C$ independent of $f$, where $\\mathcal{F}^{-1}$ denotes the inverse Fourier transform. In the case $p=1$, this is equivalent to $\\widehat {m}$ being a finite measure. If $\\alpha-\\delta\\le 0$, it is easy to see that this is the case with $m=m_\\delta$. If $\\alpha-\\delta>0$, then we have\n$$\\widehat{m}_\\delta(x)=c\\cdot\\left (\\hat{\\chi}*{|\\cdot|^{\\alpha-\\delta-1}}\\right )(x)\\cdot \\hat{\\mu}(x)$$\nfor some constant $c$. Thus, ${m}_\\delta$ is an $L^1$-Fourier multiplier if and only if \n\\begin{align*}\n\\int_{\\mathbb{R}} |\\widehat m_\\delta(x)|dx\n&=\\int_{|x|\\le 3} |\\widehat m_\\delta(x)|dx + \\sum_{k=1}^\\infty \\int_{3^k<|x|\\le 3^{k+1}} |\\widehat m_\\delta(x)|dx\\\\\n&\\approx 1 + \\sum_{k=1}^\\infty 3^{(\\alpha-\\delta-1)k} \\int_{3^k}^{3^{k+1}} \\prod_{j=1}^\\infty |{\\cos(\\pi 3^{-j} x)}|dx\\\\\n&<\\infty\n\\end{align*}\nwhere we have used \n$$\\hat{\\chi}*{|\\cdot|^{\\alpha-\\delta-1}}(x)\\approx |x|^{\\alpha-\\delta-1},\\ \\text{as } |x|\\rightarrow\\infty$$\nand \\eqref{fourier-cantor}. On the other hand, notice that\n\\begin{align*}\n\\int_{3^k}^{3^{k+1}} \\prod_{j=1}^\\infty |{\\cos(\\pi 3^{-j} x)}|dx\n&=3^k \\int_{1}^{3} \\prod_{j=1}^\\infty |{\\cos(\\pi 3^{k-j} x)}|dx\\\\\n&=3^k \\int_{1}^{3} |\\hat \\mu(x)| \\prod_{j=0}^{k-1} |{\\cos(\\pi 3^{j} x)}| dx\\\\\n&\\approx 3^k \\int_{0}^{1} \\prod_{j=0}^{k-1} |{\\cos(\\pi 3^{j} x)}| dx\n\\end{align*}\nwhere in the last line we have used periodicity and the fact that $|\\hat\\mu(x)|$ is bounded below on the interval $[2,3]$. Now by Theorem \\ref{L:t4}(b), we know that\n$$\\int_{0}^{1} \\prod_{j=0}^{k-1} |{\\cos(\\pi 3^{j} x)}| dx\\approx c(1)^k.$$\nTherefore \n$$\\int_{\\mathbb{R}} |\\widehat m_\\delta(x)|dx<\\infty$$\nif and only if\n$$\\sum_{k=1}^\\infty 3^{(\\alpha-\\delta-1)k} 3^k c(1)^k<\\infty,$$\nwhich is equivalent to\n$$\\delta>\\frac{\\log 2}{\\log 3}+\\frac{\\log c(1)}{\\log 3}.$$\nThis completes the proof.\n\\end{proof}\n\nSince $m_\\delta$ is compactly supported, we can choose $f\\in L^p(\\mathbb{R})$ in \\eqref{eq:multiplier} such that $\\hat f\\equiv 1$ on the support of $m=m_\\delta$, and get $\\widehat m_\\delta\\in L^p(\\mathbb{R})$ as a necessary condition for $m_\\delta$ to be an $L^p$-Fourier multiplier. By the same argument as above, this leads us to\n$$\\delta>\\delta(p):=\\frac{\\log 2}{\\log 3}-1+\\frac{1}{p}+\\frac{\\log \\big(c(p)^{1\/p}\\big)}{\\log 3}.$$\n\n\\begin{figure}[h]\n\\includegraphics[scale=0.4]{deltap.png}\n\\caption{A graph of $\\delta(p)$ as a function of $1\/p\\in (0,1)$.}\n\\end{figure}\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe nature of dark matter and dark energy is one of the most important issues today in physics. There are strong observational evidences in astrophysics and cosmology for the existence of these two components of the cosmic energy budget, indicating that about $95\\%$ of the Universe is composed by dark matter (about $25\\%$) and by dark energy (about $70\\%$), but no direct detection has been reported until now. The usual candidates \\textbf{for} dark matter (neutralinos and axions, for example) and dark energy (cosmological constant, quintessence, etc.) lead to very robust scenarios, but at same time they must face theoretical and observational issues. For recent reviews on the subject, see for example \\cite{Padmanabhan:2002ji, Peebles:2002gy, Sahni:2004ai, Bertone:2004pz, Sahni:2006pa, Copeland:2006wr, Frieman:2008sn, Martin:2008qp, Caldwell:2009ix, Li:2011sd}.\n\nThe strongest issue is perhaps the one regarding dark energy as the vacuum expectation value of some quantum field, which would be a natural candidate, but whose correct theoretical value could be predicted only in the framework of a complete theory of quantum gravity, which still we do not possess. Nevertheless, it is possible, at least, to guess some of the features of this theory. In particular, the holographic principle \\cite{'tHooft:1993gx, Susskind:1994vu, Bousso:2002ju} may shed some light on the dark energy problem. According to this principle, in presence of gravity the number of the degrees of freedom of a local quantum system would be related to the area of its boundary, rather than to the volume of the system (as expected when gravity is absent). Following this \\textbf{idea}, in \\cite{Cohen:1998zx} the authors suggested an entanglement relation between the infrared and ultraviolet \\textbf{cutoffs} due to the limitation set by the formation of a black hole, which sets an upper bound for the vacuum energy. We can then interpret the ultraviolet cutoff as the vacuum density value, but still we need an ansatz for the infrared cutoff. As a candidate for such distance, in \\cite{Li:2004rb, Huang:2004ai} the authors propose and investigate the future event horizon, tested against type Ia supernovae data and cosmic microwave background anisotropies in \\cite{Zhang:2005hs, Zhang:2007sh}. We shall present more detail on this in Sec.~\\ref{Sec:HolDE}. \n\nAdding new components of dark energy to the whole energy budget in order to explain the current observation is a way, but not the only one. Since General Relativity has been thoroughly tested up to solar system scales, it may be possible that the Einstein-Hilbert action contain corrections on larger, cosmological, scales thereby candidating as possible explanation of the evolution of the universe. Such modifications should be, in principle, composed by higher order curvature invariant terms (such as $R^2$, $R_{\\mu\\nu}R^{\\mu\\nu}$, etc) but also by non-trivial coupling between matter or fields and geometry. See for example \\cite{Nojiri:2006ri, Nojiri:2010wj, Amendola:2006kh, Amendola:2006we, Capozziello:2007ec, Sotiriou:2008rp, DeFelice:2010aj} for some reviews on the subject (especially on $f(R)$ theory). It is also worth pointing out that these terms should naturally emerge as quantum corrections in the low energy effective action of quantum gravity or string theory \\cite{Buchbinder:1992rb}.\n\nIn this paper we connect these two approaches, considering a $f(R,T)$ theory of gravity, where $R$ is the Ricci scalar, whereas $T$ is the trace of the stress-energy momentum. This modified gravity theory has been recently introduced in \\cite{Harko:2011kv}, where the authors derived the field equations and considered several cases, relevant in cosmology and astrophysics. As for the former, $f(R,T)$ models have been constructed describing the transition from the matter dominated phase to the late times accelerated one \\cite{Houndjo:2011tu}. \n\nOur task here, is to find out which form the function $f(R,T)$ has to have in order to reproduce the same properties of the holographic dark energy proposed in \\cite{Li:2004rb}. To this purpose, we employ the same reconstruction scheme proposed and employed in \\cite{Capozziello:2005ku, Nojiri:2006gh, Nojiri:2006jy, Nojiri:2006be, Wu:2007tn}. For reference, in order to track the contribution of the $T$ part of the action in the reconstruction, we consider two special $f(R,T)$ models: in the first instance, we investigate the modification $R + 2f(T)$, i.e. the usual Einstein-Hilbert term plus a $f(T)$ correction. In the second instance we consider a $f(R)+\\lambda T$ theory, i.e. a $T$ correction to the renown $f(R)$ gravity. In both cases, we consider dark energy accompanied by a pressureless matter component (which would determine $T$).\n\nThe paper is organised as follows. In Sec.~\\ref{Sec:HolDE}, the equations of motion are established and the holographic dark energy introduced. In Sec.~\\ref{Sec:Simpl} and \\ref{Sec:ComplCase} the above mentioned cases are analysed. Finally, Sec.~\\ref{Sec:DiscConcl} is devoted to discussion and conclusions.\n\nWe use $8\\pi G = c = 1$ units and adopt the metric formalism, i.e. \\textbf{the variation of the action is considered with respect to the metric quantities.}\n\n\n\\section{$f(R,T)$ gravity and holographic Dark Energy}\\label{Sec:HolDE}\n\nIn \\cite{Harko:2011kv}, the following modification of \\textbf{Einstein's} theory is proposed:\n\\begin{equation}\\label{actionfRT}\n S = \\frac{1}{2}\\int f(R,T) \\sqrt{-g}\\;d^4x + \\int L_{\\rm m} \\sqrt{-g}\\;d^4x\\;,\n\\end{equation}\nwhere $f(R,T)$ is an arbitrary function of the Ricci scalar $R$ and of the trace $T$ of the energy-momentum tensor, defined as\n\\begin{equation}\n T_{\\mu\\nu} = -\\frac{2}{\\sqrt{-g}}\\frac{\\delta\\left(\\sqrt{-g}L_{\\rm m}\\right)}{\\delta g^{\\mu\\nu}}\\;,\n\\end{equation}\nwhere $L_{\\rm m}$ is the matter Lagrangian density. We assume the matter lagrangian to depend on the metric, so that\n\\begin{equation}\n T_{\\mu\\nu} = g_{\\mu\\nu}L_{\\rm m} - 2\\frac{\\partial L_{\\rm m}}{\\partial g^{\\mu\\nu}}\\;.\n\\end{equation}\nVarying action \\eqref{actionfRT} with respect to the metric $g^{\\mu\\nu}$, one obtains \\cite{Harko:2011kv}\n\\begin{equation}\\label{Eqmod}\n f_R(R,T) R_{\\mu\\nu} - \\frac{1}{2}f(R,T)g_{\\mu\\nu} + \\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R,T) = T_{\\mu\\nu} - f_T(R,T)T_{\\mu\\nu} - f_T(R,T)\\Theta_{\\mu\\nu}\\;,\n\\end{equation}\nwhere the \\textbf{subscripts} $R$ or $T$ \\textbf{imply} derivation with respect that quantity and we have also defined\n\\begin{equation}\n \\Theta_{\\mu\\nu} \\equiv g^{\\alpha\\beta}\\frac{\\delta T_{\\alpha\\beta}}{\\delta g^{\\mu\\nu}}\\;.\n\\end{equation}\n\\textbf{Planning} a cosmological application, we assume matter to be described by a perfect fluid energy-momentum tensor\n\\begin{equation}\n T_{\\mu\\nu} = \\left(\\rho + p\\right)u_\\mu u_\\nu - p g_{\\mu\\nu}\\;,\n\\end{equation}\nand that $L_{\\rm m} = -p$, so that we have \n\\begin{equation}\n \\Theta_{\\mu\\nu} = -2T_{\\mu\\nu} - p g_{\\mu\\nu}\\;,\n\\end{equation}\nand Eq.~\\eqref{Eqmod} simplifies as\n\\begin{equation}\\label{Eqmod2}\n f_R(R,T) R_{\\mu\\nu} - \\frac{1}{2}f(R,T)g_{\\mu\\nu} + \\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R,T) = T_{\\mu\\nu} + f_T(R,T)T_{\\mu\\nu} + p f_T(R,T) g_{\\mu\\nu}\\;.\n\\end{equation}\nIn order to compare \\textbf{it} with \\textbf{Einstein's}, we cast the above equation as follows:\n\\begin{eqnarray}\\label{Eqmod3}\n G_{\\mu\\nu} &=& \\frac{1 + f_T(R,T)}{f_R(R,T)}T_{\\mu\\nu} + \\frac{1}{f_R(R,T)}p f_T(R,T) g_{\\mu\\nu} - \\frac{1}{f_R(R,T)}\\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R,T)\\nonumber\\\\ &+& \\frac{1}{2f_R(R,T)}f(R,T)g_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu} R\\;,\n\\end{eqnarray}\nwhere $G_{\\mu\\nu} \\equiv R_{\\mu\\nu} - Rg_{\\mu\\nu}\/2$ is the Einstein tensor. Now we can identify\n\\begin{equation}\\label{Effmatt}\n \\tilde{T}_{\\mu\\nu}^{(m)} = \\frac{1 + f_T(R,T)}{f_R(R,T)}T_{\\mu\\nu} + \\frac{1}{f_R(R,T)}p f_T(R,T) g_{\\mu\\nu}\\;,\n\\end{equation}\nas the \\textit{effective} matter energy-momentum tensor and\n\\begin{equation}\\label{Effgeom}\n \\tilde{T}_{\\mu\\nu}^{(geom)} = - \\frac{1}{f_R(R,T)}\\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R,T) + \\frac{1}{2f_R(R,T)}f(R,T)g_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu} R\\;,\n\\end{equation}\nas the energy-momentum tensor of a ``geometric'' matter component. \n\nWe now assume a \\textbf{background} described by the Friedmann-Lema\\^{\\i}tre-Robertson-Walker metric\n\\begin{equation}\\label{RWmet}\n ds^2 = dt^2 - a(t)^2\\delta_{ij}dx^idx^j\\;, \n\\end{equation}\nwith spatially flat hypersurfaces, and find a form for the function $f(R,T)$ which is able to reconstruct \\textbf{holographic dark energy}. \n\n\\subsection{Holographic Dark Energy}\n\nAccording to the holographic principle \\cite{'tHooft:1993gx, Susskind:1994vu, Bousso:2002ju} an entanglement relation between the infrared (IR) and ultraviolet (UV) cut-offs of a quantum theory, due to the limitation set by the formation of a black hole, sets an upper bound for the vacuum energy \\cite{Cohen:1998zx}:\n\\begin{equation}\\label{vacen}\n \\rho_{\\rm v} = \\frac{3b^2}{L^2}\\;,\n\\end{equation}\nwhere $b$ is a free parameter and the IR (large scales) cutoff $L$ needs to be specified by an ansatz. We are interested in the one proposed in \\cite{Li:2004rb}:\n\\begin{equation}\\label{anshol}\n L = R_{\\rm h} = a\\int_t^\\infty \\frac{dt'}{a(t')} = a\\int_a^\\infty \\frac{d\\bar{a}}{H(\\bar{a})\\bar{a}^{2}}\\;,\n\\end{equation}\ni.e. the future event horizon, that is the distance covered by a photon from now until the remote future. Note that the very presence of a vacuum energy component makes the above integration finite. We consider a model composed by holographic dark energy plus ordinary pressureless matter, i.e.\n\\begin{equation}\\label{Friede}\n 3H^2 = \\rho_{\\rm v} + \\rho_{\\rm m} = \\rho_{\\rm v} + \\rho_{\\rm m0}(1 + z)^3\\;,\n\\end{equation}\nwhere $H \\equiv \\dot{a}\/a$ is the Hubble parameter and the dot denotes derivation with respect to the cosmic time. Introduce the critical energy density $\\rho_{\\rm cr} := 3H^2$\\textbf{, we define}\n\\begin{equation}\\label{oliver15}\n\\Omega_{\\rm v} := \\frac{\\rho_{\\rm v}}{\\rho_{\\rm cr}}=\\frac{b^2}{R^2_{\\rm h}H^2}\\;,\n\\end{equation}\nUsing Eqs.\\eqref{vacen} and \\eqref{anshol}, it is easy to show that \n\\begin{equation}\\label{oliver16}\n\\dot{R}_{\\rm h} = \\frac{b}{\\sqrt{\\Omega_{\\rm v}}} - 1\\;.\n\\end{equation}\nThe holographic dark energy density $\\rho_{\\rm v}$ evolves according to the conservation law\n\\begin{equation}\\label{oliver17}\n\\dot{\\rho}_{\\rm v} + 3H\\rho_{\\rm v}\\left(1 + w_{\\rm v}\\right) = 0\\;,\n\\end{equation}\nbecause in Eq.~\\eqref{Friede} we have implicitly assumed the matter component to conserve separately. Now, using Eqs.~\\eqref{vacen}, \\eqref{anshol} and \\eqref{oliver16}, one can find\n\\begin{equation}\\label{oliver18}\n\\dot{\\rho}_{\\rm v} = -\\frac{2}{R_{\\rm h}}\\left(\\frac{b}{\\sqrt{\\Omega_{\\rm v}}}-1\\right)\\rho_{\\rm v}\\,.\n\\end{equation}\nComparing \\eqref{oliver18} with \\eqref{oliver17} one can read off\n\\begin{equation}\\label{oliver19}\nw_{\\rm v} = -\\left(\\frac{1}{3} + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{3b}\\right)\\,\\,.\n\\end{equation}\nMoreover, combining Eq.~\\eqref{anshol} with Eqs.~\\eqref{vacen} and \\eqref{Friede}, the evolution for $\\Omega_{\\rm v}$ is determined by the following equation:\n\\begin{equation}\\label{OmegavEvo}\n \\Omega_{\\rm v}' = -\\left(1 + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{b}\\right)\\frac{1}{1 + z}\\Omega_{\\rm v}\\left(1 - \\Omega_{\\rm v}\\right)\\;,\n\\end{equation}\nwhere the prime denotes derivation with respect to the redshift $z$. Testing this model against type Ia supernovae and cosmic microwave background anisotropies, $b$ turns out to be constrained around unity, with the case $b < 1$ favoured \\cite{Zhang:2005hs, Zhang:2007sh}. Note that, from Eq.~\\eqref{oliver19}, $b < 1$ means that the universe will end up in a phantom phase. For more comprehensive analysis of holographic dark energy models, we refer the reader to \\cite{Pavon:2005yx, delCampo:2011jp}.\n\nIn the next section we investigate a simple case of reconstruction of $f(R,T)$.\n\n\\section{A simple case}\\label{Sec:Simpl}\n\nWe now consider a single perfect fluid model with density $\\rho$ and pressure $p$, together with the following ansatz (one of the first considered in \\cite{Harko:2011kv}):\n\\begin{equation}\n f(R,T) = R + 2f(T)\\;,\n\\end{equation}\ni.e. the action is given by the same Einstein-Hilbert one plus a function of $T$. This is a particularly interesting choice since, from Eqs.~\\eqref{Effmatt} and \\eqref{Effgeom}, we get\n\\begin{equation}\\label{Tmtil}\n \\tilde{T}_{\\mu\\nu} = \\left(1 + 2f_T\\right)T_{\\mu\\nu} + 2p f_T g_{\\mu\\nu} + f(T)g_{\\mu\\nu}\\;.\n\\end{equation}\nFor $p = 0$ one has $T = \\rho$ and, choosing $f(T) = \\lambda T$ one can construct a model with an effective cosmological constant \\cite{Poplawski:2006ey}. From Eq.~\\eqref{Tmtil} one can read off the effective energy density and pressure of the universe content:\n\\begin{eqnarray}\n\\label{rhotot} 3H^2 &=& \\rho_{\\rm eff} = \\left(1 + 2f_T\\right)\\rho + 2p f_T + f(T)\\;,\\\\\n\\label{ptot} -2\\dot{H} - 3H^2 &=& p_{\\rm eff} = p - f(T)\\;,\n\\end{eqnarray}\nand therefore a dark energy component may appear, even if we are considering a single perfect fluid model. From Eqs.~\\eqref{rhotot} and \\eqref{ptot}, it is clear that we can pick out a ``fictitious'' component, due to $f(T)$, described by\n\\begin{eqnarray}\n \\rho_{f} &=& 2f_T\\rho + 2p f_T + f(T)\\;,\\\\\n p_{f} &=& - f(T)\\;,\n\\end{eqnarray}\nand, provided $f$ positive, it may well describe a dark energy component, since its pressure is negative. In order to reconstruct the function $f$ starting from the holographic principle, we note that the equation of state parameter of the dark component induced by $f$ is\n\\begin{equation}\n w_{f} = -\\frac{f(T)}{2(\\rho + p)f_T + f(T)}\\;,\n\\end{equation}\nand we identify it with $w_{\\rm v}$, the one provided by the holographic dark energy in Eq.~\\eqref{oliver19}:\n\\begin{equation}\\label{EqfT}\n \\frac{f(T)}{2(\\rho + p)f_T + f(T)} = \\frac{1}{3} + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{3b}\\;.\n\\end{equation}\nFor the standard model given in Eq.~\\eqref{Friede}, consider the fluid component to be pressureless matter, i.e. $p = 0$. We are left to solve the following system of equations:\n\\begin{eqnarray}\n\\label{fevo} \\rho\\frac{df(\\rho)}{d\\rho} &=& f(\\rho)\\frac{b - \\sqrt{\\Omega_{\\rm v}}}{b + 2\\sqrt{\\Omega_{\\rm v}}}\\;,\\\\\n\\label{Omevo} \\frac{d\\Omega_{\\rm v}}{d\\rho} &=& -\\frac{1}{3\\rho}\\left(1 + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{b}\\right)\\Omega_{\\rm v}\\left(1 - \\Omega_{\\rm v}\\right)\\;,\n\\end{eqnarray}\nwhere we have used $T = \\rho$, because we are considering pressureless matter. In order to solve the above system, we have to fix some initial conditions. Clearly, $\\Omega_{\\rm v}(\\rho = \\rho_0) = 1 - \\Omega_{\\rm m0}$, and we choose $\\Omega_{\\rm m0} = 0.3$, accordingly with current cosmological observation. For the initial condition on $f$, from Eq.~\\eqref{rhotot} (with $p = 0$) we write\n\\begin{equation}\\label{friedeqf0}\n \\left[1 + 2f_{T}(\\rho_0)\\right]\\Omega_{\\rm m0} + \\frac{f(\\rho_0)}{3H_0^2} = 1\\;.\n\\end{equation}\nEvaluating Eq.~\\eqref{fevo} today and combining it with Eq.~\\eqref{friedeqf0} we find the following algebraic equation determining the initial condition on $f$:\n\\begin{equation}\\label{f0cond}\n f(\\rho_0)\\left(2\\frac{b - \\sqrt{\\Omega_{\\rm v0}}}{b + 2\\sqrt{\\Omega_{\\rm v0}}} + 1\\right) = 3H_0^2\\Omega_{\\rm v0}\\;.\n\\end{equation}\nAs we expected, when $\\Omega_{\\rm v0} = 0$, then $f(\\rho_0) = 0$ and Eq.~\\eqref{fevo} implies that $f$ is identically vanishing.\\\\\n\\begin{figure}[htbp]\n  \\includegraphics[width=0.45\\columnwidth]{Fig1.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig2.eps}\n\\caption{Left panel: evolution of $f(\\rho)$. Right panel: evolution of $\\Omega_{\\rm v}$. The cases considered are $b = 0.6, 0.8, 1.0, 1.2$ (solid black, dashed red, dot-dashed blue and dotted green, respectively). We have chosen as initial conditions in $\\Omega_{\\rm m0} = 0.3$ the values $\\Omega_{\\rm v0} = 1 - \\Omega_{\\rm m0} = 0.7$ and $f_0$ given by Eq.~\\eqref{f0cond}. Note that $f$ and $\\rho$ are normalised to $3H_0^2$. The vertical lines in the plots represent $\\rho = \\rho_0$, i.e. the present instant.}\n\\label{Fig1}\n\\end{figure}\\\\\nIn \\figurename{ \\ref{Fig1}} we plot the solution of the system of differential equations \\eqref{fevo} and \\eqref{Omevo}. Note that we normalise $f$ and $\\rho$ to $3H_0^2$. As expected from Eq.~\\eqref{fevo}, for large values of $\\rho$, i.e. far in the past, $f \\propto \\rho$ because the dark energy component is subdominant. The actual difference among the various choices of $b$ takes place at late times, for small values of $\\rho$. Again from inspection of Eq.~\\eqref{fevo}, we can see that for large values of $b$ the linear evolution $f \\propto \\rho$ is again solution. That is why in the left panel of \\figurename{ \\ref{Fig1}} the curve seems to ``straighten up'' for increasing $b$. \n\nIn \\figurename{ \\ref{Fig2}} we plot the same quantities, but as functions of the redshift, in order to make clearer their cosmological evolution.\n\\begin{figure}[htbp]\n  \\includegraphics[width=0.45\\columnwidth]{Fig3.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig4.eps}\n\\caption{Same as \\figurename{ \\ref{Fig1}}, but with the redshift $z$ as independent variable.}\n\\label{Fig2}\n\\end{figure}\n\nA final remark about the future evolution. It is clear from Eq.~\\eqref{fevo}, that when $\\Omega_{\\rm v} \\to 1$, in the remote future, the solution for $f$ gets the asymptotic form\n\\begin{equation}\n f(\\rho) \\propto \\rho^{\\frac{b - 1}{b + 2}}\\;.\n\\end{equation}\nTherefore, we would have a future singularity for $-2 < b < 1$, as it appears for the relevant cases of \\figurename{ \\ref{Fig1}} and \\figurename{ \\ref{Fig2}}. The special case $b = 1$ implies an asymptotically constant $f$.\n\nWe now turn our discussion on a more general case, where the curvature $R$ \\textbf{comes into} the action \\textbf{as} a function to be determined.\n\n\n\\section{A more complicated case}\\label{Sec:ComplCase}\n\nNow we turn our attention to the special case\n\\begin{equation}\\label{complcase}\n f(R,T) = f(R) + \\lambda T\\;,\n\\end{equation}\ni.e. a $T$-linear correction to the class of $f(R)$ theories. With the ansatz \\eqref{complcase} the matter content \\eqref{Effmatt} is ``corrected'' as follows:\n\\begin{equation}\\label{Effmattcomplcase}\n \\tilde{T}_{\\mu\\nu}^{(m)} = \\frac{1 + \\lambda}{f_R(R)}T_{\\mu\\nu} + \\frac{1}{f_R(R)}\\lambda p g_{\\mu\\nu}\\;,\n\\end{equation}\nwhereas the geometry induced stress-energy tensor is\n\\begin{equation}\\label{Effgeomcomplcase}\n \\tilde{T}_{\\mu\\nu}^{(geom)} = - \\frac{1}{f_R(R)}\\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R) + \\frac{1}{2f_R(R)}[f(R) + \\lambda T]g_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu} R\\;.\n\\end{equation}\nOur aim is now to reconstruct the form of the $f(R)$ which is able to reproduce the holographic dark energy paradigm. We again consider a pressureless perfect fluid with density $\\rho$ and again assume metric \\eqref{RWmet}. From Eqs.~\\eqref{Effmattcomplcase} and \\eqref{Effgeomcomplcase} the effective density and pressure are the following:\n\\begin{eqnarray}\n\\label{rhovcomplcase} 3H^2 &=& \\rho_{\\rm eff} = \\frac{\\rho}{f_R} + \\frac{3\\lambda}{2f_R}\\rho - \\frac{R}{2} + \\frac{f}{2f_R} - 3H\\frac{\\dot{f}_R}{f_R}\\;,\\\\\n\\label{pvcomplcase} -2\\dot{H} - 3H^2 &=& p_{\\rm eff} =  \\frac{1}{f_R}\\left(\\ddot{f}_R + 2H\\dot{f}_R\\right) - \\frac{f}{2f_R} - \\frac{\\lambda}{2f_R}\\rho + \\frac{R}{2}\\;.\n\\end{eqnarray}\nFrom Eq.~\\eqref{rhovcomplcase} it appears that the energy density of the perfect fluid is rescaled by a factor $1\/f_R$. Looking at Eq.~\\eqref{Friede}, we can extract a form for $\\rho_{\\rm v}$ in the following way:\n\\begin{equation}\\label{rhovcomplcase2} \n\\rho_{\\rm v} = \\frac{\\rho}{f_R} - \\rho + \\frac{3\\lambda}{2f_R}\\rho - \\frac{R}{2} + \\frac{f}{2f_R} - 3H\\frac{\\dot{f}_R}{f_R}\\;,\n\\end{equation}\nwhereas the form of $p_{\\rm v}$ is already given in Eq.~\\eqref{pvcomplcase}, since our fluid is pressureless. From Eqs.~\\eqref{pvcomplcase} and \\eqref{rhovcomplcase2} we can write the following differential equation for $f_R$:\n\\begin{equation}\\label{freqcompl}\n \\ddot{f}_R - H \\dot{f}_R - \\left[\\rho + \\rho_{\\rm v}\\left(1 + w_{\\rm v}\\right)\\right]f_R = -\\rho(1 + \\lambda)\\;.\n\\end{equation}\nNote, as a cross-check, that for $\\rho_{\\rm v} = \\lambda = 0$ the above equation simplifies to\n\\begin{equation}\\label{freqcompl2simpl}\n \\ddot{f}_R - H\\dot{f}_R - \\rho f_R = - \\rho\\;,\n\\end{equation}\nwhich possesses the particular solution $f_R = 1$, i.e. $f = R + \\Lambda$, the original Einstein-Hilbert action plus an integration ``cosmological'' constant. We expect this solution to be the only one, otherwise there would exist an alternative $f(R)$ theory which would behave exactly as general relativity. Let us speculate a bit more on this point. If $\\rho_{\\rm v} = \\lambda = 0$, i.e. for a pure Einstein-de Sitter universe, we have from Eq.~\\eqref{Friede}\n\\begin{equation}\n H = \\frac{2}{3t}\\;, \\qquad \\rho = \\rho_0\\frac{t^2_0}{t^2}\\;,\n\\end{equation}\nwhere $t_0$ is the present cosmic time (i.e. the age of the universe). Considering the homogeneous part of Eq.~\\eqref{freqcompl2simpl} and looking for a solution of the form $f_R \\propto t^n$, we find:\n\\begin{equation}\\label{freqcompl2simplhom}\n n(n - 1) - \\frac{2}{3}n - \\rho_0 t_0^2 = 0\\;,\n\\end{equation}\nwhich gives:\n\\begin{equation}\\label{freqcompl2simplhomsol}\n n_{1,2} = \\frac{5}{6} \\pm \\sqrt{\\frac{25}{36} + \\rho_0 t_0^2}\\;,\n\\end{equation}\nand the general solution of Eq.~\\eqref{freqcompl2simpl} can be written as:\n\\begin{equation}\\label{freqcompl2simplhomsol2}\n f_R = 1 + C_1\\;t^{n_1} + C_2\\;t^{n_2}\\;.\n\\end{equation}\nNow, the initial conditions we adopt here are $f_R(t_0) = 1$ and $\\dot{f}_R(t_0) = 0$. The reason is essentially not spoiling the agreement between general relativity and solar system tests, see \\cite{Capozziello:2005ku, Nojiri:2006gh, Nojiri:2006jy, Nojiri:2006be, Wu:2007tn}. However, we stress here that these two conditions also imply $C_1 = C_2 = 0$ and therefore restore the general relativity limit $f_R = 1$ when $\\rho_{\\rm v} = \\lambda = 0$. \n\nChanging the variable to the redshift and employing Eqs.~\\eqref{Friede} and \\eqref{OmegavEvo} one can recast Eq.~\\eqref{freqcompl} in the following compact form:\n\\begin{eqnarray}\\label{freqcompl2}\n (1 + z)^2f''_R + \\frac{1 + z}{2}\\left[7 - \\Omega_{\\rm v}\\left(1 + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{b}\\right)\\right]f_R' - 3\\left[1 - \\Omega_{\\rm v}\\left(\\frac{1}{3} + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{3b}\\right)\\right]f_R = \\nonumber\\\\ - 3(1 + \\lambda)(1 - \\Omega_{\\rm v})\\;,\n\\end{eqnarray}\nwhere again the prime denotes derivation with respect to the redshift $z$. The curvature $R$ can be easily found as\n\\begin{equation}\n R = -6(\\dot{H} + 2H^2) = -\\frac{3H_0^2\\Omega_{\\rm m0}(1 + z)^3}{1 - \\Omega_{\\rm v}}\\left[1 + \\Omega_{\\rm v}\\left(1 + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{b}\\right)\\right]\\;.\n\\end{equation}\nThe initial conditions $f_R(t_0) = 1$ and $\\dot{f}_R(t_0) = 0$, translated to the redshift variable, are\n\\begin{eqnarray}\n \\left.\\frac{d^2f}{dz^2}\\right|_{z = 0} = \\left.\\frac{d^2R}{dz^2}\\right|_{z = 0}\\;, \\qquad \\left.\\frac{df}{dz}\\right|_{z = 0} = \\left.\\frac{dR}{dz}\\right|_{z = 0}\\;,\\\\\n\\end{eqnarray}\nFinally, the initial condition on $f$ can be extracted by Eq.~\\eqref{rhovcomplcase}, being that $\\rho_{\\rm v0} = 3H_0^2 - \\rho_{\\rm m0}$. Thus, we have\n\\begin{equation}\nf(z = 0) = R(z = 0) + 6H_0^2\\left(1 - \\Omega_{\\rm m0} - \\frac{3}{2}\\lambda\\Omega_{\\rm m0}\\right)\\;.\n\\end{equation}\nNote the correction due to the $\\lambda T$ term.\n\\begin{figure}[htbp]\n  \\includegraphics[width=0.45\\columnwidth]{Fig5.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig6.eps}\n\\caption{Evolution of $f$ as a function of the curvature $R$. Left panel: $\\lambda = 0$, i.e. we are considering a pure $f(R)$ theory, and $b = 0.6, 0.8, 1.0, 1.2$ (solid black, dashed red, dot-dashed blue and dotted green, respectively). Right panel: $b = 1.0$ and $\\lambda = -0.2, 0, 0.2, 0.4$ (solid black, dashed red, dot-dashed blue and dotted green, respectively. The curves appear superposed.). We have chosen as initial conditions in $\\Omega_{\\rm m0} = 0.3$ the values $\\Omega_{\\rm v0} = 1 - \\Omega_{\\rm m0} = 0.7$. Note that $f$ and $R$ are normalised to $3H_0^2$ and the redshift interval chosen is $0 < z < 10$.}\n\\label{Fig3}\n\\end{figure}\nIn \\figurename{ \\ref{Fig3}} we plot the solution for $f$. Note that we normalise $f$ and $R$ to $3H_0^2$. In the left panel we fix $\\lambda = 0$, i.e. we are actually considering a pure $f(R)$ theory, and vary $b$. In the right panel, on the other hand, we consider positive and negative values of $\\lambda$. As one may note, $\\lambda$ has a poor influence on the evolution of $f$. We could expect this from inspection of Eq.~\\eqref{freqcompl2}. Indeed, $\\lambda$ only enters the source term on the right hand side and therefore, when $\\Omega_{\\rm v}$ grows to unity, its impact on the evolution of $f$ is weak. On the other hand, larger values of $\\lambda$ may have a relevant effect at early times, determining the slope of $f$.\n\\begin{figure}[ht]\n  \\includegraphics[width=0.45\\columnwidth]{Fig7.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig8.eps}\n\\caption{Same as \\figurename{ \\ref{Fig3}}, but with $-0.9 < z < 1$. Note, in the left panel, that the evolution of $f$ starts from below for all the cases.}\n\\label{Fig4}\n\\end{figure}\n\\newpage\nIn \\figurename{ \\ref{Fig4}} and \\figurename{ \\ref{Fig5}} we display the future evolution of $f$, both as function of $R$ or of $z$. For the same reason stated above, the effect of $\\lambda$ is not relevant.\n\\begin{figure}[ht]\n  \\includegraphics[width=0.45\\columnwidth]{Fig9.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig10.eps}\n\\caption{Same as \\figurename{ \\ref{Fig4}}, but with $f$ as function of $z$.}\n\\label{Fig5}\n\\end{figure}\n\\newpage\nFinally, in \\figurename{ \\ref{Fig6}} we plot the solution for $\\Omega_{\\rm v}$. Again, its evolution appears to be independent of $\\lambda$.\n\\begin{figure}[ht]\n  \\includegraphics[width=0.45\\columnwidth]{Fig11.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig12.eps}\n\\caption{Evolution of $\\Omega_{\\rm v}$ as a function of the redshift. Left panel: $\\lambda = 0$, i.e. we are considering a pure $f(R)$ theory, and $b = 0.6, 0.8, 1.0, 1.2$ (black, dashed red, dash-dotted blue and dotted green, respectively). Right panel: $b = 1.0$ and $\\lambda = -0.2, 0, 0.2, 0.4$ (black, dashed red, dash-dotted blue and dotted green, respectively). Note that the curves are superposed. We have chosen as initial conditions in $\\Omega_{\\rm m0} = 0.3$ the values $\\Omega_{\\rm v0} = 1 - \\Omega_{\\rm m0} = 0.7$.}\n\\label{Fig6}\n\\end{figure}\n\n\\newpage\n\n\\section{Discussion and Conclusions}\\label{Sec:DiscConcl}\n\nIn this work we have investigated a description of holographic dark energy in terms of suitably reconstructed $f(R, T)$ gravity theories. The latter have been recently introduced as modifications of Einstein's theory possessing some interesting solutions \\textbf{which are} relevant in cosmology and astrophysics \\cite{Harko:2011kv}.\n\nWe have \\textbf{considered two special} types of models: $f(R,T) = R + 2f(T)$, i.e. a correction to the Einstein-Hilbert action depending on the matter content, and $f(R,T) = f(R) + \\lambda T$, i.e. a simple $T$-linear correction to the class of $f(R)$ theories. \n\nSince we have assumed the matter content to be a pressureless perfect fluid, then $T = \\rho$, i.e. the corrections assumed are directly dependent on the energy density of the universe content. We \\textbf{have} constructed differential equations for the function $f$ under investigation and numerically solved \\textbf{them}, physically specifying the required initial conditions. Our simple analysis shows that holographic dark energy models are contained in the larger class of $f(R,T)$ theories, at least considering a given background evolution of the universe. \n\nIt would be interesting to investigate how the evolution of matter perturbations would change, depending on the description of dark energy. We expect, in principle, different results when using holographic dark energy or its $f(R,T)$ reconstruction and therefore there is possibility for discriminating between the two descriptions. For example, it would be interesting to adapt the recently proposed scheme for perturbations in $f(R)$ theories \\cite{Bertacca:2011wu} to the broader $f(R,T)$ class. We leave this as a future work.\n\n\n\\section*{Acknowledgements}\n\nMJSH and OFP thank Professor S. D. Odintsov for useful comments and also CNPq (Brazil) for partial financial support. \n\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\n\nCOMPASS is a fixed target experiment on the CERN SPS that uses tertiary high energy and high intensity muon or hadron beams with the aim of studying the nucleon spin structure and hadron spectroscopy. It is taking data since 2002, around seven months per year, and has shutdown periods in between, in which operations of maintenance and preparation of the following data taking period take place. The experimental setup is described in detail in \\cite{COMPASS01}.\n\nThe detector devices and the experiment's environmental parameters are monitored and controlled using an experiment-wide DCS. This system must ensure a coherent, safe and efficient operation of the experiment, by providing clear and prompt information for the shift crew and detector experts in the COMPASS control room. Some complex subsystems of the experiment have dedicated stand-alone control systems. These systems communicate with the DCS, providing it the most relevant parameters. Since 2003, the COMPASS DCS has been an exclusive responsibility of the LIP-Lisbon group participating in the collaboration. This structure and organization is at contrast with the one of the big LHC experiments, which have a hierarchical structure, with distributed responsibilities~\\cite{ATLAS,CMS,ALICE,LHCb}.\n\nThe DCS provides a graphical user interface for the shift crew in the COMPASS control room and detector experts to have access to all the relevant parameters monitored, their state (normal or in alert - indicated visually, by use of a color code, and acoustically) and their history, and a straightforward way to change their state, their settings, and the thresholds that define their state of alert.\n\nThe architecture of the DCS is shown in Fig.~\\ref{DCSscheme}. In the following sections, its different layers are described in detail: the supervisory layer, the front-ends layer and the devices layer.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.8 \\textwidth]{dcs-schema-3.eps}\n\\caption{The COMPASS Detector Control System architecture, comprising a devices layer, a front-ends layer and a supervision layer. The technologies used in each layer are indicated in the rightmost column.} \n\\label{DCSscheme}\n\\end{figure*}\n\n\\noindent\\section{\\uppercase{The Supervisory Layer}}\n\\label{Supervisory} \n\nOn the supervisory layer, all the data collected, managed and published by the different kinds of servers or made available in databases is gathered, analysed and displayed to the end user. This includes visual and sound alarms in case of states of alarm. It also provides archiving of the data in an external database. Settings and alarm limits are also managed by the system.\n\nPVSS-II\\footnote{Pro\\-zess\\-teue\\-rung und Pro\\-zess\\-vi\\-sua\\-li\\-sie\\-rung. See http:\/\/www.etm.com}, is the commercial SCADA system that was chosen by CERN to use in the LHC experiments, after a thorough evaluation process. Some of the aspects taken into account were: openess, scalability, cross-platform capability (i.e. to run in Windows and Linux) and long term support.\n\nThe COMPASS experiment has adopted PVSS early in its development phase and has been a benchmark for other experiments at CERN. In fact, before the LHC starting in 2010, COMPASS was the biggest experiment operating at CERN. Over the years, several versions of PVSS were used in COMPASS, that had been previously tested and validated both by CERN and by the COMPASS DCS group.\nThe installation of these patches during the data-taking periods requires a careful evaluation.\n\nThe JCOP Framework~\\cite{JCOP} is a CERN project to develop common software tools for High Energy Physics related equipment and operations, to be used with PVSS. It provides templates of datapoint types, panels, functions and mass configuration tools for different classes of equipments or functionalities, providing, for instance, tools for the management of priviledges, or for trending plots.\n\nThe objects provided by PVSS and the JCOP Framework have sometimes to be adapted to meet COMPASS' needs. In addition, other solutions had to be developed independently for non-supported custom devices. This includes the control of custom devices,\naccessed using their serial (RS232) interfaces or their web servers; \nor the monitoring of items from dedicated control systems, such as (EPICS, LabView, etc.\\footnote{See http:\/\/www.aps.anl.gov\/epics, http:\/\/www.ni.com\/labview}), which are made available by various means, including mySQL and Oracle databases. \n\nThe PVSS production system in use is both distributed and scattered. Historically, it started as a scattered project, meaning that it was constituted by a main PVSS project running in a Linux machine, and 3 associated PVSS projects running on Windows machines, that had PVSS processes running as OPC\\footnote{OLE (Object Linking and Embedding) for Process Control. See http:\/\/www.opcfoundation.org} clients (7 clients in total). As the DCS developed, the main project was split into two distributed projects, for performance reasons. \n\nPVSS works with objects called datapoints, which are structures, {\\it i.e.} they have a tree structure that can include branches and where the leaves are the the monitored and controlled parameters (and can have different types, such as floats, integers, booleans, strings, etc., or the corresponding array types). \n\nPresently, the project comprises over 20000 datapoints. Close to 17000 parameter values have alert handling, whereas almost 19000 parameters have their values archived. \n\nThe polling rates are adapted to the rate of variation of the parameters, and range from one value per parameter read every 1.5 seconds (for fast varying parameters, sensitive to the beam, such as high voltage channels' actual values) to 2 minutes (for slowly varying parameters, such as high voltage channels' settings, or detector positions). For any given type of equipment, the items are grouped in PVSS subscription data sets according to these rates.\n\nThe access to the PVSS project is made available upon login. There is a general user name for the shift crew, user names for each of the detector experts, and a username for guests. For each login, there is an authorizations policy associated: certain operations are restricted (such as switching on or off the high voltage channels for guest users), or specifically allowed (such as saving recipes or reference files of high voltage settings; see \ndetails later in this text).\n\nThe graphical user interface (UI) is the main mean for users to interact with the DCS. It is composed of multiple subpanels, organised in a hierarchical way, as can be seen in Fig.~\\ref{ui}. One can see, on the top, the alert table and, on the left, the buttons to access dedicated detector panels and, below them, a table with the summary status of the experiment. In the larger area of the panel, a synoptic view of the spectrometer is displayed. This area is also used for navigation in the subsystems controlled and to display the actual data.\n\n\nThe datapoints history is made available online. In fact, PVSS trending plots (namely values over time) are one of the more useful and more used features of the DCS. In the user interface panels, customised buttons are created for the items that have numerical values (generally, floating point), so that their history can be easily accessed by users. The JCOP Framework provides its own trending plot widget, that was further customised for COMPASS, to make it simple to use. Users can choose the time range they want to visualise, change the scales or zoom in or out the abscissa or the ordinate by using the mouse scroll button, or choosing a rectangular region for zooming in by simply selecting two opposite vertices of the region. \nTemplates can be created to allow, with one click, to see related parameters and adjust the ranges for each.\nIt is also possible for the users to include additional parameters to a predefined plot as well as to print the trends being displayed, or to save them to a file, not only as an image, but also in ascii format (comma separated values or CVS).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7 \\textwidth]{dcsUI.eps}\n\\caption{The graphical user interface (UI) of the COMPASS DCS. One can see, on the top, the alert panel and, on the left, the buttons to access dedicated detector panels and, below them, a table with the summary status of the experiment.}\n\\label{ui}\n\\end{figure*}\n\nOne of the most important functionalities of the DCS is the display of visual and audible alarms, when predefined conditions are met, namely, when parameter values beyond predefined thresholds are reached for datapoints with numerical values or when devices send alarm flags. The visual display of alarms follows a color code that indicates its severity. For the most relevant parameters, it is important to assure that the operator didn't fail to notice the alarm condition, even if it has disapeared in the meantime; in this case, it is requested that the alarms are acknowledged by the operator in the graphical user interface.\nUpon activation of an alarm, detector experts are warned by email or SMS of states of alert in their detectors. \n\nSince the DCS has both a relatively fast knowledge of the state of the parameters it monitors and the ability to send commands to the devices, it is used to ensure software-wise protections to several equipments.\nFor instance, some detectors have components that are sensitive to magnetic field gradients; thus, when a trip of one of the spectrometer magnets (SM1 or SM2) occurs, or when these are switched off with the detectors high voltage channels still switched on, the DCS issues a switch-off command, so that the time interval during which the gradient is felt is minimised.\nIn addition, for some detectors, the high voltage channels should only be switched on or off in pairs; hence, if a trip is detected in only one member of the pair, the DCS sends a switch-off command to the other member of the pair.\n\nFurthermore, some detectors have front-end cards that are refrigerated by a water circuit. When this circuit stops for some reason, the temperature of the cards increases and can reach a value above a predefined threshold. If this happens, the DCS issues a switch-off command to the low voltage power supplies that power them, thus preventing these sensitive and sophisticated cards to be damaged. A hardware interlock is activated at a higher temperature, but the recovery from the interlocked state implies access to the experimental area, and therefore represents greater beam time losses and is therefore to be avoided. \n\nA configurations database associated to the project was implemented. It is based in a JCOP Framework package, which was adapted for the COMPASS needs. It saves and retrieves data in an independent Oracle database. This database has two main purposes. On the one hand, it allows to save and to retrieve the so called ``recipes\", {\\it i.e.} sets of thresholds for alarms of groups of items. The recipes can be created, or its values committed in the PVSS project, using the DCS UI, provided that the user has the privileges to change the respective detector items.\n\nThe second important purpose of the configuration database is to store so cal\\-led ``con\\-fi\\-gu\\-ra\\-tions\", {\\it i.e.} the mapping of hardware names vs.\\ logical names ({\\it i.e.} PVSS datapoint element names and respective aliases), for snapshots of stable states of the PVSS project. These are used to keep track of changes of the aforementioned mapping. These changes can happen either because, for instance, a high voltage channel gets broken and the same part of the detector ({\\it i.e.} same alias) is then powered by a different channel ({\\it i.e.} different datapoint), or because channels are reused when switching between the muon and the hadron Physics program data-taking.\n\nFor storage of settings of high voltage channels (set voltage, maximum current allowed, ramp up speed, ramp down speed and trip time), ascii files are used, for convenience. Experts can access the files, edit them, and send the values to be used by the equipment. The shift crew can use these reference file to recover the normal state of the equipment in case of problems.\n\nOnly a subset of all the data that PVSS receives and manages is actually saved. For this to happen, the PVSS datapoints need to have an archiving policy defined. This is chosen according to the known changes of each datapoint and the relevance of its history. For instance, it may be useful to store the readings of a temperature every ten minutes or if the change with respect to the previous reading exceeds one degree. This smoothing condition, called dead-band, is adjusted for each datapoint group or even per datapoint, if needed. The generic rates of archiving range from one value for every $\\sim$40 seconds (corresponding to the beam supercycle time interval) for beam-related quantities, to one value for every half an hour (for the positions of detectors). \n\nCurrently, there are around $2\\times 10^9$ values stored ({\\it i.e.} around 300 GB of data, including indexes), comprising the project history since 2006.\nThe DCS historical data that had been saved in PVSS internal format (during its first years of operation) was copied to a CERN central Oracle database, and the new data produced was all stored in this database. This way of storing data has all the advantages of Oracle and makes their access independent of PVSS. The data is continuously replicated to a second database, to ensure that the access to the data never compromises the performance of the production database. The data can also be provided in other formats, such as ascii or ROOT~\\cite{ROOT} trees.\n\nThe DCS data is very important for studies of detector performances.\nSome particularly relevant parameters for offline Physics data analysis are regularly copied to the experiment's mySQL conditions database, using a cron job.\nThe history of alerts of all the datapoint elements that have alert handling is also saved and made available by PVSS. This includes the timestamps of their arrival and departure, and of eventual acknowledgements done, among other information.\n\nThe knowledge of malfunctioning of parts of the experiment relies substantially on the DCS, namely on the display of alarms. Hence, it is important to assure its integrity and availability, ideally, at all times. Some of the mechanisms used are heart-beats, watch-dogs, back-ups, a security policy and the issuing of regular ping commands.\n\nThe managers of PVSS's main project, which are independent processes running in Linux, may, for a number of reasons, either block or stop running. On the other hand, the servers -- either OPC, \\cite{DIM}, or other -- may stop delivering meaningful data. For this reason, for each manager in PVSS that acts as a client, a heart-beat item was created, that gives the timestamp of the last meaningful data it received.\n\nMoreover, to verify that PLCs\\footnote{Programmable logic controllers.} are sending meaningful data at all times, a mechanism of watchdogs is implemented. The OPC server marks as invalid values sent by the PLC in case the values of the items published for this purpose (which have, during normal operation, varying integer values), stop begin updated.\n\nThe communication with individual VME\\footnote{VERSAmodule Eurocard bus.} crates or power supplies is also monitored, by continuously checking for selected equipment items that the timestamp of the latest value read is more recent than a predefined time interval.\n\nTo ensure the integrity of the project if a software corruption occurs in the PVSS project and associated software, the data is copied every twenty-four hours to a central repository. Furthermore, local copies are periodically made. \n\nA thorough security policy is implemented. All the computers that integrate the DCS belong to a dedicated experimental domain, that communicates with the CERN network using dedicated gateways. All the PCs in use have firewalls implemented.\nIn addition, all the user interfaces, with the exception of the one in the control room that should be permanently accessible, have an auto-logout after one hour.\n\nThe project should be available in the network at all times, for instance to diagnose of eventual DCS problems remotely. For this to happen, the gateways of the COMPASS domain have to be switched on and accessible via the CERN network. To check that this is the case at all times, regular ping commands are issued (every fifteen minutes) from an external server and the response is monitored; a notification is sent to the DCS experts in case those gateways are not reachable. \n\n\\noindent\\section{\\uppercase{The Front-ends Layer}}\n\\label{FrontEnds}\n\nThe experiment devices that are monitored and controlled by the DCS are spread over nearly two hundred meters, including the spectrometer and the beam tunnel. To communicate with all the devices, different field buses and communication protocols are \nused, namely CAN\\footnote{Controller area network. ISO standard 11898, see e.g.\\ www.iso.org} bus (8 daisy-chains), CAENet (6 daisy-chains), ModBus\\footnote{See http:\/\/www.modbus.org}, Profibus\\footnote{See http:\/\/www.profibus.com} (4 daisy-chains) and Ethernet.The general baud rate used for monitoring in the COMPASS CAN buses is 125 kbaud ($\\simeq 34$ kbits\/s), which is the recommended baud rate for the length of the daisy-chains used. These field buses transmit the information about the measurements of sensors to the front-end PCs (and commands to actuators in the opposite direction). In the front-end PCs, standard PCI\\footnote{Peripheral Component Interconnect.} cards are installed to collect the information carried by the field buses. The data is transmitted to the supervisory layer using a server-client model. An exception to this model is the three-layer model which is used when a database is included as an intermediate between the server and PVSS. This happens for the monitoring of the calorimeters, beam and trigger rates, and part of the polarised target system.\n\nIn ad\\-di\\-tion, spe\\-cialised de\\-vi\\-ces are used as intermediates between the devices and some of the front-end PCs, namely ELMBs (Embedded Local Monitor Boards) and PLCs.\n\nThe ELMB, described in \\cite{ELMB}, is a multi-purpose multiplexed ADC with 64-analog input channels with 16 bit-resolution which was developed by the ATLAS experiment. The communication of the ELMBs with the front-end PCs is done with the CAN field bus, using the CANopen protocol. The ELMB was designed and tested to be radiation- and magnetic field tolerant: its tolerance ranges up to about 5 Gy and $3\\times 10^10$ neutrons\/cm$^2$ for a period of 10 years and to a magnetic field up to 1.5 T.\n\nThe PLCs (Programmable Logic Controllers) are stand-alone, very robust, reliable and relatively fast devices that allow, among other operations, to regulate flows of gases and their percentage in mixtures, according to predefined settings and tolerance intervals, as well as to regulate cryogenic systems. The measurement of gas flows or gas percentages in mixtures is provided by the PLCs by ModBus to the DCS front-end PCs. \n\nManufacturer's OPC servers are used when available and stable. This is the case for the modern CAEN equipment and for Iseg equipment. In order to communicate with PLCs, an OPC server from Schneider\\footnote{See http:\/\/www.schneider-electric.com}, is used. Moreover, an OPC server was developed at CERN to control relatively old Wiener equipment, as the one used in COMPASS. To communicate with ELMBs, a CANopen OPC server, described in \\cite{CANopen}, is used.\n\nThe Distributed Information Management system (DIM) was developed at CERN and allows the implementation of a server-client model of publishing of lists of items and their actual values. The SLiC\\footnote{See http:\/\/j2eeps.cern.ch\/wikis\/display\/EN\/SLiC} DIM server developed at CERN, allows the control of the six CAENet lines used for the older type of CAEN crates. Each server has different groups of items with individually tunable speed reading cycles, thus permitting the separation of fast reading cycles (comprising voltages, currents and channel status) with reading frequencies as low as 1 Hz, thus allowing a fast detection of high voltage trips and failures; and of slow cycles, used for the read-back of settings.\n\t\t\t\t\nThe DIM protocol is also used to monitor other systems, namely temperatures and disk occupancy of servers, and processes of data transfer from the DAQ machines to CASTOR. \n\nDIP\\footnote{See http:\/\/en-dep.web.cern.ch\/en-dep\/Groups\/ICE\/Services\/DIP} is a protocol developed at CERN, based on DIM, but allowing exclusively read-only parameters, which are, in practical terms, those related to the CERN infrastructure (such as beam line magnet currents and the last beam file loaded, the primary target head inserted, the parameters to allow the monitoring of the CEDAR detectors, and data relative to the liquid nitrogen supply).\n\nOne PLC from the polarized target system is monitored using the S7 driver provided by PVSS, thus avoiding the use of an OPC server.\n\nPVSS provides functions to access relational databases such as mySQL and Oracle. This allows the access of information from the experiment conditions database (a set of mySQL databases), such as the calorimeter calibration event amplitudes, the beam and trigger rates, and parameters related with the polarised target.\n\nThe high voltage system of some of the detectors (Micromegas and the so called Saclay Drift Chambers) have special requirements with regards to its monitoring, and thus have a dedicated control system based on EPICS. This system publishes the most relevant data, which is read by a specially developed PVSS API (Application Programming Interface).\n\nThe Profibus protocol is used to transmit the data coming from the PLCs that monitor the detector gas systems to the PC that runs the Schneidar OPC server.\n\nMoreover, the magnetic field of the SM2 is measured with an NMRmeter that comes with a serial interface which, by use of the Profibus protocol, allows the communication with a standard PC, where a custom c program reads the information transmited, writes it in an ascii file and thereby makes it available for a PVSS API that collects the values and writes them into a datapoint.\n\n\n\\noindent\\section{\\uppercase{The Devices Layer}}\n\\label{Devices}\n\nMany different types of devices need to be controlled or simply monitored by the DCS, from high and low voltage crates and VME crates, to gas systems, sensors of temperature, humidity, pressure and magnetic fields.\n\nCOMPASS uses CAEN crates of different models to power most of its high voltage channels and for part of its low voltage channels. About 20 CAEN crates of older models (with CAENet interface) are in use, and six crates of newer models (with Ethernet interface).\nSeventeen Iseg high voltage modules are also in use and integrated in the DCS by use of their CAN interfaces. \nIn addition, fourteen Wiener low voltage power supplies are controlled, of which four are of type UEP6000, eight of type PL6021 and two of type PL508L. Nineteen VME crates are integrated in the DCS, both of older models (power supplies of type UEP5021) and newer models (power supplies of type UEP6021), the former being the majority. Both the power supplies and the VMEs are controlled by use of their CAN interface.\n\nIn subsystems where PLCs are used, the DCS only monitors the values that are published by them. This happens for the detector gas systems, the CEDAR detectors, and for systems that have dedicated control systems, namely the cryogenic systems of the polarized target, liquid hydrogen target and cold silicon detectors, see \\cite{Cesar, Anibus}.\n\nA wide range of devices are monitored by use of the ELMBs.\n\nHundreds of sensors are installed to monitor specific components of detectors or the experimental hall environment. For temperature monitoring, PT100 sensors in a 4-wire configuration are extensively used, whose output is read using the ELMBs.\n\nSome of the low voltage power supplies used in the experiment only have an interface for monitoring channel voltages or currents by means of voltage signals proportional to the values to be monitored, which is also read by ELMBs. In such cases, a calibration formula is used in the configuration of the CANopen OPC server, to provide the conversion to the real values delivered by the channels.\n\nMoreover, two of the most important magnets of the experiment, SM1 and Bend6, have their magnetic field monitored by Hall probes, whose output signals are read by ELMBs.\n\nIn the case of the second dipole magnet of the experiment, an NMRmeter is used. The NMRmeter has a serial interface, which, by use of the Profibus protocol, allows the communication with a PC.\n\nA custom power switch is also controlled, by use of the web server and driver provided with the equipment.\n\nThe DCS has an indirect monitoring of the powering system plus read-out chain of the four calorimeters of the experiment, based on the calibration signals of a laser system (for ECAL1) or a LED system (for the remaining three calorimeters). A component of the DAQ system calculates a spill-average amplitude of the signal read-out by each of the $\\sim$4500 channels, see \\cite{Konopka}, and saves this information in a mySQL conditions database, that is subsequently accessed by a PVSS script. In the DCS, a reference is chosen by the detector experts; afterwards, the DCS calculates, for each beam spill (using a synchronization scheme with the DAQ), the state of alert of each channel, based on the relative difference of the actual amplitude of the calibration amplitudes with respect to the reference values. The conditions to indicate alarms in the main panel are specified for the total of channels with a given alert state.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.4 \\textwidth]{ecal2.eps}\n\\caption{Panel for monitoring of the high voltage system of ECAL2, comprising around 3000 channels.}\n\\label{ECAL2}\n\\end{figure}\n\nThe main power supplies powering the electromagnetic calorimeters, their monitoring systems, and the subgroups distributor voltages are monitored by use of ELMBs.\n\nThe positions and movement of the electromagnetic calorimeters are controlled by CAMAC\\footnote{Computer Automated Measurement And Control.} modules, from whose readings the detector positions are calculated and read by the DCS.\n\nA recent integration in the main PVSS project is the monitoring of the most relevant parameters of the complex Polarized Target system. The communication with the devices required the usage of different protocols and front-end solutions: PLC S7, ModBus, DIP and ODBC (for MySQL and Oracle database connection). \n\nThe rates of the different triggers of the experiment, mo\\-ni\\-to\\-red online by the shift crew, are stored in a mySQL conditions database read by the DCS, which calculates rates normalised to the beam flux, and triggers alarms when those normalised rates fall outside predefined ranges.\n\nThe servers used in the DAQ run DIM servers to publish data related to internal temperatures, occupancy of their disks and status of important processes.\n\n\n\nThe beam line M2 belongs to CERN's infrastructure and thus is monitored and controlled by dedicated programs. The most important parameters, such as magnet currets, collimator positions, the primary target head or the beam file loaded are made available via DIP, thus providing alarms and and historical values of these parameters.\n\nMoreover, the CEDAR detectors (\\v{C}Erenkov Differential counters with Achromatic Ring focus), used in the hadron program of the experiment, are a responsability of CERN, and its relevant parameters are published using a DIP server. For the operation of these detectors, the density of the gas used must be within a predefined range. When this doesn't happen, the DCS displays a state of alarm, allowing the shift crew to start a procedure to refill the detectors. The high voltage system and the motors are also monitored.\n\n\\noindent\\section{\\uppercase{Conclusions}}\n\\label{Conclusions}\n\nThe DCS of the COMPASS experiment at CERN was presented in detail. This is a centralized system that displays to the end user in a homogeneised graphical user interface many different subsystems that use very different devices and thus require the use of a wide range of front-end solutions.\n\n\\section*{Acknowledgements}\n\nWe gratefully acknowledge the Controls group of CERN (IT\/CO and, later, EN\/ICE) and CERN's PhyDB for their constant and efficient support. This work was supported by FCT.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{table}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{llll}\n\\multicolumn{4}{c}{\\textbf{List of Abbreviations}}                                                               \\\\ \\hline\n\\rowcolor[HTML]{EFEFEF} \n\\textbf{AMI} & Advanced Metering Infrastructure & \\textbf{HMS}  & Head-End Management Server                     \\\\\n\\textbf{CA}  & Certificate Authority            & \\textbf{MAC}  & Medium Access Control                          \\\\\n\\rowcolor[HTML]{EFEFEF} \n\\textbf{CRL} & Certificate Revocation List      & \\textbf{OCSP} & Online Certificate Status Protocol             \\\\\n\\textbf{DHT} & Distributed Hash Table           & \\textbf{NIST} & National Institute of Standards and Technology \\\\\n\\rowcolor[HTML]{EFEFEF} \n\\textbf{DMZ} & Demilitarized Zone               & \\textbf{PKI}  & Public Key Infrastructure                      \\\\\n\\textbf{HES} & Head-End System                  & \\textbf{RSA}  & Rivest\u2013Shamir\u2013Adleman \n\\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\n\nThe existing power grid is currently going through a major transformation to enhance its reliability, resiliency, and efficiency by enabling networks of intelligent electronic devices, distributed generators, and dispersed loads \\citep{farhangi2010path}, which is referred to as \\textit{Smart(er) Grid}. Advanced Metering Infrastructure (AMI) network is one of the renewed components of Smart Grid that helps to collect smart meter data using a two-way communication\\citep{saputro2012}. Smart meters and integrated Internet-of-Things (IoT) devices are typically connected via a wireless mesh network with a gateway (or access point) serving as a relay between the meters and the utility company.\n\nThe security requirements for the AMI network are not different from the conventional networks as confidentiality, authentication, message integrity,  access control, and non-repudiation are all needed to secure the AMI. Confidentiality is required to prevent exposure of customer's private data to unauthorized parties while integrity is necessary to ensure that power readings are not changed for billing fraud. Furthermore, authentication is crucial to prevent any compromised smart meters communicating with other smart meters. On the other hand, the National Institute of Standards and Technology (NIST) urges to use Public Key Infrastructure (PKI) for providing the security requirements of AMI \\citep{nist2014}. As an example, companies such as Landis\\&Gyr and Silver Spring Networks already use PKI to provide security for millions of smart meters in the US\\citep{landisgyr}. In such a PKI, the public-keys for smart meters and utilities are stored in \\textit{certificates} which are issued by Certificate Authorities (CAs). The employment of PKI in AMI requires management of certificates which include the creation, renewal, distribution and revocation. In particular, the certificate revocation and its association with smart meters are critical.\n\\vspace{10pt}\n\\begin{thisnote}\n\\subsection{Problem Description and Existing Solutions}\nSeveral reasons necessitate revoking certificates, such as key compromise, certificate compromise, excluding malicious meters, renewing devices, etc.  Besides, if there is a vulnerability in the algorithms or libraries that are used in certificate generation, a massive number of revocations may additionally occur. For instance, a recent discovery of a chip deficiency on RSA key generation caused revocation of more than 700K certificates of devices that deployed this specific chip \\citep{2017ccsnemec} and renowned heartbleed vulnerability caused the revocation of millions of certificates, immediately \\citep{durumeric2014matter}.  Thus, to establish secure communication, a smart meter should check the status of the other smart meter's certificate against a certificate revocation list (CRL) that keeps all revoked certificates. Considering the large number of smart meters in an AMI and the fact that the expiration period can be even lifelong in particular applications \\citep{landisgyr}, the CRL size will be huge. Consequently, revocation management becomes a burden for the AMI infrastructure which is typically restricted in terms of bandwidth. This overhead is particularly critical since the reliability and efficiency of AMI data communication are crucial for the functionality of the Smart Grid. Considering the potential impact on the performance of AMI applications \\citep{mahmoud2015investigating}, handling the overhead of revocation management is essential.\n\nCertificate revocation management is commonly handled by utilizing CRL that is stored in the smart meter. The status of a smart meter is determined by checking whether its certificate is listed in the CRL or not. An alternative method would be to store the CRL in a remote server as in the case of Online certificate status protocols (OCSPs) \\citep{galperin2013x}\\citep{pettersen2013transport}. In OCSP, an online and interactive certificate status server stores revocation information. Thus, each time a query is sent to the server to check the status of the certificate. While OCSP-like approaches can be advantageous on Internet communications, employing them for AMI is not attractive since it will require access to a remote server for each time. In this regard, another alternative would be to use OCSP \\textit{stapling} \\citep{pettersen2013transport} where the smart meters query the OCSP server at certain intervals and obtain a signed timestamped OCSP response which\nis included (\"stapled\") in the certificate.  Again, this approach also needs frequent access to a remote server. Moreover,  the 'stapled' certificates should be downloaded frequently by smart meters to ensure security, and this will create additional traffic overhead on the AMI which affects applications such as demand response or outage management. \n\n\\subsection{Our Approach and Contributions}\nIn this paper, we propose a communication-efficient revocation or CRL mananegment scheme for AMI networks by using RSA accumulators\\citep{camenisch2002dynamic}. RSA accumulator is a cryptographic tool which is able to represent a set of values with a single accumulator value (i.e., digest a set into a single value). Also, it provides a mechanism to check whether an element  is in the set or not which implicitly means that cryptographic accumulators can be used for efficient membership testing. Due to the attractiveness of size, in this paper, we adapt RSA accumulators for our needs by introducing several novel elements as following:\n\n\\begin{itemize}\n\\item An accumulator manager is introduced within the utility company (UC) that is tasked with collection of CRLs from CAs and accumulating these CRLs (i.e., revoked certificates' serial numbers) to a single accumulator value which will then be distributed to the smart meters. \n\\item We also introduce a non-revoked proof tuple for allowing a smart meter to check whether another meter's certificate is revoked without referring to the CRL file.\n\\item We defined additional entities within AMI and assign functions to them to govern an accumulator based revocation management.\n\\item We introduced several security countermeasures against possible attacks to a accumulator-based scheme.\n\\end{itemize}\n\nThe computation and communication related aspects of the proposed approach is assessed via simulations in ns3 network. In addition, we built an actual testbed using in-house smart meters to assess the performance realistically. We compared our approach with the other methods that use conventional CRL schemes and Bloom-filters  \\citep{akkaya2014efficient}. The results show that the proposed approach significantly outperforms the other existing methods in terms of reducing the communication overhead that is measured with the completion time. The overhead in terms of computation is not major and can be handled in advance within the utility that will not impact the smart meters.\n\nThis paper is organized as follows: In the next two sections, we summarize the related work and the background. Section IV introduces the threat model. Section V presents the proposed approach with its features. Section VI and VII are dedicated to evaluation criteria and experimental validation. Section VIII analyzes the security of the approach. Section IX discusses the benefits and limitations. The paper is concluded in Section X. \n\n\\end{thisnote}\n\n\\label{intro}\n\n\\section{Related Work} \\label{section2}\n\n\\begin{thisnote}\nDue to increasing threats towards Smart Grid, there has been a number of efforts to adapt PKI for Smart Grid communication infrastructure. For instance, Metke et al.\\citep{metke2010security} surveyed the existing key security technologies in Smart Grid domain by mainly focusing on PKI. On the other hand, the study  \\citep{mahmoud2013efficient} stressed the importance of revocation overhead of PKI in Smart Grid. Beyond directly related studies on the PKI and Smart Grid relation, we also focus on studies about cryptographic accumulators and membership management. In this section, we examine the relation between this study and previous studies and highlight major differences.\n\n\\subsection{Revocation Management in AMIs}\nThe studies \\citep{mahmoud2015investigating} investigated different revocation management aspects such as short-lived-certificate scheme, tamper-proof device scheme, Online Certificate Status Protocol (OCSP), conventional CRL, and compressed CRL. However, this study just hypothetically analyzed the applicability of existent revocation solutions for AMI.  The first offered approach that focused on reducing the revocation management overhead for AMI was based on Bloom Filters \\citep{rabieh2015scalable}. They provided a Bloom Filters based scheme particularly to reduce the size CRL. \\end{thisnote} However, since Bloom Filters suffer from false positives, the approach\nrequires accessing the CA to check the validity of a certificate. Our proposed scheme, on the other hand, never\nrequires accessing a remote server and provides a better reduction\non CRL size. The study in \\citep{cebe2017efficient} use distributed hash tables (DHT) to reduce the CRL size again. Although this study provides a reduction in CRL size, it suffers from additional inter-meter communication overhead for accessing the CRL information. \n\\begin{thisnote}\nWe would like to note that a very preliminary version of this work was published in \\citep{cebe2018efficient}. In this work, we improved the various aspects of the previous one. First, we improved computation performance utilizing Euler's Theorem. Second, we extended our threat\nmodel to new attack types that were not considered in the conference version. In this regard, we changed our approach in several ways: We proposed to use an initial secret during accumulation. We then introduced a non-revoked proof concept that was not used before in any of the revocation works. This required major changes to the accumulation process which was not in \\citep{cebe2018efficient}. We finally proposed an extensive certificate verification protocol as countermeasures to the new threats. This also required proposing a new secure multi-level AMI architecture as opposed to the monolithic architecture used in \\citep{cebe2018efficient}. In addition, we added several new experiments with accumulator computation overhead under various assumptions.\n\\end{thisnote}\n\\subsection{Cryptographic Accumulators}\nBenalog and DeMare \\citep{benaloh1993one} first introduced cryptographic accumulators. After their first appearance, there have been studies \\citep{camenisch2002dynamic,reyzin2016efficient,baldimtsi2017accumulators} offering to use them for membership testing. However, these studies solely focused on building the cryptographic fundamentals of accumulators, and thus, omit application-specific issues and security features when deploying them. Besides, these studies are offering to use accumulators for membership testing by accumulating a valid list. Considering AMI, accumulation of valid smart meter's certificates to provide a revocation mechanism would constitute a significant overhead due to the fact that revocation frequency is less than that of creating new certificates {(i.e., no need to update the accumulator each time when a new smart meter is added to AMI)}. Furthermore, since the number of revoked certificates is also less than the number of valid certificates which affects the required computation time significantly\\citep{durumeric2014matter}. \nOur approach mitigates these drawbacks by addressing security and application-specific issues and offering to use CRLs instead of valid certificates.\n\n\n\\section{Preliminaries}\n\\label{section3}\n\\begin{thisnote}\nBefore explaining  our  approach  we  provide  some  cryptographic background of accumulators and its particular form as RSA accumulators. In addition, to help the reader grasp a general idea of revocation management through CRLs, we explain the CRL and delta-CRL notions.\n\\end{thisnote}\n\\subsection{Background on Cryptographic Accumulators}\nBenaloh and De Mare\\citep{benaloh1993one} introduced the cryptographic accumulator concept which is a one-way hash function with a special property of being \\emph{quasi-commutative}. A quasi-commutative function is a special function $\\mathcal{F}$ such that $y_0,y_1,y_2 \\in \\mathbb{Y}:$\n\\begin{equation}\n\\mathcal{F}(\\mathcal{F}(y_0,y_1),y_2)=\\mathcal{F}(\\mathcal{F}(y_0,y_2),y_1)  \\label{quasi}\n\\end{equation}\nThe properties of this function can be summarized as follows: \\textit{1)} it is a one-way function, i.e., hard to invert; \\textit{2)} it is a hash function for obtaining a secure digest $\\mathcal{A}$ (i.e., accumulator value) where $\\mathcal{A} = \\mathcal{F}(\\mathcal{F}(\\mathcal{F}(y_0,y_1),y_2),...,y_n)$ for a set of values $\\{y_0,y_1,y_2, . . . , y_n\\} \\in \\mathbb{Y}$; \\textit{3)} it is a \\emph{quasi-commutative} hash function which is different from other well-known hash functions such that the accumulator value $\\mathcal{A}$ does not depend on the order of $y_i$ accumulations. \n\nThese properties allow cryptographic accumulators to be used for a condensed representation of a set of elements. In addition, since the resulting accumulated hashes of $y_i$ ($\\mathbb{Y}=\\{y_i;~0<i<n\\}$) \nstays the same even if the order of hashing is changed, it can be used for efficient membership testing by using a special value called witness value $w_i$. For instance, the witness $w_i$ of corresponding $y_i$ is calculated by accumulating all $y_j$ except the case where $i \\neq j$ (e.g., $w_i = \\mathcal{F}(\\mathcal{F}(\\mathcal{F}(y_0,y_1),...,y_{j-1},y_{j+1}...,y_n)$). Then, when necessary any of the members can check whether $y_i$ is also a member of the group by just verifying whether $\\mathcal{F}(w_i,y_i)=\\mathcal{A}$. Note that, because $\\mathcal{F}$ is a one-way function, it would be computationally infeasible to obtain $w_i$ from $y_i$ and $\\mathcal{A}$. \nHowever, there is a risk for collusion in this scheme when an adversary can come up with ${w_i}^{'}$ and ${y_i}^{'}$ pairs where ${y_i}^{'} \\notin \\mathbb{Y}$ to obtain the same accumulator value: $\\mathcal{F}({w_i}^{'},{y_i}^{'})=\\mathcal{A}$. In the literature, there is already a cyrptographic accumulator, namely the RSA construction \\citep{baric1997collision} which guarantees that finding such pairs is computationally hard by restricting the inputs to the accumulator function to be prime numbers only. This scheme is known as collision-free accumulator that enables secure membership testing (i.e., without any collision). Therefore, in this paper, we chose to employ RSA construction which is elaborated next. \n\\subsection{RSA Accumulator}\nRSA accumulator\\citep{baric1997collision} has a RSA modulus $\\mathcal{N}=pq $, where $p$ and $q$ are strong primes. The RSA accumulation value $\\mathcal{A}$ is calculated on consecutive modular exponentiation of prime numbers set $\\mathbb{Y} = \\{y_1,...,y_n\\}$ and $g$ is quadratic residue of $\\mathcal{N}$ as follows:\n\\begin{equation}\n\\mathcal{A} = g^{y_1,...,y_n}~(mod~\\mathcal{N})   \\label{eq:accumulator}\n\\end{equation}\nThe witness $w_i$ of corresponding $y_i$ is calculated by accumulating all values\nexcept $y_i$:\n\\begin{equation}\nw_i = g^{y_1,...,y_{i-1},y_{i+1},...,y_n}~(mod~\\mathcal{N})   \\label{eq:witness}\n\\end{equation}\nThen, the membership testing can be done via a simple exponential operation by comparing the result with the accumulator value $\\mathcal{A}$:\n\\begin{equation}\n    w_i^{y_i} \\leftrightarrow \\mathcal{A}\n    \\label{eq:authenticate}\n   \n\\end{equation}\nThe described accumulator scheme so far basically allows generation\nof a ``witnesses\" to prove that an item is in the set. A more advanced accumulator would\noffer proofs of non-membership which proves that an item is \\textbf{NOT} in the set \\citep{li2007universal}. For this scheme, let us assume any $x \\notin \\mathbb{Y} = \\{y_1,...,y_n\\}$. In a nutshell, the non-witness values can be computed by the following steps: Let $u$ denote $\\prod_{i=1}^ny_i$, the scheme finds  non-witness $nw_1,b$ value pairs of $x$ by solving the equation of $nw_1\\times u+b \\times x = 1$ using the Extended Euclidean algorithm. Then, the scheme computes an additional value $nw_2$ such that:\n\\begin{equation}\n{nw_2}=g^{-b}~(mod~\\mathcal{N})\n\\label{eq:witnessCalculation}\n\\end{equation}\nAfter these steps, the item $x$ will have cryptographic proof values $nw_1$ \\& $nw_2$ which can be used to ensure that the item $x$ is \\textbf{NOT} in the set $\\mathbb{Y}$. Then, any third party that posses the $\\mathcal{A}$ value can do the non-membership test of $x$ via a simple exponential operation by checking whether the following equation holds:\n\\begin{equation}\n\\mathcal{A}^{nw_1}  \\leftrightarrow {nw_2}^x \\times g~(mod~\\mathcal{N})\n \\label{eq:authenticate2}\n\n\\end{equation}\nBesides, if a new value $y^{'}$ is added to list, the accumulator value is updated by using the previous accumulator value $\\mathcal{A}$:\n\\begin{equation}\n\\mathcal{A}^{'}  = \n\\mathcal{A}^{y^{'}}~(mod~\\mathcal{N})\n \\label{eq:updateAccumulator}\n\n\\end{equation}\n\n\n\n\n\\subsection{Certificate, CRL and Delta CRLs}\n\nAs we deal with certificates, we would like to also provide some basic background on certificates and their management. Certificates are issued by a CA with a planned lifetime to an expiration date and have unique serial numbers.\nOnce issued, these certificates are valid until their expiration date. However, there are various reasons that cause a certificate to be revoked before the expiration date. These reasons include but not limited to compromise of the corresponding private key, changing the underlying device infrastructure, etc. \n\nRevocation causes each CA regularly issued a signed list called a CRL which is a time-stamped list consisting of serial numbers of revoked certificates and revocation dates. When a PKI-enabled system uses a certificate (for example, for verifying the integrity of a message), that system should not only check the time validity of the certificate, but an additional check is required to determine a certificate's revocation status during the integrity check. To do so, CRL can be checked to determine the status of the certificate.\n\nThere are two main types of CRL: \\textit{full CRLs} and \\textit{delta CRLs}. A full CRL contains the status of all revoked certificates which are not expired yet. \\textit{Delta CRLs}, which is a concept defined in in RFC 5280  \\citep{cooper2008internet}, contain only the status of newly revoked certificates that have been revoked after the issuance of the last \\textit{full CRL} and before the new release of it. Therefore, a \\textit{full CRL} is issued for a limited time frame and should be updated regularly. Until next update time, \\textit{delta CRLs} help keeping track of the newly revoked certificates. When \\textit{delta CRLs} are enabled, the CA can distribute \\textit{full CRLs} at longer intervals (for reducing distribution overhead) and delta CRLs at shorter intervals. An important point about delta CRL concept is that it does not eliminate the requirement of full CRL distribution. The full CRL must still be re-distributed when the previous full CRL expires since CRL has also a lifetime period as certificates and the lifetime period of delta CRLs are dependent on the lifetime of the previous full CRL. This means both the \\textit{full CRL} and \\textit{delta CRL} should be updated regularly by all the potential nodes that will be using them. In the case of AMI, these CRLs may contain thousand of revoked certificate IDs due to longer expiration dates of issued certificates (even lifelong \\citep{landisgyr}) and need to be distributed to the each smart meters which will cause a huge overhead due to their size as will be shown in the experiments.\n\n\n\n\n\n\\section{System and Threat Model}\n\\label{sec:threat}\n\n\\begin{thisnote}\n\n\\begin{figure}[!ht]\n\\begin{centering}\n\\fbox{\n\\includegraphics[scale=0.4]{GeneralViewAccumulatorAMI.png}}\n\\caption{System Model}\n\\vspace{-0mm}\n\\label{fig:initialsystem}\n\\end{centering}\n\\end{figure}\n\nIn this paper, we build a revocation management scheme for a typical AMI infrastructure. Basically, revocation information is collected by the utility company in the forms of CRL files. Each CRL file contains revoked certificates IDs issued by different CAs. Then, all these revocation information are disseminated to AMI through a 4G\/LTE and AMI mesh communication infrastructure. A sample system model is shown in Fig. \\ref{fig:initialsystem}. \n\nThe security of the proposed revocation management scheme depends on the secure implementation of the proposed accumulator-based system. Therefore, we consider the following threats to the security of the proposed approach and identified the relevant security goals. Note that in our attack model, we assume that both the accumulation process within the perimeter and smart meters (outside the perimeter) can be compromised. Besides,  the communication between UC and smart meters is happening on a non-secure medium which means an adversary can eavesdrop the communication both actively and passively. This threat model is very strong and adequate to represent the increasing threats to Smart Grid.\nHowever, a single counter-measure against this threat model would not be sufficient when considering the broad and diverse attack surface of it. To ensure the security of AMI against adversaries, the utility company needs to deploy intrusion prevention systems and proper attack prevention tools as well. Thus, we assume that a PKI inspection system along with an intrusion detection system (IDS) is already deployed and provides device-level controls to protect PKI keys and informs UC in case of any infiltration.\n\\end{thisnote}\n\n\\begin{itemize}[label=\\null]\n\\begin{thisnote}\n\\item \\ballnumber{1} \\textbf{Compromising Smart Meters:} In an attacker's perspective, the meter\/gateway is the entry point to the AMI. The attacker can use a compromised smart meter or impersonate the gateway to apply various attacks.\n\\item \\ballnumber{2} \\textbf{Compromising the UC Servers:} Apparently, compromising the servers within perimeters of UC provides lots of attack opportunities to adversary. The adversary can target AMI by directly attacking revocation management through compromising servers that governs revocation operations.\n\\item \\ballnumber{3} \\textbf{Compromising the Communication:} When UCs are deploying AMI systems, they generally opt-out enabling encryption since IEEE standards does not enforce the UCs to deploy encryption due to various reasons \\citep{IEEE1815}. It makes AMI open to adversaries who can easily eavesdrop whole AMI traffic or a portion of it. This can also pose a threat to revocation management.\n\\end{thisnote}\n\\end{itemize}\n\n\\section{Proposed Approach}\n\\label{sec:proposed}\n\\subsection{Overview}\nThe proposed approach basically eliminates the need to store and distribute CRLs when the devices communicate in a secure manner. Instead of keeping a CRL file for verification of revocation status of certificates, our approach dictates to store at each device (e.g., smart meter, gateway, HES, etc.)  only an accumulator value and a proof which proves the validity of the device's certificate. The accumulator value and proof can be computed at the utility company and distributed to devices in advance. Any updates regarding revoked certificates trigger re-computation of these values. Keeping just two integer values for revocation management brings a lot of efficiency in terms of storage and distribution overhead as will be shown in the Experiments section. In the next subsections, we will explain the details of our approach\n\\subsection{Adaptation of RSA Accumulator for Our Case}\n\\begin{thisnote}\nTo apply the cryptographic accumulators for revocation management, the revocation management needs to be viewed holistically from the lens of systems thinking to ensure security. We took a bottom-up approach while adapting the accumulator scheme to our approach. First, we modified the CRL inputs to meet the requirements for constructing a secure accumulator setup.  Second, we improved the performance of accumulator calculation. Third, the accumulation process was divided into different functions and  their tasks were defined. Then, we introduced new entities to AMI and assigned tasks to them. Lastly, we constructed a revocation check protocol that utilizes the produced accumulator solution. This section covers how we accomplished all these steps in details.\n\\end{thisnote}\n\\begin{itemize}\n\n\\item[\\textbf{a.}] {\\textbf{Integration of CRL and non-witness Concept:}}\n    \nIn the traditional CRL approach, when a smart meter presents its certificate to the recipient meter, that meter needs to verify that the presented certificate is \\textbf{NOT} in the CRL. \nTo be able to employ the accumulator approach, we generate \\textit{non-witness values} for the presenter to prove that it is not in the list.\nWe accumulate the revocation information (stored in CRLs) into a single accumulator value and produce non-membership witnesses for the non-revoked smart meters\n\n\n\n\\item[\\textbf{b.}] {\\textbf{Reducing the Complexity of Accumulator Computation:}}\n\nWhile computing the accumulator value using Eq.~\\ref{eq:accumulator}, the exponent needs to be computed as\n$\\prod_{i=1}^ny_i$ before doing the modular exponentiation. This becomes infeasible when the size of $\\mathbb{Y}$ increases since $u=\\prod_{i=1}^ny_i$ will be $n \\times k$ bits assuming each $y_i$ is a $k$-bit integer. In our approach, we decided to use Euler's Theorem \\citep{rosen2005elementary} to cope with this complexity. With access to the totient of $\\mathcal{N}$ (i.e., $\\phi(\\mathcal{N})$), the exponent of $g$ in accumulation computation will be $u^{'}=\\prod_{i=1}^ny_i~mod~ \\phi(\\mathcal{N}$). Thus, with the knowledge of the totient, it becomes more efficient to compute the required values via reducing the $u$ by $\\phi(\\mathcal{N})$.\n\n\\item[\\textbf{c.}] {\\textbf{Generating Prime Inputs for the Accumulator:}}\n\nFor accumulation, we can use the certificate IDs ($c_{id}$) which are generated by the CAs. However, to ensure a collision-free accumulator, we need to use only prime numbers as dictated by the RSA accumulator. Since CRLs contain arbitrary serial numbers for certificate IDs, it is necessary to compute a prime representative for each certificate ID as an input to the RSA accumulator. Thus, we used the random oracle based prime number generator described in \\citep{papamanthou2008authenticated} to obtain prime representatives of certificates from their serial numbers ($c_{id}$). The scheme basically has a random oracle $\\ohm()$ function which produces a random number $r$ for an input $c_{id}$. We use $\\ohm()$ to find a \\textit{256-bit} number, $d$, which causes the result of the following equation to be a prime number:\n\\begin{equation}\n    y = 2^{256}\\times\\ohm(c_{id})+d\n  \n\\end{equation}\nBy solving this equation, we generate a prime representative $y$ for a revoked certificate. The reader is referred to \\citep{papamanthou2008authenticated} for security proof details of the method.\n\n\n\\item[\\textbf{d.}] {\\textbf{Defining Functions of Revocation Management:}} \\\\\nAfter preparing the inputs, we compiled and modified the offered accumulator structure and proposed the following functions to construct revocation management for AMI. Our RSA accumulator uses the following input sets: $\\mathbb{Y}$ \\textit{is the set of prime representatives of revoked certificates' serial numbers} and  $\\mathbb{X}$ \\textit{is set of prime representative of valid certificates' serial numbers} where $x \\in \\mathbb{X}$ :\n\n\\begin{itemize}\n\\item $aux_{info}, \\mathcal{N} \\leftarrow Setup(k)$: This function is to setup the parameters of the accumulator. It takes $k$ as an input which represents the length of the RSA modulus in bits (e.g., 2048, 4096, etc.) and generates modulus $\\mathcal{N}$ along with $aux_{info}$ which is basically Euler's totient $\\phi(\\mathcal{N})$.\n\n\n\n\\item $\\mathcal{A} \\leftarrow ComputeAcc(\\mathbb{Y},r_k,aux_{info})$: This is the actual function which accumulates revocation information by taking prime representatives of serial numbers set $\\mathbb{Y}$.\nWhile computing the accumulator value, we propose to use an initial random secret prime number $r_k$ as a first exponent ($g^{r_k}$) in Eq.~\\ref{eq:accumulator}.\n\\item $nr_{proof} \\leftarrow ComputeNonRevokedProof(aux_{info},\\mathbb{Y},x)$: This function first computes a pair of non-witness values represented as $(nw_1,nw_2)$ for a valid certificate whose prime representative is $x$. Then, the UC concatenates the non-witness value pair with $x$ and the serial number of the certificate creating a 4-tuple called $nr_{proof}$.\n\n\\item ${0,1} \\leftarrow RevocationCheck(\\mathcal{A},nr_{proof})$: When a smart meter which has a prime representative $x$ wants to authenticate itself to another party, the other one uses $nr_{proof}$ and $\\mathcal{A}$ to verify that $x$ is \\emph{not} in the accumulated revocation list by checking Eq.~\\ref{eq:authenticate2}.\n\\item $\\mathcal{A}^t \\leftarrow UpdateAcc(\\mathcal{A}^{t-1},\\mathbb{Y}^t)$: This function is for updating the accumulator value $\\mathcal{A}$ when the revocation information is updated via deltaCRLs. It  takes a set of prime representatives of corresponding newly revoked certificates $\\mathbb{Y}^t$ and latest accumulator value $\\mathcal{A}^{t-1}$, and returns the new accumulator value $\\mathcal{A}^t$ by utilizing Eq.~\\ref{eq:updateAccumulator}.\n\n\\item ${nr_{proof}}^t \\leftarrow UpdateNonRevokedProof(\\mathcal{A}^{t},\\mathbb{Y}^t,x)$: This function is for updating the non-revoked proof of corresponding valid smart meters when the revocation information is updated via deltaCRLs. It takes a set of prime representatives of corresponding newly revoked certificates $\\mathbb{Y}^t$, the updated accumulator value $\\mathcal{A}^t$, and the prime representative $x$  and returns non-revoked proof ${nr_{proof}}^t$ of smart meter after some additional certificates are revoked by utilizing the process for Eq.~\\ref{eq:witnessCalculation}.\n\n\\end{itemize} \n\n\\end{itemize}\n\n\n\n\nNext, we define the components of the proposed framework. \n\\subsection{Components of Revocation Management System}\nWe propose the system architecture shown in Figure~\\ref{fig:AMI_Inf} to enable the proposed revocation management and to define its interaction with the deployed AMI components. In addition, the newly introduced components of this architecture and their roles in executing the above defined functions are described below: \n\\begin{figure}[ht]\n   \\centering\n   \\includegraphics[width=0.95\\textwidth]{images\/AMI_Infrastructure6.png}\n   \\vspace{-10pt}\n    \\caption{The structure of proposed revocation management.}\n    \\label{fig:AMI_Inf}\n    \\vspace{-10pt}\n\\end{figure}\n\\begin{itemize}\n\\item \\emph{Smart Meters and Gateway:} The smart meters and gateway can directly communicate with each other and with Head-end System (HES) over LTE. Thus, to ensure the security of applications, these devices need to run the $RevocationCheck()$ function and carry the latest $\\mathcal{A}$ and the corresponding $nr_{proof}$.\n\\item \\emph{Head-End System:} HES is an interface between the utility operations center and smart meters, and it is located in  a demilitarized zone (DMZ). The primary function of the HES is collecting the power data from smart meters and transfer them to head-end management servers (HMS).\nSince it has two-way communication with smart meters, it needs to run the $RevocationCheck()$ function and carry the latest $\\mathcal{A}$ and its $nr_{proof}$.\n\\item \\emph{CRL Collector:} The CRL collector plays one of the key roles in our revocation management system. It basically collects CRLs from various CAs and feeds them to the Accumulator Manager. Since it has an open interface to the outside network (communicating with other CAs), it is placed in DMZ area.\n\\item \\emph{Accumulator Manager:} Accumulator Manager is the core of our revocation management scheme. It gets CRL information from the CRL Collector and accumulates them to obtain latest accumulator value. It implements the \\emph{Setup()}, \\emph{ComputeAcc()}, \\emph{ComputeNonRevokedProof()}, \\emph{UpdateAcc()}, and \\emph{UpdateNonRevokedProof()}  functions. Whenever a new accumulator value is calculated at a time $t$, it sends the accumulator value $\\mathcal{A}^t$ and updated ${nr_{proof}}^t$ to the HMS which then forwards them to HES for distributing to the smart meters.\n\n\n\\item \\emph{Head End Management Server:} The collected data is managed within HMS. It basically monitors activity logs, identifies new devices and manages incident response processes. As mentioned, the HMS collects the newly generated $\\mathcal{A}$ and $nr_{proof}$ values and sends them to HES for distribution.\n\\end{itemize}\n\\subsection{Revocation and Certificate Verification Processes}\nIn this section, we describe the proposed revocation scheme and the protocol for certificate verification.  \n\n\\subsubsection{Accumulating the CRL}\nThis process includes two phases namely the setup phase and the update phase which are described below.\n\\begin{itemize}\n\\item \\textbf{The setup phase:}\nIn this phase of our approach, the Accumulator Manager in the UC basically accumulates the revoked certificate IDs in \\emph{full CRLs}. This process works as follows: The \\emph{full CRL} files are read, and each certificate ID and its issuer's public key are concatenated to obtain a unique string that will be input to the accumulator. Note that the issuer's public key is concatenated on purpose to eliminate any duplicates in serial numbers that may come from different CAs. Then, the Accumulator Manager calculates prime representatives for each concatenated string and accumulates these prime representatives to obtain the accumulator value.\nFinally, the Accumulator Manager generates non-revoked proofs (i.e., the 4-tuple $nr_{proof}$) for each end-device (smart meter, gateway, HES, etc.) by using $ComputeNonRevokedProof()$ function.\n\\item \\textbf{The update phase:} This phase is for revocation information updates that can be done through \\textit{delta CRLs}. Due to such updates, the accumulator value $\\mathcal{A}$ and $nr_{proof}$ values should be updated. To update these values, the Accumulator Manager first prepares the prime representatives for the newly revoked certificates (i.e., the ones that are included in the \\textit{delta CRLs}) by following the same approach in the setup phase. It then updates the previously computed accumulator value, $\\mathcal{A}^{t-1}$, by using the $UpdateAcc()$ function to obtain $\\mathcal{A}^t$ which is then used to generate new $nr_{proof}$ tuples for the end devices by using the $UpdateNonRevokedProof()$ function.\n\\end{itemize} \n\\begin{figure}[ht]\n\\vspace{-4pt}\n   \\centering\n   \\includegraphics[scale=.35]{images\/mutual2.png}\n   \\vspace{-10pt}\n    \\caption{Certificate Verification Protocol Scheme.}\n    \\label{fig:mutual}\n   \n\\end{figure}\n\\subsubsection{Certificate Verification Protocol} \nWhen two meters communicate by sending\/receiving signed messages, the signatures in these messages need to be verified. To be able to start the verification process, a receiving device needs to use the public key (for signature verification) presented in the certificate sent to itself. To ensure that this certificate is not revoked, then it needs to initiate a process which we call as certificate verification protocol.  Figure~\\ref{fig:mutual} shows an overview of this process.\nBasically, the receiving device checks the corresponding $nr_{proof}$ tuple's signature to ensure that it is produced by the UC. Once the signature is verified, it then checks whether the the serial number within the tuple is same as the serial number of the provided certificate (i.e., either EndDevice\\#1.cer). For additional security, it also checks the length of the $nw_1\\&nw_2$ to see whether it is equal to the first accumulation setup parameter $k$. Next, it perfoms $RevocationCheck()$ function amd checks whether the provided $nr_{proof}$ is correct. Finally, the signature of connection request message is checked to ensure the integrity and authenticity of the request. If all these steps are successful, the end-device has successfully complete the certificate verification protocol. Note that, without carrying the $nr_{proof}$, a smart meter can not be authenticated even if it has a valid certificate.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Evaluation of the Approach and its Objectives}\n\\begin{thisnote}\nThe main objective of our work is to decrease the dissemination overhead of revocation information on AMI. However,  although any reduction in this overhead is important for the general health of Smart Grid, achieving this goal without sacrificing the security is vital.  Thus, we have determined the following general measures to evaluate the proposed approach in terms of security, communication, computation and storage.\n\\begin{itemize}\n\\item First, we will evaluate distributing process of non-revoked proofs to smart meters to assess the communication-related overhead of our approach.\n\\item Second, since our approach requires computational resources to calculate the non-revoked proofs and accumulator value,  we will evaluate computational costs on UC servers. \n\\item Considering the limited computation resource of a typical smart meter, it is essential to evaluate computational aspects on smart meters as well. Moreover, we will evaluate storage space requirement of our approach for smart meters. \n\\item Finally, we will assess the security of the proposed approach against threats that are defined in Section~\\ref{sec:threat}.\n\\end{itemize}\nWe evaluate the communication, computation and storage overhead of our approach by using the following metrics:\n\\end{thisnote}\n\\begin{thisnote}\n\n\\begin{itemize}\n\\item \\textit{Completion Time}: This metric is defined for communication overhead assessment, which indicates the total elapsed time to complete the distribution of accumulator value and non-revoked proofs to the smart meters from the HES. This metric hints on the communication overhead of revocation management in terms of assessing how it keeps the communication channels busy which are critical for carrying other information.\n\\item \\textit{Computation Time}: This is the metric to measure the total time for completing the required computations such as computation of accumulator value, prime representatives, and revocation check time, etc.\n\n\\item \\textit{Storage}: This metrics indicates the amount of space for storing the CRL information in the meters. \n\\end{itemize}\nFor comparison to our approach, we used two other baselines from the literature: \n\\begin{itemize}\n\\item \\textit{Traditional CRL Method}: Each smart meter keeps the whole CRL \\citep{mahmoud2013efficient} locally which is distributed by the UC. \n\\item \\textit{Bloom Filter Method}: A Bloom filter \\citep{akkaya2014efficient} is used to store revoked certificates information. Note that, we employed \\textit{murmur} hash function, which is a non-cryptographic hash function suitable for \\textit{fast} hash-based lookup, to build this Bloom filter. In this case, the Bloom Filter is distributed to each meter by the UC. \n\\end{itemize}\n\n\\end{thisnote}\n\\section{Security Analysis}\n\\label{sec:security}\n\\begin{thisnote}\nThe security of the AMI depends on the secure implementation  of our approach. Therefore, we considered the threats in Section~\\ref{sec:threat} against the security of the proposed approach and identified the relevant security goals. \n\n\\begin{itemize}[label=\\null]\n\\item \\ballnumber{1} \\textbf{Compromised smart meter attack:} \nAn adversary can accomplish this attack in two different ways. For the first one, the adversary compromises the smart meter, and then the smart meter may be used to perform various attacks to Smart Grid. The UC is responsible for detecting malicious activity by utilizing different tools and sources. After detection, the UC puts the serial number of smart meter's certificate to the accumulation process. The updated $nr_{proof}$ and $\\mathcal{A}$ values are distributed to the other smart meters, HES and gateways. This process basically detaches the compromised smart meter from the AMI. Every other non-compromised components which may interact with this smart meter are no longer be able to interact due to the our revocation check protocol. Once the information of the compromised smart meter is accumulated to obtain a new accumulator value, the attacker can not successfully bypass the revocation check mechanism by using the compromised smart meter itself or its stolen private key and certificate in the future. \n\nFor the second one, the attacker can abuse a vulnerability in the certificate issuing process or steak smart meters' private keys from manufacturers. This time, the CA revokes certificates of the corresponding devices and publish new CRLs. Those CRLs collected by our CRL collectors. These newly revoked certificates are then accumulated, and the corresponding non-revocation proofs are disseminated to AMI. With the updated accumulator values, an adversary can not interact with any of the smart meters, HES, or gateways within AMI by using the revoked certificates.\n\n\\item \\ballnumber{2} \\textbf{Compromising the UC Servers:} In the event of an attack, the adversary's first target will be to compromise the accumulator manager to attack our revocation management. In our scheme, the accumulator manager plays a critical role but can be missed out easily because it is located within UC perimeters. However, the accumulator manager is the Achilles' heel of our approach and should be protected thoroughly. Thus, we investigate three possible attack scenarios and corresponding countermeasures within our revocation system.\n\\begin{enumerate}[label=\\alph*)]\n\\item First, through the architectural design in Figure~\\ref{fig:AMI_Inf} \\emph{accumulator manager} is protected from any attacks by not allowing direct communication from outside of the network through two different firewalls. For instance, the second firewall configuration just allows incoming traffic, which is directly started by the Accumulator Manager to collect CRLs. Thus, it is not easy to access to accumulator manager from outside of the perimeter.\n\\item  Considering the ever-increasing threat environment and improved skills of adversaries, no matter what level of protection our system has, the accumulator manager can get compromised by breaking a path the proposed defense layers. In such a case, the key factor will be how quickly our approach responds to the incident. After detection of the compromise (e.g., attacker steals RSA setting parameters such as $aux\\_info$ and $p\\&q$), the accumulator manager can be migrated to another server, and new $nr_{proof}$ and $\\mathcal{A}$  is computed from scratch by using different RSA primes $p'$ and $q'$ within minutes as shown in Section~\\ref{sec:accmulatormanager}. Then, updated proofs are distributed to smart meters to prevent any further damage. So, our approach offers a pretty easy recovery capability which is important considering critical operations in AMI.\n\\item Third, our scheme is also allowing computation of $nr_{proof}$ without keeping critical security parameters of RSA accumulator settings  RSA (i.e., $aux_{info}$ and $p\\&q$), since stolen $aux_{info}$ and $p\\&q$ values enable a malicious actor to prove any arbitrary statements. These parameters can be deleted once they are used in the  setup phase. In such a case, compromising the accumulator manager does not give any advantage to the adversary to attack revocation management by abusing these parameters. Moreover, the computation of $nr_{proof}$ can still be accomplished for new smart meters or\/and in case of updated revocation information, but it is more computationally intensive as shown in the Experiments Section.\n\\end{enumerate}\n\n\\item \\ballnumber{3} \\textbf{Compromising the Communication:}\nAs stated before, the traffic of AMI is generally unencrypted and causes additional attack surface to our approach. In this subsection, we investigate two possible attack scenarios and countermeasures against them as follows:\n\\begin{enumerate}[label=\\alph*)]\n\\item \\textbf{Accumulator freshness attack:}\nAn eavesdropping attack of AMI traffic poses a unique threat to our approach by combining public revocation information and circulating accumulator values. An attacker may perform a targeted attack if the UC has not updated the accumulator value properly by pinpointing the smart meters that use old accumulator values. However, our approach is robust to this attack since while computing the accumulator value, we use a secret prime number $r_k$ as a first exponent ($g^{r_k}$) in Eq.~\\ref{eq:accumulator}. This prevents inferring the freshness of accumulator value by combining publicly known revocation information and circulated values in unencrypted traffic.\n\n\\item \\textbf{Stolen non-witness attack:} One possible attack can be performed by using $nw_{1\\&2}$ values to masquerade a valid smart meter since it can be obtained easily by eavesdropping. Our protocol is protected from this threat, even if the corresponding non-witness values $nw_{1\\&2}$ of a smart meter are tried to be abused by attacker, the authentication during revocation check will still fail due to the multi-level signature checks. Thus, we relieve the attacks by abusing of stolen $nw_{1\\&2}$ values through a multi-level authentication which combines the signature check with the accumulator check.\n\\end{enumerate}\n\\end{itemize}\n\n\\end{thisnote}\n\n\\section{Performance Evaluation}\n\\label{sec:result}\n\\subsection{Experimental Setup}\nTo assess the performance of the proposed approach, we implemented it in C++ by using FLINT \\citep{hart2011flint}, which is the fastest library for number theory and modular arithmetic operations over large integers.\nFor the RSA modulus generation and prime representatives computation, we used Crypto++ library since it allows thread-safe operations. \nWe prepared a binary-encoded \\textit{full CRL} and \\textit{delta CRL} that have been digitally signed according to RFC 5280 standard and \ncontained 30,000 and 1000 revoked certificates for \\textit{full CRL} and \\textit{delta CRL} respectively. The \\textit{full CRL} was used to compute $\\mathcal{A}$ and  $nr_{proof}$ tuples during the setup phase while the \\textit{delta CRL} ws used for updating both $\\mathcal{A}$ and $nr_{proof}$ tuples.\n\nFor communication overhead assessment, we used the well-known ns-3 simulator \\citep{ns3} \nwhich has a built-in implementation of IEEE 802.11s mesh network standard. The underlying MAC protocol used was 802.11g.\nWe created two different AMI grid topologies that consist of 81 and 196 smart meters. Even though the number of smart meters in our simulation setup is less than a real AMI setup, it still represents a practical setup in terms of the number of hops due to limited transmission range of 802.11g which leads to multiple hops to reach a smart meter from the gateway (e.g., for 81 nodes the average hop count is 6 and for 196 setup average hop count is 9). In a typical AMI setup in the wild, utilities are able to use 900MHz frequency bands \\citep{TinyMesh} which helps to reach thousand of smart meters through a few hops due to the extended transmission range. Unfortunately, ns-3 does not support those frequencies to build a mesh network, and thus we created a simulation environment which reflects similar number of hops as in the wild.    \n\\begin{figure}[!ht]\n\\label{fig:smartmeter12}\n\\begin{minipage}{0.30\\columnwidth}\n\\includegraphics[width=\\columnwidth,height=2.2cm]{images\/smartMeter.png}\n\\\\[3mm]\n\\includegraphics[width=\\columnwidth,height=1.6cm]{images\/protonix.jpg}\n\\hspace*{10pt}\\small{\\textbf{(a)} Smart meter}\n\\end{minipage}\n\\begin{minipage}{0.7\\columnwidth}\n\\includegraphics[width=\\columnwidth,height=42mm]{images\/testbed18.png}\n\\hspace*{40pt} \\small{\\textbf{(b)} Testbed topology}\n\\end{minipage}\n\\caption{AMI Testbed}\n\\vspace{-6pt}\n\\end{figure}\n\n\\begin{thisnote}\nAlthough ns-3 provides very good simulation environment in terms of signal propagation, it still lacks to reflect the effects of real conditions on the signal such as path attenuation, refraction and diffraction while it propagates in wild. To see the effects of such conditions, we also built an IEEE 802.11s-based mesh network comprised of 18 Protronix  Wi-Fi  dongles attached  to  Raspberry-PIs which are integrated with the in-house meters  as shown in Fig. 4a. We carefully dispersed the meters on the floor as shown in Fig.~4b and build the shown multi-hop routing structure among meters by limiting transmission range by decreasing Tx-Power up to by a factor of 16 \\citep{tourrilhes1997wireless}. By such positioning and decreased Tx-Power, we strive to mimic realistic conditions on signal propagation and its effects on multi-hop routing in a real AMI setup.\n\\end{thisnote}\n\n\n\\subsection{Communication Overhead}\n\\subsubsection{Distribution Overhead} In this subsection, we report on the completion time for the non-revoked proofs distribution of our approach with respect to other baselines both in simulation and testbed environments. The results which are shown in Fig.~\\ref{fig:Completion_Time} indicate the accumulator approach significantly reduces the completion time compared to local CRL and bloom filter approaches due to condense accumulating. Even with respect to Bloom filter, which is touted as one of the most efficient methods in the literature, our approach reduced the completion time in approximately more than 10 orders of magnitude. \n\n\\begin{wrapfigure}[9]{l}{0.65\\textwidth}\n\\centering\n  \\vspace{-16pt}\n   \\centering\n   \\includegraphics[width=0.65\\textwidth]{images\/CRLDist.png}\n    \\vspace{-32pt}\n    \\caption{CRL distribution overhead}\n    \\label{fig:Completion_Time}\n\\end{wrapfigure}\nAnother critical observation from the simulation results is the scalability capabilities of our approach. While especially for the local CRL approach, the completion time increases significantly, this is not the case for our approach.\nThis can be attributed to the fact that the accumulator value is independent of the revoked CRL size while the overhead of other methods is proportional to the CRL size. The main overhead of our approach is directly related to the accumulator setting which was 2048 bits in our case. Therefore, even for very large-scale deployments that can have millions of meters, the overhead will not be impacted. In analyzing the experiments results for the testbed, we observe that the completion time takes more time even though the network size is much smaller. This is mainly because of the signal propagation issues such as path attenuation, refraction, interference from other devices, etc. within the building which does not exist in ns-3 simulations. Such issues cause more errors and packet loss and thus increase the re-transmissions to complete all packet distributions. In fact, the AMI infrastructure might have a similar challenge depending on the geographical location (e.g., urban vs rural environments) and thus the distribution of CRL will become even more critical. Therefore, our approach will be more suitable for such environments to reduce the impact from the wild. \n \n\\subsubsection{Update Overhead}\nIn this subsection, we conducted experiments to assess the overhead\nof CRL updates assuming that such updates are done regularly using the \\textit{delta CRL} concept. Fig.~\\ref{fig:updateOverhead} shows revocation update overhead in terms of  the completion time.\n\\begin{wrapfigure}[9]{l}{0.66\\textwidth}\n\\centering\n  \\vspace{-14pt}\n   \\centering\n    \\includegraphics[width=0.65\\textwidth]{images\/deltaCRLOverhead1.png}\n  \\vspace{-18pt}\n  \\caption{CRL update overhead}\n  \\label{fig:updateOverhead}\n\\end{wrapfigure}\nAs in the case of full CRL, our approach significantly outperforms others due to of the size of the delta CRL. However, the results for the Bloom filter approach shows a different trend this time. It performs worse than the local CRL approach. This can be explained as follows:\nFor each updated revocation information, the Bloom filters must be created from scratch to carry both previous and newly revoked certificates. As a result, updating the CRL will take slightly more time than the whole CRL distribution for Bloom filter and thus will take more time than the local CRL approach. Note that the overhead of CRL distribution is proportional to the size of the delta CRL and thus the completion time follows a similar trend with the results in Fig. \\ref{fig:Completion_Time}.\n\nFor the testbed results, we observe a similar which consistent with the simulations.  Again, the completion time is more due to signal propagation and interference issues. \n\\subsection{Computation Overhead}\\label{sec:benchmark}\nWe have demonstrated in the previous subsection that our approach significantly reduces the communication overhead. But, we need to also assess whether such a reduction introduces any major computational overhead. Thus, in this subsection, we investigated a detailed computational overhead of our approach. Specifically, we conducted two types of experiments: 1) We assessed the overhead of the computations due to the accumulation process in the Accumulator Manager. These experiments were conducted on a computer which has 64-bit 2.2GHz CPU with 10 hardware cores, and 32 GB of RAM assuming that these are reasonable assumptions for the computer that will act as the Accumulator Manager. Moreover, we also investigated whether some of these computations can be parallelized to reduce the computation times through multi-thread implementations further; and 2) We assessed the computation time for the $RevocationCheck()$ function in meters by implementing it in a Raspberry Pi (smart meter).\n\n\\subsubsection{Overhead Results for the Accumulator Manager}\\label{sec:accmulatormanager}\nIn this subsection, we present and discuss the overhead at the Accumulator Manager by considering the functions  below:\n\n\\noindent \\textbf{Computing Prime Representatives}:\nTo assess the computational overhead of prime representative generation, we computed prime representatives for different set sizes. Note that since both the valid and revoked certificates' serial numbers are used in our approach, the input size can become huge when AMI scales. \n\\begin{wrapfigure}[9]{l}{0.65\\textwidth}\n\\vspace{-15pt}\n\\centering\n\\includegraphics[width=0.65\\textwidth]{images\/primeRep.png}\n\\vspace{-35pt}\n\\caption{Prime representative computation}\n\\label{fig:primeOverhead}\n\\end{wrapfigure}\nTherefore, we also conducted a benchmark test by using threads to show the parallelization ability of our approach. The results are shown in Fig.~\\ref{fig:primeOverhead}.\nAs can be seen, the computational complexity of the prime representative generation is not overwhelming. $10^5$ representatives can be computed nearly in 1 minute even\nusing a single core. Parallelization reduces the computational complexity by roughly 10 folds which allows computational times in the order of seconds. \n\n\\noindent \\textbf{Computing the Accumulator Value}: \nNext, we benchmark the computation cost of accumulator value according to different CRL sizes as used in the previous experiment. In addition, we also conducted tests to assess the computational difference between our setting (i.e., the Accumulator Manager has all $aux_{info}$ information) and the case where the Accumulator Manager does not have $aux_{info}$ as discussed in Section IV.C.\n\\begin{wrapfigure}[9]{l}{0.65\\textwidth}\n\\vspace{-17pt}\n\\centering\n\\includegraphics[width=0.65\\textwidth]{images\/accumulator.png}\n\\vspace{-36pt}\n\\caption{Accumulator computation}\n\\label{fig:accumulatorOverhead}\n\\end{wrapfigure} \nNote that for the computation of the accumulator value, a parallel implementation was not possible since each step in the computation depends on the previous operation. As seen in Fig.~\\ref{fig:accumulatorOverhead}, the accumulator value is calculated under a minute for $10^5$ revoked certificates even without using $aux_{info}$. However, the availability of $aux_{info}$ significantly reduces the computation time making it possible to finish it milliseconds regardless of the size of the CRL.\n\n\\noindent \\textbf{Computing Non-Revoked Values}: \nFinally, we assessed the overhead of the computation of non-revoked proofs for both the first setup phase by using \\textit{full CRL} and the update phase by  using \\textit{delta CRL}. Again, we conducted tests based on the availability\/lack of  $aux_{info}$ and parallelization ability.Fig.~\\ref{fig:nonwitness1} shows the computation overhead of this function according to different AMI sizes. As seen, $aux_{info}$ makes a significant difference in this case. \n\\begin{wrapfigure}[9]{l}{0.65\\textwidth}\n\\vspace{-16pt}\n\\centering\n\\includegraphics[width=0.65\\textwidth]{images\/nonWitness.png}\n\\vspace{-38pt}\n\\caption{$nr_{proof}$ computation for \\textit{full CRL}}\n\\label{fig:nonwitness1}\n\\end{wrapfigure}\nEven with parallelization, the computational times are still in the order of days which may not be acceptable in an AMI setting. The results indicate that $aux_{info}$ needs to be available for efficient computations. We repeated the same experiment for the $UpdateNonRevokedProof()$ function and observed the same trends since the only change was the size of the CRL (i.e., delta CRL is much smaller). These results were not shown due to space constraints. \n\n\n\\subsubsection{Overhead Results for Smart Meter}\n\\noindent \\textbf{Revocation Check Overhead}:  We looked at the computational time of revocation check operations in smart meter based on the three approaches compared. This is an important experiment to understand the computation overhead of our approach on the smart meter, considering the fact that it has limited resources.\nAs can be seen in Table~\\ref{tbl:CRLCheck}, the elapsed time for a single revocation check is around 10 milliseconds in our approach. Comparing with the other methods, the Bloom Filter has the best results as expected  \nbecause it enables faster checking by efficient hash operations. However, Bloom filter suffers from false-positives which degrades its efficiency by requiring access to the server \\citep{akkaya2014efficient}. Our approach does not have such a problem. While our approach doubles the revocation check time compared to the local CRL method, the time is still pretty fast as it is in the order of milliseconds which does not impact any other operation. This is a negligible overhead given that it brings a considerable space-saving benefit which affects both distribution and storage overhead.\n\\begin{table}[h]\n\\setlength{\\abovecaptionskip}{0cm}\n\\setlength{\\belowcaptionskip}{0cm}\n\\renewcommand{\\arraystretch}{1.1}\n\\caption{Elapsed Revocation Check Time}\n\\label{tbl:CRLCheck}\n\\centering\n\\begin{tabular}{ |m{3.0cm}|m{1.1cm}|m{1.1cm}|m{2.4cm}|  }\n \\hline\n & \\bf{Local CRL} & \\bf{Bloom Filter}& \\bf{Accumulator Approach}\\\\\n \\hline \n Average Time (ms) & 4.1 & 0.06 & 9.8 \\\\\n \\hline \n  \\end{tabular}\n\\end {table}\n\n\\noindent \\textbf{Storage Overhead}: To compare the storage requirements, we identified the needed revocation information size for our approach and compared it with the other approaches, as shown in Table~\\ref{tbl:storage}. As expected, accumulator has a superior advantage since smart meters just need to store a small accumulator value and non-revoked proof value. Local CRL, on the other hand, keeps  the whole CRL list and depending on the number of revoked certificates, it can be huge. For our scenario, the CRL size is around 0.7MB for 30K revoked certificates. While Bloom filter's performance is also promising, it is still not better than our approach and it suffers from false positives as discussed. \n\\begin{table}[h]\n\\setlength{\\abovecaptionskip}{0cm}\n\\setlength{\\belowcaptionskip}{0cm}\n\\renewcommand{\\arraystretch}{1.1}\n\\caption{CRL Storage Overhead}\n\\label{tbl:storage}\n\\centering\n\\begin{tabular}{ |m{3.5cm}|m{1.1cm}|m{1.1cm}|m{2.4cm}|  }\n \\hline\n & \\bf{Local CRL} & \\bf{Bloom Filter}& \\bf{Accumulator Approach}\\\\\n \\hline \n Required Space (MB) & 0.690 & 0.046 & 0.001 \\\\\n \\hline \n  \\end{tabular}\n\\end {table}\n\n\n\n\n\\begin{comment}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[scale=.33]{RR}\n    \\caption{Retransmission Ratio Comparison under varying \\# of meters.}\n    \\label{fig:RR}\n\\end{figure}\n\n\\subsubsection{Communication Overhead} We compared the performance of the approaches in terms of packet retransmission ratio in order to investigate their overhead. Note that for our scenario, the compressed CRL is sent to a smart meter by using 125 CoAP post messages (i.e., the file fits into 125 messages). Bloom filter needs to send nearly 15 CoAP post messages while accumulator needs to send just one CoAP post message to update the revocation information. Thus, the overhead of our approach is much less in terms of  packet count. Figure~\\ref{fig:RR} shows the comparison of these approaches in terms of re-transmission ratio when these post messages are sent. As expected, Bloom filter and accumulator have less retransmission count since there will be less traffic congestion to complete the distribution. Again our approach significantly outperforms Bloom filter in reducing the retransmissions. However, for CRL, this is not the case. Due to large number of packets sent, this increases congestion and causes more re-transmissions.   \n\n\n\\subsubsection{Storage Overhead}\nTo compare the storage requirements, we identified the needed revocation information size for our approach and compared it with the other approaches, as shown in Table~\\ref{tbl:CRLCheck}. As expected, accumulator has a superior advantage since smart meters just needs to store a small accumulator value and non-membership witness value. Local CRL, on the other hand, keeps  the whole CRL list and depending on the number of revoked certificates, it can be huge. For our scenario, the CRL size is nearly 0.1MB for 4.5K revoked certificates. While Bloom filter's performance is also promising, it is still not better than our approach and it suffers from false positives as discussed before. \n\n\\begin{table}[h]\n\\setlength{\\abovecaptionskip}{-0.1cm}\n\\setlength{\\belowcaptionskip}{0cm}\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{CRL Storage Overhead}\n\\label{tbl:CRLCheck}\n\\centering\n\\begin{tabular}{ |p{2.3cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|  }\n \n \\hline\n & \\bf{Local CRL} & \\bf{Bloom Filter}& \\bf{Accumulator}\\\\\n \\hline \n Required Space (kb)   & 102.2 &   6.1  & 0.7 \\\\\n \\hline \n  \\end{tabular}\n\\end {table}\n\n\\end{comment}\n\n\n\n\n\n\n\\section{Benefits and Limitations}\n\\begin{thisnote} \n\nThere are several benefits associated with the use of the proposed approach for revocation management in AMI. Also, we highlighted some challenges and limitations of the approach.\n\n\n\\subsection{Benefits}\n\\begin{enumerate}\n \\item \\textit{Low overhead}: Our approach imposes minimal to no overhead to the smart meters deployed in the AMI and very low overhead to the central servers supporting the revocation management. In general, a revocation check contains at most one additional modular arithmetic operations if compared with the other revocation check methods (considering investigated methods realizes at least one modular arithmetic operation for signature check).  Also, the overhead imposed by disseminating the revocation information to the smart meters is very low. \n \\begin{table}[htb]\n\\label{tbl:applicability}\n\\centering\n\\caption{High-level Comparison of Revocation Management Schemes.}\n\\resizebox{0.95\\textwidth}{!}{%\n\\begin{tabular}{lcccc}\n\\toprule\n\\multicolumn{1}{c}{{\\color[HTML]{000000} }} &\n  {\\color[HTML]{000000} \\textbf{Applicability}} &\n  {\\color[HTML]{000000} \\textbf{\\begin{tabular}[c]{@{}c@{}}Storage\\\\ Overhead\\\\Advantage\\end{tabular}}} &\n  {\\color[HTML]{000000} \\textbf{\\begin{tabular}[c]{@{}c@{}}Communication\\\\\n  Overhead\\\\Advantage\\end{tabular}}} &\n  {\\color[HTML]{000000} \\textbf{Security}} \\\\ \\midrule\n\\rowcolor[HTML]{EFEFEF} \n{\\color[HTML]{000000} OCSP}         & \\Circle & \\CIRCLE & \\Circle & \\CIRCLE \\\\\n{\\color[HTML]{000000} OCSP-Staple}  & \\Circle & \\CIRCLE & \\Circle & \\CIRCLE \\\\\n\\rowcolor[HTML]{EFEFEF} \n{\\color[HTML]{000000} CRL}          & \\LEFTcircle  & \\Circle & \\Circle & \\CIRCLE \\\\\n{\\color[HTML]{000000} Delta CRL}    & \\LEFTcircle & \\CIRCLE & \\LEFTcircle & \\CIRCLE \\\\\n\\rowcolor[HTML]{EFEFEF} \n{\\color[HTML]{000000} Bloom Filter} & \\LEFTcircle & \\CIRCLE & \\LEFTcircle & \\CIRCLE \\\\\n{\\color[HTML]{000000} Our Approach} & \\CIRCLE & \\CIRCLE & \\CIRCLE & \\CIRCLE \\\\\n\\bottomrule\n\\multicolumn{5}{c}{\\small {\\normalsize \\CIRCLE} $=$ offers the benefit; {\\normalsize \\LEFTcircle} $=$ almost offers the benefit; {\\normalsize \\Circle} $=$ does not offer the benefit.} \n\\end{tabular}%\n}\n\\end{table}\n \\item \\textit{Applicability and Security}: The steps used in our approach can be easily implemented to the current AMI infrastructure with few adjustments. We showed that which components of the current AMI setup will be affected and need to be updated with new functionality.  To compare the applicability of our work with its alternatives, we determined four key benefits in total as shown in Table~3.  Our approach collects the revoked certificates information without interrupting the current smart grid operational network setup. However, unlike OCSP or OCSP-stapled methods, it requires extra communication overhead to distribute revocation information. Still, AMI communication infrastructure is not natural to an off-the-shelf OCSP-based solution due to the frequent query requirement, so obviously, it does not carry any advantage for decreasing the communication overhead.\nOn the other hand, our solution outperforms all other methods in terms of introduced distribution overhead. In brief, our conclusion from this comparative evaluation shows that our approach offers the same security benefits as other notable methods while keeping the overhead at the minimum level. \n\n\\item \\textit{A General Revocation Framework for Smart Grid}: Smart Grid is equipped with a myriad of various smart devices and sensors. This represents a new domain for security that is far beyond the traditional air-gapped operational network technology (OT) needs because of investments in distribution technologies such as renewable energy sources like rooftop solars and wind turbines.\nIn this context,  our approach emerges as the first comprehensive solution that adapts the cryptographic accumulators to instrument lightweight revocation management and can be applied different domains in smart grid beyond AMI.\n\\end{enumerate}\n\n\\subsection{Limitations}\n\\begin{enumerate}\n\\item \\textit{Tight Synchronization Requirement}: Cryptographic accumulators are powerful tools for short set representation and secure non-membership proofs.\nHowever, a disadvantage of using an accumulator-based revocation scheme is that the non-revoked proof and accumulator value has to be synchronized between smart meters.\nThis might occur in two ways. First, the accumulator value at the verifier's site is out-of-date, but the non-revoked proof of the prover is updated and vice versa.\nAsynchronous non-revoked proofs and accumulators between communicating smart meters may hinder the authentication operations; thus, AMI should ensure that all smart meters are updated and start to use the new proofs at the same time. Although the requirement for strict synchronization seems prohibitive, the AMI is a well-managed and synchronized network. Because of this characteristic of AMI, the synchronization requirement can be met easily.\n\n\\item \\textit{Not-allow to use Unreliable Distribution Methods}: \nAnother limitation related to synchronization requirement is that an attacker may selectively drop the packets to cause a synchronization problem between smart meters. Thus, any unreliable method for the distribution of non-revoked proofs should be avoided and the UC should ensure that required values are completely reached to smart meters.\n\\end{enumerate}\n\n\\end{thisnote}\n\\section{Conclusion}\nConsidering the overhead of certificate and CRL management in AMI networks, in this paper, we proposed a one-way cryptographic accumulator based approach for maintaining and distributing the revocation information. The framework condenses the CRLs into a short accumulator value and builds a \\emph{secure}, efficient and lightweight revocation mechanism in terms of communication overhead. The approach is inspired by cryptographic accumulators and adopted based on the requirements of AMI. \nThe experiment results indicate that the proposed approach can reduce the distribution completion time significantly for compared to CRL and Bloom filter approaches while introducing only minor additional computational overhead which is handled by the UC. There is no overhead imposed to smart meters.  \n\\begin{thisnote}\nAs future works, we first aim to incorporate an improved accumulator scheme to relax the tight synchronization requirement. Since different smart meters may use different proofs and accumulator values, the proposed method may require some architectural modifications to enable relaxed revocation check but still ensures security.\n\\end{thisnote}\nSecond, instead of using our in-house testbed, we will consider to use a real testbed such as EPIC \\citep{adepu2018epic} to be able to test our approach on a realistic AMI infrastructure with integrated power grid components.\n\n\n\\section{Responses to Reviewer 1}\n\n\n\\begin{itemize}\n\\item \\textit{Comment 1:  It is stated in the paper that the proposed approach is more efficient than traditional CRLs. How is the CRL Update Overhead measured? In its simplest form a CRL update is simply the addition of an entry to the list. Each device then checks the CRL for revoked certificates. This seems to be much more efficient in terms of communication and computation than sending all the messages and performing all the computations needed for the proposed approach. It would also be beneficial for the reader and the conclusion of the paper if the mechanism of traditional CRLs is described in detail and directly compared to the proposed approach with respect to the particular setup.}\n\n\\added{Response: We thank the reviewer for this valuable comment. As pointed out, the the overhead of CRL was not clearly described. To address this issue, Section III.C contains the following discussion in the revised manuscript:}\n\n\\begin{itemize}\n\\item There are two main types of CRL: \\textit{full CRLs} and \\textit{delta CRLs}. A full CRL contains the status of all revoked certificates which are not expired yet. Delta CRLs contain only the status of newly revoked certificates that have been revoked after the issuance of the last full CRL and before the new release of it. Therefore, the most recent version of the CRL or delta CRLs is made available to all the potential nodes that will be using it. \\textcolor{red}{In the case of AMI, these CRLs (e.g., which may contain numerous number of revoked certificates) need to be distributed to the each smart meters which will cause additional overhead due to size of \\textit{full CRLs} or \\textit{delta CRLs} as will be shown in the experiments}.\n\\end{itemize}\n\n\\item \\textit{Comment 2: Section IV is based on a non-witness value rather than the basic approach presented in the previous section. Some of the cryptographic key aspects should be summarized in the paper rather than just referring the reader to reference [21] for all the details}\n\n\\added{Response: In light of the reviewer's comment, we re-structured the Section IV  and add a summary of non-witness concept. We believe that this improvement makes tha manuscript self-explanatory and aligned with the reviewer's comment as following:}\n\n\\begin{itemize}\n\\textcolor{red}{The so far described accumulator scheme  basically allow generation \nof \"witnesses\" to prove that an item is in the set, a certain more advanced accumulator also\noffer proofs of non-membership which proves an item is \\textbf{NOT} in the set \\cite{li2007universal}. For this scheme, let consider for any  $x \\notin \\mathbb{Y} = \\{y_1,...,y_n\\}$, the non-witness values are computed by following steps in a nutshell. Let $u$ denote $\\prod_{i=1}^ny_i$, the scheme finds  non-witness $nw_1,nw_2$ value pairs of $x$ by solving the $nw_1\\times u+nw_2 \\times x = 1$ equation by using extended Euclidean algorithm. Then, the scheme computes a value $d$ such that $d=g^{-nw_2}~(mod~\\mathcal{N})$. After these steps, the non-membership testing can be done via a simple exponential operation by checking whether following equation holds:\n\\begin{equation}\n\\mathcal{A}^{nw_1}  \\leftrightarrow d^x \\times g~(mod~\\mathcal{N})\n \\label{eq:authenticate2}\n\\end{equation}\n}\n\\end{itemize}\n\n\\item \\textit{Comment 3: In Section V.A., please provide a reference for your statement about the choice for Tx-Power values and the positioning of the meters. It is not obvious to the reader that these values are representative.}\n\n\n\\added{Response: The process is a custom process to obtain the multihop topology which is shown in Figure 3.b. In light of the reviewer's comment, Section V.A is  revised as follows for a better description:}\n\n\\textcolor{red}{To obtain the shown multi-hop routing structure among meters}, we decreased the Tx-Power \\textcolor{red}{up to} by a factor of 16 to limit their transmission range. \\textcolor{red}{By such positioning and decreased Tx-Power, we strive to mimic realistic conditions that reflect multi-hop routing in a real AMI setup.}\n\n\\item \\textit{Comment 4:  Section V requires thorough proofreading and improvements to the language.)}\n\n\\added{Response: Based on the reviewer's comment, we will proofread again.}\n\n\n\\end{itemize}\n\n\n\n\\par\\bigskip\\hrul\n\n\n\n\n\\section{Responses to Reviewer 2}\n\n\\begin{itemize}\n\\item \\textit{Comment 1:  What is the difference between this paper and the conference paper given below. I do not think there is a sufficient amount of novel contribution}\n\n\\added{Response: We will solve the difference file issue.}\n\n\\end{itemize}\n\n\n\n\\par\\bigskip\\hrul\n\n\\section{Responses to Reviewer 3}\n\n\\begin{itemize}\n\\item \\textit{Comment 1: The main technical contribution in this paper has already been published in ICC 2018 conference by the same authors: \"M. Cebe and K. Akkaya, \"Efficient Public-Key Revocation Management for Secure Smart Meter Communications Using One-Way Cryptographic Accumulators,\" 2018 IEEE International Conference on Communications (ICC), Kansas City, MO, 2018, pp. 1-6.\"\nThis reference is absent from the state of the art in the paper. There is no reason why this reference should not have been included..}\n\n\\added{Response: We will solve the difference file issue :}\n\n\\item \\textit{Comment 2: A new section on the threat model that is taken into account needs to be added. In this new section, the authors need to lay out the detailed assumptions that are taken into account about the capability and access of the attacker. In this case, the security of the proposed approach can be more clearly assessed under these assumptions.}\n\n\\added{Response: We thank the reviewer for the comment. As suggested, we  now included the lack of Threat Model and Security Analysis sections to the revised manuscript.}\n\n\\begin{itemize}\n\\item \\textcolor{red}{\\textbf{\\textit{Threat Model:}}\nThe security of the proposed approach depends on the secure implementation of the revocation management system. Therefore, we considered the following threats to the security of the proposed approach.  \\ballnumber{1} In an attacker's perspective, the meter\/gateway is the entry point to the AMI. The attacker can use a compromised certificate to force a malicious smart meter to connect to the AMI network or impersonate the gateway to apply different attacks.\n\\ballnumber{2} Compromising the servers that performing accumulator operations could threaten all AMI structure.\n\\ballnumber{3} If an attacker obtains the accumulator value $\\mathcal{A}$ and combines it with the public knowledge of CRL information, he\/she can easily determine which certificates are accumulated in the list. This might lead to deducing the freshness of the accumulator value; as a result, the attacker can perform a chosen attack by using a compromised device which has not been accumulated yet. \n\\ballnumber{4} The security of accumulator approach depends on the security of witness\/non-witness values. If those values are stolen, an attacker can use those values to authenticate itself. \n}\n\\item \\textcolor{red}{\\textbf{\\textit{Security Analysis:}}\n\\label{sec:security}\nOur proposed approach addresses all the threats mentioned in Section \\ref{sec:threat}.\n\\ballnumber{1} \\textbf{Compromised certificate attack:}  The proposed scheme is robust to this type of attack since any compromised certificate information in CRL will be transferred by $nr_{proof}$ values to the AMI. This will prevent authentication of any revoked devices to the AMI.\n\\ballnumber{2} \\textbf{Compromising accumulator manager attack:}\nFirst, through the architectural design in Figure~\\ref{fig:AMI_Inf} the core of our revocation mechanism (i.e \\emph{accumulator manager}) is protected from any attacks by not allowing a direct communication from outside of the network and protected through two different firewalls. Second, our scheme is also allowing computation of $nr_{proof}$ without keeping critical security parameters of RSA accumulator settings  RSA (i.e., $aux_{info}$ and $p\\&q$), since auxiliary data enables a malicious actor to prove arbitrary statements. These parameters can be deleted once they are used initially. In such case, the computation of $nr_{proof}$ can still be accomplished but it may be much more computationally intensive as will be shown in the Experiments Section.\n\\ballnumber{3} \\textbf{Accumulator freshness attack:}\nWhile computing the accumulator value, we use a secret prime number $r_k$ as a first exponent ($g^{r_k}$) in Eq.~\\ref{eq:accumulator}. This prevents an adversary from making a guess about accumulated serial numbers. \n\\ballnumber{4} \\textbf{Stolen non-witness attack:}\nEven if the corresponding non-witness value $nw$ of a smart meter is exposed, the authentication will fail while checking the signatures of the message by the proposed certificate verification protocol in Fig.2..\n}\n\\end{itemize}\n\n\n\\item \\textit{Comment 3: Page 3, section 3.B: There are no constraints provided for the value of \"g\". Is this always the case or does \"g\" needs to satisfy some constraints?}\n\n\\added{Response: We thank the reviewer for the valuable comment. We missed to put the required constraint for \"g\" into the previous text. Now Section III.B of the revised manuscript contains the \"g\" requirement as follows:}\n\\begin{itemize}\n    \\item The RSA accumulation value $\\mathcal{A}$ is calculated on consecutive modular exponentiation of prime numbers set $\\mathbb{Y} = \\{y_1,...,y_n\\}$ \\textcolor{red}{and $g$ is quadratic residue of $\\mathcal{N}$} as follows:\n\\end{itemize}\n\n\n\\item \\textit{Comment 4:  Page 4, section 4.4: The authors need to provide the space and time complexity for each of the functions, especially UpdateAcc() and UpdateNonRevokedProof().}\n\n\\added{Response: Because of the limit on number of pages, I removed all the complexity discussion in the text.}\n\n\n\\item \\textit{Comment 5: Page 7, section 5.D.1: The authors evaluate the effect of the absence of $aux_{info}$ on the computation time of all $nr_{proof}$ of smart meters. Is there any reason why the $aux_{info}$ will be absent?.}\n\n\\added{Response: We thank the reviewer for the comments. $aux_{info}$ has a special importance in order to protect the security of proposed revocation approach according to our threat model which assumes the compromising of Accumulator Manager. As opposed to the fact that the the Accumulator Manager is protected with two firewalls and not allowing any outgoing traffic by our proposed network design, we introduced additional countermeasure by calculating non-witness values without using $aux_{info}$ that might protect the revocation system even in case of a breach.}\n\n\\item \\textit{Comment 6: Page 8, section 5.D.2: \"We observe that our approach has comparable results with the local CRL method which requires a simple text search over complete full CRL file. \" -> In fact, it is double the amount of time as shown in Table 1. The authors should not shy away from pointing this fact, while stressing on the other hand the space saving of their proposed approach}\n\n\\added{Response: We thank the reviewer for the comments.We updated the text accordingly}\n\\begin{itemize}\n    \\item We observe that our approach has comparable results with the local CRL method which requires a simple text search over complete \\textit{full CRL} file, \\textcolor{red}{yet providing huge space saving benefit which effects both distribution and storage overhead.}\n\\end{itemize}\n\n\\end{itemize}\n\n\\item \\textit{Comment 7: Page 8, section 5. D.2: While the results in Table 1 show that the accumulator approach has higher revocation check time, it is hard to get a sense of the significance of the values provided in the table. It is not clear from the text whether the tests were done on the smart meters (and in this case, the technical characteristics of the smart meters are not provided), or on another device.}\n\n\\added{Response: We clarified the experiment is accomplished on the smart meter by adding following text to the revised manuscript}\n\n\\begin{itemize}\n    \\item Finally, we looked at the computational time overhead for checking whether a certificate is revoked or not based on the three approaches compared. \\textcolor{red}{ This is an important experiment to understand the computation overhead our approach on the smart meter, which is shown in Figure 3.a, considering the fact that it has limited resources.}\n\\end{itemize}\n\n\\item \\textit{Comment 8: -A small typo in page 1: \"we focus efficient handling of this issue in this paper.\" -> focus \"on\" efficient\u2026\n}\n\\added{Response: The typo was relieved.}\n\n\\par\\bigskip\\hrul\n\n\\section{Responses to Reviewer 4}\n\\item \\textit{Comment 1: In section 4, it presented a high-level design of the approach but computation details of all algorithms are not given. \n}\n\n\\added{Response: We now give references to the equitations which are introduced in Background section}\n\n\\item \\textit{ Comment 2: In section 3.A, the introduction of one-way accumulator is slightly different from the paper they cited where $y_0$ and $y_1$, $y_2$ comes from different domains. Similarly, in section 3.B, domains of g and other variables are not clearly claimed.. \n}\n\n\\added{Response: This is the same issues raised by Reviewer #1 in Comment #2 and Reviewer #3 in Comment #3. We resolved these issues accordingly}\n\n\\item \\textit{ Comment 3: Since the details of the algorithms and variables are incomplete, the security of the approach cannot be evaluated.\n}\n\n\\added{Response: The revised manuscript now contains additional computational details. Moreover, it also contains Threat Model and Security Analysis sections. }\n\n\\item \\textit{Comment 4. To improve efficiency, CRL is used instead of valid list in the approach. However, there is still possibility that certificates presented by attackers which are not in CRL end up to be invalid ones. Further explanation is needed.}\n\n\\added{Response: We think Section V.D explains enough the authentication process. Nevertheless, I have added following text to help relieving confusions.}\n\n\\begin{itemize}\n\\item When two meters communicate by sending\/receiving signed messages, the signatures in these messages need to be verified. To be able to start the verification process, a receiving device needs to use the public key (for signature verification) presented in the certificate sent to itself. To ensure that this certificate is not revoked, then it needs to initiate a process which we call as certificate verification protocol.  Figure~\\ref{fig:mutual} shows an overview of this process.\nBasically, the receiving device checks the corresponding $nr_{proof}$ tuple's signature to ensure that it is produced by the UC. Once the signature is verified, it then checks whether the the serial number within the tuple is same as the serial number of the provided certificate (i.e., either EndDevice\\#1.cer). For additional security, it also checks the length of the $nw_1\\&nw_2$ to see whether it is equal to the first accumulation setup parameter $k$. Finally, by performing  $RevocationCheck()$ function, it checks whether the provided $nr_{proof}$ is correct. If all these steps are successful, the end-device has successfully complete the certificate verification protocol. \\textcolor{red}{Note that, without carrying the $nr_{proof}$ a smart meter can not be authenticated even if it has a valid certificate.}\n\\end{itemize}\n\n\\item \\textit{Comment 5: The implementation in simulation network consist of 81 and 196 meters which is far less than the scale in real smart grid, thus the results may not be credibly applied.\n}\n\n\\added{Response: Thanks to the reviewer for pointing out the fact that the produced simulation environment not exactly represent the real AMI size. However, to enable a larger AMI simulation, we should create our simulation environment by using  Parallel Programming MPI + OpenMP and some sort of the cluster hardware. We believe that it will be out of scope of this study since the results clearly shows the scalibility of our approach by providing a revocation check mechanism which is independent from the size of CRL. In addition, the results shows that  }\n\n\\item \\textit{Comment 6: There are some typos in section 2 and section 3. For example, \"collusion\" in line 21, section 3.A might be \"collision\".\n}\n\n\\added{Response: The typo is resolved.}\n\n\\par\\bigskip\\hrul\n\n\\section{Responses to Reviewer 5}\n\\item \\textit{Comment 1: About the use of CRLs, why ask the smart meter to store CRLs? I think the CRLs should be maintained by CAs. Whenever a smart meter MA needs to verify the validity of another smart meter MB's certificate, it just needs to download the latest certificate of the CA who signs MB's certificate. That is, the smart meter does not need to store the CRLs. If this is the case, then the motivation of this paper is not strong enough.}\n\n\\added{In light of the reviewer's comment, we think that could not clearly give the motivation of the problem in the Introduction section. Thus, we added following paragraph to the Introduction section. We hope this will help to unravel the motivation of the problem more clearly.}\n\n\\begin{itemize}\n    \\item \\textcolor{red}{An alternative method would be to store the CRL in a remote server where an online and interactive certificate status server stores the revocation information as in the case of Online Certificate Status Protocol(OCSP) \\cite{galperin2013x}. By this way, a query can be sent to  the server to check the status of a certificate whenever two meters would like to communicate. While OCSP can be advantageous for systems where Internet connections are always on, employing it for AMI is not an attractive idea since it will require access to a remote server for each interaction. In this regard, another alternative would be to use OCSP \\textit{stapling} \\cite{pettersen2013transport} where the smart meters query the OCSP server at certain intervals and obtains a timestamped OCSP response which is directly signed by the CA. This response is included (i.e., \"stapled\") in the certificate as a proof that it is not revoked.  However, again, this approach requires frequent remote access. Furthermore, in order to perform authentication properly, the \"stapled\" smart meters' certificates should be download frequently by all the smart meters even in case there is no revocation incident, which will create enormous traffic on the AMI network. \n}\n\\end{itemize}\n\n\\item \\textit{Comment 2: In Section II.B, the authors claimed that their approach mitigated drawbacks of existing accumulator schemes by addressing security and application specific issues and offering to use CRLs instead of valid certificates in sense that the size of CRLs is much smaller than that of valid certificates. However, if the accumulator outputs a constant size of data, then it does not matter whether you accumulate a valid list or accumulate the CRLs.}\n\n\\added{Response: The reviewer is right about accumulating either valid or non-valid certificates would not be different  in terms of accmulator size. However, the computation overhead and the number of times to update the accumulator value would be different. To clarify this issue, we modified the revised manuscript as follow:}\n\n\\begin{itemize}\n\\item Considering AMI, accumulation of valid smart meter's certificates to provide a revocation mechanism\nwould constitute a significant overhead due to the fact that revocation frequency is less than that of creating new certificates \\textcolor{red}{(i.e., no need to update accumulator for each new smart meter addition to AMI)} and number of revoked certificates is also less than the number of valid certificates \\textcolor{red}{ (i.e in terms of computation time)}\\cite{durumeric2014matter}.\n\\end{itemize}\n\n\\item \\textit{Comment 3:  The proposed approach relies on the modification of RSA accumulator for the AMI case. However, the modifications inherit existing schemes [10] and the second part reducing the complexity of accumulator computation by using Euler's Theorem is actually common knowledge and thus not new. So what are the design challenges?}\n\n\\added{Response: As the reviewer pointed out, we are adapting existing cryptographic methods for revocation management. However, the adaptation of the existing cryptographic operations implicitly carry a number of different challenges which we resolved in our approach. The first challenge as pointed out is reducing the computation overhead by using Euler's theorem. The second is related to the adaption of certificate serial numbers to RSA accumulator. RSA accumulator requires special type of input to proper cryptographic operations. We implemented the method in [21] to produce a proper input to RSA accumulator from  certificate serial numbers. Since the security of the accumulator is critical, we proposed a network topology and related additional components. Fourth, as we described in our Threat Model, to mitigate a possible attack regarding the \\textit{freshness of accumulator values}, we introduced a random secret $r_k$ accumulation.  Fifth, we introduced $nr_{proof}$ concept and revocation check mechanism which is described in Figure 2 to mitigate any \\textit{stolen non-witness attack}. Without these modification, current accumulation approach can only mitigate a \\textit{comprised  certificate attack}. In addition,  naive application of the accumulator concept to the AMI for revocation management would be lack of many practical aspects as well who will compute accumulator value, how accumulator values should be updated to ensure the security of the mechanism through existing components, etc,.}\n\n\\item \\textit{Comment 4: The proposed algorithms are vaguely described, so it is not clear how these algorithms work.\n}\n\n\\added{Response: We have added additional description to Background section and its relation with our algorithm such as which equation used in which function in the Proposed appraoch section}\n\\item \\textit{Comment 5: The security analysis of the proposed protocol is missing.\n}\n\n\\added{Response: Now the revised text contains both Threat Model and Security Analysis.}\n\n\\section*{Acknowledgement}\nThis material is based upon work supported by  the Department of Energy under Award Number DE-OE0000779.\n\\vspace{2\\baselineskip}\n\n\\begin{comment}\n\\begin{wrapfigure}[5]{l}{0.20\\textwidth}\n\\vspace{-9pt}\n\\centering\n\\includegraphics[width=0.20\\textwidth]{mumin.jpg}\n\\vspace{-34pt}\n\\end{wrapfigure}\n\\noindent Mumin Cebe is a PhD student in the Department of Electrical and Computer Engineering at Florida International University. He works at the Advanced Wireless and Security Lab (ADWISE) which is lead by Prof. Kemal Akkaya. He conducts research in the areas of wireless networking and security\/privacy that relates to Internet-of-Things (IoT) and Cyber-physical Systems (CPS), particularly in Smart Grids and Vehicular Networks.\n\n\\vspace{2\\baselineskip}\n\n\\begin{wrapfigure}[6]{l}{0.20\\textwidth}\n\\vspace{-9pt}\n\\centering\n\\includegraphics[width=0.20\\textwidth]{Akkaya.jpg}\n\\vspace{-34pt}\n\\end{wrapfigure}\n\\noindent Kemal Akkaya is a professor in the Department of Electrical and Computer Engineering at Florida International University. His current research interests include security and privacy, internet-of-things, and cyber-physical systems. He is the area editor of Elsevier Ad Hoc Network Journal and serves on the editorial board of IEEE Communication Surveys and Tutorials. He has published over 120 papers in peer reviewed journal and conferences. He has received ``Top Cited'' article award from Elsevier.\n\n\\end{comment}\n\n\\section*{References}\n\n\n\\section{Responses to Reviewer 1}\n\n\\begin{itemize}\n\n\\item \\textit{Comment 1: The paper is interesting, motivation is clear, the manuscript is well organized, and the authors able to convey their intended message. The paper explains the results in a clear manner.}\n\n\\added{\\textbf{Response}: We would like to thank the reviewer for his\/her appreciating comments.}\n\n\\item \\textit{Comment 2: It would be nice to list the limitations of current work precisely. If possible, discuss possible ways to attacks on the proposed certification revocation mechanism.}\n\n\\added{\\textbf{Response}: We thank the reviewer for this comment. In light of this comment, we further expanded our manuscript with a ``Benefits and Limitations'' Section to demonstrate the limitations of our approach where we discuss a possible side attack utilizing those limitations along with it precautions. Please refer to Subsection 9.2 for corresponding changes in the revised manuscript. (Page 32-33)}\n\n\n\\end{itemize}\n\n\\par\\bigskip\\hrul\n\n\n\\section{Responses to Reviewer 2}\n\n\\begin{itemize}\n\\item \\textit{The paper addresses the challenge of distributing and storing the certificate revocation list of smart metering systems under consideration of security protection. To avoid space limitations the paper proposes the use of cryptographic accumulators which shall allow reducing the size of needed revocation information. POINTS FOR IMPROVEMENT:}\n\\end{itemize}\n\\added{\\textbf{Response}: Thank you for the kind feedback. We addressed those issues in the revised manuscript as detailed below:}\n\\bigskip\n\n\\begin{itemize}\n\\item \\textit{Comment 1: FIU: Please introduce all abbreviations first. In the abstract try to avoid abbreviations at all. \nIn the abstract, the reader does not need to know that this has been applied at the FIU as this is obvious from the author affiliations and can be clarified in the detailed sections of the paper. Keep the Abstract short.}\n\n\\added{\\textbf{Response}: We thank the reviewer for this comment. We edited the abstract and removed all abbreviations and made abstract more succinct in the revised manuscript. In addition, we added a ``List of Abbreviations'' table at the end of the second page. \n}\n\n\n\\item \\textit{Comment 2: It does not become obvious in the introduction what the contribution will be. Do lines 73-80 introduce a second contribution? Make the contribution(s) of the paper clear, list them explicitly.\nHow does the proposed solutions relate to the problem introduced? This does not become clear. It might be advisable to restructure the introduction with clear subsections, i.e. background and motivation, problem statement, contribution, outline.}\n\n\\added{\\textbf{Response}: Thank you for the valuable suggestion. In light of the reviewer's comments, we clarified our position by restructuring the Introduction completely and precisely described the problem, motivation and our contribution in different subtitles. Please refer to pages 2-5 in the revised manuscript.}  \n\n\n\\item \\textit{Comment 3: The related work is in need of a brief introduction. The reader does not get to know which related work is explained, why this is related and how this relates to the remainder of the paper.}\n\n\\added{\\textbf{Response}: In light of the reviewer's comment, related work is revised by adding a brief introduction and revising some portions to better describe the relation of our work with the previous ones. Please refer to pages 5-7 in the revised manuscript.}\n\n\n\\item \\textit{Comment 4: In line 115 it is discussed that this work provides 60 more content than in previous work. While this is an important information for the editor, it is not for the reader. The reader needs to know how this work relates to the previous work. Why have things been changed etc.}\n\n\\added{\\textbf{Response}: We thank the reviewer for this comment. We removed that part from the manuscript and re-structured the related work accordingly to resolve raised issues. Please see pages 5-6 in the revised manuscript.}\n\n\\item \\textit{Comment 5: Section 3 is also missing a brief introduction. Please explain to the reader what will happen in this section, why this section is needed and where these foundations are taken from. (Page 7)}\n\n\\added{\\textbf{Response}:\nTo help the reader, we provided a brief introduction to the Section 3 in the revised manuscript.}\n\n\n\\item \\textit{Comment 6: Also for Section 4, explain to the reader why this section is needed and how it relates to the contribution of the paper. How do the 4-points of the treat model relate to the approach. Are they already part of the approach? In this case integrate them with the next section.}\n\n\\added{\\textbf{Response}: We thank the reviewer for the comment. We are sorry to see that our explanations in the paper created a confusion about our threat model that is highly coupled with the approach. Actually, this was not our intention. To clarify this, we have completely modified the threat model section in the revised manuscript. Basically, in the threat model, we are now providing a more generic model where an adversary can both compromise the devices and communication. In addition, we simplified the system model so that it does not also get into approach description. Please refer to pages 10-12 for these changes. }\n\n\\item \\textit{Comment 7: For Section 5.2 please explain what this section is doing.}\n\\added{\\textbf{Response}: \nIn light of the reviewer's comments, we have updated the section and put an introduction to frame our bottom-up approach in Section 5.2 page 13 of the revised manuscript.}\n\n\\item \\textit{Comment 8:  Between Section 5 and 6 a Section explaining the evaluation setup is needed. Please make clear what should be evaluated, why this shall be evaluated, and how this will show that the approach is feasible }\n\n\\added{\\textbf{Response}:  We agreed that a section about evaluation would give a broad idea to reader on our evaluation criteria. Thus, we have added the requested section named \\textit{``Evaluation of the Approach and its Objective''} in the revised manuscript. (Pages 19-20)}\n\n\n\\item \\textit{Comment 9: Section 6 only lists loose proposals why the approach supports the threats from section 4. It is in need of deeper argumentation to explain or show why this is feasible. }\n\n\\added{\\textbf{Response}:  Thanks for this valuable comment.  This feedback help us to realize missing points related to the security analysis of our work. Thus, we now provided a deeper analysis and discussion for security analysis in the revised manuscript. Please refer to the \\textit{''Security Analysis''} for revised version. (Pages 28-31)}\n\n\n\\item \\textit{Comment 10: A discussion section is missing. Particularly, a discussion section should summarize the findings, discuss the limitations\/TTV, and deduce insights that go beyond the concrete technique and contribute to the state of the art in general. }\n\n\\added{\\textbf{Response}: In light of reviewer's comments, we added a new section to summarize our findings and also pointed out potential limitations of the proposed approach. Please refer to ``Benefits and Limitations'' section of the revised manuscript. (Pages 31-33)}\n\n\\item \\textit{Comment 11: Line 49: revoked certificates serial numbers -> revoked certificates' serial numbers\n}\n\n\\added{\\textbf{Response}: Corrected accordingly.}\n\n\\item \\textit{Comment 12: Line 89: remove one \"aspects such as\"\n}\n\n\\added{\\textbf{Response}: Corrected accordingly.}\n\n\\end{itemize}\n\n\\newpage\n\\section{Responses to Reviewer 5}\n\n\\begin{itemize}\n\n\\item \\textit{Comment 1: The objectives and potential impact of the research is not clearly presented. It should be included as a separate section in \"5 Proposed approach\"}\n\n\\added{\\textbf{Response}: Thanks for this valuable comment. This feedback help us to realize missing points related to the main objective and potential impact. As a result, we made some changes and now clearly state our objective and potential impacts in two different sections. The objective of our approach along with evaluation criteria are given in Section 6 after Section 5 as a separate section (page 19). The potential impacts of study are given in Section 9.1 on pages 31-32 of the revised manuscript.}\n\n\n\n\\item \\textit{Comment 2: The experimental simulation should be compared with a real world experiment and potential differences should be pointed out. Please also explain what is the meaning of \"realistic results\" in the paragraph: \"Finally, for more realistic results, we built an IEEE 802.11s-based mesh\"}\n\n\\added{\\textbf{Response}: The reviewer is right in pointing out what realistic refers to as we did not provide details about it. Basically, we built an actual mesh network at FIU by using Raspberry PIs and IEEE 802.11  antennas. Therefore, we updated manuscript accordingly to clarify the goal behind building such a testbed environment on pages 21-22 of the revised manuscript.}\n\n\\item \\textit{Comment 3: A security analysis is presented in section 6 but these aspects need to be addressed extensively in section 7 as well.}\n\n\\added{\\textbf{Response}: In light of reviewer's comment, we extended the security analysis section and moved it after experiments to be able emphasize obtained results and their affects on security. Please refer to Section 8 in the revised manuscript (Pages 28-31).  }\n\n\\item \\textit{Comment 4: The added value of the research in comparison to other methods is not clearly addressed in the conclusion section. Some aspects are addressed in each of sub-sections of section 7. A suggestion is to address these aspects in a structured manner, pointing out the advantages and potential disadvantages in a separate section before conclusions.}\n\n\\added{\\textbf{Response}:  Thank you the reviewer to stressing this important shortfall of the manuscript. As a result, we introduced a new ``Benefits and Limitations'' section. Please refer pages 31-34 for details in the revised manuscript.}\n\n\\end{itemize}\n\\end{document}\n\n\n\\section{Responses to Reviewer 1}\n\n\\begin{itemize}\n\n\\item \\textit{Comment 1: All the reviewer comments were addressed in the revised manuscript.}\n\n\\added{\\textbf{Response}: We thank the reviewer again for his\/her feedback which helped us to significantly improve the paper.}\n\n\\item \\textit{Comment 2:  would recommend authors to add \"how the scalability or any other issues needs to handle when applied to the real testbeds such as EPIC [1]  or real city-scale AMI metering infrastructure? \"\n}\n\n\\added{\\textbf{Response}: We thank the reviewer informing us  a real testbed which is available to researchers. In light of this comment, we further expanded our future work where we discuss possibility of using a real testbed to assess our approach. Please refer to Section 10 for corresponding changes in the revised manuscript. (Page 34)}\n\n\n\\end{itemize}\n\n\\par\\bigskip\\hrul\n\n\n\\section{Responses to Reviewer 2}\n\n\\begin{itemize}\n\\item \\textit{Comment 1: The manuscript has considerably been revised. A great effort has been made to address all reviewer comments. The paper is in a state where it can be accepted as is. }\n\n\\added{\\textbf{Response}: We would like to thank the reviewer for his\/her appreciating comments.\n}\n\n\n\\item \\textit{Comment 2: There is one minor remark the authors might want to reconsider. The introduction section got rather long, which is not a bad thing in case of this paper. However, it might be advisable to use subsection headings within the introduction to guide the reader a bit more }\n\n\\added{\\textbf{Response}: Thank you for the valuable suggestion. In light of the reviewer's comments, we edited the introduction by adding a proper subsections introduction. Please refer to pages 2-5 in the revised manuscript.}  \n\n\\end{itemize}\n\n\\newpage\n\\section{Responses to Reviewer 5}\n\n\\begin{itemize}\n\n\\item \\textit{Comment 1: The authors have addressed the comments\"}\n\n\\added{\\textbf{Response}: Thanks again for valuable comments during revision process to help us to realize missing points of our manuscript.}\n\n\n\n\\end{itemize}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTopological crystalline phases in non-interacting, clean structures have attracted a great deal of recent theoretical and experimental attention\\cite{Fu2011,Hsieh2012,ando2015topological,Hughes11,Turner2010,Turner2012}. \nFrom the discovery of helical edge states in $\\mathbb{Z}_2$ topological insulators\\cite{Kane04,bernevig2006quantum,konig2007quantum}, to surface Dirac cones protected by time-reversal or crystal symmetries\\cite{fukanemele,xia2009observation,Hsieh2012,Hourglass}, the experimental manifestations of band topology have come primarily through the exploration of surface states. \nThe recent theoretical prediction of higher-order topological insulators\\cite{hotis,Po2017,khalaf2018higher,benalcazar2017quantized} has triggered a wave of materials predictions\\cite{NaturePaper,bigmaterials,bigmaterials-china,ashvin-materials,xu2020high} and experimental efforts to observe their predicted gapped surfaces but gapless corners (in 2D) or hinges (in 3D). \nAt a theoretical level, topological band insulators can be classified by exploiting the constraints of symmetry, relating the topology of bands to the transformation properties of Bloch functions under crystal symmetries\\cite{Kruthoff2016,NaturePaper,Po2017,Fu2007,po2020symmetry,cano2020band,MTQC,watanabe2018structure,Zhang_TMCI_corep_RPB,Slager_magnetic_spacegroup_rep}. \nIn the simplest cases, the symmetry eigenvalues of occupied electronic wavefunctions at different crystal momenta in the Brillouin zone can be used to deduce the absence of an exponentially localized, position space description of the occupied states, and hence the presence of non-trivial topology. \nProgress along these lines has led to a full, predictive classification of topological band structures both with and without time-reversal symmetry. \nEssential to these efforts is the presence of discrete translation symmetry, which ensures that localized electronic functions are identical in each unit cell, and hence allows the symmetry properties of the system to be described as a function of momentum. \n\nAt the same time, the interplay between topological bands and symmetry-breaking order has started to attract a great deal of attention. \nIt has been argued theoretically that in topological systems with charge-density wave (CDW) order, the collective phason mode of the CDW may inherit topological properties from the Fermi sea, such as an induced axion coupling to electromagetic fields\\cite{wang2013chiral,you2016response,BurkovCDW,zyuzin2012weyl,maciejko2014weyl}. \nSignatures of this axion coupling have been recently experimentally detected in (TaSe$_4$)$_2$I\\cite{gooth2019evidence,shi2019charge}. \nAdditionally, the quantum anomalous Hall phase in the Dirac semimetal ZrTe$_5$ can be understood as originating from a magnetic-field induced CDW transition\\cite{tang2019three,qin2020theory,song2017instability,zhang2017transport}. \nBecause CDW order is in general incommensurate with the underlying lattice, a full understanding of the interplay between mean-field CDW order and band topology requires us to examine topology of incommensurately modulated electronic systems. \nSuch a study would also yield insights into topology in artificially modulated photonic\\cite{ozawa2018topological,ozawa2016synthetic}, metamaterial\\cite{grinberg2020robust}, and cold-atomic lattice systems\\cite{Ane1602685}, which have become a focus of recent research due to their tunability and experimental accessibility.\n\n\n\nNaively, the breaking of translational and point group symmetries implied by incommensurate modulation would seem to prohibit the application of symmetry-based tools which have been so successful in identifying and classifying topological crystalline systems. \nHowever, it is often possible to view the single-particle dynamics in an incommensurately modulated system as describing the behavior of a particle in a larger number of dimensions, the phase offsets of the incommensurate modulations playing the role of momenta in the extra ``synthetic'' dimensions. \nThe canonical example of this mapping is the 1D Harper (Aubry-Andr\\'{e}) model with incommensurate on-site potential. \nAs was shown some time ago\\cite{hofstadter-original}, the Hamiltonian for the Harper model is equivalent to the Hamiltonian for a 2D square-lattice system coupled to a background magnetic field\\cite{equivalence_Fibo_Harper_2012,Kraus_1D_QC_to_2D_QHE}. \nThe phase of the on-site modulation plays the role of the momentum in the second, synthetic dimension, while the wavevector of the modulation plays the role of the magnetic flux per plaquette in the 2D lattice. \nBands in the enhanced, 2D system can be characterized by a Chern number, which mandates the presence of gapless chiral modes at the edges of the system. \nReducing back to 1D, these two-dimensional edge states manifest as boundary states of a 1D wire which appear and disappear as a function of the phase of modulation, thus realizing a Thouless pump\\cite{Thouless_pump_original_paper,niu1984quantised,Topologically_quantized_current_PRR}. \n\\textcolor{black}{Recent studies also show that certain generalization of the 1D Harper model allows for the investigation of higher-order topological phases~\\cite{Zeng_generalized_AAH_model_HOTI_PRB}.} \n\n\nIn this work, we will extend the connection beyond 1D, to show how modulated systems in 2D and 3D can be related to topological crystalline phases in higher dimensions. \nWe will first review a general method for representing a modulated system as a higher dimensional system coupled to a background gauge field\\cite{RiceMele,Thouless_pump_original_paper,Kraus_1D_QC_to_2D_QHE,equivalence_Fibo_Harper_2012}.\nFor systems with negligible spin-orbit coupling and spin-independent modulation, the gauge field will be a $U(1)$ magnetic field; for spin-dependent modulations we will show that there can also be induced $SU(2)$ gauge fields. \nWe will exploit the fact that both $U(1)$ and $SU(2)$ gauge fields with constant field strength preserve inversion symmetry to show that 2D modulated systems can realize higher-order chiral ($U(1)$) and helical ($SU(2)$) topological phases in one extra synthetic dimension. \nWe show how the hinge states of these synthetic higher-order topological insulators (HOTIs) manifest as corner modes in 2D, with energies that can be tuned by changing the phase of the modulation. \nGoing further, we use the mapping to synthetic dimensions to bring order to the complex landscape of eigenstates of the modulated system, showing how the states can be interpreted as bulk and surface Landau level (LL) wavefunctions in synthetic dimensions. Finally, we also revisit a 3D minimal model for a Weyl semimetal (WSM) with (generally incommensurate) CDW order\\cite{dynamical_axion_insulator_BB}, and show how it realizes a 4D nodal line semimetal gapped into a phase with a non-trivial second Chern number. We will verify our conclusions with a combination of exact numerical results and approximate low-energy analytic calculations.\nWe will also exploit the fact that the phase of a (charge- or spin-) density wave (DW) order parameter can be shifted with an applied electromagnetic field, by exciting the (nominally gapless, but sometimes pinned) sliding mode\\cite{gruner1988dynamics}. \nThis will allow us to make predictions about topological pumping of boundary states in modulated structures, driven by the sliding mode of the DW.\nIn contrast to other recent proposals for topological pumping in synthetic dimensions, the coupling of the DW sliding mode to electromagnetic fields allows for tunability of synthetic dimensions in modulated structures. \nWe will comment on potential experimental realizations in condensed matter, photonic, and cold-atom systems throughout. \nThis work will enable new avenues for exploring higher-order topological phenomena which, with the exception of some promising results in Bismuth\\cite{hsu2019topology,nayak2019resolving,schindler2018higher}, have not been unambiguously identified in crystalline electronic systems.\n\n\nThe rest of the paper is organized as follows. \nIn Sec.~\\ref{sec_review_thouless_pump_Rice_Mele}, we review how the Thouless pump in a 1D Rice-Mele [Su-Schrieffer-Heeger (SSH)] chain is realized by the sliding of a CDW, and we review its connection to topology by promoting the model to a 2D $\\pi$-flux lattice. \nIn Sec.~\\ref{sec_Dimension_promotion} we next develop a general method to compute the $U(1)$ gauge fields that are coupled to a higher dimensional models promoted from a low-dimensional modulated system. \nIn Secs.~\\ref{sec_chiral_HOTI} and~\\ref{sec_helical_HOTI_sliding_modes}, we construct 2D modulated systems that can be promoted to 3D chiral and helical HOTIs coupled to $U(1)$ and $SU(2)$ gauge fields, respectively. \nWe demonstrate the pumping of corner modes by the sliding of DWs in these systems via numerical calculations of the energy spectra. \nWe examine the properties of wave functions in these 2D modulated systems by constructing low energy theories coupled to gauge fields in 3D. \nWe show how the evolution of bulk, edge, and corner states in 2D can be understood from the perspective of the low energy theory in 3D. \nIn Sec.~\\ref{sec:Weyl_CDW}, we turn to a model for a 3D WSM gapped by a CDW. We show that this model can be promoted to a 4D nodal line system gapped by a $U(1)$ gauge field. \nWe derive the corresponding low energy theory in 4D, and use it to explain both the existence of QAH surface states and the interpolation between topologically distinct QAH phases at the two inversion-symmetric values of the CDW phase in this 3D system.\nFinally, in Sec.~\\ref{sec:outlook}, we give an outlook as to how our work may extend the search of (higher-order) topological insulators in higher dimensions and enable simulations of $SU(2)$ gauge physics in higher dimensions. \nSome details of our models, further numerical results, and detailed derivations of the low energy theories are presented in the Supplementary Material (SM)\\cite{SM}. \n\nThroughout this paper, we use units where $\\hbar = c = |e| = 1$, and where the electron has charge $-|e| = -1$. \nFurthermore, the Einstein summation convention will not be used; whenever there is a summation over an index, we will write the summation explicitly.\n\n\n\n\\section{\\label{sec_review_thouless_pump_Rice_Mele}Review - Thouless Pump as Sliding Mode }\n\n\nIn this section, we review the CDW picture of the Rice-Mele (SSH) model\\cite{RiceMele,ssh1979}, and the interpretation of the Thouless pump\\cite{Thouless_pump_original_paper} as a CDW sliding mode. \nConsider the following Hamiltonian for a 1D chain\n\\begin{align}\n    H_{\\text{Rice-Mele}} = \\sum_{n} &  \\left( t + \\delta t (-1)^{n} \\cos{\\phi} \\right)c^{\\dagger}_{n+1}c_{n} + \\text{h.c.} \\nonumber\\\\\n    & + \\sum_{n} (-1)^{n+1} \\Delta  \\sin{\\phi} c^{\\dagger}_{n}c_{n}, \n    \\label{eq:1DRiceMele}\n\\end{align}\nwhere $c^{\\dagger}_{n}$ is the creation operator for an electron at site $n$. \nThe nearest-neighbor hopping and on-site potential are modulated with periodicity $2$, and their relative strength is related to the phase $\\phi$ of the modulation. \nWe thus identify $\\phi$ as the phase of this CDW modulation. In this paper, we use the terms \"CDW sliding phase\" and \"phases of the mean-field CDW order parameter\" interchangeably to refer to $\\phi$. \nFor suitable choices of $t$, $\\delta t$ and $\\Delta$, the spectrum of this Hamiltonian is gapped for all $\\phi \\in [0,2\\pi)$. \nFocusing on the half-filled insulating ground state in this parameter regime, the occupied-band Wannier centers\\cite{Kohn59,Brouder2007,Marzari2012,NaturePaper,shockley1939surface} will be pumped by a length of one unit cell (two sites) as the phase $\\phi$ adiabatically slides from $0$ to $2\\pi$, leading to a quantized change of bulk polarization\\cite{xiao2010berry,Aris2014,RiceMele,ksv}. \nThis quantization has a topological origin: If we regard $\\phi$ as a crystal momentum along a second, synthetic dimension which we call $y$, Eq.~(\\ref{eq:1DRiceMele}) is equivalent to a 2D square lattice model with a $U(1)$ $\\pi$-flux (equivalent to half flux quantum $\\Phi_{0} = 2\\pi \\hbar \/ |e|$ where electron has charge $-|e|$) per plaquette, and with a fixed crystal momentum $k_{y}$ along $y$. \nThe quantized polarization change is then identified as the Chern number\\cite{tknn,niu1984quantised,niu1985quantized,Aris2014,bernevigbook} of the occupied bands in 2D.\nWe provide further details, including numerical verification of charge pumping, and the explicit construction of the dimensional promotion to 2D, in the SM\\cite{SM}.\n\n\nWe see from this example that promoting the dimension of a modulated system to a higher dimensional lattice coupled to gauge fields can help explain the topological origin of low-dimensional properties, including charge transport and boundary modes. \nA general method for dimensional promotion will thus be helpful in dealing with various topological modulated systems in more than 1D. \nIn what follows, we will show that the dimensional promotion approach can be extended to higher dimensions, and to cases where the modulation is incommensurate with the underlying lattice periodicity.\n\n\n\\section{\\label{sec_Dimension_promotion}Dimensional Promotion Procedure}\n\n\nIn this section, we will generalize the $1$D-to-$2$D dimensional promotion of the Rice-Mele chain to general dimensions. \nTo begin, let us consider a $d$-dimensional ($d$D) electronic model on a cubic lattice with $N$ mutually \nincommensurate on-site modulations\\cite{Earliest_dimension_promotion_superspace,Kraus_1D_QC_to_2D_QHE,equivalence_Fibo_Harper_2012,2D_QC_4D_QHE,time_periodic_1,time_periodic_2}, described by the Hamiltonian\n\\begin{equation}\n    H_{\\text{low-dim}} = \\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m}} \\left[{H}_{\\vec{m}}\\right] {\\psi}_{\\vec{n}} + \\sum_{\\vec{n}} \\sum_{i=1}^{N}{\\psi}^{\\dagger}_{\\vec{n}} \\left[{V}^{(i)}_{\\vec{n}}\\right] {\\psi}_{\\vec{n}}.\\label{eq:h_low_dim}\n\\end{equation}\nHere both $\\vec{n} = (n_{1},\\cdots,n_{d})$ and $\\vec{m} = (m_{1},\\cdots,m_{d}) \\in \\mathbb{Z}^{d}$ are vectors in the $d$D cubic lattice, and ${\\psi}^{\\dagger}_{\\vec{n}}$ is the electron creation operator for an electron at position $\\vec{n}$ with a given set of spin and orbital degrees of freedom.\nWe denote by $\\left[{H}_{\\vec{m}}\\right]$ the hopping matrix connecting position $\\vec{n}$ to $\\vec{n}+\\vec{m}$, and by $\\left[{V}^{(i)}_{\\vec{n}}\\right]$ the matrix representing $i^{\\text{th}}$ modulated on-site energy at position $\\vec{n}$ ($i = 1 ,\\ldots, N$), with matrix indices encoding the spin and orbital dependence of the hopping\\footnote{throughout this work, we will use square brackets to denote matrices and matrix-valued functions}. \nNote that hermiticity of the Hamiltonian requires that $\\left[{H}_{\\vec{m}}\\right] = \\left[{H}_{-\\vec{m}} \\right]^\\dag$ {{and $\\left[{V}^{(i)}_{\\vec{n}}\\right]^{\\dagger}=\\left[{V}^{(i)}_{\\vec{n}}\\right]$}}. \nWe further assume that $\\left[{V}^{(i)}_{\\vec{n}}\\right] = \\left[f^{(i)}\\left(2\\pi\\vec{q}^{(i)}\\cdot\\vec{n} + \\phi^{(i)} \\right)\\right]$ with $\\left[f^{(i)}(x)\\right] = \\left[f^{(i)}(x+2\\pi)\\right]$, where $\\vec{q}^{(i)}$ is the $i^{\\text{th}}$ modulation wave vector and $\\phi^{(i)}$ is the sliding phase associated with the $i^{\\text{th}}$ modulation.\nFor the cubic system with unit lattice vectors we are discussing here, each component $q^{(i)}_{j}$, ($j = 1 ,\\ldots, d$) of $\\vec{q}^{(i)}$ is defined within $[0,1)$; that is, each $2\\pi \\vec{q}^{(i)}$ lies within the primitive Brillouin zone of the unmodulated system.\nSince each $\\left[{V}^{(i)}_{\\vec{n}}\\right]$ is a periodic function, they can be expanded in terms of Fourier series as\n\\begin{equation}\n     \\left[{V}^{(i)}_{\\vec{n}}\\right] =\\sum_{p_{i}\\in \\mathbb{Z}} \\left[{V}^{(i)}_{p_{i}}\\right] e^{ip_{i}\\left( 2\\pi\\vec{q}^{(i)}\\cdot\\vec{n} + \\phi^{(i)} \\right)}, \\label{eq:expand_FT_V}\n\\end{equation}\nwhere $\\left[{V}^{(i)}_{p_{i}}\\right]$ is the matrix-valued ${p_{i}}^{\\text{th}}$ Fourier component of $\\left[{V}^{(i)}_{\\vec{n}} \\right]$. \nNote that $\\left[{V}^{(i)}_{p_{i}}\\right] = \\left[{V}^{(i)}_{-p_{i}} \\right]^{\\dagger}$ due to hermiticity of the Hamiltonian.\n\nTo perform the enhancement of dimensions, we first insert the expansion Eq.~($\\ref{eq:expand_FT_V}$) into the Hamiltonian Eq.~(\\ref{eq:h_low_dim}). \nWe then regard each $\\phi^{(i)}$ as the $i^{\\text{th}}$ crystal momentum $k^{i}$ along one of the additional $N$ synthetic dimensions. \nWe then promote the $d$D model to a $(d+N)$D space by summing over $\\vec{k} = (k^{1},\\cdots,k^{N}) \\in \\mathbb{T}^{N}$ (where $\\mathbb{T}^N$ denotes the $N$-dimensional torus), which yields the Hamiltonian in $(d+N)$D as\n\\begin{align}\n    H_{\\text{high-dim}} &= \\sum_{\\vec{n},\\vec{m},\\vec{k}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{k}} \\left[{H}_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{k}} \\nonumber  \\\\\n    & + \\sum_{\\vec{n},\\vec{k},i,p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{k}} \\left[{V}^{(i)}_{p_{i}}\\right] e^{ip_{i}k^{i}} e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\vec{n}}{\\psi}_{\\vec{n},\\vec{k}}. \\label{eq:d_n_Bloch}\n\\end{align}\nEach physically distinct configuration of $\\{\\phi^{(i)}\\}$ can be recovered by restricting the Hamiltonian Eq.~(\\ref{eq:d_n_Bloch}) to a single $\\vec{k}$-point. \nOnce we sum over $\\vec{k}$, however, we can reinterpret the Hamiltonian in a $(d+N)$D space. \nAs we will see below, adiabatic pumping of the phases $\\phi^{(i)}$ by an external field will allow us to explore dynamics in the full $d+N$ dimensional space. \n\nTo obtain the $(d+N)$D model in position-space, we perform an inverse Fourier transform of ${\\psi}^{\\dagger}_{\\vec{n},\\vec{k}}$, yielding\n\\begin{align}\n    H_{\\text{high-dim}} &= \\sum_{\\vec{n},\\vec{m},\\vec{\\nu}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu}} \\left[{H}_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{\\nu}} \\nonumber  \\\\\n    & + \\sum_{\\vec{n},\\vec{\\nu},i,p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[{V}^{(i)}_{p_{i}}\\right] e^{i2\\pi p_{i}\\vec{q}^{(i)}\\cdot \\vec{n} }\\psi_{\\vec{n},\\vec{\\nu}}, \\label{eq:general_n_plus_d_model}\n\\end{align}\nwhere $\\vec{\\nu} = (\\nu_{1},\\cdots,\\nu_{N}) \\in \\mathbb{Z}^{N}$ and $\\hat{\\nu}_{i}$ is the unit vector along the $i^{\\text{th}}$ additional dimension, such that $\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}= (\\nu_{1},\\cdots,\\nu_{i}-p_{i},\\cdots,\\nu_{N})$. \nEq.~(\\ref{eq:general_n_plus_d_model}) can be viewed as the Hamiltonian for a system on a $(d+N)$D {{cubic lattice}} whose lattice sites are located at $(\\vec{n},\\vec{\\nu}) = (n_{1},\\cdots,n_{d},\\nu_{1},\\cdots,\\nu_{N}) \\in \\mathbb{Z}^{d+N}$. \nThe system is coupled to a continuous $U(1)$ gauge field\n\\begin{align}\n    \\vec{A} = (\\underbrace{\\vec{0}}_\\text{$d$D},\\underbrace{2\\pi \\vec{q}^{(1)}\\cdot \\vec{r},\\cdots,2\\pi \\vec{q}^{(N)}\\cdot \\vec{r}}_\\text{$N$D}) \\label{eq:expression_A}\n\\end{align}\nthrough a Peierls substitution\\cite{Peierls_substitution}, explaining the appearance of the phase factors multiplying $\\left[{V}^{(i)}_{p_{i}}\\right]$ in Eq.~(\\ref{eq:general_n_plus_d_model}). \nNote $\\vec{r} \\in \\mathbb{R}^{d}$ is a vector in the original $d$D space.\n\n\nAs the vector potential in Eq.~(\\ref{eq:expression_A}) is linear in position $\\vec{r}$, the antisymmetric field strength $F_{\\mu \\nu} = \\partial_{\\mu}A_{\\nu} - \\partial_{\\nu}A_{\\mu}$ is constant in space. \nIn particular, Eq.~(\\ref{eq:expression_A}) implies that the nonzero components of $F_{\\mu \\nu}$ are given by\n\\begin{align}\n    F_{i,j+d} = \\partial_{i}A_{j+d} - \\partial_{j+d}A_{i} = \\partial_{i}A_{j+d} = 2\\pi q^{(j)}_{i}, \\label{eq:F_i_j_plus_d}\n\\end{align}\nwhere $i = 1 ,\\ldots, d$ and $j = 1 ,\\ldots, N$. \nDue to the antisymmetry of the field strength, $F_{i+d,j}$ with $i = 1 ,\\ldots, N$, $j = 1 ,\\ldots, d$ is also nonzero and given by $F_{i+d,j} = -2\\pi q^{(i)}_{j}$. \nTherefore the (nonzero) constant field strength is proportional to the magnitude of the modulation wave vectors.\n\n\n\nThis shows that that a $d$D modulated system with phase offset $\\{\\phi^{(i)}\\}$ is equivalent to the Bloch Hamiltonian (see Eq.~(\\ref{eq:d_n_Bloch})) of the promoted $(d+N)$D lattice model with fixed crystal momenta $\\vec{k}$, once we identify $\\phi^{(i)}$ as $k^{i}$.\nIn practice, the modulation $\\left[{V}^{(i)}_{\\vec{n}}\\right]$ can be induced by a set of DW modulations. \nThe phase offset $\\{\\phi^{(i)}\\}$ is then regarded as the phason degrees of freedom, namely the phase of the  $i^{\\text{th}}$ mean-field DW order parameter. \nBy applying electric fields that depin the DWs and make them slide\\cite{gooth2019evidence,Zakphase,gruner1988dynamics}, we may sample the whole spectrum of the $(d+N)$D model. \nIn particular, and as we will explore in subsequent sections, non-trivial topology in the $(d+N)$D lattice model--which may support localized boundary states--will manifest in the response of the $d$D model to adiabatic sliding of the DW phase mode(s). \n\\textcolor{black}{We emphasize here that in our dimensional promotion procedure for a DW system, there are no emergent electric fields in the promoted $(d+N)$D space. The electric fields mentioned here are external and serve as a way to depin the DW in order to vary $\\{\\phi^{(i)}\\}$ adiabatically. This allows for the sampling of the entire spectrum of the $(d+N)$D model as a function of $\\{\\phi^{(i)}\\}$, namely the additional crystal momenta.}\n\n\nBefore we move on to consider the band topology of promoted lattice models, let us make a few general comments about our dimensional promotion procedure. \nFirst, note that the dimensional promotion procedure places no constraints on the modulation vectors $\\vec{q}^{(i)}$; in particular, they need not be commensurate with the underlying lattice. \nIn the case of incommensurate modulation, the dimensional promotion procedure allows us to write the $d$D incommensurate model in terms of a periodic $(d+N)$D model with an irrational $U(1)$ flux per plaquette. \nWe will see below how we can use this to explore the topology of systems with incommensurate modulation. \n\\textcolor{black}{We emphasize that the dimensional promotion procedure is independent of whether in the original $d$D space the system is infinite or finite. When we promote the dimension of a $d$D system to $(d+N)$D space, the $(d+N)$D system is inherently infinite along the additional $N$ dimensions, as it allows a Fourier transformation to obtain the Bloch Hamiltonian with fixed $N$ additional crystal momenta. From this viewpoint there are two ways to utilize the dimensional promotion procedure. If we promote the dimension of an infinite $d$D system, we will obtain an infinite $(d+N)$D system that allows us to discuss the non-trivial bulk topology in the promoted $(d+N)$D space. If we instead promote the dimension of a finite $d$D system, we will obtain a $(d+N)$D system which is finite along the original $d$ dimensions and infinite in the additional $N$ dimensions. This allows us to compute the energy spectrum to examine whether there are boundary states protected by the non-trivial bulk topology in $(d+N)$D space.}\n\n\nSecond, although here we consider only dimensional promotion of a $d$D cubic lattice model with only on-site modulations \\textcolor{black}{and all orbitals located at the lattice points labelled by $\\vec{n} \\in \\mathbb{Z}^{d}$} to a $(d+N)$D cubic lattice model, we may generalize our method to $d$D models with modulations in both on-site and hopping matrix elements, together with non-orthogonal lattice vectors \\textcolor{black}{and arbitrary orbital positions}. \nWe show how to systematically promote the dimensions of such $d$D models to $(d+N)$D and compute the corresponding $U(1)$ gauge fields in the SM\\cite{SM}. \nWe also give several examples in the SM\\cite{SM}, including the dimensional promotion of: (1) the 1D Rice-Mele chain in Sec.~\\ref{sec_review_thouless_pump_Rice_Mele} to a 2D square lattice with $\\pi$-flux, (2) 1D lattices with modulation in both on-site energies and hopping terms to 2D hexagonal lattices under a perpendicular magnetic field, and (3) 2D modulated systems with hexagonal lattice to 3D systems also with hexagonal lattices coupled to a $U(1)$ gauge field. \nThe $U(1)$ gauge fields will take a slightly different form from Eq.~(\\ref{eq:expression_A}) when we consider a system with non-orthogonal lattice vectors. \nHowever, the vector potentials will still be linear in $\\vec{r} \\in \\mathbb{R}^{d+N}$, and hence will still produce constant field strengths $F_{\\mu \\nu}$. \nFurthermore, note that although we considered for simplicity models where the electrons were localized to the origin of each unit cell, this is not essential for the application of our formalism.\n\n\nThird, we emphasize that no additional parameters are used in the above derivation. \nThe hopping matrices connecting $(\\vec{n},\\vec{\\nu})$ to $(\\vec{n}+\\vec{m},\\vec{\\nu})$ and $(\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ are given by $\\left[{H}_{\\vec{m}}\\right]$ and $\\left[{V}^{(i)}_{p_{i}}\\right]$, respectively, in the $(d+N)$D model. \nThe phase $\\phi^{(i)}$ corresponds to the $i^{\\text{th}}$ crystal momentum along the $i^{\\text{th}}$ additional dimensions. \nFurther, the modulation wave vectors $\\vec{q}^{(i)}$ specify the strength of the $U(1)$ gauge field in $(d+N)$D, see Eq.~(\\ref{eq:expression_A}). \nNotice that the on-site modulations $\\left[{V}^{(i)}_{\\vec{n}}\\right]$ only lead to hopping parallel to $\\hat{\\nu}_{i}$ in $(d+N)$D. \nIf we also consider modulated hopping matrices in $d$D, upon dimensional promotion we will get hopping along $\\vec{m} + p_{i}\\hat{\\nu}_{i}$ in $(d+N)$D\\cite{equivalence_Fibo_Harper_2012,Oded_4D_CI_to_2D_HOTI}, which we show in the SM\\cite{SM}. \nNotice that the index $i$ is not summed over in $p_{i}\\hat{\\nu}_{i}$. \nRecall also that $\\vec{m}$ and $\\hat{\\nu}_{i}$ are vectors in the original $d$D and additional $N$D space, respectively. \nAn example that demonstrates this is the 1D Rice-Mele model\\cite{RiceMele} in Sec.~\\ref{sec_review_thouless_pump_Rice_Mele}. \nIn the SM\\cite{SM} we promote Eq.~(\\ref{eq:1DRiceMele}) to a 2D lattice with $\\pi$-flux per plaquette in which the electrons can hop along $\\hat{x}+\\hat{y}$ (where $\\hat{x}$ and $\\hat{y}$ are in the original 1D and additional 1D space, respectively).\n \nNext, our construction provides a way to compute the promoted $(d+N)$D model and the $U(1)$ gauge field to which it is coupled. \nAs a $U(1)$ gauge field breaks time-reversal-symmetry (TRS), this dimensional promotion procedure is suitable to investigate non-trivial topological phases in $(d+N)$D space without TRS. \nBelow we will also consider a dimensional promotion to $(d+N)$D space with an $SU(2)$ gauge field, which preserves TRS and allows us to explore non-trivial topological phases protected by TRS\\cite{Kane04,bernevig2006quantum,ryu2010topological,KitaevClassify}.\nIn order to construct a low dimensional modulated model equivalent to a higher dimensional lattice coupled to an $SU(2)$ gauge field, we adopt a top-down approach. \nWe will in Sec.~\\ref{sec_helical_HOTI_sliding_modes} present a 2D modulated model which is obtained from a 3D model coupled to one $SU(2)$ gauge field with a fixed crystal momentum.\n\n\nIn the following sections, we explore various 2D and 3D modulated systems that admit a dimensional promotion to a higher dimensional topological phases coupled to either $U(1)$ or $SU(2)$ gauge fields. \nWe will show how an analysis of the higher-dimensional models can shed light on the eigenstates and boundary state dynamics of incommensurate DWs.\n\n\n\n\\section{\\label{sec_chiral_HOTI}Chiral Higher-Order Topological Sliding Modes}\nIn this section, we will show how the dimensional promotion procedure can be used to realize 3D chiral HOTIs in 2D density wave (DW) materials. \nWe will first construct a Hamiltonian for an insulating 2D modulated system that is inversion-symmetric for special values of the DW sliding phase $\\phi$. \nThen, we will show how, after dimensional promotion, the Hamiltonian corresponds to a 3D inversion-symmetric chiral HOTI coupled to a $U(1)$ gauge field. \nWe will explore the connection between hinge states of the 3D system and corner states of the 2D system using a combination of numerical diagonalization and a 3D low-energy $\\vec{k}\\cdot\\vec{p}$ theory.\n\n\\subsection{Dimensionally Promoted Chiral Model}\n\nConsider the following 2D Hamiltonian for electrons on a square-lattice, with one modulated on-site potential $[V(\\vec{q},\\vec{n},\\phi)]$:\n\\begin{equation}\n\\begin{aligned}\nH = {} & \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}+\\hat{x}} [H_{+\\hat{x}}]\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}+\\hat{y}} [H_{+\\hat{y}}]\\psi_{\\vec{n}} + \\text{h.c.}\\\\\n&+ \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}}\\left( [H_{\\text{on-site}}] + [V(\\vec{q},\\vec{n},\\phi)] \\right)\\psi_{\\vec{n}},\n\\end{aligned}\\label{chiral_modulated_2}\n\\end{equation}\nwhere the unmodulated hoppings and on-site energies are\n\\begin{align}\n    & [H_{+\\hat{e}_{i}}] = \\frac{J_{i}}{2}\\tau_{z}\\sigma_{0} - \\frac{\\lambda_{i}}{2i}\\tau_{x}\\sigma_{i}, \\label{eq:chiral_xy_hopping}\\\\\n    & [H_{\\text{on-site}}] = M \\tau_{z}\\sigma_{0}+\\tau_{0} \\vec{B_0} \\cdot \\vec{\\sigma} . \\label{eq:chiral_on_site}\n\\end{align}\nWe use $\\hat{e}_{i}$ to denote the unit vector along the $i^{\\text{th}}$ ($i = 1, 2$) direction. \nThe Pauli matrices $\\vec{\\tau} = (\\tau_{x},\\tau_{y},\\tau_{z})$ and $\\vec{\\sigma} = (\\sigma_{x},\\sigma_{y},\\sigma_{z})$ denote respectively  orbital (for example $s$ and $p$ orbitals) and spin degrees of freedom. \nThis Hamiltonian is inversion-symmetric, with inversion symmetry represented by $\\tau_z$. \nFurthermore, when $\\vec{B}_0=0$ the model is also time-reversal (TR) symmetric, with the TR operator represented as $i\\sigma_y\\mathcal{K}$ (where $\\mathcal{K}$ is the complex conjugation operator).\n\nWe assume that both orbital degrees of freedom are located at the lattice sites.\nThe hopping matrices $[H_{+\\hat{e}_{i}}]$, and $M \\tau_{z}\\sigma_{0}$ give rise to, at low energy, four-component massive Dirac fermions, allowing us to access various topological phases\\cite{ryu2010topological,haldanemodel,bernevigbook}.\nPhysically, we can interpret $M \\tau_{z}\\sigma_{0}$ as the on-site energy difference for different orbitals, and $\\tau_{0}\\vec{B_0} \\cdot \\vec{\\sigma}$ as a ferromagnetic potential which splits the spin degeneracy of bands\\cite{wieder2018axion}. \nThe modulated on-site potential, which can arise from a density wave modulation, is\n\\begin{align}\n    [V(\\vec{q},\\vec{n},\\phi)] =&  J_{z}\\cos\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{z}\\sigma_{0} + \\lambda_{z}\\sin\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{x}\\sigma_{z}, \\label{eq:chiral_modulated_V}\n\\end{align}\nwhere $\\theta_{\\vec{q},\\vec{n},\\phi} = 2\\pi \\vec{q}\\cdot \\vec{n} + \\phi$, $\\vec{q} = (q_{x},q_{y})$ is the modulation wave vector in 2D, $\\vec{n} \\in \\mathbb{Z}^{2}$ is the lattice position, and $\\phi$ is the sliding phase. \nThe first term in Eq.~(\\ref{eq:chiral_modulated_V}) modulates the mass $M \\tau_{z}\\sigma_{0}$ in Eq.~(\\ref{eq:chiral_on_site}), while the second modulation denotes an on-site spin-orbit coupling between $s$ and $p$ orbitals. \nNote that the modulation $J_{z}\\cos\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{z}\\sigma_{0}$ is a TR-even charge ordering, while $\\lambda_{z}\\sin\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{x}\\sigma_{z}$ is a TR-odd spin ordering. \nTo see this, note that TR maps $(\\tau_{0},\\tau_{x},\\tau_{y},\\tau_{z}) \\to (\\tau_{0},\\tau_{x},-\\tau_{y},\\tau_{z})$ and $(\\sigma_{0},\\sigma_{x},\\sigma_{y},\\sigma_{z}) \\to (\\sigma_{0},-\\sigma_{x},-\\sigma_{y},-\\sigma_{z})$. \nIn addition, the modulations $J_{z}\\cos\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{z}\\sigma_{0}$ and $\\lambda_{z}\\sin\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{x}\\sigma_{z}$ are both inversion-symmetric when $\\phi=0$, $\\pi$.\n\n\nDenoting the third, synthetic dimension as $z$ and identifying $\\phi$ as the corresponding crystal momentum $k_{z}$, we may use our general procedure in Sec.~\\ref{sec_Dimension_promotion} to promote this 2D modulated system to a 3D lattice model. \nWe first expand the modulations in terms of Fourier series as\n\\begin{align}\n    & J_{z}\\cos\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{z}\\sigma_{0} = \\frac{J_{z}}{2}\\left(e^{i\\theta_{\\vec{q},\\vec{n},\\phi}} + e^{-i\\theta_{\\vec{q},\\vec{n},\\phi}} \\right) \\tau_{z}\\sigma_{0}, \\label{eq:J_expand} \\\\\n    & \\lambda_{z}\\sin\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{x}\\sigma_{z} = \\frac{\\lambda_{z}}{2i}\\left( e^{i\\theta_{\\vec{q},\\vec{n},\\phi}} - e^{-i \\theta_{\\vec{q},\\vec{n},\\phi}} \\right)\\tau_{x}\\sigma_{z}. \\label{eq:lambda_z_expand}\n\\end{align}\nAccording to Eqs.~(\\ref{eq:expand_FT_V}) and (\\ref{eq:general_n_plus_d_model}), the hopping along $+\\hat{z}$ can be identified with the terms associated with $e^{-i \\theta_{\\vec{q},\\vec{n},\\phi}}$ in Eqs.~(\\ref{eq:J_expand}) (\\ref{eq:lambda_z_expand}). \nTherefore, the hopping along $+\\hat{z}$ in the promoted 3D space reads\n\\begin{equation}\n[H_{+\\hat{z}}]=\\frac{J_{z}}{2}\\tau_{z}\\sigma_{0} -\\frac{\\lambda_{z}}{2i}\\tau_{x}\\sigma_{z}.\n\\end{equation} \nFrom Eq.~(\\ref{eq:expression_A}) we can also identify the vector potential in the promoted 3D space as\n\\begin{align}\n    \\vec{A} = (0,0,2\\pi \\vec{q} \\cdot \\vec{r}) = (0,0,2\\pi q_{x} x + 2\\pi q_{y}y), \\label{A_U1}\n\\end{align}\nwhere $\\vec{r} = (x,y) \\in \\mathbb{R}^{2}$. \nTherefore, we have that the lattice Hamiltonian in the promoted 3D space is given by\n\\begin{widetext}\n\\begin{align}\n    H = \\sum_{\\vec{n}} & \\left[ \\left( \\psi^{\\dagger}_{\\vec{n}+\\hat{x}} [H_{+\\hat{x}}]\\psi_{\\vec{n}} +\\psi^{\\dagger}_{\\vec{n}+\\hat{y}} [H_{+\\hat{y}}]\\psi_{\\vec{n}} + {\\psi}^{\\dagger}_{\\vec{n}+\\hat{z}} [H_{+\\hat{z}}] e^{-i2\\pi (q_{x}n_{x}+q_{y}n_{y})} {\\psi}_{\\vec{n}} + \\text{h.c.} \\right)+ \\psi^{\\dagger}_{\\vec{n}} [H_{\\text{on-site}}] \\psi_{\\vec{n}} \\right], \\label{eq:lattice_model_chiral_sliding}\n\\end{align}\n\\end{widetext}\nwhere the vector potential Eq.~(\\ref{eq:expression_A}) is coupled to the system through a Peierls substitution\\cite{Peierls_substitution}, and\n$[H_{+\\hat{x}}]$, $[H_{+\\hat{y}}]$, $[H_{+\\hat{z}}]$ and $[H_{\\text{on-site}}]$ are given by Eqs.~(\\ref{eq:chiral_xy_hopping}) and (\\ref{eq:chiral_on_site}), respectively. \nHereafter, we will set $J_{x} = J_{y} = J_{z} = J$ for simplicity. \nIf we Fourier transform Eq.~(\\ref{eq:lattice_model_chiral_sliding}) along $z$ and regard $k_{z}$ (the wavenumber along $z$) as the sliding phase $\\phi$, we can obtain the 2D modulated system in Eq.~(\\ref{chiral_modulated_2}). \n\nWe will now use Eq.~(\\ref{eq:lattice_model_chiral_sliding}) to analyze the topological properties of the higher-dimensional model, in order to infer the properties of the low-dimensional modulated system.\nThis approach can also be employed in other low-dimensional modulated systems provided the corresponding higher-dimensional models are constructed. \nFor $q_{x}=q_{y}=0$ and $\\vec{B_0}=0$, Eq.~(\\ref{eq:lattice_model_chiral_sliding}) describes a TR and inversion-symmetric insulator whose inversion operation is represented by $\\tau_{z}$ (note that inversion symmetry acts to flip the sign on the synthetic momentum $k_z$). \nWe can employ the theory of symmetry-based indicators of band topology~\\cite{khalaf,Po2017,song2017,xu2020high,MTQC,Wieder_spin_decoupled_helical_HOTI,dynamical_axion_insulator_BB,Kruthoff2016} to compute the $\\mathbb{Z}_4$ indicator\n\\begin{align}\n    z_{4} = \\frac{1}{4}\\sum_{\\vec{k}_{a} \\in \\text{TRIMs}} \\left( n^{a}_{+} - n^{a}_{-} \\right) \\text{ mod }4,\n\\end{align}\nwhere $n^{a}_{+}$[$n^{a}_{-}$] is the number of positive[negative] parity eigenvalues in the valence band at the time-reversal invariant momentum (TRIM) $\\vec{k}_{a} $. \nWe find that for $|M\/J|>3$, $|M\/J|<1$, $1<M\/J < 3$, $-3<M\/J<-1$, the $\\mathbb{Z}_{4}$ symmetry-based indicator is given by $z_{4} = 0$, $0$, $1$ and $3$, respectively. \nThe regimes where $z_{4} \\mod 2 =1$ give a strong TRS-invariant topological insulator (TI). \nFor non-zero $\\vec{B_0}$ which breaks TRS but does not induce additional band inversions, the magnetic $\\mathbb{Z}_{4}$ symmetry-based indicator~\\cite{xu2020high,NaturePaper,MTQC,Po2017,watanabe2018structure,khalaf2018higher,wieder2018axion,Zhang_TMCI_corep_RPB,Slager_magnetic_spacegroup_rep,JiabinZhidaAXIDirac,MurakamiAXI1,MurakamiAXI2,dynamical_axion_insulator_BB} \nis given by \n\\begin{align}\n    \\tilde{z}_{4} = \\frac{1}{2}\\sum_{\\vec{k}_{a} \\in \\text{TRIMs}} \\left( n^{a}_{+} - n^{a}_{-} \\right) \\text{ mod }4,\n\\end{align}\nsuch that for $|M\/J|>3$, $|M\/J|<1$, $1<|M\/J| < 3$ we have $\\tilde{z}_{4} = 0$, $0$ and $2$. \nThe corresponding weak indices are all necessarily trivial.\nTherefore, for $1 < |M\/J| <3$ with $q_{x}=q_{y}=0$, the system described by Eq.~(\\ref{eq:lattice_model_chiral_sliding}) gives a strong TI with $\\vec{B_0}=0$ and a chiral HOTI (axion insulator)\\cite{MTQC} with $\\vec{B_0}\\ne0$, where the gapless surface states of the strong TI are gapped by the inversion-preserving ferromagnetic potential $\\tau_{0} \\vec{B_0} \\cdot \\vec{\\sigma}$. \nTherefore, Eq.~(\\ref{eq:lattice_model_chiral_sliding}) with $\\vec{q} \\ne 0$ describes an inversion-symmetric chiral HOTI\\cite{pozo2019quantization} coupled via a Peierls substitution to a 3D $U(1)$ gauge field given by the $\\vec{A}$ in Eq.~(\\ref{eq:expression_A}).\nThis $\\vec{A}$ produces a constant $U(1)$ magnetic field \n\\begin{align}\n    \\nabla \\cross \\vec{A}  = (2\\pi q_{y},-2\\pi q_{x},0), \\label{B_U1}\n\\end{align} \nwhich preserves the inversion symmetry represented by $\\tau_{z}$ in 3D, up to a gauge transformation (see SM\\cite{SM}). \nTherefore, for a suitable choice of parameters, as long as the $U(1)$ gauge field does not close the bulk gap in 3D, the insulating ground state will be in the same inversion symmetry-protected non-trivial chiral HOTI phase. \nThis implies that our model should exhibit the characteristic boundary modes of a chiral HOTI in 3D. \nIn particular, our promoted model will support odd numbers of sample-encircling chiral hinge modes in rod geometries which respect inversion symmetry~\\cite{pozo2019quantization,wieder2018axion,dynamical_axion_insulator_BB}.\n\n\\begin{figure}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig_main_text_chiral_sliding_change_ticking_freq.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the chiral 2D model in Eq.~(\\ref{chiral_modulated_2}) with parameters given in the text. \n(b) Probability distribution of corner modes in the gap-crossing bands at $\\phi = 0.5\\pi$ and $E=-0.1368$. \n(c) $\\&$ (d) Probability distribution of edge and bulk modes at $\\phi = 0.9\\pi$ and $E = -0.2508$ and $E = 0.278$, respectively. \nThe darker (black) color in (b)--(d) implies higher probability density. \\textcolor{black}{(b), (c) and (d) correspond to the corner mode, edge-confined mode and bulk-confined mode discussed in Sec.~\\ref{sec:chiral_sliding_corner_modes}, \\ref{sec:chiral_sliding_confined_edge_modes} and \\ref{sec:chiral_sliding_confined_bulk_modes}, respectively.}\nIn (b), (c) and (d), the $x$- and $y$-coordinate both range from $-15 ,\\ldots, +15$.}\n\\label{fig:chiral_sliding_main_text}\n\\end{figure}\n\n\\subsection{\\label{sec:chiral_sliding_corner_modes}Corner states}\n\nRecalling that in our case the $\\hat{z}$ direction is conjugate to the phase $\\phi$ of the sliding mode (regarded as the crystal momentum $k_z$), it is natural for us to consider inversion-symmetric rod geometries which are finite in the $\\hat{x}$ and $\\hat{y}$ directions, and infinite in the $\\hat{z}$ direction.\nIn our 2D system, this corresponds to considering the properties of a finite system as a function of the phase $\\phi$.\nWe can thus compute the energy spectrum of our 2D system in an open geometry with size $L_{x} \\times L_{y}$ as a function of $\\phi$ to obtain the energy dispersion along $k_{z}$ in the promoted model. \nIn the following, we call this kind of calculation the {\\it $\\phi$-sliding spectrum}, since the variation of $\\phi$ can be obtained by electromagnetically exciting the sliding mode of the underlying DW.\nFig.~\\ref{fig:chiral_sliding_main_text} (a) shows the $\\phi$-sliding spectrum of Eq.~(\\ref{chiral_modulated_2}) with parameters $J = 1$, $M =2 $, $\\lambda_{i}= 1$, $(\\vec{B}_{0})_{i}= 0.5\/\\sqrt{3}$\\cite{pozo2019quantization}, and $\\vec{q} = (0,q_{y})$, where $q_{y} = 0.11957$ is comparable with the experimental CDW wave vectors in (TaSe$_4$)$_2$I\\cite{shi2019charge} and is incommensurate with the underlying 2D square lattice in Eq.~(\\ref{chiral_modulated_2}).\nThe system size is $31 \\times 31$. \nAs we can see the spectrum contains modes which, as a function of $\\phi$, traverse the bulk spectral gap. \nExamining the wave functions of these ``gap-crossing modes,'' we see that they are localized to the corners of our $2$D sample, as shown in Fig.~\\ref{fig:chiral_sliding_main_text} (b). \nThe gap-crossing modes with opposite slopes correspond to states at inversion-related corners; in our example one mode is localized at the corner $(x_{\\text{corner}},y_{\\text{corner}})=(L\/2,-L\/2)$ (Fig.~\\ref{fig:chiral_sliding_main_text} (b)) and the other at $(x_{\\text{corner}},y_{\\text{corner}})=(-L\/2,L\/2)$ where $L = 30$.\nIf we start in a half-filled insulating ground state (with Fermi level $E_{F}=0$), then as $\\phi$ slides from $0$ to $2\\pi$, we realize charge pumping as one corner mode merges into the occupied-state subspace while the inversion-related counterpart flows into the unoccupied state subspace. \nThe ground states at the two inversion-symmetric values $\\phi=0,\\pi$ differ in electron number by $1$, demonstrating a ''filling anomaly''\\cite{benalcazar2018quantization,wieder2020strong}. \\tabularnewline\nBecause these corner modes originate as hinge modes in the $3$D dimensionally promoted system (where, recall, $\\phi$ is the momentum $k_z$), their existence is mandated by the non-trivial higher-order topology of the model Eq.~(\\ref{eq:lattice_model_chiral_sliding}).\n\nBy analyzing the low energy theory of the 3D hinge modes, we will now derive the dynamics of the 2D corner modes as a function of $\\phi$. \nIn 3D, the corresponding low energy 1D hinge Hamiltonian\\cite{hasan2010colloquium,khalaf,hotis} with a chiral mode as a function of $k_{z}$ is given by\n\\begin{align}\n    H_{\\text{hinge}} = \\xi v_{F} \\left( k_{z} + 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{y}y_{\\text{hinge}} \\right) \\right). \\label{eq:chiral_low_energy_hinge_H_1}\n\\end{align}\nWe have assumed that for the hinge along $z$ at position $(x_{\\text{hinge}},y_{\\text{hinge}})$ there is only one chiral mode with Fermi velocity $\\xi v_{F}$ where $v_{F} > 0$. \nWe have introduced $\\xi = \\pm 1$ to denote whether the chiral mode has positive or negative velocity. \nFollowing our dimensional promotion procedure, Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) is then minimally coupled to a $U(1)$ gauge field in Eq.~(\\ref{A_U1}) through the Peierls substitution $k_{z} \\to k_{z} + 2\\pi (q_{x}x+q_{y}y)$, where $x = x_{\\text{hinge}}$ and $y = y_{\\text{hinge}}$ are fixed. \n\nTo map Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) in 3D to the corner mode dispersion in 2D, it is helpful to first compute the $\\phi$-sliding spectrum for Eq.~(\\ref{chiral_modulated_2}) with $\\vec{q} = (0,0)$, as shown in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a). \nIf we identify $\\phi$ as $k_{z}$ in the hinge theory (modulo a constant offset that we will fix later), Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a) is the $\\hat{z}$-directed rod band structure for Eq.~(\\ref{eq:lattice_model_chiral_sliding}) without coupling to any vector potential. \nAs we can see, there are linear dispersing hinge modes spanning the bulk gap, which cross each other at $k_{z} = \\pi$. \nThis will be used below in Eq.~(\\ref{eq:low_E_corner_chiral}) to complete the mapping from Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) to 2D. \nFig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a) will also serve as a reference calculation when we examine the response of the $\\phi$-sliding spectrum as we increase the magnitude of $\\vec{q}$, which will confirm our low energy analysis.\n\n\nWe now use Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) to construct a low energy description of the corner modes in Fig.~\\ref{fig:chiral_sliding_main_text} (a) for Eq.~(\\ref{chiral_modulated_2}). \nUpon projecting from 3D to 2D, the fixed hinge mode position $(x_{\\text{hinge}},y_{\\text{hinge}})$ becomes the fixed corner mode position $(x_{\\text{corner}},y_{\\text{corner}})$, and the hinge modes become corner modes. \nSince the gap-crossing modes in the $q=0$ system shown in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a) intersect at $\\phi = \\pi$, we replace $k_{z}$ in the hinge theory by $\\Delta \\phi = \\phi - \\pi$. \nThus, we obtain an effective low energy description of the corner modes as\n\\begin{align}\n    H_{\\text{corner}} = \\xi v_{F} \\cdot \\left(\\Delta \\phi+2\\pi \\left(q_{x}x_{\\text{corner}}+q_{y}y_{\\text{corner}}\\right)\\right).\n    \\label{eq:low_E_corner_chiral} \n\\end{align}\nWe now verify Eq.~(\\ref{eq:low_E_corner_chiral}) by numerically computing the $\\phi$-sliding spectrum shown in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b) with same parameters as Fig.~\\ref{fig:chiral_sliding_main_text} (a) but with $q_{y}$ changed to $0.02$. This small value of $q_y$ gives a smooth modulation--and hence a low flux per plaquette in the dimensionally-promoted model--and is thus a suitable platform to examine the low energy theory with minimal coupling. \nWe observe gap-crossing modes with negative and positive slopes corresponding to corner modes at $(-L\/2,L\/2)$ and $(L\/2,-L\/2)$ where $L =30$, respectively. \nThese are shown in Figs.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (c) and (d) at $\\phi = 0.4\\pi$ and $1.6\\pi$, respectively. \nUsing Eq.~(\\ref{eq:low_E_corner_chiral}), we have the low energy descriptions for these two corner modes governed by the Hamiltonians\n\\begin{align}\n    & H_{\\text{corner 1}} = -v_{F}\\left( \\Delta \\phi + \\pi q_{y}L \\right), \\label{eq:eff_corner_chiral_1} \\\\\n    & H_{\\text{corner 2}} = +v_{F}\\left( \\Delta \\phi - \\pi q_{y}L \\right), \\label{eq:eff_corner_chiral_2}\n\\end{align}\nwhere we have used $q_{x} = 0$. \nThus, if we ramp up $q_{y}$ from $0$ to some non-zero value, we expect to see the corner mode dispersion shift along the $\\phi$-axis. \nThis is demonstrated in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b), which is to be compared with Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a). \nIn fact, a careful examination of Figs.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a) and (b) shows that the dispersions of the two corner modes shift in opposite directions as a function of $\\Delta\\phi$, as indicated in Eq.~(\\ref{eq:eff_corner_chiral_1}) and Eq.~(\\ref{eq:eff_corner_chiral_2}), with the shift given by $\\pi q_{y}L \\approx 0.6 \\pi$ for $q_{y} = 0.02$ and $L = 30$. \nWe thus see that the corner mode dispersion in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b) can be explained by Eq.~(\\ref{eq:low_E_corner_chiral}). \nThis demonstrates the origin of the corner modes in the 2D modulated system as higher dimensional hinge modes minimally coupled to a $U(1)$ gauge field. \nIf we consider larger $q_{y}$, such as in Fig.~\\ref{fig:chiral_sliding_main_text} where we have $q_{y} = 0.11957$, then the shift of the corner mode dispersion is predicted to be $\\pi q_{y} L \\approx 3.5871 \\pi$. This lies outside the first Brillouin zone and needs to be folded back into the range $\\phi = [0,2\\pi)$. \nThis occurs because, in passing from low energy continuum theory to a lattice model, the periodicity of $\\phi$--which in the promoted dimension is the continuous wavenumber $k_{z}$--is restored.\nAdditionally, note that Eq.~(\\ref{eq:low_E_corner_chiral}) implies that we may tune the range of $\\phi$ where the corner mode energies emerge from the bulk continuum by varying the periodicity of the modulation $\\sim 1\/|\\vec{q}|$. \nAs shown in Eq.~(\\ref{A_U1}) and Eq.~(\\ref{B_U1}), tuning $\\vec{q}$ is equivalent to changing the direction and strength of the $U(1)$ gauge field and the corresponding magnetic field in 3D.\n\n\n\\subsection{\\label{sec:chiral_sliding_confined_edge_modes}Edge states}\n Having accounted for the low energy description of the corner modes, we observe that in Fig.~\\ref{fig:chiral_sliding_main_text} (a), there are additional modes with {\\it flat dispersion}. \n These non-dispersing modes describe states confined either to the bulk or edge of the system, as shown in Figs.~\\ref{fig:chiral_sliding_main_text} (c) and (d). \n We now use low energy theories to demonstrate that these states originate from the $U(1)$ Landau quantization of the surface and bulk electrons in the promoted 3D chiral HOTI. \n We will revisit Figs.~\\ref{fig:chiral_sliding_main_text} (c) and (d) after we complete the low energy theory analysis using relatively small $q_{y}$.\n\n\n\nWe start with the edge-confined modes. \nSince a chiral HOTI can be obtained by gapping out the surface of a 3D inversion and TR-symmetric TI with a TR-breaking mass term, the generic surface theory reads\\cite{khalaf,wieder2018axion,vanderbiltaxion}\n\\begin{align}\n    H_{\\text{surf}} = \\left( \\vec{p} \\times \\vec{\\sigma}' \\right)\\cdot \\hat{n} + m \\hat{n} \\cdot \\vec{\\sigma}',\n\\end{align}\nwhere $\\vec{\\sigma}'$ are Pauli matrices that act in the basis of low-energy surface states and which capture their spin and orbital texture, $\\vec{p}$ is the momentum operator, and $\\hat{n}$ is the surface normal vector. \nThe time-reversal operator in this surface theory is given by $\\mathcal{T} = i\\sigma_{y}'\\mathcal{K}$ such that $\\mathcal{T} \\vec{\\sigma}' \\mathcal{T}^{-1} = -\\vec{\\sigma}'$. \nThe momentum dependent term $\\left( \\vec{p} \\times \\vec{\\sigma}' \\right)\\cdot \\hat{n}$ describes a helical surface Dirac cone, while $m \\hat{n} \\cdot \\vec{\\sigma}'$ is the TR-breaking mass term. \nAs shown in Eq.~(\\ref{B_U1}), if $q_{x} = 0$, which is the case we consider in Fig.~\\ref{fig:chiral_sliding_main_text} and Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp}, we have that $\\nabla \\cross \\vec{A}$ is parallel to $\\hat{x}$. \nWe then consider a surface theory on the $yz$-plane coupled to a perpendicular magnetic field $B \\hat{x}$ generated by a Landau-gauge $U(1)$ gauge field $\\vec{A} = (0,0,By)$.\nThe corresponding surface Hamiltonian with the $U(1)$ gauge field reads ${{H_{\\text{surf}} = p_{y}\\sigma_{z}' - \\left(p_{z}+By \\right)\\sigma_{y}' + m\\sigma_{x}',}}$ \nwhere we have made a Peierls substitution such that $p_{z} \\to p_{z} + By$, and we have assumed that both $B$ and $m$ are positive. \nTo facilitate the derivation, we perform a basis transformation through a $-2\\pi\/3$ radian spin rotation $U$ along the $[1,1,1]$ axis such that $U^{\\dagger} (\\sigma_{x}',\\sigma_{y}',\\sigma_{z}') U = (\\sigma_{z}',\\sigma_{x}',\\sigma_{y}')$. \nThe transformed Hamiltonian then reads\n\\begin{align}\n    H_{\\text{surf}} = p_{y}\\sigma_{y}' - \\left(p_{z}+By \\right)\\sigma_{x}' + m\\sigma_{z}'. \\label{eq:surf_H_chiral_HOTI_transformed}\n\\end{align}\nFourier transforming Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}) to replace {{$p_{z}$}} by the wavenumber $k_{z}$, and defining a $k_{z}$-dependent ladder operator\n\\begin{align}\n    & {{a^{\\dagger}_{k_{z}} = \\frac{1}{\\sqrt{2B}}\\left(\\left(k_{z}+By \\right)-ip_{y} \\right),}}  \\label{eq:U1_ladder}\n\\end{align}\nwhere $[a_{k_{z}},a^{\\dagger}_{k_{z}}]=1$, we can rewrite Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}) as\n\\begin{align}\n    {{H_{\\text{surf}}(k_{z}) = \\begin{bmatrix}\n    m & -\\sqrt{2B} a_{k_{z}} \\\\\n    -\\sqrt{2B} a^{\\dagger}_{k_{z}} & -m\n    \\end{bmatrix}.}}\n    \\label{eq:chiral_reexpress_H_surf}\n\\end{align}\nWe can solve for the eigenstates and energy eigenvalues of Eq.~(\\ref{eq:chiral_reexpress_H_surf}) to find\n\\begin{widetext}\n\\begin{align}\n    & {{\\psi^{-}_{k_{z},n=0} = e^{ik_{z}z} \\begin{bmatrix}\n    0 \\\\ \\ket{0,k_{z}}\n    \\end{bmatrix},\\  E^{-}_{k_{z},n=0} = -m,}} \\nonumber \\\\\n    & {{\\psi^{-}_{k_{z},n>0} = e^{ik_{z}z} \\begin{bmatrix}\n    \\alpha_{-}(n) \\ket{n-1,k_{z}}\\\\ \\ket{n,k_{z}}\n    \\end{bmatrix},\\  E^{-}_{k_{z},n>0} = - \\sqrt{m^{2} + 2Bn},}}  \\nonumber \\\\\n    & {{\\psi^{+}_{k_{z},n>0} = e^{ik_{z}z} \\begin{bmatrix}\n    \\alpha_{+}(n) \\ket{n-1,k_{z}}\\\\ \\ket{n,k_{z}}\n    \\end{bmatrix},\\  E^{+}_{k_{z},n>0} = + \\sqrt{m^{2} + 2Bn},}} \\nonumber \\\\\n    & {{\\text{where } \\alpha_{\\pm}(n) = \\frac{-1}{\\sqrt{2Bn}}\\left(\\pm\\sqrt{m^{2}+2Bn}+m \\right).}} \\label{eq:U1_surface_EE_EV}\n\\end{align}\n\\end{widetext}\nHere $n$ is a non-negative integer labelling the $U(1)$ Landau levels (LLs), and $\\ket{n,k_{z}}$ is the $n^{\\text{th}}$ simple harmonic oscillator (SHO) eigenstate localized along $y$ defined by the $a^{\\dagger}_{k_{z}}$ in Eq.~(\\ref{eq:U1_ladder}). \nNotice that the energies $E^{-}_{k_{z},n=0}$, $E^{-}_{k_{z},n>0}$ and $E^{+}_{k_{z},n>0}$ of these LLs shown in Eq.~(\\ref{eq:U1_surface_EE_EV}) are all independent of {{$k_{z}$}}. \nAs before, we now construct the low energy description of the edge-confined modes in the 2D modulated system from the above low energy surface theory in Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}). \nWe identify $k_{z}$ in the surface theory as $\\Delta \\phi = \\phi - \\pi$, since we have flat bands as a function of $\\phi$ in our 2D modulated system. \nWe also identify $B$ with $2\\pi q_{y}$ since in our examples of Fig.~\\ref{fig:chiral_sliding_main_text} (a) and Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b), we have $q_{x} = 0$ and the corresponding vector potential is $\\vec{A} = (0,0,2\\pi q_{y}y)$. \nWhen we project down to the 2D model, the surface electrons correspond to states confined in the left and right edges, as shown in Fig.~\\ref{fig:chiral_sliding_main_text} (c) and Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (e)--(h). \nWe again use $q_{y} = 0.02$ to demonstrate the low energy theory.\n\n\n\n\\begin{figure*}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\linewidth]{main_text_chiral_sliding_low_energy_theory.pdf}\n\\caption{(a) $\\&$ (b) $\\phi$-sliding spectrum of the chiral 2D model with the same parameters as Fig.~\\ref{fig:chiral_sliding_main_text} (a) but with $q_{y} = 0$ and $0.02$, respectively. \n(c) $\\&$ (d) Probability distribution of the corner modes in (b) at $\\phi = 0.4 \\pi$ and $1.6 \\pi$ and both with $E = 0.0365$. \n(e)--(g) Average of the probability distribution for the doubly degenerate edge-confined modes in the flat bands in (b) at $\\phi=\\pi$. \nThe double degeneracy is due to the pair of opposite edges related by inversion symmetry. \n(e)--(g) are edge-confined modes at $\\phi = \\pi$ with energies $E=-0.4705$, $-0.2235$ and $0.3811$, which are marked orange, green and red respectively in (b). \nThe corresponding energy eigenvalues are $E^{-}_{k_{z}=0,n=1}$, $E^{-}_{k_{z}=0,n=0}$ and $E^{+}_{k_{z}=0,n=1}$ in Eq.~(\\ref{eq:U1_surface_EE_EV}). \n(h) Edge-confined mode at $\\phi = 0.9 \\pi$ with energy $E = -0.2235$ corresponding to $E^{-}_{k_{z}=-0.1\\pi,n=0}$ in Eq.~(\\ref{eq:U1_surface_EE_EV}). \nThe darker (black) color in (c)--(h) implies higher probability density. \nThe inset in (e)--(h) is the probability distribution integrated over all $x$ coordinates. \nIn (c)--(h), the $x$- and $y$-coordinate both range from $-15 ,\\ldots, +15$.}\n\\label{fig:chiral_sliding_low_energy_main_text_temp}\n\\end{figure*}\n\n\n\nWe now remark on the implications of our low-energy analysis. \nFirst, the spectrum in Eq.~(\\ref{eq:U1_surface_EE_EV}) breaks particle-hole symmetry as there is a $-m$ energy eigenvalue but no $+m$ energy eigenvalue. \nThis can be observed in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b), where there are no flat bands of edge-confined modes around $E \\approx +0.2$, which corresponds to $E = +m$. \nWe thus identify the flat bands in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b) marked by red, green and orange as $E^{+}_{k_{z},n=1}$, $E^{-}_{k_{z},n=0}$ and $E^{-}_{k_{z},n=1}$ in Eq.~(\\ref{eq:U1_surface_EE_EV}). \n\n\nSecond, from Eq.~(\\ref{eq:U1_surface_EE_EV}), the probability distributions for the states $\\psi^{-}_{k_{z},n=0}$ and $\\psi^{\\pm}_{k_{z},n=1}$ are given by\n\\begin{align}\n    & {{|\\psi^{-}_{k_{z},n=0}|^2 \\propto\\left|\\varphi_{0,B}(y+k_{z}\/B) \\right|^{2}}} \\\\\n    & {{|\\psi^{\\pm}_{k_{z},n=1}|^2 \\propto\\left| \\alpha_{\\pm}(1) \\right|^{2}\\left|\\varphi_{0,B}(y+k_{z}\/B) \\right|^{2} + \\left|\\varphi_{1,B}(y+k_{z}\/B) \\right|^{2}}}\n\\end{align}\nup to a normalization factor, where $\\varphi_{n,B}(y)$ is the $n^{\\text{th}}$ eigenstate of an SHO localized along {{$y$}}. \nNotice that we have indicated the explicit $B$-dependence on $\\varphi_{n,B}(y)$ since the cyclotron frequency and the localization of the wave function depend on the strength of magnetic field. \nThis implies that $\\psi^{-}_{k_{z},n=0}$ has a pure Gaussian distribution. Furthermore, we expect that $\\psi^{-}_{k_{z},n=1}$ is more characteristic of an SHO first excited state than $\\psi^{+}_{k_{z},n=1}$ since $\\left| \\alpha_{-}(1) \\right|^{2} = \\left( -\\sqrt{m^{2} + 2B} + m \\right)^{2}\/(2B) < \\left( \\sqrt{m^{2} + 2B} + m \\right)^{2}\/(2B) = \\left| \\alpha_{+}(1) \\right|^{2}$,\nas we have assumed both $B$ and $m$ are positive. \nFigs.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (e)--(g) show the 2D wave function probability distributions at $\\phi = \\pi$ for edge confined modes in different LLs in our lattice model, together with the insets showing the integrated wave function probability over all $x$-coordinates. \nWhile both Figs.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (e) and (g) corresponds to $n = 1$ LL, the former is at the negative energy branch and the latter is at the positive energy branch.\nTherefore Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (e) shows split peaks characteristic of the SHO first excited state, more so than Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (g). \nIn contrast, Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (f), corresponding to the $n=0$ LL wave function, shows the Gaussian probability distribution characteristic of the SHO ground state. \nWe see that the qualitative properties of the wave functions are all consistent with the low energy surface theory.\n\nThird, the definition of the ladder operator in Eq.~(\\ref{eq:U1_ladder}) implies that the center of the wave functions will be shifted by $-k_{z}\/B$ from $y = 0$. \nIdentifying $k_{z}$ in the low energy theory as $\\Delta \\phi = \\phi - \\pi$ and $B$ as $2\\pi q_{y}$, we deduce that the distance $l$ that the edge-confined mode gets shifted along $y$ in the lattice model will be $l = -\\Delta \\phi\/({2\\pi q_{y}})$. \nNotice that the edge-confined mode in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (h) at $\\phi = 0.9\\pi$ ($\\Delta \\phi = -0.1\\pi$) is shifted by $\\approx +2.5$ lattice constants along $y$ comparing with Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (f), which is at $\\phi = \\pi$ ($\\Delta \\phi = 0$). \nThis is consistent with our prediction, as $l$ will  be $+2.5$ when $\\Delta \\phi = -0.1\\pi$ and $q_{y} = 0.02$.\n\nFourth, although Eq.~(\\ref{eq:U1_surface_EE_EV}) predicts non-degenerate energy levels for a single surface with a perpendicular $U(1)$ magnetic field, in Fig.~\\ref{fig:chiral_sliding_main_text} (a) the flat band corresponding to the $E^{-}_{k_{z},n=0}$ level is highly degenerate. \nThis is due to zone-folding effects, similar to what we observed for the corner mode dispersion in Fig.~\\ref{fig:chiral_sliding_main_text} (a). \nAs the gap-crossing modes are shifted outside $\\phi = [0,2\\pi)$, they get folded back to $\\phi = [0,2\\pi)$ together with the flat bands connected to them. \nUp to the degeneracy due to zone folding, the universal feature is that the edge-confined modes appearing in our 2D chiral DW system originate from the projection of surface electrons in a chiral HOTI with $U(1)$ Landau quantization.\n\n\nBefore moving on, let us remark on the robustness of our low-energy predictions to perturbations of the model. \nIf we consider a more complicated modulated system with, for example, long-range and anisotropic hopping terms together with other on-site potentials, as long as the promoted 3D system still preserves inversion symmetry and the gap is not closed, the 3D system will still be in the same chiral HOTI phase. \nHowever, the low energy theories that we have constructed might be modified. \nFor example, the low energy theory at the surfaces, which we model with Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}), may be modified as \n\\begin{align}\n    H_{\\text{surf}}  =& \\alpha_{y}p_{y}\\sigma_{y}' - \\alpha_{z}\\left(p_{z}+By \\right)\\sigma_{x}' + m\\sigma_{z}' + \\Delta \\sigma_{0}' \\nonumber \\\\\n    & + \\mathcal{O}(p_{y}^{2},p_{z}^{2},p_{y}p_{z}).\\label{eq:higher_order_terms}\n\\end{align}\nDifferences between $\\alpha_{x}$ and $\\alpha_{y}$ can lead to an anisotropic gapped Dirac cone. \nA nonzero $\\Delta$ induces unequal masses in different subspace of $\\vec{\\sigma}'$ which can shift the entire energy spectrum. \n$\\mathcal{O}(p_{y}^{2},p_{z}^{2},p_{y}p_{z})$ represents higher-order terms in the low energy theory which might cause nonlinearity in the band dispersion in Eq.~(\\ref{eq:higher_order_terms}) without minimal coupling.\nBy the same reasoning, we might also have non-linear hinge mode energies with a quadratic momentum correction in Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}).\nAll of these additional terms will change the energetic feature of the system, such as energy spectra, Fermi velocities, together with the detailed form of the wave functions, which will be inevitably different from Eq.~(\\ref{eq:U1_surface_EE_EV}). \nNevertheless, the following features are universal: (1) There will be electrons confined to the surface that undergo $U(1)$ Landau quantization, and therefore there will be states that are confined along some directions. \nUpon projecting down to the 2D modulated system, we will still obtain edge-confined modes. \n(2) There will be (non-)linear hinge mode dispersion that will be shifted along $k_{z}$ due to the minimal coupling. \nTherefore the statement that we can tune the range of $\\phi$ where the gap-crossing corner modes appear by tuning the magnitude of the modulation wave vectors, will still hold.  \nWe use the low energy theories Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) and Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}) since these allow us to uncover the relation between the states in the promoted dimension and those in the original low dimensional modulated system in an analytically tractable way.\n\n\n\\subsection{\\label{sec:chiral_sliding_confined_bulk_modes}Bulk states}\n\nThe above analysis on corner- and edge-confined modes shows that the corresponding higher dimensional description of our modulated system is a 3D chiral HOTI minimally coupled to a $U(1)$ gauge field. \nTo complete our analysis, we will now focus on the bulk states.\nAs expected, the low energy description of the bulk-confined modes, shown in Fig.~\\ref{fig:chiral_sliding_main_text} (d), will correspond to the low energy theory of bulk electrons in a 3D chiral HOTI minimally coupled to a $U(1)$ gauge field. \nWe start with the Bloch Hamiltonian of the promoted 3D chiral HOTI (Eq.~(\\ref{eq:lattice_model_chiral_sliding}) with $q_{x} = q_{y} = 0$) expanded around the $\\Gamma$ point\\cite{pozo2019quantization}, \n\\begin{align}\n    H_{\\text{bulk}} = m_{\\text{bulk}} \\tau_{z}\\sigma_{0} + \\tau_{x}\\vec{p} \\cdot \\vec{\\sigma} + \\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}. \\label{eq:3D_chiral_HOTO_bulk}\n\\end{align}\nWe have defined several parameters to make Eq.~(\\ref{eq:3D_chiral_HOTO_bulk}) compact for later convenience, and introduced $\\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}$ where $\\vec{M} = (M,M,M)$ corresponding to the ferromagnetic potential in Eq.~(\\ref{eq:chiral_on_site}). \nWe now couple this $H_{\\text{bulk}}$ to $\\vec{A} = By \\hat{z}$, which is equivalent to Eq.~(\\ref{A_U1}) with $q_{x} = 0$. \nThis can be done via the minimal substitution $p_{z} \\to p_{z} + By$. \nFourier transforming along $x$ and $z$ to replace $p_{x}$ and $p_{z}$ by wavenumbers $k_{x}$ and $k_{z}$, and defining the $k_{z}$-dependent ladder operator as\n\\begin{align}\n    a^{\\dagger}_{k_{z}} = \\frac{1}{\\sqrt{2B}}\\left( k_{z} + By - ip_{y} \\right), \\label{eq:3D_chiral_bulk_U1_ladder}\n\\end{align}\nwe can rewrite Eq.~(\\ref{eq:3D_chiral_HOTO_bulk}) coupled to $\\vec{A} = By \\hat{z}$ in terms of $a_{k_{z}}$ and $a^{\\dagger}_{k_{z}}$ as\n\\begin{widetext}\n\\begin{align}\n    H_{\\text{bulk}}(k_{x},k_{z}) = m_{\\text{bulk}} \\tau_{z}\\sigma_{0} + \\tau_{x} \\begin{bmatrix}\n    \\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}+a^{\\dagger}_{k_{z}} \\right) & k_{x} -\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}-a^{\\dagger}_{k_{z}} \\right) \\\\\n    k_{x} +\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}-a^{\\dagger}_{k_{z}} \\right) & -\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}+a^{\\dagger}_{k_{z}} \\right)\n    \\end{bmatrix} + \\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}. \\label{eq:3D_bulk_LL_Hamiltonian}\n\\end{align}\n\\end{widetext}\nWe have numerically shown in SM\\cite{SM} that the effective theory in Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) captures several properties of the flat bulk bands in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b) with relatively small $q_{y} = 0.02$, such as energy asymmetry with respect to $E = 0$ and the confinement direction of the bulk states due to $U(1)$ Landau quantization.\n\nFrom the above analysis on corner-, edge-, and bulk-confined modes, we conclude that we can characterize this topological 2D modulated system with chiral sliding modes in terms of a 3D chiral HOTI coupled to a $U(1)$ gauge field. \nIn addition, such 2D modulated systems provide a platform to examine the properties of a 3D chiral HOTI, by sliding the DW order parameter $\\phi$.\n\n\n\n\\section{\\label{sec_helical_HOTI_sliding_modes}Helical Higher-Order Sliding Modes and $SU(2)$ Gauge Fields }\n\n\n\n\nNext, we will generalize our formalism to time-reversal invariant spinful systems. \nIn doing so, we will see that incommensurate modulations induce coupling to $SU(2)$ gauge fields in the dimensionally promoted models.\n$SU(2)$ gauge fields can be used to represent spin-orbit coupling\\cite{YiLi_SU2_Hofstadter}, which is ubiquitous in topological states of matter. \nFor example, $SU(2)$ gauge fields in 3D and 4D generates $SU(2)$ LLs that give rise to 3D TIs and 4D QHEs\\cite{YiLi_TI_SU2_LL,zhang2001four}. \nA non-Abelian $SU(2)$ Peierls phase in 2D and 3D lattices can also lead to 2D and 3D TIs\\cite{SU2_gauge_in_2D_Goldman,YiLi_SU2_Hofstadter}. \nIn addition, in response to a bulk $SU(2)$ gauge flux insertion, a 2D TI can bind various quasi-particle excitations such as spinons, holons and chargeons\\cite{qispincharge}. \nIn this section, we present a 2D modulated system that allows us to simulate a 3D helical HOTI coupled to an $SU(2)$ gauge field.\n\n\\begin{figure}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig_main_text_helical_sliding_revised_change_ticking_freq.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the 2D helical model in Eq.~(\\ref{eq:H_helical_sliding}) with parameters given in the text. \n(b) Summation of the probability density of the doubly-degenerate corners modes at $\\phi = 0.4\\pi$ and $E=0.0146$. \nThe two corner modes at the same $\\phi$ are related to each other by the $\\mathcal{I}\\mathcal{T}$-symmetry, and hence they are localized at inversion-related corners and have opposite spins. \n(c) $\\&$ (d) Probability distribution of edge and bulk modes at $\\phi = 0.9\\pi$ and $E = -0.1459$ and $E = 0.5227$, respectively. \nThe darker (black) color in (b)--(d) implies higher probability density. \nIn (b), (c) and (d), the $x$- and $z$-coordinate both range from $-15 ,\\ldots, +15$.}\n\\label{fig:helical_sliding_main_text}\n\\end{figure}\n\n\\subsection{Dimensionally Promoted Helical Model}\n\nWe start by considering the following 2D Hamiltonian on a square lattice with one modulated on-site potential $[V(\\vec{q},\\vec{n},\\phi)]$:\n\\begin{equation}\n\\begin{aligned}\nH = {} & \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}+\\hat{x}} [H_{+\\hat{x}}]\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}+\\hat{z}} [H_{+\\hat{z}}]\\psi_{\\vec{n}} + \\text{h.c.}\\\\\n&+ \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}}\\left( [H_{\\text{on-site}}] + [V(\\vec{q},\\vec{n},\\phi)] \\right)\\psi_{\\vec{n}}, \\label{eq:H_helical_sliding}\n\\end{aligned}\n\\end{equation}\nwhere the unmodulated couplings are are\n\\begin{align}\n    & [H_{+\\hat{x}}] = \\frac{v_{x}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{u_{x}}{2i}\\tau_{y}\\mu_{y}\\sigma_{0}, \\\\\n    & [H_{+\\hat{z}}] = \\frac{v_{z}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{u_{z}}{2i}\\tau_{x}\\mu_{0}\\sigma_{0}, \\\\\n    & [H_{\\text{on-site}}] = m_{1}\\tau_{z}\\mu_{0}\\sigma_{0}+m_{2}\\tau_{z}\\mu_{x}\\sigma_{0} + m_{3}\\tau_{z}\\mu_{z}\\sigma_{0} \\nonumber \\\\ \n    & +m_{v_{1}}\\tau_{0}\\mu_{z}\\sigma_{0} + m_{v_{2}}\\tau_{0}\\mu_{x}\\sigma_{0}. \\label{eq:helical_H_on_site}\n\\end{align}\nThe matrices $\\vec{\\tau}$, $\\vec{\\mu}$ and $\\vec{\\sigma}$, are Pauli matrices and denote the orbital, sub-lattice and spin degrees of\nfreedom, respectively. \nThe hopping matrices $[H_{+\\hat{x}}]$ and $[H_{+\\hat{z}}]$, together with the on-site potential $[H_{\\text{on-site}}]$ respect both inversion and TR symmetries. \nThe inversion and TR operations are represented by $\\tau_{z}$ and $i\\tau_{z}\\sigma_{y}\\mathcal{K}$, respectively\\cite{Wieder_spin_decoupled_helical_HOTI}. \nThese hoppings give rise to low energy four-component Dirac fermions in each spin subspace, realizing a topological critical point. \nThe modulated on-site energy is given by\n\\begin{eqnarray}\\nonumber\n[V(\\vec{q},\\vec{n},\\phi)]&=& v_{y}\\tau_{z}\\mu_{0}\\begin{bmatrix}\n    \\cos\\theta^{+}_{\\vec{q},\\vec{n},\\phi} & 0 \\\\ 0 & \\cos\\theta^{-}_{\\vec{q},\\vec{n},\\phi}\n    \\end{bmatrix}\\\\\n&& + v_{H} \\tau_{y}\\mu_{z}\\begin{bmatrix}\n    \\sin\\theta^{+}_{\\vec{q},\\vec{n},\\phi} & 0 \\\\ 0 & \\sin\\theta^{-}_{\\vec{q},\\vec{n},\\phi}\n    \\end{bmatrix},\n    \\label{eq:helical_sliding_modulation_H}\n\\end{eqnarray}\nwhere $\\theta^{\\pm}_{\\vec{q},\\vec{n},\\phi} = 2\\pi \\vec{q}\\cdot \\vec{n}\\pm\\phi$, $\\vec{q} = (q_{x},q_{z})$ is the modulation wave vector in 2D, $\\vec{n} \\in \\mathbb{Z}^{2}$ is the lattice position, and $\\phi$ is the sliding phase.\nThe first term in Eq.~(\\ref{eq:helical_sliding_modulation_H}) modulates the mass $m_{1}\\tau_{z}\\mu_{0}\\sigma_{0}$ in Eq.~(\\ref{eq:helical_H_on_site}), which may represent unequal on-site energy for $s$ and $p$ orbitals, with forward ($-\\phi$) and backward ($+\\phi$) sliding phase in each spin subspace\\cite{SU2_gauge_in_2D_Goldman,SU2_gauge_2D_to_1D_Goldman}. \nThe second term in Eq.~(\\ref{eq:helical_sliding_modulation_H}) describes a modulation of the on-site energy which mixes $s$ and $p$ orbitals with unequal strength for different sublattices. \nSimilarly, we have forward and backward sliding phases in different spin subspaces for the second term. \nSince the modulation in Eq.~(\\ref{eq:helical_sliding_modulation_H}) has opposite phase offsets in each spin subspace, it may be induced from spin-orbit coupled spin ordering. \nThis modulation is TR- and inversion-symmetric only when $\\phi=0$, $\\pi$. \nNote, however, that the product of inversion and TR symmetry, which we will denote $\\mathcal{I}\\mathcal{T}$-symmetry, is preserved for all values of $\\phi$.\nIf we denote the third, synthetic dimension as $y$, this 2D model is equivalent to the inversion and TR symmetric 3D helical HOTI model of Ref.~\\onlinecite{Wieder_spin_decoupled_helical_HOTI}, coupled to an $SU(2)$ gauge field given by\n\\begin{align}\n    \\vec{A} =  (0,2\\pi (q_{x}x+q_{z}z)\\sigma_{z},0). \\label{SU2_A}\n\\end{align}\nThis matrix-valued $\\vec{A}$ produces a constant $SU(2)$ magnetic field\\cite{Estienne_2011} $\\vec{B} = \\vec{\\nabla} \\cross \\vec{A} - i \\vec{A} \\cross \\vec{A}$, determined from the field strength\\cite{eguchi1980gravitation} $F_{\\mu \\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu} - i \\left[ A_{\\mu},A_{\\nu}\\right]$, and given by\n\\begin{align}\n    \\vec{B} = (-2\\pi q_{z}\\sigma_{z},0,2\\pi q_{x} \\sigma_{z} ). \\label{SU2_B}\n\\end{align}\nThis constant $SU(2)$ field strength preserves both inversion and TR symmetry in 3D, up to a spin-dependent gauge transformation (see SM\\cite{SM}). \nNotice that Eq.~(\\ref{SU2_B}) implies that the $SU(2)$ magnetic field in this example can be interpreted as a $U(1)$ magnetic field with opposite sign for spin-up and spin-down electrons\\cite{SU2_gauge_in_2D_Goldman,SU2_gauge_2D_to_1D_Goldman}. \nWe then expect that, for a suitable choice of parameters such that the $SU(2)$ gauge field does not close the bulk gap in 3D, the insulating ground-state will be in the same non-trivial helical HOTI phase as the model with $\\vec{q}=0$\\cite{khalaf,hotis,Po2017}.\nTherefore, in 3D, our promoted model will support an odd number of pairs of sample-encircling helical hinge modes respecting inversion and TR symmetries\\cite{khalaf,Wieder_spin_decoupled_helical_HOTI}.\nUpon projected back to 2D, the helical hinge modes in 3D become $\\mathcal{I}\\mathcal{T}$-related pairs of corner modes at the same $\\phi$ in the 2D modulated system. \nIn the SM\\cite{SM}, we give the form of the 3D dimensionally-promoted model in position-space. \n\n\\subsection{Calculation of the Spectrum}\n\nLet us now numerically verify these conclusions. \nFig.~\\ref{fig:helical_sliding_main_text} (a) shows the $\\phi$-sliding spectrum of Eq.~(\\ref{eq:H_helical_sliding}) with parameters $m_{1} = -3$, $m_{2} = 0.3$, $m_{3} = 0.2$, $m_{v_{1}} = -0.4$, $m_{v_{2}} = 0.2$, $v_{x}=v_{z}=u_{x}=u_{z} = 1$, $v_{y} = 2$, $v_{H} = 1.2$\\cite{Wieder_spin_decoupled_helical_HOTI}, and $\\vec{q} = (0,q_{z})$ where $q_{z} = 0.11957$ \\cite{shi2019charge}. \nThe system size is $31 \\times 31$. \nThere are doubly-degenerate pairs of states which cross the gap as a function of $\\phi$, where the degeneracy is protected by $\\mathcal{IT}$-symmetry. \nWe see from the wave functions that these are corner modes related by $\\mathcal{IT}$-symmetry, as shown in Fig.~\\ref{fig:helical_sliding_main_text} (b) for the branch with negative slope around $\\phi \\approx 0.4\\pi$. \nIn the other branch of doubly-degenerate gap-crossing states with positive slope, the corner modes are the inversion-symmetric counterpart (where recall that inversion symmetry leaves spin invariant) to those in Fig.~\\ref{fig:helical_sliding_main_text} (b). \nTherefore, as $\\phi$ slides from $0$ to $2\\pi$, this model realizes a $\\mathbb{Z}_{2}$ pump\\cite{fu2006time,teo2010topological} as one of the pairs of corner states will merge into the occupied state subspace (with Fermi level $E_{F}=0$) while the other pair will flow out. \nIn our specific examples, the two states in each  $\\mathcal{I}\\mathcal{T}$-related pair at the same $\\phi$ are spin eigenstates and therefore in this case the $\\mathbb{Z}_2$ pump is a spin pump; our conclusions, however, hold even when spin is not conserved.\n\n\nAs mentioned earlier, and in analogy with our chiral HOTI model, the corner modes here are equivalent to hinge modes along $y$ in 3D. \nThe corresponding low energy theory for these corner modes is \n\\begin{align}\n    H_{\\text{corner}}= v_{F} \\left( \\phi \\sigma_{z}' + 2\\pi \\left( q_{x}x_{\\text{corner}} + q_{z}z_{\\text{corner}} \\right) \\sigma_{0}' \\right), \\label{eq:H_helical_hinge_1}\n\\end{align}\nwhere $v_{F}$ is the group velocity of the hinge modes in 3D. \nWe use the Pauli matrices $\\vec{\\sigma}'$ to denote the effective basis where in each subspace the states have opposite spin together with some orbital and sub-lattice textures. \nWe have assumed without loss of generality that there is only one pair of helical hinge modes at the hinge along $y$ in the promoted 3D system. \nBy denoting $\\phi$ as $k_{y}$, which is the crystal momentum along $y$, we recognize Eq.~(\\ref{eq:H_helical_hinge_1}) as the hinge mode dispersion $H(k_{y}) = v_{F}k_{y}\\sigma_{z}'$ in 3D minimally coupled to an $SU(2)$ gauge field described by Eq.~(\\ref{SU2_A}).\nSimilar to Sec.~\\ref{sec_chiral_HOTI}, as we vary $\\vec{q}$--which is equivalent to changing the strength and (spatial) direction of the $SU(2)$ gauge field in Eqs.~(\\ref{SU2_A}) and (\\ref{SU2_B})--the dispersion of the spin-polarized corner modes will shift along the $\\phi$-axis. \nIn the SM\\cite{SM} we present a complete low energy theory analysis for the corner modes with the same structure as Sec.~\\ref{sec_chiral_HOTI}.\n\nIn addition, we show in Figs.~\\ref{fig:helical_sliding_main_text} (c) and (d) the probability density for the edge- and bulk-confined modes in the flat bands of Fig.~\\ref{fig:helical_sliding_main_text} (a). \nSimilar to the corner modes, these can be respectively understood in terms of 3D low energy surface and bulk theories minimally coupled to an $SU(2)$ gauge field, leading to an $SU(2)$ Landau quantization\\cite{YiLi_TI_SU2_LL,YiLi_SU2_Hofstadter}. \nThe relevant surface theory describes a time-reversed pair of Chern insulators. \nThe relevant bulk theory is the $\\vec{k}\\cdot\\vec{p}$ expansion around $\\Gamma$ of the promoted 3D helical HOTI Hamiltonian\\cite{Wieder_spin_decoupled_helical_HOTI}. \nWe provide further details in the SM\\cite{SM}.\nTogether with the corner mode analysis, we see that this topological 2D modulated system with helical sliding modes can be characterized by a 3D lattice model coupled to an $SU(2)$ gauge field. \nIn addition, we have shown how 2D modulated systems can provide a platform to examine $SU(2)$ gauge physics in higher dimensions, by sliding the phase $\\phi$ of the DW order parameter.\n\n\n\\section{\\label{sec:Weyl_CDW}Weyl-CDWs and 4D topological modes}\n\n\nAs a final demonstration of our dimensional promotion formalism and its utility to investigating physics in more than 3D, we consider the mean-field state of a correlated inversion-symmetric 3D Weyl semimetal with CDW distortion (Weyl-CDW)\\cite{dynamical_axion_insulator_BB,gooth2019evidence,shi2019charge,wang2013chiral,CDW_Weyl_Sehayek,CDW_in_Q1D_Cohn,Monopole_CDW_in_Weyl_Yi_Li,yu2020dynamical}. \nIt has been shown that such a system can realize various topological phases. \nDepending on the phase $\\phi$ of the CDW order parameter, the system can interpolate between quantum anomalous Hall (QAH) and {\\it obstructed} QAH (oQAH) phase\\cite{dynamical_axion_insulator_BB}. \nThis is due to the $\\pi$ mod $2\\pi$ axion angle difference $\\delta \\theta_{\\phi} = \\theta(\\phi = \\pi) - \\theta(\\phi = 0)$ for the system with $\\phi = 0$ and $\\phi = \\pi$, in the thermodynamic limit. \nPhysically, this leads to a Hall conductance difference\n\\begin{align}\n    \\left|G_{xy}(\\phi = \\pi) - G_{xy}(\\phi = 0) \\right| = e^{2}\/h \\text{ mod } 2e^{2}\/h \\label{eq:main_G_xy_eqn}\n\\end{align}\nfor a semi-infinite slab [see also Eq.~(\\ref{eq:agnostic}) below, as well as Refs.~\\onlinecite{olsen2020gapless,2020_Axion_coupling_Vanderbilt}]. \nIn this section, we analyze a minimal model of a 3D inversion-symmetric magnetic Weyl-CDW system, which admits a dimensional promotion to 4D with a $U(1)$ gauge field. \nWe will explain the origin of the background QAH response and the interpolation between QAH and oQAH phases using the corresponding 4D theory. \nIn the following, we will denote a sample infinite along $x$ and $y$ with finite thickness $L_z$ along the $z$ direction as an $xy$-slab. \nSimilarly, we will use the term $y$-rod to denote a sample infinite along $y$ and finite along $x$ and $z$ with size $L_{x}\\times L_{z}$.\n\n\n\\subsection{3D Weyl-CDW Model and Dimensional Promotion}\n\nTo begin, we consider electrons on a 3D cubic-lattice with Hamiltonian $H = H_{0} + H_{CDW}(\\phi) $. \nHere $H_0$ is a periodic tight-binding Hamiltonian given by\n\\begin{align}\n    H_0&=\\left(\\sum_{\\vec{n}}\\left[-it_x\\psi^\\dag_{\\vec{n}+\\hat{x}}\\sigma_x \\psi_{\\vec{n}}-it_y \\psi^\\dag_{\\vec{n}+\\hat{y}}\\sigma_y \\psi_{\\vec{n}}+t_z\\psi^\\dag_{\\vec{n}+\\hat{z}}\\sigma_z \\psi_{\\vec{n}}\\right]\\right. \\nonumber \\\\\n    &+\\sum_{\\vec{n}}\\frac{m}{2}\\left(\\psi^\\dag_{\\vec{n}+\\hat{x}}\\sigma_z \\psi_{\\vec{n}} + \\psi^\\dag_{\\vec{n}+\\hat{y}}\\sigma_z \\psi_{\\vec{n}}  -2 \\psi^\\dag_{\\vec{n}}\\sigma_z \\psi_{\\vec{n}} \\right) \\nonumber \\\\\n    &\\left.-\\sum_{\\vec{n}} t_z \\left(\\cos (\\pi q)\\right) \\psi^\\dag_{\\vec{n}}\\sigma_z c_{\\vec{n}}\\right) +\\mathrm{h.c.}\n\\end{align}\nin position space. \nThe corresponding Bloch Hamiltonian is\n\\begin{align}\n    &H_0(\\vec{k})= -2[t_x\\sin (k_x)\\sigma_x +t_y\\sin (k_y)\\sigma_y]  \\nonumber \\\\\n    &  -m[2-\\cos (k_x) - \\cos (k_y)]\\sigma_{z} +2t_z[\\cos (k_z) -\\cos (\\pi q)]\\sigma_{z} \\label{eq:Bloch_H_0}  , \n\\end{align}\nwith $m\/2 \\ge t_{x},-t_{y},t_{z} >0$. \nWe take for the on-site modulation\n\\begin{align}\n    H_{CDW}(\\phi)&=2|\\Delta|\\sum_{\\vec{n}}\\cos(2\\pi q n_{z}+\\phi)\\psi^\\dag_{\\vec{n}}\\sigma_z \\psi_{\\vec{n}}.\n\\end{align}\nHere $2|\\Delta|$ is the strength of the CDW modulation, $2\\pi q$ is the magnitude of the modulation wave vector $2\\pi \\vec{q}=(0,0,2\\pi q)$ and $\\phi$ is the phase of CDW order parameter. \nWe again use $\\vec{\\sigma}$ to denote the Pauli matrices, which here index an orbital degree of freedom. \nThe inversion and TR operation are represented by  $\\sigma_{z}$ and $\\mathcal{K}$, respectively (note that this is a model of spinless electrons).\nThe Hamiltonian $H_{0}$ then describes a TR-breaking, inversion-symmetric magnetic Weyl semimetal (WSM) with Weyl nodes at $\\vec{k} = (0,0,\\pm \\pi q)$, see Eq.~(\\ref{eq:Bloch_H_0})\\cite{mccormick2017minimal}.\nThe perturbation $H_{CDW}(\\phi)$ is the CDW modulation that couples these two Weyl nodes and opens a gap in the bulk spectrum~\\cite{dynamical_axion_insulator_BB}. \nNote that in this simple model, we have chosen the modulation wavevector to be exactly equal to the Weyl node separation vector for simplicity of analysis. \nEven though the bulk is gapped, the surface of this 3D Weyl-CDW is gapless, due to the presence of QAH surface states. \nIn Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a) we show the $\\phi$-sliding spectrum for a $y$-rod of $H_{0}+H_{CDW}(\\phi)$ at $k_{y} = 0$ with size $L_{x} \\times L_{z} = 25 \\times 25$, $t_{x} = - t_{y} = t_{z} = 1$, $m = 2$, $2|\\Delta|=0.75$ and $q = 1\/5$. \nThis corresponds to a commensurate Weyl-CDW system. \nThe mid-gap zero modes in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a) correspond to the QAH surface states. \nIn Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (b) we show the probability distribution of the 10 zero modes at $\\phi = 0$. \nTogether with Wilson loop and Berry curvature calculation in the SM\\cite{SM}, we verify that the corresponding $xy$-slab with $L_{z} = 25$ carries a slab Hall conductance $ G_{xy}(\\phi = 0)  = -5 e^{2}\/h$. \nWe can then identify the weak Chern number\\cite{qi2008topological,kohmoto1992diophantine,halperin1987possible,fukanemele} of the 3D periodic system with $5=25\/5$ unit cells (since $q= 1\/5$) as $\\nu_{z} = -1$.\n\n\n\\begin{figure*}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\linewidth]{Q_2pi_over_5_tx_1_ty_-1_tz_1_m_2_D_0_75_Lz_25.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the Weyl-CDW model in a $y$-rod geometry at $k_{y} = 0$ with size $L_{x}\\times L_{z} = 25 \\times 25$, $t_{x} = -t_{y} = t_{z} = 1$, $m = 2$, $2|\\Delta| = 0.75$ and $q = 1\/5$. \n(b) The average probability distribution of the 10 zero modes at $\\phi = 0$ in (a). \nThese zero modes correspond to QAH surface states. \n(c) The average probability distribution of the 5 non-trivial states at $\\vec{k} = \\Gamma$ of the $xy$-slab at $\\phi = 0$, which in total lead to $G_{xy}(\\phi = 0) = -5 e^{2}\/h$. \n(d) The average probability distribution of the 8 zero modes at $\\phi = \\pi$ in (a). \nThese zero modes correspond to QAH surface states. \n(e) The average probability distribution of the 4 non-trivial states at $\\vec{k} = \\Gamma$ of the $xy$-slab at $\\phi = \\pi$, which in total lead to $G_{xy}(\\phi = \\pi) = -4 e^{2}\/h$. \nThe darker (black) color in (b)--(e) implies higher probability density. \nIn (b) and (d), the $x$- and $z$-coordinate both range from $-12 ,\\ldots, +12$. \nIn (c) and (e), the $z$-coordinate ranges from $-12 ,\\ldots, +12$.}\n\\label{fig:temporary_3d_weyl_cdw_fig}\n\\end{figure*}\n\n\n\nAs in Sec.~\\ref{sec_chiral_HOTI} and~\\ref{sec_helical_HOTI_sliding_modes}, we identify $\\phi$ with the crystal momentum $k_{w}$ along a fourth, synthetic direction denoted by $w$. \nUsing the dimensional promotion procedure in Sec.~\\ref{sec_Dimension_promotion}, we can promote $H_{0} + H_{CDW}(\\phi)$ to a 4D nodal line system coupled to a $U(1)$ gauge field. \nIn the SM\\cite{SM} we give the explicit form of the promoted model in 4D position space. \nThe corresponding 4D nodal line system (with $q=0$) has a Bloch Hamiltonian\n\\begin{align}\n    &H(\\vec{k})= -2[t_x\\sin (k_x)\\sigma_x +t_y\\sin (k_y)\\sigma_y] + 2|\\Delta|\\cos{(k_{w})}\\sigma_{z}  \\nonumber \\\\\n    & -m[2-\\cos (k_x) - \\cos (k_y)]\\sigma_{z} +2t_z[\\cos (k_z) -\\cos (\\pi q)]\\sigma_{z}. \n\\label{eq:ham0}\n\\end{align}\nThe spectrum of this Hamiltonian features nodal lines at $k_{x}=k_{y} = 0$ defined by the implicit equation\n\\begin{align}\n    t_{z}\\cos{(k_{z})}  +  |\\Delta| \\cos{(k_{w})} = t_{z} \\cos{(\\pi q)}.\n\\end{align}\nAccording to Eq.~(\\ref{eq:expression_A}), we then couple this Hamiltonian to a 4D $U(1)$ gauge field given by\n\\begin{align}\n    \\vec{A} = (0,0,0,2\\pi q z), \\label{eq:U1_4D}\n\\end{align}\nsince $2\\pi \\vec{q} = 2\\pi q \\hat{z}$ in this system. \nThis $\\vec{A}$ only produces non-zero field strength threading the $zw$ plane,\n\\begin{align}\n    F_{zw} = -F_{wz}= \\partial_{z}A_{w} - \\partial_{w}A_{z}=2 \\pi q, \\label{eq:Fzw}\n\\end{align}\nwhere all other components of $F_{\\mu \\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}$ are zero. \nWe are now in a position to reinterpret the existence of a background QAH response and QAH surface states when the bulk gap is opened due to the CDW. \nWe will see how these features emerge from the low energy approximation for this 4D system minimally coupled to Eq.~(\\ref{eq:U1_4D}).\n\n\\subsection{Low Energy Theory Analysis}\n\n\\begin{widetext}\n\nWe start from the 4D Bloch Hamiltonian in Eq.~(\\ref{eq:ham0}). \nExpanding around $\\vec{k} = \\vec{0}$, we have\n\\begin{align}\n    H(\\vec{k}) \\approx -2[t_x k_{x} \\sigma_{x} +t_y  k_{y} \\sigma_{y}] + 2t_{z}  \\left( 1 - \\frac{k_{z}^{2}}{2} - \\cos (\\pi q) \\right)\\sigma_{z} + 2|\\Delta| \\left(  1 - \\frac{k_{w}^{2}}{2}\\right) \\sigma_{z}. \\label{eq:ham0_low_energy}\n\\end{align}\nThe nodal line in this low energy theory is an ellipse in the $k_{z}$-$k_{w}$ plane with $k_{x}=k_{y}=0$, defined by\n\\begin{align}\n    t_{z}k_{z}^{2} +  |\\Delta|k_{w}^{2} =  2t_{z}\\left[ 1 - \\cos{ (\\pi q)} \\right] + 2|\\Delta| > 0.\n\\end{align}\nReplacing the 4D wave vector $\\vec{k}=(k_{x},k_{y},k_{z},k_{w})$ by the 4D momentum operator $\\vec{p}=(p_{x},p_{y},p_{z},p_{w})$ using the so-called  envelope function approximation\\cite{Envelope_function_approximation_1,Envelope_function_approximation_2,Envelope_function_approximation_3,Envelope_function_approximation_4,Envelope_function_approximation_5,Qi_Zhang_RMP,hasan2010colloquium,bernevig2006quantum}, the Hamiltonian governing the low energy dynamics reads\n\\begin{align}\n    H = -2[t_x p_{x} \\sigma_{x} +t_y p_{y} \\sigma_{y}] + 2t_{z}  \\left( 1 - \\frac{p_{z}^{2}}{2} - \\cos (\\pi q)\\right) \\sigma_{z} + 2|\\Delta| \\left(  1 - \\frac{p_{w}^{2}}{2}\\right) \\sigma_{z}. \\label{eq:ham0_low_energy_p}\n\\end{align}\nNext, let us minimally couple Eq.~(\\ref{eq:ham0_low_energy_p}) to a {{4D}} $U(1)$ gauge field $\\vec{A} = (0,0,0,2\\pi qz)$ via a Peierls substitution such that $p_{w} \\to p_{w} + 2\\pi q z$. \nEq.~(\\ref{eq:ham0_low_energy_p}) then becomes\n\\begin{align}\n    H = -2[t_x p_{x} \\sigma_{x} +t_y p_{y} \\sigma_{y}] + 2 \\left( t_{z}\\left[ 1 - \\cos{(\\pi q)} \\right] + |\\Delta| \\right) \\sigma_{z}  - \\left( t_{z}p_{z}^{2} +|\\Delta| \\left( p_{w} +  2 \\pi q z \\right)^{2} \\right) \\sigma_{z},  \\label{eq:after_4D_U1_couple}\n\\end{align}\nwhere we have assumed that the particle carries $-1$ charge. \nFourier transforming along $x$, $y$ and $w$, we may replace $p_{x}$, $p_{y}$ and $p_{w}$ by the corresponding wavenumbers $k_{x}$, $k_{y}$, $k_{w}$, such that\n\\begin{align}\n    H(k_{x},k_{y},k_{w}) = -2[t_x k_{x} \\sigma_{x} +t_y k_{y} \\sigma_{y}] +2 \\left( t_{z}\\left[ 1 - \\cos{(\\pi q)} \\right] + |\\Delta| \\right) \\sigma_{z}  - \\left( t_{z}p_{z}^{2} + |\\Delta| \\left( k_{w} + 2\\pi q z \\right)^{2} \\right) \\sigma_{z}. \\label{eq:FT1_4D_U1_LL}\n\\end{align}\n\\end{widetext}\n\nNotice that the coefficient of $\\sigma_z$ in the final term in the Hamiltonian,\n\\begin{align}\n    t_{z}p_{z}^{2} + |\\Delta| \\left( k_{w} + 2 \\pi q z \\right)^{2}, \\label{eq:4D_U1_SHO_H}\n\\end{align}\nis an SHO Hamiltonian along $z$ which can be diagonalized as\n\\begin{align}\n    4 \\pi q \\sqrt{t_{z}|\\Delta|} \\left( n + \\frac{1}{2} \\right).\n\\end{align}\nHere $n$ is a non-negative integer and the eigenvalue of the number operator $ a^{\\dagger}_{k_{w},q} a_{k_{w},q}$ with\n\\begin{align}\n    a^{\\dag}_{k_{w},q} = \\frac{1}{\\sqrt{ 4\\pi q}} \\left( \\frac{t_{z}}{|\\Delta|} \\right)^{\\frac{1}{4}}\\left[ \\left({\\frac{|\\Delta|}{t_{z}}}\\right)^{\\frac{1}{2}} \\left(k_{w}+ 2\\pi q z \\right) - ip_{z} \\right]. \\label{eq:4D_U1_LL_ladder}\n\\end{align}\nThe quantum number $n$ is the 4D $U(1)$ LL index. \nBy restricting to a subspace of the full Hilbert space with fixed $n$ and $k_{w}$, we see that the 4D low energy Hamiltonian Eq.~(\\ref{eq:after_4D_U1_couple}) may be decomposed into a direct sum of 2D low energy Chern insulators (CIs) in $xy$-plane parameterized by $n$ and $k_{w}$.\nThe Hamiltonian for these 2D CIs is given by\n\\begin{align}\n    H_{\\text{2D CI}}(n,k_{w}) = -2[t_x p_{x} \\sigma_{x} +t_y p_{y} \\sigma_{y}] + 2 m \\sigma_{z}, \\label{eq:2D_CI_subspace}\n\\end{align}\nwhere\n\\begin{align}\n    m = t_{z}\\left(1 - \\cos{(\\pi q)} \\right)+ |\\Delta| - 2\\pi q \\sqrt{t_{z}|\\Delta|}\\left( n + \\frac{1}{2} \\right). \\label{eq:mass_term_emerge_CIs}\n\\end{align}\nSince we have restricted to the subspace with fixed $n$ and $k_{w}$ in Eq.~(\\ref{eq:2D_CI_subspace}), according to Eq.~(\\ref{eq:4D_U1_SHO_H}) the wave function along $z$ and $w$ will be SHO eigenstates centered at $z = -k_{w} \/ (2\\pi q)$ multiplied by a plane wave $e^{ik_{w} w}$. \nNotice that the $k_{w}$-dependence of Eq.~(\\ref{eq:2D_CI_subspace}) is due to the integer $n$ in Eq.~(\\ref{eq:mass_term_emerge_CIs}) which is an eigenvalue of the number operator $a_{k_{w},q}^{\\dagger}a_{k_{w},q}$. \nTherefore, the eigenstates in the low energy approximation take the form of plane waves in $w$, and Chern insulator eigenstates as a function of $(x,y)$ localized at different constant-$z$ planes for different $k_{w}$.\nThis provides a four-dimensional interpretation of the layer construction of the Weyl-CDW presented in Refs.~\\onlinecite{dynamical_axion_insulator_BB,CDW_Weyl_Sehayek}.\n\nAs in a 3D nodal ring system with a perpendicular magnetic field~\\cite{Nodal_ring_perpendicular_B}, Eq.~(\\ref{eq:2D_CI_subspace})  can yield a gapped 4D bulk spectrum provided that $m \\ne 0$ $\\forall n \\ge 0$. \nThis insulating ground state will then carry non-trivial topology inherited from the nodal line system, since in Eq.~(\\ref{eq:2D_CI_subspace}) we found that the gapped 4D continuum theory is composed of low energy 2D CIs. \nWe then expect that there will be CI layers in the $xy$-plane of the corresponding 4D lattice model (see SM\\cite{SM}). \nThe CI layers will also be separated along $z$ by $2\\pi \/ (2\\pi q) = 1\/q$ for a fixed $k_{w}$, due to the $2\\pi$ periodicity of $k_{w}$ in the lattice model. \nIn our current example this separation is $5$ since $q  =1\/5$. \nNotice that $k_{w}$ is now interpreted as the crystal momentum along the $4^{\\text{th}}$ dimension. \nTo connect these observation in 4D back to the physical 3D Weyl-CDW system with Hamiltonian $H_{0} + H_{CDW}(\\phi)$, we notice that each 3D Weyl-CDW system with a fixed $\\phi$ corresponds to the 4D theory with a fixed $k_{w}$. \nFocusing on the $xy$-slab with $\\phi = 0$ and thickness $L_{z} = 25$, in which $ G_{xy}(\\phi = 0)  = -5 e^{2}\/h$, we show wavefunctions corresponding to the only $5$ layers of non-trivial CIs separated from each other by $5$ lattice constants along $z$ in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (c). \nEach of these CI layers carries Chern number $C = -1$ and contributes one chiral edge mode along $y$ in the ${y}$-rod, shown in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (b). \nIn the SM\\cite{SM} we provide technical details on identifying the non-trivial CI layers using hybrid Wannier function, Berry phases and Berry curvature calculations for an $xy$-slab. \nWe can thus regard the CDW-induced gap opening and the existence of background QAH response as the results of $U(1)$ Landau quantization in the 4D nodal line system. \n\n\nNext, we address the interpolation between the QAH phase at $\\phi = k_w=0$ and the oQAH phase at $\\phi = k_w=\\pi$ using the 4D theory. \nBefore we turn to the 4D low energy theory, we begin with the observation that in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a), the number of mid-gap zero modes corresponding to QAH surface states decreases by $2$ as $\\phi$ slides from $0$ to $\\pi$; one state is lowered into the valence band, while one state is elevated to the conduction band.\nThis is consistent with the change in Hall conductance Eq.~(\\ref{eq:main_G_xy_eqn}), which is derived in the thermodynamic limit where the 2D slab thickness $L_{z} \\to \\infty$ with infinitesimal but non-zero $2|\\Delta|$~\\cite{dynamical_axion_insulator_BB}. \nThe ambiguity modulo $2e^{2}\/h$ in the change of Hall conductance is due to the axion angle $\\theta$, which is only well-defined mod $2\\pi$, as shown below in Eq.~(\\ref{eq:agnostic}). \nTaking $L_{z} \\to \\infty$ with infinitesimal $2|\\Delta|$ ensures that the only effect the CDW modulation has is to open the gap at the Weyl points without inverting bands at other high-symmetry points in the 3D Brillouin zone. \nWe have also verified that our choice for the parameters in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a) is adiabatically connected to this condition by increasing $L_{z}$ and decreasing $2|\\Delta|$. \nThe slab Hall conductance $G_{xy}$ of the $xy$-slab contains both an extensive contribution from the bulk QAH phase through the weak Chern number $\\nu_{z}$, and an intensive contribution from axion angle $\\theta$, which collectively gives\\cite{vanderbiltaxion,2020_Axion_coupling_Vanderbilt}\n\\begin{align}\n    G_{xy} = \\frac{e^2}{h}(\\nu_{z} l_{z} + \\theta \/ \\pi), \\label{eq:agnostic}\n\\end{align}\nwhere $l_{z}$ is the number of unit cells in the slab. \nIn our examples for $q = 1\/5$, $l_{z}$ will be given by $L_{z}\/5$. \nRecall also that as we slide $\\phi$ from $0$ to $\\pi$, the bulk gap of the 3D Weyl-CDW system never closes, hence the $\\nu_{z}$ is unchanged during the process. \nPutting this all together, we see that Eq.~(\\ref{eq:main_G_xy_eqn}) implies that there is a $\\pi$ mod $2\\pi$ change in the axion angle between $\\phi = 0$ and $\\phi = \\pi$. \nTo be more specific, in our current examples we have $G_{xy}(\\phi = 0) = -5 e^{2}\/h$ and $G_{xy}(\\phi = \\pi) = -4 e^{2}\/h$. \nThis quantized change of $G_{xy}$ or $\\theta$ can be explained again using the 4D low energy theory, as we now show. \n\n\nGoing back to the 4D low energy theory, Eq.~(\\ref{eq:4D_U1_SHO_H}) predicts that if we shift $k_{w}$ to $k_{w} + \\Delta k_{w}$, the corresponding CI layers described by the Hamiltonian in Eq.~(\\ref{eq:2D_CI_subspace})--which are localized around $z = -k_{w}\/(2\\pi q)$--will be shifted in the $z$ direction by $\\Delta z = -\\Delta k_{w} \/ (2\\pi q)$. \nConnecting this observation back to the physical 3D Weyl-CDW system, it implies that as we slide $\\phi$ from $0$ to $\\pi$, all the CI layers will be shifted by $\\Delta z = -\\pi \/ (2\\pi q) = -1\/(2q)$; for our choice of $q= 1\/5$ this gives a shift of $\\Delta z = -2.5$. \nWe demonstrate this numerically in Figs.~\\ref{fig:temporary_3d_weyl_cdw_fig} (d) and (e) which show the probability distribution of the $8$ QAH zero modes and the corresponding 4 non-trivial CI layers (with Chern number $C = -1$) for $\\phi = \\pi$. \nThe physical interpretation of Eq.~(\\ref{eq:main_G_xy_eqn}) is now clear: As we slide $\\phi$ from $0$ to $\\pi$, the non-trivial CI layers will be shifted by $\\Delta z = -2.5$ unit cells, all in the same direction. \nTherefore, the bottom non-trivial CI layer at $\\phi = 0$ and $z = -10$ depicted in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (c) will be shifted outside the finite sample and hence will not appear when $\\phi=\\pi$. \nAt $\\phi = \\pi$, there will be only $4$ non-trivial CI layers remaining. \nThis leads to a change in the Hall conductance by $e^2\/h$, as indicated by Eq.~(\\ref{eq:main_G_xy_eqn}). \nSimultaneously, the number of QAH zero modes in the $y$-rod decreases by $2$ when we slide $\\phi$ from $0$ to $\\pi$. \nPhysically, these two QAH zero modes are pushed toward the boundary of the system, due to the shifting of the bottom non-trivial CI layer. \nTherefore, their energies will be pushed toward the bulk continuum, leading to the inevitable appearance of gap-crossing bands as shown in Fig~\\ref{fig:temporary_3d_weyl_cdw_fig} (a). \nNumerically, we have observed that in all of our examples (Figs.~\\ref{fig:temporary_3d_weyl_cdw_fig} and~\\ref{fig:temporary_3d_weyl_cdw_fig_2}), the zero modes in the band structure of the $y$-rod only appear at $k_{y} = 0$. \nTherefore, as far as the zero modes are concerned, we can focus on the energy spectrum of the $y$-rod at $k_{y}=0$, as in Figs.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a) and \\ref{fig:temporary_3d_weyl_cdw_fig_2} (a). \nAnalytically, this can be understood from the Hamiltonian of the low energy Chern insulator Eq.~(\\ref{eq:2D_CI_subspace}) for each $n$ and $k_{w}$, which has zero energy edge modes only at $k_{y} = 0$\\cite{hasan2010colloquium,Qi_Zhang_RMP,bernevigbook,Jackiw_Rebbi}.\n\n\n\nTo summarize, we have shown that the identity Eq.~(\\ref{eq:main_G_xy_eqn}) can be regarded as a consequence of the $U(1)$ Landau quantization of a 4D nodal line system in which the localization centers along $z$ of the states are directly related to $k_{w}$.\nWe then identified $k_w$ with the sliding phase $\\phi$ through our dimensional promotion formalism in Sec.~\\ref{sec_Dimension_promotion}. \nThe change in conductance as a function of $\\phi$ can thus be regarded as a physical manifestation of the {\\it Chern number polarization}, which can alternatively be computed in terms of $z$-localized hybrid Wannier centers~\\cite{2020_Axion_coupling_Vanderbilt,zilberberg2018photonic,dynamical_axion_insulator_BB,yu2020dynamical,olsen2020gapless}.\n\n\\begin{figure*}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\linewidth]{Q_taupi_over_2_tx_1_ty_-1_tz_1_m_2_D_2_Lz_21.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the Weyl-CDW model in a $y$-rod geometry at $k_{y} = 0$ with size $L_{x}\\times L_{z} = 21 \\times 21$, $t_{x} = -t_{y} = t_{z} = 1$, $m = 2$, $2|\\Delta| = 2$ and $q = \\tau \/4$ where $\\tau = (1+\\sqrt{5})\/2$. \n(b) The average probability distribution of the 18 zero modes at $\\phi = 0$ in (a). \nThese zero modes correspond to QAH surface states. \n(c) The average probability distribution of the 9 non-trivial states at $\\vec{k} = \\Gamma$ of the $xy$-slab at $\\phi = 0$, which in total lead to $G_{xy}(\\phi = 0) = -9 e^{2}\/h$. \n(d) The average probability distribution of the 16 zero modes at $\\phi = \\pi$ in (a). \nThese zero modes correspond to QAH surface states. \n(e) The average probability distribution of the 8 non-trivial states at $\\vec{k} = \\Gamma$ of the $xy$-slab at $\\phi = \\pi$, which in total lead to $G_{xy}(\\phi = \\pi) = -8 e^{2}\/h$. \nThe darker (black) color in (b)--(e) implies higher probability density. \nIn (b) and (d), the $x$- and $z$-coordinate both range from $-10 ,\\ldots, +10$. \nIn (c) and (e), the $z$-coordinate ranges from $-10 ,\\ldots, +10$.}\n\\label{fig:temporary_3d_weyl_cdw_fig_2}\n\\end{figure*}\n\nHaving demonstrated the utility of our dimensional promotion formalism for a 3D Weyl-CDW system coupled to a commensurate CDW with $q = 1\/5$, we next explore the case of incommensurate modulations which are prevalent in nature\\cite{gruner1988dynamics}. \nIn particular, the experimentally intriguing Weyl-CDW (TaSe$_4$)$_2$I is incommensurate\\cite{shi2019charge,tasei_original,heeger_tasei,tournier2013electronic,zhang2020first,shi2019charge}. \nWe still consider $H_{0}+H_{CDW}(\\phi)$ with $t_{x}=-t_{y}=t_{z}=1$, $m=2$, $2|\\Delta|=2$. \nHowever, we now choose the modulation $q = \\tau \/4$ where $ \\tau = (1+\\sqrt{5})\/2$ is the golden ratio. \nFor an $xy$-slab we choose $L_{z} = 21$ and for $y$-rod we choose $L_{x} \\times L_{z} = 21\\times 21$. \nSince $q=\\tau\/4$ is an irrational number, the modulation $H_{CDW}(\\phi)$ is incommensurate with $H_{0}$. \nCrucially though, we can use our dimensional promotion procedure regardless of whether or not the modulation is commensurate with the underlying lattice. \nThe $U(1)$ gauge field to which the 4D nodal line system is coupled still takes the form in Eq.~(\\ref{eq:U1_4D}).\nThe main difference is that now, the 4D system has an irrational flux $2\\pi q = \\pi\\tau\/2$ per plaquette in the $zw$ plane. \nWe have verified that for the $xy$-slab we have $G_{xy}(\\phi = 0) = -9 e^{2}\/h$ and $G_{xy}(\\phi = \\pi) = -8 e^{2}\/h$, consistent with Eq.~(\\ref{eq:agnostic}). \nWe also show in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2} (a) the $\\phi$-sliding spectrum of the $y$-rod at $k_{y}=0$. \nWe see that there are $2$ fewer QAH zero modes at $\\phi = \\pi$ than at $\\phi = 0$. \nThe 18 and 16 QAH zero modes for $\\phi = 0$ and $\\phi = \\pi$ are shown in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2} (c) and (e), respectively. \nWe again identify 9 and 8 non-trivial states in the $xy$-slab at $\\vec{k} = \\Gamma$ for $\\phi = 0$ and $\\phi = \\pi$, and show their probability distributions in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2} (b) and (d), respectively. \nIn the SM\\cite{SM} we present the details of the numerical methods for identifying non-trivial states in the $xy$-slab. \nThe existence of non-zero QAH response and QAH zero modes can again be attributed to 4D $U(1)$ Landau quantization which gaps the 4D nodal line system, yielding a topologically non-trivial insulating ground state. \nIn particular, we also have $\\left| G_{xy}(\\phi = \\pi) - G_{xy}(\\phi = 0) \\right| = e^{2}\/h$ mod $2e^{2}\/h$. \nThis can again be understood from the shifting of non-trivial CI layers. \nIn this case, as $\\phi$ slides from $0$ to $\\pi$, all the non-trivial CI layers will be shifted downward by $\\Delta z = -\\pi \/ (2\\pi q) = -2 \/ \\tau \\approx -1.236$ lattice constants. \nThe non-trivial CI layer at the bottom ($z = -10$) of Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2} (b) will be shifted outside the finite size system and thus the absolute value of slab Hall conductance will be changed by $-1$. \nConsequently, the number of QAH zero modes in the $y$-rod at $k_{y}=0$ will be decreased by $2$. \nTherefore, together with the examples in Sec.~\\ref{sec_chiral_HOTI} and Sec.~\\ref{sec_helical_HOTI_sliding_modes}, we see that our dimensional promotion procedure provides a way to understand topological properties of systems with incommensurate modulations.\n\n\n\\subsection{Weyl-CDW and 4D Chern Number}\n\nWe can also understand the topological properties of the Weyl-CDW model from the perspective of 4D response theory. \nCombining the field strength in Eq.~(\\ref{eq:Fzw}) with our analysis of the Hall conductance above allows us to formulate a $(4+1)$D field-theoretical description of the QAH response in a 3D Weyl-CDW system. \nThe corresponding action is that of the $(4+1)$D Chern-Simon theory\\cite{qi2008topological,zhang2001four}\n\\begin{align}\n    S = \\frac{C_{2}}{24 \\pi^{2}} \\sum_{\\mu\\nu\\lambda\\rho\\sigma}\\int d^{5}x \\epsilon^{\\mu \\nu \\lambda \\rho \\sigma} A_{\\mu} \\partial_{\\nu} A_{\\lambda} \\partial_{\\rho} A_{\\sigma}, \\label{eq:action}\n\\end{align}\nwhere $C_{2}$ is the second Chern number, $A_{\\mu}$ is the electromagnetic gauge potential and $\\epsilon^{\\mu \\nu \\lambda \\rho \\sigma}$ is the Levi-Civita symbol in $(4+1)$D.\nThe Greek indices here are taken to run over all $4+1$ dimensions. \nEq.~(\\ref{eq:action}) gives the electromagnetic response through\n\\begin{align}\n    J^{\\mu} = \\frac{\\delta S}{\\delta A_{\\mu}} =\\frac{C_{2}}{32 \\pi^{2}}  \\sum_{\\nu\\lambda\\rho\\sigma}\\epsilon^{\\mu \\nu \\lambda \\rho \\sigma} F_{\\nu \\lambda} F_{\\rho \\sigma},\n\\end{align}\nwhere $J^{\\mu}$ is the current density along the $\\mu$ direction. \nSince we have $F_{zw} = 2\\pi q$, an electric field $E^{y}$ along $y$ (implying $F_{ty} = E_{y}$) \nwill induce a Hall current density along $x$ through\n\\begin{align}\n    J^{x} = q \\frac{C_{2}}{2\\pi}  E^{y}. \\label{eq:Hall_current_x}\n\\end{align}\nIntegrating this along the $z$ direction, we find then that, with non-zero $C_{2}$, the Hall conductance $G_{xy}$ is proportional to $qL_z$. \nThis is consistent with Eq.~(\\ref{eq:agnostic}) and the the recent calculation\\cite{dynamical_axion_insulator_BB} showing that the Hall conductance $G_{xy}$ of a 3D Weyl-CDW system is given by \n\\begin{align}\n    G_{xy} =  \\left( |\\vec{Q}|L_{z}+2\\theta \\right) \\cdot e^{2} \/ (2\\pi h), \\label{eq:Eq_from_dynamical_axion_insulator_preprint}\n\\end{align}\nwhere $L_{z}$ is the thickness of the $xy$-slab, $\\vec{Q}$ is the CDW wave vector along $z$, which in our specific model system is $\\vec{Q} = 2\\pi q \\hat{z}$, and $\\theta$ is the bulk axion angle computed from the inversion-symmetric unit cell. \nAs we take the thermodynamic limit $L_{z} \\to \\infty$, the axion angle contribution to $G_{xy}$ becomes negligible and thus $G_{xy}$ can also be regarded as proportional to the magnitude of CDW wave vector, which is consistent with Eq.~(\\ref{eq:Hall_current_x}).\nTherefore, the field strength in Eq.~(\\ref{eq:Fzw}) indeed allows a sensible construction of higher dimensional continuum theory.\n\nTo see concretely that the 3D Weyl-CDW system indeed emulates a 4D system with non-zero $C_{2}$, we notice that for both examples in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} and Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2}, the system can be deformed into a limit where we have layers of decoupled Chern insulators localized along $z$. \nIn the decoupled-layer limit, for the commensurate case, for example, $q = 1\/5$, where we consider the single nontrivial band in each unit cell, this implies that $C_{2}$, which is defined through\\cite{2D_QC_4D_QHE,zilberberg2018photonic,ozawa2016synthetic,4D_QHE_ultracold_atom,qi2008topological}\n\\begin{align}\n    C_{2} = \\frac{1}{4\\pi^{2}} \\int_{\\mathbb{T}^{4}} d^{4}k  \\left( \\Omega_{xy}\\Omega_{zw} + \\Omega_{wx}\\Omega_{zy} + \\Omega_{zx} \\Omega_{yw} \\right)\n\\end{align}\nbecomes\n\\begin{align}\n    C_{2} = \\frac{1}{4\\pi^{2}} \\left(\\int_{\\mathbb{T}^{2}} dk_{x}dk_{y}  \\Omega_{xy}\\right)\\left(\\int_{\\mathbb{T}^{2}} dk_{z}dk_{w}\\Omega_{zw}\\right)\n\\end{align}\nin this limit, where $\\Omega_{\\mu \\nu}$ is the Abelian Berry curvature in the $k_{\\mu}$-$k_{\\nu}$ plane. \nFor both examples in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} and Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2}, we have identified the weak Chern number $\\nu_{z} = -1$, implying that both systems have $\\frac{1}{2 \\pi} \\int_{\\mathbb{T}^{2}} dk_{x}dk_{y}  \\Omega_{xy} = -1$. \nIn fact, for 3D Weyl-CDW system it have been shown that there will always be background QAH response in the $xy$ plane\\cite{dynamical_axion_insulator_BB}, implying that in the limit of decoupled Chern insulators we have $\\frac{1}{2 \\pi} \\int_{\\mathbb{T}^{2}} dk_{x}dk_{y}  \\Omega_{xy} \\ne 0$. \nFurthermore, as we shift the CDW sliding phase $\\phi$, which is equivalent to shifting the momentum $k_{w}$, by $2\\pi$, all the Chern insulating layers will be shifted by $\\Delta z = -\\Delta k_{w}\/(2\\pi q) = -2\\pi \/(2\\pi q) = -1\/q$, implying a non-trivial Thouless charge pump along $z$. \nSpecifically, for Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} with $q = 1\/5$, all the Chern insulating layers will be shifted by $\\Delta z = -5$, which is equal to the unit cell length along $z$, implying $\\left| \\frac{1}{2 \\pi} \\int_{\\mathbb{T}^{2}} dk_{z}dk_{w}\\Omega_{zw} \\right|=1$. \nThe fact that the 3D Weyl-CDW system can be viewed as layers of Chern insulators\\cite{dynamical_axion_insulator_BB} and the expression $\\Delta z = -1\/q$ governing the charge pumping along $z$ as we vary $k_{w}$ by $2\\pi$ collectively predict a non-zero $\\frac{1}{2 \\pi} \\int_{\\mathbb{T}^{2}} dk_{z}dk_{w}\\Omega_{zw}$. \nTherefore, for a 3D Weyl-CDW system with QAH surface states\\cite{dynamical_axion_insulator_BB}, the corresponding 4D theory is described by a (4+1)D Chern-Simon theory in Eq.~(\\ref{eq:action}) with non-zero $C_{2}$. \nFurthermore, this result holds even as we deform away from the decoupled-layer limit, provided no energy gaps close. \nThus the 3D Weyl-CDW system serves as a platform to study higher-dimensional topological field theories.\n\n\nLet us conclude with two remarks. \nFirst, from the above analysis, we see that a 3D Weyl-CDW system with QAH surface states provides a platform to examine a 4D nodal line system gapped by a $U(1)$ gauge field and carries non-zero second Chern number $C_{2}$. \nSecondly, as opposed to Secs.~\\ref{sec_chiral_HOTI} and~\\ref{sec_helical_HOTI_sliding_modes} where we have in higher dimensions a gapped topological phase coupled to $U(1)$ or $SU(2)$ gauge fields, in the 4D model promoted from a 3D Weyl-CDW system it is precisely the coupling to a $U(1)$ gauge field that opens up a bulk gap, inducing emergent CI layers, QAH surface states and non-zero $C_{2}$.\n\n\n\\section{\\label{sec:outlook}Outlook}\n\n\nTo conclude, we have shown in Secs.~\\ref{sec_chiral_HOTI} and \\ref{sec_helical_HOTI_sliding_modes} that higher-order topology in 3D can be probed in 2D DW systems. \nFurthermore, we showed in Sec.~\\ref{sec:Weyl_CDW} how 3D Weyl-CDW systems with background QAH response can be used to study 4D topology. \nThe next and natural step is to identify 3D systems with modulations coexisting with hinge or corner modes. \nThis will be a platform for studying 4--or even higher--dimensional higher-order topology. \nOur dimensional promotion procedure in Sec.~\\ref{sec_Dimension_promotion} can also be used together with the topological classification based on crystalline symmetries\\cite{Po2017,khalaf2018higher,Chiu2016,NaturePaper} in the promoted dimensions, in order to explore topological crystalline phases in higher dimensions. \nWith suitably chosen modulated systems, we may either study (1) how topological crystalline insulators diagnosed by symmetry-based indicators\\cite{Po2017,comment,khalaf,ashvin-materials,po2020symmetry,watanabe2018structure} in the promoted dimensions respond to a background $U(1)$ or $SU(2)$ gauge fields, or (2) how topological semimetals\\cite{armitage2018weyl} in the promoted dimensions can be gapped by background $U(1)$ or $SU(2)$ gauge fields. \nWith the dimensional promotion procedure, we may also extend our studies of topological materials to those with space groups beyond 3D, known as superspace groups\\cite{superspace1,superspace2,superspace3,superspace4,superspace5}. \nTo extract the full information in higher dimension, a way to control the phase offset $\\{\\phi^{(i)} \\}$ experimentally is needed, and currently applying electromagnetic fields to depin the (charge- or spin-)density waves is one practical approach\\cite{gooth2019evidence,gruner1988dynamics}. \nIn addition, since we can systematically compute the background continuous gauge field coupled to the dimensionally-promoted model, we can again use low dimensional modulated systems to study the low energy dynamics in higher dimensions, by minimally coupling the low energy theory to the known continuous gauge fields as in Sec.~\\ref{sec:Weyl_CDW}. \nAs our dimensional promotion procedure can be carried out for both commensurate and incommensurate modulations, this approach can be used to study topological properties of system with quasi-periodic potentials~\\cite{Earliest_dimension_promotion_superspace,2D_QC_4D_QHE,zilberberg2018photonic,equivalence_Fibo_Harper_2012,Kraus_1D_QC_to_2D_QHE} where conventional band theory is not applicable. \nThe general procedure will be to promote the dimension of these quasi-periodic systems and examine the response of possible topological phase in higher dimensions to a gauge field producing an irrational flux per plaquette. \nThese techniques can be applied to analyze the DW phases in material systems of interest such as (TaSe$_4$)$_2$I\\cite{gooth2019evidence,shi2019charge,zhang2020first,tournier2013electronic} and ZrTe$_5$\\cite{tang2019three,qin2020theory,song2017instability,zhang2017transport}. \nThis can also lead to interesting studies on the higher-dimensional Hofstadter butterfly, complementing the recent studies of Refs.~\\onlinecite{Bernevig_fragile_LL,Herzog_Hof_topo}. \n\\textcolor{black}{Another interesting direction is to introduce dynamics to the DW modulation. This can happen, for example, when the phase offsets $\\{ \\phi^{(i)}\\}$ acquire non-adiabatic time-dependence and become $\\{ \\phi^{(i)}(t)\\}$. Previous studies have focused on promoting the dimension of a periodically-driven system to a Floquet lattice, which under certain conditions can lead to topologically-protected quantized energy pump~\\cite{time_periodic_1,time_periodic_2,Refael_Topo_freq_conversion_cavity_PRB}. We expect that richer phenomena in higher-dimensional space can be investigated when the DW modulations are not only periodic in real-space but also (1) periodic in time or (2) have general time-dependence.} Finally, we have shown in Sec.~\\ref{sec_helical_HOTI_sliding_modes} the simplest case of how $SU(2)$ gauge field physics may be studied through a 2D modulated system. \nRecently, the spin-orbit-coupled Hofstadter models induced by non-Abelian $SU(2)$ gauge fields have also been studied both in 2D\\cite{2D_SU2_butterfly_Joannopoulos} and 3D\\cite{3D_SU2_butterfly_Joannopoulos}, where Dirac points with up to 16-fold degeneracy and various topological insulating states were found. \nWe expect that 3D DW materials with different types of spin-orbit coupled modulations may enable simulation of various aspects of the physics of $SU(2)$ gauge fields in 4D or higher dimensions, including topological states and $SU(2)$ Hofstadter butterflies~\\cite{YiLi_SU2_Hofstadter,2D_SU2_butterfly_Joannopoulos,3D_SU2_butterfly_Joannopoulos}. \nWe hope that this work will lay the groundwork for the exciting future investigations mentioned above, and extend the search for exotic topological phases beyond 3D. \nIn particular, there are many possible defects that one can imagine in a spin-orbit coupled density wave order parameter, each of which may correspond to a non-trivial response to $SU(2)$ gauge field defects in the higher-dimensional system.\n\n\n\n\n\\begin{acknowledgments}\nThe authors would like to thank Y. Li and B. Wieder for fruitful discussions. \nThis work was supported by the Alfred P. Sloan Foundation, and by the National Science Foundation under grant DMR-1945058. \nNumerical computations made use of the Illinois Campus Cluster, a computing resource that is operated by the Illinois Campus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA) and which is supported by funds from the University of Illinois at Urbana-Champaign. \nNumerical calculations in this work employed the open-source PythTB package\\cite{PythTB}.\n\\end{acknowledgments}\n\n\n\n\\section{\\label{sec:General_dimension_promotion}General dimensional promotion procedure}\n\nIn this section, we develop a general procedure to promote a $d$-dimensional ($d$D) modulated system to a $(d+N)$D lattice model coupled to a $U(1)$ gauge field. \nGeneralizing Sec.~III in the main text, here we allow for both on-site energy and hopping modulations, general (potentially non-cubic) lattice structures in the higher-dimensional model, as well as arbitrary orbital positions. \nSec.~\\ref{subsec:formalism} is devoted to the development of the method. \nSpecific examples are given in Sec.~\\ref{subsec:examples_general_dim_promotion}.\n\n\n\\subsection{\\label{subsec:formalism}Formalism}\n\n\\subsubsection{\\label{subsubsec:__dimensional_promotion}Dimensional promotion}\nSuppose we have a $d$D periodic electronic system with lattice vectors $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d} \\}$, which form a general $d$D Bravais lattice. \nNote that the $\\vec{a}_i$ need not be mutually orthogonal. \nWe add $N$ mutually incommensurate on-site and hopping modulations with modulation wave vectors $\\{\\vec{q}^{(1)},\\cdots,\\vec{q}^{(N)}\\}$, such that the Hamiltonian is given by\n\\begin{equation}\n    H_{\\text{low-dim}} = \\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m}} \\left[H_{\\vec{m}}\\right] {\\psi}_{\\vec{n}} + {\\sum_{\\vec{n},\\vec{l}}}  \\sum_{i=1}^{N} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l}} \\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right] {\\psi}_{\\vec{n}} + \\sum_{\\vec{n}} \\sum_{i=1}^{N}{\\psi}^{\\dagger}_{\\vec{n}} \\left[ V^{(i)}_{\\vec{n}} \\right] {\\psi}_{\\vec{n}}. \\label{eq:H_low_general}\n\\end{equation}\nHere $\\vec{n} = (n_{1},\\cdots,n_{d})$, $\\vec{m} = (m_{1},\\cdots,m_{d})$ and $\\vec{l} = (l_{1},\\cdots,l_{d})\\in \\mathbb{Z}^{d}$ index lattice positions in reduced coordinates. \nFor example, the vector $\\vec{n}$ uniquely denotes the lattice site $\\sum_{j=1}^d n_j\\vec{a}_j$. \nThe multi-component operator ${\\psi}^{\\dagger}_{\\vec{n}}$ creates an electron at site $\\vec{n}$ with a given set of spin and orbital degrees of freedom.\nThe matrix $\\left[H_{\\vec{m}}\\right]$ is the unmodulated hopping matrix connecting lattice position $\\vec{n}$ to $\\vec{n}+\\vec{m}$, with matrix indices encoding the spin and orbital dependence of the hopping. \nSimilarly, $\\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right]$ is the $i^{\\text{th}}$ ($i = 1 ,\\ldots, N$) modulated hopping matrix connecting lattice position $\\vec{n}$ to $\\vec{n}$+$\\vec{l}$, and $\\left[ V^{(i)}_{\\vec{n}} \\right]$ is the $i^{\\text{th}}$ ($i = 1 ,\\ldots, N$) modulated on-site energy at lattice position $\\vec{n}$. \n\nEach $\\vec{q}^{(i)}$ is associated with a set of modulated couplings $\\left[V^{(i)}_{\\vec{n}} \\right]$ and $\\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right]$. \nNote it is possible that for a given $\\vec{q}^{(i)}$, $\\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right]$ is only non-zero for some $\\vec{l}$.\nHere we sum over all $\\vec{l}$, and set $\\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right] =0$ if for a given $\\vec{q}^{(i)}$ there is no modulated hopping matrix with the given $\\vec{l}$. \nIn most practical cases of interest, $\\vec{l}$ will be restricted to a nearest-neighbor hopping. \nNote that in the situations considered in the main text we took $\\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right]=0$ for all $\\vec{l}$. \nAnalogous considerations let us analyze systems with modulated hoppings but no modulated on-site potentials by taking $[V_{\\vec{n}}^{(i)}]=0$.\n\nWe require $H_{\\text{low-dim}}$ to be Hermitian, such that\n\\begin{align}\n    & \\left[ H_{-\\vec{m}} \\right]^{\\dagger} = \\left[H_{\\vec{m}} \\right],  \\label{eq:Hermitian_1}  \\\\\n    & \\left[ H^{(i)}_{-\\vec{l},\\vec{n}+\\vec{l}} \\right]^{\\dagger} = \\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right], \\label{eq:Hermitian_2} \\\\\n    &  \\left[ V^{(i)}_{\\vec{n}} \\right]^{\\dagger} = \\left[ V^{(i)}_{\\vec{n}} \\right].\\label{eq:Hermitian_3} \n\\end{align}\nEq.~(\\ref{eq:Hermitian_2}) is another statement that the hopping process from $\\vec{n}$ to $\\vec{n}+\\vec{l}$ is the conjugate of the hopping process from $\\vec{n}+\\vec{l}$ to  $\\vec{n}$. \nWe assume that the modulations are periodic with functional dependence\n\\begin{align}\n    & \\left[ H_{\\vec{l},\\vec{n}}^{(i)} \\right] = \\left[ g^{(i)}_{\\vec{l}} \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left( n_{1}\\vec{a}_{1}+\\cdots+n_{d}\\vec{a}_{d} \\right) + \\phi^{(i)} \\right)\\right], \\label{eq:H_form_1} \\\\\n    & \\left[ V^{(i)}_{\\vec{n}} \\right] = \\left[f^{(i)}  \\left( 2\\pi \\vec{q}^{(i)} \\cdot  \\left( n_{1}\\vec{a}_{1}+\\cdots+n_{d}\\vec{a}_{d} \\right) + \\phi^{(i)} \\right)\\right], \\label{eq:H_form_2}\n\\end{align}\nsuch that\n\\begin{align}\n    & \\left[ g^{(i)}_{\\vec{l}} (x)\\right] = \\left[ g^{(i)}_{\\vec{l}} (x + 2\\pi)\\right], \\\\\n    & \\left[f^{(i)} (x)\\right] = \\left[f^{(i)} (x+2\\pi)\\right].\n\\end{align}\nTherefore, we can expand $\\left[ H_{\\vec{l},\\vec{n}}^{(i)} \\right]$ and $\\left[ V^{(i)}_{\\vec{n}} \\right]$ in terms of Fourier components,\n\\begin{align}\n    & \\left[ H_{\\vec{l},\\vec{n}}^{(i)} \\right] = \\sum_{p } \\left[ H_{\\vec{l},p}^{(i)} \\right] e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) }, \\label{eq:general_FT_1} \\\\\n    &  \\left[ V^{(i)}_{\\vec{n}} \\right] = \\sum_{p } \\left[ V^{(i)}_{p} \\right] e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) }, \\label{eq:general_FT_2}\n\\end{align}\nwhere $p \\in \\mathbb{Z}$. The matrices $\\left[ H_{\\vec{l},p}^{(i)} \\right]$ and $\\left[ V^{(i)}_{p} \\right]$ are the $p^{\\text{th}}$ (matrix-valued) Fourier components of $\\left[ H_{\\vec{l},\\vec{n}}^{(i)} \\right]$ and $\\left[ V^{(i)}_{\\vec{n}} \\right]$, respectively. \nEqs.~(\\ref{eq:Hermitian_2}) and (\\ref{eq:Hermitian_3}) also imply\n\\begin{align}\n    & \\left[ H^{(i)}_{-\\vec{l},-p} \\right]^{\\dagger} e^{ip 2 \\pi \\vec{q}^{(i)} \\cdot \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}} = \\left[ H^{(i)}_{\\vec{l},p} \\right], \\label{eq:Hermitian_4}\\\\\n    & \\left[ V^{(i)}_{-p} \\right]^{\\dagger} = \\left[ V^{(i)}_{p} \\right]. \\label{eq:Hermitian_5}\n\\end{align}\n\nAlthough Eq.~(\\ref{eq:Hermitian_4}) involves several different indices, it will be crucial in forming our dimensionally-promoted Hamiltonian. \nAs such, it is illuminating to prove it as follows. \nStarting from Eq.~(\\ref{eq:general_FT_1}), we can obtain\n\\begin{align}\n    \\left[ H_{-\\vec{l},\\vec{n}}^{(i)} \\right] = \\sum_{p } \\left[ H_{-\\vec{l},p}^{(i)} \\right] e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) }\n\\end{align}\nby changing $\\vec{l} \\to -\\vec{l}$. \nWe next shift $\\vec{n} \\to \\vec{n} + \\vec{l}$ such that\n\\begin{align}\n    \\left[ H_{-\\vec{l},\\vec{n}+\\vec{l}}^{(i)} \\right] = \\sum_{p } \\left[ H_{-\\vec{l},p}^{(i)} \\right] e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) } e^{ip 2\\pi \\vec{q}^{(i)} \\cdot \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}}. \\label{eq:general_before_Hermitian_1}\n\\end{align}\nTaking Hermitian conjugate of Eq.~(\\ref{eq:general_before_Hermitian_1}) and setting $p \\to -p$, we have\n\\begin{align}\n    \\left[ H_{-\\vec{l},\\vec{n}+\\vec{l}}^{(i)} \\right]^{\\dagger} = \\sum_{p } \\left[ H_{-\\vec{l},-p}^{(i)} \\right]^{\\dagger} e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) } e^{ip 2\\pi \\vec{q}^{(i)} \\cdot \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}}. \\label{eq:pf_1}\n\\end{align}\nIn order for Eq.~(\\ref{eq:pf_1}) being consistent with the Hermiticity constraint Eq.~(\\ref{eq:Hermitian_2}), we deduce that Eq.~(\\ref{eq:Hermitian_4}) must hold.\n\nTo continue with our development of the dimensional promotion procedure, we plug Eq.~(\\ref{eq:general_FT_1}) and Eq.~(\\ref{eq:general_FT_2}) into Eq.~(\\ref{eq:H_low_general}) such that Hamiltonian of the $d$D modulated system can be written as\n\\begin{align}\n    H_{\\text{low-dim}} & = \\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m}} \\left[ H_{\\vec{m}}\\right] {\\psi}_{\\vec{n}}  \\nonumber \\\\\n    & + {\\sum_{\\vec{n},\\vec{l}}}  \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l}} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right] e^{i  p_{i} \\left( 2\\pi \\vec{q}^{(i)} \\cdot  \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) } {\\psi}_{\\vec{n}} \\nonumber \\\\\n    & + \\sum_{\\vec{n}} \\sum_{i=1}^{N} \\sum_{p_{i}}{\\psi}^{\\dagger}_{\\vec{n}} \\left[ V^{(i)}_{p_{i}} \\right] e^{i  p_{i} \\left( 2\\pi \\vec{q}^{(i)} \\cdot  \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) } {\\psi}_{\\vec{n}}. \\label{eq:H_low_general_2}\n\\end{align}\nNotice that for each $i$ (which indexes the modulation wave vectors) we sum over all $p_{i} \\in \\mathbb{Z}$. \nTo promote Eq.~(\\ref{eq:H_low_general_2}) to a $(d+N)$D space, let us introduce a set of additional lattice vectors $\\{\\vec{c}_{1},\\cdots,\\vec{c}_{N} \\}$ such that, together with $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d} \\}$, we have a linearly independent basis spanning a $(d+N)$D lattice. \nNotice that this $(d+N)$D lattice is not necessarily orthorhombic, since we do not require the lattice vectors $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d} \\}$ and $\\{\\vec{c}_{1},\\cdots,\\vec{c}_{N} \\}$ be pairwise orthogonal. \nNext, we introduce the corresponding reciprocal lattice vectors $\\{\\vec{g}_{1},\\cdots,\\vec{g}_{d}\\}$ and $\\{\\vec{G}_{1},\\cdots,\\vec{G}_{N}\\}$ in the $(d+N)$D reciprocal space such that\n\\begin{align}\n    & \\vec{g}_{i} \\cdot \\vec{a}_{j} = 2 \\pi \\delta_{ij},\\ i = 1 ,\\ldots, d,\\ {\\text{and }}j = 1 ,\\ldots, d, \\label{eq:reciprocal_1} \\\\\n    & \\vec{g}_{i} \\cdot \\vec{c}_{j} = 0,\\ i = 1 ,\\ldots, d,\\ {\\text{and }}j = 1 ,\\ldots, N, \\label{eq:reciprocal_2} \\\\\n    & \\vec{G}_{i} \\cdot \\vec{a}_{j} = 0,\\ i = 1 ,\\ldots, N,\\ {\\text{and }}j = 1 ,\\ldots, d, \\label{eq:reciprocal_3} \\\\\n    & \\vec{G}_{i} \\cdot \\vec{c}_{j} = 2\\pi \\delta_{ij},\\ i = 1 ,\\ldots, N,\\ {\\text{and }}j = 1 ,\\ldots, N. \\label{eq:reciprocal_4}\n\\end{align}\nWe can now identify $\\phi^{(i)} \\in [0,2\\pi)$ with $\\vec{k} \\cdot \\vec{c}_{i}$, which is $2\\pi$ times the coefficient of $\\vec{k}$ along $\\vec{G}_{i}$ (see Eq.~(\\ref{eq:reciprocal_4})); the periodicity of $\\phi^{(i)}$ is reflected in the periodicity of the $(d+N)$D Brillouin zone. \nWe then promote the $d$D model to a $(d+N)$D space by summing over $\\{ \\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N} \\} \\in \\mathbb{T}^{N}$, where $\\mathbb{T}^N$ denotes the $N$-torus represented as a parallelepiped with boundary spanned by $\\{\\vec{G}_{1},\\cdots,\\vec{G}_{N} \\}$, and opposite edges identified. \nWe further label the original creation and annihilation operators by additional symbols $\\{ \\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N} \\} \\in \\mathbb{T}^{N}$, such that\n\\begin{align}\n    H_{\\text{high-dim}} =&\\sum_{\\vec{k}\\cdot\\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N}} \\sum_{\\vec{n},\\vec{m}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} \\left[H_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}}   \\nonumber \\\\\n    & + \\sum_{\\vec{k}\\cdot\\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N}} {\\sum_{\\vec{n},\\vec{l}}}  \\sum_{i=1}^{N} \\sum_{p_{i}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}}  \\left[H_{\\vec{l},p_{i}}^{(i)}\\right] e^{ip_{i}\\vec{k}\\cdot\\vec{c}_{i}} e^{i  p_{i}  2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)   } {\\psi}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} \\nonumber \\\\\n    & +  \\sum_{\\vec{k}\\cdot\\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N}}\\sum_{\\vec{n}}  \\sum_{i=1}^{N} \\sum_{p_{i}}  {\\psi}^{\\dagger}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} \\left[V^{(i)}_{p_{i}}\\right] e^{ip_{i}\\vec{k}\\cdot\\vec{c}_{i}} e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}{\\psi}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}}, \\label{eq:non_ortho_3}\n\\end{align}\nwhere the summation of each $\\vec{k} \\cdot \\vec{c}_{i}$ is from $0$ to $2\\pi$. \nNotice that the summation over $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k} \\cdot \\vec{c}_{N} \\}$ is outside of $\\sum_{i=1}^{N} \\sum_{p_{i}} $ over all integer $p_{i}$ for each modulation $i = 1 ,\\ldots, N$. \nThis means that we are summing over $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots, \\vec{k}\\cdot\\vec{c}_{N} \\}$ for each term in Eq.~(\\ref{eq:H_low_general_2}). \nAs mentioned in the main text, the physical motivation to sum over $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N} \\}$ is because a single set of $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N} \\}$ does not contain the full information of the lattice model in the promoted $(d+N)$D space. \nOnly when we consider all values of $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N} \\}$, can we obtain the exact form of the promoted lattice model, shown in Eq.~(\\ref{eq:non_ortho_4}) below. \nWe remind the readers that repeated indices are not implicitly summed.\n\nNote that ${\\psi}^{\\dagger}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}}$ is the Fourier transform of $\\psi^{\\dagger}_{\\vec{n},\\vec{\\nu}}$ through\n\\begin{align}\n    {\\psi}^{\\dagger}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} = \\frac{1}{\\sqrt{L}} \\sum_{\\vec{\\nu}} e^{i \\vec{k}\\cdot\\left( \\nu_{1}\\vec{c}_{1}+ \\cdots + \\nu_{N}\\vec{c}_{N} \\right)} \\psi^{\\dagger}_{\\vec{n},\\vec{\\nu}}, \\label{eq:FT_general}\n\\end{align}\nwhere $\\vec{\\nu} = (\\nu_{1},\\cdots,\\nu_{N}) \\in \\mathbb{Z}^{N}$ and $L$ is the size of the system in the additional dimensions (taken to infinity at the end of the calculation). We can thus take an inverse Fourier transform to find\n\\begin{align}\n    H_{\\text{high-dim}} =& \\sum_{\\vec{n},\\vec{m},\\vec{\\nu}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu}} \\left[ H_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{\\nu}}   \\nonumber \\\\\n    & + {\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}}  \\sum_{i=1}^{N} \\sum_{p_{i}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}}  \\left[H_{\\vec{l},p_{i}}^{(i)}\\right] e^{i  2\\pi p_{i}   \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)   } {\\psi}_{\\vec{n},\\vec{\\nu}} \\nonumber \\\\\n    & + \\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[V^{(i)}_{p_{i}}\\right]  e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}{\\psi}_{\\vec{n},\\vec{\\nu}}, \\label{eq:non_ortho_4}\n\\end{align}\nwhere ${\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}}$ denotes the electron creation operator for an electron at the lattice position $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j}$, and ${\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu} - p_{i}\\hat{\\nu}_{i}}$ denotes the electron creation operator for an electron at the lattice position $\\sum_{j=1}^{d}(n_{j} + l_{j})\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} - p_{i} \\vec{c}_{i}$. \nNotice that a $d$D modulated system with phase offsets $\\{\\phi^{(1)},\\cdots,\\phi^{(N)}\\}$ is described by the Bloch Hamiltonian (see Eq.~(\\ref{eq:non_ortho_3})) of the promoted $(d+N)$D lattice model with fixed crystal momenta $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N} \\}$ via our identification of $\\vec{k} \\cdot \\vec{c}_{i}$ with $\\phi^{(i)}$. \nFor later convenience, we can also rewrite Eq.~(\\ref{eq:non_ortho_4}) as\n\\begin{align}\n    H_{\\text{high-dim}} =& \\sum_{\\vec{n},\\vec{m},\\vec{\\nu}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu}} \\left[H_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{\\nu}}    \\label{eq:non_ortho_5_1} \\\\\n    & + {\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}} \\sum_{i=1}^{N} \\sum_{p_{i}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}}  \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]e^{-i \\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d} l_{j}\\vec{a}_{j} \\right) }e^{i 2\\pi p_{i}   \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}\\left( n_{j} + \\frac{1}{2}l_{j} \\right)\\vec{a}_{j} \\right)   } {\\psi}_{\\vec{n},\\vec{\\nu}} \\label{eq:non_ortho_5_2} \\\\\n    & + \\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[V^{(i)}_{p_{i}}\\right]  e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}{\\psi}_{\\vec{n},\\vec{\\nu}}, \\label{eq:non_ortho_5_3}\n\\end{align}\nwhere we have used\n\\begin{align}\n    \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} = \\sum_{j=1}^{d}\\left( n_{j} + \\frac{1}{2}l_{j} \\right)\\vec{a}_{j} - \\sum_{j=1}^{d} \\frac{1}{2}l_{j} \\vec{a}_{j}.\n\\end{align}\nThis will prove useful when we go to identify the gauge fields appearing in the hopping matrix elements. \nWe remind the readers again that the product $p_{i}\\vec{q}^{(i)}$ in Eqs.~(\\ref{eq:non_ortho_5_1}--\\ref{eq:non_ortho_5_3}) does not imply a summation over $i$, rather it denotes the product of integer $p_{i}$ and the $i^{th}$ modulation wave vector $\\vec{q}^{(i)}$. \nEqs.~(\\ref{eq:non_ortho_5_1}--\\ref{eq:non_ortho_5_3}) can then be interpreted as a $(d+N)$D lattice model with Hamiltonian\n\\begin{align}\n    H_{\\text{high-dim}} =& \\sum_{\\vec{n},\\vec{m},\\vec{\\nu}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu}} \\left[H_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{\\nu}}    \\label{eq:non_ortho_6_1} \\\\\n    & + {\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}} \\sum_{i=1}^{N} \\sum_{p_{i}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}}  \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]e^{-i \\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d} l_{j}\\vec{a}_{j} \\right) }{\\psi}_{\\vec{n},\\vec{\\nu}} \\label{eq:non_ortho_6_2} \\\\\n    & + \\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[V^{(i)}_{p_{i}}\\right] {\\psi}_{\\vec{n},\\vec{\\nu}}, \\label{eq:non_ortho_6_3}\n\\end{align}\ncoupled through a Peierls substitution\\cite{Peierls_substitution} to a $U(1)$ gauge field \n\\begin{align}\n    \\vec{A} = \\frac{1}{2\\pi} \\sum_{i=1}^{N} \\sum_{j=1}^{d} \\vec{G}_{i} \\left( \\vec{q}^{(i)}\\cdot \\left[ \\left( \\vec{r} \\cdot \\vec{g}_{j} \\right) \\vec{a}_{j} \\right] \\right), \\label{eq:non_ortho_A}\n\\end{align}\nwhere $\\vec{r} \\in \\mathbb{R}^{d+N}$. \nNotice that Eqs.~(\\ref{eq:non_ortho_6_1}--\\ref{eq:non_ortho_6_3}) represent the $(d+N)$D model {\\it without} $U(1)$ gauge fields.\nAlthough there are complex phases $e^{-i \\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d} l_{j}\\vec{a}_{j} \\right) }$ in Eq.~(\\ref{eq:non_ortho_6_2}), they do not depend on the reduced coordinates $(\\vec{n},\\vec{\\nu}) \\in \\mathbb{Z}^{d+N}$ in the dimensionally-promoted lattice and so may be regarded as inherent phase factors in the $(d+N)$D model without $U(1)$ gauge field. \nTo validate the identification Eq.~(\\ref{eq:non_ortho_A}), we compute the various Peierls phase factors in the next section. \n\n\\subsubsection{\\label{sec:computing_Peierls_phase_1}Computation of Peierls phases}\n\nLet us consider the terms in Eq.~(\\ref{eq:non_ortho_6_3}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j}\n\\end{equation}\nto \n\\begin{equation}\n \\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} - p_{i} \\vec{c}_{i}. \n\\end{equation}\nThe Peierls phase can be computed from a straight line integral from $\\vec{r}_{i}$ to $\\vec{r}_{f} \\in$ $\\mathbb{R}^{d+N}$ through\n\\begin{align}\n    \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}},\n\\end{align}\nwhere we have worked in unit $\\hbar = c= |e| = 1$ and the electron has charge $-|e| = -1$. \nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n    \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} - p_{i} \\vec{c}_{i}t,\n\\end{align}\nwhere $ t\\in [0,1]$ and the corresponding infinitesimal displacement vector is $-p_{i}\\vec{c}_{i}dt$. \nThe line integral can then be computed as follows:\n\\begin{align}\n    \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} & =  \\exp{i \\int_{0}^{1}dt  p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{l=1}^{d}n_{l}\\vec{a}_{l} + \\sum_{l=1}^{N}\\nu_{l}\\vec{c}_{l} - p_{i} \\vec{c}_{i}t \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_1} \\\\\n    & = \\exp{i \\int_{0}^{1}dt  p_{i}  \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\delta_{ij} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{l=1}^{d}n_{l}\\vec{a}_{l}  \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_2} \\\\\n    & = \\exp{i 2\\pi \\int_{0}^{1}dt  p_{i}   \\sum_{k=1}^{d} \\left( \\vec{q}^{(i)}\\cdot \\left[ \\sum_{l=1}^{d}n_{l}\\delta_{lk}    \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_3} \\\\\n    & = \\exp{i 2\\pi \\int_{0}^{1}dt  p_{i}   \\left( \\vec{q}^{(i)}\\cdot \\sum_{k=1}^{d}  n_{k}  \\vec{a}_{k}  \\right) } \\label{eq:Peierls_1_4} \\\\\n    & = \\exp{i 2\\pi  p_{i}    \\vec{q}^{(i)}\\cdot \\sum_{k=1}^{d}  n_{k}  \\vec{a}_{k}  }. \\label{eq:Peierls_1_5}\n\\end{align}\nWe used Eq.~(\\ref{eq:reciprocal_2}) and Eq.~(\\ref{eq:reciprocal_4}) in going from Eq.~(\\ref{eq:Peierls_1_1}) to Eq.~(\\ref{eq:Peierls_1_2}), Eq.~(\\ref{eq:reciprocal_1}) in going from Eq.~(\\ref{eq:Peierls_1_2}) to Eq.~(\\ref{eq:Peierls_1_3}) and then finally do an integral $\\int_{0}^{1}dt=1$ to obtain Eq.~(\\ref{eq:Peierls_1_5}). \nEq.~(\\ref{eq:Peierls_1_5}) is exactly the additional phase factor in Eq.~(\\ref{eq:non_ortho_5_3}) compared to Eq.~(\\ref{eq:non_ortho_6_3}). \n\n\nNext, consider Eq.~(\\ref{eq:non_ortho_6_2}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j}\n\\end{equation}\nto \n\\begin{equation}\n\\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} - p_{i} \\vec{c}_{i}.\n\\end{equation} \nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n    \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\left(  \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} - p_{i} \\vec{c}_{i} \\right) t,\n\\end{align}\nwhere $ t= [0,1]$ and the corresponding infinitesimal displacement vector is $\\left(  \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} - p_{i} \\vec{c}_{i} \\right) dt$. \nThe line integral can then be computed as follows:\n\\begin{align}\n    & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n    & =  \\exp{-i \\int_{0}^{1}dt  \\left(  \\sum_{r=1}^{d}l_{r}\\vec{a}_{r} - p_{i} \\vec{c}_{i} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j,k} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\sum_{r=1}^{N}\\nu_{r}\\vec{c}_{r} + \\left(  \\sum_{r=1}^{d}l_{r}\\vec{a}_{r} - p_{i} \\vec{c}_{i} \\right) t \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_1} \\\\\n    & =  \\exp{i \\int_{0}^{1}dt  p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j,k} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{r=1}^{d}\\left( n_{r} + l_{r}t \\right)\\vec{a}_{r}   \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_2} \\\\\n    & =  \\exp{i 2\\pi \\int_{0}^{1}dt  p_{i}   \\sum_{j,k} \\delta_{ij} \\left( \\vec{q}^{(j)}\\cdot \\left[  \\sum_{r=1}^{d}\\left( n_{r} + l_{r}t \\right)\\delta_{rk}  \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_3} \\\\\n    & =  \\exp{i 2\\pi \\int_{0}^{1}dt  p_{i}   \\sum_{k=1}^{d}  \\left( \\vec{q}^{(i)}\\cdot \\left[  \\left( n_{k} + l_{k}t   \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_4} \\\\\n    & =  \\exp{i 2\\pi  p_{i}  \\vec{q}^{(i)}\\cdot \\left( \\sum_{k=1}^{d}   \\left( n_{k} + \\frac{1}{2}l_{k}   \\right) \\vec{a}_{k} \\right) }. \\label{eq:Peierls_2_5}\n\\end{align}\nWe used Eq.~(\\ref{eq:reciprocal_2}) and Eq.~(\\ref{eq:reciprocal_3}) in going from Eq.~(\\ref{eq:Peierls_2_1}) to Eq.~(\\ref{eq:Peierls_2_2}), Eq.~(\\ref{eq:reciprocal_1}) and Eq.~(\\ref{eq:reciprocal_4}) in going from Eq.~(\\ref{eq:Peierls_2_2}) to Eq.~(\\ref{eq:Peierls_2_3}) and then finally do integrals $\\int_{0}^{1}dt=1$ and $\\int_{0}^{1}tdt=\\frac{1}{2}$ to obtain Eq.~(\\ref{eq:Peierls_2_5}). \nEq.~(\\ref{eq:Peierls_2_5}) is exactly the additional phase factor in Eq.~(\\ref{eq:non_ortho_5_2}) compared to Eq.~(\\ref{eq:non_ortho_6_2}). \nCrucially, we see that our redefinition Eq.~(\\ref{eq:non_ortho_A}) accounts for the factor of $1\/2$ arising in the line integral of the vector potential.\n\n\nFinally, consider Eq.~(\\ref{eq:non_ortho_6_1}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j}\n\\end{equation}\nto\n\\begin{equation} \n \\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\sum_{j=1}^{d}m_{j}\\vec{a}_{j}.\n\\end{equation}\nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n    \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\left(  \\sum_{j=1}^{d}m_{j}\\vec{a}_{j} \\right) t,\n\\end{align}\nwhere $ t= [0, 1]$ and the corresponding infinitesimal displacement vector is $\\left(  \\sum_{j=1}^{d}m_{j}\\vec{a}_{j} \\right) dt$. \nThe line integral can then be computed as follows:\n\\begin{align}\n    & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n    & =  \\exp{-i \\int_{0}^{1}dt  \\left(  \\sum_{r=1}^{d}m_{r}\\vec{a}_{r} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j,k} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\sum_{r=1}^{N}\\nu_{r}\\vec{c}_{r} + \\left(  \\sum_{r=1}^{d}m_{r}\\vec{a}_{r} \\right) t \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_3_1} \\\\\n    & = 1. \\label{eq:Peierls_3_2}\n\\end{align}\nWe used Eq.~(\\ref{eq:reciprocal_3}) in going from Eq.~(\\ref{eq:Peierls_3_1}) to Eq.~(\\ref{eq:Peierls_3_2}). \nThis means that no additional phase factors arise if we compare Eq.~(\\ref{eq:non_ortho_5_1}) and Eq.~(\\ref{eq:non_ortho_6_1}). \nThus, we see that our dimensionally-promoted Hamiltonian is consistent with Eqs.~(\\ref{eq:non_ortho_6_1}--\\ref{eq:non_ortho_6_3}) coupled via a Peierls substitution to the vector potential Eq.~(\\ref{eq:non_ortho_A}).\n\n\\subsubsection{\\label{sec:remarks_consistency_check_and_summary}Remarks, consistency checks and summary}\n\nLet us briefly recap what we have developed. \nWe began with a $d$D system with modulated hoppings and on-site energies, together with lattice vectors $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d}\\}$. \nUpon dimensional promotion, we can choose {\\it any} linearly independent additional lattice vectors $\\{\\vec{c}_{1},\\cdots,\\vec{c}_{N}\\}$ which together with $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d}\\}$ span the promoted $(d+N)$D space. \nThe corresponding $U(1)$ gauge field is given in Eq.~(\\ref{eq:non_ortho_A}), which has linear dependence on $\\vec{r}$, the position vector in the promoted $(d+N)$D space. \nThis implies the field strength $F_{\\mu \\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}$ is constant in space. \nIf we choose $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d},\\vec{c}_{1},\\cdots,\\vec{c}_{N} \\}$ to be orthogonal unit vectors in the $(d+N)$D Cartesian coordinates, and if the model only has modulated on-site energies, the general dimensional promotion procedure present here will reduce to the one present in Sec.~III of the main text.\n\nIn this more general construction in terms of non-orthogonal lattice vectors with modulated hopping terms, the mapping from the $d$D modulated system to the promoted $(d+N)$D lattice coupled to a $U(1)$ gauge field also requires no additional parameters. \nThe Hamiltonians before and after the dimensional promotion, together with the convention for the Fourier series expansions are summarized in Table~\\ref{tab:model_summary}. \nThe hopping matrices in $(d+N)$D space, and the corresponding Peierls phases are summarized in Table~\\ref{tab:hopping_1}. \nTo use Table~\\ref{tab:hopping_1} we multiply the hopping matrix entry and the Peierls phases to obtain the Hamiltonian matrix elements (with background $U(1)$ gauge fields) in the promoted $(d+N)$D space in Eqs.~(\\ref{eq:non_ortho_5_1}--\\ref{eq:non_ortho_5_3}). \nWe remind the readers that there are phase factors $e^{-i p_{i} \\pi \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} \\right)}$ in the hopping matrix from $(\\vec{n},\\vec{\\nu})$ to $(\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$, as shown in Table~\\ref{tab:hopping_1}. \nThis means that these phase factors are included in the definition of the hopping matrices in the $(d+N)$D model, in addition to the Peierls phase factor. \nIn addition, we reemphasize that the phases $\\phi^{(i)}$ correspond to $\\vec{k}\\cdot \\vec{c}_{i}$ where $\\vec{k}$ is the crystal momentum $\\in \\mathbb{T}^{N}$ in the dimensionally-promoted Brillouin zone.\nThe modulation wave vectors $\\vec{q}^{(i)}$ enter the definition of the $(d+N)$D $U(1)$ gauge fields in Eq.~(\\ref{eq:non_ortho_A}).\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nOriginal $d$D modulated system & Eq.~(\\ref{eq:H_low_general}) \\\\\n\\hline\nFourier expansion convention of on-site and hopping modulations & Eqs.~(\\ref{eq:general_FT_1}--\\ref{eq:general_FT_2})\\\\\n\\hline\nPromoted $(d+N)$D system with $U(1)$ gauge fields & Eqs.~(\\ref{eq:non_ortho_5_1}--\\ref{eq:non_ortho_5_3}) \\\\\n\\hline\nPromoted $(d+N)$D system without $U(1)$ gauge fields & Eqs.~(\\ref{eq:non_ortho_6_1}--\\ref{eq:non_ortho_6_3}) \\\\\n\\hline\n\\end{tabular}\n\\caption{Relevant equations in the general dimensional promotion formalism.}\n\\label{tab:model_summary}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $(\\vec{n},\\vec{\\nu})$ to & Hopping matrices & Peierls phases  \\\\\n\\hline\n\\hline\n$(\\vec{n}+\\vec{m},\\vec{\\nu})$ & $\\left[H_{\\vec{m}}\\right]$ & $1$ \\\\\n\\hline\n$(\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ & $ \\left[V^{(i)}_{p_{i}}\\right] $ & $e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}$ \\\\\n\\hline\n$(\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ & $\\left[H_{\\vec{l},p_{i}}^{(i)}\\right]e^{-i \\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d} l_{j}\\vec{a}_{j} \\right) }$ & $e^{i 2\\pi p_{i}   \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}\\left( n_{j} + \\frac{1}{2}l_{j} \\right)\\vec{a}_{j} \\right)   }$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping terms in the promoted $(d+N)$D model and the corresponding Peierls phases, expressed in terms of parameters from the $d$D modulated system in Eq.~(\\ref{eq:H_low_general}). \nNotice that $p_{i}\\hat{\\nu}_{i}$ does not imply a summation over $i$.}\n\\label{tab:hopping_1}\n\\end{table}\n\nBefore moving on, some additional remarks are in order. \nFirst, the vector potential in Eq.~(\\ref{eq:non_ortho_A}) satisfies\n\\begin{align}\n    \\vec{A}\\left( \\vec{r} \\right) = \\vec{A}\\left( \\vec{r} + \\sum_{j=1}^{N} \\nu_{j} \\vec{c}_{j} \\right)\n\\end{align}\nfor any $\\{\\nu_{1},\\cdots,\\nu_{N} \\} \\in \\mathbb{Z}^{N}$, which is a direct consequence of Eq.~(\\ref{eq:reciprocal_2}), and corresponds to a generalized Landau gauge condition. \nThis allows us to recover our low dimensional modulated system, by  Fourier transforming the higher-dimensional model Eq.~(\\ref{eq:non_ortho_4}) along $\\vec{k}$ in the subspace spanned by $\\{\\vec{G}_{1},\\cdots,\\vec{G}_{N} \\} \\in \\mathbb{T}^{N}$ and regarding $\\vec{k} \\cdot \\vec{c}_{i}$ as $\\phi^{(i)}$. \nAny fixed value of $\\{ \\phi^{(1)},\\cdots,\\phi^{(N)} \\}$ then describes a lower-dimensional modulated system with fixed phase offsets. \nIn other words, we can use a $d$D modulated system with controllable phase offsets $\\{ \\phi^{(1)},\\cdots,\\phi^{(N)} \\}$ to map out, by varying the values of $\\phi^{(i)}$, the whole spectrum of the promoted $(d+N)$D model coupled to a $U(1)$ gauge field. \nSecond, the constraints Eq.~(\\ref{eq:Hermitian_1}), Eq.~(\\ref{eq:Hermitian_4}) and Eq.~(\\ref{eq:Hermitian_5}) ensures that $H_{\\text{high-dim}}$ in Eq.~(\\ref{eq:non_ortho_4}) with Peierls phases, as well as Eqs.~(\\ref{eq:non_ortho_6_1}--\\ref{eq:non_ortho_6_3}) without Peierls phases are Hermitian. \nWe will make use of this in the subsequent examples and only list non-redundant matrix elements of the Hamiltonian. \nThird, we have assumed so far that  for a given modulation $\\vec{q}^{(i)}$, the phase offsets of the on-site and hopping modulation are the same (see Eqs.~(\\ref{eq:H_form_1}) and (\\ref{eq:H_form_2})). \nHowever, it is possible for a system to develop incoherence between the modulation of the on-site energy and hopping terms. \nIn these cases, we can relax our requirement that all $\\vec{q}^{(i)}$ are mutually incommensurate, and regard the modulated hopping terms and on-site energy as being described by two distinct but numerically equal modulation wave vectors. \nThis will increase the number of additional dimensions necessary to represent the system.\nWe demonstrate how this situation can be handled using the 1D Rice-Mele chain later in Sec.~\\ref{sec:1D_RM_chain_incoherence}. \nFourth, in Refs.~\\onlinecite{LL_fragile_Lian_Biao,Herzog_Hof_topo}, it is shown that in general the line integral of the Peierls phase should be taken on a piecewise linear path along which the Wannier functions have greatest overlap, as opposed to the linear path we used here. \nIn our present situation, this means we are implicitly assuming that the total Hilbert space (occupied plus unoccupied states) for our dimensionally promoted model can be described by topologically trivial Wannier functions. \nThe topological properties of the Wannier functions in the promoted $(d+N)$D space are an interesting direction for future investigation, including investigating whether topologically non-trivial $(d+N)$D Wannier functions can imply any physical properties in the low dimensional modulated system. \nLastly, we emphasize that our interpretation of the promoted models in $(d+N)$D, given in Eq.~(\\ref{eq:non_ortho_4}) is not unique. \nOur interpretation is fixed by the requirement that the $U(1)$ gauge fields (Eq.~(\\ref{eq:non_ortho_A})) have a spatially constant field strength $F_{\\mu\\nu}$ in $(d+N)$D. \nTherefore, this allows us to easily construct the continuum theory describing low energy dynamics in $(d+N)$D, as we know the underlying $U(1)$ gauge field, written as a function of $\\vec{r} \\in \\mathbb{R}^{d+N}$. \nThis completes our description on how to compute the $U(1)$ gauge field in a $(d+N)$D lattice with non-orthogonal lattice vectors, and the corresponding lattice model in $(d+N)$D space.\n\n{\n\\subsubsection{Generalization to systems with arbitrary orbital positions in the original $d$D space}\n\nIn Sec.~\\ref{subsubsec:__dimensional_promotion}--\\ref{sec:remarks_consistency_check_and_summary}, we have implicitly chosen to promote the dimension of a $d$D modulated lattice model whose orbitals are located exactly at the $d$D lattice points $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j}$ constructed from $\\{ \\vec{a}_{1},\\ldots,\\vec{a}_{d}\\}$. \nThis choice makes the emergence of the $U(1)$ gauge field and the identification of Peierls phases transparent. \nHowever, in many models of practical interest, not all of the orbitals are located at the $d$D lattice points. \nAs the computation of Peierls phases depends on the actual orbital positions, we now examine the effect of orbital positions in our dimensional promotion method.\n\nWe consider a $d$D modulated system whose $\\alpha^{\\text{th}}$ orbital is located at the position $\\vec{r}_{\\alpha}$ measured from the origin of the unit cell. \nWe will assume that $\\vec{r}_{\\alpha}$ can be written as a linear combination of the $d$D lattice vectors $\\{\\vec{a}_{1},\\ldots,\\vec{a}_{d}\\}$, namely\n\\begin{align}\n    \\vec{r}_{\\alpha} = \\sum_{j=1}^{d} x^{j}_{\\alpha} \\vec{a}_{j}, \\label{eq:r_alpha_expansion}\n\\end{align}\nwhere $x^{j}_{\\alpha} \\in [0,1)$ denotes the fractional component of $\\vec{r}_{\\alpha}$ along $\\vec{a}_{j}$ (this excludes certain quasi-$d$-dimensional models). \nTherefore, the actual position of the $\\alpha^{\\text{th}}$ orbital in the unit cell labelled by $\\vec{n} \\in \\mathbb{Z}^{d}$ is $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\vec{r}_{\\alpha}$. \nWhen all the orbitals are located at the lattice points $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j}$, we will have $\\vec{r}_{\\alpha} = 0$ for all $\\alpha$. \nThe generic Hamiltonian for our $d$D system is still given by Eq.~(\\ref{eq:H_low_general}). \nFor later convenience, here we rewrite Eq.~(\\ref{eq:H_low_general}) with an explicit summation over the orbital components as\n\\begin{equation}\n    H_{\\text{low-dim}} = \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\alpha} \\left[H_{\\vec{m}}\\right]_{\\alpha,\\beta} {\\psi}_{\\vec{n},\\beta} + \\sum_{\\alpha,\\beta}{\\sum_{\\vec{n},\\vec{l}}}  \\sum_{i=1}^{N} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\alpha} \\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right]_{\\alpha,\\beta} {\\psi}_{\\vec{n},\\beta} + \\sum_{\\alpha,\\beta}\\sum_{\\vec{n}} \\sum_{i=1}^{N}{\\psi}^{\\dagger}_{\\vec{n},\\alpha} \\left[ V^{(i)}_{\\vec{n}} \\right]_{\\alpha,\\beta} {\\psi}_{\\vec{n},\\beta}, \\label{eq:H_low_general_alpha}\n\\end{equation}\nwhere ${\\psi}^{\\dagger}_{\\vec{n},\\alpha}$ and ${\\psi}_{\\vec{n},\\alpha}$ are the creation and annihilation operator of the $\\alpha^{\\text{th}}$ orbital at the unit cell labelled by $\\vec{n} \\in \\mathbb{Z}^{d}$. \nAs before, $\\left[H_{\\vec{m}}\\right]_{\\alpha,\\beta}$, $\\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right]_{\\alpha,\\beta}$ and $\\left[ V^{(i)}_{\\vec{n}} \\right]_{\\alpha,\\beta}$ are the $(\\alpha,\\beta)$ entries for the unmodulated Hamiltonian $\\left[H_{\\vec{m}}\\right]$, the hopping modulations $\\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right]$ and the on-site modulations $\\left[ V^{(i)}_{\\vec{n}} \\right]$, respectively. \nThe dimensional promotion procedure is identical to that in Sec.~\\ref{subsubsec:__dimensional_promotion} and thus we obtain the same $(d+N)$D model given by Eq.~(\\ref{eq:non_ortho_4}). \nAgain, for later convenience we rewrite Eq.~(\\ref{eq:non_ortho_4}) with an explicit summation over the orbital components as\n\\begin{align}\n    H_{\\text{high-dim}} =& \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{m},\\vec{\\nu}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu},\\alpha} \\left[ H_{\\vec{m}}\\right]_{\\alpha,\\beta} \\psi_{\\vec{n},\\vec{\\nu},\\beta}   \\label{eq:H_high_dim_normal_hopping_alpha} \\\\\n    & + \\sum_{\\alpha,\\beta}{\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}}  \\sum_{i=1}^{N} \\sum_{p_{i}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha}  \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]_{\\alpha,\\beta} e^{i  2\\pi p_{i}   \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)   } {\\psi}_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_hopping_modulation_alpha} \\\\\n    & + \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[V^{(i)}_{p_{i}}\\right]_{\\alpha,\\beta}  e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}{\\psi}_{\\vec{n},\\vec{\\nu},\\beta}. \\label{eq:H_high_dim_onsite_modulation_alpha}\n\\end{align}\nWe then take the actual position of the $\\alpha^{\\text{th}}$ orbital in the unit cell labelled by $(\\vec{n},\\vec{\\nu}) \\in (\\mathbb{Z}^{d},\\mathbb{Z}^{N})$ to be $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\alpha}$ where we have assumed that there are no components of $\\vec{r}_\\alpha$ along $\\vec{c}_{j}$ and Eq.~(\\ref{eq:r_alpha_expansion}) still holds. \nHowever, due to the generic position $\\vec{r}_{\\alpha}$ of the orbitals, the separation of phase factors in Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha}--\\ref{eq:H_high_dim_onsite_modulation_alpha}) between periodic hopping and the Peierls phase due to the gauge field in Eq.~(\\ref{eq:non_ortho_A}) needs to be modified compared with those in Sec.~\\ref{sec:computing_Peierls_phase_1}.\n\nWe now compute the various Peierls phases associated with terms in Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha}--\\ref{eq:H_high_dim_onsite_modulation_alpha}) due to the gauge field in Eq.~(\\ref{eq:non_ortho_A}). First, we consider the terms in Eq.~(\\ref{eq:H_high_dim_onsite_modulation_alpha}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta}\n\\end{equation}\nto \n\\begin{equation}\n \\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} - p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha}. \n\\end{equation}\nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n    \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta} + \\left(- p_{i} \\vec{c}_{i} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t,\n\\end{align}\nwhere $ t\\in [0,1]$ and the corresponding infinitesimal displacement vector is $\\left(- p_{i} \\vec{c}_{i} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)dt$. The corresponding Peierls phase with a line integral can then be computed as follows:\n\\begin{align}\n    & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n    & =  \\exp{-i \\int_{0}^{1}dt  \\left(- p_{i} \\vec{c}_{i} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\begin{bmatrix}\n    \\sum_{l=1}^{d}n_{l}\\vec{a}_{l} + \\sum_{l=1}^{N}\\nu_{l}\\vec{c}_{l} + \\vec{r}_{\\beta}  \\\\\n    +\\left(- p_{i} \\vec{c}_{i} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n    \\end{bmatrix}\n    \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_1_alpha}\\\\\n    & =  \\exp{+i \\int_{0}^{1}dt   p_{i} \\vec{c}_{i}   \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left(\n    \\left(\n    \\sum_{l=1}^{d}n_{l}\\vec{a}_{l} + \\vec{r}_{\\beta} + \\left(  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n    \\right)\n    \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_2_alpha} \\\\\n    & =  \\exp{+i \\int_{0}^{1}dt   p_{i} \\vec{c}_{i}   \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left(\n    \\left(\n    \\sum_{l=1}^{d}\\left( n_{l} + x_{\\beta}^{l} + t x_{\\alpha}^{l} - t x_{\\beta}^{l} \\right)\\vec{a}_{l} \n    \\right)\n    \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_3_alpha}\\\\\n    & =  \\exp{+i2\\pi \\int_{0}^{1}dt   p_{i}  \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\delta_{ij} \\left( \\vec{q}^{(j)}\\cdot \\left[ \n    \\sum_{l=1}^{d}\\left(n_{l} + x_{\\beta}^{l}+tx^{l}_{\\alpha} - t x^{l}_{\\beta}\\right)\\delta_{lk} \n   \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_4_alpha} \\\\\n   & =  \\exp{+i2\\pi \\int_{0}^{1}dt   p_{i}  \\left( \\vec{q}^{(i)}\\cdot \\left[ \n    \\sum_{l=1}^{d}\\left(n_{l} + x_{\\beta}^{l}+tx^{l}_{\\alpha} - t x^{l}_{\\beta}\\right)\n     \\vec{a}_{l} \\right] \\right) } \\label{eq:Peierls_1_5_alpha} \\\\\n    & =  \\exp{+i2\\pi   p_{i}  \\vec{q}^{(i)}\\cdot \\left[ \n    \\sum_{l=1}^{d}\\left(n_{l} + x_{\\beta}^{l}+\\frac{1}{2}x^{l}_{\\alpha} - \\frac{1}{2} x^{l}_{\\beta}\\right)\n     \\vec{a}_{l} \\right]  }\\label{eq:Peierls_1_6_alpha} \\\\\n    & =  \\exp{+i2\\pi   p_{i}  \\vec{q}^{(i)}\\cdot \\left[ \n    \\sum_{l=1}^{d}\\left(n_{l}  + \\frac{x^{l}_{\\alpha} + x^{l}_{\\beta}}{2} \\right)\n     \\vec{a}_{l} \\right]  } \\label{eq:Peierls_1_7_alpha} .\n\\end{align}\nTo go from Eq.~(\\ref{eq:Peierls_1_1_alpha}) to Eq.~(\\ref{eq:Peierls_1_2_alpha}) we have used Eq.~(\\ref{eq:reciprocal_2}) and the fact that $(\\vec{r}_{\\alpha} - \\vec{r}_{\\beta}) \\cdot \\vec{G}_{j} = 0$ because of Eqs.~(\\ref{eq:reciprocal_3}) and (\\ref{eq:r_alpha_expansion}). \nWe then use Eq.~(\\ref{eq:r_alpha_expansion}) to go from Eq.~(\\ref{eq:Peierls_1_2_alpha}) to Eq.~(\\ref{eq:Peierls_1_3_alpha}). \nTo go from Eq.~(\\ref{eq:Peierls_1_3_alpha}) to Eq.~(\\ref{eq:Peierls_1_4_alpha}), we have used Eqs.~(\\ref{eq:reciprocal_1}) and (\\ref{eq:reciprocal_4}). \nPerforming the summation over $j$ and $k$ we obtain Eq.~(\\ref{eq:Peierls_1_5_alpha}). \nUsing $\\int_{0}^{1}dt  = 1$ and $\\int_{0}^{1}tdt  = \\frac{1}{2}$ we arrive at Eqs.~(\\ref{eq:Peierls_1_6_alpha}--\\ref{eq:Peierls_1_7_alpha}).\n\nNext, let us consider the terms in Eq.~(\\ref{eq:H_high_dim_hopping_modulation_alpha}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta}\n\\end{equation}\nto \n\\begin{equation}\n\\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} - p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha}.\n\\end{equation}\nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n    \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta} + \\left( \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}- p_{i} \\vec{c}_{i} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t,\n\\end{align}\nwhere $ t\\in [0,1]$ and the corresponding infinitesimal displacement vector is $\\left( \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}- p_{i} \\vec{c}_{i} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)dt$. \nThe corresponding Peierls phase with a line integral can then be computed as follows:\n\\begin{align}\n    & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n    & =  \\exp{-i \\int_{0}^{1}dt  \\left( \\sum_{r=1}^{d}l_{r}\\vec{a}_{r}- p_{i} \\vec{c}_{i} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\begin{bmatrix}\n    \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\sum_{r=1}^{N}\\nu_{r}\\vec{c}_{r} + \\vec{r}_{\\beta}  \\\\\n    + \\left( \\sum_{r=1}^{d}l_{r}\\vec{a}_{r}- p_{i} \\vec{c}_{i} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n    \\end{bmatrix}\n    \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_1_alpha}\\\\\n    & =  \\exp{+i \\int_{0}^{1}dt    p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\begin{bmatrix}\n    \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\vec{r}_{\\beta}  \\\\\n    + \\left( \\sum_{r=1}^{d}l_{r}\\vec{a}_{r} +  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n    \\end{bmatrix}\n    \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_2_alpha}\\\\\n    & =  \\exp{+i \\int_{0}^{1}dt    p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{r=1}^{d}\\left( n_{r} + tl_{r} + x^{r}_{\\beta} + t x^{r}_{\\alpha} - t x^{r}_{\\beta} \\right)\\vec{a}_{r}  \\right)\n    \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_3_alpha}\\\\\n    & =  \\exp{+i2\\pi \\int_{0}^{1}dt    p_{i}  \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\delta_{ij} \\left( \\vec{q}^{(j)}\\cdot \\left[  \\sum_{r=1}^{d}\\left( n_{r} + tl_{r} + x^{r}_{\\beta} + t x^{r}_{\\alpha} - t x^{r}_{\\beta} \\right)\\delta_{rk}   \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_4_alpha}\\\\\n    & =  \\exp{+i2\\pi \\int_{0}^{1}dt    p_{i}    \\vec{q}^{(i)}\\cdot \\left[  \\sum_{r=1}^{d}\\left( n_{r} + tl_{r} + x^{r}_{\\beta} + t x^{r}_{\\alpha} - t x^{r}_{\\beta} \\right) \\vec{a}_{r} \\right] } \\label{eq:Peierls_2_5_alpha}\\\\\n    & =  \\exp{+i2\\pi  p_{i}    \\vec{q}^{(i)}\\cdot \\left[  \\sum_{r=1}^{d}\\left( n_{r} + \\frac{1}{2}l_{r} + x^{r}_{\\beta} + \\frac{1}{2} x^{r}_{\\alpha} - \\frac{1}{2} x^{r}_{\\beta} \\right) \\vec{a}_{r} \\right] } \\label{eq:Peierls_2_6_alpha}\\\\\n    & =  \\exp{+i2\\pi  p_{i}    \\vec{q}^{(i)}\\cdot \\left[  \\sum_{r=1}^{d}\\left( n_{r} + \\frac{l_{r} +x^{r}_{\\alpha} + x^{r}_{\\beta} }{2}  \\right) \\vec{a}_{r} \\right] }. \\label{eq:Peierls_2_7_alpha}\n\\end{align}\nTo go from Eq.~(\\ref{eq:Peierls_2_1_alpha}) to Eq.~(\\ref{eq:Peierls_2_2_alpha}) we have used Eqs.~(\\ref{eq:reciprocal_2}--\\ref{eq:reciprocal_3}), and again the fact that $(\\vec{r}_{\\alpha} - \\vec{r}_{\\beta}) \\cdot \\vec{G}_{j} = 0$ because of Eqs.~(\\ref{eq:reciprocal_3}) and (\\ref{eq:r_alpha_expansion}).\n We then use Eq.~(\\ref{eq:r_alpha_expansion}) to go from Eq.~(\\ref{eq:Peierls_2_2_alpha}) to Eq.~(\\ref{eq:Peierls_2_3_alpha}). \n To go from Eq.~(\\ref{eq:Peierls_2_3_alpha}) to Eq.~(\\ref{eq:Peierls_2_4_alpha}), we have used Eqs.~(\\ref{eq:reciprocal_1}) and (\\ref{eq:reciprocal_4}). \n Performing the summation over $j$ and $k$ we obtain Eq.~(\\ref{eq:Peierls_2_5_alpha}). \n Using $\\int_{0}^{1}dt  = 1$ and $\\int_{0}^{1}tdt  = \\frac{1}{2}$ we arrive at Eqs.~(\\ref{eq:Peierls_2_6_alpha}--\\ref{eq:Peierls_2_7_alpha}).\n\nFinally, let us consider the terms in Eq.~(\\ref{eq:H_high_dim_normal_hopping_alpha}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta}\n\\end{equation}\nto \n\\begin{equation}\n\\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\sum_{j=1}^{d}m_{j}\\vec{a}_{j} + \\vec{r}_{\\alpha}.\n\\end{equation}\nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n    \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta} + \\left( \\sum_{j=1}^{d}m_{j}\\vec{a}_{j}+  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t,\n\\end{align}\nwhere $ t\\in [0,1]$ and the corresponding infinitesimal displacement vector is $\\left( \\sum_{j=1}^{d}m_{j}\\vec{a}_{j}+  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)dt$. \nThe corresponding Peierls phase with a line integral can then be computed as follows:\n\\begin{align}\n    & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n    & =  \\exp{-i \\int_{0}^{1}dt  \\left( \\sum_{r=1}^{d}m_{r}\\vec{a}_{r}+  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\begin{bmatrix}\n    \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\sum_{r=1}^{N}\\nu_{r}\\vec{c}_{r} + \\vec{r}_{\\beta}  \\\\\n    + \\left( \\sum_{r=1}^{d}m_{r}\\vec{a}_{r}+  \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n    \\end{bmatrix}\n    \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_3_1_alpha}\\\\ \n    & = 1. \\label{eq:Peierls_3_2_alpha}\n\\end{align}\nTo go from Eq.~(\\ref{eq:Peierls_3_1_alpha}) to Eq.~(\\ref{eq:Peierls_3_2_alpha}) we have used Eq.~(\\ref{eq:reciprocal_3}) and again the fact that $(\\vec{r}_{\\alpha} - \\vec{r}_{\\beta}) \\cdot \\vec{G}_{j} = 0$ because of Eqs.~(\\ref{eq:reciprocal_3}) and (\\ref{eq:r_alpha_expansion}).\n\n\nWith this knowledge of the Peierls phases in Eqs.~(\\ref{eq:Peierls_1_7_alpha}), (\\ref{eq:Peierls_2_7_alpha}), and (\\ref{eq:Peierls_3_2_alpha}), we can rewrite $H_{\\text{high-dim}}$ in Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha}--\\ref{eq:H_high_dim_onsite_modulation_alpha}) as \n\\begin{align}\n    H_{\\text{high-dim}} =& \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{m},\\vec{\\nu}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu},\\alpha} \\left[ H_{\\vec{m}}\\right]_{\\alpha,\\beta} \\psi_{\\vec{n},\\vec{\\nu},\\beta}   \\label{eq:H_high_dim_normal_hopping_alpha_split} \\\\\n    & + \\sum_{\\alpha,\\beta}{\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}}  \\sum_{i=1}^{N} \\sum_{p_{i}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha}  \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]_{\\alpha,\\beta} e^{-i\\pi  p_{i}    \\vec{q}^{(i)}\\cdot  \\left(\\sum_{j=1}^{d}\\left(  l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta}  \\right) \\vec{a}_{j} \\right) } e^{i2\\pi  p_{i}    \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}\\left( n_{j} + \\frac{l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} }{2}  \\right) \\vec{a}_{j} \\right) }  {\\psi}_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_hopping_modulation_alpha_split} \\\\\n    & + \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[V^{(i)}_{p_{i}}\\right]_{\\alpha,\\beta}  e^{-i\\pi   p_{i}  \\vec{q}^{(i)}\\cdot \\left(\n    \\sum_{j=1}^{d}\\left( x^{j}_{\\alpha} + x^{j}_{\\beta} \\right)\n     \\vec{a}_{j} \\right)  } \n    e^{i2\\pi   p_{i}  \\vec{q}^{(i)}\\cdot \\left( \n    \\sum_{j=1}^{d}\\left(n_{j}  + \\frac{x^{j}_{\\alpha} + x^{j}_{\\beta}}{2} \\right)\n     \\vec{a}_{j} \\right)  } \n    {\\psi}_{\\vec{n},\\vec{\\nu},\\beta}. \\label{eq:H_high_dim_onsite_modulation_alpha_split}\n\\end{align}\nEqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha_split}--\\ref{eq:H_high_dim_onsite_modulation_alpha_split}) can then be interpreted as a $(d+N)$D lattice model with Hamiltonian\n\\begin{align}\n    H_{\\text{high-dim}} =& \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{m},\\vec{\\nu}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu},\\alpha} \\left[ H_{\\vec{m}}\\right]_{\\alpha,\\beta} \\psi_{\\vec{n},\\vec{\\nu},\\beta}   \\label{eq:H_high_dim_normal_hopping_alpha_no_gauge} \\\\\n    & + \\sum_{\\alpha,\\beta}{\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}}  \\sum_{i=1}^{N} \\sum_{p_{i}}  {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha}  \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]_{\\alpha,\\beta} e^{-i\\pi  p_{i}    \\vec{q}^{(i)}\\cdot  \\left(\\sum_{j=1}^{d}\\left(  l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta}  \\right) \\vec{a}_{j} \\right) }  {\\psi}_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_hopping_modulation_alpha_no_gauge} \\\\\n    & + \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[V^{(i)}_{p_{i}}\\right]_{\\alpha,\\beta}  e^{-i\\pi   p_{i}  \\vec{q}^{(i)}\\cdot \\left(\n    \\sum_{j=1}^{d}\\left( x^{j}_{\\alpha} + x^{j}_{\\beta} \\right)\n     \\vec{a}_{j} \\right)  } \n    {\\psi}_{\\vec{n},\\vec{\\nu},\\beta}, \\label{eq:H_high_dim_onsite_modulation_alpha_no_gauge}\n\\end{align}\nwhich is periodic with lattice vectors $\\{\\vec{a}_{1},\\ldots,\\vec{a}_{d},\\vec{c}_{1},\\ldots,\\vec{c}_{N} \\}$, coupled through a Peierls substitution\\cite{Peierls_substitution} to a $U(1)$ gauge field given in Eq.~(\\ref{eq:non_ortho_A}). \nNotice that we have regarded\n\\begin{align}\n    e^{i2\\pi  p_{i}    \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}\\left( n_{j} + \\frac{l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} }{2}  \\right) \\vec{a}_{j} \\right) } \n\\end{align}\nand\n\\begin{align}\n    e^{i2\\pi   p_{i}  \\vec{q}^{(i)}\\cdot \\left( \n    \\sum_{j=1}^{d}\\left(n_{j}  + \\frac{x^{j}_{\\alpha} + x^{j}_{\\beta}}{2} \\right)\n     \\vec{a}_{j} \\right)  } \n\\end{align}\nin Eqs.~(\\ref{eq:H_high_dim_hopping_modulation_alpha_split}) and (\\ref{eq:H_high_dim_onsite_modulation_alpha_split}) as the Peierls phases (see Eqs.~(\\ref{eq:Peierls_2_7_alpha}) and (\\ref{eq:Peierls_1_7_alpha})) due to the $U(1)$ gauge field in Eq.~(\\ref{eq:non_ortho_A}). \nNotice that there are no Peierls phases induced from the gauge field in Eq.~(\\ref{eq:H_high_dim_normal_hopping_alpha_split}). \n\nWe have thus generalized our dimensional promotion method to $d$D modulated lattice models whose orbitals are not located at the $d$D lattice points $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j}$. \nSimilar to Tables~\\ref{tab:model_summary} and \\ref{tab:hopping_1}, we have summarized the Hamiltonian before and after the dimensional promotion in Table~\\ref{tab:model_summary_general_orbital_positions}, and the hopping matrix elements together with the Peierls phases in Table~\\ref{tab:hopping_1_general_orbital_positions}. \nA notable feature is that now both the hopping matrix elements of the $(d+N)$D lattice model {\\it without} the gauge field and the Peierls phases due to the gauge field in Eq.~(\\ref{eq:non_ortho_A}) encode information of the orbital positions. \nWhen all the orbitals are located right at the lattice points we have $\\vec{r}_{\\alpha} = 0$ for all $\\alpha$, namely $x^{j}_{\\alpha}=0$ for all $j$ and $\\alpha$ in Eq.~(\\ref{eq:r_alpha_expansion}). \nIn such cases, Table~\\ref{tab:hopping_1_general_orbital_positions} effectively reduces to Table~\\ref{tab:hopping_1}.\n\n\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nOriginal $d$D modulated system & Eq.~(\\ref{eq:H_low_general_alpha}) \\\\\n\\hline\nPromoted $(d+N)$D system with $U(1)$ gauge fields & Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha_split}--\\ref{eq:H_high_dim_onsite_modulation_alpha_split}) \\\\\n\\hline\nPromoted $(d+N)$D system without $U(1)$ gauge fields & Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha_no_gauge}--\\ref{eq:H_high_dim_onsite_modulation_alpha_no_gauge}) \\\\\n\\hline\n\\end{tabular}\n\\caption{Relevant equations in the general dimensional promotion formalism with arbitrary orbital positions.}\n\\label{tab:model_summary_general_orbital_positions}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $\\beta^{\\text{th}}$ orbital in unit cell $(\\vec{n},\\vec{\\nu})$ to & Hopping matrix elements & Peierls phases  \\\\\n\\hline\n\\hline\n$\\alpha^{\\text{th}}$ orbital in unit cell $(\\vec{n}+\\vec{m},\\vec{\\nu})$ & $\\left[H_{\\vec{m}}\\right]_{\\alpha,\\beta}$ & $1$ \\\\\n\\hline\n$\\alpha^{\\text{th}}$ orbital in unit cell $(\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ & $ \\left[V^{(i)}_{p_{i}}\\right]_{\\alpha,\\beta}  e^{-i\\pi   p_{i}  \\vec{q}^{(i)}\\cdot \\left(     \\sum_{j=1}^{d}\\left( x^{j}_{\\alpha} + x^{j}_{\\beta} \\right)      \\vec{a}_{j} \\right)  }  $ & $e^{i2\\pi   p_{i}  \\vec{q}^{(i)}\\cdot \\left(      \\sum_{j=1}^{d}\\left(n_{j}  + \\frac{x^{j}_{\\alpha} + x^{j}_{\\beta}}{2} \\right)      \\vec{a}_{j} \\right)  } $ \\\\\n\\hline\n$\\alpha^{\\text{th}}$ orbital in unit cell $(\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ & $\\left[H_{\\vec{l},p_{i}}^{(i)}\\right]_{\\alpha,\\beta} e^{-i\\pi  p_{i}    \\vec{q}^{(i)}\\cdot  \\left(\\sum_{j=1}^{d}\\left(  l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta}  \\right) \\vec{a}_{j} \\right) } $ & $e^{i2\\pi  p_{i}    \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}\\left( n_{j} + \\frac{l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} }{2}  \\right) \\vec{a}_{j} \\right) } $ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping terms in the promoted $(d+N)$D model with arbitrary orbital positions and the corresponding Peierls phases, expressed in terms of parameters from the $d$D modulated system in Eq.~(\\ref{eq:H_low_general}). \nNotice that $p_{i}\\hat{\\nu}_{i}$ does not imply a summation over $i$.}\n\\label{tab:hopping_1_general_orbital_positions}\n\\end{table}\n\n}\n\n\\subsection{\\label{subsec:examples_general_dim_promotion}Examples}\n\nTo demonstrate our general construction, we will use Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} to consider four examples: \n(1) promoting the 1D Rice-Mele chain to a 2D square lattice with $\\pi$-flux, \n(2) promoting the 1D Rice-Mele chain with incoherent phase offsets in on-site and hopping modulations to a 3D cubic lattice coupled to a $U(1)$ gauge field, \n(3) promoting a 1D modulated system to a 2D hexagonal lattice with a perpendicular magnetic field, and \n(4) promoting a 2D modulated system with hexagonal lattice to a 3D hexagonal lattice coupled to a $U(1)$ gauge field.\n\n\\subsubsection{\\label{sec:ex_1D_RM_chain} 1D Rice-Mele chain $\\to$ 2D square lattice with $\\pi$-flux}\n\nThe Hamiltonian of the 1D Rice-Mele\\cite{RiceMele} chain oriented along the $x$-axis is given by\n\\begin{align}\n    H_{\\text{Rice-Mele}} = \\sum_{n}  \\left( t + \\delta t (-1)^{n} \\cos{\\phi} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} + \\sum_{n} (-1)^{n+1} \\Delta  \\sin{\\phi} \\psi^{\\dagger}_{n}\\psi_{n},\n    \\label{eq:supp_Rice_Mele_1}\n\\end{align}\nwhere $\\psi^{\\dagger}_{n}$ is the creation operator for an electron at position $n$, and $\\text{h.c.}$ denotes the Hermitian conjugate of all terms before it. \nWe can rewrite Eq.~(\\ref{eq:supp_Rice_Mele_1}) as\n\\begin{align}\n    H_{\\text{Rice-Mele}} & = \\sum_{n}  \\left( t + \\delta t \\cos{(\\pi n + \\phi)} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} - \\sum_{n} \\Delta  \\sin{(\\pi n + \\phi)} \\psi^{\\dagger}_{n}\\psi_{n} \n    \\label{eq:supp_Rice_Mele_2} \\\\\n    & = \\sum_{n}  \\left( t + \\delta t \\cdot \\frac{e^{i\\left( \\pi n + \\phi\\right)} + e^{-i\\left( \\pi n + \\phi\\right)}}{2} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} - \\sum_{n} \\Delta  \\cdot \\frac{e^{i\\left( \\pi n + \\phi\\right)} - e^{-i\\left( \\pi n + \\phi\\right)}}{2i} \\psi^{\\dagger}_{n}\\psi_{n} .\\label{eq:supp_Rice_Mele_3}\n\\end{align}\nThus we can see that 1D Rice-Mele chain is a 1D modulated system with modulation wave vector $q = 1\/2$ ($2\\pi q = \\pi$). \nWe now choose the second, synthetic lattice vector to be $\\hat{y}$. \nThe promote 2D system will then have\n\\begin{align}\n    \\vec{a}_{1} = (1,0),\\ \\vec{c}_{1} = (0,1),\\ \\vec{g}_{1} = 2\\pi(1,0) \\text{ and } \\vec{G}_{1} = 2\\pi (0,1),\n\\end{align}\nin terms of the notation from Eqs.~(\\ref{eq:reciprocal_1}--\\ref{eq:reciprocal_4}). \nNotice that $\\vec{a}_{1}$ and $\\vec{c}_{1}$ describe a square lattice. \nUsing the Fourier expansion convention in Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} with $\\vec{q} = (1\/2,0)$ and Eq.~(\\ref{eq:supp_Rice_Mele_3}), we obtain the hopping terms summarized in Table~\\ref{tab:hopping_2}. \nMultiplying the entries for hopping matrices and Peierls phases, the Hamiltonian for the promoted 2D model is\n\\begin{align}\n    H_{\\text{2D}} = \\sum_{n,m} \\left( t\\psi^{\\dagger}_{n+1,m}\\psi_{n,m}- \\frac{i\\delta t}{2}e^{i\\pi \\left(n+\\frac{1}{2} \\right)} \\psi^{\\dagger}_{n+1,m-1}\\psi_{n,m}+ \\frac{i\\delta t}{2}e^{-i\\pi \\left(n+\\frac{1}{2} \\right)} \\psi^{\\dagger}_{n+1,m+1}\\psi_{n,m}-  \\frac{i\\Delta}{2}e^{-i\\pi n} \\psi^{\\dagger}_{n,m+1}\\psi_{n,m} + \\text{h.c.}  \\right), \\label{eq:1D_Rice_Mele_to_2D_2}\n\\end{align}\nwhere $\\psi^{\\dagger}_{n,m}$ is the creation operator for an electron at position $(n,m)$, and the vector potential to which this 2D model is coupled is \n\\begin{align}\n    \\vec{A} = \\frac{1}{2\\pi}  \\vec{G}_{1} \\left( \\vec{q}\\cdot \\left[ \\left( \\vec{r} \\cdot \\vec{g}_{1}\\right) \\vec{a}_{1} \\right] \\right) = (0,\\pi x).\n\\end{align}\nThis $\\vec{A}$ produces a perpendicular magnetic field $\\vec{B} = \\pi \\hat{z}$ in the promoted 2D system such that there is a $\\pi$-flux per plaquette. \n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $(n,m)$ to & Hopping matrices & Peierls phases  \\\\\n\\hline\n\\hline\n$(n+1,m)$ & $t$ & $1$ \\\\\n\\hline\n$(n,m+1)$  & $ \\Delta \/ (2i) = -i\\Delta \/ 2$ & $e^{-i \\pi n}$  \\\\\n\\hline \n$(n+1,m+1)$ & $(\\delta t\/ 2) \\cdot e^{i \\pi \/2} = i\\delta t\/ 2 $  & $e^{-i \\pi \\left( n + \\frac{1}{2} \\right)}$ \\\\\n\\hline\n$(n+1,m-1)$ & $(\\delta t\/ 2) \\cdot e^{-i \\pi \/2} = -i\\delta t\/ 2 $ & $e^{i \\pi \\left( n + \\frac{1}{2} \\right)}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping matrices and the corresponding Peierls phases for the promoted $2$D model from a 1D Rice-Mele chain.\nThe hopping matrices along the opposite directions of those listed here are omitted and can be obtained through Hermitian conjugation.}\n\\label{tab:hopping_2}\n\\end{table}\n\n\\subsubsection{\\label{sec:1D_RM_chain_incoherence}1D Rice-Mele chain with phase offset incoherence $\\to$ 3D cubic lattice coupled to a $U(1)$ gauge field}\n\nConsider again a 1D Rice-Mele chain oriented along the $x$-axis as in Sec.~\\ref{sec:ex_1D_RM_chain}. \nNow, however, we assume that the phase offsets in the hopping and on-site modulation can be different.\nThe $1$D Hamiltonian then reads\n\\begin{align}\n    H_{\\text{Rice-Mele}} = \\sum_{n}  \\left( t + \\delta t (-1)^{n} \\cos{\\phi^{(1)}} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} + \\sum_{n} (-1)^{n+1} \\Delta  \\sin{\\phi^{(2)}} \\psi^{\\dagger}_{n}\\psi_{n},\n    \\label{eq:Rice_Mele_incoherence}\n\\end{align}\nwhere $\\psi^{\\dagger}_{n}$ is the creation operator for an electron at position $n$, and $\\text{h.c.}$ means the Hermitian conjugate of all terms before it. \nThis situation arises when we are able to tune the phase offsets of the on-site and hopping modulations independently. \nWe can write Eq.~(\\ref{eq:Rice_Mele_incoherence}) as\n\\begin{align}\n    H_{\\text{Rice-Mele}} & = \\sum_{n}  \\left( t + \\delta t \\cos{(\\pi n + \\phi^{(1)})} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} - \\sum_{n} \\Delta  \\sin{(\\pi n + \\phi^{(2)})} \\psi^{\\dagger}_{n}\\psi_{n} \n    \\label{eq:Rice_Mele_incoherence_2} \\\\\n    & = \\sum_{n}  \\left( t + \\delta t \\cdot \\frac{e^{i\\left( \\pi n + \\phi^{(1)}\\right)} + e^{-i\\left( \\pi n + \\phi^{(1)}\\right)}}{2} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} - \\sum_{n} \\Delta  \\cdot \\frac{e^{i\\left( \\pi n + \\phi^{(2)}\\right)} - e^{-i\\left( \\pi n + \\phi^{(2)}\\right)}}{2i} \\psi^{\\dagger}_{n}\\psi_{n}. \\label{eq:Rice_Mele_incoherence_3}\n\\end{align}\nWe now choose the second and third synthetic lattice vectors to be $\\hat{y}$ and $\\hat{z}$. \nThe promote 3D system will then have\n\\begin{align}\n    & \\vec{a}_{1} = (1,0,0),\\ \\vec{c}_{1} = (0,1,0),\\ \\vec{c}_{2} = (0,0,1) \\\\\n    & \\vec{g}_{1} = 2\\pi(1,0,0),\\ \\vec{G}_{1} = 2\\pi(0,1,0),\\ \\vec{G}_{2} = 2\\pi(0,0,1),\n\\end{align}\nin terms of the notation from Eqs.~(\\ref{eq:reciprocal_1}--\\ref{eq:reciprocal_4}). \nNotice that $\\{\\vec{a}_{1},\\vec{c}_{1},\\vec{c}_{2} \\}$ describe a 3D cubic lattice. \nUsing the Fourier expansion convention in Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} with $\\vec{q}^{(1)}= \\vec{q}^{(2)} = (1\/2,0,0)$ and Eq.~(\\ref{eq:Rice_Mele_incoherence_3}), we obtain the hopping terms in this case summarized in Table~\\ref{tab:hopping_2_incoherence}. \nMultiplying the entries for hopping matrices and Peierls phases, the Hamiltonian for the promoted 3D model is\n\\begin{equation}\n\\hspace*{-0cm}\n    H_{\\text{3D}} = \\sum_{n,m,l} \\left( t\\psi^{\\dagger}_{n+1,m,l}\\psi_{n,m,l}- \\frac{i\\delta t}{2}e^{i\\pi \\left(n+\\frac{1}{2} \\right)} \\psi^{\\dagger}_{n+1,m-1,l}\\psi_{n,m,l}+ \\frac{i\\delta t}{2}e^{-i\\pi \\left(n+\\frac{1}{2} \\right)} \\psi^{\\dagger}_{n+1,m+1,l}\\psi_{n,m,l}-  \\frac{i\\Delta}{2}e^{-i\\pi n} \\psi^{\\dagger}_{n,m,l+1}\\psi_{n,m,l} + \\text{h.c.}  \\right),\n\\end{equation}\nwhere $\\psi^{\\dagger}_{n,m,l}$ is the creation operator for an electron at position $(n,m,l)$, and the vector potential to which this 3D model is coupled is \n\\begin{align}\n    \\vec{A} = \\frac{1}{2\\pi} \\left( \\vec{G}_{1} \\left(\\vec{q}^{(1)}\\cdot\\left[ \\left( \\vec{r}\\cdot \\vec{g}_{1} \\right)\\vec{a}_{1} \\right] \\right) +  \\vec{G}_{2} \\left(\\vec{q}^{(2)}\\cdot\\left[ \\left( \\vec{r}\\cdot \\vec{g}_{1} \\right)\\vec{a}_{1} \\right] \\right) \\right) = (0,\\pi x, \\pi x).\n\\end{align}\nThis $\\vec{A}$ produces a magnetic field $\\vec{B} = (0,-\\pi,\\pi)$ in the promoted 3D system such that there is a $\\pi$-flux threading through the plaquettes in $zx$- and $xy$-planes. \nIn contrast to Sec.~\\ref{sec:ex_1D_RM_chain} where the promoted 2D model is a Chern insulator (see Sec.~\\ref{sec:Thouless_pump_1D_Rice_Mele}), the 1D Rice-Mele chain with phase offset incoherence promotes to a 3D gapless model. \nWe can see this by noting that with $\\phi^{(1)} = \\pi \/2$ and $\\phi^{(2)} = 0$, Eq.~(\\ref{eq:Rice_Mele_incoherence}) becomes\n\\begin{align}\n    \\sum_{n} \\left( t  \\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.}  \\right),\n\\end{align}\nwhich is the Hamiltonian for the 1D Rice-Mele chain at the gapless critical point.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $(n,m,l)$ to & Hopping matrices & Peierls phases  \\\\\n\\hline\n\\hline\n$(n+1,m,l)$ & $t$ & $1$ \\\\\n\\hline\n$(n,m,l+1)$  & $ \\Delta \/ (2i) = -i\\Delta \/ 2$ & $e^{-i \\pi n}$  \\\\\n\\hline \n$(n+1,m+1,l)$ & $(\\delta t\/ 2) \\cdot e^{i \\pi \/2} = i\\delta t\/ 2 $  & $e^{-i \\pi \\left( n + \\frac{1}{2} \\right)}$ \\\\\n\\hline\n$(n+1,m-1,l)$ & $(\\delta t\/ 2) \\cdot e^{-i \\pi \/2} = -i\\delta t\/ 2 $ & $e^{i \\pi \\left( n + \\frac{1}{2} \\right)}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping matrices and the corresponding Peierls phases for the promoted $3$D model for a $1$D Rice-Mele chain with an incoherence between the phase offsets of on-site and hopping modulations. \nThe hopping matrices along the opposite directions of those listed here are omitted and can be obtained through Hermitian conjugation.}\n\\label{tab:hopping_2_incoherence}\n\\end{table}\n\n\\subsubsection{\\label{sec:1D_to_2D_hexa}1D modulated system $\\to$ 2D hexagonal lattice under a perpendicular magnetic field}\n\nLet us now explore what happens when our dimensionally-promoted lattice vectors are non-orthogonal. \nConsider a $1$D modulated system with both on-site and nearest-neighbor hopping modulations, with Hamiltonian\n\\begin{align}\n    H_{1D} = & \\sum_{n_{1}} \\left( \\psi^{\\dagger}_{n_{1}+1} [H_{\\vec{a}_{1}}] \\psi_{n_{1}} + \\psi^{\\dagger}_{n_{1}+1} e^{i \\left( 2\\pi \\vec{q} \\cdot \\left( n_{1} + \\frac{1}{2} \\right)\\vec{a}_{1} + \\phi \\right)} [H_{\\vec{a}_{1}-\\vec{a}_{2}}] \\psi_{n_{1}} + \\text{h.c.} \\right)  \\label{eq:1D_hexagonal_1} \\\\\n    & + \\sum_{n_{1}} \\left(  \\psi^{\\dagger}_{n_{1}} [H_{0}] \\psi_{n_{1}} + \\psi^{\\dagger}_{n_{1}} [V_{n_{1}}] \\psi_{n_{1}} \\right), \\label{eq:1D_hexagonal_2}\n\\end{align}\nwhere $\\text{h.c.}$ denotes hermitian conjugation, and $\\psi^{\\dagger}_{n_{1}}$ is the creation operator for an electron at position $n_{1}\\vec{a}_{1}$. \nAdditionally, $[V_{n_{1}}]$ is a modulated on-site interaction which can be decomposed into\n\\begin{align}\n    [V_{n_{1}}] = e^{-i \\left( 2\\pi \\vec{q} \\cdot n_{1}\\vec{a}_{1} + \\phi\\right)} [H_{\\vec{a}_{2}}] +  e^{i \\left( 2\\pi \\vec{q} \\cdot n_{1}\\vec{a}_{1} + \\phi \\right)} [H_{\\vec{a}_{2}}]^{\\dagger}, \\label{eq:1D_hexagonal_3}\n\\end{align}\nand $\\vec{a}_{1} = (1,0)$ and $\\vec{q} = (q,0)$. \nWe also have a modulated nearest-neighbor hopping term from site $n_{1}$ to $n_{1} + 1$ given by the second term\n\\begin{equation}\ne^{i \\left( 2\\pi \\vec{q} \\cdot \\left( n_{1} + \\frac{1}{2} \\right)\\vec{a}_{1} + \\phi \\right)} [H_{\\vec{a}_{1}-\\vec{a}_{2}}]\n\\end{equation}\nin Eq.~(\\ref{eq:1D_hexagonal_1}). \nThe $\\vec{a}_{2}$ in Eq.~(\\ref{eq:1D_hexagonal_1}--\\ref{eq:1D_hexagonal_3}) will become useful when we promote the dimension: \nAt this stage, $\\vec{a}_{2}$ is an unspecified label, though we have anticipated that it will be identified with the synthetic direction in the promoted lattice. \n\nWe promote the dimension of this 1D modulated system to 2D and choose the second, synthetic lattice vector as $\\vec{a}_{2} = (1\/2, \\sqrt{3}\/2)$. \nThus, $\\{\\vec{a}_{1},\\vec{a}_{2} \\}$ forms a 2D hexagonal lattice. \nThe reciprocal lattice vectors in this promoted 2D space are then\n\\begin{align}\n    & \\vec{g}_{1} = 2\\pi \\left( 1 , -\\frac{1}{\\sqrt{3}} \\right), \\\\\n    & \\vec{g}_{2} = 2\\pi \\left( 0,\\frac{2}{\\sqrt{3}}\\right),\n\\end{align}\nwhere we bear in mind that, if compared with our previous notation from Eqs.~(\\ref{eq:reciprocal_1}--\\ref{eq:reciprocal_4}), we should identify $\\vec{a}_{2} \\leftrightarrow \\vec{c}_{1}$ and $\\vec{g}_{2} \\leftrightarrow \\vec{G}_{1}$. \nUsing the Fourier expansion convention in Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} with $\\vec{q} = (q,0)$, and Eqs.~(\\ref{eq:1D_hexagonal_1}--\\ref{eq:1D_hexagonal_3}), we obtain the hopping terms in this case in Table~\\ref{tab:hopping_3}. \nMultiplying the entries for hopping matrices and Peierls phases, the Hamiltonian for the promoted 2D model is\n\\begin{align}\n    H_{2D}= &  \\sum_{n_{1},n_{2}} \\left( \\psi^{\\dagger}_{n_{1}+1,n_{2}} [H_{\\vec{a}_{1}}] \\psi_{n_{1},n_{2}} + \\psi^{\\dagger}_{n_{1},n_{2}+1} [H_{\\vec{a}_{2}}]e^{-i 2\\pi \\vec{q} \\cdot n_{1}\\vec{a}_{1}} \\psi_{n_{1},n_{2}} + \\psi^{\\dagger}_{n_{1}+1,n_{2}-1} [H_{\\vec{a}_{1}-\\vec{a}_{2}}] e^{i 2\\pi \\vec{q} \\cdot \\left( n_{1} + \\frac{1}{2} \\right)\\vec{a}_{1}} \\psi_{n_{1},n_{2}} + \\text{h.c.} \\right)  \\\\\n    & + \\sum_{n_{1},n_{2}} \\psi^{\\dagger}_{n_{1},n_{2}} [H_{0}] \\psi_{n_{1},n_{2}},\n\\end{align}\nwhere $\\psi^{\\dagger}_{n_{1},n_{2}}$ is the creation operator for an electron at position $n_{1}\\vec{a}_{1}+ n_{2}\\vec{a}_{2}$, and the vector potential to which this 2D model is coupled is \n\\begin{align}\n    \\vec{A} = \\frac{1}{2\\pi}  \\vec{g}_{2} \\left( \\vec{q}\\cdot \\left[ \\left( \\vec{r} \\cdot \\vec{g}_{1}\\right) \\vec{a}_{1} \\right] \\right) = \\left( 0 , \\frac{4\\pi q}{\\sqrt{3}} \\left( x - \\frac{y}{\\sqrt{3}} \\right) \\right).\n\\end{align}\nThis $\\vec{A}$ reproduces a perpendicular magnetic field $\\vec{B} = \\frac{4\\pi q}{\\sqrt{3}} \\hat{z}$ in the promoted 2D system. \nWe thus see that this 1D modulated system may be used to map out the Hofstadter spectrum of a hexagonal lattice with an irrational  magnetic flux.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $n_{1}\\vec{a}_{1} + n_{2}\\vec{a}_{2}$ to & Hopping matrices & Peierls phase  \\\\\n\\hline\n\\hline\n$n_{1}\\vec{a}_{1} + n_{2}\\vec{a}_{2}$ & $[H_{0}]$ & $1$ \\\\\n\\hline\n$(n_{1}+1)\\vec{a}_{1} + n_{2}\\vec{a}_{2}$ & $[H_{\\vec{a}_{1}}]$ & $1$ \\\\\n\\hline\n$n_{1}\\vec{a}_{1} + (n_{2}+1)\\vec{a}_{2}$  & $[H_{\\vec{a}_{2}}]$  & $e^{-i 2\\pi \\vec{q} \\cdot n_{1}\\vec{a}_{1}}$ \\\\\n\\hline\n$(n_{1}+1)\\vec{a}_{1} + (n_{2}-1)\\vec{a}_{2}$ & $e^{i 2\\pi \\vec{q} \\cdot \\frac{1}{2}\\vec{a}_{1}} [H_{\\vec{a}_{1}-\\vec{a}_{2}}]e^{-i \\pi \\vec{q}\\cdot \\vec{a}_{1}} = [H_{\\vec{a}_{1}-\\vec{a}_{2}}]$   & $e^{i 2\\pi \\vec{q}\\cdot \\left( n_{1} + \\frac{1}{2} \\right)\\vec{a}_{1}}$\\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping matrices and the corresponding Peierls phases for the promoted $2$D model for a 1D modulated chain described by Eqs.~(\\ref{eq:1D_hexagonal_1}--\\ref{eq:1D_hexagonal_3}). \nThe hopping matrices along the opposite directions for those listed here are omitted and can be obtained through Hermitian conjugate.}\n\\label{tab:hopping_3}\n\\end{table}\n\n\\subsubsection{2D hexagonal lattice $\\to$ 3D hexagonal lattice coupled to a $U(1)$ gauge field}\n\nConsider a 2D modulated system with one on-site modulation, hexagonal lattice and Hamiltonian\n\\begin{align}\n    H_{2D}= \\sum_{\\vec{n},\\vec{m}} \\psi^{\\dagger}_{\\vec{n}+\\vec{m}} [H_{\\vec{m}}] \\psi_{\\vec{n}} + \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}} [V_{\\vec{n}}] \\psi_{\\vec{n}}, \\label{eq:2D_hexa_to_3D_hexa_1}\n\\end{align}\nwhere $\\vec{n} = (n_{1},n_{2})$, $\\vec{m} = (m_{1},m_{2})$, $\\vec{a}_{1} = \\left(1\/2,\\sqrt{3}\/2\\right)$, $\\vec{a}_{2} = \\left(-1\/2,\\sqrt{3}\/2\\right)$, and $\\psi^{\\dagger}_{\\vec{n}}$ is the creation operator for an electron at lattice position $n_{1} \\vec{a}_{1} + n_{2} \\vec{a}_{2}$.\nThe matrix $[H_{\\vec{m}}]$ describes hopping terms from position $n_{1} \\vec{a}_{1} + n_{2} \\vec{a}_{2}$ to $(n_{1}+m_{1}) \\vec{a}_{1} + (n_{2}+m_{2})\\vec{a}_{2}$ which can include long range hopping terms. \nThe on-site modulation can be expanded as \n\\begin{align}\n    [V_{\\vec{n}}] = \\sum_{p} [V_{p}] e^{i p \\left( 2\\pi \\vec{q} \\cdot \\left(\\sum_{j=1}^{2}n_{j}\\vec{a}_{j} \\right) + \\phi \\right) }, \\label{eq:2D_hexa_to_3D_hexa_2}\n\\end{align}\nwhere $p \\in \\mathbb{Z}$, $[V_{p}]$ is the $p^{\\text{th}}$ Fourier coefficient of $[V_{\\vec{n}}]$ and $\\vec{q} = (q_{x},q_{y})$ is the modulation wave vector parallel to the 2D system. \nWe next promote the dimension of this 2D system to 3D and choose the third, synthetic lattice vector $\\vec{a}_{3}$ as $(0,0,1)$. \nThe lattice vectors in the promoted 3D space are then\n\\begin{align}\n    \\vec{a}_{1} = \\left( \\frac{1}{2},\\frac{\\sqrt{3}}{2},0 \\right),\\  \\vec{a}_{2} = \\left( -\\frac{1}{2},\\frac{\\sqrt{3}}{2},0 \\right),\\ \\vec{a}_{3} = \\left( 0,0,1 \\right),\n\\end{align}\nwhich describes a 3D hexagonal lattice. \nThe corresponding reciprocal lattice vectors are\n\\begin{align}\n    \\vec{g}_{1} = 2\\pi \\left( 1, \\frac{1}{\\sqrt{3}},0 \\right),\\ \\vec{g}_{2} = 2\\pi \\left( -1, \\frac{1}{\\sqrt{3}},0 \\right),\\ \\vec{g}_{3} = 2\\pi (0,0,1).\n\\end{align}\nAs in Sec.~\\ref{sec:1D_to_2D_hexa}, we bear in mind that in comparing with Eqs.~(\\ref{eq:reciprocal_1}--\\ref{eq:reciprocal_4}), we should identify $\\vec{a}_{3} \\leftrightarrow \\vec{c}_{1} $ and $\\vec{g}_{3} \\leftrightarrow \\vec{G}_{1}$. \nUsing the Fourier expansion convention in Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} with $\\vec{q} = (q_{x},q_{y})$, and Eqs.~(\\ref{eq:2D_hexa_to_3D_hexa_1}--\\ref{eq:2D_hexa_to_3D_hexa_2}), we obtain the hopping terms given in Table~\\ref{tab:hopping_4}. \nMultiplying the entries for hopping matrices and Peierls phases, the Hamiltonian for the promoted 3D model is\n\\begin{align}\n    H_{3D} = \\sum_{\\vec{n},\\vec{m}} \\psi^{\\dagger}_{\\vec{n} + \\vec{m}} [H_{\\vec{m}}] \\psi_{\\vec{n}} + \\sum_{\\vec{n},p} \\psi^{\\dagger}_{\\vec{n} - (0,0,p)} [V_{p}] e^{i 2\\pi p \\vec{q} \\cdot \\left( \\sum_{j=1}^{2}n_{j}\\vec{a}_{j} \\right)} \\psi_{\\vec{n}},\n\\end{align}\nwhere $\\vec{n} = (n_{1},n_{2},n_{3})$, $\\vec{m }= (m_{1},m_{2},0)$, $\\psi^{\\dagger}_{\\vec{n}}$ is the creation operator for an electron at lattice position $\\sum_{j=1}^{3}n_{j}\\vec{a}_{j}$, and the vector potential to which this 3D model is coupled is \n\\begin{align}\n    \\vec{A} = \\frac{1}{2\\pi}  \\vec{g}_{3} \\left( \\vec{q}\\cdot \\left[ \\left( \\vec{r} \\cdot \\vec{g}_{1}\\right) \\vec{a}_{1} + \\left( \\vec{r} \\cdot \\vec{g}_{2}\\right) \\vec{a}_{2} \\right] \\right) = \\left( 0,0,2\\pi\\left(q_{x}x + q_{y}y \\right) \\right).\n\\end{align}\nThis $\\vec{A}$ produces a magnetic field $\\vec{B} = \\left(2\\pi q_{y},-2\\pi q_{x},0 \\right)$ in the promoted 3D space.\n\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $\\sum_{j=1}^{3}n_{j}\\vec{a}_{j}$ to & Hopping matrices & Peierls phase  \\\\\n\\hline\n\\hline\n$\\sum_{j=1}^{3}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{2} m_{j}\\vec{a}_{j}$  & $[H_{\\vec{m}}]$ & $1$ \\\\\n\\hline\n$\\sum_{j=1}^{3}n_{j}\\vec{a}_{j} - p \\vec{a}_{3}$ & $[V_{p}]$ & $e^{i2\\pi p \\vec{q} \\cdot \\left(\\sum_{j=1}^{2}n_{j}\\vec{a}_{j} \\right)}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping matrices and the corresponding Peierls phases for the promoted $3$D model from a 2D hexagonal lattice. \nThe hopping matrices along the opposite directions of those listed here are omitted and can be obtained through Hermitian conjugation.}\n\\label{tab:hopping_4}\n\\end{table}\n\n\n\n\\section{\\label{sec:Thouless_pump_1D_Rice_Mele}Thouless pump and dimensional promotion for 1D Rice-Mele model}\n\nHere we review how topological properties such as the Thouless pump\\cite{Thouless_pump_original_paper} in the 1D Rice-Mele model\\cite{RiceMele} (SSH chain\\cite{su1979solitons}) can be attributed to a 2D lattice model coupled to a $U(1)$ gauge field, via our dimensional promotion procedure. \nConsider the 1D Rice-Mele model\n\\begin{align}\n    H_{\\text{Rice-Mele}} = \\sum_{n}  \\left( t + \\delta t (-1)^{n} \\cos{\\phi} \\right)c^{\\dagger}_{n+1}c_{n} + \\text{h.c.} + \\sum_{n} (-1)^{n+1} \\Delta  \\sin{\\phi} c^{\\dagger}_{n}c_{n},\n    \\label{eq:supp_Rice_Mele}\n\\end{align}\nwhere $c^{\\dagger}_{n}$ is the creation operator for an electron at 1D sites $n \\in \\mathbb{Z}$, and $\\text{h.c.}$ means the Hermitian conjugate of all terms before it. \nThis Hamiltonian describes a $1$D chain with a twofold (Peierls) CDW distortion. \nSuppose we prepare the ground state (with Fermi level $E_{F}=0$) for $H_{\\text{Rice-Mele}}$ with $t = 1$, $\\delta t=-0.1$, $\\Delta = 0.5$ at $\\phi = 0$. \nBefore dimensional promotion, let us first review the properties of this 1D Hamiltonian. \nAs the phase $\\phi$ adiabatically changes from $0$ to $2\\pi$, the single valence band Wannier center\\cite{Kohn59,Brouder2007,Marzari2012,ksv,fu2006time} in the bulk is pumped by one unit cell (two lattice sites), as shown in Fig.~\\ref{Fig_Rice_Mele} (c). \nThe bulk polarization then changes by $(-1)\\times d = -d$. \nThis bulk polarization change is quantized\\cite{ksv,resta1994macroscopic} in units of the unit cell length $d=2$. \nThis is the classic realization of the Thouless pump\\cite{Thouless_pump_original_paper} in 1D. \nIn Fig.~\\ref{Fig_Rice_Mele} (a) we show the $\\phi$-sliding spectrum (defined in Sec.~IV of the main text) for a finite chain with size $100$ and positions $n = 1 ,\\ldots, 100$. \nWe see that in addition to the gapped bulk states, there are two linearly dispersing modes that cross the bulk gap. \nThis indicates that during the pumping process, localized charges at the right and left end (see Figs.~\\ref{Fig_Rice_Mele} (b) and (d)) of the chain flow out of and into the occupied state subspace, respectively. \nThis is the celebrated topological origin of the quantized polarization change in the bulk, and the existence of boundary states crossing the energy gap.\n\\begin{figure}[h]\n  \\includegraphics[scale=0.4]{Fig_1D_Rice_Mele.pdf}\n  \\caption{(a) $\\phi$-sliding spectrum of the Rice-Mele model with $t = 1$, $\\delta t=-0.1$ and $\\Delta = 0.5$. \n  (c) The shifting of the valence band Wannier center (blue dots) as a function of $\\phi$. \n  The vertical black lines denote the boundary of the unit cell, which has length $2$. \n  The orange dots denote the positions of the tight-binding basis orbitals. \n  The origin of the $x$-axis is placed at the middle of the bond with hopping $t - \\delta t \\cos{\\phi}$ where $t = 1$ and $\\delta t = -0.1$. \n  In other words, the unit cell is formed by the sites $n=1$ and $n=2$ in Eq.~(\\ref{eq:supp_Rice_Mele}) and the origin is placed at the midpoint between these two sites. \n  The inversion centers at $\\phi=0$ and $\\pi$ lie at integer values of $x$, and as such the Wannier centers at these two $\\phi$ are located either at the center of the unit cell ($\\phi=0$) or the boundaries between unit cells ($\\phi=\\pi)$\\cite{Aris2014}. \n  Also, this unit cell choice is commensurate with the finite size system in (a), (b) and (d), where we choose $n = 1 ,\\ldots, 100$. \n  (b) $\\&$ (d) Probability distribution of localized modes around the two ends at $\\phi = 0.9 \\pi$ and energy $E = -0.1545$ $\\&$ $+0.1545$. \n  Notice that during the adiabatic pumping from $\\phi =0$ to $2\\pi$, it is the boundary state at the left [right] end, see (d) [(b)], that flows into [out of] the subspace of occupied states. \n  This is consistent with (c) where the Wannier center flows toward the right and reappear at the left boundary of the unit cell at $\\phi = \\pi$.}\n  \\label{Fig_Rice_Mele}\n\\end{figure}\n\nLet us now see how these properties emerge in our 2D dimensionally-promoted picture. \nIdentifying $\\phi$ as the crystal momentum $k_{y}$ along the second, synthetic dimension $y$, $H_{\\text{Rice-Mele}}$ is equivalent to the Bloch Hamiltonian of a 2D model (see Sec.~\\ref{sec:ex_1D_RM_chain} for the detailed derivation) \n\\begin{align}\n    H_{\\text{2D}} = \\sum_{n,m} \\left( tc^{\\dagger}_{n+1,m}c_{n,m}- \\frac{i\\delta t}{2}e^{i\\pi \\left(n+\\frac{1}{2} \\right)} c^{\\dagger}_{n+1,m-1}c_{n,m}+ \\frac{i\\delta t}{2}e^{-i\\pi \\left(n+\\frac{1}{2} \\right)} c^{\\dagger}_{n+1,m+1}c_{n,m}-  \\frac{i\\Delta}{2}e^{-in\\pi} c^{\\dagger}_{n,m+1}c_{n,m} \\right)+ \\text{h.c.}, \\label{eq:1D_Rice_Mele_to_2D_1}\n\\end{align}\nwith a fixed $k_{y}$. Here $c^{\\dagger}_{n,m}$ is the creation operator for an electron at 2D site $(n,m)$ on a square lattice. \nIf, during the adiabatic process, $H_{\\text{Rice-Mele}}$ is always gapped, the quantized polarization change is determined by the Chern number\\cite{tknn,niu1984quantised,niu1985quantized,Aris2014,fu2006time,bernevigbook} of the occupied bands of $H_{\\text{2D}}$. \nWe can thus characterize the topological properties shown in Fig.~\\ref{Fig_Rice_Mele} by a Chern number $C=1$. \nWe notice that in addition to hopping terms from site $(n,m)$ to $(n\\pm1,m)$ and $(n,m\\pm1)$, $H_{\\text{2D}}$ also has hopping terms going from site $(n,m)$ to $(n\\pm1,m+1)$ and $(n\\pm1,m- 1)$. \nThis is in contrast to the standard textbook correspondence between the SSH chain and a 2D Chern insulator, where a re-embedding of orbitals within the enlarged unit cell is typically used to obtain a 2D model with only perpendicular hoppings along the $\\hat{x}$ and $\\hat{y}$ directions.\nNote also that $H_{\\text{2D}}$ is a 2D lattice model coupled to a $U(1)$ gauge field $\\vec{A} = (0,\\pi x)$ through the Peierls substitution\\cite{Peierls_substitution} assuming the electron has charge $-1$. \nThis $\\vec{A}$ produces a uniform $U(1)$ magnetic field threading $\\pi$-flux per plaquette, since the magnetic field $\\vec{B} = \\vec{\\nabla} \\cross \\vec{A} = \\pi \\hat{z}$. \nReinserting factors of $\\hbar$ and $|e|$, this corresponds to half a flux quantum $\\Phi_{0} = 2\\pi \\hbar \/ |e|$ per plaquette. \nThe localized and mid-gap states in $H_{\\text{Rice-Mele}}$ are then identified as the chiral edge modes due to the quantum Hall effect in $H_{\\text{2D}}$.\n\n\\section{\\label{sec:promoted_lattice_model} Promoted lattice models}\n\nFor completeness, in this section we give the promoted lattice models in position space corresponding to the 2D modulated system with helical sliding modes and the 3D Weyl semimetal with mean-field charge-density waves (CDWs) in Secs.~V and VI of the main text, respectively.\n\n\\subsection{\\label{sec:promoted_helical_lattice_model}2D modulated system with helical sliding modes}\n\nThe promoted 3D lattice model\\cite{Wieder_spin_decoupled_helical_HOTI} for the 2D modulated system with helical sliding modes in the main text is\n\\begin{align}\n    H = \\sum_{\\vec{n}} & \\left( \\psi^{\\dagger}_{\\vec{n}+\\hat{x}} [H_{+\\hat{x}}]\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}-\\hat{x}} [H_{+\\hat{x}}]^{\\dagger}\\psi_{\\vec{n}} +\\psi^{\\dagger}_{\\vec{n}+\\hat{z}} [H_{+\\hat{z}}]\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}-\\hat{z}} [H_{+\\hat{z}}]^{\\dagger}\\psi_{\\vec{n}}+ \\psi^{\\dagger}_{\\vec{n}} [H_{\\text{on-site}}] \\psi_{\\vec{n}} \\nonumber \\right. \\\\\n    & \\left. + \\psi^{\\dagger}_{\\vec{n}+\\hat{y}} [H_{+\\hat{y}}] e^{-i\\text{A}_{\\vec{n}+\\hat{y},\\vec{n}}}\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}-\\hat{y}} e^{i\\text{A}_{\\vec{n},\\vec{n}-\\hat{y}}}[H_{+\\hat{y}}]^{\\dagger} \\psi_{\\vec{n}}  \\right) \\label{eq:lattice_model_helical_sliding} \n\\end{align}\nwith\n\\begin{align}\n    & [H_{+\\hat{x}}] = \\frac{v_{x}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{u_{x}}{2i}\\tau_{y}\\mu_{y}\\sigma_{0},\\\\\n    & [H_{+\\hat{y}}] = \\frac{v_{y}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{v_{H}}{2 i} \\tau_{y}\\mu_{z}\\sigma_{z},\\\\\n    & [H_{+\\hat{z}}] = \\frac{v_{z}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{u_{z}}{2i}\\tau_{x}\\mu_{0}\\sigma_{0}, \\\\\n    & [H_{\\text{on-site}}] = m_{1}\\tau_{z}\\mu_{0}\\sigma_{0}+m_{2}\\tau_{z}\\mu_{x}\\sigma_{0} + m_{3}\\tau_{z}\\mu_{z}\\sigma_{0}+m_{v_{1}}\\tau_{0}\\mu_{z}\\sigma_{0} + m_{v_{2}}\\tau_{0}\\mu_{x}\\sigma_{0}.\n\\end{align}\nThe $SU(2)$ lattice gauge field is given by\n\\begin{align}\n    \\text{A}_{\\vec{n}+\\hat{y},\\vec{n}} =2\\pi \\left( q_{x}n_{x}+q_{z}n_{z} \\right) \\tau_{0}\\mu_{0}\\sigma_{z}. \\label{eq:lattice_SU2_A}\n\\end{align}\nThe $SU(2)$ gauge field in the continuous coordinate representation is then (dropping the $\\tau_{0}\\mu_{0}$)\n\\begin{align}\n    \\vec{A} = (0,2\\pi(q_{x}x+q_{z}z)\\sigma_{z},0). \\label{eq:A_SU2}\n\\end{align}\nThe corresponding $SU(2)$ magnetic field $\\vec{B} = \\vec{\\nabla} \\cross \\vec{A} - i \\vec{A} \\cross \\vec{A}$ is\\cite{Estienne_2011}\n\\begin{align}\n    \\vec{B} = (-2\\pi q_{z}\\sigma_{z},0,2\\pi q_{x}\\sigma_{z}). \\label{eq:B_SU2_1}\n\\end{align}\nEq.~(\\ref{eq:lattice_model_helical_sliding}) describes a helical higher-order topological insulator (HOTI) coupled to a $SU(2)$ gauge field. \nIf we Fourier transform Eq.~(\\ref{eq:lattice_model_helical_sliding}) along $y$ and regard $k_{y}$ (wavenumber along $y$) as the sliding phase $\\phi$, we can obtain the 2D modulated system in the main text.\n\n\\subsection{\\label{sec:promoted_4D_lattice_model}3D Weyl semimetal with mean-field charge density waves}\n\nThe promoted 4D lattice model for the 3D Weyl semimetal with mean-field CDW order\\cite{dynamical_axion_insulator_BB} in the main text is\n\\begin{align}\n    H&=\\left(\\sum_{\\vec{n}}\\left[-it_x\\psi^\\dag_{\\vec{n}+\\hat{x}}\\sigma_x \\psi_{\\vec{n}}-it_y \\psi^\\dag_{\\vec{n}+\\hat{y}}\\sigma_y \\psi_{\\vec{n}}+t_z\\psi^\\dag_{\\vec{n}+\\hat{z}}\\sigma_z \\psi_{\\vec{n}} + |\\Delta| e^{-i2 \\pi q n_{z}} \\psi^\\dag_{\\vec{n}+\\hat{w}} \\sigma_{z} \\psi_{\\vec{n}} \\right]\\right. \\nonumber \\\\\n    & \\left. +\\sum_{\\vec{n}}\\frac{m}{2}\\left(\\psi^\\dag_{\\vec{n}+\\hat{x}}\\sigma_z \\psi_{\\vec{n}} + \\psi^\\dag_{\\vec{n}+\\hat{y}}\\sigma_z \\psi_{\\vec{n}}  -2 \\psi^\\dag_{\\vec{n}}\\sigma_z \\psi_{\\vec{n}}  \\right) -\\sum_{\\vec{n}} t_z \\left(\\cos (\\pi q)\\right) \\psi^\\dag_{\\vec{n}}\\sigma_z \\psi_{\\vec{n}}\\right) +\\mathrm{h.c.}, \\label{eq:lattice_model_4D_nodal_line}\n\\end{align}\nwhere the phase factors correspond to Peierls substitution of a 4D $U(1)$ gauge field\n\\begin{align}\n    \\vec{A} = (0,0,0,2\\pi q z).\n\\end{align}\nThe only non-zero components of the $U(1)$ field strength $F_{\\mu \\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}$ is\n\\begin{align}\n    F_{zw} = -F_{wz} = \\partial_{z}A_{w} - \\partial_{w}A_{z}=2 \\pi q, \\label{eq:4D_U1_Fzw}\n\\end{align}\nwhich threads through the $zw$ plane. \nWithout coupling to a 4D $U(1)$ gauge field, the Bloch Hamiltonian of Eq.~(\\ref{eq:lattice_model_4D_nodal_line}) with $q=0$ is\n\\begin{align}\n    H(\\vec{k})=& -2[t_x \\sin (k_x) \\sigma_x +t_y \\sin (k_y) \\sigma_y] +2t_z[\\cos (k_z) -\\cos (\\pi q)] \\sigma_{z} -m[2-\\cos (k_x) - \\cos (k_y)] \\sigma_{z} + 2|\\Delta|\\cos{(k_{w})}\\sigma_{z},\n\\label{eq:lattice_model_4D_nodal_line_bloch}\n\\end{align}\nwhich has a nodal line in the $k_{z}$-$k_{w}$ plane (with $k_{x}=k_{y}=0$) defined by\n\\begin{align}\n    t_{z}\\cos(k_{z}) +  |\\Delta| \\cos(k_{w}) = t_{z}\\cos{(\\pi q)}. \\label{eq:lattice_nodal_line}\n\\end{align}\nTherefore, Eq.~(\\ref{eq:lattice_model_4D_nodal_line}) describes a 4D nodal line system coupled to a 4D $U(1)$ gauge field where the corresponding field strength Eq.~(\\ref{eq:4D_U1_Fzw}) threads through the area enclosed by the nodal line defined in Eq.~(\\ref{eq:lattice_nodal_line}). \nIf we Fourier transform Eq.~(\\ref{eq:lattice_model_4D_nodal_line}) along $w$ and regard $k_{w}$ as the sliding phase $\\phi$, we obtain the model for a 3D Weyl semimetal with mean-field CDW order given in the main text.\n\n\\section{\\label{sec:inv_sym_gauge_tr}Inversion symmetry up to a gauge transformation}\n \nIn this section we will show that if a lattice has inversion symmetry, such as the 3D models promoted from our examples of 2D modulated systems with chiral and helical sliding modes, then upon coupling to a $U(1)$ or $SU(2)$ gauge field producing a constant magnetic field, the inversion symmetry is still preserved {\\it up to a gauge transformation}. \nWe will do this in details in the simplest case, which is a 2D square lattice coupled to a perpendicular magnetic field. \nWe will briefly mention the generalization to the 3D cases, which corresponds to the dimensional promotion from 2D modulated systems.\n\n\\subsection{2D system with inversion symmetry}\n\nLet us consider a 2D square lattice with only one degree of freedom within each unit cell labelled by $(n,m) \\in \\mathbb{Z}^{2}$\n\\begin{align}\n    H = -t\\sum_{n,m} \\left( \\psi^{\\dagger}_{n+1,m}\\psi_{n,m}  +\\psi^{\\dagger}_{n-1,m}\\psi_{n,m}+ e^{-i2\\pi bn}\\psi^{\\dagger}_{n,m+1}\\psi_{n,m}+ e^{+i2\\pi bn}\\psi^{\\dagger}_{n,m-1}\\psi_{n,m} \\right). \\label{eq:inv_sym_gauge_tr_1}\n\\end{align}\nEq.~(\\ref{eq:inv_sym_gauge_tr_1}) is coupled to a $U(1)$ gauge field $\\vec{A} = 2\\pi bx \\hat{y}$ through Peierls substitution where we have assumed the particle carries charge $-1$. \nThis $U(1)$ gauge field produces a constant perpendicular magnetic field $\\vec{B} = 2\\pi b \\hat{z}$. \nIf we turn off the $U(1)$ gauge field by setting $b = 0$, Eq.~(\\ref{eq:inv_sym_gauge_tr_1}) will have inversion symmetry, where the inversion operators are defined by \n\\begin{align}\n    & \\mathcal{I}_{n_{c},m_{c}} \\psi^{\\dagger}_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m}, \\label{eq:inv_sym_gauge_tr_2} \\\\\n    & \\mathcal{I}_{n_{c},m_{c}} \\psi_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi_{2n_{c}-n,2m_{c}-m}. \\label{eq:inv_sym_gauge_tr_3}\n\\end{align}\nHere $\\mathcal{I}_{n_{c},m_{c}}$ is a unitary operator and $(n_{c},m_{c})$ is the inversion center (which can be any lattice point). We then have\n\\begin{align}\n    \\mathcal{I}_{n_{c},m_{c}} H \\mathcal{I}_{n_{c},m_{c}}^{-1} = H\\  \\forall (n_{c},m_{c})\n\\end{align}\nprovided that the summation over $n$ and $m$ in Eq.~(\\ref{eq:inv_sym_gauge_tr_1}) goes from $- \\infty$ to $+\\infty$. \nWhen $b \\ne 0$, we must modify our unitary inversion operations in Eq.~(\\ref{eq:inv_sym_gauge_tr_2}) and Eq.~(\\ref{eq:inv_sym_gauge_tr_3}) to be\n\\begin{align}\n    & \\mathcal{I}_{n_{c},m_{c}} \\psi^{\\dagger}_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m}e^{i4\\pi b n_{c}(m-m_{c})}, \\label{eq:inv_sym_gauge_tr_4} \\\\\n    & \\mathcal{I}_{n_{c},m_{c}} \\psi_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi_{2n_{c}-n,2m_{c}-m}e^{-i4\\pi b n_{c}(m-m_{c})}, \\label{eq:inv_sym_gauge_tr_5}\n\\end{align}\nwhich acts as inversion through the center $(n_{c},m_{c})$ together with a gauge transformation.\nWith these modified inversion operations, the following three identities can be proved: \n\\begin{align}\n    & \\mathcal{I}_{n_{c},m_{c}} H \\mathcal{I}_{n_{c},m_{c}}^{-1} = H \\ \\forall (n_{c},m_{c}), \\label{eq:inv_sym_gauge_tr_6} \\\\\n    & \\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2} \\psi^{\\dagger}_{n,m} \\left( \\mathcal{I}_{n_{c},m_{c}}^{-1} \\right)^{2} = \\psi^{\\dagger}_{n,m}, \\label{eq:inv_sym_gauge_tr_7}\\\\\n    & \\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2} \\psi_{n,m} \\left( \\mathcal{I}_{n_{c},m_{c}}^{-1} \\right)^{2} = \\psi_{n,m}. \\label{eq:inv_sym_gauge_tr_8}\n\\end{align}\nWe prove Eq.~(\\ref{eq:inv_sym_gauge_tr_6}) as follows:\n\\begin{align}\n    \\mathcal{I}_{n_{c},m_{c}} H \\mathcal{I}_{n_{c},m_{c}}^{-1} & = -t \\sum_{n,m} \\begin{bmatrix}\n    \\psi^{\\dagger}_{2n_{c}-n-1,2m_{c}-m} e^{i 4\\pi b n_{c}(m-m_{c})} \\psi_{2n_{c}-n,2m_{c}-m} e^{-i 4\\pi b n_{c}(m-m_{c})}  \\\\\n    + \\psi^{\\dagger}_{2n_{c}-n+1,2m_{c}-m} e^{i 4\\pi b n_{c}(m-m_{c})} \\psi_{2n_{c}-n,2m_{c}-m} e^{-i 4\\pi b n_{c}(m-m_{c})} \\\\\n    + e^{-i2\\pi b n } \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m-1}e^{i 4\\pi b n_{c}(m+1-m_{c})} \\psi_{2n_{c}-n,2m_{c}-m} e^{-i 4\\pi b n_{c}(m-m_{c})}  \\\\\n    + e^{+i2\\pi b n } \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m+1}e^{i 4\\pi b n_{c}(m-1-m_{c})} \\psi_{2n_{c}-n,2m_{c}-m} e^{-i 4\\pi b n_{c}(m-m_{c})}\n    \\end{bmatrix} \\label{eq:pf_inv_1}\\\\\n    &  = -t \\sum_{n,m} \\begin{bmatrix}\n    \\psi^{\\dagger}_{2n_{c}-n-1,2m_{c}-m}   \\psi_{2n_{c}-n,2m_{c}-m}   \\\\\n    + \\psi^{\\dagger}_{2n_{c}-n+1,2m_{c}-m} \\psi_{2n_{c}-n,2m_{c}-m}  \\\\\n    + e^{-i2\\pi b n } \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m-1}e^{i 4\\pi b n_{c}} \\psi_{2n_{c}-n,2m_{c}-m}   \\\\\n    + e^{+i2\\pi b n } \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m+1}e^{-i 4\\pi b n_{c}} \\psi_{2n_{c}-n,2m_{c}-m} \n    \\end{bmatrix} \\label{eq:pf_inv_2} \\\\\n    &  = -t \\sum_{n,m} \\begin{bmatrix}\n    \\psi^{\\dagger}_{n-1,m}   \\psi_{n,m}   \\\\\n    + \\psi^{\\dagger}_{n+1,m} \\psi_{n,m}  \\\\\n    + e^{-i2\\pi b (-n+2n_{c}) } \\psi^{\\dagger}_{n,m-1}e^{i 4\\pi b n_{c}} \\psi_{n,m}   \\\\\n    + e^{+i2\\pi b (-n+2n_{c}) } \\psi^{\\dagger}_{n,m+1}e^{-i 4\\pi b n_{c}} \\psi_{n,m} \n    \\end{bmatrix}  \\label{eq:pf_inv_3} \\\\\n    & = -t \\sum_{n,m} \\begin{bmatrix}\n    \\psi^{\\dagger}_{n-1,m}   \\psi_{n,m}   \\\\\n    + \\psi^{\\dagger}_{n+1,m} \\psi_{n,m}  \\\\\n    + e^{+i2\\pi bn } \\psi^{\\dagger}_{n,m-1}\\psi_{n,m}   \\\\\n    + e^{-i2\\pi bn } \\psi^{\\dagger}_{n,m+1} \\psi_{n,m} \n    \\end{bmatrix}  \\label{eq:pf_inv_4} \\\\\n    & = H. \\label{eq:pf_inv_5}\n\\end{align}\nIn Eq.~(\\ref{eq:pf_inv_1}) we apply the inversion operation to each of the creation and annihilation operators according to Eq.~(\\ref{eq:inv_sym_gauge_tr_4}) and Eq.~(\\ref{eq:inv_sym_gauge_tr_5}). \nThe first to fourth terms in Eq.~(\\ref{eq:pf_inv_1}) correspond to the (transformed) first to fourth terms in Eq.~(\\ref{eq:inv_sym_gauge_tr_1}). \nIn Eq.~(\\ref{eq:pf_inv_2}) we cancel out redundant exponential phase factors and reindex $n \\to -n + 2n_{c}$ and $m\\to -m+2m_{c}$ in Eq.~(\\ref{eq:pf_inv_3}). \nEq.~(\\ref{eq:pf_inv_4}) and Eq.~(\\ref{eq:pf_inv_5}) then show that the transformed Hamiltonian is the same. \nWe can also prove Eq.~(\\ref{eq:inv_sym_gauge_tr_7}) as follows:\n\\begin{align}\n    \\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2} \\psi^{\\dagger}_{n,m} \\left( \\mathcal{I}_{n_{c},m_{c}}^{-1} \\right)^{2} & = \\mathcal{I}_{n_{c},m_{c}} \\mathcal{I}_{n_{c},m_{c}} \\psi^{\\dagger}_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1}  \\mathcal{I}_{n_{c},m_{c}}^{-1}  \\\\\n    & =  \\mathcal{I}_{n_{c},m_{c}} \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m}e^{i4\\pi b n_{c}(m-m_{c})} \\mathcal{I}_{n_{c},m_{c}}^{-1}\\\\\n    & = \\psi^{\\dagger}_{2n_{c}-(2n_{c}-n),2m_{c}-(2m_{c}-m)} e^{i4\\pi b n_{c}((2m_{c}-m)-m_{c})} e^{i4\\pi b n_{c}(m-m_{c})}\\\\\n    & = \\psi^{\\dagger}_{n,m} e^{i4\\pi b n_{c}(m_{c}-m)} e^{i4\\pi b n_{c}(m-m_{c})} \\\\\n    & = \\psi^{\\dagger}_{n,m} .\n\\end{align}\nWe have used Eq.~(\\ref{eq:inv_sym_gauge_tr_4}) twice above, first acting on $\\psi^{\\dagger}_{n,m}$ and then on $\\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m}$. \nEq.~(\\ref{eq:inv_sym_gauge_tr_8}) can also be proved in a similar way as follows:\n\\begin{align}\n    \\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2} \\psi_{n,m} \\left( \\mathcal{I}_{n_{c},m_{c}}^{-1} \\right)^{2} & = \\mathcal{I}_{n_{c},m_{c}} \\mathcal{I}_{n_{c},m_{c}} \\psi_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1}  \\mathcal{I}_{n_{c},m_{c}}^{-1}  \\\\\n    & =  \\mathcal{I}_{n_{c},m_{c}} \\psi_{2n_{c}-n,2m_{c}-m}e^{-i4\\pi b n_{c}(m-m_{c})} \\mathcal{I}_{n_{c},m_{c}}^{-1}\\\\\n    & = \\psi_{2n_{c}-(2n_{c}-n),2m_{c}-(2m_{c}-m)} e^{-i4\\pi b n_{c}((2m_{c}-m)-m_{c})} e^{-i4\\pi b n_{c}(m-m_{c})}\\\\\n    & = \\psi_{n,m} e^{-i4\\pi b n_{c}(m_{c}-m)} e^{-i4\\pi b n_{c}(m-m_{c})} \\\\\n    & = \\psi_{n,m} .\n\\end{align}\nEq.~(\\ref{eq:inv_sym_gauge_tr_6}) implies that if the $H$ in Eq.~(\\ref{eq:inv_sym_gauge_tr_1}) has non-zero $b$ with the summation over $n$ and $m$ going from $- \\infty$ to $+\\infty$, then $H$ is invariant under the inversion operation $\\mathcal{I}_{n_{c},m_{c}}$ defined by Eqs.~(\\ref{eq:inv_sym_gauge_tr_4}) and (\\ref{eq:inv_sym_gauge_tr_5}). \nAlso, since $\\psi^{\\dagger}_{n,m}$ and $\\psi_{n,m}$ denote the creation and annihilation operators for electronic states spanning the whole Hilbert space, Eq.~(\\ref{eq:inv_sym_gauge_tr_7}) and Eq.~(\\ref{eq:inv_sym_gauge_tr_8}) imply that $\\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2}$ is the identity operation. \n\nTo complete the proof, we give a construction for the unitary operator $\\mathcal{I}_{n_c,m_c}$.\nThe matrix representation of $\\mathcal{I}_{n_{c},m_{c}}$ in Eq.~(\\ref{eq:inv_sym_gauge_tr_5}) for sites at $(n,m)$ and $(2n_{c}-n,2m_{c}-m)$ is given by\n\\begin{align}\n    \\begin{bmatrix}\n    0 & e^{-i4\\pi b n_{c}(m-m_{c})} \\\\\n    e^{-i4\\pi b n_{c}(m_{c}-m)} & 0\n    \\end{bmatrix}, \\label{eq:mat_I_nc_mc}\n\\end{align}\nsince\n\\begin{align}\n    & \\mathcal{I}_{n_{c},m_{c}} \\psi_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi_{2n_{c}-n,2m_{c}-m}e^{-i4\\pi b n_{c}(m-m_{c})}, \\\\\n    & \\mathcal{I}_{n_{c},m_{c}} \\psi_{2n_{c}-n,2m_{c}-m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi_{n,m}e^{-i4\\pi b n_{c}(m_{c}-m)}.\n\\end{align}\nAs we can see Eq.~(\\ref{eq:mat_I_nc_mc}) is in fact a unitary matrix. \nTherefore, even though in this 2D lattice we have a constant perpendicular magnetic field which couples to the lattice through a Peierls substitution, there still exist unitary inversion operators with inversion centers at every lattice site which square to the identity and commute with the Hamiltonian.\n\n\\subsection{3D system with inversion symmetry}\n\nWe have shown that by defining unitary inversion operations up to a gauge transformation in Eq.~(\\ref{eq:inv_sym_gauge_tr_4}) and Eq.~(\\ref{eq:inv_sym_gauge_tr_5}), the inversion symmetry of Eq.~(\\ref{eq:inv_sym_gauge_tr_1}) is preserved. \nLet us now discuss how this extends to our $3$D promoted systems. \nSince our $3$D model of a chiral HOTI from the main text is also coupled to a constant $U(1)$ gauge field given by $\\vec{A} = (0,0,2\\pi(q_{x}x+q_{y}y))$, we can construct the proper unitary inversion operators squaring to the identity, given by\n\\begin{align}\n    & \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}} \\psi^{\\dagger}_{n_{x},n_{y},n_{z}} \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}}^{-1} = \\psi^{\\dagger}_{2n_{x}^{c}-n_{x},2n_{y}^{c}-n_{y},2n_{z}^{c}-n_{z}} [\\mathcal{I}]^{-1} e^{i 4\\pi \\left( q_{x}n_{x}^{c} + q_{y}n_{y}^{c} \\right) \\left( n_{z} - n_{z}^{c} \\right)}  , \\\\\n    & \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}} \\psi_{n_{x},n_{y},n_{z}} \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}}^{-1} = [\\mathcal{I}] \\psi_{2n_{x}^{c}-n_{x},2n_{y}^{c}-n_{y},2n_{z}^{c}-n_{z}} e^{-i 4\\pi \\left( q_{x}n_{x}^{c} + q_{y}n_{y}^{c} \\right) \\left( n_{z} - n_{z}^{c} \\right)}.\n\\end{align}\nIn these expressions $\\psi^{\\dagger}_{n_{x},n_{y},n_{z}}$ is the 4-component creation operator for an electron at site $(n_{x},n_{y},n_{z})$, $(n_{x}^{c},n_{y}^{c},n_{z}^{c})$ is the inversion center at any lattice point, and $[\\mathcal{I}] = \\tau_{z}\\sigma_{0}$ is the unitary inversion matrix which also squares to the identity and acts on the degrees of freedom within a unit cell. \nFor the case of our helical model, recall that our example of Eq.~(\\ref{eq:lattice_model_helical_sliding}) in Sec.~\\ref{sec:promoted_helical_lattice_model} is spin-decoupled. \nThus the $SU(2)$ gauge field given by $\\vec{A} = (0,2\\pi (q_{x}x+q_{z}z)\\sigma_{z},0)$ to which the model is coupled acts effectively as oppositely oriented $U(1)$ magnetic fields for spin up and down electrons. \nWe can thus define the unitary inversion operations squaring to identity up to a spin-dependent ($SU(2)$) gauge transformation\n\\begin{align}\n    & \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}} \\psi^{\\dagger}_{n_{x},n_{y},n_{z},\\sigma} \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}}^{-1} = \\psi^{\\dagger}_{2n_{x}^{c}-n_{x},2n_{y}^{c}-n_{y},2n_{z}^{c}-n_{z},\\sigma} [\\mathcal{I}]^{-1} e^{i 4\\sigma\\pi \\left( q_{x}n_{x}^{c} + q_{z}n_{z}^{c} \\right) \\left( n_{y} - n_{y}^{c} \\right)}  , \\\\\n    & \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}} \\psi_{n_{x},n_{y},n_{z},\\sigma} \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}}^{-1} = [\\mathcal{I}] \\psi_{2n_{x}^{c}-n_{x},2n_{y}^{c}-n_{y},2n_{z}^{c}-n_{z},\\sigma} e^{-i 4\\sigma\\pi \\left( q_{x}n_{x}^{c} + q_{z}n_{z}^{c} \\right) \\left( n_{y} - n_{y}^{c} \\right)},\n\\end{align}\nwhere $\\psi^{\\dagger}_{n_{x},n_{y},n_{z},\\sigma}$ is the 4-component creation operator for an electron at site $(n_{x},n_{y},n_{z})$ for a fixed spin $\\sigma = \\pm$ and $[\\mathcal{I}] = \\tau_{z}\\mu_{0}$. \nTherefore, the 3D chiral (helical) HOTI coupled to a $U(1)$ ($SU(2)$) gauge field in our examples preserves inversion symmetry.\n\n\n\\section{\\label{bigsec:chiral_sliding}Low energy bulk theory of chiral higher-order topological sliding modes}\n\nIn this section, we consider the low energy bulk theory of electrons in a 3D chiral HOTI minimally coupled to a $U(1)$ gauge field, shown in the main text and verify that it can capture several qualitative properties in numerical results of the lattice model calculation with $q_{y} = 0.02$.\n\nAs mentioned in the main text, the relevant low energy theory is \n\\begin{align}\n    H_{\\text{bulk}} = m_{\\text{bulk}} \\tau_{z}\\sigma_{0} + \\tau_{x}\\vec{p} \\cdot \\vec{\\sigma} + \\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}. \\label{eq:3D_chiral_HOTO_bulk}\n\\end{align}\nCoupling this $H_{\\text{bulk}}$ to a vector potential $\\vec{A} = By \\hat{z}$ such that $p_{z} \\to p_{z} + By$, and defining \n\\begin{align}\n    a^{\\dagger}_{k_{z}} = \\frac{1}{\\sqrt{2B}}\\left( k_{z} + By - ip_{y} \\right), \\label{eq:3D_chiral_bulk_U1_ladder}\n\\end{align}\nwe can rewrite the Hamiltonian as \n\\begin{align}\n    H_{\\text{bulk}}(k_{x},k_{z}) = m_{\\text{bulk}} \\tau_{z}\\sigma_{0} + \\tau_{x} \\begin{bmatrix}\n    \\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}+a^{\\dagger}_{k_{z}} \\right) & k_{x} -\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}-a^{\\dagger}_{k_{z}} \\right) \\\\\n    k_{x} +\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}-a^{\\dagger}_{k_{z}} \\right) & -\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}+a^{\\dagger}_{k_{z}} \\right)\n    \\end{bmatrix} + \\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}. \\label{eq:3D_bulk_LL_Hamiltonian}\n\\end{align}\n\nFig.~\\ref{SM_chiral_bulk_low_energy} (b) shows the numerically computed energy spectrum $E_{k_{x},k_{z}}$ for Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) as a function of $k_{x}$. \nNote that $E_{k_{x},k_{z}}$ does not depend on $k_{z}$, giving rise to flat Landau levels (LLs) along $k_{z}$. \nWe now relate this low energy bulk theory to the $\\phi$-sliding spectrum of the 2D modulated system by identifying $k_{z}$ in Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) with $\\Delta \\phi = \\phi - \\pi$, as we have done in the main text. \nTo assist the following discussion, we also show again the $\\phi$-sliding spectrum of the 2D modulated system which can be promoted to a 3D chiral HOTI coupled to a $U(1)$ gauge field with $\\vec{q} = (0,q_{y})=(0,0.02)$ in Fig.~\\ref{SM_chiral_bulk_low_energy} (a). \nThe flat bands corresponding to bulk-confined modes in Figs.~\\ref{SM_chiral_bulk_low_energy} (c) and (d) are marked in green ($E = -0.5144$) and orange ($E = +0.4656$).\n\n\nFirst, the band edges for the valence and conduction in Fig.~\\ref{SM_chiral_bulk_low_energy} (b) are at energy $E = -0.5248$ and $+0.4661$.\nThis indicates that the particle-hole symmetry in the bulk of a 3D chiral HOTI is generally broken upon $U(1)$ Landau quantization.\nThis is also reflected in the energy eigenvalues of the bulk flat bands  in the lattice model Fig.~\\ref{SM_chiral_bulk_low_energy} (a), with probability density shown in Figs.~\\ref{SM_chiral_bulk_low_energy} (c) and (d). \nWe see that the energies ($E = -0.5144$ for Fig.~\\ref{SM_chiral_bulk_low_energy} (c) and $E = +0.4656$ in Fig.~\\ref{SM_chiral_bulk_low_energy} (d)) of these states are not symmetric about 0. \n\n\nSecond, the Hamiltonian in Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) has eigenstates that are extended along $x$ and $z$ while confined along $y$. \nThe confinement along $y$ can be understood through the definition of the ladder operator in Eq.~(\\ref{eq:3D_chiral_bulk_U1_ladder}), which creates simple harmonic oscillator (SHO) states along $y$. \nProjecting the 3D eigenfunction to the 2D modulated system, only the probability distributions along $x$ and $y$ are physically meaningful. \nTherefore in 2D we expect to see flat bands along $k_{z}$, which is identified as $\\Delta \\phi  = \\phi - \\pi$ in the low energy model. The wave functions with these energies are confined along $y$ while extended along $x$. \nThis is consistent with Figs.~\\ref{SM_chiral_bulk_low_energy} (a) and (c)--(d). \nThe above discussion shows that our low energy bulk theory in 3D does qualitatively explain the existence of bulk confined modes in 2D and their flat dispersion along the $\\phi$-axis.\n\n\n\\begin{figure}[h]\n\\includegraphics[scale=0.4]{SM_chiral_bulk_low_energy_add_horizontal_line_at_pm_0_5_new.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the 2D modulated system that can be promoted to a 3D chiral HOTI\\cite{pozo2019quantization} coupled to a $U(1)$ gauge field with $\\vec{q} = (0,q_{y})$ and $q_{y} = 0.02$. \n(b) Bulk Landau levels of Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) for the low energy model of a 3D chiral HOTI with $m_{\\text{bulk}} = 0.5$, $B = 0.2$ and $\\vec{M} = (0.03,0.03,0.03)$. \nThe band edges are given by $E = -0.5248$ and $0.4661$. \nThe orange, black and green horizontal lines correspond to energies $E = +0.5$, $0$ and $-0.5$, respectively. \n(c) $\\&$ (d) Probability distribution of bulk-confined modes in the flat bands of (a) at $\\phi = \\pi$ with energies $-0.5144$ and $0.4656$ marked by green and orange respectively.\nWe note that (c) and (d) share similar probability distributions, for example both are confined along $y$ and centered around the middle of the finite sample. This is consistent with the low energy theory prediction of energy-asymmetry in Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) and (b). The darker (black) color in (c)--(d) implies higher probability density. \nIn (c) and (d), the $x$- and $y$-coordinate both range from $-15, \\ldots, +15$.}\n\\label{SM_chiral_bulk_low_energy}\n\\end{figure}\n\n\n\\section{\\label{bigsec:helical_sliding}Low energy theory of helical higher-order topological sliding modes}\n\nIn this section, we construct various low energy theories for the dimensionally-promoted 3D helical HOTI coupled to an $SU(2)$ gauge field. \nWe use these low energy models to explain corner, edge and bulk states in the 2D modulated system with helical sliding modes.\n\n\\subsection{2D corner modes $\\leftrightarrow$ 3D hinge modes with $SU(2)$ gauge field}\n\nIn our promoted 3D model, different spin ($\\vec{\\sigma}$) subspaces are decoupled, see Sec.~\\ref{sec:promoted_helical_lattice_model}. \nTherefore, all eigenstates are spin-polarized.\nThe 2D spin-polarized corner modes are the projection of the helical hinge modes in the 3D lattice. \nIf we denote the third, synthetic dimension as $y$ and the corresponding crystal momentum as $k_{y}$, the low energy theory of the helical hinge mode will be\n\\begin{align}\n    H_{\\text{hinge}}= v_{F} \\left( k_{y}\\sigma_{z}' + 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\sigma_{0}' \\right), \\label{3D_helical_hinge_mode_Hamiltonian}\n\\end{align}\nwhere we have minimally coupled the Hamiltonian $v_{F}k_{y}\\sigma_{z}'$ to a $SU(2)$ gauge field \n\\begin{align}\n    \\vec{A} =  (0,2\\pi (q_{x}x+q_{z}z)\\sigma_{z}',0). \\label{SU2_A}\n\\end{align}\nNotice that we denote our basis as $\\vec{\\sigma}'$. \nAlthough the eigenstates of $\\sigma'_z$ have opposite spins, they might contain non-trivial orbital and sub-lattice textures. \nAlso notice that we have $2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\sigma_{0}'$ in Eq.~(\\ref{3D_helical_hinge_mode_Hamiltonian}) instead of $2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\sigma_{z}'$. \nSince as we replace $k_{y}$ by $k_{y} + 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\sigma_{z}'$ through the minimal coupling, we will have \n\\begin{align}\n   v_{F}k_{y}\\sigma_{z}' = v_{F}\\begin{bmatrix}\n    k_{y} & 0 \\\\ \n    0 & -k_{y}\n    \\end{bmatrix} & \\to v_{F}\\begin{bmatrix}\n    \\left( k_{y} + 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\right) & 0 \\\\ \n    0 & -\\left( k_{y} - 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\right)\n    \\end{bmatrix} \\\\\n    & = v_{F}k_{y}\\sigma_{z}' + v_{F} \\cdot 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right)\\sigma_{0}'.\n\\end{align}\nIn Eq.~(\\ref{3D_helical_hinge_mode_Hamiltonian}), we have assumed that: \n(1) there is only one pair of helical hinge modes along this hinge,\n(2) the magnitude of the group velocity is $v_{F}$, \n(3) the electron has charge $-1$, and \n(4) the fixed position of the hinge along $y$ is at $(x_{\\text{hinge}},z_{\\text{hinge}})$ which is set by the coordinate system. \nEq.~(\\ref{SU2_A}) again implies that the modulation wave vector $\\vec{q} = (q_{x},q_{z})$ enters the definition of the $SU(2)$ gauge field. \nAs we can see in Eq.~(\\ref{3D_helical_hinge_mode_Hamiltonian}), if we tune $\\vec{q}$, we are effectively shifting the hinge mode dispersion along the $k_{y}$-axis for spin up [down] electrons by an amount $-2\\pi \\left(q_{x}x_{\\text{hinge}}+q_{z}z_{\\text{hinge}}\\right)$ [$+2\\pi \\left(q_{x}x_{\\text{hinge}}+q_{z}z_{\\text{hinge}}\\right)$]. \nSince we use $q_{x}=0$ and $q_{z}\\ne0$ in the main text for helical sliding modes, the following discussion will focus on this case. \nThe generalization to other combinations of $q_{x}$ and $q_{z}$ follows the same procedure.\n\n\nTo connect the low energy theory of the $\\phi$-sliding spectrum to the shifting of spin-polarized corner mode dispersion, we identified $k_{y}$ in Eq.~(\\ref{3D_helical_hinge_mode_Hamiltonian}) as $\\phi$, since the center of the $\\phi$-sliding spectrum is at $\\phi = 0$, as shown in Fig.~\\ref{SM_Fig_5} (a) for $q_{z} = 0$. \nUpon projection to 2D, the position $(x_{\\text{hinge}},z_{\\text{hinge}})$ of the hinge along $y$ again becomes the position $(x_{\\text{corner}},z_{\\text{corner}})$ of the 2D corner. \nWe thus obtain\n\\begin{align}\n    H_{\\text{corner}}= v_{F} \\left( \\phi \\sigma_{z}' + 2\\pi \\left( q_{x}x_{\\text{corner}} + q_{z}z_{\\text{corner}} \\right) \\sigma_{0}' \\right) \\label{eq:H_helical_hinge_1}\n\\end{align}\nfrom the main text. \nWe now examine this low energy theory through numerical simulations. \nWe will be focusing on how the doubly-degenerate gap-crossing bands corresponding to spin-polarized corner modes respond as we increase magnitude of $\\vec{q}$.\n\nWe show in Fig.~\\ref{SM_Fig_5} (b) the $\\phi$-sliding spectrum for $q_{z} = 0.02$. \nComparing Fig.~\\ref{SM_Fig_5} (b) with Fig.~\\ref{SM_Fig_5} (a), we see that as we turn on $q_{z}$, the two doubly-degenerate bands cross the gap with opposite slopes. \nTo explain this, we notice that each of the doubly-degenerate gap-crossing states corresponds to a localized pair of modes at inversion-related corners with opposite spins, as they are related by the $\\mathcal{I}\\mathcal{T}$-symmetry (see main text Sec.~V) and shown in Figs.~\\ref{SM_Fig_5} (c) and (d). \nIn our coordinate system, the corner modes are localized at positions $(x_{\\text{corner}},z_{\\text{corner}})$ = $\\pm(L\/2,L\/2)$, where $L=30$. \nThe corresponding 3D helical hinge mode dispersion relations are\n\\begin{align}\n    & H_{\\text{hinge 1}}= v_{F} \\left( k_{y}\\sigma_{z}' - \\pi q_{z}L \\sigma_{0}' \\right), \\\\\n    & H_{\\text{hinge 2}} = -v_{F} \\left( k_{y}\\sigma_{z}' + \\pi q_{z}L \\sigma_{0}' \\right),\n\\end{align}\nwhere inversion symmetry requires that the eigenstate of $H_{\\text{hinge 1}}$ and $H_{\\text{hinge 2}}$ with same spins have opposite group velocities. \nIdentifying $k_{y}$ as $\\phi$, we have that the Hamiltonians of the bands crossing the gap are given by\n\\begin{align}\n    & H_{\\text{corner 1}}= v_{F} \\left( \\phi \\sigma_{z}' - \\pi q_{z}L \\sigma_{0}' \\right), \\label{eq:corner_1_SU2}\\\\\n    & H_{\\text{corner 2}} = -v_{F} \\left( \\phi \\sigma_{z}' + \\pi q_{z}L \\sigma_{0}' \\right). \\label{eq:corner_2_SU2}\n\\end{align}\nWe then see that as we increase the $q_{z}$, the band in $H_{\\text{corner 1}}$ along $\\phi$ with slope $+v_{F}$ and spin $\\uparrow$ moves in the same direction as the band in $H_{\\text{corner 2}}$ with group velocity $+v_{F}$ and spin $\\downarrow$, which is consistent with Fig.~\\ref{SM_Fig_5} (b). \nIn fact, a detailed comparison between Figs.~\\ref{SM_Fig_5} (a) and (b) shows that the corner mode dispersion shifts along $\\phi$ by $\\pi q_{z}L \\approx 0.6\\pi $ for $q_{z} = 0.02$ and $L = 30$, implying that our low energy theory describes both the 2D corner modes and the corresponding 3D helical hinge modes. \nThis confirms that we can tune the range of $\\phi$ where the spin-polarized corner modes emerge from the bulk bands by modifying the periodicity of the modulation, as we have stated in the main text. \nSimilar to the chiral sliding modes in the main text, when  $\\pi q_{z}L > 2\\pi$, the gap-crossing bands will be folded back within the range $\\phi = [0,2\\pi)$. \nThis happens in Fig. 3 (a) of the main text, where $q_{z} = 0.11957$.\n\n\\begin{figure}[h]\n  \\includegraphics[scale=0.4]{SM_Fig_5_change_ticking_freq_add_color.pdf}\n  \\caption{(a) $\\phi$-sliding spectrum of Eq.~(35) in the main text with parameters $m_{1} = -3$, $m_{2} = 0.3$, $m_{3} = 0.2$, $m_{v_{1}} = -0.4$, $m_{v_{2}} = 0.2$, $v_{x}=v_{z}=u_{x}=u_{z} = 1$, $v_{y} = 2$, $v_{H} = 1.2$ and $\\vec{q} = (0,q_{z})$ where $q_{z} = 0$. \n  This reduces to the $y$-rod band structure of a 3D helical HOTI\\cite{Wieder_spin_decoupled_helical_HOTI} without $SU(2)$ gauge fields. \n  Notice the Rashba-like shifting of the surface band away from $\\phi = 0$, near $E \\approx \\pm 0.2$. \n  (b) $\\phi$-sliding spectrum with the same parameters as (a) but with $q_{z} = 0.02$. \n  We identify the flat dispersion in (b) marked by orange, green and red as $E^{-}_{\\sigma',k_{y},n = 1}$, $E^{-}_{\\sigma',k_{y},n =0}$ and $E^{+}_{\\sigma',k_{y},n = 1}$ in Eq.~(\\ref{wvfn_top}), respectively. \n  (c)$\\&$(d) Summation of probability distribution of the doubly-degenerate corners modes related to each other by the $\\mathcal{I}\\mathcal{T}$-symmetry at $\\phi = 0.6\\pi$ and $1.4\\pi$ with both energies equal to $-0.0124$. \n  The two corner modes in each of (c) and (d) are localized at inversion-related corners and have opposite spins. \n  The darker (black) color in (c)--(d) implies higher probability density. \n  In (c) and (d), the $x$- and $z$-coordinate both range from $-15,\\ldots,+15$.}\n  \\label{SM_Fig_5}\n\\end{figure}\n\n\\subsection{\\label{sec:surface_SU2}2D edge-confined modes $\\leftrightarrow$ 3D surface modes with $SU(2)$ gauge field}\n\nAs stated in the main text, we may understand the flat bands corresponding to edge-confined modes using a low energy theory in the promoted 3D lattice. \nIn this subsection we construct a surface theory minimally coupled to an $SU(2)$ gauge field to describe the projected 2D edge states in the modulated system.\n\nDue to the shifted band edge of the surface bands in the corresponding 3D helical HOTI (see Fig.~\\ref{SM_Fig_5} (a) and its caption), we may consider the surface theory by stacking a low energy Chern insulator with spin $\\uparrow$ and its time-reversal counterpart, with a relative shift in momentum space. \nThe low energy theory of a Chern insulator is given by $H_{\\text{Chern}}=\\vec{p}_{\\parallel}\\cdot \\vec{\\tau}_{\\parallel}' - m\\hat{n} \\cdot \\vec{\\tau}'$, where $\\vec{\\tau}'$ is the basis describing spin $\\uparrow$ electron with some orbital and sub-lattice textures, $\\parallel$ is the parallel component along the surface, $\\hat{n}$ is the surface normal vector and $m$ is the mass term. \nAs shown in Eq.~(\\ref{eq:B_SU2_1}), if $q_{x} = 0$, which is the case we consider in Fig.~\\ref{SM_Fig_6} below, we have $\\vec{B} = \\vec{\\nabla} \\cross \\vec{A} - i \\vec{A} \\cross \\vec{A}$ parallel to $\\hat{x}$. \nWe then consider a surface theory on the $yz$-plane whose surface normal is $\\hat{x}$. \nThe corresponding surface Hamiltonian without an $SU(2)$ gauge field reads $H_{\\text{surf}} = p_{y}\\tau_{y}'\\sigma_{0}'+p_{z}\\tau_{z}'\\sigma_{z}'+(\\Delta k_{y})\\tau_{y}'\\sigma_{z}'-m\\tau_{x}'\\sigma_{0}'$, where $\\vec{\\sigma}'$ again denotes the spin degrees of freedom, and $\\Delta k_{y}$ is a {\\it real constant} denoting the shift of low energy surface band minima from $\\vec{k}=\\Gamma$. \nTo facilitate the subsequent analysis, we perform a basis transformation through a $-2\\pi\/3$ radian rotation $U$ along the $[1,1,1]$ axis in the {\\it orbital} space $\\vec{\\tau}'$ such that $U^{\\dagger}(\\tau_{x}',\\tau_{y}',\\tau_{z}')U = (\\tau_{z}',\\tau_{x}',\\tau_{y}')$. \nThe transformed surface Hamiltonian then reads\n\\begin{align}\n    H_{\\text{surf}} = p_{y}\\tau_{x}'\\sigma_{0}'+p_{z}\\tau_{y}'\\sigma_{z}'+(\\Delta k_{y})\\tau_{x}'\\sigma_{z}'-m\\tau_{z}'\\sigma_{0}'.\n    \\label{eq:top_surf_SU2_helical_transformed}\n\\end{align}\nNotice that this surface theory is spin-decoupled. \nIn a general helical HOTI the spins are coupled. \nHowever, the general procedure will be the same: we first construct a low energy surface theory, and then couple it to the desired $SU(2)$ gauge field. \n\nWe now couple Eq.~(\\ref{eq:top_surf_SU2_helical_transformed}) to an $SU(2)$ gauge field $\\vec{A} = (0,Bz \\sigma_{z}',0)$ which produces a $SU(2)$ magnetic field $\\vec{B} = (-B\\sigma_{z}',0,0)$. \nTherefore opposite spins experience opposite magnetic fields. \nThe minimally-coupled surface theory is\n\\begin{align}\n    H_{\\text{surf}} = \\begin{bmatrix}\n    \\left(p_{y}+\\Delta k_{y} + Bz \\right)\\tau_{x}' + p_{z}\\tau_{y}' - m \\tau_{z}' &  0 \\\\ 0 & \\left(p_{y}-\\Delta k_{y} - Bz \\right)\\tau_{x}' - p_{z}\\tau_{y}' - m \\tau_{z}'\n    \\end{bmatrix}, \\label{eq:helical_surface_before_rewrite}\n\\end{align}\nwhere we have assumed that both $B$ and $m$ are positive. \nFourier transforming to replace $p_{y}$ by the wavenumber $k_{y}$, and defining spin- and $k_{y}$-dependent ladder operators\n\\begin{align}\n    a^{\\dagger}_{k_{y},\\uparrow} = \\frac{1}{\\sqrt{2B}}\\left(k_{y} + \\Delta k_{y} + Bz - ip_{z} \\right) \\text{ and } a^{\\dagger}_{k_{y},\\downarrow} = \\frac{1}{\\sqrt{2B}}\\left(k_{y} - \\Delta k_{y} - Bz + ip_{z} \\right), \\label{eq:top_ladders_SU2}\n\\end{align}\nwe can rewrite Eq.~(\\ref{eq:helical_surface_before_rewrite}) in each spin subspace ($\\sigma'=\\pm$) as\n\\begin{align}\n    H_{\\text{surf}}(k_{y},\\sigma') = \\begin{bmatrix}\n    -m & \\sqrt{2B} a_{k_{y},\\sigma'}^{\\dagger} \\\\\n    \\sqrt{2B} a_{k_{y},\\sigma'} & m\n    \\end{bmatrix}.\n\\end{align}\nWe can solve for the eigenstates and energies to find\n\\begin{align}\n    & \\psi^{-}_{\\sigma',k_{y},n=0} = e^{ik_{y}y}\\begin{bmatrix}\n    \\ket{0,k_{y},\\sigma'} \\\\ 0\n    \\end{bmatrix},\\   E^{-}_{\\sigma',k_{y},n=0} = -m, \\nonumber \\\\\n    & \\psi^{-}_{\\sigma',k_{y},n>0} = e^{ik_{y}y}\\begin{bmatrix}\n    \\ket{n,k_{y},\\sigma'} \\\\  \\alpha_{-}(n)\\ket{n-1,k_{y},\\sigma'}\n    \\end{bmatrix},\\   E^{-}_{\\sigma',k_{y},n>0} = -\\sqrt{m^{2} + 2Bn}, \\nonumber \\\\\n    & \\psi^{+}_{\\sigma',k_{y},n>0} = e^{ik_{y}y}\\begin{bmatrix}\n    \\ket{n,k_{y},\\sigma'} \\\\ \\alpha_{+}(n)\\ket{n-1,k_{y},\\sigma'}\n    \\end{bmatrix},\\   E^{+}_{\\sigma',k_{y},n>0} = +\\sqrt{m^{2} + 2Bn}, \\nonumber \\\\\n    & {\\color{black}{\\text{where }}} \\alpha_{\\pm}(n) = \\frac{1}{\\sqrt{2Bn}}\\left(\\pm \\sqrt{m^{2} + 2Bn} + m \\right). \\label{wvfn_top}\n\\end{align}\nHere $\\sigma' = \\pm$ is the spin quantum number, $k_{y}$ is the wavenumber along $y$, $n$ is an non-negative integer {\\color{black}{labelling}} the $SU(2)$ LLs, and $\\ket{n,k_{y},\\sigma'}$ is the $n^{\\text{th}}$ simple harmonic oscillator (SHO) eigenstate along $z$ defined using $a^{\\dagger}_{k_{y},\\sigma'}$. \nNotice that the coefficient $\\alpha_{\\pm}(n)$ does not depend on which spin sector we are considering. \nWe now connect the above surface theory to the flat bands and corresponding wavefunctions in the $\\phi$-sliding spectrum of the 2D modulated system by identifying $k_{y}$ as $\\phi$ and $B$ as $2\\pi q_{z}$. \nThis is because in our 3D promoted lattice Eq.~(\\ref{eq:lattice_model_helical_sliding}) and the numerical examples, we have $q_{x} = 0$ such that $\\vec{A} =  (0,2\\pi q_{z}z\\sigma_{z}',0)$ produces an $SU(2)$ magnetic field with strength $2\\pi q_{z}$. \nWe will mainly focus on the two [one] flat bands in Fig.~\\ref{SM_Fig_5} (b) with negative [positive] energy closest to the zero. \nThese correspond to $E^{-}_{\\sigma',k_{y},n \\le 1}$ [$E^{+}_{\\sigma',k_{y},n=1}$].\n\nFirst, notice that the spectrum in Eq.~(\\ref{wvfn_top}) breaks particle-hole symmetry in both spin sectors, as there are no $+m$ energy eigenvalues. \nThis can be observed in Fig.~\\ref{SM_Fig_5} (b) where there are no flat bands of edge-confined modes around $E\\approx +0.2$ corresponding to $E = +m$. \nWe thus identify the flat bands in Fig.~\\ref{SM_Fig_5} (b) marked by orange, green and red as $E^{-}_{\\sigma',k_{y},n = 1}$, $E^{-}_{\\sigma',k_{y},n =0}$ and $E^{+}_{\\sigma',k_{y},n = 1}$ in Eq.~(\\ref{wvfn_top}), respectively.\n\nNext, the probability densities $|\\psi^{-}_{\\sigma',k_{y},n=0}|^2$ and $|\\psi^{\\pm}_{\\sigma',k_{y},n=1}|^2$ are respectively proportional to $\\left|\\varphi_{0,B}(z+(\\Delta k_{y} + \\sigma' k_{y})\/B) \\right|^{2}$ and $\\left| \\alpha_{\\pm}(1) \\right|^{2}\\left|\\varphi_{0,B}(z+(\\Delta k_{y} + \\sigma' k_{y})\/B) \\right|^{2} + \\left|\\varphi_{1,B}(z+(\\Delta k_{y} + \\sigma' k_{y})\/B) \\right|^{2}$, where $\\varphi_{n,B}(z)$ is the $n^{\\text{th}}$ SHO eigenstate along $z$. \nNotice that we have put explicit $B$-dependence on $\\varphi_{n,B}(z)$ since the cyclotron frequency and the localization of wave functions depend on the field strength.\nThis implies that: \n(1) the probability density computed from $\\psi^{-}_{\\sigma',k_{y},n=0}$ has a pure Gaussian distribution along $z$, and \n(2) $\\psi^{-}_{\\sigma',k_{y},n=1}$ has a larger contribution from the SHO first excited state than $\\psi^{+}_{\\sigma',k_{y},n=1}$, since $\\left| \\alpha_{-}(1) \\right|^{2} < \\left| \\alpha_{+}(1) \\right|^{2}$ and we have assumed both $B$ and $m$ are positive. \nFigs.~\\ref{SM_Fig_6} (a)--(c) show the 2D probability distribution at $\\phi = 0$ for edge-confined modes in different LLs together with the insets showing the probability integrated over non-negative $x$-coordinates. \nWhile both Figs.~\\ref{SM_Fig_6} (a) and (c) corresponds to $n = 1$ LL, Fig.~\\ref{SM_Fig_6} (a) is from the negative energy branch and Fig.~\\ref{SM_Fig_6} (c) is from the positive energy branch. \nTherefore Fig.~\\ref{SM_Fig_6} (a) is more characteristic of the SHO first excited state than Fig.~\\ref{SM_Fig_6} (c). \nIn contrast, Fig.~\\ref{SM_Fig_6} (b), being the $n=0$ LL wave function, shows a Gaussian probability distribution characteristic of the SHO ground state. \nWe notice that, if we restrict ourselves to a single edge, the wave function in Fig.~\\ref{SM_Fig_6} (a) contains a slightly asymmetric SHO first excited state. \nThis is due to the complicated on-site and hopping energies in our model, which break extraneous symmetries, such that the wave function in Fig.~\\ref{SM_Fig_6} (a) has nonzero penetration to bulk. \nThis can be compared with Fig.~\\ref{SM_Fig_6} (b) where the edge-confined modes are much more localized on the edges. \nThus, the surface theory present above should be recognized as an effective theory we use to extract out qualitative properties of wave functions, and it suffices to identify the SHO first excited state character of Fig.~\\ref{SM_Fig_6} (a) corresponding to $\\psi^{-}_{\\sigma',k_{y},n=1}$ with energy level $E^{-}_{\\sigma',k_{y},n = 1}$ in Eq.~(\\ref{wvfn_top}).\n\nThird, the definition of the ladder operator in Eq.~(\\ref{eq:top_ladders_SU2}) predicts that even when we have $\\phi = 0$ (which corresponds to $k_{y} = 0$) the center of the SHO eigenstate will be shifted from the center of the coordinate system by a distance $|\\Delta k_{y}\/ B|$ along $z$. \nRecall that $\\Delta k_{y}$ is the slight Rashba-like shift of the surface bands away from $\\Gamma$ (see Fig.~\\ref{SM_Fig_5} (a) and its caption). \nIn the 2D modulated system, this is equivalent to saying that the edge-confined modes at $\\phi = 0$ will not have wavefunctions centered at the middle of the edge along $z$. \nThe probability distribution at $\\phi = 0$ shown in Fig.~\\ref{SM_Fig_6} (a) to (c) confirms this, as no states are centered around the middle of the edges.\nGoing further, we also expect from Eq.~(\\ref{eq:top_ladders_SU2}) that the center of wave functions will be shifted along $z$ by $(-\\Delta k_{y} - \\sigma' k_{y})\/B$ for a given $k_{y}$. \nIdentifying $k_{y}$ as $\\phi$ and $B$ as $2\\pi q_{z}$, we deduce that the shifting of the edge-confined modes from the center of the edges is spin-dependent. \nThe center of each probability distribution is given by $l_{\\sigma'} = (-\\Delta k_{y} - \\sigma'  \\phi)\/(2\\pi q_{z}) $. \nNotice that the edge-confined modes in Figs.~\\ref{SM_Fig_6} (d) and (e) are shifted by $\\approx \\pm 2.5$ lattice constants comparing with Fig.~\\ref{SM_Fig_6} (b). \nThis is because Figs.~\\ref{SM_Fig_6} (d) and (e) correspond to $\\Delta \\phi = \\phi - 0 = 0.1\\pi$, $q_{z} = 0.02$, and wave functions with opposite spins will be shifted in the opposite directions, according to the expression of $l_{\\sigma'}$ given above and $a^{\\dagger}_{k_{y},\\sigma'}$ in Eq.~(\\ref{eq:top_ladders_SU2}). \nTo be more precise, we estimate the center of the wave functions in Figs.~\\ref{SM_Fig_6} (b), (d) and (e) to be around $z \\approx  1.5$, $4$ and $-1$, respectively.\n\nFinally, similar to the chiral sliding modes in the 2D modulated system, we also have an additional degeneracies in the flat band states due to zone-folding (see Sec.~IV of the main text). \nUp to the degeneracy due to zone folding, the universal property we expect in these 2D helical sliding systems is that as we vary $\\phi$, which corresponds to sliding of density waves (DWs) with spin-orbit coupled interactions, there will be flat bands together with edge-confined modes that are projected from the surface of a helical HOTI with $SU(2)$ Landau quantization. \nWe have seen that the qualitative properties of the wave functions are all consistent with the low energy surface theory.\n\n\\begin{figure}[h]\n\\includegraphics[scale=0.4]{SM_helical_sliding_edge_modes.pdf}\n\\caption{(a)--(c) Average of the probability distribution for the four-fold degenerate edge-confined modes in the flat bands shown in Fig.~\\ref{SM_Fig_5} (b). \nThe four-fold degeneracy comes considering the two edges and two spins. \n(a)--(c) are edge-confined modes at $\\phi = 0$ with $E = -0.4227$, $-0.1711$ and $0.3675$ which are marked orange, green and red in Fig.~\\ref{SM_Fig_5} (b). \nThe corresponding energy levels are $E^{-}_{\\sigma^\\prime,k_{y}=0,n=1}$, $E^{-}_{\\sigma^\\prime,k_{y}=0,n=0}$ and $E^{+}_{\\sigma^\\prime,k_{y}=0,n=1}$in Eq.~(\\ref{wvfn_top}). \n(d)$\\&$(e) Edge modes at $\\phi = 0.1 \\pi$ with $E = -0.1711$ confined at the right edge.\nThey correspond to $E^{-}_{\\sigma^\\prime,k_{y}=0.1\\pi,n=0}$ with $\\sigma^\\prime = \\pm$. \nWe notice that in addition to (d) and (e) there are two other edge-confined modes with nearly identical energies localized on the left edge.\nThe darker (black) color in (a)--(e) implies higher probability density. \nThe inset in (a)--(e) is the probability distribution integrated over non-negative $x$-coordinates, from $x= 0,\\ldots, L\/2$ where $L=30$. \nIn (a)--(e), the $x$- and $z$-coordinate both range from $-15,\\ldots, +15$.}\n\\label{SM_Fig_6}\n\\end{figure}\n\n\n\\subsection{2D bulk-confined modes $\\leftrightarrow$ 3D bulk modes with $SU(2)$ gauge field}\n\nHaving accounted for the edge-confined modes in the 2D modulated system with helical sliding, we move on to consider the flat bands corresponding to bulk-confined modes. \nSimilar to Sec.~\\ref{sec:surface_SU2}, we will construct a low energy theory and couple it to an $SU(2)$ gauge field.\n\nWe consider the Bloch Hamiltonian of the promoted 3D helical HOTI around the $\\Gamma$ point (see Sec.~\\ref{sec:promoted_helical_lattice_model}), which is\n\\begin{align}\n    H_{\\text{bulk}} = & m_{\\text{bulk}} \\tau_{z}\\mu_{0}\\sigma_{0} + m_{2} \\tau_{z}\\mu_{x}\\sigma_{0} + m_{3} \\tau_{z}\\mu_{z}\\sigma_{0} + m_{v_{1}}\\tau_{0}\\mu_{z}\\sigma_{0} + m_{v_{2}} \\tau_{0}\\mu_{x}\\sigma_{0} \\nonumber \\\\\n    & + u_{x} p_{x} \\tau_{y}\\mu_{y}\\sigma_{0} + u_{z}p_{z}\\tau_{x}\\mu_{0}\\sigma_{0} + v_{H}p_{y} \\tau_{y}\\mu_{z}\\sigma_{z}. \\label{eq:3D_helical_HOTI_bulk}\n\\end{align}\nWe have redefined several parameters in the original model in Sec.~\\ref{sec:promoted_helical_lattice_model} for convenience. \nFor example, $m_{\\text{bulk}}$ is effectively $m_{1} +v_{x} + v_{y} + v_{z}$ from Eq.~(A1) \nin Ref.~\\onlinecite{Wieder_spin_decoupled_helical_HOTI}. \nWe now couple this $H_{\\text{bulk}}$ to the $SU(2)$ vector potential $\\vec{A} = Bz \\tau_{0}\\mu_{0}\\sigma_{z} \\hat{y}$, which is equivalent to Eq.~(\\ref{SU2_A}) with $q_{x} = 0$. \nTherefore, the term $v_{H}p_{y} \\tau_{y}\\mu_{z}\\sigma_{z}$ becomes\n\\begin{align}\n    v_{H}p_{y} \\tau_{y}\\mu_{z}\\sigma_{z} \\to v_{H}\\tau_{y}\\mu_{z} \\begin{bmatrix}\n    p_{y} + Bz & 0 \\\\ 0 & -(p_{y}-Bz)\n    \\end{bmatrix},\n\\end{align}\nwhere the matrix acts in spin space. \nFourier transforming along $x$ and $y$ and defining the $k_{y}$- and spin($\\sigma = \\pm$)-dependent ladder operators\n\\begin{align}\n    a^{\\dagger}_{\\sigma,k_{y}} = \\frac{1}{\\sqrt{2B}}\\left( \\sigma k_{y} + Bz - ip_{z} \\right), \\label{eq:ladder_bulk_helical_SU2}\n\\end{align}\nwe can rewrite Eq.~(\\ref{eq:3D_helical_HOTI_bulk}) coupled to $\\vec{A} = Bz \\tau_{0}\\mu_{0}\\sigma_{z} \\hat{y}$ in different spin sectors as\n\\begin{align}\n    H_{\\sigma,k_{x},k_{y}} &= m_{\\text{bulk}} \\tau_{z}\\mu_{0} + m_{2} \\tau_{z}\\mu_{x} + m_{3}\\tau_{z}\\mu_{z} + + m_{v_{1}} \\tau_{0}\\mu_{z} + m_{v_{2}}\\tau_{0}\\mu_{x} \\nonumber \\\\\n    & + u_{x}k_{x}\\tau_{y}\\mu_{y} + v_{H}\\sqrt{\\frac{B}{2}} \\left( a_{\\sigma,k_{y}} + a^{\\dagger}_{\\sigma,k_{y}} \\right)\\tau_{y}\\mu_{z} -i u_{z} \\sqrt{\\frac{B}{2}}\\left( a_{\\sigma,k_{y}} - a^{\\dagger}_{\\sigma,k_{y}} \\right)\\tau_{x}\\mu_{0}. \\label{eq:3D_bulk_SU2_LL_Hamiltonian}\n\\end{align}\nIn Fig.~\\ref{SM_Fig_7} (a) we show the numerically computed spectrum $E_{\\sigma,k_{x},k_{y}}$ as a function of $k_{x}$. \nNote that $E_{\\sigma,k_{x},k_{y}}$ does not depend on $\\sigma$ and $k_{y}$, giving rise to flat Landau levels as a function of $k_{y}$. \nWe relate this low energy bulk theory to the $\\phi$-sliding spectrum of the 2D modulated system by identifying $k_{y}$ in Eq.~(\\ref{eq:3D_bulk_SU2_LL_Hamiltonian}) with $\\phi$. \nWe now examine the consequences of this correspondence.\n\n\nFirst, examining the band edges for the valence and conduction bands in Fig.~\\ref{SM_Fig_7} (a) we notice that there is no particle-hole symmetry in the low energy spectrum of helical HOTI with $SU(2)$ Landau quantization.\nThis is also reflected in the energy eigenvalues and eigenstate probability distribution for the flat bands in Fig.~\\ref{SM_Fig_5} (b) corresponding to bulk-confined modes shown in Figs.~\\ref{SM_Fig_7} (b) and (c), which are in the bulk flat continuum directly below and above the edge flat dispersion marked in orange and red in Fig.~\\ref{SM_Fig_5} (b). \nThe figure caption of Fig.~\\ref{SM_Fig_7} gives the asymmetric energies of the two bulk states in Figs.~\\ref{SM_Fig_7} (b) and (c).\n\nSecond, we note that the Hamiltonian in Eq.~(\\ref{eq:3D_bulk_SU2_LL_Hamiltonian}) has eigenstates that are extended along $x$ and $y$ while confined along $z$. \nThe confinement along $z$ can also be understood through the definition of the spin-dependent ladder operators in Eq.~(\\ref{eq:ladder_bulk_helical_SU2}), which create SHO states along $z$. \nProjecting this 3D wave function to the 2D modulated system, only the probability distribution along $x$ and $z$ is preserved. \nTherefore in 2D we expect to see flat bands along $k_{y}$--which is identified as $\\phi$--and with corresponding wave functions confined along $z$ while extended along $x$. \nThis is consistent with Figs.~\\ref{SM_Fig_7} (b) and (c). \nThe above discussion shows that our low energy bulk theory in 3D qualitatively explains the existence of bulk confined modes in 2D. \nThe universal property of these types of 2D helical sliding systems, even with larger $|\\vec{q}|$ presented in the main text, is that there will be flat bands as we vary $\\phi$ with bulk-confined modes that are projected from the bulk of helical HOTI with $SU(2)$ Landau quantization.\n\n\n\\begin{figure}[h]\n  \\includegraphics[scale=0.4]{SM_Fig_7_add_horizontal_pm_0_6_new.pdf}\n  \\caption{(a) Bulk $SU(2)$ Landau levels of Eq.~(\\ref{eq:3D_bulk_SU2_LL_Hamiltonian}) for the low energy model of a 3D helical HOTI\\cite{Wieder_spin_decoupled_helical_HOTI} with $m_{\\text{bulk}} = 1$, $m_{2} = 0.3$, $m_{3} = 0.2$, $m_{v_{1}} = -0.4$, $m_{v_{2}} = 0.2$, $u_{x} = 1$, $v_{H}=1.2$, $u_{z} = 1$ and $B = 0.2$. \n  The band edges occur at $E = -0.6618$ and $0.601$. \n  The green, black and orange horizontal lines correspond to energy $+0.6$, $0$ and $-0.6$, respectively. \n  Notice that there is in general no particle-hole symmetry in the bulk $SU(2)$ LL spectrum of a 3D helical HOTI. \n  (b)$\\&$(c) Probability distribution of bulk-confined modes in the flat band of Fig.~\\ref{SM_Fig_5} (b) at $\\phi = 0$ with energy $-0.5393$ and $0.4835$ respectively with similar wave function confinement. \n  These two modes are at the flat bulk continuum right below and above those flat bands  for {\\it edge-confined} modes corresponding to $E^{-}_{\\sigma',k_{y},n=1}$ and $E^{+}_{\\sigma',k_{y},n=1}$ in Eq.~(\\ref{wvfn_top}), which we have marked in orange and red in Fig.~\\ref{SM_Fig_5} (b).\n  We note that both (b) and (c) show modes confined along $z$, which is consistent with the ladder operators defined in Eq.~(\\ref{eq:ladder_bulk_helical_SU2}). \n  However, the absolute value of energies for (b) and (c) are not same, indicating the spectrum is not particle-hole symmetric, and the minimal continuum model calculation in (a) captures this trend. \n  The darker (black) color in (b)--(c) implies higher probability density. \n  In (b) and (c), the $x$- and $z$-coordinate both range from $-15,\\ldots, +15$.}\n  \\label{SM_Fig_7}\n\\end{figure}\n\n\n\n\\section{\\label{sec:numerical_method}Numerical methods to identify non-trivial Chern insulating layers in 3D Weyl-CDW systems}\n\n\nIn this section, we describe the numerical methods we used to identify layers (hybrid Wannier functions) and corresponding Bloch states in our 3D Weyl-CDW model. \nIn the following, we use the term $xy$-slab to denote a sample of 3D Weyl-CDW system infinite along the $x$ and $y$ directions and finite along $z$ with size $L_{z}$. \nIn addition, we use $y$-rod to denote a sample of 3D Weyl-CDW system infinite along $y$ and finite along $x$ and $z$ with size $L_{x} \\times L_{z}$. \nIn this section, we will present the numerical methods based on Berry phase, Berry curvature, and hybrid Wannier functions to identify the non-trivial bands in the $xy$-slab for $q = 1\/5$, $L_{z} = 25 $, $t_{x} = -t_{y} = t_{z} = 1$, $m=2$, $2|\\Delta|=0.75$ with $\\phi = 0$ and $\\phi = \\pi$ in the main text. \nThe numerical methods can also be applied to systems such as incommensurate CDWs with $q = \\tau \/4$ where $\\tau = (1+\\sqrt{5})\/2$ is the golden ratio (see main text).\n\nAs mentioned in the main text, we can understand the existence of quantum anomalous Hall (QAH) surface states by viewing the $xy$-slab as composed of a layered stack of Chern insulators (CIs). \nWhen we cut the $xy$-slab into the corresponding $y$-rod, those states in the $xy$-slab with non-zero Chern number will each contribute a number of chiral edge modes, depending on the Chern number of the bands. \nThese chiral edge modes collectively form the QAH surface states. The number of chiral edge modes in the $y$-rod on each edge is equal to the Chern number of occupied bands in the corresponding $xy$-slab. Specifically, the low energy theory in 4D predicts that topologically non-trivial bands in the $xy$-slab have a layered structure in position space, in which non-trivial bands will be localized along $z$ and separated by some amount of lattice constants\\cite{NicoDavidAXI2,dynamical_axion_insulator_BB}. \nIn the following, we will use a combination of Berry phase, Berry curvature and hybrid Wannier function calculations to identify those non-trivial bands in the $xy$-slab. \nThe goal is to find non-trivial bands whose probability distributions along $z$ match the probability distribution of chiral edge modes (or QAH surface states) in the $y$-rod {\\color{black}{shown in Figs.~4 and 5 in the main text}}.\n\nThe first step of our analysis is to compute the hybrid Wannier bands for the occupied states in the $xy$-slab. \nThis involves diagonalizing the position operator $z$ projected into the space of occupied bands at each $\\vec{k}$-point of the 2D Brillouin zone (BZ)\\cite{PythTB,Marzari2012}. \nNotice that, as we have finite size along $z$, the position operator $z$ is well-defined. \nWe thus obtain hybrid Wannier bands, which are the eigenstates of the projected $z$ operator as a function of $k_x$ and $k_y$. \nIn our calculation we find that for $q = 1\/5$, $t_{x}=-t_{y}=t_{z}=1$, $m = 2$, $2|\\Delta|=0.75$, $\\phi = 0$ and $L_{z} = 25$ the hybrid Wannier bands are non-degenerate.\nThis allows us to compute the Berry phase integrated along $k_{y}$ as a function of $k_{x}$ for each hybrid Wannier band, as shown in the first column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}. \nWe denote such Berry phases as $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$, where $m$ indexes the hybrid Wannier band, $\\vec{G}_{2}$ indicates that we are integrating along $k_{y}$, and $k_{x}$ denotes the functional dependence on $k_{x}$. \nDue to the non-degeneracy of the hybrid Wannier bands, the Berry phases $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ coincide with the eigenvalues of the $\\hat{y}$-directed non-Abelian slab Berry phase (Wilson loop) $\\text{arg}(\\mathcal{W})$\\cite{wieder2018axion,Wilson_2,Yu11,Alexandradinata16}. \nFrom the negative winding of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ shown in the first column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}, we find that each of the $m = 2$, $7$, $12$, $17$ and $22$ hybrid Wannier bands (the first hybrid Wannier band is labelled by $m=0$) carries Chern number $C = -1$. \nImportantly, at $\\Gamma$ point, the hybrid Wannier functions for these 5 hybrid Wannier bands are localized at $z_\\Gamma \\approx -10$, $-5$, $0$, $5$ and $10$, see the second column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0} and the second column of Table~\\ref{tab:hwfc_table}. \nThis is consistent with our deduction in the main text that the CIs will be separated by $1\/q$ along $z$, which is $5$ lattice constants in this case. \nThis also supports our identification that this system can be viewed as layered stack of CIs.\nThe discrepancy between $z_\\Gamma$ and $\\langle z\\rangle$ (reported in the third column in Table~\\ref{tab:hwfc_table}), the average of the hybrid Wannier center over all momenta, can be attributed to the fact that our present model has nonzero coupling along the $z$-direction, and hence nonzero coupling between CI layers. \n\nFor those hybrid Wannier bands with non-zero winding of the Berry phase, we compute the decomposition of hybrid Wannier functions in terms of Bloch states at $\\vec{k} = \\Gamma$: \n\\begin{align}\n\t\\psi^{\\text{hybrid}}_{m,\\Gamma} = \\sum_{n=1}^{N_{\\text{occ}}} c_{m,\\Gamma}^{n}  \\psi^{\\text{Bloch}}_{n,\\Gamma}, \\label{eq:wvfn_decomposition}\n\\end{align}\nwhere $N_{\\text{occ}}$ is the number of occupied bands. \nThe values of $n$ for those non-zero $ c_{m,\\Gamma}^{n} $ then directly tell us the index of the topologically non-trivial Bloch bands (not hybrid Wannier bands) at $\\vec{k} = \\Gamma$. \nHereafter, we will use $m$ to denote hybrid Wannier band indices and $n$ to denote Bloch band indices.\nIn our specific examples, the decomposition of $\\psi^{\\text{hybrid}}_{m,\\Gamma}$ in the Bloch basis where $m$ is the index of non-trivial hybrid Wannier bands are given in the third column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}. \nWe can then identify that the Bloch band with index $n = 18,\\ldots, 22$ at $\\vec{k} = \\Gamma$ are non-trivial. \nImportantly, we notice that for the hybrid Wannier bands at $\\Gamma$ with $m \\ne 2$, $7$, $12$, $17$ and $22$, the decomposition coefficient $c^{n}_{m,\\Gamma}$ are zero for $n = 18,\\ldots, 22$, showing that the hybrid Wannier bands at $\\Gamma$ with $m = 2$, $7$, $12$, $17$ and $22$ are truly spanned only by Bloch bands with $n = 18,\\ldots, 22$. \nThese states contribute to the chiral QAH surface states. \nAlthough the band structure of $xy$-slab is complicated with various band entanglements, we can track the topologically non-trivial bands by starting from the non-trivial bands at $\\vec{k} = \\Gamma$. \nWhenever we encounter band crossing, we choose to proceed along the direction where there is no discontinuity of the bands. \nFor example, we can identify the non-trivial valence bands at $\\vec{k} = \\Gamma$ and $\\vec{k} = (0,\\pi) = Y $, as shown in Fig.~\\ref{slab_band_and_wvfn_Q_0.4pi_Lz_25_D_0.75_phi_0_and_pi} marked by orange and green respectively. \nWe also plot the  probability distribution of these non-trivial states in Fig.~\\ref{slab_band_and_wvfn_Q_0.4pi_Lz_25_D_0.75_phi_0_and_pi}. \nThe probability distribution along $z$ for these non-trivial states are exactly the same as the QAH zero modes for the corresponding $y$-rod, which we have shown in the main text, and the hybrid Wannier function at $\\Gamma$ for $m = 2$, $7$, $12$, $17$ and $22$, shown in the middle column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}.\nFurthermore, we have computed the Berry curvature of each Bloch band (not hybrid Wannier band) around the $\\Gamma$ point of the $xy$-slab in Fig.~\\ref{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_0}. \nThose non-trivial bands with band index $n = 18,\\ldots, 22$ show negative values of Berry curvature around $\\Gamma$, which is distinct from the other Bloch bands having positive values of Berry curvature around $\\Gamma$.\nThis negative Berry curvature contributes to the total Chern number $C = -5$ of occupied bands. \nThis is consistent with $G_{xy}(\\phi = 0) = -5 e^{2}\/h$, as we stated in the main text.\n\nThe above process can be also applied to the case with $\\phi = \\pi$. \nWe also identify the non-trivial bands for the $xy$-slab, and plot their probability distribution at $\\vec{k} = \\Gamma$ and $\\vec{k} = Y$ in Fig.~\\ref{slab_band_and_wvfn_Q_0.4pi_Lz_25_D_0.75_phi_0_and_pi}. \nAgain, the probability distribution along $z$ for these non-trivial states are exactly the same as the QAH zero modes for the corresponding $y$-rod, which we have shown in the main text, and the hybrid Wannier functions at $\\Gamma$ for $m = 4$, $9$, $15$ and $20$, shown in the second column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi}. \nThe similar analysis of Berry phase for hybrid Wannier bands, wave function decomposition of Eq.~(\\ref{eq:wvfn_decomposition}) and Berry curvature around $\\vec{k} = \\Gamma$ are also shown in Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi} and Fig.~\\ref{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_pi}. \nImportantly, we can see that in the second column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi} and the fifth column of Table~\\ref{tab:hwfc_table}, the hybrid Wannier functions at $\\Gamma$ corresponding to the non-trivial bands are localized at $z_\\Gamma \\approx -7.5$, $-2.5$, $2.5$ and $7.5$. \nAs above, we attribute the discrepancy between $z_\\Gamma$ and $\\langle z \\rangle$ (the average of the hybrid Wannier center over the Brillouin zone, reported in column six of Table~\\ref{tab:hwfc_table}) to the nonvanishing interlayer coupling in our model. \nThis is again consistent with our stacked layer identification. \nAs mentioned in the main text, as we vary $\\phi$ by $\\pi$, each of the CIs will be shifted by $-\\Delta \\phi \/ (2\\pi q) = -\\pi\/(2\\pi q) = -1\/(2q)$, which in our current example is equal to $-2.5$ lattice constants. \nComparing this with the second column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}, which shows that the non-trivial CIs are at $z \\approx -10$, $-5$, $0$, $5$ and $10$, we can see that the stacked layers in Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi} can be identified as those in Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0} with a shift of $-2.5$ lattice constant along $z$. \nWe notice that in this case, where we have used $q = 1\/5$, $t_{x}=-t_{y}=t_{z}=1$, $m = 2$, $2|\\Delta|=0.75$, $\\phi = \\pi$ and $L_{z} = 25$, the hybrid Wannier bands are nearly degenerate at $\\Gamma$ for $m = 9$, $10$ and $m = 14$, $15$, which result in a dramatic (though still continuous) change of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ for $m = 9$ and $15$ around $k_{x} = 0$, as shown in the first column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi}. \nNevertheless, we are able to separate the hybrid Wannier bands.\n\n\nWe conclude this section by making several remarks. \nFirst, since the above process does not involve computing the band structure in a 3D BZ, there is no problem in generalizing the above method to 3D Weyl-CDW systems with incommensurate density waves with modulation wave vector along $z$. \nWe have carried out the same numerical analysis for $q = \\tau \/ 4$ (where $\\tau = (1+\\sqrt{5})\/2$ is the golden ratio).\nThe results are shown in the main text in which we can identify non-trivial states corresponding to CI layers and these states have exactly the same probability distribution along $z$ as the QAH zero modes in the $y$-rod at $k_{y} = 0$. \nWe shall in here briefly summarize some numerical results for $q = \\tau \/ 4$, $t_{x} = -t_{y} = t_{z} = 1$, $m=2$, $2|\\Delta|=2$: \n(1) The $xy$-slab Hall conductances with $L_{z} = 21$ are $G_{xy}(\\phi = 0)=-9 e^{2}\/h$ and $G_{xy}(\\phi = \\pi)=-8 e^{2}\/h$\\cite{dynamical_axion_insulator_BB}. \n(2) There are 9 and 8 non-trivial hybrid Wannier bands in the $xy$-slab for $\\phi = 0$ and $\\pi$, respectively. \nEach of these non-trivial hybrid Wannier bands carries Chern number $C = -1$. \n(3) Similar to Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0} and Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi}, we have identified 9 and 8 non-trivial Bloch bands of the $xy$-slab using non-zero values of $c^{n}_{m,\\Gamma}$ in Eq.~(\\ref{eq:wvfn_decomposition}). \nThe Berry curvature of these 9 and 8 non-trivial Bloch bands around $\\Gamma$ are negative, which is distinct from other Bloch bands having positive values of Berry curvature around $\\Gamma$, and contribute to $G_{xy}(\\phi = 0)=-9 e^{2}\/h$ and $G_{xy}(\\phi = \\pi)=-8 e^{2}\/h$. \nThis is similar to the cases of $q = 1\/5$ in Fig.~\\ref{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_0} and Fig.~\\ref{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_pi}. \nSecondly, the above procedure to filter out non-trivial CI layers might break down when we have strong interlayer coupling, causing complicated hybridization between layer states. \nWe emphasize again that the hybrid Wannier center $\\left\\langle z \\right\\rangle$ averaged over the 2D Brillouin zone for a fixed non-trivial hybrid Wannier band can differ slightly from the hybrid Wannier center $z_{\\Gamma}$ at $\\vec{k} = (k_{x},k_{y})=(0,0)=\\Gamma$ where our low energy theory works. \nFor example, as shown in Table~\\ref{tab:hwfc_table}, for $\\phi = 0$, the hybrid Wannier band with index $m = 2$ has $z_{\\Gamma} = -9.9245$ while $\\left\\langle z\\right\\rangle = -9.9997 \\pm 0.0026$. \nThis can be viewed as the consequence of hybridization between layers. \nWe expect that if the layers are completely decoupled from each other, there will be no difference between $z_{\\Gamma}$ and $\\left\\langle z \\right\\rangle$. \nNotice that for $\\phi = 0$ the inversion center of the $xy$-slab is at $z = 0$, which pins both $z_{\\Gamma}$ and $\\left\\langle z \\right\\rangle$ to $z = 0$ for the $m=12$ hybrid Wannier band. \nThis is consistent with Refs.~\\onlinecite{song2017,MTQC}, where it was emphasized that the distinction between QAH and oQAH states is due to the presence or absence of a non-trivial CI layer exactly at the inversion center $\\left\\langle z \\right\\rangle = 0$, respectively. \nWe have also verified this for the incommensurate case of $q=\\tau \/4$ where $\\tau = (1+\\sqrt{5})\/2$, which indicates that the difference of a non-trivial CI layer at the inversion center $\\left\\langle z \\right\\rangle=0$ between QAH and oQAH states holds for generic CDW wave vectors. \nOur method can be viewed as a way to map a system with interlayer coupling to another system with decoupled CI layers.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c||c|c|c|}\n\\hline\n$m$ for $\\phi = 0$ & $z_{\\Gamma}$ & $\\left\\langle z\\right\\rangle$ & $m$ for $\\phi = \\pi$ & $z_{\\Gamma}$ & $\\left\\langle z\\right\\rangle$ \\\\\n\\hline\n2 & $-9.9245$ & $-9.9997 \\pm 0.0026$ & 4 & $-7.5467$ & $-7.9963 \\pm 0.0167$\\\\\n\\hline\n7 & $-4.9972$ & $-5.0 \\pm 0.0001$ & 9 & $-2.5023$ & $-2.9963 \\pm 0.0176$ \\\\\n\\hline\n12 & $0.0$ & $0.0 \\pm 0.0$ & 15 & $2.5023$ & $2.9963 \\pm 0.0176$ \\\\\n\\hline\n17 & $4.9972$ & $5.0 \\pm 0.0001$ & 20 & $7.5467$ & $7.9963 \\pm 0.0167$ \\\\\n\\hline\n22 & $9.9245$ & $9.9997 \\pm 0.0026$ &  &  &  \\\\\n\\hline\n\\end{tabular}\n\\caption{Hybrid Wannier centers along $z$ for different hybrid Wannier bands with non-zero winding of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ and $q = 1\/5$.\n$m$ denotes the non-trivial hybrid Wannier band index. \nThe first and last three columns summarize the results for $\\phi = 0$ and $\\phi = \\pi$, respectively. \nThere are $5$ and $4$ non-trivial hybrid Wannier bands for $\\phi = 0$ and $\\pi$ with $m = 2$, $7$, $12$, $17$, $22$ and $m = 4$, $9$, $15$, $20$, respectively.\n$z_{\\Gamma}$ is the hybrid Wannier center along $z$ for the hybrid Wannier functions at $\\vec{k} = (k_{x},k_{y}) = (0,0) = \\Gamma$.\n$\\left\\langle z \\right\\rangle$ is the hybrid center along $z$ averaged over the 2D Brillouin zone with grid size $100 \\times 100$. \nWe also report the standard deviation of $\\left\\langle z \\right\\rangle$ in the same column.}\n\\label{tab:hwfc_table}\n\\end{table}\n\n\n\\begin{figure}[ht]\n  \\includegraphics[width=\\columnwidth]{grid_100_hwf_winding_decomposition_orbitals_bloch_Q_0_4pi_Lz_25_D_0_75_phi_0.pdf}\n  \\caption{Berry phase winding of non-trivial hybrid Wannier bands (first column), hybrid Wannier functions at $\\vec{k} = \\Gamma$ decomposed in terms of orbital (second column), and Bloch bases (third column) for an $xy$-slab of the Weyl-CDW model with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$ and $\\phi = 0$. \nThe $z$-coordinate ranges from $-12, \\ldots, +12$. \nThere are 5 non-trivial hybrid Wannier bands for the occupied states, each of them displaying $-2\\pi$ winding of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ as $k_{x}$ goes from $0$ to $2\\pi$, such that the Chern number for each of the CI layers is $C = -1$. \nFrom the second column, we can deduce that the non-trivial states are localized at $z \\approx -10$, $-5$, $0$, $+5$, $+10$. \nFrom the third column, where we plot $|c^{n}_{m,\\Gamma}|^{2}$, we can deduce that the non-trivial Bloch band indices at $\\vec{k} = \\Gamma$ are $ n =18,\\ldots, 22$ (the first band is labelled by $n=0$).}\n  \\label{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}\n\\end{figure}\n\n\n\n\n\\begin{figure}[ht]\n  \\includegraphics[scale=0.4]{Berry_curvature_around_Gamma_Q_0_4pi_Lz_25_D_0_75_phi_0.pdf}\n  \\caption{The Berry curvature around $\\vec{k} = \\Gamma$ for the 25 valance Bloch bands (not hybrid Wannier bands) of the Weyl-CDW model for an $xy$-slab with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$ and $\\phi = 0$.\n   All the figures have $k_{x}$ and $k_{y}$ in the range range $2\\pi [-0.003,0.003]$.\n    We notice that the Bloch bands with indices $n = 18,\\ldots, 22$ have negative Berry curvature around $\\vec{k} = \\Gamma$ contributing to the total Chern number $C = -5$ of this $xy$-slab, leading to $G_{xy} = -5 e^{2}\/h$.}\n  \\label{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_0}\n\\end{figure}\n\n\\begin{figure}[ht]\n  \\includegraphics[width=\\columnwidth]{grid_100_hwf_winding_decomposition_orbitals_bloch_Q_0_4pi_Lz_25_D_0_75_phi_pi.pdf}\n  \\caption{Berry phase winding of non-trivial hybrid Wannier bands (first column), hybrid Wannier functions at $\\vec{k} = \\Gamma$ decomposed in terms of orbital (second column), and Bloch basis (third column) for an $xy$-slab of the Weyl-CDW model with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$ and $\\phi = \\pi$.\nThe $z$-coordinate ranges from $-12, \\ldots, +12$. \nThere are 4 non-trivial hybrid Wannier bands for the occupied states, each of them displaying $-2\\pi$ winding of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ as $k_{x}$ goes from $0$ to $2\\pi$, such that the Chern number for each of the CI layers is $C = -1$. \nFrom the second column we can deduce that the non-trivial states are localized at $z \\approx -7.5$, $-2.5$, $+2.5$, $+7.5$. \nFrom the third column we can deduce that the non-trivial Bloch band indices at $\\vec{k} = \\Gamma$ are $n = 17$, $18$, $20$ and $21$ (the first band is labelled by $n=0$).}\n  \\label{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n  \\includegraphics[scale=0.4]{Berry_curvature_around_Gamma_Q_0_4pi_Lz_25_D_0_75_phi_pi.pdf}\n  \\caption{The Berry curvature around $\\vec{k} = \\Gamma$ for the 25 valance Bloch bands (not hybrid Wannier bands) for the $xy$-slab of the Weyl-CDW with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$ and $\\phi = \\pi$. \n  All the figures have $k_{x}$ and $k_{y}$ in range $2\\pi [-0.003,0.003]$. \n  We notice that the Bloch bands with indices $n=17$, $18$, $20$ and $21$ have negative Berry curvature around $\\vec{k} = \\Gamma$ contributing to the total Chern number $C = -4$ of this $xy$-slab, leading to $G_{xy} = -4 e^{2}\/h$.}\n  \\label{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_pi}\n\\end{figure}\n\n\n\n\\begin{figure}[ht]\n  \\includegraphics[scale=0.6]{slab_band_and_wvfn_Q_0_4pi_Lz_25_D_0_75_phi_0_and_pi.pdf}\n  \\caption{The $xy$-slab valence band structure of the Weyl-CDW model plotted along the path $-Y \\to \\Gamma \\to Y$ with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$, (a) $\\phi = 0$ and (b) $\\phi = \\pi$.\n   The (a) 5 and (b) 4 non-trivial bands around $\\vec{k}=\\Gamma$ and $Y$ are marked by orange and green. \n   The summation of the probability distribution for the (a) 5 and (b) 4 non-trivial states at $\\vec{k}=\\Gamma$ and $Y$ are also plotted on the right of each $xy$-slab band structure for (a) $\\phi = 0$ and (b) $\\pi$. \n The $z$-coordinate ranges from $-12,\\ldots, +12$. \n As the non-trivial states at $\\vec{k} = \\Gamma$ and $\\vec{k} =Y$ have exactly same probability distribution along $z$, they lie in the same Landau level index $n$ (see Sec.~VI of the main text) subspace. \n This also confirms that our identification of non-trivial bands is consistent.}\n  \\label{slab_band_and_wvfn_Q_0.4pi_Lz_25_D_0.75_phi_0_and_pi}\n\\end{figure}\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $g\\in L^2(\\Bbb R^d)$,  and let $\\Lambda\\subset \\Bbb R^{2d}$ be  a countable subset. We define the {\\it Gabor system} (also known as {\\it Weyl-Heisenberg system}) $\\mathcal G(g, \\Lambda)$ with respect to $g$ and $\\Lambda$  to be  the  collection of functions $\\pi(a, b)g$ defined by  combined time and frequency shifts of $g$:\n$$\n\\pi(a,b)g(x)= M_{b}T_{a}g =  e^{2\\pi i \\langle b, x\\rangle} g(x-a) \\quad (a,b)\\in \\Lambda .\n$$\n$\\Lambda$ is also known as  {\\it time-frequency set}  and\nthe frequency shift is also called modulation. We say $g$ is an {\\it orthonormal Gabor window function}  with respect to $\\Lambda$, or simply a window function, if  $\\mathcal G(g, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$. See e.g. \n\\cite{OleBook} or \\cite{Groechenig-book}. \n\n\n\n\nWe  call $\\Lambda$ {\\it separable} if it is of the form of $\\Lambda = {\\mathcal T}\\times \\Gamma$ for some countable subsets ${\\mathcal T}$ and $\\Gamma$ in ${\\mathbb R}^d$. Gabor systems  have been introduced for the first time in 1946 by Gabor \\cite{Gabor46} and are now  fundamental objects in applied and computational harmonic analysis. Moreover, for $\\mathcal G(g, \\Lambda)$ to be a Gabor orthonormal basis,  the (Beurling) density of $\\Lambda$, denoted by dens$(\\Lambda)$, must be $1$ \\cite{RS}.\n\n\n\n\\medskip\n\n\nThe existence of a window function for a given lattice  has been investigated  for several  special cases of $M$. The question of existence has been  completely answered\nby Han  and Wang \\cite{HanW1}  for separable lattices of the form  $\\Lambda= \\mathcal J\\times \\mathcal T$  with  dens$(\\Lambda)=1$. They answered the question by showing the existence of a common fundamental domain for two different lattices. Later, the same authors  partially answered the question for non-separable lattices (i.e. the lattices of not of the form of $\\mathcal J \\times \\mathcal T$)  for special cases of matrix $M$ \\cite{HanW4}. Indeed, they proved that,  when  for example $M$ is a block triangular matrix, a  window function $g$ exists  and it can be chosen so  that $|g|$ or $|\\hat g|$ is the scalar multiple of a characteristic function. They also showed the existence of a window function with compact support for rational matrices  $M$.\n\n\\medskip\n\nGiven a subset $K\\subset \\Bbb R^d$,\nwe denote by $\\chi_K$ the indicator function of $K$ and by $|K|$ its Lebesgue measure.\nThe main focus of this paper  is  the following. Suppose that $\\Lambda = M({\\mathbb Z}^{2d})\\subset \\Bbb R^{2d}$ is a full lattice with Beurling density dens$(\\Lambda)= |\\det(M)|^{-1} =1$   and suppose that $\\mathcal G(|K|^{-1\/2} \\chi_K, \\Lambda)$ forms a Gabor orthonormal basis for $L^2(\\Bbb R^d)$. What can we say about the structure of $K$? This question is related to the study of spectral sets and translational tiles which we will call the {\\it Fuglede-Gabor Problem} later on.\n\n\\medskip\n\n\n\\begin{definition}[Spectral and tiling sets]\\label{Spectral and tiling sets}   A Lebesgue measurable set\n  $K\\subset \\Bbb R^d$  with positive  and finite measure is a  {\\it spectral set} in $\\Bbb R^d$ if there is a countable set $B\\subset \\Bbb R^d$ (not necessarily unique) such that\n  exponentials $\\{e_b(x):= e^{2\\pi i \\langle b,x\\rangle} : b\\in B, x\\in K\\}$ constitute an orthogonal basis for   $L^2(K)$, i.e., the exponentials are mutual orthogonal and complete in $L^2(K)$.  In this case $B$ is called a {\\it spectrum} for $K$.\n\n  \\medskip\n\n We say $K$ {\\it multi-tiles} $\\Bbb R^d$ by its translations  if there\n is a countable set ${\\mathcal J}\\subset \\Bbb R^d$ and integer $N\\ge 1$ such that\n\n\\begin{equation}\\label{multitile-generic}\n\\sum_{t\\in{\\mathcal J}} \\chi_{K} (x+t)=N\\quad a.e. \\ x\\in \\Bbb R^d .\n\\end{equation}\nIf $N=1$, then $K$ {\\it tiles} ${\\mathbb R}^d$ and  the set ${\\mathcal J}$ is called {\\it tiling set} for $K$ (For more details about multi-tiles, see e.g. \\cite{Kol}).\n \\end{definition}\n\n\\medskip\n\n  Spectral sets   have been studied extensively in the recent years and their study has been reduced to the study of  tiling sets by the Fuglede Conjecture or Spectral Set Conjecture \\cite{Fug74}   which asserts: {\\it A set $K\\subset \\Bbb R^d$ with positive and finite measure is a spectral set if and only if $K$ tiles $\\Bbb R^d$ by translations. }\n Fuglede proved the conjecture in his celebrated 1974 paper \\cite{Fug74} for the case  when $K$ tiles by a lattice or $K$ has a spectrum which is a lattice. The Fuglede Conjecture led to considerable activity in the past three decades. In\n  2004, Tao \\cite{T04} disproved the Fuglede conjecture for dimension $5$ and higher, followed by Kolountzakis and Matolcsi\\rq{}s result \\cite{FMM,KM06}  where they proved  that  the conjecture fails in dimensions $3$ and higher. For more recent results and historical comments see e.g. \\cite{BHM16,IMP17}.\n\n\n\\medskip\nSpectral sets and tiles appear naturally in the Gabor setting.  Indeed, let $\\Lambda= \\mathcal J \\times \\mathcal T$ be a separable countable set (not necessarily a lattice) with  dens$(\\Lambda)=1$ and let  $\\Omega$ be a compact set  in $\\Bbb R^d$  which tiles by $\\mathcal J$ and is spectral for $\\mathcal T$.  For example, take $\\Omega = [0,1]^2$ and ${\\mathcal J} = {\\mathcal T} = {\\mathbb Z}^2$. Let $g$ be a   function supported in $\\Omega$.  Then an easy calculation shows that the Gabor system\n $\\mathcal G(g, \\Lambda)$ is an orthonormal basis  if   $|g(x)|=|\\Omega|^{-1\/2} \\chi_\\Omega(x)$\n (see also \\cite{LW03}, Lemma 3.1).  We call such Gabor bases {\\it standard}.  Liu and Wang \\cite{LW03} conjectured the converse of this result that  for  a  compactly supported function $g$ and  a countable separable set $\\Lambda= \\mathcal J \\times \\mathcal T$, if $\\mathcal G(g, \\Lambda)$  is an orthonormal basis for $L^2(\\Bbb R^d)$, then there is a compact set $\\Omega\\subset \\Bbb R^d$ such that\n $|g|$ is a constant multiple of $\\chi_\\Omega$,    $\\Omega$ tiles by $\\mathcal J$  and is a spectral set for $\\mathcal T$. Liu and Wang proved their conjecture when the support of $g$ is an interval.\n  Dutkay and the first listed author recently proved that the Liu and Wang\\rq{}s conjecture is affirmative if $g$ is non-negative \\cite[Theorem 1.8]{Dutkay-Lai}. \n\n\n \n\nHowever, the conjecture  is still unsolved for general compactly supported $g$.\n\n\n\n\n\\iffalse\n\\begin{Dutkay-Lai}\\label{Dutkay-Lai}(\\cite{Dutkay-Lai}, Theorem 1.8 resp. Theorem 6.2) Suppose that $g$ is a non-negative function with bounded and measurable support $K$ in $\\Bbb R^d$. Suppose that    $\\Lambda=\n \\mathcal J\\times \\mathcal T$ is a countable set with $D(\\Lambda)=1$. If  $\\mathcal G(g, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$, then the following hold.\n\\begin{itemize}\n\\item[(i)] $K$ tiles $\\Bbb R^d$ by translations by $\\mathcal J$.\n\\item[(ii)] $g=  |K|^{-1\/2}\\chi_K$ a.e. on $K$.\n\\item[(iii)] $K$ is a spectral with respect to $\\mathcal T$.\n\\end{itemize}\n\\end{Dutkay-Lai}\n\n\n\nWe shall sketch the proof of Theorem A here. The completeness of the Gabor system $\\mathcal G(g, \\Lambda)$ proves that $\\cup_{j\\in \\mathcal J} K+j$ is a cover for $\\Bbb R^d$. The orthogonality implies that the collection of sets $\\{K+j: \\ j\\in \\mathcal J\\}$ are mutual disjoint up to a Lebesgue measure zero set, thus  $K$ is a tiling set with respect to $\\mathcal J$. Further, the measure  $d\\mu(x):=g(x)^2 dx$ has a spectrum by proving an equivalent statement that\n$$\\sum_{p\\in \\mathcal T} |\\widehat{\\mu} (x-p)|^2 =1.$$\nThis implies that the measure $\\mu$  is spectral, thus   $|g|^2$ must be a constant multiple of a characteristic function by\nCorollary 1.4 of \\cite{Dutkay-Lai}. This forces  $g= |K|^{-1\/2}\\chi_K$ a.e. on its support $K$, hence  the   conclusions of (ii) and (iii).  The converse holds by a direct calculation.\n\\fi\n\n\n\n\nThe following problem links the study of window functions associated with Gabor orthonormal bases to the tiling and spectral properties of sets.\n\\medskip\n\n\\begin{problem}\\label{our conjecture1}(Fuglede-Gabor Problem)\nLet    $K\\subset \\Bbb  R^d$ be a measurable subset  with positive and finite measure, and let   $\\Lambda \\subset \\Bbb R^{2d}$ be a countable subset. If the  Gabor family $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$, then $K$ tiles and is a spectral set.\n \\end{problem}\n\n\n\nSince the indicator function of a set is non-negative, {\\it Problem \\ref{our conjecture1} is already affirmative if the time-frequency set is a separable countable sets} using the results of Dutkay and the first listed author.\n Therefore we only focus on the case when the time-frequency set is non-separable. Moreover, although the problem does not require any boundedness assumption of $K$, our interest will mainly be focused on the set $K$ being bounded. \n\n\n   It is hard to speculate whether Problem \\ref{our conjecture1}  is true or not in its full generality. But from   the point of view of Fuglede's result for lattices, we still hope that   the Fuglede-Gabor problem is true for  non-separable lattices  as well. Unfortunately, after our intensive study, we found out that, similar to many notoriously difficult problems in Gabor analysis (see e.g. \\cite{Groechenig-mystery}),  \n    the Fuglede-Gabor problem for lattices appears to be uneasy. This paper     gives a partial answer towards the full solution together with some unexpected examples, as we explain below. \n\n\n\n\\noindent{\\bf Main Results of the paper.} Our main results will mostly be focused on the lower triangular block matrices since  most    matrices can be reduced to the lower triangular form:\n   \\begin{equation}\\label{lower_block}\n   \\Lambda = \\left(\n                                      \\begin{array}{cc}\n                                        A & O \\\\\n                                        C & B \\\\\n                                      \\end{array}\n                                    \\right)({\\mathbb Z}^{2d}), \\ \\mbox{and} \\ |\\det(AB)|=1\n                                    \\end{equation}\n    (i.e. dens$(\\Lambda)=1$). We will use $B^{-t}$ to denote the inverse  transpose of  the matrix $B$.  Our first general key lemma is as follows, it will serve as a key step for our further analysis.\n\n    \\medskip\n\n    \\begin{lemma}[Key Lemma]\\label{Th_union of FD} Let  $\\Lambda = M({\\mathbb Z}^{2d})$ with $M$ an  $2d\\times 2d$ invertible  lower triangular block matrix of the form (\\ref{lower_block}).\nSuppose that ${\\mathcal G}\\left({|K|^{-1\/2}}\\chi_K,\\Lambda\\right)$ is a Gabor orthonormal basis. Then there exists an integer $N\\ge 1$ such that\n                                    $$\n                                    K = \\bigcup_{j=1}^N D_j=\\bigcup_{j=1}^N E_j\n                                    $$\nwhere $D_j$\\rq{}s are fundamental domains of $B^{-t}(\\Bbb Z^d)$ and $E_j$\\rq{}s are almost disjoint fundamental domains of $A(\\Bbb Z^d)$ with $|D_i\\cap D_j| = 0$ and $|E_i\\cap E_j| =0$ for all $i\\ne j$. (i.e. K  multi-tiles  ${\\mathbb R}^d$ simultaneously by $A({\\mathbb Z}^d)$ and $B^{-t}(\\Bbb Z^d)$.)\n\\end{lemma}\n\n If we can prove that $N=1$, then $K$ will be a common fundamental domain for $A(\\Bbb Z^d)$ and $B^{-t}(\\Bbb Z^d)$ and this will imply that the Fuglede-Gabor problem  holds. In particular, this is true when $A^tB$  is an integer matrix and $K$ is a bounded set, as our next result confirms.\n\n\\medskip\n\n\n\\begin{theorem}\\label{lower triangle}  Let $K$ be  a bounded measurable subset of $ \\Bbb R^d$  with  positive measure, and let  $\\Lambda\\subset \\Bbb R^{2d}$ be a lower triangular  lattice in (\\ref{lower_block}). Suppose that\n ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^d)$ and  $A^tB$  is an integer matrix.  Then $K$ tiles and is spectral. More precisely, $K$ is a common fundamental domain for  $A(\\Bbb Z^d)$ and $B^{-t}(\\Bbb Z^d)$, $K$ tiles by $A(\\Bbb Z^d)$ and is spectral with spectrum $B(\\Bbb Z^d)$.\n\\end{theorem}\n\n\n\n\\medskip\n\n\n\n\nAs a consequence of  Theorem \\ref{lower triangle},  we resolve the  Fuglede-Gabor Problem \\ref{our conjecture1}  in dimension one  for rational matrices  and $K$ is bounded.\n\n\\vskip.124in\n\n\\begin{theorem}\\label{rational_dim1} Suppose that $K\\subset \\Bbb R$ is a bounded set with positive Lebesgue measure. Suppose that   $\\Lambda$   is a rational lattice in $\\Bbb R^{2}$ with $dens(\\Lambda)=1$.  If\n ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R)$,  then $K$ tiles and is spectral.\n\\end{theorem}\n\n\n\\medskip\n\nWe  also have the following result for upper triangular block matrices using Theorem \\ref{lower triangle}.\n\n                                    \\vskip.124in\n\n\\begin{theorem}\\label{UT} Suppose that $K\\subset \\Bbb R^d$ is a bounded set with positive Lebesgue measure. Supposes that  $\\Lambda\\subset \\Bbb R^{2d}$ is a lattice such that\n $\\Lambda = \\left(\n                                      \\begin{array}{cc}\n                                        A & D \\\\\n                                        O & B \\\\\n                                      \\end{array}\n                                    \\right)({\\mathbb Z}^{2d})$ with  $dens(\\Lambda)=1$,\n                                     $A^{-1}D$  symmetric rational matrix and $A^tB=I$. If\n ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^d)$, then $K$ tiles and is spectral.\n \\end{theorem}\n\n \\medskip\n\nTheorem \\ref{lower triangle} may also be consider as a converse of \\cite[Lemma 4.1]{HanW4}, which states that  if   $K$ is a  common fundamental domain for the lattice $A({\\mathbb Z}^d)$ and $B^{-t}({\\mathbb Z}^d)$, then for any  matrix $C$, the system  ${\\mathcal G}(|K|^{-1\/2}\\chi_K,\\Lambda)$ is an Gabor orthonormal basis. Therefore one may  naturally expect that $N=1$  in Theorem \\ref{Th_union of FD} is always the case. However, we will show that  {\\it $N>1$ can actually happen with a suitable choice of $C$ if $A^tB$ is a rational matrix (see Example \\ref{mutli-tile K})}. This poses additional difficulty to solve the Fuglede-Gabor Problem for rational matrices in higher dimension, as we shall discuss it  later.  Finally, for a general matrix containing irrational entries, the answer to  Fuglede-Gabor problem is completely open.  We will discuss  this in detail  in Section \\ref{Open problems}.\n\n\n\n\n %\n\n\n\n \\medskip\n\n\n\n\n\n {\\bf Outline of the paper.} We organize the paper as follows: After some definitions and recalling some known and basic facts about  lattices and Gabor analysis in Section \\ref{notations},  in Section \\ref{proof of Theorem 1.3}  we prove Theorem \\ref{Th_union of FD}.\n The proof of Theorems \\ref{lower triangle} are \\ref{rational_dim1} are presented in\n   Section \\ref{thm:lower triangle}.  In  Section \\ref{thm:rational_dim1 and UT} we prove Theorem \\ref{UT}.   Section \\ref{Examples} is devoted to   examples illustrating the possibility for $N> 1$.\n   We conclude the paper with a series of open problems in Section  \\ref{Open problems} both  for rational and irrational lattices as well as the full generality of the Fuglede-Gabor Problem.  In our exposition, we discover that a new notion of completeness, which we will call  {\\it exponential completeness},  is crucial in studying the Fuglede-Gabor problem, we will give a short study in Appendix A. In Appendix B, we will show that the octagon will not produce any Gabor orthonormal basis using rational matrices.  \n\n\n\n\n\n\\iffalse\n A classical example of a set  in dimension $d$ which tiles and is spectral is the unit cube $[0,1]^d$. It is known that   $[0,1]^d$ tiles $\\Bbb R^d$ with integer shifts with $\\Bbb Z^d$ and $\\{e_n(x):=e^{2\\pi in \\langle n, x\\rangle } : n\\in \\Bbb Z^d\\}$  is an orthonormal basis for $L^2([0,1]^d)$, also known as Fourier basis.\n  It is also well-known that the Gabor system $\\mathcal G(\\chi_{[0,1]^d}, \\alpha\\Bbb Z^d\\times\\beta\\Bbb Z^d)$ is an orthonormal basis for $L^2(\\Bbb R^d)$ when $\\beta=\\alpha^{-1}$ (see e.g. \\cite{Groechenig-book}).   This simple observation suggests that there is a link between the study of the Gabor orthonormal basis and the study of tiling and spectral properties. The connection was  observed in the earliest by   Liu and Wang \\cite{LW03} in 2003, who conjectured the following for separable time-frequency shifts $\\Lambda$:\n  \\medskip\n\n  \\fi\n\n\n \\iffalse\n The link between the study of the Gabor orthonormal basis and the study of tiling and spectral properties was   observed in more generality   by   Liu and Wang \\cite{LW03} in 2003. They  conjectured  that:\n\n {\\bf Liu and Wang\\rq{}s Conjecture:} (\\cite{LW03}) {\\it Given a compactly supported function $g(\\neq 0) \\in L^2(\\Bbb R^d)$ with finite and positive measure support $\\Omega$ and given a separable countable set $\\Lambda=\\mathcal J\\times \\mathcal T \\subseteq \\Bbb R^{2d}$, the Gabor family $\\mathcal G(g, \\Lambda)$ is an orthonormal basis if and only if the following hold: \\\\\n (1) $\\Omega$ tiles $\\Bbb R^d$ with translations  with tiling set  $\\mathcal J$. \\\\\n (2) $|g|=|\\Omega|^{-1\/2} \\chi_\\Omega$. \\\\\n (3) $\\Omega$ is a spectral set with spectrum  $\\mathcal T$.}\n\nWhen (1) and (3) hold, then  for any  function $g$ satisfying (2) the  Gabor family $\\mathcal G(g, \\Lambda)$ constitutes an  orthonormal basis for $L^2(\\Bbb R^d)$ (\\cite{LW03}, Lemma 3.1).\nThe other direction of the conjecture  has not been solved  in general but only in some special cases.  Recently,\n Dutkay and the first listed author (\\cite{Dutkay-Lai}, Theorem 6.2), proved that if for the separable countable set  $\\Lambda=\\mathcal J\\times \\mathcal T$, the Gabor system $\\mathcal G(g, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$, and if the window function $g\\neq 0$ is non-negative on its support $K$,  a   bounded and measurable set, then $g$ is a constant multiple of  characteristic function of $K$, and $K$ tiles by $\\mathcal J$ and is spectral in $\\Bbb R^d$ with respect to $\\mathcal T$.  This result provides a partial affirmative answer to the Liu and Wang\\rq{}s Conjecture in separable case.\n\n \\fi\n\n\n\n\n\n \\section{Preliminaries}\\label{notations}\n\nIn this section, we will collect several basic definitions and results required for the rest of the paper.  A {\\it full-rank lattice}\n $\\Lambda\\subset \\Bbb R^d$ is a discrete and countable subgroup of $\\Bbb R^{d}$ with compact quotient group  $\\Bbb R^{d}\/\\Lambda$. A full-rank lattice  in $\\Bbb R^d$ is  given by $\\Lambda = {M}({\\mathbb Z}^{2d})$ for some\n  $2d\\times 2d$ invertible matrix $M\\in GL(2d, \\Bbb R)$. The density of $\\Lambda$ is given by   dens$(\\Lambda)=|\\det(M)|^{-1}$.\n\n Let $\\Lambda$ be a lattice in $\\Bbb R^d$.\n   The {\\it dual lattice} of $\\Lambda$ is  defined as\n $$\\Lambda^\\perp:= \\{ x\\in \\Bbb R^{2d} : \\ \\langle \\lambda, x\\rangle\\in \\Bbb Z,  \\ \\forall \\lambda\\in \\Lambda\\} .$$\nA direct calculation shows that  $\\Lambda^\\perp=  M^{-t}(\\Bbb Z^{d})$.\n\nA fundamental domain of a lattice $\\Lambda$ is a measurable set $\\Omega$ in $\\Bbb R^d$ which contains distinct representatives (mod $\\Lambda$) in $\\Bbb R^d$, so that the any intersection of $\\\n\\Omega$ with any coset $x+\\Lambda$ has only one element. For the existence of a fundamental domain see   \\cite[Theorem 1]{Feld-Green68}.  It is also evident that $\\Omega$ tiles $\\Bbb R^d$ with translations by $\\Lambda$  and\nany other tiling set differs from  $\\Omega$ at most for a zero measure set.\n\n\n\\noindent{\\bf 1. A reduction lemma.} For an invertible $d\\times d$  matrix\n $A$, the operator    ${\\mathcal D}_A:L^2(\\Bbb R^d)\\to L^2(\\Bbb R^d)$ defined by\n $${\\mathcal D}_Ag(x):=|\\det(A)| ^{1\/2} g(Ax).$$\n %\n is unitary, i.e, ${\\mathcal D}_A$ is onto and isometry $\\|{\\mathcal D}_Ag\\|= \\|g\\|$. The following lemma follows from a simple computation and is in general known. We will omit the detail of the proof. \n\n\\vskip.124in\n\n \\begin{lemma}\\label{lemma 3-prim}\n Let ${\\Lambda} $ be a lattice such that\n %\n\\begin{align}\\label{block lattice}\n\\Lambda= \\left(\n                    \\begin{array}{cc}\n                       A & D\\\\\n                       C & B  \\\\\n                    \\end{array}\n                  \\right)({\\mathbb Z}^{2d})\n                   \\end{align}\n                   where  $A$ is an invertible $d\\times d$  matrix. Then\n$$\\mathcal G(g, \\Lambda) = {\\mathcal D}_{A^{-1}}\\mathcal G({\\mathcal D}_Ag, \\widetilde{\\Lambda})$$\n where  $$\n\\widetilde{\\Lambda} = \\left(\n                    \\begin{array}{cc}\n                       I & A^{-1}D\\\\\n                       A^tC & A^tB  \\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d}).\n $$\n\nConsequently,  $\\mathcal G(g, \\Lambda) $ is a Gabor orthonormal basis if and only if $\\mathcal G({\\mathcal D}_Ag, \\widetilde{\\Lambda})$ is a Gabor orthonormal basis.\n   \\end{lemma}\n\n\n\nIn Lemma \\ref{lemma 3-prim},  if we let\n   $D=O$ and $g(x)=|K|^{-1\/2} \\chi_K$, then the conclusion of the lemma shows that   ${\\mathcal G}(|K|^{-1\/2} \\chi_K, \\Lambda)$ is an orthonormal basis  for $L^2(\\Bbb R^d)$ if and only if  ${\\mathcal G}(|A^{-1}K|^{-1\/2} \\chi_{A^{-1}K}, \\widetilde{\\Lambda})$ is an orthonormal basis with\n$$\n\\widetilde{\\Lambda} = \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       A^tC & A^tB  \\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d}).\n$$\n We shall use this observation   later.\n\n \\iffalse\n\\begin{proof}\nLet $g = |K|^{-1\/2} \\chi_K$. For each $\\lambda = (Am,Cm+Bn),\\lambda' =(Am',Cm'+Bn')\\in \\Lambda$. Note that by a change of variable $x \\rightarrow Ax$\n$$\n\\begin{aligned}\n\\langle\\pi(\\lambda)g,\\pi(\\lambda')g\\rangle =&  |K|^{-1} \\int \\chi_K (x-Am)\\chi_K (x-Am') e^{-2\\pi i \\langle C(m-m')+B(n-n'),x\\rangle}dx\\\\\n=& |K|^{-1}\\int \\chi_K (x-Am)\\chi_K (x-Am') e^{-2\\pi i \\langle C(m-m')+B(n-n'),x\\rangle}dx\\\\\n=&  |\\det A||K|^{-1}\\int \\chi_K (Ax-Am)\\chi_K (Ax-Am') e^{-2\\pi i \\langle C(m-m')+B(n-n'),Ax\\rangle}dx\\\\\n=& |A^{-1}K|^{-1} \\int \\chi_{A^{-1}K} (x-m)\\chi_{A^{-1}K} (x-m')e^{-2\\pi i \\langle A^tC(m-m')+A^tB(n-n'),x\\rangle}dx.\n\\end{aligned}\n$$\nThis shows that ${\\mathcal G}(|K|^{-1\/2} \\chi_K, \\Lambda)$ is mutually orthogonal if and only if ${\\mathcal G}(|A^{-1}K|^{-1\/2} \\chi_{A^{-1}K}, \\widetilde{\\Lambda})$ is mutually orthogonal. In a similar change of variable, we see that the Parseval identities of these Gabor system are actually equivalent. Hence, the statement of the lemma follows.\n\\end{proof}\n\\fi\n\n\\medskip\n\n\\noindent{\\bf 2. Orthogonality implies completeness.} The following proposition says that completeness automatically holds for a lattice of density one if we can establish the mutually orthogonality. \n\n\n\\medskip\n\n                   \\begin{proposition}\\label{complt}\n                   Let $g\\in L^2(\\Bbb R^d)$, $\\|g\\|=1$ and $\\Lambda\\subset \\Bbb R^{2d}$ be a lattice with density $\\mbox{dens}(\\Lambda)=1$. Assume that\n                   $\\mathcal G(g,\\Lambda)$ is  an orthonormal set. Then $\\mathcal G(g,\\Lambda)$ is complete.\n                  \\end{proposition}\n\n\\medskip\n                                     For the  proof of Proposition \\ref{complt} we require  the following lemma.\n                                  Note that for a positive Borel measure $\\mu$,\n                                  $$\n                                  f\\ast\\mu (x) = \\int f(x-y)d\\mu(y),\n                                  $$\n                                  given that the integral is well-defined. If\n $\\mu = \\sum_{\\lambda\\in\\Lambda}\\delta_{\\lambda}$, then $\\chi_K\\ast\\mu = 1$ ($\\le 1$) if and only if $K$ tiles (packs) ${\\mathbb R}^d$ by $\\Lambda$\\footnote{Recall that $K$ packs ${\\mathbb R}^d$ by ${\\mathcal J}$ if $\\sum_{t\\in{\\mathcal J}}\\chi_{K}(x-t)\\le1$, a.e. $x\\in \\Bbb R^d$}. With this introduction we recall the following  result.\n\n\\medskip\n\n                   \\begin{lemma}\\label{GLW}(\\cite[Theorem 2.1]{GLW})\n                 Suppose that $f,g\\in L^1({\\mathbb R}^d)$ are non-negative functions such that $\\int f(x) dx = \\int g(x) dx= 1$. Suppose that for positive Borel measure $\\mu$  on $\\Bbb R^d$\n$$\nf\\ast\\mu \\le 1  \\ \\mbox{and} \\ g\\ast\\mu\\le 1.\n$$\nThen $f\\ast\\mu = 1$  if and only if $ g\\ast \\mu= 1$.\n\\end{lemma}\n\n\n Given $f,g\\in L^2({\\mathbb R}^d)$, the {\\it short time Fourier transform} is defined by\n        \\begin{align}\\label{STFT}\nV_gf(t,\\xi)=\\int f(x)\\overline{g(x-t)}e^{-2\\pi i \\langle\\xi,x\\rangle}dx, \\quad (t,\\xi)\\in \\Bbb R^{2d}\n \\end{align}\nand it is  a continuous function on ${\\mathbb R}^{2d}$ \\cite{Groechenig-book}.\n\n\n                  \\begin{proof}[Proof of Proposition \\ref{complt}]\n                  The mutual orthogonality of $\\mathcal G(g, \\Lambda)$ implies the Bessel inequality of the system:\n\\begin{align}\\label{Bessel}\n\\sum_{(t,\\xi)\\in \\Lambda} |V_gf(t, \\xi)|^2\\leq \\|f\\|^2 \\ , \\quad \\forall f\\in L^2(\\Bbb R^d).\n\\end{align}\nLet $ s,\\xi  \\in \\Bbb R^d$. The inequality (\\ref{Bessel}) for $e^{2\\pi i \\langle \\nu, x\\rangle} f(x-s)$ in the place of $f$ yields\n\\begin{align}\\notag\n\\sum_{(t,\\xi)\\in \\Lambda} |V_gf(t-s, \\xi-\\nu)|^2\\leq \\|f\\|^2  \\quad \\forall \\ (s, \\nu)\\in \\Bbb R^{2d}.\n\\end{align}\nHence, $|V_gf|^2\\ast \\delta_\\Lambda \\leq \\| f\\|^2$. Take $G:= \\| f\\|^{-2}|V_gf|^2$. Then $\\int_{\\Bbb R^{2d}}  G(z) dz=1$ and $G\\ast \\delta_\\Lambda\\leq 1$. On the other hand, $\\Lambda$ is a lattice with  density $1$. Let $\\Omega\\subset \\Bbb R^{2d}$ be any fundamental domain for $\\Lambda$. Then  $|\\Omega|=1$ and it tiles $\\Bbb R^{2d}$ by  $\\Lambda$. Therefore $\\chi_\\Omega\\ast \\delta_\\Lambda=1$. Now Lemma \\ref{GLW}  implies that    $G\\ast \\delta_\\Lambda =1$. But this  is equivalent to the completeness of the system $\\mathcal G(g, \\Lambda)$ and we are done.\n                  \\end{proof}\n\n\\medskip\n\n\n\n\\noindent{\\bf 3. Some reduction to lower triangular block matrices.} The following result  is due to Han and Wang which states that any  invertible integer matrix   can be converted into a  lower triangular integer  matrix. We will need it in later sections. An integer matrix $P$ is called {\\it unimodular}  if $\\det P=1$.\n\n\n\\medskip\n\n\\begin{lemma}\\label{HanW1}(\\cite{HanW4}, Lemma 4.4) Let $M$ be an $d\\times d$ invertible integer matrix. Then there is an  $d\\times d$  unimodular  matrix $P$ such that $MP$ is a  lower triangular integer matrix.\n\\end{lemma}\n\n\n\n\nAs a corollary of Lemma  \\ref{HanW1} we can show that any rational matrix can be represented as a  lower triangular rational matrix.\n\n\\medskip\n\n\\begin{corollary}\\label{M=N}\nLet $M$ be an $d\\times d$ invertible rational matrix. Then there is a    lower triangular rational matrix $N$ such that $M(\\Bbb Z^d) = N(\\Bbb Z^d)$.\n\\end{corollary}\n\nHenceforth, we shall say matrix $M$ is {\\it equivalent} to $N$ if $M(\\Bbb Z^d) = N(\\Bbb Z^d)$. \n\n\n\\medskip\n\n\\noindent{\\bf 4. Exponential Completeness.}\\label{exponential completeness} Recall that a collection of functions $\\{\\varphi_n\\}$ is said to be {\\it complete} in $L^2(\\Omega)$ if $\\langle f, \\varphi_n\\rangle =0$ for all $n$ implies that $f=0$ a.e. on $L^2(\\Omega)$. Given $f\\in L^2({\\mathbb R}^d)$, the Fourier transform of $f$ is defined to be $\\widehat{f}(\\xi)= \\int_{{\\mathbb R}^d}f(x)e^{-2\\pi i \\langle\\xi, x\\rangle}dx$. In our study, we will need to following weaker notion of the completeness property. \n\n\\medskip\n \n\\begin{definition}\\label{exponential completeness} \n Let $\\Lambda$ be a countable set and let $\\Omega$ be a Lebesgue measurable set with positive finite measure. We say that the set of exponentials $\\{e^{2\\pi i \\langle\\lambda,x\\rangle}:\\lambda\\in\\Lambda\\}$ (or  $\\Lambda$) is {\\it exponentially complete} for $L^2(\\Omega)$ if there does not exist any $\\xi\\in{\\mathbb R}^d$ such that \n\n $$\n \\widehat{\\chi_{\\Omega}}(\\lambda-\\xi)=\\int_{\\Omega}  e^{2\\pi i \\langle \\xi ,x\\rangle}  e^{-2\\pi i \\langle \\lambda, x\\rangle}  dx =0, \\ \\forall \\lambda\\in\\Lambda.\n $$\n \\end{definition}\n\n\\medskip\n\n\n\\begin{remark}\\label{remark2}\nThroughout the paper, we will see that exponential completeness plays an important role in constructing Gabor orthonormal basis using non-separable lattices.  If a countable set of exponentials is complete for $L^2(\\Omega)$, then it must be exponentially complete (otherwise $e^{2\\pi i \\langle \\xi,x\\rangle}$ will be orthogonal to all $e^{2\\pi i \\langle\\lambda,x\\rangle}$ contradicting completeness). However, the converse is not true.\n For example, the set of exponentials associated to the lattice $\\Lambda=\\sqrt{2}{\\mathbb Z}$ is exponentially complete in $L^2([0,1])$, but it is not complete in it (see Lemma \\ref{lemma 4}).   In Appendix 1, we will give a short study about the exponential completeness for lattices in $L^2[0,1]^d$.\n\\end{remark}\n\n\n\n\n\\section{Proof of Lemma \\ref{Th_union of FD} - Union of fundamental domains}\\label{proof of Theorem 1.3} \n We now prove our Theorem \\ref{Th_union of FD}. It follows from two theorems in two separate fields. The first one is taken from the study of Fuglede's problems.  It was first proved by Jorgensen and Pedersen \\cite[Theorem 6.2 (b)]{JoPe}  and then Lagarias and Wang \\cite[Theorem 2.1]{LW97} gave a simpler proof.\n\n\\medskip\n\n\\begin{theorem}\\label{thJoPeLgWa}\nLet $\\Omega\\subset{\\mathbb R}^d$ be a  Lebesgue measurable set with positive finite measure. Suppose $\\Gamma$ is a full-rank lattice such  that $\\Gamma \\subseteq \\{\\xi: \\widehat{\\chi_{\\Omega}}(\\xi) = 0\\} \\cup \\{0\\}$. Then\n$\n\\Omega = \\bigcup_{j=1}^N D_j$, up to measure zero, where $D_j$ are  fundamental domains for the dual lattice $\\Gamma^\\perp$, $|D_i\\cap D_j|=0, \\ i\\neq j$ ~ and $N=|\\Omega|\/|D_j|$.\n \\end{theorem}\n\n\n\n\\medskip\n\nGiven a lattice  $\\Lambda=M(\\Bbb Z^{2d})$, the   {\\it adjoint lattice} $\\Lambda^\\circ$ is   a lattice such that\n                  $$J(\\Lambda^\\circ)=  \\Lambda^\\perp $$\n                  where\n             $J=  \\left(\n                    \\begin{array}{cc}\n                      O & -I \\\\\n                       I & O \\\\\n                    \\end{array}\n                  \\right)$. In other words, $\\Lambda^\\circ= J^{-1}M^{-t}(\\Bbb Z^d)$.\n\nThe  Ron-Shen duality theorem \\cite{RS-duality97} is well-known in Gabor analysis. It was first proved over symplectic lattices, it is known to be true over any lattice (see e.g \\cite[Theorem 2.3]{Groechenig-mystery} for a proof by Poisson Summation Formula). We will need the following version of duality theorem.\n\n\\medskip\n\n\\begin{theorem}\\label{thRS} ${\\mathcal G}(g,\\Lambda)$ is a Gabor orthonormal basis if and only if ${\\mathcal G}(g,\\Lambda^{\\circ})$ is a Gabor orthonormal basis.\n\\end{theorem}\n\n\\begin{proof}[Sketch of Proof.] This statement is  well-known. Here we provide a simple proof based on \\cite[Theorem 2.3]{Groechenig-mystery} and Proposition \\ref{complt}. Since $(\\Lambda^{\\circ})^{\\circ} =\\Lambda$, both sides of the statements are symmetric and we just need to prove one side of the equivalence. Suppose that ${\\mathcal G}(g,\\Lambda)$ is a Gabor orthonormal basis. Then $g$ is the only dual window with the property that \n$$\n\\langle g, \\pi (\\mu)(g)\\rangle = 0 \\ \\forall \\mu\\in\\Lambda^{\\circ}\\setminus \\{0\\}\n$$\n(by \\cite[Theorem 2.3]{Groechenig-mystery}). This means that for all distinct $\\mu,\\mu'\\in\\Lambda^{\\circ}$,  $\\langle \\pi(\\mu)g, \\pi (\\mu')(g)\\rangle = c \\langle g, \\pi (\\mu-\\mu')(g)\\rangle = 0$ ($c$ is some unimodular constant). Thus ${\\mathcal G}(g,\\Lambda^{\\circ})$ is mutually orthogonal. As ${\\mathcal G}(g,\\Lambda)$ is a Gabor orthonormal basis, $\\|g\\|=1$ and dens$(\\Lambda^{\\circ})$ = dens$(\\Lambda)$ =1, by Proposition \\ref{complt}, ${\\mathcal G}(g,\\Lambda^{\\circ})$ is complete and is thus an orthonormal basis.\n\\end{proof}\n\nFor a lower triangular lattice $\\Lambda=  \\left(\n                                      \\begin{array}{cc}\n                                        A & O \\\\\n                                        C & B \\\\\n                                      \\end{array}\n                                    \\right){\\mathbb Z}^{2d}$,  the adjoint lattice  $\\Lambda^\\circ$  is also a lower triangular and it can be calculated as follows:\n %\n$$\n\\Lambda^{\\circ} = \\left(\n                    \\begin{array}{cc}\n                      O & I \\\\\n                      -I & O \\\\\n                    \\end{array}\n                  \\right)\\left(\n                    \\begin{array}{cc}\n                      A^{-t} & -A^{-t}C^tB^{-t} \\\\\n                      O & B^{-t} \\\\\n                    \\end{array}\n                  \\right)  ({\\mathbb Z}^{2d}) = \\left(\n                    \\begin{array}{cc}\n                      O & B^{-t} \\\\\n                      -A^{-t} & A^{-t}C^tB^{-t} \\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d}).\n$$\n\nFrom the other hand we can write  $$\\left(\n                    \\begin{array}{cc}\n                      O & B^{-t}\\\\\n                      -A^{-t}& A^{-t}C^tB^{-t} \\\\\n                    \\end{array}\n                  \\right)  =  \\left(\n                    \\begin{array}{cc}\n                       B^{-t} & O\\\\\n                       A^{-t}C^tB^{-t} & A^{-t}  \\\\\n                    \\end{array}\n                  \\right)\n                  \\left(\n                    \\begin{array}{cc}\n                      O & I \\\\\n                      -I & O\\\\\n                    \\end{array}\n                  \\right)({\\mathbb Z}^{2d}).\n$$\nBut  $\\left(\n                    \\begin{array}{cc}\n                      O & I \\\\\n                      -I & O\\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d})={\\mathbb Z}^{2d}$, therefore we have\n$$\n\\Lambda^{\\circ}= \\left(\n                    \\begin{array}{cc}\n                       B^{-t} & O\\\\\n                       A^{-t}C^tB^{-t} & A^{-t}  \\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d}).\n$$\n\n\n\\begin{proof}[Proof of Theorem \\ref{Th_union of FD}]\nThe orthogonality of the Gabor system  implies that\n \\begin{align}\\label{ortho}\n \\int_K e^{-2\\pi i \\langle  Bn, x\\rangle} dx = 0 \\quad \\forall \\ n\\in \\Bbb Z^d\\setminus\\{0\\} .\n \\end{align}\n By Theorem \\ref{thJoPeLgWa}, (\\ref{ortho}) implies      that  $K$ can be written as\n $\n K = \\bigcup_{j=1}^N D_j,\n $\n where $D_j$  is a fundamental domain  for $B^{-t}({\\mathbb Z}^d)$. On the other hand,  by the duality Theorem \\ref{thRS},  ${\\mathcal G}({|K|^{-1\/2}}\\chi_K,\\Lambda^{\\circ})$ is a Gabor orthonormal basis, too.  Similarly,   the exponentials $\\{e^{2\\pi i  \\langle A^{-t}n, x\\rangle}: n\\in {\\mathbb Z}^d\\}$ are mutually orthogonal  in $L^2(K)$. Hence, by Theorem \\ref{thJoPeLgWa}, we have\n $\n K = \\bigcup_{j=1}^M E_j\n $\n where $E_j$\\rq{}s are fundamental domains for $A({\\mathbb Z}^d)$.  Since $\\det(AB)=1$  we conclude that  $|D_i|=|E_j|, \\forall \\ i, j$. Since $|K|<\\infty$, the latter forces that $M=N$,  hence the proof of the theorem is completed.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorems \\ref{lower triangle} and \\ref{rational_dim1} - Lower triangular matrices}\\label{thm:lower triangle}\n\nTo prove Theorem \\ref{lower triangle},  first we shall apply  some preliminary reductions to  the theorem, as follows.   Due to   Lemma  \\ref{lemma 3-prim}  and  the hypothesis of  the theorem  on the matrices $A$ and $B$,   for the proof  it is sufficient\n  to  assume that\n$\n\\Lambda = \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       C & B\\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d}) ,\n $\n where $B$ is an invertible matrix with integer entries (since originally $A^tB$ has integer entries by the assumption of Theorem \\ref{lower triangle}). Notice by the density condition dens$(\\Lambda) = 1$,  we have $|\\det(B)|=1$. Thus $B^{-1}$ is also an integral matrix with determinant 1  and we have\n  $\\Bbb Z^{2d} = \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       O & B^{-1}\\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d})$. Thus\n      we can rewrite $\\Lambda$ as follows:\n $$\n\\Lambda = \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       C & B\\\\\n                    \\end{array}\n                  \\right) \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       O & B^{-1}\\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d}) = \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       C & I\\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d}).\n                  $$\nTherefore to prove Theorem \\ref{lower triangle}   it suffices to consider $\\Lambda = \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       C & I\\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d})$. In this case  if ${\\mathcal G}\\left(|K|^{-1\/2}\\chi_K,\\Lambda\\right)$ is an orthonormal basis,  then  by Theorem \\ref{Th_union of FD} we must have\n $\nK = \\bigcup_{j=1}^N E_j\n $\nwhere $E_j$\\rq{}s are disjoint fundamental domains of ${\\mathbb Z}^d$. In particular, $K$ is a multi-tile for $\\Bbb R^d$ with respect to ${\\mathbb Z}^d$, i.e.,\n\\begin{equation}\\label{multitile}\n\\sum_{k\\in{\\Bbb Z}^d} \\chi_K(x+k) = N \\ \\mbox{a.e.} \\ x\\in [0,1)^d.\n\\end{equation}\n Our goal is to show that $N=1$. For this, the following proposition will serve a key role.\n  \\medskip\n\n\n\\begin{proposition}\\label{prop2} Suppose $K$ is a bounded set which multi-tiles $\\Bbb R^d$ with respect to $\\Bbb Z^d$ at level $N$,  i.e. (\\ref{multitile}) holds.  Suppose that $N>1$. Then there exists $m \\in{\\mathbb Z}^d$ such that\n\\begin{enumerate}\n\\item $K\\cap (K+m)$ has positive Lebesgue measure.\n\\item $ K\\cap (K+m)$ consists of distinct representative   (mod ${\\mathbb Z}^d$).\n\\item $K\\cap (K+m)$ is a packing by ${\\mathbb Z}^d$, i.e.\n$$\n\\sum_{n\\in {\\mathbb Z}^d} \\chi_{K\\cap (K+m)} (x+n)\\le 1. \\ a.e.\n$$\n\\end{enumerate}\n\\end{proposition}\n \\begin{proof}\nPut  $Q := [0,1)^d$. The identity (\\ref{multitile}) means that for a.e. $x\\in Q$, there are exactly $N$ integers $n_1,...,n_N$ such that  $x+n_i\\in K$ $i=1,...,N$.  Using this observation we will  decompose $K$ as follows. For each $x\\in Q$, put\n$$\nK(x) := \\{k\\in \\Bbb Z^d: x+ k\\in K\\}.\n$$\nThen $|K(x)|=N$.\n Let $S \\subset{\\mathbb Z}^d$ and $|S| = N$. Define\n$$\nK_S = \\{x\\in Q: K(x) = S\\}.\n$$\nSince $K$ is a multi-tile of level $N$, we have\n$$\nK = \\bigcup_{|S|=N} (K_S+S), \\ \\mbox{and} \\ \\bigcup_{|S|=N} K_S = Q = [0,1)^d,\n$$\nwhere the union runs through all possible subsets $S\\in{\\mathbb Z}^d$ of cardinality $N$. Furthermore,\n $K_S \\cap K_{S'} =\\emptyset, \\forall S\\ne S'\n$, since there are exactly $N$ integers $n$ for which $x+n\\in K$. By the boundedness of $K$, there are only finitely many possible $S\\in{\\mathbb Z}^d$ with $|S|=N$ such that $|K_S|>0$. Thus, we can enumerate those $S$ as $S_1,...,S_r$ so that\n\\begin{equation}\\label{decomp}\nK = \\bigcup_{i=1}^r (K_{S_i}+S_i).\n\\end{equation}\n\n(Notice the   decomposition (\\ref{decomp}) also holds for any multi-tile bounded set $K$  with respect to any lattice $\\Gamma$ in place of $\\Bbb Z^d$ and\n any bounded fundamental set of  $\\Gamma$ in the place of $Q$.)\n\n\n\nWe order ${\\mathbb Z}^d$ by the natural lexicographical ordering. We then enumerate all possible elements in $S_i$, $1\\le i\\leq r$, by\n$$\n n_1^{S_i}<....<n^{S_i}_{N}\n$$\nLet\n$$\n m_{S_i} =  n_N^{S_i}- n_1^{S_i}, \\ \\mbox{and} \\  m= \\max_{i=1,...,r} m_{S_i}.\n$$\nNote that for any $i\\ne j$, since $K_{S_i}$ and $K_{S_j}$ are distinct subsets in $Q$, the intersection of  $K_{S_i}+{\\mathbb Z}^d$ and $K_{S_j}+{\\mathbb Z}^d$ has zero Lebesgue measure. As  $K_{S_i}+S_i\\subset K_{S_i}+{\\mathbb Z}^d$ and $K_{S_j}+S_j+ m\\subset K_{S_j}+{\\mathbb Z}^d$, the set $(K_{S_i}+S_i)\\cap (K_{S_j}+S_j+ m)$ has zero Lebesgue measure for $i\\neq j$. Therefore, up to Lebesgue measure zero, we have\n$$\n\\begin{aligned}\nK\\cap(K+ m) =& \\left(\\bigcup_{i=1}^r (K_{S_i}+S_i)\\right)\\cap\\left(\\bigcup_{j=1}^r (K_{S_j}+S_j+m)\\right)\\\\\n=&\\bigcup_{i=1}^r (K_{S_i}+S_i)\\cap (K_{S_i}+S_i+ m)\\\\\n\\end{aligned}\n$$\nNote that $(K_{S_i}+S_i)\\cap (K_{S_i}+S_i+ m)$ has positive Lebesgue measure if and only if $S_i\\cap (S_i+{m})\\ne \\emptyset.$ By our construction of $ m$,\n$$\nS_i\\cap (S_i+{m}) = \\left\\{\n                  \\begin{array}{ll}\n                    \\emptyset, & \\hbox{if $m_{S_i}<m$;} \\\\\n                    n_N^{S_i}, & \\hbox{if $m_{S_i}= m$.}\n                  \\end{array}\n                \\right.\n$$\nHence,\n$$\nK\\cap(K+m) = \\bigcup_{\\{i: m_{S_i} = m\\}} (K_{S_i}+  n_N^{S_i}).\n$$\nNote that $\\{i :m_{S_i}= m\\}$ is non-empty by our construction, thus $K\\cap (K+m)$ has positive Lebesgue measure. On the other hand, $K_{S_i}$ consists of distinct representatives in $Q$. Therefore $K\\cap(K+m)$ has distinct representatives (mod ${\\mathbb Z}^d$) in $Q$. This implies (2). The conclusion (3) follows directly  from (2)  and  by the  definition of packing.\n\\end{proof}\n\n\n\\medskip\nWe remark that Proposition \\ref{prop2} does not hold if $K$ is unbounded (see Example \\ref{example3}).  The following well-known lemma also hold  (see \\cite[Lemma 2.10]{GLi}).\n\n \\medskip\n\n\n\\begin{lemma}\\label{lemma 4}\nLet $B$ be an $d\\times d$ invertible matrix and $\\Omega\\subset \\Bbb R^d$ be  a measurable set with positive and finite measure. The set $\\{e^{2\\pi i \\langle B n, x\\rangle}: n\\in{\\mathbb Z}^d\\}$ is complete in $L^2(\\Omega)$ if and only if $\\Omega$ is a packing by $B^{-t}{\\mathbb Z}^d$, i.e.\n$$\n\\sum_{n\\in {\\mathbb Z}^d} \\chi_{\\Omega} (x+B^{-t}n)\\le 1. \\ \\  a.e.\\  x\\in B^{-t}(Q) .\n$$\nHere, $Q=[0,1)^d$.\n\\end{lemma}\n\n\nWe are now ready to prove  Theorem \\ref{lower triangle}.\n\n\n \\begin{proof}[Proof of Theorem \\ref{lower triangle}] As we mentioned above,\nby   Lemma  \\ref{lemma 3-prim} and the assumption that $A^tB$ is an integral matrix with $\\det(A^tB)=1$,  it is sufficient to prove the theorem for lattices  in the form of\n$$\n\\Lambda = \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       C & I  \\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d}).\n$$\nIn this case, by the orthogonality of $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$,\nthe exponentials $\\{e^{2\\pi i \\langle n, x\\rangle}: \\ n\\in \\Bbb Z^d\\}$  are mutual  orthogonal in $L^2(K)$. Thus,\nby Theorem \\ref{Th_union of FD}, $K = \\bigcup_{j=1}^N E_j$, where $E_j$\\rq{}s are fundamental domains of ${\\mathbb Z}^d$. \n\n\\medskip\n\nWe claim that $N=1$. Suppose that $N>1$.   By  Proposition \\ref{prop2}, there exists ${m}\\in{\\mathbb Z}^d$ such that  $K\\cap (K+{m})$ has positive Lebesgue measure and $K\\cap (K+{ m})$ is a packing  in $\\Bbb R^d$ by ${\\mathbb Z}^d$. Thus  by Lemma \\ref{lemma 4}, the set $\\{e^{2\\pi i \\langle  n,x\\rangle}: n\\in{\\mathbb Z}^d\\}$ is complete in $K\\cap (K+{m})$. By Remark \\ref{remark2}, ${\\mathbb Z}^d$ is exponentially complete in $L^2(K\\cap (K+{m}))$. Obviously, ${ m}=0$ cannot satisfy the packing property (3) in Proposition \\ref{prop2} since we have assumed $N>1$. Thus  ${m}\\ne 0$. On the other hand, the orthogonality of the Gabor system implies that for any $n\\in \\Bbb Z^d$ we must have\n\\begin{align}\\label{eq}\n   \\widehat{\\chi_{K\\cap (K+{m})}} (C{m}+n)= \\int_{\\Bbb R^d} \\chi_K(x)\\chi_{K}(x-{m}) e^{-2\\pi i \\langle C { m}, x\\rangle}e^{-2\\pi i \\langle  n, x\\rangle}dx  =0\n \\end{align}\n for all $n\\in {\\mathbb Z}^d$. \nThis contradicts the exponential completeness of ${\\mathbb Z}^d$. Therefore, the assumption $N>1$ cannot hold,  and $K$ is thus a fundamental domain of ${\\mathbb Z}^d$. This completes the proof.\n\\end{proof}\n\n\n\\begin{remark} In proving Theorem \\ref{lower triangle}, once $\\Lambda$ is reduced to the lattice \n$${\\mathcal A}({\\mathbb Z}^{2d}), \\ \\mbox{where} \\  {\\mathcal A}=  \\left(\n                    \\begin{array}{cc}\n                       I & O\\\\\n                       C & I  \\\\\n                    \\end{array}\n                  \\right),\n$$\none may suspect that we can apply a metaplectic transformation to further reduce the lattice to the separable lattice ${\\mathbb Z}^{d}\\times {\\mathbb Z}^d$ and hence the problem is trivially solved. Unfortunately, this approach does not seem to work. To be precise, one can look up \\cite{Groechenig-book} for the metaplectic transformation in this case. We have that ${\\mathcal G}(|K|^{-1\/2}\\chi_K,{\\mathcal A}({\\mathbb Z}^{2d}))$ is a Gabor orthonormal basis if and only if ${\\mathcal G}( \\tilde{g},{\\mathbb Z}^{d}\\times{\\mathbb Z}^d)$ is a Gabor orthonormal basis, where \n$$\n\\tilde{g} = e^{-\\pi  i\\langle x,Cx\\rangle}|K|^{-1\/2}\\chi_K.\n$$\nAlthough the lattice is separable, the window funciton $\\tilde{g}$ is now complex-valued, we cannot conclude that $K$ has  no overlap in the time domain as it were the case when $C=O$. \n\n\\smallskip\n\nNonetheless,  metaplectic transformation and symplectic matrices seems to provide some strong tools that may lead to a progess in the Fuglede-Gabor problem,  readers may refer to \\cite{Folland,deGosson,Groechenig-book} for details about these tools.\n\\end{remark}\n\n\\medskip\n\n\n\n Theorem \\ref{rational_dim1} is  now straightforward. We prove  it   here for the sake of completeness. \n\n\\begin{proof}[Proof of Theorem \\ref{rational_dim1}]\nLet $\\Lambda= M(\\Bbb Z^2)$ where $M=\\left(\n                                      \\begin{array}{cc}\n                                       a & d \\\\\n                                        c & b \\\\\n                                      \\end{array}\n                                    \\right)$  is a rational matrix with $\\det M=1$. Let $q$ be the least common multiple of $a, b, c $ and $d$. Then we can write $\\Lambda$ as  $\\Lambda= q^{-1} \\tilde M(\\Bbb Z^2)$ where  $\\tilde M$ is an integer matrix. By Lemma \\ref{HanW1},  we can find a  unimodular integer matrix  $P$  such that $\\tilde M P$ is the lower triangular integer matrix.  By the unimodularity of the matrix $P$  we have\n                                     $q^{-1} \\tilde M(\\Bbb Z^2) = q^{-1}  \\tilde M P(\\Bbb Z^2)$. Therefore, $\\Lambda = q^{-1}  \\tilde M P(\\Bbb Z^2)$ and  $q^{-1}  \\tilde M P$ is a lower triangular rational matrix. We can therefore write\n                                     $$\n                                     \\Lambda = \\left(\n                                      \\begin{array}{cc}\n                                       \\alpha & 0 \\\\\n                                        \\gamma & \\beta \\\\\n                                      \\end{array}\n                                    \\right)(\\mathbb Z^2)$$\n                                    for some $\\alpha, \\beta, \\gamma\\in \\Bbb Q$.\n                                    Notice that the density of $\\Lambda$  equals $1$, meaning that $\\alpha\\beta = 1$. Thus, all assumptions of Theorem \\ref{lower triangle} are satisfied. Hence, $K$ is a translational tile with tiling set $\\Bbb Z$ and is  a spectral set. \n\\end{proof}\n\n\n \\section{  Proof of Theorem \\ref{UT} - Upper triangular matrices}\\label{thm:rational_dim1 and UT}\nWe will discuss a case of upper triangular matrices which can be converted into  the lower one. Then we will use   Theorem \\ref{lower triangle} to prove the theorem.  First we need few lemmas.\n\\medskip\n\n\\begin{lemma}\\label{Gamma-D}\n    Let $D$  be  a $d\\times d$  rational matrix.\n    Then there is an integer  lattice $\\Gamma$ such that $D\\gamma\\in \\Bbb Z^d$ for $\\gamma\\in \\Gamma$.\n    \\end{lemma}\n    \\begin{proof}\n Define\n$$\n\\Gamma := \\{k\\in{\\mathbb Z}^d: Dk\\in{\\mathbb Z}^d\\}.\n$$\nIt is easy to check that $\\Gamma$ is a lattice contained in $\\Bbb Z^d$.  Moreover, $\\Gamma$  contains $p\\Bbb Z^d$ where\n $p$  is the least common multiple of the denominators of   entries of $D$.  Thus, $\\Gamma$ is  a full-rank lattice and has the form of\n$\n\\Gamma = M({\\mathbb Z}^d),\n $\nfor some  invertible $d\\times d$ matrix  $M$ with integer entries.\n\\end{proof}\nObserve that according to the  Lemma \\ref{Gamma-D},  for any given rational matrix $D$, there is an integer $M$ such that $DM$ is integer.\n   With this observation, we have the following result.\n\n \\medskip\n\n\\begin{lemma}\\label{complete residue classes} Let $D$ be a rational matrix, and  let   $\\Gamma$  and $M$   be given  as in Lemma \\ref{Gamma-D} and  $\\det M=n$.\nSuppose that $\\{\\gamma_1,\\cdots, \\gamma_n\\}$ be a complete representative (mod $M({\\mathbb Z}^d)$) in ${\\mathbb Z}^d$.  If $D$ is symmetric, then   $\\{M^tD\\gamma_1,\\cdots,$ $ M^tD\\gamma_n\\}$ is a complete representative (mod $M^t{\\mathbb Z}^d$) in ${\\mathbb Z}^d$.\n\\end{lemma}\n\\begin{proof} We saw above that by\n  the structure of $M$ and $\\Gamma$, $DM$ is an integer matrix, therefore $DM(\\mathbb Z^d)\\subseteq{\\mathbb Z}^d$. We also have  $(DM)^t{\\mathbb Z}^d\\subseteq{\\mathbb Z}^d$. This implies that\n$$M^tD({\\mathbb Z}^d) = (D^tM)^t({\\mathbb Z}^d )\\subseteq{\\mathbb Z}^d.$$\nThus, $M^tD\\gamma_i$ are all integer vectors for all $i = 1,\\cdots,n$.\n\n\\medskip\n\nNext, we show that $\\{M^tD\\gamma_1,...,M^tD\\gamma_n\\}$ consists of distinct representative (mod $M^t({\\mathbb Z}^d)$) in $\\Bbb Z^d$. Suppose that this is not the case. Then for some $i\\neq j$ we must have  $M^tD\\gamma_i - M^tD\\gamma_j \\in M^t({\\mathbb Z}^d)$. This implies that $D\\gamma_i-D\\gamma_j\\in {\\mathbb Z}^d$, which means that  $\\gamma_i$ and $\\gamma_j$ belong  to the same representative class (mod $M({\\mathbb Z}^d)$) in ${\\mathbb Z}^d$ which is a contradiction.\n\n\\medskip\n\nFinally, we show that $\\{M^tD\\gamma_1,...M^tD\\gamma_n\\}$ is complete.  This follows immediately by counting the  number of\ncosets   present in ${\\mathbb Z}^d\/M^t({\\mathbb Z}^d)$ which is  $|\\det(M^t)| = |\\det(M)|= n$. Hence, $\\{M^tD\\gamma_1,...M^tD\\gamma_n\\}$ is complete.\n\\end{proof}\n\nIn the following constructive lemma we  shall present   a class of   upper triangle lattice  which can be converted  into a lower triangular lattice.\n\n  \\begin{lemma}\\label{a technical lemma}\n            For   $\\Lambda=\n \\left(\n                                      \\begin{array}{cc}\n                                        I & D \\\\\n                                        O & I \\\\\n                                      \\end{array}\n                                    \\right)(\\Bbb Z^{2d})$  with $D$  rational and symmetric, there are integer matrices $E$ and $X$ such that\n         $\\Lambda = \\left(\n                                      \\begin{array}{cc}\n                                       E^{-t} & O \\\\\n                                        X & E\\\\\n                                      \\end{array}\n                                    \\right)({\\mathbb Z}^{2d})$.\n                                    \\end{lemma}\n\n                                    \\begin{proof}\n                                     Let $\\{{\\bf e_i}: \\ i=1, \\cdots, d\\}$ denote the standard basis in $\\Bbb Z^d$. Associated to this $D$ let\n                                      $\\Gamma$ and $M$ be given as in Lemma \\ref{Gamma-D}.\n Then by Lemma \\ref{complete residue classes}, for any $i$ there exists  $z_i\\in \\Bbb Z^d$ and $\\gamma^i\\in \\{\\gamma_1, \\cdots, \\gamma_n\\}$ such that ${\\bf e_i}= M^tz_i + M^tD\\gamma^i$.  Put $Z:=[z_1 \\cdots z_d]$\n and $X=[\\gamma_1\\cdots \\gamma_d]$.  It is clear that $Z$ and $X$ are integer matrices.\n\n\n  A direct calculation shows that $(Z+DX)(\\Bbb Z^d)= M^{-t}(\\Bbb Z^d)$.\n\n \\medskip\n\nPut $P:=\\left(\n                                      \\begin{array}{cc}\n                                     Z& -DM \\\\\n                                        X & M\\\\\n                                      \\end{array}\n                                    \\right)$. Then $P$ is an integer matrix with  $\\det P=1$ and   $P(\\Bbb Z^{2d}) = \\Bbb Z^{2d}$. Recall that\n $DM$ is an integer matrix. So  we can  write\n                                    $$\\Lambda=\n \\left(\n                                      \\begin{array}{cc}\n                                        I & D \\\\\n                                        O & I \\\\\n                                      \\end{array}\n                                    \\right)(\\Bbb Z^{2d}) = \\left(\n                                      \\begin{array}{cc}\n                                        I & D \\\\\n                                        O & I \\\\\n                                      \\end{array}\n                                    \\right)P(\\Bbb Z^{2d}) = \\left(\n                                      \\begin{array}{cc}\n                                        M^{-t} & O \\\\\n                                       X &  M\\\\\n                                      \\end{array}\n                                    \\right)(\\Bbb Z^{2d}). $$ Now take $E=M$, and we are done.\n\\end{proof}\n\\medskip\n\nAt this point  we are ready to complete the proof of Theorem \\ref{UT}.\n\n\\begin{proof}[Proof of Theorem \\ref{UT}]\n Let $\\Lambda=\n \\left(\n                                      \\begin{array}{cc}\n                                        A & D \\\\\n                                        O & B \\\\\n                                      \\end{array}\n                                    \\right)(\\Bbb Z^{2d})$ where  $B= A^{-t}$ and $\\widetilde{D}: = A^{-1}D$ is rational and symmetric.  By Lemma \\ref{lemma 3-prim},  the  associated matrix  can be reduced to a block matrix   of the form $\\left(\n                                      \\begin{array}{cc}\n                                        I &  \\widetilde{D} \\\\\n                                        O & I \\\\\n                                      \\end{array}\n                                    \\right)$ where $\\widetilde{D}$ is   rational and symmetric. Therefore, it is sufficient to prove the theorem for  a lattice of the form\n            $\\Lambda=\n \\left(\n                                      \\begin{array}{cc}\n                                        I &D \\\\\n                                        O & I \\\\\n                                      \\end{array}\n                                    \\right)(\\Bbb Z^{2d})$  where $D$  is rational and symmetric.  By Lemma \\ref{a technical lemma}, there are integer matrices $E$ and $X$ such that $\\Lambda=  \\left(\n                                      \\begin{array}{cc}\n                                       E^{-t} & O \\\\\n                                        X & E\\\\\n                                      \\end{array}\n                                    \\right)({\\mathbb Z}^{2d})$.    The   rest of the proof is  now a conclusion of   Theorem \\ref{lower triangle}  and we have completed the proof.\n                \\end{proof}\n\n            \\medskip\n\n In \\cite{HanW4}, the authors  proved that for  any lattice $\\Lambda$ formed by an upper triangular matrix exists a window $g$ such that ${\\mathcal G}(g,\\Lambda)$ is a Gabor orthonormal basis and such $g$ satisfies $\\widehat{g} = \\chi_K$, where $K$ is the common fundamental domain for the diagonal block matrices $A$ and $B^{-t}$.  However, their proof does not  provide any  constructive technique   for producing a compactly supported window. The proof of Theorem \\ref{UT}    provides a technique to    construct a large class of examples of   sets  $K$ forming a Gabor orthonormal basis with respect to the lattice generated by upper triangular matrices. We explain this next.\n\n\\medskip\n\n\\begin{proposition}\\label{converse of UT} Assume that the lattice $\\Lambda$ and matrices $A$, $B$ and $D$ are given as in Theorem \\ref{UT} and satisfying\n the hypotheses of the theorem. Then there is a set $K$ such that ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^d)$. \n\\end{proposition}\n\n \\begin{proof}\n Let $A$, $B$ and $D$ be given. Notice that \n  by Lemma \\ref{lemma 3-prim} and the hypotheses of the proposition we know that for any given set $K$, the system ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$  is an orthonormal basis  if and only if ${\\mathcal G}(|A^{-1}(K)|^{-1\/2}\\chi_{A^{-1}(K)}, \\tilde \\Lambda)$  is an orthonormal basis where $\\tilde\\Lambda= \\left(\\begin{array}{cc} I & A^{-1}D\\\\ O & I\\end{array}\\right)(\\Bbb Z^{2d})$.  And, by Lemma \\ref{a technical lemma} we also know that for $\\tilde\\Lambda$  there   are integer matrices  $X$ and $E$    such\n  $$E(\\Bbb Z^d) = \\{n\\in \\Bbb Z^d: ~ A^{-1}Dn\\in \\Bbb Z^d\\}$$ \n  and \n $\\widetilde\\Lambda=    \\left(\\begin{array}{cc}  {E}^{-t} & O\\\\  X &   E \\end{array}\\right)(\\Bbb Z^{2d})$.\n  By appealing to Lemma \\ref{lemma 3-prim} one more time,\n  ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis if and only if\n  %\n $${\\mathcal G}\\left(|K|^{-1\/2}\\chi_{K},  \\left(\\begin{array}{cc} A{E}^{-t} & O\\\\   A^{-t} X & A^{-t} E \\end{array}\\right)(\\Bbb Z^{2d})\\right)$$\n  is an orthonormal basis. Now take $K$ as a fundamental domain of  $AE^{-t}(\\Bbb Z^d)$ and we are done. \n \\end{proof}\n\n\\medskip\n\nThe following gives a more  explicit example.\n\n\\begin{example}\\label{D not I} Let $A = \\left(\n                                      \\begin{array}{cc}\n                                        \\frac{1}{2} & 0\\\\\n                                   0 & 2 \\\\\n                                      \\end{array}\n                                    \\right)$, $B = \\left(\n                                      \\begin{array}{cc}\n                                      2 & 0 \\\\\n                                   0 &   \\frac{1}{2} \\\\\n                                      \\end{array}\n                                    \\right)$. Take $D= \\left(\n                                      \\begin{array}{cc}\n                                       1 & 0\\\\\n                                   0 &  \\frac{1}{3} \\\\\n                                      \\end{array}\n                                    \\right)$.   Then    $A^{-1}D$  is   rational  and symmetric.  Let $\\Gamma= \\{n\\in \\Bbb Z^2: \\ A^{-1}Dn\\in \\Bbb Z^2\\}$.   $\\Gamma$ is a full lattice and a simple  calculation shows that $\\Gamma= E(\\Bbb Z^2)$ where  $E=     \\left(\n                                   \\begin{array}{cc}\n                                1& 0\\\\\n                              0 & 6 \\\\\n                                       \\end{array}\n                                       \\right)$.\n                                      Now let $K$   be any   fundamental domain of the lattice $AE^{-t} ({\\mathbb Z}^2)= \\left(\n                                        \\begin{array}{cc}\n                                1\/2& 0\\\\\n                              0 & 1\/3 \\\\\n                                       \\end{array}\n                                       \\right)(\\Bbb Z^2)$. Then by Proposition \\ref{converse of UT}  the system\n $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^2)$   for the lower triangular lattice $\\Lambda$ with diagonal matrices  $AE^{-t}$ and $A^{-t}E$ and any matrix $C$.\n                                       \\end{example}\n\n\n\\iffalse\n \\begin{example}\\label{D=I} Let $A = \\left(\n                                      \\begin{array}{cc}\n                                        \\frac{1}{2} & 0\\\\\n                                   0 & 2 \\\\\n                                      \\end{array}\n                                    \\right)$, $B = \\left(\n                                      \\begin{array}{cc}\n                                      2 & 0 \\\\\n                                   0 &   \\frac{1}{2} \\\\\n                                      \\end{array}\n                                    \\right)$. Take $D=I$.  Then   $A^{-1}D=A^{-1}$ is  rational  and symmetric.  Let $\\Gamma= \\{n\\in \\Bbb Z^2: \\ A^{-1}n\\in \\Bbb Z^2\\}$.   $\\Gamma$ is a full lattice and we can see  that $\\Gamma= E(\\Bbb Z^2)$ where  $E=     \\left(\n                                   \\begin{array}{cc}\n                                1& 0\\\\\n                              0 & 2 \\\\\n                                       \\end{array}\n                                  \\right)$.\n Now  take $K=[0,1\/2]\\times [0,1]$. Then $K$ is  a fundamental domain for the lattice $AE^{-t} ({\\mathbb Z}^2)$\nand $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^2)$ by Proposition \\ref{converse of UT}.\n \\end{example}\n\\fi\n\n \\section{Examples}\\label{Examples}\nConsider $\\Lambda=L(\\Bbb Z^{2d})$ where $L= \\left(\\begin{array}{cc}\n                                        A & O \\\\\n                                        C & B \\\\\n                                      \\end{array}\n                                    \\right)$ is  rational.  In Theorem \\ref{lower triangle} we proved that for a set $K$ if $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$ and    $A^tB$ is an integer matrix, then  $K$ is a fundamental domain of $A(\\Bbb Z^d)$.  By Theorem \\ref{Th_union of FD} this means $N=1$.  Thus  $K$ tiles by $A({\\mathbb Z}^d)$ and  is  spectral with spectrum $B^{-t}(\\mathbb Z^d)$.  In this section   we will provide examples showing that $N>1$ can also happen in Theorem \\ref{Th_union of FD} if $A^tB$ is not an integer matrix and yet the system $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$. \n\n\n\n\n\n\n                                      \n                                      \n                                      \n                                       %\n                                       %\n                  \n\n\n\n\n                                      \\medskip\n\n                                    We are now ready to present our example of  a  set $K$ which is the union of fundamental domains of  lattice  $A(\\Bbb Z^d)$ and  the union of fundamental domains of lattice  $B^{-t}(\\Bbb Z^d)$,    $\\chi_K$ is a  window function for a possible Gabor orthonormal basis, and $A^tB$ is {\\it not an integer matrix}. \n\n\n\\begin{example}\\label{mutli-tile K}\n\nLet $K = [0,2]\\times [0,1]$ and\n$\\Lambda = \\left(\n                                      \\begin{array}{cccc}\n                                        I& O \\\\\n                                       C & B \\\\\n                                      \\end{array}\n                                    \\right)({\\mathbb Z}^{4})$\nwhere\n                                    $B=\n  \\left(\n                                     \\begin{array}{cc}\n                                        1\/2 & 0  \\\\\n                                         0&  2\n                                      \\end{array}\n                                    \\right)$ and $C=  \\left(\n                                     \\begin{array}{cc}\n                                        c_{11}& c_{12}  \\\\\n                                         c_{21}&  c_{22}\n                                      \\end{array}\n                                    \\right)$.    The system $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis  if\n                         $c_{21}$ is  an odd number.\n                                   \\end{example}\n\n                                   \\begin{proof}\n Observe  that\n\\begin{align}\\notag\n[0,2]\\times [0,1] &= [0,1]^2 + \\{(0,0), (1,0)\\} \\\\\\notag\n &= [0,2]\\times[0,1\/2]+ \\{(0,0), (0,1\/2)\\}.\n \\end{align}\n\nThis shows that   $K$ is   union of two fundamental domains of $\\Bbb Z^2$ and union of two fundamental domains of $B^{-t}(\\Bbb Z^2)$, respectively.\n\n  Let $({m}, n) \\in {\\mathbb Z}^4$ with ${m},{ n}  \\in{\\mathbb Z}^2$ and $({m},{ n})\\neq (0,0)$.  Put\n %\n\\begin{align}\\notag\n\\mathcal I: = &\\int \\chi_{K}(x-{m})\\chi_{K}(x) e^{-2\\pi i \\langle C({m}), x\\rangle} e^{-2\\pi i \\langle B({n}), x\\rangle}dx\\\\\\notag\n=&\\int \\chi_{(K+{\\bf m})\\cap K}(x) e^{-2\\pi i \\langle C({ m}), x\\rangle} e^{-2\\pi i \\langle B({n}), x\\rangle}dx .\n\\end{align}\n Mutual orthogonality of the Gabor system will follow if we can show that $\\mathcal I=0$ for any $( m, n)\\neq (0,0)$. Note that\n$$\n(K+m)\\cap K \\simeq \\left\\{\n                                         \\begin{array}{ll}\n                                           K, & \\hbox{${m} = 0$;} \\\\\n                                                  {[0,1]}^2, & \\hbox{${m} = (\\pm 1,0)$;} \\\\\n                                           \\emptyset , & \\hbox{otherwise.}\n                                         \\end{array}\n                                       \\right.\n$$\n\n(Here, we write $A\\simeq B$ if $A = B+k$ for some $k\\in{\\mathbb Z}^d$.)\n\n\n  Since $K$ is a union of fundamental domains for $B^{-t}(\\Bbb Z^2)$, then  if ${m}=0$, we automatically have   $\\mathcal I=0. $\n\n\n                                   If ${ m}= (\\pm 1,0)$,  for any ${ n} = (n_{1},n_{2})$, $\\mathcal I$ is equal to the following integral up to a unimodular constant:\n\\begin{equation}\\label{eq6.3}\n\\begin{aligned}\n\\int_{[0,1]^2} e^{-2\\pi i \\langle C((\\pm 1, 0)^t), x\\rangle}  e^{-2\\pi i \\langle B({n}), x\\rangle}dx=&\\int_{[0,1]}\\int_{[0,1]} e^{-2\\pi i (\\pm c_{11}x_1\\pm c_{21}x_2)}  e^{-2\\pi i (\\frac{1}{2}n_{1}x_1+2n_{2}x_2)}dx_1dx_2 \\\\\n=& \\widehat{\\chi_{[0,1]}}(\\pm c_{11}+\\frac{1}{2} n_1) \\widehat{\\chi_{[0,1]}}(\\pm c_{21}+2 n_2).\n\\end{aligned}\n\\end{equation}\n                             It is obvious that the last line   equals to zero for all $(n_1, n_2)$ only  if  $c_{21}$ is an odd number. For other cases of $m$ it is trivial that $\\mathcal I=0$. Thus the orthogonality is obtained and\n                                the completeness\n                              is a direct conclusion of  Proposition \\ref{complt}.\n\\end{proof}\n\nWe notice that the previous example exploited the fact that $B({\\mathbb Z}^2)$ is exponentially incomplete  for $L^2[0,1]^2$ in (\\ref{eq6.3}). The following   example  illustrates a case where, contrary to the previous example,    the  finite union of fundamental domains  cannot form  a Gabor orthonormal basis for any choice of matrix  $C$.\n\n\\medskip\n\n                  \\begin{example}\\label{example2}\n                  Let $K=[0,2]^2$ and let  $B$ be the matrix as in Example \\ref{mutli-tile K}. Then $K$ is   union of $4$ fundamental domains of  the  lattice  $\\Bbb Z^2$ and union of $4$ fundamental domains of  $B^{-t}(\\Bbb Z^2)$. However,  there is  no  matrix $C$ for which $\\chi_K$ is a   window function yielding an orthonormal basis for the lower triangular  lattice $\\Lambda=\\left(\\begin{array}{cc} I & O\\\\ C & B\\end{array}\\right)(\\Bbb Z^{4})$.    \\end{example}\n\n \n\n\\begin{proof}\nThe fact that $K$  is union of  fundamental domains of $\\Bbb Z^d$ and $B^{-t}(\\Bbb Z^d)$ is straight forward. In short,\n$$\n\\begin{aligned}\nK =& [0,1]^2  + \\{(0,0), (1,0), (0,1), (1,1)\\}\\\\\n  =& [0,2]\\times [0,1\/2] + \\{(0,0), (0,1\/2), (0,1), (0,3\/2)\\}.\n\\end{aligned}\n$$\nSuppose that there exists a matrix $C$ such that $\\chi_K$ is a   window function for the lower triangular  lattice $\\Lambda=\\left(\\begin{array}{cc} I & O\\\\ C & B\\end{array}\\right)(\\Bbb Z^{4})$. Thus,    for any $(0,0)\\neq (m, n)\\in \\Bbb Z^{4}$ we must have\n$$\n 0 = {\\mathcal I}:  =\\int \\chi_{(K+{m})\\cap K}(x) e^{-2\\pi i \\langle C({ m}), x\\rangle} e^{-2\\pi i \\langle B({n}), x\\rangle}dx.\n$$\nFrom the other side, the only non-empty intersection sets $(K+ m)\\cap K$ are\n$$\n(K+m)\\cap K \\simeq \\left\\{\n  \\begin{array}{ll}\n    K, & \\hbox{if }{m}=0 \\\\\n    {[0,1]\\times[0,2]}, & \\hbox{if } {m} = (\\pm1,0) \\\\\n    {[0,2]\\times[0,1]}, & \\hbox{if }  m = (0,\\pm1) \\\\\n    {[0,1]}^2, & \\hbox{if } m = (\\pm1,\\pm1).\n  \\end{array}\n\\right.\n$$\nFrom the last three intersections, we obtain that  if ${\\mathcal I}=0$, then  the following three equations must hold for all integer vectors $(n_1,n_2)$, respectively:\n\n\n  \\begin{align}\\notag\n    \\widehat{\\chi_{[0,1]}}(\\pm c_{11}+\\frac{1}{2} n_1) \\widehat{\\chi_{[0,2]}}(\\pm c_{21}+2 n_2) &=0  \\\\\\notag\n         \\widehat{\\chi_{[0,2]}}(\\pm c_{12}+\\frac{1}{2}n_1) \\widehat{\\chi_{[0,1]}}(\\pm c_{22}+2 n_2)\n         &=0\\\\\\notag\n       \\widehat{\\chi_{[0,1]}}(\\pm c_{11}\\pm c_{12}+ \\frac{1}{2} n_1) \\widehat{\\chi_{[0,1]}}(\\pm c_{21}\\pm c_{22}+2 n_2) &=0\n \\end{align}\n\n\\medskip\n\nWe claim that if   the first equation holds, then $\\widehat{\\chi_{[0,2]}}(\\pm c_{21}+2 n_2) =0$ for all integers $n_2$ and $c_{21}\\in 2{\\mathbb Z}+\\{\\frac{1}{2} ,1,\\frac{3}{2} \\}$.\n\n\\medskip\n\nTo justify the claim, suppose that there exists an integer $n_2$ such that $\\widehat{\\chi_{[0,2]}}(\\pm c_{21}+2 n_2) \\ne0$. Then $\\widehat{\\chi_{[0,1]}}(\\pm c_{11}+\\frac{1}{2} n_1)=0$ for all integers $n_1$. However, this would imply the existence of an exponentials $e^{-2\\pi i c_{11} x}$ such that it is orthogonal to all $\\{e^{2\\pi i (\\frac{1}{2}  n) x}: n\\in{\\mathbb Z}\\}$ in $L^2[0,1]$. This is impossible since the exponentials set $\\{e^{2\\pi i (\\frac{1}{2}  n) x}: n\\in{\\mathbb Z}\\}$ is complete in $L^2[0,1]$. Hence, we have only $\\widehat{\\chi_{[0,2]}}(\\pm c_{21}+2 n_2) =0$ for all integers $n_2$. Finally, since the zero set for $\\widehat{\\chi_{[0,2]}}$ is $\\frac{1}{2} {\\mathbb Z}$ except zero, then we must have  $c_{21}\\in 2{\\mathbb Z}+\\{\\frac{1}{2} ,1, \\frac{3}{2} \\}$, as desired.\n\n\\medskip\n\nSimilarly, the second equation  implies that $\\widehat{\\chi_{[0,1]}}(\\pm c_{22}+2 n_2) =0$ for all integers $n_2$ and thus $c_{22}\\in 2{\\mathbb Z}+1$ must be an odd integer.\n\n\\medskip\n\nThe third equation implies that\n\\begin{equation}\\label{eq5.4}\n\\widehat{\\chi_{[0,1]}}(\\pm c_{21}\\pm c_{22}+2 n_2) =0\n\\end{equation}\nfor any integers $n_2$. Now, we write $c_{21} = 2k_1+ \\frac{1}{2} j$, for some $j = 1,2,3$ and $c_{22} = 2k_2+1$. Then $\\pm c_{21}\\pm c_{22}\\in 2{\\mathbb Z}+ \\{\\frac{3}{2},2,\\frac{5}{2}\\}$. In the case of fraction, (\\ref{eq5.4}) cannot be zero. If $\\pm c_{21}\\pm c_{22} = 2m+2$, we take $n_2 = -m-1$. Then  (\\ref{eq5.4}) will  imply $\\widehat{\\chi_{[0,1]}}(0) = 1$, which  is impossible.  Thus, the third equation can never be zero. This implies that  such $C$ does not exist.\n\\end{proof}\n\n\\medskip\nThe following example  proves that the boundedness property for the set $K$ is necessary in Proposition \\ref{prop2}. \n\n                  \\begin{example}\\label{example3} Let $I_0 = [0,1)$ and $I_n = \\left[1-\\frac{1}{2^{n-1}},1-\\frac{1}{2^{n}}\\right)$ for $n\\ge1$. Define \n                  $$\n                  K = \\bigcup_{k\\in{\\mathbb Z}} (k+ I_{|k|}) \n                  $$\n                  The set $K$ is unbounded and we have the following. \n                  \n\\begin{enumerate}\\item $K$ multi-tiles ${\\mathbb R}$ by ${\\mathbb Z}$ at level 3. However, for any $m\\ne 0$, $K\\cap( K+m)$ is not a packing of ${\\mathbb R}$. Therefore, Proposition \\ref{prop2} does not hold if $K$ is unbounded.\n \\item Nonetheless, $K$ cannot form a Gabor orthonormal basis using lattices of the form $\n  \\left(\n                                     \\begin{array}{cc}\n                                        1 & 0  \\\\\n                                         c&  1\n                                      \\end{array}\n                                    \\right) ({\\mathbb Z}^2)$.  \\end{enumerate}\n\\end{example}\n\n\n\\begin{proof}\nThe fact that $K$ mult-tiles ${\\mathbb R}$ by ${\\mathbb Z}$ at level 3 follows from a direct observation, so  we omit the details here. We note that for any $m\\ne 0$,\n$$\nK\\cap (K+m)\\supset I_{|m|} \\cup (I_{|m|}+m).\n$$\nHence, for all $x\\in I_{|m|}$, $\\sum_{n\\in{\\mathbb Z}} \\chi_{K\\cap(K+m)}(x+n)\\ge 2$. Therefore, it is never a packing for any $m\\ne 0$. To show  the last statement, notice \n\n$$\nK\\cap (K+1) = \\left[0,\\frac12\\right)\\cup\\left[1,\\frac32\\right), \\ K\\cap (K+2) = \\left[\\frac12,\\frac34\\right)\\cup\\left[1,\\frac32\\right)\\cup\\left[\\frac52,\\frac{11}{4}\\right).\n$$\nSuppose that $K$ forms a Gabor orthonormal basis using some lattice of the form $\n  \\left(\n                                     \\begin{array}{cc}\n                                        1 & 0  \\\\\n                                         c&  1\n                                      \\end{array}\n                                    \\right) ({\\mathbb Z}^2)$. Then $\\widehat{\\chi_{K\\cap(K+1)}}(c+m) = 0$ and $\\widehat{\\chi_{K\\cap(K+2)}}(c+m) = 0$ for all $m\\in{\\mathbb Z}$. In particular, \n $$\n \\widehat{\\chi_{K\\cap(K+1)}} (c+m) = (1+e^{2\\pi i (c+m)})\\widehat{\\chi_{[0,1\/2]}}(c+m) =0, \\forall m\\in{\\mathbb Z}.\n                                    $$\nWe see that the only possibility of the above is that $c\\in\\frac12+{\\mathbb Z}$. We now consider \n$$\n\\widehat{\\chi_{K\\cap(K+2)}} (\\xi) = e^{2\\pi i \\frac12\\xi}\\left(1+e^{2\\pi i \\frac12\\xi}+e^{2\\pi i \\frac34\\xi}+e^{2\\pi i \\frac2\\xi}\\right)\\widehat{\\chi_{[0,1\/4)}}(\\xi).\n$$\nIf $c \\in\\frac12+{\\mathbb Z}$ and $\\widehat{\\chi_{K\\cap(K+2)}} (c) =0$, we must have \n$$\n0=1+e^{2\\pi i \\frac12c}+e^{2\\pi i \\frac34c}+e^{2\\pi i 2c} = 2+e^{2\\pi i \\frac12c}+e^{2\\pi i \\frac34c}.\n$$ \nThis forces $e^{2\\pi i \\frac12c} = e^{2\\pi i \\frac34c}=-1$. Thus, \n$$c\\in 2(1\/2+{\\mathbb Z})\\cap \\frac43(1\/2+{\\mathbb Z}) = (1+2{\\mathbb Z})\\cap \\left(\\frac23+\\frac43{\\mathbb Z}\\right)\\subset 1+2{\\mathbb Z}$$\nThis is a contradiction since $c\\in\\frac12+{\\mathbb Z}$ is never an  integer. This completes the proof.\n \\end{proof}\n\n\n\n\n\n\n\n\n \n   \n     \n       \n         \n           \n             \n               \n\n\n\n\n\n\n\n\\section{Discussions and Open problems}\\label{Open problems}\n\nThis paper investigates the Fuglede-Gabor Problem \\ref{our conjecture1} over the lattices. We believe that this  problem should be true for all lattices. We solved the problem completely in dimension one when the lattice  is rational and in higher dimensions when the lattice is integer.   In what follows we shall explain how to resolve Problem \\ref{our conjecture1}  in full generality for any  lattices  $\\Lambda = \\left(\n                                      \\begin{array}{cc}\n                                       A  & D \\\\\n                                        C & B \\\\\n                                      \\end{array}\n                                    \\right)({\\mathbb Z}^{2d})$. In fact,\n it is sufficient  to solve the following two cases.\n\\begin{enumerate}\n\\item[(1)]\\label{item 1} {\\bf Rational case:} After converting a rational  lattice into a lower triangular rational matrix by Corollary \\ref{M=N} and   reducing the  matrix   where $A=I$  by    Lemma \\ref{lemma 3-prim},   $\\Lambda$ is a lattice of the form of   $  \\left(\n                                      \\begin{array}{cc}\n                                        I  & O \\\\\n                                        C & B \\\\\n                                      \\end{array}\n                                    \\right)({\\mathbb Z}^{2d})$, where $B$ and $C$ are $d\\times d$ rational matrices but  $B$ is not necessarily an integral matrix. \n\n\n                                    \\medskip\n\n                    \n\n\n\n\\item[(2)] {\\bf Irrational case:}  After a reduction process  by Lemma \\ref{lemma 3-prim}, $\\Lambda$ is a lattice of the form of  \n$$\\left(\n                    \\begin{array}{cc}\n                       I & D\\\\\n                       C & B  \\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d})$$  which   contains  irrational entries.\n\\end{enumerate}\n\n\n\\medskip\nIn what follows, we shall discuss these two cases in more details.\n\n\n\n\\subsection{Rational Case}\nLet  $\\Lambda= \\left(\n\\begin{array}{cc}\nA & O\\\\\nC & B\n\\end{array}\\right)(\\Bbb Z^{2d})$ be a lower triangular rational matrix.\n Example \\ref{mutli-tile K}  tells us that when $A^tB$ is  a non-integer matrix, $K$ is  not a  fundamental domain for the lattices  $A(\\Bbb Z^d)$ and $B^{-t}(\\Bbb Z^d)$ but the union of their fundamental domains. However, in Example \\ref{example2} we see that there exists no $C$ such that $K$, as the  union of fundamental domains, is a set whose characteristic function generates a  Gabor orthonormal basis associated to the given matrices $A$ and $B$. We predict that this failure is due to the  number of decompositions of $K$ into fundamental domains of $B^{-t}(\\Bbb Z^d)$.\n In this concern and in relation to  the examples illustrated  in Section \\ref{Examples},\n  we conjecture the following problem  for the lattices\n$\\Lambda= \\left(\n                    \\begin{array}{cc}\n                       I & D\\\\\n                       C & B  \\\\\n                    \\end{array}\n                  \\right) ({\\mathbb Z}^{2d})$.\n \\medskip\n\n\n\n  \\medskip\n\n  \\begin{conjecture}\\label{our conjecture}   Let $K\\subset \\Bbb R^d$ and $B$ be a  rational matrix with  $\\det(B)=1$. Let\n  $s$ be  the least common multiple of the denominators of the  matrix entries  $(b_{ij})$.\n There  is a matrix $C$ such that $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a  Gabor orthonormal  basis  for $L^2(\\Bbb R^d)$  if and only if $K$ tiles   and\n  $$K= \\bigcup_{i=1}^sE_i= \\bigcup_{i=1}^sF_i $$\n  where $E_i$  and $F_i$ are\n   fundamental  domains  of $\\Bbb Z^d$   and  $B^{-t}(\\Bbb Z^d)$, respectively.\n \\end{conjecture}\n\nIt is obvious that the conjecture automatically holds when $B$ is an  integer matrix which means  $s=1$.  Indeed,  this is  the result of Theorem \\ref{lower triangle}.\n\n\n\n\n\n\n \\medskip\n\n Observe that if  $K$ is as  in Conjecture \\ref{our conjecture},   then for any non-zero $n\\in \\Bbb Z^d$ we have $\\int_K e^{2\\pi i \\langle Bn, x\\rangle} dx=0$. And, there are only finitely many $m\\in \\Bbb Z^d$ such that $|K\\cap (K+m)|\\neq 0$, as $K$ is a finite union of fundamental domains of $\\Bbb Z^d$.  To prove the orthogonality,  one must first show the existence of  a matrix $C$ such that for  all   $n\\in \\Bbb Z^d$\n\\begin{equation}\\label{eq6}\n\\int_{K\\cap K+m}  e^{-2\\pi i \\langle Cm, x\\rangle}  e^{-2\\pi i \\langle Bn, x\\rangle}  dx=0,\n\\end{equation}\nwhich is equivalent to the exponential incompleteness of the lattices $B(\\Bbb Z^d)$ for $L^2(K\\cap K+m)$. It appears that we need to study the exponential completeness of the lattices over different domains.  In fact, the following problem has not yet had a definite answer.\n  \n  \\medskip\n\n\\begin{problem} Given a set $K$ with positive and finite measure,  classify all invertible $d\\times d$ matrices $B$ for which \n the exponents $\\{ e^{-2\\pi i \\langle Bn, x\\rangle} :  n\\in \\Bbb Z^d\\}$ are  exponentially complete in   $L^2(K)$. \n\\end{problem}\n\n\\medskip\n\nProposition \\ref{prop_G} provides a sufficient condition for matrices $B$  when $K=[0,1]^d$. Unfortunately, the converse of the proposition does not hold in dimensions $d=2$ and higher, as the counterexample following Proposition \\ref{prop_G} shows. However,  in Proposition \\ref{dimension one} we obtain a full characterization of exponential completeness for $[0,1]$ in dimension $d=1$ for integer lattices in $\\Bbb Z$.    \n\n\n\\medskip \n\nAs mentioned earlier, the exponential completeness of a set does not imply the completeness of the set in general. For a recent developments in study of \n the completeness, frame and Riesz bases properties of  exponentials we refer the reader to the paper by  De Carli and her co-authors \\cite{De Carli}. \n\n\n\n \n\n\n\n\n\\subsection{Irrational Cases} The case for irrational lattices is more challenging and complicated. It appears that the lower and upper triangular case is asymmetric.  We have seen that in Theorem \\ref{lower triangle}, the lower triangular block matrix $C$ is not involved in the statement.  Thus, irrational entries are allowed for lower triangular matrices. On the other hand, Han and Wang \\cite{HanW4} implicitly conjectured the following problem in their paper  \\cite{HanW1}:\n\n\\medskip\n\n {\\bf Han and Wang\\rq{}s Conjecture:} {\\it  Let $\\Lambda = \\left(\n                                      \\begin{array}{cc}\n                                        1 & \\alpha \\\\\n                                        0 & 1 \\\\\n                                      \\end{array}\n                                    \\right)({\\mathbb Z}^{2})$ where $\\alpha$ is irrational. Then there doesn't exist compactly supported window $g$ such that ${\\mathcal G}(g,\\Lambda)$ forms a Gabor orthonormal basis for $L^2({\\mathbb R})$.}\n\n \\medskip\n\n Observe that if $\\Lambda=  \\left(\n                                      \\begin{array}{cc}\n                                     1& \\alpha \\\\\n                                   0 &   1 \\\\\n                                      \\end{array}\n                                    \\right)$, with $\\alpha$ irrational, our method applied to construct Example \\ref{D not I}  would not work for the existence of $K$ since in that case $\\Gamma=\\{0\\}$. This observation predicts that  Han and Wang Conjecture  \\cite{HanW1}  might be true, although we do not have a proof for it now.  However, a simple calculation shows that the function $\\chi_{[0,1]}$ can not be a window function for this  lattice. \n\n                                    \\medskip\n\n\n \n\n\\subsection{Full generality of Fuglede-Gabor Problem \\ref{our conjecture1}} \nIt is known that non-symmetric convex bodies   as well as  convex sets with a point of non-vanishing Gaussian curvature    have no basis of exponentials and yet they do not tile \\cite{K99,IKT01}. Recently, similar results were also   proved for Gabor bases.  Indeed, the authors in \\cite{IM17} proved that in dimensions $d\\neq1$    (mod 4),  convex sets with a point of non-vanishing Gaussian curvature cannot generate any Gabor orthonormal basis with respect to any countable time-frequency set $\\Lambda$. Also, the authors in \\cite{CL} proved that non-symmetric convex polytopes do not produce any Gabor orthonormal basis with respect to any  countable time-frequency set. However, the result for non-symmetric  convex domains is not known yet.  The existent  results predict that  Fuglede-Gabor problem will  still be true to some  extent.\n\n\n \n\n\n\\medskip\n\n\n\n One may notice that there are also \n  examples of  spectral sets which do not tile by translations (see e.g.\n \\cite{T04}, \\cite{Matolcsi} and \\cite{KM06}). Therefore, by a result of Dutkay and the first listed author \\cite{Dutkay-Lai}, it is known that the indicator function of these sets can not serve as    Gabor orthonormal window with resepct to  any separable time-frequency set. However, they may still produce a  Gabor orthonormal basis using some non-separable and countable  time-frequency sets. \n  This   requires some input of new ideas.\n\n\n\\medskip\n\nFinally, it is known that octagon does not tile ${\\mathbb R}^2$ by translations, but it is a multi-tile by ${\\mathbb Z}^2$. The following example tells us that it does not form Gabor orthonormal basis using any lattices, confirming that the Fuglede-Gabor problem  holds up to some extent.\n\n\\medskip\n\n\n\n\n\n                                      \\begin{example}\\label{Octagon} Let ${\\mathcal O}_8\\subset \\Bbb R^2$ be the octagon symmetrically centred at the origin with integer vertices $\\{(\\pm 1, \\pm 2), (\\pm 2, \\pm 1)\\}$. ${\\mathcal O}_8$ multi-tiles $\\Bbb R^2$ with $\\Bbb Z^2$  and it is the  union of  $s=14$ fundamental domains of $\\Bbb Z^2$.  Yet there doesn't exist any $\\Lambda=   \\left(\n                                      \\begin{array}{cc}\n                                        I & O \\\\\n                                        C & B \\\\\n                                      \\end{array}\\right) (\\Bbb Z^4)$ with rational matrix $B$  for which   $\\mathcal G(|{\\mathcal O}_8|^{-1\/2}\\chi_{{\\mathcal O}_8}, \\Lambda)$ forms an  orthonormal basis for $L^2(\\Bbb R^2)$ .\n                                      \\end{example}\n                                      \n                                      \n                                      The proof of this example is involved and so it will be provided in the next section.    \n                                      \n                                                                      \n\n\n                                         \n                                           \n \n \n\\iffalse\n \\begin{lemma}\nLet $K$ be a measurable set in $\\Bbb R^d$ with positive and finite measure. Let $\\Lambda$ be the lower triangular lattice with diagonal block matrices $A=I$ and $B$ and $C\\neq 0$.\n\n  Assume that there is a non-zero $m\\in \\Bbb Z^2$ such that $K\\cap K+m$ is a packing by  $B^{-t}(\\Bbb Z^d)$. Then $\\mathcal G(\\chi_K, \\Lambda)$ is not an orthogonal set.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is a direct consequence of Lemma \\ref{lemma 4}.\n\\end{proof}\n\\fi\n\n\\section{Gabor orthonormal bases on Octagon and Proof of Example \\ref{Octagon}}\\label{Appendix Octagon}\n\nIn this appendix, we will prove that the octagon symmetrically centered at the origin with integer vertices $\\{(\\pm 1, \\pm 2), (\\pm 2, \\pm 1)\\}$ cannot admit any Gabor orthonormal basis  with  $\\Lambda=   \\left(\n                                      \\begin{array}{cc}\n                                        I & O \\\\\n                                        C & B \\\\\n                                      \\end{array}\\right) (\\Bbb Z^4)$, where  $B$ is  a rational matrix. For this we need the following lemma.\n                                      \n                                      \n\\begin{lemma}\\label{lemma_shear}\nLet $M = \\left(\n                                      \\begin{array}{cc}\n                                        \\alpha & 0 \\\\\n                                        0 & 1\/\\alpha \\\\\n                                      \\end{array}\\right)$ and $M_{\\beta} = \\left(\n                                      \\begin{array}{cc}\n                                        \\alpha & 0 \\\\\n                                        \\beta & 1\/\\alpha \\\\\n                                      \\end{array}\\right)$, $\\alpha\\neq 0$. Then $\\Omega$ is a multi-tile by $M_{\\beta}({\\mathbb Z}^2)$ if and only if  $\\Omega$ is a multi-tile by $M({\\mathbb Z}^2)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\alpha>0$ and put $Q_{\\alpha} = [0,\\alpha)\\times[0,1\/\\alpha)$. Then $Q_{\\alpha}$ is a fundamental domain for the lattice $M(\\Bbb Z^2)$. We claim that $Q_{\\alpha}$ is a fundamental domain for $M_{\\beta}(\\Bbb Z^2)$ for any $\\beta\\in\\Bbb R$. Indeed, this follows from a direct calculation: For almost every $(x,y)$, we have \n$$\n\\begin{aligned}\n\\sum_{m,n\\in{\\mathbb Z}}\\chi_{Q_{\\alpha}}(x-\\alpha m,y-\\beta m-n\/\\alpha) =& \\sum_{m\\in{\\mathbb Z}}\\chi_{[0,\\alpha)}(x-\\alpha m)\\left(\\sum_{n\\in{\\mathbb Z}}\\chi_{[0,1\/\\alpha)}(y-\\beta m-n\/\\alpha)\\right) \\\\\n=&\\sum_{m\\in{\\mathbb Z}}\\chi_{[0,\\alpha)}(x-\\alpha m)=1.\\\\\n\\end{aligned}\n$$\n \n\nLet $\\Omega$ be any set which is a tile by $M({\\mathbb Z}^2)$. Let $\\{E_{(u,v)} : (u,v)\\in \\Bbb Z^2\\}$ be a  partition of $Q_{\\alpha}$  such that \n %\n$$\n\\Omega = \\bigcup_{(u,v)\\in{\\mathbb Z}^2} (E_{(u,v)}+(\\alpha u,v\/\\alpha)) .$$ \n \n \n  Then set $\\Omega$ is a tile by $M_\\beta(\\Bbb Z^2)$. Indeed,  \n$$\n\\begin{aligned}\n\\sum_{m,n\\in{\\mathbb Z}}\\chi_{\\Omega}(x-\\alpha m,y-\\beta m-n\/\\alpha)=&\\sum_{m,n\\in{\\mathbb Z}} \\sum_{(u,v)\\in{\\mathbb Z}^2}\\chi_{E_{(u,v)+(\\alpha u,v\/\\alpha)}} (x-\\alpha m,y-\\beta m-n\/\\alpha)\\\\\n=&\\sum_{(u,v)\\in{\\mathbb Z}^2}\\sum_{m,n\\in{\\mathbb Z}} \\chi_{E_{(u,v)}} (x-\\alpha (m+u),y-\\beta m-(n+v)\/\\alpha)\\\\\n=&\\sum_{(u,v)\\in{\\mathbb Z}^2}\\sum_{m,n\\in{\\mathbb Z}} \\chi_{E_{(u,v)}} (x-\\alpha m,y-\\beta m-n\/\\alpha)\\\\\n=&\\sum_{m,n\\in{\\mathbb Z}}\\sum_{(u,v)\\in{\\mathbb Z}^2} \\chi_{E_{(u,v)}} (x-\\alpha m,y-\\beta m-n\/\\alpha)\\\\\n=&\\sum_{m,n\\in{\\mathbb Z}} \\chi_{Q_{\\alpha}} (x-\\alpha m,y-\\beta m-n\/\\alpha)=1.\\\\\n\\end{aligned}\n$$\nNote that above we used an application of Fubini\\rq{}s theorem.   \nNow let \n $\\Omega$ be   a multi-tile with respect to the lattice $M(\\Bbb Z^2)$. Then $\\Omega = \\bigcup_{i=1}^N\\Omega_i$, where $\\Omega_i$ are tiles by $M({\\mathbb Z}^2)$. Thus, by the previous  results  above, each $\\Omega_i$ is a tile by $M_\\beta(\\Bbb Z^2)$, hence we have the following. \n$$ \n\\sum_{m,n\\in{\\mathbb Z}}\\chi_{\\Omega}(x-\\alpha m,y-\\beta m-n\/\\alpha)=\\sum_{i=1}^N\\sum_{m,n\\in{\\mathbb Z}}\\chi_{\\Omega_i}(x-\\alpha m,y-\\beta m-n\/\\alpha)=N.\n$$\n\n The converse can be obtained by a similar calculation. \n \n\\end{proof}\n\n   \\begin{proof}[Proof of Example \\ref{Octagon}]\n                                        Before we prove our claim, note that  if the family\n                                 ${\\mathcal G}(|{\\mathcal O}_8|^{-1\/2}\\chi_{{\\mathcal O}_8}, \\Lambda)$   is a Gabor basis and $B$ is an integer matrix, then according to Theorem \\ref{lower triangle}   the set ${\\mathcal O}_8$ must  tile which is impossible. Thus, we assume that $B$ has some non-integer rational entries.   Let ${\\bf m}_0=(3,2)$. Then  ${\\mathcal P}:=K\\cap (K+{\\bf m}_0)$,   $K=\\mathcal O_8$, is the parallelogram   with vertices\n                                             $\\{(1,2), (2,1), (1,1), (2,0)\\}$. Hence,\n                                             $$\n                                             {\\mathcal P} = Q[0,1]^2+(1,1)^t, \\ \\mbox{where} \\  Q = \\left(\n                                      \\begin{array}{cc}\n                                        1 & 0 \\\\\n                                        -1 & 1 \\\\\n                                      \\end{array}\\right).\n                                             $$\n                                              The  Fourier transform of $\\chi_{\\mathcal P}$ at $\\xi  = (\\xi_1,\\xi_2)$ is given by \n                                             $$\n                                              \\widehat{\\chi_{{\\mathcal P}}}(\\xi) = c ~ \\widehat{\\chi_{[0,1]^2}}(Q^{T}\\xi)\n                                             $$\n                                              where $c:=c(\\xi)$ is some unimodular constant.\n                                      \n                    \n                                      \\medskip\n                                      \n                                      \n                                   By  the mutual orthogonality of the element $( m_0, n)$ with $(0, 0)$    for all $ n\\in{\\mathbb Z}^2$, we obtain \n                      \n                                              \\begin{align}\\label{integral-octagon}\n                                     0= I:= &   \\int_{K\\cap (K+{m}_0)} e^{-2\\pi i \\langle C m_0, x\\rangle} e^{-2\\pi i \\langle Bn,x\\rangle} dx  \n                                      = c\\cdot\\widehat{\\chi_{[0,1]^2}}(Q^TCm_0+Q^TB n) . \n                                               \\end{align}\n                                          \n                                         \n                                         If $Cm_0=0$, by putting ${n}=0$, then one must have  $c=0$, which is a contradiction . Otherwise, (\\ref{integral-octagon})               shows that $Q^TB({\\mathbb Z}^2)$ is exponentially incomplete for  $L^2[0,1]^2$. By Theorem \\ref{Theorem_Appendix1} in the appendix, $Q^TB$ is equivalent to  one of the following two forms.\n                                        %\n                               %\n                                        \\begin{align}\\label{matrices forms}\n                                        \\left(\n                                      \\begin{array}{cc}\n                                        \\frac{1}{q'} & 0 \\\\\n                                        r' & q' \\\\\n                                      \\end{array}\\right) \\  \\  \\mbox{or} \\ \\  \\left(\n                                      \\begin{array}{cc}\n                                        p' & 0 \\\\\n                                        \\frac{r''}{s''} & \\frac{1}{p'} \\\\\n                                      \\end{array}\\right)\n                              \\end{align}\n                                      for some integers $p',q', r'>1$, gcd of $r'$ and $q'$ is strictly greater than 1 and  $(r'',s'')$ is relatively prime. \n                                      \n                                      \\medskip\n                                            \n                                         \n                                          We now prove that $B$ also is equivalent to  the same desired form in (\\ref{matrices forms}). We are going to establish the first case, the second case is similar. Recall that $Q =  \\left(\n                                      \\begin{array}{cc}\n                                        1& 0 \\\\\n                                        -1 & 1 \\\\\n                                      \\end{array}\\right)$. If $Q^TB\\tilde U = \\left(\n                                      \\begin{array}{cc}\n                                        \\frac{1}{q} & 0 \\\\\n                                        r & q \\\\\n                                      \\end{array}\\right)$ for some unimodular integer matrix $\\tilde U$,  then $B\\tilde U= \\left(\n                                      \\begin{array}{cc}\n                                        \\frac{1+rq}{q} & q \\\\\n                                        r & q \\\\\n                                      \\end{array}\\right)$. Note that $1+rq$ and $q^2$ is relatively prime. Thus there exist co-prime  integers $u,v$ such that $(1+rq)u+q^2v = 1$. Define $ U = \\left(\n                                      \\begin{array}{cc}\n                                        u & -q^2 \\\\\n                                        v & 1+rq \\\\\n                                      \\end{array}\\right)$. Then $\\det(U)=1$ and the lattice $B\\tilde U(\\mathbb Z^2) = B\\tilde UU({\\mathbb Z}^2)$. Note that \n                                      $$\n                                      B\\tilde UU = \\left(\n                                      \\begin{array}{cc}\n                                        \\frac{1+rq}{q} & q \\\\\n                                        r & q \\\\\n                                      \\end{array}\\right)\\left(\n                                      \\begin{array}{cc}\n                                        u & -q^2 \\\\\n                                        v & 1+rq \\\\\n                                      \\end{array}\\right) = \\left(\n                                      \\begin{array}{cc}\n                                        \\frac{1}{q} & 0 \\\\\n                                        ru+vq & q\\\\\n                                      \\end{array}\\right).\n                                      $$ \n                                          This shows that $B$ is equivalent to the desired form in (\\ref{matrices forms}). \n                              For the rest, we prove that the  neither of these forms   can  form an Gabor orthonormal basis.   By the Lemma \\ref{lemma_shear} and Theorem \\ref{Th_union of FD}, we just need to prove that the octagon ${\\mathcal O}_8$ is not a multi-tile for the matrices $\\left(\\begin{array}{cc}\n                                        p& 0 \\\\\n                                        0& 1\/p \\\\\n                                      \\end{array}\\right)$ and $\\left(\\begin{array}{cc}\n                                        1\/p& 0 \\\\\n                                        0 & p \\\\\n                                      \\end{array}\\right)$ where $p>1$ is an integer. By the symmetry of the octagon, we just need to prove that ${\\mathcal O}_8$ is not a multi-tile for the first one.  Let $B_p:=\\left(\\begin{array}{cc}\n                                        p& 0 \\\\\n                                        0& 1\/p \\\\\n                                      \\end{array}\\right)$. To prove this, we first note that an elementary calculation shows that the area of ${\\mathcal O}_8$ is 14. If it   multi-tiles by $B_p({\\mathbb Z}^2)$, then for almost every  $x\\in{\\mathbb R}^2$, the cardinality of the set $(x+B_p({\\mathbb Z}^2))\\cap {\\mathcal O}_8$ is $l=14$.\n                                      To obtain a contradiction \n                                      consider the rectangle $R_p:=[0,1)\\times[0,1\/p)$ for $p>1$. It is a simple observation that $R_p$ can be covered  $12p$ by translations of $\\mathcal O_8$ by the matrix $B_p$. This is a contradiction to the level of multi-tiling $l=14$,  thus we have completed the proof.\n                                      \n                                      \n                                      \\iffalse \n                                      \n                                      \n                                      \\medskip\n                                      \n                                      If $p=2$, we consider the upper corner of  triangle formed by the vertices $(1,2), (1,3\/2)$ and $(3\/2,3\/2)$ and denote it by $\\Delta$, then for any $x\\in \\Delta$, \n                   $$ \n                   x+\\left\\{(k,j\/2): k\\in\\{0,-2\\},j \\in\\{0,-1,-2,-3,-4,-5,-6,-7\\}\\right\\}\\in (x+B_2({\\mathbb Z^2}))\\cap {\\mathcal O}_8.\n$$\nThere are 16 elements and $\\Delta$ has positive measure. This shows that ${\\mathcal O}_8$ cannot be a multi-tile by $B_2({\\mathbb Z}^2)$.\n\n\\medskip\n\nIf $p\\ge3$, observe that  the rectangle $[0,1)\\times[0,1\/p)$ is covered $12p$ times by $B_p(\\Bbb Z^2)$, and $p\\geq 3$. \n\n   $ \\{(0,j\/p): -2p\\leq  j\\leq 2p-1\\}$  exactely $4p$ times.  Since $4$ does not divide $14$, thus we have completed the proof.\n   \\fi \n\n\\end{proof}\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n$ {\\tt MergeSort} $ is one of the fundamental sorting algorithms that is being taught in undergraduate Computer Science curricula across the U.S. and elsewhere. Its worst-case performance, measured by the number of comparisons of keys performed while sorting them, is optimal for the class of algorithms that sort inductively\\footnote{Inductive sorting of $ n $ keys sorts a set of $ n-1 $ of those keys first, and then ``sorts-in'' the remaining $ n $-th key.} by comparisons of keys.\\footnote{In its standard form analyzed in this paper, $ {\\tt MergeSort} $ is not an inductive sorting algorithm. However, its worst-case performance, measured by the number of comparisons of keys performed while sorting them, is equal to the worst-case performance of the \\textit{binary insertion sort} first described by Steinhaus in \\cite{stein:matsnap} that is worst-case optimal in the class of inductive sorting algorithms that sort by comparisons of keys; see \\cite{knu:art} page 186.} Historically, it\\footnote{A bottom-up version of it, invented by John Neumann.} was the first sorting algorithm to run in $ O(n \\lg n) $ time\\footnote{In the worst case.}.\n\n\\medskip\n\nSo it seems only fitting to provide an exact formula for $ {\\tt MergeSort} $'s worst-case performance \\textit{and} derive it precisely.\nUnfortunately, many otherwise decent texts offer  unnecessarily imprecise\\footnote{Notable exceptions in this category are \\cite{baavan:ana} and \\cite{sedfla:ana} that derive almost exact formulas, but see Section~\\ref{sec:oth} page~\\pageref{sec:oth} for a brief critique of the results and their proofs offered there.} variants of it, and some with quite convoluted, incomplete, or incorrect proofs. Due to these imperfections, the fact that the worst-case performance of $ {\\tt MergeSort} $\nis the same as that of another benchmark sorting algorithm, the  \\textit{binary insertion sort}  of \\cite{stein:matsnap}, has remained unnoticed\\footnote{Even in \\cite{knu:art}.}.\n\n\\medskip\n\n\nIn this paper, I present two outlines\\footnote{The detailed derivations can be found in \\cite{suc:worstmergeMS}.} of elementary yet precise and complete derivations of an exact formula\n \\[ W(n) = \\sum_{i=1} ^{n} \\lceil \\lg i \\rceil = n \\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} + 1 \\]\nfor the maximum number $ W(n) $ \\footnote{Elementary derivation of an exact formula for the \\textit{best}-case performance $ B(n) $ of $ {\\tt MergeSort} $, measured by the number of comparisons of keys performed while sorting them, has been done in \\cite{suc:bestmerg}; see Section~\\ref{sec:best} page~\\pageref{sec:best} of this paper.} of comparisons of keys performed by \\linebreak $ {\\tt MergeSort} $ on an $ n $-element array. The first of the two, due to its structural regularity, is well worth carefully studying in its own right. \n\n\\smallskip\n\nUnlike some other basic sorting algorithms\\footnote{For instance, $ {\\tt Heapsort} $; see \\cite{suc:wcheap} for a complete analysis of its worst-case behavior.} that run in $ O(n \\lg n) $ time, $ {\\tt MergeSort} $ exhibits a remarkably regular\\footnote{As revealed by Theorem~\\ref{thm:mersorbounds}, page~\\pageref{thm:mersorbounds}.} worst-case behavior, \nthe elegant simplicity of which\nhas been mostly lost on its rough analyses. In particular,\n$ W(n) $ is linear\\footnote{See Figure \\ref{fig:boundsMergeSort} page \\pageref{fig:boundsMergeSort}.} between the points $ n = 2^{\\lfloor \\lg n \\rfloor} $ and it linearly interpolates its own lower bound $ n \\lg n - n + 1 $ \\footnote{Given by the left-hand side of the inequality~\\eqref{eq:mersorbounds} page~\\pageref{eq:mersorbounds}.} between these points.\n\n\\bigskip\n\nWhat follows is a short version (SV) of a manuscript dated January 20, 2017, of the full version version \\cite{suc:worstmergeMS} of this paper that has been posted at: \\bigskip \\\\\n\\verb|http:\/\/csc.csudh.edu\/suchenek\/Papers\/Analysis_of_MergeSort.pdf| \\bigskip \\\\\nThe derivation of the worst case of $ {\\tt MergeSort} $ presented here is roughly the same\\footnote{Except for the present proof of Lemma~\\ref{thm:sumCeilLog} which I haven't been using in my class.} as the one I have been doing in my undergraduate Analysis of Algorithms class. \\ref{sec:slides} shows sample class notes from one of my lectures. \n\n\n\n\\section{Some Math prerequisites}\n\n\nA manuscript of the full version \\cite{suc:worstmergeMS} of this paper contains a clever derivation of a well-known\\footnote{See \\cite{knu:art}.} closed-form formula for\n$ \\sum_{i=1} ^{n} \\lceil \\lg i \\rceil $. It proves insightful in my worst-case analysis of $ {\\tt MergeSort} $ as its right-hand side will occur on page~\\pageref{eq:result} in the fundamental equality~\\eqref{eq:result} and serve as an instrument to derive the respective exact formula for $ {\\tt MergeSort} $'s worst-case behavior.\n\n\\begin{lem} \\label{thm:sumCeilLog}\nFor every integer $ n\\geq 1 $,\n\\begin{equation} \\label{eq:sumCeilLog} \n\\sum_{i=1} ^{n} \\lceil \\lg i \\rceil = \\sum _{y=0}^{\\lceil \\lg n \\rceil-1} ( n - 2^y) .\n\\end{equation}\n\\end{lem}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\\medskip\n\n\nFrom this one can easily conclude that:\n\n\n\\begin{cor} \\label{cor:sumCeilLog}\nFor every integer $ n\\geq 1 $,\n\\begin{equation} \\label{eq:sumCeilLog2} \n\\sum_{i=1} ^{n} \\lceil \\lg i \\rceil = n \\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} + 1 .\n\\end{equation}\n\\end{cor}\n\n\n\\section{$ {\\tt MergeSort} $ and its worst-case behavior $ W(n) $} \\label{sec:mergsor}\n\n\nA call to $ {\\tt MergeSort} $ inherits an $ n $-element array $ {\\tt A} $ of integers and sorts it non-decreasingly, following the steps described below.\n\n\\begin{algMergeSort} \\label{alg:mersor}\nTo \nsort an $ n $-element array $ {\\tt A} $ do:\n\n\\begin{enumerate}\n\\item \\label{alg:mersor:item1} If $ n \\leq 1 $ then return $ {\\tt A} $ to the caller,\n\\item \\label{alg:mersor:item2} If $ n \\geq 2 $ then\n\\begin{enumerate}\n\\item \\label{alg:mersor:item2:it1} pass the first $ \\lfloor \\frac{n}{2} \\rfloor $ elements of $ {\\tt A} $ to a recursive call to $ {\\tt MergeSort} $,\n\\item pass the last $ \\lceil \\frac{n}{2} \\rceil $ elements of $ {\\tt A} $ to another recursive call to $ {\\tt MergeSort} $,\n\\item \\label{alg:mersor:item2:it2} linearly merge, by means of a call to $ {\\tt Merge} $, the non-decreasingly sorted arrays that were returned from those calls onto one non-decreasingly sorted array  $ {\\tt A}^{\\prime} $,\n\\item \\label{alg:mersor:item2:it3} return $ {\\tt A}^{\\prime} $ to the caller.\n \\end{enumerate}  \n\\end{enumerate}\n\\end{algMergeSort}\n\n\n\nA Java code of \n$ {\\tt Merge} $ is shown on the Figure~\\ref{fig:Merge}.\\footnote{A Java code of \n$ {\\tt MergeSort} $ is shown in \\ref{sec:appCode} Figure~\\ref{fig:Sort} page~\\pageref{fig:Sort}.}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{.\/Merge1} \n\\includegraphics[width=0.7\\linewidth]{.\/Merge2}\n\\caption{A Java code of $ {\\tt Merge} $, based on a pseudo-code from \\cite{baa:ana}. Calls to $ {\\tt Boolean} $ method $ {\\tt Bcnt.incr()} $ count the number of comps for the purpose of experimental verification of the worst-case analysis of $ {\\tt MergeSort} $.}\n\\label{fig:Merge}\n\\end{figure}\n\n\\medskip\n\nA typical measure of the running time of $ {\\tt MergeSort} $ is the number of \\textit{comparisons of keys}, which for brevity I call \\textit{comps}, that it performs while sorting array $ {\\tt A} $. \n\n\\begin{df}\nThe worst-case running time \n\\[W(n)\\]\nof $ {\\tt MergeSort} $ is defined as the maximum number of comps it performs while sorting an array of $ n $ distinct\\footnote{This assumption is superfluous for the purpose of worst-case analysis as the mere presence of duplicates does not force $ {\\tt MergeSort} $ to perform more comps.} elements. \n\\end{df}\n\n\\label{pag:n>0}Clearly, if $ n=0 $ then $ W(n) = 0 $. From this point on, I am going to assume that $ n \\geq 1 $.\\footnote{This assumption turns out handy while using expression $ \\lg n $.}\n\n\n\\medskip\n\nSince no comps are performed outside $ {\\tt Merge} $, $ W(n) $ can be computed as the sum of numbers of comps performed by all calls to $ {\\tt Merge} $ during the execution of  $ {\\tt MergeSort} $. The following classic results will be useful in my analysis.\n\n\n\\begin{theorem} \\label{thm:merge_n-1}\nThe maximum number of comps performed by $ {\\tt Merge} $ on two sorted list of total number $ n $ of elements is $ n-1 $.\n\\end{theorem}\n\\noindent \\textit{Proof} (constructive, with Java code that generates worst cases shown in the \\ref{sec:gen_worst_cases}) in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\\medskip\n\n\\label{pag:mergopt} Moreover, if the difference between the lengths of merged list is not larger than 1 then no algorithm that merges sorted lists by means of comps beats $ {\\tt Merge} $ in the worst case, that is, has a lower than $ n-1 $ maximum number of comps.\\footnote{Proof in \\cite{knu:art}, Sec. 5.3.2 page 198; the worst-case optimality of $ {\\tt Merge} $ ($ n-1 $ comps) was generalized in \\cite{stoyao:optmerge} over lists of lengths $ k $ and $ m $, with $ k \\leq m $, that satisfy $ 3k \\geq 2m-2 $.}  This fact makes $ {\\tt MergeSort} $ optimal in the intersection of the class of sorting algorithms that sort by merging two sorted lists of lengths' difference not larger than 1 \\footnote{Or, by virtue of the above-quoted result from \\cite{stoyao:optmerge}, with the difference not larger than the half of the length of the shorter list plus 1.}\nwith the class of sorting algorithms that sort by comps.\n\n\\section{An easy yet precise derivation of $ W(n) $} \\label{sec:easy}\n\n$ {\\tt MergeSort} $  is a recursive algorithm. If $ n \\geq 2 $ then it spurs a cascade of two or more recursive calls to itself.  A rudimentary analysis of the respective recursion tree $ T_n $, shown on Figure~\\ref{fig:rectre}, yields a neat derivation of the exact formula for the maximum number $ W(n) $ of comps that $ {\\tt MergeSort} $ performs on an $ n $-element array.\n\n\\begin{figure} [h]\n\\includegraphics[scale=.175]{RecTreeMergeSort_sketch3.jpg} \n\\caption{\\label{fig:rectre} A sketch of the recursion 2-tree $ T_n $ for $ {\\tt MergeSort} $ for a sufficiently large $ n $, with level numbers shown on the left and the numbers of nodes in the respective level shown on the right. The nodes correspond to calls to $ {\\tt MergeSort} $ and show sizes of (sub)arrays passed to those calls. The last non-empty level is $ h $. The empty levels (all those numbered $ > h $) are not shown. The root corresponds to the original call to $ {\\tt MergeSort} $. If a call that is represented by a node $ p $ executes further recursive calls to $ {\\tt MergeSort} $ then these calls are represented by the children of $ p $; otherwise $ p $ is a leaf. The wavy line \n$ \\photon $ \nrepresents a path in $ T_n $.}\n\\end{figure}\n\n\\medskip\n\n\n\n\n\n\nThe idea behind the derivation\nis strikingly simple. It is based on the observation\\footnote{Which I  prove in \\cite{suc:worstmergeMS} as Theorem~4.6, page~14.} that for every  $ k \\in \\mathbb{N} $, the maximum number $ C_k $ of comps performed at each level\\footnote{Empty or not.} $ k $  of $ T_n $ is given by this neat formula:\\footnote{It is a simplification of formulas used in derivation presented in \\cite{baavan:ana} and discussed in Section~\\ref{sec:oth} page~\\pageref{sec:oth}; in particular, it does not refer to the depth $ h $ of the decision tree $ T_n $.}\n\\begin{equation} \\label{eq:main_level_comps_formula}\nC_k = \\max \\{n - 2^k, 0\\} .\n\\end{equation}\nSince  \n\\begin{equation} \\label{eq:hight_rec_tree}\nn - 2^k > 0 \\mbox{ if, and only if, }  \\lceil \\lg n \\rceil - 1 \\geq k ,\n\\end{equation}\nthe Corollary~\\ref{cor:sumCeilLog} will allow me to conclude from \\eqref{eq:main_level_comps_formula} and \\eqref{eq:hight_rec_tree} the main result of this paper\\footnote{This is how I have been deriving it in my undergraduate Analysis of Algorithms class for some 15 years or so, now.}:\n\\begin{equation} \\label{eq:result}\nW(n)  = \\sum \n_{k \\in \\mathbb{N}}\n C_k = \\sum _{k=0}^{\\lceil \\lg n \\rceil-1} (n - 2^k) = n\\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} + 1 = \\sum_{i=1} ^{n} \\lceil \\lg i \\rceil.\n\\end{equation}\n\n\n\n\n\\bigskip \n\nThe missing details\\footnote{Which I did not show in my Analysis of Algorithms class.} in the above sketch are in \\cite{suc:worstmergeMS}. Naturally, their only purpose is to prove the equality \\eqref{eq:main_level_comps_formula} for all  $ k \\in \\mathbb{N} $,\nas the rest, shown in \\eqref{eq:result}, easily follows from it. \nIn particular, we get:\n\n\n\\begin{mth} \\label{thm:w-cMS}\nThe number $ W(n) $ of comparisons of keys that $ {\\tt MergeSort} $ performs in the worst case while sorting an $ n $-element array is\n\\begin{equation} \\label{eq:levComp2} \nW(n) = \\sum_{i=1} ^{n} \\lceil \\lg i \\rceil = n \\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} + 1 .\n\\end{equation}\n\\end{mth}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\n\n\n\\medskip\n\nFrom that we can conclude a usual rough characterization of $ W(n) $: \n\\[ W(n) \\leq n (\\lg n + 1) - 2^{ \\lg n } + 1  =  n \\lg n + n - n + 1 = n \\lg n  + 1\\]\nand\n\\[ W(n) \\geq n \\lg n  - 2^{ \\lg n + 1 } + 1  =  n \\lg n - 2n + 1. \\]\nTherefore,\n\\[ W(n) \\in \\Theta (n \\log n). \\]\n\n\n\n\\medskip\n\nThe occurrence of $  \\sum_{i=1} ^{n} \\lceil \\lg i \\rceil $ in \\eqref{eq:levComp2} allows to conclude that $ W(n) $ is exactly equal\\footnote{\\cite{knu:art} contains no mention of that fact.} to the number of comparisons of keys that the \\textit{binary insertion sort}, considered by H. Steinhaus in \\cite{stein:matsnap} and analyzed in \\cite{knu:art}, performs in the worst case. Since the \\textit{binary insertion sort} is known to be worst-case optimal\\footnote{With respect to the number of comparisons of keys performed.} in the class of algorithms that perform incremental sorting, $ {\\tt MergeSort} $ is worst-case optimal in that class\\footnote{Although it is not a member of that class.}, too. From this and from the observation at the end of Section~\\ref{sec:mergsor}, page~\\pageref{pag:mergopt}, I conclude that no algorithm that sorts by merging two sorted lists and only by means of comps is worst-case optimal in the class of algorithms that sort by means of comps as it must perform 8 comps in the worst case while sorting 5 elements\\footnote{They can be split in two: 1 plus 4, and follow the \\textit{binary insertion sort}, or 2 plus 3, and follow $ {\\tt MergeSort} $.}, while one can sort 5 elements by means of comps with no more than 7 comps.  \n\n\n\\section{Close smooth bounds on $ W(n) $}\n\n\nOur formula for $ W(n) $ contains a function ceiling that is harder to analyze than arithmetic functions and their inverses. In this Section, I outline a derivation of close lower and upper bounds on $ W(n) $ that are expressible by simple arithmetic formulas. I show that these bounds are the closest to $ W(n) $ in the class of functions of the form $  n \\lg n + c n + 1 $, where $ c $ is a real constant.\n\\label{pag:epsilon} \nThe detailed derivation and missing proofs can be found in \\cite{suc:worstmergeMS}.\n\n\\medskip\n\nUsing the function $ \\varepsilon $ (analyzed briefly in \\cite{knu:art} and \\cite{suc:wcheap}), a form of which is shown on Figure~\\ref{fig:eps}, \ngiven by:\n\\begin{equation} \\label{eq:eps}\n\\varepsilon  = 1 + \\theta\n- 2^{\\theta} \\mbox{ and } \\theta =  \\lceil \\lg \\, n \\rceil -  \\lg \\, n ,\n\\end{equation}\none can conclude\\footnote{\\label{foo:proofdelta}See \\cite{suc:wcheap}, Thm. 12.2 p. 94 for a proof.} that, for every $ n > 0 $,\n\\begin{equation} \\label{eq:eps_basic_formula}\n n \\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} = n(\\lg n + \\varepsilon - 1) ,\n\\end{equation}\nwhich yields\n\\begin{equation} \\label{eq:W_eps}\n W(n) =  n( \\lg n+ \\varepsilon - 1) + 1 =  n \\lg n + (\\varepsilon - 1) n + 1 . \n\\end{equation}\n\\begin{figure} [h]\n\\center\n\\includegraphics[scale=.34]{Alpha_epsilon-1.pdf}  \n\\caption{\\label{fig:eps} Graph of $  \\varepsilon - 1 $ as a function of $ \\lg  n $.}\n\\end{figure}\n\n\n\\begin{property} \\label{prop:delta}\nFunction $ \\varepsilon $ given by \\eqref{eq:eps} is a continuous function of $ n $ \non the set of reals $ > 0 $. It assumes the minimum $ 0 $ for every  $ n = 2^{\\lfloor \\lg n \\rfloor} $ and the maximum \n\\begin{equation} \\label{eq:Erdos}\n\\delta = 1 - \\lg e + \\lg \\lg e \\approx 0.0860713320559342 ,\n\\footnote{The constant $ 1 - \\lg e + \\lg \\lg e $\nhas been known as the {\\em Erd\\\"{o}s constant} $ \\delta $. Erd\\\"{o}s used it around 1955 in order to establish an asymptotic upper bound for the number $ M(k) $ of different numbers in a multiplication table of size $ k \\times k $ by means of the following limit:\n\\[\\lim _{k \\rightarrow \\infty} \\frac{\\ln \\frac{k \\times k}{M(k)}}{\\ln \\ln (k \\times k)} =  \\delta .\n\\]}\n\\end{equation}\nfor every\n\\begin{equation} \\label{eq:n_max_eps}\n n  =  2^{\\lfloor \\ln n + \\lg\\lg e \\rfloor} \\ln 2 \n\\end{equation}\nand only such $ n $.\nThe function \n$ \\varepsilon $ restricted to integers never reaches the value $\\delta$. However,  $\\delta$ is the  \\textit{supremum} of $ \\varepsilon $ restricted to integers. \n\\end{property}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\nCharacterization \\eqref{eq:W_eps} and Property~\\ref{prop:delta} yield close smooth bounds of $ W(n) $. They are both of the form $  n \\lg n + c n + 1 $ and they sandwich tightly $ W(n) $ between each other. If one sees $ W(n) $ as an infinite polygon\\footnote{Which it is.}, its lower bound circumscribes it and its upper bound inscribes it.\n\n\\begin{theorem} \\label{thm:mersorbounds} \n$ W(n) $ is a continuous concave function, linear between the points $ n =   2^{\\lfloor \\lg n  \\rfloor}$, that for every $ n > 0 $ satisfies this inequality:\n\\begin{equation} \\label{eq:mersorbounds} \n n \\lg n - n + 1 \\leq W(n) \\leq n \\lg n - (1-\\delta) n + 1 <  n \\lg n - 0.913 n + 1 , \n\\end{equation}\nwith the left $ \\leq $ becoming $ = $ for every $ n = 2^{\\lfloor \\lg n \\rfloor} $ and  the right $ \\leq $ becoming $ = $ for every $ n = 2^{\\lfloor \\lg n + \\lg \\lg e \\rfloor} \\ln 2 $, and only for such $ n $. Moreover, the graph of $ W(n) $ is tangent to the graph of $ n \\lg n - (1-\\delta) n + 1 $ at the points $ n = 2^{\\lfloor \\lg n + \\lg \\lg e \\rfloor} \\ln 2 $, and only at such points.\n\\end{theorem}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\t\\begin{figure} [h] \\center\n\\includegraphics[scale=.34]{Worst_Case_Mergesort_bounds4}\n\t\t\\caption{\\label{fig:boundsMergeSort} $ W(n) = n\\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} +1 $ (the middle line) and its bounds $ n \\lg n - n + 1 $ and $ n \\lg n - (1-\\delta) n + 1 $ $ \\approx $ $ n \\lg n - 0.913 n + 1 $, all three treated as functions of a positive real variable $ n $, plotted for $ n \\in [1,6] $. $ W(n) $ is linear between the points $ n = 2^{\\lfloor \\lg n \\rfloor} $ and it linearly interpolates its lower bound $ n \\lg n - n + 1 $ between these points. Its upper bound $ n \\lg n - (1-\\delta) n + 1 $ inscribes it and is tangent to it at the points $  n $ $ = $  $ 2^{\\lfloor \\lg n + \\lg\\lg e \\rfloor} \\ln 2 $.}\n\t\\end{figure}\n\t\n\t \\medskip\n\t \n\t \n\t \n\t The bounds given by (\\ref{eq:mersorbounds}) are really close\\footnote{The distance between them is less than $  \n\t \\delta n \\approx 0.0860713320559342 n $ for any positive integer $ n $.} to the exact value of $ W(n) $, as it is shown on Figure~\\ref{fig:boundsMergeSort} page~\\pageref{fig:boundsMergeSort}. The exact value $  n\\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} +1 $ is a continuous function (if $ n $ is interpreted as a real variable) despite that it incorporates discontinuous function \\textit{ceiling}. \n\t \n\t\\medskip\n\t\n\t\\begin{note}\n\t\\label{pag:interesting_interpolation}  It seems interesting that $ W(n) = n\\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} +1 $  (whether $ n $ is interpreted as a real variable or an integer variable) is linear between points $ n = 2^{\\lfloor \\lg n \\rfloor} $ and linearly interpolates its own lower bound $ n \\lg n - n + 1 $ between these points.\n\t\\end{note}\n\t\n\t \n\t \\medskip\n\t\nFor $ n $ restricted to positive integers, the inequality \\eqref{eq:mersorbounds} can be slightly enhanced by replacing the $ \\leq $ symbol with $ < $, with the following result.\n\\begin{theorem} \\label{thm:minconst}\n$ 1-\\delta $ is the greatest constant $ c $ such that for every integer $ n \\geq 1 $, \n\\begin{equation} \\label{eq:minconst}\n W(n) < n \\lg n - c n + 1.\n\\end{equation}\n\\end{theorem}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\\medskip\n\nTheorem \\ref{thm:minconst} can be reformulated as follows.\n\n\n\\begin{cor} \\label{cor:minconstinf}\n\\begin{equation} \\label{eq:minconstinf}\n\\inf \\{c \\in \\mathbb{R} \\mid \\forall n \\in \\mathbb{N} \\setminus \\{0\\} , W(n) < n \\lg n - c n + 1 \\}  = 1 - \\delta .\n\\end{equation}\n\\end{cor}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\nNo upper bound of $ W(n) $ that has a form $ n \\lg n - c n + 1 $ can coincide with $ W(n) $ at any integer $ n $, as the following fact ascertains.\n\n\\begin{cor} \\label{cor:minconst_neq}\nThere is no constant $ c $ such that for every integer $ n \\geq 1 $, \n\\begin{equation} \\label{eq:minconst_neq1}\n W(n) \\leq n \\lg n - c n + 1\n\\end{equation}\nand for some integer $ n \\geq 1 $, \n\\begin{equation} \\label{eq:minconst_neq2}\n W(n) = \\lg n - c n + 1.\n\\end{equation}\n\\end{cor}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\\medskip\n\nIn particular\\footnote{Note the $ \\leq $ symbol in \\eqref{eq:minconstinf2}.},\n\\begin{equation} \\label{eq:minconstinf2}\n\\inf \\{c \\in \\mathbb{R} \\mid \\forall n \\in \\mathbb{N} \\setminus \\{0\\}, W(n) \\leq n \\lg n - c n + 1 \\}  = 1 - \\delta .\n\\end{equation}\n\n\\medskip\n\nMoreover, we can conclude from Theorem \\ref{thm:minconst} the following fact.\n\n\\begin{cor} \\label{cor:minconst}\n$ 1-\\delta $ is the greatest constant $ c $ such that for every integer $ n \\geq 1 $, \n\\begin{equation} \\label{eq:cor:minconst}\n W(n) \\leq \\lceil n \\lg n - c n \\rceil.\n\\end{equation}\n\\end{cor}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\n\n\\medskip\n\nSince for any integer $ n \\geq 1 $, $ W(n) $ is integer, the lower bound given by \\eqref{eq:mersorbounds}\nyields\n\\begin{equation} \\label{eq:mersorbounds2} \n \\lceil n \\lg n \\rceil - n + 1 \\leq W(n) \\leq \\lceil n \\lg n - 0.913 n \\rceil . \n\\end{equation}\nBy virtue of Corollary \\ref{cor:minconst}, for some integers $ n \\geq 1 $,\\footnote{For instance, for $ n=11 $.}\n\\begin{equation} \\label{eq:wrongbound}\nW(n) > \\lceil n \\lg n - 0.914 n \\rceil .\n\\end{equation}\nAlthough the bounds given by \\eqref{eq:mersorbounds2} \\footnote{Almost the same bounds were given in \\cite{baavan:ana}; see Section~\\ref{sec:oth} for more details on this.} are tighter than those given by \\eqref{eq:mersorbounds}, they nevertheless involve the discontinuous \\textit{ceiling} function, so that they may not be as easy to visualize or analyze as some differentiable functions, thus losing their advantage over the precise formula $ W(n) = n\\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} +1 $. Therefore, the bounds given by \\eqref{eq:mersorbounds} appear to have an analytic advantage over those given by \\eqref{eq:mersorbounds2}.\n\n\n\\section{Other properties of the recursion tree $ T_n $}\n\n\nThis sections contains some well-known auxiliary facts that I didn't need for the derivation of the exact formula for $ W(n) $ but am going to derive from the Main Lemma 4.1 of \\cite{suc:worstmergeMS} for the sake of a thoroughness of my analysis of the decision tree $ T_n $.\n\n\\begin{thm} \\label{thm:depthRectree}\nThe depth $ h $ of the recursion tree $ T_n $ is \n\\begin{equation} \\label{eq:height}\nh = \\lceil \\lg n \\rceil .\n\\end{equation}\n\\end{thm}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\\medskip\n\n\\begin{note}\nTheorem~\\ref{thm:depthRectree} allows for quick derivation of fairly close upper bound on the number of comps performed by $ {\\tt MergeSort} $ on an $ n $-element array. Since at each level of $ T_n $ less than $ n $ comparisons are performed by $ {\\tt Merge} $ and at level $ h $ no comps are performed, and there are $ h $ $ = $ $ \\lceil \\lg n \\rceil  $ levels below level $ h $, the total number of comps is not larger than\n\\begin{equation} \\label{eq:UBapprox} \n(n-1)h = (n-1)(\\lceil \\lg n \\rceil) < (n-1) (\\lg n + 1)   \\in O(n \\log n).\n\\end{equation}\n\\end{note}\n\n \\medskip\n\n\n\n\n\\medskip\n\nA \\textit{cut} of a tree $ T_n $ is a set $ \\Gamma $ of nodes of $ T $ such that every branch\\footnote{A maximal path.} in $ T_n $ has exactly one element in $ \\Gamma $.\n\n\\begin{thm} \\label{thm:cut}\nThe sum of values shown at the elements of any cut of $ T_n $ is $ n $.\n\\end{thm}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\n\n\n\\begin{thm} \\label{thm:sum_leaves}\nThe number of leaves in the recursion tree $ T_n $ is $ n $.\n\\end{thm}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\nThe following corollary provides some statistics about recursive calls to $ {\\tt MergeSort} $.\n\n\\begin{cor} \\label{note:calls_stats}\nFor every integer $ n > 0 $,\n\\begin{enumerate}\n     \\renewcommand\\labelenumi{\\theenumi}\n     \\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item \\label{note:calls_stats:1} $ T_n $ has $ 2n - 1 $ nodes.\n\n\\item  \\label{note:calls_stats:2} The number or recursive calls spurred by $ {\\tt MergeSort} $ on any $ n $-element array is $ 2(n - 1) $.\n\n\\item  \\label{note:calls_stats:3}\nThe sum $ S_n $ of all values shown in the recursion tree $ T_n $ on Figure~\\ref{fig:rectre} is equal to:\n\\begin{equation} \\label{eq:sum_nodes}\nS_n =  n \\lceil \\lg n \\rceil - 2^{\\lceil \\lg n \\rceil} + 2n  = n(\\lg n + \\varepsilon + 1).\n\\end{equation}\n\\item \\label{note:calls_stats:4}\nThe average size $ A_n $ of array passed to any recursive call to $ {\\tt MergeSort} $ while sorting an $ n $-element array is:\n\\begin{equation} \\label{eq:avg_nodes}\nA_n =  \\frac{1}{2}(1+\\frac{1}{n-1}) (\\lg n + \\varepsilon) \\approx \\frac{1}{2} (\\lg n + \\varepsilon).\n\\end{equation}\n\\end{enumerate}\n\\end{cor}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\\medskip\n\n\n Here is a very insightful property. It states that $ {\\tt MergeSort} $ is splitting its input array fairly evenly\\footnote{The sizes of the sub-arrays passed to recursive calls at any non-empty level $ k $ of the decision tree $ T_n $ above the last non-empty level $ h $ are the same as the sizes of the elements of the maximally even partition of an $ n $-element set onto $ 2^k $ subsets.} so that at any level of the recursive tree, the difference between the lengths of the longest sub-array and the shortest sub-array is $ \\leq 1. $ This fact is the root cause of \ngood worst-case performance\nof $ {\\tt MergeSort} $.\n\n\\begin{property} \\label{pro:level} \nThe difference between values shown by any two nodes in the same level of of the recursion tree $ T_n $ is $ \\leq 1 $.\n\\end{property} \n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\\label{pag:merg_opti_for_mersor}Property \\ref{pro:level} has this important consequence that $ {\\tt Merge} $ is, by virtue of the observation on page~\\pageref{pag:mergopt} after the Theorem~\\ref{thm:merge_n-1} page~\\pageref{thm:merge_n-1}, worst-case comparison-optimal while merging any two sub-arrays of the same level of the recursion tree. Thus the worst-case of $ {\\tt MergeSort} $ cannot be improved just by replacing $ {\\tt Merge} $ with some tricky merging $ {\\tt X} $ as long as $ {\\tt X} $ merges by means of comparisons of keys.\n\n\\begin{cor} \\label{cor:merg_opti_for_mersor}\nReplacing  {\\tt Merge} with any other method that merges sorted arrays by means of comps will not improve the worst-case performance of $ {\\tt MergeSort} $ measured with the number of comps while sorting an array.\n\\end{cor}\n\\begin{proof}\nProof follows from the  above observation.\n\\end{proof}\n\n\n\\medskip\n\nSince a parent must show a larger value than any of its children, the Property~\\ref{pro:level} has also the following consequence.\n\\begin{cor} \\label{cor:last2}\nThe leaves in the recursion tree $ T_n $ can only reside at the last two non-empty levels of $ T_n $. \n\\end{cor}\n\\begin{proof}\nProof follows from the Property~\\ref{pro:level} as the above observation indicates.\n\\end{proof}\n\nAs a result, one can conclude\\footnote{Cf. \\cite{knu:art}, Sec. 5.3.1 Ex. 20 page 195.} that the recursion tree $ T_n $ has the miminum \\textit{internal} and \\textit{external path length}s among all binary trees on $ 2n - 1 $ nodes.\n\n\\medskip\n\nSince all nodes at the level $ h $ of the recursion tree $ T_n $ are leaves and show value $ 1 $, no node at level $ h-1 $ can show a value $ >2 $. Indeed, level $ h-1 $ may only contain leaves, that show value $ 1 $, and parents of nodes of level $ h $ that show value $ 1+1=2 $. This observation and the previous result allow for easy characterization of contents of the last two non-empty levels of tree $ T_n $.\n\n\n\\noindent \\begin{cor} \\label{cor:last2char}\nFor every $ n\\geq 2  $: \n\n\\begin{enumerate}\n     \\renewcommand\\labelenumi{\\theenumi}\n     \\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item \\label{cor:last2char:item1} there are $ 2^h - n $ leaves, all showing value $ 1 $, \nat the level $ h-1 $,\n\\item \\label{cor:last2char:item2} there are $  n - 2^{h-1} $ non-leaves, all showing value $ 2 $, \nat the level $ h-1 $,\nand\n\\item \\label{cor:last2char:item3} there are $  2n - 2^{h} $ \\footnote{This value shows in the lower right corner of Figure~\\ref{fig:rectre} page~\\pageref{fig:rectre} of a sketch of the recursion tree $ T_n $; it was not need needed for the derivation of the main result \\eqref{eq:levComp2} page~\\pageref{eq:levComp2}, included for the sake of completeness only.} nodes, all leaves showing value $ 1 $, at the level $ h $\n\\end{enumerate}\n\\noindent of the recursion tree $ T_n $, where $ h $ is the depth\\footnote{The level number of the last non-empty level of $ T_n $.} of $ T_n $.\n\\end{cor}\n\\noindent \\textit{Proof} in \\cite{suc:worstmergeMS}.\n\\hfill $ \\Box $\n\n\\section{A derivation of $ W(n) $ without references to the recursion tree}\n\n\nIn order to formally prove Theorem~\\ref{thm:w-cMS} without any reference to the recursion tree, I use here the well-known\\footnote{For instance, derived in \\cite{baa:ana} and \\cite{baavan:ana}.} recurrence relation\n\\begin{equation} \\label{eq:recmergesort1} \nW(n) = W(\\lfloor \\frac{n}{2} \\rfloor) + W(\\lceil \\frac{n}{2} \\rceil) + n - 1 \\mbox{ if } n \\geq 2\n\\end{equation}\n\\begin{equation} \\label{eq:recmergesort2} \nW(1) = 0\n\\end{equation}\nthat easily follows from the description (Algorithm~\\ref{alg:mersor} page~\\pageref{alg:mersor}) of $ {\\tt MergeSort} $, steps~\\ref{alg:mersor:item2:it1}, \\ref{alg:mersor:item2:it2} and Theorem~\\ref{thm:merge_n-1}. I am going to prove, by direct inspection,  that the function $ W(n) $ defined by (\\ref{eq:levComp2}) satisfies equations (\\ref{eq:recmergesort1}) and (\\ref{eq:recmergesort2}).\n\n\\medskip\n\nThe details of the proof are in \\cite{suc:worstmergeMS}.\n\n\\section{Other work} \\label{sec:oth}\n\nAlthough some variants of parts of the formula \\eqref{eq:levComp2} appear to have been known for quite some time now, even otherwise precise texts offer derivations that leave room for improvement. For instance, the recurrence relation for $ {\\tt MergeSort} $ analyzed in \\cite{sedfla:ana} asserts that the least number of comparisons of keys performed outside the recursive calls, if any, that suffice to sort an array of size $ n $ is $ n $ rather than $ n-1 $.  This seemingly inconsequential variation results in a solution $ W(n) =  \\sum_{i=1} ^{n-1} (\\lfloor \\lg i \\rfloor + 2)$ \\footnote{I saw $ W(n) =  \\sum_{i=1} ^{n-1} \\lfloor \\lg i \\rfloor  $ on slides that accompany \\cite{sedfla:ana}.} on page 2, Exercise 1.4, rather than the correct formula (\\ref{eq:result})  $ W(n) =  \\sum_{i=1} ^{n} \\lceil \\lg i \\rceil $ derived in this paper. (Also, the relevant derivations presented in  \\cite{sedfla:ana}, although quite clever, are not nearly as precise and elementary as those presented in this paper.) As a result, the fact that $ {\\tt MergeSort} $ performs exactly the same number of comparisons of keys as does another classic, \\textit{binary insertion sort}, considered by H. Steinhaus and analyzed in \\cite{knu:art}, remains unnoticed.\n\n\\bigskip\n\n\\bigskip\n\nPages 176 -- 177 of \\cite{baavan:ana} contain \nan early sketch of proof of \n\\begin{equation} \\label{eq:baavan_result}\nW(n) = n h - 2^{h} + 1,\n\\end{equation}\nwhere $ h $ is the depth of the recursion tree $ T_n $, with remarkably close\\footnote{Although not 100 percent correct.} bounds given by~\\eqref{eq:baavan_bound} page~\\pageref{eq:baavan_bound}. It is similar\\footnote{The idea behind the sketch of the derivation in \\cite{baavan:ana} was based on an observation that \\[W(n) = \\sum _{i = 0} ^{h-2} (n - 2^i) + \\frac{n - B}{2} , \\] where $ B $ was the number of leaves at the level $ h-1 $ of the decision tree $ T_n $; it was sketchily derived from the recursion tree shown on Figure~\\ref{fig:RecTreeBaaVan} and properties stated in the Corollary~\\ref{cor:last2char} page~\\pageref{cor:last2char} (with only a sketch of proof in \\cite{baavan:ana})  not needed for the derivation presented in Section~\\ref{sec:easy}.}  to a simpler \nderivation based on the equality \\eqref{eq:main_level_comps_formula}, presented in this paper in Section~\\ref{sec:easy} and outlined in~\\eqref{eq:result} page~\\pageref{eq:result} (except for the $ \\sum_{i=1} ^{n} \\lceil \\lg i \\rceil $ part), which it predates by several years. \n\n\\medskip\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=1\\linewidth]{.\/RecTreeBaaVan}\n\\caption{A snapshot from \\cite{baavan:ana}, page 177, showing a decision tree for $ {\\tt MergeSort}$. \\textit{\\textbf{Note}}: This picture is copyrighted by \\textit{Addison Wesley Longman} (2000). It was reproduced here from \\cite{baavan:ana} for \\textit{criticism} and \\textit{comment} purposes only, and not for any other purpose, as prescribed by U.S. Code Tittle 17 Chapter 1 para 107 that established the ``fair use'' exception of copyrighted material. }\n\\label{fig:RecTreeBaaVan}\n\\end{figure}\n\n\nThe \\cite{baavan:ana}'s version of the decision tree $ T_n $ (Figure 4.14 page 177 of \\cite{baavan:ana}, shown here on Figure~\\ref{fig:RecTreeBaaVan}) was a re-use of a decision tree for the special case of $ n = 2^{\\lfloor \\lg n \\rfloor} $, with an ambiguous, if at all correct\\footnote{It may be interpreted as to imply that for any level $ k $, all the left-child sizes at level $ k $ are the same and all the right-child sizes at level $ k $ are the same, neither of which is a valid statement.}, comment in the caption that ``[w]henever a node size parameter is odd, the left child size parameter is rounded up\\footnote{Should be: \\textit{down}, according to \\eqref{eq:recmergesort1} page~\\pageref{eq:recmergesort1}.} and the right child size is rounded down\\footnote{Should be: \\textit{up}, according to \\eqref{eq:recmergesort1} page~\\pageref{eq:recmergesort1}.}.'' The proof of the fact, needed for the derivation in \\cite{baavan:ana}, that $ T_n $ had no leaves outside its last two levels (Corollary~\\ref{cor:last2} page~\\pageref{cor:last2}, not needed for the derivation presented in Section~\\ref{sec:easy}) was waved with a claim ``[w]e can\\footnote{This I do not doubt.} determine that [...]''\n\n\\medskip\n\n\nAlthough $ h $ was claimed in \\cite{baavan:ana} to be equal to $ \\lceil \\lg (n+1) \\rceil $~\\footnote{Which claim must have produced an incorrect formula $ n \\lceil \\lg (n+1) \\rceil - 2^{\\lceil \\lg (n+1) \\rceil} + 1 $ for $ W(n) $ and precluded concluding the neat characterization $ W(n) = \\sum_{i=1} ^{n} \\lceil \\lg i \\rceil $.} (and not to the correct $ \\lceil \\lg n \\rceil $ given by the equality~\\eqref{eq:height} page~\\pageref{eq:height}, a fact not needed for the derivation presented in Section~\\ref{sec:easy}), somehow the mostly correct conclusion\\footnote{Almost identical with \\eqref{eq:mersorbounds2} page~\\pageref{eq:mersorbounds2}, except for the constant $ 0.914 $.}\nwas inferred from it, however, with no details offered - except for a mention that a function $ \\alpha $ that satisfies $ h = \\lg n + \\lg \\alpha $, similar to function $ \\varepsilon $ shown on Figure~\\ref{fig:eps} page~\\pageref{fig:eps}, was used. It stated that (Theorem 4.6, page 177, in \\cite{baavan:ana}):\n\\begin{equation} \\label{eq:baavan_bound}\n \\lceil n \\lg n - n + 1 \\rceil \\leq W(n) \\leq \\lceil n \\lg n - 0.914 n \\rceil .\n\\end{equation}\n\t\t\nIt follows from \\eqref{eq:wrongbound} page \\pageref{eq:wrongbound} that the constant $ 0.914 $ that appears in \\eqref{eq:baavan_bound} is incorrect. It was a rounding error\\footnote{Of  $ 1 - \\delta $, where $ \\delta $ is given by \\eqref{eq:Erdos} page~\\pageref{eq:Erdos}.}, I suppose, that produced a false upper bound\\footnote{For instance, if $ n = 11 $ then $ {\\tt MergeSort} $ performs 29 comparisons of keys while the value of the upper bound $ \\lceil n \\lg n - .914 n \\rceil $ given in \\cite{baavan:ana}, Theorem 4.6. p. 177, is 28; this is a significant error as 28 or less comps while sorting any 11-element array beats the \\textit{binary insertion sort} that requires $ \\sum_{i=1} ^{11} \\lceil \\lg i \\rceil = 29$ comps in the worst case.}.   \n\n\\section{Best-case analysis of $ {\\tt MergeSort} $} \\label{sec:best}\n\nIt turns out that derivation of minimum number $ B(n) $ of comps performed by $ {\\tt MergeSort} $ on an $ n $-element array is a bit more tricky. A formula\n\\begin{equation} \\label{eq:best}\n \\frac{n}{2}(\\lfloor \\lg n \\rfloor + 1) -  \\sum _{k=0} ^{\\lfloor \\lg n \\rfloor} 2^k  \\mbox{\\textit{Zigzag}}\\,(\\frac{n}{2^{k+1}}),\n\\end{equation}\n where\n \\[ \\mbox{\\textit{Zigzag}} (x)  =  \\min (x - \\lfloor x \\rfloor, \\lceil x \\rceil - x), \\]\n has been derived and thoroughly analyzed in \\cite{suc:bestmerg}. It has been also demonstrated in \\cite{suc:bestmerg} that there is no closed-form formula for $ B(n) $.\n \n \\medskip\n \n Incidentally, as it was pointed out in \\cite{suc:bestmerg} $, B(n) $ is equal to the sum $ A(n,2) $ of bits in binary representations of all integers $ < n $.\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt has been nearly a century since the first nuclear reaction was intentionally measured in the laboratory ($^{14}\\rm{N}(\\alpha,p)$ in 1919 by Rutherford~\\cite{Ruth35}) and a half century since the existence of the neutron star crust was proposed by Ruderman~\\cite{Rude68}. These milestones, and the later discovery of the first X-ray burst in the source 3U~1820-30~\\cite{grindlay75}, laid the foundation for a collaborative effort between nuclear physics and astrophysics to advance our understanding of dense matter. \nIt was soon postulated that X-ray bursts on accreting neutron stars arise from the unstable ignition of accreted hydrogen and helium material \\cite{Woos76,joss77}, an idea later supported by numerical models~\\cite{fujimoto81,ayasli1982,Woos04}. Burst models now serve as an important probe of nuclear reactions~\\cite{Cybu16} because nuclear burning in X-ray bursts proceeds through a combination of the triple-$\\alpha$ reaction, rapid proton-capture (rp)-process, and $\\alpha$-capture proton-emission ($\\alpha \\rm{p}$)-process \\cite{Wall81}. Moreover, burst models reveal the thermal structure of the neutron star's outer layers where the bursts originate. \n\nAs accreted material accumulates on the neutron star \nsurface it compresses underlying material to greater depths and higher mass densities. Before the ashes of hydrogen and helium burning join the crust, further nuclear burning takes place. For example, any $^4$He still present will be captured on heavier isotopes. Runaway $^{12}$C+$^{12}$C fusion powering superbursts is the most dramatic instance of burning in the ashes layer \\cite{cumming2001,Stro02}. These superbursts are a thousand times more energetic, a thousand times longer, and occur much deeper than the standard X-ray bursts. These rare events allow one to probe deeper layers of the stellar envelope.\n\nCooling neutron stars provide another avenue to explore the thermal properties of dense matter and the nuclear reactions therein. In transient systems, the neutron star accretes material in episodes lasting days to years (see Reference~\\cite{Dege15} for a summary) and the accretion-induced compression\nof the neutron star crust triggers non-equilibrium nuclear reactions~\\cite{bisnovatyi1979,Sato79} that heat the neutron star crust~\\cite{Haen90,Haen01}. When an accretion episode ends, the neutron star cools on observable timescales~\\cite{ushomirsky2001,rutledge2002} and the cooling trend probes the microphysics of the interior~\\cite{Shte07,Brow09}. In addition to constraints on the thermal properties of the outer layers, the observed cooling reveals the strength and location of nuclear reaction heating, and by extension the details of the nuclear reactions themselves. For example, observations of cooling neutron stars can be explained by models including \n$e^{-}$-capture heating~\\cite{Gupt07} and pycnonuclear fusion heating~\\cite{haensel2008} in the neutron star crust. Furthermore, crust cooling has the potential to reveal strong Urca cooling layers~\\cite{Scha14,Deib16} that are suspected to be present in crusts enriched by X-ray burst ashes~\\cite{Meis17}.\n\nExperimental nuclear data are critical physics input for neutron star models. Dense matter constraints derived from neutron star observations are therefore limited by the precision of nuclear physics input in addition to other systematics. \nThe reduction of nuclear physics uncertainties, \nas well as the development of more sophisticated neutron star models, will improve future observational constraints on dense matter. For example, current investigations of the critical reaction rates in X-ray bursts~\\cite{Cybu16,Scha17}, studies of key properties of nuclei in the accreted crust~\\cite{Deib16,Meis17}, and nuclear reaction network calculations of accreted crust compositions~\\cite{Gupt07,Scha14,Lau18}, all promise to improve the observational constraints on dense matter derived from accreting neutron stars. \n\nWe begin in section~\\ref{section:structure} by briefly outlining the neutron star structure. Sections~\\ref{subsection:pristine} and \\ref{section:accretion} describe the original composition of the neutron star crust and the accretion process which drives the system from equilibrium. In section~\\ref{section:production}, we discuss the nuclear burning that can occur on the surface of accreting neutron stars and the nuclei produced during the different possible burning regimes. In actively accreting neutron stars, the ashes of prior surface nuclear burning are compressed to greater depths by newly accreted material. We discuss in section~\\ref{section:interaction} the nuclear interactions involving the ashes that take place as the ambient mass density increases. In section~\\ref{section:impact}, we investigate the impact of interior nuclear interactions on observable neutron star phenomena. We summarize and discuss prospects for future work in section~\\ref{section:summary}. \n\n\\section{Neutron Star Structure} \\label{section:structure}\nA neutron star is born in the death of a massive star (from 8 and up to 50\\ $\\rm{M}_{\\odot}$ \\cite{Heger2003,Pejcha2015ApJ}, where $M_\\odot=1.99\\times 10^{33}\\,\\mathrm{g}$ is the mass of the Sun), when the lack of sufficient outward pressure from the core to balance the inward compression from gravity leads to a dramatic collapse. The iron-group nuclei in the progenitor core are rapidly transmuted by electron-captures and photodisintegration, transforming the core into a hot and extremely dense neutron-rich sphere~\\cite{Woos05,Burr86}. The collapse halts when the central density approaches the mass density of atomic nuclei, $\\rho_0=2.8\\times 10^{14}$~g~cm$^{-3}$ (baryon density $n_{0}=0.16~\\rm{fm}^{-3}$), and the neutron degeneracy pressure and the repulsion from the strong nuclear force bounce back the infalling matter. This can finally result in a core collapse supernova event, leaving a dense compact remnant -- a neutron star  \n-- behind \\cite{Heger2003,Pejcha2015ApJ}. \n\nAs a consequence, neutron stars \ncontain $\\sim1\\textrm{--}2~\\rm{M}_{\\odot}$ in a sphere of radius $R\\sim10\\textrm{--}15 \\, \\mathrm{km}$~\\cite{steiner2010,steiner13} and have average mass densities of several $\\rho_0$. Almost all (99 per cent) of the mass is concentrated in the bulky core composed of uniform nuclear (or possibly more exotic) matter. The equation of state and even the composition of the neutron star core are unknown and their elucidation are some of the fundamental problems of neutron star astrophysics \\cite{HPY2007}.  \n\nAt mass densities $\\rho\\lesssim 0.5\\, \\rho_0$ the uniform nuclear matter is unstable and arranges into \nnuclear clusters, which at the low densities are familiar albeit neutron-rich nuclei \\cite{Peth95}. These outer layers of the neutron star, though comprising only roughly one percent of the \nmass, provide the settings for all of the astronomical observables used to characterize accreting neutron stars,\nand are the primary subjects of the present review. \n\nMuch like a terrestrial planet, the neutron star outer layers consist, from the outside-in, of an atmosphere and liquid ocean (collectively referred to as the envelope), solid crust (outer and inner), and the mantle, roughly 1~km thick in total as schematically shown in figure~\\ref{figure:schematic}. \nIn contrast to planetary envelopes, matter in these layers is \nunder extreme conditions, for example, the enormous gravity $g=(GM\/R^2)(1+z)$, where $G$ is the gravitational constant, $M$ is the neutron star mass, $(1+z)\\equiv \\left(1-2GM\/(Rc^2)\\right)^{-1\/2}$ is the gravitational redshift at the neutron star surface, and $c$ is the speed of light. For a canonical neutron star of $M=1.4 M_\\odot$ and $R=10$~km, $g=2.44\\times 10^{14}$~cm~s$^{-2}$ and $1+z= 1.31$. Large gravity ensures that General Relativity effects are important when neutron star phenomena are studied. Since the outer layers are thin, the gravity and the redshift can be set constant and equal to the surface values. \n\nThroughout this paper we will use various physical quantities to specify the current position in the global structure of the envelope depending on the aspects of the problem discussed. The equation of state and composition of dense matter mainly depend on the mass density $\\rho$ or baryon number density $n_B$. However, these quantities may be discontinuous at composition changes (see below) and thus may not be the appropriate variables to follow the crustal structure. Instead, the pressure $P$ or the column density $y\\equiv P\/g$ are continuous and increase monotonically, and are convenient measures of the depth. The column density is especially useful when accretion phenomena are studied, since the total baryon mass above a layer with $y=\\mathrm{constant}$ is $\\Delta M_B(y)=4\\pi R^2 y$. Note that the gravitational mass of the same layer is smaller, $\\Delta M = \\Delta M_B\/(1+z)$, because of the gravitational binding energy.\nOne also distinguishes between the radial coordinate $r$ (so that the surface area is $4\\pi r^2$) and the proper depth measured by a local observer in the envelope $\\zeta=\\int_r^R (1+z(r))\\mathrm{d}r\\approx (1+z)(R-r)$. \n\n\n\\begin{figure} [ht]\n\\centering\n\\includegraphics[scale=0.7]{ns_structure}\n\\caption{Schematic of the outer layers of a neutron star.}\n\\label{figure:schematic}\n\\end{figure}\n\n\nThe outer neutron star envelope -- atmosphere and ocean -- contains a non-ideal plasma of electrons and ions, and except in the outermost layers, the plasma is fully ionized. There is usually no phase transition between the gaseous and liquid state, except for some special cases \\cite{Potekhin2014PhyU}. The atmosphere is distinguished as the region where the neutron star surface emission is formed \nand extends up to a density of \n$\\sim 10^{-4}-10^{6}~\\rm{g}~\\rm{cm}^{-3}$ \\cite{Potekhin2010PhyU}, depending on the physical conditions. Both the atmosphere and ocean may host convective processes, driven by gradients of temperature and chemical potentials~\\cite{Malone2011,Medin2011}. \nIn most of the envelope, electrons are strongly degenerate, relativistic (at $\\rho\\gg 10^6$~g~cm$^{-3}$), and provide the main contribution to the pressure. For ultrarelativistic electrons, their contribution to the pressure is $P_e=\\mu_e^4\/(12\\pi^2 \\hbar^3 c^3)$, where $\\mu_e$ is the electron chemical potential and  $\\hbar$ is the reduced Planck constant. Thus $\\mu_e$ is directly related to the column depth $y$ (such that $\\mu_e\\propto y^{1\/4}$) and provides another useful measure of the depth. $\\mu_e$ is a particularly convenient coordinate to use when the crust composition \nand nuclear reactions are studied (sections~\\ref{subsection:pristine}, \\ref {section:interaction}). \nFor convenience, we give the relation between various measures of depth in the neutron star outer envelope and crust, where the dominant pressure is from degenerate electrons:\n\\begin{eqnarray}\ny&\\approx& 7.2\\times 10^9\\, \\mathrm{g}\\,\\mathrm{cm}^{-2} \\left(\\frac{\\mu_e} {1~\\mathrm{MeV}}\\right)^4\\frac{2.44\\times 10^{14}~\\mathrm{cm}~\\mathrm{s}^{-2}}{g},\\label{eq:y-mu}\\\\\n\\rho&\\approx&7.2\\times 10^6\\, \\mathrm{g}\\,\\mathrm{cm}^{-3} \\left(\\frac{\\mu_e} {1~\\mathrm{MeV}}\\right)^3\\frac{1}{Y_e},\\label{eq:rho-mu}\\\\\n\\zeta&\\approx&40~\\mathrm{m}\\, \\frac{\\mu_e} {1~\\mathrm{MeV}}\\frac{2.44\\times 10^{14}~\\mathrm{cm}~\\mathrm{s}^{-2}}{g} Y_e \\label{eq:z-mu},\n\\end{eqnarray}\nwhere $Y_e$ is the electron fraction, \nand in equation~(\\ref{eq:z-mu}) it is a depth-averaged quantity.\n\n\nThe properties of ions are determined by the ratio of their Coulomb energy to the thermal energy. For a plasma containing ions of one species with charge number $Z$, the plasma coupling parameter $\\Gamma=Z^2 e^2\/ (a k_\\mathrm{B} T)$, where $a=[3\/(4\\pi n_i)]^{1\/3}$ is the ion-sphere (Wigner-Seitz) radius, $n_i$ is the ion number density, $e$ is the elementary electric charge, and $k_\\mathrm{B}$ is the Boltzmann constant. When $\\Gamma\\gtrsim 1$, ion correlations are important so they form a non-ideal Coulomb liquid.\nIn the region where electrons are ultrarelativistic,\n\\begin{equation}\n\\Gamma\\approx 105\\, \\left(\\frac{Z}{26}\\right)^{5\/3} \\left(\\frac{T}{10^8~{\\rm K}}\\right)^{-1} \\frac{\\mu_e}{1~\\mathrm{MeV}}\\approx 361 \\left(\\frac{Z}{26}\\right)^{5\/3} \\left(\\frac{T}{10^8~{\\rm K}}\\right)^{-1} y_{12}^{1\/4},\n\\label{eq:Gamma}\n\\end{equation}\nwhere $y_{12}$ is the column depth measured in units of $10^{12}$~g~cm$^{-2}$ and the canonical value of $g$ is used. When there are multiple ion species $Z_j$, which is the case for the accreted crust, partial $\\Gamma_j$ are introduced, and an average $\\langle\\Gamma_j\\rangle$ can be used to describe the state of the whole mixture. An important \nquantity is the so-called impurity parameter which describes variance\nof the charge mixture around the mean charge $\\langle Z\\rangle$. It is defined as\n\\begin{equation}\nQ_{\\rm imp} \\equiv \\frac{1}{n_{\\rm ion}} \\sum_j n_j (Z_j - \\langle Z \\rangle)^2 \\ ,\n\\label{equation:Qimp}\n\\end{equation}\nwhere $n_{\\rm ion}$ is the total local number density of ions, and $n_j$ is the local number density of nuclei species $j$.\nThe impurity parameter is important when the transport properties such as thermal conductivity are discussed (see section~\\ref{section:impact}).\n\nAt a depth of several tens of meters, the ocean solidifies~\\cite{Rude68,Medi10}. The pure Coulomb plasma solidifies at about $\\Gamma\\approx175$ \\cite{Potekhin2000} \nand forms a perfect crystal with a body-centered cubic lattice.  As follows from equation~(\\ref{eq:Gamma}), for a typical temperature $T=10^8$~K solidification occurs at $\\mu_e\\approx1.7$~MeV ($y\\approx6\\times 10^{10}$~g~cm$^{-2}$, $\\zeta\\approx 140$~m) for $Z=26$ (iron). This point corresponds to mass density $\\rho\\approx 7\\times 10^7$~g~cm$^{-3}$ and shifts deeper inside the neutron star for decreasing $Z$.\n\nThe solidification of the multi-component mixture is thought to occur at a similar order of magnitude of the average $\\langle \\Gamma \\rangle$ \\cite{Mcki16}. However, the exact structure of the  multicomponent solid crust and solidification layer at the ocean-crust boundary are not well understood. There is strong theoretical and observational evidence that the solid mixture also arranges in a regular but impure crystalline structure (see sections \\ref{subsection:impact_inner} and \\ref{sec:cooling_transients}). \n\nA few-hundred meters deeper, the rising electron chemical potential results in $\\beta$-equilibrium nuclei closer to the neutron dripline, until ultimately the neutron separation energy for nuclei in the plasma becomes negative.\nAt this point it is energetically favorable for neutrons to drip out of nuclei and form a neutron gas~\\cite{Chamel2008LRR,HPY2007} \nmarking the upper boundary of the inner crust. This neutron drip occurs at densities $\\rho_{\\rm{nd}}\\sim 4-10 \\times 10^{11}$~g~cm$^{-3}$ \ndepending on the composition and the theoretical model used. For the accreted crust, generally, $\\rho_{\\rm{nd}}$ is on the high end of the aforementioned range.\nThus in the inner crust an ion lattice coexists with a gas of free neutrons which soon becomes degenerate and starts giving the dominant contribution to the pressure. At this point, equations~(\\ref{eq:y-mu})--(\\ref{eq:z-mu}) are now inapplicable.\n\nDue to the attractive component of the strong interaction, the degenerate neutron gas is subject to a pairing instability. It is believed that in the crust paring occurs in the singlet $^1S_0$ channel and the critical temperature $T_\\mathrm{cn}$ is density-dependent with the maximal value of order of $\\sim 5\\times 10^9$~K \\cite{Lombardo2001LNP}.\nThus a large portion of the crustal neutrons is thought to be superfluid, although the precise form of the $T_\\mathrm{cn}(\\rho)$ profile is very model-dependent. At large densities the nuclear interaction in the $^1S_0$ channel becomes repulsive and superfluidity ceases at the lower bound of the inner crust. However, in some models, $^1S_0$ superfluidity penetrates the core. We \ndiscuss the observable impact of neutron superfluidity in section~\\ref{sec:quasipers}.\n\nWhen the density increases further in the inner crust, groupings\nof protons and neutrons are no longer accurately considered as nuclei \nand can rather be called nuclear clusters. Eventually, the clusters become so large and closely spaced that the competition between the nuclear surface energy and Coulomb energy distorts the nuclei in into complex shapes called nuclear pasta \\cite{ravenhall1983}.\nMolecular dynamics simulations demonstrate that nuclear pasta can begin to appear near mass densities of $\\rho \\gtrsim 8\\times 10^{13} \\ \\mathrm{g\\,cm^{-3}}$ \\cite{hashimoto1984}.\nAs a consequence,  this `mantle' layer can contain half the mass of the whole crust. \nNote that the pasta layer does not exist for all equations of state \\cite{Douchin2000PhLB,Oyamatsu2007PhRvC}. Finally, when proton clustering ceases at $\\rho\\gtrsim 0.5\\rho_0$, the transition to a liquid core occurs.\n\nThe structure and properties of the crust as described above can be modified by the presence of fast rotation or high magnetic fields. For instance, rotation deforms the crust, making it non-spherical. The presence of a strong magnetic field in the outer layers can result in the absence of the ocean and a first order phase transition between the thin gaseous atmosphere and the condensed neutron star surface. These effects are outside the scope of this paper  (see, e.g., References~\\cite{Chamel2008LRR,Potekhin2014PhyU}). \n\n\n\n\nFor the upper reaches of the inner crust and above, these regions are composed of nuclei which can be made in current and near-future nuclear physics laboratories. As such, accreting neutron stars\nprovide a medium to explore many interesting multi-physics questions through astronomical observations and theory, as well as nuclear physics theory and experiment. The following sections will explore the dominant nuclear physics processes occurring in accreting neutron stars'  \nouter layers and the astronomical observables these processes impact. But first, a brief discussion of the crust composition in the absence of accretion is beneficial in order to provide context for the dramatic transformation accretion induces.\n\n \\section{Pristine Crust Composition}\n \\label{subsection:pristine}\n  \nThe extreme temperatures ($\\gtrsim 10^{11}$~K) achieved during the supernova collapse drive \nmatter in the unaccreted crust to its ground state in terms of nuclear and $\\beta$ equilibrium, as determined by the local environment conditions and nuclear masses~\\cite{Beth70,Buch71,Baym71a,Baym71b}. \nThis means that the unaccreted outer crust is stratified into several layers, each of which is comprised of a single species of nuclide. \nHere we discuss the cold-catalyzed crust, where matter is in its absolute ground state~\\cite{Baym71a}. \nIn principle, the composition may freeze-in at a higher temperature, modifying the equilibrium abundance distribution due to thermal fluctuations (see, e.g., Reference~\\cite{Gori11}).\n\nIn the outer crust, it is energetically favorable for nuclei to arrange themselves in a Coulomb lattice within an electron Fermi gas~\\cite{Rude68}. As such, beyond minimizing the usual liquid drop terms describing the nuclear mass in vacuum, the additional energy contributions from the lattice and the electron gas must be considered~\\cite{Buch71,Baym71a}. The dominant liquid drop terms include the volume energy, which can be modified due to compression by the dense environment; the surface energy, which generally favors large-$A$ nuclides;\nthe Coulomb energy, which favors low-$A$ nuclides; and the symmetry energy, which favors nuclides with neutron number $N=Z$. The lattice energy favors the existence of large-$A$ nuclei, providing a Coulomb energy of opposite sign to its liquid-drop partner. The resulting total energy per nucleon is\n$E_{\\mathrm{total}}\/A = E_{\\mathrm{nuclear}}\/A + E_{e^{-}\\mathrm{-gas}}\/A + E_{\\mathrm{lattice}}\/A$,\nwhose minimum defines the equilibrium nucleus. In practice, the equilibrium composition is solved for at a given pressure, which is a proper continuous variable. Therefore, it is generally the Gibbs free energy per nucleon that is minimized~\\cite{Baym71b,Haen01,Roca08}.\n  \n  \\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.5]{EquilibriumNucleusMuE.pdf}\n\\caption{Equilibrium composition for a cold-catalyzed neutron star outer crust calculated using a liquid drop model for nuclear masses (thick lines). To demonstrate the impact of shell effects, the equilibrium composition obtained using experimental masses when available and the BSk8 Skyrme model otherwise, as reported by Reference~\\cite{Rust06}, is also shown (thin lines). See figure~\\ref{figure:ECcrustcomposition} to compare to the accreted crust composition.}\n\\label{figure:pristinecomp}\n\\end{figure}\n  \nAt relatively low density near the neutron star \nsurface, the nuclear energy term will dominate and the nucleus with the minimum mass per nucleon, $^{56}\\rm{Fe}$, is optimal. At deeper depths, the increasing electron chemical potential $\\mu_e$ creates an energetic incentive to lower the electron fraction, $Y_{e}=Z\/A$, favoring more neutron-rich nuclides, in competition with the nuclear symmetry energy which favors $Y_{e}=1\/2$. A decent estimate of the equilibrium nuclide at a given depth can be obtained by employing the semi-empirical mass formula for the nuclear binding energy in conjunction with the $e^{-}$-gas chemical potential and lattice energy, resulting in the total energy per nucleon\n\\begin{eqnarray}\n\\label{eqn:CrustE}\nE_{\\rm{total}}\\left(A,Z,\\mu_{e}\\right)\/A= &m_\\mathrm{p}c^{2}Y_{e}+m_\\mathrm{n}c^{2}\\left(1-Y_{e}\\right)-a_{v}+\\frac{a_{s}}{A^{1\/3}}+a_{c}A^{2\/3}Y_{e}^{2}\\\\\n&+a_{a}\\left(1-2Y_{e}\\right)^{2}+\\frac{3}{4}\nY_{e}\n\\mu_{e}-C_{\\ell}A^{2\/3}\nY_{e}^{5\/3}\\mu_{e},\\nonumber\n\\end{eqnarray}\nwhere $m_\\mathrm{p}$ and $m_\\mathrm{n}$ are the proton and neutron masses, respectively; $a_{v}$, $a_{s}$, $a_{c}$, and $a_{a}$ are the volume, surface, Coulomb, and asymmetry coefficients of the semi-empirical mass formula;  \nand $C_{\\ell}$ is a (relatively small) constant scale factor for the Coulomb lattice energy~\\cite{Roca08}. This assumes an ultrarelativistic form for the electron Fermi gas contribution.  \nThe equilibrium nuclide is found by minimizing equation~(\\ref{eqn:CrustE}) with respect to $Y_{e}$\nand separately with respect to $A^{1\/3}$, which gives $Z$ and $A$ at the minimum.\n \nTo demonstrate how liquid-drop nuclear binding, electron gas, and Coulomb lattice energy contributions impact the equilibrium nucleus, an approach similar to Reference~\\cite{Roca08} is followed here. We fit the nuclear binding contribution from equation~(\\ref{eqn:CrustE}) to the experimental masses of the 2012 Atomic Mass Evaluation~\\cite{Audi12}, though including an additional term for the nuclear pairing energy that goes as $i{a_{p}}{A^{-3\/2}}$~\\cite{Mart09}, where $i=+1,-1,$ or $0$ to enhance, penalize, or leave alone nuclear binding for even-even, odd-odd, and even-odd nuclides, respectively. For such a fit, $a_{v}$=15.302, $a_{s}$=16.518, $a_{c}$=0.687, $a_{a}$=22.243, and $a_{p}$=5.898, each with units of MeV. Using these parameters and $C_{\\ell}$=3.40665$\\times10^{-3}$ (for a body-centered cubic lattice, e.g.  Reference~\\cite{Roca08}), simultaneous minimization of $A^{1\/3}$ and $Y_{e}$ results in the equilibrium nucleus trend shown with thick lines in figure~\\ref{figure:pristinecomp}.\n\nThe trend in composition for a liquid drop model is modified by the presence of nuclear shell closures present in more realistic nuclear mass models, as shown with thin lines in figure~\\ref{figure:pristinecomp}. \nThough models disagree on the detailed composition depending on which nuclear mass model is employed, generally, the equilibrium nuclide is near the $Z=28$ shell closure at shallow depths \n($n_B\\sim$$10^{-7}$~fm$^{-3}$, i.e. $\\sim$50 m), transitioning to $N=50$ nuclides ($n_B\\sim$$10^{-5}$~fm$^{-3}$, i.e. $\\sim$150 m), and finally resulting in $N=82$ nuclides prior to neutron-drip ($n_B\\sim$$10^{-4}$~fm$^{-3}$, i.e. $\\sim$300 m)~\\cite{Baym71b,Roca08,Pear11,Rust06}.\nNuclear mass measurements at radioactive ion beam facilities are working their way toward more exotic nuclides to resolve existing discrepancies in theoretical predictions, where experimental constraints are presently available down to a depth of $\\sim$200~m~\\cite{Wolf13}. \nBeyond this, theoretical estimates are necessary~\\cite{Utam17}.\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{minipage}{0.4\\textwidth}\n\\includegraphics[width=\\textwidth]{inner.pdf}\n\\end{minipage}\n\\hspace{0.05\\textwidth}\n\\begin{minipage}{0.5\\textwidth}\n\\includegraphics[width=\\textwidth]{nuc_prof.pdf}\n\\end{minipage}\n\\end{center}\n\\caption{{\\it (a)}: Composition in the inner crust in the BSk21 model after Reference \\cite{Pote13}. Solid, dashed, and dash-dotted lines show charge number $Z$, mass number in the cluster $A$, and total mass number in the Wigner-Seitz cell $A'$, respectively. {\\it Right:} Neutron and proton number density distributions  in the Wigner-Seitz cell. {\\it (b)}: nucleon density profiles near the top of the inner crust, at $\\mu_\\mathrm{e}=30$~MeV. Relevant nuclear quantities are given in the plot. {\\it (c)}: the same for  $\\mu_\\mathrm{e}=70$~MeV, close to the nuclear pasta onset. }\\label{fig:inner}\n\\end{figure}\n\nBy contrast, the pristine inner crust composition is far more uncertain. The presence of the neutron gas makes describing the composition in terms of nuclei less adequate  and, instead, it is more precise to use the concept of proton clusters~\\cite{Chamel2008LRR}. Example equilibrium ``nuclei\"  are $^{1269}\\rm{Zr}$ and $^{633}\\rm{Ca}$~\\cite{Bald06}, where the total mass number per cluster is used to mark the isotopes. In this respect, one usually wants to distinguish between the total number of nucleons $A'$ per cluster and the number of nucleons inside the cluster $A$. Such a distinction is not a well-defined procedure and one usually takes the nucleon density far from the cluster center as the `gas' phase density.\nOne approach to describe this region is the classical approach employing a compressible liquid drop model~\\cite{Baym71a}. Conceptually, this technique is similar to the method described above for the outer crust, however, extra terms are included in the total energy to account for contributions from the neutron gas and modifications to the liquid-drop semi-empirical mass formula to account for nuclear compressibility. Instead, semi-classical approaches describe the inner crust using energy density functionals, e.g. via the Thomas-Fermi approximation~\\cite{Onsi08}. The most sophisticated and computationally-intensive approach is a quantum mechanical solution of the wave function for the cluster existing at each depth~\\cite{Bald06}. Each of the three approaches feature compositions dominated by proton shells $Z=20, 40,$ and $50$, though disagreement remains as to which shell the composition converges (see tables 3 and 4 of Reference~\\cite{Chamel2008LRR}). The example of the composition resulting from calculations in the BSk21 model \\cite{Pote13} is shown in the left panel of figure~\\ref{fig:inner} where we plot it until the density where the possible pasta phase would appear. In this model, the proton shell $Z=40$ is found everywhere in the inner crust. \nThe concept of a ``cluster\" as opposed to a nucleus is illustrated in the right panels in figure~\\ref{fig:inner} where we plot the nucleon density profiles in the same model  for $\\mu_e=30$~MeV (panel (b)) and near the bottom of the crust $\\mu_e=70$~MeV (panel (c)). In the second case there is a significant free neutron fraction \n$Y_\\mathrm{n}\\approx 0.8$ and a broad neutron distribution is clearly seen.  We refer the reader to References~\\cite{Chamel2008LRR,HPY2007} and more recent works \\cite{Sharma2015A&A,Pote13,Lim2017PhRvC,Pastore2017JPhG} for a detailed discussion of the inner crust composition in the absence of accretion. \n\n\n\n\\section{Accretion onto the Neutron Star Surface}\\label{section:accretion}\n\nThe neutron star \ncrust undergoes a significant transformation from the pristine state due to accretion.\nNeutron stars \nin binary systems with a lower-mass companion (low-mass X-ray binaries) can accrete material onto their surfaces through Roche-lobe overflow. Matter from the outer layers of the companion is transferred to an accretion disk surrounding the neutron star.\nThere it loses angular momentum and subsequently falls onto the neutron star. \nAccretion can proceed continuously and we refer to the system as a ``persistent X-ray source'', or can proceed episodically in ``transient'' sources. A transient outburst may have any duration from weeks or months to years (section~\\ref{sec:cooling_transients}). \nThe composition of the accreted material is that of the outer layers of the companion star, which is often assumed to be similar to that of the Sun with mass fractions of $0.739$\\,$^1$H, $0.247$\\,$^4$He, and the rest heavier isotopes \\cite{Lodders2010}. In Ultra-Compact X-ray Binaries~\\cite{intZand2007} with a binary period shorter than $80$ minutes, the companion star lacks a hydrogen-rich envelope, and the accretion composition is mostly $^4$He. The signatures of hydrogen and helium have been detected in the optical spectra of low-mass X-ray binaries, but it is challenging to determine the mass fractions precisely~\\cite{Nele06}.\n\nWhen the companion mass is higher than the mass of the compact object, the accretion via Roche-lobe overflow is not possible. However, the neutron star in the binary system with massive ($>10\\,M_\\odot$) early-type companion (high-mass X-ray binaries) can accrete matter from the stellar wind or Be-star decretion disk. Neutron stars in high-mass X-ray binaries usually have relatively large magnetic fields ($\\sim 10^{12}$~G) and show regular X-ray pulsations, while the magnetic fields of neutron stars in low-mass X-ray binaries are low ($10^{8}-10^{9}$~G). We do not consider high-mass X-ray binaries below and focus on low-mass X-ray binary sources, since the latter provide the main observational manifestations of nuclear processes during accretion. \n\n\nWhen an accreted nucleon finally settles at the neutron star surface, it releases $z\/(1+z)\\times m_u c^2\\approx 220$~MeV of the gravitational energy (as measured in a distant observer's frame), where  $m_u$ is the nucleon mass unit \\cite{ShaprioTeukolsky1986}. Part of this energy is radiated from the accretion disk and the remaining part is assumed to be radiated away from the surface. Therefore the estimate for the total accretion luminosity (mostly in X-rays), as seen by the distant observer, is\n\\begin{equation}\nL_A^\\infty=\\frac{z}{1+z}\\dot{M} c^2,\n\\end{equation}\nwhere $\\dot{M}$ is the mass accretion rate as seen at infinity (the mass accretion rate as seen by the observer on the neutron star surface is $\\dot{M}(1+z)$). Thus a measurement of the persistent accretion flux in principle allows one to estimate $\\dot{M}$. This is usually done by assuming spherically-symmetric accretion and introducing the appropriate bolometric correction factor to convert from the observed X-ray luminosity $L_X^\\infty$ to bolometric luminosity $L_A^\\infty$. \nIt is customary to quantify $\\dot{M}$ in fractions of the Eddington mass accretion rate $\\dot{M}_\\mathrm{Edd}$ at which the radiation pressure of the emitted X-ray photons balances the gravitational pull on the infalling material. The critical Eddington luminosity is\n\\begin{equation}\\label{eq:Ledd}\nL^\\infty_\\mathrm{Edd}=\\frac{4\\pi G M c}{\\kappa_\\mathrm{es}}\/(1+z_\\mathrm{ph}),\n\\end{equation}\nwhere $\\kappa_{\\rm es}=0.2\\,(1+X)$~g$^{-1}$~cm$^2$, with $X$ being the hydrogen mass fraction in the infalling material, is the electron scattering opacity and  $(1+z_\\mathrm{ph})$ is the gravitational redshift of the emission region (photosphere); this region \ncan be high above the neutron star surface. \nUsually the Eddington luminosity without the latter factor is used, which is the truly maximal limiting luminosity a distant observer can see \\cite{ShaprioTeukolsky1986}. Alternatively, one sometimes sets $z_\\mathrm{ph}=z$ in equation~(\\ref{eq:Ledd}). For a 10~km Newtonian neutron star (neglecting all General Relativity corrections) and a solar composition,\none obtains $\\dot{M}_\\mathrm{Edd}=1.72\\times 10^{-8}\\,M_\\odot$~yr$^{-1}$. \n\nDuring the time interval $\\Delta t$, a neutron star \naccretes $\\Delta M_B = \\dot{M}\\Delta t$  baryon mass. According to section~\\ref{section:structure}, the base of this slab will be at the column density $y=\\dot{M}\\Delta t\/(4\\pi R^2)$. When accretion continues, this layer is compressed to higher $y$. In this sense, by following the crust structure in section~\\ref{section:structure} \nwith increasing column depth we at the same time are following the journey\nof a given accreted fluid element deep through the crust, where it undergoes various nuclear transformations, as discussed in the following sections, to finally merge with the neutron star core. It is clear {\\it a fortiori} that the composition of the  accreted layers shall be very different from the pristine crust composition discussed in section~\\ref{subsection:pristine}. A neutron star accreting steadily at $0.01\\dot{M}_\\mathrm{Edd}$, will replace its entire crust with the accreted crust in $10^8$~yr.\n\n\\section{Production of Nuclei on Neutron Star Surfaces} \\label{section:production}\n\nA variety of processes are responsible for nucleosynthesis on and near the surface of accreting neutron stars.  \nThough the products of these burning processes, the ashes, are likely unable to escape the neutron star\ngravitational potential and contribute to the cosmic abundances~\\cite{Wall81}, they \nhave a significant impact on the accreted crust thermal and compositional structure. Here we discuss the primary mechanisms through which nuclei are produced in and on accreting neutron stars. \n\n\n\\begin{figure} [ht]\n\\centering\n\\includegraphics[scale=0.8]{AshComparison2.pdf}\n\\caption{Ash abundances predicted for stable surface burning at near-Eddington acretion rate \n(black-lined unfilled histogram), superbursts (gray-filled histogram), and Type-I X-ray bursts for a nominal reaction rate library (red-filled translucent histogram) and, to highlight the impact of nuclear physics uncertainties, for the $^{59}\\rm{Cu}(p,\\gamma)$ rate reduced by a factor of 100 (red-lined unfilled histogram). Calculation details are described in Reference~\\cite{Meis17}. General ash properties are listed in Table~\\ref{table:AvgZAQTable}.\\label{figure:ashes}}\n\\end{figure}\n  \n  \\subsection{Ash Production from Type I X-ray Bursts and Other Hydrogen\/Helium Burning Regimes}\n   The most prominent departure from the single-species-per-depth composition of pristine crusts is realized for accreting neutron stars \nexhibiting Type I X-ray bursts (see figure~\\ref{figure:ashes})~\\cite{Meis17}. Here we describe the key features of X-ray burst nucleosynthesis, focusing on helium-ignited hydrogen and helium burning bursts, as this case requires the most extensive nuclear physics input as compared to other burning regimes. For completeness, we first summarize the key observational and theoretical aspects of X-ray bursts before discussing the nuclear physics sensitivities in detail (see also the recent review \\cite{GallowayKeek2017}).\n   \n\n\\subsubsection{Observation} \n\\label{sec:burst_observation}\n \\hfill\\\\\n     \nOver $100$ X-ray bursting systems are currently known in our Galaxy, including both persistent and transient sources\\footnote{\\url{https:\/\/personal.sron.nl\/~jeanz\/bursterlist.html}}; figure~\\ref{figure:LCobs} shows examples of bursting systems. For many systems the mass accretion rate is observed to be time variable, and the bursting behavior changes accordingly, such that for a specific source different kinds of bursts may be observed over time. X-ray bursts were first reported in 1974 from observations with the Astronomical Netherlands Satellite (ANS; \\cite{grindlay75}). At present several thousands of bursts have been observed with a wide variety of durations, recurrence times, and energetics \\cite{Galloway2008catalog}. Notable instruments include the {\\it European X-ray Observatory Satellite} (EXOSAT; active from 1983 to 1986) and the {\\it BeppoSAX Wide-Field Cameras} (WFCs; active from 1996 to 2002),\nwhich observed large burst samples from a range of sources, and demonstrated how the bursting behavior changes as a function of the accretion rate \\cite{Paradijs1988,Cornelisse2003}. The majority of bursts are observed at accretion rates $\\dot{M}\\simeq 0.01-1.00\\ \\dot{M}_\\mathrm{Edd}$, with bursts lasting tens-to-hundreds of seconds and recurring every few hours-to-days. Typically, the burst rate increases with $\\dot{M}$: bursts repeat faster when their fuel is replenished more rapidly. Around $0.1-0.3\\ \\dot{M}_\\mathrm{Edd}$, however, the burst rate reaches a maximum, and decreases at higher $\\dot{M}$. For most systems, no bursts are observed near $\\dot{M}_\\mathrm{Edd}$. In that case, it is thought that the accreted material burns \nin a stable way, \nunlike a thermonuclear runaway that would produce a burst.\nNote that we do not consider Type-II X-ray bursts, which generally have much shorter recurrence times (tens of seconds to minutes)\nand $\\sim$100 times greater time-averaged luminosities than Type-I bursts~\\cite{Lewi93}. While Type-I bursts are attributed to nuclear burning (see section~\\ref{sec:burst_theory}), Type-II bursts are attributed to impulses of accretion onto the neutron star~\\cite{Hoffman1978}.\n\n\\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.8]{LCobs.pdf}\n\\caption{Bolometric luminosity light curves observed with the RXTE\/PCA\nfor sources 4U~1820$-$303, SAX~J1808.4$-$3658, GS~1826$-$24 (panel (a)), and 4U~1636$-$536 (panel (b)), reproduced from Reference~\\cite{Gall17}. These bursts are thought to be pure-He Type-I, He-rich Type-I, H-rich Type-I, and a superburst, respectively. \nNote that the precursor burst for 4U~1636$-$536~\\cite{Stro02} is not visible in panel (b) due to the coarse time binning.\nThe gap in the data for panel (b) is due to occultation by the Earth. Data are courtesy of the Multi-Instrument Burst Archive (MINBAR): \\url{https:\/\/burst.sci.monash.edu\/minbar\/}.\n\\label{figure:LCobs}}\n\\end{figure}\n\nThe most detailed burst observations have been performed with the Proportional Counter Array (PCA) on-board the {\\it Rossi X-ray Timing Explorer} (RXTE; active from 1995 to 2012) \\cite{Strohmayer2006,Galloway2008catalog}. With its large collecting area, RXTE detected detailed burst light curves that can be compared to nucleosynthesis models \\cite{Hege07}. For example, the contribution of the rp-process to the burst tail has been quantified for a large sample \\cite{Zand17}, and the burst rise is significantly shaped by flame spreading across the neutron star surface \\cite{Maurer2008}. Recently, a \nreference sample of RXTE bursts has been created for different fuel compositions (mixed hydrogen and helium, pure helium, and carbon) as a benchmark for nucleosynthesis calculations \\cite{Gall17}.\n\nIn many astrophysical sites, for example novae and supernovae, the ash composition can be inferred from spectral lines or edges from the debris of the explosion. Unfortunately this is not the case for most X-ray bursts. The strong surface gravity binds the ashes to the neutron star, \nand they are quickly covered by newly accreted material. An exception is a small group of bursts with strong photospheric radius expansion.\nAt the peak of these powerful bursts, the luminosity reaches the Eddington limit. Potentially, the upper layers of the star are blown off, and the burst ashes are exposed for a few seconds. Observations of such bursts hint at the presence of spectral edges, but the data quality has been insufficient to identify the ions involved \\cite{Zand2013,Barr15,Kaja17}. Another possibility is that the ashes could be ejected in the photospheric radius expansion wind~\\cite{Wein06}.\n\nRXTE has ceased operations, but at present, similar high quality burst observations are performed with the {\\it Nuclear Spectroscopic Telescope Array} (NuSTAR \\cite{Harrison2013}), ASTROSAT \\cite{Verdhan2017} and the {\\it Neutron Star Interior Composition Explorer} (NICER \\cite{Gendreau2012NICER}).\nThe latter is sensitive in a lower energy band, which will allow for a more accurate separation of the X-ray flux from nuclear burning and from the surrounding environment.\n\n    \\subsubsection{Theory} \n    \\label{sec:burst_theory}\n     \\hfill\\\\\nThe first X-ray burst observations occurred just after the publication of a study that predicted shell flashes from neutron stars\n\\cite{Hansen1975}, such that the thermonuclear nature was quickly established \\cite{Woos76,Maraschi1977,Lamb1978}. Simple one-zone ignition models were used to map out the different burning regimes as a function of $\\dot{M}$ \\cite{fujimoto81} (see also \\cite{Fushiki1987ApJ,Nara03}), and one-dimensional multi-zone models with increasingly sophisticated nuclear networks have been used to simulate the nuclear burning during bursts in great detail (e.g., \\cite{joss77,Wall81,Taam1996,Woos04,Fisker2008,Jose10}). Many of the observed features of bursts and other burning modes are reproduced, including a burst rate that increases with $\\dot{M}$ and stable burning at high $\\dot{M}$. For those systems that accrete a mixture containing both hydrogen and helium, the burning of both is investigated separately. Unstable hydrogen burning via the CNO cycle produces bursts at ignition temperatures of $T\\lesssim0.7\\times 10^8\\ \\mathrm{K}$. These conditions are reached at the lowest accretion rates, $\\dot{M}\\lesssim 0.004\\dot{M}_\\mathrm{Edd}$. If hydrogen burning ignites helium as well, a mixed H\/He burst results. However, for $\\dot{M}\\gtrsim0.001\\dot{M}_\\mathrm{Edd}$ this is not the case: helium continues to pile up during several (relatively weak) hydrogen bursts until a powerful helium burst ignites. The weak hydrogen bursts have not been identified in observations. \nAt these low accretion rates, there is sufficient time for gravitational settling to separate the ions in the accreted composition, which may have an effect on the boundaries of these regimes \\cite{Peng2007}.\n\nFor $\\dot{M}\\gtrsim0.004\\dot{M}_\\mathrm{Edd}$, hydrogen burns to helium via the ``hot'' or ``$\\beta$-limited'' CNO cycle, where the burning time scale is set by the combined half lives of $^{14}$O and $^{15}$O \\cite{Wall81}. This burning rate is independent of temperature, hence hydrogen burning cannot run away, and instead proceeds stably. Bursts in this regime are, therefore, produced by unstable helium burning, starting with runaway $3\\alpha$ burning. For $\\dot{M}\\lesssim0.1\\dot{M}_\\mathrm{Edd}$ there is sufficient time to burn all hydrogen prior to the helium runaway. The burst then ignites in a layer where all hydrogen has burned to helium: a so-called pure-helium burst. Runaway $3\\alpha$ burning of helium to carbon raises the temperature and enables a series of $\\alpha$-captures on carbon, creating a chain of $\\alpha$ elements up to calcium. Interestingly, when $T\\gtrsim 1$~GK,  $(\\alpha,\\rm{p})$ reactions take place which create a small amount of protons \\cite{Wein06}. This expands the accessible reactions with proton-captures, even though no hydrogen was present at ignition. The by-pass of  $^{12}\\mathrm{C}(\\alpha,\\gamma)$ by the faster $^{12}\\mathrm{C}(\\rm{p},\\gamma)$, has been suggested as the explanation for observed bursts with exceptionally short rise times of $\\sim 1$\\,ms \\cite{Wein06,Zand2014a}.\n\nAt mass accretion rates in excess of $0.1\\dot{M}_\\mathrm{Edd}$, some hydrogen remains when helium burning ignites, and again\na mixed hydrogen\/helium burst results. In the presence of hydrogen, the $3\\alpha$ runaway is more complicated. Helium burns to carbon, which boosts the burning of hydrogen in the $\\beta$CNO cycle. In turn, the CNO-cycle burning increases the helium abundance as well as the temperature, which further boosts $3\\alpha$. This interplay continues until at a temperature of $5\\times 10^8$~K breakout from the CNO cycle via $^{15} \\mathrm{O}(\\alpha,\\gamma)$ and via $^{18} \\mathrm{Ne}(\\alpha,\\rm{p})$ at $6\\times 10^8$~K becomes efficient~\\cite{Wies86}. The break-out reactions open the door for two long reaction chains. First the $\\alpha \\rm{p}$-process: a series of $(\\alpha,\\rm{p})(\\rm{p},\\gamma)$ reactions. These are fast reactions that mostly take place at the burst onset. This process feeds into the rp-process: a series of $(\\rm{p},\\gamma)$ reactions and $\\beta^+$-decays. The proton captures are typically fast, and the timescale of the rp-process is set by the half lives of the decays. Therefore, the initial part of the reaction chain goes fast, until the first slow decay is reached at $^{30}$S \\cite{Woos04,Fisker2008} within roughly a second. \nA small amount of the material in high-temperature regions enables this waiting point to be bypassed, but the reaction sequence nonetheless stalls at $^{56}\\rm{Ni}$~\\cite{Fisker2008}. The rest of the rp-process burning takes place in the tail of the X-ray burst, where it powers the light curve. The length of the reaction chain depends on the amount of hydrogen that is still present at this time and on the maximum temperature that is reached. The end of the rp-process lies at the closed SnSbTe cycle \\cite{Scha01,Jose10}, but for typical burst conditions the  composition of ashes peaks at lighter elements near Ge and Ga \\cite{Woos04}.\n\nThe rp-process can follow multiple branches in the nuclear reaction network. \nWhich branch is dominant depends on the reaction rates, which are often only weakly constrained by theoretical calculations (e.g., \\cite{Raus00}). Experimental constraints on the rates are crucial for accurate predictions of the ash composition. Changes to the reaction path that substantially change the ashes often also produce observable changes in the X-ray light curve (e.g., \\cite{Heger2005}). Large numerical studies identify which reactions are most important in this respect and require new nuclear physics experiments to improve their accuracy \\cite{Pari13,Cybu16,Scha17}.\n\nBoth the pure-helium and mixed hydrogen\/helium bursts as described by theory have been observed. Furthermore, theory predicts that helium burning becomes stable at high mass accretion rates $\\dot{M}\\geq\\dot{M}_\\mathrm{Edd}$.\nObservations see stable burning already at a three times lower rate. Moreover, the observed reduced burst rate is not predicted by theory. Possibly a hot crust \\cite{Keek2009,Zamfir2014} or mixing processes \\cite{Piro2007,Keek2009,Cave17} in the envelope stabilize nuclear burning already at lower accretion rates. In addition, the reaction rates of the CNO breakout reactions influence the stability transition \\cite{Fisker2006,Keek14}. A further observational constraint is that hydrogen and helium burning must produce sufficient carbon to power the rare superbursts (Section \\ref{sec:superbursts}). Whereas burst models do not create enough carbon, stable burning and shallow heating may be of importance here \\cite{Stev14,Reic17}. \n\nHere we discussed one-dimensional models, which resolve the neutron star \nenvelope in the radial direction. They employ an approximation for turbulent mixing, which hints at the fact that convective mixing of ashes and fresh fuel has an important effect \non the burst ignition and the resulting ash composition (``compositional inertia\" \\cite{Woos04}). In recent years 2D and 3D hydrodynamics models have been created to study convection at the burst onset \\cite{Fryx1982,Zingale2001,Malone2011,Malone2014,Zingale2015} and flame spreading across the stellar surface \\cite{Spit2002,Cavecchi2013,Cavecchi2015,Cavecchi2016}. The computational demands on such simulations are substantial, and only small approximate nuclear networks are employed that lack the $\\alpha \\rm{p}$- and rp-processes.\n\n    \\subsubsection{Nuclear Sensitivities and Recent Uncertainty Reduction Efforts}\\label{sec:rpprocess}\n     \\hfill\\\\\n The nuclear reaction sequence powering Type-I X-ray bursts involves more than a thousand reactions on over three-hundred nuclei, posing a daunting experimental challenge at first glance. Fortunately, not all nuclei and their associated reaction rates are of equal importance. Model calculations of X-ray burst light curves and nucleosynthesis have played and continue to play a critical role in identifying nuclear physics uncertainties that are of the highest priority. The key nuclear physics uncertainties associated with Type-I X-ray bursts are summarized here. Since X-ray burst nucleosynthesis is covered extensively elsewhere~\\cite{Lewi93,Stro03,Scha06,Pari13,Jose16}, we will focus on past highlights and important results from roughly the past decade.\n \n Detailed calculations of the rp-process demonstrate that, of the large number of reactions present, only a handful play a significant role in X-ray bursts~\\cite{Scha98,Wies00,Woos04}. Broadly speaking, these reactions can be categorized into groups (1) ignition\/breakout, (2) branch points, (3) waiting points, and (4) cycles. For a rough orientation on the nuclear chart, (1) and (2) are generally located within the HCNO and $\\alpha \\rm{p}$-process portions of figure~\\ref{figure:RPprocess}, respectively, whereas (3) and (4) are generally located within the rp-process reaction sequence.\n \n  \n\\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.6]{RPprocess.png}\n\\caption{Nuclear reaction sequence powering Type I X-ray bursts~\\cite{Scha01} with colored lines indicating rates driving particular parts of the X-ray burst light-curve~\\cite{Lewi93}.\n\\label{figure:RPprocess}}\n\\end{figure}\n \n \n Group (1) reactions are responsible for triggering the thermonuclear runaway that initiates bursts. Namely, these are reaction rates that connect and break-out of the CNO cycles via $\\alpha$-capture~\\cite{Wies99}. $\\alpha$-capture enables material to flow along the valley of $\\beta$-stability in the nuclear landscape and avoid unbound nuclides that easily photodisintegrate. The important $\\alpha$-captures occur on nuclides with large equilibrium abundances in the $\\beta$-limited CNO cycles, i.e. those whose destruction while in the cycle happens via the relatively slow $\\beta^{+}$-decay \n process~\\cite{Wies99}. \n \n Group (2) reactions occur at nuclides whose competing destruction rates are of a similar order of magnitude, meaning the details of the competition determine the subsequent reaction network flow. Branch points are especially prevalent during the burst rise, when $(\\alpha,\\rm{p})$ and $(\\rm{p},\\gamma)$ reactions on relatively low-$Z$ nuclides compete~\\cite{Fisk04}. These reactions are all located below $^{40}\\rm{Ca}$, as the Coulomb barrier is too large at and above this $Z$ for $\\alpha$-capture to be competitive.\n \n Group (3) reactions are nuclides without a swift destruction process, which temporarily brings most energy generation to a stand-still until a sufficient quantity of material has been transmuted beyond that point. Generally, a waiting-point nucleus has a low or negative proton-capture $Q$-value $Q_{\\rm{p},\\gamma}$, meaning the photodisintegration reaction \n is competitive with the corresponding radiative captures. \n Therefore, the rp-process flow is funneled into much slower $\\beta^{+}$-decay \n or $e^{-}$-capture decay, while a small percentage of the flow bypasses the waiting-point via proton-capture on the equilibrium abundance of the first proton-capture daughter~\\cite{Gorr95,Scha98}. The waiting-point nuclide location and equilibrium abundance are defined by the local proton and photon densities, $n_{\\rm{p}}$ and $n_{\\gamma}$, respectively, and the environment temperature $T$. Waiting points occur when $Q_{\\rm{p},\\gamma}$\n is such that the photodisintegration rate $\\lambda_{\\gamma,\\rm{p}}$ is related to the radiative proton-capture rate $\\lambda_{\\rm{p},\\gamma}$ via\n \\begin{equation}\n \\label{eqn:equilibrium}\n \\frac{\\lambda_{\\gamma,\\rm{p}}}{\\lambda_{\\rm{p},\\gamma}}=\\frac{n_{\\rm{p}}}{n_{\\gamma}}\\left(\\frac{\\mu_\\mathrm{red} c^{2}}{k_\\mathrm{B}T}\\right)^{3\/2}\\mathrm{exp}{\\left(-\\frac{Q_{\\rm{p},\\gamma}}{k_\\mathrm{B}T}\\right)}\\gtrsim1,\\\\\n \\end{equation}\n where \n $\\mu_\\mathrm{red}$ is the reduced mass of the $(\\rm{p},\\gamma)$ reaction, which, for most cases of relevance, $\\mu_\\mathrm{red}\\approx1$.\n From the Planck distribution, $n_{\\gamma}=\\pi\\left(k_{B}T\\right)^{3}\/(13c^{3}\\hbar^{3})$. %\n Using typical rp-process conditions, $\\rho\\approx10^{6}$~g~cm$^{-3}$,  $X(\\mathrm{H})\\approx0.7$, and $T\\approx1$~GK, relation~(\\ref{eqn:equilibrium}) is satisfied for $Q_{\\mathrm{p},\\gamma}\\lesssim$1~MeV.\n \nGroup (4) refers to reaction sequences that operate collectively like a single-nucleus waiting point~\\cite{VanW94}. The most well known of which is probably the SnSbTe cycle which marks the rp-process end-point~\\cite{Scha01}. In these cases, $(\\rm{p},\\alpha)$ and $(\\rm{p},\\gamma)$ reactions compete, where the former continues the cycle and the latter enables the matter flow to break free.\n \nEarly X-ray burst calculations explored the impact of reaction rates on features of the burst light-curve and ashes. For instance, $(\\rm{p},\\gamma)$ reactions beyond $^{56}\\rm{Ni}$ strongly impact the late time light curve in single-zone calculations~\\cite{Hana83}, as \nconfirmed later through calculations with higher-fidelity nuclear physics input~\\cite{Koik99} and multiple zones~\\cite{Woos04}. These results prompted efforts to improve nuclear physics inputs, such \nas implementing Hauser-Feshbach calculations and empirically-based direct capture and resonant rates computed for \nastrophysical calculations~\\cite{VanW94}.\n \nThe impact of nuclear physics uncertainties on X-ray burst ashes has been explored by several groups. \nEarly \nstudies \ntested different reaction rate~\\cite{Koik99} and mass~\\cite{Clem03} libraries. Due to limited computing power, early multi-zone work was restricted to modifying half-lives to approximate the behavior of waiting-points~\\cite{Woos04}. Until recently, detailed sensitivity studies investigating individual reaction rates were necessarily limited to post-processing studies, which lack feedback between nuclear energy generation and environmental conditions, but enabled exploring the impact of rates for a variety of astrophysical conditions~\\cite{Pari08,Pari09}. To date only one X-ray burst sensitivity study has been performed for the whole rp-process with feedback between energy generation and the environment~\\cite{Cybu16}. Even in this case, multi-zone reaction rate variations were limited to cases highlighted as important by a single-zone study, as a train of a dozen or so bursts used to assess rate sensitivity required \na week or more to calculate (see Reference~\\cite{GallowayKeek2017} for a discussion of burst models with different dimensionality). The \nfirst single-zone mass sensitivity study with coupled energy generation and hydrodynamics was only \nrecently  performed~\\cite{Scha17}. \nNote that progress in high-precision mass measurements (e.g. References within~\\cite{Kank17}) means that \nonly a handful of masses with insufficient precision remain. Half-life sensitivities need not be investigated, as almost all relevant half-lives are experimentally well-constrained~\\cite{Scha06,Scha17}. That said, uncertainties remain in the modifications to terrestrial half-lives that are required to account for high-temperature and high-density effects, such as thermal population of excited states, inhibition of bound state electron capture, and continuum electron capture~\\cite{Full1982,Oda1994,Prue2003,Lau18b}.\n \n A cautionary point regarding what the ``most important'' nuclear sensitivities are is that this will depend on the astrophysical conditions. For instance, it is known that the rp-process reaction network path depends on metallicity~\\cite{Jose10} and very different bursts result from models with varied accretion rates~\\cite{Woos04,Lamp16}. \n Additionally, at least one model~\\cite{Davi11} has found discrepant sensitivity with respect to other models~\\cite{Fisker2006,Cybu16} for $^{15}\\rm{O}(\\alpha,\\gamma)$, indicating more cross-code comparisons (e.g., \\cite{Meis17,Meis18}) are needed.\n \n Efforts to remove or reduce the most critical nuclear physics uncertainties have primarily consisted of indirect measurements to constrain nuclear reaction rates. These include measurements of nuclear masses, spectroscopy of low-lying nuclear excited states, and determinations of statistical properties for more highly excited compound nuclei. While direct measurements are preferable, the necessary radioactive beams at astrophysically relevant energies and sufficient intensities are not yet available. \n \n\nWe continue with a \nbrief summary of recent uncertainty reduction efforts for relevant rp-process reaction rates grouped by reaction categories defined above.\n\n\\paragraph{Ignition\/Breakout: } Experimental efforts have primarily focused on the two key break-out rates, $^{18}\\rm{Ne}(\\alpha,p)$ and $^{15}\\rm{O}(\\alpha,\\gamma)$, and the $^{14}\\rm{O}(\\alpha,p)$ rate which links the CNO cycles on the way to breakout. $^{14}\\rm{O}(\\alpha,p)$ has most recently been summarized in References~\\cite{Alma12b} and \\cite{Hu14}. Stated briefly, this rate may be known to sufficient precision. However, further experimental investigations are merited in the $\\sim6-7$~MeV excited state energy region of the $^{18}\\rm{Ne}$ compound nucleus to confirm important resonance information. Furthermore, theoretical confirmation is required to assess the impact of $^{14}\\rm{O}(\\alpha,p)$ in more sophisticated X-ray burst calculations. $^{15}\\rm{O}(\\alpha,\\gamma)$ has most recently been evaluated in Reference~\\cite{Davi11}, though the most recent experimental results come from References~\\cite{Tan07,Tan09,Lund16,Wred17}. For this case the uncertainty stems from the unknown resonance strength for capture into the $\\sim 4$~MeV excited state of $^{19}\\rm{Ne}$. The small $\\alpha$-branching ratio dominates the rate uncertainty and has thus far evaded direct measurement. An example of this rate's impact on the X-ray burst light curve is shown in figure~\\ref{figure:modelLC}. The state of $^{18}\\rm{Ne}(\\alpha,p)$ is summarized in Reference~\\cite{Mohr14}, relying primarily on complementary measurements from References~\\cite{Mati09,Salt12,He13,Zhan14}. Further progress for this rate would require experimental resonance strength determinations in the $\\sim9-11$~MeV excitation energy region in $^{22}\\rm{Mg}$.  A less significant but nonetheless notable development is the recent set of experimental constraints placed on $^{19}\\rm{Ne}(p,\\gamma)$, which connects $^{15}\\rm{O}(\\alpha,\\gamma)$ to the rp-process, including the first direct measurement of capture onto a radioactive ion excited state~\\cite{Wilk17,Bela16}.\n \n  \n\\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.7]{RateVariationExample.pdf}\n\\caption{Averaged X-ray burst light curves and asymmetric 1-$\\sigma$ uncertainties corresponding to sequences of $>$10 bursts, for multi-zone X-ray burst calculations performed with MESA~\\cite{Paxt11,Paxt13,Paxt15}. Calculations employ a solar accreted composition at $\\dot{M}$=0.17$\\dot{M}_\\mathrm{Edd}$ onto an 11.2~km neutron star with $0.1$~MeV per accreted nucleon of base heating, where the distance and redshift of GS~1826$-$24 reported in Reference~\\cite{Gall17} are used to convert luminosity (from the calculation results) to flux. Results are shown using the ReacLibV2.2 \nrate library~\\cite{Cybu10} (gray band), with a reduced $^{15}\\rm{O}(\\alpha,\\gamma)$ rate (blue band), and a reduced $^{59}\\rm{Cu}(p,\\gamma)$ rate (red band). All other calculation details are as described in Reference~\\cite{Meis17}.\n\\label{figure:modelLC}}\n\\end{figure}\n \n \\paragraph{Branch Points: } Rates in the $\\alpha \\rm{p}$-process of particular importance are those involving nuclides where destruction via $(\\alpha,\\rm{p})$ is competitive with the $(\\rm{p},\\gamma)$ or $\\beta^{+}$-decay alternatives. Chief among these are $^{26}\\rm{Si}(\\alpha,\\rm{p})$, $^{30}\\rm{S}(\\alpha,\\rm{p})$ and its competitor $^{30}\\rm{S}(\\rm{p},\\gamma)$, and $^{34}\\rm{Ar}(\\alpha,\\rm{p})$. A single measurement is the source of experimental constraints for $^{26}\\rm{Si}(\\alpha,\\rm{p})$, providing \n level energies but largely lacking certain spin assignments and $\\alpha$-widths~\\cite{Alma12a}. However, the relevant level densities are large enough that a statistical reaction rate approach may be adequate. $^{30}\\rm{S}(\\alpha,\\rm{p})$ has been the subject of more investigations, including direct measurements of the time-reversed reaction~\\cite{Deib11} and spectroscopic studies of the compound nucleus $^{34}\\rm{Ar}$~\\cite{Obri09,Kahl18}. As with the previous case, absent a direct measurement of the forward reaction,\n $\\alpha$-widths are the source of the main experimental uncertainty.  The competing reaction $^{30}\\rm{S}(\\rm{p},\\gamma)$ involves lower-lying excitation energies in the compound nucleus $^{31}\\rm{Cl}$ and has thus been more amenable to spectroscopic constraints, resulting in a relatively well-constrained reaction rate~\\cite{Wred09,Lang14b}. In the past few years $^{34}\\rm{Ar}(\\alpha,\\rm{p})$ has been the subject of a series of complementary direct and indirect measurements. These include a spectroscopic measurement using the $(\\rm{p},t)$ mechanism~\\cite{Long17}, a direct measurement above the Gamow-window~\\cite{Schm17}, and another announced work on spectroscopy of the compound nucleus using $\\rm{p},\\rm{p}'$~\\cite{Laue16}.\nA comprehensive analysis of these works will likely vastly improve the situation regarding this branch-point nucleus.\n \n\\paragraph{Waiting Points: }\nThe nuclides which satisfy\nrelation~(\\ref{eqn:equilibrium})\nand have several-second half-lives\nare $^{56}$Ni, $^{64}$Ge, $^{68}$Se, $^{72}$Kr, and $^{100}$Sn. For these cases, $Q_{\\rm{p},\\gamma}$ for the waiting-point nucleus is essential to determine the equilibrium abundance of the proton-capture daughter. $Q_{\\rm{p},\\gamma}$ for the first proton-capture daughter and the structure of the second proton-capture daughter are equally important in order to determine the flow of material through the waiting-point. For waiting-points away from the proton drip-line, the possibility of material flowing around the waiting-point also needs to be investigated. $^{56}$Ni falls into the latter category, where experimental constraints suggest a strong waiting-point~\\cite{Lang14}, but \nit is possible this waiting-point could be bypassed \nat high temperatures and densities~\\cite{Ong17,Valv18}. $^{64}$Ge has been declared to not be a waiting-point~\\cite{Wing93,Tu11}, but this claim was likely premature. The large remaining uncertainty in the nuclear masses of $^{65}$As and $^{66}$Se leave open the possibility for a strong waiting-point, as do uncertain properties of relevant resonances for the $^{64}\\rm{Ge}(\\rm{p},\\gamma)^{65}\\rm{As}(\\rm{p},\\gamma)^{66}\\rm{Se}$ sequence~\\cite{Lam16}.  Independent measurements have determined a rather negative proton-separation energy of $^{69}\\rm{Br}$, resulting in $<$13\\% of material flowing through proton-capture on $^{68}$Se, solidifying $^{68}$Se as a strong waiting-point~\\cite{Blan95,Pfaf96,Roge11,DelS14}. Similar recently completed measurements will soon resolve the waiting-point status of $^{72}$Kr, but results are not yet published~\\cite{Roge17}.\nStudies of $^{100}\\rm{Sn}$ are limited to a confirmation of its long half-life~\\cite{Bazi08}, but the proximity of this reaction to the rp-process end-point limits its importance for all but the most energetic X-ray bursts~\\cite{Scha01}.\n\n\\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.8]{NiCuZnGaCyclesV2.png}\n\\caption{Portion of the rp-process reaction sequence featuring the NiCu and ZnGa reaction cycles. $^{59}\\rm{Cu}(p,\\gamma)$, $^{61}\\rm{Ga}(p,\\gamma)$, and $^{63}\\rm{Ga}(p,\\gamma)$ (red arrows) significantly affect the reaction flow in this region.\n\\label{figure:cycles}}\n\\end{figure}\n\n\\paragraph{Cycles: }\nFor cases where the $\\alpha$ emission threshold is lower than the proton threshold,  $(\\rm{p},\\gamma)$ and $(\\rm{p},\\alpha)$ reactions can compete~\\cite{Remb97}. In the vicinity of $Z=28-31$, just beyond $^{56}\\rm{Ni}$ in the rp-process path, this leads to the NiCu and ZnGa cycles~\\cite{VanW94}. For these cycles, radiative proton-capture onto $^{59}\\rm{Cu}$, $^{61}\\rm{Ga}$, and $^{63}\\rm{Ga}$ are of particular importance, as indicated in figure~\\ref{figure:cycles}. Proton-capture onto $^{59}\\rm{Cu}$ either returns the cycle to $^{56}\\rm{Ni}$ or breaks-out via $^{59}\\rm{Cu}(\\rm{p},\\gamma)$, depending on the $(\\rm{p},\\gamma)\/(\\rm{p},\\alpha)$ rate ratio. Once out of the NiCu cycle, the several minute half-life of $^{60}\\rm{Zn}$ stalls the rp-process unless it is bypassed by proton-capture. The $\\sim250$~keV $Q$-value for $^{60}\\rm{Zn}(p,\\gamma)^{61}\\rm{Ga}$ enables an equilibrium abundance of $^{61}\\rm{Ga}$ to be built-up, which can then bypass $^{60}\\rm{Zn}(\\beta^{+})$ via $^{61}\\rm{Ga}(\\rm{p},\\gamma)$. $^{63}\\rm{Ga}$ plays the analogous role in the ZnGa cycle to the role of $^{59}\\rm{Cu}$ in the NiCu cycle. At present these rates are primarily determined by theoretical estimates~\\cite{Fisk01,Raus00} since the distance from the valley of $\\beta$-stability makes even indirect measurements challenging.\n\n\\paragraph{}\nThe majority of rp-process reaction rates lack any experimental constraints beyond nuclear masses and rough structure details. In their stead, the most common approach is to employ shell model rates when available (and applicable), particularly in the mid-mass region~\\cite{Fisk01}, and statistical model rates otherwise, e.g.~\\cite{Raus00}. A commonly used reaction library which follows this strategy is the JINA-CEE ReacLib database~\\cite{Cybu10}.\n\nHowever, rapid experimental progress is anticipated in the near future. Reaccelerated beams at the Facility for Rare Isotope Beams studied with devices such as the JENSA gas-jet target~\\cite{Bard16} and SECAR recoil separator~\\cite{Berg17} will enable direct measurements of several interesting reaction rates at astrophysically relevant energies. Nonetheless, indirect measurements will continue to guide direct measurement studies and to cover the large number of cases for which direct measurements will still not be possible. Newer techniques, such as the $\\beta$-Oslo method~\\cite{Spyr14}, are poised to significantly grow the ranks of experimentally-constrained reaction rates for nuclides far off stability. \n\nThough only discussed tangentially here, much of the discussion above also pertains to nuclosynthesis during stable burning on the neutron star \nsurface. For near Eddington and super-Eddington accretion rates, nuclear reaction sequences often resemble the rp-process~\\cite{Scha99,Stev14,Keek16,Meis17}, resulting in burst-like ash distributions (see figure~\\ref{figure:ashes}). However, the detailed abundance distribution depends on the astrophysical conditions, and so a wide variety of ash abundances can be produced in stable burning.\nFor instance, recently a regime of stable burning has been proposed around $0.1\\dot{M}_\\mathrm{Edd}$, with carbon-rich ashes \\cite{Keek16}.\nThe exact reactions of interest for these processes will depend on the hydrogen\/helium composition\nof the accreted material and the accretion rate\n~\\cite{Nara03,Keek14}. \n \n \\subsection{X-ray Superbursts:}\n\\label{sec:superbursts}\n\n  \\subsubsection{Observation}\n   \\label{sssec:SuperburstObs}\n     \\hfill\\\\\nSuperbursts were first observed in 1996 with the BeppoSAX\/WFCs \nand RXTE~\\cite{Cornelisse2000,Stro02}. To date, 26 superbursts have been detected from 15 neutron stars \\cite{Zand17}. Superbursts reach a peak flux that is similar to normal bursts, but their decay lasts several hours, and the total emitted energy is $\\sim 10^3$ times larger than for hydrogen\/helium flashes \nwhich justifies the ``super'' designation \\cite{wijnands2001}.\nThe X-ray observatories that detected superbursts are in low Earth orbits of $\\sim 90$ minutes, which is shorter than the typical superburst duration. Therefore, superburst observations are interrupted by data gaps of up to $30$ minutes, when the view of the X-ray source is blocked by the Earth (see  figure~\\ref{figure:LCobs}, bottom panel).\nOften the start of the superburst falls in a data gap, and it is challenging to accurately determine the burst's properties and confirm its superburst nature, so it is instead referred to as a ``superburst candidate\"~\\cite{Altamirano2012}. \nMost superbursts have been observed with wide-field or all-sky instruments, which produce data of modest quality. Only in two cases have detailed observations been performed with RXTE\/PCA \\cite{Stro02,Strohmayer2002a}.\n\nThe observed superburst light curves are fit with numerical models of cooling envelopes to determine the ignition conditions. From the tail of the light curve the ignition column depth is measured, \nand the fluence constrains the energetics of the fuel \\cite{2004CummingMacBeth,Cumming2006}. The inferred parameters  suggest that unstable carbon burning in the neutron star ocean ignites superbursts \\cite{cumming2001,Stro02}. Ignition column\ndepths are found to be in the range of $y_\\mathrm{ign}\\simeq(0.2-3)\\times 10^{12}\\ \\mathrm{g\\ cm^{-2}}$, and the fuel energetics are equivalent to a carbon mass fraction of $\\sim 20\\%$ \\cite{Cumming2006,Zand17}. The rise of the light curve is shaped by the temperature profile left behind by the carbon flame (measured only once \\cite{Keek16}).\n\nAll superbursting sources also exhibit normal (short) bursts. Most have mass accretion rates in the range of $0.1-0.3\\ \\dot{M}_\\mathrm{Edd}$, where the burst rate drops and a substantial fraction of the accreted hydrogen\/helium burns in a stable manner \\cite{Zand2004}. Both explosive and stable hydrogen\/helium burning may be necessary for superbursts. As the normal bursts do not produce sufficient carbon, stable burning is likely needed to create the carbon fuel (e.g., \\cite{Stev14,Keek16}; Section~\\ref{sec:burst_theory}). However, the bursts produce heavy isotopes (iron group or heavier), which increase the opacity, allowing the base of the ocean to reach the temperature required for runaway carbon fusion \\cite{cumming2001}.\n\nBecause of their long recurrence times (typically a year) \\cite{Zand17}, superbursts are rare, and each new observation brings new insight. \n\n   \\subsubsection{Theory}\n      \\hfill\\\\\nThe nuclear processes that power superbursts show a resemblance to those in models of Type Ia supernovae. The runaway is initiated by $^{12}$C+$^{12}$C burning. One of the dominant channels is $\\mathrm{^{12}C(^{12}C,\\alpha)^{20}Ne}$. A fraction of the $\\alpha$ particles capture on carbon to form oxygen, enabling the follow-up reactions $^{12}$C+$^{16}$O and $^{16}$O+$^{16}$O. The ashes of these reactions are rich in $^{28}$Si. At sufficiently high temperatures ($T\\gtrsim 10^9$~K) photodisintegration of silicon occurs (``silicon melting''). A large number of $\\alpha$ particles is \ncreated, the majority of which are quickly captured again to form a composition that is dominated by $^{56}$Ni. The precise ash composition is set by the nuclear statistical equilibrium of photodisintegration and capture reactions. Any isotopes in the hydrogen\/helium ashes substantially heavier than nickel will also photodisintegrate, and this can contribute up to 50\\% of the superburst energy~\\cite{Scha03}. From the carbon runaway, all these burning processes take less than a second to complete. Afterwards, as the layer cools, electron captures transform $^{56}$Ni into $^{56}$Fe, making $A=56$ the primary component of the ashes as seen in figure~\\ref{figure:ashes}. \n\nSimilar to the hydrogen\/helium bursts (section~\\ref{sec:burst_theory}), one-dimensional multi-zone models are employed to study superburst ignition and the various burning processes. Self-consistent simulations should model hundreds of hydrogen\/helium flashes to build up a carbon layer and produce a superburst, but this approach suffers from two problems. First, it is computationally expensive, and studies with reduced rp-process networks are likely not accurate (compare Reference~\\cite{Taam1996} to Reference~\\cite{Woos04}). Second, as noted in section~\\ref{sssec:SuperburstObs}, current models of hydrogen\/helium burning do not produce enough carbon. \nTherefore, superburst models typically directly accrete a carbon-rich fuel  \nto study \nignition conditions \\cite{Keek11}. Shortly before ignition, the accretion composition can be switched to hydrogen\/helium-rich material to study the effect of a superburst on the atmosphere \\cite{Keek2012}. Importantly, the heated ocean after a superburst quenches short bursts in the atmosphere: for several days hydrogen and helium burning becomes stable.\n\nNo multi-dimensional models have yet been created for superbursts. One-dimensional simulations show, however, that the hydrodynamics of the carbon flame is important for shaping the observable X-ray light curve. Initially, a convective region forms around the ignition depth, until \nheating by thermonuclear burning in one  zone becomes faster than cooling by convection.\nA local runaway ensues, burning all carbon in that zone. The burning time scale becomes shorter than the sound-crossing time scale, raising the question whether the flame will spread as a detonation \\cite{Weinberg2006sb,Weinberg2007}, or whether convection can spread the flame to lower depths as a deflagration. Either of these processes can generate a shock that produces a short precursor burst at the start of the superburst light curve \\cite{Keek2012}. \nThe carbon flame leaves behind a temperature profile in the envelope, which shapes the observable light curve. It encodes information about the ignition depth, fuel energetics, and the flame spreading (e.g., whether the flame reached the surface or stalled) \\cite{2004CummingMacBeth,Cumming2006,Keek16}.\n\nThe ignition depths inferred from observations are smaller than those predicted by theory \\cite{Keek11}, leading to questions about carbon fusion as the process that powers superbursts.\nThe problem of carbon ignition became more pressing with the discovery of superbursts from transient sources \\cite{Keek08,Altamirano2012,Chenevez2011ATel,Serino2016}. The neutron star \nenvelope heats up during accretion, but in \nmost of these cases accretion was active for mere days or weeks, which was thought to be insufficient to reach the temperature needed for runaway carbon fusion \\cite{Keek08}. Therefore, stronger heating of the base of the crust is required on the relatively short time scales of the transient accretion events. This may be related to the shallow heat source inferred for \nquasi-persistent transients\nwhen they cool after accretion ceases (section~\\ref{sec:cooling_transients}). Simulations confirm that the ignition depth is reduced with the addition of extra heating \\cite{Keek11}. Interestingly, additional shallow heating has also been proposed for enhancing carbon production during stable burning~\\cite{Reic17}.\n\n\n   \\subsubsection{Nuclear Sensitivities and Recent Uncertainty Reduction Efforts}\n   \\label{sssec:superburstnuc}\n     \\hfill\\\\\n     The several gigakelvin temperature achieved in superbursts is more than sufficient to drive material to nuclear statistical equilibrium~\\cite{Scha03}.\n     As such, the final abundance distribution is largely determined by the nuclear masses, environment temperature, and environment \n     $Y_{e}$. In principle the abundance distribution will be modified by light charged-particle capture after nuclear statistical equilibrium freeze-out, but low proton and $\\alpha$ abundance predictions result in a negligible modification~\\cite{Scha03}. \nSince the majority of the relevant nuclear masses are well known, uncertainties which may affect $Y_{e}$ are of interest. Namely, these are the reactions modifying the ashes of the stable burning and X-ray bursts \nalready discussed above.\n\nThough inconsequential in terms of the resulting crust composition, the  $^{12}$C$+$$^{12}$C fusion rate is nonetheless of interest due to its role in triggering superburst ignition~\\cite{Coop09}. The relatively low energy of interest for the large Coulomb barrier involved (see section~\\ref{sec:pycno})\nhave made advances in this area extremely challenging~\\cite{Cost09}. Only very recently have direct \nlaboratory measurements pushed into the astrophysical energy region\n~\\cite{Buch15,Jian18}. Of particular interest is the possible existence of a resonance in $^{12}$C$+$$^{12}$C at a center of mass energy of $\\sim$1.5~MeV~\\cite{Coop09}. This resonance has been posited to explain the discrepancy between the observed and predicted superburst ignition depth, much in the way the Hoyle state was predicted to explain cosmic carbon production~\\cite{Hoyl54}.\nHowever, present experimental constraints are contradictory and rely on theoretical extrapolations~\\cite{Buch15b}.\n\n\\section{Interaction of Nuclei Within the neutron star Ocean and Crust} \\label{section:interaction}\n\nThe ashes of surface burning experience compression under the pressure of the continuously infalling material (section~\\ref{section:accretion}). As they move to higher mass densities (and greater electron chemical potentials), they undergo a variety of non-equilibrium reactions.\nThese reactions, primarily $e^{-}$-captures, cause the compressed nuclei to become increasingly neutron-rich. Near the neutron drip point, the  reaction mechanisms operating in the crust transitions from  $e^{-}$-capture reactions (with only $\\nu$ and $\\gamma$ emission) \nto neutron emission reactions \\cite{Gupt07,Gupt08,Lau18}. The presence of pycnonuclear (density-driven) fusion reactions~\\cite{Came59,Harr64,Salp69} was also found in the inner crust~\\cite{Sato79}. These reactions\ndirectly modify the crust composition, but more importantly, they can drastically alter the thermal structure of the crust, impacting thermonuclear processes on and near the surface \nand related observables~(\\cite{Mira90,Keek11,steiner2012,Scha14,Deib16,Meis17}; section~\\ref{section:impact}). \n\n\nNuclear reactions deposit heat in the neutron star  \ncrust during an accretion episode~\\cite{Haen90,Haen03,haensel2008} and an accurate accounting of crustal heating during active accretion is required to match the neutron star's \nsurface temperature when accretion ends~\\cite{Brow09}. The impact of $e^{-}$-captures on heating was substantially increased when it was realized that nuclear de-excitations following electron captures increase heat deposition~\\cite{Gupt07,Gupt08,Zhan15}. Experimental investigations along these lines include References~\\cite{Estr11,Meis15}. \n\nNon-equilibrium reactions may also cool the neutron star  \ncrust. Urca cooling\\textrm{---} cycles of $e^{-}$-capture and $\\beta^{-}$-decay that\ngenerate neutrino emission\\textrm{---} was introduced for white dwarf stars~\\cite{Gamo40,Gamo41}. \nOnly recently was it realized that the finite temperature of the neutron star ocean and crust allows sufficient phase-space for Urca cooling to exist~\\cite{Gupt07,Scha14}. \nShortly after, this was limited to odd-A nuclides~\\cite{Meis15} and efforts have been made to employ more experimentally-based nuclear data and identify the presence of Urca cooling in the neutron star's \nouter layers \nbased on observations of neutron star crustal cooling~\\cite{Deib16,Meis17}. \n\n   \n    \\subsection{$e^{-}$-Capture Reactions}\n    \\label{ssec:ECrxns}\nThe first transmutation ashes experience as they are buried by accretion of additional material is a sequence of $e^{-}$-captures.\nThe rising electron chemical potential \n$\\mu_e$ with increased depth makes it energetically possible for $e^{-}$-capture to proceed for nuclei with steadily more negative $e^{-}$-capture $Q$-values, $Q_{\\rm{EC}}$. Once $\\mu_e>|Q_{\\rm{EC}}|$, $e^{-}$-capture immediately proceeds.\nThe end result is that the crust becomes steadily more neutron-rich with increasing depth\nas shown in figure~\\ref{figure:ECcrustcomposition}. Surface values of $\\langle Z\\rangle$, $\\langle A\\rangle$, and $Q_{\\rm{imp}}$ for the ashes shown in figure~\\ref{figure:ashes} are listed in table~\\ref{table:AvgZAQTable}.\n\n   \n\\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.65]{CrustCompositionFromECMuE.pdf}\n\\caption{Evolution of the average proton number $\\langle Z\\rangle$, average neutron number $\\langle N\\rangle$, and impurity $Q_\\mathrm{imp}$ (see equation~\\ref{equation:Qimp}) \ndue to $e^{-}$-capture for X-ray burst and superburst ashes (see figure~\\ref{figure:ashes}), using experimental masses when available~\\cite{Audi12} and the WS3 global mass model otherwise~\\cite{Wang10}. The superburst impurity is multiplied by 10 for visibility. \nThis plot does not account for neutron emission and fusion reactions (sections \\ref{sec:nuc_emission} and \\ref{sec:pycno}).\nSee figure~\\ref{figure:pristinecomp} to compare to the pristine crust composition.\n\\label{figure:ECcrustcomposition}}\n\\end{figure}\n   \n   \\begin{table}\n\\caption{Properties on the neutron star surface of the ashes shown in figure~\\ref{figure:ashes}.}\n\\begin{center}\n\\begin{tabular}{c c c c  }\n\\hline\nQuantity & X-ray Burst & Superburst & Stable Burning \\\\\n\\hline\n$\\langle Z\\rangle$ & 24 & 25 & 26 \\\\\n$\\langle A\\rangle$ & 52 & 55 & 60 \\\\\n$Q_{\\mathrm{imp}}$ & 69 & 3 & 101 \\\\\n\\hline\n \\label{table:AvgZAQTable}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nThe interesting pattern in the impurity shown for X-ray burst and superburst ashes in figure~\\ref{figure:ECcrustcomposition} deserves some discussion. The overall decline in impurity with increasing depth is due to the fact that higher-$Z$ elements near stability undergo $e^{-}$-capture at lower $\\mu_e$ relative to their lower-$Z$ counterparts because the nuclear mass surface is more shallow near the valley of $\\beta$-stability at high $A$.\nThe jumps in impurity are due to $e^{-}$-captures on the few isotopes which are the most abundant (see figure~\\ref{figure:ashes}). For superbursts, the dominant species are $A=52,54$, and $56$, while for X-ray bursts $A=60,64$, and $68$ dominate the composition with significant but less influential contributions from $A=28$ and $32$. This can be confirmed \nby a comparison of the location for jumps in the impurity in figure~\\ref{figure:ECcrustcomposition} and $|Q_{\\rm{EC}}|$ for neutron-rich isotopes of the aforementioned isobars.\n\n   \\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.6]{ECschematic.png}\n\\caption{Scenarios for $e^{-}$-capture in the accreted crust at a depth indicated by the electron chemical potential $\\mu_{e}=|Q_{\\rm{EC}}|$, resulting in heating (a and b), which happen in steps indicated by the encircled numbers, or cooling (c and d), which are cyclic.\nFor $e^{-}$-capture on an odd-$A$ nucleus, a subsequent $e^{-}$-capture cannot occur until the nucleus has been buried to a larger $\\mu_{e}$. Therefore whether heating (a) or cooling (c) occurs depends on whether $\\beta^{-}$-decay from the first $e^{-}$-capture daughter is favorable. For $e^{-}$-capture on an even-$A$ nucleus, odd-even mass staggering generally enables an immediate subsequent $e^{-}$-capture, resulting in heating (b). However, special circumstances, namely a low-lying (relative to $k_\\mathrm{B}T$) low-$J^{\\pi}$ isomer in the odd-odd daughter of the first $e^{-}$-capture, can enable cooling via $e^{-}$-capture-$\\beta^{-}$-cycling to proceed instead (d). \n\\label{figure:ECschematic}}\n\\end{figure}\n\n\n$e^{-}$-capture reactions are the richest in terms of potential for experimental nuclear physics constraints with present and near-future facilities. Depending on the nuclear masses and structure of the nuclei involved in an $e^{-}$-capture reaction sequence,\none of four scenarios can take place once $\\mu_e\\approx|Q_{\\rm{EC}}|$, as depicted in figure~\\ref{figure:ECschematic}. For these cases, the two-step $e^{-}$-captures will generally create local heat sources~\\cite{Gupt07}, whereas the $e^{-}$-captures that can be reversed via $\\beta^{-}$-decays can create local cooling sources~\\cite{Scha14}.  The heating predominantly comes from radiative de-excitation of states populated in $e^{-}$-capture, whereas cooling is due to neutrinos produced in the $e^{-}$-capture and $\\beta^{-}$-decay escaping the neutron star  \ncrust. \n\nThe majority of $e^{-}$-capture heating comes from the scenario depicted in panel (b) of figure~\\ref{figure:ECschematic}, where odd-even mass staggering makes it energetically favorable for an $e^{-}$-capture on an even-even nucleus to be immediately followed by $e^{-}$-capture on the odd-odd daughter of the first reaction. For this case, $\\mu_e\\approx|Q_{\\rm{EC}}(Z,A)|>|Q_{\\rm{EC}}(Z-1,A)|$. The first $e^{-}$-capture happens near threshold (unless there is a large change in $J^{\\pi}$ required for a ground-state to ground-state transition), \nso little heating or cooling is achieved. In the second $e^{-}$-capture, roughly one-quarter of the surplus energy will be deposited into the crust~\\cite{Haen03}. \nUsually the second $e^{-}$-capture proceeds into an excited state of energy $E_\\mathrm{xs}$, and all of the de-excitation energy \nis deposited into the crust, resulting in the lion's share of the heating.\nThe energy deposited for a two-step $e^{-}$-capture starting on isotope $Z,A$ at a depth $\\mu_e=|Q_{\\rm{EC}}(Z,A)|$ is \n\\begin{equation}\nE_{\\rm{heat}}=\\eta\\left(|Q_{\\rm{EC}}(Z,A)|-|Q_{\\rm{EC}}(Z-1,A)|-E_\\mathrm{xs}\\right)+E_\\mathrm{xs},\n\\label{equation:NucHeating}\n\\end{equation}\nwith $1\/6\\lesssim\\eta\\lesssim1\/4$, where the exact pre-factor $\\eta$ requires an explicit calculation of $e^{-}$-capture to excited states~\\cite{Haen03,Gupt07}.\nIn the case depicted in figure~\\ref{figure:ECschematic}a, $\\mu_e$ is not large enough for the second e$^-$-capture to occur immediately. However, the $\\beta^{-}$-decay from the first $e^{-}$-capture daughter is much slower than the accretion timescale. Therefore the second $e^{-}$-capture happens only after the  chemical potential has increased enough for the next $e^{-}$-capture to proceed. Little heating is produced in this case.\nEvidently the nuclear masses and low-lying excited states of neutron-rich isobars of nuclides predicted to be abundant in the ashes of surface burning processes are of significant interest for $e^{-}$-capture heating in the accreted neutron star outer crust. \nThe amount of heating produced by $e^{-}$-capture reactions in the outer crust depends on the ashes composition.  For X-ray burst ashes, the total heating is on the order of $0.1-0.15$~MeV\/u, or about $10^{35}$~erg~s$^{-1}$ at Eddington accretion rate \\cite{Gupt07}. Notice that this is more than five times larger than the earlier estimates for a single-component composition \\cite{Haen90} which missed the role of excited states shown in figure~\\ref{figure:ECschematic}b.  Significant heating mostly results from predominant nuclides with $X(A)\\gtrsim 10\\%$. It was found that when the heating through excited states is included, the simple single-component model give similar $e^{-}$-capture heating as compared to results of more complete multi-component reaction networks \\cite{haensel2008}.\n\n$e^{-}$-capture cooling via neutrino-emission primarily occurs from scenario (c) in figure~\\ref{figure:ECschematic}, where the odd-even mass staggering eliminates the possibility of an immediate subsequent $e^{-}$-capture following $e^{-}$-capture on an odd-$A$ nucleus. Instead, a $\\beta^{-}$-decay may proceed (as opposed to case (a), where we assumed a much slower rate for the weak transition), \nresulting in an $e^{-}$-capture\/$\\beta^{-}$-decay cycle known as an Urca process. Nominally such a scenario is possible for an even-even nucleus, however a specific set of circumstances is required. \nNamely, as depicted in scenario (d) of figure~\\ref{figure:ECschematic}, a low $J$ ($\\sim0$$-$1) state with $E_\\mathrm{xs}\\sim k_\\mathrm{B}T$ must be present for an odd-odd nucleus with a high $J$ ($\\gtrsim2$) ground state. The only promising candidate so far for scenario (d) has been ruled-out~\\cite{Meis15}.\n\nThe strength of Urca cooling is quantified by the luminosity of neutrinos produced by the pair of nuclides involved in the $e^{-}$-capture\/$\\beta^{-}$-decay Urca cycle. In its essence, the neutrino luminosity is determined by the quantity of nuclides in an Urca pair, which is \nset by the isobaric abundance and the Urca shell thickness, multiplied by the Urca cycle rate,\nwhich in turn is determined by the weak transition rate. The shell thickness is proportional to the temperature $T$, since the Urca cycle can operate within the window $\\mu_e\\approx|Q_{\\rm{EC}}|\\pm k_\\mathrm{B}T$, and inversely proportional to the local gravity $g$,\nas follows from differentiating equation~(\\ref{eq:z-mu})~\\cite{Scha14}. The weak transition rates for the Urca cycle are  \nmostly determined by the integral over the momentum phase-space~\\cite{Beth47}. \nDue to the electron-degeneracy, the phase-space is limited to the small \nthermal window around $E_{e^{-}}\\approx \\mu_e$, which is several $k_\\mathrm{B}T$ wide~\\cite{Tsur70}.\n\nThe above considerations result~\\cite{Tsur70,Deib16} in a neutrino cooling luminosity\n\\begin{equation}\n\\label{equation:Lnu}\nL_{\\nu}(Z,A) \\approx L_{34}(Z,A)\\times10^{34}\\,{\\rm{erg~s}}{}^{-1}X(A)T_{9}^{5}\\left(\\frac{g_{14}}{2}\\right)^{-1}\\left(\\frac{R}{10~\\mathrm{km}}\\right)^{2}. \\\\\n\\end{equation}\nHere $X(A)$ is the mass-fraction of the $e^{-}$-capture parent nucleus in the composition, $T_{9}$ is the temperature of the Urca shell in units of $10^{9} \\, \\mathrm{K}$, \nand $g_{14}\\equiv g\/(10^{14}~\\rm{cm}~\\rm{s}^{-2})$.\nThe intrinsic cooling strength of the Urca pair, $L_{34}(Z,A)$, is given by\n\\begin{equation}\n\\label{equation:L34}\nL_{34}(Z,A)\\approx 0.87\\left(\\frac{10^{6}~{\\rm{s}}}{ft}\\right)\\left(\\frac{56}{A}\\right)\\left(\\frac{Q_{\\rm{EC}}(Z,A)}{4~{\\rm{MeV}}}\\right)^{5}\\left(\\frac{\\langle F\\rangle^{*}}{0.5}\\right), \\\\\n\\end{equation}\nwhere $\\langle F\\rangle^{*}\\equiv\\langle F\\rangle^{+}\\langle\nF\\rangle^{-}\/(\\langle F\\rangle^{+}+\\langle F\\rangle^{-})$, the Coulomb factor $\\langle F\\rangle^{\\pm}\\approx2\\pi\\alpha_f\nZ\/|1-\\exp(\\mp2\\pi\\alpha_f Z)|$, and $\\alpha_f\\approx1\/137$ is the\nfine-structure constant. $ft$ is the comparative half-life, which should be implemented as $ft=({ft_{e^{-}\\rm{-capture}}+ft_{\\beta^{-}}})\/{2}$, since the degeneracy of the parent state impacts the transition rate. However, $ft$ from the $e^{-}$-capture and $\\beta^{-}$-decay generally agree within a factor of a few~\\cite{Paxt15e}, so using an estimate for one or the other transitions results in a negligible uncertainty compared to other contributions.\n\nThe cyclic nature of the  $e^{-}$-capture\/$\\beta^{-}$-decay enables the energy release from cooling to exceed energy deposition from the inherently one-way $e^{-}$-capture-only cases by more than an order of magnitude~\\cite{Meis15}. We stress that this is \nplausible when the crust temperature is of the order of $1~$GK, given the $T^{5}$ dependence of equation (\\ref{equation:Lnu}). For a typical outer crust $Q_{\\rm{EC}}\\approx12$~MeV and a not unreasonable estimate of ${\\rm log}_{10}(ft)\\approx5$, \nequation (\\ref{equation:L34}) results in $L_{34}\\approx10^{3}$. Similar $L_{34}$ values \nare obtained for $Q_{\\rm{EC}}\\approx8$~MeV and ${\\rm log}_{10}(ft)\\approx4$ or $Q_{\\rm{EC}}\\approx15$~MeV and ${\\rm log}_{10}(ft)\\approx5.7$.\nFor such nuclei, $L_{\\nu}$ will be significant for $X(A)\\gtrsim1$\\%~\\cite{Meis17} (compare with the $e^-$-capture heating estimate above). \nTherefore, it is critical for model calculations of near-surface nuclear burning to follow abundance evolutions even for relatively low-abundance nuclides. This requires multi-species and multizone nuclear reaction networks with precise nuclear physics input~\\cite{Cybu16}. \nFurthermore, it is worth noting that cooling by  $e^{-}$-capture\/$\\beta^{-}$-decay cycling does not require continued accretion, whereas $e^{-}$-capture heating only occurs while accretion is active.\n\n\nFrom equations~(\\ref{equation:Lnu}) and (\\ref{equation:L34}), it is evident that nuclear masses and low-lying nuclear structure of neutron-rich isobars of abundant surface-burning ashes are required to determine the strength of Urca cooling in accreted neutron star crusts. Ash abundances (discussed in detail in section~\\ref{section:production}) are also critical, as $L_{\\nu}(Z,A)$ scales linearly with the abundance of the Urca pair $X(A)$. Nuclear masses impact the location and strength of Urca cooling since $Q_{\\rm{EC}}$ is \nrelated to atomic mass excesses $\\rm{ME}$ via $Q_{\\rm{EC}}=\\rm{ME}(Z,A)-\\rm{ME}(Z-1,A)$. Low-lying structure of Urca nuclides is key as $ft$ is directly related to the change in spin $\\Delta J$ and parity $\\Delta\\pi$ required for a weak transition to occur~\\cite{Sing98}.\n\nIn the absence of direct measurements, nuclear masses are estimated using global mass models. Popular choices for crust model calculations include the Finite Range Droplet Model (FRDM)~\\cite{Moll12}, one of the many Hartree-Fock-Bogoliubov variations (e.g. HFB-21~\\cite{Gori10}), and one of the Weisz\\\"{a}cker-Skyrme variations (e.g. WS3~\\cite{Wang10}). Globally, the trend in the odd-even mass staggering for increasing neutron-richness is of particular interest. Large odd-even mass-differences along an isobar will result in greater $e^{-}$-capture heating, but will eliminate the possibility of Urca cooling for even-$A$ nuclides~\\cite{Meis15}. Due to this effect, the largest crustal heating is predicted for FRDM, whereas HFB-21 \nresults in the most significant cooling~\\cite{Scha14}. Nonetheless, the sensitivity of $L_{34}(Z,A)$ to $Q_{\\rm{EC}}(Z,A)$ highlights the need for experimental constraints. Recently, such constraints have been the province of time-of-flight mass measurements, as this technique is able to access the most exotic nuclides as compared to alternatives~\\cite{Meis13}. Already such measurements have constrained the heating for some of the strongest heating sources~\\cite{Estr11,Meis16} and ruled-out Urca cooling for $^{56}\\rm{Ti}\\leftrightarrow^{56}\\rm{Sc}$~\\cite{Meis15}, which was once thought to be the strongest cooling source~\\cite{Scha14}.\n\nPresently, experimental data for $ft$ of weak transitions between neutron-rich nuclides are limited.\nFor several cases near to stability, $ft$ is available for ground-state to ground-state transitions and ground-state to excited-state (which would be $e^{-}$-capture parent excited state to $e^{-}$-capture daughter ground-state) from $\\beta^{-}$-decay measurements. Developments are ongoing to enable $(d,^{2}\\rm{He})$ charge-exchange measurements in inverse kinematics with radioactive ion beams, which would provide $ft$ for $e^{-}$-capture parent ground-state to $e^{-}$-capture daughter ground and excited states. However, the latter technique requires a radioactive beam rate of $\\sim10^{6}$~Hz (e.g. Ref.~\\cite{Noji15}), limiting the applicable distance from stability. For the large number of remaining cases lacking direct $ft$ measurements, one must resort to theoretical calculations or estimates derived from data-based systematics~\\cite{Scha14,Deib16,Meis17}.\n\nAmong theoretical estimates, $ft$ from shell-model calculations tend to compare most favorably to experiment~\\cite{Cole12}; however, the absence of large-scale calculations results in limited coverage of the nuclear chart~\\cite{Sull16}. As such, predictions from quasi-random phase-approximation (QRPA) calculations have frequently been adopted instead~\\cite{Gupt07,Scha14}. A more recent, alternative procedure has been to employ $ft$-values consistent with systematics based on the change in spin and parity $\\Delta J^{\\Delta\\pi}$ for a weak transition~\\cite{Deib16,Meis17}. Such a compilation is available in Reference~\\cite{Sing98}. This \nmeans that improved predictions can be provided by spectroscopic measurements, which determine $E_{xs}$ and $J^{\\pi}$ for low-lying states of interest. This is advantageous since spectroscopic measurements require far fewer statistics than measurements of $ft$ via $\\beta^{-}$-decay. \nTherefore constraints can be provided for more exotic nuclides.\n\nGiven the dearth of available data, it is clear that many additional direct and indirect experimental constraints on $ft$ are necessary. In particular, measurements are critical for neutron-rich isobars of odd-$A$ nuclides predicted to have surface-burning ash abundances on the percent-level. Current estimates implicate $^{33}\\rm{Al}\\leftrightarrow^{33}\\rm{Mg}$ and $^{55}\\rm{Sc}\\leftrightarrow^{55}\\rm{Ca}$ as the most significant Urca pairs\nfor X-ray burst and superburst ashes, respectively~\\cite{Meis17}.  \n\n\n \\subsection{Neutron Emission}\\label{sec:nuc_emission}\n \nWith increasing $\\mu_e$, $e^{-}$-captures shift the nuclear composition along isobaric chains towards the neutron drip line where the neutron separation energy $S_\\mathrm{n}$ becomes negative. In that case, $(e^{-},x\\mathrm{n})$ reactions \n(here $x$ means the number of neutrons emitted)\nlead to the buildup of a free neutron abundance. In fact, neutron emission in this process can occur before the neutron drip line if excited states with $E_\\mathrm{xs}> S_\\mathrm{n}$ \nare populated during electron captures \\cite{Gupt08}, as shown in case (b) in figure~\\ref{figure:ECschematic} (at the second $e^{-}$-capture in double capture, or if the first allowed transition at the first step is to the excited state). After the neutron drip line, some neutron emission can proceed in rapid sequences called superthreshold electron-capture cascades (SEC) \\cite{Gupt08}. In these sequences the products of $(e^{-}, x\\mathrm{n})$ reactions can be highly unstable to more $e^{-}$-captures which can occur in a cascade faster than neutron captures.\nThis situation can repeat many times resulting in a diverse reaction sequence with large neutron exchange between isotopic chains. SEC are set by a competition between the $e^{-}$-capture and $(\\mathrm{n},\\gamma)-(\\gamma, \\mathrm{n})$ rates. Note that neutron capture reactions strongly depend on the \nneutron degeneracy and, since radiative processes are involved, can be enhanced by  plasma physics effects  \\cite{Shternin2009PhRvD,Shternin2012PhRvC}. Some free neutrons can appear (to be immediately recaptured by the most abundant nuclei) from $(e^{-},x\\mathrm{n})$ reactions as early as densities of $\\mu_e\\gtrsim 14$~MeV, but do not significantly alter the reaction sequences~\\cite{Lau18}. The SEC processes are only found in multicomponent reaction networks \\cite{Gupt08,Lau18} and do not appear in simplified one-species treatments \\cite{Haen90,Haen03,haensel2008}. Neutron emission processes are an important source of heat since emitted neutrons are rapidly thermalized by mutual collisions. The presence of SEC shifts this type of heating to shallower depths \\cite{Gupt08} than found in one-component models.\n\n\n\\subsection{Pycnonuclear Reactions}\\label{sec:pycno}\n\nFusion reactions become significant when the reaction rate is faster than the rate at which matter is buried to a depth where another $e^{-}$-capture and\/or neutron emission is\/are energetically favorable~\\cite{Sato79,Haen90}. As nuclei are charged, \nfusion reactions require tunneling through the Coulomb barrier. In dense stellar matter the probability of the barrier penetration, and so the rates of fusion reactions, are strongly affected by the environmental conditions. Only when ions are in the weak coupling regime, $T\\gtrsim T_\\mathrm{l}$, where $T_\\mathrm{l}$ corresponds to $\\Gamma=1$ (see section~\\ref{section:structure}), the standard thermonuclear burning (regime I in figure~\\ref{fig:pycno}a) operates. In this regime, the reaction rate is determined solely by the temperature and reacting nuclide abundances. In neutron star crusts most fusion reactions operate in the opposite, pycnonuclear limit, where the possibility to overcome the Coulomb barrier is driven by zero-point oscillations of the ions around their equilibrium positions. The rates of the pycnonuclear reactions are subject to huge uncertainties. In order to discuss those we first briefly address fusion in the familiar thermonuclear regime.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{minipage}[t]{0.45\\textwidth}\n\\includegraphics[width=\\textwidth]{Cpycno.pdf}\n\\end{minipage}\n\\hspace{0.05\\textwidth}\n\\begin{minipage}[t]{0.45\\textwidth}\n\\includegraphics[width=\\textwidth]{MgNepycno.pdf}\n\\end{minipage}\n\\end{center}\n\\caption{Fusion burning in dense stellar matter. {\\it (a)}: Phase diagram for carbon burning adapted from \\cite{Chugunov2007PhRvD}. Black solid lines show relevant temperatures (marked in the plot) separating different burning regimes (I--V). The dash-dotted line shows solidification temperature in a classical liquid ($\\Gamma=175$). \nUpper and lower red solid curves show the loci of the burning time $\\tau_\\mathrm{burn}=1\\,\\mathrm{s}$ and $10^{10}\\,\\mathrm{yr}$, respectively. Thin dashed lines bracket the uncertainty in calculations as discussed in the text. Dotted lines  represent the pure thermonuclear regime neglecting all medium effects. Values of $\\mu_e$ are shown on the upper horizontal scale. {\\it (b)}: Pycnonuclear reaction rates for $^{34}$Ne fusion (upper, red curves) and $^{40}$Mg fusion (lower, blue curves) in the neutron star crust assuming respective one-species composition and body-centered cubic lattice. Solid lines give the optimal reaction rates, while thin dashed lines bracket the theoretical uncertainties in Coulomb barrier penetration calculations. Symbols (open circles for $^{34}$Ne and open triangles for $^{40}$Mg) indicate in each case the depths where $\\tau_\\mathrm{burn}=1$~yr, 1~d, and 1~s. }\\label{fig:pycno}\n\\end{figure}\n\n\nThe key nuclear physics quantity for estimating a fusion reaction rate is the astrophysical $S$-factor~\\cite{Mart09} \nrelated to the reaction cross section $\\sigma(E)$ for charged-particle reactions by  $\\sigma(E)= S(E) E^{-1}\\,\\rm{exp}\\left(-2\\pi\\eta\\right)$, where $E$ is the center of mass energy for the reaction,\n$\\eta=\\alpha_f Z_{1}Z_{2}\\sqrt{{\\mu_{\\rm{red}}c^{2}}\/{(2E)}}$, is the Sommerfeld parameter, \n$Z_{i}$ and $\\mu_{\\rm{red}}c^{2}$ are the nuclear charges and reduced mass for the fusing nuclei, respectively. The advantage of introducing the $S$-factor is that it is a relatively smooth function of energy, where the main part of the strong energy-dependence resulting from the Coulomb tunneling probability is factored out by $\\rm{exp}\\left(-2\\pi\\eta\\right)$. For a structureless (i.e. non-resonant) reaction, which most of the relevant reactions here are thought to be (though see the remark in section~\\ref{sssec:superburstnuc} $^{12}$C$+$$^{12}$C), \nthe counterbalance between the exponential increase in tunneling probability towards higher energies and the exponential decrease of the number of energetic nuclei results in that only nuclei in the vicinity of the \nGamow peak energy $E_\\mathrm{pk}$ in the tail of the thermal distribution actually fuse. The Gamow peak energy is given by\n\\begin{equation}\\label{eq:Gamow_th}\nE_\\mathrm{pk}=\\left(\\frac{\\pi^2 \\mu_\\mathrm{red} Z_1^2 Z_2^2 e^4 k_\\mathrm{B}^2 T^2}{2\\hbar^2}\\right)^{1\/3}\\approx 0.5\\,\\mathrm{MeV}\\ \\left(\\frac{A}{12}\\right)^{1\/3} \\left(\\frac{Z}{6}\\right)^{4\/3} \\left(\\frac{T}{10^8\\,\\mathrm{K}}\\right)^{2\/3},\n\\end{equation}\nwhere the second equality is for the fusion of like nuclei. Under the assumption of weak $S(E)$ dependence, the thermonuclear rate becomes \n\\begin{equation}\\label{eq:fus_rate_therm}\n\\lambda_\\mathrm{th}= 4\\frac{n_i^2}{2} \\sqrt{\\frac{2 E_\\mathrm{pk}}{3\\mu_\\mathrm{red}}}\\frac{S(E_\\mathrm{pk})}{k_\\mathrm{B} T} \\mathrm{exp}\\left(-3E_\\mathrm{pk}\/(k_\\mathrm{B} T)\\right).\n\\end{equation}\n\nDirect measurements of fusion between two neutron-rich nuclides is not possible, as this would require high-intensity low-energy beam-beam collisions between radioactive nuclides and no such facility exists. As such, experimental studies focus on fusion involving stable nuclides, which are useful in terms of benchmarking theoretical models~\\cite{Bass77,Yako06a,Bear10}. Nonetheless, measurements of $S(E)$ for more systems are welcome, particularly in light of the fact that, while theoretical predictions and experimental results agree for fusion reactions involving some neutron-rich nuclides~\\cite{Carn14}, intriguing discrepancies remain for other cases~\\cite{Rudo12,Stei14,Sing17}.\n\nThe energies relevant for neutron stars are smaller than the current experimental measurements reach. \nHence, determinations of the $S$-factor are either performed by extrapolations from fits to data measured at much higher energies or by theoretical calculations based on tunneling through a barrier corresponding to a theoretical nuclear potential~\\cite{Gasq05,Yako06a,Yako06b,Bear10,Horo08}. Theoretical \nstudies amount to calculating the tunneling probability for some angular momentum transfer $\\ell$, as represented by the so called transmission coefficient $T_{\\ell}$ ($\\leq1$). The cross section results from the semi-classical relation $\\sigma(E)=\\pi\\mbox{\\makebox[-0.5ex][l]{$\\lambda$} \\raisebox{0.7ex}[0pt][0pt]{--}}^{2}\\Sigma_{\\ell=0}^{\\ell,\\rm{max}}(2\\ell+1)T_{\\ell}$,\nwhere $\\mbox{\\makebox[-0.5ex][l]{$\\lambda$} \\raisebox{0.7ex}[0pt][0pt]{--}}$ is the reduced de Broglie wavelength. $\\ell_{\\rm{max}}={R_{\\rm{max}}}\/{\\mbox{\\makebox[-0.5ex][l]{$\\lambda$} \\raisebox{0.7ex}[0pt][0pt]{--}}}$ corresponds to the angular momentum at which the impact parameter is equal to the sum of the participating nuclear\nradii $R_{\\rm{max}}=r_{0}(A_{\\rm{1}}^{1\/3}+A_{\\rm{2}}^{1\/3}$) ($r_{0}\\sim1.2$~fm), though generally the few lowest $\\ell$ provide the dominant contributions~\\cite{Gasq05}. \nA large set \nof astrophysical $S$-factors relevant to neutron star studies was collected recently in Reference \\cite{Afanasjev2012PhRvC}, where an economical analytical parameterization was proposed.\n\nUsing $S$-factors from Reference~\\cite{Afanasjev2012PhRvC}, in figure~\\ref{fig:pycno}a we plot with dotted lines the locations in $T-\\rho$ plane where the thermonuclear burning time $\\tau_\\mathrm{burn}\\equiv n_\\mathrm{i}\/\\lambda_\\mathrm{th}$ is equal to $10^{10}$~yr and $1$~s for the $^{12}$C+$^{12}$C  reaction in a one-species carbon plasma. However, at large densities and\/or low temperatures (when $\\Gamma\\gtrsim 1$), these rates become irrelevant and the effects of ion correlations start to play the dominant role. The respective lines of $\\tau_\\mathrm{burn}=\\mathrm{constant}$ are shown with solid red curves and are clearly very different from the dotted lines. One finds five regimes of nuclear burning (figure~\\ref{fig:pycno}a) \\cite{Salp69}. In the regime of strong ion coupling $\\Gamma\\gtrsim 1$, screening of the Coulomb interaction eases barrier penetration enhancing the reaction rates as seen in figure~\\ref{fig:pycno}a. This burning regime II, called thermonuclear with strong plasma screening, operates until quantum effects in ion motion become important at $T\\lesssim T_\\mathrm{pi}$. Here $T_\\mathrm{pi}$ is the \nion plasma temperature  (for a one-species plasma)\n\\begin{equation}\\label{eq:Tpi}\n T_\\mathrm{pi}\\equiv \\frac{\\hbar \\omega_\\mathrm{pi}}{k_\\mathrm{B}} = \\frac{\\hbar}{k_\\mathrm{B}} \\sqrt{\\frac{4 \\pi Z^2 e^2 n_i}{A m_u}}\\approx1.9\\times 10^9\\,\\mathrm{K}\\ \\sqrt{\\frac{Z}{A}}\\left(\\frac{\\mu_e}{20\\,\\mathrm{MeV}}\\right)^{3\/2},\n\\end{equation}\nwhere $\\omega_\\mathrm{pi}$ is the ion plasma frequency.\nThe pycnonuclear regime V, already mentioned above, operates for the lowest temperatures $T\\lesssim T_\\mathrm{q}=0.5\\, T_\\mathrm{pi}\/\\ln\\left(T_\\mathrm{l}\/T_\\mathrm{pi}\\right)$. The remaining regimes are the thermally enhanced pycnonuclear regime IV ($0.5T_\\mathrm{pi}\\gtrsim T\\gtrsim T_\\mathrm{q}$) and the thermopycnonuclear regime III ($T_\\mathrm{pi}\\gtrsim T\\gtrsim 0.5T_\\mathrm{pi}$), which is the most uncertain. We do not discuss intermediate regimes II--IV here, see detailed discussions in References \\cite{Gasques2005PhRvC,Yako06a,Chugunov2007PhRvD,Pote12,Chugunov2009PhRvC}.\n\nWe now briefly describe the pycnonuclear regime mainly following the discussion in References \\cite{Gasques2005PhRvC,Yako06a}. More details can be found there and in References \\cite{Salp69,Yako06b,Afanasjev2012PhRvC,Chugunov2007PhRvD}. For simplicity we first address the single-species composition and then discuss the more relevant case of mixtures. \nThe rate in the pycnonuclear regime $T\\lesssim T_\\mathrm{q}$ is temperature-independent. In analogy to equation (\\ref{eq:fus_rate_therm}) it can be written\n\\begin{equation}\\label{eq:fus_rate_pycno}\n\\lambda_\\mathrm{pyc} =C_\\mathrm{pyc} \\frac{n_i^2}{2} S(E_\\mathrm{pk}) \\frac{\\hbar}{m_i Z^2 e^2}\\, \\xi^\\beta \\exp\\left(-\\alpha \\xi \\right),\n\\end{equation}\nwhere the typical interaction energy is $E_\\mathrm{pk}\\sim \\hbar \\omega_\\mathrm{pi}$ [cf. equation~(\\ref{eq:Gamow_th})]. Constants $C_\\mathrm{pyc}$, $\\alpha\\sim 0.7$, and $\\beta$ depend on the model for the Coulomb barrier penetration via zero-point vibrations, and the parameter \n$\\xi=d^2\/r_\\mathrm{rms}^2$ characterizes the relative amplitude of these vibrations. Here $r_\\mathrm{rms}^2=\\hbar\/(2\\mu_\\mathrm{red}\\omega_\\mathrm{pi})$ is the mean-square displacement of the oscillating ion and $d$ is the equilibrium distance between the closest reacting neighbors. It is assumed that ions in the one-species plasma form a body-centered cubic lattice. Then $d=(3\\pi^2)^{1\/6} a$, where $a$ is the ion-sphere radius defined in section~\\ref{section:structure}. Putting numbers in,\n\\begin{equation}\n\\xi\\approx 0.2 A^{1\/2} Z^{7\/6} \\left(\\frac{\\mu_e}{20\\,\\mathrm{MeV}}\\right)^{-1\/2}.\n\\end{equation}\nUnder conditions relevant for the neutron star crust, $\\xi$ is large and gradually decreases with depth. For pure $^{12}$C at $\\mu_e=20$~MeV, $\\xi=5.7$, and for $^{34}$Ne, the first species to fuse in the one-component model of Reference \\cite{Haen90},  $\\xi=17$ at the same depth. The pre-exponential factor in equation (\\ref{eq:fus_rate_pycno}) is usually large so the pycnonuclear reactions start to be important at $\\xi\\gg1 $. Such reaction rate is very sensitive to variations in $\\xi$ and in the exponential prefactor $\\alpha$. Different models of tunneling resulting in small variations in $\\alpha$ transform to huge (several orders of magnitude) variations in $\\lambda_\\mathrm{pyc}$ \\cite{Gasq05}. If the type of the crystalline lattice is other than body-centered cubic, the value of $d$ can change producing huge variations again (see the MCP case below). To illustrate these uncertainties,  in figure~\\ref{fig:pycno}b we plot the zero-temperature pycnonuclear reaction rates as a function of $\\mu_e$ for two potentially important pycnonuclear reactions for the neutron star inner crust, $^{34}$Ne+$^{34}$Ne \\cite{Haen90} and $^{40}$Mg+$^{40}$Mg, using $S$-factors from Reference~\\cite{Afanasjev2012PhRvC}. Solid lines are the `optimal' rates, while thinner dashed lines correspond to maximal and minimal rates, see Reference~\\cite{Gasq05} for an extensive discussion. Triplets of open symbols on each rate curve give depths where $\\tau_\\mathrm{burn}=1\\,\\mathrm{yr},\\,1\\,\\mathrm{d},\\,\\mathrm{and}\\, 1\\,\\mathrm{s}$. Note, that we use $\\mu_e$ instead of $\\rho$ here, since in the inner crust the latter depends on the amount of free neutrons and hence on the equation of state. As seen, the uncertainty in the rate can be as large as ten orders of magnitude. Therefore, depending on a model, a given pycnonuclear reaction can start at various points, can be delayed and start after the accretion ceased \\cite{Yako06b}, or does not operate at all. \nThe same uncertainties are shown with dashed lines in the left panel in figure~\\ref{fig:pycno}a for carbon burning where all regimes are considered. It is clear that the theoretical uncertainty increases with regime number.\n\nThe situation is even more complicated in a multi-species plasma \\cite{Yako06a}. \nThe most important point is the availability of close neighbors. In the most popular uniform mixing model, the mean equilibrium distance is taken to be the same as in the one-component model. Then the rates are calculated in a similar way as in the one-component plasma (with proper renormalization of the model parameters, and this procedure is also not very certain \\cite{Yako06a}). \nIt is unclear, however, if the formalism  applicable for a one-component body-centered cubic lattice is equally good for potentially amorphous structures, or for less abundant chaotic impurities (if they have smaller charges they may start to fuse first). One can expect more frequent close encounters for this case than in a regular lattice, increasing the reaction rate. On the other hand, consider a regular lattice of two intermittent species. The closest distance $d$ for two nuclei of the same species will be a factor of 1.155 larger than in the one-component or uniformly mixed case, greatly suppressing the pycnonuclear rate due to exponential behavior in $d^2$ \\cite{Yako06a}. Additional complications arise when the structural changes caused during the course of pycnonuclear burning are considered \\cite{Salp69}. When two nuclei in a lattice fuse, this clearly creates a defect. What will happen next, how the lattice will react to a number of such defects and in what respect this can affect the burning rates, is basically unexplored.\n\nNuclear physics enters the pycnonuclear reaction rate (equation (\\ref{eq:fus_rate_pycno})) through the astrophysical factor $S(E_\\mathrm{pk})\\approx S(0)$. As the pycnonuclear reactions are thought to occur in the inner crust of the neutron star,  \nthe presence of free neutrons provides another potential source of uncertainty. It is not unreasonable to assume that the neutron gas can alter the properties of the Coulomb barrier, making it lower and\/or thicker. It is impossible to study this experimentally. In Reference~\\cite{Afanasjev2012PhRvC}, the authors addressed this problem by introducing phenomenological modifications of the barrier shape. They found, taking $^{34}$Ne as an example, that even slight perturbations of the barrier shape lead to another ten orders of magnitude variations in the $S$-factor at small energies and, as a consequence, in the reaction rate. \n\nThe onset of pycnonuclear burning is thus strongly model-dependent. The lighter nuclei can burn even in the ocean \\cite{Horo08}. However the main pycnonuclear fusion is expected to occur in the inner crust \\cite{Haen90,haensel2008,steiner2012}. The importance of the pycnonuclear burning is that it is the main deep crustal heating source. For instance, in a one-species-per-depth model of \\cite{Haen90,Haen03,haensel2008}, the first pycnonuclear reaction ($^{34}$Ne+$^{34}$Ne) releases about 0.5~MeV\/u, or $25-35$ percent of the total heating. In some cases, similar to superthreshold electron capture cascades discussed in the previous section, the product of the pycnonuclear fusion may become immediately unstable to a cascade of electron capture reactions which drives it back to the initial nucleus by a chain of neutron emission reactions. In other words, one of the nuclei acts as a catalyst for the conversion of the other one into free neutrons \\cite{steiner2012,Lau18}. Note, that like superthreshold electron capture cascades this scenario is only possible when multi-component reaction networks are employed. \n\nAt first glance, the strong variation in the properties of pycnonuclear burning make impossible any conclusions about the nuclear transformations and heat release in the inner crust. However, this is not the case. As it has been shown \\cite{haensel2008}, the total heat released in the crust is remarkably independent of the particular sequence of nuclear reactions an accreted element encounters. Indeed, close to the crust-core boundary the accreted crust composition is expected to merge with the cold-catalyzed one. This means that the energy released in the accreted crust is determined by the heat reservoir the initial ashes have with respect to the ground (cold-catalyzed) state. During compression, this extra energy is released in one way or another, but the net result is the same \\cite{haensel2008}. The total heat release in the accreted crust is found to be about $2$~MeV\/u. Looking from the other side, the total heating is governed by the properties of the cold-catalyzed state. Reference~\\cite{steiner2012} analyzed several equations of state in the inner crust and found that some of them have lower ground states predicting twice the heating that is assumed in traditional models. Unfortunately, the pycnonuclear uncertainties strongly impact the heating sources' distribution, which may considerably affect the crustal thermal state during and after an accretion outburst.  \n\n\n\\section{Observable Impact of Nuclei in the neutron star Ocean and Crust} \\label{section:impact}\n\n \\subsection{Thermal Timescale of the Accreted Crust }\\label{subsection:impact_inner}\n\nAs shown above, nuclear transformations during accretion modify the composition of the crust and are the source of non-equilibrium heating\/cooling processes. Below we discuss the importance of these processes for observable systems  -- cooling X-ray transients and for ignition conditions and lightcurves of X-ray superbursts. To begin, we consider the heat propagation in the crust.\n\nThe thermal diffusion timescale  \nbetween a column depth $y$  \nand the surface is %\n\\cite{Henyey1969ApJ}\n\\begin{equation}\\label{eq:tau_diff}\n\\tau = \\frac{1+z}{4}\\left[\\int_0^y \\left(\\frac{c_P}{\\rho K}\\right)^{1\/2} \\mathrm{d } y' \\right]^2,\n\\end{equation}\nwhere $c_P$ is the specific heat capacity per unit mass\\footnote{Under neutron star crust conditions, the heat capacity at constant pressure $c_P$ and heat capacity at constant volume $c_V$ are the same, $c_P\\approx c_V$ \\cite{HPY2007}.} and $K$ is the thermal conductivity. The heat capacity in the ocean and outer crust is set by ions, except for the outer ocean at high temperatures, where the electrons dominate. In the inner crust, free neutrons dominate the heat capacity unless they are superfluid (see section~\\ref{section:structure}). In the superfluid regions, the neutron contribution is suppressed and $c_P$ is determined by ions and electrons. \n\nDegenerate electrons provide the dominant contribution to the thermal conductivity and it is mainly set by electron-ion scattering:\n\\begin{equation}\\label{eq:kappa_e}\nK=K_e=\\frac{\\pi^2 c^2 k_\\mathrm{B}^2 T n_e}{3 \\mu_e \\nu_{\\rm ei}},\n\\end{equation}\nwhere $\\nu_{\\rm ei}$ is the effective electron-ion collision frequency. The character of the electron-ion collisions depends strongly on the phase state of the ionic system. In the multicomponent liquid ocean, under the linear mixing rule approximation (e.g., \\cite{pote99})\n\\begin{equation}\\label{eq:nuei_liq}\n\\nu_{\\rm ei} = \\frac{4\\alpha_f^2}{3\\pi} \\frac{\\mu_{\\rm e}}{\\hbar} \\frac{\\langle Z^2 \\Lambda_{\\rm ei}\\rangle }{\\langle Z\\rangle}, \n\\end{equation}\nwhere $\\Lambda_{\\rm ei}\\sim 1$ \nis the Coulomb logarithm of the species in mixture. For estimates, the mean ion model with $\\langle Z^2 \\Lambda_{\\rm ei}\\rangle\\approx \\langle Z \\rangle^2$ can be used \\cite{pote99}. Consider, for example, thermal wave propagation in the hot ocean, where the heat capacity is set by electrons, $c_P=\\pi^2 \\langle Z\\rangle k_\\mathrm{B}^2 T\/(\\langle A \\rangle m_u \\mu_e)$. The integrand in equation~(\\ref{eq:tau_diff}) is then temperature-independent and behaves like $y^{-5\/8}$ (neglecting the composition dependence on $y$). Thus the resulting estimate is\n\\begin{equation}\\label{eq:time_ocean}\n\\tau \\approx 16.5\\ {\\rm hr} \\left(\\frac{y}{10^{12}~\\mathrm{g}~\\mathrm{cm}^{-2}}\\right)^{3\/4}\n\\left(\\frac{g_{14}}{2.44}\\right)^{-5\/4} \\frac{\\langle Z\\rangle \\langle Z^2 \\Lambda_{\\rm ei}\\rangle}{26^3} \\left(\\frac{56}{\\langle A\\rangle}\\right)^2\n\\end{equation}\nfor $1+z=1.31$.\nThis gives \nan order of magnitude estimate of the timescale for superburst decay \\cite{cumming2001,2004CummingMacBeth,Stro02}.\n\nThe situation is different in the crust. If a perfect crystal is formed (as expected for the pristine crust, section~\\ref{subsection:pristine}), the thermal conductivity is set by electron-phonon scattering. The corresponding collision frequency in the classical limit ($T\\gtrsim 0.2 T_\\mathrm{pi}$) in the single-phonon approximation is \\cite{Yakovlev1980,Schmitt17}\n\\begin{equation}\\label{eq:nuei_phon}\n\\nu_{\\rm ei}=\\nu_{\\rm e-ph}\\approx 13\\;\\alpha_f \\frac{k_\\mathrm{B}T}{\\hbar}.\n\\end{equation}\nNote that this expression neglects various corrections (e.g., \\cite{pote99}) but is sufficient for the qualitative discussion here. Clearly, since $\\nu_{\\rm e-ph}\\propto T$, it is much smaller than $\\nu_{\\rm e i}$ in the liquid phase [equation~(\\ref{eq:nuei_liq})]; accordingly, the thermal conductivity is high. The reason is that the elastic (Bragg) part of the scattering off the ordered crystalline lattice leads to renormalization of the electron states to Bloch waves, and as such does not lead to dissipation.\n\nConsider for example the diffusion timescale of the crystalline outer crust. This is relevant for studies of crustal cooling in young ($\\lesssim 100$~yr) neutron stars \n(e.g., \\cite{Gnedin2001MNRAS,Shternin2008AstL}) and \nin the so-called quasi-persistent transients (section \\ref{sec:quasipers}). \nFor a classical crystal,\nthe ion heat capacity is $c_P=3 k_\\mathrm{B}\/(A m_u)$ and from equation~(\\ref{eq:tau_diff}) one obtains \\cite{Brow09}\n\\begin{equation}\\label{eq:time_crust}\n\\tau\\approx 40\\ \\mathrm{days}\\times y_{14}^{3\/4} \\left(\\frac{g_{14}}{2.44}\\right)^{-5\/4} \\left(\\frac{56}{\\langle A \\rangle}\\right)^2 \\frac{\\langle Z \\rangle}{26},\n\\end{equation}\nwhere now $y_{14}$ is the column density measured in units of $10^{14}~\\mathrm{g}~\\mathrm{cm}^{-2}$ and again $1+z=1.31$. The pure crust cools relatively fast. However, if weakly correlated impurities are present, they contribute to the charge fluctuations that electrons scatter off. It is customary to write the total conductivity using  Mathiessen's rule $\\nu_{\\rm ei}=\\nu_{\\rm e-ph}+\\nu_{\\rm e-imp}$, where $\\nu_{\\rm e-imp}$ is given by equation~(\\ref{eq:nuei_liq}) with the substitution $\\langle Z^2 \\Lambda_{\\rm ei}\\rangle \\to Q_\\mathrm{imp} \\Lambda_\\mathrm{imp}$, where $\\Lambda_\\mathrm{imp}\\sim 1$ is the Coulomb logarithm for uncorrelated impurities scattering. \nComparing $\\nu_{\\rm e-imp}$ with  $\\nu_{\\rm e-ph}$ we find that the former dominates if\n\\begin{equation}\nQ_\\mathrm{imp} \\Lambda_\\mathrm{imp}\\gtrsim 31\\ \\frac{\\langle Z \\rangle}{26} \\frac{30\\ \\mathrm{MeV}}{\\mu_e}\\, \\frac{T}{10^8~\\mathrm{K}}.\n\\end{equation}\nThus, the impurity scattering becomes more important in the inner crust (large $\\mu_e$) at low temperatures (since impurity scattering is elastic and thus temperature-independent). Moreover, at $T\\lesssim 0.2 T_\\mathrm{pi}$, the electron-phonon scattering is further suppressed by quantum effects roughly as $\\propto T\/(0.2 T_\\mathrm{pi})$ (e.g., \\cite{Schmitt17}), making the impurity contribution dominant even for a ``small'' impurity parameter $Q_\\mathrm{imp}\\sim 1$. These estimates show that large impurity content would increase the thermal diffusion time. \n\nThe large impurity parameter of the ashes from most surface burning processes (see figure~\\ref{figure:ECcrustcomposition} and table~\\ref{table:AvgZAQTable}) raises doubts about the applicability of the simple prescription of the pure crystal with uncorrelated impurities to the multicomponent mixture of the accreted crust. Most of the recent molecular dynamics studies suggest that crystallization occurs even for large $Q_{\\rm imp}$ in a way that ions of large charge occupy lattice sites (of a body-centered cubic lattice), while smaller charge nuclei are found in interstitial regions \\cite{Horo09,Horowitz2007PhRvE,Hughto2011PhRvE,Horowitz2009PhRvE}. The appearance of the Bragg structure in scattering is clearly seen in simulations. In addition, it was found that the impurities are actually correlated. Calculations of the thermal conductivity \\cite{Horowitz2007PhRvE,Horo09,Rogg16} have shown that the uncorrelated impurities limit underestimates the impurity contribution. Recently, the authors of Reference~\\cite{Rogg16} used the Path Integral Monte Carlo approach to find that the electron-impurity scattering can be adequately approximated by $\\nu_\\mathrm{e-imp}$, where the {\\it effective} impurity parameter $\\tilde{Q}_\\mathrm{imp}=L(\\Gamma) Q_\\mathrm{imp}$ is used in place of $Q_\\mathrm{imp}$. \nAccording to~\\cite{Rogg16},  \nthe correction factor $L(\\Gamma)$ is about $1.5-2$ at $\\Gamma\\sim 300-500$ and increases to $3-4$ at lower temperatures (higher densities) where $\\Gamma\\sim 10^{4}$. \n\nFinally, \nif the mixture is so diverse that no crystal is formed and the solid phase is in an amorphous state, then no regular lattice exists and equation~(\\ref{eq:nuei_liq}) is applicable as well (e.g., \\cite{Daligault2009}). This is thought to be a lower limit for the thermal conductivity and results in much longer diffusion timescales than given by equation~(\\ref{eq:time_crust}) (e.g., \\cite{Brow00}). Nevertheless, it has been shown that diffusion in a multicomponent Coulomb plasma is sufficiently fast to relax an initially amorphous structure  to a crystalline state \\cite{Hughto2011PhRvE}.\n\nThe presence of a pasta layer at the base of the inner crust (section~\\ref{section:structure}) with anisotropic nuclear clusters can drastically change the thermal behavior of the inner crust. Transport properties of the pasta were studied by several authors during the last decade (e.g., \\cite{horowitz2008,Schn16,NandiSchramm2018ApJ,Yakovlev2015MNRAS}), but a consistent picture has not been developed yet. It is expected that transport in the pasta phase is anisotropic, with conductivity along one of the symmetry directions being much more effective than along another \\cite{Yakovlev2015MNRAS,Schn16}. The resulting thermal conductivity will depend on the orientation of the pasta domains in the star. If these domains are oriented chaotically, or the lower-conduction axis is aligned with the radius,  the conductivity can be significantly reduced. The pasta phase is thought to have a regular structure within one domain, but can contain topological defects that can act like impurities \\cite{Schn16}. This can make pasta a highly resistive phase. Taking into account these complications, one usually describes thermal conductivity in the pasta phase by introducing the phenomenological (usually large) impurity parameter  $Q_\\mathrm{imp,\\mathrm{pasta}}$. Note that some studies do not find reduced conductivity in pasta (e.g., \\cite{NandiSchramm2018ApJ}).\n\nAn additional uncertainty that affects thermal relaxation in the inner crust is contained in the heat capacity $c_P$. If neutrons are superfluid (section \\ref{section:structure}), they do not contribute to the heat capacity. However, in regions where neutrons are normal, they \ncomprise the dominant contribution to $c_P$. As follows from equation~(\\ref{eq:tau_diff}), the presence of normal neutrons will delay thermal relaxation; additionally, they can store more heat. Thus, the results are sensitive to the profile of the neutron singlet $^1$S$_0$ pairing gap. This is especially important if the pairing gap does not penetrate the core and closes at lower mass densities than the crust-core transition (e.g., \\cite{gandolfi2008}). In this case a layer with a high abundance of normal neutrons would be present at the base of the inner crust. In the regions where normal neutrons exist,  the neutron thermal conductivity $K_\\mathrm{n}$ may play a role. It was found, however, that generally $K_{\\rm n} < K_e$ throughout \nthe inner crust~\\cite{Bisnovaty1982,Deib17}. Still, the neutron thermal conductivity can be important near the crust-core interface and is worthy of future study  \\cite{Deib17}.\n\n\n\\subsection{Cooling Transients}\n\\label{sec:cooling_transients}\n\nAccreting neutron star \ntransients in low-mass X-ray binaries (section~\\ref{section:accretion}) are\nbright ($L_\\mathrm{X}\\sim 10^{36}-10^{39}$~erg~s$^{-1}$) \nX-ray sources during accretion outbursts (see table 1 of Reference~\\cite{Dege15} for a summary). \n\nAs described in section~\\ref{section:interaction}, non-equilibrium \nreactions \ndeposit $\\approx 1 \\textrm{--} 2 \\, \\mathrm{MeV}$ per accreted nucleon in the neutron star \ncrust during an accretion episode. When an accretion outburst ends, the X-ray luminosity drops by several orders of magnitude, the neutron star \nenters quiescence, and the fainter ($L_\\mathrm{X}\\sim 10^{31}-10^{34}$~erg~s$^{-1}$) \nthermal emission from the now cooling surface can be measured by sensitive X-ray observatories. The subsequent evolution depends on the duration of the accretion episode. One distinguishes normal transients, where accretion lasts for several weeks, and \nquasi-persistent transients, where accretion can proceed for years or even decades. A recent detailed discussion of the properties of cooling transients can be found in Reference~\\cite{Wijnands2017JApA}. \nHere we briefly describe the properties of transients necessary to understand the nuclear physics impact on these objects.\n\n\\subsubsection{Normal transients}\nBefore  the start of an accretion episode, the neutron star is isothermal inside (except a thin outer \nheat blanketing envelope). Energy release from nuclear reactions during accretion breaks this equilibrium and heats the crust creating an inward and outward energy flux from the heating regions. During a relatively short accretion outburst, the crust is not heated strongly and quickly relaxes to thermal equilibrium with the core after accretion ceases. In this process, a fraction $f\\lesssim 1$ of the total heat release enters the neutron star core, typically $\\approx 90\\,\\%$~\\cite{Brow00}, and a smaller fraction $1-f$ of heat diffuses toward the surface and is radiated away during crustal cooling and by increased neutrino emission from the heating regions. Many cycles of accretion\/quiescence result in secular heating of the whole star with the secular heating rate (as measured by a distant observer) of $L_\\mathrm{heat}^\\infty=fQ \\langle \\dot{M} \\Delta t\\rangle\/(m_u t_\\mathrm{rec} (1+z))$,\nwhere $Q$ is the deep crustal heating power, $\\langle\\dot{M} \\Delta t\\rangle$ is the mass accreted during an outburst, on average, and $t_\\mathrm{rec}$ is the average recurrence time. \nThis heating is enough to balance the energy loss due to surface emission $L_\\gamma^\\infty$ and neutrino cooling $L_\\nu^\\infty$ from the bulk. If the transient accretion behavior persists for a long time, the neutron star is found in a steady state set by the condition $L_\\mathrm{heat}^\\infty=L_\\nu^\\infty+L_\\gamma^\\infty$, with the temperature much higher than expected for a passively cooling neutron star of comparable age \\cite{BBR1998}. This steady state point is thus determined by the (unknown) neutrino emission rate from the core. For higher rates, the steady state will be reached at lower temperatures (and thus surface luminosities) for a given $L_\\mathrm{heat}^\\infty$. Measurements of the quiescent luminosity with sensitive X-ray observatories\nthus in principle  allow one to constrain the neutrino emission mechanisms operating in the neutron star core \\cite{Yakovlev2004ARA&A,Wijnands2013,Wijnands2017JApA}, as well as the core heat capacity~\\cite{Cumm17}.\nAt the moment, the strongest constraint comes from the coldest (in quiescence) \ntransient in the observed sample, SAX~J1808.4$-$3658 \\cite{Heinke2009ApJ}, whose conditions suggest that the powerful direct Urca neutrino emission process is operating in the core of the neutron star in this binary. In addition, the need for a direct Urca process in about 1 percent of the core was recently proposed to explain the low inferred core temperature in the quasi-persistent transient (section \\ref{sec:quasipers}) MXB~1659$-$29 \\cite{Brown2018arXiv}. However, this case is more uncertain as the accretion duty cycle is not known.\n\nThe main nuclear input here is the \npower of the deep crustal heating $Q$; however, a robust inference is plagued by other uncertainties such as distance measurements,  stability of accretion duty cycles, and composition of the \nheat blanketing envelope of the neutron star. The latter strongly affects the relation between the measured surface temperature and the internal temperature \nwhich determines\n$L_\\nu^\\infty$. \nThis uncertainty does not apply for hot and weakly accreting systems where $L_\\gamma^\\infty\\gg L_\\nu^\\infty$ so that the steady state is determined by the directly measurable neutron star surface emission (e.g., IGR~J00291+5934, \\cite{Wijnands2017JApA}). Such systems in principle have a potential to constrain $Q$, but the observational uncertainties are still large. \n\n\n\\subsubsection{Quasi-persistent transients}\\label{sec:quasipers}\nDuring the long accretion outbursts in quasi-persistent transients,\nthe  heat deposited in the crust is so large that it brings the crust out of thermal equilibrium with the core~\\cite{Page13}. When an accretion outburst ends, the neutron star\nenters quiescence, \nthe crust cools toward thermal equilibrium with the core, and the neutron star's \nsurface thermal emission powers an X-ray light curve~\\cite{ushomirsky2001,rutledge2002}. The shapes of quiescent cooling curves depend on \nthe thermal structure of the neutron star's \nouter layers at the \nbeginning of quiescence. Confronting  thermal evolution models of the neutron star's \nouter layers \nwith observations leads to constraints on crustal properties such as thermal conductivity and specific heat of dense matter, as discussed in section~\\ref{subsection:impact_inner}. \n\nActive studies of crust cooling in quasi-persistent transients started about two decades ago when the first source KS~1731$-$260 ended a $\\gtrsim 12.5$~yr outburst and was observed in quiescence with Chandra \\cite{wijnands2001c,rutledge2002}. The continuous monitoring of this source in quiescence revealed the crust cooling towards thermal equilibrium with the core on a $\\sim$yr timescale \\cite{cackett2006}. This relatively short cooling timescale \nsuggests a high thermal conductivity in the crust consistent with an ordered lattice  \ncontaminated by a small impurity content \\cite{cackett2006,Shte07,Brow09}. An amorphous crust with low thermal conductivity, which would have a much longer thermal time (section~\\ref{subsection:impact_inner}), contradicts observations.\n\nSince 2001, quiescent cooling curves have been observed in several low-mass X-ray binaries and provide important data on the thermal evolution of the neutron star outer layers across accretion regimes and  \ngravities \\cite{Wijnands2017JApA}. Interestingly, crustal cooling is observed now not only in the quasi-persistent sources with long $>1$~yr outbursts, but also for several `ordinary' transients with shorter accretion outbursts. The results of these observations and crustal light curve modeling are extensively reviewed elsewhere (e.g., \\cite{Wijnands2017JApA}). Here we summarize only the main points:\n\n\\paragraph{The crust is pure:} The analyses of all cooling sources generally suggest that a high thermal conductivity and hence a low effective impurity parameter $\\tilde{Q}_{\\rm imp}\\lesssim 4-7$ throughout the crust is favored (see, however, Reference~\\cite{Degenaar2014}). Taking into account the results of Reference~\\cite{Rogg16}, this indicates even smaller actual impurity content $Q_{\\rm imp}\\lesssim 1-3$. By contrast, the distribution of ashes produced from X-ray bursts and stable surface burning can have $Q_{\\mathrm{imp}} \\sim 70-100$ (see figure~\\ref{figure:ECcrustcomposition} and table~\\ref{table:AvgZAQTable}) suggesting that a purification mechanism exists in the crust. \nWe illustrate the impact of the impurity parameter on cooling curves in figure~\\ref{figure:NScool_Qimp}. This crust cooling model uses the accreted crust composition from Reference~\\cite{Haen90} and contains nuclear heating of $Q = 0.3 \\, \\mathrm{MeV \\ u^{-1}}$ in the outer crust, and deep crustal heating of $Q = 1.5 \\, \\mathrm{MeV \\ u^{-1}}$ in the inner crust. As can be seen in figure~~\\ref{figure:NScool_Qimp}, large values of $Q_{\\rm imp}$ lower the thermal conductivity of the inner crust and increase the thermal time. Therefore, it takes longer for heat to diffuse out of the inner crust during quiescence and for the crust to reestablish thermal equilibrium with the core.\n\nThe estimates obtained in this way are more relevant for the inner crust where the impact of impurities is more pronounced (section~\\ref{subsection:impact_inner}). In addition, the constraints on the outer crust properties are contaminated by the mysterious shallow heating source (discussed momentarily). \nThe impurity parameter in most models and in our illustrative calculations is set constant. Reference~\\cite{Page13} allowed for a variable $Q_{\\rm imp}$ with a higher value in the outer crust, lower value in the inner crust, and a smooth transition between them. This allowed the authors  \nto explain the observations of the source XTE~J1701$-$462; the adopted model is also consistent with observations of other transients. \n\n\\paragraph{Neutron superfluidity is favored:} The intermediate part of the transient cooling curve is usually fit better when neutron superfluidity is taken into account (e.g., \\cite{Shte07,Brow09}), which lowers the heat capacity in the inner crust causing it to store less heat and cool more rapidly (section~\\ref{subsection:impact_inner}). \n\n\\paragraph{Extra shallow heating is required:} It has been found that the standard heat release from deep crustal heating models is usually insufficient to heat the crust to the high effective temperatures observed at the early phase of quiescence \\cite{Brow09,Turl15}. Although the uncertainties in the nuclear physics may vary the heat deposited in the inner crust~\\cite{steiner2012}, analyses of quiescent light curve shapes demonstrate that the additional heating source must be located at densities $\\rho_\\mathrm{shallow}\\lesssim 10^{11}$~g~cm$^{-3}$. For a standard model of deep crustal heating, the strength of this additional source is found to be on the $Q_\\mathrm{shallow}\\sim 1 \\, \\mathrm{MeV}$ level. However, this estimate is model-dependent, varies from source to source, and may vary from one outburst to another in the same source~\\cite{Turl15,Pari17}. \n\n\\paragraph{Nuclear pasta is favored:} It was found that the cooling curve of MXB~1659$-$29 is better explained if a low-conducting region is included in the inner crust, that can be attributed to the pasta layer \\cite{Horo15,Deib17}. We illustrate the impact of the presence of  pasta in figure~\\ref{figure:NScool_pasta}, where the thermal conductivity in the pasta is modeled by a large impurity parameter $Q_{\\mathrm{imp},\\mathrm{pasta}}$ for $\\rho>8\\times10^{13}$~g~cm$^{-3}$ and the model uses the $^1$S$_0$ neutron singlet pairing gap from Reference~\\cite{gandolfi2008}. \nThe presence of the heat insulator in the pasta layer may explain the continuous cooling observed in MXB~1659$-$29 after  11 yr in quiescence \\cite{Cackett2013ApJ}. If the nuclear pasta layer has a low thermal conductivity, some regions of the inner crust may remain above the neutron superfluid critical temperature depending on the choice for the pairing gap \\cite{Deib17}. As discussed in section~\\ref{subsection:impact_inner},  there may be normal neutrons present in the pasta layer and the heat stored by normal neutrons will impact the late-time cooling behavior of neutron star \ntransients which remain in quiescence for more than $ > 1000 \\, \\mathrm{days}$~\\cite{Deib17}. Note, however, that the temperature drop observed in  MXB~1659$-$29 may not be caused by continuous crustal cooling, but could be a result of an increase in the absorption column density prior to the last observation \\cite{Cackett2013ApJ}.\n\n\\paragraph{}\nIt is important to note that strong constraints on the composition of the crust are currently difficult to obtain because the shape of the cooling light curve is degenerate with other neutron star \nparameters, such as the  \ngravity and the core temperature, which are unknown a priori. Also, accurate accounting for a variable accretion rate during a long outburst can be important \\cite{Ootes2016MNRAS}.\nFinally, the observations, especially of the early stages of crustal cooling, may be contaminated by residual accretion.\n\n \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=1.2]{NScooling_Qimp.pdf}\n\\caption{Crust cooling models, showing the temperature decline of a neutron star's surface in quiescence,\nemploying various choices of $Q_{\\mathrm{imp}}$ for the entire crust. This crust cooling model uses a neutron star mass of $M= 1.4 \\, \\mathrm{M}_{\\odot}$, neutron star radius of $R = 12 \\, \\mathrm{km}$, and a core temperature of $T_{\\rm core} = 3 \\times 10^{7} \\, \\mathrm{K}$.\n\\label{figure:NScool_Qimp}}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=1.2]{NScooling_pasta.pdf}\n\\caption{Crust cooling models, showing the temperature decline of a neutron star's surface in quiescence, employing various choices of $Q_{\\mathrm{imp}}$ for the crust and the pasta layer at $\\rho \\gtrsim 8 \\times 10^{13} \\, \\mathrm{g \\ cm^{-3}}$. This crust cooling model uses a neutron star mass of $M= 1.4 \\, \\mathrm{M}_{\\odot}$, neutron star radius of $R = 12 \\, \\mathrm{km}$, and a core temperature of $T_{\\rm core} = 3 \\times 10^{7} \\, \\mathrm{K}$. \\label{figure:NScool_pasta}}\n\\end{figure}\n\nThe hottest neutron star \ntransient MAXI~J0556$-$332 \\cite{matsumura2011,Homa11,sugi13,homan14} is an excellent test bed for observational signatures of Urca cooling nuclei pairs \\cite{Deib15} because of the strong ($T^5$) temperature dependence of Urca neutrino cooling (see equation~(\\ref{equation:Lnu})). The inferred surface temperature at the onset of quiescence is nearly twice as large as the next hottest transient observed~\\cite{homan14}, and it is the only transient thought to have a hot enough crust the Urca cooling could produce an appreciable $L_{\\nu}$. It was found that MAXI~J0556$-$332 likely does not have Urca cooling operating in its crust because the quiescent cooling trend would follow a different behavior than what is observed \\cite{Deib15}.\n\nThe authors of Reference~\\cite{Meis17} reexamined the quiescent cooling of MAXI~J0556$-$332 and self-consistently added Urca cooling nuclei to the ocean and crust composition of a thermal relaxation model. \nThey found that Urca cooling would be\napparent for X-ray burst or superburst ashes, but not for the ashes of stable burning. Interestingly, stable burning ashes produce percent-level $X(A)$ for odd-$A$ nuclides in the $70\\lesssim$$A$$\\lesssim80$ range, but these isobars appear to lack significant Urca pairs due to their relatively large $ft$~\\cite{Scha14,Meis17}, see equations~(\\ref{equation:Lnu})-- (\\ref{equation:L34}). Because of the large ocean and crust temperature in MAXI~J0556$-$332, it is indeed likely that stable burning has occurred \nin the outer layers. As such, the absence of an Urca cooling signature in the MAXI~J0556$-$332 light curve, as shown in figure~\\ref{figure:urcaLC}, is consistent with model calculations. Further consistency checks will require the remaining nuclear physics uncertainties, particularly $ft$-values (section~\\ref{ssec:ECrxns}) of abundant nuclides, to be determined experimentally. Spectroscopy studies at the limits of experimental accessibility are underway for this purpose.\n\n  \\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.5]{MAXIlc.png}\n\\caption{Light curve of MAXI J0556$-$332 after the end of an accretion outburst~\\cite{homan14} compared to {\\tt dStar}~\\cite{Brow15} model calculations with (dashed line) and without (solid line) Urca cooling expected for $^{33}\\rm{Al}\\leftrightarrow^{33}\\rm{Mg}$~\\cite{Meis17}. Models employ $Q_{\\rm{shallow}}=8$~MeV per accreted nucleon,  $Q_{\\rm{imp}}=1$ throughout the crust, and an outburst duration and accretion rate that match observations~\\cite{homan14}. The inset shows the residuals to the no-cooling calculation.\\label{figure:urcaLC}}\n\\end{figure}\n\nWell-constrained estimates of Urca cooling strengths, coupled with models of cooling transient light curves, enable surface nuclear burning to be constrained over vast timescales \nfor neutron stars \nwith high-temperature crusts. By analyzing cooling transient light curves for signatures from Urca cooling, the composition of the accreted neutron star crust\ncan be constrained by determining the presence or absence of Urca pairs~\\cite{Meis17}. As the composition is determined by\npast nucleosynthetic activity on the neutron star  \nsurface, this method provides a theoretical\napproach to constrain surface nuclear burning on accreting neutron stars \nover the past millenia.\n\n   \n\\subsection{X-ray Superbursts}\\label{subsection:obs_superbursts}\nRecall that  X-ray superbursts are thermonuclear runaways triggered by carbon ignition in the accreted neutron star \nocean (section~\\ref{sec:superbursts}). \nTheir long duration is set by the thermal time scale of deep neutron star ocean layers, indicating an ignition column depth of $10^{11}-10^{12}\\ \\mathrm{g\\ cm^{-2}}$. This depth is relatively close to the outer crust. Therefore, the occurrence of superbursts provides a measure of the thermal properties at the crust-ocean interface as a function of depth. This is illustrated in figure~\\ref{figure:NStemp} where the dashed line shows temperature required to ignite carbon at a given column depth. Its position is sensitive to the $^{12}$C$+$$^{12}$C reaction rate, and enhancements in this rate are of particular interest, as they may help to explain the discrepancy between carbon ignition depths inferred from observations and required in simulations (section~\\ref{sssec:SuperburstObs}).\nFurthermore, superbursts quantify the carbon content of the hydrogen\/helium burning ashes, which cannot be inferred directly from observations of short bursts (section \\ref{sec:burst_observation}).\n\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{CrustEquilibriumTemperature2.png}\n\\caption{Steady-state ocean temperature as a function of column depth (solid line) for a neutron star with $\\dot{M}=0.3~\\dot{M}_{\\rm{Edd}}$, $M=1.4~M_{\\odot}$, $R=10$ km, and\n$T_{\\rm{core}}=3\\times10^{7}$ K. A shallow heating of 5~MeV per nucleon is assumed, resulting in $\\sim$0.25~MeV per accreted nucleon in the ocean. The dashed black line indicates the temperature and density at which ignition of unstable\ncarbon burning in a mixed iron-carbon ocean with $X_{\\rm{Fe}}=0.8$ and $X_{\\rm{C}}=0.2$ will ensue. \nThe open arrows indicate the regions where $e^{-}$-capture heating\/cooling and pycnonuclear heating are active, where the region of pycnonuclear heating extends to depths beyond the figure extent.\n(Adapted from Reference~\\cite{Deib16}).\n\\label{figure:NStemp}}\n\\end{figure}\n\nThe thermal profile around the ignition layer is set by the heating sources in the ocean and the crust.\nThe impact of  the nuclear reactions discussed in section~\\ref{section:interaction} \nis highlighted in figure~\\ref{figure:NStemp}, where the locations for $e^{-}$-capture and pycnonuclear reactions are shown. \nHeat from these reactions, which is typically produced at depths below carbon ignition,\ndiffuses upwards, raising the ocean temperature and enabling carbon to ignite at some depth~\\cite{Brow04,Keek11}.\nThis heat is insufficient, though, to raise the ocean temperature to the level required to explain observations. The deep crustal heating uncertainty of $\\sim 2$~MeV\/u, discussed above \\cite{steiner2012}, modifies the heating in the `pycno' region just beyond the right bound of figure~\\ref{figure:NStemp}. \nThis heat is mainly conducted inwards to the core due to high conductivity and provides insufficient impact on the ignition layer temperature. The temperature profile shown by a solid line in figure~\\ref{figure:NStemp}, which crosses the ignition curve (dashed line) at reasonable densities, is obtained by adding a strong ($\\sim 5$~MeV\/u) shallow heat source at $y_\\mathrm{shallow}\\sim 10^{15}$~g~cm$^{-2}$ \\cite{Deib16}.\nThe need for such a shallow heat source is in line with the findings for the quasi-persistent transients discussed in section~\\ref{sec:quasipers}. Some heating can also come from compositionally-driven convection proposed in the ocean~\\cite{Medin2011} (see section~\\ref{sec:pure}). \n\nAdditional complications come from the \nUrca pairs that can be present  \nin the neutron star's ocean and \ncrust.\nThey form a thermal barrier between heat released deeper in the crust and the carbon ignition layer. The strong temperature sensitivity of the neutrino luminosity from Urca cooling (see equation~(\\ref{equation:Lnu})) means that even \nstrong deep crustal heating will not raise the ocean temperature if such a barrier exists. \n\nA recent study investigated the interplay between heating and cooling reactions in the accreted ocean and crust and the carbon ignition depth~\\cite{Deib16}.\nThe authors self-consistently calculated the Urca cooling nuclei pairs \\cite{Scha14} one would expect \nfrom the compressed \nashes of Type I X-ray bursts and superbursts \\cite{Scha99,Scha03,Keek08}. Urca cooling layers were then implemented into a superburst ignition model \\cite{Pote12} and it was found that Urca cooling in the neutron star's \ncrust lowers the ocean's steady state temperature during an accretion outburst, while the ocean Urca pairs produce little effect. \nAs a consequence, superburst ignition occurs deeper than it would otherwise (see dashed curve in figure~\\ref{figure:NStemp}).\n\nThe highest mass-fraction odd-$A$ Urca nuclides produced in superbursts are of interest for this scenario, i.e. $A=53,55$, and $57$ (see figure~\\ref{figure:ashes}), assuming recurring superbursts would erase the signature of X-ray burst ashes. Of these, $A=55$ is by far the most significant due to the predicted percent-level abundance. Theoretical estimates vary largely \nfor $L_{\\nu}$ for $A=55$ nuclides, where the primary uncertainty comes from the assumptions used to determine the values of ${\\rm log}_{10}(ft)$ for the $^{55}$Ti$\\rightarrow^{55}$Sc$\\rightarrow^{55}$Ca $e^{-}$-capture sequence~\\cite{Scha14,Deib16}. As such, measurements are required to resolve this issue. Should ${\\rm log}_{10}(ft)\\lesssim5$ be found for either of these transitions, this would imply that the corresponding \nthermal barrier \nprevents a significant heat flow from deep crustal heating \nto the ocean. \nThis would deepen the mystery of the superburst ignition problem. Furthermore, the presence of such an Urca layer would \nlimit any shallow heating source, discussed above, to locations above existing Urca layers.\n\n\\subsection{Accreting neutron stars as Sources of Gravitational Waves}\\label{sec:gw}\nThe first detection of the gravitational wave transient GW~150914 from a double black hole coalescence with LIGO in 2015 \\cite{Abbott2016PhRvLGW150914} has started the era of gravitational wave astronomy. Only two years after the first detection, a double neutron star \nmerger GW~170817 was reported \\cite{Abbott2017PhRvLGW170817} opening a new window for neutron star \nastrophysics and providing exciting new insights into the physics of these objects.\n\nNot only binary neutron stars \ncan emit\ngravitational waves. In fact, fast spinning solitary neutron stars \nhave long been considered as possible {\\it persistent} \ngravitational wave sources if they possess some degree of asymmetry; for a recent review see Reference~\\cite{Glampedakis2017arXiv}. This can result from the certain types of oscillation modes possibly excited in the star, or because of the presence of static density inhomogeneities (``mountains'') that the neutron star crust can hold \\cite{Glampedakis2017arXiv}. Let us focus on the latter case. If the mountains result in neutron star ellipticity $\\varepsilon$, a star rotating with a frequency $\\nu$ would emit \ngravitational waves mainly with a frequency $f=2\\nu$ and the energy loss is\n\\begin{equation}\\label{eq:E_gw}\n\\dot{E}_\\mathrm{GW}=2\\pi\\nu N_\\mathrm{GW} = - \\frac{32}{5} \\frac{G(2\\pi \\nu)^6 (I\\varepsilon)^2}{c^5},\n\\end{equation}\nwhere $I$ is the neutron star moment of inertia and $N_\\mathrm{GW}$ is the associated braking torque. Thus, the emission of \ngravitational waves results in effective spin down which strongly scales with frequency. The maximal ellipticity that crustal mountains can produce is quite large $\\varepsilon_\\mathrm{max}\\sim (0.1-1)\\times10^{-4}\\, \\sigma_\\mathrm{max}$ \\cite{Johnson2013,Ushomirsky2000MNRAS,Haskell2006MNRAS} where \n$\\sigma_\\mathrm{max}\\lesssim 0.1$ is the maximal strain \\cite{Horowitz2009PhRvL,Chugunov2010MNRAS} the crust can sustain.\nThe largest mountains could produce a detectable signal. However no \ngravitational wave emission has been detected yet from known pulsars and current upper limits on ellipticity with advanced LIGO are of the order of $10^{-7}$ with a minimum value of few of $10^{-8}$ \\cite{Abbott2017ApJaLIGO}. \n\nMountains can be built up in accreting neutron stars \nonce some asymmetry in the accretion process is assumed. One of the proposed mechanisms is directly related to nuclear reactions in the crust (for other possibilities such as ``magnetic'' mountains see, e.g., Reference~\\cite{Glampedakis2017arXiv}). The idea is that if lateral thermal or compositional gradients are present in the crust, this radially shifts the positions of the electron capture layers and hence the associated density jumps \\cite{Bildsten1998ApJL,Ushomirsky2000MNRAS}. The physics of the temperature sensitivity of the $e^{-}$-capture layers is based on the observation that at sufficiently high temperatures $T\\gtrsim 2\\times 10^{8}$~K the $e^{-}$-capture rates become faster than the accretion timescale considerably before the corresponding threshold. Calculations show that in fact, most $e^{-}$-capture transitions occur subthreshold at $|Q_\\mathrm{EC}|-\\mu_e=\\Upsilon k_\\mathrm{B} T$, where $\\Upsilon\\approx 10-20$ and depends logarithmically on $T$, comparative half-life $ft$, \nthreshold $Q_\\mathrm{EC}$, and local accretion rate \n$\\dot{m}\\equiv \\dot{M}\/(4\\pi R^2)$ \\cite{Bildsten1998ApJL,Ushomirsky2000MNRAS,Bild98}. \nThe value of $\\Upsilon$ gives a measure of the thermal sensitivity of the position of the $e^{-}$-capture layer. The radial amplitude of the layer variation is then $\\Delta \\zeta \\approx \\Upsilon Y_e k_\\mathrm{B} \\delta T\/(m_u g) $, where $\\delta T$ is the amplitude of lateral temperature variations \\cite{Ushomirsky2000MNRAS}. If these variations are quadrupole, the resulting ellipticity is proportional to $\\Delta \\zeta \\Delta\\rho$, where in the outer crust the density jump at a given $e^{-}$-capture layer is $\\Delta \\rho\/\\rho= 2\/Z$. Based on this, the authors of Reference~\\cite{Ushomirsky2000MNRAS} obtain the following fiducial estimate\n\\begin{equation}\\label{eq:ellipt_fid}\n\\varepsilon \\approx 1.7\\times 10^{-9}\\, \\frac{\\Upsilon}{Z} \\frac{\\delta T}{10^7\\,\\mathrm{K}} \\frac{2.44}{g_{14}}\\, \\left(\\frac{R}{10~\\mathrm{km}}\\right)^4 \\,I_{45}^{-1}\\left(\\frac{|Q_\\mathrm{EC}|}{30\\,\\mathrm{MeV}}\\right)^3,\n\\end{equation}\nwhere $I_{45}$ is the neutron star moment of inertia in units of $10^{45}$~g~cm$^{2}$.\nThis estimate is strictly speaking applicable for the outer crust, but numerical calculations give similar orders of magnitude also when the reactions in the inner crust are considered (there the capture layers are thicker and their structure is modified since degenerate neutrons dominate the pressure). In addition, equation~(\\ref{eq:ellipt_fid}) does not account for the elastic response of the crust which in fact can result in an order of magnitude smaller values \\cite{Ushomirsky2000MNRAS}.\nSimilar $\\Delta \\zeta$ can be provided by compositional gradients instead of thermal ones, if, for instance, the ashes that are compressing are not symmetrically distributed. The source of the temperature asymmetry can stem from the asymmetry of deep crustal heating. Therefore the largest gradients can be expected from the regions \nof the largest heat release  -- i.e. in the inner crust where pycnonuclear reaction or superthreshold electron-capture cascades operate. Notice that too large of gradients would result in flux variations that are currently constrained at the level of $\\delta T\/T\\lesssim 0.1$ \\cite{Haskell2015MNRAS}. \n\nCould nuclear mountains be detected with gravitational observatories? When accretion ceases, the thermal gradients and thus mountains are thought to be erased on the thermal timescale (section~\\ref{subsection:impact_inner}) \\cite{Bildsten1998ApJL,Haskell2015MNRAS}. Thus, although the mountains in the inner crust can be sustained longer, the chance of the gravitational wave detection from transient sources is small. However, for the persistent accretion sources the situation is more optimistic with the future instrumentation \\cite{Haskell2015MNRAS}. There is also a possibility that the mountains are frozen in the crust and build up incrementally. However, this can lead to too large of a spin-down rate due to gravitational wave emission which contradicts observations for at least some sources \\cite{PatrunoWatts2012}. See Reference~\\cite{Haskell2015MNRAS} for a detailed discussion.\n\nEven if they are undetected by gravitational wave observatories, crustal mountains have important astrophysical implications. Initially the gravitational emission from accreting neutron stars \nwas\ninvoked as a mechanism for limiting the neutron star spin-up due to accretion \\cite{Bildsten1998ApJL}. The fastest rotator among the low-mass X-ray binaries is 4U~1608$-$52 with a frequency measured from bursts oscillations of 619~Hz \\cite{Hartman2003HEAD} and the fastest  radio millisecond pulsar spins at  716~Hz \\cite{Hessels2006Sci}  while basically one expects that the accretion torque can spin a star up until the stability limiting frequency $\\sim 1$~kHz \\cite{HPY2007}. The strong frequency dependence of gravitational wave spin-down torque in equation~(\\ref{eq:E_gw}) thus provides a natural explanation of the cutoff existence. Estimating the accretion torque as $N_a=\\dot{M}\\sqrt{ G M R}$, we get the ellipticity which allows gravitational wave torque to balance $N_a$: \n\\begin{equation}\\label{eq:ellip_a}\n\\varepsilon_a\\approx 8\\times 10^{-9} I_{45}^{-1} \\left(\\frac{600\\,\\mathrm{Hz}}{\\nu}\\right)^{5\/2} \\frac{\\dot{M}}{10^{-9}\\, M_\\odot\\, \\mathrm{yr}^{-1}}\n\\end{equation}\nfor $M=1.4 M_\\odot$ and $R=10$~km.\nIn fact, the accretion torque at high frequencies may be smaller, so the estimate from equation~(\\ref{eq:ellip_a}) is the upper limit (see Reference~\\cite{Patruno2017ApJ} for further discussion). This estimate is generally consistent with equation~(\\ref{eq:ellipt_fid}) if the mountains are built in the inner crust (note also that equation~(\\ref{eq:ellipt_fid}) is for a single capture layer so many asymmetric capture layers can combine in larger mountains). \nAlthough it is not clear now that \ngravitational wave emission is even needed for the explanation of the spin-up limit and, instead, \nthe physics of accretion possibly plays the major role \\cite{Patruno2017ApJ,D'Angelo2017MNRAS}, the bimodal distribution of the low-mass X-ray binaries over spin frequencies reveals a sharp spike clustering around $600$~Hz which is hard to explain without assuming that gravitational wave \nemission becomes important around this frequency \\cite{Patruno2017ApJ}.\n\nAnother piece of evidence came recently from observations of the spin down of the transient millisecond pulsar PSR~J1023$+$0038 which showed a transition between radiopulsar and low-mass X-ray binary stages. According to Reference~\\cite{Haskell2017PhRvL},\nthe PSR~J1023$+$0038 spin-down rate is faster in the low-mass X-ray binary state by $\\dot{\\nu}_\\mathrm{diff}=-6.428\\times 10^{-16}$~Hz~s$^{-1}$. This could be the first evidence that a mountain is built up during accretion and the \ngravitational wave torque provides additional spin down. The required ellipticity (estimated from equation~(\\ref{eq:E_gw})) is about $5\\times10^{-10}$, in line with the distortions nuclear mountains can provide according to equation~(\\ref{eq:ellipt_fid}). If this is indeed the case, then the gradual decrease of the spin-down rate on thermal timescales is expected in the radiopulsar stage. On the other hand, if continuous monitoring of X-ray oscillations during the low-mass X-ray binary stage reveals a spin-down rate increase, this could indicate mountain(s) build-up and the nuclear mountain mechanism can be tested.\n\n  \n\\section{Summary and Outlook} \\label{section:summary}\n\nRemarkable progress has been made in the 50 years since the neutron star \ncrust was first proposed. We now know the outer layers of accreting neutron stars \nhost a variety of nuclear phenomena, many of which can be studied in terrestrial laboratory experiments. Decades of efforts with X-ray telescopes have yielded a number of complementary observations which, with insight provided by astrophysics model calculations, have enabled \nthe construction of the tomographic picture of the accreted neutron star \nouter layers that we have today. These advances and current efforts in observation, experiment, and theory have been \ndiscussed\nin the previous sections. Here we summarize major remaining challenges and planned efforts to address them.\n\n\\subsection{What is Made In Surface Burning Processes?}\nSurface burning processes provide the seeds for crust processes, determining major features of the crust composition. Therefore they indicate which nuclear reaction heating\/cooling processes are relevant for a particular neutron star. \nHowever, major uncertainties remain as to the ashes produced in various burning regimes. \n Models struggle to reproduce observational signatures of surface burning processes when employing the inferred environment conditions.\nFor example, among H-rich H\/He-burning X-ray bursters, only GS~1826$-$24 has had its light curve successfully modeled and for that case models need a higher accretion rate than is inferred from observations (possibly due to a high inclination angle of this source)~\\cite{Hege07,Meis18}.\n\nType-I X-ray burst ash compositions from model calculations are sensitive to a large number of poorly constrained reaction rates~(section \\ref{sec:rpprocess}). \nRates of particular interest are those that lead to global changes in $\\langle Z\\rangle$, which would alter the depth at which pycnonuclear fusion could proceed~\\cite{Wijn13}, and rates affecting odd-$A$ abundances, which may result in significant Urca cooling in the crust~\\cite{Meis17}.\nWhile some studies have \ninvestigated which rates these may be~\\cite{Pari08,Pari09,Cybu16,Scha17}, a relatively small range of astrophysical conditions has been explored in self-consistent calculations. In principle the rates which impact superburst ash production, namely rates operating after freeze-out from nuclear statistical equilibrium, would be of interest, but computational challenges have made sensitivity studies prohibitively \nexpensive.\n\nThe range in which different burning regimes operate is also not clear. One challenge facing the study of Type I X-ray bursts is a discrepancy between theory and observations for the boundary between stable and unstable nuclear burning (section~\\ref{sec:burst_theory}). Observational surveys of thousands of Type I X-ray bursts find a peak burst rate near a mass accretion rate of  $\\dot{M} \\approx 0.3 \\, \\dot{M}_{\\rm Edd}$~\\cite{Paradijs1988,Cornelisse2003,galloway08} after which the burst rate decreases with increasing $\\dot{M}$. Numerical models, however, predict an increase in burst rate with increasing $\\dot{M}$. Furthermore, studies of stable and explosive burning fail to produce adequate amounts of carbon needed to ignite superbursts \nseen in some objects exhibiting Type-I bursts~\\cite{Stev14}.\n\n\\subsection{What are the Major Heating\/Cooling Sources in the Accreted Crust?}\nThe thermal structure of the crust influences the transitions between burning regimes, the depth at which fuel is ignited for unstable burning, and the rate of cooling after accretion turns off. Therefore identifying and quantifying the most significant heat sources and heat sinks is paramount to constructing accurate models of near-surface phenomena on accreting neutron stars.\nWhile the general features of nuclei that make them susceptible to generating substantial heating or cooling are known, identifying specific nuclei is presently \nproblematic (see section~\\ref{section:interaction}).\n\nThe masses, low-lying structure, and weak transition rates for neutron-rich nuclides with $A\\lesssim100$ lack sufficient constraints to definitively quantify the presence and strengths of $e^{-}$-capture heating and cooling. For pycnonuclear fusion, the properties of the nuclear potential for low-$Z$ nuclides far from stability needs further study to establish whether observed enhancement in fusion rates, e.g. Reference~\\cite{Sing17}, are the exception or the rule. The myriad of theoretical uncertainties associated with estimating pycnonuclear rates (section~\\ref{sec:pycno}) need to be reconciled if the region of deep crustal heating is to be determined. More modeling efforts for crust composition evolution need to be coupled to surface-burning models for the same objects in order to remove degrees of freedom and more rigorously test our understanding of heating and cooling reactions in the crust.\n\nA related and even \nless certain issue is the identity of the shallow heating mechanism required in model-observation comparisons for nearly all accreting neutron star \nobservables. Heating beyond the scale of nuclear physics uncertainties (as we understand them) appears to be necessary to reproduce observed properties of some Type-I X-ray bursts~\\cite{Keek17},\ncooling transients~\\cite{Turl15,Deib15}, and possibly superbursts~\\cite{Keek11,Reic17}. Better constraints on reaction-based heating and establishing the presence or absence of Urca cooling layers will be key to restricting the strength and location, and therefore mechanism, of shallow heating in accreting neutron stars. Furthermore, a proper accounting of all possible heat sources needs to be done~\\cite{Fattoyev2017arXiv}.\n\n\\subsection{How does the Crust Become So Pure?}\\label{sec:pure}\nInferences of the accreted crust thermal conductivity from observed cooling indicate that \nminimal crust impurities are allowed (section ~\\ref{sec:cooling_transients}).\nHowever, Type-I X-ray burst\nand stable burning ashes have more than 10 times larger \n$Q_{\\rm{imp}}$ (table~\\ref{table:AvgZAQTable} and figure~\\ref{figure:ashes}). \nModels suggest that phase separation in the ocean and crust can be crucial for \nformation of a purified crust. \nMolecular dynamics simulations\n\\cite{Horowitz2007PhRvE,Horowitz2007PhRvE,Schneider2012PhRvE,Hughto2012PhRvE,Capl17} and semi-analytical calculations \\cite{Medi10,Mcki16} show \nthat at the crystallization interface in most cases the solid phase is enriched in heavier (higher $Z$) elements, while lighter elements remain in the liquid phase. In some cases, however, (when the ashes average charge is low) the inverse situation was found \\cite{Mcki16,Capl17}. \n\nThe consequences of phase separation are not entirely clear. It was proposed that enrichment of the ocean with light elements will \nresult in compositionally-driven convection~\\cite{Medin2011,Medin2014ApJ,Medin2015ApJ}.\nHowever once a steady state is achieved, the composition of the freezing solid  should match on average the composition of matter entering\nthe top of the ocean \\cite{Medin2011}. \nAlternatively, the accumulation of light nuclei in the ocean eventually results in their solidification so that alternating crystalline layers or crystalline domains of different composition can form, reducing considerably $Q_{\\mathrm{imp}}$ in each layer or domain \\cite{Capl17}\n\nAnother possibility is that nuclear reactions can purify the ashes as they sink deep in the crust (section~\\ref{section:interaction}). \nRecall that the observational constraints on impurities are more relevant for the inner crustal regions (section~\\ref{sec:cooling_transients}). Pycnonuclear reactions burn lighter nuclei before the neutron drip point and neutron emission reactions around neutron drip simplify the composition towards a few closed-shell nuclei. Interestingly, this results in lowering the impurity for rp-process ashes, but on the contrary increases the diversity of species if one starts from a pure one-element composition \\cite{Gupt08,Lau18,steiner2012}.  Furthermore, very recent network calculations that follow the \nevolution of ashes beyond neutron drip indicate that these multicomponent configurations are quickly destroyed and all matter at $\\rho\\gtrsim 1.5\\times 10^{12}$~g~cm$^{-3}$ is converted into a single nucleus and a neutron gas, regardless of the initial surface abundances~\\cite{Lau18}. The only exception is the case when heavy nuclei ($A\\ge 106$) are present in the initial ashes. In this case the $N=82$ nuclei are locked and still present in the inner crust; the impurity parameter remains high in this case \\cite{Lau18}.\n\nWe also note, that binary (or multinary) crystalline structures can form, for instance, if two species dominate the composition in a certain proportion \\cite{Chamel2017JPhCS}. \nThen the impurity parameter can formally  be high (depending on the charges in the mixture) but the pure crystalline structure will lead to a high conductivity. Indeed, the compositions dominated by two or few species in similar fractions are found at some point in the reaction network evolved to the inner crust \\cite{Lau18}.\n\nClearly, a solid understanding of the accreted crust structure and composition does not exist yet. This would require time-dependent calculations which couple models \nof the structural changes in the crust (e.g. molecular dynamics) \nto a nuclear reaction network, following the reaction sequence all the way to the crust-core transition, including all relevant nuclear transitions. \n\n\\begin{figure} [h]\n\\centering\n\\includegraphics[scale=0.6]{ReAProductionRP.pdf}\n\\caption{Predicted production rates in particles per second, \nas indicated by the color,\nfor ReA3 beams at the Facility for Rare Isotope Beams~\\cite{Boll11} compared to the rp-process path~\\cite{Scha06}. See Reference~\\cite{Meis16b} for a similar figure with anticipated fast beam rates.\n\\label{figure:reaproduction}}\n\\end{figure}\n\n\\subsection{What is Being Done to Solve These Problems?}\nOngoing efforts in experiment, observation, and theory are constantly advancing the frontier of what is known about the nuclear processes occurring in accreted neutron star \ncrusts and their observable impacts. Recent and near-future advances in instrumentation and the open-source software movement promise to accelerate the pace of progress.\n\nX-ray observations, on which neutron star \nstudies rely, are \nperformed with\na number of relatively new and advanced telescopes. These include stalwarts like the {Neil Gehrels Swift Observatory}~\\cite{swift}, INTEGRAL~\\cite{integral}, the Chandra X-ray observatory~\\cite{chandra}, XMM-Newton~\\cite{newton}, \nand { MAXI}~\\cite{maxi} and newcomers {ASTROSAT}~\\cite{astrosat}, {NUSTAR}~\\cite{nustar}, and {NICER}~\\cite{nicer}. The problems discussed here would benefit from additional observational data of Type-I X-ray bursts, superbursts, and cooling transients. For instance, more regular bursters like GS~1826$-$24 would provide extra tests for models that successfully reproduce observations~\\cite{Hege07,Meis18,Gall17}. Superbursts and cooling transients have small populations, making our conclusions about these phenomena susceptible to small sample biases. Another cooling transient with a hot crust like MAXI~J0556$-$332 would provide \na new \ntest for the presence of Urca cooling in the accreted crust. The holy grail from this perspective would be a source with a variable accretion rate that exhibits Type-I X-ray bursts and\/or superbursts prior to an extended accretion outburst at high (near Eddington) accretion rate. Further progress will also require next-generation telescopes, e.g. the planned missions eXTP~\\cite{extp} and Strobe-X~\\cite{strobex}, in order to enable high-precision observations that will obviate the need for light-curve averaging and possibly allow time-resolved spectroscopy.\n\nNuclear physics experimental efforts are beginning to benefit from newly-developed techniques and will soon benefit from next-generation facilities. Improved constraints on masses, structure, and reaction rates are required from nearly dripline to dripline on the nuclear chart for $A\\lesssim100$. A consequence of the large number of nuclei involved is that many useful indirect measurements are left to be done at presently existing stable and radioactive ion beam facilities. The reach of these facilities is being expanded by studies leveraging recently developed capabilities such as in-ring reactions~\\cite{Mei15}, the $\\beta$-Oslo technique~\\cite{Spyr14}, phase-imaging penning trap mass measurements~\\cite{Elis15}, and ion beams produced in projectile fragmentation that have been stopped and re-accelerated to astrophysical energies coupled to windowless gaseous targets~\\cite{Bard16}. Nonetheless, the next-generation facilities FRIB~\\cite{frib} and FAIR~\\cite{fair} will provide unprecedented access to nuclei participating in surface burning and crust reactions (see figure~\\ref{figure:reaproduction}).\n\nMost aspects of the accreted neutron star \natmosphere, ocean, and crust have been modeled, including various regimes of nuclear burning, phase separation, and exotic reaction processes. Some of the most valuable ongoing and near-future efforts are and will be those which seek to combine the major aspects of several models, e.g. time-dependent accretion, phase separation, and a nuclear reaction network. More rigorous tests of crust models could be achieved by engaging in consistent multi-observable modeling for sources such as KS~1731$-$260 which exhibits Type-I X-ray burst, superburst, and cooling transient phases. Aside from reproducing observables, more parameter studies assessing the sensitivity of models to nuclear processes are desired to move beyond the small set of conditions that have been explored thus far~\\cite{Pari08,Pari09,Cybu16,Scha17}. Open-source astrophysics model codes, such as {\\tt MESA}~\\cite{Paxt11} and {\\tt dStar}~\\cite{Brow15} are invaluable resources for these studies by enabling more members of the nuclear astrophysics community to contribute.\n\nIt \ngoes without saying that efforts in observation, experiment, and theory will benefit tremendously from cross-discipline collaboration. In this regard the nuclear astrophysics community\nis performing exceedingly well. Continuing and strengthening these bonds (through focal points such as the Joint Institute for Nuclear Astrophysics) will be key to solving the many remaining issues of nuclear physics in the outer layers of accreting neutron stars and the affected astronomical observables.\n\n\\section*{Acknowledgements}\nThe authors thank M. Prakash, H. Schatz, and D.~G. Yakovlev for useful discussions. Z.~M. was supported by the U.S. Department of Energy under grant No. DE-FG02-88ER40387. P.~S. was supported by the BASIS Foundation.\nThis work was supported in part by the U.S. National Science Foundation under grant No. PHY-1430152 (Joint Institute for Nuclear Astrophysics--Center for the Evolution of the Elements).\n\n\n\\section*{References}\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we focus on parametric estimation of interacting particle system of the form \n\\begin{equation}\n\\begin{cases}\n d X_t^{\\theta,i, N} = b \\big(\\theta_1, X_t^{\\theta,i, N}, \\mu_t^{\\theta, N} \\big) dt + a \\big( \\theta_2, X_t^{\\theta,i, N}, \\mu_t^{\\theta, N} \\big) d W_t^i,  \\qquad i = 1, ... , N, \\quad t\\in [0, T], \\\\[1.5 ex]\n \\mathcal{L} \\big( X_0^{{\\theta,} 1, N}, ... , X_0^{{\\theta,} N, N} \\big) : = \\mu_0 \\times ... \\times \\mu_0.\n\\end{cases}\n \\label{eq: model}\n\\end{equation}\nHere the unknown parameter $\\theta\n:= (\\theta_1, \\theta_2)$ belongs to the set $\\Theta := \\Theta_1 \\times \\Theta_2$, where $\\Theta_j\\subset \\mathbb{R}^{p_j}${, $j=1,2$,} are compact {and convex} sets; we set \n$p: = p_1 + p_2$. The processes $(W^i_t)_{t \\in [0, T]}$, $i=1,\\dots,N$, are  independent $\\mathbb{R}$-valued Brownian motions, independent of the initial {value $(X^{\\theta,1,N}_0, \\dots, X^{\\theta,N,N}_0\n$ of the system}\nand $\\mu_t^{\\theta, N}$ is the empirical measure of the system at time $t$, i.e.\\\n\\[\n\\mu_t^{\\theta, N} := \\frac{1}{N} \\sum_{i = 1}^N \\delta_{X_t^{\\theta, i, N}}.\n\\]\nThe model coefficients are functions $b: \\Theta_1 \\times \\mathbb{R} \\times \\mathcal{P}_2 \\rightarrow \\mathbb{R}$ and $a: \\Theta_2 \\times \\mathbb{R} \\times \\mathcal{P}_2 \\rightarrow \\mathbb{R}$, where $\\mathcal{P}_2$ denotes the set of probability measures on $\\mathbb{R}$ with a finite second moment, endowed with the Wasserstein 2-metric\n\\begin{equation}\nW_2(\\mu, \\nu) := \\Big( \\inf_{m \\in \\Gamma (\\mu, \\nu)} \\int_{\\mathbb{R}^2} |x - y|^2 m(dx, dy) \\Big)^{\\frac 1 2},\n\\label{eq: wass}\n\\end{equation}\nand $\\Gamma(\\mu, \\nu)$ denotes the set of probability measures on $\\mathbb{R}^2$ with marginals $\\mu$ and $\\nu$. The underlying observations\nare\n$\n\\big(X_{t_{j,n}}^{\\theta,i,N}\\big)_{j= 1, \\dots, n}^{i=1, \\ldots, N}, $$\nwhere \n$t_{j,n} := {T j\/n}$ and $\\Delta_n := {T\/n}$ is the discretization\n{step}.\nWe assume \nthat the time horizon $T$ is fixed, and\n${N,n \\to \\infty}$.\n\nThe interacting particle system is naturally associated to its\nmean field equation as $N \\rightarrow \\infty$. The latter is described by  the 1-dimensional McKean-Vlasov SDE\n\\begin{equation}\nd \\bar{X}_t^{\\theta} = b \\big(\\theta_1, \\bar{X}_t^{\\theta}, \\bar{\\mu}_t^{\\theta} \\big) dt + a \\big(\\theta_2, \\bar{X}_t^{\\theta}, \\bar{\\mu}_t^{\\theta} \\big) d W_t,  \\quad t\\in [0, T],\n\\label{eq: McK}\n\\end{equation}\nwhere $\\bar{\\mu}_t^{\\theta}$ is the law of $\\bar{X}_t^{\\theta}$ and $(W_t)_{t \\in [0, T]}$ is a standard Brownian motion{, independent of the initial value $\\bar X^\\theta_0$ having the law $\\bar \\mu_0^\\theta := \\mu_0$}.\nThis equation is non-linear in the sense of McKean, see e.g.\\ \\cite{60imp,61imp,79imp}. It means, in particular, that the coefficients {depend not only on the current state but also on the current distribution of the solution.}\nIt is well known that, under appropriate assumptions on the coefficients $a$ and $b$, it is possible to obtain a phenomenon commonly named \\textit{propagation of chaos} (see e.g.\\ \\cite{79imp}). It implies that the empirical law $\\mu_t^{\\theta, N}$ weakly converges to $\\bar{\\mu}_t^{\\theta}$ as $N \\rightarrow \\infty$. The McKean-Vlasov SDE in \\eqref{eq: McK} links to a non-linear non-local partial differential equation on the space of probability measures (see e.g.\\ \\cite{Cat08}), which naturally arises in several applications in statistical physics. Indeed, stochastic systems of interacting particles and the associated McKean non-linear Markov processes have been introduced in 1966 in \\cite{60imp} starting from statistical physics, to model the dynamics of plasma. Their importance has increased in time, and a huge number of probabilistic tools have been progressively developed in this context (see  \\cite{Cat08,Fer97,Mal01,Mel96}, just to name a few). \n\n\nOn the other hand, however, statistical inference in this framework remained out of reach for many years (except for the early work of Kasonga in \\cite{Kas90}), mainly as microscopic particle systems derived from statistical physics are not directly observable. Later on, McKean-Vlasov models found applications in several other fields, in which the data is observable. Nowadays, these models are used in finance (smile calibration in \\cite{smile}; systemic risk in \\cite{Fou13}) as well as social sciences (opinion dynamics in \\cite{Cha17}) or mean-field games (see e.g.\\ \\cite{Car19, Zanella, Giesecke}). Moreover, some applications in neuroscience and population dynamics can be found respectively in \\cite{Bal12} and \\cite{Mog99}. At the same time, the interest in analysis of statistical models related to PDEs has gradually increased. A clear illustration of that is provided by the works on nonparametric Bayes and uncertainty quantification for inverse problems, as in \\cite{AbrNic,Nickl1,Nickl2}. \n\n\nMotivated by the increasing interest in statistical inference for McKean-Vlasov processes, we aim at estimating jointly the parameters ${\\theta_1, \\theta_2}$ starting from the discrete observations of the interacting particle systems \\eqref{eq: model} over a fixed time interval $[0, T]$. \nDespite recent interest in the study of the McKean-Vlasov SDEs, the problem of parameter estimation for this class has received relatively little attention. In \\cite{Wen} the authors established asymptotic consistency and normality of the maximum likelihood estimator for a class of McKean-Vlasov SDEs with constant\n{diffusion coefficient}, based on the continuous observation of the trajectory. This has been extended to the path dependent case in \\cite{Liu}. The mean field regime has been firstly considered by Kasonga in \\cite{Kas90}, who studied a system of interacting diffusion processes depending linearly in the drift coefficient on some unknown parameter. Starting from continuous observation of the system over a fixed time interval $[0, T]$, he showed that the MLE is consistent and asymptotically normal as $N \\rightarrow \\infty$. This has been extended in \\cite{Sharrock} to the case where the parametrisation is not linear, while Bishwal \\cite{Bis} extended it to the case where only discrete observations of the system are available and the parameter to be estimated is a function of time. In \\cite{Giesecke} the authors develop an asymptotic inference approach based on the approximation of the likelihood function for mean-fields models of large interacting financial systems. Moreover, Chen \\cite{24 imp} has established the optimal convergence rate for the MLE in the large $N$ and large $T$ case. Even in this work the drift coefficient is linear and the diffusion coefficient is constant. \n\n\nLet us also mention the works  \\cite{GenLar1, GenLar2}, where parametric inference for a particular class of nonlinear self-stabilizing SDEs is studied, starting from continuous observation of the non-linear diffusion. Some different asymptotic regimes are considered, such as the small noise and the long time horizon. The problem of the semiparametric estimation of the drift coefficient starting from the observation of the particle system at time $T$, for $T \\rightarrow \\infty$ is studied in \\cite{Vyt}, while \\cite{Marc} considers non-parametric estimation of the drift term in a McKean-Vlasov SDE, based on the continuous observation of the associated interacting particle system over a fixed time horizon. \n\nNone of these works, however, consider the problem of the joint estimation of the drift and\n{diffusion} coefficients. Moreover, not only we are not aware of any work about parameter estimation for interacting particle system where the\n{diffusion} coefficient can depend on the solution and on the law of the solution itself, but in the majority of the above mentioned work the\n{diffusion coefficient} is directly assumed to be constant. We consider a more general model, as in \\eqref{eq: model}, motivated by several applications in which the\n{diffusion coefficient} depends on the law. For example, this is the case in mathematical finance for the calibration of local and stochastic volatility models, with applications connected to the Dupire's local volatility function (see \\cite{Bos,Gyo,Lac}). Moreover, they are used to capture the diversity of a financial market, as in \\cite{Alm}. \n\nWe underline that the joint estimation of the two parameters introduces some significant difficulties: since the drift and\nthe {diffusion coefficient}\nparameters are not estimated at the same rate, we have to deal with asymptotic properties in two different regimes. Another challenge comes from the fact that both  coefficients depend on the empirical law of the process. This introduces some complexity compared to the case where $a$ is constant. \n\nA natural approach to estimation of unknown parameters in our context would be to use a maximum likelihood\nestimation. However, the likelihood function based on the discrete sample is not tractable in this setting, since\nit depends on the transition densities of the process, which are not explicitly known. To overcome this difficulty several methods have been developed, in the case of high frequency estimation for discretely observed classical SDEs. A widely-used method is to consider a pseudo likelihood\nfunction, for instance based on the high frequency approximation of the dynamic of the process\nby the dynamic of the Euler scheme, see for example \\cite{FloZmi,Kes97,Yos92}. \n\nOur statistical analysis is based upon minimisation of a contrast function, which is similar in spirit to the methods \\cite{FloZmi, Kes97,Yos92} that have been proposed in the setting of classical SDEs.\nThe main result of the paper is the consistency and asymptotic normality of the resulting estimator, which is showed by using a central limit theorem for martingale difference triangular arrays. The convergence rates for estimation of the two parameters are different,  which leads us to the study of the asymptotic properties of the contrast function in two different asymptotic schemes.\nWe emphasize that our inference is made on the time horizon $[0, T]$ with $T$ being fixed. It is well known that it is impossible to estimate the drift parameter of a classical SDE on a finite time horizon. However, due to increasing number of particles, we are able  to consistently estimate the drift even when $T$ is fixed.\n\n\nThe outline of the paper is as follows. In Section \\ref{s: ass} we present the estimation approach,\nlist the required assumptions and demonstrate some examples. Section \\ref{s: main} is devoted to  main results of the paper, which include consistency and asymptotic normality of the estimator. In Section \\ref{s: tec} we provide the technical lemmas we will use in order to show our main results. The proofs of the main results are collected in Section \\ref{s: proof main} while the technical results are shown in Section \\ref{s: proof technical}.\n\n\n\n\\subsection*{Notation}\n\nThroughout the paper all positive constants are denoted by $C$ or $C_q$ if they depend on an external parameter $q$. \n{All vectors are row vectors, $\\| \\cdot \\|$ denotes the Euclidean norm for vectors\n}\n{ We write $f(\\theta) = f(\\theta_1,\\theta_2)$ for $\\thet\n= (\\theta_\n,\\theta_\n)$.}\n{For $r=0,1,\\dots$, we denote by $C^{r}(X;\\mathbb{R})$ the set of\n$r$ times continuously differentiable functions $f: X \\to \\mathbb{R}$. We denote by $\\partial_x f$ the partial derivative of a function $f(x,y,\\dots)$ with respect to $x$. We denote by $\\nabla_{\\theta_j} f$ the vector $(\\partial_{\\theta_{j,1}} f, \\dots, \\partial_{\\theta_{j,p_j}} f)$, $j=1,2$, and $\\nabla_\\theta f = (\\nabla_{\\theta_1} f, \\nabla_{\\theta_2} f)$.}\nWe say that a function $f:\\mathbb{R} \\times \\mathcal{P}_l \\to \\mathbb{R}$ has \\textit{polynomial growth} if \n\\begin{align} \\label{eq: pol growth}\n|f(x,\\mu)| \\le C (1 +|x|^k +W_2^l(\\mu,\\delta_0) )\n\\end{align}\nfor some {$k,l =0,1,\\dots$ and all $(x,\\mu)\\in \\mathbb{R}\\times \\mathcal{P}_l$, where $\\mathcal{P}_l$ denotes the set of probability measures with a finite $l$-th absolute moment}. In the sequel, we suppress the dependence \nof several objects on the true parameter $\\theta_0$. In particular, we write $\\P := \\P^{\\theta_0}$, $\\mathbb{E} := \\mathbb{E}^{\\theta_0}$,  \n$X_t^{i, N}:=X_t^{\\theta_0, i, N}$, $\\bar X_t := \\bar X^{\\theta_0}_t$, $\\mu_t^N:=\\mu_t^{\\theta_0, N}$ and \n$\\bar \\mu_t := \\bar \\mu^{\\theta_0}_t$.\nFurthermore, we denote by $\\xrightarrow{\\mathbb{P}}$, $\\xrightarrow{\\mathcal{L}}$, \n$\\xrightarrow{L^p}$ the convergence in probability, in law, in $L^p$ respectively. We also denote by $c(\\theta_2,x,\\mu)$ the value $a^2(\\theta_2,x,\\mu)$. \n\n\n\n\n\n\n\\section{Minimal contrast estimator, assumptions and examples}{\\label{s: ass}}\n\nWe aim at estimating the unknown parameter $\\theta_0 \n= (\\theta_{0,1}, \\theta_{0,2}) \n\\in \\Theta^{\\circ}$ given equidistant discrete observations of the system introduced in \\eqref{eq: model}.\nWe study the asymptotic regime $N,n\\to \\infty$. \n\nThe estimator we propose is based upon a contrast function, which originates from the Gaussian quasi-likelihood. \nStarting from discrete observations of the model there are  difficulties due to the fact that the transition density of the process is unknown. A common way to overcome this issue is to base the inference on a discretization of the continuous likelihood (see for example \\cite{Gen90}, \\cite{Kes97} and \\cite{Yos92} where classic SDEs are considered). \nThis motivates us to consider the following contrast function:\n\\begin{align}\nS^N_n (\\theta)\n:= \\sum_{i=1}^N \\sum_{j=1}^n \\Bigg\\{ &\\frac{\\big(X^{i,N}_{t_{j,n}} - X^{i,N}_{t_{j-1,n}} - \\Delta_n b\\big(\\theta_1, X^{i,N}_{t_{j-1,n}}, \\mu^N_{t_{j-1,n}}\\big)\\big)^2}{\\Delta_n c \\big(\\theta_2,X^{i,N}_{t_{j-1,n}}, \\mu^N_{t_{j-1,n}}\\big)}\n+\\log c \\big(\\theta_2, X^{i,N}_{t_{j-1,n}}, \\mu^N_{t_{j-1,n}}\\big) \\Bigg\\}.\n\\label{eq:def contrast}\n\\end{align}\n{for $\\theta = (\\theta_1,\\theta_2)$.} The estimator $\\hat{\\theta}_n^N = (\\theta^N_{n,1},\\theta^N_{n,2})$ of $\\theta_0$\nis obtained as \n\\begin{equation*}\n\\hat{\\theta}_n^N \\in \\mathop{\\mathrm{arg\\,min}}_{\\theta \\in \\Theta} S_n^N(\\theta).\n\\end{equation*}\nIn order to study the asymptotic properties of $\\hat{\\theta}_n^N$\nwe need to introduce a set of conditions. \nThe first four conditions are regularity assumptions on the coefficients $a$ and $b$.\n\\begin{assumption} \\label{as1}\n(\\textit{Boundedness of moments})\n\\textit{For all $k \\ge 1$,}\n$$\n\\int_{\\mathbb{R}} |x|^k \\mu_0 (d x) \\le C_k.\n$$\n\\end{assumption} \n\n\\begin{assumption} \\label{as2}\n(\\textit{Lipschitz condition}) \\textit{The drift and\n{diffusion} coefficients are Lipschitz continuous {in $(x,\\mu)$}, i.e.\\ {for all $\\theta$ there exists $C$ such that} for all $(x,\\mu), (y,\\nu) \\in \\mathbb{R} \\times {\\cal P}_2$,\n$$|b(\\theta_1, x, \\mu) - b(\\theta_1, y, \\nu)| + |a(\\theta_2, x, \\mu) - a(\\theta_2, y, \\nu)|  \\le C ( |x - y| + W_2(\\mu, \\nu) ).$$\n}\n\\end{assumption}\n\\begin{assumption} \\label{as3}\n(\\textit{Regularity of the diffusion coefficient})\n\\textit{The diffusion coefficient is uniformly bounded away from $0$:\n$$\\inf_{(\\theta_2, x, \\mu) \\in \\Theta_2 \\times \\mathbb{R} \\times {\\cal P}_2} c(\\theta_2, x, \\mu) >0.\n$$}\n\\end{assumption}\n\\begin{assumption} \\label{as4}\n(\\textit{Regularity of the derivatives}) \\textit{(I) {For all $(x,\\mu)$}, the functions $b(\\cdot, x,\\mu)$,\n$a(\\cdot, x,\\mu)$ are in $C^3(\\Theta_1;\\mathbb{R})$,\n$C^3(\\Theta_2;\\mathbb{R})$ respectively. Furthermore, all their {partial} derivatives up to order three have polynomial growth, in the sense\nof \\eqref{eq: pol growth},\n uniformly in $\\theta$.\n \\\\\n(II) The {first and} second order derivatives in $\\theta$ are locally Lipschitz in $(x,\\mu)$ with polynomial weights, i.e.\\ {for all $\\theta$ there exists $C >0$, $k,l =0,1,\\dots$ such that for all $r_1 + r_2 =1, 2$, $h_1, h_2 = 1, ... , p_1$, $\\tilde{h}_1, \\tilde{h}_2 = 1, ... , p_2$,\n$(x,\\mu), (y,\\nu) \\in \\mathbb{R} \\times {\\cal P}_2$,}\n\\begin{align*}\n&\\big|\\partial_{\\theta_{1,h_1}}^{r_1} \\partial_{\\theta_{1,h_2}}^{r_2} b (\\theta_1, x, \\mu)- \\partial_{\\theta_{1,h_1}}^{r_1} \\partial_{\\theta_{1,h_2}}^{r_2} b (\\theta_1, y, \\nu) \\big| + \\big|\\partial_{\\theta_{2,\\tilde{h}_1}}^{r_1} \\partial_{\\theta_{2,\\tilde{h}_2}}^{r_2} a (\\theta_2, x, \\mu)-\\partial_{\\theta_{2,\\tilde{h}_1}}^{r_1} \\partial_{\\theta_{2,\\tilde{h}_2}}^{r_2} a (\\theta_2, y, \\nu) \\big|\n\\\\%[1.5 ex]\n&\\qquad \\le C (|x-y| + W_2 (\\mu,\\nu)) \\big( 1 + |x|^k + |y|^k + W_2^l(\\mu,\\delta_0) + W_2^l(\\nu,\\delta_0) \\big).\n\\end{align*}\n} \n\\end{assumption}\n\n\n\\begin{remark} \\rm\n(i) It is possible to relax assumption \\ref{as2} on the drift coefficient to allow for a locally Lipschitz condition in $x$ with polynomial weights, cf.\\ \\cite[Assumption 2.1]{DosReis}. In this setting \nthe boundedness of moments shown in our Lemma \\ref{l: moments} can  be replaced by \\cite[Theorem 3.3]{AAP} and the propagation of chaos needed in order to prove Lemma \\ref{l: Riemann} would follow from \\cite[Proposition 3.1]{DosReis}. As a consequence the main results of this paper still hold. \\\\ \\\\\n(ii) \n\\ref{as4}(I) \nis sufficient to show consistency of the estimator $\\hat{\\theta}_n^N$. We require the additional condition (II) of \\ref{as4} to prove the asymptotic normality. \\qed\n\\end{remark}\n\n\\noindent\nWe now state an assumption on the identifiability of the model\nand some further conditions that are required to prove the asymptotic normality. For this purpose we define the functions $I : \\Theta \\to \\mathbb{R}$, $J : \\Theta_2 \\to \\mathbb{R}$ as\n\\begin{align}\\label{def:I}\nI (\\theta)\n&:= \\int_0^{{T}} \\int_{\\mathbb{R}} \\frac{\\left(b(\\theta_1,x, \\bar \\mu_t) - b(\\theta_{0,1}x, \\bar \\mu_t)\\right)^2}{c(\\theta_2,x, \\bar \\mu_t)} \\bar \\mu_t (d x) d t,\\\\[1.5 ex] \\label{def:J}\nJ (\\theta_2) &:= \\int_0^{{T}} \\int_{\\mathbb{R}} \\Big( \\frac{c(\\theta_{0,2},x, \\bar \\mu_t)}{c (\\theta_2,x, \\bar \\mu_t)} + \\log c (\\theta_2,x, \\bar \\mu_t) \\Big) \\bar \\mu_t (d x) d t.\n\\end{align}\nThe next set of conditions are the following assumptions.\n\\begin{assumption} \\label{as5}\n \\textit{(Identifiability)} \\textit{The functions $I, J$ defined above satisfy that for every $\\varepsilon > 0$,\n$$\n\\inf_{\\theta \\in \\Theta: \\|\\theta_1-\\theta_{0,1}\\|\\ge \\varepsilon} I(\\theta)>0\\qquad \\mbox{and } \\inf_{\\theta_2 \\in \\Theta_2: \\|\\theta_2-\\theta_{0,2}\\|\\ge \\varepsilon} (J(\\theta_2)-J(\\theta_{0,2}))>0.\n$$}\n\\end{assumption}\n\\begin{assumption}\\label{as6}\n (\\textit{Invertibility}) \\textit{We define a $p \\times p$ block diagonal matrix \n$\\Sigma(\\theta_0) {:}=\n\\operatorname{diag}(\\Sigma^{(1)}(\\theta_0),\\Sigma^{(2)}(\\theta_0))$ whose main-diagonal blocks \n$\\Sigma^{(j)}(\\theta_0) = (\\Sigma^{(j)}_{kl}(\\theta_0))$\nare defined via\n \\begin{align*}\n\\Sigma^{(j)}_{kl} (\\theta_0) {:}= \\begin{cases}\n\\displaystyle 2 \\int_0^{{T}} \\int_\\mathbb{R} \\frac{\\partial_{\\theta_{1, k}} b(\\theta_{0, 1}, x, \\bar{\\mu}_s) \\, \\partial_{\\theta_{1, l}} b(\\theta_{0, 1}, x, \\bar{\\mu}_s)}{c(\\theta_{0, 2}, x, \\bar{\\mu}_s)} \\bar{\\mu}_s(dx) ds, \\qquad &j=1, \\, k, l = 1, \\dots, p_1,\n\\\\\n\\displaystyle \\int_0^{{T}} \\int_\\mathbb{R} \\frac{\\partial_{\\theta_{2, k}}c(\\theta_{0, 2}, x, \\bar{\\mu}_s) \\, \\partial_{\\theta_{2, l}}c(\\theta_{0, 2}, x, \\bar{\\mu}_s)}{c^2(\\theta_{0, 2}, x, \\bar{\\mu}_s)}  \\bar{\\mu}_s(dx) ds, \\qquad &j=2, \\, k,l = 1,\\dots, p_2.\n\\end{cases}\n\\end{align*}\nWe assume that $\\operatorname{det}(\\Sigma^{{(j)}} (\\theta_0)) \\neq 0${, $j=1,2$.\n}\n}\n\\end{assumption}\n\n\n \n\n\\begin{assumption} \\label{as7}\n \\textit{\n\n \n \n (\\textit{Integral condition on the diffusion coefficient}) {At $\\theta_{0,2}$ for all $(x,\\mu)$ t}he diffusion coefficient takes the form \n $$\n a(\\theta_{{0,}2}, x, \\mu) {:} = \\Tilde{a}\n \\Big(\n x, \\int_{\\mathbb{R}} \n (x, y) \\mu(dy)\\Big)\n $$\n {for some functions \n\n\n\n $\\tilde a, K \\in C^2(\\mathbb{R}^2; \\mathbb{R})$,\n which satisfy $|\\partial^{r_1}_x \\partial_y^{r_2} \\tilde \n (x,y)| + |\\partial^{r_1}_x \\partial^{r_2}_y \n (x,y)| \\le C (1+|x|^{k}+|y|^l)$ for some $k,l = 0,1,\\dots$ and all $r_1+r_2\n 1,2$, \n $(x,y) \n \\in \\mathbb{R}^2\n $.\n }\n }\n\n\\end{assumption}\n\n\\noindent\nAssumptions \\ref{as1}-\\ref{as5} are required to prove consistency of our estimator, while additional conditions \\ref{as6}-\\ref{as7} are need to obtain the central limit theorem. We also remark that, in the case where the unknown parameter $\\theta$ appears only in the drift coefficient, there is no need to add a further assumption on the derivatives of the diffusion coefficient to estimate it, even if the diffusion coefficient still depends on the law of the process. \n\n\\begin{example} \\rm\nA number of interacting particle models (and associated mean field equations) have been analyzed in the literature. We highlight a few here to illustrate the scope of our paper. \\\\\nWe start by considering some examples where the diffusion coefficient is a constant on a compact set that does not include the origin. This case has several applications (see (i) and (ii)). After that, some more general examples are presented. \\\\\n\\\\\n{\n(i) The Kuramoto model is the most classical model for synchronization phenomena in large populations of coupled oscillators such as a clapping crowd, a population of fireflies or a system of neurons (see Section 5.2 of \\cite{Review prop} and references therein). Let $N$ oscillators be defined by $N$ angles $X_t^{i,N}$, $i=1,\\dots,N$ (defined modulo $2 \\pi$, in this way they can actually be considered as elements of the circle), evolving in $t \\in [0,T]$ according to}\n$$d X_t^{i, N}=-\\frac{\\theta_{0, 1}}{N}\\sum_{j=1}^N \\sin \\big(X_t^{i, N} - X_t^{j, N} \\big) d t + \\theta_{0,2}d W^{{i}}_t.$$\n{This variant of the model satisfies our assumptions.\\\\ \n\\\\\n(ii) A popular model for opinion dynamics (see e.g.\\ \\cite{Cha17,65imp}) takes the form\n$$d X_t^{i, N}=-\\frac{1}{N}\\sum_{j=1}^N  \\varphi_{\\theta_{0, 1}} \\big( \\big| X_t^{i, N} - X_t^{j, N} \\big| \\big) \\big( X_t^{i, N} - X_t^{j, N} \\big) d t + \\theta_{0,2}d W^{{i}}_t \n$$\nfor $i=1,\\dots,N$, $t \\in [0,T]$, where $\\varphi_{\\theta_{0,1}} (x) := \\theta_{0,1,1} \\mathds{1}_{[0,\\theta_{0,1,2}]} (x)$, $x \\in \\mathbb{R}$, is the influence function which acts on the ``difference of opinions'' between agents. To have our regularity assumptions hold true in practice we can replace the function $\\varphi_{\\theta_{0,1}}$ by its infinitely differentiable approximation as it is done in Section 5.2 of \\cite{Sharrock}. In \\cite{Sharrock} we also note that the proxy of $\\varphi_{\\theta_{0,1}}$ depends non-linearly on the parameter $\\theta_{0,1,2}$.\\\\\n\\\\\n(iii) Another example is\n$$\nd X^{i,N}_t = \\Big( \\theta_{0,1,1} + \\frac{\\theta_{0,1,2}}{N} \\sum_{j=1}^N X^{j,N}_t - \\theta_{0,1,3} X^{i,N}_t \\Big) d t + \\theta_{0,2} \\sqrt{ 1+\\big( X^{i,N}_t \\big)^2 } d W^i_t\n$$\nfor $i=1,\\dots,N$, $t \\in [0,T]$. We note that in the case $\\theta_{0,1,2}=0$ the interacting particle system reduces to $N$ independent samples of a special case of the Pearson diffusion, which has applications in finance, see \\cite{For08} and references therein.\\\\\n\\\\\n(iv) We consider the dynamic of the system\n$$\nd X^{i,N}_t = \\Big( \\theta_{0,1,1} + \\frac{\\theta_{0,1,2}}{N} \\sum_{j=1}^N X^{j,N}_t - \\theta_{0,1,3} X^{i,N}_t \\Big) d t + \\Big( \\theta_{0,2,1} + \\theta_{0,2,2} \\sqrt{ \\frac{1}{N} \\sum_{j=1}^N \\big( X^{j,N}_t \\big)^2 } \\Big) d W^i_t\n$$\nfor $i=1,\\dots,N$ in $t\\in [0,T]$, where both the coefficients $b$ and $a$ depend on the law argument. We remark that the mean field limit of the above interacting particle system is a time-inhomogeneous Ornstein-Uhlenbeck process. See \\cite{Kas9\n} for the case\n$\\theta_{0,1,1} = \\theta_{0,2,2} = 0$. }\n\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\\section{Main results}{\\label{s: main}}\nOur main results demonstrate the consistency and the asymptotic normality of the estimator $\\hat{\\theta}_n^N$. \n\n\\begin{theorem}{(Consistency)}\nAssume that \\ref{as1}-\\ref{as5} hold, with only condition (I) in \\ref{as4}. Then the estimator $\\hat{\\theta}_n^N$ is consistent in probability:\n$$\\hat{\\theta}_n^N \\xrightarrow{\\mathbb{P}} \\theta_0 \\quad \\mbox{as } n, N \\rightarrow \\infty.$$\n\\label{th: consistency}\n\\end{theorem}\n\\noindent\nIn order to obtain the asymptotic normality of our estimator we need to add an assumption on the relation between the rates $N$ and $\\Delta_n$. In particular, we require that $N{\\Delta_n} \\rightarrow 0$ as $N, n \\rightarrow \\infty$.\n\n\\begin{theorem}{(Asymptotic normality)}\nAssume that \\ref{as1}-\\ref{as7} hold. If $N {\\Delta_n} \\rightarrow 0$ then \n$$\\big(\\sqrt{N}(\\hat{\\theta}_{n,1}^N - \\theta_{0,1}), \\sqrt{N\n\/ {\\Delta_n}}(\\hat{\\theta}_{n,2}^N - \\theta_{0,2}) \\big)\n\\xrightarrow{\\mathcal{L}} {\\cal N}\\big(0, 2 ( \\Sigma(\\theta_0) )^{-1}\\big) \\quad \\mbox{as } n, N\\rightarrow\\infty,$$\nwhere \n$$ 2 ( \\Sigma(\\theta_0) )^{-1} {:}= \n2 \\operatorname{diag} \\big( ( \\Sigma^{(1)}(\\theta_0) )^{-1}, ( \\Sigma^{(2)}(\\theta_0) )^{-1} \\big)$$\nwith $\\Sigma^{(j)} (\\theta_0)${, $j=1,2$,} being defined in \\ref{as6}. \n\n\\label{th: normality}\n\\end{theorem}\n\n\n\\noindent\nAs common in the literature on contrast function based methods, understanding the asymptotic behaviour of $S^N_n (\\theta_1,\\theta_2)$ and its derivatives is key to obtain the statements of Theorems \\ref{th: consistency} and \\ref{th: normality}. In particular, we show that, under proper normalisation,  the first derivative of \n$S^N_n (\\theta_1,\\theta_2)$ \nconverges to a Gaussian law with mean $0$ and covariance matrix $2 \\Sigma(\\theta_0)$ (see Proposition \\ref{p: norm L}), while the second derivative converges in probability to the matrix $\\Sigma(\\theta_0)$ defined in \n\\ref{as6} (see Proposition \\ref{p: second derivatives contrast}). These results lead to the statement of Theorem \\ref{th: normality}. \n\nThe condition on relation between the two convergence rates have been discussed in details in the framework of classical SDEs. Therein only the discrete trajectory of one particle is observed with discretization step $\\Delta_n\\to 0$ and terminal time $T:=n\\Delta_n \\to \\infty$. In \\cite{FloZmi} the corresponding condition was $T \\Delta_n = n \\Delta_n^2 \\rightarrow 0$\nas $n\\to \\infty$, which has been later improved to $n \\Delta_n^3 \\rightarrow 0$  in \\cite{Yos92} thanks to a correction introduced in the contrast function. Finally, Kessler \\cite{Kes97} proposed  a contrast function based on a Gaussian approximation of the transition density, which allowed him to consider a weaker condition $n \\Delta_n^p \\rightarrow 0$ for an arbitrary integer $p$. Similar developments have been made in the setting of classical SDEs with jumps in \\cite{Sjs,Joint,GLM,Shi}. \n\nOne may wonder if it possible to weaken the condition on the discretization step in the setting of particle systems. On the one hand, the condition $N {\\Delta}_n \\rightarrow 0$ is needed in order to approximate the derivative of the contrast function via a triangular array of martingale increments, as it is the case for classical SDEs. In this step higher order approximation similar to \\cite{Kes97} can potentially help to obtain a weaker condition.  \nOn the other hand, our condition on the discretization step is also required due to correlation between particles and higher order approximation does not seem to solve this issue. Thus, we leave this investigation for future research. \n\nA recent paper \\cite{Hof2} establishes the \\textit{LAN property} for\ndrift estimation in $d$-dimensional McKean-Vlasov models under continuous observations and with diffusion {coefficient \nbeing a function of $(t,\\bar X_t)$ {only}. The authors show that the Fisher information matrix is given as\n$$\\Bigg(\\int_0^T \\int_{\\mathbb{R}^d} \\partial_{\\theta_k} (c^{- \\frac{1}{2}} b) (\\theta, {t}, x, \\bar \\mu_t)^{{\\top}} \\partial_{\\theta_l} (c^{- \\frac{1}{2}} b) (\\theta, {t}, x, \\bar \\mu_t) \\bar \\mu_t (dx) dt \\Bigg)_{1 \\le k, l \\le p}$$\n(cf.\\ \\cite{Sharrock} where the diffusion coefficient is {an identity matrix}). This is consistent with our Theorem \\ref{th: normality} when restricted to drift estimation. In other words, our drift estimator is asymptotically efficient. When considering joint estimation of the drift and diffusion coefficients, the LAN property has not yet been shown, although the results of Gobet \\cite{Gobet 2002} in the classical diffusion setting give some hope.   \n\n\n\n\n\n\n\n\n\n\n\\section{Technical lemmas}{\\label{s: tec}}\nBefore proving the main statistical results stated in previous section, we need to introduce some additional notations and to state some lemmas which will be useful in the sequel. \n\nDefine $\\mathcal{F}_t^N := \\sigma \\{(W_u^k)_{u \\in [0,t]}, \\, X_0^\nk, N}; \\, k= 1, ... , N \\}$ and $\\mathbb{E}_t [\\cdot] := \\mathbb{E}[\\cdot | \\mathcal{F}^N_t]$. For a set $(Y^{i,N}_{t,n})$ of random variables and $\\delta \\ge 0$, the notation \n$$\nY^{i,N}_{t,n} = R_t^i (\\Delta_n^\\delta)\n$$ \nmeans that $Y^{i,N}_{t,n}$ is $\\mathcal{F}_t^N$-measurable and the set $(Y_{t,n}^{i,N}\/\\Delta_n^\\delta)$ is \nbounded in $L^q$ for all $q \\ge 1$, uniformly in $t, i, n, N$. That is \\\n$$\n\\mathbb{E} \\big[ \\big|Y_{t,n}^{i,N}\/\\Delta_n^\\delta \\big|^q \\big]^{1\/q}  \\le C_q\n$$ \nfor all $t, i, n, N$, $q \\ge 1$. \n\n\n\n\nWe will repeatedly use some moment inequality gathered in the following lemma. \n\\begin{lemma}\nAssume \\ref{as1}-\\ref{as2}. Then, for all $p \\ge 1$, $0 \\le s < t \\le T$ such that $t-s \\le 1$, $i \\in \\{ 1, ..., N \\}$, $N \\in \\mathbb{N}$, the following\nhold true.\n\\begin{enumerate}\n    \\item $\\sup_{t \\in [0,T]} \\mathbb{E}[|X_t^\n    i, N}|^p] < C$,\n   \n   \n   \n   \n   \n    moreover, $\\sup_{t \\in [0,T]} \\mathbb{E}[W_p^q(\\mu_t^\n    N}, \\delta_0)] < C$ for $p \\le q$.\n    \\item\n    $\\mathbb{E}[|X_t^\n    i, N} - X_s^\n    i, N}|^p] \\le C (t - s)^{\\frac{p}{2}}$.\n    \\item\n    $\\mathbb{E}_s [|X_t^\n    i, N} - X_s^\n    i, N}|^p\n    ] \n    \\le C (t - s)^\\frac{p}{2}R_s^i\n    1)$. \n    \\item\n    $\\mathbb{E}[W_2^p(\\mu_t^\n    N}, \\mu_s^\n    N})] \\le C (t - s)^{\\frac{p}{2}}$.\n    \\item\n    $\\mathbb{E}_s [W_2^p(\\mu_t^\n    N}, \\mu_s^\n    N}) |\n    ] \\le C (t - s)^{\\frac{p}{2}}R_s \n    1)$.\n\\end{enumerate}\n\\label{l: moments}\n\\end{lemma}\nThe asymptotic properties of the estimator are deduced by the asymptotic behaviour of our contrast function. To study it, the following lemma will be useful. \n\n\n\\begin{lemma}\nAssume \\ref{as1}-\\ref{as2}. \nLet $f : \\mathbb{R} \\times {\\cal P}_l \\to \\mathbb{R}$ satisfy for some $C>0$, $k,l =0,1,\\dots$ and all $(x,\\mu), (y,\\nu) \\in \\mathbb{R} \\times {\\cal P}_l$,\n\\begin{equation}\\label{cond:Riemann}\n|f(x,\\mu)-f(y,\\nu)| \\le C (|x-y| + W_2(\\mu,\\nu))(1 + |x|^k + |y|^k + W_l^l(\\mu,\\delta_0) + W_l^l(\\nu,\\delta_0))\n\\end{equation}\nMoreover, let\nthe mapping $(x,t) \\mapsto f(x,\\bar \\m\n_t)$ be integrable with respect to $\\bar \\m\n_t (d x) d t$ over $\\mathbb{R} \\times [0,T]$.\nThen \n$$\n\\frac{\\Delta_n}{N} \\sum_{i=1}^N \\sum_{j=1}^n f(X^\ni,N}_{t_{j-1,n}},\\mu^\nN}_{t_{j-1,n}}) \\xrightarrow{\\mathbb{P}} \\int_0^T \\int_{\\mathbb{R}} f(x,\\bar \\m\n_t) \\bar \\m\n_t (d x) d t \\quad \\text{as } n,N \\to \\infty.\n$$\n\\label{l: Riemann}\n\\end{lemma}\n\n\nOur main results heavily rely on the study of derivatives of our contrast function and so on the moments of its numerator. We recall that, in the sequel, we will denote by $c(\\theta_2, x, \\mu)$ the\nvalue $a^2(\\theta_2, x, \\mu)$.\n\n\\begin{lemma}\nAssume \\ref{as1}-\\ref{as2}.\nThen, the following hold true. \n\\begin{enumerate}\n    \\item If also \\ref{as7} is satisfied, then\\\\ \n    $\\mathbb{E}_{t_{j,n}} [(X_{t_{j+1,n}}^\n    i, N} - X_{t_{j,n}}^\n    i, N} - \\Delta_n b(\\theta_{0,1\n    , X_{t_{j,n}}^\n    i, N}, \\mu_{t_{j,n}}^\n    N}) )^2] = \\Delta_{n} c(\\theta_{0,2\n    , X_{t_{j,n}}^\n    i, N},\\mu_{t_{j,n}}^\n    N}) + R_{t_{j,n}}^i\n    \\Delta_{n}^{2}).$\n    \\item $\\mathbb{E}_{t_{j,n}} [(X_{t_{{j+1},n}}^\n    i, N} - X_{t_{j,n}}^\n    i, N} - \\Delta_n b(\\theta_{0,1}, X_{t_{j,n}}^\n    i, N}, \\mu_{t_{j,n}}^\n    N}) )^4] = 3 \\Delta_{n}^2 c^2(\\theta_{0, 2}\n    , X_{t_{j,n}}^{i, N},\\mu_{t_{j,n}}^\n    N}) + R_{t_{j,n}}^i\n    \\Delta_{n}^{\\frac{5}{2}}).$\n    \\item $|\\mathbb{E}_{t_{j,n}} [ X_{t_{j+1,n}}^\n    i, N} - X_{t_{j,n}}^\n    i, N} - \\Delta_n b(\\theta_{0,1}, X_{t_{j,n}}^\n    i, N}, \\mu_{t_{j,n}}^\n    N}) ] |\n    = R_{t_{j,n}}^i\n    \\Delta_{n}^{\\frac{3}{2}}).$\n  \n   \n\\end{enumerate}\n\\label{l: conditional expectation}\n\\end{lemma}\nThe proof of the lemmas stated in this section can be found in Section \\ref{s: proof technical}.\n\n\n\\section{Proofs}{\\label{s: proof main}}\n\t\n\\subsection{Consistency}\t\t\n\t\n\n\nLet us prove the (asymptotic) consistency of $\\hat \\theta^N_n = (\\hat \\theta^N_{n,1},\\hat \\theta^N_{n,2})$ component-wise. Our approach is similar to that taken in the proof of \\cite[Theorem 5.7]{van98}. \nIn particular, we consider a criterion function $\\theta \\mapsto S^N_n (\\theta)$ \nas a random element taking values in $(C(\\Theta;\\mathbb{R}), \\| \\cdot \\|_\\infty)$. The uniform convergence of criterion functions is proved in the following lemma. \n\n\n\\begin{lemma}\\label{lemma:cns}\nAssume \\ref{as1}-\\ref{as3}, {\\ref{as4}(I)}, \\ref{as5}. {Then a}s $N,n \\to \\infty$,\n\\begin{align}\\label{lim:con2}\n&\\sup_{(\\theta_1,\\theta_2) \\in \\Theta} \\Big| \\frac{\\Delta_n}{N} S^N_n (\\theta_1,\\theta_2) - J (\\theta_2) \\Big| \\xrightarrow{\\mathbb{P}} 0,\\\\ \\label{lim:con1}\n&\\sup_{(\\theta_1,\\theta_2) \\in \\Theta} \\Big| \\frac{1}{N} (S^N_n (\\theta_1,\\theta_2)-S^N_n (\\theta_{0, 1},\\theta_2)) - I (\\theta_1,\\theta_2) \\Big| \\xrightarrow{\\mathbb{P}} 0,    \n\\end{align}\nwhere the functions $I, J$ are defined in \\eqref{def:I}, \\eqref{def:J} respectively.\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nIt suffices to show the following steps:\n\\begin{enumerate}\n\\item $\\frac{\\Delta_n}{N} S^N_n (\\theta_1,\\theta_2) \\xrightarrow{\\mathbb{P}} J (\\theta_2)$ for every $(\\theta_1,\\theta_2) \\in \\Theta$.\n\\item The sequence $(\\theta_1,\\theta_2) \\mapsto \\frac{\\Delta_n}{N} S^N_n(\\theta_1, \\theta_2)$ is tight in $(C(\\Theta;\\mathbb{R}), \\| \\cdot \\|_\\infty)$.\n\\item $\\frac{1}{N} (S^N_n (\\theta_1,\\theta_2)-S^N_n (\\theta_{0,1},\\theta_2)) \\xrightarrow{\\mathbb{P}} I (\\theta_1,\\theta_2)$ for every $(\\theta_1,\\theta_2) \\in \\Theta$,\n\\item The sequence $(\\theta_1,\\theta_2) \\mapsto \\frac{1}{N} (S^N_n (\\theta_1,\\theta_2)-S^N_n (\\theta_{0,1},\\theta_2))$ is tight in $(C(\\Theta;\\mathbb{R}), \\| \\cdot\\|_\\infty)$.\n\\end{enumerate}\n\n\n\n\\noindent\nLet us omit the notation for dependence on $N,n$, in particular, write $X^i_t$ for $X^{i,N}_t$, $\\mu_t$ for $\\mu^N_t$, $t_j$ for $t_{j,n}$.\nDenote $f(\\cdot, X^i_t, \\mu_t)$ by $f^i_t (\\cdot)$ for a function $f$, for example equal to\n$h$ or $g$ defined as\n\\begin{equation}\\label{def:gd}\nh (\\theta, x,\\mu) = \\frac\n(b(\\theta_{0,1}, x, \\mu) - b(\\theta_1, x, \\mu))^2}{c (\\theta_2, x, \\mu)}, \\qquad g(\\theta, x, \\mu) = \\frac\nb(\\theta_{0,1}, x, \\mu) - b(\\theta_1, x, \\mu)}{c (\\theta_2, x, \\mu)\n\\end{equation}\nfor all $\\theta = (\\theta_1,\\theta_2) \\in \\Theta_1 \\times \\Theta_2 = \\Theta$, $x \\in \\mathbb{R}$, $\\mu \\in {\\cal P}_2$. \\\\\n\n\\noindent \n$\\bullet$ Step 3. We start proving that for every $\\theta = (\\theta_1,\\theta_2) \\in \\Theta_1 \\times \\Theta_2 = \\Theta$,\n$$\n\\frac{1}{N} (S^N_n(\\theta_1,\\theta_2) - S^N_n(\\theta_{0,1},\\theta_2)) \\xrightarrow{\\P} I(\\theta) = \\int_0^{T} \\int_{\\mathbb{R}} h(\\theta, x,\\bar \\mu_t) \\bar \\mu_t (d x) d t.\n$$\nLet us first decompose the left hand side as a sum of a main term and remainder.\nWe have\n$$\nS^N_n(\\theta_1,\\theta_2) = \\sum_{i=1}^N \\sum_{j=1}^n \\frac{({H}^i_j + \\Delta_n (b^i_{t_{j-1}}(\\theta_{0,1}) - b^i_{t_{j-1}}(\\theta_1))\n)^2}{\\Delta_n c^i_{t_{j-1}} (\\theta_2) } + (\\log c)^i_{t_{j-1}}(\\theta_2),\n$$\nwhere\n${H}^i_j = X^i_{t_{j-1}} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}}(\\theta_{0,1})$\nfor all $i, j$. We decompose\n\\begin{equation}\\label{cfdeComp}\n\\frac{1}{N} (S^N_n(\\theta_1,\\theta_2) - S^N_n(\\theta_{0,1},\\theta_2)) = I^N_n (\\theta) + 2 \\rho^N_n (\\theta),\n\\end{equation}\nwhere\n\\begin{equation}\\label{def:INnrhoNn}\nI^N_n (\\theta) = \\frac{\\Delta_n}{N} \\sum_{i=1}^N \\sum_{j=1}^n\nh^i_{t_{j-1}}(\\theta), \n\\qquad \\rho^N_n (\\theta) = \\frac{1}{N} \\sum_{i=1}^N \\sum_{j=1}^n\ng^i_{t_{j-1}} (\\theta) {H}^i_j.\n\\end{equation}\nThen\n$$\nI^N_n (\\theta) \\xrightarrow{\\P} I(\\theta)\n$$\nfollows from Lemma~\\ref{l: Riemann} if the function $h(\\theta,\\cdot)$\nis locally Lipschitz with polynomial growth. To check this assumption we note\nthat the functions $\nb(\\theta_{0,1},\\cdot) - b(\\theta_1,\\cdot)$, $a(\\theta_2,\\cdot)$ are Lipschitz continuous and have linear growth by \\ref{as2}. We also recall that $\\inf_{x,\\mu} c (\\theta_2,x,\\mu) > 0$ by \\ref{as3}. Hence, $h(\\theta,\\cdot)$\nsatisfies the assumption of Lemma~\\ref{l: Riemann}.\n\n\n\n\nIt remains to show that \n\\begin{equation}\\label{lim:rhoNn}\n\\rho^N_n (\\theta) \\xrightarrow{\\P} 0.\n\\end{equation}\nWith \n${H}^i_j = B^i_j + A^i_j$, where\n$$\nB^i_j = \\int_{t_{j-1}}^{t_j} (b^i_s(\\theta_{0,1}) - b^i_{t_{j-1}} (\\theta_{0,1})) d s, \\qquad A^i_j = \\int_{t_{j-1}}^{t_j} a^i_s (\\theta_{0,2}) d W^i_s,\n$$\nfor all $i,j$, let us further decompose\n\\begin{equation}\\label{rhoNndeComp}\n\\rho^N_n (\\theta) = \\rho^N_{n,1} (\\theta) + \\rho^N_{n,2} (\\theta),\n\\end{equation}\nwhere\n$$\n\\rho^N_{n,1} (\\theta)= \\frac{1}{N} \\sum_{i=1}^N \\sum_{j=1}^n g^i_{t_{j-1}}(\\theta) B^i_j, \\qquad\n\\rho^N_{n,2} (\\theta) = \\frac{1}{N} \\sum_{i=1}^N \\sum_{j=1}^n g_{t_{j-1}}^i (\\theta) A^i_j.\n$$\nIt is enough to show that\n\\begin{equation}\\label{lim:rhoNnp}\n\\rho^N_{n,k} (\\theta) \\xrightarrow{L^k} 0, \\qquad k=1,2.\n\\end{equation}\nFirst, let us show \\eqref{lim:rhoNnp} in case $k=2$. Note that for all $i_1 = i_2$ and $j_1 \\neq j_2$, \n\\begin{equation}\\label{covgA}\n\\mathbb{E} [ g^{i_1}_{t_{j_1-1}}(\\theta) A^{i_1}_{j_1} g^{i_2}_{t_{j_2-1}}(\\theta) A^{i_2}_{j_2} ] = 0\n\\end{equation}\nfollows from $\\mathbb{E}_{t_{j_1-1}} [A^{i_1}_{j_1}] = 0$, whereas independence of Brownian motions implies \\eqref{covgA} for all $i_1 \\neq i_2$ and $j_1, j_2$.  We conclude that\n\\begin{equation}\\label{varrhoNn2}\n\\mathbb{E} [(\\rho^N_{n,2} (\\theta))^2] = \\frac{1}{N^2} \\sum_{i=1}^N \\sum_{j=1}^n \\mathbb{E} [ (g^i_{t_{j-1}}(\\theta) A^i_j)^2 ]. \n\\end{equation}\nNext, the It\\^o isometry gives \n$$\n\\mathbb{E} [ (g^i_{t_{j-1}}(\\theta) A^i_j)^2 ] = \\int_{t_{j-1}}^{t_j} \\mathbb{E} [ (g^2)^i_{t_{j-1}}(\\theta) c^i_s(\\theta_{0,2}) ] d s\n$$\nwhere $\\mathbb{E} [ (g^2)^i_{t_{j-1}}(\\theta) c^i_s(\\theta_{0,2}) ] = O(1)$ \nuniformly in $t_{j-1} \\le s \\le t_j, j,i$ thanks to  $\\inf_{x,\\mu}c(\\theta_2,x,\\mu)>0$ by \\ref{as3}, linear growth of $a(\\theta_{0,2},\\cdot)$, $b(\\theta_1,\\cdot)$\nby \\ref{as2}\nand moment bounds in  Lemma~\\ref{l: moments}(1). We conclude that $\\mathbb{E} [ (g^i_{t_{j-1}}(\\theta) A^i_j)^2 ] = O(\\Delta_n)$ uniformly in $i,j$, which in turn implies \n$$\n\\mathbb{E} [(\\rho^N_{n,2}(\\theta))^2] = O \\big(N^{-1} \\big).\n$$\nFinally, let us show \\eqref{lim:rhoNnp} in case $k=1$. For this purpose, use\n$$\n\\mathbb{E}[ |g^i_{t_{j-1}}(\\theta) B^i_j| ] \\le \\int_{t_{j-1}}^{t_j} \\mathbb{E} [|g^i_{t_{j-1}}(\\theta) (b^i_s(\\theta_{0,1}) - b^i_{t_{j-1}}(\\theta_{0,1}))|] d s\n$$ \nand then the Cauchy\u2013Schwarz inequality.\nNote $\\mathbb{E} [ (g^2)^i_{t_{j-1}} (\\theta) ] = O(1)$ uniformly in $j,i$ follows in the same way\nas \nabove. Lipschitz continuity of  $b(\\theta_{0,1},\\cdot)$ by \\ref{as2} and moment bounds in  Lemma~\\ref{l: moments}(2) and (4)\nimply $\\mathbb{E} [ (b^i_s(\\theta_{0,1}) - b^i_{t_{j-1}}(\\theta_{0,1}))^2 ] = O(\\Delta_n)$ uniformly in $t_{j-1} \\le s \\le t_j, j, i$. We conclude that \n$$\n\\mathbb{E} [|\\rho^N_{n,1}|] = O(\\Delta_n^{\\frac{1}{2}}).\n$$ \nThis completes the proof of Step 3. \\\\\n\n\\noindent\n$\\bullet$\nStep 4. \nRecall the decomposition \\eqref{cfdeComp}, \\eqref{rhoNndeComp}.\nIt is enough to show tightness of \n$$\n\\theta \\mapsto I^N_n (\\theta), \\qquad\n\\theta \\mapsto \\rho^N_{n,k} (\\theta), \\qquad k=1,2.\n$$\nOur approach to showing tightness of both sequences are based upon \\cite[Theorem 14.5]{Kall}. We need to show that for all $N,n$:\n\\begin{equation}\\label{ineq:tight1}\n\\mathbb{E} \\big[ \\sup_{\\theta} \\| \\nabla_\\theta I^N_n (\\theta) \\| \\big] \\le C, \\qquad \\mathbb{E} \\big[ \\sup_{\\theta} \\| \\nabla_\\theta  \\rho^N_{n,1} (\\theta) \\| \\big] \\le C.\n\\end{equation}\nThe above bounds follow if for all $N,n$, and $i,j,t_{j-1} \\le s \\le t_j$,\n\\begin{equation}\\label{ineq:tight10}\n\\mathbb{E} \\big[ \\sup_\\theta \\| \\nabla_\\theta h^i_{t_{j-1}}(\\theta) \\| \\big] \\le C,\n\\qquad \n\\mathbb{E} \\big[ | b^i_s(\\theta_{0,1}) | \\sup_\\theta \\|\\nabla_\\theta g^i_{t_{j-1}}(\\theta) \\|  \\big] \\le C,\n\\end{equation}\nwhere $h,g : \\Theta \\times \\mathbb{R} \\times {\\cal P}_2\\to \\mathbb{R}$ are defined by \\eqref{def:gd}. In $\\nabla_{\\theta_k} h, \\nabla_{\\theta_k} g$, $k=1,2$,\nwe note $\\nabla_{\\theta_1} (b(\\theta_{0,1},\\cdot) - b(\\theta_1,\\cdot)) = - \\nabla_{\\theta_1} b(\\theta_1,\\cdot)$. Moreover, by the mean value theorem, $|b(\\theta_{0,1},\\cdot) - b(\\theta_1,\\cdot)| \\le C \\sup_{\\theta_1} \\|\\nabla_{\\theta_1} b(\\theta_1,\\cdot)\\|$ for all $\\theta_1 \\in \\Theta_1$, since $\\Theta_1$ is convex, bounded.\nAdditionally using $\\inf_{\\theta_2,x,\\mu} {c}(\\theta_2,x,\\mu) >0$ by \\ref{as3}, we get\n$$\n\\|\\nabla_{\\theta_1} g(\\theta,\\cdot)\\| \\le C \\sup_{\\theta_1} \\|\\nabla_{\\theta_1} b(\\theta_1,\\cdot) \\|, \\qquad \\|\\nabla_{\\theta_2} g (\\theta,\\cdot)\\| \\le C \\sup_{\\theta_1} \\| \\nabla_{\\theta_1} b(\\theta_1,\\cdot) \\| \\sup_{\\theta_2} \\|\\nabla_{\\theta_2} a(\\theta_2,\\cdot)\\|,\n$$ \nand\n$$\n\\|\\nabla_{\\theta_1} h (\\theta,\\cdot)\\| \\le C \\sup_{\\theta_1} \\|\\nabla_{\\theta_1} b(\\theta_1,\\cdot) \\|^2, \\qquad \\|\\nabla_{\\theta_2} h (\\theta,\\cdot)\\| \\le C \\sup_{\\theta_1} \\| \\nabla_{\\theta_1} b(\\theta_1,\\cdot)\\|^2 \\sup_{\\theta_2} \\|\\nabla_{\\theta_2} a(\\theta_2,\\cdot)\\|\n$$\nfor all $\\theta$. We have the polynomial growth of $\\sup_{\\theta_1} \\|\\nabla_{\\theta_1}b(\\theta_1,\\cdot)\\|$, $\\sup_{\\theta_2} \\|\\nabla_{\\theta_2} a(\\theta_2,\\cdot)\\|$ thanks to assumption \\ref{as4} and linear growth of $b(\\theta_{0,1},\\cdot)$ thanks to \\ref{as2}. \nThe Cauchy-Schwarz inequality and\nmoment bounds in Lemma~\\ref{l: moments}(1) yield \\eqref{ineq:tight10} and so \\eqref{ineq:tight1}.  \n\n\nFollowing the approach of \\cite[Theorem 20 in Appendix 1]{IbrHas}, we want to show that  for all $N,n$ and $\\theta, \\theta' \\in \\Theta$,\n$$\n\\mathbb{E} [ | \\rho^N_{n,2}(\\theta) |^2 ] \\le C, \\qquad \\mathbb{E} [ |\\rho^N_{n,2}(\\theta) - \\rho^N_{n,2}(\\theta')|^2 ] \\le C \\| \\theta - \\theta' \\|^2_2.\n$$\nWe note that the second relation implies the first one because $\\rho^N_{n,2}(\\theta) = 0$ with $\\theta_1 = \\theta_{0,1}$ and $\\Theta_2$ is bounded. \nIn the same way as in \\eqref{varrhoNn2} we get \n$$\n\\mathbb{E} [ |\\rho^N_{n,2}(\\theta)-\\rho^N_{n,2}(\\theta') |^2 ] = \\frac{1}{N^2} \\sum_{i=1}^N \\sum_{j=1}^n \\mathbb{E} [ | (g^i_{t_{j-1}}(\\theta) - g^i_{t_{j-1}} (\\theta')) A^i_j |^2 ],\n$$\nwhere the It{\\^o} isometry gives\n$$\n\\mathbb{E} [ |  (g^i_{t_{j-1}}(\\theta)-g^i_{t_{j-1}}(\\theta')) A^i_j |^2 ] = \\int_{t_{j-1}}^{t_j} \\mathbb{E} [  (g^i_{t_{j-1}}(\\theta)-g^i_{t_{j-1}}(\\theta'))^2 c^i_s (\\theta_{0,2}) ] d s. \n$$\nBy the mean value theorem, \n$$\n|g(\\theta,\\cdot)-g(\\theta',\\cdot)| \\le \\| \\theta - \\theta' \\| \\sup_{\\theta} \\| \\nabla_\\theta g(\\theta,\\cdot) \\| \n$$\nsince $\\Theta$ is convex. Then \n$$\n\\mathbb{E} \\big[ \\sup_\\theta \\| \\nabla_\\theta g^i_{t_{j-1}} (\\theta) \\|^2 c^i_s (\\theta_{0,2}) \\big] \\le C\n$$ \nfor all $t_{j-1} \\le s \\le t_j, j, i$ and $N,n$ follows in a similar way as the second bound in \\eqref{ineq:tight10} does using, in addition, linear growth of $a(\\theta_{0,2},\\cdot)$, which follows from its Lipschitz continuity by \\ref{as2}. \\\\\n\n\n\\noindent\n$\\bullet$\nStep 1. \nWe want to prove that for every $\\theta \\in \\Theta$,\n\\begin{equation}\\label{lim:intJ}\n\\frac{\\Delta_n}{N} S^N_n (\\theta) \\xrightarrow{\\P} J (\\theta_2) = \\int_0^{{T}} \\int_{\\mathbb{R}} f (\\theta_2,x,\\bar \\mu_t) \\bar \\mu_t(d x) d t,\n\\end{equation}\nwhere \n$$\nf (\\theta_2, x, \\mu) = \\frac{c (\\theta_{0,2},x,\\mu)}{c (\\theta_2,x,\\mu)} + \\log c (\\theta_2,x,\\mu)\n$$\nfor every $(\\theta_2,x,\\mu) \\in \\Theta_2 \\times \\mathbb{R} \\times {\\cal P}_2$. For this purpose, in $\\Delta_n S^N_n(\\theta)$ let us decompose every term as \n\\begin{equation}\\label{def:rij}\n\\frac{(X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}} (\\theta_1))^2}{c^i_{t_{j-1}} (\\theta_2)} + \\Delta_n (\\log c)^i_{t_{j-1}} (\\theta_2) = \\Delta_n f^i_{t_{j-1}}(\\theta_2)+ r^i_j.\n\\end{equation}\nWe can decompose $r^i_j$ further with \n\\begin{equation}\\label{eq:incrX}\nX^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}} (\\theta_1) = B^i_j (\\theta_1) + A^i_j,    \n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:incrXBA}\nB^i_j (\\theta_1) = \\int_{t_{j-1}}^{t_j} b^i_s(\\theta_{0,1}) d s - \\Delta_n b^i_{t_{j-1}} (\\theta_1), \\qquad A^i_j = \\int_{t_{j-1}}^{t_j} a^i_s (\\theta_{0,2}) d W^i_s,\n\\end{equation}\nnote\n$$\n\\mathbb{E}_{t_{j-1}} [ (A^i_j)^2 ] = \\int^{t_j}_{t_{j-1}} c^i_s (\\theta_{0,2}) d s\n$$\nWe get\n\\begin{equation}\\label{def:rijk}\nr^i_j = \\sum_{k=0}^2 r^i_{j,k}, \\qquad \\text{where } r^i_{j,k} = \\frac{H^i_{j,k}}{c^i_{t_{j-1}} (\\theta_2)}, \\qquad k=0,1,2,\n\\end{equation}\nand \n\\begin{align*}\n&H^i_{j,2} = (A^i_j)^2 - \\mathbb{E}_{t_{j-1}} [ (A^i_j)^2 ],\\qquad H^i_{j,1} = 2 A^i_j B^i_j(\\theta_1) + (B^i_j (\\theta_1))^2,\\\\\n&\\qquad H^i_{j,0} = \\mathbb{E}_{t_{j-1}} [ (A^i_j)^2 ] - \\Delta_n c^i_{t_{j-1}} (\\theta_{0,2}). \n\\end{align*}\nOur proof of \\eqref{lim:intJ} consists of the following steps: \n\\begin{equation}\n\\frac{\\Delta_n}{N} \\sum_{i=1}^N \\sum_{j=1}^n f^i_{t_{j-1}} (\\theta_2) \\xrightarrow{\\P} J(\\theta_2), \\qquad \\frac{1}{N} \\sum_{i=1}^N \\sum_{j=1}^n r^i_{j,k} \\xrightarrow{L^1} 0, \\quad k=0,1,2.\\label{cns:vol}\n\\end{equation}\nLet us start from the convergence in  \\eqref{cns:vol} for $k=2$. It is enough to show that \n$\\sup_i \\mathbb{E} [ ( \\sum_j r^i_{j,2} )^2] = o(1)$. \nWe note that $\\mathbb{E} [ r^i_{j_1,2} r^i_{j_2,2} ] = 0$, $j_1 \\neq j_2$, since $\\mathbb{E}_{t_{j-1}} [ r^i_{j,2}] = 0$. We are left to show that $\\sup_i \\sum_j \\mathbb{E} [ (r^i_{j,2})^2 ] = o(1)$.\nThanks to assumption \\ref{as3}\nit reduces to showing $\\sup_i \\sum_j \\mathbb{E} [ (H^i_{j,2})^2 ] = o(1)$,\nwhere $\\mathbb{E}_{t_{j-1}} [ (H^i_{j,2})^2 ] = \\mathbb{E}_{t_{j-1}} [ (A^i_j)^4 ] -  (\\mathbb{E}_{t_{j-1}} [ (A^i_j)^2 ])^2$ leads to $\\mathbb{E} [ (H^i_{j,2})^2 ] \\le \\mathbb{E} [ (A^i_j)^4 ]$ for all $i,j$. Furthermore, by the Burkholder-Davis-Gundy inequality and Jensen's inequality,\n\\begin{equation}\\label{ineq:A4}\n\\mathbb{E} [ (A^i_j)^4 ] \\le C \\mathbb{E} \\Big[ \\Big( \\int_{t_{j-1}}^{t_j} c^i_s (\\theta_{0,2}) d s \\Big)^2 \\Big] \\le C \\Delta_n \\int_{t_{j-1}}^{t_j} \\mathbb{E} [ (c^2)^i_s (\\theta_{0,2}) ] d s = O(\\Delta_n^2)\n\\end{equation}\nuniformly in $i,j$, where the last relation follows thanks to linear growth of $a(\\theta_{0,2},\\cdot)$ by \\ref{as2} and moment bounds in Lemma~\\ref{l: moments}(1).\nWe conclude that $\\sup_{i,j} \\mathbb{E} [(R^i_{j,2})^2] = O (\\Delta_n^2)$.\n\nWe now turn to the convergence in \\eqref{cns:vol} for $k=1$. It is enough to show that $n \\sup_{i,j}  \\mathbb{E} [ | r^i_{j,1} | ] = o(1)$. Assumption \n\\ref{as3}\nimplies $\\mathbb{E} [|r^i_{j,1}|] \\le C \\mathbb{E} [ |H^i_{j,1}| ]$ for all $i,j$, where $\\sup_{i,j} \\mathbb{E} [ ( A^i_j )^2 ] = O (\\Delta_n)$ follows from \\eqref{ineq:A4}. Moreover, by Jensen's inequality,\n$$\n\\mathbb{E} [ (B^i_j (\\theta_1))^2 ] \n\\le 2 \\Delta_n \\int_{t_{j-1}}^{t_j} \\mathbb{E} [ ( b^i_s (\\theta_{0,1})  )^2 ] d s + 2 \\Delta_n^2 \\mathbb{E} [ (b^i_{t_{j-1}}(\\theta_1))^2 ] = O(\\Delta_n^2)\n$$\nuniformly in $i,j$, where the last relation follows thanks to linear growth of $b(\\theta_1,\\cdot)$ for every $\\theta_1$ by \\ref{as2} and moment bounds in  Lemma~\\ref{l: moments}(1).\nWe conclude that\n$\\sup_{i,j} \\mathbb{E} [ |r^i_{j,1}| ] = O (\\Delta_n^{\\frac 3 2})$.\n \nNext, we consider the convergence in \\eqref{cns:vol} for $k=0$. It is enough to show that $n \\sup_{i,j}  \\mathbb{E} [ | r^i_{j,0} | ] = o(1)$. Assumption \n\\ref{as3} \nimplies $\\mathbb{E} [| r^i_{j,0} |] \\le C \\mathbb{E} [ |H^i_{j,0}| ]$, where\n$$\n\\mathbb{E} [ | H^i_{j,0}| ] \\le \\int^{t_j}_{t_{j-1}} \\mathbb{E} [ | c^i_s (\\theta_{0,2}) - c^i_{t_{j-1}} (\\theta_{0,2}) | ] d s.\n$$\nLipschitz continuity of $a(\\theta_{0,2},\\cdot)$ and Lemma~\\ref{l: moments}(2) and (4) imply $\\mathbb{E} [ (a^i_s (\\theta_{0,2}) - a^i_{t_{j-1}} (\\theta_{0,2}))^2 ] = O(\\Delta_n)$ uniformly in $t_{j-1} \\le s \\le t_j,j,i$. \nFinally, linear growth of $a(\\theta_{0,2},\\cdot)$ and moment bounds in Lemma~\\ref{l: moments}(1) guarantee $\\mathbb{E} [ (a^i_s (\\theta_{0,2}) + a^i_{t_{j-1}} (\\theta_{0,2}))^2 ] = O(1)$ uniformly in $t_{j-1} \\le s \\le t_j, j, i$. We conclude\nby Cauchy-Schwarz inequality that $\\mathbb{E} [|c^i_s (\\theta_{0,2}) - c^i_{t_{j-1}} (\\theta_{0,2})|] = O(\\Delta_n^{\\frac{1}{2}})$ uniformly in $t_{j-1} \\le s \\le t_j, j,i$, whence\n$\\sup_{i,j} \\mathbb{E} [|r^i_{j,0}|] = O (\\Delta_n^{\\frac 3 2})$. \n\nThe first relation in \\eqref{cns:vol} follows from Lemma \\ref{l: Riemann} if the function $f(\\theta_2,\\cdot)$ is locally Lipschitz with polynomial growth. To check this assumption, use \n$\n|\\log y_1 - \\log y_2| \\le |y_1-y_2|\/ \\min(y_1, y_2) \n$\nfor $y_1,y_2>0$ and assumption \\ref{as3}. Note $b(\\theta_1,\\cdot)$, $a(\\theta_2,\\cdot)$ are Lipschitz continuous and have linear growth by \\ref{as2}. Hence, the function $f (\\theta_2,\\cdot)$ satisfies the assumption of Lemma~\\ref{l: Riemann}. \\\\\n\\noindent\n$\\bullet$\nStep 2. We want to prove that the sequence $\\frac{\\Delta_n}{N} S^N_n(\\theta)$ in $(C(\\Theta;\\mathbb{R}), \\| \\cdot \\|_\\infty)$ is tight. So we have to show that  for all $N,n$,\n$$\n\\frac{\\Delta_n}{N} \\mathbb{E} \\Big[\\sup_\\theta {\\sum_{k=1}^2} \\| \\nabla_{\\theta_k} S^N_n (\\theta) \\| \\Big] \\le C.\n$$\nWe have \n$$ \n\\nabla_{\\theta_k} S^N_n (\\theta)=\\sum_{i=1}^N \\sum_{j=1}^n \\zeta_{j,k}^i (\\theta), \\qquad k =1,2,\n$$\nwhere\n\\begin{align*}\n\\zeta_{j,1}^i (\\theta) &= - \\frac{2(X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_1) )}{c^i_{t_{j-1}}(\\theta_2)} \\nabla_{\\theta_1} b^i_{t_{j - 1}}(\\theta_1),\\\\    \n\\zeta_{j,2}^i (\\theta) &= - \\frac{(X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_1) )^2}{ \\Delta_n (c^2)^i_{t_{j - 1}}(\\theta_2)} \\nabla_{\\theta_2} c^i_{t_{j - 1}}(\\theta_2) + \\frac{1}{c^i_{t_{j - 1}} (\\theta_2)} \\nabla_{\\theta_2} c^i_{t_{j - 1}}(\\theta_2).\n\\end{align*}\nIt suffices to show that for all $N,n$ and $i,j$,\n\\begin{equation}\\label{ineq:Esupxi}\n\\mathbb{E} \\big[ \\sup_\\theta \\|\\zeta_{j,k}^i (\\theta)\\| \\big] \\le C, \\qquad k =1,2. \n\\end{equation}\nUsing \\ref{as3} and the Cauchy-Schwarz inequality, we get \n\\begin{align*}\n\\mathbb{E} \\big[ \\sup_\\theta \\|\\zeta_{j,1}^i (\\theta)\\|\\big] &\\le C \\big(\\mathbb{E} \\big[ \\sup_{\\theta_1} |X^i_{t_j}-X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}}(\\theta_1)|^2 \\big] \\big)^{\\frac{1}{2}} \\big( \\mathbb{E} \\big[ \\sup_{\\theta_1} \\| \\nabla_{\\theta_1} b^i_{t_{j-1}}(\\theta_1) \\|^2 \\big] \\big)^{\\frac 1 2},\\\\\n\\mathbb{E} \\big[ \\sup_\\theta \\|\\zeta_{j,2}^i (\\theta)\\|\\big] &\\le \\frac{C}{\\Delta_n} \\big( \\mathbb{E} \\big[ \\sup_{\\theta_1} |X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}} (\\theta_1)|^4 \\big] \\big)^{\\frac{1}{2}} \\big( \\mathbb{E} \\big[ \\sup_{\\theta_2} \\| \\nabla_{\\theta_2} a^i_{t_{j - 1}}(\\theta_2) \\|^2 \\big] \\big)^{\\frac{1}{2}} \\\\\n&\\qquad + C \\mathbb{E} \\big[ \\sup_{\\theta_2} \\| \\nabla_{\\theta_2} a^i_{t_{j - 1}} (\\theta_2) \\| \\big].\n\\end{align*}\nWe use polynomial growth of $\\sup_{\\theta_1}\\| \\nabla_{\\theta_1} b(\\theta_1,\\cdot)\\|$, $\\sup_{\\theta_2} \\|\\nabla_{\\theta_2} a(\\theta_2,\\cdot)\\|$ and moment bounds in Lemma~\\ref{l: moments}(1). Moreover, Lemma~\\ref{l: moments}(2) gives $\\sup_{i,j} \\mathbb{E} [|X^i_{t_j} - X^i_{t_{j-1}}|^4] = O(\\Delta_n^2)$. Finally, $b(\\theta_{0,1},\\cdot)$ has a linear growth and the mean value theorem implies \n$b(\\theta_1,\\cdot) - b(\\theta_{0,1},\\cdot) = \\int_0^1 \\nabla_{\\theta_1} b(\\theta_{0,1} + (\\theta_1-\\theta_{0,1})u,\\cdot) d u \\cdot (\\theta_1-\\theta_{0,1})$ \nfor all $\\theta_1$ in $\\Theta_1$, where $\\Theta_1$ is convex, bounded and we recall that $\\sup_{\\theta_1}\\| \\nabla_{\\theta_1} b(\\theta_1,\\cdot)\\|$ has polynomial growth. The moment bounds in Lemma~\\ref{l: moments}(1) imply $\\mathbb{E} [\\sup_{\\theta_1} |b^i_{t_{j-1}}(\\theta_1)|^4] \\le C$, completing the proof of \\eqref{ineq:Esupxi}.\n\\end{proof}\n\n\\subsubsection{Proof of Theorem \\ref{th: consistency}}\n\\begin{proof}\nAssumption \\ref{as5} implies that for every $\\varepsilon>0$ there exists $\\eta>0$ such that $J(\\theta_2) - J(\\theta_{0,2}) > \\eta$\nfor every $\\theta_2$ with $\\|\\theta_2 - \\theta_{0,2}\\| \\ge \\varepsilon$. Thus  $\\{ \\|\\hat \\theta^N_{n,2}-\\theta_{0,2}\\| \\ge \\varepsilon \\}\\subseteq \\{J(\\hat \\theta^N_{n,2}) - J(\\theta_{0,2}) > \\eta\\}$. The probability of the latter event converges to $0$ in view of \n$$\nJ (\\hat \\theta^N_{n,2}) - J (\\theta_{0,2}) = J^N_{n,0} + J^N_{n,1},\n$$\nwhere \nthe definition of $\\hat \\theta^N_n$ and \\eqref{lim:con2} imply respectively\n\\begin{align*}\nJ^N_{n,0} &:= \\frac{\\Delta_n}{N} (S^N_n(\\hat \\theta^N_{n,1}, \\hat \\theta^N_{n,2}) - S^N_n(\\hat \\theta^N_{n,1}, \\theta_{0,2})) \\le 0,\\\\\nJ^N_{n,1} &:= J (\\hat \\theta^N_{n,2}) - J (\\theta_{0,2}) - J^N_{n,0} \\le 2 \\sup_{(\\theta_1,\\theta_2) \\in \\Theta} \\Big| \\frac{\\Delta_n}{N} S^N_n(\\theta_1,\\theta_2)-J(\\theta_2) \\Big| = o_\\P(1).\n\\end{align*}\nConsistency of $\\hat \\theta^N_{n,1}$ follows in a similar way. \nAssumption \\ref{as5} implies that for every $\\varepsilon>0$ there exists $\\eta >0$ such that $I(\\theta_1,\\theta_2) > \\eta$ for every $(\\theta_1,\\theta_2)$ with $\\|\\theta_1-\\theta_{0,1}\\| \\ge \\varepsilon$. Thus $\\{\\|\\hat \\theta^N_{n,1}-\\theta_{0,1}\\| \\ge \\varepsilon\\}\\subseteq \\{I(\\hat \\theta^N_{n,1}, \\hat \\theta^N_{n,2}) > \\eta \\}$. The probability of the latter event converges to $0$ because \n$$\nI(\\hat \\theta^N_{n,1}, \\hat \\theta^N_{n,2}) = I^N_{n,0} + I^N_{n,1},\n$$\nwhere the definition of $\\hat \\theta^N_n$ and \\eqref{lim:con1} imply respectively\n\\begin{align*}\nI^N_{n,0} &:= \\frac{1}{N} (S^N_n (\\hat \\theta^N_{n,1},\\hat \\theta^N_{n,2}) - S^N_n(\\theta_{0,1},\\hat \\theta^N_{n,2})) \\le 0,\\\\\nI^N_{n,1} &:= I(\\hat \\theta^N_{n,1},\\hat \\theta^N_{n,2}) - I^N_{n,0}\n= o_\\P(1).\n\\end{align*}\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Asymptotic normality}\n\nThe proof of the asymptotic normality of our estimator is obtained following a classical route. It consists in proving the asymptotic normality of the first derivative of the contrast function \\eqref{eq:def contrast} (see for example  \\cite[Section5a]{GenJac93}). We introduce in particular the appropriate normalization matrix\n$$M_n^N := \\operatorname{diag} \\Big( \\underbrace{\\frac{1}{\\sqrt{N}}, \\dots , \\frac{1}{\\sqrt{N}}}_{\\text{$p_1$ times}}, \\underbrace{\\sqrt{\\frac{\\Delta_n}{N}}, \\dots , \\sqrt{\\frac{\\Delta_n}{N}}}_{\\text{$p_2$ times}} \\Big).\n$$\nThe proof of Theorem \\ref{th: normality} is based on the following proposition.\n\\begin{proposition}\nAssume\n\\ref{as1}-\\ref{as4}(I) and \n(II),\n\\ref{as7}. \nIf $N\\Delta_n \\to 0$ then\nas $N,n \\rightarrow \\infty$,\n$\n\\nabla_\\theta S^N_n(\\theta_0) {M^N_n}\n\\xrightarrow{\\mathcal{L}} {\\cal N}(0, 2 \\Sigma (\\theta_0)),\n$$\nwhere \n{ $\\Sigma (\\theta_0)$ is a $p \\times p$ matrix defined in \\ref{as6}}.\n\n\n\n\\label{p: norm L}\n\\end{proposition}\n\\noindent\nWe observe that, as $\\nabla_\\theta S_n^N (\\hat{\\theta}_n^N) = 0$, by Taylor's formula we obtain\n\\begin{equation}\n{(\\hat \\theta^N_n - \\theta_0)} \\int_0^1 \\nabla^2_\\theta S_n^N (\\theta_0 + s (\\hat{\\theta}_n^N - \\theta_0)) ds \n\\,  = - \\nabla_\\theta S_n^N (\\theta_0).\n\\label{eq: taylor}\n\\end{equation}\nMultiplying the equation \\eqref{eq: taylor} by $M^N_n$,\nwe obtain\n\\begin{equation}\n{(\\hat \\theta^N_n - \\theta_0) (M^N_n)^{-1}} \\int_0^1 \\Sigma_n^N (\\theta_0 + s (\\hat{\\theta}_n^N - \\theta_0)) d s\n= -\n\\nabla_\\theta S^N_n(\\theta_0) {M^N_n},\n\\label{eq: Taylor2}\n\\end{equation}\nwhere \n$$\n\\Sigma^N_n (\\theta) := M_n^N \\nabla^2_\\theta  S_n^N (\\theta) M_n^N = \\begin{pmatrix}\n\\Sigma_n^{N,(1)} (\\theta) &  \\Sigma_n^{N,(12)} (\\theta) \\\\\n\\Sigma_n^{N,(21)} (\\theta) & \\Sigma_n^{N,(2)} (\\theta)\n\\end{pmatrix}\n$$\nwith\n\\begin{equation*}\n\\begin{aligned}[c]\n\\Sigma_n^{N,(1)} (\\theta) &= (1\/N) \\nabla^2_{\\theta_1} S_n^N {(\\theta)},\\\\\n\\Sigma_n^{N,(21)}{(\\theta)} &= (\\sqrt{\\Delta_n}\/N) \\nabla_{\\theta_2} \\nabla_{\\theta_1} S_n^N (\\theta), \n\\end{aligned}\n\\qquad\n\\begin{aligned}[c]\n\\Sigma_n^{N,(12)} {(\\theta)} &= (\\sqrt{\\Delta_n}\/N) \\nabla_{\\theta_1} \\nabla_{\\theta_2} S_n^N (\\theta),\\\\ \\Sigma_n^{N,(2)}{(\\theta)}&= (\\Delta_n\/N) \\nabla^2_{\\theta_2} S_n^N (\\theta).\n\\end{aligned}\n\\end{equation*}\nThe analysis of the second derivatives of the contrast function is gathered in the following proposition, which will be proven at the end of this section. \n\\begin{proposition}\nAssume \\ref{as1}-\\ref{as5} \nwith both (I) and \n(II) in \\ref{as4}.\nThen\nas $N,n \\to \\infty$,\n\\begin{enumerate}\n\\item $\\Sigma_n^N (\\theta_0) \\xrightarrow{\\mathbb{P}} \\Sigma (\\theta_0)$,\n\\item $\\sup_{s \\in [0,1]} \\| \\Sigma^N_n (\\theta_0 + s (\\hat \\theta^N_n -\\theta_0)) - \\Sigma^N_n (\\theta_0) \\| \\xrightarrow{\\mathbb{P}} 0$, where $\\| \\cdot \\|$ refers to the operator norm on the space of $p \\times p$ matrices induced by the Euclidean norm for vectors.\n\\end{enumerate}\n\\label{p: second derivatives contrast}\n\\end{proposition}\n\n\\noindent\nBy Proposition \\ref{p: second derivatives contrast} assumption \\ref{as6} implies that the probability that\n$\n\\int_0^1 \\Sigma_n^N (\\theta_0 + s (\\hat{\\theta}_n^N - \\theta_0)) d s\n$\nis invertible tends to 1. Applying its inverse to the equation \\eqref{eq: Taylor2},\nby Proposition~\\ref{p: norm L} and the continuous mapping theorem, we get \n$$\n{ \\big( \\sqrt{N} (\\hat{\\theta}_{n,1}^N - \\theta_{0, 1} ), \\sqrt{N\/\\Delta_n} (\\hat{\\theta}_{n,2}^N - \\theta_{0, 2}) \\big) }\n=\n(\\hat \\theta^N_n -\\theta_0 ) {(M^N_n)^{-1} } \\xrightarrow{{\\cal L}} \n{\\cal N} \\big(0, 2 ( \\Sigma(\\theta_0) )^{-1} \\big). $$\n\n\n\n\n\\subsection{Proof of Proposition \\ref{p: norm L}}\n\\begin{proof}\nAs in the proof of consistency, we omit the notation for dependence on $N,n$. In particular, we write $X^i_t$ for $X^{i,N}_t$, $\\mu_t$ for $\\mu^N_t$, $t_j$ for $t_{j,n}$. Denote by $f^i_{t_{j-1}}(\\theta)$ the values of $f(\\theta, X^i_{t_{j-1}}, \\mu_{t_{j-1}})$.\nWe note that $-\n\\nabla_{\\theta} S^N_n (\\theta) {M^N_n}$ consists of\n$- \\partial_{\\theta_{1, h}} S_n^N(\\theta)\/\\sqrt{N} =:\\sum_{j = 1}^n \\xi_{j,h}^{(1)} (\\theta)$ and $- \\sqrt{\\Delta_n\/N} \\partial_{\\theta_{2, \\Tilde{h}}} S_n^N(\\theta) =: \\sum_{j = 1}^n {\\xi}_{j,\\Tilde{h}}^{(2)} (\\theta)$,\nwhere\n\\begin{align*}\n\\xi_{j,h}^{(1)} (\\theta) &{:}= \\frac{1}{\\sqrt{N}} \\sum_{i = 1}^N 2 \\frac{\\partial_{\\theta_{1, h}} b^i_{t_{j - 1}}(\\theta_1)}{c^i_{t_{j - 1}}(\\theta_2)} (X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_1) ),\\\\\n\\xi_{j,\\Tilde{h}}^{(2)} (\\theta) &{:}= \\sqrt{\\frac{\\Delta_n}{N}} \\sum_{i = 1}^N \\frac{\\partial_{\\theta_{2, \\Tilde{h}}} c^i_{t_{j - 1}}(\\theta_2)}{ \\Delta_n (c^i_{t_{j - 1}}(\\theta_2))^2} (X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_1) )^2 - \\frac{\\partial_{\\theta_{2, \\Tilde{h}}} c^i_{t_{j - 1}}(\\theta_2) }{c^i_{t_{j - 1}}(\\theta_2) }\n\\end{align*}\nfor $h = 1, \\dots, p_1$, $\\Tilde{h} = 1, \\dots, p_2$.\nTo prove the asymptotic normality of \n\n\\nabla_\\theta S^N_n (\\theta_0){M^N_n}$ we want to use a  central limit theorem for martingale difference arrays,\nin accordance with Theorems 3.2 and 3.4 of \\cite{HalHey80}.\nApproximation of $\n\\nabla_\\theta S^N_n (\\theta_0){M^N_n}$ by a martingale array follows from \n\\begin{equation}\n\\sum_{j = 1}^n \\mathbb{E}_{t_{j - 1}}[\\xi^{(1)}_{j,h} (\\theta_0)] \\xrightarrow{\\mathbb{P}} 0, \\qquad  \\sum_{j = 1}^n \\mathbb{E}_{t_{j - 1}}[\\xi^{(2)}_{j,\\Tilde{h}} (\\theta_0)] \\xrightarrow{\\mathbb{P}} 0\n\\label{e: mtg array}\n\\end{equation}\nfor $h = 1, \\dots, p_1$, $\\Tilde{h} = 1, \\dots, p_2$.\nMoreover, application of the central limit theorem requires that for some $r > 0$ the following convergences hold:\n\\begin{align}\n &\\sum_{j = 1}^n \\mathbb{E}_{t_{j - 1}}[\\xi^{(1)}_{j,h_1} (\\theta_0)\\xi^{(1)}_{j,h_2} (\\theta_0)] \\xrightarrow{\\mathbb{P}} 4 \\int_0^{{T}} \\int_{\\mathbb{R}}  \\frac{\\partial_{\\theta_{1, h_1}} b(\\theta_{0, 1}, x, \\bar{\\mu}_t)\\partial_{\\theta_{1, h_2}} b(\\theta_{0, 1}, x, \\bar{\\mu}_t)}{c(\\theta_{0, 2}, x, \\bar{\\mu}_t)}  \\bar{\\mu}_t(dx) dt,\n\\label{e: norm theta1}\\\\\n &\\sum_{j = 1}^n \\mathbb{E}_{t_{j - 1}}[{\\xi}^{(2)}_{j,\\Tilde{h}_1} (\\theta_0){\\xi}^{(2)}_{j,\\Tilde{h}_2} (\\theta_0)] \\xrightarrow{\\mathbb{P}} 2 \\int_0^{{T}} \\int_{\\mathbb{R}} \\frac{\\partial_{\\theta_{2, \\Tilde{h}_1}} c(\\theta_{0, 2}, x, \\bar{\\mu}_t)\\partial_{\\theta_{2, \\Tilde{h}_2}} c(\\theta_{0, 2}, x, \\bar{\\mu}_t)}{c^2(\\theta_{0, 2}, x, \\bar{\\mu}_t)} \\bar{\\mu}_t(dx) dt,\n\\label{e: norm theta2}\\\\\n &\\sum_{j = 1}^n  \\mathbb{E}_{t_{j - 1}}[{\\xi}^{(1)}_{j,h} (\\theta_0) \\xi^{(2)}_{j,\\Tilde{h}} (\\theta_0)]  \\xrightarrow{\\mathbb{P}} 0,\n \\label{e: norm mixed}\\\\\n &\\sum_{j = 1}^n \\mathbb{E}_{t_{j - 1}}[|{\\xi}^{(1)}_{j,h} (\\theta_0)|^{2 + r}] \\xrightarrow{\\mathbb{P}} 0, \\qquad  \\sum_{j = 1}^n \\mathbb{E}_{t_{j - 1}}[|{\\xi}^{(2)}_{j,\\Tilde{h}} (\\theta_0)|^{2 + r}] \\xrightarrow{\\mathbb{P}} 0,\n\\label{e: negl theta}\n\\end{align}\nwhere $h,h_1,h_2 = 1,\\dots,p_1$, $\\tilde h, \\tilde h_1, \\tilde h_2 = 1,\\dots,p_2.$\\\\\n\n\\noindent\n$\\bullet$ Proof of \\eqref{e: mtg array}. \\\\\nAssumptions \\ref{as3} and \\ref{as4}(I) imply that $F^i_{j,h} := 2 \\partial_{\\theta_{1, h}} b^i_{t_{j-1}}(\\theta_{0,1})(c^i_{t_{j - 1}}(\\theta_{0, 2}))^{-1}$ satisfies $|F^i_{j,h}|\\le C (1 + |X_{t_{j - 1}}^i|^{k_1} + W_2^{l_1} (\\mu_{t_{j - 1}}, \\delta_0))$. Hence, from  Lemma \\ref{l: moments}(1) it is easy to see that $F^i_{j,h} = R^i_{t_{j - 1}}(1)$.\nIf $N \\Delta_{{n}} \\to 0$ then  Lemma \n\\ref{l: conditional expectation}(3) implies \n\\begin{align*}\n\\sum_{j=1}^n \\mathbb{E}_{t_{j - 1}}[\\xi^{(1)}_{j,h} (\\theta_0)]\n&= \\frac{1}{N^{\\frac 1 2}} \\sum_{i=1}^N \\sum_{j=1}^n R^i_{t_{j - 1}}(1) R^i_{t_{j - 1}}(\\Delta_n^{\\frac 3 2}) \\xrightarrow{L^1} 0\n\\end{align*}\nand so the convergence in probability.\nIn a similar way, using  Lemma \\ref{l: conditional expectation}(1), we obtain \n\\begin{align*}\n \\sum_{j=1}^n \\mathbb{E}_{t_{j - 1}}[{\\xi}^{(2)}_{j,\\Tilde{h}} (\\theta_0)] & = \\Big( \\frac{\\Delta_n}{N} \\Big)^{\\frac 1 2} \\sum_{i=1}^N \n \\sum_{j=1}^n \\Delta_n^{-1} R^i_{t_{j-1}}(1)\n (\\Delta_n c^i_{t_{j - 1}}(\\theta_{0, 2}) + R^i_{t_{j - 1}}(\\Delta_n^2) - \\Delta_n c^i_{t_{j - 1}}(\\theta_{0, 2}))\n \\\\\n& = \\Big( \\frac{\\Delta_n}{N} \\Big)^{\\frac 1 2} \\sum_{i=1}^N \n\\sum_{j=1}^n R^i_{t_{j - 1}}(\\Delta_n), \n\\end{align*}\nwhich converges to $0$ in $L^1$ and so in probability if $N \\Delta_n \\to 0$.\\\\\n\n\\noindent\n$\\bullet$ Proof of \\eqref{e: norm theta1}. \\\\\nWe have\n\\begin{equation}\\label{eq: start norm theta1}\n\\mathbb{E}_{t_{j-1}} [\\xi_{j,h_1}^{(1)} (\\theta_0)\\xi_{j,h_2}^{(1)} (\\theta_0)] = \\frac{1}{N} \\sum_{i_1, i_2=1}^N \\mathbb{E}_{t_{j-1}} [(A^{i_1}_j + B^{i_1}_j)(A^{i_2}_j + B^{i_2}_j)] F^{i_1}_{j,h_1} F^{i_2}_{j,h_2},   \\end{equation}\nwhere \n$$\nF^i_{j,h} := 2 \\frac{\\partial_{\\theta_{1, h}} b^{i}_{t_{j-1}}(\\theta_{0,1})} {c^{i}_{t_{j-1}}(\\theta_{0,2})} = R^{i}_{t_{j-1}}(1),\n$$\nand\n\\begin{equation}\\label{def: BAnorm}\nB^{i}_j := \\int_{t_{j-1}}^{t_j} (b^{i}_s (\\theta_{0,1}) - b^{i}_{t_{j-1}}(\\theta_{0,1})) d s, \\qquad A^{i}_j := \\int_{t_{j-1}}^{t_j} a_s^{i}(\\theta_{0,2}) d W^{i}_s.\n\\end{equation}\nWe have $\\mathbb{E}_{t_{j-1}} [(B^{i}_j)^2] = R^{i}_{t_{j-1}}(\\Delta_n^3)$ and $\\mathbb{E}_{t_{j-1}} [(A^{i}_j)^2] = R^{i}_{t_{j-1}}(\\Delta_n)$, whereas\nif $i_1 \\neq i_2$ then $\\mathbb{E}_{t_{j-1}} [A^{i_1}_j A^{i_2}_j] = 0$ because of the independence of Brownian motions. \nHence, by the Cauchy-Schwarz inequality, \n$$\n\\mathbb{E}_{t_{j-1}} [(A^{i_1}_j + B^{i_1}_j)(A^{i_2}_j + B^{i_2}_j)] = \\mathbb{E}_{t_{j-1}}[(A^{i_1}_j)^2] \\mathbf{1} (i_1=i_2) + R_{t_{j-1}}^{i_1,i_2} (\\Delta_n^2).\n$$ \nWe get\n$$\n\\sum_{j=1}^n \\mathbb{E}_{t_{j-1}} [\\xi_{j,h_1}^{(1)} (\\theta_0)\\xi_{j,h_2}^{(1)} (\\theta_0)] = \\frac{1}{N} \\sum_{j=1}^n \\sum_{i=1}^N \\mathbb{E}_{t_{j-1}} [(A^i_j)^2] F^i_{j,h_1} F^i_{j,h_2} + \\frac{1}{N} \\sum_{j=1}^n \\sum_{i_1{,} i_2=1}^N R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^2),\n$$\nwhere the last sum converges to $0$ in $L^1$ and so in probability if $N \\Delta_n \\to 0$. We can therefore focus on the first sum. We decompose the term $\\mathbb{E}_{t_{j-1}}[(A^i_j)^2]$ into $\\Delta_n c^i_{t_{j-1}}(\\theta_{0,2})$\nand \n$$\n\\mathbb{E}_{t_{j-1}} [(A^i_j)^2] - \\Delta_n c^i_{t_{j-1}}(\\theta_{0,2}) = \\int_{t_{j-1}}^{t_j} \\mathbb{E}_{t_{j-1}} [c^i_s(\\theta_{0,2}) - c^i_{t_{j-1}}(\\theta_{0,2})] d s = R^i_{t_{j-1}} (\\Delta_n^{\\frac 3 2})\n$$\nThe result follows from $\\Delta_n \\to 0$ and application of Lemma \\ref{l: Riemann}.\\\\\n\n\\noindent\n$\\bullet$ Proof of \\eqref{e: negl theta}, first convergence. \\\\\nWe want to show \\eqref{e: negl theta} with $r=2$. We use the same notation as in \\eqref{eq: start norm theta1} and consider the terms\n\\begin{equation}\\label{eq:AB4}\n\\mathbb{E}_{t_{j-1}} [(A^{i_1}_j + B^{i_1}_j)(A^{i_2}_j + B^{i_2}_j)(A^{i_3}_j+B^{i_3}_j)(A^{i_4}_j +B^{i_4}_j)] F^{i_1}_{j,h} F^{i_2}_{j,h} F^{i_3}_{j,h} F^{i_4}_{j,h}.    \n\\end{equation}\nWe have $F^i_j = R^i_{t_{j-1}}(1)$, moreover, $\\mathbb{E}_{t_{j-1}} [(A^{i}_j)^4] = R_{t_{j-1}}^{i} (\\Delta_n^2)$, $\\mathbb{E}_{t_{j-1}}[(B^{i}_j)^4] = R_{t_{j-1}}^{i} (\\Delta_n^6)$ and so\n$\\mathbb{E}_{t_{j-1}} [(A^{i}_j + B^{i}_j)^4] = R^{i}_{t_{j-1}}(\\Delta_n^2)$. Application of the Cauchy-Schwarz inequality shows that the term in \\eqref{eq:AB4} is also $R^{i_1,i_2,i_3,i_4}_{t_{j-1}}(\\Delta_n^2)$.\nIn case where $i_1, i_2, i_3, i_4$ are pairwise distinct we decompose $A^i_j$ into \n\\begin{equation}\\label{def: A12norm}\nA^{i}_{j,2} := \\int_{t_{j-1}}^{t_j} (a^{i}_s(\\theta_{0,2}) - a^{i}_{t_{j-1}}(\\theta_{0,2})) d W^{i}_s,\\qquad\nA^{i}_{j,1} := \\int_{t_{j-1}}^{t_j} a^{i}_{t_{j-1}} (\\theta_{0,2}) d W^{i}_s,\n\\end{equation}\nwhich satisfy $\\mathbb{E}_{t_{j-1}} [(A^i_{j,k})^4] = R^i_{t_{j-1}}(\\Delta_n^{2k})$, $k=1,2$. In particular the independence of the Brownian motions implies \n$$\n\\mathbb{E}_{t_{j-1}} [A^{i_1}_{j,1} A^{i_2}_{j,1} A^{i_3}_{j,1} A^{i_4}_{j,k}] F^{i_1}_{j,h} F^{i_2}_{j,h} F^{i_3}_{j,h} F^{i_4}_{j,h} = 0\n$$ \nfor $k=1,2$. The\nterm {converging to $0$ at the slowest rate} in \\eqref{eq:AB4} is then, up to a permutation of the indices $i_1,i_2,i_3,i_4$, \n$$\n\\mathbb{E}_{t_{j-1}} [A^{i_1}_{j,1} A^{i_2}_{j,1} A^{i_3}_{j,2} A^{i_4}_{j,2} + A^{i_1}_{j,1} A^{i_2}_{j,1} A^{i_3}_{j,1} B^{i_4}_j] F^{i_1}_{j,h} F^{i_2}_{j,h} F^{i_3}_{j,h} F^{i_4}_{j,h} = R^{i_1,i_2,i_3,i_4}_{t_{j-1}} (\\Delta_n^3).\n$$\nWe get\n\\begin{equation}\\label{eq:negl1}\n\\sum_{j=1}^n \\mathbb{E}_{t_{j-1}} [(\\xi_{j,h}^{(1)}(\\theta_0))^4] = \\frac{1}{N^2} \\sum_{j=1}^n \\Big( \\sum_{i \\in I} R^i_{t_{j-1}} (\\Delta_n^3) + \\sum_{i \\in I^c} R^i_{t_{j-1}} (\\Delta_n^2) \\Big),\n\\end{equation}\nwhere $I$ denotes a set of all $i = (i_1,i_2,i_3,i_4) \\in \\{1,\\dots,N\\}^4$ such that $i_1, i_2, i_3, i_4$ are pairwise distinct. We note that $\\operatorname{card}(I) = O(N^4)$ and $\\operatorname{card}(I^c) = O(N^3)$. We conclude that \\eqref{eq:negl1} converges to $0$ in $L^1$ and so in probability if $N \\Delta_n \\to 0$.\\\\\n\n\\noindent\n$\\bullet$ Proof of \\eqref{e: norm theta2}. \\\\\nWe rewrite the left hand side of \\eqref{e: norm theta2} as\n\\begin{equation}\n\\frac{\\Delta_n}{N} \\sum_{j=1}^n \\sum_{i_1, i_2=1}^N \\Delta_n^{-2} C^{i_1}_{j,\\Tilde{h}_1} C^{i_2}_{j,\\Tilde{h}_2} \\mathbb{E}_{t_{j-1}} [ D^{i_1}_j D^{i_2}_j],\\label{eq: norm theta2 start}\n\\end{equation}\nwhere\n$$\nC^i_{j,\\Tilde{h}} := \\frac{\\partial_{\\theta_{2, \\Tilde{h}}} c^{i}_{t_{j-1}} (\\theta_{0,2})}{(c^{i}_{t_{j-1}}(\\theta_{0,2}))^2} = R^{i}_{t_{j-1}}(1), \\qquad D^{i}_j :=  (X^{i}_{t_j}-X^{i}_{t_{j-1}}-\\Delta_n b^{i}_{t_{j-1}}(\\theta_{0,1}))^2 - \\Delta_n c^{i}_{t_{j-1}}(\\theta_{0,2}).\n$$\nWe consider the term $\\mathbb{E}_{t_{j-1}} [D^{i_1}_j D^{i_2}_j]$ in \\eqref{eq: norm theta2 start}. By  Lemma \\ref{l: conditional expectation}(1)\nit equals\n\\begin{align}\\label{eq:cexpected22}\n&\\mathbb{E}_{t_{j-1}} [ (X^{i_1}_{t_j} - X^{i_1}_{t_{j-1}} - \\Delta_n b^{i_1}_{t_{j-1}}(\\theta_{0,1}))^2 (X^{i_2}_{t_j} - X^{i_2}_{t_{j-1}} - \\Delta_n b^{i_2}_{t_{j-1}}(\\theta_{0,1}))^2]\\\\\n&\\qquad- \\Delta_n c^{i_1}_{t_{j-1}} (\\theta_{0,2}) \\Delta_n c^{i_2}_{t_{j-1}} (\\theta_{0,2}) + R^{i_1,i_2}_{t_{j-1}}(\\Delta_n^3).\\nonumber\n\\end{align}\nIf $i_1=i_2$ then  Lemma \\ref{l: conditional expectation}(2) implies\n$$\n\\mathbb{E}_{t_{j-1}} [ (X^{i}_{t_j} - X^{i}_{t_{j-1}} - \\Delta_n b^{i}_{t_{j-1}}(\\theta_{0,1}))^4] = 3 \\Delta_n^{2} (c^{i}_{t_{j-1}}(\\theta_{0,2}))^2 + R^{i}_{t_{j-1}} (\\Delta_n^{\\frac 5 2}),\n$$\nwhence \n\\begin{equation}\\label{eq:condD2}\n\\mathbb{E}_{t_{j-1}}[(D^{i}_j)^2] = 2 \\Delta_n^2  (c^{i}_{t_{j-1}}(\\theta_{0,2}))^2 + R^{i}_{t_{j-1}} (\\Delta_n^{\\frac 5 2}).\n\\end{equation}\nIf $i_1 \\neq i_2$ then to deal with the term in \\eqref{eq:cexpected22} we decompose\n$$\nX^{i}_{t_j} - X^{i}_{t_{j-1}} - \\Delta_n b^{i}_{t_{j-1}}(\\theta_{0,1}) = A^i_{j,1} + A^i_{j,2} + B^i_j\n$$\nas in \\eqref{def: BAnorm}, \\eqref{def: A12norm}, where\n$\\mathbb{E}_{t_{j-1}} [(A^{i}_{j,k})^{4}] = R^{i}_{t_{j-1}}(\\Delta_n^{2k})$, $k=1,2$, and $\\mathbb{E}_{t_{j-1}} [(B^{i}_j)^4] = R^{i}_{t_{j-1}}(\\Delta_n^6)$.\nWe note that \n$$ \n\\mathbb{E}_{t_{j-1}}[(A^{i_1}_{j,1})^2 (A^{i_2}_{j,1})^2] = \\Delta_n c^{i_1}_j (\\theta_{0,2}) \\Delta_n c^{i_2}_j (\\theta_{0,2}). \n$$ \nMoreover, we have\n\n\\begin{align*}\n\\mathbb{E}_{t_{j-1}} [ (A^{i_1}_{j,1})^2 A^{i_2}_{j,1} A^{i_2}_{j,2} ] \n= \nc^{i_1}_{t_{j-1}} (\\theta_{0,2}) a^{i_2}_{t_{j-1}} (\\theta_{0,2}) \nV^{i_1,i_2}_j,\n\\end{align*}\nwhere independence of Brownian motions together with It\\^o isometry implies \n\\begin{align}\nV^{i_1,i_2}_j &:= \\mathbb{E}_{t_{j-1}} \\Big[ (W^{i_1}_{t_j}-W^{i_1}_{t_{j-1}})^2 \\int_{t_{j-1}}^{t_j} d W^{i_2}_s \\int_{t_{j-1}}^{t_j} (a^{i_2}_s(\\theta_{0,2}) - a^{i_2}_{t_{j-1}}(\\theta_{0,2})) d W^{i_2}_s \\Big]\\nonumber\\\\\n&= \\int_{t_{j-1}}^{t_j} \\mathbb{E}_{t_{j-1}} [ (W^{i_1}_{t_j}-W^{i_1}_{t_{j-1}})^2 (a^{i_2}_t (\\theta_{0,2}) - a^{i_2}_{t_{j-1}}(\\theta_{0,2}))] d t.\\label{eq:ri1i2j}\n\\end{align}\nAssumption \\ref{as7} allows us to apply It\\^o's lemma to $a_t^{i_2}(\\theta_{0,2})$. We get that the conditional expectation in \\eqref{eq:ri1i2j} equals\n\n\\begin{align*}\n&\\mathbb{E}_{t_{j-1}} \\Big[ (W^{i_1}_{t_j}-W^{i_1}_{t_{j-1}})^2 \\int_{t_{j-1}}^t \\sum_{k{=1}}^{{N}} \\Big( b^k_s (\\theta_{0,1}) \\partial_{x_k} a^{i_2}_s (\\theta_{0,2}) + \\frac{1}{2} c^k_s (\\theta_{0,2}) \\partial^2_{x_k} a^{i_2}_s (\\theta_{0,2}) \\Big) d s \\Big] \\\\\n&\\qquad+ \\mathbb{E}_{t_{j-1}} \\Big[ (W^{i_1}_{t_j}-W^{i_1}_{t_{j-1}})^2 \\int_{t_{j-1}}^t \\sum_{k{=1}}^{{N}} a^{k}_s (\\theta_{0,2}) \\partial_{x_k} a^{i_2}_s (\\theta_{0,2}) d W_s^k \\Big].\n\\end{align*}\nThe first term is clearly a $R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^2)$ function. Regarding the second one, for $k \\neq i_1$, the independence of the Brownian motions makes it directly equal to $0$. For $k = i_1$, instead, we have \n$$\n\\mathbb{E}_{t_{j-1}} \\Big[ (W^{i_1}_{t_j}-W^{i_1}_{t_{j-1}})^2 \\int_{t_{j-1}}^t a^{i_1}_s (\\theta_{0,2}) \\partial_{x_{i_1}} a^{i_2}_s (\\theta_{0,2}) d W_s^{i_1} \\Big],\n$$\nwhere under \\ref{as7} we obtain \n$$\n\\partial_{x_{i_1}} a^{i_2}_s (\\theta_{0,2}) := \\partial_y \\tilde \n\\Big( X^{i_2}_s, \\frac{1}{N} \\sum_{l{=1}}^{{N}} \n(X^{i_2}_s,X^l_s) \\Big) \\frac{1}{N} \\partial_y \n(X^{i_2}_s,X^{i_1}_s)\n$$\nwith\n$\\partial_y \\tilde \n$, $\\partial_y \n$ having\npolynomial growth.\nUsing the Cauchy-Schwarz inequality, it follows that the above quantity  is upper bounded by \n\\begin{align*}\n&\\Big( 3\\Delta_n^2 \\mathbb{E}_{t_{j-1}} \\Big[\\Big( \\int_{t_{j-1}}^t a^{i_1}_s (\\theta_{0,2}) \\partial_{x_{i_1}} a^{i_2}_s (\\theta_{0,2}) d W_s^{i_1}\\Big)^2 \\Big] \\Big)^\\frac{1}{2} \\\\\n&\\qquad = \\Big( 3 \\Delta_n^2\n\\int_{t_{j-1}}^t \\mathbb{E}_{t_{j-1}} [ (  a^{i_1}_s (\\theta_{0,2}) \\partial_{x_{i_1}} a^{i_2}_s (\\theta_{0,2}) )^2 ] ds \\Big)^\\frac{1}{2} \n= \\frac{1}{N} R^{i_1, i_2}_{t_{j-1}} (\\Delta_n^\\frac{3}{2}).\n\\end{align*}\nIt implies\n\\begin{equation}\\label{eq:ito}\n\\mathbb{E}_{t_{j-1}}[(A^{i_1}_{j,1})^2 A^{i_2}_{j,1} A^{i_2}_{j,2}] = R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^3)  + \\frac{1}{N} R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^\\frac{5}{2}).\n\\end{equation}\nWe conclude that\n\\begin{align*}\n&\\mathbb{E}_{t_{j-1}} [ (X^{i_1}_{t_j} - X^{i_1}_{t_{j-1}} - \\Delta_n b^{i_1}_{t_{j-1}}(\\theta_{0,1}))^2 (X^{i_2}_{t_j} - X^{i_2}_{t_{j-1}} - \\Delta_n b^{i_2}_{t_{j-1}}(\\theta_{0,1}))^2]\\\\ \n&\\qquad = \\Delta_n c^{i_1}_j (\\theta_{0,2}) \\Delta_n c^{i_2}_j (\\theta_{0,2}) + R^{i_1,i_2}_{t_{j-1}}(\\Delta_n^3)  + \\frac{1}{N} R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^\\frac{5}{2}),    \n\\end{align*}\nwhence \n\\begin{equation}\\label{eq:condD12}\n\\mathbb{E}_{t_{j-1}}[D^{i_1}_j D^{i_2}_j] = R^{i_1,i_2}_{t_{j-1}}(\\Delta_n^3) + \\frac{1}{N} R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^\\frac{5}{2})    \n\\end{equation}\nif $i_1 \\neq i_2$. Finally, we plug \\eqref{eq:condD2}, \\eqref{eq:condD12} back into \\eqref{eq: norm theta2 start}, where \nuse of the conditions $N \\Delta_n \\to 0$, $\\Delta_n \\to 0$ and Lemma \\ref{l: Riemann} completes the proof of the convergence in \\eqref{e: norm theta2}.\\\\\n\n\n\\noindent\n$\\bullet$ Proof of \\eqref{e: negl theta}, second convergence. \\\\\nWe prove it for $r = 2$. We use the same notation as in \\eqref{eq: norm theta2 start} and rewrite the left hand side of \\eqref{e: negl theta} as \n\\begin{equation}\\label{eq:sumCD}\n\\frac{\\Delta_n^2}{N^2} \\sum_{j=1}^n \\sum_{i_1, i_2, i_3, i_4=1}^N \\Delta_n^{-4} C^{i_1}_{j,\\Tilde{h}} C^{i_2}_{j,\\Tilde{h}} C^{i_3}_{j,\\Tilde{h}} C^{i_4}_{j,\\Tilde{h}} \\mathbb{E}_{t_{j - 1}} [D_j^{i_1} D_j^{i_2} D_j^{i_3} D_j^{i_4}].    \\end{equation}\nWe have $\\mathbb{E}_{t_{j-1}}[(D^{i}_j)^4] = R^{i}_{t_{j-1}}(\\Delta_n^{4})$ and $\\operatorname{card}(I^c) = O(N^3)$, where $I$ denotes a set of all $i=(i_1,i_2,i_3,i_4) \\in \\{1,\\dots,N\\}^4$ such that $i_1,i_2,i_3,i_4$ are pairwise distinct. In \\eqref{eq:sumCD} the sum over $i \\in I^c$ converges to $0$ in $L^1$ and so in probability since $N\\Delta_n \\to 0$. In case $i \\in I$  we use the decomposition \n\\begin{align*}\nD^i_j &= (A^i_{j,1} + A^i_{j,2} + B^i_j)^2 - \\Delta_n c^i_{t_{j-1}}(\\theta_{0,2})\\\\\n&= (A^i_{j,2}+B^i_j)(2 A^i_{j,1}+A^i_{j,2} + B^i_j) + (A^i_{j,1})^2 - \\Delta_n c^i_{t_{j-1}} (\\theta_{0,2}).\n\\end{align*}\nWe note that\n\\begin{equation*}\n\\begin{gathered}\n\\mathbb{E}_{t_{j-1}} [((A^i_{j,1})^2- \\Delta_n c^i_{t_{j-1}}(\\theta_{0,2}))^4] = R^i_{t_{j-1}}(\\Delta_n^4),\\\\\n\\mathbb{E}_{t_{j-1}} [(A^i_{j,k})^8] = R^i_{t_{j-1}} (\\Delta_n^{4k}), \\ k=1,2, \\qquad\n\\mathbb{E}_{t_{j-1}} [(B^i_j)^8] = R^i_{t_{j-1}} (\\Delta_n^{12}).  \n\\end{gathered}\n\\end{equation*}\nMoreover, because of the independence of Brownian motions, we have\n\\begin{align*}\n\\mathbb{E}_{t_{j-1}} \\Big[ \\prod_{k=1}^4 ((A^{i_k}_{j,1})^2- \\Delta_n c^{i_k}_{t_{j-1}}(\\theta_{0,2})) \\Big] = 0\n\\end{align*}\nand in a similar manner as in \\eqref{eq:ito} under \\ref{as7} we have\n\\begin{align*}\n&\\mathbb{E}_{t_{j-1}} \\Big[ A^{i_1}_{j,2} A^{i_1}_{j,1} \\prod_{k=2}^4 ( (A^{i_k}_{j,1})^2 - \\Delta_n c^{i_k}_{t_{j-1}} (\\theta_{0,2}) ) \\Big]\\\\ \n&\\qquad = a^{i_1}_{t_{j-1}} (\\theta_{0,2}) \\prod_{k=2}^4 c^{i_{k}}_{t_{j-1}} (\\theta_{0,2}) \\int_{t_{j-1}}^{t_j} \\mathbb{E}_{t_{j-1}} \\Big[ (a^{i_1}_s (\\theta_{0,2}) - a^{i_1}_{t_{j-1}} (\\theta_{0,2})) \\prod_{l=2}^4 ((W^{i_{l}}_{t_j} - W^{i_{l}}_{t_{j-1}})^2 - \\Delta_n) \\Big] d s\\\\ \n&\\qquad= R_{t_{j-1}}^{i_1,i_2,i_3,i_4}(\\Delta_n^5) + \\frac{1}{N} R_{t_{j-1}}^{i_1,i_2,i_3,i_4} (\\Delta_n^{\\frac{9}{2}}),\n\\end{align*}\nwhence it follows\n$$\n\\mathbb{E}_{t_{j-1}} [D^{i_1}_j D^{i_2}_j D^{i_3}_j D^{i_4}_j] = R^{i_1,i_2,i_3,i_4}_{t_{j-1}} (\\Delta_n^5) + \\frac{1}{N} R_{t_{j-1}}^{i_1,i_2,i_3,i_4} (\\Delta_n^{\\frac{9}{2}}).\n$$\n We recall that $\\operatorname{card}(I) = O(N^4)$. Since $N\\Delta_n \\to 0$, $\\Delta_n \\to 0$, the sum over $i \\in I$ in \\eqref{eq:sumCD} converges to $0$ in $L^1$ and so in probability.\\\\\n\n\\noindent\n$\\bullet$ Proof of \\eqref{e: norm mixed}. \\\\\nWe rewrite the left hand side of \\eqref{e: norm mixed} as\n\\begin{equation}\\label{eq:mixed}\n\\frac{\\Delta_n^{\\frac{1}{2}}}{N} \\sum_{j=1}^n \\sum_{i_1,i_2=1}^N \\mathbb{E}_{t_{j-1}} [ (A^{i_1}_{j,1}+A^{i_1}_{j,2}+B^{i_1}_j) D^{i_2}_j ] \\Delta_n^{-1} C^{i_2}_{j,\\Tilde{h}} F^{i_1}_{j,h},\n\\end{equation}\nwhere\n\\begin{align*}\nD^{i}_j\n&= (A^{i}_{j,2} +B^{i}_j)(2 A^{i}_{j,1} + A^{i}_{j,2} +B^{i}_j) + (A^{i}_{j,1})^2 - \\Delta_n c^{i}_{t_{j-1}} (\\theta_{0,2})\n\\end{align*}\nwith the notations introduced above. We recall that \n$F^i_{j,h} = R^{i}_{t_{j-1}}(1)$,\n$C^i_{j,\\Tilde{h}} = R^{i}_{t_{j-1}}(1)$, \n$\\mathbb{E}_{t_{j-1}} [(B^{i}_j)^4] = R^{i}_{t_{j-1}}(\\Delta_n^6)$, $\\mathbb{E}_{t_{j-1}}[(A^{i}_{j,k})^4] = R^{i}_{t_{j-1}}(\\Delta_n^{2k})$, $k=1,2$, and so $\\mathbb{E}_{t_{j-1}}[((A^{i}_{j,1})^2 - \\Delta_n c^{i}_{t_{j-1}} (\\theta_{0,2}))^2] = R^{i}_{t_{j-1}}(\\Delta_n^2)$. We note that\n$$\n\\mathbb{E}_{t_{j-1}} [A^{i_1}_{j,1}((A^{i_2}_{j,1})^2 - \\Delta_n c^{i_2}_{t_{j-1}}(\\theta_{0,2}))] = 0\n$$\nfor all $i_1,i_2$.  This is a consequence of the independence of the Brownian motions for $i_1 \\neq i_2$, while for $i_1 = i_2$ it derives from the fact that the odd moments are centered. Hence, in case $i_1=i_2 = i$ the term $\\mathbb{E}_{t_{j-1}} [(A^{i}_{j,1})^2 A^{i}_{j,2}]$ makes the main contribution to\n$$\n\\mathbb{E}_{t_{j-1}} [ (A^{i}_{j,1}+A^{i}_{j,2}+B^{i}_j) D^{i}_j ] = R^{i}_{t_{j-1}}(\\Delta_n^2).\n$$\nNow we can see that the sum over $i_1=i_2$ in \\eqref{eq:mixed} converges to $0$ in $L^1$ and so in probability. In case $i_1 \\neq i_2$ we have\n$$\n\\mathbb{E}_{t_{j-1}} [A^{i_1}_{j,2}((A^{i_2}_{j,1})^2 - \\Delta_n c^{i_2}_{t_{j-1}}(\\theta_{0,2}))] = 0. \n$$\nMoreover,\n\\begin{align*}\n\\mathbb{E}_{t_{j-1}} [A^{i_1}_{j,1} A^{i_2}_{j,1} A^{i_2}_{j,2}]\n= a^{i_1}_{t_{j-1}} (\\theta_{0,2}) a^{i_2}_{t_{j-1}} (\\theta_{0,2}) \\int_{t_{j-1}}^{t_j} \\mathbb{E}_{t_{j-1}} [(W^{i_1}_{t_j} - W^{i_1}_{t_{j-1}}) (a^{i_2}_s (\\theta_{0,2}) - a^{i_2}_{t_{j-1}} (\\theta_{0,2}) ] ds.\n\\end{align*}\nThe application of It\\^o's lemma to $a_s^{i_2}(\\theta_{0,2})$ under \\ref{as7} similarly as in the proof of \\eqref{eq:ito} provides \n$$\\mathbb{E}_{t_{j-1}} [A^{i_1}_{j,1} A^{i_2}_{j,1} A^{i_2}_{j,2}] = R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^{\\frac 5 2}) + \\frac{1}{N} R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^{2}). $$\nWe conclude that\n$$\n\\mathbb{E}_{t_{j-1}} [(A^{i_1}_{j,1}+A^{i_1}_{j,2}+B^{i_1}_j) D^{i_2}_j] = R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^{\\frac 5 2})  + \\frac{1}{N} R^{i_1,i_2}_{t_{j-1}} (\\Delta_n^{2}).\n$$\nin case $i_1 \\neq i_2$. Hence, the sum over $i_1 \\neq i_2$ in \\eqref{eq:mixed} converges to $0$ in $L^1$ and so in probability when $N \\Delta_n \\to 0$, $\\Delta_n \\to 0$. \nThis concludes the proof of the asymptotic normality of $\n\\nabla_{\\theta}S^N_n (\\theta_0) {M^N_n}$. \n\\end{proof}\n\n\\subsection{Proof of Proposition \\ref{p: second derivatives contrast}}\n\\begin{proof}\nThe proof relies on the computation of the second derivatives of the contrast function. We have that, for any $k,l = 1, ... , p_1$,\n\\begin{align*}\n\\partial_{\\theta_{1, k}} \\partial_{\\theta_{1, l}} S_n^N (\\theta) = 2 \\sum_{i = 1}^N \\sum_{j = 1}^n \\Big\\{ \\Delta_n &\\frac{\\partial_{\\theta_{1, k}} b^i_{t_{j - 1}}(\\theta_1) \\partial_{\\theta_{1, l}} b^i_{t_{j - 1}}(\\theta_1)}{c^i_{t_{j - 1}}(\\theta_2)}\\\\ \n- &\\frac{\\partial_{\\theta_{1, k}} \\partial_{\\theta_{1, l}} b^i_{t_{j - 1}}(\\theta_1)}{c^i_{t_{j - 1}}(\\theta_2)} (X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_1) )\\Big\\},    \n\\end{align*}\nwhere the last factor can further be decomposed into $\\Delta_n (b^i_{t_{j-1}}(\\theta_{0,1}) - b^i_{t_{j-1}}(\\theta_1))$ and $X^i_{t_{j}} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}}(\\theta_{0,1})$. We can see that $\\partial_{\\theta_{1, k}} \\partial_{\\theta_{1, l}} S_n^N (\\theta)\/N$ converges to\n\\begin{align}\\label{def:B11}\n\\Sigma^{(1)}_{kl} (\\theta) := 2 \\int_0^1 \\int_{\\mathbb{R}}  \\Big\\{ &\\frac{\\partial_{\\theta_{1, k}} b(\\theta_1, x,\\bar \\mu_t) \\partial_{\\theta_{1, l}} b(\\theta_1, x,\\bar \\mu_t)}{c(\\theta_2, x,\\bar \\mu_t)}  \\\\\n- &\\frac{\\partial_{\\theta_{1, k}} \\partial_{\\theta_{1, l}} b(\\theta_1, x,\\bar \\mu_t)}{c(\\theta_2, x,\\bar \\mu_t)} (b(\\theta_{0,1}, x,\\bar \\mu_t) - b(\\theta_1, x,\\bar \\mu_t))  \\Big\\} \\bar \\mu_t (d x) d t \\nonumber\n\\end{align}\nuniformly in $\\theta$ in probability. Indeed, the proof follows along the lines of the proof of \\eqref{lim:con2}. We refer to Steps 3, 4 of the proof of Lemma \\ref{lemma:cns}, where in  \\eqref{def:INnrhoNn} in $I^N_n(\\theta)$, $\\rho^N_n (\\theta)$  it is enough to replace the functions $h(\\theta,\\cdot)$ {and} $g(\\theta,\\cdot)$ with the integrand of \\eqref{def:B11} and $\\partial_{\\theta_{1, k}} \\partial_{\\theta_{1, l}} b (\\theta_1,\\cdot) \/c(\\theta_2,\\cdot)$ respectively, and to check them for the respective conditions. We note that both functions have polynomial growth. Moreover, the integrand in \\eqref{def:B11} is locally Lipschitz continuous, which allows us to apply Lemma~\\ref{l: Riemann} and yields the convergence in probability of the sequence $\\partial_{\\theta_{1,k}} \\partial_{\\theta_{1, l}} S_n^N (\\theta)\/N$ for every $\\theta$. To get tightness in $(C(\\Theta{; \\mathbb{R}}), \\| \\cdot\\|_\\infty)$, we use that uniformly in $\\theta$ the partial derivatives with respect to $\\theta_{i',j'}$,  $j'=1,\\dots,p_{i'}$, $i'=1,2$, of the two functions have polynomial growth.\n\nIn the same way as above we get that for any $k = 1, ... , p_1$, $l = 1, ... , p_2$, once multiplied by $\\sqrt{\\Delta_n}\/N$, \n$$\n\\partial_{\\theta_{1,k}} \\partial_{\\theta_{2,l}} S_n^N (\\theta) = 2 \\sum_{i = 1}^N \\sum_{j = 1}^n \\frac{\\partial_{\\theta_{1,k}} b^i_{t_{j - 1}}(\\theta_1) \\partial_{\\theta_{2,l}} c^i_{t_{j - 1}}(\\theta_2)}{(c^i_{t_{j - 1}}(\\theta_2))^2} (X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_1) ),\n$$\nconverges to $\n0$ uniformly in $\\theta$ in probability. \n\nFinally, we have that for any $k,l = 1,\\dots, p_2$, \n\\begin{align*}\n\\partial_{\\theta_{2,k}} \\partial_{\\theta_{2,l}} S^N_n (\\theta) = \\sum_{i=1}^N \\sum_{j=1}^n &\\Big\\{\n\\frac{\\partial_{\\theta_{2,k}} \\partial_{\\theta_{2,l}} c^i_{t_{j-1}}(\\theta_2) c^i_{t_{j-1}}(\\theta_2) - \\partial_{\\theta_{2,k}} c^i_{t_{j-1}}(\\theta_2) \\partial_{\\theta_{2,l}} c^i_{t_{j-1}}(\\theta_2)}{(c^i_{t_{j-1}}(\\theta_2))^2}\\\\\n&+ \\frac{2 \\partial_{\\theta_{2,k}} c^i_{t_{j-1}}(\\theta_2)\\partial_{\\theta_{2,l}} c^i_{t_{j-1}}(\\theta_2)-\\partial_{\\theta_{2,k}} \\partial_{\\theta_{2,l}} c^i_{t_{j-1}}(\\theta_2)c^i_{t_{j-1}}(\\theta_2)}{\\Delta_n (c^i_{t_{j-1}}(\\theta_2))^3}\\\\  \n&\\times (X^i_{t_j}-X^i_{t_{j-1}}- \\Delta_n b^i_{t_{j-1}}(\\theta_1))^2\n\\Big\\},\n\\end{align*}\nwhere the last factor can further be decomposed into $(X^i_{t_j}-X^i_{t_{j-1}}- \\Delta_n b^i_{t_{j-1}}(\\theta_1))^2 - \\Delta_n c^i_{t_{j-1}}(\\theta_{0,2})$ and $\\Delta_n c^i_{t_{j-1}}(\\theta_{0,2})$. We note that $ (\\Delta_n\/N) \\partial_{\\theta_{2,k}} \\partial_{\\theta_{2,l}} S^N_n(\\theta)$ converges to\n\\begin{align*}\n\\Sigma^{(2)}_{kl} (\\theta) := \\int_0^T \\int_{\\mathbb{R}} &\\Big\\{  \\frac{\\partial_{\\theta_{2,k}} \\partial_{\\theta_{2,l}} c(\\theta_2, x,\\bar \\mu_t)c(\\theta_2, x,\\bar \\mu_t)-\\partial_{\\theta_{2,k}} c(\\theta_2, x,\\bar \\mu_t)\\partial_{\\theta_{2,l}} c(\\theta_2, x,\\bar \\mu_t)}{c(\\theta_2, x,\\bar \\mu_t)^2}\\\\\n&+ \\frac{2 \\partial_{\\theta_{2,k}} c(\\theta_2, x,\\bar \\mu_t)\\partial_{\\theta_{2,l}} c(\\theta_2, x,\\bar \\mu_t) - \\partial_{\\theta_{2,k}} \\partial_{\\theta_{2,l}} c(\\theta_2, x,\\bar \\mu_t)c(\\theta_2, x,\\bar \\mu_t)}{c(\\theta_2, x,\\bar \\mu_t)^3}\\\\ &\\times c(\\theta_{0,2}, x,\\bar \\mu_t) \\Big\\} \\bar \\mu_t (d x) d t\n\\end{align*}\nuniformly in $\\theta$ in probability. We will prove the uniform in $\\theta$ convergence to the second term of \n\\Sigma^{(2)}_{kl}(\\theta)$ only: \n\\begin{equation}\\label{lim:B221}\n\\sum_{j=1}^n \\chi^N_{n,j} (\\theta) \\xrightarrow{\\P} \\tilde\n\\Sigma^{(2)}_{kl} (\\theta) := \\int_0^T \\int_{\\mathbb{R}} \\tilde f (\\theta_2, x,\\bar \\mu_t) c(\\theta_{0,2}, x,\\bar\\mu_t) \\bar \\mu_t (d x) d t,    \n\\end{equation}\nwhere \n$$\n\\chi^N_{n,j}(\\theta) = \\frac{1}{N} \\sum_{i=1}^N \\tilde f^i_{t_{j-1}} (\\theta_2) (X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}}(\\theta_1))^2\n$$\nand function $\\tilde f : \\Theta_2 \\times \\mathbb{R} \\times {\\cal P} \\to \\mathbb{R}$ is given by $(2 (\\partial_{\\theta_{2,k}} c) (\\partial_{\\theta_{2,l}} c) - (\\partial_{\\theta_{2,k}} \\partial_{\\theta_{2,l}} c)  c) \/ c^3$.\nFor every $\\theta$ the convergence in \\eqref{lim:B221} follows from\n$$\n\\sum_{j=1}^n \\mathbb{E}_{t_{j-1}} [\\chi^N_{n,j}(\\theta)] \\xrightarrow{\\P} \\tilde\n\\Sigma^{(2)}_{kl} (\\theta), \\qquad \\sum_{j=1}^n \\mathbb{E}_{t_{j-1}}[(\\chi^N_{n,j}(\\theta))^2] \\xrightarrow{\\P} 0\n$$\nby  \\cite[Lemma 9]{GenJac93}. Indeed, the above relations hold, because by  Lemma \\ref{l: conditional expectation}(1), \n$$\n\\mathbb{E}_{t_{j-1}} [\\chi^N_{n,j} (\\theta)] = \\frac{1}{N} \\sum_{i=1}^N \\tilde f^i_{t_{j-1}} (\\theta_2) (\\Delta_n c^i_{t_{j-1}} (\\theta_{0,2}) + R^i_{t_{j-1}}(\\Delta_n^{3\/2})),\n$$ \nby Jensen's inequality and Lemma \\ref{l: conditional expectation}(2),\n$$\n\\mathbb{E}_{t_{j-1}} [(\\chi^N_{n,j}(\\theta))^2] \\le \\frac{1}{N} \\sum_{i=1}^N (\\tilde f^i_{t_{j-1}}(\\theta_2))^2 R^i_{t_{j-1}}(\\Delta_n^2),\n$$\nby polynomial growth of $\\partial^{i'}_{\\theta_{2,j'}} c (\\theta_2,\\cdot)$, $i'=0,1,2$, $j'=1,\\dots,p_2$, \\ref{as3} and Point 1.\\ of Lemma~\\ref{l: moments},\n$$\n(\\tilde f^i_{t_{j-1}}(\\theta_2))^2 = R^i_{t_{j-1}}(1)\n$$\nThe tightness in $(C(\\Theta{;\\mathbb{R}}), \\| \\cdot \\|_\\infty)$ follows from \n$\\mathbb{E} [ \\sup_\\theta \\|\\nabla_\\theta \\sum_{j{=1}}^{{n}} \\chi^N_{n,j}(\\theta) \\| ] = O(1)$. Indeed, we have  \n\\begin{align*}\n\\nabla_{\\theta_1} \\chi^N_{n,j} (\\theta) &= - 2 \\frac{\\Delta_n}{N} \\sum_{i=1}^N \\nabla_{\\theta_1} b^i_{t_{j-1}} (\\theta_1)  \\tilde f^i_{t_{j-1}} (\\theta_2) (X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}}(\\theta_1) ),\\\\\n\\nabla_{\\theta_2} \\chi^N_{n,j} (\\theta) &= \\frac{1}{N} \\sum_{i=1}^N \\nabla_{\\theta_2} \\tilde f^i_{t_{j-1}}(\\theta_2) (X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}}(\\theta_1))^2,\n\\end{align*}\nwhere by polynomial growth of $\\sup_{\\theta_1} \\|\\nabla_{\\theta_1} b (\\theta_1,\\cdot)\\|$, $\\sup_{\\theta_2} |\\partial_{\\theta_{2,j'}}^{i'} c (\\theta_2,\\cdot)|$, $i'=0,1,2,3$, $j'=1,\\dots,p_2$, and \\ref{as3},\n$$\n\\sup_\\theta \\|\\nabla_{\\theta_{1}} b^i_{t_{j-1}}(\\theta_1)  \\tilde f^i_{t_{j-1}} (\\theta_2) \\| = R^i_{t_{j-1}} (1), \\qquad \\sup_{\\theta_2} \\| \\nabla_{\\theta_2} \\tilde f^i_{t_{j-1}}(\\theta_2)\\| = R^i_{t_{j-1}} (1)\n$$\nand \n$$\n\\sup_{\\theta_1} |X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}}(\\theta_1)| \\le  |X^i_{t_j}-X^i_{t_{j-1}}| + \\Delta_n \\sup_{\\theta_1} |b^i_{t_{j-1}}(\\theta_1)|\n$$\nwith $\\sup_{\\theta_1} |b^i_{t_{j-1}}(\\theta_1)| = R^i_{t_{j-1}}(1)$. Finally, we have\n$\\mathbb{E} [|X^i_{t_j}-X^i_{t_{j-1}}|^4] \\le C \\Delta_n^2$ uniformly in $i,j$ and $N,n$ by Lemma~\\ref{l: moments}(2).\n\nWe conclude that the matrix \n$\\Sigma^N_n (\\theta)$ converges to $\\Sigma(\\theta) =\n\\operatorname{diag}(\\Sigma^{(1)}(\\theta),\\Sigma^{(2)}(\\theta))$ uniformly in $\\theta$ and so at $\\theta = \\theta_0$ in probability. Hence,\n\\begin{align*}\n\\| \\Sigma^N_n (\\theta_0 + s (\\hat \\theta^N_n - \\theta_0)) - \\Sigma^N_n (\\theta_0) \\| \n&\\le  o_\\P (1) + \\| \\Sigma(\\theta_0 + s (\\hat \\theta^N_n - \\theta_0)) - \\Sigma(\\theta_0) \\|,\n\\end{align*}\nwhere  the uniform convergence in probability (in $s$) of the last term to $0$ follows from continuity of $\\Sigma (\\theta)$ at $\\theta = \\theta_0$ and consistency of the estimator sequence $\\hat \\theta^N_n$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of technical results}{\\label{s: proof technical}}\n\n\\subsection{Proof of Lemma \\ref{l: moments}}\n\n\\begin{proof} \n\\noindent\nProof of Lemma \\ref{l: moments}(1). \\\\\nWe have, for any $i = 1, \\dots , N$, $0 \\le t \\le T$, $p \\ge 2$, \n\\begin{align*}\n\\mathbb{E}[|X_t^i|^p] & \\le \\mathbb{E} \\Big[ \\Big|X_0^i + \\int_0^t b_u^i(\\theta_{0,1}) du + \\int_0^t a_u^i(\\theta_{0,2}) dW_u^i \\Big|^p \\Big] \\\\\n& \\le C \\Big( \\mathbb{E}[|X_0^i|^p] + t^{p - 1} \\int_0^t \\mathbb{E}[|b_u^i(\\theta_{0,1})|^p] d u +  t^{\\frac{p}{2} - 1} \\int_0^t \\mathbb{E}[|a_u^i(\\theta_{0,2})|^p] d u \\Big),\n\\end{align*}\nwhere we have used the Burkholder-Davis-Gundy and Jensen inequalities. We observe that, as a consequence of the lipschitzianity gathered in \\ref{as2}, for the true value of the parameter both coefficients are upper bounded by $C(1 + |X_u^i| + W_2(\\mu_u, \\delta_0))$.\nDue to \nJensen's inequality, we have \n$$\\mathbb{E}[W_2^p(\\mu_u, \\delta_0)] \\le \\frac{1}{N} \\sum_{j = 1}^N \\mathbb{E} [ |X_u^j|^p ] \n= \\mathbb{E}[|X_u^i|^p].\n$$\nThe last identity follows from the fact that the particles are equally distributed.\nWe obtain\n\\begin{equation}{\\label{eq: end for p larger than 2}}\n\\mathbb{E}[|X_t^i|^p] \\le C \\Big( \\mathbb{E}[|X_0^i|^p] +  (t^{p - 1} + t^{\\frac{p}{2} - 1} ) \\Big( t +  2\\int_0^t  \\mathbb{E}[|X_u^i|^p] du \\Big)\n\\Big).\n\\end{equation}\nWe infer by Gronwall's lemma that \n$$\n\\mathbb{E}[|X_t^i|^p] \\le \nC( \\mathbb{E}[|X_0^i|^p] +  {T^{p}} +  {T^{\\frac{p}{2}}} ) \\exp ( {C' (T^{p} +  T^{\\frac{p}{2}})} ).\n$$\nAs the constants do not depend on $t \\le T$ and $\\mathbb{E}[|X_0^i|^p] < \\infty$ by \\ref{as1}, we have the wanted result for $p \\ge 2$. \nThen, by a Jensen argument and the boundedness of the moments for $p \\ge 2$, it follows the result also for $p< 2$.\n\\\\\n\n\\noindent\nProof of Lemma \\ref{l: moments}(2). \\\\\nWe have for any $0 \\le s < t \\le T$, $p \\ge 2$,\n\\begin{align*}\n\\mathbb{E}[|X_t^i - X_s^i|^p] & = \\mathbb{E} \\Big[ \\Big|\\int_s^t b_u^i(\\theta_{0,1}) du + \\int_s^t a_u^i(\\theta_{0,2}) dW_u^i \\Big|^p \\Big] \\\\\n& \\le C \\Big( (t-s)^{p - 1}\\int_s^t \\mathbb{E}[|b^i_u(\\theta_{0,1})|^p] du +  (t - s)^{\\frac{p}{2}-1} \\int_s^t \\mathbb{E}[|a^i_u(\\theta_{0,2})|^p] ds \\Big),\n\\end{align*}\nwhere we have used the Jensen and Burkholder-Davis-Gundy inequalities. Because of \\eqref{eq: pol growth} and the just shown Lemma \\ref{l: moments}(1), the result follows letting $t-s \\le 1$. \\\\\n\n\\noindent\nProof of Lemma \\ref{l: moments}(3). \\\\\nAccording to the definition of $R^i_s(1)$, we want to evaluate the $L^q$ norm of $\\mathbb{E}_s [|X^i_t-X^i_s|^p]$.\nFor any $0 \\le s < t \\le T$ such that $t-s\\le 1$ and $p \\ge 2$, $q \\ge 1$, \n$$\n\\mathbb{E} \\big[ \\big| \\mathbb{E}_s [ | X^i_t-X^i_s|^p ] \\big|^q \\big]^{\\frac 1 q} \\le \\mathbb{E} [ |X^i_t-X^i_s|^{pq} ]^{\\frac 1 q}  \\le C (t-s)^{\\frac{p}{2}}\n$$\nfollows by conditional Jensen's inequality and Lemma \\ref{l: moments}(2).\n\\\\\n\n\\noindent\nProof of Lemma \\ref{l: moments}(4).\\\\ \nThis is a straightforward consequence of\n\\begin{equation}\\label{eq: bound W2}\nW_2^p (\\mu_t,\\mu_s) \\le \\Big( \\frac{1}{N} \\sum_{j=1}^N |X^j_t-X^j_s|^2 \\Big)^{\\frac{p}{2}} \\le \\frac{1}{N} \\sum_{j=1}^N |X^j_t - X^j_s|^p\n\\end{equation}\nby Jensen's inequality for any $0 \\le s < t \\le T$ such that $t-s \\le 1$, $p \\ge 2$ and Lemma \\ref{l: moments}(2).\\\\\n\n\\noindent\nProof of Lemma \\ref{l: moments}(5).\\\\ \nIt follows directly from \\eqref{eq: bound W2}, where we use Minkowski's inequality as follows:\n$$\n\\mathbb{E} \\big[ \\big| \\mathbb{E}_s [ W^p_2 (\\mu_t,\\mu_s) ] \\big|^q \\big]^{\\frac{1}{q}} \\le \\frac{1}{N} \\sum_{j=1}^N \\mathbb{E} \\big[ \\big| \\mathbb{E}_s [ |X^j_t-X^j_s|^{pq} ] \\big| \\big]^{\\frac{1}{q}},  \n$$\nand then Lemma \\ref{l: moments}(3).\n\\end{proof}\n\n\n\\subsection{Proof of Lemma \\ref{l: Riemann}}\n\\begin{proof} \n\\noindent \nStep 1. We prove that \n$$\n\\frac{\\Delta_n}{N} \\sum_{i=1}^N \\sum_{j=1}^n f(X^{i,N}_{t_{j{-1,n}}}, \\mu^N_{t_{j{-1,n}}}) - \\frac{1}{N} \\sum_{i=1}^N \\int_0^T f(X^{i,N}_s, \\mu^N_s) d s \\xrightarrow{L^1} 0.\n$$\nHere we note $\\Delta_n = t_{j,n}-t_{j-1,n}$ and decompose the  above integral into integrals over $[t_{j-1,n},t_{j,n})$. We can see that the above convergence\nfollows from \n$$\n\\sum_{j=1}^n \\int_{t_{j-1,n}}^{t_{j,n}} \\mathbb{E} [ | f(X^{i,N}_{t_{j-1,n}}, \\mu^N_{t_{j-1,n}}) - f(X^{i,N}_s, \\mu^N_s) | ] d s \\to 0, \\qquad N,n \\to \\infty,\n$$\nfor fixed $i$,\nwhich in turn follows using\nthe condition \\eqref{cond:Riemann}, Cauchy-Schwarz inequality and moment bounds in  Lemma \\ref{l: moments}(1), (2) and (4). In particular, $\\mathbb{E} [|X^{i,N}_{t_{j-1,n}} - X^{i,N}_s|^2] \\le C \\Delta_n$ for  all $t_{j-1,n} \\le s \\le t_{j,n}$, $j$ and $n,N$. \\\\\n\n\\noindent\nStep 2. Next, let us prove that \n$$\n\\frac{1}{N} \\sum_{i=1}^N \\int_{{0}}^{{T}} f(X^{i,N}_s, \\mu^N_s) d s - \\frac{1}{N} \\sum_{i=1}^N \\int_0^T f(\\bar X^i_s, \\bar \\mu_s) d s \\xrightarrow{L^1} 0, \\qquad N \\to \\infty,\n$$\n{where each $(\\bar X^i_t)_{t\\in [0,T]}$ satisfies \\eqref{eq: McK} with $(W_t)_{t \\in [0,T]} = (W^i_t)_{t \\in [0,T]}$ and $\\bar X^i_0 = X^{i,N}_0$.}\nIt suffices to prove\n$$\n\\int_0^T \\mathbb{E} [ | f(X^{i,N}_s, \\mu^N_s ) - f(\\bar X^i_s, \\bar \\mu_s ) |] d s \\to 0,\n$$\nwhere $i$ is fixed and the integral is over a bounded interval.\nFor this purpose, let us use again the condition \\eqref{cond:Riemann} and the Cauchy-Schwarz inequality.  Following the same arguments as in the proof of Lemma \\ref{l: moments}(1) and Gronwall lemma, it is easy to show that for all $p>0$ there exists $ C_p > 0$ such that for all $s, i, N$ it holds $\\mathbb{E} [|\\bar X^i_s|^p ] <  C_p$.\n Moreover we have\n$$\n\\mathbb{E} [ |X^{i,N}_s - \\bar X^i_s|^2 ] \\le  \\frac{C}{\\sqrt{N}}\n$$ \nfor all $0 \\le s \\le {T}$ and $i, N$,  thanks to Theorem 3.20 in \\cite{Review prop}, based on Theorem 1 of \\cite{145 review}. We remark that, from the boundedness of the moments, the quantity $q$ appearing in the statement of Theorem 3.20 in \\cite{Review prop} is larger than $4$. Hence, the rate $N^{- (q - 2)\/q}$ is negligible compared to $N^{- 1\/2}$.\nThe propagation of chaos stated above implies\n$$\n\\mathbb{E} [ W^2_2 (\\mu^N_s, \\bar \\mu_s) ] \\le \\frac{C}{\\sqrt{N}}.\n$$ \nIndeed, to get the last relation, we introduce the empirical measure $\\bar \\mu^N_s = N^{-1} \\sum_{i=1}^N \\delta_{\\bar X^i_s}$ of the independent particle system at time $s$ and use the triangle inequality for $W_2$. Then\n$$\n\\mathbb{E} [ W_2^2 (\\mu^N_s, \\bar \\mu^N_s) ] \\le \\frac{1}{N} \\sum_{i=1}^N \\mathbb{E} [|X^{i,N}_s-\\bar X^i_s|^2]\n\\le  \\frac{C}{\\sqrt{N}},\n$$\nwhereas Theorem~1 of {\\cite{145 review}} implies\n$$\n\\mathbb{E} [W^2_2(\\bar \\mu^N_s,\\bar \\mu_s)] \\le \\frac{C}{\\sqrt{N}}.\n$$\n\n\\noindent\nStep 3. Finally, the law of large numbers gives \n$$\n\\frac{1}{N} \\sum_{i=1}^N \\int_0^T f(\\bar X^i_s, \\bar \\mu_s) d s \\xrightarrow{\\P} \\mathbb{E} \\Big[ \\int_0^T f(\\bar X_s, \\bar \\mu_s) d s \\Big],  \\qquad N \\to \\infty.\n$$\n\\end{proof}\n\n\n\\subsection{Proof of Lemma \\ref{l: conditional expectation}}\n\\begin{proof}\nWe use the same notation as before. \\\\ \n\\\\\nProof of Lemma \\ref{l: conditional expectation}(2).  We decompose $X^i_{t_j} - X^i_{t_{j-1}} - \\Delta_n b^i_{t_{j-1}}(\\theta_{0,1})$ into $A^i_{j,1}$ and $H^i_{j,2} := A^i_{j,2} + B^i_j$, where\n\\begin{equation}\n\\begin{gathered}\nA^i_{j,1} := \\int^{t_j}_{t_{j-1}} a^i_{t_{j-1}}(\\theta_{0,2}) d W^i_s, \\qquad \nA^i_{j,2} := \\int^{t_j}_{t_{j-1}} (a^i_s(\\theta_{0,2})-a^i_{t_{j-1}}(\\theta_{0,2})) d W^i_s,\\\\\nB^i_j := \\int^{t_j}_{t_{j-1}} (b^i_s(\\theta_{0,1})-b^i_{t_{j-1}}(\\theta_{0,1})) d s,\n\\end{gathered}\\label{eq: dynamics X}\n\\end{equation}\nare the same as in {\\eqref{def: BAnorm}, \\eqref{def: A12norm}}. \n\nFirstly, we will show that for any $p \\ge 2$,\n\\begin{equation}\\label{eq: bound R}\n\\mathbb{E} [|H^i_{j,2}|^p] \\le C \\Delta_n^p.\n\\end{equation}\nUsing Jensen's inequality and Lipschitz continuity of $b(\\theta_1,\\cdot)$ we get\n\\begin{align}\n\\mathbb{E} [|B^i_j|^p] & \\le  \\mathbb{E} \\Big[ \\Delta_n^{p - 1} \\int_{t_{j-1}}^{t_j} |b^i_{s} (\\theta_{0,1})- b^i_{t_{j - 1}} (\\theta_{0,1})|^p d s \\Big] \\nonumber \\\\\n& \\le C \\Delta_n^{p - 1} \\int_{t_{j-1}}^{t_j} ( \\mathbb{E} [|X_s^i - X_{t_{j - 1}}^i|^p] + \\mathbb{E} [W_2^p(\\mu_s, \\mu_{t_{j - 1}})] ) ds \\nonumber \\\\\n& \\le C \\Delta_n^{p-1} \\int_{t_{j-1}}^{t_j} (s-t_{j-1})^{\\frac p 2} d s = C \\Delta_n^{\\frac{3}{2}p},\\label{ineq: bound Bp} \n\\end{align}\nwhere the last inequality follows from Lemma \\ref{l: moments}(2) and (4). \nFurther use of the Burkholder-Davis-Gundy and Jensen inequalities gives  \n\\begin{align}\n\\mathbb{E} [ |A^i_{j,2}|^p] &\\le C \\mathbb{E} \\Big[ \\Big(  \\int_{t_{j-1}}^{t_j} | a^i_s (\\theta_{0,2})-a^i_{t_{j-1}}(\\theta_{0,2}) |^2 d s \\Big)^{\\frac p 2} \\Big]\\nonumber\\\\ \n&\\le C \\Delta_n^{\\frac{p}{2}-1} \\int_{t_{j-1}}^{t_j} \\mathbb{E} [ | a^i_s (\\theta_{0,2})-a^i_{t_{j-1}}(\\theta_{0,2}) |^p ] d s\\nonumber\\\\\n&\\le C \\Delta_n^p,\\label{e: bound A2}\n\\end{align}\nwhere the last inequality follows from Lipschitz continuity of $a(\\theta_2,\\cdot)$ and  Lemma~\\ref{l: moments}(2) and (4) as so does \\eqref{ineq: bound Bp}. Hence, we have\nshown \\eqref{eq: bound R}.\n\nNext, we have\n\\begin{equation}\\label{e: bound A1}\n\\mathbb{E} [|A^i_{j,1}|^p] = C \\Delta_{{n}}^{\\frac{p}{2}}  \\mathbb{E} [ |a^i_{t_{j-1}} (\\theta_{0,2})|^p ] \\le C \\Delta_n^{\\frac p 2}\n\\end{equation} \nsince we know the absolute moments of a centered normal distribution and have linear growth of $a(\\theta_{0,2},\\cdot)$, moment bounds in Lemma \\ref{l: moments}(1). In particular, we note\n$$ \n\\mathbb{E}_{t_{j-1}}[(A^i_{j,1})^4] = 3 \\Delta_n^2 (c^2)^i_{t_{j-1}}(\\theta_{0,2}).\n$$\nFinally, we have\n\\begin{equation}\\label{eq:X4}\n\\mathbb{E}_{t_{j - 1}}[(X^i_{t_j} - X^i_{t_{j - 1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_{0,1}))^4]\n= 3 \\Delta_n^2 (c^2)^i_{t_{j-1}}(\\theta_{0,2}) + \\sum_{k = 0}^3 \\binom{4}{k}  \\mathbb{E}_{t_{j - 1}} [ ( A^i_{j,1} )^k (H^i_{j,2})^{4-k} ].\n\\end{equation}\nFor any $k = 0,1,2,3$ and $q \\ge 1$, using Jensen's inequality for conditional expectation, we get\n\\begin{align*}\n\\mathbb{E} \\big[ \\big| \\mathbb{E}_{t_{j-1}} [ (A^i_{j,1})^k (H^i_{j,2})^{4-k} ] \\big|^q \\big] &\\le \\mathbb{E} \\big[ \\big| (A^i_{j,1})^k (H^i_{j,2})^{4-k} \\big|^q \\big] \n\\le C \\Delta_n^{(4-\\frac{k}{2})q},\n\\end{align*}\nwhere the last inequality follows from \\eqref{e: bound A2}, \\eqref{eq: bound R} using Cauchy-Schwarz inequality. Hence, the\nterm {converging to $0$ in $L^q$ at the slowest rate}\nis the one for which $k= 3$. \nWe therefore obtain that the remaining sum on the right hand side of \\eqref{eq:X4} is an $R^{{i}}_{t_{j - 1}}(\\Delta_n^{\\frac{5}{2}})$ function.\\\\\n\\\\\nProof of Lemma \\ref{l: conditional expectation}(3).  This follows directly from \\eqref{ineq: bound Bp} by decomposing the dynamics of $X^i$ as in \\eqref{eq: dynamics X} and remarking that the stochastic integral is centered. \\\\ \\\\\nProof of Lemma \\ref{l: conditional expectation}(1). We decompose\n$X^i_{t_j} - X^i_{t_{j - 1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_{0,1})$ into \n$$\nA^i_j := A^i_{j,1} + A^i_{j,2} = \\int_{t_{j-1}}^{t_j} a^i_{s} (\\theta_{0,2}) dW^i_s,\n$$\nand $B^i_j$ satisfying respectively $\\mathbb{E} [|A^i_j|^{2p}] \\le C \\Delta_n^{p}$ and $\\mathbb{E} [|B^i_j|^{2p}] \\le C \\Delta_n^{3p}$, whence $\\mathbb{E} [ |A^i_j B^i_j|^p ] \\le C \\Delta_n^{2p}$ for any $p \\ge 1$, see \\eqref{ineq: bound Bp}-\\eqref{e: bound A1}. We conclude that\n\\begin{align*}\n\\mathbb{E}_{t_{j - 1}}[(X^i_{t_j} - X^i_{t_{j - 1}} - \\Delta_n b^i_{t_{j - 1}}(\\theta_{0,1}))^2]\n= \\int_{t_{j-1}}^{t_j} \\mathbb{E}_{t_{j - 1}}[c^i_{s} (\\theta_{0,2})] ds + R^{{i}}_{t_{j-1}}(\\Delta_n^2).    \n\\end{align*}\nWe are left to show that we can replace $\\mathbb{E}_{t_{j - 1}}[c^i_{s} (\\theta_{0,2})]$ with $c^i_{t_{j-1}} (\\theta_{0,2})$ and that the remaining integral is an $R^{{i}}_{t_{j-1}}(\\Delta_n^2)$ function.\n\nUnder A7 we have that for any $i$, \n$$\n(x_1,\\dots,x_N) \\mapsto c \\Big(\\theta_{0,2}, x_i,\\frac{1}{N}\\sum_{j=1}^N \\delta_{x_j} \\Big) = {\\tilde a^2\n} \n\\Big( x_i, \\frac{1}{N} \\sum_{j=1}^N \n(x_i,x_j) \\Big) =: g^i(x_1,\\dots,x_N)\n$$\nis a\n{twice continuously differentiable} function\nfrom $\\mathbb{R}^N$ to $\\mathbb{R}$.\nGiven a vector \n$(X^1_s,\\dots,X^N_s)_{s \\in [0,T]}$ of processes, we denote\n$$\n(\\partial^l_{x_k} c)^i_s (\\theta_{0,2}) := \\partial^l_{x_k} g^i(X^1_s, \\dots, X^N_s).\n$$ \nWe apply the multidimensional It\\^o's formula to $g^i(X^1_s,\\dots,X^N_s) = c^i_s (\\theta_{0,2})$\nas follows: \n\\begin{align*}\nc^i_s(\\theta_{0,2}) - c^i_{t_{j-1}} (\\theta_{0,2}) \n= \\sum_{k=1}^N \\int_{t_{j-1}}^s \\Big( &(\\partial_{x_k} c)^i_u (\\theta_{0,2}) b^k_u (\\theta_{0,1}) + \\frac{1}{2} (\\partial^2_{x_k} c)^i_u (\\theta_{0,2}) c^k_u (\\theta_{0,2}) \\Big) d u\\\\\n+ \\sum_{k=1}^N \\int_{t_{j-1}}^s &(\\partial_{x_k} c)^i_u (\\theta_{0,2}) a^k_u (\\theta_{0,2}) d W^k_u.\n\\end{align*}\nSince the driving $(W^1_u, \\dots, W^N_u)_{u \\in[t_{j-1},s]}$ is independent of ${\\cal F}^{{N}}_{t_{j-1}}$, it follows that\n\\begin{align}\n\\mathbb{E}_{t_{j-1}} [&c^i_s(\\theta_{0,2})] - c^i_{t_{j-1}} (\\theta_{0,2}) \\nonumber\\\\ = \\mathbb{E}_{t_{j-1}} \\Big[ \\sum_{k=1}^N \\int_{t_{j-1}}^s \\Big( (\\partial_{x_k} &c)^i_u (\\theta_{0,2}) b^k_u (\\theta_{0,1})\n + \\frac{1}{2} (\\partial^2_{x_k} c)^i_u (\\theta_{0,2}) c^k_u (\\theta_{0,2}) \\Big) d u \\Big].\\label{eq:a2remainder}\n\\end{align}\nTo conclude, we need to bound {each} $(\\partial^l_{x_k} c)^i_u (\\theta_{0,2})$, $l=1,2$. To do that, we rely on the assumption about the dependence of the diffusion coefficient on the convolution with a probability measure gathered in \\ref{as7}.\nTo compute the derivatives with respect to $x_k$ we need to consider two different cases, depending on whether $k \\neq i$ or $k = i$. When $k \\neq i$ we have $(\\partial_{x_k} c)^i_u (\\theta_{0,2}) = 2\na^i_u (\\theta_{0,2}) (\\partial_{x_k} a)^i_u (\\theta_{0,2})$, where\n\\begin{equation}\\label{eq:partialaki}\n(\\partial_{x_k} a)^i_u (\\theta_{0,2}) := \\partial_y \\tilde \n\\Big( X^i_u, \\frac{1}{N} \\sum_{j=1}^N \n(X^i_u, X^j_u) \\Big) \\frac{1}{N} \\partial_y \n(X^i_u,X^k_u),\n\\end{equation}\nwhile for $k = i$ we have $(\\partial_{x_i} c)^i_u (\\theta_{0,2}) = 2 a^i_u (\\theta_{0,2}) (\\partial_{x_i} a)^i_u (\\theta_{0,2})$, where\n\\begin{align*}\n(\\partial_{x_i} a)^i_u (\\theta_{0,2}) := \\partial_x \\tilde \n\\Big( X^i_u, \\frac{1}{N} \\sum_{j=1}^N &\n(X^i_u, X^j_u) \\Big) + \\partial_y \\tilde \n\\Big( X^i_u, \\frac{1}{N} \\sum_{j=1}^N \n(X^i_u, X^j_u) \\Big)\\\\\n\\times \\Big( \\frac{1}{N} \\sum_{j=1}^N \\partial_x &\n(X^i_u,X^j_u) + \\frac{1}{N} \\partial_y \n(X^i_u,X^i_u) \\Big).\n\\end{align*}\nFrom polynomial growth of the $l$-th order partial derivatives of\n$K, \\tilde a$ for $l = 0, 1$, that of $b(\\theta_{0,1},\\cdot)$, moment bounds in Lemma~\\ref{l: moments}{(1)} applying Jensen's inequality it follows that \n$\\sum_{k=1}^N (\\partial_{x_k} c)^i_u (\\theta_{0,2}) b^k_u (\\theta_{0,1})$ is bounded in $L^p$ for any $p\\ge 1$ uniformly in $u,i$. We proceed similarly to compute $(\\partial_{x_k}^2 c)^i_u(\\theta_{0,2})$. Then from polynomial growth of the $l$-th order partial derivatives of\n$K, \\tilde a$ for $l=0,1,2$, moment bounds in Lemma~\\ref{l: moments}(1) applying Jensen's inequality it follows that\n$\\sum_{{k}=1}^N (\\partial^2_{x_k} c)^i_u (\\theta_{0,2}) c^k_u (\\theta_{0,2})$ is bounded in $L^p$ for any $p \\ge 1$ uniformly in $u,i$.\nFor any $p \\ge 1$, $t_{j-1} \\le s \\le t_j$, repeatedly applying Jensen's inequality to \\eqref{eq:a2remainder} we get\n$$\n\\mathbb{E} \\big[ \\big| \\mathbb{E}_{t_{j-1}} [c^i_s (\\theta_{0,2})] - c^i_{t_{j-1}}(\\theta_{0,2}) \\big|^p \\big] \\le C (s-t_{j-1})^p,\n$$\nwhence \n$$\n\\mathbb{E} \\Big[ \\Big| \\int_{t_{j-1}}^{t_j} \\big( \\mathbb{E}_{t_{j-1}} [c^i_s (\\theta_{0,2})] - c^i_{t_{j-1}} (\\theta_{0,2}) \\big) d s \\Big|^p \\Big] \\le C \\Delta_n^{2p}.\n$$\nwhich completes the proof.\n\\end{proof}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nA standard approach in noise time series simulation is based on Carson's theorem \\cite{carson31}. In fact, the superposition of randomly delayed pulses of a definite shape $f(t)$ with arbitrary coefficients a$_k$ gives rise to a pulse train \n\\begin{equation}\nn(t)=\\sum_{k} a_k f(t-t_k)\n\\label{series}\n\\end{equation}\nwhose power spectrum $N(\\omega)$ can be expressed in terms of the $f(t)$ power spectrum $P_f(\\omega)$ according to\n\\begin{equation}\nN(\\omega)=\\alpha P_f(\\omega) = \\alpha {|\\widehat F(\\omega)|^2}\n\\label{carson_law}\n\\end{equation}\nwhere $\\alpha$ is a proper normalizing constant and $\\widehat F(\\omega)$ is the Fourier transform of f(t). The delays $t_{k}$'s are distributed according to Poisson statistics and $f(t)$ is usually required to have a zero mean value so that $n(t)$ averages to zero. It can be shown that this result is independent of the density function used for the random amplitudes $a_k$, if the  variance of such distribution is fixed by the choice of the normalizing constant $\\alpha$.\\\\\n\nThe basic idea of the method consists then in searching a proper expression for $f(t)$ so that the generated time series $n(t)$ have the desired  power spectrum $N(\\omega)$. In other terms one is faced with the problem of inverting eq.~(\\ref{carson_law}) to obtain the expression of $f(t)$ given $P_f(\\omega)$. Then, $n(t)$ can be built iteratively according to eq.~(\\ref{series}) with a proper choice of the random parameters $a_k$ and $t_k$.\\\\\n\nThis approach can be quite difficult whenever the desired spectrum is different from a simple power law and analytical approaches to the determination of $f(t)$ are usually not straight-forward. In fact, the desired noise spectrum often shows complex behaviours in the measured frequency range. An example of such a common situation is shown in fig.~\\ref{menotre_spectra} (referring to the output of a few hundred microgram bolometric detector) where we have a complex forest of microphonic lines overimposed to a smoothly varying distribution. The anti-aliasing filter cut frequency is also apparent by the quick drop of the power spectrum in the 100-200 Hz region.\\\\\n\nThe method proposed here consists in the determination of an $f(t)$ which exactly matches a given noise power spectrum. This is essentially based on the choice of a particular (out of an infinite number of possibilities) function $f(t)$ which satisfies eq.~(\\ref{carson_law}) and can be applied both to the case in which $N(\\omega)$ is known analytically or experimentally measured.\\\\\n\nLet us begin with the discrete case of a sampled sequence $f[k]$. If we consider a finite number L of samples and assume ergodicity, then the power spectrum can be approximated by an ensemble average over a sufficiently large number of finite sequences $f_L[k]$ of fixed length L according to\n\\begin{equation}\nP_f[k] \\simeq <|\\mathcal{F}(f_{L}[k])|^2>\n\\label{powerspectrum}\n\\end{equation}\nwhere $\\mathcal{F}$ stands for the Discrete Fourier Transform (DFT) operator.\nSince the sequence elements $f_L[k]$ are supposed to be real, the corresponding power spectrum is a real even function and only half of its values are independent. In fact all the informations concerning the phases are lost in the quadrature and this is the reason why the inversion of eq.~(\\ref{powerspectrum}) does not admit a unique solution $f_L[k]$.\\\\ \nThe simplest solution to eq.~(\\ref{powerspectrum}) is obtained by deleting the ensemble average and inverting the resulting expression. Phases can be then randomly added assuming a flat distribution. The result represents the core of this work and can be summarized as follows:\n\\begin{equation}\nF[k] \\equiv \\sqrt{N[k]}\\cdot e^{i\\theta_k}\n\\label{met01}\n\\end{equation}\n\\begin{equation}\nf[k]=\\mathcal{F}^{-1}(F[k])\n\\label{met02}\n\\end{equation}\n\nwhere $N[k]$ is the power spectrum according to which we want to generate our time series (known a priori) and $\\theta_k$ are random numbers uniformely distributed between $0$ and $2\\pi$. In order to guarantee the condition of reality of $f[k]$ upon inverse DFT the sYmmetry constraint $F[k]=F^*[-k]$  must be imposed.\\\\\n\nNoise time series can now be built following eq.~(\\ref{series}), simply generating the delays $t_k$ according a Poisson distribution:\n\\begin{equation}\nt_k = t_{k-1} -\\ln(1-R)\/\\lambda\n\\label{delays}\n\\end{equation}\nwhere R is a random number with a uniform distribution and $\\lambda$, which represents the overlap rate of the signals $f[k]$, is the only free parameter of the method and is usually adjusted to improve the quality of the simulated time series.\\\\ \n\nAs stated above, the choice of the normalization term $\\alpha$ partially fixes the arbitraryness in the choice of the amplitudes density function by setting its variance. This can be better understood by following some of the steps in the derivation of the Carson's theorem. Actually, by substituting eq.~(\\ref{series}) into the the power spectrum definition we obtain: \n\\begin{eqnarray}\nN(\\omega) = \\left< \\left| \\mathcal{F} \\left[\\sum_{k} a_k f(t-t_k)\\right] \\right|^2 \\right>  = \\nonumber \\\\\n=\\left< \\left| F(\\omega) \\right|^2 \\cdot \\sum_{ij} a_i a_j e^{i\\omega(t_j-t_i)} \\right>\n\\label{car01}\n\\end{eqnarray}\nThe last sum can then be separated into two different terms, one for $i=j$, the other for $i\\neq j$. It can be shown that the latter is proportional to $F(0)$ and therefore vanishes since we required $f(t)$ to average to zero. We obtain therefore:\\\\\n\\begin{equation}\nN(\\omega) = \\left< \\left| F(\\omega)\\right|^2\\right>  \\left<\\sum_{k=1}^{L} |a_k|^2\\right>\n\\label{car02}\n\\end{equation}\n\n\\begin{figure}[htb]\n\\includegraphics[width=.4\\textwidth]{noisemenotre-comp.eps}\n\\caption{Superposition between a simulated (black) and measured (red) bolometric noise power spectrum obtained with a bolometric detector of O(mg) mass~\\cite{ale03}. The simulated spectrum was obtained averaging 50 finite time series. Error bars are the average standard deviations of the simulated spectra. The distribution of the $\\chi^2$\/d.o.f. for a set of simulated spectra is shown in the inset.}\n\\label{menotre_spectra}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\includegraphics[width=.4\\textwidth]{noisefet-comp.eps}\n\\caption{Superposition between a simulated (black) and measured (red) jfet noise power spectrum. The simulated spectrum was obtained averaging 50 finite time series. Error bars are the average standard deviations of the simulated spectra. The distribution of the $\\chi^2$\/d.o.f. for a set of simulated spectra is shown in the inset.}\n\\label{fet_spectra}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\includegraphics[width=.4\\textwidth]{noisemenotre-ratio.eps}\n\\caption{Ratio between the simulated and measured bolometric noise power spectra shown in Fig.\\ref{menotre_spectra}.} \n\\label{bolo_ratio}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\includegraphics[width=.4\\textwidth]{noisefet-time-real2.eps}\n\\includegraphics[width=.4\\textwidth]{noisefet-time-sim.eps}\n\\caption{Samples of simulated (top) and a real (bottom) noise time series of a jfet}\n\\label{timedomain}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\includegraphics[width=.4\\textwidth]{noisefet-ratio2.eps}\n\\caption{Ratio between a simulated and measured jfet noise power spectra shown in Fig.\\ref{fet_spectra}.}\n\\label{fet_ratio}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\includegraphics[width=.4\\textwidth]{sqrt.eps}\n\\caption{Average normalized standard deviation (dots) of the simulated power spectra as a function of the number of simulated time series used for the average process. The black line joins simulated points while the red one is an inverse square root law shown for reference.}\n\\label{chisquare}\n\\end{figure}\n\nThe averaged sum on the right hand part can be thought as the average of the squared amplitude of each single pulse times their average number $M$ of occurencies. As a result we can express the average power spectrum as: \\\\\n\\begin{equation}\nN(\\omega) = \\left< \\left| F(\\omega)\\right|^2\\right> M<\\left|a\\right|^2>\n\\label{nps_fin}\n\\end{equation}\n\nBy comparing eq.~(\\ref{nps_fin}) and eq.~(\\ref{carson_law}) we can finally obtain the relation between the average rate $\\lambda$, the normalization constant $\\alpha$ and the amplitudes variance $|a|^2$ :\n\\begin{equation}\n\\left|a\\right|^2=\\frac {\\alpha} {M}=\\frac{\\alpha}{\\lambda T}\n\\label{rate}\n\\end{equation}\nwhere $T$ is the finite length of the time series. Any arbitrary distribution with a variance given by eq.~(\\ref{rate}) can now be used to generate the random amplitudes $a_k$ and the method is fully defined.\\par\nWe can therefore summarize the most relevant steps for the method implementation as follows:\n\\\\\\\\\\\\\n\\begin{itemize}\n\\item Given the desired noise power spectrum $N(\\omega)$ select the basis function f(t) according to eqs.~(\\ref{met01}) and (\\ref{met02}). Then eq.~(\\ref{carson_law}) fixes the constant $\\alpha$ \n\\item Generate a set of increasing delays $t_k$ according to a Poisson distribution with average rate $\\lambda$. \n\\item Generate the random amplitudes $a_k$ according to an arbitrary distribution with variance fixed by eq.(\\ref{rate}). \n\\item Shift $f(t)$ by $t_k$ imposing a periodicity constraint $f(t)=f(t+T)$ and multiply the result by $a_k$.\n\\item Sum iteratively over $t_k$ while $\\sum t_k<T$\n\\end{itemize}\n\nThe method can be used also when a theoretical (analog) estimate of $F(\\omega)$ is known. This is a common situation when the performance of an electronic circuit or of a detector are studied. In this cases the discrete power spectrum F[k] can be in fact obtained by means of a proper frequency warping~\\cite{dsp75} of the analog theoretical estimate of the power spectrum.\\\\\n\n\\section{Applications}\n\nWe would like to discuss in this section a few examples of application of the method to situations characterized by a complex structure of the desired noise power spectrum. We have chosen to this end two completely different noise sources: a single particle bolometric detector~\\cite{fio84,ale99,ale03} and a jfet transistor~\\cite{arn04,arn06}. Indeed, being characterized by very different masses, frequency ranges and noise sources, they provide a clear example of the potential and versatility of the method.\\par\nThe concept of a single particle bolometric detector is very simple. It is an ideal calorimeter for which all the energy of the incident radiation is converted to heat giving rise to a temperature rise of the detector's body. Such a temperature variation is then measured by means of a proper transducer which in our case will be represented by a doped thermistor whose resistance varies with temperature. A bias circuit is therefore needed and the observed signals express the voltage change at the thermistor terminals. Working temperatures are as low as few tens of K in order to guarantee measurable temperature variations.\nIn such working conditions different noise contributions can be recognized: the electronic noise coming from the acquisition chain, the Johnson noise in bias circuit elements, the mechanical vibrations or instabilities of the of the experimental setup which may produce variation in the acquired voltage samples and finally the temperature instabilities due to thermodynamic fluctuations.  The result is a complex  noise power spectrum in which a forest of sharp lines is overimposed to a generally smooth behaviour. Detector details (eg. mass) naturally affect the specific shape of the noise spectra (range, lines, ...) but usually do not alter the general trends described above~\\cite{pir06,cuore04}.\nFollowing the common prescriptions~\\cite{dsp75} such spectra were obtained by averaging a few hundreds of randomly selected baselines (defined as time windows in the absence of a signal). \nThe same averaging procedure was followed to obtain a power spectrum of the output noise of a simple jfet transistor (fig.~\\ref{fet_spectra}).  \nFollowing the recipe described in this paper and starting from the power spectra of a small mass bolometer and a jfet, a set of 50 simulated time series was generated. The respective power spectra were obtained according to eq.\\ref{powerspectrum} by averaging their DFT's. The result is directly compared in fig.~\\ref{menotre_spectra} and \\ref{fet_spectra} with the original noise power spectra obtained experimentally. The ratio between the experimental and simulated power spectra is also reported in fig.~\\ref{bolo_ratio} and in fig.~\\ref{fet_ratio}. Experimental and simulated baselines are also compared in fig.~\\ref{timedomain} in the case of the jfet transistor.\n\nIn order to allow a more quantitative estimate of the level of agreement between the simulated and the original power spectra, a full set of simulated power spectra was generated in order to evaluate the distribution of the simulated values at each frequency. Each power spectrum was obtained averaging 50 simulated time series. The obtained standard deviations are shown in fig.~\\ref{fet_spectra} and fig.~\\ref{menotre_spectra}. The distributions of the $\\chi^2$\/d.o.f. are also shown to demonstrate the correct statistical behaviour.\nThe fluctuations of the simulated power spectra values depend of course also on the number of time series used for the average process (eq.\\ref{powerspectrum}). Such a dependence was explicitely studied by generating a different set of power spectra and evaluating the corresponding average standard deviation from the desired otput, as a function of the number of averaged time series. The result is shown in fig.~\\ref{chisquare} and found in excellent agreement with an inverse square root law.\n  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGiven a connected undirected simple graph $G=(V(G),E(G))$ with at least $3$ vertices, a nucleus of $G$ is a connected subgraph $N \\subseteq G$ such that $V(N)$ is a vertex cover of $G$. The set of nuclei is denoted by $\\mathcal{N}(G)$ \\cite{nucleus}. The $k$-th Elser number $\\mathrm{els}_k(G)$ is defined as follows\n\n\\begin{align*}\n\t\\mathrm{els}_k(G)=(-1)^{|V(G)|+1} \\sum_{N \\in \\mathcal{N}(G)} (-1)^{|E(N)|} |{V(N)}|^{k}.\n\\end{align*}\n\nElser's conjecture \\cite{Elser} says $\\mathrm{els}_k(G) \\ge 0 $ for $k \\ge 2$, $\\mathrm{els}_1(G) = 0 $ and $\\mathrm{els}_0(G) \\le 0 $. This conjecture was proved in \\cite{nucleus}. That proof made use of the $U$-nucleus complex $\\Delta^{G}_{U}$ as defined below. \n\n\n\\[ \n\\text{For } U \\subseteq V(G), \\Delta^{G}_{U}= \\{ E(G) \\setminus E(N): N \\in \\mathcal{N}(G), \\ U \\subseteq V(N) \\}. \n\\]\t\n\nThe $k$-th Elser number, $\\mathrm{els}_k(G)$ can be seen as sum of reduced Euler characteristic of $\\Delta^{G}_{U}$ as below.\n\n\\begin{align*}\n\t\\mathrm{els}_k(G)=(-1)^{|E(G)|+|V(G)|} \\sum_{U \\subseteq V(G)} \\mathrm{Sur}(k,|U|) \\ \\tilde{\\chi}(\\Delta^{G}_{U})\n\\end{align*}\n\nwhere, $\\mathrm{Sur}(a,b)$ denotes the number of surjections from a set of size $a$ to a set of size $b$. Dorpalen-Barry et al. proved Elser's conjecture by establishing analogous inequalities for the reduced Euler characteristic $\\tilde{\\chi}(\\Delta^{G}_{U})$.  They posed a general conjecture (Conjecture 9.1 in \\cite{nucleus}) about the homology of these complexes that implies the results about the reduced Euler characteristic as a corollary.  \n\n\\begin{conjecture} \\cite{nucleus}\\label{conj}\nLet $G$ be a connected graph and $U \\subseteq V(G)$. Then the reduced homology\ngroup $\\tilde{H}_k(\\Delta^{G}_{U}; \\mathbb{R})$ is nonzero only if (i) $U=\\emptyset$ and $k =|E(G)|-|V (G)|-1$, or (ii) $|U|>1$ and $k=|E(G)|-|V (G)|$.\n\t\n\\end{conjecture}\n\nWe prove the following theorem about homology of $\\Delta^{G}_{U}$ settling the conjecture in the case of $|{U}|>1$ and for $U= \\emptyset$ we provide a counterexample at the end of this section. \n\n\\begin{theorem} \\label{maintheorem}\nFor $|U|>1$, $\\tilde{H}_{k}(\\Delta^{G}_{U};\\mathbb{R}) \\cong 0$ unless $k=|E(G)|-|V(G)|$. \n\n\\end{theorem}\n\n\nOur approach to prove Theorem~\\ref{maintheorem} is inspired by Grinberg's proof of Elser's conjecture for the case $k=1$ (i.e., $\\mathrm{els}_1(G)=0$) using discrete Morse theory \\cite{Grinberg}. For $F \\subseteq E(G)$, an $F$-path is a path consisting of edges from $F$. Now given a vertex $v\\in V(G)$, we define $\\mathrm{Shade}_v (F)$ as follows. \n\n\\[ \\mathrm{Shade}_v (F):=\\{ e \\in E(G) : \\text{ There is a } F \\text{-path from an endpoint of } e \\text { to } v \\}. \\]\n\nWe consider the following abstract simplicial complex as defined in \\cite{Grinberg} \n\n\\[ \\mathcal{A}_U = \\{ F \\subseteq E(G)  :  \\mathrm{Shade}_v(F) \\subsetneq E(G) \\text{ for some } v \\in U \\}. \\]\n\n\nIt can be shown that $\\mathcal{A}_U $ is the Alexander dual of $\\Delta^{G}_{U}$. Grinberg showed that $\\mathcal{A}_U $  is collapsible when $|U|=1$ by producing an acyclic matching (also known as gradient vector field) with no unmatched simplices. By Alexander duality, it follows that in the case of $|U|=1$, $\\tilde{H}_i(\\Delta^{G}_{U}) \\cong \\tilde{H}^{|E(G)|-i-3}(\\mathcal{A}_U) \\cong \\tilde{H}_{|E(G)|-i-3}(\\mathcal{A}_U) \\cong 0  $. Therefore, $\\Delta^{G}_{U}$ has homology of a point. \\\\\n\n\nWe observe that $\\mathcal{A}_U = \\cup_{x \\in U} \\mathcal{A}_x$.  We extend Grinberg's matching to $\\mathcal{A}_U $ when $|U|>1$ and show that $\\mathcal{A}_U $ can be given an acyclic matching  with unmatched simplices in the dimension $k=|V(G)|-3$. Again by Alexander's duality we conclude that, $\\Delta^{G}_{U}$ has homology concentrated in a single homological degree $|E(G)|-|V(G)|$.\\\\\n\nHowever, we note that the Conjecture~\\ref{conj} as stated is not true for the other case $U= \\emptyset$. For example,  for any cycle $C_n$ with $n \\ge 3$,  we note that the connected vertex covers of $C_n$  are (i) any path of length $n-2$ (i.e., a path with $n-2$ edges), (ii) any path of length $n-1$, and (iii) $C_n$ itself. Then it can be verified that the nuclei complex,  $\\Delta^{C_n}_{\\emptyset}$ is homotopy equivalent to the circle and thus, $\\tilde{H}_{1}(\\Delta^{C_n}_{\\emptyset};\\mathbb{R}) \\neq 0$.\\\\\n\n\n\n\n\\textbf{Acknowledgments} The authors would like to thank Goutam Mukherjee for helpful discussions.\n\n\n\n\\section{Preliminaries}\n\n\n\\subsection{Graph theoretic notation}\n\n\\begin{itemize}\n\n\\item A (simple undirected) graph $G$ is an ordered pair $(V(G), E(G))$, where $V(G)$ is a finite set and $E(G) \\subseteq \\{ S \\subseteq V(G): \\ |S|=2 \\}$. Elements of $V(G)$ are called vertices and elements of $E(G)$ are called edges. \\\\\n\n\\item We call a vertex $y$ a \\emph{neighbor} of $x$ in $G$ if $x$ and $y$ are adjacent, i.e.,  $\\{ x,y \\} \\in E(G)$. \\\\\n\n\\item A \\emph{leaf} in $G$ is a vertex with only one neighbor.\\\\\n\n\\item A \\emph{leaf-edge} denotes an edge which is incident to (i.e., contains) a leaf.\\\\\n\n\\item For a connected graph $G$, an edge $e$ is called a \\emph{bridge} if the graph obtained after deleting $e$ is disconnected.\\\\\n\n\\item  \\emph{Edge-induced subgraph} : Let  $\\sigma \\subseteq E(G)$. We define the subgraph induced by $\\sigma$, denoted by $G_\\sigma$ as follows\n \\[ V(G_{\\sigma})= \\{ x \\in V(G) : x \\in e \\text{ for some } e \\in \\sigma \\} \\text{ and }\nE(G_{\\sigma})= \\sigma.\\]\n\nWe use $V_{\\sigma}$ to denote  $V(G_{\\sigma})$ for brevity.\\\\\n\n\\item \\emph{Vertex cover (of edges)} of $G$: $X \\subseteq V(G) \\text{ such that for all } e \\in E(G), e\\cap X \\neq \\emptyset$.\n\n\n\\end{itemize}\n\n\\subsection{Basics of discrete Morse Theory}\n\nIn the case of smooth Morse theory, existence of a Morse function is equivalent to the existence of a gradient vector field. Similarly, existence of a \\emph{discrete Morse function} on a simplicial complex is equivalent to the existence of a \\emph{discrete gradient vector field}. First we recall the notion of a \\emph{discrete vector field} or \\emph{matching} on a simplicial complex. Let $K$ be a simplicial complex. \n\n\\begin{definition}[Discrete vector field \/ matching]\n\t\n\tA discrete vector field $V$ on $K$ is a collection of pairs $\\{\\alpha^{(p)} < \\beta^{(p+1)}\\}$ of simplices of $K$ such that each simplex is in at most one pair of $V$.\n\t\n\\end{definition}   \n\nPictorially, given a discrete vector field on $K$, we can assign arrows on $K$ such that for a pair $\\{\\alpha^{(p)} < \\beta^{(p+1)}\\}$ the tail of the arrow lies in $\\beta^{(p+1)}$ and the head of the arrow lies in $\\alpha^{(p)}$.  A gradient vector field is a discrete vector field with some special properties about this arrows.  \n\n\n\\begin{definition}[Gradient vector field]\n\tA discrete vector field is called a gradient vector field if given any simplex $\\alpha$ in $K$, it satisfies exactly one the following.\n\t\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\alpha$ is the tail of exactly one arrow.\n\t\t\n\t\t\\item $\\alpha$ is the head of exactly one arrow.\n\t\t\n\t\t\\item $\\alpha$ is neither the head nor the tail of an arrow.\n\t\\end{enumerate}\n\n\\end{definition}\n\n\n\n\\begin{figure}[htbp] \n\t\\centering\n\t\\def12cm{8cm}\n\t\\input{gradientfield.pdf_tex}\n\t\\caption{An example of a discrete gradient vector field.}\\label{gradfield}\n\\end{figure}\n\n\nAccording to this notion of discrete Morse functions in terms of discrete gradient fields, a simplex is \\emph{critical} if and only if it is neither the tail nor the head of any arrow. For instance, in Figure \\ref{gradfield}, $e_2$ is a critical $1$-simplex while $e_1$ is not critical. The criteria when a discrete vector field is a gradient vector field is more straight forward once we have the notion of a \\emph{$V$-path}.\n\n\n\\begin{definition}\n\tGiven a discrete vector field $V$ on a simplicial complex $K$, a $V$-path is a sequence of simplices $$\\alpha_0^{(p)}, \\beta_0^{(p+1)}, \\alpha_1^{(p)}, \\beta_1^{(p+1)},\\cdots,\\beta_r^{(p+1)}, \\alpha_{r+1}^{(p)}$$ such that for each $i \\in \\{0,1, \\cdots, r\\}$, $\\{\\alpha_i < \\beta_i \\} \\in V$ an $\\beta_i > \\alpha_{i+1} \\neq \\alpha_i$.\n\n\t \n\\end{definition}\n\n\\noindent \n\nWe say a path is a non-trivial \\emph{closed path} if $r \\geq 0$ and $\\alpha_0 = \\alpha_{r+1}$. It can be shown that a gradient vector field can not have a non-trivial closed $V$-path (i.e., a cycle). Moreover, the other direction is also true.\n\n\\begin{theorem}[Forman \\cite{Forman}] \\label{v-path criteria}\n\tA discrete vector field $V$ is the gradient vector field of a discrete Morse function if and only if there are no non-trivial closed $V$-path.\n\\end{theorem} \n\nIn other words we say a discrete vector field $V$ is gradient vector field if the corresponding matching is acyclic (i.e., it contains no closed $V$-path).\\\\\n \n\\noindent The notion of gradient vector field is closely related to the notion of \\emph{simplicial collapse}. A simplex $\\alpha$ is called a face of a simplex $\\beta$ if $\\alpha \\subsetneq \\beta$. A simplex which is a face of exactly one simplex is called a \\emph{free face}. For example, in Figure \\ref{gradfield}, $v_1$, $e_1$ and $e_2$ are free faces while $v_2$ is not. Whenever we have a free face $\\alpha$ of a simplex $\\beta$ in a simplicial complex $K$, we can remove $\\alpha$ and $\\beta$ (but keeping faces of $\\beta$ other than $\\alpha$ unperturbed) from $K$  by a deformation retraction. This is known as \\emph{elementary collapse}. If $K \\setminus \\{ \\alpha, \\beta\\}$ is obtained from $K$ by a elementary collapse, we can extend any gradient vector field of $V$ of $K \\setminus \\{ \\alpha, \\beta\\}$ to $K$ by adding the pair $(\\alpha,\\beta) $ to $V$.\n\n\n\n\\begin{figure}\n\t\\def12cm{12cm}\n\t\\input{collapse.pdf_tex}\n\t\\caption{A sequence of elementary collapses.} \\label{collapse}\n\\end{figure}\n\nIf one can go from a simplicial complex $K_1$ to another simplicial complex $K_2$ via a sequence of elementary collapses, then we say $K_1$ collapses to $K_2$. If a gradient vector field on a simplicial complex $K$ does not have any critical faces of dimension greater than $0$, then $K$ collapses to a point.  We also say $K$ is collapsible. \\\\\n\nNow, we state the fundamental theorem of discrete Morse theory.\n\n\n\n\\begin{theorem}[Forman \\cite{Forman}]  \\label{forth}\nSuppose $K$ is a simplicial complex with a discrete Morse function. Then $K$ is homotopy equivalent to a CW complex with exactly one cell of dimension $p$ for each critical simplex of dimension $p$.\n\\end{theorem}\n\n\n\nWe also need the following crucial fact from discrete Morse theory which follows from the theorem above.\n\n\\begin{theorem}[Sphere theorem, \\cite{J}] If $K$ has only critical faces of dimension $d$ ($\\geq 1$), then $K$ is homotopy equivalent to a wedge of $d$-spheres. \n\t\n\\end{theorem}\n\n\n\\subsection{Collapsibility of $\\mathcal{A}_x$}\n\nWe consider the case $U= \\{x \\}$. Grinberg produced an acyclic matching on $\\mathcal{A}_x$ with no critical simplices (here we adopt the convention that if only one $0$-simplex is unpaired in some matching, we are allowed to pair it with $\\emptyset$). Then, we can conclude by Forman's theorem \\cite{Forman} that $\\mathcal{A}_x$ is collapsible.\\\\\n\nWe briefly explain the matching $\\Phi_0$ constructed in Grinberg's proof. We fix an arbitrary ordering in $E(G)$. For any set $F \\in \\mathcal{A}_x $ define, $\\sigma(F)$ to be the minimum edge in $E(G) \\setminus \\mathrm{Shade}_x(F)$ (this set is non-empty by definition). If $ \\sigma(F) \\notin F$, we pair off $F$ with $F \\cup \\sigma(F)$. This gives an acyclic matching in $\\mathcal{A}_x $ with no critical simplices \\cite{Grinberg}. Thus we have the following theorem.  \n\n\\begin{theorem} \\cite{Grinberg}\nFor any $x\\in V(G)$, $\\mathcal{A}_x$ is collapsible.\n\\end{theorem}\n\n\n\n \n\n\\section{Proof of the main theorem}\n\n\n\\begin{proof}[Proof of Theorem \\ref{maintheorem}]\n\nLet $E(G)=\\{ e_1, e_2,\\cdots, e_m \\}$. We fix the ordering $ e_1 < e_2 < \\cdots < e_m $ on $E(G)$. The proof is by induction. For the base case, let $U=\\{x,y\\}.$ Clearly, $\\mathcal{A}_U = \\mathcal{A}_x \\cup \\mathcal{A}_y.$ We consider the acyclic matching  $\\Phi_0$ on $\\mathcal{A}_x$ with no critical simplices as mentioned before. Our aim is to extend this matching $\\Phi_0$ to the whole simplicial complex $\\mathcal{A}_U.$ In order to achieve our goal we proceed recursively. We divide this recursive process into several steps. Note that we want to define an acyclic matching on the simplices of $\\mathcal{A}_y \\setminus\\mathcal{A}_x.$ First, we make the following observation to characterize the simplices in $\\mathcal{A}_y \\setminus\\mathcal{A}_x$. \n\n\\begin{obs} \n Let $\\sigma \\subseteq E(G)$ and $\\sigma \\neq \\emptyset$. Then, $ \\sigma \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x$ if and only if the following conditions hold \n \\begin{enumerate}\n \\item $G_\\sigma$ is a connected subgraph of $G$,\n \\item $V_\\sigma$ is a vertex cover of $G$,\n \\item $x \\in V_\\sigma$ and  $y \\notin V_\\sigma$.\n \n \\end{enumerate}\nAlso $\\sigma= \\emptyset \\in \\mathcal{A}_y \\setminus\\mathcal{A}_x$ only if $G$ is a star graph with $x$ as the central vertex.\n\\end{obs}\n\n\nNow, we proceed to our initial step of the recursive process. \n\n\n\n \\textbf{Step 1:} Consider those simplices $\\sigma$ in $\\mathcal{A}_y \\setminus\\mathcal{A}_x,$ such that, $e_1 \\in \\sigma$ and one of the following holds. \n \\begin{enumerate}\n    \\item $e_1$ is a part of a cycle (i.e., not a bridge) in $G_\\sigma$ (see Fig. \\ref{sigma1}).\n    \\item $e_1$ is a leaf-edge in $G_\\sigma$ such that \n    \\begin{enumerate}\n        \\item $e_1$ doesn't contain $x$ \\emph{as a leaf} in $G_\\sigma$ \\\\ and\n        \\item If  $z \\in V(G) \\setminus V_\\sigma$, then $e_1$  doesn't contain any neighbor of $z$ \\emph{as a leaf} in $G_\\sigma$ (see Fig. \\ref{sigma2}). \n\\end{enumerate}         \n\\end{enumerate}    \n\n\n\n\\begin{figure}[!ht]\n\\begin{minipage}{0.45\\textwidth}\n\\[\\begin{tikzpicture}[scale=1.5]\n\t\\node at (-0.3,0) {$x$};\n\t\\node at (2.3,0) {$y$};\n\t\n\t\\node at (0.5,0.2) {$e_1$};\n\t\\node at (1.2,-0.5) {$e_2$};\n\t\\node at (0.2,-0.5) {$e_3$};\n\t\n\t\\node[vertex] (x) at (0,0) {};\n\t\\node[vertex] (w) at (1,0) {};\n\t\\node[vertex] (z) at (1,-1) {};\n\t\n\t\\node[redvert] (y) at (2,0) {};\n\n\t\\path [line width=1pt]\n\t\t(x) edge (w)\n\t\t(x) edge (z)\n\t\t(w) edge (z)\n\t ;  \n\n\t\\path [red,dashed,line width=1pt]\n\t\t(w) edge (y)\n\t\t(z) edge (y)\n\t;  \n\n\\end{tikzpicture}\\]\n\\caption{ $\\sigma = \\{e_1,e_2,e_3\\}$; $e_1$ is a part of a cycle in $G_\\sigma$.} \\label{sigma1}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.45\\textwidth}\n\t\\[\\begin{tikzpicture}[scale=1.5]\n\t\\node at (-0.3,0) {$x$};\n\t\\node at (1.3,1) {$y$};\n\t\\node at (1.3,-1) {$z$};\n\t\n\t\\node at (1.5,0.2) {$e_1$};\n\t\\node at (0.5,0.2) {$e_2$};\n\t\n\t\\node[vertex] (x) at (0,0) {};\n\t\\node[vertex] (w1) at (1,0) {};\n\t\\node[vertex] (w2) at (2,0) {};\n\t\n\t\\node[redvert] (y) at (1,) {};\n\t\\node[redvert] (z) at (1,-1) {};\n\t\n\t\\path [line width=1pt]\n\t(x) edge (w1)\n\t(w2) edge (w1)\n\t;  \n\t\n\t\\path [red,dashed,line width=1pt]\n\t(x) edge (y)\n\t(x) edge (z)\n\t(w1) edge (y)\n\t(w1) edge (z)\n\t;  \n\t\n\t\n\t\\end{tikzpicture}\\]\n\t\\caption{ $\\sigma = \\{e_1, e_2\\}$; $e_1$ is a leaf-edge in $G_\\sigma$ and $e_1$ doesn't contain any neighbor of $y$ or $z$ as a leaf.} \\label{sigma2}\n\t\\end{minipage}\n\n\n\\end{figure}\n\n\n\n\n\n\nWe note that in this case $\\sigma \\setminus \\{ e_1 \\} \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x.$ We pair off $\\sigma$ with $\\sigma \\setminus \\{e_1\\}.$ We mention an exceptional pairing in the special case of $ \\{e_1 \\} \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x$ as well as $\\emptyset \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x$ ( as mentioned before possible only if $G$ is a star graph with $x$ as the central vertex). In that case we pair off  $ \\{e_1 \\}$ with $\\emptyset$.\\\\ \n\nWe observe that conversely for any $\\sigma \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x$ with $e_1 \\notin \\sigma$ is uniquely paired with $\\sigma \\cup \\{ e_1 \\}$ whenever $\\sigma \\cup \\{ e_1 \\} \\in  \\mathcal{A}_y \\setminus \\mathcal{A}_x$. Therefore, this extension of $\\Phi_0$ is well defined and is denoted by $\\Phi_1.$\n\n\n\n\n\n \n\\textbf{Step 2:} The set of all unpaired simplices after step $1$ is denoted by $\\mathcal{C}_1.$ Note that $\\mathcal{C}_1$ consists of simplices  $\\tau$ in $\\mathcal{A}_y \\setminus\\mathcal{A}_x,$ such that, either $e_1 \\notin \\tau$ (possible only if $y \\in e_1$)  or $e_1 \\in \\tau$ and one of the following holds.\n\n \\begin{enumerate}\n    \\item $e_1$  is a bridge but not a leaf-edge in $G_\\tau$ (see Fig. \\ref{sigma3}).\n    \\item $e_1$ is a leaf-edge in $G_\\tau$ such that,\n    \\begin{enumerate}\n         \\item  $e_1$ contains $x$ as a leaf in $G_\\tau$ (see Fig. \\ref{sigma4}, \\ref{sigma5})\\\\ or \n    \\item  there exists $z \\in V(G) \\setminus V_\\tau$ such that $e_1$  contains a neighbor of $z$ as a leaf in $G_\\tau$ (see Fig. \\ref{sigma5}, \\ref{sigma6}). \n\n\\end{enumerate}\n\\end{enumerate}    \n\n\n\\begin{figure}[!ht]\n\n\\begin{minipage}{0.45\\textwidth}\n\t\\[\\begin{tikzpicture}[scale=1.5]\n\t\\node at (-0.3,0) {$x$};\n\t\\node at (4.3,0) {$y$};\n\t\n\t\\node at (0.5,0.2) {$e_2$};\n\t\\node at (1.2,-0.5) {$e_5$};\n\t\\node at (0.2,-0.5) {$e_4$};\n\t\\node at (2,-0.5) {$e_1$};\n\t\\node at (2.7,-0.5) {$e_3$};\n\t\n\t\\node[vertex] (x) at (0,0) {};\n\t\\node[vertex] (w1) at (1,0) {};\n\t\\node[vertex] (w2) at (2.5,-1) {};\n\t\\node[vertex] (w3) at (2.5,0) {};\t\n\t\\node[vertex] (z) at (1,-1) {};\n\t\n\t\\node[redvert] (y) at (3.5,0) {};\n\t\n\t\\path [line width=1pt]\n\t(x) edge (w)\n\t(x) edge (z)\n\t(w1) edge (z)\n\t(w1) edge (w2)\n\t(w3) edge (w2)\n\t;  \n\t\n\t\\path [red,dashed,line width=1pt]\n\t(w3) edge (y)\n\t(w2) edge (z)\n\t(w3) edge (w1)\n\t;  \n\t\n\t\n\t\\end{tikzpicture}\\]\n\t\\caption{ $\\sigma = \\{e_1,\\ldots,e_5\\}$; $e_1$ is a bridge in $G_\\sigma$, but not a leaf-edge.}\\label{sigma3}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.45\\textwidth}\n\t\t\\[\\begin{tikzpicture}[scale=1.5]\n\t\\node at (-0.3,0) {$x$};\n\t\\node at (2.3,0) {$y$};\n\t\n\t\\node at (0.5,0.2) {$e_1$};\n\t\\node at (1.2,-0.5) {$e_2$};\n\t\n\t\\node[vertex] (x) at (0,0) {};\n\t\\node[vertex] (w) at (1,0) {};\n\t\\node[vertex] (z) at (1,-1) {};\n\t\n\t\\node[redvert] (y) at (2,0) {};\n\t\n\t\\path [line width=1pt]\n\t(x) edge (w)\n\t(w) edge (z)\n\t;  \n\t\n\t\\path [red,dashed,line width=1pt]\n\t(w) edge (y)\n\t(x) edge (z)\n\t;  \n\t\n\t\n\t\\end{tikzpicture}\\]\n\t\\caption{ $\\sigma = \\{e_1,e_2\\}$; $e_1$ is a leaf-edge in $G_\\sigma$ and $e_1$ contains $x$ as a leaf in $G_\\sigma$.}\\label{sigma4}\n\t\\end{minipage}\n\t\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\t\n\\begin{minipage}{0.45\\textwidth}\n\t\\[\\begin{tikzpicture}[scale=1.5]\n\t\\node at (-0.3,0) {$x$};\n\t\\node at (2.3,0) {$y$};\n\t\\node at (1.3,-1) {$z$};\n\t\n\t\\node at (0.5,0.2) {$e_1$};\n\t\n\t\\node[vertex] (x) at (0,0) {};\n\t\\node[vertex] (w) at (1,0) {};\n\t\n\t\\node[redvert] (y) at (2,0) {};\n\t\\node[redvert] (z) at (1,-1) {};\n\t\n\t\\path [line width=1pt]\n\t(x) edge (w)\n\t;  \n\t\n\t\\path [red,dashed,line width=1pt]\n\t(w) edge (y)\n\t(x) edge (z)\n\t(w) edge (z)\n\t;  \n\t\n\t\n\t\\end{tikzpicture}\\]\n\t\\caption{ $\\sigma = \\{e_1\\}$; $e_1$ is a leaf-edge in $G_\\sigma$ and $e_1$ contains $x$ as a leaf in $G_\\sigma$. Moreover, $e_1$ also contains neighbors of $y$ and $z$ as leaves.}\\label{sigma5}\n\t\\end{minipage}\n\\hfill\n\\begin{minipage}{0.45\\textwidth}\n\t\\[\\begin{tikzpicture}[scale=1.5]\n\\node at (-0.3,0) {$x$};\n\\node at (3.3,0) {$y$};\n\\node at (1.3,-1) {$z$};\n\n\\node at (1.5,0.2) {$e_1$};\n\\node at (0.5,0.2) {$e_2$};\n\n\\node[vertex] (x) at (0,0) {};\n\\node[vertex] (w1) at (1,0) {};\n\\node[vertex] (w2) at (2,0) {};\n\n\\node[redvert] (y) at (3,0) {};\n\\node[redvert] (z) at (1,-1) {};\n\n\\path [line width=1pt]\n(x) edge (w1)\n(w2) edge (w1)\n;  \n\n\\path [red,dashed,line width=1pt]\n(w2) edge (y)\n(x) edge (z)\n(w1) edge (z)\n(w2) edge (z)\n;  \n\\end{tikzpicture}\\]\n\\caption{ $\\sigma = \\{e_1, e_2\\}$; $e_1$ is a leaf-edge in $G_\\sigma$ and $e_1$ contains (some) neighbors of $y$ and $z$ as leaves.} \\label{sigma6}\n\\end{minipage}\t\n\t\n\\end{figure}\n\nSimilarly as before, we consider those simplices $\\tau$ in $\\mathcal{C}_1$ such that, $e_2$ is in $\\tau$ and either a part of a cycle or a leaf-edge in $G_\\tau$ such that $e_2$ doesn't contain $x$ as a leaf, and if $z \\in V(G) \\setminus V_\\tau$ then $e_2$  doesn't contain any neighbor of $z$ as a leaf in $G_\\tau$. We pair off $\\tau$ with $\\tau \\setminus \\{e_2\\}.$ This extended matching is denoted by $\\Phi_2.$\n\n As before, in the special case of $ \\{e_2 \\} \\in \\mathcal{C}_1$ and $\\{e_2 \\}, \\emptyset \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x$, we pair off $ \\{e_2 \\}$ with $\\emptyset$.\n\n\\textbf{Step 3:} The set of all unpaired simplices after step $2$ is denoted by $\\mathcal{C}_2.$ Note that $\\mathcal{C}_2$ consists of simplices  $\\eta$ in $\\mathcal{C}_1$ if\none of the following four conditions holds: \n\n\n \\begin{enumerate}\n\\item $e_2 \\in \\eta$ is a bridge but not a leaf-edge in $G_\\eta$. \n\n\\item $e_2 \\in \\eta$ is a leaf-edge in $G_\\eta$ such that, \n\n    \\begin{enumerate}\n         \\item  $e_2$ contains $x$ as a leaf  in $G_{\\eta}$ \\\\ or\n       \\item  there exists $z \\in V(G) \\setminus V_\\eta$ such that $e_2$  contains a neighbor of $z$ as a leaf in $G_\\eta$. \n\n\\end{enumerate}\n\\item $e_2 \\notin \\eta$ and $y \\in e_2.$\n\\item $e_2 \\notin \\eta, y \\notin e_2$ and $\\eta \\cup \\{e_2\\}$ contains a cycle containing $e_1.$\n\n\\end{enumerate}\n\nNow we proceed as before, i.e., consider those simplices $\\eta$ in $\\mathcal{C}_2$ such that, $e_3$ is in $\\eta$ and either  a part of a cycle or a leaf-edge in $G_\\eta$ such that $e_3$ doesn't contain $x$ as a leaf, and if $z \\in V(G) \\setminus V_\\eta$ then $e_3$  doesn't contain any neighbor of $z$ as a leaf in $G_\\eta$. We pair off $\\eta$ with $\\eta \\setminus \\{e_3\\}.$ This extended matching is denoted by $\\Phi_3.$\n\nProceeding this way, we end up with a matching $\\Phi$. To verify that $\\Phi$ is acyclic, we first observe that at the $i$-th step, we pair a simplex $\\Sigma$ with $\\Sigma \\setminus e_i$ only if $\\mathrm{Shade}_x(\\Sigma)=\\mathrm{Shade}_x(\\Sigma \\setminus e_i)=E(G) $. More generally, we make the following observation.\n\n\\begin{obs} \\label{pair}\n\nWe pair off   $\\Sigma  \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x$ with $\\Sigma \\setminus e$ only if $e$ is the minimal edge in $\\Sigma$ such that $\\mathrm{Shade}_x(\\Sigma \\setminus e)=E(G) $. \n\n\\end{obs} \n\nWe also make the following observations regarding a unpaired simplex $\\Sigma$ in this matching $\\Phi$.\n\n\\begin{obs}\n\nIf a simplex  $\\Sigma \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x $ is unpaired in $\\Phi$, then the following properties hold.\n\\begin{enumerate}\n\t\\item $G_\\Sigma$ is a spanning tree of $G \\setminus \\{ y \\}$. Therefore, it is a $(|V(G)|-3)$-simplex (i.e., it has $|V(G)|-2$ edges).\n\t\\item For any $e_j \\notin \\Sigma$ with $y \\notin e_j$, the unique cycle in $G_{\\Sigma \\cup e_j}$ has an edge $e_i$ such that $e_i<e_j$. \n\\end{enumerate} \n\n\\end{obs}\n\n\n\nFirst, we consider the following alternating path with respect to the matching  $\\Phi$ starting from $\\sigma_1 \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x$.\n\n\t\n\t\\begin{tikzcd}\n\t& \\Sigma_1 \\arrow[rd]  & & \\Sigma_2 \\arrow[rd] && \\Sigma_3 & \\cdots & \\Sigma_{k-1} \\arrow[rd]\n\t\\\\\n\t\t\\sigma_1 \\arrow[ru] & & \\sigma_1 \\arrow[ru] & & \\sigma_2 \\arrow[ru] && \\cdots & & \\sigma_{k}   \n\t\\end{tikzcd}\n\t\n\tSuppose $i$ is an integer  between $2$ and $k-1$ such that  $\\sigma_i \\in \\mathcal{A}_x$. Then, we observe that $\\sigma_j \\in \\mathcal{A}_x$ for all $j \\in \\{i, \\cdots, k \\}$ and  $\\Sigma_{j} \\in \\mathcal{A}_x$ for all $j \\in \\{i, \\cdots, k-1 \\}$. It is already known that the matching $\\Phi_0$ is acyclic in $\\mathcal{A}_x$. Therefore, it is impossible to have closed $\\Phi$-path containing simplices from both $\\mathcal{A}_y \\setminus \\mathcal{A}_x$ and $\\mathcal{A}_x$. So it suffices to show that no closed $\\Phi$-path exists containing simplices only from $\\mathcal{A}_y \\setminus \\mathcal{A}_x$.\\\\\n\n\nNow assume on the contrary that there is a directed cycle in the matching as follows.\n\n\t\\begin{tikzcd}\n\t\t\\Sigma_n \\arrow[rd] & & \\Sigma_1 \\arrow[rd] & & \\Sigma_2 \\arrow[rd] && \\cdots & & \\Sigma_n   \\\\\n\t\t & \\sigma_1 \\arrow[ru]  & & \\sigma_2 \\arrow[ru] && \\sigma_3 & \\cdots & \\sigma_n \\arrow[ru]\n\t\\end{tikzcd}\n\nHere, we assume $\\Sigma_i, \\sigma_i \\in \\mathcal{A}_y \\setminus \\mathcal{A}_x$ for each $i$. We note that for each $i$ there are edges $f(i) \\notin \\sigma_i$ and $g(i) \\in \\Sigma_i$, such that $\\Sigma_i=\\sigma_i \\cup f(i)$ and $\\sigma_{i+1}=\\Sigma_i \\setminus g(i)$.  \n It now follows that, $f(i)<g(i)$ (with respect to the ordering in $E(G)$) since otherwise it will violate Observation~\\ref{pair} . We also assume without loss of generality that $f(1)= \\min_{i} f(i) $. Since, $\\{f(i): i=1,\\cdots,n\\}=\\{g(i): i=1,\\cdots,n\\}$, there is $1 \\le j \\le n$ such that $f(1)=g(j)>f(j)$  which contradicts the fact that $f(1)$ is the minimum. Therefore, the matching $\\Phi$ is an acyclic extension of the acyclic matching $\\Phi_0$ for $\\mathcal{A}_x$. \n\n\nNow, let $U_n= \\{ x_1,\\cdots,x_n \\}$. We observe that $\\mathcal{A}_{U_n}=\\mathcal{A}_{U_{n-1}} \\cup  (\\mathcal{A}_{x_n} \\setminus \\mathcal{A}_{U_{n-1}})$. Proceeding inductively, we can construct an acyclic matching of $\\mathcal{A}_U$ where $|U|>1$ that has critical simplices at dimension $|V(G)|-3$. \n \nBy Theorem \\ref{forth}, we can conclude that $\\mathcal{A}_U$ is homotopy equivalent to a wedge of spheres of dimension $|V(G)|-3$. Since, $\\Delta^{G}_{U}$ is the Alexander dual of $\\mathcal{A}_U$, it follows by Alexander duality that $\\tilde{H}_k(\\Delta^{G}_{U}) \\cong \\tilde{H}^{|E(G)|-k-3}(\\mathcal{A}_U) \\cong \\tilde{H}_{|E(G)|-k-3}(\\mathcal{A}_U) \\cong 0 $ unless $|E(G)|-k-3=|V(G)|-3$. Therefore, $\\tilde{H}_{k}(\\Delta^{G}_{U};\\mathbb{R}) \\cong 0$ unless $k=|E(G)|-|V(G)|$.\n\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nCircinus~X-1 exhibits a dramatic 16.55-d cycle of X-ray flaring which\nis believed to be the result of enhanced mass transfer occurring near\nperiastron of a highly eccentric binary orbit (\\cite{kaluzienski76};\n\\cite{murdin80}).  Although the identification of the orbital period\nseems to be secure, there is very little direct evidence concerning\nthe masses of the two components of the binary and the other orbital\nparameters. Nonetheless, the compact component is thought to be a\nweakly magnetized neutron star on the basis of three type~I X-ray\nbursts seen with {\\it EXOSAT\\,}\\ (\\cite{tennant86}).  Further type~I bursts\nhave not been observed from Cir X-1 since the {\\it EXOSAT\\,}\\ discovery,\npossibly because the source intensity has been higher during\nsubsequent observations.  No coherent pulsations, which would be\nexpected to be present if the compact star is a strongly magnetized\nneutron star, have been detected (\\cite{dower82}; \\cite{vaughan94}).\n\nQuasi-periodic oscillations (QPOs) were seen at 1.4~Hz, 5--20~Hz,\nand 100--200~Hz in {\\it EXOSAT\\,}\\ observations of Cir~X-1 in a bright state\n(\\cite{tennant87}; \\cite{tennant88}), but other observations at lower\nintensity showed no such QPOs (\\cite{oosterbroek95}). Based on these\ndata, it was suggested that Cir~X-1 is an atoll-source low-mass X-ray\nbinary (LMXB) that can uniquely reach the Eddington accretion rate and\nexhibit normal\/flaring branch QPOs (NBOs\/FBOs) at 5--20~Hz\n(\\cite{oosterbroek95}; \\cite{klis94}). Similar QPOs were observed in\n{\\it Ginga\\,}\\ observations of Cir~X-1 (\\cite{makino93}).\n\nObservations with the All-Sky Monitor (ASM) and Proportional Counter\nArray (PCA) on the {\\em Rossi X-ray Timing Explorer} ({\\it RXTE\\,}) have\nshown that since early 1996 (from the beginning of {\\it RXTE\\,}\\ monitoring)\nthe baseline intensity level of Cir~X-1 has remained bright\n($\\sim$1~Crab, 2--12~keV; $\\sim$1060~$\\mu$Jy at 5.2~keV) at all phases\nof the 16.55-d cycle, with both dips and flares associated with phase\nzero of the cycle (Shirey et~al.\\ 1996\\nocite{shirey96},\n1998\\nocite{shirey98:feb97}, hereafter Papers I \\& II;\n\\cite{shirey98:phd}).  During non-flaring phases (intensity $\\approx$\n1~Crab) of a cycle in 1996 March, the centroid frequency of a narrow\nQPO was observed to evolve between 1.3~Hz and 12~Hz and was strongly\ncorrelated with the cutoff frequency of low-frequency noise and with\nthe centroid frequency of a broad peak ranging from 20--100~Hz, at\n$\\sim$13 times the frequency of the lower frequency QPO\n(\\cite{shirey96})\\@. During portions of a more active cycle in 1997\nFebruary--March, a similar narrow QPO evolved between 6.8--32~Hz,\nwhile during other portions of the cycle, a broad QPO was observed at\n4~Hz (\\cite{shirey98:feb97})\\@.  By correlating the timing properties\nwith fragmented branches in hardness-intensity diagrams, we identified\nhorizontal, normal, and flaring branches. Thus, we interpreted the\nnarrow 1.3--32~Hz QPOs in Cir~X-1, including the 5--20~Hz QPOs\nobserved with {\\it EXOSAT\\,}, as horizontal-branch oscillations (HBOs), and\nthe broad 4~Hz QPOs as NBOs, thus calling into question the earlier\ninterpretation of the QPOs in Cir~X-1.\n\nIn this paper we present results from a 10-day set of high-efficiency\n{\\it RXTE\\,}\\ observations, including 56\\% coverage for 7~d starting 1.5~d\nbefore phase zero. Portions of these data show a complete Z-source\ntrack for Cir~X-1. These results confirm our previous interpretation\nbased on incomplete and shifted spectral tracks constructed from\nshorter and more widely spaced observations.  We show how both the\nFourier power spectra and the energy spectra evolve along the various\nspectral tracks.  We explore possible models for the energy spectrum\nand parameterize the evolution by fitting the data at various points\nalong the hardness-intensity track to a model consisting of a\nmulti-temperature ``disk blackbody'' and a higher-temperature\nisothermal blackbody.  \n\n\\section{Observations} \n\nThe PCA light curves and a hardness ratio (``broad color'') for the\n1997 June observations are shown in Figure~\\ref{fig:june97_10d}. These\ndata show only moderate variability for the first day of\nobservations. The source entered a phase of significant dipping during\nthe half day before phase zero (day 611.5), based on the radio\nephemeris (\\cite{stewart91}). The hardness ratio shows that dramatic\nspectral evolution, both hardening and softening, occurs during these\ndips.\n\nBy phase zero the main dipping episode ends, and the transition to the\nflaring state begins. While the intensity increases by more than a\nfactor of three in the lowest energy band (2--6.3~keV), the intensity\nbetween 6.3~keV and 13~keV does not climb at all, and above 13~keV it\ndecreases by a factor of about 10 over the first 1.5~days following\nphase zero. This anti-correlation of the low and high-energy intensity\nduring the transition results in a decreased hardness ratio after\nphase zero, as is observed in {\\it RXTE\\,}\\ ASM data (Papers\nI\\nocite{shirey96} \\& II\\nocite{shirey98:feb97};\n\\cite{shirey98:phd}). \n\nAfter a relatively smooth transition toward high total intensity\n(2--18~keV) during the first day following phase zero, the intensity\nbecomes highly variable (i.e., the ``active'' or ``flaring'' state)\nfor the remaining 7 days of the observation. The intensity in the hard\nband remains much lower than before phase zero but shows strong\nvariability with a peak value that gradually increases from about 24 to\n36 counts~s$^{-1}$~PCU$^{-1}$.  Based on the results in\n\\cite{shirey98:feb97} and those discussed below, we can identify\nshort-term variability within restricted limits as motion along\nbranches in hardness-intensity diagrams and the gradual evolution of\nthe limits as shifting of the branches.\n\n\n\\section{Complete Track in Hardness-intensity and Color-color Diagrams}\n\nThe color-color and hardness-intensity diagrams (CDs\/HIDs) for all\ndata in Figure~\\ref{fig:june97_10d} are shown in\nFigure~\\ref{fig:june97_cchid_all}. Only data from three of the five\nPCA detectors, PCUs 0, 1, and 2, were included in the diagrams, since\nPCUs 3 and 4 were not operating during portions of the observations.\nThese data cover a significant portion (10~d) of an entire 16.55~d\ncycle. Data from before day 612.5 generally had soft color ${\\stackrel{>}{_\\sim}}$\n1.5 and broad color ${\\stackrel{>}{_\\sim}}$ 0.325, while the data after day 612.5\ngenerally fell below those values.\n\nThe dips seen in Figure~\\ref{fig:june97_10d} appear as prominent but\nless dense tracks with two sharp bends in the CD (initially toward the\nright of the main arc-shaped locus) and one sharp bend in the HID\n($I<2.3$~kcounts s$^{-1}$ PCU$^{-1}$).  We show in another paper\n(Shirey, Levine, \\& Bradt 1999\\nocite{shirey99:dips}) that tracks with\nthese shapes are due to a variably absorbed bright spectral component\nplus an unobscured faint component.  Brandt et~al.\\\n(1996\\nocite{brandt96}) used a similar model to explain the spectral\nchanges during an intensity transition of Cir~X-1 observed with\n{\\it ASCA\\,}\\@.  Having identified absorption dip signatures, we now focus on\nnon-dip spectral behavior, which is presumably more directly related\nto the mechanisms of X-ray production.\n\nMost of the data in Figure~\\ref{fig:june97_cchid_all} fall along a\nsingle arc-shaped locus in the CD and a more complicated structure in\nthe HID\\@.  We use spectral bands with lower energies than those used\nfor the 1997 February--March observations in \\cite{shirey98:feb97};\nthis enhances the branch structure in the CD\\@.  When the diagrams for\nthe two observations are constructed in the same manner, with the same\nhardness ratios and only three PCUs rather than five, the data from\nthese two cycles cover approximately the same extent in both\ndiagrams. The high-efficiency coverage of the current observations\nresulted in tracks that are more complete than the tracks of the\nearlier observations. Unpublished data from several other cycles\nobserved with the PCA also fall in similar regions of the diagrams as\nthe data in Figure~\\ref{fig:june97_cchid_all}.\n\nThe detailed structure within the overall locus of CD\/HID points is\nrevealed in CD\/HID plots of data divided into shorter time segments\n(${\\stackrel{<}{_\\sim}}$12~h). In particular, four time segments labelled ``A'',\n``B'', ``C'', and ``D'' in Figure~\\ref{fig:june97_10d} have been\nselected for further study. The intensity from time segments A and B\n(days 609.93--610.16 and 610.66--610.90 respectively) shows minimal\nvariability in each energy channel, and thus these time segments\nproduce small CD\/HID clusters whose locations are indicated in\nFigure~\\ref{fig:june97_cchid_all}.\nThe source was more active during time segments C and D (days\n612.625--613.125 and 616.075--616.600 respectively), and the CD\/HID\ntracks from these times show several connected branches.  Enlarged\nviews of the CD and HID for these time segments are shown in\nFigure~\\ref{fig:june97_cchid_13_17}. The full ranges of the diagrams\nof Figure~\\ref{fig:june97_cchid_13_17} are indicated by dashed\nrectangular boxes in Figure~\\ref{fig:june97_cchid_all}.\nTracks of other time segments generally each resemble some portion of\nthe entire pattern shown in Figure~\\ref{fig:june97_cchid_13_17}, but\noften with a shifted position in the diagrams. \n\nThe data from segment~C in Figure~\\ref{fig:june97_cchid_13_17} include\nsome absorption dips, which result in tracks moving off the right side\nof the CD and the left side of the HID (and far beyond the limits of\nthe plot in both cases). The timing and spectral data associated with\nthese absorption dips will be omitted from analysis of the HID\nbranches.\n\nThe HID patterns reveal the shape of the full ``Z'' track as\npreviously inferred from the fragmented tracks in\n\\cite{shirey98:feb97}\\@.  Time segments C and D both show a horizontal\nbranch, a normal branch, and a flaring branch which turns above the\nnormal branch. In addition, segment~C exhibits a long nearly-vertical\nextension on the left end of the HB, while for segment~D, there is\nonly a small hint of an upward turn at the left end of the HB\\@. The\nHIDs show significant shifts of the HB and upper NB between the time\nsegments C and D, which were separated by several days.\n\nThe branches in the CDs of Figure~\\ref{fig:june97_cchid_13_17} are\nless well-separated than those in the HIDs. This was also the case for\nthe diagrams in \\cite{shirey98:feb97}\\@. However, the flaring branch\nclearly turns above the normal branch in the lower left part of the\ncurrent CDs, and the upturned left extension of the HID horizontal\nbranch of segment~C is marked by an increase in the slope in the upper\nright part of the associated CD.\n\nThe HID for segment~C is similar to that derived from\n{\\it RXTE\\,}\\ PCA observations of the Z~source Cyg~X-2 (\\cite{smale98}). The\nCyg~X-2 HID also shows a very prominent vertical extension of the\nHB\\@.  A similar upturned HB was reported in {\\it Ginga\\,}\\ and {\\it EXOSAT\\,}\\\nobservations of GX~5$-$1 (\\cite{lewin92:gx5-1};\n\\cite{kuulkers94:gx5-1}).\nThe upturned flaring branch is similar to the flaring branch observed\nin the CD for the Z~source GX~349+2 in recent {\\it RXTE\\,}\\ PCA observations\n(\\cite{zhang98}).\n\nThe HID track for time segment~C was divided into 20 regions which\nwere used to group data for further timing and spectral analysis.  The\n20 regions have been numbered as shown in\nFigure~\\ref{fig:june97_hid20reg}, with numbers increasing from the\nvertical HB, through the NB, to the FB\\@.  We show below that region~6\ndoes not adhere to the otherwise monotonic variation of\nspectral\/temporal characteristics with region number.  This region may\nbe an indication of an upward-shifted horizontal portion of the HB.\n\nDetails of the temporal variability of the intensity, hardness ratio,\nand HID region numbers for time segment~C are shown in\nFigure~\\ref{fig:june97_lc_hr_reg}.  During this half-day segment, the\nsource generally moves from lower to higher region number as the\nobservation progresses. Thus, the time series can be divided into four\nsub-segments which predominantly correspond to each portion of the HID\ntrack: the vertical and horizontal portions of the HB, the NB, and the\nFB\\@. Absorption dips occur in all but the flaring branch during this\nparticular data set; these are easily identified by brief intensity\ndips coupled with pronounced increases in broad color. No region\nnumber was assigned to most data points from dips since the HID\nregions in Figure~\\ref{fig:june97_hid20reg} were selected to avoid\ndips.\n\nThe light curves of the different branches\n(Fig.~\\ref{fig:june97_lc_hr_reg}) exhibit the following\ncharacteristics, excluding the behavior associated with the dips.\nWhen the source is on the vertical portion of the HB, the intensity\nevolves relatively smoothly, with a slight increase in the 2--6.3~keV\nband and a decrease of almost a factor of two in the 13--18~keV\nband. On the horizontal portion of the HB, the source shows a\nsubstantial increase in soft intensity and on average shows relatively\nsteady hard intensity. On the normal branch, the intensity is high in\nthe soft band while decreasing and highly variable in the hard band.\nThe NB\/FB transition occurs at lower intensity in all bands compared\nto most of the NB\\@. The flaring branch itself is then produced by\nhigh-variability ``mini-flares'' or bursts above the NB\/FB apex level\n(region~17).\n\nAlthough the HID regions were defined such that obvious absorption\ntracks were avoided, one brief dip, at day 612.98, occurred from\nregion 12 on the normal branch and placed a few points artificially\nacross regions 9, 7, and 5. These points are easily identified in\nFigure~\\ref{fig:june97_lc_hr_reg} (bottom plot) and are thus not\nincluded in subsequent timing and spectral analysis.\n\nLikewise, the highest mini-flares on the flaring branch actually\nextend beyond region~20 and cross regions 8 and 9. In fact a few such\npoints can even be seen above region~10 in\nFigure~\\ref{fig:june97_hid20reg}. The FB points that fell into HB\nregions can also be clearly identified as points with region numbers\nof 8 or 9 in the FB portion of\nFigure~\\ref{fig:june97_lc_hr_reg}. These are not included in\nsubsequent timing and spectral analysis.\n\n\\section{Evolution of the Power Density Spectrum}\n\nFourier power density spectra (PDSs) were computed for each 16~s of\ntime segment~C\\@.  Each transform used $2^{16}$ 244-$\\mu$s\n($2^{-12}$~s) time bins and covered the full 2--32~keV energy range.\nThe expected Poisson level, i.e., the level of white noise due to\ncounting statistics, was estimated taking into account the effects of\ndeadtime~(\\cite{morgan97}; \\cite{zhang95}; \\cite{zhang96}) and\nsubtracted from each PDS; this method tends to slightly underestimate\nthe actual Poisson level. For each of the 20 HID regions defined in\nFigure~\\ref{fig:june97_hid20reg}, an average PDS was calculated from\nthe power spectra corresponding to points in that region. The average\nPDSs were then logarithmically rebinned and are shown in\nFigure~\\ref{fig:june97_20pds}.\n\nThe general features of the power spectra are similar to those\nobserved in previous PCA observations (see Papers I\\nocite{shirey96}\n\\& II\\nocite{shirey98:feb97}).  In \\cite{shirey98:feb97}, based on\nfeatures of the power spectra associated with fragmented spectral\ntracks, we identified many of these features with those of a Z-source,\nand we discussed the properties of the QPOs on the horizontal and\nnormal branches. In the current observations, the characteristics of\nthe time variability are seen to evolve smoothly during a single 12-h\nobservation (segment~C).  The narrow QPO is observed to evolve from\n12~Hz in region~1 to 25~Hz in region~7 as the HID location moves down\nthe vertical extension of the HB (region 6 may be a shifted version of\n8). Across the horizontal portion of the HB (regions 8--11), the\nnarrow QPO fades into a knee close to 30~Hz, while the broad QPO\ngradually rises near 4~Hz. The broad QPO is present near 4~Hz along\nthe normal branch (regions 12--16). It is most prominently peaked in\nthe middle of the branch and weak at the bottom of the branch\n(region~17). On the flaring branch (regions 18--20), no QPOs are\npresent and the power spectrum shows only strong very low frequency\nnoise.\n\nThe PDS properties of the horizontal portion of the HB are somewhat\nsimilar to those of the upper normal branch of some Z sources, namely\nno significant evolution of the HBO frequency and weak NBOs.  However,\nwe will continue to refer to this branch as part of the the HB since\nother Z sources also show both vertical and horizontal portions of the\nHB (see above).\n\n\\section{Evolution of the Energy Spectrum}\n\\label{sec:june97_spec_qual}\n\nThe Standard2 data mode of the PCA instrument produces 129-channel\nenergy spectra every 16~s.  A parallel background file was constructed\nusing the ``pcabackest'' program\\footnote[1]{The background model was\ndefined in three files provided by the PCA instrument team at\nNASA\/GSFC: \\nl {\\tt pca\\_bkgd\\_q6\\_e03v01.mdl}, {\\tt\npca\\_bkgd\\_xray\\_e03v02.mdl}, and {\\tt pca\\_bkgd\\_activ\\_e03v03.mdl}.}\nprovided with the FTOOLS analysis package (version 4.0).  Average\npulse-height spectra (and background spectra) were constructed for\neach of the 20 HID regions, separately for each of the five\nPCUs. Version 2.2.1 response matrices were used in the analysis of\nthese spectra.  A 1\\% systematic error estimate was added in\nquadrature to the estimated statistical error (1~$\\sigma$) for each\nchannel of the spectra to account for calibration uncertainties.\nAlthough the instrument response matrix is imperfectly known, we can\nsafely assume that any spectral features that vary during the 12-hours\nspanned by time segment~C are due to evolution of the source spectrum.\nRepresentative spectra from the hard, bright, and soft extremes\n(regions 1, 11, and 17, respectively) of the evolution along the HID\ntrack are shown in Figure~\\ref{fig:june97_spec_branches}.\n\nThe evolution of the spectrum may be studied by inter-comparison of\nratios of pulse-height spectra from each region to that of a reference\nspectrum, from region~11 (Fig.~\\ref{fig:june97_pha_ratio}).\nThe spectrum is hardest in region~1, at the top of the vertical\nextension of the horizontal branch.  Motion down the branch\n(softening, regions~1--7) corresponds to pivoting of the spectrum\nabout $\\sim$7~keV, i.e., increasing intensity below $\\sim$7~keV and\ndecreasing intensity above that energy.\nMotion to the right across the horizontal portion of the HB\n(regions~8--11) corresponds to continued increasing low-energy\nintensity with modest softening, but with a nearly constant spectrum\nabove 12 keV\\@.  Note that in the hardness ratios of\nFigure~\\ref{fig:june97_hid20reg} the high-energy channel (6.3--13 keV)\nis dominated by photons near the lower bound of the interval.\n\nWhen the source moves down the normal branch (regions 11--17), the\nflux generally decreases across the entire 2.5--25~keV band but\ndecreases most significantly at high energy from region 11 to 15 (thus\nfurther softening).  Moving down the NB, the spectrum gradually\ndevelops a dip or step above $\\sim$9~keV and a peak slightly above\n10~keV.\nMotion up the FB (regions 18--20) is produced by increasing intensity\nat intermediate energies with a relatively constant spectrum at low\nenergy, thus hardening. The peaked feature near 10~keV becomes more\nprominent moving up the FB and will be discussed below.\n\n\\section{Selection of Spectral Models}\n\\label{sec:june97_spec_models}\n\nSpectral forms (e.g., blackbody emission, a power law, etc.) for use\nin fitting the spectra from the HID regions were explored by first\napplying them to high-quality spectra from time segments A and B (see\nFig.~\\ref{fig:june97_10d}). Variability in both of these segments was\nlimited to less than 10\\% in all energy bands between 2.5--18~keV. We\nthus constructed a single (averaged) pulse-height spectrum, known\nhereafter as spectrum~A or spectrum~B, for the entire 17--19~ks of\neach segment.  Errors in these spectra are dominated by the 1\\%\nsystematics at all energies up to $\\sim$20~keV.\n\nThe timing properties measured throughout data sets A and B indicate\nthat time segment~A falls on the vertical portion of the HB (strong\nnarrow QPO at 8.4--11.5~Hz) and time segment~B falls near the HB\/NB\napex (weak narrow QPO above 30~Hz and\/or the broad 4~Hz QPO).  The\nlocations of time segments A and B in\nFigure~\\ref{fig:june97_cchid_all} indicate that the branches are\nsignificantly shifted relative to those of segments C and D.\n\nRemillard et~al.\\ (1998\\nocite{remillard98}) studied version 2.2.1 PCA\nresponse matrices using Crab nebula data and found that the response\nmodel was most accurate for PCUs 0, 1, and 4 (of the five PCA\ndetectors) and for energies between 2.5~keV and 25~keV\\@. Thus, in\nfitting spectra we only include data from PCUs 0, 1, and 4 and from\nenergy channels corresponding to 2.5--25 keV\\@.  Spectra from each of\nthese detectors are fit separately. Fit parameters reported are the\naverage values for PCUs 0, 1, and, when appropriate, 4.  Errors are\nconservatively estimated as the entire range encompassed by the 90\\%\nconfidence intervals from each of the detectors. PCU~4 consistently\ngives lower normalizations for fitted spectral components, so fit\nparameters from that detector are not included when computing the\naverage normalizations and flux values and their errors.  Spectrum~B\nwas not constructed for PCU~4 since that detector was turned off\nduring part of time segment~B.\n\nInterstellar photoelectric absorption was included in all models. The\nabsorption model used solar abundances (\\cite{anders82}) and\ncross-sections given by Morrison \\& McCammon (1983\\nocite{morrison83}).\n\nSeveral single-component models were fit to spectra~A and\nB\\@. Blackbody and power-law models fit very poorly in both cases, as\ndid a multi-temperature ``disk blackbody'' spectrum (\\cite{mitsuda84};\n\\cite{makishima86}; model ``diskbb'' in XSPEC), with reduced $\\chi^2$\n($\\chi^2_r$) values of 22--545.  A thermal bremsstrahlung model\nprovided a better fit to spectrum~B ($\\chi^2_r=4.0$), but fit\nspectrum~A poorly ($\\chi^2_r=34$).\nA relatively good fit was achieved for both spectra with a modified\nbremsstrahlung model (see Table~\\ref{tab:june97_fitsAB}) which\nincludes the effects of Compton scattering of bremsstrahlung photons\nto higher energy in an optically thick plasma cloud (\\cite{compls};\nmodel ``compLS'' in XSPEC).\n\nA number of two-component models were also fit to these two spectra.\nA model using a disk blackbody and power law did not fit well\n($\\chi^2_r$=3--5), mainly because a single power-law slope does\nnot adequately describe the spectrum at high energy.\nA blackbody with $kT {\\stackrel{>}{_\\sim}}$ 2~keV is often included in models of the\nhard X-ray emission of LMXBs thought to contain a neutron star, where\nemission from or near the surface might produce high-temperature\nblackbody emission with a small effective area.\nTwo blackbodies ($kT \\sim$ 1.1~keV and 2.2~keV) fit moderately well\n(see Table~\\ref{tab:june97_fitsAB}), but required negligible\ninterstellar absorption. The low absorption is inconsistent with\nprevious measurements from {\\it ASCA\\,}\\ and {\\it ROSAT\\,}\\ (both sensitive below\n2~keV where the absorption is most easily constrained) which were used\nto estimate the interstellar column density to be\n$N_H=$(1.8--2.4)$\\times 10^{22}$~cm$^{-2}$\n(\\cite{brandt96}; \\cite{predehl95}).\n\nTwo models commonly used to fit Z-source energy spectra are the\n``Western model'' and the ``Eastern model'' (\\cite{hasinger90};\n\\cite{asai94}). The Western model consists of blackbody emission from\nthe hot surface of the neutron star or from a boundary between the\naccretion disk and surface, plus a Boltzmann-Wien component due to\nunsaturated Comptonization of soft photons by hot electrons\n(\\cite{white86}; Schulz, Hasinger, \\& Tr\\\"{u}mper\n1989\\nocite{schulz89}; Langmeier, Hasinger, \\& Tr\\\"{u}mper\n1990\\nocite{langmeier90}; \\cite{schulz93}).  The Eastern model also\nincludes blackbody radiation emitted from or near the surface, plus\nemission from a multi-temperature accretion disk (\\cite{mitsuda84};\n\\cite{hoshi91}; \\cite{hirano95}).\n\nThe Western model, a power law with a high-energy exponential cutoff\nplus a blackbody, fit well (see Table~\\ref{tab:june97_fitsAB}), but\nthe best-fitting high-energy cutoff energy ($E_{cut}\\approx1.7$~keV)\nwas so low relative to the PCA bandpass (${\\stackrel{>}{_\\sim}} 2$~keV) that the\npower law photon index was not well constrained.\nThe Eastern model, a disk blackbody with temperatures at the inner\nedge of the disk of 1.5--1.8~keV, plus a $\\sim$2~keV blackbody, fit\nspectra A \\& B quite well (see Table~\\ref{tab:june97_fitsAB} and\nFig.~\\ref{fig:june97_specA_diskbb_bb}) and gave absorption column\ndensities roughly consistent with the {\\it ASCA\\,}\\ and {\\it ROSAT\\,}\\ values.\nAlthough a number of other two-component models also produce similar\nquality fits, the Eastern model is used below to provide a physically\nmotivated parameterization of the spectra from the HID regions.\n\nThe Eastern model fit to spectrum~A\n(Fig.~\\ref{fig:june97_specA_diskbb_bb}) shows peaked residuals at\n6--7~keV, suggesting the presence of an emission line, probably iron\nK$\\alpha$. Very similar residuals appear in most of the fits discussed\nabove for both spectra A and B\\@. Addition of a Gaussian line to the\nmodels does in fact improve the fits in almost all cases; however, the\nbest-fitting line often has an extremely large Gaussian width\n($\\sigma>1$~keV). The energy resolution of the PCA is about 1~keV FWHM\nat 6~keV; thus it is difficult to place reliable constraints on\nparameters such as the centroid and width of a narrow component.  We\nhave not included an emission line in the fits reported in\nTable~\\ref{tab:june97_fitsAB}. The presence of an emission line near\n6.4~keV is discussed in more detail in Shirey et~al.\\\n(1999\\nocite{shirey99:dips}) in conjunction with spectra of absorption\ndips, which show the line more prominently.\n\n\\section{Fits to Spectra from 20 HID Regions}\n\nA disk blackbody plus isothermal blackbody model was fit to the\naverage spectrum for each of the 20 HID regions.  Representative fits\nand residuals are shown in Figure~\\ref{fig:june97_spec_hidfits}.  The\nresulting fit parameters are listed in Table~\\ref{tab:june97_fitsHID}\nand plotted versus HID region number in\nFigure~\\ref{fig:june97_fit_params}. The distance to Cir~X-1 has been\nestimated to be about 6--10~kpc (\\cite{stewart91}; \\cite{goss77}), so\nwe adopt a value of 8~kpc in converting blackbody and disk blackbody\nnormalizations to radii.\n\nAlthough both spectra A and B are fit well by the Eastern model, the\nreduced chi-squared values in Table~\\ref{tab:june97_fitsHID} indicate\nthat none of the fits for the 20 regional spectra are formally\nacceptable.  The fit results must therefore be regarded as an\napproximate description of the spectrum and its evolution.  We\nemphasize that here, as in any case, caution is advised in drawing\nphysical conclusions from the best-fit model parameters.\n\nThe spectra along the horizontal branch (regions 1--11) were all fit\nrelatively well. The residuals for these fits (see\nFig.~\\ref{fig:june97_spec_hidfits}) are similar in structure to those\nfor spectra A and B above. Thus they also suggest the presence of an\nemission line from iron. These spectra all show column densities of\n1.8--2.3$\\times10^{22}$~cm$^{-2}$, consistent with the {\\it ASCA\\,}\\ and\n{\\it ROSAT\\,}\\ values discussed above.  \nThe temperature of the $\\sim$2.0~keV blackbody is relatively stable on\nthe HB\\@ (see Fig.~\\ref{fig:june97_fit_params}). The temperature of\nthe inner disk decreases from region 1 to region 5 (down the vertical\nportion of the branch) and then stabilizes at $\\sim$1.3 keV\\@. The\ncooling is at least in part responsible for the pivoting of the\nspectrum on the vertical portion of the HB.\n\nThe inner radius of the disk blackbody component, times a factor of\norder unity involving the inclination angle of the disk, increases\nfrom 19 to 33~km, while the radius of the blackbody remains\nbetween 3~and 4~km.  These size scales are consistent with the\nhypothesis that these components arise from emission close to a\nneutron star. In this model, an increasing inner radius of the\naccretion disk is the most significant factor in producing the HB\ntrack; however, one should use caution in interpreting this as an\nactual physical radius.  The inclination angle of the disk in Cir~X-1\nis unknown but might be high since absorption dips are observed.\n\nFrom region~1 to 11, the total 2.5--25~keV flux increases\nmonotonically, with the exception of region~6, from\n2.89$\\times10^{-8}$ to 4.35$\\times10^{-8}$~erg~cm$^{-2}$~s$^{-1}$,\ncorresponding to 1.2--1.8 times the Eddington luminosity limit for a\n1.4~${\\rm M}_\\odot$\\ neutron star at 8~kpc.\n\nAlong the normal branch, the quality of the fits decrease from\nregion~12 to region~17, as indicated by increasing $\\chi^2_r$ values\n(see Table~\\ref{tab:june97_fitsHID}). The absorption column density\ngradually decreases by a factor of two, but this may be related to the\ndecreasing fit quality.\nThe inner radius and temperature of the disk blackbody change only\nslightly on the normal branch. In contrast, the $\\sim$2~keV blackbody\nbegins to fade on the upper portion of the normal branch (regions\n12--14), as indicated by a decreasing radius for the emission area.\nThe fading blackbody is illustrated in\nFigure~\\ref{fig:june97_photon_spec}, which shows the modeled incident\nspectra and both model components for several spectral fits. By the\nmiddle of the normal branch, the $\\sim$2~keV blackbody has faded\nentirely and fits have lower $\\chi^2_r$ values without it. Thus, the\nblackbody is omitted from the fits for regions 15--20. The residuals\nbelow $\\sim$6~keV continue to appear similar to those on the HB (see\nFig.~\\ref{fig:june97_spec_hidfits}).  The peak at $\\sim$6.5~keV\nbecomes broader and more complicated than on the HB\\@, and the dip and\npeak above 8~keV become more prominent.\n\nOn the flaring branch (regions 18--20), the fit quality decreases\nfurther, accompanied by very low values for the absorption column\ndensity. A number of other spectral models were fit to the HID-region\nspectra, and all failed to satisfactorally fit the spectra from the\nlower portion of the HID track (region number 14 and greater). A\nsignificant contribution to the high $\\chi^2$ values on the flaring\nbranch is due to the feature near 10~keV.  Addition of a Gaussian line\nor an absorption edge at 9--11~keV does improve the fits somewhat, but\nthese components cannot account for all the residuals near that\nenergy.  A combination of a line {\\em and} an edge near 10~keV can\nadequately fit the residuals, but such features are difficult to\njustify physically at that energy.  Even hydrogen-like iron can be\nruled out as a possible cause due to the high energy of the feature.\nMany X-ray pulsars show cyclotron absorption features at high\nenergy. Inclusion of a cyclotron absorption component in the spectral\nmodel results in a fit similar in quality to that of an absorption\nedge. However, such features require magnetic fields of $\\sim\n10^{12}$~G, which would be expected to result in strong pulsations\nrather than Z or atoll behavior.\n\n\n\\section{Discussion}\n\nOur spectral and timing analysis of the current observations shows\nclear evidence for Z~source behavior in Cir~X-1. This is significant\nbecause Cir~X-1 was reported to exhibit atoll source behavior at lower\nintensity (\\cite{oosterbroek95}). Earlier {\\it RXTE\\,}\\ observations, each\nlasting about two hours and separated by about two days, showed\nfragments of one or two spectral branches in hardness-intensity\ndiagrams (\\cite{shirey98:feb97})\\@. In the much more extensive\nobservations presented in this paper, we have found longer 12-hour\nsegments (time intervals C and D) which clearly exhibit all three\nbranches of a Z~source.  We have demonstrated that these complete Z\ntracks shift in the HID, confirming the behavior we inferred from the\nfragmented tracks of the previous observations.\n\nThe current data also allow us to demonstrate how the timing\nproperties evolve along the complete HID track of Cir~X-1 and confirm\nour original identification, in \\cite{shirey98:feb97}\\@, of horizontal\nand normal branch QPOs. Fourier power spectra for different regions of\nthe complete HID track show continuous evolution from the narrow QPO\n(increasing in frequency from 12~Hz to 30~Hz in the current\nobservations) on the horizontal branch, to the broad 4~Hz QPO on the\nnormal branch, to only very low frequency noise on the flaring branch.\nProperties of the fast timing characteristics associated with spectral\nbranches in Cir~X-1 were discussed in \\cite{shirey98:feb97}\\@.  For\nthe remainder of this discussion we focus on the properties of the\nenergy spectrum.\n\nWe tried fitting energy spectra of Cir~X-1 with various simple models.\nThe spectra for time intervals A and B were well fit using the Western\nand Eastern models (see discussion below), but no simple spectral form\nwas found that fit the range of spectra seen during time interval~C\\@.\nWe have not attempted to go beyond simple parameterized spectral\nmodels, e.g., by computing model spectra based on the \"unified model\"\nof Lamb and collaborators (\\cite{lamb89}; Psaltis, Lamb, \\& Miller\n1995\\nocite{psaltis95}; \\cite{psaltis98}), which was proposed to\nexplain the X-ray spectra and rapid variability of Z~sources.  Such\nsophisticated models may be necessary to correctly interpret the\nspectral changes in Cir~X-1 and other Z~sources.\n\nThe fits of spectra A and B with the Western model yielded a\ncutoff energy of $\\sim$1.7 keV for the Comptonized (Boltzmann-Wien)\ncomponent.  The cutoff energy in GX 5-1 was found to be 1--3~keV\n(\\cite{asai94}), similar to our results for Cir~X-1, but was found\nto be higher, 4--6 keV, in Cyg~X-2 (\\cite{hasinger90}).  We did not\nuse the Western model in parameterizing evolution associated with the\n``Z'' track because the cutoff energy in Cir~X-1 is low relative to\nthe PCA bandpass, resulting in a poorly constrained power law index\nand absorption column density.\n\nParameters for the Eastern model were more well-constrained (see\nTable~\\ref{tab:june97_fitsAB}), and thus this model was used to\nparameterize the spectral variations associated with the\nhardness-intensity track.  In this model, motion along the HB is\nmainly associated with an increasing inner radius of the disk\n(increasing disk blackbody normalization) but also by a decreasing\ninner disk temperature. This would imply that, as the luminosity\nincreases across the HB, the inner edge of the disk is pushed further\naway from the surface. It is not clear how this is related to the\nincreasing QPO frequency, which would typically be expected to require\na {\\em decreasing} radius if the QPOs were related to Keplerian motion\nat the inner edge of the disk, e.g., through the magnetospheric beat\nfrequency model (\\cite{alpar85}; \\cite{lamb85}).  \n\nFits of the Eastern model to energy spectra from the normal branch\nindicate that the $\\sim$2~keV blackbody gradually fades away, leaving\nonly the disk blackbody. This is similar to the result obtained when\nthe Eastern model was fit to the spectrum of Cyg~X-2, where the\nblackbody luminosity decreases from the HB to the FB\n(\\cite{hasinger90}).  Furthermore, the FB of GX~5$-$1 is characterized\nby intensity dips which in the Eastern model can be explained by\ndisappearance of the blackbody component, suggesting that accretion\nflow onto the neutron-star surface is interrupted (\\cite{mitsuda84}).\n\nOn the lower NB, a feature in the spectrum develops above 10~keV. This\nfeature becomes more prominent on the flaring branch.  A very similar\nline-like feature at $\\sim$10~keV was reported in {\\it Ginga\\,}\\ observations\nof the Z~source GX~5$-$1 (\\cite{asai94}). In GX~5$-$1, as in Cir~X-1,\nthe feature was present on the lower NB and stronger on the FB; we\nthus suggest that these features of the two sources may be of similar\norigin.  Asai et~al.\\ showed that a peak near 11~keV occurs in the\ncorrelation coefficients of the time-series data of different energy\nbands versus the 1.7--4.0 keV band. This demonstrates a temporal\ncharacter in the narrow band at 11~keV different than that at adjacent\nenergies. In turn, this gives assurance that the line-like feature at\nthat energy is not the result of the continuum model used to fit the\nspectrum but is intrinsic to the source.\nWe carried out similar cross-correlation analysis for each of the 20\nHID regions in our study. The cross-correlation results, relative to\nthe 2.5--2.9~keV band, from three representative regions are shown in\nFigure~\\ref{fig:june97_cross_corr}.  We find a clear peak in the\ncross-correlation coefficient at about 11~keV in regions 19 and 20 of\nthe FB\\@, further confirming the similarity of the spectral features\nin Cir X-1 and GX~5$-$1.  Regions 16 and 17 of the lower NB show an\nabrupt drop in the cross-correlation coefficient above 8--10~keV.  The\ncross-correlation coefficients on the HB and upper NB, where the line-\nor edge-like 10~keV spectral feature is absent or weak, generally show\nmuch less remarkable behavior.\n\nAsai et al.\\ found that that the spectral feature near 10~keV in\nGX~5$-$1 was better fit with a Gaussian line rather than, e.g., an\nabsorption edge.  However, as we mentioned above, such a feature\ncannot be produced by even hydrogen-like iron. Asai et~al.\\ discuss\nseveral of the mechanisms that could possibly produce a line at\n$\\sim$10~keV. For example, emission from a heavy element such as Ni\ncould produce the line, but is unlikely since iron should be far more\nabundant than the heavier elements. Alternately, a line could be\nblue-shifted due to motion in a relativistic jet or rotation in the\naccretion disk, but extreme conditions would be required to boost the\nenergy up to $\\sim$10~keV and one might expect a red-shifted line to\nalso be observed in the X-ray spectrum.  As mentioned above, we find\nthat in Cir~X-1 an emission line component alone is insufficient to\nproduce the observed structure of the feature. Both line-like and\nedge-like components may be required to explain this unusual spectral\nfeature at high energy.\n\nCir X-1 exhibits a number of unusual features in its Z-source behavior\nin addition to the 10~keV spectral feature.  Its HBOs are observed at\nfrequencies as low as 1.3~Hz (\\cite{shirey96}), an order of magnitude\nlower than those of other Z~sources.  The highest frequency reached by\nHBOs in Cir X-1, at the HB\/NB apex, is 30--32~Hz\n(\\cite{shirey98:feb97}), about a factor of two below the extreme HBO\nfrequency in other Z sources.  Power spectra of Cir~X-1 show a broad\nhigh-frequency peak, centered at 20--200~Hz, which shifts in frequency\nmaintaining an almost constant ratio with the HBO frequency\n(\\cite{shirey96}; \\cite{tennant87}).  Cir X-1 shows a long vertical\nextension of the HB\\@.  The entire HID track shows very large\ncolor\/intensity shifts, possibly associated with the presumed\neccentric 16.55-d binary orbit (\\cite{shirey98:feb97}).  Atoll\nbehavior has been reported at lower intensity (\\cite{oosterbroek95}).\nIt is likely that some or many of these unusual features are related\nby some physical property of the system, indicating that Cir~X-1 may\nprovide important constraints on models of low-mass X-ray binaries.\n\n\\acknowledgements \n\nWe would like to acknowledge the {\\it RXTE\\,}\\ teams at MIT and GSFC for\ntheir support.  In particular we thank E.~Morgan, R.~Remillard,\nW.~Cui, and D.~Chakrabarty for useful discussions related to this\nwork.  We thank D.~Psaltis for useful discussions regarding Z-source\nspectral models. We also appreciate the detailed comments and\nsuggestions provided by the referee.  Support for this work was\nprovided through NASA Contract NAS5-30612.\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{...}\n\n\nAn important source of photons in nuclear collisions are ``prompt photons''. They are produced in initial hard parton collisions, through two mechanisms\\footnote{This division into two mechanisms is useful conceptually, although it can only be done rigorously at leading order.}\n: \n\t(i) processes such as Compton scattering ($q g \\to q \\gamma$) \n\tand quark annihilation ($q \\bar{q} \\to g \\gamma$), \n\twhere a photon is a direct final state of the hard parton interaction; and\n\t(ii) fragmentation, where a hard parton-parton collision (e.g. $q \\bar{q} \\to q \\bar{q}$) is followed by a photon being radiated from one of the final state partons.\n   \n\nPrompt photon production in proton-proton (p+p) collisions has been calculated in perturbation theory at next-to-leading order in $\\alpha_s$;\nagreement with p+p data is good\nin a wide range of center-of-mass energies~\\cite{Aurenche:2006vj}. \nPrompt photons remain an important source of direct photons in heavy ion collisions. They are generally said\nto scale with the number of binary nucleon collisions, up to modest corrections from isospin and nuclear effects on the parton distribution functions; this statement is supported by the good agreement of high $p_T^\\gamma$ photon spectra ($\\gtrsim 5-10$~GeV)  in heavy ion measurements with binary-scaled prompt photon calculations (or p+p photon data).\nAt low  $p_T^\\gamma$, the production of prompt photon in heavy ion collisions is more complex.\nThe fragmentation component\nof prompt photons, subdominant at high $p_T^\\gamma$, is the dominant source of prompt photons at low $p_T^\\gamma$. These fragmentation photons are affected by parton energy loss.\nMoreover prompt photons are not the only source of low $p_T^\\gamma$ direct photons in heavy ion collisions: additional photons originate from (i) blackbody radiation produced by the hot expanding plasma (``thermal photons''), and (ii) ``jet-medium photons'' produced in parton-plasma interactions (interactions which also lead to parton energy loss) \\cite{Turbide:2007mi}. \nMeasurements indicate that the low $p_T^\\gamma$ direct photon spectra measured in heavy ion collisions is considerably larger than binary-scaled photon spectra measurements from proton-proton collisions. This suggests that thermal and jet-medium photons more than compensate for the suppression of prompt photons resulting from parton energy loss. Additional simulations are still required to confirm this scenario\\footnote{Almost all recent comparisons with data rely on prompt photon calculations that neglect the effect of energy loss.\nUp-to-date calculations of jet-medium photons and of prompt photons that account for parton energy loss will be essential to clarify the status of model-to-data comparisons.\n}.\n\n\n\nRecent work by the PHENIX Collaboration brought the role of prompt photons back to the front of the discussion~\\cite{Adare:2018wgc}. They observed that the \\emph{centrality dependence} of the direct photon multiplicity is consistent with binary collisions scaling:\n\\begin{equation}\n\\left. d N_{direct}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma}\n\\approx N_{binary}^{\\alpha} K\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma \\right) \n\\label{eq:phenix_scaling}\n\\end{equation}\nwhere the number of binary collisions $N_{binary}$ is a \ncalculated\nfrom the Monte-Carlo Glauber model and the exponent $\\alpha$ was found to be consistent with $1$. The normalization factor $K(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma )$ depends on the center-of-mass energy $\\sqrt{s_{NN}}$, and the multiplicity cutoff $p_T^\\gamma$. However the centrality dependence itself of the multiplicity appears not to depend on the cutoff $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$ (i.e. $\\alpha\\approx 1$ independent of $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$).\nThe same conclusions had already been obtained for one center-of-mass energy in Ref.~\\cite{Adare:2014fwh}.\n\nIn what follows we provide a theoretical perspective on this binary-scaling from the point of view of thermal photons, medium-modified prompt photons and jet-medium photons.\n\n\n\\paragraph{Medium-modified prompt photons and jet-medium photons}\nThe multiplicity of sufficiently high $p_T^\\gamma$ prompt and jet-medium photons in heavy ion collisions can be written schematically as\n\\begin{align}\n\\left. d N_{j}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma} =\n\\frac{N_{binary}}{\\sigma^{inel}_{NN }} & \\int \\frac{d \\phi}{2 \\pi} \\int_{p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma} d p_T p_T \\left[ \\sum_{a, b, c} f_{a\/A} \\left(x_a,Q\\right) \n\\otimes  f_{b\/A}\\left(x_b,Q \\right) \\otimes \\, d\\hat{\\sigma}_{a b \\to c \\gamma}  \\right. \\nonumber \\\\\n& \\left. +\\sum_{a, b, c, d} f_{a\/A} \\left(x_a,Q\\right) \n\\otimes  f_{b\/A}\\left(x_b,Q \\right) \\otimes \\, d\\hat{\\sigma}_{a b \\to c d}  \\otimes D^M_{\\gamma\/c} \\left(z_c,Q\\right)\n\\right]\n\\label{eq:prompt_jetmedium}\n\\end{align}\nwhere $f(x,Q)$ are nuclear parton distribution;\n$d\\hat{\\sigma}$ are perturbative parton cross-sections; and $\\otimes$ represents a convolution over the kinematic variables. All effects from the quark-gluon plasma --- parton energy loss and jet-medium photon production --- are absorbed into a medium-modified fragmentation function $D^M_{\\gamma\/c} \\left(z_c,Q\\right)$. At a fixed collision energy, the centrality dependence of  Eq.~\\ref{eq:prompt_jetmedium} originates from the $N_{binary}$ pre-factor as well as from $D^M_{\\gamma\/c} \\left(z_c,Q\\right)$. This medium-modified fragmentation function is assumed to encode the same non-perturbative fragmentation into photons as the vacuum function. However, the perturbative sector of the vacuum fragmentation function is modified to account for the presence of a medium: it includes\\footnote{We are not aware of numerical studies of photons that include both medium-modified high and low-virtuality showering. Simulations that include only the latter have been performed previously, however (e.g. Ref.~\\cite{Turbide:2005fk}).} (i) perturbative DGLAP-like photon emission between virtuality $Q$ and a lower virtuality $Q_0$; and (ii) perturbative low-virtuality showering, which simultaneously leads to medium-induced photon emission and medium-induced parton energy loss. In vacuum, only the  DGLAP-like photon emission is present.\nThe final non-perturbative fragmentation into photons is performed after this perturbative showering. Unlike vacuum fragmentation functions, the medium-modified $D^M_{\\gamma\/c} \\left(z_c,Q\\right)$ is not universal: it must be evaluated dynamically to account for the exact profile of the medium.\n\n\nTypically the spectra of prompt and jet-medium photons fall off rapidly with $p_T^\\gamma$ and the photon multiplicity is dominated by the smallest $p_T^\\gamma$ regions of the integrand ($p_T^\\gamma \\sim p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$).\nThis presents a challenge for Eq.~\\ref{eq:prompt_jetmedium}.\nIntrinsically in collinear perturbative calculations, there is a lower scale $Q_0 \\sim 1-2$~GeV separating the perturbative and non-perturbative sectors. In Eq.~\\ref{eq:prompt_jetmedium}, the scale $Q>Q_0$ is expected to be of the order of the photon transverse momentum: $Q\\sim p_T^\\gamma$. This implies that multiplicities evaluated with small $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma\\sim Q_0$ may be dominated by $p_T^\\gamma$ regions where the perturbative calculations are the least reliable. Careful numerical simulations of medium-modified prompt and jet-medium photons will be essential to clarify the situation. Alternative approaches to calculate low momentum jet and photon production may also be necessary (see e.g. Ref.~\\cite{Hattori:2016jix} and references therein).\n\n\\paragraph{Thermal photons}\nThermal photons are calculated by convoluting a thermal emission rate $E d^3 \\Gamma\/d^3 p$ with a spacetime profile of the plasma obtained from hydrodynamics:\n\\begin{equation}\n\\left. d N_{th}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma} =\n\\int d^4X \\int \\frac{d \\phi}{2 \\pi} \\int_{p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma} d p_T p_T  \\left [E \\frac{d^3 \\Gamma}{d^3 p}\\left(P \\cdot u(X),T(X),\\pi^{\\mu\\nu}(X),\\Pi(X)\\right) \\right]\n\\label{eq:thermal}\n\\end{equation}\nwhere $T$, $u$, $\\pi^{\\mu\\nu}(X)$,$\\Pi(X)$ are the temperature, flow velocity, shear tensor and bulk pressure profiles of the plasma, and $P$ is the photon four-momentum.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=0.38\\linewidth]{thermal_pTcuts_sqrts200_Nbin}\n\t\\hspace{2.cm}\n\t\\includegraphics[width=0.39\\linewidth]{thermal_pTcuts_sqrts5020_Nbin}\n\t\\caption{Thermal photon multiplicity as a function of the number of binary nucleon collisions, obtained by varying the centrality for (a) Au-Au at $\\sqrt{s_{NN}}=200$~GeV, and (b) Pb-Pb at $\\sqrt{s_{NN}}=5020$~GeV.}\n\t\\label{fig:thermal_vs_bin}\n\\end{figure}\n\nFigure~\\ref{fig:thermal_vs_bin} shows the power law scaling of the photon multiplicity as a function of $N_{binary}$\n, for Au-Au collisions at $\\sqrt{s_{NN}}=200$~GeV and Pb-Pb collisions at $\\sqrt{s_{NN}}=5020$~GeV. Three different multiplicity cutoff  $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$ are shown: 0.5, 1 and 1.5~GeV. The results can be fitted reasonably well with a linear function; the slopes\nare indicated on the figures. Based on Figure~\\ref{fig:thermal_vs_bin}, we can write the thermal photon multiplicity as:\n\\begin{equation}\n\\left. d N_{th}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma}\n\\approx N_{binary}^{\\alpha\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma\\right)} M\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma \\right) \n\\label{eq:thermal_Nbin_scaling}\n\\end{equation}\nwhere $M(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma ) $ is a function that can be tabulated, and $\\alpha(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma)$ is an exponent for $N_{binary}$ that is approximately $1.1-1.3$. This exponent increases as the cutoff $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$ is increased. The exact numerical values of $\\alpha(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma)$ depend on the details of the hydrodynamic simulation (initial conditions, viscosities, \\ldots) as well as the photon emission rates; this exact dependence will need to be studied in greater details\\footnote{In particular, we believe an observation made in Ref.~\\cite{Shen:2013vja} --- that an increase in the photon emission rate at low temperatures leads to smaller slopes for the centrality dependence --- may not as general as previously thought.}.\nNevertheless, the $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$ \\emph{dependence} of these values is a robust prediction: there is no expected scenario where the thermal photon multiplicity is independent from this cutoff.\n\n\\paragraph{Discussion \\& Outlook}\n\n\n   \n\nAny discussion of the multiplicity scaling must be made with other photon observables in mind, in particular the photon momentum anisotropy $v_2$. Thermal photons generally have a large $v_2$ at low $p_T^\\gamma$, while medium-modified prompt and jet-medium are generally understood to have a small $v_2$~\\cite{Turbide:2005bz}. The measured photon $v_2$ is large, and is interpreted as favoring thermal photons as the dominant source of low $p_T^\\gamma$ direct photons.\n\n\nSumming thermal, prompt and jet-medium photons, we write the direct photon multiplicity:\n\\begin{equation}\n\\left. d N_{direct}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma}\n\\approx  \\left[ \n\\left. d N_{j}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma}\n+ N_{binary}^{\\alpha\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma\\right)} M\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma \\right)  \\right]\n\\end{equation}\n\nInevitably the centrality dependence of this prediction depends on the relative size of thermal photons as opposed to that of the medium-modified prompt photons and jet-medium photons. Given we do not currently have calculations of the latter two sources in a setting consistent with our thermal photon calculations, we limit our discussion to asymptotic scenarios.\n\nThe first limit is a multiplicity dominated by jet-medium and medium-modified prompt photons. To simultaneously describe the measured photon $v_2$, these jet-medium and prompt photons need to have a large $v_2$. This scenario is arguably not supported by older calculations~\\cite{Turbide:2005bz}. Whether calculations that include recent developments in hydrodynamics simulations (e.g. fluctuating initial conditions, initial flow and bulk viscosity) would produce a significantly larger momentum anisotropy for these photons will need to be studied numerically.\n\nIf thermal photons dominate the multiplicity, the centrality dependence of the photon multiplicity should be close to $N_{binary}^{\\alpha}$ with $\\alpha=\\alpha(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma)$ discussed above. Importantly measurements could rule out this possibility with improved constraints on the cutoff \\emph{independence} of the $\\alpha$ exponent.\n\nGiven that intermediate scenarios are also possible,\nadditional measurements and calculations are essential to determine if the observed centrality scaling can be explained by our current understanding of photon production, or if there is a new direct photon puzzle.\n\n\\paragraph{Acknowledgements} We are grateful to the ALICE and PHENIX photon working groups for insightful discussions. This work was supported by the U.S. Department of Energy\nunder Award Numbers DE-FG02-05ER41367 (JFP) and DE-SC0013460 (CS), and by the Natural Sciences and Engineering Research Council of Canada. SM acknowledges funding from The Fonds de recherche du Qu\\'ebec - Nature et technologies \n(FRQ-NT) \nthrough the Programme de Bourses d'Excellence pour \\'Etudiants \\'Etrangers.\nComputations were made \non the supercomputer Guillimin, managed by Calcul Qu\\'ebec and Compute Canada and funded by the Canada Foundation for Innovation (CFI), Minist\\`ere de l'\\'Economie, des Sciences et de l'Innovation du Qu\\'ebec (MESI) and FRQ-NT. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nA minor of a graph $G$ is a graph obtained from $G$ by a succession of\nedge deletions, edge contractions and vertex deletions.\nAll graphs we consider are simple, i.e. without loops or multiple edges.\nThe following theorem of Mader~\\cite{mader1} bounds the number of edges in a\n$K_r$-minor free graph.\n\\begin{thm}[Mader, 1968,~\\cite{mader1}]\nFor $3 \\leq r \\leq 7$, any $K_r$-minor free graph $G$ on $n\\ge r$\nvertices has at most $(r-2)n - {{r-1}\\choose{2}}$ edges.\n\\end{thm}\nNote that since $|E(G)| = \\frac{1}{2} \\sum_{u \\in  V(G)}\\,\\deg(u)$,\nthis theorem implies that every $K_r$-minor\nfree graph $G$, for $3 \\leq r \\leq 7$, is such that $\\delta(G)\\le\n2r-5$, where $\\delta(G)$ denotes the minimum degree of $G$.\nThis property will be of importance in the following. We are\ninterested in a sufficient condition for a graph to\nadmit a complete graph as a minor, dealing with the minimum number\nof triangles each edge belongs to. Nevo~\\cite{nevo1} already studied\nthis problem for small cliques. In the following, we assume\nthat every graph has at least one edge.\n\\begin{thm}[Nevo, 2007,~\\cite{nevo1}]\nFor $3 \\leq r \\leq 5$, any $K_r$-minor free graph $G$ has an edge that\nbelongs to at most $r - 3$ triangles.\n\\label{th:nevoleq6}\n\\end{thm}\nHe also gave a weaker version for $K_6$-minor free graphs.\n\\begin{thm}[Nevo, 2007,~\\cite{nevo1}]\nAny $K_6$-minor free graph $G$ has an edge that belongs to at\nmost $r - 3$ triangles, or is a clique-sum over $K_r$, $r \\leq 4$.\n\\label{th:nevok6}\n\\end{thm}\n\nNevo has conjectured that Theorem \\ref{th:nevok6} can be extended to\nthe case of $K_7$-minor free graphs. We improve\nTheorems~\\ref{th:nevoleq6} and \\ref{th:nevok6} in the following way.\n\n\\begin{thm}\\label{th:krtri}\nFor $3 \\leq r \\leq 7$, any $K_r$-minor free graph $G$ has\nan edge $uv$ such that $\\deg(u) \\leq 2r-5$ and $uv$ belongs to at\nmost $r-3$ triangles.\n\\end{thm}\n\nIn particular, this answers Nevo's conjecture about $K_7$-minor free graphs.\nAs pointed out by Nevo, Theorem~\\ref{th:nevok6} cannot be further extended\nto $K_8$-minor free graphs as such, since $K_{2,2,2,2,2}$\nis a $K_8$-minor free graph whose every edge belongs to $6$\ntriangles. Actually, one can obtain $K_8$-minor free graphs whose every edge\nbelongs to $6$ triangles by gluing copies of $K_{2,2,2,2,2}$ on cliques of\nany $K_8$-minor free graph. It is interesting to notice that $K_{2,2,2,2,2}$\nappears in a Mader-like theorem for $K_8$-minor free graphs~\\cite{jorg1}.\n\\begin{thm}[J\u00f8rgensen, 1994,~\\cite{jorg1}]\nEvery graph on $n \\geq 8$ vertices and at least $6n - 20$ edges either\nhas a $K_8$-minor, or is a $(K_{2,2,2,2,2}, 5)$-cockade (i.e. any\ngraph obtained from copies of $K_{2,2,2,2,2}$ by 5-clique sums).\n\\label{th:jorg}\n\\end{thm}\n\nAlthough Theorem~\\ref{th:krtri} cannot be extended to $K_8$-minor free\ngraphs, some similar conclusions can be reached by considering stronger\nhypotheses. By increasing the minimum degree of the graph or\nexcluding $K_{2,2,2,2,2}$ as an induced subgraph, we have the following\nthree theorems.\n\\begin{thm}\nAny $K_8$-minor free graph $G$ with $\\delta(G)=11$ has an edge $uv$\nsuch that $u$ has degree 11 and $uv$ belongs to at most $5$\ntriangles.\n\\label{th:k8triweak}\n\\end{thm}\n\n\\begin{thm}\nAny $K_8$-minor free graph $G$ with $\\delta(G)\\ge 9$ has an edge that\nbelongs to at most $5$ triangles.\n\\label{th:k8-deg9}\n\\end{thm}\n\n\\begin{thm}\nAny $K_8$-minor free graph $G$ with no $K_{2,2,2,2,2}$ as induced\nsubgraph has an edge that belongs to at most $5$ triangles.\n\\label{th:k8tri}\n\\end{thm}\n\nWe investigate applications of the previous results in the\nrest of the paper.\nIn Section~\\ref{sec:moytri}, we relax the hypothesis into a\nmore global condition on the overall number of triangles in\nthe graph. In particular, we prove that, for $3 \\leq k \\leq 7$\n(resp. $k=8$), if a graph has $m\\ge 1$ edges and at least\n$\\frac{k-3}{2}m$ triangles, then it has a $K_k$-minor (resp.\na $K_8$- or a $K_{2,2,2,2,2}$-minor).\nIn Section~\\ref{sec:stress}, we show applications to stress freeness\nof graphs, and settle some open problems of Nevo~\\cite{nevo1}.\nFinally, we show some applications to graph coloration\nin Section~\\ref{sec:double} and Section~\\ref{sec:coloration}.\nIn the former section, we show an application to double-critical\n$k$-chromatic graphs which settle a special case of a conjecture\nof Kawarabayashi, Toft and Pedersen~\\cite{kpt10}.\nIn the latter section, motivated by Hadwiger's conjecture,\nwe show that every $K_7$-minor free graph is $8$-colorable\nand that every $K_8$-minor free graph is $10$-colorable.\n\n\n\\section{Proof of Theorem~\\ref{th:krtri} for $r\\le 6$ : A slight improvement of Nevo's theorem}\\label{sec:proofk6}\n\nFirst note that the cases $r=3$ or $4$ are trivial.  The case $r=5$ is\nalso quite immediate, but we need a few definitions to prove it. A\n\\emph{separation} of a graph $G$ is a pair $(A,B)$ of subsets of\n$V(G)$ such that $A \\cup B = V(G)$, $A\\setminus B \\neq \\emptyset$,\n$B\\setminus A \\neq \\emptyset$, and no edge has one end in $A\n\\backslash B$ and the other in $B \\backslash A$. The \\emph{order} of a\nseparation is $|A \\cap B|$.  A separation of order $k$ will be denoted\nas a $k$-separation, and a separation of order at most $k$ as a $(\\leq\nk)$-separation. Given a vertex set $X\\subseteq V(G)$ (eventually $X$\nis a singleton) the sets $N(X)$ and $N[X]$ are respectively defined by\n$\\{y\\in V(G)\\setminus X \\ |\\ \\exists x\\in X\\ {\\rm s.t.}\\ xy\\in E(G)\\}$\nand $X\\cup N(X)$.\n\nLet us prove the case $r=5$. Consider any $K_5$-minor free graph\n$G$. According to Wagner's characterization of $K_5$-minor free\ngraphs~\\cite{w37}, $G$ is either the Wagner graph, a 4-connected\nplanar graph, or has a $(\\le 3)$-separation $(A,B)$ such that $H=G[A]$\nis either the Wagner graph or a 4-connected planar graph. If $G$ or\n$H$ is the Wagner graph, as this graph has only degree 3 vertices and\nno triangle, we are done. If $G$ (resp. $H$) is a 4-connected planar\ngraph, Euler's formula implies that  there is a vertex $v$\nof degree at most 5 in $V(G)$ (resp. in $A \\setminus B$). One can then observe\nthat, any edge around $v$ belongs to at most 2 triangles, as otherwise there would be\na separating triangle in $G$ (resp. $H$), contradicting its\n4-connectivity.\n\nLet us now focus on the case $r=6$ of Theorem~\\ref{th:krtri}.\nConsider by contradiction a $K_6$-minor free graph $G$ with at least\none edge, and such that every edge incident to a vertex of degree at\nmost $7$, belongs to at least $4$ triangles.  By Mader's theorem, we\nhave that $\\delta(G)\\le 7$.  We start by studying the properties of\n$G[N(u)]$, for the vertices $u$ of degree at most 7. First, it is\nclear that $G[N(u)]$ is $K_5$-minor free because otherwise there would\nbe a $K_6$-minor in $G$, contradicting the hypothesis.\n\n\\begin{lem}\n$\\delta(G)\\ge 6$, and for any vertex $u$ of degree at most 7, $\\delta(G[N(u)])\\ge 4$.\n\\label{lem:mindeg6}\n\\end{lem}\n\n\\begin{proof}\nFor any vertex $u$ of degree at most 7, and any vertex of $v\\in N(u)$\nthe edge $uv$ belongs to at least 4 triangles. The third vertex of\neach triangle clearly belongs to $N(u)$ and is adjacent to $v$.  Thus\n$v$ has degree at least $4$ in $G[N(u)]$.\n\nSince for any vertex $u$ of degree at most $7$ we have\n$\\delta(G[N(u)])\\ge 4$, $|N(u)|\\ge 5$ (i.e. $\\deg(u)\\ge 5$).\nFurthermore if there was a vertex $u$ of degree 5, as\n$\\delta(G[N(u)])\\ge 4$, the graph $G[N(u)]$ would be isomorphic to\n$K_5$, contradicting the fact that $G[N(u)]$ is $K_5$-minor free. Thus\n$\\delta(G)\\ge 6$.\n\\end{proof}\n\nAs observed by Nevo (Proposition 3.3,~\\cite{nevo1}), since\n$|N(u)| \\leq 7$, $\\delta(G[N(u)]) \\geq 4$ and $N(u)$ is $K_5$-minor\nfree, then $G[N(u)]$ is $4$-connected. Note that by Wagner's\ncharacterization of $K_5$-minor free graphs, every $4$-connected\n$K_5$-minor free is planar. Chen and Kanevsky~\\cite{ck1} proved that\nevery $4$-connected graph can be obtained from $K_5$ and the\ndouble-axle wheel $W_4^2$ by operations involving vertex splitting and\nedge addition. Their result implies that the only two possibilities\nfor $G[N(u)]$ are the double-axle wheels on $4$ and $5$ vertices\ndepicted in Figure~\\ref{fig:doublewheel}.  Note that theses two graphs\nhave $3|N(u)| - 6$ edges, and hence are maximal $K_5$-minor free (by\nMader's theorem).\n\n\\begin{figure}[h]\n\\centering\n\\subfigure{\n\\includegraphics[scale=1]{doubleaxlewheel4.eps}\n}\n\\subfigure{\n\\includegraphics[scale=1]{doubleaxlewheel5.eps}\n}\n\n\\caption{The double-axle wheel on $4$ and $5$ vertices.}\n\\label{fig:doublewheel}\n\\end{figure}\n\nWe need the following lemmas on the neighborhood of the vertices with\nsmall degree.\n\n\\begin{lem}\nFor any vertex $u$ of degree at most 7, every vertex $v\\in N(u)$ has a\nneighbor in $G\\setminus N[u]$.\n\\label{lem:voisink6}\n\\end{lem}\n\\begin{proof}\nRecall that $G[N(u)]$ is a double-axle wheel. Note that in a\ndouble-axle wheel, every vertex has degree at most 5, and every edge\nbelongs to exactly 2 triangles.  Thus, every vertex of $N(u)$ has\ndegree at most 6 in $G[N[u]]$, and every edge of $G[N(u)]$ belongs to\nexactly 3 triangles in $G[N[u]]$. This implies that any vertex $v\\in\nN(u)$ has either degree $>8$ in $G$, and thus at least 2 neighbors in\n$G\\setminus N[u]$, or that any of its incident edges $vw$ in $G[N(u)]$\nis contained in a fourth triangle $vwx$, with $x\\in G\\setminus N[u]$.\n\\end{proof}\n\n\\begin{lem}\nFor any vertex $u$ of degree at most $7$, and any connected component\n$C$ of $G\\setminus N[u]$, the graph $G[N(C)]$ is a clique on at most 3\nvertices.\n\\label{lem:voisin2k6}\n\\end{lem}\n\\begin{proof}\nAs $G[N(u)]$ has no clique on more than 3 vertices, let us show that\n$N(C)$ does not contain two non-adjacent vertices , say $v_1$ and\n$v_2$. There exists a path from $v_1$ to $v_2$ with inner vertices in\n$C$.  Since $G[N(u)]$ is maximal $K_5$-minor free, this path together\nwith $G[N[u]]$ induces a $K_6$ minor in $G$, a contradiction.\n\\end{proof}\n\n\\begin{lem}\nFor any vertex $u$ of degree at most $7$, and any connected component\n$C$ of $G\\setminus N[u]$, there exists a vertex $u'\\in C$ of degree at\nmost 7 in $G$.\n\\label{lem:existsk6}\n\\end{lem}\n\\begin{proof}\nSuppose for contradiction that every vertex of $C$ has degree at least\n8 in $G$.  Note that by definition, every vertex in $N(C)$ has a\nneighbor in $C$.  Thus, as by Lemma~\\ref{lem:voisin2k6} $G[N(C)]$ is a\nclique on $k\\le 3$ vertices, the vertices in $N(C)$ have degree at\nleast $k$ in $G[N[C]]$.  Thus the number of edges of $G[N[C]]$ is at\nleast\n\\[|E(G[N[C]])| \\geq \\frac{1}{2}(8|C| + k^2)  > 4(|C|+k) - 10\\]\nand by Mader's theorem, there is a $K_6$-minor in $G[N[C]]$, a\ncontradiction.\n\\end{proof}\n\n\nNow choose a vertex $u$ of degree at most 7 and a connected component\n$C$ of $G\\setminus N[u]$, in such a way that $|C|$ is minimum.  By\nLemma~\\ref{lem:existsk6}, $C$ has a vertex $v$ of degree at most 7.\n\nLet $C_u$ be the connected component of $G\\setminus N[v]$ that\ncontains $u$, and let $x\\in N(v)\\setminus N(C_u)$.  By\nLemma~\\ref{lem:voisink6}, there is a connected component $C'$ of\n$G\\setminus N[v]$ such that $x\\in N(C')$.\n\nAs $N[u]\\subset N[C_u]$, it is clear that $G[C'\\cup\\{x,v\\}]$ is a\nconnected subgraph of $G\\setminus N[u]$. We thus have that\n$C'\\subsetneq C$ and thus that $|C'|<|C|$, contradicting the choice of\n$u$ and $C$.  This concludes the proof of the case $r=6$ of\nTheorem~\\ref{th:krtri}.\n\n\n\\section{Proof of Theorem~\\ref{th:krtri} for $r=7$ : the case of $5$ triangles}\\label{sec:proofk7}\n\nConsider by contradiction a $K_7$-minor free graph $G$ with at least\non edge, and such that every edge incident to a vertex of degree at\nmost $9$ belongs to at least $5$ triangles. By Mader's theorem,\n$|E(G)| \\leq 5|V(G)| - 15$, hence there are vertices $u$ such that\n$\\deg(u) \\leq 9$.\n\nWe start by studying the properties of $G[N(u)]$, for any vertex $u$\nof degree at most 9. First, it is clear that $G[N(u)]$ is $K_6$-minor\nfree because otherwise there would be a $K_7$-minor in $G$,\ncontradicting the hypothesis.\n\n\\begin{lem}\n$\\delta(G)\\ge 7$, and for any vertex $u$ of degree at most 9, $\\delta(G[N(u)])\\ge 5$.\n\\label{lem:mindeg7}\n\\end{lem}\n\n\\begin{proof}\nFor any vertex $u$ of degree at most 9, and any vertex of $v\\in N(u)$\nthe edge $uv$ belongs to at least 5 triangles. The third vertex of\neach triangle clearly belongs to $N(u)$ and is adjacent to $v$.  Thus\n$v$ has degree at least $5$ in $G[N(u)]$.\n\nSince for any vertex $u$ of degree at most $9$ we have\n$\\delta(G[N(u)])\\ge 5$, $|N(u)|\\ge 6$ (i.e. $\\deg(u)\\ge 6$).\nFurthermore if there was a vertex $u$ of degree 6, as\n$\\delta(G[N(u)])\\ge 5$, the graph $G[N(u)]$ would be isomorphic to\n$K_6$, contradicting the fact that $G[N(u)]$ is $K_6$-minor free. Thus\n$\\delta(G)\\ge 7$.\n\\end{proof}\n\nThere is no appropriate theorem (contrarily to the previous case) to\ngenerate all possible neighbourhoods of the small degree vertices.\nInstead, we use a computer to generate all graphs with at most $9$\nvertices and minimum degree at least 5. Then we refine (by computer)\nour list of graphs, by removing the ones having a $K_6$-minor.  At the\nend, we end up with a list of $22$ graphs. A difference with the\nprevious case is that not all the $22$ graphs are maximal $K_6$-minor\nfree graphs.  We deduce two of the following lemmas from the study of\n$N(u)$ by computer~\\cite{ag1}.\n\n\\begin{lem}\nFor any vertex $u$ of degree at most $9$, any connected component $C$\nof $G\\setminus N[u]$ is such that $|N(C)| = k \\le 5$ and $|E(N(C))|\\ge\n{k \\choose 2} -3$ (i.e. $G[N[C]]$ has at most 3 non-edges).\n\\label{lem:compk7}\n\\end{lem}\n\n\\begin{proof}\nAs any connected component $C$ could be contracted into a single\nvertex, we prove the lemma by attaching a new vertex to all possible\ncombinations of $k$ vertices of $N[u]$ (as we know that $N(u)$ induces\none of the 22 graphs generated above), for any $k\\le 6$, and check\nwhen it induces a $K_7$-minor.\n\\end{proof}\n\nThis allows us to prove the following equivalent of\nLemma~\\ref{lem:existsk6}.\n\\begin{lem}\nFor any vertex $u$ of degree at most $9$, any connected component $C$\nof $G\\setminus N[u]$ has a vertex $u'$ of degree at most 9 in $G$.\n\\label{lem:existsk7}\n\\end{lem}\n\n\\begin{proof}\nLet $u$ be a vertex of $G$ of degree at most $9$ and let $C$ be a\nconnected component of $G\\setminus N[u]$ which vertices have degree at\nleast 10 in $G$. Note that by definition every vertex of $N(C)$ has at\nleast one neighbor in $C$. Lemma~\\ref{lem:compk7} implies that\n$|N(C)|=k \\leq 5$ and that $G[N(C)]$ has at most 3 non-edges. Thus,\ncontracting a conveniently choosen edge between $u$ and $N(C)$, one\nobtains that $G[N(C)]$ has at most 1 non-edge.  After this\ncontraction, we have:\n\\begin{align*}\n|E(N[C])| &\\geq \\frac{1}{2} \\Big[ 10|C| + k(k-1) - 2 + k \\Big] \\\\\n&= 5|C| + \\frac{k^2}{2} -1 > 5(|C|+k) - 15 .\n\\end{align*}\nThis contradicts the fact that $G[N[C]]$ is $K_7$-minor free, and thus\nconcludes the proof of the lemma.\n\\end{proof}\n\n\\begin{lem}\nFor any vertex $u$ of degree at most $9$, at most one vertex $v$ of\n$N(u)$ is such that $N(v) \\subseteq N[u]$.\n\\label{lem:numberk7}\n\\end {lem}\n\n\\begin{proof}\nFor every such vertex $v$, as $\\deg(v)\\le \\deg(u)\\le 9$, the edges\nadjacent to $v$ with both ends in $N(u)$ belong to at least $5$\ntriangles in $G$ (i.e. belong to at least $4$ triangles in\n$G[N(u)]$). We checked that for every graph in the list at most one\nsuch vertex satisfies this condition.\n\\end{proof}\n\nThis allows us to prove the following lemma.\n\\begin{lem}\nFor any vertex $u$ of degree at most $9$ and any connected component\n$C$ of $G\\setminus N[u]$, there exists a connected component $C'$ of\n$G\\setminus N[u]$ such that $N(C')\\setminus N(C) \\neq \\emptyset$.\n\\label{lem:numberk7bis}\n\\end {lem}\n\\begin{proof}\nAs $\\deg(u)\\ge 7$ (by Lemma~\\ref{lem:mindeg7}) and $|N(C)|\\le 5$ (by\nLemma~\\ref{lem:compk7}), there are at least 2 vertices in\n$N(u)\\setminus N(C)$. By Lemma~\\ref{lem:numberk7}, one of these 2\nvertices has a neighbor $x$ out of $N[u]$.  Thus the component of\n$G\\setminus N[u]$ containing $x$ fulfills the requirements of the\nlemma.\n\\end{proof}\n\nNow choose a vertex $u$ of degree at most 9 and a connected component\n$C$ of $G\\setminus N[u]$, in such a way that $|C|$ is minimum.  By\nLemma~\\ref{lem:existsk7}, $C$ has a vertex $v$ of degree at most 9.\nLet $C_u$ be the connected component of $G\\setminus N[v]$ that\ncontains $u$. By Lemma~\\ref{lem:numberk7bis} there exists a connected\ncomponent $C'$ of $G\\setminus N[v]$ such that $N(C')\\setminus N(C_u)\n\\neq \\emptyset$, and let $x\\in N(C')\\setminus N(C_u)$.  As\n$N[u]\\subset N[C_u]$, it is clear that $G[C'\\cup\\{x,v\\}]$ is a\nconnected subgraph of $G\\setminus N[u]$. We thus have that\n$C'\\subsetneq C$ and thus that $|C'|<|C|$, contradicting the choice of\n$u$ and $C$.  This concludes the proof of case $r=7$ of\nTheorem~\\ref{th:krtri}\n\n\\section{Proof of Theorem~\\ref{th:k8triweak}, \\ref{th:k8-deg9} and \\ref{th:k8tri} : the case of $6$ triangles}\\label{sec:proofk8}\n\nAs in the previous sections, we will consider vertices of small degree\n(and their neighborhoods) in $K_8$-minor free graphs.  We thus need the\nfollowing technical lemma that has been proven by computer~\\cite{ag1}.\n\n\\begin{lem}\nEvery $K_7$-minors free graph $H$ distinct from $K_{2,2,2,2}$,\n$K_{3,3,3}$ and $\\overline{P_{10}}$ (the complement of the Petersen\ngraph), and such that $8\\le |V(H)| \\le 11$ and $\\delta(H)\\ge 6$,\nverifies:\n\\begin{itemize}\n\\item $H$ is 5-connected.\n\\item $H$ has at most one vertex $v$ such that each of its\n  incident edges belongs to 5 triangles.\n\\item For any subset $Y\\subsetneq V(H)$ of size 7, the graph obtained\n  from $H$ by adding two vertices $x$ and $y$ such that $N(x)=V(H)$\n  and $N(y)=Y$, has a $K_8$-minor.\n\\end{itemize}\n\\label{lem:compk8}\n\\end{lem}\nNote that the second property also holds for $K_{2,2,2,2}$,\n$K_{3,3,3}$ and $\\overline{P_{10}}$. Actually any edge of these 3\ngraphs belongs to less than 5 triangles.\n\nBy Theorem~\\ref{th:jorg}, any $K_8$-minor free graph has minimum\ndegree at most 11.  Theorem~\\ref{th:k8triweak} considers the case\nwhere the minimum degree is exactly 11.  It will be used in\nSection~\\ref{sec:coloration} to color $K_8$-minor free graphs.\n\n\\begin{proofof}{Theorem~\\ref{th:k8triweak}}\nWe prove this using the same technique as in\nSection~\\ref{sec:proofk7}.  Consider by contradiction a $K_8$-minor\nfree graph $G$ with $\\delta(G)=11$, and such that every edge adjacent\nto a degree 11 vertex belongs to at least $6$ triangles.  We start by\nstudying the properties of $G[N(u)]$, for any degree 11 vertex $u$.\nFirst, it is clear that $G[N(u)]$ is $K_7$-minor free because\notherwise there would be a $K_8$-minor in $G$, contradicting the\nhypothesis.\n\n\\begin{lem}\nFor any degree 11 vertex $u$, $\\delta(G[N(u)])\\ge 6$.\n\\label{lem:mindegk8}\n\\end{lem}\n\\begin{proof}\nFor any degree 11 vertex $u$ and any vertex of $v\\in N(u)$,\nthe edge $uv$ belongs to at least 6 triangles. The third vertex of\neach triangle clearly belongs to $N(u)$ and is adjacent to $v$.  Thus\n$v$ has degree at least $6$ in $G[N(u)]$.\n\\end{proof}\n\n\\begin{lem}\nFor any degree 11 vertex $u$, any connected component $C$\nof $G\\setminus N[u]$ has a vertex $u'$ of degree at most 11 in $G$.\n\\label{lem:existsk8}\n\\end{lem}\n\\begin{proof}\nLet $u$ be a degree 11 vertex of $G$ and let $C$ be any connected\ncomponent of $G\\setminus N[u]$ which vertices have degree at least 12\nin $G$. Lemma~\\ref{lem:compk8} implies that $G[N(u)]$ is 5-connected\nand that $|N(C)|=k \\leq 6$.  Thus the lemma holds by considering the\ngraph $G[N[u]\\cup C]$ in the following Lemma~\\ref{lem:existsk8bis}.\n\\end{proof}\n\n\\begin{lem}\nA graph $H$ with a degree 11 vertex $u\\in V(H)$ and such that:\n\\begin{itemize}\n\\item[(A)] $H[N(u)]$ is 5-connected,\n\\item[(B)] $\\delta(H[N(u)])\\ge 6$,\n\\item[(C)] the set $C=V(H)\\setminus N[u]$ is non-empty, and all its vertices have degree at least 12, and\n\\item[(D)] the set $N(C) \\subseteq N(u)$ has size $k\\le 6$,\n\\end{itemize}\nhas a $K_8$-minor.\n\\label{lem:existsk8bis}\n\\end{lem}\n\\begin{proof}\nConsider a minimal counter-example $H$, that is a $K_8$-minor free\ngraph $H$ fulfilling conditions (A), (B) (C) and (D), and minimizing\n$|V(H)|$.  Note that by definition every vertex of $N(C) \\subseteq\nN(u)$ has at least one neighbor in $C$. Let us prove that actually\nevery vertex of $N(C)$ has at least 2 neighbors in $C$.  If $x\\in\nN(C)$ has only one neighbor $y$ in $C$, contract the edge $xy$ and\ndenote $G'$ the obtained graph. It is clear that $G'$ is $K_8$-minor\nfree, and fulfills conditions (A), (B) and (D). Moreover, $C\\setminus\n\\{y\\}$ is non-empty as it contains at least 6 vertices of $N(y)\\cap C$\n(as $\\deg(y)\\ge 12$ and $|N(C)|=k\\le 6$), and every vertex of\n$C\\setminus \\{y\\}$ has degree at least 12 in $H'$ as none of these\nvertices are adjacent to $x$ in $H$. So $G'$ also fulfills condition\n(C), and this contradicts the minimality of $G$. Thus every vertex of\n$N(C)$ has at least 2 neighbors in $C$.\n\nOne can easily see that every $(K_{2,2,2,2,2}, 5)$-cockade has at\nleast 10 degree 8 vertices. Thus the graph $H[N[C]]$, and any graph\nobtained from $H[N[C]]$ by adding edges, cannot be a\n$(K_{2,2,2,2,2}, 5)$-cockade as it has at most $6$ vertices of\ndegree $8$. Thus as $H[N[C]]$ has at least $\\frac{1}{2}(12|C| + 2k)$\nedges and as this is at least $6(|C|+k)-20$ for $k \\leq 4$, by\nTheorem~\\ref{th:jorg} we have that $5\\le k\\le 6$.\n\nNow suppose that $k = 5,6$. Let $v_1$ and $v_2$ be two vertices of\nsmallest degree in $H[N(C)]$. Denote $\\delta_1$ and $\\delta_2$ their\nrespective degree in $H[N(C)]$. Note that if $k=6$ then $\\delta_1\\ge\n1$ as $v_1$ has at least $6$ neighbors in $N(u)$ and as there are only\n$5$ vertices in $N(u)\\setminus N(C)$. By contracting the edge $uv_1$,\nwe have $k - 1 -\\delta_1$ additionnal edges in $H[N[C]]$. Moreover\nsince $H[N(u)]$ is $5$-connected and since $|N(C)| \\leq 6$, for every\nvertex $x\\neq v_2$ of $N(C)$ we have $|N(C)\\setminus\\{x,v_2\\}|=4$ and\nthus the graph $H[N(u)]\\setminus (N(C)\\setminus\\{x,v_2\\})$ is\nconnected.  Thus, iteratively contracting all the edges between $v_2$\nand $N(u)\\setminus N(C)$ we add at least $k - 2 - \\delta_2$ edges in\n$H[N[C]]$ (as we have potentially already added the edge $v_1v_2$ in\nthe previous step). The number of edges in the obtained graph is at\nleast\n\\[\\frac{1}{2}[(\\delta_1 +2) + (\\delta_2\n  +2)(k-1)) + 12|C|] + (k - 1 -\\delta_1) + (k - 2 -\\delta_2)\\]\nwhich is more than $6(|C|+k)-20$ (as $k\\le 6$ and as if $k=6$ then\n$\\delta_1 \\ge 1$). Thus this graph has a $K_8$-minor, and so does $H$.\nThis completes the proof of the lemma.\n\\end{proof}\n\n\n\\begin{lem}\nFor any degree 11 vertex $u$ and any connected component\n$C$ of $G\\setminus N[u]$, there exists a connected component $C'$ of\n$G\\setminus N[u]$ such that $N(C')\\setminus N(C) \\neq \\emptyset$.\n\\label{lem:numberk8bis}\n\\end {lem}\n\\begin{proof}\nAs $\\deg(u)=11$ and $|N(C)|\\le 6$ (by Lemma~\\ref{lem:compk8}), there\nare at least 5 vertices in $N(u)\\setminus N(C)$.  As $\\delta(G)=11$\none can easily derive from Lemma~\\ref{lem:compk8} that one (actually,\nat least 4) of these vertices has a neighbor $x$ out of $N[u]$.  Thus\nthe component of $G\\setminus N[u]$ containing $x$ fulfills the\nrequirements of the lemma.\n\\end{proof}\n\nNow choose a degree 11 vertex $u$ and a connected component $C$ of\n$G\\setminus N[u]$, in such a way that $|C|$ is minimum.  By\nLemma~\\ref{lem:existsk8}, $C$ has a degree 11 vertex $v$.  Let $C_u$\nbe the connected component of $G\\setminus N[v]$ that contains $u$. By\nLemma~\\ref{lem:numberk8bis} there exists a connected component $C'$ of\n$G\\setminus N[v]$ such that $N(C')\\setminus N(C_u) \\neq \\emptyset$,\nand let $x\\in N(C')\\setminus N(C_u)$.  As $N[u]\\subset N[C_u]$, it is\nclear that $G[C'\\cup\\{x,v\\}]$ is a connected subgraph of $G\\setminus\nN[u]$. We thus have that $C'\\subsetneq C$ and thus that $|C'|<|C|$,\ncontradicting the choice of $u$ and $C$.  This concludes the proof of\nTheorem~\\ref{th:k8triweak}\n\\end{proofof}\n\nlet us now prove Theorem~\\ref{th:k8tri}.  Given a counter-exemple $G$\nof Theorem~\\ref{th:k8tri}, note that adding a vertex $s$ to $G$,\nadjacent to a single vertex of $G$, one obtains a counter-exemple of\nthe following theorem, thus Theorem~\\ref{th:k8tri} is a corollary of\nthe following theorem.\n\n\\begin{thm}\nConsider a connected $K_8$-minor free graph $G$ with a vertex $s$ of\ndegree at most 7 and such that $N[s] \\subsetneq V(G)$.  If every edge\n$e \\in E(G) \\setminus E(G[N[s]])$ belongs to at least 6 triangles, then\n$G$ contains an induced $K_{2,2,2,2,2}$.\n\\label{th:k8triwiths}\n\\end{thm}\nNote that as $K_{2,2,2,2,2}$ is maximal $K_8$-minor free, any\n$K_8$-minor free graph $G$ containing a copy of $K_{2,2,2,2,2}=G[X]$,\nfor some vertex set $X\\subseteq V(G)$, is such that any connected\ncomponent $C$ of $G\\setminus X$ verifies that $N(C)$ induces a clique\nin $G[X]$.\n\\begin{proof}\nConsider a connected $K_8$-minor free graph $G$ with a vertex $s$ of\ndegree at most 7 such that $N[s] \\subsetneq V(G)$, such that $G$ does not\ncontain an induced $K_{2,2,2,2,2}$, and such that every\nedge $e \\in E(G) \\setminus E(G[N[s]])$ belongs to at least 6\ntriangles.  Assume also that $G$ minimizes the number of vertices.\nThis property implies that $G\\setminus N[s]$ is connected. Indeed,\notherwise one could delete one of the connected components in\n$G\\setminus N[s]$ and obtain a smaller counter-example.  The graph $G$\nis almost 8-connected as observed in the following lemma.\n\\begin{lem}\nFor any separation $(A,B)$ of $G$ (denote $S = A \\cap B$), we have\neither:\n\\begin{itemize}\n\\item $|S| \\geq 8$, or\n\\item $s \\notin S$ and $A\\setminus B = \\{s\\}$ (i.e. $B = V(G) \\setminus\n  \\{s\\}$), or\n\\item $s \\in S$ and $|S| \\geq 6$.\n\\end{itemize}\n\\label{lem:connectk8withs}\n\\end{lem}\n\\begin{proof}\nSuppose there exists a separation $(A,B)$ contradicting the lemma.\nNote that $|S| < 8$ and let us assume that $s\\in A$.\n\nConsider first the case where $s\\notin S = A\\cap B$, that is the case\nwhere $\\{s\\} \\subsetneq A\\setminus B$.  Assume that among all such\ncounter-examples, $(A,B)$ minimizes $|S|$.  In this case, if the\nconnected component of $A\\setminus B$ containing $s$ has more vertices\nthen, contracting this component into $s$, one obtains a proper minor\n$G'$ of $G$ such that $N[s]\\subsetneq V(G')$ (as $B\\setminus A \\neq\n\\emptyset$) and such that every edge not in $E(N[s])$ belongs to $6$\ntriangles.  This would contradict the minimality of $G$, and we thus\nassume the existence of a component $C_0=\\{s\\}$ in $G\\setminus B$.  As\n$\\{s\\} \\subsetneq A\\setminus B$, let $C_1\\neq \\{s\\}$ be some connected\ncomponent of $G\\setminus B$. Let also $C_2$ be some component of\n$G\\setminus A$.  Note that for any of these components $C_i$,\n$N(C)\\subsetneq S$. Otherwise one could contract (if needed) all the\ncomponent into a single vertex $s'$ and the graph induced by\n$\\{s\\}\\cup N[C_1]$ or by $\\{s\\}\\cup N[C_2]$ (a proper minor of $G$)\nwould be a smaller counter-example. Note now that $S = N(s)\\cup\nN(C_i)$ for $i=1$ or 2.  Indeed, otherwise the separation $(N[s]\\cup\nN[C_i], V(G)\\setminus (C_i\\cup \\{s\\}))$ would be a counter-example\ncontradicting the minimality of $S$. Finally note that\n$N(C_2)\\not\\subseteq N(C_1)$, as otherwise contracting $C_1$ into a\nsingle vertex $s'$ and considering the graph induced by\n$C_2\\cup\\{s'\\}$ one would obtain a smaller counter-example. Thus there\nexists a vertex $x\\in N(s)\\cap N(C_2)$ such that $x\\notin\nN(C_1)$. Contracting the edge $xs$ and contracting the whole component\n$C_2$ into $x$, and considering the graph induced by $C_1\\cup \\{x\\}$\none obtains a smaller counter-example (where $x$ plays the role of\n$s$).\n \nConsider now the case where $s\\in S=A\\cap B$ and note that $|S|< 6$.\nAssume that among all the separations containing $s$, $(A,B)$\nminimizes $|S|$. Note that every connected component $C$ of $G\n\\setminus S$ is such that $s\\in N(C)$. Indeed, we have seen above that\notherwise $C$ would be such that $|N(C)|\\ge 8$ (by considering the\nseparation $(V(G)\\setminus C,N[C])$ and noting that $\\{s\\}\\subsetneq\nV(G)\\setminus N[C]$), and this would contradict the fact that $|S|<\n6$. This implies that every connected component $C$ of $G \\setminus S$\nis such that $N(C)=S$. Otherwise $(V(G)\\setminus C, N[C])$ would be a\nseparation containing $s$ contradicting the minimality of $S$. We can\nassume without loss of generality that $s$ has at most as many\nneighbors in $B\\setminus A$ than in $A\\setminus B$. In particular,\nsince $\\deg(s)\\le 7$, $s$ has at most 3 neighbors in $B\\setminus\nA$. Note that $B \\not\\subseteq N[s]$ as otherwise $G \\setminus\n(B\\setminus A)$ would be a smaller counter-example. Thus there is an\nedge in $G[B\\setminus N[s]$ that belongs to at least 6 triangles, and\nthus $|B|\\ge 9$ ($s$ and the 6 triangles). Thus contracting every\ncomponent of $A \\setminus B$ on $s$, results in a proper minor $G'$\nof $G$ such that $\\deg(s) \\leq 7$ (at most 4 in $S$ and 3 in\n$B\\setminus A$), such that $N[s]\\subsetneq V(G')$ (as\n$|V(G')|=|B|\\ge 9$), and such that every edge not in $E(N[s])$\nbelongs to $6$ triangles, contradicting the minimality of $G$. This\nconcludes the proof of the lemma.\n\\end{proof}\n\nBy Theorem~\\ref{th:jorg} $G$ has at most $6n-20$ edges, and thus there\nare several vertices in $G$ with degree at most 11.  Let us prove that\nthere are such vertices out of $N[s]$.\n\n\\begin{lem}\nThere are at least 2 vertices in $V(G) \\setminus N[s]$ with degree at\nmost 11.\n\\label{lem:2small_ink8}\n\\end{lem}\n\\begin{proof}\nAssume for contradiction that every vertex of $V(G) \\setminus N[s]$\nbut one, say $x$, has degree at least 12, and recall that such vertex\nhas degree at least 8. Note that every vertex $v\\in N(s)$ has a\nneighbor in $V(G) \\setminus N[s]$, as otherwise $G\\setminus v$ would\nbe a smaller counter-exemple. Thus every vertex $v\\in N(s)$ has an\nincident edge that belongs to at least 6 triangles (without using the\nedge $sv$), which implies that $\\deg(v)\\ge 8$. This implies that the\nnumber of edges in $G$ verifies :\n\\[ 12n-42 \\ge 2|E(G)| = \\sum_{v\\in V(G)} \\deg(v) \\ge 8 + k + 8k + 12(n-k-2) \\] \nwhere $k=\\deg(s)$. This implies that $3k\\ge 26$ which contradicts\nthe fact that $k=\\deg(s) \\le 7$. This concludes the proof of the\nlemma.\n\\end{proof}\n\nAs for any vertex $u\\in V(G)\\setminus N[s]$ each of its incident edges\nbelongs to 6 triangles, the graph $G[N(u)]$ has minimum degree at\nleast 6. As $G$ does not contain $K_8$ as subgraph, this also implies\nthat $\\deg(u)\\ge 8$.  So there are at least two vertices in\n$V(G)\\setminus N[s]$ with degree between 8 and 11.  The next lemma\ntells us more on the neighborhood of these small degree vertices.\n\\begin{lem}\nFor every vertex $u \\in V(G) \\setminus N[s]$ with degree at most 11 in\n$G$, $G[N(u)]$ is isomorphic to $K_{2,2,2,2}$, $K_{3,3,3}$ or\n$\\overline{P_{10}}$.\n\\label{lem:NminDeg}\n\\end{lem}\n\\begin{proof}\nLet $u$ be any vertex of $V(G) \\setminus N[s]$ with degree at most 11\nin $G$.  As observed earlier $8\\le \\deg(u)\\le 11$ and\n$\\delta(G[N(u)])\\ge 6$.  Assume for contradiction that $N(u)$, is not\nisomorphic to $K_{2,2,2,2}$, $K_{3,3,3}$ or $\\overline{P_{10}}$. Note\nthat $|N(u)\\cap N(s)| \\le 6$, as otherwise Lemma~\\ref{lem:compk8}\nwould contradict the $K_8$-minor freeness of $G$.\n\nBy Lemma~\\ref{lem:compk8} one of the (at least two) vertices in\n$N(u)\\setminus N(s)$, say $x$, has an incident edge in $G[N(u)]$ that\nbelongs to at most 5 triangles in $G[N[u]]$.  Thus the sixth triangle\ncontaining this edge goes through a vertex $v$ of $V(G) \\setminus\n(N[u] \\cup \\{s\\})$.\n\nLemma~\\ref{lem:connectk8withs} implies that the connected component\n$C$ of $v$ in $V(G) \\setminus N[u]$ is such that $N(C)\\ge 8$. The\ngraph obtained by contracting $C$ into a single vertex has a\n$K_8$-minor (by Lemma~\\ref{lem:compk8}), a contradiction.\n\\end{proof}\n\nA $K_3$-minor rooted at $\\{a, b, c\\}$, or a $\\{a, b, c\\}$-minor, is a\n$K_3$-minor in which you can contract edges incident to $a$, $b$ or\n$c$, to obtain a $K_3$ with vertex set $\\{a, b, c\\}$.  For the rest of\nthe proof we need the following characterization of rooted $K_3$-minor.\n\n\\begin{thm}[D. R. Wood and S. Linusson, Lemma 5 of~\\cite{wl1}]\nFor distinct vertices a, b, c in a graph G, either:\n\\begin{itemize}\n\\item $G$ contains an $\\{a, b, c\\}$-minor, or\n\\item for some vertex $v \\in V(G)$ at most one of $a, b, c$ are in each component of $G \\setminus v$.\n\\end{itemize}\n\\label{th:rootedtri}\n\\end{thm}\n\n\\begin{lem}\nFor every vertex $u \\in V(G) \\setminus N[s]$ with degree at most 11 in\n$G$, the graph $G[N(u)]$ is not isomorphic to $K_{3,3,3}$.\n\\label{lem:noK333}\n\\end{lem}\n\\begin{proof}\nObserve that adding two vertex disjoint edges or three edges of a\ntriangle in $K_{3,3,3}$ yields a $K_7$-minor.  Now assume for\ncontradiction that there exists some vertex $u\\in V(G) \\setminus N[s]$\nsuch that $G[N(u)]$ is isomorphic to $K_{3,3,3}$.\n\nAs the set $N(u) \\setminus N[s]$ is non-empty (it has size at least\n$9-7$) and as every vertex $v$ in $N(u) \\setminus N[s]$ has degree at\nleast $8$, and thus has a neighbor out of $N[u]$, $G \\setminus N[u]$\nhas a connected component $C\\neq \\{s\\}$.  By\nLemma~\\ref{lem:connectk8withs} $|N(C)| \\geq 8$.\n\nIf $G \\setminus N[u]$ has another connected component $C'$ such that\n$|N(C')| \\geq 6$, one can create two vertex disjoint edges in\n$K_{3,3,3}$ by contracting two vertex disjoint paths with non-adjacent\nends in $N(u)$, one living in each component. This would contradict\nthe $K_8$-minor freeness of $G$. Thus if there is a component $C'$,\nwe should have $C'=\\{s\\}$ and $\\deg(s)\\le 5$, as by\nLemma~\\ref{lem:connectk8withs} a component $C'\\neq \\{s\\}$ would be\nsuch that $|N(C')| \\geq 8$. In the following we consider the graph\n$G' = G[N[u] \\cup C]$ (which is $G$ or $G \\setminus s$).\n\nLet $\\{a_1,a_2,a_3\\}$, $\\{b_1,b_2,b_3\\}$ and $\\{c_1,c_2,c_3\\}$ be the\nthree disjoint stables of $N(u) = K_{3,3,3}$. Without loss of\ngenerality we can assume that $\\{a_1,a_2,a_3\\}\\subset N(C)$, and that\n$a_1\\notin N(s)$. As the edges of $N(u)$ incident to $a_1$ belong to\nat least 6 triangles, $a_1$ has at least two neighbors in $G'\n\\setminus N[u]$. By Theorem~\\ref{th:rootedtri} (applied to\n$\\{a_1,a_2,a_3\\}$ in the graph $G''= G'\\setminus\n\\{u,b_1,b_2,b_3,c_1,c_2,c_3\\}$), there is a vertex $v \\in V(G'')$ such\nthat at most one of $a_1, a_2, a_3$ are in each component of $G''\n\\setminus v$. Note that since $a_1$, $a_2$ and $a_3\\in N(C)$, all the\nsets $C\\cup\\{a_i,a_j\\}$ induce a connected graph, and thus $v\\neq\na_1$, $a_2$ or $a_3$. Equivalently we have that $v\\in V(G') \\setminus\nN[u]$.  Hence $G'' \\setminus \\{v\\}$ contains at least $3$ components\n$C_1$, $C_2$ and $C_3$ with $a_i \\in C_i$, for $1 \\leq i \\leq\n3$. Since $a_1$ has at least two neighbors in $G' \\setminus N[u]$, one\nof them is distinct from $v$ and we can define $C'_1$ as a connected\ncomponent of $C_1\\setminus \\{a_1\\}$. Note that by construction\n$N(C'_1) \\subset N(u)\\cup\\{v\\}$.  Since $C'_1\\neq\\{s\\}$ (as $a_1\\notin\nN(s)$) and as we might have $v=s$, Lemma~\\ref{lem:connectk8withs}\nimplies that $N(C'_1)\\ge 6$ (including $v$ and $a_1$). So $C'_1$ has\nat least 4 neighbors in $\\{b_1,b_2,b_3,c_1,c_2,c_3\\}$ and there is a\npath with interior vertices in $C'_1$ between two vertices $b_i$ and\n$b_j$, or between two vertices $c_i$ and $c_j$. Furthermore, there is\na path with interior vertices in $C_2 \\cup \\{v\\} \\cup C_3$ between the\nvertices $a_2$ and $a_3$. This contradicts the $K_8$-minor freeness of\n$G$, and thus concludes the proof of the lemma.\n\\end{proof}\n\n\\begin{lem}\nFor every vertex $u \\in V(G) \\setminus N[s]$ with degree at most 11 in\n$G$, the graph $G[N(u)]$ is not isomorphic to $K_{2,2,2,2}$.\n\\label{lem:noK2222}\n\\end{lem}\n\\begin{proof}\nAssume for contradiction that there exists some vertex $u\\in V(G)\n\\setminus N[s]$ such that $G[N(u)]$ is isomorphic to\n$K_{2,2,2,2}$. One can check that adding two edges in $K_{2,2,2,2}$\ncreates a $K_7$-minor. Thus as $G$ is $K_8$-minor free it should not\nbe possible to add (by edge contractions) two new edges in $N(u)$.\n\n\\begin{claim}\nA vertex $v\\in V(G) \\setminus N[u]$ has at most six neighbors in\n$N(u)$.\n\\label{claim:k8-v6neighbors}\n\\end{claim}\n\\begin{proof}\nIf there was a vertex $v$ with 8 neighbors in $N(u)$, $N[u]\\cup \\{v\\}$\nwould induce a $K_{2,2,2,2,2}$, a contradiction to the definition of\n$G$. We thus assume for contradiction that there is a vertex $v$ with\nexactly 7 neighbors in $N(u)$. Note that eventually $v=s$.  Let us\ndenote $x$ the only vertex in $N(u)\\setminus N(v)$. Note that among\nthe 4 non-edges of $G[N(u)]$, only one cannot be created by\ncontracting an edge incident to $v$. So if there is a path whose ends\nare non-adjacent in $N(u)$ and whose inner vertices belong to\n$V(G)\\setminus (N[u]\\cup \\{v\\})$, then we have a $K_8$-minor, a\ncontradiction. There is clearly such path if $s\\neq v$ and if $s$ has\n5 neighbors in $N(u)$, we thus have that either $s=v$ or $s$ has at\nmost 4 neighbors in $N(u)$. Both cases imply that some edge $xy$\n(incident to $x$) does not belong to $G[N[s]]$, and thus $xy$ belongs\nto at least 6 triangles. As $xy$ belongs to only 5 triangles in\n$G[N[u]]$, this implies the existence of a vertex $w\\in V(G) \\setminus\nN[u]$ adjacent to $x$ such that $w\\neq s, v$. Let $C$ be the connected\ncomponent of $w$ in $G\\setminus (N[u]\\cup \\{v\\})$. As $C\\neq \\{s\\}$,\nLemma~\\ref{lem:connectk8withs} implies that $N(C)$ has size at least\n6. Thus $C$ has at least 5 neighbors in $N(u)$ and one can link two\nnon-adjacent vertices of $N(u)$ by a path going through $C$, a\ncontradiction.\n\\end{proof}\n\nBy Lemma~\\ref{lem:2small_ink8} there exists another vertex $u'\\in V(G)\n\\setminus N[s]$ such that $\\deg(u')\\le 11$. By Lemma~\\ref{lem:NminDeg}\nand Lemma~\\ref{lem:noK333}, $G[N(u')]$ is isomorphic to $K_{2,2,2,2}$\nor $\\overline{P_{10}}$.\n\n\\begin{claim}\nThe vertices $u$ and $u'$ are non-adjacent.\n\\end{claim}\n\\begin{proof}\nWe assume for contradiction that $u$ and $u'$ are adjacent and we\nfirst consider the case where $G[N(u')]$ is isomorphic to\n$K_{2,2,2,2}$. In this case, as $u'$ has allready 7 neighbors in\n$N[u]$, $u'$ has a exactly one neighbor $v$ in $G\\setminus N[u]$.  As\n$v$ has 7 neighbors in $N(u')$, we have that $|N(u)\\cap N(v)|\\ge 7$, a\ncontradiction to Claim~\\ref{claim:k8-v6neighbors}.\n\nIf $G[N(u')]$ is isomorphic to $\\overline{P_{10}}$, this implies that\n$G[N(u)\\cap N(u')]$ is isomorphic to $\\overline{C_6}$ (the complement\nof the 6-cycle). This is not compatible with $G[N(u)]$ being\nisomorphic to $K_{2,2,2,2}$, as this in turn implies that $G[N(u)\\cap\nN(v)]$ is isomorphic to $K_{2,2,2}$.\n\\end{proof}\n\nAs by Lemma~\\ref{lem:connectk8withs} there is no $(\\le 5)$-separator\n$(A,B)$ with $u\\in A\\setminus B$ and $u'\\in B\\setminus A$, Menger's\nTheorem implies the existence of 6 vertex disjoint paths between $u$\nand $u'$.  These paths induces $6$ disjoint paths $P_1 \\ldots P_6$\nbetween $N(u)$ and $N(u')$. Note that every vertex in $N(u) \\cap\nN(u')$ can be seen as a path of length $0$.\n\nTherefore, since $N(u)$ is isomorphic to $K_{2,2,2,2}$, there are two\nnon-edges $a_1a_2$ and $a_3a_4$ of $G[N(u)]$ such that each $a_i$ is\nthe end of the path $P_i$. We denote by $b_i$, $1 \\leq i \\leq 4$ the\nend in $N(u')$ of the path $P_i$.  Note that if $a_i \\in N(u) \\cap\nN(u')$ then $a_i = b_i$.  Moreover we can suppose that the choice of\n$a_1a_2$ and $a_3a_4$ maximizes the size of $\\{a_1,a_2,a_3,a_4\\} \\cap\nN(u')$. Since $N(u)$ is isomorphic to $K_{2,2,2,2}$ and since $|N(u)\n\\cap N(u')| \\leq 6$ (by Claim~\\ref{claim:k8-v6neighbors}), there are\nat most two vertices in $N(u) \\cap N(u')$ distinct from $a_1,a_2,a_3$,\nand $a_4$. Let $X= (N(u) \\cap N(u')) \\setminus \\{a_1,a_2,a_3,a_4\\}$.\n\nSince both $K_{2,2,2,2}$ and $\\overline{P_{10}}$ are $6$-connected\nthen $N[u']$ is $7$-connected and so $G[N[u'] \\setminus X]$ is\n$5$-connected.  Moreover $G[N[u'] \\setminus X]$ has too many edges to\nbe planar. Indeed, it has $9-|X|$ vertices and at least $32-7|X|$\nedges, which is more than $3(9-|X|)-6$ for $0 \\le |X| \\le 2$. We now\nneed the following theorem of Robertson and Seymour about vertex\ndisjoint pairs of paths.\n\n\\begin{thm}[Robertson and Seymour~\\cite{rs1}]\n\\label{th:disjointpath}\nLet $v_1 , \\ldots, v_k$ be distinct vertices of a graph $H$. Then either\n\\begin{itemize}\n\\item[(i)] there are disjoint paths of $H$ with ends $p_1$ $p_2$ and\n  $q_1$ $q_2$ respectively, so that $p_1$, $q_1$, $p_2$, $q_2$ occur\n  in the sequence $v_1, \\ldots, v_k$ in order, or\n\\item[(ii)] there is a $(\\le 3)$-separation $(A,B)$ of $H$ with $v_1,\n  \\ldots, v_k \\in A$ and $|B \\setminus A| \\geq 2$, or\n\\item[(iii)] $H$ can be drawn in a disc with $v_1 , \\ldots, v_k$ on\n  the boundary in order.\n\\end{itemize}\n\\end{thm}\n\nApplying this theorem to the graph $G[N[u'] \\setminus X]$ with\n$(v_1,\\ldots v_k) = (b_1,b_3,b_2,b_4)$ one obtains that there are two\nvertex disjoint paths in $N[u'] \\setminus X$, a path $P_{1,2}$ between\n$b_1$ and $b_2$, and a path $P_{3,4}$ between $b_3$ and $b_4$.  Theses\npaths are disjoint from $N[u]$ by construction, except possibly at\ntheir ends.  Finally, since the paths $P_i$, for $1 \\leq i \\leq 4$,\nconstructed above are disjoint from $N[u]$ and from $N[u'] \\setminus\nX$, except at their ends, there exists two disjoint paths respectively\nlinking $a_1$ with $a_2$ (through $P_1$, $P_{1,2}$ and $P_2$), and\n$a_3$ with $a_4$ (through $P_3$, $P_{3,4}$ and $P_4$). This\ncontradicts the $K_8$-minor freeness of $G$ and thus concludes the\nproof of the lemma.\n\\end{proof}\n\n\nBy Lemma~\\ref{lem:2small_ink8} there exists at least two vertices\n$u$ and $u'\\in V(G) \\setminus N[s]$ with degree at most $11$. By\nLemma~\\ref{lem:NminDeg}, Lemma~\\ref{lem:noK333}, and\nLemma~\\ref{lem:noK2222}, both $G[N(u)]$ and $G[N(u')]$ are isomorphic\nto $\\overline{P_{10}}$. The two graphs induced by $N[u]$ and $N[u']$\nare close to a $K_8$-minor as observed in the following claim.\n\\begin{claim}\n\\label{cm:2edgeP10}\nIn $\\overline{P_{10}}$, adding two edges $ab$ $cd$, such that $ab$,\n$bc$ and $cd \\notin E(\\overline{P_{10}})$, creates a $K_7$-minor.\nFurthermore adding three edges $e_1$ $e_2$ and $e_3$, such that $e_1\n\\cap e_2 \\cap e_3 =\\emptyset$ in $\\overline{P_{10}}$, creates a\n$K_7$-minor.\n\\end{claim}\n\\begin{proof}\nOne can easily check the accuracy of the first statement, by noting\nthat adding any such pair of edges $ab$ and $cd$, yields the same\ngraph, and by noting that adding the edges $u_1u_2$ and $u_3u_4$ in\n$\\overline{P_{10}}$ (notations come from Figure~\\ref{fig:P10}) the\npartition $\\{\\{0,2\\},\\{1\\},\\{3\\},\\{4\\},\\{5\\}, \\{6,7\\}, \\{8,9\\}\\}$\ninduces a $K_7$-minor.\n\nFor the second statement, we can assume that the three added edges are\nsuch that they pairwise do not correspond to the first statement.\nWithout loss of generality, assume that one of the three edges is\n$u_0u_5$, and note that the other added edges are distinct from\n$u_1u_2$, $u_1u_6$, $u_3u_4$, $u_4u_9$, $u_2u_7$, $u_7u_9$, $u_3u_8$\nand $u_6u_8$. Consider the case where one of the other added edges is\nincident to $u_0u_5$. By symmetry one can assume that this edge is\n$u_0u_1$, but this implies that the third added edge is distinct from\n$u_0u_4$ (as the three edges would intersect), and from $u_3u_4$,\n$u_6u_9$, $u_5u_7$ and $u_5u_8$ (by the first statement). There is\nthus no remaining candidate for the third edge.\nThis implies that it is sufficient to consider the case where the\nedges $u_0u_5$, $u_2u_3$ and $u_6u_9$ are added in\n$\\overline{P_{10}}$. In this case the partition $\\{\\{1\\},\\{ 4\\},\\{\n7\\},\\{ 8\\},\\{ 0,5\\},\\{ 6,9\\},\\{ 2,3\\} \\}$ induces a $K_7$-minor.\n\\end{proof}\n\\begin{figure}[h]\n\\centering\n\\includegraphics{petersen.eps}\n\\caption{The Petersen graph $P_{10}$.}\n\\label{fig:P10}\n\\end{figure}\n\nLet us list the induced subgraphs of $\\overline{P_{10}}$ of size 6.\n\\begin{claim}\n\\label{cm:k8-P10-6subgraphs}\nThere are exactly 6 distinct induced subgraphs of size 6 in\n$\\overline{P_{10}}$, including $K_{2,2,2}$. The complements of these\ngraphs are represented in Figure~\\ref{fig:P10-6subgraphs}.\nFurthermore note that every induced subgraphs of $\\overline{P_{10}}$\nof size at least 7, has a subgraph of size 6 distinct from\n$K_{2,2,2}$.\n\\end{claim}\nWe do not prove the claim here as one can easily check its accuracy.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=390px]{p10-sub.eps}\n\\caption{The complements of the subgraphs of $\\overline{P_{10}}$ of\n  size 6 (i.e. the subgraphs of $P_{10}$ of size 6).}\n\\label{fig:P10-6subgraphs}\n\\end{figure}\n\n\\begin{lem}\nThe vertices of $N(u)\\setminus N(s)$ (resp. of $N(u')\\setminus N(s)$)\nhave degree at least 12. Thus in particular, $u$ and $u'$ are\nnon-adjacent.\n\\label{lem:k8-P10-deg12}\n\\end{lem}\n\\begin{proof}\nWe assume for contradiction that $u$ has a neighbor $v$ of degree at\nmost 11.  By Lemma~\\ref{lem:NminDeg}, Lemma~\\ref{lem:noK2222}, and\nLemma~\\ref{lem:noK333}, the graph $G[N(v)]$ is isomorphic to\n$\\overline{P_{10}}$.\n\nAssume $v=u_0$ in Figure~\\ref{fig:P10}. Since $N(u_0) \\supset\n\\{u,u_2,u_3,u_6,u_7,u_8,u_9\\}$, the adjacencies in\n$G[\\{u,u_2,u_3,u_6,u_7,u_8,u_9,\\}]$ allow us to denote $u$ by $u'_0$,\nand denote $u'_1$, $u'_4$ and $u'_5$ the vertices in $N(u_0)\\setminus\nN[u]$, in such a way that these indices again correspond to\nFigure~\\ref{fig:P10}. It is now easy to see that contracting one edge\nin each of the paths $(u_2,u'_4,u_7)$ and $(u_6,u'_5,u_9)$ creates the\nedges $u_2u_7$ and $u_6u_9$ in $G[N[u]]$ and thus yields a $K_8$-minor\n(by Claim~\\ref{cm:2edgeP10} as $u_7u_9$ is a non-edge of\n$\\overline{P_{10}}$), a contradiction.\n\\end{proof}\n\nThe vertices $u$ and $u'$ are non-adjacent, however they can share\nneighbors.  Let us prove that they cannot share more than 7 neighbors.\n\\begin{lem}\n$|N(u) \\cap N(u')|\\le 7$.\n\\label{lem:k8-P10->=7common}\n\\end{lem}\n\\begin{proof}\nAssume for contradiction that $|N(u) \\cap N(u')|\\ge 8$, that is\nequivalently that $|N(u) \\setminus N(u')|\\le 2$ and $|N(u') \\setminus\nN(u)|\\le 2$.  Note that as $\\deg(s)\\le 7$ the set $(N(u) \\cap N(u'))\n\\setminus N(s)$ is non-empty, and denote $x$ one of its vertices.  By\nLemma~\\ref{lem:k8-P10-deg12}, this vertex $x$ as degree at least $12$.\nAs it has exactly 6 neighbors in $N(u)$, at most 2 neighbors in $N(u')\n\\setminus N(u)$, and as it is adjacent to both $u$ and $u'$, $x$ has\nat least two neighbors in $V(G) \\setminus (N[u] \\cup N[u'])$. Thus\nthere exists a component $C\\neq\\{s\\}$ in $G \\setminus (N[u] \\cup\nN[u'])$. As $C\\neq\\{s\\}$ and $N(C)\\subseteq N(u)\\cup N(u')$,\nLemma~\\ref{lem:connectk8withs} implies that $|N(C)|\\ge 8$. Therefore,\nas $|N(u') \\setminus N(u)|\\le 2$, $|N(C) \\cap N(u)| \\ge 6$ and there\nexist a path $P$ with inner vertices in $C$ and with non-adjacent ends\nin $N(u)$ (by Claim~\\ref{cm:k8-P10-6subgraphs}). Let us denote $x$ and\n$y$ the ends of $P$. As $|N(u) \\cap N(u')|\\ge 8$ and by\nClaim~\\ref{cm:k8-P10-6subgraphs}, there exists a vertex $z\\in N(u)\n\\cap N(u')$ such that $z\\neq x$ or $y$, and such that contracting the\nedge $zu'$ creates at least two edges in $N(u)$. As these three added\nedges ($xy$ and the edges adjacent to $z$) do not intersect,\nClaim~\\ref{cm:2edgeP10} implies that there is a $K_8$-minor, a\ncontradiction.\n\\end{proof}\n\nAs by Lemma~\\ref{lem:connectk8withs} there is no $(\\le 5)$-separator\n$(A,B)$ with $u\\in A\\setminus B$ and $u'\\in B\\setminus A$, Menger's\nTheorem implies the existence of 6 vertex disjoint paths $P_1 \\ldots\nP_6$ between $u$ and $u'$.\nBy minimizing the total length of these paths we can assume that\neach vertex in $N(u)\\cap N(u')$ corresponds to one of these paths,\nand that any of these paths intersect $N(u)$ (resp. $N(u')$) in\nonly one vertex. Contracting the inner edges (those non-incident to\n$u$ or $u'$) of these paths, and considering the graph induced by\n$N[u]\\cup N[u']$ one obtains a graph $H$ such that:\n\\begin{itemize}\n\\item $u$ and $u'$ are nonadjacent and $|N_H(u)\\cap N_H(u')| = 6$ or\n  $7$.\n\\item $\\deg_H(u)=10$, and $H[N(u)]$ contains $\\overline{P_{10}}$ as a\n  subgraph.\n\\item $\\deg_H(u')=10$, and $H[N(u')]$ contains $\\overline{P_{10}}$ as\n  a subgraph.\n\\end{itemize}\n\nIf the graph induced by $N_H(u)\\cap N_H(u')$ is isomorphic to\n$K_{2,2,2}$, then one can assume without loss of generality that\n$N(u)=\\{u_0,\\ldots,u_9\\}$ and that $N(u')=\\{u_0,u'_1, u_2,u_3,\nu'_4,u_5,u_6, u'_7,u'_8,u_9\\}$, where the indices correspond to\nFigure~\\ref{fig:P10}. Now observe that contracting the edge $u_0u'$,\nthe path $(u_6,u'_7,u'_8)$, and the path $(u_2,u'_4,u'_1)$,\nrespectively create the edges $u_0u_5$, $u_6u_9$, and $u_2u_3$.  This\nimplies by Claim~\\ref{cm:2edgeP10} that $N[u]$ contains a $K_8$-minor,\na contradiction. We can thus assume by\nClaim~\\ref{cm:k8-P10-6subgraphs} that the complement of $N_H(u)\\cap\nN_H(u')$ contains a path $(a,b,c,d)$.  As $\\overline{P_{10}}$ is\n6-connected, the graph induced by $\\{a,b\\}\\cup (N_H(u')\\setminus\nN(u))$ is connected, and thus contains a path from $a$ to $b$.  By\nClaim~\\ref{cm:2edgeP10}, this path with the path $(c,u',d)$, imply\nthat $H$ (which is a minor of $G$) contains a $K_8$-minor, a\ncontradiction. Thus there is no counter-example $G$, and this\nconcludes the proof of the theorem.\n\\end{proof}\n\nThe proof Theorem~\\ref{th:k8-deg9} is very similar. To do this one can\nprove the following variant of Theorem~\\ref{th:k8triwiths}.\n\\begin{thm}\nConsider a connected $K_8$-minor free graph $G$ with a vertex $s$ of\ndegree at most 7, such that $N[s] \\subsetneq V(G)$ and such that\n$\\min_{v\\in V(G)\\setminus N[s]}\\ge 9$. Then $G$ has an edge\n$e \\in E(G) \\setminus E(G[N[s]])$ that belongs to at most 5 triangles.\n\\end{thm}\nThe proof of this theorem is as the proof of\nTheorem~\\ref{th:k8triwiths}, except that one does not need to consider\nthe case where some vertex $u$ is such that $N(u)$ induces a\n$K_{2,2,2,2}$.\n\n\\section{Global density of triangles}\\label{sec:moytri}\n\nIn this section, we investigate the relation between the number of\ntriangles and the number of edge of a graph. Denotes by $\\rho =\n\\frac{t}{m}$ the ratio between the number of triangles $t$ and the\nnumber of edges $m$ of a graph $G$.  For each $k$, what is the minimum\nnumber $f(k)$ such that for all graph $G$ with $\\rho \\ge f(k)$, $G$\ncontains a $K_k$ minor ?\n\nIt is easy to notice that 2-trees on $n \\ge 2$ vertices have exactly\n$1 + 2(n-2)$ edges and $n-2$ triangles.  Furthermore, for $k \\ge 3$\none can notice that $k$-trees on $n \\ge k$ vertices have exactly\n$\\frac{k(k-1)}{2} + k(n-k)$ edges and $\\frac{k(k-1)(k-2)}{6} +\n(n-k)\\frac{k(k-1)}{2}$ triangles. Thus any $k$-tree, for $k \\ge 2$,\nverifies\n\\[ t = \\frac{k-1}{2} m -\\frac{1}{2}{{k+1}\\choose{3}} .\\]\nSince $k$-trees are $K_{k+2}$-minor free, for all $k \\ge 4$ there\nexists $K_k$-minor free graphs with $\\frac{k-3}{2} m\n-\\frac{1}{2}{{k-1}\\choose{3}}$ triangles.\n\nWe deduce that for all $k \\ge 4$, $f(k) \\geq \\frac{k-3}{2}$.  Indeed\nfor every $\\epsilon > 0$, there exists a number $m$ and a $K_k$-minor\nfree graph with $m$ edges such that $\\frac{k-3}{2} - \\epsilon \\leq\n\\rho < \\frac{k-3}{2}$.  In fact, for $4 \\le k \\le 7$, the following\ntheorem proves that this lower bound is best possible, so we have\n$f(k) = \\frac{k-3}{2}$.\n\n\\begin{thm}\nFor $4 \\le k \\le 7$ (resp. $k = 8$), every graph with $m \\ge 1$ edges\nand $t \\ge m(k-3)\/2$ triangles has a $K_k$-minor (resp. a $K_8$- or a\n$K_{2,2,2,2,2}$-minor).\n\\label{thm:global}\n\\end{thm}\n\n\\begin{proof}\nConsider by contradiction, a non-trivial $K_k$-minor free\n(resp. $K_8$- and $K_{2,2,2,2,2}$-minor free) graph $G$ with $t\n\\ge m(k-3)\/2$ triangles.  Among the possible graphs $G$, consider one\nthat minimizes $m$ (given that $m \\ge 1$).\n\nGiven any edge $uv \\in E(G)$ let $H_{uv} = G[N(u) \\cap N(v)]$ and\ndenote $n'$ and $m'$ its number of vertices and edges respectively.\nContracting $uv$ yields a proper minor of $G$, with exactly $1 + n'$\nedges less, and with at most $n' + m'$ triangles less. Thus by\nminimality of $G$, for every edge $uv$\n\\[ n' + m'  >  \\frac{k-3}{2} (1 + n') \\]\nwhich implies that\n\\[ m'  >  \\frac{k-3}{2} + \\frac{k-5}{2} n' .\\]\nOn the other hand we have that $\\frac{n'(n' - 1)}{2} \\ge m'$, and this\nimplies that $n'$ should verify $(n' + 1)(n' + 3 - k) > 0$, that is\nthat $n' \\ge k - 2$. In other words, every edge $uv$ of $G$ belongs to\nat least $k-2$ triangles. By Theorems~\\ref{th:krtri},\n(resp. Theorem~\\ref{th:k8tri}), this contradicts the $K_k$-minor\nfreeness (resp. $K_8$- and $K_{2,2,2,2,2}$-minor freeness) of $G$.\n\\end{proof}\n\n\n\\section{Application to stress freeness of graphs}\\label{sec:stress}\n\nThe motivation of this application is a problem that arises from the\nstudy of tension and compression forces applied on frameworks in the\nEuclidian space $\\mathbb{R}^d$. A $d$-framework is a graph $G=(V,E)$\nand an embedding $\\rho$ of $G$ in $\\mathbb{R}^d$. The reader should\nthink of a framework as an actual physical system where edges are\neither straight bars or cables and vertices are articulated joints.  A\n\\textit{stress} on a framework $(G,\\rho)$ is a function $\\omega:\nE(G)\\, \\rightarrow\\; \\mathbb{R}$ such that $\\forall v \\in V$,\n\\[\\underset{\\{u,v\\}\\in E}{\\sum}\\:\\omega(\\{u,v\\})(\\rho(v) - \\rho(u)) = 0 .\\]\nStress corresponds to some notion of equilibrium for the associated\nphysical system.  Each vertex is affected by tension and compression\nforces created by the bars and cables.  $\\omega(\\{u,v\\})$ can be\nthought of as the magnitude of such force per unit length, with\n$\\omega(\\{u,v\\}) < 0$ for a cable tension and $\\omega(\\{u,v\\}) > 0$\nfor a bar compression.  A stress is a state of the system where these\nforces cancel each other at every vertex.  We can see that every\nframework admits a \\textit{trivial} stress where $\\omega$ is\nidentically zero.  A $d$-framework admitting only the trivial stress\nis called \\textit{$d$-stress free}.\n\nTo make this notion independent of the embedding of $G$, the following\nwas introduced.  A graph $G$ is \\textit{generically $d$-stress free}\nif the set of all $d$-stress free embeddings of $G$ in $\\mathbb{R}^d$\nis open and dense in the set of all its embeddings (i.e. every\nstressed embedding of $G$ is arbitrary close to a stress free\nembedding).\n\nThis notion has been first used on graphs coming from $1$-skeletons of\n$3$-dimensional polytopes\n\\cite{cauchy-13,maxwell-64,cw-93,whiteley-84}, which are planar by\nSteiniz's theorem.  Gluck generalized the results on $3$-dimensional\npolytopes to the whole class of planar graphs.\n\\begin{thm}[Gluck, 1975,~\\cite{gluck-75}]\nPlanar graphs are generically $3$-stress free.\n\\label{th:gluckstress}\n\\end{thm}\nNevo proved that we can generalize Theorem~\\ref{th:gluckstress} for\n$K_5$-minor free graphs, and extended the result as follows.\n\n\\begin{thm}[Nevo, 2007,~\\cite{nevo1}]\nFor $2 \\leq r \\leq 6$, every $K_r$-minor free graph is\ngenerically $(r-2)$-stress free.\n\\label{th:nevostress}\n\\end{thm}\n\nHe conjectured this to hold also for $r = 7$ and noticed that the\ngraph $K_{2,2,2,2,2}$ is an obstruction for the case $r=8$. Indeed,\n$K_{2,2,2,2,2}$ is $K_8$-minor free and has too many edges to be\ngenerically $6$-stress free (a generically $\\ell$-stress free graph\nhas at most $\\ell n - {\\ell+1 \\choose 2}$ edges~\\cite{nevo1}).  We\nanswer positively to Nevo's conjecture and we give a variant for the\ngenerically $6$-stress freeness.\n\n\\begin{thm}\nEvery $K_7$-minor free graph (resp. $K_8$- and $K_{2,2,2,2,2}$-minor\nfree graph) is generically $5$-stress free (resp. $6$-stress free).\n\\label{th:genstressk78}\n\\end{thm}\n\nThe following result of Whiteley~\\cite{whiteley-89} is used to derive\nTheorem~\\ref{th:genstressk78}.\n\\begin{thm}[Whiteley, 1989,~\\cite{whiteley-89}]\nLet $G'$ be obtained from a graph $G$ by contracting an edge $\\{u,v\\}$.\nIf $u$, $v$ have at most $d - 1$ common neighbors and $G'$ is generically\n$d$-stress free, then $G$ is generically $d$-stress free.\n\\label{th:whitcontract}\n\\end{thm}\n\nNow, we prove Theorem~\\ref{th:genstressk78}.\n\\begin{proof}\nAssume that $G$ is a $K_7$-minor free graph (resp. a $K_8$- and\n$K_{2,2,2,2,2}$-minor free graph). Without loss of generality, we can\nalso assume that $G$ is connected.  Now, contract edges belonging to\nat most $4$ (resp. 5) triangles as long as it is possible and we\ndenotes by $G'$ the graph obtained. Note that by construction, every\nedge of $G'$ belongs to 5 (resp. 6) triangles. Note also that $G'$ is\na minor of $G$, and is thus $K_7$-minor free (resp.  $K_8$- and\n$K_{2,2,2,2,2}$-minor free). Theorem~\\ref{th:krtri}\n(resp. Theorem~\\ref{th:k8tri}) thus implies that $G'$ is the trivial\ngraph without any edge and with one vertex. This graph is trivially\ngenerically $5$-stress free (resp. $6$-stress free), and so by\nTheorem~\\ref{th:whitcontract}, $G$ also is generically $5$-stress free\n(resp. $6$-stress free).\n\\end{proof}\n\nWe denote by $\\mu(G)$ the Colin de Verdi\u00e8re parameter of a graph $G$.  A\nresult of Colin de Verdi\u00e8re~\\cite{cdv1} is that a graph $G$ is planar\nif and only if $\\mu(G) \\leq 3$. Lov\\'asz and Schrijver~\\cite{ls1}\nproved that $G$ is linklessy embeddable if and only if $\\mu(G) \\leq\n4$.  Nevo conjectured that the following holds.\n\n\\begin{conj}[Nevo, 2007,~\\cite{nevo1}]\nLet $G$ be a graph and let $k$ be a positive integer. If $\\mu(G) \\leq k$\nthen $G$ is generically $k$-stress free.\n\\label{conj:munevo}\n\\end{conj}\n\nThis conjecture holds for the cases $k = 5$ and $k = 6$ as a\nconsequence of Theorem~\\ref{th:genstressk78}.\n\n\\begin{cor}\nIf $G$ is a graph such that $\\mu(G) \\leq 5$ (resp. $\\mu(G) \\leq 6$) then\n$G$ is generically $5$-stress free (resp. $6$-stress free).\n\\end{cor}\n\n\\begin{proof}\nNote that $\\mu(K_r) = r - 1$ and that if the complement of an\n$n$-vertex graph $G$ is a linear forest, then $\\mu(G) \\geq n -\n3$~\\cite{lsv1}. So we have that $\\mu(K_7) = 6$, $\\mu(K_8) = 7$,\nand $\\mu(K_{2,2,2,2,2}) \\geq 7$.\n\nAs the parameter $\\mu$ is minor-monotone~\\cite{cdv1}, the graph $K_7$\n(resp. $K_8$ and $K_{2,2,2,2,2}$) is an excluded minor for the class\nof graphs defined by $\\mu(G) \\leq 5$ (resp. $\\mu(G) \\leq 6$). Hence by\nTheorem~\\ref{th:genstressk78}, these graphs are generically $5$-stress\nfree (resp. $6$-stress free).\n\\end{proof}\n\n\n\\section{Application to double-critical $k$-chromatic graphs}\\label{sec:double}\n\nA connected $k$-chromatic graph is said to be double-critical is for\nall edge $uv$ of $G$, $\\chi(G \\setminus\\{u,v\\}) = \\chi(G) - 2$. It is\nclear that the clique $K_k$ is such a graph.  The following\nconjecture, known has the Double-Critical Graph Conjecture, due to\nErd\u0151s and Lov\u00e1sz states that they are the only ones.\n\n\\begin{conj}[Erd\u0151s and Lov\u00e1sz, 1968,~\\cite{e68}]\nIf $G$ is a double-critical $k$-chromatic graph, then $G$ is\nisomorphic to $K_k$.\n\\label{conj:dcgraph}\n\\end{conj}\n\nThis conjecture has been proved for $k \\leq 5$ but remains open for $k\n\\geq 6$.  Kawarabayashi, Pedersen and Toft have formulated a relaxed\nversion of both Conjecture~\\ref{conj:dcgraph} and the Hadwiger's\nconjecture, called the Double-Critical Hadwiger Conjecture.\n\n\\begin{conj}[Kawarabayashi, Pedersen, and Toft, 2010,~\\cite{kpt10}]\nIf $G$ is a double-critical $k$-chromatic graph, then $G$ contains a\n$K_k$-minor.\n\\label{conj:dchadwiger}\n\\end{conj}\n\nThe same authors proved this conjecture for $k \\leq 7$~\\cite{kpt10},\nbut the case $k = 8$ is left as an open problem. Pedersen proved that\nevery $8$-chromatic double-critical contains $K_8^-$ as a\nminor~\\cite{ped11}. Below we prove that the conjecture also holds for\n$k=8$.\n\nThe following proposition lists some interesting properties about\n$k$-chromatic double-critical graphs :\n\\begin{prop}[Kawarabayashi, Pedersen, and Toft, 2010,~\\cite{kpt10}]\nLet $G \\neq K_k$ be a double-critical $k$-chromatic graph, then\n\\begin{itemize}\n\\item The graph $G$ does not contains $K_{k-1}$ as a subgraph,\n\\item The graph $G$ has minimum degree at least $k + 1$,\n\\item For all edges $uv \\in E(G)$ and all $(k-2)$-coloring of $G - u -\n  v$, the set of common neighbors of $u$ and $v$ in $G$ contains\n  vertices from every color class.\n\\end{itemize}\n\\label{prop:kpt}\n\\end{prop}\n\nIn particular, the last item implies that every edge belongs to at\nleast $k - 2$ triangles.\n\n\\begin{thm}\nEvery double-critical $k$-chromatic graph, for $k\\le 8$, contains\n$K_k$ as a minor.\n\\end{thm}\n\\begin{proof}\nConsider for contradiction a $K_k$-minor free graph $G$ that is\ndouble-critical $k$-chromatic. By the second item of\nProposition~\\ref{prop:kpt}, $\\delta(G)\\ge k+1$. By\nTheorem~\\ref{th:krtri} and Theorem~\\ref{th:k8-deg9}, this graph\nhas an edge that belongs to at most $k-3$ triangles. This\ncontradicts the last item of Proposition~\\ref{prop:kpt}.\n\\end{proof}\n\nLet us now give an alternative proof of the case $k=8$ that does not\nneed Theorem~\\ref{th:k8-deg9}, but uses Theorem~\\ref{th:k8tri}\ninstead. This might be usefull to prove the next case of\nConjecture~\\ref{conj:dchadwiger}.\n\nConsider for contradiction a $K_8$-minor free graph $G$ that is\ndouble-critical $8$-chromatic.  By Theorem~\\ref{th:k8tri} this graph\nhas an edge that belongs to at most $5$ triangles or contains\n$K_{2,2,2,2,2}$ as an induced subgraph.  By Proposition~\\ref{prop:kpt}\nevery edge of $G$ belongs to at least 6 triangles, thus $G$ contains\n$K_{2,2,2,2,2}$ as an induced subgraph.  Let us denote $K\\subseteq\nV(G)$ the vertex set of a copy of $K_{2,2,2,2,2}$ in $G$. As\n$K_{2,2,2,2,2}$ is maximal $K_8$-minor free, any connected component\n$C$ of $G\\setminus K$ is such that $N(C)\\subset K$ induces a clique.\nAs $G$ is double-critical $8$-chromatic, there exists a $6$-coloring\nof $G[N[C]]$, and a $6$-coloring of $G\\setminus C$. As these two\ngraphs intersect on a clique one can combine their colorings and thus\nobtain a 6-coloring of $G$, a contradiction.\n\n\n\\section{Application for coloration of $K_d$-minor free graphs}\\label{sec:coloration}\n\nHadwiger's conjecture says that every $t$-chromatic graph $G$\n(i.e. $\\chi(G) =t$) contains $K_t$ has a minor. This conjecture has\nbeen proved for $t \\leq 6$, where the case $t = 5$ is equivalent to\nthe Four Color Theorem by Wagner's structure theorem of $K_5$-minor\nfree graph, and the case $t = 6$ has been proved by Robertson, Seymour\nand Thomas~\\cite{rst1}. The conjecture remains open for $t \\geq 7$.\nFor $t = 7$ (resp. $t = 8$) the conjecture asks $K_7$-minor free graphs\n(resp. $K_8$-minor free graphs) to be $6$-colorable (resp. $7$-colorable).\nUsing Claim~\\ref{claim-alpha-Nv} and the $9$-degeneracy\n(resp. $11$-degeneracy) of these graphs, one can prove that they are\n$9$-colorable (resp. 11-colorable). We improve these bounds by one.\n\nA graph $G$ is said to be \\emph{$t$-minor-critical} if $\\chi(G) = t$\nand $\\chi(H) < t$ whenever $H$ is a strict minor of $G$.  Hadwiger's\nconjecture can thus be reformulated as follows : Every\n$t$-minor-critical graph contains $K_t$ has a minor.  In the following\n$\\alpha(S)$ means $\\alpha(G[S])$,the independence number of\n$G[S]$. The following is a folklore claim, here for completeness.\n\n\\begin{claim}\n\\label{claim-alpha-Nv}\nGiven a $k$-minor critical graph $G$, for every vertex $v\\in V(G)$ we\nhave that $\\deg(v) + 2 - \\alpha(N(v)) \\ge k$.\n\\end{claim}\n\n\\begin{proof}\nGiven a vertex $v$ and a stable set $S$ of $N(v)$, consider the graph\n$G'$ obtained from $G$ by contracting the edges between $v$ and\n$S$. Since $G'$ is a strict minor of $G$ it is\n$(k-1)$-colorable. Given such coloring of $G'$, one can $(k-1)$-color\n$G\\setminus\\{v\\}$ in such a way that all the vertices of $S$ have the\nsame color assigned. In this coloring at most $\\deg(v) +1 - |S|$\ncolors are used in $N(v)$, thus $\\deg(v) +2 - |S|$ colors\nare sufficient to color $G$, and thus $\\deg(v) + 2 - \\alpha(N(v)) \\ge k$.\n\\end{proof}\n\nA \\emph{split graph} is a graph which vertices can be partionned into\none set inducing a clique, and one set inducing an independent set.\nThese graphs are the graphs that do not contain $C_4$, $C_5$ or $2K_2$\nas induced subgraphs~\\cite{fh1}.\n\n\\begin{claim}\n\\label{claim-no-split}\nGiven a $k$-minor critical graph $G$, every separator $(A,B)$ of $G$\nis such that $G[A \\cap B]$ is not a split graph (i.e $G[A \\cap B]$\ncontains $C_4$, $C_5$ or $2K_2$ as an induced subgraph).\n\\end{claim}\n\n\\begin{proof}\nAssume by contradiction that there exists such separator $(A',B')$.\nThis implies the existence of a separator $(A,B)$ such that $S = A\\cap\nB \\subseteq A'\\cap B'$ , and such that each $G[A\\setminus S]$ and\n$G[B\\setminus S]$ have a connected component, $C_{A}$ and $C_{B}$ such\nthat $N(C_{A})=N(C_{B})=S$. Note that $G[S]$ is a split graph and let\n$I$ be one of its maximum independent sets and let $K=S \\setminus I$\nbe a clique. Let $G_A$ and $G_B$ be the graphs respectively obtained\nfrom $G[A]$ and $G[B]$ by identifying the vertices of $I$ into a\nsingle vertex $i$. By maximality of $I$, in both graphs the vertex set\n$K\\cup \\{i\\}$ induces a clique.  Furthermore, these graphs are strict\nminors of $G$ as the identification of the vertices in $I$ can be done\nby contracting edges incident to $C_B$ or $C_A$ respectively.  Thus,\nthese graphs are $(k-1)$-colorable and these colorings imply the\nexistence of compatible $(k-1)$-colorings of $G[A]$ and $G[B]$, since\nin both colorings the vertices of $I$ use the same color, and each\nvertex of $K$ uses a distinct color. This yields in a $(k-1)$-coloring\nof $G$, a contradiction.\n\\end{proof}\n\n\\begin{thm}\n$K_7$-minor free graphs are $8$-colorable.\n$K_8$-minor free graphs are $10$-colorable.\n\\end{thm}\n\n\\begin{proof}\nConsider by contradiction that there is a $K_7$-minor free graph $G$\nnon-8-colorable (resp. a $K_8$-minor free graph $G$ non-10-colorable).\nThis graph is chosen such that $|E(G)|$ is minimal, this graph is\nthus 9-minor-critical (resp. 11-minor-critical).\n\nFor any vertex $v$, since $\\alpha(N(v))$ is at least 1,\nClaim~\\ref{claim-alpha-Nv} implies that $\\deg(v) > 7$ (resp. $\\deg(v)\n> 9$).  If $\\deg(v) = 8$ (resp. $\\deg(v) = 10$), since $G$ is\n$K_7$-minor free (resp. $K_8$-minor free), we have $\\alpha(N(v)) \\ge\n2$, contradicting Claim~\\ref{claim-alpha-Nv}. Finally if $\\deg(v) = 9$\n(resp. $\\deg(v) = 11$), Claim~\\ref{claim-alpha-Nv} implies that\n$3>\\alpha(N(v))$, and since $N(v)$ cannot be a clique, $\\alpha(N(v)) =\n2$. Thus with Mader's theorem we have that $\\delta(G)= 9$\n(resp. $\\delta(G)= 11$), and that for every vertex $v$ of degree 9\n(resp. of degree 11), $\\alpha(N(v)) = 2$.  By Theorem~\\ref{th:krtri}\n(resp. Theorem~\\ref{th:k8triweak}), we consider a vertex $u$ of degree\n9 (resp. 11) such that there is an edge $uv$ which belongs to at most\n$4$ (resp. $5$) triangles. Let $H = G[N(u)]$, and recall that\n$\\alpha(H) = 2$.\n\n\\begin{claim}\n\\label{claim-no-K5}\nThe graph $H = G[N(u)]$ does not contain a $K_5$ (resp. a $K_6$).\n\\end{claim}\n\n\\begin{proof}\nAssume by contradiction that $H$ contains a $K_t$ with vertices\n$x_1,\\ldots,x_t$, for $t=5$ (resp. for $t=6$). Assume first that the\ngraph induced by $Y = N(u) \\setminus \\{x_1,\\ldots,x_t\\}$ is\nconnected. Since $\\delta(G) \\ge 9$ every vertex $x_i$ has a neighbor\nin $Y$ or a neighbor $w_i$ in $G \\setminus N[u]$. In the latter case,\ndenote $C_i$ the connected component of $w_i$ in $G\\setminus N[u]$.\nSince by Claim~\\ref{claim-no-split} (for the partition $(N[C_i], V(G)\n\\setminus C_i)$) $N(C_i)$ intersects $Y$, one can contract $Y\\cup\n(V(G) \\setminus N[u])$ into a single vertex and form a $K_{t+2}$\ntogether with vertices $u,x_1,\\ldots,x_t$, a contradiction.\n\nAssume now that the graph induced by $Y$ is not connected and let\n$y_1, y_2 \\in Y$ be non-adjacent vertices. Since $G$ is $(2t-1)$-minor\ncritical, consider a $(2t-2)$-coloring of the graph $G'$ obtained from\n$G$ by contracting $uy_1$ and $uy_2$. This coloring implies the\nexistence of a $(2t-2)$-coloring $c$ of $G\\setminus u$ such that\n$c(y_1) = c(y_2)$. As this coloring does not extends to $G$, the\n$2t-1$ vertices in $N(u)$ use all the $(2t-2)$ colors. This implies\nthat the colors used for the $x_i$ are used only once in $N(u)$, and\nthat there exists a vertex $z \\in Y$ which color is used only once in\n$N(u)$. Assume $c(x_i)=i$ and $c(z)=7$. Given two colors $a$, $b$ and\na vertex $v$ colored $a$, the \\emph{$(a,b)$-component of $v$} is the\nthe connected component of $v$ in the graph induced by $a$- or\n$b$-colored vertices. For any $1\\le i \\le t$, suppose we switch colors\nin the $(i,7)$-component of $z$. As this cannot lead to a coloring\nwhich does not use all the colors in $N(u)$, there exists a\n$(7,i)$-bicolored path from $z$ to $x_i$. This is impossible as\ncontracting these paths on $z$ would lead to a $K_{t+2}$ (with vertex\nset $\\{u,z,x_1,\\ldots,x_t\\}$). This concludes the proof of the claim.\n\\end{proof}\n\nLet $v$ be a vertex of $H$ with minimum degree in $H$.  By the choice\nof $u$ and Theorem~\\ref{th:krtri} (resp. Theorem~\\ref{th:k8triweak}),\n$\\deg_{H}(v)\\le 4$ (resp. $\\deg_{H}(v)\\le 5$).\n\n\\begin{claim}\n$\\delta(H) = \\deg_H(v) = 4$ (resp. $\\delta(H) = \\deg_H(v) = 5$).\n\\end{claim}\n\\begin{proof}\nSince $\\alpha(H)=2$, the non-neighbors of $v$ in $H$ form a\nclique. Furthermore since $H$ does not contain a $K_5$ (resp. a $K_6$)\nwe have that $9 - 1 - \\deg_{H}(v) < 5$ (resp. that $11 - 1 -\n\\deg_{H}(v) < 6$), and hence $\\deg_{H}(v) = 4$ (resp. $\\deg_{H}(v) =\n5$).\n\\end{proof}\n\nLet $y_1,\\ldots,y_t$ with $t = 4$ (resp. $t = 5$) be the neighbors of\n$v$ in $H$, and let $K$ be the $t$-clique formed by its non-neigbors.\nBy Claim~\\ref{claim-no-K5} we can assume that $y_1$ and $y_2$ are\nnon-adjacent. Note that since $\\alpha(G[N(u)]) = 2$ every vertex of\n$K$ is adjacent to $y_1$ or $y_2$. Since $G$ is $(2t+1)$-minor\ncritical, consider a $2t$-coloring of the graph $G'$ obtained from $G$\nby contracting $uy_1$ and $uy_2$. This coloring implies the existence\nof a $2t$-coloring $c$ of $G \\setminus u$ such that $c(y_1) = c(y_2)$. As\nthis coloring does not extends to $G$, the $2t + 1$ vertices in $N(u)$\nuse all the $2t$ colors. In particular, the colors used by $K$ (say\n$1,\\ldots t$) and $y_3$ (say 6) are thus used only once in $N(u)$.\nFor any $1\\le i\\le t$, suppose we switch colors in the $(i,6)$-component\nof $y_3$. As this cannot lead to a coloring which does not use all the\ncolors in $N(u)$, there exists a $(i,6)$-bicolored path from $y_3$ to\nthe $i$-colored vertex of $K$. This is impossible as contracting these\npaths on $y_3$, and contracting the edges $vy_1$ and $vy_2$ on $v$\nwould lead to a $K_{t+2}$ with vertex set $\\{u,v,y_3\\} \\cup K$. This\nconcludes the proof of the theorem.\n\\end{proof}\n\n\n\\section{Conclusion}\\label{sec:concl}\n\nTheorem~\\ref{thm:global} gives a sufficient condition for a graph to\nhave a $K_k$-minor. We wonder whether this condition is stronger than\nMader's Theorem : Is there a graph $G$ with a $K_k$-minor, for $4\\le\nk\\le 7$, that has $m\\le (k-2)n - {k-1 \\choose 2}$ edges and $t\\ge\nm(k-3)\/2$ triangles ?\n\nWe believe that our work can be extended to the next case.  Song and\nThomas~\\cite{st1} proved a Mader-like theorem, similar to\nTheorem~\\ref{th:jorg} in the case of $K_9$-minor free graphs.\n\n\\begin{thm}[Song and Thomas, 2006,~\\cite{st1}]\nEvery graph on $n \\geq 9$ vertices and at least $7n - 27$ edges either\nhas a $K_9$-minor or is a $(K_{1,2,2,2,2,2}, 6)$-cockade or is isomorphic to\n$K_{2,2,2,3,3}$.\n\\end{thm}\n\nNote that $K_{2,2,2,3,3}$ has edges that belong to exactly $6$ triangles\nand contains $K_{2,2,2,2,2,1}$ as a minor. We conjecture that we\ncan extend our main theorem as follows.\n\\begin{conj}\nLet $G$ a graph such that every edge belongs to at least $7$ triangles\nthen either $G$ has a $K_9$-minor or contains $K_{1,2,2,2,2,2}$\nas an induced subgraph.\n\\label{conj:extendk9}\n\\end{conj}\n\n\nProving this conjecture would have several consequences.\nThis would extend Theorem~\\ref{thm:global}\nas follows : Every graph $G$ with $m\\ge 1$ edges and $t\\ge 3m$ triangles\nhas a $K_9$ or $K_{1,2,2,2,2,2}$-minor.\nIt would also imply Conjecture~\\ref{conj:munevo} for the case $k = 7$,\ni.e.  $\\mu(G)\\le 7$ implies that $G$ is generically $7$-stress free.\nFinally, it would imply Conjecture~\\ref{conj:dchadwiger} for $k = 9$,\ni.e. double-critical $9$-chromatic graphs have a $K_9$-minor.\nWe also conjecture that the following holds. In particular, it would\nimply that $K_9$-minor free graphs are $12$-colorable (using the same\narguments as in Section \\ref{sec:coloration}).\n\\begin{conj}\nAny $K_9$-minor free graph $G$ with $\\delta(G)=13$ has an edge $uv$\nsuch that $u$ has degree $13$ and $uv$ belongs to at most $6$\ntriangles.\n\\label{conj:extendk9weak}\n\\end{conj}\n\nWe also believe that these structural properties on graph with edges\nbelonging to many triangles can actually be extended to\nmatroids. Graph minors can be studied in the more general context of\nmatroid minors~\\cite{o1}. A triangle is then a circuit of size\n$3$. Contrary to graphs, the case when every element of the matroid\nbelongs to $3$ triangles is already intricate. There are three\nwell-known matroids for which each element belongs to $3$ triangles :\nthe Fano matroid $F_7$, the uniform matroid $\\mathcal{U}_{2,4}$, and\nthe graphical matroid $\\mathcal{M}(K_5)$.  We conjecture that the\nfollowing holds.\n\n\\begin{conj}\nLet $\\mathcal{M}$ be a matroid where each element is contained in $3$\ntriangles, then $\\mathcal{M}$ admits $\\mathcal{M}(K_5)$, $F_7$ or\n$\\mathcal{U}_{2,4}$ as a minor.\n\\end{conj}\n\n\\bibliographystyle{plain}\n\\nocite{*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nManipulating the molecular rotational degrees of freedom in gas phase by means of laser fields remains a very attractive topic in quantum control \\cite{reviewQC1,reviewQC2} with a wide range of applications in photochemistry extending from chemical reactivity \\cite{Warren1993, Stapelfeldt2003} to nanoscale design \\cite{Seideman1997,Stapelfeldt1997}, stereochemistry \\cite{Rakitzis2004}, surface processing \\cite{Seideman1997,reuter2008}, catalysis \\cite{Bulthuis1991}, and attosecond molecular dynamics \\cite{Krausz2009}. Such phenomena play also a role in quantum computing \\cite{shapiro2003} and high-order harmonic generation \\cite{Atabek2003,Ramakrishna2007,Ramakrishna2010,houzet2013}. In this setting, molecular alignment and orientation can be identified as crucial prerequisites before exploring more complex control scenarios \\cite{reviewseideman,Stapelfeldt2003}. The alignment process \\cite{Friedrich1995} is by now a well-established concept from both the experimental and theoretical points of view with recent extensions ranging from the deflection of aligned molecules \\cite{gershnabel2010}, the introduction of planar alignment \\cite{hoque2011}, the control of molecular unidirectional motion \\cite{korech2013,steinitz2014,karras2015}, the study of molecular superrotors \\cite{korobenko2016,korobenko2014,milner2015} or the analysis of dissipation effects due to molecular collisions \\cite{ramakrishna2005,viellard1,viellard2,milner2014}. The description and the control of molecular orientation are  not currently at the same degree of improvement, in particular from the experimental point of view. Molecular orientation was achieved in the adiabatic regime \\cite{Goban2008,Ghafur2009,Filsinger2009,Takei2016,Muramatsu2009,Holmegaard2009}. If no static field is used, a rapid turn-off of the laser field allows to get orientation under field-free conditions. At low temperature, a very high degree of orientation can be obtained using such control strategies and a molecular quantum-state selection \\cite{Filsinger2009,Takei2016,Muramatsu2009,Holmegaard2009}. Control schemes in the sudden regime, where the duration of the control field is short with respect to the rotational period, have been also developed \\cite{averbukh2001,daems2005,dion2001,dion2005,ortigoso2012,Atabek2003,sugny2005,sugny2014,Lapert2012,machholm2001,henriksen1999,prasad2013,shu2013,Tehini08,zhang,Tehini2012,wu,Kanai2001}.\nAmong other techniques, we can mention the interaction of the molecule with a terahertz (THz) laser pulse and the $(\\omega,2\\omega)$- scheme. Note that a larger degree of orientation can be achieved with the two-color  mechanisms through ionization depletion \\cite{Spanner2012}. In this second regime, molecular orientation has been recently addressed experimentally for linear molecules using only a THz field \\cite{Nelson2011} or its combination with a laser field \\cite{Kitano2011}, and in \\cite{Znakovskaya2009,Kraus2014} for an excitation process with two-color laser fields.\n\nIn this work, we complement the previous experimental and theoretical works on the orientation dynamics produced by THz fields by exploring the orientation at high temperature (typically room temperature) of a symmetric top molecule, CH$_3$I. This molecule is a good candidate to achieve a high degree of orientation at room temperature due to its large permanent dipole moment and its relatively small rotational constant \\cite{Lapert2012}. The THz pulses are obtained from the excitation of a plasma by a two-color femtosecond laser field \\cite{Cook2000}, while the detection process is based on the free-induction decay (FID) emitted by the molecular sample after the THz excitation \\cite{hardprl,hard1991,hard1994,bigourd2008}. We extend the previous studies on the subject by considering the case of a symmetric top molecule at room temperature. We show that a noticeable degree of orientation can be reached. A theoretical description of the propagation of a THz field in the sample shows that the FID is not proportional to the degree of orientation but to its time derivative. A complete analytical derivation of this result is given in this paper. Note that this dependency has been already mentioned in \\cite{Nelson2011}. The relation between the FID and the degree of orientation allows us to quantitatively compare the experimental observations with the numerical simulations. A very good match is found for the first two orientation revivals. This agreement has been improved by accounting for the centrifugal distorsion and the relaxation effects in the computations \\cite{ramakrishna2005,viellard1,viellard2}. We also use this theoretical description to explore the influence of the laser parameters on the orientation dynamics.\n\nThe paper is organized as follows. The experimental setup is described in Sec.~\\ref{sec3}, with a special emphasis on the generation of THz pulses and on the detection process. We show in Sec.~\\ref{sec2p} that the FID is given at first order by the time derivative of the degree of orientation. The model system is introduced in Sec.~\\ref{sec2}. The numerical and the experimental results are discussed in Sec.~\\ref{sec4}. Conclusions and prospective views are given in Sec.~\\ref{sec5}.\n\n\\section{Experimental setup for producing molecular orientation \\label{sec3}}\nThe experimental set-up for producing and measuring molecular orientation is shown in Fig.\\ref{Set_Up}. It is based on a THz pulsed source and an electro-optical sampling device for the detection. The THz pulses are produced through plasma generation in gases with two-color femtosecond pulses \\cite{Cook2000,Kress2004,Kim2007,Kim2008,Vvedenskii2014}. A fraction of the output of a chirped pulse amplifier (CPA, 800 nm wavelength, 100 fs pulse duration, 7.5 mJ energy, 100 Hz repetition rate) is focused in dry nitrogen. A type I phase matching $\\beta$-Barium Borate (BBO) crystal  is inserted on the beam path at a given distance from the focusing point to produce the second harmonic at 400 nm. The generation of THz pulses is optimized by adjusting the phase matching angle of the frequency doubling crystal and its longitudinal position using a micrometric stage. The THz source provides pulses of typically few hundred femtoseconds pulse duration with only one cycle (see Fig.~\\ref{fig2} below for details) and covers a spectral range from 0 to 4 THz, with a maximum at about 1.5 THz. After collimation by an off-axis parabolic mirror, the two incoming beams at 800 and 400 nm are filtered out by means of a 2-mm thick plate from polytetrafluoroethylene and a thin black polyethylene sheet. The THz pulse is then focused in a cell (40 mm optical pathway) containing the sample at room temperature. The CH$_{3}$I sample (\\textit{Sigma Aldrich}, batch reference \\emph{BCBK 1300 V}, reagent grade chemicals) is initially stored in the liquid phase and vaporized by expansion into the cell under vacuum, just before the experiment so as to avoid water pollution of the sample.\nThe THz electric field in the gas sample is less than 100 kVcm$^{-1}$. The typical pressure used in this work is around 0.2-0.3 bar (\\textit{e.g.} below the saturation vapour pressure of CH$_{3}$I). For a good transmission of the THz beam through the gas cell, the windows are in polymethylpentene polymer (TPX, 38.1 mm diameter, 2.0 mm  thickness). The transmitted THz radiation is collected from the sample by another off-axis parabolic mirror and sent to the detection device. This latter is based on an electro-optical sampling of the THz pulse shape in the time domain \\cite{Winnewisser1997,Cai1998,Gallot1999,Planken2001}. The weak probe beam (typically $\\approx 5$ nJ) derived from the output of the CPA is focused and spatially overlapped with the THz radiation in a ZnTe (110) crystal. The polarization of the probe beam is modified through the Pockels effect induced by the THz beam (electro-optic detection process). This allows us to sample the THz electric field by changing the time delay between the two pulses with a motorized delay line stage. The change of the polarization state of the probe beam is measured from the combination of a quarter wave plate (QWP), a Wollaston Prism (WP), and two head-to-tail connected photodiodes so that the difference of their signals is directly obtained  (see Fig.~\\ref{Set_Up}). The difference signal is then amplified and sent to a lock-in amplifier synchronized with the laser repetition rate. The quarter wave plate is oriented so as to get a circular polarization without THz pulse and equivalent signals for the two photodiodes. The THz pathway is included in a box continuously purged (relative humidity $\\lesssim$ 7 $\\%$) with dry nitrogen to avoid absorption by water vapor.\n\\begin{figure*} [ht]\n                \\centering\n                \\includegraphics[width=12cm]{fig1}\n                \\caption{Experimental set-up. CPA: Chirped Pulse Amplifier (100 Hz repetition rate, $\\tau_L  \\approx 100~\\textrm{fs}$, $\\lambda_0 = 800~\\textrm{nm}$, $E_{100~\\textrm{Hz}}  \\approx 7.5~\\textrm{mJ}$), QWP: Quarter Wave Plate, HWP: Half Wave Plate, P: Polarizer, L: Lens, BBO:  Beta Barium Borate crystal, Filter: see the text, M: off-axis parabolic Mirror, BS: indium tin oxide Beam Splitter, ZnTe: electro-optical crystal, DL: motorized Delay Line stage,  WP: Wollaston Prism, PhD1 and PhD2: balanced photodiodes, A: integrated amplifier, PB: purged box.}\n                \\label{Set_Up}\n\\end{figure*}\nThe signal $\\Delta{V}$ recorded by the lock-in amplifier is proportional to $\\Delta{I}$, the difference between the intensities measured by the two head-to-tail connected photodiodes. $\\Delta{I}$ is given by \\cite{Planken2001}:\n\\begin{equation}\\label{firsteq}\n\\Delta{V}\\propto\\Delta{I}=I_{\\textrm{probe}} \\omega n^{3} {E(t) } r_{41} \\frac{L}{{c}},\n\\end{equation}\nwhere $I_{\\textrm{probe}}$ is the probe intensity, $\\omega$ the probe angular frequency, $n$ the refraction index at the probe frequency, $E(t)$ the THz electric field, $r_{41}$ the electro-optic coefficient of the ZnTe crystal (Pockels effect, $r_{41}$=4 pm~V$^{-1}$), $L$ the crystal length (200 $\\mu$m), and $c$ the speed of light in vacuum. The typical ratio $\\frac{{\\Delta V_{\\textrm{THz}} }}{{2V_{\\textrm{PhD}} }}=\\frac{{\\Delta{I}}}{I_{\\textrm{probe}}}$ measured in the experiment is $\\cong$ 1 - 5 $\\%$, $\\Delta V_{\\textrm{THz}}$ being the signal produced by the THz pulse  and $V_{\\textrm{PhD}}$ the signal delivered by each photodiode. The electric field $E(t)$ can be directly evaluated by using Eq.~(\\ref{firsteq}) and taking into account the transmission and reflection coefficients of the different optical components. The estimated electric field in the gas sample is typically within the range 6~-~30 kV~cm$^{-1}$. Note that $E(t)$ in Eq.~(\\ref{firsteq}) is the total electric field including the transmitted THz pulse $E_{0}(x,t)$ and the FID electric field, as discussed below. A typical recording of the electro-optical sampling signal is depicted in  Fig.~\\ref{Typical experimental signal}. It exhibits the transmitted THz pulse at zero delay and the two first orientational revivals at 67 and 134 ps. The goal of Sec.~\\ref{sec2p} is to interpret this experimental trace in terms of orientation efficiency.\n\\begin{figure} [ht]\n\\centering\n\\includegraphics[width=10cm]{fig2}\n\\caption{Experimental electro-optical sampling signal as a function of the delay between probe and THz pulses in CH$_{3}$I. The pressure in the gas cell was 0.26 bar. The transmitted THz pulse around zero delay and two first orientational revivals at 67 and 134 ps are shown.}\n\\label{Typical experimental signal}\n\\end{figure}\n\\section{Propagation of a THz field in a gaseous sample \\label{sec2p}}\nThis section is aimed at describing the production of FID and its propagation in a gaseous sample. We follow here the formalism and the approximations used in \\cite{hard1991,hard1994,hardprl,bigourd2008}.\n\nTo be more concrete, we consider a THz pulse linearly polarized along the $z$- direction and propagating along the $x$- one. This field is of the form $E_{0}(0, t)$ at $x = 0$.\nThis THz field experiences an instantaneous dipole $d(t)$ along the $z$- direction which satisfies:\n\\begin{equation}\nd(t) = \\mu_{0} \\langle\\cos\\theta\\rangle(t),\n\\end{equation}\nwhere $\\mu_0$ is the permanent dipole moment, $\\mu_0=1.6406$~D for the CH$_{3}$I molecule \\cite{gachi1989}, and $\\theta$ stands for the angle between the molecular axis defined by the C-I bond and the field polarization direction. The induced dipole $d(t)$ generates a contribution to the THz field which also propagates within the sample.\nWe introduce the polarization $p(\\omega)$ of the sample given in the frequency domain by:\n\\begin{equation}\np(\\omega) =\\frac{N}{V}d(\\omega)\n\\end{equation}\nwhere $N$ is the number of molecules and $V$ the corresponding volume. This polarization is also related to the electric field through the susceptibility parameter $\\chi(\\omega)$:\n\\begin{equation}\np(\\omega) = \\epsilon_{0}\\chi(\\omega)E(\\omega)\n\\end{equation}\nwhere $\\varepsilon_0$ is the vacuum permittivity. The spectral distribution of the field $E(\\omega)$ is defined as:\n\\begin{equation}\nE(\\omega) = \\frac{1}{2\\pi} \\int_{-\\infty}^{+\\infty} E_0(0, t) e^{i\\omega t} dt,\n\\end{equation}\nwhich leads to:\n\\begin{equation}\n\\chi(\\omega) = \\frac{Nd(\\omega)}{\\epsilon_{0}V E(\\omega)}.\n\\end{equation}\nIn a linear approximation framework where $\\chi$ does not depend on the amplitude of the field, the resolution of the Maxwell equations gives that the propagation of the field can be written as follows \\cite{hard1991,hard1994}:\n\\begin{equation}\nE(x, t) = \\int_{-\\infty}^{+\\infty} E(\\omega)e^{i[k(\\omega)x-\\omega t]}d\\omega,\n\\end{equation}\nwith $k(\\omega) = n(\\omega)\\omega\/c$, $n^{2} = 1+\\chi(\\omega)$. In the case of a dilute medium, the complex refractive index is given by $n(\\omega) \\simeq 1+\\frac{\\chi(\\omega)}{2}$. Expanding the exponential term $\\exp[\\frac{i\\omega x\\chi}{2c}]$ in Taylor series up to the second order, we arrive at:\n\\begin{eqnarray*}\nE(x, t)\n&= \\int_{-\\infty}^{+\\infty} E(\\omega)e^{i( \\omega \\frac{x}{c}-\\omega t)}d\\omega \\\\\n&+ \\int_{-\\infty}^{+\\infty}E(\\omega)e^{i( \\omega\\frac{x}{c}-\\omega t)}i\\frac{\\omega x}{c}\\frac{\\chi}{2}d\\omega ,\n\\end{eqnarray*}\nwhich leads to:\n\\begin{equation}\nE(x, t) = E_{0}(x, t) + \\int_{-\\infty}^{+\\infty} E(\\omega)e^{i( \\omega\\frac{x}{c}-\\omega t)}i \\frac{\\omega x}{c} \\frac{Nd(\\omega)}{2\\epsilon_{0}V E(0, \\omega)} d\\omega\n\\end{equation}\nwhere $E_{0}(x, t)$ is the initial THz pulse.\nWe then get:\n\\begin{equation}\nE(x,t) = E_{0}(x,t) - \\frac{xN}{2c\\epsilon_{0}V}\\frac{d}{dt}d(t-\\frac{x}{c}) ,\n\\end{equation}\nwhere we have used the fact that the time derivative of a Fourier transform ($FT$) of a function $f$ is given by: $FT[\\frac{d}{dt}f(t)] = -i\\omega FT[f(t)]$. The THz electric field can finally be written as follows\n\\begin{equation}\nE(x,t) = E_{0}(x,t) - \\alpha(x) \\frac{d}{dt} \\langle \\cos\\theta\\rangle(t-\\frac{x}{c})\n\\label{eq:A10}\n\\end{equation}\nwhere $\\alpha(x)$ is a positive scalar factor depending upon the propagation coordinate $x$. The second term of the right hand side of Eq. (\\ref{eq:A10}) is the FID electric field emitted by the molecules of the sample. This contribution is used in the detection process as described above.\n\nFrom the experimental point of view, we point out that the FID manifests itself as recurrent THz echos launched by the molecular sample after its interaction with the THz pulse. It originates from transient orientation revivals of molecules inducing, under field-free conditions, a non-zero dipole. Another way of interpreting this phenomenon is to consider that the THz pulse experiences spectral shaping during its propagation. Absorption related to $J\\to J+1$  transitions produces periodic holes in the spectrum with a spectral separation in angular frequency $\\Delta \\omega= 4 \\pi c B_e$, with $B_e$ the rotational constant (neglecting the centrifugal distortion).  A periodic modulation in the frequency domain leads to periodic replica in the time domain every $\\Delta t=2\\pi\\Delta \\omega$. We emphasize that the first term of Eq.~(\\ref{eq:A10}) does not reflect the absorption experienced by the incident pulse because of the use of first order expansion.\n\n\\section{Model system \\label{sec2}}\nWe describe in this section the model used in the numerical computations to study the control of molecular orientation of the symmetric top molecule of methyl-iodide CH$_{3}$I. We consider the rotational dynamics of this molecular system, which is assumed to be in its ground vibronic state, in interaction with a linear polarized THz field. The Hamiltonian of the system can be written as:\n\\begin{equation}\n\tH(t) = H_{0} + H_{\\textrm{int}}(t),\n\t\\label{eq:01}\n\\end{equation}\nwhere $H_0$ and $H_{\\textrm{int}}$ describe respectively the field-free Hamiltonian and the interaction with the laser field.\nThe Hamiltonian of the molecular system is given by \\cite{zare}:\n\\begin{equation}\n\tH_{0} = B_{e}J^2 + (A_{e} - B_{e})J_{Z}^{2} - D_{J}J^{4} - D_{JK}J^2J_{Z}^{2} - D_{K}J_{Z}^{4}\n\t\\label{eq:02}\n\\end{equation}\nwhere $J$ is the angular momentum operator and $J_{Z}$ the component of $J$ along the body-fixed $Z$- axis defined by the C-I bond. The energy eigenvalues $E_{JK}$ of the operator $H_{0}$ in the Wigner basis $|JKM\\rangle$ for a prolate symmetric top can be expressed as follows:\n\\begin{eqnarray}\n\tE_{JK}\n\t&= B_{e}J(J+1) + (A_{e} - B_{e})K^{2} - D_{J}J^{2}(J+1)^{2}\\nonumber \\\\\n\t& - D_{JK}J(J+1)K^{2} - D_{K}K^{4},\n\\label{eq:03}\n\\end{eqnarray}\nwhere $A_{e}$ and $B_{e}$ are the rotational constants. The states $|JKM\\rangle$ are the eigenstates of the square of the angular momentum operator $J^2$ and of its projections, $J_Z$ and $J_z$ on the body-fixed $Z$- axis and space-fixed $z$- axis, respectively \\cite{zare}. The molecular parameters of the CH$_{3}$I molecule used in the numerical computations are given in Tab.~\\ref{tab:01} \\cite{Carocci1998}.\n\\begin{table}[!htp]\n\\centering\n\\begin{tabular}{p{3cm} p{3cm}}\n\t\\hline\n\tParameters & Values in cm$^{-1}$ \\\\ \\hline\\hline\n\t$B_{e}$    & $0.25098$                 \\\\\n\t$A_{e}$    & $5.173949$                 \\\\\n\t$D_{J}$    & $2.1040012\\times 10^{-7}$   \\\\\n\t$D_{JK}$   & $3.2944780\\times 10^{-6}$   \\\\\n\t$D_{K}$    & $8.7632195\\times 10^{-5}$   \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the rotational and of the centrifugal constants of the CH$_{3}$I molecule used in the numerical computations.}\n\\label{tab:01}\n\\end{table}\nThe interaction between the molecular system and the external electromagnetic field reads:\n\\begin{equation}\n\tH_{\\textrm{int}}(t) = -\\mu_{0} E(t) \\cos\\theta,\n\t\\label{eq:04}\n\\end{equation}\nwhere the function $E(t)$ represents here the amplitude of the THz electric field. We neglect in this paper the effect of the polarizability components since the maximum intensity of the electric field remains moderate. The units used are atomic units unless otherwise specified.\n\nAt room temperature, the system is described by a density matrix $\\rho(t)$ whose dynamics is\ngoverned by the Liouville-von Neumann equation \\cite{Shapiro2012}:\n\\begin{equation}\n\ti\\frac{\\partial{\\rho(t)}}{\\partial{t}} = [H(t),\\rho(t)],\n\t\\label{eq:05}\n\\end{equation}\nwhere the initial condition $\\rho(0)$ is given by the canonical density operator at thermal equilibrium\n\\begin{equation}\n\t\\rho (0) = \\frac{1}{Z} \\sum_{J=0}^{\\infty}\\sum_{M,K=-J}^{J}  e^{-E_{JK}\/(k_{B}T)} |JKM\\rangle \\langle JKM|,\n\t\\label{eq:06}\n\\end{equation}\nwhere $Z = \\sum_{J=0}^{\\infty}\\sum_{M,K=-J}^{J} e^{-E_{JK}\/(k_{B}T)}$ is the partition function, with $T$ the temperature fixed to $T=298$~K and $k_{B}$ the Boltzmann constant.\n\nThe degree of orientation of the molecular system is given by the expectation value:\n\\begin{equation}\n\t\\langle\\cos\\theta\\rangle (t) = \\textrm{tr}[\\rho(t) \\cos\\theta].\n\t\\label{eq:07}\n\\end{equation}\nFurthermore, in order to simulate more realistic experimental conditions (see Sec.~\\ref{sec4}), we add to the model system the dissipative effects due to molecular collisions \\cite{Seideman2005,viellard1}. To limit the complexity of the numerical computations, we consider in this paper the effective approach proposed in \\cite{Seideman2005} to account for coherence relaxation described by the time $T_2\/P$, where $P$ is the pressure of the sample. We approximate the decoherence by an exponential decay such that the final degree of orientation is given by:\n\\begin{equation}\n\\langle\\cos\\theta\\rangle (t) = \\langle \\cos\\theta\\rangle _{\\textrm{ND}} e^{-\\frac{tP}{T_{2}}},\n\\end{equation}\nwhere the non-dissipative orientation, $\\langle \\cos\\theta\\rangle _{\\textrm{ND}}$, is computed from the Liouville-von Neumann equation (\\ref{eq:05}).\n\n\\section{Numerical and experimental results \\label{sec4}}\nIn this paragraph, we first investigate theoretically the degree of orientation that can be achieved with a THz laser pulse in the experimental conditions of the set-up. The Hilbert space is spanned by the Wigner's functions $|J,K,M\\rangle$, with $0\\leq J$, $-J\\leq K\\leq J$ and $-J\\leq M\\leq J$. $M$ and $K$ being good quantum numbers, the Hamiltonian of the system only depends on the angle $\\theta$. Numerically, we consider a finite dimensional Hilbert space with $J\\leq J_{\\textrm{max}}$, $J_{\\textrm{max}}=90$. From a physical point of view, this reduction can be justified by the fact that the THz excitation only transfers a finite amount of energy to the system, which thus stays in a finite dimensional subspace.\n\nThe control pulse can be approximated by a set of Hermite polynomials as follows:\n\\begin{equation}\\label{eqherm}\nE_0(t)\n= \\frac{E_1}{2}  e^{ -\\frac{t^2}{2\\sigma^2}} [ -3 D_{3} \\hat{H}_{2}(\\frac{t}{\\sigma}) + D_{1} \\hat{H}_{0}(\\frac{t}{\\sigma})]\n\\end{equation}\nwhere $D_{n}=(2^{n}n!\\pi^{1\/2})^{-\\frac{1}{2}}$, and $\\hat{H}_{n}(t)$ stands for the Hermite polynomials of order $n$, $\\hat{H}_2(t)=4 t^2-2$ and $\\hat{H}_0(t)=1$.\nThe parameter $\\sigma$ is given by the relation $16\\log(2)\\sigma^2 = \\tau^2$ with $\\tau = 1$~ps. The reasonable match between the theoretical THz pulse and the experimental one shown in Fig.~\\ref{fig2} justifies the choice made for the electric field in Eq.~(\\ref{eqherm}). Note that the peak-to-peak amplitude is experimentally of 9.4~kV~cm$^{-1}$ in this case.\n\\begin{figure} [ht]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{fig3}\n\t\t\\caption{(Color online) Experimental (red or dark gray) and theoretical (black), given by Eq.~(\\ref{eqherm}), THz pulses at delay zero. The amplitude $E_1$ has been adjusted to get the best match between the two pulses.}\n\t\t\\label{fig2}\n\\end{figure}\n\nWe start the analysis of the dynamics by giving a global picture of the time evolution of the molecular orientation as displayed in Fig.~\\ref{fig3}, where the relaxation effects are taken into account with $T_{2}= 23$~ps~atm \\cite{Hennequin1987,Roberts1968}. The pressure of the cell is $P=0.35~\\textrm{bar}$. Here, we fix the amplitude $E_1$ to 100 kV~cm$^{-1}$, which is the maximum experimental available amplitude. Revivals are observed at times multiples of the rotational period. The maximum of orientation is respectively of the order of $5\\times 10^{-4}$ and $2\\times 10^{-4}$ for the first and second revivals. Note that this maximum is larger than $10^{-3}$ in the non-dissipative case.\n\\begin{figure} [ht]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{fig4}\n\t\t\\caption{(Color online) Numerical orientation dynamics of the CH$_{3}$I molecule after an excitation with a $1~\\textrm{ps}$ pulse centered at $t=0~\\textrm{ps}$. The small insert displays the numerical evolution of the maximum degree of orientation at delay zero (blue, circle) and for the first (red, rectangle) and second (yellow, diamond) revivals as a function of the amplitude $E_1$ of the THz field.}\\label{fig3}\n\\end{figure}\nIn Fig.~\\ref{fig3}, we also study how the amplitude of the initial THz field affects the behavior of the orientation dynamics. Due to the low intensity of the field, we observe that the maximum values of each transient evolve linearly with respect to the amplitude. The effect of the pulse duration on the degree of orientation is displayed in Fig.~\\ref{fig4new}. More precisely, we consider the role of the parameter $\\tau$ as defined in Eq.~(\\ref{eqherm}). Note that the overall structure of the field is not modified by this parameter, the field is only compressed or extended in time. This modification corresponds to the currently available shaping techniques of THz pulses. In Fig.~\\ref{fig4new}, we observe a nonlinear behavior of the degree of orientation, which is maximum for the two revivals for a value of $\\tau$ of the order of 2 ps.\n\\begin{figure} [ht]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{fig5}\n\t\t\\caption{(Color online) Maximum of $|\\langle\\cos\\theta\\rangle |$ for the delay zero (blue-circle) and the first (orange-square) and the second (yellow-diamond) revivals as a function of the parameter $\\tau$ of the pulse given by Eq.~(\\ref{eqherm}). In the experiment, the parameter $\\tau$ is of the order of 1 ps.}\\label{fig4new}\n\\end{figure}\n\nAs shown in Sec.~\\ref{sec2p}, the detection process is sensitive to the time derivative of the degree of orientation, i.e. $\\frac{d[\\langle\\cos\\theta\\rangle]}{dt}$ and not directly to the orientation. In Fig.~\\ref{var_1tr}, we study the effect of the amplitude of the field on the first revival. We observe that the shape of the transient does not change when the amplitude is varied. Similar results are obtained for the second revival.\n\n\n\\begin{figure} [ht]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{fig6}\n\t\t\\caption{(Color online) Numerical evolution of the time derivative of $\\langle \\cos\\theta\\rangle$ of the first revival as a function of the peak amplitude of the electric field which is set to 20, 40, 60, 80 and 100 kV~cm$^{-1}$. The $\\alpha$ parameter is set to 1~kV~cm$^{-1}$~ps.}\n\t\t\\label{var_1tr}\n\\end{figure}\nAfter this complete theoretical description of the orientation dynamics, the goal is now to make a full comparison of the experimental and theoretical results. A scaling factor and a shift parameter along the vertical axis are determined to get the best match between the two sets of data. As shown in Fig.~\\ref{fig:Figs}, the numerical simulation reproduces quite well the experimental signal for the two first transients. The signal is due to a large number of molecules within the volume of the sample, each molecule being excited by a field of different amplitude. The fact that the shape of the revivals does not depend on the intensity of the electric field explains the good agreement observed in Fig.~\\ref{fig:Figs}. The signal is too weak after the second revival to pursue this comparison.\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.6]{fig7}\n\\caption{(Color online) Time derivative of the degree of molecular orientation of the (a) first and (b) second revivals. The red lines (light gray) represent the experimental data while the black ones correspond to the numerical results. A filtered experimental signal is plotted in blue (dark gray) to ease the comparison with the simulated dynamics. The parameter $\\alpha$ is set to 5.75~kV~cm$^{-1}$~ps.}\n\\label{fig:Figs}\n\\end{figure}\n\n\\section{Conclusion \\label{sec5}}\nIn this article we have investigated the orientation of a symmetric top molecule, namely CH$_{3}$I. We have shown the efficiency of the full-optical ultrafast set-up resolved in time to generate THz pulses and to produce molecular orientation. We provide a detailed description of the experimental set-up used and a complete numerical study of the corresponding dynamics. The analysis of the detection process shows that the degree of orientation is indirectly measured via the time derivative of the expectation value of $\\cos\\theta$.  The theoretical model reproduces accurately the experimental results up to the second orientation revival. Additional numerical simulations reveal that the orientation dynamics induced by this THz pulse is qualitatively similar for the linear molecule, OCS. It will be interesting to consider also asymmetric top molecules which have a more complex and non-periodic field-free evolution than linear or symmetric top molecules.\n\nThe results of this work can be viewed as an important step forward for the control of molecular orientation. The good match between theory and experiment will allow us to explore the efficiency of more complex strategies using for instance a pre-alignment by a laser field. Such approaches are necessary to increase the degree of orientation and reach efficiencies where molecular orientation could be useful in practice.\\\\ \\\\\n\n\n\n\\noindent\\textbf{ACKNOWLEDGMENT}\\\\\nD. Sugny acknowledges the support from the ANR-DFG research programs Explosys (ANR-14-CE35-0013-01) and Coqs (ANR-15-CE30-0023-01). This work was supported by the Conseil R\\'egional de Bourgogne under the Photcom Pari program as well as the Labex ACTION program (ANR-11-LABX-01-01) and the CoConiCs program (Contract No. ANR-13-BS08-0013). This work has been done with the support of the Technical University of Munich, Institute for Advanced\nStudy, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement 291763.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is generally agreed that two-dimensional field-theory models may provide an excellent and rich framework to test ideas in gauge theories. In fact, the interest in studying these models is basically connected to the possibility of obtaining exact solutions, which are believed to be shared by their more realistic counterparts in four dimensions.\nOf these, the Schwinger model, also known as Quantum Electrodynamics in $(1+1)$-space-time dimensions, or ${QED}_2$\\cite{Schwinger:1962tn,Coleman:1975pw} has probably enjoyed the greatest popularity due to some special features that it possesses. For example, the energy spectrum contains a massive mode in spite of the gauge invariance of the original Lagrangian, the charge is screened and confinement is enforced by the explicit occurrence of a rising Coulomb potential. To our mind, these special features represent the essential ingredients of a mechanism by which one hopes to understand the phenomenon of quark-binding into physical hadrons. These issues were first analyzed in ${QED}_2$ in Refs. \\cite{Schwinger:1962tn,Schwinger:1962tp, Casher:1973uf,Coleman:1975pw}.\n\nUnfortunately, against this suggestive two-dimensional perspective, it seems to us that a convincing analytical proof of color confinement in quantum chromodynamics (QCD) still eludes us. The root of the problem is well known: while asymptotic freedom is a well established property of the perturbative dynamics of QCD, the transition to infrared slavery is problematic because of non-perturbative effects that dominate in the large distance limit of the theory. Once this ?large distance limit? is defined in terms of some phenomenological scale of distance, the immediate problem is that of identifying the dynamical variables that operate in that limit. A hint about the nature of those hidden dynamical variables comes from the phenomenological bag models of hadrons: the partial success of those models indicate that, in the large distance limit of QCD, the spatial extension of hadrons and the bag degrees of freedom must somehow be included among those new dynamical variables. It is clear that, in order to speak meaningfully of a ?QCD-solution? of the confinement problem, one would expect that such variables should arise from the very dynamics of QCD and control the mechanisms of color confinement \\cite{Luscher:1978rn}. \nThis is where the extrapolation of results from two to four spacetime dimensions may play a significant role in the understanding of the confinement mechanism in QCD. For instance, the correspondence between the colorless topological sector of QCD and the zero-charge sector of ${QED}_2$ was noted long ago in Ref.\\cite{Aurilia:1979dw} but never fully exploited;  The extrapolation from two to four dimensions, at least for the bosonized version of the Schwinger model, was considered in \\cite{  Aurilia:1980jz} while a general \"gauge mixing mechanism for the generation of mass\" was proposed in \\cite{Aurilia:1981xg}. \n\nMotivated by these observations, the general purpose of the present discussion is to communicate a deeper understanding  of the physical content of the $(3+1)$-dimensional generalization of the Schwinger model. The many avenues of research that are open to us were outlined in a research proposal by the authors \\cite{Aurilia:2015qia}. However, it seems clear that the first line of inquiry is to explore in more detail the role of the Abelian $3$-form field among the physical observables of the model. It has long been known that this $3$-form field does not support any propagating degree of freedom, its sole physical effect consisting of a static interaction between two probe charges.This remarkable property is entirely analogous to the two dimensional case where in ${QED}_2$ there are no \"photons\" associated with the electromagnetic field \\cite{Aurilia:1979dw}. Then, if the Schwinger model has any relevance in the issue of confinement in four dimensions, then the static potential induced by the Abelian $3$-form field must also exhibit the same behavior found in the two-dimensional case. We find that this reasonable expectation is fully supported by the explicit calculation of the interaction energy between two external test charges.\n\nA second objective of this work is to elucidate the remarkable interplay between guge invariance and the appearance of mass in the physical spectrum of the Schwinger model. With hindsight, the emergence of this massive mode can be traced back directly to the dimensionality of the coupling constant in ${QED}_2$ which sets a mass scale in the model. Evidently this is not the case in ${QED}_4$ but a similar phenomenon takes place, at least in the bosonized version of the Schwinger model in $(3+1)$  dimensions. We illustrate how this same generalization of the S-model basically amounts to a Stueckelberg-like formulation of a massive gauge theory characterized by the mixing between a $U(1)$ potential and an Abelian $3$-form field.\n\nOur work is organized according to the following outline: in Section II, we recall the salient features of dualization in terms of two simple Lagrangian systems and show their equivalence to different representations of a massive Proca field. In Section III,  using a path-integral approach, we compute the interaction energy, and hence the analytic form of the static potential in the bosonized version of the Scwinger model in four spacetime dimensions. Finally, some Concluding Remarks are cast in Sec. IV. \n\nThroughout the following discussion, the signature of the metric is ($+1,-1,-1,-1$).\n\n\\section{Dualization, gauge invariance and mass generation}\n\nLet us start our considerations by recalling that the study of duality symmetry in gauge theories has been of considerable importance in \norder to provide an equivalent description of physical phenomena by distinct theories. As well-known, duality refers to a physical equivalence \nbetween two field theories which formulated in terms of different dynamical variables \\cite{Hjelmeland:1997eg}. \n\nIn order to put our discussion into context, we also recall that the dualization of Stueckelberg-like massive gauge theories \nand $ B \\wedge F$ models follows from a general $p$ dualization of interacting theories in $d$ spacetime \ndimensions \\cite{Ansoldi:1999wi,Smailagic:1999qw,Ansoldi:2000qs,Smailagic:2000hr,Smailagic:2001ch}. \nIn particular, in the case of $(3+1)$ dimensions, the following $ B \\wedge F$ models are found:\n\\begin{equation}\n{\\cal L}^{(1)} =  - \\frac{1}{4}F_{\\mu \\nu }^2\\left( A \\right) + \\frac{1}{{12}}H_{\\mu \\nu \\rho }^2\\left( B \\right) \n+ \\frac{m}{{24}}{\\varepsilon ^{\\mu \\nu \\rho \\sigma }}{B_{\\mu \\nu }}{\\partial _{[\\rho }}{A_{\\sigma ]}},  \\label{Dual05}\n\\end{equation}\n\n\\begin{equation}\n{\\cal L}^{(2)} =  - \\frac{1}{4}H_{\\mu \\nu }^2\\left( B \\right) + \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2\\left( A \\right) \n+ \\frac{m}{{24}}{\\varepsilon ^{\\mu \\nu \\rho \\sigma }}{B_\\mu }{\\partial _{[\\nu }}{A_{\\rho \\sigma ]}}, \\label{Dual10}\n\\end{equation}\n\n\\begin{equation}\n{\\cal L}^{(3)} = \\frac{1}{2}{\\left( {{\\partial _\\mu }\\varphi } \\right)^2} - \\frac{1}{{48}}F_{\\mu \\nu \\rho \\sigma }^2\\left( A \\right) \n+ \\frac{m}{{24}}{\\varepsilon ^{\\mu \\nu \\rho \\sigma }}\\varphi {\\partial _{[\\mu }}{A_{\\nu \\rho \\sigma ]}}. \\label{Dual15}\n\\end{equation}\n\nAt this point, it is instructive to make a brief re-examination of equations (\\ref{Dual05}) and (\\ref{Dual10}). For this purpose, we observe \nthat the Lagrangian density (\\ref{Dual05}) may be rewritten as\n\\begin{equation}\n{{\\cal L}^{\\left( 1 \\right)}} =  - \\frac{1}{4}F_{\\mu \\nu }^2 - \\frac{1}{2}{\\tilde H_\\sigma }{\\tilde H^\\sigma } \n- \\frac{m}{6}{\\tilde H^\\sigma }{A_\\sigma }, \\label{Dual25}\n\\end{equation}\nwhere we have made use of ${\\tilde H^\\mu } = \n{\\raise0.5ex\\hbox{$\\scriptstyle 1$}\\kern-0.1em\/\\kern-0.15em\\lower0.25ex\\hbox{$\\scriptstyle 2$}}{\\varepsilon ^{\\mu \\nu \\lambda \\rho }}{\\partial _\\nu }{B_{\\lambda \\rho }}$.\n\nNext, in order to eliminate the dual-field $H^{\\sigma}$ care must be taken, for it satisfies the constraint \n${\\partial _\\mu }{\\tilde H^\\mu } = 0$ (Bianchi identity). Thus, to take into account the constraint, we shall introduce a Lagrange \nmultiplier $\\chi$. In such a case, the corresponding effective Lagrangian density (\\ref{Dual25}) reads \n\\begin{equation}\n{{\\cal L}^{\\left( 1 \\right)}} = - \\frac{1}{4}F_{\\mu \\nu }^2 - \\frac{1}{2}{\\tilde H_\\sigma }{\\tilde H^\\sigma }\n- \\frac{m}{6}{\\tilde H^\\sigma }{A_\\sigma } + \\chi {\\partial _\\sigma }{\\tilde H^\\sigma }   .\\label{Dual30}\n\\end{equation}\nBy defining ${Z_\\sigma } \\equiv {A_\\sigma } + \\frac{6}{m}{\\partial _\\sigma }\\chi$, with $  {Z_{\\mu \\nu }} \n= {F_{\\mu \\nu }}$, we readily verify that\n\\begin{equation}\n{{\\cal L}^{\\left( 1 \\right)}} =  - \\frac{1}{4}Z_{\\mu \\nu }^2 - \\frac{1}{2}{\\tilde H_\\sigma }{\\tilde H^\\sigma } \n- \\frac{m}{6}{\\tilde H^\\sigma }{Z_\\sigma }.\\label{Dual35} \n\\end{equation}\nBy a further definition of the fields, ${W_\\sigma } \\equiv {\\tilde H_\\sigma } + \\frac{m}{6}{Z_\\sigma }$, we find that the Lagrangian \ndensity (\\ref{Dual05}) can be brought to the form\n\\begin{equation}\n{{\\cal L}^{\\left( 1 \\right)}} =  - \\frac{1}{4}Z_{\\mu \\nu }^2 + \\frac{1}{2}{\\mu ^2}Z_\\mu ^2, \\label{Dual40}\n\\end{equation}\nwith ${\\mu ^2} \\equiv {\\raise0.5ex\\hbox{$\\scriptstyle {{m^2}}$}\n\t\\kern-0.1em\/\\kern-0.15em\n\t\\lower0.25ex\\hbox{$\\scriptstyle {36}$}}$. We immediately see that the Lagrangian density (\\ref{Dual40}) exhibits a Proca-type mass term.\n\nWe now turn our attention to the Lagrangian density (\\ref{Dual10}). It is convenient to rewrite this equation in the alternative form \n\\begin{equation}\n{{\\cal L}^{\\left( 2 \\right)}} = \\frac{1}{4}\\tilde H_{\\mu \\nu }^2 + \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2 \n+ \\frac{m}{{24}}{\\tilde H^{\\rho \\sigma }}{A_{\\rho \\sigma }}, \\label{Dual45}\n\\end{equation}\nwhere ${\\tilde H^{\\mu \\nu }} = {\\raise0.5ex\\hbox{$\\scriptstyle 1$}\n\t\\kern-0.1em\/\\kern-0.15em\n\t\\lower0.25ex\\hbox{$\\scriptstyle 2$}}{\\varepsilon ^{\\mu \\nu \\lambda \\rho }}{H_{\\lambda \\rho }}$.\n\nIt is worthy to notice that the $B^{\\mu}$- field appears only through ${\\tilde H^{\\mu\\nu}}$.\nAgain, in order to eliminate the dual-field $\\tilde H^{\\mu\\nu}$ care must be taken, for it satisfies the \nconstraint ${\\partial _\\mu }{\\tilde H^{\\mu\\nu} } = 0$. As before, we shall introduce a Lagrange multiplier $\\chi_{\\nu}$. \nIt gives rise to the following Lagrangian density,\n\\begin{equation}\n{{\\cal L}^{\\left( 2 \\right)}} = \\frac{1}{4}\\tilde H_{\\mu \\nu }^2 + \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2 \n+ \\frac{m}{{24}}{\\tilde H^{\\mu \\nu }}{A_{\\mu \\nu }} - \\frac{1}{2}{\\tilde H^{\\mu \\nu }}{\\chi _{\\mu \\nu }},\\label{Dual50}\n\\end{equation}\nwhere ${\\chi _{\\mu \\nu }} = {\\partial _\\mu }{\\chi _\\nu } - {\\partial _\\nu }{\\chi _\\mu }$.\nNow, letting ${Z_{\\mu \\nu }} = {A_{\\mu \\nu }} - \\frac{{12}}{m}{\\chi _{\\mu \\nu }}$, we obtain\n\\begin{equation}\n{{\\cal L}^{\\left( 2 \\right)}} = \\frac{1}{4}\\tilde H_{\\mu \\nu }^2 + \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2 \n+ \\frac{m}{{24}}{{\\tilde H}^{\\mu \\nu }}{Z_{\\mu \\nu }}. \\label{Dual55}\n\\end{equation}\nIt should be further noted that, by defining ${W_{\\mu \\nu }} = {{\\tilde H}_{\\mu \\nu }} \n+ \\frac{m}{{12}}{Z_{\\mu \\nu }}$, equation (\\ref{Dual55}) reduces to \n\\begin{equation}\n{{\\cal L}^{\\left( 2 \\right)}} = \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2 - \\frac{1}{2}{\\mu ^2}Z_{\\mu \\nu }^2, \\label{Dual60} \n\\end{equation}\nwhere we have written, ${\\mu ^2} = \\frac{{{m^2}}}{{288}}$, and \n$F_{\\mu \\nu \\rho }^2 = Z_{\\mu \\nu \\rho }^2$. Thus ${{\\cal L}^{\\left( 2 \\right)}}$ describes a massive field of spin $1$, exactly a \nProca equation, although ${Z_{\\mu \\nu }} \\in \\left[ {\\left( {1,0} \\right) \\oplus \\left( {0,1} \\right)} \\right]$. Actually, a massive\nrank-two skew-symmetric tensor field is, on-shell, equivalent to a Proca field.\n\nIn short, equations (\\ref{Dual05}) and (\\ref{Dual10}) are equivalent; both of these equations describe a Proca field.\n\nConsidering, finally, equation (\\ref{Dual15}), we find that this model reduces to a  massless Schwinger model in $(3+1)$ dimensions, \nas we shall indicate it below.\n\n\\section{Interaction energy}\n\nInspired by the preceding observation, we shall now consider the $(3+1)$-dimensional generalization of the Schwinger model, as originally \nintroduced in Ref.\\cite{Aurilia:1979dw}. As we have already noticed, we will work out the static potential for this $(3+1)$ generalization, \nvia a path-integral approach. To this end, we consider the bosonized form of the Schwinger model in D=$(3+1)$, that is,\n\\begin{equation}\n{\\cal L} = \\frac{1}{2}{\\left( {{\\partial _\\mu }\\phi } \\right)^2} + \\frac{1}{2}m_\\phi ^2{\\phi ^2} \n+ \\frac{g}{{6\\sqrt \\pi  }}{\\partial _\\mu }\\phi\\ {\\varepsilon ^{\\mu \\nu \\rho \\sigma }}{A_{\\nu \\rho \\sigma }} \n- \\frac{1}{{48}}F_{\\mu \\nu \\rho \\sigma }^2, \\label{Scwhin3-05}\n\\end{equation}\nwhere $g$ is a coupling constant and $m_\\phi$ refers to the mass of the scalar field $\\phi$.\n\nWe readily verify that when, ${m_\\phi } \\to 0$,\nequation (\\ref{Scwhin3-05}) reduces to equation (\\ref{Dual15}).\n\nAccording to usual procedure, integrating out the $\\phi$ field induces an effective theory for the $A_{\\nu \\rho \\sigma }$ field.  \nIt is now important to recall that the $ A_{\\nu \\rho \\sigma }$ field  can also be written as ${A_{\\nu \\rho \\sigma }} \n= {\\varepsilon _{\\nu \\rho \\sigma \\lambda }}{\\partial ^\\lambda }\\xi$  \\cite{Aurilia:2004cb, Aurilia:2004fz}, \nwhere $\\xi$ refers to an another scalar field. This then leads to the following effective theory for the model under consideration:\n\\begin{equation}\n{\\cal L} = \\frac{1}{2}\\left[ {\\xi \\ \\Delta \\left( {1 + \\frac{{{\\raise0.7ex\\hbox{${{g^2}}$} \\!\\mathord{\\left\/\n\t\t\t\t\t\t{\\vphantom {{{g^2}} \\pi }}\\right.\\kern-\\nulldelimiterspace}\n\t\t\t\t\t\\!\\lower0.7ex\\hbox{$\\pi $}}}}{{\\left( {\\Delta  - m_\\phi ^2} \\right)}}} \\right)\\Delta \\ \\xi } \\right], \\label{Scwhin3-10}\n\\end{equation}\nwhere $\\Delta  = {\\partial _\\mu }{\\partial ^\\mu }$.\n\nWe are now ready to compute the interaction energy between static pointlike sources. We start off our analysis by writing down the functional \ngenerator of the Green's functions, that is,\n\\begin{equation}\nZ\\left[ J \\right] = \\exp \\left( { - \\frac{i}{2}\\int {{d^4}x{d^4}yJ(x)D(x,y)J(y)} } \\right), \\label{Scwhin3-15}\n\\end{equation}\nwhere, $D(x,y) = \\int {\\frac{{{d^4}k}}{{{{\\left( {2\\pi } \\right)}^4}}}D(k){e^{ - ikx}}}$, is the propagator. In this case, the corresponding \npropagator is given by\n\\begin{equation}\nD(k) = \\left( {1 - \\frac{{m_\\phi ^2}}{{{{\\cal M}^2}}}} \\right)\\frac{1}{{{k^2}\\left( {{k^2} + {{\\cal M}^2}} \\right)}} \n+ \\frac{{m_\\phi ^2}}{{{{\\cal M}^2}}}\\frac{1}{{{k^4}}}, \\label{Scwhin3-20}\n\\end{equation}\nwhere ${{\\cal M}^2} = m_\\phi ^2 - {\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\\kern-0.1em\/\\kern-0.15em\n\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}}$.\n\nBy means of expression $Z = {e^{iW\\left[ J \\right]}}$ and employing Eq. (\\ref{Scwhin3-15}), ${W\\left[ J \\right]}$ takes the form \n\\begin{eqnarray}\nW\\left[ J \\right] &=&  - \\frac{1}{2}\\int {\\frac{{{d^4}k}}{{{{\\left( {2\\pi } \\right)}^4}}}} {J^ * }\\left( k \\right)\n\\frac{{\\left( {1 - \\frac{{m_\\phi ^2}}{{{{\\cal M}^2}}}} \\right)}}{{{k^2}\\left( {{k^2} + {{\\cal M}^2}} \\right)}}\nJ\\left( k \\right) \\nonumber\\\\\n&-& \\frac{1}{2}\\int {\\frac{{{d^4}k}}{{{{\\left( {2\\pi } \\right)}^4}}}} {J^ * }\\left( k \\right)\\frac{{m_\\phi ^2}}{{{{\\cal M}^2}}}\n\\frac{1}{{{k^4}}}J\\left( k \\right).\n\\label{Scwhin3-25}\n\\end{eqnarray}\n\nNext, for $J({\\bf x}) = \\left[ {Q{\\delta ^{\\left( 3 \\right)}}\\left( {{\\bf x} - {{\\bf x}^{\\left( 1 \\right)}}} \\right) \n\t+ {Q^ \\prime }{\\delta ^{\\left( 3 \\right)}}\\left( {{\\bf x} - {{\\bf x}^{\\left( 2 \\right)}}} \\right)} \\right]$, we obtain that the interaction \nenergy of the system is given by\n\\begin{eqnarray}\nV &=&  - Q{Q^ * }\\int {\\frac{{{d^3}k}}{{{{\\left( {2\\pi } \\right)}^3}}}} \n\\frac{{\\left( {\\frac{{{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}}}}{{{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2}}} \\right)}}{{\\left( {{{\\bf k}^2} + {\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2} \\right)}}\n{e^{i{\\bf k} \\cdot {\\bf r}}} \\nonumber\\\\\n&+& Q{Q^ * }\\int {\\frac{{{d^3}k}}{{{{\\left( {2\\pi } \\right)}^3}}}} \\left( {\\frac{{m_\\phi ^2}}{{{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2}}} \\right)\\frac{1}{{{{\\bf k}^4}}}{e^{i{\\bf k} \\cdot {\\bf r}}}, \\label{Scwhin3-30}\n\\end{eqnarray}\nwhere ${\\bf r} = {{\\bf x}^{\\left( 1 \\right)}} - {{\\bf x}^{\\left( 2 \\right)}}$.\n\nThis, together with ${Q^ \\prime }=-Q$, yields finally\n\\begin{eqnarray}\nV &=& \\frac{{{Q^2}}}{{4\\pi }}\\frac{{{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}}}}{{{{\\left( {{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2} \\right)}^2}}}\\frac{1}{L}\\left( {1 - {e^{ - \\sqrt {{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2} L}}} \\right)  \\nonumber\\\\\n&+& \\frac{{{Q^2}}}{{4\\pi }}\\frac{{m_\\phi ^2}}{{2\\left( {{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2} \\right)}}L, \\label{Scwhin3-35}\n\\end{eqnarray}\nwhere $L = |{\\bf r}|$. One immediately sees that the above static potential profile is analogous to that encountered in the two-dimensional \nSchwinger model. Incidentally, in order to put our discussion into context it is useful to summarize the relevant aspects of the \ntwo-dimensional Schwinger model. In such a case, we begin by recalling the bosonized form of the model under consideration \\cite{Gross:1995bp}:\n\\begin{eqnarray}\n{\\cal L} &=&  - \\frac{1}{4}F_{\\mu \\nu }^2 + \\frac{1}{2}{\\left( {{\\partial _\\mu }\\phi } \\right)^2} \n- \\frac{e}{{2\\sqrt \\pi  }}{\\varepsilon ^{\\mu \\nu }}{F_{\\mu \\nu }}\\phi \\nonumber\\\\\n&+&m\\sum \\left( {\\cos \\left( {2\\pi \\phi  + \\theta } \\right) - 1} \\right), \\label{Scwhin3-40}\n\\end{eqnarray}\nwhere $\\sum  = \\left( {\\frac{e}{{2{\\pi ^{{\\raise0.5ex\\hbox{$\\scriptstyle 3$}\n\t\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle 2$}}}}}}} \\right){e^{{\\gamma _E}}}$ with ${\\gamma _E}$ the Euler-Mascheroni constant and $\\theta$ refers to \nthe $\\theta$-vacuum. \n\nConsequently, by using the gauge-invariant but path-dependent variables formalism which provides a physically-based alternative to the Wilson \nloop approach \\cite{Gaete:1999iy, Gaete:2001wh}, the static potential reduces to\n\\begin{equation}\nV = \\frac{{{Q^2}}}{2}\\frac{{\\sqrt \\pi  }}{e}\\left( {1 - {e^{ - \\frac{e}{{\\sqrt \\pi  }}L}}} \\right), \\label{Scwhin3-45}\n\\end{equation}\nfor the massless case. On the other hand, for the massive case ($\\theta=0$), the static potential then becomes\n\\begin{equation}\nV = \\frac{{{Q^2}}}{{2\\lambda }}\\left( {1 + \\frac{{4\\pi m\\sum }}{{{\\lambda ^2}}}} \\right)\\left( {1 - {e^{ - \\lambda L}}} \\right) \n+ \\frac{{{q^2}}}{2}\\left( {1 - \\frac{{{\\raise0.5ex\\hbox{$\\scriptstyle {{e^2}}$}\n\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}}}}{{{\\lambda ^2}}}} \\right)L, \\label{Scwhin3-50}\n\\end{equation}\nwhere ${\\lambda ^2} = \\frac{{{e^2}}}{\\pi } + 4\\pi m\\sum$. The above results clearly show that the $(3+1)$-D generalization of the Schwinger \nmodel is structurally identical to the $(1+1)$-D Schwinger model.\n\nIn this perspective it is worth recalling that there is an alternative way of obtaining the Lagrangian density  (\\ref{Scwhin3-10}), \nwhich provides a complementary view into the physics of confinement. In fact, we refer to a theory of antisymmetric tensor fields that \nresults from the condensation of topological defects as a consequence of the Julia-Toulouse mechanism. \nMore precisely, the Julia-Toulouse mechanism is a condensation process dual to the Higgs mechanism proposed in \\cite{Quevedo:1996uu}. \nThis mechanism describes phenomenologically the electromagnetic behavior of antisymmetric tensors in the presence of magnetic-branes \n(topological defects) that eventually condensate due to thermal and quantum fluctuations. Using this phenomenology we have discussed \nin \\cite {Gaete:2004dn,Gaete:2005am} the dynamics of the extended charges (p-branes) inside the new vacuum provided by the condensate. \nActually, in \\cite {Gaete:2004dn} we have considered the topological defects coupled both longitudinally and \ntransversally to two different tensor potentials, $A_p$ and $B_q$, such that $p+q+2=D$, where $D=d+1$ space-time dimensions.\n\nWe skip all the technical details and refer to \\cite{Gaete:2004dn} for them. Thus, after the condensation, the Lagrangian density turns out to be\n\\begin{eqnarray}\n{\\cal L} &=& \\frac{{{{\\left( { - 1} \\right)}^q}}}{{2\\left( {q + 1} \\right)!}}{\\left[ {{H_{q + 1}}\\left( {{B_q}} \\right)} \\right]^2} \n+ e{B_q}{\\varepsilon ^{q,\\alpha ,p + 1}}{\\partial _\\alpha }{\\Lambda _{p + 1}} \\nonumber\\\\\n&+& \\frac{{{{\\left( { - 1} \\right)}^{p + 1}}}}{{2\\left( {p + 2} \\right)!}}{\\left[ {{F_{p + 2}}\\left( {{\\Lambda _{p + 1}}} \\right)} \\right]^2} \n\\nonumber\\\\\n&+& \\frac{{{{\\left( { - 1} \\right)}^{p + 1}}\\left( {p + 1} \\right)!}}{2}{m^2}\\Lambda _{p + 1}^2,  \\label{Scwhin3-55}\n\\end{eqnarray}\nshowing a $B$$\\wedge$$F$ type of coupling between the $B_q$ potential with the tensor $\\Lambda_{p+1}$ carrying the degrees of freedom of the \ncondensate. Following our earlier procedure \\cite{Gaete:2004dn}, the effective theory that results from integrating out the fields representing \nthe vacuum condensate, is given by\n\\begin{equation}\n{\\cal L} = \\frac{{{{\\left( { - 1} \\right)}^{q + 1}}}}{{2\\left( {q + 1} \\right)!}}{H_{q + 1}}\\left( {{B_q}} \\right)\n\\left( {1 + \\frac{{{e^2}}}{{\\Delta  - {m^2}}}} \\right){H^{q + 1}}\\left( {{B_q}} \\right). \\nonumber\\\\\n\\label{Scwhin3-60}\n\\end{equation}  \nHence we see that this expression with $p=-1$ and $q=3$ becomes\n\\begin{equation}\n{\\cal L} = \\frac{1}{{2 \\times 4!}}{F_{\\mu \\nu \\rho \\lambda }}\\left( A \\right)\\left( {1 + \\frac{{{e^2}}}{{\\Delta  - {m^2}}}} \\right)\n{F^{\\mu \\nu \\rho \\lambda }}\\left( A \\right). \\label{Scwhin3-65}\n\\end{equation}\nIt is straightforward to verify that Eq. (\\ref{Scwhin3-65}) reduces to Eq. (\\ref{Scwhin3-10}).\n\nIn this way, we establish a new connection among different effective theories. It must be clear from this discussion that the above connections \nare of interest from the point of view of providing unifications among diverse models as well as exploiting their equivalence in explicit \ncalculations. \n\n\n\\section{Concluding Remarks}\n\nFinally, the point we wish to emphasize is that there are two generic features that are common in the four-dimensional case and their \nupper\/lower extensions, as we shall show below. First, the existence of a linear potential, leading to the confinement of static charges. \nThe second point is related to the correspondence among diverse effective theories. To see this, it should be noted that by using the \nmethodology illustrated in \\cite{  Smailagic:1999qw}, we have that one of the $ B \\wedge F$ models in $(4+1)$ dimensions is given by the \nmixing between a $U(1)$ potential and an Abelian $3$-form field by means of a topological mass term, that is,\n\\begin{eqnarray}\n{\\cal L}^{\\left( {4 + 1} \\right)} &=&  - \\frac{1}{4}{F_{\\mu \\nu }}\\left( A \\right){F^{\\mu \\nu }}\\left( A \\right) \n+ \\alpha {H_{\\mu \\nu \\kappa \\lambda }}\\left( C \\right){H^{\\mu \\nu \\kappa \\lambda }}\\left( C \\right) \\nonumber\\\\\n&+& \\beta {\\varepsilon ^{\\mu \\nu \\kappa \\lambda \\rho }}{A_\\mu }{\\partial _\\nu }{C_{\\kappa \\lambda \\rho }}, \\label{CR-05}\n\\end{eqnarray} \nwith $\\alpha  =  - \\frac{1}{{48}}$ and $ \\beta  = \\frac{\\sigma }{6}$, where the parameter $\\beta$ has mass dimension. This model was considered \nin \\cite{Cocuroci:2013bga}, and the main motivation to consider this model is based on the possible connection with dark energy.\n\nHowever, we shall start from the five-dimensional spacetime model\n\\begin{eqnarray}\n{\\cal L}^{\\left( {4 + 1} \\right)} &=&  - \\frac{1}{4}{F_{\\hat \\mu \\hat \\nu }}{F^{\\hat \\mu \\hat \\nu }} \n+ \\alpha {H_{\\hat \\mu \\hat \\nu \\hat \\kappa \\hat \\lambda }}{H^{\\hat \\mu \\hat \\nu \\hat \\kappa \\hat \\lambda }} \\nonumber\\\\\n&+& \\beta {\\varepsilon ^{\\hat \\mu \\hat \\nu \\hat \\kappa \\hat \\lambda \\hat \\rho }}{A_\\mu }{\\partial _\\nu }{C_{\\hat \\kappa \\hat \\lambda \\hat \\rho }} + \\frac{1}{{12}}m_C^2{C_{\\hat \\mu \\hat \\nu \\hat \\rho }}{C^{\\hat \\mu \\hat \\nu \\hat \\rho }}, \\nonumber\\\\\n\\label{CR-10}\n\\end{eqnarray}\nwith the additional presence of a mass term $m_C$ for the Abelian $3$-form field; this explicit mass term makes a difference: if it were not introduced, the model\ncould be reduced to nothing but a Proca-type model in $(4+1)$ dimensions. Next, we perform its dimensional reduction along the \nlines of \\cite{Cocuroci:2013bga,Gaete:2012yu}: \n${A^{\\hat \\mu }} \\to \\left( {{A^{\\bar \\mu }},{A^4}} \\right)$, ${A^4} = \\phi$, ${\\partial _4}\\left( {everything} \\right) = 0$, \n${C^{\\hat \\mu \\hat \\nu \\hat \\kappa }} = \\left( {{C^{\\bar \\mu \\bar \\nu \\bar \\kappa }},{C^{\\bar \\mu \\bar \\nu 4}}} \\right)$ \nand ${C^{\\bar \\mu \\bar \\nu 4}} = {B^{\\bar \\mu \\bar \\nu }}$. Carrying out this prescription in equation (\\ref{CR-10}), we then obtain\n\\begin{eqnarray} \n{{\\cal L}^{\\left( {3 + 1} \\right)}} &=&  - \\frac{1}{4}{F_{\\bar \\mu \\bar \\nu }}{F^{\\mu \\nu }} \n+ \\frac{1}{2}{\\left( {{\\partial _{\\bar \\mu} }\\phi } \\right)^2} + \\alpha {H_{\\bar \\mu \\bar \\nu \\bar \\kappa \\bar \\lambda }}\n{H^{\\bar \\mu \\bar \\nu \\bar \\kappa \\bar \\lambda }} \\nonumber\\\\\n&-& 4\\alpha {G_{\\bar \\mu \\bar \\nu \\bar \\kappa }}{G^{\\bar \\mu \\bar \\nu \\bar \\kappa }} - 3\\beta {\\varepsilon ^{4 \\bar \\mu \\bar \\nu \\bar \\kappa \n\t\t\\bar \\lambda }}{A_{\\bar \\mu} }{\\partial _{\\bar \\nu} }{B_{\\bar \\kappa \\bar \\lambda }} \\nonumber\\\\\n&-& \\beta {\\varepsilon ^{4 \\bar \\nu \\bar \\kappa \\bar \\lambda \\bar \\rho }}\\phi {\\partial _{\\bar \\nu} }{C_{\\bar \\kappa \\bar \\lambda \\bar \\rho }} \n+ \\frac{{m_C^2}}{{12}}{C_{\\bar \\mu \\bar \\nu \\bar \\rho }}{C^{\\bar \\mu \\bar \\nu \\bar \\rho }} \\nonumber\\\\\n&-& \\frac{{m_C^2}}{4}{B_{\\bar \\mu \\bar \\nu }}{B^{\\bar \\mu \\bar \\nu }}, \\label{CR-15}\n\\end{eqnarray}\nwhere $\\bar \\mu ,\\bar \\nu ,\\bar \\kappa ,\\bar \\lambda ,\\bar \\rho  = 0,1,2,3$. Making use of an additional dimensional reduction, that is, \n$ {A^{\\bar \\mu }} \\to \\left( {{A^\\mu },{A^3}} \\right)$, ${\\partial _3}\\left( {everything} \\right) = 0$, ${B^{\\bar \\mu \\bar \\nu }} \n= \\left( {{B^{\\mu \\nu }},{C^{\\mu }}} \\right)$\n\\begin{eqnarray}\n{{\\cal L}^{\\left( {2 + 1} \\right)}} &=&  - \\frac{1}{4}{F_{\\mu \\nu }}{F^{\\mu \\nu }} + 12\\alpha {G_{\\mu \\nu }}{G^{\\mu \\nu }} \\nonumber\\\\\n&-& 6\\beta {\\varepsilon ^{\\mu \\nu \\kappa }}{A_\\mu }{\\partial _\\nu }{C_\\kappa } \n+ \\frac{{m_C^2}}{2}{C_\\mu }{C^\\mu}, \\label{CR-20}\n\\end{eqnarray}\nwhere ${G_{\\mu \\nu }} = {\\partial _\\mu }{C_\\nu } - {\\partial _\\nu }{C_\\mu }$. Next, after performing the integration over $C_{\\mu}$, \nthe induced effective Lagrangian density is given by\n\\begin{equation}\n{{\\cal L}^{\\left( {2 + 1} \\right)}} =  - \\frac{1}{4}{F_{\\mu \\nu }}\\left( {1 + \\frac{\\sigma }{{\\left( {\\Delta  + m_C^2} \\right)}}} \\right)\n{F^{\\mu \\nu }}. \\label{CR-25}\n\\end{equation}\nAgain, by applying the gauge-invariant formalism, the corresponding static potential for two opposite charges located at ${\\bf y}$ and \n${\\bf y}\\prime$ turns out to be\n\\begin{equation}\nV =  - \\frac{{{q^2}}}{{2\\pi }}{K_0}\\left( {ML} \\right) + \\frac{{{q^2}m_C^2}}{{4M}}L, \\label {CR-30}\n\\end{equation}\nwhere $L = |{\\bf y} - {{\\bf y}^ \\prime }|$ and ${M^2} = {\\sigma ^2} + m_C^2$. In summary, then: this potential displays the conventional \nscreening part, encoded in the Bessel function, and the linear confining potential. As expected, confinement disappears whenever ${m_C} \\to 0$\nand also in the case $m_{C}$ is non-trivial, but much smaller than the topological mass parameter, $\\sigma$.\n\n\nA final consideration we would like to raise concerns the presence of some sort \nof fundamental mechanism that endows one of the gauge potentials, the $p$- or the\n$(p+1)$-form, with a Proca-type mass term: if only the usual field-strength squared and\nthe topological mass terms are present, a field reshuffling is always possible to be done \nand one of the gauge potentials can be integrated over yielding, at the end, a Proca-like \n$p$-form or $(p+1)$-form massive model; exactly like we have worked out for the Lagrangians\n(\\ref{Dual05}) and (\\ref{Dual10}). However, if a more fundamental mechanism is at work\n(like the Higgs mechanism, for example) that gives an explicit (non-topological) mass term\nto one of the gauge fields, then the simple equivalence to a $p$-form\nProca field is no longer true and a confining contribution to the static interparticle potential\nshows. We would like to conclude our work by pointing out the relationship between the\ngeneration of a non-topological mass and the confinig profile of the interparticle potential.\n\n\\section{Acknowledgments}\n\nOne of us (P. G.) wishes to thank the Abdus Salam ICTP for hospitality, the Field Theory Group of the COSMO\/CBPF for the pleasant visit with the PCI-BEV\/MCTIC support. P. G. was partially supported by Fondecyt (Chile) grant 1130426 and by Proyecto Basal FB0821.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\section{Introduction}\nQuantum mechanics is one of the most transformative physical theories of the 20th century. However, while the evolution of the quantum wave function is deterministically described by Sch\\\"odinger's equation, the outcome of a measurement is probabilistic, given by Born's rule. Despite recent progress \\cite{Ozawa_Bayes_1997,Zurek_Darwinism_2009,Shrapnel_Born_2018}, there is no consensus on how to reconcile these two viewpoints, as illustrated by the measurement paradox \\cite{Schroedinger_Cat_1935}. There are two conceptually distinct approaches: either the interpretative postulates must be modified \\cite{Adler_Quantum_2009,Zeh_Measurement_1970,Everett_Relative_1957,Heisenberg_Prinzipien_1930,Bohm_Hidden_1952}, or quantum mechanics approximates a deeper theory yet to be discovered. The later approach gives rise to collapse models \\cite{Bassi_Review_2012,Bassi_Gravitational_2017}, postulating a stochastic nonlinear modification to Schr\\\"odinger's equation. Irrespective of whether they successfully allow the reconciliation of quantum evolution and measurement theory, these collapse models are considered the only mathematically consistent, phenomenological modifications against which quantum theory can be tested \\cite{Adler_Quantum_2009,Ferialdi_Dissipative_2012}.\n\nThe most universal and well studied collapse model is Continuous Spontaneous Localization (CSL) \\cite{Pearle_Localization_1989,Ghirardi_Markov_1990},\nwhich serves as a framework to describe a variety of collapse mechanisms \\cite{Bassi_Review_2012,Bassi_Gravitational_2017,Ghirardi_gravity_1990,Adler_Nonwhite_2007,Adler_Nonwhite_2008,Smirne_Dissipative_2014,Smirne_Dissipative_2015,Bassi_Breaking_2010}.\nIn CSL, a collapse noise field is introduced which couples nonlinearly to the local mass density. In its simplest form this noise is white, and the model has two parameters --- the collapse rate $\\lambda_c$, which determines the interaction strength with the collapse noise field, and the correlation length $r_c$, which determines the spatial resolution of the collapse process \\cite{Ghirardi_Markov_1990,Adler_Quantum_2009}.\nThe correlation length is expected to be $\\sim100$~nm \\cite{Adler_Bounds_2006}, since the behaviour of larger systems is generally adequately described by classical theories, whereas quantum mechanics appears to apply on smaller scales.\nRefined dissipative and coloured models introduce two additional parameters, associating a temperature and high-frequency cut-off to the collapse noise field to ensure energy conservation and permit an identifiable physical origin of collapse \\cite{Adler_Nonwhite_2007,Adler_Nonwhite_2008,Smirne_Dissipative_2014,Smirne_Dissipative_2015,Bassi_Breaking_2010}. \nBased on the assumption that the origin is of cosmological nature, and thermalised to the photon-, neutrino-, or gravitational wave background,\nthe high-frequency cut-off is estimated to occur at $\\Omega_{\\rm csl}\/2\\pi\\sim 10^{10}-10^{11}$~Hz \\cite{Bassi_Breaking_2010}.\n\nTo date, the most stringent unambiguous upper bounds on the collapse rate at the expected correlation length are based on mechanical resonators, with signatures of spontaneous collapse expected to manifest as an anomalous temperature increase.\nHowever, the suggested lower bounds to the collapse-induced heating are lower than one phonon per day \\cite{Adler_Bounds_2006,Bassi_Breaking_2010,Ghirardi_Unified_1986}.\nThe challenge of resolving these exquisitely small collapse signatures over a large thermal noise background has precluded conclusive tests of CSL, and has also introduced significant challenges for data interpretation \\cite{Vinante_Improved_2017}.\nEven were these issues resolved, quantum backaction heating \\cite{Khalili_backaction_2012,Nimmrichter_Optomechanical_2014} would remain orders of magnitude larger than the predicted collapse signatures.\nMoreover, with micron- \\cite{Vinante_Cantilever_2015,Vinante_Improved_2017} to meter-sizes \\cite{Carlesso_Gravitational_2016,Helou_LISA_2017}, the resonators employed to-date are larger than the anticipated correlation length and have frequencies far below the expected high-frequency cut-off. As such they are unable to provide insight into the physical origin of collapse \\cite{Adler_Nonwhite_2007,Adler_Nonwhite_2008,Smirne_Dissipative_2014,Smirne_Dissipative_2015,Bassi_Breaking_2010}.\n\n  \\begin{figure}[h!]\n  \\includegraphics{Fig_Setup_nc_1.pdf}\n\\caption{{\\bf Illustration of protocol.}\nTop left: array of optomechanical cavities.\nTop right: nonlinear pair production from a signal and pump photon (frequency $\\omega_{\\rm pump}$).\nBottom: Energy level diagram for scattering of a probe photon with a phonon.\n$n_b$:  phonon number.\n}\n  \\label{Fig_Setup}\n    \\end{figure}\n\nIn this work we propose to test collapse theories with high frequency nanomechanical resonators. This offers the advantages of miniaturisation to match the expected collapse correlation length, resonance frequency around the high-frequency cut-off, and the abilities to both exponentially suppress thermal phonons via passive cryogenic cooling and apply quantum measurement techniques to improve performance.\nTo assess the approach, we develop a specific experimental implementation that makes use of phonon counting in a nanoscale mechanical resonator. \nOur proposal includes new mitigation strategies for optical, thermal, electrical and quantum back-action noise that, for the first time, provide a way to bring each of these noise sources below the expected lower bounds for collapse-induced heating. We conclude that with challenging but plausible improvements in the state-of-the-art our approach could conclusively test CSL, closing the gap between measured upper bounds and predicted lower bounds on the collapse rate, and could also potentially identify the physical mechanism underlying the collapse process.\nThis provides an experimental pathway to answer one of the longest standing questions in physics, and  \nalso opens up possibilities for laboratory tests of astrophysical models of dark matter \\cite{Riedel_undetectable_2013,Riedel_collisions_2017}, \nand other exotic particles \\cite{Riedel_diffusion_2015}.\n\n\n\n\\section*{Results}\n\n\\subsection*{Basic protocol}\n\n  \n    \nOur protocol is illustrated in Fig. \\ref{Fig_Setup}, and is based on a gigahertz nanomechanical resonator, or array thereof,\nwithin a millikelvin environment.\nAs opposed to standard optomechanical measurement, consisting of an optical cavity linearly coupled to a mechanical resonator \\cite{Aspelmeyer_Review_2014}, we propose to perform phonon counting in a three-mode optomechanical system where two optical modes\nare coupled via a mechanical resonator with resonance frequency $\\Omega$.\nThis allows collapse signatures to be spectrally distinguished from most noise sources.\nOne mode, the {\\it probe mode}, is excited by a continuous weak laser at its resonance frequency $\\omega_p$.\nIn the ideal case, the other, the {\\it signal mode} at frequency $\\omega_s=\\omega_p+\\Omega$, is only excited by resonant anti-Stokes Raman scattering between collapse induced phonons and probe photons.\nA single-photon readout scheme minimises both absorption heating  \\cite{Meenehan_millikelvin_2014} and quantum back-action heating.\nSignal photons are spectrally separated from probe photons by a filter cavity, while dark counts are suppressed by nonlinearly downconverting signal photons to pairs and performing coincidence detection.\n\n\n\nAs a concrete example, we consider using a three-mode photonic-phononic crystal optomechanical system, such as proposed in \\cite{Chang_Array_2011,BasiriEsfahani_Phonon_2012,Ludwig_TwoMode_2012,Safavi_Naeini_traveling_2011}. \nWe choose most parameters based on those achieved in \\cite{MacCabe_Ultralong_2019}, with a mechanical resonance frequency $\\Omega\/2\\pi=5.3$~GHz, a mechanical damping rate $\\Gamma\/2\\pi=108$~mHz, an effective mass $m_{\\rm eff}=136$~fg, and thermalisation to the base temperature of a dilution refrigerator ($T=10$~mK).\nWe use the theoretical scattering-limited intrinsic decay rate of $\\kappa_{p,0}=\\kappa_{s,0}=2\\pi\\cdot 9.2$~MHz calculated for these devices \\cite{Ren_Crystal_2019} for both optical modes, where the subscripts `$p$' and `$s$' distinguish the probe and signal mode throughout \\cite{Aspelmeyer_Review_2014}.\nFinally, we assume a tenfold improved single-photon optomechanical coupling rate of $g_0\/2\\pi=11.5$~MHz, as predicted to be feasible with optimized designs \\cite{Matheny_coupling_2018}.\n\n\n\n\n\\subsection*{Phonon flux induced by CSL}\n\nThe CSL phonon flux is $\\dot n_c= \\lambda_c D$, where $D$ is a geometrical factor that quantifies the susceptibility of the resonator to spontaneous collapse.\nThe requirement that CSL should resolve the measurement problem introduces lower bounds on $\\lambda_c$, and therefore on the phonon flux. \nAdler proposed  $\\lambda_c\\geq 10^{-8\\pm 2}$~s$^{-1}$ from the postulate that collapse should account for latent image formation in photography \\cite{Adler_Bounds_2006}, while Bassi {\\it et al.} proposed $\\lambda_c\\geq 10^{-10\\pm 2}$~s$^{-1}$ from the presumption that collapse should occur in the human eye \\cite{Bassi_Breaking_2010}.\nWe estimate  $D=5.1\\cdot 10^{5}$ for our proposed device \\cite{Vinante_Cantilever_2015} (see Supplemental Material \\cite{Supp}), which combined with these bounds implies minimum CSL induced phonon fluxes of $\\dot n_c=5.1\\cdot10^{-3\\pm 2}$~s$^{-1}$ and $\\dot n_c=5.1\\cdot10^{-5\\pm 2}$~s$^{-1}$, respectively. \n\n\n\n\n\\subsection*{Optomechanical dynamics and conversion efficiency}\n\nIf the oscillator is initially in its ground state,\nwith one photon in the probe mode, a phonon introduced by spontaneous collapse\nprepares the state $\\ket{n_b n_p  n_s}=\\ket{110}$, where $n_b$ is the phonon number in the mechanical resonator, while $n_p$ and $n_s$ are the photon numbers in probe- and signal-mode, respectively. \nThe optomechanical conversion efficiency $\\eta_{\\rm om}$ for this state to emit a signal photon at frequency $\\omega_s$\nis obtained by numerically solving the Born-Markov master equation taking into account that $\\Gamma\\ll\\kappa_p,\\kappa_s$ (see Methods).\nWe choose the external probe decay rate $\\kappa_{p,\\rm ex}\/2\\pi=2.2$~MHz \\cite{Aspelmeyer_Review_2014}, allowing operation at the threshold of strong coupling with $g_0\\approx\\kappa_p$. \nThis is advantageous for efficient conversion of collapse-induced phonons to signal photons and ensures low occupancy, minimising noise, as discussed later.\nWe choose the signal mode to be significantly overcoupled ($\\kappa_{s,\\rm ex}\/2\\pi=0.7\\kappa_{s}=21$~MHz) in a trade-off between optimising the conversion efficiency and suppressing noise from direct occupancy of the signal mode (see later).\nTogether, these external decay rates result in relatively high conversion efficiency of $\\eta_{\\rm om}=0.32$.\n\n \\subsection*{Noise sources}\n\n\\begin{figure}[!htbp]\n\t \\includegraphics{Fig_heating_nc_1.pdf}\n\\caption{{\\bf Heating rates of a $Q=\\Omega\/\\Gamma=10^7$ silica sphere resonator vs. mechanical frequency and sphere diameter}. Red traces: heating due to coupling to the thermal environment at temperatures 300,1 and 0.01 K. Gray shaded: Lower bounds on CSL heating rates for a sphere, according to  Adler \\cite{Adler_Bounds_2006}, Bassi {\\it et al.} \\cite{Bassi_Breaking_2010} and GRW \\cite{Ghirardi_Unified_1986}, assuming the fundamental mechanical breathing mode frequency $\\Omega=c\/R$, with sphere radius $R$ and speed of sound $c=3000$ m\/s. CSL heating rates drop once the resonator becomes smaller than the noise correlation length, which is set to $r_c=10^{-7}$~m. Green: lower bound on heating rate predicted from classical channel gravity \\cite{Kafri_Classical_2014,Khosla_Classical_2018,Altamirano__Pairwise_2018}. At high frequencies and low temperatures, collapse signatures exceed the thermal heating. Blue shaded: proposed range of $\\Omega_{\\rm csl}$.}\n  \\label{Fig_heating} \n\\end{figure}\n\nFour classes of noise can potentially imitate a collapse signal:  thermal phonons, probe photons that leak through the system, phonons introduced by the measurement process, and detector dark counts.\nPhotons that leak through the system can be efficiently filtered using a standard laser stabilisation reference cavity \\cite{Kessler_Laser_2011}\n(see Table \\ref{tab:noise comparison} and Methods), and will not be considered further here.\n\n\n\n\n{\\it Thermal phonons.}\n A collapse signature is resolvable in a thermal noise background if $\\dot n_c\/\\dot n_{\\rm th}>1$, where $\\dot n_{\\rm th}=\\Gamma (e^{\\hbar \\Omega\/k_BT}-1)^{-1}$ is the thermal phonon flux.\nThis gives a minimum testable collapse rate $\\lambda_{c,\\rm th}=\\dot n_{\\rm th}\/D$. \nExisting experiments have operated with comparatively low frequency oscillators in the high temperature limit, $k_BT \\gg \\hbar\\Omega$ \\cite{Vinante_Cantilever_2015,Vinante_Improved_2017,Carlesso_Gravitational_2016,Helou_LISA_2017}, with thermal phonon flux significantly larger than Bassi {\\it et al.}'s lower bound,\nand have sought to resolve small collapse signatures on top of this large thermal noise background.\nA significant advantage of our approach is that miniaturisation and cryogenic cooling allow access to the regime where $k_BT \\ll \\hbar\\Omega$.\nThe average thermal phonon occupation is then exponentially suppressed due to Bose-statistics, $\\dot{n}_{\\rm th} \\approx \\Gamma e^{-\\frac{\\hbar\\Omega}{k_ BT}}$.\nFig. \\ref{Fig_heating} shows this exponential suppression as a function of resonator size, and in comparison to the CSL signal, for the simple example of the fundamental breathing mode of a silica sphere (see Supplemental Material \\cite{Supp} for calculation).\nAs can be seen, for gigahertz resonators at millikelvin temperatures the exponential suppression allows thermal phonon fluxes beneath both Adler and Bassi {\\it et al.}'s lower bounds.\nFor the proposed photonic crystal device, we find $\\lambda_{c,\\rm th}=1.2\\cdot 10^{-17}$~s$^{-1}$,\nalso well beneath both bounds.\nThis provides the potential for\nunambiguous tests of collapse models.\n\n  \\begin{figure}[!htbp]\n\\includegraphics{Fig_Energy_nc_1.pdf}\n\\caption{{\\bf Signal pathways due to measurement-induced phonons.}\na) Phonon created after direct excitation of signal mode.\nb) Phonon due to counter-rotating transition.\nc) Two phonons created by counter-rotating transition followed by resonant transition. \n}\n\\label{Fig_Energy}\n      \\end{figure}\n\n{\\it Measurement-induced phonons.}\nPhonons introduced by the optomechanical measurement can imitate collapse signatures. These phonons are created by non-resonant scattering processes between the signal and probe modes, the three lowest order of which are represented in Fig. \\ref{Fig_Energy}.\nWe calculate the probability of phonon occupancy due to these processes numerically by solving the Born-Markov master equation (see Methods).\nWe find that each of these processes is suppressed by the square of the resolved sideband ratio $\\Omega\/\\kappa_p$, with predicted phonon occupancies shown in Fig. \\ref{figSOM} (a) and (b).\nPhotoabsorptive heating can also introduce phonons. \nHowever, it only adds a negligible contribution to the measurement-induced phonon occupancy (see Table \\ref{tab:noise comparison} and Methods).\n\n  \n\n\n\\begin{figure}[!htbp]\n\t\\includegraphics{Fig_OM_nc_2.pdf}%\n\n\n\n\\caption{{\\bf Numerical calculations of noise magnitude.}\na) Blue (orange): occupancy of density matrix elements containing one (two) phonon(s);\ndashed blue: contribution from direct signal mode excitation.\nFast oscillations on timescale $\\Omega^{-1}$ correspond to the counter-rotating transition $\\ket{010}\\leftrightarrow\\ket{101}$; \nslow oscillations to the resonant process $\\ket{101}\\leftrightarrow\\ket{210}$, with period $g_0^{-1}=\\kappa_p^{-1}$. \nDotted lines: asymptotic values for $\\kappa_p^{-1},\\kappa_s^{-1}\\ll t\\ll\\Gamma^{-1}$.\nb) Same as (a) for $t\\sim\\Gamma^{-1}\\gg\\kappa_p^{-1}, \\kappa_s^{-1}$. \nc) Cumulative probability $p_{\\rm om}(t)$ of a probe photon creating a photon at frequency $\\omega_s$. \nDotted line: asymptotic value for $t\\gg\\Gamma^{-1}$.\n}\n\\label{figSOM}\n\\end{figure} \n \nA measurement-induced phonon can only be converted to a collapse-imitating photon at frequency $\\omega_s$ if it scatters with a second photon entering the probe mode within the lifetime $\\Gamma^{-1}$ of the mechanical resonator (see Fig. \\ref{Fig_Energy}).\nIt is therefore possible to suppress these photons by operating with a low average photon occupancy $\\bar n_p$. Here, we choose the photon occupancy so that the probability of a photon entering the probe mode during one mechanical oscillation lifetime is $\\eta_p = \\bar n_p \\kappa_p\/\\Gamma\\sim$~1\\%.\nThis reduces the rate of measurement-induced photons by a factor of a hundred.\nThe cumulative probability of a probe photon generating a phonon, and a second probe photon then causing emission of a photon at frequency $\\omega_s$, is shown in Fig. \\ref{figSOM} (c).\nThe asymptotic probability is $p_{\\rm om}(t\\rightarrow \\infty)=8.4\\cdot 10^{-8}$ (see Supplemental Material \\cite{Supp}).\n\n\n{\\it Coincidence dark counts.}\n Detecting collapse induced phonons at the predicted rate of less than one per day necessitates very high suppression of photon dark counts, which typically occur at hertz to kilohertz rates.\nOne way to  achieve this is to nonlinearly downconvert signal photons to pairs (bottom right inset, Fig. \\ref{Fig_Setup}) \nusing a bright pump beam in a third-order nonlinear medium.\nIt has been shown that this process can convert single photons to pairs with near-unit efficiency $\\eta_\\chi$ \\cite{Langford_Conversion_2011} (see Supplemental Material \\cite{Supp}). \nA signal is recorded only if a coincidence detection event is registered.\nThe coincidence dark count rate is suppressed as the square of the single-detector dark count rate $R_{d,1}$, $R_{d,2}=R_{d,1}^2 \\cdot \\tau_c$, where $\\tau_c$ is the coincidence timing resolution. \nFor commercially available photon counters with $R_{d,1}=3.5$~s$^{-1}$ and $\\tau_c=30$~ps \\cite{Photonspot_private}, we predict $R_{d,2}\\sim 3.7\\cdot10^{-10}$~s$^{-1}$.\n\n \n \\subsection*{Minimum testable collapse rate}\n \n For $r_c=10^{-7}$~m, the rate of coincidence counts attributed to collapse is  $R_c=\\lambda_c D \\eta=5.5\\cdot 10^{2}\\lambda_c$, where the efficiency  $\\eta=\\eta_p\\eta_{\\rm om}\\eta_{\\chi}\\eta_d\\eta_{\\rm f}=1.1\\cdot10^{-3}$ quantifies the fraction of phonons in the mechanical resonator that result in a coincidence count, $\\eta_{\\chi}=0.95$, and $\\eta_d=0.64$ is the coincidence detection efficiency (see Supplemental Material \\cite{Supp}).\nThis rate must exceed the sum of the noise rates, setting the limit to the minimum observable collapse rate $\\lambda_c$.\nFor optomechanically induced phonons, probe photons leaking through the system, and thermal phonons, $R_{\\rm om}=\\kappa_{p,\\rm ex} \\bar n_pp_{\\rm om}(t\\rightarrow \\infty)\\eta_f\\eta_\\chi\\eta_d$, $R_{\\rm phot}=\\kappa_{p,\\rm ex} \\bar n_pp_f\\eta_\\chi\\eta_d$, and $R_{\\rm th}=\\eta\\dot n_{\\rm th}$, respectively, where $\\eta_f=0.56$ is the transduction efficiency through the filter and $p_f=3.5\\cdot 10^{-10}$ the probability of a probe photon leaking through the filter (see Supplemental Material \\cite{Supp}).\nThe numerical values and corresponding minimum testable collapse rates are given in Table \\ref{tab:noise comparison} (see also Supplemental Material \\cite{Supp}).\nOptomechanical measurement-induced phonons and coincidence dark counts set comparable limits on $\\lambda_c$, with negligible contributions from leaked probe photons, photoabsorption, and thermally excited phonons.  \nThe minimum testable collapse rate limited by all noise sources is $\\lambda_c=\\sum_i\\lambda_{\\rm c,i}=1.0\\cdot10^{-12}$~s$^{-1}$, sufficient to test both Bassi {\\it et al.}'s and Adler's proposals.\n\n\\begin{table}\n\\center\n\\begin{tabular}{c | c | c | c} \n\nnoise type & scaling & signal rate~[s$^{-1}$]   &$\\lambda_{c,\\rm min}$~[s$^{-1}$] \\\\\n\n\\hhline{=|=|=|=}\n\\rule{0pt}{9pt} collapse noise & $\\eta l^2\\rho^2x_0^2$\\cite{Supp} & $5.5\\cdot 10^{2}\\lambda_c$ & --- \\\\\n\\hline\n\\rule{0pt}{12pt} thermal & $e^{-\\frac{\\hbar\\Omega}{k_BT}}$ &  $6.7 \\cdot 10^{-15}$ & $1.2\\cdot 10^{-17}$ \\\\\n\\hline\n\\rule{0pt}{10pt} optom. phonons &  $\\eta_p\\kappa^2\/\\Omega^2$ & $1.9 \\cdot 10^{-10}$  & $3.5 \\cdot 10^{-13}$  \\\\\n\\hline\n\\rule{0pt}{10pt} abs. heating & $l\/(m^{\\frac{1}{4}}\\Omega)$ \\cite{Supp} & $1.4 \\cdot 10^{-14}$  & $ 2.6 \\cdot 10^{-17}$ \\\\ %\n\\hline\n\\rule{0pt}{10pt} probe photons & see \\cite{Supp}& $1.4\\cdot 10^{-12}$  &$2.6\\cdot 10^{-15}$\\\\\n\\hline\n\\rule{0pt}{9pt} dark counts & $R_{d,1}^2 \\cdot \\tau_c\/N$ &$3.7\\cdot 10^{-10}$ & $ 6.7\\cdot 10^{-13}\/N$ \\\\\n\\hline\n\\rule{0pt}{9pt} all noise & --- & $5.6\\cdot 10^{-10} $ & $ 1.0\\cdot 10^{-12}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{{\\bf Comparison of noise sources and respective testable CSL parameter $\\lambda_c$.} $m$ is the oscillator mass, $l$ its linear size, $\\rho$ its density, and $x_0$ its zero point motion.}\n\\label{tab:noise comparison}\n\\end{table}\n\n\n\n \\subsection*{Signal rate and measurement time}\nThe predicted average time required to observe one signal due to CSL-collapse is $t_{\\rm meas}=(\\lambda_c D\\eta)^{-1}$. \nFully probing both Adler's and Bassi {\\it et al.}'s proposals with a single optomechanical resonator, including their respective uncertainties, would require $t_{\\rm meas}>57$~years.\nFabricating an array of $N$ optomechanical cavities on a silicon wafer \\cite{Bekker_tuning_2018,Xu_cascaded_2006,GilSantos_cascaded_2016,Zhang_synchronisation_2011} (see Fig. \\ref{Fig_Setup}), coupled to a single filter cavity, nonlinear medium and detector  (or a small number of such elements)\ncould significantly reduce this time to $t_{\\rm meas}^{(N)}=t_{\\rm meas}\/N$, and also the dark count-limited testable collapse rate to $\\lambda_{c,\\rm det}^{(N)}=\\lambda_{c,\\rm det}\/N$.\nWe estimate that $N\\sim10^{4}$ may be feasible (see Supplemental Material \\cite{Supp}), essentially eliminating detector dark counts as a limit, and allowing a reduction of the measurement time to about two days.\n\n\n \\subsection*{Feasibility and alternative parameter regimes}\nThe only optomechanical parameters that must be improved from the current state-of-the-art \\cite{MacCabe_Ultralong_2019} to realise our protocol, are a reduced optical linewidth (by a factor of $\\sim50$), as predicted by theoretical modelling based on the device realized in \\cite{Ren_Crystal_2019}, and an enhanced single-photon coupling rate (by a factor of $\\sim10$), based on theoretical modelling in \\cite{Matheny_coupling_2018}. \nAlternatively, effective enhanced single-photon coupling could be achieved by coupling to a qubit or other highly nonlinear system, e.g. as demonstrated in \\cite{Pirkkalainen_hybrid_2013}.\nGiven the trajectory of the field,\nwe estimate these requirements to be likely achievable in the intermediate future. \nNevertheless, it is also useful to consider alternative realizations of the method.\n\n \n\n{\\it Quadratic coupling.}\nWhile here we consider phonon-counting via an optomechanical Raman interaction,\nin principle the method could be implemented with any low-noise phonon-counting method applied to a high-frequency oscillator \\cite{Oconnell_quantum_2010,Dellantonio_nondemolition_2018,Sletten_Fock_2019,Cohen_counting_2014,ArrangoizArriola_nanomechanical_2019}. \nOne promising approach may be quantum non-demolition measurement of phonon number using non-linear optomechanics \\cite{Thompson_strong_2008}. \nIn the regime of quadratic optomechanical coupling and resolved mechanical sidebands \\cite{Aspelmeyer_Review_2014}, a collapse-induced phonon imparts a frequency shift $2\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}$ on the optical resonance at frequency $\\omega$, where $\\bar n_{\\rm cav}=\\langle a^\\dagger a \\rangle$ is the average intracavity photon number with $a$ the annihilation operator for the optical cavity field, and $g_0^{(2)}$ the zero-point quadratic coupling rate \\cite{Thompson_strong_2008,Paraiso_Squared_2015,Hauer_nondemolition_2018}.\nThe shift is detectable if it is larger than the significant noise sources, which are random fluctuations in the probe frequency, absorption heating, and quantum-backaction from spurious linear coupling.\n\nConsidering Bassi {\\it et al.}'s proposed mechanism \\cite{Bassi_Breaking_2010}, taking $\\bar n_{\\rm cav}=10^2$ and assuming that the probe is shot noise limited, we find that a zero-point quadratic coupling rate of  $g_0^{(2)}\\gtrsim3.5\\sqrt{\\kappa\\Gamma}\\bar n_{\\rm cav}^{-3\/2}\\gtrsim 2\\pi\\cdot 28$~Hz would be sufficient for\nthe weakest possible collapse signal to exceed the probe frequency noise using the photonic-phononic crystal considered in the protocol above \\cite{MacCabe_Ultralong_2019}\n(see Methods).\nThis is well within experimentally achieved values in optomechanical photonic crystals (e.g. $g_0^{(2)}\/2\\pi=245$~Hz in \\cite{Paraiso_Squared_2015}).\n\nPerhaps the most significant challenge in this approach would be to engineer a strong suppression of linear optomechanical coupling, so that the phonon flux due to quantum back-action does not exceed the predicted CSL signature. \nIf using standard architectures, there is a fundamental limit to this suppression of linear coupling \\cite{Miao_Limit_2009}. Hence, either a different architecture would have to be employed \\cite{Kaviani_paddle_2015,Dellantonio_nondemolition_2018,Hauer_nondemolition_2018}, or the substantially more stringent condition $g_0^{(2)}\\geq\\kappa$ would have to be realized.\nThe phonon flux due to quantum back-action is given by  $\\dot{n}_{\\rm ba}=4g_0^2\\bar n_{\\rm cav}\/\\kappa=4g^2\/\\kappa$. \nTo resolve a potential CSL signature, $\\dot{n}_{\\rm c}$ must be greater than $\\dot{n}_{\\rm ba}$.\nAs a result, the linear optomechanical coupling would need to be suppressed to $g\\leq\\sqrt{\\lambda_c D \\kappa \/4}$. To test $\\lambda_c=10^{-12}$, we find the condition $g\/2\\pi\\lesssim 10^{-1}$~Hz, about seven orders of magnitude lower than typical linear coupling rates in photonic-phononic crystal structures  \\cite{MacCabe_Ultralong_2019}.\nWhile some architectures may in principle allow for vanishing linear coupling $g$, achieving the required suppression in practice may be challenging  \\cite{Dellantonio_nondemolition_2018,Kaviani_paddle_2015,Anetsberger_Near-Field_2009}.\n\nIn continuous operation, with currently available technology \\cite{MacCabe_Ultralong_2019}, absorption heating would exceed the expected heating from collapse by about seven orders of magnitude.\nEven with this very large heating in the continuous domain, it may be possible to resolve the problem by operating in a pulsed regime, so that each optomechanical measurement process is completed in a timescale much shorter than the time required for absorption events to create phonons.\nIn this case, the measurements would need to be sufficiently temporally spaced to allow for phonons to fully dissipate.\n\n\n \n\n\n\\subsection*{Discussion}\n\n\\begin{figure}[!htbp]\n  \\includegraphics{Fig_CSL_nc_2.pdf}\n\\caption{{\\bf Parameter diagram for CSL-model.}\nExcluded upper bounds: gravitational wave detectors (yellow shaded); cold atoms (gray shaded); microcantilevers (dashed blue line); KDTL-interferometry (dashed black); Excluded for simple CSL only: neutron stars (dashed black) and X-ray (dotted black).\nProposed lower bounds: Adler (vertical blue bars and dotted blue line) and Bassi {\\it et al.} (vertical black bar).\nRed: predicted testable parameter space using our protocol.\n}\n\n\n\\label{figCSL}\n\\label{stats}\n\\end{figure}\n\n\nFig. \\ref{figCSL} compares the predicted upper bound on the collapse rate $\\lambda_c$ from our protocol to those of existing experiments,\ntogether with Adler's and Bassi {\\it et al.}'s lower bounds, and their uncertainties.\nExisting upper bounds  are provided by the motional stability of gravitational wave interferometers  \\cite{Carlesso_Gravitational_2016,Helou_LISA_2017,Nobakht_Unraveling_2018} (yellow region); the thermalisation of ultracold cantilevers \\cite{Vinante_Cantilever_2015,Vinante_Improved_2017} (blue outlined region); Kapitza-Dirac-Talbot-Lau (KDTL)-Interferometry \\cite{Fein_25kDa_2019,Nimmrichter_Matterwave_2011,Toros_Colored_2017,Feldmann_Parameter_2012} (dashed black); spontaneous X-ray emission from Germanium \\cite{Curceanu_Xray_2016,Curceanu_LNGS_2020,Fu_Radiation_1997} (dotted black) and the observed temperature of neutron stars \\cite{Adler_Fermi_2019,Stace_Neutron_2019} (dashed black), which are valid however only for white noise CSL; and cold atom interference \\cite{Toros_Colored_2017} (gray region), though we note the controversy \\cite{StamperKurn_Comment_2016} on the actual size of the superposition reported in \\cite{Kovachy_Superposition_2015}.\n\nThe red shaded region in Fig. \\ref{figCSL} could be tested by our protocol\nas discussed above. In the case of white-noise collapse, the protocol could for the first time fully test Bassi {\\it et al.}'s proposal. \nIf collapse noise has one of the proposed physical origins \\cite{Bassi_Breaking_2010,Smirne_Dissipative_2014,Smirne_Dissipative_2015}, the envisaged protocol would also for the first time probe Adler's prediction, which is in this case not tested by X-ray emission (black dotted line in Fig. \\ref{figCSL}).\nThe resonance frequency is close to the frequency range in which a drastic frequency-dependent reduction of the collapse noise stemming from a physical origin is expected \\cite{Bassi_Breaking_2010}. \nTo identify the physical origin of collapse, and to differentiate between collapse-induced signal and technical noise, we suggest employing a number of mechanical resonators of slightly different frequencies, or one frequency-tunable resonator \\cite{Pfeifer_tunable_2016}, at frequencies around $\\Omega\/2\\pi\\sim10$~GHz, such as reported in \\cite{Ren_Crystal_2019}.\n\nWe also evaluate the capability of the protocol to constrain parameters in gravitational collapse models.\nWhile for the Di\\'osi-Penrose model \\cite{Diosi_Gravitation_1984,Diosi_Models_1989,Penrose_Reduction_1996} we find that it cannot exceed existing bounds, for the classical channel gravity model in a typical parameter range \\cite{Altamirano__Pairwise_2018,Kafri_Classical_2014,Khosla_Classical_2018} we predict about a one order-of-magnitude stronger bound than previously achieved \\cite{tbp}  (see Supplemental Material \\cite{Supp}).\n\n\nIn summary, we have proposed the concept of testing quantum linearity using high-frequency mechanical oscillators. \nThis offers the advantages of thermal noise suppression to well below expected collapse signatures, and the potential for identification of the physical origin of collapse. \nAs a possible implementation we suggest a protocol based on a dual-cavity high-frequency optomechanical device passively ground-state-cooled and operating in the strong coupling regime. This design, combined with nonlinear optical techniques to reduce dark counts, is predicted to allow measurement of the minuscule phonon-flux generated by collapse-induced heating.  While challenging, the protocol has the potential to conclusively test CSL, and thus whether collapse mechanisms can be invoked to resolve the measurement paradox. Unlike previous proposals and experiments, it is designed to allow for identification of the physical noise field underlying CSL, and for differentiation between excess technical noise and signatures of collapse.\n      \n\n\n\\section*{Methods}  \n\n\\subsection*{Born-Markov master equation}  \n\nTo model the dynamics of the three-mode optomechanical system we employ the Born-Markov framework for open quantum systems \\cite{Nunnenkamp_Single_2011,Liu_Strong_2013,BasiriEsfahani_control_2016}.\n The interaction picture Hamiltonian for our system is \\cite{BasiriEsfahani_Phonon_2012,BasiriEsfahani_control_2016,Chang_Array_2011}\n\\begin{equation}\nH_{\\rm int}=\\hbar g_0(b^{\\dagger}e^{-i\\Omega t}+be^{i\\Omega t})(a_p^{\\dagger}a_se^{i\\Omega t}+a_pa_s^{\\dagger}e^{-i\\Omega t})+\\hbar\\sqrt{\\kappa_{p,\\rm ex}}(a_p^{\\dagger}a_{\\rm in}+a_{\\rm in}^{\\dagger}a_p),\n\\end{equation}\nwhere $b$, $a_p$ and $a_s$ are annihilation operators for the mechanical mode and optical modes, respectively, and $a_{\\rm in}$ is the coherent input field. The first term describes the mechanically mediated cross-coupling of the optical modes, while the second term describes the coherent excitation \\cite{Aspelmeyer_Review_2014}. In the parameter regime of this work where $g_0\\ll \\Omega$ and $\\Gamma \\ll \\kappa_p,\\kappa_s,g_0$, the dynamics of the system can be described by the\nBorn-Markov master equation as \\cite{Nunnenkamp_Single_2011,Liu_Strong_2013}\n\\begin{equation}\n\\frac{d\\hat \\rho}{dt}=-\\frac{i}{\\hbar}\\big[H_{\\rm int},\\hat \\rho\\big] + \\kappa_p \\mathcal{D}[a_p]\\hat \\rho + \\kappa_s \\mathcal{D}[a_s]\\hat \\rho + \\Gamma(1+\\bar{n}_{\\rm th}) \\mathcal{D}[b]\\hat \\rho + (\\Gamma \\bar{n}_{\\rm th} + \\dot n_c)\\mathcal{D}[b^{\\dagger}]\\hat \\rho,\n\\label{eq: master_equation}\n\\end{equation}\nwhere $\\hat \\rho$ is the density matrix, $\\bar n_{\\rm th}$ the mechanical mean thermal occupancy, \nand $\\mathcal{D}$ the dissipating superoperator, $\\mathcal{D}[A]\\hat \\rho=A\\hat \\rho A^{\\dagger}-\\frac{1}{2}(A^{\\dagger}A\\hat\\rho+\\hat\\rho A^{\\dagger}A)$. \nA weak phonon flux due to spontaneous collapse is described by $\\dot n_c=\\lambda_c D$, independent of its origin.\nThis allows us to model the conversion of a signal phonon to a signal photon, as well as creation of noise phonons introduced by measurement (see Supplemental Material \\cite{Supp}).\n\n\n\n\n\\subsection*{Negligible sources of noise}\n\n{\\it Probe photons leaking though the system.}\n Probe photons passing directly from the laser through the optomechanical system, without a scattering event, could in principle imitate a signal, obfuscating collapse signatures.\nWe find that, using a standard laser stabilisation reference cavity as a filter \\cite{Kessler_Laser_2011}, this noise can be suppressed well below both Adler's and Bassi {\\it et al.}'s lower bounds.\nSimilarly, if a photon is created in an optomechanical conversion process and subsequently outcoupled into the signal mode, due to energy conservation it either remains at frequency $\\omega_p$, or has a frequency reduced by integer multiples $n$ of the mechanical resonance frequency, $\\omega_p-n\\Omega$.  In both cases, this noise is doubly suppressed --- first by the suppression of the direct occupation pathway, and second by the filter. \nThis makes probe photons that leak through the system a negligible source of noise (see Supplemental Material \\cite{Supp} for details).\n\n{\\it Optical absorption heating.}\nTo estimate the phonon occupancy due to optical absorption,\nwe use the model for absorption heating in silicon optomechanical crystals outlined in \\cite{Meenehan_millikelvin_2014,Ren_Crystal_2019}.\nPhotoabsorption creates an electronic excitation, which is then transferred to terahertz-frequency phonons. While radiating from the resonator to the environment with a geometry- and material-dependent rate $\\gamma_{\\rm THz}$, they also couple to lower energy phonons with a generally longer timescale, potentially exciting the mechanical resonator. \nIn \\cite{Meenehan_millikelvin_2014,Ren_Crystal_2019}, the average phonon number $\\bar n_b$ is related to the average intracavity photon number $\\bar n_{\\rm cav}$ via $\\bar n_b\\propto \\bar n_{\\rm cav}^{1\/3}$. \nWe expect this relationship to break down when the time between photoabsorption events is long enough for the generated heat to fully dissipate, $\\bar n_{\\rm cav} \\cdot \\gamma_{\\rm THz}\/\\kappa \\lesssim1$, where $\\kappa$ is the loaded optical decay rate, as any discrete photon absorption event is expected to create a fixed amount of heat.\nIn this case $\\bar n_{\\rm cav}$ determines the frequency of these events, but not the magnitude of dissipated heat.\nWe compute the average phonon number excited by of one probe photon in the mechanical resonator, $\\bar n_{\\gamma}$, due to photoabsorption for time $t_{\\rm abs}$, at which the oscillator is in thermal equilibrium with the material, but not yet with the environment, $\\Gamma^{-1}\\gg t_{\\rm abs} \\gg \\gamma_{\\rm THz}^{-1}$.\nFor the proposed setup we find $p_{\\rm abs}(t\\rightarrow\\infty)=6.1\\cdot 10^{-12}$ and  $R_{\\rm abs}=1.4\\cdot10^{-14}$~s$^{-1}$ (see Supplemental Material \\cite{Supp} for calculation details).\n\n\\subsection*{Measurement-induced phonons.}\nA probe photon can create a noise phonon by coupling directly into the signal mode instead of the probe mode (Fig. \\ref{Fig_Energy} (a)). \nThis process is suppressed by the square of the resolved-sideband ratio $\\Omega\/\\kappa_s$.\nThe corresponding occupancy is calculated by numerically solving the Born-Markov master equation (see Methods and Supplemental Material \\cite{Supp}) and shown by the dashed blue line in Fig. \\ref{figSOM} (a).\n\nA photon that does enter the probe mode, corresponding to the state $\\ket{ n_b  n_p  n_s}=\\ket{010}$, can introduce noise by undergoing the non-resonant phonon-creating transition $\\ket{010}\\rightarrow\\ket{101}$ (see Fig. \\ref{Fig_Energy} (b)).\nThe resulting state can also resonantly transition to a two-phonon state, $\\ket{101}\\rightarrow\\ket{210}$, as shown in Fig. \\ref{Fig_Energy} (c).\nSimilarly to above, noise phonons from this process are suppressed by $\\sim(\\Omega\/\\kappa_p)^2$.\nPredicted phonon occupancies are shown in Fig. \\ref{figSOM} (a) and (b).\n\n\\subsection*{Zero-point quadratic coupling rate required to fully test Bassi et al.'s lower bound}\n\nThe linearised quadratic part of the optomechanical interaction Hamiltonian is $H_{\\rm int}^{(2)}=\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}(a^{\\dagger}+a)(2b^{\\dagger}b+b^{\\dagger}b^{\\dagger}+bb)$ \\cite{Hauer_nondemolition_2018}. \nThe term proportional to $b^{\\dagger}b$ yields a per-phonon optical resonance frequency shift of $2\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}$, which is the signature of a collapse-induced phonon.\nA random fluctuation $\\delta \\omega$ in the frequency of the probe can imitate a signal if it is larger or equal to this frequency shift, and sustained over a time comparable to the phonon lifetime $\\Gamma^{-1}$. \nFor shot noise limited probe, the probability of a fluctuation larger than $2\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}$ is given by an error function of a Gaussian distribution\n\\begin{equation}\np(\\delta \\omega)=\\bigg(\\frac{1}{\\sqrt{2\\pi \\sigma^2}}\\int^\\infty_{\\delta \\omega}e^{-\\omega^2\/2 \\sigma^2}d\\omega\\bigg),\n\\label{eq:Gauss}\n\\end{equation}\nwith standard deviation of\n$\\sigma\\approx\\kappa\/\\sqrt{N}$, where $N$ is\nthe number of photons interacting with a phonon within the mechanical lifetime $\\Gamma^{-1}$, and is related to the average intracavity photon number via $N=\\bar n_{\\rm cav} \\cdot \\kappa\/\\Gamma$ for a continuous measurement. \nThe rate of spurious signals due to such fluctuations is $R_{\\delta\\omega}=\\Gamma p(\\delta\\omega)$.\nTo test a collapse-induced phonon flux of $\\dot n_{\\rm c}$, we require $\\dot n_{\\rm c}\\geq R_{\\rm \\delta\\omega}$.\nFrom Eq. \\ref{eq:Gauss} we find that, to fully exclude Bassi {\\it et. al's} lower bound using the photonic-phononic crystal considered in the protocol above \\cite{MacCabe_Ultralong_2019}, requires $\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}\\gtrsim 3.5\\sigma$.\nAssuming an average intracavity photon number of $\\bar n_{\\rm cav}=10^2$, with $\\kappa\/2\\pi=575$~MHz \\cite{MacCabe_Ultralong_2019}, leads to the condition $g_0^{(2)}\\gtrsim3.5\\sqrt{\\kappa\\Gamma}\\bar n_{\\rm cav}^{-3\/2}\\gtrsim 2\\pi\\cdot 28$~Hz.\n\nThe term proportional to $b^{\\dagger}b^{\\dagger}a$ converts a probe photon to two phonons, potentially imitating a collapse-signature. \nHowever, the shift induced by two phonons is $4\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}$ and can be clearly distinguished from the collapse-induced shift caused by one phonon.\nTherefore, two-phonon creation can only imitate a collapse signal if it coincides with a frequency fluctuation of the probe mode $-\\delta \\omega\\geq2 n_{\\rm cav}^{1\/2}g_0^{(2)}$, sustained at least over the two-phonon lifetime $(\\sqrt2\\Gamma)^{-1}$. The low probability of such a fluctuation, together with suppression on the order of $(2\\Omega\/\\kappa)^2$ due to the non-resonant nature of the interaction, make this source of noise negligible.\n\n\n\\section*{Data availability}\nThe data that support the findings of this study are available within the paper and its Supplemental Material.\nCodes for the numerical simulations are available on request from the corresponding author.\n\n\\section*{Author contributions}\nWPB provided overall leadership for the project. SF and WPB conceptualized the idea. SF, SB, MZ and KK developed the theoretical model. SF and SB performed numerical simulations.  All co-authors contributed in the development of the manuscript which was drafted by SF and WPB.\n\n\n\n\\section*{Acknowledgements}\nThe authors thank Gerard Milburn, Nathan McMahon and James Bennett for helpful discussions, and Nicolas Mauranyapin for preparing Figure 1. We also acknowledge funding by Australian Research Council grants (EQUS, CE170100009, DE180101443) and the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 663830.\n\n\n\n\\section*{Additional information}\n\n\n{\\bf Competing interests:} The Authors declare no competing interests.\n\n\n\n\\newpage\n\\footnotesize \\renewcommand{\\refname}{\\vspace*{-30pt}}  \n\\bibliographystyle{apsrev4-2} \n\n\\section*{Supplementary Note 1. Signal rates from spontaneous collapse}\n\\label{sec:collapserates}\nWhile there exists a plethora of collapse models \\cite{Bassi_Review_2012,Bassi_Gravitational_2017}, they can all be formulated in terms of a stochastic nonlinear modification to Schr\\\"odinger's equation of the general form \\cite{Ferialdi_Dissipative_2012,Adler_stochastic_2001}: \n\\begin{equation}\n\\frac{d\\Psi}{dt} = \\bigg[-\\frac{i}{\\hbar}H + \\sqrt{\\lambda}A\\xi(t)-\\frac{\\lambda}{2}(A^{\\dagger}A+A^2) \\bigg]\\Psi,\n\\label{stochastic} \n\\end{equation}\nwhere the operator $H$ is related to the standard Hamiltonian of the system, $\\xi(t)$ is defined in terms of an increment $dW(t)$ of a stochastic Wiener process $W(t)$ through\n$\\xi(t) dt = dW(t)$; \n$\\lambda$ is a coupling that sets the strength of the collapse and $A$ is the reduction operator, which is specific to the particular realization of the collapse mechanism. The requirements of norm-preservation of the state evolution and {of} no-superluminal signalling make these collapse models the only mathematically consistent, phenomenological modifications against which quantum theory can be tested in this context \\cite{Ferialdi_Dissipative_2012}.\n\n\nIn the following we give expressions for the  \ndecoherence rates due to two commonly studied collapse mechanisms. Collapse models \ndiffer in the properties and nature of the mechanisms purported to cause the collapse and can thus be classified according to the basis in which decoherence occurs, the mathematical properties of the stochastic mechanism and whether the mechanism has a quantum mechanical origin, or is due to a modification to Sch\\\"odinger's equation from a deeper-level theory. In the models discussed here decoherence acts in the position basis with Gaussian correlations in space, see \\cite{Bassi_Review_2012} and references therein. \nA generic expression for the diffusion term $\\sqrt{\\lambda}A\\xi(t)dt$ in these models is $\\sqrt{\\lambda}\\int d\\vec x \\left(\\rho(\\vec x)-\\langle{\\rho(x)}\\rangle\\right) dW(\\vec x,t)$ with  $\\rho(\\vec x)$ being the mass density, $W(\\vec x,t)$ the ensemble of Wiener processes (for different points $\\vec x$ in space) which can in general be correlated. The noise correlation length $r_c$ and the coupling rate $\\lambda$ to the noise field  are the model parameters. The mass density typically takes the form $\\rho(\\vec x) = \\sum_jm_j\\int dy G(y-x)\\psi_j^\\dagger(y)\\psi_j(y)$ with $G(y)$ a  ``smearing function'' and $\\psi_j^\\dagger(y), \\psi_j(y)$ creation and annihilation operators of the particle with mass $m_j$. \nThe stochastic Sch\\\"odinger equation Eq. \\eqref{stochastic} and the above form of the diffusion term result in a master equation for the density operator $\\hat\\rho$ where the off-diagonal elements $\\vec x', \\vec x''$ evolve as $\\frac{\\partial}{\\partial t}\\bra{\\vec x'} \\hat\\rho \\ket{\\vec x''} = -\\tilde{D}( x',  x'')\\bra{\\vec x'} \\hat\\rho \\ket{\\vec x''}$ where \n $\\tilde{D}(\\vec x', \\vec x'') = \\sum_{i,j}\\frac{\\lambda}{2}\\left[G(\\vec x_i' -\\vec x_j')+G(\\vec x_i'' -\\vec x_j'')-2G(\\vec x_i' - \\vec x_j'')\\right]$\nand describes the rate at which the  spatial coherence is suppressed. Taking as an example the typically used Gaussian smearing function of the mass density $G(\\vec x)=(4\\pi r^2_c)^{-3\/2}e^{-\\vec x^2\/4r_c^2}$, the decoherence rate reads $\\tilde{D}(\\vec x', \\vec x'')=\\lambda\\cdot(4\\pi r^2_c)^{-3\/2}\\cdot(1-e^{-|\\vec x'-\\vec x''|^2\/4 r^2_c})$. We note that this is exactly the one-particle decoherence rate of the Ghirardi-Rimini-Weber (GRW) \\cite{Ghirardi_Unified_1986} and in the Continuous Spontaneous Localization (CSL) models \\cite{Pearle_Localization_1989,Ghirardi_Markov_1990}. Furthermore, for small superpositions sizes (as compared to the correlation length $r_c$) $|\\vec x'-\\vec x''|\\ll r_c$ the decoherence rate becomes approximately $\\tilde{D}(\\vec x', \\vec x'')\\propto |\\vec x'-\\vec x''|^2\/4 r^2_c$. In this limit one obtains as expected that the collapse rate increases with the size of superposition and the corresponding Lindblad term in the master equation takes a simple form $\\propto - [\\vec x, [\\vec x, \\hat\\rho]]$. \n\nPosition-basis decoherence yields localization of macroscopic superposition states. The same Lindblad term $-\\tilde{D}[\\vec x,[\\vec x,  \\hat\\rho]]$ results in momentum diffusion of the system (this can be seen as a direct consequence of the Heisenberg Uncertainty \nPrinciple and the fact that the above decoherence process can equivalently be seen as a reduction of the position uncertainty of the system). In turn, the momentum diffusion can be interpreted as heating with the collapse-induced heating rate being directly proportional to the decoherence rate $\\tilde{D}$. \nThis allows testing the collapse models also by monitoring spontaneous heating of even isolated systems.\nFor a quantum oscillator the heating leads to an increase of the phonon number expectation value. \nThe equivalent Lindblad term reads $-\\tilde{D}(b)[ b,[ b, \\hat\\rho]]$, with $\\tilde{D}(b)=x_0^2\\tilde{D}(\\vec x', \\vec x'')$, where $x_0$ is the zero-point motion (or equivalently, for spatial superpositions of massive particles, the superposition size $\\Delta x$). \n\n\nThe most studied collapse model is the\nCSL model \\cite{Pearle_Localization_1989,Ghirardi_Markov_1990}.\nIt considers second-quantized (albeit non-relativistic) indistinguishable particles where the collapse occurs in the particle number (Fock) basis. The key consequence of the indistinguishability of the particles is that for multi-particle systems the model predicts quadratic dependence of the decoherence rate on the number of particles that are within the cutoff distance $r_c$. For a comparison, the GRW model postulates discrete in time collapse events of the wave-function of individual (and also distinguishable) particles in the positions basis which yields linear dependence of the decoherence rate on the particle number \\cite{Ghirardi_Unified_1986}.\nThe stochastic process in the CSL model is introduced in terms of a time-dependent Wiener noise at each point in space, coupling to mass-density smeared over some length scale $r_c$. As CSL is linear in the coupling rate $\\lambda_c$ of matter to the collapse noise field, we define a dimensionless decoherence operator $D$ by $\\tilde{D}_{\\rm csl}(b)=\\lambda_c D$.\nFor an oscillator with mass density $\\rho(\\vec x)$ and direction of motion along the $z$-axis, the decoherence operator in the CSL-model reads \\cite{Nimmrichter_Optomechanical_2014,Vinante_Cantilever_2015}\n\n\\begin{equation}\nD_{\\rm csl}=\\frac{(4\\pi)^{3\/2} r_c^3 x_0^2}{u^2}\\int \\frac{d^3\\vec{k}}{(2\\pi)^3}k^2_z e^{-k^2r_c^2}\\vert\\tilde{\\rho}(\\vec{k})\\vert^2,\n\\end{equation}\nwhere $\\tilde{\\rho}(\\vec{k})=\\int d^3r\\rho(\\vec{r})e^{-i\\vec{k}\\cdot\\vec{x}}$ is the Fourier transform of the mass density and $u=1.66\\cdot 10^{-27}$~kg is the atomic mass unit\n\nCSL in its original form predicts infinite energy increase as time goes to infinity \\cite{Ghirardi_Markov_1990,Bassi_Dynamical_2003,Bassi_energy_2005}. This problem can be solved by postulating a finite temperature of the noise \\cite{Smirne_Dissipative_2015}. Furthermore, the model assumes a white noise spectrum, which cannot be identified with any physical origin \\cite{Ferialdi_Dissipative_2012}. In order to generalise the model to become compatible with relativity and with observations, the noise field should have a more general, i.e.~non-white spectrum. In such a case, however,  the model becomes non-Markovian~\\cite{Adler_Nonwhite_2007, Adler_Nonwhite_2008}. While such models are difficult to study in full generality, it has been demonstrated~\\cite{Adler_Nonwhite_2007, Adler_Nonwhite_2008} that to lowest order in $\\lambda$ the qualitative features  of the model are the same as for the white-noise model.  A common assumption that helps to lift ambiguities in defining the model is that the  field underlying the collapse process has a cosmological origin. This allows one to introduce a high-frequency cutoff of the order of $\\Omega_{\\rm csl}\/2\\pi\\approx 10^{10}-10^{11}$ Hz \\cite{Bassi_Breaking_2010,Smirne_Dissipative_2015} which ensure the collapse rate is essentially as in a white-noise CSL model, but which changes the relaxation behaviour: in the coloured-noise model the system in the limit of long times thermalises to the temperature of the noise field, while in the white-noise model the system energy keeps growing with time \\cite{Pearle_Collapse_1996}.\n\n$D$ can be calculated analytically for simple geometries of composite test-systems \\cite{Nimmrichter_Optomechanical_2014,Vinante_Cantilever_2015}.\nFor a sphere of radius $R$  the decoherence operator reads\n\n\\begin{equation}\nD_{\\rm sphere}=\\frac{14\\pi^2 R^2 r_c^2\\rho^2 x_0^2}{3 u^2} \\big(1-\\frac{2r_c^2}{R^2}+e^{-R^2\/r_c^2}\\big(1+\\frac{2r_c^2}{R^2}\\big)\\big),\n\\end{equation} \nand for the case of a cuboid with constant density $\\rho$ and sidelengths $L_1$, $L_2$ and $L_3$, where $L_3$ is the direction of motion \n\\begin{equation}\nD_{\\rm cuboid}= \\frac{32 r_c^4\\rho^2 x_0^2}{u^2}\\big(1-e^{-\\frac{L_3^2}{4r_c^2}}\\big) \\big(e^{-\\frac{L_2^2}{4r_c^2}}-\\frac{\\sqrt{\\pi}L_2}{2r_c}{\\rm Erf}(\\frac{L_2}{2r_c}) -1\\big)  \\big(e^{-\\frac{L_1^2}{4r_c^2}}-\\frac{\\sqrt{\\pi}L_1}{2r_c}{\\rm Erf}(\\frac{L_1}{2r_c}) -1\\big).\n\\label{eq:dcuboid}\n\\end{equation}\nFor our proposed experiment, we estimate length, height and width of the photonic crystal beam to $L_1,L_2,L_3=1.21,0.22,0.22~\\mu$m, respectively, reproducing the effective motional mass of the relevant mechanical mode of  136~fg \\cite{MacCabe_Ultralong_2019}, where we used the density of silicon, $\\rho=2.33\\cdot10^{3}$~kg\/m$^3$.  We choose the cuboid approximation for the modeshape following \\cite{Vinante_Cantilever_2015}. From Eq. \\eqref{eq:dcuboid} we find $D=5.1\\cdot10^{5}$.  \n\n\nFor the models where the collapse is assumed to have a gravitational origin, two main types of theories can be distinguished: where decoherence arises due to an intrinsic uncertainty in the local value of the gravitational field \\cite{Diosi_Gravitation_1984,Diosi_Models_1989} or, equivalently, gravitational self-interaction \\cite{Penrose_Reduction_1996}; and where decoherence is a consequence of the assumption that gravity is fundamentally a classical channel \\cite{Kafri_Classical_2014,Khosla_Classical_2018,Altamirano_Pairwise_2018}. In both cases for small superposition size, the resulting effect has the same general form as the corresponding regime of the CSL and GRW models: For an oscillator it is proportional to the square of the zero point motion, for spatial superpositions of massive particles the effect is proportional the square of the superposition size.  Gravity-based decoherence has also been described within the framework of CSL in \\cite{Ghirardi_gravity_1990}.\n\nIn the Di\\'osi-Penrose (DP) model the decoherence rate is quantified by gravitational potential evaluated between superposed amplitudes of the system: \n$\\frac{G}{2\\hbar}[U(XX)+U(YY)-2U(XY)]$, where $U(XY)=-G\\int d^3r\\int d^3r'\\frac{\\rho_X(\\vec x)\\rho_Y(\\vec x')}{|\\vec x-\\vec x'|}$ is the gravitational interaction between mass-densities $\\rho_X, \\rho_Y$ associated with the superposed configurations $X, Y$ \\cite{Diosi_bulk_2014}, with $G=6.67\\cdot10^{-11}$~m$^3$kg$^{-1}$s$^{-2}$ being Newton's gravitational constant. For point particles the above expression gives divergent decoherence rate and thus a short-distance cutoff $r_{\\rm DP}$ is needed. The decoherence operator reads \\cite{Nimmrichter_Optomechanical_2014}\n\n\\begin{equation}\n\\tilde{D}_{\\rm DP}=\\frac{x_0^2 G}{6\\sqrt{\\pi}\\hbar}\\big(\\frac{a}{r_{\\rm DP}}\\big)m\\rho(\\vec x),\n\\label{eq:DPheating}\n\\end{equation}\nwhere $a$ is the lattice constant of the composite object.\nComparing the heating rate expected from Eq. \\eqref{eq:DPheating} to measurement-induced spurious phonons, which constitute the strongest noise source in our proposed experiment (see main text), we find that short-distance cutoffs up to $r_{\\rm DP}\\approx 3.9$~fm can be excluded. For a discussion of experimental tests of classical channel gravity \\cite{Kafri_Classical_2014,Khosla_Classical_2018,Altamirano_Pairwise_2018}, see \\cite{tbp}.\n\n\n\n\n\n\n\n\\section*{Supplementary Note 2. Calculation details: efficiency and noise levels of the optomechanical system}\n\\label{sec:optomechanics}\nThe optomechanical system is probed by a strongly attenuated coherent source, such as described in \\cite{Kessler_Laser_2011}, which can be stabilized to a sub-Hz linewidth $\\kappa_L$. Because $\\kappa_L$ is much smaller than $\\kappa_s$, $\\kappa_p$, and $\\kappa_f$, which are the linewidths of the {\\it probe mode}, the {\\it signal mode}, and the filter cavity, respectively, the field $a_{\\rm in}$ of the incoming probe laser is well approximated with a $\\delta$ - function: $a_{\\rm in}(\\omega)=a_{\\rm in}\\delta(\\omega_L)$, where $\\omega_{\\rm L}$ is the laser frequency. We set $\\omega_L=\\omega_p$ to maximize coupling into the probe mode $a_p$. \nThe mechanical decay rate $\\Gamma$, as well as the collapse rate are slow compared to timescales of the optomechanical interaction:  $D,\\Gamma \\ll g_0,\\kappa_{p\/s}$, where $g_0$ is the single photon optomechanical coupling rate \\cite{Aspelmeyer_Review_2014}. \nIn this limit, the conversion dynamics can be modelled by the reduced master equation:\n\\begin{equation}\n\\frac{d\\hat \\rho}{dt}=-\\frac{i}{\\hbar}\\big[H_{\\rm int},\\hat\\rho\\big] +\\kappa_p \\mathcal{D}[a_p]\\hat\\rho + \\kappa_s \\mathcal{D}[a_s]\\hat\\rho,\n\\label{eq:reducedME}\n\\end{equation}\nwhere $\\hat \\rho$ is the density matrix, the operators $a_p$ and $a_s$ correspond to annihilation operators for the optical probe- and signal-modes, respectively, and $H_{\\rm int}$ is the interaction Hamiltonian given in the main text.\n\n{\\it Resonant anti-Stokes scattering.} \nIf a phonon is introduced into the ground-state cooled oscillator, and a probe pulse is incident within the lifetime of the mechanical excitation, the system is in the initial state $\\ket{ n_b n_p  n_s}=\\ket{110}$. \nThe optomechanical conversion efficiency $\\eta_{\\rm om}$ is the probability of one photon in the probe mode $a_p$ scattering with one phonon in the mechanical resonator, creating a photon in the signal mode $a_s$ ($\\ket{110}\\rightarrow\\ket{001}$) via a anti-Stokes Raman process, and this photon being outcoupled to create a signal photon at frequency $\\omega_s$. \n$\\eta_{\\rm om}$ is obtained by numerically solving Eq. (\\ref{eq:reducedME}) and time-integrating over the emission from the signal mode, $\\eta_{\\rm om}=\\kappa_{s,\\rm ex}\\int_0^{\\infty}\\langle a_s^{\\dagger}(t)a_s(t)\\rangle dt$, where $\\kappa_{s,\\rm ex}=\\kappa_s-\\kappa_{s,0}$ is the external decay rate of the signal mode due to coupling, with $\\kappa_{s,0}$ and $\\kappa_s$ its intrinsic and loaded decay rates, respectively.\nSupplementary Figure \\ref{fig_hOM} (a) shows $\\eta_{\\rm om}$ as a function of the effective coupling strength $g_0\/\\kappa_p$ for critical coupling of the probe mode $\\kappa_{p,0}=\\kappa_{p,\\rm ex}$ and equal intrinsic couplings $\\kappa_{s,0}=\\kappa_{p,0}$, for critically ($\\kappa_{s,0}=\\kappa_{s,\\rm ex}$) and overcoupled ($\\kappa_{s,\\rm ex}\/\\kappa_p=2$ and $4$) signal mode.\nThe efficiency $\\eta_{\\rm om}$ is sensitive to the ratio $g_0\/\\kappa_p$, as $g_0$ sets the rate for anti-Stokes scattering, and $\\kappa_p$ sets the rate for the competing optical decay.\nFurthermore, it depends on the ratio $\\kappa_{s,\\rm ex}\/\\kappa_p$ as higher values favour outcoupling of the signal photon.\nFor our proposed experiment, $g_0=\\kappa_p$ and $\\kappa_{s,\\rm ex}\/\\kappa_p=1.8$.\n\n\n \n\n \\begin{figure}[h!]\n\n  \\includegraphics[width=\\textwidth]{Fig_Supp_hOM5.pdf}\n\\caption{Efficiencies of optomechanical transitions.\na) Cumulative probability for the output of a signal photon due to anti-Stokes scattering from the initial state $\\ket{ n_bn_p n_s}=\\ket{110}$ as a function of $g_0\/\\kappa_p$, for $\\kappa_{s,\\rm ext}\/\\kappa_p=1,2,$ and $4$.\nb) Efficiency of spurious Stokes scattering, corresponding to the phonon number occupancy expectation value $\\eta_{\\rm Stokes}=\\langle b^{\\dagger}(t)b(t)\\rangle$ after an incident photon in the signal mode, for times $\\Gamma^{-1}\\gg t \\gg \\kappa_p^{-1},\\kappa_s^{-1}$.\nc) Cumulative probability for the output of a signal photon due to anti-Stokes scattering from the two-phonon state $\\ket{n_b n_p n_s}=\\ket{210}$.\n}\n\n\\label{fig_hOM}\n\\end{figure}\n\n\nThe probability of a phonon in the mechanical resonator translating to a coincidence count, imitating a signal, is given by $\\eta=\\eta_p\\eta_{\\rm om}\\eta_{\\rm f}\\eta_{\\chi}\\eta_d=1.1\\cdot10^{-3}$, where $\\eta_p$ is the probability of a photon entering the probe mode during the mechanical excitation lifetime, $\\eta_f=0.56$ is the transduction efficiency through the filter for a signal photon at frequency $\\omega_s$, $\\eta_{\\chi}=0.95$ and $\\eta_d=0.64$ are downconversion and coincidence detection efficiencies, respectively. These efficiencies are analysed in more detail in the following paragraphs. As the rate of phonons created in the resonator due to spontaneous collapse is given by the collapse rate $\\lambda_cD$, the rate of registered collapse signatures is $R_c=\\lambda_cD\\eta=5.5\\cdot 10^{2}\\lambda_c$.\n\n{\\it Probe field occupancy.}\nThe average number of photons encountered by one phonon is given by\n\\begin{equation}\n\\bar n_{\\rm ph}=\\int_0^\\infty e^{-\\Gamma t}\\kappa_p\\bar n_p dt,\n\\label{eq:etap}\n\\end{equation}\nwhere $\\bar n_p$ is the average photon occupancy of the probe mode and $e^{-\\Gamma t}$ is the phonon occupancy at time $t$ after one phonon is created in the mechanical resonator at time $t=0$. \nIn the limit $\\bar n_p \\ll \\Gamma\/\\kappa_p$, Eq. (\\ref{eq:etap}) also quantifies the probability $\\eta_p$ of a phonon in the mechanical resonator encountering a probe photon, $\\eta_p=\\bar n_{\\rm ph}$. \nFurthermore, in this limit, $\\eta_p$ asymptotes to $\\eta_p=\\bar n_p\\kappa_p\/\\Gamma$. \nIn our protocol, the probe laser power is adjusted so that $\\eta_p=0.01$, corresponding to an average of 0.01 photons per mechanical oscillator lifetime.\nIn the steady state, the average intracavity photon number is given by \n$\\bar n_p=4\\kappa_{p,\\rm ex}\\bar n_{\\rm in}\/\\kappa_p^2$ \\cite{Aspelmeyer_Review_2014},\nand hence, to achieve a given $\\eta_p$ the input field occupancy is adjusted to $\\bar n_{\\rm in}(\\eta_p)=(\\eta_p\\Gamma\\kappa_p)\/(4\\kappa_{p,\\rm ex})$. \nBecause the signal is proportional to $\\eta_p$, and the noise background from measurement-induced noise phonons is proportional to $\\eta_p^2$, the input field occupancy gives a handle to lower the noise at the cost of longer measurement time, or vice versa.\n\n{\\it Transmission through the filter cavity.} \nThe efficiency  $\\eta_f$ of the signal passing through the filter, assuming equal input- and output coupling strengths $\\kappa_{f,\\rm ex}$, is given by \\cite{Hecht_Optics_2002}\n\\begin{equation}\n\\eta_f=\\big(1-\\frac{\\kappa_{f,0}}{\\kappa_f}\\big)^2,\n\\end{equation}\nwhere $\\kappa_{f,0}$ is the intrinsic filter linewidth and $\\kappa_f=\\kappa_{f,0}+\\kappa_{f,\\rm in}+\\kappa_{f,\\rm out}$ is the loaded filter linewidth, with $\\kappa_{f,\\rm in}$ and $\\kappa_{f,\\rm out}$ the in- and output coupling, respectively.\nUsing a typical a laser stabilisation filter cavity \\cite{Kessler_Laser_2011},  $\\kappa_{f,0}=30$~kHz, and overcoupling both at the input and output $\\kappa_{f,\\rm in}=\\kappa_{f,\\rm out}=1.5\\cdot\\kappa_{f,0}$, a transmission efficiency of $\\eta_f=0.56$ is achieved.\n\n{\\it Nonlinear downconversion.} \nAfter separation from probe light at frequency $\\omega_p$, signal photons at frequency $\\omega_s$ are downconverted to photon pairs in a nonlinear medium.\nIn order to minimize detector dark counts, we propose a nonlinear conversion process to convert signal photons to pairs. A bright classical pump beam with electric field amplitude $E$ is coupled into a medium exhibiting a third order $\\chi^{(3)}$ optical nonlinearity. This yields an effective second order interaction \n$H_{\\chi,\\rm eff}=\\gamma E a_{f,\\rm out}d_1^{\\dagger}d_2^{\\dagger}+\\gamma^* E a_{f,\\rm out}^{\\dagger}d_1d_2$, where $\\gamma$ is the nonlinear coupling strength \\cite{Langford_Conversion_2011}, $a_{f,\\rm out}$ is the mode of the signal transmitted through cavity and filter, and $d_{1\/2}$ are the modes coupled to the detectors. \nGiven the input state $\\ket{n_{f,\\rm out}n_{d1}n_{d2}}=\\ket{100}$, the time evolution in the nonlinear medium is \\cite{Langford_Conversion_2011}\n\\begin{equation}\n\\ket{\\Psi(t)}=\\cos{(\\abs{\\gamma}t\/\\hbar)}\\ket{100}+i \\sin{(\\abs{\\gamma}t\/\\hbar)}\\ket{011}.\n\\end{equation}\n \nBy setting the length of the nonlinear medium to $L=\\frac{1}{2}\\pi\\hbar c_n\\abs{\\gamma}^{-1}$, where $c_n$ is the speed of sound in the medium, the output state is $\\ket{\\Psi(t_{\\rm final})}=\\ket{011}$, corresponding to a photon in each detector mode $d_{1\/2}$.\nIt has been shown that this conversion can be performed with near-unit efficiency \\cite{Langford_Conversion_2011}, hence we assume $\\eta_\\chi=0.95$.\n\n{\\it Coincidence detection.}  \nAssuming a detection efficiency of $80\\%$ for a single detector \\cite{Photonspot_private}, the efficiency for coincidence detection is $\\eta_d=(0.80)^2=0.64$. \nThe coincidence dark count rate is $R_{\\rm coincidence}=R_{\\rm d}^2 \\cdot \\tau_c$, where $R_{\\rm d}$ is the dark count rate of a single detector and $\\tau_c$ is the coincidence timing resolution (time jitter). \nThis allows a suppression of dark counts with the square of the single-detector dark count rate. For $R_{\\rm d}=3.5$~Hz and $\\tau_c=30$~ps \\cite{Photonspot_private}, the predicted coincidence dark count rate is $R_{\\rm coincidence}=3.7\\cdot10^{-10}$~s$^{-1}$.\n \n{\\it Probe photons leaking through the system.}\nThere are two ways in which probe photons can leak through the system and potentially imitate a collapse signature.\nFirstly, optomechanical conversion processes can create photons at frequency $\\omega_p$, or at a frequency reduced by multiple integers $n$ of the mechanical resonance frequency, $\\omega_p-n\\Omega$.\nDiscussions of these processes are included in the following paragraphs on noise phonons.\nThe amplitudes of these processes are negligible due to strong suppression (see main text).\nSecondly, measurement noise is introduced by probe photons transmitted through the filter cavity and downconverted to a pair of photons in the nonlinear medium, imitating a signal. \nThe probability of a probe photon with detuning $\\Delta=\\omega_s-\\omega_p=\\Omega$ transmitting through both the near-critically coupled optomechanical system and a subsequent filter cavity of linewidth $\\kappa_f$ and free spectral range $\\omega_{\\rm fsr}$ is given by \\cite{Hecht_Optics_2002}\n\n\\begin{equation}\np_f(\\Omega)=\\eta_f\\big[ 1 + (\\frac{4}{\\kappa_f})^2 \\cdot {\\rm sin}^2{(\\frac{\\pi\\Delta}{\\omega_{\\rm fsr}})}\\big]^{-1}.\n\\label{eqFilter}\n\\end{equation}\nFor the frequency of the proposed mechanical resonator of $\\Omega\/2\\pi=5.3$~GHz \\cite{Chan_Nanomechanical_2011}, a finesse of $3.16\\cdot10^{5}$ as in a standard laser stabilisation reference cavity \\cite{Kessler_Laser_2011} and a cavity length $L$ of one centimeter, we find $\\omega_{\\rm fsr}\/2\\pi=c\/2L=15$~GHz and $p_f=3.5\\cdot 10^{-10}$.\nA noise photon is then downconverted to a photon pair and registered as a coincidence count with efficiency $\\eta_\\chi\\eta_d$. \nAs the rate of incoming probe photons coupled into the probe mode in the steady state is $4\\kappa_{p,\\rm ex}\\bar n_{\\rm in}\/\\kappa_{p}=\\bar n_p\\kappa_p=\\eta_p\\Gamma$, the rate of coincidence counts due to noise photons, imitating a signal, is \n\\begin{equation}\nR_{\\rm phot}=\\eta_p\\Gamma p_f \\eta_\\chi\\eta_d=1.4\\cdot 10^{-12}\\,{\\rm s}^{-1}.\n\\label{eqRf}\n\\end{equation}\n\n\n{\\it Noise phonons from direct occupation of the signal mode.}\nA photon from the coherent probe laser can create a phonon by coupling into the signal mode $a_s$ instead of the probe mode $a_p$, which results in a direct occupation of the signal mode $\\ket{ n_b  n_p  n_s}=\\ket{001}$, as shown in Fig. 3 (a) in the main text. \nThis process is suppressed due to the small spatiotemporal overlap $\\Theta$ of the signal mode with the coherent laser beam, $\\Theta=\\kappa_s^2\/(\\kappa_s^2+\\Omega^2)\\approx(\\kappa_s\/\\Omega)^2=3.2\\cdot 10^{-5}$, where $\\kappa_s\/2\\pi=30$~MHz is the loaded decay rate of the signal mode.\nIf the photon is outcoupled, it has a frequency of $\\omega_p$ due to energy conservation and can be efficiently filtered from a signal at frequency $\\omega_s$.\nHowever, a scattering process to the probe mode can create a noise phonon in mode $b$, imitating a decoherence-signature.\nThe efficiency of this Stokes scattering process is given by the probability of the initial state $\\ket{001}$ causing the mechanical oscillator to be in the excited state, which equals the phonon occupancy after optical decay, $\\eta_{\\rm Stokes}=\\langle b^{\\dagger}(t)b(t)\\rangle$, for times $t\\gg\\kappa_p^{-1},\\kappa_{s}^{-1}$.\n$\\eta_{\\rm Stokes}$ is shown as a function of $g_0\/\\kappa_p$, for $\\kappa_{s,\\rm ex}\/\\kappa_p =1,2,$ and $4$ in Supplementary Figure \\ref{fig_hOM} (b).\nHigher values of $\\kappa_{s,\\rm ex}$ correspond to lower probabilities for this type of noise, as the decay process of rate $\\kappa_{s,\\rm ex}$ competes with the Stokes scattering of rate $g_0$.\nFor the proposed experiment with $\\kappa_{s,\\rm ex}\/\\kappa_p=1.8$ we find $\\eta_{\\rm Stokes}=0.17$.\nThe probability of phonon creation, due to this process, \nat time $t_0$ after incidence of the probe photon is $p_{\\rm direct}=(\\kappa_p\/\\kappa_{p,\\rm ex})\\cdot\\Theta\\eta_{\\rm Stokes}=2.7\\cdot10^{-5}$.\n\n\n{\\it Noise phonons from counterrotating optomechanical processes.}\nAnother mechanism for noise phonon creation is given by probe photons anti-Stokes scattering into the signal mode, corresponding to the resonantly suppressed (counterrotating) transition   $b^{\\dagger}a_p a_s^{\\dagger}e^{-2i\\Omega t}\\ket{010}\\rightarrow\\ket{101}$. \nThe resulting state contains a photon in the signal mode as well as a phonon in the mechanical resonator, which could in principle both imitate a signal. \nHowever, the outcoupled photon has a frequency of $\\omega_p-\\Omega$ and can, therefore, be filtered from the signal at frequency $\\omega_s$.\nThe probability of phonon creation due to this process, caused by a single incident probe photon, is obtained by numerically solving Eq. (\\ref{eq:reducedME}), is $p_{\\rm counterrot,1}=1.7\\cdot 10^{-5}$, where $p_{\\rm counterrot,1}=\\bra{001}\\hat \\rho(t\\gg\\kappa_p^{-1},\\kappa_s^{-1}) \\ket{100}+\\bra{101}\\hat \\rho(t\\gg\\kappa_p^{-1},\\kappa_s^{-1}) \\ket{101}$, with $\\hat \\rho(t)$ the density matrix initialized in the state $\\hat\\rho(t=0)=\\ket{010}\\bra{010}$  (see Fig. 4 (a) and (b) in the main text).\nFurther, the state $\\ket{101}$ can resonantly transition to a state with two phonons in the mechanical resonator and one photon in the probe mode: $b^{\\dagger}a_p^{\\dagger}a_s\\ket{101}\\rightarrow\\ket{210}$.\nThe probability of one incident probe photon preparing the system in this two-phonon state $p_{\\rm counterrot,2}=$\\mbox{$\\bra{002}\\hat \\rho(t\\gg\\kappa_{p}^{-1},\\kappa_s^{-1}) \\ket{200}$}$+\\linebreak$\\mbox{$\\bra{012}\\hat \\rho(t\\gg\\kappa_p^{-1},\\kappa_s^{-1}) \\ket{210}$} (with $\\hat\\rho(t=0)=\\ket{010}\\bra{010}$) numerically obtained by solving Eq. (\\ref{eq:reducedME}) (see Fig. 4 (b) in the main text), finding $p_{\\rm counterrot,2}=1.6\\cdot 10^{-6}$.\nWe find that the occupancies $p_{\\rm counterrot,1}$ and $p_{\\rm counterrot,2}$ scale with $(\\kappa_p\/\\Omega)^2$ in the limit $g_0=\\kappa_p$, with higher order transitions suppressed by $(\\kappa_p\/\\Omega)^4$ and therefore negligible.\n\n\n{\\it Scattering of noise phonons to signal photons.}\nA spurious signal photon at frequency $\\omega_s$ is created if a noise phonon scatters with a second photon entering the probe mode within the lifetime of the mechanical excitation. \nFor timescales of the mechanical excitation lifetime,  $t \\sim \\Gamma^{-1}\\gg \\kappa_{p}^{-1},\\kappa_{s}^{-1}$, after incidence of a probe photon, it is convenient to define the `occupancy probabilities' of the one- and two-phonon states from the optomechanical processes described above: $p_{n_b=1}(t_0)=p_{\\rm direct}+p_{\\rm counterrot,1}$ and $p_{n_b=2}(t_0)=p_{\\rm counterrot,2}$, where $t_0$ is long compared to timescales of the optical decay, so that all optomechanical conversions have concluded, but short compared to the mechanical decay ($\\Gamma^{-1}\\gg t_0\\gg\\kappa_{p}^{-1},\\kappa_{s}^{-1}$). The time dynamics of these one- and two phonon occupancy probabilities are described by\n\\begin{equation}\np_{n_b=1}(t)=n_{n_b=1}(t_0) e^{-\\Gamma t}+2p_{n_b=2}(t_0)\\big( e^{-\\Gamma t}-e^{-2\\Gamma t} \\big)\n\\end{equation} \nand \n\\begin{equation}\np_{n_b=2}(t) =p_{n_b=2}(t_0)  e^{-2\\Gamma t},\n\\end{equation}\nrespectively.\nThe probability of these measurement-induced phononic states encountering a second probe photon is given by $\\eta_p\\Gamma\\int_0^\\infty p_{n_b=1}(t) dt$ and $\\eta_p\\Gamma\\int_0^\\infty p_{n_b=2}(t) dt$, respectively. The conversion efficiency of the one- and two-phonon state to a spurious photon outcoupled from the cavity is then given by $\\eta_{\\rm om}$ and $ \\eta_{\\rm om,2}$, respectively, where latter is numerically calculated, and plotted as a function of $g_0\/\\kappa_p$ for different values of $\\kappa_{s,\\rm ex}\/\\kappa_p =1,2,$ and $4$, as shown in Supplementary Figure \\ref{fig_hOM} (c).\nThe resulting probability of a noise phonon of frequency $\\omega_s$ coupled out of the cavity, due to optomechanical processes, potentially imitating a signal, is\n\\begin{equation}\np_{\\rm om}(t)= \\eta_p\\Gamma\\int_0^t p_{n_b=1}(t)\\eta_{\\rm om}+ p_{n_b=2}(t)\\eta_{\\rm om,2} dt,\n\\label{eq:pom} \n\\end{equation}\nas shown in the main text in Fig. 4 (c). \nWe find the asymptotic value $p_{\\rm om}(t\\rightarrow\\infty)=8.4\\cdot 10^{-8}$.\nIn analogy to Eq. (\\ref{eqRf}), the rate of coincidence counts due to noise is given by $R_{\\rm om}=\\eta_p\\Gamma p_{\\rm om}(t\\rightarrow\\infty)\\eta_f \\eta_\\chi\\eta_d=1.9\\cdot 10^{-10}$~s$^{-1}$.\n\n\n\n{\\it Noise phonons due to photoabsorptive heating.}\nThe number of noise phonons originating from absorption heating is estimated following \\cite{Meenehan_millikelvin_2014,Ren_Crystal_2019}.\nWe compute the average phonon number excited by one probe photon in the mechanical resonator, $\\bar n_{\\rm abs,1}$, due to photoabsorption for time $t_{\\rm abs}$, at which the oscillator is in thermal equilibrium with the material, but not yet with the environment  ( $\\Gamma^{-1}\\gg t_{\\rm abs} \\gg \\gamma_{\\rm THz}^{-1}$, with $\\gamma_{\\rm THz}$ the rate at which THz-frequency phonons radiate to the environment, see also Methods).\nWe approximate $\\bar{n}_{\\rm abs,1}$ by extrapolating from \\cite{MacCabe_Ultralong_2019,Ren_Crystal_2019}.\nIn this case, an average intracavity photon number $ \\bar n_{\\rm cav}=1$ yields an average phonon number of $\\bar n_{\\rm abs}=10$ in the mechanical resonator.\n$\\bar n_{\\rm cav}=1$ is equivalent to $\\kappa\/\\Gamma$ photons passing through the cavity within the lifetime $\\Gamma^{-1}$ of the mechanical resonator, creating in total 10 phonons.\nThe intrinsic optical decay is limited by photon absorption, $\\kappa_0\\approx\\kappa_{\\rm abs}$ and  $\\gamma_{\\rm THz} \\approx \\kappa_0\/2\\pi$, within the errors given.\nTherefore, from one photon we expect the induced occupancy $\\bar{n}_{\\rm abs,1}=10\\frac{\\Gamma}{\\kappa}=10\\cdot\\frac{108\\  \\rm mHz\/2\\pi}{575\\ \\rm MHz\/2\\pi}\\approx 1.9\\cdot 10^{-9}$.\n\n\nAs with the optomechanical heating rates, a spurious signal will only occur if the phonons resulting from absorption heating interact with another probe photon. In analogy to Eq. (\\ref{eq:pom}), the probability of a probe photon creating a spurious signal photon at frequency $\\omega_s$ due to absorption heating is \n\\begin{equation}\np_{\\rm abs}(t)=\\eta_p\\Gamma \\int_0^t\\bar n_{\\rm abs,1} e^{-\\Gamma t'}\\eta_{\\rm om} dt'.\n\\label{eq:pabs}\n\\end{equation}\nFor the proposed setup we find $p_{\\rm abs}(t\\rightarrow\\infty)=6.1\\cdot 10^{-12}$ and  $R_{\\rm abs}=1.4\\cdot10^{-14}$~s$^{-1}$.\n\nFor the quadratic coupling approach, $\\bar n_{\\rm cav}=10^2$ (see main text), and the relation $\\bar n_{\\rm abs} \\propto \\bar n_{\\rm cav}^{1\/3}$ holds \\cite{Meenehan_millikelvin_2014,Ren_Crystal_2019}, and we find an average intracavity phonon number of $\\bar n_{\\rm abs} \\approx 5$. This corresponds to a noise phonon rate of $\\dot n=\\Gamma\\bar n_{\\rm abs} \\approx 3$~s$^{-1}$, close to seven orders of magnitude higher than the lowest possible phonon flux expected from Bassi {\\it et al.}'s proposal of $\\dot n_c=5.1\\cdot10^{-7}$~s$^{-1}$.\n\n\n\n{\\it Multiplexing.}\nA photonic-phononic crystal, including suspension and phononic shield, requires an area of about 1000 $\\mu$m$^2$ \\cite{MacCabe_Ultralong_2019}. Thus it would be conceivable to fabricate a high number of them on a 4-inch-wafer with an area of $\\sim 2\\cdot 10^{9}$ $\\mu$m$^2$. We assume $N\\sim10^{4}$, allowing more than 99$\\%$ of the wafer to be reserved for waveguide coupling, fabrication tolerances, etc. In principle, all these devices to may be connected to the same filter cavity, nonlinear medium, and detector pair, or to a small number of such elements. \n\n\n\n\n\\section*{References}\n\n\\footnotesize \\renewcommand{\\refname}{\\vspace*{-30pt}}  \n\\bibliographystyle{apsrev4-2}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nComputational fluid dynamics (CFD) is a branch of fluid mechanics that deals with numerically solving and analyzing fluid flow problems such as those found in aerodynamics, geology, biology, etc. CFD simulations are known for their high computational requirements, memory usage, and run times. Because of this, there is an ever growing body of work using simulation data to create reduced order models or surrogate models that can be evaluated with significantly less resources. Towards this end, we develop a neural network approach that both compresses the computation time and memory usage of fluid simulations.\n\nWe investigate fluid simulations that contain complex time dependent turbulence. Simulations of this form are difficult because they require fluid solver to have high resolution and small times steps. Never the less, they frequently occur in nature and are an important area of study. Motivated by need for these simulations and the recent success of neural network based models in related areas \\cite{tompson2016accelerating} \\cite{guo2016convolutional} \\cite{yang2016data}, we choice this setting to test our model.\n\nThe most popular approach to modeling fluid flow is with the Navier stokes equation. The solution to this partial differential equation gives the flow velocity field for a given domain. Recently, a new method for simulating fluid flow has emerged named the Lattice Boltzmann Method (LBM). It is derived from the Boltzmann equation and grew out of Lattice Gas Automaton (LGA) in the late 80s \\cite{mcnamara1988use}. The main advantages of the LBM are its ability to run on complex geometries, its scalability to parallel architectures (particularly GPUs) and applicability to complex flows that contain phenomena such as heat transfer and chemical reactions. Our method is centered around this method of simulating flow.\n\nLat-Net works by compressing the state of a simulation while learning the dynamics of the simulation on this compressed form. The model can be broken up into three pieces, an encoder, compression mapping, and decoder. The encoder compresses both the state of the simulation as well as the given boundary conditions. The compression mapping learns the dynamics on the compressed state that correspond to the dynamics in the fluid simulation. The decoder decompresses the compressed state allowing either the whole simulation state or desired pieces to be extracted.\n\nWe focus the content of this paper on LBM fluid simulations because this is the most popular use of the LBM, however, this method of simulation is known to be able to solve a large set of partial differential equations \\cite{galindolattice}. In fact, LBM can simulate many physical systems of interest such as Electromagnetism, Plasma, Multiphase flow, Schr\u00f6dinger equation etc. \\cite{mendoza2010three} \\cite{kim2008wavelet} \\cite{zhong2006lattice} \\cite{shan1993lattice}. With this in mind, we keep our method general and show evidence our method works equally well on Electromagnetic simulations. However, because the dominate use of LBM is on fluid flow problems we center discussion on this subject.\n\nOur work has the following contributions.\n\\begin{itemize}\n  \\item It allows for simulations to be generated with less memory then the original flow solver. There is a crucial need for such methods because memory requirements grow cubic to grid size in 3D simulations. In practice, this quickly results in the need for large GPU clusters \\cite{onodera2013large} \\cite{xian2011multi}.\n  \\item Once our model is trained, it can be used to generate significantly larger simulations. This allows the model to learn from a training set of small simulations and then generate simulations as much as 16 times bigger with little effect in accuracy.\n  \\item Our method is directly applicable to a variety of physics simulations, not just fluid flow. We show this with our electromagnetic example and note that the changes to our model are trivial.\n\\end{itemize}\n\n\\section{Related Work}\n\nRecently, there have been several papers applying neural networks to fluid flow problems. Guo etc. \\cite{guo2016convolutional} proposed to use a neural network to learn a mapping from boundary conditions to steady state flow. Most related to our own work, Yang etc. \\cite{yang2016data} and Tompson etc. \\cite{tompson2016accelerating} use a neural network to solve the Poisson equation in order to accelerate Eulerian fluid simulations. The key difference between this and Lat-Net is its ability to compress the memory usage and the generality of our method to other physics simulations.\n\nThere has also been an increasing body of work applying neural networks to other physics modeling problems. For example, neural networks have been readily adopted in many chemistry applications such as predicting molecular properties from descriptors, protein contact prediction and computational material design \\cite{goh2017deep}. Very recently, neural networks have been applied to quantum mechanics problems as seen in Mills etc. \\cite{mills2017deep} and Giuseppe etc. \\cite{carleo2017solving} where neural networks are used to approximate solutions to the Schr\u00f6dinger equation. In high energy Physics, Paganini etc. \\cite{2017arXiv170502355P} uses a generative adversarial networks (GAN)\\cite{goodfellow2014generative} to model electromagnetic showers in a longitudinally segmented calorimeter. Many of these applications are relatively recent and indicate a resurgence of interest in applications of neural networks to modeling physics.\n\nReduced order Modeling is an area of research that focuses on techniques to reduce the dimensionality and computational complexity of mathematical models. A Reduced order model (ROM) is constructed from high-fidelity simulations and can subsequently be used to generate simulations for lower computation. The most popular ROM method for fluid dynamics is Galerikin projection \\cite{rowley2004model} \\cite{barone2009reduced}. This method uses Proper Orthogonal Decomposition to reduce the dimensionality of flow simulations and then finds the dynamics on this reduced space. There are other methods that build on this such as reduced basis methods and balanced truncation \\cite{veroy2005certified} \\cite{rowley2005model}. While these approaches are centered around the Navier stokes equation and thus not directly comparable to our own, we note that the compression mapping present in these methods is typically quite simple. Given the recent success neural networks have had in creating well structured encodings (such as Variational Autoencoders \\cite{kingma2013auto} \\cite{watter2015embed}), we feel our approach is well justified.\n\n\\section{Deep Neural Networks for Compressed Lattice Boltzmann}\n\nIn this section, we present our model for compressing Lattice Boltzmann simulations.\n\n\\subsection{Review: The Lattice Boltzmann Method}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.19]{.\/figs\/lattice_boltzmann.pdf}}\n\\caption{Illustration of the Lattice Boltzmann update steps}\n\\label{lattice_boltzmann}\n\\end{figure}\n\nIn a Lattice Boltzmann simulation, the domain is discretized into an equal sized Cartesian grid. Each cell of this grid contains a velocity distribution function $f_i$ that describes the velocity of flow at that point. $f_i$ has values ranging over $i$ that correspond to the $\\{ \\vec{c}_i \\}$ directions of flow. In our 2 dimensional simulations, there are 9 such directions (D2Q9 scheme). A figure showing this grid structure is seen in \\ref{lattice_boltzmann}. From the distribution function $f_i$, one can calculate the density ($p$) and velocity ($\\vec{u}$) of the fluid flow with the following equations.\n\n\\begin{equation}\n  p = \\sum_i f_i  \\qquad\\text{and}\\qquad \\vec{u} = \\sum_i \\vec{c}_i f_i \n\\end{equation}\n\nThe lattice states are updated with two separate steps, the collision step and the streaming step. The collision step mimics the flow interacting with itself and is updated in the following way,\n\n\\begin{equation}\n  f^t_i(x, t + \\delta_t) = f_i(x,t) + \\frac{1}{\\tau} (f_i^{eq} - f_i)\n\\end{equation}\n\nwhere $\\tau$ is the relaxation constant and $f_i^{eq}$ is the flow equilibrium. For our simulations, we use the $f_i^{eq}$ from the Lattice Bhatnagar-Gros-Krook (LBGK) scheme \\cite{guo2013lattice}. After the collision step is applied, the flow propagates to adjacent cells following the streaming step. \n\n\\begin{equation}\n  f_i(x + c_i, t + \\delta_t) = f^t_i(x,t + \\delta_t)\n\\end{equation}\n\nThis step will contain bounce back if one of the adjacent cells is a boundary. Figure \\ref{lattice_boltzmann} illustrates these steps for the 2 dimensional case.\n\nIt is interesting to note the simplicity of this method and its similarity to convolutional neural networks. In fact, if we treat the lattice state $f$ as a tensor of size ($n_x$,$n_y$,9), as we do for the remainder of this paper, the streaming operator can be mimicked with a 3 by 3 convolution and the collision step can be performed with a 1 by 1 convolution (D2Q9). This offers a unique way to interpret our method. In some sense, we are taking a large convolutional neural network and compressing it onto a much smaller and more memory efficient network. With this mental picture in mind, we now describe our approach.\n\n\\subsection{Proposed Architecture}\n\nFigure \\ref{fig_1} shows a sketch of the model. The figure can be understood by following the arrows starting from the flow state $f_t$ and the boundary $b$. We treat $f_t$ as a tensor with shape ($n_x,n_y,9$) for the 2D case and ($n_x,n_y,n_z,15$) for the 3D case. The boundary is treated as a binary tensor of shape ($n_x,n_y,1$) and ($n_x,n_y,n_z,1$) with the value being 1 if the cell is solid. Bellow we walk through each step of our method.\n\nFirst, we compress both the state of the fluid simulation $f_t$ and the boundary conditions $b$ using two separate neural networks $\\phi_{enc}$ and $\\phi'_{enc}$ respectively. The result from $\\phi_{enc}$ is a compressed representation of the flow $g_t$ and the result of $\\phi'_{enc}$ are two tensors $b_{mul}$ and $b_{add}$ of equal size to $g_t$. These three tensors represent the entirety of the compressed state of the simulation.\n\nIn a Lattice Boltzmann solver, the boundary conditions are used at each time-step to add bounce back to the streaming step. In a similar way, our model applies the compressed boundary to the compressed state every time-step. We do this in the following way,\n\\begin{equation}\n  g_t = (g_t \\odot b_{mul}) + b_{add}\n\\end{equation}\nThis method proved extremely successful at keeping the boundary information firmly planted through the duration of the simulation. This method of applying boundary conditions was inspired by \\cite{vondrick2016generating} where they use a similar method to combine foreground and background information in video prediction. After the boundary is applied to $g_t$, we can run the state through another neural network to emulate the dynamics, i.e. $\\phi_{comp}:g_{t} \\rightarrow g_{t+1}$. Each step of $\\phi_{comp}$ is equivalent to $n$ time-steps of the Lattice Boltzmann solver. For example, in the 2 dimensional simulation, each step of $\\phi_{comp}$ mimics 120 steps of the Lattice Boltzmann solver. Once $g_t$ is computed, we can extract out the generated state of the simulation with a decoder network $\\phi_{dec}$. \n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.28]{.\/figs\/fig_1.png}}\n\\caption{Illustration of the Lat-Net architecture}\n\\label{fig_1}\n\\end{figure}\n\n\\subsection{Network Implementation Details}\n\nWe implement 2 networks trained on 2 dimensional and 3 dimensional lattice simulations. The encoder, compression mapping and decoder pieces of the 2D network are each a series of 3 by 3 residual blocks with the sequences 4x(down res-res)-res, 4x(res), and 3x(transpose conv-res-res)-transpose conv, respectively \\cite{he2016deep}. For the 3D network, the sequences are 2x(down res)-res, 3x(res), and (transpose conv-res-transpose conv) where 3 by 3 by 3 convolutions are used. The down residual blocks are created by changing the first convolution to have kernel size 4 by 4, stride 2 and double the filter size. The up sampling is achieved with transpose convolutions of kernel size 4 by 4, stride 2 and half the filter size. For the last residual block on the 3D network, the filter size is halved once again.\n\nAs mentioned above, each network is kept entirely convolutional. Fluid flow is inherently spatially correlated so using convlutional layers allows this spatial information to be preserved. Keeping the network convolutional also allows different input sizes to be used. This is how our model is able to train on small simulations and then generate larger simulations.\n\nResidual connections have been used in many deep learning architectures with much success. Adding residual connections allows for much deeper networks to be trained, often resulting in improved results \\cite{he2016deep}. When training our model, it is necessary to unroll the compression network over several time-steps. This has the same effect as making the network deeper. For this reason, it seems logical to take advantage of this network architecture. We have seen that removing these residual connections results in much slower convergence and worse accuracy.\n\n\\subsection{Training Details}\n\nLat-Net is trained by unrolling the network and comparing the generated flow with the true. Our loss function is Mean Squared Error (MSE) with Image Gradient Difference Loss (GDL) \\cite{mathieu2015deep}. The GDL is multiplied by $\\lambda_{GDL}$ and then added to the MES. In all our experiments, $\\lambda_{GDL}$ is set to $0.2$. Removing the GDL tended to produce less accurate models. Lat-Net is unrolled 5 time-steps and then trained with the Adam optimizer \\cite{kingma2014adam}.\n\n\\section{Experiments}\n\nIn this section, we describe our experiments testing Lat-Net on a variety of problems. Our experiments are designed to test our models ability to generate large simulations as well as its transferability to new boundary geometries. We also explore computation time and working memory usage. Finally, we briefly show results applying this method to electromagnetic simulations.\n\n\\subsection{Dataset Generation}\nIn order to train and test our model, we generate sets of fluid and electromagnetic simulations. All simulations were generated with the MechSys library \\cite{mechsys}.\n\nThe train set for the 2D fluid simulations consists of 50 runs of grid size 256 by 256 and 9 directional flows in the lattice Boltzmann solver (D2Q9 scheme)\\cite{guo2013lattice}. The simulations use periodic boundary conditions on top and bottom as well as uniform inlet flow and outlet flow of 0.04 from the left and right. 8 Objects are placed randomly with height and width sizes ranging from 140 to 20 cells. The test set for the 2 dimensional simulations consists of 10 runs of size 256 by 256, and 5 runs of size 1024 by 1024 with the same boundary conditions and object densities. We also generate a test set of size 256 by 512 with vehicle cross sections as objects. There are 28 cross sections used ranging from trucks to minivans. For all 2 dimensional simulations, the ratio of network steps to Lattice Boltzmann steps is 1 to 120.\n\nThe train set for the 3D fluid simulations consists of 50 runs of grid size 40 by 40 by 160 and 15 directional flows in the lattice Boltzmann solver (D3Q15 scheme)\\cite{guo2013lattice}. Similar to the 2D simulations, periodic boundary conditions are used with same inlet and outlet flow. 4 spheres are randomly placed with height and width 24. The reason different object geometries and sizes were not explored was due to the fact that smaller objects or objects with complex geometries tended to have too course a resolution for the lattice Boltzmann solver and larger objects required too large a simulation size. The test set comprises 10 runs of 40 by 40 by 160 and 5 runs of 160 by 160 by 160 simulations with the same object density. The ratio of network steps to Lattice Boltzmann steps is 1 to 60.\n\nThe train set for the electromagnetic simulations are grid size 256 by 256 with periodic boundaries. An electromagnetic wave is initialized in the top of the simulations and proceeds to interact with randomly placed objects of different dielectric constants. When the wave hits these objects, the reflection and refraction phenomenon is seen. The test set consists of simulations of size 512 by 512 with the same object density.\n\n\\subsection{Generating Simulations}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.302]{.\/figs\/256x256_2d_flow_image.png}}\n\\subfigure{\\includegraphics[scale=0.302]{.\/figs\/1024x1024_2d_flow_image.png}}\n\\subfigure{\\includegraphics[scale=0.302]{.\/figs\/160x160x160_3d_flow_image.png}}\n\\caption{A visual comparison of flows generated by Lat-Net and the Lattice Boltzmann method. Each figure shows the Generated, True, and Difference of the flow for various time-steps.}\n\\label{2d_image_plot}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.280]{.\/figs\/256x256_2d_error_plot.png}}\n\\subfigure{\\includegraphics[scale=0.280]{.\/figs\/1024x1024_2d_error_plot.png}}\n\\subfigure{\\includegraphics[scale=0.280]{.\/figs\/40x40x160_3d_error_plot.png}}\n\\subfigure{\\includegraphics[scale=0.280]{.\/figs\/160x160x160_3d_error_plot.png}}\n\\caption{ Comparison plot of the flows generated by Lat-Net and the Lattice Boltzmann method. Each plot shows the average mean squared error of the true and generated generated along with the average divergence of the velocity vector field for both simulations. The standard deviation is also displayed. In addition, the calculated values for drag and flux are displayed for a single simulation run. For the 1024 by 1024 simulation, the flow produced by the Lattice Boltzmann solver tended to produce instabilities resulting in the chaotic divergence observed.}\n\\label{2d_error_plot}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.74]{.\/figs\/car_2d_flow_image.png}}\n\\subfigure{\\includegraphics[scale=0.28]{.\/figs\/256x512_2d_error_plot.png}}\n\\caption{Comparison of generated flows for the vehicle cross section dataset. The images show the generated and true flow at step 100 for 3 different cars in the dataset. The plot shows the same values as described in \\ref{2d_error_plot}.}\n\\label{car_dataset}\n\\end{figure}\n\n\nA key component of our model is its ability to generate larger simulations then those trained on. To test its effectiveness in doing so, we compare the accuracy of generated 2D and 3D simulations to ground truth simulations.\n\nComparing the accuracy of our simulations require some consideration. The naive approach is to compare the MSE between the generated and true simulation at various time-steps. The problem with this approach is that fluid flow is a chaotic dynamical system and small perturbations in flow quickly compound leading to dramatic differences at latter times. For this reason, we compare a variety of metrics in evaluating the generated flows accuracy. Similar to \\cite{tompson2016accelerating}, we compare the divergence of the generated and true velocity vector field to test our models stability. We also compare the computed values of drag and flux. Flow simulations are often run to calculate such values so comparing this is a strong indicator of our models real world applicability. The drag is calculated directly from the lattice state via the momentum transfer method \\cite{guo2013lattice}. The flux value is the average of the flux in each non-boundary cell. These values can be used to calculate important quantities such as the drag coefficient and Reynolds number. Lastly, we visually inspect the produced flow to check for instabilities and blurring effects.\n\nIn figure \\ref{2d_error_plot}, we see the predicted values for different grid sizes in 2D and 3D simulations. In the 2D simulations, Lat-Net is able to effectively transfer to larger domain sizes with very similar calculated values of drag and flux. The generated flow also maintains its stability even after hundreds of steps. In the 3D simulation we see that, while our model predicts realistic values for the 40x40x160 simulation, it tends to have a slight bias in the direction of flow that manifests itself in the 160x160x160 simulation.\n\nWhen visually inspecting our produced flows (figure \\ref{2d_image_plot}), we see a slight blurring effect but overall similar structure in the 2D flows. We attribute this blur effect to the dimensionality reduction and use of MSE. This can possibly be overcome with the use of generative adversarial network \\cite{goodfellow2014generative} where the loss is derived from a discriminator network. Another solution may be to craft a loss function that takes advantage of the statistical properties of flow \\cite{kim2008wavelet}. We leave these pursuits to future work.\n\nThere is a distinct difference in the generated and true flow for the 3D 160x160x160 simulation. While the generated flow appears accurate close to the objects, in regions far between objects the network tends to underestimate the flow velocity. We believe this is due to these types of regions not being present in the train set and is the probable cause for the biases seen in the drag and flux. As mentioned above, our 3D train set is limited due to memory constraints and so developing a diverse train set to overcome this proved difficult.\n\nThe boundaries used in the above evaluation are drawn from the same distribution as the train set. This motivates the question of how our model performs on drastically different geometries. To test this, we apply our model to predicting flow around vehicle cross sections. Surprisingly, even though are model is only trained on flows around simple shapes (ovals and rectangles) it can effectively generalize to this distinctly different domain. In figure \\ref{car_dataset}, we see the predicted flows are quite similar but with the same blurring effect. Calculating the same values as above, we see the flow is stable and produces similar drag and flux.\n\n\\subsection{Computation and Memory Compression}\n\n\\begin{table}[]\n\\small\n\\caption{Computation Time of Network} \\label{compute_times}\n\\centering\n\\begin{tabular}{|c|cccccc|}\n\\hline\nSimulation    & Comp. Size       & Comp. Mapping       & Full State  & Plane      & Line       & Point \\\\ \\hline\n(1024, 1024, 9)  & (64, 64, 128)                 & 2.7 ms            & 36.2 ms   & NA         & 6.7 ms   & 6.6 ms \\\\\n(160, 160, 160, 15) & (40, 40, 40, 64)                 & 23.1 ms           & 272.1 ms  & 38.2 ms  & 25.6 ms  & 24.1 ms  \n\\\\ \\hline\n\\end{tabular}\n\\label{computation_table}\n\\end{table}\n\n\nIn this section, we investigate the computational speed-up of our model. The standard performance metric for Lattice Boltzmann Codes is Million Lattice Updates per Second (MLUPs). This metric is calculated by the following equation,\n\\begin{equation}\n  MLUP = \\frac{n_x \\times n_y \\times n_z \\times 10^{-6}}{Compute \\ Time}\n\\end{equation}\n where $n_x$, $n_y$, and $n_z$ are the dimensions of the simulation. For 3 dimensional simulations like the ones seen in this paper, a speed of 1,200 MLUPS can be achieved with a Nvidia K20 GPU and single precision floats \\cite{januszewski2014sailfish}. We use this as our benchmark value to compare against.\n\nThe computation time and memory usage of the encoder can be neglected because this is a one time cost for the simulation. In addition, if the simulation is started with uniformly initialized flow as seen in our experiments, the computation to compress the flow is extremely redundant and can easily be optimized.\n\nAs seen in table \\ref{computation_table}, the computation time of the compression mapping is 23.1 ms for a 3D simulation of grid size 160 by 160 by 160. Because each step of the compression mapping is equivalent to 60 Lattice Boltzmann steps, this equates to 10,600 MLUPS and a roughly 9x speed increase (a similar speed-up is seen with the 2D simulation). This does not give a complete picture though. Once the compressed states have been generated, the flow must be extracted with the decoder. Unfortunately, this requires considerable amounts of computation and memory because it involves applying convolutions to the full state size. Fortunately, there are ways around this. In many applications of CFD, it is not necessary to to have the full state information of the flow at each time-step. For example, calculating the drag only requires integrating over the surface of the object. By using the convolutional nature of the decoder, we can extract specif pieces of the flow without needing to compute the full state. In table \\ref{computation_table}, we show computation times for extracting flow information of a plane, line and single point. While these computations can still be somewhat expensive, they do not necessarily need to be performed at every time-step and require very little working memory. \n\nUnfortunately, there are some measurements that do require the full state information to compute such as the average flux seen in our tests. Our method is currently unable to handle these without requiring high run-times and large working memory. A possible solution is training a separate neural network that takes in the compressed state and predicts the desired measurement. This would negate the need to extract out the full state and keep memory usage low. We leave this and similar ideas for future work.\n\nWhile Lat-Net compresses the simulation state size by more then an order of magnitude, the working memory requirements for the compression network must be considered. A typical GPU based Lattice Boltzmann solver requires around 1.5 times as much working GPU memory as the memory size of the lattice \\cite{januszewski2014sailfish}. For example, the maximum sized D3Q15 lattice that can fit on a GPU with 8 Gigabytes is $446^3$. We have observed that in our implementation of Lat-Net, the maximum 3D compression network we can run with an 8 Gigabyte GPU corresponds to a lattice size of $672^3$. This represents a 3.4x efficiency gain in working memory usage. While this is certainly a nontrivial gain, we feel that further improvements can be realized with a more memory efficient implementation of the compression mapping.\n\n\n\\subsection{Electromagnetic Results}\n\nFinally, we illustrate the generality of our method by applying it to electromagnetic simulations. The same neural network architecture is used as in the 2D flow simulations with the only difference being the filter size on the compression is half of that in the flow network. The loss is kept identical however the lattice values are scaled up by a factor of 10 so they are on the same range as the flow lattice values. In figure \\ref{em_dataset}, we see very similar waves formed with the same reflection and refraction.\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.28]{.\/figs\/512x512_2d_em_image.png}}\n\\subfigure{\\includegraphics[scale=0.14]{.\/figs\/512x512_2d_em_error_plot.png}}\n\\caption{ Comparison of generated Electromagnetic fields. The images show the true and generated magnetic field a various time-steps. The reflection and refraction phenomena can clearly be seen in both. The plot shows the mean squared error of the true and generated simulation.}\n\\label{em_dataset}\n\\end{figure}\n\n\n\\section{Conclusion}\n\nFluid Simulations are incredibly important for a variety of tasks however they are extremely computation and memory intensive. In this work, we have developed a unique method to overcome this using deep neural networks. We have demonstrated it is capable of accurately reconstructing a variety of simulations under different conditions with significantly less computation and memory. We have also shown that our method can be readily applied to other physics simulations such as electromagnetic simulations. While our method has proved successful on the problems in this paper, there is still significant room for improvement. A loss function that either takes into account the statistical nature of the flow or uses recent advances in GANs could produce shaper, more realistic flow. Training a network to extract desired measurements from the compressed state such as average flux would overcome the current memory limitation for such a task. We leave these and other improvements for future work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Light-cone Co-ordinates}\\label{appendixA}\nTo perform the light-cone expansion one relate the meson's 4-momentum $P_\\mu$, polarisation vector $e^{(\\lambda)}$ and the coordinate $x_\\mu$ to two light-like vectors $p_\\mu$ and $z_\\mu$. We have the usual relations\n\\begin{equation}\np^2=0, \\hspace{1in} z^2=0\\,,\n\\end{equation}\nand\n\\begin{equation}\nP^2=m_{K^*}^2, \\hspace{1in} e^{(\\lambda)}\\cdot e^{(\\lambda)} =-1,\\hspace{1in} P\\cdot e^{(\\lambda)} =0,\n\\end{equation}\nso that the limit $m_{K^*}^2\\to 0$ gives $p \\to P$ and $x^2 \\to 0$ gives $z \\to x$. From this it follows that\n\\begin{eqnarray}\\label{lccoords1}\n  z_\\mu &=& x_\\mu-P_\\mu\\,\\frac{1}{m_{K^*}^2}\\left[x\\cdot P -\\sqrt{(x\\cdot P)^2-x^2m^2_{K^*}}\\,\\right]\n= x_\\mu\\left[1-\\frac{x^2m_{{K^*}}^2}{4(z\\cdot p)^2}\\right]\n-\\frac{1}{2}p_\\mu\\,\\frac{x^2}{p\\cdot z}+ \\mbox{\\cal O}(x^4)\\,,\\nonumber\\\\\np_\\mu &=& P_\\mu-\\frac{1}{2}\\,z_\\mu\\, \\frac{m^2_{K^*}}{p\\cdot z}\\,.\n\\end{eqnarray}\nThe meson's polarization vector $e^{(\\lambda)}$ can be decomposed into projections onto the two light-like vectors and the orthogonal plane\n\\begin{eqnarray}\\label{lccoords2}\n e^{(\\lambda)}_\\mu &=& \\frac{e^{(\\lambda)}z}{p\\cdot z}\\, p_\\mu + \\frac{e^{(\\lambda)} p}{p\\cdot z}\\, z_\\mu +\n                     e^{(\\lambda)}_{\\perp\\mu} =  \\frac{e^{(\\lambda)} z}{p\\cdot z}\\left( p_\\mu -\\frac{m^2_{K^*}}{2p\\cdot z}\\, z_\\mu \\right)+e^{(\\lambda)}_{\\perp\\mu}\\,,\\nonumber\\\\\n&=&(e^{(\\lambda)} \\cdot x)\\frac{P_\\mu (x\\cdot P)-x_\\mu m^2_{K^*}}{(x\\cdot P)^2 -x^2 m^2_{K^*}}+ e^{(\\lambda)}_{\\perp\\mu}\\,.\n\\end{eqnarray}\nWe also need the projector $g_{\\mu\\nu}^\\perp$ onto the directions orthogonal to $p$ and $z$\n\\begin{equation}\ng^\\perp_{\\mu\\nu} = g_{\\mu\\nu} -\\frac{1}{p\\cdot z}(p_\\mu z_\\nu+ p_\\nu z_\\mu)\\,.\n\\end{equation}\nSome useful scalar products are\n\\begin{eqnarray}\nz\\cdot P = z\\cdot p &=& \\sqrt{(x \\cdot P)^2 - x^2 m^2_{K^*}}\\,,\\nonumber\\\\\np \\cdot e^{(\\lambda)}&=& -\\frac{m^2_{K^*}}{2 pz} z \\cdot e^{(\\lambda)}\\,,\\nonumber\\\\\nz \\cdot e^{(\\lambda)}&=&x \\cdot e^{(\\lambda)}\\,.\n\\end{eqnarray}\nWill use the notations\n\\begin{equation}\\label{note1}\na_z\\equiv a_\\mu z^\\mu, \\qquad b_p\\equiv b_\\mu p^\\mu,\\qquad \\slash{c}\\equiv \\gamma_\\mu c^\\mu,\\qquad d_\\mu^\\perp\\equiv g_{\\mu \\nu}^\\perp d^\\nu,\n\\end{equation}\nfor  arbitrary Lorentz vectors $a_\\mu$, $b_\\mu$, $c_\\mu$ and $d_\\mu$ and \n\\begin{equation}\nx^\\mu = x_- n^\\mu + x_+ \\bar{n}^\\mu +x^\\mu_\\perp\\,,\n\\end{equation}\nfor null unit vectors $n^2=\\bar{n}^2=0$ and $n \\cdot \\bar{n} =1$. The following notation is also used:\\begin{equation}\na_+=a\\cdot z\\,,\\qquad a_- =\\frac{a\\cdot p}{p\\cdot z}\\,,\\qquad a^{\\perp}_\\mu\n= a_\\mu - \\frac{a_- p_\\mu}{p\\cdot z}-a_+ z_\\mu\\,.\n\\end{equation}\n\n\n\n\n\\chapter{Useful formulas for sum rule determinations}\\label{appendixB}\n\\section{Loop Integrals}\nHere we summarise the loop integrals needed for calculating the twist-3 correlation\nfunctions in Chapter~\\ref{chapter4_det}.  At one loop, one has ($z^2=0$)\\cite{Ball:2003sc}\n\\begin{eqnarray}\n\\int \\left[d^L k\\right]  e^{i f_k k\\cdot z} \\,\\frac{(k\\cdot z)^n}{(k^2)^a\n  ((k-p)^2)^b} & = & (-1)^{a+b} \\left(-p^2\\right)^{D\/2-a-b} (p\\cdot z)^n\n  \\,\\frac{\\Gamma(a+b-D\/2)}{\\Gamma(a)\\Gamma(b)}\n\\nonumber\\\\\n&& \\times\\int_0^1 dw\\,\n  e^{i(1-w) f_k p\\cdot z}\\, w^{D\/2-1-b} (1-w)^{D\/2+n-1-a}\\,,\\nonumber\\\\ \\label{E.1}\n\\end{eqnarray}\nwhere the integration measure is defined as  $d^D k = i\/(4\\pi)^2 \\left[d^L k\\right]$ and $f_k$ is an arbitrary numerical factor, which in the cases considered in Chapter~\\ref{chapter4_det} is either $v$ or $\\bar v$. One also needs the integral\n\\begin{eqnarray}\n\\lefteqn{\\int \\left[d^L l\\right]  e^{i f_l l\\cdot z} \\,\\frac{(l\\cdot p)(l\\cdot z)^j}{(l^2)^c\n  ((l-k)^2)^d}}\n\\nonumber\\\\\n& = & (-1)^{\\frac{D-4}{2}} \\left(k^2\\right)^{D\/2-c-d} (k\\cdot p) (k\\cdot z)^j\n  \\,\\frac{\\Gamma(c+d-D\/2)}{\\Gamma(c)\\Gamma(d)}\\int_0^1 du\\,\n  e^{i(1-u) f_l k\\cdot z}\\, u^{D\/2-1-d} (1-u)^{D\/2+j-c}\n\\nonumber\\\\\n&&{}+(-1)^{\\frac{D-4}{2}} \\left(k^2\\right)^{D\/2+1-c-d} (p\\cdot z)(k\\cdot z)^{j-1}\n  \\,\\frac{\\Gamma(c+d-D\/2-1)}{2\\Gamma(c)\\Gamma(d)}\n\\nonumber\\\\\n&&{}\\times\\int_0^1 du\\,\n  e^{i(1-u) f_l k\\cdot z}\\, u^{D\/2-d} (1-u)^{D\/2-1+j-c}\\left( j + i f_l\n  (1-u) (k\\cdot z)\\right)\\,.\\label{E.2}\n\\end{eqnarray}\nTwo-loop integrals are obtained by combining the above one-loop integrals.\n\n\\section{Borel Subtraction}\nTo derive the sum rules from $\\widetilde{\\pi}_{3;V}^\\parallel$, $\\pi_{3;V}^\\parallel$ and $\\pi_{3;V}^\\perp$ we use the relation\n\\begin{equation}\n\\frac{1}{\\pi}\\textrm{Im}_s \\left[-q^2-i0\\right]^\\alpha=\\frac{s^\\alpha}{\\Gamma(-\\alpha)\\Gamma(1+\\alpha)}\\Theta(s)\\,,\n\\end{equation}\nwhere $s=-q^2$, to find the imaginary part. Using the following notation for the Borelisation and continuum subtraction procedure\n \\begin{equation}\n\\hat{\\mathcal{B}}_{sub}\\left[X\\right]=\\int^{s_0}_0ds\\,e^{-s\/M^2}\\frac{1}{\\pi} \\textrm{Im}_s X\\,,\n\\end{equation}\nand the definition of the Borel transform (\\ref{borel1}) allows one to write the required results as\n\\begin{eqnarray}\n\\qquad\\hat{\\mathcal{B}}_{sub}\\left[\\frac{1}{(q^2)^\\alpha}\\right]&=&\\frac{(-1)^\\alpha}{(\\alpha-1)! (M^2)^{\\alpha-1}} \\,,\\qquad\n\\hat{\\mathcal{B}}_{sub}\\left[\\ln(-q^2)\\right]=-M^2+\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}\\,,\\nonumber\\\\\n\\hat{\\mathcal{B}}_{sub}\\left[q^2\\ln(-q^2)\\right]&=&-M^4+\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}s\\,,\\nonumber\\\\\n\\hat{\\mathcal{B}}_{sub}\\left[\\frac{\\ln(-q^2)}{q^2}\\right]&=&\\gamma_{E}-\\ln M^2+\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}\\frac{1}{s}\\,,\\nonumber\\\\\n\\hat{\\mathcal{B}}_{sub}\\left[\\frac{\\ln(-q^2)}{q^4}\\right]&=&\\frac{1}{M^2}\\left(1-\\gamma_E +\\ln M^2\\right)+\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}\\frac{1}{s^2}\\,,\\nonumber\\\\\n\\hat{\\mathcal{B}}_{sub}\\left[\\ln(-q^2)^2\\right]&=&2M^2\\left(\\gamma_E-\\ln M^2\\right) +2\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}\\ln s\\,,\n\\end{eqnarray}\nwhere $\\gamma_E$ is Euler's constant. \n\n\\section{Input Parameters}\nFor the twist-2 and twist-3 DA parameter sum rule determinations of Chapter~\\ref{chapter4_det} we use the following input parameters:\n\\begin{table}[ht]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{|r@{\\:=\\:}l||r@{\\:=\\:}l|}\n\\hline\n\\langle \\bar q q\\rangle & (-0.24\\pm0.01)^3\\,\\mbox{GeV}^3 & \\langle \\bar s s\\rangle & (1-\\delta_3)\\,\\langle \\bar q q\\rangle\\\\\n\\langle \\bar q \\sigma g_sG q\\rangle & m_0^2\\,\\langle \\bar q q\\rangle &  \\langle \\bar s \\sigma g_sG s\\rangle & (1-\\delta_5)\\langle \\bar q \\sigma g_sG q\\rangle\\\\[6pt]\n\\displaystyle \\left\\langle \\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle & (0.012\\pm 0.003)\\, \n{\\rm GeV}^4 & \\multicolumn{2}{l|}{}\\\\[6pt]\\hline\n\\multicolumn{4}{|c|}{m_0^2 = (0.8\\pm 0.1)\\,{\\rm GeV}^2~,\\qquad \\delta_3\n  = 0.2\\pm 0.2, \\qquad \\delta_5 = 0.2\\pm 0.2}\\\\\\hline\n\\multicolumn{4}{|c|}{\\overline{m}_s(2\\,\\mbox{GeV}) = (100\\pm\n20)\\,\\mbox{MeV}\\qquad\\longleftrightarrow\\qquad\\overline{m}_s(1\\,\\mbox{GeV})\n= (133\\pm 27)\\,\\mbox{MeV}}\\\\\n\\multicolumn{4}{|c|}{\\overline{m}_q(\\mu) = \\overline{m}_s(\\mu)\/R\\,,\n  \\qquad R = 24.6\\pm 1.2}\\\\\\hline\n\\multicolumn{4}{|c|}{\\alpha_s(M_Z) = 0.1176\\pm 0.002  ~\\longleftrightarrow~ \n\\alpha_s(1\\,\\mbox{GeV}) = 0.497\\pm 0.005}\\\\\\hline\n\\end{array}\n$$\n\\renewcommand{\\arraystretch}{1}\n\\addtolength{\\arraycolsep}{-3pt}\n\\caption[Summary of input parameters for Chapter 4.]{\\small Input parameters for sum rules at the renormalisation scale $\\mu=1\\,$GeV. The value of $m_s$ is obtained from unquenched lattice calculations with $N_f=2$ flavours  as summarised in \\cite{mslatt}, which agrees with the results from QCD sum rule calculations \\cite{jamin}. $\\overline{m}_q$ is taken from chiral perturbation theory \\cite{chPT}. $\\alpha_s(M_Z)$ is the PDG  average \\cite{Yao:2006px}.}\n\\label{QCDSRinput}\n\\end{table}\n\nTo evaluate the sum rules for the three-particle twist-3 DA parameters we use the following values of the continuum threshold $s_0$\n\\begin{eqnarray}\ns_0^\\parallel (\\rho) &=& (1.3\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad \ns_0^\\parallel (K^*) = (1.3\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad\ns_0^\\parallel (\\phi) = (1.4\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad \\nonumber\\\\\ns_0^\\perp (\\rho) &=& (1.5\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad \ns_0^\\perp (K^*) = (1.6\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad\ns_0^\\perp (\\phi) = (1.7\\pm 0.3)\\,{\\rm GeV}^2\\,. \\nonumber\\\\\n\\end{eqnarray}\nThe threshold for the $\\rho$ channel is from \\cite{Shifman:1978bz}. \n\n\n\\chapter*{Introduction}\\label{chapter0_intro}\\addcontentsline{toc}{chapter}{\\protect\\numberline{Introduction\\hspace{-96pt}}} \nOne only has to ask the question ``why?'' a handful of times before one reaches the answer ``I don't know'', regardless of the topic considered and regardless of the person asked.  It is safe to say, however, almost all questions of the structure of matter at the smallest of distances leads one directly to, or at least through, the field of modern particle physics. The beginnings of our understanding of the physical world harks back to the dawn of scientific reasoning in the ancient world; logic and reasoning were applied with the aim of describing the behaviour of physical systems in terms of simple universal axioms, a philosophy which still holds strong today. Through experimentation and the language of mathematics the scientific method has driven back the edge of ignorance to frontiers unimaginable to those physicists of 100 years ago, let alone the natural philosophers of millennia ago. The present ``coal face'' is known as the Standard Model \\cite{weak, QCD} which describes three of the four known forces of nature -- electromagnetism, and the weak and strong nuclear forces -- in one unifying framework. \n\nFrustratingly, the Standard Model does not explain many of the things which it encompasses; it does not provide an origin for CP violation but only gives a parameterisation, nor does it explain why there are three generations of quarks and leptons, or their hierarchy of masses.  All attempts to bring gravity into the fold have so far failed, however, whatever theory lies beyond must yield the Standard Model as some limiting case.\n\nThe Standard Model has been scrutinised relentlessly since its inception. Remarkably, nearly without fail it has held its ground over the entire breadth of its theoretical reach and so the task of finding new ways to probe its structure requires ever more the creativity and ingenuity of both theorists and experimentalists alike.  Novel experimental signatures, against which to pit theory, must be used to maximum potential. From a theoretical standpoint there are still many challenges to be met, especially in preparation for the next generation of collider experiments now just round the corner. Particularly, the control and reduction of the theoretical uncertainty of Standard Model predictions is of paramount importance as only then can one hope to be in a position to discern signs of new physics from that of the Standard Model background.\n\nSome of its most challenging tests of the Standard Model fall in the field of heavy-flavour physics, within which $B$ physics has proven itself to be rich and fertile. Today it is an area of high activity with many success stories, including the recent measurement of the $B_s^0$-$\\bar{B}_s^0$ mass difference $\\Delta m_s$ at the Tevatron \\cite{Abulencia:2006ze}. Moreover, two dedicated ``$B$-factories'', Belle at KEK \\cite{:2000cg} and \\textsc{BaBar} at SLAC \\cite{Aubert:2001tu}, have measured a range of observables, such as branching fractions and CP asymmetries, of a vast number of $B$ decay modes. Looking to the future, the $B$ physics community eagerly await the forthcoming LHCb experiment, and beyond that so-called ``superflavour factories'' \\cite{superB} have been championed with the aim of probing rare $B$ decays to extract CP violation parameters to much higher levels of accuracy. It is imperative to find tests of the Standard Model  which may be observed in these up-and-coming experiments \\cite{Gershon:2006mt} and promising modes include the rare decays $B\\to V \\gamma$ and $B\\to K \\mu^+\\mu^-$. \n\nThe strict pattern of CP violation of the Standard Model finds its origin in the Cabbibo-Kobayashi-Maskawa (CKM) matrix \\cite{Cabibbo:1963yz,Kobayashi:1973fv}. CP violation was discovered in $B$ physics via the decay mode $B_d^0\\to J\/\\psi K_S^0$ and found to be large, in contrast to $K$ decays where the violation is tiny. The possible largeness of CP violation in $B$ decays offers promising ways to detect new physics indirectly via CP violating observables testing the CKM paradigm. \n\nTheoretically, central to the description of $B$ decays is the disentanglement of the weak decay process from strong interaction effects leading to a low-energy effective Hamiltonian in which the physics at a scale $\\mathcal{O}(M_W)$ is well under control. Achieving this goal for the wide range of $B$ decays of interest has only been possible through huge calculational effort; the availability in the literature of Wilson coefficients at next-to-leading-order, and in some cases next-to-next-to-leading-order, is testament to this.  Furthermore, the theoretical description of the matrix elements of effective $B$ decay operators has been hugely improved through QCD factorisation methods. We discuss and make use of one such framework, namely that introduced by Beneke, Buchalla, Neubert and Sachrajda \\cite{Beneke:1999br, Beneke:2000ry,Beneke:2001ev}. The so-called BBNS approach showed, to leading-order in a $1\/m_b$ expansion, that the $\\alpha_s$ corrections beyond naive-factorisation of a large class of non-leptonic $B$ decay matrix elements are calculable in terms of $B$ transition form factors and meson light-cone distribution amplitudes. Armed with the corresponding amplitudes the phenomenologist may construct observables, such as branching ratios, CP asymmetries and isospin symmetries, which may then be compared to experiment. The predictive power of the QCD factorisation framework is jeopardised by a poor understanding of both these non-perturbative QCD quantities and the impact of the generally unknown power-suppressed contributions $\\mathcal{O}(1\/m_b)$; this in part motivates the work of this thesis.\n\nIn this thesis we investigate $\\rm SU(3)_F$-breaking effects in vector meson distribution amplitudes which are crucial in differentiating between the particles $\\rho$, $K^*$ and $\\phi$. The leading non-perturbative DA parameters are determined via the method of QCD sum rules introduced by Shifman, Vainshtein and Zakharov  \\cite{Shifman:1978bz, Shifman:1978by, Shifman:1978bx}. The method provides a prescription for the systematic calculation of non-perturbative QCD parameters,  albeit with an irreducible error $\\sim 20-30\\%$, and constitutes an extremely useful theoretical tool. \n\nThe sum rule results have a direct application in the QCD factorisation description of $B$ decays to $\\rho$, $K^*$ and $\\phi$ mesons. In particular, radiative $B$ decays to vector mesons $B\\to V\\gamma$, are an excellent example of a process potentially sensitive to new physics contributions, as at leading order the decays are mediated at loop level in the Standard Model. We perform a phenomenological analysis of these decays using the QCD factorisation framework of Bosch and Buchalla \\cite{Bosch:2001gv,Bosch:2002bw} including leading power-suppressed corrections for which the updated non-perturbative distribution amplitude parameters find use.  The impact of the power-suppressed corrections on the key decay observables is discussed and leads to a better understanding of the theoretical uncertainty of the QCD factorisation predictions. \n\nAlso, we calculate important contributions to the $B\\to\\eta^{(\\prime)}$ transition form factors via a variant sum rule approach, known as light-cone sum rules, for which distribution amplitudes play a crucial role. The result of the analysis elucidates a major source of theoretical uncertainty of the $B\\to \\eta^{(\\prime)}$ form factor. The result impacts $B\\to K^* \\eta^{(\\prime)}$, for example, where the experimental data and QCD factorisation predictions of the branching ratios are inconsistent.\n\nThe thesis is structured as follows: \n\\begin{itemize}\n\\item{Chapter~\\ref{chapter1_basics} introduces some of the fundamentals of the Standard Model and its application to $B$ physics. We define the QCD Lagrangian and the CKM matrix, introduce CP violation in Standard Model $B$ decays, and briefly discuss the structure of the $\\Delta B =1$ weak effective Hamiltonian. }\n\\item{Chapter~\\ref{chapter2_DAs} covers the definitions of the light-cone distribution amplitudes of the light vector mesons $\\rho$, $K^*$ and $\\phi$. We determine their structure up to twist-3 accuracy and using the conformal expansion and QCD equations of motion express the distribution amplitudes in terms of a finite set of non-perturbative parameters. We extend previous determinations in order to fully differentiate between the three particles by including all G-parity violating contributions and $\\rm{SU}(3)_F$-breaking effects. }\n\\item{Chapter~\\ref{chapter3_SR}  discusses the QCD sum rule method and its extension light-cone sum rules.  The methods allow, amongst other things, the determination of the non-perturbative distribution amplitude parameters and transition form factors respectively, and are very widely applicable in and beyond $B$ physics.}\n\\item{In Chapter~\\ref{chapter4_det} we apply QCD sum rules to determine the leading non-perturbative distribution parameters defined in Chapter~\\ref{chapter2_DAs}. Consistency requires the inclusion of all G-parity violating contributions and $\\rm{SU}(3)_F$-breaking effects to the sum rules, and we extend previous determinations by including higher-order strange quark mass effects and $\\mathcal{O}(\\alpha_s)$ contributions to the quark condensates. We analyse the resulting sum rules and provide updated numerical results for all parameters. The results of this section find immediate application in QCD factorisation and light-cone sum rule descriptions of processes involving these vector mesons.}\n\\item{In Chapter~\\ref{chapter5_eta} we calculate the gluonic flavour-singlet contribution to the semileptonic $B\\to \\eta^{(\\prime)}$ transition form factor in the framework of light-cone sum rules. In doing so we discuss pseudoscalar meson and two-gluon distribution amplitudes. The new contribution is combined with the previous determination of the quark contribution, to complete the theoretical treatment of these form factors. The $\\eta^{(\\prime)}$ system is complicated due to large mixing effects via the $\\rm U(1)_A$ anomaly. We introduce the phenomenological framework of $\\eta$-$\\eta^{\\prime}$ mixing and connect it to the form factor calculation in a consistent manner. The results of this chapter find immediate application in the QCD factorisation description of $B\\to \\eta^{(\\prime)}$ transitions, which in turn, in principle, allow a determination of the CKM matrix element $|V_{ub}|$ from $B\\to\\eta^{(\\prime)} l \\nu$.}\n\\item{Chapter~\\ref{chapter6_QCDF}  introduces the framework of QCD factorisation, which is an important application of meson distribution amplitudes and transition form factors. We briefly discuss the BBNS approach and then go on to discuss the leading contributions to QCD factorisation in the context of $B\\to V \\gamma$ decays.}\n\\item{In Chapter~\\ref{chapter7_rad} we investigate the impact of the relevant, power-suppressed contributions to $B\\to V \\gamma$ beyond the QCD factorisation formula. We include long-distance photon emission from weak annihilation diagrams and soft gluon emission from quark loops. The non-perturbative distribution amplitude parameters determined in Chapter~\\ref{chapter4_det} find use in a light-cone sum rule estimation of the latter. The key observables are the branching ratios, isospin asymmetries and the indirect time-dependent CP asymmetry $S(V\\gamma)$  which, as has been know for some time, forms the basis of a ``null test'' of the Standard Model. Assuming no new physics contributions, we extract the ratio of CKM matrix parameters $\\left|V_{td}\/V_{td}\\right|$ to a competitive degree of accuracy.}\n\\item{We summarise and conclude in Chapter~\\ref{chapter8_conc}.}\n\\end{itemize}\nThe material of Chapters~\\ref{chapter2_DAs} and \\ref{chapter4_det} follows Ref.~\\cite{Ball:2007rt} and the material of Chapters~\\ref{chapter5_eta} and \\ref{chapter7_rad} follows Refs.~\\cite{Ball:2007hb} and \\cite{Ball:2006eu}, respectively. Some of the more bulky equations, and material not necessary in the general flow of reading the thesis, are given in two appendices. \n\\chapter{Fundamentals Of $B$ Physics}\\label{chapter1_basics}\nIn this chapter we begin with the basics of the Standard Model and then go on to  discuss two concepts which are central to the investigations of $B$ physics, and those of this thesis:\n\\begin{itemize}\n\\item{CP violation in the flavour sector, which follows a strict pattern in the Standard Model and can readily be sensitive to new physics;}\n\\item{the $\\Delta B=1$ effective weak Hamiltonian, which we briefly discuss as it is the starting point of many phenomenological studies in $B$ physics.}\n\\end{itemize}\n\n\n\\section{The Standard Model}\nThe \\textit{Standard Model} (SM) \\cite{weak,QCD} is a model of great scope and predictive power. Despite its successes, however, we know it to be incomplete; for example, the recent discovery of neutrino oscillation and the lack of conclusive evidence for the Higgs particle providing two areas of intense theoretical and experimental effort.  The SM describes three of the four known fundamental forces of nature;  the strong force, the weak force and electromagnetism. \\textit{Quantum Chromodynamics} (QCD) is a Yang-Mills gauge theory based on the gauge group $\\rm SU(3)$ and describes the fundamental interactions of the strong interaction as  interactions between quarks and gluons \\cite{Yang:1954ek,Gell-Mann:1964nj,FGM,Fritzsch:1973pi}. The basic QCD Lagrangian is\n\\begin{equation}\n\\mathcal{L}_{\\rm QCD}=\\sum_q \\bar{q}^i \\left(i \\gamma_\\mu \\left(D^\\mu\\right)_{ij}-m_q \\delta_{ij}\\right)q^j -\\frac{1}{4} G^a_{\\mu\\nu} G^{a \\mu\\nu}\\,,\n\\label{basics_eq1}\n\\end{equation}\nwith\n\\begin{equation}\n(D_\\mu)_{ij} = \\delta_{ij} \\partial_\\mu -i g_s (t^a)_{ij} A^a_\\mu\\,, \\qquad G^a_{\\mu\\nu}=\\partial_\\mu A_\\nu^a-\\partial_\\nu A_\\mu^a + g_s f^{abc}A^b_\\mu A^c_\\nu\\,,\n\\label{basics_eq2}\n\\end{equation}\nwhere the sum is over all quark flavours $q$, $i,j=\\{1,2,3\\}$ are colour indices, the $t^d$ are the $3\\times3$ colour matrices with $d=\\{1,\\dots,8\\}$ and $f^{abc}$ are the structure constants.  $G^a_{\\mu\\nu}$ is the gluonic field strength tensor, and $A^a_\\mu$ is the gluon field. We will make use of the notation $(G_{\\mu\\nu})_{ij}=G^a_{\\mu\\nu} (t^a)_{ij}$ and the relation $g_s^2=4 \\pi \\alpha_s$ (and $e^2=4 \\pi \\alpha_{\\rm QED}$). The Lagrangian can alternatively be defined with the replacement $g_s\\to-g_s$ and the sign convention matters for the applications in Chapters~\\ref{chapter4_det} and \\ref{chapter7_rad}.\n\nThe non-Abelian nature of QCD leads to the possibility of gluon self-interaction and the celebrated \\textit{asymptotic freedom} property of QCD \\cite{FGM,Fritzsch:1973pi,Politzer:1973fx,Gross:1973id1}. The coupling tends to zero, giving a theory of free quarks, at asymptotically high energy. On the other hand, at low energy, or large distances, the coupling increases.  At energies for which $\\alpha_s\\gtrsim1$ perturbation theory is not applicable, and one has to resort to \\textit{non-perturbative} methods to determine the effects of QCD. Despite the simplicity of the QCD Lagrangian (\\ref{basics_eq1}) an accurate determination of non-perturbative QCD from first principles, and hence  \\textit{confinement}, poses a major challenge. One such method, based on ideas of Wilson \\cite{Wilson:1974sk}, is that of Lattice QCD, which aims to calculate the QCD action computationally on a grid of discretised spacetime points.  An altogether different, and less rigourous, method is that of QCD sum rules, which encodes non-perturbative effects in terms of non-vanishing vacuum expectation values of operators with the quantum numbers of the vacuum. This method is central to the work in this thesis, and shall be discussed in Chapter~\\ref{chapter3_SR}.\n\nThe electroweak force is the unification of the weak nuclear force and electromagnetism given by the \\textit{Glashow-Salam-Weinberg model}. The model is based on the gauge group $\\rm SU(2)_L \\otimes U(1)_Y$, which is broken by  \\textit{spontaneous symmetry breaking} to yield $\\rm U(1)_Q$ - the gauge group corresponding to \\textit{Quantum Elecrodynamics} (QED).  The weak interaction is mediated by three massive gauge bosons $W^\\pm$ and $Z^0$ and occurs between quarks and leptons. The quarks and leptons are arranged, within the three generations, into left-handed doublets and right-handed singlets under $\\text{SU(2)}_{\\rm L}$\n\\begin{eqnarray}\n Q_{\\rm L}=  \\left(\\begin{array}{c} \n               U\\\\ \n                D \\\\ \n         \\end{array}\\right)_{\\rm L}\\,,\\quad\n          E_{\\rm L}=  \\left(\\begin{array}{c} \n               \\nu_l \\\\ \n                 l^- \\\\ \n         \\end{array}\\right)_{\\rm L} \\,;\\quad\n         U_{\\rm R}\\,,D_{\\rm R}\\,,l^-_{\\rm R}\\,,\n         \\label{basics_eq5}\n\\end{eqnarray}\nwhere the \\textit{weak eigenstates} $U=\\{u,c,t\\}$, $D=\\{d,s,b\\}$ and $l^-=\\{e^-,\\mu^-,\\tau^-\\}$ are the up-type quarks,  down-type quarks and charged leptons respectively. The subscript L (R) represents the left (right)-handed projectors $q_{\\rm L (R)}=\\frac{1}{2}(1\\mp \\gamma_5)q$ which reflect the chiral nature of the weak  interaction. The neutrinos are massless in the SM, and the right handed neutrino does not exist.  The electroweak interactions of the quarks are described by the following Lagrangian, which consists of a \\textit{charged current} ($CC$) and a \\textit{neutral current} ($NC$)\n\\begin{eqnarray}\n   \\mathcal{L}^{\\textrm{ew}} &=& \\mathcal{L}_{CC} +\n   \\mathcal{L}_{NC}\\,,\\nonumber \\\\\n    &=& \\frac{g}{\\sqrt{2}}\\left[J_\\mu^+W^{+\\mu} + J_\\mu^-W^{-\\mu}\\right]\\,,\\nonumber \\\\\n    & +&\n   e\\,\\left[J^{\\textrm{em}}_\\mu A^\\mu  \\right]+ \\frac{g}{\\cos{\\theta_W}}\\left[ \\left(J^3_\\mu - \\sin^2{\\theta_W}J^{em}_\\mu\\right) Z^\\mu   \\right]\\,.\n\\end{eqnarray}\nThe neutral current part of the Lagrangian is made up of the electromagnetic  current $J^{\\textrm{em}}_\\mu$ and neutral weak current $J^3_\\mu$:\n\\begin{equation}\n   J^{\\textrm{em}}_\\mu = Q_U\\,\\bar{U}_{\\rm L}\\gamma_\\mu U_{\\rm L}+Q_D\\,\\bar{D}_{\\rm L}\\gamma_\\mu D_{\\rm L}\\,, \\qquad\n   J^3_\\mu= \\frac{1}{2}(\\bar{U}_{\\rm L} \\gamma_\\mu U_{\\rm L}-\\bar{D}_{\\rm L}\\gamma_\\mu D_{\\rm L})\\,,\n\\end{equation}\nwhere $Q_{U(D)}=2\/3\\,(-1\/3)$ is the electric charge of the $U$ $(D)$ quarks, $\\theta_W$ is the weak mixing angle and $g$ is the electroweak coupling related to the electromagnetic coupling by $e=g \\sin \\theta_W$. Rotating to the basis of \\textit{mass eigenstates} modifies the charged current in the quark sector to\n\\begin{equation}\n   J^+_\\mu =  \\bar{U}_{\\rm L}^m \\gamma_\\mu \\,\\hat{V}_{\\rm{CKM}}\\, D_{\\rm L}^m\\,,\n\\end{equation}\nwhere $\\hat{V}_{\\mathrm{CKM}}$ is the \\textit{Cabbibo-Kobayashi-Maskawa} matrix \\cite{Cabibbo:1963yz,Kobayashi:1973fv} and the superscript $m$ denotes mass eigenstates. The CKM matrix is $3\\times3$ (for three quark generations), unitary, and its off-diagonal entries allow for transitions between the quark generations.  There are no flavour-changing neutral-currents (FCNC) at tree-level in the SM as the neutral currents $ J^{\\textrm{em}}_\\mu$ and $J^3_\\mu$ are invariant under the transformation to the mass eigenbasis, which is known as  the \\textit{Glashow-Iliopoulos-Maiani (GIM) mechanism} \\cite{Glashow:1970gm}. The entries of the CKM matrix are written as\n\\begin{equation}\n   \\hat{V}_{\\mathrm{CKM}} = \\left(\\begin{array}{ccc}\n                    V_{ud} & V_{us} & V_{ub} \\\\\n                    V_{cd} & V_{cs} & V_{cb} \\\\\n                    V_{td} & V_{ts} & V_{tb}  \n              \\end{array}\\right)\\,,\n\\label{basics_eq6}\n\\end{equation}\nand are fundamental parameters of the SM that have to be determined from experiment. Evidently, the matrix has $n^2=9$ parameters $n (n-1)\/2=3$ of which are rotation angles due to its unitarity. The six quark fields in Eq.~(\\ref{basics_eq5}) can be re-phased, up to an overall phase, leaving the Lagrangian invariant and therefore $9-5-3=1$ phase remains giving rise to complex entries -- complex coupling constants. This is the origin of \\textit{CP violation} in the quark sector of the weak interaction. The leptonic sector is described by an analogous mixing matrix which, in the absence of neutrino masses, is given by the unit matrix because all phases can be rotated away.\n\nThe CKM matrix (\\ref{basics_eq6}) is often parameterised to incorporate the constraints of unitarity.\\footnote{The ``standard'' parameterisation of the CKM matrix is in terms of the three mixing angles $\\theta_{ij}$ $(i,j=1,2,3)$ and the CP violating phase $\\delta$ \\cite{Yao:2006px}.} A very useful and convenient parameterisation is the \\textit{Wolfenstein parameterisation} \\cite{Wolfenstein:1983yz} which, along with unitarity, incorporates the experimental observations $|V_{us}|\\ll 1$, $|V_{cb}|\\sim|V_{us}|^2$ and $|V_{ub}| \\ll |V_{cb}|$. It is an expansion in $\\lambda=|V_{us}| \\approx 0.22$, and as such is only approximately unitary at a given order in $\\lambda$:\n\\begin{equation}\n  \\hat{V}_\\mathrm{CKM}=\\left(\n  \\begin{array}{ccc} \n     1-\\frac{\\lambda^2}{2} & \\lambda & A\\lambda^3 (\\rho-i\\eta) \\\\\n     -\\lambda & 1-\\frac{\\lambda^2}{2} & A\\lambda^2 \\\\\n     A \\lambda^3 (1-\\rho-i\\eta) & -A\\lambda^2 & 1\n   \\end{array}\\right)+\\mathcal{O}(\\lambda^4)\\,.\n   \\label{basics_eq7}\n\\end{equation}\nThe matrix is given in terms of the four parameters ($A,\\lambda,\\rho,\\eta$); $A$ and $\\rho^2+\\eta^2$ are order unity and the hierarchy of sizes of elements can be infered from the powers of $\\lambda$. The smallness of $V_{cb}$ and $V_{ub}$ are responsible for the relatively long lifetime of $B$ mesons (and baryons), which facilitates their experimental detection. The unitarity of the CKM matrix gives six equations that equal zero and can be represented as triangles in the complex plane. The most widely used of these relations in $B$ physics  is\n\\begin{equation}\nV_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0\\,,\n\\label{basics_eq8}\n\\end{equation}\nwhich is invariant under phase transformations and is an observable. The above relation is divided by $V_{cd}V_{cb}^*$ to give a triangle in the complex plane with a base of unit length and upper apex at the point $(\\bar\\rho,\\bar\\eta)$\\footnote{The following rescaling proves convenient to the definition of the UT: $\\rho\\to\\bar\\rho=\\rho\\,(1-\\lambda^2\/2)$ and $\\eta\\to\\bar\\eta=\\eta\\,(1-\\lambda^2\/2)$.} known as \\textit{The Unitary Triangle} (UT), see Figs.~\\ref{basics_fig1} and \\ref{basics_fig2}. The sides of the UT are given by\n\\begin{eqnarray}\nR_b &\\equiv& \\frac{|V_{ud}V_{ub}^*|}{|V_{cd}V_{cb}^*|} = \\sqrt{\\bar\\rho^2+ \\bar\\eta^2}=\n\\left(1-\\frac{\\lambda^2}{2}\\right)\\frac{1}{\\lambda}\n\\left|\\frac{V_{ub}}{V_{cb}}\\right|\\,,\\label{basics_eq88}\\\\ \nR_t &\\equiv& \\frac{|V_{td}V_{tb}^*|}{|V_{cd}V_{cb}^*|} = \\sqrt{(1-\\bar\\rho)^2+ \\bar\\eta^2}\\,.\n\\label{basics_eq9}\n\\end{eqnarray}\nThe angles are given by\n\\begin{equation}\n\\alpha\\equiv\\arg\\left(-V_{td}V_{ub}V^*_{tb}V_{ud}^*\\right)\\,,\\quad\\beta\\equiv\\arg\\left(-V_{cd}V_{tb}V^*_{cb}V_{td}^*\\right)\\,,\\quad\\gamma\\equiv\\arg\\left(-V_{ud}V_{cb}V^*_{ub}V_{cd}^*\\right)\\,.\n\\end{equation}\nThe (over) determination of the sides and angles of the UT is a major quest in understanding the SM.   To achieve this goal one must construct decay observables, which can then be matched to experimental results in order to extract values for the desired CKM (or equivalently UT) parameters. Such observables include branching ratios, which may appear simply proportional to a CKM matrix element, and CP asymmetries, which encode the effects of the SM predictions of CP violation, and can also be measured experimentally.\n\\begin{figure}[h]\n$$  \\epsfxsize=0.5\\textwidth\\epsffile{UT.eps}$$\n\\caption[The Unitary Triangle.]{\\small The Unitary Triangle. The determination of the sides $R_b$ and $R_t$ and the angles $\\alpha$, $\\beta$ and $\\gamma$ lead to stringent tests of the Standard Model.} \n\\label{basics_fig1}\n$$  \\epsfxsize=0.5\\textwidth\\epsffile{CKMfitterUT.eps}$$\n\\caption[Constraints on the angles and sides of the Unitarity Triangle.]{\\small Constraints on the angles $\\alpha$, $\\beta$, and $\\gamma$ and sides $R_b$ and $R_t$ of the Unitarity Triangle as imposed from numerous experimental sources. Complied by the CKM fitter group \\cite{global}.} \n\\label{basics_fig2}\n\\end{figure}\n\n\\section{CP Violation In $B$ Decays}\nDoes  the CKM matrix (\\ref{basics_eq6}) account for the CP violation observed in nature? Examining CP violation in $B$ decays allows one to probe the structure of the CKM matrix and is a very promising way to detect the effects of new physics, which many not be expressed through other decay observables. Consequently, the CP properties of  FCNC processes, which are characterised by their potential sensitivity to new physics effects, have been under intense theoretical and experimental investigation for many years. Prime examples of such processes include $B^0$-$\\bar B^0$ mixing (see for example Ref.~\\cite{Ball:2006xx}) and radiative $B$ decays, see Chapter~\\ref{chapter7_rad}.\n\nThe idea that the weak interaction may violate parity was first suggested many years ago by Lee and Yang \\cite{Lee:1956qn}, and quickly confirmed in the $\\beta$ decay of $^{60}$Co by Wu \\textit{et al.} \\cite{Wu:1957my}. The violation of the combined CP symmetry was first observed in the context of $K$ decays in 1964 \\cite{Christenson:1964fg} and it was not until 2001 that it was first observed outside the $K$ system in $B^0_d \\to J\/ \\psi \\,K^0_S$ decays \\cite{Aubert:2001nu,Abe:2001xe}; in both cases the CKM paradigm was upheld.  Recently discoveries in $B$ physics include the measurement by CDF of the mass difference $\\Delta m_s$ \\cite{Abulencia:2006ze}. Some of the most important sources of information about the UT from $B$ physics include: the determination of $\\sin 2 \\beta$ from the ``gold-plated'' decay $B\\to J\/\\psi \\,K_S$; the  extraction of $\\alpha$ from non-leptonic $B$ decays such as $B \\to \\pi\\pi$; the extraction of $|V_{td}|\/|V_{ts}|$ from $B$ mixing and radiative $B$ decays, such as $B\\to V \\gamma$; and the determination of $|V_{ub}|$ from $B\\to\\pi l\\nu$.\n\nThe $B^0_q$-$\\bar B^0_q$ systems, where $q=\\{d,s\\}$, exhibit the phenomenon of particle-antiparticle mixing, which, in the SM is mediated by so-called \\textit{box diagrams} whose amplitudes are $\\sim G_F^2$ and therefore very small.  We do not go into any detail about the theory of neutral state mixing and we restrict ourselves to only the formulas required in this thesis; for more information see Refs.~\\cite{Buras:1998ra,CPV}. State mixing causes, for example, an initially pure beam of $B^0$ mesons to evolve into a time-dependent linear combination of $B^0$ and $\\bar B^0$ mesons. There are four main quantities that describe the $B^0_q$-$\\bar B^0_q$ system and its decays: the width difference $\\Delta \\Gamma_q$, the mass difference $\\Delta m_q$, the CP violating mixing phase $\\phi_q$ and $\\lambda_f$ (not to be confused with the Wolfenstein CKM parameter $\\lambda\\approx 0.22$). One begins by writing the  heavy (H) or light (L) eigenstates of evolution in terms of the flavour states:\n\\begin{equation}\n\\ket{B_{\\rm H}}=p \\ket{B^0}-q \\ket{\\bar{B}^0}\\,,\\qquad \\ket{B_{\\rm L}}=p \\ket{B^0}+q \\ket{\\bar{B}^0}\\,,\n\\label{basics_eq10}\n\\end{equation}\nwith $|p|^2+|q|^2=1$. The ratio $q\/p$ is given in terms of the $B^0_q$-$\\bar B^0_q$  mixing matrix $M_{12}^q$,  by\n\\begin{equation}\n\\left.\\frac{q}{p}\\right|_{q} = \\sqrt{\\frac{(M_{12}^{q})^*}{M_{12}^{q}}} = e^{-i\\phi_{q}}\\,,\n\\label{basics_eq11}\n\\end{equation}\nunder the condition $\\Delta \\Gamma_q \\ll \\Delta m_q$.  Experimentally, there is no evidence for \\textit{mixing-indiced} CP violation in the $B^0_q$-$\\bar B^0_q$ systems, i.e. $\\left|q\/p\\right|_{d,s}\\approx 1$ \\cite{Barberio:2007cr}.   The CP violating mixing phase is given by $ \\phi_q={\\rm arg}\\left[M_{12}^q\\right]$ which in the SM and the Wolfenstein parametrisation of the CKM matrix can be written in terms of the UT angles as\n\\begin{equation}\n\\phi_d \\equiv {\\rm arg}[(V_{td}^* V_{tb})^2] = 2 \\beta\\,,\\qquad \n\\phi_s \\equiv {\\rm arg}[(V_{ts}^* V_{tb})^2] = -2 \\lambda\n\\left|\\frac{V_{ub}}{V_{cb}}\\right|  \\sin\\gamma\\,.\n\\label{basics_eq12}\n\\end{equation}\nBesides mixing-induced CP violation  there also exists \\textit{direct} and \\textit{indirect} CP violation for $B$ and $\\bar B$ decays to a common CP eigenstate $f$. The corresponding time-dependent CP asymmetry is given by\n\\begin{eqnarray}\nA_{CP}(t) &=& \\frac{\\Gamma(\\bar B^0_q(t)\\to f) - \\Gamma( B^0_q(t)\\to   \\bar   f)}{\\Gamma(\\bar B^0_q(t)\\to  f) + \\Gamma( B^0_q(t)\\to \\bar     f)} \\nonumber\\\\\n&=&\\underbrace{S(f)}_{\\rm indirect} \\sin(\\Delta m_q\\, t )-\\underbrace{C(f)}_{\\rm direct}\\cos(\\Delta m_q\\, t)\\,,\n\\label{basics_eq13}\n\\end{eqnarray}\nwhere we have neglected the width difference $\\Delta\\Gamma_q=2 {\\rm Re}\\left[M^q_{12} \\Gamma^{q*}_{12}\\right]\/|M^q_{12}|$. The oscillation frequency is set by the mass difference between the heavy and light states\n\\begin{equation}\n\\Delta m_q = m_H^q-m_L^q=2|M_{12}^{q}|\\,,\n\\label{basics_eq14}\n\\end{equation}\nand the current world averages are \\cite{Barberio:2007cr}:  \n\\begin{equation}\n\\Delta m_d =0.507\\pm 0.004 \\,{\\rm ps}^{-1}\\,,\\qquad\\Delta m_s=17.77\\pm \\overbrace{0.10}^{stat.}\\pm \\overbrace{0.07}^{sys.} {\\rm ps}^{-1}\\,.\n\\label{basics_eq15}\n\\end{equation}\nFinally, if  we define the observable quantity \n\\begin{equation}\n\\lambda_f =\\frac{q}{p} \\frac{\\bar A}{A}\\,,\n\\label{basics_eq16}\n\\end{equation}\nwhere $A$ denotes the decay amplitude, then the two CP asymmetries can be written as\n\\begin{equation}\nC(f)=\\frac{1-|\\lambda_f|^2}{1+|\\lambda_f|^2}\\,,\\qquad S(f)=\\frac{2 \\,{\\rm Im}\\left[ \\lambda_f\\right]}{1+|\\lambda_f|^2}\\,.\n\\label{basics_eq17}\n\\end{equation}\n\n\n\\section{Effective Field Theories Of Weak Decays}\\label{basics_eftowd}\nA very widely used tool in the theoretical description of $B$ decay processes is the framework of \\textit{effective field theories} \\cite{Gilman:1979bc,Buras:1998ra}. The framework simplifies the dynamics of the weak decay by relying on an \\textit{operator product expansion}  (OPE) \\cite{Wilson:1969zs} of the weak vertices to separate the short and long distance physics. The OPE yields a concise \\textit{effective Hamiltonian} $\\mathcal{H}^{eff}$ built from a set of local effective operators $Q_i$ multiplied by renormalisation-scale dependent perturbatively calculable \\textit{Wilson coefficient functions} $C_i(\\mu)$:\n\\begin{equation}\n\\left<\\mathcal{H}\\right> \\stackrel{\\rm OPE}{\\longrightarrow}\\left<\\mathcal{H}^{eff}\\right> \\sim \\sum_i C_i(\\mu) \\left<Q_i\\right>+\\mathcal{O}(k^2\/M_W^2)\\,,\\\n\\label{basics_eq18}\n\\end{equation}\nwhere $k$ is the momentum flowing through the $W$ boson propagator. The separation of energy scales stems naturally from the fact that the weak decay of the $B$ meson is governed by physics originating at well separated scales: $m_t,\\,M_W\\gg m_{b,c}\\gg \\Lambda_{\\rm QCD} \\gg m_{u,d,s}$. It is the interplay of weak and strong effects that complicates the treatment of these decays, and must be dealt with appropriately.  By taking into account radiative corrections to tree-level and penguin diagrams, ultimately one obtains the effective Hamiltonian in terms of the set of all relevant local operators, which is closed under renormalisation. The full $\\Delta B=1$ effective Hamiltonian is, for a final state containing a $D$ quark\n\\begin{equation}\n\\mathcal{H}^{eff}=\\frac{G_f}{\\sqrt{2}}\\sum_{U=u,\\,c}\\lambda_U^{(D)} \\left[C_1 Q_1^U+C_2 Q_2^U+C_{7\\gamma} Q_{7\\gamma}+C_{8g} Q_{8g}+\\sum_{i=3,\\dots,10} C_i Q_i\\right]\\,,\n\\label{basics_eq20}\n\\end{equation}\nwhere make use of the standard short-hand notation for the product of CKM matrix elements $\\lambda_U^{(D)}\\equiv V^*_{UD} V_{Ub}$.  The form of Eq.~(\\ref{basics_eq20}) is chosen by assuming the unitarity of the CKM matrix (\\ref{basics_eq8}) to explicitly remove the dependence of the top quark CKM matrix elements which originate from penguin loops.  The effective operators are\n\\begin{eqnarray}\n\\lefteqn{\\bf{Current-Current\\footnote{The literature is not consistent concerning the labelling of the two operators $Q_{1,2}$ and one should be aware that the practice of swapping of these two operators is commonplace. We use the convention that the larger Wilson coefficient belongs to $Q_2$; that is, $Q_1$ is the new operator.}:}}\\hspace{4cm}\\nonumber\\\\\nQ^U_1 &=& (\\bar D_i U_j)_{V-A}(\\bar U_j b_i)_{V-A}\\,,\n\\qquad Q^U_2 = (\\bar DU)_{V-A}(\\bar Ub)_{V-A}\\,,\\nonumber\\\\\n\\lefteqn{\\rm\\bf{QCD~Penguin:}}\\hspace{4cm}\\nonumber\\\\\nQ_3 &=& (\\bar Db)_{V-A} \\sum_q (\\bar qq)_{V-A}\\,,\n\\qquad Q_4 = (\\bar D_i b_j)_{V-A} \\sum_q (\\bar q_j q_i)_{V-A}\\,,\\nonumber\\\\\nQ_5 &=& (\\bar Db)_{V-A} \\sum_q (\\bar qq)_{V+A}\\,, \n\\qquad Q_6 = (\\bar D_i b_j)_{V-A} \\sum_q (\\bar q_j q_i)_{V+A}\\,,\\nonumber\\\\\n\\lefteqn{\\rm\\bf{Electroweak~Penguin:}}\\hspace{4cm}\\nonumber\\\\\nQ_7 &=& (\\bar Db)_{V-A} \\sum_q \\frac{3}{2} e_q (\\bar qq)_{V+A}\\,,\n\\qquad Q_8 = (\\bar D_i b_j)_{V-A} \\sum_q  \\frac{3}{2} e_q (\\bar q_j q_i)_{V+A}\\,,\\nonumber\\\\\nQ_9 &=& (\\bar Db)_{V-A} \\sum_q \\frac{3}{2} e_q  (\\bar qq)_{V-A}\\,, \n\\qquad Q_{10} = (\\bar D_i b_j)_{V-A} \\sum_q  \\frac{3}{2} e_q (\\bar q_j q_i)_{V-A}\\,,\\nonumber\\\\\n\\lefteqn{\\rm\\bf{Electromagnetic~Dipole:}}\\hspace{4cm}\\nonumber\\\\\nQ_{7\\gamma} &=& \\frac{e}{8\\pi^2}m_b\\, \n        \\bar D\\sigma^{\\mu\\nu}(1+\\gamma_5)F_{\\mu\\nu}\\,b\n         + \\frac{e}{8\\pi^2}m_D\\, \n        \\bar D\\sigma^{\\mu\\nu}(1-\\gamma_5)F_{\\mu\\nu}\\,b\\,, \\nonumber\\\\\n\\lefteqn{\\rm\\bf{Chromomagnetic~Dipole:}}\\hspace{4cm}\\nonumber\\\\\nQ_{8g} &=& \\frac{g_s}{8\\pi^2}m_b\\, \n        \\bar D\\sigma^{\\mu\\nu}(1+\\gamma_5)G_{\\mu\\nu}\\, b\n        + \\frac{g_s}{8\\pi^2}m_D\\, \n        \\bar D\\sigma^{\\mu\\nu}(1-\\gamma_5)G_{\\mu\\nu}\\, b\\,,\n        \\label{basics_eq21}\n\\end{eqnarray}\nwhere $e_q$ is the electric charge of the quark $q$ in units of $|e|$ and $F_{\\mu\\nu}$ is the photonic field strength tensor. The Wilson coefficients entering the effective Hamiltonian are essentially effective coupling constants of the local effective operators. One can view the renormalisation of the matrix elements as an equivalent renormalisation of their Wilson coefficients. One makes use of renormalisation-group techniques to sum the potentially large logarithms $\\sim \\ln M_W^2\/\\mu^2$ that appear naturally in the evolution from weak scales $\\mathcal{O}(M_W)$ to hadronic scales, such as $\\mu\\sim m_b$. The operators (\\ref{basics_eq21}) mix with each other under evolution and from the renormalisation-scale invariance of $\\mathcal{H}^{eff}$ one finds\n\\begin{equation}\n\\mu \\frac{d}{d \\mu} C_i (\\mu)=\\gamma_{ji}(\\mu)\\, C_j(\\mu)\\,, \n\\label{basics_eq22}\n\\end{equation}\nwhere $\\hat \\gamma$ is the \\textit{anomalous dimension} matrix, which can be given as an expansion in the strong coupling via the renormalisation constant $\\hat Z$\n\\begin{equation}\n\\gamma_{ji}(\\mu)= Z^{-1}_{ik}\\frac{d Z_{kj}}{d \\ln \\mu}\\,,\\qquad \\hat{\\gamma}=\\left(\\frac{\\alpha_s}{4\\pi}\\right)\\hat{\\gamma}^{(0)}+\\left(\\frac{\\alpha_s}{4\\pi}\\right)^2\\hat{\\gamma}^{(1)}+\\mathcal{O}(\\alpha_s^3)\\,.\n\\label{basics_eq23}\n\\end{equation}\nSolving  Eq.~(\\ref{basics_eq22}) yields the evolution of the Wilson coefficients via the evolution matrix $\\hat U(\\mu,\\mu_0)$\n\\begin{equation}\n C_i(\\mu)= U_{ij}(\\mu,\\mu_0)\\, C_j(\\mu_0)\\,,\\qquad\\hat{U}(\\mu,\\mu_0)=\\exp \\int^{g(\\mu)}_{g(\\mu_0)}dg^\\prime\\,\\frac{\\hat{\\gamma}^{T}(g^\\prime)}{\\beta(g^\\prime)}\\,,\n\\label{basics_eq24}\n\\end{equation}\nwhere $\\beta(g)$ is the QCD $\\beta$-function. To leading order one has\n\\begin{equation}\n\\hat{U}^{\\rm LO}(\\mu,\\mu_0)=\\left(\\frac{\\alpha_s(\\mu_0)}{\\alpha_s(\\mu)}\\right)^{\\frac{\\hat{\\gamma}^{(0)T}}{2 \\beta_0}}= \\hat V\\left[\\left(\\frac{\\alpha_s(\\mu_0)}{\\alpha_s(\\mu)}\\right)^{\\frac{\\overrightarrow{\\gamma}^{\\left(0\\right)}}{2 \\beta_0}}\\right]_D\\hat V^{-1}\\,,\n\\label{basics_eq25}\n\\end{equation}\nwhere $V$ is the matrix that diagonalises $\\hat{\\gamma}^{(0)T}$ and $\\overrightarrow{\\gamma}^{\\left(0\\right)}$ is a vector of the eigenvalues of the leading order anomalous dimension matrix $\\hat{\\gamma}^{(0)}=\\hat V\\hat{\\gamma}^{(0)T}_D\\hat V^{-1}$. At NLO we have\n\\begin{equation}\nC_i(\\mu)=C_i^{(0)}(\\mu)+\\frac{\\alpha_s(\\mu)}{4\\pi}C_i^{(1)}(\\mu)\\,,\n\\label{basics_eq26}\n\\end{equation}\nand the evolution is a bit more complicated:\n\\begin{equation}\n\\hat{U}^{\\rm NLO}(\\mu,\\mu_0)=\\left[1+\\frac{\\alpha_s(\\mu)}{4\\pi}\\hat{J}\\right]\\hat{U}^{\\rm LO}(\\mu,\\mu_0)\\left[1-\\frac{\\alpha_s(\\mu_0)}{4\\pi}\\hat{J}\\right]\\,,\n\\label{basics_eq27}\n\\end{equation}\nwith\n\\begin{equation}\n\\hat{J}=V \\hat{S}V^{-1}\\,,\\qquad S_{ij}=\\delta_{ij}\\gamma_{i}^{(0)}\\frac{\\beta_1}{2\\beta_0^2}-\\frac{G_{ij}}{2\\beta_0+\\gamma_i^{(0)}-\\gamma_j^{(0)}}\\,,\\qquad\\hat{G}=V^{-1} \\hat{\\gamma}^{(1)T}V\\,.\n\\label{basics_eq28}\n\\end{equation}\nTo NLO the required $\\beta$-function coefficients are $\\beta_1=\\frac{34}{3}N_c^2-\\frac{10}{3}N_c N_f -2 C_F N_f$ and $\\beta_0=\\frac{11}{3}N_c  -\\frac{2}{3} N_f$ with $N_f$ is the number of active flavours, $C_F=(N_c^2-1)\/(2N_c)$ and $N_c$ the number of colours. Care must be taken in evolving through ``thresholds'' where the number of active flavours $N_f$ changes; the evolution must then be taken in stages, as a change in $N_f$ changes the $\\beta$-function coefficients and the anomalous dimension matrices. If there is a flavour threshold $\\mu_{\\rm th}$  between $\\mu_0$ and $\\mu$, which changes the number of active flavours from $N_f$ to $N_f+1$, then one has to make the replacement\n\\begin{equation}\n\\hat U(\\mu,\\mu_0)\\to \\left.\\hat U(\\mu,\\mu_{\\rm th})\\right|_{N_f +1}\\cdot \\left.\\hat U(\\mu_{\\rm th},\\mu_0) \\right|_{N_f}.\n\\label{basics_eq29}\n\\end{equation} \nThe effective Hamiltonian, combined with the renormalisation-group improvement of the perturbative series forms an exceptionally powerful framework. The matrix elements of the local operators $\\left<Q_i\\right>$ are the subject of QCD factorisation theorems, such as that discussed in Chapter~\\ref{chapter6_QCDF}, which allow the calculation of $B$ decay amplitudes.  From these amplitudes one can construct observables such as branching fractions, CP asymmetries, and isospin asymmetries which can be investigated phenomenologically.\n\\chapter{Vector Meson Light-Cone Distribution Amplitudes}\\label{chapter2_DAs}\nIn this chapter we discuss light vector meson light-cone distribution amplitudes and via the (approximate) conformal symmetry of QCD present expressions for the distribution amplitudes up to twist-3. The method introduces a set of non-perturbative parameters which is reduced in size by invoking the QCD equations of motion to relate the two-particle twist-3 distribution amplitudes to the three-particle twist-3 and two-particle twist-2 distribution amplitudes. In our analysis we include all $\\rm SU(3)_F$-breaking effects and G-parity violating terms thus allowing one to fully differentiate between $\\rho$, $K^*$ and $\\phi$ mesons. Moreover, a non-zero quark mass induces a mixing between twist-2 and twist-3 parameters under a change of renormalisation scale $\\mu$. To simplify notation we explicitly consider the $K^*$ meson, with quark composition $s\\bar{q}$ where $q=\\{u,d\\}$.\\footnote{The notation in this thesis, $K^*$ being a $(s\\bar q)$   bound state, is in contrast to the standard labelling, according to  which $K^{*0}=(d\\bar s)$ and $\\bar K^{*0}=(s\\bar d)$. This is the standard notation used for light-cone distribution amplitudes where $K^*$ always contains an $s$ quark, and $\\bar K^*$ an $\\bar s$  quark. This distinction is relevant because of  a sign change of G-odd matrix elements under $(s\\bar  q)\\leftrightarrow (q\\bar s)$. This notation also applies to calculations of form factors and other matrix  elements which involve light-cone distribution amplitudes.}\n\nThere are two main applications of meson distribution amplitudes that motivate their study:\n\\begin{itemize}\n\\item{they are directly applicable to the theoretical description of exclusive decay processes via QCD factorisation theorems, which require the distribution amplitudes as a non-perturbative input, see Chapter~\\ref{chapter6_QCDF}.}\n\\item{they are also applicable to the determination of transition form factors from the light-cone sum rule approach and as such are indirectly applicable to the same QCD factorisation theorems for which the transition form factors are also required,  see Chapters~\\ref{chapter3_SR} and \\ref{chapter5_eta}.}\n\\end{itemize}\nIn Chapter~\\ref{chapter4_det} we calculate,  from QCD sum rules,  numerical values for the leading twist-2 and twist-3 distribution amplitude  parameters defined here. Standard notations used, such as the light-cone coordinates, are given in Appendix~\\ref{appendixA}. The material covered in this chapter partially follows that of Ref.~\\cite{Ball:2007rt}.\n\n\\section{Introduction}\nHadronic light-cone distribution amplitudes (DAs) of light mesons were first discussed in the ground-breaking papers of Brodsky, Lepage, and others, see Refs.~\\cite{Chernyak:1977fk, Chernyak:1980dk, Lepage:1980fj, Efremov:1979qk, Efremov:1978rn,Chernyak:1977as, Lepage:1979zb,Chernyak:1980dj} and play an essential role in the QCD description of hard exclusive processes \\cite{Chernyak:1981zz,Brodsky:1989pv}.  The amplitudes that describe such processes factorise in the asymptotic limit $Q^2\\sim 1\/x^2 \\to \\infty$ -- where $Q^2$ is the momentum transfer and $x$ the transverse separation of the partons -- and are dominated by contributions from near the light-cone. The factorisation is given by the convolution of a hard-scattering kernel, calculable in perturbation theory, and process-independent, universal, non-perturbative DAs. \n\nThe study of hadronic DAs has a long history. The simplest and first to be investigated were the twist-2 DA of the $\\pi$ \\cite{Lepage:1980fj,Efremov:1979qk,Chernyak:1977as,Lepage:1979zb}. Higher twist DAs of the $\\pi$, alongside those of the other pseudoscalar mesons followed \\cite{Ball:1998je}.   For vector mesons, the leading-twist DAs of the $\\rho$ were first investigated by Chernyak and Zhitnitsky in Ref.~\\cite{Chernyak:1983ej} and later in Refs.~\\cite{Ali:1993vd, Ball:1996tb}. The formalism of higher twist-3 and twist-4 contributions, including meson mass corrections, was investigated by Ball \\textit{et al.} in Refs.~\\cite{Ball:1998sk,Ball:1998ff,Ball:2006wn,Ball:2007zt}. \n\nThe DAs of the $K^*$  ($K$) differ to those of the $\\rho$ ($\\pi$) due to the non-zero strange quark mass which yields $\\rm SU(3)_F$-breaking and G-parity violating corrections from a number of different sources.\\footnote{Perfect $\\rm SU(3)_F$ symmetry is realised for equal $u,d,$ and $s$ quark masses.} The study of the various contributions span many publications:\n\\begin{itemize}\n\\item{explicit quark mass corrections to DAs and evolution equations are generated by the QCD equations of motion (EOM) and only affect higher twist DAs. The contributions for vector mesons were calculated in Ref.~\\cite{Ball:1998sk} up to twist-3, and those to the evolution equations for vector mesons in Ref.~\\cite{Ball:2007rt} and flavour-octet pseudoscalar mesons Ref.~\\cite{Ball:2006wn}.}\n\\item{G-parity violating contributions, which are proportional to $m_s-m_q$ and hence vanish for equal quark masses, i.e. for $\\rho$ and $\\phi$, were investigated for twist-2 DAs in Refs.~\\cite{Chernyak:1983ej,Ball:1998sk,Ball:2003sc,Braun:2004vf,Ball:2005vx,Ball:2006fz} and for twist-3 DAs in Ref.~\\cite{Ball:2007rt}.}\n\\item{$\\rm SU(3)_F$-breaking of non-perturbative hadronic parameters entering the DAs.\nThe effects for the twist-2 parameters are known from\nRefs.~\\cite{Chernyak:1983ej,Ball:1998sk,Ball:2003sc}, twist-3 from Ref.~\\cite{Ball:2007rt} and twist-4 from Ref.~\\cite{Ball:2007zt}. The twist-3 vector meson parameters are discussed in Chapter~\\ref{chapter4_det} where we include all these effects in a determination of numerical values using QCD sum rules.}\n\\end{itemize}\nThe objects which define the DAs are vacuum-to-meson matrix elements of non-local operators at strictly light-like separations $z^2=0$ \\cite{Chernyak:1983ej}. Two examples we shall encounter are\n\\begin{eqnarray}\n\\bra{0}\\bar{q}(z)\\Gamma [z,-z] s(-z) \\ket{K^*(p,\\lambda)}\\,,\\qquad \\bra{0}\\bar{q}(z) [z,v z] g_s G_{\\mu \\nu}(vz)\\Gamma [v z,-z] s(-z) \\ket{K^*(p,\\lambda)}\\,,\\nonumber\\\\\n\\label{das_eq1}\n\\end{eqnarray}\nwhere $\\Gamma$ is a general Dirac matrix, $\\lambda=\\{\\parallel,\\perp\\}$ is the polarisation of the $K^*$ meson and the quark fields are taken at symmetric separation for simplicity.\\footnote{The Dirac matrices  $\\Gamma =\\{ \\sigma_{\\mu \\nu},\\,i \\gamma_5,\\, \\bf 1\\}$ give rise to so-called \\textit{chiral-odd} distributions because they are chirality-violating. Likewise, distributions generated from $\\Gamma =\\{ \\gamma_{\\mu},\\,\\gamma_{\\mu} \\gamma_5\\}$ are \\textit{chiral-even}.}  The first (second) matrix element above corresponds to a two- (three-) particle Fock state. To render the matrix element gauge invariant the path-ordered gauge factor is included\n\\begin{equation}\n[x,y]=\\textrm{P}\\, \\exp \\left[ i g_s \\int^1_0 dt\\, (x-y)_\\mu A^\\mu (t x+(1-t)y)\\right].\n\\end{equation}\nFor convenience we work in the \\textit{fixed-point} gauge\\footnote{also known as the \\textit{Fock-Schwinger} gauge.}\n\\begin{equation}\n(x-x_0)^\\mu A_\\mu^a (x)=0\\,,\n\\label{das_eq2}\n\\end{equation}\nand by choosing $x_0=0$ we have $[x,-x]=1$. The gauge factor will be implied unless otherwise stated. The DAs are dimensionless functions of  the collinear momentum fractions of a fixed number of constituents within a meson, at zero transverse separation. For two-particle DAs the constituent strange quark and antiquark  ($\\bar{q}$) share $u$ and $\\bar{u}=1-u$ of the meson momentum $p$ respectively.  For three-particle DAs we have $\\underline{\\alpha} = (\\alpha_1, \\alpha_2, \\alpha_3 )$ corresponding to the momentum fractions carried by the strange quark, antiquark ($\\bar{q}$) and gluon, respectively.  For a minimum number of constituents, the DAs are related to the \\textit{Bethe-Salpeter wavefunction} $\\phi_{BS}$ by integration over the transverse momenta\n\\begin{equation}\n\\phi(u, \\mu) \\sim \\int^{|k_{\\perp}|<\\mu}d^2 k_{\\perp}\\,\\phi_{BS}(u,k_{\\perp})\\,,\n\\label{das_eq3}\n\\end{equation}\nwhere  $\\mu$ is the renormalisation scale.  The price to pay for integrating out $k_{\\perp}$ below $\\mu$ is a renormalisation-scale dependence of the DAs governed by renormalisation-group equations.  The DAs have to be evaluated at the scale $\\mu^2 \\sim x^{-2}$ i.e. of the order of the deviation from the light-cone \\cite{Balitsky:1987bk}.\n\nNon-local operators that appear at finite $Q^2$ or mass scales are expanded near the light-cone $x^2 \\neq 0$ as an OPE in terms of the renormalised  non-local operators on the light-cone - the \\textit{light-cone expansion} \\cite{Balitsky:1987bk}.\\footnote{The expansion is facilitated by using light-cone coordinates which are given in Appendix~\\ref{appendixA}.} After taking matrix elements the resulting Lorentz-invariant amplitudes are matched to the definitions of the DAs with the coefficient functions of the expansion taken at tree-level, to leading logarithmic accuracy.  \n\nThe structure of vector meson DAs follows the same pattern as the nucleon structure functions and can be classified in the same way \\cite{Jaffe:1991ra}. They are described by separate DAs for each polarisation and thus there are more vector meson DAs than pseudoscalar DAs. \n\nLastly, we briefly mention some other DAs. Flavour-singlet pseudoscalar meson DAs\nare complicated by the $\\rm U(1)_A$ anomaly of QCD and are discussed in Chapter~\\ref{chapter5_eta} in the context of the $B \\to \\eta^{(\\prime)}$ transition form factor \\cite{Ball:2007hb}. Much work has been done concerning the DAs of heavy mesons, such as the $B$ meson \\cite{Szczepaniak:1990dt ,Braun:2003wx}; indeed, the DAs of $B$ mesons enter the QCD factorisation framework of radiative and non-leptonic $B$ decays, as discussed in Chapter~\\ref{chapter6_QCDF}, and a variant light-cone sum rule method devised in Ref.~\\cite{Khodjamirian:2006st}. There also exist DAs of the photon which describe its ``soft'' hadronic components, along with the usual ``hard'' electromagnetic components \\cite{Ball:2002ps}. The photonic DAs can be important in, for example, $B \\to V \\gamma$ decays \\cite{Ball:2006eu} as investigated in Chapter~\\ref{chapter7_rad}, and $B \\to \\gamma e \\nu$ \\cite{Descotes-Genon:2002mw,Ball:2003fq}. Finally, the field of baryon DAs is also active and many of the tools and concepts we cover in this thesis find application there, see for example Ref.~\\cite{Braun:2006hn} for a review.\n\n\\section{The Conformal Expansion}\nThe standard determination of meson DAs  proceeds by making use of the conformal symmetry of massless QCD at tree-level.  The conformal expansion is analogous to the partial wave expansion of wave functions in quantum mechanics in spherical harmonics $\\psi(r,\\theta,\\phi) \\to R(r) \\sum_{m,l} Y^l_m (\\theta,\\phi)$. The expansion uncovers a simple multiplicative renormalisation at leading-order, and as such different partial waves, with different \\textit{conformal spin}, do not mix under  a change of renormalisation scale.  At next-to-leading-order this is not the case, because strictly speaking the conformal symmetry of a quantum theory requires its $\\beta$ function to vanish. Proximity to the conformal limit in QCD is therefore governed by the value of the strong coupling constant, becoming true as $\\alpha_s \\to 0$ and we pass to the free theory.\\footnote{It must be noted that mass terms break the conformal expansion immediately at the classical level. This does not upset the conformal expansion, however. See Ref.~\\cite{Braun:2003rp} for details.} Using the QCD equations of motion we can elucidate this mixing order-by-order in the conformal expansion. \n\nThe application of conformal symmetry to exclusive processes has recieved a lot of attention in the literature, see Refs.~\\cite{Brodsky:1980ny,Brodsky:1985ve,Ohrndorf:1981qv,Braun:1989iv,Makeenko:1980bh}. The main benefit of the conformal expansion is the systematic separation of the longitudinal and transverse degrees of freedom in meson DAs. The former correspond to the longitudinal momentum fractions and is given by irreducible representations of the relevant symmetry group, SL(2,$\\mathbb R$).  The latter are integrated out to yield a renormalisation-scale dependence of the DAs, described by renormalisation-group equations.  Here we focus on the most important points, see Ref.~\\cite{Braun:2003rp} for a detailed review. \n\n\\subsection{Conformal Group}\nThe conformal group is defined as all transformations that change only  the scale of the metric and as such preserve angles and leave the light-cone invariant $g_{\\mu\\nu}^\\prime(x^\\prime)=\\omega(x) g_{\\mu\\nu}(x)$; the spacetime interval $ds^2=g_{\\mu\\nu}(x) \\,dx_\\mu dx_\\nu$ is conserved up to scaling. These transformations form a generalisation of the Poincar\\'e group. The full conformal algebra in 4 dimensions includes fifteen generators\n\\begin{eqnarray}\n\\textbf{P}_\\mu &\\to& \\rm 4~ Translations, \\nonumber\\\\\n\\textbf{M}_{\\mu \\nu}&\\to& \\rm 6 ~Lorentz ~rotations, \\nonumber\\\\\n\\textbf{D} &\\to& \\rm 1 ~Dilatation,\\nonumber\\\\\n\\textbf{K}_\\mu &\\to& \\rm 4 ~Special ~conformal ~translations.\n\\label{das_eq4}\n\\end{eqnarray}\nOur hadronic picture is of partons moving collinearly in, say the $p_{\\mu}$ direction, existing near the light-cone. We therefore restrict the \\textit{fundamental fields} of the conformal group to the light-cone\n$\\Phi(x) \\to \\Phi(\\alpha z)$, where $\\alpha $ is a real number, and we assume fields to be eigenstates of the spin operator \n\\begin{equation}\n\\Sigma^{\\mu\\nu} \\psi = \\frac{i}{2}\\sigma^{\\mu\\nu} \\psi\\,,\n\\label{das_eq5}\n\\end{equation}\nso as to have a fixed Lorentz-spin projection $s$ in the $z_{\\mu}$ (``plus'') direction $\\Sigma_{+-} \\Phi(\\alpha z) = s \\,\\Phi(\\alpha z)$. For leading-twist operators this is automatically satisfied and for higher-twist operators projections are used to separate different spin states, as we shall discuss shortly. The full conformal symmetry (\\ref{das_eq4}) is now modified  and it turns out that the resulting group of transformations form the special linear group SL(2,$\\mathbb R$), or so-called \\textit{collinear conformal group}, given by just four generators. They are written in standard form by constructing the following linear combinations\n\\begin{eqnarray}\n\\textbf{L}_+= \\textbf{L}_1+i \\textbf{L}_2=-i \\textbf{P}_+\\,,& \\qquad& \\textbf{L}_-= \\textbf{L}_1-i \\textbf{L}_2= \\frac{i}{2} \\textbf{K}_-\\,,\\nonumber\\\\\n\\textbf{L}_0= \\frac{i}{2}(\\textbf{D}+\\textbf{M}_{+-})\\,,& \\qquad& \\textbf{E}=\\frac{i}{2}(\\textbf{D}-\\textbf{M}_{+-})\\,.\n\\end{eqnarray}\nwhich leads to the familiar relations\n\\begin{equation}\n[\\textbf{L}_0,\\,\\textbf{L}_\\mp]=\\mp \\textbf{L}_\\mp\\,, \\qquad [\\textbf{L}_-,\\,\\textbf{L}_+]=-2 \\textbf{L}_0\\,.\n\\end{equation}\nThe operators act on the fundamental fields as\n\\begin{eqnarray}\n\\left[\\textbf{L}_+,\\Phi(\\alpha z) \\right]&=&-\\partial_{\\alpha} \\Phi(\\alpha n)\\,,\n\\label{das_eq6}\\\\\n\\left[\\textbf{L}_{-},\\Phi(\\alpha z)\\right]&=&(\\alpha^2 \\partial_{\\alpha} +2 j \\alpha)\\Phi(\\alpha n)\\,,\n\\label{lower}\\\\\n\\left[\\textbf{L}_0,\\Phi(\\alpha z) \\right]&=&(\\alpha \\partial_{\\alpha}+j) \\Phi(\\alpha n)\\,,\n\\label{das_eq7}\\\\\n\\left[\\textbf{E},\\Phi(\\alpha z)\\right]&=&\\frac{1}{2} (l-s) \\Phi(\\alpha n)\\,,\n\\label{das_eq8}\n\\end{eqnarray}\nwhere $t=l-s$ is the \\textit{twist},\\footnote{strictly it is the \\textit{collinear twist} which is defined as ``dimension minus spin projection on the positive direction''. There also exists \\textit{geometric twist} which is defined as ``dimension minus spin''.} $l$ is the canonical mass dimension,\\footnote{For example, $l=3\/2$ for quarks and $l=2$ for gluons.} $s$ the Lorentz-spin projection, and $j=\\frac{1}{2}(l+s)$ the \\textit{conformal spin} of the field $\\Phi$. The conformal spin specifies the representation of the collinear conformal group. The operator $\\textbf{E}$ commutes with all $\\textbf{L}_i$ and therefore twist is a good quantum number for each conformal field. The Casimir operator commutes with all $\\textbf{L}_i$ and is given by\n\\begin{equation}\n\\sum_{i=0,1,2}[\\textbf{L}_i,[\\textbf{L}_i,\\Phi(\\alpha z)]]=j(j-1)\\Phi(\\alpha z) = \\textbf{L}^2 \\Phi(\\alpha z)\\,.\n\\end{equation}\nAt the origin of the light-cone $\\alpha=0$ and the field $\\Phi(0)$ is killed by the lowering operator $\\textbf{L}_-$ and as such has the minimum spin projection $j_{\\rm min}$ of states of conformal spin $j$. One can define a \\textit{conformal operator} $\\mathbb{O}_n=\\Phi(0)$ by requiring that it transforms just as the fundamental field, Eqs.~(\\ref{lower} - \\ref{das_eq8}), and is killed by the lowering operator $\\textbf{L}_-$. The raising operator $\\textbf{L}_+$ can be repeatedly applied to $\\Phi(0)$ to give\n\\begin{equation}\n\\mathbb{O}_{n,n+k}=\\underbrace{\\left[ \\textbf{L}_+,...,\\left[\\textbf{L}_+,\\left[\\textbf{L}_+\\right.\\right.\\right.}_{k},\\left.\\left.\\left.\\Phi(0)\\right]\\right]\\right]=\\left(-i \\partial_+\\right)^k \\mathbb{O}_n\\,,\n\\end{equation}\nwhere $\\mathbb{O}_{n,n}=\\mathbb{O}_n$ and the subscript $n$ defines the \\textit{conformal tower} of states, of conformal spin $j_{\\textrm{min}}<j_{\\textrm{min}}+k<\\infty$, generated by the collinear conformal algebra. This is an infinite dimensional representation of the collinear conformal group.\n\n\\subsection{States of Definite Spin}\nNow the main language of the collinear conformal group is defined, if one can relate the fundamental fields $\\Phi$ to the operators of hard processes in QCD one can export all the machinery above and immediately reap the benefits. To this end,  consider the non-local two-particle operator at light-like separation (\\ref{das_eq1}) and expand at small distances\n\\begin{equation}\n\\bar{q}(z)\\Gamma s(-z)= \\sum_k \\frac{1}{k!}\\, \\bar{q}(0) (\\stackrel{\\leftrightarrow}{D} \\cdot z)^k \\Gamma s(0)\\,,\n\\label{das_eq9}\n\\end{equation}\nwhere $\\stackrel{\\leftrightarrow}{D}_\\mu= \\stackrel{\\rightarrow}{D}_\\mu- \\stackrel{\\leftarrow}{D}_\\mu$. The question is, how does one express these local operators in terms of conformal operators and thus separate all the different twist contributions?  To proceed one decomposes the quark fields into definite Lorentz-spin components using the projection operators\n\\begin{equation}\n\\Pi_+ = \\frac{1}{2} \\frac{\\slash{p}\\slash{z}}{ p\\cdot z}\\,, \\qquad \\Pi_- = \\frac{1}{2} \\frac{\\slash{z}\\slash{p}}{ p\\cdot z}\\,, \\qquad \\Pi_+ +\\Pi_- =1\\,.\n\\end{equation}\nwhich project onto the ``plus'' and ``minus'' components of the spinor respectively. Using the generator of the spin rotations of a spinor field (\\ref{das_eq5}) one can show\n\\begin{equation}\n\\underbrace{\\psi_+ = \\Pi_+ \\psi}_{ \\shortstack{\\footnotesize{$s=+1\/2$} \\\\ \\footnotesize{$j=1$}}}\\,, \\qquad \\underbrace{\\psi_- = \\Pi_- \\psi}_{ \\shortstack{\\footnotesize{$s=-1\/2$} \\\\ \\footnotesize{$j=1\/2$}}}\\,.\n\\end{equation}\nThe composite operators (\\ref{das_eq9}) have  conformal spin projection $j=j_q+j_s+k$  where the subscripts correspond to the separate quark fields. The composite operators are ordered by increasing twist\n\\begin{equation}\n\\underbrace{\\bar{q}_{+} \\Gamma s_{+}}_{ \\rm t=2}\\,, \\qquad \\underbrace{\\bar{q}_{+} \\Gamma s_{-} + \\bar{q}_{-} \\Gamma s_{+}}_{ \\rm t=3}\\,, \\qquad \\underbrace{\\bar{q}_{-} \\Gamma s_{-}}_{ \\rm t=4}\\,. \n\\label{das_eq10}\n\\end{equation}\nIt can be shown that the corresponding local conformal operators are\n\\begin{eqnarray}\n\\mathbb{Q}_n^{\\rm t=2}(x) &=& (i\\partial_+)^n \\left[\\bar{q}(x) \\gamma_+ C^{3\/2}_{n} \\left(\\stackrel{\\leftrightarrow}{D}_+ \/ \\partial_+\\right) s(x) \\right]\\,,\n\\label{das_eq11}\\\\\n\\mathbb{Q}_n^{\\rm t=3}(x) &=& (i\\partial_+)^n \\left[\\bar{q}(x)\\gamma_+ \\gamma_\\perp \\gamma_-P^{(1,0)}_{n} \\left(\\stackrel{\\leftrightarrow}{D}_+ \/ \\partial_+\\right) s(x) \\right]\\,,\\label{das_eq12}\\\\\n\\mathbb{Q}_n^{\\rm t=4}(x) &=& (i\\partial_+)^n \\left[\\bar{q}(x) \\gamma_- C^{1\/2}_{n} \\left(\\stackrel{\\leftrightarrow}{D}_+ \/ \\partial_+\\right) s(x) \\right]\\,,\n\\label{das_eq13}\n\\end{eqnarray}\nwhere $\\partial_+ =\\stackrel{\\leftarrow}{D}_+ +\\stackrel{\\rightarrow}{D}_+$, $C^m_n(x)$ are \\textit{Gegenbauer polynomials} and $P^{(r,s)}_n(x)$ are \\textit{Jacobi polynomials}. There is another twist-3 operator corresponding to Eq.~(\\ref{das_eq12}) with the replacement $P^{(1,0)}_{n}(x) \\to P^{(0,1)}_{n}(x)$. One can now connect firmly to QCD with the specific example of the leading-twist operator; consider again Eq.~(\\ref{das_eq1}) and specify the twist-2 Dirac matrix $\\Gamma \\to \\gamma_\\mu$ (projected onto $z_\\mu$) and define the DA as\n \\begin{equation}\n \\bra{0} \\bar{q}(z)\\gamma_z s(-z) \\ket{K^*(p,\\lambda)}= f_{K^*}^\\parallel m_{K^*} (e^{(\\lambda)}\\cdot z) \\int^1_0 du\\, e^{-i [\\bar{u} z+ u(-z)]\\cdot  p} \\phi^{\\parallel}_{2;K^*}(u,\\mu)\\,.\n \\label{das_eq14}\n\\end{equation}\nThen, using Eq.~(\\ref{das_eq9}) and comparing the result to Eq.~(\\ref{das_eq11}), one finds\n\\begin{equation}\n\\int^1_0 du\\,  C^{3\/2}_{n}(\\xi)\\,\\phi^{\\parallel}_{2;K^*}(u,\\mu) = \\langle\\langle \\mathbb{Q}_n^{\\rm t=2}\\rangle\\rangle\\,,\n\\label{das_eq15}\n\\end{equation}\nwhere we introduce the shorthand $\\xi = u-\\bar{u} = 2u-1$, and $\\langle\\langle \\mathbb{Q}_n^{\\rm t=2}\\rangle\\rangle$ are the reduced matrix elements of the operator $\\mathbb{Q}_n^{\\rm t=2}$. The Gegenbauer polynomials form a complete set of orthogonal functions over the weight function $6 u (1-u)$ in the interval $0<u<1$\n\\begin{equation}\n\\int^1_0 du\\,  u \\bar u \\,C^{3\/2}_{n}(\\xi)\\, C^{3\/2}_{m}(\\xi)=\\delta_{m n}\\frac{(n+1)(n+2)}{4(2n+3)}\\,,\n\\end{equation}\nand so one can invert (\\ref{das_eq15}) to find\n\\begin{equation}\n\\phi^{\\parallel}_{2;K^*}(u,\\mu) = 6 u (1-u) \\left\\{1+ \\sum_{n=1}^{\\infty} a_n^{\\parallel}(\\mu) \\,C^{3\/2}_{n}(\\xi)\\right\\}\\,,\n\\label{das_eq16}\n\\end{equation}\nwhere the \\textit{Gegenbauer coefficients} $a_n^{\\parallel}(\\mu)$ are related to the reduced matrix elements, Eq.~(\\ref{das_eq15}), as\n\\begin{equation}\na_n^{\\parallel}(\\mu) =\\frac{2(2n+3)}{3(n+1)(n+2)}\\langle\\langle \\mathbb{Q}_n^{\\rm t=2}\\rangle\\rangle\\,, \\qquad a_0^{\\parallel} = 1\\,.\n\\label{das_eq17}\n\\end{equation}\nThe result is that the conformal symmetry has separated the longitudinal degrees of freedom -- as contained in the orthogonal Gegenbauer polynomials which are function of the momentum fraction $u$ for the  twist-2 distribution -- from the transverse degrees of freedom, which now show up as the renormalisation-scale dependence of the Gegenbauer coefficients $a_n^{\\parallel}$. The Gegenbauer coefficients  contain the non-perturbative information of the DA and, for the leading-twist DA, are of conformal spin $n+2$.  The higher-twist two-particle DAs are expanded analogously in $P^{(1,0)}_{n}$ and $P^{(0,1)}_{n}$ for twist-3 and $C^{1\/2}_{n}$ for twist-4, see Eqs. (\\ref{das_eq12})  and (\\ref{das_eq13}) respectively.  The explicit expression for the DA of an $m$-particle state with the lowest possible conformal spin $j=j_1+\\dots+j_m$, the so-called \\textit{asymptotic distribution amplitude}, is given by\n\\begin{equation}\n\\phi_{as}(\\alpha_1,\\alpha_2,\\cdots,\\alpha_m) =\n\\frac{\\Gamma(2j_1+\\cdots +2j_m)}{\\Gamma(2j_1)\\cdots \\Gamma(2j_m)}\\,\n\\alpha_1^{2j_1-1}\\alpha_2^{2j_2-1}\\ldots \\alpha_m^{2j_m-1}\\,,\n\\label{das_eq18}\n\\end{equation}\nwhere $\\sum_{k=1}^m \\alpha_k =1$ \\cite{Braun:1989iv}. For the twist-2 two-particle DA considered $j=1$ and we recover the weight function $6 u (1-u)$. Analogously, for a multi-particle DA, states higher in conformal spin are multiplied by polynomials orthogonal over the weight function Eq.~(\\ref{das_eq18}). The matrix element Eq.~(\\ref{das_eq14}) with the chiral-odd Dirac matrix $\\Gamma \\to \\sigma_{z p}$ starts at twist-2 also, and gives rise to the second two-particle twist-2 DA\n\\begin{equation}\n\\phi^{\\perp}_{2;K^*}(u,\\mu) = 6 u (1-u)\\left\\{1+ \\sum_{n=1}^{\\infty} a_n^{\\perp}(\\mu) \\,C^{3\/2}_{n}(\\xi)\\right\\}\\,.\n\\label{das_eq19}\n\\end{equation}\nThe expansion of DAs in terms of an infinite sum of partial waves, as in Eq. (\\ref{das_eq19}), is very general, and is valid at the level of operators. In practice the concept of \\textit{G-parity} allows one to classify which Gegenbauer coefficients contribute for specific matrix elements of those operators. The G-parity operator $\\cal{G}$ is defined as $\\mathcal{G}=\\mathcal{C} e^{-i\\pi T_2}$ where $T_2$ is the isospin generator of the $2$ axis and $\\cal{C}$ is the charge conjugation operator.  G-parity is the generalisation of charge conjugation to particle multiplets, for example, $\\mathcal{G}\\ket{\\pi^{\\pm,0}}=-\\ket{\\pi^{\\pm,0}}$ and is conserved in QCD. It effectively swaps quarks for anti-quarks and therefore for equal mass quarks $u \\leftrightarrow \\bar{u}$ and consequently, for $\\pi,\\,\\rho,\\,\\omega$ and $\\phi$, the odd Gegenbauer coefficients $a_{2n+1}$ vanish. \n\n\\section{Two-Particle Twist-2 Distribution Amplitudes}\\label{das_sec1.3}\nAs mentioned in the last section, there are two two-particle matrix elements that begin at twist-2 \\cite{Ball:1998sk}:\\footnote{The vacuum-vector meson matrix elements vanish for $\\Gamma =i \\gamma_5$ because it is impossible to construct a pseudoscalar quantity from the three available 4-vectors $p_\\mu,\\,z_\\mu$ and $e^{(\\lambda)}_\\mu$.}\n\\begin{eqnarray}\n\\lefteqn{\n\\bra{0} \\bar q(x) \\gamma_\\mu s(-x) \\ket{K^*(P,\\lambda)} = }\\hspace*{1cm}\n\\nonumber\\\\\n&&{}f_{K^*}^\\parallel m_{K^*} \\left\\{ \\frac{e^{(\\lambda)}x}{P\\cdot x}\\,P_\\mu \\int_0^1 du\\, e^{i\\xi P\\cdot x} \\left[ \\phi_{2;K^*}^\\parallel(u) +   \\frac{1}{4}\\, m_{K^*}^2 x^2  \\phi^\\parallel_{4;K^*}(u)\\right]\\right.\\nonumber\\\\\n&&{} + \\left( e^{(\\lambda)}_\\mu -\nP_\\mu\\,\\frac{e^{(\\lambda)}x}{P\\cdot x}\\right) \\int_0^1 du\\,e^{i\\xi  P\\cdot x}\\,\\phi_{3;K^*}^\\perp(u) \\nonumber\\\\\n&&\\left. - \\frac{1}{2}\\,x_\\mu \\,\\frac{e^{(\\lambda)}x}{(P\\cdot x)^2} \\, m_{K^*}^2 \\int_0^1 du\\,e^{i\\xi P\\cdot x}\\,\n\\left[ \\psi_{4;K^*}^\\parallel(u) + \\phi_{2;K^*}^\\parallel(u) - 2   \\phi_{3;K^*}^\\perp(u)\\right]\\right\\}\\,,\n  \\label{das_eq20}\\\\[10pt]\n\\lefteqn{\n\\bra{0} \\bar q(x) \\sigma_{\\mu\\nu} s(-x) \\ket{ K^*(P,\\lambda)} =}\\hspace*{1cm} \\nonumber\\\\\n&&{}i f_{K^*}^\\perp \\left\\{ (e_\\mu^{(\\lambda)} P_\\nu - e_\\nu^{(\\lambda)}   P_\\mu) \\int_0^1 du \\, e^{i\\xi P\\cdot x} \\left[ \\phi_{2;K^*}^\\perp(u) +   \\frac{1}{4}\\, m_{K^*}^2 x^2  \\phi^\\perp_{4;K^*}(u)\\right]\\right. \\nonumber\\\\\n&&{}\n+ (P_\\mu x_\\nu - P_\\nu x_\\mu)\\,\\frac{e^{(\\lambda)}x}{(P\\cdot x)^2}\\,m_{K^*}^2 \\int_0^1 du \\,\ne^{i\\xi P\\cdot x} \\left[ \\phi_{3;K^*}^\\parallel(u) -   \\frac{1}{2}\\phi_{2;K^*}^\\perp(u) -   \\frac{1}{2}\\psi_{4;K^*}^\\perp(u)\\right] \\nonumber\\\\\n&&{}\\left.\n+ \\frac{1}{2}\\,(e_\\mu^{(\\lambda)} x_\\nu - e_\\nu^{(\\lambda)}x_\\mu)\\, \\frac{m_{K^*}^2}{P\\cdot x} \\int_0^1 du \\, e^{i\\xi P\\cdot x} \\left[ \\psi_{4;K^*}^\\perp(u)-\\phi_{2;K^*}^\\perp(u)  \\right]\\right\\}\\,.\n\\label{das_eq21}\n\\end{eqnarray}\nAll other DAs in the above relations are of twist-3 or -4 and all terms in the light-cone expansion of twist-5 and higher are neglected. The twist-4 DAs are shown for completeness. The normalisation of all DAs is given by\n\\begin{equation}\n\\int_0^1 du\\, \\phi(u) = 1\\,.\n\\end{equation}\nThe conformal expansions of the leading-twist DAs $\\phi_{2;K^*}^\\parallel$ and $\\phi_{2;K^*}^\\perp$ are given by Eqs.~(\\ref{das_eq16}) and (\\ref{das_eq19}) respectively. In the local limit $x^\\mu \\to 0$ the matrix elements (\\ref{das_eq20}) and (\\ref{das_eq21}) reduce to the longitudinal $f^{\\parallel}_{K^*}$ and transverse $f^{\\perp}_{K^*}$ decay constants;\n\\begin{eqnarray}\n\\bra{0} \\bar q(0) \\gamma_\\mu s(0) \\ket{K^*(P,\\lambda)} &=&  e_{\\mu}^{(\\lambda)}  m_{K^*} f^{\\parallel}_{K^*}\\,,\n\\label{das_eq22}\\\\\n\\bra{0} \\bar q(0) \\sigma_{\\mu\\nu} s(0) \\ket{ K^*(P,\\lambda)}&=& i \\left(e_{\\mu}^{(\\lambda)} P_\\nu-e_{\\nu}^{(\\lambda)}\nP_{\\mu}\\right)f^{\\perp}_{K^*}(\\mu)\\,.\n\\label{das_eq23}\n\\end{eqnarray}\nNote that $f^\\perp_{K^*}(\\mu)$ is scale-dependent because the tensor current is not conserved. Numerical values of the decay constants are discussed in Chapter~\\ref{chapter4_det}. The above DAs are related to those defined in Refs.~\\cite{Ball:1998sk,Ball:1998ff} by\n\\begin{equation}\n\\addtolength{\\arraycolsep}{5pt}\n\\begin{array}[b]{r@{\\ =\\ }l@{\\qquad}r@{\\ =\\ }l@{\\qquad}r@{\\ =\\ }l}\n\\phi_{2;K^*}^{\\parallel(\\perp)} & \\phi_{\\parallel(\\perp)}\\,, &\n\\phi_{3;K^*}^\\parallel & h_\\parallel^{(t)}\\,, & \\psi_{4;K^*}^\\parallel &\ng_3\\,,\\\\[10pt]\n\\phi_{4;K^*}^{\\parallel(\\perp)} & {\\mathbb A}_{(T)}\\,, &\n\\phi_{3;K^*}^\\perp & g_\\perp^{(v)}\\,, & \\psi_{4;K^*}^\\perp &\nh_3\\,.\n\\end{array}\n\\addtolength{\\arraycolsep}{-5pt}\n\\label{das_eq24}\n\\end{equation}\n\n\\section{Two-Particle Twist-3 Distribution Amplitudes}\\label{das_sec1.4}\nThe two-particle twist-3 DAs $\\phi_{3;K^*}^{\\perp,\\parallel}$ have already been defined in\nEqs.~(\\ref{das_eq20}) and (\\ref{das_eq21}). There are two more two-particle DAs,\n$\\psi_{3;K^*}^{\\perp,\\parallel}$, defined as:\\footnote{In the notations of Ref.~\\cite{Ball:1998sk}, $\\psi_{3;K^*}^\\perp =   \\{1-(f_{K^*}^\\perp\/f_{K^*}^\\parallel) (m_s+m_q)\/m_{K^*}\\} g_\\perp^{(a)}$, $\\psi_{3;K^*}^\\parallel =   \\{1-(f_{K^*}^\\parallel\/f_{K^*}^\\perp) (m_s+m_q)\/m_{K^*}\\} h_\\parallel^{(s)}$.}\n\\begin{eqnarray}\n\\bra{0}\\bar q(z) \\gamma_{\\mu} \\gamma_{5}s(-z)\\ket{K^*(P,\\lambda)}\n&=& \\frac{1}{2}\\,f_{K^*}^\\parallel m_{K^*} \\epsilon_{\\mu}^{\\phantom{\\mu}\\nu \\alpha \\beta}\ne^{(\\lambda)}_{\\nu} p_{\\alpha} z_{\\beta}\\int_{0}^{1} \\!du\\, e^{i \\xi p \\cdot z} \\psi_{3;K^*}^\\perp(u)\\,,\\\\\n\\bra{0}\\bar q(z)s(-z)\\ket{K^*(P,\\lambda)} & = & {} -i f_{K^*}^\\perp(e^{(\\lambda)} \\cdot z) m_{K^*}^{2}\n\\int_{0}^{1} \\!du\\, e^{i \\xi p \\cdot z} \\psi_{3;K^*}^\\parallel(u)\\,.\n\\end{eqnarray}\nThe normalisation is given by\n\\begin{equation}\n\\int_0^1 du\\, \\psi_{3;K^*}^{\\parallel(\\perp)}(u) = 1-\\frac{f_{K^*}^{\\parallel(\\perp)}}{f_{K^*}^{\\perp(\\parallel)}}\\,\\frac{m_s+m_q}{m_{K^*}}\\,,\n\\end{equation}\nwhich differs from Ref.~\\cite{Ball:1998sk}, where all DAs were normalised to 1; here we keep the full dependence on the quark masses. \n\n\\section{Three-Particle Twist-3 Distribution Amplitudes}\\label{das_sec1.5}\nThere are also three three-particle DAs of twist-3:\n\\begin{eqnarray}\n\\bra{0}\\bar q(z) g_s \\widetilde{G}_{\\beta z}(vz)\\gamma_{z}\\gamma_5 s(-z) \\ket{K^*(P,\\lambda)} & = & f_{K^*}^\\parallel m_{K^*} (p\\cdot z)^2\ne^{(\\lambda)}_{\\perp\\beta}\n\\widetilde\\Phi_{3;K}^\\parallel(v,p\\cdot z)+\\dots\\,,\\\\\n\\bra{0}\\bar q(z) g_s G_{\\beta z}(vz)i\\gamma_{z} s(-z) \\ket{K^*(P,\\lambda)} & = & \nf_{K^*}^\\parallel m_{K^*} (p\\cdot z)^2\ne^{(\\lambda)}_{\\perp\\beta} \\Phi_{3;K}^\\parallel(v,p\\cdot z)+\\dots\\,,\\nonumber\\\\\n\\bra{0}\n\\bar q(z) g_s G_{z\\beta}(vz)\\sigma_{z\\beta} s(-z) \\ket{K^*(P,\\lambda)} & = & f_{K^*}^\\perp m_{K^*}^2\n(e^{(\\lambda)}\\cdot z)(p\\cdot z) \\Phi_{3;K^*}^\\perp(v,p\\cdot z)+\\dots\\,,\\nonumber\n\\label{das_eq25}\n\\end{eqnarray}\nwhere the dots denote terms of higher twist and we use the short-hand notation\n\\begin{equation}\n{\\cal F}(v,p\\cdot z) = \\int {\\cal D}\\underline{\\alpha}\\,e^{-ip\\cdot z(\\alpha_2-\\alpha_1+v\\alpha_3)} {\\cal F}(\\underline{\\alpha}) \n\\end{equation}\nwith ${\\cal F}(\\underline{\\alpha})$ being a three-particle DA  and the integration measure ${\\cal D}\\underline{\\alpha}$ is defined as\n\\begin{equation}\n\\int{\\cal D}\\underline{\\alpha} \\equiv \\int_0^1 d\\alpha_1 d\\alpha_2 d\\alpha_3\\, \\delta(1-\\sum \\alpha_i)\\,.\n\\label{das_eq26}\n\\end{equation}\nThe twist-3 three-particle DAs correspond to the light-cone projection $\\gamma_z G_{z\\perp}$ and $\\sigma_{\\perp z} G_{\\perp z}$, respectively, which picks up the $s=\\frac{1}{2}$ component of the\nquark fields and the $s=1$ component of the gluonic field strength tensor. According to Eq.~(\\ref{das_eq18}), the (normalised) asymptotic DA is then given by $360\\,\\alpha_1\\alpha_2\\alpha_3^2$. To NLO in the conformal expansion, each three-particle twist-3 DA involves three hadronic parameters, which we label in the following way: $\\zeta,\\kappa$ are LO and $\\omega,\\lambda$ NLO parameters.  $\\zeta$ and $\\omega$ are G-parity conserving, whereas \n$\\kappa$ and $\\lambda$ violate G-parity and hence vanish for mesons with quarks of equal mass, i.e.\\  $\\rho$, $\\omega$ and $\\phi$. We then have\n\\begin{eqnarray}\n\\Phi_{3;K^*}^\\parallel(\\underline{\\alpha})\n & = & 360\\alpha_1\\alpha_2\\alpha_3^2 \\left\\{\n\\kappa_{3K^*}^\\parallel + \\omega_{3K^*}^\\parallel (\\alpha_1-\\alpha_2) +\n\\lambda_{3K^*}^\\parallel \\frac{1}{2}\\,(7\\alpha_3 -\n3)\\right\\},\\nonumber\\\\\n\\widetilde\\Phi_{3;K^*}^\\parallel(\\underline{\\alpha}) & = & \n360\\alpha_1\\alpha_2\\alpha_3^2 \\left\\{\n\\zeta_{3K^*}^\\parallel + \\widetilde\\lambda_{3K^*}^\\parallel (\\alpha_1-\\alpha_2) +\n\\widetilde\\omega_{3K^*}^\\parallel \\frac{1}{2}\\,(7\\alpha_3 -\n3)\\right\\},\\nonumber\\\\\n\\Phi_{3;K^*}^\\perp(\\underline{\\alpha}) \n& = & 360\\alpha_1\\alpha_2\\alpha_3^2 \\left\\{\n\\kappa_{3K^*}^\\perp + \\omega_{3K^*}^\\perp (\\alpha_1-\\alpha_2) +\n\\lambda_{3K^*}^\\perp \\frac{1}{2}\\,(7\\alpha_3 -\n3)\\right\\}\\,.\n\\label{das_eq27}\n\\end{eqnarray}\nThe parameters defined above are related to those of Ref.~\\cite{Ball:1998sk} by:\n\\begin{eqnarray}\n\\zeta_{3}^A = \\zeta^\\parallel_{3},&\\qquad & \\zeta_{3}^V =\\omega_{3}^\\parallel\/14,  \\nonumber\\\\[10pt]\n\\zeta_{3}^T = \\omega_{3}^\\perp\/14,&\\qquad &\\zeta_3^\\parallel\\,\\omega^A_{1,0} =\n\\widetilde\\omega_{3}^\\parallel\\,.\n\\label{das_eq28}\n\\end{eqnarray}\nG-parity breaking terms were not considered in Ref.~\\cite{Ball:1998sk}. For equal mass quarks, $\\Phi^{\\perp,\\parallel}_{3;K^*}$ are antisymmetric under $\\alpha_1\\leftrightarrow \\alpha_2$, whereas $\\widetilde\\Phi^{\\parallel}_{3;K^*}$ is symmetric. All these parameters can be defined in terms of matrix elements of local twist-3 operators. For chiral-odd operators, for instance, one has\n\\begin{eqnarray}\n\\bra{0}\\bar q \\sigma_{z\\xi} g_sG_{z\\xi}\ns\\ket{K^*(P,\\lambda)} & = & f_{K^*}^\\perp m_{K^*}^2 \n(e^{(\\lambda)}\\cdot z)(p\\cdot z)\\kappa_{3K^*}^\\perp\\,,\\nonumber\\\\\n\\bra{0}\\bar q \\sigma_{z\\xi}  [iD_z,g_sG_{z\\xi}] s\n- \\frac{3}{7}\\, i\\partial_z \\bar q \\sigma_{z\\xi} \ng_sG_{z\\xi} s \\ket{K^*(P,\\lambda)} \n& = & f_{K^*}^\\perp m_{K^*}^2 (e^{(\\lambda)}\\cdot z)(p\\cdot z)^2\n\\,\\frac{3}{28}\\, \\lambda_{3K^*}^\\perp\\,,\n\\nonumber\\\\\n\\bra{0} \\bar q i \\!\\stackrel{\\leftarrow}{D}_z\\!\n\\sigma_{z\\xi}  g_sG_{z\\xi} s - \\bar q\n\\sigma_{z\\xi}  g_sG_{z\\xi} i\\! \\stackrel{\\rightarrow}{D}_z\\! s\n \\ket{K^*(P,\\lambda)}  & = & f_{K^*}^\\perp m_{K^*}^2 (e^{(\\lambda)}\\cdot z)(p\\cdot z)^2\n\\,\\frac{1}{14}\\, \\omega_{3K^*}^\\perp\\,;\\nonumber\n\\label{das_eq29}\n\\end{eqnarray}\nthe formulas for chiral-even operators are analogous.  In Chapter~\\ref{chapter4_det} we calculate numerical values for all the parameters in Eq.~(\\ref{das_eq27}) from QCD sum rules.\n\n\\section{Relations Between Distribution Amplitudes}\nThe QCD EOM are a crucial ingredient in simplifying the kinematic contributions of different operators, a task which is facilitated by the fact that they are preserved to all orders in the conformal expansion. The EOM relate via integral equations the two-particle twist-3 DAs, defined in Sections~\\ref{das_sec1.3} and \\ref{das_sec1.4}, to the two-particle twist-2 DAs, Eqs.~(\\ref{das_eq16}) and (\\ref{das_eq19}), and three-particle twist-3 DAs, defined in Section~\\ref{das_sec1.5}. We do not quote the EOM themselves for which we refer the reader to the literature. The framework for the procedure was developed in Ref.~\\cite{Braun:1989iv} based on deriving the EOM for non-local light-ray operators \\cite{Balitsky:1987bk}. The operator relations are then sandwiched between the vacuum and meson states and the definitions of the DAs used to convert them into integral equations, making use of partial integration to remove explicit dependence on co-ordinates and momentum 4-vectors. The resulting expressions are then solved order-by-order in the conformal expansion, see Ref.~\\cite{Braun:2003rp} for an overview. The EOM contain mass dependent contributions $\\propto m_s\\pm m_q$ that were calculated in Ref.~\\cite{Ball:1998sk}. In the present analysis,  G-parity breaking terms of the three-particle twist-3 DAs are included, which then, via the EOM, impact on the two-particle DAs \\cite{Ball:2007rt}. The resulting integral equations are:\n\\begin{eqnarray}\n\\psi_{3;K^*}^\\parallel(u) & = & \\bar{u}\\int_0^u dv\\,\\frac{1}{\\bar v}\\,\n\\Upsilon(v) + u \\int_u^1 dv\\,\\frac{1}{v}\\, \\Upsilon(v)\\,,\\nonumber\\\\\n\\phi_{3;K^*}^\\parallel(u) \n& = & \\frac{1}{2}\\,\\xi\\left[\\int_0^u dv\\,\\frac{1}{\\bar v}\\,\n\\Upsilon(v) - \\int_u^1 dv\\,\\frac{1}{v}\\, \\Upsilon(v)\\right] + \n\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\,\\frac{m_s+m_q}{m_{K^*}}\\,\n\\phi_{2;K^*}^\\parallel(u)\\nonumber\\\\\n&& {}+\\frac{d}{du}\\,\\int_0^u d\\alpha_1\\int_0^{\\bar u} d\\alpha_2\n\\frac{1}{\\alpha_3}\n\\,{\\Phi}^\\perp_{3;K^*}(\\underline{\\alpha})\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\Upsilon(u) & = & 2\\phi_{2;K^*}^\\perp(u) - \n\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\,\\frac{m_s+m_q}{m_{K^*}} \\left[\n1 - \\frac{1}{2}\\,\n\\xi\\frac{d}{du}\\right]\\phi_{2;K^*}^\\parallel(u) -\n\\frac{1}{2}\\,\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\,\n\\frac{m_s-m_q}{m_{K^*}}\\,\\frac{d}{du}\\,\\phi_{2;K^*}^\\parallel(u)\\nonumber\\\\\n& & {}+\\frac{d}{du}\\,\\int_0^u\\!\\!d\\alpha_1\\int_0^{\\bar\nu}\\!\\! d\\alpha_2\n\\,\\frac{1}{\\alpha_3}\\left(\\alpha_1\\,\\frac{d}{d\\alpha_1} +\n\\alpha_2\\,\\frac{d}{d\\alpha_2}\\, - 1\\right) \n\\Phi_{3;K^*}^\\perp(\\underline{\\alpha})\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\psi_{3;K^*}^\\perp(u)& = & \\bar{u}\\int_0^u dv\\,\\frac{1}{\\bar v}\\,\n  \\Omega(v) + u \\int_u^1 dv\\,\\frac{1}{v}\\, \\Omega(v)\\,,\\nonumber\\\\\n\\phi_{3;K^*}^\\perp(u) \n& = & \\frac{1}{4}\\left[\\int_0^u dv\\,\\frac{1}{\\bar v}\\,\n\\Omega(v) + \\int_u^1 dv\\,\\frac{1}{v}\\, \\Omega(v)\\right] + \n\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\,\\frac{m_s+m_q}{m_{K^*}}\\,\n\\phi_{2;K^*}^\\perp(u)\\nonumber\\\\\n&& {}+\\frac{d}{du}\\,\\int_0^u d\\alpha_1\\int_0^{\\bar u} d\\alpha_2\n\\frac{1}{\\alpha_3}\n\\,{\\Phi}^\\parallel_{3;K^*}(\\underline{\\alpha})\\nonumber\\\\\n&&{} + \\int_0^u d\\alpha_1\\int_0^{\\bar u}\nd\\alpha_2\\,\\frac{1}{\\alpha_3}\\left( \\frac{d}{d\\alpha_1} +\n\\frac{d}{d\\alpha_2} \\right) \n\\widetilde{\\Phi}_{3;K^*}^\\parallel(\\underline{\\alpha})\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\Omega(u)  &=&  2\\phi_{2;K^*}^\\parallel(u) + \n\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\,\\frac{m_s+m_q}{m_{K^*}}\\, \n\\xi\\,\\frac{d}{du}\\,\\phi_{2;K^*}^\\perp(u)-\n\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\,\\frac{m_s-m_q}{m_{K^*}}\\, \n\\frac{d}{du}\\,\\phi_{2;K^*}^\\perp(u)\\nonumber\\\\\n& & {}+2\\frac{d}{du}\\,\\int_0^u\\!\\!d\\alpha_1\n\\int_0^{\\bar u}\\!\\! d\\alpha_2\n\\,\\frac{1}{\\alpha_3}\\left(\\alpha_1\\,\\frac{d}{d\\alpha_1} +\n\\alpha_2\\,\\frac{d}{d\\alpha_2}\\right)\n\\Phi_{3;K^*}^\\parallel(\\underline{\\alpha})\\nonumber\\\\\n&&{} +2\\,\\frac{d}{du}\\,\\int_0^u\\!\\!d\\alpha_1\n\\int_0^{\\bar u}\\!\\! d\\alpha_2\n\\,\\frac{1}{\\alpha_3}\\left(\\alpha_1\\,\\frac{d}{d\\alpha_1} -\n\\alpha_2\\,\\frac{d}{d\\alpha_2}\\right) \n\\widetilde\\Phi_{3;K^*}^\\parallel(\\underline{\\alpha})\\,.\n\\end{eqnarray}\nUsing Eq.~(\\ref{das_eq27}), and the corresponding relations for twist-2 DAs, one obtains expressions for the twist-3 two-particle DAs, which are valid to NLO in the conformal expansion. As discussed in Ref.~\\cite{Ball:1998sk}, the structure of this expansion is complicated by the fact that  these DAs do not correspond to a fixed Lorentz-spin projection $s$ of the quark fields.  The resulting expansion is in $C^{3\/2}_n(\\xi)$ for $\\psi^{\\perp,\\parallel}_{3;K^*}$ and $C^{1\/2}_n(\\xi)$ for $\\phi^{\\perp,\\parallel}_{3;K^*}$:\n\\begin{eqnarray}\n\\phi_{3;K^*}^\\parallel(u) & = & 3\\xi^2 + \\frac{3}{2}\\,\\xi(3\\xi^2-1)\na_1^\\perp + \\frac{3}{2}\\,\\xi^2 (5\\xi^2-3)a_2^\\perp\\nonumber\\\\\n&&{} + \\left(\\frac{15}{2}\\,\\kappa_{3K^*}^\\perp -\n\\frac{3}{4}\\,\\lambda_{3K^*}^\\perp\\right) \\xi (5\\xi^2-3) +\n\\frac{5}{8}\\,\\omega_{3K^*}^\\perp (3-30\\xi^2 + 35\\xi^4) \\nonumber\\\\\n&&{}+\\frac{3}{2}\\,\\frac{m_s+m_q}{m_{K^*}}\\,\n\\frac{f^\\parallel_{K^*}}{f_{K^*}^\\perp}\n\\left\\{ 1 + 8\\xi a_1^\\parallel + 3 (7-30 u\\bar{u} )a_2^\\parallel + \\xi \\ln\n\\bar{u} (1+3 a_1^\\parallel + 6 a_2^\\parallel)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.6cm} - \\xi \\ln u (1-3\na_1^\\parallel + 6 a_2^\\parallel)\\right\\}\\nonumber\\\\\n&&{}-\\frac{3}{2}\\,\\frac{m_s-m_q}{m_{K^*}}\\,\n\\frac{f^\\parallel_{K^*}}{f_{K^*}^\\perp}\\,\n\\xi \\left\\{ 2 + 9\\xi a_1^\\parallel + 2 (11-30 u\\bar{u} )a_2^\\parallel + \\ln\n\\bar{u} (1+3 a_1^\\parallel + 6 a_2^\\parallel)\\right.\\nonumber\\\\\n&&{}\\hspace*{3.9cm}\\left. + \\ln u (1-3\na_1^\\parallel + 6 a_2^\\parallel)\\right\\},\n\\label{das_eq30}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\psi_{3;K^*}^\\parallel(u) & = & 6 u\\bar u \\left\\{ 1 + \\left(\n\\frac{a_1^\\perp}{3} + \\frac{5}{3} \\kappa_{3K^*}^\\perp\\!\\right)\nC_1^{3\/2}(\\xi) + \\left( \\frac{a_2^\\perp}{6}  + \\frac{5}{18}\n\\omega_{3K^*}^\\perp\\right) C_2^{3\/2}(\\xi) -\n\\frac{1}{20}\\lambda_{3K^*}^\\perp C_3^{3\/2}(\\xi)\\right\\}\\nonumber\\\\\n&&{}+ 3\\,\n\\frac{m_s+m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\left\\{\nu\\bar u (1 + 2 \\xi a_1^\\parallel + 3 (7-5u\\bar u) a_2^\\parallel) + \\bar\nu \\ln \\bar u (1+3 a_1^\\parallel + 6 a_2^\\parallel)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.2cm} + u \\ln u \n(1-3 a_1^\\parallel + 6 a_2^\\parallel)\\right\\}\\nonumber\\\\\n&&- 3\\,\n\\frac{m_s-m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\left\\{\nu\\bar u (9 a_1^\\parallel + 10\\xi a_2^\\parallel) + \\bar\nu \\ln \\bar u (1+3 a_1^\\parallel + 6 a_2^\\parallel)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.2cm} - u \\ln u \n(1-3 a_1^\\parallel + 6 a_2^\\parallel)\\right\\},\n\\label{das_eq31}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\psi_{3;K^*}^\\perp(u) & = & 6 u\\bar u \\left\\{ 1 + \\left(\n\\frac{1}{3} a_1^\\parallel + \\frac{20}{9} \\kappa_{3K^*}^\\parallel\\right)\nC_1^{3\/2}(\\xi) \n\\right.\\nonumber\\\\\n&&\\left.\\hspace*{0pt} + \\left( \\frac{1}{6} a_2^\\parallel +\n\\frac{10}{9}\\zeta_{3K^*}^\\parallel + \\frac{5}{12}\\,\n\\omega_{3K^*}^\\parallel-\\frac{5}{24}\\,\n\\widetilde\\omega_{3K^*}^\\parallel\\!\\right)\nC_2^{3\/2}(\\xi) +\n\\left(\\frac{1}{4}\\widetilde\\lambda_{3K^*}^\\parallel - \\frac{1}{8}\n\\lambda_{3K^*}^\\parallel\\right) C_3^{3\/2}(\\xi)\\right\\}\\nonumber\\\\\n&&{}+ 6\\,\n\\frac{m_s+m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\left\\{\nu\\bar u (2 + 3 \\xi a_1^\\perp + 2 (11-10u\\bar u) a_2^\\perp) + \\bar\nu \\ln \\bar u (1+3 a_1^\\perp + 6 a_2^\\perp)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.2cm} + u \\ln u \n(1-3 a_1^\\perp + 6 a_2^\\perp)\\right\\}\\nonumber\\\\\n&&- 6\\,\n\\frac{m_s-m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\left\\{\nu\\bar u (9 a_1^\\perp + 10\\xi a_2^\\perp) + \\bar\nu \\ln \\bar u (1+3 a_1^\\perp + 6 a_2^\\perp)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.2cm} - u \\ln u \n(1-3 a_1^\\perp + 6 a_2^\\perp)\\right\\},\n\\label{das_eq32}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\phi_{3;K^*}^\\perp(u) & = & \\frac{3}{4}\\,(1+\\xi^2) + \\frac{3}{2}\\,\\xi^3\na_1^\\parallel + \\left\\{ \\frac{3}{7}\\,a_2^\\parallel + 5\n\\zeta_{3K^*}^\\parallel \\right\\} (3\\xi^2-1) + \\left\\{5\\kappa_{3K^*}^\\parallel\n- \\frac{15}{16}\\,\\lambda_{3K^*}^\\parallel \\right.\\nonumber\\\\\n&&{} \\left. +\n\\frac{15}{8}\\,\\widetilde{\\lambda}_{3K^*}^\\parallel \\right\\} \\xi\n(5\\xi^2-3)+ \\left\\{ \\frac{9}{112}\\,a_2^\\parallel +\n\\frac{15}{32}\\,\\omega_{3K^*}^\\parallel -\n\\frac{15}{64}\\,\\widetilde\\omega_{3K^*}^\\parallel \\right\\} (35\\xi^4-30\n\\xi^2+3)\\nonumber\\\\\n&&{}+\\frac{3}{2}\\,\\frac{m_s+m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\perp}{\nf_{K^*}^\\parallel} \\left\\{ 2 + 9 \\xi a_1^\\perp + 2 (11-30 u \\bar{u})\n    a_2^\\perp\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.5cm} + \n(1-3 a_1^\\perp + 6 a_2^\\perp) \\ln u + (1+3 a_1^\\perp + 6 a_2^\\perp) \n\\ln \\bar{u} \\right\\}\\nonumber\\\\\n&&{}-\\frac{3}{2}\\,\\frac{m_s-m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\perp}{\nf_{K^*}^\\parallel} \\left\\{ 2\\xi + 9 (1-2u \\bar{u}) a_1^\\perp + 2 \\xi (11-20 u \\bar{u})\n    a_2^\\perp\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.5cm} + \n(1+3 a_1^\\perp + 6 a_2^\\perp) \\ln \\bar{u} - (1-3 a_1^\\perp + 6 a_2^\\perp) \n\\ln u \\right\\}\\,.\n\\label{das_eq33}\n\\end{eqnarray}\nThese expressions  supersede those given in Ref.~\\cite{Ball:1998sk} where G-parity violating terms in $\\kappa_3$ and $\\lambda_3$ were not included. The DAs given above now contain a minimum number of parameters which can be determined from one's favourite method, such as QCD sum rules or Lattice QCD.  We briefly mention that the matrix elements of QCD operator identities can also be used to relate twist-2 and twist-4 DA parameters to each other; such investigations were performed for the G-partiy violating twist-2  $K^*$ parameters $a_1^{\\parallel,\\perp} (K^*)$ in Refs.~\\cite{Braun:2004vf,Ball:2006fz}.\n\n\\section{Evolution Equations}\\label{evolution}\nThe scale dependence of the leading-twist DAs of Eqs.~(\\ref{das_eq16}) and (\\ref{das_eq19}) can be investigated using perturbation theory. The resulting renormalisation-group equation is the \\textit{Efremov-Radyushkin-Brodsky-Lepage (ER-BL) evolution equation} \\cite{Lepage:1980fj,Efremov:1979qk,Efremov:1978rn,Lepage:1979zb}\n\\begin{equation}\n\\mu^2 \\frac{d}{d \\mu^2}\\, \\phi(u,\\mu)=\\int^1_0dv\\,V(u,v;\\alpha_s(\\mu))\\,\\phi(v,\\mu)\\,,\n\\label{das_eq34}\n\\end{equation}\nwhich completely specifies $\\phi(u,\\mu)$ given $\\phi(u,\\mu_0)$. The kernel is given by an expansion in $\\alpha_s$\n\\begin{equation}\nV(u,v;\\alpha_s(\\mu)) = \\frac{\\alpha_s(\\mu)}{4 \\pi} V^{(0)}(u,v)+ \\left(\\frac{\\alpha_s(\\mu)}{4 \\pi}\\right)^2 V^{(1)}(u,v)+\\dots \\,.\n\\end{equation}\nThe evolution equation (\\ref{das_eq34}) can be solved readily at leading-order using the conformal expansion \\cite{Brodsky:1980ny, Ohrndorf:1981qv, Makeenko:1980bh, Brodsky:1980XX}. This amounts to finding its eigenfunctions, which we already know to be Gegenbauer polynomials, and using this fact it can be shown that the leading-order kernel can be written as\n\\begin{equation}\nV^{(0)}(u,v)=-6u \\bar{u} \\sum^\\infty_{n=0} \\frac{4(2n+3)}{(n+1)(n+2)}\\gamma_n^{(0)}C_n^{3\/2}(2u-1)C_n^{3\/2}(2v-1)\\,,\n\\end{equation}\ngiving the LO anomalous dimensions of the Gegenbauer coefficients $\\gamma_n$. The renormalisation is hence multiplicative at leading-order\n\\begin{equation}\na^{\\rm LO}_n(\\mu^2)=a_n(\\mu^2_0)L^{\\gamma^{(0)}_n \/(2 \\beta_0)}\\,,\n\\end{equation}\nwhere $L=\\alpha_s(\\mu^2)\/\\alpha_s(\\mu^2_0)$. The one-loop anomalous dimensions of the twist-2 Gegenbauer coefficients are \\cite{Gross:1973id1,Shifman:1980dk}\n\\begin{eqnarray}\n\\gamma^{\\parallel(0)}_{(n)}&=&8 C_F \\left(\\sum_{k=1}^{n+1}\\frac{1}{k} -\\frac{3}{4}-\\frac{1}{2(n+1)(n+2)}\\right)\\,,\n\\label{das_eq35}\\\\\n\\gamma^{\\perp(0)}_{(n)}&=&8 C_F\\left(\\sum_{k=1}^{n+1} \\frac{1}{k}-\\frac{3}{4}\\right)\\,.\n\\label{das_eq36}\n\\end{eqnarray}\nAt next-to-leading-order the scale dependence is more complicated \\cite{Mueller:1993hg,Mueller:1994cn}\n\\begin{equation}\na_n^{\\rm  NLO}(\\mu^2)=a_n(\\mu^2_0) E_n^{\\rm NLO}+\\frac{\\alpha_s}{4 \\pi}\\sum_{k=0}^{n-2} a_k(\\mu^2_0)L^{\n\\gamma^{(0)}_k \/(2 \\beta_0)} d_{nk}^{(1)}\\,,\n\\label{evo}\n\\end{equation}\nwhere\n\\begin{equation}\nE_n^{\\rm  NLO}=L^{\\gamma^{(0)}_k \/(2 \\beta_0)}\\left\\{1+\\frac{\\gamma_n^{(1)} \\beta_0 - \\gamma_n^{(0)} \\beta_1}{8 \\pi \\beta_0^2}\\left[\\alpha_s(\\mu^2)-\\alpha_s(\\mu^2_0)\\right]\\right\\}\\,.\n\\end{equation}\n$\\gamma_n^{(1)}$ are the diagonal two-loop anomalous dimensions which are available for the vector current \\cite{Floratos}, and the tensor current \\cite{Haya}. The mixing coefficients $d^{(1)}_{nk}$ are given in closed form in Refs.~\\cite{Mueller:1993hg,Mueller:1994cn} where the formulas are valid for arbitrary currents by substitution of the corresponding one-loop anomalous dimension. For the lowest moments $n=\\{0,1,2\\}$ one has\n\\begin{eqnarray}\n\\gamma_0^{\\parallel(1)} =0\\,, \\qquad \n\\gamma_1^{\\parallel(1)} = \\frac{23110}{243} - \\frac{512}{81}\\, N_f\\,,\\qquad  \n\\gamma_2^{\\parallel(1)} =  \\frac{34072}{243}-\\frac{830}{81}\\, N_f\\,,\\\\[10pt]\n\\gamma_0^{\\perp(1)} =\\frac{724}{9} - \\frac{104}{27}\\,N_f\\,, \\qquad \n\\gamma_1^{\\perp(1)} = 124 - 8 N_f\\,,\\qquad  \n\\gamma_2^{\\perp(1)} =  \\frac{38044}{243}-\\frac{904}{81}\\, N_f\\,,\n\\end{eqnarray}  \nand\n\\begin{equation}\n  d^{\\parallel(1)}_{20}  = \n  \\frac{35}{9}\\,\\frac{20-3\\beta_0}{50-9\\beta_0} \\left(1-L^{50\/(9\\beta_0)-1}\\right)\\,, \\quad  d^{\\perp(1)}_{20}  =   \\frac{28}{9}\\,\\frac{16-3\\beta_0}{40-9\\beta_0}  \\left(1-L^{40\/(9\\beta_0)-1}\\right)\\,.\n\\end{equation}\nIt is evident from Eqs.~(\\ref{das_eq35}) and  (\\ref{das_eq36}) that the anomalous dimensions of the Gegenbauer coefficients increase logarithmically  with conformal spin. This implies that as $\\mu \\to \\infty$ the coefficients higher in conformal spin are damped and the DA approaches its asymptotic form (\\ref{das_eq18})\n\\begin{equation}\n\\lim_{\\mu \\to \\infty} \\phi_{2;K^*}(u,\\mu^2) \\to 6 u \\bar{u}\\,.\n\\end{equation}\nThis limit offers a great simplification in that if it can be verified, experimentally of otherwise, that a given process is well described by the asymptotic form of the DA at hadronic scales, there are no non-perturbative parameters to be determined. The convergence of the conformal expansion in general has to be verified case by case and there is no \\textit{a priori} reason why it should do so. In practice one has to truncate the expansion at some order in conformal spin, usually $n=2$, and as this constitutes an approximation it thus introduces a model dependent assumption. In Ref.~\\cite{Ball:2005ei} for example,  these issues are discussed and an alternative method suggested.\n\nThe scale dependence of the three-particle twist-3 DAs can in principle be deduced from evolution equations for the non-local operators in (\\ref{das_eq25}) using the techniques of Ref.~\\cite{Balitsky:1987bk}. The evolution of the parameters in (\\ref{das_eq27}) could then be found by projecting out the desired conformal spin, as  for the leading-twist DA. Another approach is to consult the literature of results for the corresponding nucleon structure functions \\cite{andim}. The three-particle twist-3 parameters $\\zeta_{3K^*}^\\parallel$, $\\kappa_{3K^*}^{\\perp,\\parallel}$, $\\omega_{3K^*}^\\perp$ and $\\lambda_{3K^*}^\\perp$ renormalise multiplicatively in the chiral limit, and the others mix with each other. For non-zero strange quark mass, there is additional mixing with twist-2 parameters with the mass corrections featuring as $m_s \\pm m_q$ depending on the G-parity of the parameter. Here, we write down explicitly only the renormalisation-group improved relations for the above  5 parameters.   The relations can be written in compact form as\n\\begin{equation}\nP_i(\\mu^2) = L^{(\\gamma_P)_i\/\\beta_0}\\, P_i(\\mu_0^2) + \\sum_{j=1}^3\nC_{ij} \\left( L^{(\\gamma_Q)_{ij}\/\\beta_0} -\nL^{(\\gamma_P)_i\/\\beta_0}\\right) Q_{ij}(\\mu_0^2)\\,,\n\\label{das_eq37}\n\\end{equation}\nwhere the parameters are given by:\n\\begin{eqnarray}\nP & = & \\{f_{K^*}^\\parallel \\zeta_{3K^*}^\\parallel,\\,f_{K^*}^\\parallel\n\\kappa_{3K^*}^\\parallel,\\,f_{K^*}^\\perp \\kappa_{3K^*}^\\perp,\\,f_{K^*}^\\perp\n\\omega_{3K^*}^\\perp,\\,f_{K^*}^\\perp \\lambda_{3K^*}^\\perp\\}\\,,\\nonumber\\\\\nQ_{1(2)} &=& \\frac{f_{K^*}^\\perp}{m_{K^*}}\\,\\{\nm_s\\pm m_q,\\,(m_s\\mp m_q) a_1^\\perp,\\,(m_s\\pm m_q)a_2^\\perp\\}\\,,\\nonumber\\\\\nQ_{3,5} &=& \\frac{f_{K^*}^\\parallel}{m_{K^*}}\\,\\{\nm_s-m_q,\\,(m_s+m_q) a_1^\\parallel,\\,(m_s-m_q) a_2^\\parallel\\}\\,,\\nonumber\\\\\nQ_{4}& =& \\frac{f_{K^*}^\\parallel}{m_{K^*}}\\,\\{\nm_s+m_q,\\,(m_s-m_q) a_1^\\parallel,\\,(m_s+m_q) a_2^\\parallel\\}\\,,\n\\nonumber\\\\\n\\gamma_P & = & \\left\\{\n\\frac{77}{9},\\,\\frac{77}{9},\\,\\frac{55}{9},\\,\\frac{73}{9},\\,\n\\frac{104}{9}\\right\\}, \\nonumber\\\\\n(\\gamma_Q)_{1,2} & = & \\left\\{ \\frac{16}{3},\\, 8,\\,\n\\frac{88}{9} \\right\\},\\qquad\n(\\gamma_Q)_{3,4,5}  =  \\left\\{ 4,\\, \\frac{68}{9},\\,\n\\frac{86}{9} \\right\\}\\,,\n\\end{eqnarray}\n\\begin{eqnarray}\nC & = & \\left(\n\\begin{array}{r@{\\hskip10pt}r@{\\hskip10pt}r}\n\\displaystyle \\frac{2}{29} & \\displaystyle \\frac{6}{25} & \\displaystyle 0\\\\[10pt]\n\\displaystyle -\\frac{2}{29} & \\displaystyle-\\frac{6}{25} & 0\\\\[10pt]\n\\displaystyle-\\frac{4}{19} & \\displaystyle \\frac{12}{65} & 0\\\\[10pt]\n\\displaystyle \\frac{14}{37} & \\displaystyle-\\frac{42}{25} & \\displaystyle\\frac{12}{13}\\\\[10pt]\n\\displaystyle -\\frac{1}{85} & \\displaystyle -\\frac{1}{5} & \\displaystyle-\\frac{4}{15}\n\\end{array} \\right)\\,.\n\\end{eqnarray}\nWe refrain from delving into a full discussion here of the mixing including the remaining parameters $\\lambda_3^\\parallel$, $\\omega_3^\\parallel$, and guide the reader to Appendix A of Ref.~\\cite{Ball:2007rt} for details. \n\\chapter{QCD Sum Rules}\\label{chapter3_SR}\nThe original QCD sum rule approach was introduced by the revolutionary work of Shifman, Vainshtein and Zakharov in Refs.~\\cite{Shifman:1978bz, Shifman:1978by, Shifman:1978bx}, and has proven itself to be one  of the most effective tools for determining non-perturbative parameters of low-lying hadronic states. It does so in terms of a finite number of  universal non-perturbative input parameters, and as such has great predictive power. The approach has been massively successful in ascertaining  a wide range of phenomena of non-perturbative origin.  QCD sum rules are particularly advantageous for $B$ physics because the presence of an  intrinsic heavy-quark mass scale $m_b$ provides the necessary conditions required for the application of the short-distance OPE or light-cone  expansion of relevant correlation functions, from which the relevant quantities can be extracted. The heavy-quark limit $m_b\\to\\infty$  is not necessary and sum rules can be derived in full QCD for finite $m_b$. Despite its successes, the  method is limited by an inherent  irreducible systematic uncertainty of $20 - 30\\%$. However, such is the relative ease of the QCD sum rule method, as compared to, for example, Lattice QCD, that its place in the tool-box of the QCD practitioner is ensured.\n\nFirstly we  discuss step-by-step the methodology of the original QCD sum rule approach. Secondly we discuss its modification to accommodate non-local correlation functions which aides the extraction of DA parameters of beyond leading-order in conformal spin. Thirdly, we outline an extension of the original\napproach; light-cone sum rules. All three methods find application in this thesis:\n\\begin{itemize}\n\\item{in Chapter~\\ref{chapter4_det} we make use of the non-local formalism to extract numerical values for the leading twist-2 and twist-3 DA parameters defined in the Chapter~\\ref{chapter2_DAs}.}\n\\item{In Chapter~\\ref{chapter5_eta} we calculate important contributions to the semileptonic $B\\to \\eta^{(\\prime)}$ transition form factors in light-cone sum rules.}\n\\item{An example of the original method is presented in the last section of this chapter, section \\ref{example}, where we calculate the $\\alpha_s$ corrections to the gluon condensate contribution from a local correlation function.}\n\\end{itemize}\n We focus solely on the points required for future chapters. For more information on sum rules, see for example Refs.~\\cite{Colangelo:2000dp,Narison:1989aq,Shifman:1992xu,Reinders:1984sr}.\n\n\\section{SVZ Sum Rules}\nThe original sum rule method, which we refer to as \\textit{SVZ sum rules}, parameterises unknown non-perturbative QCD vacuum effects in terms of the so-called universal \\textit{vacuum condensates}. These quantities are vacuum expectation values of local operators $O_i$ that vanish in perturbation theory by definition and are ordered by their dimension $D$.\n\nThe calculation of a QCD sum rule starts from the calculation in QCD of a suitable correlation function in which the mesons are represented by interpolating currents possessing the correct quantum numbers. The method proceeds by equating two different representations of the correlation function. The first is obtained by performing a short-distance OPE, the result of which is matched to a second representation, in terms of a dispersion relation over physical hadronic states, leading to a sum rule from which various properties of the hadronic states can be extracted.\n\nThe SVZ sum rules find an important application in determining the universal hadronic parameters that appear in meson DAs.  Indeed, some of the first SVZ sum rule calculations were performed to extract decay constants $f_{\\pi, \\rho}$ \\cite{Shifman:1978bx}, and Gegenbauer coefficients  $a_n$ \\cite{Zhitnitsky:1982dd,Zhitnitsky:1983dd,Zhitnitsky:1985dd} of light meson DAs. The method can only be applied to parameters of the lowest few orders in conformal spin; parameters higher in conformal spin must be determined from other methods because the sum rules become unreliable.  One such method is that of \\textit{non-local} condensates, see for example Ref.~\\cite{Mikhailov:1986be}.\n\n\\subsection{Correlator}\nThe following \\textit{two-point correlation function} describes the propagation of a quark-antiquark pair\n\\begin{equation}\n\\Pi(q) = i \\int d^4 x \\,e^{i q \\cdot x} \\bra{0} T J_{1}(x) J_{2}(0) \\ket{0}\\,,\n\\label{sr_eq1}\n\\end{equation}\nwhere $q$ is the incoming momentum and possible Lorentz indices are omitted for simplicity. The local currents $J_i$  are chosen to have the correct quantum numbers and particle content corresponding to the particular hadronic parameters under investigation.  The physical picture of a hadron is of quarks, antiquarks and gluons confined within a typical hadronic size $R$  which is large when compared to the scale associated with perturbative effects. If one can show, however, that the correlation function is dominated by small spacial distances and time intervals\n\\begin{equation}\n|\\vec{x}| \\sim x^0 \\sim 1\/\\sqrt{Q} \\ll R\\,,\n\\label{sr_eq2}\n\\end{equation}\nfor a certain momentum configuration, then one has ensured the small size of the strong coupling $\\alpha_s$ and hence the use of perturbation theory in our calculation. One begins by noting that after contracting any Lorentz indices which may appear in the currents $J_i$ in Eq.~(\\ref{sr_eq1}), the correlation function can only depend on the interval $x^2=x^2_0-\\vec{x}^2$. By taking the Fourier transform, completing the square, and shifting the variable $x$ one finds\n\\begin{equation}\n\\Pi(q^2) = i \\int d\\kappa \\,\\int d^4 x \\,e^{i \\kappa x^2}\\,e^{i Q^2\/4 \\kappa} f(\\kappa)\\,.\n\\label{sr_eq3}\n\\end{equation}\nThe integral is dominated by the region where the arguments of the exponential vary slowly. This condition requires  $\\kappa \\sim 1\/x^2$ and $\\kappa \\sim Q^2$ which are both  fulfilled for $x^2 \\sim 1\/Q^2$; for large momentum transfer the quarks propagate near the light-cone.\\footnote{An expansion round $x^2 \\to 0$ is the basis of QCD sum rules on the light-cone -- see Section \\ref{LCSR}.} To find the true short-distance dominance one needs to dig a little further and by choosing the Lorentz frame $q_0=0$ one finds $\\vec{x}^{2} \\sim 1\/Q^2$ as required (\\ref{sr_eq2}). In the case of light quarks one needs the momentum transfer to the quarks to be large, $Q^2 \\equiv -q^2\\gg \\Lambda^2_{\\rm QCD}$. In the case of heavy quarks a large energy scale is introduced through the quark mass, for example $m_b$, which then serves to set the characteristic distances for the correlation function $|\\vec{x}|\\sim x_0\\sim 1\/(2 m_b)$; one is thus automatically in the perturbative regime.\n\n\\subsection{Short-Distance OPE}\nThe first of the two representations of the correlation function is obtained by performing the QCD calculation, valid for $Q^2 \\equiv -q^2 \\gg  \\Lambda^2_{\\rm QCD}$, using the short-distance OPE\n\\begin{equation}\n\\Pi (q^2) \\stackrel{\\rm{OPE}}{\\longrightarrow}\\sum_{i} C_i(q^2) \\left< O_i\\right> \\equiv \\Pi^{\\rm{OPE}}\\,,\n\\label{sr_eq4}\n\\end{equation}\nwhere the non-perturbative long distance effects of QCD are encoded in the condensates $\\left< O_i\\right>$ and the short-distance effects are included in the Wilson coefficient functions $C_i$ which are calculable in perturbation theory. Both the condensates and their coefficients are in general renormalisation scale dependent. Perturbative corrections to the condensates are calculated when necessary.  The perturbation theory contribution to Eq.~(\\ref{sr_eq4}) has $D=0$ and corresponds to the unit operator $\\left<O_{\\rm PT}\\right>=\\bf{1}$. The condensates play the role of power-corrections and are suppressed by inverse powers of the hard scale as $(Q^2)^{-D\/2}$. In the asymptotic limit $Q^2\\to\\infty$ only the unit operator survives, corresponding to asymptotic freedom. \n\n\\subsection{Condensates}\nThe condensates represent the effects of non-perturbative QCD and they cannot be determined from first principles due to the unknown nature of the QCD vacuum. The determination of the condensates is an industry in itself. The light quark condensate $\\bra{0} \\bar{q} q\\ket{0}$ has been known for a long time \\cite{Gell-Mann:1968rz} and it drives the breakdown of the chiral symmetry of the light quarks $q=\\{u,d\\}$ and its value can be extracted from experiment:\n\\begin{equation}\nm_\\pi^2 f_\\pi^2 \\approx-(m_u+m_d)\\left<\\bar q q\\right>\\,,\n\\end{equation}\nwhere we use the notation $\\left<O_i\\right>\\equiv\\bra{0}O_i\\ket{0}$. To define other condensates, one notes that the only vacuum expectation values of operators that can survive are those which are Lorentz invariant, spin zero, colour and flavour-singlets i.e. possess the quantum numbers of the vacuum. The complete set of condensates $\\left<O_i\\right>$ that contribute with $D\\le 6$ are\n\\begin{eqnarray}\n\\underbrace{\\left<\\textbf{1}\\right>}_{D=0}\\,, \\hspace{1cm}&\\underbrace{m_q\\left< \\bar{q} q\\right>}_{D=4}\\,,& \\hspace{1cm} \\underbrace{\\left<\\frac{\\alpha_s}{\\pi}G^2\\right>}_{D=4}\\,,\\nonumber \\\\\n\\underbrace{m_q\\left<  \\bar{q}\\sigma g_s G q \\right>}_{D=6}\\,, \\hspace{1cm}&\\underbrace{\\left<\\bar{q} \\Gamma_1q\\,\\bar{q}\\Gamma_2 q\\right>}_{D=6}\\,,&\\hspace{1cm} \\underbrace{\\left<g_s^3f G^3\\right>}_{D=6}\\,,\\nonumber\n\\end{eqnarray}\nwhere $q=\\{u,d,s\\}$ is a light quark spinor and all indices are contracted.\\footnote{The heavy quarks $c$, $b$ and $t$ do not form condensates because they are too massive to interact non-perturbatively with the QCD vacuum.}  We assume isospin symmetry  for $q=\\{u,d\\}$ and one must differentiate $q=s$ when $\\rm SU(3)_F$-breaking effects are taken into account.  Higher dimensional condensates $D> 6$ are not very well determined and generally unknown. If required, however, they can be estimated by employing the \\textit{vacuum saturation hypothesis} whereby the operator fields are simply split to form products of known condensates; for example, the quark-antiquark  $D=6$ operator can be simplified to the product of two $\\bar{q} q$ operators \\cite{PT:84,Shifman:1978by}. In practice, the OPE is truncated to a given order, and is usually justified by the stability of the resulting sum rule. The series, Eq.~(\\ref{sr_eq4}), is then given in terms of a limited number of condensates allowing sum rules to be written in terms of a small set of parameters incorporating the general features of non-perturbative QCD, while retaining its predictive power.\n\nThe procedure works in reverse, of course, where the values of condensates are deduced from sum rules for which the hadronic parameters are known from other methods;  two-point correlation functions featuring $\\bar{b} \\gamma_\\mu b$ or $\\bar{c} \\gamma_\\mu c$ currents correspond to the $\\Upsilon$ and $J\/\\Psi$ resonances respectively, of which the decay constants and masses are known. Values for the condensates are given, along with other input parameters, in Appendix~\\ref{appendixB}. Uncertainties in the values of the condensates and other input parameters constitute part of the reducible theoretical uncertainty of the sum rule approach. \n\n\\subsection{Dispersion Relation}\nTo proceed we need to relate the result of the OPE to a second representation of the correlation function which is obtained in terms of the spectrum of hadronic states in the physical region  $q^2>0$.   This is done via a \\textit{dispersion relation}, which is derived from the analytic properties of the correlation function as follows. The function $\\Pi(q^2)$ is analytic in all $q^2$ except on the real axis starting at a pole corresponding to the ground state particle. At higher energy  higher mass excited states and a continuum of many-particle states also feature. The higher mass resonances give poles above the ground state, the details of which depend on the physical spectrum of particles which possess the correct quantum numbers to couple to $\\Pi$. The continuum of many-particle states, correspond to a continuous cut, see Fig.~\\ref{sr_fig1}.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.5\\textwidth\\epsffile{analytic.eps}$$\n  \\caption[The spectral density function in the complex plane.]{\\small The general features of a spectral density function $\\rho^{\\rm had}(s)$ in the complex plane. The blob represents the pole due to the ground-state, the cross possible poles due to  higher mass resonances, and the thick line the cut due to the continuum of multi-particle states. The dotted line is the integration contour.} \n  \\label{sr_fig1}\n\\end{figure}\n\nUsing Cauchy's formula we can write\n\\begin{equation}\n\\Pi(q^2) =\\frac{1}{2 \\pi i} \\oint\\limits_{|z|=R} dz\\, \\frac{\\Pi(z)}{z-q^2}+\\frac{1}{2 \\pi i} \\int^R_0 dz\\, \\frac{\\Pi(z+i \\epsilon)-\\Pi(z- i \\epsilon)}{z-q^2}\\,,\n\\end{equation}\nwhere the region of integration is split into the parts just above and below the positive real axis and the circle of radius $R$. Provided that the correlation function vanishes at least as quickly as $q^{-2}$ as $|q^2| \\sim R \\to \\infty$ then the integral over the circle at radius $R$ goes to zero.\\footnote{If $\\Pi$ does not vanish quickly enough we subtract the first few terms in its Taylor expansion as required. We shall see that this does not matter in the end, due to the Borel transformation.} The remaining integral can be simplified using the fact that below the first pole at $q^2=s_{\\rm min}$,  $\\Pi(q^2)$ is real and above this point, according to the Schwarz reflection principle, $\\Pi(z+i \\epsilon)-\\Pi(z- i \\epsilon)= 2 i\\, \\textrm{Im}\\,\\Pi(z+ i \\epsilon)$. Hence\n\\begin{equation}\\label{disp}\n\\Pi(q^2) = \\int_{s_{\\rm min}}^{\\infty} ds\\, \\frac{\\rho(s)}{s-q^2- i \\epsilon}\\,,\n\\end{equation}\nwhere the function $\\rho(s)=\\frac{1}{\\pi} \\textrm{Im}\\, \\Pi(s)$ is the \\textit{spectral density} and describes the physical particle spectrum as a function of energy $s$. \n\n\\subsection{Unitarity Relation}\nAs we have seen, for large negative $q^2$ our correlation function is dominated by short-distance physics.  As $q^2$  becomes more positive the separation of the quarks increases. For large enough positive values of $q^2$ long-distance QCD interactions become more important and the correlation function then describes the creation of hadrons, which is the basis of its second representation. As discussed in the last section, $\\Pi$ uncovers a very complicated spectrum of states for $q^2>0$. We describe this situation by using the \\textit{unitarity relation}, which allows in insertion of a complete set of states into the correlation function\n\\begin{equation}\\label{complete}\n\\textbf{1}= \\sum_{n} \\int d \\Omega_n\\, \\ket{n(p)}\\bra{n(p)}\\,,\n\\end{equation}\nwhere $d \\Omega_n$ includes all phase-space factors and momentum conservation and the sum runs over all possible particles and polarisations, starting from the ground state $M$ of mass $m_M$. Inserting (\\ref{complete}) between the currents of our original correlation function (\\ref{sr_eq1}) yields an expression which we can relate to the hadronic spectral density\n\\begin{equation}\\label{unitarity}\n \\Pi^{\\rm had}(q^2) =   \\int \\frac{d^4 p}{(2 \\pi)^4} \\frac{1}{m^2_{M}-p^2} \\int d^4 x \\,e^{i q \\cdot x} \\bra{0} J_{1}(x)   \\ket{M(p)}\\bra{M(p)}J_{2}(0) \\ket{0} + \\dots\\,,\n\\end{equation}\nwhere the dots denote higher mass states which contribute to the continuum. We are usually interested in the ground state, and can insert the expressions for the matrix elements on the right hand side. The local matrix elements considered here can be used to extract vacuum-meson decay constants, for example.  Using the unitarity relation (\\ref{unitarity}) one can single out the ground state $M$ by comparing it to (\\ref{disp}) and writing the hadronic spectral density  as:\n\\begin{equation}\\label{gs}\n\\rho^{\\rm had}(s)=f_{M} \\,\\delta(s-m_{M}^2)+\\rho^{\\textrm{cont}}(s),\n\\end{equation}\nwhere $f_{M}$  is directly related to the matrix elements of the currents $J_1$ and $J_2$ in Eq.~(\\ref{unitarity}). For example, one could choose $J_1=J_2^\\dagger= \\bar q \\gamma_{z} s$ to extract $(f^\\parallel_{K^*})^2$ c.f. Eq.~(\\ref{das_eq22}). The exact form of the spectral density beyond the ground state is mostly unknown and the higher mass states and continuum contributions are usually lumped together in one function $\\rho^{\\textrm{cont}}(s)$. If the next highest particle above the ground state occurs at an energy not very much higher than $m_{M}$ then it is possible to explicitly include this particle as another delta-function term, analogously to the ground state. This procedure was used, for example, while investigating the leading-twist $K^*$ and $\\rho$ DA parameters for which the relevant correlators couple to the  $K_1$ and $b_1$ resonances respectively \\cite{Ball:1996tb,Ball:2005vx}. \n\n\n\\subsection{Quark-Hadron Duality}\nIt is possible to write the result of the OPE as a dispersion relation, with spectral density $\\rho^{\\textrm{OPE}}(s)$. As $\\rho^{\\textrm{cont}}(s)$ is mostly unknown  we replace it by $\\rho^{\\textrm{OPE}}(s)$ above a certain energy $s_0$\n\\begin{equation}\\label{qhd}\n\\rho^{\\textrm{cont}}(s) \\to \\rho^{\\textrm{OPE}}(s)\\,\\Theta(s-s_0)\\,.\n\\end{equation}\nThis assumption relies on the validity of  the hadronic  representation  being approximated by the partonic representation at higher energies. Inserting Eqs.~(\\ref{gs}) and  (\\ref{qhd}) into Eq.~(\\ref{disp}) one finds\n\\begin{equation}\n\\Pi^{\\rm had}(q^2)=\\frac{f_M}{m_M^2-q^2}+\\int_{s_{0}}^{\\infty} ds\\, \\frac{\\rho^{\\rm OPE}(s)}{s-q^2- i \\epsilon}\\,.\n\\end{equation}\nNow the assumption is not so strict because we only require a duality between the integrated spectral densities, not the spectral densities themselves. This is called \\textit{semi-global quark-hadron duality}. The parameter $s_0$ is called the  \\textit{continuum threshold} and its value  is specific for each particle spectrum being roughly equal to the energy of the next highest resonance above the ground state: $s_0 \\sim (m_M + \\Delta)^2$ where $\\Delta \\sim \\mathcal{O}(\\Lambda_{\\textrm{QCD}})$.  Ultimately it must be determined from the sum rule itself by requiring the numerical value of the determined quantity to be largely insensitive to its variation and this introduces the first source of systematic uncertainty to the sum rule method. We are now in a position to equate both representations\n\\begin{equation}\n\\Pi^{\\rm{had}}=\\Pi^{\\rm{OPE}}\\,, \n\\end{equation}\nto derive our sought after sum rule, however, before we do so, there is one last procedure to discuss, which greatly improves the behaviour of the sum rule.\n\n\n\\subsection{Borel Transformation And The Sum Rule}\nThe sum rule can be improved by suppressing the continuum contribution, which we have assumed to be well described by $ \\rho^{\\textrm{OPE}}(s>s_0)$ and the possible detrimental impact of this assumption is thus reduced.  We do this by performing a \\textit{Borel transformation} to both sides of the sum rule. The transformation is obtained by applying the operator\n\\begin{equation}\\label{borel1}\n\\mathcal{\\hat{B}} = \n\\lim_{\\stackrel{-q^2,n \\to \\infty}{-q^2\/n=M^2}}\\frac{(-q^2)^{(n+1)}}{n!}\\left(\\frac{d\\phantom{q^2}}{dq^2}\\right)^{n+1},\n\\end{equation}\nwhich takes a function of $q^2$ and produces a new function of  the \\textit{Borel parameter} $M^2$. One frequently encountered example is\n\\begin{equation}\n\\mathcal{\\hat{B}} \\frac{1}{(m^2-q^2)^k}= \\frac{1}{(k-1)!}\\frac{e^{-m^2\/M^2}}{(M^2)^{k}}\\,,\n\\end{equation}\nproviding an exponential suppression of the unknown continuum contributions, and a suppression of the power-corrections by factorials thus reducing the impact of neglected higher dimensional condensates. Also, as $\\mathcal{\\hat{B}} (q^2)^k=0$, any subtraction terms introduced to Eq.~(\\ref{disp}), which can only appear as polynomials in $q^2$, are killed off. The Borel transformation improves the stability and accuracy of the sum rule.\n\nThe Borel parameter $M^2$ is the second and last sum rule specific parameter to be introduced; along with $s_0$ it is required to impact very little, when varied, on the numerical value of the quantity being determined. The variation of $M^2$ changes the relative impact of the power-corrections and perturbation theory contributions.  In evaluating sum rules one looks for a \\textit{Borel window} which is  usually in the range $1\\,\\rm GeV^2\\leqslant M^2\\leqslant 2 \\,\\rm GeV^2$ for a typical mesonic DA parameter.  The sum rule should be reliable if a weak dependence (a plateau) is found, the contribution from the continuum is small, and there are no unnatural numerical cancellations.\n\nWe now equate Eqs.~(\\ref{disp}) and (\\ref{sr_eq4}) to reach the sum rule\n\\begin{equation}\\label{sr1}\nf_M\\,e^{-m_M^2\/M^2}= \\int^{s_0}_0 ds\\,e^{-s\/M^2} \\,\\rho^{\\textrm{OPE}}(s)\\,,\n\\end{equation}\nwhere the hadronic quantity $f_M$ is given as a function of the universal non-perturbative condensates, the perturbative short-distance coefficients as calculated from QCD, and the sum rule parameters $s_0$ and $M^2$. The sum rule is saturated by the ground state and higher mass states are suppressed. As the correlation function (\\ref{sr_eq1}) does not depend on the renormalisation scale, the $\\mu$ dependence of the condensates, when multiplied by their coefficient functions, must cancel in the sum of (\\ref{sr_eq4}). The sum is always truncated, however, and the residual $\\mu$ dependence will be a source of theoretical uncertainty.\n\n\\subsection{Non-local Formalism}\nOne way to gain access to parameters higher in conformal spin is to calculate sum rules involving operators  which are related to moments of DAs\n\\begin{equation}\n\\bra{0}\\bar{q}(0) (\\stackrel{\\leftrightarrow}{D} \\cdot z)^k \\Gamma s(0)\\ket{V}\\sim \\int^1_0 du\\, (2u-1)^k \\phi(u)\\equiv \\left<\\xi^k\\right>\\,.\n\\end{equation}\nFor the $K^*$ for example the first few moments of both the leading-twist DAs are $\\left<\\xi^0\\right>=1$, $\\left<\\xi^1\\right>=\\frac{3}{5} a_1(K^*)$, $\\left<\\xi^2\\right>=\\frac{1}{35}(7+12 a_2(K^*))$ and $\\left<\\xi^3\\right>=\\frac{1}{105}(27 a_1(K^*)+20 a_3(K^*))$. A more elegant method, enabling the  DA parameters to be extracted individually, relies on calculating a correlator of two currents, one of which is non-local, with fields at light-like separations ($z^2=0$) \\cite{Ball:2003sc}. Consider the following\n\\begin{equation}\n\\Pi(q\\cdot z) = i \\int d^4 x \\,e^{i q \\cdot x} \\bra{0} T J(x) \\bar{s}(0) \\gamma_{z} q(z) \\ket{0}\\,,\n\\label{nonlocalCF}\n\\end{equation}\nwhere $J(x)$ is local, and the non-local current yields the leading-twist DA (\\ref{das_eq16}). The sum rule (\\ref{sr1}) then reads\n\\begin{equation}\\label{sr2}\nf_J f^\\parallel_{K^*}\\,e^{-m_M^2\/M^2}\\int^1_0 du\\,e^{-i \\bar{u}q\\cdot z} \\phi_{2;K^*}^\\parallel= \\int^{s_0}_0 ds\\,e^{-s\/M^2} \\,\\int^1_0 du\\,e^{-i \\bar{u}q\\cdot z}\\rho^{\\textrm{OPE}}(s,u)\\,.\n\\end{equation}\nThe integration over $u$ on the right hand side naturally arises via the Feynman parameterisation used in the calculation. At this point one can exploit the orthogonality of the Gegenbauer polynomials by replacing the exponential weight function $e^{-i\\xi q\\cdot z} \\to C_n^{3\/2}(\\xi)$ on both sides to project out $a_n^\\parallel(K^*)$ via Eqs.~(\\ref{das_eq15}) and (\\ref{das_eq17}). \n\\begin{figure}[h]\n$$ \\epsfxsize=0.3\\textwidth\\epsffile{generalnonlocal.eps}$$\n  \\caption[A generic diagram for a non-local sum rule.]{\\small A generic non-local diagram. The dotted line denotes the path ordered gauge factor $[z,-z]$ between the two quark fields. The momentum $q$ is injected at point $y$ - the vertex on the right hand side.} \n  \\label{sr_fig2}\n\\end{figure}\nIn Fig.~\\ref{sr_fig2} we show the leading diagram of the non-local correlation function (\\ref{nonlocalCF}). The dotted line denotes the path ordered gauge factor $[z,-z]$ between the two quark fields. The non-local formalism allows, in principle, an extraction of  parameters of arbitrary order $n$. In practice, however, only the parameters of the lowest few orders $n$ are accessible due to instability of the resulting sum rules. One finds that the power-corrections in $\\rho^{\\textrm{OPE}}$ grow with positive powers of $n$ compared to the perturbative contribution. For high enough $n$ this behaviour upsets the hierarchy of contributions to the OPE, where non-perturbative terms are expected to be moderately sized corrections to the leading term. Hence the method is justified for low-order coefficients $n\\leq2$ where the non-perturbative coefficients describe the general features of the DA. It breaks down for higher-order coefficients $n>3$ because the local vacuum condensates appear with $\\delta$-functions which cannot accommodate the information needed to describe the more detailed shape of the DA, see Refs.~\\cite{Ball:1997rj,Ball:2003sc}. \n\n\\section{QCD Sum Rules On The Light-Cone}\\label{LCSR}\nA modification of the QCD sum rule method known as QCD sum rules on the light-cone, or \\textit{light-cone sum rules} (LCSRs)  \\cite{Balitsky:1989ry,Braun:1988qv,Chernyak:1990ag}, was developed to overcome difficulties encountered when calculating transition and electromagnetic form factors in the SVZ method.\\footnote{The term ``light-cone sum rules\" first appears in Ref.~\\cite{Ball:1991bs}.} The problems are related to the asymptotic scaling behaviour of the form factors in the heavy-quark limit $m_b \\to \\infty$. LCSRs rely on the use of DAs as their universal non-perturbative hadronic input and lead to the correct asymptotic behaviour in the heavy-quark limit. The DAs represent a  partial re-summation of the operators appearing in the condensates and appear ordered in contributions of increasing twist \\cite{Ball:1997rj}. We can view LCSRs as a marriage of the SVZ technique and the theory of hard exclusive processes \\cite{Chernyak:1977fk,Lepage:1980fj,Efremov:1979qk,Chernyak:1977as}. In the case of the ``heavy-to-light'' $B\\to M$ transition form factors, LCSRs have been applied successfully to pseudoscalar transition form factors \\cite{oldpseudo,Ball:1998tj,Ball:2001fp,Ball:2004ye} and vector transition form factors \\cite{Ball:1998kk,Ball:2004rg}. \n\nFor LCSRs to become competitive with the SVZ sum rules, a good knowledge of higher-twist DAs is required. This motivates the determination of the non-perturbative DA parameters via SVZ sum rules and via LCSR, the DAs themselves to determine other non-perturbative parameters, such as transition form factors. As with SVZ sum rules, the starting point of LCSRs is with a suitable correlation function. For the extraction of $B\\to M$ transition form factors we require a two-point correlator, this time sandwiched between the vacuum and the meson state $M$, which is the example considered in this section. One employs much of the same methodology as in the last section, although now one requires the correlation function to be expanded in an OPE on the light-cone. In doing so one finds that the correlation function factorises and can be written in terms of a convolution of hard scattering kernels and the universal non-perturbative DAs of the light-meson.   To that end, consider a correlation function of two quark currents taken between the vacuum and an on-shell meson $M$\n\\begin{equation}\\label{CFLCSR}\n\\Pi(q,p_B)=i\\int d^4x \\,e^{i q\\cdot x} \\bra{M(p)}T J_1(x) j_B^{\\dagger}(0)\\ket{0}\\,,\n\\end{equation}\nwhere $j_B = m_b\\, \\bar{q}i \\gamma_5 b$ is the \\textit{interpolating current} of the $B$ meson which defines the $B$ meson decay constant\n\\begin{equation}\n\\bra{B(p_B)}j_B\\ket{0}=m^2_B f_B\\,.\n\\label{bdecayconstant}\n\\end{equation}\nThe current $J_1(x)$ is chosen to project out the form factor of interest. The  momentum $q$ is  injected into the weak vertex, $p_B$ is the momentum of the $B$ meson and $p$ is the  momentum of $M$ with $q+p=p_B$. The correlation function is dominated by light-like distances for virtualities\n\\begin{equation}\nm_b^2 - p_B^2 \\geq \\mathcal{O}(\\Lambda_{\\rm QCD} m_b)\\,, \\qquad m_b^2 - q^2 \\geq \\mathcal{O}(\\Lambda_{\\rm QCD} m_b)\\,,\n\\end{equation}\nwhich ensures the slow variation of the exponential in Eq.~(\\ref{CFLCSR}) and its suitability for an expansion around the light-cone. The light-cone expansion results in the transverse and ``minus'' degrees of freedom being integrated out, leaving the longitudinal momenta of the partons as the relevant degrees of freedom. As a result a cutoff $\\mu$ is introduced below which the transverse momenta are included in the resulting light-mesons DAs. The contributions from momenta above this cutoff are calculable in perturbation theory. The procedure yields the \\textit{collinear factorisation} of the correlation function\n\\begin{equation}\\label{LCSR:fac}\n\\Pi(q^2,p_B^2)=\\sum_n \\int_0^1 \\,du\\, T^{(n)}(u,q^2,p_B^2,\\mu)\\,\\phi_{n;M}(u,\\mu)\\,,\n\\end{equation}\nwhere $u$ $(1-u)$ denotes the momentum fraction of the outgoing quark (antiquark) and the sum is over all twist and possible polarisation contributions. The scale dependence of the hard scattering kernels $T^{(n)}$ must cancel that of the DAs $\\phi_{n;M}$. The factorisation formula has to be verified by direct calculation and a proof to all orders in  $\\alpha_s$ does not exist. The verification relies on the cancelation of divergences, of which there are two types: the IR and UV singluarities arising from loop calculations and so-called soft singularities which appear when the convolution over $u$ does not converge at the end-point regions ($u \\sim 0~\\textrm{or}~1$) i.e. when one of the quarks is soft. In terms of kinematics there are two main contributing processes: the hard-scattering mechanism and the soft contribution or Feynman mechanism. Both mechanisms are included in the LCSR approach for which there are no soft divergences and the IR\/UV divergences can be treated in dimensional regularisation. \n\nOne can write the result of the light-cone expansion (\\ref{LCSR:fac}) as a dispersion relation in $p_B^2$\n\\begin{equation}\n\\Pi^{\\rm  LC} (p_B^2, q^2) = \\int_{m_b^2}^{\\infty} ds\\,\\frac{\\rho^{\\rm  LC} (s, q^2)}{s-p_B^2}\\,.\n\\end{equation}\nTaking the imaginary part, to obtain $\\rho^{\\rm  LC} $, is straight forward after integration over the momentum fraction $u$ is performed. The correlation function has a cut in $p_B^2$ starting at $m_b^2$ over the physical region. One now matches this calculation to the hadronic representation of the correlation function, which can also be written as a dispersion relation\n\\begin{equation}\n\\Pi^{\\rm  had} (p_B^2, q^2) = \\int_{m_B^2}^{\\infty} ds\\,\\frac{\\rho^{\\rm  had} (s, q^2)}{s-p_B^2}\\,,\n\\end{equation}\nwhere the physical spectral density is given by the ground state $B$ meson plus higher mass states forming a continuum as\n\\begin{equation}\n\\rho^{\\rm had} (s, q^2)= F_M\\, \\delta(s-m_B^2) +  \\rho^{\\rm LC } (s, q^2) \\,\\Theta(s-s_0)\\,.\n\\end{equation}\nThe quantity $F_M$ will contain the form factor we require. We perform the Borel transformation to arrive at the LCSR\n\\begin{equation}\nF_M\\, e^{-m_B^2\/M^2}  =\\int_{m_b^2}^{s_0} ds \\, e^{-s\/M^2} \\rho^{\\rm LC} (s, q^2)\\,.\n\\end{equation}\nTo extract the form factor we need to find a sets of parameters $M^2$ and $s_0$ such that the form factor is largely insensitive to their variation. As with SVZ sum rules, there is no rigourous way to do this and so the procedure introduces the irreducible source of uncertainty to the method. \n\n\\section{Example Calculation - The Gluon Condensate}\\label{example}\nHere we present an example calculation within the SVZ sum rule framework. The result of the calculation is used in the sum rule for the G-even $K$ meson three-particle twist-3 DA parameter $f_{3K}$, see Ref.~\\cite{Ball:2006wn}. We calculate the $\\alpha_s$ correction to the gluon condensate $\\left< \\frac{\\alpha_s}{\\pi} G^2 \\right>$ which proceeds from the following local correlation function\n\\begin{equation}\\label{cf2}\n\\Pi^{(G^2)}   =  i \\int d^4 y \\, e^{i q\\cdot y}\\bra{0} T \\bar q(0) \\sigma^{\\mu z}  g_sG_{\\mu  z}(0) s(0) \\bar s(y) \n\\sigma^{\\nu z} g_sG_{\\nu  z}(y) q(y)\\ket{0}\\,,\n\\end{equation}\nfor which the leading-order contribution  vanishes. A convienient way of extracting the gluon condensate is to make use of the \\textit{back-ground field technique} in which the fixed-point gauge allows the Taylor expansion of quark and gluon fields to be written in a gauge-covariant form, see  Ref.~\\cite{Novikov:1983gd} for details. The gluon field in the QCD Lagrangian (\\ref{basics_eq1}) is split into ``quantum''  and  ``classical''  (background) fields\n\\begin{equation}\\label{split}\nA^a_\\mu\\to a^a_\\mu+\\mathcal{A}^a_\\mu\\,,\n\\end{equation}\nwhere the background field $\\mathcal{A}^a_\\mu$ is taken in the fixed-point gauge at $x_0=0$. The quantum field $a^a_\\mu$ is taken to be in the Feynman gauge, thus requiring the gauge fixing term ($\\xi=1$)\n\\begin{equation}\n\\mathcal{L}^{\\rm fix}=-\\frac{1}{2}(\\partial^\\mu a_\\mu^a+g_s f^{abc}\\mathcal{A}^{b \\mu} a_\\mu^c)^2\\,,\n\\end{equation}\nto be added to the QCD Lagrangian. The quantum field propagates perturbatively and we may use the standard expression\n\\begin{equation}\n\\begC1{a^a_\\mu}\\conC{(x)\\,}\\endC1{a^b_\\nu}(y)=i \\delta^{ab}\\int \\frac{d^4 l}{(2 \\pi)^4}D_{\\mu\\nu}(l)e^{-i l\\cdot (x-y)}\\,,\\qquad D_{\\mu\\nu}(l)=\\frac{-g_{\\mu\\nu}}{l^2},\n\\end{equation}\nThe background field does not propagate perturbatively, and is the field that goes to form the condensate; it represents the low-energy, long distance modes of the gluon field that probe the non-perturbative structure of the QCD vacuum. The fixed-point gauge condition allows $\\mathcal{A}_\\mu^a(x)$ to be expressed in terms of the gluonic field strength tensor as\n\\begin{equation}\n\\mathcal{A}_\\mu^a(x)=\\sum^\\infty_{n=0}\\frac{1}{n!(n+2)} x^\\omega x^{\\omega_1}... x^{\\omega_n}\\left[ D_{\\omega_1}(0),\\left[D_{\\omega_2}(0),\\left[...\\left[D_{\\omega_n}(0),G_{\\omega\\mu}^a(0)\\right]...\\right]\\right]\\right]\\,,\n\\end{equation}\nand translating to momentum space one finds\n\\begin{equation}\\label{condfield}\n\\mathcal{A}_\\mu^a(k)=-\\frac{i}{2} G^a_{\\omega\\mu}(0)\\left[(2\\pi)^4 \\delta^{(4)}(k)\\right]\\frac{\\partial\\phantom{ k^\\omega}}{\\partial k^\\omega}+\\dots\\,,\n\\end{equation}\nwhere we only require the first term; higher order terms give rise to higher dimensional condensates which we do not consider. As we have to introduce two condensate gluons to construct $\\left<G^2\\right>$ we introduce two auxiliary vacuum momenta $k$ and $k^\\prime$ for which the fixed-point $x_0=0$ is a sink. After integration over coordinates these momenta appear in the quark and gluon propagators. The two corresponding derivatives are then taken, and the vacuum momenta set to zero. The following expression proves very useful in managing derivatives of quark propagators\n\\begin{equation}\\label{quarkderiv}\n\\frac{\\partial}{\\partial p_\\mu} S^{(q)}(p)=-S^{(q)}(p)\\gamma^\\mu S^{(q)}(p)\\,,\\qquad S^{(q)}(p)=\\frac{\\slash{p}+m_q}{p^2-m_q^2}\\,,\n\\end{equation}\nfor arbitrary quark flavour $q$. The gluon condensate is finally extracted using\n\\begin{equation}\\label{vacav}\nG^a_{\\omega\\mu}(0)G^b_{\\omega^{\\prime}\\nu}(0)=\\frac{1}{D(D-1)}\\frac{\\delta^{ab}}{N_c^2-1}\\left(g_{\\omega\\omega^{\\prime}}g_{\\mu \\nu}-g_{\\omega\\nu}g_{\\omega^{\\prime}\\mu}\\right)\\left<G^2\\right>\\,,\n\\end{equation}\nwhere $D$ is the spacetime dimension. Due to Eq.~(\\ref{split}) the expansion of $\\mathcal{L}_{\\rm QCD}$  yields ``interaction'' terms in which background fields are radiated from the propagating gluons at single or double vertices, both of which contribute to the $\\mathcal{O}(\\alpha_s)$ correction to the gluon condensate. These vertices are shown in Fig.~\\ref{sr_fig3} and the corresponding terms are \n\\begin{eqnarray}\\label{glueint}\n \\mathcal{L}^{\\mathcal{A}aa}_{int}&=&-\\frac{1}{2}g_s f^{abc}\\left[\\left(\\partial^\\mu\\mathcal{A}^{a\\nu}-\\partial^\\nu\\mathcal{A}^{a\\mu}\\right)a_\\mu^b a_\\nu^c\\right.\\nonumber\\\\\n &+&\\left.(\\partial^\\mu a^{a\\nu}-\\partial^\\nu a^{a\\mu})(\\mathcal{A}^b_\\mu a^c_\\nu+a^b_\\mu\\mathcal{A}^c_\\nu)+2(\\partial_\\mu a^{a\\mu})\\mathcal{A}^{b\\nu}a_\\nu^c\\right]\\,,\\nonumber\\\\\n \\mathcal{L}^{\\mathcal{AA}aa}_{int}&=&-\\frac{1}{2} g_s^2 f^{abc} f^{ade}\\left[\\mathcal{A}^b_\\mu \\mathcal{A}^{d\\mu} a_\\nu^e a^{c\\nu} +\\mathcal{A}^b_\\mu  a^{d\\mu} \\mathcal{A}^e_\\nu a^{c\\nu}+\\mathcal{A}^b_\\mu  a^{c\\mu} \\mathcal{A}^d_\\nu a^{e\\nu}\\right]\\,,\n\\end{eqnarray}\nwhere terms which vanish eventually via Eq.~(\\ref{vacav}) due to $f^{abc}\\delta^{bc}=0$ are omitted.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.8\\textwidth\\epsffile{gluonvertices.eps}$$\n\\caption[Interactions of the background field $\\mathcal{A}^a_\\mu$ with the quantum field $a^a_\\mu$.]{\\small  The interactions of the background field $\\mathcal{A}^a_\\mu$ (denoted by a cross) with the quantum field $a^a_\\mu$ corresponding to $\\mathcal{L}^{\\mathcal{A}aa}_{int}$ and $\\mathcal{L}^{\\mathcal{AA}aa}_{int}$ respectively.} \n\\label{sr_fig3}\n\\end{figure}\nContributions also stem directly from the gluonic field strength tensors in Eq.~(\\ref{cf2}) which give rise to gluon emission of either one or two fields from the vertices at co-ordinates $0$ and $y$. Due to the gauge condition there is no ``left-right'' symmetry and all diagrams with two gluons, of which at least one is a condensate gluon, emerging from the vertex at $x=0$ vanish due to $A_\\mu(0)=0$.  Diagrams with two condensate gluons at point $y$, which originate from the non-linear part of the gluonic field strength tensor,  also give zero due to $f^{abc}\\delta^{bc}=0$. There is an ``up-down'' symmetry where diagrams related by a  reflection in the central horizontal axis are equal. Overall we find there to be 10 distinct non-zero diagrams to be calculated which are shown in Fig.~\\ref{sr_fig4}. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.7\\textwidth\\epsffile{cond1.eps}$$\n$$\\epsfxsize=0.5\\textwidth\\epsffile{cond2.eps}$$\n\\caption[Diagrams contributing to the gluon condensate at $\\mathcal{O}(\\alpha_s)$.]{\\small The diagrams contributing to the gluon condensate at $\\mathcal{O}(\\alpha_s)$ for the SVZ sum rule of the $K$ twist-3 DA parameter $f_{3K}$ -- see Ref.~\\cite{Ball:2006wn}. For each diagram the fixed-point $x_0=0$ is at the left most vertex and the right most is at $y$.} \n\\label{sr_fig4}\n\\end{figure}\n\nSome of the diagrams are divergent, however, all divergences cancel in the sum of all diagrams.\\footnote{We use dimensional regularisation and the $\\overline{MS}$ renormalisation scheme throughout this thesis.} For an explicit example consider the last diagram in the second line of Fig.~\\ref{sr_fig4}. It is evident that we require $\\mathcal{L}^{\\mathcal{A}aa}_{int}$ to be contracted in all possible ways with quantum fields originating from the linear part of the gluonic field strength tensors at points $0$ and $y$. This, multiplied by the condensate field originating from the quark loop yields the gluonic part of the calculation\n\\begin{equation}\\label{allcon}\n\\sim\\mathcal{A}^d_\\delta(v) \\left.(\\partial_\\mu a^a_z(0)-\\partial_z a^a_\\mu(0))\\, \\mathcal{L}^{\\mathcal{A}aa}_{int}(w) \\,(\\partial_\\nu a^b_z(y)-\\partial_z a^b_\\nu(y))\\right|_{\\rm all\\,contractions}\\,,\n\\end{equation}\nwhich is eventually given in momentum space by (omitting Lorentz indices)\n\\begin{equation}\\label{gluonicpart}\n\\sim \\frac{\\partial}{\\partial k}\\frac{\\partial}{\\partial k^\\prime}\\frac{f(l,k^\\prime)}{l^2 (l-k^\\prime)^2} \\left<G^2\\right>\\,,\n\\end{equation}\nwhere the condensate gluon within $\\mathcal{L}^{\\mathcal{A}aa}_{int}(w)$ is expressed by Eq.~(\\ref{condfield}) with momentum $k^\\prime$ and $f(l,k^\\prime)$ is a function of the loop momentum $l$ and the vacuum momentum $k^\\prime$. The  quark loop yields a usual trace \n\\begin{equation}\\label{quarkpart}\n\\sim \\frac{\\textrm{tr}\\left[(\\slash{p}+\\slash{q}-\\slash{l})\\sigma^{\\mu z}(\\slash{p}+\\slash{k})\\gamma^\\delta\\slash{p}\\,\\sigma^{\\nu z}\\right]}{(p+q-l)^2(p+k)^2 p^2}\\,,\n\\end{equation}\nand after multiplying together Eqs.~(\\ref{gluonicpart}) and (\\ref{quarkpart}), performing the derivatives in $k$ and $k^\\prime$ and integrating over the momenta $p$ and $l$ we find\n\\begin{equation}\n\\Pi^{(G^2)}_{\\rm example}  = \\frac{1}{384}\\frac{\\alpha_s}{\\pi}  \\left\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle \\frac{(q\\cdot z)^4}{q^2}\\,.\n\\end{equation}\nIn this way we can include all the other diagrams shown in Fig.~\\ref{sr_fig4} to obtain the contribution to the sum rule\n\\begin{equation}\n\\Pi^{(G^2)}  = \n-\\frac{89}{5184}\\,\\frac{\\alpha_s}{\\pi}\\,\\left\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle \\,\\frac{(q \\cdot z)^4}{q^2}\\,,\n\\label{last}\n\\end{equation}\nwhich differs from the result obtained in Ref.~\\cite{Zhitnitsky:1985dd}; the logarithmic term is not reproduced:\n\\begin{equation}\n\\sim \\log \\left(\\frac{M^2}{\\mu^2}\\right) \\left< \\frac{\\alpha_s}{\\pi} \\,G^2 \\right>\\,.\n\\label{least}\n\\end{equation}\n\n\\chapter{The Determination Of Vector Meson Twist-2 And Twist-3  Parameters}\\label{chapter4_det}\nIn this chapter we determine the leading twist-2 and twist-3 two- and three-particle vector meson DA parameters using the non-local modification of SVZ sum rules. The parameters are defined in Chapter~\\ref{chapter2_DAs} and the sum rule method is outlined in Chapter~\\ref{chapter3_SR}. We express the relevant correlation functions, via the OPE, in terms of the perturbative and condensate  contributions.  Key to the analysis is the inclusion of all G-parity and  $\\rm SU(3)_F$-breaking effects which, as discussed in Chapter~\\ref{chapter2_DAs}, come from a variety of sources, and allow a consistent determination of the parameters for the $\\rho$, $K^*$, and $\\phi$. Motivation for the present analysis comes from various sources, including:\n\\begin{itemize}\n\\item{values for the decay constants and leading-twist DA Gegenbauer moments are required as input for QCD factorisation frameworks which provide a systematic method for the calculation of $B$ decay matrix elements. We discuss one such framework in Chapter~\\ref{chapter6_QCDF}.}\n\\item{Twist-2 and twist-3 DAs provide the leading non-perturbative input within the method of LCSR, as discussed in Chapter~\\ref{chapter3_SR}, and as such are applied to many problems in heavy-flavour physics, such as the calculation of $B$ transition form factors and the estimation of $B$ decay matrix elements including power-suppressed contributions to QCD factorisation frameworks, see Chapter~\\ref{chapter7_rad}.}\n\\item{A full determination of the twist-3 DA parameters, including $\\rm SU(3)_{F}$-breaking and G-parity violating effects, and the inclusion of $\\mathcal{O}(\\alpha_s)$ and $\\mathcal{O}(m_s^2)$ corrections to the quark condensate contributions to the twist-2 DA parameter sum rules are new to the present analysis, allowing $a_2^{\\parallel,\\perp}(\\phi)$ to be determined, for the first time, to the same accuracy as $a_2^{\\parallel,\\perp}(\\rho,K^*)$.}\n\\end{itemize}\nAll input parameters for the sum rules, and useful formulas, such as those required to take the imaginary parts of intermediate results, and various relevant integrals, are given in Appendix~\\ref{appendixB}. In performing the calculations we find Refs.~\\cite{PT:84,Borodulin:1995xd} very useful. The material covered in this chapter partially follows that of Ref.~\\cite{Ball:2007rt}.\n\n\\section{Twist-2}\nIn this section we focus on the determination of the twist-2 DA Gegenbauer coefficients $a_{2}^{\\parallel,\\perp}$ defined by Eqs.~(\\ref{das_eq16}) and (\\ref{das_eq19}).  The sum rules for $f_{K^*}^{\\parallel,\\perp}$, including $\\rm SU(3)_F$-breaking corrections, were calculated in Refs.~\\cite{Govaerts:1986ua,Ball:2005vx,Ball:2006fz}. Those for the G-parity violating $a_1^{\\parallel,\\perp}(K^*)$ in Refs.\\cite{Ball:2005vx,Ball:2006fz} and those for $a_2^{\\parallel,\\perp}(K^*)$ in \\cite{Ball:2003sc} apart from perturbative terms in $m_s^2$ and the $\\mathcal{O}(\\alpha_s)$ and $\\mathcal{O}(m_s^2)$ corrections to the quark condensate, which are new to the present analysis.  Motivation for including these corrections is found by examining the individual contributions to the sum rules for $a_2^{\\parallel,\\perp}(K^*)$ given in Ref.~\\cite{Ball:2003sc}. They are found to be dominated by $\\left<\\bar s s\\right>$ as we can see from the following explicit break down of contributions:\n\\begin{eqnarray}\na_2^\\parallel(K^*)&=&\\overbrace{0.05}^{\\textrm{PT}}+\\overbrace{0.08}^{\\left<\\frac{\\alpha_s}{\\pi} G^2\\right>}+\\overbrace{0.11}^{\\left<\\bar s g_s \\sigma G s\\right>}+\\overbrace{0.04}^{\\left<\\bar q q\\right>^2}-\\overbrace{0.16}^{\\left<\\bar s s\\right>}+\\overbrace{0.02}^{\\left<\\bar s s\\right>^2}-\\overbrace{0.05}^{\\left<\\bar q q\\right>\\left<\\bar s s\\right>}\\nonumber\\\\\na_2^\\perp(K^*)&=&0.06+\\,\\,0.10\\,\\,+\\,\\,\\,0.25\\,\\,\\,+0.03-0.27+0.02\\,-0\\,,\n\\end{eqnarray}\nfor the reference point $s_0=1.2\\,\\textrm{GeV}^2$, $M^2=1\\,\\textrm{GeV}^2$ and $\\mu=1\\,\\textrm{GeV}$. Moreover, for the  $\\phi$ the impact of a finite strange quark mass may be even more pronounced with respect to perturbation theory and the gluon condensate. \n\nFirstly, we give an overview of the calculation of the $\\mathcal{O}(\\alpha_s)$ and $\\mathcal{O}(m_{s}^2)$ corrections to the quark condensate $\\left<\\bar s s\\right>$; the calculations for $\\left<\\bar q q\\right>$ are analogous. We only need extract terms proportional to $m_s$  as the contributions proportional to $m_q$ are identical; we can find the contributions for $\\phi$ by simply replacing $\\left<\\bar q q\\right>\\to\\left<\\bar s s\\right>$ and doubling the terms in $m_{s} \\left<\\bar s s\\right>$, $m_{s} \\left<\\bar q q\\right>$ and $m_{s} \\left<\\bar s g_s G s\\right>$. Contributions for $\\rho$ are found by setting $m_s\\to0$. Secondly, we go on to analyse the sum rules for $a_2^{\\parallel,\\perp}(\\phi)$. We end this section by presenting the results.\n\n\\subsection{Calculation}\nFor both polarisations we begin from the diagonal correlation function\n\\begin{equation}\n\\Pi_{2;K^*}(q\\cdot z) = i \\int d^4y \\,e^{-iq\\cdot y} \\bra{0} T \\bar q(y) \\Gamma s(y) \\bar s(0) \\Gamma [0,z]q(z) \\ket{0}\\,\n\\label{C.0}\n\\end{equation}\nwhere $\\Gamma^{\\parallel}=\\gamma_z$ and $\\Gamma^{\\perp}=\\sigma_{\\mu z}$. For the longitudinal parameters the sum rule is exactly that given by Eq.~(\\ref{sr2}) with $f_J\\to f^\\parallel_{K^*}$ and for the transverse parameters the sum rule is analogous. Both polarisations have the same projections onto the DA parameters\n\\begin{eqnarray}\n\\left(f_{K^*}\\right)^2e^{-m_{K^*}^2\/M^2}\\left[1\\right]\n & = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int^1_0 du\\,\\left[1\\right]\n      \\frac{1}{\\pi}\\, {\\rm Im}_{s}\n\\pi_{2;K^*}(u)\\,,\\label{C0}\n\\\\\n\\left(f_{K^*}\\right)^2  e^{-m_{K^*}^2\/M^2}\n      \\,\\left[\\frac{9}{5}\\,a_{1}(K^*)\\right]\n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int^1_0 du\\,\n     \\left[ 3 \\xi \\right]\\frac{1}{\\pi}\\, {\\rm Im}_{s}\n \\pi_{2;K^*}(u)\\,,\n\\nonumber\\\\\n\\left(f_{K^*}\\right)^2 e^{-m_{K^*}^2\/M^2}\n      \\,\\left[\\frac{18}{7}\\,a_{2}(K^*)\\right]\n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int^1_0 du\\,\n    \\left[ \\frac{1}{2}(  15 \\xi^2-3)\\right]\\frac{1}{\\pi}\\, {\\rm Im}_{s}\n    \\pi_{2;K^*}(u)\\,, \\nonumber\n\\end{eqnarray}\nwhere ${\\rm Im}_s$ denotes taking the imaginary part with respect to $s$.  The fact that we are dealing with non-local correlation functions means that we do not integrate over the co-ordinate $z$. The resulting residual exponential function remains throughout the calculation and can contribute to the momentum integrals yielding powers of $i c (q\\cdot z)$, where $c$ is a constant. Ultimately the exponential functions can be cast into  the ``canonical form'' set by the exponential appearing in front of the leading-twist DA i.e. $e^{- i \\bar{u}q\\cdot z}$ -- see Eq.~(\\ref{sr2}). \n\\subsection*{Quark Condensate}\n\\begin{figure}[h]\n$$ \\epsfxsize=0.25\\textwidth\\epsffile{tw2_quark0_sr.eps}$$\n\\caption[Diagram contributing to the quark condensate $\\left<\\bar s s \\right>$ at leading-order.]{\\small The leading-order diagram contributing to the quark condensate $\\left<\\bar s s \\right>$.} \n\\label{det_fig1}\n\\end{figure}\nThe tree-level diagram is shown in Fig.~\\ref{det_fig1}. To extract the quark condensates to $\\mathcal{O}(m_s^2)$ we use the following expansion of the quark fields (for general quark flavour $q$)\n\\begin{eqnarray}\n\\bra{0}\\!:\\! \\bar q^i_\\alpha(x_1)\\, q^j_\\beta(x_2)\\!:\\!\\ket{0}&=&\\delta^{ij}\\frac{\\left<\\bar q q\\right>}{12}\\left\\{\\delta_{\\beta\\alpha}\\left(1-\\frac{\\Delta^2}{2D}m_q^2\\right)\\right.\\nonumber\\\\\n&-&\\left.m_q\\frac{i }{D}(\\gamma_\\lambda)_{\\beta\\alpha}\\Delta^\\lambda\\left(1-\\frac{\\Delta^2 }{2(2+D)}m_q^2\\right)\\right\\}\\,,\n\\label{quarkextract}\n\\end{eqnarray}\nwhere  $\\Delta_\\mu=(x_2-x_1)_\\mu$ and $i,j$ are colour and $\\alpha,\\beta$ spinor indices. One can deal with the co-ordinate $\\Delta_\\mu$ by trading it, via partial integration (PI), for a derivative of the trace that arises from the quark loop. A convenient way to do so is via an auxiliary momentum $Q$\n\\begin{equation}\n\\Delta_\\kappa \\stackrel{\\textrm{PI}}{\\longrightarrow} ie^{-i\\Delta\\cdot Q}\\frac{\\partial}{\\partial Q_\\kappa}\\Big|_{Q\\to 0}\\,.\n\\label{deriv1}\n\\end{equation}\n\\begin{figure}[h]\n$$ \\epsfxsize=0.8\\textwidth\\epsffile{tw2_quark1_sr.eps}$$\n\\caption[Diagrams contributing to the quark condensate $\\left<\\bar s s \\right>$ at $\\mathcal{O}(\\alpha_s)$.]{\\small Diagrams contributing to the quark condensate $\\left<\\bar s s \\right>$ at $\\mathcal{O}(\\alpha_s)$. The crossed circle $\\otimes$ depicts the emission of a gluon from the non-local gauge factor -- see Eq.~(\\ref{emissiongf}). The corresponding diagrams for $\\left<\\bar q q\\right>$ are identical but reflected top to bottom.} \n\\label{det_fig2}\n\\end{figure}\nDiagrams for the $\\mathcal{O}(\\alpha_s)$ corrections to the strange quark condensate are shown in Fig.~\\ref{det_fig2}.  Importantly there are contributions from the gauge-factor which need to be included\n\\begin{equation}\n[0,z]=\\textrm{P} \\exp{\\left\\{-i g_s \\int^1_0 dt\\,z^\\mu A_\\mu(\\bar{t}z)\\right\\}}=1-i g_s \\int^1_0 dt\\,z^\\mu A_\\mu(\\bar{t}z)+\\dots\\,.\n\\label{emissiongf}\n\\end{equation}\nCalculating $\\mathcal{O}(\\alpha_s)$ corrections leads to divergent diagrams and the dependence of the condensate on the spacetime dimension $D$ leads to $\\mathcal{O}(\\epsilon)$ contributions at tree level, that then cause finite counter-terms upon renormalisation. Also, the derivative with respect to $Q_\\kappa$ in Eq.~(\\ref{deriv1}) yields $\\gamma_\\kappa$ in the trace via Eq.~(\\ref{quarkderiv}) which can also give a finite counter-term. This happens for the vertex correction diagrams.\n\n\n\\subsection{Evaluation of The Sum Rules}\nThe new quark condensate contributions are added to the results presented in the literature, see Refs.~\\cite{Ball:2005vx,Ball:2007rt}.  For $f_{K^*}^{\\parallel,\\perp}$ the sum rules read\n\\begin{eqnarray}\n\\lefteqn{(f_{K^*}^\\parallel)^2 e^{-m_{K^*}^2\/M^2}  = \n\\frac{1}{4\\pi^2}\\int\\limits_{m_s^2}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{(s-m_s^2)^2 (s+2m_s^2)}{s^3} +\n\\frac{\\alpha_s}{\\pi}\\, \\frac{M^2}{4\\pi^2}\\left( 1 -\ne^{-s_0\/M^2}\\right)}\\nonumber\\\\[5pt]\n&&{} +\\frac{m_s\\langle \\bar s s\\rangle}{M^2}\\left(1 +\n\\frac{m_s^2}{3M^2} - \n\\frac{13}{9}\\,\\frac{\\alpha_s}{\\pi}\\right)+\n\\frac{4}{3}\\,\\frac{\\alpha_s}{\\pi} \\, \\frac{m_s\\langle \\bar q\n  q\\rangle}{M^2}+\\frac{1}{12M^2}\\,\n\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\rangle \\left(1+\\frac{1}{3}\\frac{m_s^2}{M^2}\\right)\\label{tw2sr1}\n\\nonumber\\\\[5pt]\n&&{}  -\\frac{16\\pi\\alpha_s}{9M^4}\\,\n\\langle \\bar q q\\rangle\\langle \\bar s s\\rangle +\n\\frac{16\\pi\\alpha_s}{81M^4}\\,\\left( \\langle \\bar q q\\rangle^2 +\n\\langle \\bar s s\\rangle^2 \\right),\\label{eq:fKP}\\\\ \\nonumber \\\\\n\\lefteqn{(f_{K^*}^\\perp)^2 e^{-m_{K^*}^2\/M^2}\n= \\frac{1}{4\\pi^2}\\int\\limits_{m_s^2}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{(s-m_s^2)^2 (s+2m_s^2)}{s^3} }\\nonumber\\\\[5pt]\n&&{}+ \\frac{1}{4\\pi^2}\\int\\limits_{0}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{\\alpha_s}{\\pi}\\left( \\frac{7}{9} +\n\\frac{2}{3}\\,\\ln \\frac{s}{\\mu^2}\\right) \n-\\frac{1}{12M^2}\\,\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\rangle \n\\nonumber\\\\[5pt]\n&&{}\n\\times\\left\\{ 1-\\frac{2m_s^2}{M^2}\\left( \\frac{7}{6}-\\gamma_E + {\\rm   Ei}\\left(\n-\\frac{s_0}{M^2}\\right) - \\ln\\,\\frac{\\mu^2}{M^2} +\n\\frac{M^2}{s_0}\\left( 1 - \\frac{M^2}{s_0}\\right) e^{-s_0\/M^2}\n\\right)\\right\\} \\nonumber\\\\[5pt]\n&&{} +\\frac{m_s\\langle \\bar s\n  s\\rangle}{M^2}\\left\\{1+\\frac{m_s^2}{3M^2}+\n\\frac{\\alpha_s}{\\pi}\\left(-\\frac{22}{9} + \\frac{2}{3}\n\\left[ 1-\\gamma_E + \\ln\\,\\frac{M^2}{\\mu^2} +\n  \\frac{M^2}{s_0}\\,e^{-s_0\/M^2} + {\\rm Ei}\\left(-\\frac{s_0}{M^2}\\right)\\right]\n\\right)\\right\\}\\nonumber\\\\[5pt]\n&&{} \n-\\frac{1}{3M^4}\\,m_s\\langle \\bar s\\sigma gGs\\rangle -\n\\frac{32\\pi\\alpha_s}{81M^4}\\,\\left( \\langle \\bar q q\\rangle^2 +\n\\langle \\bar s s\\rangle^2 \\right)\\,,\\label{eq:fKT}\n\\end{eqnarray}\nand for $a_2^{\\parallel,\\perp}(K^*)$\n\\begin{eqnarray}\n\\lefteqn{\na_2^\\parallel(K^*) (f_{K^*}^\\parallel)^2 e^{-m_{K^*}^2\/M^2} = }\\nonumber\\\\\n&&{}\\frac{7}{4\\pi^2}\\,m_s^4\\int\\limits_{m_s^2}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{(s-m_s^2)^2(2m_s^2-s)}{s^5} +\n\\frac{7}{72\\pi^2}\\,\\frac{\\alpha_s}{\\pi} \\,M^2 (1-e^{-s_0\/M^2})\n+\\frac{7}{36M^2}\\,\\left\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle \n\\nonumber\\\\[5pt]\n&&{}+\\frac{7}{3}\\,\\frac{m_s\\langle \\bar s\n  s\\rangle}{M^2}\\left\\{1+ \n\\frac{\\alpha_s}{\\pi}\\left[ -\\frac{184}{27} +\n\\frac{25}{18} \\left(1-\\gamma_E + \\ln\\,\\frac{M^2}{\\mu^2} +\n\\frac{M^2}{s_0}\\,e^{-s_0\/M^2} + {\\rm Ei}\\left(-\\frac{s_0}{M^2}\\right)\n\\right)\\right]\\right\\}  \\nonumber\\\\[5pt]\n&&{}+\\frac{49}{27}\\,\\frac{\\alpha_s}{\\pi} \\, \\frac{m_s\\langle \\bar q\n q\\rangle}{M^2} - \\frac{35}{18}\\, \n\\frac{m_s\\langle\\bar s \\sigma g Gs\\rangle}{M^4}\n+\\frac{224\\pi\\alpha_s}{81M^4}\\,\\left( \\langle \\bar q q\\rangle^2 +\n\\langle \\bar s s\\rangle^2 \\right) -\n\\frac{112\\pi\\alpha_s}{27M^4}\\,\\langle \\bar q q\\rangle \\langle \\bar s\ns\\rangle\\,,\\label{tw2sr3}\\\\ \\nonumber \\\\[5pt]\n\\lefteqn{\na_2^\\perp(K^*) (f_{K^*}^\\perp)^2 e^{-m_{K^*}^2\/M^2} = }\\nonumber\\\\[5pt]\n&&{}\\frac{7}{4\\pi^2}\\,m_s^4\\int\\limits_{m_s^2}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{(s-m_s^2)^2(2m_s^2-s)}{s^5} +\n\\frac{7}{90\\pi^2}\\,\\frac{\\alpha_s}{\\pi} \\,M^2 (1-e^{-s_0\/M^2})\n+\\frac{7}{54M^2}\\,\\left\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle \n\\nonumber\\\\[5pt]\n&&{}+\\frac{7}{3}\\,\\frac{m_s\\langle \\bar s\n  s\\rangle}{M^2}\\left\\{1+ \n\\frac{\\alpha_s}{\\pi}\\left[ -\\frac{206}{27} +\n\\frac{16}{9} \\left(1-\\gamma_E + \\ln\\,\\frac{M^2}{\\mu^2} +\n\\frac{M^2}{s_0}\\,e^{-s_0\/M^2} + {\\rm Ei}\\left(-\\frac{s_0}{M^2}\\right)\n\\right)\\right]\\right\\}  \\nonumber\\\\[5pt]\n&& - \\frac{49}{18}\\, \n\\frac{m_s\\langle\\bar s \\sigma g Gs\\rangle}{M^4}\n+\\frac{112\\pi\\alpha_s}{81M^4}\\,\\left( \\langle \\bar q q\\rangle^2 +\n\\langle \\bar s s\\rangle^2 \\right)\\,.\\label{tw2sr4}\n\\end{eqnarray}\nTo obtain the sum rules for $f_{\\phi}^{\\parallel,\\perp}$ and\n$a_2^{\\parallel,\\perp}(\\phi)$, one has to substitute $\\langle \\bar qq\\rangle\\to \\langle\n\\bar s s\\rangle$ and to double the terms in $m_s\\langle\n\\bar s s\\rangle$, $m_s\\langle \\bar q q\\rangle$ and $m_s \\langle \\bar s\n\\sigma g G s \\rangle$, and replace the perturbative  contribution by\n\\begin{eqnarray}\n\\mbox{for $(f_{\\phi}^{\\parallel,\\perp})^2$:~~}&&\n \\frac{1}{4\\pi^2}\\int_{4m_s^2}^{s_0} ds \\,e^{-s\/M^2} \\frac{(s+2 m_s^2)\n \\sqrt{ 1-4 m_s^2\/s}}{s}\\,,\\nonumber\\\\\n\\mbox{for $a_2^{\\parallel,\\perp}(\\phi)(f_{\\phi}^{\\parallel,\\perp})^2$:~~}&&\n -\\frac{7}{2\\pi^2}\\int_{4m_s^2}^{s_0} ds \\,e^{-s\/M^2} \\frac{m_s^4\n \\sqrt{ 1-4 m_s^2\/s}}{s^2}\\,.\n\\end{eqnarray}\nWe have derived sum rules for the decay constants $f^{\\parallel,\\perp}_V$, however, numerical values can be extracted from experiment for the longitudinal decay constants.  The perpendicular decay constants, on the other hand, must be determined from non-perturbative methods; results are available from Lattice QCD calculations and previous QCD sum rule determinations.  A detailed discussion of the latest numerical values of the decay constants can be found in Ref.~\\cite{Ball:2006eu} from which we just quote the following\n\\begin{eqnarray}\nf_{\\phi}^\\parallel=(215\\pm5)\\,\\textrm{MeV}\\,,\\qquad f_{\\phi}^\\perp =(186\\pm9)\\,\\textrm{MeV}\\,,\n\\label{decayresults}\n\\end{eqnarray}\nwhere $f_{\\phi}^\\parallel$ is an experimental result, and $f_{\\phi}^\\perp$ is from Lattice QCD \\cite{Becirevic:2003pn}. We can compare these results to the sum rules of Eqs.~(\\ref{eq:fKP}) and (\\ref{eq:fKT}) which are plotted in the upper row of Fig.~\\ref{aaa}. The sum rule determinations of  $a_2^{\\parallel,\\perp}(\\phi)$ are plotted in the lower row.  \n\\begin{figure}[h]\n$$ \\epsfxsize=0.5\\textwidth\\epsffile{fparaphi.eps}\\quad \\epsfxsize=0.5\\textwidth\\epsffile{fperpphi.eps}$$\n$$ \\epsfxsize=0.5\\textwidth\\epsffile{a2paraphi.eps}\\quad \\epsfxsize=0.5\\textwidth\\epsffile{a2perpphi.eps}$$\n  \\caption[The hadronic parameters $f^{\\parallel,\\perp}_\\phi$ and $a_2^{\\parallel,\\perp}(\\phi)$ as a function of $M^2$.]{\\small  The decay constants $f^\\parallel_\\phi$ (upper left) and $f^\\perp_\\phi$ (upper right) and the  Gegenbauer coefficients $a_2^\\parallel(\\phi)$ (lower left) and $a_2^\\perp(\\phi)$ (lower right) plotted as a function of $M^2$. The continuum thresholds are $s_0^\\parallel =1.85\\pm 0.05\\,{\\rm GeV}^2$ and $s_0^\\perp =1.40\\pm 0.05\\,{\\rm GeV}^2$ -- see text.   Solid line: central input parameters of Tab.~\\ref{QCDSRinput}. Dashed lines: variation due to the uncertainties of $m_s$ and the gluon condensate. All quantities are evaluated at $\\mu=1\\,{\\rm GeV}$.} \n  \\label{aaa}\n\\end{figure}\n\nIn all the plots the dashed line and shaded region represent the central value and uncertainty of the parameter in question. To evaluate the sum rules we use the input parameters of Tab.~\\ref{QCDSRinput}. For the continuum threshold we note that for the sum rule determination of $f_{K^*}^\\parallel$ in Ref.~\\cite{Ball:2005vx} it is taken to be $s_0^\\parallel(K^*)=1.7\\,{\\rm GeV}^2$, and we expect for $\\phi$ it to be slightly larger. Indeed, by taking $s_0^\\parallel(\\phi)=1.85\\pm 0.05\\,{\\rm GeV}^2$ we find a stable  plateau and excellent agreement with the experimental result for $f_{\\phi}^{\\parallel}$ (upper left plot). Likewise, guided by $s_0^\\perp(K^*)=1.3\\,{\\rm GeV}^2$ \\cite{Ball:2005vx} we find $s_0^\\perp(\\phi)=1.40\\pm 0.05\\,{\\rm GeV}^2$ yields a result consistent with that from Lattice QCD (upper right plot). We use these thresholds in evaluating the sum rules for $a_2^{\\parallel,\\perp}(\\phi)$ and also replace the decay constants by their sum rules, which helps reduce dependence on the Borel parameters. The results are plotted for $a_2^{\\parallel}(\\phi)$ (lower left plot) and $a_2^{\\perp}(\\phi)$ (lower right plot). It is found that the longitudinal parameters exhibit a stronger continuum threshold dependence, which is reflected in the larger uncertainty of the determined value of $a_2^\\parallel(\\phi)$.  The sum rule determinations of the other particle DA parameters follow analogously and all the numerical results  are given in Tab.~\\ref{det_tab1}.\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{| l || c | c ||  l | l || c | c |}\n\\hline\n& \\multicolumn{2}{c||}{\\rho} & \\multicolumn{2}{c||}{K^*}  &  \n\\multicolumn{2}{c|}{\\phi}\\\\\n\\cline{2-7}\n& \\mu = 1\\,{\\rm GeV} & \\mu = 2\\,{\\rm GeV} & \\mu = 1\\,{\\rm GeV} & \\mu =\n2\\,{\\rm GeV} & \\mu = 1\\,{\\rm GeV} & \\mu = 2\\,{\\rm GeV}\\\\\n\\hline\na_1^\\parallel & 0 & 0 & \\phantom{-}0.03(2) & \\phantom{-}0.02(2) & 0 & 0 \n\\\\\na_1^\\perp & 0 & 0 & \\phantom{-}0.04(3) & \\phantom{-}0.03(3) & 0 & 0\n\\\\\na_2^\\parallel & 0.15(7) & 0.10(5) & \\phantom{-}0.11(9) & \n\\phantom{-}0.08(6) & 0.18(8) & 0.13(6)\n\\\\\na_2^\\perp & 0.14(6) & 0.11(5) & \\phantom{-}0.10(8) &\n\\phantom{-}0.08(6) & 0.14(7) & 0.11(5)\n\\\\\\hline\n\\end{array}\n$$\n\\renewcommand{\\arraystretch}{1}\n\\addtolength{\\arraycolsep}{-3pt}\n\\caption[Results for the leading twist-2 distribution amplitude parameters.]{\\small Results for the twist-2 hadronic DA parameters  at the scale  $\\mu= 1\\,{\\rm GeV}$ and scaled up to $\\mu= 2\\,{\\rm GeV}$\nusing the evolution equations (\\ref{evo}). Note that  $a_1^{\\parallel,\\perp}({K^*})$ refers to a $(s\\bar q)$ bound state; for a $(q\\bar  s)$ state it changes sign.}\n\\label{det_tab1}\n\\end{table}\n\n\\section{Twist-3}\nIn this section we determine the twist-3 three-particle parameters of the DAs $\\Phi_{3;K^*}^\\perp$, $\\Phi_{3;K^*}^\\parallel$ and $\\widetilde\\Phi_{3;K^*}^\\parallel$ as defined by Eq.~(\\ref{das_eq27}).  Previous determinations of these parameters are rather few and far between, thus motivating the present analysis. The chiral-even $\\rho$ parameters $\\zeta^\\parallel_{3\\rho}$, $\\omega^\\parallel_{3\\rho}$, and $\\widetilde{\\omega}^\\parallel_{3\\rho}$ were obtained in Ref.~\\cite{Zhitnitsky:1985dd}, and  $\\omega^\\perp_{3\\rho}$ was obtained in Ref.~\\cite{Ball:1998sk}. We make a comparison with these results in Section~\\ref{section_evaluation}.\n\nFirstly, we outline the calculation of the three functions $\\pi_{3;K^*}$ which all proceed in a similar manner, and secondly we explicitly discuss the sum rules for $\\widetilde\\Phi_{3;K^*}^\\parallel$ and present the results. In the diagrams that follow, $q$ is the upper line and $s$ is the lower line.\n\n\\subsection{Calculation}\nEach DA is accessed via a correlation function featuring its defining current. The chiral-even twist-3 parameters $\\zeta_{3K^*}^\\parallel$, $\\widetilde\\omega_{3K^*}^\\parallel$, $\\widetilde\\lambda_{3K^*}^\\parallel$\ncan be determined from\n\\begin{equation}\n\\widetilde\\Pi^\\parallel_{3;K^*}(v,q\\cdot z) = \\frac{i g_{\\alpha\\mu}^{\\perp}}{(q\\cdot z)^2 (2-D)} \\int d^4y \\,e^{-iq\\cdot y} \\bra{0} T \\bar q(z) g_s\n\\widetilde G^{\\alpha z} (vz) \\gamma_z \\gamma_5 s(0) \\bar s(y)\n\\gamma^\\mu q(y) \\ket{0}\\,,\n\\label{C.1}\n\\end{equation}\nwhere the definition of $g^\\perp_{\\mu\\nu}$ is given in Appendix~\\ref{appendixA}.\\footnote{We also make use of the relation $\\gamma_\\mu\\gamma_5=\\frac{i}{6}\\epsilon_{\\mu\\lambda\\nu\\pi}\\gamma^\\lambda\\gamma^\\nu\\gamma^\\pi$ defined in $D$ dimensions.} The parameters $\\kappa_{3K^*}^\\parallel$, $\\omega_{3K^*}^\\parallel$ and $\\lambda_{3K^*}^\\parallel$ can be obtained from the correlation\nfunction $\\Pi_{3;K^*}^\\parallel$ obtained from $\\widetilde\\Pi^\\parallel_{3;K^*}$ by making the replacement\n\\begin{equation}\ng_s\\widetilde G_{\\alpha z} \\gamma_z\\gamma_5 \\to g_s G_{\\alpha z} i \\gamma_z\\,.\n\\end{equation}\nLastly for the chiral-odd operator \n\\begin{equation}\n\\Pi^\\perp_{3;K^*}(v,q\\cdot z)= \\frac{1}{ (q\\cdot z)^3 }\\int d^4y e^{-iq\\cdot y} \\bra{0}T \\bar q(z) \\sigma_{z\\mu} g_s\nG_{z\\mu}(vz) s(0) \\bar s(y) \\sigma_{qz} q(y) \\ket{0}\\,.\n\\label{C.3} \n\\end{equation}\nAll three correlation functions $\\Pi$ can be written as\n\\begin{equation}\n\\Pi_{3;K^*}(v,q\\cdot z) = \\int {\\cal D}\\underline{\\alpha}\\,e^{-i q\\cdot z (\\alpha_2 + v \\alpha_3)}\\pi_{3;K^*}(\\underline{\\alpha})\\,,\n\\label{C.4}\n\\end{equation}\nwhere the exponential function is due to the fact that we keep the correlation functions non-local. The calculation proceeds for each correlation function analogously.  Considering Eq.~(\\ref{C.1}) for instance, firstly we express it in terms of hadronic contributions\n\\begin{equation}\n\\widetilde{\\Pi}_{3;K^*}^\\parallel(v,q\\cdot z) =\n\\frac{(f_{K^*}^\\parallel)^2\n  m_{K^*}^2}{m_{K^*}^2-q^2} \\int{\\cal\n  D}(\\underline{\\alpha})\\,e^{-iq\\cdot z(\\alpha_2+v\\alpha_3)}\\,\n\\widetilde{\\Phi}_{3;K^*}^ \\parallel(\\underline{\\alpha}) + \\dots\\, ,\n\\label{C.5}\n\\end{equation}\nwhere the dots denote contributions from higher-mass states. To derive the sum rule we tread down a well worn path;  express Eq.~(\\ref{C.4}) as a dispersion relation and equate to\nEq.~(\\ref{C.5}), subtract the continuum contribution for $s>s_0$, perform the Borel transformation and project out the desired DA parameter by substitution of the relevant polynomial.  The three hadronic parameters $\\zeta_{3K^*}^\\parallel$, $\\widetilde\\omega_{3K^*}^\\parallel$, $\\widetilde\\lambda_{3K^*}^\\parallel$ are projected out like so:\n\\begin{eqnarray}\n\\left(f_{K^*}^ \\parallel\\right)^2 m_{K^*}^2 e^{-m_{K^*}^2\/M^2}\n      \\left[  \\zeta_{3K^*}^\\parallel\\right]  \n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int {\\cal D}\\underline{\\alpha}\\,\n    \\left[1\\right]  \\frac{1}{\\pi}\\, {\\rm Im}_{s}\n    \\widetilde{\\pi}^\\parallel_{3;K^*}(\\underline{\\alpha})\\,,\n\\\\\n\\left(f_{K^*}^ \\parallel\\right)^2 m_{K^*}^2 e^{-m_{K^*}^2\/M^2}\n      \\,    \\left[\\frac{1}{14}\\,\\widetilde\\lambda_{3K^*}^\\parallel\\right]  \n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int {\\cal D}\\underline{\\alpha}\\,\n         \\left[ \\alpha_1-\\alpha_2\\right]  \\frac{1}{\\pi}\\, {\\rm Im}_{s}\n      \\widetilde{\\pi}_{3;K^*}^\\parallel(\\underline{\\alpha})\\,,\n\\nonumber\\\\\n\\left(f_{K^*}^ \\parallel\\right)^2 m_{K^*}^2 e^{-m_{K^*}^2\/M^2}\n      \\,    \\left[\\frac{3}{28}\\,\\widetilde\\omega_{3K^*}^\\parallel\\right]  \n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int {\\cal D}\\underline{\\alpha}\\,\n         \\left[\\alpha_3-\\frac{3}{7}\\right]  \\frac{1}{\\pi}\\, {\\rm Im}_{s}\n      \\widetilde{\\pi}_{3;K^*}^\\parallel(\\underline{\\alpha})\\,.\\nonumber\n      \\label{C.6}\n\\end{eqnarray}\nThe formulas for the other parameters are analogous. In calculating  the functions  $\\pi_{3;K^*}$ we keep explicit mass corrections $\\mathcal{O}(m_s^2,m_q^2,m_s m_q)$ and all operators up to $D=6$ except  the triple gluon condensate $\\left<g_s^3 f G^3\\right>$ which is expected to yield a negligible contribution. By retaining all mass terms the resulting formulas for $\\pi_{3;K^*}$ can be used to derive sum rules for all the DA parameters for $K^*$, $\\rho$ and $\\phi$ by setting  $m_q=0$, $m_q= m_s=0$ and $m_q= m_s$ respectively. For $\\rho$ and $\\phi$ expressions for the three-particle twist-3 DAs are analogous to Eq.~(\\ref{das_eq27}), except that the G-parity violating parameters $\\kappa$ and $\\lambda$ vanish.\n\n\\subsection*{Perturbation Theory}\nThe perturbation theory calculation is given by the two diagrams shown in Fig.~\\ref{a}. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{pt_sr.eps}$$\n  \\caption[Diagrams contributing to perturbation theory.]{\\small Diagrams contributing to perturbation theory.} \n  \\label{a}\n\\end{figure}\nAs an example, consider the first diagram, which up to an overall factor can be written generally as\n\\begin{eqnarray}\n\\Pi^{(\\rm{PT}_1)}&=&g_s^2\\int\\frac{d^D p}{(4 \\pi)^D}\\int\\frac{d^D l}{(4 \\pi)^D}\\,\\textrm{Tr}[\\Gamma_1 S^{(s)}(\\slash{p}+\\slash{q})\\Gamma_2 S^{(q)}(\\slash{p})\\gamma^{\\beta}S^{(q)}(\\slash{p}+\\slash{l})] \\nonumber\\\\\n&&\\cdot\\,[l_\\mu D_{\\nu\\beta}(l)-l_\\nu D_{\\mu\\beta}(l)]\\,e^{iz \\cdot(l \\bar{v}+p)}\\,.\\label{part1}\n\\end{eqnarray}\nwhere the Dirac matrices $\\Gamma_{1,2}$ depend on the correlation function. In performing the two successive integrations over $l$ and $p$, Feynman parameterisation leads to shifting the variables $l\\to l-p\\bar x$ and $p\\to p-q\\bar y$ respectively. Each time the exponential in (\\ref{part1}) is also shifted. In expanding the part of the exponential that contributes to the integral, for example, for $l$ we have $e^{i  l  \\cdot z \\bar{v}}=1+i (l  \\cdot z)\\bar{v} +\\dots$, only the first two terms contribute; higher order terms are killed off either via $z^2=0$ or because integrals with odd numbers of open indices, for example $l^{\\mu_1}l^{\\mu_2}l^{\\mu_3}$, in the numerator vanish due to symmetry. After the integrations any terms $(\\mathcal{T})$ including factors of $i( q  \\cdot z)\\bar{v}$ are dealt with by trading them for derivatives of $\\mathcal{T}$ by using partial integration of the final exponential\n\\begin{equation}\ni (q\\cdot z) \\bar{v}=\\frac{1}{\\bar{y}}\\frac{\\partial}{\\partial\\bar{x}}e^{-i q  \\cdot z \\bar{y}(1-\\bar{x}\\bar{v})} \\qquad\\Rightarrow\\qquad (q\\cdot z) \\bar{v}\\, \\mathcal{T} \\stackrel{\\textrm{PI}}{\\longrightarrow} \\frac{i}{\\bar{y}}\\frac{\\partial }{\\partial\\bar{x}}\\mathcal{T} \\,,\\label{PI}\n\\end{equation}\nwhere surface terms do not contribute as they vanish for $x=\\{1,0\\}$. The exponential can be matched to the ``canonical form'' by writing\n\\begin{equation}\n\\int^1_0dx\\,\\int^1_0dy\\, e^{-i q  \\cdot z \\bar{y}(1-\\bar{x}\\bar{v})} = \\int^1_0dx\\,\\int^1_0dy\\,  \\int {\\cal D}\\underline{\\alpha}\\, \\delta(\\alpha_1-y)\\delta(\\alpha_2-\\bar{x}\\bar{y})\\delta(\\alpha_3- x\\bar{y})\\,e^{-iq\\cdot z(\\bar\\alpha_1-\\bar{v}\\alpha_3)}\\,.\n\\end{equation}\nPerforming the $x$ and $y$ integration of the whole expression gives the desired result\n\\begin{equation}\n\\int {\\cal D}\\underline{\\alpha}\\, e^{-iq\\cdot z(\\alpha_2+v\\alpha_3)}\\pi^{(\\textrm{PT}_1)}(\\underline{\\alpha})\\,.\n\\end{equation}\nThe second diagram follows analogously. Both diagrams are divergent and need to be renormalised separately. We find finite counter terms which are proportional to the quark masses. \n\n\\subsection*{Gluon Condensate}\nThe leading order contribution to the gluon condensate $\\left<\\frac{\\alpha_s}{\\pi} G^2\\right>$ is found using the background field method as outlined in Section~\\ref{example}. There are only two diagrams contributing  as depicted in Fig.~\\ref{b}. One vacuum momentum $k$, from the gluon attached to the quark line, is introduced and hence one derivative is taken. As the gluon emerging from the non-local vertex $G(vz)$ carries no momentum these diagrams are proportional to $\\delta(\\alpha_3)$ and the remaining momentum fractions are related by $1-\\alpha_1=\\alpha_2$; the identification of the momentum fractions with the Feynman parameters is therefore straightforward. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{g2_sr.eps}$$\n  \\caption[Diagrams contributing to the gluon condensate $\\left<\\frac{\\alpha_s}{\\pi} G^2\\right>$.]{\\small Diagrams contributing to the gluon condensate $\\left<\\frac{\\alpha_s}{\\pi} G^2\\right>$.}\n  \\label{b} \n\\end{figure}\nThe calculation requires the integration over one momentum $p$ and the result can simply be written unexpanded in the quark masses. \n\n\n\\subsection*{Mixed Condensate}\nThe mixed condensates $\\left<\\bar q \\sigma g_s G q \\right>$ and $\\left<\\bar s \\sigma g_s G s \\right>$ originate from the diagrams shown in Fig.~\\ref{f}. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{mixed_sr.eps}$$\n  \\caption[Diagrams contributing to the mixed condensates $\\left<\\bar q \\sigma g_s G q \\right>$ and $\\left<\\bar s \\sigma g_s G s \\right>$.]{\\small Diagrams contributing to the mixed condensates $\\left<\\bar q \\sigma g_s G q \\right>$ and $\\left<\\bar s \\sigma g_s G s \\right>$.}\n  \\label{f} \n\\end{figure}\nTo extract the mixed condensates one uses the first non-local term in the expansion  $(D=4)$ \\cite{PT:84}\n\\begin{eqnarray}\n\\lefteqn{\\bra{0}\\!:\\!\\bar{q}^i_\\alpha(x_1) g_s (G_{\\mu\\nu})_{ij}(y)q^j_\\beta(x_2)\\!:\\!\\ket{0}=}\\hspace{1.2in}\\\\\n&&\\delta^{ij}\\left[\\frac{\\left<\\bar{q} g_s \\sigma G q\\right>}{144}\\left\\{\\sigma_{\\mu\\nu}+\\frac{m_q}{2}\\left[\\Delta_\\mu \\gamma_\\nu-\\Delta_\\nu \\gamma_\\mu -i(\\Delta^\\lambda \\gamma_\\lambda)\\sigma_{\\mu\\nu}\\right]\\right\\}\\right.\\nonumber\\\\\n&&\\left.+g_s^2 \\left<6\\right>\\left\\{\\frac{i}{288}(x_2^\\xi \\sigma_{\\mu\\nu}\\gamma_\\xi-x_1^\\xi \\gamma_\\xi \\sigma_{\\mu\\nu})-\\frac{1}{216}(y_\\mu\\gamma_\\nu-y_\\nu\\gamma_\\mu)\\right\\}\\right]_{\\beta\\alpha}\\,. \\nonumber\n\\label{mixed}\n\\end{eqnarray}\nThe first $\\sigma_{\\mu\\nu}$ does not contribute, but the term $\\sim m_q$ does. The $\\Delta_\\mu$s can be expressed as derivatives of the trace via partial integration which is dealt with simply by using Eq.~(\\ref{quarkderiv}). Along with the condensate gluon, the quark condensate lines carry no momentum. There is therefore no loop integration to perform and the results are proportional to $\\delta(\\alpha_3)\\delta(\\alpha_{1,2})$.\n\n\\subsection*{Quark Condensates}\nThe diagrams of Fig.~\\ref{c} generate the condensates $m_{q,s}\\left<\\bar q q\\right>$ and $m_{q,s}\\left<\\bar s s \\right>$. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.8\\textwidth\\epsffile{quark1_sr.eps}$$\n  \\caption[Diagrams contributing to the quark  condensates $\\left<\\bar q q\\right>$ and $\\left<\\bar s s \\right>$.]{\\small Diagrams contributing to the quark  condensates $\\left<\\bar q q\\right>$ and $\\left<\\bar s s \\right>$.}\n  \\label{c} \n\\end{figure}\nWe do not consider $\\mathcal{O}(m_{q,s}^2)$ corrections, which are however of dimension six, as they are very well suppressed with respect to the other contributions. To extract all $\\mathcal{O}(m_{q,s})$ mass corrections the first non-local term in the expansion of the quark fields, given by Eq.~(\\ref{quarkextract}), is needed. There is one loop momentum to integrate over and one finds contributions from the exponential which can be dealt with via partial integration in the same way as with the perturbation theory calculation, see Eq.~(\\ref{PI}). The results are proportional to $\\delta(\\alpha_{1,2})$.  The diagrams in Fig.~\\ref{d} generate the condensate $\\left<\\bar q q\\right>\\left<\\bar s s \\right>$ which is already of dimension six, so we do not require mass corrections. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{quark2_sr.eps}$$\n  \\caption[Diagrams contributing to the quark  condensate $\\left<\\bar q q\\right>\\left<\\bar s s \\right>$.]{\\small  Diagrams contributing to the quark  condensate $\\left<\\bar q q\\right>\\left<\\bar s s \\right>$.}\\label{d} \n\\end{figure}\nThe two diagrams are of equal magnitude and cancel, however only for $\\widetilde{\\pi}^{\\parallel}_{3;K^{*}}$ they add. There is no loop integral to perform and the result is proportional to $\\delta(\\alpha_1)\\delta(\\alpha_2)$. The four quark condensate is simplified via the vacuum saturation hypothesis (VSH) \\cite{PT:84,Shifman:1978by}\n\\begin{equation}\n\\bra{0}\\!:\\!\\bar{q}_\\alpha^i (x_1) q_\\beta^j(x_2)\\bar{s}_\\gamma^k (x_3) s_\\delta^l(x_4)\\!:\\!\\ket{0} \\stackrel{\\textrm{VSH}}{\\longrightarrow}\\bra{0}\\!:\\!\\bar{q}_\\alpha^i (x_1) q_\\beta^j(x_2)\\!:\\!\\ket{0}\\bra{0}\\!:\\!\\bar{s}_\\gamma^k (x_3) s_\\delta^l(x_4)\\!:\\!\\ket{0}\\,.\n\\end{equation}\nThe diagrams in Fig.~\\ref{e} generate the condensates $\\left<\\bar q q\\right>^2$ and $\\left<\\bar s s \\right>^2$. They stem from the operator $\\left<6\\right>$ appearing in the expansion of the mixed condensate, Eq.~(\\ref{mixed}), which simplifies as\n\\begin{equation}\n \\left<6\\right>=\\langle\\bar q \\gamma_\\kappa t^a q \\sum_{u,d,s}\\bar{q} \\gamma^\\kappa t^a q\\rangle\\stackrel{\\textrm{VSH}}{\\longrightarrow} -\\frac{4}{9}\\left<\\bar q q\\right>^2\\,;\n \\end{equation}\nthus at higher order the mixed condensate also contributes to the quark condensates. The light-like co-ordinate of the gluonic field strength tensor $v z_\\mu$ simplifies the resulting trace via $z^2=0$ from Eq.~(\\ref{mixed}) and the other co-ordinates are dealt with as before. \n \\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{quark3_sr.eps}$$\n  \\caption[Diagrams contributing to the quark  condensates $\\left<\\bar q q\\right>^2$ and $\\left<\\bar s s \\right>^2$.]{\\small Diagrams contributing to the quark  condensates $\\left<\\bar q q\\right>^2$ and $\\left<\\bar s s \\right>^2$ from the expansion of the mixed condensate -- see Eq.~(\\ref{mixed}).}\n  \\label{e} \n\\end{figure}\n\\subsection*{Results}\nFor the functions $\\pi_{3;K^*}$, given by Eq.~(\\ref{C.6}), we find (dropping all terms that vanish upon taking the imaginary part):\n\\begin{eqnarray}\n\\pi^{\\perp }_{3;K^{*}}\\left(\\underline{\\alpha}\\right)\n&=&\n\\frac{\\alpha_{s}}{2\\pi^{3}}\\ln\\frac{-q^2}{\\mu^{2}}\n\\left[q^2\\alpha_{1}\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{2}}-\n\\frac{1}{\\bar{\\alpha}_{1}}\\right)\\right.\n\\nonumber \\\\\n&+&m_{s}m_{q}\\frac{\\alpha_{3}^{2}}{\\bar{\\alpha}_{1}\\bar{\\alpha}_{2}}\n\\left[\\bar{\\alpha}_{2}\\left(\\ln\\frac{\\alpha_{2}\\alpha_{3}}{\\bar{\\alpha}_{1}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)-\\left\\{\\alpha_{1}\n\\leftrightarrow\\alpha_{2}\\right\\}\\right]\n\\nonumber \\\\\n&+&m_{s}^{2}\\left\\{-\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{2}}-\n\\frac{1}{\\bar{\\alpha}_{1}}\\right)-\\frac{\\alpha_{2}\\alpha_{3}^{2}}{\n\\bar{\\alpha}_{2}^{2}}\\left(\\ln\\frac{\\alpha_{1}\\alpha_{3}}{\\bar{\\alpha}_{2}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right\\}\n-m_{q}^{2}\\left\\{\\alpha_{1}\\leftrightarrow\\alpha_{2}\\right\\}] \n\\nonumber \\\\\n&+&\\frac{1}{12}\\langle\\frac{\\alpha_{s}}{\\pi}G^{2}\\rangle\n\\frac{\\alpha_{1}\\alpha_{2}\\left(\\alpha_{1}-\\alpha_{2}\\right)\\delta\n\\left(\\alpha_{3}\\right)}{\\alpha_{1}m_{q}^{2}+\\alpha_{2}m_{s}^{2}-\n\\alpha_{1}\\alpha_{2}q^2}\n\\nonumber \\\\\n&+&\\frac{2}{3q^2}\\frac{\\alpha_{s}}{\\pi}\\left\\{\\right.\n\\frac{\\bar{\\alpha}_{3}}{2}\\left(1+\\alpha_{3}\\right)\\left(m_{q}\n\\langle\\bar{q}q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{s}\\langle\\bar{s}s\n\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right) \n\\nonumber \\\\\n&+&\\alpha_{3}\\left[1+\\alpha_{3}\\left(1+\\ln\\left(\\alpha_{3}\n\\bar{\\alpha}_{3}\\right)+\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right]\\left(m_{s}\n\\langle\\bar{q}q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{q}\\langle\\bar{s}s\n\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right)\\left.\\right\\} \n\\nonumber \\\\\n&+&\\frac{1}{6q^4}\\delta\\!\\left(\\alpha_{3}\\right)\\left\\{m_{q}\\langle\\bar{q}\n\\sigma g_s G q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{s}\\langle\\bar{s}\n\\sigma g_s G s\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right\\}\n\\nonumber \\\\\n&+&\\frac{16}{27q^4} \\pi \\alpha_{s} \\delta\\!\\left( \\alpha_{3}\\right) \n\\left\\{\\langle\\bar{q}\nq\\rangle^{2}\\delta\\!\\left(\\alpha_{2}\\right)-\\langle\\bar{s}s\\rangle^{2}\n\\delta\\!\\left(\\alpha_{1}\\right)\\right\\},\\label{corresult1}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\pi^{\\parallel}_{3;K^{*}}\\left(\\underline{\\alpha}\\right)\n&=&\n\\frac{\\alpha_{s}}{4\\pi^{3}}\\ln\\frac{-q^2}{\\mu^{2}}\\left[\\right.q^2\n\\alpha_{1}\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{2}}-\n\\frac{1}{\\bar{\\alpha}_{1}}\\right)\n\\nonumber \\\\\n&+&m_{s}m_{q}\\frac{\\alpha_{3}^{2}}{\\bar{\\alpha}_{1}\\bar{\\alpha}_{2}}\n\\left\\{\\bar{\\alpha}_{2}\\left(\\ln\\frac{\\alpha_{2}\\alpha_{3}}{\\bar{\\alpha}_{1}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)-\\left\\{\\alpha_{1}\n\\leftrightarrow\\alpha_{2}\\right\\}\\right\\}\n\\nonumber \\\\\n&+&m_{s}^{2}\\left\\{-\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{2}}-\n\\frac{1}{\\bar{\\alpha}_{1}}\\right)-\\frac{\\alpha_{2}\\alpha_{3}^{2}}{\n\\bar{\\alpha}_{2}^{2}}\\left(\\ln\\frac{\\alpha_{1}\\alpha_{3}}{\\bar{\\alpha}_{2}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right\\}\n-m_{q}^{2}\\left\\{\\alpha_{1}\\leftrightarrow\\alpha_{2}\\right\\}\\left.\\right]\n\\nonumber \\\\\n&+&\\frac{1}{24}\\langle\\frac{\\alpha_{s}}{\\pi}G^{2}\\rangle\\frac{\\alpha_{1}\n\\alpha_{2}\\left(\\alpha_{1}-\\alpha_{2}\\right)\\delta\\!\\left(\\alpha_{3}\\right)}{\n\\alpha_{2}m_{s}^{2}+\\alpha_{1}m_{q}^{2}-\\alpha_{1}\\alpha_{2}q^2}\n\\nonumber\\\\\n&+&\\frac{1}{3q^2}\\frac{\\alpha_{s}}{\\pi}\\left\\{\\right.\\frac{\n\\bar{\\alpha}_{3}}{2}\\left(1+\\alpha_{3}\\right)\\left(m_{q}\\langle\\bar{q}q\n\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{s}\\langle\\bar{s}s\\rangle\\delta\n\\left(\\alpha_{1}\\right)\\right) \n\\nonumber \\\\\n&+&\\alpha_{3}\\left[1+\\alpha_{3}\\left(\\ln\\left(\\alpha_{3}\\bar{\\alpha}_{3}\n\\right)+\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right]\\left(m_{s}\\langle\\bar{q}q\n\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{q}\\langle\\bar{s}s\\rangle\\delta\n\\left(\\alpha_{1}\\right)\\right)\\left.\\right\\}\n\\nonumber \\\\\n&+&\\frac{1}{12 q^4}\\delta\\!\\left(\\alpha_{3}\\right)\\left\\{\nm_{q}\\langle\\bar{q}\n\\sigma g_s G q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{s}\\langle\\bar{s}\n\\sigma g_s G s\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right\\}\n\\nonumber \\\\\n&+&\\frac{8}{27 q^4}\\alpha_{s}\\pi \\delta\\!\\left( \\alpha_{3}\\right)\n\\left(\\langle\\bar{q}q\\rangle^{2}\\delta\\!\\left(\\alpha_{2}\\right) -\n\\langle\\bar{s}s\\rangle^{2}\\delta\\!\\left(\\alpha_{1}\\right) \\right),\\label{corresult2}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\widetilde{\\pi}^{\\parallel}_{3;K^{*}}\\left(\\underline{\\alpha}\\right)\n&=&\\frac{\\alpha_{s}}{4\\pi^{3}}\\ln\\frac{-q^2}{\\mu^{2}}\\left[\\right.-\nq^2\\alpha_{1}\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{1}}+\n\\frac{1}{\\bar{\\alpha}_{2}}\\right)\n\\nonumber \\\\\n&+&m_{s}m_{q}\\frac{\\alpha_{3}^{2}}{\\bar{\\alpha}_{1}\\bar{\\alpha}_{2}}\n\\left\\{\\bar{\\alpha}_{1}\\left(\\ln\\frac{\\alpha_{1}\\alpha_{3}}{\\bar{\\alpha}_{2}}-\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)+\\left\\{\\alpha_{1}\n\\leftrightarrow\\alpha_{2}\\right\\}\\right\\}\n\\nonumber \\\\\n&+&m_{s}^{2}\\left\\{\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{1}}+\n\\frac{1}{\\bar{\\alpha}_{2}}\\right)+\\frac{\\alpha_{2}\\alpha_{3}^{2}}{\n\\bar{\\alpha}_{2}^{2}}\\left(\\ln\\frac{\\alpha_{1}\\alpha_{3}}{\\bar{\\alpha}_{2}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right\\}\n+m_{q}^{2}\\left\\{\\alpha_{1}\\leftrightarrow\\alpha_{2}\\right\\}\\left.\\right]\n\\nonumber \\\\\n&+&\\frac{1}{24}\\langle\\frac{\\alpha_{s}}{\\pi}G^{2}\\rangle\\frac{\\alpha_{1}\n\\alpha_{2}\\delta\\!\\left(\\alpha_{3}\\right)}{\\alpha_{2}m_{s}^{2}+\\alpha_{1}\nm_{q}^{2}-\\alpha_{1}\\alpha_{2}q^2}\n\\nonumber\\\\\n&+&\\frac{1}{3q^2}\\frac{\\alpha_{s}}{\\pi}\\left\\{\\right.\\frac{\n\\bar{\\alpha}_{3}^{2}}{2}\\left(m_{s}\\langle\\bar{s}s\\rangle\\delta\\!\\left(\n\\alpha_{1}\\right)+m_{q}\\langle\\bar{q}q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)\n\\right) \n\\nonumber \\\\\n&+&\\alpha_{3}\\left[1-\\alpha_{3}\\left(2+\\ln\\left(\\alpha_{3}\\bar{\\alpha}_{3}\n\\right)+\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right]\\left(m_{s}\\langle\\bar{q}q\n\\rangle\\delta\\!\\left(\\alpha_{2}\\right)+m_{q}\\langle\\bar{s}s\\rangle\\delta\n\\left(\\alpha_{1}\\right)\\right)\\left.\\right\\}\n\\nonumber \\\\\n&+&\\frac{1}{12 q^4}\\delta\\!\\left(\\alpha_{3}\\right)\\left\\{ m_{q}\\langle\n\\bar{q}\\sigma g_s\nGq\\rangle\\delta\\!\\left(\\alpha_{2}\\right)+m_{s}\\langle\\bar{s}\n\\sigma g_s G s\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right\\}\n\\nonumber \\\\\n&+&\\frac{8}{27 q^4}\\alpha_{s}\\pi \\delta\\!\\left( \\alpha_{3}\\right)\n\\left(\\langle\\bar{q}q\\rangle^{2}\\delta\\!\\left(\\alpha_{2}\\right)+\n\\langle\\bar{s}s\\rangle^{2}\\delta\\!\\left(\\alpha_{1}\\right) \\right)\n\\nonumber\\\\\n&+&\\frac{2}{3 q^4}\\alpha_{s}\\pi \\delta\\!\\left( \\alpha_{1}\\right) \n\\delta\\!\\left( \\alpha_{2}\\right)\\langle\\bar{q}q\\rangle\\langle\\bar{s}s\n\\rangle.\\label{corresult3}\n\\end{eqnarray}\n\n\n\\subsection{Evaluation of The Sum Rules}\\label{section_evaluation}\nIn the following we consider $\\widetilde{\\pi}_{3;K^*}^\\parallel$; the sum rules for the other DA parameters and particles $\\rho$ and $\\phi$ follow similarly. The values of the input parameters  and the continuum thresholds used for all sum rules are given in Appendix~\\ref{appendixB}. \n\nOne subtlety must be noted: upon integration over $\\alpha_i$ and subsequent\nexpansion in powers of the quark masses, the gluon condensate contribution yields\nterms in $m_{q,s}^2 \\ln (m_{q,s}^2\/(-q^2))$, which are long-distance\neffects and must not appear in the short-distance OPE of the correlation functions of Eqs~(\\ref{C.1}) and (\\ref{C.3}). The appearance of these \nlogarithmic terms is due to the fact that the expressions of Eqs.~(\\ref{corresult1}-\\ref{corresult3}) are\nobtained using Wick's theorem which implies that the condensates are normal-ordered: \n$\\langle O \\rangle =\\bra{0}\\!:\\!O\\!:\\! \\ket{0}$ \\cite{logms}. Rewriting the OPE in terms of\nnon-normal-ordered operators, all infrared sensitive terms can be\nabsorbed into the corresponding condensates. Indeed, using, \n\\begin{equation}\n\\bra{0}\\bar s g_s G s \\ket{0} = \\bra{0}\\!:\\!\\bar s g_s G s\\!:\\!\\ket{0} +\n\\frac{m_s}{2}\\, \\ln\\,\\frac{m_s^2}{\\mu^2} \\,\\bra{0}\\! :\\!\n\\frac{\\alpha_s}{\\pi}\\, G^2\\!:\\!\\ket{0}\\,,\n\\end{equation}\nand the corresponding formula for $q$ quarks, all terms in $\\ln m_{q,s}^2$ can be absorbed into the mixed quark-quark-gluon condensate and the resulting short-distance coefficients\ncan be expanded in powers of $m_{q,s}^2$. \n\\begin{figure}[h]\n$$\n\\epsfxsize=0.6\\textwidth\\epsffile{ltileven.eps}\n$$\n$$\n\\epsfxsize=0.6\\textwidth\\epsffile{otileven.eps}\n$$\n$$\n\\epsfxsize=0.6\\textwidth\\epsffile{zeven.eps}\n$$\n\\caption[Hadronic parameters of $\\widetilde\\Phi_{3;K^*}^\\parallel$ as functions of $M^2$.]{\\small  Hadronic parameters of the twist-3 distribution amplitude $\\widetilde\\Phi_{3;K^*}^\\parallel$ as functions of $M^2$.  Upper: $\\widetilde\\lambda_{3K^*}^\\parallel$, middle:  $\\widetilde\\omega_{3K^*}^\\parallel$, and lower:  $\\zeta_{3K^*}^\\parallel$. The solid curve is for central input values for $\\mu=1\\,{\\rm GeV}$ and outer curves take into consideration their uncertainties -- see Tab.~\\ref{QCDSRinput}. Horizontal dashed line is the extracted DA parameter value and shaded region its uncertainty -- see Tab.~\\ref{det_tab2}.}\n  \\label{g} \n\\end{figure}\n\nIn Fig.~\\ref{g} we plot the sum rules  for $\\widetilde\\lambda_{3K^*}^\\parallel$,  $\\widetilde\\omega_{3K^*}^\\parallel$ and $\\zeta_{3K^*}^\\parallel$, given by Eqs.~(\\ref{C.6}), which are evaluated for the central input parameters of Tab.~\\ref{QCDSRinput} and at a scale $\\mu=1\\,{\\rm GeV}$. The parameters unfortunately exhibit  very strong $M^2$ dependence, which leads to increased uncertainty of their values; we do not find a stable plateau in the region $ 1\\,\\textrm{GeV}^2\\leqslant M^2\\leqslant  2.5\\,\\textrm{GeV}^2$. On the other hand, there is only a very small $s_0$ dependence $\\approx 1 \\%$ over the range $s_0^\\parallel (K^*) = (1.3\\pm 0.3)\\,{\\rm GeV}^2$. The curves flatten at high $M^2$ which is expected, as the power corrections become negligible compared to the perturbative contribution.\\footnote{The quark condensates survive as $M^2\\to\\infty$ as $\\hat{\\mathcal{B}}\\left[q^{-2}\\right]=-1$ but perturbation theory $\\sim M^4$ -- see Appendix~\\ref{appendixB}.} The sum rules for the other parameters and particles show the same general behaviour which is fairly typical of non-diagonal correlation functions. If one were to use diagonal correlation functions then it is possible that the sum rules would be better behaved and thus the uncertainties would be reduced somewhat. The calculation of diagonal correlation functions of three-particle operators, as we saw with the gluon condensate in Chapter~\\ref{chapter3_SR}, is rather more involved, especially when calculating radiative corrections, which may very well be necessary in this case. \n\nAll the numerical results, including the uncertainties from the variation of $M^2$, $s_0$, and input parameters,  are given in Tab.~\\ref{det_tab2}.  The results are presented at the scale  $\\mu= 1\\,{\\rm GeV}$ and scaled up to $\\mu= 2\\,{\\rm GeV}$, \nusing the evolution equations, Eq.~(\\ref{evo}).  The only previous determination for  comparison is for the chiral-even $\\rho$ parameters, $\\zeta^\\parallel_{3\\rho}(1\\,\\rm{GeV})=0.033\\pm0.003$, $\\omega^\\parallel_{3\\rho}(1\\,\\rm{GeV})=0.2$, and \n$\\widetilde{\\omega}^\\parallel_{3\\rho}(1\\,\\rm{GeV})=-0.1$ \\cite{Zhitnitsky:1985dd}\nand  $\\omega^\\perp_{3\\rho}(1\\,\\rm{GeV})=0.3\\pm0.3$ \\cite{Ball:1998sk}. These results agree with ours, although we consider the uncertainty of $\\zeta^\\parallel_{3\\rho}$ to be optimistic.\n\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{| l || c | c ||  l | l || c | c |}\n\\hline\n& \\multicolumn{2}{c||}{\\rho} & \\multicolumn{2}{c||}{K^*}  &  \n\\multicolumn{2}{c|}{\\phi}\\\\\n\\cline{2-7}\n& \\mu = 1\\,{\\rm GeV} & \\mu = 2\\,{\\rm GeV} & \\mu = 1\\,{\\rm GeV} & \\mu =\n2\\,{\\rm GeV} & \\mu = 1\\,{\\rm GeV} & \\mu = 2\\,{\\rm GeV}\\\\\n\\hline\n\\zeta_{3V}^\\parallel \n& 0.030(10) & 0.020(9) & \\phantom{-}0.023(8) & \\phantom{-}0.015(6) & \n0.024(8) & 0.017(6)\n\\\\\n\\widetilde\\lambda_{3V}^\\parallel \n& 0 & 0 & \\phantom{-}0.035(15)& \\phantom{-}0.017(8) & 0 & 0\n\\\\\n\\widetilde\\omega_{3V}^\\parallel \n& -0.09(3) & -0.04(2) & -0.07(3) &  -0.03(2) & -0.045(15) & -0.022(8)\n\\\\\n\\kappa_{3V}^\\parallel \n& 0 & 0 & \\phantom{-}0.000(1) & -0.001(2) & 0 & 0\n\\\\\n\\omega_{3V}^\\parallel \n& 0.15(5) & 0.09(3) & \\phantom{-}0.10(4) & \\phantom{-}0.06(3) &\n0.09(3) & 0.06(2)\n\\\\\n\\lambda_{3V}^\\parallel \n& 0 & 0 & -0.008(4) & -0.004(2) & 0 & 0\n\\\\\n\\kappa_{3V}^\\perp \n& 0 & 0 & \\phantom{-}0.003(3) & -0.001(2) & 0 & 0 \n\\\\\n\\omega_{3V}^\\perp \n& 0.55(25) & 0.37(19) & \\phantom{-}0.3(1) & \\phantom{-}0.2(1) &  \n0.20(8) & 0.15(7)\n\\\\\n\\lambda_{3V}^\\perp \n& 0 & 0 & -0.025(20) & -0.015(10) & 0 & 0\\\\\\hline \n\\end{array}\n$$\n\\renewcommand{\\arraystretch}{1}\n\\addtolength{\\arraycolsep}{-3pt}\n\\caption[Results for the leading twist-3 distribution amplitude parameters.]{\\small Results for the leading three-particle twist-3 hadronic parameters of the DAs of Eq.~(\\ref{das_eq27}). The results are presented at the scale  $\\mu= 1\\,{\\rm GeV}$ and scaled up to $\\mu= 2\\,{\\rm GeV}$ using the evolution equations (\\ref{evo}). The sign of the  parameters corresponds to the sign convention for the strong coupling defined by the covariant derivative $D_\\mu = \\partial_\\mu - i g_s A^a_\\mu t^a$; they change sign if $g_s$ is fixed by $D_\\mu = \\partial_\\mu + i g_s A^a_\\mu t^a$.}\n\\label{det_tab2}\n\\end{table}\n\nIn Fig.~\\ref{graphs} we plot the two-particle twist-3 DAs as defined by Eqs.~(\\ref{das_eq30}- \\ref{das_eq33}). G-parity violating effects cause the small asymmetry of the $K^*$ curves. The effects of $\\rm SU(3)_F$-breaking are larger and cause the pronounced difference between $\\phi_{3}^\\parallel$ and $\\phi_{3}^\\perp$  for the $\\rho$ and $\\phi$. We notice in particular the end-point behaviour of the DAs is greatly modified. The fact that both $\\phi_{3;\\rho}^{\\parallel\\perp}$ and $\\phi_{3;K^*}^{\\parallel\\perp}$ diverge as $u\\to1$ and $\\phi_{3;\\rho}^{\\parallel\\perp}$ for $u\\to0$ is in itself not  a problem. It is only the leading-twist DA that can be considered a probability distribution and likewise there is no cause for concern that $\\phi_{3;\\rho}^{\\parallel}$ takes negative values. Moreover, in practical calculations we are only interested in convolutions of the DAs with hard scattering kernels, which are generally finite. If not, this signals a problem with the hard scattering kernel, rather than the DA, as happens with end-point divergences within the QCD factorisation framework for non-leptonic $B$ decays, see Chapter~\\ref{chapter6_QCDF}. \n\\begin{figure}[t]\n$$\n\\epsfxsize=0.5\\textwidth\\epsffile{fig1a.eps}\\quad\n\\epsfxsize=0.5\\textwidth\\epsffile{fig1b.eps}\n$$\n\\caption[The  distribution amplitudes $\\phi^\\parallel_{3;V}$ and $\\psi^\\parallel_{3;V}$ as a function of  $u$.]{\\small Left: $\\phi^\\parallel_{3}$ as a function of  $u$ for the central values of hadronic parameters, for $\\mu=1\\,$GeV.   Red line: $\\phi_{3;\\rho}^\\parallel$, green:   $\\phi_{3;K^*}^\\parallel$, blue: $\\phi_{3;\\phi}^\\parallel$. Right: same for $\\psi^\\parallel_{3}$.}\n$$\n\\epsfxsize=0.5\\textwidth\\epsffile{fig2a.eps}\\quad\n\\epsfxsize=0.5\\textwidth\\epsffile{fig2b.eps}\n$$\n\\caption[The distribution amplitudes $\\phi^\\perp_{3;V}$ and $\\psi^\\perp_{3;V}$ as a function of  $u$.]{\\small Left: $\\phi^\\perp_{3}$ as a function of $u$ for the central values of hadronic parameters, for $\\mu=1\\,$GeV.   Red line: $\\phi_{3;\\rho}^\\perp$, green:   $\\phi_{3;K^*}^\\perp$, blue: $\\phi_{3;\\phi}^\\perp$. Right: same for $\\psi^\\perp_{3}$.}\n  \\label{graphs}\n\\end{figure}\n\n\\chapter{ $B \\to \\eta^{(\\prime)}$ Form Factors in QCD}\\label{chapter5_eta}\nIn this chapter we discuss the semileptonic $B\\to\\eta^{(\\prime)}$ form factors $f_+^{B\\to\\eta^{(\\prime)}}$ in the LCSR approach. The previous LCSR determination of the $B\\to\\eta^{(\\prime)}$ form factors  presented in Ref.~\\cite{Ball:2004ye} is completed by calculating the gluonic contribution, the mechanism for which involves the annihilation of the $B$ meson to two gluons. The $\\eta^{(\\prime)}$ particles undergo pronounced mixing with each other due to the $\\rm U(1)_A$ anomaly of QCD and the $\\eta$-$\\eta^{\\prime}$ system, after many years of investigation, has succumbed to the phenomenologically motivated mixing scheme proposed by Feldmann, Kroll and Stech \\cite{Feldmann:1998vh,Feldmann:1998sh}. The consideration of this mixing scheme is central to the correct description of the $B\\to\\eta^{(\\prime)}$ form factors.\n\nMotivation to complete the calculation of  $f_+^{B\\to\\eta^{(\\prime)}}$ comes from a variety of sources, with probably the most prominent being:\n\\begin{itemize}\n\\item{the flavour-singlet contributions to the QCD factorisation framework to be discussed in Chapter~\\ref{chapter6_QCDF} were added by Beneke and Neubert in Ref.~\\cite{Beneke:2002jn}. It is found that the branching ratios of $B \\to \\eta^{\\prime} (V,P)$ are very sensitive to $f_+^{B\\to\\eta^{(\\prime)}}$  as the leading-order annihilation diagrams can be interpreted as a  gluon contribution to the $B \\to \\eta^{(\\prime)}$ form factors \\cite{Beneke:2003zv}.  Therefore a consistent estimation of the annihilation diagrams necessitates the inclusion of the gluonic contributions to the form factor.}\n\\item{There exists a ``tension'' in the determinations of $|V_{ub}|$ from  inclusive semileptonic decays $B\\to X_u l\\nu$ and their exclusive counterparts, namely from $B\\to\\pi l \\nu$. The former have led to larger values than the latter, and the reason for the discrepancy is unclear. $B\\to \\eta^{(\\prime)}$ transitions are at leading order a $b\\to u$ transition and so sensitive to $|V_{ub}|$ which can, in principle, be extracted from $B\\to \\eta^{(\\prime)} l \\nu$.  An improved calculation of $f_+^{B\\to\\eta^{(\\prime)}}$ would reduce the theoretical uncertainty of the result.}\n\\item{Finally, the observation that exclusive $B\\to \\eta^{\\prime} K$ and inclusive $B\\to \\eta^{\\prime} X$ decays have shown unexpectedly large branching ratios with respect to $B\\to\\pi$ transitions, for example, is an unresolved issue which an improved calculation of $f_+^{B\\to\\eta^{(\\prime)}}$ may help clarify.}\n\\end{itemize}\nWe begin by introducing the $\\eta^{(\\prime)}$ system and define two closely related $\\eta$-$\\eta^{\\prime}$ mixing schemes. We then discuss the calculation of the flavour-singlet contribution to the form factor before lastly we discuss the results of the LCSR analysis, the framework for which was covered in Chapter~\\ref{chapter3_SR}.  The material presented in this chapter follows that of Ref.~\\cite{Ball:2007hb}.\n\n\\section{The $\\eta$-$\\eta^{\\prime}$ System}\nThe approximate chiral symmetry of light quarks $u,d$ and $s$ in QCD seems to be broken by Nature to reveal the pseudoscalar mesons $(\\pi^0,\\pi^+,\\pi^-, K^+ ,K^-, K^0 ,\\bar{K}^0, \\eta)$ as the corresponding octet of Goldstone bosons (all massless in the \\textit{chiral limit} $m_{u,d,s}\\to 0$) of the broken $\\rm SU(3)\\otimes SU(3)$ symmetry. There is another symmetry of the QCD Lagrangian (\\ref{basics_eq1}); a global $\\rm U(1)_A$ symmetry which exists at the classical level in the chiral limit. Due to non-vanishing quark masses,  the broken $\\rm U(1)_A$ symmetry creates a Goldstone boson, but such a light particle does not appear in the physical spectrum and this embodies the \\textit{$\\rm U(1)_A$ problem}. At the quantum level, however, the $\\rm U(1)_A$ symmetry in the massless limit is broken due to the QCD anomaly and so was not present in the first place; thus a ninth state, the  $\\eta^{\\prime}$, exists as a singlet and only becomes massless in the chiral limit \\textit{and} as $N_c\\to\\infty$, causing the effects of anomaly to vanish. The situation is complicated by instanton effects, but was ultimately resolved by 't Hooft  with the same conclusion \\cite{Hooft:1976up,Hooft:1986nc}.  It has been known for a while that the $\\rm U(1)_A$ anomaly plays a decisive role in the $\\eta^{(\\prime)}$ system with the $\\eta^{\\prime}$ consisting of a large gluonic component \\cite{Witten:1978bc,Ball:1995zv}. The large mass of the $\\eta^{\\prime}$ is mostly generated by the anomaly and $\\rm SU(3)_F$-breaking effects.\\footnote{The particles $\\eta^{(\\prime)}$ have masses $m_{\\eta}=547.51 \\pm 0.18 ~\\textrm{MeV}$ and $m_{\\eta^{\\prime}}=957.78\\pm 0.14 ~\\textrm{MeV}$  and quantum numbers  $J^{PC}=0^{-+}$ \\cite{Yao:2006px}.}\n\nThe $\\eta$-$\\eta^{\\prime}$ system has been of considerable interest for a number of years \\cite{Fritzsch:1976qc,Isgur:1976qg,Novikov:1979ux}. Vast simplifications can be made in studying the low-energy particle spectrum of QCD by employing  \\textit{Chiral Perturbation Theory} (ChPT) which is an effective theory in which the heavy quarks are integrated out and the dynamically relevant  light quarks remain at a scale $\\mu\\sim\\Lambda_{\\rm QCD}$ after an expansion in powers of energies, momenta and quark masses. Alongside the $1\/N_c$ expansion, ChPT is the method of choice for analysing the light pseudoscalar mesons.\\footnote{Another interesting approach to understanding the $\\eta^{(\\prime)}$ system was given in Ref.~\\cite{Katz:2007tf}.} We do not discuss ChPT in any detail although we do quote a few of its constraints; for more details see for example \\cite{Weinberg:1978kz,Gasser:1984gg,Leutwyler:2001hn} \n\nConcerning $\\eta$-$\\eta^{\\prime}$  mixing, ChPT requires a description in terms of two mixing angles  beyond leading-order \\cite{Leutwyler:1997yr,Feldmann:1997vc}. How this is implemented in practice has caused some confusion in the past but a consistent picture has emerged \\cite{Feldmann:1998vh,Feldmann:1998sh}. Key to the phenomenological picture of the $\\eta$-$\\eta^{\\prime}$ system is the understanding that the main contributions to the mixing are due to the $\\rm U(1)_A$ anomaly of QCD, and so-called \\textit{OZI-rule violating} processes. Named after Okubo, Zweig and Iizuka the OZI-rule states that strong interaction processes that must proceed via the annihilation of all initial state quarks to gluons are suppressed \\cite{Okubo:1963fa,Iizuka:1966fk,Zweig:1964wu}. In Fig.~\\ref{eta_ozi} we show the unsuppressed process $\\phi \\to K^+ K^-$ (left) alongside the suppressed process $\\phi \\to \\pi^+\\pi^-\\pi^0$ (right) for which the rule was originally formulated.  Such processes are shown to be $\\mathcal{O}(1\/N_c)$ in a $1\/N_c$ expansion and phenomenologically they are found to be small $\\approx 10\\%$; they can be safely neglected, leaving the $\\rm U(1)_A$ anomaly as the only mixing mechanism. For the mixing schemes we discuss in the next section, this assumption has been confronted with experimental data and holds to the expected accuracy. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.6\\textwidth\\epsffile{ozi.eps}$$\n\\caption[Examples of an OZI-rule suppressed and allowed strong decays.]{\\small Examples of strong interaction decays. Left: $\\phi \\to K^+ K^-$, right: $\\phi \\to \\pi^+\\pi^-\\pi^0$. The former occurs preferentially over the latter due to the fact that the annihilation of the $\\phi$ requires all gluons to be hard, yielding a suppression via a small $\\alpha_s$ which need not be the case for the first decay. This forms the basis of the OZI-rule.}\n\\label{eta_ozi}\n\\end{figure}\n\nA schematic picture of the $\\rm U(1)_A$ anomaly at work for $B\\to\\eta^{(\\prime)}$ is shown in Fig.~\\ref{eta_u1a}., where the flavour-singlet contribution is defined as the amplitude for producing either a quark-antiquark pair in a singlet state which does not contain the $B$'s spectator quark, or two gluons, which then hadronise into an $\\eta^{(\\prime)}$.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.3\\textwidth\\epsffile{eta_schem.eps}$$\n\\caption[$B\\to \\eta^{(\\prime)}$ via the $\\rm U(1)_A$ anomaly.]{\\small $B\\to \\eta^{(\\prime)}$ via the $\\rm U(1)_A$ anomaly. The $b\\to u$ transition allows for an annihilation of the $B$ meson's quarks to two gluons, thus probing the gluonic content of $\\eta^{(\\prime)}$.}\n\\label{eta_u1a}\n\\end{figure}\n\nWhat about mixing between other pseudoscalar mesons? In $\\eta$ - $\\eta^{\\prime}$ - $\\pi^{0}$ mixing the  gluonic component present in the $\\pi^0$ is found to be at the level of a few percent  and so can be neglected \\cite{Feldmann:1998sh,Feldmann:1999uf,Kroll:2004rs}. There also exists a $c \\bar{c}$ component to $\\eta^{(\\prime)}$ ($\\eta_{c}$) which is considered in Ref.~\\cite{Feldmann:1998vh} and found to be small with the conclusion that it is not the solution to the abnormally large $B \\to K \\eta^{\\prime}$ branching ratio. Sometimes other particles are included as possible glueball candidates produced via OZI-rule suppressed processes in $J\/\\psi$ decay, see for example Refs.~\\cite{Ball:1995zv,Li:2007ky}.  Although it is unclear whether pseudoscalar mesons contain pure glueball properties, Ref.~\\cite{Kroll:2003yi}  concludes that it is unlikely. Thus the $\\eta$-$\\eta^{\\prime}$  system stands out on its own. \n\nPhenomenologically, the semileptonic decay $B \\to \\eta^{(\\prime)} l \\nu_l$ can be used to determine the size of  the CKM matrix element $|V_{ub}|$ from the spectrum\n\\begin{equation}\\label{eq:spectrum}\n\\frac{d\\Gamma}{dq^2}(B \\to P l \\nu_l) = \\frac{G_F^2 |V_{ub}|^2}{\n192\\pi^3m_B^3}\\lambda^{3\/2}_P(q^2) |f^P_+(q^2)|^2 \\,,\n\\end{equation}\nwhere $P=\\{\\eta,\\eta^{\\prime}\\}$ and $\\lambda_P(x) = (m_B^2+m_{P}^2-x)^2-4m_B^2m_{P}^2$. Alternatively,\nas we shall see, the ratio of branching ratios ${\\cal B}(B\\to\\eta^{\\prime}\n\\ell\\nu)\/{\\cal B}(B\\to \\eta\\ell\\nu)$ can be used to constrain the\ngluonic Gegenbauer moment $B_2^g$.\n\n\\section{State Mixing}\nThe first step in describing $\\eta$-$\\eta^{\\prime}$ mixing is to decompose the two physical states $\\ket{\\eta^{(\\prime)}}$ into other, more convenient orthogonal states. As proposed in Refs.~\\cite{Feldmann:1998vh,Feldmann:1998sh}  one can proceed in two ways; either by employing the singlet-octet scheme (SO) or the quark-flavour scheme (QF). The SO axial-vector currents are respectively\n\\begin{equation}\n J^{0}_{\\mu 5}=\\frac{1}{\\sqrt{3}}\\left(\\bar u \\gamma_\\mu\\gamma_5 u+\\bar d \\gamma_\\mu\\gamma_5  d +\\bar s \\gamma_\\mu\\gamma_5  s\\right)\\,,\\quad  J^{8}_{\\mu 5}=\\frac{1}{\\sqrt{6}}\\left(\\bar u \\gamma_\\mu\\gamma_5  u+\\bar d \\gamma_\\mu\\gamma_5  d -2\\bar s \\gamma_\\mu\\gamma_5  s\\right)\\,,\n\\end{equation}\nand their couplings are given by\n\\begin{equation}\n\\bra{0} J^{i}_{\\mu 5}\\ket{P(p)} = i f^{i}_{P} p_{\\mu} \\qquad (i=0,8)\\,,\n\\label{SOdc}\n\\end{equation}\nwhere $J^{8}_{\\mu 5}$ denotes the $\\rm SU(3)_F$-octet and $J^{0}_{\\mu 5}$ the $\\rm SU(3)_F$-singlet axial-vector current. The four quantities are related to the decay constants of a pure singlet or octet state $\\ket{\\eta_i}$ by two mixing angles  $\\theta_i$\n\\begin{equation}\n\\left(\n\\begin{array}{cc}\nf_\\eta ^8 & f_\\eta^0 \\\\\nf_{\\eta^{\\prime}} ^8 & f_{\\eta^{\\prime}}^0\n\\end{array}\\right)\n= \n\\left(\n\\begin{array}{cc}\n\\cos\\theta_8 & -\\sin\\theta_0 \\\\\n\\sin\\theta_8 & \\phantom{-}\\cos\\theta_0 \n\\end{array}\\right)\n\\left(\n\\begin{array}{cc}\nf_8 & 0 \\\\\n0 & f_0\n\\end{array}\n\\right).\n\\label{corr}\n\\end{equation}\nEvidently $\\rm SU(3)_F$-breaking effects cause $\\theta_i\\neq 0$ and $f_8\\neq f_\\pi$, and as such the SO scheme is very natural. In fact, at leading-order in ChPT an expansion in quark masses  and $1\/N_{c}$ gives \\cite{Leutwyler:1997yr}\n\\begin{equation}\n\\sin (\\theta_{0}-\\theta_{8})=\\frac{2 \\sqrt{2} (f_{K}^{2}-f_{\\pi}^{2})}{4f_{K}^{2}-f_{\\pi}^{2}}+\\dots\\,,\n\\label{su3}\n\\end{equation}\nwhere the dots denote neglected higher-order terms which are required to match phenomenology \\cite{Kaiser:1998ds}. The impact of the U(1)$_{\\rm A}$ anomaly is plainly localised in $f_0$ via the divergence of the singlet current $J_{\\mu 5}^0$ which can be written\n\\begin{equation}\n\\partial^\\mu J^{a}_{\\mu 5} = 2\\,\\bar{q} \\left[ t^a \\hat{m} i \\gamma_5 \\right]q + \\delta^{a 0}\\,\\frac{\\alpha_{s}}{4 \\pi} G \\widetilde{G}\\,,\n\\label{anomaly}\n\\end{equation} \nwhere $a=\\{0,1,\\dots,8\\}$, $\\textrm{Tr}[t^a t^b ] = \\frac{1}{2}\\delta^{a b}$, $t^0 = \\textbf{1}\/\\sqrt{3}$ and the mass matrix $\\hat{m}=\\textrm{diag}[m_u,m_d,m_s]$. The SO scheme diagonalises the renormalisation-scale dependence of parameters; $f_8$ and $\\theta_i$ are\nscale-independent, whereas $f_0$ renormalises multiplicatively\n\\begin{equation}\n\\mu\\,\\frac{d f_0}{d \\mu} = - N_f \\left(\\frac{\\alpha_s}{\\pi}\\right)^2 f_0 + O(\\alpha_s^3)\\,.\n\\label{scaledep}\n\\end{equation}\nIn the QF mixing scheme, on the other hand, the basic axial-vector currents are\n\\begin{equation}\nJ^q_{\\mu 5} = \\frac{1}{\\sqrt{2}} \\left(\\bar u \\gamma_\\mu \\gamma_5 u + \n\\bar d \\gamma_\\mu \\gamma_5 d \\right),\\qquad\nJ^s_{\\mu 5} = \\bar s \\gamma_\\mu \\gamma_5 s\\,,\n\\end{equation}\nand the corresponding couplings are\n\\begin{equation}\\label{6}\n\\bra{0} J^r_{\\mu 5}\\ket{P(p)} = i f_P^r p_\\mu \\quad (r=q,s)\\,.\n\\end{equation}\nThe mixing is analogous to (\\ref{corr}) with\n\\begin{equation}\n\\left(\n\\begin{array}{cc}\nf_\\eta ^q & f_\\eta^s \\\\\nf_{\\eta^{\\prime}} ^q & f_{\\eta^{\\prime}}^s \n\\end{array}\\right)\n= \n\\left(\n\\begin{array}{cc}\n\\cos\\phi_q & -\\sin\\phi_s \\\\\n\\sin\\phi_q & \\phantom{-}\\cos\\phi_s \n\\end{array}\\right)\n\\left(\n\\begin{array}{cc}\nf_q & 0 \\\\\n0 & f_s\n\\end{array}\n\\right).\n\\label{corr2}\n\\end{equation}\nBoth quark flavour states $\\ket{\\eta_{q,s}}$ have vanishing vacuum-particle matrix elements with the opposite currents\n\\begin{equation}\n\\bra{0}J^s_{\\mu 5}\\ket{\\eta_q}=\\bra{0}J^q_{\\mu 5}\\ket{\\eta_s}=0\\,,\n\\end{equation}\nwhich is an assumption that has been tested. It is in part motivated by the observation  of near ideal mixing in vector and tensor mesons. It implies that the mixing of states is the same as that of the decay constants and moreover leads to the diagonalisation of the mass matrix, which we come back to shortly. This hypothesis does not hold for the SO basis. It is found by Refs.~\\cite{Feldmann:1997vc,Feldmann:1999uf} that the difference  between the two mixing angles of the QF scheme $\\phi_q-\\phi_s$ is generated by OZI-rule suppressed processes and is not caused by SU(3)$_{\\rm F}$-breaking effects, as for the SO scheme (\\ref{su3}). While the numerical values of $\\theta_i$ differ largely, with typical values $\\theta_8\\approx -20^\\circ$ and $\\theta_0\\approx - 5^\\circ$, one finds $\\phi_s-\\phi_q\\, \\lesssim\\, 5^\\circ$, with $\\phi_q\\approx\n\\phi_s \\approx 40^\\circ$ \\cite{Feldmann:1998vh,Feldmann:1998sh,Feldmann:1997vc}. This observation led the authors of Refs.~\\cite{Feldmann:1998vh,Feldmann:1998sh} to suggest the QF scheme as an approximation to\ndescribe $\\eta$-$\\eta^{\\prime}$ mixing, based on neglecting the difference $\\phi_q-\\phi_s$ (and all other OZI-breaking effects):\n\\begin{equation}\n\\phi\\equiv \\phi_{q,s},\\qquad \\phi_q-\\phi_s\\equiv 0\\,.\n\\end{equation}\nThe state mixing is then given by\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\\ket{\\eta} \\\\ \\ket{\\eta^{\\prime}}\n\\end{array}\n\\right) \n= \n\\left(\n\\begin{array}{ll}\n\\cos\\phi & -\\sin\\phi\\\\\n\\sin\\phi & \\phantom{-}\\cos\\phi\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\ket{\\eta_q}\\\\ \\ket{\\eta_s}\n\\end{array}\n\\right)\\,.\n\\label{8}\n\\end{equation}\nThe re\\-nor\\-ma\\-li\\-sa\\-tion-scale dependence of $f_0$ given by Eq.~(\\ref{scaledep}) is not reproduced as it is induced precisely by neglected OZI-breaking terms \\cite{Feldmann:1999uf}. Numerically, this is not a problem as the scale-dependence of $f_0$ is a two-loop effect. In the case of non-local matrix elements, the DAs, this lack of scale dependence of the QF scheme is somewhat problematic. We come back to this point in the next section.\n\nReturning to the diagonalisation of the mass matrix; from Eq.~(\\ref{SOdc}) one finds the quadratic diagonal mass matrix, for example\n\\begin{equation}\\label{div}\n\\bra{0} \\partial^{\\mu}J^{s}_{\\mu 5}\\ket{\\eta (p)} = M^{2}_{\\eta} f^{s}_{\\eta}\\,,\n\\end{equation}\nwhich, via Eq.~(\\ref{anomaly}), gives the mass matrix in QF basis\n\\begin{equation}\n\\mathcal{M}_{\\rm QF}^2=\\left(\n\\begin{array}{cc}\nm_{qq}^2 +\\frac{\\sqrt{2}}{f_q}\\bra{0}\\frac{\\alpha_s}{4\\pi}G\\widetilde{G}\\ket{\\eta_q} & \\frac{1}{f_s}\\bra{0}\\frac{\\alpha_s}{4\\pi}G\\widetilde{G}\\ket{\\eta_q} \\\\\n\\frac{\\sqrt{2}}{f_q}\\bra{0}\\frac{\\alpha_s}{4\\pi}G\\widetilde{G}\\ket{\\eta_s}&m_{ss}^2+\\frac{1}{f_s}\\bra{0}\\frac{\\alpha_s}{4\\pi}G\\widetilde{G}\\ket{\\eta_s}\n\\end{array}\\right)\\,,\n\\label{mass}\n\\end{equation}\nwith the short-hand notation\n\\begin{equation}\nm_{qq}^2=\\frac{\\sqrt{2}}{f_q}\\bra{0}m_u \\bar{u}i\\gamma_5 u+m_d \\bar{d} i \\gamma_5 d\\ket{\\eta_q}\\,,\\qquad m_{ss}^2=\\frac{2}{f_s}\\bra{0}m_s \\bar{s} i \\gamma_5 s\\ket{\\eta_s}\\,.\n\\end{equation}\nFrom Eq.~(\\ref{mass}) the crucial impact of the anomaly, as the only term in the off-diagonal elements, is evident. To first order in $\\rm SU(3)_F$-breaking, the decay constants and quantities $m_{qq,ss}^2$ are fixed giving the theoretical estimate\n\\begin{eqnarray}\nf_q=f_\\pi\\,,&\\quad& f_s =\\sqrt{2 f_K^2-f^2_\\pi}\\,,\\nonumber\\\\\n m_{qq}^2=M_\\pi^2\\,,&\\quad& m_{ss}^2=2 M_K^2-M_\\pi^2\\,,\n\\end{eqnarray}\nwhich also leads to a fixed value of $\\phi$; there is no free parameter left and thus the QF scheme is totally determined \\cite{Feldmann:1998vh}. We do not work in this limit, however, and take numerical values of the decay constants and mixing angle from phenomenology. Given enough data to fix all independent parameters, there is no reason to prefer the QF over the SO scheme. The QF scheme is beneficial when considering DAs as the SO scheme leads to a proliferation of unknown parameters. For this reason we decide to use the QF scheme for the analysis. Its basic parameters have been determined as \\cite{Feldmann:1998vh,Feldmann:1998sh}\n\\begin{equation}\nf_q  =  (1.07\\pm 0.02)f_\\pi,\\qquad f_s = (1.34\\pm\n0.06)f_\\pi\\,,\\qquad\n\\phi =  39.3^\\circ\\pm 1.0^\\circ\\,.\n\\end{equation}\nThis can be translated into values for the SO parameters as\n\\begin{eqnarray}\nf_8 & = & \\sqrt{\\frac{1}{3}\\,f_q^2 + \\frac{2}{3} f_s^2} = (1.26\\pm\n0.04) f_\\pi\\,,\\nonumber\\\\\nf_0  &=&  \\sqrt{\\frac{2}{3}\\,f_q^2 + \\frac{1}{3} f_s^2} = (1.17\\pm\n0.03) f_\\pi\\,,\\nonumber\\\\\n\\theta_8 & = & \\phi-{\\rm arctan}[\\sqrt{2} f_s\/f_q] = (-21.2 \\pm\n1.6)^\\circ\\,,\\nonumber\\\\\n\\theta_0  &=&  \\phi-{\\rm arctan}[\\sqrt{2} f_q\/f_s] = (-9.2 \\pm\n1.7)^\\circ\\,,\n\\end{eqnarray}\nNote that in the QF scheme $f_{q,s}$ are scale-independent parameters, and so is $f_0$ as obtained from the above relations. The SO decay constants are related to those of the QF scheme by a change of basis\n\\begin{equation}\\label{11}\n\\left(\n\\begin{array}{cc}\nf_\\eta ^8 & f_\\eta^0 \\\\\nf_{\\eta^{\\prime}} ^8 & f_{\\eta^{\\prime}}^0 \n\\end{array}\\right)\n= \n\\left(\n\\begin{array}{cc}\n\\cos\\phi & -\\sin\\phi \\\\\n\\sin\\phi & \\phantom{-}\\cos\\phi \n\\end{array}\\right)\n\\left(\n\\begin{array}{cc}\nf_q & 0 \\\\\n0 & f_s\n\\end{array}\\right)\n\\left(\n\\begin{array}{cc}\n\\phantom{-}\\sqrt{\\frac{1}{3}} & \\sqrt{\\frac{2}{3}}\\\\\n-\\sqrt{\\frac{2}{3}} & \\sqrt{\\frac{1}{3}}\n\\end{array}\\right).\n\\end{equation}\nThe last matrix originates from the ideal mixing angle $\\theta_{\\textrm{ideal}}=\\arctan{\\sqrt{2}}$ which rotates from the QF basis to the SO basis.\n\n\\section{Pseudoscalar Meson Distribution Amplitudes}\nAs discussed in Chapter~\\ref{chapter3_SR}, the method of LCSRs relies on the non-perturbative universal light-cone DAs; specifically here we require pseudoscalar meson DAs including the two-gluon DA. At leading-twist both these DAs contribute and indeed mix with each other under renormalisation. The quark-antiquark DAs are extensions of the matrix elements given by Eqs.~(\\ref{SOdc}) and (\\ref{6}) to those of non-local operators on the light-cone. Pseudoscalar mesons' quark-antiquark DAs have been investigated previously in  Refs.~\\cite{Ball:1998je,Ball:2006wn,Braun:1989iv}. The two-gluon DAs of leading and higher twist have been investigated in Ref.~\\cite{AP03}. In this analysis we only include the effects of the leading-twist two-gluon DA, which is justified as its effects turn out to be fairly small and higher-twist DAs are estimated to have even smaller impact. Following Ref.~\\cite{Kroll:2002nt}, the twist-2 two-quark DAs of $\\eta^{(\\prime)}$ are defined as\n\\begin{equation}\n\\bra{0} \\bar\\Psi(z) {\\cal C}^i\\gamma_z \\gamma_5 [z,-z] \\Psi(-z) \\ket{P(p)} = i (p\\cdot z) f_P^i \\int_0^1 du\\, e^{i \\xi p \\cdot  z} \\phi_{2;P}^i(u) \\,.\n\\end{equation}\n$\\phi_{2;P}^i(u)$ is the twist-2 DA of the meson $P$ with respect to the current whose flavour content is given  by ${\\cal C}^i$, with $\\Psi = (u,d,s)$ the triplet of light-quark fields in flavour space. For the SO currents, one has ${\\cal C}^0 = \\mbox{\\boldmath $1$}\/\\sqrt{3}$ and ${\\cal C}^8 = \\sqrt{2}\\, t^8$, while for the QF currents ${\\cal C}^q = (\\sqrt{2} {\\cal C}^0 + {\\cal C}^8)\/\\sqrt{3}$ and  ${\\cal C}^s = ({\\cal C}^0 - \\sqrt{2} {\\cal C}^8)\/\\sqrt{3}$. Due to the positive G-parity of $\\eta$ and $\\eta^{\\prime}$, the two-quark DAs are symmetric under $u\\leftrightarrow 1-u$, and hence all odd Gegenbauer moments vanish:\n\\begin{equation}\n\\phi_{2;P}^i(u) = \\phi_{2;P}^i(1-u)\\,,\n\\end{equation}\nand the DAs are expanded in terms of Gegenbauer polynomials in exactly the same way as for the vector mesons\n\\begin{equation}\n\\phi_{2;P}^i(u) = 6 u (1-u) \\left( 1 + \\sum_{n=2,4,\\dots} a_n^{P,i}(\\mu)\nC^{3\/2}_n(\\xi) \\right) \\,\\quad (i=1,8,q,s)\\,,\n\\label{expansion}\n\\end{equation}\nwhere $a_n^{P,i}$ are the quark Gegenbauer moments. The gluonic twist-2 DA is defined as\\footnote{This definition refers to the ``$\\sigma$-rescaled'' DA $\\phi^\\sigma_g$ in Ref.~\\cite{Kroll:2002nt} with  $\\sigma = \\sqrt{3}\/C_F$. It agrees with that used in Refs.~\\cite{AP03,Charng:2006zj}, which means that we can use their results for the two-gluon Gegenbauer moment\n$B^g_2$ without rescaling.}\n\\begin{equation}\n\\bra{0} G_{\\mu z}(z) [z,-z] \\widetilde G^{\\mu z}(-z) \\ket{P(p)} = \\frac{1}{2}\\,(p\\cdot z)^2 \\frac{C_F}{\\sqrt{3}} f_P^0 \\int_0^1 du\\, e^{i\\xi p\\cdot z} \\psi_{2;P}^g(u)\\,.\n\\end{equation}\nIn order to perform the calculation of the correlation function defined in the next section, we also need the matrix element of the meson $P$ over two gluon fields. Dropping the gauge factor $[z,-z]$ one has\n\\begin{equation}\n\\bra{0} A^a_\\alpha(z) A^b_\\beta(-z)\\ket{P(p)} =\n\\frac{1}{4}\\,\\epsilon_{\\alpha\\beta\\rho\\sigma} \\,\\frac{z^\\rho\n  p^\\sigma}{p\\cdot z} \\,\\frac{C_F}{\\sqrt{3}}\\, f_P^0 \\,\\frac{\\delta^{ab}}{8}\n\\int_0^1 du\\, e^{i\\xi p\\cdot z}\\,\\frac{\\psi_{2;P}^g(u)}{u(1-u)}\\,.\n\\label{gluefields}\n\\end{equation}\nThe two-gluon asymptotic DA is $u^{2j-1}(1-u)^{2j-1}$ with $j=3\/2$ the lowest conformal spin of the operator $G_{\\mu z}$ and the expansion goes in terms of Gegenbauer polynomials $C^{5\/2}_n$, see Eq.~(\\ref{basics_eq18}). One can show that $\\psi_{2;P}^g$ is antisymmetric:\n\\begin{equation}\n\\psi_{2;P}^g(u) = - \\psi_{2:P}^g(1-u)\\,,\n\\end{equation}\nand in particular $\\int_0^1 du\\, \\psi_{2;P}^g(u) = 0$ and the local twist-2 matrix element $\\bra{0} G_{\\mu z} \\widetilde G^{\\mu z}\\ket{P}$ vanishes. The non-vanishing coupling $\\bra{0}G_{\\alpha\\beta}  \\widetilde G^{\\alpha\\beta}\\ket{P}$ induced by the U(1)$_{\\rm A}$ anomaly is a twist-4 effect. The corresponding matrix elements are discussed in Refs.~\\cite{Feldmann:1998vh,Feldmann:1998sh} and are given, in the QF scheme, by:\n\\begin{eqnarray}\n\\bra{0}\\frac{\\alpha_s}{4\\pi} G\\widetilde{G} \\ket{\\eta_q} & = & \nf_s (m_\\eta^2-m_{\\eta^{\\prime}}^2) \\sin\\phi \\cos\\phi\\,,\\nonumber\\\\\n\\bra{0}\\frac{ \\alpha_s}{4\\pi} G\\widetilde{G} \\ket{\\eta_s} & = & \nf_q (m_\\eta^2-m_{\\eta^{\\prime}}^2)\/\\sqrt{2} \\sin\\phi \\cos\\phi\\,.\n\\label{extra}\n\\end{eqnarray}\nIn taking the ratios of both sides of the above relations one can see that $\\rm SU(3)_F$-breaking in the decay constants $f_q\/f_s$ is driven by the anomaly. There are no twist-3 two-gluon DAs and the remaining twist-4 DAs also have vanishing normalisation \\cite{AP03}.  The conformal expansion of the twist-2 two-gluon DA reads\n\\begin{equation}\n\\psi_{2;P}^g(u,\\mu) = u^2 (1-u)^2 \\sum_{n=2,4,\\dots} B^{P,g}_n(\\mu)\nC^{5\/2}_{n-1}(\\xi)\\,,\n\\end{equation}\nwith the gluonic Gegenbauer moments $B^{P,g}_n$. In this analysis, we truncate both $\\phi^i_{2;P}$ and $\\psi^g_{2;P}$ at $n=2$. An estimate of the effect of higher Gegenbauer moments in $\\phi_{2;\\pi}$ on the $B\\to\\pi$ form factor $f_+^\\pi$ has been given in Ref.~\\cite{Ball:2005ei}, based on a\ncertain class of models for the full DA beyond conformal expansion. The effect of neglecting $a_{n\\geq 4}^\\pi$ was found to be very small $\\approx 2\\%$ hence we expect the truncation error from neglecing $B^g_{n\\geq 4}$ to be of similar size.\n\n$\\phi_{2;P}^0$ and $\\psi_{2;P}^g$ mix upon a change of scale $\\mu$ and as discussed in  Refs.~\\cite{Baier:1981pm,Kroll:2002nt} this amounts to a mixing of $a_2^{P,0}$ and $B^{P,g}_2$, resulting in the renormalisation-group equation to LO accuracy\n\\begin{equation}\n\\mu\\,\\frac{d}{d\\mu}\\left(\\begin{array}{c} a_2^0\\\\ B^g_2\n\\end{array}\\right)\n=\n-\\frac{\\alpha_s}{4\\pi} \\left(\\begin{array}{cc} \\displaystyle\\frac{100}{9} &\n\\displaystyle  -\\frac{10}{81}\\\\\\vphantom{\\displaystyle\\frac{100}{9}}\n -36 & 22\\end{array}\\right)\n\\left(\\begin{array}{c} a_2^0\\\\ B^g_2\n\\end{array}\\right),\n\\label{20}\n\\end{equation}\nwhere for simplicity we have dropped the superscript $P$. The solution for $a_2^0$ reads\n\\begin{eqnarray}\na_2^0(\\mu^2) & = & \\left[ \\left(\\frac{1}{2} -\n  \\frac{49}{2\\sqrt{2761}}\\right) L^{\\gamma_2^+\/(2\\beta_0)} + \n\\left(\\frac{1}{2} +\n  \\frac{49}{2\\sqrt{2761}}\\right) L^{\\gamma_2^-\/(2\\beta_0)}\\right]\n  a_2^0(\\mu_0^2) \\nonumber\\\\\n&&{}+ \\frac{5}{9\\sqrt{2761}}\\left[L^{\\gamma_2^-\/(2\\beta_0)}-\nL^{\\gamma_2^+\/(2\\beta_0)}\\right] B_2^g(\\mu_0^2)\n\\label{21}\n\\end{eqnarray}\nwith the anomalous dimensions $\\gamma_2^\\pm = (149\\pm \\sqrt{2761})\/9$. The octet Gegenbauer moment does not have another DA with which it can mix and so its evolution is simpler\n\\begin{equation}\na_2^8(\\mu^2) = L^{50\/(9\\beta_0)} a_2^8(\\mu_0^2)\\,.\n\\label{22}\n\\end{equation}\nThe mixing amongst the DAs complicates matters; as the scale dependence of the decay constants is lost in the QF scheme, one expects to have to lose scale dependence in the DAs too, and we must be careful to be consistent. The verification of the anomalous dimensions in Eq.~(\\ref{20}) from the singlet and octet parts of the form factor calculations is a crucial test of the LCSR analysis. For this reason, we discuss the implications of mixing on the twist-2 DA parameters, and only briefly cover higher-twist quark DAs which are included in the octet part; for a detailed discussion one is referred to Ref.~\\cite{Ball:2007hb}. Following Ref.~\\cite{Kroll:2002nt}, for the DAs introduced by Eq.~(\\ref{expansion}) we have, in terms of the quark valence Fock states $\\ket{q\\bar q}$ and $\\ket{s\\bar s}$\n\\begin{equation}\n\\ket{\\eta_q} \\sim \\phi_2^q (u) \\ket{q\\bar q} + \\phi_2^{\\rm\n OZI}(u) \\ket{s\\bar s}\\,,\\quad\n\\ket{\\eta_s} \\sim  \\phi_2^{\\rm OZI} (u) \\ket{q\\bar q} + \n\\phi_2^s(u) \\ket{s\\bar s}\\,,\n\\end{equation}\nwhere $q\\bar q$ is shorthand for $(u\\bar u + d\\bar d)\/\\sqrt{2}$ and \n\\begin{equation}\n\\phi_2^q = \\frac{1}{3}\\,(\\phi_2^8 + 2\\phi_2^0)\\,,\\quad\n\\phi_2^s = \\frac{1}{3}\\,(2\\phi_2^8 + \\phi_2^0)\\,,\\quad\n\\phi_2^{\\rm OZI} = \\frac{\\sqrt{2}}{3} (\\phi_2^0-\\phi_2^8)\\,.\n\\label{darel}\n\\end{equation}\nIn the QF scheme, the ``wrong-flavour'' DA  $\\phi_2^{\\rm OZI}$, which is generated by OZI-violating interactions,  is set to 0. Once this is done at a certain scale, however, the different evolution of $a_n^0$ and $a_n^8$  will generate a non-zero $\\phi_2^{\\rm OZI}$ already to LO accuracy. A consistent implementation of the QF scheme hence requires one to either set $a_n^{0,8}\\equiv 0$ and also $B^g_n\\equiv 0$, or to set $a_n^8\\equiv a_n^0$ and neglect the different \nscale-dependence of these parameters. The induced non-zero DA $\\phi_2^{\\rm OZI}$ is numerically very small for the scales relevant for our calculation, $\\mu=1\\,$GeV and $2.4\\,$GeV.\\footnote{$2.4\\,$GeV is  a typical scale in the calculation of form factors from LCSRs: $\\mu=\\sqrt{m_B^2-m_b^2}$ is chosen as an intermediate scale between $m_b$ and the typical hadronic scale $1\\,{\\rm GeV}$.} The left panel of Fig.~\\ref{eta_fig2} shows a plot of $\\Delta=100\\,| (a^0_2(\\mu)-a^8_2(\\mu))\/a^0_2(\\mu)|$ as a function of scale $\\mu$, according to Eqs.~(\\ref{21}) and (\\ref{22}),  for $a_2^8(1\\,{\\rm GeV})\\equiv a_2^0(1\\,{\\rm GeV})$ and $B_2^g=0$.  We see that $\\Delta$ is less than $0.25\\,\\%$ over the range $1\\,\\textrm{GeV}<\\mu<2.4\\,\\textrm{GeV}$.  Choosing $a_2^8(1\\,{\\rm GeV})=0.25\\pm0.15$, guided by our knowledge of twist-2 DAs of the $\\pi$; we have $a_2^8(2.4\\,{\\rm   GeV}) = 0.171$ from Eq.~(\\ref{22}), and $a_2^0(2.4\\,{\\rm GeV}) = 0.171$ for $B^g_2=0$, from Eq.~(\\ref{21}). Evidently, the impact of the different anomalous dimensions of $a_2^0$ and $a_2^8$ is negligible.\n\\begin{figure}[h]\n$$\\epsfxsize=0.45\\textwidth\\epsffile{scaling1.eps}\\qquad \\qquad\\epsfxsize=0.45\\textwidth\\epsffile{scaling2.eps}$$\n\\caption[Scale dependence of the twist-2 distribution amplitude parameters.]{\\small  Left: $\\Delta=100\\,| (a^0_2(\\mu)-a^8_2(\\mu))\/a^0_2(\\mu)|$ as a function of scale $\\mu$, according to Eqs.~(\\ref{21}) and (\\ref{22}) with $B_2^g=0$. Right: dependence of $a^0_2(2.4\\,{\\rm GeV})$ on $B^g_2(1\\,{\\rm GeV})$ for  $a^0_2(1\\,{\\rm GeV})=0.25$ according to Eq.\\,(\\ref{21})}\n\\label{eta_fig2}\n\\end{figure}\nAlso, the evolution of $a_2^0$ is not hugely different to that of $a_2^8$, for a wide range of values of $B^2_g$.  The right panel of Fig.~\\ref{eta_fig2} shows the evolution of the singlet Gegenbauer moment $a_2^0$ from $\\mu=1\\,{\\rm GeV}$ - $2.4\\, {\\rm GeV}$, from  Eq.~(\\ref{21}), for the range of gluon Gegenbauer moments $|B_2^g(1\\,{\\rm  GeV})|<20$, which is a \\textit{very} conservative estimated range, as discussed below. The mixing of $B_2^g$ into $a_2^0$ is up to $20\\%$ for $B_2^g=20$ and $40\\%$ for $B_2^g=-20$.\n\nFrom the conclusions of the above discussion we are justified in implementing the QF scheme for DAs as follows: we set $\\phi_2^0\\equiv \\phi_2^8$ at the scale $\\mu=1\\,$GeV, which, by virtue of Eq.~(\\ref{darel}), implies $\\phi_2^q\\equiv\\phi_2^s$ at the same scale. We then evolve $a_2$ according to the scaling-law for the octet Gegenbauer moment (\\ref{22}).\\footnote{This is equivalent to imposing the QF-scheme  relation $a_2^0=a_2^8$ as the scale $\\mu=2.4\\,$GeV and defining $B^g_2$ as $B^g_2(2.4\\,{\\rm GeV})$.}  We also set $\\psi_{2;\\eta}^g=\\psi_{2;\\eta^{\\prime}}^g$; again any SU(3)$_{\\rm F}$-breaking of this relation is expected to have only very small impact on $f_+^{B\\to\\eta^{(\\prime)}}$. The twist-2 parameters used in our calculation are then reduced to two: $a_2$ and $B^g_2$.  \n\nConcerning numerical values, we assume that the bulk of  SU(3)$_{\\rm F}$-breaking effects is described by the decay constants via $f_q\\neq f_\\pi$, and that SU(3)$_{\\rm F}$-breaking in Gegenbauer moments is sub-leading \\cite{Ball:2006wn}. Sum rules for $a_2^\\pi$ and $a_2^q$ would essentially be the same, with $f_\\pi\\neq f_q$ driving the SU(3)$_{\\rm F}$-breaking and any small differences in $s_0$ and $M^2$ being negligible. This motivates setting $a_2^q = a_2^\\pi$,  with $a_2^\\pi(1\\,{\\rm GeV}) = 0.25\\pm 0.15$ as an average over a large number of  calculations and fits to experimental data \\cite{Ball:2006wn}. \n\nFor $B_2^g$, however, no direct calculation is available.  Results from fits to data have been obtained from the $\\eta^{\\prime}\\gamma$ transition form factor, yielding $B_2^g(1\\,{\\rm GeV}) = 9\\pm 12$ \\cite{Kroll:2002nt}, and the combined analysis of this form factor and the inclusive decay $\\Upsilon(1S)\\to \\eta^{\\prime} X$ yielding $B_2^g(1.4\\,{\\rm GeV}) = 4.6\\pm 2.5$ \\cite{AP03}.  Caution must be taken when considering these results as they are highly correlated with the simultaneous determination of $a_2^0$ and $a_2^8$ from the same data, yielding $a_2^0(1\\,{\\rm GeV}) = -0.08\\pm 0.04$, $a_2^8(1\\,{\\rm GeV}) =\n-0.04\\pm 0.04$ and $a_2^0(1.4\\,{\\rm GeV}) = a_2^8(1.4\\,{\\rm GeV}) = -0.054\\pm 0.029$, respectively. The same analysis, applied to the $\\pi\\gamma$ form factor, returns $a_2^\\pi (1\\,{\\rm GeV}) =-0.06\\pm 0.03$ \\cite{vogt}. These results are not really compatible with those from the direct calculation of $a_2^\\pi$ from Lattice QCD and QCD sum rules; in particular the sign of $a_2^\\pi$ is unambiguously fixed as being positive. A possible reason for this discrepancy is the neglection of higher-order terms in the light-cone expansion and that, in addition, as one of the photons in the process is nearly real with virtuality $q^2\\approx 0$, one also has to take into account long-distance photon interactions, of order $1 \\sqrt{q^2}$, as discussed in Ref.~\\cite{rady}. For this reason, we assume the very conservative range $B_2^g(2.4\\,{\\rm GeV}) = 0\\pm20$ in the analysis.\n\nAs far as higher-twist quark DAs are concerned, we only need those involving currents with flavour content $\\bar q q = (\\bar u u + \\bar d d)\/\\sqrt{2}$. In line with the implementation of the QF scheme\nfor twist-2 DAs, we include SU(3)$_{\\rm F}$-breaking only via the decay constants. The precise definitions of all twist-3 and 4 DAs, as well as up-to-date numerical values of the $\\pi$'s hadronic parameters can be found in Ref.~\\cite{Ball:2006wn}. A discussion of the correct treatment of these DAs\nwithin LCSR, as modified to describe $\\eta^{(\\prime)}$, can be found in Ref.~\\cite{Ball:2007hb}. \n\n\\section{Calculation}\nWe define the $B\\to P$ form factors analogously to those of other pseudoscalar mesons as  \\cite{Ball:2004ye}\n\\begin{equation}\n\\bra{P(p)} \\bar u \\gamma_\\mu b \\ket{B(p+q)} = \\left\\{\n(2p+q)_\\mu - \\frac{m_B^2-m_P^2}{q^2}\\,q_\\mu\\right\\} \\frac{f_+^P(q^2)}{\\sqrt{2}} + \\frac{m_B^2-m_P^2}{q^2}\\,q_\\mu\\,\\frac{f_0^P(q^2)}{\\sqrt{2}}\\,.\n\\label{FF}\n\\end{equation}\nwhere the factor of $1\/\\sqrt{2}$ on the right-hand side is to ensure that in the SU(3)$_{\\rm F}$ symmetry limit, without $\\eta$-$\\eta^{\\prime}$ mixing, $f_+^\\eta = f_+^\\pi$. For semileptonic decays $B\\to \\eta^{(\\prime)} l \\nu_l$ the form factor $f_0^P$ appears proportional to $q^2\\approx m_l^2$ which is negligible for light leptons $l=\\{e,\\mu\\}$ for which only $f_+^P$ is required.  Using the LCSR method outlined in Chapter~\\ref{chapter3_SR} we extract the semileptonic form factor $f_{+}^P$  from the following correlation function\n\\begin{eqnarray}\n\\Pi^P_{\\mu}(p,q) &=& i\\int d^4x\\,e^{i q\\cdot x} \\bra{P(p)}  T [\\bar u \\gamma_\\mu b](x) j_B^{\\dagger}(0)\\ket{0}\\\\\n&=& \\Pi_+^P(q^2,p_B^2) (2p+q)_\\mu + \\dots\\,,\\nonumber\n\\label{eq:corr}\n\\end{eqnarray}\nwhere $j_B= m_b \\bar u i\\gamma_5 b$ is the interpolating current for the $B$ meson and $p_B^2=(p+q)^2$ its virtuality. In calculating the correlation function, we use Eq.~(\\ref{8}) which relates the physical states $\\ket{\\eta^{(\\prime)}}$ and the QF basis states  $\\ket{\\eta_{q,s}}$ so that\n\\begin{equation}\n\\Pi^\\eta_\\mu = \\frac{1}{\\sqrt{2}}\\left(\\Pi^q_{\\mu} \\cos\\phi - \\Pi^s_{\\mu} \\sin\\phi\\right),\\quad\n\\Pi^{\\eta^{\\prime}}_\\mu = \\frac{1}{\\sqrt{2}}\\left(\\Pi^q_{\\mu} \\sin\\phi + \\Pi^s_{\\mu} \\cos\\phi\\right)\\,.\n\\label{32}\n\\end{equation}\nThe interpolating current $\\bar u \\gamma_\\mu b$ only probes the $\\bar u u $ quark component of the $\\eta^{(\\prime)}$ so $\\Pi^s_\\mu$ vanishes to leading order in $\\alpha_s$ and at $O(\\alpha_s)$ is due only to gluonic Fock states of the meson. $\\Pi^q_\\mu$, on the other hand, receives contributions from both quark and gluon states. The final LCSR for $f_+^P$ then reads\n\\begin{equation}\\label{35}\ne^{-m_B^2\/M^2}\\,m_B^2 f_B\\, \\frac{f_+^P(q^2)}{\\sqrt{2}} = \\int_{m_b^2}^{s_0}\nds\\,e^{-s\/M^2}\\, \\frac{1}{\\pi}\\,{\\rm Im}_s\\,\\Pi^P_+(s,q^2)\\,,\n\\end{equation}\nwith the usual sum rule specific parameters $M^2$, the Borel parameter, and $s_0$, the continuum threshold.\n\\subsection*{Quark Contribution}\nThe quark contributions follow from the studies already undertaken for the $\\pi$, for more details see Ref.~\\cite{Ball:2004ye}. We briefly cover the general features of the calculation to put the singlet contribution in context. The leading quark contributions to $\\Pi_+^P$ originate from the diagrams of Fig.~\\ref{eta_fig1}, where first order $\\mathcal{O}(\\alpha_s)$ corrections are shown. The external quarks have momentum fractions $u p$ and $(1-u)p$ and are on-shell; $p^2=m_P^2$.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.85\\textwidth\\epsffile{eta_diags1.eps}$$\n\\caption[The quark contributions to $f_+^{\\eta^{(\\prime)}}(q^2)$ to $\\mathcal{O}(\\alpha_s)$.]{\\small The quark-antiquark contributions to the semileptonic $B\\to\\eta^{(\\prime)}$ form factors $f_+^{\\eta^{(\\prime)}}(q^2)$ from light-cone sum rules. The top left diagram is the leading one, the others are $\\mathcal{O}(\\alpha_s)$. The double line corresponds to the $b$ quark and the dashed lines the injection of the weak vertex momentum $q$, and the momentum of the $B$ meson $p_B$.} \n\\label{eta_fig1}\n\\end{figure}\nThe two-particle DAs are projected out by using the general spinor decomposition of quark fields\n\\begin{eqnarray}\n\\bar{q}_a q_b^\\prime &=& \\frac{1}{4} (\\textbf{1})_{ba}(\\bar q q^\\prime)-\\frac{1}{4} (i \\gamma_5)_{ba}(\\bar q i \\gamma_5  q^\\prime)+\\frac{1}{4} (\\gamma_\\mu)_{ba}(\\bar q \\gamma^\\mu q^\\prime)-\\frac{1}{4} (\\gamma_\\mu \\gamma_5)_{ba}(\\bar q\\gamma^\\mu \\gamma_5  q^\\prime)\\nonumber\\\\\n&-&\\frac{1}{8} (i \\sigma_{\\mu\\nu}\\gamma_5)_{ba}(\\bar q i \\sigma^{\\mu\\nu} \\gamma_5 q^\\prime)\\,.\n\\end{eqnarray}\nThe vacuum-meson matrix elements of each term above either vanish or yield a DA depending on the quantum numbers of the meson in question. For pseudoscalar mesons the leading-twist contribution comes from $\\gamma_\\mu\\gamma_5$, whereas $i\\gamma_5$ and $i\\sigma_{\\mu\\nu}\\gamma_5$ give two-particle twist-3 contributions, and although the two-particle twist-3 contributions appear in the sum rules as formally $1\/m_b$, they are \\textit{chirally enhanced} by numerically large factors \\cite{Ball:1998je} and so are included in typical LCSR analyses \\cite{Ball:2004ye}. Three-particle twist-3 and two- and three-particle twist-4 DAs are also included; all twist-2 and -3 contributions include $O(\\alpha_s)$ corrections twist-4 contributions are to tree level accuracy. The corresponding expressions yield $\\Pi^q_+$, with the replacement $f_\\pi\\to f_q$.  \n\n\\subsection*{Gluonic Contribution}\nIn order to obtain the gluonic contribution to $\\Pi_+^P$, one needs to calculate the diagrams shown in Fig.~\\ref{eta_diags}. The last diagram is divergent and the other two are finite.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.85\\textwidth\\epsffile{eta.eps}$$\n\\caption[The leading diagrams for the flavour-singlet contribution to $f_+^{\\eta^{(\\prime)}}(q^2)$.]{\\small The leading diagrams for the flavour-singlet contribution to the semileptonic $B\\to\\eta^{(\\prime)}$ form factors  from light-cone sum rules. The double line corresponds to the $b$ quark. The dashed lines the injection of weak vertex momentum $q$, and momentum of the $B$ meson  interpolating current $p_B$.} \n\\label{eta_diags}\n\\end{figure}\nThe gluon fields are introduced in the standard way\n\\begin{eqnarray}\n\\left.\\Pi^{P}_{\\mu}\\right|_{\\textrm{gluon}}=i\\int d^4[x,w,y] \\, e^{i q\\cdot x} \\bra{P(p)} T [\\bar{u}\\gamma_{\\mu} b](x) [m_b \\bar{b} i\\gamma_5 u](0) \\, S \\mathcal{L}^{q_1}_{g}(w)\\mathcal{L}^{q_2}_{g}(y)\\ket{0}\\,,\\nonumber\n\\end{eqnarray}\nwith the usual interaction Lagrangian $\\mathcal{L}^{q_{i}}_{g}(x) = ig_{s} [\\bar{q_i} \\gamma^{\\alpha} A_{\\alpha}^a t^a q_i](x)$ with $q_{i} = \\{u,b\\}$ and the statistical factor $S$ takes values $1$ if $q_1\\neq q_2$ and  $1\/2$ if $q_1=q_2$. The integral is over each co-ordinate separately. To extract the gluon contribution we need the projection onto the twist-2 two-gluon DA, which can be read off Eq.~(\\ref{gluefields}), which amounts to the following replacement of the gluon fields (up to the numerical factor)\n\\begin{equation}\nA_{\\alpha}^a(w) A_{\\beta}^b(y) \\stackrel{\\textrm{twist-2}}{\\longrightarrow}\\delta^{ab} \\epsilon_{\\alpha \\beta \\rho\\sigma} \\,\\frac{\\tilde{z}^\\rho p^\\sigma }{p \\cdot \\tilde{z}}\\,\\,\\int^1_0 du\\,\\frac{\\psi^g_{2;P}(u)}{u \\bar{u}} \\,e^{i p \\cdot (u w +  \\bar{u} y)}\\,,\n\\end{equation}\nwhere the separation $\\tilde{z}$ is light-like i.e. $\\tilde{z}^2 = (w-y)^2=0$. Via partial integration we can simplify the resulting expression for $\\left.\\Pi^{P}_{+}\\right|_{\\textrm{gluon}}$; the co-ordinate $\\tilde{z}$ is traded for a derivative of the hard scattering kernel with respect to the momentum of one of the emitted gluons; and the dot product $1\/ (p\\cdot \\tilde{z})$ can be traded for an integral with respect to the DA momentum fraction. As the boundary terms vanish due to the leading-twist gluon DA being antisymmetric, the calculation takes a rather simple form:\n\\begin{equation}\n\\left.\\Pi^{P}_{+}\\right|_{\\textrm{gluon}}=\\left.\\int^1_0 du  \\, \\left[\\frac{\\partial \\,T_{\\mu }^\\rho(u p) }{\\partial(u p)^\\rho}\\right] \\int^u_0 dv \\,\\frac{\\psi_{2;P}^g(v)}{v\\bar{v}}\\right|_{p_\\mu \\to \\frac{1}{2},\\,q_\\mu \\to 0}\\,,\n\\end{equation}\nwhere $T_{\\mu }^\\rho(u p) $ is the hard scattering kernel.  Both the gluonic and quark contributions are renormalisation scale dependent. The relevant term concerning the quark Gegenbauer moment $a_2$ is\n\\begin{equation}\n\\Pi^q_+ \\sim 18 f_q  a_2 \\left( 1 + \\frac{\\alpha_s}{4\\pi}\n\\,\\frac{50}{9}\\,\\ln\\,\\frac{\\mu^2}{m_b^2}\\right)F(p_B^2,q^2)\\,,\n\\label{33}\n\\end{equation}\nwhere $F(p_B^2,q^2)$ is a function of $p_B^2$ and $q^2$. The logarithmic terms in the convolution of the gluonic diagrams of Fig.~\\ref{eta_diags} with $\\psi_{2;P}^g$ are\n\\begin{equation}\n\\Pi^P_+ \\sim -\\frac{10}{9\\sqrt{3}}\\,\\frac{\\alpha_s}{4\\pi}\\, B_2^g f^0_P\n\\ln\\,\\frac{\\mu^2}{m_b^2}\\, F(p_B^2,q^2) \\,.\n\\end{equation}\nBy expressing $f_q$ via Eq.~(\\ref{11}) in terms of $f^0_\\eta$ and $f^0_{\\eta^{\\prime}}$, respectively, and inserting Eq.~(\\ref{33}) into Eq.~(\\ref{32}), one verifies that the renormalisation-group equation, Eq.~(\\ref{20}), is fulfilled.  The twist-2 two-gluon contribution to the correlation functions $\\Pi_+^P$, Eq.~(\\ref{32}), is given in terms of a spectral density as\n\\begin{equation}\n\\left.\\Pi_{+}^{P}\\right|_{\\textrm{gluon}} = \\int_{m_b^2}^\\infty ds\\,\\frac{\\rho^P_{\\textrm{gluon}}(s)}{s-p_B^2}\n\\end{equation}\nwith the result being\n\\begin{eqnarray}\n\\rho^P_{\\textrm{gluon}}(s) \n& = & B_2^g \\alpha_s f_0^P m_b\\, \\frac{5}{36\\sqrt{3}}\\, \\frac{m_b^2-s}{(s-q^2)^5} \\, \\left\\{ 59 m_b^6 + 21 q^6 - 63 q^4 s - 19 q^2 s^2 + 2 s^3\\right. \\nonumber\\\\\n&& \\hspace*{3cm}\\left. + \\,m_b^2 s (164 q^2 + 13 s) - m_b^4 (82 q^2 + 95s)\\right\\}\n\\nonumber\\\\\n&& {} + B_2^g \\alpha_s f_0^P m_b\\, \\frac{5}{6 \\sqrt{3}}\\, \\frac{(m_b^2-q^2)(s-m_b^2)}{(s-q^2)^5} \\,\\{ 5 m_b^4 + q^4 + 3 q^2 s + s^2 - 5m_b^2 (q^2+s)\\} \\nonumber\\\\\n&& \\hspace*{3cm} \\times\\left\\{ 2 \\ln\\,\\frac{s-m_b^2}{m_b^2} - \\ln\\,\\frac{\\mu^2}{m_b^2} \\right\\}.\n\\end{eqnarray}\n\n\\section{Discussion}\nFor the evaluation of the LCSR, Eq.~(\\ref{35}), as with any sum rule, optimum values of  $M^2$ and $s_0$ need to be found. The standard procedure \\cite{Ball:2004ye} is to replace $f_B$ by its sum rule, derived via SVZ sum rules, thus reducing the dependence of the LCSR on $m_b$ for which we use the one-loop pole mass $m_b=4.80\\pm0.05\\,\\rm{GeV}$ \\cite{Colangelo:2000dp}. From the $f_B$ sum rule the optimum threshold parameter  $s_0=34.2 \\pm 0.7\\,{\\rm GeV}^2$ is found, and this value is taken over to the LCSR. As mentioned before $\\mu=2.4\\,{\\rm GeV}$ is chosen as an intermediate scale between $m_b$ and $1\\,{\\rm GeV}$. The Borel parameter is taken to be $M^2>6\\,{\\rm GeV}^2$ and is varied in the range $6\\,{\\rm GeV}^2<M^2<14\\,{\\rm GeV}^2$ to reflect the corresponding uncertainty. In Fig.~\\ref{eta_diags3} we plot $f^{B\\to \\eta}_+(0)$ and $f^{B\\to \\eta^{\\prime}}_+(0)$ respectively as functions of $M^2$, making explicit the result of varying $s_0$ by $\\pm0.7\\,{\\rm GeV}^2$, $a_2$ by $\\pm0.15$ and $B_2^g$ by $\\pm10$.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.45\\textwidth\\epsffile{eta_fig4a.eps}\\qquad\\qquad \\epsfxsize=0.45\\textwidth\\epsffile{eta_fig4b.eps}$$\n\\caption[$f_+^\\eta(0)$  and $f_+^{\\eta^{\\prime}}(0)$ as a\n  function of the Borel parameter $M^2$.]{\\small $f_+^\\eta(0)$ (left) and $f_+^{\\eta^{\\prime}}(0)$ (right) as a\n  function of the Borel parameter $M^2$ and various choices of input\n  parameters. Solid curves: central values of input parameters and\n  $s_0=34.2\\,{\\rm GeV}^2$. Long-dashed curves: $s_0$ varied by\n  $\\pm 0.7\\,{\\rm GeV}^2$. Short-dashed curves: $a_2(1\\,{\\rm\n  GeV})$ varied by $\\pm 0.15$. Dash-dotted curves: $B_2^g$ varied\n  by $\\pm 10$.} \n\\label{eta_diags3}\n\\end{figure}\nAs expected, $f_+^\\eta(0)$ is not very sensitive to the gluonic twist-2 DA parameter $B^g_2$ (dashed-dotted curves), but is quite sensitive to the Gegenbauer moment $a_2$ (short-dashed\ncurves). For  $f_+^{\\eta^{\\prime}}(0)$, on the other hand, the dependence on $B^g_2$ is more pronounced than that of $a_2$. Varying all relevant parameters within their respective ranges, i.e.\\ $\\Delta m_b = \\pm\n0.05\\,{\\rm GeV}$, $\\Delta a_2(1\\,{\\rm GeV})=\\pm0.15$ and $\\Delta B^g_2=\\pm 20$, as well as all twist-3 and twist-4 parameters within the ranges given in Ref.~\\cite{Ball:2006wn}, we find\n\\begin{eqnarray}\\label{fp1}\nf_+^\\eta(0) & = & 0.229\\pm \\overbrace{0.005}^{M^2}\\pm \\overbrace{0.006}^{s_0}\\pm\n\\overbrace{0.016}^{a_2^\\eta}\\pm \\overbrace{0.007}^{B^g_2}\\pm \\overbrace{0.005}^{f_q,\\phi}\\pm \\overbrace{0.011}^{{\\rm T3}}\\pm \\overbrace{0.001}^{{\\rm T4}} \\pm \\overbrace{0.007}^{f_B,m_b}\\nonumber\\\\\n& = &\n0.229\\pm \\underbrace{0.024}_{{\\rm param.}}\\pm \\underbrace{0.011}_{{\\rm syst.}}\\,,\\\\\nf_+^{\\eta^{\\prime}}(0) & = & 0.188\\pm \\overbrace{0.004}^{M^2}\\pm \\overbrace{0.005}^{s_0}\\pm\n\\overbrace{0.013}^{a_2^{\\eta^{\\prime}}}\\pm \\overbrace{0.043}^{B^g_2}\\pm \\overbrace{0.005}^{f_q,\\phi}\n\\pm \\overbrace{0.009}^{{\\rm T3}}\\pm \\overbrace{0.005}^{{\\rm T4}} \\pm \\overbrace{0.006}^{f_B,m_b}\\nonumber\\\\\n& =&\n0.188\\pm \\underbrace{0.043}_{B^g_2} \\pm \\underbrace{0.019}_{{\\rm param.}}\\pm \\underbrace{0.009}_{{\\rm syst.}}\\,.\n\\label{fp0}\n\\end{eqnarray}\nThe entry labelled ``T4'' also contains an estimate of the possible impact\nof the local twist-4 two-gluon matrix elements (\\ref{extra}). For\nthis estimate, we exploit the fact that the asymptotic DA of the\nnon-local generalisation of Eq.~(\\ref{extra}) is the same as for the\ntwist-2 two-quark DA: $6 u (1-u)$.\\footnote{This follows from Eq.~(\\ref{das_eq18}). For\n$G_{\\perp\\perp}$, one has $l=2$ and $s=0$.} We then\nassume that the corresponding correlation function is the same as that\nfor the leading conformal wave in the two-quark twist-2 contribution,\ni.e.\\ the coefficient in the Gegenbauer moment $a_0=1$, and replace\n$a_0$ by $\\bra{0} \\alpha_s\nG\\tilde{G}\/(4\\pi)\\ket{\\eta_{q,s}}\/(f_{q,s}m_b^2)$. The factor\n$1\/m_b^2$ comes from the fact that this is a twist-4 effect and hence\nsuppressed by two powers of $m_b$ with respect to the twist-2\ncontribution. This is only a\nrough estimate, of course, as the true spectral density will be\ndifferent. The results (\\ref{fp0}) show that for small $B^g_2\\approx 2$ both twist-2 and -4\ntwo-gluon effects can indeed be of similar size. In this case,\nhowever, the total flavour-singlet contribution to $f_+^{\\eta^{\\prime}}$ will\nalso be small, $\\sim 0.008$.\nIn the third lines, we have added all uncertainties from the input\nparameters (param.) in quadrature and the sum-rule specific\nuncertainties from $M^2$ and $s_0$ (syst.) linearly. For\n$f_+^{\\eta^{\\prime}}(0)$, we have displayed the dependence on $B^g_2$\nseparately. The new result for $f_+^\\eta(0)$ is, within errors, in agreement with\nthe previous one from LCSR, $f_+^\\eta(0) = 0.275\\pm 0.036$, obtained in\nRef.~\\cite{Ball:2004ye}. That for $f_+^{\\eta^{\\prime}}(0)$ is new to the present analysis.\nThe results agree well with those obtained in Ref.~\\cite{Charng:2006zj},\nfrom pQCD,\n$f_+^\\eta(0) = 0.208$ and $f_+^{\\eta^{\\prime}}(0) = 0.171$, including a\nrescaling by a factor $\\sqrt{2}$ to bring their definition of the form\nfactors into agreement with Eq.~(\\ref{FF}). We confirm the finding of\nRef.~\\cite{Charng:2006zj} that the range of the singlet contribution to the form\nfactor estimated in Ref.~\\cite{Beneke:2002jn} is likely to be too large, unless\n$B_2^g$ assumes extreme values $\\sim 40$.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.55\\textwidth\\epsffile{eta_fig5.eps}$$\n\\caption[$f_+^{\\eta^{\\prime}}(0)\/f_+^\\eta(0)$ as a\n  function of the Borel parameter $M^2$.]{\\small $f_+^{\\eta^{\\prime}}(0)\/f_+^\\eta(0)$ as a\n  function of the Borel parameter $M^2$ and various choices of input\n  parameters. Solid line: central values of input parameters,\n  which corresponds to $f_+^{\\eta^{\\prime}}(0)\/f_+^\\eta(0)\\equiv \\tan\\phi =\n  0.814$. Dot-dashed curves: $B_2^g$ varied\n  by $\\pm 10$. Dotted curves: $a_2^{\\eta,\\eta^{\\prime}}(1\\,{\\rm GeV})$\n  varied independently: $a_2^\\eta=0.1$, $a_2^{\\eta^{\\prime}}=0.4$ and\n  $a_2^{\\eta^{\\prime}}=0.4$, $a_2^\\eta=0.1$.} \n\\label{eta_fig5}\n\\end{figure}\n\nIn Fig.~\\ref{eta_fig5} we plot the ratio $f^{\\eta^{\\prime}}_+(0)\/f^{\\eta}_+(0)$\nas a function of the Borel parameter. In the ratio, many uncertainties\ncancel, in particular that on $f_B$. As we have chosen $B_2^g=0$ as\ncentral value, $f^{\\eta^{\\prime}}_+(0)\/f^{\\eta}_+(0)\\equiv \\tan\\phi = 0.814$ exactly,\nsee Eq.~(\\ref{32}). The figure also illustrates the change of the\nresult upon inclusion of a non-zero $B^g_2$ (dot-dashed curves). The ratio is actually rather sensitive to that parameter. While the dependence on $a_2$ largely cancels when $a_2^\\eta$ and\n$a_2^{\\eta^{\\prime}}$ are set equal, there is a considerable residual dependence on\n$a_2^\\eta - a_2^{\\eta^{\\prime}}\\neq 0$ (dotted curves). While\n$|a_2^\\eta - a_2^{\\eta^{\\prime}}|=0.3$ as illustrated by these curves is\nrather unlikely, and would signal very large OZI-breaking\ncontributions (recall that $a_2^\\eta\\neq a_2^{\\eta^{\\prime}}$ or,\nequivalently, $a_2^1\\neq a_2^8$ signals the presence of\n``wrong-flavour'' contributions to the $\\eta_{q,s}$ DAs and is set to 0 in\nthe QF mixing scheme), one should nonetheless keep in mind that moderate\ncorrections of this type are not excluded and compete with the\nOZI-allowed corrections in $B^g_2$.\n\\begin{figure}\n$$\\epsfxsize=0.55\\textwidth\\epsffile{eta_fig8.eps}$$\n\\caption[${\\cal    B}(B\\to\\eta^{\\prime} e\\nu)\/{\\cal B}(B\\to\\eta e\\nu)$ as a function of the    singlet-parameter $B_2^g$.]{\\small The ratio of branching ratios    $R_{\\eta\\eta^{\\prime}} = {\\cal    B}(B\\to\\eta^{\\prime} e\\nu)\/{\\cal B}(B\\to\\eta e\\nu)$ as a function of the    singlet-parameter $B_2^g$. Solid curve: central values of    input parameters; Long-dashed curves: including variation of extrapolation model parameters, See Ref.~\\cite{Ball:2007hb}. Short-dashed curves: theoretical uncertainty of $R_{\\eta\\eta^{\\prime}}$ for $B^g_2=0$, for $a_2^{\\eta,\\eta^{\\prime}}(1\\,{\\rm GeV})$ varied independently, as in Fig.~\\ref{eta_fig5}.\n    }\\label{eta_fig8}\n\\end{figure}\n\nFinally, in Fig.~\\ref{eta_fig8} we show the dependence of the ratio of\nbranching ratios $R_{\\eta\\eta^{\\prime}} = {\\cal\n    B}(B\\to\\eta^{\\prime} e\\nu)\/{\\cal B}(B\\to\\eta e\\nu)$ on\n$B_2^g$. The advantage of this observable is that all hadronic effects\nare encoded in the form factors and that $|V_{ub}|$ cancels.\nThe solid curve corresponds to the branching ratios obtained\nfrom the central values of input parameters; the long-dashed curves illustrate the dependence on parameters originating from the model  used to extrapolate the $q^2$ dependence of the form factor from beyond the limit of the LCSR approach, in this case $q^2=16\\,{\\rm GeV}^2$, to the maximum possible value $q_{\\rm max}^2=(m_B-m_{\\eta^{(\\prime)}})^2$. It may be noted that the dependence on these parameters is very small. We do not go into detail about the extrapolation procedure and refer the reader to Ref.~\\cite{Ball:2007hb}.  On the other hand, $R_{\\eta\\eta^{\\prime}}$ also depends on $a_2^\\eta\\neq\na_2^{\\eta^{\\prime}}$. This dependence is shown by the short-dashed\ncurves. The conclusion is that large values of $B^g_2$, $|B^g_2|>5$,\ncan be distinguished from the OZI-breaking parameter\n$|a_2^\\eta-a_2^{\\eta^{\\prime}}|$, once an accurate experimental value of\n$R_{\\eta\\eta^{\\prime}}$ is available, but that for smallish $B^g_2$ and\nunknown $|a_2^\\eta-a_2^{\\eta^{\\prime}}|$ only mutual constraints on these\nparameters can be extracted from the data. In this case  also twist-4 gluonic DAs can become important. \n\n\n\n\n\\chapter{QCD Factorisation}\\label{chapter6_QCDF}\nIn this chapter we discuss the framework of QCD factorisation which was introduced  in the context of exclusive two-body non-leptonic $B$ decays by Beneke, Buchalla, Neubert and Sachrajda in Refs.~\\cite{Beneke:1999br, Beneke:2000ry}. We shall refer to the the original implementation of the framework as the BBNS approach.  We also focus on its application to the radiative $B$ decays $B \\to V \\gamma$, as presented by Bosch and Buchalla in Refs.~\\cite{Bosch:2001gv,Bosch:2002bw}.  \n\nQCD factorisation allows a rigourous determination of the $B$ decay matrix elements of the weak effective Hamiltonian (\\ref{basics_eq20}) to leading order in the heavy-quark limit of QCD  $m_b\\gg \\Lambda_{\\rm QCD}$, and yields a neat factorisation formula. It relies on the factorisation of hadronic matrix elements into universal non-perturbative hadronic parameters, given by transition form factors and meson light-cone DAs,  and process dependent hard-scattering kernels, calculable in perturbation theory. The validity of the QCD factorisation formula, to all orders in $\\alpha_s$, and the impact of generally unknown power corrections, formally suppressed by powers of $1\/m_b$, must be addressed case by case. The introduction of the QCD factorisation framework has made more discerning phenomenological studies of exclusive $B$ decays possible whereby key observables, such as branching ratios, CP and isospin asymmetries, can be calculated and confronted with experimental data.\n\nThe dependence of the factorisation formula on meson DAs, either directly or via LCSR calculations of the transition form factors, greatly motivates their study, with their better determination reducing the theoretical uncertainty of the QCD factorisation predictions, and aiding the quest to discover new physics effects from decay observables. \n\nWe begin with a short introduction, in the context of $B\\to M_1 M_2$ decays, of the general features of QCD factorisation, and in particular, discuss the appearance of meson DAs. We then discuss the framework as applied to the radiative $B$ decays $B \\to V \\gamma$. We postpone all discussions of phenomenology to Chapter~\\ref{chapter7_rad} in which we perform an analysis of the decays $B_{u,d} \\to (\\rho, \\omega, K^*)\\gamma$ and $B_{s} \\to (\\bar{K}^*,\\phi)\\gamma$ using QCD factorisation, augmented by the inclusion of the dominant power-suppressed corrections.\n\n\\section{Introduction}\n\\textit{QCD factorisation} (QCDF) \\cite{Beneke:1999br, Beneke:2000ry} was introduced in the context of the ``heavy-to-light'' decays $B \\to \\pi\\pi$ where the factorisation of the relevant QCD matrix elements was shown to apply, to leading order in a $1\/m_b$ expansion, to a large class of non-leptonic $B$ decays.   Consequently, QCDF has opened up the rich and varied landscape of $B$ decays to a more complete quantitative analysis. The existence of factorisation in non-leptonic decays is non-trivial and complicated by the possible gluonic interactions amongst the initial and final states. Conversly, leptonic and semi-leptonic decays factorise much more easily into the product of a quark current and a leptonic current, which cannot interact via gluon exchange.\n\nPhenomenologically,  QCDF has been remarkably successful, especially given the range of processes for which the method holds. After its introduction, it was swiftly generalised to encompass $\\pi K$ final states \\cite{Beneke:2001ev}, pseudoscalar-vector final states \\cite{Beneke:2003zv} and vector-vector meson final states \\cite{Kagan:2004uw}. The gluonic flavour-singlet contributions to $B \\to K^{(*)} \\eta^{(\\prime)} $ decays were added by Ref.~\\cite{Beneke:2002jn}. To date, the framework has been extended to many other processes, including for example, (double) radiative $B$ decays $B \\to \\gamma (\\gamma, V)$ \\cite{Bosch:2002bv,Bosch:2002bw} and $B \\to \\gamma l \\nu$ \\cite{Descotes-Genon:2002mw}.  Also,  other factorisation frameworks have since been developed and applied to the same problems:\n\\begin{itemize}\n\\item{\\textit{Soft Collinear Effective Theory} (SCET) \\cite{SCET1, SCET2, SCET3, SCET4}  makes a careful distinction between a hierarchy of ``hard'' $(m_b)$, ``hard-collinear'' ($\\sqrt{\\Lambda_{\\rm QCD} m_b}$) and ``collinear'' ($\\Lambda_{\\rm QCD}$) scales via contributions of internal quark and gluon lines. Details of the differences between the SCET and BBNS approaches to QCD factorisation can be found in Refs.~\\cite{Bauer:2004tj,Beneke:2004bn,Bauer:2005wb}.}\n\\item{The \\textit{Perturbative QCD}  (pQCD) approach \\cite{PQCD}, which yields a factorisation formula that depends on the mesons' transverse momenta.}\n\\item{The method of LCSRs, although having existed before the advent of QCDF, was applied to $B \\to \\pi \\pi$, both to the matrix elements which exhibit factorisation and also a class of power corrections, providing some useful complementary insights, see Refs.~\\cite{Khodjamirian:2000mi, Khodjamirian:2005wn}.}\n\\end{itemize}\nWe now go on to discuss the general features of QCDF. \n\\section{General Structure}\nConsider the case of non-leptonic decays where the $B$ meson decays into two mesons. The simplest way of dealing with the resulting matrix elements is to employ \\textit{naive factorisation} \\cite{Fakirov:1977ta,Cabibbo:1977zv}. Simply put, naive factorisation splits each local operator $Q_i$ of the effective Hamiltonian into two colour-singlet currents, whose matrix elements are proportional to a decay constant and a transition form factor respectively. For example, consider the four-quark operator $Q_2^U =(\\bar{D} U)_{\\rm V-A}(\\bar U b)_{\\rm V-A}$ then\n \\begin{equation}\n\\bra{M_1 M_2}(\\bar{D} U)_{\\rm V-A}(\\bar U b)_{\\rm V-A}\\ket{B} \\,\\stackrel{\\rm{NF}}{\\longrightarrow}\\,\\underbrace{\\bra{M_2}(\\bar{D} U)_{\\rm V-A}\\ket{0}}_{f_{M_2}}\\underbrace{\\bra{M_1}(\\bar U b)_{\\rm V-A}\\ket{B}}_{F^{B \\to M_1}}\\,.\n\\label{qcdf_1}\n\\end{equation}\nThe motivation for factorising in this way comes from the \\textit{colour transparency} argument \\cite{Bjorken:1988kk}. It follows that a major shortcoming of naive factorisation is that it assumes the exchange of gluons of virtualites $\\mu \\lesssim m_b$ to be negligible and hence rescattering between the decay products is not considered; there is then no mechanism for the generation of strong phase effects between different amplitudes. Also, the matrix elements (\\ref{qcdf_1})  do not display the correct renormalisation-scale dependence.\n\nThe framework of QCDF allows the calculation of $\\mathcal{O}(\\alpha_s)$ corrections to naive factorisation, which occur at scales $\\mu\\lesssim m_b$. It is constructed by observing the cancelation of infrared (IR) and collinear divergences, via consistent power-counting arguments, allowing the use of perturbation theory to describe the hard-gluon exchanges. The resulting intuitive factorisation formula thus presents a massive simplification of the long-distance QCD effects, with QCDF recovering naive factorisation in the limit $m_b\\to \\infty$. In terms of two-body non-leptonic $B$ decays to light pseudoscalar mesons $B\\to M_1 M_2$ the factorisation formula, as presented in Ref.~\\cite{Beneke:1999br}, reads schematically as\n\\begin{eqnarray}\n\\bra{M_1 M_2} Q_i\\ket{B}&=& F^{B \\to M_1}\\int^1_0 du\\,T^{I}_i (u)\\, \\phi_{2;M_2}(u) + (M_1 \\leftrightarrow M_2)\\nonumber \\\\\n&+&\\int^1_0 d\\xi\\,du\\,dv\\,T^{II}_i (\\xi,u,v)\\,\\phi_B(\\xi) \\,\\phi_{2;M_1}(v)\\,\\phi_{2;M_2}(u) \\nonumber \\\\\n&+&\\mathcal{O}(\\Lambda_{\\rm QCD}\/m_b)\\,\n\\label{qcdf_2}\n\\end{eqnarray}\nwhere $F^{B \\to M_1}$ is the relevant form factor, $T^{I,II}_i$ are the hard-scattering kernels, $\\phi_{B}$ is one of the leading-twist DAs of the $B$ meson and $\\phi_{2;P}$ the leading-twist DA of the final state meson $P$, and the $Q_i$ are the operators of the effective Hamiltonian. The matrix elements are given as the convolution of the universal DAs and the process dependent hard-scattering kernels, with respect to the meson momentum fractions. Since the transition form factor and the DAs are real functions, all strong phases are generated by the hard-scattering kernels and are suppressed by powers of $\\alpha_s$. Factorisation has be proven to one-loop for ``light-light'' final states and two-loop for ``heavy-light'' final states \\cite{Beneke:2000ry}. It has be proven to all orders in $\\alpha_s$ for $B\\to D \\pi $ using SCET \\cite{SCET2}.\n\nThe ability of QCDF to accurately describe $B$ decay processes is limited by two main considerations; firstly, by the nature of the factorisation formula itself, which is valid up to power corrections $\\mathcal{O}(1\/m_b)$ and to a given order in $\\alpha_s$; and secondly by uncertainties of the necessary input parameters, such as the DAs, the transition form factors, the strange quark mass, the $B$ meson decay constant $f_B$ etc.  Whether a discrepancy between experiment and QCDF predictions can be put down to new physics, or not, requires an estimation of neglected power corrections; certainly the $b$ quark mass is not asymptotically large $m_b\\sim 5 \\,{\\rm GeV}$ and power corrections are therefore expected to feature at the level of $\\mathcal{O}(\\Lambda_{\\rm QCD}\/m_b) \\sim 10\\%$. The size and nature of power corrections can be probed via phenomenology, however, the task is not straight forward; even the initial focus of the approach, the decays $B \\to \\pi (K,\\pi)$, which stands as a crucial test, has not been resolved satisfactorily, see for example Ref.~\\cite{Feldmann:2004mg} and Refs.~\\cite{Fleischer:2005vz,Fleischer:2007wd}. Better determined input parameters will nevertheless shed light, case by case, on whether power corrections are important, and the QCDF predictions must be used to determine or constrain CKM matrix elements (UT angles), or detect signs of new physics, with that in mind.  \n\n\n\\section{Light-Cone Distribution Amplitudes}\nTo leading-order in the heavy-quark limit the leading-twist final state meson DAs contribute to the factorisation formula and can be safely truncated after the second Gegenbauer moment $a_2$. For pseudoscalar meson final states the two-particle twist-3 DAs come with large normalisation factors $r^P_\\chi$ and are said to be \\textit{chirally enhanced}, and are therefore included even though they are formally $1\/m_b$ suppressed. The vector mesons  do not have the same large normalisation factors but their two-particle twist-3 DAs are included in the BBNS approach for consistency. For a pseudoscalar  or vector meson, with valence quark content $\\bar q q^{\\prime}$, the normalisation factors are respectively\n\\begin{equation}\nr_\\chi^P(\\mu) = \\frac{2 m_P^2}{m_b(\\mu)(m_q+m_{q^\\prime})(\\mu)} \\sim\\frac{\\Lambda_{\\rm QCD}}{m_b}\\,,\\qquad r^V_\\chi(\\mu)=\\frac{2 m_V}{m_b(\\mu)}\\frac{f_V^\\perp(\\mu)}{f_V^\\parallel}\\,.\n\\label{qcdf_3}\n\\end{equation}\nThree-particle twist-3 DAs are neglected because they do not come with large normalisations.  The inclusion of the chirally enhanced DAs leads to end-point divergences from the convolutions of the two-particle twist-3 pseudoscalar DAs with the corresponding hard-scattering kernels originating from both the hard-spectator scattering and annihilation contributions. The resulting divergent integrals signal the breakdown of factorisation and are parameterised by two universal unknown parameters $X_{H,A}$, introducing a source of theoretical uncertainty to the BBNS approach \\cite{Beneke:1999br}. \n\nAt leading-twist the $B$ meson is described by two DAs, only one of which is required as input for Eq.~(\\ref{qcdf_2}) and appears in the hard-spectator diagrams contributing to $T^{II}_i$. The DAs of the $B$ mesons are complicated by the fact that the momentum of the meson is shared in a highly antisymmetric way: the $b$ quark has most of it. The $B$ meson DAs are given, at leading-order in $1\/m_b$, by\n\\begin{equation}\n\\bra{0}\\bar{q}_\\alpha(0) b_{\\beta}(z)\\ket{B(p_B)}=i \\frac{f_B}{4} \\left[(\\slash{p}_B+m_b) \\gamma^5\\right]_{\\beta\\gamma}\\int^1_0 d\\xi\\,e^{-i\\xi (p_B)_+  z_-}\\left[\\Phi_{B1}(\\xi)+\\slash{n}_- \\Phi_{B2}(\\xi)\\right]_{\\gamma \\alpha}\\,,\n\\label{qcdf_4}\n\\end{equation}\nwith the decay constant $f_B$ given by Eq.~(\\ref{bdecayconstant}). With a careful choice of $n_-=(1,0,0,-1)$ only the following normalisation conditions are required\n\\begin{equation}\n\\int^1_0d\\xi\\, \\Phi_{B1}(\\xi)=1\\,,\\qquad\\int^1_0d\\xi\\, \\Phi_{B2}(\\xi)=0\\,,\n\\label{qcdf_5}\n\\end{equation}\nalong with the first inverse moment of $ \\Phi_{B1}$ which is parameterised as\n\\begin{equation}\n\\int^1_0 d\\xi\\,\\frac{\\Phi_{B1}(\\xi)}{\\xi}\\equiv \\frac{m_B}{\\lambda_B}\\,,\n\\label{qcdf_6}\n\\end{equation}\nand the numerical value of $\\lambda_B$ is a source of uncertainty in the QCDF framework for both $B\\to M_1 M_2$ and $B\\to V \\gamma$. We now discuss the radiative decays $B\\to V \\gamma$ within QCDF.\n\n\\section{Radiative $B$ decays to Vector Mesons}\\label{qcdf_rad}\nWe consider the leading contributions to the $B\\to V \\gamma$ QCDF factorisation formula  as of Refs.~\\cite{Bosch:2001gv,Bosch:2002bw,Bosch:2004nd,Beneke:2001at} in which a model independent framework is presented.  Contributions that are power-suppressed by one power of $1\/m_b$ or more \\textit{and} are $\\mathcal{O}(\\alpha_s)$ are not considered. At the quark level the decays are $b\\to D \\gamma$ transitions, where $D=\\{s,d\\}$. If otherwise not stated, in the following we refer to $\\bar{B}\\to V\\gamma$ decays where $\\bar{B}$ ($V$) denotes a $b \\bar{q}$ ($D \\bar{q}$)  bound state. For $B\\to V \\gamma$ decays the matrix element of each relevant local operator in the effective Hamiltonian factorises as\n\\begin{equation}\n\\bra{V\\gamma}Q_i\\ket{B}=e^* \\cdot \\left[ T_1^{B\\to V} (0) \\,T^{I}_i+\\int^1_0 d\\xi du\\,T^{II}_i(\\xi,u)\\phi_B(\\xi) \\phi_{2;V}^{\\perp}(u)\\right]+\\mathcal{O}(1\/m_b)\\,,\n\\label{qcdf_7}\n\\end{equation}\nwhere $e_\\mu$ is the photon polarisation vector and $T_1^{B \\to V}(0)$ is the relevant form factor. $\\phi_{2;V}^{\\perp}$ the leading-twist DA of the perpendicularly polarised final state vector meson (\\ref{das_eq19});  contributions from $\\phi^{\\parallel}_{2;V}$ are power-suppressed in the heavy-quark limit. Problems of end-point divergences are not encountered in $B\\to V\\gamma$ decays and the twist-3 vector meson DA does not feature -- the $B$ meson DAs (\\ref{qcdf_6}) do however. The factorisation formula is accurate up to corrections suppressed by powers of $1\/m_b$, as shown, and was proven to hold to all orders in $\\alpha_s$ in SCET \\cite{Becher:2005fg}.  The form factor $T_1^{B \\to V}(0)$ has been calculated, for example, from LCSR in Ref.~\\cite{Ball:2004rg}. \n\nThe $B\\to V \\gamma$ decay produces either left- or right-handed photons, which therefore constitute, in principle,  two separate observable processes. In practise the direct measurement of the photon's helicity is very difficult; indirectly, however, it can be accessed by measurement of the time-dependent CP asymmetry in $\\bar B^0\\to V^0\\gamma$, which vanishes if one of them is absent, see Chapter~\\ref{chapter7_rad}.  We define the two amplitudes as\n\\begin{equation}\n\\bar{\\cal A}_{L(R)} = {\\cal A}(\\bar B\\to V\\gamma_{L(R)})\\,,  \\qquad\n{\\cal A}_{L(R)} = {\\cal A}(B\\to \\bar V \\gamma_{L(R)})\\,.\n\\label{qcdf_8}\n\\end{equation}\nFor ($B$) $\\bar B$  decays the production of the (left-) right-handed photon is suppressed by $1\/m_b$ with respect to the opposite helicity. The decays are dominated by the electromagnetic dipole operator $Q_{7\\gamma}$, and as such are penguin mediated and so loop-suppressed. The operators $Q_{7\\gamma}^{L(R)}$ are given by\n\\begin{equation}\nQ_{7\\gamma}^{L(R)} = \\frac{e}{8\\pi^2}\\, m_b \\bar D \\sigma_{\\mu\\nu}\\left(1 \\pm \\gamma_5\\right)b F^{\\mu\\nu}\\,,\n\\label{qcdf_9}\n\\end{equation} \nand generate left- (right-) handed photons.  Their matrix elements can be parameterised in terms of the form factor $T_1^{B\\to V}$ as\n\\begin{eqnarray}\n\\lefteqn{\\bra{V(p,\\eta) \\gamma_{L(R)}(q,e)} Q_{7\\gamma}^{L(R)} \\ket{\\bar\nB}}\\hspace*{1cm}\\nonumber\\\\\n&=& -\\frac{e}{2\\pi^2}\\, m_b T_1^{B\\to V}(0) \\left[\n\\epsilon^{\\mu\\nu\\rho\\sigma} e_\\mu^* \\eta_\\nu^* p_\\rho q_\\sigma \\pm i\n\\{ (e^* \\cdot \\eta^*) (p \\cdot  q) - (e^* \\cdot p)(\\eta^* \\cdot q)\\}\\right]\n\\nonumber\\\\\n&\\equiv& -\\frac{e}{2\\pi^2}\\, m_b T_1^{B\\to V}(0) S_{L(R)}\\,,\n\\label{qcdf_10}\n\\end{eqnarray}\nwhere $S_{L,R}$ are the helicity amplitudes corresponding to left- and right-handed photons, respectively, and $e_\\mu$ $(\\eta_\\mu)$ is the polarisation four-vector of the photon (vector meson).   The leading-order diagram is given in Fig.~\\ref{qcdf_fig1} which is also the leading diagram for the form factor $T_1^{B\\to V}$.\n\\begin{figure}[h]\n$$\\epsfxsize=0.2\\textwidth\\epsffile{rad_leading.eps}$$\n\\caption[The leading contribution to $B \\to V\\gamma$.]{\\small The leading contribution to $B \\to V\\gamma$ due to the  electromagnetic dipole operator $Q_{7\\gamma}$. }\\label{qcdf_fig1}\n\\end{figure}\nThe factorisation formula (\\ref{qcdf_10}) is therefore trivial to leading order in $\\alpha_s$ and the heavy-quark limit; the matrix element given by the standard form factor, the scattering kernel $T^I_7$ by a purely kinematical function and $T^{II}_7$ does not feature.  The electroweak penguin operators $Q_{7,\\dots,10}$ appear at higher-order and safely neglected in the analysis. All other operators begin to contribute at $\\mathcal{O}(\\alpha_s)$. The hard-vertex corrections contribute to $T^I_i$ yielding functions of $m_{u,c}^2\/m_b^2$ and originate from penguin contractions of the operators $Q_{1,\\dots,6}$ and the chromomagnetic operator $Q_{8g}$ as shown in Fig.~\\ref{qcdf_fig2}. \n\\begin{figure}[h]\n$$\\epsfxsize=0.5\\textwidth\\epsffile{rad_vertexcorr.eps}$$\n\\caption[Contributions to the hard-scattering kernel $T^{I}_i$ for $B\\to V \\gamma$ decays.]{\\small Penguin contractions of $Q_{1,\\dots,6}$ (top line) and the chromomagnetic dipole operator $Q_8$ (bottom line) contributing to the hard-vertex corrections of $T^{I}_i$ at $\\mathcal{O}(\\alpha_s)$. Crosses denote possible photon emission vertices.}\n\\label{qcdf_fig2}\n\\end{figure}\nThe hard-spectator scattering diagrams of Fig.~\\ref{qcdf_fig3}, in which the spectator quark of the $B$ meson participates, contribute to  $T^{II}_i$ and involve the same operators as the hard-vertex corrections. The hard-gluon exchange probes the momentum distribution of the $B$ and vector mesons and so requires the introduction of the mesons' light-cone DAs, as suggested by the factorisation formula; it is in these contributions that the $B$ meson DA parameter $\\lambda_B$ and decay constants $f_B$ and $f_V^\\perp$ appear. \n\\begin{figure}[h]\n$$\\epsfxsize=0.5\\textwidth\\epsffile{rad_spec.eps}$$\n\\caption[Contributions to the hard-scattering kernel $T^{II}_i$ for $B\\to V \\gamma$ decays.]{\\small Penguin contractions of $Q_{1,\\dots,6}$ (left) and the chromomagnetic dipole operator $Q_{8g}$ (right) contributing to the hard-scattering kernel  $T^{II}_i$ at $\\mathcal{O}(\\alpha_s)$. Crosses denote possible photon emission vertices at leading order. Photon emission from the other quark lines power-suppressed. Photon emission from the final state meson for $Q_{8g}$ breaks factorisation.}\n\\label{qcdf_fig3}\n\\end{figure}\nAlso, the dominant power-suppressed  \\textit{weak annihilation} (WA) contributions, shown in Fig.~\\ref{qcdf_fig4}, are calculable in the QCDF approach, and involve the operators $Q_{1,\\dots,6}$. WA contributions are\n$O(1\/m_b)$; photon emission from the $b$ quark and the quarks in the vector meson is further suppressed and $O(1\/m_b^2)$ -- unless the weak interaction operator is $Q_{5,6}$, which can be Fierz transformed into $(\\bar D (1+\\gamma_5) q) (\\bar q (1-\\gamma_5) b)$ and picks up an additional factor $m_B$ from the projection onto the $B$ meson DA thus resulting in this contribution being $O(1\/m_b)$. Consequently, due to the large Wilson coefficients $C_{1,2}$ these contributions are sizeable and important phenomenologically, see Chapter~\\ref{chapter7_rad}. \n\\begin{figure}[h]\n$$\\epsfxsize=0.2\\textwidth\\epsffile{rad_ann.eps}$$\n\\caption[Weak annihilation contributions to $B\\to V \\gamma$.]{\\small Weak annihilation contributions, which are suppressed by one power of $1\/m_b$. Crosses denote possible photon emission vertices at leading order. The dominant mechanism for $Q_{1,\\dots,4}$ is the emission of the photon from the light quark in the $B$ meson and for $Q_{5,6}$ it is the emission from the final state vector meson quarks. Other possible emissions are either vanishing or more strongly suppressed.}\n\\label{qcdf_fig4}\n\\end{figure}\n\nThe decay amplitude is then given by\n\\begin{equation}\n\\mathcal{A}(\\bar{B}\\to V \\gamma_{L(R)})=\\frac{G_F}{\\sqrt{2}}\\left(\\lambda_u^D a_{7 L(R)}^u(V)+ \\lambda_c^D a_{7 L(R)}^c(V)\\right)\\bra{V\\gamma_{L(R)}}Q_{7\\gamma}^{L(R)}\\ket{\\bar{B}}\\,,\n\\label{qcdf_11}\n\\end{equation}\nwhere the left-handed coefficients are given, to leading order in QCDF, by\n\\begin{equation}\na_{7L}^{U,{\\rm QCDF}}(V)=C_7+\\mathcal{O}(\\alpha_s,1\/m_b)\\,,\n\\label{qcdf_12}\n\\end{equation} \nand the right-handed parameters, for a $b\\to D$ transition, by \\cite{Ball:2006cv}\n\\begin{equation}\na_{7R}^{U,{\\rm QCDF}} = C_7\\,\\frac{m_D}{m_b}+\\mathcal{O}(1\/m_b,\\alpha_s\/m_b)\\,.\n\\label{qcdf_13}\n\\end{equation}\nExplicit expressions for the $\\mathcal{O}(\\alpha_s)$ corrections to the left-handed coefficients can be found in Refs.~\\cite{Bosch:2001gv,Bosch:2002bw} and will be considered in Chapter~\\ref{chapter7_rad}, alongside the dominant power-suppressed corrections.\n\\chapter{$B \\to V \\gamma$ Beyond QCD Factorisation}\\label{chapter7_rad}\nIn this chapter we perform a phenomenological analysis of the exclusive radiative $B$ decays to vector mesons. We make use of the QCDF framework outlined in Chapter~\\ref{chapter6_QCDF} and investigate the impact of the leading power-corrections on the branching ratios, CP asymmetries and isospin asymmetries for all $b\\to D$ transitions; $B_{u,d} \\to (\\rho, \\omega, K^*)\\gamma$ and  $B_{s} \\to (\\phi, \\bar{K}^*)\\gamma$.  Weak annihilation effects, although power-suppressed, are calculable in QCDF, and are included for all decay modes in this analysis.  The other power-suppressed contributions ``beyond QCDF''  considered are; soft photon emission from the soft $B$ spectator quark \\cite{Ball:2003fq}; and long-distance contributions from heavy quark loops \\cite{Ball:2006cv} and light quark loops \\cite{Ball:2006eu} which have been estimated from LCSR.  The estimation of the light quark loop contribution is new to the present analysis. Whereas the branching ratios are generally dominated by the leading contributions, and power-suppressed contributions play a minor role, the same cannot be said for the CP and isospin asymmetries for which the impact of power-corrections is in fact crucial.\n\nThe motivation to study radiative $B$ decays stems from a variety of sources:\n\\begin{itemize}\n\\item{as loop-induced, penguin mediated decays, they allow the extraction of the CKM matrix element $|V_{t,(d,s)}|$ complimentarily to the determination from $B$ mixing and also that from the SM UT analysis based on the tree-level observables $|V_{ub}\/V_{cb}|$ and the angle $\\gamma$.}\n\\item{They are sensitive to new physics contributions, which may occur within the penguin loops, with the time-dependent CP asymmetry a very promising avenue of investigation. They are also subject to large short-distance QCD corrections, which now approach next-to-next-to-leading-order accuracy, see Refs.~\\cite{misiak,NNLObsgamma}.}\n\\item{The decay rates are of order $G_F^2 \\alpha_{\\rm QED}$ and are enhanced with respect to other loop-induced non-radiative rare decays which are of order $G_F^2 \\alpha_{\\rm QED}^2$. Also, the $b\\to s$ modes are CKM-favoured. Consequently there exist good experimental results for the exclusive branching ratios; $B\\to K^* \\gamma$ is known to 5\\%, but the $b\\to d$ transitions are not so well known.}\n\\end{itemize}\n\nAs discussed in Chapter~\\ref{chapter6_QCDF}, the QCDF framework for $B\\to V\\gamma$ relies on the leading-twist vector meson DA $\\phi_{2;V}^\\perp$. Moreover, the LCSR calculations of the form factors $T_1^{B\\to V}$ and the parameters entering expressions for the soft-quark contributions rely also on the higher-twist DAs of the vector mesons and thus we find immediate use for the results of the twist-2 and twist-3 DA parameters of Chapter~\\ref{chapter4_det}, as presented in Tab.~\\ref{det_tab1} and Tab.~\\ref{det_tab2}.\\footnote{The analysis presented in Ref.~\\cite{Ball:2006eu} used preliminary input for the DA parameters, values for which were later finalised in Ref.~\\cite{Ball:2007rt}. The conclusions and numerics of the analysis are unaffected, due somewhat to the large errors attributed to the soft quark loop calculations in which the twist-3 DA parameters feature.}\n\nWe begin with an introduction, and then go on to discuss the power-suppressed contributions and investigate their impact on the decay observables. We extract the CKM parameter $|V_{t,d}\/V_{ts}|$ from the branching ratio results, assuming no new physics contributions, and discuss possible new physics contributions to the CP and isospin asymmetries. The material covered in this chapter follows that of Ref.~\\cite{Ball:2006eu}.\n\n\\section{Introduction}\n$B \\to V \\gamma$ decays are a very rich and promising probe of flavour physics. Both the inclusive decay $B\\to X_s \\gamma$ and the exclusive decays $B\\to (K^*,\\rho)\\gamma$ have been under scrutiny for many years, see for example Refs.~\\cite{Neubert:2002ku,Kagan:1998bh}.  The experimental results for $B\\to (\\rho,\\omega,K^*)\\gamma$ are shown in Tab.~\\ref{rad_tab1}.  For $B_s\\to\\phi\\gamma$  only an upper bound ${\\cal B}(B_s\\to\\phi\\gamma)<120\\times 10^{-6}$ exists and no experimental information is available for $B_s\\to \\bar K^*\\gamma$ \\cite{Yao:2006px}. \n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.4}\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l||c|c||l|c}\n\\hline\n{\\cal B} \\times 10^6 & \\mbox{{\\sc BaBar} \\cite{babar_rad}} & \\mbox{Belle\n  \\cite{belle_rad}} & {\\cal B} \\times 10^6 & \\mbox{HFAG \\cite{Barberio:2007cr}}\n\\\\\\hline\nB\\to (\\rho,\\omega)\\gamma & 1.25^{+0.25}_{-0.24}\\pm 0.09 &\n                           1.32^{+0.34}_{-0.31}{}^{+0.10}_{-0.09} \n                         &\nB^+\\to K^{*+}\\gamma & 40.3\\pm 2.6\n                          \n\\\\\nB^+\\to \\rho^+\\gamma &      1.10^{+0.37}_{-0.33}\\pm 0.09 &\n                           0.55^{+0.42}_{-0.36}{}^{+0.09}_{-0.08} \n                          \n                    &\nB^0\\to K^{*0}\\gamma &  40.1\\pm 2.0\n\\\\\nB^0\\to\\rho^0\\gamma  &      0.79^{+0.22}_{-0.20}\\pm0.06 &\n                           1.25^{+0.37}_{-0.33}{}^{+0.07}_{-0.06} \n                          \n\\\\\nB^0\\to\\omega\\gamma  &      <0.78 &\n                           0.96^{+0.34}_{-0.27}{}^{+0.05}_{-0.10}\n                          \n\\\\\\hline\n\\end{array}\n$$\n\\caption[Experimental branching ratios of exclusive $b\\to (d,s)\\gamma$\n  transitions.]{\\small Experimental branching ratios of exclusive $b\\to (d,s)\\gamma$\n  transitions. All entries are CP averaged. The first error is statistical, the second\n  systematic. $B\\to (\\rho,\\omega)\\gamma$ is the CP   average of the\n  isospin average over $\\rho$ and $\\omega$ channels:\\\\  $\\overline{\\cal B}(B\\to\n  (\\rho,\\omega)\\gamma) = \\frac{1}{2}  \\left\\{ \\overline{\\cal B}(B^\\pm\\to \\rho^\\pm\\gamma) +\n  \\frac{\\tau_{B^\\pm}}{\\tau_{B^0}} \\left[ \\overline{\\cal B}(B^0\\to \n  \\rho^0\\gamma) +  \\overline{\\cal B}(B^0\\to \\omega \\gamma)\\right]\\right\\}$.}\n  \\label{rad_tab1}\n\\end{table}\n\nIn the SM the decays are flavour-changing-neutral-current (FCNC) $b\\to D\\gamma$ transitions, mediated by penguin diagrams; they are therefore loop-suppressed and potentially very sensitive to new physics. To determine the relative sizes of contributions to the decays one must consider the following points:\n\\begin{itemize}\n\\item{the leading term is loop-suppressed $\\sim 1\/(4 \\pi)^2$ and proportional to $C_7\\sim-0.3$.}\n\\item{Evidently from Eq.~(\\ref{qcdf_11}) for each mode there are two amplitudes proportional to different CKM factors $\\lambda^{(D)}_{u,c}$. For $b\\to d$ transitions both $\\lambda^{(d)}_u$ and $\\lambda^{(d)}_c$ are $\\sim\\lambda^3$, however, for $b\\to s$ transitions $\\lambda^{(s)}_u\\sim\\lambda^4$ and $\\lambda^{(s)}_c\\sim\\lambda^2$; there is a relative CKM suppression of the up-quark contribution.}\n\\item{Power suppressed corrections from WA are formally $\\sim 1\/m_b$ although come with  large Wilson coefficients $C_1\\sim-0.3$ and $C_2\\sim 1$ and are not loop suppressed. The WA contributions drive the isospin asymmetries.}\n\\item{The production of ``wrong'' helicity photons is suppressed by $m_D\/m_b$ (\\ref{qcdf_13}). The interplay of both helicity amplitudes generates the time-dependent CP asymmetries, which are small in the SM due to this suppression.}\n\\end{itemize}\n\n\\section{Wilson Coefficients}\nConsiderable effort has gone into calculating the Wilson coefficients to NLO accuracy. Using the expressions for the NLO anomalous dimension matrices available in the literature we employ the renormalisation techniques of Eqs.~(\\ref{basics_eq22}-\\ref{basics_eq29}) to calculate the Wilson coefficients at the required scales. Numerical values of all the NLO Wilson coefficients $C_i$ used in the analysis are given in Tab.~\\ref{rad_tab2}. The situation is complicated by the fact that the QCDF results of Ref.~\\cite{Bosch:2002bw} are given in terms of two bases. The first, the so-called BBL basis named after the authors of Ref.~\\cite{Buchalla:1995vs}, is that of Eqs.~(\\ref{basics_eq20}) and (\\ref{basics_eq21}) except with $Q_1$ and $Q_2$ exchanged with respect to the basis of Ref.~\\cite{Bosch:2001gv}. The second is the so-called CMM basis of Ref.~\\cite{munz, buras}. The two bases differ except for $Q_{7(8)}^{\\rm BBL}=Q_{7(8)}^{\\rm CMM}$. Following Ref.~\\cite{Bosch:2002bw}, the CMM set is used for calculating hard-vertex  corrections to the QCDF formulas and the BBL set at the lower scale $\\mu_h\\sim \\sqrt{\\Lambda_{h} \\,\\mu}$ (with $\\lambda_h\\sim 0.5\\,{\\rm GeV}$ and $\\mu= \\mathcal{O}(m_b)$)  is used to calculate hard-spectator corrections. Power corrections are calculated from the BBL set at scale $m_b$. \n\nNLO accuracy is mandatory only for $C_7$, as it is for this term only that the hadronic matrix element is also known to NLO accuracy. We evaluate all $\\mathcal{O}(\\alpha_s)$ and power-suppressed corrections using both LO and NLO scaling for Wilson coefficients and hadronic matrix elements and include the resulting scale dependence in the theoretical uncertainty.\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{2pt}\n$$\n\\begin{array}{c|c|c|c|c|c|c}\nC^{\\rm CMM}_1(m_b) & C^{\\rm CMM}_2(m_b) & C^{\\rm CMM}_3(m_b) & \nC^{\\rm CMM}_4(m_b) & C^{\\rm CMM}_5(m_b) & C^{\\rm CMM}_6(m_b) &\nC^{\\rm CMM}_7(m_b)\n\\\\\\hline\n-0.322 & 1.009 & -0.005 & -0.087 & 0.0004 & -0.001 & -0.309 \n\\\\\\hline\\hline\nC^{\\rm BBL}_1(m_b) & C^{\\rm BBL}_2(m_b) & C^{\\rm BBL}_3(m_b) & \nC^{\\rm BBL}_4(m_b) & C^{\\rm BBL}_5(m_b) & C^{\\rm BBL}_6(m_b) \n& C^{\\rm CMM}_8(m_b) \n\\\\\\hline\n-0.189 & 1.081  & 0.014 & -0.036 & 0.009  & -0.042 & -0.170\n\\\\\\hline\\hline\nC^{\\rm BBL}_1(\\mu_h) & C^{\\rm BBL}_2(\\mu_h) & C^{\\rm BBL}_3(\\mu_h) & \nC^{\\rm BBL}_4(\\mu_h) & C^{\\rm BBL}_5(\\mu_h) & C^{\\rm BBL}_6(\\mu_h) &\nC^{\\rm CMM}_8(\\mu_h)\n\\\\\\hline\n-0.288 & 1.133 & 0.021 & -0.051 & 0.010 & -0.065 & -0.191\n\\end{array}\n$$\n\\caption[Numerical values of the next-to-leading-order Wilson coefficients.]{\\small NLO Wilson coefficients to be used in the analysis, at the scales $m_b=4.2\\,$GeV and $\\mu_h=2.2\\,$GeV. The coefficients labelled BBL correspond to the operator  basis of Ref.~\\cite{Buchalla:1995vs} and given in Eq.~(\\ref{basics_eq21}),   whereas CMM denotes the basis of Ref.~\\cite{munz}. We use   $\\alpha_s(m_Z) = 0.1176$ \\cite{Yao:2006px} and ${m}_t({m}_t) =   163.6\\,$GeV \\cite{mt}. Note that $C_1^{\\rm BBL}$ and $C_2^{\\rm BBL}$   are exchanged with respect to the basis of Ref.~\\cite{Bosch:2001gv} and  that $C_{7(8)}^{\\rm BBL}=C_{7(8)}^{\\rm CMM}$. Following Ref.~\\cite{ Bosch:2002bw}, the CMM set is used for calculating hard-vertex  corrections to the QCDF formulas and the BBL set at the lower scale  $\\mu_h$ is used to calculate hard-spectator corrections. The BBL set at  scale $m_b$ is used for the calculation of power-corrections.}\n\\label{rad_tab2}\n\\end{table}\n\n\n\n\\section{Leading and Power Suppressed Contributions}\nIt proves convenient to split to the coefficients in Eq.~(\\ref{qcdf_11}) into three contributions which we will investigate separately:\n\\begin{eqnarray}\na_{7L}^U( V) &=& a_{7L}^{U,{\\rm QCDF}}( V) + a_{7L}^{U,{\\rm ann}}(  V) + a_{7L}^{U,{\\rm soft}}( V)+\\dots\\,,\\nonumber\\\\\na_{7R}^U( V) &=& a_{7R}^{U,{\\rm QCDF}}( V) + a_{7R}^{U,{\\rm ann}}(V)+  a_{7R}^{U,{\\rm soft}}( V)+\\dots\\,,\n\\label{asplit}\n\\end{eqnarray} \nwhere the leading term in the $1\/m_b$ expansion is given by Eq.~(\\ref{qcdf_12}) and all other terms are suppressed by at least one power of $m_b$. The dots denote terms of higher order in $\\alpha_s$ and further $1\/m_b$ corrections to QCDF, most of which are incalculable. We only include those power-suppressed terms that are either numerically large or relevant for isospin and CP asymmetries.  \n\n\n\n\\subsection{Leading Contributions}\nThe diagrams giving the leading QCDF contributions are given in Chapter~\\ref{chapter6_QCDF}. It turns out that, at the level of two decimal places, all $a_{7L}^{c,{\\rm QCDF}}$ are equal and so are\n$a_{7L}^{u,{\\rm QCDF}}$.\\footnote{Explicit formulas for  $a_{7L}^{U,{\\rm QCDF}}$, complete to $\\mathcal{O}(\\alpha_s)$, can be found in Ref.~\\cite{Bosch:2002bw}.} For central values of the input parameters of Tab.~\\ref{rad_tab8} we obtain\n\\begin{eqnarray}\na^{c,{\\rm QCDF}}_{7L}(V) & = & -\\overbrace{(0.41+0.03i)}^{\\shortstack{{\\rm \\footnotesize Vertex}\\\\{\\rm \\footnotesize Corrections}}} - \n\\overbrace{(0.01+0.01i)}^{\\shortstack{{\\rm \\footnotesize Hard-Spectator}\\\\{\\rm \\footnotesize Corrections}}}\\,,\\nonumber\\\\\na^{u,{\\rm QCDF}}_{7L}(V) & = & -(0.45+0.07i) + (0.02-0i)\\,.\n\\label{10}\n\\end{eqnarray}\nThe size of the hard-spectator corrections is set by the factor\n\\begin{equation}\nh_V = \\frac{2 \\pi^2}{9}\\,\\frac{f_B f_V^\\perp}{m_B  \nT_1^{B\\to V}(0)\\lambda_B}\\,.\n\\end{equation}\nFor $B_s$ decays one has to set $f_B\\to f_{B_s}$ and correspondingly for the other $B$ meson parameters. We estimate the value of  $\\lambda_{B_s}$, the first inverse moment of the twist-2 $B$-meson light-cone DA, from $\\lambda_{B_d}$ by a simple scaling argument:\n\\begin{equation}\n\\frac{m_{B_s}}{\\lambda_{B_s}}\\,(\\Lambda_{\\rm QCD}+m_s) = \n\\frac{m_{B_q}}{\\lambda_{B_q}}\\,\\Lambda_{\\rm QCD}\\,,\n\\label{rad_bs}\n\\end{equation}\nwhich follows from the assumption that the $B_q$ DA peaks at the spectator momentum $k_+ = \\Lambda_{\\rm QCD}$, whereas that of $B_s$ peaks at $\\Lambda_{\\rm QCD}+m_s$. Its numerical value is given, along with all the other input parameters, in Tab.~\\ref{rad_tab8}.\n\n\n\n\\subsection{Weak Annihilation}\n $a_{7L}^{U,{\\rm ann}}$ encodes the $\\mathcal{O}(1\/m_b)$ contribution of the WA diagram of Fig.~\\ref{rad_fig1}(a) which drives  the isospin asymmetries and has been  calculated in QCDF in  Ref.~\\cite{Bosch:2002bw} with $\\alpha_s$ corrections given in Ref.~\\cite{Kagan:2001zk} for $\\rho$ and $K^*$  and in Ref.~\\cite{Bosch:2004nd} for $\\omega$. WA receives contributions from the current-current operator $Q_2^u$, which for $b\\to s$ transitions is doubly CKM suppressed, and QCD penguin operators $Q_{3,\\dots,6}$, which are not CKM suppressed. Formulas for $a_{7L}^{U,{\\rm ann}}(\\rho,K^*)$ in QCDF can be found in Refs.~\\cite{Bosch:2002bw,Bosch:2004nd};  in this approximation, there is no contribution to  $a_{7R}^{U,{\\rm ann}}$. \n\\begin{figure}[h]\n$$\\epsfxsize=0.6\\textwidth\\epsffile{rad_power.eps}$$\n\\caption[Diagrams for weak annihilation and soft-gluon emission from a quark loop.]{\\small (a) Weak annihilation diagram where photon emission from the $B$ meson spectator quark is power-suppressed. The crosses denote possible photon emission vertices for $Q_{5,6}$ only. (b) soft-gluon emission from a quark loop, where there is also a second diagram in which the gluon is picked up by the $B$ meson.}\n\\label{rad_fig1}\n\\end{figure}\n\nPreliminary results for the $\\mathcal{O}(\\alpha_s)$ corrections to WA in $B\\to\\rho \\gamma$ were presented in Ref.~\\cite{chamonix}.   In QCDF,  the $a_{7L}^{U,{\\rm ann}}$  are expressed in terms of the hadronic quantities\n\\begin{equation}\nb^V = \\frac{2\\pi^2}{T_1^{B\\to V}(0)} \\,\\frac{f_B m_V f_V}{m_B m_b\n  \\lambda_B}\\,, \\qquad\nd^V_{v} = -\\frac{4\\pi^2}{T_1^{B\\to V}(0)} \\,\\frac{f_B f_V^\\perp}{m_B\n  m_b} \\,\\int_0^1 dv\\,\\frac{\\phi_{2;V}^\\perp(v)}{v}\n\\end{equation}\nand $d^V_{\\bar v}$, obtained by replacing $1\/v\\to 1\/\\bar v$ in the integrand; $\\phi_{2;V}^\\perp$ is the twist-2 DA of a transversely polarised vector meson, (\\ref{das_eq19}). Numerically, one finds, for instance for the $\\rho$, $b^\\rho = 0.22$ and $d^\\rho = -0.59$, at the scale $\\mu = 4.2\\,$GeV. As $T_1\\sim 1\/m_b^{3\/2}$ and $f_B\\sim m_b^{-1\/2}$ in the heavy-quark limit, these terms are $\\mathcal{O}(1\/m_b)$, but not numerically small because of the tree-enhancement factors of $\\pi^2$.\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l||c|c|c|c|c}\n\\mbox{WA} &  \nB^-\\to K^{*-} & \\bar B^0\\to K^{*0} & B\\to (\\rho,\\omega) & B_s\\to\\phi & \nB_s\\to \\bar  K^*\\\\\\hline\n\\mbox{induced by} & \\mbox{C (and P)} & \\mbox{P} & \\mbox{C and P} &\n\\mbox{P} & \\mbox{P}\\\\\n\\mbox{CKM} & \\lambda^2 \\mbox{~(and 1)} & 1 & 1 & 1 & 1\n\\end{array}\n$$\n\\caption[Parametric size of the weak annihilation contributions.]{\\small Parametric size of WA  contributions to $B\\to V\\gamma$. C denotes the charged-current operators $Q_{1,2}$, P  the penguin operators $Q_{3,\\dots,6}$; their Wilson coefficients are   small -- see Tab.~\\ref{rad_tab2}. CKM denotes the order in  the Wolfenstein parameter $\\lambda$ with respect to the dominant amplitude induced by $Q_7$.}\n\\label{rad_tab3}\n\\end{table}\n\nFor $\\omega$, $\\bar K^*$ and $\\phi$ we obtain\n\\begin{eqnarray}\n\\left. a_{7L}^{u,{\\rm ann}}(\\omega)\\right|_{\\rm QCDF}\n & = & Q_d b^\\omega (a_1 + 2 (a_3+a_5)\n+ a_4) + Q_d (d^\\omega_v + d^\\omega_{\\bar v}) a_6\\,,\\nonumber\\\\\n\\left. a_{7L}^{c,{\\rm ann}}(\\omega)\\right|_{\\rm QCDF} \n& = & Q_d b^\\omega (2 (a_3+a_5)\n+ a_4) + Q_d (d^\\omega_v + d^\\omega_{\\bar v}) a_6\\,,\\nonumber\\\\\n\\left. a_{7L}^{U,{\\rm ann}}(\\phi)\\right|_{\\rm QCDF} \n& = & Q_s b^\\phi (a_3+a_5) + \n         Q_s (d^\\phi_v + d^\\phi_{\\bar v}) a_6\\,,\\nonumber\\\\\n\\left. a_{7L}^{U,{\\rm ann}}(\\bar K^*)\\right|_{\\rm QCDF} \n& = & Q_s b^{\\bar K^*} a_4 + \n         Q_s (d^{\\bar K^*}_v Q_d\/Q_s + d^{\\bar K^*}_{\\bar v}) a_6\\,,\n\\label{15}\n\\end{eqnarray}\nwith $a_1 = C_1+C_2\/3$, $a_3 = C_3+C_4\/3$, $a_4 = C_4+C_3\/3$, $a_5 = C_5+C_6\/3$, $a_6 = C_6+C_5\/3$.\\footnote{Note that $a_1\\leftrightarrow a_2$ as compared to \\cite{Bosch:2002bw} as in our operator basis  (i.e.\\ the BBL basis) $Q_1$  and $Q_2$ are exchanged.} The expressions for $\\phi$ and $\\bar K^*$ are new; for $\\omega$, we do not agree with \\cite{Bosch:2004nd}.  Apart from for $\\rho$ and $\\omega$, all the WA coefficients are numerically small and do not change the branching ratio significantly; the terms in $a_6$, however, are relevant  for the isospin asymmetries. \n\nIn Tab.~\\ref{rad_tab3} we show the relative weights of these diagrams in terms of CKM factors and Wilson coefficients. The numerically largest contribution occurs for $B^\\pm\\to \\rho^\\pm\\gamma$: it comes with the large combination of  Wilson coefficients $C_2+C_1\/3=1.02$ and is not CKM suppressed. For $B^0\\to (\\rho^0,\\omega)\\gamma$ it comes with the factor $C_1+C_2\/3 = 0.17$\ninstead and an additional suppression factor $1\/2$ from the electric charge of the spectator quark ($d$ instead of $u$). For all other decays, WA is suppressed by small (penguin) Wilson coefficients. Apart from $B\\to(\\rho,\\omega)\\gamma$, WA is not relevant so much for the total values of $a_{7L}$, but rather for isospin breaking, which is set by photon emission from the spectator quark. WA is the only mechanism to contribute to isospin asymmetries at tree-level; see Ref.~\\cite{Kagan:2001zk} for $\\mathcal{O}(\\alpha_s)$ contributions.\n\nIn view of the large size of $a_{7L}^{u,{\\rm ann}}(\\rho)$ it is appropriate to have a look at further  corrections. The most  obvious ones are $\\mathcal{O}(\\alpha_s)$ corrections to the QCDF expressions, shown in Fig.~\\ref{rad_fig2}.\n\\begin{figure}\n$$\\epsfxsize=\\textwidth\\epsffile{rad_fig2.eps}$$\n\\caption[Example radiative corrections to \nweak annihilation.]{\\small Example radiative corrections to \nweak annihilation. The corrections to the $B$ vertex in (a) are known \n\\cite{bellnu,Descotes-Genon:2002mw} and those to the $V$ vertex in (b) are included in $f_V$. \nFor the non-factorisable corrections in (c) only preliminary results\nare available \\cite{chamonix}.}\n\\label{rad_fig2}\n\\end{figure}\nAs it turns out, the corrections to the $B$ vertex in Fig.~\\ref{rad_fig2}(a) are known: they also enter the decay $B\\to\\gamma \\ell\\nu$ and were calculated in Ref.~\\cite{bellnu,Descotes-Genon:2002mw}. Numerically, they are at the level of 10\\%.  Fig.~\\ref{rad_fig2}(b) shows the vertex corrections to the $V$ vertex, which are actually included in the decay constant $f_V$. For the non-factorisable corrections shown in Fig.~\\ref{rad_fig2}(c) preliminary results have been reported in Ref.~\\cite{chamonix} according to which these corrections are of a size similar to the $B$ vertex corrections. \n\nIn Ref.~\\cite{Kagan:2001zk} also another class of $1\/m_b$ corrections to $B\\to K^*\\gamma$ was calculated, namely $\\mathcal{O}(\\alpha_s)$  corrections to the isospin asymmetry in this decay. As these corrections break factorisation (require an infra-red cut-off in the  momentum  distribution of the valence quarks in the $K^*$ meson) and are numerically small, we do not include them in our analysis. \n\n\\subsection{Long-Distance Photon Emission}\nAnother class of corrections is suppressed by one power of $m_b$ with respect to the QCDF contributions and is  due to  long-distance photon emission from the soft $B$ spectator quark. A first calculation of this effect was attempted in Ref.~\\cite{WA} and was corrected and extended in Ref.~\\cite{Ball:2003fq}. The long-distance photon emission from a soft-quark line requires the inclusion of higher-twist terms in the expansion of the quark propagator in a photon background field, beyond the leading-twist (perturbative) contribution; a comprehensive discussion of this topic can be found in Ref.~\\cite{Ball:2002ps}. The quantity calculated in Ref.~\\cite{Ball:2003fq} is\n\\begin{eqnarray}\n\\lefteqn{\n\\bra{\\rho^-(p)\\gamma(q)} (\\bar d u)_{V-A} (\\bar u b)_{V-A}\\ket{B^-(p+q)} =}\\hspace*{3cm}\\nonumber\\\\\n& = & e\\,\\frac{m_\\rho f_\\rho}{m_B} \\eta^*_\\mu\n\\left\\{F_V \\epsilon^{\\mu\\nu\\rho\\sigma} e^*_\\nu p_{\\rho}\n  q_\\sigma - i F_A [e^{*\\mu} (p \\cdot q) - q^\\mu\n    (e^* \\cdot p)]\\right\\}\\nonumber\\\\\n& = & -e \\,\\frac{m_\\rho f_\\rho}{m_B} \\left\\{ \\frac{1}{2}\\,\n  F_V (S_L+S_R) + \\frac{1}{2}\\, F_A (S_L-S_R)\\right\\}\n  \\label{problem}\n\\end{eqnarray}\nin terms of the photon-helicity amplitudes $S_{L,R}$.\\footnote{Eq.~(\\ref{problem}) differs from the one given in \\cite{Ball:2003fq} by an overall sign, which is due to the different convention used in \\cite{Ball:2003fq} (and in \\cite{Ball:2002ps}) for the covariant derivative: $D_\\mu = \\partial_\\mu\n- i e Q_f A_\\mu$ instead of $D_\\mu = \\partial_\\mu + i e Q_f A_\\mu$ as in this analysis.}\nIn QCDF, $F_{A,V}$ are given by $Q_u f_B\/\\lambda_B$ and induce a term $Q_u a_2 b^\\rho$ in  $a_{7L}^{u,{\\rm    ann}}(\\rho^-)$. The long-distance photon contribution to $F_{V,A}$ was found to be \\cite{Ball:2003fq}\n\\begin{equation}\nF^{\\rm soft}_A = -0.07\\pm 0.02 \\equiv Q_u G_A\\,,\\qquad \nF^{\\rm soft}_V = -0.09\\pm\n0.02 \\equiv Q_u G_V\\,.\n\\label{Fsoft}\n\\end{equation}\nwith $G_A+G_V = -0.24\\pm 0.06$ and $G_V-G_A = -0.030\\pm 0.015$.\\footnote{Again, there is a relative sign with respect to the results in \\cite{Ball:2003fq}. This comes from the fact that the product $e F_{A,V}^{\\rm soft}$ is independent of the sign convention for $e$, and as we have changed the overall sign of (\\ref{problem}) with respect to \\cite{Ball:2003fq}, we also have to change the sign of $F_{A,V}^{\\rm soft}$. Stated differently: the relative sign between $F_{A,V}^{\\rm soft}$ and $F_{A,V}^{\\rm hard}$ in \\cite{Ball:2003fq} is wrong because of a mismatch in sign conventions for $e$ in the covariant derivative.}\nIn order to obtain concise expressions for $a_{7L(R)}^{U,{\\rm ann}}$, it proves convenient to define one more hadronic quantity:\n\\begin{equation}\ng^\\rho_{L,R} = \\frac{\\pi^2}{T_1^\\rho}\\,\\frac{m_\\rho f_\\rho}{m_b m_B}\\,\n(G_V\\pm G_A)\n\\end{equation}\nand correspondingly for other mesons. $g_L$ is $\\mathcal{O}(1\/m_b^2)$ as $G_V+G_A$ has the same power scaling in $m_b$ as $T_1$, i.e.\\ $\\sim m_b^{-3\/2}$, as one can read off from the explicit expressions in \\cite{WA}. The difference $G_V-G_A$, on the other hand, is a twist-3 effect due to three-particle light-cone DAs of the photon and is suppressed by one more power of $m_b$, i.e.\\ $g_R\\sim\n1\/m_b^3$. This quantity will enter the CP asymmetry. Our final expressions for $a_{7L(R)}^{U,{\\rm ann}}$ then read:\n\\begin{eqnarray}\na_{7L}^{U,{\\rm ann}}(V) & = & \\left. a_{7L}^{U,{\\rm\n    ann}}(V)\\right|_{\\rm QCDF} (b^V\\to b^V + g^V_L)\\,,\\nonumber\\\\\na_{7R}^{U,{\\rm ann}}(V) & = & \\left. a_{7L}^{U,{\\rm\n    ann}}(V)\\right|_{\\rm QCDF} (b^V\\to g^V_R, d^V\\to 0)\\,.\n    \\label{20A}\n\\end{eqnarray}\nNumerically, one has $g^{\\rho}_L\/b^\\rho = -0.3$, so these corrections, despite being suppressed by one more power in $1\/m_b$, are not small numerically and larger than the known $\\mathcal{O}(\\alpha_s)$\ncorrections to QCDF from $B\\to\\gamma\\ell\\nu$. Based on this, we feel justified in including these long-distance corrections in our analysis, while dropping the radiative ones of Figs.~\\ref{rad_fig2}(a) and (c). For central values of the input parameters we find the following numerical values for the various WA and long-distance photon contributions, including in particular those to which $Q_{1,2}$ contribute (with no Cabibbo suppression):\n\\begin{eqnarray}\na_{7L}^{c,{\\rm ann}}(K^{*0})  &=&  -0.013-0.001\\, {\\rm LD}\\,,\n\\qquad  a_{7L}^{c,{\\rm ann}}(K^{*-}) =  0.004+0.001\\, {\\rm\n  LD}\\,,\\nonumber\\\\\na_{7L}^{u,{\\rm ann}}(\\rho^0) &=&  -0.001-0.004\\, {\\rm LD}\\,,\n \\qquad ~~ a_{7L}^{u,{\\rm ann}}(\\rho^-) =  0.149-0.043\\, {\\rm\n  LD}\\,,\\nonumber\\\\\n a_{7L}^{u,{\\rm ann}}(\\omega)  &=&  -0.024+0.003\\, {\\rm LD}\\,.\n \\label{LDcont}\n\\end{eqnarray}\nThe contribution from the long-distance photon emission is labelled ``LD'' (LD$\\to 1$ at the end). \nThe unexpectedly small $a_{7L}^{u,{\\rm ann}}(\\rho^0)$ is due to a numerical cancellation between the charged-current and  penguin-operator contributions. Comparing these results with those from QCDF,\nEq.~(\\ref{10}), it is evident that WA is, as expected,  largely irrelevant for the branching ratios, except for $B^\\pm\\to \\rho^\\pm\\gamma$.\n\n\n\\subsection{Soft Quark Loops}\n$a_{7L(R)}^{U,{\\rm soft}}$ encodes soft-gluon emission from a (light or heavy quark) loop as shown in Fig.~\\ref{rad_fig1}(b). Soft-gluon emission from a charm loop was first considered in  Ref.~\\cite{KRSW97} as a potentially relevant long-distance contribution to the branching ratio of $B\\to K^*\\gamma$, however, the same diagram also contributes dominantly to the time-dependent CP asymmetry in $B^0\\to K^{*0}\\gamma$ \\cite{grin05}.  As for $a_{7R}^U$, the dominant contributions to $a_{7R}^c(K^*)$ were calculated in Ref.~\\cite{Ball:2006cv} and new to this analysis is their generalisation to the other vector mesons and the inclusion of contributions from light-quark loops. Motivation to include light quark loops stems from the fact that they are doubly CKM-suppressed for $b\\to s\\gamma$ transitions, but not for $b\\to d\\gamma$, for which they are on an equal footing as the heavy quark loops. The quark loop contributions are suppressed by one power of $m_b$ with respect to $a_{7L}^{U,{\\rm QCDF}}$, but they also induce a right-handed photon amplitude which is of the same order in $1\/m_b$ as $a_{7R}^{U,{\\rm QCDF}}$ (\\ref{qcdf_13}), and this amplitude induces the time-dependent CP asymmetry.  The asymmetry is expected to be very small in the SM and $\\propto m_D\/m_b$ due to the chiral suppression of the leading transition (\\ref{qcdf_13}), but could be drastically enhanced by new physics contributions -- thus constituting an excellent ``null test'' of the SM \\cite{Gershon:2006mt,Ball:2006cv}.  It was noticed in Refs.~\\cite{grin04,grin05}  that the chiral suppression is relaxed by emission of a gluon from the quark loop and contributes dominantly to the time-dependent CP asymmetry in $B^0\\to K^{*0}\\gamma$, which motivates the inclusion of these contributions. The task of the present analysis, however, is not so much to calculate these contributions to high accuracy, but to exclude the possibility of  {\\em  large} contributions to the CP asymmetry.  With this in mind, the theoretical uncertainties of the results are very generously estimated  --- which is somewhat unavoidable due to the current uncertainties of the relevant hadronic input parameters.\n\nPotentially the most important contribution to the soft-gluon emission  diagram in Fig.~\\ref{rad_fig1}(b) \ncomes from the charged-current operator $Q_2^U$ with the  large Wilson coefficient $C_2\\sim 1$; it vanishes for $Q_1^U$  by gauge invariance. In addition, the penguin operators $Q_{3,4,6}$ give a non-zero contribution. Details of the derivation of $a_7^{U,{\\rm soft}}$ can be found in Ref.~\\cite{Ball:2006eu} in which the following expression is obtained:\n\\begin{eqnarray}\na_{7L(R)}^{U,{\\rm soft}}(V) & = &  \\frac{\\pi^2}{m_b T_1^{B\\to V}(0)} \\left\\{ Q_U C_2 (l_U\\pm \\tilde l_U)(V) + Q_D C_3 (l_D\\pm \\tilde l_D)(V)\\right.\\nonumber\\\\\n&&\\left. + \\sum_q Q_q (C_4-C_6) (l_q\\pm \\tilde l_q)(V)\\right\\}.\n\\label{24}\n\\end{eqnarray}\nHere the sum over $q$ runs over all five active quarks $u,d,s,c,b$. The contribution from $Q_5$ is proportional to $m_D$ and hence helicity suppressed and neglected.  Assuming $\\rm SU(3)_F$-flavour symmetry for the light quark loops, one has $l_u=l_d=l_s$, and ditto for $\\tilde l_{u,d,s}$, which causes a cancellation of these contributions in the last term of Eq.~(\\ref{24}). ${\\rm SU(3)_F}$-breaking effects are estimated to be around 10\\% \\cite{Ball:2006eu}.  The parameters  $l_c(K^*)$ and $\\tilde{l}_c(K^*)$ were first calculated from three-point sum rules in Ref.~\\cite{KRSW97} and were re-calculated in the more suitable method of LCSR via a local OPE in Ref.~\\cite{Ball:2006cv}. The analysis therein as been updated and extended to $l_b,\\, \\tilde{l}_b$ and the other particles $\\rho,\\,\\omega ,\\,\\bar{K}^*,\\,\\phi$ for the present analysis \\cite{Ball:2006eu}. The results for $l_c$ and $\\tilde l_c$ are given in the upper table of Tab.~\\ref{rad_tab5}. Those for $l_b$ and $\\tilde l_b$ are obtained as\n\\begin{equation}\nl_b = \\frac{m_c^2}{m_b^2}\\, l_c\\,,\\qquad \\tilde l_b = \\frac{m_c^2}{m_b^2}\\, \\tilde l_c\\,.\n\\end{equation}\nFor light-quark loops the photon is almost at threshold and the local OPE does not apply. In Ref.~\\cite{Ball:2006eu} a method was developed for calculating these contributions via LCSRs.  Similar to the method of Ref.~\\cite{Khodjamirian:2000mi}  used for the calculation of soft-gluon contributions to $B\\to\\pi\\pi$, a dispersion relation approach is used to connect the off-shell matrix element to the physical regime $q^2=0$.  The results are presented in the lower table of Tab.~\\ref{rad_tab5}.\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l ||r|r|r|r}\n& \\multicolumn{1}{c|}{l_c} & \\multicolumn{1}{c|}{\\tilde l_c} \n& \\multicolumn{1}{c|}{l_c-\\tilde l_c} & \\multicolumn{1}{c}{l_c +\n    \\tilde l_c} \\\\\n \\hline\n B \\to K^*   & -355 \\pm 280 &  -596 \\pm 520 & 242 \\pm 370 & -952 \\pm 800  \\\\\n B \\to (\\rho,\\omega)   & -382 \\pm 300 & -502\\pm  430 & 120 \\pm 390 & \n-884\\pm 660  \\\\\n B_s \\to \\bar K^*   &  -347\\pm 260 & -342\\pm 400 & -4\\pm 300 & -689\\pm 600 \\\\\n B_s \\to \\phi   & -312 \\pm  240 & -618 \\pm 500 & 306 \\pm 320 & -930 \\pm\n750 \n\\end{array}\n$$\\\\\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l ||r|r|r|r}\n& \\multicolumn{1}{c|}{l_u} & \\multicolumn{1}{c|}{\\tilde l_u} &   \n\\multicolumn{1}{c|}{l_u-\\tilde l_u} & \\multicolumn{1}{c}{l_u + \\tilde l_u} \\\\\n \\hline\n B \\to K^*   & 536 \\pm 70\\% &  635 \\pm 70\\% & -99 \\pm 300 & 1172 \\pm 70 \\% \\\\\n B \\to (\\rho,\\omega)   & 827 \\pm 70\\% & 828\\pm 70\\% & -1\\pm 300&\n 1655\\pm 70\\%  \\\\\n B_s \\to \\bar K^*   &  454\\pm 70\\% & 572\\pm 70\\% & -118\\pm 300 & 1025\\pm\n70\\% \\\\\n B_s \\to \\phi   & 156\\pm 70\\% & 737\\pm 70\\% & -581\\pm 300 & 893\\pm 70\\% \\\\\n\\end{array}\n$$\n\\caption[Soft-gluon contributions from $c$-quark and $u$-quark  loops in units KeV. ]{\\small Soft-gluon contributions from $c$-quark (upper table) and $u$-quark (lower table) loops in units KeV. The quantities $l_{c,u}$ and $\\tilde l_{c,u}$ are defined in Ref.~\\cite{Ball:2006eu}. We assume equal parameters for $\\rho$ and $\\omega$. $l_b$ is obtained as $l_b = l_c m_c^2\/m_b^2$ and correspondingly for $\\tilde l_b$. The uncertainty for $l_u-\\tilde l_u$  is given in absolute numbers because of cancellations. In the $\\rm SU(3)_F$-flavour limit assumed in this calculation one has  $l_u = l_d = l_s  \\equiv l_q$}\n\\label{rad_tab5}\n\\end{table}\n\n\\section{Phenomenological Results}\nIn this section we combine the different contributions to the factorisation coefficients $a_{7L(R)}^U$ and give results for the observables, namely the branching ratios, the isospin asymmetries and the time-dependent CP asymmetries. \n\\subsection{Branching Ratios}\\label{rad_brs}\nThe (non-CP-averaged) branching ratio of the $b\\to D\\gamma$ decay \n$\\bar B\\to V\\gamma$ is given by\n\\begin{eqnarray}\n{\\cal B}(\\bar B\\to V\\gamma) & = & \\frac{\\tau_B}{c_V^2}\\,\n\\frac{G_F^2\\alpha_{\\rm QED} m_B^3  m_b^2}{32 \\pi^4} \\left(1-\\frac{m_V^2}{m_B^2}\\right)^3 \\left[T_1^{B\\to V}(0)\\right]^2\\nonumber\\\\\n&&\\times \\left\\{ \\left| \\sum_{U=u,c} \\lambda_U^{(D)} a_{7L}^U(V)\\right|^2 +  \\left| \\sum_{U=u,c} \\lambda_U^{(D)}\n  a_{7R}^U(V)\\right|^2\\right\\}\n  \\label{BR}\n\\end{eqnarray}\nwith the isospin factors $c_{\\rho^\\pm,K^*,\\phi}=1$  and $c_{\\rho^0,\\omega} = \\sqrt{2}$. The branching ratio for the CP-conjugated channel $B\\to \\bar V\\gamma$ ($\\bar b\\to \\bar D\\gamma$ at parton level) is obtained by replacing $\\lambda_U^{(D)}\\to (\\lambda_U^{(D)})^*$.  With the input parameters from Tab.~\\ref{rad_tab8} and the lifetimes given in Tab.~\\ref{rad_tab9} we find the following CP-averaged branching ratios  for $B\\to K^*\\gamma$, making explicit various sources of uncertainty:\n\\begin{eqnarray}\n\\overline{\\cal B}(B^- \\to K^{*-}\\gamma) & = & (53.3\\pm \\overbrace{13.5}^{T_1}\n\\pm \\overbrace{4.8}^{\\mu}\n\\pm \\overbrace{1.8}^{V_{cb}}\\pm \\overbrace{1.9}^{l_{u,c}} \\pm \\overbrace{1.3}^{\\mbox{other}})\\times\n10^{-6}\\nonumber\\\\\n& =& (53.3\\pm \\underbrace{13.5}_{T_1}\\pm 5.8)\\times 10^{-6}\\,,\n\\nonumber\\\\\n\\overline{\\cal B}(\\bar B^0 \\to K^{*0}\\gamma) & = & (54.2\\pm \\overbrace{13.2}^{T_1}\n\\pm \\overbrace{6.0}^{\\mu}\n\\pm \\overbrace{1.8}^{V_{cb}}\\pm \\overbrace{1.8}^{l_{u,c}} \\pm \\overbrace{1.4}^{\\mbox{other}})\\times\n10^{-6}\\nonumber\\\\\n& = & (54.2\\pm \\underbrace{13.2}_{T_1}\\pm 6.7)\\times 10^{-6}\\,.\n\\label{50}\n\\end{eqnarray}\nWe have added all individual uncertainties in quadrature, except for that induced by the form factor. The uncertainty in $\\mu$ is that induced by the renormalisation-scale dependence, with  $\\mu= m_b(m_b)\\pm 1\\,$GeV.  ``Other'' sources of uncertainty include the dependence on the parameters in Tab.~\\ref{rad_tab7}, on the size of LD WA contributions and the replacement of NLO by LO Wilson coefficients.  The above results agree, within errors, with the experimental ones given in Tab.~\\ref{rad_tab1}, within the large theoretical uncertainty induced by the form factor.\n\nAs the uncertainties of all form factors in Tab.~\\ref{rad_tab8} are of roughly the same size, one might conclude that the predictions for all branching ratios will carry uncertainties similar to those in\n(\\ref{50}). This is, however, not the case: the accuracy of the theoretical predictions can be improved by making use of the fact that the {\\em ratio} of form factors is known much better than the individual form factors themselves. The reason is that the values given in Tab.~\\ref{rad_tab8}, which were calculated using the same method, LCSRs, and with a common set of input parameters,  include common systematic uncertainties (dependence on $f_B$, $m_b$ etc.) which partially cancel in the ratio. In Ref.~\\cite{Ball:2006nr}  the ratio of the $K^*$ and $\\rho$ form factors was found to be\n\\begin{equation}\n\\xi_\\rho \\equiv \\frac{T_1^{B\\to K^*}(0)}{T_1^{B\\to \\rho}(0)} = 1.17\\pm\n0.09\\,.\n\\label{xirho}\n\\end{equation}\nThe uncertainty is by a factor 2 smaller than if we had calculated $\\xi_\\rho$\nfrom the entries in Tab.~\\ref{rad_tab8}; analogously for $\\omega$ one finds\n\\begin{equation}\n\\xi_\\omega \\equiv \\frac{T_1^{B\\to K^*}(0)}{T_1^{B\\to \\omega}(0)} = 1.30\\pm\n0.10\\,.\n\\label{xiomega}\n\\end{equation}\nThe difference between $\\xi_\\rho$ and $\\xi_\\omega$ is mainly due to the difference between $f_\\omega^\\perp$ and $f_\\rho^\\perp$.  For the $B_s$ form factors, we also need the ratio of decay constants $f_{B_s}\/f_{B_d}$. The status of $f_B$ from Lattice QCD was reviewed in  Ref.~\\cite{Onogi}; the present state-of-the-art calculations are unquenched with $N_f=2+1$ active flavours \\cite{unquenchedfB},  whose average is $f_{B_s}\/f_{B_d}=1.23\\pm 0.07$. Again, this ratio is fully consistent with that quoted in Tab.~\\ref{rad_tab8}, but has a smaller uncertainty. One then finds the following ratios for $B_s$ form factors:\n\\begin{equation}\n\\xi_\\phi \\equiv \\frac{T_1^{B\\to K^*}(0)}{T_1^{B_s\\to\\phi}(0)} = 1.01\\pm 0.13\n\\,,\\qquad\n\\xi_{\\bar K^*} \\equiv \\frac{T_1^{B\\to K^*}(0)}{T_1^{B_s\\to\\bar K^*}(0)} \n= 1.09\\pm 0.09\\,.\n\\label{xis}\n\\end{equation}\nThe uncertainty of $\\xi_{\\bar K^*}$ is smaller than that of $\\xi_\\phi$ because the input parameters for $K^*$ and $\\bar K^*$ are the same (except for G-odd parameters like $a_1^\\perp$) and cancel in the ratio; the uncertainty is dominated by that of $f_{B_s}\/f_{B_d}$. To benefit from this reduced theoretical uncertainty in predicting branching ratios, one has to calculate ratios of branching ratios, which mainly depend on $\\xi_V$ and only mildly on  $T_1$  itself: in addition to the  overall normalisation, $T_1$ also enters hard-spectator interactions and power-suppressed corrections, whose size is set by hadronic quantities $\\propto 1\/T_1$. As these corrections are subleading (in  $\\alpha_s$ or $1\/m_b$),  however, a small shift in $T_1$ has only very minor impact on the  branching ratios. The absolute scale for the branching ratios is set by the CP- and isospin-averaged  branching ratio with the smallest experimental uncertainty, i.e.\\ $B\\to K^*\\gamma$; from  Tab.~\\ref{rad_tab1}, one finds:\n\\begin{equation}\n\\overline{\\cal B}(B\\to K^*\\gamma) = \\frac{1}{2}\\left\\{ \\overline{\\cal\n  B}(B^\\pm \\to K^{*\\pm}\\gamma) + \\frac{\\tau_{B^\\pm}}{\\tau_{B^0}}\\, \n  \\overline{\\cal B}(\\bar B^0 \\to K^{*0}\\gamma)\\right\\} = (41.6\\pm\n  1.7)\\times 10^{-6}\\,.\n  \\label{x}\n\\end{equation}\nThat is, we obtain a theoretical prediction for $\\overline{\\cal B}(B\\to\nV\\gamma)$ as \n\\begin{equation}\n\\left.\\overline{\\cal B}(B\\to V\\gamma)\\right|_{\\rm th}= \\left[ \\frac{\\overline{\\cal B}(B\\to V\\gamma)}{\n\\overline{\\cal B}(B\\to K^*\\gamma)}\\right]_{{\\rm th}} \\, \\left.\n\\overline{\\cal B}(B\\to K^*\\gamma)\\right|_{\\rm exp}\\,,\n\\end{equation}\nwhere $\\left[\\dots\\right]_{\\rm th}$ depends mainly on $\\xi_V$ and only in subleading terms on the individual form factors $T_1^{B\\to K^*}$ and $T_1^{B\\to V}$. It is obvious that, except for these subleading terms, this procedure is equivalent to extracting an {\\em effective form factor} $\\left.T_1^{B\\to  K^*}(0)\\right|_{\\rm eff}$ from $B\\to K^*\\gamma$ and using $\\left.T_1^{B\\to V}(0)\\right|_{\\rm eff} = \\left.T_1^{B\\to   K^*}(0)\\right|_{\\rm  eff}\/\\xi_{V}$ for calculating the branching ratios for $B\\to V\\gamma$. From (\\ref{x}) we find\n\\begin{equation}\n\\left. T_1^{B\\to K^*}(0)\\right|_{\\rm eff} = 0.267\\pm \\overbrace{0.017}^{{\\rm th}} \\pm\n\\overbrace{0.006}^{{\\rm exp}} = 0.267\\pm 0.018\\,,\n\\end{equation}\nwhere the theoretical uncertainty  follows from the second uncertainty given in (\\ref{50}). Eqs.~(\\ref{xirho}), (\\ref{xiomega}) and (\\ref{xis}) then yield\n\\begin{eqnarray}\n\\left. T_1^{B\\to \\rho}(0)\\right|_{\\rm eff} &=& 0.228 \\pm 0.023\\,, \\qquad\n\\left. T_1^{B\\to \\omega}(0)\\right|_{\\rm eff} = 0.205 \\pm\n0.021\\,,\\nonumber\\\\\n\\left. T_1^{B_s\\to \\bar K^*}(0)\\right|_{\\rm eff} &=& 0.245 \\pm\n0.024\\,, \n\\qquad\n\\left. T_1^{B_s\\to \\phi}(0)\\right|_{\\rm eff} = 0.260 \\pm\n0.036\\,.\n\\label{56}\n\\end{eqnarray}\nNote that all effective form factors agree, within errors, with the results from LCSRs given in Tab.~\\ref{rad_tab8}, which confirms the results obtained from this method; the crucial point, however, is that the uncertainties are reduced by a factor of 2 (except for $T_1^{B_s\\to \\phi}$). We would like to stress that the motivation for this procedure is to achieve a reduction of the theoretical uncertainty of the\npredicted branching fractions in $B\\to (\\rho,\\omega)\\gamma$ and $B_s$ decays.  The effective form factors do {\\em not} constitute a new and independent theoretical determination, but are derived from the experimental results for $B\\to K^*\\gamma$ under the following assumptions:\n\\begin{itemize}\n\\item there is no new physics in $B\\to K^*\\gamma$;\\footnote{Which is motivated by the\n  results from inclusive $B\\to X_s \\gamma$ decays \\cite{misiak}.}\n\\item QCDF is valid with no systematic uncertainties;\n\\item LCSRs can reliably predict the ratio of form factors at zero\n  momentum transfer.\n\\end{itemize}\n From (\\ref{BR}) and (\\ref{56}), we then predict the following CP-averaged branching ratios:\n\\begin{eqnarray}\n\\overline{\\cal B}(B^-\\to \\rho^-\\gamma) & = & (1.16\\pm \\overbrace{0.22}^{T_1}\\pm \\overbrace{0.13}^{\\rm Other})\\times 10^{-6}\\,, \\nonumber\\\\\n\\overline{\\cal B}(B^0\\to \\rho^0\\gamma) & = & (0.55\\pm 0.11\\pm \n0.07)\\times 10^{-6}\\,, \\nonumber\\\\\n\\overline{\\cal B}(B^0\\to \\omega\\gamma) & = & (0.44\\pm 0.09 \\\n\\pm 0.05)\\times 10^{-6}\\,,\\nonumber\\\\\n\\overline{\\cal B}(B_s\\to \\bar K^*\\gamma) & = & (1.26\\pm 0.25\\pm \n0.18)\\times 10^{-6}\\,, \\nonumber\\\\\n\\overline{\\cal B}(B_s\\to \\phi\\gamma) & = & (39.4\\pm 10.7 \\ \\pm 5.3)\\times 10^{-6}\\,,\n\\label{57}\n\\end{eqnarray}\nwhere the first uncertainty is induced by the effective form factors and  the second includes the variation of all inputs from Tab.~\\ref{rad_tab8}  except for the angle $\\gamma$ of the UT, which is fixed at\n$\\gamma=53^\\circ$.\\footnote{The value of the UT angle\n$\\gamma$ in Tab.~\\ref{rad_tab8} comes from Belle's \nDalitz-plot analysis of the CP asymmetry in $B^-\\to (K_S^0 \\pi^+\\pi^-)\nK^-$, with $K_S^0 \\pi^+\\pi^-$ \\cite{Bellegamma} being a three-body final state common\nto both $D^0$ and $\\bar D^0$. Other determinations all come with theoretical uncertainties and\/or possible contamination by unresolved new physics, so we take this result as a reference point.} The total uncertainty in each channel is $\\sim 20\\%$, except for $B_s\\to \\phi\\gamma$, where it is 30\\%. The results for $\\rho$ and $\\omega$ agree very well with those of {\\sc  BaBar}, Tab.~\\ref{rad_tab1}, but less so with the Belle results, although present experimental and theoretical uncertainties preclude a firm conclusion. Our prediction for $B_s\\to \\phi\\gamma$ is well below the current experimental bound $120\\times 10^{-6}$ \\cite{Yao:2006px}. A branching ratio of the size given in (\\ref{57}) implies that $\\mathcal{O}(10^3)$ $B_s\\to\\phi\\gamma$  events will be seen within the first few years of the LHC.\n\nIn Tab.~\\ref{rad_tab6} we detail the contributions of individual terms to the branching ratios. In all cases ${\\cal B}$ is dominated by the QCDF contribution, with WA most relevant for $B^-\\to \\rho^-\\gamma$. This is expected as WA enters with the large Wilson coefficient $C_2\\sim 1$. The effect is extenuated by long-distance (LD) photon emission, which itself is compensated by soft-gluon emission. The other channels follow a similar pattern, although the size of the effects is smaller.\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l||c|c|c|c}\n\\mathcal{B}\\times 10^{6}& \\mbox{QCDF} & \\mbox{+ WA (no LD)} & \\mbox{+ WA (inc.\\ LD)}& \n\\mbox{+ soft gluons}\\\\\\hline\nB^-\\to \\rho^-\\gamma & 1.05&1.17 & 1.11&1.16\n\\\\\nB^0\\to \\rho^0\\gamma & 0.49&0.53& 0.53&0.55\n\\\\\nB^0\\to \\omega\\gamma & 0.40& 0.42&0.42 &0.44\n\\\\\nB^-\\to K^{*-}\\gamma &39.7&38.4& 38.3&39.4\n\\\\\nB^0\\to K^{*0}\\gamma & 37.1&39.7 &39.9 &41.0\n\\\\\nB_s^0\\to \\bar K^{*0}\\gamma & 1.12& 1.22& 1.23&1.26\n\\\\\nB_s^0\\to \\phi\\gamma &34.6 & 38.2& 38.3&39.4\n\\end{array}\n$$\n\\caption[Contributions to CP-averaged branching ratios.]{\\small Contributions to CP-averaged branching ratios, using effective form factors and  central values of all other input parameters, Tab.~\\ref{rad_tab8} (in particular $\\gamma=53^\\circ$). LD stands for long-distance photon-emission contributions. Each column  labelled ``+X'' includes the contributions listed in the previous  column plus the contribution induced by X. The entries in the last  column are our total central values.}\n  \\label{rad_tab6}\n\\end{table}\n\\begin{figure}[h]\n$$ \\epsfxsize=0.48\\textwidth\\epsffile{fig7a.eps}\\quad\\epsfxsize=0.48\\textwidth\\epsffile{fig7b.eps}$$\n$$ \\epsfxsize=0.48\\textwidth\\epsffile{fig7c.eps}$$\n  \\caption[CP-averaged branching ratios of $B\\to(\\rho,\\omega)\\gamma$ as function   of angle $\\gamma$.]{\\small CP-averaged branching ratios of $B\\to(\\rho,\\omega)\\gamma$ as function   of UT angle $\\gamma$, using the effective form factors and central values of   other input parameters. (a): $B^\\pm \\to \\rho^\\pm\\gamma$, (b):   $B^0\\to \\rho^0\\gamma$, (c): $B^0\\to\\omega\\gamma$. The boxes indicate the 1$\\sigma$ experimental results from {\\sc BaBar} \\cite{babar_rad}, Tab.~\\ref{rad_tab1}.  Note that the resulting value of $\\gamma$   from the average of all three channels is $\\gamma =  (61.0^{+13.5}_{-16.0}({\\rm exp})^{+8.9}_{-9.2})^\\circ$ -- see Section~\\ref{ckmextract}.} \n  \\label{rad_fig3}\n\\end{figure}\n\nWe would like to close this section by making explicit the  dependence of the three $B\\to (\\rho,\\omega)\\gamma$ branching ratios on $\\gamma$.  In Fig.~\\ref{rad_fig3}  we plot these branching ratios, for central values of the input parameters, as functions of $\\gamma$. We also indicate the present experimental results from {\\sc BaBar} \\cite{babar_rad}, Tab.~\\ref{rad_tab1}, within their 1$\\sigma$ uncertainty. \n\n\\subsection{Isospin Asymmetries}\\label{isosec}\nThe asymmetries $A_I(\\rho)$, $A_I(K^*)$, and   $A(\\rho,\\omega) $ are given by\n\\begin{eqnarray}\nA(\\rho,\\omega) & = &  \\frac{\\overline{\\Gamma}(B^0\\to \\omega   \\gamma)}{\\overline{\\Gamma}(B^0\\to \\rho^0   \\gamma)}-1\\,, \n  \\label{rad_arw} \\\\\nA_{I}(\\rho) & = & \\frac{2\\overline{\\Gamma}(\\bar B^0\\to    \\rho^0 \\gamma)}{\\overline{\\Gamma}(\\bar B^\\pm\\to    \\rho^\\pm \\gamma)} - 1\\,,\n  \\label{rad_air}\\\\\nA_{I}(K^*) & = & \\frac{\\overline{\\Gamma}(\\bar B^0\\to    K^{*0} \\gamma) - \\overline{\\Gamma}(B^\\pm\\to \n  K^{*\\pm} \\gamma)}{\\overline{\\Gamma}(\\bar B^0\\to    K^{*0} \\gamma) + \\overline{\\Gamma}(B^\\pm\\to \n  K^{*\\pm} \\gamma)}\\,,\n  \\label{rad_aik}\n\\end{eqnarray}\nwhere the partial decay rates are CP-averaged. Let us first discuss $A(\\rho,\\omega)$ and $A_{I}(\\rho)$ which are  relevant for the experimental determination of $\\overline{\\cal  B}(B\\to(\\rho,\\omega)\\gamma)$,  which in turn is used for the determination of $|V_{td}\/V_{ts}|$ (or $\\gamma$), see Section~\\ref{ckmextract}. The present experimental statistics for $b\\to d\\gamma$ transitions is rather low, so the experimental value of $\\overline{\\cal B}(B\\to(\\rho,\\omega)\\gamma)$ is obtained under the explicit assumption of perfect symmetry, i.e.\\ $\\overline{\\Gamma}(B^\\pm\\to \\rho^\\pm \\gamma) = 2 \\overline{\\Gamma}(B^0\\to \\rho^0 \\gamma) = 2 \\overline{\\Gamma}(B^0\\to \\omega \\gamma)$. In reality, the symmetry between $\\rho^0$ and $\\omega$ is broken by different values of the form factors, and isospin symmetry between neutral and charged $\\rho$ is broken by photon emission from the spectator quark, the dominant mechanism of which is WA. From the formulas for individual branching ratios, Eq.~(\\ref{BR}),  and the various contributions to the factorisation coefficients $a_{7L(R)}^U$, we find\n\\begin{equation}\nA(\\rho,\\omega) = -0.20\\pm \\overbrace{0.09}^{{\\rm th.}}\\,.\n\\label{eq:AIrw}\n\\end{equation}\nThe uncertainty is dominated by that of the  form factor ratio  $T_1^{B\\to\\omega}(0)\/T_1^{B\\to\\rho}(0)=0.90\\pm 0.05$.\\footnote{Note that this result is dominated by the ratio of  decay constants given in Tab.~\\ref{rad_tab8} and discussed in Ref.~\\cite{Ball:2006eu}. The experimental results entering these averages have a  large spread which may cast a shadow of doubt on the averaged final  branching ratios for $(\\rho^0,\\omega)\\to e^+ e^-$ quoted by PDG  \\cite{Yao:2006px}.} The dependence on all other input parameters is marginal. The result (\\ref{eq:AIrw}) is not compatible with the strict isospin limit $A(\\rho,\\omega) =0$.  $A_{I}(\\rho)$, on the other hand, is very sensitive to $\\gamma$, whereas the form factors drop out. It is driven by the WA contribution and, in the QCDF framework, vanishes if WA is set to zero. In Fig.~\\ref{rad_fig4}(a) we plot $A_{I}(\\rho)$ as  function of $\\gamma$, including the theoretical uncertainties. \n\\begin{table}\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{c||c|c|c|c}\n\\gamma & 40^\\circ & 50^\\circ & 60^\\circ & 70^\\circ\\\\\\hline\nA_I(\\rho) & -(5.3\\pm 6.9)\\% & (0.4\\pm 5.3)\\% & (5.7\\pm 3.9)\\% &\n(10.5\\pm 2.7)\\%\n\\end{array}\n$$\n\\caption[Isospin asymmetry $A_I(\\rho)$ for different values  of $\\gamma$.]{\\small Isospin asymmetry $A_I(\\rho)$  for different values  of $\\gamma$.}\n  \\label{rad_tab7}\n\\end{table}\nAs suggested by the findings of Ref.~\\cite{chamonix}, these results are  not expected to change considerably upon inclusion of  the non-factorisable radiative corrections of  Fig.~\\ref{rad_fig2}(c). In Tab.~\\ref{rad_tab7}, we give the corresponding  results for several values of $\\gamma$, together with the theoretical  uncertainty. Our result agrees very well with that  obtained by the {\\sc BaBar} collaboration: $A_I(\\rho)_{\\rm BaBar} =  0.56\\pm 0.66$ \\cite{babar_rad}.\n\n$A_I(K^*)$ was first discussed in Ref.~\\cite{Kagan:2001zk}, including power-suppressed $\\mathcal{O}(\\alpha_s)$ corrections which unfortunately violate QCDF, i.e.\\ are divergent. It is for this reason that we decide to drop these corrections and include only leading-order terms in $\\alpha_s$. We then find\n\\begin{eqnarray}\nA_I(K^*) &=& (5.4\\pm \\overbrace{1.0}^{\\mu} \\pm \\overbrace{0.6}^{{\\rm NLO}\\leftrightarrow{\\rm\n  LO}} \\pm \\overbrace{0.6}^{f_B} \\pm \\overbrace{0.6}^{{\\rm other}})\\%\\nonumber\\\\\n&=& (5.4\\pm 1.4)\\%\\,,\n\\label{62}\n\\end{eqnarray}\nwhere ${\\rm NLO}\\leftrightarrow{\\rm LO}$ denotes the uncertainty induced by switching from NLO to LO accuracy in the Wilson coefficients and ``other'' summarises all other sources of theoretical uncertainty. \nAs can be inferred from the entries in Tab.~\\ref{rad_tab1}, the  present experimental result is $A_I(K^*)_{\\rm exp}=(3.2\\pm 4.1)\\%$.  In Ref.~\\cite{Kagan:2001zk} it was pointed out that $A_I(K^*)$\nis very sensitive to the values of the Wilson coefficients $C_{5,6}^{\\rm BBL}$ in the combination $a_6\\equiv C_{5}^{\\rm BBL}+C_6^{\\rm   BBL}\/3$. In the SM, varying the renormalisation scale as $\\mu=m_b(m_b)\\pm 1\\,{\\rm GeV}$ and switching between LO and NLO accuracy for the Wilson coefficients, one has $a_6= -0.039\\pm 0.008$, which actually induces the bulk of the uncertainty in Eq.~(\\ref{62}). In Fig.~\\ref{rad_fig4}(b) we plot $A_I(K^*)$ as function of $a_6\/a_6^{\\rm SM}$, with $a_6^{\\rm SM}=-0.039$.\n\\begin{figure}[tb]\n$$\\epsfxsize=0.45\\textwidth\\epsffile{fig8a.eps}\\quad\n  \\epsfxsize=0.45\\textwidth\\epsffile{fig8b.eps}$$\n\\caption[$A_I(\\rho)$  as function of the angle $\\gamma$ and $A_I(K^*)$ as function of $r\\equiv a_6\/a_6^{\\rm     SM}$.]{\\small Left panel: isospin asymmetry $A_I(\\rho)$, Eq.~(\\ref{rad_air}),  as function of the UT angle $\\gamma$. Solid line: central values of input parameters; dashed lines: theoretical uncertainty. Right panel: $A_I(K^*)$, Eq.~(\\ref{rad_aik}), in percent,  as function of the ratio $r\\equiv a_6\/a_6^{\\rm     SM}$ of the combination of penguin Wilson coefficients   $a_6\\equiv C_6+C_5\/3$. Solid line: central value of input parameters,  dashed lines: theoretical uncertainty. The box indicates the  present experimental uncertainty and the straight black lines the  theory uncertainty in $r$.}\n    \\label{rad_fig4}\n\\end{figure}\nThe figure clearly indicates that, although there is presently no\ndiscrepancy between theoretical prediction and experimental result,\na reduction of the experimental uncertainty\nof $A_I(K^*)$ may well reveal some footprints of new physics\nin this observable.\n\n\n\\subsection{CP Asymmetries}\\label{cpsec}\nThe time-dependent CP asymmetry in $\\bar B^0\\to V^0\\gamma$ is given analogously to Eq.~(\\ref{basics_eq13}) as\n\\begin{equation}\nA_{CP}(t)  = S(V\\gamma) \\sin(\\Delta m_D\\, t ) - C(V\\gamma) \n\\cos(\\Delta m_D\\, t)\\,.\n\\label{rad_cpa}\n\\end{equation}\nThe above equation is technically only valid for $\\Delta \\Gamma =0$ and while this is a good assumption for $B^0_d$ decays, it is not so for $B_s^0$ decays. Although Eq.~(\\ref{rad_cpa})  can easily be adapted to non-zero $\\Delta \\Gamma_s$ we refrain from doing so: the whole point in calculating the CP asymmetry is not so much to give precise predictions for $S$ and $C$, but rather to exclude the possibility of large corrections to the naive expectation $S\\sim m_D\/m_b$. With this is mind, small corrections from a non-zero $\\Delta\\Gamma_s$ are irrelevant. The time-dependent CP asymmetries are given in terms of the left- and right-handed photon amplitudes (\\ref{qcdf_8}) by\n\\begin{equation}\nS(V\\gamma) \n =  \\frac{2 \\,{\\rm Im}\\,\\left(\\frac{q}{p}({\\cal A}_L^* \\bar{\\cal A}_L + \n                                       {\\cal A}_R^* \\bar{\\cal A}_R)\\right)}{\n        |{\\cal A}_L|^2 + |{\\cal A}_R|^2 + |\\bar{\\cal A}_L|^2 + |\\bar{\\cal\n                                       A}_R|^2}\\,,\n\\quad\nC(V\\gamma)  =   \\frac{|{\\cal A}_L|^2 + |{\\cal A}_R|^2 - |\\bar{\\cal A}_L|^2 - \n               |\\bar{\\cal A}_R|^2}{\n        |{\\cal A}_L|^2 + |{\\cal A}_R|^2 + |\\bar{\\cal A}_L|^2 + |\\bar{\\cal\n                                       A}_R|^2}\\,.\n\\label{54}\n\\end{equation}\nWith ${\\cal A}_{L,R}$ and $\\bar{\\cal A}_{L,R}$ as given in (\\ref{qcdf_11}). The indirect CP asymmetry $S(V\\gamma)$ relies on the interference of both left- and right-helicity amplitudes and vanishes if one of them is absent; it thus probes indirectly the photon helicity. The direct CP asymmetry\n$C(V\\gamma)$ is less sensitive to $\\bar{\\cal A}_R$, but very sensitive to the    strong phase of $\\bar{\\cal A}_L$ and vanishes if the radiative    corrections to $a_{7L}^{U,{\\rm QCDF}}$, Eq.~(\\ref{10}), are\n     neglected. As the accuracy of the prediction of strong phases in      QCDF is subject to discussion, and in any case $C(V\\gamma)$ is      less sensitive to new physics than $S(V\\gamma)$, we shall\n  not consider direct CP asymmetries in this analysis.\n\nLet us briefly discuss the reason for the expected smallness of $S$. In the process $b\\to D\\gamma$, in the SM, the emitted photon is  predominantly  left-handed in $b$, and right-handed in $\\bar b$ decays. This  is due to the fact that the dominant contribution to the  amplitude comes from the chiral-odd dipole operator $Q_7$. As only left-handed quarks participate in the weak  interaction, an effective operator of this type necessitates, in the SM, a helicity flip on one of the external quark lines, which results in a factor $m_b$ (and a left-handed photon) in $b_R\\to D_L\\gamma_L$ and a factor $m_D$ (and a right-handed photon) in $b_L\\to D_R\\gamma_R$. Hence, the emission of right-handed photons is suppressed by  a factor $m_D\/m_b$, which leads to the QCDF prediction (\\ref{qcdf_13}) for $a_{7R}^U$.  The interesting point is not the smallness of the CP asymmetry {\\em   per se}, but the fact that the helicity suppression can easily be alleviated  in a large number of new physics scenarios where the spin flip occurs on an internal line, resulting in a factor $m_i\/m_b$ instead of $m_D\/m_b$. A prime example is left-right symmetric models \\cite{LRS}, whose impact on the photon polarisation was discussed in\nRefs.~\\cite{alt, grin04,grin05}.  These models also come in a supersymmetric version whose effect on $b\\to s\\gamma$ was investigated in Ref.~\\cite{frank}. Supersymmetry with no left-right symmetry can also provide large  contributions to $b\\to D\\gamma_R$, see Ref.~\\cite{susy} for recent studies. Other\npotential sources of large effects  are warped extra dimensions \\cite{warped} or anomalous right-handed top couplings \\cite{anomalous}.  Unless the amplitude for $b\\to D\\gamma_R$ is of the same order as the SM prediction for $b\\to D \\gamma_L$, or the enhancement of $b\\to D \\gamma_R$ goes along with a suppression of $b\\to D \\gamma_L$, the impact on the branching ratio is small,  as the two helicity amplitudes add incoherently. This implies there can be a substantial contribution of new physics to $b\\to D\\gamma$ escaping detection when only branching ratios are measured.\n\nWe can calculate $S$ directly from (\\ref{54}) and obtain, making explicit the contributions from  different sources:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{array}[b]{rll}\nS(\\rho\\gamma) = \n& \\phantom{-}(\\overbrace{~0.01~}^{m_D\/m_b}+\\overbrace{~0.02~}^{\\rm\n  LD~WA}+\\overbrace{~0.20~}^{\\shortstack{\\footnotesize{soft} \\\\  \\footnotesize{gluons}}}~\\pm~ 1.6)\\%\n& = \\phantom{-}(0.2\\pm 1.6)\\%\\,,\\\\\nS(\\omega\\gamma) =\n& \\phantom{-}(0.01-0.08+0.22\\pm 1.7)\\% \n& = \\phantom{-}(0.1\\pm 1.7)\\%\\,,\\\\\nS(K^*\\gamma) =\n& -(2.9-0+0.6\\pm 1.6)\\%\n& = -(2.3\\pm 1.6)\\%\\,,\\\\\nS(\\bar K^*\\gamma) =\n& \\phantom{-}(0.12+0.03+0.11\\pm 1.3)\\%\n& = \\phantom{-}(0.3\\pm 1.3)\\%\\,,\\\\\nS(\\phi\\gamma) =\n& \\phantom{-}(0+0+5.3\\pm8.2)\\times 10^{-2}\\,\\%\n& =  \\phantom{-}(0.1\\pm 0.1)\\%\\,.\n\\label{eq:SVgamma}\n\\end{array}\n\\end{equation}\nIncluding only the helicity-suppressed contribution, one expects, for $B\\to K^*\\gamma$, neglecting the doubly Cabibbo suppressed amplitude in $\\lambda_u^{(s)}$\n\\begin{equation}\n\\left.S(K^*\\gamma)\\right|_{\\mbox{\\footnotesize no soft gluons}} \n =  -2\\, \\frac{m_s}{m_b}\\,\\sin\\,\\phi_d\\approx -2.7\\%\\,.\\label{75}\n\\end{equation}\nFor $B_s\\to\\phi\\gamma$, one expects the CP asymmetry to vanish if the decay amplitude is proportional to $\\lambda_t^{(s)}$, which, at tree-level, precludes any contributions of type $\\sin(\\phi_s) m_s\/m_b$ and also any contribution from WA. This is     because the mixing angle $\\phi_s$ is given by ${\\rm\n arg}[(\\lambda_t^{(s)})^2]$, Eq.~(\\ref{basics_eq12}),  and the interference of amplitudes in    (\\ref{54}) also yields a factor $(\\lambda_t^{(s)})^2$,  if the individual amplitudes are proportional to  $\\lambda_t^{(s)}$ or \n$(\\lambda_t^{(s)})^*$, respectively; this is indeed the case for the helicity-suppressed term $m_s\/m_b$ induced by the operator $Q_7$ and the WA contributions to $a_{7R}^U(\\phi)$,\nEqs.~(\\ref{15}) and (\\ref{20A}), so that the phases cancel in (\\ref{54}).\n\n\nThe actual results in (\\ref{eq:SVgamma}) disagree with the above\nexpectations because of the contributions from soft-gluon emission,\nwhich enter $a_{7R}^U$. Moreover, for $S(\\phi\\gamma)$ this is because the\nsoft-gluon emission from quark loops is different for $u$ and $c$ loops so that $a_{7R}^c\\neq a_{7R}^u$ and hence  $\\bar {\\cal A}_{R}$ (${\\cal A}_{L}$) is not proportional to   $\\lambda_t^{(s)}$ ($(\\lambda_t^{(s)})^*$). Note that a substantial enhancement of $S(\\phi\\gamma)$ by new physics requires\nnot only an enhancement of $|\\bar{\\cal A}_R|$ (and $|{\\cal\n  A}_L|$), but also the presence of a large phase in (\\ref{54}); \nthis could be  either\na large $B_s$ mixing phase which will also manifest itself in\na sizable CP violation in, for instance, $B_s\\to J\/\\psi \\phi$, see\nRef.~\\cite{BF06,Ball:2006xx}; or it could be a new weak phase in $\\bar{\\cal A}_{R}$\n(and ${\\cal A}_L$); or it could be a non-zero strong phase in\none of the $a_{7R}^{c,u}$ coefficients. Based on the light quark loop results there is not much scope for\na large phase in $a_{7R}^{u}$ (whose contribution is, in addition,\ndoubly Cabibbo suppressed), but the situation could be different for\n$a_{7R}^{c,{\\rm soft}}$, where only the\nleading-order term in a $1\/m_c$ expansion is included, which does not carry a\ncomplex phase \\cite{Ball:2006eu}. It is not excluded that a\nresummation of higher-order terms in this expansion will generate a\nnon-negligible strong phase --- which is not really relevant for our\nresults in Eq.~(\\ref{eq:SVgamma}), but could be relevant for the\ninterpretation of any new physics to be found in that observable. For\n$S(K^*\\gamma)$, on the other hand, no new phases are required, and\nany enhancement of $ |\\bar{\\cal A}_R|$ (and $|{\\cal A}_L|$) by new physics will\nresult in a larger value of $S(K^*\\gamma)$.\n\nFor all $S$ except $S(K^*\\gamma)$,\nthe uncertainty is entirely dominated by that of the soft-gluon emission\nterms $l_{u,c}-\\tilde l_{u,c}$, whose uncertainties have been doubled with\nrespect to those given in Tab.~\\ref{rad_tab5}. The smallness of\n$S((\\rho,\\omega)\\gamma)$ is due to the fact that the helicity\nfactor is given by $m_d\/m_b$ (we use $m_{u,d}\/m_s = 1\/24.4$ from\nChPT). For $\\bar K^*$,\nthe suppression from the small mixing\nangle is relieved by the fact that both weak amplitudes in\n$\\lambda_U^{(d)}$ contribute\nso that the CP asymmetry is comparable\nwith that of $\\rho$ and $\\omega$. Despite the generous uncertainties, it is\nobvious that none of these CP symmetries is larger than\n4\\% in the SM, which makes these observables very interesting\nfor new physics searches. The present experimental result from the $B$\nfactories, $S(K^*\\gamma)=-0.28\\pm 0.26$ \\cite{Barberio:2007cr}, certainly encourages\nthe hope that new physics may manifest itself in that\nobservable. While a measurement of the $b\\to d$ CP asymmetries is\nprobably very difficult even at a super-flavour factory,\n$S(K^*\\gamma)$ is a promising observable for $B$ factories \\cite{superB}, but\nnot for the LHC.\\footnote{$K^*$ has to be traced via its decay into a CP\neigenstate, i.e.\\ $K_S\\pi^0$. Neutrals in the final state are not\nreally LHC's favourites.} $B_s\\to \\phi(\\to K^+K^-)\\gamma$, on the\nother hand, will be studied in detail at the LHC, and in particular at\nLHCb, and any largely enhanced value of $S(\\phi\\gamma)$ \nwill be measured within the first years of running. \n\n\\section{Extraction Of CKM Parameters}\\label{ckmextract}\n\nLet us now turn to the determination of CKM parameters from the\nbranching ratios determined in Section~\\ref{rad_brs}. In this context, two particularly interesting\nobservables are \n\\begin{equation}\\label{58}\nR_{\\rho\/\\omega}\\equiv\\frac{\\overline{\\cal B}(B\\to (\\rho,\\omega)\\gamma)}{\n\\overline{\\cal B}(B\\to K^*\\gamma)}\\,,\\qquad\nR_{\\rho}\\equiv\\frac{\\overline{\\cal B}(B\\to \\rho\\gamma)}{\n\\overline{\\cal B}(B\\to K^*\\gamma)}\\,,\\qquad\n\\end{equation}\ngiven in terms of the CP- and isospin-averaged branching ratios of\n$B\\to(\\rho,\\omega)\\gamma$ and $B\\to \\rho\\gamma$, respectively, \nand $B\\to K^*\\gamma$ decays. $R_{\\rho\/\\omega}$\nhas been measured by both {\\sc BaBar} and Belle \\cite{babar_rad,belle_rad}, a first\nvalue of $R_\\rho$ has been given by {\\sc BaBar} \\cite{babar_rad}. \nThe experimental determinations\nactually assume exact isospin symmetry, i.e.\\\n$\\overline{\\Gamma}(B^\\pm\\to \\rho^\\pm\\gamma) \\equiv\n2 \\overline{\\Gamma}(B^0\\to \\rho^0\\gamma)$, and also\n$\\overline{\\Gamma}(B^0\\to \\rho^0\\gamma) \\equiv \n\\overline{\\Gamma}(B^0\\to \\omega\\gamma)$; and as we have seen in Section~\\ref{isosec}, these relations are not in fact exact. Hence, the present\nexperimental results for $R_{\\rho\/\\omega}$ are theory-contaminated. \nAs the isospin asymmetry between the charged and neutral $\\rho$ decay \nrates turns out to be smaller than the asymmetry\n between $\\rho^0$ and $\\omega$, \nit would actually be preferable, from an experimental point of view,\nto drop the $\\omega$ channel and\nmeasure $R_\\rho$ instead of $R_{\\rho\/\\omega}$, as done in the most\nrecent {\\sc BaBar} analysis on that topic \\cite{babar_rad}. \nWe will give numerical results and\ntheory uncertainties for both $R_{\\rho\/\\omega}$ and $R_{\\rho}$.\n\nOne parametrisation of $R_{\\rho\/\\omega}$ often quoted, \nin particular in experimental papers, is\n\\begin{equation}\\label{Brat}\nR_{\\rho\/\\omega} = \\left|\\frac{V_{td}}{V_{ts}}\\right|^2\n\\left(\\frac{1-m_{\\rho}^2\/m_B^2}{1-m_{K^*}^2\/m_B^2}\\right)^3\n\\frac{1}{\\xi^2_{\\rho}} \\left [ 1 +  \\Delta R\\right],\n\\end{equation}\nwith $\\Delta R=0.1\\pm 0.1$ \\cite{BVga1} and again assuming isospin\nsymmetry for $\\rho$ and $\\omega$. This parametrisation creates the impression\nthat $\\Delta R$ is a quantity completely unrelated to\nand with a fixed value independent of\n$|V_{td}\/V_{ts}|$. We would like to point out here that this \nimpression is {\\em wrong}: $\\Delta R$ contains both QCD\n(factorisable and non-factorisable) effects and such from weak\ninteractions. In Ref.~\\cite{Ball:2006nr}  $\\Delta R$ is expressed in\nterms of the factorisation coefficients $a_{7L}^U$, assuming isospin\nsymmetry for $\\rho^0$ and $\\omega$, as\n\\begin{eqnarray}\n1+\\Delta R & = & \\left|\n  \\frac{a_{7L}^c(\\rho)}{a_{7L}^c(K^*)}\\right|^2 \\left( 1 +\n  {\\rm Re}\\,(\\delta a_\\pm + \\delta a_0) \\left[\\frac{R_b^2 - R_b\n  \\cos\\gamma}{1-2 R_b \\cos\\gamma + R_b^2}\\right]\\right.\\nonumber\\\\\n& & \\left. + \\frac{1}{2}\\left( |\\delta a_\\pm|^2 + |\\delta a_0|^2\\right)\n  \\left\\{ \\frac{R_b^2}{1-2 R_b \\cos\\gamma + R_b^2}\\right\\} \\right)\n\\label{delR}\n\\end{eqnarray}\nwith $\\delta a_{0,\\pm}=\na_{7L}^u(\\rho^{0,\\pm})\/a_{7L}^c(\\rho^{0,\\pm})-1$. Eq.~(\\ref{delR}) shows explicitly that $\\Delta R$ depends both on QCD\n ($\\delta a_{\\pm,0}$) and CKM parameters ($R_b,\\gamma$).\nThe point we would like to make is that the calculation of $\\Delta R$\nrequires input values for $R_b$ and $\\gamma$. Once these parameters\n(and the Wolfenstein parameter $\\lambda$)\nare fixed, however, $|V_{td}\/V_{ts}|$ is also fixed and given by\n\\begin{equation}\\label{61}\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = \\lambda \\sqrt{1-2 R_b \\cos\\gamma\n  + R_b^2} \\left[ 1 + \\frac{1}{2}\\,( 1 - 2 R_b \\cos\\gamma) \\lambda^2 +\n  \\mathcal{O}(\\lambda^4)\\right]\\,.\n\\end{equation} \nHence, as $|V_{td}\/V_{ts}|$ and $(R_b,\\gamma)$ are not independent\nof each other, \nit is {\\em impossible} to extract $|V_{td}\/V_{ts}|$ from (\\ref{Brat})\nwith a fixed value of $\\Delta R$. Of course $R_{\\rho\/\\omega}$ and $R_\\rho$ of (\\ref{58}) \n{\\em can} be used to extract information\nabout CKM parameters, but in order to do so one has to settle for a set\nof truly independent parameters. Based on (\\ref{61}), one can\nexchange, say, $\\gamma$ for $|V_{td}\/V_{ts}|$.\\footnote{Strictly speaking, (\\ref{61}) only\n  fixes $\\cos\\gamma$ as function of $|V_{td}\/V_{ts}|$, leaving\n  a twofold degeneracy of $\\gamma$. Eq.~(\\ref{delR}), however, only\n  depends on $\\cos\\gamma$, so that indeed one can unambiguously \nreplace $\\gamma$ by $|V_{td}\/V_{ts}|$.}  So we can either consider $R_V$ as a\nfunction of the CKM parameters $R_b$ and $\\gamma$ (let us call this\nthe $\\gamma$ set of parameters) or as a function of $R_b$ and\n$|V_{td}\/V_{ts}|$ (to be called the $|V_{tx}|$ set). Using the\n$\\gamma$ set, a measurement of $R_V(\\gamma,R_b)$ allows a\ndetermination of $\\gamma$, whereas \n$R_V(|V_{td}\/V_{ts}|,R_b)$ allows the\ndetermination of $|V_{td}\/V_{ts}|$. \nIn either case, the simple quadratic relation\n(\\ref{Brat}) between $R_V$ and $|V_{td}\/V_{ts}|$ becomes more\ncomplicated.\n\n\n In Figs.~\\ref{rad_fig5} and \\ref{rad_fig6} we plot the resulting values of $|V_{td}\/V_{ts}|^2$ and $\\gamma$, respectively,  as a  function of $R_V$. Although the curve in Fig.~\\ref{rad_fig5}(a) looks like a straight line, as naively expected from (\\ref{Brat}), this is not exactly the case, because of the dependence of $\\Delta R$ on $|V_{td}\/V_{ts}|$. In  Fig.~\\ref{rad_fig5}(b) we plot $\\Delta R$ for the $|V_{tx}|$ set of parameters. The dependence of $\\Delta R$ on $|V_{td}\/V_{ts}|$ is rather strong. Apparently indeed $\\Delta R=0.1\\pm 0.1$ in the expected range $0.16<|V_{td}\/V_{ts}|<0.24$, but this estimate does not reflect  the true theoretical uncertainty which is indicated by the dashed lines in the figure. \n\\begin{figure}\n$$\n\\epsfxsize=0.45\\textwidth\\epsffile{rad_fig4a.eps}\\qquad\\epsfxsize=0.45\\textwidth\\epsffile{rad_fig4b.eps}\n$$\n\\caption[$|V_{td}\/V_{ts}|^2$ as function of   $R_{\\rho\/\\omega}$ and $\\Delta R$ as function of   $|V_{td}\/V_{ts}|$.]{\\small Left panel: $|V_{td}\/V_{ts}|^2$ as function of   $R_{\\rho\/\\omega}$, Eq.~(\\ref{58}),   in the $|V_{tx}|$ basis -- see   text.  Solid line: central values. Dash-dotted lines: theoretical   uncertainty induced by $\\xi_\\rho = 1.17\\pm 0.09$, (\\ref{xirho}).  Dashed lines: other   theoretical uncertainties, including those induced   by $|V_{ub}|$, $|V_{cb}|$ and the hadronic parameters of   Tab.~\\ref{rad_tab8}. Right panel: $\\Delta R$ from   Eq.~(\\ref{delR}) as function of\n  $|V_{td}\/V_{ts}|$   in the $|V_{tx}|$ basis. Solid   line: central values. Dashed   lines: theoretical uncertainty.}\\label{rad_fig5} \n  $$\\epsfxsize=0.45\\textwidth\\epsffile{rad_fig5.eps}$$\n\\caption[The UT angle $\\gamma$ as function of   $R_{\\rho\/\\omega}$.]{\\small The UTangle $\\gamma$ as function of   $R_{\\rho\/\\omega}$ in the $\\gamma$ set of CKM parameters. Solid   lines: central values of input parameters. Dash-dotted lines: theoretical   uncertainty induced by $\\xi_\\rho = 1.17\\pm 0.09$. Dashed lines: other   theoretical uncertainties.}\\label{rad_fig6}\n$$\\epsfxsize=0.45\\textwidth\\epsffile{rad_fig6.eps}$$\n\\caption[Central values of  $R_{\\rho\/\\omega}$ and $R_{\\rho}$ as functions of $|V_{td}\/V_{ts}|$]{\\small Central values of  $R_{\\rho\/\\omega}$ (solid line) and $R_{\\rho}$   (dash-dotted line) as functions of $|V_{td}\/V_{ts}|$.}\\label{rad_fig7}\n\\end{figure}\n\nIt is now basically a matter of choice whether to use\n$R_{\\rho\/\\omega}$ to determine $|V_{td}\/V_{ts}|$ or $\\gamma$. Once one of\nthese parameters is known, the other one follows from Eq.~(\\ref{61}). In\nFig.~\\ref{rad_fig6} we plot $\\gamma$ as a function of\n$R_{\\rho\/\\omega}$, together with the theoretical uncertainties. In\nFig.~\\ref{rad_fig7} we also compare the central values of\n$R_{\\rho\/\\omega}$ with those of $R_{\\rho}$, as a function of\n$|V_{td}\/V_{ts}|$. Although the difference is small, $R_{\\rho}$ is\nexpected to be larger than $R_{\\rho\/\\omega}$.  $R_{\\rho\/\\omega}$ and $R_{\\rho}$ are dominated by the  uncertainties of $\\xi_{\\rho}$ and as discussed in Ref.~\\cite{Ball:2006nr}, a reduction of this uncertainty  would require a reduction of the uncertainty of the transverse decay constants $f_V^\\perp$ of $\\rho$ and $K^*$. With the most recent results from {\\sc BaBar}, $R_{\\rho\/\\omega} = 0.030\\pm 0.006$ \\cite{babar_rad}, and from Belle, $R_{\\rho\/\\omega} = 0.032\\pm 0.008$ \\cite{belle_rad}, we then find\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{array}[b]{l@{\\quad}l@{\\quad\\leftrightarrow\\quad}l}\n\\mbox{{\\sc BaBar}:} & \\displaystyle\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = 0.199\\overbrace{^{+0.022}_{-0.025}}^{{\\rm exp}}\\pm\n\\overbrace{0.014}^{{\\rm th}} &\\displaystyle\n\\gamma = (61.0\\overbrace{^{+13.5}_{-16.0}}^{{\\rm exp}}\\overbrace{^{+8.9}_{-9.3}}^{\n{\\rm th}})^\\circ\\,,\\\\[10pt]\n\\mbox{Belle:} & \\displaystyle\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = 0.207\\,^{+0.028}_{-0.033}\\,\n^{+0.014}_{-0.015} &\\displaystyle\n\\gamma = (65.7\\,^{+17.3}_{-20.7}\\,^{+8.9}_{-9.2})^\\circ\\,.\n\\end{array}\n\\label{63}\n\\end{equation}\nThese numbers  compare well with the Belle result  \\cite{Bellegamma} \nfrom tree-level processes, $\\gamma=(53\\pm 20)^\\circ$, quoted in\nTab.~\\ref{rad_tab8}, and results from  global fits\n\\cite{global}. We also would like to point out that the above\ndetermination of $\\gamma$ is actually a determination of\n$\\cos\\gamma$, via Eq.~(\\ref{61}), and implies, in principle, a twofold\ndegeneracy $\\gamma\\leftrightarrow 2\\pi-\\gamma$. This is in contrast to the\ndetermination from $B\\to D^{(*)} K^{(*)}$ in \\cite{Bellegamma}, which\ncarries a twofold degeneracy\n$\\gamma \\leftrightarrow \\pi+\\gamma$. Obviously these two\ndeterminations taken together remove the degeneracy and \nselect $\\gamma\\approx 55^\\circ<180^\\circ$. If \n$\\gamma\\approx 55^\\circ+180^\\circ$ instead, one would have \n$|V_{td}\/V_{ts}|\\approx 0.29$ from\n(\\ref{61}), which is definitely ruled out by data. Hence, the result\n(\\ref{63}) confirms the SM interpretation of $\\gamma$ from \nthe tree-level CP asymmetries in $B\\to D^{(*)} K^{(*)}$.\n\n\\begin{table}[ht]\n$$\n\\begin{array}{c|c|c}\n\\tau_{B^0} & \\tau_{B^\\pm}\/\\tau_{B^0} & \\tau_{B_s^0}\/\\tau_{B^0}\n\\\\\\hline\n1.530(9)\\,{\\rm ps} & 1.071(9) & 0.958(39)\n\\end{array}\n$$\n\\caption[$B$ lifetimes from HFAG.]{\\small $B$ lifetimes from HFAG \\cite{Barberio:2007cr}.}\n\\label{rad_tab9}\n\\addtolength{\\arraycolsep}{3pt}\n\\renewcommand{\\arraystretch}{1.3}\n$$\n\\begin{array}{c|c|c|c|c|c}\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{CKM parameters and couplings}}\\\\\\hline\n\\lambda \\mbox{~\\cite{Yao:2006px}} & |V_{cb}| \\mbox{~\\cite{inclmoments}} & \n|V_{ub}| & \\gamma \\mbox{~\\cite{Bellegamma}} & \\alpha_s(m_Z)\n\\mbox{~\\cite{Yao:2006px}} & \\alpha_{\\rm QED}\\\\\\hline\n0.227(1) & 42.0(7)\\times 10^{-3} & 4.0(7)\\times\n10^{-3} & (53\\pm 20)^\\circ & 0.1176(20) & 1\/137\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{B parameters}}\\\\\\hline\nf_{B_q}\\mbox{~\\cite{Onogi}} & f_{B_s}\\mbox{~\\cite{Onogi}} &\n    \\lambda_{B_q}(\\mu_h) \\mbox{~\\cite{Ball:2006nr}} & \\lambda_{B_s}(\\mu_h)\n& \\mu_h     \\\\\\hline\n200(25)\\,{\\rm MeV} & 240(30)\\,{\\rm MeV} & 0.51(12)\\,{\\rm GeV} &\n0.6(2)\\,{\\rm GeV} & 2.2\\,{\\rm GeV}\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{$\\rho$ parameters}}\\\\\\hline\nf_{\\rho} & f_{\\rho}^\\perp & a_1^\\perp({\\rho}) &\na_2^\\perp({\\rho}) & T_1^{B\\to\\rho}(0)\\\\\\hline\n216(3)\\,{\\rm MeV} & 165(9)\\,{\\rm MeV} & 0 & 0.14(6) & 0.27(4)\n\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{$\\omega$  parameters}}\\\\\\hline\nf_{\\omega} & f_{\\omega}^\\perp & a_1^\\perp({\\omega}) &\na_2^\\perp({\\omega}) & T_1^{B\\to\\omega}(0)\\\\\\hline\n187(5)\\,{\\rm MeV} & 151(9)\\,{\\rm MeV} & 0 & 0.15(7) & 0.25(4)\n\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{$K^*$ parameters}}\\\\\\hline\nf_{K^*} & f_{K^*}^\\perp & a_1^\\perp({K^*})\\mbox{~\\cite{Ball:2005vx}} & \na_2^\\perp({K^*}) & T_1^{B_q\\to K^*}(0) & \nT_1^{B_s\\to \\bar  K^*}(0)\\\\\\hline\n220(5)\\,{\\rm MeV} & 185(10)\\,{\\rm MeV} & 0.04(3) & 0.15(10) &\n0.31(4) & 0.29(4)\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{$\\phi$ parameters}}\\\\\\hline\nf_{\\phi} & f_{\\phi}^\\perp & a_1^\\perp({\\phi}) &\na_2^\\perp({\\phi}) & T_1^{B_s\\to\\phi}(0)\\\\\\hline\n215(5)\\,{\\rm MeV} & 186(9)\\,{\\rm MeV} & 0 & 0.2(2) & 0.31(4) & \\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{quark masses}}\\\\\\hline\n\\multicolumn{2}{c|}{m_s(2\\,{\\rm GeV})\\mbox{~\\cite{ms}}} \n& m_b(m_b)\\mbox{~\\cite{inclmoments}} & \nm_c(m_c)\\mbox{~\\cite{czakon}} & \\multicolumn{2}{c}{\nm_t(m_t)\\mbox{~\\cite{mt}}}\\\\\\hline\n\\multicolumn{2}{c|}{100(20)\\,{\\rm MeV}} \n& 4.20(4)\\,{\\rm GeV} & 1.30(2)\\,{\\rm GeV} & \n\\multicolumn{2}{c}{163.6(2.0)\\,{\\rm GeV}} \\\\\\hline\\hline\n\\end{array}\n$$\n\\caption[Summary of input parameters.]{\\small Summary of input parameters. The  value of $|V_{ub}|$ is an average over inclusive and exclusive  determinations and the result from UTangles Refs.~\\cite{Barberio:2007cr,global,Vub}. None of our results is very sensitive  to $|V_{ub}|$. For an explanation of our choice of the value of the UT angle  $\\gamma$, see text. $\\lambda_{B_s}$ is obtained from   $\\lambda_{B_q}$, see Eq.~(\\ref{rad_bs}).  The vector meson  decay constants $f_V$, $f_V^\\perp$ are discussed in Ref.~\\cite{Ball:2006eu}; the  values of the Gegenbauer moments $a_i^\\perp$ are compiled from  various sources   \\cite{Ball:2006nr,Ball:1996tb,Ball:1998sk,Ball:2003sc} and include only small ${\\rm SU(3)_F}$-breaking,  in line with the findings for pseudoscalar mesons \\cite{Ball:2006wn}. The form factors $T_1$  are \nupdates of previous LCSR results  \\cite{Ball:2004rg}, including the updated values of the decay constants\n $f_{\\rho,\\omega,\\phi}$ and of $a_1^\\perp({K^*})$ \\cite{Ball:2005vx,Ball:2006fz}. \n All scale-dependent quantities are given at the scale $\\mu=1\\,$GeV unless stated otherwise.}\n\\label{rad_tab8}\n\\end{table}\n\\chapter{Summary and Conclusions}\\label{chapter8_conc}\nThis thesis has consisted of three main analyses centred on the investigations and determinations of meson light-cone distribution amplitudes.  We have seen how the determinations of decay observables in $B$ decays are reliant on the sound understanding of both theoretical and experimental uncertainties with the work presented in this thesis striving towards the former. To summarise:\n\nWe began, in Chapter~\\ref{chapter1_basics}, with a brief introduction defining the QCD Lagrangian, discussing CP violation and the $\\Delta B =1$ effective Hamiltonian.\n\nIn Chapter~\\ref{chapter2_DAs} we investigated the structure of vector mesons distribution amplitudes to twist-3 accuracy. We included all $\\rm SU(3)_F$-breaking and G-parity violating effects. The QCD equations of motion were implemented to unpick the interwoven relations between the distribution amplitudes ultimately expressing the two-particle twist-3 distribution amplitudes in terms of the three-particle twist-3 and two-particle twist-2 distribution amplitudes. The equations of motion result in integral equations which are readily solved order-by-order in conformal spin and to the order considered all the distribution amplitudes are then expressed by a small number of non-perturbative parameters. Finite quark mass effects appear in the equation of motion and therefore impact the two-particle twist-3 distribution amplitudes (\\ref{das_eq31}-\\ref{das_eq33}). Such effects also cause mixing between the twist-3 hadronic parameters under renormalisation scale evolution, see Eq.~(\\ref{das_eq37}).\n\nIn Chapter~\\ref{chapter3_SR} we discussed the methods of QCD sum rules (the SVZ method) and QCD sum rules on the light-cone. We outlined the procedures with example correlation functions and ended the chapter with an example calculation of the $\\alpha_s$ corrections to the gluon condensate contribution to a $K$ meson sum rule. The calculation made use of the background field technique and served to illustrate the calculation of radiative corrections to -- and extraction of -- vacuum condensates in the SVZ method. The result of the calculation is in conflict with that in the literature, see Eqs.~(\\ref{last}) and (\\ref{least}).\n\n\nIn Chapter~\\ref{chapter4_det}  we determined the leading hadronic parameters defined in Chapter~\\ref{chapter2_DAs} via SVZ sum rules. We calculated the three-particle twist-3 parameters to NLO in conformal spin, also including all G-parity violating terms and finite strange quark mass effects. The determination of the twist-3 parameters is new for $K^*$ and $\\phi$. The results for the $\\rho$ agree within uncertainties  with previous determinations and are presented in Tabs.~\\ref{det_tab1} and \\ref{det_tab2}. We also calculate $\\mathcal{O}(\\alpha_s)$ and $\\mathcal{O}(m_s^2)$ corrections to the quark condensate for the sum rules for $a_n^{\\parallel,\\perp}(V)$, which for $n=2$ is the first non-trivial Gegenbauer coefficient of the G-even particles $\\rho$ and $\\phi$. We add this contribution to the existing sum rules taken from the literature and update the value of $a_2^{\\parallel,\\perp}(\\phi)$ which we find to be consistent with that found for $K^*$ and $\\rho$;  $a^\\perp_{1,2}(V)=a^\\parallel_{1,2}(V)$ within uncertainties. The results find direct application in QCD factorisation descriptions of $B\\to V$ decays, and the light-cone sum rule analyses of $B\\to V$ transition form factors.\n\nIn Chapter~\\ref{chapter5_eta} we calculated the form factors of $B\\to \\eta^{\\prime}$\nsemileptonic transitions from light-cone sum rules,\nincluding the gluonic singlet contributions. We built upon the previous light-cone sum rule determination of the $B\\to \\eta$ form factor by casting the calculation consistently within the phenomenologically motivated $\\eta$-$\\eta^{\\prime}$ mixing scheme of Refs.~\\cite{Feldmann:1998vh,Feldmann:1998sh}. We found that, as\nexpected, these contributions are more relevant for $f_+^{\\eta^{\\prime}}$ than\nfor $f_+^\\eta$ and can amount up to 20\\% in the former, \ndepending on the only poorly\nconstrained leading Gegenbauer moment $B^g_2$ of the gluonic twist-2\ndistribution amplitude of $\\eta^{\\prime}$.  The numerical results, with each contribution listed separately, are given by Eqs.~(\\ref{fp1}) and (\\ref{fp0}). Consequently, it seems unlikely that the large exclusive $B\\to \\eta^{\\prime} K$ and inclusive $B\\to \\eta^{\\prime} X$ branching ratios can be explained by a large $B^g_2$, as it would have to assume a very extreme value. We also found that the form factors\nare sensitive to the values of the twist-2 two-quark Gegenbauer\nmoments $a_2^{\\eta,\\eta^{\\prime}}$ which, given the uncertainty of independent\ndeterminations, we have set equal to $a_2^\\pi$, see Fig.\\ref{eta_diags3}.\n\nThe ratio\nof branching ratios ${\\cal B}(B\\to\\eta^{\\prime} e\\nu)\/{\\cal B}(B\\to\\eta e\\nu)$\nis sensitive to both $a_2$ and $B^g_2$ and may be used to constrain\nthese parameters, once it is measured with sufficient accuracy, see Fig.~\\ref{eta_fig8}. The\nextraction of $|V_{ub}|$ from these semileptonic decays, in particular\n$B\\to\\eta e\\nu$, with negligible singlet contribution, although\npossible in principle, at the moment is obscured by the lack of\nknowledge of $a_2$. We\nwould also like to stress that, in the framework of the quark-flavour\nmixing scheme for the $\\eta$-$\\eta^{\\prime}$ system as used in this analysis,\n$B\\to \\eta^{\\prime}$ transitions probe only the $\\eta_q$ component of these\nparticles. The $\\eta_s$ component could be probed directly for\ninstance in the $b\\to s$ penguin transition $B_s\\to \\eta^{\\prime}\n\\ell^+\\ell^-$, although such a measurement would also be sensitive\nto new physics in the penguin diagrams.\n\nIn Chapter~\\ref{chapter6_QCDF} we discussed the QCD factorisation (QCDF) approach of Refs.~\\cite{Beneke:1999br, Beneke:2000ry} and its application to the radiative $B$ decays $B \\to V \\gamma$ of Refs.~\\cite{Bosch:2001gv,Bosch:2002bw}.   We discussed the appearance of distribution amplitudes in the factorisation formulas and focused on the leading contributions to the $B\\to V \\gamma$ decays.\n\n\nIn Chapter~\\ref{chapter7_rad} we performed a phenomenological analysis of the radiative $B$ decays to vector mesons $B\\to V \\gamma$, using the framework discussed in Chapter~\\ref{chapter6_QCDF}. We investigated the most relevant power-suppressed corrections to the QCDF predictions for the radiative decays $B_{u,d} \\to (\\rho, \\omega, K^*)\\gamma$ and  $B_{s} \\to (\\phi, \\bar{K}^*)\\gamma$. We use the QCDF framework presented in Refs.~\\cite{Bosch:2001gv,Bosch:2002bw} in which we find use for the twist-2 DA parameters determined in Chapter~\\ref{chapter4_det}. Besides the leading QCDF contributions we included long-distance photon emission and soft-gluon mission from quark loops. These effects, although formally $\\sim 1\/m_b$ with respect to the leading contributions, augment the QCDF predictions for the branching ratios, CP and isospin asymmetries. \n\nThe impact of the power-suppressed corrections on the branching ratios is found to be very small, with the exception of the weak annihilation contributions to $B^\\pm\\to \\rho^\\pm \\gamma$ which are large due to a large combination of Wilson coefficients $C_2+C_1\/3=1.02$ and no CKM-suppression. Moreover, long-distance photon emission also impacts most here, see Eq.~(\\ref{LDcont}). An explicit break down  of the results are given in Tab.~\\ref{rad_tab6}. \n\nThe isospin asymmetries $A(\\rho,\\omega)$, $A_I(\\rho)$ and $A_I(K^*)$  are driven by weak annihilation and long-distance photon emission contributions. We found a non-zero asymmetry $A(\\rho,\\omega)=-0.20\\pm0.09$ which suggests the  explicit assumption of perfect symmetry, i.e.\\ $\\overline{\\Gamma}(B^\\pm\\to \\rho^\\pm \\gamma) = 2 \\overline{\\Gamma}(B^0\\to \\rho^0 \\gamma) = 2 \\overline{\\Gamma}(B^0\\to \\omega \\gamma)$ used to obtain the experimental value of $\\overline{\\cal B}(B\\to(\\rho,\\omega)\\gamma)$ is not so well justified. We found $A_I(\\rho)$ to depend strongly on the UT angle $\\gamma$, as shown in Tab.~\\ref{rad_tab7}. With our central value of $\\gamma=53^\\circ$ (see Tab~\\ref{rad_tab8}) our result agrees very well with the {\\sc BaBar} result $A_I(\\rho)_{\\rm BaBar} =  0.56\\pm 0.66$ \\cite{babar_rad}. For $A_I(K^*)$ we found a result consistent with the experimental result $A_I(K^*)_{\\rm exp}=(3.2\\pm4.1)\\%$ and, via its sensitivity to the Wilson coefficient combination $C_5+C_6\/3$ conclude that a reduction in the experimental uncertainty may uncover signs of new physics contributing to these Wilson coefficients, see Fig.~\\ref{rad_fig4}.\n\nThe  indirect CP asymmetries $S(V\\gamma)$ are caused by the interference between the amplitudes describing the production of left and right-handed photons, see Eqs.~(\\ref{qcdf_8}) and (\\ref{54}). The right-handed amplitude is suppressed by $m_D\/m_b$ with respect to the left-handed one for $\\bar B =b \\bar q$ decays (and vice versa for $B$ decays). Due to this natural suppression in  the SM we expect the CP asymmetries to be small, and this suppression can be relieved by many new physics senarios. We investigated the soft-gluon effects arising from soft heavy and soft quark loops. The calculation of these contributions makes use of the three-particle twist-3 DA parameters determined in Chapter~\\ref{chapter4_det}. They contribute to both the left and right-handed amplitudes, and so may also relieve to SM suppression. We found that although they do divert the results from the values naively expected, there is no scope for a large enhancement due to these power-suppressed contributions. The results are given in Eq.~(\\ref{eq:SVgamma}).\n\nFinally, using the most recent results from {\\sc BaBar} and Belle,  we extracted the CKM parameter ratio $|V_{td}\/V_{ts}|$ and equivalently the UT angle $\\gamma$ from the  ratio of branching ratios $R_{\\rho\/\\omega}$. The results are \n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{array}[b]{l@{\\quad}l@{\\quad\\leftrightarrow\\quad}l}\n\\mbox{{\\sc BaBar}:} & \\displaystyle\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = 0.199\\overbrace{^{+0.022}_{-0.025}}^{{\\rm exp}}\\pm\n\\overbrace{0.014}^{{\\rm th}} &\\displaystyle\n\\gamma = (61.0\\overbrace{^{+13.5}_{-16.0}}^{{\\rm exp}}\\overbrace{^{+8.9}_{-9.3}}^{\n{\\rm th}})^\\circ\\,,\\\\[10pt]\n\\mbox{Belle:} & \\displaystyle\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = 0.207\\,^{+0.028}_{-0.033}\\,\n^{+0.014}_{-0.015} &\\displaystyle\n\\gamma = (65.7\\,^{+17.3}_{-20.7}\\,^{+8.9}_{-9.2})^\\circ\\,.\n\\end{array}\n\\end{equation}\nand agree well with the Belle result $\\gamma=(53\\pm20)^\\circ$ obtained from tree-level processes, and results from global fits \\cite{global}. The result confirms the SM interpretation of $\\gamma$ from \nthe tree-level CP asymmetries in $B\\to D^{(*)} K^{(*)}$.\n\n\\chapter*{Acknowledgements}\n\nFirst and foremost, I would like to thank my supervisor Patricia Ball for all her help and guidance over the last three years. It has been a great opportunity to work with her, and a fantastic learning experience. I must also thank Roman Zwicky for always finding the time to quell my confusions, and with whom it was a pleasure to collaborate. Also, I thank Angelique Talbot for all her friendly discussions, and I wish Aoife Bharucha all the best with her future projects.\n\nI also thank my office mates Ciaran Williams, Karina Williams, Kemal Ozeren, Martyn Gigg and Stefan Hoeche, and the many other friends who have made my time in Durham and the IPPP so enjoyable.\n\nTo those whose support cannot be appreciated enough; I must thank my parents. I thank my brother too for all the discussions and debates we had over coffee, and finally, I must also thank my grandparents. \n\nThis work was supported by a PPARC studentship which is gratefully acknowledged.\n\n\\chapter*{Declaration}\nI declare that no material presented in this thesis has previously been submitted\nfor a degree at this or any other university. The research described in this thesis has been carried out in collaboration with Prof.~Patricia Ball and Dr.~Roman Zwicky and has been published as follows:\n\\begin{itemize}\n  \\item{``$B \\to V \\gamma$ beyond QCD factorisation,''\\newline\n  P.~Ball, G.~W.~Jones and R.~Zwicky, Phys.\\ Rev.\\  D {\\bf 75} (2007) 054004,\\newline [arXiv:hep-ph\/0612081].}\n\n    \\item{``Twist-3 distribution amplitudes of $K^*$ and $\\phi$ mesons,''\\newline\nP.~Ball and G.~W.~Jones, JHEP {\\bf 03} (2007) 069, \\newline [arXiv:hep-ph\/0702100].}\n   \n    \\item{``$B \\to \\eta^{(\\prime)}$ Form Factors in QCD,''\\newline\nP.~Ball and G.~W.~Jones, JHEP {\\bf 08} (2007) 025, \\newline  arXiv:0706.3628 [hep-ph].}\n   \n\\end{itemize}\nThe copyright of this thesis rests with the author.  No quotation from it should be published without their prior written consent and information derived from it should be acknowledged. \n\\begin{flushright}\ngarethwarrenjones@gmail.com\n\\end{flushright}\n\\tableofcontents\n\\listoffigures\n\\listoftables\n\\input{chapter0_intro.tex}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $K$ be a complete discretely valued field of mixed characteristic with perfect residue field $k$. Fix a separable closure of $\\overline{K}$ of $K$ and let $G_K$ be the absolute Galois group of $K$. The study of stable lattices in crystalline representations of $G_K$ plays an important role in number theory. For example, in many modularity lifting results, one wants to understand liftings of mod $p$ representations of the Galois group of a number field $F$ to Galois representations over $\\mathbb Z_p$-lattices with nice properties when restricted to the Galois groups of $F_v$ for all places $v$ of $F$. And a reasonable property at places over $p$ is that the representation of the Galois group of the local field is crystalline. There are various theories about characterizing $G_K$-stable lattices in crystalline representations, for example, theory of strongly divisible lattices of Breuil(cf. \\cite{BreuilIntegral}), Wach modules(cf. \\cite{Wach96} and \\cite{Berger}), Kisin modules(cf. \\cite{KisinFcrystal}), Kisin-Ren's theory(cf. \\cite{KisinRen}) and the theory of $(\\varphi, \\widehat{G})$-modules(cf. \\cite{liu-notelattice}). The theories above state that one can describe lattices in crystalline representations using certain linear algebraic data over certain commutative rings $A$. \n\nIn a recent work of Bhatt-Scholze\\cite{BS2021Fcrystals}, they give a different characterization of the category of lattices in crystalline representations. To explain their result, let $\\O_K$ be the ring of integers in $K$, and they consider the absolute prismatic site $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, which is defined as the opposite category of all bounded prisms over $\\O_K$ and equipped with the faithfully flat topology. Let $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$ be the structure sheaf over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, and $\\mathcal{I}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\subset \\O_{\\mathlarger{\\mathbbl{\\Delta}}}$ be the ideal sheaf of the Hodge-Tate divisor, then $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$ carries a $\\varphi$-action coming from the $\\delta$-structures. A prismatic $F$-crystal in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is defined as a crystal $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$ over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules together with an isomorphism $(\\varphi^\\ast \\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}})[1\/\\mathcal{I}_{{\\mathlarger{\\mathbbl{\\Delta}}}}] \\simeq \\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/\\mathcal{I}_{{\\mathlarger{\\mathbbl{\\Delta}}}}]$. The main result in \\cite{BS2021Fcrystals} is the following:\n\n\\begin{theorem}\\label{thm-intro-main-1}(\\cite[Theorem 1.2]{BS2021Fcrystals} and Theorem~\\ref{Thm-main-1})\nThere is an equivalence of the category of prismatic $F$-crystals in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ and the category of Galois stable lattices in crystalline representations of $G_K$.\n\\end{theorem}\n\nTo relate the result of Bhatt-Scholze with previous works of characterizing lattices in crystalline representations using linear algebraic data, one should first realize the base rings $A$ used in those theories as certain prisms $(A,I)$ over $\\O_K$. Then one should expect that evaluating the prismatic $F$-crystals on $(A,I)$ should recover the corresponding theory. For example, in the theory of Kisin \\cite{KisinFcrystal}, he uses the base ring $A=\\mathfrak{S}:=W(k)[\\![u]\\!]$ with $\\delta(u)=0$, and if one fixes a uniformizer $\\varpi$ of $\\O_K$ which is a zero of an Eisenstein polynomial $E \\in W(k)[u]$, then it is well-known that $(A,(E))$ is the so-called Breuil-Kisin prism which is inside $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$. And Kisin was able to attach any lattice $T$ in a crystalline representation of $G_K$ a finite free $A$-module together with an isomorphism $(\\varphi^\\ast \\mathfrak{M})[1\/E] \\simeq \\mathfrak{M}[1\/E]$. Now, if $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is the prismatic $F$-crystal attaching to $T$ under Theorem~\\ref{thm-intro-main-1}, then Bhatt-Scholze show that the evaluation of $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$ on $(A,(E))$ recovers Kisin's theory (cf. Theorem 1.3 of $loc.cit.$). \n\nThe first question answered in this paper is whether and how one can recover the theory of $(\\varphi,\\hat{G})$-modules from the prismatic $F$-crystals characterization of Bhatt-Scholze. The category of $(\\varphi,\\hat{G})$-modules, roughly speaking, consisting of pairs $((\\mathfrak{M},\\varphi_{\\mathfrak{M}}),\\hat{G})$, where $(\\mathfrak{M},\\varphi_{\\mathfrak{M}})$ is a Kisin module, and $\\hat{G}$ is a $G_K$-action on $\\mathfrak{M}\\otimes_{\\mathfrak{S},\\varphi} \\widehat {\\mathcal R}$ that commutes with $\\varphi_{\\mathfrak{M}}$ and satisfying some additional properties. Here $\\widehat {\\mathcal R}$ is a subring of $\\Ainf$ that is stable under $\\varphi$ and $G_K$, where $\\Ainf=W(\\O_{\\overline{K}}^\\flat)$ introduced by Fontaine, and there is a surjection $\\theta: \\Ainf:=W(\\O_{\\overline{K}}^\\flat) \\to \\widehat{\\O_{\\overline{K}}}$. However, the period ring $\\widehat {\\mathcal R}$ introduced by Liu is not known to be $p$-adically complete or not, and it is even harder to determine whether it can be shown up as a prism. So in order to relate the theory of $(\\varphi,\\hat{G})$-modules with the category of prismatic $F$-crystals of Bhatt-Scholze, we develop a theory of prismatic $(\\varphi,\\hat{G})$-modules, in which theory the ring $\\widehat {\\mathcal R}$ is replaced by $A^{(2)}_{\\st}$, a subring of $\\Ainf$ constructed as certain prismatic envelope in \\S \\ref{subsec-Ast}. \n\nThe first result of this paper is about the theory of prismatic $(\\varphi,\\hat{G})$-modules. We can show similar to the classical $(\\varphi,\\hat{G})$-module theory, there is an equivalence between the category of prismatic $(\\varphi,\\hat{G})$-modules and lattices in semi-stable representations of $G_K$. Moreover, $(A^{(2)}_{\\st},(E))$ is indeed a prism in $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, it admits a map $(A,(E)) \\to (A^{(2)}_{\\st},(E))$ of prisms, and carries an action of $G_K$. For a $G_K$-stable lattice $T$ in a crystalline representation, if $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is the prismatic $F$-crystal attaches to $T$, then evaluating $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ on the morphism $(A,(E)) \\to (A^{(2)}_{\\st},(E))$ recovers the prismatic $(\\varphi,\\hat{G})$-module attaches to $T$. We can also show the map $A^{(2)}_{\\st} \\to \\Ainf \\xrightarrow{\\varphi} \\Ainf$ factor through $\\widehat {\\mathcal R}$, so the theory of prismatic $(\\varphi,\\hat{G})$-modules recovers the classical theory. The ring $A^{(2)}_{\\st}$ is simpler than $\\widehat {\\mathcal R}$ in many ways, although it is still very complicated and non-noetherian, it is more explicitly described and is $p$-adic complete. In particular, our new theory can be used to fix the gap \\cite{liu-Fontaine} indicated by \\cite[Appendix B]{gao2021breuilkisin}.\n\nThe second attempt made in this paper is to provide a new approach to the equivalence between the category of prismatic $F$-crystals and the category of lattices in crystalline representation established by Bhatt and Scholze as in Theorem~\\ref{thm-intro-main-1}. That is, using the known equivalence between lattices in semi-stable representations and prismatic $(\\varphi, \\hat G)$-modules, we will establish a functor from the category of prismatic $(\\varphi, \\hat G)$-modules that correspond to crystalline representations to prismatic $F$-crystals, and show this functor is an equivalence. \n\nTo be more precise, let $T$ be a $G_K$-stable lattice in a crystalline representation with positive Hodge-Tate weights, let $(A, E)$ be the Breuil-Kisin prism, and let $(A^{(2)},(E))$ (resp. $(A^{(3)},(E))$) be the self-product (self-triple-product) of $(A, (E))$ in $(\\O_K)_{\\mathlarger{\\mathbbl{\\Delta}}}$. Then evaluating prismatic $F$-crystals on the diagram $(A,(E)) \\xrightarrow{i_1} (A^{(2)},(E)) \\xleftarrow{i_2} (A,(E))$ induces an equivalence of the category of prismatic $F$-crystals and Kisin modules with descent data, that is pairs $((\\mathfrak{M},\\varphi_{\\mathfrak{M}}),f)$ where $(\\mathfrak{M},\\varphi_{\\mathfrak{M}})$ is a Kisin module and \n$$\nf: \\mathfrak{M}\\otimes_{\\mathfrak{S},i_1} A^{(2)} \\simeq \\mathfrak{M}\\otimes_{\\mathfrak{S},i_2} A^{(2)}\n$$\nis an isomorphism of $A^{(2)}$-modules that is compatible with $\\varphi$ and satisfies cocycle condition over $A^{(3)}$. Using this, to establish an equivalence between prismatic $(\\varphi, \\hat G)$-modules that correspond to crystalline representations and prismatic $F$-crystals, it remains to find certain correspondence between the $\\hat{G}$-action and the descent isomorphism $f$. We will show the descent isomorphism can be obtained by taking the $G_K$-action of the $(\\varphi,\\widehat{G})$-module at a specific element. To be more precise, fix a Kummer tower $K _\\infty = \\bigcup_{n = 1}^\\infty K (\\varpi _n )$ used in the theory of Kisin, where $\\{\\varpi _n\\}_{n}$ is a compatible system of $p^n$-th roots of $\\varpi_0=\\varpi$, and let $L$ be the normalization of $K _\\infty$ inside $\\overline{K}$. Choose $\\tau\\in \\hat{G}:=\\Gal(L\/K)$ satisfying $\\tau(\\varpi_n)=\\zeta_{p^n}\\varpi_n$ such that $\\{\\zeta_{p^n}\\}$ is a compatible system of primitive $p^n$-th roots of $1$, then our slogan is that the descent isomorphism corresponds to the $\\tilde{\\tau}$-action on the Kisin module $\\mathfrak{M}$ inside $T^{\\vee}\\otimes \\Ainf$ where $\\tilde{\\tau}\\in G_K$ is any lifting of $\\tau$ under the quotient map $G_K \\to \\widehat{G}$. \n\nTo sketch our idea, first we have the maps $u\\mapsto [{\\varpi}^\\flat]$ and $u\\mapsto [{\\tau}({\\varpi}^\\flat)]$ defines two morphisms of $(A,(E))$ to $(\\Ainf,\\Ker\\theta)$. By the universal property of $(A^{(2)},(E))$, these two maps induce a morphism $(A^{(2)},(E)) \\to (\\Ainf,\\Ker\\theta)$. We can show this map is injective, and the embedding factors through $A^{(2)}_{\\st}$, which is the base ring used in our prismatic $(\\varphi, \\hat G)$-module theory. That is,  we have a chain of subrings $A\\subset A^{(2)} \\subset A^{(2)}_{\\st}$ of $\\Ainf$, such that $\\tilde{\\tau}(A)$ is also contained in $A^{(2)}$. We can show a prismatic $(\\varphi, \\hat G)$-module corresponds to a crystalline representation if and only if the coefficients of the $\\tilde{\\tau}$-action on $\\mathfrak{M}$ in $T^{\\vee}\\otimes \\Ainf$ lie inside $A^{(2)}$. And once this is proved, the $\\tilde{\\tau}$-action will induce an isomorphism:\n$$\nf_{\\tau}: \\mathfrak{M}\\otimes_{\\mathfrak{S},\\tau} A^{(2)} \\simeq \\mathfrak{M}\\otimes_{\\mathfrak{S}} A^{(2)}.\n$$\nWe will see $f_{\\tau}$ gives the descent isomorphism. As a result, we give a new proof for Theorem~\\ref{thm-intro-main-1}.\n\nAn advantage of our approach is that our new method can be easily generalized to the semi-stable representations cases. It turns out that the prism $(A^{(2)}_{\\st},(E))$ is isomorphic to the self-coproduct of $(A,(E))$ in the category of logarithmic prisms over $\\O_K$ defined by Koshikawa\\cite{Koshikawa2021log-prism}. Using the equivalence between prismatic $(\\varphi, \\hat G)$-modules and lattices in semi-stable representations of $G_K$. we will show in \\S\\ref{sec-logprismandsemistablereps} the following generalization of Theorem~\\ref{thm-intro-main-1} for semi-stable representations. \n\n\\begin{theorem}\\label{thm-intro-log-main}(Theorem~\\ref{thm-log-main-1})\nThere is an equivalence of the category of prismatic $F$-crystals in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ and the category of Galois stable lattices in semi-stable representations of $G_K$.\n\\end{theorem}\n\nAnother interesting and natural question one can ask is whether Theorem~\\ref{thm-intro-main-1} and Theorem~\\ref{thm-intro-log-main} can accommodate more general base rings. Motivated by our strategy, it seems to us that the answer should be affirmative if a suitable theory of $(\\varphi, \\hat G)$-module can accommodate more general base rings, for example, if the base ring $R$ is a complete DVR with \\emph{imperfect} residue field that admits a finite $p$-basis. We are working on such direction and hopefully will report our progress in the future. So part of our paper, for example, \\S~\\ref{sec-ring-strcuture} do allow specific general base rings.  \n\n\\subsection*{Acknowledgments} It is our pleasure to thank Hui Gao,  Wansu Kim, Teruhisa Koshikawa, Zeyu Liu,  Yong Suk Moon, Peter Scholze, Koji Shimizu, Yupeng Wang,  Zhiyou Wu and Min Yu for comments and conversations during the preparation of this paper. \n\n\\section{Ring Structures on certain prismatic envelope} \\label{sec-ring-strcuture}\n\nRecall that $K$ is a completed discrete valuation field in mix characteristic $(0 , p)$ with ring of integers of $\\O_K$ and prefect residue field $k$.   Write $W= W(k)$. Let $\\varpi\\in \\O_K$ be a uniformizer and $E= E(u)\\in W[u]$ be the Eisenstein polynomial of $\\varpi$. \nLet $\\mathbb C_p$ be the $p$-adic completion of $\\overline{K}$, and $\\O_{\\mathbb C_p}$ be the ring of integers. Let $R_0$ be a $W(k)$-algebra which admits Frobenius lift $\\varphi : R_0 \\to R_0 $. Set $R: = R_0 \\otimes_{W(k)}\\O_K$. We make the following assumptions for $R_0$ and $R$: \n\\begin{enumerate}\n\\item Both $R_0$ and $R$ are $p$-adically complete integral domains, and $R_0 \/ p R_0= R\/ \\varpi R$ is an integral domain;  \n    \\item Let $\\Breve{R}_0 =W\\langle t _1,  \\dots , t _m \\rangle $. $R_0$ is a $\\Breve{R}_0 $-\\emph{formally \\'etale} algebra with $p$-adic topology;   \n\\item $\\breve{R}_0$ admits a Frobenius lift such that $\\breve{R_0} \\to R_0$ defined in (2) is $\\varphi$-equivalent. \n\n\\item The $k$-algebra $R_0 \/ p R_0$ has finite $p$-basis in the sense of \\cite[Definition 1.1.1]{deJong}.\n\\end{enumerate}\nOur main example is $R_0= \\breve R_0 =  W(k). $ We will not use the finite $p$-basis assumption until \\S4. The following are other examples of  $R_0$:  \n\\begin{example}\\label{Eg-1} \n\\begin{enumerate}\n    \\item $R_0 = W(k) \\langle t _1^{\\pm 1} ,  \\dots , t _m ^{\\pm 1}\\rangle$ with $\\varphi (t_j) = t ^p_j$\n\\item $ R_ 0 = W(k) [\\![t]\\!]$ with $\\varphi (t) = t^p$ or $(1+t)^p -1 $.\n\\item $ R_0$ is an unramified complete DVR with imperfect field $\\kappa$ with finite $p$-basis. See  \\S\\ref{subsec-baserings} for more discussions. \n\\end{enumerate}\n\\end{example} \n\nWe reserve $\\gamma_i(\\cdot)$ to denote $i$-th divided power. \n\\subsection{Construction of \\texorpdfstring{$A^{(2)}$}{A(2)}} \\label{subsrc-construct-A2}\nLet $A=\\mathfrak{S}=R_0[\\![u]\\!]$ and extend $\\varphi : A \\to A$ by $\\varphi (u)= u^p$. It is well-known that $(A, E)$ is a prism and we can define a surjection $\\theta: A \\to R$ via $u\\mapsto \\varpi$. We have $\\Ker\\theta = (E(u))$.  Let $\\breve A := \\breve R_0 [\\![u]\\!]$ and define $\\varphi$ and $\\breve \\theta:  \\breve A \\to \\breve R : = \\O_K \\otimes _W \\breve R_0$ similarly. \nWe set \\[ A ^{\\ho 2}: = A [\\![y -x, s_1 - t_1, \\dots , s_m - t_m]\\!], \\ A^{\\ho 3}: = A[\\![ y -x, w-x , \\{ s_i - t _i , r_i - t_i\\}_{j= 1, \\dots , m}]\\!].\\] \nNote that $A ^{\\ho 2}$  (resp. $ A^{\\ho 3}$) is $\\breve A \\otimes_{\\mathbb Z_p} \\breve A $(resp. $\\breve A \\otimes_{\\mathbb Z_p} \\breve A \\otimes_{\\mathbb Z_p} \\breve A$)-algebra by $ u \\otimes 1 \\mapsto x$, \n$1\\otimes u \\mapsto y$ and $1 \\otimes t_i \\mapsto s_i$ (resp. $1\\otimes 1 \\otimes u \\mapsto w$ and $1 \\otimes 1 \\otimes  t_i \\mapsto r_i$). So in this way, we can extend Frobenius $\\varphi$ of $A$, which is compatible with that on $\\breve A$ to $A ^{\\ho 2}$ and $A^{\\ho 3}$. \nSet $J ^{(2)}= (E, y -x , \\{s_i- t _i \\}_{i = 1, \\dots,  m} )\\subset A^{\\ho 2}$ and $J ^{(3)} = (E, y-x , w-x  , \\{s_i-t_i , r_i - t_i\\}_{i = 1, \\dots , m}) \\subset A ^{\\ho 3}.$ Clearly, we have $A^{\\ho i}\/ J ^{(i)}\\simeq R$ for $i = 2, 3$. And we have $A^{\\ho 2}\/(p,E)$ (resp. $A^{\\ho 3}\/(p,E)$) is a formal power series ring over the variables $\\bar{y}-\\bar{x}, \\{\\bar{s}_i-\\bar{t}_i\\}_{i = 1, \\dots,  m}$ (resp. $\\bar{y}-\\bar{x} , \\bar{w}-\\bar{x}  , \\{\\bar{s}_i-\\bar{t}_i , \\bar{r}_i - \\bar{t}_i\\}_{i = 1, \\dots , m}$), so $(A,(E)) \\to (A^{\\ho i}, J ^{(i)})$ satisfies the requirements of  in \\cite[Prop. 3.13]{BS19}, and we can construct the prismatic envelope with respect to this map, which will be denoted by $A^{(i)}$. More precisely,  $A^{(i)}\\simeq A^{\\ho i}\\left \\{\\frac{J ^{(i)}}{E}\\right\\}_\\delta^{\\wedge}$, here $\\{\\cdot\\}_\\delta^{\\wedge}$ means freely adjoining elements in the category of $(p, E(u))$-completed $\\delta$-$A$-algebras. We will see  $A^{(i)}$, $i = 2 ,3$ are the self product and triple product of $A$ in category $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ in \\S  \\ref{subsec-pris-crystal}. \n\n\\subsection{The ring \\texorpdfstring{$A^{(2)}_{\\max}$}{A2max}} Now we set $t_0 = x$, $s_0 = y$ and \n\\[ z_j = \\frac{s_i - t _i}{E} \\textnormal{ and } z_0 = z=  \\frac{y -x }{E}= \\frac{s_0 - t_0}{E}. \\]\nNote that $A^{(i)}$ are  $A$-algebras via $u \\mapsto x$.\n\\begin{definition}\nLet $\\Omax$ be the $p$-adic completion of the $A$-subalgebra of $A[\\frac{1}{p}]$ generated by $p^{-1}E$. And let $A_{\\max}^{(2)}$ be the $p$-adic completion of the $A$-subalgebra of $ A [z_j ,  \\frac{1}{p}; j = 0 , \\dots , m ]$ generated by $p^{-1}E$ and $\\{\\gamma_i(z_j)\\}_{i\\geq 1, j = 0 , \\dots , m}$. \n\\end{definition}\nWe first note that $A^{(2)}_{\\max}$ is an $A^{\\widehat \\otimes 2}$-algebra via $ (s_j - t_j ) = E z_j, j =0 ,\\dots,  m$. Write $\\iota: A^{\\ho 2} \\to A^{(2)}_{\\max}$ for the structure map. \nBy construction, it is easy to see that $A^{(2)} _{\\max}\\subset R_0[\\frac 1 p] [\\![ E, z_j, j = 0, \\dots , m]\\!]$. In particular, $A^{(2)}_{\\max}$ is a domain and    \nany element $b\\in A^{(2)}_{\\max}$ can be \\emph{uniquely} written as \n$\\sum\\limits_{i_0= 0}^\\infty\\cdots \\sum\\limits_{i_m= 0 }^\\infty  b_{i_1 , \\dots, i _m} \\prod\\limits_{j= 0}^m\\gamma_{i_j} (z_{j})$ with $b _{i_0 , \\dots , i_m} \\in \\Omax$ and $b_{i_0, \\dots , i_m}\\to 0$ $p$-adically when $i_0 + \\cdots+  i_m \\to \\infty$.  \nOur next aim is to define $\\varphi$ on $A^{(2)}_{\\max}$. For this, we need a little preparation. \n\\begin{lemma}\\label{lem-Omax}\n$c: = \\frac{\\varphi(E)}{p}\\in \\Omax$ and $ c^{-1} \\in \\Omax$. \n\\end{lemma}\n\\begin{proof}\nWe have $A$ is a $\\delta$-ring, and $E$ is a distinguished element, so in particular\n$$\n\\varphi(E)\/p=c_0+E^p\/p\n$$\nwhere $c_0=\\delta(E)\\in A^\\times$. So $c = \\varphi(E)\/p\\in \\Omax$, and \n$c ^{-1} = c_0 ^{-1} \\sum\\limits_{i = 0}^\\infty \\frac {(- c_0^{-1} E^p ) ^i }{p ^i }\\in \\Omax. $\n\\end{proof}\n\nNow we define $\\varphi(z)=\\varphi(z_0)= \\frac{y^p-x^p}{\\varphi(E)}$ and $\\varphi (z_j) = \\frac{\\varphi(s_j)  - \\varphi(t_j) }{\\varphi (E)}$. Since \n\\begin{IEEEeqnarray*}{+rCl+x*}\n\\varphi(z)=\\frac{y^p-x ^p}{\\varphi(E)}=c^{-1}\\frac{y^p-x^p}{p}=c^{-1}\\frac{(x+ Ez)^p-x^p}{p} &= & c^{-1}\\sum_{i=1}^px^{p-i}(Ez)^i\\binom{p}{i}\/p\\\\\n&=& c^{-1}\\sum_{i=1}^{p}a_iz^i,\n\\end{IEEEeqnarray*}\nwhere $a_i\\in W(k)[\\![x]\\!][\\frac{E^p}{p}]\\subset \\Omax\\subset A^{(2)}_{\\max}$ and $c$ is a unit in $\\Omax$, we have $ \\varphi (z) \\in A^{(2)}_{\\max}$.\nThen \n$$\n \\gamma_n(\\varphi(z))=\\frac{\\varphi(z)^n}{n!}=\\frac{z^n}{n!}(c^{-1}\\sum_{i=1}^{p}a_iz^{i-1})^n\n$$\nis in $A^{(2)}_{\\max}.$  The argument for $\\varphi (z_j)$ for $j >1$ need a little more details. Note that $\\varphi (t_j) = t_j ^p + p \\delta (t_j)$ with $\\delta(t_j) \\in \\breve R_0$ by our assumptions. It is clear that $\\delta(s_j)-\\delta (t_j) = (s_j - t_j) \\lambda_j$ with $\\lambda_j \\in A ^{\\ho  2}$. Using that $(s_j -t_j) = E z_j$, so\n\\begin{equation}\\label{eqn-special-shape}\n\\varphi (z_j) = c^{-1} (\\frac{s^p _j - t^p_j}{p} + E z_j \\lambda_j)    \n\\end{equation}\nThe same argument as that for $\\varphi (z_0)$ also shows that $\\gamma_n (z_j)\\in A^{(2)}_{\\max}$, for $j =1 , \\dots , m$. \n\nSince any element $b\\in A^{(2)}_{\\max}$ can be uniquely written as $\\sum\\limits_{i_0= 0}^\\infty\\cdots \\sum\\limits_{i_m= 0 }^\\infty  b_{i_1 , \\dots, i _m} \\prod\\limits_{j= 0}^m\\gamma_{i_j} (z_{j})$ with $b _{i_0 , \\dots , i_m} \\in \\Omax$ and $b_{i_0, \\dots , i_m}\\to 0$ $p$-adically when $i_0 + \\cdots+  i_m \\to \\infty$, this allows to extend Frobenius map  $\\varphi $ on $ A$ to a \\emph{ring} map $\\varphi: A^{(2)}_{\\max} \\to A^{(2)}_{\\max}$ by sending $u \\mapsto u ^p$,  $z \\mapsto \\frac{y ^p -x^p}{\\varphi(E)}$,  $\\varphi (z_j) = \\frac{\\varphi(s_j)  - \\varphi(t_j)}{\\varphi (E)}$,  and $\\gamma_i (z_j) \\mapsto \\gamma_i (\\varphi (z_j))$ as the above.\n\n\\begin{remark}\\label{rem-not-Frob-lift} The ring map $\\varphi: A^{(2)}_{\\max}\\to A^{(2)}_{\\max}$ is \\emph{not} a Frobenius lift of $A^{(2)} _{\\max}\/ p$ because $\\varphi (E\/p)- (E\/p) ^p \\not \\in p A^{(2)}_{\\max}$. In particular, $A^{(2)}_{\\max}$ is not a $\\delta$-ring. \n\\end{remark}\n\nRecall that $A^{(2)}_{\\max}$ is an $A^{\\ho 2}$-algebra via map  $\\iota : A^{\\ho 2}\\to A^{(2)}_{\\max}$. The above construction of Frobenius $\\varphi$ on $A^{(2)} _{\\max}$ is obviously compatible with  $\\iota$. \n\nOur next goal is to show that $\\iota$ induces a map $A^{(2)}\\to A^{(2)}_{\\max}$ so that $A^{(2)} $ is a subring of $A^{(2)}_{\\max}$ which is compatible with $\\varphi$-structures and filtration. We need a little preparation. \nWrite $\\mathfrak z_{n}= \\delta^n (z)$ with $\\delta_0(z)= z= \\mathfrak z_0$, and $A_0 = W(k) [\\![u]\\!]$. \n\\begin{lemma}\\label{lem-delta-n} $$\\delta^n (Ez) = b_n \\mathfrak z_{n} + \\sum_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i.  $$\nwhere $a^{(n)}_i \\in A_0 [\\mathfrak z_0, \\dots, \\mathfrak z_{n -2}]$ so that $a^{(n)}_p\\in A_0 ^\\times$ and for $0 \\leq i \\leq p-1$ each monomials of  $a^{(n)}_i $ contains a  factor $\\mathfrak z_{j}^p$  for some  $ 0 \\leq j\\leq n -2$. Furthermore,  $b_{n+1 } =p\\delta (b_n ) + b^p_n $ and $b_1 = p \\delta (E) + E^p$. \n\\end{lemma}\n\\begin{proof} Given  $f \\in A_0[x_1 , \\dots , x_m]$, if each monomials of $f$  contains $x_j^l$ for some $j$ and $l \\geq p$ then we call $f$  \\emph{good}.  For example, $f= x_1^p x_2 + 2 x_ 1x_2^{p +3}.$ So we need to show that $a^{(n)}_i\\in A_0[\\mathfrak z_0, \\dots, \\mathfrak z_{n -2}]$ is good.  Before making induction on $n$, we discuss some properties of good polynomial. It is clear that the set of good polynomials is closed under addition and multiplications. Note that \n\\begin{equation}\\label{eqn-delta}\n\\delta(\\mathfrak z_{l}^i)= \\frac 1 p (\\varphi (\\mathfrak z_{l}^i) - \\mathfrak z_{l} ^{p i})= \\frac 1 p \\big( (p\\mathfrak z_{l +1} + \\mathfrak z_{l } ^p)^i - \\mathfrak z_{l}^{p i}\\big) = \\sum \\limits_{j  = 1 }^{i} \\binom{i}{j}(p^{j -1} \\mathfrak z_{l }^{p(i-j)} ) \\mathfrak z_{l+1} ^{j }. \n\\end{equation}\nIn particular, given an $f\\in A_0[\\mathfrak z_0 , \\dots , \\mathfrak z_{m}]$,  $\\delta(\\mathfrak z_{m}^p f)= f^p \\delta (\\mathfrak z_{m} ^p) +  \\mathfrak z_{m}^{p ^2}\\delta (f) + p \\delta (\\mathfrak z_{m} ^p)\\delta (f)$  is  a good polynomial in $A[\\mathfrak z_0 , \\dots, \\mathfrak z_{m+1}]$. Using the fact that \n$\\delta (a+b)=\\delta(a) + \\delta (b) + F(a, b)$ where  $F(X, Y) = \\frac 1 p ( X^p + Y^p - (X+Y)^p) = - \\sum\\limits_{i = 1}^{p-1} \\binom{p}{i}\/p X^i Y^{p-i}$, together with the above argument of $\\delta(\\mathfrak z_l^p f)$, it is not hard to show that if $g\\in A_0[\\mathfrak z_0 , \\dots , \\mathfrak z_{m}]$ is good  then $\\delta (g) \\in A_0[\\mathfrak z_0 , \\dots , \\mathfrak z_{m}, \\mathfrak z_{m +1}]$ is also good. \n\nNow we make induction on $n$. When $n =1$, we have \n$$\\delta (Ez)= E^p \\mathfrak z_{1} + z^p \\delta(E) + p \\delta (E) \\mathfrak z_{1}= (p \\delta (E) + E^p) \\mathfrak z_{1} + \\delta(E) z^p.$$ \nThen $b_1 = p \\delta (E) + E^p $,  $a^{(1)}_p= \\delta(E)\\in A_0 ^\\times$ and $a^{(1)}_i = 0$ for $1 \\leq i \\leq p-1$ are required. Now assume the formula is correct for $n$, then \n$$\\delta ^{n +1} (Ez) = \\delta (b_n \\mathfrak z_{n} + \\sum_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i) = \\delta (b_n \\mathfrak z_{n}) + \\delta (\\sum_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i)) + F(b_n \\mathfrak z_{n}, \\sum_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i)), $$\nClearly, $ F(b_n \\mathfrak z_{n}, \\sum\\limits_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i))  = \\sum\\limits_{j = 1} ^{p-1} \\tilde a^{(n)}_j \\mathfrak z_{n} ^j$ with $\\tilde a^{(n)}_j$ being good. An easy induction shows that \n$\\delta (\\sum\\limits_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i) = \\sum\\limits_{i = 0} ^p \\delta (a^{(n)}_i \\mathfrak z_{n -1} ^i) + f$ with $f \\in A_0[\\mathfrak z_0, \\dots, \\mathfrak z_{n -1}]$ being good. Since \n$\\delta (a^{(n)}_i \\mathfrak z_{n -1} ^i)= (a^{(n)}_i)^p \\delta (\\mathfrak z_{n -1}^i) + (\\mathfrak z_{n-1}^{pi})\\delta (a_i ^{(n)}) + p \\delta (\\mathfrak z_{n -1}^i  ) \\delta (a_i^{(n)})$, by using formula of $\\delta(\\mathfrak z_{n-1}^i)$ in \\eqref{eqn-delta} and that $a^{(n)}_i$ is good implies that \n$\\delta (a_i^{(n)}) $ is also good, we conclude that for $0 \\leq i \\leq p-1$,  $$\\sum\\limits_{i = 0} ^{p-1} \\delta (a^{(n)}_i \\mathfrak z_{n -1} ^i) = \\sum_{i =0}^{p-1} \\alpha_i \\mathfrak z_{n} ^i$$ with $\\alpha_i \\in A_0 [\\mathfrak z_0, \\dots, \\mathfrak z_{n-1}]$ being good polynomials. Using that $a _p^{(n)} \\in A_0 ^\\times$, we compute that  $\\delta (a_p^{(n)}\\mathfrak z^p_{n-1}) = \\sum \\limits_{i = 0}^p \\beta_i \\mathfrak z_{n}^i$ with $\\beta_p \\in p A_0$ and $\\beta_j\\in A_0 [\\mathfrak z_0 , \\dots , \\mathfrak z_{n-1}]$ being good for $1\\leq j \\leq p-1$. \n Now we only need to analyze $\\delta (b_n \\mathfrak z_{n})$, which is \n $\\delta (b_n ) \\mathfrak z_{n}^p + b_n^p \\mathfrak z_{n+1} + p \\delta (b_n)\\mathfrak z_{n+1}$. So $b_{n+1} = p \\delta(b_n) + b_n ^p$ and $a_p^{(n+1)} = \\delta (b_n) + \\beta_p$. Since $\\delta(b_n) \\in A_0^\\times$, we see that $a_p^{(n+1)}= \\delta(b_n) + \\beta_p \\in A_0^\\times$ as required. \n\\end{proof}\n\nLet $\\widetilde A^{(2)} := A^{\\ho 2} [z_j]_\\delta= A^{\\ho2} [\\delta ^n (z_j), n \\geq 0, j =0 , \\dots , m]$ and natural map $\\alpha : \\widetilde A^{(2)} \\to \\widetilde A^{(2)} [\\frac 1 p]$ (we do not know $\\alpha$ is injective at this moment).  \n\\begin{lemma}\\label{lem:gamma(z)-polynomial-in-E\/p}\nFor $i\\geq 0$ and $j=0,1,\\ldots,d$, there exists $f_{ij}(X) \\in \\widetilde A^{(2)} [X]$ such that, as elements of $\\widetilde A^{(2)}[\\frac 1 p] $ via $\\alpha: \\widetilde A^{(2)} \\to \\widetilde A^{(2)} [\\frac 1 p]$, \n\\[\n\\gamma_i(z_j) = f_{ij}\\Bigl(\\frac{E}{p}\\Bigr).\n\\]\n\\end{lemma}\n\\begin{proof}\nWrite $z = z_{j}$ for simplicity, and let $\\tilde{\\gamma}(z)= \\frac{z^p}{p}$ and $\\tilde{\\gamma}^n = \\underbrace{\\tilde{\\gamma} \\circ \\tilde{\\gamma} \\cdots \\circ \\tilde{\\gamma}}_n$. It suffices to show that for each $n \\geq 1$, we have $\\tilde{\\gamma}^n(z) = f_n(\\frac{E}{p})$ inside $\\widetilde A^{(2)}[\\frac 1 p]$ for some $f_n(X) \\in \\widetilde A^{(2)} [X]$. For an element $x \\in A[\\delta^i(z)]_{i \\geq 0}$, we say that $x$ has \\emph{$\\delta$-order $\\leq n$}  if $x\\in \\sum_{0\\leq j\\leq n}A[\\{\\delta^i(z)\\}_{0 \\leq i \\leq n }] \\delta^j(z)$, namely, if $x$ can be written as a sum of monomials such that each term is divisible by $\\delta^j(z)$ for some $0 \\leq j \\leq n$. \n\nWe claim that the following two equations hold for each $n \\geq 1$:\n\\begin{enumerate}\n\\item We have\n\\begin{equation} \\label{eq:delta(z)}\n    \\delta^n(z) = \\nu_n \\tilde{\\gamma}^n(z)+P_n\\Bigl(\\frac{E}{p}\\Bigr)+\\frac{E^p}{p}d_n\\delta^n(z)\n\\end{equation}\nfor some $\\nu_n \\in A^{\\times}$, $d_n \\in A$, and $P_n(X) \\in (A[\\delta^i(z)]_{i \\geq 0})[X]$ such that each coefficient of $P_n(X)$ has $\\delta$-order $\\leq n-1$. \n\n\\item We have\n\\begin{equation} \\label{eq:gamma(delta(z))}\n    \\tilde{\\gamma}(\\delta^{n-1}(z)) = \\mu_{n-1}\\tilde{\\gamma}^n(z)+Q_{n-1}\\Bigl(\\frac{E}{p}\\Bigr)\n\\end{equation}\nfor some $\\mu_{n-1} \\in A^{\\times}$ and $Q_{n-1}(X) \\in (A[\\delta^i(z)]_{i \\geq 0})[X]$ such that each coefficient of $Q_{n-1}(X)$ has $\\delta$-order $\\leq n-1$.\n\\end{enumerate}\n\nWe prove claims (1) and (2) by induction. For $n = 1$, since\n\\[\n\\delta(Ez) = z^p\\delta(E)+(p\\delta(E)+E^p)\\delta(z)\n\\]\nand $\\delta(E) \\in \\mathfrak{S}^{\\times}$, we have\n\\[\n\\delta(z) = -\\tilde{\\gamma}(z)+\\delta(E)^{-1}\\frac{\\delta(Ez)}{p}-\\delta(E)^{-1}\\frac{E^p}{p}\\delta(z). \n\\]\nBy easy induction, we also have $\\delta^i(Ez) \\in (Ez)A$ for each $i \\geq 1$. So claim (1) holds. Claim (2) holds for $n = 1$ trivially with $Q_0(X) = 0$.\n\nSuppose that claims (1) and (2) hold for $1 \\leq n \\leq m$. We will verify claims (1) and (2) for $n = m+1$. We first consider claim (2). Since each coefficient of $P_m(X)$ has $\\delta$-order $\\leq m-1$, $\\frac{E^p}{p}=p^{p-1}\\bigl(\\frac{E}{p}\\bigr)^p$, and Equations \\eqref{eq:delta(z)} and \\eqref{eq:gamma(delta(z))} hold for $1\\leq n \\leq m$, applying $\\tilde{\\gamma}(\\cdot)$ to Equation \\eqref{eq:delta(z)} for $n = m$ yields\n\\[\n\\tilde{\\gamma}(\\delta^m(z)) = \\nu_m^p \\tilde{\\gamma}^{m+1}(z)+Q_m\\Bigl(\\frac{E}{p}\\Bigr)\n\\]\nfor some $Q_m(X) \\in (\\mathfrak{S}[\\delta^i(z)]_{i \\geq 0})[X]$ such that each coefficient of $Q_m(X)$ has $\\delta$-order $\\leq m$. This proves the claim (2) for $n = m+1$.\n\n\nWe now consider claim (1) for $n = m+1$. By the above Lemma for $n = m+1$ and that $b_n= p\\alpha_n +\\beta_n  E^p$\nfor some $\\alpha_n \\in A^{\\times}$ and $\\beta_n \\in  A$ (via an easy induction on $n$), we have \n\\[\n\\alpha_{m+1}\\delta^{m+1}(z) = \\frac{\\delta^{m+1}(Ez)}{p}-\\beta_{m+1}\\frac{E^p}{p}\\delta^{m+1}(z)-a_p^{(m+1)}\\tilde{\\gamma}(\\delta^m(z))-\\frac{1}{p}\\sum_{j=0}^{p-1} a_j^{(m+1)}(\\delta^{m}(z))^j.\n\\]\nAs noted above, we have $\\delta^{m+1}(Ez) \\in (Ez)A$. Furthermore, by the condition on $a_j^{(m+1)}$, the last term $\\frac{1}{p}\\sum_{j=0}^{p-1} a_j^{(m+1)}(\\delta^{m}(z))^j$ is a linear combination of terms involving $\\tilde{\\gamma}(\\delta^l(z))=\\frac{1}{p}(\\delta^l(z))^p$ for some $0\\leq l\\leq m-1$.\nThus, by applying Equations~\\eqref{eq:delta(z)} and \\eqref{eq:gamma(delta(z))} for $1 \\leq n \\leq m$, we see that claim (1) also holds for $n = m+1$ with $\\nu_{m+1} = -\\alpha_{m+1}^{-1}a_p^{(m+1)}\\mu_m$ and $d_{m+1} = -\\alpha_{m+1}^{-1}\\beta_{m+1}$.\nThis completes the induction and prove the lemma .\n\\end{proof}\n\n\\begin{remark}\nIn the above proof, by equation~\\eqref{eq:gamma(delta(z))}, we even have for each $i,j\\geq 0$, $\\gamma_i(\\delta^j(z))=f(\\frac{E}{p})$ for some $f\\in \\widetilde A^{(2)} [X]$. \n\\end{remark}\n\nAn easy induction by \\eqref{eq:delta(z)} implies that $\\alpha (\\delta^n (z) ) \\in A^{\\ho 2}[\\{\\gamma_i (z_j )\\}_{i \\geq 0 , j =1 , \\dots , m}, \\frac E p]\\subsetA^{(2)}_{\\max}$, which satisfies equations in Lemma \\ref{lem-delta-n} by replacing \n$\\mathfrak z_n$ by $\\alpha (\\delta^n (z))$ inside $A^{(2)}_{\\max}$. It is clear that $\\iota$ is still Frobenius compatible (because both $A ^{\\ho 2}$ and $A^{(2)}_{\\max}$ are domains). Since $E = p \\frac E p$, $\\iota$ is a continuous for $(p , E)$-topology on $\\widetilde A^{(2)} $ and $p$-topology on $A^{(2)}_{\\max}$. Finally, we construct a ring map $\\iota: A^{(2)} \\to A^{(2)}_{\\max}$ so that $\\iota$ is compatible with Frobenius.  \n\nOur next goal is to show that $\\iota$ is injective. Define ${\\textnormal{Fil}} ^i A^{(2)}_{\\max}[\\frac 1 p]:  = E^i A^{(2)}_{\\max}[\\frac 1 p]$. For any subring $B \\subset A^{(2)}_{\\max}[\\frac 1 p]$, set \n$${\\textnormal{Fil}} ^i B : = B \\cap {\\textnormal{Fil}} ^i A^{(2)}_{\\max}[\\frac 1 p]= B \\cap E^i A^{(2)}_{\\max}[\\frac 1 p].$$\nLet $D_z$ be the $p$-adic completion of $R [\\gamma_i (z_j), i \\geq 0;  j = 0, \\dots , m]$. \n\\begin{proposition} \\label{prop-key-property}\n\\begin{enumerate}\n    \\item $\\widetilde A^{(2)}\/ E = R [\\gamma_i (z_j ), i \\geq 0; j = 0, \\dots , m]$.\n    \\item $ A^{(2)}\/ E \\simeq D_z$.\n    \\item $\\iota$ is injective.\n    \\item ${\\textnormal{Fil}} ^1 A^{(2)}= E A ^{(2)}$.\n    \\item $A^{(i)}$ are flat over $A$ for $i = 2, 3$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n(1)  By definition, $\\widetilde A^{(2)}  = A ^{\\ho  2}[ z^{(n)}_j , n\\geq 0; j = 0 , \\dots, m ]\/ J $ where $\\mod J  $ is equivalent the following relations (note that $z_0 = z$):  $E z= (x-y), E z_j = s_j - t_j, \\delta  (z^{(n)}_j)= z^{(n+1)}_{j}, \\delta ^n (Ez) = \\delta^n (y -x) , \\delta ^n (Ez_j)= \\delta ^n (s_j-t _j). $ Since $\\delta (x-y)= \\frac{(x^p - y ^p)- (x-y)^p}{p}$ and $\\delta(s_j - t_j) = \\frac{\\varphi (s_j - t_j)- (s_j - t_j)^p}{p}$, \nit is easy to prove by induction that $\\delta^n (x-y)$  and $\\delta^n (s_j - t_j)$ always contains a factor $(x-y)$, $s_j - t_j$ and hence $\\delta^n (x-y), \\delta(s_j - t_j)\\equiv 0 \\mod E$. Therefore $\\delta ^n (Ez_j) \\equiv 0 \\mod E$. By Lemma \\ref{lem-delta-n}, we see that \n$$p \\mu_n z^{(n )}_{j} = -\\sum_{i = 0} ^p  \\overline{a^{(n)}_i} (z^{(n -1)}_j) ^i  \\mod E \\text{ and } pz^{(1)}_j = z_j^p \\mod E $$\nwhere $\\overline {a^{(n)}_i} = a^{(n)}_i \\mod E$ and $\\mu_n = \\frac{\\delta (b_n)}{p}\\mod E \\in \\O_K^\\times$. Using that $a_p^{(n)} \\in A_0^\\times$, and $a_i^{(n)}, 1 \\leq i \\leq p-1$ are good in the sense that they contains factor of $(z^{(l)}_j) ^p$ for some $l = 0 , \\dots, n-2$, we easily see by induction that \n$\\widetilde A^{(2)} \/E = R [\\widetilde \\gamma^n (z_j ), n \\geq 0; j = 0 , \\dots , m  ]$. But it is well-known that $R [\\widetilde \\gamma^n (z_j), n \\geq 0; j = 0 , \\dots , m ] = R [\\gamma_n (z_j), n \\geq 0; j = 0 , \\dots , m]. $\n\nNow we show that the natural map $\\iota:  \\widetilde A^{(2)}\\to A^{(2)}_{\\max}[\\frac  1 p]$ induced by $\\alpha (\\delta ^n(z_j))$ is injective. Note that $\\widetilde A^{(2)} $ is the direct limit of $\\widetilde A^{(2)}_n : =  A ^{\\hat \\otimes 2}[\\{ \\delta^i (z_j)\\}_{i = 1 , \\dots , n, j = 0 , \\dots , m}]$. A similar argument similar as above show that $\\widetilde  A^{(2)}_n \/ E $ injects to $A^{(2)}_{\\max}[\\frac 1 p]\/E = D_z [\\frac 1 p]$. \nSince $\\widetilde A^{(2)}_n$ is $E$-separate and  $A^{(2)}_{\\max}$ is a domain, this implies that $\\widetilde A^{(2)}_n$ injects to $A^{(2)}_{\\max}[\\frac 1 p]$. So \n$\\widetilde A^{(2)}$ injects to $A^{(2)}_{\\max}$ via $\\iota$. \n\n(2) Since $A^{(2)}$ is $(p, E)$-completion of $\\widetilde A^{(2)}$ \\footnote{Indeed, $A^{(2)}$ is \\emph{derived} $(p, E)$-completion. Since $\\widetilde A^{(2)}\/ E$  is $\\mathbb Z_p$-flat, then derived completion coincides with the classical completion, which is used here.}, we have a natural map from $\\bar \\iota: A^{(2)}\/E \\to D_z$. The surjectivity of $\\bar \\iota$ is straightforward as $A^{(2)}$ is also $p$-complete. To see injectivity, given an sequence $f_n $ so that $f_{n +1}- f _n \\in  (p , E)^n \\widetilde A^{(2)} $ and $ f_n = E g_n$ for all $n$, we have to show that $g_n$ is a convergent sequence in $A^{(2)}$.  Since $E (g_{n +1} - g_n) = \\sum_{i = 0} ^n p ^i E ^{n - i} h_i$ with $h _i \\in \\widetilde A^{(2)} $. Then $ E|p ^n h_n $. Since $\\widetilde A^{(2)} \/E$ has no $p$-torsion,\nwe have $E | h_n $ and write $h_n  = E h'_n $. Since $\\widetilde A^{(2)} $ is a  domain as it is inside the fraction field of $A ^{\\ho 2}$, we see that \n$ g_{n +1}- g_n = p ^n h'_n + \\sum\\limits_{i = 0} ^{n -1} p ^i E^{n - i - 1} h_i $. Hence $g_n$ converges in $ A^{(2)}$ as required. \n \n(3) It is clear that $A^{(2)}_{\\max}[\\frac 1 p]\/ E \\simeq D_z [\\frac 1 p]$. So the map $\\iota \\mod E(u)$ induces an injection $D_z \\hookrightarrow D_z [\\frac 1 p]$. So for any $x\\in \\Ker (\\iota)$, we see that $x= Ea$ for some $a \\in A^{(2)}$. As $A^{(2)}_{\\max}$ is $E$-torsion free and $A^{(2)}$ is $E$-complete, we see that $x= 0$ as required.  \n \n(4) By the definition of ${\\textnormal{Fil}} ^1 A^{(2)}  $, we see that $E A ^{(2)} \\subset {\\textnormal{Fil}} ^1 A ^{(2)}$ and $ A^{(2)} \/ {\\textnormal{Fil}} ^1 A^{(2)} $ injects to $A^{(2)}_{\\max}[\\frac 1 p ]\/E= D_z[\\frac 1 p]$. But we have seen that $A^{(2)}\/E = D_z$ injects to $D_z$. Then ${\\textnormal{Fil}} ^1 A^{(2)} = E A^{(2)}$. \n\n(5) Both $A^{(2)}$ and $A^{(3)}$ are obtained by the construction of \\cite[Proposition 3.13]{BS19}, which implies that they are $(p,E)$-complete flat over $A$. Since $A$ is Noetherian, by \\cite[Tag 0912]{stacks-project}, we have both $A^{(2)}$ and $A^{(3)}$ are $A$-flat.\n\\end{proof}\n\\begin{corollary}\\label{cor-filtration-shape}\n\\begin{enumerate}\n    \\item ${\\textnormal{Fil}} ^i A ^{(2)} = E^i A^{(2)}. $\n    \\item $A^{(i)}$ are bounded prisms for $i = 2, 3$. \n\\end{enumerate}\n\\end{corollary} \n\\begin{proof}These follow that $A^{(2)} \/ E A^{(2)} \\simeq D_z$ which is $\\mathbb Z_p$-flat. For (2), we have $A^{(2)}$ and $A^{(3)}$ are $(p,E)$-complete flat over $A$, so boundedness follows from (2) in \\cite[Lemma 3.7]{BS19}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem-subring} $A^{(2)}$ is a closed subset inside $A^{(2)}_{\\max}$.  \n\\end{lemma}\n\\begin{proof} We need to show the following statement:  Given $ x \\in \\widetilde A^{(2)} $, if $x= p ^n y$ with $y \\in A^{(2)}_{\\max}$ then $x = \\sum\\limits_{i = 0}^n p ^{n-i} E^i x_i$ with $x_i \\in \\widetilde A^{(2)} . $ Indeed, since $A^{(2)}\/E \\simeq A^{(2)}_{\\max}\/ {{\\textnormal{Fil}} ^1}$, there exists $x_0, w_1 \\in \\widetilde A^{(2)} $ so that $x= p ^n x_0 + E w_1$. Then $Ew_1 \\in p ^n  A^{(2)}_{\\max}$. Write $ E w_1= p ^n \\sum \\limits_{i =0} ^\\infty\\sum\\limits_{j =0}^m f_{ij} \\gamma_i (z_j)$, we see that $f_{ij}= \\sum_{l \\geq 1} a_{ijl}  \\frac{E^l}{p^l}\\in {\\textnormal{Fil}} ^1 \\Omax$. So it is easy to see that $p ^n E^{-1}f_{ij} \\in p ^{n -1} \\Omax$ and then \n$w_1 = p ^{n -1} x_1 $ with $x_1 \\in A^{(2)}_{\\max}$. Then we may repeat the above argument to $w_1$, and finally $x= \\sum\\limits_{i = 0}^n p ^{n-i} E^i x_i$ with $x_i \\in \\widetilde A^{(2)}$ as required. \n\\end{proof}\n\nNow we realize $A^{(2)}$ as a subring of $A^{(2)}_{\\max}$ via $\\iota$. We need to introduce some auxiliary rings. By the description of elements in $A^{(2)}_{\\max}$, we define ${\\widetilde S}_0$ be the subring of $A^{(2)}_{\\max}$ as follow\n$$\n\\widetilde{S} :=  A^{(2)} [\\![ \\frac {E ^p}{p} ]\\!] := \\{ \\sum_{i \\geq 0} a_i (\\frac{E^p}{p})^i \\mid a_i\\in A^{(2)} \\}.\n$$\nAnd when $p=2$, we define \n$\n\\widehat{S} := A^{(2)} [\\![\\frac{E^4}{2}]\\!]\n$ simiarly. \nWe will have $\\widehat{S} \\subset \\widetilde{S} \\subset A^{(2)}_{\\max}$. Viewing $\\widetilde S$ and $\\widehat{S}$ as subrings of $A^{(2)}_{\\max}$, we give them the filtration induced from $A^{(2)}_{\\max}$. The following lemma is crucial for later applications and we thank Yong Suk Moon for many useful comments to improve many details in the proof.  \n\n\\begin{lemma}\\label{lem-auxiliaryrings}\nFix $h \\in \\mathbb N$, then we have\n\\begin{enumerate}\n    \\item We have $\\varphi(A^{(2)}_{\\max}) \\subset \\widetilde{S} \\subset A^{(2)}_{\\max}$, and when $p=2$, we have $\\varphi(\\widetilde{S}) \\subset \\widehat{S} \\subset \\widetilde{S}$;\n    \\item $x \\in {\\textnormal{Fil}} ^h \\widetilde S $ if and only if $x$ can be written as\n    $$\n    x = \\sum\\limits_{i \\geq h } a_i \\frac {E ^i}{p ^{\\lfloor \\frac i p\\rfloor}} \n    $$\n    with $a_i\\in A^{(2)}$. \n    \\item when $p>2$, there is a $h_0>h$ such that $\\varphi ({\\textnormal{Fil}} ^m \\widetilde S ) \\subset A ^{(2)} + E^h{\\textnormal{Fil}}^{m +1} \\widetilde S$ for all $m > h_0$;\n    \\item when $p=2$, then $x \\in {\\textnormal{Fil}} ^h \\widehat S $ if and only if $x$ can be written as\n    $$\n    x = \\sum\\limits_{i \\geq h } a_i \\frac {E ^i}{2 ^{\\lfloor \\frac i 4\\rfloor}} \n    $$\n    with $a_i\\in A^{(2)}$;\n    \\item when $p=2$, there is a $h_0>h$ such that $\\varphi ({\\textnormal{Fil}} ^m \\widehat{S}  ) \\subset A ^{(2)} + E^h{\\textnormal{Fil}}^{m +1} \\widehat{S} $ for all $m > h_0$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFor $(1)$, any $a\\in A^{(2)}_{\\max}$, we can write \n$$\na = \\sum _{i_0 = 0}^\\infty \\cdots \\sum_{i _m = 0}^\\infty \\sum_{l = 0}^\\infty  a_{i_0 , \\dots , i_m, l } \\left (\\frac E p\\right ) ^l \\prod_{j = 0}^m \\gamma_{i_j} (z_j)\n$$\nwhere $a_{i_0 , \\dots , i_m, l}\\in A$ and $a_{i_0 , \\dots , i_m, l}\\to 0$ $p$-adically when $\\sum_j i _j + l \\to \\infty$. Thanks for Lemma \\ref{lem:gamma(z)-polynomial-in-E\/p}, we see that $b_{i_0 , \\dots , i_m , l}: = \\varphi \\left  (\\left (\\frac E p\\right ) ^l \\prod_{j = 0}^m \\gamma_{i_j} (z_j) \\right)\\in \\widetilde S$. So $\\varphi (a) = \\sum a_{i_0 , \\dots , i_m , l} b_{i_0 , \\dots , i_m , l}$ converges in $\\widetilde S$. \n\nFor the claim in $(1)$ for $p=2$, we have $\\varphi(\\frac{E^2}{2})=(E^2+2b')^2\/2= \\frac{E^4}{2} + 2b$ for some $b,b'\\in A$. And for $a = \\sum_{i \\geq 0} a_i (\\frac{E^p}{p})^i \\in \\widetilde{S}$, we have \n$$\n\\varphi(a) = \\sum_{i \\geq 0} \\varphi(a_i) (\\frac{\\varphi(E^2)}{2})^i= \\sum_{i \\geq 0} \\varphi(a_i) \\sum_{j=0}^{i} c_{ij}(2b)^{i -j} (\\frac{E^4}{2})^j = \\sum_{j \\geq 0} \\left (\\sum_{i=j}^{\\infty} \\varphi(a_i)c_{ij} (2b)^{i-j} \\right ) (\\frac{E^4}{2})^j\n$$\nfor some $c_{ij} \\in \\mathbb Z$. So we have $\\varphi(a) \\in \\widehat{S}$.\n\nFor $(2)$, the if part is trivial. For the other direction, any $x \\in {\\textnormal{Fil}} ^h \\widetilde S $, we have \n$$\nx = \\sum\\limits_{i \\geq 0 } a_i \\frac {E^{i}}{p^{\\lfloor \\frac i p  \\rfloor}} \n$$\nas element in $\\widetilde S$. And if we also have $x \\in {\\textnormal{Fil}} ^h A^{(2)}_{\\max}[\\frac  1 p ] = E^h A^{(2)}_{\\max}[\\frac 1 p]$, this implies for $\\tilde a_0=\\sum\\limits_{0\\leq i \\leq h} a_i \\frac {E^{i}}{p^{\\lfloor \\frac i p \\rfloor}}$ is in ${\\textnormal{Fil}} ^h A^{(2)} [\\frac 1 p]$. This implies $p^{\\lfloor \\frac h p\\rfloor}\\tilde a_0 \\in {\\textnormal{Fil}} ^h A^{(2)}= E^h A^{(2)}$. That is $\\tilde a_0={p^{-\\lfloor \\frac h p\\rfloor}}{E^h} b$ for some $b \\in A^{(2)}$. So $x$ is of the given form. The proof for $(4)$ is similar.\n\nFor $(3)$, we have by $(2)$, $x \\in {\\textnormal{Fil}} ^m \\widetilde S$, $x$ can be written as\n$$\nx = \\sum\\limits_{i \\geq m } a_i \\frac {E ^i}{p ^{\\lfloor \\frac i p\\rfloor}}.\n$$\nAnd use the fact $\\varphi(E)=E^p+pb$ for some $b \\in A^{(2)}$, we have \n$$\n\\varphi(x) = \\sum\\limits_{i \\geq m } \\varphi(a_i) \\sum_{j=0}^{i} \\frac {c_{ij}E^{p(i-j)} p^j}{p ^{\\lfloor \\frac i p\\rfloor}} = \\sum_{i \\geq m} \\sum_{ j \\geq \\lfloor \\frac i p\\rfloor}^{i} \\frac {b_{ij}E^{p(i-j)} p^j}{p ^{\\lfloor \\frac i p\\rfloor}} + \\sum_{i \\geq m} \\sum_{0 \\leq j < \\lfloor \\frac i p\\rfloor} E^h\\frac {b_{ij}E^{p(i-j)-h} p^j}{p ^{\\lfloor \\frac i p\\rfloor}}\n$$    \nwith  $b_{ij} \\in A^{(2)}$.\n\nIn particular, we have $\\sum_{i \\geq m} \\sum_{ j \\geq \\lfloor \\frac i p\\rfloor}^{i} \\frac {b_{ij}E^{p(i-j)} p^j}{p ^{\\lfloor \\frac i p\\rfloor}}$ is inside $A^{(2)}$. To prove $(3)$, it is amount to find $h_0$ such that whenever $m>h_0$, $i \\geq m$ and $0 \\leq j < \\lfloor \\frac i p\\rfloor$, we have \n$$\n\\sum_{i \\geq m} \\sum_{0 \\leq j < \\lfloor \\frac i p\\rfloor} \\frac {b_{ij}E^{p(i-j)-h} p^j}{p ^{\\lfloor \\frac i p\\rfloor}} \\in {\\textnormal{Fil}}^{m +1} \\widetilde S.\n$$\nThe claim follows if we can find $h_0 > h$ such that $\\frac {E^{p(i-j)-h} p^j}{p ^{\\lfloor \\frac i p\\rfloor}} \\in \\widetilde S$ and $p (i-j)- h \\geq m +1$ for all $m>h_0$, $i \\geq m$ and $0 \\leq j < \\lfloor \\frac i p\\rfloor$. That is $\\lfloor\\frac{p (i -j)-h}{p} \\rfloor +j \\geq \\lfloor \\frac i p \\rfloor $ and $p(i-j)-h\\geq m+1$ for all $i,j,m$ in this range. And solve this we have it is enough to choose $h_0 > \\max \\{h, \\frac{p(h+1)+1}{p(p-2)}\\}$, which is valid for $p>2$.\n\nStatement in $(5)$ is similar to $(3)$. Any $x \\in {\\textnormal{Fil}} ^m \\widehat S$, $x$ can be written as\n$$\nx = \\sum\\limits_{i \\geq m } a_i \\frac {E ^i}{2^{\\lfloor \\frac i 4\\rfloor}}.\n$$\nWe have $\\varphi(E)=E^2+2b$ for some $b \\in A^{(2)}$, so \n$$\n\\varphi(x) = \\sum\\limits_{i \\geq m } \\varphi(a_i) \\sum_{j=0}^{i} \\frac {c_{ij}E^{2(i-j)} 2^j}{2 ^{\\lfloor \\frac i 4\\rfloor}} = \\sum_{i \\geq m} \\sum_{ j \\geq \\lfloor \\frac i 4\\rfloor}^{i} \\frac {b_{ij}E^{2(i-j)} 2^j}{2 ^{\\lfloor \\frac i 4\\rfloor}} + \\sum_{i \\geq m} \\sum_{0 \\leq j < \\lfloor \\frac i 4\\rfloor} E^h\\frac {b_{ij}E^{2(i-j)-h} 2^j}{2 ^{\\lfloor \\frac i 4\\rfloor}}.\n$$ \nSimilar to the argument in $(3)$, it is amount to find $h_0$ such that whenever $m>h_0$, $i \\geq m$ and $0 \\leq j < \\lfloor \\frac i 4\\rfloor$, we have $\\lfloor (i -j)-\\frac h 2 \\rfloor + j  \\geq \\lfloor \\frac i  4 \\rfloor $ and $2(i-j)-h \\geq m +1$. It is enough to choose $h_0 > 2(h+2)$. \n\\end{proof}\n\n\nIf  $A$ is a ring then we denote by  ${\\rm M} _d (A)$ the set of $d\\times d$-matrices with entries in $A$.\n\n\\begin{proposition}\\label{prop-desecnt} Let $Y \\in {\\rm M}_d (A^{(2)}_{\\max})$ so that $E^ h Y = B  \\varphi (Y) C$ with $B$ and $C$ in ${\\rm M}_d (A ^{(2)})$ then   $Y$ is in ${\\rm M}_d (A ^{(2)}[\\frac 1 p])$. \n\\end{proposition}\n\\begin{proof} First, we claim that there is a constant $s$ only depends on $h$, such that the entries of $p^s Y$ is in $\\widetilde S$. By $(1)$ of Lemma~\\ref{lem-auxiliaryrings}, entries of  $E ^ h Y$ are in $\\widetilde S$. So for each entry $a$ of $Y$, we can write $E^h a = \\sum \\limits_{i = 0}^\\infty a_i \\frac{ E^{pi}}{p^i}$ with $a_i \\in A ^{(2)}$. It is clear that $E^h p ^h a = a' + E^h \\sum\\limits_{i \\geq h} a_j \\frac {E^{pi-h }}{p ^i} $ so that $a' \\in A^{(2)}$. Therefore, $a' \\in {\\textnormal{Fil}}^h A^{(2)} = E^h A ^{(2)}$ by Corollary \\ref{cor-filtration-shape}. So write $a' =  E^h b$, we have $ p ^h a = b' + \\sum\\limits_{i \\geq h} a_j \\frac {E^{pi-h }}{p ^i}$. In particular, we see that $p ^{2h} a \\in \\widetilde S$, this proves our claim. When $p=2$, then we may repeat the above argument and we can assume $p^s Y$ is in ${\\rm M} _d (\\widehat{S})$.\n\nLet $R=\\widetilde S$ when $p>2$ and $R=\\widehat{S}$ when $p=2$, then we may assume $Y$ is inside ${\\rm M} _d (R)$. Then we claim there is another constant $r$ only depends on $h$, such that for each entry $a$ of $Y$, there is a sequence $\\{b_i\\}_{i\\geq 1}$ in $A^{(2)}$ such that we have $a - \\sum\\limits_{i = 0} ^m b _i E^ i \\in {\\textnormal{Fil}} ^{m +1} R$. Note that once this is known, we will have $\\sum\\limits_{i = 0} ^m b _i E^ i$ converges to an element $b$ in $A^{(2)}$, and $a-b=0$ since it is in ${\\textnormal{Fil}} ^{m} R$ for all $m\\in \\mathbb N$. \n\nSo it remains to show our claim. When $p>2$, let $h_0$ be the integer in $(3)$ of Lemma~\\ref{lem-auxiliaryrings}, then it is easy to show there is a constant $r$ only depends on $h_0$ (so only on $h$) and sequence $\\{b_i\\}_{i=1}^{h_0}$ such that for each entry $a$ of $Y':=p^rY$, we have\n$$\na - \\sum\\limits_{i = 0} ^{h_0} b _i E^ i \\in {\\textnormal{Fil}} ^{h_0 +1} R.\n$$\nNow we show our claim by induction, assume for each entry $a$ in $Y'$, there is a sequence $\\{b_i\\}_{i=1}^{m}$ such that,\n$$\na - \\sum\\limits_{i = 0} ^{m} b _i E^ i \\in {\\textnormal{Fil}} ^{m +1} R.\n$$\nfor some $m \\geq h_0$.  So we can write $Y'$ as\n$$\n\\sum_{i = 0}^m Y_i E^i + Z_{m+1},   \n$$\nwith $Y_i \\in {\\rm M}_d (A^{(2)})$ and $Z_{m+1} \\in {\\rm M}_d ({\\textnormal{Fil}} ^{m +1} R)$. Writing $X_{m}= \\sum_{i = 0}^m Y_i E^i$, then $E^h Y' = B \\varphi (Y') C$ implies \n$$\nE^hZ_{m+1} = B\\varphi(X_m)C -E^hX_m + B\\varphi(Z_{m+1})C.\n$$\nBy $(3)$ in Lemma~\\ref{lem-auxiliaryrings}, we have $B\\varphi(Z_{m+1})C = A_{m+1} + E^h B_{m+1}$, with $A_{m+1} \\in {\\rm M}_d (A^{(2)})$ and $B_{m+1} \\in {\\rm M}_d ({\\textnormal{Fil}} ^{m +2} R)$. One can check $B\\varphi(X_m)C -E^hX_m + A_{m+1} \\in {\\rm M}_d ({\\textnormal{Fil}}^{h +m +1 } A^{(2)} )$, so $B\\varphi(X_m)C -E^hX_m + A_{m+1}=E^{h+m+1} Y_{m+1}$ with $Y_{m+1} \\in {\\rm M}_d (A^{(2)})$. And we have $Y - \\sum_{i = 0}^{m+1} Y_i E^i = B_{m+1} \\in {\\rm M}_d ({\\textnormal{Fil}} ^{m +2}R)$ as required.\n\nAt last when $p=2$. We know we can assume $Y$ is inside ${\\rm M}_d (\\widehat{S})$. Then repeat the above arguments by replacing $(3)$ in Lemma~\\ref{lem-auxiliaryrings} with $(5)$, we can also prove our claim.\n\\end{proof}\n\n\\subsection{The ring \\texorpdfstring{$A^{(2)}_{\\st}$}{A(2)st}}\\label{subsec-Ast} We assume that $R = \\O _K$ in the following two subsections. \nFor our later use for semi-stable representations, we construct $A^{(2)}_{\\st}$ as the following: Define $\\varphi$ on $W(k)[\\![x, \\mathfrak y]\\!]$ by $\\varphi(x)= x^p$ and $\\varphi (\\mathfrak y ) = (1+\\mathfrak y)^p -1$ and set $w = \\frac{\\mathfrak y}{ E}$. Set $ A^{(2)}_{\\st}: = W(k)[\\![x,\\mathfrak y ]\\!]\\{w\\}_\\delta^\\wedge$ where $\\wedge$ means $(p, E)$-completion. Similarly, we define $A^{(3)}_{\\st}=W(k)[\\![x,\\mathfrak y, \\mathfrak z ]\\!]\\{\\frac{\\mathfrak y}{E},\\frac{\\mathfrak z}{E}\\}^\\wedge_{\\delta}$, with the $\\delta$-structure on $W(k)[\\![x,\\mathfrak y, \\mathfrak z ]\\!]$ given by $\\delta(x)=0$, $\\varphi(\\mathfrak y)=(\\mathfrak y+1)^p-1$ and $\\varphi(\\mathfrak z)=(\\mathfrak z+1)^p-1$. Define $A^{(2)}_{\\st,\\max}$ to be the $p$-adic completion of $W(k)[\\![x, \\mathfrak y]\\!][w, \\frac E p , \\gamma_i (w), i \\geq 0].$ It is clear that for any $f \\in A^{(2)}_{\\st ,\\max}$ can be written uniquely $a = \\sum\\limits_{i= 0}^\\infty f_i \\gamma_i (w) $ with $f_i \\in \\Omax$ and $f_i \\to 0$ $p$-adically. For any subring $B\\subset A^{(2)}_{\\st , \\max}[\\frac 1 p]$, we set ${\\textnormal{Fil}} ^i B : = B \\cap E^i A^{(2)}_{\\st , \\max}[\\frac 1 p]$ and $D_w $ the $p$-adic completion of $\\O_K [\\gamma_i(w), i \\geq 0]$. \n\nIt turns out that $A^{(2)}$ and $A^{(2)}_{\\st}$ share almost the same properties by replacing $z$ with $w$. \nSo we summarize all these properties in the following: \n\\begin{proposition}\\label{prop-Ast-properties}\n\\begin{enumerate}\n \\item One can extend Froebnius from $A$ to $A^{(2)}_{\\st, \\max}$.  \n    \\item There exists an embedding $\\iota : A^{(2)}_{\\st} \\hookrightarrow A^{(2)}_{\\st , \\max}$ so that $\\iota$ commutes with Frobenius. \n    \\item $A^{(2)}_{\\st} \\cap E ^ i A^{(2)}_{\\st , \\max}[\\frac 1 p] = E A^{(2)} _{\\st}$. \n    \\item $A ^{(2)}_{\\st}\/E \\simeq D_w = A ^{(2)}_{\\st , \\max}\/ {\\textnormal{Fil}} ^1 A^{(2)}_{\\st , \\max}.$\n    \\item $A^{(2)}_{\\st}$ is closed in $A^{(2)}_{\\st, \\max}$. \n    \\item $A^{(2)}_{\\st}$ and $A^{(3)}_{\\st}$ are flat over $A$, and in particular they are bounded. \n    \\item Proposition \\ref{prop-desecnt} holds by replacing $A^{(2)}_{\\max}$ and $A^{(2)}$ by \n    $A^{(2)}_{\\st} $ and $A^{(2)}_{\\st, \\max}$ respectively. \n\\end{enumerate}\n\n\\end{proposition}\n\\begin{proof}\nAll previous proof applies by noting the following difference \n$$\\varphi(w)= \\varphi ( \\frac {\\mathfrak y}{E}) = c ^{-1} \\frac {1}{p} \\sum_{i =1}^p \\binom{p}{i} \\mathfrak y ^i= c ^{-1 }\\sum _{i =1}^{p-1} \\mathfrak y ^i\\binom{p}{i}\/ p  + c^{-1}\\frac{E^p w^p}{p}.   $$\nAlso $\\delta (\\mathfrak y) = \\sum\\limits _{i =1}^{p-1} \\mathfrak y ^i\\binom{p}{i}\/ p$ always contains $\\mathfrak y$-factor and this is a key input for the analogy of Lemma \\ref{lem:gamma(z)-polynomial-in-E\/p}. \n\nFor the boundedness of $A^{(3)}_{\\st}$, we have \n$$ \nW(k)[\\![x,\\mathfrak y, \\mathfrak z]\\!]\/(p,E)\\simeq (\\O_K\/p)[\\![\\bar{\\mathfrak y}, \\bar{\\mathfrak z}]\\!]\n$$ so $\\{\\mathfrak y, \\mathfrak z\\}$ form a $(p,E)$-complete regular sequence, and by \\cite[Proposition 3.13]{BS19}, $A^{(3)}_{\\st}$ is also $A$-flat, and this implies $A^{(3)}_{\\st}$ is bounded by (2) in Lemma 3.7 of $loc.cit.$. \n\\end{proof}\n\nNote that $A^{\\ho 2}= W(k) [\\![x, y]\\!]\\subset W(k)[\\![x,\n \\mathfrak y]\\!]$ via $y = x(\\mathfrak y +1)$ or equivalently $\\mathfrak y = \\frac y x -1 $. It is clear that this inclusion is a map of $\\delta$-rings. By the universal property of prismatic envelope to construct $A^{(2)}$, the inclusion induces a map of prisms $\\alpha: A^{(2)}\\to A^{(2)}_{\\st}$. Since $z= xw$, we easily see that $A^{(2)}_{\\max} \\subset A^{(2)}_{\\st, \\max}$. So $A^{(2)}\\subset A^{(2)} _{\\st}$ via $\\alpha$. We will see that $A^{(2)}$ (resp. $A^{(2)}_{\\st}$) is the self product of $A$ in category $X_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ (resp. $(X, M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\text{log}}}$) in \n \\S \\ref{subsec-pris-crystal} and \\S \\ref{sec-logprismandsemistablereps}. Then the existence of $\\alpha: A^{(2)} \\to A^{(2)}_{\\st}$ can be explained by the universal property of self product. See \\S\\ref{sec-logprismandsemistablereps} for details. \n\nTo simplify our notation, let $B^{(2)}_{\\st}$ (resp. $B^{(3)}_{\\st}$, $B^{(2)}$, $B ^{(3)}$) be the $p$-adic completion of $ {A^{(2)}_{\\st}} [\\frac 1 E]$ (resp. $A^{(3)}_{\\st}[\\frac 1 E]$, $A^{(2)} [\\frac 1 E]$, $A^{(3)}[\\frac 1 E]$). \n\\begin{lemma}\\label{lem-intersection} \\begin{enumerate}\n\\item $ A^{(i)}_{\\st}  \\subset B^{(i)}_{\\st}\\subset B^{(i)}_{\\st}[\\frac 1 p]$ and $ A^{(i)} \\subset B ^{(i)} \\subset B ^{(i)}[\\frac 1 p]$ for $i = 2, 3$. \n    \\item $B^{(2)}_{\\st} \\cap {A^{(2)}_{\\st}} [\\frac 1 p] = A^{(2)}_{\\st}$ and $B ^{(2)} \\cap {A^{(2)}} [\\frac 1 p] = A^{(2)}$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} Here we only prove the case $A^{(2)}$ while the proofs for $A^{(2)}_{\\st}$, $A^{(3)}$ and $A^{(3)}_{\\st}$ are almost the same. \n\nBy Proposition \\ref{prop-key-property}, $A^{(2)}$ is a subring of $A^{(2)} _{\\max}\\subset K_0  [\\![x, z]\\!]$. So $A^{(2)}$ and hence $A^{(2)} [\\frac 1 E]$\nis an integral domain. Then $B^{(2)} $ has no $p$-torsion: Assume that $x \\in B^{(2)}$ so that $p x = 0 $. Suppose that $x_n \\in A^{(2)} [\\frac 1 E]$ so that $x \\equiv x _n \\mod p ^n$. Then $p x_n \\equiv 0 \\mod p ^n A^{(2)} [\\frac 1 E]$. Since $A^{(2)} [\\frac 1 E]$ is domain,  $x_{n}\\equiv 0 \\mod p ^{n -1}$. Hence  $x = 0 $. As $B^{(2)}$ has no $p$-torsion, we see that $B^{(2)}\\subset B^{(2)}[\\frac 1 p]$. \nTo see the natural map $A^{(2)} \\to B^{(2)}$ is injective, it suffices to show that $A^{(2)} \/ p A^{(2)} $ injects to $ A^{(2)} \/ pA^{(2)} [\\frac 1 u]= B ^{(2)}\/ pB^{(2)}$. Clearly, this is equivalent to that $A^{(2)} \/ pA^{(2)}$ has no $u$-torsion. Note that $A^{(2)}$ is obtained by taking prismatic envelope of $A^{\\ho 2}= W(k)[\\![x, z]\\!]$ for the ideal $ I = (z)$. As mentioned before, we can apply \\cite[Prop. 3.13]{BS19} to our situation. So $A^{(2)}$ is flat over $A$ and hence $A^{(2)} \/ p A^{(2)}$ has no $u$-torsion as desired. \n\n\nNow we can regard $B^{(2)}$ and $A^{(2)} [\\frac 1 p]$ as  subrings of $B^{(2)}[\\frac 1 p]$. In particular, $ B^{(2)} \\cap A^{(2)} [\\frac 1 p ]$ makes sense and contains $A^{(2)}$. For any $x\\in B^{(2)} \\cap A^{(2)} [\\frac 1 p ]$, if $x\\not \\in A^{(2)}$ but $p x \\in A^{(2)}$. Then the image of $y = px  $ inside $A^{(2)}\/ p A^{(2)}$ is nonzero but the image of $y$ in $B ^{(2)}\/ p B ^{(2)}$ is zero. This contradicts to that $A^{(2)} \/ p A^{(2)} $ injects to $B^{(2)}\/ p B^{(2)}$. So such $x$ can not exist and we have $B ^{(2)} \\cap {A^{(2)}} [\\frac 1 p] = A^{(2)}$ as required. \\end{proof}\n\n\nBy \\cite[Lem. 3.9]{BS19}, any prism $(B, J)$ admits its perfection $(B,J)_{\\perf}=(B_{\\perf}, JB _{\\perf})$. \n\\begin{remark}\nIn \\cite{BS19}, the underlying $\\delta$-ring of $(B,J)_{\\perf}$ is denoted by $(B_{\\infty},JB_{\\infty})$, and $B_{\\perf}$ is defined as the direct perfection of $B$ in the category of $\\delta$-rings. In this paper, we write $B_{\\perf}$ as the $(p,J)$-adic completion of $\\mathrm{colim}_{\\varphi} B$, which also coincides with the derived $(p, I)$-completion of $\\mathrm{colim}_{\\varphi} B$ (cf. Lemma 3.9 of $loc.cit.$).\n\\end{remark}\n\n\\begin{lemma}\\label{lem-perfisflat}\nWe have $(A^{(2)})_{\\perf}$ and $(A^{(2)}_{\\st})_{\\perf}$ are $A$-flat.\n\\end{lemma}\n\\begin{proof} We have seen that $A^{(2)}$ is $A$-flat via $i_1$. And it is easy to see $\\varphi$ on $A$ is flat. Since $i_1$ is a $\\delta$-map, so we have $\\varphi^n\\circ i_1 =i_1 \\circ \\varphi^n$ which is flat. So $\\mathrm{colim}_\\varphi A^{(2)}$ is flat over $A$. In particular, we will have $A_{\\perf}$ is $(p , E)$-complete flat over $A$. Now since $A$ is Noetherian, by \\cite[Tag 0912]{stacks-project}, we have $(A^{(2)})_{\\perf}$ is $A$-flat. The proof for $(A^{(2)}_{\\st})_{\\perf}$ is the same. \n\\end{proof}\n\n\\subsection{Embedding \\texorpdfstring{$A^{(2)}$}{A(2)} and \\texorpdfstring{$A^{(2)}_{\\st}$}{A(2)st} to \\texorpdfstring{$\\Ainf$}{Ainf}}\\label{subsec-embedding}\nLet $\\Ainf=W(\\O_{\\mathbb C_p}^\\flat)$, then there is a surjection $\\theta: \\Ainf \\to \\O_{\\mathbb C_p}$ and $\\Ker\\theta=(E)$. And let $\\BdR^+$ be the $\\ker\\theta$-adic completion of $\\Ainf[\\frac{1}{p}]$.\n\n\\begin{definition}\nLet $\\A_{\\max}$ be the $p$-adic completion of the $\\Ainf$-subalgebra of $\\BdR^+$ generated by $E\/p$. \n\\end{definition}\nIt can be easily seen that $\\varphi(E\/p):=\\varphi(E)\/p\\in A_{\\mathrm{cris}}\\subset \\A_{\\max}$ is well-defined and it extends the Frobenius structure on $\\Ainf$ to an endomorphism on $\\Amax$.\n\nLet $\\{\\varpi_n\\}_{n\\geq 0}$ be a compatible system of $p^n$-th roots of $\\varpi_0=\\varpi$ and $\\{\\zeta_n\\}_{n\\geq 0}$ be a compatible system of $p^n$-th roots of 1. Write $\\varpi^\\flat  : = \\{\\varpi_n\\}_{n\\geq 0}, \\zeta^\\flat : = \\{\\zeta_n\\}_{n\\geq 0}\\in \\O_{\\mathbb C_p}^\\flat$ and  let $u=[\\varpi^\\flat ]$, $\\epsilon=[\\zeta^\\flat]$, $v=\\epsilon u$ and $\\mu=\\epsilon-1$ be elements inside $\\Ainf$. We can regard $W(k)[\\![x,y]\\!]$ as a subring of  $\\Ainf$ via $x \\mapsto u$ and $y\\mapsto v$. Consider $z' = \\frac{u-v}{E}\\in \\Ainf [\\frac 1 E]$. Since $u -v = u (\\epsilon -1)$ is clearly inside $\\Ker (\\theta )$ and $ \\Ker (\\theta) = E \\Ainf$, we conclude that $ z' \\in \\Ainf$. Hence we have a natural map (of $\\delta$-rings) $\\iota_A : \\widetildeA^{(2)} \\to \\Ainf$ via $z\\mapsto z'$,  which naturally extends to $\\iota _A : A^{(2)}\\to \\Ainf$ because $(p, E)$-topology of $A^{(2)}$ matches with the weak topology of $\\Ainf$. Similarly, we have map of $\\delta$-rings $\\iota_{\\st}: A^{(2)}_{\\st} \\to A_{\\inf}$ via $ x \\mapsto u$ and $\\mathfrak y \\mapsto \\epsilon-1$ and $w\\mapsto \\frac{\\epsilon-1}{E}$. \n\\begin{remark}\\label{rem-embedding-depend} Once we know that $A^{(2)}$ is self-product of $A$ inside $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ with $X= \\Spf (\\O_K)$ as explained in \\S \\ref{subsec-pris-crystal}. The map $\\iota_A$ can be constructed as the following: First we fix an embedding $A\\to \\Ainf$ by sending $x \\mapsto u =  [\\varpi^\\flat]$. Then $A\\to \\Ainf$ by $x \\to v= \\epsilon u$ is another map of prisms. By universal property of $A^{(2)}$, these two maps extends to a map $\\iota_A : A^{(2)} \\to \\Ainf$. Clearly, the map $\\iota_A : A^{(2)} \\to \\Ainf$ depends on choice of ${\\varpi}^\\flat = (\\varpi_n)_{n \\geq 0}$ and ${\\zeta}^\\flat = (\\zeta_n)_{n \\geq 0}$. Also $\\iota_A$ is a special case of $\\iota^{(2)}_\\gamma$ defined by \\eqref{equ-diagram-prisms} in \\S\\ref{subsec-pris-crystal-proof}. Indeed if $\\gamma ([w^\\flat])= [\\zeta^\\flat][w ^\\flat]$ then $\\iota_A = \\iota^{(2)}_{\\gamma}$. Similarly comment also applies to $\\iota_{\\st}$. \n\\end{remark}\n\n\n\\begin{proposition}\nThere is a unique embedding \n$\n\\begin{tikzcd}\nA_{\\max}^{(2)} \\arrow[r, hookrightarrow] & \\A_{\\max}\n\\end{tikzcd}\n$ such that\n$$\n\\begin{tikzcd}\nW(k)[\\![x,y]\\!] \\arrow[r, hookrightarrow]\\arrow[d, hookrightarrow] & \\Ainf \\arrow[d, hookrightarrow]  & \\\\\nA_{\\max}^{(2)} \\arrow[r, hookrightarrow] & \\A_{\\max}  \\arrow[r, hookrightarrow] & \\BdR^+\n\\end{tikzcd}\n$$\ncommutes. Furthermore, ${\\textnormal{Fil}} ^ i \\BdR^+ \\cap A^{(2)}_{\\max}= {\\textnormal{Fil}} ^i A^{(2)}_{\\max}$. The same result holds when $A^{(2)}_{\\max}$ is replaced by $A^{(2)}_{\\st , \\max}$.  \n\\end{proposition}\n\\begin{proof} In the following, we only treat the case of $A^{(2)}_{\\st , \\max}$ while the proof of $A^{(2)}_{ \\max}$ is the same by noting that $z= u w$ in $A_{\\inf}$. \n\nThe uniqueness is clear. To show the existence of the embedding, it is enough to show $\\gamma_i(w)\\in\\Amax$ for all $i\\geq 1$. \n\nIt is a well-known fact that $\\Amax$ is isomorphic to the $p$-adic completion of $\\Ainf[\\frac{u^e}{p}]$, and $\\Amax[1\/p]$ is a Banach $\\mathbb Q_p$-algebra, which is the completion of $\\Ainf[1\/p]$ under the norm $\\lvert \\cdot \\rvert_{p^{-1}}$ such that\n$$\n\\lvert x \\rvert_{p^{-1}} = \\sup_{n} \\{p^{-n}\\lvert x_n \\rvert_{\\O_{C}^\\flat}\\}\n$$\nwhere $x=\\sum_{n\\gg 0} [x_n]p^n \\in \\Ainf[1\/p]$. And we have for $x\\in \\Amax[1\/p]$, $x\\in \\Amax$ if and only if $\\lvert x \\rvert_{p^{-1}} \\leq 1$. Moreover $\\lvert \\cdot \\rvert_{p^{-1}}$ is multiplicative. So now it is enough to show for $x=\\gamma_i(w)$ considered as an element inside $ \\Amax[1\/p]$, we have $\\lvert x^{p-1} \\rvert_{p^{-1}}\\leq 1$. To show this, we have by \\cite[Proposition 3.17]{BMS1}, $\\xi:=\\mu\/\\varphi^{-1}(\\mu)$ is a generator of $\\Ker\\theta$ with $\\mu = \\epsilon-1$. In particular, $w=\\mu\/E = a  \\varphi^{-1} (\\mu) \\in \\Ainf$ with $a\\in \\Ainf^\\times$. \nAnd we can check $\\overline{w}^{p-1} = c \\overline{u}^e$ inside $\\O_C^\\flat=\\Ainf\/p\\Ainf$, with $c$ a unit. So $w^{p-1}=au^e+bp$ with $a,b\\in \\Ainf$, and\n$$\nx^{p-1}=\\frac{(au^e+bp)^i}{(i!)^{p-1}}.\n$$\nUsing the fact $v_p(i!)< \\frac{i}{p-1}$, one can show each term in the binomial expansion on the right hand side of the equation has $\\lvert \\cdot \\rvert_{p^{-1}}$-norm less or equal to $1$, so in particular, $\\lvert x^{p-1} \\rvert_{p^{-1}}\\leq 1$.\n\nTo prove that ${\\textnormal{Fil}} ^i \\BdR^+ \\cap  A ^{(2)}_{\\st, \\max} = {\\textnormal{Fil}} ^i A ^{(2)}_{\\st , \\max}$, it suffices to show that $E \\BdR ^+ \\cap A ^{(2)}_{\\st, \\max}[\\frac 1 p] = E A ^{(2)}_{\\st , \\max}[\\frac 1 p]$. By  Proposition  \\ref{prop-key-property},  we reduces to prove that the map $$\\theta : D_w= A ^{(2)}_{\\st , \\max}[\\frac 1 p ]\/E \\to \\BdR^+ \/ E= \\mathbb C_p$$ is injective. Let $f(w) = \\sum_{i \\geq 0} a_i \\gamma_i (w)\\in \\Ker \\theta$ with $a_i \\in \\O_K$ limits to $0$ $p$-adically. Then $f(w_0) = 0$ with $w_0: = \\theta (w)= \\theta (\\frac{\\epsilon-1}{E}) \\in \\mathbb C_p$. Note  $v_p (w_0)\\geq \\frac{1}{p-1}$ because it is well-known  $ \\frac{\\epsilon-1}{\\varphi^{-1}(\\epsilon) -1}$ is another generator of kernel $\\theta: A_{\\inf}  \\to \\O_{\\mathbb C_p}$ and then  \n$v_p (w_0) = v_p (\\theta (\\varphi^{-1} (\\epsilon)-1))= \\frac{1}{p-1}$. \nSince we are aiming to show that $f= 0$, without loss of generality, we can assume that $K$ contains $p_1= \\sqrt[p-1]{p}$. Note that \n$v_p (i !)\\leq \\frac{1}{p-1}$, we conclude that $\\frac{w_0}{p_1}$ is a root of $f(p_1 w)$ which is in $\\O_K\\langle w\\rangle$. By Weierstrass preparation theorem, $w_0$ is algebraic over $K$ unless $f=0$.  By Lemma below,  $w_0: = \\theta (w) \\in \\mathbb C_p$ is transcendental over $K$ and hence $f= 0$. \n\\end{proof}\n\\begin{lemma}\\label{lem-transcendental}\n$w_0 = \\theta (\\frac{\\epsilon-1}{E})$ is transcendental over $K$. \n\\end{lemma}\n\\begin{proof}\nIf $w_0$ is contained in an algebraic extension $L$ over $K$, we define $L_{0,\\infty}=\\bigcup_n L(\\varpi_n)$. For $g\\in G_{L_{0,\\infty}}$, we will have \n$$\n\\theta(g(\\frac{\\epsilon-1}{E}))=g(w_0)=w_0=\\theta(\\frac{\\epsilon-1}{E}).\n$$\nSince $G_{L_{0,\\infty}}$ fix $E$,  $\\theta(\\frac{g(\\epsilon-1)-(\\epsilon-1)}{E})=0$. This implies $g(\\epsilon-1)-(\\epsilon-1)\\in {\\textnormal{Fil}}^2\\BdR^+$. Recall for $t=\\log \\epsilon$, $t-(\\epsilon-1)\\in {\\textnormal{Fil}}^2\\BdR^+$, so we have $g(t)-t \\in {\\textnormal{Fil}}^2\\BdR^+$. But this can't be true. Since $L_{0,\\infty}$ can only contain finitely many $p^n$-th roots of $1$, for $g\\in G_{L_{0,\\infty}}$, $g(t)=c(g)t$ satisfying $c(g) \\in \\mathbb Q_p$ and $c(g)\\neq 1$. This implies $g(t)-t = (c(g)-1)t \\in {\\textnormal{Fil}}^1\\BdR^+ \\setminus {\\textnormal{Fil}}^2\\BdR^+$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-inj}\nThe natural maps $\\iota _ A : A ^{(2)} \\to A_{\\inf}$ and $\\iota_{\\st} : A^{(2)}_{\\st} \\to A_{\\inf}$ are injective. \n\\end{corollary}\n\nTo summarize, we have the following commutative diagram of rings inside $\\BdR^+$:\n$$\n\\begin{tikzcd}\nA^{(2)}  \\arrow[d, hookrightarrow] \\arrow[r, hookrightarrow] & A^{(2)}_{\\st}\\arrow[r, hookrightarrow] \\arrow[d, hookrightarrow]  & \\Ainf \\arrow[d, hookrightarrow] \\\\\nA^{(2)}_{\\max} \\arrow[r, hookrightarrow] & A^{(2)}_{\\st, \\max} \\arrow[r, hookrightarrow] & \\Amax.  \n\\end{tikzcd}\n$$\n\n \\section{Application to semi-stable Galois representations}\nIn this section, we assume that $R= \\O_K$. We explain how to use the period ring $A^{(2)}$ and $A^{(2)}_{\\st}$ to understand lattices in crystalline and semi-stable representations. Roughly speaking, we are going to use $A^{(2)}$ and $A^{(2)}_{\\st}$ to replace $\\widehat{\\mathcal R}$ in the theory of $(\\varphi , \\hat G)$-modules developed in \\cite{liu-notelattice}. \n\nLet $K _\\infty =\\bigcup_{n = 1}^\\infty K (\\varpi _n )$, $G_\\infty: = {\\rm Gal } (\\overline  K \/ K_\\infty)$ and $G_K: = {\\rm Gal } (\\overline K  \/ K)$.  Recall that $A= \\mathfrak S = W(k)[\\![u]\\!]$. Let $S $  be the $p$-adic completion of $ W(k) [\\![u , \\frac{E^i}{i !}, i \\geq 1]\\!]$, which is the  PD envelope of $W(k)[u]$ for the ideal $(E)$. It is clear that $S\\subset \\Omax$. We define $\\varphi$ and ${\\textnormal{Fil}} ^i$ on $S$ induced that from those on $\\Omax$, in particular, ${\\textnormal{Fil}} ^i S = S \\cap E ^i \\Omax[\\frac 1 p]$. Note that $A$ embeds to $\\Ainf$ via $u \\mapsto [\\varpi^\\flat ]$ is not stable under $G_K$-action but only on $G_\\infty$-action. For any $g \\in G_K$, define ${\\underline{\\varepsilon}} (g)= \\frac{g(u)}{u}$. It is clear that ${\\underline{\\varepsilon}} (g) = \\epsilon ^{a(g)}$ with $a(g) \\in \\mathbb Z_p$. We define \\emph{two}  differential operators $N_S$ and $\\nabla_S$ on $S$ by $N_S(f) = \\frac{d f}{du}u$ and $\\nabla_S (f) = \\frac{ df }{du}$. We need $\\nabla_S$ to treat crystalline representations.  \n\n\\subsection{Kisin module attached to semi-stable representation}\\label{subsec-Kisin-st} Fix $h \\geq 0$, \na \\emph{Kisin module of height $h$} is a finite free $A$-module $\\mathfrak{M} $ with a semi-linear endomorphism $\\varphi_{\\mathfrak{M}}: \\mathfrak{M} \\to \\mathfrak{M}$ so that $\\coker (1 \\otimes \\varphi_\\mathfrak{M})$ is killed by $E^h$, where $1 \\otimes \\varphi_{\\mathfrak{M}} : \\mathfrak{M} ^* : = A \\otimes _{\\varphi, A}\\mathfrak{M} \\to \\mathfrak{M} $ is linearization of  $\\varphi_\\mathfrak{M}$. Note here we are using classical setting of Kisin modules used in \\cite{liu-notelattice} but it is good enough for this paper. The following summarizes the results on Kisin modules attached to $G_K$-stable $\\mathbb Z_p$-lattices in semi-stable representations. The details and proofs of these facts can be found in \\cite{liu-notelattice}. \n\nLet $T$ be a $G_K$-stable $\\mathbb Z_p$-lattice inside a semi-stable representation $V$ of $G_K$ with Hodge-Tate weights in $\\{0, \\dots , h\\}$. Let $D: = D^*_{\\st} (V)= \\mathrm{Hom}_{\\mathbb Q_p, G_K} (V , B_{\\st})$ be the filtered $(\\varphi , N)$-module attached to $V$ and $D_K : = K \\otimes_{K_0} D$. Then there exists a unique Kisin module $\\mathfrak{M} : = \\mathfrak{M} (T) $ of height $h$ attached to $T$ so that \n\\begin{enumerate}\n    \\item $\\mathrm{Hom}_{\\varphi , A} (\\mathfrak{M} , \\Ainf)\\simeq T|_{G_\\infty}$. \n    \\item There exists an $S$-linear isomorphism  \n    $$\\iota_S : S [\\frac 1 p] \\otimes _{\\varphi, A}\\mathfrak{M} \\simeq D \\otimes _{W(k)} S $$ so that $\\iota_S$ is compatible with $\\varphi$ on the both sides. \n\\item $\\iota_S$ also induces an isomorphism ${\\textnormal{Fil}}^h (S [\\frac 1 p] \\otimes _{\\varphi, A}\\mathfrak{M}) \\simeq {\\textnormal{Fil}} ^h (D\\otimes_{W(k)} S) $. The filtration on the both sides are defined as following: \n\\[{\\textnormal{Fil}}^h (S [\\frac 1 p] \\otimes _{\\varphi, A}\\mathfrak{M}): =\\left  \\{ x \\in S [\\frac 1 p] \\otimes _{\\varphi, A}\\mathfrak{M}| (1\\otimes \\varphi_\\mathfrak{M} (x)) \\in {\\textnormal{Fil}} ^h S[\\frac 1 p ] \\otimes_A \\mathfrak{M} \\right \\}.  \\] \nTo define \nfiltration on ${\\mathcal D} : = S \\otimes _ {W(k)} D$, we first extend the monodromy operator $N_{\\mathcal D}$ (resp. $\\nabla_{\\mathcal D}$) on $D$ to ${\\mathcal D}$ by $N_{{\\mathcal D}}= 1 \\otimes N_D + N_S \\otimes 1$ (resp. $\\nabla_{\\mathcal D} = 1 \\otimes N_D + \\nabla_S \\otimes 1$). Then we define ${\\textnormal{Fil}} ^i {\\mathcal D}$ by induction: set ${\\textnormal{Fil}}^0 {\\mathcal D} = {\\mathcal D}$ and \n\\[ {\\textnormal{Fil}} ^i{\\mathcal D}: = \\{x \\in {\\mathcal D}|  N_{{\\mathcal D}}(x) \\in {\\textnormal{Fil}}^{i-1}{\\mathcal D}, f_\\varpi (x) \\in {\\textnormal{Fil}} ^i D_K\\}\\]\nwhere $f_\\varpi : {\\mathcal D} \\to D_K$ is induced by $S\\to \\O_K$ via $u \\mapsto \\varpi. $\n\\end{enumerate}\n\\begin{remark}[Griffith transversality]\\label{rem-GT} From the construction of ${\\textnormal{Fil}} ^i {\\mathcal D}$, we see that $N_{{\\mathcal D}} ({\\textnormal{Fil}}  ^i{\\mathcal D}) \\subset {\\textnormal{Fil}}^{i -1}{\\mathcal D}$. This property is called Griffith transversality. \n\nWe only use $\\nabla_{\\mathcal D}$ when $N_D = 0$, that is, when $V$ is crystalline. In this case, it is clear that $N_{\\mathcal D} = u \\nabla_{{\\mathcal D}}$. So it is clear that $\\nabla_{\\mathcal D} ({\\textnormal{Fil}} ^i{\\mathcal D}) \\subset {\\textnormal{Fil}} ^{i-1}{\\mathcal D}$. \n\\end{remark}\nFor ease of notations, we will write $N = N_{\\mathcal D}$ and $\\nabla = \\nabla_{\\mathcal D}$ in the following. \nLet $T^\\vee: = \\mathrm{Hom}_{\\mathbb Z_p} (T , \\mathbb Z_p)$ and $V ^\\vee : = T^\\vee \\otimes_{\\mathbb Z_p}\\mathbb Q_p$ denote the dual representations.  \nThen there exists an $A_{\\inf}$-linear injection \n\\begin{equation}\\label{eqn-iota-A}\n\\iota_\\mathfrak{M}: A_{\\inf} \\otimes _A \\mathfrak{M} \\to  T ^\\vee \\otimes_{\\mathbb Z_p} \\Ainf, \n\\end{equation}\nwhich is compatible with $G_\\infty$-actions ($G_\\infty$ acts on $\\mathfrak{M}$ trivially) and $\\varphi$ on both sides. Applying $S \\otimes_{\\varphi, A}$ and using $\\iota _S: =S \\otimes_{\\varphi, A} \\iota_\\mathfrak{M} $, we obtain the following commutative diagram \n$$\n\\xymatrix@C=5em{ \\Acris[\\frac 1 p] \\otimes_{\\varphi , A}\\mathfrak{M} \\ar[d]_\\wr  ^{\\Acris \\otimes_S \\iota _S} \\ar[r] ^{S \\otimes_{\\varphi, A} \\iota _\\mathfrak{M}} & V^\\vee \\otimes_{\\mathbb Z_p} \\Acris \\ar@{=}[d]\\\\   \\Acris \\otimes_{W(k)} D \\ar[r]^{\\alpha} & V^\\vee \\otimes _{\\mathbb Z_p} \\Acris}\n$$\nwhere the second row  $\\alpha$ is built by the classical comparison $$B_{\\st} \\otimes_{K_0} D^*_{\\st}(V) \\simeq   V^\\vee \\otimes_{\\mathbb Q_p} B_{\\st},   $$\nand  $\\alpha$ is $G_K$-stable on the both sides. The left side of $\\alpha$ is defined by \n\\[ \\forall x \\in D, \\forall g \\in G_K, g (x) = \\sum_{i = 0}^\\infty N^i (x) \\gamma_i (\\log ({\\underline{\\varepsilon}}(g))) \\]\nTherefore, if we regard $\\mathfrak{M}^* : = A \\otimes_{\\varphi, A} \\mathfrak{M}$ as an  $A$-submodule of  \n$ V^\\vee \\otimes_{\\mathbb Z_p} \\Acris$ via injection $\\iota^* : = S \\otimes_{\\varphi, A} \\iota_A$, one can show that: \n\\begin{equation}\\label{eqn-G-action}\n\\forall g \\in G_K, x \\in \\mathfrak{M}^*, g(x) = \\sum_{i = 0}^\\infty N_{\\mathcal D}^i (x) \\gamma_i (\\log ({\\underline{\\varepsilon}}(g))).  \n\\end{equation}\n When $V$ is crystalline, or equivalently, $N_D = 0$, we have (\\cite[\\S8.1]{LL2021comparison})\n \\begin{equation}\\label{eqn-G-action-2}\n\\forall g \\in G_K, x \\in \\mathfrak{M}^*, g(x) = \\sum_{i = 0}^\\infty \\nabla_{\\mathcal D}^i (x) \\gamma_i (u {\\underline{\\varepsilon}}(g)).  \n\\end{equation}\n\n\\subsection{Descent of the \\texorpdfstring{$G_K$}{GK}-action}\\label{subsec-G-image}\n\nLet us first discuss the  $G_K$-action on $\\mathfrak{M} \\subset T ^\\vee \\otimes_{\\mathbb Z_p}\\Ainf$ via $\\iota_\\mathfrak{M}$ in \\eqref{eqn-iota-A} in more details.  \nWe select an $A$-basis $e_1, \\dots , e_d$ of $\\mathfrak{M}$ so that $\\varphi (e_1, \\dots, e_d)= (e_1, \\dots , e_d )\\mathfrak A$ with $\\mathfrak A \\in {\\rm M}_d (A)$.  Then there exists a matrix $B\\in {\\rm M}_d (A)$ so that $\\mathfrak A B = B \\mathfrak A = E^h I_d$. For any $g \\in G_K, $ let $X_g$ be the matrix so that \n\\[ g (e_1, \\dots , e_d) = (e_1, \\dots , e_d) X_g. \\]\nIn this section, we are interested in where the entries of $X_g$ locates. \n\\begin{theorem}\\label{Thm-1}The entries of $X_g$ are in $A^{(2)}_{\\st}$. If $V$ is crystalline and $g(u)- u = Ez$ then $X_g \\in {\\rm M}_{d} (A^{(2)}). $\n\\end{theorem}\n\nFirst,  it is well-known that $ W (\\mathbb C_p^\\flat) \\otimes_{\\Ainf} \\iota_\\mathfrak{M} $ is an isomorphism. So $X_g\\in {\\rm M}_d (W (\\mathbb C_p^\\flat))$. Since $G_K$-actions and $\\varphi$-commutes, we have $$ \\mathfrak A \\varphi(X_g)= X_g g (\\mathfrak A)  .$$  \n Define $${\\textnormal{Fil}} ^h \\mathfrak{M} ^* : = \\{ x \\in \\mathfrak{M}^* | (1 \\otimes \\varphi_{\\mathfrak{M}}) (x) \\in E^h \\mathfrak{M}\\}. $$\nSince $\\mathfrak{M}$ has $E$-height $h$, it is easy to show that ${\\textnormal{Fil}} ^h \\mathfrak{M}^*$ is a finite free $A$-module and ${\\textnormal{Fil}}^h {\\mathcal D}$ is generated by ${\\textnormal{Fil}} ^h \\mathfrak{M}^*$. \n\nTo be more precise, let $\\{e^*_i : =1 \\otimes e_i, i =1 , \\dots , d\\}$ be an $A$-basis of $\\mathfrak{M}^*$. It is easy to check that $(\\alpha_1, \\dots , \\alpha_d)= (e_1^* , \\dots , e_d^*) B$ is an $A$-basis of ${\\textnormal{Fil}} ^h \\mathfrak{M}^*$,  and it is  also an $S[\\frac 1 p]$-basis of ${\\textnormal{Fil}} ^h {\\mathcal D}$. \nSo for any $g \\in G_K$, we have $g (\\alpha_j) = \\sum\\limits_{i = 0}^\\infty N ^i (\\alpha_j) \\gamma _i (\\log ({\\underline{\\varepsilon}} (g))) $. By Griffith transversality in Remark \\ref{rem-GT}: \n$N ({\\textnormal{Fil}} ^i {\\mathcal D})\\subset {\\textnormal{Fil}}^{i-1}{\\mathcal D},   $\n we have, \n \\begin{equation}\\label{eqn-action-g}\n g (\\alpha_j)= \\sum_{i = 0}^h N ^i (\\alpha_j) E^i \\gamma _i (\\frac {\\log ({\\underline{\\varepsilon}} (g))}{E}) + \\sum_{i > h}^\\infty N^i (\\alpha_j) \\gamma_i(E) (\\frac{\\log ({\\underline{\\varepsilon}} (g))}{E})^i.      \n \\end{equation}\nSince $N^i (\\alpha_j) E^i \\in {\\textnormal{Fil}} ^h {\\mathcal D}$,  $\\gamma_i (E)$ in $\\Omax$ and $w ^n \\to 0$ inside $A^{(2)}_{\\st, \\max}$,  we see that $g (\\alpha_1, \\dots , \\alpha_d) = (\\alpha_1 , \\dots , \\alpha_d) Y_g $ with $Y_g \\in A^{(2)}_{\\st , \\max}[\\frac 1 p ]. $\n\nIn the case that $V$ is crystalline, using \\eqref{eqn-G-action-2}, we have \n$$g (\\alpha_j)= \\sum_{i = 0}^h \\nabla ^i (\\alpha_j) E^i \\gamma _i (\\frac {u{\\underline{\\varepsilon}} (g)}{E}) + \\sum_{i >  h}^\\infty \\nabla^i (\\alpha_j) \\gamma_i(E) (\\frac{u{\\underline{\\varepsilon}} (g)}{E})^i $$\n\\emph{If $g $ is chosen so that  $ g (u)-u = Ez$} then, a similar argument can shows that $g (\\alpha_1, \\dots , \\alpha_d) = (\\alpha_1 , \\dots , \\alpha_d) Y^\\nabla_g $ with $Y^\\nabla_g \\in A^{(2)}_{\\max}[\\frac 1 p]. $\n\nNow $g (e_1^*, \\dots , e_d^*) = (e_1^*, \\dots, e_d ^*) \\varphi (X_g)$.  Using similar arguments, we see that $\\varphi(X_g)$'s entry are in $A^{(2)}_{\\st , \\max}[\\frac 1 p ]$ and $A^{(2)}_{\\max}[\\frac 1 p]$ respectively. Since $(\\alpha_1 , \\dots , \\alpha_d) = (e_1^*, \\dots, e_d^*)B$, we conclude that $$ \\varphi(X_g) g (B) = B Y_g.$$ \nUsing the formula that $\\mathfrak A \\varphi (X_g) = X_g g (\\mathfrak A)$ and $\\mathfrak A B = B \\mathfrak A = E^h I_d$, we conclude that \n$Y_g = (\\frac{g (E)}{E}) ^h X_g$. Write  $r= \\frac{g (E)}{E}$. We claim that $r$ is a unit in $A^{(2)}_{\\st}$. Indeed, \n$\\frac{g(E)}{E}= \\frac{E (u \\epsilon ^{a(g)})}{E(u)}= \\sum\\limits_{i =0}^e E^{(i)} (u) \\frac{ u ^i (\\epsilon^{a(g)}-1)^i}{E i !}$ is again inside $A_{\\st}^{(2)}$, where $E^{(i)}$ means the $i$-th derivative of $E$. And it is easy to show $g(E)$ is also a distinguished element $A_{\\st}^{(2)}$, so by \\cite[Lemma 2.24]{BS19}, $r$ is a unit. Similarly, when $g(u)-u=Ez$, we will have $r=\\frac{g (E)}{E}\\in (A^{(2)})^\\times$. Hence  \n\\begin{equation}\\label{eqn-key-eqn}\nE^h X_g  = r ^{-h} \\mathfrak A  \\varphi (X_g) g (B).\n\\end{equation}\n\n\n\nNow we can apply Proposition \\ref{prop-desecnt} and Proposition \\ref{prop-Ast-properties} (5) to the above formula, we conclude that for $g\\in G_K$ (resp. $g\\in G_K$ such that $g(u)-u=Ez$ and $V$ is crystalline), we have $X_g$ has entries in $A^{(2)}_{\\st}[\\frac 1 p]$ (resp. $A^{(2)}[\\frac 1 p]$). \n\nTo complete the proof of Theorem \\ref{Thm-1}, it suffices to show that  entries $X_g$ are in $A^{(2)}_{\\st}$ (resp. $A^{(2)}$). Unfortunately, the proof to remove ``$\\frac 1 p$\" is much harder, which needs \\S \\ref{subsec-phi-tau} and \\S \\ref{subsec-pris-crystal-proof}. For the remaining of this subsection, we only show that the proof of Theorem \\ref{Thm-1} can be reduced to the case that $g = \\tilde \\tau$ for a special selected $\\tilde \\tau \\in G_K$.  \n\nLet $L = \\bigcup\\limits_{n =1}^\\infty K_{\\infty} (\\zeta_{p ^n})$, $K_{1^\\infty}: = \\bigcup_{n =1}^\\infty K (\\zeta_{p ^n})$,  $\\hat G : = \\Gal(L \/K) $ and $H_K : = \\Gal (L \/ K _\\infty)$. \nIf $p > 2$ then it is known that $\\hat G \\simeq \\Gal (L\/ K_{1 ^\\infty}) \\rtimes H_K $ with \n$\\Gal (L\/ K_{1 ^\\infty}) \\simeq \\mathbb Z_p$. Let $\\tau$ be a topological generator of $\\Gal (L\/ K_{1 ^\\infty}) $. We have $\\tau (u) = \\epsilon^a u$ with $a\\in \\mathbb Z_p ^\\times$. Without loss of generality, we may assume that $\\tau (u) = \\epsilon u$. If $p=2$ then we can still select $\\tau \\in \\hat G $ so that $\\tau (u)= \\epsilon u$ and $\\tau, H_K$ topologically generate $\\hat G$. Pick $\\tilde \\tau \\in G_K$ a lift of $\\tau$. Clearly, we have $\\tilde \\tau (u ) - u = E z$. \n\n\\begin{proposition}\\label{thm-1prime} \nFor $g = \\tilde \\tau, $ the entries of $X_g$ are in $A^{(2)}_{\\st}$, and if further  $V$ is crystalline, then $X_g \\in {\\rm M}_{d} (A^{(2)}).$\n\\end{proposition}\n\\begin{lemma}\\label{lem-equivalenceofthm}\nProposition ~\\ref{thm-1prime} is equivalent to Theorem~\\ref{Thm-1}.\n\\end{lemma}\n\\begin{proof}\nSince  $\\hat{G}$ is topologically generated by $\\tau$ and $H_K$. So $G_K$ is topologically generated by $G_\\infty$ and $\\tilde{\\tau}$. And we have $\\tau(u)-u=(\\epsilon-1)u=Ez$. Now if $X_{\\tilde{\\tau}}$ has coefficient in $A^{(2)}_{\\st}$ and $X_g=I_d$ for all $g\\in G_\\infty$  then to show that $X_g \\in A^{(2)}_{\\st}$ for all $g \\in G_K$, it suffices to show that $X_{\\tilde \\tau^{p ^n }}$ converges to $I_d$ inside ${\\rm M}_d (A^{(2)}_{\\st})$. Since $A^{(2)}_{\\st}$ is closed in $A^{(2)}_{\\st, \\max}$ by Proposition \\ref{prop-Ast-properties} (5), it suffices to show that $X_{\\tilde \\tau^{p ^n}}$ converges inside $A^{(2)}_{\\st, \\max}$. Since $X_g= (\\frac{E}{g (E)})^r Y_g$ and $Y_g$ is defined by \\eqref{eqn-action-g}, we easily check that $X_{\\tilde \\tau^{p ^n}}$ converges to $I_d$ in $A^{(2)}_{\\st,\\max}$ by using that ${\\underline{\\varepsilon}}(\\tilde \\tau^{p ^n})$ converges to $0$ in $(p , \\epsilon-1)$-topology. The proof for the crystalline case is similar by replacing $A^{(2)} _{\\st}$ with $A^{(2)}$.\n\\end{proof}\nSo it remains to prove Proposition ~\\ref{thm-1prime} to complete the proof of  Theorem~\\ref{Thm-1}. We will prove Proposition~\\ref{thm-1prime} in \\S\\ref{subsec-pris-crystal-proof}. Briefly speaking, for $g = \\tilde \\tau$, we have shown that the linearization of the $g$-action defines a $\\varphi$-equivariant isomorphism:\n$$\nf_g: \\mathfrak{M}\\otimes_{A,\\iota_g} A_{\\st}^{(2)}[\\frac{1}{p}] \\simeq \\mathfrak{M}\\otimes_{A} A_{\\st}^{(2)}[\\frac{1}{p}]\n$$\nof $A_{\\st}^{(2)}[\\frac{1}{p}]$-modules, and since $g(u)- u = Ez$ and $V$ is crystalline, $f_g$ defines a $\\varphi$-equivariant isomorphism:\n$$\nf_g: \\mathfrak{M}\\otimes_{A,\\iota_g} A^{(2)}[\\frac{1}{p}] \\simeq \\mathfrak{M}\\otimes_{A} A^{(2)}[\\frac{1}{p}]\n$$\nof $A^{(2)}[\\frac{1}{p}]$-modules. Here $\\iota_g: A \\to A^{(2)}_{\\st}$ (resp. $\\iota_g: A \\to A^{(2)})$) is defined by $u \\to g(u)$. On the other hand, by \\cite[Theorem 5.6]{wu2021galois}, we will see the $g$-action on $T^\\vee \\otimes W(\\mathbb C_p^\\flat)$ also descent to a $\\varphi$-equivariant morphism $c$ of $B^{(2)}$-modules, and recall that $B^{(2)}$ the is $p$-adic completion of $A^{(2)} [\\frac 1 E]$. Then by comparing $c$ and $f_g$ using the technique developed in \\S\\ref{subsec-phi-tau}, we will deduce Proposition~\\ref{thm-1prime} from Lemma~\\ref{lem-intersection}.\n \n\\begin{remark}\\label{rem-inputofWu} Our original strategy to prove Theorem \\ref{Thm-1} is to show $A^{(2)}_{\\st} [\\frac 1 p] \\cap W ({\\mathbb C^\\flat_p}) = A^{(2)}_{\\st}$ (resp. $A^{(2)} [\\frac 1 p] \\cap W (\\O^\\flat_{\\mathbb C_p}) = A^{(2)}$).  This is equivalent to that $ A^{(2)}\/ p , A^{(2)}_{\\st}\/ p$ injects in $\\mathbb C_p^\\flat$. Unfortunately, it does not work out though we can show  $ \\widetilde A^{(2)}\/ p , \\widetilde {A^{(2)}_{\\st}}\/ p$ injects in $\\mathbb C_p^\\flat$.\n\n\\end{remark}\n\n\n\\subsection{Relation to \\texorpdfstring{$(\\varphi, \\hat G)$}{(phi,Ghat)}-modules}\\label{subsec-phiGhatmodules} In this subsection, we show that the base ring $\\widehat{{\\mathcal R}}$ for the theory of $(\\varphi, \\hat G)$-modules can be replaced by $A^{(2)}_{\\st}$. To this end, this builds a new theory of $(\\varphi, \\hat G)$-modules  with new base ring $A^{(2)}_{\\st}$. Since the idea of this new theory is almost the same as that of the old one, We will use \\emph{classical} to indicate we are using the theory over $\\widehat {\\mathcal R}$. For example, when we say classical $(\\varphi, \\hat G)$-module, it means a $(\\varphi , \\hat G)$-module over $\\widehat {\\mathcal R}$. \nRecall $L = \\bigcup\\limits_{n =1}^\\infty K_{\\infty} (\\zeta_{p ^n})$, $\\hat G : = \\Gal(L \/K) $ and $H_K : = \\Gal (L \/ K _\\infty)$. Let $\\mathfrak m $  be the maximal ideal of $\\O_{\\mathbb C_p}^\\flat$ and set $I_+ = W(\\mathfrak m)$ so that $\\Ainf\/ I _+ = W(\\bar k)$. For any subring $B\\subset \\Ainf$ set $ I_+ B = B \\cap I_+$. Let $t = \\log \\epsilon$,  $t ^{(i)} = t ^{r(i)} \\gamma_{\\tilde q(i)}(\\frac{t ^{p-1}}{p})$ where $ i = (p-1) \\tilde q (i) + r(i)$ with $ 0 \\leq r(i)<p-1. $\n Recall that $\\widehat {\\mathcal R} : = \\Ainf \\cap {\\mathcal R}_{K_0 }$ where \n$${\\mathcal R}_{K_0}:=\\left  \\{ \\sum_{i = 0}^\\infty f_i t ^{(i)}, f_i\\in S[\\frac 1 p], f_i \\to 0\\  p{\\text{-adically}}  \\right \\}.$$\n\n\n\\begin{lemma} \\label{lem-properties-Ast}\n\\begin{enumerate}\n    \\item As a subring of $\\Ainf$, $A^{(2)}_{\\st}$ is stable under $G_K$-action and the $G_K$-action factors through $\\hat G$. \n\\item $A^{(2)}_{\\st}\/ I_+A^{(2)}_{\\st} = W(k)$. \n\\item $I_+ A^{(2)} \\subset u A^{(2)}_{\\st}$. \n\\item $\\varphi (A^{(2)}_{\\st} ) \\subset \\widehat {\\mathcal R}$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}(1) It is clear that the $G_K$-action is stable on $W(k)[\\![u , \\epsilon -1]\\!]$. Since $A^{(2)}_{\\st}$ is $(p , E)$-completion of $W(k)[\\![u , \\epsilon -1]\\!] [\\delta^i (w), i \\geq 0]$, to show that $A^{(2)}_{\\st}$ is $G_K$-stable, it suffices to show that $g (w)\\in A^{(2)}_{\\st}$ (because $g$ and $\\delta$ commutes, if $g(x)\\in A^{(2)} _{\\st}$ then so is $g (\\delta (x))$). Now $E w = \\epsilon-1$, we have $g (E) g (w) = g (\\epsilon -1)= \\epsilon ^{a(g)} -1 $. Then $g (w) = \\frac{E}{g(E)} \\frac {\\epsilon^{a(g)} -1}{E}$. By \\cite[Lemma 2.24]{BS19}, $E\/g(E)$ is a unit in $A^{(2)}_{\\st}$, then $g (w)\\in A^{(2)}_{\\st}$. \n\n(2) It is clear that both $u , \\epsilon-1 $ are in $I_+$. Hence $w \\in I_+$ because $E w = \\epsilon-1 \\in I_+$ and $E \\equiv p \\mod I_+$. For any $x=\\sum\\limits_{i = 0}^\\infty  p ^i[x_i] \\in \\Ainf$, $x \\in I_+$ if and only of $x_i \\in \\mathfrak m$. Then it is easy to check that $\\delta (I_+)\\subset I_+$, and consequently  all $\\delta^i (w) \\in I_+$. So $I_+ A^{(2)}_{\\st}$ is topologically generated by $u, y=\\epsilon-1 , \\delta ^i (w), i \\geq 0$ and hence $A^{(2)}_{\\st}\/ I_+A^{(2)}_{\\st} = W(k) $ as required. \n\n(3) $I_+ A^{(2)}$ is topologically generated by $u, v=\\epsilon u, \\{\\delta ^i (z)\\}, i \\geq 0$. And (3) follows from $z= u w$ and $\\delta^n (z)= u^{p ^n} \\delta^n (w)$.  \n\n(4) Since $A^{(2)}_{\\st} \\subset A^{(2)}_{\\st , \\max}$, it suffices to show that \n$\\varphi (A^{(2)}_{\\st , \\max}) \\subset {\\mathcal R}_{K_0}$. Since $\\varphi(\\Omax) \\subset A[\\![\\frac{E^p }{p}]\\!]\\subset S$, it suffices to show that $\\varphi (\\gamma_n (w) ) \\in {\\mathcal R}_{K_0}$. Note that $\\varphi (E) = p \\nu$ with $\\nu \\in A[\\![\\frac{E^p }{p}]\\!] ^\\times$ and $\\gamma _i (\\epsilon -1) \\in {\\mathcal R}_{K_0}$. And we have\n$$\n\\varphi(w)=\\varphi (\\frac{(\\epsilon -1)}{E})= \\nu^{-1} (\\epsilon-1) \\sum_{i = 1}^p \\left ( \\binom{p}{i} \/ p \\right ) (\\epsilon -1)^{i -1} \n$$\nwhich is a polynomial with coefficients in $\\mathbb Z$ and in variables $\\nu^{-1}$ and $\\gamma_i(\\epsilon -1)$'s. In particular $\\varphi (\\gamma_n (w) ) \\in {\\mathcal R}_{K_0}$ by basic properties of divided powers. \n\\end{proof}\n\n\\begin{definition} A (finite free) $(\\varphi, \\hat G)$-module of height $h$ is a (finite free) Kisin module $(\\mathfrak{M} , \\varphi_{\\mathfrak{M}})$ of height $h$ together with an $A^{(2)}_{\\st}$-semi-linear $\\hat G$-action on $\\widehat \\mathfrak{M} : = A^{(2)}_{\\st} \\otimes _A \\mathfrak{M}$ so that \n\\begin{enumerate}\n    \\item The actions of $\\varphi$ and $\\hat G$ on $\\widehat \\mathfrak{M}$ commutes; \n    \\item $\\mathfrak{M} \\subset \\widehat \\mathfrak{M} ^{H_K}$; \n    \\item $\\hat G$-acts on $\\widehat \\mathfrak{M} \/ I^+ A^{(2)}_{\\st}$ trivially. \n\\end{enumerate}\n\\end{definition}\n\nThe category of $(\\varphi, \\hat G)$-modules consists of the above objects and morphism of two $(\\varphi, \\hat G)$-modules is morphism of Kisn modules that commutes with actions of $\\hat G$. Given a $(\\varphi, \\hat G)$-modules $\\widehat \\mathfrak{M} : = (\\mathfrak{M} , \\varphi, \\hat G)$, we define a $\\mathbb Z_p$-representation of $G_K$, \n$$\\widehat T (\\widehat \\mathfrak{M}) : = \\mathrm{Hom}_{A^{(2)}_{\\st}, \\varphi} (A^{(2)}_{\\st} \\otimes _A \\mathfrak{M}, \\Ainf).$$\n\n\nSince $\\varphi (A^{(2)}_{\\st})\\subset \\widehat {\\mathcal R}$, given a $(\\varphi, \\hat G)$-module $\\widehat \\mathfrak{M} : = (\\mathfrak{M}, \\varphi, \\hat G)$-defined as the above, $(\\mathfrak{M}, \\varphi)$ together $\\hat G$-action on $\\widehat{\\mathcal R} \\otimes_{\\varphi, A}\\mathfrak{M}$ is a \\emph{classical} $(\\varphi, \\hat G)$-modules $\\widehat \\mathfrak{M}_c$. It is easy to check that $\\widehat T (\\widehat \\mathfrak{M} ) = \\widehat T(\\widehat \\mathfrak{M}_c): =\\mathrm{Hom}_{\\widehat {\\mathcal R}, \\varphi} (\\widehat {\\mathcal R} \\otimes _{\\varphi , A} \\mathfrak{M}, \\Ainf). $ \n\n\n\\begin{theorem}\\label{thm-2}\nThe functor $\\widehat T$ induces an anti-equivalence between the category of $(\\varphi, \\hat G)$-modules of height $h$ and the category of $G_K$-stable $\\mathbb Z_p$-lattices in semi-stable representations with Hodge-Tate weights in $[0 , \\dots , h]$. \n\\end{theorem}\n\\begin{proof}\nGiven an $(\\varphi, \\hat G)$-module $\\widehat \\mathfrak{M}= (\\mathfrak{M}, \\varphi, \\hat G)$, $\\widehat\\mathfrak{M}_c $ is a classical $(\\varphi, \\hat G)$-module. So $\\widehat T (\\widehat \\mathfrak{M})= \\widehat T (\\widehat \\mathfrak{M}_c)$ is a lattice inside semi-stable representation with Hodge-Tate weights in $[0, \\dots , h]$. Conversely, given a lattice in semi-stable representation $T$ with Hodge-Tate weights in $[0, \\dots , h]$, following the proof for the existence of classical $(\\varphi, \\hat G)$-module $\\widehat \\mathfrak{M} $ so that $\\widehat T (\\widehat \\mathfrak{M} ) = T$, it suffices to show that for any $g \\in G_K$, \n$g (\\mathfrak{M}) \\subset A^{(2)}_{\\st} \\otimes _A \\mathfrak{M}$, here $\\mathfrak{M} $ and $ A^{(2)}_{\\st} \\otimes _A \\mathfrak{M}$ are regarded as submodules of $T^\\vee \\otimes _{\\mathbb Z_p} \\Ainf$ via $\\iota_\\mathfrak{M}$ in \\eqref{eqn-iota-A} and uses the $G_K$-action on $T^\\vee \\otimes _{\\mathbb Z_p} \\Ainf$. This follows Theorem \\ref{Thm-1}. \n\\end{proof}\n\nNow let us discuss when $\\widehat T (\\widehat \\mathfrak{M})$ becomes a crystalline representation. Recall that $\\tau$ is a selected topological generator of $\\Gal (L\/ K_{1 ^\\infty}) $, and we have $\\tau (u) = \\epsilon u$ and $\\tau, H_K$ topologically generate $\\hat G$. \n\n\n\n\\begin{corollary}\\label{cor-crystalline} Select $\\tau \\in \\hat G$ as the above. Then $\\widehat T (\\widehat \\mathfrak{M} )$ is crystalline if and only if $\\tau (\\mathfrak{M}) \\subset A^{(2)} \\otimes _{A} \\mathfrak{M}$. \n\\end{corollary}\n\\begin{proof} Clearly for the selected $\\tau$, we have $\\tau (u) - u = Ez$. If $T: = \\widehat T (\\widehat \\mathfrak{M})$ is crystalline then Theorem \\ref{Thm-1} proves that $\\tau (\\mathfrak{M}) \\subset A^{(2)} \\otimes _{A} \\mathfrak{M}$. Conversely,  Suppose $\\tau (\\mathfrak{M}) \\subset A^{(2)} \\otimes _{A} \\mathfrak{M}$. \nThen we see that $(\\tau-1)\\mathfrak{M} \\subset u \\Ainf \\otimes_A \\mathfrak{M}$ by Lemma \\ref{lem-properties-Ast} (3). And we have this is enough to show that $\\widehat T (\\widehat \\mathfrak{M})$ is crystalline. For example, We will have $\\mathfrak{M}\\otimes_{A}(\\Ainf[\\frac{1}{p}]\/\\mathfrak{p})$ has a $G_K$-fixed basis given by a basis of $\\mathfrak{M}$, where the ideal $\\mathfrak{p}$ is defined as $\\mathfrak{p}:=\\cup_{n\\in \\mathbb N} \\varphi^{-n}(u)\\Ainf[\\frac{1}{p}] \\subset \\Ainf[\\frac{1}{p}]$. Then one can prove by the same method in \\cite[Thm. 3.8]{Ozeki} or directly use \\cite[Theorem 4.2.1]{Duthesis} that $T$ is crystalline.  \n\\end{proof}\n\\begin{remark}\\label{rem-Ast-is-good} Though $A^{(2)}_{\\st}$ is still complicated, for example, it is not noetherian, $A^{(2)}_{\\st}$ is still better than old $\\widehat {\\mathcal R}$: at least it has explicit topological generators. Furthermore, $A^{(2)}_{\\st}$ is $p$-adic complete. This can help to close the gap in \\cite{liu-Fontaine} mentioned in \\cite[Appendix B]{gao2021breuilkisin}. Indeed, as indicated by Remark B.0.5 \\emph{loc.cit.}, if $\\widehat {\\mathcal R}$ can be shown to be $p$-adic complete then the gap in \\cite{liu-Fontaine} can be closed. So by replacing $\\widehat {\\mathcal R}$ by $A^{(2)}_{\\st}$, we close the gap of \\cite{liu-Fontaine} (\\cite{gao2021breuilkisin} provides another similar way to close the gap). \n\\end{remark}\n\n\\section{Crystalline representations and prismatic \\texorpdfstring{$F$}{F}-crystals}\nIn this section, we reprove the theorem of Bhatt and Scholze on the equivalence of prismatic $F$-crystal and lattices in crystalline representations of $G_K$ and complete the proof of Theorem \\ref{Thm-1}. We start to discuss some  general facts on the absolute prismatic site (which allows general base rings). \n\n\\subsection{Prismatic \\texorpdfstring{$F$}{F}-crystals in finite projective modules}\\label{subsec-pris-crystal}\nLet $R=R_0\\otimes_W \\O_K=R_0[u]\/E$ as in the beginning of \\S2 and $X=\\Spf(R)$ with the $p$-adic topology.\n\n\\begin{definition}\nThe (absolute) prismatic site $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ of $X$ is the opposite of the category of bounded prisms $(A, I)$ that are $(p,I)$-completed together with a map $R \\to A\/I$, and a morphism of prisms $(A, I) \\to (B, J)$ is a covering map if and only if $A \\to B$ is $(p, I)$-completely faithfully flat. \\\\\n\nDefine the following functors: \n$$\n\\O_{\\mathlarger{\\mathbbl{\\Delta}}}: (A,I)\\mapsto A,\n$$ \nand for all $h\\in \\mathbb N$, let\n$$\n\\mathcal{I}_{\\mathlarger{\\mathbbl{\\Delta}}}^h: (A,I)\\mapsto I^h.\n$$\nIt is known in \\cite{BS19} that these are sheaves on $X_{\\mathlarger{\\mathbbl{\\Delta}}}$. We will also use $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/\\mathcal{I}_{{\\mathlarger{\\mathbbl{\\Delta}}}}]^\\wedge_p$ to denote functor assign $(A,I)$ to the $p$-adic completion of $A$ with $I$ inverted.\n\\end{definition} \n\nNow we verify $A^{(2)} $(resp.  $A^{(3)})$  constructed in \\S\\ref{subsrc-construct-A2} is indeed self (resp. triple) product of $A$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$. We mainly discuss the situation of $A^{(2)}$ while the proof of $A^{(3)}$ is almost the same. Recall that $\\breve A = \\breve R_0 [\\![u]\\!]= W\\langle t_1, \\dots , t_m \\rangle[\\![u]\\!]$. \n\nFirst we want to make a remark on the existence of nonempty self-coproduct in the category of prisms over $R$. We thank Peter Scholze for answering our question on Mathoverflow. And we just repeat his answer here. Let $(A_i,I_i)$ for $i=1,2$ that are prisms over $R$, let $A_0=A_1\\hat{\\otimes}_{\\mathbb Z_p} A_2$ where the completion is taken for the $(p,I_1,I_2)$-adic topology. Let $J$ be the kernel of the map:\n$$\nA_0 \\to A_1\/I_1\\otimes_{R} A_2\/I_2.\n$$\nLet $(A,I)$ be the prismatic envelope of $(A_1,I_1)\\to (A_0,J)$, one can  check this is the initial object in the category of prisms over $R$ that admits maps from $(A_i,I_i)$ such that the two $R \\to A_i\/I_i \\to A\/I$ agree. Also we want to note that in general, we don't know if the boundedness of $(A_1,I_1)$ and $(A_2,I_2)$ will imply the boundedness of their coproduct. But we have seen $A^{(2)}$ and $A^{(3)}$ are indeed bounded by Corollary \\ref{cor-filtration-shape}.\n \nTo start, note that there exists a $W$-linear map $\\breve i_2: \\breve A \\to A^{\\ho 2}$ induced by $u \\mapsto y$ and $t_i \\mapsto s_i$. We claim that $\\breve i_2$ uniquely extends to $ i_2 : A \\to A^{\\ho 2}$ which is compatible with $\\delta$-structures. Indeed, consider the following commutative diagram\n$$\\xymatrix@C=45pt{ A \\ar[r] ^-{\\overline i_2}\\ar@{-->}[rd]^{i_{2, n}} & A^{\\ho 2} \/ (p , J^{(2)} )  \\\\ {\\breve A}  \\ar[u]\\ar[r]^-{\\breve i_{2, n} } & A^{\\ho 2} \/ (p , J ^{(2)})^n\\ar[u]}  $$\nHere $\\breve i_{2, n}= \\breve i_2 \\mod (p , J ^{(2)})^n$ and $\\overline{i}_2$ is induced by $A \\to A\/(p, E)  \\simeq A^{\\ho 2}\/ (p , J ^{(2)}) $. Since $\\breve i_2 (u) = y = x+ (y -x)$ and $\\breve i_2 (t_i) = s_i = t _i + (s_i - t_i)$, we see that the above (outer) diagram commutes. Since $A$ is formally \\'etale over $\\breve A $ by $(p, u)$-adic topology, we conclude that there exists a unique map $i_{2, n} : A \\to A^{\\ho 2} \/ (p , J^{(2)})^n$ so that the above diagram commutes. Since $A ^{\\ho 2}$ is $(p, J ^{(2)}  )$-complete,  there uniquely exists  $i _2 : A \\to A^{\\ho 2}$ which extends $\\breve i_2$. To see $i_2$ is compatible with $\\delta$-structures. it suffices to show that $\\varphi \\circ i _2 = i_2 \\circ \\varphi$. But both of $\\varphi \\circ i _2$  and $ i_2 \\circ \\varphi$ extend $ \\breve A \\overset \\varphi \\to \\breve A \\to A^{\\ho 2}$.  Again by formally \\'etaleness of $A$ over $\\breve A$, we see that $\\varphi \\circ i _2 = i_2 \\circ \\varphi$. Hence we obtain a map $ 1 \\otimes i_2: A \\otimes _{\\mathbb Z_p }A \\to A^{\\ho 2}$. Define $\\theta^{\\otimes 2}: A\\otimes_{\\mathbb Z_p} A \\to R$ via $\\theta^{\\otimes 2} (a \\otimes b)= \\theta (a) \\theta (b)$. By the construction of $i _2$, we have the following commutative diagram\n\\[ \\xymatrix{ A \\otimes _{\\mathbb Z_p} A \\ar[r] ^{1 \\otimes i _2} \\ar[d]^{\\theta ^{\\otimes 2 }} & A ^{\\ho 2}\\ar[d]  \\\\ R \\ar[r]^- \\sim & A^{\\ho 2}\/ J ^{(2)}  }\\]\nLet $\\widehat {A ^{\\otimes 2}} $ be the $(p , \\ker (\\theta ^{\\otimes 2}))$-completion of $A^{\\otimes 2}: = A \\otimes_{\\mathbb Z_p} A$. \nHence $1 \\otimes i_2$ induces a map $\\hat i_2$ from the $\\widehat {A ^{\\otimes 2}}$ to $ A ^{\\ho 2}$ because $A ^{\\ho 2}$ is clearly $(p , J ^{(2)})$-complete. To treat $A^{\\ho 3}$, we construct $ i _3: A \\to A ^{\\ho 3}$ by extending $\\breve i _3: A \\to A^{\\ho 3}$ by sending $u \\mapsto w $ and $t _j \\mapsto r_j$. The same method shows that $i_3$ is compatible with $\\delta$-structure and we obtain a map $1 \\otimes i _2 \\otimes i_3 : A^{\\otimes 3} \\to A^{\\ho 3}$ with $A ^{\\otimes 3}: A \\otimes_{\\mathbb Z_p} A \\otimes_{\\mathbb Z_p}A$. Similarly, we obtain a natural map $\\hat i _3 : \\widehat {A ^{\\otimes 3}} \\to A ^{\\ho 3} $. \n\\begin{lemma} For $s= 2, 3$, $\\hat i_s : \\widehat {A ^{\\otimes s}} \\to A ^{\\ho s}$ are isomorphisms. \n\\end{lemma}\n\\begin{proof}\nWe need to construct an inverse of $\\hat i_s$. We only show for $\\hat i _2$ and the proof for $\\hat i _3$ is the same. \nLet $g: A^{\\ho 2} \\to \\widehat {A ^{\\otimes 2}}$ be the $A$-linear map by sending $y -x \\mapsto 1 \\otimes u - u \\otimes 1 $ and $s_j - t _j  \\mapsto  1 \\otimes t_j  - t_j \\otimes 1$.  Clearly $g$ is well-defined because $ 1 \\otimes u - u \\otimes 1$ and $1 \\otimes t_j  - t_j \\otimes 1$ are in $\\Ker (\\theta ^{\\otimes 2})$. Since $i_2 (u) = y$ and $i_2 (t_j) = s_j$,  $\\hat i _2 \\circ g $ is identity on $A ^{\\ho 2}$. Now it suffices to show that $h : = g \\circ \\hat i_2 $ is identity. Write $K = (p , \\Ker (\\theta ^{\\otimes 2}))$. Note that we have map $ A \\otimes_{\\mathbb Z_p} \\breve A \\to  \\widehat {A ^{\\otimes 2}} \\overset h \\to   \\widehat {A ^{\\otimes 2}}$ induced by $h  $ which we still call it $\\breve h $. \nNow  we  have the following commutative diagram  \n$$\\xymatrix@C=55pt{ A\\otimes_{\\mathbb Z_p} A  \\ar[r] ^-{\\mod K }\\ar@{-->}[rd]^{\\mod K^n}_{h_{ n} } & (A\\otimes_{\\mathbb Z_p} A ) \/ K  \\\\  A \\otimes_{\\mathbb Z_p}{\\breve A}  \\ar[u]\\ar[r]^-{\\breve h \\mod K^n  } & (A \\otimes_{\\mathbb Z_p} A )\/ K ^n\\ar[u]},   $$\nwhere $h_n$ is induced by $h \\mod K^n$. \nWe see that both $ h_n $ and $\\mod K^n$ on the dashed arrow can make the diagram commute. Then by the formal \\'etaleness of $A$ over $\\breve A$, we conclude that $h_n  = \\mod K^n$ and $h$ is the identity map. \n\\end{proof}\n\\begin{proposition}\\label{prop-selfproduct} $A^{(2)}$ and $A^{(3)}$ is self-product  and triple product of  $A$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$. \n\\end{proposition}\n\\begin{proof} \nIn the following, we only treat the case of $A^{(2)}$ while the proof for $A^{(3)}$ is the same. \nWe need to prove that  for any $B = (B,J)$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$, \n$$\n\\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A^{(2)},B)=\\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A, B ) \\times \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A,B).\n$$\nBy the above lemma, we have natural maps $A \\otimes_{\\mathbb Z_p} A \\to \\widehat{A^{\\otimes 2}} \\simeq A^{\\ho 2}$. Combined with natural map $A^{\\ho 2}\\to A^{(2)}$ as $A^{(2)}$ is the prismatic envelope of $ A^{\\ho 2}$ for the ideal $J^{(2)}$, we have map $\\alpha : A \\otimes_{\\mathbb Z_p} A \\to A^{(2)}$ which is compatible with $\\delta$-structures. Then $\\alpha$ induces map $$\n\\beta: \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A^{(2)},B)\\to \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A, B ) \\times \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A,B).\n$$\nTo prove the surjectivity of $\\beta$, given $f_i \\in  \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}}(A, B )$ for $i = 1,2$, we obtain a map  $f _1 \\otimes f _2: A \\otimes_{\\mathbb Z_p} A \\to B$. It is clear that $(f_1 \\otimes f_2) (\\Ker (\\theta ^{\\otimes 2})) \\subset J$. Since $B$ is $(p, J)$-derived complete, $f \\otimes f_2$ extends to a map $ f _1 \\ho f_2 : \\widehat{A ^{\\otimes 2}}\\simeq A ^{\\ho 2} \\to B$ which is compatible with $\\delta$-structures, Hence $f_1 \\ho f _2$ is a morphism of $\\delta$-algebra. Finally, by the universal properties of prismatic envelope, $f _1 \\ho f_2 $ extends to   a map of prisms $ f_1 \\ho_{{\\mathlarger{\\mathbbl{\\Delta}}}} f_2: A^{(2)} \\to B$ as required. \n\nFinally, we need to show that $\\beta$ is injective. It suffices to show that $A$-algebra structure map $i _1 : A\\to A^{(2)} $ and $i'_2: A \\overset{i_2}{\\to } A^{\\ho 2} \\to A^{(2)}$ both are  injective. \nSince all rings here are $(p, E)$-complete integral domains, it suffices to check that $i_1 , i_2' \\mod (p, E)$ are injective. By Proposition \\ref{prop-key-property}, we see that $i_1 \\mod (p, E)$ is $R\/ pR \\to R\/pR [\\{\\gamma _i (z_j)\\}] $, so it is injective. By the construction $i'_2$ and $i_2$, we see that $i'_2 \\mod (p, E)$ is the same as $A\/(p, E) \\to A ^{\\ho 2}\/ (p , J ^{(2)}) \\to A^{(2)} \/(p, E)$, which is same as $R\/ pR \\to R\/pR [\\{\\gamma _i (z_j)\\}] $.  So it is injective. \n\\end{proof}\n\n\\begin{remark}\\label{rem-Astprelog}\n    When $R=\\O_K$ is a complete DVR with perfect residue field $k$, we know a priori, the self-product $A^{(2)}$ of $(A,(E))$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ can be constructed as the prismatic envelope of $(A,(E))\\to (B,I)$, where $B$ is the $(p,E(u),E(v))$-adic completion of $W(k)[\\![u]\\!] \\otimes_{\\mathbb Z_p} W(k)[\\![v]\\!]$ and $I$ is the kernel of the map:\n    $$\n    B \\to A\/(E)\\otimes_{R} A\/(E)=R.\n    $$\n    On the other hand,  $W(k)$ is formally \\'etale over $\\mathbb Z_p$ for the $p$-adic topology, so for all $(C,J)\\in X_{\\mathlarger{\\mathbbl{\\Delta}}}$, the map $W(k)\\to R \\to C\/J$ lifts uniquely to a map $W(k) \\to C$. In particular, for all $(C,J)\\in X_{\\mathlarger{\\mathbbl{\\Delta}}}$, $C$ has a natural $W(k)$-algebra structure. So when we construct the self-product, we can also consider $A^{(2)}$ as the prismatic envelope of $(A,(E))\\to (C,J)$, where $C$ is the $(p,E(u),E(v))$-adic completion of $A\\otimes_{W(k)} A$ and $J$ is the kernel of the map:\n    $$\n    C \\to A\/(E)\\otimes_{R} A\/(E)=R.\n    $$\n    We  have $C\\simeq W(k)[\\![u,v]\\!]$, $J=(E(u),u-v)$ and $A^{(2)}=W(k)[\\![u,v]\\!]\\{\\frac{u-v}{E}\\}^\\wedge_\\delta$.\n\\end{remark}\n\n\\begin{definition}\\label{def-Fcrystal}\n\\begin{enumerate}\n    \\item \nA prismatic crystal over $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules (resp. $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/I]^\\wedge_p$-modules) is a finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-module (resp. $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/I]^\\wedge_p$-module) $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ such that for all morphisms $f: (A, I) \\to (B, J)$ of prisms, it induces an isomorphism:\n$$\nf^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},A} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,I))\\otimes_A B \\simeq \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},B} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B,J))\n$$\n$$\n(resp.\\quad f^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},A} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,I))\\otimes_{A[1\/I]^\\wedge_p} B[1\/I]^\\wedge_p \\simeq \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},B} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B,J))).\n$$\n\n\\item A prismatic $F$-crystal over $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ of height $h$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules is a prismatic crystal $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules together with a $\\varphi_{\\O_{\\mathlarger{\\mathbbl{\\Delta}}}}$-semilinear endomorphism $\\varphi_{\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}}$ of the $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-module $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}: \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\to \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ such that the cokernel of the linearization $\\varphi^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\to \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is killed by $\\mathcal{I}^h$.\n\\end{enumerate}\n\n\\end{definition}\n\n\\begin{proposition}\\label{prop-cover-final-object}\nIf the sheaf represented by $(B,I)$ in $\\Shv(X_{\\mathlarger{\\mathbbl{\\Delta}}})$ covers the final object $\\ast$ in $\\Shv(X_{\\mathlarger{\\mathbbl{\\Delta}}})$, i.e., for any $(C,J)$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$, there is a $(P, J)$ lies over $(B,I)$ and covers $(C,J)$. Also assume that the self-coproduct $B^{(2)}$ and self-triple-coproduct $B ^{(3)}$ of $(B,I)$ are inside $X_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, i.e., they are bounded. Then a prismatic crystal $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ over $X$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules is the same as a finite projective module $\\mathfrak{M}$ over $B$ together with a descent data $\\psi: \\mathfrak{M}\\otimes_{i_1,B} B^{(2)}\\simeq \\mathfrak{M}\\otimes_{i_2,B} B^{(2)}$ satisfies the cocycle condition. Here  $i _j : B \\to B^{(2)}$ $(j=1,2)$ are the two natural maps.\n\\end{proposition}\n\n\\begin{proof}\nFirst let $\\mathfrak{M}$ be a prismatic crystal in finite projective modules. Define $\\mathfrak{M}= \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B,I))$, and the descent data comes from the crystal property:\n$$\n\\psi:\\mathfrak{M}\\otimes_{i_1,B} B^{(2)}\\simeq  \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B^{(2)},I)) \\simeq \\mathfrak{M}\\otimes_{i_2,B} B^{(2)}.\n$$\nNow  given $(\\mathfrak{M}, \\psi)$, then for any $(C,J)$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$, we need to construct a finite projective module over $C$. We choose the $(P, J)$ as in the assumption, let $\\mathfrak{M}_P=\\mathfrak{M}  \\otimes_B P$, and consider the following diagram:\n$$\n\\begin{tikzcd}\nC \\arrow[r] & P \\arrow[rr,\"f_1\"] & & P^{(2)}_C \\\\\n & B \\arrow[u] \\arrow[r,\"i_1\"] & B^{(2)} \\arrow[ur,\"f\"] & \\\\\n & & B \\arrow[u,\"i_2\"] \\arrow[r] & P \\arrow[uu,\"f_2\"] \\\\\n & & & C \\arrow[u]\n\\end{tikzcd}\n$$\nHere $(P^{(2)}_C,J)$ is the self-coproduct of $(P,J)$ in the category of prisms over $(C,J)$, and the existence of $(P^{(2)}_C,J)$ is from \\cite[Corollary 3.12]{BS19}, where they also show that $P^{(2)}_C$ is the derived $(p, J)$-completion of $P\\otimes^\\mathbb L_C P$ and $(P^{(2)}_C,J)$ is bounded. As a bounded prism over $(C,J)$, $(P^{(2)}_C,J)$ is naturally inside $X_{\\mathlarger{\\mathbbl{\\Delta}}}$, so $f$ exists by the universal property of $B^{(2)}$. So if we take the base change of $\\psi$ along $f$, we get \n$$\nf^\\ast\\psi: (\\mathfrak{M}\\otimes_{i_1,B} B^{(2)})\\otimes_{B^{(2)},f}P^{(2)}_C \\simeq (\\mathfrak{M}\\otimes_{i_2,B} B^{(2)})\\otimes_{B^{(2)},f}P^{(2)}_C\n$$\nwhich is the same as an isomorphism:\n$$\n\\psi_C: \\mathfrak{M}_{P}\\otimes_{P,f_1}P^{(2)}_C \\simeq \\mathfrak{M}_{P}\\otimes_{P,f_2}P^{(2)}_C.\n$$\nSimilar arguments will show $\\psi_C$ satisfies the cocycle condition. And $\\mathfrak{M}_{P}$ descents to a finite projective module over $C$ by \\cite[Proposition A.12]{ALB}.\n\\end{proof}\n\n\\begin{remark}\nWe want to note that the structures of finite nonempty coproducts in the category of bounded prisms over a prism $(A,I)$ is much simpler compared with the structure of finite nonempty products in the category $(R\/A)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ (cf. \\cite[Lecture V, Corollary 5.2]{BhaNotes18}).\n\\end{remark}\n\n\\begin{lemma}\\label{lem-AEcoversfinal}\nThe prism $(A,(E))$ defined in \\S\\ref{subsrc-construct-A2} covers the final object $\\ast$ in $\\Shv(X_{\\mathlarger{\\mathbbl{\\Delta}}})$ in the sense of Proposition~\\ref{prop-cover-final-object}. And $A^{(2)}$ and $A^{(3)}$ are bounded.\n\\end{lemma}\n\\begin{proof}\nThe proof is similar to \\cite[Lemma 5.2.8]{ALB}, we need to show for $R$ defined as in \\S\\ref{subsrc-construct-A2}, there exists a quasi-syntomic perfectoid cover of $R$. We will construct this perfectoid cover similar to \\cite[\\S 7.1]{Kim12}.\n\nFirst recall we have $R=\\O_K\\otimes_{W}R_0$, and we fix a compatible system $\\{\\varpi_n\\}_{n\\geq 0}$ of $p^n$-th roots of a uniformizer $\\varpi_0$ of $\\O_K$ inside $E$. Let $\\widehat K_\\infty$ be the $p$-adic completion of $\\cup_n K(\\varpi_n)$, we know $\\widehat K_\\infty$ is perfectoid. Use $\\overline{R}_0[\\![u]\\!]$ to denote $A\/(p)=R\/(\\varpi)=R_0\/(p)[\\![u]\\!]$, and let $\\overline{R}_0[\\![u]\\!]_{\\rm perf}^\\wedge$ to be the $u$-adic completion of the direct perfection of $\\overline{R}_0[\\![u]\\!]$, it can be checked directly that $(\\overline{R}_0[\\![u]\\!]_{\\rm perf}^\\wedge[1\/u],\\overline{R}_0[\\![u]\\!]_{\\rm perf}^\\wedge)$ is a perfectoid affinoid $\\widehat K_\\infty^\\flat$-algebra, by tilt equivalence, there is a corresponded perfectoid affinoid $\\widehat K_\\infty$-algebra. More explicitly, let $\\tilde{R}_\\infty = W(\\overline{R}_0[\\![u]\\!]_{\\rm perf}^\\wedge)\\otimes_{W(\\O_{\\widehat K_\\infty}^\\flat),\\theta} \\O_{\\widehat K_\\infty}$. Then  $\\tilde{R}_\\infty$ is naturally an  $R$-algebra, and we claim it is a quasi-syntomic cover of $R$.\n\nTo show this, by \\cite[\\S 7.1.2]{Kim12}, we have\n$$\n\\tilde{R}_\\infty = (R_0\\widehat {\\otimes}_{W}\\O_{\\widehat K_\\infty})\\widehat {\\otimes}_{\\mathbb Z_p} \\mathbb Z_p\\langle T_i^{-p^\\infty}\\rangle\n$$\nwhere $T_i \\in R_0$ is any lift of a $p$-basis of $R_0\/(p)$. We have $\\O_K\\to \\O_{\\widehat K_\\infty}$ is a quasi-syntomic cover so by (2) of \\cite[Lemma 4.16]{BMS2}, $R \\to R_0\\widehat {\\otimes}_{W}\\O_{\\widehat K_\\infty}$ is also a quasi-syntomic cover. And we have $S=\\mathbb Z_p\\langle T_i^{-p^\\infty}\\rangle$ is a quasi-syntomic ring, this can be seen by constructing  a perfectoid quasi-syntomic covering of it, so by Lemma 4.34 of $loc.cit.$, we have the complex $\\mathbb{L}_{S\/\\mathbb Z_p} \\in D(S)$ has $p$-complete Tor amplitude in $[-1,0]$. In particular, $\\mathbb Z_p \\to \\mathbb Z_p\\langle T_i^{-p^\\infty}\\rangle$ is also a quasi-syntomic cover, so applying (1) in Lemma 4.16 of $loc. cit.$,  $R \\to \\tilde{R}_\\infty$ is a quasi-syntomic perfectoid cover.\n\nThe  boundedness of $A^{(2)}$ and $A ^{(3)}$ is from (2) in Corollary~\\ref{cor-filtration-shape}.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-crystal-descentdata}\nAssume the the base $X=\\Spf(R)$ satisfies the condition in \\S\\ref{sec-ring-strcuture}, and let $A$, $A^{(2)}$ and $A^{(3)}$ be  defined as in \\S\\ref{subsrc-construct-A2}, then a prismatic $F$-crystal $(\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}, \\varphi_{\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}})$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules of height $h$ over $X$ is the same as a Kisin module $(\\mathfrak{M},\\varphi_\\mathfrak{M})$ of height $h$ over $A$ with a descent datum\n$$\nf: \\mathfrak{M} \\otimes_{A,i_1} A^{(2)} \\simeq \\mathfrak{M} \\otimes_{A,i_2} A^{(2)}\n$$\nthat compatible with the $\\varphi$-structure and satisfies the cocycle condition over $A^{(3)}$.\n\\end{corollary}\n\n\n\\begin{theorem}(\\cite[Theorem 1.2]{BS2021Fcrystals})\\label{Thm-main-1}  \nLet $T$ be a crystalline representation of $G_K$ over a $\\mathbb Z_p$-lattice of Hodge-Tate weights in $[0,h]$, then there is a prismatic $F$-crystal $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}(T)$ over $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ of height $h$ over $X$ such that $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,E))$ is the Kisin module associated to $T$. Moreover, the association of $T\\mapsto \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}(T)$ induces an equivalence of the above two categories. \n\\end{theorem}\nWe will prove this theorem in \\S \\ref{subsec-pris-crystal-proof}. \n\n\\begin{remark}\nTheorem~\\ref{Thm-main-1} was first established by Bhatt-Scholze in \\cite[Theorem 1.2]{BS2021Fcrystals}. The harder direction of \\cite[Theorem 1.2]{BS2021Fcrystals} is to show for all $\\mathbb Z_p$-lattices inside crystalline representations of $G_K$, one can attach a prismatic $F$-crystal. Using the theory of $(\\varphi,\\hat{G})$-modules, we have shown in \\S\\ref{subsec-G-image}, given a crystalline representation of $G_K$ over a $\\mathbb Z_p$-lattices $T$, we can attach a Kisin module $\\mathfrak{M}$ and a descent data\\footnote{Strictly speaking, \\S\\ref{subsec-G-image} only constructs an isomorphism but have not checked that  it satisfies cocycle condition, which will be proved in \\S \\ref{subsec-pris-crystal-proof}.} \n$$\nf_{\\tilde{\\tau}}: \\mathfrak{M}\\otimes_{A,i_1} A^{(2)}[\\frac{1}{p}] \\simeq \\mathfrak{M}\\otimes_{A,i_2} A^{(2)}[\\frac{1}{p}]\n$$\ncomes from the $\\tau$-action. We just show this is a $\\varphi$-equivariant isomorphism, and we need to show it gives rise to a descent data over $A^{(2)}$. As we have mentioned in Remark~\\ref{rem-inputofWu}, we can not find a direct ring theoretic proof of this. Our idea is to use result of \\cite{wu2021galois} or \\cite[Corollary 3.7]{BS2021Fcrystals}: the underlying Galois representation $T$ gives a descent data over $A^{(2)}[\\frac{1}{E}]^\\wedge_p$. To finish our proof, we need to compare this descent data with $f_{\\tilde{\\tau}}$ over $A^{(2)}[\\frac{1}{E}]^\\wedge_p[\\frac{1}{p}]$. This lead us to develop a ``prismatic\" $(\\varphi,\\tau)$-module theory in the next subsection, where we will have Lemma~\\ref{lem-evaluation-1} and Lemma~\\ref{lem-evaluation-2} to help us compare descent data over $A^{(2)}[\\frac{1}{E}]^\\wedge_p$ and $A^{(2)}[\\frac{1}{E}]^\\wedge_p[\\frac{1}{p}]$ via an evaluation map to $W(\\O_{\\hat{L}}^\\flat)$.\n\\end{remark}\n\n\n\\subsection{\\texorpdfstring{$(\\varphi,\\tau)$}{(phi,tau)}-modules and prismatic \\texorpdfstring{$F$}{F}-crystals}\\label{subsec-phi-tau} In this subsection, we make some preparations to prove Proposition \\ref{thm-1prime} and Theorem \\ref{Thm-main-1}. So we restrict to the case that $R=\\O_K$ is a complete DVR with perfect residue field. \n\\begin{definition}\nAn \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ is a pair $(\\mathcal M, \\varphi_\\mathcal M)$ such that $\\mathcal M$ is a finite free module over $A[1\/E]^\\wedge_p$, and $\\varphi_\\mathcal M$ is an isomorphism\n$$\n\\varphi_\\mathcal M: \\varphi^\\ast \\mathcal M: = A[1\/E]^\\wedge_p\\otimes_{\\varphi , A[1\/E]^\\wedge_p} \\mathcal M  \\simeq \\mathcal M\n$$\nof $A[1\/E]^\\wedge_p$-modules. And we define an \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p[1\/p]$ to be a $\\varphi$-module over $A[1\/E]^\\wedge_p[1\/p]$ such that it is obtained from an \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ by base change.\n\nAn \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ (resp. $A[1\/E]^\\wedge_p[1\/p]$) with descent data is a triple $(\\mathcal M, \\varphi_\\mathcal M, c)$, such that $(\\mathcal M, \\varphi_\\mathcal M)$ is an \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ (resp. $A[1\/E]^\\wedge_p[1\/p]$), and $c$ is an isomorphism\n$$\nc: \\mathcal M \\otimes_{A[1\/E]^\\wedge_p,i_1} B^{(2)} \\simeq \\mathcal M \\otimes_{A[1\/E]^\\wedge_p,i_2} B^{(2)}\n$$\n$$\n(\\text{resp. }c: \\mathcal M \\otimes_{A[1\/E]^\\wedge_p[1\/p],i_1} B^{(2)}[1\/p] \\simeq \\mathcal M \\otimes_{A[1\/E]^\\wedge_p[1\/p],i_2} B^{(2)}[1\/p])\n$$\nthat compatible with the $\\varphi$-structure and satisfies the cocycle condition over $B^{(3)}$ (resp. $B^{(3)}[\\frac 1 p]$). Here for $j=1,2$, $i_j: A[1\/E]^\\wedge_p \\to B^{(2)}$ is the map induced from $i_j: (A,(E)) \\to (A^{(2)},(E))$. \n\\end{definition}\n\n\\begin{remark}\\label{rmk-Wuandevaluation}\nIt is the main result in \\cite{wu2021galois} and \\cite[\\S2]{BS2021Fcrystals} that there is an equivalence of the category of lattices in representations of $G_K$ and the category of prismatic $F$-crystals in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/I]^\\wedge_p$-modules over $\\O_K$. Also by \\cite[Proposition 2.7]{BS2021Fcrystals}, one can show prismatic $F$-crystals in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/I]^\\wedge_p$-modules is the same as \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ with descent data. \n\\end{remark}\n\n\nThe aim of this subsection is to use the ideas in \\cite{wu2021galois} and \\cite[\\S 5.5]{KedlayaLiu-relativeII} show that \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ (resp. $A[1\/E]^\\wedge_p[1\/p]$) with descent data are equivalence to $\\mathrm{Rep}_{\\Z_p}(G_K)$ (resp. $\\mathrm{Rep}_{\\Q_p}(G_K)$). More importantly, for all $\\gamma \\in \\hat{G}$, we will construct an evaluation at $\\gamma$ map\n$$\ne_\\gamma: B^{(2)} \\to W(\\hat{L}^\\flat)\n$$\nand use it to study $\\varphi$-equivariant morphisms between finite free $B^{(2)}$ and $B^{(2)}[1\/p]$-modules. We will see the evaluation at $\\tau$ map will play a crucial role in our proof of Proposition~\\ref{thm-1prime} and the Theorem~\\ref{Thm-main-1} below.\n\n\nRecall in \\S\\ref{subsec-phiGhatmodules}, we define $L = \\bigcup\\limits_{n =1}^\\infty K_{\\infty} (\\zeta_{p ^n})$, $\\hat G : = \\Gal(L \/K) $ and $H_K : = \\Gal (L \/ K _\\infty)$. Moreover, we define $\\widehat K_{1^\\infty}$ to be the $p$-adic completion of $\\cup_{n\\geq 0} K(\\zeta_{p^n})$, and we let $\\hat{L}$ to be the $p$-adic completion of $L$. It is clear that $A[1\/E]_p^\\wedge\\subset W(\\hat L ^\\flat)^{H_K}$.\nRecall the following definition and theorem in \\cite{Caruso-phitau}:\n\n\\begin{theorem}\\label{thm-caruso}\nAn \\'etale $(\\varphi,\\tau)$-module is a triple $(\\mathcal M, \\varphi_{\\mathcal M}, \\hat{G})$ where\n\\begin{itemize}\n    \\item $(\\mathcal M, \\varphi_{\\mathcal M})$ is an \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$;\n    \\item $\\hat{G}$ is a continuous $W(\\hat{L}^\\flat)$-semi-linear $\\hat{G}$-action on $\\hat{\\mathcal M}:=W(\\hat{L}^\\flat)\\otimes_{A[1\/E]^\\wedge_p}\\mathcal M$, and $\\hat{G}$ commutes with $\\varphi_{\\mathcal M}$;\n    \\item regarding $\\mathcal M$ as an $A[1\/E]^\\wedge_p$-submodule of $\\hat{\\mathcal M}$, we have $\\mathcal M\\subset \\hat{\\mathcal M}^{H_K}$.\n\\end{itemize} \nThen there is an anti-equivalence of the category of \\'etale $(\\varphi,\\tau)$-modules and $\\mathrm{Rep}_{\\Z_p}(G_K)$, such that if $T$ corresponds to $(\\mathcal M, \\varphi_{\\mathcal M}, \\hat{G})$, then\n$$\nT^\\vee = (\\hat{\\mathcal M}\\otimes_{W(\\hat{L}^\\flat)}W(\\mathbb C_p^\\flat))^{\\varphi=1}.\n$$\n\\end{theorem}\n\nOne of the basic facts used in the theory of \\'etale $(\\varphi,\\tau)$-modules developed in \\cite{Caruso-phitau} is that $\\Gal(\\hat{L}\/\\widehat K_{1^\\infty})\\simeq \\mathbb Z_p$, and we write $\\tau$ to be a topological generator of $\\Gal(\\hat{L}\/K_{1^\\infty})$ determined by $\\tau(\\varpi_n)=\\zeta_{p^n}\\varpi_n$ as the discussion before Corollary \\ref{cor-crystalline}. Also  $\\hat{G}$ is topologically generated by $\\tau$ and $H_K$, so in particular,  the $\\hat{G}$-action on $\\hat{\\mathcal M}$ is determined by the action of $\\tau$ on $\\mathcal M$ inside $\\hat{\\mathcal M}$. As discussed before, we will provides a direct correspondence of the category of \\'etale $(\\varphi,\\tau)$-modules and the category of \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ with descent data. Moreover, we will construct an evaluation at $\\tau$ map:\n$$\ne_\\tau: B^{(2)} \\to W(\\hat{L}^\\flat),\n$$\nand show that the $\\tau$-action on $\\mathcal M$ inside $\\hat{\\mathcal M}$ is given by the base change of the descent data along $e_\\tau$. \n\n\\begin{remark}\nIn \\cite[Theorem 5.2]{wu2021galois}, they prove a similar equivalence but for \\'etale $(\\varphi,\\Gamma)$-modules. The theory of \\'etale $(\\varphi,\\Gamma)$-module is defined for the cyclotomic tower $K_{1^\\infty}$ over $K$ while the theory of \\'etale $(\\varphi,\\tau)$-modules is defined using the Kummer tower $K_{\\infty}$. We will use a lot of ideas and results developed in \\cite{wu2021galois} when proving our claims in this subsection. The main difficulty in our situation is that the Kummer tower $K_\\infty$ is not a Galois tower over $K$. To deal with this, we have to use the idea in \\cite[\\S 5.5]{KedlayaLiu-relativeII}. Roughly speaking, we will take the Galois closure $L$ of $K_\\infty$,  then prove results over $\\hat{L}$, then descent back to $K_\\infty$ using the fact $\\widehat K_\\infty=\\hat{L}^{H_K}$. \n\nOne should be able to construct the evaluation map in the content of \\cite{wu2021galois} the same way as we define in this subsection. This map will give a more direct correspondence of the descent data and the $\\Gamma$-actions on \\'etale $(\\varphi,\\Gamma)$-modules.\n\\end{remark}\n\nBy \\cite[Lem 3.9]{BS19}, any prism $(B , J)$ admits a map into its perfection $(B_{\\perf}, JB_{\\perf})$. The following theorem (\\cite[Thm 3.10]{BS19}) is the key to understand perfect prisms.  \n\\begin{theorem}\\label{thm-perfectprismandperfectoidring}\n$(A,I)\\to A\/I$ induces an equivalence of the category of perfect prisms over $\\O_K$ with the category of integral perfectoid rings over $\\O_K$.\n\\end{theorem}\n \nLet $(A,(E))$ be the Breuil-Kisin prism defined in \\S\\ref{subsrc-construct-A2}, we have\n\\begin{lemma}\\label{lem-perfectionofA} $A_{\\perf}\\simeq W(\\O_{\\widehat K_\\infty}^\\flat)$.\n\\end{lemma}\n\\begin{proof}\nExactly the same as the proof of \\cite[Lemma 2.17]{wu2021galois}\n\\end{proof}\n\n\\begin{lemma}\nLet $\\Perfd_K$ be the category of perfectoid $K$-algebras, then $\\Perfd_{K}$ admits finite non-empty coproducts. \n\\end{lemma}\n\\begin{proof}\nLet $R$ and $S$ be two perfectoid $K$-algebras, it follows from \\cite[Corollary 3.6.18]{KedlayaLiu-relativeI} that the uniform completion $(R\\otimes_K S)^u$ of the tensor product $(R\\otimes_K S)$ is again a perfectoid $K$-algebra, and it is easy to show this is the coproduct of $R$ and $S$ in the category of perfectoid $K$-algebras.\n\\end{proof}\n\nFor $i\\in \\mathbb N_{>0}$, let $(A^{(i)},(E))$ (resp. $(\\Ainf(\\O_{\\hat{L}})^{(i)},(E))$) denote the $i$-th self-coproduct of $(A,(E))$ (resp. $(\\Ainf(\\O_{\\hat{L}}),(E))$) in the category of prisms over $\\O_K$, where $\\Ainf(\\O_{\\hat{L}}):=W(\\O_{\\hat{L}}^\\flat)$. The following is  a description of $(A^{(i)})_{\\perf}[1\/E]^\\wedge_p$ and $(\\Ainf(\\O_{\\hat{L}})^{(i)})_{\\perf}[1\/E]^\\wedge_p$. \n\n\\begin{lemma}\\label{lem-Aiperf}\nLet $\\widehat K_\\infty^{(i)}$ (resp. $\\hat{L}^{(i)}$) be the $i$-th self-coproduct of $\\widehat K_\\infty$ (resp. $\\hat{L}$) in $\\Perfd_K$, then $(A^{(i)})_{\\perf}[1\/E]^\\wedge_p \\simeq W((\\widehat K_\\infty^{(i)})^\\flat)$ (resp. $(\\Ainf(\\O_{\\hat{L}})^{(i)})_{\\perf}[1\/E]^\\wedge_p \\simeq W((\\hat{L}^{(i)})^\\flat)$).\n\\end{lemma}\n\\begin{proof}\nWe will only prove the lemma for $(A^{(i)})_{\\perf}[1\/E]^\\wedge_p$, and the case for $(\\Ainf(\\O_{\\hat{L}})^{(i)})_{\\perf}[1\/E]^\\wedge_p $ is similar.\n\nWe use similar arguments as in \\cite[Lemma 5.3]{wu2021galois}. Fix $i$, first we can show $(A^{(i)})_{\\perf}$ is the $i$-th self-coproduct of $(A_{\\perf}, (E))$ in the category of perfect prisms over $\\O_K$, i.e. $(A^{(i)})_{\\perf}=(A_{\\perf})^{(i)}_{\\perf}$. By Theorem~\\ref{thm-perfectprismandperfectoidring}, Lemma~\\ref{lem-perfectionofA} and \\cite[Proposition 2.15]{wu2021galois}, if we let $S=(A^{(i)})_{\\perf}\/E$, then $S[1\/p]$ is the $i$-th self-coproduct of $\\widehat K_\\infty$ in the category of perfectoid $K$-algebras. Now we have \n$$\n(A^{(i)})_{\\perf}[1\/E]^\\wedge_p\\simeq W(S^\\flat)[1\/[\\varpi^\\flat ]]^\\wedge_p=W(S^\\flat[1\/\\varpi^\\flat ])=W((S[1\/p])^\\flat)\\simeq W((\\widehat K_\\infty^{(i)})^\\flat).\n$$\n\\end{proof}\n\n\\begin{remark}\\label{rem-diamonds}\nThere is another way to view $\\widehat K_\\infty^{(i)}$ in terms of diamonds over $\\mathrm{Spd}(K,\\O_K)$ which is used in the proof of \\cite[Lemma 5.3]{wu2021galois}, that there exist a ring of integral elements $\\widehat K_\\infty^{(i),+}$ in $\\widehat K_\\infty^{(i)}$, such that we have \n\\begin{equation}\\label{eq-diamondKi}\n    \\Spa(\\widehat K_\\infty^{(i)},\\widehat K_\\infty^{(i),+})^\\diamond \\simeq \\underbrace{\\Spa(\\widehat K_\\infty,\\widehat K_\\infty^{+})^\\diamond \\times_{\\mathrm{Spd}(K,\\O_K)} \\ldots \\times_{\\mathrm{Spd}(K,\\O_K)}\\Spa(\\widehat K_\\infty,\\widehat K_\\infty^{+})^\\diamond}_{i\\text{-copies of } \\Spa(K_\\infty,K_\\infty^{+})^\\diamond}.\n\\end{equation}\nAnd similar results hold for $\\hat{L}$. Using this description and the fact that functor from perfectoid spaces over $\\Spa(K, \\O_K)$ to diamonds over $\\mathrm{Spd}(K, \\O_K)$ is an equivalence, we have $\\hat{L}^{(i)}$ has a natural action of $\\hat{G}^i$ coming from the action on the diamond spectrum. Since $\\hat{L}^{H_K}=\\widehat K_\\infty$, we have\n$$\n\\Spa(\\widehat K_\\infty^{(i)},\\widehat K_\\infty^{(i),+})^\\diamond \\simeq \\left (\\Spa(\\hat{L},\\O_{\\hat{L}})^\\diamond \\times \\ldots \\times_{\\mathrm{Spd}(K,\\O_K)}\\Spa(\\hat{L},\\O_{\\hat{L}})^\\diamond \\right )^{H_K^i} \\simeq (\\Spa(\\hat{L}^{(i)},\\hat{L}^{(i),+})^\\diamond  )^{H_K^i}.\n$$\nThat is, $(\\hat{L}^{(i)})^{H_K^i}=\\widehat K_\\infty^{(i)}$.\n\\end{remark}\n\nNow we use ideas in \\cite{wu2021galois} and \\cite[\\S 5.5]{KedlayaLiu-relativeII} to study \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ with descent data. We will show this category is the same as generalized $(\\varphi,\\Gamma)$-modules in the work of Kedlaya-Liu. The following is a quick review of Example 5.5.6 and 5.5.7 in \\cite{KedlayaLiu-relativeII}.\n\nFirstly, one has $\\hat{L}^{(i)}\\simeq \\Cont(\\hat{G}^{i-1}, \\hat{L})$, here $\\Cont$ means the set of continuous functions. One can see this fact from the proof of \\cite[Theorem 5.6]{wu2021galois}. When $i=2$, we choose the two canonical maps $i_1,i_2:\\hat{L} \\to \\hat{L}^{(2)}$, corresponds to $j_1,j_2: \\hat{L} \\to \\Cont(\\hat{G}, \\hat{L})$ given by \n\\begin{equation}\\label{eq-j1j2}\n   j_1(x): \\gamma \\mapsto \\gamma (x) \\quad \\text{ and } \\quad j_2(x): \\gamma \\mapsto x.\n\\end{equation}\n\nFrom Remark~\\ref{rem-diamonds}, there is a natural action of $\\hat{G}^2$ on $\\hat{L}^{(2)}$. One can check this corresponds to the $\\hat{G}^2$-action on $\\Cont(\\hat{G},\\hat{L})$ given by:\n$$\n(\\sigma_1,\\sigma_2)(f)(\\gamma)=\\sigma_2 f(\\sigma_2^{-1}\\gamma\\sigma_1).\n$$\n\n\\begin{remark}\nWe interchange the roles of $j_1$ and $j_2$ comparing with the isomorphism defined in \\cite[Example 5.5.6]{KedlayaLiu-relativeII}, so the $\\hat{G}^2$-action is different from that in Example 5.5.7 of $loc. cit.$, we will see this definition is more convenient when relating the descent data with the semilinear group actions. \n\\end{remark}\n\nOne can show $\\Cont(\\hat{G},-)$ commutes with tilting and the Witt vector functor, as been discussed in \\cite[Lemma 5.3]{wu2021galois}, so in particular, we have \n$$\nW((\\hat{L}^{(i)})^\\flat) \\simeq \\Cont(\\hat{G}^{i-1}, W(\\hat{L}^\\flat)).\n$$\nFor $i=2$, we still use $j_1$ and $j_2$ to represent the two canonical maps from $W(\\hat{L}^\\flat)$ to $\\Cont(\\hat{G}, W(\\hat{L}^\\flat))$ that comes from \\eqref{eq-j1j2}. The above isomorphism also is compatible with the action of $\\hat{G}^2$, so we have\n\\begin{equation}\\label{eq-K(2)andhatGaction}\nW((\\widehat K_\\infty^{(2)})^\\flat) \\simeq \\Cont(\\hat{G}, W(\\hat{L}^\\flat))^{H_K^2}\n\\end{equation}\nWe prove the following lemma for our later use.\n\nNow let $\\mathcal M$ be an \\'etale $\\varphi$-module over $W(\\widehat K_{\\infty}^\\flat)$ with a descent data: \n$$\n\\psi: \\mathcal M \\otimes_{W(\\widehat K_\\infty^\\flat),j_1} W((\\widehat K_\\infty^{(2)})^\\flat) \\simeq \\mathcal M \\otimes_{W(\\widehat K_\\infty^\\flat),j_2} W((\\widehat K_\\infty^{(2)})^\\flat)\n$$\nas \\'etale $\\varphi$-modules over $W((\\widehat K_\\infty^{(2)})^\\flat)$ and satisfies cocycle condition over $W((\\widehat K_\\infty^{(3)})^\\flat)$. Using \\eqref{eq-K(2)andhatGaction}, we have $\\psi$ is the same as a descent data:\n\\begin{equation}\\label{eq-descentdata-1}\n\\hat{\\psi}: {\\mathcal M} \\otimes_{W(\\widehat K_\\infty^\\flat),j_1} \\Cont(\\hat{G}, W(\\hat{L}^\\flat))^{H_K^2} \\simeq {\\mathcal M} \\otimes_{W(\\widehat K_\\infty^\\flat),j_2} \\Cont(\\hat{G}, W(\\hat{L}^\\flat))^{H_K^2}.\n\\end{equation}\n\nFor each $\\gamma \\in \\hat{G}$, we have an evaluation map $\\tilde{e}_\\gamma: \\Cont(\\hat{G}, W(\\hat{L}^\\flat)) \\to W(\\hat{L}^\\flat)$ given by evaluating at $\\gamma$. Using \\eqref{eq-j1j2}, one can check $\\tilde{e}_\\gamma \\circ j_2: W(\\widehat K_\\infty^\\flat) \\to W(\\hat{L}^\\flat)$ is given by the natural embedding and $\\tilde{e}_\\gamma \\circ j_1: W(\\widehat K_\\infty^\\flat) \\to W(\\hat{L}^\\flat)$ is given by $x\\mapsto \\gamma(x)$. So for each $\\gamma \\in \\hat{G}$, if we tensor \\eqref{eq-descentdata-1} against the evaluation map $\\tilde{e}_\\gamma$, we get an isomorphism:\n$$\n\\psi_\\gamma: {\\mathcal M}\\otimes_{W(\\widehat K_\\infty^\\flat),\\gamma} W(\\hat{L}^\\flat) \\simeq {\\mathcal M}\\otimes_{W(\\widehat K_\\infty^\\flat)} W(\\hat{L}^\\flat). \n$$\nAnd similar to the classical Galois descent theory, the cocycle condition for $\\psi$ implies $\\{\\psi_\\gamma\\}_\\gamma$ satisfies \n$$\n\\psi_{\\sigma \\gamma} = \\psi_\\sigma \\circ \\sigma^\\ast \\psi_\\gamma.\n$$\nHence $\\{\\psi_\\gamma\\}_\\gamma$ defines a continuous semilinear action of $\\hat{G}$ on $\\hat{\\mathcal M}:=\\mathcal M\\otimes_{W(\\widehat K_\\infty^\\flat)} W(\\hat{L}^\\flat)$. One can check for $\\gamma \\in H_K$, we have the composition\n$$\nW(\\widehat K_\\infty^\\flat) \\xrightarrow{j_k} W((\\widehat K_\\infty^{(2)})^\\flat) \\to \\Cont(\\hat{G}, W(\\hat{L}^\\flat)) \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat)\n$$\nis the natural embedding $W(\\widehat K_\\infty^\\flat) \\hookrightarrow W(\\hat{L}^\\flat)$ for $k=1,2$. And using the cocycle condition, one can show $\\psi_\\gamma=\\id$ for $\\gamma \\in H_K$, so in particular, $\\mathcal M \\subset \\hat{\\mathcal M}^{H_K}$. Conversely, given a semilinear action of $\\hat{G}$ on $\\hat{\\mathcal M}$ such that $\\mathcal M \\subset \\hat{\\mathcal M}^{H_K}$, $\\{\\psi_\\gamma\\}_\\gamma$ defines a descent data $\\psi$ over $\\Cont(\\hat{G}, W(\\hat{L}^\\flat))^{H_K^{2}}$ if and only if the semilinear action is continuous. In summary, we have\n\n\\begin{theorem}\\label{thm-evaluation-1}\n\\begin{enumerate}\n    \\item The category of \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ with descent data over $A^{(2)}[1\/E]^\\wedge_p$ is equivalent to the category of \\'etale $(\\varphi,\\tau)$-modules over $A[1\/E]^\\wedge_p$;\n    \\item Given a descent data $f$ of an \\'etale $\\varphi$-module $\\mathcal M$ over $A[1\/E]^\\wedge_p$, and $\\gamma\\in \\hat{G}$, we can define the evaluation $f_\\gamma$ of $f$ at $\\gamma$, defined by the base change of $f$ along \n$$\ne_{\\gamma}: A^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat),\n$$\nwhich defines an isomorphism:\n$$\nf_\\gamma: \\mathcal M \\otimes_{A[1\/E]^\\wedge_p,\\tilde{\\iota}_\\gamma} W(\\hat{L}^\\flat) \\simeq \\mathcal M \\otimes_{A[1\/E]^\\wedge_p} W(\\hat{L}^\\flat)\n$$\nwhere $\\tilde{\\iota}_\\gamma: A[1\/E]^\\wedge_p \\to W(\\hat{L}^\\flat) \\xrightarrow{\\gamma} W(\\hat{L}^\\flat)$. Suppose that  $(\\mathcal M,f)$ corresponds to a $\\mathbb Z_p$-representation $T$ of $G_K$, then $f_\\gamma$ corresponds to the semilinear action of $\\gamma$ on $\\mathcal M$ inside $\\mathcal M \\otimes_{A[1\/E]^\\wedge_p} W(\\mathbb{C}_p^\\flat)\\simeq T^\\vee \\otimes W(\\mathbb{C}_p^\\flat)$. Moreover, two descent data $f , g$ are equal if and only if $f_{\\tau} = g_{\\tau}$.\n\\end{enumerate} \n\\end{theorem}\n\\begin{proof} The discussion above the theorem establishes the equivalence between the category of \\'etale $\\varphi$-modules over $A_{\\perf}[1\/E]^\\wedge_p$ with descent data over $(A^{(2)})_{\\perf}[1\/E]^\\wedge_p$ is equivalent to the category of \\'etale $(\\varphi,\\tau)$-modules over $A[1\/E]^\\wedge_p$. Now  (1) follows \\cite[Theorem 4.6]{wu2021galois} which shows that the category of \\'etale $\\varphi$-modules over $B[\\frac 1 I]^\\wedge_p$ is equivalent to the category of \\'etale $\\varphi$-modules over $B_{\\perf}[\\frac 1 I]^\\wedge_{p}$ for bounded prism $(B, I)$ satisfying $\\varphi (I) \\mod p$ is generated by a non-zero\ndivisor in $B\/p$. Then it just remains to prove the last statement in (2). Actually one can check (2) by chasing all the functors used in (1), and use the fact that for any \\'etale $(\\varphi,\\tau)$-module, the $\\hat{G}$-action on $\\hat{\\mathcal M}$ is determined by the $\\tau$-action on $\\mathcal M$. However, this can also been seen directly from the following lemma.\n\\end{proof}\n\n\\begin{lemma}\\label{lem-evaluation-1}\nGiven two finite free \\'etale $\\varphi$-modules $\\mathcal M,\\mathcal{N}$ over $A^{(2)}[1\/E]^\\wedge_p$ and two morphisms $f, g: \\mathcal M \\to \\mathcal{N}$ of \\'etale $\\varphi$-modules over $A^{(2)}[1\/E]^\\wedge_p$. Let $f_\\tau,g_\\tau$ be the base changes of $f,g$ along the map \n$$\ne_{\\tau}: A^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\tau} W(\\hat{L}^\\flat).\n$$\nThen $f=g$ if and only if $f_\\tau=g_\\tau$.\n\\end{lemma}\n\n\\begin{proof}\nWe take the natural base change of $f$ and $g$ along $A^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p$, we get two morphisms $\\psi$ and $\\psi'$ between \\'etale $\\varphi$-modules over $(A^{(2)})_{\\perf}[1\/E]^\\wedge_p$. Since the base change functor between \\'etale $\\varphi$-modules over $A^{(2)}[1\/E]^\\wedge_p$ and $(A^{(2)})_{\\perf}[1\/E]^\\wedge_p$ is an equivalence of categories, it reduces to show that $\\psi=\\psi'$ if and only if their base change along \n$$\n\\tilde{e}_{\\tau}: (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{} W(\\hat{L}^\\flat)\n$$\nis equal. Since $\\mathcal M$ and $\\mathcal{N}$ are finite free, it is enough to show the evaluation map:\n$$\n\\tilde{e}_{\\tau}: \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\to W\\big((\\hat{L}^{(2)})^\\flat\\big)\n$$\nis injective. Suppose $h\\in \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}}$ satisfies $h(\\tau)=0$, then \n$$\n(\\sigma_1,\\sigma_2)(h)(\\tau)=\\sigma_2 h(\\sigma_2^{-1}\\tau\\sigma_1)=0\n$$\nfor $(\\sigma_1,\\sigma_2)\\in H_K^{2}$. Since $\\hat{G}$ is topologically generated by $H_K$ and $\\tau$, we get $h\\equiv 0$.\n\\end{proof}\n\nNow we give the $\\mathbb Q$-isogeny versions of Theorem \\ref{thm-evaluation-1} and Lemma \\ref{lem-evaluation-1}. \nRecall that  the \\'etale $(\\varphi,\\tau)$-modules over $A[1\/E]^\\wedge_p[\\frac{1}{p}]$ is equivalent to the category of $\\mathbb Q_p$-representations of $G_K$, and recall the following definition of \\'etale $(\\varphi,\\tau)$-modules over $B[1\/J]^\\wedge_p[\\frac{1}{p}]$ for a prism $(B,J)\\in X_{\\mathlarger{\\mathbbl{\\Delta}}}$.\n\n\\begin{definition}\\label{def-etalephimodule-2}\nAn (globally) \\'etale $\\varphi$-module $\\mathcal M$ over $B[1\/J]^\\wedge_p[\\frac{1}{p}]$ is a (finite projective) $\\varphi$-module over $B[1\/J]^\\wedge_p[\\frac{1}{p}]$ that arises by base extension from an \\'etale $\\varphi$-module $B[1\/J]^\\wedge_p$.\n\\end{definition}\n\nFrom this definition, we immediately deduce the following result from \\cite[Theorem 4.6]{wu2021galois}\n\\begin{proposition}\nFor any prism $(B,J)\\in X_{\\mathlarger{\\mathbbl{\\Delta}}}$ satisfying $\\varphi (J) \\mod p$ is generated by a non-zero\ndivisor in $B\/p$, the base change functor defined by $B[1\/J]^\\wedge_p[\\frac{1}{p}]\\to B_{\\perf} [1\/J]^\\wedge_p[\\frac{1}{p}]$ induces an equivalence between the category of \\'etale $\\varphi$-modules over $B[1\/J]^\\wedge_p[\\frac{1}{p}]$ and the category of \\'etale $\\varphi$-modules over $B_{\\perf}[1\/J]^\\wedge_p[\\frac{1}{p}]$.\n\\end{proposition}\n\nAnd similar to Theorem~\\ref{thm-evaluation-1} and Lemma~\\ref{lem-evaluation-1}, we have\n\\begin{theorem}\\label{thm-evaluation-2}\nThe category of \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p[\\frac{1}{p}]$ with descent data over $A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}]$ is equivalent to the category of \\'etale $(\\varphi,\\tau)$-modules over $A[1\/E]^\\wedge_p[\\frac{1}{p}]$. Moreover, \n$$\n\\Cont\\big(\\hat{G}, W(\\hat{L}^\\flat)[\\frac{1}{p}]\\big)^{H_K^{2}} \\simeq W(\\widehat K_\\infty^{(2)})^\\flat[\\frac{1}{p}]. \n$$\nFor  $\\gamma\\in \\hat{G}$, we can define the evaluation map\n$$\n\\tilde{e}_\\gamma: \\Cont\\big(\\hat{G}, W(\\hat{L}^\\flat)[\\frac{1}{p}]\\big) \\to W(\\hat{L}^\\flat)[\\frac{1}{p}].\n$$\nAnd given a descent data $f$ of an \\'etale $\\varphi$-module $\\mathcal M$ over $A[1\/E]^\\wedge_p[\\frac{1}{p}]$, and $\\gamma\\in \\hat{G}$, we can define the evaluation $f_\\gamma$ of $f$ at $\\gamma$, defined by the base change of $f$ along \n$$\ne_\\gamma: A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}] \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p[\\frac{1}{p}] \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat)[\\frac{1}{p}],\n$$\nwhich defines an isomorphism:\n$$\n\\mathcal M \\otimes_{A[1\/E]^\\wedge_p[1\/p],\\tilde{\\iota}_\\gamma} W(\\hat{L}^\\flat)[\\frac{1}{p}] \\simeq \\mathcal M \\otimes_{A[1\/E]^\\wedge_p[1\/p]} W(\\hat{L}^\\flat)[\\frac{1}{p}]\n$$\nwhere $\\tilde{\\iota}_\\gamma: A[1\/E]^\\wedge_p[\\frac{1}{p}] \\to W(\\hat{L}^\\flat)[\\frac{1}{p}] \\xrightarrow{\\gamma} W(\\hat{L}^\\flat)[\\frac{1}{p}]$. If $(\\mathcal M,f)$ corresponds to a $\\mathbb Q_p$-representation $V$ of $G_K$, then $f_\\gamma$ corresponds to the semilinear action of $\\gamma$ on $\\mathcal M $ inside $ V^\\vee \\otimes W(\\mathbb{C}_p^\\flat)[1\/p]$. Moreover, two descent data $f , g$ are equal if and only if $f_{\\tau} = g_{\\tau}$.\n\\end{theorem}\n\n\\begin{lemma}\\label{lem-evaluation-2}\nGiven two finite free \\'etale $\\varphi$-modules $\\mathcal M,\\mathcal{N}$ over $A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}]$ and two morphisms $f, g: \\mathcal M \\to \\mathcal{N}$ of \\'etale $\\varphi$-modules over $A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}]$. Let $f_\\tau,g_\\tau$ be the base changes of $f,g$ along the map \n$$\ne_\\tau:A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}] \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p[\\frac{1}{p}] \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat[\\frac{1}{p}]\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\tau} W(\\hat{L}^\\flat)[\\frac{1}{p}].\n$$\nThen $f=g$ if and only if $f_\\tau=g_\\tau$.\n\\end{lemma}\n\n\\begin{proof}\nThe proofs are exactly the same as the proof of Theorem~\\ref{thm-evaluation-1} and Lemma~\\ref{lem-evaluation-1}, plus the following fact that\n$$\n\\Cont\\big(\\hat{G}, W(\\hat{L}^\\flat)[\\frac{1}{p}]\\big) = \\Cont\\big(\\hat{G}, W(\\hat{L}^\\flat)\\big)[\\frac{1}{p}],\n$$\nwhich can be shown by the compactness of $\\hat{G}$. \n\\end{proof}\n\n\\subsection{Proofs of Proposition~\\ref{thm-1prime} and Theorem \\ref{Thm-main-1}}\\label{subsec-pris-crystal-proof} We keep the assumption that  $R=\\O_K$ is a mixed characteristic complete DVR with perfect residue field in this subsection, and keep our notations in \\S 2.1.\n\nLet us first prove Proposition~\\ref{thm-1prime} using Lemma~\\ref{lem-intersection} and results in \\S\\ref{subsec-phi-tau}. First, we give a different interpretation of the ``evaluation map\":\n$$\ne_\\gamma: A^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat)\n$$\nin Theorem~\\ref{thm-evaluation-1} when restricted on $A^{(2)}$ . Recall that we fix a compatible system $\\{\\varpi_n\\}_n$ of $p^n$-th roots of a uniformizer $\\varpi \\in \\O_K$, this defines a map of prisms $\\iota: (A,(E)) \\to (\\Ainf,(E))$ maps $u$ to $[{\\varpi}^\\flat ]$, and given a $\\gamma \\in G_K$, we define $\\iota_{\\gamma}$ to be the composition of $\\iota$ with $\\gamma: (\\Ainf,(E)) \\to (\\Ainf,(E))$ where the second map is defined as $a \\mapsto \\gamma(a)$. Since $(E)\\subset \\Ainf$ is equal to $\\Ker(\\theta)$ and $\\theta$ is  $G_K$-equivariant,  $\\gamma$ is a well-defined map of $\\delta$-pairs. By the universal property of $A^{(2)}$, we can define a map of prisms $\\iota_{\\gamma}^{(2)} : (A^{(2)},(E)) \\to (\\Ainf,(E))$ so that the following diagram commutes:  \n\\begin{equation}\\label{equ-diagram-prisms}\n\\begin{tikzcd}\n(A,(E)) \\arrow[rr,\"i_1\"] \\arrow[ddrr,\"\\iota_\\gamma\",swap] & & (A^{(2)}, (E) )\\arrow[dd,\"\\iota^{(2)}_\\gamma\"] & & (A,(E)) \\arrow[ll,\"i_2\",swap] \\arrow[ddll,\"\\iota\"]\\\\\n& & & &\\\\\n& & (\\Ainf, (E)) & &\n\\end{tikzcd}  \n\\end{equation}\nWe have $\\iota^{(2)}_{\\gamma}$ induces a morphism $\\tilde{\\iota}^{(2)}_{\\gamma}: A^{(2)}[1\/E]^\\wedge_p \\to W(\\mathbb C_p^\\flat)$. We claim for all $\\gamma \\in G_K$, $\\tilde{\\iota}^{(2)}_{\\gamma}$ is the same as the \n$$\nA^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat) \\hookrightarrow W(\\mathbb C_p^\\flat).\n$$\nTo see this, by the universal property of direct perfection, we have \\eqref{equ-diagram-prisms} factorizes as:\n$$\n\\begin{tikzcd}\n(A,(E)) \\arrow[d]\\arrow[rr,\"i_1\"]  & & (A^{(2)}, (E) )\\arrow[d] & & (A,(E)) \\arrow[d] \\arrow[ll,\"i_2\",swap]\\\\\n(A_{\\perf},(E)) \\arrow[rr,\"i'_1\"] \\arrow[ddrr,\"\\iota'_\\gamma\",swap] & & ((A^{(2)})_{\\perf}, (E) )\\arrow[dd,\"\\iota'^{(2)}_\\gamma\"] & & (A_{\\perf},(E)) \\arrow[ll,\"i'_2\",swap] \\arrow[ddll,\"\\iota'\"]\\\\\n& & & &\\\\\n& & (\\Ainf, (E)) & &\n\\end{tikzcd}  \n$$\nSo $\\tilde{\\iota}^{(2)}_\\gamma$ has a factorization\n$$\nA^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\to W(\\mathbb C_p^\\flat).\n$$\nWe just need to check $\\iota'^{(2)}_\\tau$ induces the evaluation map \n$$\n(A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\tau} W(\\hat{L}^\\flat) \\xhookrightarrow{} W(\\mathbb{C}_p^\\flat).\n$$\nAnd this follows from the isomorphism of $(A^{(2)})_{\\perf}[1\/E]^\\wedge_p\\simeq W((K^{(2)}_\\infty)^\\flat)$, then one check directly for $j_1,j_2$ defined in \\eqref{eq-j1j2}, $\\tilde{e}_\\gamma\\circ j_1: A_{\\perf}[1\/E]^\\wedge_p \\to W(\\hat{L}^\\flat)$ is equal to the map induced from $\\iota'_\\gamma$ and $\\tilde{e}_\\gamma\\circ j_2: A_{\\perf}[1\/E]^\\wedge_p \\to W(\\hat{L}^\\flat)$ is equal to the map induced from $\\iota'$. In particular, we have a commutative diagram:\n\\begin{equation}\\label{eq-iotaandevaluation}\n\\begin{tikzcd}\nA^{(2)} \\arrow[d, hook] \\arrow[rrr, \"\\iota^{(2)}_\\gamma\"] &&& \\Ainf \\arrow[d, hook]\\\\\nA^{(2)}[1\/E]^\\wedge_p \\arrow[r] & (A^{(2)})_{\\perf}[1\/E]^\\wedge_p  \\arrow[r,\"\\tilde{e}_\\gamma\",hook] & W(\\hat{L}^\\flat) \\arrow[r,hook] & W(\\mathbb C_p^\\flat).\n\\end{tikzcd}  \n\\end{equation}\nNow we can prove Proposition~\\ref{thm-1prime}.\n\n\\begin{proof}[Proof of Proposition~\\ref{thm-1prime}]\nFirst we pick $\\gamma=\\tilde{\\tau}$ that is a preimage of $\\tau$ under the map $G_K \\to \\hat{G}$, we have $\\gamma(u)-u=Ez$ and $\\iota^{(2)}_{\\gamma}$ defined as above is the embedding defined in \\S\\ref{subsec-embedding} by Remark~\\ref{rem-embedding-depend}. In particular, composing the embedding $A^{(2)} \\hookrightarrow \\Ainf$ defined in \\S\\ref{subsec-embedding} with $\\Ainf \\hookrightarrow W(\\mathbb C_p^\\flat)$, one get the evaluation map \n$$\n(A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\tau} W(\\hat{L}^\\flat) \\xhookrightarrow{} W(\\mathbb{C}_p^\\flat).\n$$\nrestricted on $A^{(2)}$.\n\nKeep the notations as in \\S\\ref{subsec-G-image}, and let $\\mathcal M_{\\Ainf}=W(\\mathbb C_p^\\flat)\\otimes_A \\mathfrak{M}$ and $\\mathcal M_A \\simeq \\mathfrak{M}\\otimes_A A[1\/E]^\\wedge_p$. By  Theorem~\\ref{thm-evaluation-1} and Theorem~\\ref{thm-caruso}, recall we use $B^{(2)}= A^{(2)} [\\frac 1 E]^\\wedge_p$ and $B^{(2)}_{\\st}= A^{(2)}_{\\st} [\\frac 1 E]^\\wedge_p$ to simplify our notations, we have there is a descent data \n$$\nc: \\mathcal M_A \\otimes _{A[1\/E]^\\wedge_p, \\tilde{i}_1} B^{(2)} \\to \\mathcal M_A \\otimes_{A[1\/E]^\\wedge_p, \\tilde{i}_2} B^{(2)} \n$$\nof $\\mathcal M_A$ over $B^{(2)}$ that corresponds to the representation $T$. And the semilinear action of $\\gamma=\\tilde{\\tau}$ on $\\mathcal M_{\\Ainf}$ is given by the evaluation $c_\\tau$, that is,  we have the linearization of the $\\tilde{\\tau}$-action is defined by\n$$\nc_\\tau: W(\\mathbb C_p^\\flat) \\otimes_{\\tilde{\\iota}_\\gamma,A[1\/E]^\\wedge_p} \\mathcal M_A \\simeq W(\\mathbb C_p^\\flat) \\otimes_{\\tilde{\\iota} , A[1\/E]^\\wedge_p} \\mathcal M_A.\n$$\nBy  base change $c$ along $B^{(2)} \\to B^{(2)}[\\frac{1}{p}]$, we get a $B^{(2)}[\\frac{1}{p}]$-linear $\\varphi$-equivariant morphism:\n$$\nc': \\mathcal M_A \\otimes _{A[1\/E]^\\wedge_p, \\tilde{i}_1} B^{(2)}[\\frac{1}{p}] \\to \\mathcal M_A \\otimes_{A[1\/E]^\\wedge_p, \\tilde{i}_2} B^{(2)}[\\frac{1}{p}]. \n$$\nOn the other hand, from the discussions after Proposition~\\ref{thm-1prime}, $\\tilde{\\tau}$-action also defines a $\\varphi$-equivariant morphism \n$$\nf_{\\tilde{\\tau}}: \\mathfrak{M}\\otimes_{A,\\iota_{\\tilde{\\tau}}} A_{\\st}^{(2)}[\\frac{1}{p}] \\simeq \\mathfrak{M}\\otimes_{A} A_{\\st}^{(2)}[\\frac{1}{p}].\n$$\nWe will see in Proposition~\\ref{prop-descentBsttoB2} below that $f_{\\tilde{\\tau}}$ actually descents to a $B^{(2)}[1\/p]$-linear morphism. Assuming this fact, then if we base change $f_{\\tilde{\\tau}}$ along $A^{(2)}[\\frac{1}{p}] \\to W(\\mathbb C_p^\\flat)[\\frac{1}{p}]$, we will have $f_{\\tilde{\\tau}}\\otimes W(\\mathbb C_p^\\flat)[\\frac{1}{p}]=c_\\tau$ since the way we define $f_{\\tilde{\\tau}}$ is by taking the ${\\tilde{\\tau}}$-action. From the discussion at the beginning of the proof and Lemma~\\ref{lem-evaluation-2}, we have $f_{\\tilde{\\tau}}=c'$ as a $B^{(2)}[\\frac{1}{p}]$-linear isomorphism between $\\mathcal M_A \\otimes _{A[1\/E]^\\wedge_p, \\tilde{i}_1} B^{(2)}[\\frac{1}{p}] $ and $ \\mathcal M_A \\otimes_{A[1\/E]^\\wedge_p, \\tilde{i}_2} B^{(2)}[\\frac{1}{p}]$.\n\nWe fix a basis $\\{e_i\\}$ of $\\mathfrak{M}$, for $j=1,2$ let $\\{e^j_i\\}$ be the basis of $\\mathcal M_A \\otimes _{A, \\tilde{i'}_j} B^{(2)}[\\frac{1}{p}] $ defined by $e^j_i=e_i\\otimes 1$ and the tensor is via $A \\to A[1\/E]^\\wedge_p \\xrightarrow{\\tilde{i}_j} B^{(2)}[1\/p]$. So we can interpret $f_{\\tilde{\\tau}}=c'$ as matrix using this two basis, this matrix is $X_{\\tilde{\\tau}}$ from this definition, so it has coefficients inside $A_{\\st}^{(2)}[\\frac{1}{p}]$ by the discussion before \nProposition~\\ref{thm-1prime}. On the other hand, $X_{\\tilde{\\tau}}$ has coefficients in $B^{(2)}\\subset B_{\\st}^{(2)}$ since $c'$ is defined by the $B^{(2)}$-linear map $c$. So by Lemma~\\ref{lem-intersection}, we have $X_{\\tilde{\\tau}}$ has coefficients inside $A_{\\st}^{(2)}$. The same argument shows when $T$ is crystalline, then $X_{\\tilde{\\tau}}$ has coefficients inside $A^{(2)}$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop-descentBsttoB2}\nBase change along $B^{(2)} \\to A^{(2)}_{\\st}[1\/E]^\\wedge_p$ defines an equivalence of categories of \\'etale $\\varphi$-modules over $B^{(2)}$ and $A^{(2)}_{\\st}[1\/E]^\\wedge_p$ and an equivalence of categories of \\'etale $\\varphi$-modules over $B^{(2)}[1\/p]$ and $A^{(2)}_{\\st}[1\/E]^\\wedge_p[1\/p]$.\n\\end{proposition}\n\\begin{proof}\nBy \\cite[Theorem 4.6]{wu2021galois}, we just need to show the same result after perfections, we will show $(A^{(2)})_{\\perf}= (A^{(2)}_{\\st})_{\\perf}$ in Lemma~\\ref{lem-perfectionofA2andAst2} using the logarithmic prismatic site.\n\\end{proof}\n\nNow, let us prove Theorem~\\ref{Thm-main-1} by first producing a functor $\\mathcal T$ from prismatic $F$-crystals in finite $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules to lattices inside a crystalline representation. For prism $A$, we use $i_k: A \\to A^{(2)}$ or $A^{(3)}$ for natural map from $A$ to $k$-th factor of $A^{(2)}$ or $A^{(3)}$. The notation $i_{kl}: A ^{(2)} \\to A ^{(3)}$ has the similar meaning. \n\nBy Corollary \\ref{cor-crystal-descentdata}, given a prismatic $F$-crystal ${\\mathfrak{M}}_{\\mathlarger{\\mathbbl{\\Delta}}}$, we obtain a Kisin module $(\\mathfrak{M} , \\varphi _{\\mathfrak{M}})$ of height $h$ together with descent data\n$f: \\mathfrak{M} \\otimes _{A, i_1} A^{(2)} \\to \\mathfrak{M} \\otimes_{A, i_2}A^{(2)} $ so that $f$ satisfies the following cocycle condition $ i _{13} \\otimes f = (i _{23} \\otimes f) \\circ (i _{12} \\otimes f ) $, where $i_{kl} \\otimes f$ is the base change of $f$ along $i_{kl}$, and $f$ also compatible with the $\\varphi$-structure on the both sides of $f$. Note that the existence of $f$ follows from the crystal property of   $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$: \n\\begin{equation}\\label{eqn-cocyclefromcrystal}\nf: \\mathfrak{M} \\otimes _{A, i_1} A^{(2)} \\simeq \\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}((A^{(2)},(E))) \\simeq \\mathfrak{M} \\otimes_{A, i_2}A^{(2)} \n\\end{equation}\n\nWe let $\\mathcal M=\\mathfrak{M}\\otimes_A A[1\/E]^\\wedge_p$ and  $c=f\\otimes_{A^{(2)}} B^{(2)}$, then  $(\\mathcal M,c)$ is an \\'etale $\\varphi$-module with descent data, which corresponds to a $\\mathbb Z_p$-representation of $G_K$. Moreover the semilinear action of $G_K$ on $\\mathfrak{M}\\otimes_A W(\\mathbb{C}_p^\\flat)$ comes from $\\{c_\\gamma\\}_{\\gamma\\in G_K}$ using the evaluation maps. If we define \n$$\nf_\\gamma: \\Ainf \\otimes_{\\iota_\\gamma,A} \\mathfrak{M} \\to \\Ainf \\otimes_{\\iota , A} \\mathfrak{M}\n$$\nas the base change of $f$ along $\\iota_{\\gamma}^{(2)}$, then by \\eqref{eq-iotaandevaluation}, we have $c_\\gamma=f_\\gamma$. The $G_K$-semilinear action commutes with $\\varphi$ as $f$ does. For any $\\gamma \\in G_K$, we have $\\gamma (A) \\subset W(k)[\\![u , \\epsilon-1]\\!] \\subset A^{(2)}_{\\st} \\subset \\Ainf$. Therefore, the $G_K$-action on the $\\Ainf \\otimes_A \\mathfrak{M}$ defined the above factors through $A^{(2)}_{\\st} \\otimes_A \\mathfrak{M}$. We claim that $G_K$-action on $\\widehat \\mathfrak{M} : = A^{(2)}_{\\st}\\otimes_A \\mathfrak{M}$ defines a $(\\varphi, \\hat G)$-module which corresponds to a crystalline representation.\n\nFirst, for $\\gamma \\in G_\\infty$, $\\gamma(A) = A$ in $\\Ainf$, we conclude $\\iota^{(2)}_\\gamma : A^{(2)} \\to \\Ainf$ satisfies $\\iota^{(2)}_{\\gamma}\\circ i_1=\\iota^{(2)}_{\\gamma}\\circ i_2$. In particular, for any $\\gamma \\in G_\\infty$ and $j=1,2$, using \\eqref{eqn-cocyclefromcrystal} and the crystal property of $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$,  $f_\\gamma$ comes from the base change of \\eqref{eqn-cocyclefromcrystal} along $\\iota^{(2)}_\\gamma : A^{(2)} \\to \\Ainf$, in particular, we have \n$$\nf_\\gamma: \\mathfrak{M} \\otimes _{A, \\iota^{(2)}_{\\gamma}\\circ i_1} \\Ainf \\simeq \\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}((\\Ainf,\\Ker\\theta)) \\simeq \\mathfrak{M} \\otimes_{A, \\iota^{(2)}_{\\gamma}\\circ i_2}\\Ainf. \n$$\nSince $\\iota^{(2)}_{\\gamma}\\circ i_1=\\iota^{(2)}_{\\gamma}\\circ i_2$, we have $f_\\gamma={\\rm id}$ which means $\\mathfrak{M} \\subset (\\widehat \\mathfrak{M}) ^{G_\\infty}$. Similarly, $G_K$ acts on $\\widehat \\mathfrak{M}\/ I_+$ corresponds the base change of $f$ along \n$$A^{(2)} \\xrightarrow{\\iota^{(2)}_\\gamma} \\Ainf \\to W(\\bar{k}) $$\nwhere the last arrow is the reduction modulo $W(\\mathfrak m)$ ($\\mathfrak m$ is the maximal ideal of $\\O_{\\mathbb C_p}^\\flat$). One can check for all $\\gamma\\in G_K$ and $j=1,2$, we have \n$$\nA \\xrightarrow{i_j} A^{(2)} \\xrightarrow{\\iota^{(2)}_\\gamma} \\Ainf \\to W(\\bar{k})\n$$\nare all equal to $A \\to W(k) \\hookrightarrow W(\\overline{k})$ with the first arrow given by $u \\mapsto 0$. The above map induces a morphism of prisms $(A,(E)) \\to (W(k),(p))$, then using \\eqref{eqn-cocyclefromcrystal} and the crystal condition of $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$, we can similarly prove that $G_K$ acts on $\\widehat \\mathfrak{M}\/ I_+$-trivially, so $(\\mathfrak{M}, \\varphi_{\\mathfrak{M}}, G_K)$ is a $(\\varphi, \\hat G)$-module. Furthermore, $\\widehat T (\\widehat \\mathfrak{M})$ is crystalline by Corollary \\ref{cor-crystalline} and Theorem~\\ref{Thm-1}. \n\n\\begin{remark}\nIn \\S\\ref{sec-logprismandsemistablereps}, we will consider a category consisting of modules with descent data, and similar arguments about the triviality of the Galois actions can be shown directly using the cocycle condition of the descent data. We summarize this fact in the following easy fact.\n\\end{remark}\n\\begin{lemma}\nLet $q:(A^{(2)},(E)) \\to (B,J)$ be a map of prisms satisfying $q\\circ i_1 =q\\circ i_2$, then for any descent data $f$ over $A^{(2)}$, the base change of $f$ along $q$ is the identity map.\n\\end{lemma}\n\nTo show the fully faithfulness of this functor, first let $(\\mathfrak{M}, f)$, $(\\mathfrak{M}', f')$ be two Kisin modules with descent data $f , f'$ respectively. Suppose that there exists a map $ \\alpha: \\mathcal T ((\\mathfrak{M} , f)) \\to \\mathcal T ((\\mathfrak{M}' , f'))$ as lattices of crystalline representations, then from our construction of $\\mathcal{T}$ and Theorem~\\ref{thm-2}, $\\alpha$ is induced from a map $\\hat{\\alpha}: (\\mathfrak{M} , \\varphi_{\\mathfrak{M}}, \\hat G_{\\mathfrak{M}}) \\to (\\mathfrak{M}' , \\varphi _{\\mathfrak{M}'}, \\hat G_{\\mathfrak{M}'})$ between $(\\varphi, \\hat G)$-modules. The faithfulness of $\\mathcal{T}$ follows the fact that $A \\to A[1\/E]^\\wedge_p$ induces a fully faithful functor between Kisin modules over $A$ and \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ from \\cite[Proposition 2.1.12]{KisinFcrystal}. On the other hand,  $\\hat{\\alpha}$ gives morphisms $\\hat{\\alpha}_1: \\mathfrak{M} \\otimes_{A,i_1} A^{(2)} \\to \\mathfrak{M}' \\otimes_{A,i_1}A^{(2)}$ and $\\hat{\\alpha}_2: \\mathfrak{M} \\otimes_{A,i_2} A^{(2)} \\to \\mathfrak{M}' \\otimes_{A,i_2}A^{(2)}$. If we view $A$ and $A^{(2)}$ as subrings of $\\Ainf$ using diagram \\eqref{equ-diagram-prisms}, then the following diagram commutes by the fact that $\\hat{\\alpha}: \\widehat{\\mathfrak{M}} \\to \\widehat{\\mathfrak{M}'}$ is compatible with $\\tau$-action.\n$$\n\\begin{tikzcd}\n\\mathfrak{M} \\otimes_{A,i_1} A^{(2)} \\arrow[r,\"f\"] \\arrow[d,\"\\hat{\\alpha}_1\"] & \\mathfrak{M} \\otimes_{A,i_2} A^{(2)} \\arrow[d,\"\\hat{\\alpha}_2\"] \\\\\n\\mathfrak{M}' \\otimes_{A,i_1} A^{(2)} \\arrow[r,\"f'\"] & \\mathfrak{M}' \\otimes_{A,i_2} A^{(2)}  \\\\\n\\end{tikzcd}\n$$\nThus we produces a morphism between $(\\mathfrak{M}, f)$ and $(\\mathfrak{M}', f')$, i.e. $\\mathcal{T}$ is also full. \n\nIt remains to show the functor $\\mathcal{T}$ is essential surjective. Given a lattice $T$ in a crystalline representation of $G_K$, let $\\mathfrak{M}$ be the corresponded Kisin module, it suffices to construct a descent data of $\\mathfrak{M}$ over $A^{(2)}$. We have shown in our proof of Proposition~\\ref{thm-1prime} that if we view $A^{(2)}$ as a subring of $\\Ainf$ via $\\iota^{(2)}_{\\tilde{\\tau}}$, then $X_{\\tilde{\\tau}}$ defines a $\\varphi$-equivariant isomorphism $f: \\mathfrak{M}\\otimes_{A,i_1} A^{(2)} \\simeq \\mathfrak{M}\\otimes_{A,i_2} A^{(2)}$ of $A^{(2)}$-modules. We also show the base change of $f$ along $A^{(2)} \\to B^{(2)}$ is equal to the descent data  $c$ of the \\'etale $\\varphi$-module $\\mathcal M_A=\\mathfrak{M}\\otimes_A A[1\/E]^\\wedge_p$ that corresponds to $G_K$-action on $T$. In particular, $c: \\mathfrak{M}\\otimes_{A,i_1} B^{(2)} \\simeq \\mathfrak{M}\\otimes_{A,i_2} B^{(2)}$ satisfies the cocycle condition. By Lemma \\ref{lem-intersection}, $A^{(2)}$ (resp. $A^{(3)}$) injects into $B^{(2)}$ (resp. $B^{(3)}$), so we have $f$ also satisfies the cocycle condition. In particular, $(\\mathfrak{M},f)$ together produce a primatic $F$-crystals in finite free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-module by Corollary~\\ref{cor-crystal-descentdata}.\n\n\\begin{remark}\nGiven an \\'etale $\\varphi$-module $(\\mathcal M_A,\\varphi_{\\mathcal M_A}, c)$ over $A[1\/E]^\\wedge_p$ with descent datum $c$, we call $(\\mathcal M_A,\\varphi_{\\mathcal M_A}, c)$ is \\emph{of finite $E$-height} if $\\mathcal M_A$ is of finite $E$-height, i.e., if there is a finite free Kisin module $(\\mathfrak{M},\\varphi_{\\mathfrak{M}})$ of finite height and defined over $A$ such that $\\mathfrak{M}\\otimes_A A[1\/E]^\\wedge_p \\simeq \\mathcal M_A$ as $\\varphi$-modules. Since $(\\mathcal M_A, \\varphi_{\\mathcal M_A})$ is the \\'etale $\\varphi$-module for $T|_{G_\\infty}$, our definition of finite $E$-height is compatible with the one given by Kisin under the equivalence in (1) of Theorem~\\ref{thm-evaluation-1}. \n\nWe expect same arguments in the proof of Proposition~\\ref{thm-1prime} will be used to study representations of finite $E$-height. Similar result has been studied using the theory of $(\\varphi,\\tau)$-modules by Caruso. For example, in the proof of \\cite[Lemma 2.23]{Caruso-phitau}, Caruso shows for representations of finite $E$-height, the $\\tau$-actions descents to $\\mathfrak{S}_{u\\text{-np},\\tau}$, which is a subring of $\\Ainf$ closely related to $\\tilde{\\iota}^{(2)}_{\\tilde\\tau}(B^{(2)})\\cap \\Ainf$, where $\\tilde{\\tau}$ is a preimage of $\\tau$ in $G_K$. \n\\end{remark}\n\n\\begin{remark}\nWe can also establish the compatibility of our Theorem~\\ref{Thm-main-1}, the theory of Kisin and \\cite[Theorem 1.2]{BS2021Fcrystals}. Given a lattice $T$ in a crystalline representation of $G_K$ with non-negative Hodge-Tate weight, and let $\\mathfrak{M}$ be the Kisin module corresponds to $T$ in \\cite{KisinFcrystal}, and let $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ (reso. $\\mathfrak{M}'_{{\\mathlarger{\\mathbbl{\\Delta}}}}$) be the prismatic $F$-crystal corresponds to $T^\\vee$ under \\cite[Theorem 1.2]{BS2021Fcrystals} (resp. $T$ under Theorem~\\ref{Thm-main-1}). Note that we need to take $T^\\vee$ since in the work of Bhatt-Scholze, the equivalence is covariant. By our construction of $\\mathfrak{M}'_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, we have $\\mathfrak{M}'_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,(E)))\\simeq \\mathfrak{M}$. By \\cite[Remark 7.11]{BS2021Fcrystals}, $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,(E)))\\simeq \\mathfrak{M}$. Next we need to show the descent data over $A^{(2)}$ constructed respectively are the same. By Corollary~\\ref{cor-inj}, we just need to show they are the same as descent data of \\'etale $\\varphi$-modules over $A^{(2)}[1\/E]^\\wedge_p$, but they are the same by our $\\tau$-evaluation criteria in Lemma~\\ref{lem-evaluation-1}.  \n\\end{remark}\n\n\n\\section{Logarithmic prismatic \\texorpdfstring{$F$}{F}-crystals and semi-stable representations}\\label{sec-logprismandsemistablereps}\nIn this section, we will propose a possible generalization of Theorem~\\ref{Thm-main-1} to semi-stable representations using the absolute logarithmic prismatic site. The main reference of this subsection is \\cite{Koshikawa2021log-prism}. We will restrict ourselves to the base ring $R=\\O_K$, a complete DVR with perfect residue field. And we give $R$ the log structure associated to the prelog structure $\\alpha: \\mathbb N \\to R$ such that $\\alpha(1)=\\varpi$ is a uniformizer in $R$, i.e., let $D=\\{\\varpi=0\\}$, then the log structure on $X=\\Spf(R)$ is defined by \n$$\nM_X=M_D \\hookrightarrow \\O_X \\text{ where } M_D(U):=\\{f\\in \\O_X(U) \\,|\\, f|_{U\\backslash D}\\in \\O^\\times(U\\backslash D) \\}.\n$$\nLet us introduce the absolute logarithmic site over $(X,M_X)$.\n\\begin{definition}\\cite[Definition 2.2 and Definition 3.3]{Koshikawa2021log-prism}\n\\begin{enumerate}\n    \\item A $\\delta_{\\log}$-ring is a tuple $(A,\\delta, \\alpha:M\\to A, \\delta_{\\log}:M\\to A)$, where $(A,\\delta)$ is a $\\delta$-pair and $\\alpha$ is a prelog-structure on $A$. And $\\delta_{\\log}$ satisfies:\n    \\begin{itemize}\n        \\item $\\delta_{\\log}(e)=0$,\n        \\item $\\delta(\\alpha(m))=\\alpha(m)^p\\deltalog(m)$,\n        \\item $\\deltalog(mn)=\\deltalog(m)+\\deltalog(n)+p\\deltalog(m)\\deltalog(n)$\n        for all $m,n\\in M$. And we will simply denote it by $(A,M)$ if this is no confusion. Morphisms are morphisms of $\\delta$-pairs that compatible with the perlog structure and $\\deltalog$-stucture.\n    \\end{itemize}\n    \\item A $\\delta_{\\log}$-triple is $(A,I,M)$ such that $(A,I)$ is a $\\delta$-pair and $(A,M)$ is a $\\delta_{\\log}$-ring.\n    \\item A $\\delta_{\\log}$-triple $(A,I,M)$ is a prelog prism if $(A,I)$ is a prism, and it is bounded if $(A,I)$ is bounded.\n    \\item A bounded prelog prism is a log prism if it is $(p, I )$-adically log-affine (cf. \\cite[Definition 3.3]{Koshikawa2021log-prism}). \n    \\item A bounded (pre)log prism is integral if $M$ is an integral monoid.\n    \\item A $\\delta_{\\log}$-triple $(A,I,M)$ is said to be over $(R,\\mathbb N)$ if $A\/I$ is an $R$-algebra and there is a map $M\\to \\mathbb N$ of monoids such that the following diagram commutes.\n    $$\n    \\begin{tikzcd}\n    M \\arrow[rr] \\arrow[d] & & A \\arrow[d] \\\\\n    \\mathbb N \\arrow[r] & R \\arrow[r] & A\/I\n    \\end{tikzcd}\n    $$\n    All $\\delta_{\\log}$-triples over $(R,\\mathbb N)$ form a category. Similarly, we can define the category of prelog prisms over $(R,\\mathbb N)$ and the category of bounded log prisms over $(R,\\mathbb N)^a$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\nIf $A$ is an integral domain, or more general if $\\alpha(M)$ consists of non-zero divisors, then $\\deltalog$ is uniquely determined by $\\delta$ if exists. In particular, morphisms between such $\\delta_{\\log}$-rings are just morphisms of $\\delta$-rings.\n\\end{remark}\n\n\\begin{remark}\nNote that in this paper, for a $\\delta$-pair $(A,I)$, we always assume $A$ is $(p,I)$-adic complete, but in \\cite{Koshikawa2021log-prism}, non-$(p,I)$-adic completed $\\delta_{\\log}$-triples are also been studied. By Lemma 2.10 of loc.cit., we can always take the $(p,I)$-adic completions of the $\\delta$-pair $(A,I)$ and the $\\delta_{\\log}$-structure will be inherited. \n\\end{remark}\n\n\\begin{proposition}\\cite[Corollary 2.15]{Koshikawa2021log-prism}\nGiven a bounded prelog prism $(A,I,M)$, one can associate it with a log prism\n$$\n(A,I,M)^a=(A,I,M^a)\n$$\n\\end{proposition}\n\\begin{remark}\nWhen we deal with log prisms in this paper, we will always take it as the log prism associated with some prelog prism. And by the above proposition, we know taking the associated log prism does not change the underlying $\\delta$-pair. Moreover, it is a general fact that $(A,I,M)^a$ is integral if $(A,I,M)$ is a integral.\n\\end{remark}\n\n\\begin{definition}\\label{def-logprism}\nThe absolute logarithmic prismatic site $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ is the opposite of the category whose objects are \n\\begin{enumerate}\n    \\item bounded log prisms $(A,I,M_A)$ with \\textit{integral} log structure,\n    \\item maps of formal schemes $f_A: \\Spf(A\/IA) \\to X$,\n    \\item the map $f_A$ satisfies\n$$\n(\\Spf(A\/IA),f_A^\\ast M_X) \\to (\\Spf(A),M_A)^a\n$$\ndefines an exact closed immersion of log formal schemes.\n\\end{enumerate}\nA morphism  $(A,I,M_A) \\to (B,I,M_B)$ is a cover if and only if $A \\to B$ is $(p,I)$-complete faithfully flat and the pullback induces an isomorphism on log structure. We define the structure sheaf $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ on $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ by $(A,I,M_A) \\mapsto A$.\n\\end{definition}\n\nThere is a variant of the about definition that we will also use in this subsection, we define $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$ be the full subcategory of $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ whose  objects are $(A,I,M_A)$ with $A$ perfect.\n\n\\begin{remark}\nOur definition of the absolute logarithmic prismatic site is different from \\cite[Definition 4.1]{Koshikawa2021log-prism}. First, we need to consider the absolute prismatic site, not the relative one. Furthermore, we use the $(p,I)$-complete faithfully flat topology compared with the $(p,I)$-complete \\'etale topology. Also we require the log-structures to be integral. \n\\end{remark}\n\n\\begin{proposition}\\label{prop-logprismisasite}\n$(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ forms a site.\n\\end{proposition}\n\\begin{proof}\nSimilar to \\cite[Corollary 3.12]{BS19}, we need to show for a given diagram \n$$\n\\begin{tikzcd}\n(C,I,M_C) & (A,I,M_A) \\arrow[l,\"c\",swap] \\arrow[r,\"b\"] & (B,I,M_B)\n\\end{tikzcd}\n$$\nin $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ such that $b$ is a cover, then the pushout of $b$ along $c$ is a covering. From the argument in $loc. cit.$, we known for the underlying prisms, the pushout of $b$ along $c$ is the $(p,I)$-completed tensor product $D=C\\widehat {\\otimes}_A B$, and $(D,I)$ is a bounded prism covers $(C,I)$ in the $(p,I)$-complete faithful flat topology. And we give $D$ the log structure $M_D$ defined by viewing $\\Spf(D)$ as the fiber product via \\cite[Proposition 2.1.2]{Ogus_logbook}, then $(C,M_C)\\to (D,M_D)$ is strict morphism by Remark 2.1.3 of $loc.cit.$,  so in particular, $M_D$ is integral since $M_C$ is. For the same reason, \n$$\n(\\Spf(D\/ID),f_D^\\ast M_X) \\to (\\Spf(D),M_D)^a\n$$ is strict since it is the base change of a strict morphism. It is an exact closed immersion since pushout of a surjective map of monoids is again surjective.\n\\end{proof}\n\n\\begin{example}\\cite[Example 3.4]{Koshikawa2021log-prism}\\label{exa-logprism}\n\\begin{enumerate}\n    \\item Let $(A,(E))$ be the Breuil-Kisin prism, then we can define a perlog structure to $(A,(E))$ given by $\\mathbb N \\to A; n\\mapsto u^n$, one have $(A,(E),\\mathbb N)^a$ is in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$, where (3) in Definition~\\ref{def-logprism} follows from the prelog structures $\\mathbb N \\to R \\to A\/(E)$ and $\\mathbb N \\to A \\to A\/(E)$ induce the same log structure.\n    \\item For any prism $(B,J)$ over $(A,(E))$, it has a natural prelog structure $\\mathbb N \\to A \\to B$, and similar to $(1)$, $(B,J,\\mathbb N)^a$ is in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$.\n    \\item A special case of (2) is that $(B,J)=(A_{\\perf},(E))$, the perfection of $(A,(E))$. One has the prelog structure in (2) can be directly defined as $1\\mapsto [\\varpi^\\flat]$. And $(A,(E),\\mathbb N)^a \\to(B,J,\\mathbb N)^a$ is a covering of log prisms in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$.\n    \\end{enumerate}\n\\end{example}\n\nActually, all logarithmic structures of log prisms in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ is the log structure associated to a prelog structure defined by $\\mathbb N$. We thank Teruhisa Koshikawa for letting us know the following lemma.\n\n\\begin{lemma}\\label{lem-Nchart}\nFor any log prism $(B,J,M_B)$ inside $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$, $(B,M_B)^a$ admits a chart $\\mathbb N \\to B$ defined by $n \\mapsto u_B^n$ for some $u_B\\in B$ satisfying $u_B \\equiv \\varpi \\mod J$. \n\\end{lemma}\n\\begin{proof}\nFor any log prism $(B,J,M_B)$ inside $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$, we have \n$$\n(\\Spf(B\/J),f_B^\\ast M_X) \\to (\\Spf(B),M_B)^a\n$$\ndefines an exact closed immersion of log formal schemes. So by the proof of \\cite[Proposition 3.7]{Koshikawa2021log-prism}, if we let $N^a_{B\/J}:=\\Gamma(\\Spf(B\/J),\\underline{\\mathbb N}^a)$ for the prelog structure $\\mathbb N \\to \\O_K \\to B\/J$ induced from the given prelog structure on $\\O_K$, then the fiber product $M_B \\times_{N^a_{B\/J}} \\mathbb N$ is a chart for $(B,M_B)^a$. Moreover, since we assume $M_B$ to be integral, we have $(\\Spf(B\/J),f_B^\\ast M_X) \\to (\\Spf(B),M_B)^a$ is a log thickening with ideal $J$ in the sense of \\cite[Definition 2.1.1.]{Ogus_logbook}, and one can show $M_B \\times_{N^a_{B\/J}} \\mathbb N \\simeq \\mathbb N \\times (1+J)$. Now $(1+J)^\\times =(1+J)$, so \n$$\n\\mathbb N \\to \\mathbb N \\times (1+J) \\simeq M_B \\times_{N^a_{B\/J}} \\mathbb N \\to B\n$$\nis also a chart for $(B,M_B)^a$. And the prelog structure given by $n \\mapsto u_B^n$ for some $u_B\\in B$ satisfying the image of $u_B$ in $B\/J$ coincides with the image of $\\varpi$ under $\\O_K \\to B\/J$.\n\\end{proof}\n\nIn the rest of this subsection, we will try to generalize results we proved in \\S\\ref{subsec-pris-crystal}-\\S\\ref{subsec-pris-crystal-proof} for the logarithmic prismatic site. \n\n\\begin{lemma}\\label{lem-log-nonemptyproduct}\n\\begin{enumerate}\n    \\item For $(A,I_A,M_A)^a, (B,I_B,M_B)^a\\in (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ such that $M_A,M_B$ are integral and $(A,M_A)\\to (A\/I_A,\\mathbb N)$ and $(B,M_B)\\to (B\/I_B,\\mathbb N)$ are exact surjective, there is a prelog prism $(C,I_C,M_C)$ with integral log structure that is universal in the sense that the diagram \n    $$\n    \\begin{tikzcd}\n    (A,I_A,M_A) \\arrow[r] & (C,I_C,M_C) & (B,I_B,M_B) \\arrow[l] \n    \\end{tikzcd}\n    $$\n    is initial in the category of diagrams \n    $$\n    \\begin{tikzcd}\n    (A,I_A,M_A) \\arrow[r] & (D,I_D,M_D) & (B,I_B,M_B) \\arrow[l] \n    \\end{tikzcd}\n    $$\n    of prelog prisms over $(R,\\mathbb N)$, and $(D,M_D)\\to (D\/I_D,\\mathbb N)$ is an exact surjective.\n    \\item If $(C,I_C)$ in (1) is bounded, then $(C,I_C,M_C)^a$ is the product of $(A,I_A,M_A)^a$ and $(B,I_B,M_B)^a$ inside $ (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$.\n    \\item If $(A,I_A,M_A)^a, (B,I_B,M_B)^a$ in (1) are in $ (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$, and let $(C_{\\perf},I_C)$ be the perfection of $(C,I_C)$ defined in (1). Let $(C_{\\perf},I_C,M_C)$ be the prelog prism with prelog structure induced from $C$. Then $(C_{\\perf},I_C,M_C)^a$ is the product of $(A,I_A,M_A)^a$ and $(B,I_B,M_B)^a$ in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $(A,I_A,M_A),(B, I_B, M_B)\\in (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$, define $C_0$ to be the $(p,I_A,I_B)$-adic completion of $A\\otimes_{W(k)}B$ and let $J$ be the kernel of\n$$\nC_0 \\to A\/I_A \\widehat {\\otimes}_R B\/I_B.\n$$\nThen $(C_0,J,M_A\\times M_B)$ is a $\\deltalog$-triple over $(A,I_A,M_A)$. And we have $(C_0,J,M_A\\times M_B) \\to (C_0\/J,\\mathbb N)$ is surjective. Then we can apply \\cite[Proposition 3.6]{Koshikawa2021log-prism} to get a universal prelog prism $(C,I_C,M_C)$ over $(A,I_A,M_A)$ and $(B, I_B, M_B)$ and satisfies $(C,M_C)\\to (C\/J,\\mathbb N)$ is exact surjective. Just recall in the proof of \\cite[Proposition 3.6]{Koshikawa2021log-prism}, we first construct a $\\deltalog$-triple $(C',J',M_C')$ which is universal in the sense that it is a $\\deltalog$-triple over both $(A,I_A,M_A)$ and $(B, I_B, M_B)$ satisfying $C'\/J'$ is over $A\/I_A$ and $B\/I_B$ as $R$-algebra and $(C',M_C')\\to (C'\/J',\\mathbb N)$ is exact surjective. Then we take the prismatic envelope with respect to $(A,I_A) \\to (C',J')$ to get $(C,I_C)$. Then we can check such $(C,I_C,M_C)$ satisfies the universal property. For (2), when $(C,I_C)$ is bounded, the fact that $(C,I_C,M_C)^a$ is the product of $(A,I_A,M_A)^a$ and $(B,I_B,M_B)^a$ inside $ (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ follows from Proposition 3.7 of $loc.cit.$. For (3), we have $(C_{\\perf},I_C)$ is automatic bounded, and one can check $(C_{\\perf},I_C)$ is universal using exactly the same proof of Proposition 3.7 of $loc.cit.$.\n\\end{proof}\n\nWe thank Koji Shimizu for the following lemma on $A^{(2)}_{\\st}$. \n\n\\begin{lemma}\\label{lem-Ast2islogprism}\nLet $(A,I,\\mathbb N)^a$ be the Breuil-Kisin prism defined in $(1)$ of Example~\\ref{exa-logprism}, then the self-product (resp. self-triple product) of $(A,I,\\mathbb N)^a$ in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ exist. Moreover, if we let $(A^{\\langle 2 \\rangle},I,M^2)^a$ (resp. $(A^{\\langle 3 \\rangle},I,M^3)^a$) be self-product (resp. self-triple product) of $(A,I,\\mathbb N)^a$, then $A^{\\langle i \\rangle}\\simeq A^{(i)}_{\\st}$ for $i=2,3$.\n\\end{lemma}\n\\begin{proof}\nBy our construction in Lemma~\\ref{lem-log-nonemptyproduct}, $(A^{\\langle 2 \\rangle},I,M)$ is the prelog prismatic envelope $(C,I_C,M_C)$ with respect to\n    $$\n    (A,(E),\\mathbb N) \\to (C_0,J,\\mathbb N^2) \\text{ and } (C_0\/J,\\mathbb N^2)\\to (R,\\mathbb N)\n    $$\n    where $C_0=W[\\![u,v]\\!]$, $J=(E(u),u-v)$ with the prelog structure given by $\\beta: (1,0)\\mapsto u, (0,1)\\mapsto v$. The prelog prismatic envelope is constructed using the technique of exactification: consider $\\pi: (C_0,\\mathbb N^2)\\to (R=C\/J,\\mathbb N)$ where the map between log structures is given by $\\pi_{\\log}: \\mathbb N\\times \\mathbb N \\to \\mathbb N;(m,n)\\mapsto m+n$, here $\\pi_{\\log}$ is surjective but not exact, so to constructsthe exactification of $\\pi: (C,\\mathbb N^2)\\to (R,\\mathbb N)$ (cf. \\cite[Construction 2.18]{Koshikawa2021log-prism}), first we have the exactification of $\\pi_{\\log}$ is \n    $$\n    \\alpha: M^2 \\to \\mathbb N \\quad \\text{ given by } \\quad (m,n) \\mapsto m+n,\n    $$\n    where $M^2=\\{(m,n)\\in \\mathbb Z\\times \\mathbb Z \\,|\\, m+n\\in \\mathbb N \\}$. Since $M^2$ is generated by $(-1,1)$, $(1,-1)$, $(0,1)$ and $(1,0)$, one has the exactification of $\\pi$ is \n    $$\n    \\Big( W(k)[\\![u,v]\\!]\\big[\\frac{v}{u},\\frac{u}{v}\\big]^\\wedge_{(p,J')}, J',M^2; \\alpha: (1,0)\\mapsto {u}, (0,1)\\mapsto v, (1,-1)\\mapsto \\frac{u}{v}, (-1,1)\\mapsto \\frac{v}{u} \\Big)\n    $$\n    where $J':=\\ker(W(k)[\\![u,v]\\!]\\big[\\frac{v}{u},\\frac{u}{v}\\big] \\to R)$. \n    \n    We have the $(p,J')$-adic completion of $W(k)[\\![u,v]\\!]\\big[\\frac{v}{u},\\frac{u}{v}\\big]$ is $W(k)[\\![u,\\frac{v}{u}-1]\\!]$. Then take prismatic envelope of \n    $\n    (A,(E))\\to (W(k)[\\![u,\\frac{v}{u}-1]\\!], (E,\\frac{v}{u}-1)).\n    $ One can check \n    $$ W(k)[\\![u,\\frac{v}{u}-1]\\!]\\big\\{\\frac{v\/u-1}{E(u)}\\big\\}^\\wedge_\\delta \\simeq A_{\\st}^{(2)}$$ \n    directly from the definition of $A_{\\st}^{(2)}$.\n    \n    Similarly, we can show $A^{\\langle 3 \\rangle}\\simeq A^{(3)}_{\\st}$ which is also bounded. \n\\end{proof}\n\nThe following is one of our key observations.\n\\begin{lemma}\\label{lem-perfectionofA2andAst2}\nWe have $(A^{\\langle 2 \\rangle})_{\\perf} \\simeq (A^{(2)})_{\\perf}$.\n\\end{lemma}\n\\begin{proof}\nLet $u_1,u_2$ be the image of $u$ under the two natural maps $i_{j}: A_{\\perf} \\to (A^{(2)})_{\\perf}$ for $j=1,2$. We claim that $u_2\/u_1$ is inside $(A^{(2)})_{\\perf}$.\n\nFirstly, we have already shown $A_{\\perf}\\simeq W(\\widehat{\\O}_{K_\\infty}^\\flat)$ and $u=[\\varpi^\\flat]$, here $\\varpi^\\flat=(\\varpi_n)$ with $\\{\\varpi_n\\}_{n\\geq 0}$ being a compatible system of $p^n$-th roots of $\\varpi$ inside $\\O_{\\widehat{K}_\\infty}$, and $(\\varpi_n) \\in \\O_{\\widehat K_\\infty}^\\flat$ via the identification $\\O_{\\widehat K_\\infty}^\\flat \\simeq \\lim_{x \\mapsto x^p} \\O_{\\widehat K_\\infty}$. Let $S=(A^{(2)})_{\\perf}\/(E)$, this is an integral perfectoid ring over $\\O_K$ in the sense of \\cite{BMS1}. We have $S^\\flat\\simeq (A^{(2)})_{\\perf}\/(p)$. For $j=1,2$, define $\\varpi_j^\\flat=u_j \\mod (p) \\in S^\\flat$, then we have $u_j=[\\varpi_j^\\flat]$ for $j=1,2$.\n\nRecall in \\S~\\ref{subsrc-construct-A2}, we have $z = \\frac{y -x}{E(x)}$ in $A^{(2)}$. Since  $ E(x) \\equiv  x^e \\mod p$, we have $ x  (1 + x^{e-1} z) \\equiv  y \\mod p$. If we denote $\\iota : A^{(2)} \\to (A^{(2)})_{\\perf} $ the natural map, then $\\iota(x)=u_1$ and $\\iota(y)=u_2$ in our definition, and $u _1 (1 + u_1^{e-1} \\iota(z)) \\equiv u _2 \\mod p$ inside $S^\\flat=A^{(2)}_{\\perf}\/(p)$. This is the same as $\\varpi_1^\\flat \\mu =  \\varpi_2^\\flat$ with $\\mu = (1 + u_1^{e-1} \\iota(z)) \\mod p$ in $S ^\\flat$. So we have $ [\\mu] u _1 = [\\mu] [\\varpi_1^\\flat] = [\\varpi_2^\\flat]=  u_2$, which proves our claim.\n\n\nNow by symmetry, $u_1\/u_2$ is also inside $(A^{(2)})_{\\perf}$, so $u_1\/u_2$ is a unit in $(A^{(2)})_{\\perf}$. So we can give $(A^{(2)})_{\\perf}$ a prelog structure\n$$\n\\alpha: M^2 \\to (A^{(2)})_{\\perf} \\text{ with } (1,-1)\\mapsto \\frac{u_1}{u_2}, (-1,1)\\mapsto \\frac{u_2}{u_1}, (1,0)\\mapsto {u_1}, (0,1)\\mapsto {u_2}\n$$\nwith the monoid $M^2$ defined as in the proof of Lemma~\\ref{lem-Ast2islogprism}, then  $((A^{(2)})_{\\perf},(E),M^2)^a$ is in $X_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$. \n\nOne can check the maps $i_1,i_2: (A,(E)) \\to (A^{(2)},(E)) \\to ((A^{(2)})_{\\perf},(E))$ induce $i_1,i_2: (A_{\\perf},(E),\\mathbb N) \\to ((A^{(2)})_{\\perf},(E),M^2)$ of prelog prisms. So by Lemma~\\ref{lem-Ast2islogprism}, there is a unique map $(A^{\\langle 2 \\rangle},I,M^2)\\to ((A^{(2)})_{\\perf},(E),M^2)$, which factors through $((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)$.  So it induces a map $((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2) \\to ((A^{(2)})_{\\perf},(E),M^2)$ inside $X_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$. On the other hand, by the universal property of $A^{(2)}$, we know there is a map $(A^{(2)})_{\\perf} \\to (A^{\\langle 2 \\rangle})_{\\perf}$ fits into the coproduct diagram in $X_{{\\mathlarger{\\mathbbl{\\Delta}}}}^{\\perf}$, which is the full subcategory of $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ containing perfect prisms.\n\nOne can check the composition $\\eta: ((A^{(2)})_{\\perf},(E)) \\to ((A^{\\langle 2 \\rangle})_{\\perf},(E)) \\to ((A^{(2)})_{\\perf},(E))$ satisfies $\\eta\\circ i_j= i_j \\circ \\eta$ for $i_1,i_2:(A_{\\perf},(E))\\to ((A^{(2)})_{\\perf},(E))$. Such a map is unique inside $X_{{\\mathlarger{\\mathbbl{\\Delta}}}}^{\\perf}$, so $\\eta=\\id_{((A^{(2)})_{\\perf},(E))}$. \n\nOn the other hand, the composition \n$$\n\\eta': ((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)^a \\to ((A^{(2)})_{\\perf},(E),M^2)^a \\to ((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)^a\n$$ satisfies $\\eta\\circ i'_j= i'_j \\circ \\eta$ for $i'_1,i'_2:(A_{\\perf},(E),\\mathbb N)^a\\to ((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)^a$ induced from $i'_1,i'_2:(A,(E),\\mathbb N)\\to (A^{\\langle 2\\rangle},(E),M^2)$. Such map is also unique inside $X_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$, so $\\eta'=\\id_{((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)^a}$. So in particular we have $(A^{\\langle 2 \\rangle})_{\\perf}\\simeq (A^{(2)})_{\\perf}$.\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm-log-phitau}\nThe category of \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ with a descent data over $A_{\\st}^{(2)}[1\/E]^\\wedge_p$ is equivalent to the category of lattice in representations of $G_K$. Moreover, for all $\\gamma\\in\\hat{G}$, we can define the evaluation map\n$$\ne_\\gamma: A_{\\st}^{(2)}[1\/E]^\\wedge_p \\to W(\\hat{L}^\\flat)\n$$\nsuch that Lemma~\\ref{lem-evaluation-1} is still valid. Moreover, the $\\mathbb Q$-isogeney version of this theorem also holds.\n\\end{theorem}\n\n\\begin{remark}\nThe above theorem should be related to the \\'etale comparison theorem in the log prismatic settings, which has not been studied in \\cite{Koshikawa2021log-prism} yet.\n\\end{remark}\n\nMoreover, we have a log version of Lemma~\\ref{lem-AEcoversfinal} also holds. We thank Teruhisa Koshikawa for hints of the following result.\n\n\\begin{proposition}\\label{prop-logcoverfinalobject}\nThe sheaf represented by $(A,(E),\\mathbb N)^a$ covers the final object $\\ast$ in in $\\Shv((X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}})$.\n\\end{proposition}\n\\begin{proof}\nFor any log prism $(B,J,M_B)$, by Lemma~\\ref{lem-Nchart}, we can assume $(B,J,M_B)^a=(B,J,\\mathbb N)^a$, with prelog structure defined by $n \\mapsto u_B^n$ with $u_B \\equiv \\varpi \\mod J$.\n\nUsing deformation theory, we have there is a unique $W(k)$-algebra structure for $B$, and we define $C=B[\\![u]\\!][\\frac{u_B}{u},\\frac{u}{u_B}]\\{\\frac{u_B\/u-1}{J}\\}^\\wedge_\\delta$, where the completion is taken for the $(p,J)$-adic topology. Similar to the proof of Lemma~\\ref{lem-Ast2islogprism}, we have $(C,JC,\\mathbb N)^a$ is the product of $(A,(E),\\mathbb N)^a$ and $(B,J,\\mathbb N)^a$ inside $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$. Moreover, we have $B \\to C$ is $(p, J)$-complete flat by \\cite[Proposition 3.13]{BS19}. It remains to show that $(B,J) \\to (C,J)$ is a covering, i.e., $B \\to C$ is $(p, J)$-complete faithfully flat. Let \n$$\nC^{nc}:=B[\\![u]\\!][\\frac{u_B}{u},\\frac{u}{u_B}]\\{\\frac{u_B\/u-1}{J}\\}_{\\delta}\n$$\nbe the non-complete version of $C$ that we have the $(p,J)$-adic completion of $C^{nc}$ is $C$. Now we just need to show the flat ring map $B\/(p,J) \\to C\/(p,J)=C^{nc}\/(p,J)$ is also faithful. \n\nWe claim that $C\/ (p , J)$ is free over $B \/ (p , J)$. One has $JC=E(u)C$, and $(p , J)= (p , E) = (p , J , E)$ in $C$. So $C\/ (p , J)=C^{nc}\/(p,J)$ is equal to \n\\[ B [\\![u ]\\!][\\frac{u_B}{u}, \\frac{u}{u _B}][\\delta^i(z), i \\geq 0 ]\/ \\left (p , J , E , Ez = \\frac{u_B}{u }-1 , \\delta^i (\\frac{u_B}{u }-1))= \\delta^i (Ez), i \\geq 1 \\right ).\\]\nAfter modulo $(p,J)$, the above is the direct limit of\n\\[B \/ (p , J)[\\delta ^i (z)]\/ \\left (\\delta^i (\\frac{u_B}{u }-1))= \\delta^i (Ez) \\mod (p, E , J) \\right )\\]\nfor $i\\geq 0$.\n\nNow we use Lemma \\ref{lem-delta-n} to compute $\\delta^i (\\frac{u_B}{u }-1)= \\delta^i (Ez) \\mod (p, E , J)$. We keep the notations in Lemma \\ref{lem-delta-n}, by induction, we have $b_n = 0 \\mod (p , E)$. Using that $a_p^{(j)}\\in A_0^\\times$, $\\delta^i (\\frac{u_B}{u }-1)= \\delta^i (Ez) \\mod (p, E , J)$ gives a relation $ (z_{i -1}) ^ p =   \\sum\\limits_{j= 0}^{p -1} \\tilde a_j^{(i)} (z_{i-1}) ^j$ where $z_i = \\mathfrak z_i \\mod (p , J , E)$ and $\\tilde a_j^{(i)} \\in B \/ (p , J)[z_0, z_1, \\dots, z_{i-2}]$. In summary, we have \n$$C\/ (p , J) = B\/(p, J)[z_i, i \\geq 0]\\Bigg\/ \\left ((z_{i}) ^ p -   \\sum\\limits_{j= 0}^{p -1} \\tilde a_j^{(i)} (z_{i}) ^j, i\\geq 1 \\right )$$\nwhich is free over $B \/(p, J)$. \n\\end{proof}\n\n\\begin{definition}\\label{def-logFcrystal}\n\\begin{enumerate}\n    \\item \nA prismatic crystal over $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ in finite locally free $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-modules is a finite locally free $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-module $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ such that for all morphisms $f: (A, I,M_A) \\to (B, J,M_B)$ of log prisms, it induces an isomorphism:\n$$\nf^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},A} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,I,M_A))\\otimes_A B \\simeq \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},B} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B,J,M_B))\n$$\n\n\\item A prismatic $F$-crystal over $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ of height $h$ (in finite locally free $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-modules) is a prismatic crystal $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ in finite locally free $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-modules together with a $\\varphi_{\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}}$-semilinear endomorphism $\\varphi_{\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}}$ of the $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-module $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}: \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\to \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ such that the cokernel of the linearization $\\varphi^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\to \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is killed by $\\mathcal{I}_{\\mathlarger{\\mathbbl{\\Delta}}}^h$.\n\n\\end{enumerate}\n\\end{definition}\n\nIn particular, with help of Theorem~\\ref{thm-log-phitau} and Proposition~\\ref{prop-logcoverfinalobject}, a direct translation of proofs in \\S\\ref{subsec-pris-crystal-proof} with $A^{(2)}$ replaced by $A^{(2)}_{\\st}$ shows the following theorem.\n\\begin{theorem}\\label{thm-log-main-1}\nThe category of prismatic $F$-crystals over $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ of height $h$ is equivalent to the category of lattices in semi-stable representations of $G_K$ with Hodge-Tate weights between $0$ and $h$.\n\\end{theorem}\n\n\\section{Some discussions on base rings}\\label{subsec-baserings} In this section, we show that our base ring assumed at the beginning of \\S \\ref{sec-ring-strcuture} covers many situations of base rings used in \\cite{Kim12} and \\cite{Brinon}. \n\nLet $K$ be complete DVR with perfect residue field $k$, and let $K_0=W[\\frac{1}{p}]$ with $W=W(k)$, fix a uniformizer $\\varpi\\in \\O_K$ and $E(u)\\in W[u]$ a minimal polynomial of $\\varpi$ over $K_0$. Let $R$ be a normal domain and satisfies that $R$ is a $p$-complete flat $\\O_K$-algebra that is complete with respect to $J$-adic topology, for an ideal  $J=(\\varpi, {t_1},\\ldots,{t_d})$ of $R$ containing $\\varpi$. We also assume $\\overline{R}=R\/(\\varpi)$ is a finite generated $k$-algebra with \\emph{finite $p$-basis} discussed in \\cite[\\S 1.1]{deJong}.\n\n\\begin{lemma}[\\cite{Kim12} Lemma 2.3.1 and lemma 2.3.4]\\label{Kim-lemma}\n\\begin{enumerate}\n    \\item In the above setting, there is a $p$-adic formally smooth flat $W$-algebra $R_0$ equipped with a Frobenius lift $\\varphi_0$ such that $\\overline{R}: = R_0\/(p)$. Moreover let $J_0$ be the preimage of $\\overline{J}$ inside $R_0$, then $R_0$ is $J_0$-adically complete, and under this topology, $R_0$ is formally smooth. \n    \\item $R_0\/(p)\\xrightarrow{\\sim}R\/(\\varpi)$ lifts to a $W$-algebra morphism $R_0 \\to R$ and the induced $\\O_K$-algebra morphism $\\O_K\\otimes_W R_0 \\to R$ is an isomorphism. Moreover this isomorphism is continuous with respect to the $J_0$-adic topology.\n\\end{enumerate}\n\\end{lemma}\n\nLet $(R_0, \\varphi_{R_0})$ denote a flat $W$-lift of $R\/(\\varpi)$ obtained from the above lemma. And we will have $J_0=(p, t_1, \\ldots, t_d)\\in R_0$, and we write $\\overline{J}=(\\overline{t_1},\\ldots, \\overline{t_d})\\subset \\overline{R}$. \n\n\\begin{definition}\\label{RAE}\nLet $R_0$ be a $p$-complete $\\mathbb Z_p$-algebra, we say $R_0$ satisfies the ``refined almost \u00e9talenes\" assumption, or simply RAE assumption, if $\\hat{\\Omega}_{R_0}=\\oplus_{i=1}^m R_0 dT_i$ with $T_i\\in R_0^\\times$. Where $\\hat{\\Omega}_{R_0}$ is the module of of $p$-adically continuous K\\\"ahler differentials.\n\\end{definition}\nThe following are examples of $R_0$ and $R$ which satisfy assumptions of Lemma \\ref{Kim-lemma} and RAE assumption. \n\\begin{example}\n\\begin{enumerate}\n    \\item If $R\/(\\varpi)$ is a completed noetherian regular local ring with residue field $k$, then Cohen structure theorem implies\n     $R\/(\\varpi)=k[\\![\\overline{x_1},\\ldots,\\overline{x_d}]\\!]$. In this case, $R_0=W[\\![x_1,\\ldots, x_d]\\!]$ and $J_0=(p,x_1,\\ldots, x_d)$. Then $R=W[\\![x_1,\\ldots, x_d]\\!][u]\/E$, with $E\\in W[u]$ is a Eisenstein polynomial.\n    \\item Let $R_0 = W(k) \\langle t _1^{\\pm 1} ,  \\dots , t _m ^{\\pm 1}\\rangle$ and $J_0=(p)$, in this example, $\\overline{R}=k[\\overline{t}_1^{\\pm 1} ,  \\dots , \\overline{t}_m ^{\\pm 1}]$ is not local.\n    \\item An unramified complete DVR $(R_0 , p)$ with residue field $k$ so that $[k : k ^p]<\\infty$. \n    \\item Note the the Frobenius liftings in Lemma ~\\ref{Kim-lemma} is not unique. In (2) we can choose $\\varphi_{R_0}(t_i)=t_i^p$. In (1), we can choose the $\\varphi_{R_0}(x_i)=x_i^p$ or $\\varphi_{R_0}(x_i)=(x_i+1)^p-1$.\n\\end{enumerate}\n\\end{example}\nLet $R_0$ be $p$-complete algebra which satisfies the RAE assumption, Set $\\breve R_0 = W\\langle t_1 , \\dots , t_m \\rangle$ and $f : \\breve R_0 \\to R_0$ by sending $t_i$ to $T_i$. \n\\begin{proposition}\\label{prop-fetale} Assume that $R_0$ is a $p$-complete integral domain which admits finite $p$-basis and satisfies RAE assumption.  \nThen $f$ is formally \\'etale $p$-adically. \n\\end{proposition}\n\\begin{proof} We thanks for Wansu Kim providing the following proof. By standard technique using \\cite[Ch.III, Corollaire 2.1.3.3]{Illusie1} (e.g., see the proof in \\cite[Lem. 2.3.1]{Kim12}), it suffices to show that the cotangent complex \n$\\mathbb L_{R_0 \/ \\breve R_0}$ is acyclic. Since both $R_0$ and $\\breve R_0$ are $\\mathbb Z_p$-flat, it suffice to show that $\\mathbb L_{R_1 \/ \\breve R_1}$ is acyclic where $R_1 = R_ 0 \/ p R_0$ and $\\breve R_1 = \\breve R_0 \/ p \\breve R_0$. Since $R_0$ has finite $p$-basis, by \\cite[Lem. 1.1.2]{deJong}, $\\mathbb L_{R_1 \/k}\\simeq \\Omega_{R_1\/k}$. Note that maps $k \\to \\breve R_1 \\to R_1$ induces a fiber sequence \n\\[ \\mathbb L_{\\breve R_1 \/k}\\otimes^{\\mathbb L}_{\\breve R_1} R_1 \\to \\mathbb L _{R_1 \/ k} \\to \\mathbb L_{R_1 \/ \\breve R_1}\\]\nSince that $ \\mathbb L_{\\breve R_1 \/k} \\simeq \\Omega_{\\breve R_1\/k}$ and $\\Omega_{\\breve R_1\/k}\\simeq \\Omega_{R_1\/k}$ by RAE condition, we conclude that $\\mathbb L_{R_1\/ \\breve R_1}= 0$ as required. \n\\end{proof}\nLet us end with a discussion about our base rings and the base rings used in \\cite{Brinon}. As explained in the beginning of \\cite[Chap. 2]{Brinon}, his base ring $R_0$ in \\cite{Brinon} is  obtained from $W\\langle t_1^{\\pm 1}, \\ldots, t_m^{\\pm 1}\\rangle$ by a finite number of iterations of certain  operations and is also assumed to satisfy certain properties.  By Prop. 2.0.2 \\emph{loc. cit.}, we see that $R_0$ has finite $p$-basis and satisfies RAE assumption. So the base ring $R_0$ in \\cite{Brinon} also satisfies the requirement that $f:W\\langle t_1, \\ldots, t_m\\rangle \\to R_0$ is formally \\'etale by Proposition \\ref{prop-fetale}. \n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\newbox\\ncinttbox\n\t\\setbox0=\\hbox{$-$}\n\t\\setbox2=\\hbox{$\\displaystyle\\int$}\n\t\\setbox\\ncintdbox=\\hbox{\\rlap{\\hbox\n\t\tto \\wd2{\\hskip-.125em \\box2\\relax\\hfil}}\\box0\\kern.1em}\n\t\\setbox0=\\hbox{$\\vcenter{\\hrule width 4pt}$}\n\t\\setbox2=\\hbox{$\\textstyle\\int$}\n\t\\setbox\\ncinttbox=\\hbox{\\rlap{\\hbox\n\t\tto \\wd2{\\hskip-.175em \\box2\\relax\\hfil}}\\box0\\kern.1em}\n\\newcommand{\\ncint}{\\mathop{\\mathchoice{\\copy\\ncintdbox}%\n\t\t\t\t\t{\\copy\\ncinttbox}{\\copy\\ncinttbox}%\n\t\t\t\t\t{\\copy\\ncinttbox}}\\nolimits}\n\n\\newcommand{\\commentA}[1]{\\textcolor{red}{\\textsf{A: #1}}}\n\\newcommand{\\commentJ}[1]{\\textcolor{blue}{\\textsf{J: #1}}}\n\\newcommand{\\commentS}[1]{\\textcolor{green}{\\textsf{S: #1}}}\n\n\\newcommand{\\raisebox{-2pt}{$\\overset{\\cdotp\\,\\,\\cdotp}{\\frown}$}}{\\raisebox{-2pt}{$\\overset{\\cdotp\\,\\,\\cdotp}{\\frown}$}}\n\\newcommand{\\raisebox{-2pt}{$\\overset{\\cdotp\\,\\,\\cdotp}{\\smile}$}}{\\raisebox{-2pt}{$\\overset{\\cdotp\\,\\,\\cdotp}{\\smile}$}}\n\n\n\\hyphenation{geo-me-try ma-ni-fold ma-ni-folds pro-duct pro-ducts}\n\n\n\n\\begin{document}\n\n\\maketitle\n\n\\vspace{-2pc}\n\n\n\n\\begin{abstract}\nLet $B$ be a $C^{*}$-algebra, $X$ a Hilbert $C^{*}$-module over $B$ and $M,N\\subset X$ a pair of complemented submodules. We prove the $C^{*}$-module version of von Neumann's alternating projections theorem: the sequence $(P_{N}P_{M})^{n}$ is Cauchy in the $*$-strong module topology if and only if $M\\cap N$ is the complement of $\\overline{M^{\\perp}+N^{\\perp}}$. In this case, the $*$-strong limit of $(P_{M}P_{N})^{n}$ is the orthogonal projection onto $M\\cap N$. We use this result and the local-global principle to show that the cosine of the Friedrichs angle $c(M,N)$ between any pair of complemented submodules $M,N\\subset X$ is well-defined and that $c(M,N)<1$ if and only if $M\\cap N$ is complemented and $M+N$ is closed. \n\\end{abstract}\n\n{\\bf Keywords:}\n{\\small two projections, von Neumann's alternating projection theorem, Friedrichs angle, Hilbert $C^{*}$-module, local-global principle.}\n\n{\\bf MSC2020:} {\\small 46L08, 47A46}\n\n\\parskip=6pt\n\\parindent=0pt\n\\allowdisplaybreaks\n\\section*{Introduction}\n\nIn this note we offer a new and general approach to the two projection problem in Hilbert $C^{*}$-modules.\nAs an application we extend and improve upon several of the main results in the recent work of \\cite{Luo} by giving new proofs that allow for the removal of a key hypothesis. \n\nBriefly, we begin by proving the Hilbert $C^{*}$-module version of von Neumann's alternating projections theorem, which computes the projection onto $M\\cap N$ for a concordant pair of complemented submodules $M,N$ (see below). We then proceed to use this result to define the Friedrichs angle between an arbitrary pair of complemented submodules. The angle is realised as a function on the space of representations of the coefficient algebra of the module. The properties of the Friedrichs angle give necessary and sufficient conditions for the sum and intersection of two complemented submodules to again be complemented.\nWe now give a little more detail on these results. \n\nGiven two closed subspaces $M,N$ of a Hilbert space $H$ there is an orthogonal direct sum decomposition\n\\begin{equation}\n\\label{directsum}\nH=(M\\cap N)\\oplus \\overline{(M^{\\perp}+N^{\\perp})}.\n\\end{equation}\nA fundamental result of von Neumann, the \\emph{method of alternating projections}, states that the projection $P_{M\\cap N}$ onto $M\\cap N$ can be obtained as the $*$-strong limit\n\\[P_{M\\cap N}=s-\\lim_{n\\to\\infty} (P_{M}P_{N})^{n}=s-\\lim_{n\\to\\infty} (P_{N}P_{M})^{n}.\\]\nThe (cosine) of the \\emph{Friedrichs angle between $M$ and $N$} is the quantity \n$$\nc(M,N):=\\|P_{M}P_{N}-P_{M\\cap N}\\|,\n$$ \nand the subspace $M^{\\perp}+N^{\\perp}$ is closed if and only if $c(M,N)<1$.\n\nIn this paper we consider a pair $(M,N)$ of complemented submodules of a Hilbert $C^{*}$-module $X$ over a $C^{*}$-algebra $B$. It is well-known that closed submodules of Hilbert $C^*$-modules need not be orthogonally complemented. This one technical constraint necessitates the discussion of adjointable endomorphisms and regular (unbounded) operators for these modules, \\cite{FL,Lance}. \n\nThe complementability issue does not arise for finite dimensional vector spaces of course, but already in the case of finite rank, locally trivial vector bundles on compact Hausdorff base spaces we see examples \nof non-complementability of intersections. Classically the issue gives rise to the notion of a strict homomorphism of vector bundles \\cite[Section 1.3]{Atiyah}, and we relate the vector bundle situation to the complementability problem in Remarks \\ref{eg:VB-unstrict}, \\ref{discontangle} and \\ref{Atiyah} below.\n\nIn Theorem \\ref{vN} we show that the pair $(M,N)$ induces a direct sum decomposition like \\eqref{directsum} of the Hilbert $C^*$-module $X$ if and only if von Neumann's theorem on alternating projections is valid for this pair of submodules. We call such pairs \\emph{concordant} and characterise them in terms of their Hilbert space localisations in Theorem \\ref{locharm}. Our results have implications for the Hilbert module version of the two projection problem, \\cite{Luo}. The Hilbert space version first gained prominence in the work of Halmos \\cite{H}, and has since had numerous incarnations and applications: for a recent survey see \\cite{BS}.\n\nIn \\cite{Luo}, the Friedrichs angle between complemented submodules has been defined under the constraint that $M\\cap N$ is complemented. In Section \\ref{sec:angle} of this note we remove this hypothesis and extend the definition of the Friedrichs angle to arbitrary pairs of complemented submodules via the local-global principle of \\cite{Pierrot}. We interpret the Friedrichs angle as a function on the space $\\widehat{B}$ of irreducible representations of $B$ and prove that $c(M,N)=c(M^{\\perp},N^{\\perp})$. We deduce that $c(M,N)<1$ if and only if the sequence $(P_{N}P_{M})^{n}$ is Cauchy for the operator norm if and only if $M\\cap N$ is complemented and $M^{\\perp}+N^{\\perp}$ is closed.\n\n{\\bf Notation.} For a Hilbert $C^{*}$-module $X$ over a $C^{*}$-algebra $B$ we denote by $\\End^{*}_{B}(X)$ the unital $C^{*}$-algebra of adjointable operators on $X$ and by $\\mathbb{K}(X)\\subset \\End^{*}_{B}(X)$ the ideal of compact operators. The symbols $\\otimes^{\\textnormal{alg}}_{B}, \\widehat{\\otimes}_{B}$ and $\\otimes_{B}$ denote the balanced algebraic, projective and $C^{*}$-module tensor products, respectively.\n\n{\\bf Acknowledgements.} We thank Marcel de Jeu for helpful conversations and Michael Frank for valuable correspondence.\n\\section{Concordant submodules}\nLet $X$ be a Hilbert $C^{*}$-module over the $C^{*}$-algebra $B$.\nGiven two complemented submodules $M$ and $N$ of $X$, we write $P_{M},P_{N}$ respectively for the corresponding projections in $\\End^{*}_{B}(X)$. The intersection $M\\cap N$ is a closed submodule of $X$, and there is an inclusion\n\\[M^{\\perp}+N^{\\perp}\\subset (M\\cap N)^{\\perp}.\\]\nThe submodule $M^{\\perp}+N^{\\perp}$ need not be closed, but since $(M\\cap N)^{\\perp}$ is closed,\n\\[\\overline{M^{\\perp}+N^{\\perp}}\\subset (M\\cap N)^{\\perp},\\]\nas well. In case $X$ is a Hilbert space there is an equality (see (\\cite[Theorem 4.6.4]{Dbook})\n\\begin{equation}\n\\label{concordant}\n \\overline{M^{\\perp}+N^{\\perp}}= (M\\cap N)^{\\perp},\n\\end{equation} \nand thus the projections $P_{M\\cap N}$ and $P_{\\overline{M^{\\perp}+N^{\\perp}}}$ exist and satisfy $1-P_{M\\cap N}=P_{\\overline{M^{\\perp}+N^{\\perp}}}$. \n\nIn general, the projections do not exist unless the submodules are complemented.  To our knowledge, it is an open question whether the intersection of complemented submodules is again complemented. In \\cite[Section 3]{Luo} it was shown that even in case all the projections exist, \\eqref{concordant} need not not hold (see Remark \\ref{LMXcounter} below).\n\\begin{defn}\nLet $M$ and $N$ be complemented submodules of a Hilbert $C^{*}$-module $X$. The pair $(M,N)$ is \\emph{concordant} if $X=(M\\cap N)\\oplus\\overline{(M^{\\perp}+N^{\\perp})}$. If the pair $(M,N)$ is not concordant, we say it is \\emph{discordant}. \n\\end{defn}\nThe pair $(M,N)$ is concordant if their intersection $M\\cap N$ is complemented and its complement is $\\overline{M^{\\perp}+N^{\\perp}}$.\n\\begin{rmk}\\label{LMXcounter} The pair $(M,N)$ being concordant is strictly stronger than the requirement that $M\\cap N$ be complemented. In \\cite[Section 3]{Luo} it is shown that for $X=B=C([0,\\frac{\\pi}{2}], M_{2}(\\mathbb{C}))$, the submodules\n\\[M=\\textnormal{Ran } \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix},\\quad N= \\textnormal{Ran } \\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix},\\]\nsatisfy $M\\cap N =0$, which is complemented, whereas $\\overline{M^{\\perp}+N^{\\perp}}\\neq X$ so $(M,N)$ is not concordant.\n\\end{rmk}\n\n\\begin{rmk} \nNote that $(M,N)$ is \\emph{harmonious} in the sense of \\cite[Definition 4.1]{Luo} if each of the submodules\n\\[\n\\overline{M+N},\\,\\, \\overline{M+N^{\\perp}},\\,\\,\\overline{M^{\\perp}+N},\\,\\, \\overline{M^{\\perp}+N^{\\perp}} \n\\]\nis complemented. In this case the respective complements are\n\\[\nM^{\\perp}\\cap N^{\\perp},\\,\\,M^{\\perp}\\cap N,\\,\\, M+N^{\\perp},\\,\\, M\\cap N,\n\\]\nas explained in the discussion after \\cite[Definition 4.1]{Luo}. Thus $(M,N)$ is harmonious if and only if each of the pairs $(M,N)$, $(M,N^{\\perp})$, $(M^{\\perp},N)$ and $(M^{\\perp},N^{\\perp})$ is concordant.\n \\end{rmk}\n \\begin{rmk} If $M+N$ is closed, then by \\cite[Proposition 4.6]{LSX} $M^{\\perp}+N^{\\perp}$ is closed and $X=(M\\cap N)\\oplus (M^{\\perp}+N^{\\perp})$. In particular, $M+N$ is closed if and only if $M^{\\perp}+N^{\\perp}$ is closed and in this case both $(M,N)$ and $(M^{\\perp},N^{\\perp})$ are concordant (see Proposition 3.10 below).\n \\end{rmk}\n \\begin{rmk}\n \\label{rmkuniversal} \n In \\cite{RS} it was shown that the universal $C^{*}$-algebra $C^{*}(p,q)$ generated by two projections $p$ and $q$ admits the following concrete model\n\\[\nC^{*}(p,q)\\simeq\\left\\{A(t)\\in C([0,\\pi\/2], M_2(\\mathbb{C})): A(0)\\ \\mbox{and}\\ A(\\pi\/2)\\ \\mbox{diagonal}\\right\\},\n\\]\nwith the isomorphism is determined by\n\\[\np\\mapsto P:=\\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix}, \\quad q\\mapsto Q\n:=\\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix}.\n\\]\nFrom this point of view, the counterexample of \\cite[Section 3]{Luo} discussed in Remark \\ref{LMXcounter} above arises from the universal example. This shows that specific properties such as being concordant or harmonious hold in some representations of $C^{*}(p,q)$, but not in all of them.\n\\end{rmk}\n\nWe will now characterise concordant pairs by looking at their Hilbert space localisations.\n\n\nLet $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ be a representation of $B$ on the Hilbert space $H_{\\pi}$ and write $X_{\\pi}:=X\\otimes_{B}H_{\\pi}$. There is a representation\n\\begin{equation}\n\\widehat{\\pi}:\\End^{*}_{B}(X)\\to \\mathbb{B}(X_{\\pi}),\\quad T\\mapsto T\\otimes 1.\n\\label{eq:bob}\n\\end{equation}\nWrite $M_{\\pi}:=M\\otimes_{B}H_{\\pi}\\subset X\\otimes_{B}H_{\\pi}$, and similarly for $N$. Then $M_{\\pi}$ and $N_{\\pi}$ are closed subspaces of the Hilbert space $X_{\\pi}$ and we have $P_{M_{\\pi}}:=\\widehat{\\pi}(P_{M})=P_{M}\\otimes 1$, as well as  $P_{N_{\\pi}}:=\\widehat{\\pi}(P_{N})=P_{N}\\otimes 1$. Since the subspace $M_{\\pi}\\cap N_{\\pi}$ is closed, there is a projection $P_{M_{\\pi}\\cap N_{\\pi}}\\in\\mathbb{B}(X_{\\pi})$ that projects onto $M_{\\pi}\\cap N_{\\pi}$. In general, the equality $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$ need not hold, even if $M\\cap N$ is complemented. \nWe recall the following fact.\n\\begin{prop}[Local-global principle for complemented submodules \\cite{Pierrot}]\n\\label{locglob} \nLet $\\Omega\\subset X$ be a closed submodule. Then $\\Omega$ is complemented if and only if for every irreducible representation $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ there is an equality $(\\Omega_{\\pi})^{\\perp}=(\\Omega^{\\perp})_{\\pi}$.\n\\end{prop}\n\\begin{proof}\nBy \\cite[Corollaire 1.17]{Pierrot}, we have that $X=\\Omega\\oplus \\Omega^{\\perp}$ if and only if for every irreducible representation $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ there is an equality\n\\[\nX_{\\pi}=X\\otimes_{B}H_{\\pi}=(\\Omega\\oplus \\Omega^{\\perp})\\otimes_{B}H_{\\pi}=\\Omega\\otimes_{B}H_{\\pi}\\oplus \\Omega^{\\perp}\\otimes_{B}H_{\\pi}=\\Omega_{\\pi}\\oplus (\\Omega^{\\perp})_{\\pi}.\n\\]\nSince  $(\\Omega^{\\perp})_{\\pi}\\subset (\\Omega_{\\pi})^{\\perp}$, this holds if and only if $(\\Omega^{\\perp})_{\\pi}=(\\Omega_{\\pi})^{\\perp}$.\n\\end{proof}\nA weaker form of this result was proved independently, though several years later, in \\cite{KL12}. There, the local side of the equivalence involved \\emph{all} representations of the $C^{*}$-algebra $B$. The two results are equivalent because the proof of the implication $\\Rightarrow$ in Proposition \\ref{locglob} holds verbatim for an arbitrary representation of the $C^{*}$-algebra $B$, see \\cite{KL17}. We will use both instances of the result.\n\\begin{lemma}\n\\label{inclusion} \nLet $X$ be a Hilbert $C^{*}$-module over $B$, $M,N$ complemented submodules and $\\pi:B\\to\\mathbb{B}(H_{\\pi})$ a representation. Then there is an equality of closed subspaces\n\\[\n\\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}=\\big(\\,\\overline{M^{\\perp}+N^{\\perp}}\\,\\big)_{\\pi}.\n\\]\n\\end{lemma}\n\\begin{proof}\nThe inclusion of subspaces\n\\begin{align*}\n(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}\\subset \\Big(\\overline{M^{\\perp}+N^{\\perp}}\\Big)_{\\pi}\n\\end{align*}\nshows that we have an inclusion of closed linear subspaces\n\\begin{equation*}\n\\overline{(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}}\\subset \\left(\\overline{M^{\\perp}+N^{\\perp}}\\right)_{\\pi}.\\end{equation*} \n The subspace $(M^{\\perp}+N^{\\perp})\\otimes^{\\textnormal{alg}}_{B}H_{\\pi}$ is dense in $(\\,\\overline{M^{\\perp}+N^{\\perp}}\\,)_{\\pi}$ and since\n \\[\n (M^{\\perp}+N^{\\perp})\\otimes^{\\textnormal{alg}}_{B}H_{\\pi}\\subset(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}\\subset \\overline{(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}}\\subset \\left(\\overline{M^{\\perp}+N^{\\perp}}\\right)_{\\pi},\\]\n it follows that $\\overline{(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}}=\\left(\\overline{M^{\\perp}+N^{\\perp}}\\right)_{\\pi}$. Since $M$ and $N$ are complemented we have $(M_{\\pi})^{\\perp}=(M^{\\perp})_{\\pi}$ and $(N_{\\pi})^{\\perp}=(N^{\\perp})_{\\pi}$ and thus $\\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}=\\left(\\overline{M^{\\perp}+N^{\\perp}}\\right)_{\\pi}$.\n\\end{proof}\n\\begin{thm}\n\\label{locharm}\nLet $X$ be a Hilbert $C^{*}$-module over $B$ and $M$ and $N$ complemented submodules. Then \nthe pair $(M,N)$ is concordant if and only if for every irreducible \nrepresentation $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ there is an equality of closed subspaces $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$.\n\\end{thm}\n\\begin{proof} \nSuppose that $M$ and $N$ are concordant so that\n\\[\nX=(M\\cap N)\\oplus \\left(\\overline{M^{\\perp}+N^{\\perp}}\\right).\n\\]\nTherefore Proposition \\ref{locglob} and Lemma \\ref{inclusion} give\n \\[\n ((M\\cap N)_{\\pi})^{\\perp}=((M\\cap N)^{\\perp})_{\\pi}=(\\overline{M^{\\perp}+N^{\\perp}})_{\\pi}=\\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}.\n \\]  \n\nTaking orthogonal complements we find \n$(M\\cap N)_{\\pi}=\\Big(\\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}\\Big)^{\\perp}=M_{\\pi}\\cap N_{\\pi}$.\n\nConversely, suppose that $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$ for all irreducible \nrepresentations $\\pi$. \nBy Lemma \\ref{inclusion} and Equation \\eqref{concordant} we have \n\\[\n(\\overline{M^{\\perp}+N^{\\perp}})_{\\pi}= \\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}=(M_{\\pi}\\cap N_{\\pi})^{\\perp},\n\\] \n\nand we deduce that\n\\begin{align*}\n(M\\cap N)_{\\pi}\\oplus (\\overline{M^{\\perp}+N^{\\perp}})_{\\pi}&= (M_{\\pi}\\cap N_{\\pi})\\oplus (M_{\\pi}\\cap N_{\\pi})^{\\perp}=X_{\\pi}.\n\\end{align*}\n\nBy Proposition \\ref{locglob} we conclude that $X=(M\\cap N)\\oplus \\overline{M^{\\perp}+N^{\\perp}}.$\n\\end{proof}\nIn line with the local-global principle, Proposition \\ref{locglob}, we obtain the same result when we consider all representations of the base algebra $B$.\n\\begin{corl}\n\\label{locharmcor}\nLet $X$ Hilbert $C^{*}$-module over $B$ and $M$ and $N$ complemented submodules. Then $(M,N)$ is concordant if and only if for every representation $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ there is an equality of closed subspaces $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$.\n\\end{corl}\n\\begin{proof} \nThe proof of $\\Rightarrow$ in Theorem \\ref{locharm} shows that $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$ for every representation whenever $(M,N)$ is concordant.\n\\end{proof}\n\\begin{rmk}\n\\label{eg:VB-unstrict}\nConsider $B=C([0,\\frac{\\pi}{2}])$, $X=C([0,\\frac{\\pi}{2}], \\mathbb{C}^{2})$ and consider the submodules\n\\[\nM=\\textnormal{Ran }\\!\\!\\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix},\\quad N\n= \\textnormal{Ran }\\!\\!\\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix}.\n\\] We have $M\\cap N=0$ and for the irreducible representations given by $t\\in[0,\\pi\/2]$ we have\n\\[\nM_t\\cap N_t=\\begin{cases} 0 &  t=0\\\\ \\C & t\\neq 0,\\end{cases}\n\\]\nso $(M,N)$ is discordant by Theorem \\ref{locharm}.\n\\end{rmk}\n\n\n\\section{Von Neumann's theorem of alternating projections}\nLet $P,Q\\in \\End^{*}_{B}(X)$ be projections\n\\[\nP^{*}=P^{2}=P,\\quad Q^{*}=Q^{2}=Q.\n\\]\nThe submodules $\\textnormal{Ran } P$ and $\\textnormal{Ran } Q$ are complemented in $X$, and every complemented submodule is the range of an adjointable projection. As noted before, it is an open question whether the intersection $\\Omega:=\\textnormal{Ran } P\\cap \\textnormal{Ran } Q$, which is a closed submodule, is complemented. In case $B=\\mathbb{C}$ and $X$ is a Hilbert space this is true and thus there is a projection $P_{\\Omega}$ with $\\textnormal{Ran } P_{\\Omega}=\\Omega$. For $n\\geq 0$, write\n\\[\n(P,Q)_{n}:=\\cdots PQPQ,\\quad \\textnormal{the product of exactly } n \\,\\,\\textnormal{alternating factors ending in } Q.\n\\]\nVon Neumann proved the following well-known theorem.\n\\begin{thm}[{\\cite[Lemma 22]{vN}}]\n\\label{vN} \nLet $H$ be a Hilbert space, $M,N\\subset H$ closed subspaces and $\\Omega:=M\\cap N$. Let $P=P_{M}$ and $Q=P_{N}$ be the orthogonal projections onto $M$ and $N$ respectively. The orthogonal projection $P_{\\Omega}$ onto $\\Omega$ can be obtained as the strong limit of any of the sequences\n\\begin{equation}\n\\label{alternatingseq}\n(PQ)^{n},\\quad (QP)^{n}, \\quad (P,Q)_{n},\\quad (Q,P)_{n},\n\\end{equation}\nor any of their subsequences. \n\\end{thm}\nIn a Hilbert $C^{*}$-module $X$, the analogue of the $*$-strong topology is  defined by the seminorms\n\\[\n\\|T\\|_{x}:=\\max\\{\\|Tx\\|,\\|T^{*}x\\|\\},\\quad x\\in X,\n\\]\nand we refer to this topology as the $*$-\\emph{strong module topology}.\nOn bounded sets the $*$-strong module topology coincides with the \n\\emph{strict topology} on $\\End_{B}^{*}(X)$ relative to the ideal $\\mathbb{K}(X)$, \\cite[Proposition 5.5.9]{Troitsky}.\nThe following fact is well-known.\n\\begin{lemma}\\label{complete}\nThe $*$-strong module topology is complete on bounded sets.\n\\end{lemma}\n\\begin{proof} Let $T_{n}\\in\\End^{*}_{B}(X)$ be a sequence that is Cauchy for the seminorms $\\|\\cdot \\|_{x}$, $x\\in X$. By the Uniform Boundedness Principle, the operators \\[Tx:=\\lim_{n\\to\\infty} T_{n}x, \\quad \\textnormal{and} \\quad T^{*}x:=\\lim_{n\\to\\infty} T^{*}_{n}x,\\] are well-defined, bounded and mutually adjoint.\n\\end{proof}\n\\begin{lemma}\\label{reductionlemma} Let $P,Q\\in\\End^{*}_{B}(X)$ be projections. Then $(PQ)^{n}$ and $(QP)^{n}$ are $*$-strongly Cauchy if and only if $(PQP)^{n}$ and $(QPQ)^{n}$ are $*$-strongly Cauchy  if and only if $(P,Q)_{n}$ and $(Q,P)_{n}$ (as defined in \\eqref{alternatingseq}) are $*$-strongly Cauchy. The same statement holds for the norm topology.\n\\end{lemma}\n\\begin{proof} \nSince\n\\[\n(P,Q)_{n}=\\left\\{\\begin{matrix}  (PQ)^{\\frac{n}{2}} & n\\quad \\textnormal{even}\\\\ (QPQ)^{\\frac{n-1}{2}} & n\\quad  \\textnormal{odd},\\end{matrix}\\right.\\quad (Q,P)_{n}=\\left\\{\\begin{matrix}  (QP)^{\\frac{n}{2}} & n\\quad \\textnormal{even}\\\\ (PQP)^{\\frac{n-1}{2}} & n\\quad  \\textnormal{odd},\\end{matrix}\\right. \n\\]\nit suffices to prove that $(PQ)^{n}$ and $(QP)^{n}$ are $*$-strongly Cauchy if and only $(PQP)^{n}$ and $(QPQ)^{n}$ are $*$-strongly Cauchy. The same holds for the norm topology.\n\nAny projection $P$ satisfies $\\langle Px,Px\\rangle\\leq \\langle x,x\\rangle$ and $Q(PQ)^{n}=(QPQ)^{n}$ so that\n\\begin{align}\n\\label{sandwich}\n\\langle (PQ)^{n}x,(PQ)^{n}x\\rangle&=\\langle (Q(PQ)^{n}+(1-Q)(PQ)^{n})x,(Q(PQ)^{n}+(1-Q)(PQ)^{n})x\\rangle\\nonumber\\\\\n&\\geq \\langle (QPQ)^{n}x,(QPQ)^{n}x\\rangle\\nonumber\\\\\n&=\\langle (P(QPQ)^{n}+(1-P)(QPQ)^{n})x,(P(QPQ)^{n}+(1-P)(QPQ)^{n})x\\rangle\\nonumber\\\\\n&\\geq \\langle (PQ)^{n+1}x,(PQ)^{n+1}x\\rangle.\n\\end{align}\nNow for $m>n$ we have \n\\[\n(PQ)^{n}-(PQ)^{m}=(PQ)^{n}(1-(PQ)^{m-n}),\\quad  ((QPQ)^{n}-(QPQ)^{m})=(QPQ)^{n}(1-(QPQ)^{m-n}),\n\\]\nwhich, together with \\eqref{sandwich} gives\n\\begin{align*}\n\\langle ((PQ)^{n}-(PQ)^{m})x ,((PQ)^{n}&-(PQ)^{m})x\\rangle =\n\\langle (PQ)^{n}(1-(PQ)^{m-n})x,(PQ)^{n}(1-(PQ)^{m-n})x\\rangle \\\\ &\\geq \\langle (QPQ)^{n}(1-(PQ)^{m-n})x,(QPQ)^{n}(1-(PQ)^{m-n})x\\rangle\\\\\n&= \\langle ((QPQ)^{n}-(QPQ)^{m})x,((QPQ)^{n}-(QPQ)^{m})x\\rangle\\\\\n&= \\langle (QPQ)^{n}(1-(QPQ)^{m-n})x,(QPQ)^{n}(1-(QPQ)^{m-n})x\\rangle\\\\\n&\\geq \\langle (PQ)^{n+1}(1-(QPQ)^{m-n})x,(PQ)^{n+1}(1-(QPQ)^{m-n})x\\rangle\\\\\n&\\geq \\langle( (PQ)^{n+1}-(PQ)^{m+1})x,((PQ)^{n+1}-(PQ)^{m+1})x\\rangle.\n\\end{align*}\nThis proves that $(PQ)^{n}$ is pointwise Cauchy if and only if $(QPQ)^{n}$ is pointwise Cauchy. Thus $(PQ)^{n}$ and $(QP)^{n}$ are both $*$-strongly Cauchy if and only if $(PQP)^{n}$ and $(QPQ)^{n}$ are both $*$-strongly Cauchy. The statements for the norm topology follow from the same inequalities. This completes the proof.\n\\end{proof}\n\\begin{prop} \n\\label{Cauchy} \nSuppose that $(PQ)^{n}$ is $*$-strongly Cauchy. Then so are $(QP)^{n}$, $(PQP)^{n}$, $(QPQ)^{n}$, $(Q,P)_{n}$ and $(P,Q)_{n}$. The $*$-strong limits of each of these sequences is a projection $P_{\\Omega}$ with range $\\Omega:=\\textnormal{Ran } P \\cap \\textnormal{Ran } Q$. In particular $\\Omega$ is complemented.\n\\end{prop}\n\\begin{proof} \nSince $((PQ)^{n})^{*}=(QP)^{n}$, the first statement follows from Lemma \\ref{reductionlemma}. \nWe will prove that $s-\\lim_{n\\to\\infty} (PQP)^n=s-\\lim_{n\\to \\infty} (QPQ)^{n}$ and that this operator is a projection $P_{\\Omega}$ with range $\\Omega$. It then follows that $\\Omega$ is complemented and that \n\\[\nP_{\\Omega}=s-\\lim_{n\\to\\infty}(P,Q)_{n}=s-\\lim_{n\\to\\infty}(Q,P)_{n},\n\\]\nsince $(PQP)^{n}$ is a subsequence of $(Q,P)_{n}$ and $(QPQ)^{n}$ is a subsequence of $(P,Q)_{n}$. Then $(PQ)^{n}$ and $(QP)^{n}$ are subsequences of $(P,Q)_{n}$ and $(Q,P)_{n}$, respectively it follows that\n\\[\nP_{\\Omega}=s-\\lim_{n\\to\\infty}(PQ)^{n}=s-\\lim_{n\\to\\infty}(QP)^{n},\n\\]\nas well.\n\nBy Lemma \\ref{complete} the $*$-strong limit $\\tilde{P}:=\\lim (PQP)^{n}$ exists, is self-adjoint and $\\|\\tilde{P}\\|\\leq 1$. To prove that $\\tilde{P}$ is a projection let $x\\in X$ and $\\varepsilon >0$. Choose $N$ such that for all $k\\geq N$ we have\n\\[\n\\|\\tilde{P}x-(PQP)^{k}x\\|<\\varepsilon.\n\\]\nNow consider\n\\begin{align*}\n\\|\\tilde{P}^{2}x-\\tilde{P}x\\|\n&=\\|\\tilde{P}(PQP)^{k}x-\\tilde{P}x\\|+ \\|\\tilde{P}(\\tilde{P}-(PQP)^{k})x\\|\\\\\n&\\leq \\|\\tilde{P}(PQP)^{k}x-\\tilde{P}x\\|+ \\|(\\tilde{P}-(PQP)^{k})x\\|\\\\\n&< \\|\\tilde{P}(PQP)^{k}x-\\tilde{P}x\\|+ \\varepsilon\\\\\n&=\\lim_{n\\to\\infty}\\|(PQP)^{n+k}x-\\tilde{P}x\\|+\\varepsilon=\\varepsilon,\n\\end{align*}\nand as $\\varepsilon$ was arbitrary, it follows that $\\tilde{P}^{2}x=\\tilde{P}x$.\n\nTo prove that $\\textnormal{Ran }\\tilde{P}=\\Omega$, first observe that if $x\\in \\Omega$ then $$x=Px=Qx=PQPx,$$ so $\\tilde{P}x=x$ and $\\Omega\\subset \\textnormal{Ran } \\tilde{P}$. \n\nFor the reverse inclusion we will show that $\\tilde{P}=P\\tilde{P}=Q\\tilde{P}$. The equalities \n\\[\nP\\tilde{P}x=\\tilde{P}x,\\quad \\textnormal{and}\\quad PQ\\tilde{P}x=\\tilde{P}x,\n\\] \nhold by construction. Now for any $x\\in X$ we have\n\\[\n\\langle Px, Px\\rangle\\leq \\langle x,x\\rangle,\\quad\\langle Qx, Qx\\rangle\\leq \\langle x,x\\rangle,\n\\]\n from which we deduce that\n \\[\n \\langle\\tilde{P}x,\\tilde{P}x\\rangle=\\langle PQ\\tilde{P}x,PQ\\tilde{P}x\\rangle\\leq \\langle Q\\tilde{P}x,Q\\tilde{P}x\\rangle\\leq  \\langle\\tilde{P}x,\\tilde{P}x\\rangle.\n \\]\nTherefore $\\langle Q\\tilde{P}x,Q\\tilde{P}x\\rangle= \\langle\\tilde{P}x,\\tilde{P}x\\rangle$ and $\\langle (1-Q)\\tilde{P}x,(1-Q)\\tilde{P}x\\rangle=0$. It follows that $(1-Q)\\tilde{P}x=0$ so $Q\\tilde{P}x=\\tilde{P}x$. This shows that $Q\\tilde{P}=\\tilde{P}$ and thus $\\textnormal{Ran } \\tilde{P}\\subset\\Omega$. Therefore $\\Omega$ is complemented and $P_{\\Omega}=\\tilde{P}=s-\\lim (PQP)^{n}$ in the $*$-strong module topology. By exhanging the r\\^oles of $P$ and $Q$, we find that $P_{\\Omega}=s-\\lim (QPQ)^{n}$ as well.\n\\end{proof}\nIn order to address the appropriate converse to Proposition \\ref{Cauchy}, we need a description of the Banach space dual $X^{*}:=\\mathbb{B}(X,\\mathbb{C})$ of bounded linear functionals on a Hilbert $C^{*}$-module $X$. To this end we first recall the \\emph{dual or conjugate $C^{*}$-module}. \n\nThe space of compact operators $\\mathbb{K}(X,B)$ from $X$ to $B$ is a left $B$-module via $(b\\cdot K)(x):=bK(x)$ and carries a natural left $B\\simeq \\mathbb{K}(B,B)$ valued inner product $\\langle K, L\\rangle:=KL^{*}$. The \\emph{conjugate module} $\\overline{X}$ is defined to be the set $X$ with the conjugate $\\mathbb{C}$-vector space structure, and we write elements of $\\overline{X}$ as $\\overline{x}$ with $x\\in X$. The left $B$-module structure and inner product\n\\[b\\cdot \\overline{x}:=\\overline{xb^{*}},\\quad \\langle \\overline{x},\\overline{y}\\rangle:=\\langle x,y\\rangle.\\]\nThese left Hilbert $C^{*}$-modules over $B$ are isomorphic, by the following well-known theorem \\cite[page 13]{Lance}.\n\\begin{prop}[Riesz-Fr\\'echet theorem for Hilbert $C^{*}$-modules]\nThe map \n\\[\nT:\\overline{X}\\rightarrow \\mathbb{K}(X,B),\\quad \\overline{x}\\mapsto T_{x},\\quad T_{x}(y):=\\langle x,y\\rangle, \\quad x,y\\in X, \n\\]\nis a unitary isomorphism of left Hilbert $C^{*}$-modules over $B$.\n\\end{prop}\nThe dual Banach space of the $C^{*}$-algebra $B$, $B^{*}:=\\mathbb{B}(B,\\mathbb{C})$, is a right Banach $B$-module via\n\\[\n(\\varphi\\cdot b) (a):=\\varphi(ba),\\quad a,b\\in B.\n\\]\nLastly, for a right Banach $B$-module $V$ and a left Banach $B$-module $W$, we denote by $V\\widehat{\\otimes}_{B}W$ the balanced Banach space projective tensor product of $V$ and $W$. We are now ready to recall a result of Schweizer, \\cite[Proposition 3.1]{Schweizer},  giving a complete description of the dual Banach space $X^{*}$ of the module $X$.\n\\begin{prop}\\label{Schweizer} \nLet $X$ be a Hilbert $C^{*}$-module, $\\overline{X}:=\\mathbb{K}(X,B)$ the conjugate module and $X^{*}=\\mathbb{B}(X,\\mathbb{C})$ the dual Banach space of $X$. The  map $\\psi:B^{*}{\\otimes}_{B}^{\\textnormal{alg}}\\overline{X}\\to X^{*}$ given by\n\\[\n\\psi(\\phi\\otimes\\overline{y})(x):=\\phi(\\langle y,x\\rangle), \\quad \\phi\\in B^{*},\\quad x,y\\in X,\n\\]\nextends to an isometric isomorphism $B^{*}\\widehat{\\otimes}_{B}\\overline{X}\\to X^{*}$ of Banach spaces. \n\\end{prop}\nFor a Banach space $W$, the \\emph{weak topology} on $W$ is the locally convex topology defined by the seminorms $\\|w\\|_{\\varphi}:=\\|\\varphi(w)\\|$. In general the weak topology is \\emph{not} complete, that is, weak Cauchy sequences need not have a weak limit in $X$. However, we do have the following fundamental result for weakly convergent sequences. \n\\begin{thm}[{\\cite[Chap II, Section 38]{RSz}}]\n\\label{Mazur}\nLet $W$ be a Banach space and $C\\subset W$ a convex set. Then the weak closure of $C$ coincides with the norm closure of $C$. In particular, if $w_j\\to w$ in the weak topology, then there exists a sequence of convex combinations $y_j:=\\sum_{k=j}^{n_{j}} t_j w_j$ such that $\\|y_j-w\\|\\to 0$.\n\\end{thm}\nIn the sequel we will freely use the following computational tool.\n\\begin{lemma}\n\\label{powers} \nLet $P,Q\\in\\End^{*}_{B}(X)$ be projections such that $\\Omega:=\\textnormal{Ran } P\\cap \\textnormal{Ran } Q$ is complemented. Then for all $k\\geq 1$ we have\n\\begin{align*}\n(PQ-P_{\\Omega})^{k}&=(PQP)^{k}-P_{\\Omega},\\quad (QP-P_{\\Omega})^{k}=(QPQ)^{k}-P_{\\Omega},\\\\\n(PQP-P_{\\Omega})^{k}&=(PQP)^{k}-P_{\\Omega},\\quad (QPQ-P_{\\Omega})^{k}=(QPQ)^{k}-P_{\\Omega}.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nThe statement holds for $k=1$. Since $P_{\\Omega}=P_{\\Omega}P=PP_{\\Omega}=P_{\\Omega} Q=QP_{\\Omega}$ we have \n\\[\n(PQ)^{k+1}-P_{\\Omega}=(PQ-P_{\\Omega})((PQ)^{k}-P_{\\Omega}),\n\\quad (QP)^{k+1}-P_{\\Omega}=(QP-P_{\\Omega})((QP)^{k}-P_{\\Omega}),\n\\]\nand\n\\[(PQP)^{k+1}-P_{\\Omega}=P((QP)^{k+1}-P_{\\Omega}),\\quad (QPQ)^{k+1}-P_{\\Omega}=Q((PQ)^{k+1}-P_{\\Omega}),\\]\nso the result follows by induction on $k$.\n\\end{proof}\nWe are now ready to prove our main theorem.\n\n\\begin{thm}\n\\label{complementedalternating} \nLet $M,N$ be complemented submodules of a Hilbert $C^{*}$-module $X$. Then $(M,N)$ is a concordant pair if and only if the sequence $(P_{N}P_{M})^{n}$ is Cauchy in the $*$-strong module topology on $\\End^{*}_{B}(X)$.\n\\end{thm}\n\\begin{proof} \nWe write $P=P_{M}$, $Q=P_{N}$ and $\\Omega:=M\\cap N$.\n\n\n$\\Leftarrow$ In Proposition \\ref{Cauchy} it was proved that $\\Omega$ is complemented and $\\lim (PQ)^{n}x=P_{\\Omega}x$. Now if $\\pi:B\\to\\mathbb{B}(H_{\\pi})$ is an irreducible representation then\n\\begin{align*}\nP_{M_{\\pi}\\cap N_{\\pi}}(x\\otimes h)&=\\lim_{n\\to\\infty}(P_{M_{\\pi}}P_{N_{\\pi}})^{n}(x\\otimes h)=\\lim_{n\\to\\infty}\\widehat{\\pi}(P_{M}P_{N})^{n}(x\\otimes h)\\\\\n&=\\lim_{n\\to\\infty}((PQ)^{n}x)\\otimes h=P_{\\Omega}x\\otimes h=\\widehat{\\pi}(P_{\\Omega})(x\\otimes h),\n\\end{align*}\nso $P_{M_{\\pi}\\cap N_{\\pi}}=\\widehat{\\pi}(P_{\\Omega})$ and thus $\\Omega_{\\pi}=M_{\\pi}\\cap N_{\\pi}$, so $(M,N)$ is concordant by Theorem \\ref{locharm}.\n\nFor the converse, assume that $(M,N)$ is concordant and write $P_{\\Omega}$ for the projection onto $\\Omega$. By Lemma \\ref{reductionlemma} it suffices to prove that $(PQP)^{n}x\\to P_{\\Omega} x$ and $(QPQ)^{n}x\\to P_{\\Omega} x$ for all $x\\in X$. \n\nWe first prove that $(PQP)^{n}x$ converges to $P_{\\Omega}x$ in the weak topology on $X$. To this end observe that since $\\|(PQP)^{n}\\|\\leq \\|PQP\\|^{n}\\leq 1$ the sequence $(PQP)^{n}x$ is bounded in norm. Therefore, by Proposition \\ref{Schweizer}  it suffices to show that $(\\phi\\otimes \\overline{y}) ( (PQP)^{n}x)\\to (\\phi\\otimes \\overline{y})(P_{\\Omega}x)$ for all $\\phi\\in B^{*}$ and $y\\in X$, as such functionals generate the weak topology. Since every functional on the $C^{*}$-algebra $B$ is a linear combination of four states (see \\cite{T}), we may restrict ourselves to states $\\sigma\\in B^{*}$. In the universal representation $H_{u}$ of $B$, every state $\\sigma$ arises as a vector state associated to a unit vector $h_{\\sigma}\\in H_{u}$. Denote by $\\pi_{\\sigma}$ the GNS-representation associated to the state $\\sigma$. Then by Theorem \\ref{vN} we find\n\\begin{align*}\n(\\sigma\\otimes \\overline{y})((PQP)^{n}x)&=\\sigma(\\langle y, (PQP^{n})x\\rangle )=\\langle h_{\\sigma}, \\langle y, (PQP)^{n}x\\rangle h_{\\sigma}\\rangle\\\\\n&=\\langle y\\otimes h_{\\sigma}, (PQP)^{n}x\\otimes h_{\\sigma}\\rangle \\to \\langle y\\otimes h_{\\sigma}, P_{\\Omega_{\\sigma}} (x\\otimes h_{\\sigma})\\rangle.\n\\end{align*}\nBy Corollary \\ref{locharmcor}, $M_{\\pi_\\sigma}\\cap N_{\\pi_\\sigma}=\\Omega_{\\pi_\\sigma}$ so $P_{\\Omega_{\\sigma}}=P_{\\Omega}\\otimes 1=\\widehat{\\pi}_{\\sigma}(P_{\\Omega})$,\nand\n$(PQP)^{n}x\\otimes h_{\\sigma}\\to P_{\\Omega} x\\otimes h_{\\sigma}$ in the Hilbert space $X\\otimes_{B} H_{u}$. Therefore $(PQP)^{n}x\\to P_{\\Omega}x$ weakly in $X$. \n\nBy Theorem \\ref{Mazur}, there is a sequence of convex combinations $y_{k}=\\sum_{i=k}^{n_{k}} t_{i}(PQP)^{i}x$ such that $y_k\\to P_{\\Omega}x$ in norm in $X$. Since for all $n$ we have\n\\[\nP_{\\Omega}(PQP)^{n}=(PQP)^{n}P_{\\Omega}=P_{\\Omega},\\quad (PQP)^{m}\\leq (PQP)^{n},\\quad m\\geq n,\n\\]\nwe can estimate\n\\begin{align*}\n\\langle (y_{k}-P_{\\Omega})x,  (y_{k}-P_{\\Omega})x\\rangle&=\n\\left\\langle \\left(\\sum_{i=k}^{n_{k}} t_{i}(PQP)^{i}-P_{\\Omega}\\right)x , \\left(\\sum_{i=k}^{n_{k}} t_{i}(PQP)^{i}-P_{\\Omega}\\right)x\\right\\rangle\\\\ \n&=\\left\\langle \\left(\\sum_{i=k}^{n_{k}} t_{i}(PQP)^{i}-P_{\\Omega}\\right)^{2}x , x\\right\\rangle\\\\\n&=\\left\\langle \\left(\\sum_{i,j=k}^{n_{k}}t_{i}t_{j}(PQP)^{i+j}-P_{\\Omega}\\right)x, x\\right\\rangle \\\\\n&\\geq \\left\\langle \\left(\\sum_{i,j=k}^{n_k}t_{i}t_{j}(PQP)^{2n_{k}}-P_{\\Omega}\\right)x, x\\right\\rangle \\\\\n&=\\langle ((PQP)^{2n_k}-P_{\\Omega})x, x\\rangle \\\\%-\\langle P_{\\Omega}x, P_{\\Omega}\\rangle\n&=\\langle ((PQP)^{n_k}-P_{\\Omega})x, ((PQP)^{n_k}-P_{\\Omega})x\\rangle,\n\\end{align*}\nwhere the last step follows using Lemma \\ref{powers}. \nTherefore it follows that the subsequence $(PQP)^{n_{k}}$ is such that for all $x\\in X$  we have norm convergence $(PQP)^{n_{k}}x\\to P_{\\Omega}x$ as $k\\to \\infty$. Since for any $m\\geq n$ we have \n\\[\n\\langle ((PQP)^{n}-P_{\\Omega})x, ((PQP)^{n}-P_{\\Omega})x\\rangle \\geq\\langle( (PQP)^{m}-P_{\\Omega})x,( (PQP)^{m}-P_{\\Omega})x\\rangle,\n\\]\nwe find that \n\\begin{align*}\n\\|((PQP)^{n}-P_{\\Omega})x\\|\\geq \\|((PQP)^{m}-P_{\\Omega})x\\|.\n\\end{align*}\nThus it follows that $\\lim_{n\\to\\infty} \\|((PQP)^{n}-P_{\\Omega})x\\|\\to 0$. By swapping the r\\^oles of $P$ and $Q$ we find that $\\lim_{n\\to\\infty} \\|((QPQ)^{n}-P_{\\Omega})x\\|\\to 0$ as well. This completes the proof.\n\\end{proof}\n\n\\section{Angle, sum and intersection}\n\\label{sec:angle}\nWe now consider the applications of our main result to various problems concerning pairs of complemented submodules of Hilbert $C^*$-modules.\n\\subsection{The Friedrichs angle between complemented submodules}\nIn \\cite{Luo}, the following definition for the Friedrichs angle between complemented submodules was given, which we now recall. Let $M,N\\subset X$  be complemented submodules such that $M\\cap N$ is complemented and write $P_{M},P_{N}$ and $P_{M\\cap N}$ respectively for the corresponding projections. The quantity\n\\begin{equation}\n\\label{moduleangle}\nc(M,N):=\\|P_{M}P_{N}(1-P_{M\\cap N})\\|=\\|P_{M}P_{N}-P_{M\\cap N}\\|,\n\\end{equation}\nis called the (cosine of the) \\emph{Friedrichs angle between $M$ and $N$}. \n\nFor the above definition, the existence of the projection $P_{\\Omega}$ seems necessary. This is undesirable and ideally the angle should be an invariant associated to any pair $(M,N)$ of complemented submodules. We propose the following generalisation, based on Hilbert space localisation.  \n\\begin{defn} \nLet $M,N\\subset X$  be complemented submodules. Let $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ be a representation of $B$ on $H_{\\pi}$. The quantity\n\\begin{equation}\\label{localangle}\nc_{\\pi}(M,N):=c(M_{\\pi},N_{\\pi})=\\|P_{M_{\\pi}}P_{N_{\\pi}}(1-P_{M_{\\pi}\\cap N_{\\pi}})\\|=\\|P_{M_{\\pi}}P_{N_{\\pi}}-P_{M_{\\pi}\\cap N_{\\pi}}\\|,\n\\end{equation}\nis called the (cosine of the) \\emph{local Friedrichs angle between $M$ and $N$ at $\\pi$}. \n\\end{defn}\n\n\\begin{prop}\n\\label{independent} Suppose that $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ is faithful. Then\n\\begin{enumerate}\n\\item If $(M,N)$ is concordant, then $c_{\\pi}(M,N)=c(M,N)$;\n\\item If $(M,N)$ is discordant, then $c_{\\pi}(M,N)=1$.\n\\end{enumerate}\nIn particular the (cosine of the) local Friedrich angle $c_{\\pi}(M,N)$ is independent of the choice of faithful representation $\\pi$.\n\\end{prop}\n\\begin{proof} Suppose $(M,N)$ is concordant, so that by Corollary \\ref{locharmcor}, $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$ and $P_{M_{\\pi}\\cap N_{\\pi}}=\\widehat{\\pi}(P_{M\\cap N})$. \n Since $\\pi$ is faithful, the representation $\\End^{*}_{B}(X)\\to \\mathbb{B}(X_{\\pi})$ is faithful and hence isometric. Therefore \n\\[c_{\\pi}(M,N)=\\|\\widehat{\\pi}(P_{N}P_{M}-P_{M\\cap N})\\|=\\|P_{N}P_{M}-P_{M\\cap N}\\|=c(M,N),\\]\nwhich proves {\\em1}.\n\nClearly $0\\leq c_{\\pi}(M,N)\\leq 1$, so suppose that $c_{\\pi}(M,N)<1$ and write $P=P_{M}$ and $Q=P_{N}$. We will show that the sequence $(PQ)^{n}$ is Cauchy for the norm topology. Then by Theorem \\ref{complementedalternating}, $(M,N)$ is concordant, which proves {\\em2}. So for $m\\geq n$ recall the representation $\\widehat{\\pi}$ from Equation \\eqref{eq:bob} and consider\n\\begin{align*}\n\\|(PQ)^{n}-(PQ)^{m}\\|&=\\|\\widehat{\\pi}((PQ)^{n}-(PQ)^{m})\\|\\\\\n&\\leq \\|\\widehat{\\pi}(PQ)^{n}-P_{M_{\\pi}\\cap N_{\\pi}}\\|+\\|\\widehat{\\pi}(PQ)^{m}-P_{M_{\\pi}\\cap N_{\\pi}}\\|\\\\\n&=\\|(P_{M_\\pi}P_{N_\\pi})^{n}-P_{M_{\\pi}\\cap N_{\\pi}}\\|+\\|(P_{M_\\pi}P_{N_\\pi})^{m}-P_{M_{\\pi}\\cap N_{\\pi}}\\|\\\\\n&=\\|(P_{M_\\pi}P_{N_\\pi}-P_{M_{\\pi}\\cap N_{\\pi}})^{n}\\|+\\|(P_{M_\\pi}P_{N_\\pi}-P_{M_{\\pi}\\cap N_{\\pi}})^{m}\\|\\quad(\\textnormal{by Lemma \\ref{powers}})\\\\\n&\\leq \\|P_{M_\\pi}P_{N_\\pi}-P_{M_{\\pi}\\cap N_{\\pi}}\\|^{n}+\\|P_{M_\\pi}P_{N_\\pi}-P_{M_{\\pi}\\cap N_{\\pi}}\\|^{m}\\\\\n&=c_{\\pi}(M,N)^{n}+c_{\\pi}(M,N)^{m}\\to 0,\n\\end{align*}\nsince $c_{\\pi}(M,N)<1$. This completes the proof.\n\\end{proof}\nWe denote by $\\widehat{B}$ the space of unitary equivalence classes of irreducible representations of the $C^{*}$-algebra $B$, by $\\mathcal{P}(B)$ the pure state space of $B$ and by $\\pi_{\\sigma}$ the $GNS$-representation associated to the state $\\sigma$. We can view the local Friedrichs angles as a function $\\widehat{B}\\to [0,1]$ and via the composition $\\mathcal{P}(B)\\to \\widehat{B}$, also as a function on $\\mathcal{P}(B)$.\n\\begin{corl} \\label{localglobalangle} The Friedrichs angle \\eqref{moduleangle} and the local Friedrichs angles \\eqref{localangle} are related by $c(M,N)=\\sup_{\\pi\\in\\widehat{B}}c_{\\pi}(M,N)=\\sup_{\\sigma\\in\\mathcal{P}(B)}c_{\\pi_{\\sigma}}(M,N)$.\n\\end{corl}\n\\begin{proof}\nThe representations $\\widehat{H}=\\bigoplus_{\\pi\\in\\widehat{B}}H_{\\pi}$ and $H_{\\mathcal{P}}:=\\bigoplus_{\\sigma\\in\\mathcal{P}(B)}H_{\\pi_{\\sigma}}$ are faithful.\n\\end{proof}\nIn view of Proposition \\ref{independent} and Corollary \\ref{localglobalangle}, we \\emph{define} the Friedrichs angle between an arbitrary pair of complemented submodules to be  $c(M,N):=c_{\\pi}(M,N)$, with $\\pi$ faithful. \nIt was shown in \\cite{Luo} that \n\\begin{equation}\n\\label{Deutschangle}\nc(M,N)=c(M^{\\perp},N^{\\perp}),\n\\end{equation}\nprovided that $M\\cap N$ and $M^{\\perp}\\cap N^{\\perp}$ are complemented. In particular, the equality holds for any pair of subspaces of a Hilbert space, \\cite[Theorem 2.16]{D}. \nWe will now show that the equality \\eqref{Deutschangle} holds for an arbitrary pair of complemented submodules. This gives an extension, and a different proof, of \\cite[Theorem 5.12]{Luo}.\n\\begin{thm}\n\\label{anglesymmetry}\nLet $X$ be a Hilbert $C^{*}$-module and $M,N\\subset X$ complemented submodules. Then $c(M,N)$ is well-defined and  $c(M,N)=c(M^{\\perp},N^{\\perp})$.\n\\end{thm}\n\\begin{proof}\nFor any representation $\\pi:B\\to\\mathbb{B}(H_{\\pi})$ there is an equality of submodules $(M_{\\pi})^{\\perp}=(M^{\\perp})_{\\pi}$ whenever $M$ is complemented. Moreover Equation \\eqref{Deutschangle} holds for the subspaces $M_{\\pi},N_{\\pi}$ of the Hilbert space $X_{\\pi}$. Thus by Proposition \\ref{independent} we have\n\\begin{align*}\nc(M,N)&=c_{\\pi}(M,N)=c(M_{\\pi},N_{\\pi})\\\\ &=c((M_{\\pi})^{\\perp},(N_{\\pi})^{\\perp})=c((M^{\\perp})_{\\pi},(N^{\\perp})_{\\pi})\\\\&=c_{\\pi}(M^{\\perp},N^{\\perp})=c(M^{\\perp},N^{\\perp}),\\end{align*}\nas claimed.\n\\end{proof}\nNow we further analyse the properties of the local Friedrichs angles as a function on $\\widehat{B}$.\n\\begin{prop}\n\\label{continuity}\nSuppose $(M,N)$ is concordant. Then the map\n\\[\\widehat{B} \\to [0,1],\\quad\\pi\\mapsto c_{\\pi}(M,N),\\]\nis lower semi-continuous. If $X$ is full and $\\widehat{B}$ is Hausdorff, $\\pi\\mapsto c_{\\pi}(M,N)$ is continuous.\n\\end{prop}\n\\begin{proof} Let $J:=\\left\\langle B,B\\right\\rangle$ and $\\widehat{B}\\to\\widehat{J}$ the restriction map, which is continuous. The $C^{*}$-algebras $J$ and $\\mathbb{K}(X)$ are Morita equivalent, so by the Rieffel correspondence \\cite{RieffelInd} the map $\\pi\\mapsto\\widehat{\\pi}$ is a homeomorphism $\\widehat{J} \\to \\widehat{\\mathbb{K}(X)}$. Since $\\mathbb{K}(X)\\subset \\End^{*}_{B}(X)$ is an essential ideal, there is a continuous inclusion $\\widehat{\\mathbb{K}(X)}\\to \\widehat{\\End^{*}_{B}(X)}$, see \\cite[Section 2]{Dauns}. When $(M,N)$ is concordant the map $\\pi\\mapsto c_{\\pi}(M,N)$ can be written as a composition\n\\[\\pi\\mapsto\\widehat{\\pi}\\mapsto \\|\\widehat{\\pi}(P_{M}P_{N}-P_{M\\cap N})\\|,\\]\nand is thus lower semicontinuous by \\cite[Lemma A.30]{RW}. For $X$ full and $\\widehat{B}$ Hausdorff, continuity follows by \\cite[Lemma 5.2]{RW}.\n\\end{proof}\n\\begin{corl} \nSuppose $X$ is full, $B$ is unital, $\\widehat{B}$ is Hausdorff and $(M,N)$ is concordant. Then $c(M,N)<1$ if and only if $c_{\\pi}(M,N)<1$ for every irreducible representation $\\pi$.\n\\end{corl}\n\\begin{proof} \nSince $\\widehat{B}$ is compact Hausdorff and the Friedrichs angle is continuous, \nthe pointwise estimate  $c_{\\pi}(M,N)<1$ implies that $c(M,N)=\\sup_{\\pi\\in\\widehat{B}} c_{\\pi}(M,N) <1$.\n\\end{proof}\n\\begin{rmk}\n\\label{discontangle}\nIn Proposition \\ref{continuity}, the condition that $(M,N)$ be concordant cannot be relaxed. Consider $B=C([0,\\frac{\\pi}{2}])$, $X=C([0,\\frac{\\pi}{2}], \\mathbb{C}^{2})$ and consider the submodules\n\\[\nM=\\textnormal{Ran } \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix},\\quad N\n= \\textnormal{Ran } \\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix}.\n\\]\nFor $t\\in [0,\\pi\/2]$ we write $c_{t}(M,N)$ for the Friedrichs angle at $t$. For $0<t\\leq \\frac{\\pi}{2}$ we have $M_{t}\\cap N_{t}=\\{0\\}$ whereas at $t=0$ we have $M_0=N_0$. We thus have\n\\[\nc_{t}(M,N)^{2}=\\|P_{M_{t}}P_{N_{t}}\\|^{2}=\\left\\|\\begin{pmatrix}\\cos^{2}t & \\sin t \\cos t \\\\ 0 & 0\\end{pmatrix}\\right\\|^{2}\n=\\cos^{2}t <1,\\quad 0<t\\leq \\frac{\\pi}{2},\n\\]\nand $c_{t}(M,N)=|\\cos t|$, whereas $c_0(M,N)=0$. Thus the angle is discontinuous at $0$ and in particular we conclude once more that $(M,N)$ is discordant. This is another instance where the universal example (see Remark \\ref{rmkuniversal}) provides a counterexample to a specific property. \n\n\\end{rmk}\n\n\n\\subsection{Sum and intersection}\n\\label{sec: sumint}\nWith our extended definition of the Friedrichs angle we now examine the case $c(M,N)<1$ in more detail. Our results \nare all derived from results in \\cite{Luo}, where complementability of $M\\cap N$ is an assumption. We first recall the following well-known fundamental result and a relevant corollary.\n\\begin{thm}\\label{range} Let $T\\in \\End^{*}_{B}(X)$, then\n\\[\\overline{\\textnormal{Ran}(T)}=\\overline{\\textnormal{Ran}(TT^{*})},\\]\nand $T$ has closed range if and only if $T^{*}$ has closed range. If $T$ has closed range then\n\\[\\textnormal{Ran}(T)=\\textnormal{Ran}(TT^{*}),\\]\nand $\\textnormal{Ran}(T)$ is a complemented submodule of $X$ with $\\textnormal{Ran}(T)^{\\perp}=\\ker T^{*}$.\n\\end{thm}\n\\begin{proof}\nSee \\cite[Theorem 3.2, Proposition 3.7]{Lance}. \n\\end{proof}\n\\begin{corl}[\\cite{Luo}, Remark 5.8]\\label{sumrange}\nLet $P,Q$ be adjointable projections on a Hilbert $C^{*}$-module $X$, then \n\\[\\overline{\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)}=\\overline{\\textnormal{Ran}(P+Q)}.\\]\nIn particular\n$\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)$ is closed if and only if $\\textnormal{Ran}(P+Q)$ is closed and in that case\n\\[\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)=\\textnormal{Ran}(P+Q),\\]\nwhich is a complemented submodule of $X$.\n\\end{corl}\n\\begin{proof}\nBy the following well-known observation\n\\[\n\\begin{pmatrix} 0 & 0 \\\\ 0 & P+Q\\end{pmatrix}=\\begin{pmatrix}0 & 0 \\\\ P & Q\\end{pmatrix}\\cdot \\begin{pmatrix}0 & P \\\\ 0 & Q\\end{pmatrix}=\\begin{pmatrix}0 & 0 \\\\ P & Q\\end{pmatrix}\\cdot \\begin{pmatrix}0 & 0 \\\\ P & Q\\end{pmatrix}^{*},\n\\]\nas operators on $X\\oplus X$, it follows that $\\overline{\\textnormal{Ran}(P+Q)}=\\overline{\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)}$ by Theorem \\ref{range}. Therefore,\n $\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)$ is closed if and only if $\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)=\\textnormal{Ran}(P+Q)$ is closed.\n\\end{proof}\n\\begin{lemma}\n\\label{kernel}\nLet $P,Q$ be adjointable projections on a Hilbert $C^{*}$-module $X$, then \n\\[\\ker(1-PQ)=\\ker(1-QP)=\\textnormal{Ran } P\\cap \\textnormal{Ran } Q.\\]\n\\end{lemma}\n\\begin{proof} It is clear that $\\textnormal{Ran } P\\cap \\textnormal{Ran } Q\\subset \\ker (1-PQ)\\cap \\ker (1-QP)$. We prove the reverse inclusion.\nSuppose that $x\\in \\ker(1-PQ)$, so that $x=PQx$. Then clearly $x=Px$ and \n\\[\\langle Qx, Qx\\rangle=\\langle PQx, PQx\\rangle+\\langle (1-P)Qx,(1-P)Qx\\rangle\\geq \\langle PQx,PQx\\rangle=\\langle x,x\\rangle,\\]\nand thus $(1-Q)x=0$ so $x=Qx$. By reversing the roles of $P$ and $Q$ this shows that\n\\[\\ker(1-PQ)=\\ker(1-QP)=\\{x\\in X: x=Px=Qx\\}=\\textnormal{Ran } P\\cap \\textnormal{Ran } Q,\\]\nas claimed.\\end{proof}\nLastly we recall the following remarkable result from \\cite{LSX}, concerning the case where the internal sum $M+N$ is closed.\n\\begin{prop}[{\\cite[Proposition 4.6]{LSX}}]\\label{closedsum} Let $M$ and $N$ be complemented submodules of $X$, such that $M + N$ is closed in $X$. Then both $M + N$ and\n$M \\cap N$ are  complemented and $X=(M\\cap N)\\oplus (M^{\\perp}+N^{\\perp})$.\n\\end{prop}\nIn the above result, complementability of $M\\cap N$ is concluded as opposed to assumed, and we now characterise closedness of $M+N$ in terms of our extended definition of $c(M,N)$, as well as in terms of properties of the operators $1-P_{M}P_{N}$ and $1-P_{N}P_{M}$.\n\\begin{prop}\n\\label{prop: closed range}\nLet $X$ be a Hilbert $C^{*}$-module over a $C^{*}$-algebra $B$, and $M$ and $N$ complemented submodules. \nThen the following are equivalent: \n\\begin{enumerate}\n\\item $c(M,N)<1$;\n\\item The sequence $(P_{M}P_{N})^{n}$ is Cauchy for the operator norm;\n\\item $M\\cap N$ is complemented and $ \\| P_{M}P_{N}-P_{M\\cap N} \\| = \\|P_{N}P_{M}-P_{M\\cap N}\\|<1$;\n\\item $M\\cap N$ is complemented and the operators $1-P_{M}P_{N}:(M\\cap N)^{\\perp}\\to (M\\cap N)^{\\perp}$ and $1-P_{M}P_{N}:(M\\cap N)^{\\perp}\\to (M\\cap N)^{\\perp}$ are bijective; \n\\item $\\textnormal{Ran}(1-P_{M}P_{N})$ and $\\textnormal{Ran}(1-P_{N}P_{M})$ are closed;\n\\item $X=(M\\cap N)\\oplus (M^{\\perp}+N^{\\perp})$;\n\\item $M^{\\perp}+N^{\\perp}$ is closed;\n\\item $X=(M^{\\perp}\\cap N^{\\perp})\\oplus (M+N)$;\n\\item $M+N$ is closed;\n\\item $M\\cap N$ is complemented and $(M\\cap N)^{\\perp}\\cap M+(M\\cap N)^{\\perp}\\cap N$ is closed.\n\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nWe write $P=P_{M}$, $Q=P_{N}$ and $\\Omega:=M\\cap N$.\n\n\n$1.\\Rightarrow 2.$ This was shown in the proof of Proposition \\ref{independent}.\n\n$2.\\Rightarrow 3.$ By Theorem \\ref{complementedalternating}, $\\Omega$ is complemented and $(PQ)^{n}\\to P_{\\Omega}$ in norm. By Lemma \\ref{reductionlemma}, $(QPQ)^{n}\\to P_{\\Omega}$ in norm as well. Thus for $n$ sufficiently large $\\|(QPQ)^{2^n}-P_{\\Omega}\\|<1$. Then \napplying Lemma \\ref{powers} and the $C^{*}$-identity we find\n\\begin{align*}\n\\|(QPQ)^{2^n}-P_{\\Omega}\\|=\\|(QPQ-P_{\\Omega})^{2^{n}}\\|=\\|QPQ-P_{\\Omega}\\|^{2^{n}},\n\\end{align*}\nand it follows that $\\|QPQ-P_{\\Omega}\\|<1$. Now, again by the $C^{*}$-identity, \\[\\|PQ-P_{\\Omega}\\|^{2}=\\|(QP-P_{\\Omega})(PQ-P_{\\Omega})\\|=\\|QPQ-P_{\\Omega}\\|,\\] so we find that $\\|PQ-P_{\\Omega}\\|<1$.\n\n$3.\\Rightarrow 4.$ Since $\\| PQ-P_{\\Omega}\\|<1$. We find that the series\n\\[\n\\sum_{n=0}^{\\infty}(PQ-P_{\\Omega})^{n}=P_{\\Omega}+\\sum_{n=0}^{\\infty}((PQ)^{n}-P_{\\Omega})\\quad\\textnormal{(by Lemma \\ref{powers})}, \n\\]\nis norm convergent to $(1+P_{\\Omega}-PQ)^{-1}$. Since $\\Omega=\\ker (1-PQ)$ by Lemma \\ref{kernel}, $1+P_{\\Omega}-PQ$ is bijective and commutes with $P_{\\Omega}$, it follows that it maps $\\Omega^{\\perp}$ bijectively onto itself. The same argument applies to $QP$.\n\n\n$4.\\Rightarrow 5.$ Since $X=\\Omega\\oplus \\Omega^{\\perp}$ and, by Lemma \\ref{kernel}, $\\Omega=\\ker (1-PQ)=\\ker (1-QP)$ it follows that \n\\[\\textnormal{Ran}(1-PQ)=\\textnormal{Ran}(1-QP)=\\Omega^{\\perp},\\] is closed.\n\n$5.\\Rightarrow 6.$\nSince $\\textnormal{Ran}(1-PQ)$ and $\\textnormal{Ran}(1-QP)$ are closed, they are complemented by Theorem \\ref{range}. Since $(1-PQ)^{*}=1-QP$, we have\n\\[\\textnormal{Ran}(1-QP) =\\ker(1-PQ)^{\\perp}=\\Omega^{\\perp}=\\ker(1-QP)^{\\perp}=\\textnormal{Ran}(1-PQ),\\]\nand by Lemma \\ref{kernel} $\\Omega=\\ker(1-PQ)=\\ker(1-QP)$.\nSince\n\\[1-PQ=(1-P)Q+1-Q,\\quad 1-QP=(1-Q)P+1-P,\\]\nit follows that \n\\[\n\\Omega^{\\perp}\n=\\textnormal{Ran}(1-PQ)\\subset \\textnormal{Ran}(1-P)+\\textnormal{Ran}(1-Q)\n\\subset \\Omega^{\\perp}.\n\\]\nTherefore\n\\[\n\\Omega^{\\perp}=\\textnormal{Ran}(1-P)+\\textnormal{Ran}(1-Q)=M^{\\perp}+N^{\\perp}.\n\\]\n\n$6.\\Rightarrow 7.$ Since $\\Omega^{\\perp}$ is closed, $M^{\\perp}+N^{\\perp}$ is closed.\n\n$7.\\Rightarrow 8.$ By Proposition \\ref{closedsum} $M^{\\perp}\\cap N^{\\perp}$ is complemented with complement $(M+N)^{\\perp}$\n\n$8.\\Rightarrow 9.$ Since $(M^{\\perp}\\cap N^{\\perp})^{\\perp}$ is closed $M+N$ is closed. \n\n$9.\\Rightarrow 10.$ Since $\\Omega$ is complemented, this follows from \\cite[Lemma 5.11]{Luo}, and Corollary \\ref{sumrange}  and Proposition \\ref{closedsum} above. \n\n$10.\\Rightarrow 1.$ By \\cite[Lemma 5.11]{Luo} we have $c(M^{\\perp}, N^{\\perp})<1$, so by Theorem \\ref{anglesymmetry} we have  $c(M,N)<1$.\n\\end{proof}\n\\begin{corl} Let $M,N\\subset X$ be complemented submodules. Then $c(M,N)<1$  if and only if $M\\cap N$ is complemented and $M+N$ is closed if and only if $M^{\\perp}\\cap N^{\\perp}$ is complemented and $M^{\\perp}+N^{\\perp}$ is closed.\n\\end{corl}\n\\begin{rmk}\n\\label{Atiyah}\nLet $E$ and $F$ be vector bundles over a compact Haursdorff space $Y$. In \\cite[Definition 1.3.2]{Atiyah} a vector bundle morphism $\\varphi:E\\to F$ is called \\emph{strict} if the map $y\\mapsto \\dim\\ker\\varphi_{y}$ is continuous, hence locally constant, on $Y$. By \\cite[Proposition 1.32]{Atiyah}, if $\\varphi$ is strict then $\\bigcup_{y\\in Y}\\ker \\varphi_{y}$ is a subbundle of $E$, and thus its module of sections is complemented.\n\nAs in Remarks \\ref{eg:VB-unstrict} and \\ref{discontangle} consider $B=C([0,\\frac{\\pi}{2}])$, $X=C([0,\\frac{\\pi}{2}], \\mathbb{C}^{2})$ and consider the submodules\n\\[\nM=\\textnormal{Ran } \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix},\\quad N\n= \\textnormal{Ran } \\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix}.\n\\]\nWe obtain two globally trivial rank one vector bundles over $[0,\\pi\/2]$ whose modules of sections are $M$ and $N$ respectively. \nThe vector bundle homomorphism $\\varphi:X\\to X$ defined by $\\varphi:=(1-P_{M}P_{N})$\nis not strict since\n\\[\n\\ker\\varphi_t=\\ker (1-P_{M_t}P_{N_t})=M_{t}\\cap N_{t},\n\\]\nso $t\\mapsto\\dim\\ker\\varphi_{t}$ is discontinuous at $0$ by Remark \\ref{eg:VB-unstrict}. \n\\end{rmk}\nFor a commutative unital $C^{*}$-algebra $B$ the following corollary to Proposition \\ref{prop: closed range} corresponds to the situation where cosine of the angle between the corresponding sub-vector bundles is $<1$ in which case the bundle endomorphism $1-P_{M}P_{N}$ is strict.\n\\begin{corl}If the $C^{*}$-algebra $B$ is unital and $X$ is finitely generated and projective over $B$, then for any pair $M,N\\subset X$ of complemented submodules with $c(M,N)<1$ both $M\\cap N$ and $M+N$ are finitely generated projective $B$-modules.\n\\end{corl}\n\\begin{proof}\nIf $X$ is finitely generated and projective over the unital algebra $B$, then any complemented submodule is finitely generated and projective, so in particular $M\\cap N$ is. Since $M+N$ is finitely generated and closed in $X$, it is a Hilbert $C^{*}$-module and hence is projective by \\cite[Theorem 1.4.6]{Troitsky}.\n\\end{proof}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe questions raised by the impending observation of\ngravitational radiation can be divided into three broad classes:\ndata analysis, ``anticipatory'' astrophysics, and\n``contributory'' astrophysics. By data analysis I mean the\ntechniques that convert the output of a detector system into\nstatements regarding the likelihood that a source of radiation\nhas been observed and characterizing the nature of the source. In\norder to carry out the data analysis we must draw upon our\nknowledge of the (astro)physics of the sources. The questions so\nposed ({\\em e.g.,\\\/} what are the waveforms, where are the\nsources located, how many sources are there?)  are those I\ncategorize as anticipatory astrophysics.  Finally, the direct\ndetection of gravitational radiation will allow us to add to our\nunderstanding of the physics of gravitation ({\\em e.g.,\\\/} tests\nof general relativity), sources, and astrophysics generally. I\ncategorize the astrophysics questions that can be posed of the\nobservations as contributory astrophysics.\n\nQuestions of anticipatory and contributory astrophysics are\ndefined by what is needed for data analysis (in the first case)\nand what it is possible to learn given the detector output (in\nthe second). Consequently, our first goal must be to understand\nthe data analysis process well enough to define what we need to\ncarry it out and what it is possible to learn given our detection\nsystem.\n\nThe output of a detector is a time series $g(t)$. In any given\ntime interval, $g(t)$ consists either of detector noise alone or\ndetector noise in combination with a gravitational wave signal:\n\\begin{equation}\ng(t) = \\left\\{\\begin{array}{ll}\nn(t)&\\mbox{detector noise;}\\\\\nn(t)+s(t;\\mbox{\\boldmath$\\mu$})&\\mbox{noise + signal.}\n\\end{array}\\right.\n\\end{equation}\nI assume here that the detector response to a source of\nradiation superposes linearly with the detector noise.  The\ndetector response to the gravitational radiation from a\nparticular class of sources, such as inspiraling binaries, is\nrepresented by $s(t;\\mbox{\\boldmath$\\mu$})$. Finally, the\n(vector-valued) parameter {\\boldmath$\\mu$} fully characterizes\nthe radiation from the particular source ({\\em e.g.,\\\/} the\ndistance to the source, the masses of the binary components, {\\em\netc.}). {\\em The goal of data analysis is to determine the\nconditional probability density $P[s(\\mbox{\\boldmath$\\mu$})|g]$ that\nthe signal $s(\\mbox{\\boldmath$\\mu$})$ is present given the detector\noutput $g$.}\n\nUsing Bayes law of conditional probabilities we can express\n$P[s(\\mbox{\\boldmath$\\mu$})|g]$ in terms of $g(t)$, the signal model\n$s(\\mbox{\\boldmath$\\mu$})$, the power spectral density of the detector\nnoise $S_n(f)$, and two {\\em a priori\\\/} probabilities: $P(s)$,\nthe {\\em a priori\\\/} probability that a signal of the class\nrepresented by $s$ is present in $g(t)$, and\n$P(\\mbox{\\boldmath$\\mu$}|s)$, the {\\em a priori\\\/} probability that a\nrandom source of type $s$ is characterized by {\\boldmath$\\mu$}\n(for details, see \\cite{finn92a} and references therein).\n\n\\section{Anticipatory astrophysics for binary inspiral}\n\nThe anticipatory astrophysics questions that must be addressed\nare those that lead to the determination of the {\\em a priori\\\/}\nprobabilities $P(s)$ and $P(\\mbox{\\boldmath$\\mu$}|s)$. In the case of\nbinary inspiral, $s(\\mbox{\\boldmath$\\mu$})$ represents the detector\nresponse to the binary inspiral waveform; thus, the first\nchallenge to theorists is to determine the inspiral radiation\nwaveform $h(\\mbox{\\boldmath$\\mu$})$.\n\nThe waveform $h(\\mbox{\\boldmath$\\mu$})$ is estimated via a\npost-Newtonian (weak field, slow-motion) perturbation expansion\nof the solution to the field equations of general relativity. The\ndetermination of the waveform for coalescing binary systems to\ngreater and greater levels of accuracy has been the focus of Will\nand collaborators (cf.\\ \\cite{wiseman93a} and references\ntherein). An unsettled question is how far the perturbation\nexpansion must be taken before increasing refinements of the\nwaveform are beyond the capability of the detector to\ndistinguish. At present it appears that physical effects that\nenter at the post${}^{5\/2}$Newtonian level will be discerned with\nreasonable probability using the advanced LIGO instrumentation\n\\cite{cutler94a}; consequently, the perturbation expansion needs\nto be carried out to a higher level in order to take the greatest\nadvantage of binary inspiral observations.\n\nTo determine $P(s)$ we are called upon to estimate the rate that\nbinary inspiral occurs throughout the universe. Estimates of this\nrate are largely empirical and based upon our observation of the\nvery few observed binary pulsar systems in our own galaxy and its\nglobular clusters; consequently, on top of the small number\nstatistics of the observed binaries, we must account also for the\nselection effects that lead to the {\\em observed\\\/} binary pulsar\nsystems as an unknown fraction of the total number of binary {\\em\npulsar\\\/} systems, that are (in turn) an unknown fraction of the\ntotal number of binary {\\em neutron star\\\/} systems. To turn this\nrate density into a rate we must integrate over the past light\ncone of the detector. Since LIGO will be sensitive to inspirals\nat cosmological distances, we are also forced to confront our\nuncertain knowledge of source evolution and the kinematics of the\nexpanding universe \\cite{finn93a,chernoff93a}.  Thus, our\nextrapolation from the local neighborhood is very uncertain;\npresently, our best estimate of the rate density of neutron star\n- neutron star inspiral is\n$8\\times10^{-8}\\mbox{Mpc}^{-3}\\mbox{yr}^{-1}$\n\\cite{narayan91a,phinney91a}.\n\nFinally, $P(\\mbox{\\boldmath$\\mu$}|s)$ represents a summary of our\nunderstanding of the range and distribution of neutron star\nmasses (and, to a lesser extent, spin) as they occur in binary\nsystems, as well as the space density of coalescing binaries as a\nfunction of luminosity distance and redshift. Thus we are forced\nto confront and summarize our rather thin knowledge of the\ndistribution of neutron star masses.\n\nOne question that can be addressed without knowledge of either\n$P(\\mbox{\\boldmath$\\mu$}|s)$ or $P(s)$ is the anticipated {\\em\nprecision\\\/} of {\\boldmath$\\mu$} measurements {\\em in the limit\nof high signal-to-noise ratio $\\rho$\\\/} \\cite{finn92a,finn93a}.\nOne conclusion worthy of note is that the combination of binary\ncomponent masses and system redshift ${\\cal\nM}\\equiv\\mu^{3\/5}M^{2\/5}(1+z)$ can be determined to a phenomenal\nprecision: better than a part in $10^{-2}\/\\rho$ for $\\rho\\geq10$\n\\cite{finn93a,cutler94a}.\n\n\\section{Contributory astrophysics}\n\nObservation of binary inspiral in gravitational radiation is an\nespecially exciting prospect. {\\em Exactly because the\nastrophysical issues that we are forced to address in preparing\nfor the data analysis touch upon areas of deep uncertainty and\nignorance, binary inspiral observations will yield an incredible\nwealth of new knowledge.} The impending observations will measure\nthe rate of expansion of the universe entirely independent of any\nexisting techniques and free of the uncertainties that plague the\ncosmic distance ladder \\cite{schutz86a,chernoff93a,markovic93a}.\nNeutron star mass determinations, now possible with high\nprecision only for the handful of binary pulsar systems where\nrelativistic effects are important, will be made for every binary\ninspiral observed --- and, corresponding to the best estimate\nbinary inspiral rate density, $\\sim100\\,\\mbox{yr}^{-1}$ neutron\nstar masses will be measured by the advanced LIGO instruments\nwith signal-to-noise ratio greater than 8. These two examples\nscratch only the surface of the possibilities inherent in LIGO\nobservations.\n\n\\section{Conclusions}\n\nThe study of how the direct detection of gravitational radiation\ncan be exploited to yield a greater understanding of the universe\nhas not yet received the attention it deserves: the technical\nchallenges of detecting the radiation and analyzing the data\nuntil recently have consumed most of the effort of the small\ncommunity of relativists --- both theoretical and experimental\n--- who have been active in confronting that challenge. Binary\ninspiral is the most likely of the anticipated sources to be\nobserved --- given our current understanding of the\nuniverse. Other anticipated sources may be observed first; the\nmore adventurous among us expect earlier observations of entirely\nunanticipated sources.  Regardless, we can all look forward with\nconfidence to the direct detection of gravitational radiation,\nand beyond that to the greater knowledge of the universe that\nthose observations will bring.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\nThe theoretical prediction of gravitational waves (GWs) originates\nin 1893 when Heaviside first discussed the possibility of their\nexistence. In 1916 Einstein predicted their existence\n\\cite{Einstein:1916cc,Einstein:1918btx} in the context of general\nrelativity. In the linearized weak-field approximation, he found\nthat his equations had transverse wave solutions travelling at the\nspeed of light \\cite{Ellis:2016rrr} produced by the time\ndependence of the mass quadrupole moment of the source\n\\cite{LeTiec:2016sgy}. Einstein realized that GW amplitudes would\nbe small and up until 1957, there was debate about the physical\nrelevance of their existence \\cite{Saulson:2010zz}.\\\n\nNevertheless, the discovery of the binary pulsar system PSR\nB1913+16 by Hulse and Taylor \\cite{Hulse:1974eb} and subsequent\nobservations of its energy loss by Taylor and Weisberg\n\\cite{Taylor:1982zz} demonstrated indirectly the existence of GWs.\nThis discovery, along with subsequent related analysis\n\\cite{Press:1972am}, led to the recognition that a possible direct\ndetection and analysis of GWs could reveal interesting properties of various relativistic systems and could also provide new tests of\ngeneral relativity, especially in\nthe strong-field regime.\\\\\n\nRecently, Abbot et al\n\\cite{TheLIGOScientific:2016agk,Abbott:2016blz}, reported the\nfirst direct detection of GWs emitted by\na binary black hole (BBH) system merging to form a single black hole (BH).\nTheir observation provides a direct window to the properties of\nspace time in the strong-field limit and is consistent with\npredictions of general relativity for the nonlinear dynamics of\nhighly disturbed BHs. The announced beautiful discovery is the\nresult of great efforts for a century by several scientists\n\\cite{Abbott:2016blz} (and references therein). It is a great\ninvestigation, because we have now one more window to the Universe\nand one more confirmation of the theory of general relativity.\nSuch GW observations can be used to test the equivalence principle\n\\cite{Kahya:2016prx,Wu:2016igi,Liu:2016edq,Yunes:2016jcc}, test\nthe propagation of GWs\n\\cite{Yunes:2016jcc,Branchina:2016gad,Schreck:2016qiz,Arzano:2016twc,Bicudo:2016pps,Collett:2016dey,Blas:2016qmn,Garcia-Bellido:2016zmj},\ntest the validity of general relativity\n\\cite{Konoplya:2016pmh,Moffat:2016gkd,Vainio:2016qas}, constrain\nearly cosmological phase transitions\n\\cite{Dev:2016feu,Jaeckel:2016jlh}, probe the quantum structure of\nblack holes \\cite{Giddings:2016tla}\nor the connection between dark matter and primordial black holes \\cite{Sasaki:2016jop,Bird:2016dcv,Clesse:2016vqa}. \\\\\n\nThe recent direct discovery of the GWs has been achieved by the\nLIGO\/Virgo collaboration associating the GW 150914 event\n\\cite{TheLIGOScientific:2016agk,Abbott:2016blz,Maselli:2016ekw}\nto the coalescence of a BBH. This binary detection suggests that\nBBH masses and merging rates may be higher than estimated\npreviously. The rates however, are in agreement with more recent\nestimates, obtained with a population synthesis approach,\npredicting the early formation of detectable BBH\n\\cite{Dominik:2012kk,Dominik:2014yma}. Thus, the\nstochastic gravitational waves background (SGWB) produced by merging cosmological BBH sources\ncould be larger than previously assumed \\cite{Maselli:2016ekw} (and references therein) and\nmay be detectable by advanced detectors\n\\cite{TheLIGOScientific:2016wyq}.\\\\\n\nA\nstochastic background of relic gravitational waves (RGWs) is predicted by inflationary models  \\cite{Nakama:2015nea,Nakama:2016enz} and\nhas been well studied \\cite{Starobinsky:1979ty,Abbott:1985cu}. The power spectrum of\nrelic gravitational wave background reflects the physical\nconditions in the early Universe thus providing valuable\ninformation for cosmology \\cite{Dufaux:2007pt}. This spectrum\nis determined by the early stage of inflation as well as by the expansion properties of the subsequent\nepochs, including the current one. The calculation of the spectrum\n\\cite{Allen:1987bk,Henriques:2003ga} was initially performed for a\ncurrently decelerating universe. However, it is now well known\nthat the universe expansion is currently accelerating\n\\cite{Riess:1998cb,Perlmutter:1998np} and since the evolution of\nRGWs depends on the expanding background space time,  the spectrum of RGWs should be modified accordingly. This\nmodification was confirmed and studied in Refs\n\\cite{Zhang:2005nw,Izquierdo:2004gk} using the well-known\nformulation of GWs in an expanding Universe\n\\cite{Grishchuk:1974ny} and an approximation of the scale\nfactor $a(\\tau)$ in the context of a sequence of successive expansion epochs,\nincluding the current stage of accelerating expansion. It was\nfound that the current accelerating expansion induces\nmodifications in both the shape and the amplitude of the RGW\nspectrum.\n\nSince existence of RGWs is a key prediction of the\ninflationary models, their detection could provide\n evidence that inflation actually took place. Thus,\nit is important to accurately calculate the expected detailed form\nof the RGW spectrum. Calculations related to RGWs in an\naccelerating Universe have been performed \\cite{Zhao:2006eb} and a\nnumerical method has been developed to calculate the power\nspectrum of the RGWs. Late evolution of RGWs in coupled dark\nenergy models has been examined extensively in Ref. \\cite{Almazan:2013bea}.\\\\\n\nEven though the effects of cosmological expansion on GWs have been\ninvestigated mainly in the context of RGWs, these effects are\nrelevant in all cases when the source is located at cosmologically\nlarge distances from the observer (redshift $z\\geq 0.1$). The GW\n150914 ($z=0.09$) event is in the limit of such distances and\ntherefore, the effects of cosmic expansion may be relevant.  Thus\na wide range of studies have investigated the effects of\ncosmological expansion of GWs from a variety of viewpoints\nincluding the effects of expansion on the GW group and phase\nvelocities  \\cite{Balek:2007hh,Brillouin}, mathematical aspects\nand exact solutions\n\\cite{Fabris:1998ak,Alekseev:1995uw,Tauber:1985yp,Waylen:1978dx,Tamayo:2015qla},\nquantum and thermodynamic properties of GWs\n\\cite{Arzano:2016twc},\\cite{IzquierdoSaez:2005ja}, general\nproperties \\cite{Hartnett:1900zzc,Waylen:1978dx,Jackson:2008yv}\nnonlinear effects \\cite{Ikeda:2015hqa}, properties of the GW\nenergy momentum tensor \\cite{Su:2012hf}, collision of GWs with\nelectromagnetic waves\n\\cite{Alekseev:2015wbn}, evolution of GWs in gravitational plasma \\cite{Baptista:1982dc} etc .\\\\\n\nEven though these studies have properly taken into account the\nexpansion of the background metric, they have not taken into\naccount the effects of the gravitational field of  mass\ndistributions on the evolution of the GWs. Such a gravitational\nfield combined with the expanding background may induce new\nobservable effects on the spectrum of propagating gravitational\nwaves affecting the amplitude and the frequency of such waves (due\nto gravitational time delay) \\cite{Zeldovich:1992if}.\n\nAssuming spherical symmetry, the background metric around a point\nmass embedded in an expanding Friedmann-Lemaitre-Robertson-Walker(FLRW) cosmological background\nis well approximated by the McVittie \\cite{McVittie:1933zz,9780511535185} spacetime. Such a metric is\nfurther simplified in the Newtonian limit and has been used as the background metric for the investigation of bound system geodesics\nin phantom and quintessence cosmologies \\cite{Nesseris:2004uj,Antoniou:2016obw,Faraoni:2007es,Nolan:2014maa}.\nThis metric can also be used as a background for the propagation of GWs in order to investigate the influence of a mass\ndistribution of a GW propagating in an expanding cosmological background.\n\nIn the present analysis we address the following question: 'What\nare the weak field effects of a point mass on a multipole spherical wave\ncomponent of a GW evolving in an expanding background in the\nvicinity of the mass?'. In particular we numerically solve the\ndynamical equation for the evolution of GWs in the background of\nthe Newtonian McVittie metric and identify the effects induced by\nthe point mass on the amplitude and frequency of the evolving\nGW as a function of the parameters determining the mass for fixed\nbackground expansion rate. As a test of our analysis, in the zero\nmass limit, our numerical solution reduces to the well known\nanalytic solution of a GW evolving in an expanding background.\n\nThe structure of this paper is the following: In the next section\nwe review the wave equation and the behavior of the GW in a\nhomogeneous-isotropic expanding background. We also derive the\ngravitational wave equation in a background metric corresponding\nto the Newtonian limit of the McVittie metric. In section $III$ we\nsolve numerically the wave equation in the Newtonian McVittie\nbackground and identify the new features induced in the GW by the\npresence of the point mass. Finally in section $IV$ we conclude,\nsummarize and discuss possible extensions on this analysis.\n\n\n\n\\section{Gravitational Waves in Expanding Universe in the Presence of Point Mass}\nWe first briefly review the propagation evolution of a plane GW in the\n$\\hat{z}$ direction (the direction of the wavevector $\\vec{k}$)\nwith tensor perturbations in the $x-y$ plane. The perturbations to\nthe metric are described by two functions, $h_+$ and $h_\\times$,\nassumed small. We use the FRW metric in cartesian coordinates with\nthe components $g_{00}=1$, zero space-time components $g_{0i}=0$,\n and set $c=1$. The spatial part of the\nmetric is of the form: \\be g_{ij}= -a^2(t)\\begin{pmatrix}\n  1+h_+ & h_\\times & 0 \\\\\n  h_\\times & 1-h_+ & 0 \\\\\n  0 & 0 & 1\\\\\n\\end{pmatrix}\n\\label{metric1}\\ee\n\nThe perturbation tensor $H_{ij}$ is symmetric, divergenceless,\ntraceless and has the form:  \\be H_{ij}= \\begin{pmatrix}\n  h_+ & h_\\times & 0 \\\\\n  h_\\times & -h_+ & 0 \\\\\n  0 & 0 & 1\\\\\n\\end{pmatrix}\n\\label{tensor}\\ee From the Einstein equations for tensor\nperturbations, it is easy to derive a set of equations governing\nthe evolution of the tensor variables $h_+$ and $h_\\times$. We\nwrite the FRW metric in cartesian coordinates and in conformal time $\\tau$  (defined by $dt=a d\\tau$), in the form \\be\nds^2=a^2(\\tau)[d\\tau^2-(\\delta_{ij}+H_{ij})dx^{i}dx^{j}]\n\\label{metrconf}\\ee The dynamical equation determining the\nevolution of the GWs is of the form: \\be\n \\Box H_{ij}= \\partial_{\\mu}(\\sqrt{-g}\\partial^{\\mu}H_{ij}(\\vec{r},\\tau)) =0 \\label{waveeq}\\ee\nSince all components of the tensor perturbations evolve in accordance with the same wave equation (\\ref{waveeq}) we may set $H_k\\equiv H_{ij}$.\nWithout loss of generality we assume propagation in the $z$\ndirection and thus we use the ansatz:\n \\be  H_{k}(\\tau,z)=h_{k}(\\tau) e^{\\pm ikz} \\label{waveans}\\ee  Using eq. (\\ref{waveans}) in (\\ref{waveeq}) we find the dynamical equation for the evolution of gravitational waves in conformal time in an FRW background as\n\\be h''_{k}+2\\frac{a'}{a}h'_{k}+k^2h_{k}=0 \\label{waveeq1}\\ee\nwhere the prime $'$ denotes the derivative with respect to\nconformal time. Notice that  all of the perturbation tensor\ncomponents obey the same equation. We introduce a rescaling of\nconformal time as $\\bar{\\tau}=k\\tau$ and thus it becomes clear\nthat \\be h_k(\\tau)=h(k\\tau) \\label{hresc} \\ee The rescaling\nexpressed by eq. (\\ref{hresc}) can only be made in conformal time\nprovided that the scale factor is a power law $a(\\tau)\\sim\n\\tau^\\alpha$. In the radiation dominated epoch we have $\\alpha=1$\nand during the matter dominated era $\\alpha=2$. The wave solution\n(\\ref{waveans}) can be written in spherical coordinates as: \\be\nH_k(\\tau,\\rho,\\theta)=h(k\\tau)e^{\\pm ik\\rho\\cos\\theta}\n\\label{hsphercoord}\\ee The spectrum of the GWs may be obtained as\n~\\cite{Zhang:2006mja}: \\be P(k,\\tau)=\\frac{4\\; l_{Pl}}{\\sqrt{\\pi}}k\n\\mid h(k\\tau)\\mid \\label{powersp}\\ee\nwhere $l_{Pl}\\sim \\sqrt{G}$ is the Planck length.\nThe plane wave of equation (\\ref{hsphercoord}) can be expanded in\nspherical waves as: \\be e^{i k \\rho\\cos{\\theta}}=\\sum_{l=0}^{\\infty}i^l(2l+1)j_{l}(k\\rho)P_l(\\cos{\\theta})\\label{planeexp}\\ee where $j_l(x)$ are the\nspherical Bessel functions and $P_l(\\cos{\\theta})$\nare Legendre's polynomials. Thus, the partial spherical GW is (at\norder $l$) \\be H_{ij}(\\tau,\\rho,\\theta) \\sim\nh(k\\tau)j_{l}(k\\rho) P_l(\\cos{\\theta})\\label{planeordl}\\ee\n\n\nAfter rescaling, the dynamical equation (\\ref{waveeq1}) is written\nas: \\be\n h''(k\\tau)+2\\frac{a'}{a}h'(k\\tau)+h(k\\tau)=0 \\label{rescwaveeq}\\ee\nwhere the prime $'$ now denotes differentiation with respect to\nthe rescaled conformal time $k\\tau$.\n\nAssuming a power law for the background scale factor as\n$a(\\tau)\\sim \\tau^\\alpha$, the solution of the wave equation\n(\\ref{rescwaveeq}) can be written in terms of incoming and outgoing waves (Hankel functions) as \\be\nh(k\\tau)=\\frac{1}{a(\\tau)}(\\tilde{A}_{k}\\sqrt{k\\tau}H^{(1)}_{\\alpha-1\/2}(k\\tau)+\\tilde{B}_{k}\\sqrt{k\\tau}H^{(2)}_{\\alpha-1\/2}(k\\tau))\\label{wavesol1}\\ee\nwhere $H^{(1)}$, $H^{(2)}$ are the Hankel functions and\n$\\tilde{A}_{k}$, $\\tilde{B}_{k}$ are arbitrary constants which may\ndepend on $k$ and are determined by the initial conditions. This\nsolution may also be written in terms of standing waves as \\be\nh(k\\tau)=(k\\tau)^{\\frac{1}{2}-\\alpha}(A_{k}J_{\\alpha-1\/2}(k\\tau)+B_{k}Y_{\\alpha-1\/2}(k\\tau))\\label{wavesol2}\\ee\nwhere $J$,$Y$ are the Bessel functions of first and second kind\nrespectively and $A_k=\\tilde{A}_k k^\\alpha$, $B_k=\\tilde{B}_k\nk^\\alpha$. Thus, for a power law scale factor the spherical GW is\n\\begin{eqnarray}\n&&H_k(\\tau,\\rho,\\theta)=(k\\tau)^{\\frac{1}{2}-\\alpha}(A_{k}J_{\\alpha-1\/2}(k\\tau)+B_{k}Y_{\\alpha-1\/2}(k\\tau))\\nonumber\\\\&&j_{l}(k\\rho)P_{l}(cos\\theta)\\label{wave}\\end{eqnarray}\n\nAs a warm up exercise before the introduction of a point mass in\nthe metric, we now rederive the solution (\\ref{wavesol2}),\n(\\ref{wave}) starting from the FRW metric in spherical coordinates\n\\begin{equation}\\label{frwsphercoords}\nds^2=a(\\tau)^2\\{d\\tau^2-[d\\rho^2+\\rho^2(d\\theta^2+\\sin^2{\\theta}d\\phi^2)]\\}\n\\end{equation}\n\nBecause of the azimuthal symmetry, the solution doesn't depend on\nthe variable $\\phi$ we will seek solutions of the Eq.\n(\\ref{waveeq}) of the form\n\\begin{equation}\\label{waveansspher} H_k(\\tau,\\rho,\\theta,\\phi)=\nf(\\tau)R(\\rho)P_{l}(cos\\theta) \\end{equation} From the Eqs.\n(\\ref{waveeq}),(\\ref{waveansspher}), after separation of variables\nwe find\n\\begin{equation}\\label{rdiffeqsep}\n\\frac{1}{R}[\\frac{d^2 R}{d \\rho^2}+\\frac{2}{\\rho}\\frac{d R}{d\n\\rho}-\\frac{l(l+1)}{\\rho^2}]=-k^2\n\\end{equation}\nand\n\\begin{equation}\\label{fdiffeqsep}\n\\frac{1}{f}[\\frac{d^2 f}{d \\tau^2}+\\frac{2a'}{a}\\frac{d f}{d\n\\tau}]=-k^2\n\\end{equation}\nwhere $k^2$ is arbitrary constant. As expected eq.\n(\\ref{fdiffeqsep}) is identical to eq. (\\ref{waveeq1}) while eq.\n({\\ref{rdiffeqsep}) is the spherical Bessel equation  with\nacceptable solution\n\\begin{equation}\\label{reqjl}\nR_{l}(\\rho)=A_{l}j_{l}(k\\rho)\\end{equation}  We thus reobtain\nthe general solution in spherical coordinates (\\ref{wave}).\n\nWe can now generalize the above analysis to investigate the\nbehavior of GWs when they interact with a point mass $M$. In the\npresence of a point mass and cosmological expansion, the\nappropriate background metric is the McVittie metric. In the\nNewtonian limit, using comoving coordinates the McVittie metric is\n\\cite{Antoniou:2016obw,Nesseris:2004wj}:\n\\begin{equation}\\label{nmcvitttiemetric} ds^2=(1-\\frac{R_s}{\\rho\na(t)})dt^2-a(t)^2[d\\rho^2+\\rho^2(d\\theta^2+\\sin^2{\\theta}d\\phi^2)]\n\\end{equation}\nwhere  $R_s=2GM$. The angular variable $\\theta$ separates and thus\nwe use the perturbation ansatz\n\\begin{equation}\\label{hanzmass} H_k(t,\\rho,\\theta,\\phi)=\nQ(t,\\rho)P_{l}(cos\\theta)\n\\end{equation}\n\n\\begin{figure*}[ht] \\centering\n\\begin{center}\n$\\begin{array}{@{\\hspace{-0.10in}}c@{\\hspace{0.0in}}c}\n\\multicolumn{1}{l}{\\mbox{}} & \\multicolumn{1}{l}{\\mbox{}} \\\\\n[-0.2in] \\epsfxsize=3.4in \\epsffile{fig0a.eps} & \\epsfxsize=3.4in\n\\epsffile{fig0b.eps} \\\\\n\\end{array}$\n\\end{center}\n\\vspace{0.0cm} \\caption{\\small a: A superposition of the analytic\nsolution with the numerical simulation initial condition taken at\n$\\tau=1$. The spherical wave with $l=6$ and scale factor\n$a(\\tau)=\\tau$ is shown.   b: The numerically evolved solution (dashed red line) for\nthe gravitational wave $Q(\\tau,\\rho)$ at $\\tau=16.5$ with $R_s=0$, is in\nexcellent agreement with the corresponding analytic evolved\nsolution (blue line). This is a test of the quality of the numerical\nsolution.} \\label{fig1}\n\\end{figure*}\n\n\nUsing the background metric (\\ref{nmcvitttiemetric}) and the\nansatz (\\ref{hanzmass}) in the gravitational wave equation\n(\\ref{waveeq}) we obtain the dynamical equation for $Q(t,\\rho)$ as\n\\begin{eqnarray}\\label{qpde}\n&&\\frac{\\partial^2 Q}{\\partial \\rho^2}+\\frac{2}{\\rho}(1-\\frac{3 R_s}{4a\\rho})(1-\\frac{R_s}{a\\rho})^{-1}\\frac{\\partial Q}{\\partial \\rho}-\\frac{l(l+1) Q}{\\rho^2}=\\nonumber\\\\\n&&\\frac{a^2}{1-\\frac{R_s}{a\\rho}}[\\frac{\\partial^2 Q}{\\partial\nt^2}+\\frac{3\\dot{a}}{a}(1-\\frac{7\nR_s}{6a\\rho})(1-\\frac{R_s}{a\\rho})^{-1}\\frac{\\partial Q}{\\partial\nt}] \\end{eqnarray}\n Assuming that\n\\begin{equation}\\label{smallrs}\n\\frac{R_s}{a\\rho}\\ll 1\n\\end{equation}\nand keeping terms $R_s\/a\\rho$ only in first order we can write eq.\n(\\ref{qpde}) in conformal time as\n\\begin{eqnarray}\\label{qpdeconf}\n&&(1+\\frac{R_s}{a\\rho})\\frac{\\partial^2 Q}{\\partial\n\\tau^2}+\\frac{2a'}{a}(1+\\frac{3R_s}{4a\\rho})\\frac{\\partial\nQ}{\\partial \\tau}\\nonumber\\\\\n&&=\\frac{\\partial^2 Q}{\\partial\n\\rho^2}+\\frac{2}{\\rho}(1+\\frac{R_s}{4a\\rho})\\frac{\\partial\nQ}{\\partial \\rho}-\\frac{l(l+1) Q}{\\rho^2}\n\\end{eqnarray}\n\nEq. (\\ref{qpdeconf}) is not separable and it is not tractable\nanalytically in a simple manner. As expected, in the limit of zero\nmass it separates and reduces to eqs. (\\ref{rdiffeqsep}) and\n(\\ref{fdiffeqsep}).\n\nIn the next section we integrate eq. (\\ref{qpdeconf}) numerically\nand investigate the dependence of the solution on the values of\nthe parameter $R_s$. It will be seen that as the wave approaches\nthe point mass it experiences two types of distortion\n\\begin{itemize}\n\\item gravitational time delay and increase of its period in conformal cosmological time. \\item\nIts amplitude increases in comparison to the amplitude it would\nhave in the absence of the point mass. \\end{itemize}\n\nAccording to general relativity the expected period of the wave at\na comoving distance $\\rho$ from the point mass, as measured by an\nobserver at infinity, is\n \\be\nT=\\frac{T_0}{\\sqrt{1-\\frac{R_s}{a\\rho}}}\\label{periodatrho}\\ee\nwhere $T_{0}$ is the corresponding period at infinity (or in the\nabsence of the mass). For small mass or large distance from the\nsource the increase of the period is \\be \\frac{\\Delta\nT}{T_0}=\\frac{1}{2a\\rho}R_s\\label{periodincrease}\\ee where $\\Delta\nT=T-T_0$ is the difference of the wave periods with and without\nthe presence of the mass. The validity of eq.\n(\\ref{periodincrease}) for the GW in the vicinity of a point mass\nwill be demonstrated numerically in the next section.\n\n\\section{Numerical Analysis}\n\\label{sec:Section 3}\n\nIn order to keep the analogy with the massless case $R_s=0$ we\nrescale the dynamical equation (\\ref{qpdeconf}) to dimensionless\nform using the wavenumber $k$ defining $k\\rho=\\bar{\\rho}$,\n$k\\tau=\\bar{\\tau}$. In this case we have an additional physical\ndimensionless parameter: \\be \\bar{R_s}=k R_s \\label{rsdim}\\ee In\nthe numerical analysis that follows we only use dimensionless\nquantities even though we will omit the bar in what follows.\n\n\\begin{figure*}[ht]\n\\centering\n\\begin{center}\n$\\begin{array}{@{\\hspace{-0.10in}}c@{\\hspace{0.0in}}c}\n\\multicolumn{1}{l}{\\mbox{}} & \\multicolumn{1}{l}{\\mbox{}} \\\\\n[-0.2in] \\epsfxsize=3.4in \\epsffile{fig1a.eps} & \\epsfxsize=3.4in\n\\epsffile{fig1b.eps} \\\\\n\\epsfxsize=3.4in \\epsffile{fig1c.eps} & \\epsfxsize=3.4in\n\\epsffile{fig1d.eps}\n\\end{array}$\n\\end{center}\n\\vspace{0.0cm} \\caption{\\small The evolution of the profile of a\npartial spherical gravitational wave with $l=6$, $R_s=5$ (red\ndashed line) in comparison with the corresponding free solution\n($R_s=0$, blue continuous line).\n The free wave reaches its maximum first (Fig. 1b) while the wave in the presence of the point mass shows a\n  delay in reaching its maximum (Fig. 1c) due to gravitational redshift. The wave in the presence of the mass\n  has a higher amplitude (compare 1b with 1c) in the vicinity of the mass and the phase difference increases\n  with time in the vicinity of the mass (Fig. 1d).}\n\\label{fig2}\n\\end{figure*}\n\nWe solve numerically eq. (\\ref{qpdeconf}) with initial conditions\ncorresponding to a standing gravitational wave evolving in a homogeneous\nFRW spacetime ($R_s=0$) using eq. (\\ref{wavesol2}) with $B_k=0$\nstarting the evolution at $\\tau=1$. This is equivalent to assuming\nthat the point mass appears at $\\tau=1$. For definiteness we set\n$\\alpha=1$ or $a(\\tau)\\sim \\tau$ corresponding to an expanding\nbackground in the radiation era. The boundary conditions are\nimposed for $Q$ and for its first derivative at large $\\rho$ where\nthe effects of the point mass are negligible and also correspond\nto a standing GW evolving in a homogeneous FRW spacetime ($R_s=0$) using\neq. (\\ref{wavesol2}) with $B_k=0$. \n\nThe Bessel function boundary condition (\\ref{wavesol2})-(\\ref{wave}) we\nhave used for the wave equation at large distance from the source, describes a standing\nGW which however can be expressed as a superposition of two propagating modes (Hankel functions). The asymptotic behaviour of Hankel functions, which is proportional to $e^{i\\tau}$, corresponds to a propagating GW, while the asymptotic behavior of Bessel functions is proportional to $\\cos(\\tau)$ and corresponds to a standing GW.\n\nWe stress that since we have made the Newtonian approximation our results are reliable in\nregions where the weak field condition (\\ref{smallrs}) is\nsatisfied.\n\nWe thus construct numerically the solution $\nQ(\\rho,\\tau,R_s,l,\\alpha)$ and compare with the corresponding\nanalytical solution  $ Q(\\rho,\\tau,R_s=0,l,\\alpha)$. We have\ntested our numerical evolution by verifying that the numerical\nsolution for $R_s=0$ agrees with the corresponding analytical\nsolution at a level better that $1\\%$ (Fig. \\ref{fig1}).\n\n\nFollowing the above comments about the boundary condition we fix\n$Q$ (and its derivative) at the boundary $\\rho_{end}>500$ using eq. (\\ref{wave}) with $\\alpha=1$, $A_k=1$ and $B_k=0$ as \\be\nQ(\\tau,\\rho_{bound})=\\sqrt{\\frac{2}{\\pi}}\\frac{\\sin(\\tau)}{\\tau}j_{l}(\\rho_{bound})\\label{boundary}\\ee\nSimilarly, the initial conditions set at $\\tau_i=1$ are: \\be\nQ(\\tau_i,\\rho)= \\sqrt{\\frac{2}{\\pi}}\\frac{\\sin(\\tau_i)}{\\tau_i}j_{l}(\\rho)\\label{initial1}\\ee\nand \\be \\frac{\\partial Q(\\tau_i,\\rho)}{\\partial\n\\tau}=\\frac{\\partial}{\\partial \\tau}[\\sqrt{\\frac{2}{\\pi}}\\frac{\\sin(\\tau)}{\\tau}j_{l}(\\rho)]|_{\\tau=\\tau_i}\\label{initial2}\\ee\nIn addition to the test of the validity of numerical solution\npresented in Fig. \\ref{fig1} we have performed other tests\nincluding the verification of the independence of the numerical\nsolution from the location of the boundary for $\\rho_{bound}>200$.\n\n\n\nWe have solved the partial differential equation (\\ref{qpdeconf})\nfor various values of $l$ with results that are qualitatively\nsimilar. For definiteness we present in Fig. \\ref{fig2} the\nsolution corresponding to $l=6$ for $R_s=5$ superposed with the\ncorresponding solution for $R_s=0$ in order to identify the new\nfeatures introduced in the evolution of the GW by the presence of\nthe point mass. \n\nThere are three main features to observe in Fig.\n\\ref{fig2}. First, the waves are practically identical far away\nfrom the point mass as expected. Second, there is a time delay for\nthe wave in the presence and in the vicinity of the point mass\n(Fig. 2b upper right). Third, the amplitude of the wave in the\npresence and in the vicinity of the mass increases (compare Fig.\n2b (upper right) with Fig. 2c (lower left)).\n\nThe main effect of the expansion is to reduce the amplitude of the\ngravitational wave by a factor proportional to the scale factor in the absence of the\nmass. This is shown in Fig. \\ref{fig3} which shows that the amplitude multiplied by the\nscale factor remains constant in the absence of the mass (blue oscillating line has\nconstant amplitude) for the particular time dependence of the scale factor considered\n($a(\\tau)\\sim \\tau$). In the presence of the mass however, the decrease of the amplitude due to\nthe expansion is less efficient (red line) and the product of the amplitude times the\nscale factor increases slowly with time.\n\nThe gravitational wave time evolution shown in Fig. \\ref{fig3} corresponds to $\\rho=7.9$ (closest\nmaximum amplitude to the mass for $l=6$) for $R_s=5$ (red dashed\nline) and is superposed with the corresponding evolution for $R_s=0$\n(blue continuous line). This plot demonstrates the relative\n(linear) increase of the  amplitude with time, as well as the\nincreased period of the wave in the presence of the mass. It also \ndemonstrates (as discussed above) the well known fact that the wave amplitude in the\nabsence of the mass ($R_s=0$) is inversely proportional to the\nscale factor (the blue wave has a constant amplitude).\n\n\\begin{figure}[!t]\n\\centering\n\\vspace{0cm}\\rotatebox{0}{\\vspace{0cm}\\hspace{0cm}\\resizebox{0.49\\textwidth}{!}{\\includegraphics{fig2.eps}}}\n\\caption{ The time evolution of the first spatial maximum (at\n$\\rho=7.9$) of the  partial spherical wave with $l=6$, $R_s=5$\n(red line)  in comparison with the corresponding free solution\n($R_s=0$, blue continuous line) at the same spatial point. The\nfree wave reaches its maximum first (Fig. 2b) while the wave in\nthe presence of the point mass shows a delay in reaching its\nmaximum (Fig. 2c) due to gravitational redshift. The wave in the\npresence of the mass has an amplitude that increases with time as\nindicated with the dashed red line that in tangent to the\ngravitational wave maxima. As expected, the product $a(\\tau) Q(\\tau)$ is constant for the free wave in an expanding background. } \\label{fig3}\n\\end{figure}\n\nThe effects of the gravitational time delay on the evolution of\nthe wave may also be demonstrated by plotting the power spectrum\nobtained by a Fourier series expansion of the evolving in\nconformal time numerical solution at $\\rho=7.9$ in harmonic waves.\n\nThe finite time interval power spectrum may be defined through the expansion\n\\be\nQ(\\tau,\\rho)=\\frac{a_0}{2}+\\sum \\limits_{i=1}^n(a_n\n\\cos(n\\tau)+b_n\\sin(n\\tau))\\label{coef}\\ee\nas\n\\be P(n)\\equiv \\log\\sqrt{a_n^2+b_n^2}\\label{spectrum}\\ee\n\nWe used a time interval of approximately  two\ncomplete oscillations which corresponds to a time interval\n$\\tau \\in [1,20]$ ($\\tau_i=1, \\tau_{max}=20$ as shown in Fig. \\ref{fig4}).\n\nAs shown in Fig. \\ref{fig4} the presence of the mass (red continuous line)\nleads to an increase of the amplitude of low harmonics and\ndecrease of the amplitude of higher harmonics which is consistent\nwith the effects of gravitational time delay.  The exact\nform of the spectrum clearly depends on the time interval\nconsidered, however the qualitative feature of higher amplitudes\nfor lower frequencies persists for all time intervals. This\nfeature is more prominent for lower values of $\\rho$.\n\n\\begin{figure}[!b]\n\\centering\n\\vspace{0cm}\\rotatebox{0}{\\vspace{0cm}\\hspace{0cm}\\resizebox{0.49\\textwidth}{!}{\\includegraphics{fig3.eps}}}\n\\caption{The time power spectra of the gravitational wave in the\npresence (red line) and in the absence (blue line) of the mass. Notice that\nlower frequencies have a higher amplitude for the wave in the\npresence of the mass as expected due to the gravitational time\ndelay.} \\label{fig4}\n\\end{figure}\n\nIn accordance with eq. (\\ref{periodincrease}) the increase of the\nperiod of the wave at a given distance from the mass is\nproportional to the mass in the weak field approximation. This is\nconsistent with our numerical solution as shown in Fig. \\ref{fig5} where we\nshow the relative increase of the period of the wave $\\Delta\nT\/T_0$ at given distances $\\rho$ \nfrom the mass ($\\rho=7.9$ and $\\rho=15.89$) for various values the parameter $R_s$ (points in\nplot). In order to evaluate the relative change of the period $\\Delta\nT\/T_0$ we use the time evolution of the wave perturbation as shown in Fig. \\ref{fig3} to obtain the period of the wave in the\npresence of the mass and the corresponding period in the absence of the mass. Superposed in Fig. \\ref{fig5} \nis the best fit straight line in each case. As\nis theoretically expected there is a linear relationship in\naccordance with eq. (\\ref{periodincrease}). The correlation\ncoefficients of the points with the corresponding best fit straight line are\nequal to $0.99$ indicating an excellent quality of fit.\n\nThe theoretically predicted slope is $\\frac{1}{2a\\rho}$ where the scale factor can\nbe taken as approximately constant and equal to its average value\nduring the wave period used to evaluate $\\Delta\nT\/T_0$.\n\n\\begin{figure}[!t]\n\\centering\n\\vspace{0cm}\\rotatebox{0}{\\vspace{0cm}\\hspace{0cm}\\resizebox{0.49\\textwidth}{!}{\\includegraphics{fig5.eps}}}\n\\caption{The relative difference of the wave periods $\\Delta\nT\/T_0$, where $T$ is the period in the presence of mass and $T_0$\nis the period in the absence of mass, as a function of $R_s$. It\nis clear that as the value of the variable $\\rho$ increases, the\nstatistical slope of the curve decreases. This is an anticipated\nresult due to theoretical slope of the curve $\\frac{1}{2a\\rho}$.}\n\\label{fig5}\n\\end{figure}\n\n\nIn order to estimate the theoretical value of the scale factor, we\ncalculate the mean value $\\bar{a}(\\tau)$, in the time interval\n$\\tau_{1}-\\tau_{2}$ of a single period, through the formula: \\be\n\\bar{a}(\\tau)=\\frac{1}{\\tau_{2}-\\tau_{1}}\\int_{\\tau_1}^{\\tau_2}a(\\tau)d\\tau=\\frac{\\tau_2+\\tau_1}{2}\\label{bara}\\ee\n The observed\ndeviations by about $20\\%$ between theoretically expected slope\nand numerically obtained can be attributed to the approximations\nwe have made which include, the weak field assumption  ($R_s \\ll\na\\rho$ while in the cases considered $\\frac{R_s}{a\\rho} \\leq\n0.1$), the assumed constant scale factor for\nthe evaluation of the slope etc. As shown in Figs 2 and 3 the amplitude of the wave also increases\nas the point mass is approached. A quantitative estimate of this\neffect is shown in Fig. \\ref{fig6} where we show the ratio of the\namplitudes of the waves $A\/A_0$ in the presence of a mass ($A$)\nand in the absence of the mass ($A_0$) for various values of the\nparameter $R_s$, when $\\rho=7.9$ and $\\rho=15.89$. The best fit\nstraight line is also superposed on the points showing that a\nlinear relationship between $A\/A_0$ and $R_s$ is a good\napproximation.\n\\begin{figure}[!b]\n\\centering\n\\vspace{0cm}\\rotatebox{0}{\\vspace{0cm}\\hspace{0cm}\\resizebox{0.49\\textwidth}{!}{\\includegraphics{fig6.eps}}}\n\\caption{The ratio of the amplitudes of the waves $A\/A_0$ in the\npresence of a mass ($A$) and in the absence of the mass ($A_0$) as a function of\nthe parameter $R_s$, when $\\rho=7.9$ and $\\rho=15.89$.}\n\\label{fig6}\n\\end{figure}\nThe amplitude increases up to $10\\%$ when $\\rho=7.9$ and $R_s=10$,\nwhile for $\\rho=15.89$ and the same value of $R_s$, the increase is about $5\\%$. Thus the\namplitude increase appears to vary inversely proportional with\n$\\rho$ which is consistent with the fact that the GW gains energy\nas it enters regions of space with higher curvature.\n\n\n\\section{Conclusion}\n\\label{sec:Section 4}\n\nThe effects of a point mass on a GW evolving in an expanding\nuniverse are determined by the mass $M$ and the physical distance\n$a \\rho$ of the wave from the mass through the expression\n$\\frac{R_s}{a\\rho}\\equiv\\frac{2GM}{a\\rho}$. In the context of a\nperturbative weak field analysis we have demonstrated that a point\nmass tends to increase the amplitude and the period of the GW\nlinearly with respect to $\\frac{R_s}{a\\rho}$. This result is\nconsistent with expectations based on gravitational time delay and\nenergy considerations.\n\nEven though our numerical results were presented for the special\ncase of a radiation dominated cosmological background ($a(\\tau)\\sim \\tau$) and a\nspecific multipole component of the wave ($l=6$) we have checked that their qualitative features persist for\nall multipole components and cosmological backgrounds provided that\nthe weak field condition (\\ref{smallrs}) is respected. Thus, even though we have considered specific spherical waves in this analysis, we anticipate that our\nresults can also describe a plane wave when expressed as a superposition of spherical\nwaves.\n\nThe time slicing we considered corresponds to the coordinate\ntime of the particular metric we used. This coordinate time is particularly interesting\nand generic as it corresponds to the proper time of a static observer located far away\nfrom the point mass or in the absence of the point mass. This is the standard cosmic\nobserver whose observations are consistent with the cosmological principle. Clearly a \ndifferent choice of time slicing would correspond to a different observer and would lead to a different metric and thus different\nresults.\n\nFrom the results shown in Fig. \\ref{fig5} and Fig. \\ref{fig6}, we conclude that $T=T_0(1+\\mu R_s)$ and $A=A_0(1+\\nu R_s)$ where $\\mu$ and $\\nu$ are the slopes of the curves which are approximately equal. Thus we have demonstrated that the energy density of GWs which is proportional to $\\omega^2 A^2$ has a weak dependence on $R_s$ in the context of our weak field approximation as long as the slopes $\\mu$ and $\\nu$ are approximately equal.\n\nOur result has interesting implications for the calculation of the\nRGW spectrum which currently assumes\n\\cite{Koh:2009cy,Corda:2009bx,Grishchuk:2007uz,Zhang:2008zx,Gogoberidze:2007an,Buonanno:1996xc}\na smooth homogeneous cosmological background and ignores the\npresence of mass concentrations which as shown in the present\nanalysis would tend to modify both the magnitude and the shape of\nthis spectrum. A proper stochastic analysis including the effects\nof mass concentrations on the relic GW spectrum is therefore an\ninteresting extension of the present work.\n\nA distortion of the RGW spectrum is expected due to the presence\nof point masses on various scales due to the increase of each mode amplitude and\ndecrease of each mode frequency. The effect will be stronger in regions of higher\nmass concentrations. On scales larger than the galactic scales the role of the point\nmass could be played by a galaxy while on scales of the solar system the role of the point mass could be played by a planet. \nIn the solar system the effect is expected to be rather weak and beyond the sensitivity of current experiments. \n\n\nAn additional interesting extension could be the drop of the weak\nfield approximation and the use of the full McVittie metric\n\\cite{McVittie:1933zz} for the study of GW evolution in an\nexpanding background and in the vicinity of a black hole allowing\nfor strong gravitational field.\n\nEven though our numerical analysis has been well tested and\nprovides detailed quantitative information on the GW evolution\nin the presence of expansion and a point mass, an analytical\nperturbative solution describing this evolution would provide\nfurther physical insight and appears to be a tractable useful\nextension of the present work.\n\n\\textbf{Numerical Analysis Files}: The mathematica files used for the production of the figures, as well as the figures may be downloaded from \\href{https:\/\/drive.google.com\/open?id=0B7rg6X3QljQXck5YSmQ5Rl9HeUU}{here} or upon request from the authors.\n\n\\section*{Acknowledgements}\nD. Papadopoulos would like to thank the Department of Physics of\nthe University of Ioannina for hospitality during the period when\npart of this work was in progress. We also thank K. Kleidis for useful comments.\n\n\\raggedleft\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1.1}\n\n\\begin{figure}\n\\vspace{1cm}\n\\includegraphics [width=0.5\\linewidth]{WSFig1.eps}\n\\caption{Number of papers on the subject ``anomalous diffusion'' published\nper year according to the ISI Web of Science, Thomson-Reuters. \nNotice the ``phase transition''-like rise during 1990.}\n\\label{Fig1}\n\\end{figure}\n\nAnomalous diffusion becomes an increasingly popular subject with the number of papers\npublished per year growing fast over last twenty years after a distinct\nrise occurred in 1990 \n(cf. Fig. \\ref{Fig1}). Since then, \nit spreads  from such typical physical applications as\ncharge transport in disordered solids and hot plasmas to biophysical\napplications and even quantitative finance \n\\cite{Shlesinger1,Scher,Bouchaud,Hughes,Shlesinger,\nMetzler,West,West2,Zaslavsky,Saxton,Seisenberg,Tolic,GH04,Golding,Wilhelm,\nAmblard,Szymanski,Kou,Mizuno,Reuveni}. \n\nThere are several sufficiently generic \nphysical mechanisms and accompanying theoretical approaches to describe the complexity\nof anomalous transport processes. One approach is intrinsically based on the physical\npicture of a stochastic time clock \n\\cite{Shlesinger1,Scher,Hughes,Shlesinger,Metzler,Sokolov1}. It models random \nsojourns of a travelling\nparticle in trapping domains of a disordered solid, e.g. due to energy disorder.\nAfter a random time spent in some spatially located trapping domain the particle jumps \nto a neighboring domain, or maybe farther, and such a jumping process\ncontinues in time. The jump directions and their\nlengths are not correlated from jump to jump and the next clock period\nis not correlated with those passed (semi-Markovian assumption \n\\footnote{This does not exclude infinite range memory\nleading to a weak ergodicity breaking \\cite{Barkai}.}). Such a random\nclock is completely characterized by the probability density of clock periods \n$\\psi(\\tau)$.\nIn a given time interval $t$ there will\nbe a random number of jumps $n$, or \nstochastic clock \nperiods completed. However, if the mean clock \nperiod $\\langle \\tau\\rangle$ exists,\nthe probability distribution $p(n,t)$ of ``ticking'' $n$ times within the \nobserver time window $t$  becomes for large $n$ a very sharp function \\cite{Hughes} around \n$n^*(t)=t\/\\langle \\tau\\rangle$, as characterized by the \nrelative dispersion, $\\langle \\delta n^2(t) \\rangle^{1\/2}\/\n\\langle n(t) \\rangle$ (see Appendix A). \nIn the continuum medium \napproximation the trapping domains shrink to points. \nThen, $\\langle \\tau \\rangle$ can be made arbitrarily small and $n$ becomes \nquasi-continuous variable for a finite $t$.\nCorrespondingly, the probability density of intrinsic time, \n$\\tau(t)=n(t)\\tau_{\\rm sc}$,\nwhere $\\tau_{\\rm sc}$ is a time-scaling parameter, an intrinsic clock \ntime unit equal to the\nduration of time period for the regular clock, \nassumes a delta-function,\n$p(\\tau)=\\delta(\\tau-t)$, and the stochastic clock is not different\nfrom the regular one.\n\nThe situation changes dramatically if the mean of\nsojourns does not exist, or better to say, it exceeds largely a \ntypical time required to\ndiffuse across  the physical medium of a finite size. \nThen, $p(n,t)$ is not a sharp\nfunction around the mean number $\\langle n(t)\\rangle$.\nThis is the case, for example, when $\\psi(\\tau)$ possesses\na long tail, $\\psi(\\tau)\\propto \n(\\tau\/\\tau_{\\rm sc})^{-1-\\alpha}$ for $\\tau\/\\tau_{\\rm sc}\n\\to \\infty$ with $0<\\alpha<1$.\nThis implies a divergent mean $\\langle \\tau\\rangle \\to\\infty$, and\ndiverging higher moments as well.\nNevertheless,  $\\langle n(t)\\rangle \\propto (t\/\\tau_{\\rm sc})^{\\alpha}$ \nexists for any finite $t$ and it scales \nsublinearly with the physical time $t$ (see in Appendix A). In terms of \nthe one-sided Levy distribution density ${\\cal L}_{\\alpha}(z)$, \n$p(n,t)=(t\/\\tau_{\\rm sc}){\\cal L}_{\\alpha}(n^{-1\/\\alpha}\nt\/\\tau_{\\rm sc})\/(\\alpha n^{1\/\\alpha+1})$. \n\nConsider now an ensemble of particles.\nUntil time $t$, each particle \nhas accomplished an {\\it individual} number of intrinsic time periods corresponding\nto the intrinsic time $\\tau(t)$ which becomes a random\nvariable broadly distributed: all the particles have their own history, \nmaintaining individuality \nand avoiding the fate of self-averaging even in the strict limit $t\\to\\infty$. \nOnly an additional ensemble averaging\nsmears out this principal randomness\\cite{Sokolov1,He,Lubelski}. \nThe {\\it unbiased} diffusion becomes anomalously slow and nonergodic with the spatial \nvariance of a {\\it cloud} of particles \ngrowing sublinearly, \n$\\langle \\delta x^2(t)\\rangle \\propto \\langle n(t)\\rangle \\propto \nt^{\\alpha}$. \nSuch a nonergodic approach\nto subdiffusion seems appropriate for disordered solids, e.g. thin amorphous \nfilms \\cite{Scher,Hughes}, with a more recent example \nprovided by ${\\rm Ti O_2}$ nanocrystalline electrodes in \nthe Gr\\\"atzel's\nphotovoltaic cell elements \\cite{Nelson}. For charged carriers within \nsuch media one can create\na potential energy profile $U(x)$ by applying an electrical field of spatially \ndistributed fixed charges and a static external electrical field. Then in the continuum\napproximation subdiffusion can be described by the fractional Fokker-Planck\nequation (FFPE) \\cite{Metzler,Metzler99,Barkai01}\n\\begin{eqnarray}\\label{FFPE1}\n \\frac{\\partial}{\\partial t} P(x,t) = \\sideset{_0}{_t}{\\mathop{\\hat\nD}^{1-\\alpha}} \\left [- \\frac{\\partial}{\\partial x} \n\\frac{f(x)}{\\eta _\\alpha} + \\kappa _\\alpha \\frac{\\partial ^2}{\\partial x^2}\n\\right ] P(x, t) \\,, \n\\end{eqnarray}\nwhere $f(x)=-d U(x)\/dx$ is the force,  \n$\\eta _\\alpha$ is the fractional friction coefficient related to the\nfractional subdiffusion coefficient $\\kappa_{\\alpha}$ by the generalized Einstein\nrelation, $\\eta_{\\alpha}=k_BT\/\\kappa_{\\alpha}$, at temperature $T$ and \n\\begin{equation} \\label{RL}\n\\sideset{_{t_0}}{_t}{\\mathop{\\hat D}^{\\gamma}} P (x, t) =\\frac{1}{\n\\Gamma(1-\\gamma)} \\frac{\\partial}{\\partial t} \\int_{t_0}^{t}\n\\mathrm{d} t' \\, \\frac{P(x, t')}{(t-t')^{\\gamma}} \\, ,\n\\end{equation}\nis the Riemann-Liouville operator of the fractional derivative \n\\cite{Mainardi}, where $0<\\gamma<1$ and $\\Gamma(x)$ is the gamma-function.\nThe FFPE (\\ref{FFPE1}) can be derived within the above continuous time random walk \n(CTRW) framework. It can\nalso be written in the form using the \nCaputo fractional derivative\n\\cite{Mainardi}\n\\begin{eqnarray}\n\\sideset{_{t_0}}{_{*}}{\\mathop{D}^{\\gamma}}P(x,t):=\\frac{1}{\\Gamma(1-\\gamma)}\n\\int_{t_0}^t dt' \\frac{\\partial P(x,t')\/\\partial t'}{(t-t')^\\gamma}\n\\end{eqnarray}\nacting on the left hand side, yielding~\\cite{GH06}\n\\begin{eqnarray}\\label{FFPE2}\n\\sideset{_{0}}{_{*}}{\\mathop{D}^{\\alpha}}P(x,t)= \n\\kappa_{\\alpha} \\frac{\\partial }{\\partial\nx} \\left ( e^{-\\beta U(x)} \\frac{\\partial}{\\partial x} \\, e^{\\beta\nU(x)} P(x,t)  \\right )=-\\frac{\\partial J(x,t)}{\\partial x} \\,,\n\\end{eqnarray}\nin the transport form.\nHere, $\\beta=1\/(k_BT)$ is inverse temperature and \n\\begin{eqnarray}\nJ(x,t)=-\\kappa_{\\alpha}\ne^{-\\beta U(x)} \\frac{\\partial}{\\partial x} \\, e^{\\beta\nU(x)} P(x,t)\n\\end{eqnarray}\nis the subdiffusive flux.\n It should be emphasized that a non-Markovian Fokker-Planck equation  never defines\nthe corresponding non-Markovian process completely \\cite{HT77,Grabert}. \nIt allows to find merely the \nsingle-time, conditional, and double-time probability densities, but never the \nmulti-time probability densities.\nHowever, the FFPE dynamics can be nicely \nsimulated from the underlying CTRW with the nearest neighbors jumps only \n\\cite{GH06,H06}.   \n \nA quite different approach to subdiffusion is associated with \nthe fractional Brownian motion \\cite{Mandelbrot}.\nHere, the principal issue is the long-range anticorrelations in the particle \ndisplacements, positive increments follow with a greater probability by negative \nincrements and {\\it vice versa}\\cite{Goychuk09,Jeon}, which can reflect e.g. the phenomenon of viscoelasticity in complex \nglass-forming liquids above, but close to the glass transition. The Brownian \nparticle\nis temporally trapped in a trap (cage effect) by an elastic force with spring constant\n$G(t)$ which decays to zero\nin time releasing the particle. Let's assume that the motion starts\nat $t_0$, $v(t)=\\dot x(t)=0$ for $t<t_0$. \nIn the linear approximation, on the particle  acts a \nviscoelastic force $F_{\\rm v-el}(t)=-\\int_{t_0}^t G(t-t')\n\\dot x(t')dt'$, where $\\dot x(t)$ is the particle's instant velocity. \nThe first theory of \nviscoelasticity has been proposed by J. Clerk Maxwell \\cite{Maxwell} in 1867. \nIt corresponds to $G(t)$ exponentially decaying\nin time, $G(t)=G_0\\exp(-t\/\\tau_0)$, with a relaxation time constant $\\tau_0$. \nDeparting from the phenomenon of elasticity in solids \nMaxwell derived the phenomenon of viscosity in liquids\nin the limit where the decay of elastic modulus is very fast on the time scale of\n$v(t)$ change, which corresponds to \n$G(t)=2\\eta_0 \\delta(t)$, with $\\eta_0=G_0\\tau_0$ \nbeing the viscous friction coefficient. \nIn the theory of generalized Brownian motion, $G(t)$\nis interpreted as the frictional memory kernel $\\eta(t)$, rather than\na decaying elastic force constant. Here, one departs from\nthe phenomenon of viscosity and viscous Stokes memoryless friction and considers \nthe emerging \nelasticity in complex fluids or viscoelastic bodies. Both view points are\nessentially equivalent for a positive $\\eta(t)>0$ departing just from different \nstanding points \\footnote{$\\eta(t)$ can also be negative, e.g. accounting for\nthe hydrodynamic memory or in the case\nof superdiffusion. Therefore, the memory-friction interpretation is, \nin fact, more general.}. \nThe anticorrelations in the particle's displacements are due to the\nelastic restoring force component. \n\nIn complex media, the memory function $G(t)$ is better described\nby a sum of exponentials reflecting a viscoelastic response with multiple time scales. \nMoreover, \nin 1936 A. Gemant \\cite{Gemant} found that some viscoelastic\nbodies are better described by a $G(t)$ relaxing in accordance with a\npower law, $G(t)\\propto t^{-\\alpha}$,  rather than a single-exponential \nand introduced\na fractional integro-differential in the viscoelasticity theory. Using the notion of\nfractional Caputo derivative \nsuch a visco-elastic force can be short-handed, written as \n\\begin{equation}\\label{model2}\n F_{\\rm v-el}(t)=-\\eta_{\\alpha}\n \\sideset{_{t_0}}{_{*}}{\\mathop{D}^{\\alpha}} x(t)\\;.\n\\end{equation}\nIndeed, such and similar viscoelastic responses are measured \\cite{Wilhelm,Mizuno,Mason} \nusing the microrheology\nmethods  \\cite{rheo}. The Brownian motion never stops and the frictional \nloss of energy must be compensated on average by the energy gain\nprovided by a zero-mean random force of environment so that at the thermal equilibrium \nthe equipartition theorem holds, in accordance with\nthe classical fluctuation-dissipation theorem. \nWithin the considered model of a linear \nmemory-friction such a force must be Gaussian \\cite{Reimann} \n(but not necessarily so beyond the linear friction model). \nAs a result, the Brownian motion\nof a particle of mass $m$ is described by the Fractional Langevin Equation (FLE)\n\\cite{Lutz,Coffey,Goychuk07a,Burov} \n\\begin{equation}\\label{fle}\nm\\ddot x+ \\eta_{\\alpha}\n\\sideset{_{0}}{_{*}}{\\mathop{D}^{\\alpha}} x(t)=f(x)+\\xi(t) \\;,\n\\end{equation}\n(from now on we fix $t_0=0$) which is a particular case of the celebrated \nGeneralized Langevin Equation (GLE) \\cite{Kubo1,Kubo2,Zwanzig,HTB90,WeissBook}\n\\begin{equation}\\label{gle}\nm\\ddot x+\\int_{0}^{t}\\eta(t-t')\\dot x(t')dt'=\nf(x)+\\xi(t) \\;,\n\\end{equation}\nwith the memory kernel $\\eta(t)=\\eta_{\\alpha}t^{-\\alpha}\/\\Gamma(1-\\alpha)$ \nand the noise autocorrelation \nfunction obeying the fluctuation\ndissipation relation \n\\begin{equation}\\label{fdt} \\langle\n\\xi(t)\\xi(t')\\rangle=k_BT\\eta(|t-t'|) \\;.\n\\end{equation}\nSuch a GLE can be derived also from a Hamiltonian model for a \nparticle bilinearly \ncoupled with coupling constants $c_i$\nto a thermal bath of harmonic oscillators with masses $m_{i}$ and frequencies\n$\\omega_i$, $H_{B,\\rm int}(p_i,q_i,x)=(1\/2)\\sum_i \\{p_i^2\/m_i+m_i\\omega_i^2\n[q_i-c_i x\/(m_i\\omega_i^2)]^2\\}$.\nThe total effect of the bath oscillators,\nwhich are initially canonically distributed with $H_{B,\\rm int}$ \nat temperature $T$ and fixed $x=x(0)$, \nis characterized\nby the bath spectral density\n\\begin{equation}\nJ(\\omega) = \\frac{\\pi}{2} \\sum_i\n\\frac{c_i^2}{m_{i} \\omega_{i}} \\delta(\\omega-\\omega_i).\n\\label{J}\n\\end{equation}\nThe memory kernel is $\\eta(t)=(2\/\\pi)\\int_0^{\\infty}J(\\omega)\\cos(\\omega t)d\\omega$ \nin terms of $J(\\omega)$ and the subdiffusive FLE corresponds to a \nsub-Ohmic, or fracton thermal bath \nwith $J(\\omega)=\\eta_{\\alpha}\\sin(\\pi\\alpha\/2)\n\\omega^{\\alpha}$ \\cite{WeissBook}. \nWithout frequency cutoffs such a model presents a \nclear idealization.\nThere always exists a highest frequency of the thermal bath and this leads to \na small time  regularization of the memory kernel, i.e. a short-time cutoff. \nPhysically, this takes into account the medium's granularity beyond the continuum\napproximation. Moreover, in the case of a finite-size medium \nthere always exists also a smallest frequency of the medium's oscillators corresponding\nto the inverse size of the medium. These facts become especially clear if one evaluates\nthe spectral density of low-frequency ``fracton'' oscillators in proteins, \nsee in Refs. \\cite{Reuveni}.\nThis leads also to a cutoff at large times \nin the memory kernel and the dynamics can be\nsubdiffusive on the time scale smaller than the corresponding memory cutoff.\nThe latter one can be but prominently large which makes the considered \nidealization relevant. Important is also the result that an overdamped\nFLE description  of subdiffusion can be derived \nfrom a broad class of phenomenological continuum elastic \nmodels \\cite{Taloni}.     \n\n\n\nIn the inertialess limit with $m\\to 0$, \none can conceive the idea that FFPE (\\ref{FFPE1},\n\\ref{FFPE2})\nis the fractional Fokker-Planck equation corresponding to  the FLE (\\ref{fle}). \nThis idea is but wrong \\cite{Goychuk07a}. \nThe non-Markovian Fokker-Planck equation (NMFPE) which corresponds\nto the GLE (with arbitrary kernel) \\cite{Adelman,Hanggi78,Hynes} and \nto the FLE, in particular\\cite{Goychuk07a},  \nis a different one. Presently, its explicit form\nis known only for constant or\nlinear forces $f(x)$\\cite{Adelman,Hanggi78,Hynes}. \nThis resulting NMFPE \nhas the form of \nFokker-Planck equation with \ntime-dependent kinetic coefficients. This time-dependence\nis not universal and it heavily depends on the form of potential. In turn, \nthe Langevin equation which corresponds to the above FFPE is known and it has \nthe form of a Langevin equation which is local in the stochastic time $\\tau(t)$\nand describes thus a doubly stochastic process \\cite{DoublyStoch}.\nHere lies also the profound mathematical difference between these two approaches\nto subdiffusion. The physical differences are also immense. In particular, the \nGLE and FLE\napproaches are asymptotically mostly ergodic\nas they are not based on the concept of \nfractal stochastic time\nwith divergent mean period and the mean residence time in a finite spatial\ndomain remains finite. \nBefore we discuss the striking differences in more detail, \nlet us start from some apparent, but misleading similarities.\n\n  \n\\section{Free subdiffusion and constant bias}\n\nFree subdiffusion, as well as diffusion biased by a constant force $F$ can readily be \nsolved in both approaches using the method \nof Laplace-transform. First one finds the Laplace-transform of the mean \nensemble-averaged displacement $\\langle \\delta x(t)\\rangle $, and of the position \nvariance\n$\\langle \\delta x^2(t)\\rangle= \\langle x^2(t)\\rangle-\n\\langle x(t)\\rangle^2$, starting from a delta-peaked distribution at $x=0$ and \n$t_0=0$. Then, one transforms back to the time domain.  This gives\\cite{Metzler} \n\\begin{eqnarray} \\label{mean}\n\\langle \\delta x(t)\\rangle=\\mu_{\\alpha} F t^{\\alpha}\/\\Gamma(1+\\alpha)\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{variance}\n\\langle \\delta x^2(t)\\rangle=2\\kappa_{\\alpha}t^{\\alpha}\/\\Gamma(1+\\alpha)\n\\end{eqnarray}\nwith the generalized mobility $\\mu_{\\alpha}=1\/\\eta_{\\alpha}$\nrelated to the subdiffusion coefficient at $F=0$ by the generalized Einstein \nrelation $\\mu_{\\alpha}=\\kappa_{\\alpha}\/(k_BT)$. \nWithin the FLE approach these results are valid in the strict inertialess \nlimit $m\\to 0$. Furthermore, the Eq. (\\ref{variance}) is still valid  then for \n{\\it arbitrary} $F\\neq 0$. However, within the FFPE approach the Eq. (\\ref{variance})\nis valid {\\it only} for $F=0$, which is the first striking difference, see also below. \nFurthermore, both results are also valid\nasymptotically, $t\\to\\infty$, within the FLE  for a finite $m\\neq 0$.\n\n Generally, the GLE results\ncan be obtained for arbitrary memory kernel $\\eta(t)$. Assuming the particles\nbeing initially Maxwellian distributed, i.e. thermalized with\nthermal r.m.s. velocities \n\\begin{eqnarray}\nv_T=\\sqrt{k_BT\/m}, \n\\end{eqnarray}\none can\nobtain for the Laplace-transformed stationary velocity (fluctuation) \nautocorrelation function (VACF)\n$K_v(\\tau)=\\langle \\delta v(t+\\tau)\\delta v(t)\\rangle$, \n$\\delta v(t)=v(t)-\\langle v(t)\\rangle$,\n\\begin{eqnarray}\n\\tilde K_v(s)=\\frac{k_BT}{ms+\\tilde \\eta(s)},\n\\end{eqnarray}  \nwhere $\\tilde \\eta(s)$ is the Laplace-transform of $\\eta(t)$.\nThis is a well-known result which was obtained first by Kubo\\cite{Kubo1,Kubo2} \nin the Fourier space.\nFor the FLE with $\\tilde \\eta(s)=\\eta_{\\alpha}s^{\\alpha-1}$ it \nyields by the inversion to the time-domain \\cite{Lutz}\n\\begin{eqnarray}\\label{vfle}\nK_v(\\tau)=v_T^2 E_{2-\\alpha}\n[-(\\tau\/\\tau_{v})^{2-\\alpha}]\n\\end{eqnarray}\nwith $\\tau_{v}=(m\/\\eta_{\\alpha})^{1\/(2-\\alpha)}$ being the anomalous \nvelocity relaxation time constant. In (\\ref{vfle}), \n$E_{\\gamma}(z)$ is the Mittag-Leffler\nfunction, $E_{\\gamma}(z)=\\sum_{n=0}^{\\infty}z^n\/\\Gamma(n\\gamma+1)$ \\cite{Metzler}.\nFor $0<\\alpha<1$, $K_v(\\tau)$ is initially positive reflecting \nballistic persistence due to\ninertial effects\nand then becomes negative (anti-persistence due to decaying elastic cage force). \nIn the limit $m\\to 0$, the VACF undergoes a jump starting from \n$v_T^2$ at $\\tau=0$ and\nthen becoming negative, \n$K_v(\\tau)\\propto -1\/\\tau^{2-\\alpha}$ for $\\tau>0$, corresponding\nto purely anti-persistent motion. The position variance is given by the \ndoubly-integrated VACF. Its Laplace-transform therefore reads,\n\\begin{eqnarray}\\label{x2}\n\\langle \\widetilde{\\delta x^2(s)}\\rangle =\\frac{2k_BT}{s^2[ms+\\tilde \\eta(s)]}.\n\\end{eqnarray}\nMoreover, \n\\begin{eqnarray}\n\\langle \\widetilde{\\delta x(s)}\\rangle =\\frac{F}{s^2[ms+\\tilde \\eta(s)]},\n\\end{eqnarray}\nfor arbitrary kernel, which can also be easily shown from the GLE,\n and therefore \\footnote{For nonequilibrium\ninitial preparations this result holds asymptotically in any asymptotic \nergodic case, \nincluding the FLE dynamics. \nThe relaxation to the asymptotic regime, or aging, can be but very\nslow \\cite{Pottier,Deng} which is a general feature of subdiffusive GLE dynamics \nalso in periodic potentials.} \n\\begin{eqnarray}\\label{normal}\n\\frac{\\langle \\delta x(t) \\rangle}{\n\\langle \\delta x^2(t) \\rangle}=\\frac{F}{2k_BT} \\;\n\\end{eqnarray}\nfor the thermally equilibrium initial preparation.\nFor the FLE with a finite $m$ the inversion of Eq. (\\ref{x2}) to the time domain\nyields\\cite{Lutz}, \n\\begin{eqnarray}\n\\langle \\delta x^2(t) \\rangle=2v_T^2t^2 E_{2-\\alpha,3}\n[-(t\/\\tau_{v})^{2-\\alpha}], \n\\end{eqnarray}\nwhere $E_{\\gamma,\\beta}(z)=\n\\sum_{n=0}^{\\infty}z^n\/\\Gamma(n\\gamma+\\beta)$ is the generalized Mittag-Leffler\nfunction. One recovers Eq. (\\ref{variance}) in the limit $m\\to 0$. \n \n\nHowever, for the subdiffusive CTRW\nand FFPE dynamics the behavior of the ensemble-averaged variance is very different \nfrom Eq. (\\ref{variance}) under a non-zero bias\n$F\\neq 0$. Then, the Eq. (\\ref{variance}) is not valid anymore. \nThis fact is ultimately related to the properties of the stochastic\nclock. The point is that starting from a CTRW picture it is easy \nto show (see Appendix A) \nthat the growing ensemble-averaged \nvariance $\\langle \\delta x^2(t)\\rangle$ depends in the asymmetric case\n(the probabilities to jump left and right are different) not only on the\nmean number $\\langle n(t)\\rangle$ of the stochastic clock periods passed, \nbut also on their variance $\\langle \\delta n^2(t)\\rangle$.  For $\\alpha=1$\n(regular clock), $\\langle \\delta n^2(t)\\rangle$=0. However, for\n$0<\\alpha<1$, $\\langle \\delta n^2(t)\\rangle \\propto t^{2\\alpha}$ and this\ndramatically changes the character of anomalous CTRW and FFPE diffusion in the\npresence of bias. It becomes asymptotically \n$\\langle \\delta x^2(t)\\rangle\\propto F^2 t^{2\\alpha}$, while \n$\\langle \\delta x(t)\\rangle\\propto F t^{\\alpha}$. Notice that for $1\/2<\\alpha<1$\nthe subdiffusion at $F=0$ transforms into superdiffusion for $F\\neq 0$, i.e. a cloud\nof particles spreads out anomalously fast relative to its center of mass. \nThis yields a remarkable\nscaling for the ensemble-averaged quantities\n\\begin{eqnarray} \\label{anomal}\n\\lim_{t \\to \\infty} \\frac{\\langle \\delta x^2(t) \\rangle}{\n\\langle \\delta x(t) \\rangle^2} = \n\\lim_{t \\to \\infty} \\frac{\\langle \\delta n^2(t) \\rangle}{\n\\langle n(t) \\rangle^2}=\\frac{2\n\\Gamma ^2(\\alpha + 1)}{\\Gamma(2 \\alpha + 1)} - 1 \\, .\n\\end{eqnarray}\nThis scaling, which was observed first in Refs. \\cite{Shlesinger1,Scher} for a CTRW\nsubdiffusion in the absence of any additional potential $U(x)$, has been shown to be \n{\\it universal} within the FFPE description \nalso for arbitrary tilted washboard\npotentials and temperature \\cite{GH06,H06}. Recently, this astounding fact has been\nrelated to the universal fluctuations\nof anomalous mobility and weak ergodicity breaking \\cite{Sokolov}. \nUltimately, this is just \nthe property of the stochastic clock and it reflects the scaling between the variance\nand the mean number of stochastic periods passed within the external observed time $t$.\nSurprisingly, the viscoelastic GLE subdiffusion also exhibits a {\\it universal}\nasymptotical scaling in tilted washboard potentials. In the $t\\to\\infty $ limit\nit is the {\\it same} as in Eq. (\\ref{normal}).\nAstonishingly, it works both for a vanishingly small $F$, and for an arbitrary\nstrong bias. Moreover, both the diffusion and drift in the\ntilted washboard potentials\ndo not depend {\\it asymptotically} on the amplitude and the form of the periodic potential \nin the case of GLE subdiffusion and are given by \nEqs. (\\ref{variance}) and (\\ref{mean}), correspondingly. \nThis again is very much different from the FFPE case, where Eq. (\\ref{normal})\ncan be used only to calculate the anomalous flux response at a \nvanishingly small $F$ from the equilibrium $\\langle \\delta x^2(t) \\rangle_{F=0}$\nat $F=0$.  Also, given $\\langle \\delta x(t)\\rangle$ at $F\\neq 0$\none can calculate $\\langle \\delta x^2(t) \\rangle_{F=0}$ using Eq. (\\ref{normal}) \nand the corresponding \nsubdiffusion coefficient in periodic potentials in the limit $F\\to 0$,\nfor details see in the work\\cite{H07} and below.    \n\n\n\\section{Other similarities}\n    \nOne more similarity emerges for the relaxation of mean fluctuation from\nequilibrium in harmonic potentials, $U(x)=k x^2\/2$.  Then, both the FFPE\napproach and the FLE approach (in the limit $m\\to 0$) yield the same\nrelaxation law \\cite{Kou,Goychuk07a,Metzler99}, \n$\\langle \\delta x(t)\\rangle = \n\\langle \\delta x(0)\\rangle E_{\\alpha}[-(t\/\\tau_r)^{\\alpha}]$ with the\nultraslow position relaxation time constant $\\tau_r=(\\eta_{\\alpha}\/k)^{1\/\\alpha}$. \nAsymptotically, this relaxation follows a power-law, \n$\\langle \\delta x(t)\\rangle \\propto t^{-\\alpha}$. \n\nThe asymptotic \ndistributions of the residence times within a half-infinite spatial \ndomain (or the first \nreturn times to the origin in the infinite domain)\nin the case of free subdiffusion\nare also similar, following the same scaling law \\cite{Ding,Taloni,GH04} \n$\\Psi(\\tau) \\propto 1\/\\tau^{2-\\alpha\/2}$.\nHowever, here the similarities end. The asymptotics for a finite-size domain\ncannot be same. In particular, the mean residence time \nin any finite-size domain within the subdiffusive GLE description is finite \n\\cite{Goychuk09},\nwhereas within the FFPE description is not, except for the case of \ninjection of diffusing particles on the normal radiative boundary, where they\ncan be immediately absorbed\\cite{GH04}. Moreover, the GLE (for arbitrary $\\eta(t)$, \nincluding FLE) describe a Gaussian process for constant and linear forces $f(x)$\n\\footnote{This is just by the linearity of the transformation from the\nGaussian noise $\\xi(t)$ to the stochastic process $x(t)$ as\ndescribed by Eqs. (\\ref{fle}) and (\\ref{gle}).}, whereas the FFPE does not correspond\nto a Gaussian process in these cases, see\nin Ref.\\cite{Metzler}.   \n\n \n\\section{Diffusion and transport in washboard potentials}\n\nLet us proceed with the case of washboard potentials, where the differences\nbetween the two discussed approaches to subdiffusion become particularly transparent.\nWe consider the tilted potential $U(x)=V(x)-xF$, where $V(x+L)=V(x)$ is a periodic\npotential with the spatial period $L$.\n\n\\subsection{FFPE dynamics} \n\nIn this case, one can find exact analytical results for the ensemble-averaged\nnonlinear mobility $\\mu_{\\alpha}(F)$ using Eq. (\\ref{mean}) asymptotically\nalso in washboard potentials. First, one finds the exact analytical expression\nfor the ensemble-averaged subvelocity $v_{\\alpha}(F)=\\mu_{\\alpha}(F)F$.\nThe FFPE in the form (\\ref{FFPE2}) is more convenient for this purpose. \nIndeed, it has the form of a fractional-time continuity \nequation with the flux $J(x)$. For the sake of\ngenerality we consider its further generalization with a spatially-dependent\nsubdiffusion coefficient $\\kappa_{\\alpha}(x)$, \n\\begin{eqnarray}\nJ(x,t)=-\\kappa_{\\alpha}(x)\ne^{-\\beta U(x)} \\frac{\\partial}{\\partial x} \\, e^{\\beta\nU(x)} P(x,t)\n\\end{eqnarray}\nwhich is assumed to be periodic with the same period $\\kappa_{\\alpha}(x+L)\n=\\kappa_{\\alpha}(x)$, and the generalized Einstein relation\nis fullfield locally at any $x$, $\\kappa_{\\alpha}(x)=k_BT\/\\eta_{\\alpha}(x)$. \nWe proceed similarly to the case of normal \ndiffusion \\cite{Stratonovich,ReimannRev,HM09},\n$\\alpha=1$. A spatial period averaged density $\\hat P(x,t)=\\sum_{k=-M}^{M} P(x+kL,t)\/(2M+1)$\nshould attain a steady-state regime (corresponding to a non-equilibrium\nsteady state for $F\\neq 0$) in the limit $M\\to\\infty$, $t\\to\\infty$ \nand that becomes periodic with the period $L$, $\\hat P_{\\rm st}(x+L)=\\hat P_{\\rm st}(x)$.\nThe corresponding subdiffusive flux $\\hat J(x)$, defined with $\\hat P_{\\rm st}(x)$, \nbecomes a constant $J_\\alpha$ in the steady state: \n\\begin{eqnarray}\\label{1}\nJ_\\alpha=-\\kappa_{\\alpha}(x)\ne^{-\\beta U(x)} \\frac{d}{d x} \\, e^{\\beta\nU(x)} \\hat P_{\\rm st}(x)\\;.\n\\end{eqnarray}\nThen, the dynamics of the averaged mean displacement follows as \n\\begin{eqnarray}\\label{aux}\n\\sideset{_{0}}{_{*}}{\\mathop{D}^{\\alpha}}\\langle x(t)\\rangle =LJ_\\alpha\\;,\n\\end{eqnarray}\nwhich can be shown akin to the normal diffusion case\\cite{ReimannRev}. The appearance of the fractional\nCaputo time derivative in the lhs of Eq. (\\ref{aux}) is the only mathematical\ndifference as compared with the normal diffusion case.\nThe solution of (\\ref{aux}) yields for the mean excursion\n\\begin{eqnarray}\n\\langle x(t)\\rangle = v_{\\alpha}^{({\\rm wb})}(F) t^{\\alpha}\/\\Gamma(1+\\alpha)\\;,\n\\end{eqnarray}\nwith  $v_{\\alpha}^{({\\rm wb})}(F)=LJ_\\alpha $ \nbeing the subvelocity in the washboard potential. \n\n\nOne finds $J_{\\alpha}$ and $v_{\\alpha}^{({\\rm wb})}(F)$ by multiplying \nEq. (\\ref{1}) with\n$e^{\\beta U(x)}\/\\kappa_{\\alpha}(x)$ and integrating the result within one\nspatial period. Taking into account the spatial periodicity of \n$V(x)$ and $\\kappa_{\\alpha}(x)$ this yields:\n\\begin{eqnarray}\\label{2}\nJ_\\alpha\\int_y^{y+L}\\frac{e^{\\beta U(x)}}{\\kappa_{\\alpha}(x)}dx&& = -e^{\\beta\nU(y+L)} \\hat P_{\\rm st}(y+L)+e^{\\beta\nU(y)} \\hat P_{\\rm st}(y)\\nonumber \\\\\n&& =(1-e^{-\\beta FL})e^{\\beta U(y)}\\hat P_{\\rm st}(y) \\;.\n\\end{eqnarray}\nNext, multiplying (\\ref{2}) with $e^{-\\beta U(y)}$, integrating over $y$ within\n$[0,L]$, and using the normalization $\\int_{0}^L\\hat P_{\\rm st}(y)dy=1$ one finds \nthe main result\n\\begin{eqnarray}\\label{result}\nv_\\alpha^{({\\rm wb})}(F)=\\frac{(1-e^{-\\beta FL})L}{\n\\int_0^L e^{-\\beta U(y)}dy\\int_y^{y+L}\n\\frac{e^{\\beta U(x)}}{\\kappa_{\\alpha}(x)}dx}\\;.\n\\end{eqnarray}\nAccordingly, the nonlinear anomalous mobility is $\\mu^{(\\rm wb)}_{\\alpha}(F)=\nv_\\alpha^{({\\rm wb})}(F)\/F$. This presents a further generalization of\nthe result for subvelocity in Refs. \\cite{GH06,H06} \nto a spatially-dependent subdiffusion coefficient $\\kappa_{\\alpha}(x)$. \nThe subdiffusion coefficient in the \nunbiased washboard potential for $F=0$ can also be found using\nthe generalized Einstein relation $\\kappa^{(\\rm wb)}_{\\alpha}(F=0)=k_BT \n\\mu^{(\\rm wb)}_{\\alpha}(F=0)$. It reads,\n\\begin{eqnarray}\\label{result2}\n\\kappa_\\alpha^{({\\rm wb})}(F=0)=\\frac{L^2}{\n\\int_0^L e^{-\\beta V(y)}dy\\int_0^{L}\n\\frac{e^{\\beta V(x)}}{\\kappa_{\\alpha}(x)}dx}\\;,\n\\end{eqnarray}\nand for $\\kappa_{\\alpha}=const$ this is the result  of the work\\cite{H07}.\nFor constant $\\kappa_{\\alpha}$ and a number of different potentials $V(x)$, \ntemperatures $T$\nand biasing forces $F$, \nthese two general results were beautifully confirmed by numerical simulations\nof the underlying CTRW \\cite{GH06,H06,H07} on a lattice from which the FFPE in the form \n(\\ref{FFPE2}) was derived in the work\\cite{GH06}. These simulations also confirmed \nthe universality of the scaling relation (\\ref{anomal}) within the FFPE approach. \nSurprisingly, it remains invariant\nalso in the presence of a driving which is periodic in time, \nin the biased case $F\\neq 0$ \\cite{H09}, featuring thus the universality class of\nsubdiffusion governed by a stochastic clock with divergent mean period\nand characterized by the only parameter $\\alpha$. The above $v_{\\alpha}^{(\\rm wb)}$\nis the ensemble-averaged result. The subvelocities of individual particles\nremain randomly distributed in the limit $t\\to\\infty$ and \nthey follow a universal subvelocity distribution which reflects the distribution of\nrandom individual time of travelling particles, as it has been clarified in\nRef. \\cite{Sokolov}.  Both the weak ergodicity breaking and the universal fluctuations of\nanomalous mobility within the FFPE approach are \nultimately related to this remarkable\nproperty of the stochastic time.\n\n\\subsection{GLE dynamics in periodic potentials}\n\nThe GLE subdiffusion distinctly differs in the physical mechanism and\nthis leads to quite different results for washboard potentials\\cite{Goychuk09,G10}. \nFirst of\nall, it is  {\\it asymptotically} ergodic and self-averaging over a single\ntrajectory yields a quite  definite non-random result \\cite{Goychuk09}. \nNo additional\nensemble averaging is required. Moreover, it turns out that both the\nparticle anomalous mobility $\\mu^{(\\rm wb)}_{\\alpha}$  and the\nsubdiffusion coefficient  $\\kappa^{(\\rm wb)}_{\\alpha}$ do not\ndepend {\\it asymptotically} neither on the potential $V(x)$, nor on the\nbias $F$ being universal and the same as for biased GLE subdiffusion\nin the absence of periodic potential, obeying the generalized Einstein\nrelation. The transition to this asymptotic regime is, however, very\nslow and it strongly depends on the amplitude of the periodic\npotential $V_0$ and  the temperature $T$. Because of this slowness of the \ntransient aging,  this asymptotic regime\nwill not necessarily be relevant on a finite time scale \nfor anomalous transport in finite-size systems. This is  especially so if the periodic\npotential amplitude exceeds the thermal energy by many times. However, this\nremarkable property features the very mechanism of the GLE subdiffusion, which\nis based on the long-range velocity and displacement correlations and not on\ndiverging mean residence time within a potential well, in clear contrast to\nthe CTRW subdiffusion with independent increments. \nIt outlines a quite different universality class\nof subdiffusion. This is the long-range anti-persistence\nwhich limits asymptotically the GLE subdiffusion and transport processes in the \nwashboard potentials. Since the mean residence time in a potential well \nis finite \\cite{Goychuk09}, a coarse graining over the potential period, \nwhich makes the sojourns in the\ntrapping potential wells irrelevant, becomes {\\it asymptotically} possible.\nIn fact, upon increasing the potential height the escape kinetics out \nof a potential well (being asymptotically stretched-exponential) becomes ever closer\nto the normal exponential kinetics \\cite{Goychuk09}, where it becomes  described \nby the non-Markovian rate theory\\cite{HTB90,HM82}. \nThis does not mean, however, that the diffusion spreading over many spatial periods \nbecomes normal. As a matter of fact, in the unbiased periodic potentials the\ndiffusion cannot become faster than the free subdiffusion and this is a reason why \nthe asymptotic limit of free subdiffusion is attained.  A signature of this \nuniversality has been revealed theoretically for quantum transport in sinusoidal\npotentials for the case of sub-Ohmic thermal bath which classically corresponds\nto the considered case of fractional sub-diffusive friction. Technically this\nwas done by\nusing two different approaches, one perturbative\\cite{Chen} and one non-perturbative\nbased on a quantum duality transformation between the quantum dissipative washboard\ndynamics coupled to a sub-Ohmic bath and a quantum dissipative tight-binding dynamics\ncoupled to a super-Ohmic bath\\cite{WeissBook}. In the quantum case, there are \nalso tunneling\nprocesses which are accounted for. Our numerical results for the classical\nBrownian dynamics indicate, however, that this feature is purely \nclassical and, moreover, it is universal, i.e. is beyond the particular case of\nsinusoidal potentials\\cite{G10}.  \nIt is {\\it not} caused by the quantum-mechanical effects.\n\nOur numerical simulation approach is also insightful and it can be considered\nas an independent theoretical route to model anomalous diffusion and transport processes.\nThe idea is to approximate the non-Markovian GLE dynamics with a power-law kernel\nby a {\\it finite-dimensional} Markovian dynamics of a sufficiently high \ndimensionality $D$ \\cite{Marchesoni,Kupferman,Goychuk09,Siegle}. Here,\n\"sufficient\" means the following: having subdiffusion \nextending over $r$ time-decades one finds a $D$-dimensional Markovian dynamics\nwhose projection on the $(x,v)$ plane approximates the GLE dynamics over the required\ntime range within the\naccuracy of stochastic simulations, as it can be checked for the cases where\nan exact solution of the GLE dynamics is available \n(free or  biased subdiffusion, subdiffusion\nin harmonic potentials). Increasing $D$ one can cover larger $r$ of experimental\ninterest and the embedding\ndimension $D$ turns out to be finite \nto arrive at the asymptotic results\nvalid for the strict power law kernel. Needless to say that the practically\nobserved cases of anomalous diffusion hardly extend over more than 6 time decades\n(typically several only) which underpins the practical value of our approach.\n \nWe expand the power law kernel into a sum of exponentials\n\\begin{eqnarray}\\label{expansion}\n\\eta(t)=\\frac{\\eta_{\\alpha}}{\\Gamma(1-\\alpha)}C_{\\alpha}(b)\n\\sum_{i=1}^{N}\\nu_i^\\alpha \\exp(-\\nu_i t)\n\\end{eqnarray}\nobeying a fractal scaling with $\\nu_i=\\nu_0\/b^{i-1}$, where $b>1$ is a scaling \nparameter, $\\nu_0>0$ is high-frequency (short-time) cutoff corresponding to\nthe fastest time scale $\\tau_0=1\/\\nu_0$ in the hierarchy of the \nrelaxation time constants,\n$\\tau_i=\\tau_0b^{i-1}$, of viscoelastic memory kernel. \n$C_{\\alpha}(b)$ is  a numerical constant to provide\na best fit to $\\eta(t)=\\eta_{\\alpha}t^{-\\alpha}\/\\Gamma(1-\\alpha)$ in the\ninterval $[\\tau_0,\\tau_0 b^{N-1}]$. In the theory of anomalous relaxation\nsimilar expansions are well-known \\cite{Hughes,Palmer}. In the present context, \nthe approach corresponds to an approximation of the\nfractional Gaussian noise by a sum of uncorrelated \nOrnstein-Uhlenbeck (OU) noises, \n$\\xi(t)=\\sum_{i=1}^{N}\\zeta_i(t)$, with \nautocorrelation functions, $\\langle\\zeta_i(t)\\zeta_j(t')\\rangle=k_BT \\kappa_i\n\\delta_{ij}\\exp(-\\nu_i|t-t'|)$. This idea is also known in the theory of\n$1\/f$ noise \\cite{Weissman}. \nFor $t>\\tau_0 b^{N-1}$ the tail of (\\ref{expansion}) is \nexponential and the diffusion becomes normal for $t\\gg \\tau_0 b^{N-1}$.\nHowever, by increasing $N$ one can enlarge the corresponding time scale\nand even make it practically irrelevant.\nThe subdiffusion can be modelled in this way over $r=N\\log_{10} b-2$ time decades\nand the corresponding embedding dimension, $D=N+2$, can be rather small. Such fits\nare known to exhibit logarithmic oscillations superimposed on the power law \\cite{Hughes}.\nHowever, their amplitude can be made negligibly small if to choose $b$\nsufficiently small, e.g. for $b=2$ they become already barely detectable. \nNevertheless, even the decade scaling with $b=10$\nsuffices to arrive at excellent (within the statistical errors of Monte Carlo\nsimulations) approximation of the FLE dynamics by a {\\it finite-dimensional} Markovian\ndynamics over a huge range of time scales.\nWeak logarithmic sensitivity of $r$ to $b$  and linear dependence  on $N$ allows one \nto improve the quality of\nMarkovian embedding at a moderate computational price. The choice of \nMarkovian embedding which corresponds to (\\ref{expansion}) is not \nunique\\cite{Kupferman,Siegle}. \nA particular one is the following \\cite{Goychuk09}: \n\\begin{eqnarray}\\label{embedding1}\n\\dot x&=& v\\;,\\nonumber \\\\\nm\\dot v & =& f(x,t)+\n\\sum_{i=1}^{N}u_i(t) \\;,\\nonumber \\\\\n\\dot u_i& = &-k_i v-\\nu_iu_i+\\sqrt{2\\nu_i k_i k_BT}\\xi_i(t) \\;,\n \\end{eqnarray}\nwhere $k_i=C_\\alpha(b)\\eta_\\alpha\\nu_i^\\alpha\/\\Gamma(1-\\alpha)>0$ and \n$\\xi_i(t)$ are independent unbiased white Gaussian noise sources, \n$\\langle \\xi_i(t)\\xi_j(t')\\rangle=\\delta_{ij}\\delta(t-t')$. \nIndeed, integrating out the auxiliary force variables $u_i$ in Eq. (\\ref{embedding1})\nit follows that the resulting dynamics is equivalent to\nthe GLE (\\ref{gle}), (\\ref{fdt}) with the kernel (\\ref{expansion}), provided that $u_i(0)$ are unbiased\nrandom Gaussian variables with variances $\\langle u_i^2(0)\\rangle=k_ik_BT$.\nThe latter condition ensures the stationarity of $\\xi(t)$ in the GLE (\\ref{gle}),\nas well as validity of the FDR (\\ref{fdt}) {\\it for all times}.\nUsing different non-thermal preparations of $u_i(0)$ one can study the influence\nof {\\it initial} non-stationarity of the noise $\\xi(t)$ in the GLE \non the Brownian dynamics\\cite{Siegle}. In this aspect, our approach is even more flexible and \nmore general than the standard GLE approach.\n\nThe auxiliary variables $u_i$\ncan be interpreted as elastic forces, $u_i=-k_i(x-x_i)$, exerted \nby some overdamped particles \nwith positions $x_i$, which are coupled to the central Brownian particle with elastic\nspring constants $k_i$ and are subjected to viscous friction with frictional\nconstants $\\eta_i=k_i\/\\nu_i=C_{\\alpha}(b)\n\\eta_{\\alpha}\\tau_i^{1-\\alpha}\/\\Gamma(1-\\alpha)$ \nand the thermal random forces of environment. This corresponds to\nmotion of $N+1$ particles in a potential $U(x,\\{ x_i\\})=U(x,t)+(1\/2)\n\\sum_{i=1}^N k_i(x-x_i)^2$. The Brownian particle is massive (inertial effects\nare generally included), all other ``particles'' are overdamped \n(massless, $m_i\\to 0$). For example, one can imagine \nthat some coordination spheres\nof the viscoelastic environment stick to the Brownian particle and are co-moving.\nTheir influence can be effectively represented by $N$ ``quasi-particles''.  \nIn this insightful physical interpretation, \nour embedding scheme is equivalent to: \n\\begin{eqnarray}\\label{embedding2}\nm\\ddot x & =& f(x,t)-\n\\sum_{i=1}^{N}k_i(x-x_i) \\;,\\nonumber \\\\\n\\eta_i\\dot x_i& = &k_i (x-x_i)+\\sqrt{2\\eta_ik_BT}\\xi_i(t) \\;.\n \\end{eqnarray}\nIt worth to notice that in this approach the mass of the Brownian particle\nand therefore the inertial effects are important. In order to perform an overdamped\nlimit $m\\to 0$, one has to include the viscous frictional force $-\\eta_0 \\dot x$ \nacting directly on \nthe particle and the corresponding random force. Then, in the limit $m\\to 0$,\none obtains\n\\begin{eqnarray}\\label{embedding3}\n\\eta_0 \\dot x & =& f(x,t)-\n\\sum_{i=1}^{N}k_i(x-x_i)+ \\sqrt{2\\eta_0k_BT}\\xi_0(t) \\;,\\nonumber \\\\\n\\eta_i\\dot x_i& = &k_i (x-x_i)+\\sqrt{2\\eta_ik_BT}\\xi_i(t) \\;,\n \\end{eqnarray} \nwhere $\\xi_0(t)$ is a zero-mean Gaussian random force of unit intensity which is not\ncorrelated with the set $\\{ \\xi_i(t) \\}$. \nHowever, it was noticed \\cite{Burov} that the inertial effects are\ngenerally important for the subdiffusive GLE dynamics and therefore we take them\ninto account. Of  course, here emerges one more difference with the alternative \ndescription of subdiffusion within the FFPE (\\ref{FFPE1}), (\\ref{FFPE2}). \n \n\\begin{figure}\n\\includegraphics [width=0.5\\linewidth]{WSFig2.eps}\n\\caption{Anomalous diffusion in the potential $U(x)=-V_0\\sin(2\\pi x\/L)-Fx$ for various $V_0$\nand $F$ at $T=0.1$ for $\\alpha=0.5$. Notice an excellent agreement (differences\npractically cannot be detected in this plot) \nof simulations\nwith the exact FLE result, $\\langle \\delta x^2(t)\\rangle\n=2v_T^2t^2E_{2-\\alpha,3}[-(t\/\\tau_v)^{2-\\alpha}] $, \nin the absence of periodic potential $V_0=0$. Scaling: time in \n$\\tau_v=(m\/\\eta_{\\alpha})^{1\/(2-\\alpha)}$, distance in $L$, energy in \n$m(L\/\\tau_v)^2$, force in $mL\/\\tau_v^2$ and temperature in \n$mL^2\/(\\tau_v^2k_B)$.}\n\\label{Fig2}\n\\end{figure} \n \nA proper fractal scaling of coefficients $k_i$ and $\\eta_i$ with $i$ (see above) \nallows one to \nmodel viscoelastic subdiffusion\nover arbitrary time scales of the experimental interest. \nOne can numerically solve these stochastic differential equations (\\ref{embedding1})\ne.g. with a standard stochastic Heun method\\cite{Gard} (second order Runge-Kutta method)\nas done in Refs. \\cite{Goychuk09,G10}. \nAn example of such simulations is given in Figs. \\ref{Fig2}, \n\\ref{Fig3} for $\\alpha=0.5$, $\\nu_0=100$, $b=10$, $C_{\\alpha}(b)=1.3$, \n$N=12$ and $k_B T=0.1$. The following scaling is used: \ntime in the units of $\\tau_v$ \\footnote{This is a natural \nscaling of the velocity autocorrelation function in time. \nOther scalings are but also possible\\cite{Goychuk09,G10}. \nThey are more suitable to consider dynamical regimes close to overdamped.},\ndistance in the units of $L$. All the energy units are then scaled in \n$\\Delta E=m(L\/\\tau_v)^2$ and the force units in $mL\/\\tau_v^2$.\nStochastic Heun method is used to integrate Eq. (\\ref{embedding1}) with\na time step $\\Delta t=(1-5)\\cdot 10^{-3}$ until $t_{\\rm max}=2\\cdot 10^5$ \nand $n=10^4$ trajectories are used for\nthe ensemble averaging. Stochastic numerics are compared against the exact\nresults for the free subdiffusion and  for the mean displacement \nunder a constant biasing force. The agreement is excellent. The considered\nparticular  embedding\nstill works as an approximation to the FLE dynamics until $t=10^8$. If one needs to\ndescribe subdiffusion on an even longer time scale one can increase $N$. If one needs\na better precision of approximation one can make $b$ smaller. \nInitially all the particles are localized at the origin, $x=0$, with the velocities\nthermally distributed at the temperature $T$. \nFor the time span \n$t\\lesssim\\tau_v$ the motion\nis always ballistically persistent (superdiffusion). This reflects the\ninertia of the Brownian particle. It assumes the subdiffusive \ncharacter for $t\\gg \\tau_v$, when the VACF\nis negative. The presence of a periodic potential\n$V(x)=-V_0\\sin(2\\pi x\/L)$ dramatically changes\nboth subdiffusion, $\\langle x^2(t)\\rangle-\\langle x(t)\\rangle^2$,\nas well as subdiffusive transport, $\\langle x(t)\\rangle$, on  intermediate time\nscale. However, the long time asymptotics\nof free or biased subdiffusion are  gradually attained. \nThe initial behavior still within one\npotential well remains ballistic.  One can \nconclude that both subdiffusion and subdiffusive transport are indeed asymptotically\ninsensitive to the presence of periodic potential within the GLE approach. \nThis finding is in\na striking contrast with the FFPE approach. \nHowever, the transient to\nthis asymptotic regime can be very slow, depending on the amplitude of the periodic\npotential and temperature. \n\nAn interesting phenomenon is also accelerated subdiffusion occurring on \nan intermediate time scale in tilted washboard potentials, as compared with the \nfree subdiffusion. It can be detected in Fig. \\ref{Fig2} for a strong yet subcritical\nbias $F=1<F_{\\rm cr}=2\\pi V_0\/L\\approx 1.2566...$. This calls to \nmind  the acceleration of normal diffusion in tilted \nwashboard potentials\\cite{Reimann3}. However, this accelerated\nsubdiffusive phenomenon\noccurs only on an intermediate time scale because  asymptotically the GLE \nsubdiffusion is not sensitive to the presence of the potential.\nOne more interesting effect occurs for the initially ballistic\ntransport. It first seems paradoxical that in the trapping potential \nthe initial transport becomes  \nfaster than in the absence of potential \nand not vice versa, see in Fig. \\ref{Fig3}. The result \ncan be understood in view of the fact that\nthe minimum of the potential under the strong bias $F$ is essentially\ndisplaced in the direction of biasing force and the particles are {\\it initially} \naccelerated \nby the additional to $F$ force  stemming from the periodic potential. \\\\ \\vspace{0.5cm}\n\n\n\n\n\n\n\\begin{figure}\n\\vspace{0.5cm}\n\\includegraphics [width=0.5\\linewidth]{WSFig3.eps}\n\\caption{Anomalous transport for $F>0$, see Fig.\\ref{Fig2} for details.}\n\\label{Fig3}\n\\end{figure}\n\n\\section{Summary and conclusions}\n\nWith this Chapter, we reviewed and scrutinized \ntwo different approaches, the FFPE approach and the FLE approach, to anomalously \nslow diffusion and transport in nonlinear force fields with a focus\non applications  in tilted periodic \npotentials. In spite of some similarities in the case of constant or linear forcings\nit was shown that the nonlinear dynamics radically differ, obeying\nasymptotically two different universality classes. A first one reflects the\nuniversal fluctuations of intrinsic time clock and is closely tight\nto a weak ergodicity breaking. \nIn contrast, within the GLE and FLE approach the long-range \nantipersistence of the velocity and position fluctuations renders the asymptotic dynamics\nergodic. One approach seems more appropriate for the disordered solids,\nor glass-forming liquids below the glass-forming \ntransition, as characterized by the nonergodic \nglass phase. Another one seems more appropriate for the regime \nabove but close to the glass\ntransition, or for crowded viscoelastic environments like cytosols in  biological\ncells. We have left out further pronounced differences between the \nFFPE and FLE approaches in the\ncase of time-dependent fields\\cite{SK06,HPRL07,G07,H09,West2}. \nWe are confident that  \nour results not only shed light on the origin of profound differences, but also\nwill stimulate a further development of both approaches to subdiffusion, and\npossibly other interrelationships emerging in random potentials.\n\n\n\n\n\\acknowledgements\n\nWe would like to thank  E. Heinsalu, M. Patriarca, G. Schmid, and P. Siegle for\na very fruitful collaboration on anomalous transport in washboard potentials. \nThis work was supported by the Deutsche Forschungsmeinschaft, grant \nNo. GO 2052\/1-1 (I.G.) and through Nanosystems Initiative Munich (P.H.).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Theory for heat flow, thermal response, and heat production during EDL buildup}\nThe classical understanding of the EDL is in terms of an ideally polarizable, flat electrode whose charge is screened via a compact, salt-concentration independent Stern or Helmholtz layer within \\aa ngstr\\\"{o}ms of the surface, together with a Gouy-Chapman layer that extends into the solution on the scale of the salt-concentration dependent  Debye length. Although the charging behavior of the water-immersed porous carbon electrodes of the main text is poorly described by Gouy-Chapman theory, \nit does aid the interpretation of the experiments of the main text. In particular, we will derive explicit expressions for the isothermal heat flow $\\mathbb{Q}$ (Appendix~\\ref{heatflow}) and the reversible heat production $\\Pi_{\\rm rev}$ (Appendix~\\ref{heatproduction}). A Gouy-Chapman approximation for the adiabatic temperature rise $\\Delta T_{\\rm adiab}$ (Appendix~\\ref{sec_adiabatictemperaturerise}) was previously derived in the supplemental material of Ref.~\\cite{janssen2017reversible}. To set the stage, we first review the known Gouy-Chapman expressions for the grand potential energy cost of EDL formation.\n\n\\subsection{Gouy-Chapman grand potential}\nWe consider a single electrode of area $A$ and an adjacent 1:1 electrolyte solution with a dielectric constant $\\varepsilon$, at temperature $T$ and at a bulk salt concentration $\\rho_{\\mathrm{s}}$. \nAt a finite electrode potential $\\Psi$, the surface obtains a surface charge density $\\sigma=Q\/(Ae)$, with $e$ the elementary charge, such that $\\sigma$ has the dimension of inverse area. To screen the surface charge, an inhomogeneous charge density profile develops in the electrolyte phase, characterized by the cationic and anionic density profiles $\\rho_{+}(z)$ and $\\rho_{-}(z)$, respectively. Here, the $z$-coordinate is defined as the direction out of the electrode into the solution, which runs from $z=0$ at the electrode to $z=\\infty$ far into the bulk, where the local electrostatic potential $\\psi(z)$ is zero and the ion densities take their bulk values $\\rho_{+}(\\infty)=\\rho_{-}(\\infty)=\\rho_{\\mathrm{s}}$.\nThe ionic density profiles can be obtained via classical density functional theory from an auxiliary functional $\\Omega_{\\mathcal{V}}$\nconsisting of an ideal-gas term and a mean-field Coulomb interaction term,\n\\begin{eqnarray}\\label{eq:Ftotal}\n&& \\frac{\\Omega_{\\mathcal{V}}[\\rho_{+},\\rho_{-}]}{Ak_{\\rm B}T} =  \\int_{0}^{\\infty}\\bigg\\{\\sum_{\\alpha=\\pm}\\rho_{\\alpha}(z) \\Big[ \\ln\\big(\\rho_{\\alpha}(z)\\Lambda_{\\alpha}^3\\big) - 1-\\beta\\mu_{\\alpha} \\Big] \\nonumber\\\\\n&&\\hspace{3.5cm}+ \\frac{1}{2}q(z) \\phi(z) \\bigg\\}dz, \n\\end{eqnarray}\nwith $k_{B}$ Boltzmann's constant,  $\\Lambda_{\\alpha}$ the thermal wavelength, $\\mu_{\\alpha}$ the ionic chemical potential, $\\phi(z)=e \\psi(z)\/k_{\\rm B}T$ the local dimensionless electrostatic potential, and $q(z)=\\sigma\\delta(z)+\\rho_{+}(z)-\\rho_{-}(z)$ the local unit charge density.  \nThe ion density profiles follow from the Euler-Lagrange equation $\\delta \\Omega\/\\delta \\rho_{\\pm}=0$ and amount to $ \\rho_{\\pm}(z)=\\rho_{\\mathrm{s}} \\exp{[\\mp\\phi(z)]}$ after setting the chemical potential to $\\mu_{\\pm}=k_{\\rm B}T\\ln \\rho_{\\mathrm{s}} \\Lambda_{\\pm}^{3}$. Inserting $ \\rho_{\\pm}(z)$ into the Poisson equation, $\\phi''(z) = -4\\pi {\\lambda_{\\mathrm{B}}} q(z)$, results in the Poisson-Boltzmann equation.\nThe main result of Gouy-Chapman theory is the analytic solution to that equation: \n\\begin{align} \\label{eq:10}\n\\phi=4  \\tanh^{-1}\\left(\\exp\\left[-\\lambda_{\\mathrm{D}}^{-1}z\\right] \\tanh{\\frac{\\Phi}{4}}\\right),\n\\end{align}\nwith $\\Phi=e\\Psi\/k_{\\rm B}T$ the dimensionless surface potential. Moreover, $\\lambda_{\\mathrm{D}}=(8\\pi\\lambda_{\\mathrm{B}}\\rho_{\\mathrm{s}})^{-1\/2}$ is the Debye length, in terms of the Bjerrum length $\\lambda_{\\mathrm{B}}=e^2\/(4\\pi \\varepsilon_{0} \\varepsilon k_{\\rm B}T)$, and the vacuum permittivity $\\varepsilon_{0}$. From Eq.~(\\ref{eq:10}), various other quantities can be derived, for instance the relation between the surface charge and potential\n\\begin{equation} \\label{eq:gouy_chapman}\n\\sigma=\\bar{\\sigma} \\sinh\\frac{\\Phi}{2},\n\\end{equation}\nwith $\\bar{\\sigma}=4\\rho_{\\mathrm{s}}\\lambda_{\\mathrm{D}}$ the crossover surface charge density.\n\n\nAccording to density functional theory, we find the grand potential $\\Omega(T,V_{\\mathrm{el}},\\mu_{+},\\mu_{-},Q)= \\Omega_{\\mathcal{V}}[\\rho_{+},\\rho_{-}]$ (with $V_{\\mathrm{el}}$ the electrolyte volume) by evaluating the auxiliary functional at the equilibrium ion concentration profiles. Within Gouy-Chapman theory this yields\n\\begin{widetext}\n\\begin{align}\\label{eq:equilibriumgrandpotential}\n \\Omega^{\\mathrm{GC}} &= \\underbrace{2 A \\rhosk_{\\rm B}T\\int_{0}^{\\infty}dz\\left\\{ \\phi(z) \\sinh \\phi(z)-\\cosh\\phi(z)+1\\right\\}}_{ \\displaystyle\\equiv  \\Omega^{\\mathrm{GC}}_{\\mathrm{ent}}}+  \\underbrace{\\frac{Ak_{\\rm B}T}{2}\\int_{0}^{\\infty}dz\\left[ q(z)\\phi(z)\\right]}_{  \\displaystyle\\equiv  \\Omega^{\\mathrm{GC}}_{\\mathrm{el}}},\n\\end{align}\n\\end{widetext}\nwhere we subtracted the bulk grand potential $-pV_{\\mathrm{el}}$ of the electrolyte with osmotic pressure $p=2\\rhosk_{\\rm B}T$. The term $\\Omega^{\\mathrm{GC}}_{\\mathrm{ent}}$, the integrand of which goes to zero in the bulk, is associated with the excess entropy of the double layer. Both integrals in Eq.~(\\ref{eq:equilibriumgrandpotential}) were solved in Ref.~\\cite{overbeek1990role} by employing the Gouy-Chapman solution Eq.~(\\ref{eq:10}). The resulting expression for the entropic contribution reads\n\\begin{align}\\label{eq:omegaent}\n\\Omega_{\\mathrm{ent}}^{\\mathrm{GC}}&=  A\\bar{\\sigma} k_{\\rm B}T\\left[\\Phi \\sinh\\frac{\\Phi}{2} -3 \\cosh\\frac{\\Phi}{2}+3\\right],\n\\end{align}\nand, likewise\n\\begin{align}\\label{eq:omegael}\n\\Omega^{\\mathrm{GC}}_{\\mathrm{el}}&=A\\bar{\\sigma} k_{\\rm B}T\\left[\\cosh{\\frac{\\Phi}{2}}-1\\right],\n\\end{align}\nwas found for the electrostatic contribution.\n\n\\subsection{Isothermal heat flow}\\label{heatflow}\nThe amount of heat that is required to flow into a capacitor during isothermal ($dT=0$) charging at temperature $T$ from an uncharged state to a charged state with charge $Q$ and potential $\\Psi$ was derived in Ref.~\\cite{hartel2015heat}:\n\\begin{equation}\\label{eq:thermodynamicidentity}\n\\mathbb{Q}= -T\\int_{0}^{Q}\\left(\\frac{\\partial \\Psi(Q',T)}{\\partial T}\\right)_{Q'}dQ'.\n\\end{equation}\nThe coulometric experiment determines temperature dependence of the equilibrium charge, $\\left(\\partial Q\/\\partial T\\right)_{\\Psi}$, which is related to the temperature dependence of the equilibrium potential via a cyclic reciprocity relation  \n\\begin{equation}\\label{eq_experimentalthermalvoltagerise}\n\\left(\\frac{\\partial \\Psi}{\\partial T}\\right)_{Q}=-\\left(\\frac{\\partial \\Psi}{\\partial Q}\\right)_{T}\\left(\\frac{\\partial Q}{\\partial T}\\right)_{\\Psi}.\n\\end{equation}\nHence, we can equivalently express the isothermal heat flow as\n\\begin{equation}\\label{eq:isothermalheatflow2}\n\\mathbb{Q}= T\\int_{0}^{\\Psi}\\left(\\frac{\\partial Q(\\Psi',T)}{\\partial T}\\right)_{\\Psi'}d \\Psi'.\n\\end{equation}\nIn the coulometric experiment of the main text we found  $\\left(\\partial Q\/\\partial T\\right)_{\\Psi}=\\alpha\\Psi$, with $\\alpha=-4.0\\pm0.1$~mC~V$^{-1}$~K$^{-1}$. Inserting this into the above equation we find $\\mathbb{Q}=\\alpha T\\Psi^2\/2$: the result stated in the main text.\n \nWithin Gouy-Chapman the heat flow reads [after inserting Eq.~(\\ref{eq:gouy_chapman})]\n\\begin{align}\\label{eq:gouychapmanisothermalheat}\n\\mathbb{Q}^{\\rm GC}\n&=-2Ak_{\\rm B}T\\int_{0}^{\\sigma }\\frac{\\partial }{\\partial T}\\left[T\\sinh^{-1} \\frac{\\sigma'}{\\bar{\\sigma}}\\right]d\\sigma'\\nonumber\\\\\n&= -2Ak_{\\rm B}T\\bigg[\\sigma\\sinh^{-1} \\frac{\\sigma}{\\bar{\\sigma}}-\\sqrt{\\bar{\\sigma}^{2}+\\sigma^{2}}+\\bar{\\sigma}\\bigg],\n\\end{align}\nwhere we ignored the temperature dependence of $\\bar{\\sigma}$.\nIn terms of the dimensionless surface potential $\\Phi$, and comparing to Eqs.~(\\ref{eq:omegaent}) and (\\ref{eq:omegael}), we find\n\\begin{align}\\label{eq:}\n\\mathbb{Q}^{\\rm GC}=& -A\\bar{\\sigma}k_{\\rm B}T \\bigg[2-2\\cosh\\frac{\\Phi}{2}+\\Phi\\sinh\\frac{\\Phi}{2}\\bigg]\\nonumber\\\\\n=&-\\Omega_{\\mathrm{ent}}^{\\mathrm{GC}}.\n\\end{align}\nThermodynamically, this equivalence is understood as follows. The differential $d\\Omega=-SdT+\\Psi dQ$ of the grand potential (at fixed $\\mu_{\\pm}$ and $V_{\\mathrm{el}}$) of the electric double layer capacitor\nimplies that the isothermal grand potential change during charging is equal to (minus) the performed work, $ \\Omega=-W=\\int\\Psi dQ$.\nMeanwhile, the isothermal heat Eq.~(\\ref{eq:thermodynamicidentity}) is equal and opposite, $\\mathbb{Q}=-\\int \\Psi dQ$, {\\it if} the charging behavior is such that $\\left(\\partial \\Psi\/\\partial T\\right)_{Q}=\\Psi\/T$, which is precisely what we assumed in this section by neglecting $\\partial \\bar{\\sigma}\/\\partial T$ in Eq.~(\\ref{eq:gouychapmanisothermalheat}). This implies moreover that the internal energy change is zero for the Gouy-Chapman model. In general, more advanced EDL models will have a nonzero internal energy change upon charging, in which case the heat flow is not equal to the grand potential  change. \n\n\\subsection{Adiabatic temperature rise}\\label{sec_adiabatictemperaturerise}\nSlowly charging a thermally insulated electric double layer capacitor leads to a temperature rise $\\Delta T_{\\rm adiab}$, which is the capacitive analogue of the temperature increase upon compressing a gas in a thermally insulated container \\cite{Janssen:2014aa}. Here, $\\Delta T_{\\rm adiab}$ is found by integrating the relation \n\\begin{equation}\\label{eq:adiabatic_Tincrease}\ndT=\\frac{T}{\\mathbb{C}_{p}}\\left(\\frac{\\partial Q(\\Psi,T)}{\\partial T}\\right)_{\\Psi}d \\Psi,\n\\end{equation}\nthat follows from the total differential of $S(T,\\Psi)$, together with the $dS=0$ condition on isentropic processes.  Using once more $\\left(\\partial Q\/\\partial T\\right)_{\\Psi}=\\alpha\\Psi$ (with $\\alpha=-4.0\\pm0.1$~mC~V$^{-1}$~K$^{-1}$) as obtained with the coulometric experiment of the main text, we find a small adiabatic temperature rise,\n\\begin{equation}\n\\Delta T_{\\rm adiab}=\\frac{\\alpha T\\Psi^2}{2\\mathbb{C}_{p}},\n\\end{equation}\nupon increasing the potential from 0 to $\\Psi$, with $T$ the initial temperature at the uncharged state.\n\n\n\n\n\\subsection{Reversible heat production}\\label{heatproduction}\nOverbeek pointed out that the work performed  by the electric field on ions in a solution during quasistatic EDL buildup equals the entropic contribution to the grand potential  $\\Omega_{\\mathrm{ent}}$ \\cite{overbeek1990role}. Meanwhile, the source term ${\\bf I}\\cdot{\\bf E}$ appearing in the heat equation~\\eqref{eq:microheatequation} is associated precisely with the power delivered locally by the electric field for out of equilibrium settings. This suggests the following relation\n\\begin{equation} \\label{eq:3}\n\\Pi_{\\rm rev}^{\\mathrm{ch}}\\equiv\\lim_{\\mathcal{T}\\to\\infty}\\int_{0}^{\\mathcal{T}} dt\\int_{\\mathcal{V}_{\\rm el}} d{\\bf x}~{\\bf I}\\cdot{\\bf E}=\\Omega_{\\mathrm{ent}},\n\\end{equation}\nwhere $\\mathcal{T}$ is the duration of the slow charging process.\nTo test  Eq.~(\\ref{eq:3}), we consider a very slow charging process of an electrode from an uncharged state, to some final state at charge $Q$. We study the electrode as described in the introduction of this section, and assume that the instantaneous density profiles are given as in Gouy-Chapman theory [Eq.~(\\ref{eq:10})] and moreover that the temperature $T(z,t)$ is homogeneous throughout the electrolyte volume at each time $t$. Under these conditions, the integrand of Eq.~(\\ref{eq:3}) simplifies to $IE$, with $E=-\\partial_{z}\\psi$ the local electric field, and $I=eJ_{\\mathrm{\\mathrm{ch}}}=e(J_{+}-J_{-})$ the ionic current density (in A m$^{-2}$). \nThe time integral in Eq.~(\\ref{eq:3}) can be transformed into an integral over the unit surface charge \\mbox{density $\\sigma$}\n\\begin{align} \\label{eq:4}\n\\Pi_{\\rm rev}^{\\mathrm{ch, GC}}=-Ak_{\\rm B}T\\int_{0}^{\\sigma} \\left[ \\int_0^\\infty \\frac{J_\\mathrm{\\mathrm{ch}}}{J_\\text{tot}}\\> \\frac{\\partial \\phi}{\\partial z} \\> dz \\right] d\\sigma',\n\\end{align}\nwhere we implemented \n\\begin{equation} \\label{eq:5}\n\\frac{\\partial \\sigma}{\\partial t}=J_\\text{tot},\n\\end{equation}\nwith $J_{\\text{tot}}=J_{\\mathrm{\\mathrm{ch}}}+J_{\\mathrm{M}}$ the total current density, which is equal to the electronic current entering the electrode. Here, $J_{\\text{tot}}$, invariant with position in a Cartesian one-dimensional geometry, includes the Maxwell or displacement current $J_{\\mathrm{M}}=-\\varepsilon_{0}\\varepsilon\\partial_{t}\\partial_{z}\\psi$, which is proportional to the variation of the electric field in time \\cite{van2010diffuse}.  As the Maxwell current vanishes in the charge neutral bulk, $J_\\text{tot}$ is equal to the ionic current $J_{\\mathrm{ch}}$ far from the electrode. For capacitive charging of the electrode, the ionic current $J_{\\mathrm{ch}}$ is zero at the electrode surface and increases with $z$ to reach $J_\\text{tot}$ outside the EDL. Thus, in general $J_{\\mathrm{ch}}$ is a fraction of $J_\\text{tot}$, i.e., $0<J_{\\mathrm{ch}}\/J_\\text{tot}<1$. \nThe ionic current density $J_{\\mathrm{ch}}$ relates to the local ionic charge density $q$ according to the continuity equation\n\\begin{equation} \\label{eq:6}\n\\frac{\\partial J_{\\mathrm{ch}}}{\\partial z}=-\\frac{\\partial q}{\\partial t}.\n\\end{equation}\nWithin Gouy-Chapman  theory $q=-2 \\rho_{\\mathrm{s}} \\sinh{\\phi}$ and thus Eqs.~(\\ref{eq:5}) and (\\ref{eq:6}) yield\n\\begin{equation} \\label{eq:9}\n \\frac{\\partial}{\\partial z}\\left(\\frac{J_{\\mathrm{ch}}}{J_\\text{tot}}\\right)=2 \\rho_{\\mathrm{s}} \\cosh{\\phi}\\frac{\\partial \\phi}{\\partial \\Phi}\\frac{\\partial \\Phi}{\\partial \\sigma}. \n\\end{equation}\nWe perform one partial integration in the bracketed integral in Eq.~(\\ref{eq:4}) and insert Eq.~(\\ref{eq:9}), such that\n\\begin{align} \\label{eq:13}\n\\frac{\\Pi_{\\rm rev}^{\\mathrm{ch, GC}}}{Ak_{\\rm B}T}&=-\\int_{0}^{\\sigma} \\left[\\int_0^\\infty \\frac{\\partial }{\\partial z}\\left(\\frac{J_{\\mathrm{ch}}}{J_\\text{tot}}\\right) \\phi \\> dz \\right] d\\sigma' \\nonumber\\\\\n&=-2 \\rho_{\\mathrm{s}} \\int_{0}^{\\sigma} \\left[\\int_0^\\infty  \\phi \\cosh{\\phi}\\frac{\\partial \\phi}{\\partial \\Phi}\\frac{\\partial \\Phi}{\\partial \\sigma}  \\> dz \\right] d\\sigma' \\nonumber\\\\\n&=-2 \\rho_{\\mathrm{s}} \\int_{0}^{\\Phi} \\left[\\int_0^\\infty  \\phi \\cosh{\\phi}\\frac{\\partial \\phi}{\\partial \\Phi}  \\> dz \\right] d\\Phi' \\nonumber\\\\\n&=-2 \\rho_{\\mathrm{s}} \\int_{0}^{\\Phi} \\left[\\int_{\\Phi'}^0  \\phi \\cosh{\\phi}\\frac{\\partial \\phi}{\\partial \\Phi}  \\frac{\\partial z}{\\partial \\phi} \\> d\\phi \\right] d\\Phi' ,\n\\end{align}\nwhere in the second to last step we took $\\left(\\partial \\Phi\/\\partial \\sigma\\right)$ out of the spatial integral since it does not depend on~$z$. \nThe integrand of Eq.~(\\ref{eq:13}) can be simplified considerably, in particular by converting the relation $\\phi(\\Phi,z)$ [Eq.~(\\ref{eq:10})] into $\\Phi(\\phi,z)$. This gives the following term that can be taken out of $\\phi-$integral in Eq.~(\\ref{eq:13}),\n\\begin{align}\n\\frac{\\partial \\phi}{\\partial \\Phi}  \\frac{\\partial z}{\\partial \\phi}\n&=\\left(\\frac{\\partial \\Phi}{\\partial z}\\right)^{-1}\n=\\frac{\\lambda_{\\mathrm{D}}}{4}\\frac{\\left[1-\\exp\\left[2z\/\\lambda_{\\mathrm{D}}\\right] \\tanh^{2}\\frac{\\phi}{4}\\right]}{\\exp\\left[z\/\\lambda_{\\mathrm{D}}\\right] \\tanh\\frac{\\phi}{4}}\\nonumber\\\\\n&=\\frac{\\lambda_{\\mathrm{D}}}{4}\\frac{\\left[1-\\tanh^{2}\\frac{\\Phi}{4}\\right]}{\\tanh\\frac{\\Phi}{4}}\n=\\frac{\\lambda_{\\mathrm{D}}}{2\\sinh\\frac{\\Phi}{2}},\n\\end{align}\nwhere, going to second line, we eliminated $z$ via Eq.~(\\ref{eq:10}). We can now evaluate Eq.~(\\ref{eq:13}), \n\\begin{align}\n\\frac{\\Pi_{\\rm rev}^{\\mathrm{ch, GC}}}{Ak_{\\rm B}T}&=-\\rho_{\\mathrm{s}}\\lambda_{\\mathrm{D}} \\int_{0}^{\\Phi}\\frac{1}{\\sinh\\frac{\\Phi'}{2}}\\left[\\int_{\\Phi'}^{0}\\phi\\cosh\\phi \\> d\\phi\\right]  \\> d\\Phi'\\nonumber\\\\\n&=-\\rho_{\\mathrm{s}} \\lambda_{\\mathrm{D}}\\int_{0}^{\\Phi}\\frac{\\cosh\\Phi'-\\Phi'\\sinh\\Phi' -1}{\\sinh\\frac{\\Phi'}{2}}d\\Phi'\\nonumber\\\\\n&=-2\\rho_{\\mathrm{s}} \\lambda_{\\mathrm{D}}\\int_{0}^{\\Phi}\\left[\\sinh\\frac{\\Phi'}{2}-\\Phi'\\cosh\\frac{\\Phi'}{2}\\right] d\\Phi'\\nonumber\\\\\n&=\\bar{\\sigma} \\left[3-3\\cosh\\frac{\\Phi}{2}+\\Phi\\sinh\\frac{\\Phi}{2}\\right] .\n\\end{align}\nComparing with Eq.~(\\ref{eq:omegaent}), we see indeed that \\mbox{$\\Pi_{\\rm rev}^{\\mathrm{ch, GC}}=\\Omega_{\\mathrm{ent}}^{\\mathrm{GC}}$}: precisely the hypothesis we set out to test. \n\n\n\\section{The heat transfer coefficient $K$}\\label{sec:calibration}\nThe heat transfer coefficient $K$ appearing in Eq.~\\eqref{eq:heatequation} was determined by two calibration methods. \n\n\\subsection{Calibration with heating elements}\\label{sec:alternativecalorimetry}\nExperiments were carried out with the same cell as described in the main text, but with the electrodes and current collectors replaced by heating elements (positioned at the same respective locations as the electrodes) consisting of resistive wire glued as a flat spiral onto a glass disk (total resistance: 413~$\\Omega$). \nMeasurements were conducted with the cell filled with water and with the cell filled with air. Depending on the current $I$ [see Fig.~\\ref{fig_sup2}(a)] of up to 10~mA, different voltages $\\Psi_{\\rm R}$ [Fig.~\\ref{fig_sup2}(b)] over the resistive wire were measured, and different plateau values $\\Delta T_{\\rm max}$ of the temperature difference $\\Delta T$  [Fig.~\\ref{fig_sup2}(c)] between the cell and the thermostatic bath were reached. \n\n\\begin{figure}\n\\def0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}{0.45\\textwidth}{\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%\n    \\renewcommand\\color[2][]{}%\n  }%\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%\n    \\renewcommand\\transparent[1]{}%\n  }%\n  \\providecommand\\rotatebox[2]{#2}%\n  \\ifx0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n    \\setlength{\\unitlength}{515.84695303bp}%\n    \\ifx\\svgscale\\undefined%\n      \\relax%\n    \\else%\n      \\setlength{\\unitlength}{\\unitlength * \\real{\\svgscale}}%\n    \\fi%\n  \\else%\n    \\setlength{\\unitlength}{0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}}%\n  \\fi%\n  \\global\\let0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n  \\global\\let\\svgscale\\undefined%\n  \\makeatother%\n  \\begin{picture}(1,0.83)%\n    \\put(0,0.05){\\includegraphics[width=\\unitlength]{Fig_sup2.pdf}}%\n    \\put(0.92,0.77){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(a)}}}%\n    \\put(0.92,0.54){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(b)}}}%\n    \\put(0.92,0.31){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(c)}}}%\n    \\put(-0.000,0.43){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{\\begin{rotate}{90} $\\displaystyle \\Psi_{\\mathrm{R}}$~[V]\\end{rotate}}}}%\n    \\put(0.00,0.67){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{\\begin{rotate}{90} $\\displaystyle I$~[A]\\end{rotate}}}}%\n    \\put(0.0,0.15){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{\\begin{rotate}{90} $\\displaystyle \\Delta T$~[K]\\end{rotate}}}}%\n     \\put(0.5,0.0){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{ $t$~[s]}}}%\n  \\end{picture}%\n  }\n\\caption{\\label{fig_sup2}%\nIn the calibration with the heating elements, the current $I$ (a) was cyclically switched on and off to different values, each lasting 400 s. The voltage $\\Psi_{\\rm R}$ (b) over the resistive wire, and temperature difference $\\Delta T$ (c) between the cell and the thermostatic bath  were measured simultaneously. Shown are data for the air-filled cell configuration.}\n\\end{figure}\n\nThe occurrence of $\\Delta T$-plateaus indicates that a steady state is reached where Joule heating is balanced by heat flowing out the cell. From Eq.~\\eqref{eq:heatequation} it then follows that the heat transfer coefficient $K$ can be determined as the slope of a $\\Delta T_{\\rm max}$-versus-$I\\Psi_{\\mathrm{R}}$ plot. In Fig.~\\ref{fig_sup3}(a) we see that $\\Delta T_{\\rm max}$ is proportional to the heating power $I\\Psi_{\\mathrm{R}}$. We find $K = 0.174\\pm0.001$~J~s$^{-1}$~K$^{-1}$ and $K = 0.184\\pm0.001$~J~s$^{-1}$~K$^{-1}$ for the air-filled and water-filled cell, respectively.\n\n\\begin{figure}\n\\def0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}{0.49\\textwidth}{\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%\n    \\renewcommand\\color[2][]{}%\n  }%\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%\n    \\renewcommand\\transparent[1]{}%\n  }%\n  \\providecommand\\rotatebox[2]{#2}%\n  \\ifx0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n    \\setlength{\\unitlength}{610.97107821bp}%\n    \\ifx\\svgscale\\undefined%\n      \\relax%\n    \\else%\n      \\setlength{\\unitlength}{\\unitlength * \\real{\\svgscale}}%\n    \\fi%\n  \\else%\n    \\setlength{\\unitlength}{0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}}%\n  \\fi%\n  \\global\\let0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n  \\global\\let\\svgscale\\undefined%\n  \\makeatother%\n  \\begin{picture}(1,0.46)%\n    \\put(0,0.032){\\includegraphics[width=\\unitlength]{Fig_sup3.pdf}}%\n    \\put(0.18,0.00){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\Delta T_{\\rm max}$ [K]}}}%\n    \\put(0.67,0.00){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\int\\Delta T dt$ [K s]}}}%\n    \\put(0.55,0.17){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{\\begin{rotate}{90} $\\int\\>I\\Psi_{\\mathrm{R}}\\>dt$ [J]\\end{rotate}}}}%\n    \\put(0.025,0.15){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{\\begin{rotate}{90} $I\\Psi_{\\mathrm{R}}$ [mJ s$^{-1}$]\\end{rotate}}}}%\n    \\put(0.16,0.375){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{water}}}%\n    \\put(0.16,0.33){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{air}}}%\n      \\put(0.0,0.45){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(a)}}}%\n    \\put(0.49,0.45){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(b)}}}%\n  \\end{picture}%\n  }\n\\caption{\\label{fig_sup3}%\nThe heat transfer coefficient $K$ is determined both from the plateau value $\\Delta T_{\\rm max}$ (a), as well as from the time-integrated temperature peaks $\\int\\Delta T dt$  (b), both for a water-filled cell (blue circles) and an air-filled cell (green triangles). Dotted lines indicate linear fits through these data.}\n\\end{figure}\n\nTaking a time integral over Eq.~\\eqref{eq:heatequation}, its left hand side vanishes for any time interval $t_{a}<t<t_{b}$ which starts ($t_{a}$) and ends ($t_{b}$) at $\\Delta T=\\SI{0}{\\degreeCelsius}$. The heat transfer coefficient can then alternatively be determined from the slope of a plot of $\\int_{t_{a}}^{t_{b}}\\>I\\Psi_{\\mathrm{R}}\\>dt$ versus $\\int_{t_{a}}^{t_{b}}\\>\\Delta T\\>dt$ [Fig.~\\ref{fig_sup3}(b)]. \nWe found $K = 0.173\\pm0.001$~J~s$^{-1}$~K$^{-1}$ for the air-filled cell and $K = 0.179\\pm0.001$~J~s$^{-1}$~K$^{-1}$ for the water-filled cell.\nClearly, these values agree well with those determined from Fig.~\\ref{fig_sup3}(a); we think that the second method is more reliable because it does not require a determination of the time at which $\\Delta T_{\\rm max}$ is reached.\n\nOnce the applied current was switched off, $\\Delta T$ decreased exponentially with time [see Figs.~\\ref{fig_sup2}(c) and \\ref{fig_sup2b}]; from Eq.~\\eqref{eq:heatequation}  follows $\\Delta T(t) = \\Delta T_{0} \\exp(-t\/\\tau)$, where $\\Delta T_{0}$ is the starting value of $\\Delta T$, $t$ is the time that passed since the current was switched off, and $\\tau = \\mathbb{C}_{p}\/K$ is the thermal time constant, with $\\mathbb{C}_{p}$ the heat capacity of the cell. From the measured time scales $\\tau = 34.0\\pm0.2$~s for the air-filled cell and $\\tau = 61.1\\pm0.3$~s for the water-filled cell, we calculate heat capacities of $\\mathbb{C}_{p} = K\\tau = 6.1\\pm0.1$~J~K$^{-1}$ and $10.9\\pm0.1$~J~K$^{-1}$, respectively. The difference between the two $\\mathbb{C}_{p}$ values agrees within~5\\% with a cell volume of approximately 1.1 mL and the specific heat capacity of water, 4.2~J~K$^{-1}$~g$^{-1}$. This consistency supports the validity of the heat quantities obtained. \n\\begin{figure}\n\\def0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}{0.46\\textwidth}\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%\n    \\renewcommand\\color[2][]{}%\n  }%\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%\n    \\renewcommand\\transparent[1]{}%\n  }%\n  \\providecommand\\rotatebox[2]{#2}%\n  \\ifx0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n    \\setlength{\\unitlength}{543.16897759bp}%\n    \\ifx\\svgscale\\undefined%\n      \\relax%\n    \\else%\n      \\setlength{\\unitlength}{\\unitlength * \\real{\\svgscale}}%\n    \\fi%\n  \\else%\n    \\setlength{\\unitlength}{0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}}%\n  \\fi%\n  \\global\\let0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n  \\global\\let\\svgscale\\undefined%\n  \\makeatother%\n  \\begin{picture}(1,0.63)%\n    \\put(0.04,0.02){\\includegraphics[width=\\unitlength]{Fig_sup2b.pdf}}%\n    \\put(0.7,0.53){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{water}}}%\n    \\put(0.7,0.46){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{air}}}%\n    \\put(0.03,0.25){\\color[rgb]{0,0,0}\\rotatebox{89.7025396}{\\makebox(0,0)[lb]{\\smash{$\\ln\\left[\\Delta T(t)\/\\Delta T_{0}\\right]$}}}}%\n    \\put(0.55,0.00){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{t [s]}}}%\n  \\end{picture}%\n\\caption{\\label{fig_sup2b}%\nThe heat capacity $\\mathbb{C}_{p}$ of the cell is determined from the exponential decay of  $\\Delta T$ after switching off the current $I$. Dotted lines indicate linear fits through these data.}\n\\end{figure}\n\nIn the main text we determined the adiabatic temperature rise $\\Delta T_{\\rm adiab}$ associated with charging of thermally insulated porous carbon electrodes via two independent routes. We note that both routes used $\\mathbb{C}_{p}=10.9\\pm0.1$~J~K$^{-1}$ as determined above, that is, with the heating elements. This means that the true heat capacity of the cell might differ slightly since it contains carbon electrodes instead of resistive wires.  \n\n\\subsection{Calibration with porous carbon}\\label{sec:carboncalibration}\nThe heat transfer coefficient $K$ can also be determined using the Joule heat that is generated during EDL formation in experiments with porous carbon electrodes. Employing the same arguments as presented in Appendix~\\ref{sec:alternativecalorimetry}, a time integral over Eq.~\\eqref{eq:heatequation} will leave its left hand side zero for any time interval that starts  and ends at $\\Delta T=\\SI{0}{\\degreeCelsius}$.\nIn particular, for a full cycle of charging and discharging as shown in Fig.~\\ref{fig3}  we have\n\\begin{align}\\label{eq:Kdetermination}\nK\\int_{t_{0}}^{t_{2}}\\Delta T dt&=\\Pi_{\\rm irr}^{\\mathrm{ch}}+\\Pi_{\\rm irr}^{\\mathrm{dis}}\\equiv \\Pi_{\\rm irr},\n\\end{align} \nwhere we used Eq.~\\eqref{eq:productionsplitting} and the fact that the reversible heat $\\Pi_{\\rm rev}^{\\mathrm{dis}}$ that is produced upon discharging will exactly cancel the reversible heat $\\Pi_{\\rm rev}^{\\mathrm{ch}}$ produced upon charging ($\\Pi_{\\rm rev}^{\\mathrm{ch}}=-\\Pi_{\\rm rev}^{\\mathrm{dis}}$). \nThus, the only heat production appearing in Eq.~(\\ref{eq:Kdetermination}) is Joule heat dissipated from the resistive parts of the system, \nwhich is exothermic regardless of the opposite signs of voltages and currents during charging and discharging.\nTo calculate this Joule heat, we time-integrate the dissipated power $\\dot{\\Pi}_{\\rm irr}=I(t)\\Psi_{\\mathrm{R}}(t)$ over the two time intervals. \nHere, $\\Psi_{\\mathrm{R}}(t)=\\Psi-\\Psi_{\\mathrm{EDL}}(t)$ is the voltage drop across the resistive elements that dissipate heat, \nwhich we can access experimentally if we assume a voltage drop $\\Psi_{\\mathrm{EDL}}(t) = Q(t)\/C$ over the electric double layer (EDL), with $Q(t)$ the charge in the EDL and the capacitance $C=\\lim_{t\\to\\infty}Q(t)\/\\Psi$ is assumed constant. \nInterestingly, we find that equal amounts of Joule heat are produced during charging ($\\Pi_{\\rm irr}^{\\mathrm{ch}}$) and discharging ($\\Pi_{\\rm irr}^{\\mathrm{dis}}$), despite the difference in $I(t)$ and $\\Psi_{\\mathrm{R}}(t)$ during these processes. The heat transfer coefficient $K$ can now be determined from Fig.~\\ref{fig_sup1}, where we plot $\\Pi_{\\rm irr}=\\int_{t_{0}}^{t_{2}} I(t)\\Psi_{\\mathrm{R}}(t)dt$ versus $\\int_{t_{0}}^{t_{2}}\\Delta T dt$, obtained for different salt concentrations $\\rho_{\\mathrm{s}}$ and different voltage steps $\\Psi$. A straight line is observed, whose slope corresponds to $K = 0.239\\pm 0.003$~J K$^{-1}$~s$^{-1}$. Considering only the data at $\\rho_{\\mathrm{s}}=1$~M, the linear fit has a slope $K = 0.232\\pm 0.005$~J~K$^{-1}$~s$^{-1}$, hence, $K$ depends very weakly on the salt concentration and can be regarded constant. \n\n\\begin{figure}\n\\def0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}{0.45\\textwidth}{ \n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%\n    \\renewcommand\\color[2][]{}%\n  }%\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%\n    \\renewcommand\\transparent[1]{}%\n  }%\n  \\providecommand\\rotatebox[2]{#2}%\n  \\ifx0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n    \\setlength{\\unitlength}{543.40262003bp}%\n    \\ifx\\svgscale\\undefined%\n      \\relax%\n    \\else%\n      \\setlength{\\unitlength}{\\unitlength * \\real{\\svgscale}}%\n    \\fi%\n  \\else%\n    \\setlength{\\unitlength}{0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}}%\n  \\fi%\n  \\global\\let0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n  \\global\\let\\svgscale\\undefined%\n  \\makeatother%\n  \\begin{picture}(1,0.73)%\n    \\put(-0.02,0.04){\\includegraphics[width=\\unitlength]{Fig_sup1.pdf}}%\n    \\put(-0.0000,0.2){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{\\begin{rotate}{90}$\\displaystyle \\Pi_{\\mathrm{irr}}=\\int_{t_{0}}^{t_{2}} I\\Psi_{\\mathrm{R}}\\>dt$ [J]\\end{rotate}}}}%\n    \\put(0.4,0.0){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\displaystyle\\int_{t_{0}}^{t_{2}}\\Delta T dt$~[K~s]}}}%\n    \\put(0.14,0.585){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\rho_{\\mathrm{s}}$ [mM]}}}%\n  \\end{picture\n}\n\\caption{\\label{fig_sup1}%\nIn the calibration with porous carbon electrodes, the  heat transfer coefficient $K^{\\mathrm{c.e.}} = 0.239\\pm 0.003$~J K$^{-1}$~s$^{-1}$ is determined from the slope of the parametric plot of time-integrated $\\Delta T$ peaks against the Joule heat $\\Pi_{\\mathrm{irr}}$ for several salt concentrations, indicated with different colors (see legend). Different symbols belong to different duplicate measurements at various cell potentials up to 1 V.}\n\\end{figure} \n\n\n\n\n\\subsection{Discussion}\\label{sec:discussion}\nIn the presence of carbon electrodes (c.e.), the heat transfer coefficient $K^{\\mathrm{c.e.}}=0.239$~J~K$^{-1}$~s$^{-1}$ is a bit higher than in the case of the  413~$\\Omega$ heating elements (h.e.), where we found $K^{\\mathrm{h.e.}} = 0.179$~J~s$^{-1}$~K$^{-1}$. Both experiments have their pros and cons, which we now discuss.\n\n\n\\textbf{Materials:} the calibration with the porous carbon has the benefit that it does not alter the cell setup. The different materials in the experiment with the resistive wires might affect its $K$-value. \n\n\\textbf{Joule heat determination:}\nBoth calibration methods use the Joule heat produced over a time interval that starts and ends at $\\Delta T=\\SI{0}{\\degreeCelsius}$. While the resistive wire has a precisely-known resistance, allowing for accurate Joule heat determination, the Joule heat in the porous carbon experiments is prone to larger uncertainty, since we assumed a constant capacitance in its determination. Relaxing this assumption is not straightforward without additional assumptions. \n\n\\textbf{Geometry:}\nIn both calibration methods, reversible heat can be found only after deducing $K$ from \\mbox{$\\Delta T$-signals}, which is done via Eq.~\\eqref{eq:heatequation} with an (approximately) known amount of Joule heat. We note that this equation relies on the assumption that the temperature is uniform throughout the electrochemical cell ($\\mathcal{V}_{\\rm s}$ and $\\mathcal{V}_{\\rm el}$), for which the Biot number Bi, comparing heat conduction within the cell to the heat loss to the environment, is commonly used as a benchmark. \nUnfortunately, even when the requirement of small Bi is satisfied (for our cell we estimate Bi~$\\approx0.2$, a bit higher than Bi~$\\approx0.1$ which is deemed sufficiently low \\cite{incropera2007fundamentals}), the assumption of a uniform temperature cannot be correct for far-out-of-equilibrium charging, since heat is generated quickly at specific locations within the cell.\n\nThese considerations have the following repercussions on our experiments.\nIn both calibration methods, the temperature probe was placed halfway between the two electrodes and heating elements, respectively. Meanwhile, the Joule heat is generated at different locations in the two calibration methods. In the heating elements experiment, Joule heat is generated in a resistive wire positioned at the exact same location as the electrodes in the porous carbon experiment; hence, the location of reversible heat production.\nMeanwhile, the weak salt-concentration dependence of the total resistance of the electrochemical cell (that was mentioned in the main text) indicates that in the porous carbon experiment, Joule heat is probably produced at the electrical contacts [see Fig.~\\ref{fig1}(a)], which is further away from the temperature probe than in the heating elements experiment.\nIn line with our experimental findings, we note that a unit amount of Joule heat, generated at increasingly large separations from the temperature probe, leads to ever smaller $\\Delta T$-signals, resulting thereby into ever higher $K$-values. Clearly, employing the $K^{\\mathrm{c.e.}}$--- calibrated with Joule heat generated at the electrical contacts--- in a calculation determining the reversible heat generated at a location closer to temperature probe, will overestimate its value. For the main text, we therefore chose to prioritize  $K^{\\mathrm{h.e.}}$.\n\nThe above calibration uncertainty would not have been present if the cell had had a uniform internal temperature at each instance. \nTo accomplish that, instead of a sudden step, one should \\textit{slowly} apply the external potential, such that the cell can equilibrate internally while being heated. Such experiments with the current cell were unsuccessful because heat flowed out of the cell too quickly; the measured temperature differences were too small compared to the equilibrium temperature fluctuations to separate signal from noise. \nHence, resolving the calibration uncertainty poses an experimental challenge, which will require the development of a better insulated and an even more sensitive calorimetric setup. \n\nAll above-mentioned reservations aside, the obtained heat transfer coefficients $K^{\\mathrm{c.e.}}$ and $K^{\\mathrm{h.e.}}$ still only differ by approximately $30\\%$.\n\n\\section{Time scales of current and temperature-difference decay}\\label{sec:app_timescales}\nFigure~\\ref{fig3}(b) and (c) exhibit a decaying current and temperature difference that we now discuss in more detail.\n\nIn general, electric double layer capacitors (EDLCs) are known to display a richer charging dynamics than conventional RC-circuits because  both the capacitance and electrical resistance of an EDLC depend on its charging state. For instance, model calculations on a parallel plate EDLC (plate separation $L$) showed that\nsurface charge is screened on a time scale $\\lambda_{\\mathrm{D}} L\/D$ (with $D$ the diffusion constant), but corrections also introduce other time scales: $\\lambda_{S}L\/D$, $\\lambda_{S}\\lambda_{\\mathrm{D}}\/D$, $\\lambda_{\\mathrm{D}}^{2}\/D$ (with $\\lambda_{S}$ the thickness of a Stern layer) \\cite{bazant2004diffuse}. At later times ($L^2\/D$) salt diffusion takes over.\n\nThe electrodes of this Letter, made from porous carbon, have a hierarchical structure in which  more length scales must come into play. One can therefore expect to find an even richer charging dynamics. As such, it comes as no surprise that data of the left panel of Fig.~\\ref{fig3}(b) --- redrawn in Fig.~\\ref{fig_sup5} with blue circles on linear (a) and log-linear (b) scale--- \n confirm the picture sketched above of the absence of a single time scale to the decay of the current; clearly, the data is poorly fitted by a single decaying exponential (red line) $I_{\\#1}(t)=I_{1}\\exp[-t\/\\tau_{1}]$ with $I_{1}=0.080$ A and $\\tau_{1}=73$ s. Adding a second exponential $I_{\\#2}(t)=I_{1}\\exp[-t\/\\tau_{1}]+I_{2}\\exp[-t\/\\tau_{2}]$ with $I_{0}=0.071$ A, $I_{1}=0.028$ A, $\\tau_{0}=31$~s, and $\\tau_{0}=171$~s) fits the data better (blue line) at short times but still fails at long times. \n\n\\begin{figure}\n\\def0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}{0.49\\textwidth}{\n \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%\n    \\renewcommand\\color[2][]{}%\n  }%\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%\n    \\renewcommand\\transparent[1]{}%\n  }%\n  \\providecommand\\rotatebox[2]{#2}%\n  \\ifx0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n    \\setlength{\\unitlength}{611.42745019bp}%\n    \\ifx\\svgscale\\undefined%\n      \\relax%\n    \\else%\n      \\setlength{\\unitlength}{\\unitlength * \\real{\\svgscale}}%\n    \\fi%\n  \\else%\n    \\setlength{\\unitlength}{0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}}%\n  \\fi%\n  \\global\\let0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n  \\global\\let\\svgscale\\undefined%\n  \\makeatother%\n  \\begin{picture}(1,0.48)%\n    \\put(0,0){\\includegraphics[width=\\unitlength]{Fig_sup4.pdf}}%\n    \\put(0.01,0.23101035){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{\\begin{rotate}{90} $I$ [A]\\end{rotate}}}}%\n    \\put(0.23253116,-0.02){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{t [s]}}}%\n    \\put(0.73177572,-0.02){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{t [s]}}}%\n          \\put(0.0,0.45){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(a)}}}%\n    \\put(0.5,0.45){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(b)}}}%\n  \\end{picture}%\n}\n\\caption{\\label{fig_sup5}%\n The current relaxation data of the left panel of Fig.~\\ref{fig3}(b)  replotted on linear (a) and log-linear scale (b) with blue circles (only one tenth of the data is shown). Fits $I_{\\#1}(t)$ (red line) and $I_{\\#2}(t)$ (blue line) as introduced in the text are shown as well.}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}\n\\def0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}{0.49\\textwidth}{\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%\n    \\renewcommand\\color[2][]{}%\n  }%\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%\n    \\renewcommand\\transparent[1]{}%\n  }%\n  \\providecommand\\rotatebox[2]{#2}%\n  \\ifx0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n    \\setlength{\\unitlength}{606.60228725bp}%\n    \\ifx\\svgscale\\undefined%\n      \\relax%\n    \\else%\n      \\setlength{\\unitlength}{\\unitlength * \\real{\\svgscale}}%\n    \\fi%\n  \\else%\n    \\setlength{\\unitlength}{0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}}%\n  \\fi%\n  \\global\\let0.45\\textwidth}{ \\input{Fig_sup1.pdf_tex}}\\undefined%\n  \\global\\let\\svgscale\\undefined%\n  \\makeatother%\n  \\begin{picture}(1,0.48)%\n    \\put(0,0){\\includegraphics[width=\\unitlength]{Fig_sup5.pdf}}%\n    \\put(0.01,0.2){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{\\begin{rotate}{90} $\\Delta T$ [K]\\end{rotate}}}}%\n    \\put(0.19785768,-0.02){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{t [s]}}}%\n    \\put(0.70107342,-0.02){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{t [s]}}}%\n              \\put(0.0,0.45){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(a)}}}%\n    \\put(0.5,0.45){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{(b)}}}%\n  \\end{picture}%\n}\n\\caption{\\label{fig_sup6}%\n The thermal response data of the left panel of Fig.~\\ref{fig3}(c) replotted on linear (a) and log-linear scale (b) with black circles.\n\n Fits to these data are obtained with Eq.~(\\ref{eq.sol}) for different assumptions about $\\dot{\\Pi}_{\\rm tot}$: Joule heat only (red line), reversible heat only (blue line), and both Joule and reversible heat (black line).}\n\\end{figure}\n\nIn Fig.~\\ref{fig_sup6} we reproduce (with black circles) the data of the left panel of Fig.~\\ref{fig3}(c) on linear (a) and log-linear (b) scale. To model these data, we note that Eq.~\\eqref{eq:heatequation} allows for the following general solution\n\\begin{equation}\\label{eq.sol}\n\\Delta T(t)=\\exp[-t\/\\tau]\\left[\\int^{t}_{0}\\exp[t'\/\\tau]\\frac{\\dot{\\Pi}_{\\rm tot}(t')}{\\mathbb{C}_{p}}dt' \\right],\n\\end{equation}\nwith $\\dot{\\Pi}_{\\rm tot}(t)$ the heating rate and $\\tau=\\mathbb{C}_{p}\/K=61.1\\pm0.3$~s the thermal time constant as determined in the previous section. Clearly, the time scales of thermal response to charging are dictated both by $\\tau$ and by the time scales with which $\\dot{\\Pi}_{\\rm tot}$ decays. In the main text we argued that $\\dot{\\Pi}_{\\rm tot}$ contains Joule heat contributions ($\\sim I^2$) and reversible contributions ($\\sim I$):\nwe consider  the following general form  $\\dot{\\Pi}_{\\rm tot}(t)=\\left[\\zeta_{\\rm irr}I(t)^2+\\zeta_{\\rm rev}I(t)\\right]$ where $\\zeta_{\\rm irr}$ and  $\\zeta_{\\rm rev}$ are parameters of dimension $\\Omega$ and $V$, respectively.\nFrom the above discussion we know that the current decay is characterized by multiple time scales. However, to proceed, we approximate $I(t)$ with the fit found above: $I(t)=\\Theta(t) I_{\\#2}(t)$, with $\\Theta(t) $ a Heaviside step function. Equation (\\ref{eq.sol}) can now be solved analytically: a lengthy expression with two free parameters $\\zeta_{\\rm irr}$ and $\\zeta_{\\rm rev}$.\n\nTo start, we consider the case of only Joule heat by setting $\\zeta_{\\rm rev}=0$ V. The best fit (red line) is then found for $\\zeta_{\\rm irr}=1.1$~$\\Omega$. Conversely, considering the case of only reversible heat (setting $\\zeta_{\\rm irr}=0$~$\\Omega$) leads to a best fit (blue) at $\\zeta_{\\rm rev}=0.051$~V. The short-time $\\Delta T$ response is decently described by the case of only Joule heat, while the case of only reversible heat  performs poorly in that region.\nFigure~\\ref{fig_sup6}(b) shows that both choices do not reproduce the long-time relaxation accurately: they relax either too rapidly (only Joule heat) or too slowly (only reversible heat).\n\n\nAs can be expected on the basis of Ref.~\\cite{janssen2017reversible} and this Letter, the physical situation is that of reversible and irreversible heat {\\it simultaneous} at play.\nIndeed, the best fit through the data in Fig.~\\ref{fig_sup6} is achieved for $\\zeta_{\\rm irr}=0.76$~$\\Omega$ and  $\\zeta_{\\rm rev}=0.016$ V; hence, a mixture of reversible and reversible contributions. We conclude that, in order to reproduce the long-time relaxation correctly, it seems necessary to account for both reversible and irreversible heat. \n\n\n\nSo far, we only considered $\\zeta_{\\rm rev\/irr}$ to be time-independent. However, for the slit geometry of Ref.~\\cite{janssen2017reversible}, $\\zeta_{\\rm rev}$ takes the form $\\zeta_{\\rm rev}=k_{\\rm B}T (\\partial_{z}q)\/[e (\\rho_{+}+\\rho_{-})]$; hence, there is no reason to believe $\\zeta_{\\rm rev}$ is truly time independent. Relaxing this assumption could lead to an even better reproduction of the measured temperature data. \n\n\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n   \\label{sec:sec1}\n\nThe aim of weather forecasting is to give a good prediction of the\nfuture states of the atmosphere on the basis of present observations\nand mathematical models describing the dynamics (physical behaviour)\nof the atmosphere.    \nThese models consist of sets of non-linear partial differential\nequations which have only numerical solutions. The problem with these\nnumerical weather prediction models is that the solutions highly\ndepend on the initial conditions which are always in a way or in\nanother not fully accurate. A\npossible solution to address this problem is to run the model with different\ninitial conditions and produce ensembles of forecasts. With the help\nof ensembles one can estimate the distribution of future weather\nvariables which leads us to probabilistic weather forecasting \\citep{gr}.\nThe ensemble\nprediction method was proposed by \\citet{leith} and since its first\noperational implementation \\citep{btmp,tk} it became a widely used\ntechnique all over the world. However, despite e.g. the ensemble\nmean gives a better estimate of a meteorological quantity than most or\nall of the ensemble members, the ensemble is usually under-dispersive\nand in this way, uncalibrated. This phenomena was observed at several\noperational ensemble prediction systems, for an overview see\ne.g. \\citet{bhtp}. \n\nThe Bayesian model averaging (BMA) method for\npost-processing ensembles in order to calibrate them was introduced by\n\\citet{rgbp}. The basic idea of BMA is that to each ensemble member\nforecast corresponds a conditional probability\ndensity function (PDF) that can be interpreted as the\nconditional PDF of the future weather quantity provided the considered\nforecast is the best one. Then the BMA predictive PDF of the future\nweather quantity is the weighted sum of the individual PDFs\ncorresponding to the ensemble members and the weights are based on the\nrelative performances of the ensemble members during a given training period.\nIn \\citet{rgbp} the BMA\nmethod was successfully applied to obtain 48 hour forecasts of surface\ntemperature and sea level pressure in the North American Pacific\nNorthwest based on the 5 members of the University of Washington\nMesoscale Ensemble \\citep{gm}. These weather quantities can be\nmodeled by normal distributions, so the predictive PDF is a Gaussian mixture.\nLater \\citet{srgf} developed a discrete-continuous BMA model for precipitation\nforecasting, where the discrete part corresponds to the event of no\nprecipitation, while the cubic root of the precipitation amount\n(if it is positive) is modeled by a gamma distribution. In\n\\citet{sgr10} the BMA method was used for wind speed forecasting and \nthe component PDFs follow gamma distribution. Finally, using von Mises\ndistribution to model angular data \n\\citet{bgrgg} introduced a BMA scheme to predict surface wind\ndirection.\n\nIn the present work we apply the BMA method for calibrating ensemble\nforecasts of wind speed produced by the operational Limited\nArea Model Ensemble  \nPrediction System (LAMEPS) of the Hungarian Meteorological Service\n(HMS) called ALADIN-HUNEPS \\citep{hagel, horanyi}. ALADIN-HUNEPS\ncovers a large part of Continental Europe with a horizontal resolution\nof 12 km and it is obtained by dynamical downscaling (by the ALADIN\nlimited area model) of the global\nARPEGE based PEARP system of M\\'et\\'eo France \\citep{hkkr,dljn}. The\nensemble consists of 11 members, 10 initialized from perturbed initial\nconditions and one control member from the unperturbed analysis. This\nconstruction implies that the ensemble contains groups of exchangeable\nforecasts (the ensemble members cannot be distinguished), so for\npost-processing one has to use the modification of \nBMA as suggested by \\citet{frg}. \n\n\n\n\\section{Data}\n  \\label{sec:sec2}\n\nAs it was mentioned in the Introduction, BMA post-processing\nof ensemble predictions was applied for wind speed data obtained from\nthe HMS. The \ndata file contains 11 member ensembles (10 forecasts started from perturbed\ninitial conditions and one control) of 42 hour \nforecasts for 10 meter wind speed (given in m\/s)\nfor 10 major cities in \nHungary (Miskolc, Szombathely, Gy\\H or, Budapest, Debrecen, Ny\\'\\i regyh\\'aza,\nNagykanizsa, P\\'ecs, Kecskem\\'et, Szeged) produced by the\nALADIN-HUNEPS system of the HMS, together with the corresponding\nvalidating observations, for the period between October 1, 2010 and March 25,\n2011. The forecasts are initialized at 18 UTC, the startup speed of\nthe anemometers measuring the validating observations is $0.1$\nm\/s. The data set is fairly complete, since there are only two days\n(18.10.2010 and 15.02.2011) when three ensemble members are missing\nfor all sites and one day (20.11.2010) when no forecasts are available. \n\n\\begin{figure}[t]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=verrankn.eps,height=9cm, angle=-90}\n\\caption{Verification rank histogram of the 11-member ALADIN-HUNEPS\n  ensemble. Period: October 1, 2010 -- March 25, 2011.} \n\\label{fig:fig1}\n\\end{center}\n\\end{figure}\nFigure \\ref{fig:fig1} shows the verification rank histogram of the raw\nensemble, that is the histogram of ranks of validating\nobservations with respect to the corresponding ensemble\nforecasts. This histogram is \nfar from the desired uniform distribution, in most of the cases the\nensemble members either underestimate, or overestimate the validating\nobservations (the ensemble range contains the observed wind speed only\nin $61.21\\%$ of the cases). Hence, the ensemble is under-dispersive\nand in this way it is uncalibrated.\n\n\n\n\\section{The model and diagnostics}\n   \\label{sec:sec3}\n\n\n\nTo obtain a probabilistic forecast of wind speed  the modification of \nBMA gamma model of \\citet{sgr10} for ensembles with\nexchangeable members \\citep{frg} was used. The first idea is to have two\nexchangeable groups: one contains \nthe control denoted by \\ $f_c$, \\ the other one the 10 ensemble members\ncorresponding to the different perturbed initial conditions which are denoted by\n\\ $f_{\\ell,1},\\ldots ,f_{\\ell,10}$, \\ respectively. \\ In this way we assume\nthat the probability density function (PDF) of the forecasted\nwind speed \\ $x$ \\ equals:    \n\\begin{align}\n  \\label{eq:eq3.1}\np(x | f_c,f_{\\ell,1},\\ldots ,\nf_{\\ell,10};b_0,b_1,c_0,c_1)=&\\, \\omega\ng(x;f_c,b_0,b_1,c_0,c_1) \\\\ &+  \\frac \n{1-\\omega}{10} \\sum_{j=1}^{10}  g(x;f_{\\ell,j},b_0,b_1,c_0,c_1), \\nonumber \n\\end{align} \nwhere \\ $\\omega\\in [0,1]$, \\ and \\  $g$ \\\nis the conditional PDF corresponding to the ensemble members.\nAs we are working with wind speed data, \n\\ $g(x;f,b_0,b_1,c_0,c_1)$ \\ is a gamma PDF with mean\n\\ $b_0+b_1 f$ \\ and standard deviation \\ $c_0+c_1 f$. \\ Here we restrict both\nthe mean and the standard deviation parameters to constant values for all\nensemble members, which reduces the number of parameters and\nsimplifies calculations. \nMean parameters \\ $b_0,\\ b_1$ \\ are estimated with\nthe help of linear regression, while weight  \\ $\\omega$ \\ and standard deviation\nparameters \\ $c_0, \\ c_1$, \\ by maximum likelihood method, using training\ndata consisting of ensemble members and verifying observations from\nthe preceding \\ $n$ \\ days (training period). In order to handle the problem\nthat the wind speed values under 0.1 m\/s are considered to be zero, the\nmaximum likelihood (ML) method for gamma distributions suggested by\n\\citet{wilks1} is \napplied, while the maximum of the likelihood\nfunction is found with the help of EM algorithm \\citep{mclk}. For more\ndetails see \\citet{sgr10,frg}. Once the estimated \nparameters for a given day are available,  one can\nuse either the mean or the median of the predictive PDF\n\\eqref{eq:eq3.1} as a point forecast. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=plumedebn.eps,height=10cm}\n\\caption{Plume diagram of ensemble forecast of 10 meter wind speed\n  for Debrecen initialized at 18 UTC, 22.10.2010.} \n\\label{fig:fig2}\n\\end{center}\n\\end{figure}\nBased on a more careful look on the ensemble members there are some\ndifferences in the generation of the ten \nexchangeable ensemble members. To obtain them only five perturbations\nare calculated and then they are added to (odd numbered members) and\nsubtracted from (even numbered members) the unperturbed initial\nconditions \\citep{horanyi}. Figure \\ref{fig:fig2} shows the plume\ndiagram of ensemble forecast of 10 meter wind speed\nfor Debrecen initialized at 18 UTC, 22.10.2010. (solid line: control;\ndotted line: odd numbered members, dashed line: even numbered\nmembers). This diagram clearly illustrates that the behaviour of\nensemble member groups \\ $\\{f_{\\ell,1}, \\ \nf_{\\ell,3}, \\ f_{\\ell,5}, \\ f_{\\ell,7}, \\ f_{\\ell,9}\\}$ \\ and \\ $\\{f_{\\ell,2}, \\\nf_{\\ell,4}, \\ f_{\\ell,6}, \\ f_{\\ell,8}, \\ f_{\\ell,10}\\}$ \\ really differ\nfrom each other. Therefore, in this way one can also consider a model with three\nexchangeable groups: control, odd numbered exchangeable members and even\nnumbered exchangeable members. This idea leads to the following PDF\nof the forecasted wind speed \\ $x$:\n\\begin{align}\n  \\label{eq:eq3.2}\nq(x | f_c,f_{\\ell,1},\\ldots ,\nf_{\\ell,10};&\\,b_0,b_1,c_0,c_1)= \\omega_c\ng(x;f_c,b_0,b_1,c_0,c_1) \\\\ &+ \n\\sum_{j=1}^{5} \\big(\\omega_o g(x;f_{\\ell,2j-1},b_0,b_1,c_0,c_1)+ \n\\omega_e g(x;f_{\\ell,2j},b_0,b_1,c_0,c_1)\\big), \\nonumber \n\\end{align} \nwhere for weights \\ $\\omega_c,\\omega_o,\\omega_e\\in[0,1]$ \\ we have  \\\n$\\omega_c+5\\omega_o+5\\omega_e=1$, \\ while PDF \\ $g$ \\ and parameters \\\n$b_0,b_1,c_0,c_1$ \\ are the same as for the model\n\\eqref{eq:eq3.1}. Obviously, both the weights and the parameters \ncan be estimated in the same way as before.  \n \n\n\\begin{figure}[t!]\n\\begin{center}\n\\leavevmode\n\\hbox{\n\\epsfig{file=winddebpdf1g2n.eps,height=8cm, angle=-90} \\quad\n\\epsfig{file=winddebpdf2g2n.eps,height=8cm, angle=-90}}\n\n\\centerline{\\hbox to 9 truecm {\\scriptsize (a) \\hfill (b)}}\n\n\\hbox{\n\\epsfig{file=winddebpdf1g3n.eps,height=8cm, angle=-90} \\quad\n\\epsfig{file=winddebpdf2g3n.eps,height=8cm, angle=-90}}\n\n\\centerline{\\hbox to 9 truecm {\\scriptsize (c) \\hfill (d)}}\n\\caption{Ensemble BMA PDFs (overall: thick black line; control: red\n  line; sum of exchangeable members on (a) and (b): light blue line;\n  on (c) and (d): green (odd members) and blue (even members) lines),\n  ensemble members (circles with the same colours as the corresponding\n  PDFs), ensemble BMA median forecasts (vertical black line),\n  verifying observations (vertical orange line) and the first and last\ndeciles (vertical dashed lines) for wind speed in Debrecen for\n  models  \\eqref{eq:eq3.1}: (a) 30.12.2010, (b) 17.03.2011; and\n  \\eqref{eq:eq3.2}: (c) 30.12.2010, (d) 17.03.2011.} \n\\label{fig:fig3}\n\\end{center}\n\\end{figure}\nAs an illustration we consider the data and forecasts for \nDebrecen for two\ndifferent dates 30.12.2010 and 17.03.2011 for models \\eqref{eq:eq3.1}\nand \\eqref{eq:eq3.2}.\nFigures \\ref{fig:fig3}a and \\ref{fig:fig3}b show the  PDFs of\nthe two groups in model \\eqref{eq:eq3.1}, the overall \nPDFs, the median forecasts,\nthe verifying observations, the first and last\ndeciles  and the ensemble members. The same functions and\nquantities can be seen on Figures \\ref{fig:fig3}c and \\ref{fig:fig3}d,\nwhere besides the overall PDF we have three component PDFs and three\ngroups of ensemble members. On 30.12.2010 the spread of the ensemble\nmembers is quite fair and the ensemble range contains the validating\nobservation (3.2 m\/s). In this case the ensemble mean (3.5697 m\/s)\noverestimates, while BMA median forecasts corresponding to the two- and\nthree-group models (3.2876 m\/s and 3.2194 m\/s, respectively) are\npretty close to the true wind speed.  A different situation is\nillustrated on Figures \\ref{fig:fig3}b and \\ref{fig:fig3}d, where the\nspread of the ensemble is even higher, but all\nensemble members underestimate the validating observation (6.1\nm\/s). Obviously, the same holds for the ensemble mean (3.2323 m\/s) and\ndue to the bias correction the BMA median forecasts corresponding to\nmodels \\eqref{eq:eq3.1} and \n\\eqref{eq:eq3.2} also give bad results (3.3409 m\/s and 3.0849 m\/s,\nrespectively).\n\n\\begin{figure}[t!]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=avcov2gn.eps,height=15cm, angle=-90}\n\\caption{Average widths and coverages of $66.7\\,\\%$ and $90\\,\\%$\n BMA  prediction intervals \n  corresponding to two-group model \\eqref{eq:eq3.1} for various training period\n  lengths.}   \n\\label{fig:fig4}\n\\end{center}\n\\end{figure}\nTo check the performance of probabilistic forecasts based on\nmodels \\eqref{eq:eq3.1} and \\eqref{eq:eq3.2} and the\ncorresponding point forecasts, as a reference we use the ensemble mean\nand the ensemble median. We compare the mean absolute errors (MAE) and\nthe root mean square errors (RMSE) of\nthese point forecasts and also the mean continuous ranked probability scores\n(CRPS) \\citep{wilks2,grjasa} and the coverages and average widths of\n$66.7\\,\\%$ and $90\\,\\%$ prediction intervals of the BMA predictive probability\ndistributions and of the raw ensemble. We remark that for MAE and RMSE\nthe optimal point forecasts are the median and the mean, respectively\n\\citep{gneiting11, pinhag}.  Further, given a cumulative distribution\nfunction (CDF) \\ $F(y)$ \\ and a real number \\ $x$, \\ the CRPS is defined as\n\\begin{equation*}\n\\crps\\big(F,x\\big):=\\int_{-\\infty}^{\\infty}\\big (F(y)-{\\mathbbm \n  1}_{\\{y \\geq x\\}}\\big )^2{\\mathrm d}y.\n\\end{equation*}\nThe mean CRPS of a probability forecast is the average of the CRPS values\nof the predictive CDFs and corresponding validating observations taken\nover all locations and time points considered. For the raw ensemble\nthe empirical CDF of the ensemble replaces the predictive CDF. The\ncoverage of a \\ $(1-\\alpha)100 \\,\\%, \\ \\alpha \\in (0,1),$ \\ prediction\ninterval is the proportion \nof validating observations located between the lower and upper \\\n$\\alpha\/2$ \\ quantiles of the predictive distribution. For a\ncalibrated predictive PDF this value should be around \\ $(1-\\alpha)100\n\\,\\%$.   \n\n\n\\section{Results}\n   \\label{sec:sec4}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=crpsmae2gn.eps,height=15cm, angle=-90}\n\\caption{CRPS of BMA predictive distribution, MAE values of BMA median and RMSE\n  values of BMA mean forecasts\n  corresponding to two-group model \\eqref{eq:eq3.1} for various training period\n  lengths.}   \n\\label{fig:fig5}\n\\end{center}\n\\end{figure}\nData analysis provided below was performed with the help of the {\\tt\n  ensembleBMA} package of R \\citep{frgs,frgsb}. As a first step the\nlength of the \nappropriate training period was determined, then the performances\nof the BMA post-processed ensemble forecasts corresponding to models\n\\eqref{eq:eq3.1} and \\eqref{eq:eq3.2} were analyzed.  \n\n\\subsection{Training period}\n   \\label{sec:sub4.1}\n\n\nAccording to the results of e.g. \\citet{rgbp} to determine the length\nof the training period to be used we compare the MAE values\nof BMA median forecasts, the RMSE values of BMA mean forecasts, the\nCRPS values of BMA predictive \ndistributions and the coverages and average\nwidths of  $90\\,\\%$ and $66.7\\,\\%$ BMA prediction intervals for\ntraining periods of \nlength \\ $10,11, \\ldots, 60$ \\ calendar days. In order to ensure\nthe comparability of \nthe results we consider verification results from 02.12.2010 to\n25.03.2011 (114 days).\n\n\\begin{figure}[t!]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=avcov3gn.eps,height=15cm, angle=-90}\n\\caption{Average widths and coverages of $66.7\\,\\%$ and $90\\,\\%$\n BMA  prediction intervals \n  corresponding to three-group model \\eqref{eq:eq3.2} for various\n  training period \n  lengths.}   \n\\label{fig:fig6}\n\\end{center}\n\\end{figure}\nConsider first the two-group model \\eqref{eq:eq3.1}. On Figure\n\\ref{fig:fig4} the \naverage widths and coverages of $66.7\\,\\%$ and $90\\,\\%$  BMA prediction\nintervals are plotted against the length of the training period. The\naverage widths of the prediction intervals  show an \nincreasing trend, so shorter training periods yield sharper forecasts.\nCoverages of $66.7\\,\\%$ and $90\\,\\%$ prediction intervals are not monotonously\nincreasing, too. For short training periods the coverage of the\n$66.7\\,\\%$  prediction interval oscillates around the\ncorrect $66.7\\,\\%$, but for training periods not shorter than 17 days it\nstays above this level. The coverage of the $90\\,\\%$  prediction\ninterval stabilizes above the correct $90\\,\\%$ for training periods\nlonger than 24 days. Hence, to have calibrated forecasts, one should\nchoose a training period not shorter than 25 days.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=crpsmae3gn.eps,height=15cm, angle=-90}\n\\caption{CRPS of BMA predictive distribution, MAE values of BMA median and RMSE\n  values of BMA mean forecasts\n  corresponding to three-group model \\eqref{eq:eq3.2} for various\n  training period lengths.}   \n\\label{fig:fig7}\n\\end{center}\n\\end{figure}\nFigure \\ref{fig:fig5} shows\nCRPS values of BMA predictive distribution, MAE values of BMA\nmedian forecasts and RMSE values of BMA mean forecasts as functions of\nthe training period length. CRPS and RMSE both take their minima at 28\ndays, the corresponding values are $0.7388$ and $1.3675$,\nrespectively. MAE\ntakes its minimum of $1.0472$ at 30 days, while the second smallest value\n($1.0476$) is obtained with a training period of length 28 days.\nThis means that for model\n\\eqref{eq:eq3.1} a 28 days training period seems to be reasonable and\ntraining periods longer than 30 days cannot be taken into consideration.\n\nSimilar conclusions can be drawn from Figures \\ref{fig:fig6} and\n\\ref{fig:fig7}  for the three-group model \\eqref{eq:eq3.2}. In this case the\n$66.7\\,\\%$ and $90\\,\\%$ prediction intervals are slightly narrower\nthan the corresponding intervals of model \\eqref{eq:eq3.1}, their\ncoverages stabilize above the correct\n$66.7\\,\\%$ and $90\\,\\%$ for training periods longer than 17 and 24\ndays, respectively. CRPS and MAE plotted on Figure \\ref{fig:fig7} both\nreach their minima of $0.7372$ and $1.0452$, respectively, at 30 days,\nwhile values $0.7376$ and $1.0456$ corresponding to training period of\nlength 28 days are both the fourth smallest ones. RMSE takes its\nminimum of $1.3632$ at 27 days, and increases afterwards. The fourth\nsmallest value ($1.3644$) again corresponds to 28 days, while the RMSE\ncorresponding to 30 days is significantly larger ($1.3664$). Moreover,\n$66.7\\,\\%$ and $90\\,\\%$ prediction intervals corresponding to 28 days\nare sharper than the appropriate prediction intervals calculated using training\nperiod of length 30 days ($2.5813$ and $4.4340$ vs. $2.5831$ and\n$4.4378$). Hence, we suggest the use of a training period of length 28\ndays for both BMA models.\n\n\n\n\\subsection{Predictions using BMA post-processing}\n\\label{sec:sub4.2}\n\nAccording to the results of the previous subsection, to test the performance\nof BMA post-processing on the 11 member ALADIN-HUNEPS ensemble we use a\ntraining period of 28 calendar days. In this way ensemble members,\nvalidating observations and BMA models are available for 146 calendar\ndays (on 20.11.2010 all ensemble members are missing). \n\n\\begin{figure}[t!]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=pitsn.eps,height=15cm, angle=-90}\n\\caption{PIT histograms for BMA post-processed forecasts using two-group\n  \\eqref{eq:eq3.1} and three-group \\eqref{eq:eq3.2} models.}   \n\\label{fig:fig8}\n\\end{center}\n\\end{figure}\n\nFirst we check the calibration of BMA post-processed forecasts with the\nhelp of probability integral transform (PIT) histograms. The PIT is\nthe value of the BMA predictive cumulative distribution evaluated at\nthe verifying observations \\citep{frg}. The closer the histogram to the\nuniform distribution, the better is the calibration. On Figure\n\\ref{fig:fig8} the PIT histograms corresponding to two- and\nthree-group BMA models\n\\eqref{eq:eq3.1} and \\eqref{eq:eq3.2} are displayed. Compared to the\nverification rank histogram of the raw ensemble (see  Figure\n\\ref{fig:fig1}) one can observe a large improvement with the use of\ncalibration. However, these PIT histograms are still not perfect,\ne.g. Kolmogorov-Smirnov test rejects uniformity both for the two- and\nfor the three-group model. The corresponding $p$-values are $0.0222$\nand $0.0187$, respectively, so the PIT of the two-group model is\nslightly better.\n\n\n\n\\begin{table}[b]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|c|} \\hline\n\\multicolumn{1}{|c|}{}&\n\\multicolumn{2}{|c|}{Coverage ($\\%$)}&\n\\multicolumn{2}{|c|}{Average Width}\\\\ \\cline{2-5}\nInterval&$66.7\\,\\%$ interval&$90.0\\,\\%$ interval&$66.7\\,\\%$\ninterval&$90.0\\,\\%$ interval\\\\ \\hline \nRaw ensemble&$38.70$&$55.14$&$1.4388$&$2.2001$\n\\\\ \nBMA model\n\\eqref{eq:eq3.1}&$68.08$&$90.34$&$2.6359$&$4.5297$\n\\\\ \nBMA model\n\\eqref{eq:eq3.2}&$68.36$&$90.21$&$2.6153$&$4.4931$\n\\\\ \n\\hline \n\\end{tabular} \n\\caption{Coverage and average width of prediction intervals.} \\label{tab:tab1}\n\\end{center}\n\\end{table}\nTable \\ref{tab:tab1} gives the coverages and average widths of\n$66.7\\,\\%$ and $90.0\\,\\%$ prediction intervals calculated using models\n\\eqref{eq:eq3.1} and \\eqref{eq:eq3.2}, and the corresponding measures calculated\nfrom the raw ensembles. In the latter case the ensemble of forecasts\ncorresponding to a given location and time is considered as a\nstatistical sample. The BMA prediction intervals calculated from\nboth models are approximately twice as wide, as the corresponding\nintervals of the raw ensemble. This comes from the small dispersion of\nthe raw ensemble, see the verification rank histogram of Figure\n\\ref{fig:fig1}. Concerning calibration one can observe that the\ncoverages of both  \nBMA prediction intervals are rather close to the right coverages,\nwhile the coverages of the prediction intervals calculated from the\nraw ensemble are quite poor. This also shows that BMA post-processing\nhighly improves  calibration. Further, BMA model \\eqref{eq:eq3.2}\nyields slightly sharper predictions but there is no big difference\nbetween the coverages of the two BMA models.\n\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|} \\hline\n\\multicolumn{1}{|c|}{}&\n\\multicolumn{1}{|c|}{Mean CRPS}&\n\\multicolumn{2}{|c|}{MAE}&\n\\multicolumn{2}{|c|}{RMSE}\\\\ \\cline{3-6}\n&&median&mean&median&mean\n\\\\ \\hline \nRaw ensemble&$0.8599$&$1.1215$&$1.1090$&$1.4634$&$1.4440$\n\\\\ \nBMA model\n\\eqref{eq:eq3.1}&$0.7577$&$1.0678$&$1.0763$&$1.4213$&$1.4067$\n\\\\ \nBMA model\n\\eqref{eq:eq3.2}&$0.7556$&$1.0643$&$1.0749$&$1.4153$&$1.4018$\n\\\\\\hline \n\\end{tabular} \n\\caption{Mean CRPS of probabilistic, MAE and RMSE of\ndeterministic forecasts.} \\label{tab:tab2}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=weightsn.eps,height=15cm, angle=-90}\n\\caption{BMA weights of two-group\n  \\eqref{eq:eq3.1} and three-group \\eqref{eq:eq3.2} models.}   \n\\label{fig:fig9}\n\\end{center}\n\\end{figure}\nOn Table \\ref{tab:tab2} the verification results of the model\nfit are given. Verification measures of probabilistic forecasts and point\nforecasts calculated using BMA models \\eqref{eq:eq3.1} and\n\\eqref{eq:eq3.2} are compared to the corresponding measures calculated\nfor the raw ensemble. Examining these results one can clearly observe\nthe advantage of BMA post-processing which resulted a significant\ndecrease in all verification scores. Further, the BMA median forecasts\nyield slightly lower MAE values than the BMA mean forecasts for both\nmodels, while in the case of RMSE values the situation is just the\nopposite, which is a perfect illustration of the theoretical results\nof \\cite{gneiting11} about the optimality of these verification scores. \nFinally, model  \\eqref{eq:eq3.2} distinguishing three exchangeable\ngroups of ensemble forecasts slightly outperforms model \\eqref{eq:eq3.1}.\n\n\nFigure \\ref{fig:fig9} shows the BMA weights corresponding to models\n\\eqref{eq:eq3.1} and \\eqref{eq:eq3.2}. Examining the behaviour of \nweight \\ $\\omega$ \n\\ of the control member of the ensemble in the two-group model\n\\eqref{eq:eq3.1}, one can \nobserve that in $84.56\\,\\%$ of the cases there is a real mixture of\ngamma distributions. The values of \\ $\\omega$ \\ which are close to $1$\ncorrespond to a\ncontinuous time interval 17.11.2010 -- 09.12.2010,  when the control\nmember of the ensemble gives much\nbetter forecasts than the ten exchangeable ensemble members. This can\nclearly be seen from Table \\ref{tab:tab3} where the MAE and RMSE\nvalues of the particular ensemble members are given for the above\nmentioned period. In all of these 23 subsequent days \\ $\\omega>0.995$ \\\nbut on 20.11.2010, when \\ $\\omega=0.9873$. \\ However, as it was\nmentioned earlier, on this particular day all ensemble \nforecasts are missing from the data set. The situation is quite\ndifferent in the case of the three-group model \\eqref{eq:eq3.2}, where\nthe weight \\ $\\omega_c$ \\ of the control is close to $1$ (greater than\n$0.98$) only on 7 days, so in the remaining cases ($95.30\\,\\%$)  a\nreal mixture of gamma distributions present. Further, observe that\nthere are 55 days \\ ($36.91\\,\\%$) \\ when all BMA weights\nare positive, the even numbered exchangeable members have nearly zero weights \\\n(less than $0.001$) \\ in 45 cases \\ ($30.20\\,\\%$) \\ at the beginning of the\nconsidered time period, while the odd numbered exchangeable members\nare almost zero in 53 cases \\ ($35.57\\,\\%$), \\ mainly at the end of\nit.\n\n  \n\\begin{table}[b]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|} \\hline\n\\multicolumn{1}{|c|}{}&\n\\multicolumn{1}{|c|}{Control}&\n\\multicolumn{10}{|c|}{Exchangeable members}\n\\\\ \\cline{2-12}\n&$f_c$&$f_{\\ell,1}$&$f_{\\ell,2}$&$f_{\\ell,3}$&$f_{\\ell,4}$&$f_{\\ell,5}$&$f_{\\ell,6}$&$\nf_{\\ell,7}$&$f_{\\ell,8}$&$f_{\\ell,9}$&$f_{\\ell,10}$ \\\\ \\hline\nMAE&$1.32$&$1.60$&$1.46$&$1.52$&$1.68$&$1.51$&$1.49$&$1.56$&$\n1.42$&$1.41$&$1.65$\\\\\nRMSE&$1.69$&$2.16$&$1.86$&$1.96$&$2.26$&$1.92$&$1.95$&$2.05$&$\n1.89$&$1.81$&$2.23$\\\\ \\hline\n\\end{tabular} \n\\caption{MAE and RMSE of the control and exchangeable ensemble\n  forecasts for the period  17.11.2010 -- 09.12.2010.} \\label{tab:tab3}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[t]\n\\begin{center}\n\\leavevmode\n\\epsfig{file=parsn.eps,height=16.5cm, angle=-90}\n\\caption{Parameters of two-group model\n  \\eqref{eq:eq3.1} and differences in standard deviation parameters\n  between three- and two-group models.}     \n\\label{fig:fig10}\n\\end{center}\n\\end{figure}\nFinally, on Figure \\ref{fig:fig10} common bias parameters \\ $b_0, \\ b_1$ \\\nof both BMA models investigated and standard deviation parameters \\\n$c_0, \\ c_1$ \\ \nof the two-group model \\eqref{eq:eq3.1} are plotted, together with the\ndifferences in standard deviation parameters of three- and two-group\nmodels. Bias parameters are rather\nstable, the relative standard deviations of \\ $b_0$ \\ and \\ $b_1$ \\\nare $25.44\\,\\%$ and $9.97\\,\\%$, respectively. Hence, the BMA mean\nforecast of a particular day is mainly determined by the corresponding\nensemble forecasts. The standard deviation \nparameters show more variability, for  \\ $c_0$ \\\nand \\ $c_1$ \\ the relative standard deviations are equal to\n$23.41\\,\\%$ and $41.27\\,\\%$ for \nmodel \\eqref{eq:eq3.1}, and $22.64\\,\\%$ and $36.73\\,\\%$ for\nmodel \\eqref{eq:eq3.2}.\n\n\n\n\\section{Conclusions}\n   \\label{sec:sec5}\n\nIn the present study the BMA ensemble post-processing method is applied \nfor the 11 member ALADIN-HUNEPS ensemble of the HMS to obtain 42 hour\npredictions for 10 meter wind speed. Two different BMA models are\ninvestigated, one assumes two groups of exchangeable members (control\nand forecasts from perturbed initial conditions), while the other\nconsiders three (control and forecasts from perturbed initial\nconditions with positive and negative perturbations).  For both models\na 28 days training period is suggested. The comparison of the\nraw ensemble and of the probabilistic forecasts shows that the mean\nCRPS values \nof BMA post-processed forecasts are considerably lower than the mean\nCRPS of the raw ensemble. Further, the MAE and RMSE values of BMA point\nforecasts (median and mean) are also lower than the MAEs and RMSEs of the\nensemble median and of the ensemble mean. The calibrations of BMA\nforecasts are nearly perfect, the coverages of $66.7\\,\\%$ and $90.0\\,\\%$\nprediction intervals are very close to the right values. The\nthree-group BMA model slightly outperforms the two-group one and in\nalmost all cases yields a real mixture of gamma distributions.\n\nIn this way one can conclude that BMA post-processing of ensemble\nforecasts of wind speed data of the HMS significantly improves the\nprecision and calibration of the forecasts, its operational application\nis worth considering.\n\n\n\n\n\\bigskip\n\\noindent\n{\\bf Acknowledgments.} \\  \\ Research was supported by \nthe Hungarian  Scientific Research Fund under Grants No. OTKA\nT079128\/2009 and OTKA NK101680 and by the T\\'AMOP-4.2.2.C-11\/1\/KONV-2012-0001\nproject. The project has been supported by the European Union,\nco-financed by the European Social Fund.  The authors are indebted to Tilmann\nGneiting for his useful suggestions and remarks and to M\\'at\\'e Mile and \nMih\\'aly Sz\\H ucs from the HMS for providing the data.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nRecently, several intriguing threads relating derived categories and\narithmetic geometry have emerged and motivated general structure questions \nfor $k$-linear triangulated categories for arbitrary\nfields $k$.  Such exploration has yielded many nice applications as well as \nfurther enticing problems, see as a sampling \\cite{AKW, AAGZ, ADPZ, HT, Honigs, LiebMaulSnow}. \nMeanwhile over $\\mathbb{C}$, structural results for derived\ncategories seem to have deep implications for the underlying birational geometry, e.g.\n\\cite{AT, ABB, BB, BMMSS, KuznetsovCubic4fold, Vial}. Taking these together,\nderived categories become an important invariant for studying birational\ngeometry over a general field \\cite{AB}. A further benefit of this\nnoncommutative approach is direct utility for solving problems in algebraic $K$-theory,\nfor example \\cite{MerkPan}.\n\nWith such tantalizing ties, one would like a fertile testing ground for\nquestions. In this paper, we begin a systematic study of one such area:\nderived categories of arithmetic toric varieties. This area has the\nfollowing nice features:\n\n\\begin{itemize}\n \\item rationality issues are deep in general but tractable in examples,\n \\item robust tools already exist to investigate derived categories over\nthe separable closure,\n \\item and specific questions are often amenable to computational\nexperimentation.\n\\end{itemize}\n\nOne of the best tools for understanding a derived category is an\n\\emph{exceptional collection} consisting of \\emph{exceptional objects}.\nAs originally conceived in \\cite{Beilinson}, an exceptional object of a $k$-linear derived category is one whose endomorphism algebra\nis isomorphic to the base field $k$.\nWhen $k$ is not algebraically closed, this definition is too restrictive and\ninstead we use the existing notion: an object of $\\mathsf{D^b}(X)$ is \\emph{exceptional} if its endomorphism algebra is a division algebra. \nDetails are discussed in Section~\\ref{sect:galdesc} below.\n\nWe illustrate this more general notion for two arithmetic toric\nvarieties.\nThe real conic $X = \\{x^2 + y^2 + z^2 = 0\\} \\subset \\mathbb{P}^2_{{\\mathbb R}}$\nhas an exceptional collection $\\{ \\O, \\mathcal{F} \\}$, where\n$\\text{End}(\\mathcal{F})$ is isomorphic to the quaternion algebra\n$\\mathbb{H}$.\nOver ${\\mathbb C}$, we have $X_{\\mathbb C} \\simeq {\\mathbb P}^1_{\\mathbb C}$ and $\\mathcal{F}\\otimes_{{\\mathbb R}} {\\mathbb C} \\simeq \\O(1)^{\\oplus 2}$.\nAs another example, consider the Weil restriction $Y$ of ${\\mathbb P}^1_{\\mathbb C}$ over\n${\\mathbb R}$ (``${\\mathbb P}^1({\\mathbb C})$ viewed as an ${\\mathbb R}$-variety'').\nHere $Y$ has an exceptional collection $\\{ \\O, \\mathcal{G},\n\\mathcal{H}\\}$ where $\\text{End}(\\mathcal{G}) \\simeq {\\mathbb C}$ and\n$\\text{End}(\\mathcal{H})\n\\simeq {\\mathbb R}$.\nOver ${\\mathbb C}$, we have $Y\\otimes_{{\\mathbb R}} {\\mathbb C} \\simeq {\\mathbb P}^1 \\times {\\mathbb P}^1$\nwith\n$\\mathcal{G} \\otimes_{{\\mathbb R}} {\\mathbb C}\n\\simeq \\O(1,0) \\oplus \\O(0,1)$ and $\\mathcal{H} \\otimes_{{\\mathbb R}} {\\mathbb C}\\simeq  \\O(1,1)$,\nwhere $\\O(i, j) = \\pi_1^*\\O(i) \\otimes \\pi_2 ^*\\O(j)$.\n\nA central question for derived categories of arithmetic toric varieties is the following: \n\n\\begin{question}\\label{quest:main}\n Let $X$ be a smooth projective toric variety over an arbitrary field.  Does $X$ admit a full exceptional collection? If so, does $X$ possess a full exceptional collection of sheaves?\n\\end{question}\n \nOver an algebraically-closed field of characteristic zero, there is always a full exceptional collection of objects \\cite{Kawamata,Kawamata2} while the question of a full exceptional collection of sheaves is due to Orlov. \nMaking allowances for different language,\nthe question is also known to have a positive answer for\nSeveri-Brauer varieties~\\cite{AB,Bernardara},\nminimal toric surfaces~\\cite{BSS},\nand smooth projective toric varieties with absolute Picard rank at most\n$2$~\\cite{Yan}.\n\nIn this article, we provide further evidence for a positive answer to\nQuestion~\\ref{quest:main},\ntreating cases with low dimension or a high degree of symmetry.\n\n\\begin{thm} \\label{thm:examples}\n The following possess full exceptional collections of sheaves:\n \\begin{itemize}\n  \\item smooth toric surfaces (Proposition~\\ref{prop:surface}),\n  \\item smooth toric Fano 3-folds (Proposition~\\ref{prop:3fold}), \n  \\item all forms of 43 of the 124 smooth split toric Fano 4-folds (Section~\\ref{sect:4fold}),\n  \\item all forms of centrally symmetric toric Fano varieties (Corollary~\\ref{cor:centsym}), and\n  \\item and all forms in characteristic zero of toric varieties corresponding to Weyl fans of root systems of type $A$ (Proposition~\\ref{prop:X(An)excpcoll}).\n \\end{itemize}\n\\end{thm}\n\nOur results leverage extant work in the algebraically closed case\nsuch as \\cite{Uehara} for 3-folds and\\cite{Prabhu} for 4-folds. We use Castravet and Tevelev's recently discovered exceptional collection for $X(A_n)$ \\cite{CT}. \nFor the centrally symmetric toric Fano varieties (which are\nproducts of ``generalized del Pezzo varieties'' and projective lines\n\\cite{VosKly}), we use an explicit exceptional\ncollection (see also \\cite{BDMdP}) closely related to the one found in \\cite{CT}.\nUp to a twist by a line bundle, the authors had independently\ndiscovered the exact same collection!\nThis suggests that symmetry imposes strong conditions on the\npossible exceptional collections, which paradoxically makes them easier to find.\n\nTo study arithmetic exceptional collections, we establish\nan effective Galois descent result for general exceptional collections.\nThis applies to general varieties, not just toric ones.\n\n\\begin{thm}[Theorem~\\ref{thm:descblocks}, Lemma~\\ref{lem:Galconverse}]\n Let $X$ be a $k$-scheme and $L\/k$ a $G$-Galois extension.  Then $X_L$ admits a full (resp. strong) $G$-stable exceptional collection of objects of $\\mathsf{D^b}(X_L)$ (resp. sheaves, resp. vector bundles) if and only if $X$ admits a full (resp. strong) exceptional collection of objects of $\\mathsf{D^b}(X)$ (resp. sheaves, resp. vector bundles).\n\\end{thm}\n\nWe highlight one corollary of a positive answer to\nQuestion~\\ref{quest:main}. Arithmetic toric varieties are also studied\nin \\cite{MerkPan}, which focused on computing their algebraic $K$-groups\nvia decompositions in a certain noncommutative motivic category of\n$K_0$-correspondences. They showed that for an arithmetic toric\n$k$-variety $X$ with $G = \\text{Gal}(k^s\/k)$, the group $K_0(X_{k^s})$\nis a direct summand of a \\emph{permutation $G$-module} (there exists a\n${\\mathbb Z}$-basis permuted by $G$).\n\n\\begin{question}[Merkurjev-Panin \\cite{MerkPan}]\\label{quest:MP}\nLet $X$ be an arithmetic toric variety over $k$ and $G = \\operatorname{Gal}(k^s\/k)$.  Is $K_0(X_{k^s})$ always a permutation $G$-module?\n\\end{question}\n\nQuestion~\\ref{quest:main} can be viewed as a categorification of Question~\\ref{quest:MP} as any such exceptional collection over $k$ immediately gives a permutation basis. \n\nIn order to show that every toric variety has a full exceptional collection\nover ${\\mathbb C}$, the main tool used in \\cite{Kawamata,Kawamata2} was the\nminimal model program (MMP) in birational geometry.\nThe basic building blocks are toric stacks with Picard rank one,\nwhich always have full strong exceptional collections of line bundles.\nIndeed, runs of the MMP can be leveraged to effectively produce\nexceptional collections \\cite{BFK}.\n\nOver a non-closed field, one hopes to use the Galois-equivariant MMP,\nbut the situation is more complicated.\nThe most basic building blocks in this framework are those varieties $X$\nwhich have $\\rho^G = \\text{rank} (\\text{Pic}(X)^G )= 1$.\nBased on the results above and the hope of using the MMP in the\narithmetic situation, we ask the following\nquestion in the vein of \\cite{King,BH,CM-RFrob}:\n\n\\begin{question}\\label{quest:invpicrank1}\nLet $X$ be a smooth toric $k$-variety and $L\/k$ a $G$-Galois splitting\nfield.  If $\\operatorname{Pic}(X_L)^G$ is of rank 1, does $X_L$ possess\na full strong $G$-stable exceptional collection consisting of line\nbundles?\n\\end{question}\n\n\\subsection*{Acknowledgements}\n\nThe first author was partially supported by NSF DMS-1501813. He would\nalso like to thank the Institute for Advanced Study for providing a\nwonderful research environment. These ideas were developed during his\nmembership.  The first author benefited from discussions with Alicia\nLamarche.\nThe second author was partially supported by NSA grant\nH98230-16-1-0309.\nThe third author would like to thank the Hausdorff Institute for their\nhospitality and lively research environment during the \\emph{K-theory\nand related fields} trimester program. A large portion of this\nmanuscript was drafted during his time in Bonn.\nAll authors also thank Fei Xie for pointing out that, due to an editing\nerror, in a previous version of this paper,\nProposition~\\ref{prop:surface} stated that all smooth toric surfaces\nhave strong collections of vector bundles instead of a not-necessarily\nstrong collection of sheaves as was proven.\nThe authors would also like to thank an anonymous referee for useful\ncomments.\n\n\n\\subsection*{Organization}\n\nSection~\\ref{sect:galdesc} treats Galois descent of exceptional\ncollections consisting of objects on (possibly non-toric) varieties. In\nSection~\\ref{section:toric}, we recall appropriate definitions of\narithmetic toric varieties and establish additional descent results\nwhich are specific to toric varieties.  In\nSection~\\ref{section:minimal}, we consider a range of examples.  We begin by treating toric surfaces, followed by toric Fano 3-folds.  For toric Fano 4-folds, we give partial results. We conclude by investigating the class of centrally symmetric toric Fano varieties, including the generalized del Pezzo varieties, and handling toric varieties associated to root systems of type $A$. \n\n\n\\subsection*{Notation} Throughout, $k$ denotes an arbitrary field and\n$k^s$ a separable closure. A \\emph{variety} is a geometrically integral\nseparated scheme of finite type over $k$. All our schemes will be\nquasi-compact and quasi-separated. For a $k$-scheme $X$ and field\nextension $L\/k$, we write $X_L : = X \\times _{\\operatorname{Spec} k} \\operatorname{Spec} L$.  If $A$\nis a $k$-algebra, we write $A_L = A\\otimes _{k} L$.\nWe use $\\mathsf{D^b}(X)$ to denote the bounded derived category\n$\\mathsf{D^b}(\\text{Coh}(X))$. For an $\\O_X$-algebra $A$, we write\n$\\mathsf{D^b}(A)$ for the bounded derived category of complexes of\n$A$-modules which are coherent $\\O_X$-modules.\n\n\n\\section{Galois descent and exceptional collections}\\label{sect:galdesc}\n\nIn this section we develop Galois descent for exceptional collections (in a generalized sense). We begin by recalling some definitions and conventions concerning structure theory of derived categories of schemes.  We then give our main descent results for $G$-stable exceptional collections (Theorem~\\ref{thm:descblocks}). We complete the section by collecting a few useful consequences to be used in the sequel.\n\n\n\\subsection{Exceptional collections}\n\nWe give some conventions for semiorthogonal decompositions of derived categories and in particular exceptional collections.  Such collections have been widely studied over algebraically closed fields but have recently been treated in more generality \\cite{AAGZ, AB, ABB, Bernardara, BSS, Elagin, Xie, Yan}.  We refer the reader to Remarks~\\ref{rem:descSOD} and \\ref{rem:elagin} for added details on some of these results.  \n\nFor a triangulated category $\\mathsf{T}$, we use the standard notation $\\text{Ext}^n_{\\mathsf{T}}(A, B) = \\hom _{\\mathsf{T}} (A, B[n])$.  For objects $A, B $ of $\\mathsf{D^b}(X)$, we use  $\\text{End}_X(A)$ and $\\text{Ext}_X^n(A, B)$ to denote $\\text{End}_{\\mathsf{D^b}(X)}(A)$ and $\\text{Ext}^n_{\\mathsf{D^b}(X)}(A, B)$, respectively.\n\n\\begin{defn}[see \\cite{BK}]\nLet $\\mathsf{T}$ be a triangulated category.  A full triangulated subcategory of $\\mathsf{T}$ is \\emph{admissible} if its inclusion functor admits left and right adjoints.  A \\emph{semiorthogonal decomposition} of $\\mathsf{T}$ is a sequence of admissible subcategories $\\mathsf{C}_1, ..., \\mathsf{C}_s$ such that \n\\begin{enumerate}\n\\item $\\hom _{\\mathsf{T}}(A_i, A_j) = 0$ for all $A_i \\in \\operatorname{Ob} (\\mathsf{C}_i)$, $A_j \\in \\operatorname{Ob} (\\mathsf{C}_j)$ whenever $i > j$.\n\\item For each object $T$ of $\\mathsf{T},$ there is a sequence of morphisms $0 = T_s \\to \\cdots \\to T_0 = T$ such that the cone of $T_i \\to T_{i-1}$ is an object of $\\mathsf{C}_i$ for all $i = 1,..., s$.\n\\end{enumerate}\nWe use  $\\mathsf{T} = \\langle \\mathsf{C}_1,..., \\mathsf{C}_s\\rangle$ to denote such a decomposition.\n\\end{defn}\n\nParticularly nice examples of semiorthogonal decompositions are given by exceptional collections, the study of which goes back to Beilinson \\cite{Beilinson}.\n\n\\begin{defn}\\label{def:exceptional}\nLet $\\mathsf{T}$ be a $k$-linear triangulated category.  An object $E$ in $\\mathsf{T}$ is \\emph{exceptional} if the following conditions hold:\n\\begin{enumerate}\n\\item $\\text{End}_{\\mathsf{T}}(E)$ is a division $k$-algebra.\n\\item $\\text{Ext}^n_{\\mathsf{T}}(E, E) = 0$ for $n \\neq 0$.\n\\end{enumerate}\nA totally ordered set $\\mathsf{E} = \\{E_1, ..., E_s\\}$ of exceptional objects is an \\emph{exceptional collection} if $\\text{Ext}^n_{\\mathsf{T}}(E_i, E_j) = 0$ for all integers $n$ whenever $i >j$.  An exceptional collection is $\\emph{full}$ if it generates $\\mathsf{T}$, i.e., the smallest thick subcategory of $\\mathsf{T}$ containing $\\mathsf{E}$ is all of $\\mathsf{D^b}(X)$.  An exceptional collection is \\emph{strong} if $\\text{Ext}^n_{\\mathsf{T}}(E_i, E_j) = 0$ whenever $n \\neq 0$.  An \\emph{exceptional block} is an exceptional collection $\\mathsf{E} = \\{ E_1, ..., E_s\\}$ such that $\\text{Ext}^n_{\\mathsf{T}}(E_i, E_j) = 0$ for every $n$ whenever $i \\neq j$. Given an exceptional collection $\\mathsf{E} = \\{E_1, ..., E_s\\}$, we denote by $\\langle \\mathsf{E} \\rangle$ the category generated by the objects $E_i$.\n\\end{defn}\n\n\\begin{rem}\nOur notion of exceptional object generalizes the classical one, where item $(1)$ of Definition~\\ref{def:exceptional} is replaced by:  $\\text{End}_{\\mathsf{T}}(E) = k$ \\cite[$\\S$2]{Bondal}.  Over algebraically or separably closed fields, these definitions agree.  Over non-closed fields, the classical definition is too restrictive to allow for the use of interesting arithmetic invariants in the study of exceptional collections on twisted forms, e.g., Brauer classes.\n\\end{rem}\n\n\\begin{prop}[Thm. 3.2 \\cite{Bondal}]\\label{prop:exctosod}\nLet $X$ be a $k$-scheme with exceptional collection $\\{E_1,..., E_s\\}$.  If $\\mathscr{E}_i $ is the category generated by $E_i$, there is a semiorthogonal decomposition $\\mathsf{D^b}(X) = \\langle \\mathscr{E}_1,..., \\mathscr{E}_s, \\mathsf{A} \\rangle$, where $\\mathsf{A}$ is the full subcategory with objects $A$ such that $\\operatorname{Hom}_X(A, E_i) = 0$ for all $i$.\n\\end{prop}\n\n\\begin{rem}\n The reference assumes smoothness and projectivity but the conclusion is independent of this. Note further that if $X$ admits a full exceptional collection then it is automatically smooth and proper by \\cite[Propositions 3.30 and 3.31]{OrlovNCstuff}.\n\\end{rem}\n\nThe existence of an exceptional collection on a scheme $X$ provides a means of studying derived geometry of $X$ in purely algebraic terms.  Indeed, in such a situation, one may identify an ``underlying\" $k$-algebra which is derived equivalent to $X$.  For exceptional blocks, one obtains a similar but slightly stronger fact.\n\n\\begin{prop}[Thm. 6.2 \\cite{Bondal}]\\label{thm:tilting}\nLet $X$ be a smooth projective $k$-scheme and let $\\{E_1,..., E_n\\}$ be a full strong exceptional collection on $\\mathsf{D^b}(X)$.  Let $\\mathcal{E}  = \\bigoplus E_i$ and $A = \\operatorname{End}(\\mathcal{E})$.  Then $\\mathsf{R} \\hom _{\\mathsf{D^b}(X)}(\\mathcal{E}, -) : \\mathsf{D^b}(X) \\to \\mathsf{D^b}(A)$ is a $k$-linear equivalence.\n\\end{prop}\n\n\\begin{prop}\nIf $\\mathsf{E} = \\{E_1,..., E_s\\}$ is an exceptional block with $\\operatorname{End}(E_i) = D_i$, there is a $k$-algebra isomorphism $\\operatorname{End}(\\bigoplus E_i) \\simeq D_1 \\times \\cdots \\times D_s$, and hence a $k$-linear equivalence $\\langle \\mathsf{E} \\rangle \\simeq  \\mathsf{D^b}(D_1 \\times \\cdots \\times D_n)$.\n\\end{prop}\n\nThe object $\\mathcal{E} = \\oplus E_i$ of Proposition~\\ref{thm:tilting} is usually called a \\emph{tilting object}.  If each $E_i$ is a sheaf (resp. vector bundle), then $E$ is called a \\emph{tilting sheaf} (resp. \\emph{tilting bundle}).  Until recently, the theory of tilting objects has served as the main tool for extending the study of exceptional collections to non-closed fields.  The results above show that any exceptional collection gives rise to both a tilting object and a semiorthogonal decomposition, and thus the admission of such a collection is a particularly special property of a given triangulated category.  Our aim in the following subsection is to extend descent results for semiorthogonal decompositions and tilting objects to (our more general notion of) exceptional collections. We give a formal definition of tilting object for completeness.\n\n\\begin{defn}\nA \\emph{tilting object} for a $k$-scheme $X$ is an object $\\mathcal{E}$ of $\\mathsf{D^b}(X)$ which satisfies the following conditions:\n\\begin{enumerate}\n\\item $\\text{Ext}_X ^n (\\mathcal{E}, \\mathcal{E}) = 0$ for $n > 0$.\n\n\\item $\\mathcal{E}$ generates $\\mathsf{D^b}(X)$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{rem}[$K$-theory and motivic decompositions]\n\nExceptional collections have a particularly interesting manifestation in the realm of noncommutative motives.  Indeed, an exceptional collection $\\{E_1,..., E_s\\}$ on a smooth projective variety $X$ yields a decomposition $U(X) \\simeq \\bigoplus _i U(D_i)$ of its corresponding universal additive invariant \\cite[$\\S$2.3]{Tabuada}, where $D_i = \\text{End}(E_i)$. This defines a motivic decomposition by viewing $X$ as an object in the Merkurjev-Panin category of $K$-motives \\cite{MerkPan} or Kontsevich's category of noncommutative Chow motives \\cite[Thm. 6.10]{Tabuada2} via its associated dg-category of perfect complexes.\n\nOne nice consequence is that this decomposition is detected by algebraic $K$-groups \\cite[Prop. 1.10]{AB} in addition to a slew of other additive invariants in the sense of Tabuada \\cite[$\\S$2.2]{Tabuada}. Such invariants include algebraic $K$-theory with coefficients, homotopy $K$-theory, \\'{e}tale $K$-theory, (topological) Hochschild homology, and (topological) cyclic homology.\n\\end{rem}\n\n\n\\subsection{Galois descent} We develop Galois descent for exceptional\ncollections consisting of objects in the derived category\n$\\mathsf{D^b}(X)$ of a (smooth projective) variety $X$.  Throughout this\nsection, pushforward and pullback functors are understood to be derived.\nFor a $k$-scheme $X$ and finite Galois extension $L\/k$, any element $g\n\\in \\text{Gal}(L\/k)$ defines a morphism of $k$-schemes $g: X_L \\to X_L$\nwhich in turn defines the functor $g^*: \\mathsf{D^b}(X_L) \\to \\mathsf{D^b}(X_L)$.\n\n\\begin{defn}\nLet $X$ be a scheme with an action of a group $G$.  A $G$-\\emph{stable\nexceptional collection} on $X$ is an exceptional collection $\\mathsf{E}\n= \\{E_1, ..., E_s\\}$ of objects in $\\mathsf{D^b}(X)$ such that\nfor all $g \\in G$ and $1 \\leq i \\leq s$ there exists $E \\in \\mathsf{E}$\nsuch that $g^*E_i \\simeq E$.\nWe say a $G$-stable exceptional collection $\\mathsf{E}$\nis a \\emph{$G$-orbit} if, for every pair of objects\n$E,E' \\in \\mathsf{E}$, there exists a $g \\in G$ such that\n$g^*E \\simeq E'$.\n\\end{defn}\n\n\\begin{rem}\\label{rem:invariant}\nA simple example of a $G$-stable exceptional collection is a\n$G$-\\emph{invariant} exceptional collection, i.e., an exceptional\ncollection $\\{E_1,..., E_s\\}$ such that $g^*E_i \\simeq E_i$ for all $1 \\leq i \\leq s$.  It is often the case that toric varieties admit exceptional collections consisting of line bundles.  If it is also the case that a group $G$ acts trivially on $\\text{Pic}(X)$, such a collection is automatically $G$-invariant, and hence $G$-stable (see Lemma~\\ref{lem:picinv}).\n\\end{rem}\n\n\\begin{lem}\\label{lem:collectiontoblocks}\nAny $G$-stable exceptional collection may be written as a collection of $G$-stable exceptional blocks (after possibly reordering).\n\\end{lem}\n\n\\begin{proof}\nThe decomposition of a $G$-stable exceptional collection into its $G$-orbits gives the desired exceptional blocks.  Let $\\mathsf{E}$ be a $G$-stable exceptional collection and for elements $E, E' \\in \\mathsf{E}$, we write $E \\leadsto E'$ if $\\text{Ext}^n(E, E') \\neq 0$ for some $n$.\n\n Let $\\mathsf{A} \\subset \\mathsf{E}$ be a $G$-orbit. To see that\n$\\mathsf{A}$ is an exceptional block, suppose that $E  \\leadsto E'$ for\n$E, E' \\in \\mathsf{A}$. Since $\\mathsf{A}$ is a $G$-orbit, $E' \\simeq g^*\nE$ for some $g \\in G$.  Thus, $E  \\leadsto g^* E$, and acting again by\n$g$, we have $g^*E  \\leadsto (g^2)^*E$. Since $A$ is finite, we have $E\n\\leadsto g^*E  \\leadsto \\cdots  \\leadsto (g^s)^*E  \\leadsto E$ for some\npositive integer $s$.  Thus,\nthere is no ordering of the elements of $\\mathsf{A}$\nsuch that they form a subset of an exceptional collection --- a\ncontradiction.\n\nIf $\\mathsf{B}$ is another $G$-orbit (distinct from $\\mathsf{A}$), we\nwould like to see that these blocks can be ordered to form an\nexceptional collection. We claim that for any $E \\in \\mathsf{A}$ and $F\n\\in \\mathsf{B}$, one has $E  \\leadsto F$ only if \n$\\mathsf{A}$ precedes\n$\\mathsf{B}$ in the collection $\\mathsf{E}$ (i.e., $\\text{Ext}^n(B, A) =\n0$ for all $n$ and all $A \\in \\mathsf{A}$, $B \\in \\mathsf{B}$).\nTo see this, assume that $E  \\leadsto\nF$ and $F  \\leadsto E'$ for some $E' \\in \\mathsf{A}.$ As $\\mathsf{A}$ is\na $G$-orbit, $E' \\simeq g^*E$ for some $g \\in G$. Hence, just as above,\nwe have a sequence $E  \\leadsto F  \\leadsto g^*E \\leadsto g^* F\n\\leadsto \\cdots  \\leadsto (g^s)^* F  \\leadsto E.$  Thus, there is no\nordering of the elements of $\\mathsf{A}$ and $\\mathsf{B}$ which forms an\nexceptional collection, contradicting the exceptionality of\n$\\mathsf{E}$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:galconj}\nLet $X$ be a Noetherian $k$-scheme, $L\/k$ a finite Galois extension with group $G$, and $\\pi: X_L \\to X$ the natural projection map.  For any object $M $ in $\\mathsf{D^b}(X_L)$ there is a natural isomorphism $\\displaystyle \\pi^* \\pi_*( M) \\simeq \\bigoplus _{g\\in G} g^*M.$  \n\\end{lem}\n\n\\begin{proof}\n\nAs $\\pi$ is flat and affine, every coherent sheaf on $X$ is acyclic for $\\pi^\\ast$ and every coherent sheaf on $X_L$ is acyclic for $\\pi_\\ast$. Hence, the derived functors coincide with the application of $\\pi^\\ast$ or $\\pi_\\ast$ component-wise to a complex. Thus, it suffices to establish a natural isomorphism at the level of coherent sheaves. \n\nFor any object $M$ of $\\text{Coh}(X_L)$, we have $ \\pi_*M \\simeq \\pi_*\ng^* M $, and adjunction yields a natural transformation $ \\pi^* \\pi_*\n\\to g^*$.  Summing over all $g \\in G$ provides the transformation\n$\\alpha:   \\pi^* \\pi_* \\to \\oplus g^* $. We show this is an\nisomorphism.\n\nIt suffices to check that $\\alpha$ is an isomorphism on any affine patch, $\\operatorname{Spec} R$, of $X$. Passing to modules, we abuse notation and let $M$ be a finitely-generated module over $R_L = R \\otimes_k L$. Choose a presentation of $M$\n\\begin{displaymath}\n R_L^{\\oplus m} \\to R_L^{\\oplus n} \\to M \\to 0\n\\end{displaymath}\nand evaluate $\\alpha$ on the sequence to get the commutative diagram \n\\begin{center}\n \\begin{tikzpicture}\n  \\node at (-4,0.75) (pr1) {$R^{\\oplus m} \\otimes_k \\left( L \\otimes_k L \\right)$};\n  \\node at (-4,-0.75) (gr1) {$R^{\\oplus m} \\otimes_k \\left( \\oplus_g \\Gamma_g(L) \\right)$};\n  \\node at (0,0.75) (pr0) {$R^{\\oplus n} \\otimes_k \\left( L \\otimes_k L\\right) $};\n  \\node at (0,-0.75) (gr0) {$R^{\\oplus m} \\otimes_k \\left( \\oplus_g \\Gamma_g(L)\\right) $};\n  \\node at (3,0.75) (pm) {$M \\otimes_R R_L$};\n  \\node at (3,-0.75) (gm) {$\\oplus_g g^\\ast M$};\n  \\node at (5,0.75) (p0) {$0$};\n  \\node at (5,-0.75) (g0) {$0$};\n  \\draw[->] (pr1) -- (pr0);\n  \\draw[->] (pr0) -- (pm);\n  \\draw[->] (pm) -- (p0);\n  \\draw[->] (gm) -- (g0);\n  \\draw[->] (gr1) -- (gr0);\n  \\draw[->] (gr0) -- (gm);\n  \\draw[->] (pr1) -- node[left] {$\\alpha_{R^{\\oplus m}}$} (gr1);\n  \\draw[->] (pr0) -- node[left] {$\\alpha_{R^{\\oplus n}}$} (gr0);\n  \\draw[->] (pm) -- node[left] {$\\alpha_M$} (gm);\n \\end{tikzpicture}\n\\end{center}\nwhere $\\Gamma_g(L)$ denotes the graph of $g$ in $L \\otimes_k L$. The left and middle maps are isomorphisms, so the right map must also be an isomorphism. \n\\end{proof}\n\n\\begin{prop}[Descent for orbits]\\label{prop:objblockdescent}\nLet $X$ be a $k$-scheme, $L\/k$ a finite $G$-Galois extension,\nand $\\pi: X_L \\to X$ the natural projection map. If $\\mathsf{E} = \\{E_1,\n\\ldots, E_s\\}$ is a $G$-orbit forming an exceptional collection on $X_L$,\nand if $E$ is any element of $\\mathsf{E}$, then there is an exceptional\nobject $F$ in $\\mathsf{D^b}(X)$ such that $\\pi_*E \\simeq F^{\\oplus m}$\nand $\\pi^*F$ generates the category $ \\langle \\mathsf{E} \\rangle$.\n\\end{prop}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:collectiontoblocks}, exceptional $G$-orbits are\ncompletely orthogonal (and by definition carry a transitive action of\n$G$), which will be used throughout the proof. Fix an element $E \\in\n\\mathsf{E}$, so that $E = E_i$ for some $i$. Lemma~\\ref{lem:galconj}\ngives $$\\pi^*\\pi_*E \\simeq \\bigoplus _{g \\in G} g^*E$$  We claim that\n$\\text{End}(\\pi_*E)$ is a matrix algebra over a division algebra, and\nprove this by first showing that it is semisimple. Indeed, using\n$\\text{End}_X(M) \\otimes _k L \\simeq \\text{End}_{X_L} (\\pi^*M)$ for any $M\n\\in \\mathsf{D^b}(X)$ \\cite[Rem. 2.1]{AB}, we have\n\\[ \\text{End}_X(\\pi_*E) \\otimes _k L \\simeq\n\\text{End}_{X_L}(\\pi^* \\pi_* E) \\simeq\n\\text{End}_{X_L}\\left(\\bigoplus _{ g\\in G} g^*E\\right).\n\\]  Each $g^*E$ is exceptional so that\n$\\text{End}_{X_L}(g^*E) =: D_g$ is a division algebra for each element $g\n\\in G$.  Let  $H \\leq G$ be the subgroup consisting of elements $h$\nsatisfying $h^*E \\simeq E$. For any system of coset representatives $g\n\\in G\/H$, we have $\\text{End}_X(\\pi_*E)_L \\simeq \\prod _{g \\in\nG\/H}M_m(D_g)$, where $m = |H|$. This product of matrix algebras over\ndivision algebras is semisimple, i.e., the Jacobson radical\n$\\text{rad}(\\text{End}_X(\\pi_*E)_L) = 0$.  We then have $0 =\n\\text{rad}(\\text{End}_X(\\pi_*E)_L) = \\text{rad}(\\text{End}_X(\\pi_*E))_L$\nby ~\\cite[Thm.~1]{Amitsur}, and hence $\\text{rad}(\\text{End}_X(\\pi_*E))\n= 0$. Thus, $\\text{End}_X(\\pi_*E)$ is semisimple and so must also\nbe a product of matrix algebras over division algebras by the\nArtin-Wedderburn Theorem.  \n\nLet $Z$ be the center of $\\text{End}_X(\\pi_*E)$ and $Z_L$ the center of $\\text{End}_X(\\pi_*E)_L$.\nNote that $Z$ is an \\'{e}tale $k$-algebra, and to show that $\\text{End}(\\pi_*E)$ is a matrix algebra, it suffices to show that $Z$ has no zero divisors, and is thus a field.  There is an embedding $Z \\hookrightarrow Z_L  = \\prod_{g \\in G\/H}  L_g$, where $L_g$ is the center of the division algebra $D_g$.  The transitive action of $G$ on $\\{E_1,..., E_s\\}$ implies that $G$ acts transitively on a basis of $Z_L$, so that $Z = (Z_L)^G$ has no zero divisors.\n\nWe produce the object $F$ using the identification $\\text{End}_X(\\pi_*E)\n\\simeq M_n(D)$, where $D$ is a division algebra.\nLet $e_i = e_{ii}$ denote the usual idempotent matrices, so that\n$\\{e_i\\}$ is a complete set of primitive orthogonal idempotents.\nNotice that $F_i:= \\operatorname{Im} (e_i)$ is a simple $\\text{End}_X(\\pi_*E)$-submodule of $\\pi_*E$ for each $i$, and hence $F_i \\simeq F_j$ for each $i, j$, and $\\text{End}_X(F_i) \\simeq D$. Define $F = \\operatorname{Im} (e_1) \\subset \\pi_*E$, included as a direct summand.  We note that $\\pi_*E \\simeq \\bigoplus F_i \\simeq F^{\\oplus n}$.  \n\nWe now show that $F$ is an exceptional object on $X$.  As stated above, $\\text{End}_X(F)$ is a division algebra, so it suffices to show that $\\text{Ext}^n_X(F, F) = 0$ for $ n \\neq 0$.  Using Lemma~\\ref{lem:galconj} and $(\\pi^*, \\pi_*)$-adjunction, we have $$\\text{Ext}^n_X(\\pi_*E, \\pi_*E) = \\bigoplus _{g\\in G} \\text{Ext}^n _{X_L}(g^*E, E).$$  For $n \\neq 0$, each summand of the right-hand side is 0, which follows from the mutual orthogonality of the exceptional block $\\mathsf{E}$ (when $g^*E \\not\\simeq E$) and from exceptionality of $E$ (when $g^*E \\simeq E$).  Since $F$ is a direct summand of $\\pi_*E$, it follows that $\\text{Ext}^n_X(F, F)$ is a summand of $\\text{Ext}^n_X(\\pi_*E, \\pi_*E) = 0$.\n\nLastly, we show that $\\pi^*F$ generates the category $\\langle \\mathsf{E}\n\\rangle$.   Since $ F^{\\oplus m} \\simeq \\pi_*E$, extending scalars to $L$\ngives $ (\\pi^*F)^{\\oplus m} = \\pi^*(F^{\\oplus m}) \\simeq \\pi^*\\pi_*E\n\\simeq \\bigoplus g^*E$.  Thus, $$\\langle \\pi^*F \\rangle = \\langle (\\pi ^*F )^{\\oplus m} \\rangle = \\langle \\bigoplus g^* E \\rangle = \\langle g^*E \\rangle _{g \\in G} = \\langle \\mathsf{E} \\rangle.$$\\end{proof}\n\n\n\\begin{rem}\\label{rem:descSOD}\nProposition~\\ref{prop:objblockdescent} provides a very specific case of descent\nfor triangulated categories, the main advantage of which is that it\nallows one to identify a specific exceptional object that base extends\nto the given orbit.\nMoreover, a $G$-orbit which forms an exceptional collection consisting of vector bundles (resp. sheaves) descends to an exceptional collection consisting of vector bundles (resp. sheaves).  Compare to the following descent result for semiorthogonal decompositions, which generalizes \\cite[Cor. 2.15]{Toen}.  Although this result is useful for descending semiorthogonal decompositions, it does not identify exceptional objects.\n\n\\begin{prop}[Prop. 2.12, \\cite{AB}]\\label{prop:ABdesc}\nLet $\\mathsf{T}$ be a $k$-linear triangulated category such that $\\mathsf{T}_{k^s}$ is $k^s$-equivalent to $\\mathsf{D^b}(k^s, (k^s)^n)$.  Then there exists an \\'{e}tale algebra $K$ of degree $n$ over $k$, an Azumaya algebra $A$ over $K$, and a $k$-linear equivalence $\\mathsf{T} \\simeq \\mathsf{D^b}(K\/k, A)$.\n\\end{prop}\n\n\\noindent Let $X$, $\\mathsf{E}$, and $F$ be as in Proposition~\\ref{prop:objblockdescent}, and note that taking $\\mathsf{T} = \\langle F \\rangle$, we have $\\mathsf{T}_{k^s} = \\langle \\pi^* F \\rangle_{k^s} = \\langle \\mathsf{E} \\rangle _{k^s}$. Since $\\mathsf{E} = \\{ g^*E\\}_{g \\in G}$ is a full exceptional collection for $\\langle \\mathsf{E} \\rangle$, the bundle $\\mathcal{E} : = \\oplus (g^*E)_{k^s}$ is a tilting object for $\\langle \\mathsf{E}\\rangle_{k^s}$.  This defines an equivalence $$ \\mathsf{T}_{k^s} \\simeq \\langle \\mathsf{E} \\rangle_{k^s} \\simeq \\mathsf{D^b}(k^s, \\text{End}(\\mathcal{E})) = \\mathsf{D^b}(k^s, (k^s)^n).$$ Proposition~\\ref{prop:ABdesc} yields an \\'{e}tale extension $K\/k$, an Azumaya $K$-algebra $A$, and an equivalence $\\mathsf{T} \\simeq \\mathsf{D^b}(K\/k, A)$.  In this case, since $\\mathsf{T} = \\langle F \\rangle$, we see that $A = \\text{End}_X(F)$ is an Azumaya algebra over its center $Z$ (using the notation found in the proof of Proposition~\\ref{prop:objblockdescent}), which is simply a field extension of $k$.\n\\end{rem}\n\n\\begin{thm}[Descent for stable collections]\\label{thm:descblocks}\nLet $X$ be a $k$-scheme, $L\/k$ a finite $G$-Galois extension, and $\\pi: X_L \\to X$ the natural projection map.  If $X_L$ admits a full $G$-stable exceptional collection $\\mathsf{E}$ of objects of $\\mathsf{D^b}(X_L)$, then $X$ admits a full exceptional collection $\\mathsf{F}$ of objects of $\\mathsf{D^b}(X)$.  If $\\mathsf{E}$ is strong, so is $\\mathsf{F}$.  If the elements of $\\mathsf{E}$ are vector bundles (resp. sheaves), the elements of $\\mathsf{F}$ are vector bundles (resp. sheaves).\n\\end{thm}\n\n\\begin{proof}\n\nBy Lemma~\\ref{lem:collectiontoblocks}, we may write $\\mathsf{E} =\n\\{\\mathsf{E}^1,..., \\mathsf{E}^s\\}$ as a collection of $G$-stable\nblocks, where each block is given by a $G$-orbit.\nProposition~\\ref{prop:objblockdescent} then associates to each block\n$\\mathsf{E}^i$ an exceptional object $F_i$ on $X$, and we show that\n$\\mathsf{F} = \\{F_1,..., F_s\\}$ is a full exceptional collection on $X$.\nWe first show that $\\text{Ext}^n_{X}(F_i, F_j) = 0$ for all $n$ whenever\n$i > j$.  Let $E^i$ and $E^j$ be elements of the collections\n$\\mathsf{E}^i$ and $\\mathsf{E}^j$, respectively.  We then have\n\\begin{equation}\n\\text{Ext}^n_X(\\pi_*E^i, \\pi_*E^j) \\simeq\n\\bigoplus _{g\\in G} \\text{Ext}^n _{X_L}(g^*E^i, E^j).\n\\label{eq:1}\n\\end{equation}\nSince\n$E^i$ and $E^j$ are elements of the exceptional collection $\\mathsf{E}$\nand $i < j$, each summand is 0 for all $n$, so that\n$\\text{Ext}_X^n(\\pi_*E^i, \\pi_*E^j) = 0$ for all $n$.  The objects $F_i$\nand $F_j$ are direct summands of $\\pi_*E^i$ and $\\pi_*E^j$,\nrespectively, and it follows that $\\text{Ext}^n_X(F_i, F_j) = 0$ for all\n$n$.\n\nBy Proposition~\\ref{prop:exctosod}, the exceptional collection $\\{ F_1,\n\\ldots, F_s\\}$ yields a semiorthogonal decomposition\n\\[\n\\mathsf{D^b}(X) = \\langle \\mathscr{F}_1, \\ldots, \\mathscr{F}_s, \\mathsf{A}\\rangle,\n\\]\nwhere\n$\\mathscr{F}_i = \\langle F_i \\rangle$ and $\\mathsf{A}$ is the full\nsubcategory of objects $A$ with $\\text{Hom}_{\\mathsf{D^b}(X)}(A, F_i) =\n0$ for all $i$.  In particular, the subcategories $\\mathscr{F}_i$ are\nadmissible.  Extending scalars to $L$, we have $(\\mathscr{F}_i)_L =\n\\langle \\mathsf{E}^i\\rangle$, as both categories are generated by\n$\\pi^*F$ by Proposition~\\ref{prop:objblockdescent}. The exceptional collection\n$\\mathsf{E} =\\{\\mathsf{E}^1,\\ldots, \\mathsf{E}^s \\}$ is full, hence we have\na semiorthogonal decomposition $$ \\mathsf{D^b}(X_L) =  \\langle\n(\\mathscr{F}_1)_L, \\ldots, (\\mathscr{F}_s)_L \\rangle.$$  Since our\nadmissible subcategories $\\mathscr{F}_i$ base extend to a semiorthogonal\ndecomposition, \\cite[Lem. 2.9]{ABB} gives a semiorthogonal decomposition\n$\\mathsf{D^b}(X) = \\langle \\mathscr{F}_1, \\ldots, \\mathscr{F}_s \\rangle$.\nIn particular, the collection $\\{F_1, \\ldots, F_s\\}$ generates\n$\\mathsf{D^b}(X)$, so this collection is full.\n\nIf $\\mathsf{E}$ is strong, the right side of \\eqref{eq:1} vanishes for $i \\neq j$ (and any $n$).  It follows exactly as above that $\\text{Ext}^n_{X}(F_i, F_j) = 0$ for all $n$ when $i \\neq j$, so that $\\mathsf{F}$ is strong.\n\\end{proof}\n\n\\begin{rem}\\label{rem:elagin}\nSimilar descent results for collections of sheaves are given by Elagin in the algebraically closed case (i.e., $k = \\bar{k}$) using the framework of equivariant exceptional collections in equivariant derived categories \\cite{Elagin}.  Indeed, for a variety $X$ with an action of a finite group $G$ and a $G$-invariant exceptional collection (see Remark~\\ref{rem:invariant}) consisting of sheaves, this descent result is given in terms of $\\alpha$-twisted representations of $G$ (see Theorem 2.2 of loc. cit.).  For a $G$-stable exceptional collection consisting of sheaves, results are in terms of coinduced twisted representations of $G$ (see Theorem 2.3 of loc. cit.).\n\\end{rem}\n\n\\begin{lem} \\label{lem:Galconverse}\nLet $X$ be a $k$-scheme and $L\/k$ a finite $G$-Galois extension.  If $X$ admits an exceptional collection, then $X_L$ admits a $G$-stable exceptional collection.\n\\end{lem}\n\n\\begin{proof}\n Let $E_1,\\ldots,E_s$ be the exceptional collection on $X$ and consider $\\pi^\\ast E_1, \\ldots, \\pi^\\ast E_s$ on $X_L$. To compute morphisms, we note that \n \\begin{displaymath}\n  \\operatorname{Hom}_{X_L} (\\pi^\\ast E_i , \\pi^\\ast E_j ) = \\operatorname{Hom}_X( E_i, \\pi_\\ast \\pi^\\ast E_j) = \\operatorname{Hom}_X(E_i, E_j \\otimes_k L) = \\operatorname{Hom}_X(E_i,E_j) \\otimes_k L.\n \\end{displaymath}\n This vanishes if $j > i$. Let $A_i = \\operatorname{Hom}_X(E_i, E_i)$. We can split $A_i \\otimes_k L$ as a product of matrix algebras over division algebras $A_{i,j} = M_{N_{i,j}}(D_{i,j})$ and correspondingly decompose\n \\begin{displaymath}\n  \\pi^\\ast E_i = \\bigoplus F_{i,j}^{N_{i,j}}\n \\end{displaymath}\n with \n \\begin{displaymath}\n  \\operatorname{Hom}_{X_L}( F_{i,j} , F_{i,j} ) = D_{i,j}.\n \\end{displaymath}\n Note that $F_{i,j}$ and $F_{i,j^\\prime}$ are orthogonal for $j \\not = j^\\prime$. Thus, we have an exceptional collection.\n\\end{proof}\n\n\\begin{lem}\\label{lem:picinv}\nLet $X$ be a $k$-scheme and $L\/k$ a finite extension with Galois group $G$.  If $G$ acts trivially on $\\operatorname{Pic}(X_L)$ and $X_L$ admits an exceptional collection of line bundles, then $X$ admits an exceptional collection of vector bundles.\n\\end{lem}\n\n\\begin{proof}\n The collection on $X_L$ is automatically $G$-stable pointwise. Hence we can apply Theorem~\\ref{thm:descblocks}. \n\\end{proof}\n\n\\begin{rem}\n Note that while we may start with a collection of line bundles,\nthe descended collection may not consist only of line bundles.\nAn example of this is the real conic discussed in the introduction.\n\\end{rem}\n\n\\begin{lem}\\label{lem:blowup}\nLet $X$ be a smooth $k$-variety and $L\/k$ a $G$-Galois extension.  Let $Y_1, ..., Y_s$ be a $G$-orbit of smooth transversal subvarieties of $X_L$. Let $Y_I = \\cap_{i \\in I} Y_i$ and let $H_I$ be the normalizer of $Y_I$. If each $Y_I$ admits a full $H_I$-stable exceptional collection, then $\\tilde{X}$ admits an exceptional collection, where $\\tilde{X}_L$ is the iterated blow up of $X_L$ at the $Y_i$ (in any order).\n\\end{lem}\n\n\\begin{proof}\n This is an iterated application of Orlov's Theorem, see \\cite[Lemma 7.2]{CT}.\n\\end{proof}\n\n\n\n\\section{Arithmetic toric varieties}\\label{section:toric}\n\nWe introduce toric varieties over arbitrary fields. Such varieties, also known as \\emph{arithmetic toric varieties}, have been treated in \\cite{Duncan, ELFST, MerkPan, VosKly}.  \n\\begin{defn}\n\nA \\emph{torus} (over $k$) is an algebraic group $T$ (over $k$) such that $T _{k^s} \\simeq \\mathbb{G}_{m} ^n$. A torus is \\emph{split} if $T \\simeq \\mathbb{G}_{m} ^n$.  A field extension $L\/k$ satisfying $T_L \\simeq \\mathbb{G}_{m} ^n$ is called a \\emph{splitting field} of the torus $T$.  Any torus admits a finite Galois splitting field. \n\\end{defn}\n\n\\begin{defn}\nGiven a torus $T$, a \\emph{toric} $T$-\\emph{variety} is a normal variety with a faithful $T$-action and a dense open $T$-orbit.  A toric $T$-variety is \\emph{split} if $T$ is a split torus. A \\emph{splitting field} of a toric $T$-variety is a splitting field of $T$. A variety is a \\emph{toric variety} if it is a toric $T$-variety for some torus $T$.\n\\end{defn}\n\n\n\\begin{defn}\nLet $X$ be a toric $T$-variety whose dense open $T$-orbit contains a $k$-rational point.  Then we say $X$ is \\emph{neutral} \\cite{Duncan} (or a \\emph{toric} $T$-\\emph{model} \\cite{MerkPan}).\nAn orbit of a split torus always has a $k$-point, so a split toric\nvariety is neutral; but the converse is not true in general.\n\\end{defn}\n\n\\begin{rem}\nIn what follows, we will use the term \\emph{toric variety} to mean toric $T$-variety for some fixed torus $T$, even though such a variety may have a toric structure for various tori. In fact, the choice of torus does not affect our analysis of toric varieties given below, and we refer interested readers to \\cite{Duncan} for such considerations.\n\nRecall that a $k$-\\emph{form} of a $k$-variety $X$ is a $k$-variety $X'$ such that $X_L \\simeq X_L '$ for some field extension $L\/k$.  Any $k$-form of a toric variety is a toric variety \\cite{Duncan}.\n\\end{rem}\n\n\n\\subsection{The split case}  Let us begin by recalling some facts concerning toric varieties with $T \\simeq \\mathbb{G}_{m} ^n$ (e.g., when $k = {\\mathbb C}$ or $k= k^s$), which are studied in terms of combinatorial data, e.g., lattices, cones, fans.  Good references for toric varieties over ${\\mathbb C}$ include \\cite{Fulton, CLS}, and many results hold generally in the split case.\n\nLet $N$ be a finitely generated free abelian group of rank $n$ and $M = \\hom (N, {\\mathbb Z})$.  A subsemigroup $\\sigma \\subset N_{{\\mathbb R}}$ is a \\emph{cone} if ($\\sigma ^{\\vee})^{\\vee} = \\sigma$, where $\\sigma ^{\\vee} = \\{ u \\in M \\mid u(v) \\geq 0 \\text{ for all } v \\in \\sigma\\}$.  A subsemigroup $\\tau$ is a \\emph{face} of $\\sigma$ if it is of the form $\\tau = \\{v \\in \\sigma \\mid u(v) = 0 \\text{ for all } u \\in S \\}$ for some $S \\subseteq \\sigma ^{\\vee}$.  A cone $\\sigma$ is \\emph{pointed} if 0 is a face of $\\sigma$, and in this case $\\sigma^{\\vee}$ generates $M_{{\\mathbb R}}$.  Given a pointed cone $\\sigma$, we associate the affine $k$-scheme $U_{\\sigma} = \\operatorname{Spec} k[\\sigma ^{\\vee}]$, and for any face $\\tau \\subset \\sigma$ the induced map $U_{\\tau} \\hookrightarrow U_{\\sigma}$ is an open embedding.\n\nA \\emph{fan} $\\Sigma \\subset N_{{\\mathbb R}}$ is a finite collection of pointed cones such that (1) any face of a cone in $\\Sigma$ is a cone in $\\Sigma$ and (2) the intersection of any two cones in $\\Sigma$ is a face of each.  To any fan $\\Sigma$ we associate a $k$-variety $X_{\\Sigma}$ which is obtained by gluing the affine schemes $U_{\\sigma}$ along common subschemes $U _{\\tau}$ corresponding to faces.\n\nOn the other hand, beginning with a split torus $T \\simeq \\mathbb{G}_{m} ^n$ and toric $T$-variety $X$ with fixed embedding $T \\hookrightarrow X$, we recover $M$ as the character lattice $\\hom (T, \\mathbb{G}_{m})$ of $T$ and $N$ as the cocharacter lattice $\\hom (\\mathbb{G}_{m}, T)$.  The association $\\Sigma \\mapsto X_{\\Sigma}$ defines a bijective correspondence between fans $ \\Sigma \\subset N_{{\\mathbb R}}$ and toric $T$-varieties $X$ (we remind the reader that here we assume $T$ is a split torus;  in general, fans $\\Sigma$ admitting an action by $\\text{Gal}(k^s\/k)$ are in bijection with neutral toric $T$-varieties).\n\nLet $\\Sigma(\\ell)$ denote the collection of cones in $\\Sigma$ of dimension $\\ell$.  Let $\\text{Div}_T(X)$ denote the free abelian group generated by the \\emph{rays} of $\\Sigma$, i.e., elements of $\\Sigma (1)$.  By the Orbit-Cone Correspondence \\cite[Thm. 3.2.6]{CLS}, $\\text{Div}_T(X)$ is isomorphic to the group of $T$-invariant Weil divisors of $X$.  For $X$ a (split) smooth projective toric variety, we have natural identifications $\\text{Pic}(X) = \\text{Pic} (X_{k^s}) = \\text{Cl}(X_{k^s}) = \\text{Cl}(X)$ which yield an exact sequence $$0 \\to M \\to \\text{Div}_T(X) \\to \\text{Pic}(X) \\to 0.$$  In particular, if $X$ is of dimension $n$ and $m$ is the number of rays in $\\Sigma$, the Picard rank of $X$ is $\\rho = m-n$.\n\n\\begin{defn}\nA variety $X$ is \\emph{Fano} (resp. \\emph{weak Fano}) if its anticanoncial class $-K_X$ is ample (resp. nef and big).  If $X$ is a normal variety, a Cartier $D$ divisor on $X$ is \\emph{nef} (``numerically effective\" or ``numerically eventually free\") if $D \\cdot C \\geq 0$ for every irreducible curve $C \\subset X$.  A divisor $D$ is \\emph{very ample} if $D$ is base point free and $\\varphi_D : X \\to \\mathbb{P}(\\Gamma(X, \\O_X(D))^{\\vee})$ is an embedding. A divisor $D$ is \\emph{ample} if $\\ell D$ is very ample for some $\\ell \\in {\\mathbb Z}^+$.  A line bundle $\\O_X(D)$ is nef or (very) ample if the corresponding divisor $D$ is nef or (very) ample.  A Cartier divisor is \\emph{numerically trivial} if $D\\cdot C =0$ for every irreducible complete curve $C \\subset X$.  Let $N^1(X)$ be the quotient group of Cartier divisors by the subgroup of numerically trivial divisors.  The \\emph{nef cone} $\\text{Nef}(X)$ is the cone in $N^1(X)$ generated by the nef divisors, and the \\emph{anti-nef cone} is the cone $-\\text{Nef}(X) \\subset N^1(X)$. A line bundle $\\O_X(D)$ is nef (ample) if $D$ is nef (ample).  \n\\end{defn}\n\n\\begin{prop}\\label{prop:nef}\nA Cartier divisor $D$ on a split proper toric variety $X$ is nef (resp. ample) if and only if $D\\cdot C \\geq 0$ (resp. $D\\cdot C > 0$) for all torus-invariant integral curves $C \\subset X$.  \n\\end{prop}\n\n\\begin{proof}\nWhen $k$ is algebraically closed, these are Theorems 3.1 and 3.2 of \\cite{Mustata}. One can see that the arguments remain valid in the split case more generally.\n\\end{proof}\n\n\n\\subsection{The not necessarily split case}\n\nHere we provide a ``black box''\nfor producing exceptional collections on arbitrary forms of toric\nvarieties by identifying certain special exceptional collections on\na \\emph{split} toric variety.\nThis reduces an arithmetic question to a completely geometric question.\n\nWe begin by reviewing how to obtain arbitrary forms of toric varieties\nfrom the split case\n(see, for example, \\cite{Vos82Projective,ELFST}).\nLet $T$ be the split torus of a split smooth projective toric variety\n$X$ with fan $\\Sigma$ in the space $N \\otimes {\\mathbb R}$ associated to the\nlattice $N$.\nLet $\\operatorname{Aut}(\\Sigma)$ denote the subgroup of elements $g \\in \\operatorname{GL}(N)$\nsuch that $g(\\sigma) \\in \\Sigma$ for every cone $\\sigma \\in \\Sigma$.\nThere is a natural inclusion of $T \\rtimes \\operatorname{Aut}(\\Sigma)$\ninto $\\operatorname{Aut}(X)$ as the subgroup leaving the open orbit $T$-invariant.\n\nLet $k^s$ be the separable closure of $k$.\nThe Galois cohomology set $H^1(k^s\/k,\\operatorname{Aut}(X)(k^s))$ is in bijective\ncorrespondence with the $k$-forms of $X$.\nThe natural map\n\\[ H^1(k^s\/k, T(k^s) \\rtimes \\operatorname{Aut}(\\Sigma)) \\to H^1(k^s\/k, \\operatorname{Aut}(X)(L)) \\]\nin Galois cohomology is surjective;\nthe failure of this map to be a bijection amounts to the fact that there\nmay be several non-isomorphic toric variety structures\non the same variety (see \\cite{Duncan} for more details).\n\nSuppose $X'={}^\\gamma X$ is a twisted form of a split toric variety\nfor a cocycle $\\gamma$ representing a class in\n$H^1(k^s\/k, T(k^s) \\rtimes \\operatorname{Aut}(\\Sigma))$.\nThere is a ``factorization'' $X'={}^\\alpha({}^\\beta X)$\nwhere $\\beta$ represents a class in $H^1(k^s\/k, \\operatorname{Aut}(\\Sigma)$\nand $\\alpha$ represents a class in $H^1(k^s\/k, ({}^\\beta T)(k^s) )$.\nInformally, $\\beta$ changes the torus that acts on $X$,\nwhile $\\alpha$ changes the torsor of the open orbit in $X$.\n\nSuppose $X$ is a toric $T$-variety.\nWe say that an object $E \\in \\mathsf{D^b}(X)$ is \\emph{$T$-equivariant}\nif $E$ is in the image of the forgetful functor from\n$\\mathsf{D^b}(\\operatorname{Coh}_T(X))$\n(see \\S{2}~of~\\cite{BFK2}).\nIn particular, this implies that $t^\\ast E \\simeq E$ for all $t \\in\nT(k)$.\n\n\\begin{prop} \\label{prop:blackbox}\nLet $X$ be a split toric $T$-variety over a field $k$ and let $\\Sigma$\nbe the associated fan.  Suppose that $X$ admits an\n$\\operatorname{Aut}(\\Sigma)$-stable full exceptional collection $\\mathsf{E}$\nsuch that each object is $T$-equivariant.\nThen any $k$-form $X'$ of $X$ admits a full exceptional collection\n$\\mathsf{E'}$.\nMoreover, $\\mathsf{E'}$ is strong (resp. consists of vector bundles,\nconsists of sheaves) as soon as $\\mathsf{E}$ is strong\n(resp. consists of vector bundles, consists of sheaves).\n\\end{prop}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:Galconverse}, there exists a $G$-stable\nexceptional collection $\\mathsf{F}$ on $X_L$.\nFrom the proof of that lemma, the objects $F$ of $\\mathsf{F}$ are direct\nsummands of $\\pi^\\ast E$ for each object $E \\in \\mathsf{E}$,\nwhere each isomorphism class of simple direct summand is represented\nby exactly one $F$.\nSince $\\mathsf{E}$ is $\\operatorname{Aut}(\\Sigma)$-stable and each object\nis $T$-equivariant, we may conclude that $\\mathsf{F}$ is\n$(T(L) \\rtimes \\operatorname{Aut}(\\Sigma)) \\rtimes G$-stable.\n\nLet $X'$ be a $k$-form of $X$; there exists a finite Galois\nextension $L\/k$ with Galois group $G$ such that $X'_L \\simeq X_L$.\nFrom Theorem~5.1~of~\\cite{Duncan}, the natural map\n\\[ H^1(L\/k, T(L) \\rtimes \\operatorname{Aut}(\\Sigma)) \\to H^1(L\/k, \\operatorname{Aut}(X)(L)) \\]\nin Galois cohomology is surjective.\nThus, we may assume that $X' ={}^cX$ is the \\emph{twist}\nby a cocycle $c: G \\to T(L) \\rtimes \\operatorname{Aut}(\\Sigma)$.\nRecall that the cocycle condition is that\n$c(gh)=c(g){}^gc(h)$ for all $g,h \\in G$\nwhere ${}^gc(h)$ denotes the Galois action of $g$ on $T(L) \\rtimes \\operatorname{Aut}(\\Sigma)$.\n\nIdentifying $X_L=X'_L$, twisting gives $\\sigma'(g) = c(g) \\sigma(g)$\nwhere $\\sigma$ is the action of $G$ induced from $X$ and\n$\\sigma'$ is induced from $X'$.\nThe punchline is that the action $\\sigma'$ factors through the image of\n$(T(L) \\rtimes \\operatorname{Aut}(\\Sigma)) \\rtimes G$ described above.\nThus the exceptional collection $\\mathsf{F}$ is $G$-stable for the $X'$\naction as well.\nThe proposition now follows by Theorem~\\ref{thm:descblocks}.\n\\end{proof}\n\n\\begin{cor} \\label{cor:toricLB}\nLet $X$ be a split toric $T$-variety over a field $k$ and let $\\Sigma$\nbe the associated fan.\nIf $X$ admits an $\\operatorname{Aut}(\\Sigma)$-stable full (strong) exceptional collection of\nline bundles, then every $k$-form of $X$ admits a full (strong) exceptional\ncollection of vector bundles.\n\\end{cor}\n\n\\begin{proof}\nRecall that every line bundle is isomorphic to a $T$-equivariant line\nbundle by standard results on toric varieties.\nThe corollary now follows by Proposition~\\ref{prop:blackbox}.\n\\end{proof}\n\n\\begin{lem}\\label{lem:flips}\n Let $X$ and $Y$ be smooth projective toric varieties over $k$. Let $G = \\operatorname{Gal}(k^s\/k)$. Assume we have a $K$-positive toric flip $X \\dashrightarrow Y$ such that over $k^{s}$ the flipping loci $F_i$ are disjoint and permuted by $G$. Let $H_i$ be the normalizer of $F_i$. If $X_L$ admits a full $G$-stable exceptional collection and $Y_i$ admits a full $H_i$-stable exceptional collection, then $Y$ admits a full exceptional collection.\n\\end{lem}\n\n\\begin{proof}\n Passing to $k^{s}$ we are free to use \\cite{BFK} giving semi-orthogonal decompositions for the flip over each $Y_i$. Since the $Y_i$ are disjoint, we can concatenate these collections to get a $G$-stable collection. \n\\end{proof}\n\n\\subsection{Products of toric varieties}\n\nRecall that, given groups $G,H$ along with a homomorphism\n$\\rho : H \\hookrightarrow S_n$,\nthe \\emph{wreath product} $G \\wr H$ is the group $G^n \\rtimes H$\nwhere $H$ acts on $G^n$ by permuting the copies of $G$.\nWe say a toric variety $X$ is \\emph{indecomposable} if it cannot be written as\na product $X_1 \\times X_2$ where $X_1$ and $X_2$ are\npositive-dimensional toric varieties.\n\n\\begin{lem}\\label{lem:autprod}\nSuppose $Z= X_1^{n_1} \\times \\cdots \\times X_r^{n_r}$ is a product of\nproper split toric varieties $X_1, \\ldots, X_r$, where $X_i \\not\\simeq X_j$\nfor $i \\ne j$ and each $X_i$ is indecomposable.\nThen\n\\[ \\operatorname{Aut}(\\Sigma) \\simeq (\\operatorname{Aut}({\\Sigma_1}) \\wr S_{n_1}) \\times \\cdots \\times\n(\\operatorname{Aut}({\\Sigma_r}) \\wr S_{n_r}) , \\]\nwhere $\\Sigma$ is the fan of $Z$ and $\\Sigma_1, \\ldots, \\Sigma_r$ are\nthe fans of $X_1, \\ldots, X_r$.\n\\end{lem}\n\n\\begin{proof}\nFirst, consider $Z=X_1 \\times X_2$ where\n$X_1,X_2$ are proper split toric varieties.\nLet $N$ (resp. $N_1, N_2$) be the cocharacter lattice and\n$\\Sigma$ (resp. $\\Sigma_1, \\Sigma_2$) be the fan of\n$Z$ (resp. $X_1, X_2$).\nHere $N = N_1 \\oplus N_2$ and $\\Sigma$ is the set of cones\nof the form $\\sigma_1 \\times \\sigma_2$ where $\\sigma_1 \\in \\Sigma_1$\nand $\\sigma_2 \\in \\Sigma_2$.\nThe faces of a cone $\\sigma_1 \\times \\sigma_2$ are precisely\nthe cones of the form $\\sigma_1' \\times \\sigma_2'$ where\n$\\sigma_1'$ is a face of $\\sigma_1$ and $\\sigma_2'$ is a face of\n$\\sigma_2$.\nThe fan $\\Sigma_1$ can be canonically identified with the subfan of\n$\\Sigma$ via the bijection $\\sigma \\mapsto \\sigma \\times \\{0\\}$.\n\nNow, suppose also that $Z= Y \\times W$ is a product of proper split\ntoric varieties where $Y$ is indecomposable.  Let $\\Sigma_Y$ be the fan\nof $Y$, which we can canonically identify with a subfan of $\\Sigma_Z$.\nEvery cone of $Y$ is of the form $\\sigma_1 \\times \\sigma_2$ where\n$\\sigma_1 \\in \\Sigma_1$ and $\\sigma_2 \\in \\Sigma_2$.\nSince fans are closed under taking faces, $\\sigma_1 \\times \\{0\\}$\nand $\\{0\\} \\times \\sigma_2$ are also cones in $\\Sigma_Y$.\nThus every cone in $\\Sigma_Y$ is a product of cones in the intersections\n$\\Sigma_Y \\cap \\Sigma_1$ and $\\Sigma_Y \\cap \\Sigma_2$.\n\nIn particular, since $X$ is proper, we have that the space\n$N_Y \\otimes {\\mathbb R}$ is the direct sum of\n$(N_Y \\otimes {\\mathbb R}) \\cap (N_1 \\otimes {\\mathbb R})$ and\n$(N_Y \\otimes {\\mathbb R}) \\cap (N_2 \\otimes {\\mathbb R})$,\nand $\\Sigma_Y$ is a product of the fans $\\Sigma_Y \\cap \\Sigma_1$\nand $\\Sigma_Y \\cap \\Sigma_2$.\nSince $Y$ is indecomposable, one of these fans is indecomposable\nand $\\Sigma_Y$ must be a subfan of either $\\Sigma_1$ or $\\Sigma_2$.\n\nReturning to the general case, we conclude that the decomposition\n$\\Sigma= \\Sigma_1^{n_1} \\times \\cdots \\times \\Sigma_r^{n_r}$\nis unique up to ordering.\nThe description of the automorphism group is immediate.\n\\end{proof}\n\n\\begin{lem}\\label{lem:stabprod}\nLet $Z$ be a proper toric $k$-variety with splitting field $L\/k$.\nSuppose $Z_L = \\prod_{i=1}^n X_i$ where each $X_i$ is an indecomposable\nsplit proper toric $L$-variety admitting a full (strong)\n$\\operatorname{Aut}(\\Sigma_i)$-stable exceptional collection of line bundles,\nwhere $\\Sigma_i$ is the fan of $X_i$.\nThen $Z$ has a full (strong) exceptional collection of vector bundles.\n\\end{lem}\n\n\\begin{proof}\nIt is a well known that the exterior product collection is an\nexceptional collection.\nFor each isomorphism class among the $X_i$ fix a full (strong)\n$\\operatorname{Aut}(\\Sigma_{X_i})$-stable exceptional collection of line bundles.\nThis ensures that the exterior product collection is\nstable under the action of\n$(\\operatorname{Aut}(\\Sigma_{X_1}) \\wr S_{a_1}) \\times \\cdots \\times (\\operatorname{Aut}(\\Sigma_{X_r})\n\\wr S_{a_r})$.\nSince this group is $\\operatorname{Aut}(\\Sigma)$ by Lemma~\\ref{lem:autprod},\nthe exterior product collection descends by Corollary~\\ref{cor:toricLB}.\n\\end{proof}\n\n\n\\section{Low dimension or high symmetry}\\label{section:minimal}\n\nWe provide exceptional collections for smooth toric surfaces, Fano 3-folds,\nsome Fano 4-folds, centrally-symmetric toric varieties, and\ntoric varieties corresponding to root systems of type $A$. \n\n\n\\subsection{Surfaces}\\label{sect:surfaces}\nHere we prove that every toric surface has a full exceptional\ncollection.\nWe begin by recalling the (classical) minimal model program for surfaces\nover non-closed fields.\n\nSuppose $f : X \\to X'$ is a birational morphism of smooth projective\nsurfaces over a field $k$.\nIf $k$ is separably closed, then by Proposition~5~of\\cite{Coombes}\nthe morphism factors into a sequence\n\\[\nX= X_0 \\to X_1 \\to \\cdots \\to X_r = X'\n\\]\nwhere each morphism $X_i \\to X_{i+1}$ is the blowup of a point on\n$X_{i+1}$.\nOver a non-closed field $k$, we can factor $f : X \\to X'$ into a\nsequence where each morphism $X_i \\to X_{i+1}$ is defined over $k$\nand is a blowup of a\n(necessarily finite) Galois orbit of $k^s$-points on $X_{i+1}$.\n\nBlowing up a point produces an exceptional curve: a smooth rational\ncurve with self-intersection $-1$.  By Castelnuovo's contractibility\ncriterion, such a curve can always be obtained as the result of a blow-up.\nIf one finds a skew Galois orbit of such curves on $X$, then there\nexists a birational morphism $f : X \\to X'$ contracting these curves.\nRepetition of this procedure eventually terminates.\n\n\\begin{defn}\nA \\emph{minimal surface} $X$ is a smooth projective surface over a field $k$\nsuch that every birational morphism $X \\to X'$ to a smooth projective\nsurface $X'$ is an isomorphism.\n\\end{defn}\n\nAny smooth projective surface can be obtained by iteratively blowing up\nGalois orbits of separable points starting from a minimal model.\nA toric variety is geometrically rational.\nMinimal geometrically rational surfaces were classified by\nManin~\\cite{Manin} and Iskovskikh~\\cite{Iskovskikh}.\nOne checks that the toric surfaces in their collection are the\nfollowing (see also a direct proof in~\\cite{Xie}):\n\n\\begin{lem}\\label{lem:classification}\nA minimal smooth projective toric surface\nis a $k^s\/k$-form of one of the following:\n\\begin{enumerate}\n\\item $\\mathbb{P}^2$, $\\operatorname{Aut}(\\Sigma) = S_3$.\n\\item ${\\mathbb P}^1 \\times {\\mathbb P}^1$, $\\operatorname{Aut}(\\Sigma) = D_8$.\n\\item $\\mathbb{F}_{a} = \\operatorname{Proj}(\\O_{{\\mathbb P}^1} \\oplus \\O_{{\\mathbb P}^1} (a))$,\n$a \\geq 2$, $\\operatorname{Aut}(\\Sigma) = C_2$.\n\\item $\\mathsf{dP}_6 = $ del Pezzo surface of degree 6, $\\operatorname{Aut}(\\Sigma) = D_{12}$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nA minimal geometrically rational surface is either a del Pezzo surface\nor has a conic bundle structure \\cite{Manin,Iskovskikh}.\nOver the separable closure, a del Pezzo surface is either\n${\\mathbb P}^1 \\times {\\mathbb P}^1$ or a blow up of ${\\mathbb P}^2$ at up to $8$ points in\ngeneral position.\nBlowing up only one or two points never results in a minimal surface, and\nno more than three points can be simultaneously torus invariant and\nin general position.  Thus every del Pezzo surface is a $k^s\/k$-form\nof ${\\mathbb P}^2$, ${\\mathbb P}^1 \\times {\\mathbb P}^1$ or $\\mathsf{dP}_6$.\nOver the separable closure, a conic bundle structure has at most $2$\nsingular fibers since their images must be torus invariant points on the\nbase ${\\mathbb P}^1$.\nA minimal conic bundle with at most two singular fibers over the\nseparable closure must be either a del Pezzo surface or a minimal ruled\nsurface.\n\\end{proof}\n\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}\n   [scale=.4, vertex\/.style={circle,draw=black!100,fill=black!100, inner sep=0.5pt,minimum size=0.5mm}]\n   \\filldraw[fill=black!20!white,draw=white!100]\n     (-5,5) -- (5,5) -- (5,-5) -- (-5,-5) -- (-5,5);\n   \\draw (0,0) -- (5,0);\n   \\draw (0,0) -- (0,5);\n   \\draw (-5,-5) -- (0,0);\n  \\foreach \\x in {-5,-4,...,5}\n   \\foreach \\y in {-5,-4,...,5}\n   {\n     \\node[vertex] at (\\x,\\y) {};\n   }\n\\node at (0,-6,0) {\\text{$\\mathbb{P}^2$}};\n \\end{tikzpicture}\n\\hspace{1cm}\n \\begin{tikzpicture}\n   [scale=.4, vertex\/.style={circle,draw=black!100,fill=black!100, inner sep=0.5pt,minimum size=0.5mm}]\n   \\filldraw[fill=black!20!white,draw=white!100]\n     (-5,5) -- (5,5) -- (5,-5) -- (-5,-5) -- (-5,5);\n   \\draw (0,0) -- (5,0);\n   \\draw (0,0) -- (0,5);\n   \\draw (0,0) -- (0,-5);\n   \\draw (-5,0) -- (0,0);\n  \\foreach \\x in {-5,-4,...,5}\n   \\foreach \\y in {-5,-4,...,5}\n   {\n     \\node[vertex] at (\\x,\\y) {};\n   }\n\\node at (0,-6,0) {\\text{${\\mathbb P}^1 \\times {\\mathbb P}^1$}};\n \\end{tikzpicture}\n\n\\vspace{0.5cm}\n\n\n \\begin{tikzpicture}\n   [scale=.4, vertex\/.style={circle,draw=black!100,fill=black!100, inner sep=0.5pt,minimum size=0.5mm}]\n   \\filldraw[fill=black!20!white,draw=white!100]\n     (-5,5) -- (5,5) -- (5,-5) -- (-5,-5) -- (-5,5);\n   \\draw (0,0) -- (5,0);\n   \\draw (0,0) -- (0,5);\n   \\draw (0,0) -- (0,-5);\n   \\draw (-2.5,5) -- (0,0);\n  \\foreach \\x in {-5,-4,...,5}\n   \\foreach \\y in {-5,-4,...,5}\n   {\n     \\node[vertex] at (\\x,\\y) {};\n   }\n\\node at (0,-6,0) {\\text{$\\mathbb{F}_a$}};\n \\end{tikzpicture}\n\\hspace{1cm}\n \\begin{tikzpicture}\n   [scale=.4, vertex\/.style={circle,draw=black!100,fill=black!100, inner sep=0.5pt,minimum size=0.5mm}]\n   \\filldraw[fill=black!20!white,draw=white!100]\n     (-5,5) -- (5,5) -- (5,-5) -- (-5,-5) -- (-5,5);\n   \\draw (0,0) -- (5,0);\n   \\draw (0,0) -- (0,5);\n   \\draw (0,0) -- (0,-5);\n   \\draw (0,0) -- (-5,0);\n   \\draw (5,5) -- (0,0);\n   \\draw (-5,-5) -- (0,0);\n  \\foreach \\x in {-5,-4,...,5}\n   \\foreach \\y in {-5,-4,...,5}\n   {\n     \\node[vertex] at (\\x,\\y) {};\n   }\n\\node at (0,-6,0) {\\text{$\\mathsf{dP}_6$ }};\n \\end{tikzpicture}\n\\vspace{-.3cm}\n\\caption{Fans for minimal toric surfaces}\\label{fig:fans}\n\\end{center}\n\\end{figure}\n\nHere we exhibit full strong exceptional collections consisting of\n$G$-stable blocks for each minimal toric surface exhibited above\n(none of these collections are original).\nThe fans associated to the split forms of these surfaces are given in Figure~\\ref{fig:fans}. In each case, we fix a torus $T$ which gives $X$ the structure of a toric $T$-surface.  As remarked above, this gives a homomorphism $G \\to \\text{Aut}(\\Sigma)$ as well as an action of $G$ on $\\text{Pic}(X_L)$, where $L$ is a splitting field of $T$, $G = \\text{Gal}(L\/k)$, and $\\Sigma$ is the fan corresponding to the split toric surface $X_L$. We produce $G$-stable exceptional collections in each case by exhibiting $\\text{Aut}(\\Sigma)$-stable collections.\n\n\\begin{ex}\\label{ex:p2}\nLet $X$ be a toric $T$-surface whose split form is $\\mathbb{P}^2$ with\n$\\text{Aut}(\\Sigma) = S_3$. The $ S_3$-action on\n$\\text{Pic}(\\mathbb{P}^2) = {\\mathbb Z}$ is clearly trivial, so that the\nexceptional collection $\\{ \\O, \\O(1), \\O(2)\\}$, given in\n\\cite{Beilinson} yields a full strong $\\text{Aut}(\\Sigma)$-stable\nexceptional collection.  By Corollary~\\ref{cor:toricLB},\n$X$ admits a full strong exceptional collection.\n\\end{ex}\n\n\\begin{ex}\\label{ex:p1p1}\nLet $X$ be a toric surface whose split form is ${\\mathbb P}^1 \\times {\\mathbb P}^1$ with $\\text{Aut}(\\Sigma) = D_8$, and consider the natural projections $p_1, p_2: {\\mathbb P}^1 \\times {\\mathbb P}^1 \\to {\\mathbb P}^1$.  Let $\\O(p, q) = p_1 ^*\\O (p) \\otimes p_2^* \\O (q)$.  By \\cite{KvichanskyNogin}, the collection $\\{\\O, \\O(1, 0), \\O(0, 1), \\O(1,1) \\}$ on ${\\mathbb P}^1 \\times {\\mathbb P}^1$ is exceptional since $\\{ \\O, \\O(1)\\} $ is an exceptional collection for ${\\mathbb P}^1$. The $ D_8$-action preserves this collection, with orbits given by the blocks $\\mathsf{E}^0 = \\{ \\O \\}$, $\\mathsf{E}^1 = \\{ \\O(1, 0), \\O(0,1) \\} $, and $\\mathsf{E}^2= \\{ \\O(1,1)\\}$.  In particular, this collection above is $\\text{Aut}(\\Sigma)$-stable, and Corollary~\\ref{cor:toricLB} yields an exceptional collection on $X$.\n\\end{ex}\n\n\\begin{ex}\nLet $X$ be a toric surface whose split form is the Hirzebruch surface\n$\\mathbb{F}_a$; here $\\text{Aut}(\\Sigma) = C_2$.\nLet $e_1, e_2$ be the standard basis for ${\\mathbb Z}^2$.\nAs in \\cite[Ex. 4.1.8]{CLS},\nlet $u_1  =-e_1 + ae_2$, $u_2 = e_2$, $u_3 = e_1$, and $u_4\n= -e_2$ be the generators of $\\Sigma(1)$ with corresponding toric\ndivisors $D_i$.  The Picard group of $\\mathbb{F}_a$ is freely generated\nby $\\{D_1, D_2\\}$ and $D_1$ is linearly equivalent to $D_3$.\nThe only nontrivial fan automorphism $\\sigma$ takes $e_1 \\mapsto -e_1+ae_2$\nand $e_2 \\mapsto e_2$.\nThus $\\sigma$ leaves $D_2,D_4$ fixed and interchanges $D_1$ and $D_3$.\nWe conclude the action of $C_2$ on\n$\\text{Pic}(\\mathbb{F}_a)$ is trivial, and thus, any exceptional\ncollection is necessarily $G$-stable (see Lemma~\\ref{lem:picinv}).\nAn exceptional collection for $\\mathbb{F}_a$ is given by $\\{\\O, \\O(D_3),\n\\O(D_4), \\O(D_3 + D_4)\\}$ \\cite{KvichanskyNogin}.\nCorollary~\\ref{cor:toricLB} then gives an exceptional collection on $X$.\n\\end{ex}\n\n\\begin{ex}\\label{ex:dp6}\nLet $X$ be a toric surface whose split form is $\\mathsf{dP}_6$; here\n$\\text{Aut}(\\Sigma) = D_{12}$. Viewing $\\mathsf{dP}_6$ as the blowup of\n$\\mathbb{P} ^2$ at 3 non-colinear points, let $H$ be the pullback of the\nhyperplane divisor on $\\mathbb{P}^2$ and $E_i$ the exceptional divisors, $i =\n1, 2, 3$.  As shown in \\cite[Prop. 6.2(ii)]{King}, the collection\n$$\\{\\O, \\O(H - E_1), \\O(H - E_2), \\O(H - E_3), \\O(H), \\O(2H - (E_1 + E_2\n+ E_3)) \\}$$ gives an exceptional collection for $\\mathsf{dP}_6$,\nwhich is $\\text{Aut}(\\Sigma)$-stable.\n\nLet us rephrase this in the notation of \\cite{BSS}.\nThere are two morphisms $\\mathsf{dP}_6 \\to \\mathbb{P}^2$ realizing\n$\\mathsf{dP}_2$ as a blowup of $\\mathbb{P}^2$, and we denote the collection of\nall six exceptional divisors by $L_i$ and $M_i$, with $i = 1, 2, 3$.\nLet $H$ and $H'$ denote the pullbacks of the hyperplane divisors on\n$\\mathbb{P}^2$ under the maps contracting $M_i$ and $L_i$, respectively, where\nwe identify $H$ with the divisor given in King's collection above (and\nthus we also identify $E_i$ with $M_i$).\nThen $H = L_1 + M_2 + M_3$, and it follows that $$2H - (E_1 + E_2 + E_3)\n= L_1 + L_2 + M_3 = H'$$ using the relation $L_i + M_j = L_j + M_i$.\nFurthermore, one checks that $ H - E_1 =  L_2 + M_3$, $H-E_2 = L_1 +\nM_3$, and $H- E_3  = L_1 + M_2$.\nAs described in \\cite[$\\S$2]{BSS}, the element $\\sigma$ in  $S_3 \\times\nC_2 = D_{12}$ which cyclically permutes the six lines $L_i, M_i$ also\nsatisfies $\\sigma (H) = H'$ and $\\sigma^2(H) = H$.\nWe arrange the exceptional collection above into blocks $\\mathsf{E}^0 =\n\\{\\O \\}$, $\\mathsf{E}^1 = \\{ \\O(H - E_1), \\O(H - E_2), \\O(H - E_3)\\}$\nand $\\mathsf{E}^2 = \\{\\O(H), \\O(2H - (E_1 + E_2 + E_3))\\}$.\nIn particular, the exceptional collection given above is\n$\\text{Aut}(\\Sigma)$-stable, and by Corollary~\\ref{cor:toricLB} we have an exceptional\ncollection on $X$.\n\\end{ex}\n\n\\begin{prop}\\label{prop:surface}\nEvery toric surface admits a full exceptional collection of sheaves.\n\\end{prop}\n\n\\begin{proof}\nThere is a sequence of blowups $X = X_0 \\to \\cdots \\to X_s = X'$ where\n$X'$ is minimal, so must be one of the varieties given in\nLemma~\\ref{lem:classification}.\nBy Examples~\\ref{ex:p2}-\\ref{ex:dp6},\n$X'$ admits a full strong exceptional collection of vector bundles, and\nthus $X'_L$ admits a $G$-stable exceptional collection.  By\nLemma~\\ref{lem:blowup}, $X_L$ admits a $G$-stable exceptional\ncollection.\n\\end{proof}\n\n\\begin{rem}\nThe authors would like to thank F.~Xie for pointing out a mistake\nin the statement of a previous version of\nProposition~\\ref{prop:surface}.\nXie also discusses exceptional collections of toric surfaces in\n\\cite{Xie}, although her definition of exceptional object is not the\nsame as ours. \nIn the second arXiv version of that paper, Xie sketched in Remark~8.8\nhow one might construct an exceptional collection for toric surfaces.\nAfter the authors posted a preliminary version of this paper to the\narXiv, Xie updated her preprint with Corollary~8.8, which proves the analog of\nthe above proposition for collections of vector bundles but using her notion of exceptional collection.\n\\end{rem}\n\n\n\\subsection{The toric Frobenius and toric Fano 3-folds}\\label{sect:3fold}\nIn Table~\\ref{tab:3-folds} we present the classification of smooth toric Fano 3-folds given in \\cite{Batyrev, Watanabe}, adopting Batyrev's enumeration. For each $X = X_{\\Sigma}$, we record the following invariants:\n\\begin{itemize}\n\n\\item $\\sigma(1) = | \\Sigma (1) |$ is the number of rays of $\\Sigma$ \\cite{BT}.\n\n\\item $k_0$ is the rank of the Grothendieck group $K_0(X)$, which coincides with the number of maximal cones in the fan $\\Sigma$ \\cite{BT}.\n\n\\item $\\operatorname{Aut}(\\Sigma) $ is the automorphism group of the (lattice $N$ which preserves the) fan $\\Sigma$ corresponding to $X$.\n\n\\item $\\rho$ is the Picard rank of $X$ \\cite{Watanabe}.\n\n\\item $\\rho ^G$ is the $\\operatorname{Aut}(\\Sigma)$-invariant Picard rank of $X$,\ni.e., the rank of $\\text{Pic}(X)^{\\operatorname{Aut}(\\Sigma)}$.\n\n\\item $\\mathfrak{fr} = | \\mathsf{Frob}(X) |$ is the number of\nisomorphism classes of line bundles produced by the push forward of the structure sheaf under the Frobenius morphism \\cite{BT, Uehara}.\n\n\\item $ \\mathfrak{fr}^- = |\\mathsf{Frob}(X) \\cap -\\text{Nef}(X)|$ is the\nnumber of isomorphism classes of line bundles in $\\mathsf{Frob}(X)$ which lie in the anti-nef cone of $X$ \\cite{Uehara}.\n\\end{itemize}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lrlccccccc}\n\\toprule\n  & &Toric Fano 3-fold $X$ & $\\sigma(1)$ &$k_0$ & $\\operatorname{Aut}(\\Sigma)$ & $\\rho$ & $\\rho^G$ & $\\mathfrak{fr}$ & $\\mathfrak{fr}^- $\\\\\n\\midrule\n\n& 1. & $\\mathbb{P}^3$ & 4 &  4 & $S_4$ & 1 & 1 & 4 & 4  \\\\\n\n&2.  &$\\mathbb{P}_{\\mathbb{P}^2}(\\O \\oplus \\O(2))$ & 5 & 6 & $S_3$ & 2 & 2 & 7 & 6 \\\\\n\n &3.  &$\\mathbb{P}_{\\mathbb{P}^2}(\\O \\oplus \\O(1))$ & 5 & 6 & $S_3$ & 2 & 2 & 6 & 6\\\\\n \n&4.& $\\mathbb{P}_{\\mathbb{P}^1}(\\O \\oplus \\O \\oplus \\O(1))$ & 5 & 6 & $C_2 \\times C_2$ & 2 & 2 & 6 & 6\\\\\n\n &5. & $\\mathbb{P}^2\\times \\mathbb{P}^1$ & 5 & 6 & $D_{12}$ & 2 & 2 & 6 & 6\\\\\n \n &6.  &$\\mathbb{P}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(\\O\\oplus \\O(1,1))$ & 6 & 8 & $D_8$ & 3 & 2 & 8 & 8\\\\\n \n &7.  &$\\mathbb{P}_{\\mathsf{dP}_8}(\\O\\oplus \\O(l))$, $l^2=1$ on $\\mathsf{dP}_8$ & 6 & 8 & $D_8$ & 3 & 3 & 8 & 8 \\\\\n\n&8.  &$\\mathbb{P}^1\\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ & 6 & 8 & $C_2 \\times S_4$ & 3 & 1 & 8 & 8\\\\\n\n &9.  &$\\mathsf{dP}_8\\times \\mathbb{P}^1 $ & 6 & 8 &$ C_2 \\times C_2$ & 3 & 3 & 8& 8 \\\\\n \n&10. &$\\mathbb{P}_{\\mathbb{P}^1\\times\\mathbb{P}^1}(\\O\\otimes \\O(1,-1))$& 6 & 8 & $D_8$ & 3 & 2 & 8 & 8\\\\\n\n &11.  & $\\text{Bl}_{\\mathbb{P}^1}(\\mathbb{P}_{\\mathbb{P}^2}(\\O \\oplus \\O(1)))$& 6 & 8 & $C_2$ & 3 & 3 & 9 & 8 \\\\\n \n &12. & $\\text{Bl}_{\\mathbb{P}^1}(\\mathbb{P}^2\\times \\mathbb{P}^1)$& 6 & 8 & $C_2$ & 3 & 3 & 8 & 8 \\\\\n\n& 13.  & $\\mathsf{dP}_7-$bundle over $\\mathbb{P}^1$ & 7 & 10 & $C_2$ & 4 & 4 & 10 & 10 \\\\\n\n & 14. & $\\mathsf{dP}_7-$bundle over $\\mathbb{P}^1$ & 7 & 10 & $C_2 \\times C_2$ & 4 & 3 & 10 & 10\\\\\n\n& 15. & $\\mathsf{dP}_7\\times \\mathbb{P}^1$& 7 & 10 & $C_2 \\times C_2$ & 4 & 3 & 10 & 10\\\\\n\n& 16. & $\\mathsf{dP}_7-$bundle over $\\mathbb{P}^1$& 7 & 10 & $C_2$ & 4 & 4 & 10 & 10\\\\\n\n & 17.  &$\\mathsf{dP}_6\\times \\mathbb{P}^1$& 8 & 12 & $C_2 \\times C_2 \\times S_3$ & 5 &  2 & 12 & 12\\\\\n\n & 18. & $\\mathsf{dP}_6-$bundle over $\\mathbb{P}^1$& 8 & 12 & $C_2 \\times C_2$  & 5 & 4 & 12 & 12\\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{.2cm}\n\\caption{Toric Fano 3-folds}\\label{tab:3-folds}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Toric Frobenius}\\label{section:frob}\n\nLet $X$ be a split toric variety of dimension $n$ with fixed torus embedding $T \\hookrightarrow X$ and take $\\ell \\in {\\mathbb Z}^+$.  Define the $\\ell^{\\text{th}}$ Frobenius map on $T = \\mathbb{G}_{m}^n$ to be $(x_1,..., x_n) \\mapsto (x_1^{\\ell},..., x_n^{\\ell})$.  The unique extension to $X$ will be denoted $F_{\\ell}$ and called the \\emph{$\\ell^{th}$ Frobenius morphism}.  Alternatively, if $\\Sigma \\subset N$ is the fan associated to $X$, define a lattice $N' = \\frac{1}{\\ell} N$.  The inclusion $N \\subset N'$, which sends a cone in $N_{{\\mathbb R}}$ to the cone with the same support in $N'_{{\\mathbb R}}$, induces a finite surjective morphism which is precisely the $\\ell^{\\text{th}}$ Frobenius morphism $F_{\\ell}: X \\to X.$ \n\nThe sheaf $(F_{\\ell})_*(\\O_X)$ splits into line bundles and Thomsen\nprovides an algorithm for computing its direct summands  \\cite{Thomsen}.\nWe let $\\mathsf{Frob}(X)$ denote the union of all isomorphism classes of line bundles arising as direct summands of $(F_{\\ell})_*(\\O_X)$ as $\\ell$ varies over ${\\mathbb Z}^+$.  Note that $\\mathsf{Frob}(X)$ is a finite set. \n\n\\begin{conj}[Bondal \\cite{Bondal2}]\\label{conj:Bondal}\nIf $X$ is a smooth proper toric variety then the collection $\\mathsf{Frob}(X)$ generates $\\mathsf{D^b}(X)$.\n\\end{conj}\n\nFor a toric variety $X$ in which Bondal's Conjecture is true, we will\nsay that \\emph{the Frobenius generates the derived category of} $X$.\nIn loc. cit., Bondal proves that if all summands of $\\mathsf{Frob}(X)$ are nef, one actually gets a full strong exceptional collection, so that Conjecture~\\ref{conj:Bondal} is true in this case. He also notes his arguments work for all but two (isomorphism classes of) toric Fano threefolds. To cover all toric Fano threefolds, \nUehara noticed that discarding line bundles which do\nnot lie in the set $-\\text{Nef}(X)$ yields a full strong\nexceptional collection \\cite{Uehara}.\n\n\\begin{lem}\\label{lem:FrobNef}\nLet $X$ be a toric variety over $k$ with splitting field $L$.\nSuppose $\\mathsf{E}$ is a full (strong) exceptional collection for\n$\\mathsf{D^b}(X_L)$\nwhere either $\\mathsf{E} = \\operatorname{\\mathsf{Frob}}(X_L)$ or\n$\\mathsf{E} = \\operatorname{\\mathsf{Frob}}(X_L) \\cap - \\operatorname{Nef}(X_L)$.\nThen there exists a full (strong) exceptional collection for\n$\\mathsf{D^b}(X)$.\n\\end{lem}\n\n\\begin{proof}\nBoth $\\operatorname{\\mathsf{Frob}}(X_L)$ and $\\text{Nef}(X_L)$ are\ncanonical constructions and thus are $\\text{Aut}(X_L)$-stable.\nIn particular, $\\mathsf{E}$ is $\\text{Aut}(\\Sigma)$-stable\nand so Corollary~\\ref{cor:toricLB} applies.\n\\end{proof}\n\n\\begin{prop}\\label{prop:3fold}\nLet $X$ be a smooth projective toric Fano 3-fold over a field $k$.  Then $X$ admits a full strong exceptional collection consisting of vector bundles.\n\\end{prop}\n\n\\begin{proof}\nLet $X_L$ be the associated split toric Fano 3-fold.  The main result of \\cite{Uehara} guarantees that the set $\\mathsf{E} = \\mathsf{Frob}(X_L) \\cap - \\text{Nef}(X_L)$ defines a full strong exceptional collection on $X$. Lemma~\\ref{lem:FrobNef} completes the proof.\n\\end{proof}\n\n\n\\subsection{Toric Fano 4-folds}\\label{sect:4fold}\n\nThere are 124 split smooth toric Fano 4-folds,\nwhich were first classified in \\cite{Batyrev}\n(a missing case was added in~\\cite{Sato}).\nIn \\cite{Prabhu}, Prabhu-Naik exhibits full strong exceptional\ncollections for all 124 of these 4-folds.\nHowever, it is not clear that these collections are\n$\\operatorname{Aut}(\\Sigma)$-stable, so they do not necessarily lead to full strong\nexceptional collections in the arithmetic case.\n\nAll collections obtained\nusing Method 1 of loc. cit. produce $\\operatorname{Aut}(\\Sigma)$-stable collections (note that\nthis is precisely the method used in \\cite{Uehara} for toric Fano\n3-folds, and we will refer to this as the \\emph{Bondal-Uehara Method}).\nTogether with Lemmas~\\ref{lem:stabprod} and \\ref{lem:FrobNef}, this gives stable exceptional collections for 43 of the 124 smooth toric Fano 4-folds.  However, there are examples when the\nBondal-Uehara Method fails to produce an exceptional collection.\nIn this case, all is not lost (see Section~\\ref{sec:centsym}).\n\nMore precisely, the varieties (61), (62), (63), (64), (77), (105),\n(107), (108), (110), (122), and (123) of \\cite{Prabhu} are shown to have\nexceptional collections using the Bondal-Uehara Method.\nHence, they admit exceptional collections which are $\\operatorname{Aut}(\\Sigma)$-stable\nand thus provide exceptional collections for the arithmetic forms. Secondly, for the varieties (109), (114), and (115), the set\n$\\mathsf{Frob}(X)$ is a full exceptional collection, which is $G$-stable\nby Lemma~\\ref{lem:FrobNef}. Lastly, Lemma~\\ref{lem:stabprod} guarantees the existence of exceptional collections on products.  Hence, the following varieties also admit stable exceptional collections: (0), (4), (9), (17), (24), (25), (26), (27), (45), (52), (53), (54), (55), (56), (58), (67), (73), (88), (90), (92),\n(93), (97), (103), (111), (112), (113), (118), (119), (120).\n\n\n\n\\subsection{Centrally symmetric toric Fano varieties} \\label{sec:centsym}\n\nPolytopes with the highest degree of symmetry are the \\emph{centrally\nsymmetric} polytopes, i.e., $-P = P$.\nThe smooth split toric varieties $X$ whose anti-canonical polytope is full-dimensional and centrally symmetric were classified in \\cite{VosKly}. It was shown that any such variety (which we refer to as a \\emph{centrally symmetric toric Fano varieties}) is isomorphic to a product of projective lines and \\emph{generalized del Pezzo varieties} $V_n$ of dimension $n = 2m$.  Note that $V_2 = \\mathsf{dP}_6$ and $V_4$ is the missing (116)\nfrom the list in Section~\\ref{sect:4fold} (this is (118) in the enumeration found in \\cite{Batyrev}). The goal of this section is to exhibit full stable exceptional collections on $V_n$, which in turn yields stable exceptional collections for any centrally symmetric toric Fano variety, in light of Lemma~\\ref{lem:stabprod}.\n\nIn \\cite[Theorem 6.6]{CT}, Castravet and Tevelev found $\\operatorname{Aut}(\\Sigma)$-stable full\nstrong exceptional collections for the varieties $V_n$.\nThe authors of this paper had independently discovered the same\nexceptional collection (up to a twist by a line bundle).\nNevertheless, the perspective here may be of independent interest, so we\nsketch the argument. The authors give a more detailed analysis in \\cite{BDMdP}.\n\nThe variety $V_{n}$ with $n= 2m$ has rays given by\n$$\\begin{array}{rl}\ne_1 &= (1,0,\\cdots,0)\\\\\ne_2 &= (0,1,\\cdots,0)\\\\\n&\\vdots\\\\\ne_n &= (0,0,\\cdots,1)\\\\\ne_{n+1} &= (-1,-1,\\cdots,-1)\\\\\n\\end{array}\n\\hspace{.4cm}\n\\begin{array}{rl}\n\\bar{e}_1 &= (-1,0,\\cdots,0)\\\\\n\\bar{e}_2 &= (0,-1,\\cdots,0)\\\\\n&\\vdots\\\\\n\\bar{e}_n &= (0,0,\\cdots,-1)\\\\\n\\bar{e}_{n+1} &= (1,1,\\cdots,1)\n\\end{array}$$ and whose maximal cones are given as follows.  Among the rays $e_1,..., e_{n+1}$, omit a single $e_i$.  From the remaining $n = 2m$ rays, choose $\\frac n 2$ of them and take their antipodes \\cite[Proof of Thm. 5]{VosKly}. Note that $V_2 = \\mathsf{dP}_6$ (whose fan is given in Figure~\\ref{fig:fans}).  The number of maximal cones $c(n)$ of $V_n$ is given by\n\\[\nc(n)=\\frac{(n+1)!}{(\\frac n 2)!^2} = \\frac{(2m +1)! }{m!^2}\\ .\n\\]\nThere's a natural action of $S_{n+1} \\times C_2$, where $S_{n+1}$\npermutes $e_1,\\ldots,e_{n+1}$ and $\\bar{e}_1\\ldots\\bar{e}_{n+1}$ in the\nobvious way.  The $C_2$-action is simply the antipodal map on the\ncocharacter lattice --- we will refer to it as ``the involution.''\nClearly, the involution interchanges $e_i$ and $\\bar{e_i}$. \n\nThe variety $V_n$ is of importance in birational geometry due its\nappearance in the factorization of the standard Cremona transformation\nof $\\mathbb{P}^n$. In fact, as is well-known, $V_n$ can be explicitly obtained\nfrom $\\mathbb{P}^n$ as follows. First blow up the torus fixed points, then\nflip the (strict transforms) of the lines through these points, then\nflip the (strict transforms) of planes through these points, \\ldots, up\nuntil, and not including, the half-dimensional linear subspaces. The\nresulting variety is $V_n$. For more, see \\cite{Casagrande}. \n\nSince $V_n$ and the blow up of $\\mathbb{P}^n$ at its torus fixed points are\nisomorphic in codimension $1$, they have isomorphic Picard groups. We\nuse a basis $H,E_1, \\ldots, E_{n+1}$ for $\\operatorname{Pic}(V_n)$,\nwhich correspond to the hyperplane section and the\nexceptional divisors of the blown up $\\mathbb{P}^n$.\nWe have\n\\[\n[e_i] = E_i, \\quad\n[\\bar{e}_i] = (H-\\sum_{j=1}^{n+1}E_j)+E_j\n\\]\nwhere $S_{n+1}$ permutes the $E_i$ leaving $H$ fixed, and the involution\nis represented by the following matrix\n\\[\n\\begin{pmatrix}\nn & 1 & 1 & \\cdots & 1 \\\\\n1-n & 0 & -1 & \\cdots & -1 \\\\\n1-n & -1 & 0 & \\cdots & -1 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\\n1-n & -1 & -1 & \\cdots & 0 \\\\\n\\end{pmatrix} \\ .\n\\]\n\nFor each $c \\in {\\mathbb Z}$ and $J \\subset \\{1, \\ldots, n+1\\}$, define \n\\[\n F_{c,J} := c\\left(\\sum_{i=1}^{n+1}E_i - H\\right) - \\sum_{j \\in J} E_j .\n\\]\nNote that the involution takes $F_{c,J}$ to $F_{|J|-c,J}$. Then, \n\n\\begin{prop} \\label{prop:Vn}\n The set of $F_{c,J}$ with\n \\begin{enumerate}\n  \\item $\\displaystyle{|J|-\\frac{n}{4} \\le c \\le \\frac{n}{4}}$, or\n  \\item $\\displaystyle{\\frac{n+2}{4} \\le c \\le |J| - \\frac{n+2}{4}}$.\n \\end{enumerate}\nform a full strong $(S_{n+1} \\times C_2)$-stable exceptional collection\non $V_n$ under any ordering of the\nblocks such that $|J|$ is (non-strictly) decreasing.\n\\end{prop}\n\n\\begin{sproof}\nThis collection is the same as that of \\cite[Theorem 6.6]{CT} up to a\ntwist by a line bundle.\nThus, we only sketch an argument here (expanded in \\cite{BDMdP}).\nOne checks that the description of ``forbidden cones'' given by\nBorisov and Hua in \\cite{BH} shows that relevant cohomology groups\nvanish --- this shows that it is a strong exceptional collection.\nTo prove generation, one considers the series of flips required to reach\n$\\mathbb{P}^n$ blown up at $n+1$ points.\nUsing the description of the semi-orthogonal decompositions in\n\\cite{BFK}, the line bundles can be shown to generate the necessary\nadmissible subcategories of each intermediate birational model.\n\\end{sproof}\n\nSince any centrally symmetric toric Fano variety is a product of projective lines and the varieties $V_n$, Lemma~\\ref{lem:stabprod} yields the following:\n\n\\begin{cor}\\label{cor:centsym}\nAny form of a centrally symmetric toric Fano variety admits a full\nstrong exceptional collection consisting of vector bundles.\n\\end{cor}\n\n\\subsection{Toric varieties from the Weyl fans of type A} \\label{sec:X(An)}\n\nOne method for identifying toric varieties with large symmetry groups is to start with root systems. Let $R$ be a root system in a Euclidean space $E$. The ${\\mathbb Z}$-lattice generated by $R$ is denoted $M(R)$, while its dual in $E^\\vee$ is denoted by $N(R)$. For every set $S$ of simple roots in $E$, we have the dual cone corresponding to a closed Weyl chamber\n\\begin{displaymath}\n \\sigma_S := \\{ f \\in E^\\vee \\mid \\langle f, \\alpha \\rangle \\geq 0 \\ , \\ \\forall \\alpha \\in S\\}.\n\\end{displaymath}\nThe cones $\\sigma_S$ are the maximal cones for a fan $\\Sigma_R$ in $E^\\vee$. We denote the associated toric variety by $X(R)$. Recall that an automorphism of $R$ is an element of $\\operatorname{GL}(E)$ preserving $R$. Let $W(R)$ be the Weyl group and $\\Gamma(R)$ the symmetry group of the Dynkin diagram of $R$. It is well-known that\n\\begin{displaymath}\n \\text{Aut}(R) \\simeq W(R) \\rtimes \\Gamma(R).\n\\end{displaymath}\nAny automorphism of $R$ induces an action on the fan $\\Sigma(R)$, which yields a homomorphism $\\phi: \\text{Aut}(R) \\to \\text{Aut}(\\Sigma(R))$.\n\n\\begin{lem} \\label{lem:X(R)FanAut}\n The map $\\phi:  \\operatorname{Aut}(R) \\to \\operatorname{Aut}(\\Sigma(R))$ is an isomorphism.\n\\end{lem}\n\n\\begin{proof}\n First note that the set $R$ can be reconstructed from $\\Sigma(R)$ by taking the union of the extremal rays generating the dual cones $\\sigma_S^\\vee$ for all $\\sigma_S$. Thus any symmetry of the fan induces a symmetry of $R$. This gives the inverse map to $\\phi$.\n\\end{proof}\n\nHere we focus on the case $R = A_n$. In \\cite{LosevManin}, the authors showed that $X(A_n)$ is a moduli space of rational curves with $(n+1)$ marked points and $2$ poles. Another useful proof appeared in \\cite{BatyrevBlume}. \n\nUsing this perspective, Castravet and Tevelev exhibited an exceptional\ncollection on $X(A_n)$ that is stable under the action of permuting the\nmarked points and flipping the poles, i.e., an $(S_{n+1} \\rtimes\nC_2)$-stable collection. Here we demonstrate that Castravet and\nTevelev's exceptional collection satisfies the conditions of\nProposition~\\ref{prop:blackbox} and hence descends to an exceptional\ncollection on any form of $X(A_n)$ (in characteristic $0$). \n\nTo do this requires a bit of translating divisors and actions from the moduli-theoretic language to the toric language. We recall the moduli-theoretic languge. \n\n\\begin{defn}\n Let $N$ be a set of order $n$. A \\emph{chain of polar} ${\\mathbb P}^1$'s is a $\\left(\\{0,\\infty\\} \\cup N\\right)$-marked linear nodal chain of $\\mathbb{P}^1$'s with $0$ on the left tail and $\\infty$ on the right tail. A chain of polar ${\\mathbb P}^1$'s is \\emph{stable} if \n \\begin{enumerate}\n  \\item marked points do not coincide with nodes,\n  \\item only $N$-marked points are allowed to coincide,\n  \\item each component of the chain has at least three special points (nodes or marked points). \n \\end{enumerate}\n We write $LM_N$ for the corresponding moduli space. We also use $LM_n$ depending on the context.  Note that the universal curve over $LM_n$ is isomorphic to $LM_{n+1}$.\n\\end{defn}\n\n\n\\begin{thm}\\label{thm:univcurve}\n The toric variety $X(A_{n-1})$ is isomorphic to $LM_n$. Moreover, if we fix an embedding $A_{n-1} \\to A_n$, the corresponding map $X(A_n) \\to X(A_{n-1})$ is the universal curve.\nMoreover, $X(A_n) \\to X(A_{n-1})$ is a toric morphism.\n\\end{thm}\n\n\\begin{proof}\nThis is \\cite[Theorem 2.6.3]{LosevManin}. See also \\cite[Theorem 3.19]{BatyrevBlume}. \nThe map is consequently toric by \\cite[Proposition~1.4]{BatyrevBlume}.\n\\end{proof}\n\nUnder this isomorphism, the closures of the torus orbits on $X(A_n)$ have the\nfollowing moduli-theoretic description.\nFix a partition $N_1 \\sqcup N_2 = N$ and let $\\delta_{N_1}$ denote the\ndivisor parametrizing polar chains of length exactly $2$ having the\nfirst marked by $N_1$ and the last marked by $N_2$.\nFor a partition with more parts, $N_1 \\sqcup N_2 \\sqcup \\cdots \\sqcup N_t = N$,\none has the locus $Z_{N_1,\\dots,N_t}$ parametrizing polar chains of length exactly $t$,\nwhere the $i$-th ${\\mathbb P}^1$ is marked by $N_i$.\nThese loci are precisely the proper torus orbit closures on $X(A_n)$.\n\nNote that each loci is a complete intersection\n\\begin{displaymath}\n Z_{N_1,\\dots,N_t} := \\delta_{N_1} \\cap \\delta_{N_1 \\cup N_2} \\cap \\cdots \\cap \\delta_{N_1 \\cup \\cdots \\cup N_{t-1}}. \n\\end{displaymath}\nMoreover, we have an isomorphism \n\\begin{displaymath}\n Z_{N_1,\\dots,N_t} \\simeq LM_{N_1} \\times LM_{N_2} \\times \\cdots \\times LM_{N_t}\n\\end{displaymath}\nwhere the left node of each ${\\mathbb P}^1$ is marked with $0$ and the right node is marked with $\\infty$.\nThus, we have toric morphisms \n\\begin{displaymath}\n  i_{N_1,\\ldots,N_t} : LM_{N_1} \\times LM_{N_2} \\times \\cdots \\times LM_{N_t} \\to LM_N \\ .\n\\end{displaymath} \nAlso, for each subset $K \\subset N$, we get a forgetful map $\\pi_K: LM_N\n\\to LM_K$, which is a toric morphism since it is a composition of maps\nfrom Theorem~\\ref{thm:univcurve}.\n\nRecall there is a set of line bundles $\\mathbb{G}_N$ on $LM_N$ \\cite[Definition 1.5]{CT}, and one generates a larger set $\\mathsf{H}_N$ of sheaves via\n\\begin{displaymath}\n \\mathsf{H}_N := \\left \\lbrace \\left(i_{N_1,\\ldots,N_t}\\right)_*\n(G_{l_1} \\boxtimes \\ldots \\boxtimes G_{l_t})\n\\mid \\forall N_1 \\cup \\cdots \\cup N_t = N \\ , \\ G_{l_j}\n\\in \\mathbb{G}_{N_j} \\right \\rbrace,\n\\end{displaymath}\nwhere $i_{N_1,\\ldots,N_t}: Z_{N_1, \\ldots, N_t} \\hookrightarrow LM_N$ is the inclusion. \n\n\\begin{thm}\\label{thm:CTexcp}\n There is an ordering on the set \n \\begin{displaymath}\n  \\mathsf{CT}_N :=\n\\mathsf{H}_N \\cup \\left(\n\\bigcup_{K \\subsetneq N} \\{ \\pi_K^\\ast E \\mid \\ E \\in \\mathsf{H}_K \\} \\right)\n\\cup \\{ \\mathcal{O} \\}\n \\end{displaymath}\nmaking it into an $(S_N \\rtimes C_2)$-stable exceptional collection under permutations of the two sets of markings.\n\\end{thm}\n\n\\begin{proof}\n This is \\cite[Proposition 1.5]{CT}.\n\\end{proof}\n\n\\begin{prop}\\label{prop:CTaut=rootAut}\n The action of $S_{n+1} \\rtimes C_2$ given by permuting the two sets of marked points corresponds to the action of $\\operatorname{Aut}(A_n)$ on $X(A_n)$. \n\\end{prop}\n\n\\begin{proof}\n We use the standard presentation of the root system for $A_n$ as $e_i - e_j$ for $1 \\leq i < j \\leq n+1$ and follow \\cite[Construction 3.6]{BatyrevBlume}. The embedding $A_n \\hookrightarrow A_{n+1}$ gives the universal curve $X(A_{n+1}) \\to X(A_n)$. For $i \\in \\{1,\\ldots,n\\}$, we take the $(n+1)$ projections $A_{n+1} \\to A_n$, whose kernels are generated by $e_i - e_{n+1}$ for $1 \\leq i \\leq n+1$. These give sections $s_i : X(A_n) \\to X(A_{n+1})$. Finally, for the polar sections, we have the dual vector $v_{n+2}$. The vectors $v_{n+2}$ and $-v_{n+2}$ give toric invariant divisors which are isomorphic to $X(A_n)$ \\cite[Proposition 1.9]{BatyrevBlume}. The isomorphisms give the other sections $s_0$ and $s_{\\infty}$. \n \n The Weyl group is the permutation group of the $e_i$, and hence of the $e_i - e_{n+2}$. In particular, it permutes the $s_i$. The outer involution acts on the fan by negation and thus exchanges the cone corresponding to $v_{n+2}$ with the cone corresponding to $-v_{n+2}$.\n\\end{proof}\n\n\\begin{cor}\\label{cor:CTisstable}\n The set $\\mathsf{CT}_N$ is $\\operatorname{Aut}(\\Sigma(A_n))$-stable. \n\\end{cor}\n\n\\begin{proof}\n This is an immediate corollary of Lemma~\\ref{lem:X(R)FanAut} and Proposition~\\ref{prop:CTaut=rootAut}. \n\\end{proof}\n\n\\begin{prop}\\label{prop:CTistoric}\nEach object in the collection $\\mathsf{CT}_N$ is torus-equivariant. \n\\end{prop}\n\n\\begin{proof}\nLine bundles are always isomorphic to torus-equivariant line bundles,\nso all objects in $\\mathbb{G}_N$ are torus-equivariant.\nThere is a canonical equivariant structure on tensor products and on\npullbacks by equivariant morphisms (see \\S{2}~of~\\cite{BFK2});\nthus each object $G_1 \\boxtimes \\ldots \\boxtimes G_n$\nis torus-equivariant for $G_{l_j} \\in \\mathbb{G}_{N_j}$.\nLet $i : Z \\to X$ be shorthand for some map $i_{N_1,\\ldots,N_t}$.\nThere is a splitting of tori $T = S \\times S'$ where $Z$ is an $S$-toric\nvariety and $S'$ acts trivially on $i(Z)$.\nLet $\\psi : T \\to S$ denote the projection. \nWe have a composition of functors\n\\[\n\\mathsf{D^b}(\\operatorname{Coh}_S Z) \\to\n\\mathsf{D^b}(\\operatorname{Coh}_T Z) \\to\n\\mathsf{D^b}(\\operatorname{Coh}_T X)\n\\]\nwhere the first map is the functor $\\operatorname{Res}_\\psi$\n(\\S{2.9}~of~\\cite{BFK2}) and the second map is the $T$-equivariant\npushforward (\\S{2.5}~of~\\cite{BFK2}).\nThis composition reduces to the ordinary pushforward\n$i_\\ast : \\mathsf{D^b}(Z) \\to \\mathsf{D^b}(X)$ when the equivariant\nstructure is forgotten.  We conclude that each object of\n$\\mathsf{H}_K$ is torus-equivariant\nand the result follows.\n\\end{proof}\n\nWe now prove the main result of this section:\n\n\\begin{prop}\\label{prop:X(An)excpcoll}\n Let $k$ be a field of characteristic zero and $X$ a form of $X(A_n)$ over $k$. Then $X$ admits a full exceptional collection of sheaves. \n\\end{prop}\n\n\\begin{proof}\n Combining Theorem~\\ref{thm:CTexcp}, Corollary~\\ref{cor:CTisstable}, and\nProposition~\\ref{prop:CTistoric} allows us to appeal to\nProposition~\\ref{prop:blackbox} and conclude that $\\mathsf{CT}_N$\ndescends to an exceptional collection of sheaves on $X$. \n\\end{proof}\n\n\\begin{rem}\n To remove the characteristic zero assumption one needs to extend generation results of \\cite{CT} to nonzero characteristic. This could conceivably be done by reversing the flow of reasoning in \\cite{CT}, using the fact we know the collections for $V_n$ in any characteristic. We do not pursue this. \n\\end{rem}\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{} \n\nGalaxy bimodality, described by the red sequence and blue cloud, has been central to our understanding of galaxy evolution since the turn of this century \\citep[e.g.,][]{2004ApJ...600..681B,2004ApJ...608..752B,2009ARA&A..47..159B}. Passive galaxies follow a narrow color-magnitude relation while star-forming galaxies in the blue cloud have a broader range of optical colors, resulting from a range of stellar populations, star formation rates and dust obscuration. Although star-forming galaxies are diverse, they do fall along well-established correlations with mass, including the mass-metallicity relation \\citep[e.g.,][]{2004ApJ...613..898T}, star formation rate versus mass \\citep[e.g.,][]{2007ApJ...660L..43N} and declining dust content with decreasing mass \\citep[e.g.,][]{2006ApJ...639..157W}. These relations manifest themselves in broadband photometry, albeit outside the optical wavelength range, as illustrated by the dependence of infrared colors on galaxy type \\citep[e.g.,][]{2011ApJ...735..112J}.\n\nThe far-ultraviolet and mid-infrared are both star formation rate tracers, with the former tracing massive stars while the latter traces blackbody emission from warm dust. While far-ultraviolet luminosity is directly proportional to star formation rate, for $\\lesssim L^*$ galaxies mid-infrared luminosity is proportional to star formation rate to the power of $\\sim 1.3$ \\citep[e.g.,][]{2015AA...584A..87C,2017ApJ...847..136B}, which is a consequence of dust content varying with galaxy mass. We thus expect a blue sequence to be present in ultraviolet-infrared color-magnitude diagrams.\n\nTo measure the ultraviolet-infrared color-magnitude relation, we use the local galaxy sample of \\citet{2014ApJS..212...18B, 2017ApJ...847..136B} and their multiwavelength matched aperture photometry (in AB magnitudes). We limit the sample to galaxies with $m_{W2}-m_{W3}>-0.5$, which excludes passive galaxies from the \\citet{2014ApJS..212...18B} sample, and we remove active galactic nuclei with the emission line ratio criterion of \\citet{2003MNRAS.346.1055K}. To correct the GALEX $FUV$ photometry for internal dust obscuration we use $A_{FUV}  \\propto (M_{FUV} - M_{NUV})$, leaving the constant as a free parameter that we use to minimize the scatter of the color-magnitude relation. \n\nIn Figure~\\ref{fig:thefigure} we present the ultraviolet-infrared color-magnitude plot of $z\\sim 0$ star-forming galaxies, using the $FUV$, $NUV$, and WISE $W3$ photometry. We find the best relation is produced when $A_{FUV}  = 2.6 (M_{FUV} - M_{NUV})$, which is shallower than the dust extinction relation of \\citet{2011ApJ...741..124H}, where $A_{FUV}  = (3.83\\pm 0.48) [M_{FUV} - M_{NUV} - (0.022\\pm 0.024)]$. As a cross check of our results, in Figure~\\ref{fig:thefigure} we also plot photometry of $z<0.05$ GAMA galaxies \\citep{2016MNRAS.460..765W} with WISE $m_{W2}-m_{W3}>-0.5$, and we find good agreement although GAMA spans a smaller range of $M_{W3}$ than \\citet{2014ApJS..212...18B, 2017ApJ...847..136B}. We also observe similar relations when we replace WISE $W3$ with WISE $W4$ or {\\it Spitzer} $24~{\\rm \\mu m}$, albeit with more scatter. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth,angle=0]{blueseqplot.pdf}\n\\includegraphics[width=0.49\\textwidth,angle=0]{gH.pdf}\n\\caption{The ultraviolet-infrared (left) and $g-H$ (right) color-magnitude relations for star-forming galaxies. Some of the outliers in the $g-H$ are labelled, and these are often merging galaxies rather than spirals. As the \\citet{2014ApJS..212...18B,2017ApJ...847..136B} sample deliberately selected galaxies to span parameter space, it shows more scatter than the magnitude limited GAMA sample.  \\label{fig:thefigure}}\n\\end{center}\n\\end{figure}\n\nThe best-fit color-magnitude relation is given by \n\\begin{equation}\nM_{W3} = -14.8 - 2.1 \\times \\left[ 2.6 (M_{FUV} - M_{NUV}) - M_{W3} \\right].\n\\end{equation}\nUsing the  \\citet{2014ApJS..212...18B, 2017ApJ...847..136B} sample, we find the $1\\sigma$ scatter of $M_{W3}$ about the relation is $\\sigma_{W3} =1.6~{\\rm mag}$. If ultraviolet - infrared color was used as a distance indicator then the 68\\% scatter of the distance would be a factor of $\\sim 2$. \n\nWe note color-magnitude relations for blue galaxies have been identified previously, including the median optical color of blue galaxies varying with magnitude \\citep{2004ApJ...600..681B}. Furthermore, \\citet{1982ApJ...257..527T} identified a tight $B-H$ color-magnitude relation for spiral galaxies, and in the right panel of Figure~\\ref{fig:thefigure} we reproduce this relation for star-forming galaxies using SDSS $g$ and 2MASS $H$-band photometry.  \\citet{1982ApJ...257..527T} attributed this relation to specific star formation rate, chemical abundances and\/or initial mass function varying with mass. Interestingly, we do see some outliers in the $g-H$ versus $M_H$ diagram, including merging starbursts. These outliers are not unexpected, given $g$ and $H$ trace different galaxy properties, whereas the ultraviolet and mid-infrared are both (primarily) star formation rate tracers. \n\nIn this note we have identified and characterized the ultraviolet-infrared color-magnitude relation of star-forming galaxies. The ultraviolet to mid-infrared flux ratios of star-forming galaxies span over two orders of magnitude and show a clear dependence on absolute magnitude from $M_{W3}\\sim -13$ to $M_{W3}\\sim -25$, which may present problems for models of galaxy spectral energy distributions that have been largely verified on $\\sim L^*$ galaxies. The color-magnitude relation of star-forming galaxies illustrates the (broadband) spectral diversity of star-forming galaxies that results from established correlations between the physical properties and mass, including the mass-metallicity relation.\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Background}\n\\label{sec:introduction}\n\nConventional dynamic random-access memory (DRAM)\nscaling has reached a critical tipping point as the miniaturization of the DRAM cell has plateaued in recent years. Feature size scaling below the 20 $\\text{nm}$ technology node is met with numerous challenges such as\nshorter retention times, higher leakage currents, and increased fault rates~\\cite{park2015technology}.\nSolutions to address these concerns include improved DRAM fault detection and recovery~\\cite{wang2017improving}, as well as architectural techniques to enhance DRAM scaling~\\cite{kim2015architectural}. \n\nA promising solution to the memory scaling problem is to realize the main memory system using non-volatile technologies~\\cite{mutlu2015main}. \nExamples of emerging non-volatile memories (NVMs) include spin-transfer torque magnetoresistive random-access memory (STT-MRAM), ferroelectric random-access memory (FeRAM), resistive random-access memory (ReRAM), and phase-change memory (PCM).\nInterest in the commercial application of such NVMs has increased significantly. For instance, Intel's current line of 3D XPoint memory systems utilize PCM-based NVM technology~\\cite{wyrwas2017proton}, and IBM and Everspin's solid-state drive comes with STT-MRAM write caches~\\cite{everspin}. While NVMs offer attractive features, such as high density, low leakage, and non-volatile data retention, they also suffer from poor endurance and high access latency in their current implementation.\n\nMemory security has come under more scrutiny over the years.\nThis is because of attacks such as \\textit{Spectre}~\\cite{kocher2019spectre} and \\textit{Meltdown}~\\cite{lipp2018meltdown}, which targets the side-channels associated with speculative execution and out-of-order execution, respectively, have exposed severe vulnerabilities in a wide array of currently deployed processors and their memory architectures. \nIn the case of NVMs, data remanence after power-down\npresents a severe threat to data confidentiality, as attackers aiming to steal private data can do so easily by mounting cold-boot attacks~\\cite{halderman2009lest} or other removal attacks like stealing the memory module (DIMM)~\\cite{young2015deuce}. \nMoreover, magnetic memories like STT-MRAM are highly sensitive to stray magnetic fields. \nAs such, magnetic field-based attacks~\\cite{jang2015self} can be used to corrupt the stored data or compromise the memory's functional integrity, resulting in a denial-of-service (DoS) attack. \nHence, such security vulnerabilities pose a significant impediment to the pervasive and large-scale proliferation of NVMs in the memory industry. \n\n\\subsection{Related work in Memory Security}\nPrior works on securing NVMs have focused mainly on memory encryption schemes, which are necessary to prevent attackers from exploiting data remanence in the off-state. \nChhabra \\textit{et al.} proposed an incremental encryption scheme~\\cite{chhabra2011nvmm} for NVMs where only inert memory pages, which have not been accessed for several clock cycles, are encrypted selectively. \nThe working set of the memory (which is in current use) is in plaintext and, hence, incurs no encryption overhead on access. Such a selective encryption ensures that the majority of the main memory content (but not all) remains encrypted at all times, without overly compromising the performance. \nHowever, it requires dedicated hardware, inert page prediction, and scheduling for its implementation. \nA sneak-path encryption (SPE) scheme was demonstrated for memristor-based NVMs in~\\cite{kannan2014secure}, wherein sneak paths in the memristor crossbar array are exploited to apply encryption pulses to change the resistances of the memory cells, and hence, encrypt the stored data. \n\nIn~\\cite{young2015deuce}, the authors proposed DEUCE, a dual counter encryption for PCM memories, which significantly reduces the number of modified bits per writeback, to improve performance and lifetime of the memory. \nThis scheme aims to mitigate the impact of the avalanche effect~\\cite{mandal2012performance} occurring during memory encryption, by re-encrypting and writing back only the modified words during any write operation. Swami~\\textit{et al.} took this concept forward and proposed SECRET~\\cite{swami2016secret}, a smart encryption scheme for NVMs, which integrates word-level re-encryption and zero-based partial writes to reduce memory write operations. They also demonstrate write optimization through the use of\n``energy masks'' (i.e., bit templates XORed with ciphertext to obtain lower energy dissipation)\nin the encryption XOR logic, which minimizes the bit flips in the encryption process, thereby reducing the total write energy.\nAn advanced counter-mode encryption (ACME) was presented in~\\cite{swami2018acme}, which utilizes the write leveling architecture inherent in PCM memories, to perform counter-write leveling. \nACME helps to avoid \\textit{Rowhammer}-type attacks by preventing the counter associated with any single cache line from overflowing.\n\nThe impact of contactless tampering on STT-MRAMs using external magnetic fields was highlighted in~\\cite{jang2015self}. \nUsing micromagnetic simulations, the authors of~\\cite{jang2015self}\nshowed how magnetic field-based attacks could corrupt the contents of STT-MRAM cells. Techniques to protect against contactless attacks proposed in~\\cite{jang2015self} included (i) an on-chip sensor to detect magnetic field-based incursions, and (ii) error correction modules to compensate cell failures arising due to magnetic field attacks. However, these techniques incur large energy and area penalties due to the additional hardware imposed by the magnetic field sensor and the error correction scheme.\n\n\\subsection{Contributions}\nIn this paper, we present an alternative to conventional NVMs such as STT-MRAM and PCM, in the form of \\textit{SMART: A Secure Magnetoelectric Antiferromagnet-Based Tamper-Proof Non-Volatile Memory}. \nSMART memory leverages the room-temperature linear magnetoelectric (ME) effect in antiferromagnets (AFMs) like chromia~\\cite{rado1961observation}, which can be switched solely using voltage pulses, without the use of electric currents, leading to ultra-low energy ($\\sim$ pico-Joules) operation. \nFurther, the intrinsic dynamics of AFMs is typically in the terahertz regime ($\\sim 10^{12}$ Hz)~\\cite{khymyn2017antiferromagnetic}, which could enable picosecond time-scale reversal of the AFM domain. \nIn addition to its energy and latency benefits, SMART memory offers a significant advancement in terms of secure and tamper-proof data storage. \nFor example, AFMs do not exhibit a magnetic signature since they do not have a net external magnetic moment, unlike ferromagnets (FM). \nHence, the SMART memory cannot be probed or switched with external magnetic fields, unlike the way STT-MRAMs can. \nThis, in turn, eliminates the possibility of magnetic field attacks undermining data integrity or aiming to induce DoS. \nTo address the post-shutdown data remanence of SMART memory, we demonstrate an in-memory encryption scheme employing  ME-AFM transistor-based controlled-NOT (CNOT) logic. \nWe discuss the resilience of the SMART memory against attacks aiming to undermine data confidentiality and data fidelity, in both powered-on and powered-off states. \nThe main contributions of this work can be summarized as follows:\n\n\\begin{enumerate}\n    \n\\item We discuss the design of SMART, a secure ME-AFM-based NVM and implement its SPICE circuit model to simulate the memory performance. \n    \n\\item We demonstrate the resilience of SMART memory against magnetic field and temperature attacks, which can affect other NVMs like STT-MRAM. We explore the implications of various side-channel attacks on the SMART memory.\n    \n\\item We present an in-memory encryption scheme with ME-AFM transistor-based CNOT gates, called \\textit{Memcryption}, to protect the data stored in SMART memory against cold-boot and stolen DIMM attacks, while incurring low encryption latency overheads.\nWe like to mention here that \\textit{Memcryption} is specifically tailored for the ME-AFMRAM, not for a generic NVM. Also, it does not secure the memory system against \\textit{bus snooping} attacks; such attacks are beyond the scope of this work.\n\n\\end{enumerate}\n\nIn the next section, we describe the modeling, implementation and benchmarking of the proposed ME-AFM memory both at cell- and array-level, before proceeding to evaluate its security properties in Section~\\ref{sec:security}.\n\n\\section{Device model and functionality}\n\\label{Modeling}\n\\subsection{The magnetoelectric effect}\nThe linear ME effect~\\cite{agyei1990linear} represents the coupling between applied magnetic field and induced polarization or between applied electric field and induced magnetization in non-centrosymmetric crystals like chromia ($\\text{Cr}_2\\text{O}_3$). Compared to the STT-based magnetization reversal of FMs requiring electric currents on the order of $\\sim10^6$ A\/cm$^2$ and incurring associated Joule heating, the ME effect provides an energy-efficient, all-electrical switching of the roughness-insensitive boundary magnetization of chromia~\\cite{echtenkamp2013electric}. Additionally, chromia is an AFM; hence, the net bulk magnetic moment\n(i.e., the difference of the sublattice magnetization vectors) vanishes and becomes imperceptible externally. \nHowever, the boundary magnetization is strongly coupled to the AFM order parameter. That is, the electrical switching of the AFM order results in reversal of the boundary magnetization~\\cite{wu2011imaging}, which is used to encode the information in ME-AFM memories.\n\nThe uncompensated surface moments at the (0001) surface of chromia result in an equilibrium boundary magnetization, which could be in one of the two oppositely aligned,\ndegenerate domain states. \nThe degeneracy between the domains is lifted through ME annealing, which allows the preferential selection of one of the states~\\cite{he2010robust}. That is, the ME annealing polarizes the surface and results in\na single-domain surface moment. \nIsothermal switching between these single domain states using an electric field $E$ and a small, symmetry-breaking\nDC magnetic field $H$ has been demonstrated experimentally~\\cite{he2010robust, fallarino2015magnetic}. \nThe critical condition for such ME switching is that the magnitude of the $E\\cdot H$ product must exceed the ME threshold energy barrier, which was shown experimentally to be as low as $\\approx$ 1 J\/m$^3$~\\cite{brown1969domain, martin1966antiferromagnetic}.\n\n\\subsection{ME-AFMRAM : Working principle}\nThe chromia-based ME-AFMRAM, which is at the heart of our SMART memory, is shown in Fig.~\\ref{fig:AFMRAM}. Experimentally demonstrated by Kosub \\textit{et al.}~\\cite{kosub2017purely}, the ME-AFMRAM has a bottom gate electrode (Platinum gate in the figure) for applying the gate voltage $V_G$ and providing the necessary electric field to write data into the memory. A small, symmetry-breaking magnetic field ($\\approx$ 30 mT) is provided by the stray field of a permanent magnet. A positive voltage $V_G$ will orient the bulk order and, hence, put the surface magnetization in one domain (with surface moments pointing up), whereas a negative voltage will result in the surface magnetization relaxing to the opposite domain (with surface moments pointing down). These two states correspond to binary levels `1' ($V_G > 0$) and `0' ($V_G < 0$), respectively. A gate voltage of 0 V corresponds to the `hold' mode of the memory cell. Note that the cell serves as non-volatile memory in all gate-voltage ranges, not only for $V_G = 0$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.28]{figures\/AFMRAM.pdf}\n\\caption{Chromia-based magnetoelectric antiferromagnetic random-access memory. Data (1\/0) is written by applying a voltage ($+\/-$) to the bottom gate electrode. Read-out is achieved using an anomalous Hall bar electrode placed on top, by applying a Hall bias.}\n\\label{fig:AFMRAM}\n\\end{figure}\n\nThe read-out is achieved using an anomalous Hall (AH) bar electrode setup, which discerns the boundary magnetization of chromia by sensing the proximity effect-induced magnetization in the nearby Platinum (Pt) electrode, thereby producing a \nproportional Hall voltage $V_{\\text{xy}}$ (or $V_{\\text{AHE}}$)~\\cite{kosub2015all}. Traditionally, the order parameter of AFMs is read-out via an exchange bias arrangement~\\cite{toyoki2015magnetoelectric} in another FM attached adjacently to the AFM surface. However, the exchange bias and the FM's hysteresis increase the coercive voltage required to overcome the ME barrier and, hence, impact the write energy negatively. To avoid this effect, Kosub \\textit{et al.}~\\cite{kosub2017purely} proposed the use of an exclusively ME-AFM setup with an AH read-out of the surface magnetization, thereby eliminating the need for an FM.\nAt the time of writing this paper, a complete physical understanding of the read-out mechanism for the boundary magnetization in chromia is lacking. While the authors in~\\cite{kosub2017purely} have considered an AH-based read-out in their device, recent experiments by C. Binek's group at the University of Nebraska-Lincoln have revealed the contribution of spin-Hall magnetoresistance (SMR) to the read-out signal, which is currently being investigated.\nHowever, note that the magnitude of the signal levels is the same in both cases (AH versus SMR) and also the circuit models developed would remain the same, though with different input parameters. \nFor the purposes of this paper, we consider that the read-out signal is due to the AH effect in the proximal heavy metal, as also discussed in prior experimental work.\n\n\\subsection{Performance modeling}\nThe ME reversal mechanism in chromia can be classified broadly into two categories, depending on the size of the film compared to the characteristic domain-wall (DW) width. For chromia, the typical DW width $\\lambda =\\sqrt{A\/\\mathcal{K}}\\sim$ 50-100 nm, where $A$ is the exchange stiffness constant and $\\mathcal{K}$ is the uniaxial anisotropy energy~\\cite{belashchenko2016magnetoelectric}. \nIf the sample is much smaller than the DW width, the sample reverses via coherent rotation upon application of the ME pressure. For sample dimension comparable to the DW width, ME reversal occurs via DW nucleation and propagation, which is an incoherent switching process.\nFor both coherent rotation and DW propagation, the reversal could be thermally activated for applied ME pressure lower than the energy barrier between the stable domain states. Otherwise, the domain reversal proceeds in the `flow' regime~\\cite{parthasarathy2019dynamics}.\nME-AFMRAM devices currently fabricated have dimensions in the $\\mu$m range, rendering DW propagation the favorable ME reversal mechanism. To characterize the functionality and performance of chromia ME-AFMRAM, we develop circuit models that represent DW-based reversal in both the thermally activated and the flow regimes. We also provide perspectives and future\npotential concerning dimensional scaling of the device, which could enable ultra-fast, coherent, rotation-based reversal. \n\n\\subsubsection{DW reversal of chromia ME-AFMRAM}\nConsider a chromia sample, where the applied ME pressure creates a pressure difference of $\\mathcal{F} =|2\\alpha_{\\text{ME}} E H|$ between the two domains. Here, $\\alpha_\\mathrm{ME}$ is the linear ME coefficient.\n\nIf $\\mathcal{F}> \\mathcal{F}_d$ (i.e., for DW de-pinning pressure), the DW propagates as a viscous flow with velocity given as~\\cite{parthasarathy2019dynamics}\n$$\\nu_{\\text{flow}} = \\frac{\\alpha_{\\text{G}}\\gamma \\lambda}{\\alpha_+\\xi^2}\\Big( \\frac{\\mathcal{F}-\\mathcal{F}_{\\text{d}}}{M_{\\text{s}}} \\Big),$$\nwhere $\\alpha_{\\text{G}}$ is the Gilbert damping constant, $\\gamma$ is the gyromagnetic ratio of electron, $M_{\\text{s}}$ is the sublattice saturation magnetization, and $\\xi=\\frac{\\alpha_{\\text{ME}}E}{\\mu_0 M_{\\text{s}}}$. \nFor a mean free path of $l$ of the DW, the time-scale of ME reversal due to viscous DW propagation is $\\tau_{\\text{flow}}=l\/\\nu_{\\text{flow}}$.\n\nIf $\\mathcal{F}<\\mathcal{F}_{\\text{d}}$, the DW undergoes thermal creep to overcome the de-pinning barrier, with a time-scale~\\cite{parthasarathy2019dynamics}\n$$\\tau_{\\text{creep}}=\\sqrt{\\frac{\\sigma \\mathcal{S}^3}{kT}}\\Big(\\frac{\\mathcal{F}_{\\text{d}}-\\mathcal{F}}{2\\pi\\epsilon}\\Big)\\exp\\Big[{\\frac{\\mathcal{S}^2(\\mathcal{F}_{\\text{d}}-\\mathcal{F})^2}{4\\pi kT\\epsilon}\\Big]},$$\nwhere $kT$ is the thermal energy (25 meV at 300 K), $\\epsilon$, $\\sigma$, and $\\mathcal{S}$\nare the energy, areal density, and surface area, respectively, of the DW. The DW de-pinning pressure is determined by the DW energy, its surface area, and the radius of the non-magnetic de-pinning center.\n\nTo write `1' (`0') into the memory cell, a positive (negative) electric field, $E_\\mathrm{app}$, with a magnitude greater than the critical electric field, $E_\\mathrm{crit}$, is required, in order to meet the DW propagation criteria of $\\mathcal{F}>\\mathcal{F}_d$. In this case, the time to write data into the memory is equal to $\\tau_\\mathrm{flow}$. When $E_\\mathrm{app}$ is less than $E_\\mathrm{crit}$ (i.e., $\\mathcal{F}<\\mathcal{F}_d$), the memory cell is in the hold mode and the retention time is specified by $\\tau_\\mathrm{creep}$. For typical parameters of chromia, we find $\\tau_\\mathrm{creep}\\gg \\tau_\\mathrm{flow}$, which ensures that the memory cell is thermally stable when it is not accessed. Here, the stability of the cell is determined by $\\tau_{\\text{creep}}$, since longer data retention requires the time constant in the hold mode to be larger. The retention time of the cell can be further improved by enlarging the cell dimensions.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.19]{figures\/chromia_circuit_final.pdf}\n\\caption{Equivalent circuit for the chromia ME-AFMRAM cell. $I_{\\text{int}}$, derived from the bit line, writes data on to the node $V_{\\text{ME}}$. The time constant of the write operation is $\\tau_{\\text{flow}}$ ($\\tau_{\\text{creep}}$) if the applied voltage is greater (smaller) than the critical voltage. Read-out is achieved through an AH setup, modeled with a voltage-controlled voltage source. $\\text{C}_\\text{EL}$ is the electrostatic capacitance of the chromia dielectric.}\n\\label{fig:chromia_RC}\n\\end{figure*}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.37]{figures\/MERAM_transient.pdf}\n\\caption{Transient simulations showing write operations on the chromia ME-AFMRAM cell. Note that for writing a `1' the write pulse is positive, and for writing a `0' the write pulse is negative. In this simulation, a series of `1's (0.3 V) and `0's (-0.3 V) are being written to the cell, and then finally `0' is retained once Write Enable is switched off.}\n\\label{fig:AFMRAM_timing}\n\\end{figure}\n\nWe construct a SPICE circuit model to functionally capture the ME reversal dynamics of chromia. The time constant for reversal of the magnetization of chromia due to an applied ME pressure is represented as $R_{\\text{eq}}\\times C_{\\text{eq}}$. Without loss of generality, the circuit model uses $R_{\\text{eq}} = 1$ $\\Omega$, while $C_{\\text{eq}}$ is either $\\tau_{\\text{flow}}$ or $\\tau_{\\text{creep}}$. \nTo construct the full ME-AFMRAM cell, we combine the RC model of the ME response of chromia with the peripheral read\/write circuitry in \\textit{Cadence Virtuoso} using the 15-nm CMOS FreePDK technology. Figure~\\ref{fig:chromia_RC} shows the equivalent circuit of the ME-AFMRAM cell. \nThe write pulse, used to charge the chromia dielectric and switch its\nmagnetization $M$, is provided through the current source $I_{\\text{int}}$ (derived from the bit line) in the write setup. \nFor parameters of chromia listed in Table~\\ref{tab:params}, \n$C_{\\text{flow}}=\\tau_{\\text{flow}}\\sim0.223$ nF, $C_{\\text{creep}}=\\tau_{\\text{creep}}\\sim1$ mF, and $V_{\\text{crit}}= 0.2$ V. For $|V_G| > 0.2$ V, $V_{\\text{ME}}$ tracks $V_G$ and data is written into the cell after a write access latency of $\\tau_\\mathrm{flow}$. \nWhen $|V_G| = 0$ V, data is retained for a time interval of $\\tau_{\\text{creep}}$. Since $\\tau_{\\text{creep}}$ is very large, the response in retention\/creep mode is extremely slow as compared to write\/flow mode. The transient response of the ME-AFMRAM cell is shown in Fig.~\\ref{fig:AFMRAM_timing}, to highlight the write operation. The write latency of the ME-AFMRAM cell is obtained as $\\sim 0.63$ ns, and the energy-per-bit for one write operation is $\\sim 0.063$ pJ, including the energy required to charge the electrostatic capacitance of chromia.\nGiven relative dielectric permittivity of 11 and dimensions noted in Table~\\ref{tab:params}, the electrostatic capacitance of chromia is calculated as $5.8$ aF.\n\n\\subsubsection{Anomalous Hall read-out}\nTo evaluate the read cycle, we set the signals WE to 0 and RE to 1 in Fig.~\\ref{fig:chromia_RC}. The read setup is designed to sense the boundary magnetization of chromia through an AH arrangement, which transduces the magnetization into a voltage signal. This transduction process is modeled using a voltage-controlled voltage source\n(VCVS). Typically, a heavy metal such as Pt is used to sense the proximity effect-induced moment from the coupled chromia layer~\\cite{kosub2017purely}.\n\nThe AH voltage sensed from the Hall bar arrangement is given as~\\cite{griffiths2017anomalous}\n$$V_{\\text{AHE}}=\\Big(\\frac{\\mu_0R_{\\text{s}}}{t_{\\text{Hall}}}I_{\\text{Hall}}\\Big)M_{\\text{z}},$$\nwhere $\\mu_0$ is the vacuum permeability, $R_{\\text{s}}$ is the AH coefficient, $I_{\\text{Hall}}$ is the Hall bias current, $t_{\\text{Hall}}$ is the thickness of the Hall layer and $M_{\\text{z}}$ is the proximity effect-induced magnetization. In the case of Pt\/Cr$_2$O$_3$, $R_{\\text{s}}$ is only about $\\sim5$ p$\\Omega$m\/T for $t_{\\text{Pt}}=10$ nm and $T=300$ K~\\cite{meyer2015anomalous}. This results in an AH signal $V_{\\text{AHE}}\\sim$ 0.3 $\\mu$V, considering a Hall bias of 2 mA and a magnetoelectric node voltage $V_{\\text{ME}}=0.3$ V. The Hall signal can be raised to $\\sim$ 1 $\\mu$V by increasing $V_{\\text{app}}$ to 1 V, and further enhanced by applying a larger Hall bias. However, doing so would negatively impact the energy consumed in the read operation. Sensing such a low $\\mu$V-range AH signal would require sophisticated instrumentation sense amplifiers that are area- and power-prohibitive\n(e.g., 2.5 mm$^2$ area and $\\sim$mW-range power~\\cite{witte2008current}).\n\nThis problem can be addressed by exploring other material systems with much higher interfacial spin-orbit coupling (SOC), resulting in larger AH coefficients. \nIn~\\cite{zhang2014effective}, a Pt\/Co\/Pt tri-layer is shown to exhibit $R_{\\text{s}}\\sim7.3\\times 10^{-10}$ $\\Omega$m\/T at 300 K for $t_{\\text{Co}}\\sim$ 10 nm, resulting in $V_{\\text{AHE}}\\sim$ 43.8 $\\mu$V at a Hall bias of 2 mA and $V_{\\text{ME}}=0.3$ V. \nMagnetic semiconductors like EuTiO$_3$ possess higher $R_{\\text{s}}\\sim$ $8\\times 10^{-9}$ $\\Omega$m\/T for $t_{\\text{EuTiO}_3}=$ 25 nm~\\cite{takahashi2018anomalous}. However, AH signals in such samples have been detected only at very low temperatures, of 2K, at which the ME effect in Cr$_2$O$_3$ vanishes.\n \nThe Hall signal could be improved in a topological insulators (TI) due to the presence of high SOC-enhanced surface states. \nFor example, the Bi$_2$Se$_3$\/LaCoO$_3$ stack considered in~\\cite{zhu2018proximity} demonstrates $R_{\\text{s}}$ as high as $\\sim1.59$ $\\mu\\Omega$m\/T at 100 K for $t_{\\text{Bi}_2\\text{Se}_3}\\sim$ 20 nm. \nThis results in a substantial improvement in the AH signal generated (i.e., $\\sim47.7$ mV). \nThe AH effect in the Bi$_2$Se$_3$\/LaCoO$_3$ interface is ascribed to the exchange coupling between the Bi$_2$Se$_3$ layer and the ferromagnetic LaCoO$_3$ layer via the proximity effect, and is enhanced by the high interfacial SOC. \nSimilarly, the (BiSb)$_2$Te$_3$\/TIG system considered in~\\cite{tang2017above} achieves a mV-range AH signal, though much closer to room temperature.\nA comparison of $R_{\\text{s}}\/t$ in various material systems is illustrated in Fig.~\\ref{fig:R_AHE}. \nAs can be inferred, TIs are an ideal material candidate to implement the AH read-out layer with Cr$_2$O$_3$ due to the potential of a $\\sim$mV-range AH signal, which can be easily read-out using a normal current latch sense amplifier~\\cite{kobayashi1993current}, i.e., without the need for sophisticated sensing equipment.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.33]{figures\/R_AHE_new.pdf}\n\\caption{Comparison of the AH coefficient per unit thickness and AH signal magnitude in different material systems. The AH signal $V_{\\text{AHE}}$ is calculated for a Hall bias of 2 mA and a magnetoelectric node voltage $V_{\\text{ME}}\\sim$ 0.3 V. TIs with high interfacial SOC exhibit greater AH coefficients and can generate large AH signals, capable of being detected by conventional current sense amplifiers.}\n\\label{fig:R_AHE}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.37]{figures\/MERAM_cell_comparison.pdf}\n\\caption{Benchmarking the ME-AFMRAM cell considered in this work against current state-of-the-art ME-AFMRAM technology, and trends in other emerging non-volatile storage devices from~\\cite{wong2016stanford}. Some important data points in this plot, representing the advances in various NVMs, include~\\cite{jan2012high,gajek2012spin,liu2010ultrafast} for STT-MRAM, \\cite{aratani2007novel,lin2010novel,vianello2012sb} for CBRAM, \\cite{zhao2014ultrathin,sekar2014technology,goux2014role} for RRAM, and~\\cite{matsui2006ta2o5,kim2010high,xiong2013self} for PCM, respectively.\nThe future potential of ME-AFMRAM lies in achieving ultra-fast, coherent rotation-based reversal (sub-100 ps write delay and fJ write energy) through a combination of dimensional scaling and strain-augmentation.}\n\\label{fig:cell_comparison}\n\\end{figure}\n\n\\subsubsection{Coherent rotation-based reversal}\nThe $\\sim$ns-range write latency of the ME-AFMRAM cell can be improved drastically if the chromia order can be switched through coherent rotation. In this case, the entire chromia sample undergoes reversal homogeneously, rather than following the incoherent DW propagation. For $\\mathcal{F}_{d} > 4\\mathcal{K}$, the order parameter switches via damping of gyromagnetic precessions~\\cite{parthasarathy2019dynamics}. However, if $\\mathcal{F}_{d} < 4\\mathcal{K}$, magnetization could switch due to thermal activation.\nHere, the switching time is exponentially dependent on the energy barrier of the sample.\nIn any case,\nit is thermal activation that leads to retention errors.\n\nTo realize coherent rotation in chromia, the applied ME pressure must exceed $4\\mathcal{K} = 2.92\\times 10^4$ J\/m$^3$. For a magnetic field of 0.5 T and $\\alpha_\\mathrm{ME} = 3.1$ ps\/m, the electric field required for coherent rotation is $1.18 \\times 10^{10}$ V\/m. Unfortunately, such a high electric field could lead to dielectric breakdown of chromia, given that the breakdown strength of chromia is $\\sim 2\\times 10^8$ V\/m~\\cite{sun2017local}.\nA potential solution to this challenge is to reduce the effective anisotropy of the sample such that the required threshold electric field scales down. This can be achieved through a variety of techniques, including substitutional alloying and the application of mechanical strain~\\cite{mu2019influence}. It is estimated that the write latency of a strain-augmented ME-AFMRAM cell can reach as low as a few 10's of ps. A comparison of the current state-of-the-art in ME-AFMRAM technology and its future potential versus trends in other emerging storage devices is presented in Fig.~\\ref{fig:cell_comparison}.\n\n\\subsubsection{Material and geometrical parameters of the chromia ME-AFMRAM cell}\n\nThe simulation parameters used in our SPICE models for the chromia ME-AFMRAM are listed in the following Table~\\ref{tab:params}.\n\n\\begin{table}[ht]\n\\centering\n\\footnotesize\n\\setlength{\\tabcolsep}{1mm}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{*{3}{c}}\n\\hline\n\\textbf{Parameter} & \\textbf{Value} & \\textbf{Ref.}  \\\\\n\\hline\nSaturation magnetization of Cr$_2$O$_3$, $M_{\\text{s}}$ & $2.6\\times 10^5$ A\/m & \\cite{artman1965magnetic} \\\\ \\hline\nMagnetoelectric coefficient of Cr$_2$O$_3$, $\\alpha_{\\text{ME}}$ & $3.1\\times 10^{-12}$ s\/m & \\cite{hehl2008relativistic}\\\\ \\hline\nUniaxial anisotropy energy of Cr$_2$O$_3$, $\\mathcal{K}$ & $7300$ J\/m$^3$  & \\cite{foner1963high} \\\\ \\hline\nGilbert damping constant of Cr$_2$O$_3$, $\\alpha_{\\text{G}}$ & $2\\times 10^{-4}$  & \\cite{belashchenko2016magnetoelectric}\\\\ \\hline\nThreshold ME pressure to depin DW, $\\mathcal{F}_{\\text{d}}$ & $25$ J\/m$^3$ & \\cite{parthasarathy2019dynamics} \\\\ \\hline\nApplied magnetic field, $H_{\\text{app}}$ & $0.5$ T & \\\\ \\hline\nApplied voltage, $V_{\\text{G}}$ & $0.3$ V & \\\\ \\hline\nLength of cell, $l$ & $60$ nm & \\\\ \\hline\nWidth of cell, $w$ & $60$ nm & \\\\ \\hline\nThickness of cell, $t$ & $10$ nm & \\\\ \\hline\nTemperature, $T$ & $292$ K & \\\\ \\hline\n$\\tau_\\mathrm{creep}$ (@ $\\mathcal{F} = 0$) & $\\sim1$ ms & \\\\ \\hline\n$\\tau_\\mathrm{flow}$ (@ $\\mathcal{F} = 74.2$ J\/m$^3$) & $\\sim0.22$ ns & \\\\ \\hline\n\\end{tabular}\n\\caption{Simulation parameters considered for the ME-AFMRAM cell.}\n\\label{tab:params}\n\\end{table}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.6]{figures\/AFMRAM_organization.pdf}\n\\caption{64KB ME-AFMRAM organization with 4$\\times$1 banks, 2$\\times$1 mats, 4$\\times$2 sub-arrays, and 128$\\times$64 bit cell arrays. Here, the word length is 128 bit. The memory organization is leveraged from~\\cite{dong2012nvsim}.}\n\\label{fig:MERAM_organization}\n\\end{figure*}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/AFMRAM_array.pdf}\n\\caption{Construction of the ME-AFMRAM cell array used in the memory architecture.\nThe signals BL$_{\\text{i,in}}$ serve to write data into the cells when Write Enable (WE) is on, and signals BL$_{\\text{i,out}}$ serve to read data from the cells when Read Enable (RE) is on.}\n\\label{fig:MERAM_array}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.43]{figures\/comparison_table.pdf}\n\\captionof{table}{Performance comparison of various memory technologies, from~\\cite{yang2013memristive, micron_nandflash, everspin_feram, chang2016resistance, kent2015new}. \nThe write and read latencies for ME-AFMRAM (DW model) are quoted for a 64KB memory with a 128-bit word line, simulated using NVSim~\\cite{dong2012nvsim}. \nThe energy-per-bit metric is for a single bit write onto a cell.}\n\\label{tab:Comparison_table}\n\\end{figure}\n\n\\subsection{ME-AFMRAM array}\n\nTo evaluate the system-level performance of ME-AFMRAM in the context of existing memory technologies, we simulate a 64KB DW-based ME-AFMRAM chip on NVSim, a standard tool for estimating the performance metrics of emerging NVMs~\\cite{dong2012nvsim}. \nThe organization of this 64KB memory, as leveraged from~\\cite{dong2012nvsim}, is shown in Fig.~\\ref{fig:MERAM_organization}. \nThe internal architecture of the ME-AFMRAM cell array, along with the peripheral decoders, drivers and sense amplifiers, constructed at the 15-nm CMOS node, is highlighted in Fig.~\\ref{fig:MERAM_array}. \nThe total write latency of the 64KB ME-AFMRAM, including the parasitics and peripheral latency (133.9 ps) and the dominant cell switching time ($\\sim$630 ps), is obtained as 763.9 ps from NVSim~\\cite{dong2012nvsim}. \nThe write latency can be improved by an order of magnitude via coherent rotation of the order parameter. \nThe total read latency of the chip, obtained from NVSim~\\cite{dong2012nvsim}, is $\\sim$2.3 ns. \nThis includes contributions from the sense amplifier (1.45 ns), bit-line parasitics (3.5 ps), decoders and other peripherals ($\\sim$150 ps), and the dominant AH measurement delay in the Bi$_2$Se$_3$ layer ($\\sim$0.7 ns)~\\cite{kikuchi2016anomalous}. \nState-of-the-art pulsed AH measurement schemes like~\\cite{kikuchi2016anomalous} are capable of operating in the GHz regime.\n\nThe output bit-line sensing can be achieved using a conventional current latch amplifier if a large-SOC material such as a TI is used to generate an AH signal in the range of tens of mV.\nThe read\/write endurance of the ME-AFMRAM is expected to be similar to that of STT-MRAM. A comparison of the performance metrics of the ME-AFMRAM with other memory technologies at the chip-level is presented in Table~\\ref{tab:Comparison_table}. It can be seen that the ME-AFMRAM offers some competitive advantages over other NVMs as well as over conventional memory systems.\n\n\n\\section{Application as Secure Memory}\n\\label{sec:security}\n\nAfter conducting cell- and array-level modeling and benchmarking of the chromia-based ME-AFMRAM, we continue with the implementation of the proposed SMART memory using the ME-AFMRAM.\n\n\\subsection{Threat model}\n\nFirst, we discuss the threat model, defining the strengths and capabilities of attackers, as well as the objectives and consequences of a successful attack. Most but not all attack scenarios presented here are\nspecific to NVMs.\n\n\\begin{itemize}\n\n\\item Attackers can launch cold-boot attacks~\\cite{halderman2009lest}.\nDuring power-down, there is some latency after the power-down sequence initiates until the moment when memory contents are completely secured. An attacker might use this gap to read out memory contents. To circumvent such attacks, memory encryption is typically employed~\\cite{chhabra2011nvmm,swami2018acme}.\n\n\\item Attackers could leverage properties like sensitivity to magnetic fields and temperature fluctuations to corrupt the data or induce a DoS~\\cite{jang2015self}. \nThey may forcibly write specific data patterns \nto memory, which accelerates aging and \ncauses memory failures.\n\n\\item With access to failure analysis equipment, attackers can also resort to advanced invasive attacks. \nThe majority of such attacks\ntarget at the back-end-of-line (BEOL), approaching from the top-most metal layer, which is also referred to as front-side attacks. \nVarious countermeasures have been proposed to protect the front-side, which include protective meshes, shields, and sensors~\\cite{lee19_shield,weiner18}.\nIn any case, \\textit{bus snooping} attacks are considered beyond the scope of this work.\n\n\\item Power-dissipation signatures when reading\/writing `0' and `1' within the NVM can be exploited for side-channel attacks to infer the data, through techniques like differential power analysis (DPA)~\\cite{kocher1999differential} and correlation power analysis (CPA)~\\cite{brier2004correlation}.\n\n\\end{itemize}\n\n\\subsection{Magnetic field and temperature attacks}\n\\label{sec:Mag}\n\nSTT-MRAMs have FM-based MTJs as their basic building blocks. FMs possess a macroscopic magnetization (or magnetic signature) that can be probed or inferred with using an external magnetic field. \nHence, magnetic fields can be used to infer or tamper with the stored data or even cause malfunctions in STT-MRAMs~\\cite{jang2015self}.\nStray magnetic fields as small as 10 mT could cause an unintended bit flip in STT-MRAM cells. Figure~\\ref{fig:STTMRAM} shows the magnetic field-induced bit flip in a representative FM, obtained by solving the Landau-Lifshitz-Gilbert equation for the FM dynamics~\\cite{ament2016solving}.\n\n\\begin{figure}[ht]\n\\centering\n\\subfigure[Trajectory for magnetic field-induced switching of a FM.]{%\n\\label{fig:STTMRAM_traj}%\n\\includegraphics[scale=0.18]{figures\/STTMRAM_traj.pdf}}%\n\\hspace{1ex}\n\\subfigure[Components for magnetic field-induced switching of a FM.]{%\n\\label{fig:STTMRAM_switching}%\n\\includegraphics[scale=0.23]{figures\/STTMRAM_switching.pdf}}%\n\\caption{The FMs in an STT-MRAM can be switched easily using external magnetic fields.}\n\\label{fig:STTMRAM}\n\\end{figure}\n\nAFMs, on the other hand, exhibit no external magnetic signature since their equal and opposite sublattice moments cancel each other out. \nHence, the bulk order parameter cannot be affected by external magnetic fields. \nTo switch the bulk order, staggered fields (opposite sign on opposite sublattices) must be applied on both the sublattice moments, as illustrated in Fig.~\\ref{fig:AFMRAM_field} inset. \nHowever, an external, homogeneous magnetic field is unable to provide such a staggered field arrangement, and hence, ends up canting the sublattice moments in a way wherein the torque due to the external field is exactly balanced by the exchange torque exerted by one sublattice moment on the other~\\cite{baltz2018antiferromagnetic}. \nSince external magnetic fields are unable to reorient the AFM order parameter, the SMART ME-AFMRAM is expected to be resistant to magnetic field attacks described in~\\cite{jang2015self}. \nWe note that switching the ME-AFM surface magnetization state using a combination of $E$ and $H$ fields would require an exact knowledge of the write cycles and the prior state of the surface, as well as\nmeans to control the electric field explicitly, which is to be concealed from an attacker. \n\nWith regards to temperature fluctuation-based attacks, an adversary might increase the ambient temperature of the ME-AFMRAM in an attempt to alter the stored data. \nNote that the N\\'{e}el temperature of pure chromia is 308 K~\\cite{shi2009magnetism}, above which the AFM ordering is destroyed. Hence, the attacker may corrupt the memory by heating it above the N\\'{e}el temperature. \nTo counter this, we consider Boron-doped chromia, whose N\\'{e}el temperature is demonstrated experimentally to be $\\sim400$ K~\\cite{street2014increasing}. \nHence, Boron-doped chromia can increase the resilience of SMART memory against temperature fluctuations. \nThat is because such larger temperature fluctuations (above 400 K) are easier to detect, and countermeasures like interception of such attacks become more feasible.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.3]{figures\/AFM_field_combined.pdf}\n\\caption{The application of a magnetic field is unable to switch the AFM order parameter, even when increasing the field magnitude. \nInset: (a) an external, homogeneous magnetic field may cant the sublattice moments, but it is incapable of rotating the AFM order; \n(b) staggering fields on the sublattice moments produce staggered tangential torques, which can reorient the AFM order.}\n\\label{fig:AFMRAM_field}\n\\end{figure}\n\n\\subsection{Data confidentiality attacks}\\label{Encrypt}\nAs with all NVMs, data remanence in the SMART memory could be exploited by attackers to steal sensitive information. The most effective countermeasure against such data confidentiality attacks, including cold-boot and stolen memory-modules attacks, is to encrypt the data using a secure encryption scheme before storing it in the memory. Advanced memory encryption techniques like counter mode encryption (CME) use block ciphers such as Advanced Encryption Standard (AES) to encrypt a seed using a secret key, in order to generate a one-time pad (OTP). \nThe seed for each write on a memory line consists of a secret key, the line address, and a counter value associated with that line, which is incremented with each subsequent write to the same line. Hence, the generated OTP is unique for each line address, and also for each write operation to the same address. \nThe OTP is then XOR-ed with the plaintext to obtain the ciphertext, which is stored in the non-volatile main memory.\nNote that the secret key used in the AES core is considered inaccessible to the attacker.\n\nDirectly applying XOR-based CME scheme to the SMART memory would result in large encryption overheads. This is because the CME scheme is tailored for NVMs like PCM and STT-MRAM, whose write time is on the order of $\\sim$ns. The access latency of ME-AFMRAM is sub-ns for DW-based propagation and few 10's of ps for coherent rotation. A general encryption scheme for SMART memory, switching either via DW propagation or coherent rotation, must be such that the overall memory access latency is not adversely affected. Existing encryption solutions based on CMOS XOR gates with 10's of ps delay are rendered ineffective as their encryption time is comparable to the memory write time, resulting in idle clock cycles.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.3]{figures\/Memcryption_scheme.pdf}\n\\caption{(a) CME uses AES to generate an OTP, using the memory line address, a counter, and a secret key. The encryption and decryption is performed outside the non-volatile main memory (NVMM).\n(b) \\textit{Memcryption} uses a secret key and the line address as seed for AES, to generate an encryption pulse. \nThat pulse is used to control the bitwise operation of CNOT gates, and is embedded in the data path within the NVMM.}\n\\label{fig:Memcryption_scheme}\n\\end{figure}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.42]{figures\/Memcryption.pdf}\n\\caption{SMART memory architecture with \\textit{Memcryption}.\nThe CNOT layer for decryption is not shown for simplicity.}\n\\label{fig:Memcryption}\n\\end{figure*}\n\nHere, we propose to use in-memory encryption, or \\textit{Memcryption}, using bitwise CNOT (i.e., controlled-NOT) gates constructed from ME-AFM-based logic. \nBy tying the encryption pulse to the control signals of CNOT gates, one can achieve such \\textit{Memcryption}.\nSpin devices like the ME-AFM transistor~\\cite{dowben2018towards} are able to implement polymorphic logic gates, which can provide inverting or non-inverting functionality based on a control signal~\\cite{patnaik2018advancing,patnaik2019spin}. \nHence, the ME-AFM transistor is used to realize the CNOT gate. Further, the ME-AFM transistor is shown to exhibit delays as small as $\\sim10$ ps, which is substantially faster than CMOS XOR gates and compatible with the SMART memory write-times.\nSuch homogeneity in the technology and materials by using only ME-AFM for both the memory cells and the CNOT gates will ease the fabrication. \nIn \\textit{Memcryption}, we embed ME-AFM transistor-based CNOT gates directly in the data path connected to the memory array; hence, the encryption is in-memory, as opposed to prior works using a separate encryption block. \nThis integration of encryption and memory array is not detrimental to the memory density since ME-AFM transistors have a footprint that is substantially smaller than that of CMOS XOR gates.\nFigure~\\ref{fig:Memcryption_scheme} contrasts our \\textit{Memcryption} scheme with prior CME techniques. \n\nThe SMART memory architecture with \\textit{Memcryption} is shown in Fig.~\\ref{fig:Memcryption}.\nA trusted 128-bit key, provided and stored within a secure processing module (SPM) along with the processor, is concatenated with the memory address and used as seed for AES.\nThe AES core, which is to be integrated on the NVM chip,\\footnote{Heterogeneous spin-CMOS integration is not prohibitive since the underlying AFM technology is compatible with \nCMOS processes in the BEOL. \nIn general, hybrid spin-CMOS designs have been explored in prior works~\\cite{yogendra2015domain}.} thus produces an encryption pulse whose bits are used as the control bits for the CNOT gates of the in-memory encryption layer.\nDepending on the control bits, the encryption layer flips bits selectively in the plaintext before performing a memory-write. \nDuring decryption, the same encryption pulse is generated again and used to perform bitwise CNOT operations on the ciphertext (read from memory), to obtain the plaintext.\n\nA comparison of the \\textit{Memcryption} scheme versus CME (when also applied to ME-AFMRAM) is presented in Table~\\ref{overhead_comparison}. \nThe array considered is a 128-bit ME-AFMRAM, while the AES and CMOS peripherals are synthesized using the 15nm \\textit{NanGate} technology.\nWe observe that Memcryption with SMART memory \nhas a better encryption latency than CME, which utilizes regular CMOS XORs.\nWe also note that \\textit{Memcryption} helps reduce the encryption latency but is similar to CME with respect to the energy overheads. \nThat is because energy dissipation is dominated by the AES core in any case.\nWe also reiterate that \\textit{Memcryption} is tailored specifically as a memory-side scheme \nfor ME-AFMRAM, to achieve low encryption latency, owing to the homogeneous delays of the memory array and the encryption layer. \nHowever, it may not serve well as an efficient implementation for any generic NVM.\n\nWith regards to the reliability and lifetime of the ME-AFMRAM used to construct the SMART memory, its endurance is comparable to that of STT-MRAM. \nHowever, it also suffers from the same errors that plague the STT-MRAM, i.e., faults in the peripheral CMOS circuitry including the access transistors~\\cite{chintaluri2016analysis}. \nTo address these faults and ensure the correctness of the stored data, standard error correction techniques for NVMs~\\cite{swami2017reliable} like the error correction pointer (ECP) and other advanced schemes based on ECP, including ``Pay-As-You-Go''~\\cite{qureshi2011pay} and ``Zombie memory''~\\cite{azevedo2013zombie}, can be implemented memory-side and integrated on the ME-AFMRAM array. \nThe ECP memory can be realized using homogeneous spintronics technology, including the STT-MRAM or the ME-AFMRAM itself, or by leveraging heterogeneous spin-CMOS integration.\n\n\\begin{table}[ht]\n\\centering\n\\footnotesize\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{*{3}{c}}\n\\hline\n\\textbf{Encryption technique}\n& \\textbf{Latency} & \\textbf{Energy} \\\\\n\\hline\nCME~\\cite{chhabra2009making} &  299.23 ps (2.99$\\times$) & 17.371 pJ \\\\ \\hline\nMemcryption &  273.46 ps (2.73$\\times$) & 17.370 pJ \\\\ \\hline\n\\end{tabular}\n\\caption{Comparison for latency and energy when applying the CME and \\textit{Memcryption} schemes to a 128-bit ME-AFMRAM array. The baseline latency for the unencrypted array is $\\sim 100$ ps.}\n\\label{overhead_comparison}\n\\end{table}\n\n\\subsection{Power side-channel attacks}\n\\label{Power}\n\nAsymmetric read\/write characteristics in NVMs like STT-MRAM make them susceptible to side-channel attacks which exploit the different signatures incurred when reading\/writing `1's \nand `0's bits. STT-MRAMs employ MTJs with a fixed FM reference layer, \nwith another free layer either oriented parallel or anti-parallel to that reference layer. Depending on the relative orientation of these two layers,\nthe MTJ falls into a low or high resistance state; the low or high state corresponds to logic `0' or logic `1' state, respectively. Hence, the currents drawn for read\/write operations are different depending on reading\/writing a `0' or a `1'. \nThus, an attacker could attach a resistor in a voltage-divider configuration with\nthe MTJ cell, monitor the voltage drops across that resistor, and perform DPA to recover the data being written to or read from the cell. In fact, such an attack was showcased against an STT-MRAM-based cache in~\\cite{khan2017side}.\n\nFor the SMART memory, recall that writing is achieved using electrical fields, not currents. Further, the electric-field magnitude required for writing `0's and `1's is equivalent; see write voltage and polarization voltage traces in Fig.~\\ref{fig:AFMRAM_timing}. This is because there is no reference layer or tunneling magnetoresistance in the ME-AFMRAM, which would cause asymmetricity. As for the read operation, the proximity effect-induced moment in the Pt electrode is slightly different for reading `0' or `1'. \nHowever, this imbalance in the Hall signals can be compensated for by introducing appropriate offsets in the Hall measurement setup, as demonstrated in~\\cite{kosub2017purely}. Hence, the SMART memory can achieve symmetric signatures for both read and write and for both `0$\\rightarrow$1' and `1$\\rightarrow$0' \ntransitions, thus thwarting any DPA-based power side-channel attacks.\n\n\\subsection{Photonic side-channel and \nbackside attacks}\n\\label{Photonic}\n\nLeveraging the photonic side-channel (PSC) to circumvent the security guarantees provided by cryptographic algorithms like AES and RSA has been demonstrated recently~\\cite{ferrigno2008aes,schlosser2013simple}. Simple Photonic Emission Analysis (SPEA) or Differential Photonic Emission Analysis (DPEA) can be carried out using photo-emission equipment available for similar cost as that of power-analysis equipment. The essence of the PSC is to observe photo-emissions emanating for switching of CMOS transistors.\nFor SRAM- or DRAM-based memories, this emission can then be correlated with the data being programmed into the memory. In~\\cite{ferrigno2008aes}, the PSC was found to originate when kinetic energy gained by charge carriers in the transistor channel is transferred to photons, which are visible through photo-detectors. In~\\cite{schlosser2013simple}, the authors leveraged this \ninformation to perform a side-channel attack, ultimately recovering the full AES key. Modern-day chips use several metal layers, which interfere with the emission of photons from the frontside of any integrated circuit (IC); therefore, a natural direction is to observe the photon emission from the backside of ICs. \n\nWhile CMOS-based memory technologies like SRAM and DRAM are prone to such PSC attacks, the SMART memory is AFM-based and involves no photonic emissions emanating from transistor channels. Data read-out in the SMART memory can only be accomplished through an AH measurement setup. Further, even if an advanced attacker is able to isolate the SMART memory cell and gain access to the AH setup from the frontside, they would only be able to recover the encrypted ciphertext (as described in Sec.~\\ref{Encrypt}).\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we present \\textit{SMART: A Secure Magnetoelectric Antiferromagnet $\\!$-Based Tamper-Proof Non-Volatile Memory}, by utilizing the unique properties of ME-AFMs. \nThe ME-AFMRAM, which is at the core of the SMART memory, has an access latency of sub-1 ns (for DW-based switching) down to only 10's of ps (for coherent rotation switching) with an energy-per-bit of $\\sim$ 0.13 pJ. \nBesides its superior performance as compared to prior NVMs like STT-MRAM and PCM, the SMART memory exhibits no sensitivity to external magnetic fields, which makes it resilient to magnetic field-based data tampering and denial of memory service attacks that commonly plague other ferromagnets-based NVMs. \nTo solve the security vulnerability of data remanence (after power-down) in the SMART memory, we demonstrate a new encryption technique called \\textit{Memcryption}. \nThis scheme employs emerging ME-AFM-based logic to implement a CNOT-centric in-memory encryption, which is particularly tailored to reduce the encryption and decryption latency in the SMART memory. \nFurther, symmetric read and write signatures for `0' and `1' bits render prominent side-channel attacks like the differential power attack futile against the SMART memory. \nAdvanced photonic side-channel attacks, which are powerful threats against any CMOS IC by observing all internal transistor activity from the frontside or backside, are ineffective against the SMART memory due to the fundamentally different switching mechanism as well as the proposed \\textit{Memcryption} safeguard.\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nLet $C$ be a smooth complex curve of genus $g>1$,\ndenote by $C^2$ the cartesian product of\n$C$ with itself, by $C^{(2)}$ the \nsecond symmetric product of $C$  \nand by $\\pi\\colon C^2\\to C^{(2)}$ the\ndouble cover $(p,q)\\mapsto p+q$.\nLet $\\Delta$ be the image\nof the diagonal via $\\pi$ and  $K$ be\na canonical divisor of $C^{(2)}$.\nIn this paper we are interested\nin the finite generation of the {\\em extended\ncanonical ring}:\n\\[\n R(\\Delta,K)\n \\, :=\\,\n \\bigoplus_{(a,b)\\in{\\mathbb Z}^2}H^0(C^{(2)},a\\Delta+bK).\n\\]\nIt is not difficult to show that finite generation\nis equivalent to ask that the two-dimensional \ncone $\\Nef(\\Delta,K)$ consisting of classes \nof nef divisors within the rational vector space \nspanned by the classes of $\\Delta$ and $K$ \nmust be generated by two semiample classes\n(see proof of Theorem~\\ref{teo-1}).\nThe Kouvidakis conjecture~\\cite{CK} states that,\nfor the very general curve, if $\\Nef(\\Delta,K)$ is \nclosed then $g$ must be a square, \ni.e. $g = (k-1)^2$ for $k>1$ integer. \nWe thus focus our attention to\nthe case of square genus and assume $C$\nto be a complete intersection of a \nquadric $Q$ with a degree $k>2$ hypersurface.\nWe denote by $\\eta_C$ the class of the\ndifference of the two $g_k^1$ of $C$\ninduced by the two rulings of $Q$, which\nis trivial if $Q$ is a cone.\nOur first result is the following.\n\\begin{introthm}\n\\label{teo-1}\nLet $C\\subseteq{\\mathbb P}^3$ be a smooth complete \nintersection of a quadric with a degree $k>2$\nsurface. Then the following are equivalent.\n\\begin{enumerate}\n\\item\n$R(\\Delta,K)$ is finitely generated.\n\\item\n$\\eta_C$ is torsion non-trivial.\n\\end{enumerate}\nMoreover if both the above conditions are satisfied\nthen $\\eta_C$ has order at least $k$.\n\\end{introthm}\nLet $\\mathcal F_k$ be the  open subset \nof the Hilbert scheme of curves of bi-degree $(k,k)$ \nof ${\\mathbb P}^1\\times{\\mathbb P}^1$ consisting of smooth curves\nand let $\\mathcal F_k^{\\rm tor}\\subseteq\\mathcal F_k$\nbe the subset consisting of curves $C$ such that\nthe class $\\eta_C$ of the difference between the\ntwo $g_k^1$ is torsion.\nOur second theorem is the following.\n\\begin{introthm}\n\\label{teo-2}\nThe locus $\\mathcal F_k^{\\rm tor}$ is \na countable union of subvarieties of complex dimension \n$\\geq 4k-1$ and the set of subvarieties of\ndimension $4k-1$ is dense in $\\mathcal F_k$\nin the analytic topology.\n\\end{introthm}\nThe paper is organized as follows.\nIn Section~\\ref{sym}, after recalling some basic facts\nabout the symmetric product of a curve,\nwe prove Theorem~\\ref{teo-1}. In Section~\\ref{grid-fam}\nwe introduce the grid family consisting of \ncurves of bi-degree $(k,k)$ on a smooth quadric\nwhich pass through a complete intersection\nof type $(k,0),(0,k)$. We show that  the\ngrid family is exactly the subvariety  \nof  $\\mathcal F_k^{\\rm tor}$ corresponding to \ntorsion of order $k$ and \nhas the expected dimension $4k-1$.\nSection~\\ref{density} is devoted to the proof of Theorem~\n\\ref{teo-2}. In Section~\\ref{hyp} we prove a density\ntheorem for hyperelliptic curves, providing\nan alternative proof for Theorem~\\ref{teo-2}\nin case $g=4$ (see Corollary~\\ref{cor}). \nThis result has an independent \ninterest and is proved in the spirit of Griffiths \ncomputations of the infinitesimal invariant~\\cite{Gr}.\nFinally, in Section~\\ref{exa} \nwe consider examples\nof curves $C$ with $\\eta_C$ torsion.\n\nIn all the paper we work over the field of complex \nnumbers except for Section~\\ref{sym} (see Remark~\\ref{ch}).\n\n\n\\section{The second symmetric product}\n\\label{sym}\nLet $C$ be a smooth projective \ncurve of genus $g>1$ defined over an algebraically \nclosed field ${\\mathbb K}$ of characteristic $0$.\n\\begin{proposition}\\label{pic}\nThe diagonal embedding \n$\\imath\\colon C\\to C^{(2)}$\ninduces an isomorphism \n$\\imath^*\\colon \\Pic^0(C^{(2)})\\to\n\\Pic^0(C)$ of abelian varieties.\n\\end{proposition}\n\\begin{proof}\nTo prove the statement we explicitly construct\nthe inverse map of $\\imath^*$. Given a point\n$p\\in C$ let $H_p$ be the curve of $C^{(2)}$\nwhich is the image of $\\{p\\}\\times C$ via $\\pi$.\nDefine the map ${\\rm Div}(C)\\to {\\rm Div}(C^{(2)})$ by\n$\\sum_in_ip_i\\mapsto\\sum_i n_iH_{p_i}$ and\nobserve that it maps principal divisors to principal \ndivisors. The induced map of Picard groups  \nrestricts to a homomorphism $\\Pic^0(C)\\to\n\\Pic^0(C^{(2)})$ which is easily seen to be a \nright inverse of $\\imath^*$.\nSince the two abelian varieties\n$\\Pic^0(C)$ and $\\Pic^0(C^{(2)})$ have \nthe same dimension we conclude that\n$\\imath^*$ is an isomorphism.\n\\end{proof}\nObserve that $\\Delta$ is the branch divisor\nof the double cover $\\pi$ and thus \nits class is divisible by $2$ in $\\Pic(C^{(2)})$.\nMoreover the following linear equivalences\n\\[\n \\Delta|_\\Delta\n \\sim\n -2K_\\Delta\n \\qquad\n \\qquad\n K|_\\Delta\n \\sim\n 3K_\\Delta\n\\]\ncan be proved by passing to $C^2$ \nand calculating the restriction of $K_{C^2}$\nto the diagonal.\nBy the Riemann-Hurwitz formula we get the equalities\n$2(2g-2)^2 = K_{C^2}^2 = 2 (K + \\frac{\\Delta}{2})^2$ \nfrom which we deduce the following \n\\begin{equation}\n \\label{intersections}\n K^2 = (g-1)(4g-9)\n \\qquad\n K\\cdot\\Delta = 6(g-1)\n \\qquad\n \\Delta^2 = -4(g-1).\n\\end{equation}\nIn particular the classes of $\\Delta$ and $K$ are \nindependent in the N\\'eron-Severi group of ${C^{(2)}}$.\nWe let $\\langle \\Delta, K\\rangle$ be the rational \nvector subspace of $\\Pic({C^{(2)}})\\otimes_{\\mathbb Z}{\\mathbb Q}$\ngenerated by the classes of $\\Delta$ and $K$\nand form the following cone\n\\[\n {\\rm Nef}(\\Delta,K)\n =\n \\{D\\in\\langle \\Delta, K\\rangle\\, :\\, D\\text{ is nef}\\}.\n\\]\nThis cone is related to the Kouvidakis conjecture\nwhich predicts which ones are the extremal rays of \n${\\rm Nef}(\\Delta,K)$. In case the genus is a square,\ni.e. $g = (k-1)^2$, the conjecture is known to be\ntrue~\\cite{CK} for a very general curve $C$ and it \nholds as well if $C$ has an irreducible $g_k^1$,\nthat is the curve~\\eqref{eq:gamma} defined below\nis irreducible.\nIn this case the extremal rays of ${\\rm Nef}(\\Delta,K)$ \nare spanned by the classes of $2K+3\\Delta$ and \n$2K+(5-2k)\\Delta$.\n\n\\begin{proposition}\n\\label{ray-1}\nLet $C$ be a smooth curve of genus at least\ntwo. Then the divisor \n$2K+3\\Delta$ of $C^{(2)}$ is semiample.\n\\end{proposition}\n\\begin{proof}\nObserve that the divisor $2K+3\\Delta$\nis big since $K$ is ample and $\\Delta$\nis effective. Moreover it is nef since \n$(2K+3\\Delta)\\cdot\\Delta = 0$.\nWe have an exact sequence of sheaves\n\\[\n \\xymatrix@1{\n  0\\ar[r]\n  &\n  {\\mathcal O}_{C^{(2)}}(2K+2\\Delta)\\ar[r]\n  &\n  {\\mathcal O}_{C^{(2)}}(2K+3\\Delta)\\ar[r]\n  &\n  {\\mathcal O}_\\Delta\\ar[r]\n  &\n  0.\n }\n\\]\nSince $2K+2\\Delta = N + K+\\frac{1}{2}\\Delta$,\nwith $N=\\frac{1}{2}(2K+3\\Delta)$ nef and big,\nthen by the Kawamata-Viehweg vanishing\ntheorem and the long exact sequence in \ncohomology of the above sequence we \nconclude that $\\Delta$ is not contained \nin the base locus of $|2K+3\\Delta|$.\nThe statement follows by the ampleness \nof $K$ and the Zariski-Fujita theorem\n~\\cite[Remark 2.1.32]{La}.\n\\end{proof}\n\n\nAssume now that $C$ is a smooth curve\nof genus $g = (k-1)^2>1$ which admits a \n$g_k^1$ and define the following curve \nof $C^{(2)}$:\n\\begin{equation}\n\\label{eq:gamma}\n \\Gamma \\, :=\\, \\{p+q : g_k^1-p-q\\geq 0\\}.\n\\end{equation}\nIt can be easily proved that $\\Gamma$ is irreducible \nif the $g_k^1$ does not contain a $g_r^1$ with $r<k$.\nIn particular this holds if the $g_k^1$ is simple.\nObserve that the $g_k^1$ defines a morphism\n$\\Gamma\\to{\\mathbb P}^1$ of degree $\\frac{1}{2}k(k-1)$\nwhose branch points are exactly those\nof the $g_k^1$. Thus if the $g_k^1$ is simple\nwe deduce that\n\\begin{equation}\n\\label{genus}\n 2g(\\Gamma)-2 = -k(k-1)+2(k-2)(k-1)k,\n\\end{equation}\nwhere $g(\\Gamma)$ is the genus of $\\Gamma$.\nBy continuity the genus of $\\Gamma$ stays the \nsame even if the $g_k^1$ is not simple.\n\n\\begin{lemma}\\label{gamma}\nIf $C$ admits a $g_k^1$ then the divisor\n$2K+(5-2k)\\Delta$ is numerically equivalent to \n$(4k-8)\\Gamma$.\nMoreover if $C\\in\\mathcal F_k$, the curves  \n$\\Gamma$ and $\\Gamma'$ corresponding to the\ntwo distinct $g_k^1$ on $C$ are disjoint.\n\\end{lemma}\n\\begin{proof}\nLet $H$ be the curve of $C^{(2)}$ defined by\n$\\{p+q : q\\in C\\}$. By the Riemann-Hurwitz formula one\nhas the numerical equivalence\n\\[\n K\n \\equiv\n (2g-2)H-\\frac{1}{2}\\Delta.\n\\]\nThus by the genus formula $2g(\\Gamma)-2\n= \\Gamma^2+(2g-2)\\Gamma\\cdot H-\\frac{1}{2}\n\\Delta\\cdot\\Gamma$, by Equation~\\eqref{genus}\nand the equalities $\\Gamma\\cdot H = k-1$,\n$\\Delta\\cdot\\Gamma = 2g-2+2k$ we deduce\n$\\Gamma^2=0$. Since the intersection matrix\nof the divisors $\\Gamma,K,\\Delta$\n\\[\n \\left(\n  \\begin{array}{rrr}\n   0 & (2k-5)(k-1)k & 2(k-1)k\\\\\n   (2k-5)(k-1)k & (2k-5)(k-2)k(2k+1) & 6(k-2)k\\\\\n   2(k-1)k & 6(k-2)k & -4(k-2)k\n  \\end{array}\n \\right)\n\\]\nhas rank two, by the Hodge Index theorem $\\Gamma$ is\nnumerically equivalent to a rational linear\ncombination of $K$ and $\\Delta$.\nMoreover, being the kernel of the above matrix\ngenerated by the vector $(4k - 8, -2, 2k - 5)$, we\nget the first statement. \nObserve that  $\\Gamma$ and $\\Gamma'$  \nhave no common component since \notherwise the map from $C$ to the quadric \ngiven by the two $g_k^1$ \nwould not be an embedding.\nThus the second statement \nimmediately follows from the equality\n$\\Gamma\\cdot\\Gamma' = \\Gamma^2=0$.\n\\end{proof}\n\n \n \n\\begin{lemma}\\label{fib}\nIf $f\\colon C^{(2)}\\to Y$ is a fibration with connected \nfibers onto a smooth curve, then $Y$ has genus \nat most one. If moreover a fiber of $f$ is numerically \nequivalent to a rational linear combination of \n$\\Delta$ and $K$ then $Y$ is rational.\n\\end{lemma}\n\\begin{proof}\nTo prove the first statement observe that if\n$Y$ had genus $\\geq 2$, then there would\nbe two linearly independent holomorphic \n$1$-forms $w_1,w_2$ of $C^{(2)}$,   \nobtained by pull-back of $1$-forms of $Y$, \nsuch that $w_1\\wedge w_2 = 0$. On the other hand \nwe have a commutative diagram of isomorphisms\n\\[\n \\xymatrix{\n  {\\bigwedge^2 H^0(C^{(2)},\\Omega^1)}\n  \\ar[rr]^-{w_1\\wedge w_2\\mapsto w_1\\wedge w_2}\n  \\ar[rd]_\\alpha &&\n  H^0(C^{(2)},\\Omega^2)\\\\\n  & {\\bigwedge^2 H^0(C,\\Omega^1)}\\ar[ru]_\\beta \n }\n\\]\nwhere $\\alpha$ is induced by pull-back via the \ndiagonal embedding $C\\to C^{(2)}$ and $\\beta$\nis defined by $w_1\\wedge w_2\\mapsto \n\\frac{1}{4}(\\pi_1^*w_1\\wedge\\pi_2^*w_2+\n\\pi_2^*w_1\\wedge\\pi_1^*w_2)$, with $\\pi_i:C^2\\to C$ \nthe projections onto the two factors. \nThus such $1$-forms cannot exist and $Y$\nmust have genus at most one.\n\nTo prove the second statement, assume\nthat a fiber $F$ of $f$ is numerically equivalent \nto a rational linear combination of \n$\\Delta$ and $K$.\nIf $Y$ has genus one then we have the \nfollowing commutative diagram\n\\[\n \\xymatrix{\n C\\ar[r]^-\\imath\\ar[d]^-{\\alb_C} & C^{(2)}\\ar[r]^-f\\ar[d]^-{\\alb_{C^{(2)}}} & Y\\ar[d]^-{\\alb_Y}_-\\cong\\\\\n \\Alb(C)\\ar[r]^-{\\alb_\\imath}_-\\cong & \\Alb(C^{(2)})\\ar[r]^-{\\alb_f} & \\Alb(Y)\\\\ \n }\n\\]\nwhere $\\alb_\\imath$ and $\\alb_f$\nare induced by the universal property of \nthe Albanese morphism and $\\alb_\\imath$\nis an isomorphism by Proposition~\\ref{pic}.\nLet $\\Theta$ be the theta divisor of $\\Alb(C^{(2)})$.\nRecall that both $\\alb_{C^{(2)}}(\\Delta)$ and \n$\\alb_{C^{(2)}}(K)$ are numerically equivalent to\na multiple of $\\Theta^{g-1}$.\nBy our assumption on $F$ we deduce that\n$\\alb_{C^{(2)}}(F)\\equiv c\\,\\Theta^{g-1}$. Moreover\n$c$ is non-zero since $F$ cannot be \ncontracted by $\\alb_{C^{(2)}}$.\nBut this gives a contradiction, since \n$\\alb_f\\circ\\alb_{C^{(2)}}$ contracts $F$, while\n$\\alb_f(\\Theta^{g-1})$ cannot be a point, being the \nimage of $\\alpha_Y\\circ f\\circ\\imath$.\n\\end{proof}\n\nFor a proof of the following lemma see also\n~\\cite{BHPV}*{Lemma III.8.3}.\n\n\\begin{lemma}\n\\label{covering}\nLet $f\\colon S\\to\\mathbb {\\mathbb C}$ be a proper \nmorphism and let $nF$ be a multiple fiber \nof $f$ with multiplicity $n>1$. \nThen ${\\mathcal O}_F(F)$ is torsion non-trivial.\n\\end{lemma}\n\n\\begin{proof}\nWe first show that $\\mathcal L={\\mathcal O}_S(F)$ is not trivial.\nConsider the closure of the graph of $f$ \nin $S\\times \\mathbb P^1$, let \n$\\bar S$ be its minimal resolution and \n$\\bar f:\\bar S\\to \\mathbb P^1$ \nbe the fibration given by the projection \nto the second factor. \nAssume by contradiction that $F\\sim 0$ \nin $S$. Thus $F\\sim \\alpha F'$ in $\\bar S$,\nwhere $F'$ is a divisor with support contained \nin $\\bar S-S=\\bar f^{-1}(\\infty)$ and $\\alpha$ \nis a positive integer. \nSince $h^0(\\bar S, nF)=2$ with $n>1$, then  \n$h^0(\\bar S, F)=1$, giving a contradiction.\n\n\nThe line bundle $\\mathcal L$ thus defines a\nnon-trivial \\'etale cyclic covering \n$\\eta\\colon S'\\to S$.\nBy taking the Stein factorization of $f\\circ\\eta$\nwe get a commutative diagram\n\\[\n \\xymatrix{\n  S'\\ar[r]^-\\eta\\ar[d]^-{f'} & S\\ar[d]^-f\\\\\n  B\\ar[r]^-\\nu & {\\mathbb C},\n }\n\\]\nwhere $f'$ is a morphism with connected \nfibers and $\\nu$ is a finite map.\nIf $\\mathcal L|_{F}={\\mathcal O}_F(F)$ is trivial, then \n$\\nu$ is an\n\\'etale covering of $\\mathbb {\\mathbb C}$, \nsince the restriction of $\\mathcal L$ \nto any fiber of $f$ is trivial.\nThus  $\\nu$ is the trivial covering and $B$ \nhas $n$ connected components,\na contradiction since $\\eta$ is non-trivial.\n\nSince $\\mathcal L^{\\otimes n}$ is trivial,\nthen clearly its restriction to $F$ is trivial.\nThis concludes the proof. \n\\end{proof}\n\n\n\\begin{lemma}\\label{triv}\nLet $C$ be a non-hyperelliptic \ncurve of genus $g = (k-1)^2>1$ \nwhich is complete intersection \nof a quadric cone with a degree $k$\nsurface of ${\\mathbb P}^3$. Then the divisor\n$\\Gamma$ of $C^{(2)}$, \ncorresponding to the $g_k^1$ \nof $C$ defined by the ruling of \nthe cone, is not semiample.\n\\end{lemma}\n\\begin{proof}\nWe first show that the line bundle\n${\\mathcal O}_\\Gamma(\\Gamma)$ is trivial.\nIndeed let $Q_t\\subseteq{\\mathbb P}^3\\times\n\\mathbb A^1$ be a family of quadrics\nwhose central fiber $Q_0$ is the cone\ncontaining $C$ and whose general fiber \nis a smooth quadric. Let $\\mathcal D$\nbe a divisor of ${\\mathbb P}^3\\times\\mathbb A^1$\nwhich cuts out on the general fiber $Q_t$\na smooth curve $C_t$ of type $(k,k)$ with two\nsimple $g_k^1$ and $C$ on $Q_0$.\nThe family $\\mathcal C\\to\\mathbb A^1$\nof curves $C_t$ gives a family \n$\\mathcal C^{(2)}\\to\\mathbb A^1$\nwhose general fiber is $C_t^{(2)}$.\nOn any such fiber there are two \ncurves $\\Gamma_t$, $\\Gamma_t'$ \ncorresponding to the two $g_k^1$\non $C_t$. The line bundle \n${\\mathcal O}_{\\Gamma_t}(\\Gamma_t')$ is\ntrivial, by Lemma~\\ref{gamma},\nand its limit is ${\\mathcal O}_\\Gamma(\\Gamma)$,\nwhich proves the claim.\n\nAssume now, by contradiction, that \n$\\Gamma$ is semiample. \nSince $\\Gamma^2=0$,\na multiple $n\\Gamma$ defines a morphism\n$f\\colon C^{(2)}\\to B$, where $B$ is a curve. Moreover,\nafter possibly normalizing, we can assume $B$ \nto be smooth. Now, let $f=\\nu\\circ\\varphi$ be \nthe Stein factorization of $f$, where \n$\\varphi\\colon C^{(2)}\\to Y$ is a morphism with\nconnected fibers. By Lemma~\\ref{covering}\nand the fact that ${\\mathcal O}_\\Gamma(\\Gamma)$ is\ntrivial we deduce that $\\Gamma$ is a union \nof fibers of $\\varphi$. Moreover \nboth the hypotheses of Lemma~\\ref{fib} are satisfied, \nthus $Y$ must be a smooth rational curve.\nLet $H$ be the curve of\n$C^{(2)}$ which is the image of the curve \n$\\{p\\}\\times C$ via $\\pi$.\nThe equality $\\Gamma\\cdot H = k-1$ implies\nthat $\\varphi|_H$ is a covering of $Y$ whose degree \n$d$ divides $k-1$. \nThus $C\\cong H$ would admit two maps to \n${\\mathbb P}^1$ of degrees $k$ and $d$, respectively.\nBeing the degrees coprime, the curve $C$ \nwould be birational to a curve of bi-degree $(k,d)$\nof ${\\mathbb P}^1\\times{\\mathbb P}^1$, whose genus is smaller\nthan $g$, a contradiction.\n\\end{proof}\n\n \\begin{remark}\\label{rem1}\n If $D$ is a prime divisor on a projective \n surface $X$ such that $|D|$\n has dimension $0$ then $D$ is semiample \n if and only if $\\dim |nD| > 0$ for some $n>1$.\n Indeed the ``only if'' part is obvious, while \n the other implication follows from the fact\n that the fixed divisor of $|nD|$ is $mD$ \n for some $m<n$ and thus the base locus of\n $|(n-m)D|$ is at most zero-dimensional and\n one concludes by Zariski-Fujita theorem\n ~\\cite[Remark 2.1.32]{La} \n \\end{remark}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{teo-1}]\nWe show that $(i)\\Rightarrow (ii)$ holds.\nAssume that the extended canonical ring\n$R := R(\\Delta,K)$ is finitely generated.\nWe begin to show that any divisor \nwhose class is in $\\Nef(\\Delta,K)$ \nis semiample. By Proposition~\\ref{ray-1}\nit suffices to show that $2K+(5-2k)\\Delta$\nis semiample, or equivalently that the divisor\n$\\Gamma$ defined by a $g_k^1$ of $C$\nis semiample, by Lemma~\\ref{gamma}.\nLet $f_1,\\dots,f_r$ be a minimal system of \nhomogeneous generators of $R$\nwith respect to the ${\\mathbb Z}^2$-grading\nand let $w_i := \\deg(f_i)\\in{\\mathbb Z}^2$ be\nthe degree of $f_i$ for any $i$.\nLet $f_1\\in R$ be a generator whose \ndegree belongs to the ray generated by \n$[\\Gamma]$ and let $D$ be a\nvery ample divisor of the form $a\\Gamma+b K$, with $a,b>0$\nwhose class $w$ lies in the interior of the cone  \n\n\n\\begin{minipage}{0.5\\textwidth}\n\\[\n \\bigcap_{i=2}^r{\\rm cone}(w_1,w_i)\n \\cap {\\rm cone}(w_1,[K])\n \\]\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n \\begin{tikzpicture}[scale=0.65]\n  \\draw[-,thick] (0,0) -- (2,0) node[below]{$w_r$};\n  \\draw[-,thick] (0,0) -- (-2.2,1.1) node[left] {$w_1$};\n  \\draw[-,color=blue] (0,0) -- (0,1.8); \n  \\node[right] at (-0.6,2.2){$[K]$};\n  \\foreach \\x\/\\y in {-1\/1.5,-1.5\/1.3,-0.5\/1.7} \\draw[-,color=blue] (0,0) to (\\x,\\y);\n  \\foreach \\x\/\\y in {1\/1.5,1.5\/1.3, 1.8\/0.8} \\draw[-,color=blue] (0,0) to (\\x,\\y);\n  \\fill[black] (0,0) circle (2pt);\n  \\fill[black] (2,0) circle (2pt);\n    \\fill[black] (-2.2,1.1) circle (2pt);\n      \\fill[blue] (-1,1.5) circle (2pt);\n        \\fill[blue] (-1.5,1.3) circle (2pt);\n          \\fill[blue] (-0.5,1.7) circle (2pt);\n            \\fill[blue] (1,1.5) circle (2pt);\n              \\fill[blue] (1.5,1.3) circle (2pt);\n                \\fill[blue] (1.8,0.8) circle (2pt);\n                  \\fill[blue] (0,1.8) circle (2pt);\n\\node[above] at (-1,1.5) {};\n  \\node[above, red] at (-1.7,1.3) {$w$};\n   \\node[above] at (-1,1.5) {$w_2$};\n     \\node[above] at (1.7,1.3) {$w_i$};\n\\end{tikzpicture}\n\\end{center}\n\\end{minipage}\n\n\\noindent If $\\Gamma$ is not semiample, then\nany section of $R_w = H^0(C^{(2)},D)$\nis divisible by $f_1$, a contradiction.\nThus we showed that $\\Gamma$ is semiample\nand as a consequence of Lemma~\\ref{triv}\nthe curve $C$ has two $g_k^1$. We denote\nthe corresponding curves of $C^{(2)}$ by \n$\\Gamma$ and $\\Gamma'$.\nA multiple of $\\Gamma$\ndefines a morphism $f\\colon C^{(2)}\\to B$\nonto a smooth curve $B$ whose Stein factorization is \nthe following \n\\[\n \\xymatrix{\n C^{(2)}\\ar[r]^{\\varphi}\\ar[rd]_{f} & Y\\ar[d]\\\\\n & B.\n }\n\\]\nTwo fibers of $\\varphi$ are $n\\Gamma$ and $m\\Gamma'$\nfor some positive rational numbers $n,m$. Since\n$\\Gamma$ is numerically equivalent to $\\Gamma'$ \nby Lemma~\\ref{gamma}, then $n=m$. \nMoreover by Lemma~\\ref{fib} the curve $Y$\nis rational, so that $n\\Gamma$ is linearly equivalent\nto $n\\Gamma'$. This implies that the class\nof $\\Gamma-\\Gamma'$ is torsion non-trivial\nin $\\Pic^0(C^{(2)})$\nand thus the same holds for \n\\begin{align*}\n 2\\eta_C \n & =\n \\text{ramification of $g_k^1$ - ramification of ${g_k^1}'$}\\\\\n & =\n \\imath^*(\\Gamma-\\Gamma').\n\\end{align*}\nWe now show that $(ii)\\Rightarrow (i)$ holds.\nFirst of all observe that if $\\eta_C$ is torsion\nnon-trivial, then the same holds for\n$\\Gamma-\\Gamma'$ by the above equalities\nand the fact that $\\imath^*$ is an isomorphism.\nIn this case $n\\Gamma\\sim n\\Gamma'$ for\nsome positive integer $n$ and this implies\nthat $|n\\Gamma|$ is base point free, being\n$\\Gamma^2=0$. Thus $\\Gamma$ is semiample.\nHence, if $L\\subseteq{\\mathbb Z}^2$ is the submonoid\ngenerated by the integer points of the cone \n${\\rm cone}(2K+3\\Delta,2K+(5-2k)\\Delta)$,\nthe following subalgebra \n\\[\n  S := \\bigoplus_{(a,b)\\in L}H^0(C^{(2)},a\\Delta+bK)\n\\]\nof $R$ is finitely generated \nby~\\cite[Lemma 4.3.3.4]{ADHL}.\nThe homogeneous elements of $R$ \nwhich do not belong to $S$ are sections\nof Riemann-Roch spaces \n$H^0(C^{(2)},D)$ with $D\\cdot\\Delta < 0$.\nHence any such section is divisible by\nthe generator $f_\\Delta$ of $R$ which is a defining\nsection for $\\Delta$. Thus we conclude that \n$R$ is generated by $S$ and $f_\\Delta$\nand the statement follows.\n\\end{proof}\n\n\n \n\\begin{remark}\n\\label{ch}\nOver an algebraically closed field ${\\mathbb K}$ of positive \ncharacteristic the statement of Theorem~\\ref{teo-1}\nmust be modified as follows: the algebra\n$R(\\Delta,K)$ is finitely generated if and \nonly if $\\eta_C$ is torsion. Indeed if $\\eta_C$ \nis trivial, then ${\\mathcal O}_\\Gamma(\\Gamma)$ is trivial\nas well as shown in the proof of Lemma~\\ref{triv},\nthus $\\Gamma$ is semiample by~\\cite[Theorem 0.2]{Ke}.\nMoreover, if ${\\mathbb K}$ is the algebraic closure\nof a finite field, then $\\eta_C$ is always\ntorsion and thus $R(\\Delta,K)$ is always \nfinitely generated.\n\nThe conclusion of Lemma~\\ref{covering}\nis no longer true in positive characteristic. Indeed in\nthis case the algebraic fundamental group of the\naffine line ${\\mathbb K}$ is not trivial and thus there is no\ncontradiction. For example, in characteristic $p>0$,\nif $C$ is the curve of ${\\mathbb K}^2$ defined by the equation \n$x_1^p-x_1=x_2$, then the projection onto the second\nfactor defines a non-trivial \\'etale covering of ${\\mathbb K}$.\n\\end{remark}\n\n\\begin{remark}\nWe observe that the locus of smooth curves of genus\n$(k-1)^2 > 1$ which admit two $g_k^1$ has one component\nof maximal dimension which consists of curves of \ntype $(k,k)$ on a smooth quadric $Q$. \nIndeed by~\\cite{AC} the only\nother component which could have bigger dimension \nwould consist of curves $C$ admitting an involution. \nA parameter count shows that for our curves this \ncomponent has smaller dimension.\n\nMoreover any smooth curve $C$ of type\n$(k,k)$ on $Q$ admits exactly two $g_k^1$.\nIndeed let $S = \\{p_1,\\dots,p_k\\}$ be a set of\n$k$ distinct points with $p_1+\\dots+p_k\\in g_k^1$. \nBy the Riemann-Roch theorem $S$ is in Cayley-Bacharach \nconfiguration with respect to the curves of \ntype $(k-2,k-2)$. It follows that all the \npoints of $S$ are collinear. Indeed, let\n$\\ell$ be the line through the first two points\n$p_1,p_2$, let $H$ be a general hyperplane \nwhich contains $\\ell$ and let $q\\in S\\setminus\\{p_1,p_2\\}$.\nTake a union $\\Lambda$ of $k-3$\nhyperplanes through all the points of \n$S\\setminus\\{p_1,p_2,q\\}$ and such that\n$q\\notin\\Lambda$.\nThen $H\\cup \\Lambda$ cuts out on $Q$ \na curve of type\n$(k-2,k-2)$ which, by the Cayley-Bacharach\nconfiguration, must pass through $q$.\nHence $q\\in H$ and by the generality\nassumption on $H$ we deduce $q\\in\\ell$.\n\\end{remark}\n\n\n\n\\section{The grid family}\n\\label{grid-fam}\n\nIn this section we study families of curves of type $(k,k)$\non a smooth quadric $Q = {\\mathbb P}^1\\times{\\mathbb P}^1$\nsuch that the class of the difference of the two\n$g_k^1$ induced by the two rulings is \n$n$-torsion. We prove that $n\\geq k$\nand construct the family with $n=k$.\n\n\n\\begin{definition}\n\\label{grid}\nGiven two effective \ndivisors $L_1$ and $L_2$ of $Q$ of type\n$(k,0)$ and $(0,k)$ respectively, the {\\em grid \nlinear system} defined by $L_1$ and $L_2$ is \nthe linear system of curves of $Q$ of \nbi-degree $(k,k)$ which pass through the \ncomplete intersection $L_1\\cap L_2$.\nThe {\\em grid family} \n\\[\n \\mathcal G_k\n \\subseteq\\mathcal F_k\n\\]\nis the family \nof all curves of type $(k,k)$ which belong to some \ngrid linear system.\n\\end{definition}\nObserve that if $C$ is a smooth curve in $\\mathcal G_k$\nand $\\eta_C\\in \\Pic^0(C)$ is the class of the difference of the \ntwo $g_k^1$ cut out by the two rulings then $\\eta_C^{\\otimes k}$\nis trivial. This justifies the inclusion\n$\\mathcal G_k\\subseteq\\mathcal F_k^{\\rm tor}$.\nAny curve $C$ in $\\mathcal G_k$\nadmits an equation of the form\n\\begin{equation}\n\\label{equ-fam}\n f_1(x_0,x_1)g_2(y_0,y_1)+g_1(x_0,x_1)f_2(y_0,y_1)\n =\n 0,\n\\end{equation}\nwhere $f_1, f_2, g_1$ and $g_2$ are homogeneous\npolynomials of degree $k$. Indeed it is enough to\nprove the claim for a curve $C$ in a grid family\nwhere both $L_1$ and $L_2$ consist of\ndistinct lines and then conclude by specialization \nthat the same holds for any\ncurve $C$ of $\\mathcal G_k$.\nLet $h=0$ be an equation for $C$ and\nlet $f_i = 0$ be an equation for $L_i$, for $i=1,2$.\nBy the equality\n\\[\n V(h,f_1) = V(f_1,f_2),\n\\]\nthe fact that the two ideals $(h,f_1)$ and $(f_1,f_2)$ are both\nradical and saturated with respect to the\nirrelevant ideal $(x_0,x_1)\\cap (y_0,y_1)$ of $Q$,\nwe deduce that the equality $(h,f_1) = (f_1,f_2)$\nholds. The claim follows since $h$ has bi-degree $(k,k)$.\nObserve that the general element of $\\mathcal G_k$ is\nirreducible and smooth.\n\n\\begin{proposition}\nThe {\\em grid family} $\\mathcal G_k$\nhas dimension $4k-1$.\n\\end{proposition}\n\\begin{proof}\n\nThe projectivization of the set of homogeneous polynomials of\nbidegree $(k,k)$ of type $f(x_0,x_1)g(y_0,y_1)$ can be identified \nwith the image $\\mathcal S$ of the Segre embedding of \n${\\mathbb P}^k\\times{\\mathbb P}^k\\to{\\mathbb P}^N$,\nwhere $N={k^2+2k}$.\nThus, a curve in $\\mathcal G_k$ can be identified \nwith a point of the $1$-secant variety ${\\rm Sec}(\\mathcal S)$\nof $\\mathcal S$.\nBy~\\cite[Theorem 1.4]{CS} the dimension of\n${\\rm Sec}(\\mathcal S)$ is $4k-1$ and the \nstatement follows.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{composed}\nLet $C\\in\\mathcal F_k$ and let $D$ be a divisor\nof $C$ cut out by one of the two rulings. Then,\nfor any $1\\leq n\\leq k-1$, the linear system $|nD|$ \nis composed with the pencil $|D|$.\n\\end{lemma}\n\\begin{proof}\nTo prove the statement it is enough to show that\n$h^0(C,nD) = n+1$ for $1\\leq n\\leq k-1$. \nLet $Q = {\\mathbb P}^1\\times{\\mathbb P}^1$. Without loss of generality\nwe can assume that $D$ is cut out by the first ruling\nof $Q$, so that we have the following exact sequence \nof sheaves\n\\[\n \\xymatrix@1{\n  0\\ar[r] \n  &\n  {\\mathcal O}_Q(-k+n,-k)\\ar[r]\n  &\n  {\\mathcal O}_Q(n,0)\\ar[r]\n  &\n  {\\mathcal O}_C(nD)\\ar[r]\n  &\n  0.\n }\n\\]\nTaking cohomology, using the vanishing of the higher\ncohomology groups of the middle sheaf, the Serre's \nduality theorem and the hypothesis on $n$ we deduce \nthe following equalities: \n\\[\nh^1(C,nD) = h^0(Q,{\\mathcal O}_Q(k-n-2,k-2))= (k-n-1)(k-1).\n\\]\nBy the adjunction formula $C$ has genus $(k-1)^2$.\nThus by the above and the Riemann-Roch formula \nwe conclude\n\\[\n h^0(C,nD)\n =\n nk+1-(k-1)^2+(k-n-1)(k-1)\n =\n n+1\n\\]\nand the statement follows.\n\\end{proof}\n\n\\begin{proposition}\n\\label{torsion}\nLet $C$ be a smooth curve of  type\n$(k,k)$ on $Q$ and let $\\eta_C$\nbe the class of the difference of the two $g_k^1$. \nThen $\\eta_C$ has order $\\geq k$ and the \nfollowing are equivalent:\n\\begin{enumerate}\n\\item\n$\\eta_C$ has order $k$;\n\\item\nthe curve $C$ belongs to the grid family $\\mathcal G_k$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\nWe first show that $\\eta_C$ has order $\\geq k$.\nLet $D_1$ be a divisor in the first $g_k^1$ and let  \n$D_2$ be a divisor in the second $g_k^1$. \nLet $n<k$ be a positive integer.\nBy Lemma~\\ref{composed} the linear system\n$|nD_i|$ is composed with $|D_i|$ for any $i$. \nThus the linear equivalence $nD_1\\sim nD_2$ would \nimply that for any set of $n$ lines of the first ruling \nthere are $n$ lines of the second ruling which cut \nout the same set of points on $C$.  This cannot be\nsince such set of $nk$ points would lie on a grid \nwith $n^2$ points.\nThis proves the claim.\n\nWe now show that $(i)\\Rightarrow (ii)$ holds.\nBy the exact sequence of sheaves\n\\[\n \\xymatrix@1{\n  0\\ar[r] \n  &\n  {\\mathcal O}_Q(0,-k)\\ar[r]\n  &\n  {\\mathcal O}_Q(k,0)\\ar[r]\n  &\n  {\\mathcal O}_C(kD_1)\\ar[r]\n  &\n  0\n }\n\\]\nthe fact that $h^0 = 0$, $h^1=k-1$ for the first sheaf\nand $h^0 = k+1$, $h^1 = 0$ for the second sheaf  we\nget the equality $h^0(C,{\\mathcal O}_C(kD_1)) = 2k$. \nBy hypothesis $|kD_1| = |kD_2|$ holds and by the above \ncalculation the linear system is a projective space of dimension\n$2k-1$. Let $\\mathcal H_i\\subseteq |kD_1|\\cong{\\mathbb P}^{2k-1}$ \nbe the projectivization of the $k$-th symmetric\npower of the vector space $H^0(C,D_i)$.\nSince both $\\mathcal H_1$ and $\\mathcal H_2$\nhave dimension $k$, their intersection \n$\\mathcal H_1\\cap\\mathcal H_2$ is at least\none-dimensional. A point of this intersection \ncorresponds to a divisor $D$ which is cut out \non $C$ by $k$ lines of the first ruling\nand by $k$ lines of the second ruling. \nThis proves the claim.\n\nThe implication $(ii)\\Rightarrow (i)$ is obvious.\n\\end{proof}\n\n\n\n\\section{Density of the torsion locus}\n\\label{density}\n\nWe define $\\mathcal F_k$ \nto be the open subset of the Hilbert scheme of the smooth\ncurves of type $(k,k)$ of ${\\mathbb P}^1\\times{\\mathbb P}^1$.\nLet $\\pi\\colon \\mathcal C\\to\\mathcal F_k$ be the universal\nfamily, let $\\mathcal H = R^1\\pi_*{\\mathbb C}$ be the associated\nHodge bundle whose fiber\nover a point $C\\in\\mathcal F_k$ is the\ncohomology group $H^1(C,{\\mathbb C})$ and\nlet $\\mathcal H^{1,0}$ be the subbundle \nof $\\mathcal H$ whose fiber over\n$C$ is $H^{1,0}(C)$. We recall that\nthe jacobian family $\\mathcal J\\to\\mathcal F_k$\ncan be defined as \n\\begin{equation}\n\\label{jac}\n \\mathcal J\n =\n \\frac{\\mathcal H}{\\mathcal H^{1,0}+R^1\\pi_*{\\mathbb Z}}.\n\\end{equation}\nGiven a point $t\\in\\mathcal F_k$ we denote by\n$C_t$ the fiber of $\\pi$ over $t$ and \nby $\\nu_t = \\mathcal L_1\\otimes\\mathcal L_2^{-1}\n\\in J(C_t)$ the class of the difference \nof the two $g_k^1$ cut out by the two rulings.\nThis gives a normal function $\\nu\\colon\\mathcal F_k\n\\to \\mathcal J$ defined by $t\\mapsto\\nu_t$.\nWe now consider a $\\mathcal C^\\infty$ trivialization \n$\\varphi$\nof the jacobian family over an open, simply connected\nsubset $\\mathcal U$ of $\\mathcal F_k$:\n\\[\n \\xymatrix{\n \\mathcal J\n  \\ar[r]^-\\varphi & \n  \\mathcal U\\times\\dfrac {R^1\\pi_\\ast\\mathbb R} {R^1\\pi_\\ast \\mathbb Z}\n  = \n  \\mathcal U\\times\\mathbb T,\n }\n\\]\nwhere $\\mathbb T$ is the real torus \n${\\mathbb R}^{2g}\/{\\mathbb Z}^{2g}$.\nWe let ${\\rm pr}_2\\colon\\mathcal U\\times\\mathbb T\n\\to\\mathbb T$ be the projection on the second factor\nand let \n\\[\n \\eta\\colon{\\mathcal  U}\\to\\mathbb T\n \\qquad\n t\\mapsto {\\rm pr}_2(\\varphi(\\nu(t))).\n\\]\nWe will show that, for any $k\\geq 3$, the image \nof $\\eta$ contains a non-empty open subset.\nOur strategy will be to prove that \nthe fibers of $\\eta$ are complex subvarieties \nof $\\mathcal U$, that $\\eta$ has a fiber \nof the expected real dimension \n$8k-2 = \\dim \\mathcal F_k - \\dim\\mathbb T$\nand we conclude by Proposition~\\ref{fiber}\n(see also the argument in the proof of~\\cite[Proposition 3.4]{CPP}).\n\n\\begin{proposition}\n\\label{fiber}\nLet $V\\subseteq X$ be an inclusion of complex manifolds, \n$Y$ be a real manifold and \n$\\eta\\colon X\\to Y$ be a $\\mathcal C^\\infty$\nmap whose fibers are complex subvarieties\nof $X$. Assume that there is a point $p\\in V$\nsuch that the following hold:\n\\begin{enumerate}\n\\item\nthe differential $d\\eta_p$ is surjective,\n\\item\nthe fiber of the restriction $\\eta|_V$ over $\\eta(p)$\nhas dimension $\\dim V - \\dim Y\\geq 0$.\n\\end{enumerate}\nThen $\\eta(V)$ contains\na non-empty open subset of $Y$.\n\\end{proposition}\n\\begin{proof}\nLet $F$ be the fiber of $\\eta$ over $\\eta(p)$ \nand let $U$ be a coordinate neighbourhood of $p$\nsuch that the restriction $\\eta|_U$ is a submersion.\nLet $d = \\dim V - \\dim Y$. After possibly \nintersecting $V$ with $d$ general smooth \ncomplex hypersurfaces passing through $p$\nwe can reduce to the case $d = 0$.\nThus, after possibly shrinking $U$ \nwe can assume $F\\cap V\\cap U = \\{p\\}$.\nFor $q$ in a sufficiently small neighbourhood\n$W$ of $\\eta(p)$ the intersection number \n$\\eta^{-1}(q)\\cap V$ does not change\nby~\\cite[Pag.\\,664]{GH}, so that the intersection is\nnon-empty. Then $\\eta(V)$ contains $W$\nand the statement follows.\n\\end{proof}\n\n\\begin{proposition}\nThe fibers of the map $\\eta$ are complex varieties.\n\\end{proposition}\n\\begin{proof}\nWe restrict the Hodge and the jacobian bundle \nto the simply connected open subset $\\mathcal U$\nof $\\mathcal F_k$,\nwhere the family is topologically trivial.\nLet $t_0\\in \\mathcal U$ be a distinguished  point\nand let $C = C_{t_0}$ be the corresponding\ncurve. We construct the following\ncommutative diagram\n\\[\n \\xymatrix{\n  \\mathcal U\\ar@{=}[d]\\ar[r]^-{\\tilde\\nu}\n  & \\mathcal H_{\\mathcal U}\\ar[r]^-\\cong\\ar[d]\n  & \\mathcal U\\times H^1(C,{\\mathbb C})\\ar[d]\\ar[r]\n  & H^1(C,{\\mathbb C})\\ar[d]\\\\\n  \\mathcal U\\ar[r]^-{\\nu|_\\mathcal U}\n  & \\mathcal J_{\\mathcal U}\\ar[r]^-\\varphi\n  & \\mathcal U\\times\\mathbb T\\ar[r]\n  & \\mathbb T\n }\n\\]\nwhere $\\tilde\\nu$ is a lifting of the restriction\n$\\nu|_{\\mathcal U}$ and the maps in the top row\nare complex analytic morphisms.\nObserve that, given $x\\in \\mathbb T$, we have \n\\[\n\\eta^{-1}(x)=\\{t\\in\\mathcal U: \\tilde \\nu(t)-\\tilde x+c\\in\\mathcal H^{1,0}\\},\n\\]\nwhere the projection of $\\tilde x$ to $\\mathcal J$ is $x$ \nand $c\\in H^1(C,{\\mathbb Z})$.\nThis implies that $\\eta^{-1}(x)$ \nis a complex subvariety of $\\mathcal U$ \nsince $\\mathcal H^{1,0}$ is a \ncomplex subbundle of $\\mathcal H$.\n\\end{proof}\n\n \n \n\\begin{proof}[Proof of Theorem~\\ref{teo-2}]\nWe will start proving that the image of the map $\\eta$ \ncontains a non-empty open subset of $\\mathbb T$.\nWe consider the map\n\\[\n \\phi\\colon\\mathcal J_{\\mathcal U}\\to \\mathbb T\n\\]\nwhich is composition of the trivialization $\\varphi$\nwith the projection onto the second factor.\nObserve that the differential of $\\phi$ has\nmaximal rank at any point. By applying\nProposition~\\ref{fiber} to the map\n$\\phi$, the subvariety \n$V = \\nu|_{\\mathcal U}(\\mathcal U)$ of $\\mathcal J$\nand the fiber $\\mathcal G_k$ of the restriction\n$\\phi|_V$ we deduce that $\\eta(V)$ contains\na non-empty open subset.\nObserve that the trivialization $\\varphi$\ncan be taken real analytic, since it is\nobtained by taking the real part of the \ncomplex analytic isomorphism coming\nfrom the trivialization of the Hodge bundle.\nBy Sard theorem the set of non-critical values\nof $\\phi$ is an open subset in the analytic \ntopology. Since $\\phi$ admits an analytic continuation\non any simple arc in $\\mathcal J$, its set\nof non-critical points is dense in the analytic\ntopology.\nWe conclude by observing that the set\nof torsion points of $\\mathbb T$ is dense\nand $\\mathcal F_k^{\\rm tor}$ is the preimage\nof this set via $\\eta = \\phi\\circ\\nu$.\n\\end{proof}\n\n\n\\section{Hyperelliptic curves}\n\\label{hyp}\n\nIn this section we prove a density\ntheorem for hyperelliptic curves. This result \nhas an independent interest and is proved in \nthe spirit of Griffiths computations of the \ninfinitesimal invariant~\\cite{Gr}.\nAs an application to our problem, this result\nprovides an alternative proof for Theorem~\\ref{teo-2}\nin case $g=4$.\n\nLet $\\pi\\colon\\mathcal C\\to\\mathcal U$ be a versal family \nof hyperelliptic curves of genus $g > 1$, where $\\mathcal U$\nis simply connected of dimension $2g-1$.\nWe denote by $j\\in\\Aut(\\mathcal C)$ the hyperelliptic\ninvolution and by\n$\\mathcal J\\to\\mathcal U$ the jacobian\nfamily. We consider the Abel-Jacobi map \n\\[\n \\nu\\colon \\mathcal C \\to \\mathcal J,\\quad  \n x\\mapsto\\int_{j(x)}^x.\n\\]\nIf we take $\\pi^\\ast \\mathcal J\\to \\mathcal C$ the pull-back \nof the Jacobian family on $\\mathcal C,$ \nwe may consider $\\nu$ as a normal function.\nAs in Section~\\ref{density} we consider a $\\mathcal C^\\infty$\ntrivialization of the jacobian family\n$\\mathcal J\\cong \\mathcal U\\times \\mathbb T$,\nwhere $\\mathbb T\\cong J(C)$, to construct a map\n\\[\n \\gamma\\colon\\mathcal C\\to\\mathbb T.\n\\]\n\\begin{theorem}\n\\label{diff}\nIf $p\\in C$ is not a Weierstrass point, then the\ndifferential of $\\gamma$ at $p$ is surjective.\n\\end{theorem}\n\nOur strategy is as follows. For any holomorphic\nform $\\omega\\in H^0(C,\\Omega_C)$ we show that\nthere is a curve $r(t)$ in $\\mathcal C$ such that\n$r(0) = p$ and $d\\gamma_p(r'(0))\\cdot\\omega$\nis non-zero. To this aim we produce $r(t)$\naccordingly to the order $n$ of vanishing of $w$ \nat $p$. Since the divisor ${\\rm div}(w)$ is $j$-invariant, \nthen it is natural to consider $D = p+j(p)$.\nWe thus have a filtration \n\\begin{equation}\n\\label{filt}\n H^0(C,\\Omega_C)\n =\n L^0\\supseteq L^1\\supseteq...\\supseteq L^{g-1}\\supseteq L^g=0,\n\\end{equation}\nwhere $L^k$ is the Riemann-Roch space \n$H^0(C,\\Omega_C(-kD))$. Given $\\omega\\in L^k\n\\setminus L^{k+1}$, with $k>0$, we construct $\\zeta = \\partial(f)\n\\in H^1(C,T_C)^j$ as in Subsection~\\ref{def}, where $f\\in\nH^0(Z,{\\mathcal O}_Z)$. The one-dimensional family\n\\[\n \\mathcal C_\\zeta\\to\\Delta\n\\]\ndefined by $\\zeta$, plus a choice of a smooth\nsection through the point $p$ in the family, defines a curve \n$r(t)$ in $\\mathcal C$. We show that the family\nis equipped with a $\\mathcal C^\\infty$\n $1$-form $\\Theta$ such that the restriction \nof the $(1,0)$-part $\\Theta^{1,0}$\nto the central fiber $C$\nadmits a local expansion at $p$ of the\nform $w + \\tilde f(z)dt + o(t)$, where $z$ is a coordinate\nin $C$, $\\tilde f|_Z=f$ and $t\\in\\Delta$. We finally prove \n\\[\n d\\gamma_p(r'(0))\\cdot\\omega\n =\n \\lim_{t\\to 0}\\frac{1}{t}\\left(\\int_{\\Gamma_t}\\Theta_t-\\int_{\\Gamma_0}\\omega\\right)\n =\n 2f(p)\\neq 0,\n\\]\nwhere $\\Gamma_t$ is a path between $r(t)$ \nand $j(r(t))$. When $k=0$ we choose $r(t)$\nto be a path within the central fiber $C$,\nwe write $\\omega$ locally as $h(z)dz$\nand prove the equality\n\\[\n d\\gamma_p(r'(0))\\cdot\\omega\n =\n \\lim_{t\\to 0}\\frac{1}{t}\\int_{p}^{r(t)}\\omega\n =\n h(0)\\neq 0.\n\\]\n\n\n\\subsection{Deformation of curves}\n\\label{def}\nWe recall first a result on deformation and on \nextension of line bundles \nwhich will be applied to hyperelliptic curves (see also~\\cite{CP} and~\\cite{R}).\nLet $C$ be a smooth curve of genus $g>1$ and \nlet $T_C$ and $\\Omega_C$ be respectively the holomorphic \ntangent bundle and the canonical line bundle of $C$. \nFix a non trivial $\\omega\\in H^0(C,\\Omega_C)$ and \nlet $Z$ be the canonical divisor associated to $\\omega.$\nThe form $\\omega$ defines the following exact sequence\n\\[\n\\xymatrix@1{\n0\\ar[r] & T_C\\ar[r]^{\\omega}& {\\mathcal O}_C\\ar[r]& {\\mathcal O}_Z\\ar[r]& 0.\n}\n\\]\nPassing to the long exact sequence in cohomology \nwe obtain\n\\[\n\\xymatrix@C=20pt{\n0\\ar[r]& {\\mathbb C}\\cong H^0(C,{\\mathcal O}_C)\\ar[r]& \nH^0(Z,{\\mathcal O}_Z)\\ar[r]^\\partial& H^1(C,T_C)\n\\ar[r]^{\\omega}& H^1(C,{\\mathcal O}_C).\n}\n\\]\nGiven an element $f\\in H^0(Z,{\\mathcal O}_Z)$\nits image $\\zeta=\\partial(f)$ defines an\nextension of ${\\mathcal O}_C$ by $T_C$ via the\nisomorphism $H^1(C,T_C)\\cong \n{\\rm Ext}^1({\\mathcal O}_C,T_C)$\n\\[\n\\xymatrix@1{\n0\\ar[r] & T_C \\ar[r] & E_{\\zeta}\\ar[r] & {\\mathcal O}_C\\ar[r] & 0.\n}\n\\]\nTaking tensor product with $\\Omega_C$ \nand recalling that $T_C$ is dual with $\\Omega_C$\nwe get the following exact sequence\n\\begin{equation} \\label{zeta}\n\\xymatrix@1{\n0\\ar[r] &{\\mathcal O}_C\\ar[r] & E_{\\zeta}\\otimes \\Omega_C\\ar[r] &\n \\Omega_C\\ar[r]& 0,\n }\n \\end{equation}\nwhich passing to the long exact sequence in cohomology\ngives the following sequence \nwhose coboundary  \nis the cup product with $\\zeta$:\n\\[\n \\xymatrix@C=20pt{\n0\\ar[r] & {\\mathbb C}\\cong H^0(C,{\\mathcal O}_C)\\ar[r]&\n H^0(C,E_\\zeta\\otimes \\Omega_C)\\ar[r]^-k & \n H^0(C,\\Omega_C)\\ar[r]^-{\\zeta}\\ar[r]&\n  H^1(C,{\\mathcal O}_C).\n  }\n\\]\nSince $\\zeta\\in\\ker(\\omega)$, or equivalently\nthe cup product $\\zeta\\cdot \\omega$ vanishes,\nthere exists an element $\\Omega\\in \nH^0(C,E_\\zeta\\otimes \\Omega_C)$ such that \n$k(\\Omega)=\\omega$.\nNow we consider the commutative diagram\n\n\\[\n \\xymatrix@1{\n  & 0\\ar[d] & 0\\ar[d] & H^0(C,{\\mathcal O}_C)\\ar[d]^-\\zeta\\\\\n  0\\ar[r]\\ar[d]\n  & H^0(C,{\\mathcal O}_C)\\ar[r]\\ar[d] \n  & H^0(Z,{\\mathcal O}_Z)\\ar[d]^-\\rho\\ar[r]^-\\partial \n  & H^1(C,T_C)\\ar[d] \\\\\n  0\\ar[r]\\ar[d]\n  & H^0(C,E_\\zeta\\otimes \\Omega_C)\\ar[r]^-r\\ar[d]^-k\n  & H^0(Z,E_\\zeta\\otimes \\Omega_C|_Z)\\ar[d]\\ar[r]\n  & H^1(C,E_\\zeta)\\ar[d]\\\\\n  H^0(C,{\\mathcal O}_C)\\ar[r]^-\\omega\n  & H^0(C,\\Omega_C)\\ar[r]\n  & H^0(Z,\\Omega_C|_Z)\\ar[r]\n  & H^1(C,{\\mathcal O}_C)\n }\n\\]\n \\vspace{0.2cm}\n \nThe restriction of the lifting $\\Omega$ to $Z$ \ngives an element $r(\\Omega)\\in H^0(Z,E_\\zeta\\otimes \\Omega_C|_Z)$ \nthat by construction is in the image of the map\n$\\rho$, that is $\\Omega=\\rho(g)$ \nfor some $g\\in H^0(Z,{\\mathcal O}_Z)$.\nA diagram chase proves indeed that\n$\\partial (g)=\\zeta$ holds.\nSince $\\ker(\\partial)$ is isomorphic to ${\\mathbb C}$,\nwe conclude that $f$ equals $g$ up to a constant.\nThis means that we can realize the function $f$ \nby a unique suitable lifting $\\Omega$ of $\\omega.$\nWe collect the discussion in the following lemma.\n\n\\begin{proposition}\\label{coord}\nLet $\\omega$ be a non-zero element in  \n$H^0(C,\\Omega_C)$ and let $Z$ be the \ndivisor of $\\omega$. \nThen for any $f\\in H^0(Z,{\\mathcal O}_Z)$ \nthere is a unique \n$\\Omega\\in H^0(C,E_{\\zeta}\\otimes \\Omega_C)$ \nsuch that $r(\\Omega)=\\rho (f)$. \n\\end{proposition}\n\nWhen we interpret $H^1(C,T_C)$ \nas the space of first order deformations of $C$,\nso that $\\zeta$ corresponds to a family\n\\[\n\\mathcal C_\\zeta\\to {\\rm Spec}\\, {\\mathbb C}[\\varepsilon],\n\\]\nthe isomorphism $H^1(C,T_C)\\to\n{\\rm Ext}^1({\\mathcal O}_C,\\Omega_C)$\ngives the identifications \n$E_{\\zeta}\\cong T_{\\mathcal C}|_{C}$ \nand $E_{\\zeta}\\otimes \\Omega_C\\cong\\Omega_{\\mathcal C}|_{C}.$ \nTherefore the sequence~\\eqref{zeta}\nis the cotangent sequence of the first order deformation.\nIn coordinates we may write \n\\begin{equation}\n\\label{Omega}\n \\omega= h(z)dz,\\qquad \\Omega=h(z)dz+\\tilde f(z)dt,\n\\end{equation}\nwhere $dt$ is the global section \nof the cotangent $\\Omega_{\\mathcal C}|_{C}$ \nand $f=\\tilde f|_Z$ is a section of $H^0(Z,{\\mathcal O}_Z)$ \nsuch that $r(\\Omega)=\\rho (f)$.\n\n\n\\subsection{The normal function}\nIn this subsection we will specialize the previous\nconstruction to hyperelliptic families.\nFirst of all, given a hyperelliptic curve $C$ of genus $g>1$,\nwe consider the $j$-invariant subspace\n\\[\n H^1(C,T_C)^{j}\n \\subseteq\n H^1(C,T_C),\n\\]\nwhich corresponds to the directions that are preserved\nby the hyperelliptic involution $j$.\nSince $j$ acts on $H^0(Z,{\\mathcal O}_Z)$\nas $f\\mapsto f\\circ j$\nand it acts as $-1$ on $H^0(C,\\Omega_C)$, \nthen $\\partial(f) = \\zeta$ is $j$-invariant if and only if\n$f\\circ j$ equals $-f$ up to a constant.\n\nLet $p$ be a non-Weierstrass point of $C$ and\n$\\omega\\in H^0(C,\\Omega_C)$\nbe a holomorphic form which vanishes \nwith order $k>0$ at $p$,\nthat is $\\omega\\in L^k\\setminus L^{k+1}$.\nWe now take $f_{\\omega}\\in H^0(Z,{\\mathcal O}_Z)$, where $Z \n= k(p+j(p)) + Z' = {\\rm div}(\\omega)$,\nsuch that\n\\[\n f_{\\omega}(p)=1,\\quad f_{\\omega}(j(p))=-1,\\quad f_{\\omega}(p')=0 \\text{ for } p'\\in Z'.\n\\]\nGiven $\\zeta_{\\omega}=\\partial(f_{\\omega})$,\nby the previous remark\nwe have $j(\\zeta_{\\omega})=\\zeta_{\\omega}.$ \nThus there exists  a smooth family of hyperelliptic curves\n\\begin{equation}\n \\pi\\colon \\mathcal C_\\omega\\to \\Delta \\label{curvamod},\n\\end{equation} \n where $\\Delta\\subset \\mathcal U$ is a disk, such that \n$\\pi^{-1}(0)=C$ and such that \nthe Kodaira-Spencer \nclass of the family is $\\zeta_{\\omega}$. \nLet $\\Omega_{\\omega}$ be the section of \n$E_{\\zeta_{\\omega}}\\otimes \\Omega_C$ \nassociated to $f_{\\omega}$ as in Lemma ~\\ref{coord}.\n By means of a trivialization of \n the family we can construct a closed \n differential $1$-form $\\Theta$\n on $\\mathcal C_{\\omega}$ which is invariant \n with respect to the involution $j$ and\n such that the restriction of the $(1,0)$-part\n $\\Theta^{(1,0)}$ to the central fiber equals \n $\\Omega_{\\omega}$. \nIn local coordinates we can write  \n  \\[\n  \\Theta(z,t)^{(1,0)}=\\Omega_{\\omega}+ o(t)=\\omega+f_{\\omega}(z)dt+o(t).\n  \\]\n\nWe also assume to have a holomorphic section $r$ \nof $\\pi_{\\omega}$ such that $r(0)=p$ and define $r'= j(r).$\nMoreover, we fix a differentiable map\n\\[\n\\Gamma(t,s):\\Delta\\times [0,1]\\to \\mathcal C_{\\omega}\n\\] \nsuch that $\\Gamma(t,s)\\in \\pi^{-1}(t)$,\n $\\Gamma(t,0)=r'(t)$ and  $\\Gamma(t,1)=r(t).$ \n Observe that $\\Gamma$ is a family of sections \n connecting $r'$ and $r$.\n We define the function\n\\[\n g\\colon\\Delta\\to {\\mathbb C},\\quad \n t\\mapsto \n \\int_{\\Gamma_t} \\Theta_{t}.\n\\]\n Following Griffiths~\\cite[(6.6)]{Gr} and using the fact that \n the Gauss-Manin connection vanishes on $\\Theta$,\n we have that the derivative of $g$ at $0$ \n equals\n$\nd\\gamma_p(r'(0))\\cdot \\omega.\n$\n \n\n\\begin{proof}[Proof of Theorem~\\ref{diff}]\nWith the previous notation, \ngiven any $\\omega\\in H^0(C,\\Omega_C)$ \nvanishing of order $k>0$ at $p$, \nwe consider the family $\\pi_{\\omega}$, \nwith its sections $r=r_{\\omega}$ and $r'=j(r)$, \nand $\\Theta$ the corresponding differential form on \n$\\mathcal C_{\\omega}$.\nBy the previous remark we have that \n\\[\nd\\gamma_p(r'(0))\\cdot \\omega=g'(0).\n\\]\nWe now compute the latter term:\n\\[\ng(t)-g(0)=\\int_{\\Gamma_t} \\Theta_{t}-\\int_{\\Gamma_0} \n\\Theta_{0}=\\int_{\\Gamma_t} \\Theta_{t}-\\int_{\\Gamma_0} \\omega.\n\\]\nWe call $r_t$ and $r'_t$ \nthe arcs $r([0,t])$ and $r'([0,t])$ respectively.\nSince $\\Theta$ is closed, then $\\Gamma^\\ast(\\Theta)$ is\nexact and we have \n$0= \\int_{r'_t}\\Theta_t+\\int_{\\Gamma_t}\\Theta_t - \\int_{r_t}\\Theta_t-\\int_{\\Gamma_0}\\Theta_t$, \nhence\n\\[\ng(t)-g(0)=  \\int_{r_t}\\Theta_t- \\int_{r'_t}\\Theta_t=2\\int_{r_t}\\Theta_t,\n\\]  \nwhere the last equality is due to the fact \nthat $j^\\ast(\\Theta)=-\\Theta.$\nFinally, since $\\Theta = \\omega+f_{\\omega}(z)dt + o(t)$,\nby the fundamental theorem of calculus \nwe get\n\\[\n \\lim_{t\\to 0}\\frac{1}{t}\\int_{r_t}\\Theta_t\n =\n \\lim_{t\\to 0}\\frac{1}{t}\\int_{r_t}\\Theta_t^{(1,0)}\n =\n f_{\\omega}(p)\\not=0.\n\\]\nThus $g'(0)\\not=0$.\nIf $k=0$, that is $\\omega$ does not vanish \nat $p$, we will choose a loop $r(t)$ in $C$ with\n$r(0)=p$ \nand we will compute the derivative \nof the Abel-Jacobi map ${\\rm AJ}$ on $C$. \nFirst note that if we take a \nWeierstrass point $q$ of $C$ \nwe have \n\\[\n{\\rm AJ}(r(t)-j(r(t)))=2{\\rm AJ}(r(t)-q).\n\\]\nTake a coordinate\n$z$ centered at $p$  \nsuch that $\\omega(z)=h(z)dz$ with $h(0)\\neq 0.$ \nFix a loop $r(t)$ such that $z(r(t))=t$, then\n\\[\n\\lim_{t\\to0}\\frac{1}{t} \\int_{q}^{r(t)}\\omega= \n\\lim_{t\\to 0}\\frac{1}{t}\\int_0^th(z)dz=h(0)\n\\] \nand we complete our result.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor}\nThe locus of curves $C$ in $\\mathcal M_4$ \nsuch that $\\eta_C$ is a non-trivial torsion point is a \ncountable union of subvarieties \nof complex dimension $\\geq 5$ \nand the set of subvarieties of\ndimension $5$ is dense in $\\mathcal M_4$\nin the analytic topology.\n\\end{corollary}\n\n\\begin{proof}\nLet $\\mathcal U$ be an open subset of $\\mathcal M_4$\nwhich intersects the hyperelliptic locus and let $\\tilde{\\mathcal U}$ \nbe the moduli space of pairs $(C,g_3^1)$,\nwhere $[C]\\in \\mathcal U$. \nLet $\\imath\\colon\\mathcal H\\to\\tilde{\\mathcal U}$ be the \nsubvariety containing pairs where $C$ is hyperelliptic. \nGiven a universal family \n$\\pi: \\mathcal C\\to  \\tilde{\\mathcal U}$,\nwe construct the following commutative diagram\n \\[\n \\xymatrix{\n  \\imath^*\\mathcal C\\ar[rrr]^-{(C,p)\\mapsto [j(p)-p]}\\ar[d]\n  &&& \\mathcal J\\ar@{=}[d]\\ar[rd]\\\\\n  \\mathcal C\\ar[d]^-\\pi\\ar[rrr]^-{(C,g_3^1)\\mapsto \\eta_C} \n  &&& \\mathcal J\\ar[r]^-{\\varphi} \n  &  \\tilde{\\mathcal U}\\times\n      \\mathbb T \\ar[d]^-{{\\rm pr}_2}\\\\\n \\tilde{\\mathcal U}\n  \\ar[rrrr]^-{[C]\\mapsto{\\rm pr}_2(\\varphi(\\eta_C))}\n  &&&& \\mathbb T\n }\n\\]\nwhere $\\eta_C={g'}_3^1-g_3^1=K_C-2g_3^1$ and $\\varphi$ is a $\\mathcal C^\\infty$-trivialization\n(defined after possibly shrinking $\\mathcal U$).\nThe commutativity of the top square comes from\nthe fact that on a hyperelliptic curve $C$ we have\n$g_3^1 = p + g_2^1$ and $K_C - 2g_3^1 = j(p)-p$,\nwhere $j\\in{\\rm Aut}(C)$ is the hyperelliptic involution.\nBy Theorem~\\ref{diff} the differential \nof  the map $\\gamma\\colon\\imath^*\\mathcal C\\to \\mathbb T$ \nobtained composing the maps in the diagram \nis surjective at any point $p$ which is not Weierstrass. \nThis implies that the map $\\eta: \\tilde{\\mathcal U}\\to \\mathbb T$ \nis locally a submersion at any point corresponding to a hyperelliptic \ncurve, in particular its image contains an open subset of $\\mathbb T$.\nWe thus conclude as in the last part of the \nproof of Theorem~\\ref{teo-2} given in section~\\ref{density}.\n\\end{proof}\n\n\n\\section{Examples}\n\\label{exa}\n\nIn this section we will provide further examples \nof curves having two $g_k^1$'s whose \ndifference is a torsion element in the Jacobian.\nIn particular we will show how to use automorphism\ngroups to construct new examples (see\nExample~\\ref{exa:new}).\n\n\n\\begin{proposition}\\label{auto}\nLet $C$ be a curve in $\\mathcal F_k$.\nIf $G$ is an automorphism group  of $C$\nof order $n$ which preserves each \n$g_k^1$ of $C$ and such that $C\/G$ \nhas genus zero, then the order of \n$\\eta_C$ divides $n$.\n\\end{proposition}\n\\begin{proof}\nLet $\\pi:C\\to C\/G\\cong {\\mathbb P}^1$ be \nthe quotient morphism, let \n$D=p_1+p_2+\\dots+p_k$ be an \nelement of the first $g_k^1$\nand let $q_i=\\pi(p_i)$. Then\nthe following linear equivalences\nhold\n\\[\n nD\n \\sim\n \\sum_{\\sigma\\in G} \\sigma^*(D)\n =\n \\pi^*(q_1)+\\pi^*(q_2)+\\dots+\\pi^*(q_k)\n \\sim\n kF, \n\\]\nwhere $F$ is a fiber of $\\pi$ and the first equivalence \nis due to the fact that $G$ preserves the \nlinear series $g_k^1$.\nSince the same property holds for an element\n$D'$ of the second $g_k^1$, the linear equivalence\n$nD\\sim nD'$ follows.\n\\end{proof}\n\n \n\\begin{example}\nLet $\\sigma$ be the order $k$ automorphism\nof ${\\mathbb P}^1\\times{\\mathbb P}^1$ defined by\n\\[\n \\sigma(x_0,x_1,y_0,y_1)=(\\zeta_k x_0,x_1, y_0,y_1),\n\\]\nwhere $\\zeta_k$ is a primitive $k$-th root of unity.\nWe now show that a curve $C\\in\\mathcal F_k$ \nwhich is $\\sigma$-invariant\nadmits an equation of the form\n\\[\n x_0^kg_2(y_0,y_1)+x_1^kf_2(y_0,y_1)=0,\n\\]\nwhere $f_2, g_2$ are homogeneous of degree \n$k$ in $y_0,y_1$. In particular the quotient\n$C\/\\langle\\sigma\\rangle$ \nhas genus zero. Thus  $C\\in\\mathcal F_k^{\\rm tor}$\nby either Proposition~\\ref{auto} or Proposition~\\ref{torsion}.\nThe automorphism $\\sigma$ preserves \neach ruling of the quadric and acts identically \non one of the two rulings.\nConsider a point in $\\mathcal H_1\\cap\\mathcal H_2$,\nwith the notation in the proof of Proposition~\\ref{torsion}, \nwhich corresponds to a $\\sigma$-invariant grid.\nThe lines of the grid which belong to the first ruling  \nare defined by either \n$x_0^k-x_1^k=0$ or $x_0^k=0$.\nAn equation of $C$ in such coordinates \nis then of the form\n\\[\n (x_0^k-\\mu x_1^k)h_1+(x_0^k+\\lambda x_1^k)h_2\n =\n x_0^k(h_1+h_2)+x_1^k(\\lambda h_2-\\mu h_1)\n =\n 0,\n\\]\nfor $\\mu\\in \\{0,1\\}$, $\\lambda\\in {\\mathbb C}$ and $h_1,h_2$ \nhomogeneous of degree three in $y_0,y_1$.\n\\end{example}\n\n\\begin{example}\n\\label{exa:new}\nThe moduli space of non-hyperelliptic \ncurves $C$ of genus four having an order five \nautomorphism $\\sigma$ such that \n$C\/\\langle\\sigma\\rangle$ \nhas genus zero is a 1-dimensional subvariety \nof $\\mathcal F_3^{\\rm tor}$.\nMoreover, any such $C$ is isomorphic to a \ncurve in the following family \n\\[\n x_0x_1^2y_1^3\n +\\alpha x_0^2x_1y_0^3\n +\\beta x_0^3y_0y_1^2\n +\\gamma x_1^3y_0^2y_1=0,\n\\]\nwhere $\\sigma(x_0,x_1,y_0,y_1)\n=(\\zeta_5x_0,x_1,\\zeta_5^3y_0,y_1)$ \nand $\\zeta_5$ is a primitive fifth root of unity.\nThe family contains curves which pass through the points \nof a grid of type $(5,5)$, for example the curve with\n\\[\n\\alpha=-\\zeta_5,\\ \\beta=\\zeta_5^3+\\zeta_5^2+\\zeta_5,\\ \\gamma=\\zeta_5^2+\\zeta_5. \n\\]\nHowever, the general element of the family is not \nof grilled type.\nThis means that if $D_i$ is a divisor of the $i$-th $g_3^1$ \nand $\\mathcal H_i\\subseteq |5D_1|\\cong{\\mathbb P}^{11}$ \nis the projectivization of the fifth symmetric \npower of $H^0(C,D_i)$, then the intersection\n$\\mathcal H_1\\cap\\mathcal H_2$ is empty.\nFor example this holds for the curve with $\\alpha=-1, \\beta=\\gamma=1$.\nThe statements for both curves can be checked \nby means of the Magma~\\cite{Magma} program\navailable here~\\url{http:\/\/www2.udec.cl\/~alaface\/software\/semiample\/aut}.\n\\end{example}\n\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{AC}{article}{\n   author={Arbarello, Enrico},\n   author={Cornalba, Maurizio},\n   title={Footnotes to a paper of Beniamino Segre: ``On the modules of\n   polygonal curves and on a complement to the Riemann existence theorem''\n   (Italian) [Math. Ann. {\\bf 100} (1928), 537--551;\\ Jbuch {\\bf 54}, 685]},\n   note={The number of $g^{1}_{d}$'s on a general $d$-gonal curve, and\n   the unirationality of the Hurwitz spaces of $4$-gonal and $5$-gonal\n   curves},\n   journal={Math. Ann.},\n   volume={256},\n   date={1981},\n   number={3},\n   pages={341--362},\n   issn={0025-5831},\n   review={\\MR{626954 (83d:14016)}},\n   doi={10.1007\/BF01679702},\n}\n\n\\bib{ADHL}{book}{\n    AUTHOR = {Arzhantsev, Ivan},\n    AUTHOR = {Derenthal, Ulrich},\n    AUTHOR = {Hausen, J\\\"urgen},\n    AUTHOR = {Laface, Antonio},\n    TITLE = {Cox rings},\n    series={Cambridge Studies in Advanced Mathematics},\n    volume={144},\n   publisher={Cambridge University Press, Cambridge},\n   date={2014},\n   pages={530},\n   isbn={9781107024625},\n}\n\n\n\\bib{BHPV}{book}{\n   author={Barth, Wolf P.},\n   author={Hulek, Klaus},\n   author={Peters, Chris A. M.},\n   author={Van de Ven, Antonius},\n   title={Compact complex surfaces},\n   series={Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A\n   Series of Modern Surveys in Mathematics [Results in Mathematics and\n   Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]},\n   volume={4},\n   edition={2},\n   publisher={Springer-Verlag, Berlin},\n   date={2004},\n   pages={xii+436},\n   isbn={3-540-00832-2},\n   review={\\MR{2030225 (2004m:14070)}},\n   doi={10.1007\/978-3-642-57739-0},\n}\n\n\\bib{Magma}{article}{\n    AUTHOR = {Bosma, Wieb},\n    AUTHOR = {Cannon, John},\n    AUTHOR = {Playoust, Catherine},\n     TITLE = {The {M}agma algebra system. {I}. {T}he user language},\n      NOTE = {Computational algebra and number theory (London, 1993)},\n   JOURNAL = {J. 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A Series of Modern Surveys in Mathematics]},\n   volume={48},\n   note={Classical setting: line bundles and linear series},\n   publisher={Springer-Verlag, Berlin},\n   date={2004},\n   pages={xviii+387},\n   isbn={3-540-22533-1},\n   review={\\MR{2095471 (2005k:14001a)}},\n   doi={10.1007\/978-3-642-18808-4},\n}\n\n\\bib{R}{article}{\n   author={Raviolo, Emanuele},\n   title={A note on Griffiths infinitesimal invariant for curves},\n   journal={Ann. Mat. Pura Appl. (4)},\n   volume={193},\n   date={2014},\n   number={2},\n   pages={551--559},\n   issn={0373-3114},\n   review={\\MR{3180933}},\n   doi={10.1007\/s10231-012-0290-x},\n}\n\n\n \\end{biblist}\n \\end{bibdiv}\n\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\\IEEEPARstart{W}{e}  study the problem of extracting a prescribed number of random bits by reading the smallest possible number of symbols from imperfect stochastic processes. For perfect stochastic processes, including processes with known accurate distributions or perfect biased coins, this problem has been well studied. It dates back to von Neumann [9] who considered the problem of generating random bits from a biased coin with unknown probability. Recently, in \\cite{Zhou12_Streaming},  we improved von Neumann's scheme and introduced an algorithm that generates `random bit streams' from biased coins, uses bounded space and runs in expected linear time. This algorithm can generate a prescribed number of random bits with an asymptotically optimal efficiency. On the other hand, efficient algorithms have also been developed for extracting randomness from any known stochastic process (whose distribution is given). In \\cite{Knuth1976}, Knuth and Yao presented a simple procedure for generating sequences with\narbitrary probability distributions from an unbiased coin (the probability of H and T is $\\frac{1}{2}$). In \\cite{Abrahams1996}, Abrahams considered\na source of biased coin whose distribution is an integer power of a noninteger.\nHan and Hoshi \\cite{Han1997}  studied the general problem and proposed an interval algorithm that generates a prescribed number of random bits from any known stochastic process and achieves the information-theoretic upper bound on efficiency.  However, in practice, sources of stochastic processes have inherent correlations and are affected by measurement's noise, hence, they are not perfect. Existing algorithms for extracting randomness from perfect stochastic processes cannot work for imperfect stochastic processes, where uncertainty exists.\n\nTo extract randomness from an imperfect stochastic process, one approach is to apply a seeded or seedless extractor to a sequence generated by the process that contains a sufficient amount of  randomness, and we call this approach as a fixed-length extractor for stochastic processes since all the possible input sequences have the same fixed length. Efficient constructions of seeded or seedless extractors have been extensively studied in last two decades, and it shows that the number of random bits extracted by them can approach the source's min-entropy asymptotically \\cite{Dvir08, Nis96, Sha02,Rao2007,Kamp11}. Although fixed-length extractors can generate random bits with good quality from imperfect stochastic processes,\ntheir efficiencies are not close to the optimality. Here, we define the \\emph{efficiency} of an extractor for stochastic processes as the asymptotic ratio\nbetween the number of extracted random bits and the entropy of its input sequence (the entropy of its input sequence is proportional to the expected input length if the stochastic process is stationary ergodic), which is upper bounded by $1$ since the process of extracting randomness does not increase entropy. Based on this definition, we can conclude that the efficiency of a fixed-length extractor is upper bounded by the ratio between the min-entropy and the entropy of the input sequence, which is usually several times smaller than $1$. So fixed-length extractors are not very efficient in extracting randomness from stochastic processes. The intuition is that, in order to minimize the expected number of symbols read from an imperfect stochastic process, the length of the input sequence should be adaptive, not being fixed.\n\nThe concept of min-entropy and entropy are defined as follows.\n\\begin{Definition}\nGiven a random source $X$ on $\\{0,1\\}^n$, the \\emph{min-entropy} of $X$ is defined as\n$$H_{\\min}(X)=\\min_{x\\in \\{0,1\\}^n } \\log \\frac{1}{P[X=x]}.$$\nThe \\emph{entropy} of $X$ is defined as\n$$H(X)=\\sum_{x\\in \\{0,1\\}^n } P[X=x] \\log \\frac{1}{P[X=x]}.$$\n\\end{Definition}\n\nThe following example is constructed for comparing entropy with min-entropy for a simple random variable.\n\\begin{Example}\nLet $X$ be a random variable such that $P[X=0]=0.9$ and $P[X=1]=0.1$, then\n$H_{\\min}(X)=0.152$ and $H(X)=0.469$. In this case, the entropy of $X$ is about three times its min-entropy.\\hfill\n\t$\\Box$\n\\end{Example}\n\nIn this paper, we focus on the notion and constructions of variable-length extractors (short for variable-to-fixed length extractors), namely, extractors with variable input length and fixed output length. (Note that the interval algorithm proposed by Han and Hoshi \\cite{Han1997} and the streaming algorithm proposed by us \\cite{Zhou12_Streaming} are special cases of variable-length extractors). Our goal is to extract a prescribed number of random bits in the sense of statistical distance while minimizing the expected input cost, measured by the entropy of the input sequence (whose length is variable). To make this precise, we let\n$d(\\mathcal{R},\\mathcal{M})$ be the difference between two known stochastic processes $\\mathcal{R}$ and $\\mathcal{M}$, defined by\n$$d(\\mathcal{R},\\mathcal{M})=\\limsup_{n\\rightarrow\\infty} \\max_{x\\in \\{0,1\\}^n} \\frac{\\log_2\\frac{P_{\\mathcal{R}}(x)}{P_{\\mathcal{M}}(x)}}{\\log_2\\frac{1}{P_{\\mathcal{M}}(x)}},$$\nwhere $P_{\\mathcal{R}}(x)$ is the probability of generating $x$ from $\\mathcal{R}$ when the sequence length is $|x|$, and $P_{\\mathcal{M}}(x)$ is the probability of generating $x$ from $\\mathcal{M}$ when the sequence length is $|x|$.\n\nA few models of imperfect stochastic processes are introduced and investigated, including,\n\\begin{itemize}\n  \\item Let $\\mathcal{M}$ be a known stochastic process, we consider an arbitrary stochastic process $\\mathcal{R}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a constant $\\beta$.\n  \\item We consider $\\mathcal{R}$ as an arbitrary stochastic process such that\n  $\\min_{\\mathcal{M}\\in \\mathcal{G}_{s.e.}}d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for\n  a constant $\\beta$, where $\\mathcal{G}_{s.e.}$ denotes the set consisting of all stationary ergodic processes.\n\\end{itemize}\n\nGenerally, given a real slight-unpredictable source $\\mathcal{R}$, it is not easy to estimate the exact value of $d(\\mathcal{R},\\mathcal{M})$ for\na stochastic process $M$. But its upper bound, i.e., $\\beta$, can be easily obtained.\nThe parameter\n$\\beta$ describes how unpredictable the real source $\\mathcal{R}$ is, so we call it the \\emph{uncertainty} of $\\mathcal{R}$.\nWe prove that it is impossible to construct an extractor that achieves efficiency strictly larger than $1-\\beta$\nfor all the possible sources $\\mathcal{R}$ with uncertainty $\\beta$. Then we introduce several constructions of variable-length extractors, and show that their efficiencies can reach $\\eta \\geq 1-\\beta$; that is, the constructions are asymptotically optimal. The proposed variable-length extractors have two benefits: (i) they are generalizations of algorithms for perfect sources to  address general imperfect sources; and (ii) they bridge the gap between min-entropy and entropy on efficiency.\n\nThe following example is constructed to compare the performances of a variable-length extractor and\na fixed-length extractor when extracting randomness from a slightly-unpredictable independent process.\n\\begin{Example} Consider an independent process $x_1x_2x_3...$ such that $P[x_i=1]\\in [0.9, 0.91]$, then it can be obtained that $\\beta\\leq 0.0315$.\nFor this source, a variable-length extractor can generate random bits with efficiency at least $1-\\beta=0.9685$ that is very close to the upper bound $1$. In comparison, fixed-length extractors can only reach the efficiency at most $0.3117$.\n\\end{Example}\n\n\nThe remainder of this paper is organized as follows. Section \\ref{sec_pre} presents background and related results.\nIn Section \\ref{var_sec_efficiency}, we demonstrate that one cannot construct a variable-length extractor with efficiency strictly larger than $1-\\beta$ when the source has uncertainty $\\beta$. Then we focus on the seeded constructions of variable-length extractors, namely, we use a small number of additional truly random bits as the seed (catalyst).\nThree different constructions are provided and analyzed in Section \\ref{var_section_tech1}, Section \\ref{var_section_tech2} and Section \\ref{var_section_tech4}\nseparately. All these constructions have efficiencies lower bounded by $1-\\beta$, implying their optimality.\nFinally, we discuss seedless constructions of variable-length extractors for some types of random sources in Section \\ref{sec_randomnessextraction}, followed by the concluding remarks.\n\n\n\\section{Preliminaries}\n\\label{sec_pre}\n\n\\subsection{Statistical Distance}\n\n\\emph{Statistical Distance} is used in computer science to measure the difference between two distributions. Let $X$ and $Y$ be two random sequences with range $\\{0,1\\}^m$, then the statistical distance between $X$ and $Y$ is defined as\n$$\\|X-Y\\|=\\max_{T:\\{0,1\\}^m\\rightarrow \\{0,1\\}} |P[T(X)=1]-P[T(Y)=1]|$$\nover a boolean function $T$.\nWe say that $X$ and $Y$ are $\\epsilon$-close if $\\|X-Y\\|\\leq \\epsilon$. According to this definition, we can also write\n$$\\|X-Y\\|=\\frac{1}{2}\\sum_{x\\in \\{0,1\\}^m}|P[X=x]-P[Y=x]|\\leq \\epsilon.$$\nIt is equivalent to the former expression.\n\nLet $U_m$ denote the uniform distribution on $\\{0,1\\}^m$. In order to let a sequence $Y$ to be able to take place of the truly random bits in a randomized application, we let $Y$ be $\\epsilon$-close to $U_m$, where $\\epsilon$ is small enough. In this case,\nthe extra probability error introduced by this replacement is at most $\\epsilon$.\nIn this paper, we want to extract $m$ almost-random bits such that they form a sequence $\\epsilon$-close to the uniform distribution $U_m$ on $\\{0,1\\}^m$ with specified small $\\epsilon>0$, i.e.,\n$$\\|Y-U_m\\|\\leq \\epsilon.$$\n\n\\subsection{Seeded Extractors}\n\nIn 1990, Zuckerman introduced a general model of weak random sources, called $k$-sources, namely whose min-entropy is at least $k$ \\cite{Zuc90}.\nIt was shown that given a source on $\\{0,1\\}^n$ with min-entropy $k<n$, it is impossible to devise a single function that extracts even one bit of randomness. This observation led to the introduction of seeded extractors, which use a small number of additional truly random bits as the seed (catalyst). When simulating a probabilistic algorithm, one can simply eliminate the requirement of truly random bits by enumerating all possible strings for the seed and taking a majority vote on the final results. There are a variety of very efficient constructions of seeded extractors, summarized in \\cite{Dvir08, Nis96, Sha02}. Mathematically, a seeded extractor\nis a function,\n$$E: \\{0,1\\}^n \\times \\{0,1\\}^d \\rightarrow \\{0,1\\}^m,$$\nsuch that for every distribution $X$ on $\\{0,1\\}^n$ with $H_{\\min}(X)\\geq k$, the distribution $E(X,U_d)$ is $\\epsilon$-close to\nthe uniform distribution $U_m$. Here, $d$ is the seed length, and we call such an extractor as a $(k,\\epsilon)$ extractor.\nThere are a lot of works focusing on efficient constructions of seeded extractors. A standard application of the probabilistic method \\cite{Radhakrishnan2000} shows that there exists a seeded extractor which can extract asymptotically $H_{\\min}(X)$ random bits with $\\log(n-H_{\\min}(X))$ additional truly random bits. Recently, Guruswami, Umans and Vadhan \\cite{Guruswami07} provided an explicit construction of seeded extractors, whose efficiency is  very close to the bound obtained based on the probabilistic method. Their main result is described as follows:\n\n\\begin{Lemma} \\emph{\\cite{Guruswami07}} \\label{lemma_seededextractor} For every constant $\\alpha>0$, and all positive integers $n, k$ and all $\\epsilon>0$, there is an explicit construction of a $(k,\\epsilon)$ extractor $E: \\{0,1\\}^n\\times \\{0,1\\}^d\\rightarrow\\{0,1\\}^m$ with $d\\leq \\log n+ O(\\log(k\/\\epsilon))$ and $m\\geq (1-\\alpha)k$.\n\\end{Lemma}\n\nThe above result implies that given any source $X \\in \\{0,1\\}^n$ with min-entropy $k$, if $\\geq (1+\\alpha)m$ with $\\alpha>0$, we can always construct\na seeded extractor to generates a random sequence $Y\\in \\{0,1\\}^m$ that is $\\epsilon$-close to the uniform distribution. In this case,\nthe seed length $d\\leq \\log n + O(\\log(k\/\\epsilon))$ depends on the input length $n$ and the parameter $\\epsilon$.\n\n\\subsection{Seedless Extractors}\n\nIn the last decade, the concept of seedless (deterministic) extractors has attracted renewed interests, motivated by the reduction of the computational complexity for simulating probabilistic algorithms as well as some requirements in cryptography \\cite{Dodis00}. Several specific classes of sources have been studied, including independent sources, which can be divided into several independent parts containing certain amount of randomness \\cite{Barak06, Rao2007, Raz2005}; bit-fixing sources, where some bits in a binary sequence are truly random and the remaining bits are fixed \\cite{Cohen89, Gabizon06, Kamp06}; samplable sources, where the source is generated by a process that has a bounded amount of computational resources like space \\cite{Kamp11, Trevisan00}. For example, suppose that we have multiple independent sources with the same length $n$. It is known how to extract from two sources when\nthe min-entropy in each is $\\geq 0.5n$ \\cite{Raz2005} or slightly less than $0.5n$ \\cite{Bou05}, how to extract from $O(1\/\\gamma)$ sources if the min-entropy in each is at least $n^\\gamma$ \\cite{Rao06}. All these constructions have exponentially small error, and they are able to extract $\\Theta(k)$ random bits.\n\nBoth seeded extractors and seedless extractors described above have fixed input length, fixed seed length ($d=0$ for seedless extractors) and fixed output length. So we call them fixed-length extractors. To apply fixed-length extractors in extracting randomness from a stochastic process, it needs to first read a sequence of fixed length, whose min-entropy is strictly larger than the number of random bits that we need to generate. Fixed-length extractors can generate random bits of good quality from imperfect stochastic processes,\nbut they usually consume more incoming symbols than what are necessarily required. To increase information efficiency,\nwe let the length of input sequences be adaptive, hence, we have the concept of `variable-length extractors'.\n\n\\subsection{Variable-Length Extractors}\n\nA variable-length extractor is an extractor with variable input length and fixed output length. When applying a variable-length extractor to\na stochastic process, it reads incoming symbols one by one until the whole incoming sequence meets certain criterion, then it maps the incoming sequence into a binary sequence of fixed length as the output. Depending on the sources, the construction may require a small number of additional truly random bits as the seed. Hence, we have seeded variable-length extractors and seedless variable-length extractors.\n\nA seeded variable-length extractor is a function,\n$$V_E: S_p\\times\\{0,1\\}^d\\rightarrow\\{0,1\\}^m,$$\nsuch that given a real source $\\mathcal{R}$, the output sequence is $\\epsilon$-close to\nthe uniform distribution $U_m$. Here, $S_p$ is the set consisting of all possible input sequences, called the input set. It is complete and prefix-free. The input sequence is compete, that means, any infinite sequence has a prefix in the set; so when reading symbols from any source, we can always meet a sequence in the set. Then we stop reading and map this sequence into a binary sequence of length $m$. Being prefix-free is not very necessary; it ensures that all the sequences in $S_p$ are possible to read.\n\nA general procedure of extracting randomness by using variable-length extractors can be divided into three steps:\n\n\\begin{enumerate}\n  \\item Determining an input set $S_p$ such that its min-entropy based on the real source $\\mathcal{R}$ is at least $k$, namely,\n$$\\min_{x\\in S_p}\\log_2\\frac{1}{P_\\mathcal{R}(x)}\\leq k,$$\nwhere $k\\geq (1+\\alpha)m$ for any $\\alpha>0$.\n  \\item We construct an injective function\n$$V:S_p\\rightarrow \\{0,1\\}^n,$$\nto map the sequences in $S_p$ into binary sequences of length $m$. We read symbols from the source $\\mathcal{R}$ one by one until\nthe current incoming sequence matches one in $S_p$. This incoming sequence is then mapped to a binary sequence of length $n$ based on function $V$. As a result, we get a random sequence $Z$ with length $n$ and min-entropy $k$ (since $V$ is injective).\n  \\item Since $k=(1+\\alpha)$ with an $\\alpha>0$, according to Lemma \\ref{lemma_seededextractor}, we can always find a seeded extractor,\n  $$E:\\{0,1\\}^n \\times\\{0,1\\}^d\\rightarrow\\{0,1\\}^m$$\n  that can extract $m$ almost-random bits from a source with min-entropy $k$. By applying  this seeded extractor $E$ to the sequence $Z$,\n  we get a random sequence of length $m$ that is $\\epsilon$-close to the uniform distribution $U_m$. Here, the seed length $d\\leq \\log n + O(\\log(k\/\\epsilon))$.\n\\end{enumerate}\n\nWe can see that the construction of a variable-length extractor is a cascade of\na function $V$ and a seeded extractor $E$, i.e.,\n$$V_E=E\\bigotimes V.$$\n\nNote that our requirement is to extract a sequence of $m$ almost-random bits that is $\\epsilon$-close to the uniform distribution $U_m$. The key of constructing variable-length extractors is to find the input set $S_p$ with min-entropy $k$, even the distribution of the real source $\\mathcal{R}$ is slightly unpredictable, such that the expected length of the sequences in $S_p$ is minimized. For stationary ergodic processes, minimizing the expected length is equivalent to minimizing the entropy of the sequences in $S_p$ asymptotically (this will be discussed in this section).\n\nFor some specific types of sources, including independent sources and samplable sources, by applying the ideas in \\cite{Rao2007} and \\cite{Kamp11} we can remove\nthe requirement of truly random bits without degrading the asymptotic performance. As a result, we have seedless variable-length extractors.\nFor example, if the source $\\mathcal{R}$ is an independent process, we can first apply the method in \\cite{Rao2007} to extract $d$ almost-random bits from the first $\\Theta(\\log \\frac{m}{\\epsilon})$ bits, and then use them as the seed of a seeded variable-length extractor to extract randomness from the rest of the process. The detailed discussions will be given in Section \\ref{sec_randomnessextraction}.\n\n\\section{Efficiency and Uncertainty}\n\\label{var_sec_efficiency}\n\n\\subsection{Efficiency}\n\nTo consider the performance of an extractor, we define its \\emph{efficiency} as\nthe asymptotical ratio between the output length and the total entropy of all its inputs. So the efficiency of an extractor can be written as\n$$\\eta=\\lim_{m\\rightarrow\\infty} \\frac{m}{H_{\\mathcal{R}}(X_m)+d},$$\nsuch that the output sequence is $\\epsilon$-close to the uniform distribution $U_m$\non $\\{0,1\\}^m$, where $\\epsilon$ is small, $d$ is the seed length, $m$ is the output length, and $H_{\\mathcal{R}}(X_m)$ is the entropy of the input sequence $X_m$ with range on $S_p$.\nIn our constructions,  $d\\leq \\log n + O(\\log(m\/\\epsilon))$, which is ignorable compared to $H_{\\mathcal{R}}(X_m)$ when $m\\rightarrow\\infty$. Hence,\nwe can write\n$$\\eta=\\lim_{m\\rightarrow\\infty} \\frac{m}{H_{\\mathcal{R}}(X_m)}.$$\nIn the definition, we use the entropy of the input sequence rather than the expected input length, because the source that we considered may not be stationary ergodic.\nIt needs to mention that, in seeded constructions, the value of $d$ is also an important parameter although it is much smaller than $m$. The problem of minimizing the seed length $d$ can be studied separately from minimizing the entropy of the input sequence, and it will be addressed in this paper.\n\nFirst, we demonstrate that if a distribution is $\\epsilon$-close to the uniform distribution $U_m$, then\nthe entropy of this distribution is asymptotically $m$ for any $\\epsilon<1$.\n\n\\begin{Lemma}\\label{var_lemma2}\nLet $X$ be a random sequence on $\\{0,1\\}^m$ that is $\\epsilon$-close to the uniform distribution $U_m$, then\n$$ m- \\log_2\\frac{1}{1-\\epsilon}\\leq H(X)\\leq m.$$\n\\end{Lemma}\n\n\\proof Since there are totally $2^m$ possible assignments for $X$, it is easy to get $H(X)\\leq m$.\nSo we only need to prove that\n$$H(X)\\geq m- \\log_2\\frac{1}{1-\\epsilon}.$$\n\nLet $p(x)$ denote $P[X=x]$ for $x\\in \\{0,1\\}^m$.\nSince $X$ is  $\\epsilon$-close to the uniform distribution $U_m$, we have\n$$\\frac{1}{2} \\sum_{x\\in \\{0,1\\}^m} \\|p(x)-2^{-m}\\|\\leq \\epsilon.$$\n\nThen the lower bound of $H(X)$ can be written as\n$$\\min_{p} \\sum_{x\\in \\{0,1\\}^m} p(x) \\log_2\\frac{1}{p(x)}$$\nsubject to\n$$p(x)\\geq 0, \\forall x\\in \\{0,1\\}^m;$$\n$$\\sum_{x\\in\\{0,1\\}^m}p(x)=1;$$\n$$\\sum_{x\\in \\{0,1\\}^m} \\|p(x)-2^{-m}\\|\\leq 2\\epsilon.$$\n\nObviously, the optimal solution of the above  problem happens at\n$$\\sum_{x\\in \\{0,1\\}^m} \\|p(x)-2^{-m}\\|=2\\epsilon.$$\n\nTo solve the problem based on Lagrange Multipliers, we let\n$$\\lambda(p)= \\sum_{x\\in \\{0,1\\}^m}\\ p(x) \\log_2\\frac{1}{p(x)}+\\lambda_1(\\sum_{x\\in\\{0,1\\}^m}p(x)-1)$$\n$$+\\lambda_2 (\\sum_{x\\in \\{0,1\\}^m} \\|p(x)-2^{-m}\\|-2\\epsilon).$$\n\nIf $p(x)\\geq 0$ with $x\\in\\{0,1\\}^m$ is a solution of the above question, then\n$$\\frac{\\partial \\lambda }{\\partial(p(x))}=0,$$\ni.e.,\n$$ \\left\\{ \\begin{array}{cc}\n          \\frac{\\ln p(x)+1}{\\ln 2}+ \\lambda_1+\\lambda_2=0    &  \\textrm{ if } 2^{-m}\\leq p(x)\n          \\leq 1, \\\\\n           \\frac{\\ln p(x)+1}{\\ln 2}+ \\lambda_1-\\lambda_2=0   &  \\textrm{ if } 0\\leq p(x)\\leq 2^{-m}.\n           \\end{array}\n\\right.$$\n\nSo there exists two constants $a$ and $b$ with $0\\leq a\\leq 2^{-m}\\leq b\\leq 1$, such that,\n$$ \\left\\{ \\begin{array}{cc}\n          p(x)=a   &  \\textrm{ if } 2^{-m}\\leq p(x)\\leq 1, \\\\\n          p(x)=b   &  \\textrm{ if } 0\\leq p(x)\\leq 2^{-m}.\n           \\end{array}\n\\right.$$\n\nAssume that there are $t$ assignments of $x$ with $p(x)=a$, then there are $2^m-t$ assignments of $x$ with $p(x)=b$.\nHence, the problem is converted to the one over $a,b,t$, i.e.,\n$$\\min_{a,b,t} t a\\log \\frac{1}{a} +(2^m-t) b\\log \\frac{1}{b},$$\nsubject to\n$$0\\leq t\\leq 2^m;$$\n\\begin{equation}ta +(2^m-t)b=1;\\label{var_equ_entropy1}\\end{equation}\n\\begin{equation}t(2^{-m}-a)+(2^m-t)(b-2^{-m})=2\\epsilon.\\label{var_equ_entropy2}\\end{equation}\nFrom Equ.~(\\ref{var_equ_entropy1}) and (\\ref{var_equ_entropy2}), we get\n$$a=2^{-m}-\\frac{\\epsilon}{t}, \\quad b=2^{-m}+\\frac{\\epsilon}{2^m-t}.$$\n\nSo the question is finding the optimal $t$ that minimizes\n$$-t(2^{-m}-\\frac{\\epsilon}{t})\\log_2 (2^{-m}-\\frac{\\epsilon}{t})$$\n$$-(2^m-t)(2^{-m}+\\frac{\\epsilon}{2^m-t})\\log_2(2^{-m}+\\frac{\\epsilon}{2^m-t}),$$\nsubject to\n$$0\\leq t\\leq \\frac{\\epsilon}{2^{-m}}.$$\n\nThe optimal solution is $t^*=\\frac{\\epsilon}{2^{-m}}$. In this case, the entropy of $X$ is\n$$H(X)= \\log (2^m-t)=m-\\log_2\\frac{1}{1-\\epsilon},$$\nwhich is the lower bound.\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nIn the following lemma, we show that for any extractor, its efficiency is upper bounded by $1$. The reason is that the amount of information, i.e., entropy, does not increase\nduring the process of randomness extraction.\n\n\\begin{Lemma}\\label{lemma_efficiency1}\nFor any extractor with seed length $d$ and output length $m$, if $d=o(m)$, its efficiency $\\eta\\leq 1$.\n\\end{Lemma}\n\n\\proof We consider fixed-length extractors as a special case of variable-length extractors, and consider seedless extractors as a special case of seeded extractors when $d=0$. So  our proof only focus on seeded variable-length extractors.\n\nA main observation is that for any extractor, the entropy of its output sequence is bounded\nby the entropy of the input sequence plus the entropy of the seed, since the process of extracting randomness cannot create new randomness.\n\nFor the output sequence, denoted by $Y$, it is $\\epsilon$-close to the uniform distribution $U_m$. According to\nLemma \\ref{var_lemma2}, its entropy is $$H_{\\mathcal{R}}(Y)\\geq m-\\log_2\\frac{1}{1-\\epsilon}.$$\n\nThe total entropy of the inputs is $H_{\\mathcal{R}}(X_m)+d$. Hence,\n$$H_{\\mathcal{R}}(Y)\\leq H_{\\mathcal{R}}(X_m)+d.$$\n\nAs a result, the efficiency of the extractor is\n$$\\eta=\\lim_{m\\rightarrow\\infty} \\frac{m}{H_{\\mathcal{R}}(X_m)}=\\lim_{m\\rightarrow\\infty}\\frac{H_{\\mathcal{R}}(Y)}{H_{\\mathcal{R}}(X_m)+d}\\leq 1.$$\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nIf $\\mathcal{R}$ is a stationary ergodic process, we define its entropy rate as\n$$h(\\mathcal{R})=\\lim_{l\\rightarrow\\infty}\\frac{H(X^l)}{l},$$\nwhere $X^l$ is a random sequence of length $l$ generated from the source $\\mathcal{R}$.\nIn this case, the entropy of the input sequence on $S_p$ is proportional to the expected input length.\n\n\\begin{Lemma} \\label{var_theorem_stionaryergodiclength} Given a stationary ergodic source $\\mathcal{R}$, let $X_m$ be the input sequence of a variable-length extractor that has an output length $m$. Then\n$$\\lim_{m\\rightarrow\\infty} \\frac{H_{\\mathcal{R}}(X_m)}{E_{\\mathcal{R}}[|X_m|]}=h(\\mathcal{R}),$$\nwhere $E_{\\mathcal{R}}[|X_m|]$ is the expected input length.\n\\end{Lemma}\n\n\\proof $X_m$ is a random sequence from $S_p$ based on the distribution of $\\mathcal{R}$.\nLet $l_1$ be the minimum length of the sequences in $S_p$, as $m\\rightarrow\\infty$, $l_1\\rightarrow\\infty$.\nNow, we define $$l_i=l_1+(i-1)\\log l_1 \\textrm{ for all }i\\geq 1.$$ Based on them, we divide all the sequences in $S_p$ into subsets\n$$S_i=\\{x|x\\in S_p, l_i\\leq |x|\\leq l_{i+1}-1\\}$$\nfor $i\\geq 1$.\n\nLet $p_i=P_{\\mathcal{R}}(X_m\\in S_i)$, then\n$$H_{\\mathcal{R}}(X_m)\n   \\geq \\sum_{i}[(\\sum_{j>i} p_j)H_{\\mathcal{R}}(X_{l_{i-1}+1}^{l_i}|X_{1}^{l_{i-1}}, |X_m|\\geq l_i)],$$\nwhere $l_0=0$, $\\sum_{j>i} p_j$ is the probability that $|X_m|\\geq l_i$, and\n$X_a^b$ is a sequence of $X_m$ from the $a$th element to the $b$th element.\n\nSince $X_m$ is generated from a stationary ergodic process, and $l_i-{l_{i-1}}\\rightarrow\\infty$ as $m\\rightarrow\\infty$, we can get\n$$H_{\\mathcal{R}}(X_{l_{i-1}+1}^{l_i}|X_{1}^{l_{i-1}},  |X_m|\\geq l_i)\\rightarrow (l_i-l_{i-1})h(\\mathcal{R}).$$\n\nAs a result, as $l_1\\rightarrow\\infty$, we have\n\\begin{eqnarray*}\nH_{\\mathcal{R}}(X_m)&\\geq& (1-\\epsilon) \\sum_{i}(\\sum_{j>i} p_j) (l_i-l_{i-1})h(\\mathcal{R})\\\\\n&=&(1-\\epsilon)\\sum_i p_i l_i h(\\mathcal{R}),\n\\end{eqnarray*}\nfor an arbitrary $\\epsilon>0$.\n\nAlso considering the other direction, we can get\nthat as $l_1\\rightarrow\\infty$,\n\\begin{eqnarray*}\n H_{\\mathcal{R}}(X_m)\n&\\leq &(1+\\epsilon)\\sum_i p_i l_{i+1} h(\\mathcal{R})\\\\\n&=& (1+\\epsilon)\\sum_i p_i (l_i+\\log l_1) h(\\mathcal{R}),\n\\end{eqnarray*}\nfor an arbitrary $\\epsilon>0$.\n\nFor the expected input length, i.e., $E_{\\mathcal{R}}[|X_m|]$, it is easy to show that\n$$\\sum_i p_i l_i\\leq E_{\\mathcal{R}}[|X_m| ] \\leq \\sum_i p_i l_{i+1}=\\sum_i p_i( l_i+\\log l_1)  .$$\n\nSo as $m\\rightarrow \\infty$, i.e., $l_1\\rightarrow\\infty$, it yields\n$$\\lim_{m\\rightarrow\\infty} \\frac{H_{\\mathcal{R}}(X_m)}{E_\\mathcal{R}[|X_m|]}=\\lim_{m\\rightarrow\\infty}\\frac{\\sum_i p_i l_i h(\\mathcal{R})}{\\sum_i p_i l_i}$$\n$$=h(\\mathcal{R}).$$\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\n\\subsection{Sources and Uncertainty}\n\nGiven a source $\\mathcal{R}$, if its distribution is known, we say that this source is a known stochastic process, and its uncertainty is $0$. In this paper,\nwe mainly focus on those imperfect processes whose distributions are slightly unpredictable due to many factors like\nthe existence of external adversaries.\n\nFirst, given two known stochastic processes $\\mathcal{R}$ and $\\mathcal{M}$, we let\n$d(\\mathcal{R},\\mathcal{M})$ be the difference between $\\mathcal{R}$ and $\\mathcal{M}$.  Here, we define $d(\\mathcal{R},\\mathcal{M})$ as\n$$d(\\mathcal{R},\\mathcal{M})=\\limsup_{n\\rightarrow\\infty} \\max_{x\\in \\{0,1\\}^n} \\frac{\\log_2\\frac{P_{\\mathcal{R}}(x)}{P_{\\mathcal{M}}(x)}}{\\log_2\\frac{1}{P_{\\mathcal{M}}(x)}},$$\nwhere $P_{\\mathcal{R}}(x)$ is the probability of generating $x$ from $\\mathcal{R}$ when the sequence length is $|x|$, and $P_{\\mathcal{M}}(x)$ is the probability of generating $x$ from $\\mathcal{M}$ when the sequence length is $|x|$. Although there are some existing ways such as normalized Kullback-Leibler divergence to measure the difference between two sources, with them it is not easy to estimate the uncertainty of a source and it is\nnot easy to analyze the performances of constructed variable-length extractors.\n\nIn the rest of this paper, we investigate a few models of unpredictable sources. Most natural source can be well described in those ways.\n\n\\begin{enumerate}\n  \\item The source $\\mathcal{R}$ is an arbitrary stochastic process such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a constant $\\beta\\in [0,1]$ and\n  a known stochastic process $\\mathcal{M}$.\n  \\item $\\mathcal{R}$ is an arbitrary stochastic process such that there exists a stationary ergodic process $\\mathcal{M}$ (whose distribution is unknown)\nand $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$; that is, $\\min_{\\mathcal{M}\\in \\mathcal{G}_{s.e.}}d(\\mathcal{R},\\mathcal{M})\\leq \\beta$,  where $\\mathcal{G}_{s.e.}$ denotes the set consisting of all stationary ergodic processes.\n\\end{enumerate}\n\nIn both the models, we call $\\beta$ as the \\emph{uncertainty} of the source $\\mathcal{R}$. In the real world,\n$\\beta$ can be easily estimated without knowing the distribution of the processes.\nIt just reflects how unpredictable the real source $\\mathcal{R}$ is.\n\nTo construct variable-length extractors, we only care about the possible input sequences, namely, those in $S_p$. Hence, for the case\nof finite length, $d_p(\\mathcal{R},\\mathcal{M})$ is a more important parameter for us, defined by\n$$d_p(\\mathcal{R},\\mathcal{M})= \\max_{x\\in S_p} \\frac{\\log_2\\frac{P_{\\mathcal{R}}(x)}{P_{\\mathcal{M}}(x)}}{\\log_2\\frac{1}{P_{\\mathcal{M}}(x)}},$$\n\nAs the number of required random bits $m$ increases, $d_p(\\mathcal{R},\\mathcal{M})$ quickly converge to $d(\\mathcal{R}, \\mathcal{M})$.\nAnd we can write\n$$d_p(\\mathcal{R},\\mathcal{M})=d(\\mathcal{R}, \\mathcal{M})+\\epsilon_p$$\nfor a very small constant $\\epsilon_p$. As $m\\rightarrow\\infty$, $\\epsilon_p\\rightarrow 0$. In this case, the upper bound of $d_p(\\mathcal{R},\\mathcal{M})$ or\n$\\min_{\\mathcal{M}\\in \\mathcal{G}_{s.e.}}d_p(\\mathcal{R},\\mathcal{M})$ is\n$$\\beta_p=\\beta+\\epsilon_p.$$\n\n\\begin{Example} Let $x_1x_2...\\in \\{0,1\\}^*$ be a sequence generated from an independent source $\\mathcal{R}$ such that $$\\forall i\\geq 1, P[x_i=1]\\in [0.8,0.82].$$\nIf we let $\\mathcal{M}$ be a biased coin with probability $0.8132$, then\n$$\\beta=\\max_{\\textrm{possible }\\mathcal{R}} d(\\mathcal{R},\\mathcal{M})$$\n$$=\\max(\\frac{\\log_2 \\frac{0.2}{0.1868}}{\\log_2 \\frac{1}{0.1868}}, \\frac{\\log_2 \\frac{0.82}{0.8132}}{\\log_2 \\frac{1}{0.8132}})=0.0405.$$ \\hfill  $\\Box$\n\\end{Example}\n\nAccording to our definition, $d(\\mathcal{M},\\mathcal{R})\\leq \\beta$ if and only if\n$$P_{\\mathcal{R}}(x)\\leq P_{\\mathcal{M}}(x)^{1-\\beta}$$\nfor all $x\\in\\{0,1\\}^\\infty$ with  $|x|\\rightarrow\\infty$. This is a condition that is very easy to be satisfied by many natural stochastic processes for a small $\\beta$.\n\n\\begin{Lemma}  If $d(\\mathcal{R},\\mathcal{M})\\rightarrow 0$, we have\n$$P_{\\mathcal{R}}(x)\\rightarrow P_{\\mathcal{M}}(x) $$\nfor all $x\\in \\{0,1\\}^*$.\n\\end{Lemma}\n\n\\subsection{Efficiency and Uncertainty}\n\nIn this subsection, we investigate the relation between the efficiency and uncertainty. We show that given\na stochastic process $\\mathcal{R}$ with uncertainty $\\beta$, as described in the previous subsection, one cannot construct a variable-length extractor with efficiency strictly larger than $1-\\beta$ for all the possibilities of $\\mathcal{R}$.\n\nLet us first consider a simple example: let $X$ be a random sequence with the uniform distribution on $\\{0,1\\}^n$ and let $Y$\nbe an arbitrary random sequence on $\\{0,1\\}^n$ such that $$\\frac{\\log_2 \\frac{ P[Y=x]}{P[X=x]}}{\\log_2\\frac{1}{P[X=x]}}\\leq \\beta, \\forall x\\in \\{0,1\\}^n.$$\nNow, we show that from the source $Y$, one cannot construct an extractor with efficiency strictly larger than $1-\\beta$.\nTo see this, we consider an extractor $f$ with output length $m$, and a source $Y$ with\n$$P[Y=y]\\in \\{0, 2^{-n(1-\\beta)}\\}, \\forall y\\in \\{0,1\\}^n.$$\nFor this a source $Y$, its entropy is $H(Y)=n(1-\\beta)$. In order to make sure the output sequence of $f$, denoted by $Z$,\nis $\\epsilon$-close to $U_m$, it has\n$$\\lim_{m\\rightarrow\\infty} \\frac{m}{n(1-\\beta)}\\leq \\lim_{m\\rightarrow\\infty} \\frac{H(Z)+\\log_2\\frac{1}{1-\\epsilon}}{H(Y)}\\leq 1.$$\nSo we cannot generate more than $n(1-\\beta)$ random bits asymptotically. In this case, if we apply the seeded extractor $f$\nto the random sequence $X$, which is a possibility of $Y$, then the efficiency is\n$$\\eta=\\lim_{m\\rightarrow\\infty}\\frac{m}{H(X)}=\\lim_{m\\rightarrow\\infty}\\frac{m}{n}\\leq 1-\\beta.$$\nSo there does not exist a seeded extractor that can extract randomness from an arbitrary $Y$ and its efficiency is strictly larger than $1-\\beta$.\nHere, $\\beta$ is the uncertainty of the source.\n\n\n\\begin{Theorem} \\label{var_theorem_lowerbound1} Let $\\mathcal{M}$ be a known stochastic process, and $\\mathcal{R}$ be an arbitrary stochastic process such that\n$d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then one cannot construct a variable-length extractor whose efficiency\nis strictly larger than $1-\\beta$ for all possible $\\mathcal{R}$.\n\\end{Theorem}\n\n\\proof Let $f$ be a variable-length extractor whose input sequence is a random sequence $X_m$ on $S_p$ and\nits output sequence is a random sequence $Y$ on $\\{0,1\\}^m$. Assume that as $m\\rightarrow\\infty$, $f$ can extract from an arbitrary $\\mathcal{R}$ such that the output sequence  $Y$ is $\\epsilon$-close to $U_m$.\n\nLet $h=H_{\\mathcal{M}}(X_m)$ be the entropy of the input sequence based on the distribution of $\\mathcal{M}$,\nthen we want to show that there exists a process $\\mathcal{R}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ and\n$H_{\\mathcal{R}}(X_m)\\leq h(1-\\beta)$ as $m\\rightarrow\\infty$.\n\nTo find such a process $\\mathcal{R}$, we order all the elements in $S_p$ as\n$x_1, x_2, x_3, ...$\nsuch that\n$$P_{\\mathcal{M}}(x_1)\\geq P_{\\mathcal{M}}(x_2)\\geq P_{\\mathcal{M}}(x_3)\\geq ...$$\n\nThen we divide all these elements into groups,\n$$\\{x_1, x_2, ..., x_{i_1}\\}, \\{x_{i_1+1},x_{i_1+2},...,x_{i_2}\\},...$$\nsuch that the total probability of the elements in each group is almost the probability of its first element to the power of $1-\\beta$, i.e.,\n$$0\\leq P_{\\mathcal{M}}(x_{i_j+1})^{1-\\beta}-\\sum_{k=i_j+1}^{i_{j+1}} P_{\\mathcal{M}}(x_k)\\leq P_{\\mathcal{M}}(x_{i_j+1}),$$  for all  $j\\geq 0$,\nwhere $i_0=0$.\n\nLet $A=\\{x_{1}, x_{i_1+1}, x_{i_2+1}, ...\\}$ be the set consisting of the first elements of all the groups.\nNow, we consider a possibility of $\\mathcal{R}$ in the following way: for all $x\\in \\{x_{1}, x_{i_1+1}, x_{i_2+1}, ...\\}$,\nits probability is\n$$P_{\\mathcal{R}}(x)=\\sum_{k=i_j+1}^{i_{j+1}} P_{\\mathcal{M}}(x_k), \\textrm{ if } x=x_{i_j+1};$$\nFor all $x\\in S_p\/A=S_p\/\\{x_{1}, x_{i_1+1}, x_{i_2+1}, ...\\}$, its probability is\n$$P_{\\mathcal{R}}(x)=0.$$\n\nFor this source $\\mathcal{R}$, the entropy of the input sequence is\n$$H_{\\mathcal{R}}(X_m)=\\sum_{x\\in S_p} P_{\\mathcal{R}}(x) \\log_2 \\frac{1}{P_{\\mathcal{R}}(x)}.$$\n\nAs $m\\rightarrow\\infty$, we have\n\\begin{eqnarray*}\n&&H_{\\mathcal{R}}(X_m)\\\\\n   &= & \\sum_{x\\in A} P_{\\mathcal{R}}(x) \\log_2 \\frac{1}{P_{\\mathcal{R}}(x)}  \\\\\n &\\rightarrow& (1-\\beta) \\sum_{x\\in A}  P_{\\mathcal{R}}(x) \\log_2 \\frac{1}{P_{\\mathcal{M}}(x)}\\\\\n&=& (1-\\beta)  \\sum_{j\\geq 0} \\sum_{k=i_j+1}^{i_{j+1}} P_{\\mathcal{M}}(x_k) \\log_2 \\frac{1}{P_{\\mathcal{M}}(x_{i_j+1})}\\\\\n&\\leq & (1-\\beta)  \\sum_{j\\geq 0}  \\sum_{k=i_j+1}^{i_{j+1}} P_{\\mathcal{M}}(x_k) \\log_2 \\frac{1}{P_{\\mathcal{M}}(x_k)}\\\\\n&=& (1-\\beta) H_{\\mathcal{M}}(X_m)\\\\\n&=& (1-\\beta)h.\n\\end{eqnarray*}\n\nAccording to Lemma \\ref{var_lemma2}, as $m\\rightarrow\\infty$, $\\frac{m}{H_{\\mathcal{R}}(Y)}\\rightarrow 1$.\nFurthermore, we can get $$\\lim_{m\\rightarrow\\infty}\\frac{H_{\\mathcal{R}}(Y)}{H_{\\mathcal{R}}(X_m)} \\leq 1, $$\nit implies that\n$$\\lim_{m\\rightarrow\\infty} \\frac{m}{(1-\\beta)h}\\leq 1,$$\notherwise, the output sequence cannot be $\\epsilon$-close to the uniform distribution $U_m$.\n\nIf we apply the extractor $f$ to the source $\\mathcal{M}$, which is also a possibility for $\\mathcal{R}$, then its efficiency is\n$$\\eta=\\lim_{m\\rightarrow\\infty}\\frac{m}{h}\\leq 1-\\beta.$$\n\nSo it is impossible to construct a variable-length extractor with efficiency strictly larger than $1-\\beta$ for all the possibilities of the source $\\mathcal{R}$. This completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nWith the same proof, we can also get the following theorem.\n\n\\begin{Theorem} Let $\\mathcal{R}$ be an arbitrary stochastic process such that\n$d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a stationary ergodic process $\\mathcal{M}$ with unknown distribution,\n, then one cannot construct a variable-length extractor whose efficiency\nis strictly larger than $1-\\beta$ for all possible $\\mathcal{R}$.\n\\end{Theorem}\n\nThe above theorems show that one cannot construct an extractor whose efficiency is strictly larger than $1-\\beta$ for all the possible source $\\mathcal{R}$. Here, $\\beta$ is an important parameter that measures the uncertainty of a real source $\\mathcal{R}$, either to\na known process or to the nearest stationary ergodic process.\nIn the next a few sections, we will present a few constructions for efficiently extracting randomness from the sources described in this section.\nWe show that their efficiency $\\eta$ satisfies\n$$1-\\beta\\leq \\eta\\leq 1.$$\nThat means the bound $1-\\beta$ is actually achievable and the constructions proposed in this paper are asymptotically optimal on efficiency.\n\n\n\\section{Construction I: Approximated by Known Processes}\n\\label{var_section_tech1}\n\nIn this section, we consider those sources which can be approximated by a known stochastic process $\\mathcal{M}$, namely, an arbitrary process\n$\\mathcal{R}$ with $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a known process $\\mathcal{M}$.\nWe say that a stochastic process $\\mathcal{M}$ is known if its distribution is given, i.e., $P_{\\mathcal{M}}(x)$ can be easily calculated for any $x\\in \\{0,1\\}^*$.\nNote that this process $\\mathcal{M}$ is not necessary to be stationary or ergodic. For instance, $\\mathcal{M}$ can be an independent process\n$z_1z_2...\\in \\{0,1\\}^*$ such that $$\\forall i\\geq1, P_{\\mathcal{M}}(z_i=1)= \\frac{1+sin(i\/10)}{2}.$$\n\n\\subsection{Construction}\n\nOur goal is to extract randomness from an imperfect random source $\\mathcal{R}$. The problem is that\nwe do not know the exact distribution of $\\mathcal{R}$, but we know that it can be approximated by a known process $\\mathcal{M}$.\nSo we can use the distribution of $\\mathcal{M}$ to estimate the distribution of $\\mathcal{R}$. As a result, we have the following procedure to extract $m$ almost-random bits.\n\nThe idea of the procedure is first producing a random sequence of length $n$ and min-entropy $k=m(1+\\alpha)$ with $\\alpha>0$, from which\nwe can further obtain a sequence $\\epsilon$-close to the uniform distribution $U_m$ by applying a $(k,\\epsilon)$  seeded extractor.\nAccording to the results of seeded extractors, this constant $\\alpha>0$ can be arbitrarily small.\n\n\\begin{Construction}\\label{const:1}\nAssume the real source $\\mathcal{R}$ is an arbitrary stochastic process such that $d(\\mathcal{R}, \\mathcal{M})\\leq \\beta$\nfor a known process $\\mathcal{M}$. Then we extract $m$ almost-random bits from $\\mathcal{R}$ based on the following procedure.\n\n\\begin{enumerate}\n  \\item Read input bits one by one from $\\mathcal{R}$ until we get an input sequence $x\\in \\{0,1\\}^*$ such that\n  $$\\log_2 \\frac{1}{P_{\\mathcal{M}}(x)}\\geq \\frac{k}{1-\\beta_p},$$\n  where  $\\beta_p=\\beta+\\epsilon_p$ with $\\epsilon_p>0$ and $k=m(1+\\alpha)$ with $\\alpha>0$. The small constant $\\epsilon_p$ has value depending on the input set $S_p$; as\n  $m\\rightarrow\\infty$, $\\epsilon_p\\rightarrow 0$. The constant $\\alpha$ can be arbitrarily small.\n  \\item  Let $n$ be the maximum length of all the possible input sequences, then\n  {$$n=\\arg\\min_{l}\\{l\\in \\mathbb{N}|\\forall y\\in \\{0,1\\}^l, $$\n  $$\\log_2\\frac{1}{P_\\mathcal{M}(y)}\\geq \\frac{k}{1-\\beta_p}\\}.$$}\n  If $|x|<n$, we  extend the length of $x$ to $n$ by adding $n-|x|$ trivial zeros at the end.\n Since $x$ is randomly generated, from the above procedure we get a random sequence $Z$ of length $n$. And it can be proved\n that this random sequence has min-entropy $k$.\n  \\item Applying a $(k,\\epsilon)$ extractor to $Z$ yields a binary sequence of length $m$ that is $\\epsilon$-close to the uniform distribution $U_m$.  \\hfill$\\Box$\n\\end{enumerate}\n\\end{Construction}\n\nThe following example is provided for comparing this construction with fixed-length constructions.\n\n\\begin{Example} Let $\\mathcal{M}$ be a biased coin with probability $0.8$ (of being $1$). If $\\frac{k}{1-\\beta_p}=2$, then we can get the input set\n$$S_p=\\{0, 10, 110, 1110, 11110,111110, 1111110, 1111111\\}.$$\nIn this case, the expected input length is strictly smaller than $7$.\nFor fixed-length constructions, to get a random sequence with min-entropy at least $2$, we have to read $7$ input bits independent of the context.\nIt is less efficient than the former method.\\hfill\n\t$\\Box$\n\\end{Example}\n\n\\begin{Theorem} Construction \\ref{const:1} generates a random sequence of length $m$ that is $\\epsilon$-close to $U_m$.\n\\end{Theorem}\n\n\\proof We only need to prove that given a source $\\mathcal{R}$ and a model $\\mathcal{M}$ with $d_p(\\mathcal{R},\\mathcal{M})\\leq \\beta_p$, it yields a random sequence $Z$ with min-entropy at least $k$.\n\nAccording to the definition of $d_p(\\mathcal{R},\\mathcal{M})$, for all $x\\in S_p$,\n$$\\frac{\\log_2\\frac{P_\\mathcal{R}(x)}{P_\\mathcal{M}(x)}}{\\log_2 \\frac{1}{P_\\mathcal{M}(x)}}\\leq \\beta_p.$$\n\nBased on the construction, for all $x\\in S_p$\n$$\\log_2\\frac{1}{P_\\mathcal{M}(x)}\\geq \\frac{k}{1-\\beta_p}.$$\n\nThe two inequalities above yield that\n$$\\log_2 \\frac{1}{P_\\mathcal{R}(x)}\\geq k,$$\nfor all $x\\in S_p$.\n\nSince the second step, i.e., adding trivial zeros, does not reduce the min-entropy of $S_p$. As\na result, we get a random sequence $Z$ of length $n$ and with min-entropy at least $k$.\n\nSince $k=m(1+\\alpha)$ with $\\alpha>0$, according to Lemma \\ref{lemma_seededextractor},\nwe can construct a seeded extractor that applies to the sequence $Z$ and generates a binary sequence $\\epsilon$-close to the uniform distribution $U_m$.\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\n\\subsection{Efficiency Analysis}\n\nNow, we study the efficiency of Construction \\ref{const:1}. According to our definition, given\na construction, its efficiency is\n$$\\eta=\\lim_{m\\rightarrow\\infty}\\frac{m}{H_{\\mathcal{R}}(X_m)}.$$\n\n\\begin{Theorem}\\label{var_theorem1_2} Given a real source $\\mathcal{R}$ and a known process $\\mathcal{M}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then the efficiency of Construction \\ref{const:1} is\n$$1-\\beta\\leq \\eta\\leq 1.$$\n\\end{Theorem}\n\n\\proof Since $\\eta$ is always upper bounded by $1$, we only need to show that $\\eta\\geq 1-\\beta$.\n\n According to Lemma \\ref{lemma_seededextractor}, as $m\\rightarrow \\infty$,\nwe have\n$$\\lim_{m\\rightarrow\\infty}\\frac{k}{m}=1.$$\n\nNow, let us consider the number of elements in $S_p$, i.e., $|S_p|$. To calculate $|S_p|$, we let\n$$S_p'=\\{x[1:|x|-1] |x\\in S_p\\},$$\nwhere $x[1:|x|-1]$ is the prefix of $x$ of length $|x|-1$,\nthen for all $y\\in S_p'$, $$\\log_2\\frac{1}{P_\\mathcal{M}(y)}\\leq \\frac{k}{1-\\beta_p}.$$  Hence,\n$$\\log_2|S_p'|\\leq \\frac{k}{1-\\beta_p}.$$\n\nIt is easy to see that $|S_p|\\leq 2|S_p'|$, so\n$$\\log_2|S_p|\\leq \\frac{k}{1-\\beta_p}+1.$$\n\nLet $X_m$ be the input sequence, then\n$$\\lim_{k\\rightarrow\\infty}\\frac{H_{\\mathcal{R}}(X_m)}{k}\\leq \\lim_{k\\rightarrow\\infty}\\frac{\\log_2|S_p|}{k}$$\n$$\\leq \\lim_{k\\rightarrow\\infty}\\frac{1}{1-\\beta_p}=\\frac{1}{1-\\beta}.$$\n\nFinally, it yields\n$$\\eta=\\lim_{m\\rightarrow\\infty}\\frac{m}{H_{\\mathcal{R}}(X_m)}\\geq 1-\\beta.$$\n\nThis completes the proof.\\hfill\\IEEEQED\\vspace{0.1in}\n\nWe see that the efficiency of the above construction is between $1-\\beta$ and $1$. As shown in Theorem \\ref{var_theorem_lowerbound1}, the\ngap $\\beta$, introduced by the uncertainty of the real source $\\mathcal{R}$, cannot be smaller.  Our construction is asymptotically optimal in the sense that we cannot find a variable-length extractor with efficiency definitely larger than $1-\\beta$.\n\n\\begin{Corollary} Given a real source $\\mathcal{R}$ and a known process $\\mathcal{M}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then as $\\beta\\rightarrow 0$, the efficiency of Construction \\ref{const:1} is\n$$\\eta\\rightarrow 1.$$\n\\end{Corollary}\n\nIn this case, the efficiency of the construction can achieve Shannon's limit.\n\nIf $\\mathcal{R}$ is a stationary ergodic process, we can also get the following result.\n\n\\begin{Corollary}  Given a stationary ergodic process$\\mathcal{R}$ and a known process $\\mathcal{M}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, for the expected input length of Construction \\ref{const:1}, we have\n$$\\frac{1}{h(\\mathcal{R})}\\leq \\lim_{m\\rightarrow\\infty} \\frac{E[|X_m|]}{m}\\leq \\frac{1}{(1-\\beta)h(\\mathcal{R})},$$\nwhere $h(\\mathcal{R})$ is the entropy rate of the source $\\mathcal{R}$.\n\\end{Corollary}\n\n\\proof This conclusion is immediate following Lemma \\ref{var_theorem_stionaryergodiclength}  and Theorem \\ref{var_theorem1_2}.\n\\hfill\n\\IEEEQED\\vspace{0.1in}\n\n\\section{Construction II: Approximately Biased Coins}\n\\label{var_section_tech2}\n\nIn this section, we use a general ideal model such as a biased coin or a Markov chain to approximate the real source $\\mathcal{R}$.\nHere, we do not care about the specific parameters of the ideal model. The reason is, in some cases, the source $\\mathcal{R}$ is\nvery close to an ideal source but we cannot (or do not want to) estimate the parameters accurately.\nAs a result, we introduce a construction by exploring the characters of biased coins or Markov chains. For simplicity,\nwe only discuss the case that\nthe ideal model is a biased coin, and the same idea can be generalized when the ideal model is a Markov chain. Specifically,\nlet $\\mathcal{G}_{b.c.}$ denote the set consisting of all the models of biased coins with different probabilities, and we\nconsider $\\mathcal{R}$ as an arbitrary stochastic process such that\n$$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta.$$\n\n\\subsection{Construction}\n\nThe idea of the construction is similar as Construction \\ref{const:1}, i.e., we  first produce a random sequence of length $n$ and with min-entropy $k=m(1+\\alpha)$ for $\\alpha>0$, from which\nwe can further obtain a sequence $\\epsilon$-close to the uniform distribution $U_m$ by applying a $(k,\\epsilon)$  seeded extractor.\n\n\\begin{Construction}\\label{const:2}\nAssume the real source $\\mathcal{R}$ is an arbitrary stochastic process such that $$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta$$ for a constant $\\beta$.   Then we extract $m$ almost-random bits from $\\mathcal{R}$ based on the following procedure.\n\\begin{enumerate}\n\\item Read input bits one by one from $\\mathcal{R}$ until we get an input sequence $x\\in \\{0,1\\}^*$ such that\n     $$\\log_2 {\\nchoosek{k_0+k_1}{ \\max(1,\\min(k_0,k_1))}}\\geq \\frac{k}{1-\\beta_p},$$\n     where $k_0$ is the number of zeros in $x$, $k_1$ is the number of ones in $x$,\n    $\\beta_p=\\beta+\\epsilon_p$ with $\\epsilon_p>0$ and $k=m(1+\\alpha)$ with $\\alpha>0$. The small constant $\\epsilon_p$ has value depending on the input set $S_p$; as\n  $m\\rightarrow\\infty$, $\\epsilon_p\\rightarrow 0$. The constant $\\alpha$ can be arbitrarily small.\n\\item Since the input sequence $x$ can be very long, we map it into a sequence $z$ of fixed length $n$ such that\n$$z=[I_{(k_0\\geq k_1)}, \\min(k_0,k_1), r(x)],$$\nwhere $I_{(k_0\\geq k_1)}=1$ if and only if $k_0\\geq k_1$, and $r(x)$ is the rank of $x$  among all the permutations of $x$ with respect to the lexicographic order.\nSince $x$ is randomly generated, the above procedure leads us to a random sequence $Z$ of length $n$.\n\\item Applying a $(k,\\epsilon)$ extractor to $Z$ yields a random sequence of length $m$ that is $\\epsilon$-close to $U_m$.\\hfill$\\Box$\n\\end{enumerate}\n\\end{Construction}\n\nTo see that the construction above works, we need to show that the random sequence $Z$ obtained after the second step has min-entropy at least $k$, and its length $n$ is well bounded.\n\n\\begin{Lemma}\n Given a source $\\mathcal{R}$ with $\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta$, Construction \\ref{const:2} yields a random sequence $Z$ with length\n $$n\\leq 1+\\lceil\\log_2(\\frac{k}{1-\\beta_p}+1)\\rceil+ \\lceil\\frac{2k}{1-\\beta_p}\\rceil.$$\n\\end{Lemma}\n\\proof\n1) $I_{(k_0\\geq k_1)}$ can be represented as $1$ bit.\n\n2) Without loss of generality, we assume $k_0\\leq k_1$. According to our construction,\n$$ \\log_2 {\\nchoosek{k_0+k_1-1}{ k_0-1}}< \\frac{k}{1-\\beta_p} \\textrm{ for } k_0>1,$$\nand\n$$ \\log_2 {\\nchoosek{k_1}{1}} < \\frac{k}{1-\\beta_p} \\textrm{ for } k_0=0 \\textrm{ or } k_0=1.$$\n\nThen\n\\begin{eqnarray*}\nk_0-1&\\leq &\\log_2 {\\nchoosek{2k_0-1}{ k_0-1}}\\\\\n&\\leq& \\log_2 {\\nchoosek{k_0+k_1-1}{k_0-1}}\\\\\n&<& \\frac{k}{1-\\beta_p}.\n\\end{eqnarray*}\n\nSo $\\min(k_0, k_1)$ can be represented as\n$\\lceil\\log_2 (\\frac{k}{1-\\beta_p} +1 )\\rceil$ bits.\n\n3) Let us consider the number of permutations of $x$, denoted by $N(x)$.\nIf $k_0>1$, then\n\\begin{eqnarray*}\n\\log_2 N(x)&=&\\log_2 {\\nchoosek{k_0+k_1}{ k_0}}\\\\\n&\\leq &\\log_2  {\\nchoosek{k_0+k_1-1}{ k_0-1}} +\\log_2\\frac{k_0+k_1}{k_0}\\\\\n&\\leq &\\frac{k}{1-\\beta_p} +\\log_2\\frac{k_0+k_1}{k_0}.\n\\end{eqnarray*}\n\nIf $k_0=1$, then\n$$\\log_2 N(x) \\leq \\log_2  {\\nchoosek{k_1 }{1}} +\\log_2 \\frac{k_1+1}{k_1}.$$\n\nIf $k_0=0$, then\n$$\\log_2 N(x)=0.$$\n\nBased on the analysis above, we can get $$\\log_2 N(x)\\leq \\frac{2k}{1-\\beta_p}.$$\n\nHence, $r(x)$ can be represented as $\\lceil\\frac{2k}{1-\\beta_p}\\rceil$ bits.\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nLet $\\mathbf{1}^a$ denote the all-one vector of length $a$, then we get the following result.\n\\begin{Theorem}\n Construction \\ref{const:2} generates a random sequence of length $m$ that is $\\epsilon$-close to $U_m$ if\n $P_{\\mathcal{R}}(\\mathbf{1}^a)\\leq 2^{-k}, P_{\\mathcal{R}}(\\mathbf{0}^a)\\leq 2^{-k}$ for $a=2^{\\lfloor\\frac{k}{1-\\beta_p}\\rfloor}$.\n\\end{Theorem}\n\n\\proof Since the mapping in the second step is injective, it will not affect the min-entropy; we only need to prove that the input sequence has min-entropy $k$, i.e.,\n$$\\log_2 \\frac{1}{P_{\\mathcal{R}}(x)}\\geq k, \\forall x\\in S_p,$$\nwhere $S_p$ is the set consisting of all the possible input sequences.\n\nIt is not hard to see that if $\\min(k_0,k_1)\\geq 1$,\n$$P_{\\mathcal{M}}(x)\\leq \\frac{1}{{\\nchoosek{k_0+k_1}{ k_0}}},$$\nwhich leads to\n$$\\log_2 \\frac{1}{P_{\\mathcal{M}}(x)}\\geq \\frac{k}{1-\\beta_p}.$$\n\nFurthermore, based on the definition of $d_p(\\mathcal{R},\\mathcal{M})$, we can get if $\\min(k_0,k_1)\\geq 1$,\n$$\\log_2 \\frac{1}{P_{\\mathcal{R}}(x)}\\geq k.$$\n\nIf $\\min(k_0,k_1)=0$, according to the condition in the lemma, we can also have the same result.\n\nSince $k=m(1+\\alpha)$ with $\\alpha>0$, according to Lemma \\ref{lemma_seededextractor},\nwe can construct a seeded extractor that applies to the sequence $Z$ and generates a binary sequence $\\epsilon$-close to the uniform distribution $U_m$.\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nActually, the idea above can be easily generalized if $\\mathcal{M}$ is a Markov chain that best approximates the real source $\\mathcal{R}$.\nThe idea follows the main lemma in \\cite{Zhou_Markov} that shows how to generate random bits with optimal efficiency from an arbitrary Markov chain.\n\n\\subsection{Efficiency Analysis}\n\nFor the efficiency of the construction, we can get the same bounds as Construction \\ref{const:1}.\n\n\\begin{Theorem}\nGiven an arbitrary source $\\mathcal{R}$ such that $$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}}d (\\mathcal{R},\\mathcal{M}) \\leq \\beta,$$\nif there exists a model $\\mathcal{M}\\in \\mathcal{G}_{b.c.}$ with probability $p\\leq \\frac{1}{2}$ of being $1$ or $0$ and\n$$p>\\sqrt{d(\\mathcal{R},\\mathcal{M})\\log_2\\frac{1}{p}\\frac{\\ln 2}{2}},$$ then the efficiency of Construction \\ref{const:2} is\n$$1-\\beta \\leq \\eta\\leq 1.$$\n\\end{Theorem}\n\n\\proof Let $N_{k_0, k_1}$ denote the number of input sequences with $k_0$ zeros and $k_1$ ones in $S_p$, and let $p_{k_0,k_1}$ be the probability based on $\\mathcal{R}$ of generating such a sequence. Let us define\n$$A=\\{(k_0,k_1)| N_{k_0,k_1}>0\\},$$\nthen we can get\n$$\nH_{\\mathcal{R}}(X_m)\\leq H(\\{p_{k_0,k_1}|(k_0,k_1)\\in A\\})$$\n$$+\\sum_{(k_0,k_1)\\in A}p_{k_0,k_1}\\log_2 N_{k_0,k_1}.\n$$\n\nAccording to the proof in the above theorem, $\\min(k_0,k_1)\\leq \\frac{k}{1-\\beta_p}+1$. So there are totally at most $2(\\frac{k}{1-\\beta_p}+1)$ available pairs of $(k_0, k_1)$. Hence\n$$H(\\{p_{k_0,k_1}|(k_0,k_1)\\in A\\})\\leq \\log_2(2+(\\frac{k}{1-\\beta_p}+1))=o(k).$$\n\nNow, we write $n=k_0+k_1$. According to our method, if $\\min(k_0,k_1)\\geq 1$,\n$$\\nchoosek{k_0+k_1}{\\min(k_0,k_1)}\\geq 2^{\\frac{k}{1-\\beta_p}},$$\n$$\\nchoosek{k_0+k_1-1}{\\min(k_0,k_1)-1}<2^{\\frac{k}{1-\\beta_p}}.$$\n\nHence, given $n$, we get an upper bound for $\\min(k_0,k_1)$, which is\n\\begin{equation}\\label{var_equ_lemma1_1}\nt_n=\\max\\{i\\in\\{0,1,...,n\\}| {\\nchoosek{n-1}{ i-1}}<2^{\\frac{k}{1-\\beta_p}}\\}.\n\\end{equation}\n\nNote that if $\\nchoosek{n-1}{\\frac{n}{2}-1}\\geq 2^{\\frac{k}{1-\\beta_p}}$, then $t_n$ is a nondecreasing\nfunction of $n$. Using the Stirling bounds on factorials yields\n$$ \\lim_{n\\rightarrow \\infty }\\frac{1}{n}\\log_2 {\\nchoosek{n}{ \\rho n}}=H(\\rho),$$\nwhere $H$ is the binary entropy function. Hence, following (\\ref{var_equ_lemma1_1}), we can get\n\\begin{equation}\\label{var_equ_lemma1_3} \\lim_{n\\rightarrow \\infty} H(\\frac{t_n}{n})=\\lim_{n\\rightarrow\\infty}\\frac{k}{(1-\\beta_p) n}.\\end{equation}\n\nLet $P_n$ denote the probability of having an input sequence of length at least $n$ based on the distribution of $\\mathcal{R}$. In this case,\n$P_n$ is a nonincreasing function of $n$. Let $Q_n$ denote the probability of having an input sequence of length at least $n$ based on the distribution of $\\mathcal{M}\\in \\mathcal{G}_{b.c.}$\nwhose probability is $p\\leq \\frac{1}{2}$. Since for all binary sequence $x\\in \\{0,1\\}^n$,\n$$\\log_2 \\frac{1}{P_\\mathcal{M}(x)}\\leq n\\log_2\\frac{1}{p},$$\nwe can get\n$$\\log_2\\frac{P_{\\mathcal{R}(x)}}{P_\\mathcal{M}(x)}\\leq d n\\log_2 \\frac{1}{p},$$\nwhere $d=d_p(\\mathcal{R},\\mathcal{M})$.\n\nSince\n$P_n=\\sum_{x\\in S} P_{\\mathcal{R}}(x)$ and $Q_n=\\sum_{x\\in S}\nP_{\\mathcal{M}}(x)$ for some $S\\subset \\{0,1\\}^n$, it is not hard to prove that\n\\begin{equation}\\label{equ_lemma1_13}\\log_2\\frac{P_n}{Q_n}\\leq d n \\log_2\\frac{1}{p}.\\end{equation}\n\nAccording to Hoeffding's inequality, we can get\n\\begin{eqnarray*}\nQ_n&\\leq &2P[k_1\\leq t_n]\\\\\n&\\leq& 2P[\\frac{k_1}{n}-p \\leq \\frac{t_n}{n}-p] \\\\\n&\\leq& 2e^{-2n(p-\\frac{t_n}{n})^2}.\n\\end{eqnarray*}\n\nHence\n\\begin{equation}\\label{equ_lemma1_14}P_n \\leq 2^{- d n\\log_2 p} Q_n \\leq 2 e^{-\\log_2 p \\ln 2 \\cdot d n  -2 n (p-\\frac{t_n}{n})^2}.\\end{equation}\n\nFrom this inequality, we see that $P_n\\rightarrow 0$ as $n\\rightarrow 0$ if\n\\begin{equation}\\label{equ_lemma1_15}-d \\log_2 p \\ln 2 -2 (p-\\frac{t_n}{n})^2 <0.\\end{equation}\n\nBased on (\\ref{var_equ_lemma1_3}) and (\\ref{equ_lemma1_15}), we can get that $P_n\\rightarrow 0$ as $n\\rightarrow 0$ if\n$$\\frac{n}{k}\\geq \\frac{1}{(1-\\beta_p)H(p-\\sqrt{d\\log_2\\frac{1}{p}\\frac{\\ln 2}{2}})}.$$\n\nNow, let $a=\\frac{1+\\epsilon}{(1-\\beta_p)H(p-\\sqrt{d\\log_2\\frac{1}{p}\\frac{\\ln 2}{2}})}$ with $\\epsilon>0$, we can write\n\\begin{eqnarray*}\nH_{\\mathcal{R}}(X_m) &\\leq &o(k)+ \\sum_{k_0,k_1: k_0+k_1\\geq a k} p_{k_0,k_1}\\log_2 N_{k_0,k_1}\\\\\n&&+ \\sum_{k_0,k_1: k_0+k_1<a k} p_{k_0,k_1}\\log_2 N_{k_0,k_1}.\n\\end{eqnarray*}\n\nAccording to our analysis, if $k_0+k_1\\geq a k$, as $k\\rightarrow \\infty$, $$P_n=\\sum_{k_0,k_1:k_0+k_1\\geq a k} p_{k_0,k_1}\\rightarrow 0$$ and\n$\\log_2 N_{k_0,k_1}\\leq 2\\frac{k}{1-\\beta_p}$.  If $k_0+k_1\\leq a k$, then $$\\log_2 N_{k_0,k_1}\\leq \\frac{k}{1-\\beta_p}+\\log_2 \\frac{k_0+k_1}{\\min (k_0,k_1)}\\leq \\frac{k}{1-\\beta_p} +o(k).$$ As a result, we can get\n\\begin{eqnarray*}\nH_{\\mathcal{R}}(X_m)& \\leq &o(k)+ o(1)\\frac{2k}{1-\\beta_p} + (\\frac{k}{1-\\beta_p}+o(k))\\\\\n&\\leq& \\frac{k}{1-\\beta_p}+o(k).\n\\end{eqnarray*}\n\nSo\n$$\\lim_{k\\rightarrow\n\\infty}\\frac{k}{H_{\\mathcal{R}}(X_m)}\\geq 1-\\beta.$$\n\nFurthermore, based on the fact that $\\lim_{m\\rightarrow\\infty}\\frac{k}{m}=1$, we can get $\\eta\\geq 1-\\beta$.\nIt is known that $\\eta\\leq 1$, so it concludes the theorem.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nSimilar to Construction \\ref{const:1}, this construction is also asymptotically optimal in the sense\nthat we cannot find a variable-length extractor with efficiency definitely larger than $1-\\beta$, as shown in Theorem \\ref{var_theorem_lowerbound1}.\n\n\\begin{Corollary} Given an arbitrary source $\\mathcal{R}$ such that $$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta,$$ then as $\\beta\\rightarrow 0$, the efficiency of Construction \\ref{const:2} is\n$$\\eta\\rightarrow 1.$$\n\\end{Corollary}\n\nIt is easy to see that as $\\beta\\rightarrow 0$, Construction \\ref{const:2} reaches the Shannon's limit on efficiency. If\n$\\mathcal{R}$ is a stationary ergodic process, we can also get the following corollary.\n\n\\begin{Corollary}\nGiven an arbitrary stationary ergodic source $\\mathcal{R}$ such that $$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta,$$\nif there exists a model $\\mathcal{M}\\in \\mathcal{G}_{b.c.}$ with probability $p\\leq \\frac{1}{2}$ of being $1$ or $0$ and\n$$p>\\sqrt{d(\\mathcal{R},\\mathcal{M})\\log_2\\frac{1}{p}\\frac{\\ln 2}{2}},$$ then for the expected input length of Construction \\ref{const:2}, we have\n$$\\frac{1}{h(\\mathcal{R})}\\leq \\lim_{m\\rightarrow\\infty} \\frac{E[|X_m|]}{m}\\leq \\frac{1}{(1-\\beta)h(\\mathcal{R})},$$\nwhere $h(\\mathcal{R})$ is the entropy rate of $\\mathcal{R}$.\n\\end{Corollary}\n\n\n\n\n\n\\section{Construction III: Approximately Stationary Ergodic Processes}\n\\label{var_section_tech4}\n\n\nIn this section, we consider imperfect sources that are approximately stationary and ergodic.\nHere, we let  $\\mathcal{R}$ be an arbitrary stochastic process such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a stationary ergodic process $\\mathcal{M}$.\nFor these sources, universal data compression can be used to `purify' input sequences, i.e., shortening their lengths while maintaining their entropies.\nIn \\cite{Visweswariah98}, Visweswariah, Kulkarni and Verd\\'{u} showed that optimal variable-length source codes asymptotically achieve optimal variable-length random bits generation in the sense of normalized divergence. Although\ntheir work only focused on ideal stationary ergodic processes and generates `weaker' random bits, it motivates us to combine\nuniversal compression with fixed-length extractors for efficiently generating random bits from noisy stochastic processes.\nIn this section, we will first introduce Lempel-Ziv code and then present its application in constructing variable-length extractors.\n\n\\subsection{Construction}\n\nLempel-Ziv code is a universal data compression scheme introduced by Ziv and Lempel \\cite{Ziv78}, which is simple to implement and can achieve the asymptotically optimal rate\nfor stationary ergodic sources. The idea of Lempel-Ziv code is to parse the source sequence into strings that have not appeared so far, as demonstrated by the following example.\n\n\\begin{Example} Assume the input is $010111001110000...$, then we parse it as strings\n$$0,1,01,11,00,111,000,...$$\nwhere each string is the shortest string that never appear before. That means all its prefixes have occurred earlier.\n\nLet $c(n)$ be the number of strings obtained by parsing a sequence of length $n$. For each string, we\ndescribe its location with $\\log c(n)$ bits. Given a string of length $l$, it can described by (1)\nthe location of its prefix of length $l-1$, and (2) its last bit. Hence, the code for the above sequence is\n{ $$(000,0), (000,1), (001,1), (010,1), (001,0), (100,1), (101,0),...$$}\nwhere the first number in each pair indicates the prefix location and the second number is\nthe last bit of the string.\n\\end{Example}\n\\vspace{-0.25cm}\n\\hfill$\\Box$\n\nTypically, Lempel-Ziv is applied to an input sequence of fixed length. Here, we are interested in\nLempel-Ziv code with fixed output length and variable input length. As a result, we can apply\na single fixed-length extractor to the output of Lempel-Ziv code for extracting randomness.\nIn our algorithm, we read raw bits one by one from an imperfect source until the length of the output\nof a Lempel-Ziv code reaches a certain length.\nIn another word,\nthe number of strings after parsing is a predetermined number $c$. For example, if\nthe source is $1011010100010...$ and $c=4$, then after reading $6$ bits, we can parse them into\n$1, 0, 11, 01$. Now, we get an output sequence\n$(000,1), (000,0), (001, 1), (010, 1)$, which can be used as the input of a fixed-length extractor.\nWe call this Lempel-Ziv code as a variable-length Lempel-Ziv code.\n\nLet $Z$ be a random sequence obtained based on variable-length Lempel-Ziv code such that its length is\n$$|Z|=(\\log c+1) c,$$\nfor a predetermined $c$. Then $Z$ is very close to truly random bits in the term of min-entropy if the source $\\mathcal{R}$ is stationary ergodic.\nAs a result, we have the following construction for variable-length extractors.\n\n\\begin{Construction}\\label{const:3}\nAssume the real source is $\\mathcal{R}$ and there exists a stationary ergodic process $\\mathcal{M}$ such that $d(\\mathcal{R},M)\\leq \\beta$.\nThen we extract $m$ almost random bits from $\\mathcal{R}$ based on the following procedure.\n\\begin{enumerate}\n\\item Read input bits one by one based on the variable-length Lempel-Ziv code until we get an output sequence $Z$ whose length reaches $$n=\\frac{k}{1-\\beta_p}(1+\\varepsilon),$$\n    where $\\varepsilon>0$ is a small constant indicating the performance gap between the case of finite-length and that of infinite-length\n    for Lempel-Ziv code; as $m\\rightarrow\\infty$, we have $\\varepsilon\\rightarrow 0$. Similar as above,\n    $\\beta_p=\\beta+\\epsilon_p$ with $\\epsilon_p>0$ and $k=m(1+\\alpha)$ with $\\alpha>0$. The small constant $\\epsilon_p$ has value depending on the input set $S_p$; as\n  $m\\rightarrow\\infty$, $\\epsilon_p\\rightarrow 0$. The constant $\\alpha$ can be arbitrarily small.\n    Then we get a random sequence $Z$ of length $n$ and with min-entropy $k$.\n\\item Applying a $(k,\\epsilon)$ extractor to $Z$ yields a random sequence of length $m$ that is $\\epsilon$-close to $U_m$.\\hfill$\\Box$\n\\end{enumerate}\n\\end{Construction}\n\nWe show that the min-entropy of $Z$ is at least $k$ as $m\\rightarrow\\infty$.  If $m$ is not very large, by adjusting the parameter $\\varepsilon$,\nwe can make the min-entropy of $Z$ be at least $k$.\nSo we can continue to apply\nan efficient fixed-length extractor to `purify' the resulting sequence. Finally, we can get $m$\nrandom bits that satisfy our requirements on quality in the sense of statistical distance.\n\n\\begin{Theorem}\\label{theorem_3_1}\nWhen $m\\rightarrow\\infty$, Construction \\ref{const:3} generates a random sequence of length $m$ that is $\\epsilon$-close to $U_m$.\n\\end{Theorem}\n\n\\proof Let $x$ be an input sequence. According to theorem 12.10.1 in \\cite{Cover2006}, for the stationary ergodic process $\\mathcal{M}$,\nwe can get\n$$\\frac{1}{|x|}\\log_2 \\frac{1}{ P_{\\mathcal{M}}(x)}\\geq \\frac{c}{|x|}\\log_2 c -\\frac{c}{|x|} H(U,V),$$\nwhere  $$\\frac{c}{|x|}H(U,V)\\rightarrow 0 \\textrm{ as } |x|\\rightarrow 0.$$\n\nAs a result, if $k=\\Theta(n)$,\n\\begin{eqnarray*}\n\\lim_{k\\rightarrow\\infty}\\frac{1}{k}\\log_2 \\frac{1}{ P_{\\mathcal{R}}(x)}&\\geq& \\lim_{k\\rightarrow\\infty}(1-\\beta_p)\\frac{1}{k}\\log_2\\frac{1}{P_\\mathcal{M}(x)}\\\\\n& \\geq &\\lim_{k\\rightarrow\\infty} \\frac{(1-\\beta_p)c\\log_2 c}{k} \\\\\n&=&  \\lim_{k\\rightarrow \\infty}\\frac{(1-\\beta_p)n}{k}\\\\\n&=& \\lim_{k\\rightarrow\\infty} 1+\\varepsilon\\\\\n&=& 1.\n\\end{eqnarray*}\n\nFinally, we can get that\n$$\\lim_{k\\rightarrow\\infty} \\frac{H_{\\min}(Z)}{k}=\\lim_{k\\rightarrow\\infty} \\frac{H_{\\min}(X_m)}{k}\\geq 1.$$\nThis implies that as $m\\rightarrow \\infty$, i.e., $k\\rightarrow \\infty$, the min-entropy of $Z$ is at least $k$.\n\n\nSince $k=m(1+\\alpha)$ for an $\\alpha>0$,  we can continue to apply a $(k,\\epsilon)$ extractor to extract $m$ almost-random bits from $Z$.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\n\\subsection{Efficiency Analysis}\n\nNow, we study the efficiency of the construction based on variable-length Lempel-Ziv codes.\n\n\\begin{Theorem}\nGiven a real source $\\mathcal{R}$ such that there exists a stationary ergodic process $\\mathcal{M}$ with $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then the efficiency of Construction \\ref{const:3} is \\vspace{-0.05cm}\n$$1-\\beta\\leq \\eta\\leq 1.$$\n\\end{Theorem}\n\n\\proof Similar as above, we only need to prove that $\\eta\\geq 1-\\beta$.\n\nSince there are at most $n=2^{c(\\log_2c+1)}$ distinct input sequences,\ntheir entropy\n$$H_{\\mathcal{R}}(X_m)\\leq c(\\log_2 c+1)=n.$$\n\nAccording to the proof in Theorem \\ref{theorem_3_1}, we have that the random sequence $Z$ has min-entropy at least $k$, and it satisfies\n$$\\lim_{m\\rightarrow\\infty}\\frac{n}{k}=\\frac{1}{1-\\beta}.$$\n\nBased on the construction of seeded extractors, we can also get\n$$\\lim_{m\\rightarrow\\infty}\\frac{m}{k}=1.$$\n\nAs a result,\n$$ \\eta= \\lim_{m\\rightarrow\\infty}\\frac{m}{H_{\\mathcal{R}}(X_m)} \\geq 1-\\beta.$$\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nAlthough Construction \\ref{const:3} has the same efficiency as the other constructions, when $m$ is not large,\nit is  less efficient than the other constructions because the Lempel-Ziv code does not always have the best performance when the input sequence is not long.\nBut its advantage is that it can manage more general sources without accurate estimations. In the above theorem, the gap $\\beta$ represents how far the source $\\mathcal{R}$ is from being stationary ergodic. In general, the efficiency loss introduced by the uncertainty of sources\nis a part that cannot be avoid.\n\n\\begin{Corollary} Given a real source $\\mathcal{R}$ such that there exists a stationary ergodic model $\\mathcal{M}$ with $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then as $\\beta\\rightarrow 0$, the efficiency of Construction \\ref{const:3} is\n$$\\eta\\rightarrow 1.$$\n\\end{Corollary}\n\nIt shows that as $\\beta\\rightarrow 0$, Construction \\ref{const:3} reaches the Shannon's limit on efficiency.\n\n\\begin{Corollary} Given a stationary ergodic source $\\mathcal{R}$ (assume we do not know that it is stationary ergodic), for the expected input length of Construction \\ref{const:3}, we have\n$$\\frac{1}{h(\\mathcal{R})}\\leq \\lim_{m\\rightarrow\\infty} \\frac{E[|X_m|]}{m}\\leq \\frac{1}{(1-\\beta)h(\\mathcal{R})},$$\nwhere $h(\\mathcal{R})$ is the entropy rate of $\\mathcal{R}$.\n\\end{Corollary}\n\n\n\\section{Seedless Constructions}\n\\label{sec_randomnessextraction}\n\nTo simulate seeded constructions of variable-length extractors in randomized applications, we have to enumerate all possible assignments of the seed, hence,\nthe computational complexity will be increased significantly. In real applications, we prefer seedless constructions rather than seeded constructions. It motivates us to study the seedless constructions of variable-length extractors in this section.\n\n\\subsection{An Independent Source}\n\nLet us first consider a simple independent source described in the introduction.  This type of sources have been widely studied in seedless constructions of fixed-length extractors.\n\n\\begin{Example} Let $x_1x_2...\\in\\{0,1\\}^*$ be an independent sequence generated from a source $\\mathcal{R}$ such that\n$$P[x_i=1]\\in [0.9,0.91] \\quad \\forall i\\geq i.$$\n\\end{Example}\n\\vspace{-0.4cm} \\hfill$\\Box$\n\\vspace{0.2cm}\n\nWe see that the existing methods for generating random bits from ideal sources (like biased coins or Markov chains) cannot be applied here, since\nthe probability of each bit is slightly unpredictable. Some seedless extractors have been developed for extracting randomness from such sources.\nIn particular, there exists seedless extractors which are able to extract as many as $H_{\\min}(X)$ random bits from a independent random sequence $X$ asymptotically. In order to extract $m$ random bits in the above example, it needs to read $\\frac{m}{\\log_2\\frac{1}{0.91}}$ input bits as\n$m\\rightarrow\\infty$. In this case, the entropy of the input sequence is in $$[H(0.9)\\frac{m}{\\log_2\\frac{1}{0.91}}, H(0.91)\\frac{m}{\\log_2\\frac{1}{0.91}}].$$ From which, we can get the efficiency of an optimal fixed-length  extractor, which is\n$$\\eta_{fixed}\\in [0.2901, 0.3117],$$\ni.e., about only $0.3$ of the input entropy is used for generating random bits, which is far from optimal\n\nIn the above example, we let $\\mathcal{M}$ be a biased coin model with probability $p=0.9072$. In this case,\n$$\\beta\\leq d(\\mathcal{R},\\mathcal{M})=0.0315.$$\nAccording to the constructions in the previous sections, there exists seeded variable-length extractors such that\ntheir efficiencies are\n$$\\eta_{variable}\\in [1-\\beta, 1]\\subseteq [0.9685,1],$$\nwhich are near Shannon's limit.\n\nBased on the fact that the source is independent, we can eliminate the requirement of truly random bits as the seed, hence, we have seedless\nvariable-length extractors.\nTo construct a seedless variable-length extractor, we first apply a seedless fixed-length extractor $E_1$ (which may not be very efficient)\nto extract a random sequence of length $d$ from input bits.\nUsing this random sequence as the seed, we continue to apply a seeded variable-length extractors $E_2$ to\nextract $m$ almost-random bits from extra input bits. So seedless variable-length extractors can be constructed as cascades of seedless fixed-length extractors and seeded variable-length extractors. Since the input length of $E_1$ is much shorter (it is ignorable) than the input length\nof $E_2$, the efficiency of the resulting seedless extractor, i.e., $E=E_2\\bigotimes E_1$, is dominated by the efficiency of $E_2$.\nSo the efficiency of the seedless extractor $E$ is in $[0.9685,1]$,\nwhich is very close to the optimality.\n\nThis example demonstrates a simple construction of seedless variable-length extractors for independent sources, and it shows\nthe significant performance gain of variable-length extractors compared to fixed-length extractors.\n\n\\subsection{Generalized Sources}\n\n\nHere we consider a generalization of independent processes. Given a system, we use $\\lambda_i$ denote the complete system status at time $i$.\nFor example, in a system that generates thermal noise, the system status can include the value of the noise signal, the temperature, the environmental effects, etc. Usually, the evolution of such a system has a Markov property, namely,\n$$P[\\lambda_{i+1},\\lambda_{i+2},...|\\lambda_i, \\lambda_{i-1},..., \\lambda_1]=P[\\lambda_{i+1},\\lambda_{i+2},...|\\lambda_i],$$\nfor all $i\\geq 1$. Let $X=x_1x_2...\\in \\{0,1\\}^n$ be the binary sequence generated from this system, then for any $1< k<n-1$,\n\\begin{equation}\\label{equ_seedless_1}P[X_1^{k-1},X_{k+1}^{n}|\\lambda_k]=P[X_1^{k-1}|\\lambda_k]P[X_{k+1}^{n}|\\lambda_k],\\end{equation}\nwhere $X_a^b=x_ax_{a+1}...x_{b}$. In some sense, the source $X$ that we consider is a hidden Markov process, but the number of hidden states can be infinite ($\\lambda_i$ can be discrete or continuous).\n\n\\begin{Example} One example of the above sources is the one studied in \\cite{Kamp11}, called a space $s$ source.\nA space $s$ source is basically a source generated by a width $2^s$ branching program. At each step, the state of the process\ngenerating the source is in one of $2^s$ states, and the bit generated is a function of the current state. Unlike perfect Markov chains,\nthe transition probabilities can be different at each step. In this example, the system status $\\lambda_i$ is the content of space $s$ at time $i$, that is, one of the $2^s$ states, and $x_i\\in \\{0,1\\}$ is a function of $\\lambda_i$.\n\\end{Example}\n\nSpace $s$ sources are very general in that most other classes of sources that have been considered\npreviously can be computed with a small amount of space \\cite{Kamp11}. The model that we consider, as described by (\\ref{equ_seedless_1}),\nis a natural generalization of space $s$ sources. This model has a very nice feature: from such a source, we can get a group of sequences conditionally independent of each other. Namely,\ngiven system statues at some time points\n$$[\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}]=[\\lambda_{a}, \\lambda_{2a}, ..., \\lambda_{\\gamma a}],$$\nthe subsequences\n$$[X^{(1)},X^{(2)},...,X^{(\\gamma)}, X^{(\\gamma+1)}]$$\n$$=[X_1^{a-1}, X_{a+1}^{2a-1}, ..., X_{(\\gamma-1)a+1}^{\\gamma a-1}, X_{\\gamma a+1}^\\infty]$$\nare conditionally independent of each other. Based on this condition, we have the following seedless construction of\nvariable-length extractors.\n\n\\begin{Construction}\\label{const:4}\nGiven a source $\\mathcal{R}$ described by (\\ref{equ_seedless_1}), we can construct a seedless variable-length extractor $E$ in the following way:\n\\begin{enumerate}\n  \\item Suppose that $$H_{\\min}(X^{(i)}|\\lambda^{(i)}, \\lambda^{(i+1)})\\geq k_{d}, \\forall 0\\leq i\\leq \\gamma.$$\n  We construct a $\\gamma$-source extractor \\cite{Rao2007} $E_1:[\\{0,1\\}^{a-1}]^\\gamma \\rightarrow \\{0,1\\}^d$ such that if each source has\n  min-entropy $k_d$, it can extract $d$ almost-random bits which are $\\epsilon_1$-close to the uniform distribution on $\\{0,1\\}^d$.\n  \\item We construct a seeded variable-length extractor $E_2: S_p\\times \\{0,1\\}^d\\rightarrow\\{0,1\\}^m$ such that\n  with condition on $\\lambda^{(\\gamma)}$,\n  it can extract $m$ almost-random bits from $X^{(\\gamma+1)}$ and these $m$ almost-random bits are $\\epsilon_2$-close to the uniform distribution on\n  $\\{0,1\\}^m$ if the seed is truly random.\n  \\item The seedless variable-length extractor $E$ is a cascade of $E_1$ and $E_2$:\n  Let $$D=E_1(X^{(1)}, X^{(2)},..., X^{(\\gamma)}),$$\n  then we apply $D$ as the seed of $E_2$ to generate $m$ almost-random bits from $X^{(\\gamma+1)}$; that is,\n  $$E(X)=E_2(X^{(\\gamma+1)}, E_1(X^{(1)}, X^{(2)},..., X^{(\\gamma)})).$$\n  \\end{enumerate}\n\\end{Construction}\n\nFor this construction, we have the following theorems.\n\n\\begin{Theorem} In Construction \\ref{const:4}, the $m$ almost-random bits generated by the seedless variable-length extractor $E$ are $(\\epsilon_1+\\epsilon_2)$-close to the uniform\ndistribution on $\\{0,1\\}^m$.\n\\end{Theorem}\n\n\\proof According to the construction, we can let the parameter $a=|X^{(i)}|+1$ with $1\\leq i\\leq \\gamma$ be large enough, so given $\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}$,\n$$\\|D-U_d\\|\\leq \\epsilon_1.$$\n\nLet $X_m$ be the input sequence of $E_2$ that read from $X^{(\\gamma+1)}$, then given  $\\lambda^{(\\gamma)}$,\nwe have\n$$\\|E_2(X_m, U_d)-U_m\\|\\leq \\epsilon_2.$$\n\nFrom the two inequalities above, given $\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}$,\nwe have\n$$\\|E_2(X_m, D)-U_m\\|\\leq \\epsilon_1+\\epsilon_2.$$\n\nSince it is true for any assignments of  $\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}$, we can get\n\\begin{eqnarray*}\n  &&  \\|E_2(X_m, D)-U_m\\|\\\\\n   &=& \\sum_{\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}}P[\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}] (\\epsilon_1+\\epsilon_2)\\\\\n   &\\leq & \\epsilon_1+\\epsilon_2.\n\\end{eqnarray*}\n\nHence, the $m$ almost-random bits extracted by $E$ is also $(\\epsilon_1+\\epsilon_2)$-close to $U_m$.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nIn the following theorem, we show that the seedless variable-length extractor $E$ has the efficiency as\nthe seeded variable-length extractor $E_2$.\n\n\\begin{Theorem} In Construction \\ref{const:4}, suppose that\n$$H_{\\min}(X^{(i)}|\\lambda^{(i)}, \\lambda^{(i+1)})=\\Theta(|X^{(i)}|), \\forall 0\\leq i\\leq \\gamma.$$\nLet $\\eta_{E}$ denote the efficiency of the resulting seedless variable-length extractor $E$, and\nlet $\\eta_{E_2}$ denote the efficiency of the $E_2$, then\n$$\\eta_E=\\eta_{E_2}.$$\n\\end{Theorem}\n\n\\proof According to the construction of $E_1$, we can get that\n$$d=\\Theta(a),$$\nwhere $a=|X^{(i)}|+1$ for $1\\leq i\\leq \\gamma$.\n\nIf $\\epsilon_2$ is a constant, then\n$$d=O(\\log m)=o(m).$$\n\nAs a result, $$\\lim_{m\\rightarrow\\infty}\\frac{a\\gamma}{m}=0.$$\n Let $H$ denote the entropy of the input sequence of $E_2$,\nthen\n$\\eta_{E_2}=\\lim_{m\\rightarrow\\infty}\\frac{m}{H}$, and\n$$\\lim_{m\\rightarrow\\infty}\\frac{m}{a\\gamma + H} \\leq \\eta_E \\leq \\lim_{m\\rightarrow\\infty}\\frac{m}{H}.$$\n\nHence, $\\eta_E=\\eta_{E_2}$.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nThe theorem above shows that the efficiency of seedless variable-length extractors can be very close to optimality. For many sources, such as biased coins with noise, or Markov chains with noise, the existing algorithms for ideal sources (e.g., perfect biased coins or perfect Markov chains)\ncannot generate high-quality random bits from them. At the same time, the traditional approaches of fixed-length extractors are not very efficient. The gap between their efficiency and the optimality is determined by the bias of the source. Seedless variable-length extractors take the advantages of both, as a result, they can approach the information-theoretic upper bound on efficiency while being capable to combat noise in the sources.\n\n\n\\section{Conclusion and Discussion}\n\nIn this paper, we introduced the concept of the variable-length extractors, namely, those extractors with variable input length and\nfixed output length. Variable-length extractors are generalizations of the existing algorithms for ideal sources to manage general stochastic processes. They are also improvements of traditional fixed-length extractors to fill the gap between min-entropy and entropy of the source\non efficiency. The key idea of constructing variable-length extractors is to approximate the source using a simple model, which is a known process,\na biased coin, or a stationary ergodic process. Depending on the model selected, we proposed and analyzed three seeded constructions\nof variable-length extractors. Their efficiency is lower bounded by $1-\\beta$ and upper bounded by $1$ (optimality), where\n$\\beta (0\\leq \\beta \\leq 1)$ indicate the uncertainty of the real source. We also show that our constructions are asymptotically optimal, in the sense that one cannot find a construction whose efficiency is always strictly larger than $1-\\beta$.\n In addition, we demonstrated how to construct\nseedless variable-length extractors by cascading seeded variable-length extractors with seedless fixed-length extractors. They can work for many (but not all) natural sources such as those based on noise signals.\n\nThere are certain connections between variable-length extractors and a whole family of variable-to-fixed length source codes \\cite{Lawrence1977,Tjalkens1992,Merhav1992,Savari1997,Tjalkens1987,Tunstall1967,Visweswariah2001}. With a variable-to-fixed length code, an infinite sequence is parsed into variable-length phases, chosen from some finite set $\\mathcal{D}$ of phases. Each phase is then\ncoded into a binary sequence of fixed length $m$. The set $\\mathcal{D}$ of phases is complete, i.e., every infinite sequence has a prefix in $\\mathcal{D}$. The key of constructing a good variable-to-fixed length source code is to find the best set $\\mathcal{D}$ that consists of at last $2^m$ prefix-free phases and maximizes the expected phase length. As comparison, the key of constructing a variable-length extractor is to find\nthe best input set $S_p$ that consists of sequences with probability at most $2^{-k}$ for each and minimizes the expected sequence length.\nAlthough their goals are different, some common ideas can be used to construct both the phase set $\\mathcal{D}$ and the input set $S_p$.\nFor example, in \\cite{Visweswariah2001}, Visweswariah et al.  defined  the phase set $\\mathcal{D}$ by $x^*\\in \\mathcal{D}$ if $P(x^*)\\leq c$ and no prefix of $x^*$ satisfies this property. The same idea is applied in our construction I.  In \\cite{Lawrence1977,Tjalkens1992},\nthe phase set $\\mathcal{D}$ is determined by the number of ones and zeros in the phase, so is our construction II.  In some sense,\nan optimal variable-to-fixed length code can result in a fixed-length binary sequence whose min-entropy is close to its length. However,\nvariable-to-fixed length source codes do not always work well in constructing variable-length extractors, because\n(1) the designing criteria are different and they may degrade the performance; (2) variable-to-fixed length source codes take both encoding and decoding in consideration, hence, they are more complex in computation than what we require (decoding is not necessary) for constructing variable-length extractors; and (3) the sources that we considered for variable-length extractors are unpredictable, which are more general than the ones considered in variable-to-fixed length source codes.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Discussion} \\label{section_discussion}\n\nIn this work, the Integrate-and-Fire model has been simulated and trained to achieve dog posture classification and showed promising results with relatively low energy expenditure when implemented with electronic components compared to the use of embedded logistic regression. The designed model implemented as hardware can be integrated into remote systems for embedded and energy-efficient pattern recognition, reducing the amount of data to transmit and thus reducing the number of transmissions, leading to low energy consumption.\nThe simulation of the IF neuron is faithful to the electronic implementation which makes it possible to train using the gradient descent algorithm. Once trained, the resistances that do not contribute to the pattern detection \u2013 i.e. those that converge to the highest value \u2013 are removed from the final circuit and the remaining are implemented with the final hardware. This hardware implementation has been done with only a few passive components (resistors, diodes and capacitors) and one active component (N-MOSFET transistors) which all have low costs. It has been implemented using prototyping boards but can be miniaturised on Printed Circuit Boards (PCBs) with Surface Mount Technology (SMT) that provides miniature components to produce a version of the hardware small enough to be integrated into small devices.\n\\par\nIn terms of power, the measured consumptions are almost identical due to the power demand of the micro-controller. However, the lifetime of battery-powered devices is very important and no aspect of the entire device should be overlooked, including the power usage of data processing. Therefore, by disregarding consumption of the micro-controller, the IF model consumes four times less when it is electronically implemented than a trained logistic regression running on the micro-controller. With this setup, the battery life-time is improved by 3.75\\%, but it can be enhanced even more by using a low-powered micro-controller. Moreover, an implementation of the model with spike trains should significantly reduce energy consumption. Therefore, it would be wise to rethink the way of communicating features given to the model using spike trains to further reduce the power consumption of the circuit.\n\\par\nOne main issue of the IF approach is the time dependence of the inference. As the stimulation time of synapses varies according to the input values, the inference time is also variable. Thus, the higher the inputs, the longer the inference time will be. This maximum inference time can be calculated by summing the maximum stimulation time of inputs or can be compensated by varying the capacitance value of units. Another issue of the IF model is that the leak channel, specific to the LIF neuron, has been removed and the time dimension disappeared. This model is thus no longer able to process animal dynamics to infer its activity and only the posture \u2013 i.e. static patterns \u2013 can be classified. To achieve this task, the LIF model should be used which involves transforming features into spike trains. However, due to the non-continuity of spike trains in spiking neural networks, algorithms based on differentiation \u2013 such as the gradient descent algorithm used in this work \u2013 cannot be applied for training.\n\n\\end{document}\n\\section{Future work}\nIn future works, the time capabilities of the Leaky Integrate-and-Fire model must be exploited to classify time-series patterns using spike trains. As the gradient descent algorithm is not suitable to train such models, other training algorithms must be explored to find new ways to classify patterns or compress sensory data into a spike code generated by a spiking neural network. Recent advances in neurosciences permitted the development of unsupervised learning algorithms such as Spike Time Dependent Plasticity (STDP) which is a biologically plausible Hebbian learning rule that adjusts the strength of connections between neurons in the brain \\cite{memristor_emulator_with_stdp,spike_time_dependence_of_plasticity,SNN_single_neurons_populations_plasticity}. Based on the timing of pre and post-synaptic spikes, STDP allows neurons to learn time-dependent correlations in spike trains and thus a relevant representation of input features \\cite{memristor_emulator_with_stdp,spike_time_dependence_of_plasticity,SNN_single_neurons_populations_plasticity}. Therefore, such algorithms may be able to find correlations between some sensory inputs and achieve a compression of recorded data.\n\n\\end{document}\n\\section{Results} \\label{section_results}\n\nThe capacitor is an electronic passive component that creates a potential difference between two conductive plates, analogous to the difference of electric potential of the biological neuron membrane created by ions that flow in and out of the cell. Therefore, the capacitor is often used in computational models of spiking neural networks to reproduce membrane potentials of the biological neurons. Connected in series or parallel with a resistance, the capacitor forms two circuits with distinct charge and discharge properties respectively called series and parallel Resistor-Capacitor circuits (RC). Thus, the IF neuron is mainly composed of passive components: a capacitor that reproduces the membrane potential and resistors that charge (excite) or discharge (inhibit) the neuron.\n\n\\subsection{Series RC circuit for excitatory stimulations} \\label{series_rc}\n\\begin{figure}\n\\centering\n    \\begin{circuitikz}[scale=0.7,transform shape] \\draw\n    (-3,1) node[left]{$V_{\\mathrm{in}}$}\n    to[R, o-, l=$R_e$] (0,1)\n    to[C, l=C] (2,1)\n    node[ground] {}\n    ;\n    \\end{circuitikz}\n    \\caption{Electric diagram of the series Resistor-Capacitor (RC) circuit. The circuit is composed of a voltage supplier $V_{in}$, a resistor $R_e$ that is analog to the excitatory synapses of the neuron and a capacitor $C$ that reproduces the membrane potential.}\n    \\label{fig:series_rc}\n\\end{figure}\nThe series RC circuit is defined by a successive resistor $R_e$ which represents the excitatory synapses of the biological neuron and a capacitor $C$ which reproduces the membrane potential \u2013 see Figure \\ref{fig:series_rc}. Taking into consideration Ohm's law ($I = \\frac{V}{R}$), the fact that the current $I_{R_e}$ flowing through the resistor $R_e$ is equal to the current $I_C$ flowing through the capacitor $C$ ($I_{R_e} = I_C$) and that the capacitor component theoretically does not produce any resistance, the current flowing through the circuit depends only on the input voltage $V_{in}$ and the excitatory resistor $R_e$. Thus, the resistor can be seen as a weight defined as $w = \\frac{1}{R_e}$ which scales the input value $V_{in}$ such as $I_{R_e} = wV_{in}$. Consequently, the higher the value of the resistor, the lower the current will flow through the capacitor and vice versa. Knowing that the total voltage $V_{in}$ of the circuit is defined as the sum of the voltages $V_R$ and $V_C$ respectively across the resistor and the capacitor ($V_{in} = V_R + V_C$), and the Ohm's law, we can define the following equation:\n\\begin{equation} \\label{series_rc_eq_1}\n    \\begin{aligned}\n    V_{in} = R_eI_{R_e} + V_{C}\\\\\n    \\Leftrightarrow I_{R_e} = \\frac{V_{in} - V_C}{R_e}\n    \\end{aligned}\n\\end{equation}\nEquation \\ref{series_rc_eq_1} shows that the current flows through the excitatory resistor does not only depend on the input voltage and the resistance but also depends on the voltage across the capacitor. Therefore, the higher the voltage across the capacitor, the lower the current flowing in the circuit will be. To describe the dynamic of the capacitor, the instantaneous rate of voltage change $\\frac{dV}{dt}$ of the capacitor is introduced as the current $I$ flowing through the capacitor divided by the capacitance $C$ ($\\frac{dV}{dt} = \\frac{I}{C}$). Equation \\ref{series_rc_eq_1} can be reformulated as:\n\\begin{equation}\\label{series_rc_eq_2}\n    \\begin{aligned}\n    \\tau_e\\frac{dV_C}{dt} = V_{in} - V_C\\\\\n    \\end{aligned}\n\\end{equation}\nwhere $\\tau_e = R_e C$ is the time constant of the series RC circuit which represents the number of seconds needed to reach approximately 63.2\\% of the input voltage $V_{in}$ \u2013 this value is explained below. For a constant input voltage and a given initial voltage $V_C(t)$ at time $t$, the capacitor voltage $V_C(t + \\Delta t)$ after an amount of time $\\Delta t$ can be found by integrating equation \\ref{series_rc_eq_2}:\n\\begin{equation} \\label{series_rc_eq_3}\n    \\begin{aligned}\n    V_C(t + \\Delta t) = V_{in} - (V_{in} - V_C(t)) e^{-\\frac{\\Delta t}{\\tau_e}}\n    \\end{aligned}\n\\end{equation}\nThe fact that the time constant $\\tau_e$ represents the amount of time to reach a voltage of approximately 63.2\\% of the input voltage is due to of the exponential property of equation \\ref{series_rc_eq_3}. Indeed, with an initial voltage of 0, the voltage $V_C(\\tau_e)$ reached by the capacitor after a stimulation of $\\tau_e$ seconds with an input voltage $V_{in}$ is $V_{in}(1 - e^{-1})$ where $1 - e^{-1} \\approx 0.632$.\n\n\\subsection{Parallel RC circuit for inhibition}\n\\begin{figure}\n\\centering\n    \\begin{circuitikz}[scale=0.7,transform shape] \\draw\n    (-2,1) node {}\n    to[short](0,1) {}\n    to[C, l=C] (0,-3) {}\n    to[short](-2,-3) {}\n    (-2,-2) node[nigfete] (fet) {}\n    (-4,-2.25) node[left]{$V_{in}$}\n    to[short, o-] (fet.G) {}\n    (-2, 1) to[R, l=$R_i$] (fet.D) {}\n    (fet.S) node {}\n    to[short](-2,-3) {};\n    \\end{circuitikz}\n    \\caption{Electric diagram of the parallel Resistor-Capacitor (RC) circuit controlled by a N-Channel MOSFET transistor.}\n    \\label{fig:parallel_rc}\n\\end{figure}\n\nInhibitory neurons represent 10\\% to 20\\% of brain population and their activity plays a major role in cognition \\cite{from_hiring_to_firing_activation_inhibitory_neurons}. By producing stop signals of excitation and therefore decreasing the membrane potentials of neurons receiving inhibitory stimulus, inhibitory neurons can be seen as regulators of firing rates by maintaining neurons to sub-threshold regimes.\nIn the IF neuron, an inhibitory connection is implementable with a controlled leakage \u2013 similar to the leak of the leaky-integrate-and-fire neuron (LIF). As Figure \\ref{fig:parallel_rc} shows, the parallel Resistor-Capacitor (RC) of the LIF neuron circuit can be improved with an N-Channel MOSFET transistor to control the current flowing out of the capacitor. In the parallel RC circuit, the current $I_C$ flowing through the capacitor is equal to the current $I_{R_i}$:\n\\begin{equation} \\label{para_rc_eq_1}\n    \\begin{aligned}\n    I_C=I_{R_i}=\\frac{V_C}{R_i}\n    \\end{aligned}\n\\end{equation}\nKirchhoff's voltage law states that the voltage of the capacitor is equal to the voltage drop across the resistor $R_i$ \u2013 and the transistor \u2013 is equivalent to the voltage $V_C$ of the capacitor:\n\\begin{equation} \\label{para_rc_eq_2}\n    \\begin{aligned}\n    V_{R_i} + V_C = 0\\\\\n    \\Leftrightarrow I_{R_i}R_i = -V_C\\\\\n    \\Leftrightarrow I_{C}R_i = -V_C\n    \\end{aligned}\n\\end{equation}\nFinally, as mentioned in section \\ref{series_rc}, the instantaneous rate of voltage change $\\frac{dV_C}{dt}$ of the capacitor can replace the capacitor's current term in the previous equation:\n\\begin{equation} \\label{para_rc_eq_3}\n    \\begin{aligned}\\tau_{i}\\frac{dV_C}{dt} = -V_C\n    \\end{aligned}\n\\end{equation}\nAs for the series RC circuit, the time constant $\\tau_i = R_i C$ is introduced which also represents the time required by the discharged capacitor to lose approximately 63.2\\% of its voltage. Thus, the previous equation can be integrated to obtain the capacitor's voltage $V_C(t+\\Delta t)$ after a stimulation time $\\Delta t$:\n\\begin{equation} \\label{para_rc_eq_4}\n    \\begin{aligned}\n    V_C(t + \\Delta t) = V_C(t) e^{-\\frac{\\Delta t}{\\tau_i}}\n    \\end{aligned}\n\\end{equation}\n\n\\subsection{Integrate-and-Fire neuron}\nBiological neurons receive several stimuli (excitatory and inhibitory) at their dendrites and having multiple inputs is a necessary condition to allow the IF model to compute separations of multi-dimensional spaces.\nBoth excitatory and inhibitory can be combined to obtain several inputs. In some specific situations, no excitation is provided by inputs and, for this reason, a bias connection \u2013 i.e. a connection always set to 1, as in rate-based models \u2013 is introduced to provide a constant stimulation. This allows a permanent charge of the capacitor and the neuron can become excited even if no pattern is provided. In such configuration, the total resistance of parallel resistors is not a simple sum of all the resistances but the inverse of the total resistance is the sum of all inverted resistances ($\\frac{1}{R_{total}} = \\sum_{i}^{n} \\frac{1}{R_i}$). For this reason, computation of the IF model can become complex due to the differences of input stimulation times. For a lack of simplicity, inputs are stimulated one by one and as the capacitor must be charged to allow inhibition of the membrane potential inhibitory stimulations must follow excitatory ones.\nTherefore, the inference of the IF model becomes sequential and can be represented under a recurrent form where synapses are stimulated independently.\n\n\\subsection{Integrate-and-Fire neuron as a recurrent model} \\label{IF_as_recurrent}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{images\/results\/recurrent_if.png}\n\\caption{Recurrent representation of the inference of the integrate and fire model. Green states represent excitatory stimulations and red states represent inhibition.}\n\\label{fig:recurrent_model}\n\\end{figure}\nSequential data are sequences with chronological order. In the deep learning field, this type of data is processed using recurrent units which are feedforward neural networks augmented with the inclusion of internal states of units, introducing a time dimension to the model \\cite{critical_review_rnn_sequence_learning}. At each step $t$ of the inference of a recurrent neural network, the states at $t-1$ of the neurons are integrated into the computation. Intrinsically, the integration of stimulus in the IF model depends on the capacitor voltage \u2013 see equations  \\ref{series_rc_eq_3} and \\ref{para_rc_eq_4} \u2013 and can be expressed as a recurrent form where each step is a precise synapse stimulation. As presented in Figure \\ref{fig:recurrent_model}, each step represents the stimulation of a synapse (excitatory ones first) and the hidden state is the potential of the neuron at time $t-1$ with an initial voltage of 0 \u2013 i.e. fully discharged capacitor. Thus, the final hidden state represents the membrane potential of the inferred neuron that can be compared with the voltage threshold to determine if the unit must release a spike or not \u2013 this step is achieved by the micro-controller controlling the circuit. The IF model defined as a recurrent form is a continuous and differentiable function which makes it suitable for the gradient descent algorithm. Therefore, some particular set of resistance values makes the IF neuron reach sub-threshold or super-threshold regimes for specific input and this behavior is exploited to achieve classification of patterns. To find the right combinations of resistances, optimisation algorithms can be used such as the well known gradient descent \\cite{gradient_descent_for_deep_learning}. The loss of the model can be defined as the Mean Squared Error (MSE) between output membrane potentials and target potentials and the gradient used in the algorithm is computed with respect to resistance values.\n\n\\subsection{Dataset, network architecture and training}\n\nTo demonstrate our model, we generated an artificial dataset of dog postures and trained a network of three IF neuron on it. It has been generated by using the average inclination vector for each class \u2013 i.e. we determined the average tilt of the device for each class \u2013 and created many samples by augmenting these vectors with random noise. The tilt of the device can be computed using both accelerometer and gyroscope data from inertial sensors \\cite{precise_tilt_angle_detection} which gives a three-dimension vector (pitch, roll and yaw). In this work, three distinct classes of dog postures have been used: stand, sit and lay on the side. The average tilt vectors of each class can be determined by only the pitch and roll axis as following: $\\begin{pmatrix}0 & 0\\end{pmatrix}$ for stand, $\\begin{pmatrix}0 & 0.25\\end{pmatrix}$ for sit and $\\begin{pmatrix}0.5 & 0\\end{pmatrix}$ for lying \u2013 the maximum value for each axis is 1. The yaw axis is ignored because it corresponds to the horizontal angle of the device and is irrelevant in this case. From these average tilt vectors, we can generate new input samples with a normally distributed random noise $\\epsilon \\sim \\mathcal{N}(0, 0.04^2)$. \n\\par\nThe model has been implemented as a 3 neuron network \u2013 one per class \u2013 with both excitatory and inhibitory connections for every input to allow the model to have both types of connection and be flexible enough to achieve correct separations of the input space. The chosen capacitance value for the capacitors is $1e{-6}$ which is small enough to have a low charging time, but high enough to have fine control of the charge, to limit noise and voltage dissipation when implemented as hardware. The maximum stimulation time per input is defined as 50 milliseconds \u2013 e.g. an input of value 1.0 will stimulate the corresponding synapse during 50 milliseconds and an input of 0.5 during 25 milliseconds. Finally, the output class is determined by the unit with the highest membrane potential using an argmax operator and the model has been trained using the gradient descent algorithm with a learning rate $\\alpha=5e^{-4}$.\n\\par\nDuring the training, the algorithm did not converge properly due to the scale of resistance values (between $10^3$ and $10^6$) which produces exceedingly large gradients. As the resistances are large and computed into gradients, the scale of gradients also becomes large. This very well known problem is known as \\textit{exploding gradient} in machine learning \\cite{difficulty_training_recurrent_neural_network}. Many solutions exist to solve the exploding gradient problem such as gradient clipping \\cite{difficulty_training_recurrent_neural_network}. However, the gradient clipping method makes gradients too small to converge in an acceptable amount of time \u2013 again due to the large scale of resistance values in the model. Another solution has been found to solve the issue: reduce the scale of resistances (between $1^{-3}$ and $1$) and compensate with the capacitance value $C$ of the unit. As the charge and discharge are driven by RC time constants $\\tau=RC$, decreasing the resistance $R$ can be balanced by increasing the capacitance $C$. Thus, by scaling down the resistance value, calculated gradients become small enough to obtain stable learning.\n\n\\subsection{Weights selection and hardware validation}\n\\begin{table}\n\\caption{Resistance values (weights) of stand, lie and sit units. \\textit{Excit.} is for \\textit{Excitatory} and \\textit{Inhib.} is for \\textit{Inhibitory}. All values are given in kilohms ($k\\Omega$).} \\label{model_weights_table}\n\\begin{tabular}{| c | c c c | c c c |} \n\\hline\nOutput neuron & Excit. x & Excit. y & Excit. bias & Inhib. x & Inhib. y & Inhib. bias \\\\\n\\hline\\hline\nStand & 20.33 & 101.47 & 1.53 & 9.77 & 6.65 & 1000.00 \\\\\nLie & 7.61 & 1000.00 & 1000.00 & 1000.00 & 22.44 & 1000.00 \\\\\nSit & 1000.00 & 5.42 & 1000.00 & 19.57 & 1000.00 & 1000.00 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\nTable \\ref{model_weights_table} presents the weights of the model after training. In the IF neuron, a low resistance gives high weight to the input because it lets more current flow in or out of the capacitor and thus has a high contribution in its charge or discharge. Therefore, the contribution of very high resistances is insignificant and can be ignored. For this reason, all resistance values that converged to the maximum resistance ($1000 k\\Omega)$ can be ignored in the trained model presented in Table \\ref{model_weights_table} and consequently only 9 synapses remain out of the 18.\n\\par\n\\begin{figure}\n    \\centering\n     \\includegraphics[width=0.7\\textwidth]{images\/results\/setup_diagram.png}\n    \\caption{Diagram of the experimental setup. The micro-controller collects the raw accelerometry and gyroscopic data from the inertial sensor, pre-process it to obtain tilt vectors that are sent to a network of IF neuron implemented as hardware. The class inferred by the network can be read by the micro-controller before being sent to the serial display (or a transmitter in real situations of remote systems).}\n    \\label{fig:setup_diagram}\n\\end{figure}\nAfter a weight selection (i.e. removing weights that converged to the maximum value), the three IF units for dog posture classification have been implemented as hardware to validate the training. The microcontroller used in this work is an ATmega328P on an Arduino Uno to ease its programming. An inertial unit (MPU-6050) is used to obtain accelerometry and gyroscopic data. Therefore the accelerometry data is used by the microcontroller to determine the gravity vector and the gyroscope data is integrated and combined with the previously computed vector to obtain the precise orientation of the device. Then, the microcontroller stimulates the synapses one by one during variable times depending on the pitch and roll of the device. The synapses charge (excitatory) then discharge (inhibitory) the capacitors using the digital pins. Finally, the microcontroller can read the membrane potential of each neuron by reading the voltage of the capacitors. See Figure \\ref{fig:setup_diagram} for a diagram of the setup.\n\\par\nThe model has been validated by sending all the possible inputs to the simulation and the hardware and comparing their responses. To achieve this, the hardware has not been tested using the inertial sensor but stimulated with the same tilt vectors as used in the simulation. Therefore, a mapping of the units' responses for both the simulation and the hardware has been generated \u2013 see Figure \\ref{fig:comparison}. It appears that the behaviour of the electronic implementation is close to the simulation and the slight variations in voltage are due to noise and rounding of resistance values \u2013 e.g. a resistance of 3230$\\Omega$ in the simulation is rounded to 3000$\\Omega$ in the electronic implementation. Once the hardware is implemented and the model accuracy is validated, the power consumption of the device can be measured and compared to the use of simulated artificial neural networks.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{images\/results\/voltage_comparison.png}\n    \\caption{Comparison of neural responses between simulated and electronic IF neurons. Each simulated unit (i.e. Stand, Sit and Lie) is compared with its corresponding electronic implementation by measuring the capacitor voltage for all possible inputs (pitch and roll). The hardware implementation of the model is very close to the simulation behavior and only varies due to electric noise and rounded resistance values.}\n    \\label{fig:comparison}\n\\end{figure}\n\n\\subsection{Power consumption analysis} \\label{theorical_energy_consumpion}\n\\begin{table}\n\\caption{Comparison of average power consumptions of the micro-controller only, a Logistic Regression model running on the micro-controller and the micro-controller with the designed IF neurons. The values are given with and without the micro-controller power consumption to ease understanding.}\n\\label{power_consumption}\n\\begin{tabular}{|c|c|c|} \n\\hline\nSetup & Average power consumption & Average power consumption \\\\\n& with the micro-controller & without the micro-controller \\\\\n& (in Watt) & (in Watt) \\\\\n\\hline\n\\hline\nMicro-controller only & 0.2155 & -\\\\ \n\\hline\nLogistic Regression & 0.2265 & 0.011 \\\\\non the micro-controller && \\\\\n\\hline\nMicro-controller + IF circuits & \\textbf{0.218} & \\textbf{0.0025} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\nTo measure the power efficiency of the hardware, the current consumed by the device (i.e. the micro-controller, inertial sensors and IF circuits together) has been recorded while performing real-time classification of dog postures. The measures were done using a power analyzer and power supply (Otii ARC) and performed on the micro-controller running alone, on the micro-controller classifying postures with the IF neurons implemented as hardware and on the micro-controller classifying postures with logistic regression. The following average power consumptions have been determined and written in Table \\ref{power_consumption}. Most of the power consumed by the device is due to the micro-controller, but the results show that the use of the IF circuit for embedded classification consumes less than a simple logistic regression performed on the micro-controller. If the power consumption of the micro-controller is ignored, the implemented hardware consumes 4.4 times less than the logistic regression method, which is significant. Therefore, the designed circuit is faithful to its simulation and able to recognize patterns in presented inputs with less energy demand than simulated ANNs.\n\n\\end{document}\n\\section{Introduction}\n\nEmbedded systems acquire physical measurements of the real world from sensors before performing simple computations \\cite{pattern_recognition_embedded_systems}. From signal acquisition, these systems often require a transformation of the data to make decisions or compress the information for transmission. Pattern recognition is an important area in the emergence of intelligent systems the classification of patterns from sensory information into categories is necessary to achieve a goal \\cite{pattern_recognition_embedded_systems}. For example, recent years have seen the development of new animal-attached devices called \\textit{Biologgers} which are used to monitor the environment, track locations and quantify the behaviour of certain species \\cite{biomarkers_for_animal_health}. These devices sometimes use transmission technologies such as Very High Frequency (VHF), acoustic telemetry or, more recently, orbiting satellites to monitor certain species over a long period. However, data transmission has a high cost not only financially, but also in terms of energy. This can be problematic on battery-powered devices. Thus, to optimise the lifetime of remote devices, the number of transmissions must be minimized. As some sensors often run at a high sampling frequency \u2013 typically between 10Hz and 1000hz for inertial sensors \u2013 the amount of collected data becomes so large that transmission becomes difficult without any compression or processing. To reduce this amount of data, embedded classifiers can directly process the sensor values, which significantly reduces the information to transmit. For example, some methods have been used on biologgers to classify animal activities from inertial data using machine learning approaches, especially Artificial Neural Networks \\cite{identification_of_behavior_freely_moving_dogs,using_tri_axial_acceleration_to_identify_behavior,identification_animal_movement_patterns_tri_axial_magnetometry}\n\\par\nArtificial Neural Networks (ANNs) are one of the most powerful methods to solve classification problems. ANNs try to mimic the behaviour of biological neurons to find complex relationships between input signals and desired outputs. However, the computation of these artificial neurons is computationally expensive due to complex operations that require substantial amounts of energy or sometimes the use of Graphics Processing Units (GPUs) which makes them unsuitable for battery-powered devices \\cite{optimizing_energy_consumption_snn}. Contrary to the abstracted models used in Deep Learning, Spiking Neural Networks (SNN) are biologically plausible artificial neuron models \\cite{SNN_single_neurons_populations_plasticity} that transmit information through discrete electrical signals called spike trains \\cite{SNN_single_neurons_populations_plasticity,optimizing_energy_consumption_snn}. Spiking neurons integrate synaptic events only when they occur and fire action potentials when the membrane potential reaches a defined threshold \\cite{SNN_single_neurons_populations_plasticity}. This event integration property makes them relatively easy to simulate and can also be implemented as energy-efficient dedicated hardware (called neuromorphic chips) \\cite{memristor_emulator_with_stdp,loihi_neuromorphic_processor,digital_spiking_neurmorphic_processor,spinnaker_project}. To the best of our knowledge, there is no hardware implementation of SNNs embedded in small remote devices such as biologgers \u2013 mainly because of the size of the current neuromorphic hardware. Therefore, it is necessary to bring new simple and non-energy-consuming solutions for embedded pattern recognition in remote systems.\n\\par\nIn this paper, we present a simple neuron circuit that can be used for basic pattern recognition in remote systems. This model developed is the \\textit{Integrate and Fire} (IF) which is easily implementable as energy-efficient hardware with low-cost components. It integrates successive currents during different amounts of time \u2013 according to the inputs \u2013 and exploits the charge and discharge capabilities of capacitors to create a trainable and electronically implementable neuron. The model has both excitatory and inhibitory synapses and we introduce it as a recurrent form which makes it suitable for gradient descent optimisations. This model has been chosen for the simplicity of its hardware implementation and its simulation. To validate it, we trained three neurons to classify dog postures using inclination vectors (calculated from inertial data) and implemented them with electronic components to compare the hardware and its simulation.\n\n\\end{document}\n\n\n\\section{Discussion} \\label{section_discussion}\n\nIn this work, the Integrate-and-Fire model has been simulated and trained to achieve dog posture classification and showed promising results with relatively low energy expenditure when implemented with electronic components compared to the use of embedded logistic regression. The designed model implemented as hardware can be integrated into remote systems for embedded and energy-efficient pattern recognition, reducing the amount of data to transmit and thus reducing the number of transmissions, leading to low energy consumption.\nThe simulation of the IF neuron is faithful to the electronic implementation which makes it possible to train using the gradient descent algorithm. Once trained, the resistances that do not contribute to the pattern detection \u2013 i.e. those that converge to the highest value \u2013 are removed from the final circuit and the remaining are implemented with the final hardware. This hardware implementation has been done with only a few passive components (resistors, diodes and capacitors) and one active component (N-MOSFET transistors) which all have low costs. It has been implemented using prototyping boards but can be miniaturised on Printed Circuit Boards (PCBs) with Surface Mount Technology (SMT) that provides miniature components to produce a version of the hardware small enough to be integrated into small devices.\n\\par\nIn terms of power, the measured consumptions are almost identical due to the power demand of the micro-controller. However, the lifetime of battery-powered devices is very important and no aspect of the entire device should be overlooked, including the power usage of data processing. Therefore, by disregarding consumption of the micro-controller, the IF model consumes four times less when it is electronically implemented than a trained logistic regression running on the micro-controller. With this setup, the battery life-time is improved by 3.75\\%, but it can be enhanced even more by using a low-powered micro-controller. Moreover, an implementation of the model with spike trains should significantly reduce energy consumption. Therefore, it would be wise to rethink the way of communicating features given to the model using spike trains to further reduce the power consumption of the circuit.\n\\par\nOne main issue of the IF approach is the time dependence of the inference. As the stimulation time of synapses varies according to the input values, the inference time is also variable. Thus, the higher the inputs, the longer the inference time will be. This maximum inference time can be calculated by summing the maximum stimulation time of inputs or can be compensated by varying the capacitance value of units. Another issue of the IF model is that the leak channel, specific to the LIF neuron, has been removed and the time dimension disappeared. This model is thus no longer able to process animal dynamics to infer its activity and only the posture \u2013 i.e. static patterns \u2013 can be classified. To achieve this task, the LIF model should be used which involves transforming features into spike trains. However, due to the non-continuity of spike trains in spiking neural networks, algorithms based on differentiation \u2013 such as the gradient descent algorithm used in this work \u2013 cannot be applied for training.\n\n\\end{document}\n\\section{Future work}\nIn future works, the time capabilities of the Leaky Integrate-and-Fire model must be exploited to classify time-series patterns using spike trains. As the gradient descent algorithm is not suitable to train such models, other training algorithms must be explored to find new ways to classify patterns or compress sensory data into a spike code generated by a spiking neural network. Recent advances in neurosciences permitted the development of unsupervised learning algorithms such as Spike Time Dependent Plasticity (STDP) which is a biologically plausible Hebbian learning rule that adjusts the strength of connections between neurons in the brain \\cite{memristor_emulator_with_stdp,spike_time_dependence_of_plasticity,SNN_single_neurons_populations_plasticity}. Based on the timing of pre and post-synaptic spikes, STDP allows neurons to learn time-dependent correlations in spike trains and thus a relevant representation of input features \\cite{memristor_emulator_with_stdp,spike_time_dependence_of_plasticity,SNN_single_neurons_populations_plasticity}. Therefore, such algorithms may be able to find correlations between some sensory inputs and achieve a compression of recorded data.\n\n\\end{document}\n\\section{Results} \\label{section_results}\n\nThe capacitor is an electronic passive component that creates a potential difference between two conductive plates, analogous to the difference of electric potential of the biological neuron membrane created by ions that flow in and out of the cell. Therefore, the capacitor is often used in computational models of spiking neural networks to reproduce membrane potentials of the biological neurons. Connected in series or parallel with a resistance, the capacitor forms two circuits with distinct charge and discharge properties respectively called series and parallel Resistor-Capacitor circuits (RC). Thus, the IF neuron is mainly composed of passive components: a capacitor that reproduces the membrane potential and resistors that charge (excite) or discharge (inhibit) the neuron.\n\n\\subsection{Series RC circuit for excitatory stimulations} \\label{series_rc}\n\\begin{figure}\n\\centering\n    \\begin{circuitikz}[scale=0.7,transform shape] \\draw\n    (-3,1) node[left]{$V_{\\mathrm{in}}$}\n    to[R, o-, l=$R_e$] (0,1)\n    to[C, l=C] (2,1)\n    node[ground] {}\n    ;\n    \\end{circuitikz}\n    \\caption{Electric diagram of the series Resistor-Capacitor (RC) circuit. The circuit is composed of a voltage supplier $V_{in}$, a resistor $R_e$ that is analog to the excitatory synapses of the neuron and a capacitor $C$ that reproduces the membrane potential.}\n    \\label{fig:series_rc}\n\\end{figure}\nThe series RC circuit is defined by a successive resistor $R_e$ which represents the excitatory synapses of the biological neuron and a capacitor $C$ which reproduces the membrane potential \u2013 see Figure \\ref{fig:series_rc}. Taking into consideration Ohm's law ($I = \\frac{V}{R}$), the fact that the current $I_{R_e}$ flowing through the resistor $R_e$ is equal to the current $I_C$ flowing through the capacitor $C$ ($I_{R_e} = I_C$) and that the capacitor component theoretically does not produce any resistance, the current flowing through the circuit depends only on the input voltage $V_{in}$ and the excitatory resistor $R_e$. Thus, the resistor can be seen as a weight defined as $w = \\frac{1}{R_e}$ which scales the input value $V_{in}$ such as $I_{R_e} = wV_{in}$. Consequently, the higher the value of the resistor, the lower the current will flow through the capacitor and vice versa. Knowing that the total voltage $V_{in}$ of the circuit is defined as the sum of the voltages $V_R$ and $V_C$ respectively across the resistor and the capacitor ($V_{in} = V_R + V_C$), and the Ohm's law, we can define the following equation:\n\\begin{equation} \\label{series_rc_eq_1}\n    \\begin{aligned}\n    V_{in} = R_eI_{R_e} + V_{C}\\\\\n    \\Leftrightarrow I_{R_e} = \\frac{V_{in} - V_C}{R_e}\n    \\end{aligned}\n\\end{equation}\nEquation \\ref{series_rc_eq_1} shows that the current flows through the excitatory resistor does not only depend on the input voltage and the resistance but also depends on the voltage across the capacitor. Therefore, the higher the voltage across the capacitor, the lower the current flowing in the circuit will be. To describe the dynamic of the capacitor, the instantaneous rate of voltage change $\\frac{dV}{dt}$ of the capacitor is introduced as the current $I$ flowing through the capacitor divided by the capacitance $C$ ($\\frac{dV}{dt} = \\frac{I}{C}$). Equation \\ref{series_rc_eq_1} can be reformulated as:\n\\begin{equation}\\label{series_rc_eq_2}\n    \\begin{aligned}\n    \\tau_e\\frac{dV_C}{dt} = V_{in} - V_C\\\\\n    \\end{aligned}\n\\end{equation}\nwhere $\\tau_e = R_e C$ is the time constant of the series RC circuit which represents the number of seconds needed to reach approximately 63.2\\% of the input voltage $V_{in}$ \u2013 this value is explained below. For a constant input voltage and a given initial voltage $V_C(t)$ at time $t$, the capacitor voltage $V_C(t + \\Delta t)$ after an amount of time $\\Delta t$ can be found by integrating equation \\ref{series_rc_eq_2}:\n\\begin{equation} \\label{series_rc_eq_3}\n    \\begin{aligned}\n    V_C(t + \\Delta t) = V_{in} - (V_{in} - V_C(t)) e^{-\\frac{\\Delta t}{\\tau_e}}\n    \\end{aligned}\n\\end{equation}\nThe fact that the time constant $\\tau_e$ represents the amount of time to reach a voltage of approximately 63.2\\% of the input voltage is due to of the exponential property of equation \\ref{series_rc_eq_3}. Indeed, with an initial voltage of 0, the voltage $V_C(\\tau_e)$ reached by the capacitor after a stimulation of $\\tau_e$ seconds with an input voltage $V_{in}$ is $V_{in}(1 - e^{-1})$ where $1 - e^{-1} \\approx 0.632$.\n\n\\subsection{Parallel RC circuit for inhibition}\n\\begin{figure}\n\\centering\n    \\begin{circuitikz}[scale=0.7,transform shape] \\draw\n    (-2,1) node {}\n    to[short](0,1) {}\n    to[C, l=C] (0,-3) {}\n    to[short](-2,-3) {}\n    (-2,-2) node[nigfete] (fet) {}\n    (-4,-2.25) node[left]{$V_{in}$}\n    to[short, o-] (fet.G) {}\n    (-2, 1) to[R, l=$R_i$] (fet.D) {}\n    (fet.S) node {}\n    to[short](-2,-3) {};\n    \\end{circuitikz}\n    \\caption{Electric diagram of the parallel Resistor-Capacitor (RC) circuit controlled by a N-Channel MOSFET transistor.}\n    \\label{fig:parallel_rc}\n\\end{figure}\n\nInhibitory neurons represent 10\\% to 20\\% of brain population and their activity plays a major role in cognition \\cite{from_hiring_to_firing_activation_inhibitory_neurons}. By producing stop signals of excitation and therefore decreasing the membrane potentials of neurons receiving inhibitory stimulus, inhibitory neurons can be seen as regulators of firing rates by maintaining neurons to sub-threshold regimes.\nIn the IF neuron, an inhibitory connection is implementable with a controlled leakage \u2013 similar to the leak of the leaky-integrate-and-fire neuron (LIF). As Figure \\ref{fig:parallel_rc} shows, the parallel Resistor-Capacitor (RC) of the LIF neuron circuit can be improved with an N-Channel MOSFET transistor to control the current flowing out of the capacitor. In the parallel RC circuit, the current $I_C$ flowing through the capacitor is equal to the current $I_{R_i}$:\n\\begin{equation} \\label{para_rc_eq_1}\n    \\begin{aligned}\n    I_C=I_{R_i}=\\frac{V_C}{R_i}\n    \\end{aligned}\n\\end{equation}\nKirchhoff's voltage law states that the voltage of the capacitor is equal to the voltage drop across the resistor $R_i$ \u2013 and the transistor \u2013 is equivalent to the voltage $V_C$ of the capacitor:\n\\begin{equation} \\label{para_rc_eq_2}\n    \\begin{aligned}\n    V_{R_i} + V_C = 0\\\\\n    \\Leftrightarrow I_{R_i}R_i = -V_C\\\\\n    \\Leftrightarrow I_{C}R_i = -V_C\n    \\end{aligned}\n\\end{equation}\nFinally, as mentioned in section \\ref{series_rc}, the instantaneous rate of voltage change $\\frac{dV_C}{dt}$ of the capacitor can replace the capacitor's current term in the previous equation:\n\\begin{equation} \\label{para_rc_eq_3}\n    \\begin{aligned}\\tau_{i}\\frac{dV_C}{dt} = -V_C\n    \\end{aligned}\n\\end{equation}\nAs for the series RC circuit, the time constant $\\tau_i = R_i C$ is introduced which also represents the time required by the discharged capacitor to lose approximately 63.2\\% of its voltage. Thus, the previous equation can be integrated to obtain the capacitor's voltage $V_C(t+\\Delta t)$ after a stimulation time $\\Delta t$:\n\\begin{equation} \\label{para_rc_eq_4}\n    \\begin{aligned}\n    V_C(t + \\Delta t) = V_C(t) e^{-\\frac{\\Delta t}{\\tau_i}}\n    \\end{aligned}\n\\end{equation}\n\n\\subsection{Integrate-and-Fire neuron}\nBiological neurons receive several stimuli (excitatory and inhibitory) at their dendrites and having multiple inputs is a necessary condition to allow the IF model to compute separations of multi-dimensional spaces.\nBoth excitatory and inhibitory can be combined to obtain several inputs. In some specific situations, no excitation is provided by inputs and, for this reason, a bias connection \u2013 i.e. a connection always set to 1, as in rate-based models \u2013 is introduced to provide a constant stimulation. This allows a permanent charge of the capacitor and the neuron can become excited even if no pattern is provided. In such configuration, the total resistance of parallel resistors is not a simple sum of all the resistances but the inverse of the total resistance is the sum of all inverted resistances ($\\frac{1}{R_{total}} = \\sum_{i}^{n} \\frac{1}{R_i}$). For this reason, computation of the IF model can become complex due to the differences of input stimulation times. For a lack of simplicity, inputs are stimulated one by one and as the capacitor must be charged to allow inhibition of the membrane potential inhibitory stimulations must follow excitatory ones.\nTherefore, the inference of the IF model becomes sequential and can be represented under a recurrent form where synapses are stimulated independently.\n\n\\subsection{Integrate-and-Fire neuron as a recurrent model} \\label{IF_as_recurrent}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{images\/results\/recurrent_if.png}\n\\caption{Recurrent representation of the inference of the integrate and fire model. Green states represent excitatory stimulations and red states represent inhibition.}\n\\label{fig:recurrent_model}\n\\end{figure}\nSequential data are sequences with chronological order. In the deep learning field, this type of data is processed using recurrent units which are feedforward neural networks augmented with the inclusion of internal states of units, introducing a time dimension to the model \\cite{critical_review_rnn_sequence_learning}. At each step $t$ of the inference of a recurrent neural network, the states at $t-1$ of the neurons are integrated into the computation. Intrinsically, the integration of stimulus in the IF model depends on the capacitor voltage \u2013 see equations  \\ref{series_rc_eq_3} and \\ref{para_rc_eq_4} \u2013 and can be expressed as a recurrent form where each step is a precise synapse stimulation. As presented in Figure \\ref{fig:recurrent_model}, each step represents the stimulation of a synapse (excitatory ones first) and the hidden state is the potential of the neuron at time $t-1$ with an initial voltage of 0 \u2013 i.e. fully discharged capacitor. Thus, the final hidden state represents the membrane potential of the inferred neuron that can be compared with the voltage threshold to determine if the unit must release a spike or not \u2013 this step is achieved by the micro-controller controlling the circuit. The IF model defined as a recurrent form is a continuous and differentiable function which makes it suitable for the gradient descent algorithm. Therefore, some particular set of resistance values makes the IF neuron reach sub-threshold or super-threshold regimes for specific input and this behavior is exploited to achieve classification of patterns. To find the right combinations of resistances, optimisation algorithms can be used such as the well known gradient descent \\cite{gradient_descent_for_deep_learning}. The loss of the model can be defined as the Mean Squared Error (MSE) between output membrane potentials and target potentials and the gradient used in the algorithm is computed with respect to resistance values.\n\n\\subsection{Dataset, network architecture and training}\n\nTo demonstrate our model, we generated an artificial dataset of dog postures and trained a network of three IF neuron on it. It has been generated by using the average inclination vector for each class \u2013 i.e. we determined the average tilt of the device for each class \u2013 and created many samples by augmenting these vectors with random noise. The tilt of the device can be computed using both accelerometer and gyroscope data from inertial sensors \\cite{precise_tilt_angle_detection} which gives a three-dimension vector (pitch, roll and yaw). In this work, three distinct classes of dog postures have been used: stand, sit and lay on the side. The average tilt vectors of each class can be determined by only the pitch and roll axis as following: $\\begin{pmatrix}0 & 0\\end{pmatrix}$ for stand, $\\begin{pmatrix}0 & 0.25\\end{pmatrix}$ for sit and $\\begin{pmatrix}0.5 & 0\\end{pmatrix}$ for lying \u2013 the maximum value for each axis is 1. The yaw axis is ignored because it corresponds to the horizontal angle of the device and is irrelevant in this case. From these average tilt vectors, we can generate new input samples with a normally distributed random noise $\\epsilon \\sim \\mathcal{N}(0, 0.04^2)$. \n\\par\nThe model has been implemented as a 3 neuron network \u2013 one per class \u2013 with both excitatory and inhibitory connections for every input to allow the model to have both types of connection and be flexible enough to achieve correct separations of the input space. The chosen capacitance value for the capacitors is $1e{-6}$ which is small enough to have a low charging time, but high enough to have fine control of the charge, to limit noise and voltage dissipation when implemented as hardware. The maximum stimulation time per input is defined as 50 milliseconds \u2013 e.g. an input of value 1.0 will stimulate the corresponding synapse during 50 milliseconds and an input of 0.5 during 25 milliseconds. Finally, the output class is determined by the unit with the highest membrane potential using an argmax operator and the model has been trained using the gradient descent algorithm with a learning rate $\\alpha=5e^{-4}$.\n\\par\nDuring the training, the algorithm did not converge properly due to the scale of resistance values (between $10^3$ and $10^6$) which produces exceedingly large gradients. As the resistances are large and computed into gradients, the scale of gradients also becomes large. This very well known problem is known as \\textit{exploding gradient} in machine learning \\cite{difficulty_training_recurrent_neural_network}. Many solutions exist to solve the exploding gradient problem such as gradient clipping \\cite{difficulty_training_recurrent_neural_network}. However, the gradient clipping method makes gradients too small to converge in an acceptable amount of time \u2013 again due to the large scale of resistance values in the model. Another solution has been found to solve the issue: reduce the scale of resistances (between $1^{-3}$ and $1$) and compensate with the capacitance value $C$ of the unit. As the charge and discharge are driven by RC time constants $\\tau=RC$, decreasing the resistance $R$ can be balanced by increasing the capacitance $C$. Thus, by scaling down the resistance value, calculated gradients become small enough to obtain stable learning.\n\n\\subsection{Weights selection and hardware validation}\n\\begin{table}\n\\caption{Resistance values (weights) of stand, lie and sit units. \\textit{Excit.} is for \\textit{Excitatory} and \\textit{Inhib.} is for \\textit{Inhibitory}. All values are given in kilohms ($k\\Omega$).} \\label{model_weights_table}\n\\begin{tabular}{| c | c c c | c c c |} \n\\hline\nOutput neuron & Excit. x & Excit. y & Excit. bias & Inhib. x & Inhib. y & Inhib. bias \\\\\n\\hline\\hline\nStand & 20.33 & 101.47 & 1.53 & 9.77 & 6.65 & 1000.00 \\\\\nLie & 7.61 & 1000.00 & 1000.00 & 1000.00 & 22.44 & 1000.00 \\\\\nSit & 1000.00 & 5.42 & 1000.00 & 19.57 & 1000.00 & 1000.00 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\nTable \\ref{model_weights_table} presents the weights of the model after training. In the IF neuron, a low resistance gives high weight to the input because it lets more current flow in or out of the capacitor and thus has a high contribution in its charge or discharge. Therefore, the contribution of very high resistances is insignificant and can be ignored. For this reason, all resistance values that converged to the maximum resistance ($1000 k\\Omega)$ can be ignored in the trained model presented in Table \\ref{model_weights_table} and consequently only 9 synapses remain out of the 18.\n\\par\n\\begin{figure}\n    \\centering\n     \\includegraphics[width=0.7\\textwidth]{images\/results\/setup_diagram.png}\n    \\caption{Diagram of the experimental setup. The micro-controller collects the raw accelerometry and gyroscopic data from the inertial sensor, pre-process it to obtain tilt vectors that are sent to a network of IF neuron implemented as hardware. The class inferred by the network can be read by the micro-controller before being sent to the serial display (or a transmitter in real situations of remote systems).}\n    \\label{fig:setup_diagram}\n\\end{figure}\nAfter a weight selection (i.e. removing weights that converged to the maximum value), the three IF units for dog posture classification have been implemented as hardware to validate the training. The microcontroller used in this work is an ATmega328P on an Arduino Uno to ease its programming. An inertial unit (MPU-6050) is used to obtain accelerometry and gyroscopic data. Therefore the accelerometry data is used by the microcontroller to determine the gravity vector and the gyroscope data is integrated and combined with the previously computed vector to obtain the precise orientation of the device. Then, the microcontroller stimulates the synapses one by one during variable times depending on the pitch and roll of the device. The synapses charge (excitatory) then discharge (inhibitory) the capacitors using the digital pins. Finally, the microcontroller can read the membrane potential of each neuron by reading the voltage of the capacitors. See Figure \\ref{fig:setup_diagram} for a diagram of the setup.\n\\par\nThe model has been validated by sending all the possible inputs to the simulation and the hardware and comparing their responses. To achieve this, the hardware has not been tested using the inertial sensor but stimulated with the same tilt vectors as used in the simulation. Therefore, a mapping of the units' responses for both the simulation and the hardware has been generated \u2013 see Figure \\ref{fig:comparison}. It appears that the behaviour of the electronic implementation is close to the simulation and the slight variations in voltage are due to noise and rounding of resistance values \u2013 e.g. a resistance of 3230$\\Omega$ in the simulation is rounded to 3000$\\Omega$ in the electronic implementation. Once the hardware is implemented and the model accuracy is validated, the power consumption of the device can be measured and compared to the use of simulated artificial neural networks.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{images\/results\/voltage_comparison.png}\n    \\caption{Comparison of neural responses between simulated and electronic IF neurons. Each simulated unit (i.e. Stand, Sit and Lie) is compared with its corresponding electronic implementation by measuring the capacitor voltage for all possible inputs (pitch and roll). The hardware implementation of the model is very close to the simulation behavior and only varies due to electric noise and rounded resistance values.}\n    \\label{fig:comparison}\n\\end{figure}\n\n\\subsection{Power consumption analysis} \\label{theorical_energy_consumpion}\n\\begin{table}\n\\caption{Comparison of average power consumptions of the micro-controller only, a Logistic Regression model running on the micro-controller and the micro-controller with the designed IF neurons. The values are given with and without the micro-controller power consumption to ease understanding.}\n\\label{power_consumption}\n\\begin{tabular}{|c|c|c|} \n\\hline\nSetup & Average power consumption & Average power consumption \\\\\n& with the micro-controller & without the micro-controller \\\\\n& (in Watt) & (in Watt) \\\\\n\\hline\n\\hline\nMicro-controller only & 0.2155 & -\\\\ \n\\hline\nLogistic Regression & 0.2265 & 0.011 \\\\\non the micro-controller && \\\\\n\\hline\nMicro-controller + IF circuits & \\textbf{0.218} & \\textbf{0.0025} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\nTo measure the power efficiency of the hardware, the current consumed by the device (i.e. the micro-controller, inertial sensors and IF circuits together) has been recorded while performing real-time classification of dog postures. The measures were done using a power analyzer and power supply (Otii ARC) and performed on the micro-controller running alone, on the micro-controller classifying postures with the IF neurons implemented as hardware and on the micro-controller classifying postures with logistic regression. The following average power consumptions have been determined and written in Table \\ref{power_consumption}. Most of the power consumed by the device is due to the micro-controller, but the results show that the use of the IF circuit for embedded classification consumes less than a simple logistic regression performed on the micro-controller. If the power consumption of the micro-controller is ignored, the implemented hardware consumes 4.4 times less than the logistic regression method, which is significant. Therefore, the designed circuit is faithful to its simulation and able to recognize patterns in presented inputs with less energy demand than simulated ANNs.\n\n\\end{document}\n\\section{Introduction}\n\nEmbedded systems acquire physical measurements of the real world from sensors before performing simple computations \\cite{pattern_recognition_embedded_systems}. From signal acquisition, these systems often require a transformation of the data to make decisions or compress the information for transmission. Pattern recognition is an important area in the emergence of intelligent systems the classification of patterns from sensory information into categories is necessary to achieve a goal \\cite{pattern_recognition_embedded_systems}. For example, recent years have seen the development of new animal-attached devices called \\textit{Biologgers} which are used to monitor the environment, track locations and quantify the behaviour of certain species \\cite{biomarkers_for_animal_health}. These devices sometimes use transmission technologies such as Very High Frequency (VHF), acoustic telemetry or, more recently, orbiting satellites to monitor certain species over a long period. However, data transmission has a high cost not only financially, but also in terms of energy. This can be problematic on battery-powered devices. Thus, to optimise the lifetime of remote devices, the number of transmissions must be minimized. As some sensors often run at a high sampling frequency \u2013 typically between 10Hz and 1000hz for inertial sensors \u2013 the amount of collected data becomes so large that transmission becomes difficult without any compression or processing. To reduce this amount of data, embedded classifiers can directly process the sensor values, which significantly reduces the information to transmit. For example, some methods have been used on biologgers to classify animal activities from inertial data using machine learning approaches, especially Artificial Neural Networks \\cite{identification_of_behavior_freely_moving_dogs,using_tri_axial_acceleration_to_identify_behavior,identification_animal_movement_patterns_tri_axial_magnetometry}\n\\par\nArtificial Neural Networks (ANNs) are one of the most powerful methods to solve classification problems. ANNs try to mimic the behaviour of biological neurons to find complex relationships between input signals and desired outputs. However, the computation of these artificial neurons is computationally expensive due to complex operations that require substantial amounts of energy or sometimes the use of Graphics Processing Units (GPUs) which makes them unsuitable for battery-powered devices \\cite{optimizing_energy_consumption_snn}. Contrary to the abstracted models used in Deep Learning, Spiking Neural Networks (SNN) are biologically plausible artificial neuron models \\cite{SNN_single_neurons_populations_plasticity} that transmit information through discrete electrical signals called spike trains \\cite{SNN_single_neurons_populations_plasticity,optimizing_energy_consumption_snn}. Spiking neurons integrate synaptic events only when they occur and fire action potentials when the membrane potential reaches a defined threshold \\cite{SNN_single_neurons_populations_plasticity}. This event integration property makes them relatively easy to simulate and can also be implemented as energy-efficient dedicated hardware (called neuromorphic chips) \\cite{memristor_emulator_with_stdp,loihi_neuromorphic_processor,digital_spiking_neurmorphic_processor,spinnaker_project}. To the best of our knowledge, there is no hardware implementation of SNNs embedded in small remote devices such as biologgers \u2013 mainly because of the size of the current neuromorphic hardware. Therefore, it is necessary to bring new simple and non-energy-consuming solutions for embedded pattern recognition in remote systems.\n\\par\nIn this paper, we present a simple neuron circuit that can be used for basic pattern recognition in remote systems. This model developed is the \\textit{Integrate and Fire} (IF) which is easily implementable as energy-efficient hardware with low-cost components. It integrates successive currents during different amounts of time \u2013 according to the inputs \u2013 and exploits the charge and discharge capabilities of capacitors to create a trainable and electronically implementable neuron. The model has both excitatory and inhibitory synapses and we introduce it as a recurrent form which makes it suitable for gradient descent optimisations. This model has been chosen for the simplicity of its hardware implementation and its simulation. To validate it, we trained three neurons to classify dog postures using inclination vectors (calculated from inertial data) and implemented them with electronic components to compare the hardware and its simulation.\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nThe Electromagnetic Mass Analyser (EMMA) \\cite{davids05} has been installed at the ISAC-II facility of TRIUMF \\cite{laxdal14}. EMMA is a vacuum-mode recoil mass spectrometer designed to separate the recoils of nuclear reactions from the primary beam, focus them in energy and angle, and disperse them in a focal plane according to their mass\/charge $(m\/q)$ ratios. The spectrometer is fixed at 0$^\\circ$ with respect to the beam axis and is mounted on a common support platform with 1.5 m of longitudinal travel, allowing for the positioning of various detector arrays at the target position, including the $\\gamma$-ray spectrometer TIGRESS \\cite{hackman14} and the Si charged particle detector array SHARC \\cite{diget11}. As depicted in Figure \\ref{layout}, EMMA uses a symmetric configuration of two electrostatic deflectors and a dipole magnet to focus reaction products in kinetic energy\/charge $(E\/q)$. Angular focusing is achieved via quadrupole doublets at the entrance and exit of the spectrometer, the latter of which enables variable $m\/q$ dispersion. A photograph of the spectrometer taken in December 2016 is shown  in Figure \\ref{photo}.\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=\\linewidth]{schematic}\n\\caption{\\label{layout}Schematic side view of EMMA, showing the target chamber, quadrupole and dipole magnets,  electrostatic deflectors, and focal plane detector chamber surrounded by $\\gamma$-ray detectors. The spectrometer is mounted on a platform capable of 1.5 m of travel along the beam direction.}\n\\end{figure}\n\n  \n\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.75\\linewidth, angle=270]{photo}\n\\caption{\\label{photo}Photograph of EMMA taken in December 2016.}\n\\end{figure}\n\n\\section{Ion Optics}\\label{optics}\n\nThe ion optical design of EMMA is similar to those of the Rochester RMS \\cite{cormier81, cormier83}, CAMEL at Legnaro \\cite{spolaore85, signorini94, spolaore95}, the Oak Ridge RMS \\cite{cormier90, cole92, gross00}, HIRA at IUAC \\cite{sinha94,nath07}, the JAERI RMS \\cite{ikezoe96,ikezoe97,kuzumaki99}, and the Argonne Fragment Mass Analyzer (FMA) \\cite{davids89,davids92,back96}; it was optimized to provide large acceptance without unduly compromising the resolving power necessary to study transfer reactions in inverse kinematics as well as fusion evaporation reactions. To that end, EMMA features electrodes with larger bending radii and a shorter first quadrupole magnet than the FMA. The EMMA electrodes have the same bending radii as those of HIRA and are smaller than those of the Oak Ridge RMS. For ions of a given electrostatic rigidity, larger electrode bending radii allow operation at lower voltages. The ion optics code GIOS \\cite{wollnik87} was used to design the spectrometer, both initially and again after the electromagnetic elements were fabricated, to take account of the differences between their specified and as-built properties. The standard achromatic ion optical tune has a vertical crossover in the centre of the dipole magnet and $m\/q$ dispersion $(x|\\delta_m) \\equiv \\frac{\\partial x}{\\partial \\delta_m}$ of 10 mm\/\\%. Here, $x$ is the horizontal displacement with respect to the optic axis in the focal plane and $\\delta_m \\equiv \\frac{m\/q-m_0\/q_0}{m_0\/q_0}$ is the fractional $m\/q$ deviation with respect to that of the central reference trajectory, $m_0\/q_0$. Two quadrupole doublets permit   variable angular focussing modes; by changing the fields in the second doublet, the $m\/q$ dispersion can be varied continuously between 0 and 20 mm\/\\%.\n\nTo first order, given the bending angles and radii of curvature of the electrostatic deflectors and the dipole magnet as well as the edge angles of the latter, EMMA achieves energy focussing on account of the 1225~mm separation between the effective field boundaries of the dipole magnet and the electrostatic deflectors on either side of it. Mathematically this is expressed as $(x|\\delta_E) \\equiv \\frac{\\partial x}{\\partial \\delta_E} = 0$ and $(a|\\delta_E) = 0$. The fractional kinetic energy\/charge deviation with respect to that of the central reference trajectory, $E_0\/q_0$, is defined as $\\delta_E=\\frac{E\/q-E_0\/q_0}{E_0\/q_0}$ and $a \\equiv p_x\/p_0\\approx\\theta$, the horizontal angle with respect to the optic axis. Here $p_x$ is the horizontal projection of the ion momentum and $p_0$ is the total momentum of the reference trajectory. In the horizontal direction the angular acceptance is defined by the gap of the first electrostatic deflector and in the vertical direction it is defined by the vacuum chamber of the first quadrupole magnet. The standard distance from the target to the effective field boundary of the first quadrupole is 25~cm and the focal plane position is variable but in the standard tune lies 32 cm from the effective field boundary of the fourth quadrupole. \n\n\\begin{table}\n \\caption{\\label{dimensions}As-Built EMMA Dimensions and Maximum Fields}\n \\begin{tabular}{llll}\n \\\\\nLength from target to focal plane (m) & 9.143 & &\\\\\n \\hline\nDipoles & MD & ED1, ED2 &\\\\\n\\hline\nRadius of curvature (m) & 1.0 & 5.0 &\\\\\nDeflection angle ($^\\circ$) & 40.11 & 20.05 &\\\\\nEntrance and exit inclination angles ($^\\circ$) & 7.93, 8.67 & --- &\\\\\nEffective field boundary radii (m) & 3.472 &--- &\\\\\nPole gap (cm) & 12 & 12.5 &\\\\\nMaximum field & 1.0 T & 40 kV\/cm &\\\\\nMaximum rigidity & 1.0 T m & 20 MV &\\\\\n\\hline\nMagnetic lenses & Q1 & Q2, Q3 & Q4\\\\\n\\hline\nBore diameter (cm) & 7 & 15 & 20\\\\\nEffective length (cm) & 14.0 & 30.0 & 40.2\\\\\nMaximum pole tip field (T) & 1.21 & 0.84 & 0.80\\\\\nMaximum field gradient (T\/m) & 35 & 12 & 8.1\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Electromagnetic Elements}\\label{em_elements}\n\nThe quadrupole  and dipole magnets, their power supplies, and most of the components of the electrostatic deflectors  were fabricated by Bruker BioSpin, S.\\ A.\\ S.\\ of Karlsruhe, Germany, whereas the other custom hardware was constructed at TRIUMF, including the high voltage power supplies for the deflectors. Each of the magnetic elements was mapped by the manufacturer with a Hall effect magnetometer prior to shipment. All of the fringing fields were mapped along with the central fields. The as-built properties of the electromagnetic elements are given in Table \\ref{dimensions}.\n\n\\subsection{Quadrupole Lenses} \\label{quads}\n\nEMMA's first quadrupole lens (Q1) was designed to have high pole tip fields and a short effective length to minimize chromatic aberrations. Its cantilevered support was built so as not to interfere with the placement of 12 of the 16 TIGRESS high purity germanium (HPGe) detectors around the target position while simultaneously keeping the target to Q1 effective field boundary separation small, thereby maximizing angular acceptance. The quadrupoles are arranged in two doublets, Q1-Q2 and Q3-Q4. Both Q2 and Q3 were fabricated according to the same design, while Q1 is smaller; Q4 is larger in order to transmit $m\/q$-dispersed ions. Field clamps installed on the upstream and downstream sides of each doublet limit the extent of the fringing fields, which is particularly important given the presence of HPGe detector photomultiplier tubes nearby at the target and focal plane positions. At the factory, the higher multipole components, effective field boundaries, and deviations between the mechanical and magnetic axes were measured with the field clamps in place and found to meet our specifications. \n\nA transverse Hall probe inserted just below the bore of each quadrupole is used as a reference to set and monitor its field. The Hall effect magnetometers used in all the magnets are model FM-3000-BB-10 Teslameters produced by Projekt Elektronik Mess- und Regelungstechnik GMBH. The field gradient was measured as a function of the reference Hall probe voltage. For each quadrupole, a cylindrical holder with multiple Hall probe positions was precisely machined and aligned, via a laser tracker, to make these measurements. Corrections due to the $<0.1$ mm offset between the mechanical and magnetic axes obtained from the factory field map were neglected. This method produced a calibration that allows the field gradient to be inferred from the reference probe voltage to a precision of $0.1\\%$. Accurate calibration of the Hall effect magnetometers was confirmed by comparing with an NMR magnetometer using a uniform dipole reference field.\n\n\\subsection{Dipole Magnet} \\label{md}\n\nThe dipole magnet is a $40^{\\circ}$ homogeneous field magnetic sector that was specified to have effective field boundaries (EFBs) inclined by $8.3^{\\circ}$ with respect to normal incidence. In order to reduce the horizontal focussing of the sector and increase its vertical focussing, these inclinations are such that ions following trajectories with large bending radii pass through less field and those on small-radius trajectories pass through more field than they would in the case of normal incidence. The poles of the magnet were designed in a three-piece arrangement with small pole edge inserts on either side of a large central piece. These pole edge inserts were machined and re-machined until the measured inclinations of the entrance and exit EFBs were $7.93^{\\circ}$ and $8.67^{\\circ}$, respectively. The average of the two is $8.3^{\\circ}$; the effect of these distinct entrance and exit angles on the overall ion optical performance of the spectrometer was studied using GIOS and found to be compatible with the design requirements after a slight ($<1$~mm) adjustment to the separations between the effective field boundaries of the magnet and the EDs. Each effective field boundary is curved with a radius of 3.47 m to provide a second order correction designed to minimize the $(x|\\delta^2_E)$ term. An 8 mm vertical gap between the bottom pole and the vacuum chamber was preserved to allow the placement of thin correction coils on the pole face should they be deemed necessary. A decision to fabricate and install such coils would be made only after a detailed study of ion optical aberrations, which is not currently planned. The field is measured by a Hall probe fixed to the bottom pole piece at a location well within the uniform field region.  \n\nThe dipole magnet vacuum chamber has rectangular entrance and exit apertures that measure 205 mm horizontally and 92 mm vertically. In the central region the chamber expands to a maximum horizontal extent of 491 mm. Two straight-through ports aligned with the incoming and outgoing beam axes proved useful when aligning the slit systems located at the entrance and exit of the chamber. Sheets of aluminum honeycomb cores made by Plascore line the vertical walls and bottom surface of the vacuum chamber to minimize scattering of ions on grazing trajectories. The honeycomb cells are 6.35 mm in diameter and the sheets are 3.2 mm thick. \n\n\\subsection{Electrostatic Deflectors}\n\nEach of the two electrostatic deflectors (EDs) includes a pair of polished solid titanium electrodes backed by an array of Ti ribs intended to provide structural stability. The ribs and the 25 mm thick electrode of each cathode were machined from a single piece of Ti, as were the 20 mm thick, rib-backed anodes, which were bolstered from behind with an additional ribbed piece of Ti. Each of the four electrodes is supported by four cylindrical ceramic insulators. The insulators were brazed into Ti feet that attach to adjustable, polished Al mounting blocks connected to an Al support frame that rests in a stainless steel vacuum vessel. The vacuum-insulator-conductor triple points are shielded by Al corona rings and all sharp corners and edges are concealed behind polished electrostatic shields fashioned from Al. Figure \\ref{edpic} shows the interior of the ED2 vacuum tank. All the components subject to high electric fields were polished extensively using a combination of fine grit sandpaper and a succession of five varying grades of diamond pastes from 9 $\\mu$m down to 0.25 $\\mu$m grit sizes. For the rounded electrode surfaces that sustain the highest fields, an additional phase of polishing with a colloidal silica suspension was employed.\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.75\\linewidth, angle=270]{EDpic}\n\\caption{\\label{edpic}Photograph of the interior of the 2nd electrostatic deflector vacuum vessel.}\n\\end{figure}\n\nThe electrode support structures were assembled in a clean room to minimize dust deposition on the surfaces. Alignment of the electrode pairs was done to a precision of 0.1 mm inside the clean room using a FARO coordinate measuring machine with an articulating arm and a Leica laser tracker. During the alignment procedure a significant discrepancy was discovered between the measured and specified radii of curvature of the anodes; they are listed for comparison in Table \\ref{eds}. The electrodes were positioned such that the gap between them is 125 mm in the centre, which results in a reduction of the gap at the electrode edges compared with the design due to the smaller radii of curvature of the anodes. This manufacturing defect means that the electric fields in the gap are not as homogeneous as specified and that unanticipated second- and higher-order ion optical aberrations are present in the system; it also implies that the first order energy dispersion $(x|\\delta_E)$ is not quite as small as planned. These flaws were highly impractical to correct, as the polished, cleaned, and aligned electrodes would have had to be removed, re-machined, and re-polished.\n\n \\begin{table}\n \\centering\n \\caption{\\label{eds}EMMA Electrode Radii of Curvature in mm}\n \\begin{tabular}{llll}\n \\\\\nElectrode & ED1 & ED2 & Specification\\\\\n\\hline\nAnode & 5007.3(55) & 4977.1(55) & 5062.5\\\\\nCathode & 4953.3(55) & 4952.0(55) & 4937.5\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nHigh voltage is provided to the electrodes via internal, 40 stage full-wave Cockcroft-Walton DC multipliers built at TRIUMF. The multipliers are driven by external 1~kW Glassman High Voltage Inc.\\ PG010K100JD2DRV DC supplies that allow for constant voltage or constant current operation. Each multiplier has been tested without a load to a maximum potential difference of at least 325~kV. To prevent breakdown the multipliers are housed within re-entrant ceramic vessels pressurized with 3 bar of electrically insulating SF$_{6}$ gas. As bias is applied during conditioning, steady state load currents can reach as much as 35~$\\mu$A, leading to the emission of X-rays. For shielding purposes, the entire surface of the stainless steel dipole vacuum vessels are covered with 6.35~mm thick lead sheets cut and pounded into various shapes that were affixed using epoxy and contact cement. All the ports and windows are covered with caps containing the same thickness of Pb. Thus far, the first and second electrostatic deflectors have been conditioned to maximum stable potential differences of 340~kV and 440~kV, respectively. We plan to condition them further to a potential difference of 500~kV.\n\nEach electrostatic deflector vacuum tank is pumped by an Agilent TV1001 1000 L\/s turbomolecular pump, an Oxford Instruments CryoPlex 8 1500 L\/s cryopump, and a Gamma Vacuum TiTan 600L 500 L\/s ion pump. The ion pumps run on a diesel-generator-backed uninterruptible electric power circuit in order to ensure high vacuum continuously through electric power bumps and failures. Pressures in the mid $10^{-9}$ Torr range are typically observed with only two pumps running in each isolated vessel.\n\n\\section{Experimental Apparatus}\n\nTwo 10\" OD ConFlat vacuum crosses mounted symmetrically between the electrostatic deflectors and the dipole magnet house horizontal slit systems that can block ions on unwanted trajectories. The two 1.6~mm thick stainless steel plates that make up each slit system can be driven independently to create an opening from 0-180~mm wide. The opening need not be centred on the optic axis. One cross has a 1500 L\/s cryopump and the other has a 1000 L\/s turbomolecular pump. When either of these pumps is used to evacuate the isolated MD vacuum chamber section a pressure in the low $10^{-9}$ Torr range is typically observed.\n\nSeveral experimental systems have been designed and installed at EMMA while others are still under construction. These include target and focal plane detector systems and their associated chambers and accessories. Notably, some are being developed with off-site collaborators, such as those at the University of York in the UK.\n\nA 20~cm diameter spherical target chamber couples to the vacuum chamber of the first quadrupole magnet. The chamber is designed to accommodate 12 TIGRESS detectors in a closely packed configuration. It contains a rotary target mechanism that allows for the manual positioning of up to three target foil positions into the beam path. The target chamber also houses an integral, suppressed Faraday cup on a separate rotary actuator; a Ta aperture plate at its entrance lies in the same plane as the target foils. The Ta plate has a 1~mm diameter aperture through which the beam is tuned by maximizing the current on the Faraday cup while minimizing that on the aperture plate.\n\nThe target chamber has provisions for mounting two 150~mm$^2$ silicon surface barrier detectors centred at 20$^{\\circ}$ angles with respect to the beam axis downstream of the target. They are used to monitor the beam flux and target condition via elastic scattering and are primarily intended for the normalization of reaction cross section measurements. Additionally, a highly-segmented, annular silicon detector can be mounted 33~mm upstream or downstream of the target position to detect light charged particles. A thin C foil can be positioned 116~mm downstream of the target position in order to restore the charge state distributions of transmitted ions to equilibrium through charge-changing collisions following possible internal conversion decays. Similarly, an energy degrader foil can be mounted 68 mm downstream of the target. To exclude ions on trajectories outside of the angular acceptance of the spectrometer, a circular aperture at the exit of the target chamber defines a cone of half-angle $4.2^{\\circ}$ and a solid angle of 17~msr; an additional, optional $\\pm2^{\\circ}$ aperture can be used to restrict the horizontal angular acceptance when high $m\/q$ resolving power is required, as described in Ref.\\  \\cite{davids05}.\n\nA separate target chamber designed to accommodate highly-segmented rectangular and annular silicon detectors as well as 12 TIGRESS detectors, dubbed SHARC-II, has been designed at the University of York and is in the final stages of fabrication. It can be positioned so that the target is separated from the Q1 effective field boundary by the standard 25 cm.\n\nThe focal plane station has a modular design in which detectors may be inserted and removed easily according to the experimental requirements. A position-sensitive parallel grid avalanche counter (PGAC) and an energy-sensitive ionization chamber (IC) make up the standard complement of focal plane detectors. They are mounted in separate vacuum chambers that can be joined together, allowing for the use of the PGAC without the IC if only $m\/q$ information is required. In the current configuration, the installation of the PGAC is mandatory while the IC is optional. A 3000~mm$^2$ ion-implanted Si detector can be mounted directly behind the PGAC, as can a double-sided Si strip detector. The latter can also be mounted behind or inside the IC. All of the detectors are read out using a version of the MIDAS data acquisition system.\n\nBoth the PGAC and the IC are filled with isobutane as an ionization medium and have an active area of 154 mm by 54 mm, with a larger extent in the horizontal, dispersive direction than in the vertical direction. The PGAC, which measures 73 mm between entrance and exit foils, operates at pressures between 2 and 4 Torr while the 40~cm long, 16~anode segment IC is designed to operate at pressures of $10-100$ Torr. Provisions have been made to mount up to 4 clover-shaped HPGe detectors of the type used in the GRIFFIN spectrometer \\cite{garnsworthy18} at the focal plane to study isomers and delayed activities.\n\nThe focal plane station has a set of 4 independently actuated 1.6~mm thick stainless steel plates just upstream of the PGAC that together constitute a slit system. Two of the plates are mounted on combined rotary and linear motion feedthroughs that present different profiles to the incident ions depending on their angle. In this way the focal plane slit system can obscure the entire focal plane except for $1-3$ openings of continuously variable position and width, enabling the simultaneous transmission of up to 3 charge states.\n\n\\section{Initial Measurements}\\label{measurements}\n\nDue to a flurry of last minute activity on the focal plane station vacuum and control systems just prior to its first scheduled beam time, there was no opportunity to first test the spectrometer with an alpha source. Hence initial focussing and dispersion tests were carried out by bombarding a 4.5~$\\mu$m thick Au foil with an 80~MeV $^{36}$Ar beam. The spectrometer was initially set to transmit multiply scattered, 19~MeV $^{36}$Ar$^{13+}$ ions. The foil was sufficiently thick that the multiply scattered Ar ions filled the angular and energy acceptances of the spectrometer. After observing a single, well-defined $m\/q$ peak we set the fields for the same energy but charge state 13.5$^+$ and transmitted the 13$^+$ and 14$^+$ charge states simultaneously. As shown in Figure \\ref{firstspectrum}, the peaks of the two charge states were separated by the expected 66~mm distance corresponding to the design $m\/q$ dispersion of 10 mm\/\\%. The different peak heights reflect the charge state distribution of the transmitted $^{36}$Ar ions.\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=\\linewidth]{firstspectrum}\n\\caption{\\label{firstspectrum}First $m\/q$ spectrum obtained with EMMA; it resulted from bombarding a 4.5 $\\mu$m Au foil with an 80~MeV $^{36}$Ar beam. Multiply-scattered Ar ions filled both the angular and energy acceptances of the spectrometer. The two detected charge states are separated by the 66~mm expected from the tuned $m\/q$ dispersion of 10~mm\/\\%.}\n\\end{figure}\n\nFollowing this initial in-beam test, we carried out a series of ion optical studies with a $^{148}$Gd $\\alpha$ source to study the angular focussing, $m\/q$ dispersion, and energy dispersion cancellation. These tests will be described in a forthcoming publication. The spectrometer was further tested when we bombarded a 900~$\\mu$g~cm$^{-2}$ natural Cu target foil with both stable and radioactive Na beams and set the spectrometer to detect fusion products. In these measurements the $\\pm2^{\\circ}$ horizontal aperture was in place, the PGAC was operated at a pressure of 2 Torr, and the 3000~mm$^2$ Si detector was positioned directly behind the PGAC to aid in distinguishing fusion products from scattered beam ions. We first bombarded the Cu target with an 84~MeV $^{23}$Na beam from the ISAC offline ion source \\cite{jayamanna14} and set the spectrometer for 18.4~MeV, 81~u, $10^+$ recoils.\n\nAfter we obtained $m\/q$ spectra of the fusion products with several overlapping spectrometer settings, the operations group delivered an 87~MeV, radioactive $^{24}$Na beam produced in a SiC target by TRIUMF's 500~MeV proton beam and doubly ionized by a forced electron beam induced arc discharge ion source \\cite{bricault14}. The operators succeeded in tuning more than 90\\% of the radioactive beam through the 1~mm aperture of EMMA's Faraday cup. Beam intensities on target ranged from 1 to $4\\times10^7$~s$^{-1}$ and spectra were measured at several field settings. Figure \\ref{ribmassspectrum} shows that obtained when the spectrometer was set for 17.1~MeV, 82~u, $10^+$ recoils. Beam suppression at these settings exceeded a factor of $10^9$ and the $m\/q$ resolving power was measured to be as large as 240 (FWHM), which is more than adequate to resolve masses around $A=90$.\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=\\linewidth]{run564}\n\\caption{\\label{ribmassspectrum}First $m\/q$ spectrum obtained using EMMA to detect products of reactions induced by a radioactive beam. A 900~$\\mu$g~cm$^{-2}$ natural Cu foil was bombarded with an 87~MeV $^{24}$Na beam and the spectrometer was tuned to transmit 17.1~MeV, 82~u, $10^+$ recoils. This position spectrum from the PGAC was gated to include only events with an energy signal in the Si detector corresponding to fusion residues.}\n\\end{figure}\n\n\\section{Status and Future Plans}\\label{plans}\n\nHigh voltage conditioning is expected to be completed within the next 2 months, at which point the spectrometer will have been fully commissioned. Four EMMA experiments and two letters of intent have been approved by the TRIUMF subatomic physics experiment evaluation committee. All are motivated by nuclear astrophysics and all but one of them require the installation of the TIGRESS $\\gamma$-ray spectrometer around the EMMA target position to provide spectroscopic information. This installation is planned to be completed in the spring of 2019, at which point the EMMA scientific program can begin in earnest.\n\n\\section{Acknowledgements}\n\nBD acknowledges generous support from the Natural Sciences and Engineering Research Council of Canada and is grateful to F.~Cifarelli for his mechanical design work. OSK acknowledges support from the Villum Foundation. TRIUMF receives federal funding via a contribution agreement through the National Research Council of Canada. The UK authors acknowledge the support of the Science and Technology Facilities Council.\n\n\n\n\n\\bibliographystyle{model1-num-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCellular massive multiple-input multiple-output (MIMO) is currently considered as a key wireless access technology for 5G because it can provide high spectral efficiency (SE) and high energy efficiency (EE) with simple signal processing \\cite{ngo2013energy,bjornson2017massive}. In cellular massive MIMO, the BS with massive antenna arrays simultaneously serves all users in its cell on the same time-frequency resource \\cite{marzetta2010noncooperative,larsson2013massive,bjornson2015massive,marzetta2016fundamentals}.  \n\nSince cellular massive MIMO is based on cellular topology, its inherent limitation is inter-cell interference. To overcome this limitation, cell-free massive MIMO is introduced \\cite{ngo2017cell}. Cell-free massive MIMO can be considered as a useful and scalable version of network MIMO \\cite{Shamai2001Enhancing, Venkatesan2007Network} (much in the same way as cellular Massive MIMO\nis scalable version of multi-user MIMO). \nIn cell-free massive MIMO, a large number of access points (APs), which are geographically distributed over a large area, coherently serve all users on same time-frequency resource \\cite{ngo2017cell,interdonato2018ubiquitous}. Cell-free massive MIMO can reap all benefits of massive MIMO (favorable propagation, and channel hardening when using multiple antennas at APs \\cite{chen2018Chann}) and network MIMO (increased macro-diversity gain), and hence, it can offer  very high SE, EE, and coverage probability. These benefits can be achieved with simple signal processing and local channel acquisition at each AP.  With the cell-free topology, the excessive handover issue in small-cell systems can be resolved.  Moreover,  poor cell-edge performance, which is typical in small-cell networks, can be resolved in cell-free massive MIMO network by geographically distributing the APs \\cite{interdonato2018ubiquitous}. Compared with small cells, cell-free massive MIMO can provide up to a ten-fold improvement in 95\\%-likely SE \\cite{ngo2017cell,nayebi2017precoding}.\nThus, cell-free massive MIMO has attracted a lot of research interest recently \\cite{Ngo2017totalenergy,zhang2017spectral,basharcell,Mai2018Pilot,Mai2019Uplink}. \n\n\nMost of previous works exploit the performance of cell-free massive MIMO with single-antenna users. However, in practice, many user's devices of moderate physical size (e.g. laptops, tablets, and smart vehicles) can be equipped with several antennas to increase the multiplexing gain, and to improve system reliability due to the diversity gain. Thus, it is important to evaluate the performance of cell-free massive MIMO with multiple antennas at the users. Moreover, the effect of equipping multiple antennas at the users needs to be well understood to design the systems.  Downlink channel estimation has already investigated in \\cite{Ngo2013MU-MIMO}. But \\cite{Ngo2013MU-MIMO} considered collocated massive MIMO systems with single-antenna users, orthogonal pilot sequences, and no power control. In  \\cite{Interdonato2019disguise,Interdonato2016How}, the authors studied downlink channel estimation of cell-free massive MIMO. But in \\cite{Interdonato2019disguise,Interdonato2016How}, each AP has only one antenna, and matched filtering detection is used.\n\nInspired by the above discussion, in this paper, we analyze the performance of cell-free massive MIMO systems with multiple antennas at both APs and users. In cell-free massive MIMO, each users can be close to several APs, and thus Rician channel model is more reasonable in many scenarios \\cite{ozdogan2019performance}. However, in rich scattering environments, the Rayleigh fading model is still reasonable \\cite{ngo2017cell,nayebi2017precoding,chen2018Chann,Ngo2017totalenergy,zhang2017spectral,basharcell,Mai2018Pilot,Mai2019Uplink}. In addition, Rayleigh fading model is analytically tractable which helps us to obtain initial and important insights. Therefore, in this paper, we consider cell-free massive MIMO systems using independent, identically distributed (i.i.d.) Rayleigh fading channels. The closed-form expression of downlink SE is derived with taking account of non-orthogonal pilot sequences. As the space between adjacent antennas at the same user is very small, it may causes huge interference to each others. Therefore, in this paper, orthogonal pilot sequences are assigned for antennas at the same user, and those pilot sequences can be reused at antennas of other users. Moreover, the effects of the number of antennas at APs and users on the SE are analyzed through the use of max-min fairness power control. We evaluate the system performance of two protocols for the downlink data transmission of cell-free massive MIMO. In the first protocol, the system operates with two phases: uplink channel estimation and downlink data transmission. As a result, only statistical channel state information (CSI) is available at the users. Note that the statistical CSI depends on large-scale fading which changes very slowly, and it may stay constant for a duration of some 40 small-scale fading coherence intervals \\cite{ngo2017cell}. Whereas, in the second protocol, as the level of channel hardening in cell-free massive MIMO is lesser than the one in collocated massive MIMO \\cite{chen2018Chann}, we use the downlink channel estimation to improve the system performance. Therefore, in the second protocol, the system operates with three phases: uplink channel estimation, downlink channel estimation and downlink data transmission. As a result, estimated CSI is available at the users. To improve the system performance, both protocols use minimum mean-squared error-based successive interference cancellation (MMSE-SIC) detectors at the users. The computational complexity of MMSE-SIC detectors relates to the inverse operations of the $N\\times N$ effective channel gain matrix. Since $N$ is small (several antennas per user), this complexity is low. The second protocol is a generalization of that in previous work on cell-free massive MIMO \\cite{Interdonato2016How}, where we consider multi-antennas at users. In both protocols, we also compare system performances using MMSE detectors with the ones using MMSE-SIC detectors. In this paper, to reduce the fronthaul and backhaul requirements, the conjugate beamforming technique is used for both the protocols since it can be implemented in a distributed manner \\cite{ngo2017cell}. Other linear processing techniques such as MMSE and zero-forcing (ZF) are better than the conjugate beamforming technique  in terms of the system performance \\cite{bjornson2019making}. However, MMSE and ZF need huge fronthaul and backhaul requirements, as we need to send the channel state information to the CPUs, and signal processing is mainly done at the CPUs \\cite{bjornson2019making, buzzi2017cell,liu2019spectral}. So it is hard to implement MMSE or ZF in large cell-free massive MIMO networks. Recently, \\cite{interdonato2019local} proposed a method which can implement ZF in a distributed manner. But this scheme requires a very large number of antennas at the access points. Thus, cell-free massive MIMO with conjugate beamforming techniques has still received a lot of research attention recently \\cite{van2020joint,papazafeiropoulos2020performance,nikbakht2020uplink}.\nThe main contributions of this paper are as follows.\n\\begin{itemize}\n\\item The details of two transmission protocols with and without downlink pilots are presented and analyzed. The channel estimation with non-orthogonal pilot sequences and MMSE-SIC detectors are taken into account.\n\\item We derive a general formula for the SE with arbitrary side information at the users. Based on this result, analytical expressions for the SE of different transmission protocol are derived. \n\\item Max-min fairness power control (PC) is applied for the first protocol to improve the SE of the system. For the second protocol, due to the high complexity associated with computation\nfor the closed-form of SE, we apply the power control coefficients of the first protocol to the second protocol. Numerical results show that, with those power control coefficients, the SE improves significantly.\n\\item We investigate the effects of number antennas at both the APs and users.\n\\item We propose the framework for achieving the sub-optimal system performance regardless of the number of users.\n\\end{itemize}\n\n\nThe rest of this paper is organized as follows. Section \\ref{Sysmol} defines the system model for the downlink cell-free massive MIMO for both\ndata transmission and channel estimation. Next, Section \\ref{SpecEff} derives the achievable downlink SE of cell-free massive MIMO. Then, Section \\ref{max-min-pc} derives max-min fairness power control for SE. Section \\ref{Numsec}\nevaluates the system performance by using numerical results. Finally, the conclusion is drawn in Section \\ref{Concl}.\n\n\n\n\\textit{Notation}: The superscripts $()^\\ast$, $()^T$, and $()^H$ stand for the conjugate, transpose, and conjugate-transpose, respectively. The Euclidean norm, the expectation operators, and the determinant of matrix are denoted by $\\|\\cdot \\|$, $\\mathbb {E}\\left \\{{\\cdot }\\right \\}$, and $|.|$, respectively. $\\text{var}(.)$ denotes variance. In addition, ${z} \\sim \\mathcal {CN}\\left ({{0},{\\sigma ^{2}}}\\right )$ denotes a circularly symmetric complex Gaussian random variable (RV) $z$ with zero mean and variance $\\sigma ^{2}$, and $\\mathbb {C}^{L \\times N}$ denotes the ${L \\times N}$ matrix. Finally, ${z} \\sim \\mathcal {N}(0,\\sigma ^{2})$ denotes a real-valued Gaussian RV.\n\\section{System Model}\\label{Sysmol}\nWe consider a cell-free massive MIMO system operating in time division duplex (TDD) mode with $M$ APs and $K$ users randomly located within a large area. Each AP has $L$ antennas, whereas each user has $N$ antennas. \nLet $\\textbf{G}_{mk} \\in \\mathbb {C}^{L \\times N}$ be the channel response matrix between the $k$-th user and the $m$-th AP. Then, \n\\begin{equation}\n\\textbf{G}_{mk} = \\beta^{1\/2}_{mk}\\textbf{H}_{mk},\n\\end{equation}\nwhere $\\beta_{mk}$ is large-scale fading between the $k$-th user and the $m$-th AP, and $\\textbf{H}_{mk}$ is the $L\\times N$ small-scale fading matrix whose elements are assumed to be independent and identically distributed (i.i.d.) $\\mathcal {CN}\\left ({{0},{1}}\\right )$ RVs. In this work, we focus on the downlink transmission, and hence, the uplink data transmission is neglected. Specifically, we consider two transmission protocols. The first protocol has two phases: the uplink channel estimation and the downlink data transmission. The second protocol has three phases: the uplink channel estimation, the downlink channel estimation and the downlink data transmission. System model of the two protocols will be presented in detail in the rest of this section. \n\n\\subsection{Transmission Protocol 1 - No Downlink Pilots}\nThis transmission protocol is commonly used in previous studies of cell-free massive MIMO systems. Each user relies on the channel hardening property of massive MIMO technology to detect the desired signals. So there is no downlink channel estimation phase \\cite{ngo2017cell}.\n\\subsubsection{Uplink Channel Estimation}\\label{UCE_P1}\nIn this phase, all users will send pilot signals to the APs. Then each AP will estimate its channels to all users using the received pilot signals. Let $\\tau_{\\text{u}}$ be the length of the uplink training duration per coherence interval, and $\\pmb{\\Phi}_{\\text{u},k} \\in \\mathbb {C}^{\\tau_{\\text{u}} \\times N}$, where its $n$-th column satisfies  $\\|\\pmb{\\phi}_{\\text{u},k,n} \\| = 1$, $\\forall n \\in N$, be a pilot matrix of the $k$-th user.\nThen, the received signal at the $m$-th AP is\n\\begin{equation} \n\\textbf{Y}_{\\text{u}, m}=\\sum_{k=1}^{K}\\sqrt{\\tau_{\\text{u}}\\rho_{\\text{u}}}\\textbf{G}_{mk}\\pmb{\\Phi}_{\\text{u},k}^H+\\textbf{W}_{\\text{u},m}, \n\\end{equation}\nwhere $\\rho_{\\text{u}}$ is the normalized signal-to-noise ratio (SNR) of each uplink pilot symbol and $\\textbf{W}_{\\text{u},m}$ is the $L\\times\\tau_{\\text{u}}$ matrix of additive noise at the $m$-th AP. We assume that the elements of $\\textbf{W}_{\\text{u}, m}$ are i.i.d. $\\mathcal {CN}\\left ({{0},{1}}\\right )$ RVs. A projection of $\\textbf{Y}_{\\text{u}, m}$ onto $\\pmb{\\Phi}_{\\text{u}, k}$ is\n\\begin{align} \n\\textbf{Y}_{\\text{u},mk} &=\\textbf{Y}_{\\text{u},m}\\pmb{\\Phi}_{\\text{u},k}\\notag \\\\ \n &= \\sum_{i=1}^{K}\\sqrt{\\tau_{\\text{u}}\\rho_{\\text{u}}}\\textbf{G}_{mi}\\pmb{\\Phi}_{\\text{u},ik} + \\textbf{W}_{\\text{u},mk},\n\\end{align}\nwhere $\\pmb{\\Phi}_{\\text{u},ik} \\triangleq \\pmb{\\Phi}_{\\text{u},i}^H\\pmb{\\Phi}_{\\text{u},k}$, and $\\textbf{W}_{\\text{u},mk} \\triangleq \\textbf{W}_{\\text{u},m}\\pmb{\\Phi}_{\\text{u},k}$.\nBy stacking all columns of $\\textbf{Y}_{\\text{u},mk}$ on top of each other, we have\n\\begin{align} \n\\text{vec}(\\textbf{Y}_{\\text{u},mk}) &= \\sqrt{\\tau_{\\text{u}}\\rho_{\\text{u}}}\\sum_{i=1}^{K}(\\pmb{\\Phi}_{\\text{u},ik}^T \\otimes \\textbf{I}_L) \\text{vec}(\\textbf{G}_{mi})+ \\text{vec}(\\textbf{W}_{\\text{u},mk})\\notag\\\\\n&=\\sqrt{\\tau_{\\text{u}}\\rho_{\\text{u}}}\\sum_{i=1}^{K}\\tilde{\\pmb{\\Phi}}_{ik} \\text{vec}(\\textbf{G}_{mi})+ \\text{vec}(\\textbf{W}_{\\text{u},mk}),\n\\end{align}\nwhere vec(.) is the vectorization operation, and $\\tilde{\\pmb{\\Phi}}_{\\text{u},ik} \\triangleq \\pmb{\\Phi}_{\\text{u},ik}^T \\otimes \\textbf{I}_L$. Since the distance between adjacent antennas at the same user is very small, non-orthogonal pilots may cause huge interference to each others. To mitigate interference between antennas of the same user, we assume that orthogonal pilot sequences are assigned for antennas of each user, but these pilot sequences can be\nreused in other users. Then, MMSE estimation of $\\text{vec}(\\textbf{G}_{mk})$ given $\\text{vec}(\\textbf{Y}_{\\text{u},mk})$ is expressed by \\cite{kay1993fundamentals}\n\\begin{align}\\label{vecchann} \n\\text{vec}(\\hat{\\textbf{G}}_{mk})&= \\sqrt{\\tau_{\\text{u}}\\rho_{\\text{u}}}\\beta_{mk}\\left(\\tau_{\\text{u}}\\rho_{\\text{u}}\\sum_{i=1}^{K}\\tilde{\\pmb{\\Phi}}_{\\text{u},ik}\\beta_{mi}\\tilde{\\pmb{\\Phi}}^H_{\\text{u},ik}+ \\textbf{I}_{LN}  \\right)^{-1}\\notag\\\\\n&\\quad\\times\\text{vec}(\\textbf{Y}_{\\text{u},mk}).\n\\end{align}\n\\begin{lemma}\\label{lemma1} \nThe estimate of the channel matrix $\\textbf{G}_{mk}$ is \n\\begin{align}\\label{estchan1}\n\\hat{\\textbf{G}}_{mk} = \\textbf{Y}_{\\text{u},mk} \\textbf{A}_{mk}, \n\\end{align}\nwhere \n\\begin{equation} \n\\textbf{A}_{mk} \\triangleq \\sqrt{\\tau_{\\text{u}}\\rho_{\\text{u}}}\\beta_{mk}\\left(\\tau_{\\text{u}}\\rho_{\\text{u}}\\sum_{i=1}^{K}\\beta_{mi}\\pmb{\\Phi}^H_{\\text{u},ik}\\pmb{\\Phi}_{\\text{u},ik}+ \\textbf{I}_N\\right)^{-1}.\n\\end{equation}\n\\end{lemma}\n\\begin{IEEEproof}\n\tSee Appendix \\ref{AppendixA}.\n\\end{IEEEproof}\n\n\\subsubsection{Downlink data transmission}\\label{DDT_P1}\nIn this phase, each AP uses the channel estimates in the uplink channel estimation phase together with the conventional conjugate beamforming technique to precode the desired symbols \\cite{ngo2017cell}. Then the precoded signal will be sent to all users. The $L\\times 1$ transmitted  signal from the $m$-th AP is \n\\begin{equation} \n\\textbf{x}_m= \\sqrt{\\rho}\\sum_{k=1}^{K}\\eta^{1\/2}_{mk}\\hat{\\textbf{G}}_{mk}\\textbf{q}_{k} ,\n\\end{equation}\nwhere $\\textbf{q}_{k}$, with $ \\mathbb {E} \\{\\textbf{q}_{k}\\textbf{q}^{H}_{k} \\} = \\textbf{I}_{N}$, is the vector of symbols intended for the $k$-th user, $\\rho$ is the normalized transmit SNR constraint at the $m$-th AP, and $\\eta_{mk}$ is power control coefficient corresponding to the $k$-th user.\nThe power control coefficients $\\eta_{mk}$ are chosen to satisfy the power constraint at each AP:  $\\text{E}\\{\\Arrowvert \\textbf{x}_m \\Arrowvert^2\\}\\le \\rho$ which is equivalent to \n\\begin{equation}\\label{powcontraint} \n\\sqrt{\\tau_{\\text{u}}\\rho_{\\text{u}}}\\sum_{k=1}^{K}\\eta_{mk}\\beta_{mk}\\Tr \\left(\\textbf{A}_{mk}\\right) \\le \\frac{1}{L}.\n\\end{equation}\nThe received signal at the $k$-th user is\n\\begin{align}\\label{data_recei_eq} \n\\textbf{r}_k &= \\sum_{m=1}^{M}\\textbf{G}^H_{mk}\\textbf{x}_m + \\textbf{n}_k\\notag\\\\\n&= \\sqrt{\\rho}\\sum_{k'=1}^{K}\\textbf{D}_{kk'}\\textbf{q}_{k'} + \\textbf{n}_k, \n\\end{align}\nwhere $\\textbf{D}_{kk'} \\triangleq \\sum_{m=1}^{M}\\eta_{mk'}^{1\/2}\\textbf{G}^H_{mk}\\hat{\\textbf{G}}_{mk'}$ denotes the effective downlink channel for the $k'$-th user and $\\textbf{n}_k$ is the noise vector. The elements of $\\textbf{n}_k$ are assumed to be i.i.d. $\\mathcal {CN}\\left ({{0},{1}}\\right )$.\n\n\\subsection{Transmission Protocol 2 - With Downlink Pilots}\nDifferent from cellular massive MIMO, cell-free massive MIMO offers lesser channel hardening. Therefore it is good to estimate channel at the users via the downlink pilots. Here for the analysis simplicity, we assume that orthogonal pilot sequences are used for the downlink channel estimation phase.\n\\subsubsection{Uplink Channel Estimation}\nThis phase is the same as that of Transmission Protocol 1. See Section~\\ref{UCE_P1}.\n\n\\subsubsection{Downlink Channel Estimation}\nFrom the received signal (\\ref{data_recei_eq}), to detect the desired signal, each user does not need to estimate all channel matrices $\\textbf{G}_{mk}$. Instead, it needs to estimate only the effective channel gain matrices $\\textbf{D}_{kk'}$ which have much lower dimension. To do this, the pilot sequences will be precoded before being sent to all users \\cite{Interdonato2016How}.\nLet $\\tau_{\\text{d}}$ be the length of the downlink training duration per coherence interval, and $\\pmb{\\Phi}_{\\text{d},k} \\in \\mathbb {C}^{\\tau_{\\text{d}} \\times N}$ be a pilot matrix for the $k$-th user that satisfies\n\\begin{equation*}\n\\pmb{\\Phi}^H_{\\text{d},k} \\pmb{\\Phi}_{\\text{d},k'} =\n\\begin{cases}\n\\textbf{I}_N & \\text{if } k = k',\\\\\n0 & \\text{if } k \\neq k'.\n\\end{cases}\n\\end{equation*}\nThen, precoded pilot matrix which is transmitted from the $m$-th AP is\n\\begin{equation} \n\\textbf{X}_{\\text{d}, m}=\\sqrt{\\tau_{\\text{d}}\\rho_{\\text{d}}}\\sum_{k'=1}^{K}\\eta_{mk'}^{1\/2}\\hat{\\textbf{G}}_{mk'}\\pmb{\\Phi}_{\\text{d},k'}^H,\n\\end{equation}\nwhere $\\rho_{\\text{d}}$ is the normalized signal-to-noise ratio (SNR) of the $m$-th AP. Then, the received pilot signal at the $k$-th user is \n\\begin{align} \n\\textbf{Y}_{\\text{d},k} &=\\sqrt{\\tau_{\\text{d}}\\rho_{\\text{d}}}\\sum_{m=1}^{M}\\sum_{k'=1}^{K}\\eta_{mk'}^{1\/2}\\textbf{G}^H_{mk}\\hat{\\textbf{G}}_{mk'}\\pmb{\\Phi}_{\\text{d},k'}^H+\\textbf{W}_{\\text{d},k},\\notag \\\\ \n&=\\sqrt{\\tau_{\\text{d}}\\rho_{\\text{d}}}\\sum_{k'=1}^{K}\\textbf{D}_{kk'}\\pmb{\\Phi}_{\\text{d},k'}^H + \\textbf{W}_{\\text{d},k}.\n\\end{align}\n\\begin{lemma}\\label{lemma2} \nMMSE estimation of $\\textbf{D}_{kk'}$, $\\forall k, k' = 1, 2,\\hdots, K$, given $\\textbf{Y}_{\\text{dp},k}$ is $\\hat{\\textbf{D}}_{kk'}$, whose $ij$-th element is\n\\begin{align}\\label{eq_d_kkface}\n\\hat{d}_{kk',ij}=\\begin{cases}\n\\frac{\\sqrt{\\tau_{\\text{d}}\\rho_{\\text{d}}}\\left(\\xi_{kk,i} + \\kappa_{kk,i}^2 \\right) y_{\\text{d},k,ij}  + \\kappa_{kk,i}}{\\tau_{\\text{d}}\\rho_{\\text{d}}\\left(\\xi_{kk,i} + \\kappa_{kk,i}^2 \\right) + 1}  & \\text{if } k = k', i = j, \\\\\n\\frac{\\sqrt{\\tau_{\\text{d}}\\rho_{\\text{d}}}\\xi_{kk',j}y_{\\text{d},k,ij} }{\\tau_{\\text{d}}\\rho_{\\text{d}}\\xi_{kk',j} + 1} & \\text{otherwise},\n\\end{cases}\n\\end{align}\nwhere $y_{\\text{d},k,ij}$ is $ij$-element of the matrix $\\textbf{Y}_{\\text{dp},k}$, $\\xi_{kk,i} = L\\sum^M_{m=1}\\eta_{mk}\\beta_{mk}\\gamma_{mk,i}$, $\\gamma_{mk,i} = \\bigg[ \\tau_{\\text{u}}\\rho_{\\text{u}}\\beta^2_{mk} \\left(\\tau_{\\text{u}}\\rho_{\\text{u}}\\sum_{i=1}^{K}\\tilde{\\pmb{\\Phi}}_{\\text{u},ik}\\beta_{mi}\\tilde{\\pmb{\\Phi}}^H_{\\text{u},ik}+ \\textbf{I}_{LN}  \\right)^{-1} \\bigg ]_{(i-1)L + l}$, $\\kappa_{kk,i} = L\\sum_{m=1}^{M}\\eta^{1\/2}_{mk}\\gamma_{mk,i} $, and $\\xi_{kk',i} = L\\sum^M_{m=1}\\eta_{mk'}\\beta_{mk}\\gamma_{mk',i}$.\n\\end{lemma}\n\\begin{IEEEproof}\n\tSee Appendix \\ref{AppendixB}.\n\\end{IEEEproof}\nNote that $\\textbf{D}_{kk'}$ is an  $N \\times N$ matrix. Since $N$ is small, the corresponding complexity of the MMSE estimation is low. \n\\subsubsection{Downlink data transmission}\nThe downlink transmission of this protocol is the same as that of Protocol 1, see Section \\ref{DDT_P1}. But in this protocol, since each user estimates the effective channel gain matrices, it will use this information to detect the desired symbols.\n\\section{Spectral Efficiency}\\label{SpecEff}\nIn this section, we derive analytical expressions for the SEs of transmission protocols 1 and 2 assuming that each user uses the MMSE-SIC scheme to detect the desired symbols. The SE under the assumption that the users have perfect CSI is derived as a benchmark. In addition, for the comparison,  we derive the SE expression for case that the users use the simple linear MMSE detector instead of the MMSE-SIC detector. Note that the computational complexity of MMSE-SIC detectors which relates to the inverse operations of the $N\\times N$ effective channel gain matrix, is low. Since different SEs correspond to the different of side information available at the users, we first provide a general SE expression with side information at the users as in the following theorem.\n\\begin{theorem}\\label{theo_SE}\nThe achievable downlink SE of the $k$-th user with MMSE-SIC detection scheme given the received signal $\\textbf{r}_{k}$ in (\\ref{data_recei_eq}) and side information $\\mathbf{\\Theta}_{k}$ (assuming that $\\mathbf{\\Theta}_{k}$ is independent of $\\textbf{q}_{k}$) is given by\n\\begin{align}\\label{R_k1}  \nR_k = (1 - \\tau_{\\text{tot}}\/\\tau_{\\text{c}})\\mathbb {E}\\left\\{ \\log_{2} \\left| \\textbf{I}_{N} + \\mathbf{\\Upsilon}^a_{kk} \\right|\\right\\},\n\\end{align}\nwhere $\\tau_{\\text{tot}}$ is the total training duration per coherence interval $\\tau_{\\text{c}}$,\n$\\mathbf{\\Upsilon}^a_{kk} \\triangleq \\rho \\mathbb {E}\\{\\textbf{D}^H_{kk}|\\mathbf{\\Theta}_{k}\\} \\left(\\mathbf{\\Psi}^a_{kk} \\right)^{-1} \\mathbb {E}\\{\\textbf{D}_{kk}|\\mathbf{\\Theta}_{k}\\}$,\nand $\\mathbf{\\Psi}^a_{kk} \\triangleq \\textbf{I}_{N} + \\mathbb {E}\\{(\\rho\\sum^{K}_{k'=1}\\textbf{D}_{kk'}\\textbf{D}^H_{kk'}|\\mathbf{\\Theta}_{k})\\} - \\rho \\mathbb {E}\\{\\textbf{D}_{kk}|\\mathbf{\\Theta}_{k}\\} \\mathbb {E}\\{\\textbf{D}^H_{kk}|\\mathbf{\\Theta}_{k}\\}.$\n\n\\end{theorem}\n\\begin{IEEEproof}\n\t See Appendix \\ref{Appendix_theo_SE}.\n\\end{IEEEproof}\n\\subsection{Achievable Downlink SE for Protocol 1}\\label{Scheme1}\nFor this transmission procotol, there are no downlink pilots. Each user uses ony the statistic CSI for signal detection. This means that  $\\mathbf{\\Theta}_{k} = \\bar{\\textbf{D}}_{kk} \\triangleq \\mathbb {E}\\{\\textbf{D}_{kk}\\}$.\nWith $\\mathbf{\\Theta}_{k} = \\bar{\\textbf{D}}_{kk}$, the achievable downlink SE of the $k$-th user in (\\ref{R_k1}) becomes\n\\begin{align}\\label{R_k_statistic}  \nR^{\\text{St-SIC}}_k = (1 - \\tau_{\\text{u}}\/\\tau_c) \\log_{2} \\left| \\textbf{I}_{N} + \\rho\\bar{\\textbf{D}}^H_{kk} \\left(\\mathbf{\\Psi}^b_{kk}\\right)^{-1} \\bar{\\textbf{D}}_{kk} \\right|,\n\\end{align}\nwhere \n\\begin{align}\n\\mathbf{\\Psi}^b_{kk} = \\textbf{I}_{N} + \\mathbb {E}\\left\\{\\left(\\rho\\sum^{K}_{k'=1}\\textbf{D}_{kk'}\\textbf{D}^H_{kk'}\\right)\\right\\} - \\rho\\bar{\\textbf{D}}_{kk}\\bar{\\textbf{D}}^H_{kk}. \\notag\n\\end{align}\nBy deriving all expectations, we obtain the closed-form expression for the achievable SE (\\ref{muin1}) as in the following theorem.\n\\begin{theorem}\\label{theo_closedform}\nGiven statistic CSI, says $\\mathbf{\\Theta}_{k} = \\bar{\\textbf{D}}_{kk}$, and using MMSE-SIC detectors, the achievable downlink SE for the $k$-th user can be represented\nin closed-form as\n\\begin{align}\\label{muin1} \nR^{\\text{St-SIC}}_k = (1 - \\tau_{\\text{u}}\/\\tau_c) \\log_{2} \\left| \\textbf{I}_{N} + \\rho\\bar{\\textbf{D}}^H_{kk} \\left(\\mathbf{\\Psi}^b_{kk}\\right)^{-1} \\bar{\\textbf{D}}_{kk} \\right|,\n\\end{align}\nwhere\n\\begin{align} \n\\bar{\\textbf{D}}_{kk}=L\\sqrt{\\tau_{\\text{u}}\\rho_{\\text{u}}}\\sum_{m=1}^{M}\\eta^{1\/2}_{mk}\\beta_{mk}\\textbf{A}_{mk},\n\\end{align} \nand\n\\begin{align}\\label{seq} \n&\\mathbf{\\Psi}^b_{kk}\\notag\\\\\n&= L\\tau_{\\text{u}}\\rho_{\\text{u}}\\rho\\sum_{m=1}^{M}\\Bigg(\\sum_{k'=1}^{K}\\eta_{mk'}\\beta^2_{mk}\\textbf{C}_{mkk'} - L\\eta_{mk}\\beta^2_{mk}\\textbf{A}_{mk}\\textbf{A}^H_{mk} \\notag\\\\ \n&\\quad + L\\sum_{n \\ne m}^{M}\\sum_{k'\\ne k}^{K}\\eta^{1\/2}_{mk'}\\eta^{1\/2}_{nk'}\\beta_{mk}\\beta_{nk} \\Phi_{kk'}\\textbf{A}_{mk'}\\textbf{A}^H_{nk'}\\Phi^H_{kk'}\\notag\\\\\n&\\quad +\\sum_{k'=1}^{K}\\sum_{i \\ne k}^{K}\\eta_{mk'}\\beta_{mk}\\beta_{mi}\\Tr(\\Phi_{ik'}\\textbf{A}_{mk'}\\textbf{A}^H_{mk'}\\Phi^H_{ik'})\\textbf{I}_{N} \\Bigg) \\notag\\\\ \n&\\quad+ L\\rho\\sum_{m=1}^{M}\\sum_{k'=1}^{K}\\beta_{mk}\\eta_{mk'}\\Tr(\\textbf{A}_{mk'}\\textbf{A}^H_{mk'})\\textbf{I}_{N} + \\textbf{I}_{N},\n\\end{align}\nwhere $\\textbf{C}_{mkk'}$ is a $N \\times N$ diagonal matrix that $\\left[\\textbf{C}_{mkk'}\\right]_{ii} = \\sum_{n = 1}^{N} b_{mkk',nn} + Lb_{mkk',ii}$ with $b_{mkk',ii} \\triangleq \\left[\\textbf{B}_{mkk'}\\right]_{ii}= \\left[\\Phi_{kk'}\\textbf{A}_{mk'}\\textbf{A}^H_{mk'}\\Phi^H_{kk'}\\right]_{ii}$.\n\\end{theorem}\n\\begin{IEEEproof}\n\tSee Appendix \\ref{AppendixE}.\n\\end{IEEEproof}\n\\begin{remark}\nNote that, a minor correction is updated in (\\ref{seq}) when comparing with that in conference version in \\cite{Mai2018Cell}.  \n\\end{remark}\nThe lower bound of the downlink SE in Theorem~\\ref{theo_closedform} can be achieved by using per-user-basis MMSE-SIC detector while treating co-user interference plus noise as uncorrelated Gaussian noise with the assumption that $\\textbf{q}_{k} \\thicksim \\mathcal {CN}\\left ({{0},{\\textbf{I}_{N}}}\\right )$ .\n\\begin{remark}\nIn the special case that all APs and users have a single antenna, i.e., $L=N=1$, the spectral efficiency (\\ref{muin1}) is identical to the one in \\cite{ngo2017cell}.\n\\end{remark}\n\n\n\\subsection{Achievable Downlink SE for Protocol 2}\\label{Scheme2}\nFor this transmission protocol, each user acquires the estimates of the effective channel gains via the downlink pilots. More precisely, we have $\\mathbf{\\Theta}_{k} = \\{\\hat{\\textbf{D}}_{ki}\\}$, $\\forall i \\in K$.\nUsing Theorem 1, we obtain the following achievable SE:\n\\begin{align}\\label{se_sch2}  \nR_k = \\left(1 - \\frac{\\tau_{\\text{u}} + \\tau_{\\text{d}}}{\\tau_{\\text{c}}}\\right)\\mathbb {E}\\left\\{ \\log_{2} \\left| \\textbf{I}_{N} + \\mathbf{\\Upsilon}^c_{kk} \\right|\\right\\},\n\\end{align}\nwhere \n$\\mathbf{\\Upsilon}^c_{kk} \\triangleq \\rho \\mathbb {E}\\left\\{\\textbf{D}^H_{kk}|\\{\\hat{\\textbf{D}}_{ki}\\}\\right\\} \\left(\\mathbf{\\Psi}^c_{kk}\\right)^{-1}  \\mathbb {E}\\left\\{\\textbf{D}_{kk}|\\{\\hat{\\textbf{D}}_{ki}\\}\\right\\}$,\nand $\\mathbf{\\Psi}^c_{kk} \\triangleq \\textbf{I}_{N} + \\mathbb {E}\\left\\{(\\rho\\sum^{K}_{k'=1}\\textbf{D}_{kk'}\\textbf{D}^H_{kk'}|\\{\\hat{\\textbf{D}}_{ki}\\})\\right\\} - \\rho \\mathbb {E}\\left\\{\\textbf{D}_{kk}|\\{\\hat{\\textbf{D}}_{ki}\\}\\right\\} \\mathbb {E}\\left\\{\\textbf{D}^H_{kk}|\\{\\hat{\\textbf{D}}_{ki}\\}\\right\\}.$ Since the elements of $\\textbf{D}_{kk'}$ are not Gaussian distributed, the elements of their MMSE estimate $\\hat{\\textbf{D}}_{kk'}$ and corresponding elements of estimation error $\\tilde{\\textbf{D}}_{kk'}$ are uncorrelated, but not independent. This makes (\\ref{se_sch2}) hard to be computed in closed form. However,  elements of $\\textbf{D}_{kk'}$  are very close to Gaussian, especially when $M$ is large. This is shown in the Lemma \\ref{lemma3} as follows.\n\\begin{lemma}\\label{lemma3} \nThe elements of downlink effective channels $\\textbf{D}_{kk'}, (\\forall k, k' = 1,2, \\ldots, K)$ converge in distribution to Gaussian distribution as follows:\n\\begin{align} \n&d_{kk,ij}\\overset{d}{\\rightarrow} \\mathcal{CN}(0, \\xi_{kk,j}),\\text{as}\\ M\\rightarrow\\infty, \\text{ and } \\forall i \\neq j,\\notag\\\\\n&d_{kk,ii}\\overset{d}{\\rightarrow} \\mathcal{N}\\left(L\\sum_{m=1}^{M}\\sqrt{\\eta_{mk}}\\gamma_{mk,i},L\\sum_{m=1}^{M}\\eta_{mk}\\gamma_{mk,i}^{2}\\right),\\notag\\\\\n&\\quad\\quad\\quad\\quad \\text{as}\\ M\\rightarrow\\infty,\\text{ and}\\notag\\\\\n&d_{kk',ij}\\overset{d}{\\rightarrow} \\mathcal{CN}(0, \\xi_{kk',i}),\\text{as}\\ M\\rightarrow\\infty,\\forall k' \\ne k,  \\text{ and } \\forall i, j \\in N, \\notag\n\\end{align}\nwhere $\\overset{d}{\\rightarrow}$ denotes convergence in distribution.\n\\end{lemma}\n\\begin{IEEEproof}\n\tSee Appendix \\ref{Appendix_proflemma3}. \n\\end{IEEEproof}\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{d_ii.eps}\\\\%\\vspace{1} \n\\includegraphics[width=0.23\\textwidth]{d_ij_re.eps}\n\\includegraphics[width=0.23\\textwidth]{d_ij_im.eps}\n\\caption{True (Empirical) v.s. approximated (Gaussian) pdf of $d_{kk,ij})$, $\\forall i,j = 1, 2, \\ldots, N$.}\n\\label{d_kk}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.23\\textwidth]{d_kkface_ij_re.eps}\n\\includegraphics[width=0.23\\textwidth]{d_kkface_ij_im.eps}\n\\caption{True (Empirical) vs approximated (Gaussian) pdf of $d_{kk',ij}), \\forall k' \\ne k$, and $\\forall i,j = 1, 2, \\ldots, N$. }\n\\label{d_kkface}\n\\end{figure}\nFig. \\ref{d_kk} and Fig. \\ref{d_kkface} verify the Gaussian approximation in Lemma \\ref{lemma3}.  Fig. \\ref{d_kk} shows that the probability density functions (pdfs) of the empirical and the Gaussian distribution of $d_{kk,ij}$ ($\\forall i,j= 1, 2, \\ldots, N$) are very close. Moreover, with the high probability, the imaginary part of $d_{kk,ii}$ is much smaller than the real part, so it can be neglected. In Fig. \\ref{d_kkface}, it is clear that the pdfs of both the real part and the imaginary part of $d_{kk',ij}$ ($k' \\neq k, \\forall i,j= 1, 2, \\ldots, N$) are very close to the pdfs of their Gaussian approximations. In both Fig. \\ref{d_kk} and Fig. \\ref{d_kkface}, apart from the imaginary part of $d_{kk,ii}$ which can be neglected,  the probability of mismatch between approximated and enprirical Gaussian pdfs is very small.\n\nBy using Gaussian approximation of $\\textbf{D}_{kk}$, we obtain the following approximating closed-form expression of the SE.\n\\begin{theorem}\\label{theo_SE_protocol2_UC}\nGiven estimated CSI, says $\\mathbf{\\Theta}_{k} = \\{\\hat{\\textbf{D}}_{k1}, \\hat{\\textbf{D}}_{k2}, \\ldots , \\hat{\\textbf{D}}_{kK}\\} $, and using MMSE-SIC detectors, the achievable downlink SE of the $k$-th user in (\\ref{R_k1}) can be rewritten as\n\\begin{align}\\label{I_qrThe2}  \nR^{\\text{Et-SIC}}_k \\!= \\!\\left(\\!1 - \\frac{\\tau_{\\text{u}} \\!+\\! \\tau_{\\text{d}}}{\\tau_{\\text{c}}}\\!\\right)\\! \\mathbb {E} \\left \\{ \\log_{2} \\left|\\textbf{I}_{N} \\!+\\! \\rho\\hat{\\textbf{D}}^H_{kk} \\left(\\mathbf{\\Psi}^{c'}_{kk}\\right)^{-1} \\hat{\\textbf{D}}_{kk} \\right| \\right \\},\n\\end{align}\nwhere \n\\begin{align}\\label{psi_ckk_final}\n\\mathbf{\\Psi}^{c'}_{kk} = \\rho\\sum^{K}_{k'\\neq k}\\hat{\\textbf{D}}_{kk'}\\hat{\\textbf{D}}^H_{kk'} + \\rho \\sum^{K}_{k'=1}\\tilde{\\textbf{D}}^{\\text{var}}_{kk'}  +  \\textbf{I}_{N}, \n\\end{align}\nwith\n\\begin{align}\\label{} \n\\tilde{\\textbf{D}}^{\\text{var}}_{kk'} =\\colvec{\\sum\\limits^{N}_{j=1} \\text{var}(\\tilde{d}_{kk',1j}) & 0  & \\hdots & 0  \\\\\n0 & \\sum^{N}_{j=1} \\text{var}(\\tilde{d}_{kk',2j})  & \\hdots & 0\\\\\n\\vdots\n\t\t& & \\ddots  &\\\\\n   0 & 0 & \\hdots & \\sum^{N}_{j=1} \\text{var}(\\tilde{d}_{kk',Nj}) }, \n\\end{align}\nwhere \n\\begin{equation}\\label{}\n\\text{var}(\\tilde{d}_{kk',ij}) =\n\\begin{cases}\n\\frac{\\xi_{kk,i} + \\kappa_{kk,i}^2}{\\tau_{\\text{d}}\\rho_{\\text{d,p}}(\\xi_{kk,i} + \\kappa_{kk,i}^2) + 1} & \\text{if } k = k',\\text{and } i = j \\\\\n\\frac{\\xi_{kk',j}}{\\tau_{\\text{d}}\\rho_{\\text{d,p}}\\xi_{kk',j} + 1}  & \\text{otherwise }.\n\\end{cases}\n\\end{equation}\n\\end{theorem}\n\\begin{IEEEproof}\n\tSee appendix \\ref{Appendix_proftheo_SE_protocol2_UC}.  \n\\end{IEEEproof}\n\\subsection{Achievable Downlink SE with Linear MMSE Detectors} \nIn this section, we derive the downlink SE using linear MMSE detectors instead of MMSE-SIC detectors to evaluate the difference of the system performance between them. Note that compare with the MMSE-SIC detection, the linear MMSE detection is simpler for the implementation. \n\\subsubsection{Protocol 1 with Linear MMSE Detectors} \nAt each user, linear MMSE detectors detect $N$ data streams, which are intended for $N$antennas, independently. Then, the linear MMSE detector for the $n$-th data stream of the $k$-th user is given by \\cite{li2016massive, ngo2013energy}\n\\begin{align}\n\\textbf{f}_{k,n} = \\mathbf{\\Psi}^{b'}_{kk} \\bar{\\textbf{d}}_{kk,n},\n\\end{align}\nwhere $\\mathbf{\\Psi}^{b'}_{kk} = (\\mathbf{\\Psi}^b_{kk})^{-1} + \\bar{\\textbf{D}}_{kk}\\bar{\\textbf{D}}^H_{kk}$ and $\\bar{\\textbf{d}}_{kk,n}$ denotes the $n$-th column of $\\bar{\\textbf{D}}_{kk}$.\n\nGiven MMSE detectors at the users, the achievable downlink SE of the $k$-th user can be calculated as\n\\begin{align}\\label{}  \nR^{\\text{St-MMSE}}_k = \\left(1 - \\frac{\\tau_{\\text{u}}}{\\tau_c}\\right) \\sum^{N}_{n=1}\\mathbb {E} \\left \\{ \\log_{2} (1 + \\zeta^{\\text{St-MMSE}}_{k,n}) \\right \\},\n\\end{align}\nwhere\n\\begin{align}\\label{}  \n\\zeta^{\\text{St-MMSE}}_{k,n} = \\frac{\\left|\\textbf{f}_{k,n} \\bar{\\textbf{d}}_{kk,n} \\right|^{2}}{\\textbf{f}^{H}_{k,n} { \\mathbf{\\Psi}^{b'}_{kk}  } \\textbf{f}_{k,n}   - \\left|\\textbf{f}_{k,n} \\bar{\\textbf{d}}_{kk,n} \\right|^{2}}.\n\\end{align}\n\n\\subsubsection{Protocol 2 with Linear MMSE Detectors}\nLet us denote $\\hat{\\textbf{d}}_{kk,n}$ as $n$-th column of $\\hat{\\textbf{D}}_{kk}$. Then, the linear MMSE detector for the $n$-th data stream of the $k$-th user is given by\n\\begin{align}\n\\textbf{p}_{k,n} = \\mathbf{\\Psi}^{c'}_{kk} \\hat{\\textbf{d}}_{kk,n},\n\\end{align}\nwhere $\\mathbf{\\Psi}^{c'}_{kk} = (\\mathbf{\\Psi}^c_{kk})^{-1} + \\hat{\\textbf{D}}_{kk}\\hat{\\textbf{D}}^H_{kk}$.\n\nGiven MMSE detectors at the users, the achievable downlink SE of the $k$-th user can be calculated as\n\\begin{align}\\label{I_qrMMSE1}  \nR^{\\text{Et-MMSE}}_k  =\\left(1 - \\frac{\\tau_{\\text{u}} + \\tau_{\\text{d}}}{\\tau_c}\\right) \\sum^{N}_{n=1}\\mathbb {E} \\left \\{ \\log_{2} (1 + \\zeta^{\\text{Et-MMSE}}_{k,n}) \\right \\},\n\\end{align}\nwhere \n\\begin{align}\\label{}  \n\\zeta^{\\text{Et-MMSE}}_{k,n} = \\frac{\\left|\\textbf{p}_{k,n} \\hat{\\textbf{d}}_{kk,n} \\right|^{2}}{\\textbf{p}^{H}_{k,n} { \\mathbf{\\Psi}^{c'}_{kk}  } \\textbf{p}_{k,n}   - \\left|\\textbf{p}_{k,n} \\hat{\\textbf{d}}_{kk,n} \\right|^{2}}.\n\\end{align}\nIn the section \\ref{Numsec}, we will compare the performance of the cell-free massive MIMO system when using MMSE-SIC detectors and the one using MMSE detectors at the users. \n\\subsection{Achievable Downlink SE given Perfect CSI at the Users}\\label{UpperBound}\nIn this section, we consider a cell-free massive MIMO system with perfect CSI at the users. Although, this is impractical, its performance is considered as the upper bound for the performance of protocol 2. Given perfect CSI, i.e., $\\mathbf{\\Theta}_{k} = \\{\\textbf{D}_{k1}, \\textbf{D}_{k2}, \\ldots , \\textbf{D}_{kK}\\}$, then the achievable downlink SE of the $k$-th user in (\\ref{R_k1}) when using MMSE-SIC detectors at the users is \n\\begin{align}\\label{I_qrThe3}  \nR^{\\text{up}}_k = \\left(1 - \\frac{\\tau_{\\text{u}} + \\tau_{\\text{d}}}{\\tau_c}\\right) \\mathbb {E} \\left \\{ \\log_{2} \\left| \\textbf{I}_{N} + \\rho\\textbf{D}^H_{kk} \\left(\\mathbf{\\Psi}^d_{kk}\\right)^{-1} \\textbf{D}_{kk} \\right| \\right \\},\n\\end{align}\nwhere \n\\begin{align}\n\\mathbf{\\Psi}^d_{kk} = \\rho\\sum^{K}_{k'\\neq k}\\textbf{D}_{kk'}\\textbf{D}^H_{kk'} + \\textbf{I}_{N}. \\notag\n\\end{align}\n\n\\section{Max-min Power Control}\\label{max-min-pc}\nIn this section, max-min fairness PC is applied for the first protocol to improve the SE of the system. This power control is recomputed on the large-scale fading time scale which changes very slowly. For the second protocol, as the closed-form expression of the SE is very complicated, we apply the power control coefficients of the optimization problem in the first protocol to achieve a sub-optimal solution for the achievable downlink SE of this protocol.\n\\subsection{Max-min Power Control for Protocol 1}\\label{max-min-pc-p1}\nIn this part, we consider that mutual orthogonal pilot sequences are used in the uplink channel estimation phase. To achieve the fairness good SE for all users in the system, max-min power control is applied for all users to optimize the downlink SE. The max-min fairness optimization problem can be written as\n\\begin{align}\\label{max_min_SE}\n&\\max _{\\{\\eta _{mk}\\}}~\\min \\limits _{k=1, \\cdots , K} R^{\\text{St-SIC}}_k \\\\\n&\\text {s.t.}~\\sum _{k=1}^{K} \\eta _{mk}\\gamma _{mk} \\leq \\frac{1}{LN}, ~ m=1,\\ldots , M \\notag\\\\\n&\\qquad \\qquad \\quad ~ \\eta _{mk} \\geq 0, ~ k=1,\\ldots , K, ~ m=1,\\ldots , M, \\notag\n\\end{align}\nwhere the first constraint of (\\ref{max_min_SE}) is the power constraint in (\\ref{powcontraint}) when using the mutual orthogonal pilot sequences for the uplink channel estimation phase, and $\\gamma _{mk} = \\frac{\\tau_{\\text{u}}\\rho_{\\text{u}}\\beta^2_{mk}}{\\tau_{\\text{u}}\\rho_{\\text{u}}\\beta_{mk}+ 1}$. \n An equivalent form of (\\ref{max_min_SE}) is\n\\begin{align*}\n&\\max _{\\{\\eta _{mk}\\}}~\\min \\limits _{k=1, \\cdots , K} \\frac { \\left ({\\sum _{m=1}^{M} \\gamma _{mk}\\varsigma _{mk} }\\right )^{2} }{\\frac{N}{L}\\sum \\limits _{m=1}^{M}\\beta _{mk} \\sum \\limits _{k'=1}^{K} \\gamma _{mk'}\\varsigma _{mk'}^{2} +\\frac {1}{ \\rho L^2} } \\\\[0.5pt]\n&\\text {s.t.}~\\sum _{k=1}^{K} \\eta _{mk}\\varsigma^2 _{mk} \\leq \\frac{1}{LN}, ~ m=1,\\ldots , M \\\\[0.5pt]\n&\\qquad \\eta _{mk} \\geq 0, \\quad k=1,\\ldots , K, ~ m=1,\\ldots , M, \n\\end{align*}\nwhere $\\varsigma _{mk} \\triangleq \\eta^{1\/2} _{mk}$. By introducing the slack variable $\\vartheta$, we reformulate (\\ref{max_min_SE}) as follows:\n\\begin{align}\\label{max_min_SNR}\n&\\max _{\\{\\varsigma _{mk}, \\vartheta _{m}\\}} ~\\min _{k=1, \\cdots , K} \\frac { \\left ({\\sum _{m=1}^{M} \\gamma _{mk}\\varsigma _{mk} }\\right )^{2}}{\\frac{N}{L} \\sum \\limits _{m=1}^{M}\\!\\!\\beta _{mk} \\vartheta _{m}^{2} +\\frac {1}{ \\rho L^2}} \\\\\n&\\text {s.t.} ~\\sum _{k'=1}^{K} \\gamma _{mk'}\\varsigma _{mk'}^{2} \\leq \\vartheta _{m}^{2} , ~ m=1,\\ldots , M \\tag{32a}\\\\\n&\\qquad \\qquad \\quad ~ 0 \\leq \\vartheta _{m} \\leq \\frac{1}{\\sqrt{LN}}, ~ m=1,\\ldots , M \\tag{32b}\\\\\n&\\qquad \\qquad \\quad ~ \\varsigma _{mk} \\geq 0, ~ k=1,\\ldots , K, ~ m=1,\\ldots , M. \\tag{32c} \n\\end{align}\nBy introducing the slack variable $t$, optimization problem (\\ref{max_min_SNR}) can be rewritten as \n\\begin{align}\\label{max_min_SNR_t}\n&\\max _{\\{\\varsigma _{mk}, \\vartheta _{m}, t\\}} t \\\\ \n&\\text {s.t.} ~ t \\leq \\frac { \\left ({\\sum _{m=1}^{M} \\gamma _{mk}\\varsigma _{mk} }\\right )^{2}}{\\frac{N}{L} \\sum \\limits _{m=1}^{M}\\!\\!\\beta _{mk} \\vartheta _{m}^{2} +\\frac {1}{ \\rho L^2}}, ~ k=1,\\ldots , K,\\notag\\\\\n& (32a), (32b), (32c).\\notag\n\\end{align}\nOptimization problem (\\ref{max_min_SNR_t}) is quasi-concave, as fixing $t$, the problem is second-order cone concave. Therefore, the problem (\\ref{max_min_SNR}) is quasi-concave optimization problem and it can be solved effectively by bisection algorithm \\cite{boyd2004convex}. \n\\subsection{Max-min Power Control for Protocol 2}\\label{max-min-pc-sch2}\nIn this section, similarly to the section \\ref{max-min-pc-p1}, we also consider the max-min fairness optimization problem by applying PC. However, it is very difficult (may be impossible) and complicated to obtain the optimal solution for max-min power control of protocol 2, due to the intractable form of the spectral efficiency (\\ref{I_qrThe2}). To alleviate such difficulty, we use the power control coefficients from (\\ref{max_min_SE}) for protocol 2 which results in sub-optimal performance. \n\n\\section{Numerical results and Discussion}\\label{Numsec}\nIn this section, we provide the numerical results to verify our analytical results and evaluate the performance of cell-free massive MIMO for multiple antennas at both the APs and the users, with and without downlink pilots. Firstly, Gaussian approximations in Lemma \\ref{lemma3} are verified and illustrated numerically. Then, we compare the performances of two protocols based on the different CSI available at the users: statistical CSI and estimated CSI. Moreover, the performances of both protocols using MMSE-SIC detectors are also compared with those in systems using MMSE detectors. Finally, the effects of the number of antennas per user, per AP and number of users are analyzed to propose the framework for achieving sub-optimal system performance.\n\n\\subsection{Simulation setup}\nWe assume that the locations of the $M$ APs and $K$ users are uniformly distributed at random within a square of size $1\\times1$ km$^2$. Wrapped around technique is used to avoid the boundary effects. In all examples we assume that $\\tau_{\\text{u}}= \\tau_{\\text{d}}=K\\times N$ and mutually orthogonal pilot sequences are used for both the uplink and the downlink training phase. We also use the same simulation setup with the one in \\cite{ngo2017cell}. More specifically, we use i.i.d. Rayleigh fading channels with the three-slope path loss model and shadowing correlation model. The carrier frequency is 1.9 GHz and $\\tau_c$ is 300 samples.  \n\n\\subsection{Closed-form expression and power control}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{closed_form.eps}\n\\caption{Cummulative distribution function (cdf) of per-user SE with $M$ = 50, $K$ = 10, $N$ = 2, $L$ = 4.}\n\\label{closed_form}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fig_full.eps}\n\\caption{Cdf of per-user SE with $M$ = 50, $K$ = 5, $N$ = 4, $L$ = 4.}\n\\label{fig_full}\n\\end{figure}\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.49\\textwidth]{upper_bound.eps}\n\t\\caption{Cdf of per-user SE with $M$ = 20, $K$ = 5, $N$ = 1, $L$ = 1.}\n\t\\label{fig_uppbound}\n\\end{figure}\nFig. \\ref{closed_form} numerically proves the correction of our derived closed-form expression in Theorem \\ref{theo_closedform}. For transmission protocol 1, this figure shows the perfect match between the SEs, which are derived analytically (closed-form expression), and the SEs, which are conducted by simulation of (\\ref{R_k_statistic}), regardless of using PC or not.\nIn this paper, max-min fairness power control is applied to provide the good uniform service to all users in the systems. By using closed-form of the downlink SE of protocol 1 given statistical CSI at the users, the power control of the downlink SE can be solved effectively by bisection algorithm. Due to the computation complexity of SE optimization problems in the second protocol, we apply the power control coefficients of the first protocol to the second protocol. Interestingly, Fig. \\ref{closed_form} shows that, in term of 95\\% likely per-user SE, not only the SE of  protocol 1 increases dramatically, by about 80\\% with PC, but also the SE of protocol 2 improves significantly, by about 60\\%. This verifies that the optimal PC coefficients of protocol 1 can be applied effectively to PC problem of protocol 2. \n\nFigure \\ref{fig_full} shows that the performance of protocol 2 is very close to its upper bound, when perfect channel\ninformation is available at the users. This implies that our PC method applies\neffectively on protocol 2. Moreover, this figure also compares the performance of two protocols using different detection techniques at the users, i.e., MMSE-SIC and MMSE. In protocol 1, there is no difference in SE between the system using MMSE-SIC detectors and the one using MMSE detectors. This completely agrees with the conclusion in \\cite{li2016massive}, as only statistical CSI is available at the users. However, in protocol 2, as SIC works effectively with estimated CSI at the users, the SE of the system using MMSE-SIC increases noticeably compared to the one using MMSE. This gap varies depending on the number of users, $K$, and the number of antennas per user, $N$, which is shown in Fig. \\ref{N_change}. Specially, when $N = 1$, the performances of this protocol are exactly identical regardless either using MMSE-SIC or MMSE detectors.   \n\n\nTo further see the performance limit as well as how well our sub-optimal power control in Section~\\ref{max-min-pc-sch2}  is, we compare the SE of protocol 2 using our sub-optimal power control with the SE of the ideal case where users have perfect CSI and optimal power control is performed. For simplicity, we consider a special case where each user\/AP is equipped with a  single-antenna, and mutual orthogonal pilot sequences are used in both the uplink and downlink channel estimation phases. From (\\ref{I_qrThe3}), and by using the approximation   $\\mathbb {E} \\left \\{\\log_2(1+X\/Y)\\right\\} \\approx \\log_2\\left(1 + \\mathbb {E} \\left \\{X\\right\\}\/\\mathbb {E} \\left \\{Y\\right\\} \\right)$ \\cite{zhang2014power}, the spectral efficiency with perfect CSI at user $k$ can be approximated by\n\\begin{align}\\label{eq_up3}  \n&R^{\\text{up-approx}}_k =  \\left(1 - \\frac{\\tau_{\\text{u}} + \\tau_{\\text{d}}}{\\tau_c}\\right)\\times\\notag\\\\\n&\\quad \\times \\log_{2} \\left(1 + \\frac { \\left ({\\sum _{m=1}^{M} \\gamma _{mk}\\varsigma _{mk} }\\right )^{2} + \\sum \\limits _{m=1}^{M}\\beta _{mk} \\gamma _{mk}\\varsigma _{mk}^{2}}{\\sum \\limits _{m=1}^{M}\\beta _{mk} \\sum \\limits _{k'\\ne k}^{K} \\gamma _{mk'}\\varsigma _{mk'}^{2} +\\frac {1}{ \\rho} }\\right).\n\\end{align}\nTherefore, the corresponding max-min power control can be efficiently solved  by using the\nsuccessive approximation technique \\cite{tran2012fast}. \n\nFigure~\\ref{fig_uppbound} shows the per-user SE of protocol 2 with sub-optimal power control in section IV-B, and the one with perfect CSI at the user and optimal power control. We can see that the performance gap is quite small. This verifies that the  sub-optimal power control works very well.\n\\subsection{Effects of the number of users, number of antennas per APs and per users}\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{N_changes.eps}\n\\caption{95$\\%$-likely per-user downlink SE v.s. number of antennas per user.}\n\\label{N_change}\n\\end{figure}\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.49\\textwidth]{N_change.eps}\n\t\\caption{95$\\%$-likely per-user downlink SE v.s. number of antennas per user  with $M$ = 20, $K$ = 30, $L$ = 1.}\n\t\\label{N_change_p2}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{L_change.eps}\n\\caption{95$\\%$-likely per-user downlink SE with $M$ = 50, $L$ = 4, $N$ = 4.}\n\\label{L_change}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{k_change.eps}\n\\caption{95$\\%$-likely per-user downlink SE v.s number of users. Here, $M$ = 50, $L$ = 4, $N$ = 2.}\n\\label{K_change}\n\\end{figure}\nFirst, in Fig. \\ref{N_change}, the effect of number of antennas per user is investigated by using two scenarios with different number of APs and number of antennas per APs.  In scenario 1, with small number of users in comparison with number of APs and number of total antennas at the APs, the per-user downlink SE increases proportionally with the number of antennas per user as the increasing of independent channels (or degrees of freedom) per user. Using multiple antennas at the users greatly enhances the per-user SE, especially with small number of users in the system. With $K=5$, by using 3 antennas per user and PC, we can double the 95$\\%$-likely SE of both protocols, compared to single-antenna user systems. However, in scenario 2, the per-user downlink SE first increases when the number of antennas per user increases with increasing independent channels (or degrees of freedom) per user. Then, this SE will reach the maximum value and then it will decrease when the number of antennas per user increases. Especially, when the number of users is large,  single-antenna user setups  outperform multi-antenna user setups. The reason is that although the number of independent channels per user increases, the channel estimation overhead  also increases. This channel estimation overhead largely dominates when $N$ is large.\n\nNext, we evaluate the effect of AP antennas. In Fig. \\ref{L_change}, 95$\\%$-likely per-user achievable downlink SE of two protocols with PC are shown with arbitrary number of antennas per APs. As expected, the performances of all protocols increase proportionally with the number of antennas per AP, especially when the total number of antennas at the users is small in comparison with the total number of antennas at the APs (here, $M \\times L = 200$). This comes from the fact that when $L$ increases, the channel is more favorable, and hence, the inter-user interference reduces. At the same time, the array gain increases. \n\nFinally, 95$\\%$-likely per-user achievable downlink SE of two protocols are shown in Fig. \\ref{K_change} with arbitrary number of users and $N = 2$. At first, when the number of user is small, protocol 2 achieves higher SE than protocol 1. However, when the number of users increases, protocol 1 achieves higher SE than protocol 2, as a large proportion of coherence duration is assigned for the downlink training. These above insights are very important for us to design the framework for achieving the sub-optimal SE of the system, in the next section.    \n\n\n\\subsection{Framework for achieving the sub-optimal SE}\nNormally, the number of APs and number of antennas per APs are fixed. However, the number of users in the system, which strongly affects on the system performance, is unknown. Therefore, we propose the framework that automatically chooses suitable protocol as well as the number of active antennas per users, based on the number of users in the systems. The framework is as follows\n\n\\begin{itemize}\n\\item In the setup session, based on the real number of users, $K$, and their locations, the system can calculate its SE for each protocol and for each number of active antennas per user, $n$, $\\forall n = 1, 2, \\hdots, N$.\n\\item Choose the protocol with the number of active antennas per user, $n$, which achieves the highest SE. \n\\end{itemize}\nThe framework only need to be updated with the time frame of PC, i.e., updated infrequently, as it just depends on the large-scale fading and number of active users.\n\\begin{remark}\nThe SE, which is a result of the framework, is sub-optimal, as it is considered under following conditions: using mutual orthogonal pilot sequences, using sub-optimal power control for protocol 2, and using the same number of active antennas per users. \n\\end{remark}\n\\section{Conclusion}\\label{Concl}\nIn this paper, we evaluated the downlink SE of two transmission protocols of cell-free massive MIMO system with finite numbers of APs, users and arbitrary numbers of antennas at the APs and the users. There are no downlink pilots in the first transmission protocol, while the downlink pilots were beamformed to the used for the CSI acquisition in the second transmission protocol. Compared with the first protocol, the second protocol has higher channel estimation quality at the users, but higher channel estimation overhead. Numerical results show that no protocol always shows the advantage over the other in the performance as they both depend on the number of users and number of antennas per user in the system. Finally, by using two protocols and combining with multi-antenna users, the system can achieve the sub-optimal performance regardless of number of users in the system.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\\vspace{-2mm}\n\n\\setlength{\\abovedisplayskip}{2pt}\n\\setlength{\\abovedisplayshortskip}{2pt}\n\\setlength{\\belowdisplayskip}{2pt}\n\\setlength{\\belowdisplayshortskip}{2pt}\n\\setlength{\\jot}{1pt}\n\\setlength{\\floatsep}{1ex}\n\\setlength{\\textfloatsep}{1ex}\n\\setlength{\\intextsep}{1ex}\n\nPath planning is a fundamental problem with many real-world applications, such as robot manipulation and autonomous driving. A simple planning problem within low-dimensional state space can be solved by first discretizing the continuous state space into a grid, and then searching for a path on top of it using graph search algorithms such as $A^*$~\\citep{hart1968formal}. However, due to the curse of dimensionality, these approaches do not scale well with the number of dimensions of the state space. For \\emph{high-dimensional} planning problems, people often resort to sampling-based approaches to avoid explicit discretization. Sampling-based planning algorithms, such as probabilistic roadmaps~(PRM)~\\citep{kavraki1996probabilistic}, rapidly-exploring random trees~(RRT)~\\citep{lavalle1998rapidly}, and their variants~\\citep{karaman2011sampling} incrementally build an implicit representation of the state space using probing samples. These generic algorithms typically employ a uniform sampler which does not make use of the structures of the problem. Therefore they may require lots of samples to obtain a feasible solution for complicated problems. \nTo improve the sample efficiency, heuristic biased samplers, such as Gaussian sampler~\\citep{boor1999gaussian}, bridge test~\\citep{hsu2003bridge} and reachability-guided sampler~\\citep{shkolnik2009reachability} have been proposed. All these sampling heuristics are designed manually to address specific structural properties, which may or may not be valid for a new problem, and may lead to even worse performance compared to the uniform proposal.\n\n\n\n\nOnline adaptation in path planning has also been investigated for improving sample efficiency in current planning problem. Specifically, \\citet{hsu2005hybrid} exploits online algorithms to dynamically adapts the mixture weights of several manually designed biased samplers.  \\citet{burns2005sampling,burns2005toward} fit a model for the planning environment incrementally and use the model for planning. \\citet{yee2016monte} mimics the Monte-Carlos tree search~(MCTS) for problems with continuous state and action spaces. These algorithms treat each planning problem \\emph{independently}, and the collected data from previous experiences and built model will be simply discarded when solving a new problem. However, in practice, similar planning problems may be solved again and again, where the problems are different but sharing common structures. For instance, grabbing a coffee cup on a table at different time are different problems, since the layout of paper and pens, the position and orientation of coffee cups may be different every time; however, all these problems show common structures of handling similar objects which are placed in similar fashions.\nIntuitively, if the common characteristics across problems can be learned via some shared latent representation, a planner based on such representation can then be transferred to new problems with improved sample efficiency. \n\n\nSeveral methods have been proposed recently to learn from past planning experiences to conduct more efficient and generalizable planning for future problems.\nThese works are limited in one way or the other.~\\citet{zucker2008adaptive,zhang2018learning, Huh2018EfficientSW} treat the sampler in the sampling-based planner as a stochastic policy to be learned and apply policy gradient or TD-algorithm to improve the policy.~\\citet{finney2007predicting, bowen2014closed,ye2017guided,ichter2018learning,kim2018guiding,kuo2018deep} apply imitation learning based on the collected demonstrations to bias for better sampler via variants of probabilistic models, \\eg, (mixture of) Gaussians, conditional VAE, GAN, HMM and RNN. However, many of these approaches either rely on specially designed local features or assume the problems are indexed by special parameters, which limits the generalization ability. Deep representation learning provides a promising direction to extract the common structure among the planning problems, and thus mitigate such limitation on hand-designed features. However, existing work, \\eg, motion planning networks~\\citep{qureshi2019motion}, value iteration networks~(VIN)~\\citep{tamar2016value}, and gated path planning networks~(GPPN)~\\citep{lee2018gated}, either apply off-the-shelf MLP architecture ignoring special structures in planning problems or can only deal with discrete state and action spaces in low-dimensional settings.\n\n\nIn this paper, we present \\emph{Neural EXploration-EXploitation Tree~(NEXT)}, a \\textbf{meta neural path planning} algorithm for high-dimensional continuous state space problems. The core contribution is a novel \\textbf{attention-based neural architecture} that is capable of learning generalizable problem structures from previous experiences and produce promising search directions with automatic online \\textbf{exploration-exploitation} balance adaption. Compared to existing learning-based planners,\n\\vspace{-1mm}\n\\begin{itemize}[leftmargin=*,nolistsep,nosep]\n  \\item \\textbf{NEXT is more generic}. \n  We propose an architecture that can embed high dimensional continuous state spaces into low dimensional discrete spaces, on which a neural planning module is used to extract planning representation. These module will be learned end-to-end. \n  \\item \\textbf{NEXT balances exploration-exploitation trade-off}. We integrate the learned neural prior into an upper confidence bound~(UCB) style algorithm to achieve an online balance between exploration and exploitation when solving a new problem. \n\\end{itemize}\n\\vspace{-1mm} \nEmpirically, we show that NEXT can exploit past experiences to reduce the number of required samples drastically for solving new planning problems, and significantly outperforms previous state-of-the-arts on several benchmark tasks.\n\n\n\n\\vspace{-3mm}\n\\subsection{Related Works}\\label{sec:related_works}\n\\vspace{-3mm}\n\n\nDesigning non-uniform sampling strategies for random search to improve the planning efficiency has been considered as we discussed above. Besides the mentioned algorithms, there are other works along this line, including informed RRT$^*$~\\citep{gammell2014informed} and batch informed Trees (BIT*)~\\citep{gammell2015batch} as the representative work. Randomized $A^*$~\\citep{diankov2007randomized} and sampling-based $A^*$~\\citep{persson2014sampling} expand the search tree with hand-designed heuristics. \nThese methods incorporate the human prior knowledge via hard-coded rules, which is fixed and unable to adapt to problems, and thus, may not universally applicable.~\\citet{choudhury2018data, song2018learning} attempt to learn search heuristics. However, both methods are restricted to planning on discrete domains. Meanwhile, the latter one always employs an unnecessary hierarchical\nstructure for path planning, which leads to inferior sample efficiency and extra computation.\n\n\nThe online exploration-exploitation trade-off is also an important issue in planning. For instance, \\citet{rickert2009balancing} constructs a potential field sampler and tuned the sampler variance based on collision rate for the trade-off heuristically.~\\citet{paxton2017combining} separates the action space into high-level discrete options and low-level continuous actions, and only considered the trade-off at the discrete option level, ignoring the exploration-exploitation in the fine action space. These existing works address the trade-off in an ad-hoc way, which may be inferior for the balance.\n\nThere have been sevearl non-learning-based planning methods that can also leverage experiences~\\citep{kavraki1996probabilistic, phillips2012graphs} by utilizing search graphs created in previous problems. However, they are designed for largely fixed obstacles and cannot be generalized to unseen tasks from the same planning problems distribution.\n\n\n\\vspace{-4mm}\n\\section{Settings for Learning to Plan}\\label{sec:problem_formulation}\n\\vspace{-3mm}\n\n\nLet $\\Scal \\subseteq \\RR^q$ be the state space of the problem, \\eg, all the configurations of a robot and its base location in the workspace, $\\Scal_{obs} \\subsetneq \\Scal$ be the obstacles set, $\\Scal_{free} \\defeq \\Scal \\setminus \\Scal_{obs}$ be the free space, $s_{init} \\in \\Scal_{free}$ be the initial state and $\\Scal_{goal} \\subsetneq \\Scal_{free}$ be the goal region. Then the space of all collision-free paths can be defined as a continuous function\n$\n    \\Xi \\defeq \\cbr{ \\xi \\rbr{\\cdot}:[0, 1]\\rightarrow \\Scal_{free}}.\n$ \nLet $c\\rbr{\\cdot}:\\Xi \\mapsto \\RR$ be the cost functional over a path. The optimal path planning problem is to find the optimal path in terms of cost $c(\\cdot)$ from start $s_{init}$ to goal $\\Scal_{goal}$ in free space $\\Scal_{free}$, \\ie,  \n\\begin{align}\\label{eq:path_planning}\n    \\xi^* = \\argmin\\nolimits_{\\xi\\in \\Xi}~c\\rbr{\\xi}, \\quad\\text{s.t.}~\\xi(0) = s_{init},~\\xi(1) \\in \\Scal_{goal}.\n\\end{align}\nTraditionally~\\citep{karaman2011sampling}, the planner has  direct access to $(s_{init}, \\Scal_{goal}, c\\rbr{\\cdot})$ and the workspace map~\\citep{ichter2018learning,tamar2016value,lee2018gated}, $\\mathtt{map}(\\cdot):\\RR^2~\\text{or}~\\RR^3 \\mapsto \\cbr{0,1}$, (0: free spaces and 1: obstacles).\nSince $\\Scal_{free}$ often has a very irregular geometry (illustrated in~\\figref{fig:ex_config} in~\\appref{appendix:motion_planning}), it is usually represented via a \\emph{collision detection module} which is able to detect the obstacles in a path segment. For the same reason, the feasible paths in $\\Xi$ are hard to be described in parametric forms, and thus, the nonparametric $\\xi$, such as a sequence of interconnected path segments $[s_0,s_1],[s_1,s_2],\\dots,[s_{T-1},s_T]\\subset\\Scal$ with $\\xi(0)=s_0=s_{init}$ and $\\xi(1)=s_T$, is used with an additive cost $\\sum_{i=1}^T c\\rbr{[s_{i-1},s_{i}]}$.\n\n\nAssuming given the planning problems $\\cbr{U_i\\defeq\\rbr{s_{init}, \\Scal_{goal}, \\Scal, \\Scal_{free}, \\mathtt{map}, c\\rbr{\\cdot}}}_{i=1}^N$ sampled from some distribution $\\Ucal$, we are interested in learning an algorithm~$\\mathtt{alg}\\rbr{\\cdot}$, which can produce the (nearly)-optimal path efficiently from the observed planning problems.\nFormally, the \\emph{learning to plan} is defined as\n\\vspace{-2mm}\n\\begin{equation}\\label{eq:l2p_form}\n    \\mathtt{alg}^*\\rbr{\\cdot} = \\argmin\\nolimits_{\\mathtt{alg}\\in \\Acal} \\mathbb{E}_{U\\in\\Ucal}\\sbr{\\ell\\rbr{\\mathtt{alg}(U)}},\n\\end{equation}\nwhere \n$\\Acal$ denotes the planning algorithm family, and $\\ell\\rbr{\\cdot}$ denotes some loss function which evaluates the quality of the generated path and the efficiency of the $\\mathtt{alg}\\rbr{\\cdot}$, \\eg, size of the search tree. We elaborate each component in~\\eqnref{eq:l2p_form} in the following sections. We first introduce the tree-based sampling algorithm template in~\\secref{sec:prelim}, upon which we instantiate the $\\mathtt{alg}\\rbr{\\cdot}$ via a novel attention-based neural parametrization in~\\secref{subsec:parametrization} with exploration-exploitation balance mechanism in~\\secref{subsec:guided_expansion}. We design the $\\ell$-loss function and the meta learning algorithm in~\\secref{subsec:meta_learning}. Composing every component together, we obtain the neural exploration-exploitation trees~(NEXT) which achieves outstanding performances in~\\secref{sec:experiments}.\n\n\\begin{figure}[htp!]\n\\vspace{-5mm}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/illustration_v4.pdf}\n\\vspace{-3mm}\n\\caption{\\small{Illustration of NEXT. In each epoch, NEXT is executed on a randomly generated planning problem. The search tree grows with $\\tilde V^*$ and $\\tilde \\pi^*$ guidance. $\\{\\tilde V^*, \\tilde\\pi^*\\}$ will be updated according to the successful path.\nSuch planning and learning iteration is continued interactively.\n}}\n\\label{fig:algorithm_illustration}\n\\end{figure}\n\n\n\n\\vspace{-3mm}\n\\section{Preliminaries}\\label{sec:prelim}\n\\vspace{-2mm}\n\n\n\\setlength{\\columnsep}{14pt}%\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n\\vspace{-13mm}\n\\hspace{1mm}\n\\begin{algorithm2e}[H]\n\\caption{Tree-based Sampling Algorithm}\n\\label{alg:planner_framework}\n\\SetKwFunction{expand}{Expand}\n\\SetKwFunction{obs}{ObstacleFree}\n\\SetKwFunction{connect}{Connect}\n\\SetKwFunction{post}{Postprocess}\n{\\bf Problem:} $U= (s_{init}, \\mathcal{S}_{goal}, \\mathcal{S}, \\mathcal{S}_{free}, \\mathtt{map}, c(\\cdot))$\\;\nInitialize $\\mathcal{T} = (\\Vcal, \\Ecal)$ with $\\Vcal\\gets \\{s_{init}\\}$, $\\Ecal\\gets\\emptyset$\\;\n\\For{$t\\gets0$ \\KwTo $T$}{\n    $s_{parent}, s_{new}\\gets\\expand{$\\mathcal{T}$, U}$ \\;\n    \\If{$\\obs{$s_{parent}, s_{new}$}$}{\n        $\\Vcal\\gets \\Vcal\\cup \\{s_{new}\\}$ and $\\Ecal\\gets \\Ecal\\cup \\{[s_{parent}, s_{new}]\\}$\\;\n        $\\mathcal{T}\\gets \\post{$\\mathcal{T}$, U}$~\\Comment*[r]{Optional}\n        \\If{$s_{new}\\in \\mathcal{S}_{goal}$}{\n            \\KwRet $\\mathcal{T}$\\;\n        }\n    }\n}\n\\vspace{-1mm}\n\\end{algorithm2e}\n\\vspace{-4mm}\n\\end{wrapfigure}\n\n\nThe sampling-based planners are more practical and become dominant for high-dimensional path planning problems~\\citep{elbanhawi2014sampling}. We describe a unifying view for many existing tree-based sampling algorithms (TSA), which we will also base our algorithm upon. More specifically, this family of algorithms maintain a search tree $\\Tcal$ rooted at the initial point $s_{init}$ and connecting all sampled points $\\Vcal$ in the configuration space with edge set $\\Ecal$. The tree will be expanded by incorporating more sampled states until some leaf reaches $\\Scal_{goal}$. Then, a feasible solution for the path planning problem will be extracted based on the tree $\\Tcal$. The template of tree-based sampling algorithms is summarized in Algorithm~\\ref{alg:planner_framework} and illustrated in Fig.~\\ref{fig:algorithm_illustration}(c). A key component of the algorithm is the $\\mathtt{Expand}$ operator, which generates the next exploration point $s_{new}$ and its parent $s_{parent}\\in\\Vcal$. To ensure the feasibility of the solution, the $s_{new}$ must be {\\bf reachable} from $\\Tcal$, \\ie, $[s_{parent}, s_{new}]$ is collision-free, which is checked by a collision detection function. As we will discuss in~\\appref{appendix:more_prelim}, by instantiating different $\\mathtt{Expand}$ operators, we will arrive at many existing algorithms, such as RRT~\\citep{lavalle1998rapidly} and EST~\\citep{hsu1997path,phillips2004guided}. \n\nOne major limitation of existing TSAs is that they solve each problem independently from scratch and ignore past planning experiences in similar environments. \nWe introduce the neural components into TSA template to form the learnable planning algorithm family $\\Acal$, which can explicitly take advantages of the past successful experiences to bias the $\\mathtt{Expand}$ towards more promising regions. \n\n\n\\iffalse\n\\vspace{-3mm}\n\\paragraph{Planning networks} The VIN~\\citep{tamar2016value} and GPPN~\\citep{lee2018gated} are two representative neural architectures for planning tasks. Within the Markov decision processes, then planning problem is reduced to calculating the optimal value function $V^*\\rbr{s|U}$, which can be solved by value iteration algorithm, \\ie, recursively applying Bellman update operator until converges, $V^*\\rbr{s|U} \\gets \\min_{a} \\sum_{s'} P(s'|s, a)(c\\rbr{\\sbr{s,s'}} + V^*\\rbr{s'|U})$, where $P(s'|s, a)$ is the state transition model. \nNoticing that in $2$d state space, the $P(s'|s, a)$, $c([s, s'])$, and $V^*(s)$ can be represented as matrices and vectors, then, the Bellman update operator essentially convolves $P(s'|s,a)$ with $c([s,s'])$ and $V^*(s'|U)$, and then performs a $\\min$-pooling operation across the convolution channels, VIN construct the neural network in the same way, and the parameters standing for $P\\rbr{s'|s, a}$ and $c([s, s'])$ will be learned by imitation learning. The details of the VIN and GPPN are discussed in~\\appref{appendix:more_prelim}.\n\nThe application of planning networks are restricted in low-dimension tasks. However, their success enlightens our neural architecture for generalizable representation for high-dimension planning tasks.\n\n\n\n\n\n\\vspace{-3mm}\n\\paragraph{UCB-based algorithms} \n\nThe UCB algorithms are well-studied for balancing the exploration versus exploitation  in the multi-armed bandit problem~\\citep{auer2002finite,langford2008epoch}, which achieves {\\small$\\Ocal\\rbr{\\sqrt{T}}$} regret after $T$ rounds. \nUCB has been generalized to search tree~\\citep{kocsis2006bandit}, leading to the Monte Carlo tree search~(MCTS), \\emph{a.k.a.}, the upper confidence trees~(UCT). Progressive expansion variants of MCTS have also been proposed to deal with a large number of discrete actions~\\citep{silver2017mastering, Guez2018learning}. However, the MCTS is not directly applicable to continuous state-action spaces. \nMoreover, the MCTS restricts the exploration only from leaves states, implicitly adding an unnecessary hierarchical structure for path planning, resulting inferior exploration efficiency and extra computation in path planning tasks. We provide the details of the UCB algorithms, as well as the discussion about MCTS extensions in~\\appref{appendix:more_prelim}.\n\nAlthough these off-the-shelf algorithms are not directly applicable to our path planning setting, their successes show the importance of exploration-exploitation trade-off and will provide the principles for our algorithm for continuous state-action planning problems. \n\\fi\n\n\\vspace{-3mm}\n\\section{Neural Exploration-Exploitation Trees}\\label{sec:next}\n\\vspace{-2mm}\n\n\n\nBased on the TSA framework, we introduce a \\emph{learnable neural based} $\\mathtt{Expand}$ operator, which can balance between exploration and exploitation, to instantiate $\\Acal$ in~\\eqnref{eq:l2p_form}. With the self-improving training, we obtain the meta NEXT algorithm illustrated in~\\figref{fig:algorithm_illustration}.\n\n\n\\vspace{-3mm}\n\\subsection{Guided Progressive Expansion}\n\\label{subsec:guided_expansion}\n\\vspace{-2mm}\n\n\n\\begin{wrapfigure}[7]{r}{0.55\\textwidth}\n\\vspace{-3mm}\n\\hspace{3mm}\n\\begin{algorithm2e}[H]\n$s_{parent} \\gets \\mathop{\\mathrm{argmax}}_{s\\in \\Vcal} \\phi(s)$\n\\Comment*[r]{Selection}\n$\\{s_1,\\ldots,s_k\\} \\overset{iid.}{\\sim} \\tilde \\pi^*(s'|s_{parent}, U)$\n\\Comment*[r]{Candidates}\n$s_{new}\\gets \\mathop{\\mathrm{argmax}}_{s'\\in \\{s_1,\\ldots,s_k\\} } \\phi(s')$\\Comment*[r]{Expansion}\n\\KwRet $s_{parent},s_{new}$\\;\n\\caption{$\\mathtt{NEXT::Expand}(\\Tcal=(\\Vcal,\\Ecal), U)$}\n\\label{alg:next_expand}\n\\end{algorithm2e}\n\\end{wrapfigure}\nWe start with our design of the $\\mathtt{Expand}$. We assume having an estimated value function $\\tilde V^*(s|U)$, which stands for the optimal cost from $s$ to target in planning problem $U$, and a policy  $\\tilde \\pi^*\\rbr{s'|s, U}$, which generates the promising action $s'$ from state $s$. The concrete parametrization of $\\Vtil^*$ and $\\tilde \\pi^*$ will be explained in~\\secref{subsec:parametrization} and learned in~\\secref{subsec:meta_learning}. We will use these functions to construct the learnable $\\mathtt{Expand}$ with explicit exploration-exploitation balancing. \n\n\n\nThe $\\mathtt{Expand}$ operator will expand the current search tree $\\Tcal$ by a new neighboring state $s_{new}$ around $\\Tcal$. We design the expansion as a two-step procedure: {\\bf (i)} select a state $s_{parent}$ from existing tree $\\Tcal$; {\\bf (ii)} expand a state $s_{new}$ in the neighborhood of $s_{parent}$. More specifically, \n    \n\n\\textbf{Selecting $s_{parent}$ from $\\Tcal$ in step (i).} \nConsider the negative value function $-\\tilde V^*\\rbr{s|U}$ as the rewards $r\\rbr{s}$, step \\textbf{(i)} shares some similarity with the multi-armed bandit problem by viewing existing nodes $s\\in \\Vcal$ as arms. However, the vanilla UCB algorithm is not directly applicable, since the number of states is increasing as the algorithm proceeds and the value of these adjacent states are naturally correlated. \nWe address this challenge by modeling the correlation explicitly as in contextual bandits. \nSpecifically, we parametrize the UCB of the reward function as $\\phi\\rbr{s}$, and select a node from $\\Tcal$ according to $\\phi(s)$\n\\begin{equation}\\label{eq:parametrized_ucb}\n  s_{parent} = \\argmax\\nolimits_{s \\in \\Vcal}~~\\phi(s) := \\rbar_t\\rbr{s} + \\lambda \\sigma_t\\rbr{s},\n\\end{equation}\nwhere $\\rbar_t$ and $\\sigma_t$ denote the average reward and variance estimator after $t$-calls to $\\mathtt{Expand}$. Denote the sequence of $t$ selected tree nodes so far as $\\Scal_t=\\{s_{parent}^1,\\ldots,s_{parent}^t\\}$, then we can use kernel smoothing estimator for $\\rbar_t(s)=\\frac{\\sum_{s'\\in\\Scal_t}k\\rbr{s',s}r\\rbr{s'}}{\\sum_{s'\\in \\Scal_t}k\\rbr{s', s}}$ and $\\sigma_t(s)=\\sqrt{\\frac{\\log\\sum_{s'\\in \\Scal_t}w\\rbr{s'}}{w\\rbr{s}}}$ where $k(s',s)$ is a kernel function and $w(s) = \\sum_{s' \\in \\Scal_t} k(s',s)$. Other parametrizations of $\\rbar_t$ and $\\sigma_t$ are also possible, such as Gaussian Process parametrization in~\\appref{appendix:para_ucb}. The average reward exploits more promising states, while the variance promotes exploration towards less frequently visited states; and the exploration versus exploitation is balanced by a tunable weight $\\lambda >0$. \n\n\n\n\\textbf{Expanding a reachable $s_{new}$ in step (ii).}~Given the selected $s_{parent}$, we consider expanding a reachable state in the neighborhood $s_{parent}$ as an \\emph{infinite-armed} bandit problem. Although one can first samples $k$ arms uniformly from a neighborhood around $s_{parent}$ and runs a UCB algorithm on the randomly generated finite arms~\\citep{wang2009algorithms}, such uniform sampler ignores problem structures, and will lead to unnecessary samples. Instead we will employ a policy $\\tilde \\pi^*\\rbr{s'|s, U}$\\footnote{In the path planning setting, we use $\\tilde \\pi^*\\rbr{s'|s, U}$ and $\\tilde \\pi^*\\rbr{a|s, U}$ interchangeably as the action is next state.} for guidance when generating the candidates. The final choice for next move will be selected from these candidates with $\\max \\phi\\rbr{s}$ defined in~\\eqref{eq:parametrized_ucb}. As explained in more details in~\\secref{subsec:parametrization}, $\\tilde\\pi^*$ will be trained to mimic previous successful planning experiences across different problems, that is, biasing the sampling towards the states with higher successful probability.\n\nWith these detailed step \\textbf{(i)} and \\textbf{(ii)}, we obtain $\\mathtt{NEXT::Expand}$ in~\\algref{alg:next_expand} (illustrated in~\\figref{fig:algorithm_illustration}(b) and (c)). Plugging it into the TSA in~\\ref{alg:planner_framework}, we construct $\\mathtt{alg}\\rbr{\\cdot}\\in \\Acal$ which will be learned. \n\n\nThe guided progressive expansion bears similarity to MCTS but deals with high dimensional continuous spaces. \nMoreover, the essential difference lies in the way to select state in $\\Tcal$ for expansion: the MCTS only expands the leaf states in $\\Tcal$ due to the hierarchical assumption, limiting the exploration ability and incurring extra unnecessary UCB sampling for internal traversal in $\\Tcal$; while the proposed operation enables expansion from each visited state, particularly suitable for path planning problems. \n\n\n\n\n\\vspace{-3mm}\n\\subsection{Neural Architecture for Value Function and Expansion Policy}\\label{subsec:parametrization}\n\\vspace{-1mm}\n\n\n\\begin{figure}[t]\n\\vspace{-5mm}\n\\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={right, center},capbesidewidth=0.4\\linewidth}}]{figure}[1.2\\FBwidth]\n{\n    \\hspace{-6mm}\n\n    \\caption{\\small Our neural network model maps a $N$-link robot from the original planning space (a $(N+2)$-d configuration space) to a $3$d discrete latent planning space in which we plan a path using value iteration. The result of value iteration is then used as features for defining $\\tilde V^*\\rbr{s|U}$ and $\\tilde\\pi^*\\rbr{s'|s, U}$.}\\label{fig:model_intuition}\n\n}\n{\n    \\includegraphics[width=1\\linewidth]{.\/figures\/Intuition.pdf}\n}\n\\end{figure}\n\nIn this section, we will introduce our neural architectures for $\\tilde V^*\\rbr{s|U}$ and $\\tilde\\pi^*\\rbr{s'|s, U}$ used in $\\mathtt{NEXT::Expand}$. The proposed neural architectures can be understood as first embedding the state and problem into a {\\bf discrete} latent representation via an attention-based module in~\\secref{subsec:config_embed}, upon which the neuralized value iteration, introduced in~\\secref{subsec:neural_VI}, is performed to extract features for defining $\\tilde V^*\\rbr{s|U}$ and $\\tilde\\pi^*\\rbr{s'|s, U}$, as illustrated in~\\figref{fig:model_intuition}. \n\n\\vspace{-2mm}\n\\subsubsection{Configuration Space Embedding}\\label{subsec:config_embed}\n\\vspace{-2mm}\n\nOur network for embedding high-dimension configuration space into a latent representation is designed based on an attention mechanism. More specifically, let $s^w$ denote the workspace in state and $s^h$ denote the remaining dimensions of the state, \\ie~$s=(s^w, s^h)$. $s^w$ and $s^h$ will be embedded by different sub-neural networks and combined for the final representation, as illustrated in~\\figref{fig:model_attention}. For simplicity of exposition, we will focus on the 2d workspace, \\ie, $s^w\\in \\RR^2$. However, our method applies to 3d workspace as well.\n\n\\begin{figure}[H]\n\t\\includegraphics[width=0.9\\linewidth]{.\/figures\/attention_v4.pdf}\n\t\\vspace{-2mm}\n    \\caption{\\small Attention-based state embedding module. $s^w =(x,y)$ and $\\mathbf{z} = s^h$. The upper part is spatial attention, with the first two channels being $x$ and $y$, and the last two channels being constant templates with the row and column coordinates, as shown with a $d$ set to $3$. The bottom module learns the representation for $\\zb$. The final embedding is obtained by outer-product of these two attention parts.}\n    \\label{fig:model_attention}\n\\end{figure}\n\\begin{itemize}[leftmargin=*,topsep=0pt]\n\\vspace{-2mm}\n    \\item {\\bf Spatial attention.} The workspace information $s^w$ will be embedded as $\\mu^{w}(s^w)\\in \\RR^{d\\times d}$, $d$ is a hyperparameter related to $\\mathtt{map}$ (see remark below). The spatial embedding module (upper part in \\figref{fig:model_attention}) is composed of $k_w$ convolution layers, \\ie, \n    \\begin{align}\n        \\mu^{w}(s^w)=\\mathtt{softmax2d}(f^w_{k_w}(s^w)),\\quad\\quad f^w_{i+1}(s^w)=\\mathtt{relu}(\\theta^w_i\\oplus f^w_i(s^w)),\n    \\label{eq:spatial_attention_map}\n    \\end{align} \n    where $\\theta_i^w$ denotes the convolution kernels, $\\oplus$ denotes the convolution operator and $f^w_{i}(s^w)\\in\\RR^{d\\times d\\times d_i}$ with $d_i$ channels. The first layer $f^w_0$ is designed to represent $s^w$ into a $d\\times d\\times 4$ tensor as $ f^w_0(s^w)_{ij}=[s^w_1, s^w_2, i, j]$, $i, j=1, \\ldots,d$, without any loss of information. \n\n    \\item {\\bf Configuration attention.} The remaining configuration state information will be embedded as $\\mu^{h}(s^h)$ through $k_h$ fully-connected layers (bottom-left part in \\figref{fig:model_attention}), \\ie\n    \\begin{align\n    \\mu^h(s^h)=\\mathtt{softmax}(f^h_{k_h}(s^h)),\\quad \\quad f^h_{i+1}(s^h)=\\mathtt{relu}(\\theta^{h}_i f^h_i(s^h) + b_i),\\nonumber\n    \\end{align}\n    where $\\mu^h(s^h)\\in\\RR^{d_a}$ and $f^h_0(s^h)=s^h$.\n\\end{itemize}\n\nThe final representation $\\mu_\\theta(s)$ will be obtained by multiplying $\\mu^w(s^w)$ with $\\mu^h(s^h)$ element-wisely, \n$\n\\mu_\\theta(s)_{ijl}=\\mu^w(s^w)_{ij}\\cdot \\mu^h(s^h)_l,\n$\nwhich is a $d\\times d \\times d_a$ tensor attention map with\n$\n    \\mu_\\theta(s)_{ijl}\\geqslant 0,~~\\text{and}~~\\sum_{ijl}\\mu_\\theta(s)_{ijl} = 1 \n$ (bottom-right part in \\figref{fig:model_attention}).\nIntuitively, one can think of $d_a$ as the level of the \\emph{learned discretization} of the configuration space $s^h$, and the entries in $\\mu$ softly assign the actual state $s$ to these discretized locations. $\\theta := (\\{\\theta_i^w\\}_{i=0}^{k_w-1}, \\{\\theta_i^h, b_i\\}_{i=0}^{k_h-1})$ are the parameters to be learned. \n\n\\noindent{\\bf Remark (different map size):} To process $\\mathtt{map}$ using convolution neural networks, we resize it to a $d\\times d$ image, with the same size as the spatial attention in \\eqnref{eq:spatial_attention_map}, where $d$ is a hyperparameter.\n\n\n\\vspace{-3mm}\n\\subsubsection{Neural Value Iteration}\\label{subsec:neural_VI}\n\\vspace{-2mm}\n\n\\begin{figure}[t]\n\\vspace{-1mm}\n    \\includegraphics[width=0.9\\linewidth]{.\/figures\/architecture_v4.pdf}\n    \\caption{\\small Overall model architecture. Current and goal states are embedded through attention module. Then the embedding of the goal state is concatenated with the map to produce $\\nu^{*(0)}$ and $\\tilde R$ as the input to the planning module. The output of the planning module is aggregated with the embedding of the current state to produce feature $\\psi(s)$ for defining $\\tilde V^*$ and $\\tilde \\pi^{*}$.}\\label{fig:model_architecture}\n\\end{figure}\nWe then apply neuralized value iteration on top of the configuration space embedding to extract further planning features~\\citep{tamar2016value,lee2018gated}. Specifically, we first produce the embedding $\\mu_{\\theta}(s_{goal})$ of the center of the goal region $s_{goal}$. We execute $T$ steps of neuralized Bellman updates (planning module in \\figref{fig:model_architecture}) in the embedding space, \n\\begin{equation*}\n    \\textstyle\n    \\nu^{*(t)} = \\min \\rbr{W_1\\oplus \\sbr{\\nu^{*(t-1)}, \\tilde R}},\\quad\\text{with}\\quad(\\nu^{*(0)}, \\Rtil)= \\sigma\\rbr{W_0 \\oplus \\sbr{\\mu_\\theta(s_{goal}), \\mathtt{map}}}, \n\\end{equation*}\nand obtain $\\nu^{*(T)}\\in \\RR^{d\\times d\\times d_a \\times p}$. Both $W_0$ and $W_1$ are $3$d convolution kernels, $\\min$ implements the pooling across channels. Accordingly, $\\nu^{*(T)}$ now can be understood as a latent representation of the value function $\\tilde V^*(\\cdot)$ in learned embedding space for the problem $U$ with $s_{goal}$ in $\\mathtt{map}$. \n\n\nTo define the value function for particular state $s$, \\ie, $\\Vtil^*\\rbr{s|U}$, from the latent representation $\\nu^{*(T)}$, we first construct another attention model between the embedding of state $s$ using $\\mu_{\\theta}(s)$ and $\\nu^{*(T)}$, \\ie,\n$\\psi\\rbr{s}_k = \\sum_{ijl}\\nu^{*(T)}_{ijlk}\\cdot \\mu_\\theta\\rbr{s}_{ijl}$, for $k=1,\\ldots, p$. Finally we define\n\\begin{align}\n    \\tilde V^*\\rbr{s|U} = h_{W_2}\\rbr{\\psi(s)},\\quad\\text{and}\\quad \\tilde \\pi^*\\rbr{s'|s, U} = \\Ncal(h_{W_3}\\rbr{\\psi(s)}, \\sigma^2)\n\\end{align}\nwhere $h_{W_2}$ and $h_{W_3}$ are fully connected dense layers with parameters $W_2$ and $W_3$ respectively, and $\\Ncal(h_{W_3}\\rbr{\\psi(s)}, \\sigma^2)$ is a Gaussian distribution with variance $\\sigma^2$. Note that we \nalso parametrize the policy $\\tilde \\pi^*\\rbr{s'|s, U}$ using the embedding $\\nu^{*(T)}$, since the policy is connected to the value function via $\\pi^*\\rbr{s'|s, U} =\\argmin_{s'\\in \\Scal} c\\rbr{\\sbr{s,s'}} +  V^*\\rbr{s'|U}$. It should also be emphasized that in our parametrization, the calculation of $\\nu^{*(T)}$ only relies on the $\\mu_\\theta\\rbr{s_{goal}}$, which can be reused for evaluating $\\Vtil^*\\rbr{s|U}$ and $\\tilde \\pi^*\\rbr{s'|s, U}$ over different $s$, saving computational resources. Using this trick the algorithm runs $10\\times$-$100\\times$ faster empirically.\n\n\nThe overall model architecture in $\\mathtt{alg}\\rbr{\\cdot}$ is illustrated in \\figref{fig:model_architecture}. The parameters $W = \\rbr{W_0, W_1, W_2, W_3, \\theta}$ will be learned together by our \\emph{meta self-improving learning}. For the details of the parameterization and the size of convolution kernels in our implementation, please refer to~\\figref{fig:emb_main} in~\\appref{appendix:nn_arch}.\n\n\n\n\\vspace{-2mm}\n\\subsection{Meta self-improving learning}\\label{subsec:meta_learning}\n\\vspace{-2mm}\n\nThe learning of the planner $\\mathtt{alg}\\rbr{\\cdot}$ reduces to learning the parameters in $\\tilde V^*\\rbr{s|U}$ and $\\tilde \\pi^*\\rbr{s'|s, U}$ and is carried out while planning experiences accumulate. We do not have an explicit training and testing phase separation. Particularly, we use a mixture of $\\mathtt{RRT::Expand}$ and $\\mathtt{NEXT::Expand}$ with probability $\\epsilon$ and $1 -\\epsilon$, respectively, inside the $\\mathtt{TSA}$ framework in Algorithm~\\ref{alg:planner_framework}. The RRT$^*$ postprocessing step is used in the template. The $\\epsilon$ is set to be $1$ initially since $\\{\\tilde V^*, \\tilde \\pi^*\\}$ are not well-trained, and thus, the algorithm behaves like RRT$^*$. As the training proceeds, we anneal $\\epsilon$ gradually as the sampler becomes more efficient. \n\nThe dataset $\\Dcal_n = \\{(\\Tcal_j, U_j)\\}_{j=1}^n$ for the $n$-th training epoch is collected from the previous successful planning experiences across \\emph{multiple random problems}. We fix the size of dataset and update $\\Dcal$ in the same way as experience reply buffer~\\citep{lin1992self,schaul2015prioritized}. For an experience $(\\Tcal,U)\\in\\Dcal_n$, we can reconstruct the successful path $\\{s^i\\}_{i=1}^{m}$ from the search tree $\\Tcal$ ($m$ is the number of segments), and the value of each state $s^i$ in the path will be the sum of cost to the goal region, \\ie, $y^i\\defeq\\sum_{l=i}^{m-1} c([s^{l}, {s^{l+1}}])$. We learn $\\{\\tilde V^*, \\tilde\\pi^*\\}$ by optimizing objective\n\\begin{equation}\\label{eq:msil_opt}\n    \\min_{W}\\sum_{(\\Tcal, U) \\in \\Dcal_n}\\ell(\\tilde V^*, \\tilde\\pi^*;\\Tcal,U)\\hspace{-1mm} \\defeq\\hspace{-1mm} -  \\sum_{\\Dcal_n}\\sum_{i=1}^{m-1} \\log \\tilde\\pi^*\\rbr{s^{i+1}|s^i} + \\sum_{i=1}^{m} (\\tilde V^*\\rbr{s^i} - y^i)_2^2+ \\lambda \\nbr{W}^2.\n\\vspace{-1mm}\n\\end{equation}\n\n\\begin{wrapfigure}[18]{r}{0.5\\textwidth}\n\\vspace{-4mm}\n\\begin{algorithm2e}[H]\nInitialize dataset $\\mathcal{D}_0$\\;\n\\For{\\textnormal{epoch} $n\\gets1$ \\KwTo $N$}{\n    Sample a planning problem $U$\\;\n    $\\mathcal{T}\\gets$ $\\mathtt{TSA}(U)$ with $\\epsilon \\sim \\Ucal nif[0,1]$, and \n   \n    \\begin{equation*} \n    \\textstyle\n    \\epsilon \\cdot \\mathtt{RRT::Expand} + (1-\\epsilon) \\cdot \\mathtt{NEXT::Expand};\n   \n    \\end{equation*}\n    Postprocessing with $\\mathtt{RRT^*::Postprocess}$\\;\n    $\\Dcal_n \\gets$ $\\Dcal_{n-1} \\cup\\{(\\mathcal{T}, U)\\}$ if successful else $\\Dcal_{n-1}$\\;\n    \\For{$j\\gets0$ \\KwTo $L$}{\n        Sample $(\\mathcal{T}_j, U_j)$ from $\\Dcal_n$\\;\n        Reconstruct sub-optimal path $\\{s^i\\}_{i=1}^{m}$ and the cost of paths based on $\\mathcal{T}_j$\\;\n        Update parameters $\\textstyle W \\gets W-\\eta\\nabla_{W}\\ell(\\tilde V^*, \\tilde\\pi^*; \\mathcal{T}_j,U_j)$\\;\n    }    \n    Anneal $\\epsilon = \\alpha\\epsilon$, $\\alpha\\in(0, 1)$\\;\n}\n\\vspace{-2mm}\n\\caption{Meta Self-Improving Learning}\n\\label{alg:learning}\n\\end{algorithm2e}\n\\end{wrapfigure}\nThe loss~\\eqref{eq:msil_opt} pushes the $\\Vtil^*$ and $\\tilde\\pi^*$ to chase the successful trajectories, providing effective guidance in $\\mathtt{alg}\\rbr{\\cdot}$ for searching, and therefore leading to efficient searching procedure with less sample complexity and better solution. On one hand, the value function and policy estimation $\\{\\tilde V^*, \\tilde\\pi^*\\}$ is improved based upon the successful outcomes from NEXT itself on previous problems. On the other hand, the updated $\\{\\tilde V^*, \\tilde\\pi^*\\}$ will be applied in the next epoch to improve the performance. Therefore, the training is named as~\\emph{Meta Self-Improving Learning~(MSIL)}. Since all the trajectories we collected for learning are feasible, the reachability of the proposed samples is enforced implicitly via imitating these successful paths. \n\nBy putting every components together into the learning to plan framework in~\\eqnref{eq:l2p_form}, the overall procedure is summarized in~\\algref{alg:learning} and illustrated in Figure~\\ref{fig:algorithm_illustration}.\n\n\\vspace{-2mm}\n\\section{Experiments}\\label{sec:experiments}\n\\vspace{-2mm}\n\nIn this section, we evaluate the proposed NEXT empirically on different planning tasks in a variety of environments. Comparing to the existing planning algorithms, NEXT achieves the state-of-the-art performances, in terms of both success rate and the quality of the found solutions. We further demonstrate the power of the proposed two components by the corresponding ablation study. We also include a case study on a real-world robot arm control problem at the end of the section.\n\n\\vspace{-1mm}\n\\subsection{Experiment Setup}\n\\vspace{-1mm}\n\n\n\n{\\bf Benchmark environments.} We designed four benchmark tasks to demonstrate the effectiveness of our algorithm for high-dimensional planning. The first three involve planning in a 2d workspace with a 2 DoF (degrees of freedom) point robot, a 3 DoF stick robot and a 5 DoF snake robot, respectively. The last one involves planning a 7 DoF spacecraft in a 3d workspace. \nFor all problems in each benchmark task, the workspace maps were randomly generated from a fixed distribution; the initial and goal states were sampled uniformly randomly in the free space; the cost function $c\\rbr{\\cdot}$ was set as the sum of the Euclidean path length and the control effort penalty of rotating the robot joints.\n\n{\\bf Baselines.} We compared NEXT with RRT$^*$~\\citep{karaman2011sampling}, BIT$^*$~\\citep{gammell2015batch}, CVAE-plan~\\citep{ichter2018learning}, Reinforce-plan~\\citep{zhang2018learning}, and an improved version of GPPN~\\citep{lee2018gated} in terms of both planning time and solution quality. RRT$^*$ and BIT$^*$ are two widely used effective instances of~$\\mathtt{TSA}$ in~\\algref{alg:planner_framework}. In our experiments, we equipped RRT$^*$ with the goal biasing heuristic\nto improve its performance. BIT$^*$ adopts the informed search strategy~\\citep{gammell2015batch} to accelerate planning. CVAE-plan and Reinforce-plan are two learning-enhanced~$\\mathtt{TSA}$ planners proposed recently. CVAE-plan learns a conditional VAE as the sampler~\\citep{sohn2015learning}, which will be trained by near-optimal paths produced by RRT$^*$. Reinforce-plan learns to do rejection sampling with policy gradient methods. For the improved GPPN, we combined its architecture for $\\mathtt{map}$ with a fully-connected MLP for the rest state, such that it can be applied to high-dimensional continuous spaces. Please refer to~\\appref{appendix:exp_detail} for more details.\n\n\n{\\bf Settings.} For each task, we randomly generated 3000 different problems from the same distribution without duplicated maps.\nWe trained all learning-based baselines using the first 2000 problems, and reserved the rest for testing. The parameters for RRT$^*$ and BIT$^*$ are also tuned using the first 2000 problems. For NEXT, we let it improve itself using MSIL over the first 2000 problems. In this period, for every 200 problems, we updated its parameters and annealed $\\epsilon$ once. \n\n\n\n\\vspace{-3mm}\n\\subsection{Results and Analysis}\\label{sec:exp_analysis}\n\\vspace{-2mm}\n\n\\begin{figure*}[t]\n\\centering\n\\setlength{\\tabcolsep}{4pt}\n  \\begin{tabular}{cccc}\n    \\includegraphics[width=.24\\linewidth]{.\/figures\/Example1_RRT.pdf}&\\hspace{-3mm}\n    \\includegraphics[width=.24\\linewidth]{.\/figures\/Example1_NEXT_kde.pdf}&\\hspace{-3mm}\n    \\includegraphics[width=.24\\linewidth]{.\/figures\/Example1_NEXT_GP.pdf}&\\hspace{-3mm}\n    \\includegraphics[width=.24\\linewidth]{.\/figures\/Example1_V_PI.pdf}\\\\[-2mm]\n    \\small (a) RRT* (w\/o rewiring) &\\hspace{-3mm} \\small (b) NEXT-KS search tree &\\hspace{-3mm} \\small (c) NEXT-GP search tree  &\\hspace{-3mm} \\small (d) learned $\\Vtil^*$ and $\\tilde\\pi^*$\n  \\end{tabular}\n  \\vspace{-2mm}\n  \\caption{\n\\small Search trees and the learned $\\tilde V^*$ and $\\tilde \\pi^{*}$ produced by NEXT. Obstacles are colored in blue. The start and goal locations are denoted by orange and brown dots. In (a) to (c), samples are represented with yellow circles. In (d), the level of redness denotes the value of the cost-to-go estimate $\\tilde V^*$. The cyan arrows point from a given state $s$ to the mean of the learned policy $\\tilde \\pi^*(s'|s,U)$.\n  }\n  \\label{fig:example}\n\\end{figure*}\n\\begin{figure*}[t]\n\\vspace{-2mm}\n\\centering\n  \\begin{tabular}{ccc}\n    \\includegraphics[width=.31\\linewidth]{.\/figures\/7d_path_kde.pdf}&\n    \\includegraphics[width=.31\\linewidth]{.\/figures\/7d_search_tree_kde.pdf}&\n    \\includegraphics[width=.31\\linewidth]{.\/figures\/7d_search_tree_rrts.pdf}\\\\[-3mm]\n    \\small (a) NEXT-KS solution path & \\small (b) NEXT-KS search tree& \\small (c) RRT* search tree (w\/o rewiring)\n  \\end{tabular}\n  \\vspace{-2mm}\n  \\caption{\\small Search trees and a solution path produced in an instance of spacecraft planning. The 7 DOF spacecraft has a yellow body and two 2 DOF red arms. NEXT-KS produced a nearly minimum viable search tree while RRT* failed to find a path within limited trials.}\n  \\label{fig:7d_plot}\n\\end{figure*}\n{\\bf Comparison results.} Examples of all four environments are illustrated in~\\appref{appendix:path} and \\figref{fig:7d_plot}, where NEXT finds high-quality solutions as shown. We also illustrated the comparison of the search trees on two 2d and 7d planning tasks between NEXT and RRT* in~\\figref{fig:example} (a)-(c) and \\figref{fig:7d_plot} (b) and (c). Obviously, the proposed NEXT algorithm explores with guidance and achieves better quality solutions with fewer samples, while the RRT* expands randomly which may fail to find a solution. The learned $\\tilde V^{*}$ and $\\tilde \\pi^{*}$ in the 2d task are also shown in~\\figref{fig:example}(d). As we can see, they are consistent with our expectation, towards the ultimate target in the map. For more search tree comparisons for all four experiments, please check~\\figref{fig:2d_examples},~\\ref{fig:3d_examples},~\\ref{fig:5d_examples} and \\ref{fig:7d_examples} in~\\appref{appendix:results}. \n\n\nTo systematically evaluate the algorithms, we recorded the cost of time (measured by the number of collision checks used) to find a collision-free path, the success rate within time limits, and the cost of the solution path for each run. The results of the reserved 1000 test problems of each environment are shown in the top row of~\\figref{fig:result}. We set the maximal number of samples as $500$ for all algorithms. Both the kernel smoothing (NEXT-KS) and the Gaussian process (NEXT-GP) version of NEXT achieves the state-of-the-art performances, under all three criteria in all test environments. Although the BIT* utilizes the heuristic particularly suitable for 3d maze in 7d task and performs quite well, the NEXT algorithm still outperform every competitor by a large margin, no matter learning-based or prefixed heuristic planner, demonstrating the advantages of the proposed NEXT algorithm.  \n\n{\\bf Self-improving.}  We plot the performance improvement curves of our algorithms on the 5d planning task in the bottom row of~\\figref{fig:result}.\nFor comparison, we also plot the performance of RRT$^*$. At the beginning phase of self-improving, our algorithms are comparable to RRT$^*$. They then gradually learn from previous experiences and improve themselves as they see more problems and better solutions. In the end, NEXT-KS is able to match the performance of RRT$^*$-10k using only one-twentieth of its samples, while the competitors perform consistently without any improvements. \n\n\nDue to the space limits, we put improvement curves on other environments in~\\figref{fig:result-appendix} and the quantitative evaluation in~\\tabref{tbl:exp1},~\\ref{tbl:exp2}, and~\\ref{tbl:exp3} in~\\appref{appendix:results}. Please refer to the details there. \n\n\\vspace{-3mm}\n\\subsection{Ablation Studies}\n\\vspace{-2mm}\n\n\\begin{figure*}\n   \n   \n    \\includegraphics[width=1.\\linewidth]{.\/figures\/all_bar_charts.pdf}\n    \\includegraphics[width=1.\\linewidth]{.\/figures\/all_15_5_chart.pdf}\n\\caption{\\small First row: histograms of results, in terms of success rate, average collision checks, and average cost of the solution paths; Second row: NEXT improvement curves in the 5d experiments.\nAll algorithms are set to use up to 500 samples, except RRT$^*$-10k, which uses 10,000 samples. The value of collision checks and path costs are normalized w.r.t. the performance of RRT$^*$.}\n\\label{fig:result}\n\\end{figure*}\n\n\n{\\bf Ablation study I: guided progressive expansion.} To demonstrate the power of $\\mathtt{NEXT::Expand}$, we replace it with breadth-first search (BFS)~\\citep{kim2018guiding}, another expanding strategy, while keeping other components the same. Specifically, BFS uses a search queue in planning. It repeatedly pops a state $s$ out from the search queue, samples $k$ states from $\\pi\\rbr{\\cdot|s}$, and pushes all generated samples and state $s$ back to the queue, until the goal is reached. For fairness, we use the learned sampling policy $\\pi\\rbr{s'|s, U}$ by NEXT-KS in BFS. As shown in~\\figref{fig:result}, BFS obtained worse paths with a much larger number of collision checks and far lower success rate, which justifies the importance of the balance between exploration versus exploitation achieved by the proposed $\\mathtt{NEXT::Expand}$.\n\n{\\bf Ablation study II: neural architecture.} To further demonstrate the benefits of the proposed neural architecture for learning generalizable representations in high-dimension planning problems, We replaced our attention-based neural architecture with an improved GPPN, as explained in~\\appref{appendix:exp_detail}, for ablation study. We extended the GPPN for continuous space by adding an extra reactive policy network to its final layers. We emphasize the original GPPN is not applicable to the tasks in our experiments. \nIntuitively, the improved GPPN first produces a `rough plan' by processing the robot's discretized workspace positions. Then the reactive policy network predicts a continuous action from both the workspace feature and the full configuration state of the robot. We provide more preference to the improved GPPN by training it to imitate the near-optimal paths produced by RRT* in the training problems. During test time it is also combined with both versions of the guided progressive expansion operators. \nAs we can see, both GPPN-KS and GPPN-GP are clearly much worse than NEXT-KS and NEXT-GP, demonstrating the advantage of our proposed attention-based neural architecture in high-dimensional planning tasks.\n\n{\\bf Ablation study III: learning versus heuristic.} The NEXT algorithm in \\figref{fig:example} shows similar behavior as the Dijkstra heuristic,~\\ie~sampling on the shortest path connecting the start and the goal in workspace. However, in higher dimensional space, the Dijkstra heuristic will fail. To demonstrate that, we replace the policy and value network with Dijkstra heuristic, using the best direction in workspace to guide sampling. NEXT performs much better than Dijkstra in all but the 2d case, in which the workspace happens to be the state space.\n\n\n\\vspace{-1mm}\n\\subsection{Case Study: Robot Arm Control}\n\\vspace{-1mm}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.66\\linewidth]{.\/figures\/robot_arm_chart_06.pdf}\n    \\caption{\\small The success rate and average path cost of the different planners under varying time limits. Running NEXT for 1 second achieves the same success rate as running BIT* for 50 seconds.}\n    \\label{fig:robot_arm_result}\n\\end{figure*}\n\nWe conduct a real-world case study on controlling robot arms to move objects on a shelf. On this representative real-time task, we demonstrate the advantages of the NEXT in terms of the wall-clock. \n\n\n\n\\begin{wrapfigure}[18]{r}{0.4\\textwidth}\n\\begin{center}\n    \\vspace{-5mm}\n    \\includegraphics[width=.75\\linewidth]{.\/figures\/robot_arm_traj_1.pdf}\n    \\vspace{-2mm}\n    \\caption{\\small The collision-free path produced by NEXT for robot arm planning. The start and goal configurations have end-effectors in different bins of the shelf.}\n    \\vspace{-2mm}\n\\label{fig:robot_arm_traj_1}\n\\end{center}\n\\vspace{-1mm}\n\\end{wrapfigure}\nIn each planning task, there is a shelf of multiple levels, with each level horizontally divided into multiple bins. The task is to plan a path from a location in one bin to another, \\ie,~the end effectors of the start and goal configurations are in different bins. The heights of levels, widths of bins, and the start and goal are randomly drawn from some fixed distribution. Different from previous experiments, the base of the robot is fixed. \nWe consider the BIT* instead of RRT* as the imperfect expert in $3000$ training problems. We then evaluate the algorithm on a separated $1000$ testing problems.  \nWe compare NEXT(-KS) with the highly tuned BIT* and RRT* in OMPL, and also CVAE-plan and Reinforce-plan in \\figref{fig:robot_arm_result}. As seen from the visualization of the found paths in \\figref{fig:robot_arm_traj_1}, this is a very difficult task. Our NEXT outperforms the baselines by a large margin, requiring only $1$ second to reach the same success rate as running $50$ seconds of BIT*.\n\nDue to space limits, we put details of the experiment setups, more results and analysis in \\appref{subsec:case_study}.\n\n\n\\vspace{-2mm}\n\\section{Conclusion}\\label{sec:conclusion}\n\\vspace{-2mm}\n\nIn this paper, we propose a self-improving planner, Neural EXploration-EXploitation Trees~(NEXT), which can generalize and achieve better performance with experiences accumulated. The algorithm achieves a delicate balance between exploration-exploitation via our carefully designed UCB-type expansion operation. To obtain the generalizable ability across different problems, we proposed a new parametrization for the value function and policy, which captures the Bellman recursive structure in the high-dimensional continuous state and action space. We demonstrate the power of the proposed algorithm by outperforming previous state-of-the-art planners with significant margins on planning problems in a variety of different environments.  \n\n\\section*{Acknowledgement}\nWe thank the Google Research Brain team members for helpful thoughts and discussions as well as the anonymous reviewers for their insightful comments and suggestions. This work is supported in part by NSF grants CDS\\&E-1900017 D3SC, CCF-1836936 FMitF, IIS-1841351, CAREER IIS-1350983 to L.S, and by NSF grants BIGDATA 1840866, CAREER 1841569, TRIPODS 1740735, DARPA-PA-18-02-09-QED-RML-FP-003, an Alfred P Sloan Fellowship, a PECASE award to H.L.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Background}\n\\label{sec:intro}\n\\begin{figure}[t]\n \\begin{center}\n   \n\\includegraphics[width=0.42\\linewidth]{ images\/imagenet.pdf}\\includegraphics[width=0.42\\linewidth]{ images\/DROC.pdf}\n\\includegraphics[width=0.42\\linewidth]{ images\/CCD.pdf}\\includegraphics[width=0.42\\linewidth]{ images\/MSACL.pdf}\n\\end{center}\n    \\caption{t-SNE results of the representations on the test set of Hyper-Kvasir~\\cite{borgli2020hyperkvasir} learned by IGD~\\cite{chen2021unsupervised} after being pre-trained on ImageNet~\\cite{deng2009imagenet}, or self-supervised with DROC~\\cite{sohn2020learning}, CCD~\\cite{tian2021constrained}, and our MSACL.\n    Compared to other methods,  MSACL clusters the normal image representations (\\textcolor{blue}{blue points}) in a tighter and denser region, and separates anomalous representations into three clusters (\\textcolor{red}{red points}), which can be associated with the three classes of synthetised abnormal images formed by simulating a varying number of lesions of different sizes and appearance in the normal images. \n    } \n    \\vspace{-10pt}\n    \\label{fig:tsne}\n\\end{figure}\n\n\n\nDetecting and segmenting malignant lesions from disease screening datasets is a crucial task in medical images analysis (MIA)~\\cite{tian2019one,tian2020few,liuphotoshopping,litjens2017survey,baur2020scale,fan2020pranet,lz2020computer,liu2021self,liu2021noisy}.\nA challenging aspect of this problem is that such screening datasets~\\cite{tian2021detecting,pu2019prospective} contain a disproportionately large number of normal (or healthy) images, and a small minority of abnormal (or disease) images.\nNot only is the collection and annotation of such heavily imbalanced training sets challenging, but it is hard to acquire a representative dataset containing a reasonable number of images from all possible disease sub-classes.\nInstead of relying on a fully supervised approach requiring such labelled training set, we consider in this paper an alternative approach based on unsupervised anomaly detection (UAD)~\\cite{tian2021constrained,chen2021unsupervised,chen2020unsupervised}, which is trained exclusively with normal samples.\nThere are two advantages with the UAD strategy: 1) the acquisition of such training set is straightforward given the large proportion of normal images in screening datasets; and 2) it is not necessary to collect a representative training set containing images from all possible disease sub-classes.\n \n \n\nUAD methods are generally based on a one-class classifier (OCC) that learns a normal image distribution from the normal training images, and test image anomalies (or abnormal images) are detected based on the extent that they deviate from the learned distribution~\\cite{f-AnoGAN,seebock2019exploiting,gong2019memorizing,chen2021unsupervised,liu2019photoshopping,venkataramanan2020attention,tian2021weakly,chen2020unsupervised,tian2020few,pang2019deep,reiss2021mean}.\nUAD methods can overfit the normal class due to their strong dependence on the normal images, and a mitigating solution is based on the use of pre-training to learn representations that can be transferred to UAD. \nPre-training methods can rely on \nImageNet~\\cite{tian2021constrained}, which may not allow an effective representation transferring from natural images to medical images, or on self-supervised learning (SSL)~\\cite{hendrycks2019using,golan2018deep,bergman2020classification,simclr_paper,moco,liu2020self,tian2021constrained}, which depends on the effectiveness of the pretext tasks and the assumptions of the training process to form image classes.\nSelf-supervision pre-training for UAD methods applied to MIA screening problems have shown promising results~\\cite{tian2021constrained}, but they have been sub-optimally explored given that they were adapted from computer vision methods without using MIA domain knowledge for designing the pretext tasks or the training process. \nFor instance, in MIA, diseases can be divided into classes characterised by variations in the number and appearance of lesions. \nMoreover, previous SSL methods in UAD~\\cite{tack2020csi,tian2021constrained,sohn2020learning} extend contrastive learning~\\cite{simclr_paper} to learn representations to be transferred to multi-class classification, where representations are learned by contrasting a large number of normal images and their augmented (from geometric or appearance transformations) versions that form a large number of classes in the representation space. This is sub-optimal for MIA OCC that needs to discriminate a tight and dense cluster\nof normal images against a relatively small number of abnormal classes that lie outside the normal cluster (see Fig.~\\ref{fig:tsne}).\n\n\n\n\n\n\n\n\n\n \n\n\n\n\nIn this paper, we propose the \\underline{M}ulti-class \\underline{S}trong \\underline{A}ugmentation via \\underline{C}ontrastive \\underline{L}earning (MSACL), a new self-supervised pre-training method modelled exclusively with normal training images, and designed to learn effective image representations for different types of downstream UAD methods applied to several MIA problems. \nThe main advantage of MSACL, compared to previous self-supervised pre-training method for MIA applications~\\cite{tian2021constrained}, is that we rely on MIA domain knowledge to design the optimisation and the pretext tasks. \nIn particular, our optimisation uses contrastive learning to classify training samples into multiple tight and dense clusters in terms of Euclidean distance and cosine similarity, with one cluster to represent the normal images and the remaining ones to represent sub-classes of the disease images (see Fig.~\\ref{fig:tsne}). These disease images are obtained with our MedMix augmentations that simulate a varying number of lesions of different sizes and appearance in the normal training images (see Fig.~\\ref{fig:MedMix-example}).\nWe summarise our contributions as follows:\n\\begin{itemize}\n\\item Our MSACL is the first self-supervised pre-training method specifically designed for MIA UAD applications, where our main advantage lies in the design of a contrastive learning optimisation that learns multiple classes, one for normal images, and the others for sub-classes of disease images, which are formed by our MedMix augmentations that simulate a varying number of lesions of different sizes and appearance, and\n\\item The proposed MSACL is shown to learn effective image representations that can adapt well to different types of downstream UAD methods applied to several MIA problems. \n\\end{itemize}\nWe empirically show that MSACL pre-training significantly improves the performance of two SOTA anomaly detectors, PaDiM~\\cite{defard2020PaDiM} and IGD~\\cite{chen2021unsupervised}. \nExtensive experimental results on four different disease screening medical imaging benchmarks, namely, colonoscopy images from two datasets~\\cite{borgli2020hyperkvasir,liu2019photoshopping}, fundus images for glaucoma detection~\\cite{li2019attention} and Covid-19 Chest X-ray (CXR) dataset~\\cite{wang2020covid} show that MSACL can be used to pre-train diverse SOTA UAD methods to improve their accuracy in detecting and segmenting lesions in diverse medical images.\n\n\n\n\n\\section{Related Work}\n\nIn this section, we first discuss UAD methods, then we present self-supervised pre-training methods for computer vision and MIA problems.\n\n\\textbf{UAD approaches}~\\cite{f-AnoGAN,seebock2019exploiting,gong2019memorizing,chen2021unsupervised,liu2019photoshopping,venkataramanan2020attention,tian2021weakly,chen2020unsupervised,tian2020few,pang2019deep} can be divided into two categories: predictive-based (e.g., DSVDD~\\cite{ruff2018deep}, OC-SVM~\\cite{chen2001one}, and deviation network~\\cite{pang2019deep}), and generative-based (e.g., auto-encoder~\\cite{venkataramanan2020attention,chen2020unsupervised,chen2021unsupervised,gong2019memorizing} and GAN~\\cite{f-AnoGAN,liuphotoshopping,akcay2018ganomaly}). \nPredictive-based UAD approaches train a one-class classifier to describe the distribution of normal data, and discriminate abnormal data using their distance\/deviation to the normal data distribution; whereas generative-based UAD approaches train deep generative models to learn latent representations of normal images, and detect anomalies based on image reconstruction error~\\cite{pang2021deep}. \nA fundamental challenge in both types of UAD methods is the learning of expressive feature representations from images, which is particularly important in MIA because abnormal medical images may have subtly looking lesions that can be hard to differentiate from normal images.\nHence, if not well trained, these UAD models can overfit the normal training data and learn ineffective image representations that will fail to enable the detection and segmentation of lesions.\n\n\nThe representation challenge above has been addressed with the use of ImageNet~\\cite{deng2009imagenet} pre-trained models, but transferring representations learned from natural images to medical images is not straightforward~\\cite{tian2021constrained}. \nAlternatively, the representation challenge can also be tackled by self-supervised pre-training methods that learn auxiliary pretext tasks~\\cite{hendrycks2019using,golan2018deep,bergman2020classification,simclr_paper,moco,liu2020self}, which is a strategy that has produced effective representations for UAD in general computer vision tasks~\\cite{hendrycks2019using,golan2018deep,bergman2020classification,tack2020csi}.\nHowever, their application to MIA problems needs to be further investigated because it is not clear how to design effective optimisation functions or pretext tasks that can work well in the detection of subtle lesions in medical images.\nPrevious UAD methods relied on self-supervised pretext tasks based on the prediction of geometric transformations~\\cite{hendrycks2019using,golan2018deep,bergman2020classification} or contrastive learning using standard data augmentation techniques (e.g., scaling, cropping, etc.)~\\cite{simclr_paper,moco} to form a large number of image classes characterising similar and dissimilar pairs.\nThese pretext tasks and optimisation strategy are not specifically related to the detection of subtle anomalies in medical images that contain a normal image class and a small number of disease sub-classes, so they may even degrade the detection accuracy of downstream UAD methods~\\cite{wang2020understanding}. \n\nFor SSL UAD pre-training in MIA, the only previous work CCD~\\cite{tian2021constrained} adapts standard contrastive learning and two general computer vision pretext tasks to image anomaly detection and can be applied to multiple downstream UAD methods. Although achieving good results in many benchmarks,\nthe optimisation explored by CCD does not explore the fact that the downstream UAD methods need to recognise one class of normal images and a small number of sub-classes of disease images, and their data augmentation will not produce realistic\nsynthetised medical image anomalies -- both issues can challenge the training of  downstream UAD approaches. \n\n\n\n\n\\vspace{-7pt}\n\n\\section{Method}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0 \\linewidth]{ images\/MSACL_framework_v6.pdf}\n\\end{center}\n\\vspace{-10pt}\n   \\caption{\\textbf{MSACL}: our proposed self-supervised pre-training for UAD trains four classes of images: the normal images formed by the weak augmentations in distribution $\\mathcal{A}_0$ (blue markers) and three classes of synthetised abnormal images formed by the strong augmentation in distributions $\\{\\mathcal{A}_n\\}_{n=1}^{3}$ (green, pink and orange markers). The optimisation uses a constrained contrastive learning that trains a four-class classification problem.\n   The different types of strong augmentations are produced by MedMix that introduces a varying number of fake lesions by cutting patches from the normal training images, altering them with random color jittering, Gaussian noise and non-linear intensity transformations, and pasting them to other normal training images.}\n\\label{fig:framework}\n\\end{figure}\n\n\n\nIn this section, we introduce the proposed MSACL pre-training approach depicted in Fig.~\\ref{fig:framework}. Given a training medical image dataset $\\mathcal{D} = \\{ \\mathbf{x}_i \\}_{i=1}^{|\\mathcal{D}|}$, \nwith all images assumed to be from the $\\text{normal}$ class and $\\mathbf{x} \\in \\mathcal{X} \\subset \\mathbb{R}^{H \\times W \\times C}$ ($H$: height, $W$: width, $C$: number of colour channels), our learning strategy involves two stages: 1) the self-supervised pre-training to learn an encoding network $f_{\\theta}:\\mathcal{X} \\rightarrow \\mathcal{Z}$ (with $\\mathcal{Z} \\subset \\mathbb{R}^{Z}$), and\n2) the fine-tuning of an anomaly detector or segmentation model \nbuilt from the pre-trained $f_{\\theta}(.)$.  \nThe approach is evaluated on a testing set $\\mathcal{T} = \\{ (\\mathbf{x},y,\\mathbf{m})_i \\}_{i=1}^{|\\mathcal{T}|}$, where $y \\in \\mathcal{Y} = \\{\\text{normal}, \\text{abnormal} \\}$, and $\\mathbf{m}\\in \\mathcal{M} \\subset \\{0,1\\}^{H \\times W \\times 1}$ denotes the segmentation mask of the lesion in the image $\\mathbf{x}$. Below, we first describe the optimisation proposed for MSACL in Sec.~\\ref{sec:MSACL}, then we describe the MedMix data augmentation in Sec.~\\ref{sec:medmix}, followed by a bried description of the UAD methods in Sec.~\\ref{sec:anomaly_detection_Segmentation}.\n\n\n\\subsection{MSACL Pre-training}\n\\label{sec:MSACL}\n\nThe gist of our proposed MSACL lies in the idea of discriminating the distribution of weakly augmented samples (simulating normal images) from the distributions of different types of strongly augmented samples (simulating multiple classes of abnormal images).\nInstead of attracting and repelling samples within and between a large number of image classes~\\cite{tack2020csi,tian2021constrained,sohn2020learning}, we propose a new contrastive loss to separate samples from the normal class and samples from abnormal sub-classes, and to enforce the \\textbf{clusters representing the normal and abnormal sub-classes} to be \\textbf{dense and tight}.\nTo this end, our proposed loss is defined as: \n\\begin{equation}\n\\begin{split}\n    \\ell(\\mathcal{D};\\theta,\\beta,\\gamma)= &\n    \\ell_{ctr}(\\mathcal{D};\\theta) + \\ell_{MSACL}(\\mathcal{D};\\theta) + \\\\\n    &\\ell_{aug}(\\mathcal{D};\\beta) + \\ell_{pos}(\\mathcal{D};\\gamma), \n\\end{split}\n    \\label{eq:full_loss}\n\\end{equation}\nwhere $\\ell_{ctr}(.)$ denotes the new distribution multi-centring loss, $\\ell_{MSACL}(.)$ represents the new MSACL contrastive loss, $\\ell_{aug}(.)$ and $\\ell_{pos}(.)$ are the pretext learning losses to constrain optimisation~\\cite{tian2021constrained}, and $\\theta$, $\\beta$ and $\\gamma$ are trainable parameters. \nThe loss terms in~\\eqref{eq:full_loss} rely on \\textbf{weak data augmentation distribution}, denoted by $\\mathcal{A}_0$, and $|\\mathcal{A}|$ \\textbf{strong data augmentation distributions}, represented by $\\{ \\mathcal{A}_n \\}_{n=1}^{|\\mathcal{A}|}$, each denoting a different type of augmentation. From each of these distributions, we can sample augmentation functions $a:\\mathcal{X} \\to \\mathcal{X}$.\n\nThe multi-centring loss in~\\eqref{eq:full_loss} depends on the estimation of the mean representation for each augmentation distribution, computed as\n\\begin{equation}\n\\mathbf{c}_n = \\mathbb{E}_{\\mathbf{x} \\in \\mathcal{D},  a \\sim \\mathcal{A}_n}  [ f_{\\theta}(a(\\mathbf{x})) ],\n\\label{eq:compute_center}\n\\end{equation}\nwhere $n \\in \\{0,...,|\\mathcal{A}|\\}$, with $\\mathbf{c}_n$ being the mean representation of the training data augmented by the functions sampled from $\\mathcal{A}_n$, for $n \\in \\{0,...,|\\mathcal{A}|\\}$. Note that these mean representations are computed at the beginning of the training and frozen for the rest of the training.\nThe distribution multi-centring loss is then defined as: \n\\begin{equation}\n\\ell_{ctr}(\\mathcal{D};\\theta) = \\mathbb{E}_{\\mathbf{x} \\in \\mathcal{D},  n \\in \\{0,...,|\\mathcal{A}|\\}, a \\sim \\mathcal{A}_n  }   \\| f_{\\theta}(a(\\mathbf{x}))  - \\mathbf{c}_n  \\|^2,\n\\label{eq:center_loss}\n\\end{equation}\nwhich pulls the representations of augmented samples toward their mean representations in~\\eqref{eq:compute_center}, making the augmentation clusters dense and tight in Euclidean space. \n\nTo further enforce the separation between different clusters and the tightness within each cluster, we introduce a novel contrastive learning. \nIn our contrastive learning, we maximisise the cosine similarity of samples that belong to the same class (i.e., normal or abnormal sub-classes) and minimise the cosine similarity of samples belonging to different classes.\nAn interesting aspect of this optimisation is that samples are centred by their own cluster mean representation $\\mathbf{c}_n$ from~\\eqref{eq:compute_center}, so our contrastive learning, combined with the multi-centred loss in~\\eqref{eq:center_loss} will cluster samples of the same class not only in Euclidean space, but also in inner product space (with cosine measuring similarity between samples).\nSuch re-formulated constrastive learning, combined with the multi-centring loss~\\eqref{eq:center_loss}, results in a loss that produces multiple clusters, where cluster $n=0$ contains the normal images and the others, denoted by $n\\in\\{1,...,|\\mathcal{A}|\\}$), have the synthetised abnormal images.\nOur proposed MSACL loss is defined as:\n\\begin{equation}\n\\resizebox{0.92\\linewidth}{!}{%\n$\n\\begin{split}\n    \\ell_{MSACL}(\\mathcal{D};\\theta)=\\mathbb{E}_{\\mathbf{x} \\in \\mathcal{D},n\\in\\{0,...,|\\mathcal{A}|\\},l \\in \\{0,1\\} } \\left [ \\ell^x_{MSACL}( \\mathbf{x}^{(n,l)}, \\mathcal{D};\\theta) \\right ]\n\\end{split}\n$\n}\n \\label{eq:contrastive_loss}\n\\end{equation}\nwhere $\\mathbf{x}^{(n,l)} = a(\\mathbf{x}^{(n)})$\nrepresents one of two (indexed by $l\\in\\{0,1\\}$) augmented data obtained from the application of a weak augmentation $a \\sim \\mathcal{A}_0$ on a strongly augmented data denoted by $\\mathbf{x}^{(n)} = a(\\mathbf{x})$ with $a \\sim \\mathcal{A}_n$.\nIn~\\eqref{eq:contrastive_loss}, we have:\n\\begin{equation}\n\\resizebox{0.92\\linewidth}{!}{\n$\n\\begin{split}\n    &\\ell^x_{MSACL}( \\mathbf{x}^{(n,l)}, \\mathcal{D};\\theta)= \\\\\n    &-\\log\\frac{\\exp\\left[ \\frac{1}{\\tau} \\tilde{f_{\\theta}}(\\mathbf{x}^{(n,l)})^{\\top} \n     \\tilde{f_{\\theta}}(\\mathbf{x}^{(n,(l+1)\\,\\text{mod}\\,{2})})\\right]}\n     {\n    \n    \\sum\\limits_{\\substack{\\mathbf{x}_j \\in \\mathcal{D} \\\\ m \\in \\{0,...,|\\mathcal{A}|\\} \\\\ k \\in \\{0,1\\}}}\n     \\mathbb{I}(\\mathbf{x}_j^{(m,k)} \\neq \\mathbf{x}^{(n,l)}) \\exp\\left[ \\kappa(n, m)  \\tilde{f_{\\theta}}(\\mathbf{x}^{(n,l)})^{\\top} \\tilde{f_{\\theta}}(\\mathbf{x}_j^{(m,k)}) \\right]\n     },\n\\end{split}$}\n\\end{equation}\nwhere $\\mathbb{I}(.)$ denotes an indicator function,  $\\mathbf{x}_j^{(m,k)}$ is defined similarly as $\\mathbf{x}^{(n,l)}$ in~\\eqref{eq:contrastive_loss},\n$m \\in \\{0,...,|\\mathcal{A}|\\}$ indexes the set of strong augmentations, and $k \\in \\{0,1\\}$ indexes one of the two weak augmentations applied to the strongly augmented image.\nLastly, to further constrain the normal and strongly augmented data representations in~\\eqref{eq:contrastive_loss}, our MSACL loss minimises the distance between samples centred by their representation means computed as:\n\\begin{equation}\n\\tilde{f_{\\theta}}(\\mathbf{x}^{(n,l)}) = \\frac{f_{\\theta}(\\mathbf{x}^{(n,l)}) - \\mathbf{c}_n}{\\| f_{\\theta}(\\mathbf{x}^{(n,l)}) - \\mathbf{c}_n \\|_2} , \n    \\label{eq:temperature_calibration}\n\\end{equation}\nwhere $\\mathbf{c}_n$ is defined in~\\eqref{eq:compute_center}.\nAlso in~\\eqref{eq:contrastive_loss} to map the representations from the same distribution into a denser region of the hyper-sphere, inspired by~\\cite{cho2021masked}, we propose a temperature calibration strategy defined as: \n\\begin{equation}\n\\kappa(n,m)= \n\\begin{cases}\n1 \/(\\alpha\\tau)   &, \\text{if } n = m \\\\\n\n1 \/\\tau              &, \\text{otherwise}\n\\end{cases},\n    \\label{eq:temperature_calibration}\n\\end{equation}\nwhere $\\alpha$ is a scaling factor that controls the shrinkage level of the temperature $\\tau$.\nAs a result, Eq.~\\eqref{eq:temperature_calibration} alters the temperature for the samples belongs to the same strong augmentation distributions (i.e., when $n = m$) to a smaller value $\\alpha$, which allows smaller amount of repelling strength compared to samples that belongs to strong augmentation distributions $n != m$).\nPutting all together, the loss in~\\eqref{eq:contrastive_loss} clusters the image representations into  hyper-spheres and regions within the hyper-spheres, where each hyper-sphere and region represent a different type of augmentation.\n\nInspired by~\\cite{tack2020csi,tian2021constrained}, we further constrain the training in~\\eqref{eq:full_loss} with a self-supervised classification constraint $\\ell_{aug}(\\cdot)$ that enforces the model to classify the strong augmentation function (Fig.~\\ref{fig:framework}):\n\\begin{equation}\n\\resizebox{0.92\\linewidth}{!}{%\n$\n\\ell_{aug}(\\mathcal{D};\\beta) = -\\mathbb{E}_{\\mathbf{x} \\in \\mathcal{D}, n\\in\\{0,...,|\\mathcal{A}|\\}, a \\sim \\mathcal{A}_n } \\left [ \\log \\mathbf{a}_n^{\\top} f_{\\beta}(f_{\\theta}(a(\\mathbf{x}))) \\right ],\n$%\n}\n\\label{eq:pretext_loss}\n\\end{equation}\nwhere $f_{\\beta}:\\mathcal{Z} \\rightarrow [0,1]^{|\\mathcal{A}|}$ is a fully-connected (FC) layer, and $\\mathbf{a}_n \\in \\{0,1\\}^{|\\mathcal{A}|}$ is a one-hot vector representing the strong augmentation distribution (i.e., $\\mathbf{a}_n(j)=1$ for $j=n$, and $\\mathbf{a}_n(j)=0$ for $j \\neq n$).\n\nThe final constraint in~\\eqref{eq:full_loss} is based on the relative patch location from the centre of the training image and is adapted for local patches (see Fig.~\\ref{fig:framework}). \nThis constraint is added to learn  positional and texture characteristics of the image in a self-supervised manner. \nInspired by~\\cite{doersch2015unsupervised}, the positional constraint predicts the relative position of the paired image patches, with its loss defined as \n\\begin{equation}\n\\resizebox{0.92\\linewidth}{!}{%\n$\n\\ell_{pos}(\\mathcal{D};\\gamma) = -\\mathbb{E}_{ \\{ \\mathbf{x}_{\\omega_{1}},\\mathbf{x}_{\\omega_{2}} \\} \\sim \\mathbf{x} \\in \\mathcal{D}} \\left [ \\log \\mathbf{p}^{\\top} f_{\\gamma}(f_{\\theta}(\\mathbf{x}_{\\omega_{1}}),f_{\\theta}(\\mathbf{x}_{\\omega_{2}})) \\right ],\n$%\n}\n\\label{eq:patch_prediction}\n\\end{equation}\nwhere $\\mathbf{x}_{\\omega_{1}}$ is a randomly selected fixed-size image patch from $\\mathbf{x}$, $\\mathbf{x}_{\\omega_{2}}$ is another image patch from one of its eight neighbouring patches (as shown in `patch location prediction' in Fig.~\\ref{fig:framework}),\n$\\omega_1,\\omega_2 \\in \\Omega$ represents indices to the image lattice,\n$f_{\\gamma}:\\mathcal{Z}\\times\\mathcal{Z} \\rightarrow [0,1]^8$, \nand $\\mathbf{p} = \\{0, 1\\}^8$ is a one-hot encoding of the patch location.\nThe constraints in~\\eqref{eq:pretext_loss} and~\\eqref{eq:patch_prediction} are designed to improve training regularisation.\n\n\n\n\\subsection{MedMix Augmentation}\n\\label{sec:medmix}\n\nOur MedMix augmentation is designed to augment medical images to simulate multiple lesions. \nWe target a more effective data augmentation for MIA applications than the computer vision  augmentations in~\\cite{tian2021constrained} (e.g., permutations, rotations) that do not simulate medical image anomalies and may yield poor detection performance by downstream UAD methods.\nWe realise that anomalies in different medical domains (e.g., glaucoma and colon polyps) can be visually different, but a commonality among anomalies is that they are usually represented by an unusual growth of abnormal tissue.\nHence, we propose the MedMix augmentation to simulate abnormal tissue with a strong augmentation that ``constructs\" abnormal lesions by cutting and pasting (from and to normal images) small and visually deformed patches. \nThis visual deformation is achieved by applying other transformations to patches, such as colour jittering, Gaussian noise and non-linear intensity transformations.\nThis approach is inspired by cutmix~\\cite{yun2019cutmix}, where our contribution over cutmix is the intensification of the change present in the cropped patches by the appearance transformations above. \nThese transformations are designed to encourage the model to learn abnormalities in terms of localised image appearance, structure, texture and colour.\n\nIn practice, we design $|\\mathcal{A}|=4$ strong augmentation distributions, where \n$\\mathcal{A}_n$ includes $n\\in\\{0,...,3\\}$ abnormalities in the image, which means that $\\mathcal{A}_0$ denotes the normal image distribution and $\\mathcal{A}_{n\\in\\{1,2,3\\}}$ represent the abnormal image distributions, containing $\\{1,2,3\\}$ anomalous regions.\nTherefore, our loss targets the classification of MedMix augmentations, as shown in Fig.~\\ref{fig:MedMix-example}. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=1.0 \\linewidth]{ images\/med-mix-v2.pdf}\n\\end{center}\n\\vspace{-6pt}\n   \\caption{Examples of our MedMix data augmentation, showing  augmentation $\\mathcal{A}_0$ containing zero synthetic anomalies (leftmost column) and increasingly stronger augmentations $\\{\\mathcal{A}_n\\}_{n=1}^3$ (second to fourth columns) with different number of synthetic anomalies (from one to three).}\n\\label{fig:MedMix-example}\n\n\\end{figure}\n\n\\subsection{Anomaly Detection and Segmentation}\n\\label{sec:anomaly_detection_Segmentation}\n\n\n\n\n\nAfter pre-training $f_\\theta(\\cdot)$ with MSACL, we fine-tune it with a SOTA UAD, such as IGD~\\cite{chen2021unsupervised} or PaDiM~\\cite{defard2020PaDiM}.\nThose methods use the same training set $\\mathcal{D}$ as MSACL, containing only normal images from healthy patients. \n\n\nIGD~\\cite{chen2021unsupervised} combines three loss functions: 1) two reconstruction losses based on local and global multi-scale structural similarity index measure (MS-SSIM)~\\cite{wang2003multiscale} and mean absolute error (MAE) to train the encoder $f_{\\theta}:\\mathcal{X} \\to \\mathcal{Z}$ and decoder $g_{\\phi}:\\mathcal{Z} \\to \\mathcal{X}$, 2) a regularisation loss to train adversarial interpolations from the encoder~\\cite{berthelot2018understanding}, and 3) an anomaly classification loss to train $h_{\\psi}:\\mathcal{Z} \\to [0,1]$. The anomaly detection score of image $\\mathbf{x}$ is defined by\n\\begin{equation}\n    s_{IGD}(\\mathbf{x}) = \\xi \\ell_{rec}(\\mathbf{x},\\tilde{\\mathbf{x}}) + (1-\\xi)(1 - h_{\\psi}(f_{\\theta}(\\mathbf{x}))),\n    \\label{eq:anomaly_score_image}\n\\end{equation}\nwhere $\\tilde{\\mathbf{x}} =g_{\\phi}(f_{\\theta}(\\mathbf{x}))$,\n$h_{\\psi}(.)$ returns the likelihood that $\\mathbf{x}$ is a normal image, $\\xi \\in [0,1]$ is a hyper-parameter, and\n\\begin{equation}\n\\begin{split}\n    \\ell_{rec}(\\mathbf{x}&,\\tilde{\\mathbf{x}}) = \n    \\rho \\| \\mathbf{x} - \\tilde{\\mathbf{x}} \\|_1 + \\\\\n     &(1-\\rho) \\left ( 1-\\left( \n     \\nu m_{G}(\\mathbf{x},\\tilde{\\mathbf{x}}) + (1-\\nu)m_{L}(\\mathbf{x},\\tilde{\\mathbf{x}}) \\right ) \\right ),\n\\end{split}\n    \\label{eq:reconstruction error}\n\\end{equation}\nwith $\\rho,\\nu \\in [0,1]$, $m_{G}(\\cdot)$ and $m_{L}(\\cdot)$ denoting the global and local MS-SSIM scores from the global and local models, respectively~\\cite{chen2021unsupervised}.\nAnomaly segmentation uses~\\eqref{eq:anomaly_score_image} to compute $s_{IGD}(\\mathbf{x}_{\\omega})$, $\\forall \\omega \\in \\Omega$ using global and local models, where\n$\\mathbf{x}_{\\omega} \\in \\mathbb{R}^{\\hat{H} \\times \\hat{W} \\times C}$ is an image patch.  This forms a heatmap, where large values of $s_{IGD}(.)$ denote anomalous regions. \nThe final heatmap is formed by summing up the global and local heatmaps. \n\n\n\nPaDiM~\\cite{defard2020PaDiM} utilises the multi-layer features from the pre-trained network $f_{\\theta}(.)$ to learn a position dependent multi-variate Gaussian distribution of normal image patches.\nTraining uses samples collected from the concatenation of the multi-layer features from each patch position $\\omega \\in \\Omega$ to learn the mean and covariance of the Gaussian model denoted by  $\\mathcal{N}(\\mu_{\\omega},\\Sigma_{\\omega})$~\\cite{defard2020PaDiM}.\nAnomaly detection is based on the Mahalanobis distance between the concatenated testing patch feature $\\mathbf{x}_{\\omega}$ and the learned Gaussian distribution $\\mathcal{N}(\\mu_{\\omega},\\Sigma_{\\omega})$ at that patch position $\\omega \\in \\Omega$ to provide a score of each patch position~\\cite{defard2020PaDiM}. \nIn particular, anomaly segmentation is inferred using the following anomaly score map:\n\\begin{equation}\n    s_{PaDiM}(\\mathbf{x}_{\\omega})=  \\sqrt{(\\mathbf{x}_{\\omega} - \\mu_{\\omega})^{\\top}\\Sigma_{\\omega}^{-1} (\\mathbf{x}_{\\omega} - \\mu_{\\omega})},\n\\end{equation}\nand the final score of the whole image $\\mathbf{x}$ is defined as:\n$s_{PaDiM}(\\mathbf{x})=\\max_{\\omega \\in \\Omega} s_{PaDiM}(\\mathbf{x}_{\\omega})$.\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\n\\subsection{Datasets}\n\\label{sec:dataset}\n\n\nWe test our self-supervised pre-training MSACL on four health screening datasets, where we run experiments for both anomaly detection and localisation.\nFor the anomaly detection and localisation, the datasets are: the colonoscopy images of Hyper-Kvasir dataset~\\cite{borgli2020hyperkvasir}, and the glaucoma dataset using fundus images~\\cite{li2019attention}.\nFor the anomaly detection, the datasets are: the colonoscopy dataset~\\cite{liu2019photoshopping}, and Covid-19 chest ray dataset~\\cite{wang2020covid} -- these two datasets do not have lesion segmentation annotations, so we test anomaly detection only.\n\n\\textbf{Hyper-Kvasir} is a large multi-class public  gastrointestinal imaging dataset~\\cite{borgli2020hyperkvasir}. \nWe use a subset of the normal (i.e., healthy) images from the dataset for training. \nSpecifically, 2,100 images from `cecum', `ileum' and `bbps-2-3' are selected as normal, from which we use 1,600 for training and 500 for testing. We also take 1,000 abnormal images and their segmentation masks of polyps to be used exclusively for testing, with the size of 300 $\\times$ 300.  \n\n\\textbf{LAG} is a large scale fundus image dataset for glaucoma diagnosis~\\cite{li2019attention}. For the experiments, we use 2,343 normal (negative glaucoma) images for training, and 800 normal images and 1,711 abnormal images with positive glaucoma with annotated attention maps by ophthalmologists in glaucoma diagnosis, with the size of 500 $\\times$ 500. The attention maps is based on an alternative method for eye tracking, in which the maps are used by the ophthalmologists to explore the region of interest for glaucoma diagnosis~\\cite{li2019attention}. \n\n\n\n\n\\textbf{Liu et al.'s colonoscopy dataset} is a colonoscopy image dataset with 18 colonoscopy videos from 15\npatients~\\cite{liu2019photoshopping}. The training set contains 13,250 normal (healthy) images without polyps, and the testing set contains 967 images, with 290 abnormal images with polyps and 677 normal (healthy) images without polyps, with size of 64 $\\times$ 64.\n\n\n\\textbf{Covid-X}~\\cite{wang2020covid} has a training set with 1,670 Covid-19 positive and 13,794 Covid-19 negative CXR images. \nThe test set contains 400 CXR images, consisting of 200 positive and 200 negative images. We train the methods with the 13,794 Covid-19 negative CXR training images and test on the 400 CXR images, with the size of 299 $\\times$ 299. \n\n \n\n\\subsection{Implementation Details}\n\\label{sec:implementation_details}\n\nFor the proposed MSACL pre-training, we use Resnet18~\\cite{he2016deep} as the backbone architecture for the encoder $f_{\\theta}(\\mathbf{x})$, and similarly to previous works~\\cite{simclr_paper,sohn2020learning}, we add an MLP to this backbone as the projection head for the contrastive learning, which outputs features in $\\mathcal{Z}$ of size 128. \nAll images from the Hyper-Kvasir~\\cite{borgli2020hyperkvasir}, LAG~\\cite{li2019attention} and Covid-X~\\cite{wang2020covid} datasets are resized to 256 $\\times$ 256 pixels. \nFor the Liu et al.'s colonoscopy dataset~\\cite{liu2019photoshopping}, images are resized to 64 $\\times$ 64 pixels.  \nThe batch size is set to 32 and learning rate to 0.01 for the self-supervised pre-training on all datasets.\nThe model is trained using stochastic gradient descent (SGD) optimiser with momentum. For IGD and PaDiM fine-tuning, we use their default experimental hyper-parameters and settings. \nMore details about the implementation can refer to our supplementary material.\n\n\\subsection{Evaluation Measures}\n\\label{sec:evaluation_measures}\n\n\n\nThe anomaly detection performance is quantitatively assessed by the area under the ROC curve (AUROC). \nFor anomaly segmentation, the performance is measured by Intersection over Union (IoU) and Dice score.  \n\n\n\n\n\\begin{table}[t]\n\\centering\n\\resizebox{0.88\\linewidth}{!}{\n\\begin{tabular}{@{}c|c|c|c|c@{}}\n\\toprule \\hline\nMethods         & Hyper  & Liu et al. & LAG  & Covid \\\\ \\hline\\hline\nDAE~\\cite{masci2011stacked}             & 0.705  & 0.629      & 0.574 & 0.557  \\\\\nOCGAN~\\cite{perera2019ocgan}           & 0.813    & 0.592    & 0.534    & 0.612 \\\\\nF-anoGAN~\\cite{f-AnoGAN}        & 0.907      & 0.691   & 0.778 & 0.669  \\\\\nADGAN~\\cite{liuphotoshopping}           & 0.913     & 0.730  &  -   & -   \\\\\nMS-SSIM~\\cite{chen2021unsupervised}        & 0.917    & 0.799 & 0.823  & 0.634       \\\\\nPANDA~\\cite{reiss2021panda}   & 0.937   & 0.719   & 0.789  & 0.629 \\\\  \n\\hline\nPaDiM~\\cite{defard2020PaDiM}   & 0.923   & 0.741    & 0.688 & 0.614 \\\\ \nCCD - PaDiM   & 0.978   & 0.789 & 0.728  & 0.632 \\\\\n\\textbf{MSACL - PaDiM}   & \\textbf{0.996}   & 0.814 & 0.761  & 0.658 \\\\\\hline\nIGD~\\cite{chen2021unsupervised}            & 0.939 & 0.787 & 0.796  & 0.699     \\\\ \nCCD - IGD      & 0.972   & 0.837 & 0.874  & 0.746  \\\\ \n\\textbf{MSACL - IGD}      & 0.995    &\\textbf{0.851} &  \\textbf{0.908}  & \\textbf{0.872} \\\\\n \\hline \\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{Anomaly detection:} \\textbf{AUC} test results on Hyper-Kvasir, Liu et al.'s colonocopy, LAG and Covid-X, respectively. }\n\\label{tab:detection_auc}\n\\end{table}\n\n\n\n\n\n\n\n\\subsection{Lesion Detection Results}\n\\label{sec:lesion_detection_results}\nIn Table~\\ref{tab:detection_auc}, we show the results of anomaly detection on Hyper-Kvasir, LAG,  Liu et al.'s colonoscopy and Covid-X dataset,\nwhere we present results from baseline UAD methods, including OCGAN~\\cite{perera2019ocgan}, f-AnoGAN~\\cite{f-AnoGAN}, ADGAN~\\cite{liu2019photoshopping}, and deep autoencoder~\\cite{masci2011stacked} and its variant with MS-SSIM loss~\\cite{chen2021unsupervised}. We also show the results the current SOTA UAD baseline PANDA~\\cite{reiss2021panda}. \nAs discussed in Sec.~\\ref{sec:anomaly_detection_Segmentation}, we choose IGD~\\cite{chen2021unsupervised} and PaDiM~\\cite{defard2020PaDiM} as the anomaly detector for evaluating our proposed MSACL pre-training approach and compare it with previous SOTA SSL CCD~\\cite{tian2021constrained} approach to fine-tune IGD and PaDiM.  \n\n\n\\textbf{Detection Results on Hyper-Kvasir.}\nFor Hyper-Kvasir, comparing with the baseline UAD methods, the performance of PaDiM and IGD are improved using our MSACL pre-trained encoder by around 5\\% and 6\\% AUC, which achieves SOTA anomaly detection AUC results of 99.6\\% and 99.5\\%, respectively, on Hyper-Kvasir. \nComparing with current SOTA CCD pre-training~\\cite{tian2021constrained}, our MSACL pre-training improves the performance by 2.3\\% and 1.8\\% for PaDiM and IGD, on Hyper-Kvasir. \n\n\n\n\n\n\n\\textbf{Detection Results on LAG.}\nFor LAG dataset, our MSACL pre-training improves PaDiM and IGD AUCs by 7.3\\% and 11.2\\%, compared with their ImageNet pre-trained model, where the MSACL pre-trained IGD achieves the SOTA results of 90.8\\% AUC.  \nComparing with CCD pre-trained PaDiM and IGD~\\cite{tian2021constrained}, our proposed MSACL pre-trained PaDiM and IGD surpass them by 3.3\\% and 3.4\\% in terms of AUC. \nFor LAG, the glaucoma features are often subtle and easy to neglect, so IGD with both reconstruction and anomaly classification constraints can generally perform better than PaDiM variants. \n\n\n\n\\textbf{Detection Results on Liu et al.'s colonoscopy dataset.}\nWe further test our approach on Liu et al.'s colonoscopy dataset~\\cite{liuphotoshopping}, as shown in Table~\\ref{tab:detection_auc}. \nSimilarly to the previous results, our MSACL pre-trained PaDiM improves the ImageNet pre-trained PaDiM by 7.3\\% AUC, and CCD pre-trained PaDiM by 2.5\\% of AUC. The IGD with the MSACL pre-trained encoder achieves the SOTA result of 85.1\\% AUC, surpassing the previous CCD and ImageNet pre-trained IGD by 1.4\\% and 6.4\\% AUC, respectively. \nThe gap between PaDiM and IGD may be due to the low resolution of the images in this dataset, which hinders the PaDiM performance that requires dense intermediate feature maps. \n\n\n\n\\textbf{Detection Results on Covid-X.}\nOur MSACL pre-trained PaDiM and IGD methods achieve 65.8\\% and 87.2\\% AUC on the Covid-X dataset, significantly surpassing their ImageNet pre-trained by 4.4\\% and 17.2\\% AUC, and CCD pre-trained by 2.6\\% and 12.6\\% AUC.  \nThe small abnormal lesions in CXR images are hard to detect, so the generative-based anomaly detector IGD can learn more effectively the fine-grained appearances of normal images, leading to better generalisation to detect unseen anomalous regions during testing with the SOTA results of 87.2\\% AUC. \n\n\n\nFinally, the performances of the baseline UAD methods are well below our self-supervised MSACL pre-training with IGD and PaDiM on all four datasets. \nAll results from MSACL pre-trained PaDiM and IGD show that our proposed MedMix and MSACL loss improve the generalisation of the fine-tuning stage for anomaly detection and produce better constrained feature space of normal samples. The SOTA performance across four different medical imaging datasets indicate that our pre-training is a general pre-training approach that is adaptable to different medical domains (i.e., CXR, colonoscopy, fundus).  \nMoreover, achieving SOTA results on two different types of anomaly detectors suggests that our self-supervised pre-training can produce good representations for both generative and predictive anomaly detectors, demonstrating the applicability of MSACL to real-world clinical systems. \n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=1.0 \\linewidth]{ images\/colon_seg.pdf}\n\\end{center}\n\\vspace{-6pt}\n   \\caption{Segmentation of four abnormal images from Hyper Kvasir~\\cite{li2019attention}, with their predictions (Pred) and ground truth annotations (GT), using PaDiM  with MSACL pre-training.}\n\\label{fig:heatmap_hk}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=1.0 \\linewidth]{ images\/lag_segmentation.pdf}\n\\end{center}\n\\vspace{-6pt}\n   \\caption{\n  Segmentation of four abnormal images from LAG~\\cite{li2019attention}, with their predictions (Pred) and ground truth attention maps (GT), using IGD with MSACL pre-training. }\n\\label{fig:heatmap_lag}\n\\end{figure}\n\n\n\\subsection{Lesion Segmentation Results.}\n\n\n\n\n\n\\textbf{Segmentation results on Hyper-Kvasir.}\nWe demonstrate the anomaly segmentation performance on Hyper-Kvasir on Table~\\ref{tab:localisation_auc}. Following~\\cite{tian2021constrained}, we randomly sample 100 abnormal images from the test set and compute the mean segmentation performance over five different such groups of 100 images.\nThe proposed MSACL pre-training improves the IGD and PaDiM by 1.2\\% and 2.8\\% IoU compared with the CCD pre-training, and 8.1\\% and 6.4\\% IoU with respect to the ImageNet pre-training, respectively. \nIn addition, our MSACL pre-trained PaDiM shows the SOTA result of 40.6\\% IoU and 55.4\\% Dice, demonstrating the effectiveness of our MSACL approach for abnormal lesion segmentation. \n\n\n\n\n\\begin{table}[]\n\\centering\n\\resizebox{0.8 \\linewidth}{!}{%\n\\begin{tabular}{c|cc|cc}\n\\toprule \n\\multirow{2}{*}{Methods} & \\multicolumn{2}{c|}{Hyper} & \\multicolumn{2}{c}{LAG} \\\\ \\cline{2-5} \n                         & IoU          & Dice        & IoU        & Dice       \\\\ \\hline \\hline\nPaDiM~\\cite{defard2020PaDiM}         &  0.341 &  0.475    & 0.427 & 0.579 \\\\\nCCD - PaDiM       & 0.378   & 0.497 & 0.462   & 0.612  \\\\\n\\textbf{MSACL - PaDiM}       & \\textbf{0.406} & \\textbf{0.554}  & 0.475  & 0.643  \\\\ \\hline \nIGD~\\cite{chen2021unsupervised}       &  0.303  & 0.417  & 0.409 &  0.539  \\\\ \nCCD - IGD       &  0.372  & 0.502  & 0.509    & 0.645 \\\\ \n \\textbf{MSACL - IGD}       & 0.384  & 0.521  &  \\textbf{0.516} &\\textbf{0.667} \\\\ \\hline \\bottomrule\n\\end{tabular}%\n}\n\\vspace{-6pt}\n\\caption{\\textbf{Anomaly segmentation:} Mean IoU, Dice results on test set of \\textbf{Hyper-Kvasir} on 5 different groups of 100 images with ground truth masks, and on abnormal samples from \\textbf{LAG} test set.\nBest results for each case are highlighted.\n}\n\\label{tab:localisation_auc}\n\\end{table}\n\n\n\\textbf{Segmentation results on LAG.}\nWe further demonstrate the segmentation results on LAG dataset on Table~\\ref{tab:localisation_auc}. The MSACL pre-trained IGD achieves the SOTA result of 51.6\\% IoU, 66.7\\% Dice, showing that our model can effectively segment different types of lesions, such as colon polyps or optic disk and cup with Glaucoma. Moreover, PaDiM pre-trained with MSACL improves PaDiM pre-trained with CCD and ImageNet by 1.3\\% and 4.8\\% IoU, respectively. Also, PaDiM with MSACL pre-training achieves 64.3\\% Dice, which are comparable to the SOTA results by the MSACL pre-trained IGD. \n\n\\subsection{Qualitative Results}\n\\textbf{Visualisation of predicted segmentation.}\nThe visualisation of polyp segmentation results of PaDiM with MSACL pre-training on Hyper-Kvasir~\\cite{borgli2020hyperkvasir} is shown in Fig.~\\ref{fig:heatmap_hk}. \nNotice that our model can effectively segment colon polyps with various sizes and shapes. \nWe also show the segmentation results based on the pixel-level anomaly scores of IGD with MSACL pre-training on the LAG dataset in Fig.~\\ref{fig:heatmap_lag}. \n\n\n\\textbf{Visualisation of t-SNE results.}\nTo validate our proposed MSACL pre-training, we compare the image representations produced by ImageNet, CCD, DROC and MSACL pre-training, using t-SNE on Hyper-Kvasir. The results are shown in Fig.~\\ref{fig:tsne}. The proposed MSACL appears to cluster all the normal data into a denser and tighter region of the representation space, where the abnormal data fall outside of this region in relatively distinct three clusters. \nIn contrast, the models pre-trained with the other approaches produce a poorly clustered normal data that is likely to challenge the training of the downstream UAD method.\n\n\n\\subsection{Ablation Study}\n\n\\begin{table}[t]\n\\centering\n\\scalebox{0.75}{\n\\begin{tabular}{cccc|cc}\n\\toprule\\hline\nCCD~\\cite{tian2021constrained}  & MedMix  & $\\ell_{MSACL}$ & $\\kappa(n,m)$  &  AUC - Hyper  & AUC - LAG \\\\ \\hline \\hline\n\\checkmark       &           &                        &    &   0.978   &    0.728       \\\\\n \\checkmark      & \\checkmark         &                   &   &   0.985      &   0.739     \\\\\n     & \\checkmark   & \\checkmark             &      &    0.993 & 0.753    \\\\    \\hline\n  & \\checkmark    & \\checkmark       &\\checkmark   &  0.996  & 0.761 \\\\ \\hline\\bottomrule\n\\end{tabular}\n}\n\\vspace{-6pt}\n\\caption{\\textbf{Ablation study of the MSACL components} on test set of Hyper-Kvasir and LAG, using PaDiM~\\cite{defard2020PaDiM} as anomaly detector.}\n\\label{tab:ablation}\n\\end{table}\n\n\\textbf{Different components of MSACL pre-training.} We present an ablation study that shows the influence of each step of our proposed MSACL pre-training and PaDiM fine-tuning in Table~\\ref{tab:ablation} on Hyper-Kvasir and LAG datasets.  \nStarting from CCD~\\cite{tian2021constrained}, we notice that the use of MedMix can improve the AUC on both datasets by 1\\%.  Replacing the CCD by the MSACL loss provides another improvement of between 1\\% and 2\\%.  Then, adding the temperature calibration $\\kappa(n,m)$ from~\\eqref{eq:temperature_calibration} provides around 0.5\\% improvement.\n\n\n\n\\begin{figure}[t]\n \\centering    \n \\includegraphics[width=.48\\linewidth]{ images\/AUC_aug.pdf}\n \\includegraphics[width=.48\\linewidth]{ images\/AUC_SSL.pdf}\n       \n       \n   \n   \n   \n   \n   \n   \n   \n    \\vspace{-6pt}\n\\caption{\\textbf{Left:} Anomaly detection testing results in terms of different types of strong augmentations (i.e., Cutmix, Gaussian noise, Rotation, Permutation, and our MedMix) on Hyper-Kvasir and Covid-X, where IGD~\\cite{chen2021unsupervised} is used as the anomaly detector. \\textbf{Right:} Ablation studies with different self-supervised pre-training approaches (i.e., ImageNet, SimCLR~\\cite{simclr_paper}, Rot-Net~\\cite{golan2018deep},  CSI~\\cite{tack2020csi}, DROC~\\cite{sohn2020learning}, CCD~\\cite{tian2021constrained} and our MSACL) on Hyper-Kvasir testing set, where PaDiM~\\cite{defard2020PaDiM} is used as the anomaly detector.\n}\n    \\label{fig:ablation_transformation}\n\\end{figure}\n\n\n\n\n\\textbf{Different strong augmentations.}\nIn Fig.~\\ref{fig:ablation_transformation} - left, we explore the influence of strong augmentation strategies, represented by rotation, permutation, cutout, Gaussian noise and our proposed MedMix on the AUC results on Hyper-Kvasir and Covid-X datasets, based on our self-supervised MSACL pre-training with IGD as anomaly detector. The performance of our MedMix reaches the SOTA results of 99.5\\% and 87.2\\% on those datasets.\nThe second best AUC (96.9\\%) on Hyper-Kvasir uses random permutations, which were used in CCD pre-training~\\cite{tian2021constrained}, producing an AUC 0.2\\% worse than our MedMix. For Covid-X, rotation is the second best data augmentation approach with an AUC result that is 5.1\\% worse than MedMix. \nOther approaches do not work well with the appearance characteristics of X-ray images, yielding significantly worse results than our MedMix on Covid-X. \nThese results suggest that the use of MedMix as the strong augmentation yields the best AUC results on different medical image benchmarks. \n\n\n\n\n\n\n\n\\textbf{Different self-supervised methods.}\nIn Fig.~\\ref{fig:ablation_transformation} - right, we show the results on Hyper-Kvasir from different pre-training approaches, using PaDiM as anomaly detector. \nIt can be observed that our MSACL approach surpasses previous SOTA CCD pre-training~\\cite{tian2021constrained} by 2.2\\% AUC. Other pre-training methods proposed in computer vision (e.g., ImageNet pre-training, SimCLR~\\cite{simclr_paper}, Rot-Net~\\cite{golan2018deep}) achieve worse results than CCD and MSACL. An interesting point in this comparison is the poor result from ImageNet pre-training, suggesting that it may not generalise well for anomaly detection in medical images. Finally, our MSACL achieves better results than previous SOTA UAD SSL approaches CSI~\\cite{tack2020csi} and DROC~\\cite{sohn2020learning} by about 4\\% $\\sim$ 5\\% AUC, indicting the effectiveness of our new contrastive loss. \n\n\n\\section{Conclusions and Discussions}\n\n\nIn this paper, we proposed a new self-supervised pre-training approach, namely MSACL, for UAD methods applied to MIA problems.\nMSACL is based on a new contrastive learning optimisation to learn multiple classes of normal and abnormal images, formed with the proposed MedMix data augmentation that simulates medical abnormalities.\nAfter pre-training a UAD model using our MSACL, we fine-tune it with two SOTA anomaly detecting approaches.\nThe experimental results indicate that our MSACL pre-training can effectively improve the performance of anomaly detection and segmentation on several medical datasets for both anomaly detectors.\nIn the future, we plan to design a new anomaly detector that suits better the characteristics of our self-supervised MSACL pre-training. \n\n\n\\clearpage\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Implementation Details}\nWe investigate the impact of different strong augmentations in $\\mathcal{A}_n$, including rotation, permutation, cutout, Gaussian noise and our proposed MedMix as shown in Figure 6 (Left) of the main paper. \nFor MedMix patches, we randomly apply colour jittering, Gaussian noise and  non-linear intensity transformations (i.e., fisheye and horizontal wave transformations). \nThe weak augmentations in $\\mathcal{A}_0$ are the same as in SimCLR~\\cite{simclr_paper}, namely: colour jittering, random grey scale, crop, resize, and Gaussian blur. \n\n\n\nThe model pre-trained with MSACL is fine-tuned with  IGD~\\cite{chen2021unsupervised} or PaDiM~\\cite{defard2020PaDiM}.\nFor IGD~\\cite{chen2021unsupervised}, we pre-train the global and local models, where the patch position prediction loss is only fine-tuned for the local model.  \nFor PaDiM~\\cite{defard2020PaDiM}, we pre-train the global model and use it to fine-tune the anomaly detection and segmentation models. \nFor the training of IGD~\\cite{chen2021unsupervised} and PaDiM~\\cite{defard2020PaDiM}, we use the hyper-parameters suggested by the respective papers.\nIn our experiments, the local map for IGD is obtained by considering each 32$\\times$32-pixel patch as an instance and apply our proposed self-supervised learning to it. \nThe global map for IGD is computed based on the whole image sized as 256 $\\times$ 256 pixels for Hyper-Kvasir, LAG and Covid-X datasets. For Liu et al.'s colonoscopy dataset, we only train the model globally with the image size 64 $\\times$ 64. For the auto-encoder in IGD, we use the setup suggested in~\\cite{chen2021unsupervised}, where the global model is trained with images of size 256$\\times$256 pixels or 64 $\\times$ 64 for Liu et al.'s colonoscopy dataset, and the local model is trained with image patches of size 32$\\times$32. \nFor PaDiM~\\cite{defard2020PaDiM}, we only use the default setup in their work and compute the segmentation mask based on the images of size 256$\\times$256 pixels for Hyper-Kvasir, LAG and Covid-X datasets, and 64 $\\times$ 64 for Liu et al.'s colonoscopy dataset. \n\n\n\\section{More Experimental Results}\n\n\\subsection{Variaty in the Results}\nWe show on Table~\\ref{tab:standard_deviation} the standard deviation computed from the AUC, specificity, sensitivity and accuracy results of five different trainings based on different model initialisation of the MSACL pre-trained PaDiM detector. These results in Table~\\ref{tab:standard_deviation} should be studied together with Table 1 of the main paper, and Tables~\\ref{tab:detection_auc_HK},~\\ref{tab:detection_auc_LAG},~\\ref{tab:detection_auc_liu_etal}, and~\\ref{tab:detection_auc_covid} of this supplementary material. In general, we conclude that the differences between the methods described in the detection results can be considered significant in most cases given that \nthe standard deviation only varies from $0.4\\%$ to $1.7\\%$.\n\\begin{table}[t]\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{@{}c c c c c c c c@{}}\n\\toprule \\hline\nDataset         & AUC & Specificity & Sensitivity  & Accuracy  \\\\ \\hline\\hline\nHyper-Kvasir~\\cite{borgli2020hyperkvasir}  &   0.0084   & 0.0079   & 0.0127 & 0.0036    \\\\\nLAG~\\cite{li2019attention}        & 0.0163    & 0.0085 &  0.0105 & 0.0121  \\\\\nCovid-X~\\cite{wang2020covid}    &  0.0107  & 0.0149 & 0.0092 & 0.0171 \\\\ \n \\hline\\bottomrule\n\\end{tabular}\n}\n\\caption{The standard deviation of five-run experimental results on the Hyper-Kvasir, LAG and Covid-X based on the MSACL pre-trained PaDiM anomaly detector. This results should be studied together with the results shown in Table 1 of the main paper, and Tables ~\\ref{tab:detection_auc_HK},~\\ref{tab:detection_auc_LAG},~\\ref{tab:detection_auc_liu_etal}, and~\\ref{tab:detection_auc_covid} below. \n}\n\\label{tab:standard_deviation}\n\\end{table}\n\n\n\n\\subsection{Visualisation of predicted anomaly scores}\nThe visual anomaly detection results of IGD pre-trained with MSACL on the Hyper-Kvasir~\\cite{borgli2020hyperkvasir} test set is shown in Fig.~\\ref{fig:prediction_scores}. \n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=1.0 \\linewidth]{latex\/images\/prediction_scores.pdf}\n\\end{center}\n   \\caption{Visual detection results and anomaly scores produced by the MSACL pre-trained IGD on three different datasets: Hyper-Kvasir (top), LAG (middle), Covid-X (bottom). \n   Anomaly scores $> 0.5$ classifies the image as positive, otherwise, the image is classified as negative. \n   Correctly classified images are marked with green boxes, and incorrectly classified cases are marked with red boxes.}\n\\label{fig:prediction_scores}\n\\end{figure*}\n\n\n\\subsection{More Detection results}\n\n\\begin{table}[t]\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{@{}c c c c c @{}}\n\\toprule \\hline\nMethods         & Specificity & Sensitivity  & Accuracy  \\\\ \\hline\\hline\nDAE~\\cite{masci2011stacked}            & 0.522 & 0.756  & 0.693   \\\\\nOCGAN~\\cite{perera2019ocgan}             & 0.691 & 0.811  & 0.795\\\\\nf-AnoGAN~\\cite{f-AnoGAN}            & 0.846  & 0.915   & 0.883  \\\\\nADGAN~\\cite{liuphotoshopping}               & 0.879 & 0.946  & 0.893  \\\\\nMS-SSIM~\\cite{chen2021unsupervised}          & 0.857  & 0.925  & 0.912   \\\\\n\n\n\\hline\nPaDiM~\\cite{defard2020PaDiM}     & 0.846 & 0.929 & 0.898 \\\\ \nCCD - PaDiM      & 0.923 & 0.961  & 0.967 \\\\\n\\textbf{MSACL - PaDiM}     & \\textbf{0.966} & \\textbf{0.981}  & \\textbf{0.983} \\\\\\hline\nIGD~\\cite{chen2021unsupervised}             & 0.858 & 0.913  & 0.906     \\\\ \nCCD - IGD        & 0.934 & 0.947  & 0.956  \\\\ \n\\textbf{MSACL - IGD}         & 0.947 &  0.965  & 0.972 \\\\\n \\hline\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{Anomaly detection} testing results on \\textbf{Hyper-Kvasir} in terms of Specificity, Sensitivity and Accuracy. Best results are highlighted.}\n\\label{tab:detection_auc_HK}\n\\end{table}\n\n\n\\begin{table}[t]\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{@{}c c c c c c c@{}}\n\\toprule \\hline\nMethods         & Specificity & Sensitivity  & Accuracy  \\\\ \\hline\\hline\nMS-SSIM~\\cite{chen2021unsupervised}          & 0.257 & 0.937  & 0.774     \\\\\nf-AnoGAN~\\cite{f-AnoGAN}         & 0.565 & 0.899 & 0.763  \\\\\\hline\nPaDiM~\\cite{defard2020PaDiM}   & 0.314 & 0.809  &  0.673 \\\\ \nCCD - PaDiM      & 0.429 & 0.779  & 0.694  \\\\\n\\textbf{MSACL - PaDiM}    &  0.466 & 0.877  & 0.753 \\\\\\hline\nIGD~\\cite{chen2021unsupervised}            & 0.396 & 0.958  & 0.805     \\\\ \nCCD - IGD       & \\textbf{0.572} & 0.944    &  0.875  \\\\ \n\\textbf{MSACL - IGD}     & 0.531 & \\textbf{0.979} & \\textbf{0.884} \\\\ \\hline\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{Anomaly detection} testing results on \\textbf{LAG} in terms of Specificity, Sensitivity, Precision and Recall. Best results are highlighted.}\n\\label{tab:detection_auc_LAG}\n\\end{table}\n\n\n\n\\begin{table}[t]\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{@{}c c c c c c @{}}\n\\toprule \\hline\nMethods         & Specificity & Sensitivity  & Accuracy  \\\\ \\hline\\hline\nDAE~\\cite{masci2011stacked}             & 0.733 & 0.554  & 0.597  \\\\\nOCGAN~\\cite{perera2019ocgan}              & 0.716 &  0.534  & 0.624\\\\\nADGAN~\\cite{liuphotoshopping}               & 0.852 & 0.496  & 0.713  \\\\\nf-AnoGAN~\\cite{f-AnoGAN}           & 0.865  &  0.579  & 0.694 \\\\\n\\hline\nPaDiM~\\cite{defard2020PaDiM}    & 0.851 & 0.738  &  0.751 \\\\\nCCD - PaDiM   & 0.946 & 0.792  & 0.767  \\\\\n\\textbf{MSACL - PaDiM}   & 0.973 & 0.725  & 0.803  \\\\\n\\hline\nIGD~\\cite{chen2021unsupervised}          & 0.914 & 0.596  & 0.743   \\\\ \nCCD - IGD     & 0.985 & 0.774  & 0.815  \\\\ \n\\textbf{MSACL - IGD}      & \\textbf{0.986} & \\textbf{0.792} & \\textbf{0.829}  \\\\ \\hline\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{Anomaly detection} testing results on \\textbf{Liu et al.'s colonscopy} in terms of Specificity, Sensitivity and Accuracy.   }\n\\label{tab:detection_auc_liu_etal}\n\\end{table}\n\n\n\\begin{table}[t]\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{@{}cccc@{}}\n\\toprule \\hline\nMethods         & Specificity & Sensitivity  & Accuracy   \\\\ \\hline\\hline\nPaDiM~\\cite{defard2020PaDiM}  & 0.753 & 0.318 & 0.559    \\\\ \nCCD - PaDiM  & 0.673 & 0.569 & 0.616   \\\\\n\\textbf{MSACL - PaDiM}    & 0.749 & 0.467 & 0.615 \\\\\n\\hline\nIGD~\\cite{chen2021unsupervised}    & \\textbf{0.885} & 0.490 & 0.688            \\\\ \nCCD - IGD       & 0.851 & 0.595 & 0.722  \\\\ \n\\textbf{MSACL - IGD}      & 0.863 & \\textbf{0.775} & \\textbf{0.813}   \\\\ \\hline\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{Anomaly detection} testing results on \\textbf{Covid-X} in terms of Specificity, Sensitivity and Accuracy, respectively. Best results are highlighted.}\n\\label{tab:detection_auc_covid}\n\\end{table}\n\n\n\nTables~\\ref{tab:detection_auc_HK},~\\ref{tab:detection_auc_LAG},~\\ref{tab:detection_auc_liu_etal},~\\ref{tab:detection_auc_covid} show the comparison between our approach and other SOTA methods in terms of Specificity, Sensitivity and Accuracy. Our approach achieves the best results in most of the benchmarks. In particular, our MSACL pre-trained PaDiM achieves the best Specificity, Sensitivity and Accuracy on Hyper-Kvasir, while  MSACL pre-trained IGD achieves the best results on LAG, Covid-X and Liu et al.'s colonscopy dataset. \n\n\n\\subsection{More segmentation results}\nPer-region-overlap score (Pro-AUC)~\\cite{bergmann2020uninformed} weights the ground-truth masks of different sizes equally~\\cite{bergmann2020uninformed} to verify if both large and small abnormal lesions are accurately segmented.  Here, we investigate the performance of our approach in terms of the Pro-AUC. \nTable~\\ref{tab:pro_segmentation} shows the Pro-AUC results on Hyper-Kvasir and LAG datasets. Our MSACL pre-trained IGD achieves the best result of 69.3\\% on LAG, and competitive result of 87.6\\% on Hyper-Kvasir. For PaDiM, our pre-training boosts the performance by 5\\% and 3\\% on Hyper-Kavsir and LAG, respectively, when compared with its vanilla version using ImageNet pre-training.  \n\n\n\n\n\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{@{}c c c@{}}\n\\toprule \\hline\n Methods  & Hyper - Pro  & LAG - Pro  \\\\ \\hline \\hline\nPaDiM~\\cite{defard2020PaDiM}     & 0.803      & 0.596\\\\\nCCD - PaDiM     &\\textbf{0.881}   & 0.634 \\\\\n\\textbf{MSACL - PaDiM}       & 0.854     & 0.628 \\\\\\hline  \nIGD~\\cite{chen2021unsupervised}  & 0.794       & 0.603 \\\\\nCCD - IGD   & 0.865   &  0.677 \\\\ \n\\textbf{MSACL - IGD}    & 0.876    &  \\textbf{0.693} \\\\ \n\\hline \n \\bottomrule \\hline\n\\end{tabular}%\n\\caption{\\textbf{Anomaly segmentation:} Mean Pro-AUC testing results on abnormal samples from \\textbf{Hyper-Kvasir} and \\textbf{LAG} test set. Best results are highlighted.}\n\\label{tab:pro_segmentation}\n\\end{table}\n\n\\clearpage\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{section1}\nMagnetic systems in the presence of random field are an important class of disordered systems which have been extensively studied for last many years. These systems are governed by two energies: the exchange interaction between spins which want spins to be correlated and local random field $h$ at each site which would make the spins uncorrelated. Thermal frustrations play a secondary role in random field models, since their disordering effect \nis much weaker \\cite{dfisher,binder}. \n\nOne of the very well studied model  to understand the effect of random field is the random \nfield Ising model (RFIM), which was introduced by Imry and Ma in 1975 \\cite{imryma}. \nThe pure model (Ising model) has a continuous transition for all $d \\ge 2$. The model has been \nvery useful as it has found its physical realizations in many of the condensed matter systems \\cite{belanger}. Hence the model has attracted a lot of attention, both theoretically and  experimentally \\cite{dfisher,binder,belanger,nettermann,reiger,fytas,sourlas,picco}. The model has many interesting and intriguing properties. For example, at the mean field level the phase-diagram of RFIM is known to change its character as a function of the random field distribution function. For gaussian distribution, the transition stays continuous for all strengths of the random field, but for bimodal distribution it exhibits a tricritical point \\cite{aharony}. In three dimensions though the existence of tricritical point has been widely debated \\cite{sourlas,picco, binder,hernandez,fytas2}, with recent numerical simulations suggesting continuous transition even in the strong disorder regime\\cite{fytas,fytas2}.\n\n\nRelated model is a $p$-spin model, in which $p$-spins are connected by an exchange interaction \\cite{gross}. Pure system has a sharp first order transition for $p \\ge 3$, which can easily be  studied in the mean field limit. The random field version of this model, for $p \\ge 3$, has relatively attracted much less attention in the literature. This model is a good system to \nstudy the effect of local field randomness on sharp first order transitions \\cite{cardy1}. The \nmodel is also of relevance since it is known to have connections with the structural glasses \\cite{kirkpatrick,biroli}.\n\nEffect of quenched local randomness for systems which undergo continuous phase transition in the absence of randomness is well described by the Harris criterion \\cite{harris}. This is possible due to the universal behaviour of the correlation length. On the other hand, even though first order transitions are common in nature, effect of random local disorder on first order  transitions is much harder to study theoretically, as the correlation length remains finite and hence the universal behaviour is not expected in these systems \\cite{randomfirstorder}. \n\n\nIn this paper we look at the random field $p$-spin model on fully connected graphs using the theory of large deviations. Large deviations lend itself naturally to the study of mean field statistical mechanics models \\cite{ellis}. Recently, Lowe et al \\cite{lowe}  derived the rate function (to be defined in section 2) for the  magnetisation of random field Ising model on fully connected graphs using a method based on tilted large deviation techniques \\cite{tilted}. Their method is much simpler than the earlier approaches \\cite{comet} and can easily be generalised to many other random field problems. In this paper, we extend it to solve the random field $p$-spin model for arbitrary $p$. Note that even though calculations are a straightforward generalisation of \\cite{lowe}, they are useful as they allow us to study the random $p$-spin model fully and hence allow us to look into the issue of understanding the disorder effects near the first order transitions. Besides obtaining the phase diagram in this paper, we also derive a method to obtain the value of quantities like magnetic susceptibility from the rate function. Magnetic susceptibility is an experimentally measurable quantity, and hence is useful to make direct connection with the experimental observations. We find that in the first order regime of RFIM, magnetic susceptibility is a rapidly increasing function of the temperature as one approaches the critical point, even though it does not diverge at the transition point. For $p \\ge 3$,  it is discontinuous at the transition point, with a typical paramagnetic behaviour in the high temperature phase. \n\nThe plan of the paper is as follows: In section \\ref{sec2}, we give the details of the model and calculate the rate function. We also derive the expression for magnetic susceptibility in this section. In section \\ref{sec3}, we give the phase-diagram obtained by looking at the value of magnetisation, and in section \\ref{sec4}, we calculate the magnetic susceptibility for $p=2,3$ and $4$.  We conclude in \nsection \\ref{sec5}.\n\n\\section{Rate function for a random field p-spin model}\n\\label{sec2}\nWe consider a Hamiltonian of the form:\n\n\\begin{equation}\n\\label{pspinhamiltonian}\n{\\cal{H}} = -\\frac{1}{p! N^{p-1}} \\sum_{i_1,...i_p}^N \\prod_{k=1}^{k=p} s_{i_k} -\\sum_{i_1}^N h_{i_1} s_{i_1}- H \\sum_{i_1}^N s_{i_1}\n\\end{equation}\nwhere $s_{i_k}$ are Ising spins which take values $\\pm 1$, $H$ is the uniform external \nfield and $h_i$ is the site dependent random field taken from a distribution. The \nprobability of getting a particular configuration $\\{s_{i_k}\\}$ for a given realization of \n$\\{h_i\\}$ is given by:\n\n\\begin{equation}\n\\label{eqn_gibbs_measure}\nP_{N}(\\beta,\\{ s_{i_k} \\},\\{h_i\\})=\\frac{1}{Z_{N,\\beta}^h}\\mbox{exp}\\left( \\frac{\\beta}{2N}\\sum _{ i_1,....,i_p }^N\\prod_{k=1}^{k=p} s_{i_k} +\\beta \\sum _{i_1=1}^N(h_{i_1}+H) s _{i_1}\\right)\n\\end{equation}\nwhere,\n\\begin{equation}\nZ_{N}(\\beta,\\{h_i\\})=\\sum _{\\{s_{i_k}\\}} \\mbox{exp}\\left( \\frac{\\beta}{2N}\\sum _{ i_1,....,i_p }^N\\prod_{k=1}^{k=p} s_{i_k} +\\beta \\sum _{i_1=1}^N(h_{i_1}+H) s _{i_1}\\right)\n\\end{equation}\nis the partition function of the model for a given realization $\\{h_i\\}$ of the random field and $\\beta$ is the inverse of the temperature. In order to get the free energy, one needs to average the $\\log$ of the partition function over the distribution of random fields. This averaging over the $\\log$ is a hard problem, and has been done only for a relatively fewer models like the random energy model \\cite{derrida}. Another approach is the replica approach where the $<\\log Z>$ is approximated by the use of the identity $<Z^n>=1+n <\\log Z>+O(n^2)$ \\cite{parisi}. One more method which has been found useful to study the mean field spin systems is the large deviation technique. One can prove the existence of large deviation principle with an explicit rate function for the order parameter of the model and then use it to study the phase diagram \\cite{ellis}. Recently, for random field Ising model ($p=2$), Lowe et al \\cite{lowe} derived the rate function explictly for the probability distribution of magnetisation. For a sequence of random variables $\\{s_i\\}$, the rate function is defined as\n\n\\begin{equation}\n\\lim_{N \\rightarrow \\infty} \\frac{1}{N} P_N(\\{s_i\\} \\in A)=-I(A),\n\\end{equation}\nwhere $P_N(\\{s_i\\} \\in A)$ is the probability that the given sequence of random \nvariables ($\\{s_i\\}$), belong to the set $A$. For our model, $A$ can be defined via $x=\\frac{1}{N}\\sum_{i=1}^{N} s_i$, \\emph{i.e.}, all sequences with same value of $x$ will have the same probability, as given by the equation above. \n\nNote that trying to get the rate function directly, using say Gartner Ellis theorem \\cite{touchette}, is quite complicated as $\\sum s_i\/N$ is not a sum of $N$ independent random variables with respect to the probability measure in Eq. \\ref{eqn_gibbs_measure}. In order to calculate the rate function Lowe et al \\cite{lowe} developed a method which mainly involved the following steps:\n\n1. Prove large deviation principle (LDP) with respect to a measure $Q$, which is a measure on non-interacting spins and hence is easier to average over the quenched field.\n\n2.  Use titled LDP to calculate the rate function for the probability distribution of magnetisation in the presence of coupling term from the rate function of $Q$.\n\nThis method turns out to be very useful especially in the cases where the randomness is not in the the coupling term. Also it gives the full rate function for the probability distribution of the  magnetisation, unlike earlier approaches \\cite{salinas,hemmen}. Since the average over the spin variables is done over the $Q$ measure, the method extends straightforwardly to arbitrary $p$ for any discrete spin and for any distribution of the random fields. This is unlike other approaches which for $p>2$, were doable only for the spherical spins\\cite{haddad}.\n\nLet $Q_i(s_i)$, for every $i\\in \\mathbb{N}$, be the probability measure on $\\{ 1,-1\\}$ induced by the field $h_i+H$. Then \n\\begin{eqnarray}\nQ_i( 1) &=& \\frac{e^{\\beta (h_i+H)}}{2\\mbox{cosh}(\\beta (h_i+H))}, \\nonumber\\\\\nQ_i(-1) &=& \\frac{e^{-\\beta (h_i+H)}}{2\\mbox{cosh}(\\beta (h_i+H))}.\n\\end{eqnarray}\n\nWe denote by $Q(\\{s_i\\})$, the product measure $\\otimes _{i=1}^{\\infty}Q_i(s_i)$, which is a probability measure on $\\{1,-1\\}^n$. We first calculate an LDP with respect to $Q(\\{s_i\\})$. We can employ Gartner Ellis Theorem for the sum, $S_N=\\sum_{i=1}^Ns_i$, as it is a sum of independent random variables under the product measure $Q(\\{s_i\\})$. Gartner-Ellis Theorem yields that $S_N\/N$ satisfies LDP with respect to $Q(\\{s_i\\})$, with a rate function $R(x)=\\sup _{y\\in \\mathbb{R}}\\left\\{ xy-\\Lambda(y)\\right\\}$, where \n\\begin{eqnarray}\n \\Lambda (x)&=& \\lim _{N \\rightarrow \\infty}\\frac{1}{N}\\mbox{ln }\\mathbb{E}e^{NxS_N\/N}\\nonumber\\\\\n &=& \\lim _{N \\rightarrow \\infty}\\frac{1}{N}\\sum _{i=1}^{N}\\mbox{ln }\\left( \\frac{e^{x+\\beta h_i+\\beta H}+e^{-x-\\beta h_i-\\beta H}}{2\\mbox{cosh}(\\beta (h_i+H))}\n\\right) \\nonumber\\\\\n&=&\\lim _{N \\rightarrow \\infty}\\left( f_N(x)-f_N(0)\\right).\n\\end{eqnarray}\nNow because of the Strong Law of Large numbers, we have\n\\begin{equation}\n\\label{fn_converge}\n\\lim _{N\\rightarrow \\infty}f_N(x)\\rightarrow f(x),\n\\end{equation}\nwhere $f(x)$ is the expectation value with respect to the distribution of the random fields. For the bimodal distribution of the random field $p(h_i)$, given by \n\\begin{equation}\n\\label{eqn_prob_dist_2spin}\np(h_i)=\\frac{1}{2}\\left[ \\delta (h_i-h)+\\delta (h_i+h)\\right],\n\\end{equation}\n the function $f(x)$ comes out to be \n\\begin{eqnarray}\nf(x)=\\frac{1}{2}\\mbox{ln}[\\mbox{cosh}(x+\\beta h+\\beta H)\\mbox{cosh}(x-\\beta h+\\beta H)].\n\\end{eqnarray}\nHence we get,\n\\begin{equation}\nR(x)=\\sup _{y\\in \\mathbb{R}}\\left\\{ xy-\\Lambda(y)\\right\\}=f^*(x)+f(0)\n\\end{equation}\nwhere $f^*(x)$ is the Legendre-Fenchel transform of $f(x)$,\n\\begin{equation}\nf^*(x)=\\sup _{y\\in \\mathbb{R}}\\left\\{ xy -f(y)\\right\\}.\n\\end{equation}\nNow $P_{N,\\beta}^h(x)$, the probability that the sum $S_N\/N=x$ for a given $\\beta$, $h$ and $H$, can be writtem in terms of $Q(\\{s_i\\})$ as follows:\n\\begin{eqnarray}\nP_{N,\\beta}^h(x) &=& \\frac{\\exp(\\beta N x^p\/p!) Q(\\{s_i\\}:\\sum s_i=Nx)}{\\sum_{Nx=-N}^N \\exp(\\beta N x^p\/p!) Q(\\{s_i\\}:\\sum s_i=Nx)}\\\\&=& \\frac{\\exp(\\beta N F(x)) Q(\\{s_i\\}:\\sum s_i=Nx)}{\\sum_{Nx=-N}^N \\exp(\\beta N F(x)\/2) Q(\\{s_i\\}:\\sum s_i=Nx)}\n \\end{eqnarray}\nwhere,\n\\begin{eqnarray}\nF(x)=\\left\\{\\begin{array}{lr}\n\\beta x^p\/p!& \\mbox{ if } |x|\\le 1\\\\\n\\beta \/p! & \\mbox{ otherwise }.\n\\end{array}\\right.\n\\end{eqnarray}\nThe tilted LDP \\cite{tilted} then yields that $S_N\/N$ satisfies a LDP with respect to $P_{N,\\beta}^h(x)$ with the rate function given by\n\\begin{eqnarray}\n\\label{eqn_p2_random_rate}\nI_p(x)&=& R(x)-F(x)-\\inf _{y \\in \\mathbb{R}}\\left\\{ R(y)-F(y)\\right\\}\\nonumber \\\\\n&=& f^*(x)+f(0)-F(x)-\\inf _{y \\in \\mathbb{R}}\\left\\{ f^*(y)+f(0)-F(y)\\right\\}\\nonumber \\\\\n&=& f^*(x)-\\frac{\\beta x^p}{p!}-\\inf _{y \\in \\mathbb{R}}\\left\\{f^*(y)-\\frac{\\beta y^p}{p!}\\right\\}\\nonumber \\\\\n&=& \\sup _{y\\in \\mathbb{R}}\\left\\{ xy-f(y)\\right\\}-\\frac{\\beta}{p!}x^p-\\inf _{y\\in \\mathbb{R}}\\left\\{ \\sup _{z\\in \\mathbb{R}}\\left\\{ yz-f(z)\\right\\}-\\frac{\\beta}{p!}y^p\\right\\}.\n\\end{eqnarray}\nLet\n\\begin{eqnarray}\nW(y)=xy -f(y)&=& xy-\\frac{1}{2}\\mbox{log}[\\mbox{cosh}(2(y+\\beta H))+\\mbox{cosh}(2\\beta h)].\n\\end{eqnarray}\nThe function $W(y)$ attains maximum at\n\\begin{equation}\ny=\\frac{1}{2}\\mbox{sinh}^{-1}\\left\\{ \\frac{x}{1-x^2}(b+\\sqrt{1+x^2a^2})\\right\\}-\\beta H,\n\\end{equation}\nwhere $a=\\mbox{sinh}(2\\beta h)$ and $b=\\mbox{cosh}(2\\beta h)$.\nHence the rate function in Eq.(\\ref{eqn_p2_random_rate}) becomes\n\\begin{eqnarray}\n\\label{eqn_p2_random_rate_final}\n I_p(x) &=&\\mbox{log }2-\\frac{\\beta}{p!}x^p-\\beta x H+\\frac{x}{2}\\mbox{Arcsinh}\\left( \\frac{x}{1-x^2}(b+\\sqrt{1+x^2a^2})\\right)\\nonumber\\\\&+&\\frac{1}{2}\\mbox{log}\\left( \\frac{1-x^2}{2}\\right) -\\frac{1}{2}\\mbox{log}\\left(b+\\sqrt{1+x^2a^2}\\right)-\\inf _{y \\in \\mathbb{R}}G_1(y)\n\\end{eqnarray}\nwhere, $$G_1(y)=\\sup _{z\\in \\mathbb{R}}\\left\\{ yz-f(y)\\right\\}-\\frac{\\beta}{p!}y^p.$$\nAt a given value of $\\beta,h$ and $H$, the value of magnetisation $m$, which is the average value of $\\frac{1}{N} \\sum_{i=1}^N s_i$, averaged over all spin configurations, will be given by that value of $x$ at which $I_p(x)$ is minimum. Hence magnetisation is given by that solution of the equation $I_p'(x)=0$, which minimises $I_p(x)$. Rate function defined by Eq. \\ref{eqn_p2_random_rate_final}, has one local minimum at $x=0$ in the paramagnetic phase and develops multiple minima in the ferromagnetic phase. This  rate function actually looks very similar to the Landau free energy functional (see Fig. \\ref{ratef}). This is not a priori obvious due to the \npresence of random field. Expanding Eq. \\ref{eqn_p2_random_rate_final} in powers of $x$ for \n$h=0$ and $H=0$, gives the following expressions for $p=2$, $3$ and $4$ respectively:\n\n\\begin{equation}\nI_2(x) = (1-\\beta) x^2+\\frac{1}{6} x^4 +O(x^6),\n\\end{equation}\n\n\\begin{equation}\nI_3(x)=x^2-\\frac{\\beta}{3} x^3+\\frac{1}{6} x^4+O(x^6),\n\\end{equation}\n\nand \n\n\\begin{equation}\nI_4(x)= x^2+\\frac{(2-\\beta)}{12} x^4+\\frac{1}{15} x^6+O(x^8).\n\\end{equation}\n\nAs expected, the expansion of $I_2(x)$ in the absence of randomness gives the Landau free energy functional of the Curie Weiss model. Interestingly, for $p \\ge 3$, even in the pure case, the $\\beta$ dependence is no longer with the lowest power of $x$. Since the expansion is not really useful to get the behaviour near the first order transition (for which we need to keep terms till all orders), we will not use this method to study the phase diagram of the problem in the presence of randomness. The full rate function contains much more information than just the value of magnetisation. We can easily use it to extract the magnetic susceptibility of the system. Magnetic susceptibility is the response of the system to external magnetic field ($\\chi=\\frac{\\partial x}{\\partial H}|_{H \\rightarrow 0}$).\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=4.4cm]{rate2pbefore.eps} &\n\\includegraphics[width=4.4cm]{rate2pafter.eps}\\\\ \n\\includegraphics[width=4.4cm]{beforeh0p08.eps} &\n\\includegraphics[width=4.4cm]{afterh0p08.eps}\\\\ \n\\end{tabular}\n\\caption{The first row shows the rate function ($I_p(x)$) as it changes across a continuous transition for $p=2$. In second row we have plotted the rate function($I_p(x)$) across a first order transition for $p=3$.}\n\\label{ratef}\n\\end{figure}\n\nIf $I_p'(x)$ gives us an equation of the form:\n\\begin{equation}\n\\label{pminima_eqn}\n\\frac{\\beta}{(p-1)!}x^{p-1}+\\beta H-\\mathcal{F}(x,\\beta,h)=0,\n\\end{equation}\nwhere $\\mathcal{F}$ contains those terms of $I_p'(x)=0$, that are independent of $H$ \nand $p$. Differentiating the above expression with respect to the external field $H$ and then taking the \nlimit $H \\rightarrow 0$, we get\n\\begin{equation}\n\\frac{\\beta}{(p-2)!}\\beta x^{p-2} \\chi +\\beta =\\frac{\\partial \\mathcal{F}}{\\partial x}\\chi\n\\end{equation}\nwhich gives,\n\\begin{equation}\n\\chi =\\frac{\\beta}{\\frac{\\partial \\mathcal{F}}{\\partial x}-\\frac{\\beta}{(p-2)!} x^{p-2}} .\n\\label{sus}\n\\end{equation}\nWe will use this expression to study the fluctuations in random field $p$-spin model in Section \\ref{sec4}.\n\n\\section{Phase-diagram}\n\\label{sec3}\nFor $p=2$, the above rate function has already been studied \\cite{lowe} and we will only briefly mention the results here. For $p=2$ one can actually study a simpler\nfunction, using the fact that the global extremal\npoints of a function $F(x)-\\frac{1}{2}||x||^2$ coincides with the global maximum points of\n$\\frac{1}{2}||x||^2-F^*(x)$, where $F^*$ is the Legendre Fenchel transform of $F(x)$ \\cite{touchette1}. Hence the minima of $I_p(x)$ will coincide with the minima of a function $G(x)$, which is equal to:\n\\begin{equation}\nG(x)=\\frac{\\beta}{2}x^2-\\frac{1}{2}\\mbox{log}(\\mbox{cosh}[\\beta (x+h)]\\mbox{cosh}[\\beta (x-h)]).\n\\end{equation}\nInterestingly this matches with the expression of free energy for random field Ising model by Salinas and Wreszinski  \\cite{salinas}. Studying the minima of this function, one finds that the system undergoes a continuous transition as long as $1-\\beta \\mbox{sech}^2 (\\beta h)<0$ and a first order transition beyond it. At equality, the system has a tricritical point at $h_{tp}= \\frac{2}{3} \\mbox{cosh}^{-1} \\sqrt{3\/2}(\\approx 0.439)$ and $\\beta=3\/2$. \n\nFor $p \\ge 3$, we found that the system either has first order transition to the ferromagnetic state or no transition at all. As shown in Fig. \\ref{ratef},  the rate function develops new minima discontinuously. We find that for all $p \\ge 2$, the system fails to order into a \nferromagnetic state, even at zero temperature, for $h>1\/p!$. For example, \nfor $p=3$, beyond $h_o \\approx 0.167$, the magnetisation in the system is always \nzero. Phase diagram for $p=3$ and $p=4$ are plotted in Fig. \\ref{figpd}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=6 cm]{phasediagram_m.eps} \n\\caption{Phase diagram for $p=3$(solid line) and $p=4$(dotted line). Below the line system is in ferromagnetic phase and above it, it is in paramagnetic phase. For $h>h_o \\approx 0.167$ for $p=3$, and for $h>h_o \\approx 0.041$ for $p=4$, the model has no ferromagnetic phase, i.e, it has zero magnetisation for all values of $\\beta$ for $h > h_o$}\n\\label{figpd}\n\\end{figure}\n\n\n\\section{Magnetic susceptibility}\n\\label{sec4}\nWe want to now look at the fluctuations in the magetisation, as given by the magnetic \nsusceptibility of the system. Note that $\\chi(x,\\beta,h)$ is a function of magnetisation itself, and hence one needs to get the value of $x$ for which $I_p(x)$ is minimum, and then substitute it in the equation of $\\chi(x,h,\\beta)$ to get the physically acceptable value of magnetic susceptibility at a given $h$ and $\\beta$.\n\n\\subsection{p=2}\nFor $p=2$, in the paramagnetic phase, $\\chi(0,\\beta,h)$ has a particularly simple expression:\n\n\\begin{equation}\n\\label{chizero}\n\\chi(0,\\beta,h) = \\frac{2 \\beta}{1-2 \\beta+ \\mbox{cosh}(2 \\beta h)}.\n\\end{equation}\nFor $h=0$, the above equation gives a divergence in $\\chi$ at $\\beta_c=1$. But for \n$h \\neq 0$, it gives two values of $\\beta$ where $\\chi(0,\\beta,h)$ diverges for all $h<0.447$. We find that for $h<h_{tp}$, the lower value of $\\beta$ is the physically relevant one, as beyond that value of $\\beta$, magnetisation is no longer zero. For $h_{tp}<h<0.447$, the $\\beta$ at which the transition occurs is less than the values of $\\beta$ that one obtains by equating $\\chi(0,\\beta,h)^{-1}$ to zero.\n\nFor $p=2$, near the continuous transition (for $h<h_{tp} \\approx0.439$), we can study the magnetic susceptibility by keeping terms only till $x^3$ in Eq.(\\ref{pminima_eqn}) by expanding $\\mathcal{F}(x,\\beta, h)$ around $x=0$. We get\n\\begin{equation}\n-\\beta H +\\frac{x}{2} (1-2 \\beta+\\mbox{cosh}(2 \\beta h)) -\\frac{x^3}{3}[\\mbox{cosh}^4(\\beta h) \\{-2+\\mbox{cosh}(2 \\beta h)\\}]=0.\n\\end{equation}\nDifferentiating with respect to $H$ and taking the limit $H \\rightarrow 0$, we get:\n\\begin{equation}\n\\label{p2chi_approx}\n\\chi(0,\\beta,h) = \\frac{2\\beta}{[1-2 \\beta +\\mbox{cosh}( 2 \\beta h)]-2 x^2 \\mbox{cosh}^4(\\beta h) [\\mbox{cosh} (2 \\beta h) -2]}.\n\\end{equation}\nPutting $h=0$ in the above equation, one gets $\\chi \\sim \\frac{1}{1-\\beta}$ for $\\beta <1$ and\n$\\chi \\sim \\frac{1}{2 (\\beta-1)}$ for $\\beta >1$, as expected. Hence, in the absence of random\nfield, we reproduce the results of standard mean-field theory. \n\nFor small values of $h$, we can further expand the cosh term and in that regime, the magnetisation is given by:\n\\begin{eqnarray}\nm &=& 0 \\hspace{1.4in} \\mbox{, for} \\hspace{0.1in} \\beta <\\beta_c \\nonumber\\\\\n&=& \\sqrt{3 (-1+\\beta-\\beta^2 h^2)}  \\hspace{0.055in} \\mbox{, for} \\hspace{0.1in} \\beta>\\beta_c,\n\\end{eqnarray}\nwhere $\\beta_c= \\frac{1\\pm \\sqrt{1-4 h^2}}{2 h^2}$.\n\nSubstituting in the expression of $\\chi$, we get, for $\\beta<\\beta_c$\n\\begin{equation}\n\\chi(0,b,h) \\sim \\frac{\\beta}{1-\\beta+\\beta^2 h^2},\n\\end{equation}\nand for $\\beta>\\beta_c$\n\\begin{equation}\n\\chi(m,b,h) \\sim \\frac{\\beta}{2(1-\\beta+\\beta^2 h^2)}.\n\\end{equation}\n\n\\begin{figure}[!htb]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=4.4cm]{p2h447m.eps} &\n\\includegraphics[width=4.4cm]{p2h49.eps}\\\\ \n\\end{tabular}\n\\caption{Plot of $\\chi$ as a function of $\\beta$ for $h>h_{tp}$}\n\\label{fig3}\n\\end{figure}\nFor $h>h_{tp}(\\approx 0.439)$, the system undergoes a first order transition.  For $h>h_{tp}$, since the system has a first order transition we have to solve Eq. \\ref{pminima_eqn} and Eq. \\ref{sus} numerically to obtain the magnetic susceptibility. We have plotted magnetic susceptibility and magnetisation for a few representative values of $h$ in Fig 3. We find that for $h_{tp} < h \\le 0.447$, $\\chi$ is rapidly increasing as one approaches the transition point from the high temperature phase. On the other hand, for $h>0.447$, $\\chi$ peaks at a value of $\\beta<\\beta_c$, and then decreases as it approaches $\\beta_c$. This difference in behaviour of magnetic susceptibility arises due to the fact that for $h_{tp}<h< 0.447$, even though the system has a first order transition, the function $\\chi(0,\\beta,h)$ still shows divergence at two values of $\\beta$. In this regime, none of them is relevant as none of them correspond to the transition point and occur well inside the ordered regime. But due to their presence, for $h_{tp}<h<0.447$, $\\chi$ still increases rapidly as one approaches $\\beta_c$.\n\n\n\\subsection{$p > 2$}\n\nFor $p=3$ and $p=4$, we numerically evaluate the expression in Eq. \\ref{sus} and find that the \n$\\chi(0,\\beta,h)$ is a convergent function for all values of $\\beta$ and $h$. Magnetic susceptibility shows a discontinuity, as expected, at the transition point. In Figs. 4 and 5,  we have plotted magnetic susceptibility  and magnetisation for a few representative values of $h$ for $p=3$, and $p=4$, respectively. For low $\\beta$, below the transition, $\\chi \\propto \\beta$ as expected for a paramagnet. Interestingly for $h>1\/p!$, even though the system \ndoes not order ferromagnetically, magnetic susceptibility shows a non monotonic behaviour \nas a function of $\\beta$ as shown in Fig \\ref{fig4}. Similar behaviour of magnetic \nsusceptibility is also seen for $p=4$ beyond $h>0.041$ (Fig. \\ref{fig5}).\n\n\\begin{figure}[!htb]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=4cm]{p3chi+magh0p12.eps} &\n\\includegraphics[width=4cm]{p3chi+magh0p21.eps}\\\\ \n\\end{tabular}\n\\caption{$m$ and $\\chi$ for $p=3$ for $h<0.167$ and for $h>0.167$.}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=4.4cm]{p4chi+magh0p03.eps} &\n\\includegraphics[width=4.4cm]{p4chi+magh0p08.eps}\\\\ \n\\end{tabular}\n\\caption{$m$ and $\\chi$ for $p=4$ for $h<0.041$ and for $h>0.041$.}\n\\label{fig5}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec5}\nThe nature of phase transition of $p$-spin random field model is qualitatively different \nfor $p=2$ and for $p>2$. For $p=3$, the random field model has been solved for spherical \nspins earlier \\cite{haddad}. We solve it here for Ising spins, and find that for all $p \\ge 2$, for high enough strength of the random field, Ising spins do not order into a ferromagnetic phase, even at zero temperature. Magnetic susceptibility for these strengths of fields also shows a non monotonic behaviour, peaking at a finite $\\beta$. The method can be used starightforwardly to study other random distributions, though for $p \\ge 3$, we do not expect any qualitative difference in the behaviour of the model for different distributions.\n\n \n\nWe also notice that the nature of fluctuations near the transition point is different for the first order transition when $p=2$, from that of $p \\ge 3$. For $p=2$ there exists a range of $h$ for \nwhich even though the system undergoes a first order transition, $\\chi$ increases rapidly as one approaches the transition. It would be interesting to see if the similar behaviour continues in finite dimensions. In three dimension, RFIM is known to have strong finite size effects which makes it difficult to acertain the nature of transition for the higher strengths of the random field \\cite{fytas,sourlas}. Besides that, Eq. \\ref{eqn_p2_random_rate_final} can also give us some indications for writing down the correct field theory for the model, which in turn could be useful in understanding the behaviour of randomness near a first order phase transition \\cite{cardy1,cardy2}. It should also be possible to extend the method to study the effect of random field in other models like  Potts and Blume-Capel model \\cite{sumedha}.\n\n\\section{Acknowledgements}\nSushant K Singh would like to acknowledge the local hospitality and support received at NISER  Bhubaneswar, while part of this work was done.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nGalaxies exhibit a wide range of phenomena, some of which appear to\nhave been more important in the distant Universe than today. The\nglobal co-moving star-formation rate was higher by an order of\nmagnitude \\citep{madau96}, and so were the specific star-formation\nrates \\citep{elbaz07, daddi07}. Morphologies were increasingly\nirregular, an observation that cannot be explained with extinction\nalone \\citep{conselice08}. Processes like starburst-driven outflows\nare obvious and ubiquitous in samples of high redshift galaxies\n\\citep[e.g.,][]{shapley03, steidel10} compared to only a small fraction\nof galaxies today \\citep{lehnert96}. The merger rate is also likely to be\nmuch higher \\citep{ravel09}. The gas infall rates of nearby galaxies are\napparently low, but they were likely much higher in the early Universe\n\\citep{croton06}. Today, the Universe can be considered in the ``age\nof secular evolution'', but until about 8 Gyrs ago it was in the ``age\nof feedback and self-regulation''. It is the mix of physical processes\nduring their early evolution that shaped the characteristics of the\nensemble of the galaxies we observe today.\n\nUnderstanding this complexity and the relative contribution of the\nphysical processes that shaped distant galaxies is required before we can\nconsider our understanding of galaxies in any sense complete. With the\nhigher activity levels of galaxies at z$\\sim$1-5 relative to today it is\ndifficult to gauge the importance of each process by mere analogy. This\nrequires careful, direct observations of distant galaxies.\n\nOf significant concern in this pursuit is that obtaining detailed spatial\ninformation from distant galaxies is hampered by the large impact of\ncosmological surface brightness dimming, e.g. by a factor $\\sim$150 for\nz=2.5. This implies that only the highest surface brightness galaxies\nare spatially resolved with integral field spectroscopy or in continuum\nimaging. It also biases the samples where we can detect spatially extended\nemission line gas to galaxies which have intense star formation over many\ntens of kpc$^2$ or extended emission line regions excited by powerful\nactive galactic nuclei \\citep[e.g.,][]{nesvadba06, nesvadba08}.\n\nObservations of intensely star-forming galaxies at z$\\sim$2 with bright,\nextended line emission reveal remarkably broad line widths implying\nimportant random, and presumably (at least partially) turbulent,\ngas motions. Limited by the $\\sim$ 1-few kpc resolution of current\nspectrographs, the observed line widths reflect the blended bulk\nand turbulent gas motions on kpc scales. In galaxies where velocity\ngradients are spatially resolved, both motions have very similar\namplitudes, unlike in nearby galaxies where line widths are much\nsmaller than circular velocities \\citep{law07,law09,fs09, epinat09,\nepinat10, epinat12}. This requires either a more intense, or perhaps\nmore efficient, source or an additional source of turbulent energy at\nhigh redshift. The possible sources for  energizing the turbulence\nremain the subject of significant controversy. Some authors suggest\nthat the turbulent cascade is primarily initiated by shear and large\nscale gravitational or fluid instabilities, perhaps further energized by\ncosmological gas accretion \\citep{Brooks09,Keres09,Dekel09,elmegreen10},\nwhile others give a significant role to the energy injection from young\nstars \\citep[e.g.,][]{elmegreen04, cox05, ferrara93, norman96}. However,\nthe dichotomy between gravitational and stellar sources of energy is\nlargely artificial as both no doubt play important roles. Supersonic\nturbulence can provide global support  to the ISM, but it naturally\nproduces density enhancements that may collapse locally and may\nsubsequently form stars or dissolve. Global star formation in galaxies\nappears to be controlled by the balance between gravity and turbulence\n\\citep{maclow04}. This balance is moderated by the importance of each\nform of energy injection, which on the largest scales depends on the\nrate and scale at which it is injected, and on how rapidly the energy\ncascades and dissipates in various phases, and over what time and size\nscales \\citep[e.g.,][]{elmegreen04}. None of these variables are well\nconstrained in distant galaxies. To some extent, the global regulation\nof star formation mirrors that taking place on the smallest scales,\nwhere the dissipation of turbulent energy in self-gravitating clouds is\nimportant for regulating star formation within these clouds.\n\n\\citet{L09} found that the velocity dispersions, $\\sigma$, of optical\nemission-line gas in intensely star-forming galaxies at z$\\sim$2 scale\nwith the star-formation rate per unit area (star-formation intensity,\n$\\Sigma_{\\rm SFR}$) as $\\sigma$ $\\propto$ $\\Sigma_{\\rm SFR}^{0.5}$\n\\citep[see also][]{swinbank12, MD13}. They suggested that at a very\nsimple level, such a relationship would be expected if the lines\nwere broadened by the mechanical energy injection from young stars\n\\citep{dib06}. Subsequently, \\citet{Green10} came to similar conclusions\nfrom observing strongly star-forming galaxies at low redshifts which\nhad similar overall H$\\alpha$ luminosities as galaxies observed at\nz$\\sim$2 with integral field spectroscopy \\citep{L09}. More recently,\nand through an analytical approach, \\citet{ostriker11} found that such a\nrelationship might be the natural outcome of dense ($\\Sigma_{\\rm gas}$\n$\\sim$few 100 M$_{\\sun}$ pc$^{-2}$), star-forming gas disks which are\nregulated through the momentum input from star formation. Compared to the\nother possible sources such as gas accretion, star-formation intensity in\nhigh-z galaxies can be constrained observationally, e.g. by measuring\nH$\\alpha$ luminosities and spatial distributions, although heavily\nextincted regions may have their intensities underestimated.\n\nA number of authors subsequently pointed out that this relationship\nis also consistent with the Toomre disk stability criterion\n\\citep[e.g.,][]{krumholz10}. Although not discussed in their work,\na Toomre parameter Q$\\sim$1 characteristic of gas disks which are\nmarginally unstable against star formation leads to a relationship of\nthe type observed, $\\sigma$ $\\propto$ $\\Sigma_{\\rm SFR}^{0.6-0.7}$, if\nthese galaxies follow the Schmidt-Kennicutt relation. \\citet{burkert10}\ndevelop this idea further by arguing that disks must stay close to the\ngravitational stability line. They suggest that large scale gravitational\ninstabilities will generate density and velocity irregularities which\ndrive turbulence and heat the gaseous disk. This process saturates\nclose to the instability line and thus, \\citet{burkert10} argue that\ngravitational forcing is sufficient to explain the highly turbulent\ngas. \\citet{elmegreen10} find that their analytical model of intensely\nstar-forming, fragmenting disks can reproduce the observed velocity\ndispersions in high redshift galaxies if Q is set equal to  1, suggesting\nthat whatever the mechanism responsible for driving galaxies towards\nthe line of instability, it can reproduce the observations.\n\nThe basic difficulty is that observing velocity gradients and line\nwidths consistent with marginally Toomre-stable disks only implies\nthat distant star-forming galaxies are in a state where they can form\nstars intensely, but it does not purport to explain the source of the\nturbulence. Galaxies at z$\\sim$2 appear to maintain their high gas\nturbulence over long star-formation time scales of several $10^8$ yrs,\nand maintaining Q$\\sim$1 over similar timescales is only possible if\nturbulent dissipation is balanced by the injection of turbulent energy\nover similar length of time. Since turbulent energy dissipates rapidly,\non only a dynamical time \\citep[][]{maclow99}, this seems difficult\nto ensure with gravity alone over the extended star-formation periods\n\\citep{elmegreen10}. Moreover, gravitationally-driven processes alone\nare perhaps insufficient for producing even the relatively modest\ndispersion observed in the warm neutral and ionized medium of nearby\ngalaxies \\citep{tamburro09, leroy08}. Correlations between star-formation\nrate and \\mbox{H{\\sc i}}\\ velocity dispersions in nearby galaxies \\citep{tamburro09}\nas well as the decrease in CO line dispersions with radius in some nearby\ngalaxies \\citep{wilson11} favor an important, but perhaps not dominant,\ncontribution from the energy injection by  supernovae and stellar\nwinds in regulating the turbulence of the ISM in nearby galaxies. The\nsame also appears true for intensely star-forming relatively nearby\ngalaxies with properties similar to high redshift Lyman break galaxies\n \\citep{basu-zych09, goncalves10}.\n\nHere we present new observational data on how star formation intensities\nand gas kinematics in high-redshift galaxies are related, extending our\ninitial sample from 11 galaxies to 53, and covering a larger redshift\nrange. Apart from putting our previous claims on statistically more\nrobust grounds this also allows us to demonstrate the effects of\nsurface-brightness dimming directly. We discuss the inevitable biases\nin all studies of star formation in high-redshift galaxies related\nto surface-brightness dimming and beam-smearing from our data, and\ndemonstrate that our results are not dominated by these effects. With\nthis larger sample we confirm the results of \\citet{L09} that the ISM in\nthe observed high-redshift galaxies has very high pressures compared to\ngalaxies at low redshift. We argue that this enables star formation in\nthese galaxies to be self-regulated, and demonstrate that self-regulation\nleads very naturally to a Toomre parameter, Q$\\sim$1, generalizing the\nanalytical model first presented by \\citet{elmegreen10}. We further\npaint a picture of what the optical emission line gas may be showing\nus about the turbulent cascade and its impact on star formation in high\nredshift galaxies.\n\nThroughout the paper we adopt a flat H$_0 =$70 km s$^{-1}$ Mpc$^{-3}$\ncosmology with $\\Omega_{\\Lambda} = 0.7$ and $\\Omega_{M}= 0.3$.\n\n\\section{Source characteristics}\\label{sec:observations}\n\nOur sample of 53 galaxies was observed with SINFONI on the\nESO-VLT.  Their basic properties of the 53 galaxies are presented in\nTable~\\ref{table:properties}. They span a redshift range of z=1.3-2.7\n(Fig~\\ref{fig:histo_z}). For details of the observations of the galaxies\nin this sample see the summaries in e.g. \\citet{L09, fs06, fs09}. All were\nreduced by us in a similar way as previously described by \\citet[][]{L09}.\n\nThe data, which were retrieved from the ESO archives, were taken for\na variety of programs and the selection of the galaxies is rather\ninhomogeneous. As we are not performing a population study, for the\npurposes of this paper, this is adequate.  The key to our sample is that\nall galaxies have sufficiently high H$\\alpha$ surface brightness to obtain\nspatially resolved line maps in a few hours integration time (which is\ntypical for SINFONI observations of distant galaxies). Because of this,\nour results are only applicable to galaxies with high star-formation\nintensity (star-formation rate per unit area) which necessitates a\nprocess that is able to drive intense star-formation for 100 Myrs or\nmore \\citep{erb06} and my introduce a bias. However, it should\nbe pointed out that the characteristics of the ensemble of galaxies\nstudied here are similar to those of the general distant star-forming\ngalaxy population.  Specifically, the galaxies in our sample for which\nthe specific star formation rate has been estimated \\citep[a majority;\nsee ][and references therein]{fs09,fs11,genzel11,vergani12}, lie along\nor near the ``main sequence of star formation'' for redshifts between\n1 and 2 \\citep[e.g.][]{elbaz07, daddi07, elbaz11}.\n\n\\begin{figure}\n\\includegraphics[width=9.0cm]{histo_z.eps}\n\\caption{Redshift distribution of our sample of galaxies. The two\nobserved redshift ranges of the H$\\alpha$ and \\mbox{[N{\\sc ii}]} $\\lambda$6583 lines\n(z$<$1.8 and z$>$2)are due to atmospheric transmission and the limited\nwavelength coverage of the H- and K-bands.}\n\\label{fig:histo_z}\n\\end{figure}\n\nThe observations reach surface brightness detection limits in H$\\alpha$\nof $\\sim$2$\\times$10$^{-19}$ erg s$^{-1}$ cm$^{-2}$ pixel$^{-1}$ (for\nthe 125~mas pixel$^{-1}$ scale and averaged over 3$\\times$3 pixels;\n``mas'' is milliarcseconds). The spectral resolution is FWHM$\\sim$115 and\n$\\sim$150 km s$^{-1}$ in the K and H bands, respectively. The observed\nsurface brightnesses range from about 3.6-34 $\\times$ 10$^{-18}$ erg\ncm$^{-2}$ s$^{-1}$ arcsec$^{-2}$ and the objects have isophotal radii,\nat their surface brightness detection limits, of 1-2 arcsec$^2$,\nor on average $\\sim$7$\\pm$2 kpc (corresponding to an isophotal area\nof $\\sim$150$\\pm$40 kpc$^2$). The point-spread function FWHM of the\ndata is $\\sim$\\as{0}{6}, which at z=2 represents an area of $\\sim$20\nkpc$^2$, so we have generally $\\sim$8 spatial resolution elements per\nobject. With such low spatial resolution, it is necessary to discuss the\nimpact of beam smearing on the characteristics of our data (discussed\nin detail in \\S~\\ref{sec:BSanalysis}). In the present analysis, we will\nformally define the isophotal radius, r$_{\\rm iso}$, as r$_{\\rm iso}$ =\n(A$_{\\rm iso}$\/$\\pi$)$^{1\/2}$, where A$_{\\rm iso}$ is the isophotal area,\ni.e. the total area of all pixels above the 3-$\\sigma$ surface brightness\nlimit of the data (Table~\\ref{table:properties}).\n\nThe total H$\\alpha$ luminosities of all galaxies in our sample are\nabove 10$^{41}$ erg s$^{-1}$ and most are above $\\sim$10$^{42}$\nerg s$^{-1}$. H$\\alpha$ luminosities in this range imply an average\nstar-formation rate of $\\sim$70 M$_{\\sun}$ yr$^{-1}$. However, as these\nare not corrected for extinction our estimates are lower limits. For a\nsub-sample of these galaxies we have H$\\beta$ estimates, and the line\nratios of H$\\alpha$ to H$\\beta$ suggest relatively modest extinction\ncorrections to the star-formation rates, of a factor of few \\citep{L09}.\n\n\\section{Surface brightness limits and selection effects}\\label{sec:SBlimits}\n\nDue to cosmological surface brightness dimming, over the redshift range\nof a factor $\\sim$2 in our sample there is about a factor of 5 difference\nin the faintest rest-frame line emission surface brightness levels that\ncan be probed, whereas the physical dimensions in the rest frame, i.e.,\nlength per arcsec, changes by only about 5\\%. Given this strong impact\nof cosmological surface brightness dimming, it is worth repeating the\nconclusion from \\cite{L09} that even the lowest redshift sources in our\nSINFONI sample are extreme compared to galaxies in the local volume,\nwhere only the most intense starbursts have such high H$\\alpha$ surface\nbrightness, and this on considerably smaller physical scales only.\n\nThe total range in surface brightness probed by the SINFONI observations\nof our ensemble of galaxies is $\\approx$20 (Fig.~\\ref{fig:SBvsPos}). For\npurposes of this comparison and others, we constructed 2-dimensional\nhistograms which show the frequency of occurrence of various pairs of\nvalues in all pixels of our data sets, after smearing to the spatial\nand spectral resolution of the real SINFONI observations -- for example,\nsurface brightness versus projected distance or \\mbox{H$\\alpha$}  line width versus\nstar-formation rate intensity. Using the frequency of occurrence is a\nclearer way of visualizing various relationships than showing all single\npixels and emphasizes the continuity of the data.\n\nTo investigate the dependence of the range of surface brightnesses probed\nby our data and its influence on the observed galaxy isophotal size, we\ndivided our galaxies into 4 bins of equal galaxy number based on their\nisophotal areas (Fig.~\\ref{fig:SBvsPos_Npix}). For the 3 bins with the\nlargest average isophotal sizes ($\\geq$1.4 arcsec$^2$), the observed\ndynamic range in surface brightness goes up to $\\sim$20 and is typically\nabout 15. However, for the apparently smaller galaxies, the dynamic range\nof the observations is only about a factor of 3. The relationship between\nsurface brightnesses and projected distance is the same for all bins,\nindicating that their individual declines in surface brightness follow\nthe same trend. The observations imply two things. First, although the\neffects of beam smearing cannot be ignored, as we will discuss extensively\nin the next sections, it does not dominate the overall distribution of\nthe surface brightness (i.e., the distribution does not simply represent\nthe point-spread-function, PSF, of the observations). Second, it implies\nthat the reason the apparently small galaxies appear small is that the\nsurface brightness detection limits of the observations are too high to\ndetect their extended emission. As noted above, this is not simply due to\nthe physical scale of the galaxies changing across our redshift range,\nas this effect amounts to only $\\sim$5\\%. While we do not show it here,\nthere is a crude trend for the galaxies with the highest total \\mbox{H$\\alpha$}  fluxes\nto also have the largest isophotal sizes. Again, this suggests that we\nare limited by the overall dynamic range of the data and by the impact\nof surface brightness dimming, but not by differences in the intrinsic\nisophotal sizes of the galaxies (Fig.~\\ref{fig:surflims}). This is an\nimportant point to bear in mind when reading this paper.\n\n\\begin{figure}\n\\includegraphics[width=9.0cm]{SBvsPos_blend_all.eps}\n\\caption{A 2-dimensional histogram of the frequency of occurrence of pixel values of H$\\alpha$ surface brightness versus projected physical distance for the entire sample. As shown in the color bar, the darkest regions have the highest frequency of occurrence. The surface brightness has not been corrected for cosmological dimming. The zero of the projected distance was chosen to be the symmetry point in either the H$\\alpha$ velocity field or of 3-$\\sigma$ isophotes in the H$\\alpha$ surface brightness map.}\n\\label{fig:SBvsPos}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=17.0cm]{SBvsPos_blend_norm_Npix_multi.eps}\n\\caption{Two-dimensional histograms of the relative frequency of\noccurrence of normalized H$\\alpha$ surface brightness values versus the\nprojected distance for galaxies in 4 equal-size bins in galaxy number\nselected on their isophotal area (A$_{\\rm iso}$). The relative frequency\nof occurrence is scaled in the same way as in Fig.~\\ref{fig:SBvsPos}.\nFrom left to right, each panel represents an increasing angular size\n(i.e., A$_{\\rm iso}$ $<$ 1.4 arcsec$^2$, 1.4 arcsec$^2$ $\\geq$A$_{\\rm\niso}$ $<$ 2.5 arcsec$^2$, 2.5 arcsec$^2$ $\\geq$A$_{\\rm iso}$ $<$\n3.3 arcsec$^2$ and A$_{\\rm iso}$ $\\geq$3.3 arcsec$^2$). The values\nof r$_{\\rm iso}$, which is derived from A$_{\\rm iso}$, for individual\ngalaxies are given in Table~\\ref{table:properties}. The galaxies with\nthe smaller dynamic range (higher surface brightness detection limits)\nin the data are also smaller in isophotal size.}\n\\label{fig:SBvsPos_Npix}\n\\end{figure*}\n\nFurthermore, if the trends in surface brightness and total \\mbox{H$\\alpha$}  luminosity\nwere dominated by cosmological surface brightness dimming, which is\nproportional to (1+z)$^4$, we would expect our data to show a slope\nof about 4 in a plot of the logarithmic relation between H$\\alpha$\nluminosity and isophotal area and indeed, we find such a trend\n(Fig.~\\ref{fig:surflims}). The range in surface brightness detection\nlimits in our data trace out the ranges in size and luminosity we observe\nin our sample of galaxies. In addition, this confirms that at the high\nredshift end of our sample the intrinsic surface brightnesses are very\nhigh and that they generally decline with redshift.\n\n\\begin{figure}\n\\includegraphics[width=9.0cm]{surfacebrightnesslimits.eps}\n\\caption{Plot of H$\\alpha$ luminosities versus isophotal area for the galaxies in our sample (red circles). The size of each circle is proportional to the redshift of the galaxy, with larger circles denoting higher redshifts (range in z is 1.3--2.7). The two solid lines show the relationship between luminosity and area for the average surface brightness detection level of galaxies, 1.1$\\times$10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ (black for galaxies at z=2.2 and red for z=1.3, respectively). The dot-dashed line shows the relationship for the minimum surface brightness detection limit of 3.6$\\times$10$^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ for all galaxies with z$\\approx$2.2. See\nTable~\\ref{table:properties} for the \\mbox{H$\\alpha$}  luminosities, isophotal radii and surface brightness detection limits.}\n\\label{fig:surflims}\n\\end{figure}\n\n\\section{The effect of beam smearing}\\label{sec:BSanalysis}\n\nIn \\citet{L09}, we proposed that the H$\\alpha$ emission line velocity\ndispersions of the high surface brightness spatially-resolved galaxies\nobserved with SINFONI (and other near-infrared integral field units)\ncould be explained by the mechanical output of young stellar populations\nwithin those galaxies. In fact, we framed the argument in the context\nof an overly simplified energy injection model of the form, $\\sigma$=\n($\\epsilon$$\\Sigma_{\\rm SFR})^{1\/2}$, where $\\sigma$ is the H$\\alpha$\nvelocity dispersion, $\\epsilon$ is the coupling efficiency of the\nmechanical energy output to the interstellar medium and $\\Sigma_{\\rm SFR}$\nis the star-formation intensity. The energy due to the young stars is\ngiven by population synthesis models \\citep{leitherer99}. Using scaling\nrelations based on models and observations of nearby galaxies, we were\nable to explain the trends between dispersion and star-formation intensity\nwith no free parameters \\citep[see ][ for details of the scalings used\nin this analysis]{L09}.\n\n\\begin{figure*}\n\\setlength{\\unitlength}{6cm}\n\\begin{picture}(3,1)\n\\put(0, 0){\\special{psfile=_sigdimat_large_60kpc_smallV_sfr16_sfr__Halpha_Flux.eps hscale=50 vscale=50}}\n\\put(1, 0){\\special{psfile=_sigdimat_large_60kpc_smallV_sfr16_sfr__Halpha_Velocity.eps hscale=50 vscale=50}}\n\\put(2, 0){\\special{psfile=_sigdimat_large_60kpc_smallV_sfr16_sfr__Halpha_Dispersion.eps hscale=50 vscale=50}}\n\\end{picture}\n\\caption{Output from the simulation of a star-forming disk galaxy\nas it would appear at z=2 before it was beam smeared. Shown are the\ndistribution of the star-formation rate intensity (left panel), \\mbox{H$\\alpha$} \nvelocity field (center panel) and \\mbox{H$\\alpha$}  dispersion map (right panel). The\ngalaxy is projected at an inclination angle of 45$^{\\circ}$ relative\nto face-on. The scale on the axes is in kpc and the kinematic axis is\nhorizontal. Despite the different units (kpc vs. arc seconds), we note\nthat the sizes of these panels and those in Fig.~\\ref{fig:analogs}\nare similar, for ease of direct comparison.}\n\\label{fig:simus}\n\\end{figure*}\n\nHowever, the limited spatial resolution of integral field spectroscopy\nof z$\\sim$2 galaxies causes an artificial broadening of the emission\nlines due to large-scale kinematics, which can result in a significant\nradial velocity gradient across each pixel. While we argued that this\nwould not have an important impact on our results in \\citet{L09} based\non some simple modeling, we now feel it is important to revisit this\nissue in a more complete and detailed way.\n\n\\subsection{Construction of artificial IFU observations}\n\nIn order to study the effect of beam smearing, we used the simulation\nof an isolated galaxy already presented in \\citet{diMatteo08} which was\nmade using a Tree-SPH code \\citep [see ][ for a description]{seme02,\ndimatteo09, chili10}. This model has a high gas mass fraction (50\\%)\nand the galaxy was evolved in isolation, without companions or tidal\ninteractions \\citep{qu11}. We allowed the disk to go unstable against\nstar-formation and it developed a ``clumpy'' morphology which evolved\nwith time \\citep[see][for details of the initial conditions and evolution\nof the simulation]{diMatteo08}. Intrinsically, our simulations show\nlow gas velocity dispersions, of-order 10 km s$^{-1}$ (or less), which\nare roughly constant with radius and a rotation speed of $\\sim$220 km\ns$^{-1}$ which implies a v\/$\\sigma$$\\approx$20. From this simulation,\nwe produced maps of the star-formation intensity, radial velocity field\nand velocity dispersion (Fig.~\\ref{fig:simus}). For our primary analysis,\nthe simulation was viewed at an inclination angle of 45$^{\\circ}$ and\nscaled to have a projected rotational amplitude of 110 km s$^{-1}$,\na value derived from the average intrinsic full width at half maximum\nof the integrated spectra of our sample. This implies a rotation speed\nof about 160 km s$^{-1}$.\n\nWe then degraded the resolution of the simulations to produce synthetic\nobservations similar to what we observed with SINFONI. We adopted a\nsingle redshift (z=2) for determining the scalings in the synthetic\nobservations -- the exact redshift chosen does not matter much as the\nphysical scale per angular projected size (kpc arcsec$^{-1}$) changes\nlittle ($\\sim$5\\%) over the redshift range of our sources. We also took\ninto account the instrumental spectral resolution, the seeing using a\nFWHM for a Gaussian distributed PSF of \\as{0}{6} and added an amount\nof noise to the synthetic data consistent with the observations. The\nartificial datacubes created in this way were then analyzed using the\nsame procedures as the real SINFONI observations.\n\n\\begin{figure*}\n\\setlength{\\unitlength}{6cm}\n\\begin{picture}(3,2)\n\\put(0, 1){\\special{psfile=_sigdimat_large_60kpc_smallV_sfr16_sfr__NoisyX2_Halpha_Flux.eps hscale=50 vscale=50}}\n\\put(1, 1){\\special{psfile=_sigdimat_large_60kpc_smallV_sfr16_sfr__NoisyX2_Halpha_Velocity.eps hscale=50 vscale=50}}\n\\put(2, 1){\\special{psfile=_sigdimat_large_60kpc_smallV_sfr16_sfr__NoisyX2_Halpha_Dispersion.eps hscale=50 vscale=50}}\n\\put(0, 0){\\special{psfile=_siglehnert_large_60kpc_smallV_sfr16_sfr__NoisyX2_Halpha_Flux.eps hscale=50 vscale=50}}\n\\put(1, 0){\\special{psfile=_siglehnert_large_60kpc_smallV_sfr16_sfr__NoisyX2_Halpha_Velocity.eps hscale=50 vscale=50}}\n\\put(2, 0){\\special{psfile=_siglehnert_large_60kpc_smallV_sfr16_sfr__NoisyX2_Halpha_Dispersion.eps hscale=50 vscale=50}}\n\\end{picture}\n\\caption{Results of the simulated data sets of a star-forming disk galaxy\nat z=2 after beam smearing using different relationships for the velocity\ndispersions. Shown are the distribution of the \\mbox{H$\\alpha$}  surface brightness\n(left), velocity field (middle) and velocity dispersion map (right). In\nthe upper row, we have simply taken the velocity dispersions as estimated\nfrom the N-body\/SPH simulation, while for the lower row we assumed a\nrelationship between the velocity dispersion and star-formation intensity\nof the form $\\sigma=140\\sqrt{}\\Sigma_{\\rm SFR}$ km s$^{-1}$. The circle\nin the upper right of each plot represents the FWHM of the PSF used to\nmake the artificial data set. The scales are in arc seconds. We note\nthat we have scaled the images such that the sizes of these panels and\nthose in Fig.~\\ref{fig:simus} are similar, for ease of direct comparison.}\n\\label{fig:analogs}\n\\end{figure*}\n\nIn the following analysis, we will focus on the impact of beam smearing on\nthe interpretation of the underlying cause of the high dispersion observed\nin distant galaxies with spatially-resolved line emission observations.\n\nIn \\citealt{L09} we proposed that the dispersion is proportional to\nthe star-formation intensity. For our analysis of the effects of beam\nsmearing, we make a comparison between the analogs produced using two sets\nof scaling relations for the velocity dispersions: (1) the dispersions\ntaken directly from the simulation, $\\sigma_{\\rm sim}$, and (2) a function\nof the star formation rate intensity $\\sigma=140\\sqrt{}\\Sigma_{\\rm SFR}$\nkm s$^{-1}$-- see \\cite{L09} for details. The resulting \\mbox{H$\\alpha$}  flux and\nkinematics maps made with our artificial SINFONI data sets and applying\nthese two different types of dispersion show, as expected, that the\nsurface brightness distribution is now smoother than in the original\nsimulation (Fig.~\\ref{fig:analogs}). The star-formation intensities --\nthe star-formation rate per unit area -- are related to the \\mbox{H$\\alpha$}  surface\nbrightness by a simple scaling \\citep[e.g.][]{kennicutt98}. We use this\nscaling to produce the \\mbox{H$\\alpha$}  surface brightness distribution which is\nsubsequently beam smeared.\n\n\\subsection{Effect of beam smearing on star-formation intensities}\n\nThe areas of intrinsically high star formation intensity are\nblended after smearing the data, which lowers their intensities\n(Fig.~\\ref{fig:analogs}). Whether or not such peaks in the star-formation\nintensity can be observed in principle depends on their relative\ndistribution and peak intensity relative to their surroundings within one\nor two PSFs (cf. Figs.~\\ref{fig:simus} and \\ref{fig:analogs}). However,\nFig.~\\ref{fig:SBvsPos_comp} demonstrates that, after smearing, the\nsimulated final radial distribution of surface brightness is consistent\nwith what we observe. The similarity seen does not depend on the\nunderlying velocity dispersion distribution assumed in the construction\nof the synthetic data.\n\nThe star-formation intensities from the simulation decline by $\\stackrel{<}{_{\\sim}}$25\\%\nafter beam smearing. While the precise value of this decrease is likely\nto depend on the initial spatial distribution of the surface brightness,\nit is always going to be small. The total star-formation rate, which\nin the simulations is typically about 60 M$_{\\sun}$ yr$^{-1}$, remains\nessentially unchanged after making an artificial SINFONI data set. The\nlargest impact of smearing is on the distribution of the star-formation\nintensities and in losing the lowest surface brightness emission because\nof noise and the relatively low dynamic range of the data.\n\n\\subsection{Effect of beam smearing on velocity dispersions}\n\nSince we have assumed two quite different relationships between velocity\ndispersions and star-formation intensity, it is in the resulting velocity\ndispersions that we see the most dramatic differences. In the model\nwith $\\sigma_{\\rm sim}$, the dispersion is low (a few to 10 km s$^{-1}$)\nand almost constant as a function of radius and star-formation intensity\n(Fig.~\\ref{fig:comp}, left panel). In the synthetic observations however,\nthe beam smearing by the PSF (FWHM=\\as{0}{6}) and the low spectral\nresolution of the data broadens all lines by about $\\sim$40 $\\mbox{{\\rm km}\\,{\\rm s}$^{-1}$}$\nand the noise in the data increases the scatter about this mean rise\nin the dispersions. Even though we correct for the intrinsic resolution\nof the spectrograph, at relatively low S\/N we tend to over-estimate the\nwidths of lines that are only marginally resolved. Our results suggest\nthat for galaxies with roughly constant and low dispersions ($\\sim$40\nkm s$^{-1}$), a spectrograph like SINFONI will tend to over-estimate\nthe intrinsic widths of the lines but still produce a roughly constant\ntrend of dispersion with star-formation intensity.\n\nFor the model where the simulation dispersions are scaled as\n$\\sigma=140\\sqrt{}\\Sigma_{\\rm SFR}$ km s$^{-1}$, there are almost no\ndifferences between the mean velocity dispersions from the simulation\nitself and the artificial data sets (Fig.~\\ref{fig:comp}, right\npanel). Because of the large range of the scaled velocity dispersions,\nthere is little effect of beam smearing on the average velocity\ndispersion. Looking at the pixel-by-pixel quantities, we obviously\nlose the relatively rare regions and peaks with the highest velocity\ndispersions (over 200 $\\mbox{{\\rm km}\\,{\\rm s}$^{-1}$}$). However, as suggested by the average and\nintrinsic dispersions, the decrease in the values (ranges and averages)\nis not large, even after the smoothing to low spatial and relatively low\nspectral resolution. Again, because of the loss of the lowest surface\nbrightness regions, we also tend to lose information on the areas with\nthe lowest velocity dispersions. However, in spite of all of this, the\ngeneral trend for higher star-formation intensity to result in higher\ndispersion is maintained.\n\nWe also halved the signal-to-noise ratio and used the intrinsic\nrotation speed of the simulated galaxy keeping the inclination angle at\n45$\\deg$. In both cases, the overall trend in the data is maintained and\nthe increase in the scatter in the star-formation intensity-dispersion\nplane is not significant.\n\n\\begin{figure}\n\\includegraphics[width=9.0cm]{SBvsPos_blend_norm_sigdimat_large_60kpc_smallV_sfr16_sfr__NoisyX2__all.eps}\n\\caption{Comparison of the relative frequency of occurrence of pixel\nvalues of normalized per pixel \\mbox{H$\\alpha$}  surface brightness distributions of\nthe whole sample of galaxies (in shades of red) with superimposed an\nartificial SINFONI data set constructed from an N-body\/SPH simulation\n(blue crosses). Both observations and simulation follow similar\ndistributions.} \n\\label{fig:SBvsPos_comp} \n\\end{figure}\n\n\\subsection{Are the dispersions related to H$\\alpha$ surface brightness?}\n\nWe now compare the observed trend between star-formation intensity and\n\\mbox{H$\\alpha$}  line velocity dispersion with the models we used to make synthetic\ndata sets, including the effects of beam smearing. Fig.~\\ref{fig:blend}\nshows that the values of the simulated \\mbox{H$\\alpha$}  line dispersion $\\sigma_{\\rm\nsim}$ are too low to explain the large line widths we generally observe\nin our sample of galaxies. On the other hand, the simulation using the\nrelationship, $\\sigma\\propto\\Sigma_{\\rm SFR}^{1\/2}$, is a good match to\nthe overall distribution of the data.  We note that more properly,\na constant term should perhaps be included in the relationship, as\n$\\sigma\\propto\\Sigma_{\\rm SFR}^{1\/2}$ + constant,  which is both warranted\nby observations and expected theoretically \\citep{dib06}.  At star\nformation intensities of order 10$^{-4}$ to 10$^{-5}$ M$_{\\sun}$ yr$^{-1}$\nkpc$^{-2}$, the observed velocity dispersion of the H$\\alpha$ emission\nline are $\\sim$5-20 km s$^{-1}$ \\citep{faithi07, erroz-ferrer12}. We\nhave chosen to ignore this constant in our analysis for two reasons.\nFirst, velocities this small are below the resolution of the SINFONI\ndata. Thus we are unable to probe the characteristics of galaxies with\nlow star formation intensities.  Second, the star formation intensities\nof the galaxies we are observing are several orders of magnitude greater\nand this including a small constant term would have an insignificant\nimpact on our analysis or conclusions.\n\n\\begin{figure}\n\\setlength{\\unitlength}{4.25cm}\n\\centering\n\\begin{picture}(2,1)\n\\put(0, 0){\\special{psfile=dimat_SFRdvsSig.eps hscale=20 vscale=20}}\n\\put(1, 0){\\special{psfile=lehnert_SFRdvsSig.eps hscale=20 vscale=20}}\n\\end{picture}\n\\caption{In our analysis of the impact of beam smearing on the\nvelocity dispersion and star-formation intensity, we adopted two\nrelationshipsbetween these two quantities. One taken directly from\nthe simulations ({\\it left panel}), and another where we assumed\n$\\sigma\\propto\\Sigma_{\\rm SFR}^{1\/2}$ ({\\it right panel}). The model\nvelocity dispersions are shown in red (these are not lines but many\nindividual points). We also show the points (blue) after analyzing the\nartificial data cubes derived from the simulation which include the\neffects of beam smearing. The red $+$ (red $\\times$) sign indicates the\nmean star-formation intensity and integrated (original input) velocity\ndispersion for the intrinsic relation between velocity dispersion\nand star-formation intensity, while the black $+$ (black $\\times$)\nsign has the same meaning but now for the beam-smeared relationship.}\n\\label{fig:comp} \n\\end{figure}\n\n\nFigure~\\ref{fig:Zevol} displays the differences in the H$\\alpha$\nvelocity as a function of the star-formation intensity distribution for\nthe galaxies in two different redshift regimes, z$<$1.8 and z$>$2. The\nkey result is that the data are consistent with a simple scaling\nbetween dispersion and star-formation intensity in both samples --\nhowever, on average the z$\\sim$1.5 galaxies have lower dispersions and\nstar-formation intensities. Also, the scatter is larger for the lower\nredshift galaxies, which perhaps indicates that a single relationship of\nthe type $\\sigma\\propto\\sqrt{}\\Sigma_{\\rm SFR}$ does not explain all of\nthe trends in the lower-redshift data as well as it does for the higher\nredshift objects.\n\nAt the low end of the star-formation intensity within individual galaxies\nand over the ensemble of galaxies with lower average star-formation\nintensities, it is likely that we are over-estimating the velocity\ndispersions, and it could be that the dispersions are becoming roughly\nconstant with radius. We say this, because our artificial model-based\ndata cubes suggest that observed dispersions smaller than $\\sim$40-50\n\\mbox{{\\rm km}\\,{\\rm s}$^{-1}$}\\ are consistent with $\\stackrel{<}{_{\\sim}}$10 \\mbox{{\\rm km}\\,{\\rm s}$^{-1}$}, a level at which we simply\ncan no longer differentiate between a model where the line widths are\nproportional to $\\Sigma_{\\rm SFR}^{1\/2}$ or one where the line widths\nare consistent with what is observed in the MW and other nearby galaxies\n\\citep[see, e.g.][]{wada02,dib06,agertz09}.\n\nIn Figures~\\ref{fig:Zevol}, we plotted the simulated data against\nthe normalized distribution of star formation intensity and velocity\ndispersion. However, this does not give a sense of the differences between\ngalaxies at different redshifts in their distribution of star formation\nintensity and velocity dispersion. The distributions of the galaxies\nin the higher redshift regime are consistent with those of 11 galaxies\npresented in \\citet{L09}.  Quantitatively, the ensemble of galaxies with\nz$>$1.8 and z$<$1.8 have a mean star formation intensity of 0.73$\\pm$0.09\nand 0.16$\\pm$0.02 M$_{\\sun}$ yr$^{-1}$ kpc$^{-2}$ respectively.  Since\nboth samples contain about the same number of galaxies, the width of the\ndistribution of the higher redshift sample is significantly larger and the\ndifference in width between the samples is consistent with \\citet{L09}.\n\nMoreover, correcting individual galaxies in the sample for the average\nextinction \\citep[derived from fitting the spectral energy distribution\nwith stellar population synthesis models;][]{fs11} does not reduce the\nscatter among them.\n\n\\subsection{The equivalence of integrated and spatially resolved\nmeasurements}\n\nSo far, we have studied the spatially resolved relationship between the\n\\mbox{H$\\alpha$}  velocity dispersion and the star-formation intensity as probed by the\n\\mbox{H$\\alpha$}  emission. In \\citet{LT11a}, we argued that the relationship between\nthe integrated \\mbox{H$\\alpha$}  luminosity (or equivalently the total star-formation\nrate) and surface brightness-weighted mean dispersion of whole galaxies\nreveals the same underlying physical mechanism as the spatially resolved\nmeasurements -- namely, that the intense star-formation is responsible\nfor the dynamics of the warm ionized medium as probed through the \\mbox{H$\\alpha$} \nrecombination line \\citep[see also][]{Green10}.\n\n\\begin{figure}\n\\includegraphics[width=9.0cm]{SFRdvsSig_blend_norm_all.eps}\n\\caption{Normalized 2-dimensional histogram of the frequency of occurrence\nin the data set of \\mbox{H$\\alpha$}  line widths as function of star formation rate\nintensity. The set of points for each galaxy are normalized such that all\nof the galaxies have the same mean star-formation rate intensity (their\ndistribution of the velocity dispersion in each galaxy is unaffected). The\nmean value was chosen to match the mean value of the star-formation\nintensity in the model with $\\sigma=(\\epsilon\\Sigma_{\\rm SFR})^{1\/2}$ as\nindicated by the red $+$ sign in the right panel of Fig.~\\ref{fig:comp}\n($\\sim$0.5 \\mbox{M$_{\\sun}$ yr$^{-1}$ kpc$^{-2}$}). As in Fig.~\\ref{fig:SBvsPos}, the darkest\nregions represent the highest frequency of occurrence in the data set. The\ngreen $\\times$ and blue $+$ symbols represent our simulated data after\nbeam smearing, assuming that the two quantities plotted are related as\n$\\sigma=(\\epsilon\\Sigma_{\\rm SFR})^{1\/2}$ and $\\sigma$=$\\sigma_{\\rm sim}$,\nrespectively.} \n\\label{fig:blend} \n\\end{figure}\n\nTo further investigate this relationship, we have tested the idea\nthat the intense star-formation in these galaxies can even explain\nthe integrated measurements of the total star-formation rate and the\nintegrated velocity dispersion. This is important: if we can show that\nalso these integrated measurements reveal this underlying relationship,\nthen a greater wealth of data become available for analysis, since slit\nspectra are more commonly available than integral field spectra.\n\nWe first note that we find little difference in velocity dispersion for\nwhole galaxies if we simply sum up the \\mbox{H$\\alpha$}  emission pixel-by-pixel and\nestimate their integrated dispersion or if we weigh each measurement\nby surface brightness (or signal-to-noise). The difference between both\nmethods is only about 15 km s$^{-1}$, where flux weighting tends to give\na lower value and a scatter of about 15 km s$^{-1}$. So compared to the\nmean integrated velocity dispersion of our sample, this represents a\nsystematic offset of only about 10\\% \\citep{LT11a}.\n\nTo further test the relevance of a relationship of the form,\n$\\sigma$=($\\epsilon \\Sigma_{\\rm SFR})^{1\/2}$, where $\\epsilon$ is the\nefficiency at which the mechanical energy from a star-formation intensity\n$\\Sigma_{\\rm SFR}$ is converted into turbulence and bulk flows in the\ninterstellar medium, we applied it to the spatially resolved dispersion\nmeasurements, to estimate integrated quantities. Specifically, we used\nthe spatially resolved, pixel-by-pixel \\mbox{H$\\alpha$}  line velocity dispersion\nmeasurements to estimate the \\mbox{H$\\alpha$}  surface brightness of each pixel. We\nthen used these surface brightness estimates as weights when combining\nthe individual pixel-by-pixel velocity dispersions, to estimate the\nintegrated mean dispersion in H$\\alpha$, $\\sigma_{\\rm mean}$. We summed\nthese estimated surface brightness values to estimate the integrated \\mbox{H$\\alpha$} \nfluxes and then converted these to total luminosities using our adopted\ncosmology. The results of this analysis show that we can reproduce\nthe actual measurements reasonably well considering the crudeness\nof the model and that we only considered one coupling efficiency\n(Fig.~\\ref{fig:ha_vs_sigmean}). So not only does this simple scaling\nrelationship reproduce the spatially resolved measurements, but it\ncan also reasonably reproduce the integrated measurements. Apparently,\nunlike in many models of the ISM, the observed velocity dispersion of\nthe warm ionized medium does increase with increasing star-formation\nintensity and rate \\citep{dib06, ostriker11}.\n\n\\begin{figure*}\n\\setlength{\\unitlength}{9.0cm}\n\\begin{picture}(2,1)\n\\put(1, 0){\\special{psfile=SFRdvsSig_blend_norm_hiz.eps hscale=60 vscale=60}}\n\\put(0, 0){\\special{psfile=SFRdvsSig_blend_norm_lowz.eps hscale=60 vscale=60}}\n\\end{picture}\n\\caption{Plots of the normalized frequency of occurrence of H$\\alpha$\nvelocity dispersion versus star-formation intensity in two redshift bins,\nz$<$1.8 ({\\it left}) and z$>$2 ({\\it right}) (cf. Fig.~\\ref{fig:blend}\nfor the whole sample). The normalizations are different from that\nused in Fig.~\\ref{fig:blend} and in the left and right panels. For\nboth sub-samples, the average star-formation intensity of each\ngalaxy was shifted to the average of its ensemble of galaxies\n(0.16 \\mbox{M$_{\\sun}$ yr$^{-1}$ kpc$^{-2}$} for z$<$1.8 and 0.73 \\mbox{M$_{\\sun}$ yr$^{-1}$ kpc$^{-2}$}\nand z$>$2). Note that the two sub-samples tend to sample different\nranges in both velocity dispersion and star-formation intensity. The\ngreen $\\times$ and blue $+$ symbols represent our simulated data\nafter beam smearing, assuming that the two quantities plotted are\nrelated as $\\sigma=(\\epsilon\\Sigma_{\\rm SFR})^{1\/2}$ km s$^{-1}$ and\n$\\sigma$=$\\sigma_{\\rm sim}$, respectively. The green and blue model\npoints have been shifted to overlap with the data (the location of the\nmodeled points along the axis of $\\Sigma_{\\rm SFR}$ is set by the coupling\nefficiency of the mechanical energy output of young stars to the ISM).}\n\\label{fig:Zevol}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[width=9.0cm]{LHavssigma_paper.eps}\n\\caption{Mean H$\\alpha$ line velocity dispersion, $\\sigma_{\\rm mean}$\n(in \\mbox{{\\rm km}\\,{\\rm s}$^{-1}$}), versus total \\mbox{H$\\alpha$} \\ luminosity (in erg s$^{-1}$) for different\nsamples of local and high redshift galaxies. Local galaxies (green\ntriangles) are taken from \\citet{Green10} and the model results for\nour sample are represented by red and blue stars. The blue stars show\nthe results where we assign to each pixel in a galaxy an \\mbox{H$\\alpha$}  surface\nbrightness based on the model relationship, $\\sigma=140\\sqrt{}\\Sigma_{\\rm\nSFR}$ km s$^{-1}$ and use the observed local \\mbox{H$\\alpha$}  velocity dispersion. We\nthen estimate the total \\mbox{H$\\alpha$}  luminosity by summing up the predicted\npixel-by-pixel surface brightnesses, and estimate $\\sigma_{\\rm mean}$\nby weighting the pixel-by-pixel velocity dispersion measurements by the\n\\mbox{H$\\alpha$} \\ surface brightnesses derived from the model.}\n\\label{fig:ha_vs_sigmean}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discussion}\n\nFigure~\\ref{fig:pressure} shows that [NII]\/H$\\alpha$ ratios are\nnot arbitrary, but scale with H$\\alpha$ surface brightness. This has\nalready been noted by \\citet{L09} and is confirmed with our new, larger\nsample. The emission line ratios and \\mbox{H$\\alpha$}  surface brightnesses are related\nto the gas pressure, column density, intensity of the radiation field\nand the number of line-emitting regions along the line of sight within\neach beam. New, compared to our previous analysis of this relationship,\nis that our expanded sample includes galaxies at lower redshift, which\nin turn will allow us to probe pressures, column densities and radiation\nfields in galaxies with less extreme \\mbox{H$\\alpha$}  surface brightnesses.\n\nMeasurements of the \\mbox{[S{\\sc ii}]}\\ doublet ratio in individual galaxies in a\nsub-sample of this sample \\citep{L09} and through a stacking analysis\nof this sample \\citep{LT11b}, suggest that the typical densities of\nthe warm ionized gas are  n$_{\\rm e}$$\\sim$10s to $\\sim$500 cm$^{-3}$.\nTo compare these estimates, we ran a suite of photoionization models\nusing version 08.01 of the Cloudy code \\citep{ferland98} to show the\nrelationship between the [N{\\sc ii}]$\\lambda$6583\/\\mbox{H$\\alpha$}  line ratio and the\n\\mbox{H$\\alpha$}  surface brightness as a function of the hydrogen internal pressure,\ncolumn density and ionization parameters for a single ionized cloud\n\\cite[see][]{L09}. Unfortunately, the interpretation of such modeling is\nambiguous due to the degeneracy between surface brightness, line ratio and\nthe number of individual emission line regions per large physical beam of\nour observations. Given the crude sampling of our data, it is likely that\nthe densities are lower than predicted by a single cloud model, as would\nbe the ionization parameters. Therefore, for both the higher and lower\nredshift data sets, the likely densities are certainly $<$1000 cm$^{-3}$\nbut generally larger than a few times 10 cm$^{-3}$ and are consistent\nwith estimates of the electron density directly measured from the \\mbox{[S{\\sc ii}]}\\\ndoublet ratio in individual galaxies and a stacking analysis.\n\n\\begin{figure}[h!]\n\\setlength{\\unitlength}{9.0cm}\n\\includegraphics[width=9.0cm]{SBvsRat_lowz.eps}\n\\includegraphics[width=9.0cm]{SBvsRat_highz.eps}\n\\caption{H$\\alpha$ surface brightness (corrected for cosmological\nsurface brightness dimming) versus the logarithm of the [N{\\sc\nii}]$\\lambda$6583\/\\mbox{H$\\alpha$}  line ratio. The galaxies have been separated into\ntwo redshift bins: z$<$1.8 ({\\it top}) and z$>$2 ({\\it bottom}). The\nlines represent results for photo-ionization modeling (see text for\ndetails) for 6 sets of conditions: a range of hydrogen densities,\nlog n$_H$ (cm$^{-3}$) =1 (in green), 2 (blue) and 3 (red), and\nof column densities, log N$_H$ (cm$^{-2}$)=21 (solid lines) and 20\n(dashed lines). The ionization parameters range from log U=$-$5 to $-$1\n(increasing from left to right along the lines, meaning low ionization\nparameters have relatively high log [N{\\sc ii}]\/H$\\alpha$ and low\nH$\\alpha$ surface brightnesses). At low ionization parameters the solid\nand dashed lines for the column densities overlap until the lower column\ndensity cloud becomes density bounded, at which point the models reach\nvery low ratios of [N{\\sc ii}]$\\lambda$6583\/H$\\alpha$ (as shown by the\nsharply dropping dotted lines). For clarity, we only show the log N$_H$\n(cm$^{-2}$)=21 and log n$_H$ (cm$^{-3}$) =3 line until log U = $-$3 and\nfor log N$_H$ (cm$^{-2}$)=21 and log n$_H$ (cm$^{-3}$) =2 until log U =\n$-$2.} \n\\label{fig:pressure} \n\\end{figure}\n\nThere is an interesting difference in the regions occupied in the\n[N{\\sc ii}]$\\lambda$6583\/\\mbox{H$\\alpha$}  versus \\mbox{H$\\alpha$}  surface brightness plane by\ngalaxies at z$<$1.8 and z$>$2. Taken at face value, it would suggest\nwe are probing less dense gas in the lower redshift galaxies. However,\nthis is undoubtedly due to probing lower surface brightness gas, related\nto the strong influence of surface brightness dimming, which allows us\nto probe the very diffuse, perhaps even shock heated, gas in the lower\nredshift sample (the regions with relatively low surface brightness and\nhigh ratios of [N{\\sc ii}]$\\lambda$6583\/\\mbox{H$\\alpha$}  in Fig.~\\ref{fig:pressure}\nlikely require a contribution to their ionization from shocks). This\neffect is observational, rather than evolutionary. A complete census\nof both redshift regimes would be necessary to show that the average\npressure or number of overlapping emission line regions (i.e. thickness)\nin the warm ionized media of galaxies is declining with redshift.\n\nThe thermal pressures estimated from the electron densities are much\nhigher than in the disk of our Milky Way or other nearby normal\ngalaxies. The hydrostatic mid-plane pressure in the MW is about\n10$^{3.3-4.3}$ K cm$^{-3}$, while for other local spirals it can amount\nto about 10$^{6}$ K cm$^{-3}$, though it is typically about 10$^{4.6}$\nK cm$^{-3}$ \\citep[][and references therein]{blitz06}. In the nuclear\nregions of local starburst galaxies, thermal pressures estimated from\nthe \\mbox{[S{\\sc ii}]}\\ doublet ratio are typically about 10$^{6-7}$ K cm$^{-3}$\n\\citep{lehnert96}. The pressures we estimated from the photo-ionization\nmodels are consistent with those of nearby intensely star-forming galaxies\n\\citep{heckman90, lehnert96, wang98} and measurements of density-sensitive\nlines at high redshift \\citep[\\mbox{[S{\\sc ii}]}\\ doublet ratio;][]{LT11b, L09,\nnesvadba07}.\n\nBy comparing these thermal pressures in the WIM to that generated by\nthe radiation pressure of the ionizing radiation field, it is possible\nto constrain the general distribution of the WIM \\citep{krumholz09} .\nOur optical line ratios suggest that the gas is predominately\nphotoionized by massive stars \\citep{L09}. To estimate the impact\nof radiation pressure on an ionized fluid, it is appropriate to only\nconsider the ionizing radiation. We can use the ionization parameter to\ngauge the relative contribution of radiation pressure to  the thermal\npressure of the WIM. Simple scaling of the definition of the ionization\nparameter suggests P$_{\\rm rad}$\/P$_{\\rm thermal}$= 2.5$\\times$10$^{-2}$\nU ($<$h$\\nu$$>$\/18 eV)\/k(T\/10$^4$ K) for T=10$^4$ K and $<$h$\\nu$$>$=18\neV, where $<$h$\\nu$$>$ is the average energy of ionizing photons and k\nis the Boltzmann constant. Since the ionizing parameter is always less\nthan log U = $-$1.6 (and mostly $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} -$3), the radiation pressure is\nlower than the thermal gas pressure throughout the nebulae. Finding\nthat the thermal pressure dominates the radiation pressure and that\nthe ionization parameters are relatively low, suggests that the WIM is\nwidely distributed over the volume of the ISM in our sample of galaxies,\ntruly part of the general interstellar medium, and not simply near or\ndirectly associated with the star forming clumps or regions \\citep[see\n][ for a detailed discussion of this point]{krumholz09}. This point is\ncrucial for our subsequent analysis as it suggests that the WIM is a\ntracer of the dynamics of the general ISM in distant galaxies and not\nmerely  of the \\mbox{H{\\sc ii}}\\ regions or clumps.\n\nOther models or relationships have been proposed to explain the\nhigh dispersion in the warm ionized gas that are directly related to the\nionization state of the gas. \\citet{wisnioski12} and \\citet{swinbank12}\npropose that the relationship between star formation intensity\nand velocity dispersion can be explained by a relation between\nthe luminosity of a Hydrogen recombination line (e.g. H$\\beta$)\nand the gas velocity dispersion, as observed in local \\mbox{H{\\sc ii}}\\ regions\n\\citep[e.g.][]{terlevich81}.  The sizes of local \\mbox{H{\\sc ii}}\\ regions and\nthe relationship between line luminosity and velocity dispersion can\nbe understood simply if the gas within the \\mbox{H{\\sc ii}}\\ regions is virialized\n\\citep{terlevich81}.  In such a hypothesis, the line widths are mainly\ndue to virial motions, implies that the warm ionized medium is in\nclose proximity to the star forming clumps (within the gravitational\nsphere of influence).  Nearby HII regions, for which such a relation\nis appropriate, generally show ionization parameters that are high, log\nU$\\approx -$1 to $-$2\\citep[e.g.][]{lopez13}, but our analysis suggests\nthat distant galaxies generally have much lower ionization parameters,\nimplying that their emission line gas luminosity is dominated by more\nwidely distributed gas and not directly associated with the clumps.\n\n\\subsection{Source of the pressure in the WIM}\n\\label{pressuresource}\n\nWhat are the sources of pressure in these distant galaxies?  Is the\npressure generated by the thermalization of the collective mechanical\nenergy output of massive stars through the thermalization of stellar\nwinds and supernova explosions? Observations of nearby galaxies suggest\nthat the low ionization line emitting region is approximately in\npressure equilibrium with the X-ray emitting plasma, with thermal\npressures in the range 10$^{-9}$ to 10$^{-10}$ dyne cm$^{-2}$\n\\cite[e.g.][]{heckman90,moran99,westmoquette07, strickland09}. Similar\npressures are found in models using typical initial conditions expected\nfor galaxies exhibiting outflows \\citep[e.g.][]{suchkov96}. If these\ntwo phases are in equilibrium then we can use a simple model to estimate\nthe pressures in the hot X-ray emitting plasma to see if it agrees with\nour estimates of the thermal pressures (densities) in the warm ionized\nmedia in distant galaxies.\n\n\\citet{strickland09} has shown that the simple analytic expressions\nof \\citet{chevalier85} for the pressure, temperature and velocity of\noutflows driven by the intense energy injection from young massive\nstars agree with hydrodynamical simulations. Adopting and scaling\nthese simple relations from \\citet{chevalier85} we estimate the central\npressure in a starburst region as P$_{\\rm c}$= 3.5$\\times$10$^{-10}$\n($\\zeta$$\\dot{\\rm M}_{0.26}$)$^{1\/2}$($\\epsilon_{\\rm thermal}$$\\dot{\\rm\nE}_{\\rm 41.9}$)$^{1\/2}$ R$_{\\rm SB, 350}^2$ $\\Omega$ dyne cm$^{-2}$,\nwhere $\\zeta$ is the mass-loading factor, $\\dot{\\rm M}_{\\rm 0.26}$ is the\nmass return rate in units of 0.26 M$_{\\sun}$ yr$^{-1}$, $\\epsilon_{\\rm\nthermal}$ is the thermalization efficiency of the stellar winds and\nsupernovae, $\\dot{\\rm E}_{\\rm 41.9}$ is the energy injection rate in units\nof 7.4$\\times$10$^{41}$ erg s$^{-1}$, R$_{\\rm SB, 350}$ is the radius of\nthe starburst in units of 350 pc and $\\Omega$ is the opening angle of the\noutflow in sr. The adopted scalings for the mass and energy injection\nrates from massive stars are appropriate for a star formation rate of\n1 M$_{\\sun}$ yr$^{-1}$ \\citep{leitherer99} within a radius consistent\nwith clump sizes and \\mbox{H$\\alpha$}  disk thicknesses in high redshift galaxies\n\\citep{fs11,epinat12}. For the radius scaling, see \\citet {strickland09}.\n\n\\begin{figure}[ht!]\n\\includegraphics[width=9.0cm]{toomreQ.eps}\n\\caption{\\mbox{H$\\alpha$}  velocity dispersion, $\\sigma$ as function of stellar mass\nsurface density, $\\Sigma_{\\rm stars}$ (M$_{\\sun}$ pc$^{-2}$). The lines\nrepresent the Toomre criterion Q=1 for two rotation speeds (v=100 or\n200 km s$^{-1}$) and three radii (r=3, 5, 10 kpc), as indicated in the\nlegend. These values span the range of radii and velocity shears observed\nin these galaxies \\citep{fs06, fs09}. The stellar mass surface densities\n(red dots) are estimates within one effective radius for 6 galaxies\nin our sample with NICMOS images (see text). The red solid line shows\nthe relationship between central velocity dispersion and stellar mass\nsurface density for local early-type galaxies \\citep{shen03, taylor10}.}\n\\label{fig:toomreQ}\n\\end{figure}\n\nReasonable assumptions for the thermalization efficiency, mass loading\nfactor, and mass and energy injection rates, result in pressures of\n10$^{-9}$ to 10$^{-10}$ dyne cm$^{-2}$ in our galaxies, similar to\nwhat is estimated for nearby starburst galaxies. This simple modeling\nsuggests that the hot plasma generated by the thermalization of the\nmechanical energy output of young stars can generate sufficient pressure\nto approximately match our estimates for the thermal pressure in the\nwarm ionized medium.\n\nOne of the assumptions that is needed for the model to be appropriate is\nthat the energy injection rate is approximately constant over an outflow\ntimescale \\citep[e.g.][]{chevalier85, suchkov96,strickland09}. This has\nbeen shown to be true for local starbursts \\citep[e.g.][]{heckman90,\nlehnert99}. While we do not know the outflow timescale of the distant\ngalaxies, since their star-formation intensities are similar to nearby\nstarburst galaxies \\citep[see discussion in][]{L09}, it is logical to\nassume their outflow timescales would be similar (10s Myr or less).\nDistant galaxies similar to the ones observed here \\citep[e.g.][]{erb06,\nfs11} have star formation timescales of $\\approx$100s Myrs and are\nlikely much longer than the outflow timescales. A reasonable assumption\nis that the energy injection rate is approximately constant in distant\ngalaxies as it appears to be in nearby starbursts and thus the model of\n\\citet{chevalier85} is an appropriate approximation for estimating the\npressures \\citep[see also ][ for a discussion]{strickland09}.\n\n\\subsection{Mass and energy flow?}\\label{picture}\n\nThe aforementioned rough equality of the thermal pressure in the two ISM\nphases, the hot 10$^7$--10$^8$ K plasma and the 10$^4$ K warm ionized\nmedium, is important. It may imply that the mechanical energy injected\ninto the ISM of the galaxy couples efficiently with the recombining gas\nand perhaps drive a mass and energy flow from the WIM to the cold neutral\nand molecular medium \\citep{guillard09, guillard10}. The recombining gas\nis unstable at the pressures we observe and will quickly cool, collapse\nand become part of the cold neutral medium, CNM \\citep{wolfire95}. Since\nthis is expected to happen quickly this would imply that the CNM acquires\nthe relative kinematics of the WIM \\citep{guillard12}. If the gas is\ndusty, this recombining gas will rapidly turn molecular since the H$_2$\nformation time scales on dust grains is much shorter than the expected\ndynamical time scales of the recombining, cooling and collapsing clouds\n\\citep{guillard09}. The formation of gravitationally bound clouds releases\ngravitational energy as turbulence into the molecular gas.\n\nThis picture links the energy input from intense star formation to energy\ndissipation, the formation of molecular gas and star formation. Thus not\nonly is this a flow of mass and energy  but also one of kinetic energy,\nwhich will not only contribute to the turbulent energy of the molecular\ngas but also serve to balance dissipation. Of course, it is not a simple\none-directional flow, but a cycle or feedback loops where individual\nmolecular clouds may form stars when they become self gravitating, or\nsimply be heated and destroyed. Naturally, because all the timescales\nin the cycle are relatively short, $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}}$10 Myrs, the ISM will be in a\ndynamical equilibrium. The components of the ISM should not be viewed\nas single, isolated components but rather as an intricate balance whose\nrelative mass flow rates are determined by a competition between the\nheating and cooling rates as well as the outflow rate due to starburst\ndriven winds and the infall rate due to the cosmological accretion of\ngas through cooling gas and mergers.\n\nOne of the conditions that must be met in such a picture is that the\nmass deposition rate of the WIM should be more than sufficient to\nfuel star formation, or $\\dot{\\rm M}_{\\rm SF}\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}}\\epsilon_{\\rm star\nform} \\dot{\\rm M}_{\\rm HII}$, where $\\epsilon_{\\rm star form}$ is\nthe efficiency of star formation. The recombination rate of the \\mbox{H$\\alpha$} \\\nemitting gas is given by simple ionization balance, $\\dot{\\rm M}_{\\rm\nHII}$=1.9$\\times$10$^5$ L$_{\\mbox{H$\\alpha$}  , 43}$ M$_{\\sun}$ yr$^{-1}$. The star\nformation rates in these galaxies are $\\sim$10-300 M$_{\\sun}$ yr$^{-1}$\nand thus the WIM recombination rates can supply the gas for star formation\nas long as the star formation efficiency is $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}}$0.1\\%.  For a reasonable\n$\\epsilon_{\\rm star form}$ only a small fraction, a few  to about 10\npercent, of the recombining gas  would then be necessary to fuel this\ncycle. So at least in terms of mass flow rates, there is sufficient mass\nin the warm ionized medium to plausibly sustain such a cycle.\n\nOf course, for the energy injection to be effective in regulating\nstar formation, as we have hypothesized, it should drive the galaxy\ntowards the line of Toomre disk stability, Q$\\sim$1. This occurs if\nthe star formation can inject sufficient turbulence to roughly balance\ngravitational collapse and hydrostatic pressure.\n\n\\begin{figure}[hb!]\n\\includegraphics[width=9.0cm]{ToomreQSigmaSFR.eps}\n\\caption{\\mbox{H$\\alpha$}  velocity dispersion, $\\sigma$, as function of the\nstar-formation intensity, $\\Sigma_{\\rm SFR}$ (\\mbox{M$_{\\sun}$ yr$^{-1}$ kpc$^{-2}$}). The\nlines represent the Toomre criterion for gas, assuming both Q=1 and\nthat the Schmidt-Kennicutt relation is appropriate for a variety of\nrotation speeds and radii, as in Fig.~\\ref{fig:toomreQ} and as indicated\nin the legend (see text for details). The underlying display of the\nnormalized data (shown in various shades of red) is the same as \nin Fig.~\\ref{fig:blend} (see that figure and text for details).}\n\\label{fig:toomreQSFR}\n\\end{figure}\n\nWhile the WIM traces the overall dynamics of the ISM (see beginning\nof \\S~\\ref{sec:discussion}), its contribution to the turbulent\npressure is negligible. It is given by ${\\rm P}_{\\rm gas, turb}$ =\n$<\\rho_{\\rm gas}>\\sigma_{\\rm gas}^{2}$ where $<\\rho_{\\rm gas}>$ is the\nvolumeweighted mean density (=$\\rho_{\\rm phase} f_{\\rm V,phase}$, where\n$\\rho_{\\rm phase}$ is the intrinsic density and $f_{\\rm V,phase}$ the\nvolume filling factor of the ISM phase). The total turbulent pressure\nis actually the sum of those of all individual ISM phases, but we have\nsuppressed the summation for clarity. The \\mbox{H$\\alpha$} \\ emitting gas likely does\nnot contribute significantly to the total turbulent pressure. Assuming\nsimple case B recombination, T$_e$=10$^4$ K and equality of proton and\nelectron densities, we can crudely calculate the emitting volume needed\nto supply the \\mbox{H$\\alpha$} \\ luminosity compared to the volume over which it is\nobserved. The volume filling factor of the recombining gas is ff$_{\\rm\nV, WIM}\\sim$V$_{\\rm em}$\/V$_{\\rm n}$ = 4$\\times$10$^{-5}$ L$_{\\mbox{H$\\alpha$}  ,43}$\nn$_{\\rm e, 300}^2$ V$_{\\rm n, 100}^{-1}$, where V$_{\\rm em}$ is the\nemitting volume, V$_{n, 100}$ is the actual ISM volume in units of 100\nkpc$^3$ (the typical isophotal area is 100 kpc$^2$ and we assumed a disk\nthickness of 1 kpc), L$_{\\rm \\mbox{H$\\alpha$}  ,43}$ is the \\mbox{H$\\alpha$} \\ luminosity in units\nof 10$^{43}$ erg s$^{-1}$ and n$_{\\rm e, 300}$ is the electron density\nin units of 300 cm$^{-3}$. The filling factor of the recombining gas\nis $\\sim$10$^{-6}$.\n\nThe most significant contribution to the total turbulent pressure is\nlikely from the cold neutral and molecular medium. We can estimate\nthe density of the dense molecular gas using observations of distant\ngalaxies with star formation rates and general properties similar\nto ours. H$_2$ gas mass surface density measurements are available\nfor galaxies similar to those studied here \\citep[z$\\sim$1 and\n2;][]{daddi10,aravena10,dannerbauer09,tacconi10,combes11} which suggest\nthat at z$\\sim$2 the mean value is about 500 M$_{\\sun}$ pc$^{-2}$ and\nthat total gas fractions $\\sim$30--50\\% \\citep{daddi10,tacconi10}. The\nvolume weighted H$_2$ gas density, $<n_{H_2}>$=$\\Sigma_{\\rm gas}$\/H\nm$_{\\rm H_2}$ = 10.2 $\\Sigma_{\\rm gas, 500}$\/H$_{\\rm 1 kpc}$, where\n$\\Sigma_{\\rm gas, 500}$ is the gas mass surface density in units of\n500 M$_{\\sun}$ pc$^{-2}$ and H$_{\\rm 1 kpc}$ is the thickness of the\ngas layer in units of 1 kpc. This implies that P$_{\\rm turb, mol}$=3\n$\\times$10$^{-9}$ $<\\rho_{\\rm gas, 10}>$ $\\sigma_{\\rm H_2, 100}^{2}$,\nwhere $<\\rho_{\\rm H_2, 10}>$ is the H$_2$ volume weighted particle density\nin units of 10 cm$^{-3}$ and $\\sigma_{\\rm H_2, 100}$ is in units of 100\nkm s$^{-1}$. These estimates suggest that the turbulent pressure in the\nmolecular gas may be high \\citep{swinbank11}.\n\nFor a galaxy to lie near the line of stability, Q=1,  the pressure due to\ngravity and that due to turbulence should be roughly equal. To estimate\nthe pressure due to gravity we will assume hydrostatic equilibrium, which\nis obviously an over-simplification. However, for the present analysis,\nwe are concerning ourselves with globally averaged estimates, suggesting\nthat hydrostatic equilibrium is a reasonable assumption. It is impossible\nto get direct estimates for the hydrostatic gas pressures for our sample\nof galaxies as stellar mass surface densities are available for only a\nfew objects  \\citep{fs11}. As with the estimates of the volume-weighted\ngas mass densities, we will have to rely on results from similar samples\nof distant galaxies, which find stellar mass surface densities of 100-2000\nM$_{\\sun}$ pc$^{-2}$ \\citep[e.g.][]{barden05,yuma11,fs11}. The hydrostatic\npressure scales as ${\\rm P}_{\\rm gas, hydro}$=7$\\times$10$^{-10}$\n$\\Sigma_{\\rm gas,500} \\Sigma_{\\rm total,1500}$ dyne cm$^{-2}$, where\n$\\Sigma_{\\rm gas,500}$ is the gas mass surface density in units of 500\nM$_{\\sun}$ pc$^{-2}$ and $\\Sigma_{\\rm total,1500}$ is the stellar mass\nsurface density in units of 1500 M$_{\\sun}$ pc$^{-2}$. For this estimate\nof the hydrostatic pressure, we have assumed that the velocity dispersion\nof the gas and stars are equal, which is consistent with our general\nunderlying hypothesis that the velocity dispersions of the warm ionized\nand the cold molecular ISM phases are similar. However, this is not a\ntight constraint as lowering the dispersion in the gas would lower the\nturbulent pressure as well as the hydrostatic pressure for a fixed stellar\nvelocity dispersion. The estimated ranges of gas fraction and stellar\nsurface densities for z$\\sim$2 galaxies \\citep{daddi10,tacconi10} suggests\nhydrostatic pressures of ${\\rm P}_{\\rm gas, hydro}$$\\sim$10$^{-9}$\nto 10$^{-10}$ dyne cm$^{-2}$. Thus it appears that the hydrostatic\npressure and the turbulent pressure in the molecular gas are similar if\nthe molecular gas has kinematics similar to that of the WIM.\n\n\\subsection{Surface densities and the Toomre Q parameter}\n\nWe have argued that if the turbulence in the molecular gas were  similar\nto that of the WIM, then the distant galaxies should lie near the line\nof stability. We proposed a qualitative scenario where the kinematics of\nthe WIM may be roughly captured by the cold molecular gas and thus lead\nto high turbulence. Furthermore, we suggested that these intensities can\nbe understood in terms of self-regulation \\citep{silk97, silk01}. Such\nself-regulation would reveal itself through mass surface densities in\nthe galaxies consistent with Q=1.\n\n\\citet{krumholz10} favor disk instabilities as the source of the\nhigh velocity dispersion in distant galaxies with spatially-resolved\nkinematics and line emission \\citep[see also, e.g.][]{cacciato12}. Their\nargument is that this is the simplest explanation, because it is based\non the Toomre instability parameter, Q=$\\kappa\\sigma$\/$\\pi$G$\\Sigma$,\nwhere $\\kappa$ is the epicyclic frequency (which in our subsequent\nanalysis we have taken to be 2$^{1\/2}$v\/r, where v is the rotation\nspeed and r is the radius) and $\\Sigma$ is the mass surface density\nof stars or gas \\citep[][]{wang94, romeo11}. They suggest that the\nhigh velocity dispersions are a natural result of the need to keep the\nstar-formation intense, which implies that Q$\\sim$1. The results given\nabove suggest perhaps that the star-formation itself may also lead to\nQ$\\sim$1 \\citep[see also][]{ostriker11}. Combining the Schmidt-Kennicutt\nrelation with the Toomre instability criterion, Q=1, suggests that\n$\\sigma \\propto \\Sigma_{\\rm SFR}^{0.7}$. This is similar enough to our\nsuggested relationship as to be indistinguishable \\citep{wisnioski12,\nswinbank12}. \\citet{swinbank12} show that assuming a marginally\nstable Toomre disk combined with relationships for the largest unstable\nfluctuations in disks and a Jeans relation for the growth of unstable\nmodes, can lead to insights into why disks at high redshift may have high\ndispersions, why clumps have high mass surface densities, why their\nmasses in disk galaxies decline with redshift and why their\nmass distributions have exponential cutoffs.  However, such a solution\ndoes not explain how the observed high dispersions are maintained. This\neither requires a long dissipation time or a large source of mechanical\nenergy \\citep{maclow99, ostriker11}. What maintains the instability such\nthat Q$\\sim$1? We suggest it is the star-formation itself that maintains\nthe necessary dispersion in the gas phase, as we will now argue.\n\nThe Toomre parameter for the stellar population is\nQ=$\\kappa\\sigma$\/$\\pi$G$\\Sigma_{\\rm stars}$, where $\\Sigma_{\\rm stars}$\nis the mass surface density of stars. For the galaxies we have presented\nhere, the velocity shears are of-order 100-200 km s$^{-1}$ \\citep{fs06,\nfs09} and the radii are about a few to 10 kpc \\citep{fs11}. The mass\nsurface density, $\\Sigma_{\\rm stars}$=M$_{\\rm stars}$\/2$\\pi$(b\/a)r$_e^2$,\nwhere M$_{\\rm stars}$ is the total stellar mass, b\/a is the minor-to-major\naxis ratio and r$_e$ is the effective radius. Using NICMOS and other\nimaging data of 6 galaxies in our sample, we have made crude estimates of\nstellar mass surface densities using the results from \\citet{fs11}. We\nfind that the mass surface densities within one effective radius are\n$\\sim$100-1500 M$_{\\sun}$ kpc$^{-2}$. Assuming Q=1, we can relate\nthe velocity dispersion to the mass surface density and it appears\nthat the estimated mass surface densities are consistent with Q=1\n(Fig.~\\ref{fig:toomreQ}).\n\nWithout considering the effects of beam smearing, this simple relation\nis actually also roughly consistent with our data on the emission line\nproperties, as we can demonstrate rather simply: if we assume Q=1 for\nthe gas, and adopt the Schmidt-Kennicutt relation \\citep{kennicutt98}\nbetween star formation intensity and gas  surface density, we\nfind the relation, $\\sigma$ = ($\\pi$GQ\/$\\kappa$)($\\Sigma_{\\rm\nSFR}$\/2.5$\\times$10$^{-4}$)$^{0.7}$. For a variety of rotation\nspeeds and disk radii, we find general consistency with our data\n(Fig.~\\ref{fig:toomreQSFR}). Of course these estimates are crude and\ncould be fortuitous but they do lend additional weight to our overall\narguments. We have not used the generalized (effective) Toomre criterion,\n1\/Q$_{\\rm eff}$ = 1\/Q$_{\\rm gas}$ + 1\/Q$_{\\rm stars}$ \\citep{wang94,jog96,\nromeo11} because the crudeness of our analysis does not warrant this\nadditional complication. However, since this is a relatively small change\nin the adopted Q, it will make little difference to our conclusions\nwhich is that the galaxies generally lie near the line of stability.\n\n\\subsection{Toomre-stability of star-forming high-z\ndisks}\\label{subsec:model}\n\nCan we make a self-consistent model whereby intense star formation\nis responsible for driving high turbulence and pushing the ISM in\ndistant galaxies towards the line of stability?  \\citet{elmegreen10}\ndeveloped an analytical model to follow the evolution of gravitational\ninstabilities and star formation in disk galaxies that are built\nfrom rapid gas accretion at high, constant rates. The energy of the\naccretion is balanced by the dissipation rate and loss of energy due\nto star formation. The accretion rate is defined by accreting half of a\nfiducial mass scaling in one dynamical (orbital) time \\citep[about 280\nMyrs, ][]{elmegreen10}. Star formation only serves as an energy sink\nin their formalism, to account for gas lost to the ISM because it is\nbound in stars, but they do not include the energy return from young\nstars into the ISM in a self-consistent way. In their model, turbulent\nvelocities drop to very low values well before stellar-to-gas mass\nratios reach observed values. To overcome this, they force turbulent\nvelocities, $\\sigma$, to have values consistent with Q$=$1. They find\nthat ultimately the disk reaches an equilibrium where $\\dot{\\rm M}_{\\rm\naccretion}=\\dot{\\rm M}_{\\rm star-formation}$.\n\n\\begin{figure}[ht!]\n\\includegraphics[width=9.0cm]{model2.eps}\n\\includegraphics[width=9.0cm]{model4.eps}\n\\caption{Six normalized quantities (see legend) as a function of\ntime, where t\/t$^{\\prime}$ = 90 Myrs or about 1\/3 of a dynamical\ntime of the potential. To limit the total mass so that it does not\nbecome too large compared to the observations, the accretion ended\nat t$^{\\prime}$=6 (about 540 Myrs or two dynamical times). The top\npanel is for a relatively high assumed efficiency of coupling of the\nmechanical energy output of the stellar population to the ISM (12\\%),\nwhereas  the bottom panel is for a relatively low coupling efficiency\n(5\\%). The scalings for these normalized quantities are provided in the\ntext \\citep[see][]{elmegreen10}.}\n\\label{fig:models}\n\\end{figure}\n\nWe will now extend their model to include the effects of the energy\nrelease of star formation. This is necessary if we want to investigate\nhow star formation is regulated in galaxies whose evolution is not\nonly driven by accretion and dissipation of turbulent energy, but\nwhere star formation contributes significantly to the gas energetics,\nas indicated by our observational finding that star formation produces\npressures similar to the mid-plane pressure of the disk. In our model,\naccretion, dissipation and the energy released by star formation together\nset Q. Including star-formation feedback in this way is one significant\ndifference to the model of \\citet{elmegreen10}. The other is that we allow\nfor a cut-off in the gas accretion at some point in the evolution. Some\ncut-off or severe reduction in the initial high rates of gas supply\nmust have occurred to produce  galaxies with  masses similar to those\ngenerally observed at high redshift. This is not merely an academic\nissue either. It is often argued that the star formation rate and the gas\naccretion must be equal, and at the high star formation rates observed in\nz$\\sim$2 galaxies, the star formation time is a few 100 Myr to 1 Gyr. To\naccount for the energy from star formation, we have modified equation\n(11) from \\citet{elmegreen10} as follows:\n\n\\begin{equation}\n\\nonumber\n\\frac{{\\rm dE}^{\\prime}}{{\\rm dt}^{\\prime}} = 1 - {\\rm M}^{\\prime}_{{\\rm disk}}({\\rm E}^{\\prime}{\\rm M}^{\\prime}_{{\\rm gas}})^{1\/2} - {\\rm S}^{\\prime}{\\rm E}^{\\prime}\/{\\rm M}^{\\prime}_{{\\rm gas}}\\\\ + \\epsilon_{\\rm SF} {\\rm E}_{\\rm SF} {\\rm S}^{\\prime} {{\\rm M}^{\\prime}_{{\\rm gas}}}^{3\/2} {{\\rm E}^{\\prime}}^{1\/2}\n\\end{equation}\n\n\\noindent\nwhere ${\\rm E}^{\\prime}$, ${\\rm M}^{\\prime}_{{\\rm disk}}$, ${\\rm\nM}^{\\prime}_{{\\rm gas}}$, and ${\\rm S}^{\\prime}$ are the scaled\nenergy, disk mass, gas mass and star-formation rate, respectively\n\\citep[see][for details]{elmegreen10}. The first 3 terms are straight\nfrom \\citet{elmegreen10}. The first term is constant, because for\nsimplicity we assume a constant potential, and it is unity, because the\nenergy is normalized relative to the total accretion energy, which is\nextracted from the potential. The second term quantifies the turbulent\ndissipation. The dissipation rate is E$\\sigma$\/H, where E is the total\nenergy, $\\sigma$ is the turbulent velocity and H is the scale height of\nthe disk, which  is determined by a competition between the turbulent\nenergy and the gravitational force due to the total mass surface density,\ni.e., H=$\\sigma^2$\/$\\pi$G$\\Sigma_{disk}$. The third term is the energy\nthat is lost to the ISM when stars form. Adding a time dependence to\nthe potential is quite straight-forward and we have conducted both\na time independent and time dependent analysis. To these first three\nterms from \\citet{elmegreen10} we have added another term to incorporate\nthe energy injection rate from star formation, $\\epsilon_{\\rm SF} {\\rm\nE}_{\\rm SF} {\\rm S}^{\\prime} {{\\rm M}^{\\prime}_{{\\rm gas}}}^{3\/2} {{\\rm\nE}^{\\prime}}^{1\/2}$, where $\\epsilon_{\\rm SF}$ is the coupling efficiency\nof the mechanical energy to the interstellar medium of the galaxy and\n${\\rm E}_{\\rm SF}$ is the rate at which energy is ejected per unit of\nstar formation. The whole equation is normalized such that the accretion\nenergy is one. In addition, other than the parameters already mentioned,\nother key parameters in determining the evolution of this simple model\nare the star formation efficiency, the gas accretion rate, the fraction\nof the accretion energy that couples to the kinematics of the ISM, the\nfraction of mechanical energy produced by young massive stars that cycles\nback into the ISM and the dissipation timescale. For these parameters,\nwe adopted the  values from \\citet{elmegreen10}, although we allowed\nfor a star formation efficiency of 3\\% in the models instead of 1\\%\nthey adopted. The fraction of energy produced by massive stars that\ndynamically coupled to the ISM was a free parameter, which we would\nlike to constrain, and the dissipation timescale was determined by the\nparameters of the model, as just discussed.\n\nThe scalings within this model are quite important. Using the estimates\nof the mechanical energy per unit star formation \\citep{leitherer99} and\na rotation speed of 220 km s$^{-1}$ (which sets the potential energy of\nthe galaxy from which the accretion energy is extracted), we estimate\nthat the mechanical energy injected into the ISM by stars is several\n100 times greater than that generated by accretion. Also the time scale\nfor star formation is much shorter than that for accretion.  Thus star\nformation can play an important role in injecting energy into the gas\nand perhaps even pushing disks towards marginal stability against star\nformation (Q$\\sim$1). The scalings for the original model \\citep[taken\nfrom][]{elmegreen10} are t\/t$^{\\prime}$ = 90 Myrs (or about one-third\nof an orbital time), S\/S$^{\\prime}$ = 160 M$_{\\sun}$ yr$^{-1}$, M$_{\\rm\ngas}$\/M$_{\\rm gas}^{\\prime}$ = 1.4 $\\times$10$^{10}$ M$_{\\sun}$ and\n$\\sigma$\/$\\sigma^{\\prime}$=70 km s$^{-1}$. The scaled Toomre parameter,\nQ$^{\\prime}$ is related to the actual Toomre parameters by a factor of\n$\\approx$1\/2 to 1, depending on the efficiency at which accretion energy\nand the mechanical output from stars is converted into turbulence.\n\nThe results of this model are shown in Fig.~\\ref{fig:models}. When\nthe coupling efficiency of the mechanical energy from the young stellar\npopulation is high, around 5-15\\%, Q$\\sim$1 through most of the evolution\nand because of this, the star-formation rate is roughly constant. The\nenergy injection into the ISM from the accretion and star formation is\nhigh and therefore, dispersions are also high throughout much of the\nevolution. With the adopted scalings, the dispersions are about 50-150\nkm s$^{-1}$, and star-formation rates are about 50-200 M$_{\\sun}$\nyr$^{-1}$. All of these values are within the range observed in our\nsample. Star formation increases the Toomre parameter to Q$\\sim$1\nthroughout the evolution and stabilizes disk star formation. At higher\ncoupling efficiencies, the gas dispersion can be even higher but the\nstar formation is suppressed. \\citet{elmegreen10} set Q=1 when Q in\ntheir model dropped below one. However, since we allow the energy\ninjection from star formation to evolve with star formation, we find a\ndifferent time evolution. Unlike in the model of \\citet{elmegreen10},\nthe star-formation rate does not reach the accretion rate until well\nafter the accretion has stopped and the gas fraction has dropped below\n50\\% of the total (gas plus stellar) mass.\n\nWhen the coupling  with the mechanical energy output of the\nstellar population is weak, the disk is no longer stabilized near\nQ$\\sim$1 and Q becomes very small \\citep[similar to the results\nof][]{elmegreen10}. Consequently, star formation increases rapidly\nand exceeds the accretion rate. This high star-formation rate occurs\nagain when the gas fraction drops below about 50\\%. In this model,\nstar-formation rates are $\\sim$200 M$_{\\sun}$ yr$^{-1}$. This is high\nbut reasonable compared to the star-formation rates in our galaxies. The\nvalues in Table~\\ref{table:properties} are generally lower than this, but\nthey have not been corrected for extinction, which is likely to increase\nthem by factors of a few \\citep{L09}. We note that during the early,\ngas-rich phase of evolution in these models, the Jeans mass is relatively\nhigh which implies that the disks would be very clumpy, especially if\nthe mechanical energy from stars couples very efficiently to the gas\n\\citep[because such galaxies stay relatively gas rich; ][]{elmegreen04,\nelmegreen09a,elmegreen09,bournaud07,bournaud09, cervino10}.\n\nThe basic point illustrated by this simple model is that\nstar formation can play a key role in regulating large-scale\ninstabilities within the disks of distant galaxies. Previous studies\nhave come to the same conclusion in a somewhat different context\n\\citep[e.g.,][]{silk97,silk01,silk03, silk09}. What we have added to\nthe analysis is the use of the star formation to drive the line widths\n(which are a combination of turbulent and bulk motions). In addition,\ngiven a sufficiently high coupling efficiency, young stars can control\nthe subsequent star formation in distant galaxies. There does not need to\nbe correspondence between the instantaneous cosmological gas accretion\nrate and the star-formation rate. In fact, quite the opposite. Smooth\nconstant accretion is probably unrealistic \\citep[e.g.][]{danovich12}\nand what we have shown here is that star formation, even if initiated\nby the accretion event, does not have to be related closely in time with\naccretion. The amount of energy available from the supernovae and stellar\nwinds is much larger in comparison and very early in the evolution begins\nto control the star-formation history.\n\n\\subsection{Is the coupling unrealistically efficient?}\\label{coupling}\n\nThere are  of course the important questions of whether or not\ncoupling efficiencies as high as those that are needed to explain\nthe large line widths are realistic and if the energy output from\nthe young stellar population through SNe and stellar winds can drive\nthe necessary turbulence. Some modeling studies suggest  this is not\nthe case \\citep{d09, krumholz10}, while other suggest that it may be\n\\citep{joung09}.\n\nObservationally, there are few constraints on what the value of the\ncoupling factor might be. \\citet{krumholz10} criticized the analysis of\n\\citet{L09}, arguing that the required coupling efficiency between the\nmechanical energy from star-formation and the ambient ISM is too high. Our\nanalysis in the previous section suggested that coupling efficiencies\nof 5-15\\% are necessary to explain the approximate characteristics of\ndistant galaxies, especially the relatively high velocity dispersions. But\neven these estimates are very uncertain in that we do not really know\nthe dissipation time scales and that changes in the dissipation rate\nwill modify  the coupling factor. They must be short, approximately\none turnover time scale of turbulence (a few to tens of Myrs), as\nargued in \\citet{L09} and as suggested by the model discussed in the\nprevious section and the simulations and physical arguments presented\nin \\citep[e.g.][]{maclow99, elmegreen04, joung09}. The model suggests\nthat the dissipation time scale decreases with time and increases with\ncoupling efficiency. Since the dissipation rate is proportional to\nthe ratio of the velocity dispersion and the scale height of the disk,\nthe disk must be thick throughout much of its evolution. This agrees\nwith our finding that the gas has a relatively low ionization (and weak\nradiation pressure) so it must be widely distributed in the ISM.\n\nWhile we call this the ``coupling factor'', there are various  factors\nthat contribute to this coupling. To unwind these, we will use the well\nstudied nucleus of M82 as the analog for our distant galaxies, given that\nthe star formation intensity of M82 on a several 100 pc scale is similar\nto that of the galaxies presented here. First, the mass ejection from\nthe stellar population must be efficiently thermalized. In a study of the\nnuclear X-ray emission properties, which probes gas up to temperatures of\nabout 10$^{7.8}$K and directly probes the ``piston'' driving outflows,\n\\citet{strickland09} find that on scales of kpc, the thermalization\nefficiency is 0.3-1.0. The entrainment factor -- the amount of ambient\nISM that is swept-up, accelerated, and mixed into the hot high pressure\nplasma -- is also important. Again, the modeling of \\citet{strickland09}\nsuggests relatively low entrainment rates of 1-4 times the mass loss of\nthe stellar population in the region surrounding the nucleus of M82. In\naddition, in the optical emission line gas of M82, large line widths and\nhigh pressures are observed \\citep{west09}. This suggests that a fraction\nof the energy from the intense star formation is dissipated in the WIM.\n\nThere are several ways in which the coupling between the thermalized\nmechanical energy from young stars may be enhanced. Star formation which\nis distributed increases the coupling factor between the mechanical\nenergy and the cold\/warm ambient medium \\citep{fragile04, cooper09}.\nThis enhancement results from the fact that with distributed regions\nof star formation, say in a clumpy disk, overlapping bubbles of hot\nplasma begin to interact and cause the formation of dense rims of gas\nin this interaction zone \\citep{T-T03, T-T06}. This also has the effect\nof reducing the opening angles through which the hot plasma can flow\n\\citep{T-T03} which would thereby increase the central pressures in the\nstar forming regions, further enhancing its influence on the surrounding\nISM. Our photoionization and simple dynamical modeling implies that\nour galaxies have a geometrically ``thick'' ($\\sim$1 kpc) warm ionized\nmedium of moderate densities and low ionization parameters. Such a thick\nmedium would enhance the interaction between the mechanical luminosity\nof the young stars and the ambient ISM. The strong interaction zones,\nthe reduced opening angle of the outflows, and a thick warm ionized\nmedium would have the effect of smothering the outflow and increasing\nthe efficiency at which the mechanical energy couples to the ambient\nISM, and also likely increases the cooling of the hot plasma generated\nby the collective action of SNe and stellar winds in each star forming\nclump or region \\citep{wunsch08,hopkins12}. There is evidence that the\noptical emission line gas in distant galaxies may be geometrically thick\n\\citep[e.g.]{epinat12}. In addition, these interacting and overlapping\nregions of strong mechanical energy input may also drive gas to collapse\nand  form molecular clouds which may be sites of subsequent star formation\n\\citep{T-T05, hennebelle08}, thus aiding in completing one part of the\ngas cycle we are advocating.\n\nFurthermore, we have up to now only considered the influence of the\nmechanical energy output of the massive stars but not their radiation\nfield (except in the WIM itself). Depending on the star formation history\nand duration, a significant fraction of up to many times the momentum\nimparted by the mechanical output of a starburst could be imparted on\ndusty gas by radiation pressure \\citep{leitherer99,murray05}. Moreover,\nif the optical depth to IR photons is high, the radiation pressure may\ndominate the total momentum output \\citep{hopkins11}. However, it remains\ndifficult to judge the importance of radiation pressure in  the overall\npressure \\citep[cf.][]{murray05, silk10, ostriker11, hopkins11}. For our\ndiscussion, the important point is that radiation pressure is more likely\nto be effective in imparting momentum to the dense, dusty molecular gas\nwhereas the mechanical energy output will tend to flow around, engulf,\nand destroy the molecular clouds. Thus radiation pressure may be very\neffective in driving the large scale kinematics of the denser gas and\nthus regulate star formation globally.\n\nThe coupling efficiency needed to drive the turbulence is dependent on\nthe energy dissipation timescale of the gas. In the range of coupling\nefficiencies that lead to reasonable results compared to observations,\nthe energy dissipation timescale is in the range of a few to tens of\nMyrs or in other words, shorter than a dynamical time or the duration\nof the intense star formation, and approximately as long as one\nturnover timescale of turbulence for a 1 kpc thick \\mbox{H$\\alpha$} \\ emitting layer\n\\citep[see][]{L09}. Thus in such a model, the cycle proposed is predicted\nto be long-lived through repeated cycles of a turbulent cascade driven\nby the energy input of massive stars. Since the rotational lines of\nH$_2$ are one of the dominant coolants of warm molecular gas, and trace\nthe dissipation of supersonic turbulence \\citep[e.g.][]{guillard10},\nobservations of distant galaxies with MIRI on the JWST in the infrared\nwill be crucial in constraining the coupling efficiency and testing the\nscenario we have proposed.\n\n\\subsection{Summary}\n\nThrough our analysis of near-infrared integral field spectroscopic\nobservations we found that the optical emission line gas has a high\nthermal pressure compared to nearby galaxies, and ionization parameters\nthat range from those of giant \\mbox{H{\\sc ii}}\\ regions to those of diffuse low\ndensity gas away from star formation regions, similar to what is observed\nin nearby normal and starburst galaxies \\citep[e.g.][]{lehnert96, wang98,\nLT11b, kewley02, pellegrini11}. The broad optical emission lines observed\n($\\sigma_{\\mbox{H$\\alpha$} }$$\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}}$50-250 km s$^{-1}$) are not mainly due to beam\nsmearing but show that lines are intrinsically broader in regions of\nmore intense star formation. This suggests that the warm ionized gas\nhas a high level of turbulence and bulk motions, and participates in\noutflows \\citep{LT11b, genzel11} that are largely driven by the intense\nstar formation. The high level of turbulence and the influence of intense\nstar formation is consistent with the clumpy nature of distant galaxies\n\\citep[e.g.][]{elmegreen04,elmegreen09a,elmegreen09}.\n\nThe total pressure of the warm ionized medium appears to be dominated\nby the thermal pressure as its average volume-weighted density is very\nlow. The relatively low ionization parameters of the emission line gas\nsuggest the thermal pressure in the WIM is due to the mechanical energy\noutput of massive stars and not radiation pressure of the ionizing field.\n\nThe evidence we have presented favors the high turbulence being driven\nby mechanical output of young stars, which is the energy source that\nregulates the gas to be near the line of disk instability (Q$\\sim$1). The\nmechanical energy from the intense star-formation is likely efficiently\nthermalized \\citep{strickland09}, creating large regions of high\nturbulence. The intense radiation field from young stars may also impart\nmomentum to the gas, depending on the gas column densities and opacity\nto IR photons. The importance of the radiation pressure is that it may\ncouple to the dense dusty gas more effectively than the mechanical energy\noutput, and thus regulate star formation globally.\n\nBoth the mechanical energy (through the thermalized hot plasma generated\nby stellar winds and supernovae) and the radiation field of young stars\ngenerate turbulence, bulk motions and high turbulent pressures in the gas\nwhich are similar to the thermal pressure of the WIM and the hydrostatic\npressure. However, for the turbulent pressure to be similar to the\nhydrostatic pressure, the high velocity dispersions observed in the\nWIM have to be (at least partially) transferred to the dense molecular\ngas, which likely has a much larger mass and average volume weighted\ndensity than the recombining gas. Such transfer may happen because at\nhigh pressures the recombining gas is unstable and will quickly cool\nand collapse to a cold neutral phase \\citep{wolfire95}, and become\nmolecular if dusty \\citep{guillard09}. Since this collapse and cooling\ntime scale is much shorter than the dynamical time scale, the kinematics\nof the WIM and the CNM should be similar \\citep{guillard12}. This leads\nto the formation of gravitationally bound clouds which have dissipated\ntheir gravitational energy as turbulence in the molecular gas. Some of\nthese clouds will form stars, and some may be destroyed by the intense\nenergy output of massive stars. The ISM will quickly reach a dynamical\nequilibrium, due to the relatively short timescales for gas cooling,\nand the formation of molecular gas and gravitationally bound clouds. The\nshort timescale compared to those of star formation  indicates there is\nsufficient time for repeated cycles, allowing for the build-up of the\nstellar population in our distant galaxies.\n\nTo explain the observed star formation intensity -- line width\ndistribution, we made a simple model for the energy flow in a galaxy\nundergoing both intense gas accretion and star formation. This model\nsuggests that only a few to perhaps 15\\% of the injected energy from\nstars is  required to drive the high level of turbulence we observe. This\nrelatively  small fraction of the total energy injection is reasonable.\n\nIf the high dispersion observed in the optical emission line gas is\ndue to turbulence and if the denser phases of the ISM share such high\nturbulence, then the gas and stellar mass surface densities in z$\\sim$2\ngalaxies are consistent with Q$\\sim$1 and star formation is a plausible\nhypothesis for driving the gas towards the line of stability. If so,\nthis would be an avenue for self-regulated star formation in distant\ngalaxies. More observations, especially of the warm molecular gas in\ndistant intensely star forming galaxies  are needed to gauge whether or\nnot this picture is correct.\n\n\\begin{acknowledgements} \n\nThe work of LLT, MDL and PDM is supported by a grant from the Agence\nNationale de la Recherche (ANR) in France. MDL and NPHN wish to thank\nthe CNRS for their continuing support.\n\n\\end{acknowledgements}\n\n\\bibstyle{aa}\n\\bibliographystyle{aa}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nIn their famous paper {\\sl $L$-Functions for the Classical Groups} (in \\cite{G-PS-R}), Piatetski-Shapiro and\nRallis introduced a new type of global integral which\nrepresents the standard $L$-function for any split classical group. To\ndescribe their construction, let ${\\A}$ denote the ring of adeles\nof a global field $F$. Let $G$ denote any split classical group,\n$\\pi_1$ and $\\pi_2$ denote two irreducible cuspidal automorphic\nrepresentations of $G({\\A})$, and $\\varphi_{\\pi_1},\\varphi_{\\pi_2}$\nbe automorphic forms in the corresponding spaces. Then the global integral they introduced\nis\n\\begin{equation}\\label{global10}\n\\int\\limits_{G(F)\\times G(F)\\backslash G({\\A})\\times G({\\A})}\n\\varphi_{\\pi_1}(g_1)\\,\\varphi_{\\pi_2}(g_2)E((g_1,g_2),s)\\,dg_1\\,dg_2.\n\\end{equation}\nHere $E(h,s)$ is an Eisenstein series defined on the adelic points of another classical\ngroup $H$, which depends on the choice of $G$ and\ncontains the direct product $G\\times G$ as a subgroup.\nIn the case $G=GL_n$, a slight modification is required to handle the issue of convergence. Since the\ndomain of integration is two copies of the group $G$,\nthis type of construction has become known as the doubling method.\nMoreover,\nthe doubling method also works when\n$G=\\widetilde{Sp}_{2n}$, the two-fold metaplectic cover of the symplectic group $Sp_{2n}$.\n(This was known to Piatetski-Shapiro and Rallis.)\nIn all cases, after unfolding the integral it is easy to\nsee that \\eqref{global10} is not zero only if $\\pi_2$ is the contragredient of $\\pi_1$.\n\nThe doubling method is general in two aspects. First, as indicated\nabove, it is valid for all split classical groups.\nSecond, and maybe more important, this construction works for {\\sl{any}}\nirreducible cuspidal automorphic representation $\\pi_1$. This is a rare\nphenomenon in these types of constructions. Usually, such integrals\nunfold to some special model which is afforded by some but not all cuspidal automorphic\nrepresentations, such as the Whittaker model.\n(Although every cuspidal automorphic representation of $GL_n({\\A})$ is globally generic, this is not the case for other classical groups.)\nIn the above construction, after carrying\nout the unfolding process, one obtains the inner product\n\\begin{equation}\\label{spherical0}\n<\\pi_1(g)\\varphi_{\\pi_1},\\varphi_{\\pi_2}>=\\int\\limits_{G(F)\\backslash G({\\A})}\n\\varphi_{\\pi_1}(g_1g)\\,\\varphi_{\\pi_2}(g_1)\\,dg_1.\n\\notag\n\\end{equation}\nThis inner product is identically zero unless $\\pi_2$ is the\ncontragredient of $\\pi_1$, and then it is nonzero for some choices of data for any $\\pi_1$.\n\nIn this note we describe a generalization of the doubling method to covering groups.\nLet $G$ be a split classical group,\n$m\\ge 1$ and let $G^{(m)}({\\A})$ denote a metaplectic $m$-fold cover of $G(\\A)$. This\nis defined provided the underlying field has a full set of $m$-th roots of unity.\nWhen $m=1$, we understand $G^{(1)}$ to be the linear group $G$. \nLet $\\pi^{(m)}$ denote\na genuine irreducible cuspidal automorphic representation of $G^{(m)}({\\A})$, and let\n$\\tau^{(r)}$ denote a genuine irreducible cuspidal automorphic representation of\n$GL_k^{(r)}({\\A})$. Here $r=m$ except when $G=Sp_{2n}$\nand $m$ is even, in which case we set $r=m\/2$.\nVia doubling, we shall describe a global construction which represents the\npartial standard tensor product $L$-function attached to $\\pi^{(m)}\\times\n\\tau^{(r)}$. This partial $L$-function is the product of\nlocal $L$-functions over all finite places of $F$ for which the representations $\\pi^{(m)}$ and\n$\\tau^{(r)}$ are unramified. The local $L$-functions are Rankin-Selberg $L$-functions attached to the\nlocal Shimura lifts.  These will be described in \\eqref{tensor} below, and may also be obtained from\nthe work of Savin \\cite{Sa}.\n\nThe construction we introduce is a generalization of the construction described by\nthe integral \\eqref{global10}, keeping track of covers.  It is given by\n\\begin{equation}\\label{global11}\n\\int\\limits_{G(F)\\times G(F)\\backslash G({\\A})\\times G({\\A})}\n\\varphi_1^{(m)}(g_1)\\,\\overline{\\varphi_2^{(m)}(g_2)}\\,\nE^{U,\\psi_U}_{\\tau^{(r)}}((g_1,g_2),s)\\,dg_1\\,dg_2.\n\\end{equation}\nHere, for $i=1,2$, the functions $\\varphi_i^{(m)}$ are vectors in the space of $\\pi^{(m)}$. There \nis an implicit choice of section $G({\\A})\\rightarrow G^{(m)}({\\A})$ in \\eqref{global11}, and the choice of $r$ in terms of $m$ implies that the integral is in fact independent of this section.\nThe Eisenstein series $E$ is defined on $H^{(r)}({\\A})$\nwhere $H$ is a certain suitable split classical group,\nand $E^{U,\\psi_U}_{\\tau^{(r)}}$ is a certain Fourier coefficient of this Eisenstein\nseries. The induction data of $E$ depends on a\ncertain representation which we refer to as\na Whittaker-Speh-Shalika representation, which is defined on a cover of the general linear group.\nThis representation, which depends on $\\tau^{(r)}$, is\ndefined by means of a residue of an Eisenstein series, and when $r=1$\nit reduces to the well-known Speh representation, studied for example\nby Jacquet \\cite{J}. We discuss these representations in Section \\ref{speh} below.\n\n\nUnfortunately, and this is a weak point of this construction, while these\nresidue representations are conjectured to exist for our \ncovering groups, this is not proved in general. This is because, in the absence of a general theory\nof Shimura lifts for covers of the general linear group,\n the study of the necessary partial $L$-functions needed to analyze the Eisenstein series \nattached to automorphic representations on covers of the general linear group is not available.\nWe discuss this issue in Section \\ref{res1}, and suggest some solutions in various cases. See also Conjecture \\ref{conj1}.\n\nIn the sections below we sketch the general construction of the\nglobal integrals for all classical groups $G$. We discuss the\nsymplectic case in more detail\nand in Theorem \\ref{thloc1} we state the result of the\nunramified computations.\nWork carrying out\nthe global unfolding and the computation of the local unramified integrals\nfor other classical groups is in progress and we shall report on it in detail in a follow-up article.\n\n\n\n\n\n\n\\section{Whittaker-Speh-Shalika Representations}\\label{speh}\nIn this section we define and construct a set of representations which we will then use in the global integrals.\n\n\n\\subsection{Definition of Whittaker-Speh-Shalika representations}\\label{def}\nLet $m, a$ and $b$ denote three positive integers. Let $GL_{ab}^{(m)}({\\A})$ denote an $m$-fold metaplectic cover of $GL_{ab}({\\A})$, as in Kazhdan and Patterson \\cite{K-P}. When $m=1$ we take $GL_{ab}^{(m)}$ to be $GL_{ab}$. Recall that the cover is split over unipotent subgroups. For example, we can choose a section \n$GL_{ab}({\\A})\\rightarrow GL_{ab}^{(m)}({\\A})$ which is a splitting of $GL_{ab}(F)$ and \n$V({\\A})$, where $V$ is the subgroup of upper triangular unipotent matrices.\nTherefore notions involving unipotent orbits transfer to covering groups\n{\\it mutatis mutandis}.\n\nLet $V_{a,b}$ be the unipotent radical of the standard parabolic subgroup of $GL_{ab}$ whose Levi part is $GL_b\\times GL_b \\times\\cdots\\times GL_b$. Here $GL_b$ appears $a$ times.\nIn terms of matrices, this group consists of all\nunipotent matrices $X$ of $GL_{ab}$ of the form\n\\begin{equation}\\label{mat1} X=\n\\begin{pmatrix} I&X_{1,2}&X_{1,3}&\\dots&X_{1,a}\\\\\n&I&X_{2,3}&\\dots&X_{2,a}\\\\\n&&I&\\ddots&\\vdots\\\\ &&&\\ddots&X_{a-1,a}\\\\ &&&&I\\end{pmatrix},\n\\qqua\nX_{i,j}\\in \\text{Mat}_{b\\times b},\n\\end{equation}\nwhere $I$ is the $b\\times b$ identity matrix.   Fix a nontrivial character $\\psi$ of $F\\backslash \\A$, and\nlet $\\psi_{a,b}$ be the character of $V_{a,b}(F)\\backslash V_{a,b}(\\A)$ given by\n\\begin{equation*}\n\\psi_{a,b}(X)=\\psi(\\text{tr}(X_{1,2}+X_{2,3}+\\cdots +X_{a-1,a}))\n\\end{equation*}\nwith $X$ as in \\eqref{mat1}.\n\nSuppose $\\sigma^{(m)}$ is an automorphic representation of the group $GL_{ab}^{(m)}({\\A})$.\nIf $\\varphi_{\\sigma^{(m)}}$ is in the space of $\\sigma^{(m)}$, then the integral\n\\begin{equation}\\label{whspeh1}\nW(\\varphi_{\\sigma^{(m)}})(g)=\\int\\limits_{V_{a,b}(F)\\backslash V_{a,b}({\\A})}\n\\varphi_{\\sigma^{(m)}}(vg)\\,\\psi_{a,b}(v)\\,dv\n\\end{equation}\nis a Fourier coefficient corresponding to the unipotent orbit $(a^b)$. We refer to this  Fourier coefficient as\nthe Whittaker-Speh-Shalika coefficient of the representation  $\\sigma^{(m)}$.\nNotice that when $b=1$, the Fourier coefficient \\eqref{whspeh1} reduces to the  well-known Whittaker coefficient, and\nwhen it is nonzero we say that the representation $\\sigma^{(m)}$ is globally generic.   For each completion $F_\\nu$ of $F$\nwe have a similar character of $V_{a,b}(F_\\nu)$ which we also denote $\\psi_{a,b}$.\n\nWe make the following definition.\n\\begin{definition}\\label{def1}\nAn irreducible genuine automorphic representation  $\\sigma^{(m)}$ of the group $GL_{ab}^{(m)}({\\A})$ is a Whittaker-Speh-Shalika representation of type $(a,b)$ if:\\\\\n\\begin{enumerate}[leftmargin=*,label=\\textbf{\\arabic*})]\n\\item\\label{def:Whittaker-Speh-Shalika 1}\nThis representation has a nonzero Fourier coefficient corresponding to the unipotent orbit $(a^b)$, and\nmoreover, for all orbits which are greater than or not related to $(a^b)$, all corresponding Fourier coefficients are zero for all choices of data. In the notation of Ginzburg \\cite{G1}, this statement can be written as ${\\mathcal O}_{GL_{ab}^{(m)}}(\\sigma^{(m)})=(a^b)$.\\\\\n\\item\\label{def:Whittaker-Speh-Shalika 2}\nFor a finite place $\\nu$, let $\\sigma^{(m)}_\\nu$ denote the\nirreducible constituent of $\\sigma^{(m)}$ at $\\nu$.\nSuppose that $\\sigma^{(m)}_\\nu$ is an unramified representation. Then ${\\mathcal O}_{GL_{ab}^{(m)}}(\\sigma^{(m)}_\\nu)=(a^b)$.\n(That is, the local analogue of part~\\ref{def:Whittaker-Speh-Shalika 1} holds.) Moreover, the dimension of\n\\begin{equation}\\label{speh1}\n{\\Hom}_{V_{a,b}(F_\\nu)}(\\sigma^{(m)}_\\nu,\\psi_{a,b})\n\\end{equation}\nis one.\n\\end{enumerate}\n\\end{definition}\n\nWhen $m=b=1$, it is well known that condition\n\\ref{def:Whittaker-Speh-Shalika 2} in Definition \\ref{def1} is satisfied. However, on covering groups not every generic representation satisfies condition \\ref{def:Whittaker-Speh-Shalika 2}.\n\n\nBefore constructing examples of Whittaker-Speh-Shalika representations, we note that the\nFourier coefficient $W(\\varphi_{\\sigma^{(m)}})(g)$ enjoys an extra invariance property. Indeed, let $GL_b^{\\Delta}$\ndenote the image of $GL_b$ inside $GL_{ab}$ under the diagonal embedding $h\\mapsto h_0:=\\text{diag}(h,h,\\ldots,h)$.\nThen $GL_b^{\\Delta}$ is the stabilizer of the character\n$\\psi_{a,b}$ inside the group $GL_b\\times GL_b \\times\\cdots\\times GL_b$.\nConsider the group\n$SL_b^{\\Delta}$ embedded inside $GL_b^{\\Delta}$. Then the function\n$W(\\varphi_{\\sigma^{(m)}})(g)$ is left invariant\nby all matrices $h_0$ as above with $h\\in SL_b({\\A})$, i.e.\\\n$W(\\varphi_{\\sigma^{(m)}})(h_0g)=W(\\varphi_{\\sigma^{(m)}})(g)$\nfor all $h_0\\in SL_b^{\\Delta}({\\A})$.\nThis follows since the unipotent orbit attached to $\\sigma^{(m)}$ is $(a^b)$.\nMoreover, if we expand along any unipotent subgroup of $SL_b({\\A})$, then the nontrivial contribution to the expansion is\ntrivial. This follows since the nontrivial term of the expansion is\nassociated with a unipotent orbit which is strictly greater than $(a^b)$. See Friedberg and Ginzburg,\n\\cite{F-G}, Proposition 3, for details.\nThis means that if $m>1$, then the group $SL_b({\\A})$, embedded diagonally in $GL_{ab}^{(m)}({\\A})$,\nmust split under the $m$-fold cover. This implies $m\\mid a$.\n\n\n\n\n\\subsection{ Construction of Whittaker-Speh-Shalika representations}\n\\label{construct}\nIn this subsection we shall construct examples of Whittaker-Speh-Shalika representations, by means of residues of Eisenstein series.\n\n\nLet $k$ and $c$ be two positive integers. Denote $b=mc$.\nLet $\\tau^{(m)}$ denote a genuine irreducible cuspidal automorphic representation of the group $GL_k^{(m)}({\\A})$,\nand $\\underline{s}=(s_1,\\ldots,s_b)\\in\\C^b$. We\nconstruct an Eisenstein series $E_{\\tau^{(m)}}^{(m)}(g,\\underline{s})$\ndefined on the group $GL_{kb}^{(m)}({\\A})$ as follows. Let\n$P_{k,b}$ denote the standard  parabolic subgroup of $GL_{kb}$ whose\nLevi part is $GL_k\\times GL_k\\times\\ldots\\times GL_k$. As in \\cite{F-G}, Section 2, we construct the induced representation\n\\begin{equation}\\label{ind1}\n\\text{Ind}_{P_{k,b}^{(m)}({\\A})}^{GL_{kb}^{(m)}({\\A})}\n(\\tau^{(m)}|\\cdot|^{s_1}\\otimes \\tau^{(m)}|\\cdot|^{s_2}\\otimes\n\\cdots\\otimes \\tau^{(m)}|\\cdot|^{s_b})\\delta_{P_{k,b}}^{1\/2}.\n\\end{equation}\nWe remark that the induction process here is more complicated when $m>1$ since the $GL_k$ blocks do not commute in the covering group.\nLet $E_{\\tau^{(m)}}^{(m)}(g,\\underline{s})$ denote the Eisenstein series associated with this induced representation.\nWhen $m=1$, these\nrepresentations and their residues were studied by various authors. See for example \\cite{J}.\nWhen $m>1$, these Eisenstein series were constructed and studied in Suzuki \\cite{Su}, Section 8.\n\nWe start with the following conjecture.\n\\begin{conjecture}\\label{conj1}\nGiven  $\\tau^{(m)}$ as above, the Eisenstein series $E_{\\tau^{(m)}}^{(m)}(g,\\underline{s})$ has a simple multi-residue at the point\n$$s_1+s_2+\\cdots +s_b=0;\\ \\ \\ \\ m(s_i-s_{i+1})=1;\\ \\ \\ \\ 1\\le i\\le b-1.$$\n\\end{conjecture}\nWe shall discuss this conjecture in Section \\ref{res1} below.\n\nAssuming Conjecture \\ref{conj1}, denote by $E_{\\tau^{(m)}}^{(m)}(g)$\nthe residue of the Eisenstein series\n$E_{\\tau^{(m)}}^{(m)}(g,\\underline{s})$\nat the above point (this depends on the choice of test vector, but we suppress this from the notation).\nLet ${\\mathcal L}_{\\tau^{(m)}}^{(m)}$ denote the representation of $GL_{kb}^{(m)}({\\A})$ generated\nby all the residue functions $E_{\\tau^{(m)}}^{(m)}(g)$.\nAlso, let  $Z_\\A$ denote the center of the group $GL_{kb}^{(m)}({\\A})$.\nWe have the following two results.\n\\begin{proposition}\\label{prop1}\nThe automorphic  representation ${\\mathcal L}_{\\tau^{(m)}}^{(m)}$ lies in the discrete spectrum of the space $L^2(Z_\\A GL_{kb}^{(m)}(F)\\backslash GL_{kb}^{(m)}({\\A}))$.\n\\end{proposition}\n\n\\begin{proposition}\\label{prop2}\nWe have  ${\\mathcal O}_{GL_{kmc}^{(m)}}\n({\\mathcal L}_{\\tau^{(m)}}^{(m)})=((km)^c)$.\n\\end{proposition}\nFrom these results we deduce that the representation\n${\\mathcal L}_{\\tau^{(m)}}^{(m)}$ has at least one irreducible summand\nwhich has a nonzero Fourier coefficient corresponding to the unipotent\norbit $((km)^c)$. Denote this summand by ${\\mathcal E}_{\\tau^{(m)}}^{(m)}$. Then we have\n\\begin{theorem}\\label{thspeh1}\nThe representation ${\\mathcal E}_{\\tau^{(m)}}^{(m)}$ is a Whittaker-Speh-Shalika representation of type $(km,c)$.\n\\end{theorem}\nThe first condition of Definition \\ref{def1} for\nrepresentation ${\\mathcal E}_{\\tau^{(m)}}^{(m)}$ follows from the\nabove discussion.\nThe main content of Theorem \\ref{thspeh1} is\nthe verification of {\\bf 2)} in Definition \\ref{def1}.\n\n\n\\section{The Global Construction}\\label{global}\nIn this section we introduce the general global integral. In the next section we treat the case of $Sp_{2n}$ in detail.\n\nLet $m,\\ n$ and $k$ denote three positive integers.\nLet $G$ denote one of the split groups  $GL_n,$\n$Sp_{2n},$ $SO_{2n+1}$ and $SO_{2n}$.\nLet $c(n)=n$ if $G=GL_n$,\n $c(n)=2n$ if $G=Sp_{2n}$ or $G=SO_{2n}$, and $c(n)=2n+1$ if $G=SO_{2n+1}$.\nLet $G^{(m)}({\\A})$ denote an\n$m$-fold metaplectic cover of $G({\\A})$.\nThere is a difference  in the symplectic group case depending on the parity of $m$;\n in this section we shall assume that if $G=Sp_{2n}$ then $m$ is odd.\n\nDepending on $G$, we introduce another classical group $H$ on which we shall construct an Eisenstein series.\nLet\n\\begin{align*}\nH=\n\\begin{cases}\nGL_{2nkm}&G=GL_n,\\\\\nSp_{4nkm}&G=Sp_{2n},\\\\\nSO_{4nkm}&G=SO_{2n},\\\\\nSO_{2(2n+1)km}&G=SO_{2n+1}.\n\\end{cases}\n\\end{align*}\nLet $P$ denote the maximal parabolic subgroup of $H$ whose Levi part is\n\\begin{align*}\n\\begin{cases}\nGL_{kmc(n)}\\times\nGL_{kmc(n)}&G=GL_n,\\\\\nGL_{kmc(n)}&\\text{otherwise.}\n\\end{cases}\n\\end{align*}\nFor later use, let $U(P)$ denote the unipotent radical of $P$.\nLet $\\tau^{(m)}$ denote a genuine\nirreducible cuspidal representation of $GL_k^{(m)}({\\A})$.\nAs in the previous section, construct the residue representation ${\\mathcal E}_{\\tau^{(m)}}^{(m)}$  defined on the group $GL_{kmc(n)}^{(m)}({\\A})$. This is a Whittaker-Speh-Shalika representation of type $(km,c(n))$.\nForm the Eisenstein series $E_{\\tau^{(m)}}^{(m)}\n(h,s)$ defined on the group $H^{(m)}({\\A})$ attached to the induced\nrepresentation\n\\begin{align*}\n\\begin{cases}\n\\text{Ind}_{P^{(m)}({\\A})}^{H^{(m)}({\\A})}(\n{\\mathcal E}_{\\tau^{(m)}}^{(m)}\\otimes {\\mathcal E}_{\\tau^{(m)}}^{(m)}\n)\\delta_P^s&G=GL_n,\\\\\n\\text{Ind}_{P^{(m)}({\\A})}^{H^{(m)}({\\A})}{\\mathcal E}_{\\tau^{(m)}}^{(m)}\\delta_P^s&\\text{otherwise.}\n\\end{cases}\n\\end{align*}\n\nTo introduce the global integral, let $\\pi_1^{(m)}$ and $\\pi_2^{(m)}$ denote two genuine\n irreducible cuspidal automorphic representations defined on the group $G^{(m)}({\\A})$. When $G=GL_n$, one has to be careful with the center of the group, and also with the issue of convergence. To avoid these minor complications,\nin describing the integral we shall assume\n that $G$ is not the group $GL_n$.\nThen the global integral we consider is\n\\begin{equation}\\label{global1}\n\\int\\limits_{G(F)\\times G(F)\\backslash G({\\A})\\times G({\\A})}\n\\int\\limits_{U(F)\\backslash U({\\A})}\\varphi_1^{(m)}(g_1)\\,\\overline{\\varphi_2^{(m)}(g_2)}\\,\nE_{\\tau^{(m)}}^{(m)}(u(g_1,g_2),s)\\,\\psi_U(u)\\,du\\,dg_1\\,dg_2.\n\\end{equation}\nHere, $\\varphi_i^{(m)}$ are vectors in the spaces of $\\pi_i^{(m)}$.\n\nWe still need to check that the integral is well defined. From the description below it follows that the embeddings of the two copies of $G^{(m)}({\\A})$  in $H^{(m)}({\\A})$ commute with each other. It\nalso follows that the function\n$$(g_1,g_2)\\mapsto \\varphi_1^{(m)}(g_1)\\overline{\\varphi_2^{(m)}(g_2)}\nE_{\\tau^{(m)}}^{(m)}((g_1,g_2),s)\\ \\ \\ \\ \\ \\ \\ g_i\\in G({\\A})$$\ndoes not depend on the choice of section $G({\\A})\\rightarrow G^{(m)}({\\A})$. Hence the integral \\eqref{global1} is well\ndefined and converges absolutely for $\\text{Re}(s)$ large.\n\nFinally, we describe in some detail\nthe unipotent group $U$, the character $\\psi_U$ and the embedding of $G\\times G$ inside $H$.\nLet\n\\begin{align*}\nH_{c(n)}=\n\\begin{cases}\nGL_{2n}\\times GL_{n}\\times \\ldots \\times GL_{n}&G=GL_n,\\\\\nSp_{4n}&G=Sp_{2n},\\\\\nSO_{4n}&G=SO_{2n},\\\\\nSO_{4n+2}&G=SO_{2n+1}.\n\\end{cases}\n\\end{align*}\nWhen $G=GL_n$, the group $GL_n$ appears $mk-1$ times in $H_{c(n)}$, and we mention this case for the sake of completeness.\nLet $Q_{n,m,k}$ denote the standard parabolic subgroup of $H$ whose\nLevi part is $GL_{c(n)}\\times GL_{c(n)}\\times\\ldots GL_{c(n)}\\times\nH_{c(n)}$ where $GL_{c(n)}$ appears $mk-1$ times.\nLet $U$ denote the unipotent radical of the parabolic subgroup\n$Q_{n,m,k}$.\n\nTo define the character $\\psi_U$, consider the unipotent orbit\n$((2km-1)^{c(n)}1^{c(n)})$ associated with the group $H$. It follows from Collingwood\nand McGovern \\cite{C-M} that this is a well defined orbit for every group $H$,\nand that the stabilizer of this orbit is the group $G\\times G$.\nFrom \\cite{G1} we obtain that a Fourier coefficient associated with\nthis orbit can be constructed using the group $U$, and one can define\na character $\\psi_U$ such that the stabilizer inside the Levi part\nof $Q_{n,m,k}$ is the split group $G\\times G$. In the next section\nwe shall construct this Fourier coefficient explicitly in the symplectic group case. As mentioned above, it follows\nfrom the way the group $G\\times G$ is embedded inside $H$ that the\ncovers in integral \\eqref{global1} are compatible.\n\nHere is our main theorem. (Notation not defined above is similar to that in other doubling method computations.)\n\\begin{theorem}\\label{th10}\nThe integral \\eqref{global1} is well defined, converges absolutely for $\\text{Re}(s)$ large, and admits a meromorphic continuation to the whole complex plane. It is not identically zero only if\n$\\pi_1^{(m)}=\\pi_2^{(m)}=\\pi^{(m)}$. In this case, for $\\text{Re}(s)$ large it is equal to\n\\begin{equation}\\label{global2}\n\\int\\limits_{G({\\A})}\\int\\limits_{U_0({\\A})}\n<\\pi^{(m)}(g)\\varphi_{\\pi^{(m)}},\\overline{\\varphi_{\\pi^{(m)}}}  >f_{W({\\mathcal E}_{\\tau^{(m)}}^{(m)})}(\\delta u_0(1,g),s)\n\\,\\psi_U(u_0)\\,du_0\\,dg.\n\\end{equation}\nHere\n\\begin{equation}\\label{spherical}\n<\\pi^{(m)}(g)\\varphi_{\\pi^{(m)}},\\varphi_{\\pi^{(m)}}>=\\int\\limits_{G(F)\\backslash G({\\A})}\n\\varphi_{\\pi^{(m)}}(g_1g)\\,\\varphi_{\\pi^{(m)}}(g_1)\\,dg_1.\n\\end{equation}\nIn particular the integral \\eqref{global1} represents an Euler product.\n\\end{theorem}\n\n\n\n\n\n\n\\section{The Symplectic Group Case}\\label{sp}\nIn this section we give some details for the symplectic group. Let $G=Sp_{2n}$.\nFor this group there is a  difference depending on the parity of $m$.\nTo give a uniform construction, let $r=m$ if $m$ is odd, and $r=m\/2$ if $m$ is even. Thus, the integral we will consider is integral \\eqref{global11} which agrees with integral \\eqref{global1} when $m=r$\nis an odd number.\nLet $H=Sp_{4nrk}$. Let\n${\\mathcal E}_{\\tau^{(r)}}^{(r)}$ be the Whittaker-Speh-Shalika representation of type $(rk,2n)$, as constructed in subsection \\ref{construct}. This representation is defined on the group $GL_{2nrk}^{(r)}({\\A})$, and in the notations of Definition \\ref{def1} condition {\\bf 1)}, we have ${\\mathcal O}_{GL_{2nrk}}({\\mathcal E}_{\\tau^{(r)}}^{(r)})=((rk)^{2n})$.\n\nTo describe the group $U$ in integral \\eqref{global11}, let $Q_{n,r,k}$ denote the parabolic subgroup of $Sp_{4nrk}$ whose Levi\npart is $GL_{2n}\\times\\ldots\\times GL_{2n}\\times Sp_{4n}$. Here $GL_{2n}$ appears $rk-1$ times. Denote by $U_{n,r,k}$\nor simply by $U$\nthe unipotent radical of $Q_{n,r,k}$. We may identify the quotient $U\/[U,U]$ with the group\n$$L=\\text{Mat}_{2n}\\oplus\\ldots\\oplus \\text{Mat}_{2n} \\oplus \\text{Mat}_{2n\\times 4n}.$$\nHere $\\text{Mat}_{2n}$ appears $rk-2$ times.\nTo define the character $\\psi_U$ it is enough to specify it on $L$. For\n$(X_1,\\ldots,X_{rk-2},Y)\\in L$ define $\\psi_L$ as $\\psi(\\text{tr}(X_1+\\cdots +X_{r-1})\n+\\text{tr}'(Y))$. Here $X_i\\in \\text{Mat}_{2n}$ and $Y\\in \\text{Mat}_{2n\\times 4n}$. To define\n$\\text{tr}'(Y)$, write\n$$Y=\\begin{pmatrix} Y_1&Z_1&Y_2\\\\ Y_3&Z_2&Y_4 \\end{pmatrix},\\quad\nY_i\\in \\text{Mat}_{n\\times n};\\quad Z_j\\in\\text{Mat}_{n\\times 2n}.$$\nThen $\\text{tr}'(Y)=\\text{tr}(Y_1+Y_4)$.\nLet $\\psi_U$ denote the extension of $\\psi_L$ to $U$ which is trivial on $[U,U]$.\n It follows from \\cite{G1} that the\ncorresponding Fourier coefficient given by $U$ and $\\psi_U$ is associated with the unipotent\norbit $((2rk-1)^{2n}1^{2n})$.\n\nFinally, we specify the embedding of $(g_1,g_2)\\in Sp_{2n}\\times Sp_{2n}$ in $Sp_{4nrk}$. It is\ngiven by $\\text{diag}(g_1,\\ldots,g_1,(g_1,g_2),g_1^*,\\ldots,g_1^*)$. Here $g_1$ appears $2rk-1$ times,\nand by $(g_1,g_2)$ we mean the usual embedding inside $Sp_{4n}$, i.e.\\\n\\begin{equation}\\label{diag1}\n(g_1,g_2)\\mapsto \\begin{pmatrix} g_{1,1}&&g_{1,2}\\\\ &g_2&\\\\ g_{1,3}&&g_{1,4}\\end{pmatrix},\\qquad g_1=\\begin{pmatrix} g_{1,1}&g_{1,2}\\\\ g_{1,3}&g_{1,4}\\end{pmatrix}\n;\\quad g_{1,i}\\in Mat_{n\\times n}.\n\\end{equation}\nThe entries with asterisk are determined so that embedding is symplectic.\nIt is now easy to check that the coverings are compatible, and hence\nthat integral \\eqref{global11} is well defined.\n\nIn \\cite{G2}, Ginzburg introduced the dimension equation, which is strongly correlated with global integrals\nthat unfold to Euler products.\nRoughly speaking, this equation states that the sum of dimensions of the representations\ninvolved in the integral is equal to the sum of the dimensions of the groups involved.\nWe note that the dimension equation is satisfied in our case.\nTo see what is involved, let us verify this\nwhen $m=r$ is odd.\nThe sum of the dimensions of the representations involved\nis the sum of the dimension of the\nEisenstein series $E_{\\tau^{(m)}}^{(m)}(\\cdot,s)$ and the dimension of the\nfunctional $<\\pi^{(m)}(g)\\varphi_{\\pi^{(m)}},\\varphi_{\\pi^{(m)}}>$\nobtained after unfolding the integral.\nIt follows from  \\eqref{spherical} that the dimension of this functional is $\\text{dim}\\ Sp_{2n}$.\nThus the equation we need to verify is\n\\begin{equation}\\label{dim1}\n\\text{dim}\\ Sp_{2n} + \\text{dim}\\ E_{\\tau^{(m)}}^{(m)}(\\cdot,s)= 2\\,\\text{dim}\\ Sp_{2n} +\\text{dim}\\ U.\n\\end{equation}\nFrom \\cite{G1} we see that\n\\begin{equation*}\n\\text{dim}\\ E_{\\tau^{(m)}}^{(m)}(\\cdot,s)= \\text{dim}\\ {\\mathcal E}_{\\tau^{(m)}}^{(m)}+\\text{dim}\\ U(P)=\\frac{1}{2}\\text{dim}\\ ((km)^{2n})\n+ \\text{dim}\\ U(P).\n\\end{equation*}\n Here $\\text{dim}\\ ((km)^{2n})$ is the dimension of the unipotent orbit\n$((km)^{2n})$ and $U(P)$ is the unipotent radical of the maximal parabolic $P$ defined in\nSection \\ref{global}. The number $\\frac{1}{2}\\text{dim}\\ ((km)^{2n})$ is equal to\nthe dimension of the unipotent radical of the parabolic subgroup of $GL_{2nmk}$ whose Levi\npart is $GL_{2n}\\times\\ldots\\times GL_{2n}$ where $GL_{2n}$ appears $mk$ times. Thus its\ndimension is equal to $2n^2km(km-1)$. Comparing this with the dimension of $U(P)$,\nequation \\eqref{dim1} follows.\n\nWe end this section with a theorem regarding the unramified computation.\nTo simplify notation we shall assume that $m=r$ is odd.\nWe first define the local $L$-functions under consideration. In general, if $\\tau$ is\na local unramified representation of an $m$-fold cover of $G$, it is a constituent of an unramified principal series representation, and corresponds to some $k$-tuple $\\chi=(\\chi_1,\\ldots,\\chi_k)$, where $\\chi_i$ is an unramified character of $F^*$. When $m=1$,\nthe values of $\\chi_i$\nat a local uniformizer are simply\nthe Satake parameters of $\\tau$. For the $m$-fold cover,\nwe define local $L$-functions at unramified places by using the standard definition in the case of linear groups, but replacing each $\\chi_i$ with its $m$-th power.\n\nFor example, let $\\tau_{\\nu}^{(m)}=\n\\text{Ind}_{B^{(m)}_{GL}}^{GL^{(m)}_k}\\chi\\delta_{B_{GL}}^{1\/2}$ denote the local\nunramified component of $\\tau^{(m)}$ at a finite place $\\nu$. Here $B_{GL}$ is\nthe Borel subgroup of $GL_k$ and $\\chi=(\\chi_1,\\ldots,\\chi_k)$ is an\nunramified character. Similarly let $\\pi_{\\nu}^{(m)}=\n\\text{Ind}_{B^{(m)}_{Sp}}^{Sp^{(m)}_{2n}}\\mu\\delta_{B_{Sp}}^{1\/2}$.\nLet $p$ be a generator of the maximal ideal in the ring of integers of\nthe field $F_\\nu$ and $q=|p|_\\nu^{-1}$.\nThen the local standard tensor product\n$L$ function is defined by\n\\begin{equation}\\label{tensor}\nL(s,\\pi_{\\nu}^{(m)}\\times\\tau_{\\nu}^{(m)})=\\prod_{i=1}^{n}\\prod_{j=1}^k\n\\frac{1}{(1-\\mu_i^{m}(p)\\chi_j^m(p)q^{-s})(1-\\mu_i^{-m}(p)\\chi_j^m(p)q^{-s})\n(1-\\chi_j^m(p)q^{-s})}.\n\\end{equation}\nIn a similar way one can define\nthe local standard  $L$-function $L(s,\\tau_\\nu^{(m)})$, the local exterior\nsquare $L$-function $L(s,\\tau_\\nu^{(m)},\\wedge^2)$ and the local symmetric square $L$-function $L(s,\\tau_\\nu^{(m)},\\vee^2)$.\n\n\n\nIt follows from Theorem \\ref{th10} that for $\\text{Re}(s)$ large, integral \\eqref{global2} is a product of local integrals.\nAt a finite unramified place $\\nu$, the local integral is given by\n\\begin{equation}\\label{local1}\nZ(s,\\pi^{(m)},\\tau^{(m)})=\\int\\limits_{G}\\int\\limits_{U_0}\n\\omega_{\\pi^{(m)}}(g)f_{W}(\\delta u_0(1,g),s)\\,\\psi_U(u_0)\\,du_0\\,dg.\\notag\n\\end{equation}\nHere $\\omega_{\\pi^{(m)}}(g)$ is the spherical function of $\\pi^{(m)}$, and\n$f_W$ is the unramified function obtained at the place $\\nu$ from the\nfactorizable function $f_{W({\\mathcal E}_{\\tau^{(m)}}^{(m)})}$. (We have dropped the subscripts $\\nu$ to condense\nthe notation.) We have\n\\begin{theorem}\\label{thloc1}\nSuppose that $m=r$ is odd. Then  for $\\text{Re}(s)$ large, the integral $Z(s,\\pi^{(m)},\\tau^{(m)})$ is equal to\n\\begin{equation}\\label{local2}\n\\frac{L(\\alpha s-\\frac{\\alpha-1}{2},\\pi^{(m)}\\times\\tau^{(m)})}{L(\\alpha (s-\\frac{1}{2})+nm+\\frac{1}{2},\\tau^{(m)})\\prod_{j=1}^{nm}\nL(\\alpha (2s-1)+2j,\\tau^{(m)},\\wedge^2)\n\\prod_{j=1}^{nm-1}\nL(\\alpha (2s-1)+2j+1,\\tau^{(m)},\\vee^2)}\\notag\n\\end{equation}\nwhere $\\alpha=m(2nmk+1)$.\n\\end{theorem}\n\nWe remark that, working in the context of Brylinski-Deligne extensions, Gao \\cite{Ga} also attaches Euler products to pairs of automorphic representations on covering groups.\nHe does so by studying the constant term of Eisenstein series on covering groups, that is,\nby means of a generalization of the Langlands-Shahidi method.  However, it is not clear if his Euler products\nare the same as those obtained in this note or not.\n\n\n\\section{Residues of Eisenstein Series on Covers of the General Linear Group}\\label{res1}\nIn this section\nwe discuss Conjecture \\ref{conj1} of subsection\n\\ref{construct} above. One way to study this Conjecture is by considering the\nvarious constant terms of the corresponding Eisenstein series. This reduces\nthe problem of determining the poles to the study of the\npoles of certain\nintertwining operators.  This in turn reduces to the study of the poles of the\npartial $L$-function $L^S(s,\\tau^{(m)}\\times \\hat{\\tau}^{(m)})$,\nwhere $S$ is a finite set of places including all archimedean places such that\nthe representation $\\tau^{(m)}_\\nu$ is unramified for $\\nu\\not\\in S$.\nHere $\\hat{\\tau}^{(m)}$ is the contragredient representation of $\\tau^{(m)}$.\nThe partial $L$-function is by definition the product over $\\nu\\not\\in S$ of local $L$-functions\ndefined similarly to \\eqref{tensor}.\n\nWhen $m=1$, Jacquet, Piatetski-Shapiro\nand Shalika studied this $L$-function by means of the Rankin-Selberg\nmethod.  See  for example Gelbart and Shahidi \\cite{G-S}, Section 1.7, for an overview of\nthese constructions. Using this method one may establish that\nthe partial $L$-function has a simple pole at $s=1$. Then it\nfollows from \\cite{J}, for example, that Conjecture~\\ref{conj1} holds.\nThe key property that makes this work when $m=1$ is the uniqueness of the\nWhittaker model. Unfortunately, this uniqueness does not hold when $m>1$.\n\nSuzuki studies these Eisenstein series\nand their residues for higher covers in the last section of his paper \\cite{Su}.\nIn order to establish that the residue exists, he assumes the existence of a generalized\nShimura lifting, that is, a correspondence\nbetween irreducible cuspidal automorphic representations of the group\n$GL_n^{(m)}({\\A})$ and automorphic representations of the group\n$GL_n({\\A})$ which satisfies certain properties.\nAt the moment, such a lifting has only been proved in full\nfor covers of $GL_2$ (Flicker, \\cite{F}).\nIf one had the lift in general and if the lifted automorphic representation of $GL_n({\\A})$\nwas cuspidal, then it would follow from the case $m=1$  that the partial $L$-function\n$L^S(s,\\tau^{(m)}\\times \\hat{\\tau}^{(m)})$ has a simple pole at $s=1$. This would then\nimply Conjecture \\ref{conj1}. We summarize:\n\\begin{proposition}\\label{prop10}\nAssume that $\\pi^{(m)}$ satisfies the two conditions in \\cite{Su} ,  Section 8.5, p. 752. Then Conjecture~\\ref{conj1} holds.\n\\end{proposition}\n\nAn alternative method to study the residues of Eisenstein is given in\nJacquet and Rallis \\cite{J-R}. In\nthis method one constructs a certain global nonzero integral which involves the residue representation. The non-vanishing of the global\nperiod then implies that the residue is nonzero. When $m=2$ we propose\nthe following construction. For simplicity let us consider the case of\nthe maximal parabolic Eisenstein series.\nLet $E_{\\tau^{(2)}}^{(2)}(g,s)$ denote the Eisenstein series defined on the group $GL_{2k}\n^{(2)}({\\A})$, which is associated with the induced representation\n\\begin{equation*}\n\\text{Ind}_{P_{2,k}^{(2)}({\\A})}^{GL_{2k}^{(2)}({\\A})}\n(\\tau^{(2)}\\otimes \\tau^{(2)})\\delta_{P_{2,k}}^{s}.\\notag\n\\end{equation*}\n Let $E_{\\tau^{(2)}}\n^{(2)}(g)$ denote the residue of this series at the point\n$s=(2k+1)\/4k$. Our goal is to prove that the representation generated by these residues is nonzero.  Let\n$\\Theta^{(2)}_{Sp}$  denote the theta representation of the group $Sp_{2k}^{(2)}({\\A})$.\nIt is the minimal representation of this group. We have the following result.\n\\begin{proposition}\\label{period1}\nLet $\\theta_{Sp}^{(2)}$ be a vector in the space of $\\Theta^{(2)}_{Sp}$.  Then\nthe integral\n\\begin{equation}\\label{per1}\n\\int\\limits_{Sp_{2k}(F)\\backslash Sp_{2k}({\\A})}E_{\\tau^{(2)}}\n^{(2)}(g)\\,\\theta_{Sp}^{(2)}(g)\\,dg\\notag\n\\end{equation}\nconverges absolutely.  Moreover, for some choice of data the integral is not identically zero.\nIn particular the residue representation is nonzero.\n\\end{proposition}\nUnfortunately, at this point we do not have a way to extend this result\nto higher order covers.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and motivations}\n\\label{sec:intro}\n\nFor domain decomposition preconditioners, the use of a coarse correction as a second level is usually required to provide scalability (in the weak sense), such that the iteration count is independent of the number of subdomains, for subdomains of fixed dimension. In addition, it is desirable to guarantee robustness with respect to strong variations in the physical parameters. Achieving scalability and robustness usually relies on sophisticated tools such as spectral coarse spaces \\cite{Dolean:2015:DDM}. In particular, we can highlight the GenEO coarse space \\cite{Spillane:2014:ARC}, which has been successfully analysed and applied to highly heterogeneous positive definite elliptic problems. This coarse space relies on the solution of local eigenvalue problems on subdomains and the theory in the SPD case is based on the fact that local eigenmodes form an orthonormal basis with respect to the energy scalar product induced by the bilinear form.\n\nOur motivation here is to gain a better insight into the good performance of spectral coarse spaces even for highly indefinite high-frequency Helmholtz problems with absorbing boundary conditions, as observed in \\cite{Conen:2014:ACS} (for the Dirichlet-to-Neumann (DtN) coarse space) and more recently in \\cite{Bootland:2020:CSC} for coarse spaces of GenEO type. While a rigorous analysis for Helmholtz problems still lies beyond reach, we present here results, with analysis, showing the benefits of GenEO-type coarse spaces for the heterogeneous symmetric indefinite elliptic problem\n\\begin{subequations}\n\\label{Helmholtz}\n\\begin{align}\n\\label{HelmholtzEquation}\n- \\nabla \\cdot (A(\\vec{x}) \\nabla u) - \\kappa u & = f & & \\text{in } \\Omega,\\\\\n\\label{HelmholtzDirichletBC}\nu & = 0 & & \\text{on } \\partial\\Omega,\n\\end{align}\n\\end{subequations}\nin a bounded domain $\\Omega$ with homogeneous Dirichlet boundary conditions on $\\partial\\Omega$, thus extending the results of \\cite{Spillane:2014:ARC} to this case. The coefficient function $A$ in \\eqref{HelmholtzEquation} is a symmetric positive-definite matrix-valued function on $\\Omega \\rightarrow {\\mathbb R}^{d\\times d}$ (where $d$ is the space dimension) with highly varying but bounded values ($a_{\\rm min} |\\xi|^{2}\\leq {A}(\\vec{x})\\xi\\cdot\\xi \\leq {a_{\\rm max} |\\xi|^{2}, \\vec{x} \\in\\Omega, \\xi \\in\\mathbb{R}^{d}}$) and $\\kappa$ is an $L^\\infty(\\Omega)$ function which can have positive or negative values. We assume throughout that problem \\eqref{Helmholtz} is well-posed and that there is a unique weak solution $u \\in H^{1}_{0}(\\Omega)$, for all $f \\in L^2(\\Omega)$.\n\nWe propose two types of spectral coarse spaces, one built from local spectra of the whole indefinite operator on the left-hand side of \\eqref{HelmholtzEquation}, and the other built using only the second-order term in \\eqref{HelmholtzEquation}. For the latter, the recent analysis in \\cite{Bootland:2021:OSM} applies, while the better performance of the former for large $\\Vert \\kappa \\Vert_{\\infty}$ provides some insight into the good performance of the $\\mathcal{H}$-GenEO method introduced in \\cite{Bootland:2020:CSC} for high-frequency Helmholtz problems, even though it is not amenable to the theory in~\\cite{Bootland:2021:OSM}.\n\nThe problem \\eqref{HelmholtzEquation} involves a Helmholtz-type operator (although this term would normally be associated with the case when $\\kappa$ has a positive sign and \\eqref{HelmholtzDirichletBC} would normally be equipped with an absorbing boundary condition rather than the Dirichlet condition here). In the special case $A = I, \\kappa = k^2$ with $k $ constant, the assumption of well-posedness of the problem is equivalent to the requirement that $k^2$ does not coincide with any of the Dirichlet eigenvalues of the operator $- \\Delta $ in the domain $\\Omega$. In this case, for large $k^2$, the solution of \\eqref{Helmholtz} will be rich in modes corresponding to eigenvalues near $k^2$ and thus will have oscillatory behaviour, increasing as $k$ increases. The Helmholtz problem with $A = I$ and $\\kappa = \\omega^2 n$ (with $\\omega$ real and $n$ a function), together with an absorbing far-field boundary condition appears regularly in geophysical applications; here $n$ is the refractive index or `squared slowness' of waves and $\\omega$ is the angular frequency.\n\nTo solve \\eqref{Helmholtz}, we consider an additive Schwarz (AS) method with a GenEO-like coarse space and study the performance of this solver methodology for some heterogeneous test cases. GenEO coarse spaces have been shown theoretically and practically to be very effective for treating heterogeneous positive definite problems. Here, our main focus is to investigate how this approach performs in the indefinite case~\\eqref{Helmholtz}. We now review the underlying numerical methods that are used.\n\n~\\\\\n\n\n\\section{Discretisation and domain decomposition solver}\n\\label{sec:discretisation}\n\nWe suppose that the domain $\\Omega$ is a bounded Lipschitz polygon\/polyhedron in 2D\/3D. To discretise the problem we use the Lagrange finite element method of degree $p$ on a conforming simplicial mesh $T^{h}$ of $\\Omega$. Denote the finite element space by $V^{h} \\subset H^{1}_0(\\Omega)$. The finite element solution $u_{h} \\in V^{h}$ satisfies the weak formulation $b(u_{h},v_{h}) = F(v_{h})$, for all $v_{h} \\in V^{h}$, where\n\\begin{align}\n\\label{WeakForms}\nb(u,v) & = \\int_{\\Omega} \\left( A(\\vec{x})\\nabla u \\cdot \\nabla {v} - \\kappa u {v}\\right) \\, \\mathrm{d}\\vec{x} & \\text{and} & & F(v) & = \\int_{\\Omega} {f v} \\, \\mathrm{d}\\vec{x}.\n\\end{align}\nUsing the standard nodal basis for $V^{h}$ we can represent the solution $u_{h}$ through its basis coefficients $\\vec{u}$ and reduce the problem to solving the symmetric linear system\n\\begin{align}\\label{global}\nB\\vec{u} = \\vec{f}\n\\end{align}\nwhere $B$ comes from the bilinear form $b(\\cdot,\\cdot)$ and $\\vec{f}$ from the linear functional $F(\\cdot)$. Note that $B$ is symmetric but generally indefinite. For sufficiently small fine-mesh diameter $h$, problem~\\eqref{global} has a unique solution $\\vec{u}$; see \\cite{Schatz:1996:SNE}. To solve \\eqref{global}, we utilise a two-level domain decomposition preconditioner within a Krylov method.\n\nConsider an overlapping partition $\\lbrace \\Omega_{j} \\rbrace_{1 \\le j \\le N}$ of $\\Omega$, where each $\\Omega_{j}$ is assumed to have diameter $H_{j}$ and $H$ denotes the maximal diameter of the subdomains. For each $j$ we define $\\widetilde{V}^{j} = \\{v|_{\\Omega_{j}} : v \\in V_{h}\\}$, $V^{j} = \\{v \\in \\widetilde{V}^{j} : {\\rm supp}(v)\\subset \\Omega_{j}\\}$, and for $u$, $v \\in \\widetilde{V}^{j}$\n\\begin{align*}\nb_j(u,v) &:= \\int_{\\Omega_{j}} \\left(A(\\vec{x})\\nabla u\\cdot{\\nabla {v}} - \\kappa u {v}\\right) \\mathrm{d}\\vec{x} & \\text{and} & & a_j(u,v) &:= \\int_{\\Omega_{j}} A(\\vec{x}) \\nabla u\\cdot{\\nabla {v}} \\mathrm{d}\\vec{x}.\n\\end{align*}\n\nLet $ {\\cal R}_{j}^{T} \\colon V^{j} \\rightarrow V_{h}$, $1\\le j \\le N$, denote the zero-extension operator, let $R_j^T$ denote its matrix representation with respect to the nodal basis and set $R_j = (R_j^T)^T$. The classical one-level additive Schwarz preconditioner is\n\\begin{align}\\label{eq:2-14}\nM_{\\text{AS}}^{-1} &= \\sum_{j=1}^{N} R_{j}^{T} B_{j}^{-1} R_{j}, & \\text{where} & & B_{j} &= R_{j} B R_{j}^{T}.\n\\end{align}\nIt is well-known that one-level additive Schwarz methods are not scalable with respect to the number of subdomains in general, since information is exchanged only between neighbouring subdomains. Thus, we introduce the two-level additive Schwarz method with GenEO coarse space first proposed in \\cite{Spillane:2014:ARC}. To this end, for $1\\leq j\\leq N$, let $\\{\\phi^{j}_{1},\\ldots,\\phi^{j}_{\\tilde{n}_{j}}\\}$ be a nodal basis of $\\widetilde{V}^{j}$, where $\\widetilde{n}_{j} = \\mathrm{dim} ( \\widetilde{V}^{j} )$.\n\n\\begin{definition}[Partition of unity] \\label{def:2-0}\nLet $\\text{dof}(\\Omega_j)$ denote the internal degrees of freedom (nodes) on subdomain $\\Omega_j$. For any degree of freedom $i$, let $\\mu_{i}$ denote the number of subdomains $\\Omega_j$ for which $i$ is an internal degree of freedom, i.e., $\\mu_{i} := \\#\\{j : 1 \\leq j \\leq N,\\ i \\in \\text{dof}(\\Omega_j)\\}$. Then, for $1\\leq j\\leq N$, the \\emph{local partition of unity operator} $\\Xi_{j} \\colon \\widetilde{V}^{j}\\rightarrow V^{j}$ is defined by\n\\begin{align}\\label{eq:2-16}\n\\Xi_{j}(v) &:= \\sum_{i \\in \\text{dof}(\\Omega_j)}\\frac{1}{\\mu_i} v_{i} \\phi^{j}_{i}, & \\text{for all} & & v &= \\sum_{i=1}^{\\widetilde{n}_{j}}v_{i}\\phi^{j}_{i} \\in \\widetilde{V}^{j}.\n\\end{align}\nThe operators $\\Xi_{j}$ form a partition of unity, i.e., $\\sum_{j=1}^{N} R_{j}^{T} \\Xi_{j}(v|_{\\Omega_j}) = v$, $\\forall v \\in V_{h}$ \\cite{Spillane:2014:ARC}.\n\\end{definition}\n\n\\noindent\nFor each $j$, we define the following generalised eigenvalue problems:\n\\begin{align}\n\\label{eq:deltag}\n\\text{find} \\ p \\in \\widetilde{V}_j\\backslash \\{0\\} , \\, \\lambda \\in \\mathbb{R} : & & a_{j}(p,v) &= \\lambda \\,a_{j}(\\Xi_{j}(p), \\Xi_{j}(v)), & & \\text{for all } v \\in \\widetilde{V}^{j}, \\\\\n\\label{eq:hg}\n\\text{find} \\ q \\in \\widetilde{V}_j\\backslash \\{0\\} , \\, \\lambda \\in \\mathbb{R} : & & b_{j}(q,v) &= \\lambda \\,a_{j}(\\Xi_{j}(q), \\Xi_{j}(v)), & & \\text{for all } v \\in \\widetilde{V}^{j},\n\\end{align}\nwhere $\\Xi_{j}$ is the local partition of unity operator from Definition~\\ref{def:2-0}.\n\n\\begin{definition}[$\\Delta$-GenEO and ${\\cal H}$-GenEO coarse spaces]\n\\label{def:2-2}\nFor each $j$, $1 \\le j \\le N$, let $(p_{l}^{j})_{l=1}^{m_{j}}$ and $(q_{l}^{j})_{l=1}^{m_{j}}$ be the eigenfunctions of the eigenproblems \\eqref{eq:deltag} and \\eqref{eq:hg} corresponding to the $m_{j}$ smallest eigenvalues, respectively. Then we define the \\emph{$\\Delta$-GenEO} and \\emph{${\\cal H}$-GenEO} coarse spaces, respectively, by\n\\begin{align}\n\\label{eq:deltspace}\nV_{\\Delta}^0 &:= {\\rm span}\\{{\\cal R}_{j}^{T}\\Xi_{j}(p_{l}^{j}): \\;l=1,\\ldots,m_{j};\\;j=1,\\ldots,N\\} \\ \\ \\text{and}\\\\\n\\label{eq:hspace}\nV_{\\mathcal H}^0 &:= {\\rm span}\\{{\\cal R}_{j}^{T}\\Xi_{j}(q_{l}^{j}): \\;l=1,\\ldots,m_{j};\\;j=1,\\ldots,N\\}.\n\\end{align}\n\\end{definition}\n\nNote that here and subsequently, the subscript $\\Delta$ refers to the GenEO coarse space \\eqref{eq:deltspace} based on \\eqref{eq:deltag}, the eigenproblem with respect to the `Laplace-like' operator induced by the bilinear form $a_j$, while the subscript ${\\cal H}$ refers to the ${\\cal H}$-GenEO coarse space \\eqref{eq:hspace} based on \\eqref{eq:hg}, with the `Helmholtz-like' operator appearing in $b_j$.\n\nSince $V_{\\Delta}^0, \\, V_{\\mathcal H}^0 \\subset V_{h}$, we can introduce the natural embeddings $\\mathcal{R}_{0,\\Delta}^T \\colon V_{\\Delta}^0 \\rightarrow V_h$ and $\\mathcal{R}_{0,\\mathcal H}^T \\colon V_{\\mathcal H}^0 \\rightarrow V_h$, with matrix representations ${R}_{0,\\Delta}^T$ and ${R}_{0,\\mathcal H}^T$, respectively, and set ${R}_{0,\\Delta} = ({R}_{0,\\Delta}^T)^T$ and ${R}_{0,\\mathcal H} = ({R}_{0,\\mathcal H}^T)^T$ to obtain the following two-level extensions of the one-level additive Schwarz method \\eqref{eq:2-14}:\n\\begin{align}\\label{twolevel}\nM_{\\mathrm{AS}, \\Delta}^{-1} &= M_{AS}^{-1} + {R}_{0,\\Delta}^{T}B_{0,\\Delta}^{-1}{R}_{0,\\Delta} & \\text{and} & & M_{\\mathrm{AS}, \\mathcal H}^{-1} &= M_{AS}^{-1} + {R}_{0,\\mathcal H}^{T}B_{0, \\mathcal H}^{-1}{R}_{0,\\mathcal H},\n\\end{align}\nwhere $B_{0,\\Delta} := {R}_{0,\\Delta} {B} {R}_{0,\\Delta}^{T}$ and $B_{0, \\mathcal H} := {R}_{0,\\mathcal H}{B} {R}_{0,\\mathcal H}^{T}$.\n\n\n\\section{Theoretical results}\n\\label{sec:theory}\n\nThe theoretical properties of the preconditioner $M_{\\mathrm{AS}, \\Delta}^{-1}$ are studied in the forthcoming paper \\cite{Bootland:2021:OSM}. There, the PDE studied is a generalisation of \\eqref{HelmholtzEquation}, which also allows the inclusion of a non-self-adjoint first order convection term and more general boundary conditions. The important parameters in the preconditioner are the coarse mesh diameter $H$ and the `eigenvalue tolerance'\n\\begin{align*}\n\\Theta := \\max_{1 \\le j \\le N } \\left( \\lambda_{m_j+1}^j \\right)^{-1},\n\\end{align*}\nwhere $\\{\\lambda_{m}^j: m = 1, 2, \\ldots\\}$ are the eigenvalues of the generalised eigenproblem~\\eqref{eq:deltag}, labelled in non-decreasing order. As a special case of the results in \\cite{Bootland:2021:OSM}, we highlight the following result.\n\\begin{theorem}\n\\label{thm:main}\nLet the fine-mesh diameter $h$ be sufficiently small. Then there exist thresholds $H_0>0$ and $\\Theta_0 > 0 $ such that, for all $H \\leq H_0$ and $\\Theta \\leq \\Theta_0$, the following statements hold:\n\\begin{itemize}\n\\item[(i)] The matrices $B_j$ and $B_{0,\\Delta}$ appearing in \\eqref{eq:2-14} and \\eqref{twolevel} are non-singular.\n\\item[(ii)] If the problem \\eqref{global} is solved by GMRES with left preconditioner $M_{\\mathrm{AS}, \\Delta}^{-1}$ and residual minimisation in the energy norm $\\Vert u \\Vert_a := \\big(\\int_\\Omega \\nabla u \\cdot A \\nabla u \\big)^{1\/2} $, then there exist a constant $c \\in (0,1)$, which depends on $H_0$ and $\\Theta_0$ but is independent of all other parameters, such that we have the robust GMRES convergence estimate\n\\begin{equation}\\label{eq:2-31}\n\\Vert r_{\\ell} \\Vert^{2}_{a} \\leq \\left(1-c^2\\right)^{\\ell}\\Vert r_{0} \\Vert^{2}_{a}\\,,\n\\end{equation}\nfor $\\ell = 0, 1, \\ldots \\,$, where $r_\\ell$ denotes the residual after $\\ell$ iterations of GMRES.\n\\end{itemize}\n\\end{theorem}\nIn fact, the paper \\cite{Bootland:2021:OSM} investigates in detail how the thresholds $H_0$ and $\\Theta_0$ depend on the heterogeneity and indefiniteness of \\eqref{HelmholtzEquation}. For example, if the problem is scaled so that $a_{\\min} = 1$, then as $\\Vert \\kappa \\Vert_\\infty$ and $a_\\text{max}$ grow, $H_0$ and $\\Theta_0$ have to decrease to maintain the convergence rate of GMRES:\n\\begin{align}\nH_0 &\\lesssim \\Vert \\kappa \\Vert_\\infty^{-1} & \\text{and} & & \\Theta_0 &\\lesssim C_{\\mathrm{stab}}^{-2} \\Vert \\kappa \\Vert_\\infty^{-4} \\, a_\\text{max}\\,, \\label{rates}\n\\end{align}\nwhere $C_{\\mathrm{stab}} = C_{\\mathrm{stab}}(A,\\kappa)$ denotes the stability constant for problem \\eqref{Helmholtz}, i.e., the solution $u$ satisfies $\\Vert u \\Vert_{H^1(\\Omega)} \\leq C_{\\mathrm{stab}} \\Vert f \\Vert_{L^2(\\Omega)}$ for all $f \\in L^2(\\Omega)$ and the hidden constants are independent of $h$, $H$, $a_\\text{max}$ and $\\kappa$. Thus, as $\\Vert \\kappa \\Vert_\\infty$ gets smaller, the indefiniteness diminishes and the requirements on $H_0$ and $\\Theta_0$ are relaxed.\n\n\n\\section{Numerical results}\n\\label{sec:numerics}\n\nWe give numerical results for a more efficient variant of the two-level method described in \\S\\ref{sec:discretisation}. Instead of \\eqref{twolevel}, our preconditioner here is based on the \\emph{restricted additive Schwarz} (RAS) method and the GenEO coarse space is incorporated using the following deflation approach\n\\begin{align}\n\\label{2LevelAdaptiveDeflationPreconditioner}\nM^{-1} &= M_{\\text{RAS}}^{-1}(I - B Q_0) + Q_0, & M_{\\text{RAS}}^{-1} &= \\sum_{j=1}^{N} R_{j}^{T} D_j B_{j}^{-1} R_{j}.\n\\end{align}\nHere, $D_j$ is the matrix form of the partition of unity operator $\\Xi_{j}$. Moreover, we have $Q_0 = R_{0}^{T}B_{0}^{-1}R_{0}$ with $B_{0} = R_{0}BR_{0}^{T}$ and either $R_0 = {R}_{0,\\Delta}$ or $R_0 = {R}_{0,\\mathcal H}$, depending on whether we use $\\Delta$-GenEO or $\\mathcal{H}$-GenEO. We include all eigenvectors $p_l^j$ or $q_l^j$ in $V_{\\Delta}^0$ or $V_{\\mathcal H}^0$ corresponding to eigenvalues $\\lambda_l^j < \\lambda_\\text{max}$, for $\\Delta$-GenEO or $\\mathcal{H}$-GenEO, respectively. Note that for $\\mathcal{H}$-GenEO this includes all eigenvectors corresponding to negative eigenvalues. Unless otherwise stated, we use the eigenvalue threshold $\\lambda_\\text{max} = \\frac{1}{2}$.\n\nAs a model problem, we consider \\eqref{Helmholtz} on the unit square $\\Omega = (0,1)^2$, take $\\kappa$ constant, and define $A$ to model various layered media, as depicted in Figure~\\ref{Fig:Layers}. The right-hand side $f$ is taken to be a point source at the centre $(\\frac{1}{2},\\frac{1}{2})$. To discretise we use a uniform square grid with $n_{\\text{glob}}$ points in each direction and triangulate along one set of diagonals to form P1 elements. We further use a uniform decomposition into $N$ square subdomains and throughout make use of minimal overlap (non-overlapping subdomains are extended by adding only the fine-mesh elements which touch them). Computations are performed using FreeFem (\\url{http:\/\/freefem.org\/}), in particular using the \\texttt{ffddm} framework. When solving the linear systems we use preconditioned GMRES with residual minimisation in the Euclidean norm and a relative residual tolerance of $10^{-6}$. Note that, as in Figure \\ref{Fig:Layers}, we have fixed $a_\\text{min} = 1$ as otherwise one can rescale the variables to ensure this condition. As such, the indefiniteness is predominantly controlled by $\\kappa$, which we assume is a positive constant here.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\hspace*{\\fill}\n\t\\subfloat[Increasing layers]{\n\t\\begin{tikzpicture}[scale=1.25]\n\t\\fill[gray,opacity=0.1] (-1,1) -- (1,1) -- (1,0.8) -- (-1,0.8) -- cycle;\n\t\\fill[gray,opacity=0.2] (-1,0.6) -- (1,0.6) -- (1,0.8) -- (-1,0.8) -- cycle;\n\t\\fill[gray,opacity=0.3] (-1,0.6) -- (1,0.6) -- (1,0.4) -- (-1,0.4) -- cycle;\n\t\\fill[gray,opacity=0.4] (-1,0.2) -- (1,0.2) -- (1,0.4) -- (-1,0.4) -- cycle;\n\t\\fill[gray,opacity=0.5] (-1,0.2) -- (1,0.2) -- (1,0.0) -- (-1,0.0) -- cycle;\n\t\\fill[gray,opacity=0.6] (-1,-0.2) -- (1,-0.2) -- (1,0.0) -- (-1,0.0) -- cycle;\n\t\\fill[gray,opacity=0.7] (-1,-0.2) -- (1,-0.2) -- (1,-0.4) -- (-1,-0.4) -- cycle;\n\t\\fill[gray,opacity=0.8] (-1,-0.6) -- (1,-0.6) -- (1,-0.4) -- (-1,-0.4) -- cycle;\n\t\\fill[gray,opacity=0.9] (-1,-0.6) -- (1,-0.6) -- (1,-0.8) -- (-1,-0.8) -- cycle;\n\t\\fill[gray,opacity=1.0] (-1,-1) -- (1,-1) -- (1,-0.8) -- (-1,-0.8) -- cycle;\n\t\\draw (-1,1) -- (-1,-1) -- (1,-1) -- (1,1) -- cycle;\n\t\\label{Fig:LayersA}\n\t\\end{tikzpicture}\n\t} \\hspace*{\\fill}\n\\subfloat[Alternating layers]{\n\\begin{tikzpicture}[scale=1.25]\n\t\\fill[gray,opacity=0.1] (-1,1) -- (1,1) -- (1,0.8) -- (-1,0.8) -- cycle;\n\t\\fill[gray,opacity=1.0] (-1,0.6) -- (1,0.6) -- (1,0.8) -- (-1,0.8) -- cycle;\n\t\\fill[gray,opacity=0.1] (-1,0.6) -- (1,0.6) -- (1,0.4) -- (-1,0.4) -- cycle;\n\t\\fill[gray,opacity=1.0] (-1,0.2) -- (1,0.2) -- (1,0.4) -- (-1,0.4) -- cycle;\n\t\\fill[gray,opacity=0.1] (-1,0.2) -- (1,0.2) -- (1,0.0) -- (-1,0.0) -- cycle;\n\t\\fill[gray,opacity=1.0] (-1,-0.2) -- (1,-0.2) -- (1,0.0) -- (-1,0.0) -- cycle;\n\t\\fill[gray,opacity=0.1] (-1,-0.2) -- (1,-0.2) -- (1,-0.4) -- (-1,-0.4) -- cycle;\n\t\\fill[gray,opacity=1.0] (-1,-0.6) -- (1,-0.6) -- (1,-0.4) -- (-1,-0.4) -- cycle;\n\t\\fill[gray,opacity=0.1] (-1,-0.6) -- (1,-0.6) -- (1,-0.8) -- (-1,-0.8) -- cycle;\n\t\\fill[gray,opacity=1.0] (-1,-1) -- (1,-1) -- (1,-0.8) -- (-1,-0.8) -- cycle;\n\t\\draw (-1,1) -- (-1,-1) -- (1,-1) -- (1,1) -- cycle;\n\t\\label{Fig:LayersB}\n\\end{tikzpicture}\n} \\hspace*{\\fill}\n\\subfloat[Diagonal layers]{\n\\begin{tikzpicture}[scale=1.25]\n\t\\fill[gray,opacity=1.0] (1,-1) -- (0.6,-1) -- (1,-0.6) -- cycle;\n\t\\fill[gray,opacity=0.6] (0.6,-1) -- (0.2,-1) -- (1,-0.2) -- (1,-0.6) -- cycle;\n\t\\fill[gray,opacity=1.0] (0.2,-1) -- (-0.2,-1) -- (1,0.2) -- (1,-0.2) -- cycle;\n\t\\fill[gray,opacity=0.2] (-0.2,-1) -- (-0.6,-1) -- (1,0.6) -- (1,0.2) -- cycle;\n\t\\fill[gray,opacity=1.0] (-0.6,-1) -- (-1,-1) -- (1,1) -- (1,0.6) -- cycle;\n\t\\fill[gray,opacity=0.05] (-1,-1) -- (-1,-0.6) -- (0.6,1) -- (1,1) -- cycle;\n\t\\fill[gray,opacity=1.0] (-1,-0.6) -- (-1,-0.2) -- (0.2,1) -- (0.6,1) -- cycle;\n\t\\fill[gray,opacity=0.4] (-1,-0.2) -- (-1,0.2) -- (-0.2,1) -- (0.2,1) -- cycle;\n\t\\fill[gray,opacity=1.0] (-1,0.2) -- (-1,0.6) -- (-0.6,1) -- (-0.2,1) -- cycle;\n\t\\fill[gray,opacity=0.8] (-1,0.6) -- (-1,1) -- (-0.6,1) -- cycle;\n\t\\draw (-1,1) -- (-1,-1) -- (1,-1) -- (1,1) -- cycle;\n\t\\label{Fig:LayersC}\n\\end{tikzpicture}\n\t}\n\t\\hspace*{\\fill}\n\t\\caption{Different layer profiles for $a(\\vec{x})$ where $A(\\vec{x}) = a(\\vec{x})I$ and $a_{\\rm max}>1$ is a contrast parameter. For the darkest shade $a(\\mathbf{x}) = 1$ while for the lightest shade we reach $a(\\mathbf{x}) = a_\\text{max}$.}\n\t\\label{Fig:Layers}\n\\end{figure}\n\nIn Tables \\ref{Table:VaryAmax_AlternatingLayers_nglob400}--\\ref{Table:VaryNGlob_IncreasingLayers} we provide GMRES iteration counts for both the $\\Delta$-GenEO and $\\mathcal{H}$-GenEO coarse spaces while varying the number of subdomains $N$ along with $a_\\text{max}$ and $n_\\text{glob}$ respectively. In each case we use a different layer profile from Figure \\ref{Fig:Layers} and note that results are broadly similar for each such heterogeneous profile. We first note that both methods yield results that are robust to the contrast parameter $a_\\text{max}$, the mesh size given by $n_\\text{glob}$, and the number of subdomains $N$.\n\n\\begin{table}[t]\n\\centering\n\\caption{GMRES iteration counts using $\\Delta$-GenEO or $\\mathcal{H}$-GenEO for the alternating layers problem, varying $a_\\text{max} > 1$ and the number of subdomains $N$, with fixed $\\kappa = 100$, $n_\\text{glob} = 400$, and $\\lambda_\\text{max} = \\frac{1}{2}$.}\n\\label{Table:VaryAmax_AlternatingLayers_nglob400}\n\\tabulinesep=1mm\n\\begin{tabu}{c|ccccccccc|ccccccccc}\n\t& \\multicolumn{9}{c|}{$\\Delta$-GenEO} & \\multicolumn{9}{c}{$\\mathcal{H}$-GenEO} \\\\\n\t\\cline{2-19}\n\t$a_\\text{max} \\ \\backslash \\ N$ & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 \\\\\n\t\\hline\n\t5 & 11 & 11 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 9 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 10 \\\\\n\t10 & 10 & 9 & 10 & 9 & 9 & 9 & 9 & 9 & 10 & 9 & 8 & 9 & 9 & 9 & 9 & 9 & 9 & 9 \\\\\n\t20 & 9 & 9 & 10 & 9 & 10 & 10 & 10 & 10 & 10 & 9 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 10 \\\\\n\t50 & 8 & 8 & 9 & 8 & 9 & 9 & 9 & 9 & 9 & 8 & 8 & 9 & 8 & 9 & 9 & 9 & 9 & 9 \\\\\n\\end{tabu}\n\\end{table}\n\n\\begin{table}[t]\n\\centering\n\\caption{GMRES iteration counts using $\\Delta$-GenEO or $\\mathcal{H}$-GenEO for the increasing layers problem, varying $n_\\text{glob}$ and the number of subdomains $N$, with fixed $\\kappa = 1000$, $a_\\text{max} = 50$, and $\\lambda_\\text{max} = \\frac{1}{2}$.}\n\\label{Table:VaryNGlob_IncreasingLayers}\n\\tabulinesep=1mm\n\\begin{tabu}{c|ccccccccc|ccccccccc}\n\t& \\multicolumn{9}{c|}{$\\Delta$-GenEO} & \\multicolumn{9}{c}{$\\mathcal{H}$-GenEO} \\\\\n\t\\cline{2-19}\n\t$n_\\text{glob} \\ \\backslash \\ N$ & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 \\\\\n\t\\hline\n\t200 & 13 & 15 & 16 & 16 & 17 & 17 & 16 & 16 & 16 & 8 & 8 & 8 & 9 & 9 & 9 & 9 & 10 & 9 \\\\\n\t400 & 13 & 14 & 16 & 15 & 17 & 17 & 16 & 17 & 16 & 9 & 8 & 8 & 8 & 9 & 9 & 9 & 9 & 9 \\\\\n\t600 & 13 & 13 & 16 & 16 & 17 & 17 & 16 & 17 & 16 & 9 & 8 & 8 & 8 & 8 & 9 & 9 & 9 & 9 \\\\\n\t800 & 13 & 14 & 15 & 16 & 17 & 17 & 16 & 17 & 16 & 9 & 8 & 10 & 9 & 9 & 9 & 9 & 9 & 9\n\\end{tabu}\n\\end{table}\n\nThe indefiniteness here is controlled by $\\kappa$ and in Table \\ref{Table:VaryKappa_DiagonalLayers_nglob600} we begin to see that iteration counts with $\\Delta$-GenEO become worse for the more indefinite problems with larger $\\kappa$, as expected. On the other hand, $\\mathcal{H}$-GenEO appears much more robust to an increase in $\\kappa$. Though in this case, with a fixed mesh, there is some slight degradation for the largest value of $\\kappa$. Nonetheless, when the indefiniteness is relatively small, the $\\Delta$-GenEO method can provide broadly similar iteration counts to $\\mathcal{H}$-GenEO.\n\n\\begin{table}[t]\n\\centering\n\\caption{GMRES iteration counts using $\\Delta$-GenEO or $\\mathcal{H}$-GenEO for the diagonal layers problem, varying $\\kappa$ and the number of subdomains $N$, with fixed $a_\\text{max} = 5$, $n_\\text{glob} = 600$, and $\\lambda_\\text{max} = \\frac{1}{2}$.}\n\\label{Table:VaryKappa_DiagonalLayers_nglob600}\n\\tabulinesep=1mm\n\\begin{tabu}{c|ccccccccc|ccccccccc}\n\t& \\multicolumn{9}{c|}{$\\Delta$-GenEO} & \\multicolumn{9}{c}{$\\mathcal{H}$-GenEO} \\\\\n\t\\cline{2-19}\n\t$\\kappa \\ \\backslash \\ N$ & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 \\\\\n\t\\hline\n\t10 & 9 & 9 & 9 & 8 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 8 & 9 & 9 & 9 & 9 & 9 \\\\\n\t100 & 11 & 10 & 10 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 8 & 9 & 9 & 9 & 9 & 9 & 9 & 9 \\\\\n\t1000 & 19 & 32 & 36 & 48 & 43 & 40 & 35 & 41 & 28 & 11 & 10 & 13 & 9 & 11 & 10 & 9 & 11 & 10 \\\\\n\t10000 & 77 & 156 & 215 & 301 & 339 & 419 & 437 & 540 & 506 & 19 & 26 & 27 & 22 & 33 & 26 & 40 & 51 & 18\n\\end{tabu}\n\\end{table}\n\nThese observations align with the fact that eigenfunctions appear qualitatively similar for $\\Delta$-GenEO and $\\mathcal{H}$-GenEO when $\\kappa$ is small. As seen in Figure \\ref{Fig:Eigenplots}, once $\\kappa$ increases the $\\mathcal{H}$-GenEO eigenfunctions change: the type of eigenfunctions produced by $\\Delta$-GenEO remain, albeit perturbed, but now we have further eigenfunctions which include more oscillatory behaviour in the interior of the subdomain; such features are not found with $\\Delta$-GenEO where higher oscillations only appear on the boundary.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[$\\Delta$-GenEO - homogeneous]{ \\includegraphics[width=.47\\linewidth]{eig_delta}}\n\t\\hspace*{\\fill}\n\t\\subfloat[$\\Delta$-GenEO - diagonal layers]{ \\includegraphics[width=.47\\linewidth]{eig_diagonal_delta}}\n\t\\\\\n\t\\subfloat[$\\mathcal{H}$-GenEO - $\\kappa = 1000$ - homogeneous]{ \\includegraphics[width=.47\\linewidth]{eig_h_kappa1000}}\n\t\\hspace*{\\fill}\n\t\\subfloat[$\\mathcal{H}$-GenEO - $\\kappa = 1000$ - diagonal layers]{ \\includegraphics[width=.47\\linewidth]{eig_diagonal_h_kappa1000}}\n\t\\\\\n\t\\subfloat[$\\mathcal{H}$-GenEO - $\\kappa = 10000$ - homogeneous]{ \\includegraphics[width=.47\\linewidth]{eig_h_kappa10000}}\n\t\\hspace*{\\fill}\n\t\\subfloat[$\\mathcal{H}$-GenEO - $\\kappa = 10000$ - homogeneous]{ \\includegraphics[width=.47\\linewidth]{eig_h_kappa10000_more}}\n\t\\caption{Example eigenfunctions on the central subdomain when $N=25$, $n_\\text{glob} = 800$ and, in the heterogeneous case, $a_\\text{max} = 10$. In (a)--(e) we plot equivalent eigenfunctions to illustrate how they are affected by $\\kappa$ and heterogeneity. In (f) we plot the types of eigenfunctions not seen for $\\Delta$-GenEO.}\n\t\\label{Fig:Eigenplots}\n\\end{figure}\n\nIn Table \\ref{Table:VaryLambdaMax_DiagonalLayers} we show that, for $\\mathcal{H}$-GenEO, robustness with $\\kappa$ in this fixed-mesh example can be obtained by increasing the eigenvalue threshold $\\lambda_\\text{max}$ and thus also the coarse space size, which is tabulated in each case for comparison.\n\n\\begin{table}[t]\n\\centering\n\\caption{GMRES iteration counts and the size of the coarse space when using $\\mathcal{H}$-GenEO for the diagonal layers problem, varying the number of subdomains $N$, with fixed $n_\\text{glob} = 600$ and $a_\\text{max} = 5$. Here, we also vary the eigenvalue threshold $\\lambda_\\text{max}$ with $\\kappa$ such that we obtain relatively similar iteration counts for increasing $\\kappa$ and tabulate the size of the coarse space invoked.}\n\\label{Table:VaryLambdaMax_DiagonalLayers}\n\\tabulinesep=1mm\n\\begin{tabu}{cc|ccccccccc|ccccccccc}\n\t\\multicolumn{2}{c|}{} & \\multicolumn{9}{c|}{Iteration count} & \\multicolumn{9}{c}{Coarse space size} \\\\\n\t\\cline{3-20}\n\t$\\lambda_\\text{max}$ & $\\kappa \\ \\backslash \\ N$ & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 \\\\\n\t\\hline\n\t0.1 & 10 & 22 & 23 & 23 & 22 & 23 & 25 & 25 & 24 & 23 & 36 & 69 & 108 & 144 & 199 & 200 & 214 & 263 & 324 \\\\\n\t0.1 & 100 & 23 & 21 & 23 & 23 & 24 & 28 & 28 & 28 & 27 & 36 & 76 & 111 & 144 & 201 & 203 & 223 & 264 & 324 \\\\\n\t0.2 & 1000 & 16 & 17 & 19 & 34 & 27 & 27 & 20 & 25 & 20 & 111 & 185 & 265 & 324 & 418 & 482 & 574 & 599 & 684 \\\\\n\t0.6 & 10000 & 19 & 24 & 24 & 20 & 25 & 19 & 28 & 34 & 15 & 633 & 1173 & 1430 & 1751 & 2129 & 2383 & 2680 & 2941 & 3252\n\\end{tabu}\n\\end{table}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this work we have summarised how recently developed analysis in \\cite{Bootland:2021:OSM} can be applied to a GenEO-type coarse space for heterogeneous indefinite elliptic problems. We provide numerical evidence supporting these results and a comparison with a more effective GenEO-type method for highly indefinite problems but for which no theory is yet available. For mildly indefinite problems these two approaches perform similarly, providing the first theoretical insight towards explaining the good behaviour of the $\\mathcal{H}$-GenEO method for challenging heterogeneous Helmholtz problems.\n\n\n\\bibliographystyle{spmpsci}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDuring experimental data processing often there is a task to\nfit arbitrary distribution with the only peak to some\nfunction with a set of free parameters.\nThis can be necessary for more precise determination of peak position,\nor full width at half maximum (devise resolution),\nor for the signal form approximation using Monte Carlo events for subsequent\nanalysis of data distribution and so on.\nAvailable set of fitting functions does not always matches the experimental\nrequirements, especially for high statistics (it is difficult to achieve\nsuitable confidence level).\n\nIn practice most often the sum of the three Gaussian distributions is\nused to fit such histograms.\nIn most cases this fit provides suitable confidence level.\nAt least one can add more and more Gaussians until confidence level is\nadmissible.\nDisadvantage of this method is that every Gaussian used is symmetric,\nso in principle one can not used sum of Gaussians with common center\nto fit asymmetric distributions, and if all Gaussians have different\ncenters, then it is difficult to provide the only maximum of fitting function\nand monotony to the right and left of maximum.\n\nAnother often used function is a spline of two halves of different\nGaussians. This function is good in case that the function ``tails''\ncan be approximated with Gaussian.\nHowever for high statistics the major experimental distributions far of peak\nhave exponential or power-of-x dependence, which do not match Gaussian.\n\nLogarithmic Gaussian distribution is also often used (at SLAC it is called\n``Novosibirsk'' function):\n\\begin{equation}\nF_N(x;x_p,h,\\lambda)=A\\exp\\left[\n-\\frac{1}{2\\sigma^2}\\ln^2\\frac{x_m-x}{x_m-x_p}\n\\right],\\;\\; \\int\\limits_{\\frac{x_m-x}{x_m-x_p}>0}\\!\\!\\!\\!\\!F_N\\,\\mathrm{d} x=1,\n\\end{equation}\nwhere $x_p$ is a peak location (function maximum),\n $h$ is full width at half maximum (FWHM),\n  $\\lambda$ is asymmetry parameter,\n  $A$ is normalization factor.\n\\begin{equation}\\label{eq:xm-Expression}\nx_m=x_p-\\frac{z+\\frac{1}{z}}{2\\lambda},\n\\end{equation}\n\\begin{equation}\n\\sigma=\\frac{1}{\\sqrt{2\\ln 2}}\\ln z,\\;\\;\nA=\\sqrt{\\frac{2}{\\pi}}\\frac{\\left|\\lambda\\right|}{\\left(z+\\frac{1}{z}\\right)\\sigma}\ne^{-\\frac{\\sigma^2}{2}},\n\\end{equation}\n\\begin{equation}\nz=\\sqrt{h\\left|\\lambda\\right|+\\sqrt{h^2\\lambda^2+1}}.\n\\end{equation}\nAs is seen from~(\\ref{eq:xm-Expression}),\nfor $\\lambda>0$ the boundary coordinate $x_m<x_p$ and all $x>x_m$.\nIf $\\lambda<0$, then $x_m>x_p$ and $x<x_m$.\nUsing these notations one can rewrite this function, excluding\nthe inconvenient variable  $x_m$:\n\\begin{equation}\n\\begin{array}{l}\nF_N(x;x_p,h,\\lambda)=\\sqrt{\\frac{2}{\\pi}}\\cdot\\frac{1}{z+\\frac{1}{z}}\\cdot\n\\frac{\\left|\\lambda\\right|}{\\sigma}\n\\exp\\left[-\\frac{\\sigma^2}{2}\n-\\frac{1}{2\\sigma^2}\\ln^2\\left(1+2z\\lambda\\frac{x-x_p}{z^2+1}\\right)\n\\right].\n\\end{array}\n\\end{equation}\nFunction is equal to 0 for all  $x$,\nfor which the logarithm argument is negative.\nPossible values of parameters:\n$h>0$, $\\lambda$ is arbitrary, $x_p$ is arbitrary. For $\\lambda\\sim 0$\nthe formulae have an ambiguity of the type $0\/0$, so\nsome parameters should be expanded to the Tailor series.\n\\begin{equation}\nz\\begin{array}[t]{c}\\sim \\\\[-2mm]\\scriptstyle\\lambda\\sim 0\\end{array}\n1+\\frac{h\\left|\\lambda\\right|}{2}+\\frac{h^2\\lambda^2}{8},\n\\end{equation}\ntherefore\n\\begin{equation}\n\\frac{2\\lambda z}{z^2+1}\n\\begin{array}[t]{c}\\sim \\\\[-2mm]\\scriptstyle\\lambda\\sim 0\\end{array}\n\\lambda\\cdot\\left[1-\\frac{h^2\\lambda^2}{8}\n+\\frac{7h^4\\lambda^4}{128}\\right],\n\\end{equation}\n\\begin{equation}\n\\sigma^2\n\\begin{array}[t]{c}\\sim \\\\[-2mm]\\scriptstyle\\lambda\\sim 0\\end{array}\n\\frac{h^2\\lambda^2}{8\\ln 2}\\cdot\\left[\n1-\\frac{h^2\\lambda^2}{3}\n\\right],\\;\\;\\; \\frac{\\sigma}{\\left|\\lambda\\right|}\n\\begin{array}[t]{c}\\sim \\\\[-2mm]\\scriptstyle\\lambda\\sim 0\\end{array}\n\\frac{h}{2\\sqrt{2\\ln 2}}\\cdot\\left[1-\\frac{h^2\\lambda^2}{6}\n\\right],\n\\end{equation}\n\\begin{equation}\n\\frac{\\sigma^2}{2}\n+\\frac{1}{2\\sigma^2}\\ln^2\\left(1+2z\\lambda\\frac{x-x_p}{z^2+1}\\right)\n\\begin{array}[t]{c}\\sim \\\\[-2mm]\\scriptstyle\\lambda\\sim 0\\end{array}\n\\frac{4\\ln 2 \\left(x-x_p\\right)^2}{h^2}\\cdot\\left[\n1-\\left(x-x_p\\right)\\lambda\n\\right].\n\\end{equation}\nFor $\\lambda=0$ the function converts to\n\\begin{equation}\nF_N(x;x_p,h,0)=\\sqrt{\\frac{\\ln 2}{\\pi}}\\cdot\\frac{2}{h}\\cdot\n\\exp\\left[-\\frac{4\\ln 2 \\left(x-x_p\\right)^2}{h^2}\n\\right],\n\\end{equation}\nthat is Gaussian distribution with root mean square\n$$\n\\sqrt{<(x-x_p)^2>}=\\frac{h}{2\\sqrt{2\\ln 2}}\\approx \\frac{h}{2.3548}.\n$$\n\nThis function is convenient for fitting the distributions\nwith abrupt spectrum end.\nHowever many experimental distributions are more smooth,\nand suggested in this paper function can be more successful.\n\n\\section{Convolution of Gaussian and exponential distributions}\nIt is suggested to build the fitting function on the base of\nconvolution of Gaussian and exponential distributions,\nwhich can be easily derived:\n\\begin{equation}\\label{eq:FitCurve1Explicit}\n\\begin{array}{l}\nF_{B1}(x;x_g,\\sigma_g,\\lambda)=\n\\frac{1}{2\\left|\\lambda\\right|}\n\\left[\n1-\\mathrm{erf}\\left(-\\frac{\\left(x-x_g\\right)\\lambda}{\\sigma_g\\left|\\lambda\\right|\\sqrt{2}}\n+\\frac{\\sigma_g}{\\left|\\lambda\\right|\\sqrt{2}}\n\\right)\n\\right]\n\\times \\\\[3mm]\\rule{35mm}{0mm}\\times\n\\exp\\left[\n-\\frac{x-x_g}{\\lambda}\n+\\frac{\\sigma_g^2}{2\\lambda^2}\n\\right].\n\\end{array}\n\\end{equation}\nIntegral of this function over all $x$ equals 1.\nSuch a function was first used by the author in 2004 at SLAC\n(BaBar Note \\# 582)\nfor fitting the deposited energy distributions in calorimeter\nwith the aim of peak position and resolution determination\nfor the algorithms of absolute photon energy calibration,\nand despite some technical difficulties, this function proved to be\nenough convenient for fitting such distributions, especially for high statistics.\n\nTechnical difficulties appear when the argument of ERF function is big\n\\begin{equation}\n\\mathrm{erf}(x)=\\frac{2}{\\sqrt{\\pi}}\\int\\limits_0^xe^{-\\xi^2}\\,\\mathrm{d}\\xi\n\\end{equation}\nIf the argument of ERF function in formula~(\\ref{eq:FitCurve1Explicit})\nis denoted as $z$\n\\begin{equation}\nz=-\\frac{\\left(x-x_g\\right)\\lambda}{\\sigma_g\\left|\\lambda\\right|\\sqrt{2}}\n+\\frac{\\sigma_g}{\\left|\\lambda\\right|\\sqrt{2}},\n\\end{equation}\nthen the formula~(\\ref{eq:FitCurve1Explicit})\nlooks like:\n\\begin{equation}\\label{eq:FitCurve1Reduced}\nF_{B1}(x;x_g,\\sigma_g,\\lambda)=\n\\frac{1}{2\\left|\\lambda\\right|}\n\\left[\n1-\\mathrm{erf}\\left(z\n\\right)\n\\right]\n\\exp\\left[\\frac{z\\sigma_g\\sqrt{2}}{\\left|\\lambda\\right|}-\n\\frac{\\sigma_g^2}{2\\lambda^2}\n\\right].\n\\end{equation}\nIf $z\\to +\\infty$, then exponential index also goes to infinity,\nand so the ambiguity of the type $0\\cdot \\infty$ arises.\nBecause of finite accuracy of computer calculations, this difficulty\nappears rather fast, just for moderate values of $z\\sim 10$.\n\nTo avoid this problem one can use the asymptotic expansion\n\\begin{equation}\n\\begin{array}{l}\n1-\\mathrm{erf}\\left(z\\right)\n\\approx\n\\frac{e^{-z^2}}{z\\sqrt{\\pi}}\\left[1+\\sum\\limits_{k=1}^{k_0}\\frac{(-1)^k(2k-1)!!}{2^kz^{2k}}\n\\right]\n,\\;\\;\\; k_0\\leq z^2.\n\\end{array}\n\\end{equation}\nSubstituting this expansion to~(\\ref{eq:FitCurve1Reduced}), we obtain:\n\\begin{equation}\\label{eq:FitCurve1Asymptot}\n\\begin{array}{l}\nF_{B1}(x;x_g,\\sigma_g,\\lambda)=\n\\frac{1}{2\\left|\\lambda\\right|z\\sqrt{\\pi}}\n\\exp\\left[-\\left(z-\\frac{\\sigma_g}{\\sqrt{2}\\left|\\lambda\\right|}\\right)^2\n\\right]\n\\cdot\n\\left[1+\\sum\\limits_{k=1}^{k_0}\\frac{(-1)^k(2k-1)!!}{2^kz^{2k}}\n\\right].\n\\end{array}\n\\end{equation}\nHere for big  $z$ values no ambiguities appear,\nfunction goes to 0.\nThis very expansion allows to find the limit for $\n\\lambda\\to 0$.\nIndeed,\n\\begin{equation}\n\\lim\\limits_{\\lambda\\to 0} z=+\\infty,\\;\\;\\;\n\\lim\\limits_{\\lambda\\to 0}\n\\left(z-\\frac{\\sigma_g}{\\sqrt{2}\\left|\\lambda\\right|}\\right)=\n\\pm\\frac{\\left(x-x_g\\right)}{\\sigma_g\\sqrt{2}},\\;\\;\\;\n\\lim\\limits_{\\lambda\\to 0} \\left|\\lambda\\right|z=\\frac{\\sigma_g}{\\sqrt{2}},\n\\end{equation}\nand\n\\begin{equation}\\label{eq:FitCurve1Lam=0}\nF_{B1}(x;x_g,\\sigma_g,0)=\n\\frac{1}{\\sigma_g\\sqrt{2\\pi}}\n\\exp\\left[-\\frac{\\left(x-x_g\\right)^2}{2\\sigma_g^2}\n\\right]\n\\end{equation}\nLet us consider the limit $\\sigma_g\\to 0$.\n\n\\begin{equation}\n\\lim\\limits_{\\sigma_g\\to 0} z=-\\frac{x-x_g}{\\lambda}\\cdot \\infty,\n\\;\\;\\; \\lim\\limits_{\\sigma_g\\to 0} z\\sigma_g= -\\frac{(x-x_g)|\\lambda|}{\\lambda\\sqrt{2}},\n\\end{equation}\nand substituting this to~(\\ref{eq:FitCurve1Reduced}), we get\n\\begin{equation}\\label{eq:FitCurve1Sigma=0}\nF_{B1}(x;x_g,0,\\lambda)=\n\\left\\{\n\\begin{array}{l}\n0,\\;\\;\\frac{(x-x_g)}{\\lambda}<0,\\\\[3mm]\n\\frac{1}{\\left|\\lambda\\right|}\\exp\\left[-\\frac{(x-x_g)}{\\lambda}\\right],\n\\;\\; \\frac{(x-x_g)}{\\lambda}>0.\n\\end{array}\\right.\n\\end{equation}\nFunction $F_{B1}$ plots for several sets of parameters\nare presented in Fig.\\,\\ref{SampleOfFB1}.\n\\begin{figure}[tbp]\n\\epsfxsize=0.49\\textwidth\n\\epsfbox{samplefb1lin.eps}\n\\hfill\n\\epsfxsize=0.49\\textwidth\n\\epsfbox{samplefb1log.eps}\n\\caption{\\label{SampleOfFB1}Function $F_{B1}(x;x_g,\\sigma_g,\\lambda_1)$ plots for $\\sigma_g=1,\\;\nx_g=0$ and several values of $\\lambda_1$ }\n\\end{figure}\n\nSometimes the integral distribution can be useful:\n\\begin{equation}\n\\begin{array}{l}\n\\Phi_{B1}(x;x_g,\\sigma_g,\\lambda)\n=\\!\\!\\!\\int\\limits_{-\\infty}^x\\!\\!\\!\\,\\mathrm{d}\\xi\\;F_{B1}(\\xi;x_g,\\sigma_g,\\lambda)=\n\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{x-x_g}{\\sqrt{2}\\sigma_g}\\right)\\right]\n-\\\\[1mm] \\rule{40mm}{0mm}\n-\\frac{\\lambda}{2\\left|\\lambda\\right|}\ne^{\\frac{\\sigma_g^2}{2\\lambda^2}-\\frac{x-x_g}{\\lambda}}\\left[\n1-\\mathrm{erf}\\left(\\frac{\\sigma_g}{\\sqrt{2}\\left|\\lambda\\right|}-\n\\frac{\\left(x-x_g\\right)\\left|\\lambda\\right|}{\\sqrt{2}\\sigma_g\\lambda}\n\\right)\\right].\n\\end{array}\n\\end{equation}\n\nUsage of $F_{B1}$ to fit the distributions\nwould be more convenient, if the free parameter is $x_m$ ---\nlocation of function maximum, instead of $x_g$.\nEquation for the search for $x_m$ looks rather complicated:\n\\begin{equation}\\label{eq:PeakCondition}\\!\\!\\!\\!\n\\begin{array}{l}\n1-\\mathrm{erf}\\left(-\\frac{\\left(x_m-x_g\\right)\\lambda}{\\sigma_g\\left|\\lambda\\right|\\sqrt{2}}\n+\\frac{\\sigma_g}{\\left|\\lambda\\right|\\sqrt{2}}\n\\right)\n=\n\\frac{\\sqrt{2}\\left|\\lambda\\right|}{\\sigma_g\\sqrt{\\pi}}\\exp\\left[-\n\\left(-\\frac{\\left(x_m-x_g\\right)\\lambda}{\\sigma_g\\left|\\lambda\\right|\\sqrt{2}}\n+\\frac{\\sigma_g}{\\left|\\lambda\\right|\\sqrt{2}}\n\\right)^2\n\\right]\n\\end{array}\n\\end{equation}\nor in other notations\n\\begin{equation}\\label{eq:maxFb1}\n1-\\mathrm{erf}(z_m)=\\rho\\exp(-z_m^2),\n\\end{equation}\nwhere\n$\\rho=\\frac{\\sqrt{2}\\left|\\lambda\\right|}{\\sigma_g\\sqrt{\\pi}}>0$.\nFor $\\rho\\to \\infty$ $z_m\\to -\\infty$:\n\\begin{equation}\n\\begin{array}{l}\ne^{z_m^2}=\\frac{\\rho}{2-\\frac{e^{-z_m^2}}{|z_m|\\sqrt{\\pi}}\\cdot\\left[\n1-\\frac{1}{2z_m^2}\\right]}\n\\Longrightarrow\nz_m\\approx -\\sqrt{\\ln\\frac{\\rho}{2-\\frac{2}{\\rho\\sqrt{\\left|\\ln\\frac{\\rho}{2}\\right|}\n\\sqrt{\\pi}}\\cdot\\left[\n1-\\frac{1}{2\\ln\\frac{\\rho}{2}}\\right]}}\n\\Longrightarrow\n\\\\[7mm]\\rule{5mm}{0mm}\\Longrightarrow\n-\\frac{\\left(x_m-x_g\\right)\\lambda}{\\sigma_g\\left|\\lambda\\right|\\sqrt{2}}\n+\\frac{\\sigma_g}{\\left|\\lambda\\right|\\sqrt{2}}=\n-\\sqrt{\\ln\\frac{\\left|\\lambda\\right|}{\\sigma_g\\sqrt{2\\pi}}\n}\n\\end{array}\n\\end{equation}\nor\n\\begin{equation}\nx_m-x_g\\begin{array}[t]{c} \\approx \\\\[-2mm] \\scriptstyle \\sigma_g \\ll |\\lambda|\n\\end{array}\n\\frac{\\sigma_g^2}{\\lambda}+\\frac{\\sigma_g\\left|\\lambda\\right|\\sqrt{2}}{\\lambda}\\cdot\n\\sqrt{\\ln\\frac{\\left|\\lambda\\right|}{\\sigma_g\\sqrt{2\\pi}}\n}\\to 0.\n\\end{equation}\n\nFor $\\rho\\ll 1$ $z_m\\to +\\infty$,\nand here we also can derive approximate solution:\n\\begin{equation}\n\\frac{e^{-z_m^2}}{z_m\\sqrt{\\pi}}\\left[1+\\sum\\limits_{k=1}^{k_0}\\frac{(-1)^k(2k-1)!!}{2^kz_m^{2k}}\n\\right]\n=\\rho\\exp\\left(-z_m^2\\right)\n\\end{equation}\n\n\\begin{equation}\n\\begin{array}{l}\nz_m\\approx \\frac{1}{\\rho\\sqrt{\\pi}}\\cdot\\left(1-\\frac{\\pi \\rho^2}{2}\\right)\n\\Longrightarrow\n-\\frac{\\left(x_m-x_g\\right)\\lambda}{\\sigma_g\\left|\\lambda\\right|\\sqrt{2}}\n+\\frac{\\sigma_g}{\\left|\\lambda\\right|\\sqrt{2}}=\n\\frac{\\sigma_g}{\\sqrt{2}\\left|\\lambda\\right|}\n-\\frac{\\left|\\lambda\\right|}{\\sqrt{2}\\sigma_g}\n\\end{array}\n\\end{equation}\nor\n\\begin{equation}\nx_m-x_g\\approx  \\lambda\\to 0.\n\\end{equation}\nFrom~(\\ref{eq:PeakCondition}) one can derive more terms of Taylor\nseries:\n\\begin{equation}\nx_m-x_g\\begin{array}[t]{c} \\approx \\\\[-2mm] \\scriptstyle |\\lambda| \\ll  \\sigma_g\n\\end{array}\n\\lambda\\cdot\\left[1-\\left(\\frac{\\lambda}{\\sigma_g}\\right)^2+\n\\left(\\frac{\\lambda}{\\sigma_g}\\right)^4+\\ldots\n\\right].\n\\end{equation}\n\nLet us return to the equation~(\\ref{eq:maxFb1}),\nwhich should be solved in order to find  $F_{B1}$\nmaximum location. Let us transform the interval of\n $\\rho$ variable to the interval  $(0,1)$:\n\\begin{equation}\n\\mu=e^{-\\rho} \\Longleftrightarrow \\rho=-\\ln \\mu\n\\end{equation}\nAt the ends of the interval we know the solution:\n\\begin{equation}\n\\begin{array}{l}\n\\mu \\sim 0 \\Longrightarrow z_m\\approx -\\sqrt{\\ln\\left(-\\frac{\\ln\\mu}{2}\\right)}, \\\\[3mm]\n\\mu \\sim 1\\Longrightarrow z_m\\approx \\frac{\\sqrt{\\pi}\\ln\\mu}{2}-\\frac{1}{\\sqrt{\\pi}\\ln\\mu}\n\\end{array}\n\\end{equation}\nLet us look for approximating function in the form:\n\\begin{equation}\n\\begin{array}{l}\nz_m=F_p(\\mu)\\cdot \\frac{\\left(1-\\rho\\right)\n\\left(8+\\rho\\right)\\left(\\mu+0.13\\right)\\left(1-\\mu\\right)}{\\left(3+\\rho\\right)\n\\rho\\sqrt{\\pi}}\n\\sqrt{\\frac{1}{\\rho+1}+\\pi\\ln\\left(1+\\frac{\\rho}\n{2-\\frac{2\\ln\\frac{\\rho}{2}-1}{\\left(\\rho+2\\right)\\sqrt{\\left|\\ln\\left(1+\\frac{\\rho}{2}\\right)\n\\right|}^3\n\\sqrt{\\pi}}}\\right)}\n\\end{array}\n\\end{equation}\nFunction $F_p(\\mu)$ plot and the approximating cubic\nspline with 5 knots are presented in Fig.\\,\\ref{FpApproximation}.\n\\begin{figure}[tbp]\n\\epsfxsize=0.7\\textwidth\n\\centerline{\\epsfbox{fpapprox.eps}}\n\\caption{\\label{FpApproximation}Function $F_p(\\mu)$ plot (points)\nand approximating cubic spline}\n\\end{figure}\nRoot mean square deviation equals $4.1\\cdot 10^{-4}$,\nmaximum error of interpolation $2.2\\cdot 10^{-3}$\nis achieved at $\\mu=1.0\\cdot 10^{-5}$.\nSpline coefficients are cited in Table\\,\\ref{tab:FpSpline}.\n\\begin{table}[tbp]\n\\caption{\\label{tab:FpSpline}Spline of deficiency 2 coefficients for approximation\nof $F_p(\\mu)$ function.}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|c|}\n\\hline\nKnot coordinate & 0 & 0.25 & 0.5 & 0.75 & 1 \\\\\n\\hline\nSpline value    &1.00298 & 0.97574 & 1.05593 & 1.03058 & 0.99898 \\\\\n\\hline\nSpline derivative    & -0.29472 & 0.19545 & 0.27768 & -0.35485 & 0.33206 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nNow we can calculate the shift of peak position vs Gaussian center:\n\\begin{equation}\nx_m-x_g=\\frac{\\sigma_g^2}{\\lambda}-\\frac{\\sigma_g\\left|\\lambda\\right|\\sqrt{2}}\n{\\lambda}z_m=\\Delta X_{mg}(\\sigma_g,\\lambda).\n\\end{equation}\nThe function $\\Delta X_{mg}$\ncan be easily implemented in  any programming language,\nusing the above formulae.\n\n\\section{Build of fitting function on the base of $F_{B1}$}\nIn some cases the function  $F_{B1}$ itself can be suitable\nfor fitting.\nHowever more often this function is not enough\nflexible to provide satisfactory confidence level with\nexperimental distribution.\n\nOne could try to use the function which is the convolution of\nthe three distributions: Gaussian and two different exponential.\nSuch a function could be useful if the distribution ``tails''\nboth to the right and left from the peak do not match Gaussian\ndistribution.\n\nHowever in this case and in other difficult ones the distributions\nare fitted more successfully to the sum of different $F_{B1}$ functions\nwith the common parameter --- peak position $x_m$.\nFor the sum of two functions one can use the following expression:\n\\begin{equation}\n\\begin{array}{l}\nF_{2B1}(x;x_m,\\xi,\\sigma_1,\\lambda_1,\\sigma_2,\\lambda_2)=\n\\cos^2\\xi\\cdot F_{B1}\\left(x;x_m-\\Delta X_{mg}(\\sigma_1,\\lambda_1),\\sigma_1,\\lambda_1\\right)+\n\\\\[3mm]\\rule{51mm}{0mm}+\n\\sin^2\\xi\\cdot  F_{B1}\\left(x;x_m-\\Delta X_{mg}(\\sigma_2,\\lambda_2),\\sigma_2,\\lambda_2\\right)\n\\end{array}\n\\end{equation}\n\nSample of using such a function is presented in Fig.\\,\\ref{FitEtraM0acFB1}\n(one more parameter is added --- common factor $A$).\n\\begin{figure}[tbp]\n\\epsfxsize=0.49\\textwidth\n\\epsfbox{etap2c6netraf1.eps}\n\\hfill\n\\epsfxsize=0.49\\textwidth\n\\epsfbox{etap2c6nm0acf1.eps}\n\\caption{\\label{FitEtraM0acFB1}Distributions vs invariant masses\nof photon pairs.\nLeft --- decay photons of $\\eta$, right --- decay of $\\pi^0$.\nFit function FIT2FB1.}\n\\end{figure}\n\nIn principle for complicated cases one can use the sum of more $F_{B1}$\nfunctions.\nHowever for so many free parameters the likelihood function minimization\ncan be unstable, and one need to help MINUIT program.\nThe simplest and enough effective trick is\noptimization of parameters in turn, initially fixed at some\nreasonable values.\n\n\\section{Conclusion}\nFor fitting the smooth distributions with one peak\na function is suggested, which is the convolution of\nGaussian and exponential distributions.\n\nIn cases when this function is not enough flexible to provide\nsatisfactory confidence level, one can use\nthe sum of several functions with different parameters, but\nthe same peak position.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:1}\nThe last decade has seen an explosion of research in topologically-nontrivial states of matter, following the remarkable discovery of topological insulators (TI)~\\cite{Kane05,Zhang06,Balents07,Kane07,Molenkamp07,Hasan09,Kane10,Qi10}. \nOne normally associates topologically nontrivial properties with insulators, the spectral gap providing the rigidity and insensitivity to fluctuations, which are \ncharacteristic of these states of matter. \nOne of the most recent notable developments in this field, however, has been the realization that even gapless metallic and semimetallic states \nmay be topologically nontrivial in much the same sense as gapped insulators~\\cite{Wan11,Ran11,Burkov11,Xu11}.\nThese recent developments have partly been anticipated in the earlier pioneering work of Volovik~\\cite{Volovik}, promoting topological classification \nof all possible fermionic ground states, which shed new light on such wide-ranging phenomena as the robustness of the Fermi liquid and the hierarchy problem in particle physics. \n\nThe recent work focussed on {\\em Weyl semimetals}~\\cite{Wan11,Ran11,Burkov11,Xu11}, which, in Volovik's classification, belong to the Fermi point \nuniversality class of fermionic vacua.\nIn the condensed matter context a Weyl point or node is a point of contact between two nondegenerate bands in the first Brillouin zone (BZ). Such electronic structure features have in \nfact been noticed and studied since the earliest days of the theory of solids~\\cite{Herring,Abrikosov,Landau}, but only recently have their topological properties come into focus. \nTwo important ingredients in the appearance of Weyl node features in the electronic structure are broken time reversal (TR) or inversion (I) symmetries and \ndimensionality of space. The broken symmetry requirement is a consequence of the well-known Kramers theorem. If both TR and I symmetries are present, the band eigenstates \nin a solid must be (at least) doubly degenerate at every value of the crystal momentum in the first BZ. \nContact can then occur only between {\\em pairs} of bands, which is in general impossible without fine-tuning.\nThe contact between two nondegenerate bands is, however, possible generically in three spatial dimensions (3D). \nIndeed, near the point of contact, the momentum-space Hamiltonian must have the following form, dictated by the so-called Atiyah-Bott-Shapiro \nconstruction~\\cite{Horava}\n\\begin{equation}\n\\label{eq:1}\nH = \\pm v_F \\bsigma \\cdot {{\\bf{k}}}, \n\\end{equation}\nwhere the triplet of Pauli matrices $\\bsigma$ describes the pair of touching bands, the sign in front refers to two possible {\\em chiralities} of the band contact point, and we have subsumed any possible spatial anisotropies into the definition of the crystal momentum ${{\\bf{k}}}$. We will use $\\hbar = c =1$ units throughout this paper, except in some of the final results. \nThe point of band degeneracy occurs when all three components of the crystal momentum vanish. \nThis is why three-dimensionality is important: we must have three momentum components available as ``tuning parameters\" to create a point of degeneracy between the two bands. \nA naively analogous degeneracy point in 2D graphene in fact does not exist, as was pointed out in the seminal work of Kane and Mele~\\cite{Kane05}, \nwhich started the field of TI. \nIt only looks like a degeneracy point due the smallness of the spin-orbit (SO) interactions in graphene, which always open up a small gap and destroy the degeneracy. \n\nThe presence of Weyl degeneracy nodes is, in principle, a common feature of the electronic structure of all 3D magnetic or noncentrosymmetric materials. \nIn most cases, however, their presence is of no consequence, since they will generically occur far away from the Fermi energy~\\cite{Doron13}. \n{\\em Weyl semimetal} refers to a situation in which the Fermi level exactly coincides with the Weyl nodes and no other states are present at the Fermi energy. \nThis situation is not generic and occurs only under special, but not entirely uncommon circumstances.\n\nTo identify these, it is convenient to start from the situation in which both TR and I symmetries are present.\nTR symmetry requires that if a Weyl node is present at momentum ${{\\bf{k}}}$, another node with the same chirality must be present at $-{{\\bf{k}}}$. \nI symmetry, on the other hand, requires that if a Weyl node is present at ${{\\bf{k}}}$, a node of opposite chirality must be present at $-{{\\bf{k}}}$. \nThis implies that in the presence of both TR and I, there may exist a pair of opposite-chirality Weyl nodes at the same TR and I-invariant \ncrystal momentum. For concreteness, we will take this momentum to be at the BZ centre, i.e. at the $\\Gamma$ point. \nThe minimal ${{\\bf{k}}} \\cdot {\\bf{p}}$ Hamiltonian, describing this situation, has the form\n\\begin{equation}\n\\label{eq:2}\nH = v_F \\tau^z \\bsigma \\cdot {{\\bf{k}}} + \\Delta \\tau^x, \n\\end{equation}\nwhere $\\btau$ is an additional  set of Pauli matrices that describes the two nodes of opposite chiralities, i.e. $H$ has the form of the Hamiltonian \nof a 3D four-component Dirac fermion with ``mass\" $\\Delta$. \nThe mass term $\\Delta$ annihilates the two Weyl nodes, unless it vanishes by symmetry or is fine-tuned to zero.  \nTo realize the former situation, the states at the $\\Gamma$ point must either form a four-dimensional irreducible representation of the space group of the crystal, \nas proposed to occur in $\\beta$-cristobalite BiO$_2$~\\cite{Kane12} and, with a few modifications (Dirac Hamiltonian is replaced by Luttinger Hamiltonian in this case), in some of the pyrochlore iridate materials~\\cite{Krempa12,Balents13,Krempa14}. Or they must belong to two distinct two-dimensional representations, so that pairwise crossing, \nprotected by crystal symmetry,\nis possible, as realized in the recently \ndiscovered Dirac semimetal materials $\\textrm{Na}_3 \\textrm{Bi}$ and \n$\\textrm{Cd}_2 \\textrm{As}_3$~\\cite{Fang12,Fang13,Cava13,Shen13,Hasan13}. \n\nThe latter situation, i.e. with $\\Delta$ fine-tuned to zero, occurs at the transition point between an inversion-symmetric 3D TI and a normal insulator (NI)~\\cite{Murakami07,Burkov11,Ando11,Ando14}. In the present review we will focus exclusively on this case. \nOur results, however, do not depend on this choice and are applicable to all realizations of Dirac and Weyl semimetals. \nAs mentioned above, Weyl semimetal may be obtained from Dirac semimetal by breaking either TR or I. We will focus on the TR-breaking case here, \nsince it produces the simplest possible kind of Weyl semimetal, with only a single pair of Weyl nodes.  \nGeneralization to many pairs is in most cases straightforward, but it does complicate the technical details of course. \nFormally, this type of Weyl semimetal is obtained by adding a Zeeman spin-splitting term to the Dirac Hamiltonian Eq.~\\ref{eq:2}\n\\begin{equation}\n\\label{eq:3}\nH = v_F \\tau^z \\bsigma \\cdot {{\\bf{k}}} + \\Delta \\tau^x + b \\sigma^z, \n\\end{equation}\nwhere the spin-splitting term may arise either from an external magnetic field, or from an intrinsic magnetic ordering, and we have chosen the \n$z$-direction to be the magnetization direction. \nTo analyze this Hamiltonian, it is convenient to perform the following canonical (i.e. commutation relation preseving) transformation of the $\\bsigma$ and $\\btau$ operators\n\\begin{equation}\n\\label{eq:4}\n\\sigma^{\\pm} \\rightarrow \\tau^z \\sigma^{\\pm},\\,\\, \\tau^{\\pm} \\rightarrow \\sigma^z \\tau^{\\pm}. \n\\end{equation}\nThe Hamiltonian Eq.~\\ref{eq:3} then transforms to\n\\begin{equation}\n\\label{eq:5}\nH = v_F (\\sigma^x k_x + \\sigma^y k_y) + (b + v_F k_z \\tau^z + \\Delta \\tau^x) \\sigma^z, \n\\end{equation}\nDiagonalizing the $v_F k_z \\tau^z + \\Delta \\tau^x$ block of the Hamiltonian we obtain\n\\begin{equation}\n\\label{eq:6}\nH_{\\pm} = v_F (\\sigma^x k_x + \\sigma^y k_y) + m_{\\pm}(k_z) \\sigma^z, \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:7}\nm_{\\pm}(k_z) = b \\pm \\sqrt{v_F^2 k_z^2 + \\Delta^2}. \n\\end{equation}\nEq.~\\ref{eq:6} looks like the Hamiltonian of a pair of 2D Dirac fermions with ``masses\" $m_{\\pm}(k_z)$, which depend \non the $z$-component of the crystal momentum as a parameter. \nThe mass $m_+(k_z)$ is always positive, but $m_-(k_z)$ may vanish and change sign if $b > \\Delta$. \nIf this is the case, $m_-(k_z)$ vanishes at $k_z^{\\pm} = \\pm k_0$, where \n\\begin{equation}\n\\label{eq:8}\nk_0 = \\frac{1}{v_F} \\sqrt{b^2 - \\Delta^2}. \n\\end{equation}\nThese points along the $z$-axis in momentum space, where the ``mass\" $m_-(k_z)$ vanishes, are the locations of the Weyl nodes. \nThe two bands, which are the eigenstates of the $H_-$ block of the Hamiltonian, are degenerate at those points.  \nSince massive 2D Dirac fermions are associated with half-quantized Hall conductivity~\\cite{Fisher94}\n\\begin{equation}\n\\label{eq:9}\n\\sigma_{xy}^{2D} = \\frac{e^2}{2 h} \\textrm{sign}(m), \n\\end{equation}\nit follows that the Weyl semimetal at $b > \\Delta$ has a Hall conductivity given by\n\\begin{equation}\n\\label{eq:10}\n\\sigma_{xy} = \\frac{e^2}{h} 2 k_0, \n\\end{equation}\ni.e. is equal to the distance between the Weyl nodes in units of $e^2\/h$~\\cite{Klinkhamer}.\n\nIt is easy to show that this Hall conductivity may be associated with chiral edge states. \nIndeed, suppose the sample is finite in the $y$-direction and we will take the sample to\noccupy the $y < 0$ half-space. \nReplacing $k_y \\rightarrow -i \\partial\/ \\partial y$, the $H_-$ block of the Hamiltonian becomes\n\\begin{equation}\n\\label{eq:11}\nH_- = - i v_F \\frac{\\partial}{\\partial y} \\sigma^y + v_F \\sigma^x k_x + m_-(k_z,y) \\sigma^z. \n\\end{equation}\nLet us first set $k_x = 0$ and look for a zero-energy solution of the Schr\\\"odinger equation (SE)\n\\begin{equation}\n\\label{eq:12}\nH_- \\Psi(k_z,y) = 0, \n\\end{equation}\nlocalized at the sample boundary $y = 0$. \nWe look for a solution in the form \n\\begin{equation}\n\\label{eq:13}\n\\Psi(k_z,y) = i \\sigma^y e^{f(k_z,y)} \\phi, \n\\end{equation}\nwhere $f(k_z,y)$ is a scalar function, while $\\phi$ is a two-component spinor. \nPlugging this into the SE Eq.~\\ref{eq:12}, we obtain\n\\begin{equation}\n\\label{eq:14}\n\\left[v_F \\frac{\\partial  f(k_z,y)}{\\partial y} + m_-(k_z,y) \\sigma^x \\right] \\phi = 0. \n\\end{equation}\nThe solution of Eq.~\\ref{eq:14}, satisfying our requirements is \n\\begin{equation}\n\\label{eq:15}\nf(k_z,y) = \\frac{1}{v_F} \\int_0^y m_-(k_z, y') d y', \n\\end{equation}\nand $\\phi = | \\sigma^x = -1 \\rangle$, i.e. the eigenstate of $\\sigma^x$ with eigenvalue $-1$. \nThus we finally obtain the following result for the solution of Eq.~\\ref{eq:14}, localized at the \nsample boundary\n\\begin{equation}\n\\label{eq:16}\n\\Psi(k_z,y) = e^{\\frac{1}{v_F} \\int_0^y m_-(k_z,y') dy'} | \\sigma^x = 1 \\rangle. \n\\end{equation}\nThis solution exists as a localized edge state as long as $m_-(k_z, y \\rightarrow - \\infty) \\geq 0$, i.e. \nas long as $-k_0 \\leq k_z \\leq k_0$. \nFor this reason it is called a Fermi arc~\\cite{Wan11}. \nIt is straightforward to see that the dispersion of this state in the $x$-direction is given by $\\epsilon = v_F k_x$, i.e.\nit is chiral. \n\nAt this point we will wrap up the introductory part of this review and move on to a more detailed account of the transport \nproperties of Weyl semimetals and metals, based on a more realistic microscopic model of Weyl semimetal. \nThe rest of the paper is organized as follows. \nIn Section~\\ref{sec:2} we introduce a realistic microscopic model of a Weyl semimetal, based on a magnetically-doped\nTI-NI multilayer heterostructure. \nIn Section~\\ref{sec:3} we discuss basic electromagnetic response properties of this model Weyl semimetal and introduce \ntwo distinct components of the response, which are directly related to the nontrivial topology of Weyl nodes: the Anomalous Hall Effect (AHE)~\\cite{AHE1,AHE2} \nand the Chiral Magnetic Effect (CME). \nIn Section~\\ref{sec:4} we extend the theory of electromagnetic response of Weyl metals to diffusive transport regime. \nWe demonstrate that magnetic Weyl metals are distinguished from ordinary ferromagnetic metals by a lack of impurity-scattering \nand Fermi-surface contributions to their anomalous Hall conductivity. Instead their anomalous Hall conductivity is determined \nby the distance between the Weyl nodes in momentum space and nothing else. \nWe also show that CME manifests in the diffusive regime as a new kind of weak-field magnetoresistance, which is \nnegative and quadratic in the field. This novel magnetoresistance is expected to occur in all types of Weyl metals\nand may be regarded as their smoking-gun transport characteristic. \nWe conclude in Section~\\ref{sec:5} with a brief overview of the main results.\n\n\\section{Heterostructure model of Weyl semimetal}\n\\label{sec:2}\nAs discussed in the Introduction, perhaps the most straightforward (at least theoretically) way to realize a Weyl semimetal phase is to \nbreak TR symmetry in a material, which in its nonmagnetic state is naturally poised near the phase transition between a TI and NI. \nOne way to achieve this is to engineer a composite material, made of alternating thin layers of TI and NI, as shown in Fig.~\\ref{fig:1}. \nThis system may be regarded as a ``hydrogen atom\" of Weyl semimetals: the most elementary yet realistic system, the description of which is simple enough that purely analytical \ntheory of many phenomena is possible, as will be seen below.  \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=15cm]{multilayer}\n\\end{center}\n\\vspace{-2cm}\n\\caption{Cartoon of the heterostructure model of Weyl semimetal. Magnetized impurities are shown by arrows.}\n\\label{fig:1}\n\\end{figure}\n\nThis system may be described as a chain of 2D Dirac surface states of the TI layers, which are coupled by tunnelling matrix elements \n$\\Delta_S$ for a pair of surface states, belonging to the same TI layer, and $\\Delta_D$ for a pair belonging to nearest-neighbor TI layers. \nWe will assume, for concreteness, that $\\Delta_{S,D} \\geq 0$. \nThe corresponding Hamiltonian has the form\n\\begin{equation}\n\\label{eq:17}\nH = v_F \\tau^z (\\hat z \\times \\bsigma) \\cdot {{\\bf{k}}}_{\\perp} \\delta_{i,j} + \\Delta_S \\tau^x \\delta_{i,j} + \\frac{1}{2}\\ \\Delta_D \\left(\\tau^+ \\delta_{j, i+1} + \\tau^- \\delta_{j, i-1} \\right), \n\\end{equation}\nwhere $i,j$ label the unit cells of the superlattice in the growth ($z$) direction, ${{\\bf{k}}}_{\\perp} = (k_x, k_y)$ are the crystal momentum components, \ntransverse to the growth direction, Pauli matrices $\\btau$ act on the {\\em which surface} pseudospin degree of freedom, while $\\bsigma$ act in the spin degree of freedom. \nDiagonalizing the growth direction hopping part of the Hamiltonian by Fourier transform and performing the canonical transformation of Eq.~\\ref{eq:4}, we obtain \n\\begin{equation}\n\\label{eq:18}\nH = v_F (\\hat z \\times \\bsigma) \\cdot {{\\bf{k}}} + \\hat \\Delta(k_z) \\sigma^z, \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:19}\n\\hat \\Delta(k_z) = \\Delta_S \\tau^x + \\frac{\\Delta_D}{2} \\left(\\tau^+ e^{i k_z d} + \\textrm{h.c.} \\right), \n\\end{equation}\n$d$ being the period of the superlattice in the growth direction.\n$\\hat \\Delta(k_z)$ may now be diagonalized separately, as it commutes with the Hamiltonian. \nIts eigenvalues are given by $\\pm \\Delta(k_z)$, where \n\\begin{equation}\n\\label{eq:20}\n\\Delta(k_z) = \\sqrt{\\Delta_S^2 + \\Delta_D^2 + 2 \\Delta_S \\Delta_D \\cos(k_z d)}. \n\\end{equation}\nThe corresponding two-component spinor wavefunctions are\n\\begin{equation}\n\\label{eq:21}\n|u^t(k_z) \\rangle = \\frac{1}{\\sqrt{2}} \\left(1, t \\frac{\\Delta_S + \\Delta_D e^{-i k_z d}}{\\Delta(k_z)} \\right), \n\\end{equation}\nwhere $t = \\pm$ labels the two eigenvalues. \nThe spin block of the Hamiltonian may now be diagonalized as well, giving rise to the following eigenvalues\n\\begin{equation}\n\\label{eq:22}\n\\epsilon_{s t}({{\\bf{k}}}) = s \\epsilon_t({{\\bf{k}}}) = s \\sqrt{v_F^2(k_x^2 + k_y^2) + m^2_t(k_z)}, \n\\end{equation}\nwhere $s = \\pm$ and \n\\begin{equation}\n\\label{eq:23}\nm_t(k_z) = t \\Delta(k_z). \n\\end{equation}\nThe double degeneracy of the band eigenvalues at every ${{\\bf{k}}}$ in Eq.~\\ref{eq:22} is the Kramers degeneracy, arising due to the \npresence of both TR and I symmetries. \nThe corresponding eigenvectors are given by\n\\begin{equation}\n\\label{eq:24}\n|v^{s t}({{\\bf{k}}}) \\rangle = \\frac{1}{\\sqrt{2}} \\left(\\sqrt{1 + s \\frac{m_t(k_z)}{\\epsilon_t({{\\bf{k}}})}}, - i s e^{i \\phi} \\sqrt{1 - s \\frac{m_t(k_z)}{\\epsilon_t({{\\bf{k}}})}} \\right), \n\\end{equation}\nwhere $e^{i \\phi} = \\frac{k_x + i k_y}{\\sqrt{k_x^2 + k_y^2}}$. \nThe full four-component eigenvectors of the Hamiltonian Eq.~\\ref{eq:18} may be represented as a tensor product $|u^t(k_z)  \\rangle$ and \n$|v^{s t}({{\\bf{k}}}) \\rangle$:\n\\begin{equation}\n\\label{eq:25}\n|z^{s t}({{\\bf{k}}}) \\rangle = |v^{s t}({{\\bf{k}}}) \\rangle \\otimes |u^t(k_z) \\rangle. \n\\end{equation}\n\nThe heterostructure, described by Eq.~\\ref{eq:17}, can exist in two distinct phases: strong 3D TI and an NI.  \nThis can be easily seen by computing the $Z_2$ index, using the method of Fu and Kane~\\cite{Fu07}.  \nNamely, we compute the eigenvalues of the parity operator $\\tau^x$ (or $\\tau^x \\sigma^z$ after the canonical transformation of Eq.~\\ref{eq:4}) \nat two TR-invariant momenta in the first BZ: $\\bGamma_1=(0,0,0)$ and $\\bGamma_2=(0,0,\\pi\/d)$. \nTaking the product of the parity eigenvalues over all the occupied bands and over $\\bGamma_{1,2}$, the $Z_2$ index is found to be\n\\begin{equation}\n\\label{eq:26}\n(-1)^{\\nu} = \\textrm{sign}(\\Delta_S - \\Delta_D). \n\\end{equation}\nThus the multilayer is a strong 3D TI when $\\Delta_D > \\Delta_S$ and an NI otherwise. \nThe point $\\Delta_S = \\Delta_D$ marks the transition point between the 3D TI and NI. \nAt this point, the gap at $\\bGamma_2$ closes and the heterostructure is a Dirac semimetal. \n\nTo obtain a Weyl semimetal, we break TR symmetry in the heterostructure, assuming that it is \ntuned close to the TI-NI transition point and the band gap $|\\Delta_S - \\Delta_D|$ is not large. \nThe TR breaking may be accomplished by doping the heterostructure with magnetic impurities. \nAt sufficient concentration, the impurities will form a ferromagnetic (FM) state, and we will assume that \nthe magnetization points along the growth direction of the heterostructure (distinct, non-Weyl nodal states are \nobtained if the magnetization is in the $xy$-plane~\\cite{Burkov11-2}). \nWe will describe the FM state of the heterostructure by adding a term $b \\sigma^z \\delta_{i,j}$ to the Hamiltonian Eq.~\\ref{eq:17}. \nThis splits the degeneracy of the $t = \\pm$ Kramers doublet, modifying Eq.~\\ref{eq:23} as\n\\begin{equation}\n\\label{eq:27}\nm_t(k_z) = b + t \\Delta(k_z). \n\\end{equation}\nTaking the spin splitting $b$ to be always positive, the ``mass\" $m_+(k_z)$ is then always positive, while $m_-(k_z)$ may vanish \nand change sign as a function of $k_z$, if the spin splitting is sufficiently large.  \nThis happens when $b \\geq b_{c1} = |\\Delta_S - \\Delta_D|$. \nThe Weyl node locations along the $k_x = k_y = 0$ line in the crystal momentum space are given by the solutions of the equation\n\\begin{equation}\n\\label{eq:28}\nm_-(k_z) = b - \\Delta(k_z) = 0, \n\\end{equation}\nwhich gives $k_{z}^{\\pm} = \\pi\/d \\pm k_0$, where \n\\begin{equation}\n\\label{eq:29}\nk_0 = \\frac{1}{d} \\arccos\\left(\\frac{\\Delta_S^2 + \\Delta_D^2 - b^2}{2 \\Delta_S \\Delta_D} \\right). \n\\end{equation}\nWhen the spin splitting reaches the upper critical value $b_{c2} = \\Delta_S + \\Delta_D$, $k_0 = \\pi\/d$ and the \nWeyl nodes are annihilated again at the BZ edge. The resulting state is a 3D quantum anomalous Hall insulator \nwith a quantized Hall conductivity~\\cite{Halperin92}\n\\begin{equation}\n\\label{eq:30} \n\\sigma_{xy} = \\frac{e^2}{h d}. \n\\end{equation}\n\n\\section{Electromagnetic response of Weyl semimetals}\n\\label{sec:3}\n\\subsection{General remarks}\n\\label{sec:3.1}\nTopologically-nontrivial states of matter, such as 3D TI, and Weyl semimetals, have a distinguishing spectroscopic \nfeature: the presence of metallic edge states. \nThese surface metallic states are unusual and may only exist on the surface of a bulk 3D topological phase. \nAs discussed in the Introduction, in the Weyl semimetal state, of interest to us here, the Fermi surface of this metal is not a closed 2D curve, as it must \nbe in any regular 2D metal, but instead forms an open Fermi arc, with the end points at the locations of the bulk Weyl nodes, \nprojected onto the surface BZ~\\cite{Wan11,Potter14}. \n\nHowever, apart from such spectroscopic distinguishing features, topological phases often have unusual electromagnetic response characteristics, \nwhich are always of particular interest, as these are the manifestations of the unique quantum physics of these phases on macroscopic scales. \nIn 3D TI, this unusual electromagnetic response may be described by the so-called $\\theta$-term in the action of the electromagnetic field~\\cite{Qi10}\n\\begin{equation}\n\\label{eq:31}\nS_{\\theta} = \\frac{e^2 \\theta}{32 \\pi^2} \\int d t \\,\\, d^3 r \\epsilon^{\\mu \\nu \\alpha \\beta} F_{\\mu \\nu} F_{\\alpha \\beta} = \\frac{e^2 \\theta}{4 \\pi^2} \\int d t \\,\\, d^3 r {\\bf{E}} \\cdot {\\bf{B}}. \n\\end{equation}\nThe parameter $\\theta$ is equal to $\\pi$ in 3D TI, the only nonzero value, compatible with TR symmetry~\\cite{Qi10}. \nThe $\\theta$-term in Eq.~\\ref{eq:31}, however, is a full derivative, and thus has no effect on Maxwell's equations in the bulk of the TI. \nIts only real effect is to generate the half-quantized AHE on the sample surfaces in the presence of surface magnetization (e.g. due to magnetic \nimpurities). \n\nWhen TR is broken in the bulk, as it is in the Weyl semimetal state, a $\\theta$-term with non-constant $\\theta$ is now allowed by symmetry\n\\begin{equation}\n\\label{eq:32}\nS_{\\theta} = \\frac{e^2}{32 \\pi^2} \\int d t \\,\\, d^3 r \\theta(\\br) \\epsilon^{\\mu \\nu \\alpha \\beta} F_{\\mu \\nu} F_{\\alpha \\beta}.\n\\end{equation}\nThe only functional form, compatible with translational symmetry, is linear, i.e.\n\\begin{equation}\n\\label{eq:33}\n\\theta(\\br) = 2 {\\bf{b}} \\cdot \\br, \n\\end{equation}\nwhere ${\\bf{b}}$ is a vector, which is odd under TR and even under I and coincides, as will be seen explicitly below, with \nthe bulk spin splitting. \nIf, in addition to TR, I symmetry is violated as well, the ``axion field\" $\\theta$ may also acquire a linear time dependence\n\\begin{equation}\n\\label{eq:34}\n\\theta(\\br, t) = 2 {\\bf{b}} \\cdot \\br - 2 b_0 t, \n\\end{equation}\nwhere $b_0$ is the energy difference between the Weyl nodes, allowed if $I$ is violated. \nUnlike the $\\theta$-term in 3D TI, Eq.~\\ref{eq:32} does modify Maxwell's equations in the bulk of the Weyl semimetal and thus \nhas observable consequences, namely the already mentioned AHE and the Chiral Magnetic Effect (CME)~\\cite{Kharzeev}, both of which will be discussed \nin detail below. \n\nThe action of Eq.~\\ref{eq:32} may be described as being a consequence of chiral anomaly, an important concept in relativistic field theory~\\cite{Adler69,Jackiw69,Nielsen83},\nwhich has recently found its way into condensed matter physics and has been realized to play an important role in the theory of topologically-nontrivial \nstates of matter~\\cite{Ryu12,Nagaosa13,Wen13}.  \nTo gain a basic understanding of chiral anomaly, it is useful to go back to the generic low-energy model of Weyl semimetal, given by Eq.~\\ref{eq:3}. \nFor simplicity, we will subsume the Fermi velocity $v_F$ in Eq.~\\ref{eq:3} into the definition of momentum and set the ``mass term\" $\\Delta = 0$. \nThen we obtain\n\\begin{equation}\n\\label{eq:35}\nH = \\tau^z \\bsigma \\cdot {{\\bf{k}}} + b_0  \\tau^z + {\\bf{b}} \\cdot \\bsigma,  \n\\end{equation}\nwhere we have included an energy difference between the Weyl nodes, described by the $b_0 \\tau^z $ term, which is allowed when the I symmetry is violated, \nand oriented the TR-breaking vector ${\\bf{b}}$ along an arbitrary direction. \nIt is convenient to represent the system in terms of an imaginary time action, including possible coupling of the electrons to an electromagnetic field\n\\begin{equation}\n\\label{eq:36}\nS = \\int d \\tau d^3 r \\,\\,\\psi^{\\dag} \\left[ \\partial_{\\tau} + i e A_0 + b_0 \\tau^z + \\tau^z \\bsigma \\cdot \\left( -i \\bnabla + e {\\bf{A}} + {\\bf{b}} \\tau^z \\right) \\right] \\psi^{\\vphantom \\dag},\n\\end{equation}\nwhere $A_{\\mu} = (A_0, {\\bf{A}})$ is the electromagnetic gauge potential and $\\psi^{\\dag}, \\psi^{\\vphantom \\dag}$ are the 4-component spinor Grassman fields. We have suppressed all \nexplicit spinor indices in the Grassmann fields for brevity. \nWe now observe that in addition to the charge conservation symmetry, the imaginary time action Eq.~\\ref{eq:36} possesses an extra chiral symmetry\n\\begin{equation} \n\\label{eq:37}\n\\psi \\rightarrow e^{-i \\tau^z \\theta\/2} \\psi, \n\\end{equation} \nwhich expresses an apparent separate conservation of the number of fermions of left and right chirality. This suggests that the terms $b_0 \\tau^z$ and ${\\bf{b}} \\cdot \\bsigma$ in \nEq.~\\ref{eq:36} could be eliminated by a gauge transformation\n\\begin{equation} \n\\label{eq:38}\n\\psi \\rightarrow e^{- i \\tau^z \\theta(\\br, \\tau)\/2} \\psi,\\,\\, \\psi^{\\dag} \\rightarrow \\psi^{\\dag} e^{i \\tau^z \\theta(\\br, \\tau)\/2},\n\\end{equation}\nwhere $\\theta(\\br, \\tau) = 2 {\\bf{b}} \\cdot \\br - 2 i b_0 \\tau$ and one should keep in mind that $\\psi$ and $\\psi^{\\dag}$ are not complex conjugates of each other, but are independent variables in the fermion path integral. \nThe imaginary time action then becomes\n\\begin{equation}\n\\label{eq:39}\nS = \\int d \\tau d \\br \\,\\, \\psi^{\\dag} \\left[ \\partial_{\\tau} + i e A_0 \n + \\tau^z \\bsigma \\cdot \\left( -i \\bnabla + e {\\bf{A}} \\right) \\right] \\psi^{\\vphantom \\dag}, \n\\end{equation}\nwhich describes two Weyl nodes of opposite chirality, existing at the same point in momentum space and in energy. \nThis argument then leads one to the conclusion that the system of Weyl nodes, separated in energy and momentum, is equivalent \nto the system of two degenerate Weyl nodes and thus does not possess any special transport properties, which is incorrect. \nWhat is missing in the above argument is exactly the chiral anomaly: while the imaginary time action Eq.~\\ref{eq:36} does indeed \npossess the chiral symmetry, the gauge transformation of Eq.~\\ref{eq:38} changes not only the action itself, but also the measure of the path integral. \nThis change of the path integral measure is what gives rise to the $\\theta$-term Eq.~\\ref{eq:32}. \nThis may be shown explicitly using the Fujikawa's method~\\cite{Fujikawa,Zyuzin12}. \nWe refer the reader to Ref.~\\cite{Zyuzin12} for technical details. \n\nTo see the physical consequences of the $\\theta$-term, we integrate Eq.~\\ref{eq:32} by parts and eliminate a total derivative term. \nThis leads to the following action, which resembles the Chern-Simons term in $2+1$-dimensions\n\\begin{equation}\n\\label{eq:40}\nS_{\\theta} = - \\frac{e^2}{8 \\pi^2} \\int dt d^3 r \\,\\, \\partial_{\\mu} \\theta \\epsilon^{\\mu \\nu \\alpha \\beta} A_{\\nu} \\partial_{\\alpha} A_{\\beta}. \n\\end{equation}\nVarying Eq.~\\ref{eq:40} with respect to the vector potential, we obtain the following expression for the current\n\\begin{equation}\n\\label{eq:41}\nj_{\\nu} = \\frac{e^2}{2 \\pi^2} b_{\\mu} \\epsilon^{\\mu \\nu \\alpha \\beta} \\partial_{\\alpha} A_{\\beta}, \\,\\,\\, \\mu = 1, 2, 3, \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:42}\nj_{\\nu} = - \\frac{e^2}{2 \\pi^2} b_0 \\epsilon^{0 \\nu \\alpha \\beta} \\partial_{\\alpha} A_{\\beta}. \n\\end{equation}\nEq.~\\ref{eq:41} clearly represents the AHE,\nwhile Eq.~\\ref{eq:42} represents another effect, related to chiral anomaly in Weyl semimetals, the Chiral Magnetic Effect~\\cite{Kharzeev}, whose \nphysical meaning is somewhat subtle and will be discussed in detail later. \n\nHowever, the picture of chiral anomaly and related transport phenomena, presented above, while simple and appealing, is not fully satisfactory, for the following \nreasons. \nChiral anomaly is a sharply-defined concept in the context of relativistic field theory, where \nmassless fermions in unbounded momentum space possess exact chiral symmetry Eq.~\\ref{eq:37}, which may be violated by the anomaly. \nIn the condensed matter context, however, the situation is less clear. Even though chiral symmetry may be formally defined in a low-energy model of a Weyl semimetal \nEq.~\\ref{eq:35}, in which the band \ndispersion is approximated as being exactly linear and unbounded, no real microscopic model of Weyl semimetal actually possesses such an exact symmetry. \nIt may only appear as an approximate low-energy symmetry. \nSince the chiral symmetry is not present to begin with, it is then unclear to what extent is it meaningful to speak of its violation by chiral anomaly. \nThe role of impurity scattering, present in any real condensed matter system, and important even conceptually in any discussion of transport phenomena; as well as the role of \nfinite electron or hole density, present when the Fermi energy does not exactly coincide with the location of the Weyl nodes, \nare also completely unclear in the above field-theory discussion. \nIn the remainder of this paper we will thus develop a fully microscopic theory of chiral anomaly and its manifestations, including the contribution of impurity scattering, and finite \ncharge carrier density, based on the TI-NI heterostructure model, described in Section~\\ref{sec:2}, which does not possess chiral symmetry, yet exhibits both the AHE and CME, described above. \n\n\\subsection{Microscopic theory: Anomalous Hall Effect}\n\\label{sec:3.2}\nWe will now go back to our microscopic model of a Weyl semimetal, described in Section~\\ref{sec:2}. \nWe want to explicitly derive the topological term in the action of the electromagnetic field by integrating out \nelectron variables, coupled to the field. \nWe then start from the imaginary time action of electrons in the Weyl semimetal, coupled to electromagnetic field\n\\begin{equation}\n\\label{eq:43}\nS = \\int d \\tau d^3 r \\left\\{ \\Psi^{\\dag}(\\br, \\tau) \\left[\\partial_{\\tau} - \\mu + i e A_0 (\\br, \\tau) + \\hat H\\right] \\Psi^{\\vphantom \\dag}(\\br, \\tau)\\right\\}, \n\\end{equation}\nwhere $A_0(\\br, \\tau)$ is the scalar potential and \n\\begin{equation}\n\\label{eq:44}\n\\hat H = v_F \\tau^z (\\hat z \\times \\bsigma) \\cdot \\left(- i \\bnabla + e {\\bf{A}} \\right) + \\hat \\Delta + b \\sigma^z,\n\\end{equation}\nis the Hamiltonian of noninteracting electrons in Weyl semimetal, minimally coupled to the vector potential ${\\bf{A}}$. \nWe will ignore the $z$-component of the vector potential as it will not play any role in what follows. \n\nUsing the results of Section~\\ref{sec:2} for the eigenvalues and eigenvectors of the Weyl semimetal Hamiltonian in the absence of the electromagnetic \nfield, we can now integrate out electron variables in Eq.~\\ref{eq:43} and obtain an effective action for the electromagnetic field, induced by coupling to the electrons. \nThis action will contain two distinct kinds of contributions. The first kind will contain terms, proportional to ${\\bf{E}}^2$ and ${\\bf{B}}^2$, \nwhere ${\\bf{E}}$ and ${\\bf{B}}$ are the electric and magnetic fields. These terms describe the ordinary electric and magnetic polarizability of the material, and we will not discuss this \npart of the response here. \nThe second kind contains the ``topological\" contribution, which has \nthe form of the ``3D Chern-Simons\" term of Eq.~\\ref{eq:40}. \nAdopting the Coulomb gauge $\\bnabla \\cdot {\\bf{A}} = 0$, we obtain\n\\begin{equation}\n\\label{eq:45}\nS = \\sum_{{\\bf{q}}, i\\Omega}\\epsilon^{z 0 \\alpha \\beta} \\Pi({\\bf{q}}, i\\Omega) A_0(-{\\bf{q}}, -i\\Omega) \\hat q_{\\alpha} A_{\\beta}({\\bf{q}}, i \\Omega),\n\\end{equation} \nwhere $\\hat q_{\\alpha} = q_{\\alpha}\/ q$ and summation over repeated indices is implicit. The $z$-direction in Eq.~\\ref{eq:45} is picked out by the \nmagnetization $b$. \nThe response function $\\Pi({\\bf{q}}, i\\Omega)$ is given by\n\\begin{eqnarray}\n\\label{eq:46}\n\\Pi({\\bf{q}}, i \\Omega)&=&\\frac{i e^2 v_F}{V}  \\sum_{{{\\bf{k}}}} \\frac{n_F[\\xi_{s' t'}( {{\\bf{k}}})] - n_F[\\xi_{s t} ({{\\bf{k}}} + {\\bf{q}})]}{i \\Omega + \\xi_{s' t'}({{\\bf{k}}}) - \\xi_{s t} ({{\\bf{k}}} + {\\bf{q}})} \\nonumber \\\\\n&\\times&\\langle z^{s t}_{{{\\bf{k}}} + {\\bf{q}}} | z^{s' t'}_{{{\\bf{k}}}} \\rangle \\langle z^{s' t'}_{{{\\bf{k}}}}| \\bsigma \\cdot \\hat q | z^{s t}_{{{\\bf{k}}} + {\\bf{q}}} \\rangle,\n\\end{eqnarray}\nwhere $\\xi_{s t}({{\\bf{k}}}) = \\epsilon_{s t}({{\\bf{k}}}) - \\epsilon_F$ and summation over repeated band indices $s, t$ is implicit.  \n\nTo evaluate $\\Pi({\\bf{q}}, i\\Omega)$ explicitly it is convenient to rotate coordinate axes so that ${\\bf{q}} = q \\hat x$ and assume that $\\epsilon_F > 0$. The $\\epsilon_F < 0$ result is \nevaluated analogously. \nAs seen from Eq.~\\ref{eq:46}, there exist two kinds of contributions to the response function $\\Pi({\\bf{q}}, i \\Omega)$: {\\em interband}, with $s \\neq s'$, \nand {\\em intraband}, with $s = s' = +$. \nBoth in general contribute and need to be taken into account. \nLet us first evaluate the interband contributions. \nIn this case we can set $i \\Omega = 0$ in the denominator of Eq.~\\ref{eq:46} from the start.\nExpanding to first order in $q$ and integrating over the transverse momentum components $k_{x,y}$, we obtain\n\\begin{eqnarray}\n\\label{eq:52}\n&&\\Pi^{inter}({\\bf{q}}, i\\Omega) = \\frac{e^2 q}{8 \\pi^2} \\sum_t  \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\,\\, \\textrm{sign} [m_t(k_z)] \\nonumber \\\\\n&\\times& \\left\\{1 - \\left[1 - \\frac{|m_t(k_z)|}{\\epsilon_F} \\right] \\Theta(\\epsilon_F - |m_t(k_z)|)\\right\\}, \n\\end{eqnarray}\nwhere $\\Theta(x)$ is the Heaviside step function. \nAs clear from Eq.~\\ref{eq:52}, $\\Pi^{inter}({\\bf{q}}, i\\Omega)$ consists of two distinct terms.  \nThe first term inside the curly brackets in Eq.~\\ref{eq:52} is the contribution of the completely filled $s = -$ bands. The second term is the contribution \nof incompletely filled $s = +$ bands. \n\nTo understand the meaning of Eq.~\\ref{eq:52}, let us now evaluate the {\\em intraband} contribution to $\\Pi({\\bf{q}}, i \\Omega)$. \nIn this case we have $s = s' = +$ in Eq.~\\ref{eq:46}. It is then clear that, unlike in the case of the interband contribution, evaluated \nabove, the value of the intraband contribution depends on the order in which the limits of ${\\bf{q}}  \\rightarrow 0$ and $\\Omega \\rightarrow 0$ \nare taken. If the limit is taken so that $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow \\infty$, then one is evaluating the DC limit of a transport quantity, i.e the optical Hall \nconductivity. In this case, the intraband contribution vanishes identically. \nOn the other hand, if the limit is taken so that $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow 0$, then one is evaluating an equilibrium thermodynamic property, whose \nphysical meaning will become clear below. \nIn this case, the intraband contribution is not zero and is given by \n\\begin{eqnarray}\n\\label{eq:59}\n\\Pi^{intra}({\\bf{q}}, i\\Omega) = \\frac{i e^2 v_F}{V}\\sum_t  \\sum_{{{\\bf{k}}}} \\left.\\frac{d n_F(x)}{d x} \\right|_{x=\\epsilon_t({{\\bf{k}}}) - \\epsilon_F} \\langle z^{+ t}_{{{\\bf{k}}} + {\\bf{q}}} | z^{+ t}_{{{\\bf{k}}}} \\rangle \\langle z^{+t}_{{{\\bf{k}}}} |\\bsigma \\cdot \\hat q| z^{+ t}_{{{\\bf{k}}} + {\\bf{q}}} \\rangle. \\nonumber \\\\\n\\end{eqnarray}\nThe derivative of the Fermi distribution function in Eq.~\\ref{eq:59} expresses the important fact that $\\Pi^{intra}({\\bf{q}}, i \\Omega)$ is associated \nentirely with the Fermi surface, unlike $\\Pi^{inter}({\\bf{q}}, i\\Omega)$, to which all filled states contribute, including states on the Fermi surface.  \nExpanding to first order in $q$ as before and evaluating the integrals over $k_{x,y}$, we obtain\n\\begin{equation}\n\\label{eq:61}\n\\Pi^{intra}({\\bf{q}}, i\\Omega) = - \\frac{e^2 q}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\, \\, \\frac{m_t(k_z)}{\\epsilon_F} \\Theta(\\epsilon_F - |m_t(k_z)|).\n\\end{equation}\ni.e. the intraband contribution to $\\Pi({\\bf{q}}, i\\Omega)$ is equal to the second term in the square brackets in Eq.~\\ref{eq:52} in magnitude, \nbut opposite in sign. \nCombining the inter- and intraband contributions to $\\Pi({\\bf{q}}, i\\Omega)$ we thus obtain \n\\begin{equation}\n\\label{eq:62}\n\\Pi({\\bf{q}}, i\\Omega) = \\frac{e^2 q}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\,\\, \\textrm{sign} [m_t(k_z)]  \\left[1 - \\Theta(\\epsilon_F - |m_t(k_z)|)\\right],\n\\end{equation}\ni.e. the last term in Eq.~\\ref{eq:52} cancels out when \nthe low-frequency, long-wavelength limit is taken in such a way that $\\Omega\/v_F |{\\bf{q}}| \\rightarrow 0$. \nOn the other hand, when $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow \\infty$, the intraband contribution vanishes and \n\\begin{equation}\n\\label{eq:63}\n\\Pi({\\bf{q}}, i\\Omega) = \\Pi^{inter}({\\bf{q}}, i\\Omega).\n\\end{equation}\nThis physical difference in the kind of response the system exhibits is the basis of Streda's separation of contributions to the Hall conductivity into $\\sigma_{xy}^I$ and \n$\\sigma_{xy}^{II}$~\\cite{Streda}.\nOur analysis makes it clear that this is the most physically-meaningful separation of contributions to the intrinsic anomalous Hall conductivity, since it corresponds \nto distinct and, at least in principle, separately measurable, contributions to the response function $\\Pi({\\bf{q}}, i\\Omega)$. \n\n\\begin{figure}[t]\n\\subfigure[]{\n   \\label{fig:2a}\n  \\includegraphics[width=7cm]{bands1}}\n\\subfigure[]{\n  \\label{fig:2b}\n   \\includegraphics[width=7cm]{sigmaxy1}}\n  \\caption{(a) Plot of the band edges along the $z$-direction in momentum space.\n  The parameters are such that two Weyl nodes are present.\n  (b) Total intrinsic anomalous Hall conductivity (solid line), $\\sigma_{xy}^I$ (dashed line), and $\\sigma_{xy}^{II}$ (dotted line). Note that the van Hove-like singularities in \n  $\\sigma^I_{xy}$ and $\\sigma^{II}_{xy}$, associated with band edges, mutually cancel and the total Hall conductivity $\\sigma_{xy}$ is a smooth function of the Fermi energy.}\n    \\label{fig:2}\n\\end{figure}\n\\begin{figure}[t]\n\\subfigure[]{\n   \\label{fig:3a}\n  \\includegraphics[width=7cm]{bands2}}\n\\subfigure[]{\n  \\label{fig:3b}\n   \\includegraphics[width=7cm]{sigmaxy2}}\n  \\caption{(a) Plot of the band edges along the $z$-direction in momentum space. The spin splitting is not large enough for the \n  Weyl nodes to appear (i.e. $b < b_{c1}$) and the spectrum has a full gap. \n  (b) Total intrinsic anomalous Hall conductivity (solid line), $\\sigma_{xy}^I$ (dashed line), and $\\sigma_{xy}^{II}$ (dotted line).}\n    \\label{fig:3}\n\\end{figure}\n\nGeneralizing the above results to arbitrary sign of $\\epsilon_F$ we finally obtain\n\\begin{equation}\n\\label{eq:64}\nS = - i \\epsilon^{z 0 \\alpha \\beta} \\sigma_{xy} \\int d^3 r d \\tau A_0(\\br, \\tau) \\partial_{\\alpha} A_{\\beta}(\\br, \\tau), \n\\end{equation}\nwhere, if the low-frequency long-wavelength limit is taken so that $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow \\infty$:\n\\begin{eqnarray}\n\\label{eq:65}\n\\sigma_{xy}&=&\\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\,\\, \\left\\{\\textrm{sign} [m_t(k_z)] \\left[\\Theta(\\epsilon_F + |m_t(k_z)|) - \\Theta(\\epsilon_F - |m_t(k_z)|)\\right] \\right.\\nonumber \\\\\n&+&\\left. \\frac{m_t(k_z)}{|\\epsilon_F|} \\Theta(|\\epsilon_F| - |m_t(k_z)|) \\right\\}.\n\\end{eqnarray}\nThis expression corresponds to the full DC anomalous Hall conductivity. \nOn the other hand, when the low-frequency, long-wavelength limit is taken so that $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow 0$, we obtain\n\\begin{eqnarray}\n\\label{eq:66}\n\\sigma_{xy}&=&\\sigma_{xy}^{II}=\\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\,\\, \\textrm{sign} [m_t(k_z)] \\nonumber \\\\\n&\\times&\\left[\\Theta(\\epsilon_F + |m_t(k_z)|) - \\Theta(\\epsilon_F - |m_t(k_z)|)\\right].\n\\end{eqnarray}\nThis is precisely the Streda's $\\sigma^{II}_{xy}$ contribution to the Hall conductivity, which is a thermodynamic equilibrium quantity, \nequal to\n\\begin{equation}\n\\label{eq:67}\n\\sigma^{II}_{xy} = e \\left(\\frac{\\partial N}{\\partial B}\\right)_{\\mu}, \n\\end{equation}\nwhere $N$ is the total electron number. This relation follows immediately from Eq.~\\ref{eq:64} and the order of limits\n$\\Omega\/ v_F |{\\bf{q}}| \\rightarrow 0$, which corresponds to thermodynamic equilibrium.\nCorrespondingly, the $\\sigma^{I}_{xy}$ contribution is given by\n\\begin{equation}\n\\label{eq:68}\n\\sigma^I_{xy} = \\sigma_{xy} - \\sigma^{II}_{xy} =  \\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\frac{m_t(k_z)}{|\\epsilon_F|} \\Theta(|\\epsilon_F| - |m_t(k_z)|).\n\\end{equation}\nAs clear from the above analysis, $\\sigma^I_{xy}$ is the contribution to $\\sigma_{xy}$ that can be associated with \nstates on the Fermi surface. This contribution is nonuniversal, i.e. it depends on details of the electronic structure and \nis a continuous function of the Fermi energy. \n$\\sigma^{II}_{xy}$, on the other hand, is the contribution of all states below the Fermi energy and is a thermodynamic \nequilibrium property of the ferromagnet. It attains a universal value,  which depends only on the distance between the \nWeyl nodes, when the Fermi energy coincides with the nodes, i.e. when $\\epsilon_F = 0$\n\\begin{equation}\n\\label{eq:69}\n\\sigma_{xy}^{II} = \\frac{e^2 {\\cal K}}{4 \\pi^2},\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:70}\n{\\cal K} = \\frac{2}{d} \\arccos\\left(\\frac{\\Delta_S^2 + \\Delta_D^2 - b^2}{2 \\Delta_S  \\Delta_D} \\right), \n\\end{equation}\nis the distance between the Weyl nodes. \nWhen $b > b_{c2}$, the Weyl nodes annihilate at the edges of the Brillouin zone and a gap opens up. \nIn this case ${\\cal K} = 2 \\pi\/d$, i.e. a reciprocal lattice vector and $\\sigma^{II}_{xy}$ is quantized as long as the Fermi level is in the gap~\\cite{Halperin92}.\nBoth contributions, along with the total anomalous Hall conductivity $\\sigma_{xy}$ are plotted as a function of the Fermi energy in Figs.~\\ref{fig:2},~\\ref{fig:3}\nin two different cases: when Weyl nodes are present and when they are not. \nThe former occurs when $b_{c1} < b < b_{c2}$. \nAs can be seen from Fig.~\\ref{fig:2}, Weyl nodes provide the dominant contribution to $\\sigma^{II}_{xy}$ and to the total Hall conductivity $\\sigma_{xy}$, if the \nFermi level is not too far from the nodes. \n\nIt is interesting to note the following property, which is evident from Fig.~\\ref{fig:2}. \nBoth the total anomalous Hall conductivity $\\sigma_{xy}$ and the two distinct contributions to it, $\\sigma_{xy}^{I,II}$ appear to exhibit a \nquasi-plateau behavior when $\\epsilon_F$ is not too far from the Weyl nodes. To understand the origin of this behavior, consider the derivative \nof $\\sigma^{II}_{xy}$ with respect to the Fermi energy\n\\begin{equation}\n\\label{eq:71}\n\\frac{\\partial \\sigma^{II}_{xy}}{\\partial \\epsilon_F} = - \\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\textrm{sign}[m_t(k_z)] \\delta(\\epsilon_F - |m_t(k_z)). \n\\end{equation}\nThis is straightforward to evaluate analytically and  we obtain\n\\begin{eqnarray}\n\\label{eq:72}\n\\frac{\\partial \\sigma_{xy}^{II}}{\\partial \\epsilon_F}&=&- \\frac{e^2}{8 \\pi^2} \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\left[\\delta(\\Delta(k_z) - b + \\epsilon_F) - \\delta(\\Delta(k_z) - b -\\epsilon_F) \\right] \\nonumber \\\\ &=&\\frac{e^2}{4 \\pi^2}\\left(1\/\\tilde v_{F+} - 1\/\\tilde v_{F-} \\right), \n\\end{eqnarray}\nwhere we have assumed that $\\epsilon_F$ is sufficiently close to zero, so that only the $t = -$ bands contribute to the integral, and \n\\begin{equation}\n\\label{eq:73}\n\\tilde v_{F\\pm} = \\frac{d}{2 (b \\pm \\epsilon_F)} \\sqrt{[(b \\pm \\epsilon_F)^2 - b_{c1}^2] [b_{c2}^2 - (b \\pm \\epsilon_F)^2]}, \n\\end{equation}\nare the two Fermi velocities, corresponding to two pairs of solutions of the equation $|b - \\Delta(k_z)| = \\epsilon_F$, which \narise from the Fermi level crossing the $s=+, t=-$ band on the two sides of each Weyl node along the $z$-axis in momentum space. \nThe two Fermi velocities are nearly equal when the band dispersion near the nodes is almost perfectly linear, but start to differ significantly \nwhen deviations from linearity become noticeable. \nExplicitly, as long as $b_{c1} \\ll b \\pm \\epsilon_F \\ll b_{c2}$, both Fermi velocities are independent \nof the Fermi energy and thus $\\partial \\sigma_{xy}^{II}\/\\partial \\epsilon_F$ vanishes. \nA useful way to think about this is in terms of an approximate {\\em chiral symmetry}, which emerges  in this system in the limit of small $\\epsilon_F$, and which manifests \nin an almost perfectly linear band dispersion near the Weyl nodes. \n \nBy a nearly identical calculation it is easy to show that $\\partial \\sigma^{I}_{xy} \/ \\partial \\epsilon_F$ also vanishes when $\\epsilon_F$ is sufficiently close to zero. \nThis is the origin of the quasi-plateau behavior in Fig.~\\ref{fig:2}.\nThis result implies that the intrinsic anomalous Hall conductivity is equal to its thermodynamic equilibrium part, $\\sigma^{II}_{xy}$, not just when the Fermi \nenergy coincides with the Weyl nodes, but even away from them as long as the band dispersion may be assumed to be linear~\\cite{Burkov14}.\nThis property will be discussed in more detail in Section~\\ref{sec:4}. \n\nWhen attempting to understand the physical meaning of the two contributions to the anomalous Hall conductivity, $\\sigma^I_{xy}$ and $\\sigma^{II}_{xy}$, \nit may be tempting to say that, while $\\sigma^{I}_{xy}$ is clearly associated with states on the Fermi surface, $\\sigma^{II}_{xy}$ might perhaps be associated \nwith the chiral Fermi arc edge states. Unfortunately, such an interpretation is clearly incorrect, at least in the context of the present model, since, as can be \nseen in Fig.~\\ref{fig:3}, $\\sigma^{II}_{xy}$ may be nonzero even when Weyl nodes and thus the Fermi arc edge states are absent. More work is thus needed \nto fully understand the relation between $\\sigma^{II}_{xy}$ and the edge states. \n \n\n\n\\subsection{Microscopic theory: Chiral Magnetic Effect}\n\\label{sec:3.3}\nIn this subsection we consider the second part of the topological response in Weyl semimetals, namely the CME. \nThis arises due to the presence of an energy difference between the Weyl nodes, as in Eq.~\\ref{eq:34}. \nTo induce such an energy difference in our microscopic model, we need to find an operator that acts as an ``axial chemical potential\" term, \nshifting the nodes with different chirality in opposite directions in energy. \nThe expression for this operator may be found based on symmetry considerations. \nWe define an ``axial charge density\" operator, $\\hat n_a$, as a local operator, which is odd under inversion and $z \\rightarrow -z$ reflections, even under \ntime reversal, but odd under time reversal combined with rotation of the spin quantization axis by $\\pi$ around either $x$ or $y$ axis. \nThis leads to the following expression for the axial charge density operator\n\\begin{equation}\n\\label{eq:74}\n\\hat n_a = \\tau^y \\sigma^z. \n\\end{equation}\nAdding a term $- \\mu_a \\hat n_a$, where $\\mu_a$ is the ``axial chemical potential\", to the multilayer Hamiltonian, introduces \nan energy difference between the Weyl nodes of magnitude\n\\begin{equation}\n\\label{eq:75}\n\\Delta \\epsilon = \\frac{2 \\mu_a \\tilde v_F}{\\Delta_S d}, \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:76}\n\\tilde v_F = \\frac{d}{2 b} \\sqrt{(b^2 - b_{c1}^2)(b_{c2}^2 - b^2)}, \n\\end{equation}\nis the $z$-component of the Fermi velocity at the location of the Weyl nodes. \n\nSimilarly to the previous subsection, we now couple the electrons to electromagnetic \nfield and the axial chemical potential and integrate out the electron variables to obtain \nan induced action for the electromagnetic field. \nWe will assume, without loss of generality, that the electromagnetic field consists of a magnetic field in the $z$-direction, \nand a vector potential $A_z$, whose time derivative gives the $z$-component of the electric field $E_z = - \\partial_t A_z$. \nWe will allow for a time and $z$-coordinate dependence of the vector potential $A_z$ and of the axial chemical potential $\\mu_a$, but \nassume that the magnetic field is time-independent and uniform. \n\nFor this calculation it is convenient to use the Landau level (LL) basis of the multilayer placed in a uniform external magnetic field along the growth \ndirection, i.e. the basis of the eigenstates of the following Hamiltonian\n\\begin{equation}\n\\label{eq:77}\n{\\cal H}(k_z) = v_F \\tau^z (\\hat z \\times \\bsigma) \\cdot \\left(- i \\bnabla + e {\\bf{A}} \\right) + \\hat \\Delta(k_z) + b \\sigma^z, \n\\end{equation}\nAdopting Landau gauge for the vector potential ${\\bf{A}} = x B \\hat y$, we obtain the following expressions LL eigenstates, which have the form, typical for LLs in Dirac systems\n\\begin{equation}\n\\label{eq:78}\n| n, a, k_y, k_z \\rangle = \\sum_{\\tau} \\left[z^a_{n \\uparrow \\tau}(k_z) | n -1, k_y, k_z, \\uparrow, \\tau \\rangle + z^a_{n \\da \\tau}(k_z) | n, k_y, k_z, \\da, \\tau \\rangle \\right]. \n\\end{equation} \nHere \n\\begin{equation}\n\\label{eq:79}\n\\langle \\br | n, k_y, k_z, \\sigma, \\tau \\rangle = \\frac{1}{\\sqrt{L_z}} e^{i k_z z} \\phi_{n k_y}(\\br) | \\sigma, \\tau \\rangle, \n\\end{equation}\n$\\phi_{n k_y}(\\br)$ are the Landau-gauge orbital wavefunctions, and $\\sigma, \\tau$ are the spin and pseudospin indices \nrespectively. Finally, the four-component eigenvector $| z^a_{n}(k_z) \\rangle$ may be written as a tensor product of the two-component spin and pseudospin eigenvectors, \ni.e. $| z^a_{n}(k_z) \\rangle = | v^a_{n}(k_z) \\rangle \\otimes | u^a(k_z) \\rangle$, where \n\\begin{eqnarray}\n\\label{eq:80}\n&&|v^{s t}_{n}(k_z) \\rangle = \\frac{1}{\\sqrt{2}} \\left(\\sqrt{1 + s \\frac{m_t(k_z)}{\\epsilon_{n t}(k_z)}}, - i s \\sqrt{1 - s \\frac{m_t(k_z)}{\\epsilon_{n t}(k_z)}} \\right), \\nonumber \\\\\n&&|u^t(k_z) \\rangle = \\frac{1}{\\sqrt{2}} \\left(1, t \\frac{\\Delta_S + \\Delta_D e^{- i k_z d}}{\\Delta(k_z)} \\right),\n\\end{eqnarray}\nand the eigenstate energies are given by\n\\begin{equation}\n\\label{eq:80.1}\n\\epsilon_{n s t}(k_z) = s \\sqrt{2 \\omega_B^2 n + m_t^2(k_z)} \\equiv s \\epsilon_{n t}(k_z), \n\\end{equation}\nwhere $\\omega_B = v_F\/ \\ell_B$ is the Dirac cyclotron frequency and $\\ell_B = 1\/\\sqrt{e B}$ is the magnetic length. \n\nAs in all Dirac systems, the lowest $n = 0$ LL is special and needs to be considered separately. The $s$ quantum number is absent in this case and  \ntaking $B > 0$ for concreteness, we have $\\epsilon_{n t}(k_z) = - m_t(k_z)$, and $|v^t_{0}(k_z) \\rangle = (0,1)$. \n\nThe topological term, of interest to us, is proportional to the product of $\\mu_a$ and $A_z$. \nIntegrating out the electron variables and leaving only this term in the imaginary time action, we obtain\n\\begin{equation}\n\\label{eq:85}\nS = B \\sum_{q, i \\Omega} \\Pi(q, i\\Omega) A_z(q, i\\Omega) \\mu_a(-q, -i \\Omega)\n\\end{equation}\nwhere the response function $\\Pi(q, i\\Omega)$ is given by:\n\\begin{eqnarray}\n\\label{eq:86}\n\\Pi(q, i\\Omega)&=&\\frac{e}{2 \\pi L_z} \\sum_{n, k_z} \\frac{n_F[\\xi_{n a'}(k_z)] - n_F[\\xi_{n a}(k_z + q)]}{i \\Omega + \\xi_{n a'}(k_z) - \\xi_{n a}(k_z + q)} \\nonumber \\\\\n&\\times& \\langle z^a_{n k_z}| \\hat j_z(k_z) | z^{a'}_{n k_z} \\rangle \\langle z^{a'}_{n k_z} | \\tau^y |z^a_{n k_z} \\rangle. \n\\end{eqnarray}\nHere $n_F$ is the Fermi-Dirac distribution function, $\\xi_{n a}(k_z) = \\epsilon_{n a}(k_z) - \\epsilon_F$, and the magnetic field $B$ in Eq.~\\ref{eq:85} arises \nfrom the Landau level orbital degeneracy. \nWe have also ignored the $q$-dependence of the matrix elements in Eq.~\\ref{eq:86}, which is not important for small $q$. \n\nAt this point we will specialize to the case of an undoped Weyl semimetal, i.e. set $\\epsilon_F = 0$. \nThen it is clear from Eq.~\\ref{eq:86} that for Landau levels with $n \\geq 1$, only terms with $s \\neq s'$ contribute due to the difference of Fermi factors \nin the numerator. \nWe are interested ultimately in the zero frequency and zero wavevector limit of the response function $\\Pi(q, i\\Omega)$. \nAs already seen in the previous section in the context of AHE, the value of $\\Pi(0,0)$ depends on the order in which the zero frequency and zero wavevector \nlimits are taken.\nBelow we will consider both possibilities separately and discuss their physical meaning. \n\nBefore we proceed with an explicit evaluation of the $q \\rightarrow 0$ and $\\Omega \\rightarrow 0$ limit, let us note \nan important property of the response function $\\Pi(q, \\Omega)$.  \nIf we take into account the following symmetry properties of the matrix elements in Eq.~\\ref{eq:86}\n\\begin{eqnarray}\n\\label{eq:88}\n&&\\langle v^{+ t}_{n k_z} | v^{- t'}_{n k_z} \\rangle  = - \\langle v^{+ t'}_{n k_z} | v^{- t}_{n k_z} \\rangle, \\nonumber \\\\\n&&\\langle v^{+ t}_{n k_z} | \\sigma^z| v^{- t'}_{n k_z} \\rangle  =  \\langle v^{+ t'}_{n k_z} | \\sigma^z| v^{- t}_{n k_z} \\rangle, \n\\end{eqnarray}\nit is easy to see that when the limit $\\Omega \\rightarrow 0$ is taken, independently of the value of $q$, the $n \\geq 1$ Landau levels in fact \ndo not contribute at all, mutually cancelling due to Eq.~\\ref{eq:88}. It follows that $\\Pi(q, i\\Omega)$ at small $\\Omega$ is\ndetermined completely by the contribution of the two $n = 0$ Landau levels, whose energy eigenvalues and the corresponding \neigenvectors are independent of the magnetic field. \nThis means that in the small $\\Omega$ limit $\\Pi(q, i \\Omega)$ becomes independent of the magnetic field and the effective action in Eq.~\\ref{eq:85} \nthen depends linearly on both $\\mu_a$ and $B$, independently of the magnitude of $B$. \n\nLet us now proceed to explicitly evaluate $\\Pi(0,0)$, which determines the low-frequency and long-wavelength \nresponse of our system. \nLet us first look at the situation when we send $q$ to zero before sending $\\Omega$ to zero. \nExplicitly evaluating the matrix elements in Eq.~\\ref{eq:86} and the momentum integrals we obtain the following simple expression for $\\Pi(0,0)$\n\\begin{equation}\n\\label{eq:93} \n\\Pi(0,0) = - \\frac{e^2}{4 \\pi^2} \\frac{2 \\tilde v_F}{\\Delta_S d}.\n\\end{equation}\nThus, after Wick's rotation $\\tau \\rightarrow i t$, $\\Delta \\epsilon \\rightarrow  - i \\Delta \\epsilon$, we finally obtain the following result for the electromagnetic field action\n\\begin{equation}\n\\label{eq:94}\nS =  - \\frac{e^2 \\Delta \\epsilon}{4 \\pi^2} \\frac{2 \\tilde v_F}{\\Delta_S d} B \\int d^3 r d t \\,\\, A_z(\\br, t) \\mu_a(\\br, t),\n\\end{equation}\nwhich, taking into account Eq.~\\ref{eq:75}, has precisely the form of Eq.~\\ref{eq:40}. \nFunctional derivative of Eq.~\\ref{eq:94} with respect to $A_z$ gives the current that flows in response to magnetic \nfield and axial chemical potential, i.e. the CME\n\\begin{equation}\n\\label{eq:95}\nj_z  = - \\frac{e^2 \\Delta \\epsilon}{4 \\pi^2} B. \n\\end{equation}\n\nThe physical interpretation of Eq.~\\ref{eq:95} requires some care. \nThe issue is again the order of limits when calculating $\\Pi(0,0)$, which always arises when calculating \nresponse functions in gapless systems.\nEqs.~\\ref{eq:94},\\ref{eq:95} was obtained by sending $q \\rightarrow 0$ before $\\Omega \\rightarrow 0$. \nLet us now see what happens when $\\Omega$ is sent to zero before taking the limit $q \\rightarrow 0$. \nIn this case, in addition to the contribution to $\\Pi(0,0)$, given by Eq.~\\ref{eq:93}, which arises due to transitions \nbetween the $t = +$ and $t = -$ lowest ($n = 0$) Landau levels, there is an extra contribution due to the intra-Landau-level \nprocesses within the $t = -$ Landau level, which crosses the Fermi energy at the location of the Weyl nodes. \nThis extra contribution is given by\n\\begin{equation}\n\\label{eq:96}\n\\tilde \\Pi(0,0) = \\frac{ e}{2 \\pi L_z} \\sum_{k_z} \\left. \\frac{d n_F(\\epsilon)}{d \\epsilon} \\right|_{\\epsilon = - m_-(k_z)} \\langle z^-_{0 k_z} | \\hat j_z(k_z) | z^-_{0 k_z} \n\\rangle \\langle z^-_{0 k_z} | \\tau^y | z^-_{0 k_z} \\rangle, \n\\end{equation}\nand is easily shown to be equal to Eq.~\\ref{eq:93} in magnitude, but opposite in sign, which means that in this case $\\Pi(0,0)$ vanishes. \nThus, the final result for  $\\Pi(0,0)$ depends on the order in which the $q \\rightarrow 0$ and $\\Omega \\rightarrow 0$ limits \nare taken\n\\begin{eqnarray}\n\\label{eq:97}\n&&\\lim_{\\Omega \\rightarrow 0} \\lim_{q \\rightarrow 0} \\Pi(q, \\Omega) = - \\frac{e^2}{4 \\pi^2} \\frac{2 \\tilde v_F}{\\Delta_S d}, \\nonumber \\\\\n&&\\lim_{q \\rightarrow 0} \\lim_{\\Omega \\rightarrow 0} \\Pi(q, \\Omega) = 0. \n\\end{eqnarray}\nWhat is the physical meaning of these two distinct orders of limits in calculating $\\Pi(0,0)$? \nThis is again identical to the AHE case, discussed in the previous section. \nWhen $q$ is taken to zero first, one is calculating the low-frequency limit of response to \na time-dependent external field, in our case magnetic field along the $z$-direction. \nThis response is finite and represents CME, described by Eq.~\\ref{eq:95}. \nIf one takes $\\Omega$ to zero first, however, one is calculating a thermodynamic property, in our \ncase change of the ground state energy of the system in the presence of an additional static vector potential  \nin the $z$-direction\n\\begin{equation}\n\\label{eq:98}\nj_z = \\frac{1}{V} \\frac{\\partial E(A_z)}{\\partial A_z} = 0. \n\\end{equation}\nThis could be nonzero in, for example, a current-carrying state of a superconductor, which \npossesses phase rigidity, but vanishes identically in our case~\\cite{Franz13,Burkov13}. \n\n\\section{Diffusive transport in Weyl metals}\n\\label{sec:4}\nIn the previous section we have focused on the basic properties of electromagnetic response \nof clean Weyl semimetals and metals (doped Weyl semimetals), neglecting the effect of impurity scattering. \nIt is, however, always present, and while probably not of significant importance in an undoped Weyl semimetal\nwith density of states at the Fermi energy vanishing as $\\sim \\epsilon_F^2$, it may be expected to play an important role \nin a Weyl metal with $\\epsilon_F$ significantly different from zero.  \nIf nothing else, impurity scattering is crucial for establishing a steady state under applied electric field, and thus must be \nincluded in any serious discussion of low-frequency transport phenomena. \nIn this section we will thus generalize the theory of electromagnetic response, presented in Section~\\ref{sec:3}, to the case of diffusive transport in dirty Weyl metals.\nAs will be demonstrated below, this generalization leads to two important new results. \n\nFirst, we will demonstrate that the anomalous Hall conductivity of a Weyl metal is purely intrinsic and universal, \nlacking both the so-called extrinsic contribution due to the impurity scattering and also the part of the intrinsic \nelectronic-structure contribution, coming from incompletely filled bands. Only the purely intrinsic semi-quantized contribution, \n$\\sigma_{xy}^{II}$, arising from completely filled bands and proportional to the separation between the Weyl nodes, survives \nin the Weyl metal, as long as the Fermi energy is close enough to the Weyl nodes, such that the band dispersion at the Fermi energy may be \nassumed to be linear. \n\nSecond, we show that generalization of the chiral magnetic effect to the case of diffusive transport leads to a novel weak-field magnetoresistance \neffect, negative and quadratic in the magnetic field. We argue that this effect may be regarded as a universal smoking-gun transport characteristic \nof all Weyl metals. \n\n\\subsection{Anomalous Hall effect in dirty Weyl metals}\n\\label{sec:4.1}\nWe start again from the heterostructure model of Weyl semimetal described in detail in the previous sections. \nWe would like to evaluate the anomalous Hall conductivity of this model ferromagnetic Weyl metal, in the presence of impurity potential $V(\\br) = V_0 \\sum_a \\delta(\\br - \\br_a)$, which we will assume for simplicity to be gaussian, with only second order correlators present: $\\langle V(\\br) V(\\br ') \\rangle = \\gamma^2 \\delta(\\br - \\br ')$, where $\\gamma^2 = n_i V_0^2$ and $n_i$ is the impurity density.  \nThe higher-order correlators, which are known to be important for AHE in principle, as they give rise to the so-called skew-scattering contribution to the Hall conductivity, \ndo not in fact affect our results in a significant way. \nWe will also assume that the impurity potential is diagonal in both the pseudospin $\\btau$ and the $t = \\pm$ index, which labels the eigenstates of the\n$\\hat \\Delta(k_z)$ operator Eq.~\\ref{eq:19}.  Again, this assumption \nis used only for computational simplicity and does not affect the essence of our results. \nTo find the anomalous Hall conductivity, we will use the method of Section~\\ref{sec:3.2}, which we find to be the most convenient one for our \npurposes, as it allows to most clearly separate physically distinct contributions to the AHE. \nNamely, we imagine coupling electromagnetic field to the electrons and integrating the electron degrees of freedom out to obtain an \neffective action for the electromagnetic field, which, at quadratic order, describes the linear response of the system. \nThe part of this action we are interested in has the appearance of a Chern-Simons term, which, adopting the Coulomb gauge for the electromagnetic vector potential $\\bnabla \\cdot {\\bf{A}} = 0$, is given by\n\\begin{equation}\n\\label{eq:99}\nS = \\sum_{{\\bf{q}}, i\\Omega}\\epsilon^{z 0 \\alpha \\beta} \\Pi({\\bf{q}}, i\\Omega) A_0(-{\\bf{q}}, -i\\Omega) \\hat q_{\\alpha} A_{\\beta}({\\bf{q}}, i \\Omega),\n\\end{equation} \nwhere $\\hat q_{\\alpha} = q_{\\alpha}\/ q$ and summation over repeated indices is implicit. The $z$-direction in Eq.~\\ref{eq:99} is again picked out by the \nmagnetization $b$. As we will be interested specifically in the zero frequency and zero wavevector limits of the response function $\\Pi({\\bf{q}}, i \\Omega)$, \nwe will assume henceforth that ${\\bf{q}} = q \\hat x$, which does not lead to any loss of generality due to full rotational symmetry in the $xy$-plane. \nThe anomalous Hall conductivity is given by the zero frequency and zero wave vector limit of the response function $\\Pi({\\bf{q}}, i \\Omega)$ as\n\\begin{equation}\n\\label{eq:100}\n\\sigma_{xy} = \\lim_{i \\Omega \\rightarrow 0} \\lim_{q \\rightarrow 0} \\frac{1}{q} \\Pi({\\bf{q}}, i \\Omega).\n\\end{equation}\nA significant advantage of Eq.~\\ref{eq:100}, compared to the more standard Kubo formula for the anomalous Hall conductivity, which relates \nit to the current-current correlation function, is that Eq.~\\ref{eq:100} ties the Hall conductivity to the response of a conserved quantity, \ni.e. the particle density. This means, in particular, that the response function $\\Pi({\\bf{q}}, i \\Omega)$ must satisfy exact Ward identities, which follow from \ncharge conservation, providing a useful correctness check on the results. \n\nThe impurity average of the response function $\\Pi({\\bf{q}}, i\\Omega)$ may be evaluated by the standard methods of diagrammatic perturbation theory. \nDue to our assumption that the impurity potential is diagonal in the band index $t$, we can do this calculation separately for each pair of bands, labeled \nby $t$, and then simply sum the individual contributions. We will thus omit the $t$ index in what follows, until we come to the final results. \nThe retarded impurity averaged one-particle Green's functions have the following general form\n\\begin{equation}\n\\label{eq:101}\nG^R_{\\sigma_1 \\sigma_2}({{\\bf{k}}}, \\epsilon) = \\frac{z^s_{{{\\bf{k}}} \\sigma_1} \\bar z^s_{{{\\bf{k}}} \\sigma_2}}{\\epsilon - \\xi^s_{{\\bf{k}}} + i \/2 \\tau_s}. \n\\end{equation}\nHere $s = \\pm$, as before, labels the positive and negative energy pairs of bands (the sum over $s$ is made implicit above)\n\\begin{equation}\n\\label{eq:101.1}\n\\xi^s_{{\\bf{k}}} = s \\epsilon_{{\\bf{k}}} - \\epsilon_F = s \\sqrt{v_F^2(k_x^2 + k_y^2) + m^2(k_z)} - \\epsilon_F, \n\\end{equation}\nare the band energies, counted from the Fermi energy $\\epsilon_F$, and \n\\begin{equation}\n\\label{eq:101.2}\n| z^s_{{\\bf{k}}} \\rangle = \\frac{1}{\\sqrt{2}} \\left(\\sqrt{1 + s \\frac{m(k_z)}{\\epsilon_{{\\bf{k}}}}}, - i s e^{i \\varphi} \\sqrt{1 - s \\frac{m(k_z)}{\\epsilon_{{\\bf{k}}}}}\\right), \n\\end{equation}\nis the corresponding eigenvector.\nIn what follows we will assume, for concreteness, that $\\epsilon_F > 0$, i.e. the Weyl metal is electron-doped. \nThe impurity scattering rates $1\/\\tau_{\\pm}$ are given, in the Born approximation, by \n\\begin{equation}\n\\label{eq:102}\n\\frac{1}{\\tau_s(k_z)} = \\frac{1}{\\tau}  \\left[1 +  s \\frac{m(k_z) \\langle m \\rangle}{\\epsilon_F^2} \\right],\n\\end{equation}\nwhere $1\/\\tau = \\pi \\gamma^2 g(\\epsilon_F)$ and \n\\begin{equation}\n\\label{eq:102.1}\ng(\\epsilon_F) = \\int \\frac{d^3 k}{(2 \\pi)^3} \\delta(\\epsilon_{{\\bf{k}}} - \\epsilon_F) = \\frac{\\epsilon_F}{4 \\pi^2 v_F^2} \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\Theta(\\epsilon_F - |m(k_z)|), \n\\end{equation}\nis the density of states at Fermi energy. We have also defined the average of $m(k_z)$ over the Fermi surface as\n\\begin{equation}\n\\label{eq:103}\n\\langle m \\rangle = \\frac{1}{g(\\epsilon_F)} \\int \\frac{d^3 k}{(2 \\pi)^3} m(k_z) \\delta(\\epsilon_{{\\bf{k}}} - \\epsilon_F). \n\\end{equation}\n\\begin{figure}[t]\n\\begin{center}\n  \\includegraphics[width=12cm]{diagrams}\n\\end{center}\n\\vspace{-2cm}\n  \\caption{Graphical representation of the impurity-averaged response function $\\Pi({\\bf{q}}, \\Omega)$.}  \n  \\label{fig:3.5}\n\\end{figure} \nThe impurity averaged response function, analytically continued to real frequency as $\\Pi({\\bf{q}}, \\Omega) = \\Pi({\\bf{q}}, i \\Omega \\rightarrow \\Omega + i \\eta)$,\nis given, in the self-consistent non-crossing approximation, by the sum of ladder diagrams, as illustrated in Fig.~\\ref{fig:3.5}\n\\begin{equation}\n\\label{eq:104}\n\\Pi({\\bf{q}}, \\Omega) = \\Pi^I({\\bf{q}}, \\Omega) + \\Pi^{II}({\\bf{q}}, \\Omega), \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:105}\n\\Pi^I({\\bf{q}}, \\Omega) = 2 e^2 v_F \\Omega \\int_{-\\infty}^{\\infty} \\frac{d \\epsilon}{2 \\pi i} \\frac{d n_F(\\epsilon)}{d \\epsilon} P_{0 x}({\\bf{q}}, \\epsilon - i \\eta, \\epsilon + \\Omega + i \\eta), \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:106}\n\\Pi^{II}({\\bf{q}}, \\Omega) = 4 i  e^2 v_F \\int_{-\\infty}^{\\infty} \\frac{d \\epsilon}{2 \\pi i} n_F(\\epsilon) \\textrm{Im} P_{0 x}({\\bf{q}}, \\epsilon + i \\eta, \\epsilon + \\Omega + i \\eta). \n\\end{equation} \nThe 4$\\times$4 matrix $P$, whose $0 x$ component we are interested in, is given by \n\\begin{eqnarray}\n\\label{eq:106.1}\nP({\\bf{q}}, - i \\eta, \\Omega + i \\eta)&=&\\gamma^{-2} I^{RA}({\\bf{q}}, \\Omega) {\\cal D}({\\bf{q}}, \\Omega), \\nonumber \\\\\nP({\\bf{q}}, \\epsilon + i \\eta, \\epsilon + \\Omega + i \\eta)&=&I^{RR}(\\epsilon, {\\bf{q}}, \\Omega), \n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\label{eq:106.2}\n{\\cal D} = (1 - I^{RA})^{-1},\n\\end{equation}\nis the diffusion propagator and \n\\begin{eqnarray}\n\\label{eq:106.3}\nI^{RA}_{\\alpha \\beta}({\\bf{q}}, \\Omega)&=&\\frac{\\gamma^2}{2} \\tau^{\\alpha}_{\\sigma_2 \\sigma_1} \\tau^{\\beta}_{\\sigma_3 \\sigma_4} \\int \\frac{d^3 k}{(2 \\pi)^3} G^R_{\\sigma_1 \\sigma_3}({{\\bf{k}}} + {\\bf{q}}, \\Omega) G^A_{\\sigma_4 \\sigma_2}({{\\bf{k}}} , 0), \\nonumber \\\\\nI^{RR}_{\\alpha \\beta}(\\epsilon,{\\bf{q}}, \\Omega)&=&\\frac{1}{2} \\tau^{\\alpha}_{\\sigma_2 \\sigma_1} \\tau^{\\beta}_{\\sigma_3 \\sigma_4} \n\\int \\frac{d^3 k}{(2 \\pi)^3} G^R_{\\sigma_1 \\sigma_3}({{\\bf{k}}} + {\\bf{q}},\\epsilon + \\Omega) G^R_{\\sigma_4 \\sigma_2}({{\\bf{k}}}, \\epsilon). \n\\end{eqnarray}\nThe physical meaning of the two distinct contributions to the response function $\\Pi^{I,II}({\\bf{q}}, \\Omega)$ is clear from Eqs.~\\ref{eq:105}, \\ref{eq:106}.  \n$\\Pi^{I}({\\bf{q}}, \\Omega)$ describes the non-equilibrium part of the response that happens at the Fermi surface, as clear from the appearance of the derivative of the Fermi-Dirac distribution function in Eq.~\\ref{eq:105}. This response is \ndiffusive when $\\Omega \\tau \\ll 1$ and ballistic in the opposite limit, and is generally affected significanty by the impurity scattering.  We will discuss this in more detail below. \nIn contrast, $\\Pi^{II}({\\bf{q}}, \\Omega)$ is an equilibrium, nondissipative contribution to the overall response, to which all states below \nthe Fermi energy contribute~\\cite{Streda}, and which is essentially unaffected by the impurity scattering, as will be seen below.\n\nWe will start by evaluating the nonequilibrium part of the response function, $\\Pi^I({\\bf{q}}, \\Omega)$. \nComputation of the matrix elements $I^{RA}_{\\alpha \\beta}({\\bf{q}}, \\Omega)$ is easily done in the standard way, assuming $\\epsilon_F  \\tau \\gg 1$. \nOne obtains\n\\begin{equation}\n\\label{eq:107}\n\\Pi^I({\\bf{q}}, \\Omega) = i e^2 v_F \\Omega \\tau g(\\epsilon_F) [I^{RA}_{00} {\\cal D}_{0x} + I^{RA}_{0x} {\\cal D}_{xx} + I^{RA}_{0z} {\\cal D}_{zx}], \n\\end{equation}\nwhere we have taken into account that ${\\cal D}_{yx} = 0$ by symmetry. \nThe relevant matrix elements of the diffusion propagator can be found analytically to first order in ${\\bf{q}}$. One obtains\n\\begin{equation}\n\\label{eq:108}\n\\Pi^I({\\bf{q}}, \\Omega) = i e^2 v_F \\Omega \\tau g(\\epsilon_F) \\frac{I^{RA}_{0x} (1 - I^{RA}_{zz}) + I^{RA}_{0z} I^{RA}_{zx}}{\\Gamma (1- I^{RA}_{xx})}, \n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:108.1}\n\\Gamma({\\bf{q}}, \\Omega) = (1 - I^{RA}_{00})(1 - I^{RA}_{zz}) - I^{RA}_{0z} I^{RA}_{z0}, \n\\end{equation}\nis the determinant  of the $0z$ block of the diffusion propagator, which corresponds to the diffusion of the charge density, \na conserved quantity (this block decouples from the rest of the diffuson when ${\\bf{q}} \\rightarrow 0$). \nThis means, in particular, that $\\Gamma$ must satisfy an exact Ward identity $\\Gamma(0,0) = 0$, which is easily checked to be true. \n\\begin{figure}[t]\n\\subfigure[]{\n   \\label{fig:4a}\n  \\includegraphics[width=7cm]{bands}}\n\\subfigure[]{\n  \\label{fig:4b}\n   \\includegraphics[width=7cm]{monopoles}}\n  \\caption{(a) Plot of the band edges along the $z$-direction in momentum space for the two bands that touch at the Weyl nodes, using $\\Delta_D\/\\Delta_S = -0.9$. \n  (b) Field lines of the Berry curvature \n  in the $k_y = 0$ plane, for the same band structure as in (a). Corresponding Fermi surface section is shown by the two contours, enclosing the Weyl nodes.}\n    \\label{fig:4}\n\\end{figure} \nExplicitly, the relevant matrix elements of $I^{RA}({\\bf{q}}, \\Omega)$ to first order in ${\\bf{q}}$ are given by\n\\begin{eqnarray}\n\\label{eq:109}\n&&I^{RA}_{00} =\\left\\langle \\frac{\\tau_+\/\\tau}{1 - i \\Omega \\tau_+} \\right\\rangle, \\,\\, \nI^{RA}_{0x} = \\frac{i v_F q}{2 \\epsilon_F} \\left\\langle \\frac{m }{\\epsilon_F} \\frac{\\tau_+\/\\tau}{1 - i\\Omega \\tau_+} \\right\\rangle, \\nonumber \\\\\n&&I^{RA}_{0z} = I^{RA}_{z0} = \\left\\langle \\frac{m}{\\epsilon_F} \\frac{\\tau_+\/\\tau}{1 - i \\Omega \\tau_+}\\right\\rangle, \\nonumber \\\\\n&&I^{RA}_{zx} = \\frac{i v_F q}{4 \\epsilon_F} \\left\\langle \\left(1+ \\frac{m^2}{\\epsilon_F^2}\\right) \\frac{\\tau_+\/\\tau}{1 - i\\Omega \\tau_+} \\right\\rangle, \\nonumber \\\\\n&&I^{RA}_{zz} =\\left\\langle \\frac{m^2}{\\epsilon_F^2} \\frac{\\tau_+\/\\tau}{1 - i \\Omega \\tau_+} \\right\\rangle, \\nonumber \\\\\n&&I^{RA}_{xx} = \\frac{1}{2} \\left\\langle \\left(1- \\frac{m^2}{\\epsilon_F^2}\\right) \\frac{\\tau_+\/\\tau}{1 - i\\Omega \\tau_+} \\right\\rangle,\n\\end{eqnarray}\nwhere the average over the Fermi surface is defined in the same way as in Eq.~\\ref{eq:103}. \n\nExpanding $\\Gamma(0,\\Omega)$ to first order in $\\Omega$ and taking the limit $\\Omega \\rightarrow 0$ at fixed $\\tau$, which corresponds to the diffusive \nlimit, we obtain\n\\begin{equation}\n\\label{eq:111}\n\\Pi_{dif}^I({\\bf{q}}, 0) =  - \\frac{i q \\,e^2 v_F^2 g(\\epsilon_F)}{2 \\epsilon_F}\\left\\langle \\frac{m \\tau_+}{\\epsilon_F \\tau} \\right\\rangle F[m], \n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:112}\nF[m] = \\frac{1 + \\frac{1}{2} \\left\\langle \\left(1 - \\frac{m^2}{\\epsilon_F^2} \\right) \\frac{\\tau_+}{\\tau}\\right\\rangle}\n{1 - \\frac{1}{2} \\left\\langle \\left(1 - \\frac{m^2}{\\epsilon_F^2} \\right) \\frac{\\tau_+}{\\tau}\\right\\rangle}\n\\left[\\left.\\frac{\\partial \\Gamma(0,\\Omega)}{\\partial (\\Omega \\tau)}\\right|_{\\Omega = 0}\\right]^{-1}. \n\\end{equation} \nThe explicit form of the functional $F[m]$ is in fact not that important for our purposes, except for the evenness property,\neasily seen from Eq.~\\ref{eq:112}: $F[m] = F[-m]$. \nAs a consequence, $\\Pi_{dif}^I$ is an odd functional of $m$, which will play an important role below.\nIt is important to note that the charge conservation, whose mathematical consequence is the presence of the diffusion \npole in $\\Pi^I({\\bf{q}}, \\Omega)$, is crucial in obtaining a nonzero result in the diffusive limit in Eq.~\\ref{eq:111}. \nThe analogous quantity in the calculation of the spin Hall conductivity, for example, would vanish in the diffusive limit~\\cite{Sinova04,Halperin04,Molenkamp04,Molenkamp06}.\n\nIt is also of interest to examine the ballistic limit of $\\Pi^I$, which corresponds to the case of a clean Weyl metal. In this case \nwe send both $\\Omega$ and $1\/\\tau$ to zero in such a way that $\\Omega \\tau \\rightarrow \\infty$. \nIn this case we obtain\n\\begin{eqnarray}\n\\label{eq:113}\n\\Pi^I_{bal}({\\bf{q}},0)&=&\\lim_{\\Omega \\rightarrow 0} \\lim_{1\/\\tau \\rightarrow 0} \\Pi^I({\\bf{q}}, \\Omega) = \n\\lim_{\\Omega \\rightarrow 0} \\lim_{1\/\\tau \\rightarrow 0} i e^2 v_F \\Omega \\tau g(\\epsilon_F) I^{RA}_{0x} \\nonumber \\\\\n&=&-\\frac{i q \\, e^2 v_F^2  g(\\epsilon_F)}{2 \\epsilon_F} \\left\\langle \\frac{m}{\\epsilon_F} \\right\\rangle, \n\\end{eqnarray}\nwhich agrees with the clean Weyl metal result Eq.~\\ref{eq:68}. \nA nice feature of the present calculation, compared with the calculation in a clean Weyl metal in Section~\\ref{sec:3.2}, is that here \nit is particularly clear that $\\Pi^I({\\bf{q}}, \\Omega)$ is associated with states on the Fermi surface. \nAs seen from Eqs.~\\ref{eq:111} and~\\ref{eq:113}, the difference between $\\Pi^I_{dif}$ and $\\Pi^I_{bal}$ is \nonly quantitative. In the AHE literature, this difference is said to arise from the \nso-called side-jump processes~\\cite{Molenkamp06,Nagaosa06,MacDonald07,Sinova10,Niu11}. \n\\begin{figure}[t]\n  \\includegraphics[width=8cm]{sigma}\n  \\caption{Plot of the total anomalous Hall conductivity (solid line), $\\sigma^I_{xy}$ (dashed line) and $\\sigma^{II}_{xy}$ (dotted line) versus the Fermi energy for the same system as in Fig.~\\ref{fig:4}.\n  The plateau-like feature in $\\sigma_{xy}$ correlates with the range of the Fermi energies, for which the Fermi surface consists of two separate sheets, each enclosing \n  a single Weyl node.}\n  \\label{fig:5}\n\\end{figure}\nThe final step of the calculation is to evaluate the equilibrium part of the response function, $\\Pi^{II}({\\bf{q}}, \\Omega)$. \nIn the limit $\\epsilon_F \\tau \\gg 1$ one finds that this part of the response function is unaffected by the impurity scattering and is \ngiven by\n\\begin{eqnarray}\n\\label{eq:114}\n\\Pi^{II}({\\bf{q}}, \\Omega) = e^2 v_F \\int \\frac{d^3 k}{(2 \\pi)^3} \\langle z^{s'}_{{\\bf{k}}} | z^s_{{{\\bf{k}}} + {\\bf{q}}} \\rangle \\langle z^s_{{{\\bf{k}}} + {\\bf{q}}} | \\tau^x | z^{s'}_{{\\bf{k}}} \\rangle \\frac{n_F(\\xi^s_{{{\\bf{k}}} + {\\bf{q}}} - \\Omega) - n_F(\\xi^{s'}_{{\\bf{k}}})}{\\Omega - \\xi^s_{{{\\bf{k}}} + {\\bf{q}}} + \\xi^{s'}_{{\\bf{k}}}}, \\nonumber \\\\\n\\end{eqnarray}\nwhere summation over the band indices $s, s'$ is again implicit. \nEvaluating Eq.~\\ref{eq:114} in the small $\\Omega$ and $q$ limit gives \n\\begin{equation}\n\\label{eq:115}\n\\Pi^{II}({\\bf{q}}, 0) = \\frac{-i q \\, e^2}{8 \\pi^2} \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\textrm{sign}[m(k_z)] \\left\\{1 - \\Theta[\\epsilon_F  - |m(k_z)|]\\right\\}.\n\\end{equation}\nThe first term in Eq.~\\ref{eq:115} arises from the completely filled bands, while the second is the contribution of the incompletely filled bands. \n\nWe can now finally evaluate the anomalous Hall conductivity. We will focus on the diffusive limit results, as ballistic limit has already been discussed in Section~\\ref{sec:3}. \nAt this point we will also explicitly include the contribution of both $t = \\pm$ pairs of bands, which simply amounts to restoring the index $t$ in $m_t$, and \nsumming over $t$. \nUsing Eqs.~\\ref{eq:100},\\ref{eq:111} and \\ref{eq:115}, and remembering that $A_0 \\rightarrow i A_0$ upon Wick rotation to the real time, we obtain\n\\begin{equation}\n\\label{eq:116}\n\\sigma^I_{xy} = \\frac{e^2 v_F^2}{2 \\epsilon_F} \\sum_t g_t(\\epsilon_F) \\left\\langle \\frac{m_t \\tau_{+ t}}{\\epsilon_F \\tau_t} \\right\\rangle F[m_t], \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:117}\n\\sigma^{II}_{xy} = \\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\textrm{sign}[m_t(k_z)] \\left\\{1 - \\Theta[\\epsilon_F  - |m_t(k_z)|]\\right\\}.\n\\end{equation}\nSince $m_+(k_z)$ is positive throughout the first BZ, while $m_-(k_z)$ changes sign at the Weyl nodes, the first term \nin Eq.~\\ref{eq:117}, which comes from completely filled bands, gives a universal (almost) quantized contribution\n\\begin{equation}\n\\label{eq:118}\n\\sigma^{quant}_{xy} = \\frac{e^2 {\\cal K}}{4 \\pi^2}, \n\\end{equation}\nwhere ${\\cal K}$ is the distance between the Weyl nodes, which is the same as the clean well metal result, Eq.~\\ref{eq:69}. \nThis equation also describes the cases when the Weyl nodes are absent, \nin which case $\\sigma^{quant}_{xy}$ is truly quantized since ${\\cal K} = 0, G$, where $G= 2\\pi \/d$ is a reciprocal lattice vector. \n\nThe most important new result of this subsection comes from examining the remaining, non-quantized parts of $\\sigma_{xy}$. \nSuppose we have a situation when the Weyl nodes are present and $\\epsilon_F$, while not zero, is not too far from it, \nas shown in Fig.~\\ref{fig:4}.\nRecall that at the location of the Weyl nodes $m_-(k_z) = b - \\Delta(k_z)$ changes sign. \nThis implies that, as long as ${\\cal K} \\left.\\frac{d \\Delta}{d k_z}\\right|_{k_z = k_{z0}} \\gg \\epsilon_F$, \nwhere $k_{z0}$ is the location of a given Weyl node, the average of any odd function of $m_-(k_z)$ over the Fermi surface will vanish. \nThis means that in such a situation, which we call {\\em Weyl metal}, all contributions to the anomalous Hall conductivity, \nassociated with incompletely filled bands, will vanish and $\\sigma_{xy}$ attains a universal value, characteristic \nof Weyl semimetal $\\sigma_{xy} = \\sigma_{xy}^{quant}$, where $\\sigma_{xy}^{quant}$ is given by Eq.~\\ref{eq:118}. \nThis may be seen explicitly in Fig.~\\ref{fig:5}. \nNote that the linear dispersion sufficiently close to Weyl nodes is a topological property, \nin the sense that it follows directly and exclusively  from the existence of a nonzero topological charge by the so-called Atiyah-Bott-Shapiro construction~\\cite{Horava}.  \n\nTo understand this result physically, recall that the Weyl nodes are monopole sources of the Berry curvature $\\bOmega_{{\\bf{k}}}$. \nIn a clean metal, the anomalous Hall conductivity $\\sigma_{xy}$ is given by the integral of the $z$-component of the Berry curvature over all \noccupied states $\\sigma_{xy} = e^2 \\int \\frac{d^3 k}{(2 \\pi)^3} n_F(\\epsilon_{{\\bf{k}}}) \\Omega^z_{{\\bf{k}}}$.\nHowever, as clear from Fig.~\\ref{fig:5}, when the Fermi surface breaks up into disconnected sheets, enclosing individual nodes, \nthe contribution of the states, enclosed by the Fermi surface, to this integral will always be very small, vanishing exactly in the limit when the band dispersion away from the nodes may be taken to be exactly linear.\nA useful analogy here is with the electric field of a dipole. A pair of Weyl nodes is like a dipole of two topological charges. Its field has a well-defined and nonzero on average \n$z$-component at large distances from the dipole. At short distances, however, the field is that of individual charges, which winds around the location \nof each charge and thus any particular component of it averages to zero. \n\nWe have thus demonstrated that the AHE in Weyl metals has a purely intrinsic origin and can be associated entirely with the Weyl nodes, just \nas in the case of a Weyl semimetal, when the Fermi energy coincides with the nodes and the Fermi surface is absent~\\cite{Burkov14-1}. \nThis is in contrast to an ordinary ferromagnetic metal, in which the anomalous Hall conductivity always has both a significant Fermi surface contribution and \nan extrinsic contribution. This property of magnetic Weyl metals may be thought of as being a consequence of emergent chiral symmetry, as discussed in \nSection~\\ref{sec:3.2}. \n\n\\subsection{Magnetoresistance in Weyl metals}\n\\label{sec:4.2}\nIn this section we will discuss measurable consequences of the Chiral Magnetic Effect, discussed in Section~\\ref{sec:3.3}, in the diffusive transport regime. \nAs we will demonstrate, CME in this regime leads to a novel type of magnetoresistance: negative and quadratic in the magnetic field, first discovered by Son and Spivak~\\cite{Spivak12,Son12} in TR-symmetric Weyl semimetals. As we will show below, this novel magnetoresistance is in fact a universal feature of all types of Weyl semimetals and \nmay be regarded as their smoking-gun transport characteristic~\\cite{Burkov14-2}. \n\nWe start from the axial charge density operator, given by Eq.~\\ref{eq:74}.\nWe now ask the following question: does $\\hat n_a$ represent a conserved quantity, as it would in a low-energy model of Weyl semimetal? \nOnly when $n_a$ is conserved, or nearly conserved, may we expect it to contribute significantly to observable phenomena, at least at long times on \nlong length scales. \nTo answer this we need to evaluate the commutator of $\\hat n_a$ with the Hamiltonian ${\\cal H}({{\\bf{k}}})$. \nAs before, it is convenient at this point to apply the following canonical transformation to all the operators: $\\sigma^{\\pm} \\rightarrow \\tau^z \\sigma^{\\pm},\\,\\, \\tau^{\\pm} \\rightarrow\n\\sigma^z \\tau^{\\pm}$. Evaluating the commutator at the Weyl node locations, we now obtain\n\\begin{equation}\n\\label{eq:119}\n\\left[{\\cal H}({{\\bf{k}}}), \\hat n_a\\right]_{k^z_{\\pm}} =  i \\frac{b^2 - \\Delta_D^2 + \\Delta_S^2}{\\Delta_S}  \\tau^z \\sigma^z. \n\\end{equation}\nThis means that $n_a$ may indeed be a conserved quantity in the Weyl semimetal or weakly-doped Weyl metal, provided $\\Delta_D \\geq \\Delta_S$ and $b = \\sqrt{b_{c1} b_{c2}}$, \ni.e. the magnitude of the spin splitting is exactly the geometric mean of its lower- and upper-critical values, at which the transitions out of the Weyl semimetal phase occur. \nOtherwise, the commutator is nonzero and $n_a$ is not conserved. However, as will be shown below, the relevant relaxation time may in fact still be long, even when the above \ncondition is not exactly satisfied, in which case the axial charge density is still a physically meaningful quantity. \nThe near-conservation of the axial charge density $n_a$ may be viewed as a consequence of an emergent low-energy {\\em chiral symmetry}, which is an important \ncharacteristic feature of Weyl semimetals. Both the quadratic negative magnetoresistance, and the fully intrinsic disorder-independent AHE, discussed above, are consequences \nof this emergent symmetry.   \n\nWe now want to derive hydrodynamic transport equations for both the axial charge density $n_a(\\br, t)$ and the total charge density $n(\\br, t)$. \nAs will be shown below, what is known as chiral anomaly will be manifest at the level of these hydrodynamic equations as a coupling between \n$n_a$ and $n$ in the presence of an external magnetic field. This coupling leads to significant observable magnetotransport effects, provided the \naxial charge relaxation time, calculated below, is long enough. As we will show explicitly below, long axial charge relaxation time is a direct \nconsequence of the emergent chiral symmetry, which exists in a Weyl metal at low energies. \n\nTo proceed with the derivation, we add a constant uniform magnetic field in the $\\hat z$ direction ${\\bf{B}} = B \\hat z$ and a scalar impurity potential $V(\\br)$, \nwhose precise form will be specified later. \nAdopting Landau gauge for the vector potential ${\\bf{A}} = x B \\hat y$, the second-quantized Hamiltonian of our system may be written as\n\\begin{eqnarray}\n\\label{eq:120}\nH&=&\\sum_{n a k_y k_z} \\epsilon_{n a}(k_z) c^{\\dag}_{n a k_y k_z} c^{\\vphantom \\dag}_{n a k_y k_z}  \\nonumber \\\\\n&+&\\sum_{n a k_y k_z, n'  a'  k_y' k_z'} \\langle n, a, k_y, k_z | V | n', a', k_y', k_z' \\rangle c^{\\dag}_{n a k_y k_z} c^{\\vphantom \\dag}_{n' a' k_y' k_z'}. \n\\end{eqnarray}\nHere $\\epsilon_{n a}(k_z)$ are Landau-level (LL) eigenstate energies of a clean multilayer in magnetic field, $n=0,1,2,\\ldots$ is the main LL\nindex, $k_y$ is the Landau-gauge intra-LL orbital quantum number, $k_z$ is the conserved component of the crystal momentum \nalong the $z$-direction, and $a = (s, t)$ is a composite index (introduced for \ncompactness of notation), consisting of $s = \\pm$, which labels the electron- ($s = +$) and hole- ($s = -$) like sets of Landau levels, and $t = \\pm$, which \nlabels the two components of a Kramers doublet of LLs, degenerate at $b = 0$.\nThe LL eigenstate energies and the corresponding eigenvectors have been derived in Section~\\ref{sec:3.3}. \n\nTo proceed, we will make the standard assumption, which we also made in the previous subsection, that the impurity potential obeys Gaussian distribution, with $\\langle V(\\br) V(\\br') \\rangle = \\gamma^2 \\delta(\\br - \\br ')$.\nTo simplify calculations further we will also assume that the momentum transfer due to the impurity scattering is smaller than the size of the BZ, i.e. $| k_z - k_z' | d \\ll 1$. \nIn this case $\\langle u^t(k_z)| u^{t'}(k_z') \\rangle \\approx \\delta_{t t'}$, i.e. the $t$ quantum number may be assumed to be approximately preserved during the impurity scattering. \n\\begin{figure}[t]\n\\begin{center}\n  \\includegraphics[width=12cm]{scba}\n\\end{center}\n\\vspace{-2cm}\n  \\caption{Graphical representation of the impurity-averaged Green's function within SCBA and of the diffusion propagator.}  \n  \\label{fig:3.6}\n\\end{figure} \nWe treat the impurity scattering in the standard self-consistent Born approximation (SCBA), illustrated in Fig.~\\ref{fig:3.6}. \nThe retarded SCBA self-energy satisfies the equation\n\\begin{equation}\n\\label{eq:122}\n\\Sigma^R_{n a k_y k_z }(\\omega) = \\frac{1}{L_z} \\sum_{n' a' k_y'  k_z'} \\langle |\\langle n, a, k_y, k_z| V |n', a', k_y', k_z'\\rangle|^2 \\rangle G^R_{n' a' k_y' k_z'}(\\omega), \n\\end{equation}\nWe will assume that the Fermi energy $\\epsilon_F$ is positive, i.e. the Weyl semimetal is electron-doped, and large enough that the impurity-scattering-induced broadening of the density of states is small on the scale of the Fermi energy $\\epsilon_F$~\\cite{Biswas14}. We can then restrict ourselves to the electron-like states with $s=+$ (we will drop the $s$ index henceforth for brevity), and easily solve the SCBA equation analytically. We obtain\n\\begin{equation}\n\\label{eq:123}\n\\textrm{Im} \\Sigma^R_{n t k_z} \\equiv -\\frac{1}{2 \\tau_t(k_z)} = - \\frac{1}{2 \\tau} \\left[1 + \\frac{m_t(k_z) \\langle m_t \\rangle}{\\epsilon_F^2} \\right], \n\\end{equation}\nwhere $1\/ \\tau = \\pi \\gamma^2 g(\\epsilon_F)$ and \n\\begin{equation}\n\\label{eq:124}\ng(\\epsilon_F) = \\frac{1}{2 \\pi \\ell_B^2} \\int_{-\\pi\/d}^{\\pi\/d} \\frac{d k_z}{2 \\pi} \\sum_{n t} \\delta[\\epsilon_{n t}(k_z) - \\epsilon_F],\n\\end{equation}\nis the density of states at Fermi energy. \nWe have also introduced the Fermi-surface average of $m_t(k_z)$ as\n\\begin{equation}\n\\label{eq:125}\n\\langle m_t \\rangle = \\frac{1}{2 \\pi \\ell_B^2 g(\\epsilon_F)} \\int_{-\\pi\/d}^{\\pi\/d} \\frac{d k_z}{2 \\pi} \\sum_{n t} m_t(k_z) \\delta[\\epsilon_{n t}(k_z) - \\epsilon_F]. \n\\end{equation}\n\nWe are interested in hydrodynamic, i.e. long-wavelength, low-frequency density response of our system. \nAs is well-known~\\cite{Altland}, the relevant information is contained in the diffusion propagator, or diffuson ${\\cal D}$, given \nby the sum of ladder impurity-averaging diagrams.\nThis is evaluated in the standard manner and we obtain\n\\begin{equation}\n\\label{eq:126}\n{\\cal D}^{-1} ({\\bf{q}}, \\Omega)= 1 - I({\\bf{q}}, \\Omega), \n\\end{equation}\nwhere $I$ is a $16 \\times 16$ matrix, given by\n\\begin{eqnarray}\n\\label{eq:127}\nI_{\\alpha_1 \\alpha_2, \\alpha_3 \\alpha_4}({\\bf{q}}, \\Omega) = \\frac{\\gamma^2}{L_x L_y L_z} \\int d^3 r d^3 r' e^{-i {\\bf{q}} \\cdot (\\br - \\br')} G^R_{\\alpha_1 \\alpha_3}(\\br, \\br'| \\Omega) G^A_{\\alpha_4 \\alpha_2}(\\br', \\br | 0), \\nonumber \\\\\n\\end{eqnarray}\nwhere we have introduced a composite index $\\alpha = (\\sigma, \\tau)$ to simplify the notation. \nThe impurity-averaged Green's functions $G^{R,A}$ are given by\n\\begin{equation}\n\\label{eq:128}\nG^{R,A}_{\\alpha \\alpha'}(\\br, \\br' |\\Omega) = \\sum_{n t k_y k_z} \\frac{\\langle \\br, \\alpha | n,t,k_y,k_z \\rangle \\langle n,t,k_y,k_z | \\br', \\alpha' \\rangle}\n{\\Omega - \\xi_{n t}(k_z) \\pm i \/ 2 \\tau_t(k_z)}, \n\\end{equation}\nwhere $\\xi_{n t}(k_z) = \\epsilon_{n t}(k_z) - \\epsilon_F$. \n\nIn general, the evaluation of Eq.~\\ref{eq:127} is a rather complicated task, primarily due to the fact that the impurity scattering will mix different LLs. \nAt this point we will thus specialize to the case of transport along the $z$-direction only, as this is where we can expect \nchiral anomaly to be manifest. In this case the contributions of different LLs to Eq.~\\ref{eq:127} decouple. \nSetting ${\\bf{q}} = q \\hat z$, we obtain\n\\begin{eqnarray}\n\\label{eq:129}\n&&I_{\\alpha_1 \\alpha_2, \\alpha_3 \\alpha_4} (q, \\Omega)  = \\frac{\\gamma^2}{2 \\pi \\ell_B^2 L_z} \\sum_{n t t' k_z} \\frac{\\langle \\alpha_1| z^{t}_{n}(k_z + q\/2) \\rangle \\langle z^{t}_{n}(k_z + q\/2) | \\alpha_3 \\rangle}\n{\\Omega - \\xi_{n t}(k_z + q\/2) + i\/ 2 \\tau_t(k_z + q\/2)} \\nonumber \\\\\n&\\times&\\frac{\\langle \\alpha_4 | z^{t'}_{n}(k_z - q\/2) \\rangle \\langle z^{t'}_{n}(k_z - q\/2) | \\alpha_2 \\rangle}{-\\xi_{n t'}(k_z - q\/2) - i\/ 2 \\tau_t'(k_z - q\/2)}, \n\\end{eqnarray}\nwhich is much easier to evaluate.\n\nAs mentioned above, $I$ and ${\\cal D}^{-1}$ are large $16 \\times 16$ matrices, which contain a lot of information of no interest to us. \nWe are interested only in hydrodynamic physical quantities, with long relaxation times. All such quantities need to be identified, \nif they are expected to be coupled to each other. \nOne such quantity is obviously the total charge density $n(\\br, t)$, which has an infinite relaxation time due to the exact conservation of particle number. \nAnother is the axial charge density $n_a(\\br, t)$, which, as discussed above, may be almost conserved under certain conditions. \nOn physical grounds, we expect no other hydrodynamic quantities to be present in our case. We are thus only interested in the $2 \\times 2$ block of the matrix ${\\cal D}^{-1}$, which corresponds to the coupled evolution of the total and the axial charge densities. \nTo separate out this block, we apply the following transformation to the inverse diffuson matrix\n\\begin{equation}\n\\label{eq:130}\n{\\cal D}^{-1}_{a_1 b_1,  a_2 b_2} = \\frac{1}{2} (\\sigma^{a_1} \\tau^{b_1})_{\\alpha_2 \\alpha_1} {\\cal D}^{-1}_{\\alpha_1 \\alpha_2, \\alpha_3 \\alpha_4}\n(\\sigma^{a_2} \\tau^{b_2})_{\\alpha_3 \\alpha_4}, \n\\end{equation}\nwhere $a_{1,2}, b_{1,2} = 0, x, y, z$. The components of interest to us are $a_{1,2} = b_{1,2} = 0$ which corresponds to the total charge density, $a_{1,2} = 0, b_{1,2} = y$, \nwhich corresponds to the axial charge density, and the corresponding cross-terms.   \n\nWe will be interested, as mentioned above, in the hydrodynamic regime, which corresponds to low frequencies and long wavelengths, i.e. $\\Omega \\tau \\ll 1$ and \n$v_F q \\tau \\ll 1$. We will also assume that the magnetic field is weak, so that $\\omega_B \\ll \\epsilon_F$. \nFinally, we will assume that the Fermi energy is close enough to the Weyl nodes, so that only the $t = -$ states participate in transport and \n$\\langle m_- \\rangle \\approx 0$, since $m_-(k_z)$ changes sign at the nodes~\\cite{Burkov14}. \n\nIn accordance with the above assumptions, we expand the inverse diffusion propagator to leading order in $\\Omega \\tau$, $v_F q \\tau$ and $\\omega_B\/ \\epsilon_F$ \nand obtain after a straightforward but lengthy calculation \n\\begin{eqnarray}\n\\label{eq:131}\n{\\cal D}^{-1}(q, \\Omega) = \\left(\n\\begin{array}{cc}\n -i \\Omega \\tau + D q^2 \\tau & -i q \\Gamma \\tau \\\\\n - i q \\Gamma \\tau & -i \\Omega \\tau + D q^2 \\tau + \\tau\/ \\tau_a\n \\end{array}\n \\right). \\nonumber \\\\\n \\end{eqnarray} \nHere $D = \\tilde v_F^2 \\tau \\langle m_-^2 \\rangle\/ \\epsilon_F^2$ is the charge diffusion constant, associated with the diffusion in the $z$-direction, \n$\\Gamma = e B \/ 2 \\pi^2 g(\\epsilon_F)$ is the total charge-axial charge coupling coefficient and \n\\begin{equation}\n\\label{eq:132}\n\\frac{1}{\\tau_a} = \\frac{1 - (\\tilde v_F\/ \\Delta_S d)^2}{(\\tilde v_F\/ \\Delta_S d)^2 \\tau}, \n\\end{equation}\nis the axial charge relaxation rate. \nSeveral comments are in order here. First, an important thing to note is that the off-diagonal matrix elements in ${\\cal D}^{-1}$, proportional \nto $B$ and responsible for the total charge to axial charge coupling, come entirely from the contribution of the $n = 0$ LL. \nThe remaining matrix elements arise from the contribution of all the $n \\geq 1$ Landau levels and we have taken the limit $B \\rightarrow 0$ \nafter summing over the LLs, i.e. left only the leading term in the $\\omega_B\/ \\epsilon_F$ expansion. The next-to-leading term results in \na negative correction to the diffusion coefficient, proportional to $B^2$, which corresponds to the well-known classical positive magnetoresistance. \nWe have ignored this correction here, but will comment on its effects later. \nSecond, we note that the axial charge relaxation rate $1\/\\tau_a \\geq 0$, as it should be, and vanishes when $\\tilde v_F = \\Delta_S d$. \nIt is easy to see that this is identical to the condition of the vanishing of the commutator of the axial charge operator with the Hamiltonian Eq.~\\ref{eq:119}, \nagain as it should be.\nHenceforth we will assume that this condition is nearly satisfied so that $\\tau_a \\gg \\tau$. \nFinally, the situation when $\\tilde v_F = \\Delta_S d$ and thus $1\/\\tau_a$ appears to vanish, actually needs to be treated with some care. Namely, \nthe condition $\\tilde v_F = \\Delta_S d$ may be satisfied exactly only in the limit $\\epsilon_F \\rightarrow 0$. The Fermi velocity  depends on the Fermi energy as~\\cite{Pesin14}\n\\begin{equation} \n\\label{eq:132.1}\n\\tilde v_F(\\epsilon_F) = \\frac{d}{2 (b + \\epsilon_F)} \\sqrt{[(b + \\epsilon_F)^2 - b_{c1}^2] [b_{c2}^2 - (b + \\epsilon_F)^2]}. \n\\end{equation}\nWhen $b = \\sqrt{b_{c1} b_{c2}}$ and thus $\\tilde v_F(0) = \\Delta_S d$, the Fermi energy dependence of $\\tilde v_F$ needs to be taken into account. \nExpanding to leading non vanishing order in $\\epsilon_F$ we obtain in this case\n\\begin{equation}\n\\label{eq:132.2}\n\\frac{1}{\\tau_a} = \\frac{\\epsilon_F^2}{\\Delta_S^2 \\tau}, \n\\end{equation}\ni.e. $1\/\\tau_a$ is in fact always finite, but may be very small. We can estimate the minimal value of the axial charge relaxation rate by \nsetting $\\epsilon_F \\approx 1\/\\tau$ in Eq. ~\\ref{eq:132.2}, which gives $(\\tau\/\\tau_a)_{min} \\approx 1\/ (\\Delta_S \\tau)^2$. \n\nWe may now write down the coupled diffusion equations for the total and axial charge densities, which correspond to the propagator Eq.~\\ref{eq:131}. \nThese equations read\n\\begin{eqnarray}\n\\label{eq:133}\n\\frac{\\partial n}{\\partial t}&=&D \\frac{\\partial^2 n}{\\partial z^2} + \\Gamma \\frac{\\partial n_a}{\\partial z}, \\nonumber \\\\\n\\frac{\\partial n_a}{\\partial t}&=&D \\frac{\\partial^2 n_a}{\\partial z^2} - \\frac{n_a}{\\tau_a} + \\Gamma \\frac{\\partial n}{\\partial z}. \n\\end{eqnarray}\nManifestation of chiral anomaly in these equations is the coupling between the total and the axial charge densities, proportional to the \napplied magnetic field. \nSince the total particle number is conserved, the right-hand side of the first of Eqs.~\\ref{eq:133} must be equal to minus the divergence of the total particle current. \nThen we obtain the following expression for the density of the charge current in the $z$-direction\n\\begin{equation}\n\\label{eq:134}\nj = - \\frac{\\sigma_0}{e} \\frac{\\partial \\mu}{\\partial z} - \\frac{e^2 B}{2 \\pi^2} \\mu_a, \n\\end{equation}\nwhere $\\sigma_0 = e^2 g(\\epsilon_F) D$ is the zero-field diagonal charge conductivity, $\\mu$ and $\\mu_a$ are the total and axial electrochemical potentials \nand we have used $\\delta n = g(\\epsilon_F) \\delta \\mu$, $\\delta n_a = g(\\epsilon_F) \\delta \\mu_a$. The last relation is valid when $\\tilde v_F\/ \\Delta_S d$ is close\nto unity, as seen from Eq.~\\ref{eq:75}. \nThus chiral anomaly manifests in an extra contribution to the charge current density, proportional to the magnetic field and the axial electrochemical potential. \nThis is known as chiral magnetic effect (CME) in the literature~\\cite{Kharzeev}. \nIt is important to realize that the second term in Eq.~\\ref{eq:134} does not by any means imply that an equilibrium current may be driven by an applied \nmagnetic field, despite appearances. The axial chemical potential $\\mu_a$, appearing in Eq.~\\ref{eq:134}, is a purely nonequilibrium quantity. \nIf an equilibrium energy difference, $\\mu_{a0}$,  exists between the Weyl nodes due to explicitly broken inversion symmetry~\\cite{Zyuzin12-2}, then it is the difference $\\mu_a - \\mu_{a0}$ \nthat enters in Eq.~\\ref{eq:134}. We have explicitly considered an inversion-symmetric Weyl metal here, in which case $\\mu_{a0} = 0$. \n\nTo find measurable consequences of the CME contribution to the charge current, we consider a steady-state situation, with a fixed current density $j$ flowing through the \nsample in the $z$-direction. We want to find the corresponding electrochemical potential drop and thus the conductivity. \nAssuming the current density is uniform, we obtain from the second of Eqs.~\\ref{eq:133}\n\\begin{equation}\n\\label{eq:135}\nn_a = \\Gamma \\tau_a \\frac{\\partial n}{\\partial z}, \n\\end{equation} \nwhich is the nonequilibrium axial charge density, induced by the current and the corresponding electrochemical potential gradient. \nSubstituting this into the expression for the charge current density Eq.~\\ref{eq:134}, we finally obtain the following result\nfor the conductivity\n\\begin{equation}\n\\label{eq:136}\n\\sigma = \\sigma_0 + \\frac{e^4 B^2 \\tau_a}{4 \\pi^4 g(\\epsilon_F)}. \n\\end{equation}\nIn the limit when $\\epsilon_F$ is not far from the Weyl nodes, such that the dispersion may be assumed to be linear, \nwe have $g(\\epsilon_F) = \\epsilon_F^2\/ \\pi^2 v_F^2 \\tilde v_F$, which gives\n\\begin{equation}\n\\label{eq:137}\n\\Delta \\sigma = \\sigma - \\sigma_0 = \\frac{e^2 \\tilde v_F \\tau_a}{(2 \\pi v_F)^2} \\left(\\frac{e^2 v_F^2 B}{\\epsilon_F} \\right)^2,\n\\end{equation}\nwhich agrees with the Son and Spivak result~\\cite{Spivak12,Son12}. \nThus we see that a measurable consequence of CME is a positive magnetoconductivity, proportional to $B^2$ in the \nlimit of a weak magnetic field. \nThis of course needs to be compared with the classical negative magnetoconductivity, which is always \npresent and arises from the $B^2$ corrections to the diffusion constant $D$, which we have neglected\n\\begin{equation}\n\\label{eq:138}\n\\frac{\\Delta \\sigma_{c \\ell}}{\\sigma_0} \\sim - (\\omega_c \\tau)^2, \n\\end{equation} \nwhere $\\omega_c = e v_F^2 B\/ \\epsilon_F$ is the cyclotron frequency. \nThis gives \n\\begin{equation}\n\\label{eq:139}\n\\left|\\frac{\\Delta \\sigma}{\\Delta \\sigma_{c \\ell}}\\right| \\sim \\frac{\\tau_a\/\\tau}{(\\epsilon_F \\tau)^2}. \n\\end{equation}\nThus the CME-related positive magnetoconductivity will dominate the classical negative magnetoconductivity, provided \n$\\tau_a$ is long enough. \n\nWe have so far ignored the Zeeman effect due to the applied magnetic field. \nIts effect is to modify the spin-splitting parameter $b$ as $b \\rightarrow b + g \\mu_B B\/2$. \nIn principle, the dependence on $b$ does enter into our final results through the dependence \nof the Fermi velocity $\\tilde v_F$ on $b$. Naively, this will then generate an additional {\\em linear} magnetoconductivity, which \nmay be expected to dominate the quadratic one at small fields.\nHowever, the condition of large $\\tau_a$, which is the same as $\\tilde v_F\/ \\Delta_S d \\approx 1$, is equivalent to \nthe condition $b_{c1} \\ll b \\ll b_{c2}$, in which case the dependence of $\\tilde v_F$ on $b$ becomes negligible. \nThus, in the regime in which the positive magnetoconductivity dominates the negative classical one, and is thus observable, \none may also expect a negligible linear magnetoconductivity in any type of Weyl metal. \n\\section{Conclusions}\n\\label{sec:5}\nIn this paper we have provided an overview of transport phenomena in Weyl metals, which may be attributed to chiral anomaly, \nin particular the semi-quantized intrinsic Anomalous Hall Effect and the Chiral Magnetic Effect, which manifests in negative quadratic in the \nmagnetic field longitudinal magnetoresistance. \n\nThe AHE is generally present in any ferromagnetic metal. \nIn a generic FM metal this is a complicated phenomenon, with many physically distinct sources contributing to the \nfinal observed effect. In particular, {\\em extrinsic} contribution to AHE, arising from impurity scattering, is always at least \nof the same order of magnitude, and often significantly larger, than the more theoretically appealing and universal {\\em intrinsic} one. \nIn a Weyl metal, however, as we have demonstrated in this paper, AHE has a purely intrinsic origin. As long as the Fermi energy is \nnot too far from the Weyl nodes, such that the band dispersion may be taken to be linear and the chiral symmetry is present to a good approximation, \nthe anomalous Hall conductivity of a Weyl metal \nturns out to be given exactly by the semi-quantized expression of a pure undoped Weyl semimetal, where it simply measures separation \nbetween the Weyl nodes in momentum space in units of $e^2\/h$. \nBoth the extrinsic, and even the part of the intrinsic contribution, associated with the Fermi surface (i.e. incompletely filled bands), vanish identically. \nWe have connected this property with the geometry of the Berry curvature field in the vicinity of the Weyl nodes, which in turn is a direct consequence\nof their topology, through the Atiyah-Bott-Shapiro construction. \nFM Weyl metal is thus distinguished from other FM metals by the fact that its AHE is of purely intrinsic origin. \n\nPerhaps even more important in the context of Weyl metals is the Chiral Magnetic Effect and the associated negative quadratic \nmagnetoresistance. This may be expected to occur in any type of Weyl metal, either characterized by broken time reversal symmetry\nor broken inversion (or both). It is thus of particular importance for the experimental characterization of Weyl metals. \nThe negative quadratic longitudinal magnetoresistance should in fact be observed even in Dirac semimetals, which have already \nbeen realized experimentally. The theory, presented in Section~\\ref{sec:4.2} is directly applicable to this case, if $\\Delta_S = \\Delta_D$ \nlimit is taken and the spin-splitting $b$ is identified with the Zeeman splitting $b = g \\mu_B B \/2$. \nIn fact, this effect appears to have been observed very recently in a 3D Dirac semimetal material ZrTe$_5$~\\cite{Kharzeev14}. \n\nWhat has been presented in this paper is by no means a complete story of the possible observable manifestations of chiral anomaly \nin Weyl metals. Other related phenomena, such as linear high-field magnetoresistance~\\cite{Aji12}, plasmon-magnon coupling~\\cite{Liu13},\nnonlocal transport~\\cite{Pesin13}, anomalous density response~\\cite{Pesin14}, anomalous thermoelectric response~\\cite{Fiete14}, and others~\\cite{Parameswaran12,Grushin12,Balatsky13,Goswami13,Qi13,Xiao14,Zyuzin14,Chang14,Abanin14,Miransky14,Nomura14,Hughes14}, have been discussed in the literature. \nOne of the possible promising directions for future research in this area is the interplay of chiral anomaly and superconductivity in Weyl metals~\\cite{Balents12,Moore12,Aji14}. \n\n\\section{Acknowledgments}\nWe would like to thank L. Balents, Y. Chen, M.D. Hook, I. Panfilov, D.A. Pesin, S. Wu, and A.A. Zyuzin for collaboration on the topics, covered \nin this review, and closely related ones. We also thank Y. Ando and B.Z. Spivak for useful discussions. Financial support was provided by NSERC of Canada.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The First Section}\n\n\\section{Introduction}\nThe original Camassa-Holm equation was introduced by Fuchssteiner and Fokas \\cite{Fokas} through the method of recursion operators in 1981 and derived from physical principles by Camassa and Holm in 1999 \\cite{Camassa}.\n\nOur purpose is to investigate the modified Camassa-Holm equation (mCH)\n\t\t\t\\begin{eqnarray}\n\t\t\t\t\\label{mCH}\n\t\t\t\tu_{t}-u_{xxt} = uu_{xxx}+2u_{x}u_{xx}-3u^{2}u_{x},\\ \\ x\\in\\mathbb{R},\\ t>0\\\\\n\t\t\t\tu(x,0)=u_0(x),\\label{dado}\n\t\t\t\n\t\t\t\t\\end{eqnarray}\nobtained from modified Dullin-Gottwald-Holm (mDGH) equation \\cite{Yin}\n\t\t\t\t\t$$\\tilde{u}_{t} + \\kappa\\tilde{u}_{x}   - \\tilde{u}_{xxt} - \\kappa \\tilde{u}_{xxx} = \\tilde{u}\\tilde{u}_{xxx} + 2\\tilde{u}_{x}\\tilde{u}_{xx} - 3\\tilde{u}^{2}\\tilde{u}_{x},\\ \\ x\\in\\mathbb{R},\\ t>0,$$\nby transformation $u(x,t)\\mapsto \\tilde{u}(x+\\kappa t, t)$. The original DGH equation was obtained by Dullin, Gottwald and Holm \\cite{Dullin} for a unidirectional water wave with fluid velocity $u(x,t)$, where the constant $\\kappa\\neq 0$ is the linear wave speed for undisturbed water at rest at spatial infinity. \n\n\t\tThe mCH equation can also be obtained from the $ab$-family of equations \\cite{H1,Hakkaev,H}\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq qntds conservadas}\n\t\t\t\tu_{t} + (a(u))_{x} - u_{xxt} = \\left(b^{'}(u)\\frac{u_{x}^{2}}{2} + b(u)u_{xx}\\right)_{x}\n\t\t\t\t\\end{equation}\nwhere $a,b:\\mathbb{R}\\longrightarrow\\mathbb{R}$ are smooth function and $a(0)=0$, by considering $a(u)=u^{3}$ and $b(u)=u$. From \\cite{H1,H} it is also known that (\\ref{mCH}) has three natural invariants:\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{qntds conservadas}\n\t\t\t\tE(u)=-\\int_{\\mathbb R}\\left[\\frac{u^{4}}{8}+\\frac{uu_{x}^{2}}{2}\\right]\\ dx, \\ \\ F(u)=\\frac{1}{2}\\int_{\\mathbb R}[u^{2} +u_{x}^{2}]\\ dx\\ \\ \\ \\text{and}\\ \\ \\ V(u)= \\int_{\\mathbb R}u\\ dx.\n\t\t\t\t\\end{equation}\n\t\t\t\n\n\t\n\t\n\t\n\t\n\t\n\n\t\t\tEquation (\\ref{mCH}) has been investigated in recent years. Tian and Song in \\cite{Tian} obtained peakons composed of hyperbolic functions. In \\cite{Wazwaz,Wazwaz2}, Wazwaz employed this modified form with $k=0$ as a vehicle to explore the change in the physical structure of the solution from peakons to bell-shaped solitary wave solutions and showed that mCH equation has a $sec\\ h$ like solitary wave solution. In \\cite{Yousif}, a variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of the mCH equation. More recently in \\cite{Daros}, we guarantee results about the orbital instability for a specific class of periodic traveling wave solutions with the mean zero property and large spatial period.\n\n\nIn this paper, we prove steepening at inflection points, i.e., we consider an initial condition in $H^3(\\mathbb R)$ that has an inflection point to the right of its maximum and get  that the time dependent slope at the inflection point becomes vertical in finite time. This property means that wave breaking holds. Next we classify all traveling wave solutions, $u(x, t) = \\phi(x -ct)$, $c \\in \\mathbb{R}$, for mCH equation $(\\ref{mCH})$ using a weak formulation in $H^{1}_{loc}(\\mathbb{R})$ and obtain explicit formulas for peakons in terms of Jacobian elliptic functions. As for the mDP equation \\cite{Ding}, we also obtain some very interesting types of solutions such as kinks, cuspons, composite waves and stumpons, in addition to the more familiar smooth waves and peakons. To get an idea: the composite waves are obtained by combining cuspons and peakons into new traveling waves and the waves called stumpons are obtained by inserting intervals where $\\phi$ equals a constant at the crests of suitable cusped waves. \n\n\n\n\nIn the course of this work, we denote $\\star$ the convolution on $\\mathbb R$. We also use $(\\cdot, \\cdot)$ to represent the standard inner product in $L^2_{per}(\\mathbb R)$. For $1\\leq p \\leq \\infty$, the norm in the $L^p_{per}(\\mathbb R)$ will be denoted by $||\\cdot ||_{L^p}$, while $||\\cdot||_{s}$ will stand for the norm in the classical Sobolev spaces $H^s_{per}(\\mathbb R)$ for $s\\geq 0$. \n\n\n\n\\section{Preliminaries}\\label{preliminaries}\n\nFormally, problem (\\ref{mCH})-(\\ref{dado}) is equivalent to the hyperbolic-elliptic system \n\\begin{eqnarray}\nu_t+\\partial_x\\left(\\frac{u^2}{2}\\right)+P_x =0,\\ \\  (x,t)\\in \\mathbb{R} \\times \\mathbb R_+, \\label{H}\\\\\nP-P_{xx}=u^3+\\frac{u^2}{2}+\\frac{{u_x}^2}{2}, \\ \\ (x,t)\\in \\mathbb{R}\\times \\mathbb R_+,\\label{E}\\\\\nu(x,0)=u_0(x),\\ \\ x\\in \\mathbb R\\label{idata}\n\\end{eqnarray}\n\n \n\nThe operator $(1-\\partial_{xx}^2)^{-1}$ has a convolution structure: \n \\begin{equation}\n (1-\\partial_{xx}^2)^{-1}(f)(x)=(G\\star f)(x),\n \\end{equation}\n where  $G(x)$ is the Green function\n \\begin{eqnarray*}\n G(x)=\\frac{e^{-|x|}}{2}, \\ \\ x\\in \\mathbb R.\n \\end{eqnarray*}\n\n Hence we have \n\\begin{eqnarray}\\label{P}\nP(x,t)=G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)(x,t),  \n\\end{eqnarray} \nand (\\ref{H})-(\\ref{idata}) can be rewritten as a conservation law with a non-local flux function $F$: \n\\begin{eqnarray}\nu_t+F(u)_x =0,\\ \\  (x,t)\\in \\mathbb R\\times \\mathbb R_+, \\label{CL}\\\\\nu(x,0)=u_0(x),\\ \\ x\\in \\mathbb R\\label{idata1},\n\\end{eqnarray}\t\t\nwhere $F(u)=\\frac{u^2}{2}+G\\star (u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2})$.\n\n\\begin{definition}\\label{ws} Let $u_0\\in H^1(\\mathbb R)$ be given. A function $u:[0,T]\\times \\mathbb{R} \\rightarrow \\mathbb{R} $ is called a weak solution to (\\ref{mCH}), if $u\\in L^{loc}_{\\infty}([0,T]; H^1)$ satisfies the identity\n$$ \\int_0^T \\int_{\\mathbb R} (u\\psi_t+F(u)\\psi_x)dxdt+\\int_{\\mathbb R}u_0(x)\\psi_0(x)dx=0$$\nfor all $\\psi\\in C^{\\infty}_0([0,T]\\times \\mathbb{R})$ that are restrictions to $[0,T)\\times \\mathbb{R}$\nof a continuously differentiable function on $\\mathbb{R}^2$ with compact support contained in $(-T,T)\\times \\mathbb{R}.$\n\\end{definition}\n\t\t\n\\section{Steepening lemma}\nThe local well posedness of the Cauchy problem of Equation (\\ref{mCH}) with initial data $u_0\\in H^s(\\mathbb R)$,  $s>\\frac{3}{2}$ can be obtained by applying Kato's semigroup theory \\cite{Kato}. More precisely, we have the following well-posedness result. \n\n\\begin{theorem}\n\\label{teorema local}\nGiven $u_{0}\\in H^{s}(\\mathbb R)$, $s>\\frac{3}{2}$, there exists a maximal $t_0> 0$ and a unique solution $u(x,t)$ to problem (\\ref{mCH})-(\\ref{dado}) such that\n\t\t\t\t$$u\\in C([0,t_{0}), H^{s}(\\mathbb R))\\cap  C^{1}([0,t_{0}), H^{s-1}(\\mathbb R)).$$\nMoreover, the solution depends continuously on the initial data. For $u_{0}\\in H^{3}(\\mathbb R)$ the solution posseses the aditional regularity \n$$u\\in C([0,t_{0}), H^{3}(\\mathbb R))\\cap  C^{1}([0,t_{0}), H^{2}(\\mathbb R)). $$\n  \n\\end{theorem}\n\\begin{proof}\nSee Hakkaev, Iliev and Kirchev in \\cite{H}.\n\\end{proof}\n\\begin{remark} The solutions obtained in Theorem \\ref{teorema local} are called strong solutions to Equation (\\ref{mCH}).\n\\end{remark}\n\n\n\nAssume now that $u_0\\in H^3(\\mathbb{R})$ and let $u\\in C([0,t_0);H^3(\\mathbb{R}) )\\cap C^1([0,t_0);H^2(\\mathbb{R}) )$ be the corresponding strong solution of (\\ref{mCH})-(\\ref{dado}). \n\n\\begin{remark}\nIf $u_0\\in H^3(\\mathbb{R})$, then the $|| u(\\cdot,t)||_1$ norm is conserved in time as long as the solution exists, which implies by the Sobolev embedding $H^1(\\mathbb R)\\subset L_{\\infty}(\\mathbb R)$ that  \n\\end{remark}\n\\begin{equation}\\label{tbound}\nM:=\\sup_{t\\in [0,\\infty)} ||u(\\cdot,t) ||_{L_{\\infty}} <\\infty.\n\\end{equation}\n\n\n\n\nThe operator $(1-\\partial_{xx}^2)^{-1}$ can be represented as a convolution operator:\n$$((1-\\partial_{xx}^2)^{-1} )f(x)=\\int_{\\mathbb R} G(x-y)f(y)dy,\\ \\ f\\in L^2(\\mathbb R),$$ where \n$$  G(x)=\\frac{e^{-|x|}}{2},\\ \\ x\\in\\mathbb{R}.$$\nFrom equation (\\ref{CL}) we get\n\\begin{eqnarray}\\label{advective}\nu_t+uu_x&=&-\\partial_x\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right)\\nonumber\\\\\n&=&-\\partial_x\\int_{\\mathbb R}\\frac{1}{2}\\exp(-|x-y|)\\left(u^3(y,t)+\\frac{u^2}{2}(y,t)+\\frac{u_y^2}{2}(y,t)  \\right)dy\n\\end{eqnarray}\n in $C([0,T);H^1(\\mathbb R) )$.\n\n\nBy differentiation with respect to $x$ we get \n\\begin{eqnarray*}\nu_{tx}+u_x^2+uu_{xx}&=&-\\partial_x^2\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right)\\\\\n&=& (Q^2-Id)\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right)\\\\\n&=& \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)-\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right),\n\\end{eqnarray*}\nand therefore \n\\begin{equation}\\label{eb}\n u_{tx}+uu_{xx}=u^3+\\frac{u^2}{2}-\\frac{u_x^2}{2}-\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right)\n\\end{equation}\nin the space $C([0,T);L^2(\\mathbb R) ).$\n\n\nWe prove now the following blow-up result for (\\ref{mCH})-(\\ref{dado}):\n\n\\begin{lemma}(Steepening Lemma \\cite{Camassa,Crisan}) Suppose the initial profile of velocity $u_0\\in H^{3}(\\mathbb R)$, has an inflection point at $x=\\overline{x}$ to the right of its maximum. Moreover we assume that $u_x(\\overline x,0)<-\\sqrt{2(M^2+2M^3)}$ , where $M $ is the constant defined in (\\ref{tbound}).Then, the negative slope at the inflection point will become vertical in finite time.\n\\end{lemma}\n\\begin{proof}\nConsider the evolution of the slope at the inflection point $t \\rightarrow \\overline{x}(t)$ that starts at time $0$ from an inflection point $x=\\overline{x}$ of $u_0(x)$ to the right of its maximum so that\n$$ \\rho_0=u_x(\\overline{x}(0),0)< \\infty.$$\n\nDefine $\\rho_t=u_x(\\overline{x}(t),t), \\ \\ t\\geq 0$. \n\nEquation (\\ref{eb}) yields an equation for the evolution of $t\\rightarrow \\rho_t$. Namely, by using that $u_{xx}(\\overline{x}(t),t)=0$ $(u(t)\\in H^3(\\mathbb R)\\subset C^2(\\mathbb R))$ and (\\ref{tbound}) one finds \n\n\\begin{eqnarray}\n\\frac{d{\\rho}_t}{dt} &=& -\\frac{{\\rho}_t^2}{2}+u^3(\\overline{x}(t),t)+\\frac{u^2}{2}(\\overline{x}(t),t)\\nonumber\\\\ &-&\\int_{\\mathbb R}\\frac{1}{2}\\exp(-|\\overline{x}(t)-y|)\\left(u^3(y,t)+\\frac{u^2}{2}(y,t)+\\frac{u_y^2}{2}(y,t)  \\right)dy\\nonumber\\\\ \n&\\leq &  -\\frac{{\\rho}_t^2}{2} +M^3+ M^2+ \\int_{\\mathbb R}\\frac{1}{2}\\exp(-|\\overline{x}(t)-y|)(u_{-}^3(y,t))dy\\nonumber\\\\ \n& \\leq &-\\frac{{\\rho}_t^2}{2} + M^2+2M^3,\n\\end{eqnarray}\nwhere $u_-$ stands for the negative part of $u$. Here, we used that $ \\int_{\\mathbb R}\\frac{1}{2}\\exp(-|\\overline{x}(t)-y|)dy=1.  $\n\nLet $\\tilde{\\rho}_t$ be the solution of the equation \n\\begin{equation}\\label{edo}\n\\frac{d{\\tilde{\\rho}_t}}{dt}=-\\frac{\\tilde{\\rho}_t^2}{2} + \\overline{M}, \\ \\ \\tilde{\\rho}_0=\\rho_0,\n\\end{equation}\nwhere $\\overline{M}= M^2+2M^3.$\n\nObserve that \n\\begin{eqnarray*}\n\\frac{d}{dt}\\left((\\rho_t-\\tilde{\\rho}_t)e^{\\frac{1}{2}\\int_0^t(\\rho_s+\\tilde{\\rho}_s)ds}\\right)\\leq 0, \\ \\ \\rho_0-\\tilde{\\rho}_0=0,\n\\end{eqnarray*}\ntherefore $\\rho_t\\leq \\tilde{\\rho}_t$ for all $t>0$ for which both are well defined.  \n\nIntegrating (\\ref{edo}) we obtain\n\\begin{eqnarray}\n\\coth^{-1}\\left(\\frac{\\tilde{\\rho}_t}{\\sqrt{2\\overline{M}}}\\right)=\\frac{1}{2} \\ln \\left| \\frac{\\tilde{\\rho}_t+\\sqrt{2\\overline{M}}}{\\tilde{\\rho}_t-\\sqrt{2\\overline{M}}}\\right|=\\sqrt{2\\overline{M}}\\frac{t}{2}+k,\n\\end{eqnarray} \nwhere $k= \\ln \\left| \\frac{{\\rho}_0+\\sqrt{2\\overline{M}}}{{\\rho}_0-\\sqrt{2\\overline{M}}}\\right|<0$.  Since $ \\lim_{t\\rightarrow -2k\/ \\sqrt{2\\overline{M}}} \\tilde{\\rho}_t=-\\infty$ it follows that there is a time $\\tau\\leq -2k\/ \\sqrt{2\\overline{M}}$  by which the slope $\\rho_t = u_x(\\overline{x}(t),t)$ becomes negative and vertical.\n\\end{proof}\n\n\n\n \n \n \n\n\n\n\n\n\\section{Weak formulation}\\label{weak formulation}\n\n\t\tFor a traveling wave $u(x,t)=\\phi(\\xi)$, with $\\xi=x-ct$, the equation (\\ref{mCH}) is equivalent to\n\t\t\t\\begin{equation}\n\t\t\t\\label{169}\n\t\t\t-c\\phi^{'} +c\\phi^{'''} = \\phi\\phi^{'''} + 2\\phi^{'}\\phi^{''} - 3\\phi^{2}\\phi^{'},\n\t\t\t\\end{equation}\nwhere $'$ denotes the derivative with respect to the variable $\\xi$. \n\t\t\t\n\n\t\t\tAfter an integration with respect to the variable $\\xi$ we can rewrite (\\ref{169}) as follows\n\t\t\t\\begin{equation}\n\t\t\t\\label{171}\n\t\t\t(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a.\n\t\t\t\\end{equation}\nfor some constant of integration $a\\in\\mathbb{R}$. We see that for (\\ref{171}) to make any sense it is sufficient to require $\\phi\\in H^{1}_{loc}(\\mathbb{R})$. Thus, the following definition is plausible.\n\n\\begin{definition}\n\\label{205}\nA function $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ is a traveling wave of the mCH equation if $\\phi$ satisfies (\\ref{171}) in distribution sense for some $a\\in\\mathbb{R}$.\n\\end{definition}\n\n\n\\section{Classification of traveling waves}\\label{classification}\n\nTheorems \\ref{classificacao1} and \\ref{classificacao2} classify all bounded traveling waves $\\phi\\in H^{1}_{loc}(\\mathbb{R})$, of (\\ref{mCH}). The bounded traveling waves are parametrized by their maxima, minima and speeds of propagation.\n\n\\begin{theorem}\n\\label{classificacao1}\nConsider $z\\in\\mathbb{C}\\setminus\\mathbb{R}$ such that $\\Re(z)=-(m+M)\/2$ satisfying the equation $m^{2} + M^{2} -|z|^{2} + mM -2c = 0$. Any bounded traveling wave of the mCH equation belongs to one of the following categories:\n\n$(a)$ Smooth periodic: If $m<M<c$, there is a smooth periodic traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $M=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(b)$ Periodic peakons: If $m<M=c$, there is a periodic peaked traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $M=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(c)$ Periodic cuspons: If $m<c<M $, there is a periodic cusped traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $c=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(a^{'})$ Smooth periodic: If $c<M<m$, there is a smooth periodic traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $M=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $m=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(b^{'})$ Periodic peakons: If $c=M<m$, there is a periodic peaked traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $M=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $m=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(c^{'})$ Periodic cuspons: If $M<c<m$, there is a periodic cusped traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $c=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $m=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(d)$ Composite waves: For each $c\\in \\mathbb{R}$ fixed, with $c <0$, consider the equation\n\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\\label{303}\n\t\t\t\t\t\t\t\t\t2a = (M+m)(|z|^{2} -mM)\n\t\t\t\t\t\t\t\t\t\\end{equation}\nin space $(m,M,\\Im(z))$ with $z\\in\\mathbb{C}\\setminus\\mathbb{R}$ satisfying, less than a change of variables, the hyperboloid of two sheet \n\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{304}\n\t\t\t\t\t\\frac{m^{2}}{\\alpha^{2}} - \\frac{M^{2}}{\\beta^{2}} - \\frac{\\Im(z)^{2}}{\\gamma^{2}} = 1,\n\t\t\t\t\t\\end{equation}\nwhere $\\alpha = \\sqrt{-2c}$, $\\beta = 2\\sqrt{-c}$ and $\\gamma = \\sqrt{-c}$. For any $(m,M,c)$ $\\in \\{m <c \\leqslant M\\}\\cup\\{m >c \\geqslant M\\}$ there is a corresponding cuspon or peakon according to $(b)$ or $(c)$ and $(b^{'})$ or $(c^{'})$. A countable number of cuspons and peakons corresponding to points $(m,M,c)$ that belong to the same hyperboloid of two sheet, may be joined at their crests to form a composite wave $\\phi$. If the Lebesgue measure $\\mu(\\phi^{-1}(c)) = 0$, then $\\phi$ is a traveling wave of (\\ref{mCH}).\n\n$(e)$ Stumpons: For $a=2c^{3} -2c^{2}$ the equation (\\ref{303}) jointly with (\\ref{304}) contains the points $(m,M,\\Im(z)) = (c,c, \\pm\\sqrt{4c^{2}-2c})$. These equations correspond only to cuspons. Let $\\phi$ be a composite wave obtained by joining countably many of these cuspons with each other and with intervals where $\\phi\\equiv c$. Then $\\phi$ is a traveling wave of (\\ref{mCH}) even if $\\mu(\\phi^{-1}(c)) >0$. \n\\end{theorem}\n\n\\begin{theorem}\n\\label{classificacao2}\nConsider $z= -m-r-M$ with $r\\in\\mathbb{R}$ satisfying the equation $r^{2} + m^{2} + M^{2} +rm +rM + mM -2c = 0$. Any bounded traveling wave of the mCH equation belongs, less than an alternation of $z$ for $r$, to one of the following categories:\n\n$(a)$ Smooth periodic: If $z<r<m<M<c$, there is a smooth periodic traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $M=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(b)$ Smooth with decay: If $z<r=m<M<c$, there is a smooth traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\inf_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, $M=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $\\phi\\downarrow m$ exponentially as $\\xi\\rightarrow\\pm\\infty$.\n\n$(c)$ Kinks: If $z=m<M=r<c$, there is a kinked smooth traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\inf_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, $M=\\sup_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and, $\\phi\\uparrow M$ and $\\phi\\downarrow m$ exponentially as $\\xi\\rightarrow\\pm\\infty$, respectively.\n\n$(d)$ Periodic peakons: If $z<r<m<M=c$, there is a periodic peaked traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $M=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(e)$ Peakons with decay: If $z<r=m<M=c$, there is a peaked traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\inf_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, $M=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $\\phi\\downarrow m$ exponentially as $\\xi\\rightarrow\\pm\\infty$.\n\n$(f)$ Periodic cuspons: If $z<r<m<c<M $, there is a periodic cusped traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $c=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(g)$ Cuspons with decay: If $z<r=m<c<M$, there is a cusped traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $m=\\inf_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, $c=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $\\phi\\downarrow m$ exponentially as $\\xi\\rightarrow\\pm\\infty$.\n\n$(a^{'})$ Smooth periodic: If $c<M<m<r<z$, there is a smooth periodic traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $M=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $m=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(b^{'})$ Smooth with decay: If $c<M<m=r<z$, there is a smooth traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $M=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, $m=\\sup_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $\\phi\\uparrow m$ exponentially as $\\xi\\rightarrow\\pm\\infty$.\n\n$(c^{'})$ Kinks: If $c<r=M<m=z$, there is a kinked smooth traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $M=\\inf_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, $m=\\sup_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and, $\\phi\\downarrow M$ and $\\phi\\uparrow m$ exponentially as $\\xi\\rightarrow\\pm\\infty$, respectively.\n\n$(d^{'})$ Periodic peakons: If $c=M<m<r<z$, there is a periodic peaked traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $M=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $m=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(e^{'})$ Peakons with decay: If $c=M<m=r<z$, there is a peaked traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $M=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, $m=\\sup_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $\\phi\\uparrow m$ exponentially as $\\xi\\rightarrow\\pm\\infty$.\n\n$(f^{'})$ Periodic cuspons: If $M<c<m<r<z$, there is a periodic cusped traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $c=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $m=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$.\n\n$(g^{'})$ Cuspons with decay: If $M<c<m=r<z$, there is a cusped traveling wave $\\phi(x-ct)$ of (\\ref{mCH}) with $c=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, $m=\\sup_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ and $\\phi\\uparrow m$ exponentially as $\\xi\\rightarrow\\pm\\infty$.\n\n$(h)$ Composite waves: For each $c\\in \\mathbb{R}$ fixed, with $c >0$, the equation\n\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\\label{203}\n\t\t\t\t\t\t\t\t\t2a = -(M+m)r^{2} - (m+r)M^{2} - (M+r)m^{2} -2mrM\n\t\t\t\t\t\t\t\t\t\\end{equation}\ndescribes three planes intersecting in space $(m,M,r)$ with $r\\in\\mathbb{R}$ satisfying, less than a change of variables, the ellipsoid\n\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{204}\n\t\t\t\t\t\\frac{m^{2}}{\\alpha^{2}} + \\frac{M^{2}}{\\beta^{2}} + \\frac{r^{2}}{\\gamma^{2}} = 1,\n\t\t\t\t\t\\end{equation}\nwhere $\\alpha =\\beta = 2\\sqrt{c}$ and $\\gamma = \\sqrt{c}$. For any $(m,M,r,c)\\in \\{z<r\\leqslant m <c \\leqslant M\\}\\cup\\{z>r\\geqslant m >c \\geqslant M\\}$ there is a corresponding cuspon or peakon according to $(d)-(g)$ and $(d^{'})-(g^{'})$. A countable number of cuspons and peakons corresponding to points $(m,M,r,c)$ that belong to the same ellipsoid, may be joined at their crests to form a composite wave $\\phi$. If the Lebesgue measure $\\mu(\\phi^{-1}(c)) = 0$, then $\\phi$ is a traveling wave of (\\ref{mCH}).\n\n$(i)$ Stumpons: For $a=2c^{3} -2c^{2}$ the equation (\\ref{203}) jointly with (\\ref{204}) contains the points $(m,M,r) = (c,c,-c\\pm\\sqrt{-2c^{2}+2c})$. These equations correspond only to cuspons. Let $\\phi$ be a composite wave obtained by joining countably many of these cuspons with each other and with intervals where $\\phi\\equiv c$. Then $\\phi$ is a traveling wave of (\\ref{mCH}) even if $\\mu(\\phi^{-1}(c)) >0$. \n\\end{theorem}\n\n\t\t\tIn addition to Theorems \\ref{classificacao1} and \\ref{classificacao2} is possible to characterize the existence or no of some unbounded traveling waves of the mCH equation.\n\t\t\t\n\\begin{theorem}\n\\label{classificacao3}\nIf the polynomial $P(\\phi)$ in (\\ref{188}) has only a real zero, say $m$, then the mCH equation has not bounded solutions $\\phi$ to $c<m$ or $m<c$ with $m=\\min_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$ or $m=\\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi)$, respectively. In particular, the mCH equation has traveling wave solutions with exponential behavior.\n\\end{theorem}\n\t\n\\begin{remark}\nDifferently from the classical case, the mCH equation has not traveling wave solutions with similar behavior to a parabola because this is only possible if the polynomial $P(\\phi)$ in (\\ref{188}) has only a real zero with multiplicity equal to one.\n\\end{remark}\n\n\n\\begin{remark}\n\t\t\tFor the modified Degasperis-Procesi equation\n\t\t\t\t\t\t\t$$u_{t}-u_{xxt} = uu_{xxx} +3u_{x}u_{xx} - 4u^{2}u_{x},$$\nwe can also classify all traveling wave solutions for this equation and get similar results like the above theorems. In order to get this, we proceed the same way as for the mCH equation, by considering in the next section $p(v)=\\frac{8}{3}\\phi^{3} -2c\\phi - a$ and Lemma 1 of \\cite{Lenells2} in the proof of Lemma \\ref{lema4} in this paper. Moreover, to obtain a equation like $(\\ref{180})$ we multiply by $\\phi\\phi^{'}$ the weak formulation to mDP equation given by\n\t\t\t\t\t\t\t\t$$\\frac{8}{3}\\phi^{3} -2c\\phi = ((\\phi-c)^{2})^{''} + a,$$\nwhere $a$ is a constant of integration as in $(\\ref{171})$.\n\\end{remark}\n\n\n\n\\section{Proof of Theorems \\ref{classificacao1}, \\ref{classificacao2} and \\ref{classificacao3}}\\label{proof classification}\n\n\\begin{lemma}\n\\label{lema1}\nLet $p(v)$ be a polynomial with real coefficients. Assume that $v\\in H^{1}_{loc}(\\mathbb{R})$ satisfies \n\t\t\t$$(v^{2})^{''} = (v^{'})^{2} + p(v)\\ \\ \\text{in}\\ \\ \\mathcal{D}^{'}(\\mathbb{R}).$$\nThen, $$v^{k}\\in C^{j}(\\mathbb{R})\\ \\ \\text{para}\\ \\ k\\geqslant 2^{j}.$$\n\\end{lemma}\n\\begin{proof}\nSee \\cite{Lenells}, Lemma 1, page 404.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lema2}\nLet $f:\\mathbb{R}\\longrightarrow\\mathbb{R}$ be an absolutely continuous function. Then $f^{'}=0$ a.e. on $f^{-1}(c)$ for any $c\\in\\mathbb{R}$.\n\\end{lemma}\n\\begin{proof}\nSee \\cite{Lenells}, Lemma 2, page 405.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lema3}\nLet $f\\in W^{2,1}_{loc}(\\mathbb{R})$. Then, $f^{''}=0$ a.e. on $f^{-1}(c)$ for any $c\\in\\mathbb{R}$.\n\\end{lemma}\n\\begin{proof}\nSee \\cite{Lenells}, Lemma 3, page 405.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lema4}\nA function $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ is a traveling wave of (\\ref{mCH}) with speed $c$ if and only if\nthe following three statements hold:\n\n\\noindent\\noindent $(TW1)$ There are disjoint open intervals $E_{i}$, $i\\geqslant 1$, and a closed set $C$ such that $\\mathbb{R}\\setminus C = \\bigcup^{+\\infty}_{i=1}E_{i}$, $\\phi\\in C^{\\infty}(E_{i})$ for $i\\geqslant 1$, $\\phi(\\xi)\\neq c$ for $\\xi\\in E\\ \\dot{=}\\ \\bigcup_{i=1}^{+\\infty}E_{i}$, and $\\phi(\\xi)=c$ for $\\xi\\in C$.\n\n\\noindent\\noindent $(TW2)$ There is an $a\\in\\mathbb{R}$ such that\n\n\\ $(i)$ For each $i\\geqslant 1$, there exists $d_{i}\\in\\mathbb{R}$ such that\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{178}\n\t\t\t\t(\\phi^{'})^{2} = F(\\phi)\\ \\text{for}\\ \\xi\\in E_{i}\\ \\ \\text{and}\\ \\ \\phi\\rightarrow c\\ \\text{at any finite endpoint of}\\ E_{i},\n\t\t\t\t\\end{equation}\nwhere, \n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{179}\n\t\t\t\tF(\\phi) = \\frac{\\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi +d_{i}}{c-\\phi};\n\t\t\t\t\\end{equation}\n\t\t\t\t\n\\ $(ii)$ If $C$ has strictly positive Lebesgue measure, $\\mu(C) >0$, then $a=2c^{3} -2c^{2}$.\n\n\n\\noindent\\noindent $(TW3)$ $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$.\n\\end{lemma}\n\\begin{proof}\n\t\t\t\tApplying Lemma \\ref{lema1}, with $v=\\phi -c$ and $p(v) = 2\\phi^{3} -2c\\phi -a$, we see that \n\t\t\t\t\t\t\t\t$$(\\phi -c)^{k}\\in C^{j}(\\mathbb{R})\\ \\ \\text{para}\\ \\ k\\geqslant 2^{j}.$$\n\n\t\t\tDefine $C=\\phi^{-1}(c)$. Since $\\phi$ is continuous, $C$ is a closed set. Since every open set is a countable union of disjoint open intervals, there are disjoint open intervals $E_{i}$, $i\\geqslant 1$, such that $\\mathbb{R}\\setminus C = \\bigcup_{i=1}^{\\infty}E_{i}$. By construction, it follows that $(TW1)$ is satisfied.\n\t\t\t\n\t\t\tNow let's prove $(TW2)$. To get this, we consider $E_{i}$ one of these open intervals. Since $\\phi$ is $C^{\\infty}$ in $E_{i}$, we infer that (\\ref{171}) holds pointwise in $E_{i}$. So, multiplying (\\ref{171}) by $\\phi^{'}$ we obtain\n\t\t\t$$-2c\\phi\\phi^{'} +2c\\phi^{'}\\phi^{''} +2\\phi^{3}\\phi^{'} = 2\\phi\\phi^{'}\\phi^{''} + (\\phi^{'})^{3} + a\\phi^{'}.$$\nEquivalently, we can rewrite the above equation as\n\t\t\t\t\t\t$$-c(\\phi^{2})^{'} + c[(\\phi^{'})^{2}]^{'} + \\left(\\frac{\\phi^{4}}{2}\\right)^{'} = [(\\phi^{'})^{2}\\phi]^{'} +a\\phi^{'}$$\nand from an integration with respect to the parameter $\\xi$, we deduce\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{180}\n\t\t\t\t(\\phi^{'})^{2}(c-\\phi) = \\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi +d_{i},\\ \\xi\\in E_{i}\n\t\t\t\t\\end{equation}\nfor some constants of integration $d_{i}$. Dividing (\\ref{180}) by $c-\\phi$ we have $(\\ref{178})$. That $\\phi\\rightarrow c$ at the finite\nendpoints of $E_{i}$ it follows from the continuity of $\\phi$ and $(TW1)$. This proves $(i)$ of $(TW2)$.\n\n\t\t\t\tThe left-hand side of $(\\ref{171})$ is in $L^{1}_{loc}(\\mathbb{R})$. Hence, $((\\phi -c)^{2})^{''}\\in L^{1}_{loc}(\\mathbb{R})$ and so follows $(TW3)$.\n\t\t\t\t\n\t\t\t\tTo show $(ii)$ of $(TW2)$, let us assume $\\mu(C)>0$. Since $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ and $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$, we obtain from Lemma \\ref{lema2} and Lemma \\ref{lema3} respectively,\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{181}\n\t\t\t\t\\phi^{'}(\\xi) = 0\\ \\ \\text{and}\\ \\ ((\\phi -c)^{2})^{''}(\\xi) = 0\\ \\text{a.e. in}\\ C.\n\t\t\t\t\\end{equation}\nBy the fact that $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$, we have that (\\ref{171}) holds a.e. in $\\mathbb{R}$, i.e., \n\t\t\t\t$$(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a\\ \\ \\text{a.e. in}\\ \\mathbb{R} .$$\nIn particular, the above equation occurs a.e. in $C$. Then, from (\\ref{181}) it follows that\n\t\t\t\t$$2\\phi^{3} -2c\\phi = a\\ \\ \\text{a.e. in}\\ C.$$\nSince $\\mu(C)>0$ and $\\phi\\equiv c$ in $C$, we conclude that $a=2c^{3} -2c^{2}$. This shows that all the traveling waves of equation (\\ref{mCH}) satisfy $(TW1)-(TW3)$.\n\n\n\t\tReciprocally, note that from the differentiation of (\\ref{178}) we have\n\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{182}\n\t\t\t\t\t(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a\\ \\ \\text{in}\\ E\\ \\dot{=}\\ \\bigcup_{i=1}^{\\infty} E_{i}.\n\t\t\t\t\t\\end{equation}\nIf $\\mu(C) = 0$, then (\\ref{182}) implies that\n\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{183}\n\t\t\t\t\t(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a\\ \\ \\text{a.e. in}\\ \\mathbb{R} .\n\t\t\t\t\t\\end{equation}\nSince $((\\phi -c)^{2})^{''}\\in L^{1}_{loc}(\\mathbb{R})$ by $(TW3)$, (\\ref{183}) implies (\\ref{171}), from which we conclude that $\\phi$ is a traveling wave solution of (\\ref{mCH}).\n\t\t\n\t\t\tTo complete the demonstration it remains to show that (\\ref{183}) also occurs in the case where $\\mu(C)>0$. Suppose $\\mu(C)>0$. Since $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ and $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$, we obtain from Lemmas \\ref{lema2} and \\ref{lema3} that equation (\\ref{181}) holds. From $(ii)$ of $(TW2)$ we have $a=2c^{3}-2c^{2}$. So, since $\\phi\\equiv c$ in $C$, we deduce for a.e. $\\xi$ in $C$,\n\t\t\n\t\t\t$$(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a.$$\nJointly with (\\ref{182}), what we have just shown implies (\\ref{183}) and the result follows.\n\\end{proof}\n\t\t\t\t\t\t\n\t\t\tTo prove Theorems \\ref{classificacao1} and \\ref{classificacao2}, we will show that the set of bounded functions satisfying $(TW1)-(TW3)$ in Lemma \\ref{lema4} consists exactly of the waves presented in the statements of these theorems.\n\t\t\t\t\t\t\n\t\t\tSuppose that $\\phi$ satisfies $(TW1)-(TW2)$. From $(TW1)$ and $(TW2)$ there is a countable number of $C^{\\infty}$ wave segments separated by a closed set $C$ such that each wave segment $\\phi$ satisfies\n\t\t\t\t\\begin{eqnarray}\n\t\t\t\t\\label{184}\n\t\t\t\t&& (\\phi^{'})^{2} = F(\\phi)\\ \\text{for}\\ \\xi\\in E,\\ \\ F(\\phi) = \\frac{\\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi +d}{c-\\phi}\\nonumber\\\\\n\t\t\t\t&&\\text{and}\\ \\ \\phi\\rightarrow c\\ \\text{at any finite endpoint of}\\ E\n\t\t\t\t\\end{eqnarray}\nfor some interval $E$ and constants $a,d$. If we are able to find all solutions of $(\\ref{184})$ for different intervals $E$ and different values of $a$ and $d$, then we can join solutions defined at intervals in which the union is $\\mathbb{R}\\setminus C$ for some closed set $C$ of null measure. We will have the function that we constructed defined in $\\mathbb{R}$, satisfying $ (TW1) $ and $ (TW2) $ if, and only if, all the wave segments satisfy (\\ref{184}) with the same constant $a$. In addition, if for $a=2c^{3}-2c^{2}$ we allow $\\mu(C)>0$, this procedure will give us all the functions satisfying $(TW1)$ and $(TW2)$. Let us show that these functions we have just constructed belong to $H^{1}_{loc}(\\mathbb{R})$, satisfy $(TW3)$ and are exactly the waves in Theorems \\ref{classificacao1} and \\ref{classificacao2}.\n\nTo study (\\ref{184}), we first observe that for general equations of the form\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{185}\n\t\t\t\t(\\phi^{'})^{2} = F(\\phi),\n\t\t\t\t\\end{equation}\nwhere $F:\\mathbb{R}\\longrightarrow\\mathbb{R}$ is a rational function, Lenells in \\cite{Lenells} explored the qualitative behavior of solutions to (\\ref{185}) at points where $F$ has a zero or a pole. In resume, if $F(\\phi)$ has a simple zero at $\\phi = m$, so that $F^{'}(m) >0$, then the solution $\\phi$ of (\\ref{185}) satisfies\n\t\t\t\\begin{equation}\n\t\t\t\\label{eq polinomial}\n\t\t\t\\phi(\\xi) = m+\\frac{1}{4}(\\xi-\\eta)^{2}F^{'}(m) + O((\\xi-\\eta)^{4})\\ \\ \\text{at}\\ \\ \\xi\\uparrow\\eta,\n\t\t\t\\end{equation}\nwhere $\\phi(\\eta)=m$. Assuming that $F(\\phi)$ has a double zero at $m$, so that $F^{'}(m) = 0$ and $F^{''}(m)>0$, we get\n\t\t\t\\begin{equation}\n\t\t\t\\label{eq exp}\n\t\t\t\\phi(\\xi) - m \\approx \\alpha\\textit{exp}(-\\xi\\sqrt{F^{''}(m)})\\ \\ \\text{as}\\ \\ \\xi\\rightarrow \\infty\n\t\t\t\\end{equation}\nfor some constant $\\alpha$, then $\\phi \\downarrow m$ exponentially as $\\xi\\rightarrow \\infty$. So, whenever $F$ has two simple zeros $m, M$ and $F(\\phi)>0$ for $m<\\phi <M$ we have the periodic solutions. If $F$ has a double zero $m$, a simple zero $M$, and $F(\\phi)>0$ for $m<\\phi <M$, then there are solutions with decay. To $F$ with a simple zero at $m$ and $F(\\phi)>0$ for $m<\\phi$, then no bounded solution $\\phi$ exists.\n\nIf we include all the functions $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ that are solutions of (\\ref{185}), we will expand the class of solutions, so that they will have another qualitative behavior. If $\\phi$ approaches a simple pole $\\phi =c$ of $F$, then, if $\\phi(\\xi_{0}) = c$,\n\t\t\t\\begin{equation}\n\t\t\t\\label{funcao com polo}\n\t\t\t\\phi(\\xi) -c = \\alpha |\\xi-\\xi_{0}|^{\\frac{2}{3}} + O((\\xi-\\xi_{0})^{\\frac{4}{3}})\\ \\ \\text{as}\\ \\ \\xi\\rightarrow \\xi_{0}\n\t\t\t\\end{equation}\nfor some constant $\\alpha$, so cusped solutions occur. Also, peakons occur when the evolution of $\\phi$ according to (\\ref{185}) suddenly changes direction, that is, $\\phi^{'}\\mapsto -\\phi^{'}$.\n\n\t\t\tApplying the above discussion to the particular case\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{188}\n\t\t\t\tF(\\phi) = \\frac{P(\\phi)}{c-\\phi},\\ \\ \\text{with}\\ \\ P(\\phi) = \\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi +d , \n\t\t\t\t\\end{equation}\nwe can classify all the bounded solutions of (\\ref{184}). The proof is basically an inspection of all possible distributions of zeros and poles of $F$. Note that $P(\\phi)$ is a fourth-degree polynomial with real coefficients, then it has one, two or four real zeros. Moreover,\n\n\t\t\tTo prove Theorem \\ref{classificacao3} suppose that $P(\\phi)$ has only a real zero. This zero has double multiplicity because the polynomial $P(\\phi)$ is fourth-degree. Let $m\\in\\mathbb{R}$ this double zero and $z,\\overline{z}$ the others zeros in $\\mathbb{C}\\setminus\\mathbb{R}$. For this distribution of zeros, we can write\n\t\t\t$$P(\\phi) = -\\frac{1}{2}(\\phi -m)^{2}(\\phi -z)(\\phi -\\overline{z})$$\nand so, we obtain $F^{'}(m) = 0$ and\n\t\t\t\t$$F^{''}(m) = - \\frac{|m-z|^{2}}{c-m}.$$\nThus, we can see that the solution $\\phi$ of the mCH equation satisfies $(\\ref{eq exp})$ to $c<m< \\phi$ and, therefore, $\\phi$ is not bounded. If $\\phi <m <c$, then $- F^{''}(m) >0 $ and a similar result is obtained.\n\n\t\t\tSuppose that there are two simple real zeros, say $m$ and $M$, and we going to prove Theorem \\ref{classificacao1}. We write for $z\\in\\mathbb{C}\\setminus\\mathbb{R}$\n\t\t\t\t\t$$P(\\phi) = \\frac{1}{2}(M-\\phi)(\\phi -m)(\\phi -z)(\\phi -\\overline{z})$$ \nand comparing the coefficients of (\\ref{188}) we obtain that $\\Re(z)= -(M+m)\/2$ with $z\\in\\mathbb{C}\\setminus\\mathbb{R}$ satisfying the hyperboloid of two sheet in $(m, M, \\Im(z))$,\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{hiperboloide}\n\t\t\t\t\\frac{5}{4}m^{2} +\\frac{5}{4}M^{2} -\\Im(z)^{2} + \\frac{3}{2}mM -2c = 0.\n\t\t\t\t\\end{equation}\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\nIn addition, we have\n\t\t\t\t\t\t$$F^{'}(m) = \\frac{1}{2}\\cdot\\frac{(M-m)|m-z|^{2}}{c-m}$$\nand, from (\\ref{eq polinomial}), we deduce from that there are periodic solutions $C^{\\infty}$ for (\\ref{184}) if and only if $m<M<c$ or $c<M<m$. Checking the different cases it is shown that these are all the $C^{\\infty}$ bounded solutions of (\\ref{184}) for $P(\\phi)$ with two simple real zeros. Note that these solutions never touch the line $\\phi =c$. Thus, the interval $E$ in (\\ref{184}) must be the whole real line. In particular, these $C^{\\infty}$ solutions can never be glued together with another wave.\n\nFor the case of bounded solutions to (\\ref{184}) for which $\\phi\\rightarrow c$ at a finite end-point of $E$, gluing two of these together will give rise to a non-smooth wave. Periodic peakons occur if we leave $m<M=c$. In fact, consider $\\phi$ a solution such that $m\\leqslant\\phi\\leqslant M$. If $\\phi$ is decreasing it will reach $m$ and turn back up to reaches $M$. $\\phi$ can not stop or turn back anywhere because $\\phi$ has not singularities at points where $\\phi \\neq c$. But note that $\\phi$ increases until reaches $\\phi = c$ and it can make a sudden turn at this point. This suddenly change of direction of $\\phi$ makes $\\phi^{'}$ have a jump which gives rise to a peak. To see how periodic cuspons happen, consider $m< c <M$. In this case, $F(\\phi)$ has a simple zero at $m$, a simple pole at $c$ and $F(\\phi)>0$ for $m<\\phi<c$. Thus, from (\\ref{funcao com polo}) we have\n\t\t\t\t\t$$\\phi^{'}(\\xi) = \\left\\{\\begin{array}{ll}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\frac{2}{3}\\alpha |\\xi-\\xi_{0}|^{-\\frac{1}{3}} + O((\\xi-\\xi_{0})^{\\frac{1}{3}})\\ \\ \\text{as}\\ \\ \\xi\\downarrow \\xi_{0}&\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t-\\frac{2}{3}\\alpha |\\xi-\\xi_{0}|^{-\\frac{1}{3}} + O((\\xi-\\xi_{0})^{\\frac{1}{3}})\\ \\ \\text{as}\\ \\ \\xi\\uparrow \\xi_{0},&\n\t\t\t\t\t\\end{array}\\right.$$\nfor some constant $\\alpha$. In particular, the solution $\\phi$ has a cusp.\n\n\t\t\tThese are all nontrivial bounded solutions of (\\ref{184}) for this distribution of zeros of the polynomial $P(\\phi)$ and correspond to items $(a)-(c)$ and $(a^{'})-(c^{'})$ of Theorem \\ref{classificacao1}.\n\n\t\t\tFinally, let's determine which solutions of (\\ref{184}) have the same constant $a$ to be glued into a function satisfying $(TW3)$.\tWe can find the integration constant $a$, corresponding to a solution $\\phi$, identifying the coefficients of $\\phi$ in $(\\ref{188})$. Recalling that $\\Re(z)=-(M+m)\/2$, we obtain\n\t\t\t\t\t\t\t\t$$2a = (M+m)(|z|^{2} - mM),$$\nwhere $z\\in\\mathbb{C}\\setminus\\mathbb{R}$ satisfies the equation $(\\ref{hiperboloide})$. Moreover, the quadric (\\ref{hiperboloide}) can be rewritten, less than a change of variables, in its reduced form\n\t\t\t\t\t\t\t$$\\frac{m^{2}}{\\alpha^{2}} - \\frac{M^{2}}{\\beta^{2}} - \\frac{\\Im(z)^{2}}{\\gamma^{2}} = 1,$$\nwith $\\alpha = \\sqrt{-2c}$, $\\beta = 2\\sqrt{-c}$ and $\\gamma = \\sqrt{-c}$, for $c< 0$.\t\t\n\t\t\t\n\t\t\tThus, we obtain all bounded functions satisfying $(TW1)$ and $(TW2)$ for $P(\\phi)$ with two simple real zeros by gluing of all possible ways the solutions corresponding to the same hyperboloid of two sheet. Note that the waves resulting of this approach are exactly the waves of Theorem \\ref{classificacao1}.\n\t\t\t\n\t\t\tTo prove Theorem \\ref{classificacao2}, assume that there are four real zeros, say $m$, $M$, $z$, and $r$, and write \n\t\t\t\t\t$$P(\\phi) = \\frac{1}{2}(M-\\phi)(\\phi -m)(\\phi -z)(\\phi -r).$$\nAnalogously to the previous case we obtain $z= -m-r-M$ with $r$ satisfying the ellipsoid in $(m, M, r)$,\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{elipsoide}\n\t\t\t\tr^{2} + m^{2} + M^{2} +rm +rM + mM -2c = 0,\n\t\t\t\t\\end{equation}\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t\t\t$$F^{'}(m)=\\frac{1}{2}\\cdot\\frac{(M-m)(m-z)(m-r)}{c-m}$$\nand \n\t\t\t\t\t$$F^{''}(m) = \\frac{(M-m)[(m-r) + (m-z)] - (m-z)(m-r)}{c-m} + \\frac{(M-m)(m-z)(m-r)}{(c-m)^2}.$$\nAlso, checking the different cases we conclude that there are periodic solutions $C^{\\infty}$ for (\\ref{184}) if and only if $z<r<m<M<c$ or $c<M<m<r<z$. A bounded solution $\\phi$ with $\\phi\\downarrow m$ when $\\xi\\rightarrow\\pm\\infty$, exists for $z<r=m<M<c$. A solution $\\phi\\uparrow m$ when $\\xi\\rightarrow\\pm\\infty$, exists for $c<M<m=r<z$. A solution $\\phi$ where both above cases occur, that is, $\\phi\\uparrow M$ when $\\xi\\rightarrow +\\infty$ and $\\phi\\downarrow m$ when $\\xi\\rightarrow -\\infty$, happens for $z=m<M=r<c$ or to $c<r=M<m=z$ with $\\phi\\downarrow M$ and $\\phi\\uparrow m$ when $\\xi\\rightarrow \\pm\\infty$.\n\nMoreover, for bounded solutions to (\\ref{184}) for which $\\phi\\rightarrow c$ at a finite end-point of $E$ we have, for example, periodic peakons if $z<r<m<M=c$ and cuspons with decay if we consider $z<r=m< c <M$. In general, the different possibilities for bounded solutions of (\\ref{184}) for $P(\\phi)$ with four real zeros are those presented in $(a)-(g)$ and $(a^{'})-(g^{'})$ of Theorem \\ref{classificacao2}.\n\nAgain to determine which solutions of (\\ref{184}) have the same integration constant $a$ we identify the coefficients of $\\phi$ in (\\ref{188}) obtaining\n\t\t\t$$2a = -(M+m)r^{2} - (m+r)M^{2} - (M+r)m^{2} -2mrM$$\nwhere $z=-m-r-M $ and $r\\in\\mathbb{R}$ satisfies equation $(\\ref{elipsoide})$. So, (\\ref{elipsoide}) can be rewrite in its reduced form\n\t\t\t\t\t\t\t$$\\frac{m^{2}}{\\alpha^{2}} + \\frac{M^{2}}{\\beta^{2}} + \\frac{r^{2}}{\\gamma^{2}} = 1,$$\nwith $\\alpha =\\beta = 2\\sqrt{c}$ and $\\gamma = \\sqrt{c}$, for $c> 0$. Gluing of all possible ways the solutions corresponding to the same ellipsoid we obtain that all bounded functions satisfying $(TW1)$ and $(TW2)$ for this case are the waves of Theorem \\ref{classificacao2}.\n\t\t\t\n\t\t\tTo finalize the proof of Theorems \\ref{classificacao1} and \\ref{classificacao2} is enough to show that these waves belong to $H^{1}_{loc}(\\mathbb{R})$ and satisfy $(TW3)$. This fact is ensured by Lemma \\ref{lema importante} but in order to prove this lemma we need to establish some technical results and notation.\n\t\t\t\t\t\t\n\t\t\t\\begin{definition}\nA function $f:I\\longrightarrow\\mathbb{R}$, $I=[a,b]\\subset\\mathbb{R}$, is said to be of bounded variation if\n\t\t\t\t\t\t$$\\sup\\sum_{i=1}^{N}|f(t_{i})-f(t_{i-1})|<+\\infty ,$$\nwhere the supremum is taken over all natural number $N$ and all choices of partitions $\\{t_{i}\\}_{i=1}^{N}$ such that $a=t_{0}< t_{1}< \\cdot\\cdot\\cdot < t_{N} = b$. $BV(I)$ denote the set of functions $f:I\\longrightarrow\\mathbb{R}$ of bounded variation on $I$. $BV_{loc}(\\mathbb{R})$ denote the space of functions $f:\\mathbb{R}\\longrightarrow\\mathbb{R}$ of bounded variation for all compact intervals $I\\subset\\mathbb{R}$.\n\\end{definition}\n\n\\begin{definition}\nA function $f:X\\longrightarrow\\mathbb{R}$ defined on some set $X\\subset\\mathbb{R}$ is an $N$-function if $f$ maps sets of measure zero to sets of measure zero.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lema0}\nA function $f:I\\longrightarrow\\mathbb{R}$, $I=[a,b]\\subset\\mathbb{R}$, is absolutely continuous if and only if $f\\in W^{1,1}_{loc}(I)$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lema6}\nLet $f:I\\longrightarrow\\mathbb{R}$, $I=[a,b]\\subset\\mathbb{R}$, be a continuous function of bounded variation. Then, $f$ is absolutely continuous if and only if $f$ is an $N$-function.\n\\end{lemma}\n\t\t\n\\begin{lemma}\n\\label{lema7}\nIf $f:I\\longrightarrow\\mathbb{R}$, $I=[a,b]\\subset\\mathbb{R}$, is continuous and $|f|$ is absolutely continuous, the $f$ is absolutely continuous.\n\\end{lemma}\t\t\t\n\t\t\n\t\tThe proof of Lemmas \\ref{lema0}$-$\\ref{lema7} can be found on book \\cite{Rudin}.\n\t\t\n\\begin{lemma}\n\\label{lema importante}\nAny bounded function $\\phi$ satisfying (TW1) and (TW2) belongs to $H^{1}_{\\text{loc}}(\\mathbb{R})$ and satisfies (TW3).\n\\end{lemma}\n\\begin{proof}\n\t\t\tConsider $E_{i}$ and $C$ as in $(TW1)$. We initially prove that $\\phi$ is an $N$-function. To get this, we write\n\t\t\t\t\t\t$$\\phi(N) = \\phi(N\\cap C) \\cup \\left(\\bigcup_{i=1}^{+\\infty} \\phi(N\\cap E_{i})\\right)$$\nfor any set $N$ of null measure and then we use Lemma \\ref{lema6} for $\\phi$ restricted to $N\\cap E_{i}$ and also observe that $\\phi(N\\cap C) = \\{c\\}$.\n\n\t\t\tNow, we will show that $\\phi\\in BV_{\\text{loc}}(\\mathbb{R})$. Consider $\\{i_{k}\\}_{k=1}^{+\\infty}$, or simply $\\{j\\}_{j=1}^{+\\infty}$, a subsequence and $I$ a compact subinterval such that\n\t\t\t$$I\\subset C \\cup \\left(\\bigcup_{j=1}^{+\\infty} E_{j}\\right)\\ \\ \\text{and}\\ \\ \\sum_{j=1}^{+\\infty}|E_{j}|\\ <+\\infty,$$\nwhere $|E_{j}|$ denote the length of the interval $E_{j}$.\n\n\t\t\tSuppose, for example, the polynomial $P_{j}(\\phi) = \\phi^{2}(c-\\phi^{2}\/2) +a\\phi + d_{j}$ with four real zeros, say $z_{j}, r_{j}, m_{j}$ and $M_{j}$, and the case $z_{j} < r_{j} \\leqslant m_{j}\\leqslant \\phi < c \\leqslant M_{j}$. From $(\\ref{178})$, we see that\n\t\t\t\\begin{eqnarray}\n\t\t\t\\label{equacao 30}\n\t\t\t|E_{j}| = 2\\displaystyle\\int_{m_{j}}^{c} \\frac{\\sqrt{c-\\phi}}{\\sqrt{\\Big{|}\\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi + d_{j}\\Big{|}}}\\ d\\phi.\n\t\t\t\\end{eqnarray}\n\t\t\t\n\t\t\tNote that $P_{j}(c)\\geqslant 0$ for all $j$. Moreover, the derivative of $P_{j}$ is independent of $j$ and since $P_{j}$ has four zeros we have that the $b_{j}$'s form a bounded set. Therefore, we obtain\n\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\\label{equacao 31}\n\t\t\t\t\t\t0\\leqslant P_{j}(\\phi) \\leqslant D(c-m_{j}),\\ \\ m_{j}\\leqslant \\phi\\leqslant c,\\ j\\geqslant 1,\n\t\t\t\t\t\t\\end{equation}\nfor some positive real constant $D$ independent of $j$.\n\n\t\tSo, using $(\\ref{equacao 30})$ and $(\\ref{equacao 31})$ it follows that\n\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\\label{equacao 32}\n\t\t\t\t\t\t\t\t|E_{j}| \\geqslant D_{0}(c-m_{j}),\\ \\ j\\geqslant 1,\n\t\t\t\t\t\t\t\t\\end{equation}\nwhere $D_{0}=4\/3\\sqrt{D}$. Thus, $\\phi\\in BV_{\\text{loc}}(\\mathbb{R})$ because the total variation of $\\phi$ in $I$ is bounded above by\n\t\t\t\t\t\t\t$$2\\sum_{j=1}^{+\\infty}\\left( c- \\min_{\\xi\\in E_{j}}\\phi(\\xi)\\right) = 2\\sum_{j=1}^{+\\infty} (c-m_{j}) \\leqslant \\frac{2}{D_{0}} \\sum_{j=1}^{+\\infty}|E_{j}|\\ <+\\infty .$$\n\t\t\t\t\t\t\t\n\t\tMoreover, $\\phi$ is continuous on $I$ (see \\cite{Lenells}, page 417) and so $\\phi$ is continuous in $\\mathbb{R}$. Hence, since $\\phi$ is an $N$-function of bounded variation, we have by Lemma \\ref{lema6} that $\\phi$ is absolutely continuous. This means by Lemma \\ref{lema0} that the distributional derivative of $\\phi$ is in $L^{1}_{\\text{loc}}(\\mathbb{R})$, $\\phi$ is differentiable in the classical sense at almost all points and the two derivatives coincide. \n\t\t\n\t\tTo conclude that $\\phi \\in H^{1}_{loc}(\\mathbb{R})$, we will show that $\\phi^{'}\\in L^{2}_{loc}(\\mathbb{R})$. Using $(\\ref{178})$ and $(\\ref{equacao 31})$, we get\n\t\t\t\t\t$$ \\int_{E_{j}} (\\phi^{'})^{2}\\ d\\xi \\leqslant \\sqrt{D}\\sqrt{c-m_{j}} \\int_{m_{j}}^{c} \\displaystyle\\frac{1}{\\sqrt{c-\\phi}}\\ d\\phi = 2\\sqrt{D}(c-m_{j}),\\ \\ j\\geqslant 1$$\nand so, by $(\\ref{equacao 32})$,\n\t\t\t$$\\int_{E_{j}} (\\phi^{'})^{2}\\ d\\xi \\leqslant \\frac{3}{2}D|E_{j}|,\\ \\ j\\geqslant 1.$$\n\t\t\t\n\t\tThus, we obtain\n\t\t\t\t\t$$\\int_{\\bigcup_{j=1}^{+\\infty}E_{j}} (\\phi^{'})^{2}\\ d\\xi \\leqslant \\frac{3}{2}D \\sum_{j=1}^{+\\infty} |E_{j}|\\  <+\\infty.$$\nAs Lemma \\ref{lema2} implies that $\\phi^{'}=0$ a.e. on $C=\\phi^{-1}(c)$, it follows from the above inequality that $\\phi \\in H^{1}_{loc}(\\mathbb{R})$.\n\t\t\n\t\tFinally, it remains to prove that $(\\phi - c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$. To get this, we will show that $|(\\phi-c)\\phi^{'}|$ is a bounded variation continuous $N$-function.\n\t\t\n\t\tSince $\\phi^{'}\\in L^{2}_{loc}(\\mathbb{R})$ and $\\phi^{'}= 0$ a.e. on $C$, we have that $(\\phi - c)\\phi^{'} = 0$ a.e. on $C$. Therefore, from $(\\ref{178})$,\n\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\\label{equacao 35}\n\t\t\t\t\t\t|(\\phi -c)\\phi^{'}| = \\sqrt{P_{j}(\\phi)}\\sqrt{c-\\phi},\\ \\ c\\in E_{j},\\ \\ j\\geqslant 1\n\t\t\t\t\t\t\\end{equation}\nand, then, $|(\\phi -c)\\phi^{'}|$ is smooth on any interval $E_{j}$ and $|(\\phi -c)\\phi^{'}| \\rightarrow 0$ at finite endpoint of $E_{j}$. So, on each $E_{j}$, $|(\\phi -c)\\phi^{'}|$ is a symmetric function with respect to the  midpoint of $E_{j}$ with two humps. Thus, by $(\\ref{equacao 31})$ and $(\\ref{equacao 32})$, the total variation of $|(\\phi -c)\\phi^{'}|$ in $I$ is bounded by\n\t\t$$4\\sum_{j=1}^{+\\infty} \\sup_{\\xi\\in E_{j}} |(\\phi(\\xi) -c)\\phi^{'}(\\xi)| = 4\\sum_{j=1}^{+\\infty} \\sup_{\\xi\\in E_{j}}\\sqrt{P_{j}(\\phi)} \\sup_{\\xi\\in E_{j}}\\sqrt{c-\\phi} \\leqslant \\frac{8}{3} \\sum_{j=1}^{+\\infty} |E_{j}|\\ < +\\infty.$$\n\n\t\t\tWe see from $(\\ref{equacao 35})$ that $|(\\phi -c)\\phi^{'}|$ is continuous. Now writing,\n\t\t\t\t\t$$|(\\phi -c)\\phi^{'}|(N) = |(\\phi -c)\\phi^{'}|(N\\cap C) \\cup \\left(\\bigcup_{j=1}^{+\\infty} |(\\phi -c)\\phi^{'}|(N\\cap E_{j})\\right)$$\nand using Lemma \\ref{lema6} with $|(\\phi -c)\\phi^{'}|$ restricted to $N\\cap E_{j}$, we obtain that $|(\\phi -c)\\phi^{'}|$ is an $N$-function.\n\t\t\t\n\t\t\tAgain, we apply Lemma \\ref{lema6} to get that $|(\\phi -c)\\phi^{'}|$ is absolutely continuous. Moreover, since $(\\phi -c)\\phi^{'}$ is smooth on $E_{j}$ and, for $\\xi\\in C$, $(\\phi(\\xi) -c)\\phi^{'}(\\xi) = |(\\phi -c)\\phi^{'}|(\\xi) = 0$, we have that $(\\phi -c)\\phi^{'}$ is continuous. Thus, Lemma \\ref{lema7} implies that $(\\phi -c)\\phi^{'}$ is absolutely continuous.\n\t\t\t\n\t\t\tWe conclude, since $(\\phi - c)^{2}$ is absolutely continuous, that\n\t\t\t\t\t\t\t$$\\left[(\\phi -c)^{2}\\right]^{'} = 2(\\phi - c)\\phi^{'}\\ \\in W^{1,1}_{loc}(\\mathbb{R})$$\nand, consequently, $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$.\n\\end{proof}\t\n\t\t\n\\begin{remark}\nIn Theorems \\ref{classificacao1} and \\ref{classificacao2} the solutions are defined on the whole real line. Here we restrict the solution to the interval between two crests in the case of periodic waves, and to the part left or right of the crest in the case of decaying waves. The interval $E$ is accordingly defined to be a finite or half-infinite interval.\n\\end{remark}\t\t\t\n\t\t\t\n\t\t\t\n\\section{Dependence on parameters}\\label{regularity}\n\n\t\t\tIn addition to Theorems \\ref{classificacao1} and \\ref{classificacao2}, it is possible to characterize the relation between the traveling waves of (\\ref{mCH}) and the parameters $m$, $M$ and $c$. Namely, such traveling waves are continuously dependents on these parameters. \n\n\\begin{theorem}\n\\label{teorema2}\nLet $(m_{i}, M_{i}, c_{i})$, $i\\geqslant 1$, and $(m,M,c)$ be such that there are corresponding traveling waves of (\\ref{mCH}) according to $(a)-(c)$ of Theorem \\ref{classificacao1} or $(a)-(g)$ of Theorem \\ref{classificacao2}. Let $\\phi_{i}$, $i=1,2,...$ and $\\phi$ be these traveling waves translated so that they all have crests at $\\xi =0$. If $(m_{i}, M_{i}, c_{i})\\rightarrow (m,M,c)$, then $\\phi_{i}\\rightarrow \\phi$ in $H^{1}_{loc}(\\mathbb{R})$. In particular, we have uniform convergence on compact sets.\n\\end{theorem}\n\\begin{proof}[Sketch of proof]\n\t\t\tAssume $(m_{i}, M_{i}, c_{i})$, $i\\geqslant 0$ and $(m, M, c)$ as in the statement of Theorem \\ref{classificacao1} or \\ref{classificacao2} and $\\phi_{i}$ and $\\phi$ the correspond traveling waves with crests at zero.\n\t\t\t\n\t\t\tNote that to the mCH equation we obtain from (\\ref{178}) and (\\ref{179}) that $\\phi$ and $\\phi_{i}$ are given implicitly by\n\t\t\t\\begin{equation}\n\t\t\t\\label{6.2}\n\t\t\t\\xi=\\left\\{\\begin{array}{ll}\n\t\t\t\\xi_{0} + \\int_{\\phi_{0}}^{\\phi}\\frac{dy}{\\sqrt{F(y)}},\\ \\phi^{'} >0&\\\\\n\t\t\t\\xi_{0} - \\int_{\\phi_{0}}^{\\phi}\\frac{dy}{\\sqrt{F(y)}},\\ \\phi^{'} <0,&\n\t\t\t\\end{array}\\right.\n\t\t\t\\end{equation}\nwhere $\\phi(\\xi_{0})=\\phi_{0}$. Moreover, by choosing $\\xi_{0}=0$, we have $\\phi_{0} = \\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi) = \\min\\{M, c\\}$ and\n\t\t\t$$\\xi=\t- \\int_{\\min\\{M, c\\}}^{\\phi}\\frac{dy}{\\sqrt{F(y)}},\\ \\ 0<\\xi<\\frac{L}{2},$$\nwhere $L$ is the period of $\\phi$ given by\n\t\t\t$$L=2\\int_{m}^{\\min\\{M, c\\}}\\frac{dy}{\\sqrt{F(y)}} = 2\\int_{m}^{\\min\\{M, c\\}}\\frac{\\sqrt{c-y}}{\\sqrt{P(y)}} \\ dy .$$\n\n\t\t\tThe rest of proof follows a similar line to that of Theorem 2 in \\cite{Lenells}, by considering the different distributions of zeros to the polynomial $P(\\phi) = \\phi^{2}\\left(c-\\phi^{2}\/2\\right) +a\\phi +d$ instead of $P(\\phi)=\\phi^{2}(c-\\phi) +a\\phi +d$ in the equations (6.4), (6.13) and (6.17) of \\cite{Lenells}.\n\\end{proof}\n\n\t\t\tAlso, Theorem \\ref{teorema2} is true for waves satisfying $(a^{'})-(c^{'})$ and $(a^{'})-(g^{'})$ of Theorems \\ref{classificacao1} and \\ref{classificacao2}, respectively.\n\n\n\\section{Explicit formulas for peakons}\\label{explicit peakons}\n\t\t\tSuppose that $\\phi$ is a wave corresponding to $(d)-(e)$ of Theorem \\ref{classificacao2}. In this section we determine explicit formulas for the peaked traveling waves.\n\t\t\t\n\t\t\tFrom (\\ref{6.2}), we have\n\t\t\t\\begin{equation}\n\t\t\t\\label{7.1}\n\t\t\t|\\xi-\\xi_{0}| = \\displaystyle\\int_{\\phi_{0}}^{\\phi} \\frac{dy}{\\sqrt{F(y)}} = \\sqrt{2}\\displaystyle\\int_{\\phi_{0}}^{\\phi} \\frac{\\sqrt{c-y}}{\\sqrt{(M-y)(y-m)(y-z)(y-r)}}\\ dy.\n\t\t\t\\end{equation}\n\t\t\t\nMaking $\\phi = m+ (M-m)\\sin^{2}(\\theta)$, we rewrite (\\ref{7.1}) by\n\t\t\t\\begin{equation}\n\t\t\t\\label{7.2}\n\t\t\t|\\xi-\\xi_{0}| =\\frac{2\\sqrt{2}}{\\sqrt{M-m}} \\displaystyle\\int_{\\theta_{0}}^{\\theta} \\frac{\\sqrt{A -\\sin^{2}(t)}}{\\sqrt{B+\\sin^{2}(t)}\\sqrt{C+\\sin^{2}(t)}}\\ dt\n\t\t\t\\end{equation}\nwhere $$A=\\frac{c-m}{M-m},\\ \\ B=\\frac{m-r}{M-m}\\ \\ \\text{and}\\ \\ C=\\frac{m-z}{M-m}.$$\n\nThe period of $\\phi$ is given by\n\\begin{eqnarray}\n\\label{eq periodo}\nL &=& 2\\sqrt{2}\\displaystyle\\int_{m}^{\\min\\{M,c\\}} \\frac{\\sqrt{c-y}}{\\sqrt{(M-y)(y-m)(y-z)(y-r)}}\\ dy\\nonumber\\\\\n&=& 4\\frac{\\sqrt{2}}{\\sqrt{M-m}}\\displaystyle\\int_{\\theta_{0}}^{\\theta_{\\text{max}}} \\frac{\\sqrt{A -\\sin^{2}(t)}}{\\sqrt{B+\\sin^{2}(t)}\\sqrt{C+\\sin^{2}(t)}}\\ dt .\n\\end{eqnarray}\n\nAssume $\\phi$ a periodic peakon solution as in item $(d)$ of Theorem \\ref{classificacao1}. So, $A=1$ and $B,C>0$, with $B\\neq C$. From (\\ref{7.2}), we have\n\t\t\t\\begin{eqnarray*}\n\t\t\t|\\xi-\\xi_{0}| &=& \\frac{2\\sqrt{2}}{\\sqrt{M-m}}\\displaystyle\\int_{\\theta_{0}}^{\\theta} \\frac{\\cos(t)}{\\sqrt{B+\\sin^{2}(t)}\\sqrt{C+\\sin^{2}(t)}}\\ dt\\\\\n\t\t\t&=& \\frac{2\\sqrt{2}}{\\sqrt{M-m}}\\left[- \\frac{i}{\\sqrt{C}}\\cdot F\\left( \\text{arcsin}\\left( \\frac{i}{B}\\sin(t)\\right)\\ ; \\sqrt{\\frac{B}{C}}\\right) \\right]\\Bigg{|}_{\\theta_{0}}^{\\theta},\n\t\t\t\\end{eqnarray*}\nwhere $F$ denotes the elliptic integral of the first kind (see Appendix), since $0 < B\/C< 1$.\n\nChoosing $\\xi_{0}$ such that $\\phi(\\xi_{0}) = m$, that is, $\\xi_{0}$ is a trough, we get $\\theta_{0} = 0$. Moreover, since $M=c$ we obtain $\\sin(\\theta) = \\sqrt{\\phi -m}\/\\sqrt{c-m}$. Thus\n\t\t\t$$|\\xi-\\xi_{0}| = -\\frac{i}{D_{1}} \\cdot F\\left( i\\cdot\\text{arcsinh}\\left( \\frac{1}{B} \\sqrt{\\frac{\\phi -m}{c-m}} \\right)\\ ; k\\right),$$\nwhere $D_{1} = \\sqrt{C}\\sqrt{c-m}\/2\\sqrt{2}$ and $k=\\sqrt{B\/C}$ is the module of the elliptical integral (see Appendix).\n\nSo, by properties of the elliptic integral of the first kind \\cite{BF}, we obtain\n\t\t\t\t$$\\sin\\left(i\\cdot\\text{arcsinh}\\left( \\frac{1}{B} \\sqrt{\\frac{\\phi -m}{c-m}} \\right)\\right) = \\text{sn}(D_{1}|\\xi-\\xi_{0}|i\\ ;k),$$\nwhere $sn$ is a Jacobian elliptic function (see Appendix).\n\nTherefore, solving this equation we conclude that\n\t\t\t\t\t$$\\phi(\\xi) = m + D_{2}\\text{tn}^{2}(D_{1}|\\xi-\\xi_{0}|\\ ;k'),\\ \\ |\\xi -\\xi_{0}| \\leqslant \\frac{L}{2},$$\nwhere\n\t\t\t\t\t$$L= D_{3}\\cdot F\\left(\\text{arcsin}\\left(\\sqrt{\\frac{1\/B^{2}}{1+ 1\/B^{2}}}\\right)\\ ; k'\\right)$$\nis the period of $\\phi$ obtained from (\\ref{eq periodo}), $D_{2} = B^{2}(c-m)$, $D_{3}= 2\/D_{1}$, $k'^{2}= 1 -k^{2}$ and $tn$ is a Jacobian elliptic function (see Appendix). This is the explicit formula to the periodic peakons of the mCH equation.  \n\t\t\t\t\nIf $\\phi$ is a peakon with decay as in $(e)$ of Theorem \\ref{classificacao2}, then $A=1$, $B=0$ and $C>0$. So, (\\ref{7.2}) gives us\n\t\t\t\t$$|\\xi-\\xi_{0}| =\\frac{2\\sqrt{2}}{\\sqrt{M-m}} \\displaystyle\\int_{\\theta_{0}}^{\\theta} \\frac{\\cos(t)}{\\sin^{2}(t)\\sqrt{C+\\sin^{2}(t)}}\\ dt = \\frac{2\\sqrt{2}}{\\sqrt{C}\\sqrt{M-m}}\\Big{[}-\\text{arccsch}|t|\\Big{]}\\Big{|}_{\\frac{\\sin(\\theta_{0})}{\\sqrt{C}}}^{\\frac{\\sin(\\theta)}{\\sqrt{C}}}.$$\n\nChoosing $\\xi_{0}$ such that $\\phi(\\xi_{0}) = c$, that is, $\\xi_{0}$ at the peak, we get $\\sin(\\theta_{0}) = \\sin(\\theta_{\\text{max}}) = 1$. Moreover, since $M=c$ we obtain $\\sin(\\theta) = \\sqrt{\\phi -m}\/\\sqrt{c-m}$. Thus\n\t\t\t\t$$|\\xi-\\xi_{0}| = -\\frac{2\\sqrt{2}}{\\sqrt{C}\\sqrt{c-m}} \\left[ \\text{arccsch}\\left|\\frac{\\sqrt{\\phi -m}}{\\sqrt{C}\\sqrt{c-m}}\\right| - \\text{arccsch} \\left(\\frac{1}{\\sqrt{C}}\\right) \\right].$$\n\t\t\t\t\nNow, we use the identity\n\t\t\t$$\\text{arccsch}(x) = \\ln\\left(\\frac{1}{x} + \\sqrt{\\frac{1}{x^{2}} +1}\\right) = \\ln\\left(\\frac{1 +\\sqrt{1 + x^{2}}}{x}\\right);\\ \\ x\\in\\mathbb{R},\\ x\\neq 0$$\nin the above equation to obtain\n\t\t\t\t\t\t$$ -\\frac{\\sqrt{C}\\sqrt{M-m}|\\xi-\\xi_{0}|}{2\\sqrt{2}} = \\ln\\left(\\frac{\\frac{\\sqrt{C}\\sqrt{c-m}}{\\sqrt{\\phi -m}}\\left( 1+ \\sqrt{1+ \\frac{\\phi -m}{C(c-m)}}\\right) }{\\sqrt{C}\\left(1+ \\sqrt{1+\\frac{1}{C}} \\right)}\\right).$$\n\nThus,\n\t\t\t\t$$\\phi(\\xi) = m +\\frac{D_{4}e^{-D_{5}|\\xi-\\xi_{0}|}}{(D_{6}e^{-D_{5}|\\xi-\\xi_{0}|} -1)^{2}}$$\nis an explicit expression for the peakons with decay, where $D_{4} = 4C^{2}(c-m)(1+\\sqrt{1+1\/C})^{2}$, $D_{5} = \\sqrt{C}\\sqrt{c-m}\/\\sqrt{2}$ and $D_{6}= C(1+\\sqrt{1+1\/C})^{2}$.\n\n\n\\section{Acknowledgment}\nThe first author was supported by CAPES\/Brazil.\n\n\\section{Appendix}\nIn this appendix we will talk about some concepts used so far without further explanation. In accordance with \\cite{BF}, we start setting the \\textit{normal elliptic integral of the first kind}\n\t\t\t$$F(\\phi , k) = F(\\phi\\ |\\ k^{2} ) = F_{k}(\\phi) = \\int_{0}^{y} \\ \\frac{dt}{\\sqrt{(1-t^2)(1-k^2 t^2)}} = \\int_{0}^{\\phi} \\frac{d\\varphi}{\\sqrt{(1 - k^{2}\\sin^{2}\\varphi)}},$$\nwhere $y=\\sin\\phi$.\n\n\t\t\tThe parameter $k$ is called the \\textit{modulus of the elliptic integral} and $k'^{2} = 1-k^{2}$ its \\textit{complementary modulus}, and both may take any real or imaginary value. Here we wish to take $0 < k^{2} < 1$. Moreover, the variable $\\phi$ is called \\textit{argument} and it is usually taken belonging to $\\left[\\frac{-\\pi}{2} , \\frac{\\pi}{2}\\right]$.\n\n\t\t\tIn his algebraic form, the elliptic integral above is finite for all real (or complex) values of $y$, including infinity. When $\\phi= \\frac{\\pi}{2}$, the integral $K(k) \\dot{=} F \\left(\\frac{\\pi}{2}, k \\right)$ is said to be \\textit{complete}.\n\t\t\t\n\t\t\tWe define the \\textit{Jacobian elliptic functions} using the inverse function of the elliptic integral of the first kind. This inverse function exists because that \n$$ u(y_{1}, k) \\equiv u = \\int_{0}^{y_{1}} \\frac{dt}{\\sqrt{(1-t^{2})(1-k^{2}t^{2})}} = \\int_{0}^{\\phi} \\frac{d\\varphi}{\\sqrt{1 - k^2 \\sin^2 \\varphi}} = F(\\phi, k),$$\nis a strictly increasing function of the real variable $y_{1}$ and, in its algebraic form, this integral has the property of being finite for all values of $y_{1}$. This inverse $\\phi = \\mathrm{am}(u,k) = \\mathrm{am}u$ is called \\textit{amplitude function}.\n\n\t\tThere are several Jacobian elliptic functions that can be seen in \\cite{BF}, but here we will only define the functions $\\textit{sn}$, $\\textit{cn}$ and \\textit{tn} as follows\n\t\t\t$$\n\t\t\t\\begin{array}{lll}\n\t\t\t\\text{sn}(u,k) = \\text{sin}\\ \\mathrm{am}(u,k) = \\text{sin} \\phi,&&\\\\\n\t\t\t\\text{cn}(u,k) = \\text{cos}\\ \\mathrm{am}(u,k) = \\text{cos} \\phi,&&\\\\\n\t\t\t\\text{tn}(u,k) = \\displaystyle\\frac{\\text{sin}\\ \\mathrm{am}(u,k)}{\\text{cos}\\ \\mathrm{am}(u,k)} = \\frac{\\text{sin} \\phi}{\\text{cos} \\phi}.&&\n\t\t\t\\end{array}\n\t\t\t$$\nThese functions have a real period, namely $4 K(k)$, and some important properties summarized by the formulas given below.\n$$\n\\begin{array}{lll}\n\\text{sn}(-u) = -\\text{sn}(u), &&\\ \\ \\text{sn}^2u + \\text{cn}^2u = 1,\\\\\n \\text{cn}(-u)=\\text{cn}(u), &&\\ \\ -1 \\leqslant \\text{sn} u , \\text{cn} u \\leqslant 1,\\\\\n \\text{tn}(-u) = -\\text{tn}(u), &&\\ \\  \\text{sn}(iu,k) = i\\text{tn}(u,k').\n\\end{array}\n$$\n\\\\\n\n\n\n{\n\\fontsize{10pt}{\\baselineskip}\\selectfont\n\n\\textbf{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFrom the groundbreaking work of BPST \\cite{Brower:2006ea} till today, many articles have shown how the theoretical ideas of $AdS\/CFT$ can be used to model experimental data where pomeron exchange dominates~\\cite{Hatta:2007he, Cornalba:2008sp,   Brower:2008ix, Levin:2009vj,  Hatta:2009ra, Kovchegov:2009yj, Avsar:2009xf, Cornalba:2009ax,  Cornalba:2010vk,   Kovchegov:2010uk, Levin:2010gc, Brower:2010wf, Ballon-Bayona:2017vlm, Costa:2012fw, Costa:2013uia, Brower:2012mk, Anderson:2014jia, Nally:2017nsp, Ballon-Bayona:2015wra, Amorim:2018yod,  Hatta:2008st,  Brower:2009bh, Domokos:2009hm, Domokos:2010ma, Stoffers:2012zw, Koile:2014vca,  Kovensky:2018xxa, Lee:2018zud, Kovensky:2018gif, Mamo:2019mka, FolcoCapossoli:2020pks}.\n  These works are tailored for a specific process and\/or kinematic regime. On the other hand, we have shown in~\\cite{Amorim:2021ffr} that a holographic model with few free parameters that describes successfully several processes dominated by pomeron exchange is indeed possible. The processes analysed were sensitive to the forward scattering amplitude, i.e. for Mandelstam $t = 0$. So it remains an open question if it is possible to extend previous results to processes where amplitudes with non-zero values of $t$ are necessary. One such example is Deeply Virtual Compton Scattering (DVCS), which is the focus of this letter.\n\nDVCS is an exclusive Compton scattering process where the incoming photon has high virtuality. It has been studied throughly by the H1 and ZEUS collaborations as well at JLAB. While DIS allows the determination of  nucleon parton distribuition functions (PDFs),  e.g. the proton PDFs, DVCS data is important for the study of the generalized parton distribution functions (GDPs) \\cite{ji_deeply_1997,radyushkin_nonforward_1997}, which are related to the correlation between the transverse and longitudinal components of the momentum of quarks and gluons inside the nucleons.\n\nHolographic techniques have been already employed to describe DVCS data. In \\cite{Costa:2012fw} the conformal and hard wall pomeron models have been used, while in \\cite{Stoffers:2012ai} a holographic description of dipole-dipole scattering has been studied. DIS is connected to the forward Compton scattering amplitude via the optical theorem. Hence, our analysis of DVCS in this work is close to the one presented in \\cite{Ballon-Bayona:2017vlm}. We will show that in order to include the description of DVCS data one needs to include extra parameters that are related to the holographic wave function of the proton and to the coupling dependence with the spin of the exchanged Reggeons.\n\n\\section{Holographic computation of DVSC amplitude}\n\nWe start by deriving holographic expressions for the differential cross-section $d \\sigma \/ dt$ and total cross-section~$\\sigma$ of the\n$ \\gamma^* p \\to \\gamma p $ DVCS process. In~\\cite{Amorim:2021ffr} we computed the proton structure functions $F_2^p$ and $F_L^p$ and the total cross-section $\\sigma\\left(\\gamma p \\to X \\right)$ by determining  the forward scattering amplitude of the process $\\gamma^{*} p \\to \\gamma^{*} p$. Thus, the only difference between the DIS and the DVCS computation is that the outgoing photon is \\textit{on-shell}. The associated Witten diagram is show in figure \\ref{fig:DVCS Witten diagram}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.5]{Witten_diagram}\n  \\caption[Tree level Witten diagram for the DVCS scattering amplitude]{Tree level Witten diagram associated with the computation of the amplitude $\\mathcal{A}_J^{\\lambda_1\\lambda_3}$ of the DVCS process $\\gamma^{*}p\\to \\gamma p$. The bottom lines represent the proton modeled by a scalar $\\Upsilon$.}\n  \\label{fig:DVCS Witten diagram}\n\\end{figure}\nHence we will consider the general case of $\\gamma^* p \\to \\gamma^* p$ where the incoming photon has virtuality $Q_1^2$ while the outgoing photon has virtuality $Q_3^2$. At the end of the calculation we take $Q^3 \\equiv k_3^2 \\to 0$ and use the identity\n\\begin{equation}\n\\lim_{Q \\rightarrow 0} f_Q (z) = 1\\,, \\quad \\lim_{Q \\rightarrow 0} \\frac{\\dot{f_Q}}{Q} = 0\\,,\n\\label{eq:nonnormalizable_Q_0}\n\\end{equation}\nwhere $f_Q$ is the non-normalizable mode of the bulk U(1) gauge field dual to the current $J^\\mu = \\bar{\\psi} \\gamma^\\mu \\psi$.\nWe take for the large $s$ kinematics of  $\\gamma^*p\\to \\gamma^*p$ scattering the following momenta\n\\begin{align}\n \\label{eq:NMC:kinematics}\n&k_1=\\left(\\!\\sqrt{s},-\\frac{Q_1^2}{\\sqrt{s}} ,0\\right),\n\\qquad\n k_3=-\\left(\\!\\sqrt{s},\\frac{ q_\\perp^2 -Q_3^2}{\\sqrt{s}} , q_\\perp \\right),\n\\\\\n&k_2=\\left(\\frac{M^2}{\\sqrt{s}},\\sqrt{s} ,0\\right),\n\\qquad\n k_4=-\\left(\\frac{M^2+ q_\\perp^2}{\\sqrt{s}},\\sqrt{s} ,-q_\\perp \\right),\n\\nonumber\n\\end{align}\nwhere $k_1$ and $k_3$ are,  respectively, the incoming  and outgoing photon momenta and $k_2$ and $k_4$ are, respectively, the incoming and outgoing proton momenta. \nThe incoming and outgoing off-shell photons have the following polarization vectors\n\\begin{align}\n    &n_1=\n    \\begin{cases}\n      \\left(0,0,\\epsilon_\\lambda \\right), & \\lambda=1, 2 \\\\\n      \\frac{1}{Q_1} \\left( \\sqrt{s}, \\frac{Q_1^2}{\\sqrt{s}}, 0, 0 \\right), & \\lambda = 3\n    \\end{cases}\\,,\\\\\n    &n_3=\n    \\begin{cases}\n      \\left(0,\\frac{2 q_\\perp \\cdot \\epsilon_\\lambda}{\\sqrt{s}},\\epsilon_\\lambda\\right), & \\!\\lambda=1,2 \\\\\n      \\frac{1}{Q_3} \\left(\\sqrt{s}, \\frac{Q_3^2+q_\\perp^2}{\\sqrt{s}}, q_\\perp \\right), & \\!\\lambda = 3\n    \\end{cases}\\,,\n    \\label{eq:inPolarization}\n\\end{align}\nwhere $\\epsilon_1 =  (1,0)$ and $\\epsilon_2 =  (0,1)$. \n\nTo compute the Witten diagram in figure \\ref{fig:DVCS Witten diagram}, consider  \nthe minimal coupling between the gauge field dual to the current $J^\\mu = \\bar{\\psi} \\gamma^\\mu \\psi$ \nand the higher spin field $h_{a_1 \\dots a_J}$\n\\begin{align}\n\\label{eq:gauge_field_spin_j_coupling}\n    k_J \\int d^5 X \\sqrt{-g} e^{-\\Phi} F_{b_1 a} D_{b_2} \\dots D_{b_{J-1}} F^{a}_{b_J} h^{b_1 \\dots b_J},\n\\end{align}\nwhere the field strength $F_{ab}$ is computed from the non-normalizable mode\n\\begin{equation}\nA_a (X; k,\\lambda)= n_a(\\lambda)  f_Q(z) \\, e^{ik\\cdot x}\\,.\n\\end{equation}\nThe coupling between the higher spin field and  the scalar field is\n\\begin{align}\n\\label{eq:scalar_field_spin_j_coupling}\n  \\bar{k}_J \\int d^5  X \\sqrt{- g} e^{- \\Phi} \\Upsilon D_{b_1} \\dots  D_{b_J}  \\Upsilon  h^{b_1 \\dots b_J},\n\\end{align}\nwhere \n\\begin{equation}\n \\Upsilon(X;k) =  \\upsilon(z)\\,e^{ik\\cdot x}\\,,\n\\end{equation}\nis the normalizable mode describing  the proton state.\nThe computation of the scattering amplitude follows the same steps as the ones presented in~\\cite{Amorim:2021ffr}, so we will not repeat it here. The scattering amplitude for $\\gamma^* p \\to \\gamma p$ is\n\\begin{equation}\n  \\label{eq:amplitude gamma* gamma}\n  \\mathcal{A}^{\\gamma^* p \\to \\gamma p}(s,t) = \\sum_n  g_n(t)  s^{j_n(t)}\\int dz \\,e^{-(j_n(t)-\\frac{3}{2})A} f_Q \\,\\psi_n(j_n(t))\\,,\n\\end{equation}\nwhere $A$ is the string frame warp factor and\n\\begin{equation}\n g_n (t)  = H(j_n(t)) \\left[i + \\cot \\left(\\frac{\\pi j_n(t)}{2}\\right)\\right]  \\frac{d j_n}{d t} \\int dz \\,e^{(-j_n(t)+\\frac{7}{2})A} \\upsilon^2 {\\psi_n^{*} (j_n(t))} \\, , \\label{eq:dvcs_gn_def}\n \\end{equation}\n with\n \\begin{equation}\n H(J) = \\frac{\\pi}{2} \\frac{k_{J} \\bar{k}_{J}}{2^{J}} \\, .\n\\end{equation}\nThe  amplitude (\\ref{eq:amplitude gamma* gamma}) is the one where both photons have transverse polarizations, other combinations are subleading in $s$. To obtain this expression we  used   (\\ref{eq:nonnormalizable_Q_0}).\nThe wavefunctions $\\psi_n(j_n(t))$ are eigenfunctions of the effective Schr\\\"odinger potential~\\cite{Ballon-Bayona:2017vlm}\n\\begin{align}\nV_J(z) =  \\frac{3}{2} \\left( \\ddot{A} - \\frac{2}{3} \\ddot{\\Phi} \\right) + & \\frac{9}{4}  {\\left( \\dot{A} - \\frac{2}{3} \\dot{\\Phi} \\right)}^2 + e^{2A} (J-2) \\left[ \\frac{2}{l_s^2} \\left(1+\\frac{d}{\\sqrt{\\lambda}}\\right) + \\right. \\notag \\\\\n&\\left.+ e^{-2A} \\left( a \\ddot{\\Phi} + b\\left( \\ddot{A} - \\dot{A}^2 \\right) + c \\dot{\\Phi}^2 \\right) +  \\frac{J+2}{\\lambda^{4\/3}}  \\right] ,\n\\end{align}\nwhere $\\lambda = e^\\Phi$ is the 't Hooft coupling. The constants $l_s$, $a$, $b$, $c$ and $d$ are phenomenological parameters that will be fixed later by reproducing the best match with DVCS data. These parameters describe the analytic continuation of the equation of motion for the AdS spin $J$ field dual to twist 2 gluonic operators, in an expansion around $J=2$. This approximation gives the description of the graviton\/pomeron Regge trajectory in the strong coupling approximation, in contrast with the weak coupling BFKL pomeron, whose expansion starts at $J=1$. The details of the derivation are given in \\cite{Ballon-Bayona:2017vlm}.\n\nTo proceed we need to know the proton wavefunction in order to compute the $z$ integral in (\\ref{eq:dvcs_gn_def}). In AdS\/QCD, it is expected that baryons are dual to a configuration where three open strings are attached to a D-brane~\\cite{witten_baryons_1998, Polchinski:2000uf}. So far it is not known how to derive the baryon spectrum from such a configuration. An acceptable holographic description of the proton would consist of a spectrum that matches the mass of the proton as well of the other hadrons in the same trajectory. Here we follow a phenomenological approach by approximating the combination $e^{3 A - \\Phi}\\upsilon^2$ by the delta function $\\delta(z-z^*)$. The $z^*$ parameter is related, by dimensional analysis, to the inverse of the mass of the proton and  will be used  as a fitting parameter. By making this approximation we are assuming that  $e^{3 A - \\Phi}\\upsilon^2$  is null in the UV and the IR, and that it has a global maximum. This is expected if the spectrum of the baryons is associated with a Schrodinger problem whose ground state is the proton. This approach, as an example, has successfully described DVCS data in \\cite{Costa:2012fw}. Another issue is the unknown functional form of the function $H(J)$ and its analytic continuation. In the next section we motivate a good ansatz for this expression, at least in the range $1 \\lesssim J \\lesssim 1.2$ that is relevant for the process here considered.\n\n\\section{AdS local coupling to the graviton trajectory}\n\nTo find a good ansatz for $H(J)$ let us first recall that in~\\cite{Amorim:2021ffr}, based on the forward scattering amplitude for the process $\\gamma^*p \\to \\gamma^* p$, the holographic formula for the proton structure function $F_2^p$ is given by\n\\begin{align}\n&F_2^p(x, Q^2) = \\sum_{n} \\frac{ {\\rm Im} \\, g_n}{4 \\pi^2 \\alpha} \\, x^{1-j_n}  Q^{2 j_n} \\int dz \\,e^{-\\left(j_n-\\frac{3}{2}\\right)A}  \\left( f_Q^2  +  \\frac{\\dot{f}_Q^{2}}{Q^2}      \\right) \\psi_n(z) \\, ,\n\\end{align}\nwhere $j_n$ and \n\\begin{equation}\n{\\rm Im} \\, g_n = H(j_n(0)) \\frac{d j_n}{d t} \\int d  z  \\,e^{(-j_n(0)+\\frac{7}{2}) A}  \\upsilon^2 {\\psi}_n^{*}  (j_n(0))\n\\end{equation}\nare computed at $t=0$.\nUsing  (\\ref{eq:dvcs_gn_def}) and the approximation $e^{3 A - \\Phi}\\upsilon^2 \\sim \\delta(z-z^*)$, we can compute $H(J)$ for any value of $j_n(t)$ with the result\n\\begin{equation}\n  \\label{eq:dvcs:H(J) vs g_n(t)}\n  H(j_n (t)) = \\frac{{\\rm Im} \\, g_n(t)}{e^{(-j_n (t) + \\frac{1}{2})A(z^*)}\\psi^*_n(j_n (t)) \\frac{dj_n (t)}{dt}} \\, .\n\\end{equation}\nWe can now plot $H(J)$ as a function of $J$ using\nthe $t=0$ values of ${\\rm Im} \\, g_n$, $j_n$, $\\frac{dj_n}{dt}$ and $\\psi_n$ for the different trajectories found in \\cite{Ballon-Bayona:2017vlm} and  choosing $z^{*}$ to be around 1, since by dimensional analysis it should be of the order of the inverse mass of the proton (in GeV units). \nThe black dots  shown in figure \\ref{fig:dvcs:H(J)} represent the actual reconstructed values of $H(J)$ for each $n$.\nIn a logarithmic plot, $\\log H (J)$ can be well approximated by a quadratic curve in $J$, i.e. $\\log H(J) = h_0 + h_1 (J-1) + h_2 (J-1)^2$. We can then choose $z^{*}$ and the $h_i$ to get the best quadratic fit. For the best fit value of $z^* = 0.565$ we obtained the result presented  in figure \\ref{fig:dvcs:H(J)}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.5]{H_reconstructed.pdf}\n  \\caption[Reconstruction of the $H(J)$ function using the best parameters found in \\cite{Ballon-Bayona:2017vlm} and $z^*=0.565$]{Reconstruction of the $H(J)$ function using the best parameters found in \\cite{Ballon-Bayona:2017vlm} and $z^*=0.565$. The solid line represents the function $\\exp \\left(h_0 + h_1 (J-1) + h_2 (J-1)^2\\right)$ with $h_0=3.70$, $h_1=-30.3$ and $h_2=89.1$.}\n  \\label{fig:dvcs:H(J)}\n\\end{figure}\n\nWe can check if the resulting curve has physical meaning by considering the dependence  with $J$  of the\nAdS local couplings  $\\kappa(J)$ and $\\bar \\kappa (J)$, that define $H(J)$,\nin the range $1 \\lesssim J \\lesssim 1.2$. According to the gauge\/gravity duality such couplings are related to the OPE coefficient of a spin $J$ operator \nwith two spin 1 operators (for the coupling with the EMG current operator) or spin 0 operators (for the coupling with a scalar, as we model the unpolarized proton). \nIn particular, in the UV fixed point the OPE coefficient of two EMG current operators with the spin $J$ glueball operator associated with the pomeron \ntrajectory vanishes, as only Wick contractions contribute.\nOf course, QCD is not a free theory and perturbative corrections should be taken into account. Instead, to check if the proposed curve for $H(J)$ \nis reasonable let us consider the $O(N)$ vector model, which is also free in  the UV where the corresponding  OPE coefficient is non-vanishing \\cite{Sleight:2017fpc}. In this model we can retrieve a similar shape of $H (J)$ as the one in figure \\ref{fig:dvcs:H(J)} from the three-point bulk vertex coupling between massless higher spin fields of spin $s_1$, $s_2$  and $s_3$ in type A minimal higher-spin theory in $AdS_{d+1}$. The coupling, as a function of $s_1$, $s_2$ and $s_3$, is given by\n\\begin{equation}\ng_{s_1,s_2,s_3} = \\frac{\\pi^{\\frac{d-3}{4}} 2^{\\frac{3d-1+s_1+s_2+s_3}{2}}}{\\sqrt{N}\\Gamma(d+s_1+s_2+s_3-3)} \\sqrt{\\frac{\\Gamma(s_1 + \\frac{d-1}{2})}{\\Gamma(s_1 + 1)}\\frac{\\Gamma(s_2 + \\frac{d-1}{2})}{\\Gamma(s_2 + 1)}\\frac{\\Gamma(s_3 + \\frac{d-1}{2})}{\\Gamma(s_3 + 1)}}  \\, .\n\\end{equation}\n After approximating the gamma functions in the resulting expression with the Stirling formula and expanding up to quadratic order around $s_3=J = 1$, we obtain a \n function $H(J)$ consistent with our ansatz. Because we are comparing different theories caution should be taken. However, our point is that the overall function shapes does not change, and beyond the fixed point, corrections may be well captured in suitable redefinitions of the $h_i$ parameters in our ansatz. In the next section we test our hypothesis against DVCS experimental data.\n\n\n\\section{Data analysis and results}\nNow that we have a reasonable parameterisation of $H(J)$ we proceed to find the best values for the pomeron kernel parameters $l_s$, $a$, $b$, $c$ and $d$,\n and the constants $h_0$, $h_1$ and $h_2$ in the $H(J)$ ansatz. The optimal set of parameters is found by minimising the $\\chi^2$ statistic\n\\begin{equation}\n  \\label{eq:global chi2 definition}\n  \\chi_g^2=\\chi_{\\sigma}^2  +\\chi_{\\frac{d \\sigma}{dt}}^2 \\, ,\n\\end{equation}\nwhich is the sum of the $\\chi^2$ for $\\sigma\\left(\\gamma^* p \\to \\gamma p\\right)$ and the $\\chi^2$ for $\\frac{d\\sigma\\left( \\gamma^* p \\to \\gamma p \\right)}{dt}$. \nAs usual, for a given observable $O \\in \\{\\sigma\\left(\\gamma^* p \\to \\gamma p\\right), \\frac{d\\sigma\\left( \\gamma^* p \\to \\gamma p \\right)}{dt} \\}$, the respective $\\chi^2$ function is defined as\n\\begin{equation}\n  \\chi^2_{O} \\equiv \\sum_n \\left(\\frac{O_n^{\\text{pred}} - O_n}{\\delta O_n}\\right)^2 \\, ,\n\\end{equation}\nwith $O^{\\text{pred}}$ being the predicted theoretical value and $\\delta O$ the experimental uncertainty. The sum goes over the available experimental points.\nThe total and differential cross-section data used is the combined one from H1-ZEUS available in \\cite{aaron_deeply_2009,chekanov_measurement_2009}.\n\nThe DVCS differential cross-section is given by\n\\begin{align}\n    \\frac{d\\sigma}{dt} = \\frac{1}{16 \\pi s^2} \\frac{1}{2} \\sum_{\\lambda_1,\\lambda_3 = 1}^2 \\left|A^{\\lambda_1,\\lambda_3}\\left( s, t \\right) \\right|^2 =\\frac{1}{16 \\pi^2 s^2} \\vert \\mathcal{A}^{\\gamma^*\\gamma}(s,t) \\vert^2,\n\\end{align}\nwhere we average over the incoming photon polarization and the scattering amplitude is given by equation (\\ref{eq:amplitude gamma* gamma}).\nThe total cross-section is just the integral of the above\n\\begin{equation}\n\\label{eq:dvcs_sigma}\n  \\sigma = \\int_{-1}^0 dt \\, \\frac{d \\sigma (t)}{dt} \\,,\n\\end{equation}\nwhere the integration range $-1 \\leq t \\leq 0$ comes from the data. \n\nTo solve the $\\chi^2$ minimisation problem we developed the $\\tt{HQCDP}$ $\\tt{R}$ package.\nThe Schrodinger problem associated to the gluon kernel is solved with $N=400$ Chebyshev points. To compute the differential and total cross-sections efficiently we divided the  interval $-1 \\leq t \\leq 0$ in 20 pieces of length 0.05 and computed the differential cross-section for each point. From these values we created a spline interpolation function that can be used to predict the differential cross-section values and the total cross-section through equation (\\ref{eq:dvcs_sigma}). We also make use of the $\\tt{REDIS}$ \\textit{in memory} database to avoid redoing expensive computations. The code also makes use of multiple cores, if available, in order to compute in parallel the integrals that appear in (\\ref{eq:amplitude gamma* gamma}) for different kinematical points. The present results were found using a node in a High Performance Computing (HPC) cluster with 16 cores.\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[scale=0.5]{DVCSDSigma}\n  \\caption[Predicted vs. experimental values of the differential cross-section $\\frac{d \\sigma(t)}{dt}$ for DVCS]{Predicted vs. experimental values of the differential cross-section $\\frac{d \\sigma(t)}{dt}$ for DVCS. Different gray levels correspond to different combinations of $Q^2$ and $W$ as described in the legends. Here $Q^2$ and $t$ are in $\\text{GeV}^2$, $W$ in $\\text{GeV}$ and $\\frac{d \\sigma}{d t}$ is in $\\frac{\\text{nb}}{\\text{GeV}^2}$. }\n  \\label{fig:dvcs:DVCSDSigma}\n\\end{figure}\n\n\nThe best fit to the data we have found has a $\\chi^2_{dof} \\sim 1.5$ with the parameter values of table~\\ref{tab:DVCS DIS best values}. The individual values of $\\chi^2_{dof}$ for the total and differential cross-section experimental data are $1.8$ and $1.3$ respectively, meaning that the model offers a good description of both processes. The comparison of the theoretical predictions against the experimental data can be seen in figures \\ref{fig:dvcs:DVCSDSigma} and \\ref{fig:dvcs:DVCSSigma}. Figure~\\ref{fig:regge_trajectories} plots  the first four Regge trajectories obtained with the kernel parameters of table~\\ref{tab:DVCS DIS best values}. For the data range $0 \\lesssim-t \\lesssim1\\ ({\\rm GeV}^2)$ the Reggeon spin \nis in the  range $1 \\lesssim j_n(t) \\lesssim 1.2$.\n \n\\begin{table}[h!]\n  \\centering\n  \\begin{tabular}{|c||c||c||c||c||c|}\n    \\hline \n    Kernel parameters & Extra parameters & Intercepts\\\\\n    \\hline \n    \\hline \n    $a = -4.55$      & $h_0 = 4.74$  &  $j_0=1.24$ \\\\\n    \\hline \n    $ b = 0.980$     & $h_1 = -35.9$ &  $j_1=1.13$ \\\\\n    \\hline \n    $ c = 0.809$     & $h_2 = 142$  &  $j_2=1.08$\\\\\n    \\hline \n    $d = -0.160$     & $z^* = 0.296$ &  $j_3=1.05$\\\\\n    \\hline \n    $l_{s} = 0.153$ &--  &--  \\\\\n    \\hline \n  \\end{tabular}\n  \\caption[Best fit parameters for the DVCS data fit]{The 9 parameters for our best fit and the intercept of the  first four pomeron trajectories. All parameters are dimensionless except $z^*$ and $l_s$ which are in $\\rm{GeV}^{-1}$.}\n  \\label{tab:DVCS DIS best values}\n\\end{table}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=0.5]{DVCSSigma}\n  \\caption[Predicted vs. experimental values of the total cross-section $\\sigma$ of DVCS]{Predicted vs. experimental values of the total cross-section $\\sigma$ of DVCS. Small numbers attached to lines and points of the same grey level indicate the respective value of $Q^2$ in $\\text{GeV}$.}\n  \\label{fig:dvcs:DVCSSigma}\n\\end{figure}\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[scale=0.5]{regge_trajectories_dvcs}\n  \\caption{Regge trajectories obtained from our model and their intercepts. They were obtained using the kernel parameters of table~\\ref{tab:DVCS DIS best values}. The plot includes the kinematical range\n   $0 \\lesssim-t \\lesssim1\\ ({\\rm GeV}^2)$ as in the data used in this work.}\n  \\label{fig:regge_trajectories}\n\\end{figure}\n\n\nTo conclude, we have shown that the holographic model presented in~\\cite{Amorim:2021ffr} can be extended to include a quantitative description of total and differential cross-sections of DVCS data from H1-ZEUS.  One should also include the kernel of the twist 2 fermion operators, as suggested by Donnachie and Landshoff~\\cite{Donnachie:1998gm} and study the importance\nof the exchange of meson trajectories  on the results above. A first step towards the inclusion of the twist 2 fermion operators has been done~\\cite{Amorim:2021gat}.\n\n\n\\section*{Acknowledgments}\n\n\nThis research received funding from the Simons Foundation grant 488637  (Simons collaboration on the Non-perturbative bootstrap). \nCentro de F\\'\\i sica do Porto is partially funded by Funda\\c c\\~ao para a Ci\\^encia e a Tecnologia (FCT) under the grant\nUID-04650-FCUP.\n AA is funded by FCT under the IDPASC doctorate programme with the fellowship  PD\/BD\/114158\/2016.\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section1}\n\nSuperfluids are extraordinary fluids characterised by the absence of \nviscosity.\nThey are irrotational everywhere except at vortex lines\nwhose circulation is quantised in units of $\\kappa=h\/m$, where \n$h$ is Planck's constant and $m$ is the \nmass of the boson which composes the fluid \\cite{Annett2004,Barenghi2016}.\nFirst discovered and studied in liquid helium-4 and, decades later,\nin helium-3, superfluidity has since been observed in ultracold gases and \nphotonic systems. The constraint of quantised\nvorticity is a consequence of quantum mechanics - vorticity can only arise as $2 \\pi$ topological defects of the macroscopic single-particle wavefunction of the quantum many-body system.  These defects manifest as vortex lines through the fluid.  As well as possessing a circulating flow, the vortex lines have a core of depleted density about their axis, out to a core radius $a_0$ which is of the order of the superfluid healing length. In helium-4 and helium-3 the vortex core size is around $10^{-10}$m and $10^{-8}$m, respectively.\n\nThe textbook paradigm of superfluidity is a cylindrical\nbucket of superfluid helium\nrotating at constant angular frequency $\\Omega$. \nClassical solid-body \nrotation is forbidden by the irrotational nature of the superfluid.  At  sufficiently small values of\n$\\Omega$, the fluid remains quiescent. However, if $\\Omega$\nin increased past a critical\nvalue $\\Omega_{\\rm c}$, the presence of a vortex line\nis energetically favourable.  Using hydrodynamic arguments and up to \na logarithmic correction, it is estimated \\cite{Barenghi2016} that this\ncritical angular frequency is\n\n\\begin{equation}\n\\Omega_{\\rm c} =  \\frac{\\hbar}{mR^2} \\ln \\left( \\frac{R}{a_0} \\right),\n\\label{eqn:omega_c}\n\\end{equation}\n\n\\noindent\nwhere $m$ is the mass of a helium atom and $R$ is the radius of the bucket.  At larger values of\n$\\Omega$, two vortices become favourable, and so on. \nFor $\\Omega \\gg \\Omega_{\\rm c}$ the stationary state of the fluid is \nthe famed vortex lattice, an array of vortex lines aligned along the\naxis of rotation with areal density\n\\begin{equation}\nn_{\\rm v}=\\frac{2\\Omega}{\\kappa},\n\\label{eq:Feynman}\n\\end{equation}\nknown as Feynman's rule. The vortex lattice was  \nfirst imaged in superfluid helium by Packard {\\it et al} \\cite{Yarmchuk1979} \nand more recently by Bewley {\\it et al} \\cite{Bewley2006}.  \nThe lattice has also been observed in ultracold gaseous superfluids\ntrapped by smooth confining \npotentials \\cite{Madison2000,AboShaeer2001}. (At much higher rotation frequencies where centrifugal effects dominate, a giant or macroscopic vortex carrying many quanta of circulation can become formed in both superfluid helium \\cite{Tsakadze1964,Josserand2004} and gaseous superfluids \\cite{Kasamatsu2002,Fischer2003,Kavoulakis2003,Engels2003}; however, this regime is outside the scope of this work.)\n\n\n\nThe process in which the vortices enter the superfluid in the first\nplace is called vortex nucleation. \nBeing associated with a $2\\pi$ phase singularity of the macroscopic \nwavefunction, a vortex line is topologically protected.\nThus, starting from some initially vortex-free state, vortices must enter\nthe superfluid from the boundary. It is believed that\nvortex lines are nucleated\neither {\\it intrinsically} by the flow of the superfluid past the \nmicroscopic roughness of the bucket wall (overcoming a critical velocity)\nor {\\it extrinsically} by stretching some pre-existing vortex lines \ncalled ``remanent vortices\" which,\nunder suitably conditions, can spool additional \nvortices \\cite{Schwarz-mill}.  Remanent vortices are thought \nto arise when cooling the helium sample through the superfluid transition, \nand can be avoided by using careful\nexperimental protocols \\cite{Yano-2007}.  \n\nIndividual vortex nucleation in a rotating bucket,\neither intrinsic or extrinsic, has never been visualised in detail. \nExperimentally, it remains challenging to image the flow in \nthe vicinity of a boundary, despite progress in flow visualisation \nin the bulk \\cite{Bewley2006,Zmeev2015b,Duda2015}, more so because\nthe microscopic scale of the vortices themselves.  \nTheoretically, the nucleation problem has been addressed using\nenergy arguments \\cite{Fetter1966,StaufferFetter1968} with no insight\nin the dynamics. With few exceptions \\cite{Stagg2017},\nthe effect of microscopic boundary roughness on the vortex nucleation has not been studied. \nA related and better understood nucleation  process\ntakes place when an ion bubble is driven in liquid helium by an applied \nelectric field; compared to the bucket, the nucleation is more \ncontrolled in terms of geometry (the shape of the bubble can be\ndetermined theoretically)\nand velocity (experimentally determined by time of flight measurements).\nVortex nucleation by the ion bubble has thus received much detailed \nexperimental and theoretical attention\n\\cite{MuirheadVinenDonnelly1984,McClintockBowley1995,BerloffRoberts2000,\nWiniecki2000,Villois2018} than nucleation by the walls of the bucket \nwhich contains the helium sample.\n\nIn this work we are not concerned with the vortex nucleation as such,\nbut rather with the intermediate state between the nucleation and\nthe final vortex lattice. This intermediate stage is still unexplored, \nbut, given that the length scales and the time scales involved depend on\nthe vortex separation rather the vortex core size (i.e. they are\nmesoscopic rather than microscopic), there is prospect of \nexperimental visualisation in the near future.\nThe focus of attention is therefore not individual vortex dynamics at \nnucleation but the collective dynamics of many vortex lines in the \npresence of a boundary which is not smooth. For simplicity we consider \nthe problem at sufficiently low temperature that the normal fluid does\nnot play an important role.\n\nThe traditional method to model the dynamics of superfluid vortices is\nthe Vortex Filament Method (VFM) \\cite{Schwarz1988}, \nwhich models vortex lines as infinitesimally thin filaments \ninteracting with themselves, their neighbours and the boundary \n(via suitable images).  However, this approach is not applicable\nto our problem.\nFirstly, if the boundary varies on atomic length scales comparable \nto the vortex core (which is likely to be the case for any metal or glass \nbucket containing liquid helium),\nthen the core lengthscale can no longer be ignored compared to\nother relevant lengthscale, invalidating the \nassumptions behind the VFM.\nSecondly, the implementation of the boundary\ncondition is cumbersome to set up and\nnot simple to change from one boundary shape to another; indeed, the VFM \nhas been implemented for plane \\cite{Schwarz1985}, \nsemi-spherical \\cite{Schwarz1985,Tsubota1993}, \nspherical \\cite{Schwarz1974,Kivotides2006} and cylindrical\n\\cite{Hanninen2005,HanninenBaggaley2014} boundaries, but never\nfor irregular boundaries relevant to our problem. Thirdly, the VFM does\nnot describe vortex nucleation, but requires to initialise the calculation\nwith arbitrary seeding vortex lines.\nAn alternative approach is through the Gross-Pitaevskii equation (GPE) \n\\cite{Pitaevskii,Barenghi2016}. This is a formal description \nof a dilute weakly-interacting gas of bosons, and is equivalent \nto a continuity equation and an Euler-like equation for an inviscid \nfluid (the modification being the presence\nof a quantum pressure term).  While the GPE is an excellent quantitative \ndescription of Bose gas superfluids, it is limited to being a qualitative \ndescription of superfluid helium due to the stronger interactions\ntaking place in a liquid rather than in a gas.  \nNevertheless, its capability to describe the microscopic detail of \nsuperfluid dynamics - the finite-sized core, \nvortex interactions and reconnections, even the intrinsic nucleation - makes \nit a useful model to study superfluid flows at a boundary.  \nAn important feature is that the GPE can easily implement\nirregular boundaries.\nIndeed, recent GPE simulations have predicted the occurrence of a \nturbulent boundary layer when the superfluid flows past a locally\nrough surface \\cite{Stagg2017}:  above a critical imposed flow speed, \nvortices are nucleated from the surface features, interact and become \nentwined in a layer adjacent to the surface.  \n\nReturning to the rotating bucket of superfluid helium, \nit is natural to ask if some kind of boundary layer may similarly \nform at the boundary of the rotating bucket\nin the transient evolution to the vortex lattice. Whether disordered\nor laminar, this layer will certainly involve vortex interactions.\nIt is in fact unlikely that the vortex lines which nucleate extend\nfrom the top to the bottom of the bucket, as if the process were\nessentially two-dimensional (2D). More likely, the first vortex lines\nwhich nucleate are small, and become long only after a sequence of\ninteractions and reconnections.\nTo qualitatively explore these interactions, here we perform a series of \nnumerical experiments, based on the GPE, of a superfluid being spun-up \nin a bucket whose walls are microscopically rough.  \nThese numerical experiments allows us to build a physical picture \nof how vorticity enters the superfluid and forms a vortex lattice, \nand of the role of remanent vortices, \nsharp intrusions, rotation rate, and dimensionality.  \n\nThe plan of the paper is the following.\nIn Section~\\ref{section2} we introduce our model and details our of numerical\nsimulations.  In Section~\\ref{section3} we present our main results for the\nspin up of a quiescent superfluid. Section~\\ref{section4} explores\nthe possibility that a single strong imperfection in the shape of\na protuberance, remanent vortex lines or dimensionality may affect the main results described\nin Section~\\ref{section3}. Finally, in Section~\\ref{section5} \nwe discuss and conclude our findings.\n\n\n\n\n\n\n\n\\section{Model and method}\n\\label{section2}\n\n\\subsection{Gross-Pitaevskii equation}\n\n\nWe model the superfluid dynamics using the Gross-Pitaevskii equation. Within this model, the superfluid is parametrised by a mean-field complex\nwavefunction $\\Psi({\\bf r},t)=\\vert \\Psi({\\bf r},t)\\vert e^{iS(\\bf r,t)}$.  \nThe particle density follows as $n({\\bf r},t)=|\\Psi({\\bf r},t)|^2$ and \nthe fluid velocity as ${\\bf v}({\\bf r},t)=(\\hbar\/m)\\nabla S({\\bf r},t)$, \nwhere $\\hbar=h\/(2 \\pi)$ and $S({\\bf r},t)$ is the phase distribution \nof $\\Psi$.  The dynamics of $\\Psi({\\bf r},t)$ follows the \nGPE \\cite{Pitaevskii,Barenghi2016},\n\n\\begin{equation}\ni \\hbar \\frac{\\partial\\Psi }{\\partial t}= \\hat{\\mathcal{H}} \\Psi, \n\\label{eq:GPE}\n\\end{equation}\n\n\\noindent\nwith Hamiltonian operator,\n\n\\begin{equation}\n\\hat{\\mathcal{H}} = \n-\\frac{\\hbar^2}{2m}\\nabla^2\n+V+g\\left|\\Psi \\right|^{2}.\n\\end{equation}\n\n\n\\noindent\nHere $m$ is the particle mass, $g$ $(>0)$ is a nonlinear coefficient \ndescribing the inter-particle interactions,\n and $V({\\bf r},t)$ is the external potential acting on the fluid.  \nStationary solutions of the GPE satisfy \n$\\hat{\\mathcal{H}}\\Psi= \\mu_0 \\Psi$, \nwhere $\\mu_0$ is the chemical potential of the fluid.   \n\nWe make two physically-motivated modifications to the basic GPE above.  \nFirstly, since the GPE conserves energy, we follow other works \n\\cite{Choi1998,Tsubota2001} in introducing a phenomenological dissipation \nterm into the GPE to model, at least in a qualitative way, \nthe damping of excitations of the superfluid (for example, by their \ninteraction with the normal fluid).\nThis is achieved by replacing the left-side of Eq. (\\ref{eq:GPE}) \nwith $(i-\\gamma) \\hbar \\, \\partial \\Psi\/\\partial t$, where $\\gamma$ \nspecifies the strength of the dissipation.  \nAlthough not as accurate the friction included within the\n VFM, this phenomenological dissipation\nwill help damp out the oscillations of the vortex lines (Kelvin waves),\nwhich is the main effect of the friction which concerns us here.\n Secondly, given our rotating scenario, we work in the reference\nframe rotating at constant angular frequency $\\Omega$ about the $z$ axis; this is achieved by modifying the GPE Hamiltonian to $\\hat{\\mathcal{H}}-i \\Omega L_z$, where $L_z$ is the angular momentum operator about $z$.   In Cartesian coordinates $L_z=i \\hbar (y \\partial_x-x \\partial_y)$.\n\n\n\\subsection{Bucket set-up}\n\nWe consider the fluid to be confined within \na cylindrical bucket of radius $R$ and height $H$. The axis of the cylinder\nis the z-axis of rotation.  \nThe bucket is modelled through the potential $V({\\bf r})$: in the interior \nof the bucket we set $V=0$ while at the boundary and beyond we set \n$V \\gg \\mu_0$. In the ground state, the fluid density has the bulk value \n$n_0$ in the centre of the bucket, while close to the bucket wall \nit heals to zero density over a length scale characterised by the healing \nlength $\\xi=\\hbar\/\\sqrt{m n_0 g}$.  The healing length also characterises \nthe size of the cores of vortices in the fluid.\nNote that the chemical potential in the bulk is $\\mu_0=n_0 g$.  \nThe speed of sound in the uniform systems is\n$c=\\sqrt{n_0 g \/m}$.\n\nIt is clearly computationally impossible\nto simulate the range of length scales which are realistic for a typical\nexperiment with liquid helium in the context of the GPE model.\nThe dimensions (radius and height) of typical buckets used \nin the experiments are of the order of the\ncentimetre, which is around eight orders of magnitude larger than the \nvortex core size in helium-4, $a_0=10^{-10}\\rm m$ (in helium-3 the vortex\ncore is about 100 times larger).\nInstead, in our numerical experiments we employ buckets whose scale \nis around 2 orders of magnitude larger than the vortex core size.\nWhile this is clearly a vast scale reduction compared to real systems, \nthe separation of scales between the vortices and the bucket size \nis sufficient to give us a qualitative insight into the dynamics\nof the vortex lines.\n\n\\subsection{Surface roughness}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.9\\columnwidth]{fig1.pdf}\n\\caption{(a) An example of the 2D fractional Brownian motion function $f$, \nnormalised to the range [0,1], shown as a surface plot and a heatmap.\n(b) The rough cylindrical boundary of our bucket\nis formed by using the surface in (a) to modulate\nthe radius of the bucket boundary with an amplitude $a$. Here $a=0.1$. }\n\\label{fig1}\n\\end{figure}\n\n\n\nTo mimic the experimentally unavoidable surface roughness, \nwe modify the azimuthal face of the bucket away from a perfect \ncylindrical shape using a noisy two-dimensional (2D) function.  \nThis function is numerically generated through a two-dimensional \nfractal Brownian motion \\cite{MandelbrotVanNess}\nwith Hurst index of $0.3$, a parameter which describes the fractal dimension\nof the surface \\cite{Mandelbrot1985}.\nThe choice to model the roughness in this way is motivated by the well established fractal properties of real surfaces, including machined surfaces (of relevance to helium experiments), and the success of fractal brownian motion in modelling a wide variety of real rough surfaces \\cite{Majumdar1990}.   \nThe function is normalised between $0$ and $1$, and is mirrored \nabout its edge and recombined with itself in order to create periodicity \nacross one dimension; a single realisation of the function is \ndepicted in Fig.~\\ref{fig1}(a).  The function is mapped onto \nthe space of axial coordinate $z$ and azimuthal angle $\\theta$, \nand used to modify the radius of the bucket according to the form,\n\n\\begin{equation}\nr(z,\\phi)=R(1-a f(z,\\theta)),\n\\label{eq:rbucket}\n\\end{equation}\n\n\\noindent\nwhere $R$ is the smooth bucket radius and $a$ is the \n(dimensionless) roughness parameter.\nThis numerical procedure\ngenerates all of our rough 3D bucket shapes. \nBy computing the local curvature of the surface roughness,\nwe find that the values of the average radius of curvature corresponding to\nvalues $a=0.05$, $0.1$, $0.2$ and $0.3$ of the roughness parameter are\n$10.4 \\xi$, $5.2 \\xi$, $2.6 \\xi$ and $1.7\\xi$ respectively\n(small values of $a$ correspond to large radius of curvature, i.e. smoother\nsurface).\nFor simplicity, the top and bottom surfaces of the bucket are left\nsmooth. The reason is that, by providing the vortex lines with\npinning sites, any roughness on these surfaces \nwill act essentially as an extra friction (an effect which already\nqualitatively account for via the dissipation parameter $\\gamma$) \nslowing down the final stage of cristallization of the vortex lattice.\n\n\n\n\n\n\\subsection{Simulation set-up}\n\nThe initial condition for $\\Psi$ in all of our simulations is the \nnon-rotating ground state solution, found by the method of imaginary \ntime propagation of the GPE, supplemented with low-amplitude \nwhite noise to $\\Psi$ (amplitude $0.001$) to break any symmetries \nartificially presented in the initial condition.   \nWe then impose a constant rotation on the system for $t>0$, \nwith fixed rotation frequency $\\Omega$.  Note that $\\Omega$ \nfar exceeds the critical rotation frequency to support vortices \n$\\Omega_{\\rm c}$, such that the lowest energy state of the fluid \nis a vortex lattice.\n\n\n\nThe non-dimensionalisation of the GPE is based on the natural units \nof the homogeneous fluid \\cite{Barenghi2016}:\nthe unit of length is the healing length $\\xi$, the unit of speed\nis $c$, the unit of time is \n$\\tau=\\xi\/c=\\hbar\/\\mu_0$, the unit of energy is $\\mu_0$, \nand the unit of density is $n_0$.  Both our 3D and 2D numerical simulations \nare performed using XMDS2 \\cite{xmds2}, an open-source partial and \nordinary differential equation solver. \nThe time evolution of the dimensionless GPE is computed\nvia an adaptive fourth-fifth order Runge-Kutta integration scheme \nwith typical time step $dt=0.01 \\tau$ and grid spacing $dx = 0.4 \\xi$; \nthese discretization numbers are sufficiently small to resolve \nthe smallest spatial features (vortices and the fluid boundary layer, \nwhich are of the order of few healing lengths) and the shortest \ntimescales in the fluid.  We typically conduct our 3D simulations \non a cubic grid of size $256^3$.  Threaded parallel \nprocessing is employed using the OpenMP standard across typically $44$ \nthreads to improve processing speeds on computationally intensive simulations. \n\n\n\\section{Results}\n\\label{section3}\n\n\\subsection{Typical spin-up dynamics} \n\n\\begin{figure*}\n(a) \\hspace{3.8cm} (b) \\hspace{4cm} (c) \\hspace{5cm} \\\\\n\\includegraphics[width = 0.24\\textwidth]{fig2a.pdf}\n\\includegraphics[width = 0.24\\textwidth]{fig2b.pdf}\n\\includegraphics[width = 0.24\\textwidth]{fig2c.pdf}\n\\\\(d) \\hspace{4cm} (e) \\hspace{3.8cm} (f) \\hspace{3.8cm} \\\\\n\\includegraphics[width = 0.24\\textwidth]{fig2d.pdf}\n\\includegraphics[width = 0.24\\textwidth]{fig2e.pdf}\t\n\\includegraphics[width = 0.24\\textwidth]{fig2f.pdf}\t\t\n\\caption{\nThree-dimensional snapshots at times \n$t\/\\tau=100$ (a), $200$ (b), $500$ (c), $1000$ (d), $1500$ (e) and $3000$ (f)\nduring the spin-up of the initially quiescent fluid. \nThe vortex cores are identified by density isosurfaces;\nvortices with positive and negative circulation (as determined by \ntheir pseudo-vorticity \\cite{Villois2016} in the $z$ direction)\nare visualised in red and blue respectively. The faint yellow\nisosurface represents\nthe confining bucket. A false-color shadow is projected\nonto the bottom surface to enhance the visualisation of the 3D vortex lines.\n}\n\\label{fig2}\n\\end{figure*} \n\nWe now demonstrate the typical spin up of an initially \nquiescent fluid.  Unless otherwise indicated, we present results\nfor the following choice of parameters: \nbucket radius $R=50\\xi$, bucket height $H=100 \\xi$, \nrotation frequency $\\Omega = 0.02~ \\tau^{-1}$,\ndissipation parameter $\\gamma=0.05$, and\nroughness parameter $a=0.1$ (meaning that the irregular surface\nof the bucket extends radially from $45\\rm \\xi$ to $50~\\xi$, corresponding to\nirregular `surface bumps' of height up to 5 healing lengths).\n\n\nThe evolution of the fluid is illustrated by the\nsnapshots shown in Fig.~\\ref{fig2} in which the vortex lines are tracked\nin 3D space using a precise method introduced in Ref. \\cite{Villois2016}; movies of the evolution\nare available in Supplementary Material \\cite{movies}.  From the initial quiescent and vortex-free \nfluid, first we see the nucleation of vortex lines at the \ncylindrical boundary of the fluid [Fig.~\\ref{fig2}(a)]. These vortices are all singly-quantized; we do not detect the presence of multiply-charged vortices in any of our simulations, which is consistent with the energetic instability of multiply-charged vortices and the favourability of singly-charged vortices \\cite{Barenghi2016}.\nThe nucleation takes place at the sharpest features on the surface, as seen \nin a previous calculation over a flat rough surface \\cite{Stagg2017}: at \nthese features the local (potential) flow velocity is raised by the \ncurvature of the boundary, and exceeds the critical velocity of \nvortex nucleation, which, according to Landau's criterion,\nin a Bose gas is $v_c \\approx c$.\nSince the local flow speed around a moving obstacle always exceeds \nthe translational speed of the obstacle, Landau's criterion can be\nsatisfied by a translational speed less than $c$.\nFor example, a cylindrical obstacle moving at speed approximately\nequal to $0.4c$ will nucleate vortices \\cite{Frisch1992,Stagg2014}. \nIn our case ($\\Omega =0.02\/\\tau$ and $R=50 \\xi$), the translational \nspeed of the prominences on the rough boundary\nis approximately $\\Omega R \\approx c$, which is sufficient to exceed \nLandau's criterion and nucleate vortices.\nFigure~\\ref{fig2} (a) and (b) show that the\nvortex lines which nucleate at the rough boundary have the shape of\nsmall half-loops or handles; similar vortex shapes have been reported \nin trapped Bose-Einstein condensates\n\\cite{Aftalion2003} and turbulent superfluid helium-4 near\na heated cylinder \\cite{Rickinson2020}, and have been called respectively\n``U-vortices\" and ``handles\".  \n\nWe next consider the angular momentum of the fluid, exploring its evolution and distribution.  We define the density of the $z$-component of the angular momentum of the fluid as,\n\\begin{equation}\nL_z(x,y,z) = \\psi^* i\\hbar\\left(y\\frac{\\partial}{\\partial x}-x\\frac{\\partial}{\\partial y} \\right) \\psi.\n\\end{equation}\nFigure \\ref{fig_Lz}(a) shows the angular momentum density as a function of the radial coordinate, averaged vertically and azimuthally, $\\langle L_z \\rangle$.  At $t=0$, this is zero throughout the fluid.  At time evolves, angular momentum builds at the edge of the bucket and drifts inwards, corresponding to the nucleation and inward drift of the vortex lines.  At steady-state, the angular momentum forms a stepped curve, with each step corresponding to a concentric ring of vortex lines in the final lattice.  Note how the final distribution of the angular momentum density approximately follows the result of solid-body rotation with constant mass density, $m n_0 \\Omega r^2$.  The total $z$-component of the angular momentum $\\mathcal{L}=\\iiint L_z\\, {\\rm d}x\\,{\\rm d}y\\,{\\rm d}z$  (inset in Fig. \\ref{fig_Lz}(a)) grows in time, saturating at a final value by around $t\\sim 2000~\\tau$, which is when the vortex line have settled into the lattice configuration.  In gaseous superfluids confined within smooth potentials, recent results of merging superfluids \\cite{Kanai2018,Kanai2020} suggest that the rate of angular momentum transfer between a static and rotating state is constant; however, here the growth of the angular momentum follows a sigmoidal curve, rather than a linear one. \n\n\nWe also consider the distribution of the vortex length projected in the $z$-direction, \n$\\Lambda_z$, as a function of radius, shown in Figure \\ref{fig_Lz}(b).  At early times, vortex length exists only near the bucket edge, spreading progressively into the bulk.  Later, the vortex length converges towards falling at discrete peaks at $r=0$, $r\\approx 13 \\xi$ and $r\\approx 26 \\xi$, corresponding to the concentric arrangement of vortices in the lattice configuration.\n\n\n\n\\begin{figure}\n\\centering\n\\hspace{0.5cm}(a)\\\\\n\\includegraphics[width = 0.7\\columnwidth]{fig3a.pdf}\\\\\n\\hspace{0.5cm}(b)\\\\\n\\includegraphics[width = 0.7\\columnwidth]{fig3b.pdf}\n\\caption{(a) Angular momentum density as a function of radius at various times during the spin-up of the rough bucket (same parameters as in Fig. \\ref{fig2}). Plotted is the angular momentum density averaged over the $z$-dimension, $\\langle L_z \\rangle$.    The lines correspond to times indicated by circular markers on the inset  The distribution of the angular momentum of a solid-body of uniform density is shown by the red dashed line.   The inset shows the evolution of the total angular momentum of the fluid $\\mathcal{L}=\\iiint L_z\\, {\\rm d}x\\,{\\rm d}y\\,{\\rm d}z$.  (b)  The vortex length projected in the $z$-direction \n$\\Lambda_z$ plotted as a function of radius, at the same times as in (a).  The data is binned in radial intervals of $3\\xi$.\n}\n\\label{fig_Lz}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\hspace{0.5cm}(a)\\\\\n\\includegraphics[width = 0.6\\columnwidth]{fig4a.pdf}\\\\\n\\hspace{0.5cm}(b)\\\\\n\\includegraphics[width = 0.6\\columnwidth]{fig4b.pdf}\n\\caption{Azimuthal velocity $v_{\\theta}$ of the fluid as a function of radius $r$, for\nrotation frequencies of $\\Omega=0.02~\\tau^{-1}$ (a) and\n$\\Omega=0.06~\\tau^{-1}$ (b).  We use roughness parameter $a=0.1$.  The solid red lines \nrepresent solid body rotation $v_{\\theta}=\\Omega r$; the blue lines\nare values of $v_{\\theta}(r)$ averaged in the $\\theta$ direction;\nthe pale blue rectangles are histograms with bin size $\\Delta r=5 \\xi$\n(therefore the outer bins contain more data points). It is apparent that\nthe more rapid rotation (b) creates a vortex lattice in better agreement\nwith the solid body rotation, and that there is a vortex-free region\nnear the boundary.\n}\n\\label{fig3}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.9\\columnwidth]{fig5.pdf}\n\\caption{\nEvolution of the total vortex length, $\\Lambda$ (solid lines), as well\nas the vortex length in the $z$-direction, $\\Lambda_z$ (dashed lines) and\nthe vortex length in the $xy$-plane, $\\Lambda_{xy}$ (dot-dashed lines), \nplotted versus time $t$ for different angular velocity of rotation\n$\\Omega=0.02~\\tau^{_1}$ (black), $0.04~\\tau^{-1}$ (blue) and\n$0.06 ~\\tau^{-1}$ (red) \nachieving final values of the vortex\nlength $\\Lambda_{\\infty}=2184 \\xi$, $6007 \\xi$ and $5568 \\xi$ respectively.\nAll curves refer to roughness parameter $a=0.1$. \n}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.9\\columnwidth]{fig6.pdf}\n\\caption{\nEvolution of the total vortex length, $\\Lambda$ (solid lines), as well\nas the vortex length in the $z$-direction, $\\Lambda_z$ (dashed lines) and\nthe vortex length in the $xy$-plane, $\\Lambda_{xy}$ (dot-dashed lines), \nplotted versus time $t$ for different boundary roughness\n$a=0.05$ (red), $0.1$ (black), $0.2$ (blue), and $0.3~\\rm \\xi$ (magenta),\nachieving final values $\\Lambda_{\\infty}=1899~\\xi$, $2184~\\xi$,\n$2161~\\xi$ and $1643~\\xi$. All curves refer to the same\nangular velocity $\\Omega=0.02~\\tau^{-1}$. \n}\n\\label{fig5}\n\\end{figure}\n\nThe collection of U-vortices nucleated at the boundary\nis the superfluid's analog of a boundary layer, the region separating the\nrotating boundary from the still quiescent bulk of the fluid.\nThe U-vortices tend to be aligned along the $z$-direction, creating\na superflow in the same direction of the rotating boundary.\nThe vortex nucleation is therefore short-lived, since the nucleated\nU-vortices reduce the relative motion between the fluid and the\nboundary, suppressing further nucleations.\nIn time, the U-vortices grow in size and extend further into the \nfluid [Fig. \\ref{fig2}(b,c)], ultimately filling the bulk  \n[Fig. \\ref{fig2}(d)].  During this stage of the evolution,\nthe U-vortices also grow in vertical extent in the $z$-direction, \noccasionally connecting and merging with each other, thus increasing\ntheir vertical extent.  When the length in the $z$-direction becomes of the order \nof the bucket's height $H$, one or both vortex endpoints start sliding along\nthe smooth top and\/or bottom of the bucket.\nOnce most of the vortex lines are fully extended from the top to the bottom\nof the bucket, they quickly drift into the \nbulk of the fluid.  Although the vortex lines are aligned along the direction of\nrotation, they remain highly excited and undergo reconnection events when\nthey collide with each other.  Over time they relax towards a regular \nconfiguration of straight vortices.  A small proportion of U-vortices\nremain attached to the side of the bucket for a longer period of time \n[Fig.~\\ref{fig2}(d)]; over a longer time they detach,\nand relax to the final lattice configuration.  Some of the vortex lines\nend up diagonally across the rest of the vortex lattice\n[Fig.~\\ref{fig2}(e)]: eventually they also relax to the final\nlattice configuration  \n[Fig. \\ref{fig2}(f)].  The vortex lattice is stationary in \nthe rotating frame, representing the lowest energy state of \nthe rotating superfluid.  In this final state, the coarse-grained \nfluid velocity approximates the solid-body result \n${\\bf v} = v_{\\theta}\\, {\\bf e_\\theta}= \\Omega \\, r \\, {\\bf e_\\theta}$, \nwhere ${\\bf e_\\phi}$ is the azimuthal unit vector, as shown in Fig.~\\ref{fig3};\nas expected, the agreement improves with increasing $\\Omega$, and there is\na vortex-free region near the boundary. \n\nOur 3D results are presented for a fixed bucket size due to computational constraints of simulating a larger system.  For a larger bucket we would expect qualitatively similar dynamics; indeed our 2D results in a larger bucket presented in Section IV C support this.  The most significant change under a larger bucket is more vortices in the final state (at a fixed rotation frequency) and as a result a better approximation to solid-body rotation.\n\n\n\n\\subsection{Role of angular velocity and roughness} \n\nTo analyse the vortex dynamics further it is useful to distinguish the total vortex length, \n$\\Lambda$, from the vortex length projected in the $z$-direction, \n$\\Lambda_z$, and the vortex length projected in the $xy$-plane, \n$\\Lambda_{xy}$. In the final vortex lattice all vortex lines \nare aligned along $z$, hence we expect that, after a sufficiently long time, $\\Lambda_{xy}\\approx 0$ and\n$\\Lambda_z \\approx \\Lambda$, with $\\Lambda \\rightarrow N_v H$, where $N_v$ is the final number of straight vortex lines.  \nFigure~\\ref{fig4} displays $\\Lambda$ (solid lines), $\\Lambda_z$ (dashed lines)\nand $\\Lambda_{xy}$ (dot-dashed lines) as a function of time for different\nangular velocities of rotation, $\\Omega=0.02$, $0.04$ and $0.06$\nat the same roughness parameter $a=0.1$. \n It is apparent that in the initial stage, \na great amount of vorticity is in the $xy$-plane, before realignment \nof the vortex lines along the $z$-axis of rotation takes place. The effect\nis particularly noticeable at the largest angular velocities, for which,\nduring the initial transient, the vortex length is considerably larger\nthan the value $\\Lambda_{\\infty}$ achieved in the final vortex lattice\nconfiguration. Moreover, we see that the final vortex line length increases with $\\Omega$ due to the increasing number of vortices in the final state. \n\nFigure~\\ref{fig5} shows $\\Lambda$, $\\Lambda_z$ and $\\Lambda_{xy}$ \nplotted versus time at the same angular velocity $\\Omega=0.02~\\tau^{-1}$ \nfor different values of \nroughness parameter $a$. The largest values of the final vortex\nlength $\\Lambda_{\\infty}$ are achieved with $a=0.1 \\xi$ and $a=0.2\\xi$. \nSmoother ($a=0.05\\xi$) and rougher ($a=0.3\\xi$) boundaries generate \nless vortex length. These variations in the final line length arise to the final number of vortex lines varying by a few vortices across these cases.\nIt is not surprising that the final vortex lattice depends\non the roughness which has nucleated the initial \nvorticity. Feynman's rule [Eq.~\\eqref{eq:Feynman}] only refers to \nan idealised homogeneous system. Boundaries are known to have effects \n(e.g. missing vortex lines near the boundary) and \nit has been observed that the formation of the vortex lattice may be \nhistory-dependent and involve\nmetastability \\cite{CampbellZiff,Wood2019} and hysteresis \\cite{Mathieu}. \n\nFigure~\\ref{fig_Lz_total} compares the growth of angular momentum between the default case (blue line), the case where the rotation frequency is doubled (red line), and the case where the roughness amplitude is doubled (yellow line).  The growth behaviour is qualitatively similar in all cases.  Doubling the rotation frequency leads to a much faster rate of injection of angular momentum, and a higher final value, consistent with the faster injection rate of vortex lines from the boundary and the higher density of vortex lines in the final lattice state.  Doubling the surface roughness has little effect on the growth of the angular momentum, just slightly increasing the rate of angular momentum injection, which can be attributed to the greater injection rate of vortices from the rougher surface.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.8\\columnwidth]{fig7.pdf}\\\\\n\\caption{\nEvolution of the total angular momentum for the default parameters used in Fig. \\ref{fig2} (blue line), double the rotation frequency (red line) and double the roughness amplitude (yellow line).  \n}\n\\label{fig_Lz_total}\n\\end{figure}\n\n\\begin{figure*}\n(a) \\hspace{3.8cm} (b) \\hspace{4cm} (c) \\hspace{7cm} \\\\\n\t\\includegraphics[width = 0.24\\textwidth]{fig8a.pdf}\n        \\includegraphics[width = 0.24\\textwidth]{fig8b.pdf}\n\t\\includegraphics[width = 0.24\\textwidth]{fig8c.pdf}\n\\\\(d) \\hspace{4cm} (e) \\hspace{3.8cm} (f) \\hspace{3.8cm} \\\\\n\t\\includegraphics[width = 0.24\\textwidth]{fig8d.pdf}\n\t\\includegraphics[width = 0.24\\textwidth]{fig8e.pdf}\n\t\\includegraphics[width = 0.24\\textwidth]{fig8f.pdf}\t\n\\caption{\nDensity profile in the $xy$-plane at the half-height of the bucket, showing the final vortex configurations for the following cases:\n(a): $\\Omega=0.02~\\tau^{-1}$ and $a=0.1$ (default choice);\n(b): $\\Omega=0.04~\\tau^{-1}$ and $a=0.1$ (double rotation);\n(c): $\\Omega=0.06~\\tau^{-1}$ and $a=0.1$ (triple rotation);\n(d): $\\Omega=0.02~\\tau^{-1}$ and $a=0.05$ (half roughness);\n(e): $\\Omega=0.02~\\tau^{-1}$ and $a=0.2$ (double roughness);\n(f): $\\Omega=0.02~\\tau^{-1}$ and $a=0.1$ (single strong protuberance added,\nsee Section~\\ref{section4}).  \n}\n\\label{fig6}\n\\end{figure*} \n\n\nFigure~\\ref{fig6} illustrates some of the final vortex patterns which we have\ncomputed by plotting the superfluid density,\n$\\vert \\psi(x,y) \\vert^2$ in the $xy$-plane at\nhalf-height of the bucket.  In these pictures\nthe vortices appear as small holes; to clarify the lengthscales,\nwe recall that on the vortex axis the density is zero and that\nat distance $r=2 \\xi$ from the axis, the density recovers about $80 \\%$ of the\nbulk value at infinity.\nIt is interesting\nto compare the different final vortex configurations for halved\/doubled rotation velocity and the roughness parameter with respect\nto our default choice ($\\Omega=0.02~\\tau^{-1}$ and $a=0.1$).  While the ideal\n2D vortex lattice has a vortex at the centre, surrounded by a first row of 6 vortices,\na second row of 12 vortices, etc, the vortex configurations shown\nin Fig.~\\ref{fig6} contain slightly different vortex numbers; in\nparticular some configurations contain vortex lines which seem \nmisplaced [Fig.~\\ref{fig6}(c)]\nor lack the vortex at the centre [Fig.~\\ref{fig6}(e)];\nthese configurations are metastable states corresponding to local \nminima of the free energy in the rotating frame \\cite{CampbellZiff}. \nMoreover, at slow rotations [Fig.~\\ref{fig6}(a,d)] the predicted vortex-free region near the boundary \n\\cite{NorthbyDonnelly1970,ShenkMehl1971,StaufferFetter1968} is clearly\nvisible; this phenomenon affects the coarse-grained azimuthal velocity near the\nboundary shown previously in Fig.~\\ref{fig3}(a). The depletion of the background fluid density in the centre of the bucket - particularly evident in Fig. \\ref{fig6}(b) and (c) - is due to coarse-grained centrifugal effects, analogous to the classical rotating case \\cite{Barenghi2016}.  \n\n\n\n\n\n\n\n\n\\section{Other effects}\n\\label{section4}\n\nIn this section we\nrepeat the simulation of Section~\\ref{section3} with several \nsignificantly modifications: the presence of a single strong\nprotuberance, the presence of remanent vortex lines, and the 2D case.\nThe aim is to determine whether these effects change qualitatively the dynamics \ndescribed in Section~\\ref{section3}.\n\n\\subsection{Effect of a strong protubance}\n\n\\begin{figure*}\n\\hspace{-0.2cm}  (a) \\hspace{2.9cm} (b) \\hspace{2.9cm}(c) \\hspace{2.9cm} (d) \\hspace{2.9cm} (e) \\hspace{1cm}\\\\\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9a.pdf}\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9b.pdf}\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9c.pdf}\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9d.pdf}\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9e.pdf}\t\n\\caption{Early-time dynamics during the spin-up of the fluid in the \npresence of a single strong protuberance added to the rough\ncylindrical boundary. The snapshots,  taken at \n$t\/\\tau=26$, $50$, $100$, $200$ and $500$, are presented in the same way \nas Fig.~\\ref{fig2}. }\n\\label{fig7}\n\\end{figure*}\n\n\nFirst we consider the effect of a single strong imperfection  in the\nform of a protuberance on the cylindrical wall. The question is\nwhether, by enhancing vortex nucleation, the protuberance \ncan induce a turbulent boundary layer.\nThe protuberance is numerically created by adding a Gaussian-shaped \npotential to the existing (small-scale) roughness potential. \nEquation~\\eqref{eq:rbucket} is replaced by\n\n\n\\begin{equation}\nr(z,\\phi)=R[1-a (f(z,\\theta) + G f_G(z,\\theta))],\n\\label{eq:rbucket2}\n\\end{equation}\n\n\\noindent\nwhere $G=2$ and $f_G(z,\\theta)$ is a Gaussian-shape function taking values\nfrom $0$ to $1$ and rms width $4 \\xi$. The approximate height of the \nstrong protuberance \nin the simulation which we present is $10 \\xi$, as also visible\nin Fig.~\\ref{fig6}(f).\n\n\\noindent\nSnapshots taken during the time evolution for $\\Omega=0.02~\\tau^{-1}$\nand $a=0.1$ are shown in Fig.~\\ref{fig7}; a movie can be viewed \nin Supplementary Material  \\cite{movie_gaussian}.\nThe protuberance catalyses the local nucleation of vortices \nat early times: large vortex loops \n(of the same size order as the protuberance) are rapidly generated\n[Fig.~\\ref{fig7}(a)], leading to a downstream trail of \nloops [Fig.~\\ref{fig7}(b, c)], in addition to the slower \nnucleation of U-vortices from the rough bucket wall. \nThe vortex configuration becomes\nclearly anisotropic near the bucket edge [Fig.~\\ref{fig7}(d)].  \nHowever, once the vortices fill the bulk [Fig.~\\ref{fig7}(e)],\nmemory of this effect is lost, and the subsequent evolution is\nvery similar to the evolution without the strong protuberance.\nIn fact, the final vortex lattice is not significantly different from the\nlattices considered in Section~\\ref{section3}, as shown in Fig.~\\ref{fig6}(f). Figure~\\ref{fig10} shows the time evolution of $\\Lambda$,\n$\\Lambda_z$ and $\\Lambda_{xy}$ in the presence of the protuberance (magenta lines) and its absence (black lines).  This confirms that the protuberance accelerates the generation of vortex line length at early times, but that its effect becomes washed out at later times.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width = 0.45\\textwidth]{fig10.pdf}\n\\caption{\nTime evolution of $\\Lambda$ (solid line), $\\Lambda_z$ (dashed line), \n$\\Lambda_{xy}$ (dot-dashed line). The colours correspond to the \nsimulations with default parameters $\\Omega=0.02 \\tau^{-1}$,\n$a=0.1 \\xi$ (black),\nthe added surface protuberance (magenta),\nthe added remanent negative vortex (blue), and\nthe added remanent positive vortex (red), respectively.\n}\n\\label{fig10}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Effect of remanent vortices}\n\nSecondly, we consider the effect of remanent vortex lines.\nIn experiments with liquid helium, it is believed that so-called \n`remanent vortices' may be present in the fluid, created via the \nKibble-Zurek mechanism when cooling the helium sample through the \nsuperfluid transition to the final experimental temperature.  \nThe presence of remanent vortices may modify the vortex nucleation and the\nformation of the vortex lattice when the sample is rotated.\nTo explore this idea, we have repeated the simulations imposing\na suitable phase profile to add a vortex to the initial state during the \nimaginary-time propagation.  \nFor simplicity we position the remanent vortex along the $z$-axis \nof rotation. \n\nThe evolution of the superfluid with the standard rough cylindrical\nwall and a ``positive\" remanent vortex, that is, one whose circulation is oriented in the same direction of the\nbucket's rotation is shown through Fig.~\\ref{fig8} \nand the movie in the Supplementary Material \\cite{movie_pos_remnant}. \nCompared to Section~\\ref{section3},\nthe only significant modification is a dampening of the initial injection \nof U-vortices; the effect is visible by eye when comparing like-time \nsnapshots [Fig. \\ref{fig2}(b) and Fig.~\\ref{fig8}(a)].  \nThe remanent vortex acts in the same direction as the rotating container: \nit reduces the relative speed between the bucket's wall and the superfluid,\nand remains largely undisturbed at early times \n[Fig.~\\ref{fig8}(a)] until the \nU-vortices that are nucleated fill the bulk and interact \nwith it [Fig.~\\ref{fig8}(b)]; at this point\nthe remanent vortex becomes subsumed within the other like-signed \nvortices [Fig.~\\ref{fig8}(c)], and the \nsubsequent relaxation of the vortex configuration into a vortex\nlattice largely proceeds as if there was not any remanent vortex initially. Confirming this, we see that in Figure~\\ref{fig10} that the presence of the positive vortex (red lines) depletes the generation of vortex line length at early times, but this recovers at later times such that the system reaches the same line length as in the absence of any remanent vortices (black lines).\n\n  \n\\begin{figure*}\n\\hspace{-2.2cm} (a) \\hspace{3.7cm} (b) \\hspace{3.9cm} (c)\\\\\n\\includegraphics[width = 0.22\\textwidth]{fig11a.pdf}\n\\includegraphics[width = 0.22\\textwidth]{fig11b.pdf}\n\\includegraphics[width = 0.22\\textwidth]{fig11c.pdf}\n\\caption{Spin-up of the superfluid in the presence of a positively-charged \nremanent vortex.  Snapshots are taken at $t\/\\tau=200$, $400$ and $500$, \nand are presented in the same way as Fig.~\\ref{fig2}.}\n\\label{fig8}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\hspace{0.2cm} (a) \\hspace{3.7cm} (b) \\hspace{3.2cm} (c)\n\\hspace{0.2cm} (d) \\hspace{3.7cm} (e) \\hspace{3.2cm} \n\\includegraphics[width = 0.19\\textwidth]{fig12a.pdf}\n\\includegraphics[width = 0.19\\textwidth]{fig12b.pdf}\n\\includegraphics[width = 0.19\\textwidth]{fig12c.pdf}\n\\includegraphics[width = 0.19\\textwidth]{fig12d.pdf}\n\\includegraphics[width = 0.19\\textwidth]{fig12e.pdf}\n\\caption{Spin-up of the superfluid in the presence of a negatively-charged \nremanent vortex.  \nSnapshots are taken at $t\/\\tau=200$ (a),  $360$ (b), $386$ (c), $420$ (d)\nand $500$ (e). The images are presented in the same way as Fig.~\\ref{fig2}.  \nVortices with negative circulation are coloured blue.}\n\\label{fig9}\n\\end{figure*}\n\n\n\n\n\nIf the remanent vortex is oriented in the direction opposite to the\nrotation of the bucket, i.e. a negative vortex, the evolution proceeds differently, as seen in Fig.~\\ref{fig9} and the movie in Supplementary Material\n\\cite{movie_neg_remnant}.  \nThe remanent vortex enhances the nucleation of U-vortices\nfrom the boundary, as evident from comparing \nFig.~\\ref{fig2}(b) and Fig.~\\ref{fig9}(a).  \nThis effect is caused by the counter-flow induced by the remanent\nvortex, which increases the relative speed of the fluid over the rough \nboundary. \nOnce the other vortices drift close to the remanent vortex, the \nremanent vortex becomes excited by their interaction\n[Fig.~\\ref{fig9}(b)].  \nA series of vortex reconnections break up the remanent vortex, \nforming progressively smaller vortex loops \n[Fig.~\\ref{fig9}(c,d)].  This leads to the rapid \nremoval of vorticity of the `wrong' sign from the fluid\n [Fig.~\\ref{fig9}(e)].  \nHereafter the fluid evolves in a similar manner to when the remanent vortex in absent [Section~\\ref{section3}], albeit with a slightly higher final vortex line length [Fig.~\\ref{fig10}].\n\n\n\\subsection{2D case}\n\nFinally, we have also performed the corresponding 2D simulations of the spin-up of a 2D superfluid within a rough circular boundary; the boundary is taken from the central slice of the 3D rough bucket.  A movie showing the typical dynamics\nis available in the Supplementary Material \\cite{movie_2d_large_bucket}.\nThe 2D geometry\nallows calculations of much larger buckets, up to $R=200 \\xi$ with\na $1024^2$ numerical grid.  We observe the same qualitative behaviour as in\n3D in smaller buckets, albeit with many more vortices and without 3D effects\nsuch as vortex reconnections. Collisions of vortices of the opposite\ncirculation result in the annihilation of the vortices and the emission of\nsound pulses \\cite{Kwon2014,Stagg2015,Groszek2016}.\nIn general, we find that, in 2D, the timescales of injection, diffusion and \nlattice crystallisation are faster than in 3D.   \nA particular feature that we see in the early-time dynamics of\nthe 2D simulations is the nucleation of vortices with both positive \nand negative circulation  (i.e. with circulation which is inconsistent\nwith the imposed rotation).  We notice that some negative vortices \noriginate from localised rarefaction pulses generated \nfrom the rough boundary when the bucket is set into rotation. \nWe associate these pulses with Jones-Roberts solitons \n\\cite{Jones1982,Tsuchiya2008}, which are low energy\/momentum \nsolutions of the 2D GPE.  \nAt higher energy\/momentum, these solutions become pairs of vortices \nof opposite sign (also called vortex dipoles in the literature).\nThe conversion of Jones-Roberts solitons into vortex dipoles occurs\nif the pulse gains energy from the large positive vortex cluster\nwhich starts forming in the centre of the bucket. Occasionally,\nthe vortices which are parts of a dipole separate\nand mix with the rest of the vortices.  Over time, the vortices \nof negative circulation are lost from the system, either \ncolliding (hence annihilating) with positive vortices within the bulk,\nor by exiting the fluid at the bucket's boundary (effectively\nannihilating with their images).  \n\\vspace{0.3cm}\n\n\\section{Conclusions}\n\\label{section5}\n\nIn conclusion, we have employed simulations of the Gross-Pitaevskii equation to study the spin-up of a superfluid in a rotating bucket featuring microscopically rough walls.   Within this model, we see several key stages of the dynamics.  Firstly, vortices are nucleated at the boundary by the flow over the rough features, typically in the form of small U-shaped vortex lines.  Secondly, these U-shaped vortices interact strongly and reconnect, creating\na transient turbulent state.  This becomes increasingly polarised by the imposed rotation until the vortex configuration consists of vortices of the correct orientation extending from the top to the bottom of\nthe bucket.  Finally, the vortex lines slowly straighten and arrange themselves in the expected final vortex lattice configuration. Our results highlight the importance of vortex \nreconnections \\cite{Galantucci2019}: it is\ngenerally assumed that vortex reconnections are important in turbulence,\nbut here we have seen that reconnections are essential to create, starting\nfrom potential flow, something as simple as solid body rotation\n(the vortex lattice). \nThe addition\nof a single large protuberance or one additional remanent vortex line does not\nchange the dynamics significantly, only speeding up or slowing down the injection of vorticity.  Moreover, analogous dynamics  arise in the 2D limit. \n\nWe reiterate that the GPE is not a quantitatvely accurate model of superfluid helium and these results should be interpreted qualitatively only.   For example, the role of friction is introduced into the GPE through a widely-used phenomenological dissipation term; however, a more accurate physical model of this stage of the dynamics would be provided by the VFM.  Also, a distinctive physical property of superfluid helium is its strong non-local interactions.  This, for instance, supports a roton minimum in its excitation spectrum.  While this is absent from the GPE model we have employed, it can be introduced through an additional non-local term \\cite{Berloff2014,Reneuve2018}.  It would be interesting to see if this causes any significant departures from the dynamics we have reported.  \n\n\n\n\n\\section*{Acknowledgements}\n\nN.P., L. G. and C.F.B. acknowledge support by the Engineering and Physical \nSciences Research Council (Grant No. EP\/R005192\/1).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Backpropagation through time}\n\nThere are many approaches to training recurrent networks~\\cite{A1987,P1987},\nand the most successful one by far is backpropagation\nthrough time \\cite{W1990}.\nBackpropagation through time (BPTT) is an approach to compute\nthe gradient of a recurrent network when consuming a sequence.\nA computation graph is created based on the decoding approaches,\nand the gradients are propagated back through the computation\nvertices.  It is equivalent to unrolling the recurrent networks\nfor several time steps depending on the decoding approaches;\nhence the name backpropagation through time.\n\nThere are many variants of BPTT with the most popular one\nbeing truncated BPTT \\cite{WP1990}.  In the original\ndefinition \\cite{WP1990}, instead of propagating\ngradients all the way to the start of the unrolled chain,\nthe accumulation stops after a fixed number of time steps.\nTraining recurrent networks with truncated BPTT can be justified\nif the truncated chains are enough to learn\nthe target recursive functions.\nIn the modern setting~\\cite{SSEP2014},\ntruncated BPTT is treated as regular BPTT with batch decoding,\nand the number of unrolled steps before truncation is\nthe number of context frames in batch decoding.\nHenceforth, we will use the term BPTT with batch decoding\nto avoid confusion.\n\nBPTT with batch decoding is seldom used in practice\ndue to the high computation cost,\nwith the only exception being~\\cite{SSEP2014}\nwhere the context is set to six frames.\nA more common approach used in conjunction with BPTT and batch decoding\nis to predict a batch of frames rather than a single one \\cite{SSB2014, CYH2015}.\nFormally, for a single chain of recurrent network to\npredict $p$ consecutive frames at time $t$, we have\n$\\hat{y}_{t+i-1} = o(h_{t+\\ell+i-2})$ for $i = 1, \\dots, p$\nwhere $\\ell$ is the number of lookahead.\nNote that this decoding approach is only used for training\nand is never actually used at test time.\nFigure~\\ref{fig:online-batch} summarizes\nthe decoding approaches and the hyperparameters, including\nlookahead, context frames, and consecutive prediction.\n\nIn practice, recurrent networks are trained with a combination\nof BPTT, batch decoding, lookahead, and consecutive prediction \\cite{SSB2014, CYH2015}.\nTo speed up training, no frames in an utterance are predicted more than once,\nand many sequences are processed in batches to better\nutilize parallel computation on GPUs.\nIn addition, the hidden vectors are sometimes cached \\cite{SSEP2014}.\nApplying these heuristics, however, creates a mismatch\nbetween training and testing.\nPrevious work has not addressed this issue,\nand the distinction between online and batch decoding\nunder BPTT has only been lightly explored in~\\cite{SSEP2014, KTLK2015}.\nWe will examine these in detail in our experiments.\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tikzpicture}\n\\node (x1) at (0, 0) {$x_1$} ;\n\\node (x2) at (1, 0) {$x_2$} ;\n\\node (xdots) at (2, 0) {$\\cdots$} ;\n\\node (xT) at (3, 0) {$x_T$} ;\n\n\\node (h1) at (0, 1) {$h_1$} ;\n\\node (h2) at (1, 1) {$h_2$} ;\n\\node (hdots) at (2, 1) {$\\cdots$} ;\n\\node (hT) at (3, 1) {$h_T$} ;\n\n\\node (y1) at (0, 2) {$y_1$} ;\n\\node (y2) at (1, 2) {$y_2$} ;\n\\node (ydots) at (2, 2) {$\\cdots$} ;\n\\node (yT) at (3, 2) {$y_T$} ;\n\n\\draw[->] (x1) -- (h1);\n\\draw[->] (x2) -- (h2);\n\\draw[->] (xT) -- (hT);\n\n\\draw[->] (h1) -- (h2);\n\\draw[->] (h2) -- (hdots);\n\\draw[->] (hdots) -- (hT);\n\n\\draw[->] (h1) -- (y1);\n\\draw[->] (h2) -- (y2);\n\\draw[->] (hT) -- (yT);\n\n\\node (x2T3) at (6, 0) {$x_{t-3}$} ;\n\\node (x2T2) at (7.5, 0) {$x_{t-2}$} ;\n\\node (x2T1) at (9, 0) {$x_{t-1}$} ;\n\\node (x2T) at (10.5, 0) {$x_t$} ;\n\\node (x2Tp1) at (12, 0) {$x_{t+1}$} ;\n\\node (x2Tp2) at (13.5, 0) {$x_{t+2}$} ;\n\\node (x2Tp3) at (15, 0) {$x_{t+3}$} ;\n\n\\node (h2T3) at (6, 1) {$h_{t-3}$} ;\n\\node (h2T2) at (7.5, 1) {$h_{t-2}$} ;\n\\node (h2T1) at (9, 1) {$h_{t-1}$} ;\n\\node (h2T) at (10.5, 1) {$h_t$} ;\n\\node (h2Tp1) at (12, 1) {$h_{t+1}$} ;\n\\node (h2Tp2) at (13.5, 1) {$h_{t+2}$} ;\n\\node (h2Tp3) at (15, 1) {$h_{t+3}$} ;\n\n\\node (y2T) at (10.5, 2) {$y_t$} ;\n\\node (y2Tp1) at (12, 2) {$y_{t+1}$} ;\n\\node (y2Tp2) at (13.5, 2) {$y_{t+2}$} ;\n\n\\draw[->] (h2T2) -- (h2T1);\n\\draw[->] (h2T1) -- (h2T);\n\\draw[->] (h2T) -- (h2Tp1);\n\\draw[->] (h2Tp1) -- (h2Tp2);\n\\draw[->] (h2Tp2) -- (h2Tp3);\n\n\\draw[->] (x2T2) -- (h2T2);\n\\draw[->] (x2T1) -- (h2T1);\n\\draw[->] (x2T) -- (h2T);\n\\draw[->] (x2Tp1) -- (h2Tp1);\n\\draw[->] (x2Tp2) -- (h2Tp2);\n\\draw[->] (x2Tp3) -- (h2Tp3);\n\n\\draw[->] (h2Tp1) -- (y2T);\n\\draw[->] (h2Tp2) -- (y2Tp1);\n\\draw[->] (h2Tp3) -- (y2Tp2);\n\n\\end{tikzpicture}\n\\caption{\\emph{Left}: Online decoding for $(x_1, \\dots, x_T)$.\n    \\emph{Right}: One chain of batch decoding\n    at time $t$ with 6 context frames, a lookahead of 2 frames, and consecutive prediction\n    of 3 frames.  The chains are repeated for $t = 1, \\dots, T$.}\n\\label{fig:online-batch}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure*}\n\n\\section{Conclusion}\n\nWe study unidirectional recurrent networks, LSTMs in particular,\ntrained with two decoding approaches, online and batch\ndecoding, and explore various hyperparameters,\nsuch as lookahead, context frames, and consecutive prediction.\nOnline decoding can be as broad as recursive functions,\nwhile batch decoding can only include Markov functions.\nTraining LSTMs with the matching decoding approaches\nperforms best.  LSTMs trained with online decoding\ncan still have decent performance with batch decoding,\nbut LSTMs trained with batch decoding tend\nnot to perform well with online decoding.\nThe amount of lookahead is also critical\nfor LSTMs with online decoding to get better\ntraining errors.  The number of context frames\nstrictly limits the history that can be used\nfor prediction, while increasing the amount of consecutive\nprediction gears LSTMs closer to recursive functions.\nThe results depend on the long-term dependency in the data,\nand we confirm this by exploring subphonetic states, phonemes, and words.\nFinally, we show that the vanishing gradient phenomenon\nis a necessary condition for Markov functions\nwith respect to variables beyond the necessary history.\nIt is important to note that the same LSTM architecture\nand the same number of model parameters can\nhave drastically different results and behaviors.\nWe hope the results improve the understanding\nof recurrent networks, what they learn from\nthe data, and ultimately the understanding\nof speech data itself.\n\n\\section{Experiments}\n\nTo study how well recurrent networks model Markov\nand recursive functions, we conduct experiments on\nframe classification tasks with three different\nlinguistic units, i.e., subphonetic states, phonemes,\nand words.  We expect that subphonetic states and phonemes are\nrelatively local in time, while words require\nlonger contexts to predict.  By varying the target units, we control\nthe amount of long-term dependency in the data.\n\nExperiments are conducted on the Wall Street Journal data set\n(\\textfw{WSJ0} and \\textfw{WSJ1}), consisting of about 80 hours of\nread speech and about 20,000 unique words,\nsuitable for studying rich long-term dependency.\nWe use 90\\% of \\textfw{si284}\nfor training, 10\\% of \\textfw{si284} for development,\nand evaluate the models on \\textfw{dev93}.\nThe set \\textfw{dev93} is chosen because\nit is the only set where frame error rates are\nreported for deep networks \\cite{GJM2013}.\nThe time-aligned frame labels, including\nsubphonetic states, phonemes, and words,\nare obtained from speaker-adaptive hidden Markov models\nfollowing the Kaldi recipe~\\cite{P+2011}.\n\nIn the following experiments, we use, as input to the frame classifiers,\n80-dimensional log Mel features without appending i-vectors.\nFor recurrent networks, we use 3-layer LSTMs with 512 hidden units\nin each layer.  In addition, for the baseline we use\na 7-layer time-delay neural network (TDNN)\nwith 512 hidden units in each layer and an architecture\ndescribed in~\\cite{THGG2018,PWPK2018}.\nThe network parameters are initialized based on~\\cite{HZRS2015}.\nThe networks are trained with vanilla stochastic gradient descent\nwith step size 0.05 for 20 epochs.\nThe batch size is one utterance and the gradients are clipped\nto norm 5.  The best performing model is chosen based on\nthe frame error rates (FER) on the development set.\nUtterances are padded with zeros when lookahead or\ncontext frames outside the boundaries is queried.\n\n\\subsection{Long-term dependency in subphonetic states}\n\nWe first experiment with recurrent networks on subphonetic states.\nWe expect LSTMs to perform best when trained and\ntested with online decoding, so we first explore\nthe effect of lookahead.\nResults are shown in Table~\\ref{tbl:state-lookahead}.\nLookahead has a significant effect on the frame error rates.\nThe error rate improves as we increase the amount of lookahead\nand plateaus after 10 frames.\nThe improvement is due to better training error (not regularization or other factors),\nas shown in Figure~\\ref{fig:train}.\nIn addition, the fact that increasing the amount of lookahead does\nnot hurt performance suggests that LSTMs are able\nto retain information for at least 20 frames.\nCompared to prior work, our 3-layer LSTM is unidirectional\nand uses fewer layers than the ones used in~\\cite{GJM2013},\nbut the results are on par with theirs.\nThe best LSTM achieves 11.7\\% word error rate on \\textfw{dev93},\nsimilar to the results in~\\cite{GJM2013}.\\footnote{\nWe are aware of the state of the art on this data set \\cite{CL2015}.\nAs the results are in the ballpark,\nwe do not optimize them further.}\nThe TDNN serves as an instance of a Markov function.\nWe present its result, but more investigations are needed\nto conclude anything from it.\n\n\\begin{table}[t]\n\\caption{FERs (\\%) comparing different LSTM lookaheads\n    trained with online decoding, on subphonetic states.\n    The FER of a 7-layer TDNN is provided as a reference.}\n\\label{tbl:state-lookahead}\n\\begin{center}\n\\begin{tabular}{lll}\nlookahead & dev       & \\textfw{dev93}   \\\\\n\\hline \n1         & 45.46     & 43.45   \\\\\n5         & 33.64     & 32.43   \\\\\n10        & 30.43     & 29.83   \\\\\n15        & 29.89     & 28.69   \\\\\n20        & 29.36     & 28.59   \\\\\n\\hline\n\\hline\nTDNN 512x7\n          & 33.56     & 34.19\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=5cm]{lookahead}\n\\caption{Running average of training losses\n    for comparing different amount of lookahead for LSTMs\n    with online decoding trained on subphonetic states.}\n\\label{fig:train}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\n\nTo see how decoding approaches affect performance,\nwe examine the error rates with batch decoding \non the best LSTM in Table~\\ref{tbl:state-lookahead},\ni.e., the one trained with online decoding\nand a lookahead of 20 frames.  Results are shown in\nTable~\\ref{tbl:state-batch}.  The performance\ndeteriorates as we reduce the number of context frames.\nThis suggests that LSTMs do utilize information\n40 frames away.  On the other hand, the degradation\nis not severe, suggesting that much of\nthe long-term dependency is Markov.\n\n\\begin{table}[t]\n\\caption{FERs (\\%) comparing different context frames\n    under batch decoding\n    with the best LSTM trained with online decoding,\n    and a lookahead of 20 frames on subphonetic states.}\n\\label{tbl:state-batch}\n\\begin{center}\n\\begin{tabular}{lll}\ncontext & dev       & \\textfw{dev93}   \\\\\n\\hline\n40      & 37.65     & 36.45   \\\\\n35      & 41.73     & 40.12   \\\\\n30      & 48.79     & 46.70   \\\\\n\\hline\n\\hline\nonline  & 29.36     & 28.59\n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\nWe then examine whether LSTMs with\nbatch decoding can learn recursive functions.\nThe same LSTMs are trained with batch decoding\nof different context frames.\nConsecutive prediction is not used in this setting.\nIn other words, for an utterance of $T$ frames,\nwe create $T$ chains of LSTMs,\neach of which predicts the label of a frame.\nResults are shown in Table~\\ref{tbl:state-tbptt}.\nWith a context of 40 frames, the LSTM trained\nwith batch decoding is only slightly behind the LSTM trained\nwith online decoding.  This again suggests\nthat the class of Markov functions can perform reasonably well,\nand much of the long-term dependency is likely Markov.\nHowever, the error rate degrades significantly\nwhen we switch from batch decoding to online decoding.\nThis strongly suggests that these LSTMs do not\nbehave like recursive functions.\n\n\\begin{table}[t]\n\\caption{FERs (\\%) on \\textfw{dev93} comparing online and batch decoding\n    for LSTMs trained with batch decoding,\n    and a lookahead of 20 frames on subphonetic states.}\n\\label{tbl:state-tbptt}\n\\begin{center}\n\\begin{tabular}{lll}\ncontext & batch   & online \\\\\n\\hline\n40      & 31.30   & 80.29  \\\\\n35      & 30.99   & 84.78  \\\\\n30      & 32.80   & 85.74       \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\nTo understand what makes LSTMs\nbehave like recursive functions,\nwe train LSTMs with batch decoding\nand increase the amount of consecutive\nprediction.  To simplify the setting,\nwe allow each frame to be predicted multiple\ntimes.  In fact, if a network predicts\nconsecutively for $p$ frames,\nthen each frame gets predicted $p$ times.\nResults are shown in Table~\\ref{tbl:state-pred}.\nAs we increase the number of consecutively predicted frames,\nthe error rate for batch decoding\nstays about the same, while the one for online\ndecoding improves.  This suggests that\nit is the amount of consecutive prediction that gears\nthe behavior of LSTMs towards recursive functions.\nIn addition, as seen in Table~\\ref{tbl:state-batch},\nLSTMs can achieve reasonable performance\nwith both online and batch decoding.\nPerhaps the data is a complex mix of long-term\ndependency, or perhaps the LSTM learns to\nforget the history.  More analyses are needed\nto tease the factors apart.\n\n\\begin{table}[t]\n\\caption{FERs (\\%) on \\textfw{dev93} comparing different\n    numbers of consecutive prediction\n    for LSTMs trained with batch decoding,\n    and a lookahead of 20 frames on subphonetic states.}\n\\label{tbl:state-pred}\n\\begin{center}\n\\begin{tabular}{llll}\n\\# of prediction & batch   & online \\\\\n\\hline \n1                & 31.30   & 80.29  \\\\\n5                & 31.29   & 80.21  \\\\\n10               & 31.65   & 46.82  \\\\\n15               & 32.27   & 33.03  \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\\subsection{Long-term dependency in phonemes and words}\n\nWe repeat the same experiments with phonemes and words.\nNote that the number of classes in the case of words\nis the number of unique words in the training set.\nWe do not handle out-of-vocabulary words (OOV).\n\nFor the lookahead experiments in Table~\\ref{tbl:phone-word-lookahead},\nthe general trend is similar to that in Table~\\ref{tbl:state-lookahead},\nexcept that the frame error rate plateaus at around 15 frames\nfor words, longer than the 10 frames in subphonetic states\nand phonemes.\n\nThe fact that the TDNN achieves\na low frame error rate for phonemes in Table~\\ref{tbl:phone-word-lookahead}\nsuggests that much phonetic information\nis concentrated within a 30-frame window.\nHowever, for words we expect a larger context\nwindow to predict words, so we evaluate a 10-layer\nTDNN with an effective context window of 48 frames.\nThe 10-layer TDNN performs better than the 7-layer\none, but is still behind LSTMs.\n\nThe conclusion in Table~\\ref{tbl:phone-word-batch}\nis the same as in Table~\\ref{tbl:state-batch},\ni.e., for LSTMs trained with online decoding,\nreducing the amount of context during batch decoding\nhurts the accuracy.  However, the conclusion\nin Table~\\ref{tbl:phone-word-tbptt} is different\nfrom that in Table~\\ref{tbl:state-tbptt}.\nThe LSTMs with batch decoding are\nable to perform well with online decoding\non phonemes but not on words.\nWe suspect that phonemes are more local than subphonetic states,\nand in general the results are affected by the choice of\nlinguistic units.\n\nIn Table~\\ref{tbl:phone-word-pred}, we observe a similar trend to\nthat in Table~\\ref{tbl:state-pred}, adding consecutive prediction\nimproves the LSTMs' performance with online decoding.\nIn fact, the performance with online decoding,\ncompared to that in Table~\\ref{tbl:phone-word-lookahead},\nis fully recovered when using 15 frames of consecutive prediction.\n\n\n\\begin{table}[t]\n\\caption{FERs (\\%) comparing different lookaheads\n    for LSTMs trained with online decoding, on phonemes and words.\n    The FERs of a 7-layer TDNN and a 10-layer TDNN are provided as reference.}\n\\label{tbl:phone-word-lookahead}\n\\begin{center}\n\\begin{tabular}{l|ll|ll}\n          & \\multicolumn{2}{c}{phonemes} & \\multicolumn{2}{|c}{words} \\\\\nlookahead & dev    & \\textfw{dev93}  & dev    & \\textfw{dev93}   \\\\\n\\hline                                                          \n1         & 18.31  & 16.34           & 32.78  & 45.60            \\\\\n5         & 13.57  & 12.01           & 28.39  & 40.36            \\\\\n10        & 12.33  & 11.01           & 24.68  & 36.62            \\\\\n15        & 11.90  & 10.67           & 21.57  & 32.82            \\\\\n20        & 11.86  & 10.76           & 20.85  & 31.45            \\\\\n25        &        &                 & 21.02  & 30.43            \\\\\n\\hline\n\\hline\nTDNN 512x7\n          & 14.22  & 12.46           & 39.27  & 38.93            \\\\\nTDNN 512x10\n          &        &                 & 28.97  & 32.24            \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\\begin{table}[t]\n\\caption{FERs (\\%) for batch decoding with different context frames\n    with the best LSTM trained with online decoding,\n    and a lookahead of 20 frames on phonemes and words.}\n\\label{tbl:phone-word-batch}\n\\begin{center}\n\\begin{tabular}{l|ll|ll}\n        & \\multicolumn{2}{c}{phonemes} & \\multicolumn{2}{|c}{words} \\\\\ncontext & dev   & \\textfw{dev93} & dev   & \\textfw{dev93} \\\\\n\\hline                                                   \n40      & 16.74 & 14.99          & 48.39 & 44.84          \\\\\n35      & 19.28 & 17.10          & 53.33 & 48.55          \\\\\n30      & 23.26 & 20.59          & 58.78 & 53.00          \\\\\n\\hline                                                    \n\\hline                                                    \nonline  & 11.86 & 10.76          & 20.85 & 31.45          \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\\begin{table}[t]\n\\caption{FERs (\\%) on \\textfw{dev93} comparing online and batch decoding\n    for LSTMs trained with batch decoding,\n    and a lookahead of 20 frames on phonemes and words.}\n\\label{tbl:phone-word-tbptt}\n\\begin{center}\n\\begin{tabular}{l|ll|ll}\n         & \\multicolumn{2}{c}{phonemes} & \\multicolumn{2}{|c}{words} \\\\\ncontext  & batch & online & batch & online \\\\\n\\hline                   \n40       & 11.50 & 11.24  & 31.65 & 56.96  \\\\\n35       & 12.12 & 15.11  & 34.21 & 58.26  \\\\\n30       & 12.89 & 21.97  & 37.83 & 64.74  \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\\begin{table}[t]\n\\caption{FERs (\\%) on \\textfw{dev93} comparing different\n    numbers of consecutive prediction\n    for LSTMs trained with batch decoding,\n    and a lookahead of 20 frames on phonemes and words.}\n\\label{tbl:phone-word-pred}\n\\begin{center}\n\\begin{tabular}{l|ll|ll}\n                 & \\multicolumn{2}{c}{phonemes} & \\multicolumn{2}{|c}{words} \\\\\n\\# of prediction & batch & online & batch & online \\\\\n\\hline                            \n1                & 11.50 & 11.24  & 31.65 & 56.96  \\\\\n5                & 11.58 & 11.68  & 31.79 & 45.88  \\\\\n10               & 11.49 & 10.89  & 32.31 & 36.40  \\\\\n15               & 11.48 & 10.72  & 32.34 & 30.93  \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\n\\section{Introduction}\n\nRecurrent neural networks (RNNs) have been extensively used for speech\nand language processing, and are particularly successful at\nsequence prediction tasks,\nsuch as language modeling \\cite{MKS2018-1,MKS2018-2,KHQJ2018}\nand automatic speech recognition \\cite{SSEP2014,SSB2014,PWPK2018}.\nTheir success is often attributed to their ability to learn long-term dependencies\nin the data or, more generally, their ability to remember.\nHowever, exactly how far back recurrent networks can remember \\cite{WLS2004, KHQJ2018},\nwhether there is a limit \\cite{CSS2017}, and how the limit is characterized\nby the network architecture~\\cite{Z+2016} or by the optimization method ~\\cite{PMB2013},\nare still open questions.\nIn this paper, we examine how training and decoding approaches affect\nrecurrent networks' ability to learn long-term dependencies.\nIn particular, \nwe study the behavior of recurrent networks in the context of speech recognition\nwhen their ability to remember is constrained.\n\nThe ability to remember in recurrent networks\nis affected by many factors, such as the amount of long-term dependency\nin the data, as well as the training and decoding approaches.\nWe differentiate between two types of decoding, the online approach\nand the batch approach.  In the online case,\nthere is a single chain of recurrent network that predicts\nat all time steps, while in the batch case,\nmultiple chains of recurrent networks that start\nand end at different time points are used.\nIn the online case, a recurrent network resembles a recursive function\nwhere the prediction of the current time step depends on\nthe memory that encodes the entire history.  In the batch\ncase, a recurrent network resembles a fixed order Markov process,\nbecause the prediction at the current time step strictly depends\non a fixed number of time steps in the past.\nBy changing the decoding approach, we restrict\nthe ability of recurrent networks to model certain classes of functions.\n\nThe speech signal is particularly rich with long-term dependencies.\nDue to the causal nature of time, speech is assumed to be Markovian\n(though with a potentially high order).\nThe order of the Markov property depends on the time scale\nand the linguistic units, such as subphonetic states, phonemes, and words.\nThe Markov assumption has a strong influence in many design choices,\nsuch as the model family or the training approaches.\nPartly due to the Markov assumption and partly due to\ncomputational efficiency \\cite{KTLK2015,CYH2015}, recurrent networks\nin speech recognition \\cite{SSEP2014,SSB2014}\nand in language modeling \\cite{MKS2018-1, MKS2018-2, KHQJ2018} are commonly\ntrained with truncated backpropagation through time (BPTT) \\cite{W1990, WP1990},\nwhere a recurrent network is unfolded for a fixed number of time steps\nfor the gradients to propagate.\nThe hypothesis is that even with truncation,\nrecurrent networks are still able to learn recursive\nfunctions that are richer than fixed order Markov processes.\nIn fact, recurrent networks at test time are typically applied to sequences much\nlonger than the ones they are trained on~\\cite{SSB2014,PWPK2018}.\nUnder certain conditions, it has be shown that\ntruncation does not affect the performance of RNNs \\cite{MH2018}.\nBy our definition, these recurrent networks are trained in batch mode,\nand are used in an online fashion during testing.\nWe will examine how the number of BPTT steps affects\nthe training loss and how the decoding approaches\nat test time affect the test loss.\n\nAnother factor that impacts the ability of a recurrent network \nto remember is optimization \\cite{BSF1994, PMB2013}.\nVanilla recurrent networks are known to be difficult\nto train \\cite{HBFS2001,PMB2013}.\nPrevious work has attributed this difficulty to the\nvanishing gradient problem \\cite{H1998}.\nLong short-term memory networks (LSTMs)\nhave been proposed to alleviate the vanishing gradient problem and are\nassumed to be able to learn longer dependencies in the data \\cite{HS1997}.\nWhether these statements are true is still debatable, but we will draw a connection between the\nMarkov assumption, Lipschitz continuity, and vanishing\ngradients in recurrent networks.  The vanishing gradient phenomenon\nis in fact a necessary condition for Markov processes.\n\nThe contribution of the paper is a comprehensive set of\nexperiments comparing two decoding approaches, various numbers of BPTT steps during\ntraining, and their respective results on three types of\ntarget units, namely, subphonetic states, phonemes, and words.\nThe results have implications for studying long-term dependency\nin speech and how well they are learned by recurrent networks.\n\n\n\\section{Markov Assumption, Lipschitz Continuity, and Vanishing Gradients}\n\nFrom the experiments, it is difficult to conclude\nwhether the LSTMs really learn Markov or recursive functions.\nIn this section, we derive a necessary condition\nthat we can check empirically for Markov functions.\nA function $f$ is $G$-Lipschitz with respect to $i$-th coordinate and norm $\\|\\cdot\\|$ if\n\\begin{align}\n& \\bigg|f(a_1, \\dots, a_{i-1}, a, a_{i+1}, \\dots, a_t) \\notag \\\\\n& \\quad - f(a_1, \\dots, a_{i-1}, b, a_{i+1}, \\dots, a_t)\\bigg| \\leq G\\|a - b\\|\n\\end{align}\nfor any $a_1, \\dots, a_t$ and any $a$, $b$.\nIn words, the function does not change too much when we perturb\nthe $i$-th coordinate.\nBy definition, a Markov function does not change at all\nwhen we perturb the coordinate beyond the necessary history,\nor more formally, a $\\kappa$-th order Markov function is 0-Lipschitz with\nrespect to $x_1, \\dots, x_{t-\\kappa}$.\nThe Lipschitz property relates to the gradient through\nthe perturbation interpretation.\nA gradient of a function can be regarded as having\nan infinitesimal perturbation of the coordinates,\nso the $i$-th coordinate of a gradient has a value at most $G$\nfor a $G$-Lipschitz function with respect to the $i$-th coordinate.\nA $\\kappa$-th order Markov function should have zero gradient\nwith respect to $x_1, \\dots, x_{t-\\kappa}$,\nbecause it is 0-Lipschitz with respect to those coordinates.\nIn other words, the gradients to the input, and thus to the parameters,\nmust vanish after $\\kappa$ steps for a $\\kappa$-th order Markov function.\n\nIn practice, partly because of noise and partly\nbecause we do not know if the data\nis really Markov, it is better not to enforce vanishing\ngradients in the model.\nRegardless, this gives a necessary condition we can check empirically.\nWe take the best online and batch LSTMs trained on subphonetic states\nfrom Tables~\\ref{tbl:state-lookahead} and \\ref{tbl:state-tbptt}.\nThe norms of gradients to the input at time $t-20$\nwhen predicting at time $t$\nis shown in Figure~\\ref{fig:grad-hist}.\nThe norms of the LSTM trained with batch decoding concentrate near\nzero compared to the ones with online decoding.\nThis confirms that LSTMs trained with batch decoding behave\nlike Markov functions.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=5cm]{grad-0.eps}\n\\caption{The norms of gradients to the input $x_{t-20}$ when predicting $y_t$\n    on \\textfw{dev93}\n    comparing LSTMs trained with online and batch decoding with 40 context frames.\n    Both LSTMs are trained on subphonetic states with a lookahead of 20 frames.\n    Note that the counts are in log scale.}\n\\label{fig:grad-hist}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\n\\section{Problem Setting}\n\\label{sec:prob}\n\nWe first review the definition of recurrent networks\nand discuss the types of functions they aim to model.\n\nLet $X$ be the set of input elements, $L$ be the label set,\nand $Y = \\mathbb{R}^{|L|}$ for representing\n(log-)probabilities or one-hot vectors of elements in $L$.\nFor example, in the case of speech recognition, the set $X = \\mathbb{R}^{40}$\nif we use 40-dimensional acoustic features, and $L$ can be the set of\nsubphonetic states, phonemes, or words.\nLet $\\mathcal{X}, \\mathcal{Y}$ be the sets of sequences whose elements are in\n$X$, $Y$, respectively.\nWe are interested in finding a function that maps a sequence in $\\mathcal{X}$\nto a sequence in $\\mathcal{Y}$ of the same length.\nIn other words, the goal is to map $(x_1, \\dots, x_T) \\in \\mathcal{X}$,\nwhere each $x_i \\in X$, to $(y_1, \\dots, y_T) \\in \\mathcal{Y}$,\nwhere each $y_i \\in Y$, for $i = 1, \\dots, T$.\nWe assume we have access to a loss function\n$\\ell: Y \\times Y \\to \\mathbb{R}^+$.\nFor example, the loss function for classification at each time step $t$\ncan be the cross entropy $\\ell(y_t, \\hat{y}_t) = - y_t^\\top \\log \\hat{y}_t$,\nwhere $y_t$ is the one-hot vector\nfor the ground truth label at time $t$, and $\\log \\hat{y}_t$\nis a vector of log probabilities produced by the model.\n\nA general recurrent network is a function $s: H \\times X \\to H$\nwhere $H$ is the space of hidden vectors.\nFor an input sequence $(x_1, \\dots, x_T)$, we repeatedly apply\n\\begin{align}\nh_t = s(h_{t-1}, x_t)\n\\end{align}\nfor $t = 1, \\dots, T$\nto obtain a sequence of hidden vectors $(h_1, \\dots, h_T)$\nwhere $h_0 = 0$.\nThe hidden vectors are then used for prediction.\nNote that we are interested in the setting where the length of the\ninput sequence matches the length of the output sequence.\nSpecifically, we have a function $o: H \\to Y$ and apply\n\\begin{align}\n\\hat{y}_t = o(h_t)\n\\end{align}\nfor $t = 1, \\dots, T$ to obtain the label sequence $(\\hat{y}_1, \\dots, \\hat{y}_T)$.\nFor our purposes, it suffices to use $o(h_t) = \\text{softmax}(W h_t)$\nfor some trainable parameter matrix $W$.\n\nA vanilla recurrent neural network implements the above with\n\\begin{align}\ns(h_{t-1}, x_t) & = \\sigma(U x_t + V h_{t-1}),\n\\end{align}\nwhile a long short-term memory network (LSTM) \\cite{HS1997} uses\n\\begin{align}\n\\begin{bmatrix}\ng_t \\\\\ni_t \\\\\nf_t \\\\\no_t\n\\end{bmatrix}\n& =\n\\begin{pmatrix}\n\\tanh \\\\\n\\sigma \\\\\n\\sigma \\\\\n\\sigma\n\\end{pmatrix}\n(U x_t + V h_{t-1}) \\\\\nc_t &= i_t \\odot g_t + f_t \\odot c_{t-1} \\\\\ns(h_{t-1}, x_t) &= o_t \\odot \\tanh(c_t)\n\\end{align}\nto implement the recurrent network, where\n$\\odot$ is the Hadamard product,\n$\\sigma$ is the logistic function, $\\tanh$\nis the hyperbolic tangent, and the matrices\n$U$ and $V$ are trainable parameters.\n\nRecurrent networks can benefit from stacking on top of each other~\\cite{GMH2013}.\nWe abstract away stacking in the discussion, but will use\nstacked recurrent networks in the experiments.\n\n\\subsection{Markov and recursive functions}\n\\label{sec:markov}\n\nRecurrent networks can be seen as recursive functions\nwith a constant size memory.\nTo be precise, consider $y_t$ as a function of $x_1, \\dots, x_t$ for any $t$.\nWe say that $y_t$ is a recursive function if it satisfies\n$y_t = f(m_t)$ where $m_i = g(m_{i-1}, x_i)$\nfor $i = 1, \\dots, t$ and any function $f$ and $g$.\nThe memory is of size constant if the size of $m_t$\nis independent of $t$.\nIn contrast, we say that $y_t$ is a $\\kappa$-th order Markov function if \n$y_t$ is a function of $x_{t-\\kappa+1}, \\dots, x_t$.\nBy this definition, the set of recursive functions\nincludes all Markov functions, and the inclusion is proper.\nFor example, the sum $y_t = \\sum_{i=1}^t x_i$\nis not Markov of a fixed order, but it can be realized with\na recursive function $y_t = m_t$ where $m_i = m_{i-1} + x_i$\nfor $i = 1, \\dots, t$.\nIn the language of signal processing,\na Markov function resembles a finite-impulse response filter,\nand a recursive function resembles an infinite-impulse response\nfilter.\n\n\\subsection{Online and batch decoding}\n\nBased on the two function classes,\nwe define two decoding approaches, online and batch\ndecoding, respectively.  In online decoding, a sequence of\npredictions is made one after another,\nwhile in batch decoding, the predictions\nat different time steps are made independently\nfrom each other.\n\nUsing the notation in the previous section,\nto predict $\\hat{y}_t$ at time $t$, we define online decoding as\n$\\hat{y}_t = f(m_t)$ where $m_i = g(m_{i-1}, x_i)$ for $i = 1, \\dots, t$\nand any function $f$ and $g$.\nDuring decoding, only the vector $m_i$ at time $i$\nneeds to be stored in memory, and no history is actually maintained;\nhence the name online decoding.\nTo implement online decoding with recurrent networks,\nwe simply let $f = o$, $g = s$, and $m_i = h_i$ for all $i$.\n\nWe define batch decoding as $\\hat{y}_t = f(x_{t-\\kappa+1}, \\dots, x_t)$\nfor some function $f$ and context size $\\kappa$.\nBy this definition, batch decoding is not limited to recurrent\nnetworks, and can be used with other neural networks.\nTo implement batch decoding with recurrent networks,\nwe let $f$ compute $\\hat{y}_t = o(h_t)$, $h_t = z^t_t$, and\n$z^t_i = s(z^t_{i-1}, x_i)$ where $z^t_{t - \\kappa} = 0$,\nfor $i = t - \\kappa + 1, \\dots, t$.\n\nIn terms of computation graphs, the graph for online\ndecoding with recurrent networks is a single chain, while the graph\nfor batch decoding consists of multiple parallel chains.\nNote that in batch decoding the hidden vector of each chain\nstarts from the zero vector. In other words, $h_i$ is not a function of $h_{i-1}$\nfor any $i$, so the hidden vectors cannot be reused when\npredicting at different time points.\nThough batch decoding requires more computation and space,\nit can be parallelized.  Online decoding has to be computed\nsequentially.\n\nBy the above definition, batch decoding with $\\kappa$ context frames aims to\nrealize a $\\kappa$-th order Markov function.\nOnline decoding, however, aims to realize a recursive function.\nIt has been shown that the number of unrolled steps in\nvanilla recurrent networks controls the capacity of the model\nin terms of the Vapnik-Chervonenkis dimension \\cite{KS1996}.\nIt is unclear whether recurrent networks truly learn this class of functions.\nHowever, by changing decoding approaches, we explicitly restrict\nthe class of functions that recurrent networks can realize.\n\nFinally, one distinct property in speech recognition\nthat is absent in language modeling is the option to\nlook beyond the current time frame.  \nThe task of predicting the next word becomes meaningless\nwhen the next word is observed.\nHowever, it is useful to look a few frames\nahead to predict the word while the word is being spoken.\nFormally, we say that a recurrent\nnetwork decodes with a lookahead $\\ell$ if $\\hat{y}_t = o(h_{t + \\ell - 1})$.\nLookahead can be applied to both online and batch decoding,\nand it falls back to regular decoding when the lookahead is one frame.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{~}\nEnergy-efficient control of spin dynamics\\cite{kiselev2003microwave} and sizable spin-charge conversion\\cite{HellmanInterfaces} are central topics of spintronic research. Magnetic insulators have recently risen as promising material candidates for spintronic applications\\cite{avci2017current, bozhko2016supercurrent, yang2018fmr}. They possess low magnetic damping and thus reduce energy dissipation\\cite{wunderlich2017spintronics, collet2016generation}. The low thermal conductivity of magnetic insulators, once viewed as a challenge for heat management, bears an opportunity for spintronic devices via heat recycling. Interfacial heat flow in ultra-thin bi-layers of the ferrimagnetic insulator, yttrium iron garnet (YIG), and platinum has recently been shown to inject a spin current sufficient to induce auto-oscillations of YIG magnetization\\cite{safranski2017spin}, creating a spin-torque oscillator driven by waste heat. \n\nWhile magnetic insulators help to avoid shunting of electrical currents, they possess no intrinsic means of spin-charge conversion and rely on proximity-mediated effects\\cite{HellmanInterfaces,lee2016magnetic}. In YIG\/Pt systems, the spin Hall magnetoresistance allows for read-out of spin information. However, it is rather small (typically 10$^{-1}$-10$^{-2}$\\,\\%)\\cite{nakayama2013spin,safranski2017spin}, which presents an obstacle for integration of insulators in reliable spintronic applications.  \n\n\\begin{figure*}[ht]\n\\includegraphics[width=0.99\\textwidth]{fig1.png}\n\\caption{Magnetoresistance of the nanowire device. (a)~Scanning electron micrograph of the nanowire. $H$ is the external magnetic field, applied in the film plane at angle $\\varphi$ with respect to the nanowire axis. (b)~Device resistance as a function of field angle at $H=1.6$\\,kOe. (c)~Device resistance as a function of the magnetic field for different in-plane angles $\\varphi$. All measurements in this work are carried out at 77\\,K.}\n\\label{fig1}\n\\end{figure*}\n\nMetallic ferromagnets, on the other hand, possess intrinsic, sizable spin-charge effects based on spin-orbit interaction, which can be employed for electrical read-out of static and dynamic magnetic states\\cite{SinovaSHE, tulapurkar2005spin, baumgaertl2016magnetization}. Recently, these spin-charge effects have received much attention as tunable sources of spin currents\\cite{taniguchi2015spin, davidson2019perspectives, AminInterface}. Spin injection using anomalous Hall effect\\cite{RalphAHE, DasAHE, TulapurkarAHE}, planar Hall effect\\cite{safranski2019spin}, rotational-symmetry spin-orbit effect\\cite{HumphriesRotSym} in multilayers, and spin-orbit torques in a single ferromagnetic layer\\cite{haidar2019single} have been experimentally realized.\n\nHere, we propose to exploit the virtues of insulating and metallic magnets and fabricate nanowires from thin film bi-layers of YIG and permalloy (Py=Ni$_{80}$Fe$_{20}$). We find that spin coupling at the metal-insulator interface and anisotropic magnetoresistance (AMR) in Py allow for electrical read-out of spin dynamics in YIG. Furthermore, application of interfacial temperature gradient from ohmic heating drives the nanowire into magnetic auto-oscillations which translate into sizable microwave signals. Our findings demonstrate that nanoscale heterostructures based on a combination of metallic and insulating magnets offer both sizable spin-charge conversion and thermal spin-torques, thus enabling next-generation energy-efficient spintronic applications. The observed auto-oscillations, moreover, show an experimental realization of magnon condensation \\cite{demokritov2006bose} in a two-magnet system with hybridized spin wave modes.\n\nUsing sputtering deposition\\cite{liu2014ferromagnetic}, we prepare multilayer thin films consisting of \\ce{Gd3Ga5O12}(GGG substrate)\/YIG(20\\,nm)\/Py (5\\,nm)\/AlOx(2\\,nm). The top layer prevents Py from oxidation. By means of negative e-beam lithography and ion milling, we fabricate nanowire devices (180\\,nm width, 3.4\\,$\\mu$m  length) which fan out into sub-millimeter\\cite{safranski2017spin} large electric leads (Fig.\\,1a).\n\nWe apply a large magnetic field $H=1.6$\\,kOe in the sample plane and measure the device resistance as a function of field angle $\\varphi$ with respect to the nanowire axis (Fig.\\,1b). All measurements are carried out in a cryostat at 77\\,K thermal bath temperature. As expected for the anisotropic magnetoresistance of Py\\cite{duan2014nanowire}, the resistance follows $R=R_0+\\Delta R \\cos(2\\varphi)\/2$ behavior, where $R_0=1127\\,\\Omega$ and $\\Delta R \/ R_0 = 1.24\\%$ is the magnetoresistance ratio. Figure\\,1c shows the device resistance as a function of the magnetic field. Again as expected\\cite{duan2014nanowire}, in the hard axis (at $\\varphi=90$\\,deg due to the shape anisotropy of the nanowire) the curve is bell-shaped and in the easy axis ($\\varphi=0$\\,deg) the curve is V-shaped. However, large resistance drops near zero-field and incomplete saturation in the easy axis suggest some coupling of YIG and Py spins at the interface.\n\n\\begin{figure*}[ht]\n\\includegraphics[width=0.78\\textwidth]{fig2.png}\n\\caption{Spin-torque ferromagnetic resonance. (a)~Spin wave spectrum of the long nanowire near the hard axis. The normalized ST-FMR signal $\\langle V\\rangle$ is color-coded. (b)~Spin wave spectrum of the short nanowire near the hard axis. (c)~Linewidth of the lowest spin wave mode for the long nanowire in the easy axis at $f=6.8$\\,GHz. (d)~Linewidth of the lowest spin wave mode of the short nanowire $\\sim$20\\,deg off easy axis at $f=2.5$\\,GHz. Red curves are guides to the eye.}\n\\label{fig2}\n\\end{figure*}\n\nThe presence of interfacial spin coupling is anticipated to have an impact on the spin wave spectrum of the device\\cite{sluka2019emission,PhysRevApplied.11.034065,liu2018long,PhysRevB.34.7788,NOGUES1999203,PhysRevLett.120.127201,PhysRevLett.120.217202}. While zero coupling should leave individual spin wave spectra of YIG and Py layers nearly unchanged (except for the dipolar interaction), strong coupling should significantly hybridize and delocalize the spin waves over both, YIG and Py, layers and can lead to spin wave modes of acoustic and optical type\\cite{sluka2019emission}.\n\nTo evaluate the spin wave spectrum, we carry out spin-torque ferromagnetic resonance (ST-FMR) measurements near the hard axis with field modulation\\cite{gonccalves2013spin} on the nanowire shown in Fig.\\,1a. Figure~2a presents one group of spin waves as a function of frequency and field. This group is dominated by two modes labeled '1a' and '1b'. The spin wave spectrum is not only determined by the coupling between YIG and Py, but also by the nanowire dimensions. We thus carry out ST-FMR on a much shorter nanowire (displayed in inset of Fig.\\,2d). As shown in Fig.\\,2b, the frequency-field data presents one spin wave mode, labeled '1', similar to the spin waves observed in the longer nanowire. At higher frequencies, another spin wave mode, labeled '2' is found.\n\nTo assess the nature of these spin waves, we perform micromagnetic simulations using typical material parameters of YIG and Py (Supporting Information). We find the best agreement between experiment and micromagnetics for $A_{\\mathrm{int}}=0.4$\\,pJ\/m -- a moderately large, ferromagnetic-type coupling at the interface. The spin waves of the lower-frequency group ('1a' and '1b') are predominantly localized in the YIG layer. They correspond to the normal spin wave eigenmodes\\cite{PhysRevB.92.104424,duan2014nanowire} of a nanowire (see Supporting Information) -- the number of nodes increases with increasing frequency. The higher-frequency mode ('2') corresponds to the lowest-order eigenmode of the nanowire, but is predominantly localized in the Py layer. Due to the symmetry of excitation field in the FMR experiment (Oersted field of the Py layer), the group '2' and higher-order spin waves of the group '1' are very weakly excited\\cite{duan2014nanowire}, and thus not visible in Fig.\\,2a.  \n\nMicromagnetic analysis reveals that spin wave modes at lower frequencies, while delocalized over both layers, are more strongly excited in one layer (see Supporting Information). For instance, the mode '1a' is predominantly localized in YIG, but also drags the magnetization of Py. Such delocalization allows for detecting this mode electrically via magnetoresistance of Py. While at low frequencies, the magnetizations of YIG, $\\vec{M}_{\\textrm{YIG}}$, and Py, $\\vec{M}_{\\textrm{Py}}$, precess nearly in-phase (acoustic modes), with increasing frequency the phase difference approaches $\\pi$ (optical modes). Moreover, the delocalization increases, and the spin wave mode become strongly hybridized.\n\nTo explore the possibility of manipulating the spin dynamics of the two-magnet devices, we analyze the ST-FMR linewidth, which is representative of the effective damping. The linewidth of the lowest mode '1a' of the long nanowire is shown in Fig.\\,2c as a function of electric DC current sent through the Py layer. The linewidth decreases for both polarities of the bias current in the easy axis  $\\varphi=0\\,$deg. The observed decrease of the damping cannot be explained by any known spin-orbit torque generated by electric current in Py\\cite{davidson2019perspectives} (Supporting Information): (i)~In particular, the spin Hall effect produces a spin current with polarization $\\vec{\\sigma}$ parallel to $\\hat y$ and thus\\cite{SinovaSHE} does not contribute to damping-like torque ($\\propto\\vec{M}_{\\textrm{YIG}}\\times\\vec{M}_{\\textrm{YIG}}\\times\\vec{\\sigma}$) in easy axis. (ii)~The Anomalous Hall torque\\cite{RalphAHE, DasAHE, TulapurkarAHE} is zero for parallel orientation of electric current and Py magnetization, which is the case for easy axis. (iii)~The Planar Hall torque\\cite{safranski2019spin} is zero when magnetization lies in the film plane. (iv)~The Rotational-Symmetry torque proposed in Refs.\\cite{HumphriesRotSym,davidson2019perspectives} is zero for parallel orientation of YIG and Py magnetizations. Furthermore, spin-orbit torques are generally odd in electric current and cannot explain the symmetric behavior shown in Fig.\\,2c.\n\nOn the other hand, ohmic heating in the Py layer is nearly quadratic in electric current. Using finite-element simulations (Supporting Information), we estimate a notable temperature gradient across the layers of $\\sim0.3\\,$K\/nm at $\\sim1.8$\\,mA for the long nanowire. The temperature gradient generates an anti-damping spin-torque via spin Seebeck effect\\cite{safranski2017spin,ohnuma2015magnon,bender2014dynamic,WuSpinSeebeckPRL}, that is independent of the DC current polarity, consistent with the data in Fig.\\,2c. In the short nanowire, the heat easily dissipates into the adjacent electrical leads, which reduces the temperature gradient to $\\sim0.004\\,$K\/nm. With such a small temperature gradient, the spin Seebeck torque is negligible. The linewidth of the lowest spin wave mode in Fig.\\,2c therefore depends linearly on the electric current, which is characteristic for spin-orbit torques (described above).\n\n\\begin{figure*}[h]\n\\includegraphics[width=0.80\\textwidth]{Fig3.png}\n\\caption{Auto-oscillations of magnetization of the long nanowire. (a)~Microwave signal at 3.0\\,GHz emitted from the nanowire in the easy axis. Normalized detected voltage $\\langle V \\rangle$ is proportional to the emitted microwave power density. (b)~Microwave emission in the hard axis. At low electric currents, a weak signal of thermally populated is visible. Above a current of approximately 1.8\\,mA, microwave emission is strongly increased and the emission field shows a pronounced non-linear shift.}\n\\label{fig3}\n\\end{figure*}\n\nThe linewidth of the long nanowire in Fig.\\,2c can be extrapolated to zero at about approximately 1.6\\,mA, marking the critical current at which the intrinsic damping of the lowest spin wave mode is fully compensated by the spin Seebeck torque. Once the intrinsic damping is compensated, the magnetization transitions into auto-oscillations in the absence of external microwave drive\\cite{slavin2009nonlinear}. The auto-oscillations are translated into electric signals via the effective magnetoresistance of the nanowire\\cite{safranski2017spin}.\n\nWe supply electric DC current to the long nanowire and measure the microwave signal emitted from the device (into a microwave circuit\\cite{safranski2017spin}, as described in Supporting Information). In Fig.\\,3a, the signal detected at 3.0\\,GHz is shown as a function of the magnetic field and DC current for the easy axis. We find that at low currents, the signal is nearly zero indicating only weak thermal excitation of the spin waves. Above the current of approximately 1.8\\,mA, a sudden onset of the microwave signal is observed, which is characteristic of auto-oscillations\\cite{slavin2009nonlinear}. By comparing the frequency-field relation of spin waves from auto-oscillations and from the ST-FMR experiment, we identify the lowest modes predominantly localized in the YIG layer to participate in auto-oscillations. With increasing current (around 2\\,mA), the emission field shows a strong non-linear shift, which again is indicative of the auto-oscillatory regime\\cite{slavin2009nonlinear,safranski2017spin,evelt2018spin}.\n\nThe microwave emission data, shown in Fig.\\,3a for the easy axis and in Fig.\\,3b for the hard axis, present similar critical currents for the onset of auto-oscillations. Inversion of magnetic field direction (Supporting Information) does not notably change the critical current, as well. The isotropy of the critical current further supports our conclusion -- in the long nanowire, isotropic\\cite{safranski2017spin} thermal spin-torque due to spin Seebeck effect is strong and drives the auto-oscillations, while spin-orbit torques are significantly smaller. For the short nanowire, nearly free of the spin Seebeck effect, the critical current can be estimated to over 10\\,mA, based on the ST-FMR linewidth in Fig.\\,2c. Indeed, we observe no microwave emission from this device in the experimentally accessible current range.\n\nThe emergence of thermally driven magnetic auto-oscillations observed here can be understood as a manifestation of bosonic condensation of incoherent magnons into a coherent low-frequency magnon state. In order to reach the threshold of this instability, the thermal magnons need to be pumped to the chemical potential exceeding the natural frequency of the coherent low-frequency mode\\cite{bender2014dynamic,demokritov2006bose,du2017control,demidov2017chemical}. Raising the chemical potential in one of the ferromagnet (here, YIG) by a heat flow from another one (Py), in two-magnet heterostructures, presents a conceptually novel scenario for magnetic auto-oscillations.\n\nIn Ref.\\,\\cite{bender2016thermally}, Bender et al. proposed a theoretical model for generating thermal spin currents between two ferromagnetic layers separated by a nonmagnetic metallic spacer. A temperature gradient across the layers injects a spin flow from the warmer ferromagnet into the spacer (raising its electronic spin accumulation), which is then transmitted downstream into the colder magnet. In our bi-layer system, the spacer is missing, so that the spin angular momentum is driven directly across the interface between the magnetic layers. Magnetic and possibly also magneto-elastic hybridization across the interface could result in an enhancement of the effective magnonic spin accumulation induced in the YIG layer, facilitating the onset of the condensation instability.\n\nThe underlying transfer of the spin angular momentum from the incoherent to the coherent spin precession can formally be also understood in terms of an anti-damping torque entering in the equation of motion for the latter, in proportion to the chemical potential of the incoherent magnon gas\\cite{Flebus}. The chemical potential is zero in equilibrium but can be raised or lowered, depending on the orientation of the heat flux between the magnetic layers. We generically expect the hotter (colder) side to correspond to the decrease (increase) of the magnon chemical potential\\cite{TwoFluid}.\n\nThermal spin-torque in heterogeneous systems, such as two-magnet systems investigated here, has so far remained unexplored experimentally and theoretically. We consider two possible microscopic mechanisms contributions to the torque: (i)~Spin waves are excitations of magnetic moments residing in the \\textit{d}-electrons, which are coupled by \\textit{s-d} exchange with conduction electrons in Py\\cite{tserkovnyak2002spin}. The conduction electrons thus possess a higher temperature than spin waves which are delocalized over both, warmer and colder, layers. The conduction electrons serve as a source of energy and angular momentum and populate magnons in the nanowire beyond their thermal equilibrium, thus exerting a thermal spin-torque similar to the conventional spin Seebeck in YIG\/Pt. In the two-magnet system, however, the spin Seebeck effect is different in that a temperature difference between the magnonic and electronic sub-systems within one heterogeneous magnetic system creates the anti-damping torque. (ii)~Since the spin waves at higher frequencies show a higher degree of delocalization, they are more susceptible to thermal excitations in the Py layer. The energy and angular momentum of higher-frequency magnons redistribute to the lower-frequency magnon states, creating a population beyond thermal equilibrium and exerting thermal spin-torque\\cite{bender2014dynamic,demokritov2006bose,du2017control,demidov2017chemical}.\n\nWhile our results demonstrate thermal spin-torque to be responsible for damping modification and auto-oscillations, they do not allow for quantitative comparison of contributions due to the (i)  magnon-electron and (ii) magnon-magnon mechanisms. However, the temperature gradient needed to reach auto-oscillations in our two-magnet nanowires and YIG\/Pt nanowires of Ref.\\,\\cite{safranski2017spin} are similar. This allows for a conjecture that the magnon-magnon mechanism, which is only present in two-magnet systems, is unlikely more effective than the magnon-electron mechanism. The delocalized nature of spin waves offers an effective means for spin-charge conversion in applications based on magnetic insulators. We estimate the power spectral density of the emitted microwave signal from auto-oscillation to $\\sim$1\\,pW\\,MHz$^{-1}$ which is three orders of magnitude higher than electrical signals obtained on YIG\/Pt systems. \n\nIn conclusion, we find that heterostructures based on a combination of metallic and insulating magnets bear a great promise for next-generation spintronic applications. Via interfacial spin coupling, magnetization dynamics in magnetic insulator translates into magnetization precession of the ferromagnetic metal. Inherently large magnetoresistance in ferromagnetic metal allows converting this magnetization dynamics into sizable electric signals. With a moderate interfacial spin coupling, the lower-energy spin wave modes are predominantly localized in one layer, while higher-energy modes show an increased level of hybridization and delocalization. Heat flow due to the temperature gradient across the layers leads to magnon condensation into the lower-energy modes, driving the magnetization into auto-oscillations. We demonstrate that a two-magnet system can be used to realize thermally driven spin-torque nano-oscillators. The results show a prospect for energy-efficient spintronic devices and advance our understanding of magnon thermodynamics in heterogeneous magnetic systems.  \n\n\n\\subsection{Associated Content}\n\\subsubsection{Supporting Information}\nFig.\\,S1: Micromagnetic simulations of YIG\/Py nanowire using MuMax3 package\\cite{mumax}.\\\\\nFig.\\,S2: Finite-element simulation (in COMSOL Multiphysics) of temperature profile under ohmic heating of the Py layer.\\\\\nFig.\\,S3: Microwave emission from the long nanowire at negative fields.\\\\\nTable\\,S1: Overview of spin-orbit torques generated in ferromagnetic metals.\n\n\\subsection{Author Information}\n\\subsubsection{Corresponding Author}\n*E-mail: igorb@ucr.edu\n\n\\subsubsection{ORCID}\nBassim Arkook: 0000-0002-4968-0954\\\\\nIgor Barsukov: 0000-0003-4835-5620\\\\\nKilian Lenz: 0000-0001-5528-5080\n\n\\subsubsection{Author Contributions}\nB.A. carried out magnetic characterization and performed data analysis. B.A and C.S. prepared the samples. R.R. and T.S. carried out micromagnetic simulations. H.C. and M.W. provided magnetic insulators. I.B. supervised the project. All authors discussed the data and co-wrote the manuscript.\n\n\\subsubsection{Notes}\nThe authors declare no competing financial interest.\n\n\\subsection{Acknowledgements}\nThis work was supported by the National Science Foundation under grant No.~ECCS-1810541. The material synthesis was supported as part of the ''Spins and Heat in Nanoscale Electronic Systems'' (SHINES), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (BES) under award No.~SC0012670. Work at CSU was supported by NSF through EFMA-1641989 and ECCS\u20101915849. We thank NVIDIA Corporation for donating Titan Xp GPU used for some of the calculations. 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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nIn this paper, we consider the problem of learning a functional mapping\nfrom a high-dimensional input space with structured sparsity to a multivariate output space where responses are coupled (therefore making the estimator doubly structured), with an application for detecting genomic loci\naffecting gene expression levels, a problem known as expression quantitative trait loci (eQTL) mapping.\nIn particular, we are interested in exploiting the structural information \non both the input and output space jointly to improve the accuracy for identifying \na small number of input variables relevant to the outputs\namong a large number of candidates. \nWhen the input or output variables are highly correlated among themselves,\nmultiple related inputs may synergistically influence \nthe same outputs,\nand multiple related outputs may be synergistically influenced by the same inputs.\n\n\nThe primary motivation for our work comes from the problem of genome-wide association (GWA) mapping of eQTLs in computational genomics \\cite{kendziorski2006statistical}, of which \nthe goal \nis to detect the genetic variations, often single nucleotide polymorphisms (SNPs), across the whole genome \nthat perturb the expression levels of genes, given the data of SNP genotypes \nand microarray gene-expression traits of a study cohort. \nOne of the main challenges of this problem \nis that, typically the sample size is very small (e.g. $\\sim 1000$), whereas there are a very large number of SNPs (e.g. $\\sim$ 500,000) and expression traits (e.g., $\\sim$ 10,000).\nFurthermore, there have been numerous evidences that multiple genetic variations may interact with\neach other (a.k.a., epistasis) \\cite{segre2005modular,cho1998identification,badano2005dissection}, and the same genetic variation(s) can influence multiple genes \n(a.k.a., pleiotropy) \\cite{stearns2010one,ducrest2008pleiotropy}. \nHowever, prior knowledge of such information \nimplying complex association \nstructures are difficult to exploit in standard \nstatistical analysis of GWA mapping \\cite{wang2010analysing,moore2010bioinformatics}.\nTo enhance the statistical power for mapping of eQTLs, \nit is desirable to incorporate biological knowledge of genome and transcriptome structures into the model to guide the search for true eQTLs.\nIn this article, we focus on developing a model \nwhich can make use of structural information on both input (SNPs) and output (gene expressions) sides. In particular, we consider biological knowledge about group associations as structural information. If there exist group behaviors among the covariates in the high-dimensional input $\\mathbf{X}$, for example, multiple genetically coupled SNPs \n(e.g., in linkage disequilibrium) can jointly affect a single trait \\cite{wang2010analysing}, \nsuch group information is called an input structure;\nif multiple variables in the high-dimensional output $\\mathbf{Y}$ are jointly under the influence of a similar set of input covariates, for example, a single SNP can affect multiple functionally coupled traits (e.g., genes in the same pathway or operon) \\cite{kim-xing-PLoSG09}, \nsuch group information is called an output structure.\nThe problem of GWA mapping of eQTLs can be formulated as a model selection problem under a multitask regression model $\\mathbf{Y}=\\mathbf{B} \\mathbf{X}$ with structured sparsity, \nwhere the resultant non-zero elements in the regression coefficient matrix $\\mathbf{B}$ expose the identities of the eQTLs and their associated traits.\n\n\nVariants of this problem have been widely studied in the recent high-dimensional inference and variable selection literature, and various penalty-based or Bayesian approaches for learning a shared sparsity pattern among either multiple inputs or multiple outputs in a regression model have been proposed \n\\cite{tibshirani2005sparsity,negahban2011simultaneous,han2010multi,li2010bayesian}. \nDepending on the type of structural constraints, different penalty functions\nhave been previously considered, including\nmixed-norm (group-lasso) penalty for a simple grouping structure \n\\cite{yuan2006model,zhao2009composite}, \ntree-guided group-lasso \npenalty for a tree structure \\cite{kim2009tree}, \nor graph-guided fused lasso for a graph structure \\cite{kim-xing-PLoSG09}.\nMost previous approaches, however,\nconsidered either only the input structural constraints \nor only the output structural constraints, but not both. \nThere have been a few approaches that attempted to use both structural information,\nincluding MGOMP \\cite{lozanoblock} and ``group sparsity for multi-task learning''\n\\cite{tseng2009coordinate}.\nMGOMP proposed to select the groups of regression coefficients \nfrom a predefined set of grouped variables in a greedy fashion, and \n\\cite{tseng2009coordinate} proposed to find the groups of inputs that \ninfluence the group of outputs.\nHowever, both methods may have limits on the number or the shapes of sparsity patterns that\ncan be induced in $\\mathbf{B}$. \nFor example, given a large number of input groups $|\\mathcal{G}|$ and output groups $|\\mathcal{H}|$\n(e.g. $|\\mathcal{G}|>10^5, \\; |\\mathcal{H}|>10^3$ for genome data)\nthe scalability of MGOMP can be substantially affected since it needs to select\nthe groups of coefficients from all possible combinations of input and output groups. \nFor \\cite{tseng2009coordinate}, only disjoint block sparsity patterns are considered, hence\nit may not capture the sparsity patterns where the grouped variables overlap.\n\n\nIn this paper, we address\nthe problem of exploiting both the input and output structures in a high-dimensional \nlinear regression setting practically encountered in eQTL mapping. \nFurthermore, to detect epistatic (i.e., interaction) effects between SNP pairs,\nwe additionally expand the input space to include pairwise terms (i.e., $x_i x_j$'s) guided by biological information, which necessitates \nattentions for avoiding excessive input dimension that can make the problem computationally prohibitive.   \nOur main contributions can be summarized as follows:\n\\begin{enumerate}\n\\item \nWe propose a highly general regression model \nwith structured input-output regularizers called\n``jointly structured input-output lasso'' (SIOL)\nthat discovers structured associations between SNPs and expression traits (Section \\ref{sec:methods}).  \n\\item \nWe develop a simple and highly efficient optimization method called\n``hierarchical group thresholding'' (HiGT)\nfor solving the proposed regression problem \nunder complex sparsity-inducing penalties in a very high-dimensional space (Section \\ref{sec:optimization}). \n\\item Extending \nSIOL, we propose ``structured polynomial multi-task regression'' to efficiently model non-additive SNP-SNP interactions guided by \ngenetic interaction networks \n(Section \\ref{sec:model_interaction}). \n\\end{enumerate}\n\n\nSpecifically, given knowledge of the groupings of the inputs (i.e., SNPs) and outputs (i.e., traits) in a high-dimensional multi-task regression setting, we employ an $L_1\/L_2$ norm over such structures \nto impose a group-level sparsity-inducing penalty simultaneously over both the columns and the rows of the regression coefficient matrix $\\mathbf{B}$\n(In our setting, a row corresponds to coefficients regressing all SNP (or SNP pair) inputs to a particular trait output, \nthus reflecting possible epistatic effects;\nand a column corresponds to coefficients regressing a particular SNP (or SNP pair) input to all trait outputs in question, \nthus reflecting possible pleotropic effects). \nGiven reliable input and output structures, rich structured sparsity \ncan increase statistical power significantly since\nit makes it possible to borrow information not only within different output or input variables, but also across output and input variables. \n\nThe sparsity-inducing penalties on both the inputs and \noutputs in SIOL introduce a non-differentiable \nand non-separable objective in an extremely high-dimensional optimization space, \nwhich prevents standard optimization methods such as the interior point \\cite{nesterov1987interior}, the coordinate-descent \\cite{friedman2007pathwise}, or even the recently invented union of supports \\cite{jacob2009group} algorithms to be directly applied. \nWe propose a simple and efficient algorithm called ``hierarchical \ngroup-thresholding'' \nto optimize our regression model with complex structured regularizers.\nOur method is an iterative optimization algorithm, designed to handle complex structured regularizers \nfor very large scale problems.\nIt starts with a non-zero $\\mathbf{B}$ (e.g. initialized by ridge regression \\cite{hoerl1970ridge}), \nand progressively discards irrelevant groups of covariates using thresholding operations.\nIn each iteration, we also update the coefficients of the remaining covariates. \nTo speed up our method, we employ \na directed acyclic graph (DAG) where nodes represent\nthe zero patterns encoded by our input-output structured regularizers\nat different granularity, and edges indicate the inclusion relations\namong them.\nGuided by the DAG,\nwe could efficiently discard irrelevant covariates.\n\n\n\nAs our third contribution, \nwe consider non-additive pairwise interaction effects between the input variables, \nin a way that avoids a quadratic blow-up of the input dimension. \nIn eQTL mapping studies, it is not uncommon that the effect of one SNP \non the expression-level of a gene is dependent on the genotype of another SNP, \nand this phenomenon is known as epistasis. \nTo capture pairwise epistatic effects of SNPs on the trait variation, we \nadditionally consider non-additive interactions between the input covariates. \nHowever, in a typical eQTL mapping, as\nthe input lies in a very high-dimensional space, it is computationally and statistically infeasible\nto consider all possible input pairs. \nFor example,\nfor $J$ inputs (e.g. $500,000$ for a typical genome data set), \nwe have $O(J^2)$ candidate input pairs,\nand learning with all of them will require a significantly large sample size.\nMany of the previous approaches for learning the epistatic interactions relied\non pruning candidate pairs based on the observed data \\cite{Rat:96} or constructing candidate pairs from\nindividual SNPs that were selected based on marginal effects in the previous learning phase without modeling interactions \\cite{Taskar:10,devlin2003analysis}.\nA main disadvantage of the later approach is that it will miss\npairwise interactions when they have no or little individual effects\non outputs. \nInstead of choosing candidate SNP pairs based on only marginal effects, \nwe propose to use \ngenetic interaction network \\cite{costanzo2010genetic}\nconstructed from large-scale biological experiments \nto consider biologically plausible candidate pairs.\n\n\nThe rest of this paper is organized as follows. \nIn Section 2, we discuss \nprevious works on learning a sparse\nregression model with prior knowledge on either output or input structure. \nIn Section 3, we introduce our proposed model \n``jointly structured input-output lasso'' (SIOL). \nTo solve our regression problem, we present an efficient optimization method \ncalled ``hierarchical group-thresholding'' (HiGT) in Section 4.\nWe further extend our model to consider pairwise interactions among input\nvariables and propose ``structured polynomial multi-task regression'' in Section 5.\nWe demonstrate the accuracy of recovered structured sparsity and the speed\nof our optimization method in Section 6 via simulation study, and \npresent eQTLs having marginal and interaction effects in yeast that we identified in Section 7.\nA discussion is followed in Section 8. \n\n\n\n\n\n\n\\section{Background: Linear Regression with Structured Sparsity}\n\\label{sec:background}\n\nIn this section, we lay out the notation and then review existing sparse regression \nmethods that recover a structured sparsity pattern in the estimated regression coefficients\ngiven prior knowledge on input or output structure.\n\n\\subsection{Notation for matrix operations}\n\nGiven a matrix $\\mathbf{B} \\in \\mathbb{R}^{K \\times J}$, we denote\nthe $k$-th row by $\\bm{\\beta}_k$, the $j$-th column\nby $\\bm{\\beta}^j$, and the $(k,j)$ element \nby $\\beta_k^j$. \n$\\lVert \\cdot \\rVert_F$ denotes\nthe matrix Frobenius norm, $\\lVert \\cdot \\rVert_1$ denotes an $L_1$ norm \n(entry-wise matrix $L_1$ norm for a matrix argument), and $\\lVert \\cdot \\rVert_2$\nrepresents an $L_2$ norm. \nGiven the set of {\\it column} groups $\\mathcal{G}=\\{{\\mathbf{g}}_1, \\ldots, {\\mathbf{g}}_{|\\mathcal{G}|}\\}$ \ndefined as a subset of the power set of $\\{1, \\ldots, J\\}$, \n$\\bm{\\beta}_k^{\\mathbf{g}}$ represents the row vector with\nelements $\\{\\beta_k^j: j \\in \\mathbf{g}, \\mathbf{g} \\in \\mathcal{G} \\}$, which is\na subvector of $\\bm{\\beta}_k$ due to group $\\mathbf{g}$. \nSimilarly, for the set of {\\it row} groups $\\mathcal{H}=\\{{\\mathbf{h}}_1, \\ldots, {\\mathbf{h}}_{|\\mathcal{H}|}\\}$ \nover $M$ rows of matrix $\\mathbf{B}$, \nwe denote by $\\bm{\\beta}_{\\mathbf{h}}^{j}$ the column subvector with\nelements $\\{\\beta_k^j: k \\in \\mathbf{h}, \\mathbf{h} \\in \\mathcal{H} \\}$. \nWe also define the submatrix of ${\\mathbf{B}}_{\\mathbf{h}}^{\\mathbf{g}}$ as \na $|\\mathbf{h}| \\times |\\mathbf{g}|$ matrix with elements $\\{\\beta_k^j: k \\in \\mathbf{h}, \\; j \\in \\mathbf{g}, \\;\n\\mathbf{h} \\in \\mathcal{H}, \\; \\mathbf{g} \\in \\mathcal{G} \\}$. \n\n\\subsection{Sparse estimation of linear regression}\n\n\nLet $\\mathbf{X} \\in \\mathbb{R}^{J \\times N}$ be the input data for  $J$ inputs and $N$ individuals, \nand $\\mathbf{Y} \\in \\mathbb{R}^{K \\times N}$ be the output data \nfor $K$ outputs. We model the functional mapping from the common\n$J$-dimensional input space to the $K$-dimensional output space, using a linear\nmodel parametrized by unknown regression coefficients $\\mathbf{B} \\in \\mathbb{R}^{K \\times J}$\nas follows:\n\\begin{eqnarray}\n\\mathbf{Y} = \\mathbf{B} \\mathbf{X} + \\mathbf{E}, \\nonumber\n\\label{eq:reg}\n\\end{eqnarray}\nwhere $\\mathbf{E} \\in \\mathbb{R}^{K \\times N}$ is a matrix of noise terms whose elements\nare assumed to be identically and independently distributed as Gaussian with zero mean \nand the identity covariance matrix.\nThroughout the paper, we assume that $x_j^{i}$'s and $y_k^i$'s \nare standardized\nsuch that all rows of $\\mathbf{X}$ and $\\mathbf{Y}$ have zero mean and a constant variance, \nand consider a model without an intercept.\nIn eQTL analysis, inputs are genotypes for $J$ loci encoded as 0, 1, or 2 in terms of\nthe number of minor alleles at a given locus, and output data are given as\nexpression levels of genes measured in a microarray experiment.\nThen, the regression coefficients represent the strengths of associations \nbetween genetic variations and gene expression levels.\n\nOur proposed method for estimating the coefficients $\\mathbf{B}$ is based on\na \ngroup-structured multi-task regression approach that extends \nexisting regularized regression approaches including lasso \\cite{tibshirani1996regression}, \ngroup lasso \\cite{yuan2006model}  and multi-task lasso \\cite{obozinski2006multi}, \nwhich we briefly review below in the context of our eQTL mapping problem. \nWhen $J >> N$ and only a small number of inputs are expected\nto influence outputs, lasso has been widely used and shown effective in\nselecting the input variables relevant to outputs and setting the elements of \n$\\mathbf{B}$ for irrelevant inputs to zero \\cite{zhang2008sparsity}.\nLasso obtains a sparse estimate of regression coefficients by optimizing \nthe least squared error criterion with an $L_1$ penalty over $\\mathbf{B}$ as follows:\n\\begin{eqnarray}\n\\min_{\\mathbf{B}} \\frac{1}{2}\n\\lVert \\mathbf{Y} - \\mathbf{B} \\mathbf{X}  \\rVert_F^2 \n + \\lambda \\lVert \\mathbf{B} \\rVert_1,\n\\label{eq:lasso}\n\\end{eqnarray}\nwhere $\\lambda$ is the tuning parameter that determines the amount of \npenalization. \nThe optimal value of $\\lambda$ \ncan be determined by cross validation or via an information-theoretic test based on BIC. \nAs in eQTL analysis it is often believed that the expression level of each gene is affected by\na relatively small number of genetic variations in the whole genome, lasso provides\nan effective tool for identifying eQTLs from a large number of genetic variations.\nLasso has been previously applied to eQTL analysis \\cite{brown2011application} \nand more general genetic association\nmapping problems \\cite{wu2009genome}.\n\nWhile lasso considers the input variables independently to select relevant\ninputs with non-zero regression coefficients, we may have prior knowledge \non how related input variables are grouped together and want to perform variable\nselection at the group level rather than at the level of individual inputs.\nGrouped variable selection approach can combine the statistical strengths across\nmultiple related input variables to achieve higher power for detecting\nrelevant inputs in the case of low signal-to-noise ratio.\nAssuming the grouping structure over inputs are available as \n$\\mathcal{G} = \\{{\\mathbf{g}}_1, \\ldots, {\\mathbf{g}}_{|\\mathcal{G}|}\\}$, which is a subset\nof the power set of $\\{1, \\ldots, J\\}$, group lasso uses $L_1\/L_2$ penalization\nto enforce that all of the members in each group of input variables are jointly \nrelevant or irrelevant to each output. Group lasso obtains an estimate\nof $\\mathbf{B}$ by solving the following optimization problem:\n\\begin{eqnarray}\n\\min_{\\mathbf{B}} \\frac{1}{2} \\lVert \\mathbf{Y} - \\mathbf{B} \\mathbf{X}  \\rVert_F^2 \n+ \\lambda \\sum_{k=1}^K \\sum_{\\mathbf{g} \\in \\mathcal{G}} {\\lVert \\bm{\\beta}^{\\mathbf{g}}_{k} \\rVert_2}  \n\\label{eq:lasso_g},\n\\end{eqnarray}\nwhere $\\lambda$ is the tuning parameter. The second term in the above\nequation represents an $L_1\/L_2$ penalty over each row $\\bm{\\beta}_k$ of $\\mathbf{B}$ for \nthe $k$-th output given $\\mathcal{G}$, defined by \n$\\lVert \\bm{\\beta}_k \\rVert_{L_1\/L_2} = \\sum_{\\mathbf{g} \\in \\mathcal{G}} \\lVert \\bm{\\beta}_k^{\\mathbf{g}} \\rVert_2$.\nThe $L_2$ part of the penalty plays the role of enforcing a joint selection\nof inputs within each group, whereas the $L_1$ part of the penalty is applied\nacross different groups to encourage a group-level sparsity.\nGroup lasso can be applied to an eQTL mapping problem given biologically\nmeaningful groups of genetic variations that are functionally related.\nFor example, rather than individual genetic variations acting\nindependently to affect (or not affect) gene expressions, the variations are \noften related through pathways that consist of multiple genes participating\nin a common function. Thus, genetic variations can be grouped according to pathways \nthat contain genes carrying those genetic variations. Then, given this grouping, \ngroup lasso can be used to select groups of genetic variations\nin the same pathways as factors influencing gene expression levels \\cite{silverpathway}.\n\n\n\n\nInstead of having groups over inputs with outputs being independent as in group lasso, \nthe idea of using $L_1\/L_2$ penalty for grouped variable \nselection has also been applied to take advantage of the relatedness among outputs in multiple\noutput regression.\nIn multi-task regression for union support recovery \\cite{obozinski2006multi}, \none assumes that all the outputs share a common support of relevant input variables \nand try to recover shared sparsity patterns across multiple outputs \nby solving the following optimization problem:\n\\begin{eqnarray}\n\\min_{\\mathbf{B}} \\frac{1}{2}\n\\lVert \\mathbf{Y} - \\mathbf{B} \\mathbf{X}  \\rVert_F^2 \n+ \\lambda \\sum_{j=1}^J \\sum_{\\mathbf{h} \\in \\mathcal{H}} \\lVert \\bm{\\beta}_{\\mathbf{h}}^j \\rVert_2,  \\label{eq:multi_task2} \n\\end{eqnarray}\nwhere \n$\\lambda$ can be determined by cross-validation.\nIn eQTL mapping, as gene expression levels are often correlated for the genes that participate\nin a common function, it is reasonable to assume that those coexpressed genes \nmay be influenced by common genetic variations.\nIf gene module information is available, one can use the above model to \ndetect genetic variations influencing the expressions\nof a subset of genes within each gene module. \nThis strategy corresponds to a variation of the standard group lasso, where group is defined \nover outputs rather than inputs.\n\n\nExtending the idea of lasso and group lasso, we may have \ngroup and individual level sparsity simultaneously using \ncombined $L_1$ and $L_1\/L_2$ penalty.\nIn group lasso, if a group of coefficients is not jointly set to zero, all the members \nin the group should have non-zero values.\nHowever, sometimes it is desirable to set some members of the\ngroup to zero if they are irrelevant to outputs.\nSparse group lasso \\cite{friedman2010note} is proposed to address \nthe cases where \ngroups of coefficients \ninclude both relevant and irrelevant \nones.\nUsing convex combination of $L_1$ and $L_1\/L_2$ norms,\nit solves the following convex optimization problem:\n\\begin{eqnarray}\n\\min_{\\mathbf{B}} \\frac{1}{2}\n\\lVert \\mathbf{Y} - \\mathbf{B}  \\mathbf{X} \\rVert_F^2 \n+ \\lambda_1  \\lVert \\mathbf{B} \\rVert_1\n+ \\lambda_2  \\sum_{k=1}^K \\sum_{\\mathbf{g} \\in \\mathcal{G}} {\\lVert \\bm{\\beta}^{\\mathbf{g}}_{k} \\rVert_2},\n\\label{eq:sparse_group_lasso} \n\\end{eqnarray}\nwhere $\\lambda_1$ and $\\lambda_2$ determine the individual and\ngroup level sparsity, respectively.\nThe $L_1\/L_2$ penalty shrinks\ngroups of coefficients to zero, and at the same time,\n$L_1$ penalty  sets irrelevant coefficients to zero individually within each group.\n\nOur proposed model is motivated by \ngroup lasso, multi-task lasso\nand sparse group lasso, each of which can\nexploit pre-defined groupingness of input \nor output variables to  \nachieve better statistical power.\nIn the next section, \nwe will extend the existing models in such a way\nthat we can use\nthe groups in both input and output spaces simultaneously.\nAdopting the idea of sparse group lasso, \nwe will also support variable selection\nat individual levels.\n\n\n\n\n\n\n\\section{Jointly Structured Input-Output Lasso} \n\\label{sec:methods}\n\n\nIn this section, we propose SIOL \nthat incorporates structural constraints on both the inputs and outputs.\nThe model combines the mixed-norm regularizers for the groups of \ninputs and outputs, which leads to the following optimization problem:\n\\begin{subequations}\n\\label{equ:reg5}\n\\begin{align}\n\\min \\frac{1}{2}\n\\lVert \\mathbf{Y} - \\mathbf{B} \\mathbf{X}  \\rVert_F^2 \n&  + \\lambda_1   \n\\lVert \\mathbf{B} \\rVert_1,\n\\label{equ:reg5_b}     \\\\\n  &  + \\lambda_2  \n\\sum_{k=1}^K \\sum_{{\\mathbf{g}} \\in \\mathcal{G}} {\\lVert \\bm{\\beta}^{\\mathbf{g}}_{k} \\rVert_2}    \n\\label{equ:reg5_c}   \\\\\n  & + \\lambda_3 \n\\sum_{j=1}^J \\sum_{{\\mathbf{h}} \\in \\mathcal{H}} \\lVert \\bm{\\beta}_{\\mathbf{h}}^j \\rVert_2,  \n\\label{equ:reg5_d}\n\\end{align}\n\\end{subequations}\nwhere Eq. (\\ref{equ:reg5_c})\nincorporates the  groups of inputs $\\mathcal{G}=\\{{\\mathbf{g}}_1, \\ldots, {\\mathbf{g}}_{|\\mathcal{G}|}\\}$,\nEq. (\\ref{equ:reg5_d}) \nincorporates the groups of the outputs \n$\\mathcal{H}=\\{{\\mathbf{h}}_1, \\ldots, {\\mathbf{h}}_{|\\mathcal{H}|}\\}$, \nand Eq. (\\ref{equ:reg5_b}) allows us to select individual coefficients.\nNote that it is possible that there are overlaps between \n$\\bm{\\beta}_k^{\\mathbf{g}}$ and $\\bm{\\beta}_k^{\\mathbf{g}'}$, \nbetween $\\bm{\\beta}_{\\mathbf{h}}^j$ and $\\bm{\\beta}_{\\mathbf{h}'}^j$, and\nbetween $\\bm{\\beta}_k^{\\mathbf{g}}$ and $\\bm{\\beta}_{\\mathbf{h}}^j$, where \n$\\mathbf{g} \\neq \\mathbf{g}', \\mathbf{h} \\neq \\mathbf{h}'$ and\n$\\mathbf{g}, \\mathbf{g}' \\in \\mathcal{G}, \\; \\mathbf{h}, \\mathbf{h}' \\in \\mathcal{H}$. \nThe overlaps make it challenging to optimize Eq. (\\ref{equ:reg5}), and \nthis issue will be addressed by our optimization method in Section \\ref{sec:optimization}.\n\nLet us characterize the structural constraints imposed by the penalties in our model. \nIn our analysis, we investigate a block of coefficients involved in one output group $\\mathbf{h}$ and \none input group $\\mathbf{g}$, i.e, $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$.\nWe start with Karush-Kuhn-Tucker (KKT)\ncondition for Eq. (\\ref{equ:reg5}): \n\\begin{equation}\n(\\mathbf{y}_k-\\bm{\\beta}_k \\mathbf{X} )(\\mathbf{x}_j)^T = \\lambda_1 s_k^j + \\lambda_2 c_k^j + \\lambda_3 d_k^j,\n\\label{eq:subgradients}\n\\end{equation}\nwhere $s_k^j$, $c_k^j$, and $d_k^j$ are the subgradient of\nEq. (\\ref{equ:reg5_b}), Eq. (\\ref{equ:reg5_c}), and Eq. (\\ref{equ:reg5_d})\nwith respect to $\\beta_k^j$, respectively.\nFor simple notation, we also define \n${\\mathbf{r}}_k^j=\\mathbf{y}_k-\\sum_{l\\neq j} \\beta_k^l \\mathbf{x}_l$. \n\nFirst, we consider the case where all coefficients in $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$ become\nzero simultaneously, i.e., $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}} = \\bm{0}$.\nUsing KKT condition in Eq. (\\ref{eq:subgradients}), we\ncan see that $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}} = \\bm{0}$ if and only if\n\\begin{align}\n\\label{eq:sub1}\n\\sum_{k \\in \\mathbf{h}}\\sum_{j \\in \\mathbf{g}}  \\left\\{ \\mathbf{r}_k^j(\\mathbf{x}_j)^T  - \\lambda_1 s_k^j\\right\\}^2 \n\\leq \\sum_{k \\in \\mathbf{h}}\\sum_{j \\in \\mathbf{g}} \\left( \\lambda_2 c_k^j + \\lambda_3 d_k^j \\right)^2\n\\leq \\left( \\lambda_2 \\sqrt{|\\mathbf{h}|} + \\lambda_3 \\sqrt{|\\mathbf{g}|} \\right)^2.\n\\end{align}\nThis condition is due to Cauchy-Schwarz inequality,\n$\\sum_{j\\in \\mathbf{g}}(c_k^j)^2 \\leq 1$, and $\\sum_{k\\in \\mathbf{h}}(d_k^j)^2 \\leq 1$. \nHere if $\\lambda_1$, $\\lambda_2$ and $\\lambda_3$ are large, \n$\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$ is likely to be zero jointly. \nThis structural sparsity is useful to filter out a large number of\nirrelevant covariates since it considers both the group of\ncorrelated inputs $\\mathbf{g}$ and the group of correlated outputs $\\mathbf{h}$ simultaneously.\n\nOur model also inherits grouping effects for only input (or output) groups.\nFor the analysis of such grouping effects, we fix the groups of zero coefficients that overlap with, say, an input group $\\bm{\\beta}_k^{\\mathbf{g}}$.\nFormally speaking, let us define $\\bm{\\xi} = \\{j: (\\bm{\\beta}_{\\mathbf{h}'}^j = \\bm{0}, j \\in \\mathbf{g}, \\mathbf{h}' \\in \\mathcal{H})\n\\lor (\\bm{\\beta}_{k}^{\\mathbf{g}'} = \\bm{0}, j \\in \\mathbf{g}' \\land \\mathbf{g}) \\}$,\nand fix $\\beta_k^j$s for all $j \\in \\bm{\\xi}$.\nUsing the KKT condition in Eq. (\\ref{eq:sub1}), $\\bm{\\beta}_k^{\\mathbf{g}} = \\bm{0}$ if \n\\begin{align}\n\\sum_{j \\in \\mathbf{g} - \\bm{\\xi}}  \\left\\{ {\\mathbf{r}}_k^j(\\mathbf{x}_j)^T  - \\lambda_1 s_k^j\\right\\}^2\n\\leq \\sum_{j \\in \\mathbf{g} - \\bm{\\xi}} \\left( \\lambda_2 c_k^j + \\lambda_3 d_k^j \\right)^2\n\\leq \\lambda_2^2. \n\\label{eq:sub2}\n\\end{align}\nHere, we know that $d_k^j = 0$ for $j \\in \\mathbf{g} - \\bm{\\xi}$ ($\\beta_k^j = 0$ and $\\bm{\\beta}_{\\mathbf{h}}^j \\neq \\bm{0}$) and \n$\\lambda_2 \\sum_{j \\in \\mathbf{g}} (\\beta_k^j)^2 = \\lambda_2 \\sum_{j \\in \\mathbf{g} - \\bm{\\xi}} (\\beta_k^j)^2$,\nand hence  $\\sum_{j \\in \\mathbf{g} - \\bm{\\xi}} \\left( \\lambda_2 c_k^j + \\lambda_3 d_k^j\\right)^2 \\leq \\lambda_2^2$.\nThis technique was previously introduced in \\cite{yuanefficient} to handle overlapping group lasso.\nOne can see that if the size of $\\bm{\\xi}$ is large, \n$\\bm{\\beta}_{k}^{\\mathbf{g}}$ tends to be zero together\nsince it reduces the left-hand side of Eq. (\\ref{eq:sub2}).\nThis behavior explains the correlation effects between input and output group structures. \nWhen a group of coefficients ($\\bm{\\beta}_k^{\\mathbf{g}}$, $\\bm{\\beta}_{\\mathbf{h}}^{j}$) corresponding to an input group or an output group become zero, they\naffect other groups of coefficients that overlap with them; and \nthe overlapped coefficients are more likely to be zero.\nThese correlation effects between overlapping groups are desirable for inducing \nappropriate structured sparsity as it allows us to share information across different \ninputs and different outputs simultaneously.\nWe skip the analysis of the grouping effects for output groups as\nthe argument is the same \nexcept that the input and output group are reversed.\n\nFinally, we also have individual sparsity due to $L_1$ penalty in Eq. (\\ref{equ:reg5_b}).\nIn this case, let us assume that \n$\\bm{\\beta}_{k}^{\\mathbf{g}} \\neq \\bm{0}$ and $\\bm{\\beta}_{\\mathbf{h}}^{j} \\neq \\bm{0}$\nsince if the group of coefficients is zero, we automatically have $\\beta_k^j=0$.\nUsing the KKT condition, $\\beta_k^j=0$ if and only if\n\\begin{align}\n|{\\mathbf{r}}_k^j(\\mathbf{x}_j)^T |\n\\leq \\lambda_1.\n\\label{eq:sub3}\n\\end{align}\nIt is equivalent to the condition of lasso that sets a regression coefficient to zero. \nNote that if $\\lambda_2=\\lambda_3 = 0$, \nwe have sparsity only at the individual levels, and our model is the same as lasso.\nWhen a group of coefficients contains\nboth relevant and irrelevant ones, \nwe can set the irrelevant coefficients to zero using Eq. (\\ref{eq:sub3}).\n\n\nWe briefly mention the three tuning parameters ($\\lambda_1$, $\\lambda_2$, $\\lambda_3$) \nwhich can be determined by cross validation. \nIt is often computationally expensive to search for optimal parameters in 3-dimensional grid.\nIn practice, instead, we use the following tuning parameters:\n$\\lambda_2 = \\lambda_2'\\lambda_3'$  and \n$\\lambda_3 = (1-\\lambda_2')\\lambda_3'$.\nHere $\\lambda_2$ configures the mixing proportion of\ninput and output group structures, and \n$\\lambda_3$ is the scaling factor that determines\nthe degree of penalization for the input and output groups.\nIn this setting, we also have three regularization parameters, however,\nit helps us to reduce the search space of the \ntuning parameters as we know the range of $\\lambda_2'$ ($0\\leq \\lambda_2' \\leq 1$). \n\nLet us discuss the statistical and biological benefits of\nour model.\nFirst, our model can capture rich structured sparsity in $\\mathbf{B}$.\nThe structured sparsity patterns include zero (sub)rows, zero (sub)columns and\nzero blocks of $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$. \nIt is impossible to have such rich sparsity patterns if we\nuse one part of information on either input or output side. \nFor example, group lasso \\cite{yuan2006model} \nor multi-task lasso \\cite{obozinski2006multi} consider \nstructured sparsity patterns in either rows or columns in $\\mathbf{B}$. \nSecond, our model is robust to the \ngroups which \ncontain both relevant and irrelevant coefficients.\nIf predefined groups of inputs and outputs\nare unreliable, our model may still work \nsince the irrelevant coefficients can be set to zero individually via $L_1$ penalty \neven when their groups are not jointly set to zero.\nThird, the grouping effects induced by our model in Eq. (\\ref{eq:sub1}, \\ref{eq:sub2})\nshow that we can use \nthe correlation effects between input and output groups.\nWhen we have reliable input and output groups,\nthe advantage from the structural information will be\nfurther enhanced by the correlation effects in addition to\nthe sum of the benefits of both input and output groups.\n\nWhen applied to GWA mapping of eQTLs, our model offers a number of desirable properties.\nIt is likely that our model can detect association SNPs with low signal-to-noise ratio by \ntaking advantage of rich structural information.\nIn GWA studies, one of the main challenges \nis to detect SNPs having weak signals with limited sample size.\nIn complex diseases such as cancer and diabetes, biologists \nbelieve that multiple SNPs are jointly responsible for diseases but not necessarily with\nstrong marginal effects \\cite{mccarthy2008genome}. \nHowever,\nsuch causal SNPs are hard to detect mainly due to insufficient number of samples.\nOur model can deal with this challenge by taking advantage of both input and output group\nstructures.\nFirst, by grouping inputs (or SNPs), we can increase the signal-to-noise ratio.\nSuppose each SNP has small signal marginally, \nif a group of coefficients is relevant,\ntheir joint strength will be increased, and it is unlikely that they are jointly set to zero.\nOn the other hand, if a group of coefficients is irrelevant, \ntheir joint strength will still be small, and it is likely that they are set to zero.\nSecond, taking advantage of the output groups, we can share information across \nthe correlated outputs, \nand it decreases the sample size required for successful support recovery \n\\cite{negahban2011simultaneous}.\nOverall, to detect causal SNPs having small effects,\nour model increases signal-to-noise ratio by grouping \nthe SNPs, and simultaneously\ndecreases the required number of samples by grouping\nphenotypic traits.\n\n\n\n\nUnfortunately, the optimization problem resultant from Eq.  (\\ref{equ:reg5}) is non-trivial.\nOne may find out that each $\\beta_k^j$ appears \nin all the three penalties of Eq. (\\ref{equ:reg5_b} -- \\ref{equ:reg5_d}). \nThus, our structured regularizer is non-separable, which makes \nsimple coordinate descent optimization inapplicable. \nThe overlaps between\/within input and output groups\nadd another difficulty.\nFurthermore,\nwe must induce appropriate sparsity patterns (i.e., exact zeros)\nin addition to the minimization of Eq.  (\\ref{equ:reg5}), \ntherefore approximate methods based on merely relaxing the shrinkage functions are not appropriate.\nIn the following section, we propose ``hierarchical group thresholding'' method (HiGT)\nthat efficiently solves our optimization problem\nwith hierarchically organized thresholding operations.\n\n\n\n\n\n\\section{Optimization method}\n\\label{sec:optimization}\n\n\\begin{figure} \n\\centering\n\\subfigure[]{\\includegraphics[width=0.8\\textwidth]{all_zeros2.eps}}\n\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{opt_strategy4.eps}}\n\\caption{\nSparsity patterns of $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$ and a DAG constructed with the sparsity patterns.\nThe shaded area shows zero entries.\n(a) All possible zero patterns of $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$ \nthat can be induced by Eq. (\\ref{equ:reg5_b},\\ref{equ:reg5_c},\\ref{equ:reg5_d}) when \n$\\mathbf{g} = \\{1,2\\}$ and $\\mathbf{h} = \\{1,2\\}$.\n(b) An example of a DAG that \ncontains the zero patterns of $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$.\nThe root node contains zero pattern for $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}=\\bm{0}$,\nand the internal nodes represent the zero patterns for $\\bm{\\beta}_{\\mathbf{h}}^{j}=\\bm{0}$ (one column is zero) \nor $\\bm{\\beta}_{k}^{\\mathbf{g}}=\\bm{0}$ (one row is zero). The leaf nodes denote $\\beta_{k}^{j}=0$.\nIn the DAG, the zero pattern of children nodes should be a subset of \ntheir parent nodes' zero patterns.\n}\n\\label{fig:all_zeros}\n\\end{figure}\n\nIn this section, we propose our method to optimize Eq. (\\ref{equ:reg5}).\nWe start with a non-zero $\\mathbf{B}$ initialized by other methods (e.g. ridge regression), \nand always reduce the set of non-zero $\\beta_k^j$s using thresholding operations \nas our procedure proceeds.\nOur framework is an iterative procedure consisting \nof two steps. First, we set the groups (or individual) of regression coefficients\nto zero by checking optimality conditions (called thresholding) \nas we walk through a predefined \ndirected acyclic graph (DAG). \nWhen we walk though the nodes in the DAG, \nsome $\\beta_k^j$s might not achieve zero. \nSecond, we update only these non-zero $\\beta_k^j$s using any available optimization techniques. \n\n\n\n\n\nLet us first \ncharacterize the zero patterns induced by Eq. (\\ref{equ:reg5_b} -- \\ref{equ:reg5_d}).\nWe separately consider a block of $\\mathbf{B}$ which consists of \none input group $(\\mathbf{g}\\in \\mathcal{G})$ and one output group $(\\mathbf{h}\\in \\mathcal{H})$.\nOur observation tells us that there are grouping effects \n(to be zero simultaneously) for each $\\mathbf{g}$ and $\\mathbf{h}$: \n$\\bm{\\beta}_k^{\\mathbf{g}}=\\bm{0}$ \nand $\\bm{\\beta}_{\\mathbf{h}}^{j}=\\bm{0}$. \nWe also have $\\mathbf{B}_{{\\mathbf{h}}}^{{\\mathbf{g}}} = \\bm{0}$\nwhen $\\bm{\\beta}_k^{\\mathbf{g}} = \\bm{0}, \\; \\forall k\\in \\mathbf{h}$\nor $\\bm{\\beta}_{\\mathbf{h}}^{j} = \\bm{0}, \\; \\forall j \\in \\mathbf{g}$. \nEach covariate can also be zero, i.e, $\\beta_k^j = 0$\ndue to the $\\ell_1$ penalty in Eq. (\\ref{equ:reg5_b}).\nFigure \\ref{fig:all_zeros}(a) shows \nall the possible\nzero patterns of $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$ induced \nby Eq. (\\ref{equ:reg5_b} -- \\ref{equ:reg5_d}). \nGiven these sparsity patterns, \nto induce structured sparsity, \none might be able to check whether or not\nthese zero patterns satisfy optimality conditions \nand discard irrelevant covariates accordingly. \nHowever, this approach may be inefficient as it \nneeds to examine the large number of \nzero patterns. \nInstead, to efficiently check the zero patterns,\nwe will construct a DAG, and exploit the inclusion relationships\nbetween the zero patterns.\nThe main idea is that we want to be able to check all zero patterns \nby traversing the DAG while avoiding unnecessary optimality checks.\n\n\nIn Figure \\ref{fig:all_zeros}(b), we show an example of the DAG for\n$\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$ when $\\mathbf{g} = \\{1,2\\}$  \nand $\\mathbf{h} = \\{1,2\\}$. \nWe denote the set of all possible zero patterns of $\\mathbf{B}$ \nby $\\mathcal{Z} = \\{Z_1,\\ldots,Z_{|\\mathcal{Z}|}\\}$. \nFor example, $Z_1$ can be a zero pattern for \n$\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}} = \\bm{0}$ (the root node in Figure \\ref{fig:all_zeros}(b)).\nLet us denote $\\mathbf{B}(Z_t)$ by the coefficients of $\\mathbf{B}$ corresponding to $Z_t$'s zero pattern.\nThen we define the DAG as follows:\nA node is represented by $Z \\in \\mathcal{Z}$, \nand there exists a directed edge from \n$Z_1 \\in \\mathcal{Z}$ to $Z_2 \\in \\mathcal{Z}$ if and only if $Z_1 \\supset Z_2$\nand $\\nexists Z \\in \\mathcal{Z}: Z_1 \\supset Z \\supset Z_2$.\nFor example, in Figure \\ref{fig:all_zeros}(b), the zero patterns\nof the nodes in the second level include the zero patterns of their children.\nIn general, when we have multiple input and output groups,\nwe can generate a DAG for each $\\mathbf{B}_{{\\mathbf{h}}}^{{\\mathbf{g}}}$ separately \nand then connect all the DAGs to the root node for $\\mathbf{B} = \\bm{0}$.\nThis graph is originated from Hasse diagram \\cite{cameron1994combinatorics}\nand it was previously utilized for finding a minimal set of \ngroups for inducing structured sparsity \\cite{jenatton2009structured}. \n\nWe can readily observe that our procedure has the following properties:\n\\vspace{-0.2cm}\n\\begin{itemize}\n\\item Walking through the DAG, we can check all possible zero patterns\nexplicitly without resorting to heuristics or approximations.\n\\item If $\\mathbf{B}(Z) = \\bm{0}, \\; Z \\in \\mathcal{Z}$, we know that\nall the descendants of $Z$ are also zero due to\nthe inclusion relations of the DAG. Hence, we can ``skip''\nto check the optimality conditions that the descendants of \n$Z$ are zero.  \n\\end{itemize}\n\\vspace{-0.2cm}\n\nConsidering these properties,\nwe develop our optimization framework for the following reasons.\nFirst, we can achieve accurate zero patterns in $\\mathbf{B}$\nsince we check all possible sparsity patterns when walking through the DAG. \nSecond, if $\\mathbf{B}$ is sparse,\nour framework is very efficient since \nwe can skip the optimality checks for many zero patterns in $\\mathcal{Z}$. \nMostly we will check only nodes\nlocated at the high levels of the DAG.\nThird, our framework is simple to implement. All we need\nis to check whether each node in the DAG attains zero and \nupdate non-zero $\\beta_k^j$s only when necessary.\n\nSpecifically, our hierarchical group-thresholding employs the following procedure:\n\n\\vspace{-0.1cm}\n\\begin{enumerate}\n\\vspace{-0.1cm}\n\\item Initialize a non-zero $\\mathbf{B}$ using any available methods (e.g. ridge regression).\n\\vspace{-0.1cm}\n\\item Construct a DAG that contains all zero patterns of $\\mathbf{B}$  \nthat can be induced by the penalty in Eq. (\\ref{equ:reg5_b}, \\ref{equ:reg5_c}, \\ref{equ:reg5_d}).\n\\vspace{-0.1cm}\n\\item \n\\label{algo:iter1}\nUse depth-first-search (DFS) to traverse the DAG, and check the optimality conditions\nto see if the zero patterns at each node $Z$ achieve zero.\nIf $\\mathbf{B}(Z) = 0$ \nor $Z$ satisfies the optimality condition to be zero, \nset $\\mathbf{B}(Z) = 0$,\nskip the descendants of $Z$, and visit the next node according to the DFS order.\n\\vspace{-0.1cm}\n\\item \n\\label{algo:iter2}\nFor those of $\\beta_k^j$'s which did not achieve zero in the previous step, \nupdate the coefficients of the non-zero $\\beta_k^j$'s using \nany available optimization algorithms.\n\\vspace{-0.1cm}\n\\item Iterate step \\ref{algo:iter1} and \\ref{algo:iter2} until Eq. (\\ref{equ:reg5}) converges.\n\\end{enumerate}\n\\vspace{-0.1cm}\n\n\n\n\nBellow we briefly present the derivations of the three\ningredients of our optimization framework that include\n1) the construction of a DAG, \n2) the optimality condition of each $Z \\in \\mathcal{Z}$ in the DAG and \n3) the rule for updating non-zero regression coefficients.\nOur optimization method is summarized in Algorithm \\ref{alg:hGroupThres}.\n\n\n\n\n\n\n\\begin{algorithm}[!ht]\n\\caption{Hierarchical group-thresholding method for Eq. (\\ref{equ:reg5})}\n{\\scriptsize\n\\begin{algorithmic}\n\\label{alg:hGroupThres}\n\\STATE $\\mathbf{B} \\leftarrow \\mbox{coefficients estimated by ridge regression}$\n\\STATE $\\mathcal{G} \\leftarrow \\mbox{groups of inputs}$\n\\STATE $\\mathcal{H} \\leftarrow \\mbox{groups of outputs}$\n\\STATE $D(\\mathcal{Z},\\mathcal{E}) \\leftarrow \\mbox{DAG including all zero patterns}$\n\\STATE $\\{Z_{(1)},Z_{(2)},\\ldots,Z_{(|\\mathcal{Z}|)}\\}  \\leftarrow \\mbox{DFS order of $\\mathcal{Z}$ in $D$}$\n\\REPEAT \n\\STATE $t \\leftarrow 1$\n\\WHILE{$t\\leq |\\mathcal{Z}|$}  \n\\IF{$Z_{(t)}$ contains $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}=\\bm{0}$}  \n\\STATE c $\\leftarrow$ Eq. (\\ref{equ:rule1})\n\\ELSIF{$Z_{(t)}$ contains $\\bm{\\beta}_k^{\\mathbf{g}}=\\bm{0}$} \n\\STATE c $\\leftarrow$ Eq. (\\ref{equ:rule2})\n\\ELSIF{$Z_{(t)}$ contains $\\bm{\\beta}_{\\mathbf{h}}^{j}=\\bm{0}$} \n\\STATE c $\\leftarrow$ Eq. (\\ref{equ:rule3})\n\\ELSIF{$Z_{(t)}$ contains $\\beta_{k}^{j}=0$} \n\\STATE c $\\leftarrow$ Eq. (\\ref{equ:rule4})\n\\ENDIF\n\\IF{c holds (condition for $\\mathbf{B}(Z_{(t)}) = \\bm{0}$) or $\\mathbf{B}(Z_{(t)}) = \\bm{0}$}\n\\STATE $\\mathbf{B}(Z_{(t)}) = \\bm{0}$ (Set zero to $Z_{(t)}$'s zero pattern)\n\\STATE $t \\leftarrow $ DFS order of $t'$ such that \n$Z_{(t')}$ is not a descendant of $Z_{(t)}$, $t'>t$\nand $\\nexists t{''}: t'>t{''}>t$\n(Skip the descendants of $Z_{(t)}$)\n\\ELSIF{c $=$ Eq. (\\ref{equ:rule4})}\n\\STATE Update $\\beta_k^j$ using Eq. (\\ref{equ:update_rule})\n(Updating non-zero regression coefficients)\n\\STATE $t \\leftarrow t+1$\n\\ELSE\n\\STATE $t \\leftarrow t+1$\n\\ENDIF\n\\ENDWHILE\n\\UNTIL{convergence}\n\\end{algorithmic}\n}\n\\end{algorithm}\n\\vspace{-0.3cm}\n\n\n\n\n\n\n\\paragraph{Construction of the DAG}\nTo generate the DAG, first we define the set of nodes\n$\\mathcal{Z}$. For each block of $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$, \nwe are interested in the four types of zero patterns as follows:\n\n\n\\vspace{-0.4cm}\n\\begin{enumerate}\n\\item $\\mathbf{B}_{{\\mathbf{h}}}^{{\\mathbf{g}}}$ is zero: $\\mathbf{B}_{{\\mathbf{h}}}^{{\\mathbf{g}}} = \\bm{0}$.\n\\item One row in $\\mathbf{B}_{{\\mathbf{h}}}^{{\\mathbf{g}}}$ is zero:\n$\\bm{\\beta}_{k}^{{\\mathbf{g}}} = \\bm{0}$, $k\\in {\\mathbf{h}}$.\n\\item One column in $\\mathbf{B}_{{\\mathbf{h}}}^{{\\mathbf{g}}}$ is zero:\n$\\bm{\\beta}_{{\\mathbf{h}}}^{j} = \\bm{0}$, $j\\in {\\mathbf{g}}$.\n\\item One regression coefficient in $\\mathbf{B}_{{\\mathbf{h}}}^{{\\mathbf{g}}}$ is zero:\n$\\beta_{k}^{j} = 0$, $k\\in {\\mathbf{h}}$ and $j\\in {\\mathbf{g}}$.\n\\end{enumerate}\n\\vspace{-0.3cm}\n\nThese zero patterns of \n$\\mathbf{B}$ are shown in Figure \\ref{fig:all_zeros}(b)\nwhen $|\\mathbf{g}| = |\\mathbf{h}| = 2$.\nFor example, Case 2 and 3 correspond to the nodes at the second level \nof the DAG.\nFor all $\\mathbf{g} \\in \\mathcal{G}$ and $\\mathbf{h} \\in \\mathcal{H}$, \nwe can define nodes $Z \\in \\mathcal{Z}$ using the above zero patterns.\nThen we need to determine the edges of the DAG by \ninvestigating the relations of the nodes.\nWe can also easily see that there exists the relationship among the zero patterns:\n$\\mbox{Case } 1 \\supset \\mbox{Case } 2, \\mbox{Case } 3  \\supset \\mbox{Case } 4$.\nGiven the zero patterns and their relations, \nwe create a directed\nedge $Z_1 \\rightarrow Z_2$ \nif and only if $Z_1 \\supset Z_2$\nand $\\nexists Z \\in \\mathcal{Z}: Z_1 \\supset Z \\supset Z_2$.\nIn Figure \\ref{fig:all_zeros}(b) we show an example of the DAG. \nFinally, we make a dummy root node and generate an edge from the\ndummy node to the root of all DAGs for $\\mathbf{B}_{{\\mathbf{h}}}^{{\\mathbf{g}}}= \\bm{0}$.\n\n\n\n\\paragraph{Optimality conditions for structured sparsity patterns}\nGiven a block of $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}}$, \nhere we show optimality conditions for the four sparsity patterns:\n(1) $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}} = \\bm{0}$,\n(2) $\\bm{\\beta}_k^{\\mathbf{g}} = \\bm{0}$,\n(3) $\\bm{\\beta}_{\\mathbf{h}}^j = \\bm{0}$, and\n(4) $\\beta_k^j = 0$, \n($j \\in \\mathbf{g}, k \\in \\mathbf{h}, \\mathbf{g} \\in \\mathcal{G}, \\mathbf{h} \\in \\mathcal{H}$).\nIn Figure \\ref{fig:all_zeros}(b), the root node\ncorresponds to the first case, the nodes at the second level \ncorrespond to the second and third case, and the leaf nodes correspond to\nthe fourth case.\nOur derivation of the following optimality conditions \nuse the fact that all zero coefficients are fixed, as\nit makes it simple to deal with overlapping groups.\nWe denote the column and row indices of zero entries by \n$\\bm{\\eta} = \\{j : \\beta_{k}^j = 0, \\; \\forall j \\in \\mathbf{g}, \\; \\forall k \\in \\mathbf{h}\\}$  and\n$\\bm{\\gamma} = \\{k : \\beta_{k}^j = 0, \\; \\forall j \\in \\mathbf{g}, \\; \\forall k \\in \\mathbf{h}\\}$.\n\n\nFirst, the optimality condition for the first case is as follows: $\\mathbf{B}_{\\mathbf{h}}^{\\mathbf{g}} = \\bm{0}$ if\n\\begin{align}\n\\label{equ:rule1}\n\\sum_{k \\in \\mathbf{h} - \\bm{\\gamma}} \\sum_{j \\in \\mathbf{g} - \\bm{\\eta}} \n\\left\\{ {\\mathbf{r}}_k^j (\\mathbf{x}_j)^T  - \\lambda_1 s_k^j \\right\\}^2\n\\leq \\left(\\lambda_2 \\sqrt{|\\mathbf{h}|} \n+ \\lambda_3 \\sqrt{|\\mathbf{g}|}\\right)^2, \n\\end{align}\nwhere\n\\[s_k^j = \\left\\{ \n\\begin{array}{l l}\n  \\frac{{\\mathbf{r}}_k^j (\\mathbf{x}_j)^T  }{\\lambda_1} \n  & \\mbox{if $\\left|\\frac{ {\\mathbf{r}}_k^j (\\mathbf{x}_j)^T}{\\lambda_1}\\right| \\leq 1$}\\\\ \n\tsign\\left(\\frac{ {\\mathbf{r}}_k^j (\\mathbf{x}_j)^T}{\\lambda_1}\\right) \n  & \\mbox{if $\\left|\\frac{ {\\mathbf{r}}_k^j (\\mathbf{x}_j)^T}{\\lambda_1}\\right| > 1$}.\\\\  \n  \\end{array} \\right.\\]\nIt is derived using KKT condition in Eq. (\\ref{eq:subgradients})\nand Cauchy-Schwarz inequality.\n  \nThe second case of structured sparsity, i.e, $\\bm{\\beta}_k^{\\mathbf{g}}=\\bm{0}$ is achieved if\n\\begin{align}\n\\label{equ:rule2}\n\\sum_{j \\in {\\mathbf{g} - \\bm{\\eta}}} \\left\\{{\\mathbf{r}}_k^j(\\mathbf{x}_j)^T -\\lambda_1 s_k^j \\right\\}^2\n\\leq  \\lambda_2^2,  \n\\end{align}\nand the optimality condition for the third case, i.e, $\\bm{\\beta}_{\\mathbf{h}}^{j}=\\bm{0}$ is\n\\begin{align}\n\\label{equ:rule3}\n\\sum_{k \\in \\mathbf{h} - \\bm{\\gamma}} \\left\\{{\\mathbf{r}}_k^j(\\mathbf{x}_j)^T - \\lambda_1 s_k^j \\right\\}^2\n\\leq \\lambda_3^2.\n\\end{align}\nThese conditions can be established using KKT condition in Eq. (\\ref{eq:subgradients})\nfixing all the zero coefficients.\nFinally, assuming that \n$\\bm{\\beta}_{\\mathbf{h}}^{j} \\neq \\bm{0}$ and $\\bm{\\beta}_{k}^{\\mathbf{g}} \\neq \\bm{0}$,\nthe fourth case has the optimality condition of \n\\begin{align}\n\\label{equ:rule4}\n| {\\mathbf{r}}_k^j (\\mathbf{x}_j)^T| \\leq \\lambda_1.\n\\end{align}\n \n\\paragraph{Update rule for nonzero coefficients}\nIf all the above optimality conditions do not hold, we \nknow that $\\beta_k^j\\neq 0$.\nIn this case, the gradient of Eq. (\\ref{equ:reg5}) with respect to $\\beta_k^j$ exists, and\nwe can update $\\beta_k^j$ using any coordinate descent procedures.\nWith a little bit of algebra, we derive the following update rule: $\\beta_k^j = \\hat{\\beta}_{k,-}^{j} + \\hat{\\beta}_{k,+}^{j}$ where\n{\\footnotesize\n\\begin{align}\n\\label{equ:update_rule}\n\\hat{\\beta}_{k,-}^{j} &= \\min\\left[0,   \\left( 1 +  \n\\sum_{j\\in \\mathbf{g}} \\frac{\\lambda_2}{\\left\\|\\bm{\\beta}_k^{\\mathbf{g}}\\right\\|_2} +  \n\\sum_{k\\in \\mathbf{h}} \\frac{\\lambda_3}{\\left\\|\\bm{\\beta}_{\\mathbf{h}}^{j}\\right\\|_2} \\right)^{-1} \\left\\{\\mathbf{r}_k (\\mathbf{x}_j)^T+\\lambda_1 \\right\\}\\right],\n\\\\ \n\\hat{\\beta}_{k,+}^{j} &= \\max\\left[0, \\left( 1+  \n\\sum_{j\\in \\mathbf{g}} \\frac{\\lambda_2}{\\left\\|\\bm{\\beta}_k^{\\mathbf{g}}\\right\\|_2} +  \n\\sum_{k\\in \\mathbf{h}} \\frac{\\lambda_3}{\\left\\|\\bm{\\beta}_{\\mathbf{h}}^{j}\\right\\|_2} \\right)^{-1} \\left\\{ \\mathbf{r}_k (\\mathbf{x}_j)^T-\\lambda_1 \\right\\}\\right].\n\\nonumber\n\\end{align}\n}\n\n\nWe close this section by summarizing the desirable properties of our optimization method.\nFirst, when $\\mathbf{B}$ is sparse, our optimization procedure is very fast.\nWe take advantage of not only the hierarchical structure of the DAG, \nbut also the simple forms of the optimality conditions with residuals.\nIf we keep track of the residuals, we can efficiently check \nthe optimality conditions for each sparsity pattern.\nSecond, our thresholding operations check all possible sparsity patterns,\nresulting in appropriate structured sparsity in $\\mathbf{B}$.\nIt is important for eQTL mapping since\nthe coefficients for irrelevant SNPs can be set to exactly zero.\nThird, our optimization method can deal with overlaps between\/within \nthe coefficients for input groups ($\\bm{\\beta}_k^{\\mathbf{g}}$'s)\nand output groups ($\\bm{\\beta}_{\\mathbf{h}}^{j}$'s). \nSince input or output groups may overlap, and they must be considered simultaneously, \nthis property of our method is essential. \nFinally, unlike some previous methods\n\\cite{yuan2006model,tibshirani1996regression},\nwe make no use of the assumption that the design matrix $\\mathbf{X}$ is orthonormal ($\\mathbf{X}^T\\mathbf{X} = \\mathbf{I}$).\nThis dropping of the assumption is desirable for eQTL mapping in particular as\ncovariates (SNPs) are highly correlated due to linkage disequilibrium.\nIf one uses orthonormalization as a preprocessing step to make $\\mathbf{X}$ orthonormal,\nthere is no guarantee that the same solution for the original problem is attained \\cite{friedman2010note}.\t\n\n\n\n\\section{Dealing with structures inducing higher-order effects}\n\\label{sec:model_interaction}\n\nSo far, we have been dealing with input and output structures in the context of multi-variate and multi-task linear regression where the influences from the covariates on the responses are additive. When higher interactions take place among covariates, which is known as epistasis and is prevalent \nin genetic associations \\cite{carlson2004mapping}, a common approach to model such \neffects is polynomial regression \\cite{montgomery2001introduction}, \nwhere higher-order terms of the covariates are included as additional regressors. However, in high-dimensional problems \nsuch as the one studied in this paper, this strategy is infeasible even for 2nd-order polynomial regression because, \ngiven say, even a standard genome dataset with $\\sim 10^5$ SNPs, one is left with $\\sim 10^{10}$ regressors \nwhich is both computationally and statistically unmanageable. In this section, we briefly show how to circumvent \nthis difficulty using structured regularization based on prior information of covariate interactions. \nThis strategy is essentially a straightforward generalization \nof the ideas in Section \\ref{sec:methods} to a \npolynomial regression setting using a special type of structure encoded by a graph. \nTherefore all the algorithmic solutions developed in Section \\ref{sec:optimization} \nfor the general optimization problem in Section \\ref{sec:methods} still apply here.  \n\nFollowing common practice in GWA literature, here we consider only 2nd-order interactions between SNP pairs. \nInstead of including all SNP pairs as regressors, \nwe employ a synthetic genetic interaction network \\cite{costanzo2010genetic} \nto define a relatively small candidate set ${\\bf U}$ of interacting SNP pairs. \nA synthetic genetic interaction network is derived from biological evidences of pairwise functional interactions between genes, \nsuch as double knockout experiments~\\cite{tong2004global,koh2009drygin,costanzo2010genetic,boone2007exploring}. \nIt contains information about the pairs of genes whose mutations affect the phenotype only when the mutations are present on both genes, \nand this represents a set of {\\it ground-truth} interaction effects. \nGiven such a network, we consider only those pairs of SNPs that are physically \nlocated in the genome near the genes that interact in the network within a certain distance. \nA 2nd-order regressor set ${\\bf U}$ generated by this scheme is not only much smaller than an exhaustive pair-set, \nbut also biologically more plausible.\nNote that\nit is possible to include other sets of SNP pairs from other resources in our candidate set. \nFor example, in our experiments, we also added SNP pairs that passed two-locus epistasis test\nwith p-value $<10^{-5}$ into the set ${\\bf U}$.\n\nAfter finding the candidate SNP pairs, \nwe generate the group of SNPs or interacting SNP pairs in two steps. \nIn the first step, we find highly interconnected subgraphs (or clusters) \nfrom the genetic interaction network\nusing any graph clustering algorithms.\nIn our experiments, we used MCODE algorithm \\cite{bader2003automated}  \nfor clustering the network.  \nIn the second step, we group all the SNPs or SNP pairs that are linked to the genes\nin a cluster. We linked the genes and SNPs based on physical locations in the genome.\nFor example, if a SNP is located nearby a gene within a certain \ndistance (e.g. $<$500bp), they are linked together.\nFinally, we define individual SNPs in the $m$th group as ${\\mathbf{g}}_m \\in \\mathcal{G}$ \nand SNP pairs in the $m$th group as ${\\mathbf{l}}_m \\in \\mathcal{L}$.\nWe then look for associations between inputs\/input-pairs and outputs via Eq. (\\ref{equ:reg6}): \n\n\\begin{subequations}\n\\label{equ:reg6}\n\\begin{align}\n\\min \\frac{1}{2}\n\\sum_{k=1}^K\\sum_{i=1}^{N} \n& \\left( y_k^i - \\sum_{j=1}^J{\\beta_k^{j} x_j^i } -\n\\sum_{(r, s) \\in {\\bf U}}{\\beta_k^{rs}  x_r^i x_s^i } \n\\right)^2 \n\\label{equ:reg6_a}\n\\\\\n&  + \\lambda_1   \\sum_{k=1}^K \\sum_{j=1}^J{|\\beta_k^j|} \n\\label{equ:reg6_b}     \\\\\n  &  + \\lambda_2  \n\\sum_{k=1}^K  \\left(\\sum_{m=1}^{|\\mathcal{G}|}{\\sqrt{\\sum_{j\\in {\\mathbf{g}}_m}({\\beta_k^j})^2}} +\n\\sum_{m=1}^{|\\mathcal{L}|}{\\sqrt{\\sum_{(r, s) \\in {\\mathbf{l}}_m}{(\\beta_k^{rs}})^2}}\\right)\n\\label{equ:reg6_c}   \\\\\n  & + \\lambda_3  \\left(\\sum_{j=1}^J\\sum_{m=1}^{|\\mathcal{H}|} \n   \\sqrt{\\sum_{k \\in {\\mathbf{h}}_m}{({\\beta_k^j})^2}} +\n   \\sum_{(r, s) \\in {\\bf U}} \\sum_{m=1}^{|\\mathcal{H}|} \n   \\sqrt{\\sum_{k \\in {\\mathbf{h}}_m}{({\\beta_k^{rs}})^2}} \\right)\n  \\label{equ:reg6_d}\\\\\n&+ \\lambda_4 \\sum_{k=1}^K \\sum_{\\begin{subarray}{l}\n       (r, s) \\in {\\bf U} \n       \\end{subarray}}{|\\beta_k^{rs}|}.\n\\label{equ:reg6_e}\n\\end{align}\n\\end{subequations}\nwhere $\\mathcal{G}$ is the set of input groups for \nmarginal terms and $\\mathcal{L}$ is the set of \ninput groups for pairwise interaction terms.\nHere, we use \ntwo tuning parameters for $L_1$ penalty\ndepending on whether a covariate is modeling an individual effect ($\\lambda_1$) \nor interaction effect ($\\lambda_4$) because\nthey might need different levels of sparsity.\nNote that this problem is identical to Eq. (\\ref{equ:reg5}) if we\ntreat interaction terms $x_r^i x_s^i$ as additional covariates, and hence\nour optimization method presented in Section \\ref{sec:optimization} is applicable to Eq. (\\ref{equ:reg6}).  \nHowever, Eq. (\\ref{equ:reg6}) will be more computationally expensive than\nEq. (\\ref{equ:reg5}) since Eq. (\\ref{equ:reg6}) has\na larger number of covariates in $\\mathbf{B}$ including both marginal and interaction terms \nand additional tuning parameter $\\lambda_4$.\n\n\n\\section{Simulation Study}\n\\label{subsec:simulstudy}\n\nIn this section we validate our proposed method using simulated genome\/phenome datasets, and examine the effects of simultaneous use of input and output structures \non the detection of true non-zero regression coefficients.\nWe also evaluate the speed and the performance of our optimization method\nfor support recovery in comparison to two other alternative methods.\nFor the comparison of optimization methods,\nwe selected smoothing proximal gradient method \\cite{chen2010efficient} \nand the union of supports \\cite{jacob2009group} since both methods\nare in principle able to use input\/output structures and handle overlapping groups.\n\n\n\n\nThe simulated datasets with $J=120, K=80$, and $N=100$ are generated as follows.\nFor generating $\\mathbf{X}$, we first selected 60 input covariates from a uniform distribution over $\\{0,1\\}$\nwhich indicates major or minor genotype. \nWe then simulated 60 pairwise interaction terms $(x_j^i \\times x_{j'}^i)$ by randomly selecting input-pairs from the 60 covariates mentioned above. Pooling the 60 marginal terms and 60 pairwise interaction terms resulted in a input space of 120 dimensions. \nWe also defined input and output groups as follows (for the sake of illustration and comprehension  convenience, here our input and output groups correspond to variables to be jointly selected rather than jointly shrunk, the shrinkage penalty in our regression loss can be defined on the complements of these groups): \n\n\n\n{\\tiny\n\\begin{align}\n& \\rlap{$\\overbrace{\\phantom{5,\\ldots,9,10}}^{\\mathbf{g}_1}$}\n5,\\ldots, \\underbrace{9,10,\\ldots,15}_{\\mathbf{g}_2}, \\ldots,\n\\rlap{$\\overbrace{\\phantom{25,\\ldots,29,30,31,32}}^{\\mathbf{g}_3}$}\n25,\\ldots,\\underbrace{29,30,31,32,\\ldots, 37}_{\\mathbf{g}_4}, \\ldots,  \n\\rlap{$\\overbrace{\\phantom{50,\\ldots,54,55,56,57}}^{\\mathbf{g}_5}$}\n50,\\ldots,\\underbrace{54,55,56,57,\\ldots,60}_{\\mathbf{g}_6}, \\ldots,   \n\\rlap{$\\overbrace{\\phantom{75,\\ldots,80,\\ldots,87}}^{\\mathbf{g}_7}$}\n75,\\ldots,\\underbrace{80,\\ldots,87,\\ldots,94}_{\\mathbf{g}_8}, \\ldots,\n\\rlap{$\\overbrace{\\phantom{104,\\ldots,109,110,111}}^{\\mathbf{g}_9}$}\n104,\\ldots,\\underbrace{109,110,111,\\ldots,116}_{\\mathbf{g}_{10}} \\nonumber \\\\\n& \\rlap{$\\overbrace{\\phantom{1,\\ldots,4,5}}^{\\mathbf{h}_1}$}\n1,\\ldots,\\underbrace{4,5,\\ldots,10}_{\\mathbf{h}_2}, \\ldots,\n\\rlap{$\\overbrace{\\phantom{12,\\ldots,17,18,19,20}}^{\\mathbf{h}_3}$}\n12,\\ldots,\\underbrace{17,18,19,20,\\ldots, 25}_{\\mathbf{h}_4}, \\ldots,  \n\\rlap{$\\overbrace{\\phantom{46,\\ldots,56,\\ldots,63}}^{\\mathbf{h}_5}$}\n46,\\ldots,\\underbrace{56,\\ldots,63,\\ldots, 70}_{\\mathbf{h}_6}, \\ldots,  \n\\underbrace{75,\\ldots,80}_{\\mathbf{h}_7}, \\nonumber\n\\end{align}\n}\nwhere the numbers within a bracket represent the indices of\ninputs or outputs for an input group $\\mathbf{g}_t,\\; t=1,\\ldots,10$, \nor an output group $\\mathbf{h}_o,\\; o=1,\\ldots,7$.\nThe inputs and outputs which did not belong to \nany groups were in a group by itself.\nWe then simulated $\\mathbf{B}$, i.e, the ground truth \nthat we want to discover.  \nWe selected non-zero coefficients so that $\\mathbf{B}$ includes \nvarious cases, e.g., overlap between input and output groups, \noverlap within input groups, and overlap within output groups.\nFigure \\ref{fig:visualize_beta}(a) shows the simulated $\\mathbf{B} \\in \\mathbb{R}^{80 \\times 120}$ where\nnon-zero coefficients are represented by black blocks.\nGiven $\\mathbf{X}$ and \n$\\mathbf{B}$, we generated $K=80$ outputs \nby $\\mathbf{Y}=\\mathbf{B} \\mathbf{X} + \\mathbf{E},$ \n$\\mathbf{E} \\sim \\mathcal{N}(\\bm{0},\\mathbf{I})$.\nWe generated 20 datasets and \noptimized Eq. (\\ref{equ:reg5}) \nusing the three methods.\nWe report the average performance using precision recall curves.\n\n\n\n\n\n\\begin{figure}[htp]\n\\vspace{-0.5cm}\n\\centering\n\\hspace{-1.7cm}\n\\includegraphics[width=0.9\\textwidth]{visualize_beta2.eps}\n\\vspace{-0.5cm}\n\\caption{An example of simulation results with \n$|\\beta_k^j| = 2, N=100, J=120$, and $K=80$. \n(a) True regression coefficient matrix.\nEstimated $\\mathbf{B}$ \nby SIOL (b) with  both input and output structures\n(c) with only input structure, and \n(d) with only output structure.\nIn \n(b-d), we show the normalized values of $|\\mathbf{B}|$.}\n\\label{fig:visualize_beta}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\\subsection{Evaluation of the Effects of Using Input and Output Structures}\n\\label{subsec:simul_exp1}\n\nWe first investigate the effects of using both input and output structures on the performance of our model.\nHere we applied our optimization method (HiGT) to the following three\nmodels with different use of structural information: \n{\\footnotesize\n\\begin{enumerate}\n\t\\item Use of both input and output structures (Eq. (\\ref{equ:reg5_b}) + Eq. (\\ref{equ:reg5_c}) + Eq. (\\ref{equ:reg5_d}))\n\t\\item Use of input structures (Eq. (\\ref{equ:reg5_b}) + Eq. (\\ref{equ:reg5_c}))\n\t\\item Use of output structures (Eq. (\\ref{equ:reg5_b}) + Eq. (\\ref{equ:reg5_d}))\n\\end{enumerate}\n}\n\nWe then observed how the use of input\/output structures affect the recovery of the \ntrue non-zero coefficients and the prediction error.\nIn Figure \\ref{fig:visualize_beta}, we visualize the examples of \nestimated $\\mathbf{B}$ by the three different models.\nFigure \\ref{fig:visualize_beta}(b) shows that the model\nwith input and output structure successfully recovered true\nregression coefficients in Figure \\ref{fig:visualize_beta}(a).\nHowever, as shown in Figure \\ref{fig:visualize_beta}(c-d),\nthe models with either input or output structure \nwere less effective to suppress noisy signals, which resulted in\nmany false positives.\n\n\n\\begin{figure}[htp]\n\\hspace{-1.7cm}\n\\includegraphics[width=1.2\\textwidth]{jasa_exp1_case1_412.eps}\n\\vspace{-1.5cm}\n\\caption{Precision recall curves on the recovery of true non-zero coefficients due to SIOL with both input and output structures (input\/output struct), regression with only input structure (input struct), and with only output structure (output struct), under three different signal strengths of true regression coefficients.\n(a) $\\beta_k^j =0.4$, (b) $\\beta_k^j =1$, and (c) $\\beta_k^j =2$.\nThe simulated data were generated with $N=100, J=120$, and $K=80$. \n}\n\\label{fig:comp_case1}\n\\vspace{-0.5cm}\n\\end{figure}\n\nFigure \\ref{fig:comp_case1} shows the precision recall curves \non the recovery of true non-zero coefficients by changing the threshold $\\tau$ for choosing relevant covariates ($|\\beta_k^j| > \\tau$), \nunder different signal strengths of 0.4, 1 and 2. \nFor all signal strengths, the model with input\/output structures \nsignificantly outperformed the other models with either input or output structure.\nThe most interesting result is that when the signal strength was very small such\nas 0.4, our model still achieved good performance by taking advantage of both structural information.\n\n\n\n\\begin{figure}[htp]\n\\hspace{-0.3cm}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.3\\textwidth]{predict2_stgh4.eps}}\n\\subfigure[]{\\includegraphics[width=0.3\\textwidth]{predict2_stgh1.eps}}\n\\subfigure[]{\\includegraphics[width=0.3\\textwidth]{predict2_stgh2.eps}}\n\\vspace{-0.5cm}\n\\caption{Comparison of the prediction error of SIOL (input\/output struct),\nwith regression under only input structure (input struct), on only output structure (output struct).\n(a) $\\beta_k^j = 0.4$, (b) $\\beta_k^j = 1$, (c) $\\beta_k^j = 2$.}\n\\label{fig:exp1_pred}\n\\vspace{-0.1cm}\n\\end{figure}\n\n\nWe also compare the prediction errors on our validation data with 280 ($20 \\times 14$) samples \n(each dataset had 14 samples for validation).\nFor computing the prediction error, we first selected \nnon-zero coefficients,\nand then recomputed the coefficients of those selected covariates using linear regression without shrinkage penalty. \nUsing the unbiased coefficients of the chosen covariates, we measured the prediction error for our \nvalidation data.\nFigure \\ref{fig:exp1_pred} shows the prediction error under different signal strengths ranging from 0.4 to 2.\nFor all signal strengths, we obtained significantly \nbetter prediction error using both input and output structures.\nWhen the signal strength was large such as 1 or 2, the use of both input and output \nstructures was especially beneficial for reducing the prediction error\nsince it helped the model to find most of the true covariates relevant to the outputs.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{jasa_exp1_case_23_merged2.eps}\n\\vspace{-0.5cm}\n\\caption{Precision recall curves on the recovery of true non-zero coefficients \nfor SIOL (input\/output struct), regression with only input structure, and with only output structure, \nunder two different sizes of input and output groups.\n(a) $|g| \\in \\{2,3\\}$, \n(b) $|g| = 5, \\; \\forall g \\in \\mathcal{G}$, and\n(c) $|h| = 5$, and \n(d) $|h| = 40, \\; \\forall h \\in \\mathcal{H}$.\nWe fixed the size of output groups for (a,b) ($|h| = 10$), and fixed\nthe size of input groups for (c,d) ($|g| = 5$).\nThe simulated data were generated with $\\beta_k^j =0.5$, $N=100$, and $J=120$.\n}\n\\label{fig:comp_case23}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\n\\paragraph{The effects of the size of input and output groups}\nFigure (\\ref{fig:comp_case23}a-\\ref{fig:comp_case23}d) demonstrates the results on\nsimulated datasets with different size of input and output groups.\nFor all group sizes, our method \nsignificantly improved the performance \nby effectively taking advantage of both input and output groups.\n\n\n\n\\subsection{Comparison of HiGT to Alternative Optimization Methods}\nIn this section, we compare the accuracy and speed of our optimization method (HiGT)  \nwith those of the two alternative methods including smoothing proximal gradient method (SPG) \n\\cite{chen2010efficient}\nand union of supports \\cite{jacob2009group}.\nBoth alternatives can handle overlapping groups.\nSpecifically, the smoothing proximal gradient method is developed to efficiently deal with \noverlapping group lasso penalty and graph-guided fusion penalty using an approximation approach.\nHowever, it may be inappropriate for our model since\nthe maximum gap between the approximated penalty and the exact penalty\nis proportional to the total number of groups $R$, where \n$R = J  |\\mathcal{H}| + K  |\\mathcal{G}|$. \nThus, when dealing with high dimensional data (e.g $J \\sim 500,000$) such as genome data, \nthe gap will be large, and the approximation method can be severely affected.\nOn the other hand, ``union of supports'' finds the support of $\\mathbf{B}$ from the union\nsupport of overlapping groups.\nTo obtain the union of supports, input variables are duplicated to \nconvert the penalty with overlap into the one with disjoint groups, and \na standard optimization technique for group lasso \\cite{yuan2006model} can be applied.\nOne of disadvantages of union of supports is that the number of duplicated input variables\nincreases dramatically when we have a large number of overlapping groups.\nIn our experiment, we considered all possible combinations of overlapping input and output groups,\nand used a coordinate descent algorithm for sparse group lasso \\cite{friedman2010note}.\n\n\n\n\n\n\\begin{figure}[htp]\n\\hspace{-1.7cm}\n\\includegraphics[width=1.2\\textwidth]{jasa_exp2_accuracy2_346.eps}\n\\vspace{-1.5cm}\n\\caption{Precision recall curves \non the recovery of true non-zero coefficients using the SIOL model via HiGT, smoothing proximal gradient method (SPG), and union of supports for optimization. Three different model sizes determined by the number of input variables were tested (due to high computational cost, results of union-of-support are only available for the smallest problem sizes tested): \n(a) $J = 30$, (b) $J = 400$, and (c) $J = 600$.\nThe simulated data were generated with $\\beta_k^j =2$, $N=100$, and $K=20$.\n}\n\\label{fig:accuracy}\n\\end{figure}\n\n\n\nFigure \\ref{fig:accuracy} shows the precision recall curves \non the recovery of true non-zero coefficients under the SIOL model using the three optimization methods. The size of the problem is controlled by increasing number of input variables (from 30 to 600).\nThe simulated data set used here was identical to the data \nin Section \\ref{subsec:simul_exp1}\nexcept that we used 20 outputs ($\\mathbf{y}_{61},\\ldots, \\mathbf{y}_{80}$) and different number of input variables.\nOne can see that our method outperforms the other alternatives for all\nconfigurations. \nOur method and smoothing proximal gradient method \nshowed similar performance when the input variable is small ($J=30$) \nbut as $J$ increases, our method significantly performed better than SPG.\nIt is consistent with our claim for the maximum \ngap between the approximated penalty and the exact penalty which is\nrelated to the number of groups.\nUnion of supports did not work well even when the number of \ninput variables is small ($J=30$) since the actual number of \ninput variables considered was very large due to the duplicated covariates, which\nseverely degraded the performance. \n\\begin{figure}[htp!]\n\\vspace{-0.3cm}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.4\\textwidth]{jasa_exp2_speed4_sample}}\n\\subfigure[]{\\includegraphics[width=0.4\\textwidth]{jasa_exp2_speed4.eps}}\n\\vspace{-0.5cm}\n\\caption{Time complexity of HiGT, SPG, and union of supports.  \nAll three methods used both input and output groups. \n(a) Computational time with different number of samples, \n(b) computational time with different number of inputs. \nWe used the same tuning parameters for all the methods ($\\lambda_1 = 0.01, \\lambda_2 = \\lambda_3 = 0.1$).\nWe did not report the times for the small number of samples and inputs\nfor our method and SPG since I\/O latency was dominant.\n}\n\\label{fig:speed}\n\\vspace{-0.3cm}\n\\end{figure}\n\nWe also compared the speed of our method with the two alternatives\nof union of supports with all possible combinations of input and output groups\nand SPG that considered both input and output groups.\nFigure \\ref{fig:speed}(a,b) show that our method converged faster than\nthe other competitors, \nand was significantly more scalable than the two alternatives.\nUnion of supports was very slow compared to our method and SPG because\nof the large number of duplicated input variables.\nOur experimental results confirms that our optimization technique is not only accurate\nbut also fast, which\ncan be explained by the use of DAG and the simple forms of optimality checks.\n\n\n\n\n\n\n\n\\section{Analysis of Yeast eQTL Dataset}\n\\label{subsec:yeast_eqtl}\nWe apply our method\nto the budding yeast (Saccharomyces cerevisiae) data \n\\cite{brem2005landscape} with 1,260 unique SNPs (out of 2,956 SNPs)\nand the observed gene-expression levels of 5,637 genes.\nAs network prior knowledge,\nwe used genetic interaction network reported in\n\\cite{costanzo2010genetic} with stringent cutoff to\nconstruct the set of candidates of SNP pairs ${\\bf U}$.\nWe follow the procedure in section \\ref{sec:model_interaction}\nto make ${\\bf U}$ with an additional set of significant SNP pairs\nwith p-value $< 10^{-5}$ computed from two-locus epistasis test.\nWhen determining the set ${\\bf U}$,\nwe assumed that a SNP is linked to \na gene if the distance between them is less than 500bp.\nWe consider it a reasonable choice for cis-effect as\nthe size of intergene regions for S. cerevisiae is 515bp \non average \\cite{sunnerhagen2006comparative}.\nAs a result, we included 982 interaction terms from \nthe interaction network \nin $\\mathbf{X}$ \nwith 1,260 individual SNPs.\nThe number of SNP pairs from two-locus epistasis test\nvaried depending on the trait.\nFor generating input structures, we processed the network data as follows.\nWe started with genetic interaction data \nwhich include 74,984 interactions between gene pairs. \nWe then extracted genetic interactions with low p-values ($<$0.001).\nGiven 44,056 significant interactions, using MCODE clustering algorithm, we found 55\ngene clusters. \nUsing the gene clusters, we generated the groups of \nindividual SNPs and pairs of SNPs according to the scheme in \nsection \\ref{sec:model_interaction}.\nFor generating output structures, we \napplied hierarchical clustering to the yeast gene expression data\nwith cutoff 0.8, resulting in 2,233 trait clusters. \n\n\n\\paragraph{Marginal Effects in Yeast eQTL dataset}\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{marginal_analysis_diff.eps}\n\\caption{Manhattan plot for association between (YER160C and YJR029W) and SNPs on chromosome 7.\nThe two genes YER160C and YJR029W share the same GO category ``transposition''.\nOur method detected SNPs which affect both two genes in this region.\nHowever, single SNP analysis did not find any associated SNPs and\nlasso found SNPs  associated only with YER160C in this region. Graph were generated using the GenAMap software \\cite{curtisgenamap}. \n}\n\\label{fig:marginal_diff}\n\\end{figure}\nWe briefly demonstrate the effects of input\/output \nstructures on the detection of eQTLs with marginal effects.\nIn general, the association results for marginal effects by \nour method, lasso and single SNP analysis \n(the later two are standard methods in contemporary GWA mapping \nthat use no structural information,\nand hence included for comparison) showed similar patterns for strong associations.\nHowever, we observed differences\nfor SNPs with small or medium sized signals. \nFor example, our results had fewer nonzero regression coefficients\ncompared to lasso. One possible explanation would be  \nthat the grouping effects induced by our model with input\/output structures\nmight have removed\nfalse predictions with small or medium sized effects.\nTo illustrate eQTLs with marginal effects, \nwe show some examples of association SNPs using GenAMap \\cite{curtisgenamap}.\nFigure \\ref{fig:marginal_diff} demonstrates a Manhattan plot on chromosome 7\nfor two genes including YER160C and YJR029W.\nBoth genes have the same GO category \n``transposition''.\nAs both genes share the same GO category,\nit is likely that they are affected by\nthe same SNPs if there exist any association SNPs for both genes.\nIn our results, we could see that the same SNPs \non chromosome 7 are associated with both genes\nas shown in Figure \\ref{fig:marginal_diff}. \nHowever, single SNP analysis did not find any significant\nassociation SNPs in the region.\nLasso detected association SNPs in the region but \nthey were associated with only YER160C rather than both of them\n(lasso plot is not shown to avoid cluttered figure).\nThis observation is interesting since it supports that our method \ncan effectively detect the SNPs \njointly associated with the gene traits by taking advantage of\nstructural information.\n\n\n\\paragraph{Epistatic Effects in Yeast eQTL dataset}\n\n\n\\label{subsubsec:epi_yeast_eqtl}\n\\begin{figure}[htp]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{circos_ours05_100_new2.eps}}\n\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{circos_pt5_100_new2.eps}}\n\\caption{Hotspots with interaction effects identified by\n(a) our method and (b) two-locus epistasis test. \nThis figure represents the yeast genome\nin a circular format. In clockwise direction, from the top\nof the circles, we show 16 chromosomes, \nwhich are separated with space and different colors.\nLines indicate interaction effects\nbetween two connected locations in the genome.\nThickness of the lines is proportional to the number of traits\naffected by the interaction effects.\nHere we show interaction effects which influence \nmore than 100 gene traits. \nThe hotspots for (a) are represented in Table \\ref{tab:epistatic_hotspot}.\nIn (b), two SNP pairs are found including\nchr16:718892-chr16:890898\n(affected genes are enriched with the GO category of\nribosome biogenesis with corrected p-value $1.6\\times10^{-36})$,\nand chr8:56246-chr9:362631 \n(affected genes are enriched with the\nGO category of vacuolar protein catabolic process with corrected p-value $1.6\\times10^{-14})$.\nThis figure was generated using Circos software \\cite{krzywinski2009circos}.\n}\n\\label{fig:epi_hotspots}\n\\vspace{-0.1cm}\n\\end{figure}\n\n\n\\begin{sidewaystable}\n\\caption{Hotspots of SNP pairs having epistatic effects in yeast identified by our method. \n}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\nHotspot & SNP1  & SNP2  &  Number of  & GO category of  & Corrected p-value of \\\\ \nlabel & location &location &  affected traits & affected traits & GO category\\\\\n\\hline \\hline\n1  & chr1:154328 & chr5:350744  &  455& ribosome biogenesis & $1.2\\times10^{-36}$\\\\\n2  &chr10:380085 &   chr15:170945  &  195& ribosome biogenesis & $1.6\\times10^{-12}$\\\\\n3 &chr10:380085 &   chr15:175594  &  185 & ribosome biogenesis & $4.1\\times10^{-12}$\\\\\n4  &chr5:222998 &   chr15:108577  &   170& response to temperature stimulus & $2.9\\times10^{-6}$  \\\\\n5  &chr11:388373 &   chr13:64970  &   155& regulation of translation & $1.8\\times10^{-32}$\\\\\n6  &chr2:499012 &   chr15:519764  &   145& vacuolar protein catabolic process & $1.4\\times10^{-7}$ \\\\\n7  &chr1:41483 &   chr3:64311  &   130&  & \\\\\n8  &chr7:141949 &  chr9:277908  &   125&  &\\\\\n9  &chr3:64311 &   chr7:312740  &  115& glycoprotein metabolic process & $1.5\\times10^{-4}$\\\\\n10  &chr12:957108 &   chr15:170945  &   110 & vacuolar protein catabolic process & $7.8\\times10^{-16}$\\\\\n11 &chr4:864542 &  chr13:64970  &   105& ribonucleoprotein complex biogenesis & $3.7\\times10^{-6}$\\\\\n\\hline\n\\end{tabular}\n\\label{tab:epistatic_hotspot}\n\\end{sidewaystable}\n\n\nNow we show the benefits of using the input\/output structures\nfor detecting interaction effects among SNPs by comparing the results of our method to those of two-locus epistasis test\nperformed by PLINK \\cite{purcell2007plink} which uses no structural information.\nSpecifically, we compare the hotspots with interaction effects (i.e. SNP pairs that affect a \nlarge number of gene traits) which are identified by both methods.  \nRecall that two-locus epistasis test is the most widely used \nstatistical technique for\ndetecting interaction effects in genome-wide association studies, which\ncomputes the significance of interactions \nby comparing between the null model\nwith only two marginal effects and the \nalternative model with two marginal effects and their \ninteraction effect.\nIn the following analysis, we discarded all SNP pairs\nif the correlation coefficient between the pairs $>0.5$ to avoid trivial \ninteraction effects.\n\nWe first identified the most significant hotspots that \naffect more than 100 gene traits.\nTo make sure that we include only significant interactions,\nwe considered interaction terms \nif their absolute value of regression coefficients\nare $> 0.05$. \nFor the results of two-locus epistasis test,\nwe considered all SNP pairs with p-value $< 10^{-5}$.\nFigure \\ref{fig:epi_hotspots}(a,b) show the hotspots\nfound by our method and two-locus epistasis test. \nThe rings in the figure represent the yeast genome\nfrom chromosome 1 (located at the top of each circle) to 16 clockwise, \nand the lines show interactions \nbetween the two genomic locations at both ends.\nOne can see that our method detected 11 hotspots but\ntwo-locus epistasis test found only two \nsignificant hotspots with interaction effects.\nThis observation shows that our method can find more \nsignificant hotspots with improved statistical power \ndue to the use of the input\/output structures.\nIn Table \\ref{tab:epistatic_hotspot}, we summarized the \nhotspots found by our method.\nIt turns out that our findings are also biologically interesting\n(e.g. 9 out of 11 hotspots showed GO enrichment).\nNotably, hotspot 1 (epistatic interaction between \nchr1:154328 and chr5:350744) affects 455 genes which are enriched with \nthe GO category of ribosome biogenesis with \na significant corrected p-value $< 10^{-35}$\n(multiple testing correction is \nperformed by false discovery rate \\cite{maere2005bingo}).\nThis SNP pair was included in our candidates from the genetic interaction network.\nThere is a significant genetic interaction between NUP60 and RAD51 \nwith p-value $3 \\times 10^{-7}$ \\cite{costanzo2010genetic}, and \nboth genes are located at chr1:152257-153877 and\nchr5:349975-351178, respectively.\nAs both SNPs are closely located to NUP60 and RAD51 (within 500bp), \nit is reasonable to hypothesize that \ntwo SNPs at chr1:154328 and chr5:350744 affected the two genes,\nand their genetic interaction in turn acted on a large number of\ngenes related to ribosome biogenesis.\n\nTo provide additional biological insights,\nwe further investigated the mechanism of this significant SNP-SNP interaction.\nFrom literature survey, RAD51 (RADiation sensitive) is a strand exchange\nprotein involved in DNA repair system \\cite{sung1994catalysis}, \nand NUP60 (NUclear Pore) is the subunit of unclear pore complex involved in nuclear \nexport system \\cite{denning2001nucleoporin}. \nAlso, it has been reported that yeast cells \nare excessively sensitive to DNA damaging agents if \nthere exist mutations in NUP60 \\cite{nagai2008functional}. \nIn our results, we also found out that the SNP close to NUP60 \ndid not have significant marginal effects, and\nthe SNP in RAD51 had marginal effects.\nAccording to these facts, it would be possible to hypothesize the following.\nWhen there is no mutation in RAD51,\nthe point mutation in NUP60 cannot affect other traits since\nthe single mutation is not strong enough and \nif there exist DNA damaging agents in the environment,\nDNA repair system would be able to handle them.\nHowever, when there exist point mutations in RAD51, \nDNA damaging agents would  severely harm yeast cells with the point mutation in NUP60\nsince DNA repair system might not work properly due to the mutation in RAD51 \n(recall that the SNP in RAD51 had marginal effects). \nAs a result, \nboth mutations in NUP60 and RAD51\ncould make a large impact on many gene traits.\n\n\\begin{figure}[htp]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{circos_ours1_10_new2.eps}}\n\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{circos_pt6_10_new2.eps}}\n\\caption{Hotspots with interaction effects identified by\n(a) our method and (b) two-locus epistasis test by PLINK. \nHere we show epistatic interactions which influence \nmore than 10 gene traits.\nThis figure was generated using Circos software \\cite{krzywinski2009circos}.\n}\n\\label{fig:medium_epi_hotspots}\n\\vspace{-0.1cm}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\vspace{-0.5cm}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{corr_graph.eps}\n\\vspace{-0.3cm}\n\\caption{The scatter plot for illustrating the correlation \nbetween our epistatic hotspot 1 (chr1:154328-chr5:350744) and \nsignificant SNP pairs close to hotspot 1\ndetected by two-locus epistasis test\n(p-value $< 10^{-6}$ and the distance between \nthe pairs of SNPs and hotspot 1 is within $< 50$kb).\nEach dot represents a SNP pair (SNP1, SNP2) \nfound by two-locus epistasis test,\nand x-axis represents the correlation between SNP1 and chr1:154328\nand y-axis represents the correlation between \nSNP2 and chr5:350744. \nEach dot was perturbed by a small amount of random noise\nto avoid overlapping of the dots. \n}\n\\label{fig:corr_graph}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\n\nFurthermore, we looked at the hotspots which affect $>10$ gene traits.\nFigure \\ref{fig:medium_epi_hotspots}(a,b) show epistatic interactions \nidentified by our method and two-locus epistasis test, respectively.\nIn this figure, we show significant interactions with \nregression coefficient cutoff $>0.1$ for our method, and\np-value cutoff $<10^{-6}$ for two-locus epistasis test.\nThese cutoffs are arbitrarily chosen to make the number of hotspots\nfound by both methods similar.\nSurprisingly, two methods showed very different hotspots with \nepistatic interactions. \nFigure \\ref{fig:medium_epi_hotspots}(a) was very similar to \nFigure \\ref{fig:epi_hotspots}(a) but\nin Figure \\ref{fig:medium_epi_hotspots}(b), \nseveral hotspots emerged which were absent in \nFigure \\ref{fig:epi_hotspots}(b).\nWe will analyze these hotspots in two ways.\nFirst we will look at the hotspots with epistatic effects \nwhich appeared in both Figure \\ref{fig:medium_epi_hotspots}(a) and (b).\nThen we will investigate the differences between the two\nresults.\nFirst, we observed that both methods \nfound significant epistatic effects between chromosome 1 and 5.\nRecall that in our previous analysis of the hotspots, \nthis interaction was discussed\n(see hotspot 1 in Table \\ref{tab:epistatic_hotspot}).\nAmong all significant SNP pairs found by two-locus epistasis test,\nthere was no identical SNP pair to hotspot 1\nbut there were 30 SNP pairs close to it (within $<50$kb). \nAlso, it turns out that these 30 SNP pairs had very strong correlation \nwith hotspot 1. In Figure \\ref{fig:corr_graph}, we show scatter plot to illustrate\nthe strong correlations between hotspot 1 and these 30 SNP pairs.\nMore interestingly, the total number of affected traits by these 30 SNP pairs\nwas 416, and it is very similar to 455 that is the number of affected \ngenes by hotspot 1.\nAccording to these facts and our previous analysis\nfor the mechanism of hotspot 1, it seems that hotspot 1 is \ntruly significant, and two-locus epistasis test found \nsignificant SNP pairs that are close to the true location but \nfailed to locate the exact location of hotspot 1.\nIt supports that our algorithm could find such a significant \nhotspot affecting $>$ 400 genes by detecting exact SNP pairs.\nHowever, two-locus epistasis test was unable to \nlocate many hotspots affecting a large number of traits \ndue to insufficient statistical power.\nSecond, we investigated the differences between the two \nresults in Figure \\ref{fig:medium_epi_hotspots}(a,b). \nAs we cannot report all the results in the paper,\nwe focused on a SNP pair (chr10:87113-chr15:141621) \nin Figure \\ref{fig:medium_epi_hotspots}(a),\nand another SNP pair (chr8:63314-chr9:362631) \nin Figure \\ref{fig:medium_epi_hotspots}(b).\nFigure \\ref{fig:checks}(a,b) \nshow the average gene expression levels for each SNP pair.\nIn this figure, x-axis represents the genotype $\\in \\{0,1\\}$ \nwhich is the multiplication of two SNPs \n(SNP1 $\\times$ SNP2, where SNP1, SNP2 $\\in \\{0,1\\}$), and y-axis represents the\naverage gene expression levels \nof individuals with given genotype.\nEach line in Figure \\ref{fig:checks}(a,b) shows\nhow the average gene expression level varies as the genotype changes \nfrom 0 to 1 for each trait affected by the SNP pairs \nwith error bars of one standard deviation.\nInterestingly, in Figure \\ref{fig:checks}(a), \nwe could \nsee that there is \na consistent pattern,\nwhere for most gene traits,\nthe expression levels decreased as the genotype changed from 0 to 1.\nHowever, as shown in Figure \\ref{fig:checks}(b),\nfor the SNP pair found by two-locus epistasis test,\nwe could not find such a coherent pattern.\nIt demonstrates the differences between our method and two-locus\nepistasis test. As our model borrows information across input and output group structures, \nwe could find consistent gene expression patterns for the SNP pair.\nOn the other hand, two-locus pairwise test analyzed each SNP pair\nseparately, and each trait affected by the SNP pair showed \ndifferent gene expression patterns with different  standard deviations.\nThus, \nit seems that our method can provide interesting biological insights in terms of\ngene expression patterns in addition to the statistical significance.\n\n\\begin{figure}[htp]\n\\vspace{-0.5cm}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.4\\textwidth]{our_chk1.eps}}\n\\subfigure[]{\\includegraphics[width=0.4\\textwidth]{pt_chk1.eps}}\n\\vspace{-0.5cm}\n\\caption{Variations of gene expression levels w.r.t. to the \ngenotypes of (a) a SNP pair (chr10:87113-chr15:141621)\nfound by our method, and \n(b) a different SNP pair (chr8:63314-chr9:362631) \nfound by two-locus epistasis test.\nHere, x-axis represents genotype doses and\ny-axis shows the average expression levels of the multiple genes (denoted by multiple vertical lines) affected by the corresponding SNP pairs. \nA small noise was added to the genotypes as offsets to avoid overlapping of the error bars.}\n\\label{fig:checks}\n\\vspace{-0.8cm}\n\\end{figure}\n\n\\section{Discussions}\n\n\nIn this paper, we presented jointly structured input-output lasso to \nsimultaneously take advantage of both input and output structures.\nWe also presented an efficient optimization technique for solving our \nmulti-task regression model with structured sparsity.\nOur experiments confirmed that our model is able to significantly improve \nthe accuracy for detecting true non-zero coefficients using both input and output\nstructures. \nFurthermore, we demonstrated that our optimization method is faster and more accurate than\nthe other competitors.\nIn our analysis of yeast eQTL datasets, we identified \nimportant pairs of eQTL hotspots that potentially interact with each other.\n\n\\paragraph{Prior knowledge about input and output structures} \nIn practice, it is important to generate reliable input and output groups to maximize\nthe benefits of the structural information.\nIn our experiments, we showed that yeast genetic interaction networks \ncan be used as prior knowledge to define input and output structures.\nHowever, such reliable prior knowledge \ncannot be easily attained when we deal with human eQTL datasets.\nInstead, we have a variety of resources for human genomes\nincluding protein-protein interaction networks and pathway database.\nGenerating reliable input and output structures exploiting multiple resources\nwould be essential for the successful discovery of human eQTLs.\n\n\\paragraph{Comparison between HiGT and other optimization algorithms}\nRecently, an active set algorithm \\cite{jenatton2009structured} has been proposed developed for variable selection with structured sparsity, which can potentially be used for estimating the SIOL model. \nWe observe two key differences \nbetween our method and the active set algorithm \\cite{jenatton2009structured}.\nFirst, the active set algorithm incrementally \nincreases active sets by searching available non-zero patterns, hence\none can see that it is a ``bottom-up'' approach.\nOn the other hand, our method adopts ``top-down'' approach where\nirrelevant covariates are discarded as we walk \nthrough the DAG. \nSecond, our algorithm guarantees  to search all zero patterns\nwhile the active set algorithm needs a heuristic to select candidate\nnon-zero patterns. When $\\mathbf{B}$ is sparse, our\nalgorithm is still very fast by taking advantage of the structures of DAG. \nHowever, when $\\mathbf{B}$ is not sparse, \nour algorithm needs to search a large number of zero patterns and\nupdate many non-zero coefficients but\nthe active set algorithm still does not need to \nupdate many non-zero coefficients.\nHence, in such a non-sparse case, the active set algorithm may have \nan advantage over our optimization method.\nOther alternative optimization methods for SIOL include\nMM (majorize\/minimize) algorithm \\cite{lange2004springer}\nand generalized stage-wise lasso \\cite{zhao2007stagewise}.\nHowever, these methods did not work well for SIOL as \nthe approximated penalty by MM algorithm \nand the greedy procedure of generalized stage-wise lasso \nwere incapable of efficiently inducing complex sparsity patterns.\n\n\n\\paragraph{Future work}\nOne promising research direction would be to systematically estimate the\nsignificance of the covariates that we found.\nFor example, computing p-values of our results would be helpful\nto control the false discovery rate. \nTo handle both sparse and non-sparse $\\mathbf{B}$, \nit would be also interesting to develop an optimization method for our model\nthat can take advantage of both ``bottom-up'' and ``top-down'' strategies. \nFor example, we can select variables using ``bottom-up''\napproach and discard irrelevant variables using\n''top-down'' approach alternatively in a single framework.\nFinally, we are interested in applying our method to human disease datasets.\nIn that case, the extension of our work to handle case-control studies and\nfinding reliable structural information will be necessary.\n\n\n\n\n\n\n\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nRecent new observations of hadrons are in many cases associated with the presence of heavy quarks.\nIn this regard, a systematic study of heavy baryons is important.\nSo far there are two $\\Lambda_b$ resonances, $\\Lambda_b(5912)^0$ and $\\Lambda_b(5920)^0$,\nwhose quantum numbers are $J^P = 1\/2^-$ and $3\/2^-$, respectively~\\cite{pdg}. They are both $P$-wave bottom baryons, which were studied by Capstick and Isgur in 1986 using the relativistic quark model~\\cite{Capstick:1986bm}, and their predicted masses are in very good agreement with the LHCb and CDF results~\\cite{Aaij:2012da,Aaltonen:2013tta}. In the following 30 years, various phenomenological models have been applied to study these excited bottom baryons, including the constituent quark model~\\cite{Garcilazo:2007eh,Ortega:2012cx,Yoshida:2015tia}, the relativistic quark model~\\cite{Ebert:2007nw}, the color hyperfine interaction~\\cite{Karliner:2008sv,Karliner:2015ema},\nthe heavy quark effective theory~\\cite{Roberts:2007ni}, the chiral unitary model~\\cite{GarciaRecio:2012db,Xiao:2013yca,Liang:2014eba,Torres-Rincon:2014ffa,Lu:2014ina,Garcia-Recio:2015jsa},\nthe heavy meson effective theory with $1\/M$ corrections~\\cite{Yasui:2013iga}, the heavy quark symmetry in multihadron systems~\\cite{Yamaguchi:2014era},\nthe quark pair creation model~\\cite{Mu:2014iaa}, the Faddeev approach~\\cite{Valcarce:2014fma}, the relativistic flux tube model~\\cite{Chen:2014nyo}, and\nthe Lattice QCD~\\cite{Burch:2015pka}; see a recent review article for more details~\\cite{Crede:2013sze}.\n\nIn this paper we shall study $P$-wave bottom baryons using the method of QCD sum rule~\\cite{Shifman:1978bx,Reinders:1984sr} in the\nframework of heavy quark effective theory (HQET)~\\cite{Grinstein:1990mj,Eichten:1989zv,Falk:1990yz}, which has\nbeen successful applied to study heavy mesons containing a single heavy\nquark~\\cite{Bagan:1991sg,Neubert:1991sp,Neubert:1993mb,Broadhurst:1991fc,Ball:1993xv,Huang:1994zj,Dai:1996yw,Dai:1993kt,Dai:1996qx,Colangelo:1998ga,Dai:2003yg,Zhou:2014ytp},\nand heavy baryons also containing a single heavy\nquark~\\cite{Shuryak:1981fza,Grozin:1992td,Bagan:1993ii,Dai:1995bc,Dai:1996xv,Groote:1996em,Zhu:2000py,Lee:2000tb,Huang:2000tn,Wang:2003zp,Liu:2007fg}.\nThere are also some studies using the method of QCD sum rules but not in HQET~\\cite{Bagan:1992tp,Bagan:1991sc,Duraes:2007te,Wang:2007sqa}.\nDuring the calculations, we shall take the ${\\mathcal O}(1\/m_Q)$ corrections ($m_Q$ is the heavy quark mass) into account, and extract the chromomagnetic splitting.\n\nUntil now, $\\Lambda_b(5912)^0$ and $\\Lambda_b(5920)^0$ are the only two $P$-wave bottom baryons well observed in the experiments~\\cite{pdg,Aaij:2012da,Aaltonen:2013tta}. This is quite different from charmed baryons~\\cite{pdg,Cheng:2015rra}: the four $P$-wave charmed baryons, $\\Lambda_c(2595)$, $\\Lambda_c(2625)$, $\\Xi_c(2790)$ and $\\Xi_c(2815)$, are well observed and complete two $SU(3)$ $\\mathbf{\\bar3}$\nmultiplets; there are some other candidates, like $\\Sigma_c(2800)$, $\\Xi_c(2980)$ and $\\Xi_c(3080)$, etc.\nHowever, future experiments, such as LHCb and BelleII, are able to search for more bottom baryons, as well as more charmed baryons. Hence, it is a suitable time to carry out this study and write this paper, after we have studied $P$-wave charmed baryons in Ref.~\\cite{Chen:2015kpa} which already contains much information.\nIn this paper, we shall use the same procedures and study $P$-wave bottom baryons.\n\nThis paper is organized as follows. In Sec.~\\ref{sec:sumrule}, we briefly introduce\nthe interpolating fields for $P$-wave bottom baryons, and briefly introduce how to use them to\nperform QCD sum rule analyses at the leading order and at the ${\\mathcal O}(1\/m_Q)$\norder. The procedures are identical to what we have done in Ref.~\\cite{Chen:2015kpa}\nfor $P$-wave charmed baryons.\nIn this paper we just need to redo numerical analyses using the $bottom$ quark mass instead of the $charm$ quark mass.\nThese analyses are done in Sec.~\\ref{sec:numerical}, where we use the bottom baryon doublet $[\\Lambda_b, 1, 1, \\rho]$ as an example.\nIn Sec.~\\ref{sec:summary} we discuss the results and offer a summary.\n\n\\section{QCD Sum Rule Analyses}\n\\label{sec:sumrule}\n\n\n\\begin{figure*}[hbt]\n\\begin{center}\n\\scalebox{0.6}{\\includegraphics{pwave.eps}}\n\\end{center}\n\\caption{The notations for $P$-wave bottom baryons. See Fig.~2 of Ref.~\\cite{Chen:2015kpa} and discussions therein for details.\n\\label{fig:pwave}}\n\\end{figure*}\n\n$P$-wave heavy baryons have been systematically classified in Ref.~\\cite{Chen:2007xf}, $P$-wave charmed baryon interpolating fields have been systematically constructed in Ref.~\\cite{Chen:2015kpa}, and $P$-wave bottom baryon fields can be easily constructed by replacing the $charm$ quark field by the $bottom$ quark field. The results are briefly shown in Fig.~\\ref{fig:pwave}, which is\ntaken from Ref.~\\cite{Chen:2015kpa} and slightly modified. For the notation the reader is advised to consult the discussion in Ref.~\\cite{Chen:2015kpa}.\n\nIn this paper we also use the same notation, $J^{\\alpha_1\\cdots\\alpha_{j-1\/2}}_{j,P,F,j_l,s_l,\\rho\/\\lambda}$, to denote the $P$-wave bottom baryon field belonging to the baryon multiplet $[F, j_l, s_l, \\rho\/\\lambda]$, where $j$, $P$, and $F$ are the total angular momentum, parity and $SU(3)$ flavor representation ($\\mathbf{\\bar 3}_F$ or $\\mathbf{6}_F$) of the bottom baryons; $j_l$ and $s_l$ are the total angular momentum and spin angular momentum of the light components; $\\rho$ and $\\lambda$ denote $\\rho$--type ($l_\\rho = 1$ and $l_\\lambda = 0$) and $\\lambda$--type ($l_\\rho = 0$ and $l_\\lambda = 1$), respectively, where $l_\\rho$ is the orbital angular momentum between the two light quarks, and $l_\\lambda$ is the orbital angular momentum between the bottom quark and the two-light-quark system. Their relations are $j_l = l_\\lambda \\otimes l_\\rho \\otimes s_l$ and $j = j_l \\otimes s_Q = | j_l \\pm 1\/2 |$, where $s_Q = 1\/2$ is the spin of the bottom quark.\n\nThe explicit forms of $J^{\\alpha_1\\cdots\\alpha_{j-1\/2}}_{j,P,F,j_l,s_l,\\rho\/\\lambda}$ have been given in Eqs.~(8)--(21) of Ref.~\\cite{Chen:2015kpa}, but $J^{\\alpha_1\\alpha_2}_{j,-,\\mathbf{\\bar3}_F,2,1,\\rho}$ and $J^{\\alpha_1\\alpha_2}_{j,-,\\mathbf{6}_F,2,1,\\lambda}$ therein contain both $j=3\/2$ and $j=5\/2$ components. In this paper we give the pure $j=5\/2$ fields\n\\begin{eqnarray}\n&& J^{\\alpha_1\\alpha_2}_{5\/2,-,\\mathbf{\\bar3}_F,2,1,\\rho}\n\\\\ \\nonumber && = i \\epsilon_{abc} \\Big ( [\\mathcal{D}_t^{\\mu} q^{aT}] \\mathbb{C} \\gamma_t^{\\nu} q^b - q^{aT} \\mathbb{C} \\gamma_t^{\\nu} [\\mathcal{D}_t^{\\mu} q^b] \\Big )\n\\times \\Gamma^{\\alpha_1\\alpha_2,\\mu\\nu} h_v^c \\, ,\n\\\\ && J^{\\alpha_1\\alpha_2}_{5\/2,-,\\mathbf{6}_F,2,1,\\lambda}\n\\\\ \\nonumber && = i \\epsilon_{abc} \\Big ( [\\mathcal{D}_t^{\\mu} q^{aT}] \\mathbb{C} \\gamma_t^{\\nu} q^b + q^{aT} \\mathbb{C} \\gamma_t^{\\nu} [\\mathcal{D}_t^{\\mu} q^b] \\Big )\n\\times \\Gamma^{\\alpha_1\\alpha_2,\\mu\\nu} h_v^c \\, ,\n\\end{eqnarray}\nwhere $\\Gamma^{\\alpha_1\\alpha_2,\\mu\\nu}$ is the projection operator:\n\\begin{eqnarray}\n\\Gamma^{\\alpha\\beta,\\mu\\nu} &=& g_t^{\\alpha\\mu} g_t^{\\beta\\nu} + g_t^{\\alpha\\nu} g_t^{\\beta\\mu} - {2 \\over 15} g_t^{\\alpha\\beta} g_t^{\\mu\\nu}\n\\\\ \\nonumber && - {1 \\over 3} g_t^{\\alpha\\mu} \\gamma_t^{\\beta}\\gamma_t^{\\nu} - {1 \\over 3} g_t^{\\alpha\\nu} \\gamma_t^{\\beta}\\gamma_t^{\\mu}\n\\\\ \\nonumber && - {1 \\over 3} g_t^{\\beta\\mu} \\gamma_t^{\\alpha}\\gamma_t^{\\nu} - {1 \\over 3} g_t^{\\beta\\nu} \\gamma_t^{\\alpha}\\gamma_t^{\\mu}\n\\\\ \\nonumber && + {1 \\over 15} \\gamma_t^{\\alpha} \\gamma_t^{\\mu} \\gamma_t^{\\beta} \\gamma_t^{\\nu} + {1 \\over 15} \\gamma_t^{\\alpha} \\gamma_t^{\\nu} \\gamma_t^{\\beta} \\gamma_t^{\\mu}\n\\\\ \\nonumber && + {1 \\over 15} \\gamma_t^{\\beta} \\gamma_t^{\\mu} \\gamma_t^{\\alpha} \\gamma_t^{\\nu} + {1 \\over 15} \\gamma_t^{\\beta} \\gamma_t^{\\nu} \\gamma_t^{\\alpha} \\gamma_t^{\\mu} \\, .\n\\end{eqnarray}\nThe notations used in these equations are: $a$, $b$ and $c$ are color indices, and $\\epsilon_{abc}$ is the totally antisymmetric tensor; the superscript $T$ represents the transpose of the Dirac indices only; $\\mathbb{C}$ is the charge-conjugation operator; $q(x)$ is the light quark field at location $x$, and it can be either $u(x)$ or $d(x)$ or $s(x)$; $h_v(x)$ is the heavy quark field, and we have used the Fierz transformation to move it to the rightmost place; $D^\\mu = \\partial^\\mu - i g A^\\mu$ is the gauge-covariant derivative; $v$ is the velocity of the heavy quark; $\\gamma_t^\\mu = \\gamma^\\mu - v\\!\\!\\!\\slash v^\\mu$, $D^\\mu_t = D^\\mu - (D \\cdot v) v^\\mu$, and $g_t^{\\alpha_1\\alpha_2}=g^{\\alpha_1\\alpha_2} - v^{\\alpha_1} v^{\\alpha_2}$.\n\nWe can use these $P$-wave bottom baryon interpolating fields to perform QCD sum rule analyses. The heavy baryon belonging to the multiplet $[F, j_l, s_l, \\rho\/\\lambda]$ has the mass:\n\\begin{eqnarray}\nm_{j,P,F,j_l,s_l,\\rho\/\\lambda} = m_Q + \\overline{\\Lambda}_{F,j_l,s_l,\\rho\/\\lambda} + \\delta m_{j,P,F,j_l,s_l,\\rho\/\\lambda} \\, ,\n\\label{eq:mass}\n\\end{eqnarray}\nwhere $m_Q$ is the heavy quark mass, $\\overline{\\Lambda}_{F,j_l,s_l,\\rho\/\\lambda} = \\overline{\\Lambda}_{|j_l-1\/2|,P,F,j_l,s_l,\\rho\/\\lambda} = \\overline{\\Lambda}_{j_l+1\/2,P,F,j_l,s_l,\\rho\/\\lambda}$ is the sum rule result evaluated at the leading order, and $\\delta m_{j,P,F,j_l,s_l,\\rho\/\\lambda}$ is the sum rule result evaluated at the ${\\mathcal O}(1\/m_Q)$ order:\n\\begin{eqnarray}\n&& \\delta m_{j,P,F,j_l,s_l,\\rho\/\\lambda}\n\\label{eq:masscorrection}\n\\\\ \\nonumber && = -\\frac{1}{4m_{Q}}(K_{F,j_l,s_l,\\rho\/\\lambda} + d_{M}C_{mag}\\Sigma_{F,j_l,s_l,\\rho\/\\lambda} ) \\, .\n\\end{eqnarray}\nHere, the coefficient $d_{M} \\equiv d_{j,j_{l}}$ is defined to be\n\\begin{eqnarray}\nd_{j_{l}-1\/2,j_{l}} &=& 2j_{l}+2\\, ,\n\\\\ \\nonumber d_{j_{l}+1\/2,j_{l}} &=& -2j_{l} \\, .\n\\end{eqnarray}\nIt is directly related to the baryon mass splitting within the same multiplet, caused by the chromomagnetic interaction. The other coefficient is $C_{mag} (m_{Q}\/\\mu) = [ \\alpha_s(m_Q) \/ \\alpha_s(\\mu) ]^{3\/\\beta_0} \\approx 0.8$, where $\\beta_0 = 11 - 2 n_f \/3$.\n\nThe (analytical) formulae for $\\overline{\\Lambda}_{F,j_l,s_l,\\rho\/\\lambda}(\\omega_c, T)$, $K_{F,j_l,s_l,\\rho\/\\lambda}(\\omega_c, T)$ and $\\Sigma_{F,j_l,s_l,\\rho\/\\lambda}(\\omega_c, T)$ have been obtained in Ref.~\\cite{Chen:2015kpa}. In this paper we just need to use these equations to redo numerical analyses but using the $bottom$ quark mass. There are two free parameters in these equations: the Borel mass $T$ and the threshold value $\\omega_c$. Following the procedures used in Ref.~\\cite{Chen:2015kpa}, we can obtain a Borel window $T_{min}<T<T_{max}$ for a fixed threshold value $\\omega_c$. We shall not repeat these procedures again. The other parameter, $\\omega_c$, can be fixed by requiring that its dependence of the mass prediction be the weakest, which will be done in the next section.\n\n\n\\section{Numerical Analyses}\n\\label{sec:numerical}\n\nTo perform numerical analyses, we use the following values for various condensates and parameters~\\cite{Dai:1993kt,Dai:1996yw,Dai:1996qx,Dai:2003yg,pdg,Yang:1993bp,Hwang:1994vp,Narison:2002pw,Gimenez:2005nt,Jamin:2002ev,Ioffe:2002be,Ovchinnikov:1988gk,colangelo}:\n\\begin{eqnarray}\n\\nonumber && \\langle \\bar qq \\rangle = \\langle \\bar uu \\rangle = \\langle \\bar dd \\rangle = - (0.24 \\pm 0.01)^3 \\mbox{ GeV}^3 \\, ,\n\\\\ \\nonumber && \\langle \\bar ss \\rangle = 0.8 \\times \\langle\\bar qq \\rangle \\, ,\n\\\\ \\nonumber &&\\langle g_s^2GG\\rangle =(0.48\\pm 0.14) \\mbox{ GeV}^4\\, ,\n\\\\ \\label{condensates} && m_s = 0.15 \\mbox{ GeV} \\, ,\n\\\\\n\\nonumber && \\langle g_s \\bar q \\sigma G q \\rangle = M_0^2 \\times \\langle \\bar qq \\rangle\\, ,\n\\\\\n\\nonumber && \\langle g_s \\bar s \\sigma G s \\rangle = M_0^2 \\times \\langle \\bar ss \\rangle\\, ,\n\\\\\n\\nonumber && M_0^2= 0.8 \\mbox{ GeV}^2\\, .\n\\end{eqnarray}\nBesides them, another important parameter is the $bottom$ quark mass, for which we use the $1S$ mass $m_b = 4.66 \\pm 0.03$ GeV~\\cite{pdg}. We shall see that this value gives reasonable results.\n\n\\subsection{Sum Rule Analyses for $\\Lambda_b$}\n\nAs an example, we use the bottom baryon doublet $[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 1, \\rho]$ to perform QCD sum rule analyses. It contains two bottom baryons, $\\Lambda_b(1\/2^-)$ and $\\Lambda_b(3\/2^-)$, and the relevant interpolating field is\n\\begin{eqnarray}\n&& J_{1\/2,-,\\Lambda_b,1,1,\\rho}\n\\label{eq:current}\n\\\\ \\nonumber && = i \\epsilon_{abc} \\Big ( [\\mathcal{D}_t^{\\mu} u^{aT}] C \\gamma_t^\\nu d^b -  u^{aT} C \\gamma_t^\\nu [\\mathcal{D}_t^{\\mu} d^b] \\Big ) \\sigma_t^{\\mu\\nu} h_v^c \\, .\n\\end{eqnarray}\nWe can use this current to perform QCD sum rule analyses, and calculate $\\overline{\\Lambda}_{\\Lambda_b,1,1,\\rho}(\\omega_c, T)$, $K_{\\Lambda_b,1,1,\\rho}(\\omega_c, T)$ and $\\Sigma_{\\Lambda_b,1,1,\\rho}(\\omega_c, T)$. This has been done and the results are listed in Eq.~(A6) of Ref.~\\cite{Chen:2015kpa}, which can be used to further evaluate masses of $\\Lambda_b(1\/2^-)$ and $\\Lambda_b(3\/2^-)$ through Eqs.~(\\ref{eq:mass}) and (\\ref{eq:masscorrection}), as shown below.\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{c}\n\\scalebox{0.6}{\\includegraphics{rho3F11mb.eps}}\n\\\\\n\\scalebox{0.6}{\\includegraphics{rho3F11wc.eps}}\n\\end{tabular}\n\\caption{Variations of $m_{\\Lambda_b(1\/2^-)}$ with respect to the Borel mass\n$T$ (top) and the threshold value $\\omega_c$ (bottom), calculated using the bottom baryon doublet\n$[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 1, \\rho]$. In the top panel we take $\\omega_c = 3.1$ GeV and the Borel window is\n$0.538$ GeV $< T < 0.588$ GeV. In the bottom panel, the upper and lower\nbands are obtained by using $T_{\\rm min}$ and $T_{\\rm max}$, respectively. There are non-vanishing Borel windows as long as $\\omega_c \\geq 2.85$ GeV, but they quickly vanish around this point. The results for $\\omega_c \\leq 2.85$ GeV are also shown. For such cases we choose the Borel mass $T$ when the PC (pole contribution, defined in Eq.~(34) of Ref.~\\cite{Chen:2015kpa}) is around 20\\%, and so this curve connects to the lower band obtained by using $T_{\\rm max}$.}\n\\label{fig:rho3F11}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{c}\n\\scalebox{0.6}{\\includegraphics{rho3F11splitmb.eps}}\n\\\\\n\\scalebox{0.6}{\\includegraphics{rho3F11splitwc.eps}}\n\\end{tabular}\n\\caption{Variations of $\\Delta m_{[\\Lambda_b, 1, 1, \\rho]}$ with respect to the Borel mass\n$T$ (top) and the threshold value $\\omega_c$ (bottom), calculated using the bottom baryon doublet\n$[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 1, \\rho]$. In the top panel we take $\\omega_c = 3.1$ GeV and the Borel window is\n$0.538$ GeV $< T < 0.588$ GeV.}\n\\label{fig:rho3F11split}\n\\end{center}\n\\end{figure}\n\nFirstly, we take $\\omega_c = 3.1$ GeV and obtain a Borel window $0.538$ GeV $< T < 0.588$ GeV producing the following numerical results:\n\\begin{eqnarray}\n\\nonumber \\bar \\Lambda_{\\Lambda_b,1,1,\\rho} &=& 1.17 \\mbox{ GeV} \\, , \\,\n\\\\ K_{\\Lambda_b,1,1,\\rho} &=& -0.99 \\mbox{ GeV}^2 \\, , \\,\n\\\\ \\nonumber \\Sigma_{\\Lambda_b,1,1,\\rho} &=& 0.042 \\mbox{ GeV}^{2} \\, ,\n\\end{eqnarray}\nwhose values correspond to $T=0.563$ GeV and $\\omega_c = 3.1$ GeV. Using Eqs.~(\\ref{eq:mass}) and (\\ref{eq:masscorrection}), we can further obtain\n\\begin{eqnarray}\n\\nonumber m_{\\Lambda_b(1\/2^-)} &=& 5.87 \\mbox{ GeV} \\, , \\,\n\\\\ m_{\\Lambda_b(3\/2^-)} &=& 5.88 \\mbox{ GeV} \\, , \\,\n\\\\ \\nonumber \\Delta m_{[\\Lambda_b, 1, 1, \\rho]} &=& 11 \\mbox{ MeV} \\, ,\n\\end{eqnarray}\nwhere $(m_{\\Lambda_b(1\/2^-)},m_{\\Lambda_b(3\/2^-)})$ are the masses of the two bottom baryons belonging to the baryon doublet $[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 1, \\rho]$, and $\\Delta m_{[\\Lambda_b, 1, 1, \\rho]} \\equiv m_{\\Lambda_b(3\/2^-)} - m_{\\Lambda_b(1\/2^-)}$ is their mass splitting. We show variations of $m_{\\Lambda_b(1\/2^-)}$ and $\\Delta m_{[\\Lambda_b, 1, 1, \\rho]}$ with respect to the Borel mass $T$ in the top panel of Figs.~\\ref{fig:rho3F11} and \\ref{fig:rho3F11split}.\n\nSecondly, we change the threshold value $\\omega_c$ and redo these processes. Our criterion is to require that the $\\omega_c$ dependence of\nthe mass prediction be the weakest. We show variations of $m_{\\Lambda_b(1\/2^-)}$ and $\\Delta m_{[\\Lambda_b, 1, 1, \\rho]}$\nwith respect to the threshold value $\\omega_c$ in the bottom panel of Figs.~\\ref{fig:rho3F11} and \\ref{fig:rho3F11split}. We find that there are non-vanishing Borel windows as long as $\\omega_c \\geq 2.85$ GeV, and\nthe $\\omega_c$ dependence is weak and acceptable in the region $2.9$ GeV$<\\omega_c<3.3$ GeV. We note that this dependence is the weakest around $\\omega_c \\sim 2.9$ GeV, but we prefer to choosing a suitable region where we can know the related uncertainty. The results for $\\omega_c \\leq 2.85$ GeV are also shown, and for such cases, we choose the Borel mass $T$ when the PC (pole contribution, defined in Eq.~(34) of Ref.~\\cite{Chen:2015kpa}) is around 20\\%.\n\nFinally, we choose $2.9$ GeV$<\\omega_c<3.3$ GeV and $0.538$ GeV $< T < 0.588$ GeV as our working regions, and obtain\nthe following numerical results for the baryon doublet $[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 1, \\rho]$:\n\\begin{eqnarray}\n\\nonumber m_{\\Lambda_b(1\/2^-)} &=& 5.87 \\pm 0.12 \\mbox{ GeV} \\, , \\,\n\\\\ m_{\\Lambda_b(3\/2^-)} &=& 5.88 \\pm 0.11 \\mbox{ GeV} \\, , \\,\n\\\\ \\nonumber \\Delta m_{[\\Lambda_b, 1, 1, \\rho]} &=& 11 \\pm 4 \\mbox{ MeV} \\, ,\n\\end{eqnarray}\nwhose central values correspond to $T=0.563$ GeV and $\\omega_c = 3.1$ GeV, and the\nuncertainties are due to the Borel mass $T$, the threshold value $\\omega_c$, the bottom quark mass $m_b$ and the\nquark and gluon condensates. Our result for the mass splitting within\nthe same doublet, $\\Delta m_{[\\Lambda_b, 1, 1, \\rho]}$, does not depend much on the charm quark mass and the threshold value, so they are produced quite well,\nwith much less theoretical uncertainties, compared to their masses, $m_{\\Lambda_b(1\/2^-)}$ and $m_{\\Lambda_b(3\/2^-)}$.\nThese results are consistent with experimental masses of $\\Lambda_b(5912)^0$ ($1\/2^-$) and $\\Lambda_b(5920)^0$ ($J^P=3\/2^-$) as well as their difference~\\cite{Aaij:2012da,Aaltonen:2013tta}:\n\\begin{eqnarray}\n&& m^{\\rm exp}_{\\Lambda_b(5912)^0,1\/2^-} = 5912.11 \\pm 0.13 \\pm 0.23 \\mbox{ MeV} \\, ,\n\\\\ \\nonumber && m^{\\rm exp}_{\\Lambda_b(5920)^0,3\/2^-} = 5919.91 \\pm 0.09 \\pm 0.23 \\mbox{ MeV} \\, .\n\\end{eqnarray}\nThis suggests that the use of the $1S$ mass $m_b = 4.66 \\pm 0.03$ GeV~\\cite{pdg} is reasonable. While, if we use the $\\overline{\\rm MS}$ mass $m_b = 4.18 \\pm 0.03$ GeV~\\cite{pdg}, we would obtain significantly lower masses.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\begin{tabular}{c}\n\\scalebox{0.6}{\\includegraphics{lambda3F10mb.eps}}\n\\\\\n\\scalebox{0.6}{\\includegraphics{lambda3F10wc.eps}}\n\\end{tabular}\n\\caption{Variations of $m_{\\Lambda_b(1\/2^-)}$ with respect to the Borel mass\n$T$ (top) and the threshold value $\\omega_c$ (bottom), calculated using the bottom baryon doublet\n$[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 0, \\lambda]$. In the top panel we take $\\omega_c = 3.6$ GeV and the Borel window is\n$0.597$ GeV $< T < 0.684$ GeV and $3.4$ GeV.}\n\\label{fig:lambda3F10}\n\\end{center}\n\\end{figure}\n\nWe also use the bottom baryon doublet $[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 0, \\lambda]$ to perform QCD sum rule analyses.\nWe show the variation of $m_{\\Lambda_b(1\/2^-)}$\nwith respect to the Borel mass $T$ and the threshold value $\\omega_c$ in Fig.~\\ref{fig:lambda3F10}. We find that there are non-vanishing Borel windows as long as $\\omega_c \\geq 3.37$ GeV, but in this region the $\\omega_c$ dependence is not so weak. We still choose $3.4$ GeV $<\\omega_c <3.8$ GeV and $0.597$ GeV $< T < 0.684$ GeV as our working regions, but note that the previous results obtained using the doublet $[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 1, \\rho]$ seem to be more stable (reliable), because the $\\rho$--mode signal is clearer than the $\\lambda$--one. In fact the lower panel of Fig.~\\ref{fig:lambda3F10}, which is monotonically increasing, shows that the $\\lambda$--mode is not well observed. This does not, however, say that the $\\lambda$--mode mass is larger than the $\\rho$--mode mass in a reliable manner, though the current choice of the Borel window suggests it. We obtain the following numerical results\n\\begin{eqnarray}\n\\nonumber m_{\\Lambda_b(1\/2^-)} &=& 6.01 \\pm 0.11 \\mbox{ GeV} \\, , \\,\n\\\\ m_{\\Lambda_b(3\/2^-)} &=& 6.01 \\pm 0.11 \\mbox{ GeV} \\, , \\,\n\\\\ \\nonumber \\Delta m_{[\\Lambda_b, 1, 0, \\lambda]} &=& 5 \\pm 2 \\mbox{ MeV} \\, ,\n\\end{eqnarray}\nwhose central values correspond to $T=0.641$ GeV and $\\omega_c = 3.6$ GeV. We note again that the mass splitting within the same doublet is reproduced quite well with much less (theoretical) uncertainty than the absolute values. These results are also consistent with masses of $\\Lambda_b(5912)^0$ ($1\/2^-$) and $\\Lambda_b(5920)^0$ ($J^P=3\/2^-$) as well as their difference~\\cite{Aaij:2012da,Aaltonen:2013tta}.\n\n\\subsection{Sum Rule Analysis for $\\Sigma_b$, $\\Xi_b^{(\\prime)}$ and $\\Omega_b$}\n\nThere are two baryons, $\\Lambda_b(5912)^0$ and $\\Lambda_b(5920)^0$, which are the only P-wave bottom baryons well observed in experiments~\\cite{pdg,Aaij:2012da,Aaltonen:2013tta}, and there are no other well established candidates. To evaluate masses of other baryon multiplets, we follow the previous procedures used for $[\\Lambda_b(\\mathbf{\\bar 3}_F), 1, 1, \\rho]$\nas well as for charmed baryons~\\cite{Chen:2015kpa}:\n\n\\begin{enumerate}\n\n\\item We use the bottom baryon singlet $[\\Sigma_b(\\mathbf{6}_F), 0, 1, \\lambda]$ to perform QCD sum rule analyses. We show the variation of $m_{\\Sigma_b(1\/2^-)}$ with respect to the Borel mass $T$ and the threshold value $\\omega_c$ in Fig.~\\ref{fig:lambda6F01}. We find that there are non-vanishing Borel windows as long as $\\omega_c \\geq 3.06$ GeV, and the $\\omega_c$ dependence is weak and acceptable in the region $3.1$ GeV$<\\omega_c<3.5$ GeV. Accordingly, we choose $3.1$ GeV $<\\omega_c <3.5$ GeV and $0.527$ GeV $< T < 0.628$ GeV as our working regions, and obtain the following numerical result\n    \\begin{eqnarray}\n    m_{\\Sigma_b(1\/2^-)} &=& 6.02 \\pm 0.12 \\mbox{ GeV} \\, , \\,\n    \\end{eqnarray}\n    where the central value corresponds to $T=0.578$ GeV and $\\omega_c = 3.3$ GeV.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\begin{tabular}{c}\n\\scalebox{0.6}{\\includegraphics{lambda6F01mb.eps}}\n\\\\\n\\scalebox{0.6}{\\includegraphics{lambda6F01wc.eps}}\n\\end{tabular}\n\\caption{Variations of $m_{\\Sigma_b(1\/2^-)}$ with respect to the Borel mass\n$T$ (top) and the threshold value $\\omega_c$ (bottom), calculated using the bottom baryon singlet\n$[\\Sigma_b(\\mathbf{6}_F), 0, 1, \\lambda]$. In the top panel we take $\\omega_c = 3.3$ GeV and the Borel window is\n$0.527$ GeV $< T < 0.628$ GeV.}\n\\label{fig:lambda6F01}\n\\end{center}\n\\end{figure}\n\n\\item We use the bottom baryon doublet $[\\Sigma_b(\\mathbf{6}_F), 1, 0, \\rho]$ to perform QCD sum rule analyses. We show the variations of $m_{\\Sigma_b(1\/2^-)}$ with respect to the Borel mass $T$ and the threshold value $\\omega_c$ in Fig.~\\ref{fig:rho6F10}. We find that there are non-vanishing Borel windows as long as $\\omega_c \\geq 3.01$ GeV, and the $\\omega_c$ dependence is weak and acceptable in the region $3.1$ GeV$<\\omega_c<3.5$ GeV. Hence, we choose $3.1$ GeV $<\\omega_c <3.5$ GeV and $0.534$ GeV $< T < 0.624$ GeV as our working regions, and obtain the following numerical results\n    \\begin{eqnarray}\n    \\nonumber m_{\\Sigma_b(1\/2^-)} &=& 5.91 \\pm 0.14 \\mbox{ GeV} \\, , \\,\n    \\\\ m_{\\Sigma_b(3\/2^-)} &=& 5.92 \\pm 0.14 \\mbox{ GeV} \\, , \\,\n    \\\\ \\nonumber \\Delta m_{[\\Sigma_b, 1, 0, \\rho]} &=& 4\\pm2 \\mbox{ MeV} \\, ,\n    \\end{eqnarray}\n    whose central values correspond to $T=0.579$ GeV and $\\omega_c = 3.3$ GeV. We note again that the mass splitting within the same doublet is reproduced quite well with much less (theoretical) uncertainty.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\begin{tabular}{c}\n\\scalebox{0.6}{\\includegraphics{rho6F10mb.eps}}\n\\\\\n\\scalebox{0.6}{\\includegraphics{rho6F10wc.eps}}\n\\end{tabular}\n\\caption{Variations of $m_{\\Sigma_b(1\/2^-)}$ with respect to the Borel mass\n$T$ (top) and the threshold value $\\omega_c$ (bottom), calculated using the bottom baryon doublet\n$[\\Sigma_b(\\mathbf{6}_F), 1, 0, \\rho]$. In the top panel we take $\\omega_c = 3.3$ GeV and the Borel window is\n$0.534$ GeV $< T < 0.624$ GeV.}\n\\label{fig:rho6F10}\n\\end{center}\n\\end{figure}\n\n\\item We use the bottom baryon doublet $[\\Sigma_b(\\mathbf{6}_F), 2, 1, \\lambda]$ to perform QCD sum rule analyses. We show the variation of $m_{\\Sigma_b(1\/2^-)}$ with respect to the Borel mass $T$ and the threshold value $\\omega_c$ in Fig.~\\ref{fig:lambda6F21}. We find that there are non-vanishing Borel windows as long as $\\omega_c \\geq 2.97$ GeV, and the $\\omega_c$ dependence is weak and acceptable in the region $3.1$ GeV$<\\omega_c<3.5$ GeV. Accordingly, we choose $3.1$ GeV $<\\omega_c <3.5$ GeV and $0.537$ GeV $< T < 0.620$ GeV as our working regions, and obtain the following numerical results\n    \\begin{eqnarray}\n    \\nonumber m_{\\Sigma_b(3\/2^-)} &=& 5.96 \\pm 0.18 \\mbox{ GeV} \\, , \\,\n    \\\\ m_{\\Sigma_b(5\/2^-)} &=& 5.98 \\pm 0.18 \\mbox{ GeV} \\, , \\,\n    \\\\ \\nonumber \\Delta m_{[\\Sigma_b, 2, 1, \\lambda]} &=& 15 \\pm 7 \\mbox{ MeV} \\, ,\n    \\end{eqnarray}\n    whose central values correspond to $T=0.579$ GeV and $\\omega_c = 3.3$ GeV. We note again that the mass splitting within the same doublet is reproduced quite well with much less (theoretical) uncertainty.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\begin{tabular}{c}\n\\scalebox{0.6}{\\includegraphics{lambda6F21mb.eps}}\n\\\\\n\\scalebox{0.6}{\\includegraphics{lambda6F21wc.eps}}\n\\end{tabular}\n\\caption{Variations of $m_{\\Sigma_b(3\/2^-)}$ with respect to the Borel mass\n$T$ (top) and the threshold value $\\omega_c$ (bottom), calculated using the bottom baryon doublet\n$[\\Sigma_b(\\mathbf{6}_F), 2, 1, \\lambda]$. In the top panel we take $\\omega_c = 3.3$ GeV and the Borel window is\n$0.537$ GeV $< T < 0.620$ GeV.}\n\\label{fig:lambda6F21}\n\\end{center}\n\\end{figure}\n\n\\item For flavor partners of $\\Lambda_b$ and $\\Sigma_b$, we use $\\omega_c(\\Xi_b)- \\omega_c(\\Lambda_b) = \\omega_c(\\Omega_b) - \\omega_c(\\Xi^\\prime_b) = \\omega_c(\\Xi^\\prime_b)- \\omega_c(\\Sigma_b)$ = 0.5 GeV, following the relations obtained\/used for charmed baryons in Ref.~\\cite{Chen:2015kpa}.\n\n\\item Other bottom baryon multiplets $[\\mathbf{\\bar 3}_F, 0, 1, \\rho]$, $[\\mathbf{\\bar 3}_F, 2, 1, \\rho]$, and $[\\mathbf{6}_F, 1, 1, \\lambda]$ do not give useful QCD sum rule results, which we shall not discuss to avoid confusions.\n\n\\end{enumerate}\n\nWe summarize these results obtained using the baryon multiplets $[\\mathbf{\\bar 3}_F, 1, 1, \\rho]$, $[\\mathbf{\\bar 3}_F, 1, 0, \\lambda]$, $[\\mathbf{6}_F, 0, 1, \\lambda]$, $[\\mathbf{6}_F, 1, 0, \\rho]$ and $[\\mathbf{6}_F, 2, 1, \\lambda]$ in Table~\\ref{tab:results1}.\n\n\\begin{table*}[hbt]\n\\begin{center}\n\\caption{QCD sum rule results obtained using the baryon multiplets $[\\mathbf{\\bar 3}_F, 1, 1, \\rho]$, $[\\mathbf{\\bar 3}_F, 1, 0, \\lambda]$, $[\\mathbf{6}_F, 0, 1, \\lambda]$, $[\\mathbf{6}_F, 1, 0, \\rho]$ and $[\\mathbf{6}_F, 2, 1, \\lambda]$.}\n\\begin{tabular}{c | c | c | c | c | c c | c c | c}\n\\hline\\hline\n\\multirow{2}{*}{Multiplets} & \\multirow{2}{*}{B} & $\\omega_c$ & Working region & $\\overline{\\Lambda}$ & $K$ & $\\Sigma$ & Baryons & Mass & Difference\n\\\\ & & (GeV) & (GeV) & (GeV) & (GeV$^2$) & (GeV$^2$) & ($j^P$) & (GeV) & (MeV)\n\\\\ \\hline\\hline\n\\multirow{4}{*}{$[\\mathbf{\\bar 3}_F, 1, 1, \\rho]$} & \\multirow{2}{*}{$\\Lambda_b$} & \\multirow{2}{*}{3.1} & \\multirow{2}{*}{$0.538< T < 0.588$} & \\multirow{2}{*}{$1.17$} & \\multirow{2}{*}{$-0.99$} & \\multirow{2}{*}{$0.042$}\n& $\\Lambda_b(1\/2^-)$ & $5.87 \\pm 0.12$ & \\multirow{2}{*}{$11 \\pm 4$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Lambda_b(3\/2^-)$ & $5.88 \\pm 0.11$ &\n\\\\ \\cline{2-10}\n& \\multirow{2}{*}{$\\Xi_b$} & \\multirow{2}{*}{3.6} & \\multirow{2}{*}{$0.536< T < 0.636$} & \\multirow{2}{*}{$1.35$} & \\multirow{2}{*}{$-0.98$} & \\multirow{2}{*}{$0.035$}\n& $\\Xi_b(1\/2^-)$ & $6.06 \\pm 0.13$ & \\multirow{2}{*}{$9 \\pm 4$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Xi_b(3\/2^-)$ & $6.07 \\pm 0.13$ &\n\\\\ \\hline\n\\multirow{4}{*}{$[\\mathbf{\\bar 3}_F, 1, 0, \\lambda]$} & \\multirow{2}{*}{$\\Lambda_b$} & \\multirow{2}{*}{3.6} & \\multirow{2}{*}{$0.597< T < 0.684$}\n& \\multirow{2}{*}{$1.21$} & \\multirow{2}{*}{$-2.60$} & \\multirow{2}{*}{$0.019$} & $\\Lambda_b(1\/2^-)$ & $6.01 \\pm 0.11$ & \\multirow{2}{*}{$5 \\pm 2$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Lambda_b(3\/2^-)$ & $6.01 \\pm 0.11$ &\n\\\\ \\cline{2-10}\n& \\multirow{2}{*}{$\\Xi_b$} & \\multirow{2}{*}{4.1} &\n\\multirow{2}{*}{$0.565< T < 0.759$}\n& \\multirow{2}{*}{$1.44$} & \\multirow{2}{*}{$-3.21$} & \\multirow{2}{*}{$0.014$} & $\\Xi_b(1\/2^-)$ & $6.27 \\pm 0.13$ & \\multirow{2}{*}{$4 \\pm 2$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Xi_b(3\/2^-)$ & $6.28 \\pm 0.13$ &\n\\\\ \\hline\n\\multirow{3}{*}{$[\\mathbf{6}_F, 0, 1, \\lambda]$} & $\\Sigma_b$ & 3.3 &\n{$0.527< T < 0.628$}\n& $1.22$ & $-2.53$ & $0$ & $\\Sigma_b(1\/2^-)$ & $6.02 \\pm 0.12$ & $-$\n\\\\ \\cline{2-10}\n& $\\Xi^\\prime_b$ & 3.8 & {$0.531< T < 0.676$} & $1.43$ & $-2.77$ & $0$ & $\\Xi^\\prime_b(1\/2^-)$ & $6.24 \\pm 0.11$ & $-$\n\\\\ \\cline{2-10}\n& $\\Omega_b$ & 4.3 & {$0.547< T < 0.740$} & $1.67$ & $-3.22$ & $0$ & $\\Omega_b(1\/2^-)$ & $6.50 \\pm 0.11$ & $-$\n\\\\ \\hline\n\\multirow{6}{*}{$[\\mathbf{6}_F, 1, 0, \\rho]$} & \\multirow{2}{*}{$\\Sigma_b$} & \\multirow{2}{*}{3.3} & \\multirow{2}{*}{$0.534< T < 0.624$} & \\multirow{2}{*}{$1.19$} & \\multirow{2}{*}{$-1.16$} & \\multirow{2}{*}{$0.014$} & $\\Sigma_b(1\/2^-)$ & $5.91 \\pm 0.14$ & \\multirow{2}{*}{$4 \\pm 2$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Lambda_b(3\/2^-)$ & $5.92 \\pm 0.14$ &\n\\\\ \\cline{2-10}\n& \\multirow{2}{*}{$\\Xi^\\prime_b$} & \\multirow{2}{*}{3.8} & \\multirow{2}{*}{$0.521< T < 0.681$} & \\multirow{2}{*}{$1.38$} & \\multirow{2}{*}{$-1.27$} & \\multirow{2}{*}{$0.011$}\n& $\\Xi^\\prime_b(1\/2^-)$ & $6.11 \\pm 0.13$ & \\multirow{2}{*}{$3 \\pm 1$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Xi^\\prime_b(3\/2^-)$ & $6.11 \\pm 0.13$ &\n\\\\ \\cline{2-10}\n& \\multirow{2}{*}{$\\Omega_b$} & \\multirow{2}{*}{4.3} & \\multirow{2}{*}{$0.510< T < 0.751$} & \\multirow{2}{*}{$1.60$} & \\multirow{2}{*}{$-1.56$} & \\multirow{2}{*}{$0.009$}\n& $\\Omega_b(1\/2^-)$ & $6.34 \\pm 0.13$ & \\multirow{2}{*}{$2 \\pm 1$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Omega_b(3\/2^-)$ & $6.34 \\pm 0.13$ &\n\\\\ \\hline\n\\multirow{6}{*}{$[\\mathbf{6}_F, 2, 1, \\lambda]$} & \\multirow{2}{*}{$\\Sigma_b$} & \\multirow{2}{*}{3.3} & \\multirow{2}{*}{$0.537< T < 0.620$} & \\multirow{2}{*}{$1.17$} & \\multirow{2}{*}{$-2.58$} & \\multirow{2}{*}{$0.035$}\n& $\\Sigma_b(3\/2^-)$ & $5.96 \\pm 0.18$ & \\multirow{2}{*}{$15 \\pm 7$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Sigma_b(5\/2^-)$ & $5.98 \\pm 0.18$ &\n\\\\ \\cline{2-10}\n& \\multirow{2}{*}{$\\Xi^\\prime_b$} & \\multirow{2}{*}{3.8} & \\multirow{2}{*}{$0.525< T < 0.684$} & \\multirow{2}{*}{$1.36$} & \\multirow{2}{*}{$-2.90$} & \\multirow{2}{*}{$0.028$} & $\\Xi^\\prime_b(3\/2^-)$ & $6.17 \\pm 0.17$ & \\multirow{2}{*}{$12 \\pm 6$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Xi^\\prime_b(5\/2^-)$ & $6.18 \\pm 0.16$ &\n\\\\ \\cline{2-10}\n& \\multirow{2}{*}{$\\Omega_b$} & \\multirow{2}{*}{4.3} & \\multirow{2}{*}{$0.518< T < 0.759$} & \\multirow{2}{*}{$1.59$} & \\multirow{2}{*}{$-3.45$} & \\multirow{2}{*}{$0.022$} & $\\Omega_b(3\/2^-)$ & $6.43 \\pm 0.13$ & \\multirow{2}{*}{$9 \\pm 4$}\n\\\\ \\cline{8-9}\n& & & & & & & $\\Omega_b(5\/2^-)$ & $6.43 \\pm 0.13$ &\n\\\\ \\hline \\hline\n\\end{tabular}\n\\label{tab:results1}\n\\end{center}\n\\end{table*}\n\n\n\\section{Discussion and Conclusion}\n\\label{sec:summary}\n\nIn this paper we followed Ref.~\\cite{Chen:2015kpa} and studied the $P$-wave bottom baryons using the method of QCD sum rule in the framework of HQET. We note that we just need to redo numerical analyses using the $bottom$ quark mass. The results are summarized in Table~\\ref{tab:results1} for baryon multiplets $[\\mathbf{\\bar 3}_F, 1, 1, \\rho]$, $[\\mathbf{\\bar 3}_F, 1, 0, \\lambda]$, $[\\mathbf{6}_F, 0, 1, \\lambda]$, $[\\mathbf{6}_F, 1, 0, \\rho]$ and $[\\mathbf{6}_F, 2, 1, \\lambda]$.\n\nOur results suggest that the two observed states $\\Lambda_b(5912)^0$ ($J^P=1\/2^-$) and $\\Lambda_b(5920)^0$ ($J^P=3\/2^-$)~\\cite{Aaij:2012da,Aaltonen:2013tta} can be well described by the baryon doublet $[\\mathbf{\\bar 3}_F, 1, 1, \\rho]$. They are the bottom partners of $\\Lambda_c(2595)$ and $\\Lambda_c(2625)$~\\cite{pdg}, and also belong to the $SU(3)$ $\\mathbf{\\bar 3}_F$ multiplets of $J^P=1\/2^-$ and $3\/2^-$. Our results suggest that their $SU(3)$ flavor partners, $\\Xi_b(1\/2^-)$ and $\\Xi_b(3\/2^-)$, have masses $6.06 \\pm 0.13$ GeV and $6.07 \\pm 0.13$ GeV, respectively, with a mass splitting $9 \\pm 4$ MeV.\n\nThe QCD sum rule results obtained by using baryon doublet $[\\mathbf{\\bar 3}_F, 1, 0, \\lambda]$ are similar and also consistent with the experimental data~\\cite{Aaij:2012da,Aaltonen:2013tta}. Here we obtain that the $\\rho$ mode excitation, $[\\mathbf{\\bar 3}_F, 1, 1, \\rho]$, is lower than the $\\lambda$ one, $[\\mathbf{\\bar 3}_F, 1, 0, \\lambda]$. This is consistent with our previous analysis of the charm sector~\\cite{Chen:2015kpa}, in contrast with the quark model expectation~\\cite{Copley:1979wj,Yoshida:2015tia}. Thus, further studies are needed to clarify their nature. Particularly, it would be useful to study decays and productions of these heavy baryon excitations.\n\nWe also studied the $SU(3)$ $\\mathbf{6}_F$ multiplets by using the baryon multiplets $[\\mathbf{6}_F, 0, 1, \\lambda]$, $[\\mathbf{6}_F, 1, 0, \\rho]$ and $[\\mathbf{6}_F, 2, 1, \\lambda]$. Our results suggest that the $P$-wave bottom baryons $\\Sigma_b$, $\\Xi^\\prime_b$ and $\\Omega_b$ have (averaged) masses about 6.0 GeV, 6.2 GeV and 6.4 GeV, respectively.\n\nWhile checking the status of observed bottom baryons listed in PDG, we notice that progress have been made in the experimental observations. Especially, in the past years, experiments like LHCb and CDF announced the observations of higher radial excitations of bottom baryon~\\cite{Aaij:2012da,Aaltonen:2013tta}. We believe that higher radial and orbital excitations will be released in future experiments. With the running of LHCb at 13 TeV, experimental exploration to higher radial and orbital excitations of bottom baryons will be a penitential and intriguing research topic. The present theoretical results of the mass of $P$-wave bottom baryons can provide valuable information. In near future, the joint efforts from experimentalist and theorist will be helpful to identify more and more bottom baryons.\n\n\n\n\\section*{ACKNOWLEDGMENTS}\n\nWe thank Cheng-Ping Shen and Ke-Wei Wei for useful discussions and careful reading of the manuscript.\nH.X.C. thanks Profs. Luciano Maiani, Rinaldo Baldini and Pierluigi Campana\nfor their hospitality during his stay at INFN-LNF.\nThis project is supported by the National Natural Science Foundation\nof China under Grants No. 11205011, No. 11475015, No. 11375024, No. 11222547, No.\n11175073, and No. 11261130311, the Ministry of\nEducation of China (SRFDP under Grant No. 20120211110002 and the\nFundamental Research Funds for the Central Universities).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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}\n\\def\\mathop{\\bar\\circ}{\\mathop{\\bar\\circ}}\n\\def{\\bar\\phi\\bar\\phi}{{\\bar\\phi\\bar\\phi}}\n\n\\usepackage{tempora}\n\\usepackage{color,xcolor}\n\\usepackage{manfnt}\n\\def\\textcolor{red}{\\textcolor{red}}\n\\def\\textcolor{Purple}{\\textcolor{Purple}}\n\\def\\textcolor{green}{\\textcolor{green}}\n\\def\\textcolor{blue}{\\textcolor{blue}}\n\\def\\textcolor{Orange}{\\textcolor{Orange}}\n\\def\\textcolor{Bittersweet}{\\textcolor{Bittersweet}}\n\\def\\textcolor{JungleGreen}{\\textcolor{JungleGreen}}\n\\def\\textcolor{Rhodamine}{\\textcolor{Rhodamine}}\n\\def\\textcolor{Yellow}{\\textcolor{Yellow}}\n\\def\\textcolor{Salmon}{\\textcolor{Salmon}}\n\\def\\textcolor{Turquoise}{\\textcolor{Turquoise}}\n\\def\\textcolor{Blueviolet}{\\textcolor{Blueviolet}}\n\\def\\vskip2ex\\hskip5ex{\\vskip2ex\\hskip5ex}\n\\def\\end{document}{\\end{document}}\n\\usepackage{hyperref}\n\\hypersetup{colorlinks,%\ncitecolor=black,%\nfilecolor=black,%\nlinkcolor=black,%\nurlcolor=black}\n\\def\\alpha{\\alpha}\n\\def$L_\\infty${$L_\\infty$}\n\\def${\\mathscr A}\\ ${${\\mathscr A}\\ $}\n\\defCattaneo \\& Felder{Cattaneo \\& Felder}\n\\def\\definition{\\definition}\n\\defPropositin{Propositin}\n\\def\\{\\[{\\{\\[}\n\\def\\}\\}{\\}\\}}\n\n\\newcommand{\\newcommand}{\\newcommand}\n\\newcommand{\\lb}{\\llbracket}\n\\newcommand{\\rb}{\\rrbracket}\n\\newcommand{\\gl}{\\llbracket}\n\\newcommand{\\gr}{\\rrbracket}\n\\newcommand{\\blacktriangle}{\\blacktriangle}\n\n\n\\begin{document}\n\n\n\\title{\\Large\\bf  Poisson gauge theory}\n\n\n\n\\author{{ Vladislav G. Kupriyanov}\\thanks{Email: \\ {\\tt\n    vladislav.kupriyanov@gmail.com}}\\\\\n    \\\\\n{\\it Centro de Matem\\'atica, Computa\\c{c}\\~{a}o e\nCogni\\c{c}\\~{a}o}\\\\{\\it Universidade Federal do ABC, Santo Andr\\'e, SP, \nBrazil}\\\\ and {\\it \nTomsk State University, Tomsk, Russia }  }\n\n\n\n\n\\maketitle\n\\vskip2ex\\hskip5ex\n\\begin{abstract}\nThe Poisson gauge algebra is a semi-classical limit of complete non-commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity $\\Theta^{ab}(x)$, especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the covariant derivatives defines the Poisson field strength which is covariant under the Poisson gauge transformations and reproduces the standard $U(1)$ field strength in the commutative limit. We derive the corresponding Bianchi identities. The field equations for the gauge and the matter fields are obtained from the gauge invariant action. We consider different examples of linear in coordinates Poisson structures $\\Theta^{ab}(x)$, as well as non-linear ones, and obtain explicit expressions for all proposed constructions. Our model is unique up to invertible field redefinitions and coordinate transformations.\n\\end{abstract}\n\n\n\\section{Introduction}\n\nThe consistent definition of non-commutative gauge theory is an important step for understanding of low-energy physics of D-branes in general backgrounds of string theory \\cite{SW}-\\cite{HullSzabo}. This problem was extensively investigated during the last decades. We mention here the main approaches discussed in the literature: the covariant coordinates approach \\cite{Wess,Wess1}, the twist approach \\cite{Dimitrijevic}-\\cite{twist1}, the L$_\\infty$-bootstrap approach \\cite{BBKL} and also recently proposed approach based on the combination of the L$_\\infty$ algebras and the twist \\cite{Ciric}. Nevertheless the structure of the non-commutative field theory still requires the better understanding, especially in case when the non-commutativity $\\Theta^{ab}(x)$ is not constant. In the previous work \\cite{KS21} in collaboration with Richard Szabo we formulated a novel approach to the construction of algebra of non-commutative gauge symmetries in the semi-classical limit, based on the symplectic embeddings of (almost)-Poisson gauge structure. The aim of the present research is to construct the dynamical sector of the corresponding theory. \n\nThe associative non-commutativity of space-time is usually introduced in the theory by substituting the standard pointwise multiplication of fields $f\\cdot g$ on some manifold $M$ with the star multiplication,\n\\begin{equation}\\label{star}\nf\\star g=f\\cdot g+\\frac{i\\hbar}{2}\\{f,g\\}+{\\cal O}(\\hbar^2)\\,,\n\\end{equation}\n defined as an associative deformation of the pointwise product along the given Poisson structure,\n \\begin{equation}\\label{PB}\n\\{x^i,x^j\\}=\\alpha\\,\\Theta^{ij}(x)\\,,\n\\end{equation}\nwhere $\\alpha$ is a small parameter which we will refere to as the non-commutativity parameter. The non-commutativity field $\\Theta^{ij}(x)$ is considered to be an external field. The higher order in $\\hbar$ contributions to (\\ref{star}) are defined from the condition of the associativity, $(f\\star g)\\star h=f\\star(g\\star h)$, and can be constructed according to the Formality theorem \\cite{Kontsevich}, or also the polydifferential approach \\cite{KV08}.\n\nThe non-commutative $U(1)$ gauge transformations $\\delta^{NC}_fA_a$ are defined as transformations satisfying the following two conditions: they should close the algebra,\n\\begin{equation}\n[\\delta^{NC}_f,\\delta^{NC}_g]A_a=\\delta^{NC}_{-i[f,g]_\\star}A_a\\,,\\label{ga1}\n\\end{equation}\nwith, $[f,g]_\\star=f\\star g-g\\star f$, and should reproduce the standard $U(1)$ gauge transformations $\\delta^0_fA_a=\\partial_af$, in the commutative limit.  Reminding that the Poisson bracket is the semi-classical limit of the star commutator, $\\{f,g\\}=\\lim_{\\hbar\\to0}[f,g]_\\star\/i\\hbar$, we define the Poisson gauge transformations $\\delta_fA_a$, following \\cite{KS21}, as the semi-classical limit of the full non-commutative $U(1)$ gauge transformations. They should close the algebra,\n\\begin{equation}\n[\\delta_f,\\delta_g]A_a=\\delta_{\\{f,g\\}} A_a\\,,\\label{ga}\n\\end{equation}\ncalled Poisson gauge algebra, and reproduce the standard $U(1)$ gauge transformations in the commutative limit, $\\lim_{\\alpha\\to0}\\delta_fA_a=\\partial_af$.\n\nIf $\\Theta^{ij}$ is constant, one may easily see that the expression, $\\delta_fA_a=\\partial_af+\\{f,A_a\\}$, satisfies (\\ref{ga}). However, for non-constant $\\Theta^{ij}(x)$ the standard Leibniz rule with respect to the partial derivative is violated, $\\partial_a\\{f,g\\}\\neq\\{\\partial_af,g\\}+\\{f,\\partial_ag\\}$, therefore the same expression will not close the algebra (\\ref{ga}) anymore. To overcome this problem one has to modify the expression for the gauge transformations introducing the corrections proportional to the derivatives of the non-commutativity $\\partial_a\\Theta^{ij}(x)$ which would compensate the violation of the Leibniz rule. The problem was solved in \\cite{Kupriyanov:2019ezf} using the L$_\\infty$-bootstrap approach to the non-commutative gauge theories \\cite{BBKL}. We stress that L$_\\infty$-formalism is a powerful tool for the construction of perturbative order by order in $\\alpha$ expressions for the consistent non-commutative deformations of gauge theories \\cite{Kup27}. Though, to get an explicit all order expressions normally one needs to invoke additional considerations. \n\nThe approach to the solution of this problem proposed in \\cite{KS21} is based on the symplectic embeddings of Poisson manifolds \\cite{SE1,SE2} and is especially good for obtaining explicit form of the deformed constructions. The problem with violation of the Leibniz rule for the original Poisson bracket (\\ref{PB}) can be solved in an extended space. To each coordinate $x^i$ we introduce a conjugate variable $p_i$, in such a way that the corresponding Poisson brackets,\n\\begin{equation}\\label{PB1}\n\\{x^i,p_j\\}=\\gamma^i_j(x,p)\\,,\\qquad \\{p_i,p_j\\}=0\\,,\n\\end{equation}\nshould satisfy the Jacobi identity. In our construction we will need the vanishing bracket between $p$-variables, $\\{p_i,p_j\\}=0$, while the matrix, $\\gamma^i_j(x,p)=\\delta^i_j-\\mbox{$\\frac{\\alpha}{2}$}\\partial_j\\Theta^{ik}p_k+{\\cal O}(\\alpha^2)$, defining the Poisson bracket $\\{x^i,p_j\\}$ will be constructed in Sec. 2. For constant $\\Theta^{ij}$ this matrix is constant, $\\gamma^i_j(x,p)=\\delta^i_j$, so, $\\{f(x),p_i\\}=\\partial_if(x)$, i.e., the Poisson bracket between the function $f(x)$ on $M$ and the auxiliary variable $p_i$ is just a partial derivative of this function. In case if $\\Theta^{ij}(x)$ is  not constant the expression for $\\gamma^i_j(x,p)$ is more complicated, however the action of the operator $\\{\\,\\cdot\\,,p_i\\}$ which we will call `twisted' derivative on functions $f(x)$ is similar to that of the partial derivatives $\\partial_i$. First of all because the Jacobi identity for the Poisson brackets (\\ref{PB1}) and the fact that $\\{p_i,p_j\\}=0$ imply that these operators comute, $\\{\\{f(x),p_i\\},p_j\\}=\\{\\{f(x),p_j\\},p_i\\}$. Second, because the `twisted' derivative satisfies the Leibniz rule,\n\\begin{equation}\\label{Lr}\n\\{\\{f(x),g(x)\\},p_i\\}=\\{\\{f(x),p_i\\},g(x)\\}+\\{f(x),\\{g(x),p_i\\}\\}\\,,\n\\end{equation}\nwhich also follows from the Jacobi identity. However, the price to pay is that the expression $\\{f(x),p_i\\}$ depends also on the auxiliary non-physical $p$-variables. It turns out that the auxiliary variables can be eliminated in the consistent way by introducing the constraints, $p_a=A_a(x)$\\footnote{Note that in \\cite{KS18} the symplectic embeddings were used to construct the consistent Hamiltonian description of the electrically charged particle in the field of magnetic monopole distributions. However in that case the elimination the auxiliary $p$-variables was not possible.}. In the Sec. 3 we will prove that the gauge transformations defined by,\n\\begin{equation}\\label{gtA}\n\\delta_f A_a=\\gamma^l_a(A)\\,\\partial_lf(x)+\\{A_a(x),f(x)\\}\\,,\n\\end{equation}\nwhere, $\\gamma^l_a(A):=\\gamma^l_a(x,p)|_{p_a=A_a(x)}$, close the algebra (\\ref{ga}) and reproduce the standard $U(1)$ gauge transformations, $\\delta^0_fA_a=\\partial_af$, in the commutative limit.\n\nThe new results of the present research are related to the consistent definition of the dynamical part of field-theoretical model having the Poisson gauge algebra (\\ref{ga}) as the corresponding algebra of gauge symmetries. Working in the formalism of the symplectic embeddings we introduce the matter field $\\psi$ by postulating the corresponding gauge transformation as, $\\delta_f\\psi=\\{\\psi,f\\}$. In the Sec. 4 we construct the gauge covariant derivative ${\\cal D}_a(\\psi)$ satisfying two key requirements: it transforms covariantly under the gauge transformation, $\\delta_f{\\cal D}_a(\\psi)=\\{{\\cal D}_a(\\psi),f\\}$, and reproduces the standard partial derivative in the commutative limit, ${\\cal D}_a(\\psi)\\to\\partial_a\\psi$, as $\\alpha\\to0$. In the Sec. 5 it will be shown that the commutator of two covariant derivatives defines the Poisson field strength ${\\cal F}_{ab}$ which also transforms covariantly, $\\delta_f{\\cal F}_{ab}=\\{{\\cal F}_{ab},f\\}$, and reproduces the standard $U(1)$ field strength in the commutative limit, $\\lim_{\\alpha\\to0}{\\cal F}_{ab}=\\partial_aA_b-\\partial_bA_a$. We also define the corresponding Bianchi identity in the Sec. 5.1. The main relations are resumed by the following table, which for simplicity is given for the case of linear Poisson structures, $\\Theta^{ij}(x)=f^{ij}_kx^k$,\n\\begin{center}\n\\begin{tabular}{S|  |S} \\toprule\n    {\\mbox{Object}}  & {Identity}  \\\\ \\midrule \\midrule\n  {${\\cal D}_a(\\psi)=\\rho_a^i(A)\\left(\\gamma_i^l(A)\\,\\partial_l\\psi+\\{A_a,\\psi\\}\\right)$}  & {$\\left[{\\cal D}_a,{\\cal D}_b\\right]=\\{{\\cal F}_{ab},\\,\\cdot\\,\\}+\\left({\\cal F}_{ad}\\,\\Lambda_b{}^{de}-{\\cal F}_{bd}\\,\\Lambda_a{}^{de}\\right){\\cal D}_e$} \\\\  \\midrule\n    {${\\cal F}_{ab}=R_{ab}{}^{ij}(A)(\\gamma_i^l(A)\\,\\partial_l A_j+\\{A_i,A_j\\})$} & {${\\cal D}_a\\left({\\cal F}_{bc}\\right)-{\\cal F}_{ad}\\,\\Lambda_b{}^{de}\\,{\\cal F}_{ec}+\\mbox{cycl.}(abc)=0$}    \\\\ \\bottomrule\n\\end{tabular}\n\\end{center}\nwhere,\n\\begin{equation*}\nR_{ab}{}^{cd}=\\frac12\\left(\\rho^c_a\\rho^d_b-\\rho^c_b\\rho^d_a\\right)\\,,\\qquad\\Lambda_b{}^{de}=\\left(\\rho^{-1}\\right)_j^d\\left(\\partial^j_A\\rho_b^m-\\partial^m_A\\rho_b^j\\right)\\left(\\rho^{-1}\\right)_m^e\\,,\n\\end{equation*}\nwith, $\\partial^j_A=\\partial\/\\partial A_j$, and the matrix $\\rho_a^i(A)$ is defined by, $\\rho_a^i(A):=\\rho^i_a(x,p)|_{p_a=A_a(x)}$, where $\\rho^i_a(x,p)$ should satisfy the equation,\n\\begin{equation}\\label{rhoi}\n\\{f(x),\\rho_a^i(x,p)\\}+\\rho_a^b(x,p)\\,\\partial_p^ i\\{f(x),p_b\\}=0\\,, \\qquad \\forall f(x)\\,.\n\\end{equation}\nFor the non-linear Poisson structures $\\Theta^{ij}(x)$ there may appear additional contributions in the identities in the right hand side of the table which are given in the Sec. 5.\n\nFor arbitrary non-commutativity $\\Theta^{ij}(x)$ we provide the recurrence relations for the construction of the matrices, $\\gamma_a^l(A)$ and $\\rho_a^l(A)$  which are the building blocs of our construction. For some particular choices of the non-commutative spaces, like the rotationally invariant NC space \\cite{Kup13}-\\cite{Gere} described by the $su(2)$-like Lie Poisson structure, $\\Theta^{ij}_\\varepsilon(x)=2\\alpha\\varepsilon^{ijk}x_k$, or the $\\kappa$-Minkowski space \\cite{kappa}-\\cite{kappa3} we obtain explicit all-order expressions for $\\gamma_a^l(A)$ and $\\rho_a^l(A)$ in the Sections 2 and 4 correspondingly. In the Sec. 6 we use the gauge covariant objects ${\\cal D}_a(\\psi)$ and ${\\cal F}_{ab}$ to construct the gauge invariant action and derive the field equations for the gauge field $A_a$ and the matter field $\\psi$. In case of the rotationally invariant non-commutative space \\cite{Kup13} the field equations in the pure gauge sector read, ${\\cal D}^{\\varepsilon}_a\\left({\\cal F}^{ab}_\\varepsilon\\right)=0$. We conclude with the final remarks and discussion in the Sec. 7 and provide the useful for the calculation formulas in the Appendix.                                         \n\n\n  \n\\section{Symplectic embeddings of Poisson manifold}\n\nIn this section we will summarize the necessary ingredients from the symplectic geometry that we will use throughout the paper. All precise mathematical definitions and proofs are given in \\cite{KS21}, here our aim is to recollect them  and expose on a physics friendly language. The problem of the construction of the symplectic embedding for the given Poisson structure (\\ref{PB}) formulated in the introduction consists basically in finding the matrix $\\gamma_j^i(x,p)$ which defines the Poisson bracket $\\{x^i,p_j\\}$ in such a way that the complete algebra of Poisson brackets (\\ref{PB}) and (\\ref{PB1}) should satisfy the Jacobi identity. \n\nThe Jacobi identity involving the original coordinates only, $\\{x^i,\\{x^j,x^k\\}\\}+\\mbox{cycl.}=0$, is satisfied automatically since $\\Theta^{ij}(x)$ is a Poisson bi-vector. The Jacobi identity with two original coordinates and one $p$-variable,\n$\n\\{x^i,\\{x^j,p_k\\}\\}+\\mbox{cycl.}=0\\,,\n$\nimplies an equation\\footnote{Note that here we use different notations from \\cite{KS21}, the matrix $\\gamma_a^k$ here corresponds to $\\delta_a^k+\\alpha\\,\\gamma_a^k$ used in \\cite{KS21}.},\n\\begin{equation}\n\\gamma^l_b\\,\\partial^b_p\\gamma_a^k-\\gamma^k_b\\,\\partial^b_p\\gamma^l_a+\\alpha\\,\\Theta^{lm}\\,\\partial_m\\gamma_a^k-\\alpha\\,\\Theta^{km}\\,\\partial_m\\gamma_a^l-\\alpha\\,\\gamma^m_a\\,\\partial_m\\Theta^{lk}=0\\,,\\label{eq2}\n\\end{equation}\non the function $\\gamma^i_j(x,p)$ in terms of given Poisson structure $\\Theta^{ij}(x)$.\nThe Jacobi identity with one $x$ and two $p$--variables, $\\{x^i,\\{p_j,p_k\\}\\}+\\mbox{cycl.}=0$, is also satisfied automatically. It follows from the fact that, $\\{p_i,p_j\\}=0$, and antisymmetry of $\\{x^j,p_k\\}$. And finally the Jacobi identity involving the $p$-variables only, $\\{p_i,\\{p_j,p_k\\}\\}+\\mbox{cycl.}=0$, is trivially satisfied.\n\nSo, the matrix $\\gamma_j^i(x,p)$ is defined as a solution of the equation (\\ref{eq2}) with the condition, $\\gamma_j^i(x,p)|_{\\alpha=0}=\\delta^i_j$, to guarantee that the complete algebra of the Poisson brackets (\\ref{PB}) and (\\ref{PB1}) is a deformation in $\\alpha$ of the canonical Poisson brackets, i.e., forms the symplectic algebra. Up to the second order in $\\alpha$ the solution reads,\n\\begin{eqnarray}\n\\gamma^k_a(x,p)=\\sum_{n=0}\\,\\gamma^{k(n)}_a&=&\\delta^k_a-\\frac{\\alpha}{2}\\, \\partial_a \\Theta^{kb} p_b\\label{gps}\\\\\n&&-\\frac{\\alpha^2}{12}\\left(2\\,\\Theta^{cm}\\partial_a\\partial_m\\Theta^{bk}+\\partial_a\\Theta^{bm}\\partial_m\\Theta^{kc}\\right)p_bp_c+{\\cal O}(\\alpha^3)\\,.\\notag\n\\end{eqnarray}\nThe recurrence relations for the construction of the matrix $\\gamma^k_a(x,p)$ in any order in $\\alpha$ are given in \\cite{Kup21}.\n\n\n\\subsection{Arbitrariness}\n\nWe note that the symplectic embedding is not unique. The arbitrariness is described by the invertible transformations of the variables $(x,p)$ which leave the Poisson brackets between the coordinates and coordinates $\\{x^i,x^j\\}$, as well as, momenta and momenta $\\{p_i,p_j\\}$ unchanged, while change the brackets between the coordinates and momenta $\\{x^i,p_j\\}$. Making the transformation which change only the $p$-variables, $\\phi:\\,p\\to\\tilde p$, with $\\phi(p)|_{\\alpha=0}=p$ and leave $x$-variables unchanged we do not change neither the bracket (\\ref{PB}), nor the $\\{p_i,p_j\\}=0$, while for the bracket between the coordinates and momenta one finds,\n\\begin{eqnarray}\\label{tildegamma}\n\\tilde \\gamma^i_j(x,\\tilde p)=\\{x^i,\\tilde p_j(p)\\}_{p=p(\\tilde p)}=\\gamma^i_k(x,p)\\,\\partial^k_p\\tilde p_j(p)|_{p=p(\\tilde p)}\\,.\n\\end{eqnarray}\nThe new set of Poisson brackets,\n\\begin{eqnarray}\n\\{x^i,x^j\\}&=&\\alpha\\,\\Theta^{ij}(x)\\,,\\\\\n\\{x^i,\\tilde p_j\\}&=&\\tilde \\gamma^i_j(x,\\tilde p)\\,,\\qquad \\{\\tilde p_i,\\tilde p\\}=0\\,,\\notag\n\\end{eqnarray}\nalso represents the symplectic embedding of the Poisson structure (\\ref{PB}) since $\\tilde \\gamma^i_j(x,\\tilde p)|_{\\alpha=0}=\\delta^i_j$.\n\n\\subsection{Symplectic embeddings of Lie-Poisson structures}\n\nThe Kirillov-Kostant Poisson bracket, also called sometimes the Lie-Poisson bracket, is defined in \\cite{Alekseev} as,\n\\begin{equation}\\label{e1}\n\\{x^i,x^j\\}=f^{ij}_k\\,x^k\\,,\n\\end{equation}\nwhere $f^{ij}_k$ are the structure constants of a Lie algebra. The solution for the equation (\\ref{eq2}) in this case is given by \\cite{KS21,Gutt,Meljanac},\n \\begin{equation}\\label{e4}\n   \\gamma^i_j(p)=\\delta^i_j-\\frac12\\,f^{ij_1}_j\\,p_{j_1}+\\mathcal{X}^i_j\\left(-M\/{2}\\right)\\,,\n\\end{equation}\nwhere, \\begin{equation}\\label{e4a}\nM^{i}_l={f}^{ij_1}_k {f}^{kj_2}_lp_{j_1}p_{j_2}\\,,\n\\end{equation}\n and $\\mathcal{X}^i_l\\left(-M\/{2}\\right)$ is a matrix valued function with,\n\\begin{equation}\\label{e5}\n\\mathcal{X}(t)=\\sqrt{\\frac{t}{2}}\\cot\\sqrt{\\frac{t}{2}}-1=\\sum_{n=1}^\\infty\\frac{(-2)^n\\,B_{2n}\\,t^{n}}{(2n)!}\\,,\n\\end{equation}\nwith $B_n$ being Bernoulli numbers. We stress that the matrix $\\gamma^i_j(p)$ defined in (\\ref{e4}) does not depend on $x$-variables.\n\nLet us consider an exemple of the $su(2)$-like Lie-Poisson algebra, \n\\begin{equation}\n\\{x^k,x^l\\}=2\\,\\alpha\\,{\\varepsilon^{kl}}_m\\, x^m\\,,\\label{su2}\n\\end{equation}\nphysically corresponding to the rotationally invariant non-commutative space. In (\\ref{su2}) $\\varepsilon^{klm}$ is the Levi-Civita symbol and the factor of $2$ is just a matter of convenience. We use the Kronecker delta to raise and lower  indices,\nand  summation under the repeated indices is understood. In this case the matrix $M_\\varepsilon$ appearing as the argument of the third term in the r.h.s. of (\\ref{e4}) is given by,\n\\begin{equation}\\label{e6}\n[M_\\varepsilon]^i_l=4\\,\\alpha^2\\left(p^i\\,p_l-\\delta^i_l\\,p^2\\right)\\,,\n\\end{equation}\nwhere, $p^2=p_mp^m$. This matrix is diagonalizable, it can be written as, $M_\\varepsilon=S\\cdot D\\cdot S^{-1}$, where $D$ is the diagonal matrix with the eigenvalues of $M_\\epsilon$,  $\\lambda_1=0$, $\\lambda_2=\\lambda_3=-4\\alpha^2p^2$, on the diagonal and the matrix $S$ is constructed from the corresponding eigenvectors. Therefore following \\cite{KV15} we write,\n\\begin{eqnarray}\\label{e7}\n\\mathcal{X}^i_j\\left(-M_\\varepsilon\/{2}\\right)&=&\\left[S\\cdot \\rm{diag}\\left[ \\mathcal{X}\\left(-\\lambda_1\/{2}\\right),\\mathcal{X}\\left(-\\lambda_2\/{2}\\right),\\mathcal{X}\\left(-\\lambda_3\/{2}\\right)\\right]\\cdot S^{-1}\\right]^{i}_j\\\\\n&=& \\alpha^2\\left(\\delta^i_j\\,p^2- p^ ip_j\\right)\\chi(\\alpha^2\\,p^2)\\,,\\notag\n\\end{eqnarray}\nwhere,\n\\begin{equation}\\label{chi}\n\\chi(t)=t^{-1}\\,\\mathcal{X}(2t)=\\frac1t\\,\\left(\\sqrt{t}\\cot\\sqrt{t}-1\\right)\\,.\n\\end{equation}\nWe conclude that,\n\\begin{equation}\n[\\gamma_\\varepsilon]^k_a(p)=\\left[1+\\alpha^2p^2\\chi\\left(\\alpha^2p^2\\right)\\right]\\delta^k_a-\\alpha^2\\chi\\left(\\alpha^2p^2\\right)p_ap^k-\\alpha\\,\\varepsilon_a{}^{kl}p_l\\,,\\label{gammasu2}\n\\end{equation}\nNote that the explicit form of the function $\\chi(t)$ was obtained from the general solution (\\ref{e4}) diagonalizing the matrix $\\mathcal{X}^i_j\\left(-M_\\varepsilon\/{2}\\right)$. Alternatively one may check that the expression (\\ref{gammasu2}) satisfies the equation (\\ref{eq2}) only if $\\chi(t)$ obeys the ODE,\n\\begin{equation}\\label{w21}\n2\\,t\\,\\chi^\\prime+3\\,\\chi+t\\,\\chi^2+1=0\\,,\\qquad \\chi(0)=-\\frac13\\,,\n\\end{equation}\nwhose solution is given by (\\ref{chi}).\n\nThe second particular exemple we would like to discuss here is the $\\kappa$-Minkowski space \\cite{kappa}-\\cite{kappa3} yielding the Kirillov-Kostant structure,\n\\begin{equation}\n\\{x^k,x^l\\}=2\\left(a^i\\,x^j-a^j\\,x^i\\right)\\,,\\label{kappa}\n\\end{equation}\nwith $a^i$ being constants. For this Poisson structure the simplest solution for $\\gamma^k_a(p)$ reads \\cite{KKV},\n\\begin{equation}\n[\\gamma_\\kappa]^k_a(p)= \\left[\\sqrt{1+(a\\cdot p)^2}+(a\\cdot p)\\right]\\delta^k_a -a^k\\,p_a\\,.\\label{gammakappa}\n\\end{equation}\n\n\\subsection{Change of coordinates}\n\nIf two different Poisson structures $\\Theta^{ij}(x)$ and $\\tilde\\Theta^{ij}(\\tilde x)$ are related by the invertible coordinate transformation, $\\upsilon:\\,x\\to\\tilde x$, i.e.,\n\\begin{equation}\\label{g1}\n\\tilde\\Theta^{ij}(\\tilde x)=\\{\\tilde x^i(x),\\tilde x^j(x)\\}_{x=x(\\tilde x)}=\\partial_k\\tilde x^i\\,\\Theta^{kl}(x)\\,\\partial_l\\tilde x^j|_{x=x(\\tilde x)}\\,,\n\\end{equation}\nand the symplectic embedding $\\gamma^i_j(x,p)$ of the first one is known, then the symplectic embedding of the second one can be constructed according to,\n\\begin{equation}\\label{g2}\n\\tilde \\gamma^i_j(\\tilde x,p)=\\{\\tilde x^i(x),p_j\\}_{x=x(\\tilde x)}=\\partial_k\\tilde x^i\\, \\gamma^k_j(x,p)|_{x=x(\\tilde x)}\\,.\n\\end{equation}\nThus formally having in hand a symplectic embedding of a given Poisson structure we may generate symplectic embeddings of the whole family of the related Poisson structures. At the same time we have to be careful with the applications of this construction since the physical observables should not depend on the change of coordinates.\n\nLet us consider an example. We start with the symplectic embedding (\\ref{gammasu2}) of the $su(2)$-like Lie-Poisson structure (\\ref{su2}). The corresponding non-commutativity preserves rotational symmetry, see \\cite{Kup13}. Let us discuss its generalization, namely the non-linear Poisson structure,\n\\begin{equation}\n\\{\\tilde x^i,\\tilde x^j\\}=2\\,\\alpha\\,{\\varepsilon^{ij}}_m\\, \\tilde x^m\\,f(\\tilde x^2)\\,,\\label{su2f}\n\\end{equation}\nwhere $f(\\tilde x^2)$ is some given function. The non-commutativity (\\ref{su2f}) also preserves rotations, however it does not grows on the infinity provided that $f(\\tilde x^2)$ decreases faster then $1\/\\sqrt{\\tilde x^2}$, when $\\tilde x^2\\to\\infty$. In this case we have a kind of \"local\" rotationally invariant non-commutativity. The algebra (\\ref{su2f}) can be obtained from (\\ref{su2}) by the change of variables,\n\\begin{equation}\\label{g3}\n\\tilde x^i=g(x^2)\\,x^i\\,,\n\\end{equation}\nwhere $g(x^2)$ is a function to be determined. Note that, $\\tilde x^2=g^2(x^2)\\,x^2$. One calculates,\n\\begin{equation}\\label{g4}\n\\{\\tilde x^i,\\tilde x^j\\}=2\\,\\alpha\\,{\\varepsilon^{ij}}_m\\, x^m\\,g^2(x^2)=2\\,\\alpha\\,{\\varepsilon^{ij}}_m\\, \\tilde x^m\\,g(x^2)\\,.\n\\end{equation}\nComparing (\\ref{g4}) to (\\ref{su2f}) and taking into account (\\ref{g3}) one finds,\n\\begin{equation}\\label{g5}\nf\\left(g^2(x^2)\\,x^2\\right)=g(x^2)\\,,\n\\end{equation}\nwhich is a functional relation on $g$ provided that $f$ is some given function. Taking $f$ as the Gaussian function, $f(\\tilde x^2)=\\exp(-\\alpha\\tilde x^2\/2)$, one obtains from (\\ref{g5}),\n\\begin{equation}\\label{g6}\ng(x^2)=e^{-\\frac{W(\\alpha x^2)}{2}}\\,,\\qquad\\Rightarrow\\qquad \\tilde x^i=e^{-\\frac{W(\\alpha x^2)}{2}}\\,x^i\\,,\n\\end{equation}\nwhere $W(z)$ is the Lambert $W$ function satisfying the functional relation, $z=W(z)\\,e^{W(z)}$. One may easily verify that in this case, $\\alpha\\tilde x^2=\\exp(-W(\\alpha x^2))\\, \\alpha x^2=W(\\alpha x^2)$, and thus, $f(\\tilde x^2)=\\exp(-\\alpha\\tilde x^2\/2)=\\exp(-W(\\alpha x^2)\/2)=g(x^2)$, i.e., (\\ref{g5}) holds true. The inverse transformation is,\n\\begin{equation}\\label{g7}\nx^i=\\frac{1}{g(x^2)}\\,\\tilde x^i=e^{\\frac{\\alpha\\tilde x^2}{2}}\\,\\tilde x^i\\,.\n\\end{equation}\nNow from (\\ref{g2}), (\\ref{g3}) and (\\ref{g7}) we derive the symplectic embedding of (\\ref{su2f}),\n\\begin{equation}\n\\tilde\\gamma^i_j(\\tilde x,p)=e^{-\\frac{\\alpha\\tilde x^2}{2}}\\left(\\delta^i_k-\\frac{\\alpha\\,\\tilde x_k\\,\\tilde x^i}{1+\\alpha\\,\\tilde x^2}\\right)\\gamma^k_j(p)\\,,\n\\end{equation}\nwhere the matrix $\\gamma^k_j(p)$ was constructed in (\\ref{gammasu2}) and we remind that we do not change $p$ variables.\n\n\n\n\n\n\n\\section{Poisson gauge transformations}\n\nIn this section we prove the statement formulated in the introduction regarding the closure condition (\\ref{ga}) of the Poisson gauge transformations (\\ref{gtA}).\nFirst we observe that the (\\ref{gtA}) can be written in a more convenient form as,\n\\begin{equation}\\label{APhi}\n\\delta_f A_a=\\{f(x),\\Phi_a\\}_{\\Phi=0}\\,,\n\\end{equation}\nwhere,\n\\begin{equation}\\label{Phi}\n\\Phi_a:=p_a-A_a(x)\\,.\n\\end{equation}\nIndeed, the expression $\\{f(x),p_a\\}_{p_a-A_a(x)=0}$ means that first we calculate the Poisson bracket $\\{f(x),p_a\\}$ according to (\\ref{PB1}) and then substitute all $p$ with $A$, so,\n\\begin{equation*}\n\\{f(x),p_a\\}_{\\Phi=0}=\\gamma^l_a(A)\\,\\partial_lf(x)\\,.\n\\end{equation*}\nThe main property of the transformation (\\ref{APhi}) if formulated by the following proposition.\n{\\proposition \\label{pgt} The Poisson gauge transformations (\\ref{APhi}) close the algebra (\\ref{ga}).}\n\\proof We start with some preliminary definitions,\n\\begin{equation}\n\\delta_f F(A):=F\\left(A+\\delta_f A\\right)-F(A)\\,.\n\\end{equation}\nIn particular, \n\\begin{equation}\n\\delta_g \\left(\\{A_a, f\\}\\right)=\\{\\delta_g A_a,f\\}\\,,\n\\end{equation}\nand if $F(A)$ is a smooth function in $A$, then,\n\\begin{equation}\n\\delta_f F(A)=\\left(\\partial_A^b F\\right)\\,\\delta_f A_b\\,,\n\\end{equation}\nwhere as before we use the notation, $\\partial_A^b=\\partial\/\\partial A_b$. The composition of two gauge transformations can be written,\n\\begin{eqnarray}\n\\delta_f\\left(\\delta_g A_a\\right)&=&-\\{g(x), \\delta_f A_a\\}_{\\Phi=0}+\\left(\\partial^b_p\\{g(x),\\Phi_a\\}\\right)_{\\Phi=0}\\delta_f A_b\\\\\n&=&-\\{g(x), \\{f(x),\\Phi_a\\}_{\\Phi=0}\\}_{\\Phi=0}+\\left(\\partial^b_p\\{g(x),\\Phi_a\\}\\right)_{\\Phi=0}\\{f(x),\\Phi_b\\}_{\\Phi=0}\\,,\\notag\n\\end{eqnarray}\nsince,\n\\begin{equation}\n\\partial_A^b\\left(\\{g(x),\\Phi_a\\}_{\\Phi=0}\\right)=\\left(\\partial_p^b \\{g(x),\\Phi_a\\}\\right)_{\\Phi=0}\\,.\\label{a1}\n\\end{equation}\nAnd thus we find,\n\\begin{eqnarray}\\label{t6}\n&&\\delta_f\\left(\\delta_g A_a\\right)-\\delta_g\\left(\\delta_f A_a\\right)=\\\\\n&&-\\{g(x), \\{f(x),\\Phi_a\\}_{\\Phi=0}\\}_{\\Phi=0}+\\left(\\partial^b_p\\{g(x),\\Phi_a\\}\\right)_{\\Phi=0}\\{f(x),\\Phi_b\\}_{\\Phi=0}\\notag\\\\\n&&+\\{f(x), \\{g(x),\\Phi_a\\}_{\\Phi=0}\\}_{\\Phi=0}-\\left(\\partial^b_p\\{f(x),\\Phi_a\\}\\right)_{\\Phi=0}\\{g(x),\\Phi_b\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\nUsing the relation (\\ref{r3}) from the appendix in the right hand side of (\\ref{t6}) we represent it as,\n\\begin{equation}\n-\\{g(x), \\{f(x),\\Phi_a\\}\\}_{\\Phi=0}+\\{f(x), \\{g(x),\\Phi_a\\}\\}_{\\Phi=0}\\,.\n\\end{equation}\nFinally applying now the Jacobi identity we end up with,\n\\begin{equation}\\label{t7}\n\\delta_f\\left(\\delta_g A_a\\right)-\\delta_g\\left(\\delta_f A_a\\right)=\\{\\{f(x),g(x)\\},\\Phi_a\\}_{\\Phi=0}=\\delta_{\\{f,g\\}}A_a\\\\.\n\\end{equation}\nThe latter means that the gauge variations (\\ref{APhi}) close the Lie algebra (\\ref{ga}). $\\Box$\n\n\\subsection{Field redefinition} \n\nThe form of the Poisson gauge transformation (\\ref{APhi}) relies on the precise symplectic embedding $\\gamma^i_a(x,p)$ of given Poisson structure (\\ref{PB}). \nHowever, as it was discussed in Sec. 2.1 the symplectic embedding of (\\ref{PB}) is not unique. Different embeddings are related by the invertible change of $p$-variables, $\\phi:\\,p\\to\\tilde p$, with $\\phi(p)|_{\\alpha=0}=p$. Having the Poisson gauge transformation (\\ref{APhi}) which satisfy the relation (\\ref{ga}), one may construct another gauge transformation corresponding to the new symplectic embedding $\\tilde\\gamma^i_a(x,\\tilde p)$,\n\\begin{equation}\\label{fr1}\n\\tilde\\delta_f\\tilde A_a=\\tilde\\gamma^i_a(\\tilde A)\\,\\partial_if+\\{f,\\tilde A_a\\}\\,,\n\\end{equation}\nwhich will close the same gauge algebra (\\ref{ga}).\n\nFrom the physical perspective one may expect that the invertible field redefinitions defined by, $\\phi: A\\to\\tilde A$, and $\\tilde  f=f$,\nwill leave our construction invariant. We define the new gauge transformations $\\tilde\\delta_f$ by setting,\n\\begin{equation}\n\\tilde\\delta_f:=\\phi\\circ\\delta_f\\circ \\phi^{-1}\\,,\n\\end{equation}\nor in the other words,\n\\begin{equation}\n\\tilde\\delta_f \\tilde A_a:=\\left.\\delta_f \\tilde A_a(A)\\right|_{A(\\tilde A)}=\\left.\\partial^k_A \\tilde A_a(A) \\,\\delta_f A_k\\right|_{A(\\tilde A)}\\,.\n\\end{equation}\nTaking into account (\\ref{tildegamma}) we see that it is exactly the expression (\\ref{fr1}), meaning that the arbitrariness in our construction corresponds to the invertible field redefinition.\n\nSince the field redefinition is invertible the gauge orbits are mapped onto the gauge orbits and the Seiberg-Witten condition,\n\\begin{equation}\n\\tilde A\\left(A+\\delta_ fA\\right)=\\tilde A(A)+\\tilde\\delta_f\\tilde A(A)\\,,\n\\end{equation}\nis trivially satisfied up to the linear order in $f$. Moreover,\n\\begin{equation}\n\\left[\\tilde\\delta_f,\\tilde\\delta_g\\right]=\\tilde\\delta_f\\circ\\tilde\\delta_g-\\tilde\\delta_g\\circ\\tilde\\delta_f=\\phi\\circ\\left(\\delta_f\\circ\\delta_g-\\delta_g\\circ\\delta_f\\right)\\circ\\phi^{-1}=\\tilde\\delta_{\\{f,g\\}}\\,,\n\\end{equation}\nmeaning that the gauge algebra remains the same, as expected. In \\cite{BBKT} it was shown that the Seiberg-Witten maps correspond to L$_\\infty$-quasi-isomorphisms which describe the arbitrariness in the definition of the related L$_\\infty$ algebra in the L$_\\infty$-bootstrap approach \\cite{BBKL}.\n\n\n\n\\section{Covariant derivative}\n\nWe start with the introduction of the matter fields $\\psi$ and the definition of the non-commutative $U(1)$ gauge transformations $\\delta^{NC}_f\\psi$ which should close the same gauge algebra (\\ref{ga1}) as the corresponding gauge fields $A_a$. In case pf the associative star products there are three possibilities to define such a gauge transformation. It can be taken to be left or right star multiplication, $\\delta^{NC}_f\\psi=if\\star\\psi$ or $\\delta^{NC}_f\\psi=i\\psi\\star f$, or even as the star commutator, $\\delta^{NC}_f\\psi=-i[\\psi,f]_\\star$. Only the third possibility is compatible with the semi-classical limit which is the subject of this research. That is why we define the Poisson gauge transformation of the matter field as,\n\\begin{equation}\\label{w3}\n\\delta_f\\psi=\\{\\psi,f\\}\\,.\n\\end{equation}\nThe Jacobi identity implies that such a determined gauge transformations close the same algebra as (\\ref{ga}), i.e.,\n\\begin{equation}\n[\\delta_f,\\delta_g]\\psi=\\delta_{\\{f,g\\}}\\psi\\,.\\label{w4}\n\\end{equation}\n\nWe note that in the commutative limit the Poisson gauge variation of the mater field vanishes, $\\lim_{\\alpha\\to0}\\delta_f\\psi=0$,  meaning that in this limit the matter field $\\psi$ is 'electrically neutral' and does not interact with the gauge field $A_a$. The interaction between the gauge field and such a defined matter field is caused by the non-commutativity. At the moment we do not see a clear physical meaning of such an interaction. The field $\\psi$ appears in our construction more like an auxiliary object needed for the consistent definition of the covariant derivative, which in turn is a central object of this work. As we will see in the next Section, the commutator of the covariant derivatives will produce the gauge covariant field strength for the gauge field. Also this object is essential for the derivation of the corresponding Bianchi identity and the equations of motion.\n\nOur aim now is to construct the covariant derivative ${\\cal D}_a(\\psi)$, the object satisfying two main requirements.  \n\\begin{itemize}\n\\item It should transform covariantly under the Poisson gauge transformation $\\delta_f$ defined in (\\ref{APhi}) and (\\ref{w3}), i.e.,\n\\begin{equation}\\label{w5}\n\\delta_f\\left({\\cal D}_a(\\psi)\\right)=\\{{\\cal D}_a(\\psi),f\\}\\,.\n\\end{equation}\nThis property will be essential for the construction of the gauge covariant Lagrangian and the gauge invariant action for the Poisson gauge theory.\n\\item  In the commutative limit it should reproduce the standard partial derivative,\n\\begin{equation}\\label{w5a}\n\\lim_{\\alpha\\to0}{\\cal D}_a(\\psi)=\\partial_a\\psi\\,,\n\\end{equation}\n since the interaction between the matter field and the gauge field disappears when $\\alpha\\to 0$.\n\\end{itemize}\n\n The answer is given by the following proposition.\n\n{\\proposition \\label{pcd} The operator,\n\\begin{equation}\\label{w6}\n{\\cal D}_a(\\psi)=\\rho_a^i(A)\\,\\{\\psi,\\Phi_i\\}_{\\Phi=0}\\,,\n\\end{equation}\nsatisfies the above two conditions if the matrix, $\\rho_a^i(A):=\\rho^i_a(x,p)_{\\Phi=0}$, and $\\rho^i_a(x,p)$ obeys the equation,\n\\begin{equation}\\label{rho}\n\\{f(x),\\rho_a^i(x,p)\\}+\\rho_a^b(x,p)\\,\\partial_p^ i\\{f(x),p_b\\}=0\\,, \\qquad \\forall f(x)\\,.\n\\end{equation}}\n\\proof\nLet us calculate,\n\\begin{eqnarray}\\label{w7}\n&&\\delta_f\\left(\\rho_a^i(A)\\,\\{\\psi,\\Phi_i\\}_{\\Phi=0}\\right)=\\\\\n&&\\partial_A^b\\rho_a^i(A)\\,\\{f,\\Phi_b\\}_{\\Phi=0}\\,\\{\\psi,\\Phi_i\\}_{\\Phi=0}+\\rho_a^i(A)\\,\\{\\{\\psi,f\\},\\Phi_i\\}_{\\Phi=0}+\\notag\\\\\n&&\\rho_a^i(A)\\,(\\partial_p^ b\\{\\psi,\\Phi_i\\})_{\\Phi=0}\\,\\{f,\\Phi_b\\}_{\\Phi=0}-\\rho_a^i(A)\\,\\{\\psi,\\{f,\\Phi_i\\}_{\\Phi=0}\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\nObserve that,\n\\begin{equation}\\label{w7a}\n\\partial_A^b\\rho_a^i(A)\\,\\{f,\\Phi_b\\}_{\\Phi=0}=\\{f,\\rho_a^i(p)-\\rho_a^i(A)\\}_{\\Phi=0}\\,,\n\\end{equation}\nand by (\\ref{r3}) from the appendix,\n\\begin{equation}\n\\{\\psi,\\{f,\\Phi_i\\}_{\\Phi=0}\\}_{\\Phi=0}=\\{\\psi,\\{f,\\Phi_i\\}\\}_{\\Phi=0}-(\\partial_p^ b\\{f,\\Phi_i\\})_{\\Phi=0}\\,\\{\\psi,\\Phi_b\\}_{\\Phi=0}\\,.\n\\end{equation}\nSo the right hand side of (\\ref{w7}) becomes,\n\\begin{eqnarray}\n&&\\{f,\\rho_a^i(p)\\}_{\\Phi=0}\\,\\{\\psi,\\Phi_i\\}_{\\Phi=0}+\\{\\rho_a^i(A),f\\}_{\\Phi=0}\\,\\{\\psi,\\Phi_i\\}_{\\Phi=0}+\\\\\n&&\\rho_a^i(A)\\,\\{\\{\\psi,f\\},\\Phi_i\\}_{\\Phi=0}+\\rho_a^i(A)\\,(\\partial_p^ b\\{\\psi,\\Phi_i\\})_{\\Phi=0}\\,\\{f,\\Phi_b\\}_{\\Phi=0}+\\notag\\\\\n&&\\rho_a^i(A)\\,(\\partial_p^ b\\{f,\\Phi_i\\})_{\\Phi=0}\\,\\{\\psi,\\Phi_b\\}_{\\Phi=0}-\\rho_a^i(A)\\,\\{\\psi,\\{f,\\Phi_i\\}\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\nUsing the Jacobi identity we rewrite it as,\n\\begin{eqnarray}\\label{w8}\n&&\\left[\\{f,\\rho_a^i(p)\\}_{\\Phi=0}+\\rho_a^b(A)\\,(\\partial_p^ i\\{f,\\Phi_b\\})_{\\Phi=0}\\right]\\{\\psi,\\Phi_i\\}_{\\Phi=0}+\\\\\n&&\\{\\rho_a^i(A),f\\}_{\\Phi=0}\\,\\{\\psi,\\Phi_i\\}_{\\Phi=0}+\\rho_a^i(A)\\left[(\\partial_p^ b\\{\\psi,\\Phi_i\\})_{\\Phi=0}\\,\\{f,\\Phi_b\\}_{\\Phi=0}+\\{\\{\\varphi,\\Phi_i\\},f\\}_{\\Phi=0}\\right]\\,.\\notag\n\\end{eqnarray}\nTaking into account that by (\\ref{r3}),\n\\begin{equation}\n(\\partial_p^ b\\{\\varphi,\\Phi_i\\})_{\\Phi=0}\\,\\{f,\\Phi_b\\}_{\\Phi=0}+\\{\\{\\varphi,\\Phi_i\\},f\\}_{\\Phi=0}=\\{\\{\\varphi,\\Phi_i\\}_{\\Phi=0},f\\}_{\\Phi=0}\\,,\n\\end{equation}\nthe second line in (\\ref{w8}) becomes,\n\\begin{equation}\n\\{\\rho_a^i(A)\\,\\{\\varphi,\\Phi_i\\}_{\\Phi=0},f\\}_{\\Phi=0}=\\{{\\cal D}_a(\\varphi),f\\}\\,.\n\\end{equation}\nTherefore the relation (\\ref{w5}) holds true if the first line of (\\ref{w8}) vanishes, i.e.,\n\\begin{equation}\n\\{f(x),\\rho_a^i(x,p)\\}+\\rho_a^b(x,p)\\,\\partial_p^ i\\{f(x),\\Phi_b\\}=0\\,, \\qquad \\forall f(x)\\,.\n\\end{equation}\nSince $\\{A_b(x),f(x)\\}$ does not depend on $p$ one arrives at (\\ref{rho}). The statement regarding the commutative limit, $\\alpha\\to 0$, follows from the fact that in this limit, $\\{\\psi,\\Phi_i\\}\\to\\partial_i\\psi$, and $\\rho_a^i(x,p)\\to\\delta_a^i$. $\\Box$\n\nIn local coordinates the equation (\\ref{rho}) on the function $\\rho_a^i(x,p)$ can be written as,\n\\begin{equation}\\label{w10}\n\\gamma^j_b\\,\\partial^b_p\\, \\rho_a^i+\\rho_a^b\\,\\partial^i_p\\gamma^j_b+\\alpha\\,\\Theta^{jb}\\,\\partial_b\\rho_a^i=0\\,,\n\\end{equation}\nwhere $\\Theta^{jb}(x)$ is a given Poisson structure and $\\gamma^j_b(x,p)$ represents its symplectic embedding constructed in the Sec. 2.\nFor the arbitrary non-commutativity parameter $\\Theta^{ab}(x)$ a solution of the equation (\\ref{w10}) can be found in form of the perturbative series,\n\\begin{eqnarray}\n\\rho^i_a(x,p)=\\sum_{n=0}^\\infty\\,\\rho^{i(n)}_a&=&\\delta^i_a-\\frac{\\alpha}{2}\\, \\partial_a \\Theta^{ib} p_b+\\label{w11}\\\\\n&&\\frac{\\alpha^2}{6}\\left(2\\,\\Theta^{cm}\\partial_a\\partial_m\\Theta^{ib}-\\partial_a\\Theta^{bm}\\partial_m\\Theta^{ic}\\right)p_bp_c+{\\cal O}(\\alpha^3)\\,.\\notag\n\\end{eqnarray}\nFor some specific choices of  non-commutativity, one may also discuss the convergence of such series and exhibit a closed expressions.\n\n\\subsection{Lie-Poisson structures and generalizations}\n\nFirst we observe that for the linear Poisson bi-vectors (\\ref{e1}) there is no explicit coordinate dependance of the functions $\\gamma_a^i(p)$ and $\\rho_a^i(p)$. These functions depend only on $p$-variables $p_a$. The equation (\\ref{w10}) becomes,\n\\begin{equation}\\label{w11}\n\\gamma^j_l\\,\\partial^l_p\\, \\rho_a^i+\\rho_a^l\\,\\partial^i_p\\gamma^j_l=0\\,,\n\\end{equation}\n where $\\gamma^j_l(p)$ is given by (\\ref{e4}). In what follows we will discuss the solution of the eq. (\\ref{w11}) for two particular examples of Lie-Poisson structures.\n\n\\subsubsection{$su(2)$-like Poisson structure}\n \nLet us consider now the particular case of $su(2)$-like Lie-Poisson structure (\\ref{su2}). The perturbative calculations indicate the ansatz, \n \\begin{equation}\\label{w12}\n[ \\rho_\\varepsilon]_a^i=\\sigma\\left(\\alpha^2p^2\\right)\\delta_a^i+\\alpha^2\\,\\tau\\left(\\alpha^2p^2\\right)p^ip_a+\\alpha\\,\\zeta\\left(\\alpha^2p^2\\right)\\varepsilon_a{}^{ik}p_k\\,,\n\\end{equation}\nwith the initial condition, $\\sigma(0)=1$, to match (\\ref{w5a}). Using this ansatz in the equation (\\ref{w11}) the latter becomes,\n\\begin{eqnarray}\\label{w13}\n&&\\alpha\\left[2\\,\\sigma^\\prime+\\sigma\\,\\chi+\\zeta\\right]\\delta^i_ap^j+\\alpha\\left[\\tau+\\zeta-\\sigma\\,\\chi\\right] \\delta^{ij}p_a+\\\\\n&&\\alpha\\left[\\left(1+\\alpha^2\\,p^2\\,\\chi\\right)\\tau+2\\,\\sigma\\left(\\chi+\\alpha^2\\,p^2\\,\\chi^\\prime\\right)\\right]\\delta^j_ap^i+\\alpha\\left[\\left(1+\\alpha^2\\,p^2\\,\\chi\\right)\\zeta+\\sigma\\right]\\varepsilon_a{}^{ij}+\\notag\\\\\n&&\\alpha^3\\left[2\\,\\zeta^\\prime-2\\,\\zeta\\,\\chi\\right]p^j\\varepsilon_a{}^{ik}p_k+\\alpha^3\\left[2\\,\\zeta \\left(\\chi+\\alpha^2\\,p^2\\,\\chi^\\prime\\right)-\\tau \\right]p^i\\varepsilon_a^{jk}p_k+\\notag\\\\\n&&\\alpha^4\\left[2\\,\\tau^\\prime-2\\,\\sigma\\,\\chi^\\prime-\\tau\\,\\chi\\right]p^i\\,p^j\\,p_a=0\\,.\\notag\n\\end{eqnarray}\nThus the equation (\\ref{w11}) results in the system of seven equations on coefficient functions $\\sigma(\\alpha^2p^2)$, $\\tau(\\alpha^2p^2)$ and $\\zeta(\\alpha^2p^2)$:\n\\begin{eqnarray}\n&&2\\,\\sigma^\\prime+\\sigma\\,\\chi+\\zeta=0\\,,\\label{w16}\\\\\n&&\\tau+\\zeta-\\sigma\\,\\chi=0\\,,\\label{w17}\\\\\n&&\\left(1+\\alpha^2\\,p^2\\,\\chi\\right)\\tau+2\\,\\sigma\\left(\\chi+\\alpha^2\\,p^2\\,\\chi^\\prime\\right)=0\\,,\\label{w18}\\\\\n&&\\left(1+\\alpha^2\\,p^2\\,\\chi\\right)\\zeta+\\sigma=0\\,,\\label{w19}\\\\\n&&2\\,\\zeta^\\prime-2\\,\\zeta\\,\\chi=0\\,,\\label{w20}\\\\\n&&2\\,\\zeta \\left(\\chi+\\alpha^2\\,p^2\\,\\chi^\\prime\\right)-\\tau=0\\,\\label{w20a} \\\\\n&&2\\,\\tau^\\prime-2\\,\\sigma\\,\\chi^\\prime-\\tau=0\\,.\\label{w20b}\n\\end{eqnarray}\nHowever they are not all independent. The eq. (\\ref{w18}) is just a consequence of (\\ref{w19}) and (\\ref{w20a}). The eq. (\\ref{w17}) is satisfied due (\\ref{w19}), (\\ref{w20a}) and the equation on the function $\\chi$ (\\ref{w21}). The eq. holds true as a consequence of (\\ref{w19}), (\\ref{w20}) and (\\ref{w21}). The eq. (\\ref{w20b}) is satisfied due (\\ref{w19}), (\\ref{w20}), (\\ref{w20a}) and (\\ref{w21}).\n\nIn fact we end up with only three independent equations on $\\sigma(A)$, $\\tau(A)$ and $\\zeta(A)$: (\\ref{w19}), (\\ref{w20}) and (\\ref{w20a}). Let us first discuss the initial conditions. Taking into account the initial condition, $\\sigma(0)=1$, one finds from (\\ref{w19}) that, $\\zeta(0)=-1$. Taking into account that $\\chi(0)=-1\/3$, the equation (\\ref{w17}) then gives the initial condition for the function $\\tau$ namely, $\\tau(0)=2\/3$. The solution of the equation (\\ref{w20}) with this initial condition reads,\n\\begin{equation}\\label{w22}\n\\zeta(t)=-\\frac{\\sin^2\\sqrt{t}}{t}\\,.\n\\end{equation}\nThe equation (\\ref{w19}) gives,\n\\begin{equation}\\label{w23}\n\\sigma(t)=\\frac{\\sin2\\sqrt{t}}{2\\sqrt{t}}\\,.\n\\end{equation}\nAnd from the equation (\\ref{w20a}) one finds,\n\\begin{equation}\\label{w24}\n\\tau(t)=-t^{-1}\\left(\\frac{\\sin2\\sqrt{t}}{2\\sqrt{t}}-1\\right)\\,.\n\\end{equation}\nWe conclude that the ansatz (\\ref{w12}) with the functions $\\sigma$, $\\tau$ and $\\zeta$ defined in (\\ref{w23}), (\\ref{w24}) and (\\ref{w22}) correspondingly solve the equation (\\ref{w11}).\n\nAn explicit form of the matrix, $\\rho_a^i(A)=\\rho^i_a(x,p)_{p=A}$, reads,\n \\begin{equation}\\label{rhosu2}\n [ \\rho_\\varepsilon]_a^i(A)=\\delta_a^i+\\alpha\\,\\varepsilon_a{}^{ik}A_k\\,\\zeta\\left(\\alpha^2\\,A^2\\right)- \\alpha^2\\left(\\delta^i_a\\,A^2- A^ iA_a\\right)\\tau(\\alpha^2\\,A^2)\\,.\n\\end{equation}\nAnd thus we have obtained an explicit all orders expression for the gauge covariant derivative defined in (\\ref{w6}). It satisfies the gauge covariance condition (\\ref{w5}) and reproduces in the commutative limit the standard partial derivative (\\ref{w5a}). \n\n\\subsubsection{$\\kappa$-Minkowski}\n\nFor the $\\kappa$-Minkowski Poisson structure (\\ref{kappa}) the matrix $[\\gamma_\\kappa]^j_l(p)$ is given by (\\ref{gammakappa}). Following the same strategy as in the previous particular example we write the anzats which follows from the perturbative calculations,\n \\begin{equation}\\label{rhokappa}\n[ \\rho_\\kappa]_a^i(p)=\\delta_a^i\\,\\xi(z)+a^i\\,p_a\\,\\eta(z)\\,,\n\\end{equation}\nwhere, $z=a\\cdot p$, and $\\xi(0)=1$ to guarantee the correct commutative limit. Substituting this anzats in (\\ref{w11}) one finds,\n\\begin{eqnarray}\n\\left((\\phi -z)\\,\\xi^\\prime-\\xi \\right)\\delta_a^i\\,a^j+\\left(\\phi\\,\\eta+\\xi\\,\\phi^\\prime\\right)\\delta_a^j\\,a^i+\\left((\\phi-z)\\,\\eta^\\prime-2\\eta+\\phi^\\prime\\,\\eta\\right)a^i\\,a^j\\,p_a=0\\,,\n\\end{eqnarray}\nwhere, $\\phi(z)=\\sqrt{1+z^2}+z$, according to (\\ref{gammakappa}). Consequently one arrives at the system of three equations on the coefficient functions $\\xi$ and $\\eta$:\n\\begin{eqnarray}\\label{eqxe}\n&&(\\phi -z)\\,\\xi^\\prime=\\xi\\,,\\\\\n&&\\phi\\,\\eta+\\xi\\,\\phi^\\prime=0\\,,\\notag\\\\\n&&(\\phi-z)\\,\\eta^\\prime-2\\eta+\\phi^\\prime\\,\\eta=0\\,.\\notag\n\\end{eqnarray}\nThe solution of the first one with the initial condition, $\\xi(0)=1$, reads,\n\\begin{equation}\n\\xi(z)=\\sqrt{1+z^2}+z\\,.\n\\end{equation}\nThen from the second equation one finds,\n\\begin{equation}\n\\eta(z)=-\\frac{\\sqrt{1+z^2}+z}{\\sqrt{1+z^2}}\\,.\n\\end{equation}\nThe third of the equations (\\ref{eqxe}) just becomes an identity. \n\n\n\\subsubsection{Change of coordinates}\n\nUnder the change of coordinates, $\\upsilon:\\,x\\to \\tilde x$, described in Sec. 2.4 the solution of the equation (\\ref{rho}) transforms like,\n\\begin{equation}\\label{rhotilde}\n\\upsilon:\\, \\rho_a^i( x,p)\\to\\tilde \\rho_a^i(\\tilde x,p)= \\rho_a^i( x,p)|_{x=x(\\tilde x)}\\,.\n\\end{equation}\nFor Lie-Poisson structures it does not change, since $\\rho_a^i( p)$ does not depend on coordinates $x^i$.\n\n\\section{ Field strength}\n\nLet us first discuss the commutator relation of the covariant derivatives ${\\cal D}_a(\\psi)$ introduced in the previous section.\n{\\proposition \\label{cr} The commutator relation for the covariant derivatives defined in the proposition \\ref{pcd} reads,\n\\begin{eqnarray}\\label{cr1}\n\\left[{\\cal D}_a,{\\cal D}_b\\right]=\\{{\\cal F}_{ab},\\,\\cdot\\,\\}+\\left({\\cal F}_{ad}\\,\\Lambda_b{}^{de}-{\\cal F}_{bd}\\,\\Lambda_a{}^{de}+{\\cal K}_{ab}{}^e-{\\cal K}_{ba}{}^e\\right){\\cal D}_e\\,\n\\end{eqnarray}\nwhere,\n\\begin{eqnarray}\\label{F}\n{\\cal F}_{ab}:&=&\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0}\\\\\n&=&\\rho_a^i(A)\\,\\rho_b^j(A)\\left(\\gamma_i^l(A)\\,\\partial_lA_j-\\gamma_j^l(A)\\,\\partial_lA_i+\\{A_i,A_j\\}\\right)\\,,\\notag\\\\\n\\Lambda_b{}^{de}(A)&=&\\left(\\rho^{-1}\\right)_j^d\\left(\\partial^j_A\\rho_b^m(A)-\\partial^m_A\\rho_b^j(A)\\right)\\left(\\rho^{-1}\\right)_m^e\\,,\\label{Lambda}\\\\\n{\\cal K}_{ab}{}^e(A)&=&\\rho_a^i(A)\\,\\gamma^m_i(A)\\,\\left(\\partial_m\\rho_b^j(x,p)\\right)_{\\Phi=0}\\left(\\rho^{-1}\\right)_j^e\\,.\\label{Kappa}\n\\end{eqnarray}\n\\proof The proof is straightforward, we start writing, \n\\begin{eqnarray}\\label{f2}\n\\left[{\\cal D}_a,{\\cal D}_b\\right](\\psi)&=&\\left(\\rho_a^i(A)\\,\\{\\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}-\\rho_b^i(A)\\,\\{\\rho_a^j(A),\\Phi_i\\}_{\\Phi=0}\\right)\\{\\psi,\\Phi_j\\}_{\\Phi=0}+\\\\\n&&\\rho_a^i(A)\\,\\rho_b^j(A)\\left(\\{\\{\\psi,\\Phi_j\\}_{\\Phi=0},\\Phi_i\\}_{\\Phi=0}-\\{\\{\\psi,\\Phi_i\\}_{\\Phi=0},\\Phi_j\\}_{\\Phi=0}\\right)\\,.\\notag\n\\end{eqnarray}\nWe use the formula (\\ref{r5}) from the appendix in the second line of the right hand side of the above expression to rewrite it as,\n\\begin{eqnarray}\\label{f3}\n&&\\left(\\rho_a^i(A)\\,\\{\\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}-\\rho_b^i(A)\\,\\{\\rho_a^j(A),\\Phi_i\\}_{\\Phi=0}\\right)\\{\\psi,\\Phi_j\\}_{\\Phi=0}+\\\\\n&&\\rho_a^i(A)\\,\\rho_b^j(A)\\left(\\{\\{\\psi,\\Phi_j\\},\\Phi_i\\}_{\\Phi=0}-\\{\\{\\psi,\\Phi_i\\},\\Phi_j\\}_{\\Phi=0}\\right)\\notag\\\\\n&&-\\rho_a^i(A)\\,\\rho_b^j(A)\\left[\\left(\\partial_p^m\\{\\psi,\\Phi_j\\}\\right)_{\\Phi=0}\\,\\{\\Phi_m,\\Phi_i\\}_{\\Phi=0}-\\left(\\partial_p^m\\{\\psi,\\Phi_i\\}\\right)_{\\Phi=0}\\,\\{\\Phi_m,\\Phi_j\\}_{\\Phi=0}\\right]\\,.\\notag\n\\end{eqnarray}\nEmploying the Jacobi identity in the second line and the equation (\\ref{rho}) in the third line of (\\ref{f3}) one represents it as,\n\\begin{eqnarray}\\label{f4}\n&&\\left(\\rho_a^i(A)\\,\\{\\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}-\\rho_b^i(A)\\,\\{\\rho_a^j(A),\\Phi_i\\}_{\\Phi=0}\\right)\\{\\psi,\\Phi_j\\}_{\\Phi=0}+\\\\\n&&\\rho_a^i(A)\\,\\rho_b^j(A)\\{\\{\\Phi_i,\\Phi_j\\},\\psi\\}_{\\Phi=0}+\\notag\\\\\n&&\\left(\\rho_a^i(A)\\{\\rho_b^j(p),\\psi\\}_{\\Phi=0}- \\rho_b^i(A)\\{\\rho_a^j(p),\\psi\\}_{\\Phi=0}\\right)\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0}\\notag\n\\end{eqnarray}\nApplying one more time the eq. (\\ref{r5}) in the second line of (\\ref{f4}) we reorganize it as, \n\\begin{eqnarray}\\label{f5}\n&&\\left(\\rho_a^i(A)\\,\\{\\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}-\\rho_b^i(A)\\,\\{\\rho_a^j(A),\\Phi_i\\}_{\\Phi=0}\\right)\\{\\psi,\\Phi_j\\}_{\\Phi=0}+\\\\\n&&\\{\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0},\\psi\\}_{\\Phi=0}-\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\left(\\partial_p^m\\{\\Phi_i,\\Phi_j\\}\\right)_{\\Phi=0}\\,\\{\\Phi_m,\\psi\\}_{\\Phi=0}+\\notag\\\\\n&&\\left(\\rho_a^i(A)\\{\\rho_b^j(p)-\\rho_b^j(A),\\psi\\}_{\\Phi=0}- \\rho_b^i(A)\\{\\rho_a^j(p)-\\rho_a^j(A),\\psi\\}_{\\Phi=0}\\right)\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\nWe use the equation (\\ref{rho}) in the second term of the second line and also (\\ref{w7a}) in the last line rewriting (\\ref{f5}) as,\n\\begin{eqnarray}\\label{f6}\n&&\\rho_a^i(A)\\left(\\{\\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}-\\{A_i,\\rho_b^j(p)\\}_{\\Phi=0}\\right)\\{\\psi,\\Phi_j\\}_{\\Phi=0}\\\\\n&&-\\rho_b^i(A)\\left(\\{\\rho_a^j(A),\\Phi_i\\}_{\\Phi=0}-\\{A_i,\\rho_a^j(p)\\}_{\\Phi=0}\\right)\\{\\psi,\\Phi_j\\}_{\\Phi=0}+\\notag\\\\\n&&\\{\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0},\\psi\\}_{\\Phi=0}+\\notag\\\\\n&&\\left(\\rho_a^i(A)\\,\\partial^m_A\\rho_b^j(A)\\{\\Phi_m,\\psi\\}_{\\Phi=0}- \\rho_b^i(A)\\,\\partial^m_A\\rho_a^j(A)\\{\\Phi_m,\\psi\\}_{\\Phi=0}\\right)\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\n\nLet us calculate separately,\n\\begin{eqnarray}\\label{f7}\n&&\\{\\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}-\\{A_i,\\rho_b^j(p)\\}_{\\Phi=0}=\\\\\n&&\\partial^m_A\\rho_b^j(A)\\left(\\{A_m,p_i\\}_{\\Phi=0}-\\{A_i,p_m\\}_{\\Phi=0}-\\{A_m,A_i\\}\\right)+\\left(\\partial_m\\rho_b^j(x,p)\\right)_{\\Phi=0}\\,\\gamma^m_i(A)=\\notag\\\\\n&&-\\partial^m_A\\rho_b^j(A)\\,\\{\\Phi_m,\\Phi_i\\}_{\\Phi=0}+\\left(\\partial_m\\rho_b^j(x,p)\\right)_{\\Phi=0}\\,\\gamma^m_i(A)\\,.\\notag\n\\end{eqnarray}\nAfter the use the relation (\\ref{f7}) in the expression (\\ref{f6}) the latter becomes,\n\\begin{eqnarray}\\label{f8}\n&&\\{\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0},\\psi\\}_{\\Phi=0}\\\\\n&&+\\rho_a^i(A)\\left(\\partial^j_A\\rho_b^m(A)-\\partial^m_A\\rho_b^j(A)\\right)\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0}\\,\\{\\psi,\\Phi_m\\}_{\\Phi=0}\\notag\\\\\n&&-\\rho_b^i(A)\\left(\\partial^j_A\\rho_a^m(A)-\\partial^m_A\\rho_a^j(A)\\right)\\{\\Phi_i,\\Phi_j\\}_{\\Phi=0}\\,\\{\\psi,\\Phi_m\\}_{\\Phi=0}\\notag\\\\\n&&+\\rho_a^i(A)\\,\\gamma^m_i(A)\\,\\partial_m\\rho_b^j(A)\\,\\{\\psi,\\Phi_j\\}_{\\Phi=0}-\\rho_b^i(A)\\,\\gamma^m_i(A)\\,\\partial_m\\rho_a^j(A)\\,\\{\\psi,\\Phi_j\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\nFinally using the definitions (\\ref{F}), (\\ref{Lambda}) and (\\ref{Kappa}) as well as the formula (\\ref{w6}) we represent (\\ref{f8}) as the right hand side of (\\ref{cr1}) and thus prove the proposition \\ref{cr}. $\\Box$\n\nSome comments are in order. First of all we remind that in case of Lie-Poisson structures which are the main example of the present work, the matrix $\\rho_a^i(p)$ does not depend on coordinates. Therefore the tensor ${\\cal K}_{ab}{}^{e}$ defined in (\\ref{Kappa}) vanishes and the commutator relation becomes just,\n\\begin{eqnarray}\\label{cr2}\n\\left[{\\cal D}_a,{\\cal D}_b\\right]=\\{{\\cal F}_{ab},\\,\\cdot\\,\\}+\\left({\\cal F}_{ad}\\,\\Lambda_b{}^{de}-{\\cal F}_{bd}\\,\\Lambda_a{}^{de}\\right){\\cal D}_e\\,\n\\end{eqnarray}\nFor the canonical Poisson bracket with constant non-commutativity parameter $\\Theta^{ij}$ the matrix, $\\rho_a^i(A)=\\delta_a^i$, is also constant according to (\\ref{w11}). So, the tensor $\\Lambda_b{}^{de}$ determined in (\\ref{Lambda}) vanishes and the relation (\\ref{cr2}) becomes the 'usual' one, $\\left[{\\cal D}_a,{\\cal D}_b\\right]=\\{{\\cal F}_{ab},\\,\\cdot\\,\\}$.\n\nSecond, in the commutative limit the Poisson bracket vanishes and $$\\lim_{\\alpha\\to0}\\rho_a^i(A)=\\lim_{\\alpha\\to0}\\gamma_a^i(A)=\\delta_a^i\\,,$$ so the quantity introduced in (\\ref{F}) reproduces the $U(1)$ field strength when $\\alpha\\to0$,\n\\begin{equation}\\label{clF}\n\\lim_{\\alpha\\to0}\\,{\\cal F}_{ab}=\\partial_a\\,A_b-\\partial_b\\,A_a\\,.\n\\end{equation}}\nMoreover, this object enjoys another important property formulated by the following,\n{\\proposition The quantity (\\ref{F}) introduced in the proposition \\ref{cr} transforms covariantly under the Poisson gauge transformation (\\ref{APhi}), \n\\begin{equation}\\label{gcF}\n\\delta_f {\\cal F}_{ab}=\\{{\\cal F}_{ab},f\\}\\,.\n\\end{equation}\n\\proof The proof of this proposition is analogous to the proposition \\ref{pcd}. $\\Box$\n\nBecause of the properties (\\ref{clF}) and (\\ref{gcF}) we will call the quantity ${\\cal F}_{ab}$ defined in (\\ref{F}) as the {\\it Poisson field strength}.\n\nIn the previous works \\cite{KV20,KKV} we have derived the quasi-classical limit of the non-commutative $U(1)$ field strength as the quantity, \n\\begin{equation}\n{\\cal F}_{ab}=P_{ab}{}^{cd}\\left(A\\right)\\,\\partial_c A_d+R_{ab}{}^{cd}\\left(A\\right)\\,\\left\\{A_c,A_d\\right\\}\\,,\\label{F1}\n\\end{equation}\nwhich satisfies the same properties (\\ref{clF}) and (\\ref{gcF}). In (\\ref{F1}), $P_{ab}{}^{cd}=2\\,\\gamma^c_lR_{ab}{}^{ld}$, and the coefficient function $R_{ab}{}^{cd}$ should satisfy the equation,\n\\begin{equation}\n\\gamma^k_l\\,\\partial^l_A R_{ab}{}^{cd}+\\alpha\\,\\Theta^{kl}\\,\\partial_lR_{ab}{}^{cd}+R_{ab}{}^{cl}\\,\\partial^d_A\\gamma^k_l+R_{ab}{}^{ld}\\,\\partial^c_A\\gamma^k_l=0\\,.\\label{eqR}\n\\end{equation}\nComparing (\\ref{F}) and (\\ref{F1}) we see that,\n\\begin{equation}\\label{Rrho}\nR_{ab}{}^{cd}(A)=\\frac12\\left(\\rho^c_a(A)\\rho^d_b(A)-\\rho^c_b(A)\\rho^d_a(A)\\right)\\,.\n\\end{equation}\nSubstituting (\\ref{Rrho}) in the l.h.s. of (\\ref{eqR}) one finds,\n\\begin{eqnarray}\\label{f9}\n&&\\frac12\\left[\\gamma^k_l\\,\\partial^l_A\\, \\rho_a^c+\\rho_a^l\\,\\partial^c_A\\gamma^k_l+\\alpha\\,\\Theta^{kl}\\,\\partial_l\\rho_a^c\\right]\\rho^d_b+\\\\\n&&\\frac12\\,\\rho^c_a\\left[\\gamma^k_l\\,\\partial^l_A\\, \\rho_b^d+\\rho_b^l\\,\\partial^d_A\\gamma^k_l+\\alpha\\,\\Theta^{kl}\\,\\partial_l\\rho_b^d\\right]-(a\\leftrightarrow b)\\,.\\notag\n\\end{eqnarray}\nObserving that in the square brackets we have the l.h.s. of the eq. (\\ref{w10}) written in $A$ variables we conclude that if $\\rho_a^c$ obeys the equation (\\ref{rho}), the tensor $R_{ab}{}^{cd}(A)$ defined by the relation (\\ref{Rrho}) satisfies the equation (\\ref{eqR}). \n\nIt is easy to see that substituting the expression (\\ref{rhokappa}) for the matrix $[\\rho_\\kappa]_a^c$ corresponding to the $\\kappa$-Minkowski Lie-Poisson structure (\\ref{kappa}) in (\\ref{Rrho}) one obtains exactly the formula for $R_{ab}^\\kappa{}^{cd}$ from \\cite{KKV}. The calculation in case of the $su(2)$-like Poisson bi-vector (\\ref{su2}) is a bit less straightforward. We substitute $[\\rho_\\varepsilon]_a^c$ from (\\ref{rhosu2}) in (\\ref{Rrho}) and then use the formulas from the appendix of \\cite{Kup27} to simplify it. Thus one recovers the expression for $R_{ab}^\\varepsilon{}^{cd}$ from \\cite{KV20}. And finally we note that if the Poisson structures $\\Theta^{ij}(x)$ and $\\tilde\\Theta^{ij}(\\tilde x)$ are related by the invertible coordinate transformation, $\\upsilon:\\,x\\to\\tilde x$, described in the Sec. 2.3, then the corresponding Poisson field strengths are also related by the same transformation, $\\tilde{\\cal F}_{ab}= {\\cal F}_{ab}|_{x=x(\\tilde x)}$, as can be seen from (\\ref{g1}), (\\ref{g2}), (\\ref{rhotilde}) and (\\ref{F}).\n\nFrom the practical reasons the eq. (\\ref{rho}) on $\\rho_a^c$ is easier to solve than the eq. (\\ref{eqR}) on $R_{ab}{}^{cd}$ because it has less components. However there is more fundamental significance of the covariant derivative (\\ref{w6}), since it is essential for the definition of the Bianchi identity, the action principle and the equations of motion. It is therefore one may think of the matrices $\\rho_a^c(A)$ and $\\gamma_k^l(A)$ as a building blocs of the Poisson gauge theory. \n\n\n\\subsection{Bianchi identity}\n\n{\\proposition \\label{bi} The field strength defined in (\\ref{F}) satisfies the deformed Bianchi identity,\n\\begin{equation}\\label{bi1}\n{\\cal D}_a\\left({\\cal F}_{bc}\\right)-{\\cal F}_{ad}\\,\\Lambda_b{}^{de}\\,{\\cal F}_{ec}-\\Gamma_{ab}{}^e\\,{\\cal F}_{ec}+\\Gamma_{cb}{}^e\\,{\\cal F}_{ea}+\\mbox{cycl.}(abc)=0\\,.\n\\end{equation}}\n\\proof Again the proof is straightforward. Using the definitions (\\ref{w6}) and (\\ref{F}) one calculates,\n\\begin{eqnarray}\\label{b2}\n&&{\\cal D}_a\\left({\\cal F}_{bc}\\right)+\\mbox{cycl.}(abc)=\\\\\n&&\\rho_a^i(A)\\left[\\{ \\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}\\,\\rho_c^k(A)+\\rho_b^j(A)\\,\\{\\rho_c^k(A),\\Phi_i\\}_{\\Phi=0}\\right]\\{\\Phi_j,\\Phi_k\\}_{\\Phi=0}+\\notag\\\\\n&&\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\rho_c^(A)\\left\\{\\{\\Phi_j,\\Phi_k\\}_{\\Phi=0},\\Phi_i\\right\\}_{\\Phi=0}+\\mbox{cycl.}(abc)\\,.\\notag\n\\end{eqnarray}\nIn the last term on the right we use the formula (\\ref{r6}) from the appendix and rewrite it as,\n\\begin{eqnarray}\\label{b3}\n&&\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\rho_c^k(A)\\left\\{\\{\\Phi_j,\\Phi_k\\}_{\\Phi=0},\\Phi_i\\right\\}_{\\Phi=0}+\\mbox{cycl.}(abc)=\\\\\n&&\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\rho_c^k(A)\\left\\{\\{\\Phi_j,\\Phi_k\\},\\Phi_i\\right\\}_{\\Phi=0}-\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\rho_c^k(A)\\left(\\partial^m_p\\{\\Phi_j,\\Phi_k\\}\\right)_{\\Phi=0}\\left\\{\\Phi_m,\\Phi_i\\right\\}_{\\Phi=0}\\notag\\\\\n&&+\\mbox{cycl.}(abc)\\,.\\notag\n\\end{eqnarray}\nThe Jacobi identity implies that the first term on the right in (\\ref{b3}) vanishes,\n\\begin{eqnarray}\\label{b4}\n&&\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\rho_c^k(A)\\left\\{\\{\\Phi_j,\\Phi_k\\},\\Phi_i\\right\\}_{\\Phi=0}+\\mbox{cycl.}(abc)=\\\\\n&&\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\rho_c^k(A)\\left(\\left\\{\\{\\Phi_j,\\Phi_k\\},\\Phi_i\\right\\}_{\\Phi=0}+\\left\\{\\{\\Phi_i,\\Phi_j\\},\\Phi_k\\right\\}_{\\Phi=0}+\\left\\{\\{\\Phi_k,\\Phi_i\\},\\Phi_j\\right\\}_{\\Phi=0}\\right)=0\\,.\\notag\n\\end{eqnarray}\nRenaming the contracted indices we represent the second term on the right of (\\ref{b3}) as,\n\\begin{eqnarray}\\label{b5}\n&&-\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\rho_c^k(A)\\left(\\partial^m_p\\{\\Phi_j,\\Phi_k\\}\\right)_{\\Phi=0}\\left\\{\\Phi_m,\\Phi_i\\right\\}_{\\Phi=0}+\\mbox{cycl.}(abc)=\\\\\n&&-\\rho_a^k(A)\\,\\rho_b^m(A)\\,\\rho_c^i(A)\\left(\\partial^j_p\\{\\Phi_m,\\Phi_i\\}\\right)_{\\Phi=0}\\left\\{\\Phi_j,\\Phi_k\\right\\}_{\\Phi=0}+\\mbox{cycl.}(abc)\\,.\\notag\n\\end{eqnarray}\nNote that using the equation (\\ref{rho}) one finds,\n\\begin{eqnarray}\\label{b6}\n&&-\\rho_a^k(A)\\,\\rho_b^m(A)\\,\\rho_c^i(A)\\left(\\partial^j_p\\{\\Phi_m,\\Phi_i\\}\\right)_{\\Phi=0}=\\\\\n&&-\\rho_a^k(A)\\,\\rho_b^m(A)\\,\\rho_c^i(A)\\left(\\partial^j_p\\{A_i,p_m\\}\\right)_{\\Phi=0}+\\rho_a^k(A)\\,\\rho_b^m(A)\\,\\rho_c^i(A)\\left(\\partial^j_p\\{A_m,p_i\\}\\right)_{\\Phi=0}=\\notag\\\\\n&&\\rho_a^k(A)\\,\\{A_i,\\rho_b^j(p)\\}_{\\Phi=0}\\,\\rho_c^i(A)-\\rho_a^k(A)\\,\\rho_b^i(A)\\,\\{A_i,\\rho_c^j(p)\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\nTaking into account (\\ref{b6}) and the cyclic permutations we rewrite the r.h.s. of (\\ref{b5}) as,\n\\begin{eqnarray}\\label{b7}\n&&\\left(-\\rho_a^i(A)\\,\\{A_i,\\rho_b^j(p)\\}_{\\Phi=0}\\,\\rho_c^k(A)-\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\{A_i,\\rho_c^k(p)\\}_{\\Phi=0}\\right)\\{\\Phi_j,\\Phi_k\\}_{\\Phi=0}+\\\\\n&&\\mbox{cycl.}(abc)\\,.\\notag\n\\end{eqnarray}\n\nRemind that the expression (\\ref{b7}) represents the modification of the last term in the r.h.s. of (\\ref{b2}). Substituting (\\ref{b7}) back into (\\ref{b2}) we obtain for the r.h.s.,\n\\begin{eqnarray}\\label{b8}\n&&\\rho_a^i(A)\\left(\\{\\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}-\\{A_i,\\rho_b^j(p)\\}_{\\Phi=0}\\right)\\rho^k_c(A)\\,\\left\\{\\Phi_j,\\Phi_k\\right\\}_{\\Phi=0}+\\\\\n&&\\rho_a^i(A)\\,\\rho^j_b(A)\\left(\\{\\rho_c^k(A),\\Phi_i\\}_{\\Phi=0}-\\{A_i,\\rho_c^k(p)\\}_{\\Phi=0}\\right)\\rho^k_c(A)\\,\\left\\{\\Phi_j,\\Phi_k\\right\\}_{\\Phi=0}+\\mbox{cycl.}(abc)\\,.\\notag\n\\end{eqnarray}\nLet us calculate separately,\n\\begin{eqnarray}\\label{b9}\n&&\\{\\rho_b^j(A),\\Phi_i\\}_{\\Phi=0}-\\{A_i,\\rho_b^j(p)\\}_{\\Phi=0}=\\\\\n&&\\partial^m_A\\rho_b^j(A)\\left(\\{A_m,p_i\\}_{\\Phi=0}-\\{A_i,p_m\\}_{\\Phi=0}-\\{A_m,A_i\\}\\right)+\\partial_m\\rho_b^j(A)\\,\\gamma^m_i(A)=\\notag\\\\\n&&-\\partial^m_A\\rho_b^j(A)\\,\\{\\Phi_m,\\Phi_i\\}_{\\Phi=0}+\\left(\\partial_m\\rho_b^j(x,p)\\right)_{\\Phi=0}\\,\\gamma^m_i(A)\\,.\\notag\n\\end{eqnarray}\nUsing (\\ref{b9}) in (\\ref{b8}) the latter becomes\n\\begin{eqnarray}\\label{b10}\n&&\\left(\\rho_a^i(A)\\,\\partial^m_A\\rho_b^j(A)\\,\\rho^k_c(A)+\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\partial^m_A\\rho^k_c(A)\\right)\\{\\Phi_i,\\Phi_m\\}_{\\Phi=0}\\,\\left\\{\\Phi_j,\\Phi_k\\right\\}_{\\Phi=0}+\\\\\n&&\\left(\\rho_a^i(A)\\,\\gamma^m_i(A)\\left(\\partial_m\\rho_b^j(x,p)\\right)_{\\Phi=0}\\,\\rho^k_c(A)+\\rho_a^i(A)\\,\\rho_b^j(A)\\,\\gamma^m_i(A)\\left(\\partial_m\\rho^k_c(x,p)\\right)_{\\Phi=0}\\right)\\left\\{\\Phi_j,\\Phi_k\\right\\}_{\\Phi=0}+\\notag\\\\\n&&\\mbox{cycl.}(abc)\\,.\\notag\n\\end{eqnarray}\nRenaming the contracted indices and using the cyclic permutations we may rewrite the first line of (\\ref{b10}) as,\n\\begin{eqnarray}\\label{b11}\n\\rho_a^i(A)\\left(\\partial^m_A\\rho_b^j(A)-\\partial^j_A\\rho_b^m(A)\\right)\\rho^k_c(A)\\,\\{\\Phi_i,\\Phi_m\\}_{\\Phi=0}\\,\\left\\{\\Phi_j,\\Phi_k\\right\\}_{\\Phi=0}+\\mbox{cycl.}(abc)\\,.\n\\end{eqnarray}\nFinally using (\\ref{b11}) as well as the definitions (\\ref{F}), (\\ref{Lambda}) and (\\ref{Kappa}) in (\\ref{b10}) we rewrite the latter as,\n\\begin{eqnarray}\\label{b12}\n{\\cal F}_{ad}\\,\\Lambda_b{}^{de}(A)\\,{\\cal F}_{ec}+\\Gamma_{ab}{}^e(A)\\,{\\cal F}_{ec}-\\Gamma_{cb}{}^e(A)\\,{\\cal F}_{ea}+\\mbox{cycl.}(abc)\\,.\n\\end{eqnarray}\nThe above expression is the right hand side of (\\ref{b2}), which proves (\\ref{bi1}). $\\Box$\n\n\n\\section{Action principle and field equations}\n\nHaving in hand the covariant derivative (\\ref{w6}) and the Poisson field strength (\\ref{F}) now we are in the position to construct the gauge covariant Lagrangian, $\\delta_f{\\cal L}=\\{{\\cal L},f\\}$, and gauge invariant action, $\\delta_fS=0$. To do so, first one ought to introduce an appropriate integration mesure $\\mu(x)$, which for any two Schwartz functions $f$ and $g$ should satisfy the condition,\n\\begin{equation}\\label{measure}\n\\int\\!d^Nx\\,\\mu(x)\\,\\{f,g\\}=0\\,,\\qquad  \\Leftrightarrow \\qquad \\partial_l\\left(\\mu(x)\\,\\Theta^{lk}(x)\\right)=0\\,.\n\\end{equation}\nFor the $su(2)$-like Poisson structure (\\ref{su2}) the integration measure can be taken to be a constant $\\mu(x)=1$. For $\\kappa$-Minkowski Lie-Poisson bi-vector (\\ref{kappa}) the measure is not constant. An explicit expression for $\\mu(x)$ in this case was found in \\cite{KKV}.\n\nLet us discuss first the pure gauge sector without matter fields $\\psi$. The gauge invariant action in this case reads \\cite{KV20},\n\\begin{equation}\\label{ag}\nS_g=\\int\\!d^Nx\\,\\mu(x)\\,{\\cal L}_{g}\\,,\\qquad {\\cal L}_{g}=-\\frac14\\,{\\cal F}_{ab}\\,{\\cal F}^{ab}\\,.\n\\end{equation}\nTo derive the corresponding Euler-Lagrange equations we will need the following relation.\n{\\proposition \\label{peom}\nProvided that the matrices $\\gamma^i_l(A)$ and $\\rho_a^l(A)$ are defined by the propositions \\ref{pgt} and \\ref{pcd} correspondingly, and the measure $\\mu(x)$ satisfies (\\ref{measure}), then the following relation holds true,\n\\begin{eqnarray}\\label{tr1} \n\\partial_i\\left(\\mu(x)\\,\\rho_a^l(A)\\,\\gamma^i_l(A)\\right)+\\mu(x)\\left\\{A_l,\\rho_a^l(A)\\right\\}&=&-\\mu(x)\\,\\Lambda_a{}^{cd}(A)\\,{\\cal F}_{cd}\\\\\n&&+\\left[\\partial_i\\left(\\mu(x)\\,\\rho_a^l(p)\\,\\gamma^i_l(p)\\right)\\right]_{\\Phi=0}\\,,\\notag\n\\end{eqnarray}\nwhere the quantities $\\Lambda_a{}^{cd}$ and ${\\cal F}_{cd}$ were defined in the proposition \\ref{cr}.}\n{\\proof Using the equation (\\ref{rho}) we represent the left hand side as,\n\\begin{eqnarray*}\n\\left[\\partial_i\\left(\\mu(x)\\,\\rho_a^l(p)\\,\\gamma^i_l(p)\\right)\\right]_{\\Phi=0}+\\partial^m_A\\rho^l_a(A)\\,\\{\\Phi_l,\\Phi_m\\}_{\\Phi=0}\\,,\n\\end{eqnarray*}\nand use the definitions of $\\Lambda_a{}^{cd}(A)$ and ${\\cal F}_{cd}$ to complete the proof.} $\\Box$\n\nUsing the relation (\\ref{tr1}) one finds the Euler-Lagrange equations for (\\ref{ag}),\n\\begin{eqnarray}\n&&\\mu\\,\\rho_b^k\\,{\\cal D}_a\\left({\\cal F}^{ab}\\right)-\\mu\\,{\\cal F}^{ab}\\left(\\rho_a^k\\,\\partial^m_A\\rho_b^l+\\rho_a^m\\,\\partial^l_A\\rho_b^k+\\rho_a^l\\,\\partial^k_A\\rho_b^m\\right)\\{\\Phi_l,\\Phi_m\\}_{\\Phi=0}\\\\\n&&+{\\cal F}^{ab}\\left[\\partial_i\\left(\\mu\\,\\rho_a^l(p)\\,\\rho_b^k(p)\\,\\gamma^i_l(p)\\right)\\right]_{\\Phi=0}=0\\,.\\notag\n\\end{eqnarray}\nMultiplying by $\\left(\\rho^{-1}\\right)^c_k$ we end up with Poisson gauge equations,\n\\begin{eqnarray}\\label{ncm}\n\\mu\\,{\\cal D}_a\\left({\\cal F}^{ac}\\right)+\\frac{\\mu}{2}\\,{\\cal F}^{cb}\\,\\Lambda_b{}^{de}\\,{\\cal F}_{de}-\\mu\\,{\\cal F}^{db}\\,\\Lambda_b{}^{ce}\\,{\\cal F}_{de}+\\left(\\rho^{-1}\\right)^c_k{\\cal F}^{ab}\\partial_i\\left(\\mu\\,\\rho_a^l\\,\\rho_b^k\\,\\gamma^i_l\\right)=0\\,.\n\\end{eqnarray}\nThese equations transform covariantly under the Poisson gauge transformation (\\ref{APhi}) and reproduce the first pare of the Maxwell equations in the commutative limit $\\alpha\\to0$, if the measure $\\mu(x)$ is constant.\n\nAs it was already mentioned, for the $su(2)$-like Poisson structure (\\ref{su2}) the integration measure is constant $\\mu(x)=1$. Since, for any linear Poisson structures the matrix $\\rho_a^l(p)$ does not depend on coordinates, the last term on the left in (\\ref{ncm}) disappears. Moreover, one may check that in this case,\n\\begin{equation}\n{\\cal F}^{ab}\\left(\\rho_a^k\\,\\partial^m_A\\rho_b^l+\\rho_a^m\\,\\partial^l_A\\rho_b^k+\\rho_a^l\\,\\partial^k_A\\rho_b^m\\right)\\{\\Phi_l,\\Phi_m\\}_{\\Phi=0}\\equiv0\\,.\n\\end{equation}\nTherefore the equations of motion of the $su(2)$-like Poisson gauge theory become,\n\\begin{eqnarray}\\label{ncmsu2}\n{\\cal D}^{\\varepsilon}_a\\left({\\cal F}^{ac}_\\varepsilon\\right)=0\\,,\n\\end{eqnarray}\nwhere the superscript $\\varepsilon$ in ${\\cal D}^{\\varepsilon}_a$ and the subscript $\\varepsilon$ in ${\\cal F}^{ac}_\\varepsilon$ just indicate that these objects correspond to the $su(2)$-structure.\nFor the $\\kappa$-Minkowski case the measure is not constant, so the equations of motion will contain additional terms. \n\n\nNow let us discuss the contribution from the matter fields. First of all let us observe that due proposition \\ref{peom} one has,\n\\begin{eqnarray}\n\\int\\!d^Nx\\,\\mu\\,\\psi\\,{\\cal D}_a(\\psi)&=&-\\int\\!d^Nx\\,\\mu\\,{\\cal D}_a(\\psi)\\,\\psi+\\int\\!d^Nx\\,\\mu\\,\\psi\\,\\Lambda_a{}^{cd}\\,{\\cal F}_{cd}\\,\\psi\\\\\n&&+\\int \\!d^Nx\\left[\\partial_i\\left(\\mu(x)\\,\\rho_a^l(p)\\,\\gamma^i_l(p)\\right)\\right]_{\\Phi=0}\\psi\\,\\psi\\,.\\notag\n\\end{eqnarray}\nThe latter means that the operator ${\\cal D}_a$ is not self-adjoint. Therefore to construct real gauge invariant action for the Fermionic matter fields $\\psi$ we write,\n\\begin{equation}\\label{am}\nS_{int}=\\int\\!d^Nx\\,\\mu(x)\\,{{\\cal L}_{int}\\,,\\qquad\\cal L}_{int}=\\frac{i}{2}\\left[\\bar\\psi\\,\\Gamma^a{\\cal D}_a\\psi-\\left({\\cal D}_a\\bar\\psi\\right)\\Gamma^a\\,\\psi\\right]-m\\,\\bar\\psi\\,\\psi\\,,\n\\end{equation}\nwhere $\\Gamma^a$ are the gamma matrices. The corresponding Dirac equation reads,\n\\begin{equation}\\label{ncpsi}\n\\left[i\\,\\mu\\,\\Gamma^a{\\cal D}_a-\\frac{i\\mu}{2}\\,\\Gamma^a\\,\\Lambda_a{}^{cd}\\,{\\cal F}_{cd}-\\frac{i}{2}\\,\\Gamma^a\\,\\partial_b\\left(\\mu\\,\\rho_a^l\\,\\gamma^b_l\\right)-\\mu\\,m\\right]\\psi=0\\,.\n\\end{equation}\nTo find the contribution of the matter fields to the Poisson Gauge equations one calculates,\n\\begin{eqnarray}\n\\frac{\\delta S_{int}}{\\delta A_k}=i\\,\\mu\\,\\rho_c^k\\,\\{\\bar\\psi\\,\\Gamma^c,\\psi\\}-\\frac{i\\mu}{2}\\left(\\partial^k_A\\rho_c^l-\\partial^l_A\\rho_c^k\\right)\\left[\\{\\bar\\psi,\\Phi_l\\}_{\\Phi=0}\\,\\Gamma^c\\,\\psi-\\bar\\psi\\,\\Gamma^c\\,\\{\\psi,\\Phi_l\\}_{\\Phi=0}\\right]\\,\n\\end{eqnarray}\nMultiplying it by $\\left(\\rho^{-1}\\right)^a_k$ we obtain the right hand side (the current) of the eq. (\\ref{ncm}),\n\\begin{equation}\\label{nci}\nJ^a=-i\\,\\mu\\,\\{\\bar\\psi\\,\\Gamma^a,\\psi\\}+\\frac{i\\mu}{2}\\,\\Lambda_c{}^{ab}\\left[\\left({\\cal D}_b\\bar\\psi\\right)\\Gamma^c\\,\\psi-\\bar\\psi\\,\\Gamma^c\\,{\\cal D}_b\\psi\\right]\\,.\n\\end{equation}\nIn the commutative limit the interaction disappears, the field become electrically neutral as it was already discussed in the Sec. 4. \n\n\\section{Conclusions and discussion}\n\nWorking in the framework of the symplectic embeddings we introduce the Poisson gauge transformations (\\ref{APhi}) which close the algebra (\\ref{ga}) obtained as the semi-classical limit of the full non-commutative gauge algebra (\\ref{ga1}). The proposition \\ref{pcd} determines the gauge covariant derivative of the matter field ${\\cal D}_a(\\psi)$. The commutator relation for the covariant derivatives (\\ref{cr1}) defines the Poisson field strength (\\ref{F}) which transforms covariantly under the gauge transformations (\\ref{APhi}) and tends to the standard $U(1)$ field strength in the commutative limit. The corresponding Bianchi identity is set by the proposition \\ref{bi}. The gauge invariant action, given by (\\ref{ag}) for the pure gauge fields and by (\\ref{am}) for the interaction part with the matter fields, is constructed in the Sec. 6 from the gauge covariant objects ${\\cal D}_a(\\psi)$ and ${\\cal F}_{ab}$. From this action we derive the field equations (\\ref{ncm}) for the gauge fields without interaction with the matter field, the Dirac equation for the matter field (\\ref{ncpsi}), and the current term (\\ref{nci}) describing the interaction between the gauge and the matter fields in the equations of motion. \n\nThe symplectic embedding of given Poisson structure is not unique, the corresponding arbitrariness is described in the Sec. 2.1. So, for the same Poisson structure $\\Theta^{ab}(x)$ one may obtain different Poisson gauge transformations and other subsequent structures. However the corresponding gauge systems are related by the Seiberg-Witten map described in the Sec. 3.1. If two different Poisson structures $\\Theta^{ij}(x)$ and $\\tilde\\Theta^{ij}(\\tilde x)$ are related by the invertible coordinate transformation, $\\upsilon:\\,x\\to\\tilde x$, then the corresponding Poisson gauge theories constructed following the proposed here scheme are related by the same coordinate transformation. \n\nFor arbitrary given Poisson structure $\\Theta^{ab}(x)$ all the structures appearing in the paper can be constructed up to arbitrary order in the deformation parameter $\\alpha$. For some specific choices of the non-commutativity, like the rotationally invariant NC space (\\ref{su2}), $\\kappa$-Minkowski space (\\ref{kappa}) or the spaces obtained from these two by the coordinate transformations we obtained explicit all-order in $\\alpha$ expressions for all structures. It is interesting to work out different explicit examples of NC spaces, like e.g. the angular twist non-commutativity \\cite{angular}. It would be also interesting to obtain an explicit solution of the equation (\\ref{rho}) for any Lie-Poisson structure, $\\Theta^{ab}(x)=f^{ab}_cx^c$, just like it was done in the Sec. 2.2 for the construction of the matrix $\\gamma_a^l(x,p)$. \n\nIn the abstract it is written that the model was designed to investigate the semi-classical features of the complete non-commutative gauge theory. So, it would be interesting to understand what kind of physics may stay behind the Poisson gauge field equations obtained in the Sec. 6. It would be interesting to investigate in particular the corresponding energy-momenta dispersion relations.\n\nThe most non-trivial and at the same time interesting part of \\cite{KS21} is related to the construction of the almost-Poisson gauge algebra. There are certain configurations in string and $M$-theory like the non-geometric backgrounds \\cite{NA1}-\\cite{NA4} yielding not only non-commutative but also non-associative deformations of space-time \\cite{Szabo2}-\\cite{KS17}. In this case the corresponding algebra of brackets (\\ref{PB}) does not satisfy the Jacobi identity. But even so it is possible to construct the Lie algebra of gauge symmetries defined on such non-associative spaces. The key difference with the Poisson gauge algebra (\\ref{ga}) consists in the fact that the commutator of two gauge transformations is again a gauge transformation, however with a field-dependent gauge parameter. It is challenging to generalize the constructions obtained in the present research to the case of almost-Poisson deformations and thus to construct the semi-classical limit of non-associative gauge theory and non-associative gravity.\n\n\\subsection*{Acknowledgments}\n\nI am grateful to Dima Vassilevich for fruitful discussions. I appreciate the support from the Funda\\c c\\~ao de Amparo a Pesquisa do Estado de S\\~ao Paulo (FAPESP, Brazil).\n\n\\section*{Appendix: Useful formulae}\n\nLet us first calculate,\n\\begin{eqnarray}\\label{r1}\n&&\\{f(x),\\{g(x),p_a\\}\\}_{\\Phi=0}-\\{f(x),\\{g(x),p_a\\}_{\\Phi=0}\\}_{\\Phi=0}=\\\\\n&&\\{f(x),\\gamma^i_a(x,p)\\,\\partial_ig(x)\\}_{\\Phi=0}-\\{f(x),\\gamma^i_a(x,A(x))\\,\\partial_ig(x)\\}_{\\Phi=0}=\\notag\\\\\n&&\\left(\\partial^b_p\\{g(x),p_a\\}\\right)_{\\Phi=0}\\left(\\{f(x),p_b\\}_{\\Phi=0}-\\{f(x),A_b(x)\\}_{\\Phi=0}\\right)=\\notag\\\\\n&&\\left(\\partial^b_p\\{g(x),p_a\\}\\right)_{\\Phi=0}\\{f(x),\\Phi_b\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\nObserve that by (\\ref{PB}) and (\\ref{PB1}) the Poisson bracket of two functions $f(x)$ and $g(x)$ of coordinates only does not depend on $p$-variables, so,\n\\begin{eqnarray}\\label{r2}\n\\{f(x),g(x)\\}_{\\Phi=0}=\\{f(x),g(x)\\}\\,.\n\\end{eqnarray}\nThe combination of (\\ref{r1}) and (\\ref{r2}) implies,\n\\begin{equation}\\label{r3}\n\\left\\{f(x),\\{g(x),\\Phi_a\\}\\right\\}_{\\Phi=0}-\\left\\{f(x),\\{g(x),\\Phi_a\\}_{\\Phi=0}\\right\\}_{\\Phi=0}=\\left(\\partial^d_p\\{g(x),\\Phi_a\\}\\right)_{\\Phi=0}\\{f(x),\\Phi_d\\}_{\\Phi=0}\\,.\n\\end{equation}\n\nOne may also check that,\n\\begin{eqnarray}\\label{r5}\n&&\\{\\{f(x),\\Phi_b\\},\\Phi_a\\}_{\\Phi=0}-\\{\\{f(x),\\Phi_b)\\}_{\\Phi=0},\\Phi_a\\}_{\\Phi=0}=\\\\\n&&\\left(\\partial^d_p\\{f(x),\\Phi_b)\\}\\right)_{\\Phi=0}\\{\\Phi_d,\\Phi_a\\}_{\\Phi=0}\\,,\\notag\n\\end{eqnarray}\nand,\n\\begin{eqnarray}\\label{r6}\n&&\\{\\{\\Phi_c,\\Phi_b\\},\\Phi_a\\}_{\\Phi=0}-\\{\\{\\Phi_c,\\Phi_b)\\}_{\\Phi=0},\\Phi_a\\}_{\\Phi=0}=\\\\\n&&\\left(\\partial^d_p\\{\\Phi_c,\\Phi_b)\\}\\right)_{\\Phi=0}\\{\\Phi_d,\\Phi_a\\}_{\\Phi=0}\\,.\\notag\n\\end{eqnarray}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Field Visualization}\\label{appendix1}\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nReduced-order models are a popular solution to create a computationally cheap model useful for many-query applications.\nThe usefulness and adoption of these \\emph{quick-to-evaluate} models have increased further due to multi-fidelity techniques that use sparse high-fidelity and abundant low-fidelity data.\nSuch models tackle the trade-off between predictive accuracy and the cost of generating a sufficient amount of expensive training data. \n\nThe structural design and sizing of aircraft components at the preliminary design phase is a scenario of many-query applications. A feature of this problem is the necessity to accurately predict the high-dimensional field responses. The field responses from physics-based structural simulation models of different fidelity have disparate representations. These inconsistencies are an obstacle to the use of contemporary approaches for multi-fidelity ROMs. \n\nIn this work, we proposed the use of a recent multi-fidelity ROM method that is capable of merging fields with inconsistent dimensionalities, topologies, and features. The method projects the low- and high-fidelity data to their respective reduced subspaces. Then, a Procrustes analysis is used to align the two manifolds. The alignment allows for the two datasets to be combined into a multi-fidelity ROM. \n\nThe proposed method is demonstrated on two structural scenarios that have multi-fidelity fields of inconsistent representations. The first scenario tackled inconsistent grid sizes by combining data from a fine structural mesh with that from a coarse structural mesh. The second scenario tackled inconsistent topologies by combining data from a high-rib count structure with that from a low-rib count structure. \n\nResults showed that the method could improve the accuracy by up to 62.6\\% when compared to a single-fidelity ROM trained with a similar computational budget.\nWhile lower errors were observed for the prediction of both the von-Mises stress and the structural displacement fields, the MA-ROM was noticeably more suited to the evaluation of the latter.\nIndeed, for a set of high-fidelity samples, the proposed method would become more accurate with additional low-fidelity results, but when applied to the stress field, the lowest cost for a specified accuracy is found at $\\tau = 4$.\nThe reduced effectiveness of the MA-ROM method for the prediction of the von-Mises is explained by the higher complexity and non-linearity of that field when compared to the structural displacement.\n\nAn extension of the present work is to explore creating a multi-fidelity ROM combining results from beam and shell models.\nThe computationally efficient beam model would provide low-fidelity data owing to its inaccuracy at capturing the torsional response due to the ribs' absence. \nOn the other hand, the computationally inefficient, and generally more accurate, shell model would provide the high-fidelity data.\nThe authors also intend to extend the current method to adaptive sampling scenarios.\nThis would enable even greater cost reduction in the context of surrogate-based optimization.\n\n\n\\section{Results}\\label{sec:results}\n\nThis section presents the predictive performance of the MA-ROM method applied to the CRM wing application problem.\nThe results for both Scenarios~1 and 2, i.e., combining inconsistent grid sizes and structural topologies respectively, are presented and discussed separately.\nFor each of the fidelity levels considered, a data set of 1,000 cases\n\\footnote{The field data generated with each fidelity level are provided as supplementary material.}\nis generated using an optimized Latin hypercube sampling method~\\cite{Santner2003} and the design bounds defined in Table~\\ref{tab:ffd-param}.\nFrom the high-fidelity dataset, a subset of $n$ cases are selected to form the high-fidelity training data.\nAfterward, the same $n$ cases and an additional $m - n$ cases are randomly selected from the auxiliary dataset to form the low-fidelity training set.\nOnce the MA-ROM model is trained, the remainder of the high-fidelity data is used to compute the model prediction error.\nThe accuracy of the MA-ROM prediction is also compared to that of a conventional single-fidelity ROM, where the latter is only trained with high-fidelity data.\n\nIt should be noted that for a given $n$, the above training process is repeated at least 100 times with different subsets of the overall data, and the prediction errors of all models are averaged and reported.\nThe purpose of these repetitions is to decouple the measured error from the choice of training samples.\nIn doing so, presented results represent the expected accuracy of the MA-ROM (or single-fidelity ROM) method rather than the performance of one specific model instance.\n\nIn the current study, the field outputs of interest are the von-Mises stress distribution $\\sigma_\\text{vm}$ and the structural displacement $\\delta$, although the MA-ROM could be applied to any field responses.\nFurthermore, high-fidelity training sample sizes $n$ between 10 and 800 are considered, leaving at least 200 cases for the error calculation.\nAs for the low-fidelity training set, sample size $m$ is kept proportional to $n$ using the multi-fidelity ratio $\\tau = m\/n$, and ratio of $\\tau = \\{2, 4, 8\\}$ are considered.\nLastly, all POD bases are computed with an RIC of 99.9999\\% or greater for each fidelity level.\n\n\\subsection{Scenario~1: Inconsistent Grid Sizes}\\label{sec:results:scn-1}\n\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics{Figures\/Results\/mf-comparison-grid-VM.pdf}\n    \\caption{Prediction errors of the von-Mises stress $\\sigma_\\text{vm}$ using the MA-ROM method to combine fine and coarse grid results.}\n    \\label{fig:scn-1:stress}\n\\end{figure*}\n\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics{Figures\/Results\/mf-comparison-grid-disp.pdf}\n    \\caption{Prediction errors of the structural displacement $\\delta$ using the MA-ROM method to combine fine and coarse grid results.}\n    \\label{fig:scn-1:displacement}\n\\end{figure*}\n\nAs defined in Section~\\ref{sec:scn-1}, the first multi-fidelity scenario combines the structural results of the CRM wing obtained from a fine and coarse grid.\nThe normalized prediction errors on the von-Mises stress distribution $\\sigma_\\text{vm}$ are presented in Figure~\\ref{fig:scn-1:stress} for this scenario.\nThe blue, red, and green lines represent the results using a multi-fidelity ratio $\\tau$ of 2, 4, and 8, respectively.\nFor comparison, the results for a single-fidelity ROM trained with only the fine grid results are shown using a black line.\nThe left side of Figure~\\ref{fig:scn-1:stress} presents the errors as a function of $n$, while the right side shows the results scaled with the overall training cost of the various models in CPU-sec.\nNote that for the multi-fidelity models, the training cost is comprised of the cost for both the high- and low-fidelity training data.\nFollowing a similar presentation, Figure~\\ref{fig:scn-1:displacement} also presents the prediction errors of the MA-ROM and ROM methods, although for the structural displacement $\\delta$ instead of $\\sigma_\\text{vm}$.\nFor the reader's convenience, some of the data points in Figures~\\ref{fig:scn-1:stress} and~\\ref{fig:scn-1:displacement} are also tabulated in Table~\\ref{tab:scn-1}.\n\n\\begin{table}[]\n    \\centering\n    \\caption{Prediction error of the MA-ROM method applied to Scenario~1.}\n    \\label{tab:scn-1}\n    \\begin{tabular}{ccccc}\n    \\hline\n    $n$ & $m$ & $\\widehat{E}(\\sigma_\\text{vm})$ & $\\widehat{E}(\\delta)$ & CPU-sec \\\\\n    \\hline\\hline\n    100$^*$ & -     & 0.805\\%   & 0.587\\%   & 544.0 \\\\\n    100     & 200   & 0.616\\%   & 0.284\\%   & 664.0 \\\\\n    100     & 400   & 0.568\\%   & 0.174\\%   & 783.9 \\\\\n    100     & 800   & 0.528\\%   & 0.119\\%   & 1,023.9 \\\\\n    \\hline\n    183$^*$ & -     & 0.609\\%   & 0.321\\%   & 1,000.0 \\\\\n    150     & 300   & 0.526\\%   & 0.210\\%   & 1,000.0 \\\\\n    127     & 508   & 0.514\\%   & 0.145\\%   & 1,000.0 \\\\\n    97      & 779   & 0.536\\%   & 0.120\\%   & 1,000.0 \\\\\n    \\hline\n    \\multicolumn{5}{l}{$^*$\\footnotesize{Results for a single-fidelity ROM}}\n    \\end{tabular}\n\\end{table}\n\nAs can be expected, both Figures~\\ref{fig:scn-1:stress} and~\\ref{fig:scn-1:displacement} show that the accuracy of both the single- and multi-fidelity methods monotonically improves with additional training data.\nMore importantly, these figures also demonstrate that for all $n$ values, the MA-ROM outperforms the single-fidelity ROM by enhancing the high-fidelity data with additional low-fidelity results.\nWhen compared to a single-fidelity model trained with the same computational budget, say $1,000$~CPU-sec, a relative reduction in the error of up to 15.6\\% and 62.6\\% is observed for the prediction of $\\sigma_\\text{vm}$ and $\\delta$ respectively as shown in Table~\\ref{tab:scn-1}.\nHowever, this improvement appears to depend on the specific field quantity of interest the MA-ROM is applied to.\n\nFor the von-Mises stress field, Figure~\\ref{fig:scn-1:stress} shows a reasonable reduction in $\\widehat{E}(\\sigma_\\text{vm})$ with $\\tau=2$, yet increasing $\\tau$ further only provides a marginal improvement.\nSpecifically, for $n=100$, the normalized prediction error is 0.630\\% and 0.574\\% for $m=200$ and $800$ respectively, which represents a relative improvement of only 8.9\\% for almost double the cost.\nWhen including the additional computational cost of the low-fidelity data into the analysis, the right side of Figure~\\ref{fig:scn-1:stress} indicates that for a given prediction error, the MA-ROM can reduce the overall training cost of the model over a single-fidelity approach.\nAlthough, the results also show that predicting $\\sigma_\\text{vm}$, using $\\tau > 4$ is likely not cost-effective.\nThis suggests that the high- and low-fidelity fields don't share a strong statistical dependence and that the transfer of low-fidelity information within the multi-fidelity method saturates.\n\nOn the other hand, for the prediction of the structural displacement, the results of Figure~\\ref{fig:scn-1:displacement} exhibit a more pronounced decrease in $\\widehat{E}(\\delta)$ with additional low-fidelity data.\nFor $n=100$, the normalized prediction error is 0.275\\% and 0.112\\% for $m=200$ and $800$ respectively, which represents a relative improvement of 59.3\\%.\nEven with the additional cost of the low-fidelity samples, the right side of Figure~\\ref{fig:scn-1:displacement} presents a clear reduction in $\\widehat{E}(\\delta)$ for a given computational cost, or equivalently, a reduction in cost for a target prediction error.\nWhile the results show reduced benefits with larger $\\tau$, using more low-fidelity data still appears to be cost-effective, especially when compared to the single-fidelity results.\n\nThe better performance of the MA-ROM method for the prediction of the displacement field than for the von-Mises stress field can be explained by the higher non-linearity and complexity of the latter.\nAs one can expect, this suggests that the effectiveness of the multi-fidelity method ultimately depends on the application.\nThat being said, the results demonstrate that MA-ROM can be a suitable method to predict structural results and has the potential to offer a lower training cost than a single-fidelity method.\nFurthermore, these benefits are achieved despite the high- and low-fidelity fields being represented on grids with different discretizations. \n\n\n\\subsection{Scenario~2: Inconsistent Topologies}\\label{sec:results:scn-2}\n\nThe second scenario of this study combines the results of wing structures with different structural layouts.\nUnlike Scenario~1, the fields of the main and auxiliary datasets have inconsistent topologies.\nAs discussed in Section~\\ref{sec:results:scn-1}, the auxiliary analysis is the one using a structural layout with a low-rib count (see Figure~\\ref{fig:scn-2:layout}).\nIt is assumed that this data is readily available from a previous study, and is used to enhance the main data corresponding to a new design study.\n\nThe results in Figure~\\ref{fig:scn-2} present the prediction error obtained with the MA-ROM method applied to the current scenario.\nAs in the previous scenario, the normalized error is shown for a range of high-fidelity samples $n$ and multiple multi-fidelity ratios $\\tau$.\nThe error achieved with a single-fidelity approach is also shown for comparison.\nThe top part of Figure~\\ref{fig:scn-2} gives the prediction error for the von-Mises stress field $\\sigma_\\text{vm}$, while the bottom part shows the error for the structural displacement field.\nThe numerical values for the cases at $n=100$ are also listed in Table~\\ref{tab:scn-2} for an easier comparison.\nUnlike in Figure~\\ref{fig:scn-1:stress} and~\\ref{fig:scn-1:displacement} in Scenario~1, the computational cost is not shown here since the auxiliary data is assumed to already exist.\n\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics{Figures\/Results\/mf-comparison-topo-stacked.pdf}\n    \\caption{Prediction errors of $\\sigma_\\text{vm}$ and $\\delta$ using the MA-ROM method to combine results from different structural topologies.}\n    \\label{fig:scn-2}\n\\end{figure}\n\n\\begin{table}[]\n    \\centering\n    \\caption{Prediction error of the MA-ROM method applied to Scenario~2.}\n    \\label{tab:scn-2}\n    \\begin{tabular}{cccc}\n    \\hline\n    $n$ & $m$ & $\\widehat{E}(\\sigma_\\text{vm})$ & $\\widehat{E}(\\delta)$ \\\\\n    \\hline\\hline\n    100$^*$ & -   & 0.805\\% & 0.587\\% \\\\\n    100     & 200 & 0.630\\% & 0.275\\% \\\\\n    100     & 400 & 0.595\\% & 0.167\\% \\\\ \n    100     & 800 & 0.574\\% & 0.112\\% \\\\\n    \\hline\n    \\multicolumn{4}{l}{$^*$\\footnotesize{Results for a single-fidelity ROM}}\n    \\end{tabular}\n\\end{table}\n\nWhen compared to the results reported in Section~\\ref{sec:results:scn-1}, the current results exhibit trends in agreement with those of Scenario~1.\nOnce again, the normalized error decreases monotonically with additional training data and adding auxiliary data produces better results than using the main dataset only.\nFor $n=100$, the prediction error is up to 28.70\\% and 80.92\\% smaller for $\\sigma_\\text{vm}$ and $\\delta$ respectively, which is comparable to the results presented in Table~\\ref{tab:scn-1}.\nFurthermore, the results of Figure~\\ref{fig:scn-2} indicate once more that improving the prediction of $\\sigma_\\text{vm}$ with the MA-ROM method is more challenging than it is for $\\delta$.\nThis reinforces the suspicion that the higher complexity and non-linearity of the von-Mises stress field renders the main and auxiliary fields more difficult to fuse.\n\nThe main outcome of the results of Scenario~2 is that combining data with disparate topologies does not appear to alter the performance of the MA-ROM method.\nTherefore, the accuracy of a field prediction model can readily be increased using MA-ROM if data from a similar, yet different problem is available even if their field size or topology are inconsistent.\nFor instance, this could be useful if a ROM was previously trained for an earlier iteration of the design, and is no longer adequate due to changes to the structural layout.\nConsequently, the MA-ROM method gives renewed value to old, discarded data by using them to enhance the prediction of models trained on new, expensive data.\n\n\\subsection{Field Visualization}\n\nThis section presents a few key figures for a visual depiction of the predictive accuracy provided by the MA-ROM.\nFor reference, Figure~\\ref{fig:worst_point} shows the actual solution for the test point with the largest observed prediction error.\nRecollect that single- and multi-fidelity ROMs are constructed to predict both the stress and the displacement fields.\nIt should be noted that the test case presented corresponds to a design that would normally be infeasible due to the high stress and displacement values computed.\nHowever, other than being an extreme design, this does not impact the ROM models as they merely reproduce the analysis results.\nFigures~\\ref{fig:stress_visuals} and~\\ref{fig:displacement_visuals} then present the predicted fields and the corresponding prediction errors for both quantities of interest.\nThe results for both multi-fidelity scenarios are shown, as well as the results from a single-fidelity approach for comparison sake.\n\nComparing Figures~\\ref{fig:stress_visuals} and~\\ref{fig:displacement_visuals} to Figure~\\ref{fig:worst_point} shows that all ROM methods can recover the actual field of interest with a reasonable accuracy.\nYet, an inspection of the error fields reveals that for the prediction of both the stress and the displacement fields, the MA-ROM outperforms the single-fidelity POD-based ROM even for the worst prediction in the test set.\nThese visualizations also show that the error tends to be higher near regions of high stress (e.g., the wing root) or large displacement (e.g., the wing tip)\nOne should further note that the error levels for both multi-fidelity scenarios have similar magnitudes.\nThese observations support previous discussions and reinforce MA-ROM's efficacy when applied to high-fidelity structural analyses with grid inconsistencies and differences in topology.\n\n\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics{Figures\/Results\/hf_field_results.pdf}\n    \\caption{Exact simulation results of the von-Mises stress and the $z$ displacement fields for the test case with the highest prediction error.}\n    \\label{fig:worst_point}\n\\end{figure}\n\n\n\\begin{figure*}[p]\n    \\centering\n    \\includegraphics{Figures\/Results\/stress_field_predictions.pdf}\n    \\caption{Prediction results of the von-Mises stress field for the test case with the highest prediction error. Shown results consider a single-fidelity ROM model~(A), MA-ROM model combining inconsistent grid sizes~(B), and a MA-ROM model combining inconsistent structural topologies~(C).}\n    \\label{fig:stress_visuals}\n\\end{figure*}\n\n\n\\begin{figure*}[p]\n    \\centering\n    \\includegraphics{Figures\/Results\/disp_field_predictions.pdf}\n    \\caption{Prediction results of the $z$ displacement field for the test case with the highest prediction error. Shown results consider a single-fidelity ROM model~(A), MA-ROM model combining inconsistent grid sizes~(B), and a MA-ROM model combining inconsistent structural topologies~(C).}\n    \\label{fig:displacement_visuals}\n\\end{figure*}\n\\section{Application Problem}\\label{sec:application}\n\nThe application problem considers the analysis of a wing structure to demonstrate the relevance of MA-ROM for large-scale problems representative of an industrial design process. \nThe baseline geometry considered for this application problem is the wing of the NASA CRM~\\cite{kenway2014aerostructural}.\nThe CRM is representative of a transonic wide-body transport aircraft. \n\n\\subsection{Aero-Structural Analysis}\\label{subsec:crm_baseline}\n\nThe baseline OML\\footnote{An OpenVSP file of the baseline CRM OML is included with this paper} and flight conditions considered are given in Table~\\ref{tab:crm_baseline_oml_fc} and shown in Figure~\\ref{fig:crm_wing_oml}. \n\n\\begin{table}[ht]\n        \\centering\n        \\caption{Baseline OML \\& flight condition values}\\label{tab:crm_baseline_oml_fc}\n        \\begin{tabular}{lcc}\n            \\hline\n            \\textbf{Parameter}  & \\textbf{Value} & \\textbf{Unit}    \\\\\n            \\hline\\hline\n            Planform area       & 4444.38127     & \\si{\\square\\ft}  \\\\\n            Span                & 193.34768      & \\si{\\ft}         \\\\\n            Aspect ratio        & 8.3325         & -                \\\\\n            Taper ratio         & 0.86031        & -                \\\\\n            Leading edge sweep  & 37             & \\si{\\deg}        \\\\\n            TOGW                & 650000         & \\si{\\lb}         \\\\\n            Altitude            & 30000          & \\si{\\ft}         \\\\\n            Mach number         & 0.8            & -                \\\\\n            \\hline\n        \\end{tabular}\n\\end{table}\n\n\\begin{figure*}[]\n    \\centering\n    \\includegraphics[width=1.8\\columnwidth]{Figures\/ApplicationProblem\/crm_wing_oml.pdf}\n    \\caption{Top, rear, and side view of the CRM wing. The dimensions provided are normalized by the mean aerodynamic chord whose value is 275.8~in. The top right isometric view shows the wing in context of the full aircraft configuration.}\n    \\label{fig:crm_wing_oml}\n\\end{figure*}\n\n\nThe aerodynamic loads on the wing are obtained from a panel representation of the aircraft OML.\nThis panel model has the associated sectional airfoils, local incidence angles, etc. that can be used for creating lower order aerodynamic models for tools like AVL.\nThe baseline panel geometry for the AVL model\n\\footnote{The AVL model of the baseline CRM OML is provided as supplementary material.} is shown in Figure~\\ref{fig:pegasus_avl}.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/ApplicationProblem\/crm_avl.pdf}\n    \\caption{Panel geometry and discretization for CRM AVL model}\n    \\label{fig:pegasus_avl}\n\\end{figure}\n\n\nFigure~\\ref{fig:crm_mesh} shows the structural geometry layout and mesh for the wing, horizontal tail, and vertical tail within a fused representation of the OML. In this work, we consider only the wing whose structural topology values are given in Table~\\ref{tab:crm_baseline_strcTop}.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/ApplicationProblem\/crm_structure_full_zoom.png}\n    \\caption{Geometry and mesh of the wing and empennage of the CRM}\n    \\label{fig:crm_mesh}\n\\end{figure}\n\n\\begin{table}\n        \\centering\n        \\caption{Baseline structural topology values (cn: chord normalized and sn: span normalized).}\\label{tab:crm_baseline_strcTop}\n        \\begin{tabular}{lcc}\n            \\hline\n            \\textbf{Parameter}      &   \\textbf{Value} & \\textbf{Unit} \\\\\n            \\hline\\hline\n            Front spar location     &   0.17           & cn            \\\\\n    \t\tRear spar location      &   0.68           & cn            \\\\\n    \t\tRib spacing             &   32             & \\si{\\in}      \\\\\n    \t\tRib roll                &   0              & \\si{\\deg}     \\\\\n    \t\tRib yaw                 &   -10            & \\si{\\deg}     \\\\\n    \t\tRoot rib location       &   0.1            & sn            \\\\\n    \t\tTip rib location        &   0.995          & sn            \\\\\n            \\hline\n        \\end{tabular}\n\\end{table}\n\nThe structure is modeled using CQUAD4 shell elements in Nastran\\footnote{The Nastran model (i.e., the BDF file) of the baseline CRM OML is provided as supplementary material.} for the ribs, spars, and skins. \nAll shell panels are assigned aluminium material with a density of \\num[round-mode=places,round-precision=3]{2698.79} \\si{\\kg\\per\\meter^3}, a Young's Modulus of \\num[round-mode=places,round-precision=3]{71016000118} \\si{\\newton\\per\\meter^2}, and a Poisson's ratio of $0.33$.\nThe following assumptions are made in the structural model:\n\\begin{enumerate}\n    \\item No stiffeners and stringers\n    \\item The thicknesses of the panels are fixed. The values are obtained by averaging thicknesses along the span for a baseline sizing case where the CRM was sized for 2.5g and -1.0g static maneuvers at flight conditions given in Table~\\ref{tab:crm_baseline_oml_fc}.\n    \\begin{itemize}\n        \\item Front spar: \\num[round-mode=places,round-precision=3]{0.465988771929824} \\si{\\in}\n        \\item Rear spar: \\num[round-mode=places,round-precision=3]{0.257907054747054} \\si{\\in}\n        \\item Upper skin: \\num[round-mode=places,round-precision=3]{0.923677840755735} \\si{\\in}\n        \\item Lower skin: \\num[round-mode=places,round-precision=3]{0.657111039136302} \\si{\\in}\n        \\item Rib: \\num[round-mode=places,round-precision=3]{0.148520056980057} \\si{\\in}\n    \\end{itemize}\n\\end{enumerate}\nThese assumptions are merely simplifications to make the modeling easier.\nThe proposed multi-fidelity method's performance is neither affected nor depends on this simplification.\nStiffeners can be included by using the smeared-stiffener calculations as done by Kennedy and Martins~\\cite{kennedy2014parallel}. \n\nThe transfer of aerodynamic loads from rigid-wing aerodynamics in AVL to the structure is done by integrating pressure differentials for elements on the aerodynamic panels to reference points along the span of a wing-like component.\nThe centers of mass of the ribs are chosen as reference points along the span of the wing. \nThe forces and moments associated with each reference point are then distributed to the structure via rigid body elements (RBEs) at key intersecting nodes between the ribs and skin.\n\nThe structure is analyzed in Nastran using SOL101. Field outputs such as internal loads, stresses, strains, and displacements are obtained.\nThe proposed MA-ROM approach is used to obtain a surrogate of these field outputs as a function of the input design variables that will be described in Section~\\ref{subsec:dv_parametrization}.\n\nIt should be noted that in this work each shell element has an assumed thickness.\nThe Nastran model can be used to size the structure for the applied loads using Hypersizer~\\cite{solano2020structural,solano2021pegasus}.\nOther fields can be generated from Hypersizer outputs such as optimal sectional dimensions (skin thickness, stiffener height, etc.) and margins of safety for multiple failure criteria.\n\n\\subsection{Parametrization}\\label{subsec:dv_parametrization}\n\nIn the proposed problem, global wing parameters are considered for the design parametrization.\nThese include the wing sweep, dihedral, tip twist, and span as shown in Table~\\ref{tab:ffd-param}.\nNote that the bounds are represented as percentage increments\/decrements with respect to the original aircraft OML.\n\n\\begin{table}\n        \\centering\n        \\caption{Design parameters and bounds considered for the application problem. Values are increments to the original CRM geometry}\\label{tab:ffd-param}\n        \\begin{tabular}{lc}\n            \\hline\n            \\textbf{Parameter}      &   \\textbf{Bounds} \\\\\n            \\hline\\hline\n            Wing sweep     &   $\\pm 10^\\circ$ \\\\\n    \t\tWing dihedral      &   $\\pm 10^\\circ$  \\\\\n    \t\tWing tip twist &   $\\pm 10^\\circ$    \\\\\n    \t\tWing span &   $\\pm 20\\%$     \\\\\n            \\hline\n        \\end{tabular}\n\\end{table}\n\nWhile the MA-ROM approach can combine fields of high- and low-fidelity having different representations, it still requires each fidelity level to internally use a consistent representation.\nFor instance, a similar grid topology, in terms of size and connectivity, should be used for all the high-fidelity results, and likewise for the low-fidelity results.\nHowever, this grid topology does not need to be between both high- and low-fidelity results since any discrepancy between fidelity levels will be addressed with the manifold alignment.\nFor the aforementioned reason, a grid deformation approach is used to apply the design parameters of Table~\\ref{tab:ffd-param} on both the structural and aerodynamic grids.\n\nIn the current study, the grid is deformed using a free-form deformation (FFD) approach~\\cite{Kenway2010}.\nThis technique encases a geometry inside a lattice or box as shown in Figure~\\ref{fig:ffd-crm}.\nThe FFD box behaves as a flexible rubber-like material such that displacement of the boundary nodes is propagated to the geometry within.\nFor instance, given the current problem, the wing sweep is modified by translating backward or forward all the end nodes of the FFD box.\nIn addition to maintaining the grid topology, the FFD techniques can effectively be applied to any discretized geometry and without the need for CAD software.\nNote that this technique is popular in the context of aerodynamic shape optimization, some examples of which can be found in Lyu et al.~\\cite{Lyu2014} and Koo and Zing~\\cite{Koo2016}. \n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{Figures\/ApplicationProblem\/crm_ffd.png}\n    \\caption{Structural grid of the CRM wing encased in an FFD box (red).}\n    \\label{fig:ffd-crm}\n\\end{figure}\n\n\\subsection{Multi-Fidelity Scenarios}\n\nThe usefulness of the MA-ROM method is demonstrated using two multi-fidelity scenarios based on the CRM test case with an FFD parametrization described earlier.\nEach of these scenarios is aimed at combining multi-fidelity fields having different types of inconsistencies.\n\n\\subsubsection{Scenario~1: Inconsistent Grid Sizes}\\label{sec:scn-1}\n\nIn this scenario, structural results computed on a fine grid are enhanced with data obtained from a similar analysis, yet using a coarser grid.\nSpecifically, the high- and low-fidelity simulation use grids with a resolution of roughly 79,000 and 5,000 elements respectively as shown in Figure~\\ref{fig:scn-1:grid}.\nUsing a coarser grid, the computational cost of the low-fidelity analysis is reduced from 5.5 to 0.6 CPU-sec on a modern desktop, which represents an order of magnitude reduction.\nIn this scenario, the purpose of the multi-fidelity method is to reduce the overall training cost of the model by leveraging a computationally cheaper tool.\nAs such, the multi-fidelity method's predictive performance will be contrasted with that of a single-fidelity model constructed using the results from the fine grid exclusively.\nThis comparison will look at these models' accuracy for similar training costs, which for the multi-fidelity model, includes the cost of generating both the high- and low-fidelity data.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{Figures\/ApplicationProblem\/crm_fine_grid.png}\n    \n    A) High-fidelity grid (79,167 elements)\\vspace{0.1in}\n    \n    \\includegraphics[width=0.7\\columnwidth]{Figures\/ApplicationProblem\/crm_coarse_grid.png}\n    \n    B) Low-fidelity grid (5,062 elements)\n    \n    \\caption{Comparison the (A) high- and (B) low-fidelity grids used for Scenario~1.}\n    \\label{fig:scn-1:grid}\n\\end{figure}\n\n\\subsubsection{Scenario 2: Inconsistent Topologies}\\label{sec:scn-2}\n\nFor the second scenario, the MA-ROM method is used to combine results from similar structural simulations, yet using different structural layouts as shown in Figure~\\ref{fig:scn-2:layout}.\nUnlike Scenario~1, both simulations are solved using grids with a similar discretization, and as a result, do not follow the usual high- and low-fidelity breakdown.\nHowever, for this scenario, we assume that the results for one of the two structural layouts are readily available in abundance.\nFor instance, these can correspond to the results previously generated for a different aircraft, or similarly, generated for an earlier version of the aircraft at a point where a different layout was considered.\nTherefore, the purpose of the multi-fidelity approach is to \\emph{jump-} or \\emph{warm-start} the surrogate model generation of a new structural layout using existing data from a different, yet similar application.\nInstead of a high- and low-fidelity dataset, the training data is separated into a main and an auxiliary dataset, the former representing the fields that must be predicted by the MA-ROM.\nIn this example, the main dataset corresponds to the same structural layout used in Scenario~1, and the auxiliary dataset is a structure for the CRM wing using a lower rib count.\nThe intent behind this scenario is to demonstrate the capability of the MA-ROM to combine structural results despite having inconsistent field topologies.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{Figures\/ApplicationProblem\/crm_high_rib_count.png}\n    \n    A) Main structure (high-rib count)\\vspace{0.1in}\n    \n    \\includegraphics[width=0.7\\columnwidth]{Figures\/ApplicationProblem\/crm_low_rib_count.png}\n    \n    B) Auxiliary structure (low-rib count)\n    \n    \\caption{Comparison the structural layouts used for the (A) main and (B) auxiliary datasets in Scenario 2.}\n    \\label{fig:scn-2:layout}\n\\end{figure}\n\n\\subsection{Performance Metrics}\\label{sec:metrics}\n\nTo establish the benefits of the MA-ROM method for structural problems, its predictive performance is compared to that of a conventional single-fidelity ROM.\nThis performance is measured in terms of prediction error and is contrasted with the computational cost required to train the model.\nAs with any empirical model, one can expect MA-ROM to be more accurate when provided with more data, but for it to be cost-effective, it should achieve this accuracy with lesser resource consumption than a single-fidelity method.\n\n\\subsubsection{Field Prediction Error}\n\nThe prediction error of a ROM or MA-ROM model is typically measured as the difference between the predicted and the actual field responses on previously unseen data, i.e., on data that was left out of the training process.\nUsing $n_t$ test samples held out from the training data, we define the field prediction error of some field $\\mathbf{x} \\in \\mathbb{R}^d$ as\n\\begin{equation}\n    E(\\mathbf{x}) = \\sqrt{\\frac{\\sum_{j=1}^{n_t} \\lVert \\mathbf{x}^*_j - \\widetilde{\\mathbf{x}}_j \\rVert_2^2}{n_t}}\n\\end{equation}\nwhere $\\widetilde{\\mathbf{x}}_j \\in \\mathbb{R}^d$ is the model prediction from either a ROM or MA-ROM model, and $\\mathbf{x}^*_j \\in \\mathbb{R}^d$ is the corresponding exact solution.\nWe use the $L^2$-norm of the error field to provide an integrated measure of the prediction error for a given sample.\nThe test set sample errors are then aggregated using a root-mean-squared operation to represent the overall prediction error of a given model.\n\nNote that the magnitude of $E(\\textbf{x})$ is dependent on the field quantity being considered.\nTo compare errors between different design problems, it is useful to define a normalized prediction error such as\n\\begin{equation}\\label{eq:norm-pred-error}\n    \\widehat{E}(\\textbf{x}) = \\frac{E(\\textbf{x})}{\\sqrt{\\sum_{j = 1}^{n_t} \\lVert \\mathbf{x}_j^* - \\overline{\\mathbf{x}}\\rVert^2_2\/n_t}}\n\\end{equation}\nIt is worth noting that the normalization term used in equation~\\eqref{eq:norm-pred-error} is essentially the total standard deviation of the testing dataset.\nAs a result, the definition of $\\widehat{E}(\\textbf{x})$ represents the ratio between the variation of $\\textbf{x}$ unexplained by the ROM and the total variation of the data.\n\n\\subsubsection{Computational Training Cost}\n\nThe computational cost of a ROM can be separated into its evaluation and training costs, which are also referred to as the online and offline costs.\nThe former is designated as an indicator of the cost of predicting new results and is typically negligible, and as such, we only focus on and report the training cost.\nIn comparison, the time needed to construct the ROM or MA-ROM model once the data is available is relatively large.\n\nThe offline computational cost can be measured in terms of CPU time, usually in CPU-hr or CPU-sec, which is the total amount of computational time used by each processing unit involved.\nThe CPU time can then be divided by the number of CPUs available to obtain an estimate of the wall-time, neglecting any performance loss from the parallelization.\nThe cost of each sample is assumed to be a constant in this work, and for Scenario~1 for instance, it is estimated at 5.4402 and 0.5998 CPU-sec for the baseline and coarse grid respectively\\footnote{\nCPU times are based on an Intel Core i9-10885H CPU.\n}.\nThe total training cost of ROMs in this study is then taken as the number of training samples times the constant cost per sample.\n\nNote that the computational cost can also have a monetary meaning as most high-performance computing facilities provide a price per CPU-hr used.\nThis price is a result of the operation, maintenance, and acquisition costs of a given supercomputer.\nFor instance, at the time of writing this manuscript, the Phoenix cluster at the Georgia Institute of Technology offered a preferential rate of \\$0.0068\/CPU-hr for internal users~\\cite{PACE}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Formulation}\\label{sec:formulation}\n\nThe MA-ROM method can be interpreted as a combination of three techniques: proper orthogonal decomposition (POD), manifold alignment (MA), and hierarchical Kriging (HK).\nThe following section briefly presents each of these techniques and describes how these are combined to form the MA-ROM method.\n\n\\subsection{Proper Orthogonal Decomposition}\n\nThe POD method~\\cite{Lumley67} is a linear and unsupervised dimensionality reduction technique that is also commonly referred to as principal component analysis (PCA) in the field of machine learning.\nIt is arguably the most common dimensionality reduction method used in the context of projection-based ROM.\n\nLet us consider the data matrix $\\mathbf{X} \\in \\mathbb{R}^{d\\times n}$ that contains $n$ samples $\\mathbf{x}_j$ of dimension $d$ organized column-wise.\nLet us further assume that $\\mathbf{X}$ has been pre-processed such that its sample mean\n\\begin{equation}\n    \\overline{\\mathbf{x}} = \\frac{1}{n}\\sum_{j=1}^{n} \\mathbf{x}_j\n\\end{equation}\nis zero, meaning that the mean has been subtracted from the original data set beforehand.\n\nThe POD method attempts to construct a simpler representation of $\\mathbf{X}$ that captures most of the data variance.\nThis is achieved by finding basis vectors $\\boldsymbol{\\phi}_j \\in \\mathbb{R}^d$ (also called POD modes) that maximizes the following \\emph{Rayleigh quotient}~\\cite{Magnus1997}\n\\begin{equation}\\label{eq:Rayleigh}\n    \\max_{\\boldsymbol{\\phi}_j} \\frac{\\boldsymbol{\\phi}_j^T \\mathbf{S} \\, \\boldsymbol{\\phi}_j}{\\boldsymbol{\\phi}_j^T \\boldsymbol{\\phi}_j}\n\\end{equation}\nwhere $\\mathbf{S}= \\frac{1}{n} \\mathbf{X}\\mathbf{X}^T \\in \\mathbb{R}^{d \\times d}$ is the sample covariance matrix.\n\nThe solution of to the above objective is given by the eigendecomposition of $\\mathbf{S}$~\\cite{DeBie2005} such that\n\\begin{equation}\n    \\mathbf{S} = \\boldsymbol{\\Phi} \\boldsymbol{\\Lambda} \\boldsymbol{\\Phi}^T\n\\end{equation}\nwhere $\\boldsymbol{\\Phi} \\in \\mathbb{R}^{d \\times d}$ contains the eigenvectors, and $\\boldsymbol{\\Lambda} \\in \\mathbb{R}^{n \\times n}$ is a diagonal matrix where the entries $\\lambda_j$ are the corresponding eigenvalues.\nWe define the POD basis $\\boldsymbol{\\Phi}_k \\in \\mathbb{R}^{d \\times k}$ as a low-rank approximation of $\\boldsymbol{\\Phi}$ whose columns are composed of the $\\boldsymbol{\\phi}_j$ associated with the $k$-largest $\\lambda_j$, where $k \\ll d$.\n\nUsing $\\boldsymbol{\\Phi}_k$, the original data can be projected into the POD latent space such that\n\\begin{equation}\n    \\mathbf{Z} = \\boldsymbol{\\Phi}_k^T \\mathbf{X}\n\\end{equation}\nwhere $\\mathbf{Z} \\in \\mathbb{R}^{k \\times n}$ contains the latent coordinates $\\mathbf{z}_j \\in \\mathbb{R}^k$ (also known as POD coefficients) corresponding to $\\mathbf{x}_j$.\nConversely, the original data can be approximately reconstructed from $\\mathbf{Z}$ such that\n\\begin{equation}\n    \\Xtilde = \\boldsymbol{\\Phi}_k \\mathbf{Z}\n\\end{equation}\nNote that $\\Xtilde \\in \\mathbb{R}^{d \\times n}$ is an approximation of $\\mathbf{X}$ since only the $k$-first POD modes are used.\n\nLastly, the dimension $k$ of the POD basis is usually determined using the relative information content (RIC) criterion~\\cite{Pinnau2008,Vendl2013} defined as\n\\begin{equation}\\label{eq:ric}\n    \\text{RIC} = \\frac{\\sum_{j=1}^{k} \\lambda_j}{\\sum_{j=1}^{d} \\lambda_j}\n\\end{equation}\nThe RIC criterion essentially represents the ratio of the total variance captured by the POD basis.\nA higher RIC implies a more accurate representation of $\\mathbf{X}$, but typically results in a larger latent space dimension.\nUsers typically select $k$ such that the RIC is greater than some threshold, and values of 99.9\\% or higher are fairly common in the ROM literature.\n\n\\subsection{Manifold Alignment}\n\nComputing the POD of different datasets, say a high- and low-fidelity dataset, will clearly result in distinct POD bases and coefficients.\nThe purpose of manifold alignment~\\cite{Ham2005,Wang2009} is to uncover a common latent space, instead of distinct latent spaces obtained via the POD, where disparate data can be more easily compared.\nIn the current context, the manifold alignment allows us to combine field results having different representations.\nSpecifically, the MA-ROM method leverages the Procrustes manifold alignment~\\cite{Wang2008} method as it can readily be combined with the established POD-based ROM methodology~\\cite{Perron2020}.\n\nConsider $\\mathbf{X} \\in \\mathbb{R}^{d \\times n}$ and $\\mathbf{Y} \\in \\mathbb{R}^{q \\times m}$ as matrices containing the high- and low-fidelity samples $\\mathbf{x}_j \\in \\mathbb{R}^d$ and $\\mathbf{y}_j \\in \\mathbb{R}^q$ respectively.\nNote that in view of a realistic scenario, it is assumed there are more low- than high-fidelity samples, i.e., $ m \\gg n$.\nHowever, no assumption is made regarding the dimensionality of both sets of fields;~they can be different, i.e., $d \\neq q$.\nFurthermore, let the first $n$ samples $\\mathbf{x}_j$ and $\\mathbf{y}_j$ have a pair-wise correspondence;~meaning they are generated by evaluating the respective simulation at identical design parameter values.\nConsequently, the low-fidelity data can be partitioned as $\\mathbf{Y} = [\\mathbf{Y}_\\text{L}, \\mathbf{Y}_\\text{U}]$, where $\\mathbf{Y}_\\text{L} \\in \\mathbb{R}^{q \\times n}$ contains the solutions \\emph{linked} to $\\mathbf{X}$, while $\\mathbf{Y}_\\text{U} \\in \\mathbb{R}^{q \\times (m - n)}$ holds the \\emph{unlinked} data.\n\nThe first step of the Procrustes manifold alignment is to find the latent space, i.e., the POD basis, of both the high- and low-fidelity datasets.\nLet us begin by defining\n\\begin{align}\\label{eq:mf-pod}\n    \\mathbf{Z} &= \\boldsymbol{\\Phi}_k^T \\mathbf{X}\\\\\n    \\mathbf{W} &= \\boldsymbol{\\Psi}_k^T \\mathbf{Y}\n\\end{align}\nwhere $\\mathbf{Z} \\in \\mathbb{R}^{k \\times n}$ and $\\mathbf{W} \\in \\mathbb{R}^{k \\times m}$ are the low- and high-fidelity POD coefficients respectively, while $\\boldsymbol{\\Phi}_k \\in \\mathbb{R}^{d \\times k}$ and $\\boldsymbol{\\Psi}_k \\in \\mathbb{R}^{q \\times k}$ are the associated POD modes.\nNote that despite having the same dimension $k$, $\\mathbf{Z}$ and $\\mathbf{W}$ still reside in different subspaces.\nAs with $\\mathbf{Y}$, we split the low-fidelity data into $\\mathbf{W} = [\\mathbf{W}_\\text{L}, \\mathbf{W}_\\text{U}]$, where $\\mathbf{W}_\\text{L} \\in \\mathbb{R}^{k \\times n}$ and $\\mathbf{W}_\\text{U} \\in \\mathbb{R}^{k \\times (m - n)}$ are the linked and unlinked low-fidelity POD coefficients.\n\nThen, the POD coefficients of both datasets are aligned using Procrustes analysis~\\cite{Gower2010}.\nThe goal of this step is to find an affine transformation of $\\mathbf{W}_\\text{L}$ that would optimally align it with $\\mathbf{Z}$.\nThis is expressed by the following objective\n\\begin{equation}\n\\begin{split}\n    \\min_{s, \\mathbf{t}, \\mathbf{Q}} \\quad& \\lVert \\mathbf{Z} - s \\mathbf{Q} (\\mathbf{W}_\\text{L} -\\mathbf{t}) \\lVert_F \\\\\n    \\text{s.t.} \\quad&  \\mathbf{Q}^T \\mathbf{Q} = \\mathbf{I}\n\\end{split}\n\\end{equation}\nwhere $s$ is an isotropic scaling factor, $\\mathbf{t} \\in \\mathbb{R}^{k}$ is a translation vector, and $\\mathbf{Q} \\in \\mathbb{R}^{k \\times k}$ is an orthogonal transformation matrix.\n\nThe optimal $\\mathbf{t}$ is obtained by shifting $\\textbf{W}_\\text{L}$ such that its centroid coincides with that of $\\textbf{Z}$.\nIt is assumed that $\\mathbf{X}$ has been centered prior to computing the POD such that the mean of $\\mathbf{Z}$ is also zero.\nThus, the optimal $\\mathbf{t}$ is given by\n\\begin{equation}\n    \\textbf{t} = \\frac{1}{n}\\sum_{j=1}^{n} \\mathbf{w}_j\n\\end{equation}\nFor convenience, $\\textbf{W}'_\\text{L} = \\textbf{W}_\\text{L} - \\textbf{t}$ is defined as the shifted low-fidelity latent variable.\n\nThe remaining optimal parameters of the Procrustes analysis are then obtained by the following relations\n\\begin{align}\n    s &= \\frac{\\text{tr}(\\boldsymbol{\\Sigma})}{\\text{tr}\\left(\\mathbf{W}'_\\text{L} (\\mathbf{W}'_\\text{L})^T\\right)} \\\\\n    \\textbf{Q} &= \\textbf{V} \\textbf{U}^T\n\\end{align}\nIn the results shown above, the matrices $\\mathbf{U}$, $\\boldsymbol{\\Sigma}$, and $\\mathbf{V}$, are the outputs of the following singular value decomposition (SVD)\n\\begin{equation}\n    \\textbf{U} \\boldsymbol{\\Sigma} \\textbf{V}^T = \\textbf{W}'_\\text{L} \\textbf{Z}^T\n\\end{equation}\nFinally, the optimal transformation obtained using $\\mathbf{W}_\\text{L}$ is applied to all the low-fidelity data $\\mathbf{W}$.\nThe aligned POD coefficients are thus given by\n\\begin{equation}\\label{chap3-eq:z_align}\n    \\textbf{Z}_\\text{lo} = s \\, \\textbf{Q} \\left(\\textbf{W} - \\textbf{t}\\right)\n\\end{equation}\nwhere $\\textbf{Z}_\\text{lo} \\in \\mathbb{R}^{k \\times m}$ contains the low-fidelity POD coefficients mapped onto the high-fidelity POD basis.\nFor the sake of clarity, the high-fidelity data $\\textbf{Z}$ are referred to as $\\textbf{Z}_\\text{hi}$ from here onward.\n\n\\subsection{Hierarchical Kriging}\n\nFollowing the POD and the manifold alignment, the high- and low-fidelity data have been reduced to the latent coordinates $\\textbf{Z}_\\text{hi}$ and $\\textbf{Z}_\\text{lo}$.\nThe multi-fidelity data must then be combined into a regression model in order to predict new field solutions at unseen design parameters.\nIn practice, any multi-fidelity regression technique could be used for this purpose and the literature contains multiple options such as bridge functions~\\cite{Choi2009, Eldred2004, Tang2005} and cokriging~\\cite{Myers1982,Kennedy2000,Han2012}.\nIn this study, the hierarchical Kriging (HK) method~\\cite{Han2012a} is used to remain consistent with previous MA-ROM studies~\\cite{Perron2020,Perron2020a,Decker2021}.\n\nAs defined by Han and G\\\"{o}rtz~\\cite{Han2012a}, the HK formulation is nearly identical to cokriging~\\cite{Kennedy2000}, although the former is marginally simpler to implement and provides a smoother posterior variance prediction.\nGiven a high-fidelity function $g_\\text{hi}: \\mathbf{p} \\mapsto z_\\text{hi}$, where $\\mathbf{p} \\in \\mathbb{R}^b$ is a design parameter vector and $z_\\text{hi}$ is a high-fidelity response, the HK predictor takes the following form\n\\begin{equation}\\label{eq:hk-predictor}\n    \\gtilde_\\text{hi}(\\mathbf{p}) = \\beta\\,\\gtilde_\\text{lo}(\\mathbf{p}) + \\mathbf{w}^T \\mathbf{r}(\\mathbf{p})\n\\end{equation}\nwhere $\\beta$ is a multi-fidelity scaling factor, $\\mathbf{w} \\in \\mathbb{R}^n$ is a weight vector, and $\\mathbf{r}(\\mathbf{p})$ is a kernel function.\nIn this example, low-fidelity information is included in the form of $\\gtilde_\\text{lo}(\\mathbf{p})\\mathbf{p} \\mapsto \\tilde{z}_\\text{lo}$, which is a Kriging predictor of some low-fidelity response.\nLoosely speaking, the HK predictor of equation~\\eqref{eq:hk-predictor} can be viewed as a Kriging model where the response trend is represented by $\\beta\\,\\gtilde_\\text{lo}(\\mathbf{p})$.\nNote that for the current study, both $\\gtilde_\\text{hi}(\\mathbf{p})$ and $\\gtilde_\\text{lo}(\\mathbf{p})$ use a Mat\\'ern~3\/2 kernel and the model parameters are obtained with a maximum likelihood approach.\n\nIn the context of MA-ROM, the HK model is used to predict a new latent coordinate vector $\\textbf{z}_\\text{hi}$ given some design parameters $\\mathbf{p}$.\nSince each coordinate contains $k$ individual components, an equal number of regression models $\\gtilde_\\text{hi}(\\mathbf{p})$ must be trained in order to predict the entire $\\textbf{z}_\\text{hi}$ vector.\nThe prediction of the high-dimensional field output is then given by\n\\begin{equation}\n    \\xtilde = \\sum_{i=1}^{k} \\boldsymbol{\\phi}_i \\, \\gtilde_{i,\\text{hi}}(\\mathbf{p})\n\\end{equation}\nwhere $\\gtilde_{i,\\text{hi}}(\\mathbf{p})$ is the HK model predicting the $i$-th coordinated in the POD latent space associated with the POD mode $\\boldsymbol{\\phi}_i$.\n\n\\subsection{MA-ROM Formulation}\n\nConsider a high- and low-fidelity models defined as $f_x: \\mathbf{p} \\mapsto \\mathbf{x}$ and $f_y: \\mathbf{p} \\mapsto \\mathbf{y}$ respectively.\nThe MA-ROM formulation can be divided into an \\emph{offline} phase where the model is trained, and an \\emph{online} phase where predictions are made using the generated model.\nThe offline phase is summarized through the following steps\n\\setlist[enumerate,1]{label={\\arabic*.}}\n\\begin{enumerate}\n\\item \\textbf{Generate Linked Data:} Select a set of $n$ distinct designs $\\mathbf{P}_\\text{L} = [\\mathbf{p}_1, \\dots, \\mathbf{p}_n] \\in \\mathbb{R}^{b\\times n}$ for the linked data.\n    Then, sample both $f_x$ and $f_y$ to generate the matrices $\\mathbf{X}$ and $\\mathbf{Y}_\\text{L}$.\n\n    \\item \\textbf{Generate Unlinked Data:} Select a set of $m - n$ additional designs $\\mathbf{P}_\\text{U} = [\\mathbf{p}_{n+1}, \\dots, \\mathbf{p}_m] \\in \\mathbb{R}^{b\\times (m-n)}$ for the unlinked data.\n    Ideally, $\\mathbf{P}_\\text{U}$ should not repeat points in $\\mathbf{P}_\\text{L}$.\n    Then, sample only $f_y$ with $\\mathbf{P}_\\text{U}$ to generate the matrix $\\mathbf{Y}_\\text{U}$.\n\n    \\item \\textbf{Align Manifolds:} Apply the Procrustes manifold alignment on $\\mathbf{X}$, $\\mathbf{Y}_\\text{L}$, and $\\mathbf{Y}_\\text{U}$, to obtain $\\mathbf{Z}_\\text{hi}$ and $\\mathbf{Z}_\\text{lo}$. This is divided into the following three sub-steps:\n    \\begin{enumerate}\n        \\item Use POD to extract $\\boldsymbol{\\Phi}_k$ and $\\boldsymbol{\\Psi}_k$, as well as $\\mathbf{Z}_\\text{hi}$ and $\\mathbf{W}$.\n\n        \\item Perform Procrustes analysis on $\\mathbf{Z}_\\text{hi}$ and $\\mathbf{W}_\\text{L}$ to evaluate the optimal value of $s$, $\\mathbf{t}$, and $\\mathbf{Q}$.\n\n        \\item Apply the optimal transformation to $\\mathbf{W}$ to compute $\\mathbf{Z}_\\text{lo}$.\n    \\end{enumerate}\n\n    \\item \\textbf{Fit HK Model:} Train a set of $k$ HK models $\\gbtilde(\\mathbf{p}) = [\\gtilde_1(\\mathbf{p}), \\dots, \\gtilde_k (\\mathbf{p})]$.\n    These models combine the information from both $\\mathbf{Z}_\\text{hi}$ and $\\mathbf{Z}_\\text{lo}$, together with their associated design parameters $\\mathbf{P} = [\\mathbf{P}_\\text{L}, \\mathbf{P}_\\text{U}] \\in\\mathbb{R}^{b\\times m}$.\n\\end{enumerate}\nIt should be noted that in this study, the design samples in steps 1 and 2 are obtained using Latin hypercube sampling (LHS)~\\cite{Santner2003}.\nAs for the online phase, the process is identical to most non-intrusive ROMs and is given by\n\\begin{enumerate}[resume*]\n    \\item \\textbf{Predict Latent Variable:} Given an out-of-sample design parameter $\\mathbf{p}^*$, predict the corresponding latent variable $\\ztilde^*$ using $\\gbtilde(\\mathbf{p})$.\n\n    \\item \\textbf{Reconstruct Field:} Using $\\boldsymbol{\\Phi}_k$ and $\\ztilde^*$, compute the predicted value of the new high-fidelity field $\\xtilde^*$.    \n\\end{enumerate}\n\n\\section{Literature Review}\\label{sec:lit}\n\nSeveral authors have adopted a surrogate-based optimization strategy to perform an aero-structural optimization of an aircraft wing.\nAbdelkader and Stanislaw~\\cite{benaouali2019multidisciplinary} created radial basis function (RBF) models for scalar outputs (e.g., lift, drag, weight, and aggregated structural failure) from the high-fidelity analysis.\nBordogna et al.~\\cite{bordogna2017surrogate} created surrogate models trained with lift and drag coefficients from rigid RANS computations as constraints in optimization.\nRasmussen et al.~\\cite{rasmussen2009optimization} used a polynomial regression to explore a joined-wing aircraft's optimal regions and design trends.\nJoined-wing planform parameters and structural component thicknesses were the inputs to the surrogate model, and the sized total weight was the output.\nCipolla et al.~\\cite{cipolla2021doe} created regression models for predicting the wing structural mass, estimating the maximum von-Mises stress in each main wing, and evaluating the highest wing-tip displacement between the front and rear wings.\nSurrogate models have also been employed in other many-query applications such as reliability-based optimization of aircraft wing structural components~\\cite{li2019reliability,neufeld2010aircraft}, evaluation of certification constraints~\\cite{sarojini2020certification}, or uncertainty quantification of dynamic loads on the airframe~\\cite{duca2018effects}.\n\nIn the literature mentioned so far, surrogate models have been used to approximate scalar quantities as a function of wing planform, geometrical parameters, aerodynamic parameters, and structural thicknesses.\nWhile the wing's weight and global failure criteria provide practical metrics for overall optimization purposes, these aggregated quantities hide many details useful to a designer.\nOne such detail, for instance, is the location of maximum stress or strain and its spatial shifting as design parameters change. \nFurthermore, in an aero-structural context, the full displacement of a wing structure under loads must be provided to an aerodynamic analysis code in order to adequately compute the aerodynamic forces for a given flight shape.\nROMs also have other applications, such as real-time estimation of a vehicle's evolving structural state from sensor data~\\cite{mainini2015surrogate}.\n\nAs for the prediction of structural field quantities, Lieu et al.~\\cite{lieu2006reduced,lieu2007adaptation} proposed a POD-based ROM for CFD-based aeroelastic analysis using the F-16 aircraft as a test case.\nMainini and Willcox~\\cite{mainini2015surrogate} used the POD to construct ROMs to predict high-fidelity FEM solutions for the estimation of structural capability from sensor data.\nROMs have also been extensively leveraged for the prediction of loads.\nBekemeyer and Timme~\\cite{bekemeyer2019flexible} used a subspace projection model reduction to obtain transonic gust loads.\nRipepi et al.~\\cite{ripepi2018reduced} created a ROM of the aerodynamic influence coefficients and used steady CFD solutions to obtain loads preserving non-linearities.\nROMs have also been used to reduce the computational cost of performing flutter and limit cycle oscillations analysis~\\cite{zhang2012efficient}.\nNote that the literature referenced above pertains to single-fidelity ROMs, i.e., constructed using data from one source.\n\nThe idea of leveraging a lower-fidelity analysis to reduce the computational cost of generating surrogate models has been shared by multiple authors in the past.\nFor example, Forrester et al.~\\cite{Forrester2007} used multi-fidelity surrogate models for the aerodynamic optimization of a wing, and Toal~\\cite{Toal2015} followed a similar process for the optimization of a high-pressure compressor.\n\nWhereas multi-fidelity methods are relatively mature for scalar surrogate models, the prediction of field quantities brings additional challenges, as pointed out in Section~\\ref{sec:intro}.\nAs described by Perron~\\cite{Perron2020a}, these discrepancies can be due to differences in grid sizes, grid topologies, and field features.\nA few efforts have attempted to avoid this issue by enforcing the same discretization for both the high- and low-fidelity analysis and have achieved computational savings by using a simplified physics model.\nThis approach was used by Mifsud et al.~\\cite{Mifsud2016, Mifsud2008} for the aerodynamic analysis of a projectile and by Bertram et al.~\\cite{Bertram2018} for the aerodynamic analysis of a road vehicle.\nAlternatively, one could force the high- and low-fidelity field representations to be consistent by mapping all the results on a common grid, assuming that both fidelities have compatible field topologies.\nMalouin et al.~\\cite{Malouin2013} demonstrated this strategy for the analysis of a transonic airfoil.\nSimilarly, Benamara et al.~\\cite{Benamara2016,Benamara2017} applied this approach to the analysis of a compressor blade.\nMore recently, Perron et al.~\\cite{Perron2020a} proposed the MA-ROM method that leverages manifold alignment~\\cite{Ham2005,Wang2009} to resolve discrepancies between the high- and low-fidelity fields.\nThe effectiveness of this method was tested with the aerodynamic analysis of an airfoil and a wing.\nFurthermore, results have shown that even when fields with consistent representation are used (i.e., by mapping results to a common grid), the MA-ROM method offered an accuracy on par or superior to previous multi-fidelity ROM methods~\\cite{Perron2020}.\n\nAlthough multi-fidelity ROM methods have been successfully applied to aerodynamic problems, the viability of these approaches has not yet been established for structural applications.\nMoreover, topological differences, in terms of model representation and structural layout, are much more prevalent in structural than in aerodynamic problems.\nPrevious data suggest that the MA-ROM is especially well suited for dealing with multi-fidelity structural data with possibly inconsistent representations.\nEmpirically testing this hypothesis is the primary goal of what follows.\n\\section{Introduction}\\label{sec:intro}\n\nThe design and certification of aircraft are increasingly relying on numerical simulation instead of physical experiments owing to the former's relatively rapid turnaround time and high accuracy.\nDespite the increased reliance on numerical simulations, routinely encountered many-query scenarios such as design space exploration, optimization, uncertainty quantification, etc., require repetitive evaluation of computationally expensive codes.\nConsequently, engineering workflows with expensive numerical simulations \\emph{in-the-loop} can suffer from impractical compute times.\nSurrogate models have provided respite to this computational burden by trading the high accuracy of physics-based numerical models for faster evaluation times.\n\nWhile conventional surrogate models have mainly focused on the prediction of scalar quantities~\\cite{Queipo2005, Forrester2009, Yondo2018, Bhosekar2018}, reduced-order models (ROMs) have enabled the prediction of spatially and temporally distributed field quantities.\nThis class of methods reduces the high-dimensional nature of the problem by extracting a low-dimensional subspace (or more generally, latent space) that best captures the most dominant physical features of the solution space.\nIn a sense, by preserving and capturing the underlying spatial and temporal correlation within the high-dimensional solutions, ROMs can offer physically richer predictions.\nThis trait renders ROMs useful in enabling rapid multidisciplinary analyses where high-dimensional coupling variables must be passed between simulations to achieve consistency between, for instance, the aerodynamics and the structures disciplines.\n\nThe construction of surrogate models for both scalar and field quantities suffers from a major drawback when the underlying response is highly nonlinear and consists of many independent parameters.\nIn such cases, most methods require a large amount of training data to build a model with reasonable predictive accuracy.\nTraining data generated from high-fidelity simulations usually incur a substantial computational cost, which may arguably defeat the purpose of constructing the surrogate model.\n\n\nCompared to scalar quantities, the multi-fidelity combination of fields brings more challenges because of the potential inconsistency between results produced by different analyses~\\cite{Perron2020}.\nFor instance, it is well known that a fine mesh FEM model generally produces more accurate results than a coarse mesh FEM model but at a higher computational cost.\nAlthough fine mesh and coarse mesh FEM models represent the same structure, they have inconsistent field outputs.\nThis inconsistency non-trivializes the process of combining their high-dimensional results into a single predictive ROM.\n\nRecent work by Perron et al.~\\cite{Perron2020, Perron2020a, perron2021multi} introduced a novel ROM method based on manifold alignment (MA-ROM) to overcome the issues caused by inconsistent multi-fidelity field outputs.\nThe proposed method employs manifold alignment to find a common low-dimensional representation of two datasets composed of heterogeneous high-dimensional field outputs.\nThe method's capabilities and performance in comparison to other multi-fidelity ROM methods were demonstrated on aerodynamic problems, and the results have shown that multi-fidelity fields having distinct sizes and topologies could be readily combined into a single model with superior predictive accuracy.\nIn an effort to investigate the effectiveness of MA-ROM against the unique challenges presented by structural simulations, this work primarily focuses on empirically assessing the performance of the MA-ROM by constructing surrogates on high-dimensional outputs from related yet distinct 3D structural analyses.\n\nThe Common Research Model (CRM) wing geometry serves as a testbed to showcase the method.\nThis demonstration aims to construct multi-fidelity ROMs with wing planform shape parameters as input and structural field responses as outputs for two challenging multi-fidelity scenarios.\nThe first scenario captures the differences in the outputs arising from the coarse and fine mesh FEM representation discussed earlier.\nFew samples from the more accurate but expensive fine mesh field are combined with many coarse mesh samples.\nThe second scenario is a novel demonstration that tackles field outputs from models with inconsistent topologies;~one, a primary dataset from a geometry with a high rib count, and two, an auxiliary dataset from a low rib count wing.\nThis scenario is designed to test whether the MA-ROM can \\emph{warm-start} the training of a prediction model using existing data from a related, yet different problem.\nResults show that the proposed MA-ROM can effectively combine field outputs having inconsistent dimensions and topologies from both the scenarios described above.\nThe method produces relatively more accurate results when applied to the displacement rather than the von-Mises stress field.\n\nThe remainder of the manuscript is organized as follows.\nSection~\\ref{sec:lit} briefly discusses multi-fidelity surrogate models and ROMs in the realm of structural analysis.\nSection~\\ref{sec:formulation} reviews the conventional POD-based ROM and presents a succinct summary of the multi-fidelity ROM method based on manifold alignment.\nSection~\\ref{sec:application} introduces the details of the application problem and the two multi-fidelity scenarios considered in this work.\nFinally, Section~\\ref{sec:results} presents the results of the experiments and discusses the performance of the MA-ROM in comparison to a single-fidelity POD and interpolation-based approach.\n\\section{Introduction}\\label{sec1}\n\nThe Introduction section, of referenced text \\cite{bib1} expands on the background of the work (some overlap with the Abstract is acceptable). The introduction should not include subheadings.\n\nSpringer Nature does not impose a strict layout as standard however authors are advised to check the individual requirements for the journal they are planning to submit to as there may be journal-level preferences. When preparing your text please also be aware that some stylistic choices are not supported in full text XML (publication version), including coloured font. These will not be replicated in the typeset article if it is accepted. \n\n\\section{Results}\\label{sec2}\n\nSample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.\n\n\\section{This is an example for first level head---section head}\\label{sec3}\n\n\\subsection{This is an example for second level head---subsection head}\\label{subsec2}\n\n\\subsubsection{This is an example for third level head---subsubsection head}\\label{subsubsec2}\n\nSample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. \n\n\\section{Equations}\\label{sec4}\n\nEquations in \\LaTeX\\ can either be inline or on-a-line by itself (``display equations''). For\ninline equations use the \\verb+$...$+ commands. E.g.: The equation\n$H\\psi = E \\psi$ is written via the command \\verb+$H \\psi = E \\psi$+.\n\nFor display equations (with auto generated equation numbers)\none can use the equation or align environments:\n\\begin{equation}\n\\|\\tilde{X}(k)\\|^2 \\leq\\frac{\\sum\\limits_{i=1}^{p}\\left\\|\\tilde{Y}_i(k)\\right\\|^2+\\sum\\limits_{j=1}^{q}\\left\\|\\tilde{Z}_j(k)\\right\\|^2 }{p+q}.\\label{eq1}\n\\end{equation}\nwhere,\n\\begin{align}\nD_\\mu &=  \\partial_\\mu - ig \\frac{\\lambda^a}{2} A^a_\\mu \\nonumber \\\\\nF^a_{\\mu\\nu} &= \\partial_\\mu A^a_\\nu - \\partial_\\nu A^a_\\mu + g f^{abc} A^b_\\mu A^a_\\nu \\label{eq2}\n\\end{align}\nNotice the use of \\verb+\\nonumber+ in the align environment at the end\nof each line, except the last, so as not to produce equation numbers on\nlines where no equation numbers are required. The \\verb+\\label{}+ command\nshould only be used at the last line of an align environment where\n\\verb+\\nonumber+ is not used.\n\\begin{equation}\nY_\\infty = \\left( \\frac{m}{\\textrm{GeV}} \\right)^{-3}\n    \\left[ 1 + \\frac{3 \\ln(m\/\\textrm{GeV})}{15}\n    + \\frac{\\ln(c_2\/5)}{15} \\right]\n\\end{equation}\nThe class file also supports the use of \\verb+\\mathbb{}+, \\verb+\\mathscr{}+ and\n\\verb+\\mathcal{}+ commands. As such \\verb+\\mathbb{R}+, \\verb+\\mathscr{R}+\nand \\verb+\\mathcal{R}+ produces $\\mathbb{R}$, $\\mathscr{R}$ and $\\mathcal{R}$\nrespectively (refer Subsubsection~\\ref{subsubsec2}).\n\n\\section{Tables}\\label{sec5}\n\nTables can be inserted via the normal table and tabular environment. To put\nfootnotes inside tables you should use \\verb+\\footnotetext[]{...}+ tag.\nThe footnote appears just below the table itself (refer Tables~\\ref{tab1} and \\ref{tab2}). \nFor the corresponding footnotemark use \\verb+\\footnotemark[...]+\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{minipage}{174pt}\n\\caption{Caption text}\\label{tab1}%\n\\begin{tabular}{@{}llll@{}}\n\\toprule\nColumn 1 & Column 2  & Column 3 & Column 4\\\\\n\\midrule\nrow 1    & data 1   & data 2  & data 3  \\\\\nrow 2    & data 4   & data 5\\footnotemark[1]  & data 6  \\\\\nrow 3    & data 7   & data 8  & data 9\\footnotemark[2]  \\\\\n\\botrule\n\\end{tabular}\n\\footnotetext{Source: This is an example of table footnote. This is an example of table footnote.}\n\\footnotetext[1]{Example for a first table footnote. This is an example of table footnote.}\n\\footnotetext[2]{Example for a second table footnote. This is an example of table footnote.}\n\\end{minipage}\n\\end{center}\n\\end{table}\n\n\\noindent\nThe input format for the above table is as follows:\n\n\\bigskip\n\\begin{verbatim}\n\\begin{table}[<placement-specifier>]\n\\begin{center}\n\\begin{minipage}{<preferred-table-width>}\n\\caption{<table-caption>}\\label{<table-label>}%\n\\begin{tabular}{@{}llll@{}}\n\\toprule\nColumn 1 & Column 2 & Column 3 & Column 4\\\\\n\\midrule\nrow 1 & data 1 & data 2\t & data 3 \\\\\nrow 2 & data 4 & data 5\\footnotemark[1] & data 6 \\\\\nrow 3 & data 7 & data 8\t & data 9\\footnotemark[2]\\\\\n\\botrule\n\\end{tabular}\n\\footnotetext{Source: This is an example of table footnote. \nThis is an example of table footnote.}\n\\footnotetext[1]{Example for a first table footnote.\nThis is an example of table footnote.}\n\\footnotetext[2]{Example for a second table footnote. \nThis is an example of table footnote.}\n\\end{minipage}\n\\end{center}\n\\end{table}\n\\end{verbatim}\n\\bigskip\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{minipage}{\\textwidth}\n\\caption{Example of a lengthy table which is set to full textwidth}\\label{tab2}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}lcccccc@{\\extracolsep{\\fill}}}\n\\toprule%\n& \\multicolumn{3}{@{}c@{}}{Element 1\\footnotemark[1]} & \\multicolumn{3}{@{}c@{}}{Element 2\\footnotemark[2]} \\\\\\cmidrule{2-4}\\cmidrule{5-7}%\nProject & Energy & $\\sigma_{calc}$ & $\\sigma_{expt}$ & Energy & $\\sigma_{calc}$ & $\\sigma_{expt}$ \\\\\n\\midrule\nElement 3  & 990 A & 1168 & $1547\\pm12$ & 780 A & 1166 & $1239\\pm100$\\\\\nElement 4  & 500 A & 961  & $922\\pm10$  & 900 A & 1268 & $1092\\pm40$\\\\\n\\botrule\n\\end{tabular*}\n\\footnotetext{Note: This is an example of table footnote. This is an example of table footnote this is an example of table footnote this is an example of~table footnote this is an example of table footnote.}\n\\footnotetext[1]{Example for a first table footnote.}\n\\footnotetext[2]{Example for a second table footnote.}\n\\end{minipage}\n\\end{center}\n\\end{table}\n\nIn case of double column layout, tables which do not fit in single column width should be set to full text width. For this, you need to use \\verb+\\begin{table*}+ \\verb+...+ \\verb+\\end{table*}+ instead of \\verb+\\begin{table}+ \\verb+...+ \\verb+\\end{table}+ environment. Lengthy tables which do not fit in textwidth should be set as rotated table. For this, you need to use \\verb+\\begin{sidewaystable}+ \\verb+...+ \\verb+\\end{sidewaystable}+ instead of \\verb+\\begin{table*}+ \\verb+...+ \\verb+\\end{table*}+ environment. This environment puts tables rotated to single column width. For tables rotated to double column width, use \\verb+\\begin{sidewaystable*}+ \\verb+...+ \\verb+\\end{sidewaystable*}+.\n\n\\begin{sidewaystable}\n\\sidewaystablefn%\n\\begin{center}\n\\begin{minipage}{\\textheight}\n\\caption{Tables which are too long to fit, should be written using the ``sidewaystable'' environment as shown here}\\label{tab3}\n\\begin{tabular*}{\\textheight}{@{\\extracolsep{\\fill}}lcccccc@{\\extracolsep{\\fill}}}\n\\toprule%\n& \\multicolumn{3}{@{}c@{}}{Element 1\\footnotemark[1]}& \\multicolumn{3}{@{}c@{}}{Element\\footnotemark[2]} \\\\\\cmidrule{2-4}\\cmidrule{5-7}%\nProjectile & Energy\t& $\\sigma_{calc}$ & $\\sigma_{expt}$ & Energy & $\\sigma_{calc}$ & $\\sigma_{expt}$ \\\\\n\\midrule\nElement 3 & 990 A & 1168 & $1547\\pm12$ & 780 A & 1166 & $1239\\pm100$ \\\\\nElement 4 & 500 A & 961  & $922\\pm10$  & 900 A & 1268 & $1092\\pm40$ \\\\\nElement 5 & 990 A & 1168 & $1547\\pm12$ & 780 A & 1166 & $1239\\pm100$ \\\\\nElement 6 & 500 A & 961  & $922\\pm10$  & 900 A & 1268 & $1092\\pm40$ \\\\\n\\botrule\n\\end{tabular*}\n\\footnotetext{Note: This is an example of table footnote this is an example of table footnote this is an example of table footnote this is an example of~table footnote this is an example of table footnote.}\n\\footnotetext[1]{This is an example of table footnote.}\n\\end{minipage}\n\\end{center}\n\\end{sidewaystable}\n\n\\section{Figures}\\label{sec6}\n\nAs per the \\LaTeX\\ standards you need to use eps images for \\LaTeX\\ compilation and \\verb+pdf\/jpg\/png+ images for \\verb+PDFLaTeX+ compilation. This is one of the major difference between \\LaTeX\\ and \\verb+PDFLaTeX+. Each image should be from a single input .eps\/vector image file. Avoid using subfigures. The command for inserting images for \\LaTeX\\ and \\verb+PDFLaTeX+ can be generalized. The package used to insert images in \\verb+LaTeX\/PDFLaTeX+ is the graphicx package. Figures can be inserted via the normal figure environment as shown in the below example:\n\n\\bigskip\n\\begin{verbatim}\n\\begin{figure}[<placement-specifier>]\n\\centering\n\\includegraphics{<eps-file>}\n\\caption{<figure-caption>}\\label{<figure-label>}\n\\end{figure}\n\\end{verbatim}\n\\bigskip\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[width=0.9\\textwidth]{fig.eps}\n\\caption{This is a widefig. This is an example of long caption this is an example of long caption  this is an example of long caption this is an example of long caption}\\label{fig1}\n\\end{figure}\n\nIn case of double column layout, the above format puts figure captions\/images to single column width. To get spanned images, we need to provide \\verb+\\begin{figure*}+ \\verb+...+ \\verb+\\end{figure*}+.\n\nFor sample purpose, we have included the width of images in the optional argument of \\verb+\\includegraphics+ tag. Please ignore this. \n\n\\section{Algorithms, Program codes and Listings}\\label{sec7}\n\nPackages \\verb+algorithm+, \\verb+algorithmicx+ and \\verb+algpseudocode+ are used for setting algorithms in \\LaTeX\\ using the format:\n\n\\bigskip\n\\begin{verbatim}\n\\begin{algorithm}\n\\caption{<alg-caption>}\\label{<alg-label>}\n\\begin{algorithmic}[1]\n. . .\n\\end{algorithmic}\n\\end{algorithm}\n\\end{verbatim}\n\\bigskip\n\nYou may refer above listed package documentations for more details before setting \\verb+algorithm+ environment. For program codes, the ``program'' package is required and the command to be used is \\verb+\\begin{program}+ \\verb+...+ \\verb+\\end{program}+. A fast exponentiation procedure:\n\n\\begin{program}\n\\BEGIN \\\\ %\n  \\FOR i:=1 \\TO 10 \\STEP 1 \\DO\n     |expt|(2,i); \\\\ |newline|() \\OD %\n\\rcomment{Comments will be set flush to the right margin}\n\\WHERE\n\\PROC |expt|(x,n) \\BODY\n          z:=1;\n          \\DO \\IF n=0 \\THEN \\EXIT \\FI;\n             \\DO \\IF |odd|(n) \\THEN \\EXIT \\FI;\n\\COMMENT{This is a comment statement};\n                n:=n\/2; x:=x*x \\OD;\n             \\{ n>0 \\};\n             n:=n-1; z:=z*x \\OD;\n          |print|(z) \\ENDPROC\n\\END\n\\end{program}\n\n\n\\begin{algorithm}\n\\caption{Calculate $y = x^n$}\\label{algo1}\n\\begin{algorithmic}[1]\n\\Require $n \\geq 0 \\vee x \\neq 0$\n\\Ensure $y = x^n$ \n\\State $y \\Leftarrow 1$\n\\If{$n < 0$}\\label{algln2}\n        \\State $X \\Leftarrow 1 \/ x$\n        \\State $N \\Leftarrow -n$\n\\Else\n        \\State $X \\Leftarrow x$\n        \\State $N \\Leftarrow n$\n\\EndIf\n\\While{$N \\neq 0$}\n        \\If{$N$ is even}\n            \\State $X \\Leftarrow X \\times X$\n            \\State $N \\Leftarrow N \/ 2$\n        \\Else[$N$ is odd]\n            \\State $y \\Leftarrow y \\times X$\n            \\State $N \\Leftarrow N - 1$\n        \\EndIf\n\\EndWhile\n\\end{algorithmic}\n\\end{algorithm}\n\\bigskip\n\nSimilarly, for \\verb+listings+, use the \\verb+listings+ package. \\verb+\\begin{lstlisting}+ \\verb+...+ \\verb+\\end{lstlisting}+ is used to set environments similar to \\verb+verbatim+ environment. Refer to the \\verb+lstlisting+ package documentation for more details.\n\n\\bigskip\n\\begin{minipage}{\\hsize}%\n\\lstset{frame=single,framexleftmargin=-1pt,framexrightmargin=-17pt,framesep=12pt,linewidth=0.98\\textwidth,language=pascal\n\\begin{lstlisting}\nfor i:=maxint to 0 do\nbegin\n{ do nothing }\nend;\nWrite('Case insensitive ');\nWrite('Pascal keywords.');\n\\end{lstlisting}\n\\end{minipage}\n\n\\section{Cross referencing}\\label{sec8}\n\nEnvironments such as figure, table, equation and align can have a label\ndeclared via the \\verb+\\label{#label}+ command. For figures and table\nenvironments use the \\verb+\\label{}+ command inside or just\nbelow the \\verb+\\caption{}+ command. You can then use the\n\\verb+\\ref{#label}+ command to cross-reference them. As an example, consider\nthe label declared for Figure~\\ref{fig1} which is\n\\verb+\\label{fig1}+. To cross-reference it, use the command \n\\verb+Figure \\ref{fig1}+, for which it comes up as\n``Figure~\\ref{fig1}''. \n\nTo reference line numbers in an algorithm, consider the label declared for the line number 2 of Algorithm~\\ref{algo1} is \\verb+\\label{algln2}+. To cross-reference it, use the command \\verb+\\ref{algln2}+ for which it comes up as line~\\ref{algln2} of Algorithm~\\ref{algo1}.\n\n\\subsection{Details on reference citations}\\label{subsec7}\n\nStandard \\LaTeX\\ permits only numerical citations. To support both numerical and author-year citations this template uses \\verb+natbib+ \\LaTeX\\ package. For style guidance please refer to the template user manual.\n\nHere is an example for \\verb+\\cite{...}+: \\cite{bib1}. Another example for \\verb+\\citep{...}+: \\citep{bib2}. For author-year citation mode, \\verb+\\cite{...}+ prints Jones et al. (1990) and \\verb+\\citep{...}+ prints (Jones et al., 1990).\n\nAll cited bib entries are printed at the end of this article: \\cite{bib3}, \\cite{bib4}, \\cite{bib5}, \\cite{bib6}, \\cite{bib7}, \\cite{bib8}, \\cite{bib9}, \\cite{bib10}, \\cite{bib11} and \\cite{bib12}.\n\n\\section{Examples for theorem like environments}\\label{sec10}\n\nFor theorem like environments, we require \\verb+amsthm+ package. There are three types of predefined theorem styles exists---\\verb+thmstyleone+, \\verb+thmstyletwo+ and \\verb+thmstylethree+ \n\n\\bigskip\n\\begin{tabular}{|l|p{19pc}|}\n\\hline\n\\verb+thmstyleone+ & Numbered, theorem head in bold font and theorem text in italic style \\\\\\hline\n\\verb+thmstyletwo+ & Numbered, theorem head in roman font and theorem text in italic style \\\\\\hline\n\\verb+thmstylethree+ & Numbered, theorem head in bold font and theorem text in roman style \\\\\\hline\n\\end{tabular}\n\\bigskip\n\nFor mathematics journals, theorem styles can be included as shown in the following examples:\n\n\\begin{theorem}[Theorem subhead]\\label{thm1}\nExample theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. \nExample theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. \nExample theorem text. \n\\end{theorem}\n\nSample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.\n\n\\begin{proposition}\nExample proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. \nExample proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. \n\\end{proposition}\n\nSample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.\n\n\\begin{example}\nPhasellus adipiscing semper elit. Proin fermentum massa\nac quam. Sed diam turpis, molestie vitae, placerat a, molestie nec, leo. Maecenas lacinia. Nam ipsum ligula, eleifend\nat, accumsan nec, suscipit a, ipsum. Morbi blandit ligula feugiat magna. Nunc eleifend consequat lorem. \n\\end{example}\n\nSample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.\n\n\\begin{remark}\nPhasellus adipiscing semper elit. Proin fermentum massa\nac quam. Sed diam turpis, molestie vitae, placerat a, molestie nec, leo. Maecenas lacinia. Nam ipsum ligula, eleifend\nat, accumsan nec, suscipit a, ipsum. Morbi blandit ligula feugiat magna. Nunc eleifend consequat lorem. \n\\end{remark}\n\nSample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.\n\n\\begin{definition}[Definition sub head]\nExample definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. \n\\end{definition}\n\nAdditionally a predefined ``proof'' environment is available: \\verb+\\begin{proof}+ \\verb+...+ \\verb+\\end{proof}+. This prints a ``Proof'' head in italic font style and the ``body text'' in roman font style with an open square at the end of each proof environment. \n\n\\begin{proof}\nExample for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. \n\\end{proof}\n\nSample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.\n\n\\begin{proof}[Proof of Theorem~{\\upshape\\ref{thm1}}]\nExample for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. \n\\end{proof}\n\n\\noindent\nFor a quote environment, use \\verb+\\begin{quote}...\\end{quote}+\n\\begin{quote}\nQuoted text example. Aliquam porttitor quam a lacus. Praesent vel arcu ut tortor cursus volutpat. In vitae pede quis diam bibendum placerat. Fusce elementum\nconvallis neque. Sed dolor orci, scelerisque ac, dapibus nec, ultricies ut, mi. Duis nec dui quis leo sagittis commodo.\n\\end{quote}\n\nSample body text. Sample body text. Sample body text. Sample body text. Sample body text (refer Figure~\\ref{fig1}). Sample body text. Sample body text. Sample body text (refer Table~\\ref{tab3}). \n\n\\section{Methods}\\label{sec11}\n\nTopical subheadings are allowed. Authors must ensure that their Methods section includes adequate experimental and characterization data necessary for others in the field to reproduce their work. Authors are encouraged to include RIIDs where appropriate. \n\n\\textbf{Ethical approval declarations} (only required where applicable) Any article reporting experiment\/s carried out on (i)~live vertebrate (or higher invertebrates), (ii)~humans or (iii)~human samples must include an unambiguous statement within the methods section that meets the following requirements: \n\n\\begin{enumerate}[1.]\n\\item Approval: a statement which confirms that all experimental protocols were approved by a named institutional and\/or licensing committee. Please identify the approving body in the methods section\n\n\\item Accordance: a statement explicitly saying that the methods were carried out in accordance with the relevant guidelines and regulations\n\n\\item Informed consent (for experiments involving humans or human tissue samples): include a statement confirming that informed consent was obtained from all participants and\/or their legal guardian\/s\n\\end{enumerate}\n\nIf your manuscript includes potentially identifying patient\/participant information, or if it describes human transplantation research, or if it reports results of a clinical trial then  additional information will be required. Please visit (\\url{https:\/\/www.nature.com\/nature-research\/editorial-policies}) for Nature Portfolio journals, (\\url{https:\/\/www.springer.com\/gp\/authors-editors\/journal-author\/journal-author-helpdesk\/publishing-ethics\/14214}) for Springer Nature journals, or (\\url{https:\/\/www.biomedcentral.com\/getpublished\/editorial-policies\\#ethics+and+consent}) for BMC.\n\n\\section{Discussion}\\label{sec12}\n\nDiscussions should be brief and focused. In some disciplines use of Discussion or `Conclusion' is interchangeable. It is not mandatory to use both. Some journals prefer a section `Results and Discussion' followed by a section `Conclusion'. Please refer to Journal-level guidance for any specific requirements. \n\n\\section{Conclusion}\\label{sec13}\n\nConclusions may be used to restate your hypothesis or research question, restate your major findings, explain the relevance and the added value of your work, highlight any limitations of your study, describe future directions for research and recommendations. \n\nIn some disciplines use of Discussion or 'Conclusion' is interchangeable. It is not mandatory to use both. Please refer to Journal-level guidance for any specific requirements. \n\n\n\\section*{Declarations}\n\n\\bmhead{Supplementary information}\n\nThis article has the following accompanying material that will be made available after receiving peer reviews: 1) Computer program that was developed for constructing the predictive model; 2) Data that was used to construct the models.\n\n\\bmhead{Funding}\n\nNo funding, grants, or other monetary support was received. \n\n\\bmhead{Conflict of Interest\/Competing Interests}\n\nThe authors declare that there is no conflict of interest.\n\n\\bmhead{Availability of Data and Materials}\n\nThe data and accompanying material used to generate results presented in this work will be made available as electronic supplementary material. \n\n\\bmhead{Code Availability}\n\nThe computer program that implements the method presented in this work will be made available through a repository link. \n\n\\bmhead{Financial\/Non-Financial Interests}\n\nThe authors have no relevant financial or non-financial interests to disclose. \n\n\n\n\n\n\n\n\n\n\n\\input{Parts\/Appendices}\n\n\\bibliographystyle{sn-aps.bst}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\nJean-Michel Rabat\u00e9\n\nLES GUERRES  \nDE JACQUES DERRIDA\n\nLes Presses de l'Universit\u00e9 de Montr\u00e9al\nCatalogage avant publication de Biblioth\u00e8que et Archives nationales du Qu\u00e9bec et Biblioth\u00e8que et Archives Canada\n\nRabat\u00e9, Jean-Michel, 1949-  \nLes guerres de Jacques Derrida  \n(Humanit\u00e9s \u00e0 venir)  \nComprend des r\u00e9f\u00e9rences bibliographiques.  \nISBN 978-2-7606-3681-1  \n1. Derrida, Jacques. 2. Guerre (Philosophie). I. Titre. II. Collection: Humanit\u00e9s \u00e0 venir.  \nB2430.D484R32 2016 194 C2016-940862-0\n\nMise en pages et ePub: Folio infographie\n\nD\u00e9p\u00f4t l\u00e9gal: 3e trimestre 2016  \nBiblioth\u00e8que et Archives nationales du Qu\u00e9bec\n\n\u00a9 Les Presses de l'Universit\u00e9 de Montr\u00e9al, 2016  \nwww.pum.umontreal.ca\n\nISBN (papier) 978-2-7606-3681-1  \nISBN (PDF) 978-2-7606-3682-8  \nISBN (ePub) 978-2-7606-3683-5\n\nLes Presses de l'Universit\u00e9 de Montr\u00e9al remercient de leur soutien financier le Conseil des arts du Canada et la Soci\u00e9t\u00e9 de d\u00e9veloppement des entreprises culturelles du Qu\u00e9bec (SODEC).\n\n# TABLE DES MATI\u00c8RES\n\nLIMINAIRE\n\nREMERCIEMENTS\n\nAVANT-PROPOS\n\nINTRODUCTION\n\nI. S'aguerrir \u00e0 l'Alg\u00e9rie\n\nII. De la dr\u00f4le de guerre \u00e0 la dr\u00f4le de paix\n\nIII. Chien et chat: de Levinas \u00e0 Derrida, la bellig\u00e9rance de l'Autre\n\nIV. Messianisme de guerre ou nihilisme de paix: Une enfance de J\u00e9sus\n\nV. Ma vie a toujours \u00e9t\u00e9 future: grammatologie des temps \u00e0 venir\n\n1. F\u00fbtur, entre nom et verbe\n\n2. L'intendant infid\u00e8le: vers une grammatologie prudentielle\n\n3. Pascal et le futur de l'\u00c9criture\n\n4. \u00abCeux qui vont venir\u00bb et les pr\u00e9curseurs\n\n5. Martingales de Mallarm\u00e9\n\nDU M\u00caME AUTEUR\nCOLLECTION \u00abHUMANIT\u00c9S \u00c0 VENIR\u00bb\n\ndirig\u00e9e par Ginette Michaud et Georges Leroux\n\nEn accueillant des essais brefs et la publication de grandes conf\u00e9rences, cette collection s'engage sur les chemins qu'ouvre aujourd'hui la pens\u00e9e de ce qui vient, de ce qui arrive \u00e0 un monde sans rep\u00e8res. Au confluent de la litt\u00e9rature et de la philosophie, elle inscrit son titre dans la recherche de nouvelles Humanit\u00e9s, libres et plurielles. \n\n# LIMINAIRE\n\nDix ans apr\u00e8s la disparition de Jacques Derrida le 9 octobre 2004, nous avons souhait\u00e9 inscrire un peu autrement son nom dans l'Universit\u00e9, cette universit\u00e9 \u00absans condition\u00bb qu'il appelait de ses v\u0153ux, liant \u00e9troitement sa vocation non seulement \u00e0 la litt\u00e9rature et \u00e0 la philosophie, aux \u00abHumanit\u00e9s de demain\u00bb, mais \u00e0 la d\u00e9mocratie \u00e0 venir.\n\nNous avons donc cr\u00e9\u00e9, avec le concours pr\u00e9cieux de plusieurs instances universitaires, un cycle annuel de Grandes Conf\u00e9rences sous le signe des \u00abM\u00e9moires de Jacques Derrida\u00bb afin d'affirmer, de r\u00e9affirmer la port\u00e9e d'une \u0153uvre philosophique qui a profond\u00e9ment marqu\u00e9 de nombreux domaines de la pens\u00e9e: philosophie, litt\u00e9rature, politique, droit, th\u00e9ologie, esth\u00e9tique et architecture. Si le travail de Jacques Derrida s'est r\u00e9solument engag\u00e9 d\u00e8s ses commencements dans une relecture minutieuse de tous les grands textes de la tradition philosophique, il ne s'est pas content\u00e9 de cette relecture, aussi radicale f\u00fbt-elle: il a aussi voulu contresigner de la mani\u00e8re la plus forte et la plus audacieuse, pour l'avenir, chacune des \u0153uvres auxquelles il s'est attach\u00e9.\n\nCes \u00abM\u00e9moires de Jacques Derrida\u00bb se veulent ainsi une relance de sa pens\u00e9e, une r\u00e9ponse \u00e0 des appels multiples. Comme il l'avait fait pour tant d'auteurs auxquels il \u00e9tait remarquablement fid\u00e8le, nous cherchons ici \u00e0 r\u00e9pondre (\u00e0, de, pour), \u00e0 parler en direction de, vers Derrida. Car, comme il l'\u00e9crivait dans \u00ab\"Justices\"\u00bb,\n\nR\u00e9pondre de la responsabilit\u00e9, et de ce qui la lie et l'oblige \u00e0 la justice, c'est penser la responsabilit\u00e9 en en formulant et en en formalisant la possibilit\u00e9, autant que l'aporie. Responsabilit\u00e9 \u00e9thique (c'est-\u00e0-dire aussi juridique et politique) qui s'expose non seulement dans ce qu'on appelle la vie ou l'existence mais dans la t\u00e2che de d\u00e9chiffrement, de lecture et d'\u00e9criture.\n\nGeorges Leroux et Ginette Michaud\n\n. Jacques Derrida, L'Universit\u00e9 sans condition, Paris, Galil\u00e9e, coll. \u00abIncises\u00bb, 2001, p. 11 sq.\n\n. J. Derrida, \u00ab\"Justices\"\u00bb, dans Appels de Jacques Derrida, Danielle Cohen-Levinas et Ginette Michaud (dir.), Paris, Hermann, coll. \u00abRue de la Sorbonne\u00bb, 2014, p. 60-61.\n\n# REMERCIEMENTS\n\nNous exprimons notre reconnaissance \u00e0 Jean-Michel Rabat\u00e9 qui a g\u00e9n\u00e9reusement accept\u00e9 notre invitation. Nous remercions pour leur soutien pr\u00e9cieux la doyenne par int\u00e9rim de la Facult\u00e9 des arts et des sciences de l'Universit\u00e9 de Montr\u00e9al, Mme Tania Saba, et M. Laurent Lewis, vice-doyen \u00e0 la recherche, \u00e0 la cr\u00e9ation et \u00e0 l'innovation; les directeurs des d\u00e9partements des litt\u00e9ratures de langue fran\u00e7aise et de philosophie de l'Universit\u00e9 de Montr\u00e9al, MM. Beno\u00eet Melan\u00e7on et Louis-Andr\u00e9 Dorion; la directrice du Centre de recherche interuniversitaire sur la litt\u00e9rature et la culture qu\u00e9b\u00e9coises, Mme Martine-Emmanuelle Lapointe, et Mme H\u00e9l\u00e8ne Hotton, coordonnatrice scientifique du CRILCQ; le d\u00e9partement de philosophie de l'Universit\u00e9 du Qu\u00e9bec \u00e0 Montr\u00e9al; de m\u00eame que la librairie Olivieri pour son accueil chaleureux. Merci \u00e9galement \u00e0 Cosmin Popovici-Toma pour son aide dans la mise au point de certaines r\u00e9f\u00e9rences bibliographiques. Enfin, nous remercions vivement M. Patrick Poirier, directeur g\u00e9n\u00e9ral des Presses de l'Universit\u00e9 de Montr\u00e9al, qui a repris l'initiative de M. Antoine Del Busso en cr\u00e9ant cette collection, permettant ainsi de garder une trace, une archive vivante de ce cycle de grandes conf\u00e9rences. \n\n# AVANT-PROPOS\n\n\u00abLes guerres de Jacques Derrida\u00bb est la deuxi\u00e8me \u00abConf\u00e9rence annuelle Jacques Derrida\u00bb, tenue sous les auspices du d\u00e9partement des Litt\u00e9ratures de langue fran\u00e7aise et du Centre de recherche interuniversitaire sur la litt\u00e9rature et la culture qu\u00e9b\u00e9coises (CRILCQ) de l'Universit\u00e9 de Montr\u00e9al.\n\nCe cycle de conf\u00e9rences a \u00e9t\u00e9 inaugur\u00e9 en 2014 par la philosophe et musicologue Danielle Cohen-Levinas, avec sa conf\u00e9rence sur les rapports de Derrida et du po\u00e8te Paul Celan, conf\u00e9rence depuis publi\u00e9e comme le premier titre de la collection \u00abHumanit\u00e9s \u00e0 venir\u00bb. Nous souhaitions, dans ce cycle, garder une certaine alternance entre l'Europe et les \u00c9tats-Unis. Le nom de Jean-Michel Rabat\u00e9 s'est alors imm\u00e9diatement impos\u00e9 \u00e0 nous pour cette deuxi\u00e8me conf\u00e9rence du cycle. Professeur de litt\u00e9rature anglaise et compar\u00e9e \u00e0 l'Universit\u00e9 de Pennsylvanie, \u00e0 Philadelphie, o\u00f9 il enseigne depuis 1992, Jean-Michel Rabat\u00e9 est un \u00e9minent sp\u00e9cialiste des grands \u00e9crivains de la modernit\u00e9 auxquels il a consacr\u00e9, comme auteur ou \u00e9diteur, une quarantaine ouvrages; il est l'un des \u00e9diteurs du Journal of Modern Literature et, tr\u00e8s actif, fait aussi partie des comit\u00e9s de r\u00e9daction de plusieurs revues importantes (Interfaces, The James Joyce Quarterly, The James Joyce Annual, The European Journal of English Studies, English Text Construction et Textual Practice); il est \u00e9galement membre du conseil des presses de l'Universit\u00e9 de Pennsylvanie. L'excellence de son enseignement a \u00e9t\u00e9 salu\u00e9e par plusieurs prix, notamment le Lindback Teaching Award que lui a d\u00e9cern\u00e9 l'Universit\u00e9 de Pennsylvanie en 2011. Jean-Michel Rabat\u00e9 a \u00e9t\u00e9 \u00e9lu membre de l'American Academy of Arts and Sciences en 2008.\n\nEn plus de cet impressionnant parcours acad\u00e9mique, Jean-Michel Rabat\u00e9 s'int\u00e9resse \u00e9galement de pr\u00e8s \u00e0 l'art contemporain: il est le cofondateur de la Fondation Slought, \u00e0 Philadelphie, o\u00f9 il organise expositions et conf\u00e9rences; mentionnons au passage un tr\u00e8s int\u00e9ressant DVD qu'il a co\u00e9dit\u00e9 en 2014: Art, or Listen to the Silence, o\u00f9 l'artiste cor\u00e9enne Soun-Gui Kim s'entretient avec Jacques Derrida, Jean-Luc Nancy et John Cage.\n\nMais bien s\u00fbr Jean-Michel Rabat\u00e9 est surtout reconnu comme l'un des plus fins Joyce scholars, un \u00e9crivain qui n'a cess\u00e9 de l'accompagner depuis trente ans, comme en t\u00e9moignent ses nombreux livres, parus tant en anglais qu'en fran\u00e7ais, qui ont marqu\u00e9 le domaine des \u00e9tudes joyciennes, qu'on pense \u00e0 son Portrait de l'auteur en autre lecteur (1984), \u00e0 James Joyce: Authorized Reader (1991), \u00e0 Joyce upon the Void: the Genesis of Doubt (1991), \u00e0 son excellente introduction \u00e0 l'\u0153uvre de l'\u00e9crivain parue en fran\u00e7ais et simplement intitul\u00e9e James Joyce (Hachette, 1993), ou encore \u00e0 son \u00e9tude James Joyce and the Politics of Egoism (2001). Jean-Michel Rabat\u00e9 a particip\u00e9 \u00e0 l'\u00e9dition d'Ulysse dans la prestigieuse collection de la \u00abBiblioth\u00e8que de la Pl\u00e9iade\u00bb (1995) et il a \u00e9dit\u00e9 la nouvelle traduction d'Exils par Jean-Michel Desprats, qui est parue chez Gallimard en 2012. On lui a aussi confi\u00e9 l'\u00e9dition des Works de James Joyce parue chez Anthem Press en 2016.\n\nOutre Joyce (mais y a-t-il un \u00aboutre\u00bb Joyce?), Jean-Michel Rabat\u00e9 a \u00e9galement publi\u00e9 de nombreux essais sur Beckett, Pound, Thomas Bernhard, entre autres \u00e9crivains, de m\u00eame que sur la modernit\u00e9 en g\u00e9n\u00e9ral: citons La beaut\u00e9 am\u00e8re (1986), La p\u00e9nulti\u00e8me est morte. Spectrographies de la modernit\u00e9 (1993) et, plus r\u00e9cemment, ses analyses approfondies portant sur deux ann\u00e9es cl\u00e9s du XXe si\u00e8cle: 1913: The Cradle of Modernism (2007) et 1922: Culture, Politics and Literature (2015).\n\nUn aspect singulier de son travail r\u00e9side dans l'\u00e9coute attentive qu'il a toujours accord\u00e9e \u00e0 la psychanalyse, notamment dans son rapport intime \u00e0 la langue et \u00e0 la traduction: il est \u00e0 ce titre sans conteste l'un des meilleurs lecteurs de Lacan, ce dont t\u00e9moignent les ouvrages collectifs qu'il a dirig\u00e9s, Lacan in America (2000) et Jacques Lacan, paru chez Bayard en 2005, de m\u00eame que son \u00e9tude Jacques Lacan: Psychoanalysis and the Subject of Literature (2001). Il a \u00e9galement \u00e9t\u00e9 responsable de The Cambridge Companion to Jacques Lacan (2003) et a publi\u00e9 en 2014 The Cambridge Introduction to Literature and Psychoanalysis. Jean-Michel Rabat\u00e9 a par ailleurs \u00e9crit sur une foule d'autres sujets, toujours passionnants: sur la th\u00e9orie (The Future of Theory, 2002; Crimes of the Future: Theory and its Global Reproduction, 2014), l'image et le cin\u00e9ma (Writing the Image after Roland Barthes, 1997), les logiques du mensonge (The Ethics of the Lie, 2007) et le crime (Given: 1\u00ba Art, 2\u00ba Crime, 2006, \u00e9galement paru en fran\u00e7ais en 2010). Et, en 2016, il a publi\u00e9 deux livres: The Pathos of Distance, sur les affects des Modernes, et Think, Pig! Beckett at the Limit of the Human.\n\nQuant \u00e0 la relation que Jean-Michel Rabat\u00e9 entretient avec l'\u0153uvre de Derrida, il est l'un de ses plus fid\u00e8les et justes lecteurs, \u00e9vitant le double \u00e9cueil de l'apologie ou de la critique ironique: il sait, au contraire, toujours se tenir \u00e0 juste distance, comme en fait foi cet essai o\u00f9 il analyse, de mani\u00e8re \u00e9clairante et nuanc\u00e9e, les d\u00e9bats philosophiques parfois tendus qui ont oppos\u00e9 Derrida \u00e0 Emmanuel Levinas et \u00e0 Giorgio Agamben. Il ne serait pas exag\u00e9r\u00e9 de dire que la pens\u00e9e de Derrida a en quelque sorte \u00abtoujours d\u00e9j\u00e0\u00bb \u00e9t\u00e9 pr\u00e9sente dans la r\u00e9flexion de Jean-Michel Rabat\u00e9 puisqu'il a eu la chance d'\u00eatre form\u00e9, d\u00e8s sa th\u00e8se sous la direction d'H\u00e9l\u00e8ne Cixous, \u00e0 sa lecture rigoureuse et, entre autres, \u00e0 l'attention aux langues que commande le travail de Derrida. Parmi ses contributions \u00e0 l'\u0153uvre du philosophe, notons son \u00e9dition en 1992 des Actes du colloque de Royaumont qu'il coorganisa, L'\u00e9thique du don: Jacques Derrida et la question du don et sa participation \u00e0 plusieurs ouvrages collectifs importants, parmi lesquels le Cahier de L'Herne Derrida (2004) et Appels de Jacques Derrida (2014); il signe \u00e9galement un tr\u00e8s int\u00e9ressant texte sur Derrida dans un r\u00e9cent num\u00e9ro d'\u00c9tudes fran\u00e7aises, \u00abToucher des yeux\u00bb (51: 2, 2015), o\u00f9 la question de l'ekphrasis est consid\u00e9r\u00e9e du point de vue de la d\u00e9construction. Enfin, Jean-Michel Rabat\u00e9 est le ma\u00eetre d'\u0153uvre du vaste projet After Derrida pour les presses de l'Universit\u00e9 de Cambridge.\n\n*\n\nEn r\u00e9pondant avec son enjouement caract\u00e9ristique \u00e0 notre invitation et en choisissant d'intituler sa conf\u00e9rence \u00abLes guerres de Derrida\u00bb, Jean-Michel Rabat\u00e9 laissait d\u00e9j\u00e0 r\u00e9sonner le fameux \u00abHe war\u00bb de Finnegans Wake analys\u00e9 par Derrida dans Ulysse gramophone (o\u00f9 Jean-Michel Rabat\u00e9 figure d'ailleurs comme interlocuteur privil\u00e9gi\u00e9 du philosophe). Sans doute avait-il \u00e9galement en t\u00eate cette phrase du dernier entretien, Apprendre \u00e0 vivre enfin, dans lequel Derrida confie qu'il est \u00aben guerre contre [lui]-m\u00eame\u00bb. La magnifique conf\u00e9rence qu'il offrit le 6 novembre 2015 aux auditeurs de la Librairie Olivieri confirma cette intuition, puisque ces deux textes de Derrida comptent en effet parmi ceux qu'il commenta ce soir-l\u00e0. L'essai qu'on lira ici atteste aussi de sa g\u00e9n\u00e9rosit\u00e9, car loin de se contenter d'une simple mise en forme des pages qu'il avait prononc\u00e9es \u00e0 cette occasion, il a souhait\u00e9 leur donner une ampleur, un mouvement, une dynamique propres pour cette publication. Il d\u00e9veloppe ainsi le th\u00e8me (mot qu'on pourrait entendre ici dans sa port\u00e9e musicale) des \u00abguerres\u00bb derridiennes \u00e0 partir de plusieurs sc\u00e8nes diff\u00e9rentes: historique, philosophique et litt\u00e9raire.\n\nRappelant \u00e0 quel point les perceptions au sujet de Derrida restent contradictoires \u00e0 ce sujet \u2013 pour les uns, il \u00e9tait un philosophe fonci\u00e8rement ir\u00e9nique (Christopher Norris et Alain Badiou), pour d'autres, un bagarreur en butte \u00e0 d'incessants d\u00e9bats et provocations (le pass\u00e9 nazi de Heidegger ou l'\u00abaffaire\u00bb Paul de Man, pour n'en \u00e9voquer que deux parmi les plus importants) \u2013, Jean-Michel Rabat\u00e9 s'int\u00e9resse d'abord \u00e0 une longue lettre qu'adressa Jacques Derrida en avril 1961 \u00e0 l'historien Pierre Nora, \u00e0 la suite de la parution de son ouvrage pol\u00e9mique Les Fran\u00e7ais d'Alg\u00e9rie. Cette lettre d'une cinquantaine de pages \u00e9tait demeur\u00e9e in\u00e9dite jusqu'\u00e0 la r\u00e9\u00e9dition du livre de Pierre Nora en 2012; elle est embl\u00e9matique de la position de Derrida touchant \u00e0 l'un des points les plus sensibles de sa pens\u00e9e, son rapport \u00e0 l'Alg\u00e9rie, mais aussi, comme L'\u00c9tranger de Camus se trouve violemment pris \u00e0 partie par Nora, la place pr\u00e9pond\u00e9rante qu'occupe la question de la litt\u00e9rature et son \u00e9thique dans sa philosophie. Dans son analyse de la lettre de Derrida, Rabat\u00e9 d\u00e9c\u00e8le ce qui deviendra l'un des fils rouges de la pens\u00e9e derridienne dans toute l'\u0153uvre, \u00e0 travers les tensions et les conflits auxquels elle aura constamment \u00e0 s'affronter, \u00e0 \u00abs'aguerrir\u00bb pour reprendre son expression: \u00abC'est donc la guerre qu'il faudrait, avant tout, d\u00e9construire. Mais comment d\u00e9construire une structure pol\u00e9mique qui fait partie de la d\u00e9construction elle-m\u00eame?\u00bb\n\nDans le deuxi\u00e8me chapitre, Jean-Michel Rabat\u00e9 interroge les effets des \u00abculture wars\u00bb am\u00e9ricaines des ann\u00e9es soixante-dix et quatre-vingt alors que la d\u00e9construction donnait lieu \u00e0 de v\u00e9ritables guerres de clan, souvent cruelles. Reprenant les choses de plus loin, il montre comment, pour Levinas, les premi\u00e8res impressions que provoqua en lui la triple publication en 1967 des ouvrages de Derrida (De la grammatologie, L'\u00e9criture et la diff\u00e9rence et La voix et le ph\u00e9nom\u00e8ne) furent imm\u00e9diatement associ\u00e9es \u00e0 des sc\u00e8nes de guerre, la Grande Guerre, celle de 1914-18, et aussi la \u00abdr\u00f4le de guerre\u00bb d\u00e9clench\u00e9e par l'envahissement des Allemands. Il analyse ainsi non seulement les Carnets de captivit\u00e9 de Levinas mais aussi comment un certain rapport d'affrontement de Derrida \u00e0 Levinas se met en place, de \u00abViolence et m\u00e9taphysique\u00bb (1964) \u00e0 son ouvrage posthume, L'animal que donc je suis (2006). Relisant les Carnets de captivit\u00e9 et le roman abandonn\u00e9 Eros ou Triste opulence de Levinas, Jean-Michel Rabat\u00e9 souligne comment le \u00abcontexte\u00bb historique de la guerre est pr\u00e9sent dans ce d\u00e9bat philosophique au sujet du \u00abVisage\u00bb, de Dieu (que Derrida consid\u00e8re comme le \u00abPolemos originaire\u00bb) et de la violence du langage. Ces textes plus anciens de Levinas relancent ainsi le dialogue critique entre les deux philosophes (et, dans une moindre mesure, le d\u00e9bat tardif avec Agamben, qui s'attire aussi de s\u00e9rieuses critiques de la part de Derrida dans son dernier s\u00e9minaire La b\u00eate et le souverain) autour de la question de l'animal.\n\nCette importante question philosophique, qui occupera une place de plus en plus centrale dans l'\u0153uvre de Derrida, se retrouve au c\u0153ur du troisi\u00e8me chapitre o\u00f9 l'auteur note que le d\u00e9bat entre les deux philosophes rev\u00eat un aspect quasi tot\u00e9mique, avec, pour Derrida, la figure du chat et pour Levinas, celle du chien Bobby. Jean-Michel Rabat\u00e9 tente de percer les raisons de la s\u00e9v\u00e9rit\u00e9 \u00e9tonnante manifest\u00e9e par Derrida \u00e0 l'endroit de Levinas quant \u00e0 l'animal, alors qu'il n'h\u00e9site pas, dans L'animal que donc je suis, \u00e0 remettre en cause rien moins que la philosophie tout enti\u00e8re de Levinas, celle-ci reposant sur une \u00ab\u00e9thique de l'Autre se d\u00e9ployant \u00e0 partir du th\u00e8me du visage\u00bb. Mais Rabat\u00e9 sugg\u00e8re de mani\u00e8re tr\u00e8s fine qu'il y a aussi dans les carnets de guerre de Levinas une autre pens\u00e9e de l'animal, susceptible de rouvrir la discussion avec Derrida. De fa\u00e7on audacieuse, il trace un parall\u00e8le entre la figure du chien, abandonn\u00e9, sans ma\u00eetre, des Carnets et Albertine, figure \u00e0 la fois \u00e9vanescente et \u00abprisonni\u00e8re\u00bb de la Recherche de Proust (\u00e9crivain qui fascinait particuli\u00e8rement Levinas), qui \u00abse d\u00e9tache de toute consid\u00e9ration morale pour mieux faire sentir l'alt\u00e9rit\u00e9 d'\u00c9ros et son irr\u00e9ductible libert\u00e9\u00bb.\n\nL'importance du langage et de la litt\u00e9rature, de la fiction, se trouve donc toujours au c\u0153ur de toute cette question de la guerre pour Derrida. Les deux derniers chapitres de l'essai vont encore amplifier cet aspect. Jean-Michel Rabat\u00e9 approfondit cette probl\u00e9matique en commentant, \u00e0 partir de multiples exemples litt\u00e9raires \u2013 J. M. Coetzee, mais aussi Mallarm\u00e9, l'\u00e9vangile de Luc revu par Joyce, ou encore Montaigne et Pascal, suivis de Rilke et Heidegger \u2013 \u00able lien intime entre la guerre, l'avenir et leurs modalit\u00e9s discursives\u00bb. Le quatri\u00e8me chapitre sera ainsi consacr\u00e9 \u00e0 une analyse d\u00e9taill\u00e9e du roman de Coetzee, Une enfance de J\u00e9sus, un texte qui travaille de pr\u00e8s avec les motifs derridiens de l'hospitalit\u00e9 et de la question de la langue car, en effet, Le monolinguisme de l'autre r\u00e9sonne fortement avec les pr\u00e9occupations actuelles de l'\u00e9crivain sud-africain. Enfin, dans le dernier chapitre, l'enqu\u00eate conduit \u00e0 une investigation des sens que prennent les mots \u00abavenir\u00bb et \u00abfutur\u00bb dans la pens\u00e9e de Derrida, de m\u00eame que des diff\u00e9rentes valeurs investies dans ces mots selon les langues (le latin, le fran\u00e7ais, l'anglais, l'allemand) et les idiomes po\u00e9tiques. Cette r\u00e9flexion sur les acceptions plurielles des mots \u00abguerre\u00bb et \u00abavenir\u00bb n'aurait pu \u00eatre plus pertinente pour une collection qui, comme la n\u00f4tre, tente pr\u00e9cis\u00e9ment de r\u00e9pondre \u00e0 l'exigence des \u00abHumanit\u00e9s \u00e0 venir\u00bb. \u00abLes guerres de Jacques Derrida\u00bb nous rappellent \u00e0-propos que l'avenir doit toujours \u00eatre repens\u00e9, dans une perspective derridienne, avec le \u00abfutur ant\u00e9rieur\u00bb, ce temps qui aura toujours \u00e9t\u00e9 privil\u00e9gi\u00e9 par Jacques Derrida.\n\nGinette Michaud\n\n. Danielle Cohen-Levinas, Le devenir-juif du po\u00e8me. Double envoi: Celan et Derrida, Montr\u00e9al, Les Presses de l'Universit\u00e9 de Montr\u00e9al, coll. \u00abHumanit\u00e9s \u00e0 venir\u00bb, 2015 (\u00e0 para\u00eetre en traduction italienne, Rome, Inschibboleth Edizioni, en 2017).\n\n. J. Derrida, Apprendre \u00e0 vivre enfin. Entretien avec Jean Birnbaum, Paris, Galil\u00e9e\/Le Monde, coll. \u00abLa philosophie en effet\u00bb, 2005, p. 49.\nJe voudrais exprimer toute ma gratitude \u00e0 Ginette Michaud pour son amiti\u00e9 de longue date et pour son travail remarquable qui m'a toujours inspir\u00e9. Je dois ce livre \u00e0 sa g\u00e9n\u00e9reuse invitation qui m'a amen\u00e9 \u00e0 pr\u00e9senter une premi\u00e8re version des \u00abGuerres de Jacques Derrida\u00bb \u00e0 Montr\u00e9al le 6 novembre 2015.\n\nJ'ai repris dans ce livre, en les r\u00e9\u00e9crivant substantiellement, des passages de \u00ab\"Abandonner son ma\u00eetre pour courir apr\u00e8s n'importe qui...\" Bobby entre Derrida, Agamben et Levinas\u00bb, publi\u00e9 dans Appels de Jacques Derrida (Danielle Cohen-Levinas et Ginette Michaud (dir.), Paris, Hermann, 2014, p. 467-484), et de \u00abThe \"Mujic of the Footure\": Future, Ancient, Fugitive\u00bb paru dans Futures of Jacques Derrida (Richard Rand (\u00e9d.), Stanford, Stanford University Press, 2001, p. 179-200).\n\nLorsque des textes en langues \u00e9trang\u00e8res sont traduits sans mention du traducteur, c'est moi qui traduis.\n\nJean-Michel Rabat\u00e9\n\n# INTRODUCTION\n\nDerrida n'aura pas vraiment surpris ceux qui le connaissaient lorsqu'il d\u00e9clara dans ce qu'on a appel\u00e9 son dernier entretien que son \u00eatre m\u00eame lui semblait pris dans une contradiction perp\u00e9tuelle parce qu'il se sentait fondamentalement \u00eatre \u00aben guerre contre [lui]-m\u00eame\u00bb. Il dit ceci:\n\nEt, en effet, vous retrouverez toujours ce geste chez moi, pour lequel je n'ai pas de justification ultime, sauf que c'est moi, c'est l\u00e0 o\u00f9 je suis. Je suis en guerre contre moi-m\u00eame, c'est vrai, vous ne pouvez pas savoir \u00e0 quel point, au-del\u00e0 de ce que vous devinez, et je dis des choses contradictoires, qui sont, disons, en tension r\u00e9elle, et qui me construisent, me font vivre, et me feront mourir.\n\nDerrida nous donne une bonne d\u00e9monstration de ce que la \u00abd\u00e9construction\u00bb doit \u00e0 la \u00abconstruction\u00bb dans un effort pour garder une certaine tradition vivante tout en la bousculant de l'int\u00e9rieur.\n\n\u00c0 lire la biographie de Beno\u00eet Peeters, on garde plut\u00f4t l'impression que Derrida aura \u00e9t\u00e9 en guerre contre tout le monde. Dans cette solide \u00e9tude qui se lit comme un roman ou plut\u00f4t une riche chronique de la vie culturelle de la seconde moiti\u00e9 du vingti\u00e8me si\u00e8cle, Peeters cite quantit\u00e9 de lettres, de brouillons, de s\u00e9minaires non publi\u00e9s, toute une archive comme celles de l'IMEC et d'Irvine, ce qui nous permet de consid\u00e9rer l'\u0153uvre de Derrida \u00e0 la lumi\u00e8re de sa personnalit\u00e9 complexe. En fin de compte, Derrida appara\u00eet passionn\u00e9, tourment\u00e9, excessif, romantique m\u00eame, un h\u00e9ros de la pens\u00e9e souvent violent et emport\u00e9, toujours pr\u00eat \u00e0 en d\u00e9coudre avec ses adversaires beaucoup plus qu'un penseur occup\u00e9 \u00e0 repenser les fondements de la m\u00e9taphysique ou un simple professeur de philosophie charismatique. Ainsi, Peeters nous donne envie de d\u00e9couvrir les nombreux secrets de Derrida, non seulement ses amours mais aussi ses luttes intellectuelles. Son archive th\u00e9orique ne peut \u00eatre distingu\u00e9e de sa vie priv\u00e9e.\n\nPeeters insiste sur les multiples \u00abguerres\u00bb de Derrida dans une carri\u00e8re marqu\u00e9e par des ruptures, comme avec Sollers, Kristeva et Lacan, puis une brouille avec Foucault, l'affaire Paul de Man (ce que Derrida lui-m\u00eame appelait \u00abla guerre de Paul de Man\u00bb) et bien d'autres encore. Parmi les plus spectaculaires, on peut \u00e9voquer l'affaire de Prague avec une arrestation grotesque que les autorit\u00e9s lui inflig\u00e8rent pour des raisons purement politiques en d\u00e9cembre 1981. Il y a aussi les t\u00e9moignages courageux pour Nelson Mandela et, dans un dossier qui m'int\u00e9ressait puisque j'habite \u00e0 Philadelphie, la lutte pour la lib\u00e9ration du journaliste et activiste Mumia Abu-Jamal, accus\u00e9 du meurtre d'un policier en 1981 et cit\u00e9 \u00e0 plusieurs reprises dans le S\u00e9minaire La peine de mort de 1999-2000. Derrida lan\u00e7a une p\u00e9tition qui fut sign\u00e9e par des auteures connues comme Sonia Sanchez et Toni Morrison, et elle fut entendue, la peine de mort, prononc\u00e9e en 1982, finalement commu\u00e9e en d\u00e9tention perp\u00e9tuelle en 2011. Abu-Jamal est encore vivant aujourd'hui, m\u00eame s'il est toujours emprisonn\u00e9 \u00e0 Philadelphie.\n\nUne tout autre impression, celle de Derrida qui serait un homme de paix, a \u00e9t\u00e9 donn\u00e9e par un philosophe qui le connaissait personnellement, Christopher Norris. Dans un entretien datant de 1993, Norris confie \u00e0 Anthony Arnove qu'il trouve dommage que Derrida se soit laiss\u00e9 happer par un mode de controverse tr\u00e8s am\u00e9ricain, un monde intellectuel assez superficiel dans lequel ses id\u00e9es auraient \u00e9t\u00e9 trivialis\u00e9es, r\u00e9duites \u00e0 des pol\u00e9miques vicieuses ou d'incessantes luttes de clans. Les deux terrains de bataille, celui qui concernait le pass\u00e9 nazi de Heidegger et celui portant sur les \u00e9crits journalistiques d'extr\u00eame droite r\u00e9dig\u00e9s par Paul de Man pendant la guerre, auront fini par se recouvrir dans un amalgame qui ne suscite plus que r\u00e9pulsion pour le grand public. Je traduis ce passage:\n\nParfois je serais tent\u00e9 de penser que Derrida aurait mieux fait de rester en France, d'avoir v\u00e9cu dans une obscurit\u00e9, disons, d\u00e9cente, une obscurit\u00e9 relative certes, en tout cas sans avoir constamment le projecteur braqu\u00e9 sur lui, ce qui lui aurait permis de continuer le travail philosophique s\u00e9rieux qu'il avait accompli pendant les vingt premi\u00e8res ann\u00e9es de sa carri\u00e8re, au lieu de se laisser entra\u00eener dans autant de confrontations pol\u00e9miques qui, je pense, ne lui venaient pas naturellement. Je pense qu'il a un caract\u00e8re ir\u00e9nique qui reste \u00e9tranger aux confrontations pol\u00e9miques f\u00e9roces, esprit de lutte devenu une seconde nature pour beaucoup de professeurs et intellectuels am\u00e9ricains.\n\nPeut-on accepter cette image d'un sage et doux penseur fran\u00e7ais, fonci\u00e8rement ir\u00e9nique, qui se serait laiss\u00e9 captiver par le monde des culture wars qui dominent aux \u00c9tats-Unis? On ne pourrait \u00e0 cet \u00e9gard manquer d'\u00e9voquer la s\u00e9v\u00e8re, voire f\u00e9roce pol\u00e9mique avec le philosophe am\u00e9ricain John Searle, dispute marqu\u00e9e par des coups bas et des insultes des deux c\u00f4t\u00e9s. Raoul Moati a su donner une \u00e9valuation calme et pr\u00e9cise de ses enjeux majeurs, m\u00eame si encore aujourd'hui aux \u00c9tats-Unis on ne peut \u00e9viter de prendre parti, aussi violemment que possible, pour un camp ou l'autre. Revenant sur cette pol\u00e9mique et d'autres encore dans un entretien avec \u00c9velyne Grossman en 2003, Derrida \u00e9voque la \u00abpulsion\u00bb qui le pousserait \u00e0 dire la v\u00e9rit\u00e9 m\u00eame quand elle risque de blesser son interlocuteur:\n\nCela naturellement m'a fait beaucoup d'ennemis, c'est sans doute l'origine de cette turbulence, parfois de l'ambivalence, de l'allergie autoprotectrice, voire de la haine. Il m'est arriv\u00e9 quelquefois d'\u00e9crire des textes dont je savais qu'ils allaient heurter. Ils \u00e9taient par exemple critiques \u00e0 l'\u00e9gard de L\u00e9vi-Strauss ou de Lacan \u2013 je connais tout de m\u00eame assez bien le milieu pour savoir que cela allait faire des histoires \u2013 eh bien, il m'\u00e9tait impossible de garder cela pour moi.\n\nDerrida va plus loin, avouant qu'aucune prudence ne peut l'emp\u00eacher d'attaquer des adversaires qu'il vient de se cr\u00e9er, ceci au nom d'une passion du vrai qui le pousse en avant. Il ajoute m\u00eame: \u00ab[...] ils peuvent monter, descendre, me fusiller etc., je le dirai \u00bb. Le choix d'une v\u00e9rit\u00e9 qui blesse serait ce qui aurait conduit \u00e0 des guerres intempestives, fond\u00e9es sur des susceptibilit\u00e9s froiss\u00e9es, des rancunes personnelles, alors qu'il e\u00fbt fallu une semblable passion de la v\u00e9rit\u00e9 au-del\u00e0 des enjeux narcissiques et des effets de pouvoir ou des man\u0153uvres visant \u00e0 accaparer la visibilit\u00e9 m\u00e9diatique.\n\nCette impression d'une pulsion pol\u00e9mique port\u00e9e par un d\u00e9sir de v\u00e9rit\u00e9 au-del\u00e0 des partages id\u00e9ologiques habituels a \u00e9t\u00e9 relay\u00e9e par Alain Badiou dans son hommage \u00e0 Derrida donn\u00e9 en 2005 et publi\u00e9 dans son Petit Panth\u00e9on portatif. Repensant le concept majeur de Derrida, la diff\u00e9rence, ou encore la diff\u00e9rance, en ses propres termes, Badiou y voit une tentative obstin\u00e9e pour d\u00e9border toutes les dichotomies. Derrida d\u00e9jouerait les oppositions m\u00e9taphysiques comme celle qui clive l'\u00eatre de l'\u00e9tant chez Heidegger, mais aussi l'opposition entre d\u00e9mocratie et totalitarisme, le couple juif et arabe, afin de \u00abd\u00e9classer les affaires class\u00e9es\u00bb, ce qui d\u00e9bouche sur une recherche de la paix. Badiou f\u00e9licite Derrida de son courage:\n\nDerrida a \u00e9t\u00e9 dans toutes les questions o\u00f9 il intervenait ce que j'appelle un courageux homme de paix. Il \u00e9tait courageux parce qu'il faut toujours beaucoup de courage pour ne pas entrer dans la division telle qu'elle est constitu\u00e9e. Et homme de paix, parce que le rep\u00e9rage de ce qui s'excepte de cette opposition est, de mani\u00e8re g\u00e9n\u00e9rale, le chemin de la paix. Car toute paix v\u00e9ritable se fait par un accord, non sur ce qui existe, mais sur ce qui inexiste.\n\nEn revanche, Badiou avance l'id\u00e9e que Derrida se serait exil\u00e9 de France dans les ann\u00e9es hyper-politis\u00e9es qui ont suivi les mouvements de soixante-huit, s'effor\u00e7ant se tenir \u00e0 l'\u00e9cart des pol\u00e9miques sur la politique, m\u00eame s'il \u00abn'ignorait pas la violence de toute vraie patience\u00bb.\n\nIci, on aurait une image parall\u00e8le, mais invers\u00e9e de celle que proposait Norris. Badiou pr\u00e9sente le succ\u00e8s am\u00e9ricain de la d\u00e9construction comme un effort de se placer toujours ailleurs, \u00e0 l'\u00e9tranger, et dans un perp\u00e9tuel point de fuite, comme si Derrida avait voulu continuer \u00e0 penser sans se laisser happer par le quotidien des luttes politiques fran\u00e7aises. Une \u00abglisse infinie\u00bb dans la langue aurait \u00e9t\u00e9 la condition de sa volont\u00e9 ir\u00e9nique primordiale; mais ensuite cette glisse aurait d\u00e9rap\u00e9 avant de s'emp\u00eatrer dans les chicanes des \u00abguerres\u00bb culturelles nord-am\u00e9ricaines.\n\nOn peut dire, avec le recul du temps, que Derrida aura gagn\u00e9 la plupart de ses batailles, voire sa guerre elle-m\u00eame. Il y aurait une exception majeure, celle de la \u00abguerre de Paul de Man\u00bb. Malgr\u00e9 tous ses efforts pour sauver son ancien ami de Yale de l'opprobre g\u00e9n\u00e9ral, aujourd'hui il n'y a plus qu'une poign\u00e9e d'anciens disciples pour continuer l'\u0153uvre du grand critique belgo-am\u00e9ricain. Je ne tiens pas \u00e0 m'immiscer dans une bataille id\u00e9ologique qui a perdu de son sens avec le temps et ferai juste remarquer que les phrases les plus inexcusables que Paul de Man avait \u00e9crites dans ses textes de jeunesse, ces textes syst\u00e9matiquement d\u00e9nigr\u00e9s par la suite, concernaient la guerre \u2013 mais il s'agit de la Premi\u00e8re Guerre mondiale. De Man rejetait les expressions d'un antis\u00e9mitisme grossier qui avan\u00e7ait la th\u00e8se selon laquelle l'avant-garde europ\u00e9enne ne pouvait inclure des artistes et \u00e9crivains juifs; ce genre de th\u00e8se qui partait de l'id\u00e9e que \u00able roman et la po\u00e9sie modernes ne seraient qu'une esp\u00e8ce d'excroissance monstrueuse de la guerre mondiale\u00bb selon ces penseurs serait \u00ab\u00e0 l'origine de la th\u00e8se de la mainmise juive\u00bb et l'on pourrait alors utiliser \u00able qualificatif d'enjuiv\u00e9\u00bb. Si Derrida n'eut pas de mal \u00e0 montrer que ce passage critiquait ce vocabulaire si choquant, il ne peut \u00e9viter de remarquer que Paul de Man s'\u00e9tait approch\u00e9 de trop pr\u00e8s de cette rh\u00e9torique haineuse, flirtant ainsi lui-m\u00eame avec l'antis\u00e9mitisme vulgaire. Ce qu'on a fini par pardonner \u00e0 un C\u00e9line parce que son geste \u00e9tait histrionique et excessif n'a pas \u00e9t\u00e9 tol\u00e9r\u00e9 de Paul de Man, sans doute parce qu'il avait r\u00e9ussi \u00e0 cacher si longtemps un pass\u00e9 douteux. C'est sur cet \u00e9cueil que le vaisseau am\u00e9ricain de la d\u00e9construction s'est finalement fracass\u00e9.\n\nDerrida a d'ailleurs dit que Paul de Man n'a pas simplement occult\u00e9 \u00absa\u00bb propre guerre, sugg\u00e9rant qu'il \u00e9tait en guerre lui aussi: \u00abQuelle guerre, donc? La guerre de Paul de Man, c'est enfin [...] celle que cet homme a d\u00fb vivre et endurer en lui-m\u00eame. Il fut cette guerre. Et pendant pr\u00e8s d'un demi-si\u00e8cle, cette \u00e9preuve fut une guerre parce qu'elle ne put rester une tourmente seulement priv\u00e9e. Elle a d\u00fb marquer les gestes publics, l'enseignement et les \u00e9crits.\u00bb Ce fut, en effet, une guerre meurtri\u00e8re, et Derrida a su noter la fr\u00e9n\u00e9sie d\u00e9nonciatrice, alli\u00e9e au manque de probit\u00e9 intellectuelle de bon nombre de ses adversaires. Et lorsqu'il m\u00e9dite sur le fait que la guerre de Paul de Man avait pris effet de mani\u00e8re posthume \u2013 en effet, Ortwin de Graeff avait d\u00e9couvert les \u00e9crits journalistiques qui n'avaient jamais \u00e9t\u00e9 cach\u00e9s, mais qui scandalis\u00e8rent le public apr\u00e8s la mort du c\u00e9l\u00e8bre critique, Derrida ajoute entre parenth\u00e8ses: \u00ab[...] (et la peur, la haine, c'est-\u00e0-dire parfois l'amour, r\u00eavent aussi de tuer des morts pour atteindre des vivants)\u00bb.\n\nCeci pourrait \u00e9voquer la m\u00e9ditation de Walter Benjamin commentant des textes des amis d'Ernst J\u00fcnger qui, en 1930, publi\u00e8rent un ensemble de textes sur la Premi\u00e8re Guerre mondiale afin de glorifier le c\u00f4t\u00e9 h\u00e9ro\u00efque de la guerre. Comme le dit Benjamin de mani\u00e8re ironique, pour ces auteurs nostalgiques, c'\u00e9tait le paysage des tranch\u00e9es de 1914-18 qui se mettait \u00e0 incarner l'id\u00e9alisme allemand: un crat\u00e8re de bombe devenait une aporie, un soldat pris dans des barbel\u00e9s, une antinomie, une explosion d'obus, le d\u00e9ploiement d'une th\u00e8se m\u00e9taphysique; le ciel diurne figurait l'int\u00e9riorit\u00e9 cosmique du casque d'acier, et la nuit ne pouvait \u00eatre que la loi morale surplombant le tout. Dans ce m\u00eame essai, on voit Benjamin s'efforcer de critiquer les fondements de l'id\u00e9ologie fasciste, marqu\u00e9e par une r\u00e9serve m\u00e9taphysique face \u00e0 la technologie. Cette id\u00e9ologie ne voulait rien savoir du fait que la Grande Guerre \u00e9tait d\u00e9j\u00e0 une application syst\u00e9matique de la technologie industrielle aux bombardements group\u00e9s qui avaient remplac\u00e9 les anciens \u00absports de combat\u00bb. Mais ces id\u00e9ologues nostalgiques se rapprochaient aussi du nouveau futurisme industriel du pays. Le m\u00e9lange de futurisme et de nostalgie rendait compte de l'essor du nazisme.\n\nDans ces pages, Benjamin m\u00e9dite sur ce que signifie \u00abgagner\u00bb ou \u00abperdre\u00bb une guerre. Voulant d\u00e9passer le sens courant, il comprend le concept de \u00abgagner\u00bb comme le fait que le soi-disant vainqueur \u00abposs\u00e8de\u00bb la guerre: il l'a en lui pour en faire sa chose, tandis que celui qui l'a \u00abperdue\u00bb est comme un \u00e9tourdi qui a oubli\u00e9 un objet en chemin. Le vaincu ne sait plus o\u00f9 il a \u00e9gar\u00e9 cette guerre; le vainqueur se l'accapare, l'annexe, vit avec la guerre et s'en enrichit. En fin de compte, Derrida ferait penser \u00e0 Fortinbras tel que le pr\u00e9sente Walter Benjamin quand il r\u00e9\u00e9crit Hamlet. Benjamin imagine que Shakespeare donnerait plus de place \u00e0 Fortinbras, qu'il ouvrirait la pi\u00e8ce avec une harangue passionn\u00e9e louant la paix. Mais cette hom\u00e9lie aux accents si doux se transformerait en un appel \u00e0 la guerre si soudain et f\u00e9roce que le public entier sursauterait de surprise, et Benjamin pose la question: \u00abQuelles sont les forces puissantes et innommables qui poussent cet homme, tout empli du bonheur de la paix, \u00e0 se vouer corps et \u00e2me \u00e0 la guerre?\u00bb Derrida serait donc un Fortinbras, ou un Ulysse, un homme de paix, mais pouss\u00e9 \u00e0 faire la guerre, et ce, \u00e0 outrance, s'y jetant corps et \u00e2me...\n\nIl ne suffirait pas de repenser \u00e0 nouveaux frais la vieille maxime \u00abSi vis pacem, para bellum\u00bb, car il faut mieux comprendre la dialectique de l'histoire dans laquelle l'automate du mat\u00e9rialisme dialectique \u00abgagne \u00e0 tous les coups\u00bb, comme le souffle Benjamin dans son dernier texte, les th\u00e8ses Sur le concept d'histoire. L'automate gagne gr\u00e2ce \u00e0 l'intervention du petit nain hideux qui est cach\u00e9 \u00e0 l'int\u00e9rieur. Ce nain est un th\u00e9ologien dissimul\u00e9 dans la machine pour perp\u00e9tuer les \u00e9chos affaiblis d'un messianisme oubli\u00e9. Quelle version choisir dans ces perspectives crois\u00e9es? Je commencerai par la voie historique et critique avant d'explorer le d\u00e9bat philosophique, puis terminerai par la fiction et la po\u00e9sie.\n\n. J. Derrida, Apprendre \u00e0 vivre enfin, op. cit., p. 49.\n\n. Beno\u00eet Peeters, Derrida, Paris, Flammarion, coll. \u00abGrandes Biographies\u00bb, 2010.\n\n. B. Peeters, Derrida, op. cit., p. 483.\n\n. Je renvoie \u00e0 l'analyse presque exhaustive donn\u00e9e par Ginette Michaud du th\u00e8me de la guerre chez Jacques Derrida dans le chapitre intitul\u00e9 \u00abPenser \"la guerre qui vient\"\u00bb, dans Jacques Derrida. L'art du contretemps, Montr\u00e9al, Nota bene, coll. \u00abNouveaux essais Spirale\u00bb, 2014, p. 53-94.\n\n. J. Derrida, S\u00e9minaire La peine de mort. Volume I (1999-2000), Geoffrey Bennington, Marc Cr\u00e9pon et Thomas Dutoit (\u00e9ds), Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 2012, particuli\u00e8rement p. 20, 79, 118-119, 136, 200, 292 et 295.\n\n. Christopher Norris, \u00abTruth, Criticism, and the Politics of Theory\u00bb, dans Deconstruction After All: Reflections and Conversations, David Jonathan Y. Bayot (\u00e9d.), Brighton, Sussex Academic Press, 2015, p. 124.\n\n. Cf. le remarquable livre de Raoul Moati, Derrida et le langage ordinaire (Paris, Herman, 2014), qui retrace la controverse entre Austin, Searle et Derrida dans les ann\u00e9es soixante-dix et quatre-vingt, et la repense de mani\u00e8re philosophique, ce qui lui permet d'\u00e9viter de d\u00e9cr\u00e9ter qui aurait \u00abgagn\u00e9\u00bb la bataille. Moati note justement: \u00abMalgr\u00e9 l'\u00e2pret\u00e9 manifeste du propos, nous persistons \u00e0 penser que r\u00e9duire la controverse entre Derrida et Searle \u00e0 l'extr\u00eame violence de l'\u00e9change qu'elle occasionna de part et d'autre, sans analyser les enjeux philosophiques que leur querelle soul\u00e8ve, serait une erreur.\u00bb (Ibid., p. 9.)\n\n. J. Derrida, \u00abLa v\u00e9rit\u00e9 blessante Ou le corps \u00e0 corps des langues\u00bb, entretien avec \u00c9velyne Grossman, Europe, cahier \u00abJacques Derrida\u00bb, no 901, mai 2004, p. 21.\n\n. Ibid., p. 22. C'est Jacques Derrida qui souligne. (Sauf indications contraires, ce sont toujours les auteurs qui soulignent.)\n\n. Alain Badiou, Petit Panth\u00e9on portatif, Paris, La Fabrique, 2008, p. 128.\n\n. Ibid., p. 129.\n\n. Cf. J. Derrida, \u00abComme un bruit de la mer au fond d'un coquillage. La guerre de Paul de Man\u00bb, dans M\u00e9moires \u2013 Pour Paul de Man, Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 1988, p. 147-232.\n\n. Cf. l'article de Paul de Man, \u00abLes Jufs dans la litt\u00e9rature actuelle\u00bb, Le Soir, 10 f\u00e9vrier 1942, cit\u00e9 par J. Derrida dans \u00abComme un bruit de la mer au fond d'un coquillage. La guerre de Paul de Man\u00bb, dans M\u00e9moires \u2013 Pour Paul de Man, op. cit., p. 191.\n\n. J. Derrida, M\u00e9moires \u2013 Pour Paul de Man, op. cit., p. 154.\n\n. Ibid., p. 151.\n\n. Walter Benjamin, \u00abTheorien des deutschen Faschismus\u00bb, dans Gesammelte Schriften, III, Frankfurt am Main, Suhrkamp, 1974, p. 247. Je remercie Eduardo Cadava qui a attir\u00e9 mon attention sur ce texte.\n\n. Ibid., p. 242.\n\n. W. Benjamin, \u00abTheorien des deutschen Faschismus\u00bb, dans Gesammelte Schriften, III, op. cit., p. 246.\n\n. W. Benjamin, \u00ab\u00dcber den Begriff der Geschichte\u00bb, dans Gesammelte Schriften, I, Frankfurt am Main, Suhrkamp, 1972, p. 691. \n\n# I. S'aguerrir \u00e0 l'Alg\u00e9rie\n\nLa premi\u00e8re guerre que Derrida aura mentionn\u00e9e en tant que telle est la guerre d'Alg\u00e9rie. C'est une guerre qui, il faut le souligner, a longtemps attendu qu'on la nomme de son nom en France. Dans la lettre de plus de quarante pages que Derrida envoya le 27 avril 1961 \u00e0 son ami Pierre Nora, apr\u00e8s que ce dernier eut publi\u00e9 Les Fran\u00e7ais d'Alg\u00e9rie en mars 1961, on trouve cette phrase qui vient au cours d'une analyse de la position des \u00ablib\u00e9raux\u00bb parmi les Fran\u00e7ais d'Alg\u00e9rie: \u00abEt par rapport \u00e0 quoi auraient-ils donc pu se d\u00e9finir? Auraient-ils donc d\u00fb \u00e9prouver le sentiment national alg\u00e9rien avant m\u00eame que la guerre, et la guerre seule, comme le montre fort bien Ch.-A. Julien (p. 60-61), ne l'ait assez lentement form\u00e9 chez les Arabes eux-m\u00eames?\u00bb Derrida cite ici le professeur Julien qui avait pr\u00e9fac\u00e9 le livre de Nora. De fait, Charles-Andr\u00e9 Julien, professeur d'histoire de la colonisation \u00e0 la Sorbonne, mentionnait que le peuple alg\u00e9rien ne s'\u00e9tait \u00e9prouv\u00e9 nationaliste que dans et par la lutte contre les Fran\u00e7ais.\n\nLe livre incendiaire de Nora commen\u00e7ait sa chronique avec les ann\u00e9es de l'apr\u00e8s-guerre, en mai 1945 pour \u00eatre plus pr\u00e9cis, quand les r\u00e9voltes antifran\u00e7aises de la r\u00e9gion de Constantine furent r\u00e9prim\u00e9es dans le sang \u2013 \u00e0 la suite d'une insurrection spontan\u00e9e \u00e0 S\u00e9tif, les blind\u00e9s, les avions et la marine de l'arm\u00e9e fran\u00e7aise massacr\u00e8rent en quelques jours plus de 10 000 Arabes. \u00c0 ce moment-l\u00e0, le parti communiste applaudissait \u00e0 l'\u00e9chec de la r\u00e9bellion contre la souverainet\u00e9 de la France. Nora dresse un portrait f\u00e9roce dans Les Fran\u00e7ais d'Alg\u00e9rie des colons racistes, de l'administration coloniale incomp\u00e9tente, des Fran\u00e7ais pieds-noirs arrogants m\u00e9prisant les autochtones, et il n'\u00e9pargne pas les intellectuels de gauche ou \u00ablib\u00e9raux\u00bb d'Alg\u00e9rie.\n\nC'est ainsi que Nora \u00e9pingle Albert Camus. Camus avait commenc\u00e9 L'\u00c9tranger \u00e0 Oran, puis sa carri\u00e8re avait pris de l'ampleur avec ses chroniques journalistiques entre Alger et Oran. Pour Nora, si Camus avait lanc\u00e9 la philosophie de l'absurde, celle-ci lui semblait \u00eatre \u00abl'expression vraisemblablement sublim\u00e9e d'une situation historique r\u00e9elle, d\u00e9cant\u00e9e jusqu'\u00e0 l'\u00e9pure\u00bb. Nora reconna\u00eet que le roman de Camus, qui raconte le meurtre gratuit d'un Arabe sur une plage, est bien le \u00abchef-d'\u0153uvre de la litt\u00e9rature alg\u00e9rienne\u00bb. Dans L'\u00c9tranger, Camus d\u00e9noncerait en effet la mani\u00e8re dont les citoyens fran\u00e7ais pouvaient impun\u00e9ment tuer ou terroriser les Alg\u00e9riens. Ce qui semblait surprenant dans le contexte d'alors, c'\u00e9tait moins que Meursault tue un Arabe sans raison que le fait qu'il soit puni pour ce meurtre et m\u00eame qu'il soit ex\u00e9cut\u00e9. Sa condamnation finale n'a rien d'un proc\u00e8s kafka\u00efen, c'est l'aveu d'une culpabilit\u00e9 historique. Ce ch\u00e2timent m\u00e9rit\u00e9 en toute justice prend donc \u00abun air d'anticipation\u00bb, comme dit Nora. Selon cette lecture politique, Camus devient un sympt\u00f4me historique; homme de gauche, journaliste qui d\u00e9nonce la pauvret\u00e9 en Kabylie, il repr\u00e9sente dans sa fiction le moment o\u00f9 les Fran\u00e7ais ont commenc\u00e9 \u00e0 sentir leur domination menac\u00e9e et contest\u00e9e. Son roman donnerait forme universelle au cauchemar qui hante la mauvaise conscience des colons.\n\nOr, dans le chapitre 5 consacr\u00e9 aux \u00abLib\u00e9raux\u00bb, on constate que Nora attaque Camus plus violemment et plus directement. Dans les ann\u00e9es de la lutte pour l'Ind\u00e9pendance alg\u00e9rienne, Camus avait, on le sait, tent\u00e9 de s'\u00e9lever au-dessus de la m\u00eal\u00e9e, expliquant qu'il ne voulait \u00eatre ni victime ni tortionnaire. Pour Nora, cela signifiait qu'il voulait refuser toute solidarit\u00e9 de classe au nom d'un vague tragique de la condition humaine: \u00abCamus ne pouvait se r\u00e9volter contre l'histoire qu'\u00e0 la condition de suspendre son jugement et de se retirer du champ clos. Or son d\u00e9sengagement s'op\u00e9ra, comme la torture, au nom de l'efficacit\u00e9.\u00bb Cette phrase perfide choquera Derrida.\n\nPour Nora, la rh\u00e9torique de l'absurde que Camus avait \u00e9t\u00e9 le premier \u00e0 mettre en forme masquait la r\u00e9bellion futile de celui qui se pense au-dessus de la politique. Son drame m\u00e9taphysique cacherait ainsi le dilemme bien r\u00e9el des lib\u00e9raux de gauche qui ne voulaient ni rejoindre la lutte arm\u00e9e du FLN, ni accepter la violence raciste des pieds-noirs, ni glorifier les campagnes meurtri\u00e8res de l'arm\u00e9e fran\u00e7aise. Or c'est la suggestion que Camus aurait \u00e9t\u00e9 complice de la torture pratiqu\u00e9e r\u00e9guli\u00e8rement dans les forces arm\u00e9es fran\u00e7aises qui sembla inacceptable \u00e0 Derrida.\n\nDerrida, n\u00e9 en Alg\u00e9rie, connaissait le pays intimement tandis que Nora \u00e9tait un parisien qui avait pass\u00e9 seulement deux ans en Alg\u00e9rie. Dans sa longue lettre, on voit Derrida d\u00e9fendre Camus au nom de leur pr\u00e9caire partage, cette intime familiarit\u00e9 avec une ancienne colonie qui \u00e9tait un peu la France. Derrida approuve donc les analyses de Nora au sujet de L'\u00c9tranger et ajoute que son condisciple devait continuer et m\u00eame \u00e9crire tout un livre sur ce sujet. Il est d'accord avec Nora pour critiquer l'analyse un peu grosse de Sartre. Mais Derrida rejette cat\u00e9goriquement l'accusation de collusion avec les tortionnaires visant Camus:\n\nQuand il ne veut pas choisir entre les violences pour donner une efficacit\u00e9 \u00e0 sa condamnation de toute violence, la perfidie avec laquelle tu \u00e9cris que \u00abson d\u00e9sengagement s'op\u00e8re, comme la torture, au nom de l'efficacit\u00e9\u00bb est vraiment indigne de Camus. Et de toi. Car, enfin, ou je dors, ou ce n'est pas la m\u00eame chose, ni la m\u00eame efficacit\u00e9, ni la m\u00eame fin, ni les m\u00eames moyens.\n\nCe n'est pas que Derrida apparaisse comme un \u00ablib\u00e9ral\u00bb lui-m\u00eame, c'est plut\u00f4t qu'en 1961 il reste sceptique face au nationalisme alg\u00e9rien. Nora n'a pas ce scrupule et approuve l'\u00e9nergie \u00abpassionn\u00e9e\u00bb du nationalisme enfin vainqueur du colonialisme. Il est tout \u00e0 l'honneur de Derrida d'avoir su garder une distance critique face \u00e0 cette passion politique, pourtant compr\u00e9hensible dans le contexte historique. Derrida note de mani\u00e8re ironique qu'il semble difficile de rejeter le nationalisme fran\u00e7ais et d'exalter en m\u00eame temps le nationalisme alg\u00e9rien. Je voudrais prendre ce moment de r\u00e9flexion moins comme un signe de l'appartenance de Derrida \u00e0 un fonds de terroir local, qui en ferait presque un pied-noir juif donc, que d'y voir d\u00e9j\u00e0 un exercice de d\u00e9construction face \u00e0 un consensus facile de la gauche communiste, qui, en 1961, appuyait le nationalisme des arabes vainqueurs. M\u00eame si les accords d'\u00c9vian ne seraient sign\u00e9s qu'un an plus tard, d\u00e8s septembre 1959, le nouvellement \u00e9lu pr\u00e9sident de Gaulle reconnaissait le droit \u00e0 l'auto-d\u00e9termination du peuple alg\u00e9rien.\n\nDerrida aura eu tr\u00e8s t\u00f4t le courage de r\u00e9cuser la facilit\u00e9 de ces \u00abannexions ou ex\u00e9cutions sommaires, au nom de la \"complicit\u00e9 objective\"\u00bb qui, comme il le dit, ont plus fait pour r\u00e9duire Camus au silence que sa \u00abpr\u00e9tendue nostalgie inavouable\u00bb. De plus, Derrida se demande pourquoi Nora d\u00e9nonce de mani\u00e8re si virulente le \u00abmoralisme\u00bb de Camus, car si ce moralisme semble immoral pour Nora, en fait il le critique de mani\u00e8re tr\u00e8s morale lui-m\u00eame. Poussant encore l'analyse, Derrida d\u00e9monte un paralogisme de Nora qui avance l'id\u00e9e que les Fran\u00e7ais d'Alg\u00e9rie seraient pr\u00eats \u00e0 un g\u00e9nocide qui les d\u00e9barrasserait des Arabes g\u00eanants. Nora avait \u00e9crit:\n\nSi on leur offrait le g\u00e9nocide presse-bouton, combien de Fran\u00e7ais d'Alg\u00e9rie le refuseraient? D'o\u00f9 une double confiance \u00e0 l'arm\u00e9e. Elle rassure par sa pr\u00e9sence et fait la guerre \u00e0 leur place. Elle joue donc un double r\u00f4le: elle garantit les Fran\u00e7ais d'Alg\u00e9rie de leurs propres exc\u00e8s, et, loin des yeux, dans les djebels, elle tue par procuration. Aucun Fran\u00e7ais d'Alg\u00e9rie n'ose faire le bilan: il serait terrifi\u00e9 d'apprendre que, proportionnellement, les musulmans ont perdu au moins deux fois plus des leurs depuis 1955 que les Fran\u00e7ais pendant la Grande Guerre.\n\nCette supposition d'un g\u00e9nocide qui \u00e9liminerait une partie de la population semble gratuite et odieuse \u00e0 Derrida; il note que Nora vient de se contredire, puisqu'il a \u00e9crit plus t\u00f4t que le racisme des Fran\u00e7ais d'Alg\u00e9rie leur imposait de conserver les Arabes comme une masse docile de subalternes. Leur \u00abhyper-racisme subtil\u00bb impliquerait donc qu'ils doivent s'abstenir du \u00abpresse-bouton\u00bb de l'arme atomique totale... Derrida enfonce alors le clou et ironise: \u00abQuel dommage que les nazis ou les Am\u00e9ricains n'aient pas \u00e9t\u00e9 aussi racistes ou que leur racisme n'ait pas eu tant de ressources!\u00bb Derrida discutera plus tard le concept du presse-bouton, figure militaire de la vitesse, et des d\u00e9cisions strat\u00e9giques prises trop rapidement dans une h\u00e2te id\u00e9ologique et aussi technocratique. Cette accusation qui n'est pas fond\u00e9e viendrait en derni\u00e8re analyse de la croyance que Nora poss\u00e8de le savoir absolu et qu'il peut se permettre des \u00abcoups de sonde psychanalytiques auxquels rien n'\u00e9chappe\u00bb \u2013 pas m\u00eame les relations matrimoniales.\n\nDerrida s'excuse \u00e0 la fin de la lettre de son ton agressif, agressivit\u00e9 qu'il pense caus\u00e9e par celle qu'il reconna\u00eet chez Nora. Derrida insiste sur l'id\u00e9e qu'il faut cerner l'unit\u00e9 de l'\u0153uvre de Camus afin d'\u00e9viter le curieux dualisme du livre de Nora, plein de louanges face au roman alg\u00e9rien qu'est L'\u00c9tranger, mais plus que perfide quand il s'agit de juger une position politique que Nora pense compromise. Notons que Camus \u00e9tait d\u00e9j\u00e0 mort \u2013 mort jeune \u00e0 46 ans en janvier 1960 \u2013 au moment o\u00f9 Nora publiait son livre. L'unit\u00e9 de l'\u0153uvre ne serait pas \u00e0 chercher dans une vision m\u00e9taphysique. \u00c0 l'\u00e9vidence, Derrida ne s'int\u00e9resse pas \u00e0 l'absurde, pas m\u00eame pour y voir un sympt\u00f4me de la mauvaise conscience du colonialisme. Cette unit\u00e9 ne se d\u00e9duirait pas non plus d'un mouvement de bascule entre \u00e9thique et politique. Non, si unit\u00e9 il y a, elle serait chez Camus l'\u00e9crivain, plus pr\u00e9cis\u00e9ment dans son invention d'une nouvelle modalit\u00e9 de l'\u00e9criture.\n\nSur ce point, Derrida aurait un pr\u00e9d\u00e9cesseur, qui se trouve aussi \u00eatre le premier th\u00e9oricien de l'\u00e9criture, Roland Barthes. Dans un de ses essais de jeunesse, Barthes d\u00e9crit magistralement la mani\u00e8re dont Camus a invent\u00e9 un style de l'absurde qui est \u00abplat et profond comme une glace\u00bb. Son style de l'indiff\u00e9rence au style propose ici une premi\u00e8re mouture du \u00abdegr\u00e9 z\u00e9ro de l'\u00e9criture\u00bb, concept qui sera d\u00e9fini plus clairement dans les ann\u00e9es cinquante. D\u00e8s ce compte rendu de juillet 1944, Barthes avait bien compris que le fameux pass\u00e9 compos\u00e9 de L'\u00c9tranger n'\u00e9tait pas un exercice de style, mais bien une vision du monde impliquant un rapport \u00e9thique avec l'histoire:\n\nPeut-\u00eatre bien qu'avec L'\u00c9tranger \u2013 sans trop exag\u00e9rer l'importance de cette \u0153uvre \u2013 se l\u00e8ve un nouveau style, style du silence et silence du style, o\u00f9 la voix de l'artiste \u2013 \u00e9galement \u00e9loign\u00e9e des soupirs, des blasph\u00e8mes et des cantiques \u2013 est une voix blanche, la seule en accord avec notre d\u00e9tresse irr\u00e9m\u00e9diable.\n\nAyant \u00e0 peu pr\u00e8s le m\u00eame \u00e2ge, le Barthes de 1944 (qui a 29 ans) et le Derrida de 1961 (qui a 31 ans) sont fondamentalement d'accord dans leur lecture de L'\u00c9tranger. Tous deux acceptent la validit\u00e9 de la position \u00e9thique de Camus et y lisent de plus le d\u00e9ploiement d'une \u00e9criture autre. Son \u00e9criture blanche, neutre, lisse, serait \u00e0 m\u00eame d'apporter une r\u00e9ponse possible aux catastrophes de l'histoire r\u00e9cente qui va de la trag\u00e9die de l'holocauste aux massacres impitoyables qui marqu\u00e8rent la fin du colonialisme. Or, en 1961, Sartre et Camus avaient rompu; la majorit\u00e9 des gens de gauche s'\u00e9taient ralli\u00e9s \u00e0 Sartre et Derrida d\u00e9fendait d\u00e9j\u00e0 au nom de l'ind\u00e9cidable en mati\u00e8re de jugement \u00e9thique son compatriote alg\u00e9rien Albert Camus. Sa position serait proche de celle de Kamel Daoud dans Meursault, contre-enqu\u00eate, roman qui a su approfondir et compliquer les rapports des Fran\u00e7ais et des Alg\u00e9riens avant et apr\u00e8s la guerre de lib\u00e9ration.\n\nCette analyse des paradoxes alg\u00e9riens hantera Derrida toute sa vie, comme l'ont rappel\u00e9 Philip Armstrong et Marta Segarra dans de superbes essais. Un \u00e9cho plus r\u00e9cent pour Derrida de cette aporie ancienne \u2013 c'est un texte donn\u00e9 en 1992 et publi\u00e9 en 1996 \u2013 tente de faire le point sur l'\u00eatre fondamentalement divis\u00e9 qu'est de toute \u00e9vidence un \u00abfranco-maghr\u00e9bin\u00bb et n'\u00e9vite pas de rouvrir le chapitre de la guerre:\n\n\u00c0 supposer encore, ce qui est loin d'\u00eatre s\u00fbr, qu'il y ait quelque unit\u00e9 historique de la France et du Maghreb, le \u00abet\u00bb n'aura jamais \u00e9t\u00e9 donn\u00e9, seulement promis ou all\u00e9gu\u00e9. [...] Le silence de ce trait d'union ne pacifie ou n'apaise rien, aucun tourment, aucune torture. Il ne fera jamais taire leur m\u00e9moire. Il pourrait m\u00eame aggraver la terreur, les l\u00e9sions et les blessures. Un trait d'union ne suffit jamais \u00e0 couvrir les protestations, les cris de col\u00e8re ou de souffrance, le bruit des armes, des avions et des bombes.\n\nJe reviendrai sur l'impact de ce livre sur John Maxwell Coetzee un peu plus loin. L'\u00e9thique implique un devoir de m\u00e9moire; il s'agit de ne pas oublier les blessures de la guerre, une guerre presque infinie, qui prend des proportions \u00e9normes dans les livres et les textes de Derrida. On pourrait presque avancer l'id\u00e9e que, pour Derrida, la guerre remplacerait ce qui chez Heidegger \u00e9tait encore le Gestell ou tous les signes modernes d'une technologie envahissante. En t\u00e9moignerait ce passage du texte discutant de la possibilit\u00e9 d'une guerre nucl\u00e9aire. Derrida distingue la r\u00e9alit\u00e9 de la course aux armements et la fiction d'une apocalypse nucl\u00e9aire qui est par d\u00e9finition \u00abimpossible\u00bb puisqu'elle signifie la fin de notre monde; cette fin du monde annonc\u00e9e ne peut donc \u00eatre qu'une \u00abfable\u00bb:\n\nC'est la guerre (autrement dit, pour l'instant, la fable) qui met en marche ce fabuleux effort de guerre, cette capitalisation insens\u00e9e d'armements sophistiqu\u00e9s, cette course de vitesse en vue de la vitesse, cette pr\u00e9cipitation affol\u00e9e qui par la technoscience et avec toute l'inventivit\u00e9 techno-scientifique qu'elle motive, structure non seulement l'arm\u00e9e, la diplomatie, la politique, mais le tout du socius humain aujourd'hui, tout ce qu'on appelle des vieux mots de culture, civilisation, Bildung, skhol\u00e8, paideia.\n\nC'est donc la guerre qu'il faudrait, avant tout, d\u00e9construire. Mais comment d\u00e9construire une structure pol\u00e9mique qui fait partie de la d\u00e9construction elle-m\u00eame?\n\nKant, on le sait, proposait une solution qu'il imaginait d\u00e9finitive lorsqu'il voulait que l'on remplace l'\u00e9tat de guerre, ce conflit end\u00e9mique des nations avanc\u00e9es, par un \u00c9tat de droit qui serait fond\u00e9 de mani\u00e8re stable sur un cosmopolitisme d\u00e9mocratique. Ainsi, seulement, pourrait-on arriver \u00e0 cr\u00e9er ce bien supr\u00eame pour les hommes, la \u00abpaix perp\u00e9tuelle\u00bb. Ses phrases proph\u00e9tiques r\u00e9sonnent dans nos oreilles:\n\nOr, la raison moralement pratique \u00e9nonce en nous son veto irr\u00e9sistible: il ne doit y avoir aucune guerre; ni celle entre toi et moi dans l'\u00e9tat de nature, ni celle entre nous en tant qu'\u00c9tats... [...] Aussi la question n'est plus de savoir si la paix perp\u00e9tuelle est quelque chose de r\u00e9el ou si ce n'est qu'une chim\u00e8re et si nous ne nous trompons pas dans notre jugement th\u00e9orique, quand nous admettons le premier cas, mais nous devons agir comme si la chose qui peut-\u00eatre ne sera pas devait \u00eatre, et en vue de sa fondation \u00e9tablir la constitution (peut-\u00eatre le r\u00e9publicanisme de tous les \u00c9tats ensemble et en particulier) qui nous semble le plus capable d'y mener et de mettre fin \u00e0 la conduite de la guerre d\u00e9pourvue de salut, vers laquelle tous les \u00c9tats sans exception ont jusqu'\u00e0 maintenant dirig\u00e9 leurs pr\u00e9paratifs int\u00e9rieurs, comme vers leur fin supr\u00eame.\n\nS'agit-il d'un v\u0153u pieux, d'une utopie, d'une chim\u00e8re? L'humour de Kant se manifeste dans un autre texte sur son Projet de paix perp\u00e9tuelle quand il commence par citer un panneau annon\u00e7ant la \u00abpaix perp\u00e9tuelle\u00bb \u00e0 l'entr\u00e9e d'un cimeti\u00e8re. Je reviendrai sur la question des utopies qui sont plus des dystopies en commentant un roman de J. M. Coetzee \u00e0 la fin de ce livre. Derrida tendrait \u00e0 sugg\u00e9rer que s'il n'est pas possible de r\u00e9sister aux arguments de Kant; il s'agirait de fonder autrement son utopie l\u00e9galiste, dans laquelle nous pouvons entrevoir l'ultime \u00abfin\u00bb des Lumi\u00e8res, un avenir ou un futur hypoth\u00e9tiques, h\u00e9las toujours d\u00e9mentis par le cynisme de l'histoire.\n\n. Exactement trente-sept ans, si l'on prend 1962 comme la fin de la guerre avec les accords d'\u00c9vian et sa reconnaissance nominale par l'\u00c9tat fran\u00e7ais quand l'Assembl\u00e9e nationale vote le 10 juin 1999 un texte qui enterre le terme officiel d'\u00abop\u00e9rations de maintien de l'ordre\u00bb et parle de \u00abguerre d'Alg\u00e9rie\u00bb.\n\n. J. Derrida, \u00abMon cher Nora...\u00bb, dans Pierre Nora, Les Fran\u00e7ais d'Alg\u00e9rie, \u00e9dition revue et augment\u00e9e, pr\u00e9c\u00e9d\u00e9e de \u00abCinquante ans\u00bb et suivie d'Un document in\u00e9dit de Jacques Derrida, \u00abMon cher Nora...\u00bb, Paris, Christian Bourgois \u00e9diteur, 2012 [1961], p. 281.\n\n. P. Nora, Les Fran\u00e7ais d'Alg\u00e9rie, op. cit., p. 61.\n\n. Ibid., p. 131-132.\n\n. Ibid., p. 209.\n\n. Ibid., p. 208.\n\n. Ibid., p. 209.\n\n. P. Nora, Les Fran\u00e7ais d'Alg\u00e9rie, op. cit., p. 231-232.\n\n. J. Derrida, \u00abMon cher Nora...\u00bb, dans P. Nora, Les Fran\u00e7ais d'Alg\u00e9rie, op. cit., p. 293.\n\n. Ibid., p. 296.\n\n. J. Derrida, \u00abMon cher Nora...\u00bb, dans P. Nora, Les Fran\u00e7ais d'Alg\u00e9rie, op. cit., p. 297.\n\n. P. Nora, Les Fran\u00e7ais d'Alg\u00e9rie, op. cit., p. 206-207.\n\n. J. Derrida, \u00abMon cher Nora...\u00bb, dans P. Nora, Les Fran\u00e7ais d'Alg\u00e9rie, op. cit., p. 298.\n\n. J. Derrida, \u00abNo apocalypse, not now (\u00e0 toute vitesse, sept missiles, sept missives)\u00bb [1984], dans Psych\u00e9. Inventions de l'autre, Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 1987, p. 363-386; r\u00e9\u00e9d., t. I, Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 1998, p. 395-418.\n\n. J. Derrida, \u00abMon cher Nora...\u00bb, dans P. Nora, Les Fran\u00e7ais d'Alg\u00e9rie, op. cit., p. 298-299\n\n. Ibid., p. 299.\n\n. Roland Barthes, \u00abR\u00e9flexion sur le style de l'\u00c9tranger\u00bb [1944], dans \u0152uvres compl\u00e8tes, t. I, \u00c9ric Marty (\u00e9d.), Paris, Seuil, 2002, p. 63.\n\n. Ibid.\n\n. Kamel Daoud, Meursault, contre-enqu\u00eate, Arles, Actes Sud, 2013.\n\n. Cf., dans Appels de Jacques Derrida (op. cit.), les textes de Philip Armstrong, \u00abL'abandon et l'\"\u00eatre-avec\": Derrida, un Fran\u00e7ais d'Alg\u00e9rie\u00bb, p. 279-301, et de Marta Segarra, \u00abUne politique et une po\u00e9tique de la m\u00e9moire: Jacques Derrida et H\u00e9l\u00e8ne Cixous\u00bb, p. 303-317.\n\n. J. Derrida, Le monolinguisme de l'autre ou La proth\u00e8se d'origine, Paris, Galil\u00e9e, coll. \u00abIncises\u00bb, 1996, p. 27.\n\n. J. Derrida, \u00abNo apocalypse, not now (\u00e0 toute vitesse, sept missives, sept missiles)\u00bb, dans Psych\u00e9. Inventions de l'autre, op. cit., p. 370; r\u00e9\u00e9d., op. cit., p. 402.\n\n. Emmanuel Kant, M\u00e9taphysique de la morale. III. Droit cosmopolite, tr. fr. Alain Renaut, Paris, Flammarion, 1994, p. 229. \n\n# II. De la dr\u00f4le de guerre \u00e0 la dr\u00f4le de paix\n\nTentons de prendre encore plus de recul. Douze ann\u00e9es apr\u00e8s la mort de Jacques Derrida, le moins qu'on puisse dire est que son h\u00e9ritage est contest\u00e9. Ce n'est pas qu'il fasse l'objet de r\u00e9cusations massives, mais plut\u00f4t qu'il est d\u00e9chir\u00e9 par des rivalit\u00e9s de disciples et des r\u00e9appropriations hasardeuses. \u00c0 l'\u00e9poque o\u00f9 la d\u00e9construction fleurissait dans les universit\u00e9s am\u00e9ricaines, dans les ann\u00e9es soixante-dix et quatre-vingt du si\u00e8cle dernier, on assista \u00e0 des exc\u00e8s de ces \u00abguerres culturelles\u00bb bien \u00e9voqu\u00e9es par un auteur que j'aime lire, Walter Kirn. Comme lui, j'ai pass\u00e9 un peu de temps \u00e0 Princeton et bien que mon s\u00e9jour remont\u00e2t de 2005 \u00e0 2006, j'ai reconnu l'ambiance qu'il d\u00e9crit et qui remonte aux ann\u00e9es quatre-vingt. \u00c0 cette \u00e9poque, Kirn faisait ses \u00e9tudes en litt\u00e9rature \u00e0 Princeton et il venait de d\u00e9couvrir le recueil Deconstruction and Criticism. Dans son livre de souvenirs, Lost in the Meritocracy, Kirn d\u00e9nonce le danger qui consistait \u00e0 avoir \u00e9t\u00e9 expos\u00e9 trop t\u00f4t aux analyses d\u00e9constructrices, qu'on pensait \u00e0 l'\u00e9poque sur les campus am\u00e9ricains comme une nouvelle arme aux pouvoirs terrifiants. En fait, si la t\u00e2che propos\u00e9e \u00e9tait de s'attaquer \u00e0 une \u00abhistoire de la m\u00e9taphysique\u00bb pour la \u00abd\u00e9construire\u00bb, ce concept g\u00e9n\u00e9ral ne renvoyait \u00e0 rien de pr\u00e9cis pour lui. Pour trouver des armes contre une tradition qui lui \u00e9tait inconnue et contre ceux qu'il jugeait vieux et d\u00e9mod\u00e9s, il n'avait plus qu'\u00e0 mimer un certain jargon. L'extravagance verbale de Derrida l'impressionnait, surtout lorsqu'il utilisait des mots comme \u00abh\u00e9t\u00e9ronomie\u00bb et \u00abinvagination\u00bb. Kirn montre bien comment le vocabulaire de la d\u00e9construction \u00e9tait devenu un effet de mode, une mode superficielle qui servait avant tout \u00e0 \u00e9tablir une reconnaissance de groupe:\n\nNous nous reconnaissions les uns les autres imm\u00e9diatement. [...] Nous parlions de \u00abjeu\u00bb et de \u00abtextualit\u00e9\u00bb et concluions, avant m\u00eame d'en avoir lu moins d'un centi\u00e8me, que la tradition canonique des textes de l'Occident \u00e9tait \u00abill\u00e9gitime\u00bb, un simple voile jet\u00e9 sur de puissants groupes d'int\u00e9r\u00eats constitu\u00e9s influences occultes et que notre devoir nous dictait de subvertir. [...] Nous passions tr\u00e8s vite de l'ignorance au r\u00e9visionnisme, d\u00e9construisant des domaines de savoir litt\u00e9raire que nous n'aurions jamais pu construire auparavant.\n\nCe reproche adress\u00e9 \u00e0 la d\u00e9construction am\u00e9ricaine n'est pas sans fondement. Je me suis trouv\u00e9 dans la situation identique de devoir d'abord construire une culture occidentale pour la d\u00e9construire ensuite dans l'esprit des \u00e9tudiants am\u00e9ricains. J'ai parl\u00e9 dans The Future of Theory de cette tentation perp\u00e9tuelle de trouver des \u00abarmes conceptuelles\u00bb ou des \u00abpuissantes perceuses \u00e9lectriques\u00bb dans la Th\u00e9orie afin d'attaquer les vieilles lunes de la critique litt\u00e9raire ou de la philosophie classique.\n\nEn r\u00e9alit\u00e9, Kirn a d'abord r\u00e9ussi dans ce jeu, mais tr\u00e8s vite il eut \u00e0 en payer le prix: une crise psychotique marqu\u00e9e par l'aphasie l'obligea \u00e0 revenir sur cette attitude adolescente et ultra-critique d\u00e9constructrice. Sa \u00abd\u00e9compensation\u00bb ne l'emp\u00eacha pas de terminer ses \u00e9tudes; il est maintenant un romancier qui a r\u00e9ussi \u00e0 percer (il est l'auteur de plusieurs bons sc\u00e9narios de films), non un professeur de th\u00e9orie dans une universit\u00e9. Son t\u00e9moignage nous rappelle le prix \u00e0 payer pour la critique id\u00e9ologique \u2013 la d\u00e9nonciation na\u00efve de la tradition litt\u00e9raire et philosophique con\u00e7ues comme de grandes conjurations aux mains de quelques \u00abdead white males\u00bb appartient \u00e0 une p\u00e9riode qui n'est pas termin\u00e9e aux \u00c9tats-Unis du moins, puisque l'on assiste en ce moment \u00e0 une remont\u00e9e du \u00abpolitically correct\u00bb, qui a choisi de nouvelles cibles, ainsi les po\u00e8tes Vanessa Place et Kenny Goldsmith, r\u00e9cemment accus\u00e9s de racisme \u00e0 contresens.\n\nLe recueil Deconstruction and Criticism date de 1979 et s'il a marqu\u00e9 son temps, il se trouve aussi marqu\u00e9 par lui: il remonte \u00e0 l'\u00e9poque o\u00f9 l'on pouvait parler d'une \u00ab\u00e9cole de Yale\u00bb qui aurait \u00e9t\u00e9 inspir\u00e9e par Derrida. On le sait, le lieu s'est d\u00e9plac\u00e9 par la suite, prenant les \u00c9tats-Unis en tenaille entre la c\u00f4te Est et la c\u00f4te Ouest, passant de New York avec NYU \u00e0 Irvine, en Californie. Pour commencer en un point pr\u00e9cis, revenons \u00e0 Paris en 1967, quand une triple d\u00e9flagration secoua le monde des lettres et de la philosophie: la publication presque simultan\u00e9e de De la grammatologie, L'\u00e9criture et la diff\u00e9rence et La voix et le ph\u00e9nom\u00e8ne. J'\u00e9tais un jeune \u00e9tudiant \u00e0 cette \u00e9poque et je n'oublierai jamais le sentiment d'une imm\u00e9diate et puissante onde de choc. Ce sentiment a \u00e9t\u00e9 \u00e9voqu\u00e9 avec verve par Emmanuel Levinas qui, dans une page c\u00e9l\u00e8bre, compare l'effet de la d\u00e9construction \u00e0 la soudaine panique et au vide du pouvoir ressentis en France pendant l'exode qui suivit la victoire de l'arm\u00e9e allemande en 1940:\n\nAu d\u00e9part, tout est en place, au bout de quelques pages ou de quelques alin\u00e9as, sous l'effet d'une redoutable mise en question, rien n'est plus habitable pour la pens\u00e9e. C'est l\u00e0, en dehors de la port\u00e9e philosophique des propositions, un effet purement litt\u00e9raire, le frisson nouveau, la po\u00e9sie de Derrida. Je revois toujours en le lisant l'exode de 1940. L'unit\u00e9 militaire en retraite arrive dans une localit\u00e9 qui ne se doute encore de rien, o\u00f9 les caf\u00e9s sont ouverts, o\u00f9 les dames sont aux \u00abNouveaut\u00e9s pour dames\u00bb, o\u00f9 les coiffeurs coiffent, les boulangers boulangent, les vicomtes rencontrent d'autres vicomtes et se racontent des histoires de vicomtes, et o\u00f9 tout est d\u00e9construit et d\u00e9sol\u00e9 une heure apr\u00e8s, les maisons, ferm\u00e9es ou laiss\u00e9es portes ouvertes, se vident des habitants qu'entra\u00eene un courant de voitures et de pi\u00e9tons \u00e0 travers les rues restitu\u00e9es \u00e0 leur \u00abprofond jadis\u00bb de routes, trac\u00e9es dans un pass\u00e9 imm\u00e9morial par les grandes migrations.\n\nCette analogie historique \u00e9voque un moment de fin de monde ou de suspension de l'id\u00e9e de la France. L'image dramatise les questions par lesquelles Derrida suspendait la fondation de toutes les fondations, notre croyance en la pr\u00e9sence \u00e0 soi de la conscience.\n\nLevinas ajoute une vignette suppl\u00e9mentaire: entre Paris et Alen\u00e7on, un coiffeur \u00e0 moiti\u00e9 ivre invite les soldats qui passent \u00e0 se faire raser gratuitement:\n\nAvec ses deux compagnons, il rasait gratis et ce fut aujourd'hui. La procrastination essentielle \u2013 la future diff\u00e9rance \u2013 se r\u00e9sorbait dans le pr\u00e9sent. Le temps arrivait \u00e0 sa fin avec la fin ou avec l'int\u00e9rim de la France. \u00c0 moins que le coiffeur ne fut aussi d\u00e9lirant que la quatri\u00e8me forme du d\u00e9lire du Ph\u00e8dre o\u00f9, depuis Platon, se tient le discours de la m\u00e9taphysique occidentale. (NP, 83)\n\nDans cet essai de 1973, Levinas revient vers une sc\u00e8ne qui figure d\u00e9j\u00e0 dans son roman avort\u00e9, qu'il intitulait soit Eros, soit Triste Opulence . Il s'agit de ce moment o\u00f9 la traditionnelle blague \u00abDemain, on rase gratis!\u00bb (et l'on ne rase jamais puisque c'est toujours le lendemain) se voit invers\u00e9e ironiquement. Mais ici la r\u00e9f\u00e9rence \u00e0 la quatri\u00e8me forme de la mania qui est celle d'\u00c9ros ou d'Aphrodite, la folie de l'amour, prend la forme assez surprenante d'un amour des autres, une philia g\u00e9n\u00e9reuse de coiffeur d\u00e9sireux de faire co\u00efncider le temps pr\u00e9sent et la promesse de sa g\u00e9n\u00e9rosit\u00e9.\n\nTout le d\u00e9but du roman inachev\u00e9 se pr\u00e9sente comme une m\u00e9ditation sur la diff\u00e9rence entre la Grande Guerre, celle de 1914-18, et la guerre actuelle, une dr\u00f4le de guerre durant neuf mois, mais se terminant en un rapide et absurde exode sur les routes mitraill\u00e9es. M\u00eame si Levinas se souvient de la boucherie de la Grande Guerre, l'ordre et la raison r\u00e9gnaient encore. Ce n'est plus le cas apr\u00e8s la d\u00e9route de 1940 sous les coups de boutoir de l'arm\u00e9e allemande. Levinas \u00e9voque ces images traumatiques dans son roman:\n\nIl y eut un instant de vide total entre la disparition de la France et la reapparition [sic] de la France, un instant de d\u00e9faite o\u00f9 rien ne se refaisait encore \u2013 un vide vertigineux, un interr\u00e8gne, un hiatus, l'intervalle absolu. Tout devait se d\u00e9cider \u00e0 ce moment-l\u00e0 o\u00f9 l'encha\u00eenement des causes morales s'est interrompu, o\u00f9 les institutions n'avancent plus aucune signification, o\u00f9 chacun se trouvait dans la situation de Jeanne d'Arc se demandant o\u00f9 \u00e9tait le Roi.\n\nC'est de mani\u00e8re symptomatique, mais aussi en connaissance de cause, que Levinas parle de l'effet purement litt\u00e9raire de Derrida, de sa po\u00e9sie. Si ces termes saluent sa propre tentative pour donner forme litt\u00e9raire \u00e0 ces ann\u00e9es sombres, ces mots \u00e9crits en 1973 ne peuvent manquer de trahir une certaine r\u00e9serve ou impatience. Levinas \u00e9voque le premier texte qui lui fut consacr\u00e9 par Derrida et dans lequel celui-ci mettait \u00e0 mal toute sa tentative pour d\u00e9passer la philosophie grecque et aller vers une \u00e9thique de l'Autre dans Totalit\u00e9 et Infini.\n\nCe livre de 1961 commence par une m\u00e9ditation sur la condition de guerre et la condition de paix: \u00abLa lucidit\u00e9 \u2013 ouverture de l'esprit sur le vrai \u2013 ne consiste-t-elle pas \u00e0 entrevoir la possibilit\u00e9 permanente de la guerre? L'\u00e9tat de guerre suspend la morale; il d\u00e9pouille les institutions et les obligations \u00e9ternelles de leur \u00e9ternit\u00e9 et, d\u00e8s lors, annule, dans le provisoire, les inconditionnels imp\u00e9ratifs.\u00bb La suite de cette analyse rejette le faux r\u00e9alisme, vrai cynisme en fait, qui dicte que la guerre r\u00e9v\u00e8le la face de l'\u00eatre v\u00e9ritable. La guerre n'est que l'expression de ce concept de totalit\u00e9 qui pr\u00e9tend r\u00e9gir toutes les activit\u00e9s humaines. Si la guerre s'impose par son \u00e9vidence et sa violence destructrice, Levinas esp\u00e8re opposer une plus grande violence, celle de l'infini. La v\u00e9ritable pens\u00e9e de l'\u00eatre entra\u00eene une nouvelle subjectivit\u00e9 marqu\u00e9e par l'hospitalit\u00e9 \u00e0 cet exc\u00e8s, cette inad\u00e9quation, qui provient de l'Infini qu'on peut accueillir en soi.\n\nPour Levinas, la guerre est la g\u00e9n\u00e9ralisation du meurtre, tous deux reconnaissant la transcendance du visage. C'est que, dans Totalit\u00e9 et Infini, la postulation de la paix est premi\u00e8re: \u00abLa guerre suppose la paix, la pr\u00e9sence pr\u00e9alable et non allergique d'Autrui; elle ne marque pas le premier \u00e9v\u00e9nement de la rencontre.\u00bb Raoul Moati commente ces phrases en ces termes:\n\nQue la guerre \u00abne marque pas le premier \u00e9v\u00e9nement de la rencontre\u00bb signifie qu'il existe une dissym\u00e9trie entre la guerre et la paix au sens o\u00f9 la guerre n'est pas l'exact contraire de la paix, mais d\u00e9j\u00e0 la paix et le discours inad\u00e9quats \u00e0 eux-m\u00eames. C'est en ce sens que l'horizon de la guerre ne sature jamais exhaustivement la relation \u00e0 autrui. Il est constitutif de la guerre qu'elle n'\u00e9puise pas celle-ci, autrement dit que le discours \u2013 dont la guerre \u00e9mane \u2013 puisse s'accomplir selon son mode originaire comme paix.\n\nUn certain optimisme de la Raison, tel que le trouvons chez Kant dans le passage cit\u00e9 plus haut, soutient que la guerre n'est qu'un long d\u00e9tour dans l'erreur, un t\u00e2tonnement dans la nuit avant de comprendre que ce que le Visage de l'Autre rec\u00e8le, c'est sa puissance de d\u00e9boucher sur la paix. Lorsqu'en 1987 Levinas pr\u00e9sente Totalit\u00e9 et Infini dans sa version allemande, il insiste derechef sur ce point: \u00abLa raison jusqu'au bout ou la paix entre les hommes. [...] Le probl\u00e8me de la paix et de la raison est abord\u00e9 dans Totalit\u00e9 et Infini \u00e0 partir d'une conjoncture diff\u00e9rente et plus ancienne sans doute.\u00bb Ce que cette r\u00e9f\u00e9rence \u00e0 une pens\u00e9e plus ancienne recouvre est l'id\u00e9e centrale de Levinas que ce n'est qu'en d\u00e9passant la cat\u00e9gorie ontologique de la totalit\u00e9 que l'on peut arriver \u00e0 une appr\u00e9hension ph\u00e9nom\u00e9nologique du visage de l'autre, lequel rec\u00e8le en sa nudit\u00e9 un irr\u00e9sistible appel \u00e0 la paix: \u00abGratuit\u00e9 de la transcendance-\u00e0-l'autre interrompant l'\u00eatre toujours pr\u00e9occup\u00e9 de cet \u00eatre-m\u00eame et de sa pers\u00e9v\u00e9rance dans l'\u00eatre. Interruption absolue de l'onto-logie, mais dans l'un-pour-l'autre de la saintet\u00e9, de la proximit\u00e9, de la socialit\u00e9, de la paix.\u00bb\n\nQuand Derrida commente ce grand et beau livre, c'est avant tout pour demander comment Levinas pense vraiment qu'il va \u00e9viter, d\u00e9passer ou exc\u00e9der la m\u00e9taphysique occidentale. Comment peut-on laisser derri\u00e8re soi le discours de la philosophie grecque avec toute sa grammaire de cat\u00e9gories telles que celles de \u00absujet\u00bb ou de \u00abtotalit\u00e9\u00bb? Dans ce contexte, une s\u00e9rie de questions visent le motif de la guerre:\n\nIl n'y a de guerre qu'apr\u00e8s l'ouverture du discours et la guerre ne s'\u00e9teint qu'avec la fin du discours. La paix, comme le silence, est la vocation \u00e9trange d'un langage appel\u00e9 hors de soi par soi. Mais comme le silence fini est aussi l'\u00e9l\u00e9ment de la violence, le langage ne peut jamais que tendre ind\u00e9finiment vers la justice en reconnaissant et en pratiquant la guerre en soi. Violence contre violence. \u00c9conomie de violence. \u00c9conomie qui ne peut se r\u00e9duire \u00e0 ce que Levinas vise sous ce mot.\n\nC'est un argument que Derrida reprendra souvent, notamment \u00e0 propos de Freud, de Bataille ou de Marcel Mauss: tant que l'on peut parler d'\u00e9conomie, en effet, on reste dans le discours ontologique \u00abgrec\u00bb. Derrida conclut ce d\u00e9veloppement dans un sens \u00e0 la fois plus historique et plus litt\u00e9raire:\n\nSi [...] le mouvement de transcendance m\u00e9taphysique est histoire, il est encore violent, car, c'est l'\u00e9vidence l\u00e9gitime dont s'inspire toujours Levinas, l'histoire est violence. La m\u00e9taphysique est \u00e9conomie: violence contre violence, lumi\u00e8re contre lumi\u00e8re: la philosophie (en g\u00e9n\u00e9ral). Dont on peut dire, en transposant l'intention de Claudel, que tout y \u00abest peint sur la lumi\u00e8re comme avec de la lumi\u00e8re condens\u00e9e, comme l'air qui devient du givre.\u00bb Ce devenir est la guerre. Cette pol\u00e9mique est le langage m\u00eame. Son inscription.\n\nDerrida replace Levinas dans son polemos occult\u00e9, ce qu'il fera \u00e0 plusieurs reprises dans des textes ult\u00e9rieurs; il objecte \u00e0 Levinas que son discours pacifiste est en fait tr\u00e8s violent. L'arme secr\u00e8te de Derrida est Hegel. C'est Hegel qui montre l'exact contraire de ce qu'avance Levinas: tout discours de paix contient en germe la promesse d'une guerre \u00e0 venir: \u00abC'est pourquoi cet aveu de la guerre dans le discours, aveu qui n'est pas encore la paix, signifie le contraire d'un bellicisme; dont on sait bien \u2013 et qui l'a mieux montr\u00e9 que Hegel? \u2013 que l'ir\u00e9nisme est dans l'histoire son meilleur complice.\u00bb Toutes ces remarques et analyses tendent \u00e0 pousser la pens\u00e9e de Levinas dans ses retranchements, et son ultime ligne Maginot sera la r\u00e9f\u00e9rence \u00e0 la th\u00e9ologie, soit \u00e0 la relation \u00e0 l'infini du Tout Autre d\u00e9couverte \u00e0 travers l'alt\u00e9rit\u00e9 absolue du visage humain.\n\nNulle \u00e9chappatoire n'est possible. Derrida contourne cette ligne de d\u00e9fense en montrant que nous ne pouvons pas quitter le domaine philosophique pour nous r\u00e9fugier dans une religion qui offrirait un recours contre la violence humaine de la guerre. C'est au point o\u00f9 l'\u00e9thique se renverse en th\u00e9ologie que Derrida surprend Levinas et lui d\u00e9coche une fl\u00e8che encore plus f\u00e9roce, car il montre que l'invocation de Dieu elle-m\u00eame pr\u00e9suppose la guerre \u2013 ce qui vient contredire radicalement l'affirmation majeure de Levinas, son point de d\u00e9part en fait:\n\nC'est pourquoi Dieu seul emp\u00eache le monde de Levinas d'\u00eatre celui de la pire et pure violence, le monde de l'immoralit\u00e9 elle-m\u00eame. Les structures de l'exp\u00e9rience vive et nue que d\u00e9crit Levinas sont celles m\u00eames d'un monde o\u00f9 la guerre ferait rage \u2013 \u00e9trange conditionnel \u2013 si l'infiniment autre n'\u00e9tait pas l'infini, s'il \u00e9tait d'aventure un homme nu, fini et seul.\n\nDerrida ne s'attaque pas aussit\u00f4t \u00e0 cet \u00ab\u00e9trange conditionnel\u00bb qui d\u00e9tonne face \u00e0 notre savoir r\u00e9aliste concernant l'histoire, qu'elle soit r\u00e9cente ou universelle. Non, il choisit de frapper au c\u0153ur de cette pr\u00e9supposition religieuse et force Levinas \u00e0 se trouver bloqu\u00e9 dans une aporie g\u00e9n\u00e9rale. Derrida poursuit:\n\nMais dans ce cas, dirait sans doute Levinas, il n'y aurait m\u00eame plus de guerre, car il n'y aurait ni visage ni v\u00e9ritable asym\u00e9trie. Il ne s'agirait donc plus de l'exp\u00e9rience nue et vive o\u00f9 Dieu a d\u00e9j\u00e0 commenc\u00e9 de parler. Autrement dit, dans un monde o\u00f9 le visage serait pleinement respect\u00e9 (comme ce qui n'est pas du monde), il n'y aurait plus de guerre. Dans un monde o\u00f9 le visage ne serait absolument plus respect\u00e9, o\u00f9 il n'y aurait plus de visage, il n'y aurait plus lieu de guerre. Dieu est donc m\u00eal\u00e9 \u00e0 la guerre. Son nom est aussi, comme le nom de la paix, une fonction dans le syst\u00e8me de la guerre, le seul \u00e0 partir duquel nous puissions parler, le seul dont le langage puisse jamais parler. Sans Dieu et avec Dieu, il n'y aurait pas la guerre. Celle-ci suppose et exclut Dieu.\n\nDans cette prosopop\u00e9e de Levinas, Derrida se met \u00e0 parler \u00e0 sa place pour refuser \u00e0 Dieu son r\u00f4le de fondement pacifiste. De fait, la plupart des guerres se sont d\u00e9clench\u00e9es ou ont \u00e9t\u00e9 men\u00e9es au nom de Dieu, lequel \u00e9tait cens\u00e9 b\u00e9nir un camp plut\u00f4t que l'autre: God is on our side, In hoc signo vinces! Mais cette projection ne date pas du monde chr\u00e9tien et de ses croisades \u2013 on trouve d\u00e9j\u00e0 dans l'Iliade l'intuition forte que les h\u00e9ros qui s'affrontent ne sont que des pantins manipul\u00e9s par des dieux qui se font la guerre par leur interm\u00e9diaire. Il est temps d'en revenir au r\u00e9alisme auquel nous contraint l'histoire: \u00abLa guerre \u2013 car il y a la guerre \u2013 est donc la diff\u00e9rence entre le visage et le monde fini sans visage. Mais cette diff\u00e9rence, n'est-ce pas ce que toujours on a appel\u00e9 le Monde, dans lequel joue l'absence-pr\u00e9sence de Dieu?\u00bb\n\nIl \u00e9tait logique que le long essai de 1964 consacr\u00e9 \u00e0 Levinas se termin\u00e2t par une invocation \u00e0 James Joyce. Derrida cite Ulysses et son \u00abJewgreek is greekjew\u00bb. En 1984, Derrida reviendra vers Joyce, prenant comme exemple premier du polemos de Finnegans Wake le syntagme \u00abhe war\u00bb dans lequel il choisit de retrouver la formule joycienne du nom de Dieu. Encore une fois, Dieu serait \u00e0 penser comme un Polemos premier. Derrida per\u00e7oit dans Finnegans Wake une interrogation centrale, prenant le syntagme \u00abhe war\u00bb comme une synecdoque qui all\u00e9gorise l'ensemble du livre. \u00abHe war\u00bb en donnerait un raccourci, nous menant vers ses perspectives ultimes. Voici la conclusion de \u00abDeux mots pour Joyce\u00bb:\n\nHe war \u2013 la signature de Dieu. En donnant la loi, et la langue, c'est-\u00e0-dire les langues, il a d\u00e9clar\u00e9 la guerre. [...]\n\nHe war: la citation de cette signature rejoue toute la m\u00e9moire du monde, dans Finnegans Wake. [...]\n\nHe war, c'est une contresignature, elle confirme et elle contredit, elle efface en souscrivant. Elle dit \u00abnous\u00bb et \u00aboui\u00bb pour finir au P\u00e8re ou au Seigneur qui parle haut \u2013 il n'y a gu\u00e8re que Lui \u2013, mais laisse ici le dernier mot \u00e0 la femme qui \u00e0 son tour aura dit nous et oui.\n\nDieu contresign\u00e9, Dieu qui te signe en nous, laisse-nous rire, amen, sic, si, oc, o\u00efl .\n\nDans mon exemplaire d\u00e9dicac\u00e9 par Derrida, on distingue nettement apr\u00e8s \u00abqui te signe\u00bb un petit \u00abs\u00bb ajout\u00e9 \u00e0 l'encre bleue \u00e0 \u00absigne\u00bb. Par cet ajout manuscrit, Derrida a voulu corriger une faute d'impression. Il faut laisser \u00e0 Dieu la possibilit\u00e9 d'une deuxi\u00e8me personne, comme s'il \u00e9non\u00e7ait que son \u00absigne en nous\u00bb est bien la guerre et non la paix: \u00abTu me guerres!\u00bb\n\nSi Derrida arrive \u00e0 d\u00e9montrer que pour Levinas \u00abLa guerre est donc cong\u00e9nitale \u00e0 la ph\u00e9nom\u00e9nalit\u00e9\u00bb et que sa philosophie ne peut se passer de la ph\u00e9nom\u00e9nalit\u00e9 du visage dans la proximit\u00e9 de la deuxi\u00e8me personne, il pourra renvoyer Levinas \u00e0 un polemos premier. Ceci consisterait donc \u00e0 penser avec Levinas et contre lui \u00e0 la fois un exercice auquel Levinas lui-m\u00eame n'\u00e9tait pas \u00e9tranger. Mais il va falloir auparavant s'entendre sur ce que signifie la ph\u00e9nom\u00e9nologie du visage et, l\u00e0 encore, ce sera une pol\u00e9mique qui nous y donnera acc\u00e8s.\n\n. Harold Bloom, Paul de Man, Jacques Derrida, Geoffrey H. Hartman et J. Hillis Miller, Deconstruction and Criticism, Londres, Routledge, 1979; r\u00e9\u00e9d., New York et Londres, Continuum International Publishing Group, 1995.\n\n. Walter Kirn, Lost in the Meritocracy: The Undereducation of an Overachiever, New York, Doubleday, 2009, p. 120.\n\n. Ibid., p. 121.\n\n. Jean-Michel Rabat\u00e9, The Future of Theory, Oxford, Wiley-Blackwell, coll. \u00abBlackwell Manifestos\u00bb, 2002, p. 100-101 et p. 159, note 7.\n\n. Emmanuel Levinas, \u00abJacques Derrida\u00bb, dans Noms propres, Montpellier, Fata Morgana, 1976, p. 82-83. D\u00e9sormais abr\u00e9g\u00e9 en NP, suivi du num\u00e9ro de la page. (Le texte a d'abord paru dans L'Arc, \u00abJacques Derrida\u00bb, no 54, 1973, p. 33-37; repris sous le titre \u00abTout autrement\u00bb, dans le Cahier de L'Herne Derrida, M.-L. Mallet et G. Michaud (dir.), Paris, \u00c9ditions de L'Herne, no 83, 2004, p. 16-19.)\n\n. E. Levinas, Eros ou Triste Opulence, dans \u0152uvres 3, Eros, litt\u00e9rature et philosophie. Essais romanesques et po\u00e9tiques, notes philosophiques sur le th\u00e8me d'\u00e9ros, volume publi\u00e9 sous la responsabilit\u00e9 de Jean-Luc Nancy et Danielle Cohen-Levinas, Paris, Bernard Grasset\/IMEC, 2013, p. 79.\n\n. E. Levinas, Eros ou Triste Opulence, dans \u0152uvres 3, Eros, litt\u00e9rature et philosophie, op. cit., p. 74.\n\n. E. Levinas, Totalit\u00e9 et Infini. Essai sur l'ext\u00e9riorit\u00e9, La Haye, Martinus Nijhoff, 1961, \u00e9d. rev. 1965, p. IX.\n\n. Ibid., p. 175.\n\n. R. Moati, \u00c9v\u00e9nements nocturnes. Essai sur Totalit\u00e9 et Infini, Paris, Hermann, 2012, p. 275.\n\n. E. Levinas, \u00abTotalit\u00e9 et Infini\u00bb, pr\u00e9face \u00e0 l'\u00e9dition allemande, dans Entre nous. Essais sur le penser-\u00e0-l'autre, Paris, Grasset, 1991, p. 232.\n\n. Ibid., p. 233.\n\n. J. Derrida, \u00abViolence et m\u00e9taphysique. Essai sur la pens\u00e9e d'Emmanuel Levinas\u00bb [1964], dans L'\u00e9criture et la diff\u00e9rence, Paris, Seuil, coll. \u00abTel Quel\u00bb, 1967, p. 172.\n\n. Ibid., p. 173.\n\n. J. Derrida, \u00abViolence et m\u00e9taphysique\u00bb, dans L'\u00e9criture et la diff\u00e9rence, op. cit., p. 173.\n\n. Ibid., p. 158.\n\n. Ibid.\n\n. J. Derrida, \u00abViolence et m\u00e9taphysique\u00bb, dans L'\u00e9criture et la diff\u00e9rence, op. cit., p. 158.\n\n. Ibid., p. 228. Cf. la longue note 1 de cette page o\u00f9 Derrida commente ce passage d'Ulysses (p. 622) de Joyce.\n\n. J. Derrida, \u00abDeux mots pour Joyce\u00bb, dans Ulysse gramophone, Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 1987, p. 48 et 53.\n\n. On ne peut manquer de penser \u00e0 Proust, sur qui je reviendrai, dans la mesure o\u00f9 il donne \u00e0 entendre dans le signifiant de la famille incarnant la plus haute distinction aristocratique, objet de tous les d\u00e9sirs de r\u00e9ussite sociale de son roman, que les \u00abguerres mentent\u00bb: tel est bien l'\u00e9cho assourdi du nom presque sacr\u00e9 des Guermantes.\n\n. J. Derrida, \u00abViolence et m\u00e9taphysique\u00bb, dans L'\u00e9criture et la diff\u00e9rence, op. cit., p. 190.\n\n# III. Chien et chat: de Levinas \u00e0 Derrida,   \nla bellig\u00e9rance de l'Autre\n\nDans ce chapitre, je vais suivre de pr\u00e8s les cons\u00e9quences du polemos derridien quand il s'exerce \u00e0 l'encontre d'un philosophe qui l'a pr\u00e9c\u00e9d\u00e9 dans une certaine voie et avec un certain vocabulaire (la trace, l'alt\u00e9rit\u00e9, une ph\u00e9nom\u00e9nologie fine du corps, etc.), mais qu'il a refus\u00e9 de suivre jusqu'au bout. En m\u00eame temps, quiconque a lu l'ensemble des deux \u0153uvres ne peut manquer d'\u00eatre frapp\u00e9 par la mani\u00e8re irr\u00e9versible avec laquelle Derrida s'est approch\u00e9 de la philosophie de Levinas, au point que le tout dernier Derrida semble issu textuellement de Levinas. C'est peut-\u00eatre cette exorbitante proximit\u00e9 qui explique les attaques de plus en plus dures de Derrida \u00e0 l'endroit de Levinas: en fait, cette discussion \u00e9tait une fa\u00e7on de se faire la guerre \u00e0 lui-m\u00eame. Peut-\u00eatre vaudrait-il mieux imaginer que les deux philosophes avaient choisi des animaux tot\u00e9miques incompatibles: \u00e0 Levinas, revenait le chien, \u00e0 Derrida, le chat. Voyons pourquoi ces \u00abanimaux autobiographiques\u00bb se font la guerre.\n\nL'intervention de Jacques Derrida lors de la d\u00e9cade de Cerisy consacr\u00e9e \u00e0 \u00abL'animal autobiographique\u00bb en juillet 1997 reprit un d\u00e9bat sur les droits des animaux dans lequel les tenants d'une certaine bio\u00e9thique, tels Peter Singer, tendaient \u00e0 s'enfermer. Deux ans plus tard, le vibrant plaidoyer de J. M. Coetzee, The Lives of Animals, poussa ces investigations dans le domaine de la fiction autant que de l'\u00e9thique. Ce point m'am\u00e8nera \u00e0 reparler de son \u0153uvre un peu plus loin; pour le moment je voudrais parler d'un chien de guerre, un chien qui est le pendant du chat avec lequel Derrida commence sa conf\u00e9rence \u00e0 Cerisy.\n\nOn le sait, l'intervention de Jacques Derrida en juillet 1997 partait d'une exp\u00e9rience personnelle: que ressentir, que penser, comment se d\u00e9finir lorsque l'on se d\u00e9place nu, par exemple dans une salle de bains, face \u00e0 l'animal familier qu'est un chat? Tel est le point de d\u00e9part du livre posthume de 2006, L'animal que donc je suis. Les Actes du colloque de Cerisy, dirig\u00e9s par Marie-Louise Mallet, ont conserv\u00e9 l'ordre des interventions. Derrida ne donne pas le coup d'envoi, intervenant non pas au d\u00e9but de cette d\u00e9cade, mais apr\u00e8s une dizaine de pr\u00e9sentations. Sa longue conf\u00e9rence fut suivie de celle d'Alain David sur le chien de Levinas. Entre temps, Derrida avait pr\u00e9cis\u00e9 que son expos\u00e9 n'\u00e9tait qu'une introduction et qu'il comptait donner quatre s\u00e9minaires avec des lectures de textes de Descartes, Kant, Heidegger, Levinas et Lacan au cours des jours suivants. L'intervenant qui suivit Derrida fut Alain David, qui donna une belle lecture du texte de Levinas \u00abNom d'un chien ou le droit naturel\u00bb. Alain David commente de mani\u00e8re inspir\u00e9e un r\u00e9cit atypique pour Levinas, puisqu'il universalise une exp\u00e9rience autobiographique datant de la guerre. Le r\u00e9cit de Levinas nous ram\u00e8ne aux ann\u00e9es de guerre quand il \u00e9tait prisonnier en Allemagne, prot\u00e9g\u00e9 du g\u00e9nocide antis\u00e9mite par l'uniforme de l'arm\u00e9e fran\u00e7aise. Levinas, qui faisait partie de la dixi\u00e8me arm\u00e9e fran\u00e7aise, avait \u00e9t\u00e9 captur\u00e9 en juin 1940 puis intern\u00e9 dans un stalag r\u00e9serv\u00e9 aux soldats juifs.\n\nLes biographies qui lui sont consacr\u00e9es s'accordent toutes sur ce point: Levinas n'a pas invent\u00e9 ce chien embl\u00e9matique. Les auteurs de biographies de Levinas, Marie-Anne Lescourret, Salomon Malka et Fran\u00e7ois Poiri\u00e9, acceptent l'authenticit\u00e9 des d\u00e9tails qui figurent dans sa narration avec toutes ses troublantes co\u00efncidences. Le commando des soldats juifs portait bien le num\u00e9ro 1492, comme pour \u00e9voquer l'ann\u00e9e de l'expulsion des Juifs d'Espagne. Il comptait exactement soixante-dix prisonniers, comme les r\u00e9dacteurs de la Septante traduisant la Bible. Surtout, aux yeux des Allemands qui les gardaient et les entouraient, ce groupe n'\u00e9tait pas reconnu comme pleinement \u00abhumain\u00bb. Ceci am\u00e8ne Levinas \u00e0 commenter: \u00abNous n'\u00e9tions qu'une quasi-humanit\u00e9, une bande de singes.\u00bb\n\nUn chien errant venait jouer avec les prisonniers pour un peu de nourriture, puis se mit \u00e0 les accompagner r\u00e9guli\u00e8rement, surtout \u00e0 leur retour des corv\u00e9es de travail. Les prisonniers, par jeu, appel\u00e8rent le chien \u00abBobby\u00bb. Pour Bobby, qui semblait priv\u00e9 de ma\u00eetre et de toit, et qui survivait aux alentours du camp, les prisonniers \u00e9taient indubitablement des \u00eatres humains. Levinas conclut ce texte par une magnifique \u00e9vocation:\n\nLe chien qui reconnut Ulysse sous le d\u00e9guisement \u00e0 son retour de l'Odyss\u00e9e, \u00e9tait-il le parent du n\u00f4tre? Mais non! Mais non! L\u00e0-bas, ce fut l'Ithaque et la patrie. Ici, ce fut nulle part. Dernier kantien de l'Allemagne nazie, n'ayant pas le cerveau qu'il faut pour universaliser les maximes de ses pulsions, il descendait des chiens d'\u00c9gypte. Et son aboiement d'ami \u2013 foi d'animal \u2013 naquit dans le silence de ses a\u00efeux des bords du Nil.\n\nAlain David affirme avec raison que c'est un des plus beaux textes de Levinas. Ces beaut\u00e9s n'emp\u00eachent pas Derrida de signaler sans aucune piti\u00e9 ses aspects probl\u00e9matiques, car il voit cet essai tendre trop facilement vers l'all\u00e9gorie. Derrida r\u00e9sistera violemment \u00e0 ces identifications forc\u00e9es, surtout quand Levinas finit par rejeter une image hom\u00e9rique qu'il avait d'abord suscit\u00e9e pour placer \u00abBobby le chien\u00bb dans le contexte de l'Exode biblique.\n\nTout le texte met en sc\u00e8ne la question de la nomination d'un chien et Alain David lui-m\u00eame semble h\u00e9siter entre une d\u00e9finition du chien comme \u00abvie sans la forme, le bios sans le logos \u00bb, ce qui sugg\u00e8re qu'il y a de l'animal en chacun de nous, et une d\u00e9finition du \u00abchien selon la lettre\u00bb, un chien biblique, presque un exil\u00e9 de l'Exode. La lettre juive mettrait le chien Bobby \u00e0 l'abri de la biologie si la biologie en tant que discours implique bien la \u00abr\u00e9duction d'un bios, d'une zo\u00e9, \u00e0 un logos \u00bb. Je ne peux r\u00e9sumer ici cet essai riche et \u00e9mouvant qui prolonge et nuance les affirmations de Levinas.\n\nJe note que ce commentaire sympathique et profond ne pr\u00e9pare aucunement \u00e0 la violente attaque de Derrida contre l'essai de Levinas dans le recueil de Cerisy et ensuite dans le livre L'animal que donc je suis. Derrida commence par remarquer qu'il est \u00e9tonnant que Levinas n'ait pas fait une plus grande place \u00e0 l'animal dans sa philosophie, ce qui trahirait son enfermement dans une probl\u00e9matique du sujet humain, qui seul serait dou\u00e9 de parole et porteur de \u00abvisage\u00bb. L'animal serait d'embl\u00e9e exclu du positionnement \u00e9thique de Levinas qui refuserait de reconna\u00eetre un Autre, un prochain ou un fr\u00e8re en l'animal. La critique apparemment limit\u00e9e \u00e0 ce domaine est en fait centrale, car elle pose la question du visage, donc celle de toute la ph\u00e9nom\u00e9nologie de l'autre. Peut-on admettre que pour Levinas un animal serait d\u00e9pourvu de visage?\n\nDerrida cite un dialogue entre John Llewelyn et Levinas au cours duquel ce dernier insistait sur la diff\u00e9rence irr\u00e9ductible du visage humain et confessait ceci: \u00abJe ne sais pas si le serpent a un visage. Je ne peux pas r\u00e9pondre \u00e0 cette question. Une analyse plus sp\u00e9cifique est n\u00e9cessaire.\u00bb Notons la prudence de Levinas en ce qui concerne le serpent. Derrida cite une autre conversation avec Robert Bernasconi et David Wood dans laquelle Levinas parle d'une extension de l'\u00e9thique humaine aux animaux: nous ne voulons pas les faire souffrir inutilement parce que nous voyons en eux, par une sorte de catachr\u00e8se en quelque sorte, un prolongement de nous-m\u00eames. Derrida glose longuement l'incapacit\u00e9 dans laquelle Levinas avoue se trouver face \u00e0 cette question troublante: un serpent a-t-il ou non un visage?\n\nDerrida insiste sans piti\u00e9: \u00abCar d\u00e9clarer ne pas savoir o\u00f9 commence le droit d'\u00eatre appel\u00e9 \"visage\", c'est confesser qu'on ne sait pas au fond ce qu'est un visage, ce que veut dire ce mot, ce qui en r\u00e8gle l'usage; et c'est donc confesser qu'on ne disait pas ce que c'est que r\u00e9pondre.\u00bb C'est toute une mise en cause de la philosophie de Levinas qui est impliqu\u00e9e ici si l'on admet que sa philosophie tout enti\u00e8re repose sur une \u00e9thique de l'Autre se d\u00e9ployant \u00e0 partir du th\u00e8me du visage. Comme toujours, le geste de Derrida est celui de la philosophie m\u00eame: il faut arr\u00eater une d\u00e9monstration avant qu'elle ne s'emballe et se constitue en th\u00e8se ou discours, et ce, en posant la question de ce qui a \u00e9t\u00e9 oubli\u00e9 ou simplement pr\u00e9suppos\u00e9 dans le point de d\u00e9part initial.\n\nLoin de r\u00e9pondre \u00e0 la question sur le visage de l'animal, interrogation qu'il laisse en suspens comme si elle ne pouvait se contenter de l'affirmative ou de la n\u00e9gative, Derrida cite la c\u00e9l\u00e8bre \u00ab\u00c9bauche d'un serpent\u00bb de Paul Val\u00e9ry, et par-del\u00e0 Val\u00e9ry, tous les po\u00e8tes qui ont fait du serpent le visage de la s\u00e9duction et du mal radical. Il ne cite pas, mais pourrait tout aussi bien le faire, le passage superbe du chant X du Paradis perdu de Milton o\u00f9 l'on voit Satan narrer comment il a r\u00e9ussi \u00e0 faire chuter \u00c8ve en se d\u00e9guisant en serpent. Satan raconte comment ce serpent dont il prit la forme pour remplir ses noirs desseins a depuis laiss\u00e9 une \u00e9ternelle inimiti\u00e9 entre le genre des reptiles et la race humaine: \u00abJe lui meurtrirai le talon;\/ Sa race, on ne sait quand, me meurtrira la t\u00eate.\u00bb Rien de tr\u00e8s novateur en ceci, car Milton se borne \u00e0 reprendre une tradition m\u00e9di\u00e9vale. Son g\u00e9nie po\u00e9tique se r\u00e9v\u00e8le dans le d\u00e9veloppement d'un th\u00e8me \u00e9cul\u00e9. Se vantant beaucoup, Satan explique que la douleur de cette \u00abmeurtrissure\u00bb n'est pas un prix trop cher pay\u00e9 pour avoir r\u00e9ussi \u00e0 pr\u00e9cipiter Adam et \u00c8ve hors du paradis terrestre. Alors qu'il s'attend \u00e0 des applaudissements couronnant sa fi\u00e8re harangue, il est surpris de constater que les d\u00e9mons r\u00e9unis en un pand\u00e9monium se mettent \u00e0 le siffler! Et sous ce coup inattendu, il perd son apparence humaine, son visage s'effile et s'allonge, ses bras rentrent dans le corps, il devient \u00e0 son tour la cr\u00e9ature dont il avait pris la forme. Milton emploie exactement le terme de \u00abvisage\u00bb \u00e0 ce point:\n\nHis Visage drawn he felt to sharp and spare,\n\nHis Arms clung to his Ribs, his Legs entwining\n\nEach other till supplanted he fell\n\nA monstrous Serpent on his Belly prone... (vers 511-514)\n\nLa chute du visage entra\u00eene la perte de la parole; les autres d\u00e9mons ne peuvent que faire \u00e9cho \u00e0 ses propres sifflements:\n\n... he would have spoke,\n\nBut hiss for hiss retun'd with forked tongue\n\nTo forked tongue, for now were all transform'd\n\nAlike, to Serpents... (vers 517-520)\n\nSuit alors un catalogue de serpents monstrueux, dont la m\u00e9tamorphose est dite li\u00e9e \u00e0 la \u00abcontagion\u00bb du mal.\n\nCe passage po\u00e9tique signifie quelque chose de philosophique allant dans le sens de ce que pr\u00e9suppose Levinas, \u00e0 savoir que parole et visage vont de pair. Si nous pr\u00eatons l'une ou l'autre \u00e0 quelque animal que ce soit, ce pr\u00eat signale automatiquement une animation anthropomorphique, une prosopop\u00e9e prosth\u00e9tique. On pourrait donc dire que, pour Levinas, l'animal litt\u00e9raire \u2013 ce qui inclut par exemple la c\u00e9l\u00e8bre hu\u00eetre de Ponge, coquillage ferm\u00e9 dans lequel on d\u00e9couvre \u00abtout un monde\u00bb qui inclut le ciel et toute la mer \u2013 poss\u00e8de un visage d\u00e8s qu'elle peut \u00abse regarder\u00bb en soi, donc de mani\u00e8re all\u00e9gorique, ou encore \u00ab\u00eatre regard\u00e9e\u00bb par le po\u00e8te qui m\u00e9dite \u00e0 son sujet. Si l'on peut dire de tel ou tel animal \u00abCela me regarde\u00bb, alors il y a bien visage. Mais Derrida d\u00e9cide de d\u00e9placer la question vers une probl\u00e9matique \u00e9thique. Car si Levinas admet que \u00abl\u00e0 o\u00f9 il y a mal, il y a visage\u00bb, en revanche, pour lui, la cat\u00e9gorie du meurtre ne peut s'appliquer \u00e0 l'animal. Un animal peut toujours \u00eatre sacrifi\u00e9 selon la loi.\n\nSelon la th\u00e8se de Derrida, Levinas serait donc rest\u00e9 trop profond\u00e9ment heidegg\u00e9rien, et c'est un reproche qu'il adresse aussi indirectement \u00e0 Agamben. Mais si l'on pense \u00e0 l'autre c\u00f4t\u00e9, qui est religieux et juif, le sacrifice animal n'est pas interdit dans le juda\u00efsme, et une telle tol\u00e9rance embarrasse Derrida. C'est dans ce contexte qu'il en revient au r\u00e9cit autobiographique consacr\u00e9 \u00e0 Bobby. La lecture de Derrida vise donc \u00e0 \u00abd\u00e9griser\u00bb les lecteurs qui, comme Alain David, se laissent s\u00e9duire trop facilement par le lyrisme autobiographique du r\u00e9cit de Levinas. C'est ici que Derrida \u00e9met des objections fondamentales: la phrase de Levinas (\u00abPour lui \u2013 c'\u00e9tait incontestable \u2013 nous f\u00fbmes des hommes\u00bb) ne dit rien des conditions de possibilit\u00e9 de l'\u00abincontestable\u00bb, bref, elle ne d\u00e9finit pas ce qui fait des \u00abhommes\u00bb de ces soldats fran\u00e7ais. La phrase qui semble attribuer au chien g\u00e9n\u00e9reusement ou ironiquement baptis\u00e9 \u00abBobby\u00bb une facult\u00e9 kantienne d\u00e9bouchant sur l'universel vient aussit\u00f4t lui retirer ses capacit\u00e9s \u00e0 raisonner: le cerveau limit\u00e9 de Bobby ne pouvait lui permettre de comprendre la port\u00e9e de son geste de reconnaissance. Derrida peut conclure \u00e0 bon droit: \u00abBobby est donc tout sauf kantien. Ce kantien all\u00e9gorique ou fabuleux est tout au plus un n\u00e9o-kantien infirme, un kantien priv\u00e9 de raison, un kantien sans maximes universalisables.\u00bb La charge est dure et Derrida doit s'excuser de s'acharner ainsi sur ce pauvre Bobby. Finalement, c'est aux deux points d'exclamation du texte de Levinas qu'il s'int\u00e9resse, observant comment un double \u00abMais non!\u00bb sert \u00e0 faire passer une formidable d\u00e9n\u00e9gation.\n\nLa violence rh\u00e9torique de la n\u00e9gation chez Levinas proviendrait en derni\u00e8re analyse de la forte tension entre Bobby, le chien singulier, et le r\u00f4le all\u00e9gorique qu'il doit tenir un peu malgr\u00e9 lui. Levinas encha\u00eenerait ainsi deux rejets parall\u00e8les quand il s'\u00e9crie: \u00abMais tr\u00eave de th\u00e9ologie!\u00bb, et ensuite: \u00abMais tr\u00eave d'all\u00e9gories!\u00bb Pourtant, comme Derrida le note sans piti\u00e9, nous ne sommes jamais sortis de l'all\u00e9gorie ou de la th\u00e9ologie. C'est que le chien viendrait d'\u00c9gypte et non d'Ithaque \u2013 il n'est pas Grec, mais \u00c9gyptien, autant au moins que ne l'\u00e9tait sans doute Mo\u00efse. Il est identifi\u00e9 \u00e0 un des chiens de l'Exode accompagnant les esclaves lib\u00e9r\u00e9s du joug du Pharaon. Faut-il pour autant en conclure que le chien de Levinas n'est pas diff\u00e9rent des animaux de Descartes, c'est- \u00e0-dire une machine incapable de parler qui n'arrivera \u00e0 atteindre \u00e0 la transcendance que n\u00e9gativement? Nous devrons pour r\u00e9pondre \u00e0 cette question op\u00e9rer un d\u00e9tour \u2013 tout aussi critique \u2013 par la pens\u00e9e d'Agamben.\n\nPour que Derrida en arrive \u00e0 cette s\u00e9v\u00e8re critique des fondements th\u00e9oriques de l'\u00e9thique de Levinas, il lui aura fallu en d\u00e9coudre avec un adversaire moins coriace, mais qui se r\u00e9clame de Levinas, Giorgio Agamben. Dans le S\u00e9minaire La b\u00eate et le souverain de 2001-2002, tout commence par un certain agacement face \u00e0 la mani\u00e8re dont Agamben donne la palme \u00e0 tel ou tel penseur qui aura \u00e9t\u00e9 le premier \u00e0 \u00e9noncer telle ou telle v\u00e9rit\u00e9. Agamben affirme ainsi que Levinas aurait \u00e9t\u00e9 le \u00abpremier\u00bb philosophe \u00e0 avoir compris et d\u00e9nonc\u00e9 la complicit\u00e9 entre Heidegger et le nazisme. Levinas aurait \u00e9t\u00e9 le \u00abpremier\u00bb \u00e0 replacer la d\u00e9rive politique de Heidegger dans une perspective qui serait celle de la \u00abbio-politique moderne\u00bb. Dans ce s\u00e9minaire, on voit Derrida se moquer des pr\u00e9tentions d'Agamben \u00e0 avoir toujours \u00e9t\u00e9 le premier \u00e0 discerner des concepts originaires.\n\nLa douzi\u00e8me s\u00e9ance du s\u00e9minaire approfondit l'opposition entre zo\u00ea et bios sur laquelle se fonde toute la probl\u00e9matique d'Agamben depuis Homo sacer. Se rapportant \u00e0 Heidegger et \u00e0 Aristote, Derrida montre du doigt l'\u00e9vidence: si l'id\u00e9e de zo\u00ea concerne la \u00abvie nue\u00bb, comment se fait-il qu'Aristote d\u00e9finisse l'homme comme un z\u00f4on politikon? Par une d\u00e9monstration rigoureuse et, me semble-t-il, irr\u00e9futable, Derrida montre que cette dichotomie est bien moins stricte et contraignante que ne l'affirme Agamben. Poussant plus loin, il \u00e9cart\u00e8le la pens\u00e9e d'Agamben en une alternative dramatique: ou bien on affirme avec Michel Foucault que la politique est devenue biopolitique dans une modernit\u00e9 tardive, ou bien on soutient avec Heidegger que ces notions \u00e9taient l\u00e0 depuis les Grecs et donc sous-tendent notre \u00abm\u00e9taphysique\u00bb et notre ontologie de mani\u00e8re presque imm\u00e9moriale. C'est toute la compr\u00e9hension du biopouvoir qui est ici en jeu. Agamben se fonderait \u00e0 la fois sur le dernier Foucault et sur Levinas, qui aurait toujours d\u00e9j\u00e0 per\u00e7u et critiqu\u00e9 le lien entre le nazisme de Heidegger et ses remarques infamantes sur l'holocauste vu comme un \u00e9piph\u00e9nom\u00e8ne de l'industrialisation de l'agriculture et de la mainmise de l'homme sur la production des animaux. Ce m\u00e9lange d'arch\u00e9ologie foucaldienne et de ph\u00e9nom\u00e9nologie l\u00e9vinassienne appara\u00eet effectivement confus, pris entre des probl\u00e9matiques incompatibles.\n\nDans ce d\u00e9bat qui n'est certes pas clos, je dirais volontiers que Derrida l'a emport\u00e9. Les cat\u00e9gories grecques d\u00e9ploy\u00e9es par Agamben font l'effet d'\u00eatre une projection personnelle, fort utile et int\u00e9ressante certes, mais qui n'arrivent pas \u00e0 convaincre qu'on aurait d\u00e9gag\u00e9 de la tradition philosophique une ancienne dichotomie remontant \u00e0 Aristote. Reste \u00e0 comprendre la part qui revient \u00e0 Levinas dans ce d\u00e9bat. Pour ce faire, rapportons-nous au contexte historique de la rencontre avec Bobby. Les Carnets de captivit\u00e9 sugg\u00e8rent qu'il y aurait chez lui une autre pens\u00e9e de l'animal qui \u00e9chapperait de justesse aux coups de butoir d\u00e9constructifs de Derrida. Si l'on jette un coup d'\u0153il \u00e0 la premi\u00e8re mention du chien appel\u00e9 Bobby, on constate que le ton est assez diff\u00e9rent, plus l\u00e9ger et critique: \u00ab\"Ce n'est pas un chien. Il abandonne son ma\u00eetre pour courir apr\u00e8s n'importe qui\" \u2013 dit Tramel. \"C'est pr\u00e9cis\u00e9ment cela le chien\" \u2013 r\u00e9pondit < Mimi? >.\u00bb\n\nCette note nous lance sur une discussion platonicienne concernant la nature d'un chien. Qu'est-ce qu'un chien? Selon Tramel, c'est un serviteur qui doit fid\u00e9lit\u00e9 \u00e0 son ma\u00eetre. Selon Mimi, c'est un animal libre et errant. Ce court dialogue survient dans les notes bien longtemps avant la mention du nom donn\u00e9 au chien allemand que l'on trouve seulement dans le carnet 6: \u00abLe chien Bobby est sympathique parce qu'il nous aime sans arri\u00e8re-pens\u00e9e, en dehors de toutes nos distinctions et r\u00e8gles sociales.\u00bb (CC, 150) Si on lit les brouillons du roman Eros ou Triste Opulence, il est \u00e9vident qu'il s'agit du m\u00eame chien dans ces notes.\n\nLisons le passage du roman inachev\u00e9 pour saisir comment le contexte romanesque relie le th\u00e8me du chien \u00e0 la probl\u00e9matique du visage:\n\n< p. 77 > [...] Puis le passage \u00e0 travers une grande for\u00eat o\u00f9 des sapins gigantesques cr\u00e9aient une nuit en plein jour et o\u00f9 l'on penchait comme dans une cath\u00e9drale. Etait-ce donc possible de telles for\u00eats en pleine guerre, des relations rapports de l'homme avec la nature, comme disait l'autre. Puis le retour du Kommando o\u00f9 le petit chien allait accueillir joyeusement le Kommando. La grande joie et le grand gardien de la dignit\u00e9 du prisonnier. D'o\u00f9 cet ours Boby on ne le savait pas\n\n< p. 78 > [La signification de visage pour moi l'interlocuteur en tant qu'interlocuteur n'est pas une signification comme celle que traduisent les mots, n'est pas m\u00e9taphorique comme le sont les significations qui constituent le monde \u2013 quelle que soit les la mesure dans laquelle autrui appartient \u00e0 l'ordre du monde de par le r\u00f4le qu'il y joue, quelle que soit [...] la part de \u00abla r\u00e9citation de r\u00f4le\u00bb dans le discours par lequel je r\u00e9ponds \u00e0 mon \u00e0 la signification que l'interlocuteur a pour moi en tant qu'interlocuteur\n\n< p. 79 >, Mais ce fut le seul \u00eatre qui ne faisait pas de diff\u00e9rence entre les prisonniers et les aryens qui les gardaient. Il y eut toutes sortes d'allemands et quelques-uns furent souriants et compatissants. Mais dans les yeux de tous \u00e9tait \u00e9crit qu'ils savaient que devant eux se trouvaient des demi-hommes, que tous leurs gestes et sentiments \u00e9taient d\u00e9risoires et condamn\u00e9s et que tout ce qu'il pouvait faire, et dire, et tout ce qu'on pouvait leur dire ne figurait qu'entre guillemets. Le chien Bobby plein d'amiti\u00e9 \u00e9tait le seul \u00e0 ignorer ces guillemets son aboiement et exprimait une joie sans r\u00e9serves ni arri\u00e8re- pens\u00e9e. Il reconnaissait seul le droit de l'homme et la dignit\u00e9 de la personne de ces juifs.\n\n< p. 80 > Tramuel aurait-il eu senti cela? Je n'aime pas ce chien venu on ne sait d'o\u00f9 disait-il en le voyant. Je n'aime pas les chiens infid\u00e8les \u00e0 leur ma\u00eetre et qui s'acoquinent avec les passants.\n\nJe voudrais insister sur le d\u00e9calage entre la premi\u00e8re note et la seconde: dans la seconde, le th\u00e8me est la reconnaissance \u00abkantienne\u00bb de l'humanit\u00e9 des prisonniers par le chien; dans la premi\u00e8re, le th\u00e8me est le c\u00f4t\u00e9 anarchique, abandonn\u00e9 et abandonnant, de l'animal. Pourquoi ce chien n'a- t-il pas de ma\u00eetre? A-t-il fui un mauvais ma\u00eetre, a-t-il perdu un ma\u00eetre qui aurait pu, par exemple, avoir \u00e9t\u00e9 enr\u00f4l\u00e9 de force dans les arm\u00e9es allemandes? On ne le saura pas, mais la question demeure \u2013 un chien, ce n'est pas seulement le noble ami de l'homme, c'est aussi celui qui court apr\u00e8s n'importe qui. Il est plus que libre, car c'est un ami de l'anarchie. Si le h\u00e9ros du roman est pr\u00eat \u00e0 reconna\u00eetre un visage \u00e0 ce chien amical, Tramuel le refuse au nom d'une prudence l\u00e9galiste.\n\nL'h\u00e9sitation majeure du narrateur se lit dans la phrase interrompue: \u00abD'o\u00f9 cet ours Boby [sic] on ne le savait pas\u00bb, phrase dans laquelle est mentionn\u00e9 un autre animal, l'ours, pour qualifier l'attitude de Bobby. Cela indique-t-il que Bobby peut aussi \u00abfaire la t\u00eate\u00bb, \u00eatre grossier ou mal \u00e9lev\u00e9? Mais comment faire la t\u00eate s'il n'a pas de visage? Or la longue analyse qui suit, \u00e0 la page 78, commence par biffer le mot de \u00abvisage\u00bb pour le remplacer par un \u00abinterlocuteur\u00bb et elle semble bien exclure Bobby du monde des \u00abr\u00f4les\u00bb sociaux tout en lui reconnaissant une signification qui ne passerait pas par les mots.\n\nLa r\u00e9action n\u00e9gative de Tramuel \u2013 son nom s'est ajout\u00e9 d'un \u00abu\u00bb qui le rendrait plut\u00f4t juif, mais dans un autre passage, il est appel\u00e9 \u00abTromel\u00bb, ce qui pourrait faire de lui un Allemand \u2013 suit un paragraphe anticipant de mani\u00e8re explicite et d\u00e9velopp\u00e9e le th\u00e8me majeur de \u00abNom d'un chien\u00bb: Bobby incarne une reconnaissance qui se passe de mots. Son aboiement suffit \u00e0 exprimer \u00abune joie sans r\u00e9serves ni arri\u00e8re- pens\u00e9e\u00bb, deux termes qui pr\u00e9supposent de fait quelque chose comme une \u00abpens\u00e9e\u00bb. Si Tromel\/Tramuel est du c\u00f4t\u00e9 des Allemands, tout s'explique mieux.\n\nContrairement \u00e0 l'optimisme du narrateur, le soup\u00e7on port\u00e9 sur le chien par Tramuel ferait penser \u00e0 l'analyse canonique que Derrida a donn\u00e9e de la lettre errante et sans ma\u00eetre telle que Platon la pr\u00e9sente dans le Ph\u00e8dre. \u00c0 l'encontre de W., un autre personnage du roman, Tramuel serait au fond un platonicien. Ce que \u00abLa pharmacie de Platon\u00bb proposait comme le mod\u00e8le de toute grammatologie ult\u00e9rieure aurait donc d\u00e9j\u00e0 trouv\u00e9 son \u00e9quivalent philosophique chez le premier Levinas, dont l'analyse juxtapose le point de vue de deux prisonniers juifs en d\u00e9saccord sur le sens \u00e0 donner \u00e0 la pr\u00e9sence de ce chien parmi eux.\n\nOn retrouverait une semblable tension dans les \u00abNotes philosophiques diverses\u00bb de la m\u00eame p\u00e9riode. Sous la rubrique \u00abPlaton\u00bb, on trouve une citation du Ph\u00e8dre: \u00ab\"Suis-je un animal plus pa par hasard quelque b\u00eate plus compliqu\u00e9e et bien plus enfum\u00e9e par l'orgueil que n'est Typhon? Suis-je un animal paisible, sans autant de complications et qui, de nature, participe \u00e0 une destin\u00e9e divine o\u00f9 n'entrent point les fum\u00e9es de l'orgueil?\", Ph\u00e8dre 230 a.\u00bb (CC, 308) Levinas ne d\u00e9veloppe pas cette r\u00e9flexion, mais il esquisse une critique de la diff\u00e9rence \u00e9tablie par Platon entre l'\u00eatre humain et l'\u00eatre animal. Platon lui-m\u00eame aurait reconnu une troublante parent\u00e9, ne serait-ce que parce qu'il y a de l'animal en nous.\n\nL'animal, dans la pens\u00e9e de Levinas au moment de la captivit\u00e9 et du dur travail du stalag, figure la possibilit\u00e9 d'un \u00eatre heureux; ou du moins il s'approche du bonheur dans la mesure o\u00f9 il ne travaille pas au sens o\u00f9 l'homme travaille: \u00abL'homme c'est ce qui ne lutte pas pour la vie. C'est du moins le christianisme dans son interpr\u00e9tation de Tolsto\u00ef. La notion de travail qui remplace celle de la lutte. \"L'id\u00e9e de lutte ou travail.\" Animaux ne travaillent pas.\u00bb (CC, 84) Levinas ne pense pas aux b\u00eates de somme, mais il r\u00eave \u00e0 un \u00e9tat de nature fait de libert\u00e9 oisive. De plus, l'animal est un \u00eatre qui laisse des traces \u2013 souvent comme gibier, certes, mais aussi comme un prisonnier \u00e9vad\u00e9. Cet \u00e9cho se trouve dans trois notes successives: \u00abComme l'animal qui en fuyant pr\u00e9cis\u00e9ment laisse sur la neige immacul\u00e9e les traces qui permettront de le retrouver.\u00bb (CC, 84); \u00abLa marche sur une route o\u00f9 il n'y a aucune trace humaine, mais uniquement la trace du gibier.\u00bb (CC, 85); et enfin: \u00abCette mani\u00e8re de compter les hommes sans les voir.\u00bb (CC, 85) On saisit bien le lien \u2013 face aux multitudes de prisonniers, la comparaison avec du b\u00e9tail est immanquable. Il est vrai que ces vignettes se d\u00e9finissent le plus souvent par apport \u00e0 la pens\u00e9e de Heidegger, mais c'est pour souligner la diff\u00e9rence de la pens\u00e9e de Levinas par rapport \u00e0 la philosophie de Heidegger: \u00abUn \u00e9l\u00e9ment essentiel de ma philosophie \u2013 ce par quoi elle diff\u00e8re de la philo. de Heidegger \u2013 c'est l'importance de l'Autre. Eros comme moment central.\u00bb (CC, 134)\n\nCe dernier point est capital, car si l'animal peut participer de cette alt\u00e9rit\u00e9 et de cet Eros, la critique de Derrida devient superflue. Voici deux remarques qui se suivent: \u00abLa nation comme acc\u00e8s au r\u00e9el. Monde de Heidegger.\u00bb et \u00abLe pied humain \u2013 son apparence humble, pauvre animal. Ce sont les deux \"pattes\" qui ont chez l'homme encore la fonction de l'animal. Les mains au contraire.\u00bb (CC, 105) Le corps humain renferme une dichotomie entre des pieds animaux et des mains essentiellement humaines, en une division interne qui \u00e9vite le reproche d'un trop facile anthropomorphisme.\n\nPour Levinas, la vie animale \u00e9voque, il est vrai, un \u00e9tat de nature id\u00e9alis\u00e9 o\u00f9 la sympathie universelle domine. Il commente en ces termes le Livre de San Michele d'Axel Munthe:\n\nLe contact avec la vie \u2013 contact direct avec la vie en tant qu'entit\u00e9. \u00c0 travers le r\u00e8gne animal. [...] Livre qui met en relief l'\u00e9troitesse de la philosophie sociale. La sympathie avec le monde animal ne se loge pas dans une m\u00e9taphysique sociale. Celle-ci d\u00e9finit la place de l'homme dans le monde: les choses et les hommes s'y rangent en effet facilement. Ce sont des objets et des besoins, ou des outils \u2013 les hommes \u2013 collaborateurs, exploitants, exploit\u00e9. L'animal dans la sympathie avec lui [...] n'est que vie, ne se range pas dans ces cat\u00e9gories. Par l\u00e0, la nature elle-m\u00eame et la relation avec la nature apparaissent dans son originalit\u00e9. La nature est vie ou elle n'est qu'un objet. (CC, 127)\n\nCe qui compte est une \u00absympathie\u00bb hors langage, celle m\u00eame qui caract\u00e9rise l'\u00eatre de Bobby: cette sympathie de l'homme pour l'animal et de l'animal pour l'homme offre une r\u00e9ciprocit\u00e9 dans la non-r\u00e9ciprocit\u00e9 des langues et discours.\n\nCette r\u00e9ciprocit\u00e9 impliquerait aussi la sexualit\u00e9. Dans un passage, on voit Levinas s'inqui\u00e9ter de ce qu'il per\u00e7oit comme une horreur de la sexualit\u00e9 animale chez Tolsto\u00ef: \u00abT. a horreur de ces \u00eatres [...], de ces femmes incomp en pleine beaut\u00e9 \u2013 etc. qui ne sont que des vaches qui se mettent sur les pattes arri\u00e8re. Et < l' > horreur de toutes ces complaisances. Die < Tiere? > wollen Gl\u00fccklichsein.\u00bb (CC, 143) Admettre que les animaux, eux aussi, veulent \u00eatre heureux, cela ne les inscrit pas dans l'inhumanit\u00e9 de simples \u00abb\u00eates\u00bb, mais plut\u00f4t dans un Eros g\u00e9n\u00e9ralis\u00e9. Le conatus \u00e9voqu\u00e9 n'est pas sp\u00e9cifiquement animal; le d\u00e9sir ne leur enl\u00e8ve pas leur dignit\u00e9. Leur dignit\u00e9 leur est retir\u00e9e seulement quand l'homme intervient de mani\u00e8re maladroite ou m\u00e9chante:\n\nLe chat perd toute dignit\u00e9 quand on lui marche sur la queue. On pensait que c'est un appendice d\u00e9coratif, comme la tra\u00eene d'une robe, ou le mousqueton d'un centurion. Et voil\u00e0 qu'il y a l\u00e0 des nerfs. Et que le chat cet animal myst\u00e9rieux sorcier, \u2013 le chat de Baudelaire crie comme une vieille femme. (CC, 168)\n\nDans ses carnets de guerre, Levinas cite souvent \u00abLes Chats\u00bb de Baudelaire et mentionne \u00e0 plusieurs reprises \u00abLe Corbeau\u00bb d'Edgar Poe. Dans un passage assez litt\u00e9raire ou onirique, c'est m\u00eame sa propre famille biologique et par alliance qui se m\u00e9tamorphose en arbre g\u00e9n\u00e9alogique d\u00e9viant et peupl\u00e9 de chiens et de chats: \u00abImage: je suis \u00e0 table avec la tante par alliance. Tiens! C'est une chienne. L'oncle est un chat. Comment entrer dans leur vie de chiennes et de chats, \"leurs habitudes\" \"leurs m\u0153urs\"? \u2013 C'est tellement animal \u2013 ennemi \u2013 folie! folie! La personne m\u00eame qu'on \u00e9pouse \u2013 elle est choisie abstraite \u2013 acceptable.\u00bb (CC, 188) Dans cette vignette enjou\u00e9e, animaux et humains ne participent pas de domaines s\u00e9par\u00e9s; la distance relative qui nous s\u00e9pare de l'animal familier que les Anglo-saxons appellent \u00abpet\u00bb peut au contraire all\u00e9goriser la distance relative qui nous s\u00e9pare de ceux qu'on nomme \u00abrelatives\u00bb en anglais.\n\nVoil\u00e0 qui nous permettrait de revenir vers la sc\u00e8ne de la nomination du chien allemand et \u00abkantien\u00bb. Puisque la description de Bobby est la seule que Levinas ait jamais donn\u00e9e d'un animal dans ses textes publi\u00e9s, il faut l'entendre \u00e0 nouveau:\n\nEt voici que, vers le milieu d'une longue captivit\u00e9 \u2013 pour quelques courtes semaines et avant que les sentinelles ne l'eussent chass\u00e9 \u2013 un chien errant entre dans notre vie. Il vint un jour se joindre \u00e0 la tourbe, alors que, sous bonne garde, elle rentrait du travail. Il vivotait dans quelque coin sauvage, aux alentours du camp. Mais nous l'appelions Bobby, d'un nom exotique, comme il convient \u00e0 un chien ch\u00e9ri. Il apparaissait aux rassemblements matinaux et nous attendait au retour, sautillant et aboyant gaiement. Pour lui \u2013 c'\u00e9tait incontestable \u2013 nous f\u00fbmes des hommes.\n\nIl est tentant de comparer l'attitude de ce chien libre et perdu \u00e0 la fois, cet \u00eatre abandonn\u00e9 qui se cherche un ma\u00eetre et ne trouve qu'un ma\u00eetre collectif dans la \u00abtourbe\u00bb des prisonniers, et qui donc ne saura jamais que ses \u00abma\u00eetres\u00bb sont eux-m\u00eames des prisonniers ou des esclaves, avec l'Albertine de Proust dans La Prisonni\u00e8re. C'\u00e9tait le livre de Proust que Levinas relisait et annotait \u00e0 ce moment, pensant bien s\u00fbr \u00e0 sa propre situation d'intern\u00e9 dans le stalag. La proximit\u00e9 de deux \u00abprisonniers\u00bb qui sont aussi paradoxalement libres, Bobby le chien et Albertine la ma\u00eetresse du narrateur, est plus que troublante.\n\nLevinas savait peut-\u00eatre que Proust avait imagin\u00e9 lui aussi qu'un des personnages de la Recherche s'appellerait \u00abBobby\u00bb. Proust avait d'abord appel\u00e9 \u00abBobby Santois\u00bb le personnage complexe qu'il nommera ensuite \u00abMorel\u00bb. Le premier nom ne peut manquer d'\u00e9voquer \u00absans toit\u00bb et peut m\u00eame sugg\u00e9rer, \u00e0 la mani\u00e8re de Levinas, un \u00absans toi\u00bb. M\u00eame s'il est peu probable que Levinas ait eu connaissance des carnets de Proust o\u00f9 se trouvent ces d\u00e9tails g\u00e9n\u00e9tiques sur les noms des personnages de la Recherche , les propres Carnets de captivit\u00e9 de Levinas annoncent une \u0153uvre future, dans laquelle on d\u00e9couvre deux romans \u00e0 \u00e9crire ainsi qu'un livre de critique litt\u00e9raire consacr\u00e9 \u00e0 l'\u0153uvre de Proust (CC, 74). Dans les notes r\u00e9dig\u00e9es au Kommando, on trouve plusieurs commentaires de ce que les proustiens appellent le \u00abroman d'Albertine\u00bb, soit les nombreuses pages qui vont de La Prisonni\u00e8re \u00e0 Albertine disparue. Levinas y esquisse d\u00e9j\u00e0 l'essai qu'il publiera dans la revue Deucalion en 1947 sous le titre de \u00abL'Autre chez Proust\u00bb, son point de d\u00e9part \u00e9tant le refus de l'id\u00e9e courante selon laquelle Proust serait un grand psychologue. Levinas insiste sur le fait que ce qui int\u00e9resse Proust avant tout, c'est le social:\n\nChez Proust po\u00e9sie du social pur. L'int\u00e9r\u00eat ne tient pas \u00e0 la \u00abpsychologie\u00bb, mais au th\u00e8me: le social. Toute l'histoire d'Albertine prisonni\u00e8re \u2013 est l'histoire de la relation avec autrui. Qu'est Albertine {et ses mensonges} sinon l'\u00e9vanescence m\u00eame d'autrui, sa r\u00e9alit\u00e9 faite de son n\u00e9ant, sa pr\u00e9sence faite de son absence, la lutte avec l'insaisissable? Et \u00e0 c\u00f4t\u00e9 de cela \u2013 le calme devant [...] Albertine qui dort, devant Albertine v\u00e9g\u00e9tal. Le \u00abcaract\u00e8re\u00bb, le \u00absolide\u00bb = chose. (CC, 72)\n\nCurieusement, c'est Albertine qui d\u00e9borde de l'humain et passe par-dessus l'animal pour rejoindre le domaine v\u00e9g\u00e9tal. Or, dans l'\u00e9conomie du roman proustien, Albertine figure comme un double de Bobby Santois\/Morel, ce qui est perceptible dans l'\u00e9volution du roman de 1914 \u00e0 1922. Ces deux personnages viennent du peuple, ils sont \u00e0 la fois sexuellement attirants et arrogants, ils sont tous deux bruns, ils ont des dispositions bisexuelles et, surtout, ils mentent sans cesse afin de prot\u00e9ger leur libert\u00e9.\n\nDans ses carnets comme dans l'essai de 1947, Levinas critique les erreurs de lecture de Sartre; Levinas rejette la tentative de Sartre qui veut r\u00e9duire Proust \u00e0 sa (mauvaise) psychologie. \u00c0 la diff\u00e9rence de Sartre, Levinas analyse en revanche de mani\u00e8re positive le r\u00f4le jou\u00e9 par la perversion dans la section qui pr\u00e9c\u00e8de, Sodome et Gomorrhe: ce qui compte pour Proust, c'est d'\u00e9liminer les syst\u00e8mes de valeurs \u00e9tablies. Proust se d\u00e9tache de toute consid\u00e9ration morale pour mieux faire sentir l'alt\u00e9rit\u00e9 d'\u00c9ros et son irr\u00e9ductible libert\u00e9. Levinas commente ceci en des termes tr\u00e8s forts dans son texte \u00abL'Autre dans Proust\u00bb:\n\nIl est curieux de noter \u00e0 quel point l'amoralisme de Proust introduit dans son univers la libert\u00e9 la plus folle, conf\u00e8re aux objets et aux \u00eatres d\u00e9finis le scintillement des possibles que la d\u00e9finition n'a pas \u00e9teints. On dirait que les r\u00e8gles morales bannissent les f\u00e9eries du monde plus s\u00e9v\u00e8rement que les lois naturelles et que la magie commence, comme un Sabbat fantastique, d\u00e8s que l'\u00e9thique est finie. [...] Tout est vertigineusement possible. (NP, 151-152)\n\nSi dans l'article de 1947, Proust renoue avec un Dosto\u00efevski affirmant que si Dieu est mort, tout est possible, dans les Carnets de captivit\u00e9, c'est Bobby, qu'il soit un chien, un homme ou une femme, qui nous montre la voie. Levinas suit Proust et Bobby le chien \u2013 il nous fait sentir la positivit\u00e9 de \u00abl'animal que donc je suis\u00bb, qui nous permet de d\u00e9passer l'amoralisme de Sade ou de Nietzsche: ce qui compte dans cette lecture des traces litt\u00e9raires et animales, c'est de relier alt\u00e9rit\u00e9 et libert\u00e9. Telle est bien la le\u00e7on qui est donn\u00e9e au narrateur par Albertine. On lit ainsi dans l'essai de Deucalion: \u00abLa r\u00e9alit\u00e9 d'Albertine, c'est son \u00e9vanescence dans sa captivit\u00e9 m\u00eame...\u00bb (NP, 153) Sa mort donne paradoxalement acc\u00e8s \u00e0 \u00abEros dans sa puret\u00e9 ontologique\u00bb, qui est \u00abrelation directe avec ce qui se donne en se refusant, avec autrui en tant qu'autrui, avec le myst\u00e8re\u00bb (NP, 154). Proust nous permet de comprendre l'\u00eatre en tant que pure alt\u00e9rit\u00e9. Dans la multiplicit\u00e9 des images contradictoires d'Albertine, on pressent la volont\u00e9 de donner vie \u00e0 la plasticit\u00e9 du monde social:\n\nMarcel n'aima pas Albertine, si l'amour est une fusion avec autrui, extase d'un \u00eatre devant les perfections de l'autre ou la paix de la possession. Demain il rompra avec la jeune femme qui l'ennuie. Il fera ce voyage depuis longtemps projet\u00e9. Le r\u00e9cit de l'amour de Marcel est doubl\u00e9 d'aveux destin\u00e9s, semble-t-il, \u00e0 mettre en question la consistance m\u00eame de cet amour. Mais ce non-amour est pr\u00e9cis\u00e9ment l'amour, la lutte avec l'inconnaissable \u2013 la possession, cette absence d'Albertine \u2013, sa pr\u00e9sence. (NP, 155)\n\nSelon la fine lecture de Levinas, Proust en arrive \u00e0 rompre d\u00e9finitivement avec l'ontologie de Parm\u00e9nide, et donc avec toute l'ontologie \u00abgrecque\u00bb. Albertine ou Bobby nous montrent comment r\u00e9cuser une philosophie qui ne mettrait pas l'\u00e9thique au premier plan:\n\nMais l'enseignement le plus profond de Proust \u2013 si toutefois la po\u00e9sie comporte des enseignements \u2013 consiste \u00e0 situer le r\u00e9el dans une relation avec ce qui \u00e0 jamais demeure autre, avec autrui comme absence et myst\u00e8re, \u00e0 la retrouver dans l'intimit\u00e9 m\u00eame du \u00abJe\u00bb, \u00e0 inaugurer une dialectique qui rompt d\u00e9finitivement avec Parm\u00e9nide. (NP, 155-156)\n\nDerrida est-il pr\u00eat \u00e0 l'accompagner sur ce point: veut-il courir avec un Bobby qui reste sans ma\u00eetre ni toit? Si Bobby repousse toute Loi, il met aussi \u00e0 bonne distance le \u00abToi\u00bb par sa reconnaissance kantienne et transcendantale.\n\nLa reconnaissance de l'Autre par le narrateur de Proust face \u00e0 Albertine est complexe. Proust commente ironiquement la transformation d'Albertine: \u00e0 Balbec, elle lui apparaissait comme une jeune fille tr\u00e8s jolie, mais un peu b\u00eate; une fois devenue sa \u00abprisonni\u00e8re\u00bb \u00e0 Paris, au contact du narrateur \u00e0 qui elle semble moins belle, elle se cultive au point de sembler intelligente. Un des signes de cette transformation est le fait qu'elle se met \u00e0 appr\u00e9cier la litt\u00e9rature. Albertine en vient \u00e0 se passionner pour Dosto\u00efevski et se risque m\u00eame \u00e0 juger ses textes en termes cat\u00e9goriques: \u00abC'est int\u00e9ressant, mais, par exemple, c'est \u00e9crit comme par un cochon.\u00bb Et si l'\u00e9criture, loin d'\u00eatre all\u00e9goris\u00e9e par un chien errant en Allemagne, n'\u00e9tait au fond que de la cochonnerie? Pouvons-nous penser une \u00e9criture qui serait plus radicalement du c\u00f4t\u00e9 de l'animal impur par excellence? Y aurait-il une grammatologie porcine? Serait-il loisible de l'imaginer \u00e0 partir d'un humain devenu animal dans la bassesse, la luxure et la sensualit\u00e9 la plus d\u00e9brid\u00e9e, Eros ou la Triste Opulence de l'ordure?\n\nIl fallait revenir vers les carnets et les brouillons de romans, ce riche terreau \u00e0 partir duquel s'\u00e9labore l'\u0153uvre de Levinas qui semble hant\u00e9e par la paix et que sans cesse Derrida renvoie vers une \u00abpol\u00e9mique originaire\u00bb, une \u00abviolence transcendantale\u00bb. Si Derrida semble gagner \u00e0 tous les coups contre Levinas, est-ce une Blitzkrieg ou une \u00abdr\u00f4le de guerre\u00bb que cette lutte? On peut dire que l'histoire de la d\u00e9construction aux \u00c9tats-Unis au moins nous rappellerait qu'il y a autant de danger \u00e0 gagner une guerre trop vite qu'\u00e0 ne jamais commencer la vraie guerre. L'arm\u00e9e am\u00e9ricaine a d\u00e9couvert que ses spectaculaires victoires en Irak et Afghanistan n'avaient rien r\u00e9solu. La victoire proclam\u00e9e ne vaut que si l'on repart une fois qu'une paix \u2013 provisoire ou perp\u00e9tuelle \u2013 aura \u00e9t\u00e9 conclue. Il faudrait commenter longuement le texte sur la guerre atomique, \u00abNo apocalypse, not now\u00bb, ce texte de 1984 que j'ai d\u00e9j\u00e0 cit\u00e9 et qui me semble toujours tr\u00e8s pertinent aujourd'hui. L'ouvrage r\u00e9cent de Paul Saint-Amour, Tense Future: Modernism, Total War, Encyclopedic Form , prend cet essai comme une donn\u00e9e th\u00e9orique fondamentale pour analyser l'anticipation de la catastrophe finale et de la guerre totale entre les deux guerres. Je vais \u00e0 mon tour commenter un exemple litt\u00e9raire afin de repenser le lien intime entre la guerre, l'avenir et leurs modalit\u00e9s discursives.\n\n. L'animal autobiographique. Autour de Jacques Derrida, Marie-Louise Mallet (dir.), Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 1999, p. 264, note.\n\n. Alain David, \u00abCynesth\u00e8se: auto-portrait au chien\u00bb, dans L'animal autobiographique, op. cit., p. 303-318. Je tiens ici \u00e0 saluer Alain David dont l'amiti\u00e9 et la fr\u00e9quentation depuis pr\u00e8s de quarante ann\u00e9es m'ont permis de mieux sonder les pens\u00e9es de Levinas et de Derrida. Je renvoie \u00e0 un texte plus r\u00e9cent du m\u00eame Alain David, \u00abLecture de \"Nom d'un chien ou le droit naturel\" d'Emmanuel Levinas\u00bb, dans Reflections on Levinas, Volkan Cebeci (\u00e9d.), Monokl, Istanbul, 2010, p. 75-96.\n\n. E. Levinas, \u00abNom d'un chien ou le droit naturel\u00bb, dans Difficile Libert\u00e9, Paris, Albin Michel, 1976, p. 215.\n\n. E. Levinas, \u00abNom d'un chien ou le droit naturel\u00bb, dans Difficile Libert\u00e9, op. cit., p. 216.\n\n. A. David, \u00abCynesth\u00e8se: auto-portrait au chien\u00bb, dans L'animal autobiographique, op. cit., p. 310.\n\n. Ibid., p. 317.\n\n. Ibid., p. 312.\n\n. R\u00e9ponse de Levinas cit\u00e9e par J. Derrida dans L'animal que donc je suis, M.-L. Mallet (\u00e9d.), Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 2006, p. 149.\n\n. J. Derrida, L'animal que donc je suis, op. cit., p. 150. Cf. \u00abThe Paradox of Morality: an Interview with Emmanuel L\u00e9vinas\u00bb, dans The Provocation of Levinas: Rethinking the Other, Robert Bernasconi et David Wood (\u00e9ds), Londres, Routledge, 1988, p. 168-180.\n\n. J. Derrida, L'animal que donc je suis, op. cit., p. 151.\n\n. John Milton, Paradise Lost, Book X, vers 498-499, dans Complete Poems and Major Prose, Merritt Y. Hughes (\u00e9d.), Indianapolis, Hackett Publishing, 1977, p. 418.\n\n. Francis Ponge, Le parti pris des choses, Paris, Gallimard, 1986, p. 43.\n\n. J. Derrida, L'animal que donc je suis, op. cit., p. 152.\n\n. Ibid., p. 157.\n\n. Ibid., p. 158.\n\n. Giorgio Agamben, Homo sacer. I. Le pouvoir souverain et la vie nue, tr. fr. Maril\u00e8ne Raiola, Paris, Le Seuil, 1997, p. 164; cit\u00e9 par J. Derrida dans le S\u00e9minaire La b\u00eate et le souverain. Volume I (2001-2002), Michel Lisse, Marie-Louise Mallet et Ginette Michaud (\u00e9ds), Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 2008, p. 137.\n\n. E. Levinas, \u0152uvres 1, Carnets de captivit\u00e9 suivi de \u00c9crits sur la captivit\u00e9 et Notes philosophiques diverses, volume publi\u00e9 sous la responsabilit\u00e9 de Rodolphe Calin et Catherine Chalier, Paris, Bernard Grasset\/IMEC, 2009, p. 80. Les mots ratur\u00e9s et l'orthographe sont reproduits tels qu'ils sont dans cette \u00e9dition. D\u00e9sormais abr\u00e9g\u00e9 en CC, suivi du num\u00e9ro de la page.\n\n. E. Levinas, \u00abEros ou Triste Opulence\u00bb, dans \u0152uvres 3, Eros, litt\u00e9rature et philosophie, op. cit., p. 92-93.\n\n. \u00abTromel se penchait pour cueillait les premi\u00e8res fleurs du printemps pour sa femme. Il \u00e9tait tr\u00e8s sentimental. C'\u00e9tait le printemps 1942. Quant au chauffeur, il en avait r\u00e9uni tout un bouquet qu'il avait accroch\u00e9 sur les parois du tracteur. [...] La remorque du tracteur {avec les prisonniers rentrant du travail du Kommando} sautillait plus ridiculement encore. Ils ne comptaient gu\u00e8re, \u00e9videmment. C'\u00e9tait de plus des juifs.\u00bb (E. Levinas, \u00abEros ou Triste Opulence\u00bb, dans \u0152uvres 3, Eros, litt\u00e9rature et philosophie, op. cit., p. 83.)\n\n. Cf. le livre de Mark Shell, Children of the Earth: Literature, Politics and Nationhood, Oxford, Oxford University Press, 1993, dont le chapitre 7 traite superbement de \u00abThe family pet or the human and the animal\u00bb. La pens\u00e9e originale et exigeante de Shell enfonce un coin entre Derrida et Levinas. Partant en guerre contre l'id\u00e9al universaliste de la fraternit\u00e9 humaine, Shell montre comment dans certains moments domin\u00e9s par la guerre c'est l'impossibilit\u00e9 m\u00eame de cet id\u00e9al qui va transformer certains groupes humains en animaux, lesquels peuvent d\u00e8s lors \u00eatre massacr\u00e9s sans remords. Il analyse aussi comment nos animaux familiers nous rappellent notre appartenance \u00e0 une parent\u00e9 g\u00e9n\u00e9rique (kinship).\n\n. E. Levinas, Difficile Libert\u00e9, op. cit., p. 216.\n\n. Une bonne description de l'\u00e9volution du personnage de Bobby Santois en celui de Morel est donn\u00e9e par Alison Winston, Proust's Additions: The Making of \u00c0 la Recherche du Temps Perdu, Cambridge, Cambridge University Press, 1977, I, p. 169-207.\n\n. Cf. l'essai de J. Hillis Miller, \u00ab\"Le Mensonge, le Mensonge Parfait.\" Th\u00e9ories du mensonge chez Proust et Derrida\u00bb, dans Passions de la litt\u00e9rature. Avec Jacques Derrida, M. Lisse (dir.), Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 1996, p. 405-420.\n\n. Marcel Proust, \u00c0 la recherche du temps perdu, Jean-Yves Tadi\u00e9 (dir.), t. III, Paris, Gallimard, coll. \u00abBiblioth\u00e8que de la Pl\u00e9iade\u00bb, 1988, p. 528. Les italiques sont dans le texte.\n\n. J. Derrida, \u00abViolence et m\u00e9taphysique\u00bb, dans L'\u00e9criture et la diff\u00e9rence, op. cit., p. 161.\n\n. Ibid., p. 173.\n\n. Cf. Paul K. Saint-Amour, Tense Future: Modernism, Total War, Encyclopedic Form, Oxford, Oxford University Press, 2015, p. 25-28, 36-37. \n\n# IV. Messianisme de guerre ou nihilisme de paix:  \nUne enfance de J\u00e9sus\n\nMon point de d\u00e9part pour ce chapitre est une remarque au sujet de Derrida qui se trouve dans une lettre que J. M. Coetzee envoya \u00e0 Paul Auster. Il s'agissait du Monolinguisme de l'autre. Coetzee et Auster avaient pris l'habitude d'\u00e9changer des lettres et, en mai 2009, Coetzee cite la phrase par laquelle Derrida commence Le monolinguisme de l'autre, qui, on le sait s'ouvre par une m\u00e9ditation sur cette phrase assez paradoxale: \u00abJe n'ai qu'une langue, ce n'est pas la mienne.\u00bb Coetzee commente ce curieux aphorisme:\n\nCe qui m'a int\u00e9ress\u00e9 est l'affirmation de Derrida que bien qu'il soit monolingue en fran\u00e7ais (monolingue selon ses crit\u00e8res \u2013 son anglais \u00e9tait excellent, comme je suis s\u00fbr, son allemand, sans parler de son grec), le fran\u00e7ais n'est\/n'\u00e9tait pas sa langue maternelle. Quand j'ai lu ceci, ce qui m'a frapp\u00e9 c'est qu'il aurait pu parler de moi et de mon rapport \u00e0 l'anglais; et le lendemain, ce qui m'a encore plus frapp\u00e9 est qui ni lui ni moi ne sommes des exceptions, que beaucoup d'\u00e9crivains et d'intellectuels ont une relation distante ou interrogative au langage qu'ils parlent ou \u00e9crivent, et que, de fait, d\u00e9signer la langue qu'on utilise comme sa langue maternelle est devenu tout \u00e0 fait vieux jeu.\n\nCoetzee explicite les implications de la th\u00e8se de Derrida en faisant allusion \u00e0 son impression de n'avoir jamais senti qu'il poss\u00e9dait la langue anglaise dans laquelle il \u00e9crit. Au d\u00e9part, pour lui, l'anglais n'\u00e9tait qu'une des mati\u00e8res que l'on apprenait \u00e0 l'\u00e9cole, \u00e0 c\u00f4t\u00e9 du latin ou des math\u00e9matiques. Le jeune Coetzee \u00e9tait bon en anglais et mauvais en g\u00e9ographie. M\u00eame s'il parlait l'anglais assez couramment (c'\u00e9tait l'une des deux langues qu'on utilisait dans sa famille, avec \u00e9videmment le n\u00e9erlandais si particulier des Afrikaans), il imaginait na\u00efvement que seuls les Anglais de Grande-Bretagne parlaient bien leur langue. C'\u00e9tait donc la langue de l'Empire britannique, qui avait combattu les Boers, ces colons d'Afrique du Sud, et s'\u00e9tait empar\u00e9 du pays. Sa situation \u00e9voquerait celle de Joyce qui pr\u00eate \u00e0 Stephen Dedalus ces paroles fameuses:\n\nLe langage que nous parlons lui appartient avant de m'appartenir. Combien diff\u00e9rents sont les mots: patrie, Christ, bi\u00e8re, ma\u00eetre, sur ses l\u00e8vres et sur les miennes! Je ne puis prononcer ou \u00e9crire ces mots sans une inqui\u00e9tude spirituelle. Son idiome, si familier et \u00e9tranger \u00e0 la fois, sera toujours pour moi un langage acquis. Je n'ai ni fa\u00e7onn\u00e9 ni accept\u00e9 ses mots. Ma voix les tient aux abois. Mon \u00e2me s'exasp\u00e8re \u00e0 l'ombre de son langage.\n\nCe que ressentait Joyce dans Dublin, alors capitale de la colonie britannique qu'\u00e9tait l'Irlande, n'\u00e9tait pas fondamentalement diff\u00e9rent de la situation o\u00f9 se trouvait le jeune Coetzee en Afrique du Sud, encore un dominion britannique \u00e0 l'\u00e9poque o\u00f9 se mettait en place le syst\u00e8me de l'Apartheid. Le bilinguisme de la famille de Coetzee trancherait cependant avec le monolinguisme de la famille de Joyce qui ne parlait pas le ga\u00e9lique \u00e0 Dublin, mais ce monolinguisme fut battu en br\u00e8che apr\u00e8s le d\u00e9part pour Trieste. \u00c0 partir de ce moment, l'italien rivalisa avec l'anglais dans les \u00e9changes entre Joyce, Nora et leurs deux enfants. Joyce et Nora parlaient le dialecte triestin en famille dans les ann\u00e9es parisiennes.\n\nCe n'\u00e9tait pas le cas de la famille Derrida: aucune autre langue que le fran\u00e7ais n'\u00e9tait parl\u00e9e \u00e0 El Biar ou \u00e0 Alger. La d\u00e9couverte que Coetzee fit de l'anglais d'Angleterre lorsqu'il y partit \u00e0 l'\u00e2ge de vingt et un ans ressemble pourtant \u00e0 l'exp\u00e9rience que Derrida, juif fran\u00e7ais d'Alg\u00e9rie, eut du fran\u00e7ais et des Fran\u00e7ais quand, \u00e0 dix ans, il fut expuls\u00e9 de son \u00e9cole. Les lois raciales de Vichy avaient impos\u00e9 un quota d'\u00e9l\u00e8ves juifs, il avait \u00e9t\u00e9 d\u00e9pass\u00e9, et on le pria de quitter son lyc\u00e9e. Ceci se passait en 1942 \u00e0 Alger. Ces r\u00e8gles avaient \u00e9t\u00e9 lanc\u00e9es par l'administration fran\u00e7aise qui avait devanc\u00e9 les exigences raciales des nazis. Ce traumatisme de jeunesse eut des effets \u00e0 long terme sur Derrida, ce qui rend compte de sa position complexe face \u00e0 la langue fran\u00e7aise. Son langage, le seul dans lequel il se sentait \u00e0 l'aise, n'\u00e9tait pourtant pas une \u00ablangue maternelle\u00bb.\n\nNous pouvons lire le reste de l'histoire dans la biographie de Peeters et dans Le monolinguisme de l'autre. Quelques semaines apr\u00e8s l'exclusion raciste, les troupes am\u00e9ricaines et anglaises prirent d'assaut le Maroc, la Tunisie et l'Alg\u00e9rie. Le jeune Jackie Derrida avait \u00e9t\u00e9 envoy\u00e9 dans une \u00e9cole religieuse juive qu'il n'aimait pas. Le r\u00e9gime fran\u00e7ais local changea de politique, de Gaulle prit le pouvoir et la nationalit\u00e9 fran\u00e7aise fut redonn\u00e9e aux juifs d'Alg\u00e9rie. \u00c0 l'autre extr\u00e9mit\u00e9 du continent africain, dans la ville du Cap, Coetzee avait tout juste deux ans.\n\nCes d\u00e9tails biographiques nous permettent d'appr\u00e9hender une histoire plus globale, histoire souvent retravaill\u00e9e par les romans de Coetzee. Il se trouva face \u00e0 une double all\u00e9geance quand il se rendit compte qu'il parlait un meilleur anglais que la plupart des Britanniques, mais que d\u00e8s qu'il ouvrait la bouche, on voyait qu'il \u00e9tait un \u00e9tranger. Si eux parlaient leur langue maternelle, lui n'\u00e9tait qu'un invit\u00e9. Cette situation n'est pas rare et l'\u00e9change de lettres y revient. Coetzee se demande donc s'il est possible de consid\u00e9rer un langage quelconque comme le sien propre. \u00abS'aventurer dans un langage est toujours faire infraction. Encore plus si l'on trouve dans la langue que l'on croit ma\u00eetriser des \u00e9chos venus d'ailleurs et d'autrefois qui tombent de votre plume et vous rappellent qui poss\u00e9dait ces expressions bien avant vous !\u00bb (HN, 67)\n\nLa r\u00e9ponse amicale de Paul Auster corrobore cette impression de d\u00e9possession. De New York, Auster est bien plac\u00e9 pour comprendre ce que signifie un h\u00e9ritage linguistique: \u00abIl y a juste trois g\u00e9n\u00e9rations, mes arri\u00e8re-grands-parents parlaient le russe, le polonais et le yiddish. Le fait que je suis n\u00e9 dans un pays anglophone me semble totalement contingent, un coup du hasard.\u00bb (HN, 70) Sa r\u00e9ponse inclut une bonne histoire qu'il a entendue \u00e0 la radio. Un fondamentaliste du sud des \u00c9tats-Unis attaquait les programmes scolaires visant \u00e0 enseigner des langues \u00e9trang\u00e8res aux \u00e9l\u00e8ves am\u00e9ricains avec cet argument massue: \u00abSi l'anglais suffisait \u00e0 J\u00e9sus, l'anglais devrait nous suffire.\u00bb Cette version absurde du \u00abmonolinguisme du m\u00eame\u00bb t\u00e9moigne certes d'une insondable b\u00eatise provinciale, mais aussi rappelle, comme le souligne Auster, que \u00abnotre monde est notre langage\u00bb. Auster \u00e9crit: \u00abS'il vous semble inconcevable que le fils de Dieu ait pu parler un langage autre, c'est qu'il est le Verbe, le Verbe existe dans un seul langage, celui qui se trouve \u00eatre le v\u00f4tre.\u00bb (HN, 70). Il se peut que cette histoire dr\u00f4le ait sugg\u00e9r\u00e9 \u00e0 Coetzee de traiter par la fiction cette simple question: quel langage J\u00e9sus parlait-il, le sien ou le n\u00f4tre? Pendant des si\u00e8cles, on a admis que Dieu parlait en h\u00e9breu et son fils en aram\u00e9en, la lingua franca de la Palestine du premier si\u00e8cle. Ces questions linguistiques nous am\u00e8nent \u00e0 nous demander ce que signifie le jeu appuy\u00e9 sur les langues que l'on trouve dans le dernier roman de Coetzee, Une enfance de J\u00e9sus.\n\nSon principal protagoniste s'appelle Sim\u00f3n avec un accent sur le \u00abo\u00bb, comme pour faire \u00e9cho au jeu de mots le plus c\u00e9l\u00e8bre de l'histoire du christianisme. On le trouve chez saint Jean, qui cite le mot de J\u00e9sus: \u00abTu es Simon, le fils de Jean; tu t'appelleras C\u00e9phas \u2013 ce qui veut dire Pierre.\u00bb C\u00e9phas \u00e9tait donc le nom donn\u00e9 par J\u00e9sus \u00e0 l'ap\u00f4tre connu sous le nom de Simon qui devient alors \u00abSimon Pierre\u00bb. Le mot grec utilis\u00e9 par Jean en 1: 42, \u039a\u03b7\u03c6\u1fb6\u03c2 (K\u0113ph\u00e2s), est un surnom en aram\u00e9en pour \u00e9voquer k\u00eap\u00e2, ce qui signifie \u00abroc\u00bb ou \u00abpierre\u00bb. En grec, il faut ajouter un sigma \u00e0 la fin du mot \u00abk\u00eap\u00e2\u00bb afin qu'il ait une consonance masculine. Le sens domine et oriente le destin futur de Pierre, le premier \u00abroc\u00bb de la fondation de l'\u00c9glise \u00e0 venir. Nous acceptons depuis la traduction du grec \u03a0\u03ad\u03c4\u03c1\u03bf\u03c2 (Petros) en Pierre. Ceci peut \u00e9voquer d'autres traductions dans l'onomastique des grands auteurs, allant du nom de Joyce qui venait du fran\u00e7ais \u00abjoyeux\u00bb, du nom de famille de Freud qui avait d\u00fb \u00eatre germanis\u00e9 pour ressembler \u00e0 \u00abJoie\u00bb (Freude), au nom de Beckett qui conservait les \u00e9chos de \u00abBequet\u00bb, sans parler d'un autre immigrant, lui aussi un huguenot fran\u00e7ais comme les anc\u00eatres de Beckett, un certain Dirk Couch\u00e9, n\u00e9 en 1655, dont le nom fut chang\u00e9 en Coets\u00e9 quand il s'installa au Cap. Puis il partit vivre dans une ferme \u00e0 Stellenboch qui prit le nom de Coetzenburg.\n\nCette g\u00e9n\u00e9alogie nous rappelle l'importance de ne pas oublier les accents. Dans le roman de Coetzee, nous ne pouvons ignorer l'accent sur le \u00abo\u00bb de Sim\u00f3n, ce qui le caract\u00e9rise comme hispanique. De plus, ce nom n'est pas son vrai nom. Quand son laissez-passer est mentionn\u00e9 tout au d\u00e9but, il annonce qu'il arrive du port de Belstar avec un jeune enfant, David: \u00abC'est l\u00e0 qu'ils nous ont donn\u00e9 nos noms, nos noms espagnols.\u00bb Sim\u00f3n, le point focal du r\u00e9cit, n'est presque jamais appel\u00e9 par son nom; ce n'est que de mani\u00e8re indirecte, quand on lui parle, que nous le saisissons. De la m\u00eame mani\u00e8re, David n'est pas le nom de l'enfant et Elena remarque que Sim\u00f3n l'appelle le plus souvent par le g\u00e9n\u00e9rique, \u00able gar\u00e7on\u00bb. Sim\u00f3n explique: \u00abDavid est le nom qu'ils lui ont donn\u00e9 au camp. Il ne l'aime pas, il dit que ce n'est pas son vrai nom.\u00bb (EJ, 82)\n\nLes fondations impliquent toujours de nouveaux bapt\u00eames, une nomination qui peut \u00eatre un jeu de mots, ce qui fit dire \u00e0 Joyce que l'\u00c9glise catholique \u00e9tait moins fond\u00e9e sur Pierre que sur un jeu de mots. De l'h\u00e9breu Shim'on, qui signifie \u00abcelui qui entend\u00bb, nous sommes all\u00e9s vers Cephas, Petros, Petrus et Pierre. Mais le nom de J\u00e9sus lui-m\u00eame peut se r\u00e9\u00e9crire en \u00e9quivalents tels que \u00able Christ\u00bb, \u00able Verbe\u00bb, \u00abAlpha et Om\u00e9ga\u00bb ou \u00abFils de Dieu\u00bb. Ces nominations ont jou\u00e9 un r\u00f4le important dans les textes que nous appelons les \u00e9vangiles. Afin de poursuivre l'investigation du roman, nous allons devoir ne pas nous limiter aux \u00e9vangiles \u00abauthentiques\u00bb ou \u00abcanoniques\u00bb, mais prendre en compte une tradition plus vaste qui inclut les \u00e9vangiles apocryphes. Un de ces genres est nomm\u00e9 \u00ab\u00e9vangiles de l'enfance de J\u00e9sus\u00bb et l'un d'eux en particulier offre une source tr\u00e8s nette.\n\nDans les quatre \u00e9vangiles canoniques, bien peu de d\u00e9tails sont donn\u00e9s sur l'enfance de J\u00e9sus. Nous sautons de la nativit\u00e9 \u00e0 l'adolescence. En revanche, ces ann\u00e9es de formation sont trait\u00e9es avec force d\u00e9tails surprenants dans les \u00e9vangiles du Pseudo-Thomas et du Pseudo-Matthieu. L'\u00e9vangile de Thomas en particulier nous fait d\u00e9couvrir en J\u00e9sus un enfant assez terrifiant. Il peut tuer d'un regard, surtout quand on le contrarie. J\u00e9sus tue ainsi deux enfants qui l'ennuient et il rend aveugles tous ceux qui refusent de reconna\u00eetre sa nature divine. Ayant appris ces \u00e9v\u00e9nements dramatiques, un ma\u00eetre d'\u00e9cole nomm\u00e9 Zacchaeus veut forcer J\u00e9sus \u00e0 venir dans son \u00e9cole. Il va lui apprendre \u00e0 lire, p\u00e9dagogie qui devrait calmer son esprit rebelle. Joseph, le p\u00e8re de J\u00e9sus, proteste, ajoute que cela n'est ni n\u00e9cessaire ni r\u00e9alisable. J\u00e9sus lui-m\u00eame explique qu'il n'y a rien qu'on puisse lui apprendre: il \u00e9tait vivant bien avant la naissance du ma\u00eetre d'\u00e9cole, \u00e0 qui il peut pr\u00e9dire la date de sa mort. Tous les t\u00e9moins s'\u00e9tonnent et s'effraient: un gar\u00e7on de cinq ans qui parle comme un proph\u00e8te! Mais le professeur insiste et l'on am\u00e8ne J\u00e9sus dans la salle de classe. Voici la sc\u00e8ne:\n\nLe ma\u00eetre \u00e9crivit l'alphabet pour lui et commen\u00e7a son instruction en r\u00e9p\u00e9tant la lettre alpha de nombreuses fois. Mais l'enfant ne desserrait pas les dents et ne r\u00e9pondait pas. Naturellement, le ma\u00eetre se mit en col\u00e8re et le frappa sur la t\u00eate. L'enfant prit le coup calmement et dit: \u00abC'est moi qui vais t'apprendre quelque chose, non toi. Je sais d\u00e9j\u00e0 les lettres que tu veux m'enseigner, et je te condamne grandement. Pour toi, ces lettres sont comme un vase de bronze ou des cymbales que l'on heurte sans produire ni gloire ni sagesse, car cela n'est que du bruit. Personne ne sait l'\u00e9tendue de mon savoir.\u00bb Puis cessant d'\u00eatre en col\u00e8re, il r\u00e9cita tr\u00e8s rapidement les lettres d'alpha \u00e0 om\u00e9ga.\n\n[...]\n\nAlors J\u00e9sus regarda le ma\u00eetre et dit: \u00abSi tu ne sais pas la nature de la lettre alpha, comment veux-tu enseigner la lettre b\u00eata? Imposteur, si tu sais cela, apprends-moi la premi\u00e8re lettre alpha, et alors je te ferai confiance avec b\u00eata.\u00bb Or le ma\u00eetre n'\u00e9tait pas capable de r\u00e9pondre \u00e0 ses questions, et ne savait que dire. Tout le public \u00e9coutait. Il dit \u00e0 Zacchaeus: \u00ab\u00c9coute, ma\u00eetre, et observe l'arrangement de la premi\u00e8re lettre: comment elle a deux lignes droites qui se courbent vers un point au milieu, se rassemblent vers le haut comme en dansant entre deux et trois coins, sans antagonisme, pour rester de la m\u00eame famille (omogeneis) et donner \u00e0 alpha des lignes d'\u00e9gale mesure.\u00bb (IGJT, 117 et 119)\n\nL'\u00e9vocation brillante et obscure \u00e0 la fois de J\u00e9sus pr\u00eate un dynamisme inattendu aux moindres d\u00e9tails de la forme de la lettre grecque alpha, la premi\u00e8re de l'alphabet. Tout se passe comme si J\u00e9sus l'\u00e9crivait dans l'air et donnait un sens \u00e9sot\u00e9rique \u00e0 ses circonvolutions, comme un ruban mystique qui se nouerait sur lui-m\u00eame tout en restant ouvert. Zacchaeus comprend qu'il s'agit d'all\u00e9gories (allegorias) dont il ne peut saisir la signification. Il en vient \u00e0 d\u00e9sesp\u00e9rer de sa profession. Dans un long passage de lamentation rh\u00e9torique, il se plaint de sa disgr\u00e2ce. Un simple enfant lui a fait honte devant tout le monde et lui a montr\u00e9 son ignorance. Il r\u00e9p\u00e8te: \u00ab\"Je ne sais rien, mes amis\" (agno\u00f4, \u00f4 philoi), parce qu'il ignore l'arch\u00ea ou le telos des lettres de l'alphabet.\u00bb (IGJT, 122) La sinc\u00e8re confession de son ignorance radoucit J\u00e9sus; il rend la vue \u00e0 ceux qu'il a aveugl\u00e9s et fait revivre les enfants qu'il a tu\u00e9s.\n\nPourtant, ce n'est pas encore la fin. Cette fois, c'est Joseph qui d\u00e9sire que J\u00e9sus apprenne l'alphabet, comme s'il avait oubli\u00e9 ce qu'il avait vu ou affirm\u00e9. Il d\u00e9niche un autre ma\u00eetre rendu prudent par le r\u00e9cit de ce qui s'est pass\u00e9. Tout se r\u00e9p\u00e8te, mais J\u00e9sus est moins patient: quand le ma\u00eetre exasp\u00e9r\u00e9 le frappe sur la t\u00eate, J\u00e9sus le maudit et le ma\u00eetre tombe au sol, sans vie. Joseph en conclut que J\u00e9sus ne doit plus sortir de sa maison: \u00abCeux qui l'ennuient finissent morts.\u00bb (IGJT, 133) De mani\u00e8re encore plus surprenante, il cherche et trouve un troisi\u00e8me ma\u00eetre. J\u00e9sus ne perd plus de temps \u00e0 expliquer la forme des lettres, il prend aussit\u00f4t un livre et l'interpr\u00e8te: \u00abIl ouvrit la bouche et parla par la puissance de l'esprit saint, enseignant la loi (nomos) \u00e0 ceux qui se trouvaient l\u00e0.\u00bb (IGJT, 135) Apparemment satisfait par sa propre performance, il pardonne et ressuscite le second ma\u00eetre.\n\nDans tous ces textes, J\u00e9sus est pr\u00e9sent\u00e9 comme un enfant surdou\u00e9, mais caract\u00e9riel, et donc d'autant plus dangereux qu'il dispose de pouvoirs divins ou magiques. Le paradoxe narratif est d'arriver \u00e0 nous faire voir moins un enfant-Dieu que, plus litt\u00e9ralement, un Dieu-encore-enfant. J\u00e9sus peut donc dire: \u00abJe suis l'Alpha et l'Om\u00e9ga\u00bb, comme dans l'Apocalypse (1: 8). Il est la premi\u00e8re et la derni\u00e8re lettre de l'alphabet, et donc tient ensemble les fils du pass\u00e9 et de l'avenir. Les ma\u00eetres au savoir limit\u00e9 font de pi\u00e8tres guides: il faut leur donner une bonne le\u00e7on. Nous retrouvons ce th\u00e8me p\u00e9dagogique un peu partout dans Une enfance de J\u00e9sus. David se pr\u00e9sente comme un enfant rebelle qui refuse d'apprendre parce qu'il sait d\u00e9j\u00e0. David dit m\u00eame: \u00ab\"Yo soy la verdad\" \u2013 Je suis la v\u00e9rit\u00e9.\u00bb (EJ, 307), alors qu'il devait \u00e9crire: \u00abConviene que yo diga la verdad\u00bb (Je dois dire la v\u00e9rit\u00e9). Il d\u00e9forme la phrase pour lui donner un sens mystique et eschatologique, ce qui conduit le professeur \u00e0 conclure: \u00abCe que je dis, c'est qu'il ne peut y avoir qu'une seule source d'autorit\u00e9 dans la classe.\u00bb (EJ, 307) C'est toute la structure de l'autorit\u00e9 qui va \u00eatre contest\u00e9e ou subvertie au nom d'une annonce messianique ouverte vers l'avenir.\n\nCoetzee fait allusion au topos de l'alpha-om\u00e9ga de mani\u00e8re oblique quand David affirme que 888 est plus grand que 889 \u2013 sans doute que pour lui, trois 8 c\u00f4te \u00e0 c\u00f4te occupent un volume plus large que 8, 8, et 9. S'effor\u00e7ant d'imposer son autorit\u00e9 et aussi de lui enseigner les r\u00e8gles de base de l'arithm\u00e9tique, Sim\u00f3n r\u00e9fl\u00e9chit:\n\nComment se fait-il que cet enfant, intelligent comme il l'est, si pr\u00eat \u00e0 faire son chemin dans la vie, refuse de comprendre?\n\n\u00abTu me dis que tu as fait le tour de tous les nombres, dit-il. Alors dis-moi quel est le dernier nombre. Le tout dernier de tout. Mais ne me dis pas c'est om\u00e9ga. Om\u00e9ga ne compte pas.\n\n\u2014 C'est quoi om\u00e9ga?\n\n\u2014 Peu importe. Simplement ne dis pas \"om\u00e9ga\". Dis-moi le dernier nombre, le tout dernier.\u00bb (EJ, 206-207)\n\nDavid reste muet, se perd dans une transe et se met \u00e0 hurler quand Sim\u00f3n tente de l'en tirer.\n\nHurlant, il reproche \u00e0 Sim\u00f3n de lui avoir fait oublier sa vision en une allusion claire \u00e0 la th\u00e9orie platonicienne de la r\u00e9miniscence. Comme M\u00e9non qui poss\u00e8de un jeune esclave capable de voir les nombres dans son esprit, David allie un sens des lettres inn\u00e9, m\u00eame s'il n'est pas toujours capable de lire un livre, et une intuition des chiffres qui le plonge dans un univers platonicien. D'ailleurs, David d\u00e9forme de mani\u00e8re amusante le nom du chien de Mickey, Pluto qu'il appelle Plato (Platon en anglais). Sim\u00f3n r\u00e9ussit \u00e0 amener David \u00e0 quitter l'appartement de Daga o\u00f9 il regardait la t\u00e9l\u00e9vision: \u00abTu pourras regarder Mickey la prochaine fois, dit Daga. Je te le promets. On va le garder ici rien que pour toi.\/ Et Platon?\/ Et Platon. On peut lui garder Platon aussi, mon chou, non?\u00bb (EJ, 251; traduction modifi\u00e9e) Nous savons depuis \u00abLa pharmacie de Platon\u00bb de Derrida \u00e0 quel point la question platonicienne de l'\u00e9criture est la question m\u00eame de l'autonomie ou de l'ind\u00e9pendance des fils, ce qui inclut les chiens, comme nous l'avons vu. Parmi tant de passages de dialogues c\u00e9l\u00e8bres de Platon, celui du mythe de Teuth avanc\u00e9 dans le Ph\u00e8dre vise \u00e0 r\u00e9duire le langage \u00e9crit au statut d'un aide-m\u00e9moire dangereux parce que les mots \u00e9crits perdent le contact vivant avec leurs \u00abp\u00e8res\u00bb et possesseurs de leur sens originel.\n\nL'\u00e9criture fonctionne comme un enfant indigne, un fils prodigue qui ne revient jamais vers la demeure de ses parents. Ainsi dans la sc\u00e8ne de la t\u00e9l\u00e9vision \u00e9voquant Platon, David dit: \u00abJe n'ai pas de m\u00e8re et je n'ai pas de p\u00e8re. Je suis juste moi.\u00bb (EJ, 183) Cet \u00e9cho affaibli du \u00abJe suis qui je suis\u00bb divin implique n\u00e9anmoins une grammatologie sp\u00e9cifique. Ses implications sont nombreuses et mettent en question l'autorit\u00e9 des p\u00e8res et des p\u00e9dagogues, et aussi mobilisent le fantasme freudien d'un total auto-engendrement.\n\nUne premi\u00e8re piste, d\u00e9j\u00e0 fr\u00e9quent\u00e9e, est celle du langage. Rappelons \u00e0 grands traits les lin\u00e9aments de l'intrigue. Sim\u00f3n, qui n'est pas le p\u00e8re de David, mais fait fonction de gardien et protecteur, fait partie de r\u00e9fugi\u00e9s d'une catastrophe qui n'est jamais nomm\u00e9e et s'efforce de trouver une place dans Novilla, ville principale d'un pays nouveau. Cette terre d'accueil tour \u00e0 tour hospitali\u00e8re et d\u00e9routante sinon ferm\u00e9e dans ses rites et sa bureaucratie envahissante est un pays o\u00f9 l'espagnol est la langue officielle. Pass\u00e9 un temps d'accoutumance, ayant trouv\u00e9 un travail dur et mal r\u00e9mun\u00e9r\u00e9, Sim\u00f3n d\u00e9cide qu'In\u00e8s, une jolie jeune femme qu'il voit pour la premi\u00e8re fois, pourra \u00eatre une m\u00e8re pour David. De mani\u00e8re surprenante, elle accepte, ce qui l'oblige \u00e0 quitter le luxe relatif de la r\u00e9sidence o\u00f9 elle habitait dans une certaine oisivet\u00e9. Mais l'arriv\u00e9e d'un nouvel immigrant, jeune et violent, Daga, ravive les blessures et d\u00e9sirs du pass\u00e9. Daga et Sim\u00f3n apparaissent alors comme des rivaux, moins pour les charmes d'In\u00e8s que pour la direction de la conscience de David, qui, lui, ne se plie pas du tout aux r\u00e8gles scolaires en vigueur de Novilla. \u00c0 la fin du roman, In\u00e8s, Sim\u00f3n et David quittent Novilla pour chercher une nouvelle vie. Les parall\u00e8les avec la vie de J\u00e9sus telle que nous la connaissons sont multiples, mais leur signification n'appara\u00eet qu'\u00e0 la fin.\n\nUne des apories narratologiques d'Une enfance de J\u00e9sus concerne la langue dans laquelle le roman est \u00e9crit. Les dialogues que nous lisons en fran\u00e7ais traduisent un texte en anglais qui a l'air d'avoir \u00e9t\u00e9 traduit de l'espagnol. De temps en temps, le texte glisse le mot original \u2013 d\u00e8s la premi\u00e8re page, nous d\u00e9couvrons avec Sim\u00f3n les mots formant Centro de Reubicacion Novilla, dont il semble ignorer la signification. Il explique qu'il n'a pas appris \u00abreubicacion\u00bb au centre de Belstar dont il vient. Un lecteur de Coetzee sera attentif \u00e0 ce mot, car il a \u00e9t\u00e9 employ\u00e9, soit en Afrikaans, soit en anglais, dans Life & Times of Michael K par exemple. Michael doit passer un certain nombre de mois dans le \u00abJakkalsdrif Relocation Camp\u00bb, un centre d'h\u00e9bergement pour migrants ou vagabonds.\n\nSim\u00f3n et David ont un seul langage en commun, l'espagnol qu'ils viennent d'apprendre. Ils se sont rencontr\u00e9s sur un bateau qui les amenait vers Belstar et Novilla. On comprend que David a perdu ses papiers d'identit\u00e9, ainsi que les derni\u00e8res traces de l'identit\u00e9 de sa m\u00e8re. Sim\u00f3n et David se sont rencontr\u00e9s lors de la recherche de cette lettre de sa m\u00e8re. Cette section pourrait avoir \u00e9t\u00e9 \u00e9crite par Kafka, Sim\u00f3n prenant le r\u00f4le de l'oncle qui appara\u00eet miraculeusement dans Der Verschollene ou Amerika, un autre roman all\u00e9gorique racontant la d\u00e9couverte d'un nouveau pays en tant qu'\u00e9migrant.\n\nDavid et Sim\u00f3n doivent l'un et l'autre faire face \u00e0 un processus d'adaptation qui r\u00e9ussit pour l'adulte, mais non pour l'enfant. Encore une fois, c'est l'apprentissage de la lecture qui pose probl\u00e8me. Quand il est temps que David apprenne \u00e0 lire, on lui trouve un vieil exemplaire tout d\u00e9chir\u00e9 d'une version pour enfants de Don Quichotte, un roman avec lequel, de toute \u00e9vidence, il est familier. David se passionne tellement pour cette histoire qu'il garde le livre avec lui partout o\u00f9 il va et le consid\u00e8re comme un texte sacr\u00e9. Don Quichotte joue le r\u00f4le d'un pendant de la Bible, ou plut\u00f4t des \u00e9vangiles de J\u00e9sus. Comme les \u00e9vangiles, le roman de Cervant\u00e8s met en place un contraste entre l'ancien et le nouveau.\n\nCoetzee avait conclu son discours d'acceptation du prix de J\u00e9rusalem avec une r\u00e9f\u00e9rence \u00e0 Cervant\u00e8s: \u00ab[...] comment passer de notre monde de violents fantasmes \u00e0 un v\u00e9ritable monde vivable? C'est une \u00e9nigme que le Don Quichotte de Cervant\u00e8s r\u00e9sout facilement. Il laisse derri\u00e8re lui le monde poussi\u00e9reux, \u00e9triqu\u00e9, et ennuyeux de La Manche pour p\u00e9n\u00e9trer dans le monde du merveilleux, ce qui correspond \u00e0 un acte d\u00e9lib\u00e9r\u00e9 de l'imagination.\u00bb C'est ce merveilleux que repr\u00e9sente David dans le roman. Il refuse de se laisser enfermer dans les cat\u00e9gories du savoir officiel et se laisse guider par la magie de l'imagination. Quand il d\u00e9couvre l'\u00e9pisode de la famille Montesinos, il imite Don Quichotte qui pense avoir pass\u00e9 trois jours et trois nuits dans une grotte magique alors qu'il n'est parti qu'une heure. Dans cette grotte, le h\u00e9ros rencontre le vieux Montesinos, c\u00e9l\u00e8bre pour avoir arrach\u00e9 le c\u0153ur de son cousin Durandarte apr\u00e8s sa mort. Il devait porter ce c\u0153ur \u00e0 Belerma, la femme de Durandarte. Mais il ne put r\u00e9aliser sa promesse, car Merlin les avait tous emprisonn\u00e9s dans la grotte, non sans avoir annonc\u00e9 qu'ils seraient lib\u00e9r\u00e9s quand Don Quichotte y p\u00e9n\u00e9trerait.\n\nDans cet \u00e9pisode, Don Quichotte demande au vieux Montesinos s'il est vrai qu'il a d\u00e9coup\u00e9 le c\u0153ur de Durandarte avec une dague. Montesinos confirme qu'il a bien coup\u00e9 le c\u0153ur, mais avec un poignard: \u00abIl r\u00e9pondit que tout \u00e9tait vrai, \u00e0 l'exception de la dague (daga): car ce n'\u00e9tait pas une dague, mais un poignard brillant (pu\u00f1al buido) plus effil\u00e9 qu'un poin\u00e7on.\u00bb C'est de ce passage que Daga, pr\u00e9sent\u00e9 comme un esprit n\u00e9gateur et m\u00e9phistoph\u00e9lique, tire son nom, symbolis\u00e9 d'ailleurs par le couteau avec lequel il blesse le doux Alvaro dans une rixe (EJ, 68-69).\n\nEmilio Daga, lui aussi un nouvel arrivant, ne joue pas le jeu des lois locales et choisit de s'imposer par la force. David lui fait confiance parce que Daga le laisse regarder la t\u00e9l\u00e9vision et lui fait croire \u00e0 la magie, mais ce serait plut\u00f4t le double n\u00e9gatif de Sim\u00f3n, car il repr\u00e9sente la simonie m\u00e9di\u00e9vale: il pense pouvoir acheter l'aura, le pouvoir de fascination, le charisme qui se d\u00e9gagent de David par des cadeaux. Mais il tente aussi de s\u00e9duire In\u00e8s, \u00e0 qui il serait dispos\u00e9 \u00e0 \u00abfaire\u00bb un enfant biologique. C'est donc un dangereux rival pour Sim\u00f3n, en qui les actions transgressives de Daga ram\u00e8nent la nostalgie du vieil ordre d'avant, ordre marqu\u00e9 par le d\u00e9sir.\n\nLe choix de Don Quichotte comme Bible profane s'explique par le d\u00e9sir triangulaire, ainsi que l'analyse Coetzee en 1980. Coetzee commentait alors la th\u00e9orie de d\u00e9sir triangulaire lanc\u00e9e par Ren\u00e9 Girard dans Mensonge romantique et v\u00e9rit\u00e9 romanesque: \u00abLe d\u00e9sir triangulaire appara\u00eet et est analys\u00e9 pour la premi\u00e8re fois dans Don Quichotte, qui marque le d\u00e9but de la tradition romanesque de la fiction critique. C'est donc un ph\u00e9nom\u00e8ne sp\u00e9cifique \u00e0 la modernit\u00e9. Il se d\u00e9veloppe comme la cons\u00e9quence d'un humanisme post-religieux et prend de l'essor quand les diff\u00e9rences sociales sont abolies.\u00bb Quand le roman de Coetzee r\u00e9\u00e9crit les \u00e9vangiles apocryphes, il y greffe la probl\u00e9matique girardienne d'une \u00abv\u00e9rit\u00e9 romanesque\u00bb en nous donnant le portrait saisissant d'un enfant merveilleux qui pourrait s'av\u00e9rer n'\u00eatre qu'un simple psychotique, ou un enfant autiste \u00e0 qui ses parents adoptifs c\u00e8dent trop souvent ou trop vite.\n\n\u00c0 la fin du roman, le sch\u00e9ma mythique d'un vieux p\u00e8re qui n'est pas son p\u00e8re, d'une jeune femme qui n'est pas sa m\u00e8re, et de l'enfant visionnaire, mais na\u00eff, capable de risquer le tout pour le tout, \u00e9voque les formes traditionnelles qu'on pourrait retrouver dans la multitude de tableaux intitul\u00e9s \u00abLa fuite en \u00c9gypte\u00bb. Ce qui n'est pas conventionnel est l'h\u00e9sitation sur le statut de David dont l'aspiration vers une \u00abnouvelle vie\u00bb se met en place sur fond de la culture assagie, temp\u00e9r\u00e9e, rationnelle de Novilla, dans laquelle on reconna\u00eet le monde d'un nihilisme post-apocalyptique contemporain.\n\nCe monde de l'accueil poss\u00e8de des c\u00f4t\u00e9s positifs, puisque l'hospitalit\u00e9 face aux arrivants semble une r\u00e8gle de base. C'est sur ce concept que Coetzee fait porter sa critique politique qui fait en tous points \u00e9cho aux th\u00e8ses de Derrida sur l'hospitalit\u00e9. Le th\u00e8me de l'hospitalit\u00e9 a \u00e9t\u00e9 trait\u00e9 par Mike Marais dans Secretary of the Invisible, livre publi\u00e9 en 2009, et qui ne pouvait tenir compte d'Une enfance de J\u00e9sus. Novilla a \u00e9labor\u00e9 un syst\u00e8me d'accueil qui permet aux immigrants de trouver du travail et de se loger. Bien vite, nous nous rendons compte que cette hospitalit\u00e9 rencontre de s\u00e9rieuses limites. La premi\u00e8re nuit que David et Sim\u00f3n passent dans la ville, \u00e0 cause d'une erreur sur la clef de leur logement, ils doivent dormir \u00e0 la belle \u00e9toile chez la jeune femme qui les accueille dans son jardin, mais non dans sa maison. Ils meurent de froid et n'ont qu'un peu de pain \u00e0 manger. De plus, Ana refuse de leur pr\u00eater de l'argent le lendemain. De mani\u00e8re \u00e9trange, elle se radoucit par la suite et on comprend qu'elle voulait \u00e9viter une avance sexuelle de Sim\u00f3n. Elle sait qu'elle est belle puisqu'elle pose nue pour les cours du soir de dessin, qui sont, on comprend pourquoi, les plus populaires.\n\nCe roman semble mettre en place le double concept d'hospitalit\u00e9 \u00e9labor\u00e9 par Derrida, qui oppose toujours une hospitalit\u00e9 absolue ou inconditionnelle \u00e0 une hospitalit\u00e9 relative ou limit\u00e9e. Derrida nous am\u00e8nerait \u00e0 penser comme Cervant\u00e8s, c'est-\u00e0-dire \u00e0 voir la r\u00e9alit\u00e9 \u00e0 travers les yeux de l'id\u00e9aliste absolu qu'est Don Quichotte et les yeux du r\u00e9aliste pragmatique qu'est Sancho Pan\u00e7a. Dans la parallaxe conceptuelle \u00e9tablie par Derrida, il faut d'abord qu'il y ait une \u00abhospitalit\u00e9 pure\u00bb et elle se confronte \u00e0 une hospitalit\u00e9 relative. L'hospitalit\u00e9 pure risque d'entra\u00eener des cons\u00e9quences extr\u00eames, puisqu'elle consiste \u00ab\u00e0 laisser sa maison ouverte \u00e0 l'arrivant impr\u00e9visible, qui peut bien \u00eatre un intrus dangereux, susceptible \u00e9ventuellement de faire le mal\u00bb. Si l'on pousse cette exigence \u00e0 la limite, on peut perdre ses biens et sa libert\u00e9, ce qui est une des le\u00e7ons am\u00e8res du roman Disgr\u00e2ce. En opposition avec cette loi inconditionnelle, les lois conditionnelles limitent et r\u00e9glementent l'acc\u00e8s de l'arrivant. On trouverait ici toutes les coutumes, les r\u00e8gles, les rites qui sp\u00e9cifient comment on peut accueillir un arrivant selon des liens de parent\u00e9, d'amiti\u00e9 ou de fr\u00e9quentation sociale.\n\nLa force de l'analyse de Derrida est de toujours combiner les deux types d'hospitalit\u00e9. Comme Samir Haddad l'a bien montr\u00e9 dans Derrida and the Inheritance of Democracy, \u00abla loi inconditionnelle, en elle-m\u00eame, ne suffit pas \u00e0 donner naissance \u00e0 l'hospitalit\u00e9, parce qu'il impossible de suivre cette loi\u00bb. Mais par ailleurs, les lois conditionnelles ne fournissent que des r\u00e9alisations imparfaites de la loi inconditionnelle, sans lesquelles son hospitalit\u00e9 absolue serait vide ou impuissante. Haddad exprime ceci de mani\u00e8re concise: \u00ab[...] la loi inconditionnelle d\u00e9pend de la loi conditionnelle afin d'\u00eatre une loi, tandis que les lois conditionnelles d\u00e9pendent de la loi inconditionnelle afin d'\u00eatre hospitali\u00e8res\u00bb. Dans ce rapport asym\u00e9trique, il faut \u00e0 la fois une ouverture radicale \u00e0 l'autre et en m\u00eame temps les r\u00e8gles pratiques limitant l'acc\u00e8s \u00e0 l'intimit\u00e9 de la maison. Pour qu'elle puisse \u00eatre efficace en tant que loi, l'hospitalit\u00e9 inconditionnelle implique sa propre transgression. Mais pour qu'elles puissent \u00eatre hospitali\u00e8res, les lois conditionnelles de l'hospitalit\u00e9 exigent une semblable transgression vers le transcendant.\n\nDaga est un bon exemple de l'arrivant impr\u00e9visible qui se comporte en intrus dangereux. Il travaille dur, certes, mais cherche \u00e0 n\u00e9gocier son salaire et les cadences, et demande plus que sa part \u00e0 la fin de la journ\u00e9e. Puis il n'h\u00e9site pas \u00e0 se servir de son couteau, comme on l'a vu. Plus tard, on sait qu'il a r\u00e9ussi: il a non seulement une t\u00e9l\u00e9vision dans un monde de la survie et de la p\u00e9nurie, mais une jeune et jolie ma\u00eetresse, et un superbe appartement. Ce que Daga rejette violemment, et ce qui g\u00eane sourdement Sim\u00f3n tout au cours du r\u00e9cit, c'est le consensus mou sur lequel repose la vie collective \u00e0 Novilla. On voit ceci lors d'une discussion de Sim\u00f3n avec Elena, avec qui il entretient d'\u00e9pisodiques rapports sexuels d\u00e9nu\u00e9s de toute passion. Elena formule avec pr\u00e9cision le code de ce Nouveau Monde qui rejette les valeurs de l'ancien; Sim\u00f3n lui demande si elle serait jalouse dans le cas o\u00f9 il se mettrait \u00e0 fr\u00e9quenter une autre femme, et elle r\u00e9pond ceci:\n\nLa seule amiti\u00e9, ce n'est pas assez bon pour toi. Si elle n'est pas accompagn\u00e9e des d\u00e9cha\u00eenements de la passion, il y manque quelque chose. Pour moi, c'est une fa\u00e7on de penser d\u00e9pass\u00e9e. Avec cette ancienne fa\u00e7on de penser, si combl\u00e9 que l'on soit, il manque toujours quelque chose. [...] Cette insatisfaction sans fin, ce d\u00e9sir ardent pour le quelque chose d'autre qui nous manque toujours, est une fa\u00e7on de penser dont nous sommes bien d\u00e9barrass\u00e9s, \u00e0 mon avis. Rien ne manque. Ce rien qui te para\u00eet manquer est une illusion. (EJ, 91-92)\n\nDe fait, la vie \u00e0 Novilla est r\u00e9gl\u00e9e par une utopie rationaliste qui vise \u00e0 abolir toutes passions malsaines. Tout path\u00e9tique a disparu de leur vie. Les gens ont perdu l'envie de savoir ce qui se passe ailleurs, ils vivent dans un pur pr\u00e9sent, travaillent, ont leurs loisirs sages, et voient dans la satisfaction des besoins le but de l'effort social. Une autre sc\u00e8ne permet d'avancer dans la critique politique au nom de l'hospitalit\u00e9. Sim\u00f3n a une discussion philosophique avec les autres dockers pour savoir s'il faut vivre dans un pr\u00e9sent sans histoire ou dans une soci\u00e9t\u00e9 plus historique. Il invoque l'histoire pour proposer qu'ils utilisent une grue m\u00e9canique afin d'\u00e9viter de d\u00e9barder de lourds sacs sur leurs dos, comme ils le font toute la journ\u00e9e. Sa vision de l'histoire est marqu\u00e9e par le progr\u00e8s technologique, mais elle se heurte \u00e0 la r\u00e9sistance de tous les autres qui affirment qu'ils ne se sentent pas concern\u00e9s par l'histoire. Tous sont venus d'ailleurs, ils ont trouv\u00e9 une soci\u00e9t\u00e9 plus juste et paisible, mais de ce fait pensent que l'histoire est une fable qui a \u00e9t\u00e9 invent\u00e9e (EJ, 159). Comme le dirait Nietzsche, la cons\u00e9quence de la mort de Dieu est que le monde est devenu une fable. Cette fable d\u00e9passe le doux scepticisme de ceux qui ne veulent pas prendre parti et pr\u00f4neraient une n\u00e9cessaire tol\u00e9rance religieuse, car elle s'\u00e9tend \u00e0 l'histoire elle-m\u00eame. Eugenio prend la parole et explique qu'il peut ressentir la g\u00e9ographie de mani\u00e8re concr\u00e8te, mais qu'il ne sent pas l'histoire \u00abdans ses os\u00bb (EJ, 160).\n\nCoetzee relie ici l'analyse nietzsch\u00e9enne du nihilisme au texte c\u00e9l\u00e8bre de T. S. Eliot qui d\u00e9finit le \u00absens historique\u00bb comme la possibilit\u00e9 de sentir toute une tradition, litt\u00e9raire et culturelle, de mani\u00e8re spontan\u00e9e et organique, bref, en ressentant sa pr\u00e9sence dans son propre corps et ses os. Dans le roman, les arguments de Sim\u00f3n finissent par l'emporter, au moins par la promesse de travailler dans de meilleures conditions, mais le choix du progr\u00e8s technologique se retourne contre lui. La grue m\u00e9canique, mal dirig\u00e9e par Eugenio, le heurte et le fait tomber entre le quai et le bateau. Il manque \u00eatre broy\u00e9 et noy\u00e9. Il ne meurt pas, mais sa convalescence est douloureuse. L'id\u00e9e de la grue est abandonn\u00e9e, il perd son travail, ce qui rend l'id\u00e9e de la fuite au d\u00e9sert plus tentante. La solution messianique s'impose comme la seule alternative \u00e0 un retour au monde limit\u00e9 de l'utopie rationnelle.\n\nLa critique politique de Coetzee semble assez claire: aucune soci\u00e9t\u00e9 ne peut abandonner son lien avec l'histoire, m\u00eame avec les meilleures intentions du monde. Le syst\u00e8me r\u00e9gl\u00e9 des soins et des besoins de Novilla ne peut se passer d'un suppl\u00e9ment d'\u00e2me, qui conjoint l'histoire avec la longue chronique des errements pass\u00e9s et l'avenir avec ses espoirs d\u00e9mesur\u00e9s. La soci\u00e9t\u00e9 id\u00e9ale de Novilla limite les d\u00e9sirs et la comp\u00e9tition, mais elle s'est fig\u00e9e en totalitarisme d\u00e9mocratique et s\u00e9culaire. Coetzee d\u00e9nonce ce totalitarisme mou, montrant que sa tol\u00e9rance a des limites, vite franchies d\u00e8s qu'il s'agit de l'\u00e9ducation de David. Puisque l'enfant refuse d'apprendre \u00e0 lire et semble r\u00e9cuser l'autorit\u00e9 des professeurs, on d\u00e9cide de l'envoyer dans un coll\u00e8ge sp\u00e9cialis\u00e9 qui n'est qu'un centre de redressement pour adolescents perturb\u00e9s et enfants caract\u00e9riels. Ceci correspondrait \u00e0 ce que Lacan appelait le \u00abdiscours de l'universit\u00e9\u00bb, un discours marqu\u00e9 par le conformisme scolaire et l'imposition de lois universelles qui sont d'autant plus contraignantes qu'elles sont \u00e9dict\u00e9es au nom du bien des autres. Comme Derrida, Coetzee en appelle \u00e0 une d\u00e9mocratie \u00e0 venir, mais cette d\u00e9mocratie sera, d'une part, ancr\u00e9e dans l'histoire, et donc n'ignorera pas sa \u00abtradition\u00bb et son \u00absens historique\u00bb et, d'autre part, acceptera l'ind\u00e9termination de l'avenir.\n\nUne soci\u00e9t\u00e9 juste doit s'ancrer dans le pass\u00e9 et se d\u00e9passer vers une ouverture \u00e0 l'impr\u00e9vu et l'impr\u00e9visible; c'est ainsi que l'ind\u00e9termin\u00e9 peut se combiner avec le principe fondamental qu'est la justice. C'est pourquoi Sim\u00f3n invoque la justice dans ce passage de la discussion avec les dockers:\n\nJe n'essaie pas de vous sauver, je ne suis personne de sp\u00e9cial, je ne pr\u00e9tends pas \u00eatre le sauveur de quiconque, dit-il. Comme vous j'ai travers\u00e9 l'oc\u00e9an. Comme vous, je n'apporte aucune histoire avec moi. [...] Mais je n'ai pas abandonn\u00e9 l'id\u00e9e de l'histoire, l'id\u00e9e du changement sans commencement ni fin. On ne peut pas se d\u00e9barrasser des id\u00e9es par une grande lessive, pas m\u00eame avec le concours du temps. [...] L'id\u00e9e de la justice, par exemple. Nous d\u00e9sirons tous vivre dans un syst\u00e8me juste... (EJ, 158)\n\nM\u00eame si les dockers r\u00e9ussissent \u00e0 s'entendre sur une juste distribution des t\u00e2ches et des salaires, nous d\u00e9couvrons rapidement que la soci\u00e9t\u00e9 de Novilla n'est pas id\u00e9ale et que la justice n'y r\u00e8gne pas vraiment. Nous avons vu qu'In\u00e8s vivait dans le luxe et l'oisivet\u00e9 \u00e0 la Residencia, et que Daga avait su s'attribuer en un temps record des avantages mat\u00e9riels inconnus aux autres. Mais c'est surtout la tyrannie bureaucratique en mati\u00e8re d'\u00e9ducation qui force Sim\u00f3n, In\u00e8s et David \u00e0 chercher ailleurs une soci\u00e9t\u00e9 plus juste. Ce qui ne veut pas dire simplement une soci\u00e9t\u00e9 sans classes, mais une soci\u00e9t\u00e9 qui respecte la libert\u00e9 de penser diff\u00e9remment. Tout au long du roman, Sim\u00f3n recherche une philosophie \u00abqui vous secoue. Qui change votre vie.\u00bb (EJ, 324) Lui-m\u00eame a su changer la vie d'In\u00e8s de mani\u00e8re radicale en lui offrant de mani\u00e8re si brusque et impr\u00e9vue de devenir la m\u00e8re de David. Sim\u00f3n lui a offert le seul don possible, selon Derrida, c'est-\u00e0-dire le don de vie ou de mort. Et David est devenu, selon l'expression qu'elle r\u00e9p\u00e8te, \u00abla lumi\u00e8re de sa vie\u00bb (EJ, 263).\n\nEn conclusion de cette section, je voudrais proposer qu'Une enfance de J\u00e9sus nous fait moins voir une Imitatio Christi qu'une Projectio Christi. \u00c0 la fin du r\u00e9cit, malgr\u00e9 les parall\u00e8les \u00e9vidents, nous ne tombons pas dans l'eschatologie chr\u00e9tienne. Nous serions plus proches du concept paradoxal lanc\u00e9 par Derrida d'un \u00abmessianisme sans Messie\u00bb. Et si nous tenons absolument \u00e0 lire David comme un nouveau Messie, ce ne sera que pour autant qu'il annonce la d\u00e9mocratie \u00e0 venir. Plus que J\u00e9sus, David est un nouvel Adam \u2013 Adam, appel\u00e9 dans l'\u00e9p\u00eetre aux Romains \u00abla figure de celui qui devait venir\u00bb (Rom, 5: 14). Je reviendrai avec Pascal sur cette formule de \u00abAdam forma futuri\u00bb. L'enfant g\u00e9n\u00e9rique donne naissance \u00e0 l'avenir par-del\u00e0 le futur. Parce qu'il perp\u00e9tue la croyance aux miracles et conserve son ouverture \u00e0 l'avenir, on peut dire que l'enfance de chaque enfant est l'enfance de J\u00e9sus.\n\n. J. Derrida, Le monolinguisme de l'autre, op. cit., p. 13.\n\n. Paul Auster et J. M. Coetzee, Here and Now: Letters 2008-2011, New York, Viking, 2013, p. 65-66. D\u00e9sormais abr\u00e9g\u00e9 en HN, suivi du num\u00e9ro de la page.\n\n. James Joyce, Portrait de l'artiste en jeune homme, tr. fr. Ludmila Savitsky, r\u00e9vis\u00e9e par Jacques Aubert, dans \u0152uvres, J. Aubert (\u00e9d.), t. 1, Paris, Gallimard, coll. \u00abBiblioth\u00e8que de la Pl\u00e9iade\u00bb, 1982, p. 717.\n\n. La Bible de J\u00e9rusalem, Paris, Descl\u00e9e de Brouwer, 1975, p. 1843.\n\n. Cf. J. C. Kannemeyer, J.M. Coetzee: A Life in Writing, tr. angl. Michiel Heyns, Johannesburg, Jonathan Ball, 2012, p. 620-621.\n\n. J. M. Coetzee, Une enfance de J\u00e9sus, tr. fr. Catherine Lauga Du Plessis, Paris, Seuil, 2013, p. 11. D\u00e9sormais abr\u00e9g\u00e9 en EJ, suivi du num\u00e9ro de la page.\n\n. Je fais r\u00e9f\u00e9rence au texte bilingue de The Infancy Gospels of James and Thomas, tr. angl. Ronald F. Hock, Santa Rosa, California, The Scholar's Bible, 1995, p. 111. D\u00e9sormais abr\u00e9g\u00e9 en IGJT, suivi du num\u00e9ro de la page.\n\n. Cf. J.M Coetzee, Life & Times of Michael K, Johannesburg, Ravan Press, 1983, p. 73, 75 sq. (Michael K, sa vie, son temps, tr. fr. Sophie Mayoux, Paris, Seuil, 1985.)\n\n. J.M. Coetzee, Doubling the Point: Essays and Interviews, Cambridge, Harvard University Press, 1992, p. 98.\n\n. Miguel de Cervant\u00e8s, El Ingenioso Hidalgo Don Quijote de la Mancha, Luis Andr\u00e9s Mirillo (\u00e9d.), Madrid, Editorial Castalia, 1978, vol. II, p. 213.\n\n. Cf. J.M. Coetzee, \u00abTriangular Structures of Desire in Advertising\u00bb, dans Doubling the Point, op. cit., p. 131.\n\n. Ibid.\n\n. Mike Marais, Secretary of the Invisible: The Idea of Hospitality in the Fiction of J.M. Coetzee, Amsterdam, Rodopi, 2009.\n\n. Cf. J. Derrida et Anne Dufourmantelle, De l'hospitalit\u00e9, Paris, Calmann-L\u00e9vy, 1997.\n\n. J. Derrida et \u00c9lisabeth Roudinesco, De quoi demain... Dialogue, Paris, Fayard\/Galil\u00e9e, 2001, p. 102.\n\n. Samir Haddad, Derrida and the Inheritance of Democracy, Bloomington, Indiana University Press, 2013, p. 13.\n\n. Ibid., p. 15.\n\n. Derrida part de ce texte de Cr\u00e9puscule des idoles pour lancer les prol\u00e9gom\u00e8nes d'une \u00abHistoire du mensonge\u00bb. Cf. J. Derrida, \u00abHistoire du mensonge. Prol\u00e9gom\u00e8nes\u00bb, dans le Cahier de L'Herne Derrida, op. cit., p. 495; r\u00e9\u00e9d., Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 2012, p. 10.\n\n. Coetzee a \u00e9t\u00e9 influenc\u00e9 par son coll\u00e8gue et ami de l'Universit\u00e9 de Chicago, le philosophe Robert Pippin, qui a comment\u00e9 en profondeur ce que signifiait le nihilisme pour Nietzsche. Cf. Robert B. Pippin, Modernism as a Philosophical Problem, Oxford, Blackwell, 1999, p. 78-113, et Nietzsche, moraliste fran\u00e7ais, Paris, Odile Jacob, 2006.\n\n. T.S. Eliot, \u00abTradition and the individual talent\u00bb [1919], dans The Sacred Wood, Londres, The Waking Press, 2011, p. 30-39. \n\n# V. Ma vie a toujours \u00e9t\u00e9 future:  \ngrammatologie des temps \u00e0 venir\n\nCLOV \u2013 Tu crois \u00e0 la vie future?  \nHAMM \u2013 La mienne l'a toujours \u00e9t\u00e9.  \n(Clov sort en claquant la porte.) Pan! Dans les gencives.  \nNAGG. \u2013 J'\u00e9coute.  \nHAMM \u2013 Salopard! Pourquoi m'as-tu fait?  \nNAGG \u2013 Je ne pouvais pas savoir.  \nHAMM \u2013 Quoi? Qu'est-ce que tu ne pouvais pas savoir?  \nNAGG \u2013 Que ce serait toi.\n\nSamuel Beckett, Fin de partie.\n\nCe dernier chapitre reprend des pages lues lors d'un colloque consacr\u00e9 aux \u00abFuturs de Jacques Derrida\u00bb et m\u00e9dite sur ce que ce \u00abfutur\u00bb avait d'\u00abant\u00e9rieur\u00bb. Par exemple, je ne savais pas en 1995 que mon intervention allait me mettre sur la voie du roman de Coetzee publi\u00e9 en 2013 et discut\u00e9 dans le chapitre pr\u00e9c\u00e9dent. Ce colloque eut lieu \u00e0 Tuscaloosa en septembre 1995. Il avait \u00e9t\u00e9 organis\u00e9 par Richard Rand. Rand avait voulu que le programme, dont la typographie savante copiait celle de \u00abUn coup de d\u00e9s jamais n'abolira le hasard\u00bb de Mallarm\u00e9, ne pr\u00e9cise pas les heures de passage des orateurs. Dans les premi\u00e8res minutes, nous jou\u00e2mes aux d\u00e9s pour d\u00e9cider, selon les chiffres atteints, \u00e0 quelle heure nous allions parler. Mais une fois que les d\u00e9s nous eurent indiqu\u00e9 notre tour, au lieu de respecter ce programme al\u00e9atoire, nous nous sommes lanc\u00e9s dans un troc insens\u00e9 pour obtenir des places diff\u00e9rentes selon nos pr\u00e9f\u00e9rences, les horaires de d\u00e9parts, les rythmes des uns ou des autres ou nos affinit\u00e9s personnelles. Ceci illustre une des implications du titre choisi de mani\u00e8re si rus\u00e9e par Rand: en effet, en anglais commercial, les futures d\u00e9signent des \u00abcontrats \u00e0 terme\u00bb, soit des valeurs en bourse sur des transactions encore \u00e0 accomplir. Quelle que soit la valeur de la transaction, un engagement est pris sur des achats ou des ventes qui ne sont pas encore r\u00e9alis\u00e9es. De la m\u00eame mani\u00e8re, refusant la loi du hasard, nous tentions de calculer \u00e0 l'avance nos places selon un ordre que nous voulions aussi favorable ou plaisant que possible.\n\nAu sens commercial de \u00abfutures\u00bb, ces actions qui se n\u00e9gocient sur des valeurs anticip\u00e9es, il faudrait ajouter le sens militaire. Derrida a su \u00e9voquer la course de vitesse aux armements les plus sophistiqu\u00e9s et, en 1984, il pensait bien s\u00fbr \u00e0 la bombe atomique. Qu'aurait-il pens\u00e9 des drones, des guerres par hackers interpos\u00e9s, capables de d\u00e9r\u00e9gler les contr\u00f4les informatiques de tout un pays? Si, il y a plus de vingt ans, la guerre nucl\u00e9aire pouvait jouer avec l'impensable et l'impossible, nous sommes revenus vers le domaine du possible et du vraisemblable avec la miniaturisation des ordinateurs et cam\u00e9ras, le contr\u00f4le de l'information par l'informatique et les multiples processus de t\u00e9l\u00e9commande. Reste cependant l'interrogation fondamentale sur la possibilit\u00e9 de la paix, d'un \u00e9tat futur qui correspondrait \u00e0 la proph\u00e9tie de Coetzee, selon lequel mieux vaut choisir le polemos path\u00e9tique d'un dissensus d\u00e9sirant que de se retrouver englu\u00e9 dans la paix sans sel du nihilisme triomphant. Cette promesse que rec\u00e8le la figure de l'enfant, peut-on la trouver dans d'autres discours et d'autres textes? Mallarm\u00e9 peut servir de premier guide.\n\n### 1. F\u00fbtur, entre nom et verbe\n\nEn 1869, la femme de St\u00e9phane Mallarm\u00e9, qui \u00e9tait d'origine allemande, \u00e9crivit une lettre sous la dict\u00e9e de son mari. Elle commit alors une faute d'orthographe qui, sans \u00eatre r\u00e9dhibitoire, semble pleine de sous-entendus. Marie Mallarm\u00e9 servait de \u00absecr\u00e9taire\u00bb au po\u00e8te atteint d'une curieuse \u00abhyst\u00e9rie\u00bb qui l'emp\u00eachait de mettre la main \u00e0 la plume. C'\u00e9tait une lettre pour leur ami Cazalis. Dans sa r\u00e9daction, Marie donne \u00e0 l'adjectif f\u00e9minin pluriel \u00abfutures\u00bb un dr\u00f4le d'accent circonflexe superf\u00e9tatoire, \u00e9crivant, avec plusieurs autres fautes mineures reproduites telles quelles par Bertrand Marchal: \u00abJ'ai fait un v\u0153u, \u00e0 toute extremit\u00e9, qui est de ne pas toucher \u00e0 une plume d'ici \u00e0 P\u00e2ques. Je pourrais te dire seulement [...] que le simple acte d'\u00e9crire installer l'hyst\u00e9rie dans ma t\u00eate, ce que je veux \u00e9vit\u00e9r \u00e0 toute force pour vous, mes ch\u00e8rs amis \u00e0 qui je dois un Livre et des ann\u00e9es f\u00fbtures\u00bb. Peut-on voir dans ce \u00abf\u00fbtures\u00bb une trace de l'accent allemand de Marie? Ce circonflexe refl\u00e9tait-il l'emphase particuli\u00e8re, le ton p\u00e9n\u00e9tr\u00e9 avec lequel le po\u00e8te lui dictait sa lettre? Cet accent semble donner \u00e0 l'adjectif un aspect verbal; il \u00e9voque un subjonctif pass\u00e9: l'optatif pass\u00e9 (\u00abque ce f\u00fbt\u00bb) redouble l'annonce ambigu\u00eb d'un futur. Ce futur serait donc endett\u00e9, un futur pass\u00e9 jamais pr\u00e9sent ou un avenir qui ne peut se penser \u00e0 l'indicatif. On serait tent\u00e9 d'ajouter \u00abF\u00fbt-il\u00bb \u00e0 la s\u00e9rie des subjonctifs imparfaits qui scandent la septi\u00e8me page d'\u00abUn coup de d\u00e9s jamais n'abolira le hasard\u00bb:\n\nc'\u00e9tait\n\nissu stellaire\n\nle nombre\n\nEXIST\u00c2T-IL\n\nautrement qu'hallucination \u00e9parse d'agonie\n\nCOMMEN\u00c7\u00c2T-IL ET CESS\u00c2T-IL\n\nsourdant que ni\u00e9 et clos quand apparu\n\nenfin\n\npar quelque profusion r\u00e9pandue en raret\u00e9\n\nSE CHIFFR\u00c2T-IL\n\n\u00e9vidence de la somme pour peu qu'une\n\nILLUMIN\u00c2T-IL\n\nJ'aurai l'occasion de revenir sur le contexte pr\u00e9cis de l'\u00e9criture de Marie Mallarm\u00e9 pour sugg\u00e9rer que cette graphie d\u00e9viante n'est pas d\u00e9nu\u00e9e de fondement. Mon interrogation portera sur l'\u00eatre-verbal du futur, c'est-\u00e0-dire sur ce qui lie le futur non seulement \u00e0 une \u00e9criture, mais aussi \u00e0 un geste performatif qui modalise, rend hypoth\u00e9tique, translate d'un c\u00f4t\u00e9 \u00e0 l'autre la vectorisation temporelle, bref, sur ce qui, dans la question d'une ouverture au futur, suppose une transformation de l'indicatif des temps lin\u00e9aires en un subjonctif des modes.\n\nCette question va traverser un ensemble de textes allant de Derrida \u00e0 l'\u00e9vang\u00e9liste Luc, \u00e0 Pascal relay\u00e9 par Rilke \u00e0 Heidegger et, enfin, derechef \u00e0 Mallarm\u00e9. Une telle enqu\u00eate ne pourra \u00e9viter de poser la question du Livre ultime pour l'aborder \u00e0 partir d'un couple idiomatique qui habite la langue fran\u00e7aise. D'ailleurs, la plupart des langues d'origine latine proposent deux termes en ce qui concerne les temps \u00e0 venir: le \u00abfutur\u00bb redouble \u00e0 l'\u00abavenir\u00bb sans que l'on sache si ce choix entre deux mots ob\u00e9it \u00e0 un partage conceptuel pr\u00e9cis. Dans l'exergue de De la grammatologie, nous l'avons vu, Jacques Derrida annon\u00e7ait de mani\u00e8re proph\u00e9tique ou messianique une nouvelle pens\u00e9e de l'\u00e9criture: \u00abL'avenir ne peut s'anticiper que dans la forme du danger absolu. Il est ce qui rompt absolument avec la normalit\u00e9 constitu\u00e9e et ne peut donc s'annoncer, se pr\u00e9senter, que sous l'esp\u00e8ce de la monstruosit\u00e9. Pour ce monde \u00e0 venir et pour ce qui en lui aura fait trembler les valeurs de signe, de parole et d'\u00e9criture, pour ce qui conduit ici notre futur ant\u00e9rieur, il n'est pas encore d'exergue.\u00bb\n\nDe telles phrases pr\u00e9supposent une ligne de d\u00e9marcation entre un \u00abavenir\u00bb et un \u00abfutur ant\u00e9rieur\u00bb, ligne qui se voit pr\u00e9cis\u00e9e dans le commentaire que donne Derrida du livre intitul\u00e9 Le Mo\u00efse de Freud. Juda\u00efsme terminable et interminable de Josef Hayim Yerushalmi:\n\nOr s'il y a un seul trait sur lequel Yerushalmi reste intraitable, s'il y a une affirmation soustraite \u00e0 toute discussion (psychanalytique ou talmudique), une affirmation inconditionnelle, c'est l'affirmation de l'\u00e0-venir (je pr\u00e9f\u00e8re dire de l'avenir que du futur pour faire signe vers la venue d'un \u00e9v\u00e9nement plut\u00f4t que vers quelque pr\u00e9sent futur).\n\nL'affirmation de l'\u00e0-venir, donc: ce n'est pas une th\u00e8se positive. Ce n'est rien d'autre que l'affirmation m\u00eame, le \u00aboui\u00bb en tant qu'il est la condition de toute promesse ou de toute esp\u00e9rance, de toute attente, de toute performativit\u00e9, de toute ouverture \u00e0 l'avenir, quel qu'il soit, pour la science ou pour la religion.\n\nReprenant \u00e0 Yerushalmi le th\u00e8me de l'attente d'un \u00e9v\u00e9nement qui, pour \u00eatre v\u00e9ritablement \u00e9v\u00e9nement doit garder sa radicale nouveaut\u00e9 et donc rester sinon impossible du moins impensable dans les anciennes cat\u00e9gories, Derrida pressent l'insuffisance ou la limitation a priori du terme de \u00abfutur\u00bb. Il serait complice d'une d\u00e9clinaison logique qui aboutirait au \u00abpr\u00e9sent futur\u00bb et reconduirait le m\u00eame qu'aujourd'hui tout juste repouss\u00e9 \u00e0 plus tard, pr\u00e9f\u00e9rant l'affirmation plus radicale qu'implique le terme d'\u00ab\u00e0- venir\u00bb. Seul le vocable d'\u00abavenir\u00bb saurait garder la force th\u00e9tique de la promesse messianique, id\u00e9e confirm\u00e9e par Walter Benjamin dans son dernier texte, cit\u00e9 \u00e0 la fois par Yerushalmi et Derrida, qui mentionne l'interdiction faite aux Juifs de sp\u00e9culer sur l'avenir:\n\nOn le sait, il \u00e9tait interdit aux Juifs de pr\u00e9dire l'avenir. La Torah et la pri\u00e8re s'enseignent au contraire dans la comm\u00e9moration. Pour eux la comm\u00e9moration d\u00e9senchantait l'avenir auquel ont succomb\u00e9 ceux qui cherchent instruction chez les devins. Mais pour les Juifs l'avenir ne devint pas n\u00e9anmoins un temps homog\u00e8ne et vide. Car en lui chaque seconde \u00e9tait la porte \u00e9troite par laquelle pouvait passer le Messie.\n\nLa croyance en un savoir mantique portant sur les pr\u00e9sages et oracles doit \u00eatre laiss\u00e9e aux mages pa\u00efens; \u00e9vitant ces \u00abenchantements\u00bb, le messianisme juif se tourne vers le pass\u00e9, mais aussi il pense la venue toujours possible-impossible de Celui qui Vient. Ce que Benjamin ne peut dire en allemand, qui ne dispose que du seul terme de \u00abZukunft\u00bb, s'\u00e9claire mieux en fran\u00e7ais. Qu'en est-il donc de cette opposition apparemment intraduisible?\n\nLe \u00abfutur\u00bb provient du supin du verbe \u00ab\u00eatre\u00bb en latin: esse donne futurus; au neutre pluriel, futura d\u00e9signe \u00ables choses qui vont se passer\u00bb. Le mot \u00abfutur\u00bb devint courant en fran\u00e7ais vers le XIIIe si\u00e8cle. C'\u00e9tait l'\u00e9poque de la mise en forme judiciaire de l'eschatologie m\u00e9di\u00e9vale articul\u00e9e autour du purgatoire. Le terme d'\u00abavenir\u00bb est une cr\u00e9ation plus tardive d'un si\u00e8cle au moins: le verbe \u00abadvenir\u00bb donna un substantif qui signifiait \u00absucc\u00e8s, r\u00e9ussite dans les temps futurs\u00bb, puis vers la fin du XVe si\u00e8cle, le mot prit le sens d\u00e9finitif de \u00abtemps encore \u00e0 venir\u00bb. Littr\u00e9 tente courageusement d'apporter de l'ordre dans la confusion s\u00e9mantique:\n\nLe futur est ce qui sera; l'avenir est ce qui adviendra. Ces deux sens se confondent dans l'usage presque toujours: les si\u00e8cles \u00e0 venir ou les si\u00e8cles futurs ne pr\u00e9sentent pas d'autre nuance que celle qui est dans la notion m\u00eame d'\u00eatre ou de venir. Il n'y a que dans la langue du droit o\u00f9 futur ne peut \u00eatre remplac\u00e9 par \u00e0 venir: les futurs conjoints. On dirait cependant l'h\u00e9ritier \u00e0 venir aussi bien que le futur h\u00e9ritier.\n\nJe voudrai mieux comprendre la sp\u00e9cificit\u00e9 d'un jugement l\u00e9gal dissociant le \u00abfutur\u00bb de l'\u00abavenir\u00bb. Les dictionnaires fran\u00e7ais op\u00e8rent une opposition entre Pascal, pr\u00e9sent\u00e9 comme celui qui syst\u00e9matise la finalit\u00e9 de l'\u00abavenir\u00bb (\u00abLe pass\u00e9 et le pr\u00e9sent sont nos moyens, le seul avenir est notre fin\u00bb) et un \u00e9crivain comme Gide, pour qui les deux notions se confondent dans un \u00e9loge de la pure potentialit\u00e9 (\u00abJe confonds possible et futur\u00bb).\n\nAvec sa double d\u00e9finition, \u00able futur est ce qui sera; l'avenir ce qui adviendra\u00bb, Littr\u00e9 avait tent\u00e9 de frayer une voie pour le sortir d'une aporie linguistique, et \u00e0 cette polarit\u00e9 il associait deux noms, ceux de Montaigne et de Pascal. Dans la p\u00e9roraison qui cl\u00f4t l'essai consacr\u00e9 \u00e0 \u00abLa force de l'imagination\u00bb, Montaigne oppose les \u00e9crivains scrupuleux, qui visent \u00e0 rester proches de l'histoire, et son propre jeu litt\u00e9raire avec l'imagination, qui lui laisse la t\u00e2che d'ouvrir le possible par son \u00e9criture. Dans \u00abLa force de l'imagination\u00bb, Montaigne \u00e9voque des t\u00e9moignages fabuleux qui semblent plus possibles que vrais:\n\nAdvenu ou non advenu, \u00e0 Paris ou \u00e0 Rome, \u00e0 Jean ou \u00e0 Pierre, c'est toujours un tour de l'humaine capacit\u00e9: duquel je suis utilement avis\u00e9 par ce r\u00e9cit. Je le voy, et en fay mon profit, egalement en umbre qu'en corps. Et aux diverses le\u00e7ons, qu'ont souvent les histoires, je prends \u00e0 me servir de celle qui est la plus rare et m\u00e9morable. Il y a des autheurs, desquels la fin, c'est dire les evenements. La mienne, si j'y s\u00e7avoys advenir, seroit dire sur ce qui peut advenir.\n\nLe redoublement du verbe \u00abadvenir\u00bb oriente tout \u00e0 la fois vers les le\u00e7ons que l'on peut tirer d'\u00e9v\u00e9nements pass\u00e9s et un projet litt\u00e9raire et philosophique qui, comme le veut Aristote, ne se limite pas aux d\u00e9tails historiques, mais en donne le sens encore ouvert vers ce qui viendra, tels de nouveaux lecteurs. If faut laisser \u00abadvenir\u00bb ce sens selon un rapport pr\u00e9cis entre des r\u00e9cits d'historiens et un texte encore \u00e0 venir, et qui vise \u00e0 pr\u00e9senter ces \u00e9v\u00e9nements parfois fictifs qui \u00e9chappent aux strictes cat\u00e9gories de l'histoire.\n\nFace \u00e0 cet \u00e9tagement d'\u00abadvenirs\u00bb multiples, un sens simplifi\u00e9 du futur nous dit qu'il s'agit tout simplement de ce qui sera l\u00e0: un nouvel \u00eatre va \u00eatre l\u00e0 que l'on peut anticiper. Il \u00e9tait logique que ce soit Pascal qui nous donne un \u00e9claircissement sur ce terme, car c'est lui qui, le premier, a introduit une math\u00e9matisation du futur dans la science des probabilit\u00e9s. En m\u00eame temps, il attaquait le \u00abprobabilisme\u00bb moral de la casuistique j\u00e9suite. Et par ailleurs, il inventait la premi\u00e8re machine \u00e0 calculer, anc\u00eatre de nos ordinateurs. Renversant la dichotomie \u00e9tablie par Derrida, la strat\u00e9gie g\u00e9n\u00e9rale de Pascal repose sur une distinction entre l'\u00abavenir\u00bb pens\u00e9 comme ce qui adviendra t\u00f4t ou tard pour chacun de nous (par exemple la mort, suivie du salut ou de la damnation) et le \u00abfutur\u00bb d'une \u00ab\u00e9criture\u00bb sainte pourtant d\u00e9j\u00e0 entre nos mains. Selon Pascal, la Bible serait \u00e0 placer dans la s\u00e9rie d'ouvrages de science-fiction publi\u00e9s par Deno\u00ebl qui s'intitule \u00abPr\u00e9sence du futur\u00bb.\n\nC'est pourquoi, m\u00eame si les consid\u00e9rations eschatologiques paraissent sombres, on ne pourrait traduire sa pens\u00e9e par un d\u00e9calque de la formule anglaise courante: \u00abNo Future!\u00bb Une telle expression ne se traduirait pas par \u00abPas de futur!\u00bb, mais bien par \u00abAucun avenir!\u00bb Le fran\u00e7ais entend que l'on dispose d'un ou de plusieurs futurs (m\u00eame si c'est au sens de \u00abpromis\u00bb ou \u00abpromise\u00bb pour un mariage \u00e0 venir), tandis que l'avenir est une entit\u00e9 globale, et souvent noire... On ne dirait donc pas ais\u00e9ment en fran\u00e7ais \u00abJe travaille pour le futur\u00bb (alors que \u00abJe travaille pour l'avenir\u00bb semble correct). Je ne peux travailler pour un moment dont l'issue para\u00eet trop hypoth\u00e9tique, alors qu'un avenir semble programm\u00e9 en bien ou en mal vers son \u00abadvenir\u00bb. Joseph Addison avait eu cette boutade fameuse: \u00abPourquoi travaillerions-nous pour la post\u00e9rit\u00e9? Et d'abord, qu'a-t-elle jamais fait pour nous, la post\u00e9rit\u00e9?\u00bb Et pourtant, suivant Pascal, nous pouvons nous efforcer de parier sur la post\u00e9rit\u00e9 du futur. Le but ultime serait d'op\u00e9rer une r\u00e9duction ph\u00e9nom\u00e9nologique afin d'\u00e9valuer ce qui reste ou de calculer les chances d'un avenir \u00e0 pr\u00e9server dans sa radicale ouverture \u00e0 l'Autre. Si la post\u00e9rit\u00e9 des enfants encore \u00e0 na\u00eetre semble plus impliqu\u00e9e dans la trame de l'avenir que dans les jeux du futur ant\u00e9rieur, c'est le signe qu'il faut parier. Parier sans espoir autre qu'une communaut\u00e9 humaine, voil\u00e0 la principale justification de cette sp\u00e9culation sur les temps \u00e0 venir. Une parabole de l'\u00e9vangile de Luc pourra nous y aider.\n\n### 2. L'intendant infid\u00e8le:   \nvers une grammatologie prudentielle\n\nLa parabole de l'intendant infid\u00e8le m'avait toujours peu ou prou choqu\u00e9 lorsque je la retrouvais \u00e0 la fin d'une nouvelle de Joyce dans Dublinois. Dans \u00abLa Gr\u00e2ce\u00bb, un pr\u00e9dicateur, le p\u00e8re Purdon (le nom \u00e9voque moins le \u00abpardon\u00bb qu'une rue tr\u00e8s mal fam\u00e9e), donne un sermon lors d'une retraite spirituelle \u00e0 laquelle quelques amis ont r\u00e9ussi non sans mal \u00e0 entra\u00eener Tom Kernan, un ivrogne qu'ils veulent ramener dans le droit chemin. Dans cette parodie de la complicit\u00e9 entre l'\u00c9glise catholique irlandaise et les valeurs les plus suspectes d'une bourgeoisie bien-pensante, Joyce d\u00e9signe ce qu'il appelle la \u00absimonie\u00bb dublinoise, c'est-\u00e0-dire la commercialisation du sacr\u00e9. Il revient au p\u00e8re Purdon d'incarner l'apoth\u00e9ose de la simonie dans la p\u00e9roraison de son sermon et il cite le texte de Luc:\n\nCar les enfants de ce monde-ci sont plus avis\u00e9s avec leurs semblables que les enfants de lumi\u00e8re. En v\u00e9rit\u00e9, je vous le dis: faites-vous des amis avec l'argent malhonn\u00eate afin qu'au jour o\u00f9 il viendra \u00e0 manquer, ceux-ci vous re\u00e7oivent dans les demeures \u00e9ternelles.\n\nLe P\u00e8re Purdon d\u00e9veloppa ce texte avec une sonore assurance. C'\u00e9tait, disait-il, l'un des textes les plus difficiles de toutes les \u00c9critures \u00e0 interpr\u00e9ter correctement. C'\u00e9tait un texte qui, aux yeux d'un observateur superficiel, pouvait sembler en contradiction avec la haute moralit\u00e9 pr\u00each\u00e9e par le Christ en d'autres endroits. [...] C'\u00e9tait un texte adress\u00e9 aux hommes d'affaires et aux professions lib\u00e9rales.\n\nLe texte de Luc (16: 8-9) cit\u00e9 par Purdon s'\u00e9carte de la version habituelle: au lieu de \u00abwhen you fail\u00bb, il \u00e9crit \u00abwhen you die\u00bb, \u00abquand vous mourrez\u00bb. De plus, il s'arr\u00eate en un point strat\u00e9gique, car Purdon omet de mentionner la conclusion de l'apologue, qui est cat\u00e9gorique: \u00abVous ne pouvez servir Dieu et Mammon.\u00bb\n\nReprenons donc cette histoire, sans minimiser ce qu'elle a de potentiellement scandaleux. Un ma\u00eetre ou seigneur (adon) d\u00e9crit comme tr\u00e8s riche a entendu dire que son intendant (oikonomos ou villicus dans la Vulgate) a dilapid\u00e9 ses biens. Il lui dit alors deux choses: l'intendant doit rendre compte de sa gestion et il va \u00eatre renvoy\u00e9. Ce dernier a donc un peu de temps avant de partir. Sachant qu'il a \u00e9t\u00e9 d\u00e9nonc\u00e9, il ne cherche pas \u00e0 se d\u00e9fendre, mais convoque les d\u00e9biteurs du seigneur \u00abpour qu'ils [l]'accueillent chez eux\u00bb (Luc, 16: 4). Commence alors une curieuse sc\u00e8ne: un d\u00e9biteur doit cent barils d'huile, l'intendant lui fait \u00e9crire cinquante; un d\u00e9biteur doit cent mesures de bl\u00e9, il lui fait \u00e9crire quatre-vingt. C'est une sc\u00e8ne de dict\u00e9e redoubl\u00e9e, un th\u00e9\u00e2tre pr\u00e9cipit\u00e9 de l'\u00e9criture forc\u00e9e: \u00abPrends ton billet (gramma), assieds-toi et \u00e9cris vite (grapson) cinquante. [...] Prends ton billet et \u00e9cris quatre-vingt.\u00bb (Luc, 16: 6-7)\n\nLe r\u00e9cit embl\u00e9matique du Villicus iniquitatis nous offre une parabole reliant la dette et l'\u00e9criture. Le mod\u00e8le propos\u00e9 par l'intendant, un mod\u00e8le donn\u00e9 par d\u00e9faut en quelque sorte, est celui de l'astuce humaine dans le domaine de l'argent, par opposition au d\u00e9faut de sagesse humaine dans les affaires du salut de l'\u00e2me. L'id\u00e9e est qu'il suffit de voler \u00e0 la dette m\u00eame. Apr\u00e8s qu'on a demand\u00e9 \u00e0 l'intendant de rendre compte de sa mauvaise gestion, il se lance dans une anti-\u00e9conomie syst\u00e9matique. Lui qui a toujours servi de m\u00e9diateur entre un seigneur qui est un capitaliste et une foule de d\u00e9biteurs se met \u00e0 pervertir cette relation. Il semble de fait que r\u00e8gne un certain d\u00e9sordre, voire une anarchie compl\u00e8te dans ses r\u00e9ductions ing\u00e9nieuses, mais improvis\u00e9es: on passe ainsi de cent \u00e0 cinquante, puis de cent \u00e0 quatre-vingt, comme si tout calcul rationnel \u00e9tait devenu impossible. L'intendant prend \u00e0 la dette (en enlevant encore un peu plus \u00e0 la fortune \u00e9corn\u00e9e du ma\u00eetre) ou il donne \u00e0 la dette (en laissant un profit marginal aux d\u00e9biteurs): les deux op\u00e9rations sont interchangeables en l'occurrence. Notons qu'il n'efface pas la dette: il rature l'ancien chiffre et le remplace par un chiffre diminu\u00e9. Il ne d\u00e9truit pas les tablettes, ce qui serait peut-\u00eatre en dehors de son pouvoir et affaiblirait la d\u00e9monstration. Il faut que les d\u00e9biteurs restent attach\u00e9s \u00e0 la dette initiale pour qu'ils puissent ensuite reconna\u00eetre leur dette symbolique face \u00e0 l'intendant qui les a aid\u00e9s en la r\u00e9duisant. Cette modification frauduleuse est l'effet d'une \u00e9criture particuli\u00e8re, qui conserve une trace pass\u00e9e afin d'aller vers une lecture future.\n\nL'une des ironies de la parabole est que le ma\u00eetre, au lieu d'\u00eatre furieux ou de se sentir trahi, f\u00e9licite l'intendant et admire sa sagesse: \u00abEt le ma\u00eetre loua cet intendant malhonn\u00eate d'avoir agi de fa\u00e7on avis\u00e9e. Car les fils de ce monde-ci sont plus avis\u00e9s envers leurs propres cong\u00e9n\u00e8res que les fils de la lumi\u00e8re.\u00bb (Luc, 16: 8) Sa conduite \u00e9tait la seule qui p\u00fbt non seulement lui donner de nombreux amis, mais aussi lui permettre de retrouver son poste. Sa strat\u00e9gie reposait au d\u00e9part sur un aveu: il se sent incapable de travailler de ses mains ou de mendier, il faut donc qu'il s'en remette aux autres.\n\nLe contexte de cette louange paradoxale d'un homme injuste ou infid\u00e8le est r\u00e9v\u00e9lateur. Luc \u00e9crit au nom de J\u00e9sus afin d'attaquer les pharisiens qui \u00e9taient les gardiens intransigeants de la vieille Loi apr\u00e8s la destruction du Temple et qui abhorraient l'id\u00e9e d'\u00eatre en dette. \u00abLes pharisiens, qui sont amis de l'argent, entendaient tout cela et ils se moquaient de lui.\u00bb (Luc, 16: 14) Luc critique leur rectitude exag\u00e9r\u00e9e en termes \u00e9conomiques: ce sont eux les capitalistes de la foi, ils d\u00e9nient et la faute et la catastrophe qui ne peut manquer d'arriver. L'intendant qui \u00abdonne\u00bb aux autres en r\u00e9duisant leurs dettes, dans l'espoir d'une charit\u00e9 \u00e0 venir, est plus proche de la v\u00e9rit\u00e9. Il sait que ses comptes, comme ceux de tous les autres, sont plus ou moins faux. On ne peut \u00eatre quitte de sa dette; il vaut mieux se faire de nouveaux partenaires ou des complices que de parier sur sa propre puret\u00e9 face \u00e0 la Loi. Ceux qui se sentent en s\u00e9curit\u00e9 morale, religieuse et eschatologique sont en fait des avares. Ils croient \u00e0 tort qu'ils sont des riches, des aristocrates, des \u00e9lus \u2013 leur pire erreur est leur foi en une justice transcendante dont ils seraient les gardiens immanents.\n\nEn revanche, la position de l'intendant peut \u00eatre universalis\u00e9e: comme tous les humains, il va devoir rendre des comptes sans s'y \u00eatre pr\u00e9par\u00e9, et les comptes indiqueront pr\u00e9varications, malversations et fautes en tous genres. La catastrophe de la mort et du jugement, cette soudaine p\u00e9rip\u00e9tie redout\u00e9e, est d\u00e9j\u00e0 l\u00e0 et il n'y a pas assez de temps pour se blanchir. La seule direction dans laquelle il faut regarder est humaine et collective; ce sont les autres qui nous tireront d'affaire et non quelque jugement final. La fuite en avant d'une diss\u00e9mination att\u00e9nu\u00e9e de la dette va permettre un nouveau partage des responsabilit\u00e9s, une redistribution des all\u00e9geances. Le pari de Luc porte sur la r\u00e9ciprocit\u00e9 humaine, qui entra\u00eene ensuite le pardon du seigneur, et non sur l'id\u00e9e d'une justification morale transcendante.\n\nCeci suppose une \u00e9conomie de la dette, toujours hant\u00e9e par un sens des temps \u00e0 venir, mais aussi fond\u00e9e sur une sociabilit\u00e9 qui \u00e9tend et d\u00e9place la dette. La v\u00e9ritable \u00e9conomie n'est pas celle du livre de comptes plus ou moins exact, mais la justice humaine du r\u00e9seau social, le r\u00e9seau des d\u00e9biteurs et des sommes en retard. Dans cet apologue, Dieu est pr\u00e9sent\u00e9 comme un grand capitaliste. L'adon ou ma\u00eetre, comme Dieu, pr\u00e9f\u00e8re avoir un intendant malhonn\u00eate mais intelligent \u00e0 un intendant honn\u00eate mais stupide. Il perdra sans doute moins d'argent avec l'oikonomos malhonn\u00eate, m\u00eame si ce dernier se sert au passage: ses calculs avis\u00e9s laisseront une marge de b\u00e9n\u00e9fices \u00e0 l'adon.\n\nL'intendant offre une image \u00e9conomique de ce que rec\u00e8le d\u00e9j\u00e0 le terme de \u00abr\u00e9demption\u00bb: le \u00abrachat\u00bb des p\u00e9ch\u00e9s effectu\u00e9 par J\u00e9sus est cens\u00e9 \u00abenlever \u00e0 la dette\u00bb. Il \u00abprend\u00bb \u00e0 la vieille Loi inflexible et \u00abdonne\u00bb aux p\u00e9cheurs l'espoir du salut. Comme le Fils face au P\u00e8re, l'oikonomos face \u00e0 l'adon va mettre en place une nouvelle \u00e9conomie. Cette \u00e9conomie est plus proche d'un capitalisme financier naissant que du troc des biens, lequel suppose un \u00e9change \u00e0 terme de valeurs \u00e9quivalentes. Il s'agit d'apprendre \u00e0 jouer rapidement sur des futures, c'est-\u00e0-dire calculer des pourcentages; ainsi une r\u00e9duction de 50% ne sera pas fondamentalement diff\u00e9rente d'une r\u00e9duction de 20% si la dette est m\u00e9taphorique, comme lorsqu'on parle de \u00abdette du p\u00e9ch\u00e9\u00bb.\n\nCette parabole avait de quoi plaire aux j\u00e9suites et aux casuistes en tous genres. L'argent fonctionne comme une m\u00e9taphore de la temporalit\u00e9 dans l'\u00e9change et aussi dans une in\u00e9vitable sp\u00e9culation sur le salut. Il semble que ce soit une version n\u00e9gative du pari de Pascal qui nous est propos\u00e9e. Selon Luc, il vaut mieux parier sur le futur que de chercher \u00e0 se justifier dans le pr\u00e9sent. Une rapide \u00e9valuation des chances qui n'exclut pas une certaine pr\u00e9cipitation doit pr\u00e9parer un nouveau d\u00e9part. Ou, pour varier les m\u00e9taphores, un dossier \u00e0 moiti\u00e9 propre vaut mieux que d'attendre l'issue incertaine d'un proc\u00e8s qui peut durer ind\u00e9finiment. Il faut en tout cas prendre une d\u00e9cision vite et non attendre la sentence de la Loi. C'est donc un jugement prudentiel qui est recommand\u00e9 ici: il faut calculer son int\u00e9r\u00eat plut\u00f4t que de chercher \u00e0 v\u00e9rifier un certain \u00e9tat de fait par un jugement \u00e9videntiel. Cette grammatologie prudentielle dans laquelle un trait de plume r\u00e9duit une dette culmine dans cette traduction de la \u00abmort\u00bb comme une \u00abfaillite\u00bb: la faillite de l'humanit\u00e9 est l\u00e0, de toute origine, comme le p\u00e9ch\u00e9 originel, mais peut aussi \u00eatre rachet\u00e9e par le R\u00e9dempteur. Nous pouvons donc vivre avec une dette interminable, du moment que nous savons \u00eatre sages quant \u00e0 l'avenir. Il faut parier sur l'avenir, m\u00eame si cet avenir se noue \u00e0 un \u00e9v\u00e9nement d\u00e9j\u00e0 pass\u00e9 (la venue du Christ) qui sera r\u00e9p\u00e9t\u00e9 dans un \u00e9v\u00e9nement messianique ou eschatologique de la fin des temps. Tout ceci nous am\u00e8ne \u00e0 r\u00e9examiner l'argument du pari de Pascal.\n\n### 3. Pascal et le futur de l'\u00c9criture\n\nLe fameux argument de Pascal implique une double s\u00e9rie de calculs. Au premier niveau, d\u00e9lib\u00e9r\u00e9ment simpliste, la d\u00e9cision entre, d'un c\u00f4t\u00e9, l'existence de Dieu, de l'autre, sa non-existence, devrait \u00eatre r\u00e9gl\u00e9e tout de suite: il semble \u00e9videmment plus avantageux de parier sur l'infini que sur le n\u00e9ant. Cette r\u00e9duction \u00e0 une alternative tranch\u00e9e \u00e9veille les soup\u00e7ons et am\u00e8ne l'interlocuteur \u00e0 refuser de parier: \u00ab[...] ils sont tous deux en faute; le juste est de ne point parier.\/ \u2013 Oui, mais il faut parier. Cela n'est pas volontaire, vous \u00eates embarqu\u00e9.\u00bb C'est que la position initiale du probl\u00e8me ressemble trop \u00e0 une prime de s\u00e9duction (\u00absi vous gagnez, vous gagnez tout, et si vous perdez, vous ne perdez rien\u00bb), qui a pour mission de faire accepter un enjeu que l'on peut juger exorbitant: il s'agit de tout donner, tout, vie et raison, se trouve plac\u00e9 dans la balance. \u00ab[...] il n'y a point \u00e0 balancer, il faut tout donner. Et ainsi quand on est forc\u00e9 \u00e0 jouer, il faut renoncer \u00e0 la raison pour garder la vie plut\u00f4t que de la hasarder pour le gain infini aussi pr\u00eat \u00e0 arriver que la perte du n\u00e9ant\u00bb. Le calcul probabiliste des \u00abhasards\u00bb en balance (\u00abquand il y aurait une infinit\u00e9 de hasards dont un seul serait pour vous, vous auriez encore raison de gager un pour avoir deux, et vous agirez de mauvais sens, en \u00e9tant oblig\u00e9 \u00e0 jouer, de refuser de jouer une vie contre trois \u00e0 un jeu o\u00f9 d'une infinit\u00e9 de hasards il y en a un pour vous, s'il y avait une infinit\u00e9 de vie infiniment heureuse \u00e0 gagner [...]\u00bb) vise surtout \u00e0 forcer l'adversaire \u00e0 jouer. Qu'il accepte de se voir comme un joueur, et il a gagn\u00e9 et perdu \u00e0 la fois.\n\nUn calcul complexe sur les rapports entre le fini et l'infini est sous-tendu par un \u00e9cho faisant rimer la \u00abpartie\u00bb de jeu avec le \u00abparti\u00bb, lequel devient enfin un \u00abpari\u00bb. Qu'est-ce qu'un \u00abparti\u00bb? Le \u00abTrait\u00e9 du triangle arithm\u00e9tique\u00bb nous en donne une d\u00e9finition: la loi du \u00abparti\u00bb implique que ce que les joueurs ont pos\u00e9 sur la table ne leur appartient plus. Cette somme est vou\u00e9e au hasard et correspond \u00e0 ce que la fortune d\u00e9cidera, soit leur prendre tout, soit le redonner multipli\u00e9. Puisque cette loi est volontaire, chacun peut d\u00e9cider de \u00abpartir\u00bb et de reprendre son bien. Les calculs complexes propos\u00e9s par Pascal r\u00e9glementent ce qui doit leur appartenir selon des proportions initiales:\n\nSi deux joueurs jouent \u00e0 un jeu de pur hasard, \u00e0 condition que si le premier gagne, il lui reviendra une certaine somme, et s'il perd, il lui reviendra une moindre; s'ils veulent se s\u00e9parer sans jouer, et prendre chacun ce qui leur appartient, le parti est que le premier prenne ce qui lui revient en cas de perte, et de plus la moiti\u00e9 de l'exc\u00e8s dont ce qui lui reviendrait en cas de gain surpasse ce qui lui revient en cas de perte.\n\nQuel est le but de Pascal? Comme la plupart des commentateurs l'ont soulign\u00e9, l'argument math\u00e9matique ne risque gu\u00e8re de convaincre un agnostique qu'il vaut mieux parier sur l'existence de Dieu que sur son inexistence, car on ne joue pas aux d\u00e9s ou aux cartes avec la foi. En fait, derri\u00e8re la sophistication des calculs, toute une mise en sc\u00e8ne ontologique est impliqu\u00e9e, mais l'argument essentiel est ce qui est propos\u00e9 d\u00e8s le d\u00e9but: si ne pas parier c'est encore parier, il faut donc parier. Que je le veuille ou non, le jeu se joue d\u00e9j\u00e0. La partie est engag\u00e9e m\u00eame si je m'aveugle sur les gains possibles ou les pertes vraisemblables, c'est ma raison, ma foi et ma vie qui sont de fait les enjeux. On ne peut donc pas conclure: \u00abQui ne risque rien n'a rien.\u00bb On ne peut dire non plus: \u00abQui ne risque rien ne gagne rien (ou ne perd rien).\u00bb En fait, celui qui ne risque rien a d\u00e9j\u00e0 perdu, mais sans le savoir.\n\nCette dramatisation pourrait faire de Pascal un pr\u00e9curseur de l'existentialisme chr\u00e9tien, l'anc\u00eatre de Kierkegaard. Une telle question d\u00e9borde du cadre de l'ontologie et c'est sans doute pourquoi Heidegger cite Pascal de mani\u00e8re approbative d\u00e8s le d\u00e9but de Sein und Zeit, alors qu'on pourrait imaginer une r\u00e9sistance de sa part \u00e0 une math\u00e9matisation des choix existentiels. Heidegger cite un fragment de De l'esprit g\u00e9om\u00e9trique au premier paragraphe de son ouvrage: \u00abOn ne peut entreprendre de d\u00e9finir l'\u00eatre sans tomber dans cette absurdit\u00e9: car on ne peut d\u00e9finir un mot sans commencer par celui-ci, c'est, soit qu'on l'exprime ou qu'on le sous-entende. Donc pour d\u00e9finir l'\u00eatre, il faudrait dire c'est, et ainsi employer le mot d\u00e9fini dans sa d\u00e9finition.\u00bb Pascal d\u00e9duit de l'impossibilit\u00e9 o\u00f9 nous sommes de d\u00e9finir des notions aussi fondamentales que \u00abl'homme\u00bb, \u00able temps\u00bb ou l'\u00ab\u00eatre\u00bb que la raison serait impuissante sans un recours constant \u00e0 une sorte de foi intuitive. Voil\u00e0 pourquoi cet opuscule se transforme en un \u00abArt de persuader\u00bb avant de d\u00e9boucher sur l'id\u00e9e d'une humilit\u00e9 de la raison humaine: \u00abCe n'est pas dans les choses extraordinaires et bizarres que se trouve l'excellence de quelque genre que ce soit. On s'\u00e9l\u00e8ve pour y arriver, et on s'en \u00e9loigne: il faut le plus souvent s'abaisser. Les meilleurs livres sont ceux que ceux qui les lisent croient qu'ils auraient pu faire.\u00bb La Bible correspond-elle \u00e0 ce crit\u00e8re?\n\nOui, dans la mesure o\u00f9 l'humiliation de la raison est l'enjeu principal que vise l'argument du pari, car les dilemmes personnels ne peuvent \u00eatre r\u00e9gl\u00e9s par un calcul rationnel, mais bien par un saut logique, un bond hors des cat\u00e9gories ontologiques. C'est toute une pragmatique qui doit relayer l'investigation m\u00e9taphysique \u00e0 la Descartes. Si tout choix implique un risque, Pascal d\u00e9place le terrain de la discussion en passant de l'\u00e9tablissement d'un fait \u00e0 la justification d'une action. Voil\u00e0 qui annonce la th\u00e9orie des jeux et des d\u00e9cisions, et pose d'embl\u00e9e l'id\u00e9e d'une attente qui d\u00e9termine tout le reste. Plus que les calculs abstraits qui pr\u00e9supposent une bifurcation des chances, il faut accepter qu'un \u00abab\u00eatissement\u00bb de la raison ouvre le chemin du \u00abc\u0153ur\u00bb.\n\nLe passage m\u00eame du \u00abpari\u00bb dans les Pens\u00e9es se termine de mani\u00e8re apor\u00e9tique, dialogique et non conclusive. Comme dans l'apologue de Luc, ce sont en d\u00e9finitive les autres qui peuvent montrer l'exemple. Les protestations du pauvre libertin bouscul\u00e9 en ses derniers retranchements sont presque amusantes:\n\n[...], mais encore n'y a-t-il point moyen de voir le dessous du jeu? \u2013 Oui, l'\u00c9criture et le reste, etc. \u2013 Oui, mais j'ai les mains li\u00e9es et la bouche muette, on me force \u00e0 parier, et je ne suis pas en libert\u00e9, on ne me rel\u00e2che pas, et je suis fait d'une telle sorte que je ne puis croire. [...] apprenez de ceux, etc., qui ont \u00e9t\u00e9 li\u00e9s comme vous et qui parient maintenant tout leur bien. Ce sont des gens qui savent ce chemin que vous voudriez suivre... [...] C'est en faisant tout comme s'ils croyaient, en prenant de l'eau b\u00e9nite, en faisant dire des messes, etc. Naturellement m\u00eame cela vous fera croire et vous ab\u00eatira. \u2013 Mais c'est ce que je crains. \u2013 Et pourquoi? Qu'avez-vous \u00e0 perdre?\n\nLa circularit\u00e9 de l'argument semble \u00e9vidente. Il suffit de convaincre le libertin de faire un pas ou un saut dans le vide et d'accepter de rejoindre une communaut\u00e9 qui fait \u00abcomme si\u00bb elle croyait. Le \u00abc\u0153ur\u00bb s'accommode donc assez bien de la mascarade.\n\nOn aura crois\u00e9 \u00e0 plusieurs reprises l'argument textuel: lire la Bible serait une mani\u00e8re de voir \u00able dessous du jeu\u00bb. Qu'est-ce donc que lire? Comment lire? On se souvient de la remarque que Paul de Man cite en exergue d'All\u00e9gories de la lecture: \u00abQuand on lit trop vite ou trop doucement on n'entend rien.\u00bb La strat\u00e9gie de Pascal vise \u00e0 bousculer les certitudes mondaines en prenant l'esprit dans un \u00e9tau logique reposant sur le calcul des infinis pour promouvoir ultimement un art de persuader \u2013 une rh\u00e9torique au sens plein. Or c'est moins un art de \u00abconvertir\u00bb ou d'\u00abab\u00eatir\u00bb que de tourner l'esprit vers la lecture. Il s'agit d'une rh\u00e9torique qui va persuader l'interlocuteur d'accepter de commencer \u00e0 (bien) lire. Le texte dit exactement:\n\nDeux infinis, milieu.\n\nQuand on lit trop vite ou trop doucement, on n'entend rien.\n\nL\u00e0 encore, une \u00e9conomie pragmatique est \u00e0 l'\u0153uvre, puisqu'il s'agit de trouver un \u00abjuste milieu\u00bb pourtant donn\u00e9 comme impossible. Voici une illustration parmi beaucoup d'autres:\n\nBorn\u00e9s en tout genre, cet \u00e9tat qui tient le milieu entre deux extr\u00eames se trouve [...] en toutes nos puissances. Nos sens n'aper\u00e7oivent rien d'extr\u00eame, trop de bruit nous assourdit, trop de lumi\u00e8re \u00e9blouit, trop de distance et trop de proximit\u00e9 emp\u00eache la vue. [...] trop de plaisir incommode, trop de consonances d\u00e9plaisent dans la musique, et trop de bienfaits irritent. Nous voulons avoir de quoi surpasser la dette: \u00abBeneficia eo usque laeta sunt dum videntur exsolvi posse; ubi multum antevenere, pro gratia odium redditur.\u00bb Nous ne sentons ni l'extr\u00eame chaud, ni l'extr\u00eame froid.\n\nLa phrase de Tacite apporte un commentaire oblique sur la tactique de l'intendant infid\u00e8le qui r\u00e9duit la dette des d\u00e9biteurs sans l'abolir: \u00abLes bienfaits sont agr\u00e9ables tant qu'ils paraissent pouvoir \u00eatre pay\u00e9s de retour. S'ils d\u00e9passent de beaucoup ce pouvoir, au lieu de reconnaissance nous les payons de haine.\u00bb Pascal a trouv\u00e9 cette citation, parmi bien d'autres, au chapitre VIII du troisi\u00e8me livre des Essais de Montaigne, le premier \u00e0 citer Tacite en latin.\n\nOr l\u00e0 o\u00f9 Montaigne r\u00e9it\u00e8re des maximes de prudence en mati\u00e8re de jugement historique, Pascal radicalise le propos sur la dette. Montaigne tente de montrer qu'il faut accepter d'avoir des dettes et, par exemple, cite Philippe de Commynes qui disait: \u00abQu'il faut bien garder de faire tant de service \u00e0 son maistre, qu'on l'empesche d'en trouver la juste recompence.\u00bb Et il mentionne S\u00e9n\u00e8que qui \u00e9crivait: \u00abCar qui trouve honteux de ne pas rendre, souhaite ne rencontrer personne \u00e0 qui rendre.\u00bb Pascal vise \u00e0 d\u00e9montrer une faiblesse ou une incapacit\u00e9 constitutive. Alors m\u00eame qu'il postule un calcul, il d\u00e9montre l'impossibilit\u00e9 de lui trouver le \u00ablieu\u00bb qui serait par la m\u00eame op\u00e9ration un \u00abmilieu\u00bb. Voil\u00e0 qui rejaillit sur la question du temps et de l'avenir, comme le prouve un autre passage:\n\nNous ne nous tenons jamais au temps pr\u00e9sent. Nous anticipons l'avenir comme trop lent \u00e0 venir, comme pour h\u00e2ter son cours, ou nous rappelons le pass\u00e9 pour l'arr\u00eater comme trop prompt, si imprudents que nous errons dans des temps qui ne sont point n\u00f4tres... [...] Nous ne pensons presque point au pr\u00e9sent, et si nous y pensons ce n'est que pour en prendre la lumi\u00e8re pour disposer de l'avenir. Le pr\u00e9sent n'est jamais notre fin. Le pass\u00e9 et le pr\u00e9sent sont nos moyens; le seul avenir est notre fin.\n\nLorsqu'il joue au moraliste, Pascal conclut \u00e0 l'impossibilit\u00e9 de trouver le bonheur; lorsqu'il vise \u00e0 promouvoir son apolog\u00e9tique, il transforme le vertige existentiel en une herm\u00e9neutique des textes. C'est ce qui se v\u00e9rifie en premier lieu pour le texte de la Bible.\n\nSi nous sommes \u00abembarqu\u00e9s\u00bb de force dans une situation o\u00f9 les choix sont impossibles, la plong\u00e9e dans une pragmatique de la h\u00e2te devrait faire surgir par contraste le besoin d'une stabilit\u00e9 textuelle. Pascal vise \u00e0 montrer au libertin qu'il peut tirer de la Bible plus de profit que de tout autre texte, au moins pour \u00e9viter une labilit\u00e9 universelle. Le principe herm\u00e9neutique est que toutes les obscurit\u00e9s d'un texte s'\u00e9clairent d\u00e8s que l'on trouve le point d'\u00e9quilibre, clef qui transforme ses incongruit\u00e9s en une signification globale: \u00abTout auteur a un sens auquel tous les passages contraires s'accordent ou il n'a point de sens du tout. On ne peut pas dire cela de l'\u00c9criture et des proph\u00e8tes: ils avaient assur\u00e9ment trop de bon sens. Il faut donc en chercher un qui accorde toutes les contrari\u00e9t\u00e9s.\u00bb Pascal conclut ce passage aussit\u00f4t, et donc trop vite peut-\u00eatre, par un all\u00e9gorisme de la figure du Christ: \u00abLe v\u00e9ritable sens n'est donc pas celui des Juifs, mais en J\u00e9sus-Christ toutes les contradictions sont accord\u00e9es.\u00bb Il suffira de trouver un plan de sens qui peut harmoniser les contradictions de surface. Un mod\u00e8le sera fourni par la symbolique de saint Augustin, un autre par les lectures talmudiques que Pascal avait d\u00e9couvertes dans le trait\u00e9 de Raymond Martin, le Pugio fidei adversus Mauros et Judaeos. Cet essai du XIIIe si\u00e8cle venait d'\u00eatre r\u00e9\u00e9dit\u00e9; il contenait une critique de l'herm\u00e9neutique rabbinique. Pour simplifier ce mod\u00e8le, et en m'aidant de l'excellent ouvrage de Pierre Force sur Pascal et l'herm\u00e9neutique, on peut dire que le mod\u00e8le de ce discours proph\u00e9tique implique quatre conditions pour \u00eatre r\u00e9alis\u00e9. Il lui faut donc:\n\na) un \u00e9nonciateur conscient de parler \u00e0 un double niveau de sens;\n\nb) un message cod\u00e9 de mani\u00e8re tropologique qui m\u00eale \u00e9l\u00e9ments figuratifs et non figuratifs;\n\nc) une clef permettant de saisir les figures et permettant de distinguer entre les passages qui se comprennent au niveau litt\u00e9ral et ceux qui se comprennent au niveau figuratif;\n\nd) une prise en compte de la temporalit\u00e9 qu'implique le texte, ce qui am\u00e8ne \u00e0 distinguer entre les \u00e9v\u00e9nements qui ont d\u00e9j\u00e0 eu lieu et ceux qui sont toujours \u00e0 venir.\n\nIl ne faudrait pas comprendre Adam comme un personnage litt\u00e9ral. Pascal m\u00e9dite sur la formule de saint Paul que nous avons d\u00e9j\u00e0 rencontr\u00e9e dans l'\u00e9p\u00eetre aux Romains (Rom, 5: 14). La version de la Vulgate dit: \u00abAdam forma futuri\u00bb ou \u00abAdam est la figure de celui qui devait venir\u00bb. Pascal la commente dans un fragment \u00e0 la logique un peu d\u00e9routante:\n\n\u00abAdam forma futuri.\u00bb Les six jours pour former l'un, les six \u00e2ges pour former l'autre. Les six jours que Mo\u00efse repr\u00e9sente pour la formation d'Adam ne sont que la peinture des six \u00e2ges pour former J\u00e9sus-Christ et l'\u00c9glise. Si Adam n'e\u00fbt point p\u00e9ch\u00e9 et que J\u00e9sus-Christ ne f\u00fbt point venu, il n'y e\u00fbt eu qu'une seule alliance, qu'un seul \u00e2ge des hommes et la cr\u00e9ation e\u00fbt \u00e9t\u00e9 repr\u00e9sent\u00e9e comme faite en un seul temps.\n\nIci, Adam incarne \u00e0 la fois le principe augustinien de la felix culpa, cette faute heureuse qui a permis \u00e0 J\u00e9sus de venir pour racheter l'humanit\u00e9 du p\u00e9ch\u00e9, et l'id\u00e9e paulinienne d'une \u00abmort\u00bb qui aurait r\u00e9gn\u00e9 d'Adam \u00e0 Mo\u00efse, qui inclut l'id\u00e9e des limbes pour ceux qui ne pouvaient conna\u00eetre la Loi.\n\nLe \u00abfutur\u00bb que d\u00e9couvre Pascal dans la Bible n'est pas seulement un futur qui s'est d\u00e9j\u00e0 incarn\u00e9. Ce futur n'aura pas \u00e9t\u00e9 per\u00e7u comme tel par ceux qui auraient eu le plus de droit \u00e0 ce genre de savoir, les Juifs, ce pourquoi Pascal insiste \u00e0 plusieurs reprises sur le fait que les Juifs ont eu tort de prendre l'id\u00e9e de la circoncision au sens litt\u00e9ral: \u00abLa circoncision n'\u00e9tait qu'un signe\u00bb, et un peu plus loin: \u00abQue la circoncision du c\u0153ur est ordonn\u00e9e.\u00bb\n\nL'un des th\u00e8mes fondamentaux que Pascal emprunte \u00e0 l'herm\u00e9neutique rabbinique est sa num\u00e9rologie. Il lui faut d\u00e9montrer que les proph\u00e8tes de l'Ancien Testament annoncent clairement la venue du Christ \u2013 d'o\u00f9 un passage sur les \u00absix jours\u00bb. Un autre fragment des Pens\u00e9es dit: \u00abMal\u00e9diction des Grecs contre ceux qui comptent trois p\u00e9riodes de temps.\u00bb Le Guern r\u00e9tablit un texte diff\u00e9rent et imprime: \u00abMal\u00e9diction des {Juifs} contre ceux qui comptent trois p\u00e9riodes des temps\u00bb, car il suppose que son auteur d\u00e9nonce ici les Juifs qui n'aiment pas qu'on divise l'histoire en trois p\u00e9riodes, avant la Loi, pendant la Loi, et avec la Gr\u00e2ce. Tout se d\u00e9double dans la Bible pour peu que l'on sache lire le futur qui lui est inh\u00e9rent: \u00abDouble loi, doubles tables de la loi, double temps, double captivit\u00e9.\u00bb Ceci confirme le lien entre l'herm\u00e9neutique et la proph\u00e9tie: les \u00abpreuves\u00bb de la religion r\u00e9sident dans cette futurit\u00e9 d\u00e9j\u00e0 inscrite dans un texte opaque.\n\nLa d\u00e9monstration du \u00abhasard\u00bb existentiel dans lequel nous pr\u00e9cipitaient d\u00e9j\u00e0 les couples d'extr\u00eames oppos\u00e9s doit conduire \u00e0 une conversion moins \u00e0 la ferveur religieuse qu'\u00e0 la rigueur textuelle. L'incertitude portant sur l'avenir de l'homme \u2013 incertitude qui peut \u00eatre sublim\u00e9e ou transcend\u00e9e par l'ordre de la foi ou du c\u0153ur \u2013 donne naissance \u00e0 la certitude d'une connaissance du futur. Ce futur inscrit au c\u0153ur m\u00eame du texte sacr\u00e9, un texte qui culmine avec l'\u00e9v\u00e9nement pass\u00e9, pr\u00e9sent et futur de l'Incarnation.\n\nSi, de fait, la futurit\u00e9 de la venue du Messie s'est d\u00e9j\u00e0 r\u00e9alis\u00e9e, elle doit conserver sa part future: sans elle, point de d\u00e9chiffrage du texte, pour tous ceux qui vont na\u00eetre et venir apr\u00e8s nous. C'est un futur ant\u00e9rieur qui ouvre aussi l'espace d'un futur post\u00e9rieur, ce futur que chacun porte en lui-m\u00eame par la \u00abgr\u00e2ce\u00bb de la finesse qui s'\u00e9panouit dans un rapport au texte. Un tel rapport d\u00e9c\u00e8le la futurit\u00e9 dans la figurabilit\u00e9. Cet apprentissage de la lecture dans un temps pr\u00e9caire et labile qui nous pousse en avant n'exclut pas un l\u00e9ger persiflage envers la philosophie: \u00abLa vraie \u00e9loquence se moque de l'\u00e9loquence, la vraie morale se moque de la morale... Se moquer de la philosophie, c'est vraiment philosopher.\u00bb En un sens, Pascal serait plus pr\u00e8s de Nietzsche que de Kierkegaard, mais comme nous allons le voir en traversant un commentaire de Heidegger, cette proximit\u00e9 m\u00eame n'est pas sans poser probl\u00e8me.\n\n### 4. \u00abCeux qui vont venir\u00bb et les pr\u00e9curseurs\n\nL'ambivalence de Heidegger face \u00e0 Nietzsche ressemble \u00e0 son ambivalence face \u00e0 Pascal. Lorsque Heidegger commente un po\u00e8me de Rilke au sujet de l'Ouvert, il d\u00e9ploie les sens qui se rattachent au \u00abpari\u00bb et au \u00abrisque\u00bb, et donc rencontre la pens\u00e9e de Pascal. Cependant, il pose une \u00e9quivalence par renversement entre Descartes, pour qui \u00abla sph\u00e8re de l'int\u00e9riorit\u00e9 invisible se d\u00e9termine comme la r\u00e9gion de pr\u00e9sence des \u00eatres calcul\u00e9s\u00bb, et Pascal, pour qui cette zone se d\u00e9termine comme celle du \u00abc\u0153ur\u00bb:\n\nPresque en m\u00eame temps que Descartes, Pascal d\u00e9couvre, antith\u00e8se de la logique, de la raison calculante, la logique du c\u0153ur. L'int\u00e9rieur \u2013 l'invisible de la dimension du c\u0153ur \u2013 est non seulement plus int\u00e9rieur que l'int\u00e9rieur de la repr\u00e9sentation calculante \u2013 et, pour cela, plus invisible \u2013 mais il porte en m\u00eame temps plus loin que la r\u00e9gion des simples objets productibles. Ce n'est que dans l'intimit\u00e9 invisible du c\u0153ur que l'homme se penche vers l'aimable: les a\u00efeux, les morts, l'enfance, ceux qui vont venir.\n\nLe terme allemand \u00abdie K\u00fcnftige\u00bb (traduction mot \u00e0 mot: \u00ables futurs\u00bb) provient d'un passage d'une lettre de Rilke \u00e9crite \u00e0 Muzot le 11 ao\u00fbt 1924. Rilke \u00e9voque les morts et ceux qui sont \u00e0 venir, tous ceux qui d\u00e9sirent un espace plus intime que notre intimit\u00e9. Cet espace poss\u00e8de \u00abla dimension de profondeur de notre intimit\u00e9\u00bb: \u00abSi donc les morts, si donc ceux qui viendront ont besoin d'un s\u00e9jour, quel abri leur serait plus agr\u00e9able et plus offert que cet espace imaginaire?\u00bb Ces \u00abfuturs\u00bb substantiv\u00e9s \u00e9voquent la fin de la Neuvi\u00e8me \u00c9l\u00e9gie de Duino, que Heidegger cite en conclusion de ce paragraphe:\n\nMais l'int\u00e9rieur de cette conscience inusuelle demeure un espace intime \u00e0 l'int\u00e9rieur duquel, pour nous, toute chose a d\u00e9pass\u00e9 le num\u00e9rique du calcul, et peut ainsi, libre de limitation, s'\u00e9pancher dans le Tout d\u00e9gag\u00e9 de l'Ouvert. Un tel superflu surnum\u00e9raire na\u00eet dans l'int\u00e9riorit\u00e9 invisible du c\u0153ur. La derni\u00e8re parole de la Neuvi\u00e8me El\u00e9gie \u2013 celle qui chante l'appartenance des hommes \u00e0 l'Ouvert \u2013 est: \u00abDans le c\u0153ur me na\u00eet une existence surnum\u00e9raire.\u00bb (CH, 250)\n\nLes trois derniers vers de Rilke:\n\nSiehe, ich lebe. Woraus? Weder Kindheit noch Zukunft\n\nwerden weniger... Ueberz\u00e4hliges Dasein\n\nentspringt mir im Herzen.\n\nHeidegger reprend ici une analyse philologique qu'il avait men\u00e9e auparavant sur le terme de \u00abWagnis\u00bb (le \u00abrisque\u00bb), qu'il avait mis en rapport avec l'\u00e9tymologie de \u00abWaage\u00bb, la \u00abbalance\u00bb. Le superflu de l'existence provient d'un rapport privil\u00e9gi\u00e9 des cr\u00e9atures avec l'Ouvert, car cette Ouverture vient les projeter en avant pour les \u00abrisquer\u00bb. Ces vers sont tir\u00e9s d'une autre lettre de Muzot \u00e9crite en ao\u00fbt 1924:\n\nComme la nature abandonne les \u00eatres\n\nau risque de leur obscur d\u00e9sir et n'en prot\u00e8ge\n\naucun dans les sillons et les branches,\n\nde m\u00eame nous aussi, au tr\u00e9fonds de notre \u00eatre\n\nne sommes pas plus chers; il nous risque. Sauf que nous,\n\nplus encore que la plante ou l'animal,\n\nallons avec ce risque, le voulons, et parfois m\u00eame\n\nrisquons plus (et point par int\u00e9r\u00eat)\n\nque la vie elle-m\u00eame, d'un souffle\n\nplus... Ainsi avons-nous, hors d'abri,\n\nune s\u00fbret\u00e9, l\u00e0-bas o\u00f9 porte la gravit\u00e9\n\ndes forces pures; ce qui enfin nous sauve,\n\nc'est d'\u00eatre sans abri, et de l'avoir, cet \u00eatre,\n\nretourn\u00e9 dans l'ouvert, le voyant menacer,\n\npour, quelque part dans le plus vaste cercle,\n\nl\u00e0 o\u00f9 le statut nous touche, lui dire oui. (Cit\u00e9 dans CH, 226)\n\nCes vers \u00abimprovis\u00e9s\u00bb apportent comme une clef de la pens\u00e9e de Rilke, du moins selon le long commentaire qu'en donne Heidegger. Heidegger insiste de mani\u00e8re tr\u00e8s symptomatique sur le concept de \u00abvolont\u00e9\u00bb gliss\u00e9 au vers sept, afin de g\u00e9n\u00e9raliser la proposition contenue dans le terme de \u00abrisque\u00bb: \u00abTout \u00e9tant est risqu\u00e9. L'\u00eatre est le risque lui-m\u00eame par excellence. Il nous \"risque\", nous, les hommes. Il risque les \u00eatres vivants. L'\u00e9tant est, dans la mesure o\u00f9 il demeure ce qui toujours est \u00e0 nouveau risqu\u00e9.\u00bb (CH, 228), Mais de mani\u00e8re un peu surprenante, Heidegger identifie le \u00abrisque\u00bb avec la volont\u00e9 humaine, non pas une volont\u00e9 au sens psychologique, mais au sens m\u00e9taphysique, entendue comme l'essence de la volont\u00e9: \u00abL'\u00eatre de l'\u00e9tant est le risque. Celui-ci repose dans la volont\u00e9 qui, depuis Leibniz, s'est de plus en plus pr\u00e9cis\u00e9ment annonc\u00e9e comme l'\u00eatre de l'\u00e9tant d\u00e9voil\u00e9 en la M\u00e9taphysique.\u00bb (CH, 228) \u00c0 travers Leibniz, c'est toute l'histoire de la m\u00e9taphysique qui est vis\u00e9e ici, une histoire dans laquelle Rilke joue un r\u00f4le proche de celui de Nietzsche. Pascal, Nietzsche et Rilke affirmeraient tous les trois une primaut\u00e9 de l'essence de la volont\u00e9 qui se tourne vers l'\u00eatre pens\u00e9 comme Ouverture. Le seul risque qui \u00abrisque plus que la vie\u00bb et qui \u00e9chappa au calcul de la raison scientifique est ce qui relie le c\u0153ur au langage \u2013 condition ontologique de la po\u00e9sie, qui, comme l'a rappel\u00e9 Derrida dans Che cos'\u00e8 la poesia, exige qu'on l'apprenne par c\u0153ur.\n\nVivre en po\u00e9sie, \u00abbutiner \u00e9perdument le miel du visible, pour l'accumuler dans la grande ruche d'or de l'invisible\u00bb, comme dans les c\u00e9l\u00e8bres paroles que Rilke \u00e9crit en fran\u00e7ais (cit\u00e9 dans H, 304), est ce qui permet de transformer l'ext\u00e9rieur en l'int\u00e9rieur. Ce mouvement s'approche de l'Er-innerung, rem\u00e9moration active et int\u00e9riorisation au sens de Hegel. Heidegger, lui aussi, utilise le fran\u00e7ais pour introduire la locution cruciale de \u00abpar c\u0153ur\u00bb:\n\nLa recordation [Er-innerung] retourne notre essence qui ne veut que sur le mode de l'imposition ainsi que ses objets, vers l'Invisible intime de l'espace du c\u0153ur. Ici, tout est (r\u00e9)version int\u00e9rieure: non seulement tout reste tourn\u00e9 vers le v\u00e9ritable \u00abdedans\u00bb de la conscience, mais encore, \u00e0 l'int\u00e9rieur de ce dedans, tout se d\u00e9verse pour nous, sans aucune borne, l'un dans l'autre. La version int\u00e9rieure de l'espace intime du monde nous d\u00e9limite l'Ouvert. Ce que nous retenons ainsi int\u00e9rieurement, c'est cela seul que nous savons par c\u0153ur. (CH, 252, traduction modifi\u00e9e; H, 305)\n\nHeidegger demande qui donc aurait l'audace de prendre un tel risque ontologique. Il r\u00e9pond que celui qui affronte le Dire po\u00e9tique, le po\u00e8te, va risquer plus que l'\u00eatre, car il en vient \u00e0 risquer la langue elle-m\u00eame:\n\nParce que la langue est la demeure de l'\u00eatre, nous n'acc\u00e9dons \u00e0 l'\u00e9tant qu'en passant constamment par cette demeure. Quand nous allons \u00e0 la fontaine, quand nous traversons la for\u00eat, nous traversons toujours d\u00e9j\u00e0 le nom \u00abfontaine\u00bb, le nom \u00abfor\u00eat\u00bb, m\u00eame si nous n'\u00e9non\u00e7ons pas ces mots, m\u00eame si nous ne pensons pas \u00e0 la langue. Pensant \u00e0 partir du temple de l'\u00eatre, nous pouvons pr\u00e9sumer ce que risquent ceux qui parfois risquent plus que l'\u00eatre de l'\u00e9tant. Ils risquent l'enceinte de l'\u00eatre. Ils risquent la langue. (CH, 253)\n\nFaut-il identifier celui qui prend le risque de la langue avec le po\u00e8te? On pourrait voir en ce joueur imp\u00e9nitent aussi bien la figure tut\u00e9laire et embl\u00e9matique de l'Ange, car l'Ange op\u00e8re une transformation du Visible en l'Invisible, de l'Ext\u00e9rieur en l'Int\u00e9rieur. Heidegger voit en l'Ange rilk\u00e9en un double du Zarathoustra de Nietzsche (CH, 255). Une nouvelle lecture de po\u00e8mes lui permettra ensuite de d\u00e9passer l'homme moderne selon Nietzsche, un homme r\u00e9duit \u00e0 la volont\u00e9, \u00abcelui qui veut\u00bb et de fait reste dans l'illusion (CH, 260). Car pour Heidegger, il s'agit de retrouver le divin tel qu'il a \u00e9t\u00e9 exprim\u00e9 po\u00e9tiquement par H\u00f6lderlin: \u00abH\u00f6lderlin est le pr\u00e9d\u00e9cesseur [Vor-g\u00e4nger] des po\u00e8tes en temps de d\u00e9tresse. C'est pourquoi aucun po\u00e8te de cet \u00e2ge ne saurait le surpasser.\u00bb (CH, 261) Pour Heidegger, c'est donc H\u00f6lderlin, plus que Rilke, qui figure le po\u00e8te par excellence, avant tout parce qu'il incarne la futurit\u00e9 absolue de l'\u00e9criture: \u00abLe pr\u00e9d\u00e9cesseur, pourtant, ne s'en va pas en un avenir [Zukunft], mais au contraire, en vient, de sorte que seul dans l'advenue [Ankunft] de sa parole, l'avenir est pr\u00e9sent [allein die Zukunft anweist].\u00bb (CH, 261)\n\nUne telle advenue au pr\u00e9sent inscrit l'avenir dans le concept de destin (Geschick). Cela suppose \u00e0 la fois un \u00abenvoi\u00bb perp\u00e9tuel et un d\u00e9bordement de l'histoire qu'il contient pourtant. \u00abLa futurition de l'advenue se recueille et se reprend dans le Destin.\u00bb (CH, 261) La formule semble bien lourde et exotique en fran\u00e7ais, contrairement \u00e0 la traduction anglaise qui choisit la simplification: \u00abWhat occurs in the arrival gathers itself back into destiny.\u00bb Cette difficult\u00e9 s'estompe si l'on joue en fran\u00e7ais sur la bifurcation de l'avenir et du futur: il suffit d'admettre que le futur inscrit dans l'\u00e9criture le risque po\u00e9tique qui rejoue la langue et donc l'oriente vers l'avenir tout en reprenant le pass\u00e9 de la langue en soi. Pariant sur une volont\u00e9 de d\u00e9passement qui risque une perte radicale, le Dire po\u00e9tique advient comme un Destin qui s'\u00e9crit au futur, ou encore mieux, comme un \u00abf\u00fbtur\u00bb.\n\nPour penser ceci au plus pr\u00e8s de la langue fran\u00e7aise, en des idiomes qui font d\u00e9couler le \u00abpar c\u0153ur\u00bb po\u00e9tique moins d'une rem\u00e9moration ant\u00e9rieure que d'une r\u00e9citation ult\u00e9rieure, il faut revenir chez Mallarm\u00e9 \u00e0 cause de sa situation historique et historiale. \u00c0 mi-chemin entre la fulgurance du divin chez H\u00f6lderlin et la scintillation des parcelles d'or de l'Invisible et de l'Ouvert rilk\u00e9en, Mallarm\u00e9 a su \u00e0 la fois penser la futilit\u00e9 et la futurit\u00e9 du Destin po\u00e9tique.\n\n### 5. Martingales de Mallarm\u00e9\n\nMon enqu\u00eate a port\u00e9 jusqu'\u00e0 pr\u00e9sent sur les liens entre les divers \u00abparis\u00bb et \u00abpartis\u00bb de Pascal, Rilke et Mallarm\u00e9, sans \u00e9viter le th\u00e8me pascalien du \u00abdivertissement\u00bb. Une certaine futilit\u00e9 habite les calculs, m\u00eame tragiques et d\u00e9sesp\u00e9r\u00e9s, jouant avec un impr\u00e9vu du destin qui est \u00e9crit, mais de mani\u00e8re illisible. Roger Dragonetti a fait surgir de la po\u00e9tique du Livre de Mallarm\u00e9 une pratique du futile au quotidien, avan\u00e7ant dans Un fant\u00f4me dans le kiosque. Mallarm\u00e9 et l'esth\u00e9tique du quotidien  que la r\u00e9alisation du programme mallarm\u00e9en n'est pas \u00e0 chercher du c\u00f4t\u00e9 des notes pour le fameux grand Livre, mais bel et bien dans sa correspondance.\n\nComme pour lui donner raison, on trouve dans la correspondance de jeunesse de Mallarm\u00e9 un aveu du lien secret, mais puissant qui relie la d\u00e9couverte des jeux de hasard et la mise en place de la po\u00e9tique de l'abolition et du n\u00e9ant. En avril 1866, Mallarm\u00e9 d\u00e9crit \u00e0 sa femme qui \u00e9tait rest\u00e9e \u00e0 Tournon ses vacances sur la C\u00f4te d'Azur en compagnie de son ami Lef\u00e9bure. Il \u00e9voque son s\u00e9jour \u00e0 Monaco: \u00abJe te raconterai toutes mes heures \u00e0 mon retour, je me contente donc de te dire que l'excursion \u00e0 Monaco a \u00e9t\u00e9 d\u00e9licieuse, que j'y ai gagn\u00e9 \u00e0 la roulette quelques sous avec lesquels je t'ai achet\u00e9 une jolie petite... je ne dirai pas quoi, laquelle surprise ira \u00e0 merveille avec la robe que tu ach\u00e8teras cet \u00e9t\u00e9.\u00bb (C, 290-291) De mani\u00e8re significative, on retrouve dans ce passage une strat\u00e9gie qui vise \u00e0 postuler un futur \u2013 \u00abcet \u00e9t\u00e9\u00bb, on est en avril, \u00ab\u00e0 mon retour\u00bb, dans une semaine environ \u2013 afin d'exciter la curiosit\u00e9 ou l'imagination, et ne pas platement narrer les \u00abfaits\u00bb.\n\nM\u00eame si le po\u00e8te admet que sa femme lui manque et qu'il \u00abcherche dans le vide\u00bb sa main, son absence ne l'emp\u00eache pas de jouir des beaut\u00e9s du paysage. La d\u00e9couverte du soleil et de la mer est bien une r\u00e9v\u00e9lation confirm\u00e9e dans une lettre \u00e0 Cazalis qui suit de peu son retour \u00e0 Tournon: \u00abJe m'aper\u00e7ois que j'ai laiss\u00e9 aller ma plume, et ne t'ai rien dit de mon voyage enchant\u00e9. Lef\u00e9bure m'a lev\u00e9 le rideau qui me voilait \u00e0 jamais le d\u00e9cor de Nice, et je me suis follement enivr\u00e9 de la M\u00e9diterran\u00e9e. Ah! mon ami, que ce ciel terrestre est divin!\u00bb (C, 300) Quelques ann\u00e9es avant Nietzsche, donc, qui d\u00e9couvrit \u00e0 Nice non seulement la sant\u00e9 solaire et le rejet des arri\u00e8re-mondes de la vieille m\u00e9taphysique, mais encore la tentation de jouer \u00e0 la loterie (\u00abIl y avait une belle occasion \u00e0 saisir: le dernier tirage de la loterie \u00e0 Nice, \u2013 et pendant au moins une demi-heure, je me suis offert le petit luxe idiot de faire comme si j'\u00e9tais assur\u00e9 de gagner le gros lot\u00bb), Mallarm\u00e9 refuse les illusions d'un paradis autre que terrestre, mais ne se refuse pas une incursion dans les jeux de hasard. Le futur n'est plus celui de la promesse eschatologique du salut, seulement celui d'une partie de roulette, ou d'un \u00abrisque\u00bb radical face aux enjeux ontologiques de la langue. Comme Nietzsche, Mallarm\u00e9 a simplifi\u00e9 sa strat\u00e9gie face aux jeux du hasard; on sait que ce mot provient de l'arabe et signifie \u00abcoup de d\u00e9s\u00bb; il suffit de \u00abmartingaler\u00bb, c'est-\u00e0-dire de toujours doubler l'enjeu lorsque l'on perd.\n\nPourtant, \u00e0 la diff\u00e9rence de Nietzsche, Mallarm\u00e9 n'a pas perdu \u00e0 la roulette. Certes, il n'a pas gagn\u00e9 un demi-million, mais assez pour acheter le compl\u00e9ment d'une tenue f\u00e9minine \u2013 ce qui a d\u00fb l'obliger \u00e0 aller dans un magasin de mode et n'a pas manqu\u00e9 d'influer sur sa fameuse d\u00e9cision de reprendre et de r\u00e9diger \u00e0 lui seul, un peu plus tard, le magazine de mode f\u00e9minine connu sous le nom de La Derni\u00e8re Mode (1874). Pourtant, il a perdu quelque chose lors de ce s\u00e9jour au bord de la M\u00e9diterran\u00e9e, qui est peut-\u00eatre tout simplement l'illusion rassurante, qu'on pourrait dire \u00abm\u00e9taphysique\u00bb, celle qui aide \u00e0 vivre au jour le jour. Il s'en plaint dans une lettre \u00e0 Copp\u00e9e o\u00f9 il explique qu'il ne peut plus \u00e9crire de po\u00e9sie: \u00abPour moi, voici deux ans que j'ai commis le p\u00e9ch\u00e9 de voir le R\u00eave dans sa nudit\u00e9 id\u00e9ale, tandis que je devais amonceler entre lui et moi un myst\u00e8re de musique et d'oubli. Et maintenant, arriv\u00e9 \u00e0 la vision horrible d'une \u0152uvre pure, j'ai presque perdu la raison et le sens des paroles les plus famili\u00e8res.\u00bb (C, 380) Ce qui \u00e9tait digne de louanges et d'admiration quelques semaines apr\u00e8s ce \u00abs\u00e9jour enchant\u00e9\u00bb s'est mu\u00e9 en une mal\u00e9diction: on ne peut impun\u00e9ment \u00ablever le rideau\u00bb qui voilait le d\u00e9cor de Nice, ou s'enivrer follement de la M\u00e9diterran\u00e9e. Une autre lettre comm\u00e9more \u00e0 nouveau cet \u00e9pisode crucial:\n\n[...] du reste, je ne saurai quoi vous dire, (car je passe d'instants voisins de la folie entrevue \u00e0 des extases \u00e9quilibrantes), si ce n'est que je suis dans un \u00e9tat de crise qui ne peut durer [...] \u2013 D\u00e9cid\u00e9ment, je redescends de l'Absolu, je n'en ferai pas, suivant la belle phrase de Villiers, \u00abla Po\u00e9sie\u00bb ni ne d\u00e9roulerai \u00able vivant panorama des formes du Devenir\u00bb \u2013, mais cette fr\u00e9quentation de deux ann\u00e9es (vous vous rappelez? depuis notre s\u00e9jour \u00e0 Cannes) me laissera une marque, dont je veux faire un Sacre. (C, 384-385)\n\nUne telle \u00abredescente\u00bb prend des allures de convalescence entrecoup\u00e9e de rechutes; mais comment parler de \u00abrechute\u00bb lorsqu'il s'agit de l'Absolu? Ces rechutes am\u00e8nent le po\u00e8te \u00e0 l'id\u00e9e que seul un exercice raisonn\u00e9 de la futilit\u00e9 permettra la mobilit\u00e9 n\u00e9cessaire \u00e0 l'\u00e9criture. C'\u00e9tait d\u00e9j\u00e0 ce que Mallarm\u00e9 glissait \u00e0 Copp\u00e9e: \u00abJe donnerais les v\u00eapres magnifiques du R\u00eave et leur or vierge pour un quatrain, destin\u00e9 \u00e0 une tombe ou \u00e0 un bonbon, qui f\u00fbt \"r\u00e9ussi\".\u00bb (C, 381) Yves Bonnefoy a comment\u00e9 l'oxymore po\u00e9tologique qui effectue une \u00e9quivalence entre \u00abtombe\u00bb et \u00abbonbon\u00bb, y voyant la cons\u00e9quence de la crise de Tournon qui montrait la folie de trop \u00abcreuser le vers\u00bb. Bonnefoy \u00e9crit de mani\u00e8re tr\u00e8s perceptive: \u00abDe moindre co\u00fbt et, de ce fait m\u00eame, d\u00e9j\u00e0 de l'impersonnel: le bonbon, le fruit d\u00e9livrant les instants pendant lesquels ils se d\u00e9font dans la bouche des souvenirs et soucis et r\u00eaves de la personne, quitte pour ces tracas \u00e0 reprendre \u00e0 la fin de la jouissance. La mati\u00e8re, qui est la cause du hasard, lequel aveugle la po\u00e9sie, voici que, sucr\u00e9e, elle abolit le hasard.\u00bb Cette futilit\u00e9 de la po\u00e9sie est donc toute aussi \u00abutile\u00bb pour le po\u00e8te, et revendiqu\u00e9e sur fond de crise du capitalisme, car la pens\u00e9e politique de Mallarm\u00e9, on le sait, flirtait avec l'anarchisme.\n\nMallarm\u00e9 rejoindrait ainsi la pens\u00e9e de Hannah Arendt commentant Karl Marx en des termes \u00e0 elle:\n\nPlus la vie devient facile dans une soci\u00e9t\u00e9 de consommateurs ou de travailleurs, plus il devient difficile de rester conscient des forces de n\u00e9cessit\u00e9 auxquelles cette soci\u00e9t\u00e9 ob\u00e9it m\u00eame quand le labeur et l'effort, manifestations ext\u00e9rieures de la n\u00e9cessit\u00e9, deviennent \u00e0 peine sensibles. Une telle soci\u00e9t\u00e9, \u00e9blouie par l'abondance de sa f\u00e9condit\u00e9, prise dans le fonctionnement b\u00e9at d'un processus sans fin, ne reconna\u00eet plus sa futilit\u00e9 \u2013 la futilit\u00e9 d'une vie qui n'arrive plus \u00e0 se fixer ni \u00e0 se r\u00e9aliser en un sujet permanent qui dure apr\u00e8s que son labeur est pass\u00e9.\n\nL'espoir que l'\u0152uvre d'art absolue puisse justifier un pr\u00e9sent somme toute d\u00e9cevant pousse le po\u00e8te vers un f\u00fbtur [sic] toujours menac\u00e9 de futilit\u00e9.\n\nLa rime int\u00e9rieure qui conjoint sournoisement le futile et le futur travaille sourdement Mallarm\u00e9. On en voit un exemple irr\u00e9futable dans une lettre \u00e0 Villiers de l'Isle-Adam, son ma\u00eetre \u00e8s h\u00e9g\u00e9lianisme; elle commence par le m\u00eame constat d'impuissance et s'ach\u00e8ve de mani\u00e8re presque guillerette, avec l'attente de la publication de plusieurs po\u00e8mes en prose. Symptomatiquement, le mot \u00abAvenir\u00bb vient d\u00e9crire la ruine due \u00e0 la vision trop nue et forte, tandis que le \u00abfutur\u00bb \u00e9voque le jeu avec une attente pleine de promesses:\n\nPour l'Avenir, du moins pour le plus voisin, mon \u00e2me est d\u00e9truite. Ma pens\u00e9e a \u00e9t\u00e9 jusqu'\u00e0 se penser elle-m\u00eame et n'a plus la force d'\u00e9voquer en un N\u00e9ant unique le vide diss\u00e9min\u00e9 en sa porosit\u00e9.\n\n[...]\n\nLe miroir qui m'a r\u00e9fl\u00e9chi l'\u00catre a \u00e9t\u00e9 le plus souvent l'Horreur et vous devinez si j'expie cruellement ce diamant de Nuits innomm\u00e9es.\n\nIl me reste la d\u00e9limitation parfaite et le r\u00eave int\u00e9rieur de deux livres, \u00e0 la fois nouveaux et \u00e9ternels, l'un tout absolu \u00abBeaut\u00e9\u00bb l'autre personnel, les \u00abAll\u00e9gories somptueuses du N\u00e9ant\u00bb, mais (d\u00e9rision et torture de Tantale,) l'impuissance de les \u00e9crire \u2013 d'ici \u00e0 bien longtemps, si mon cadavre doit ressusciter. (C, 366-367)\n\nLe ton posthume de ces lettres a \u00e9t\u00e9 parodi\u00e9 avec la verve irlandaise de Beckett lorsqu'il fait dire \u00e0 un Malone qui n'en finit pas de mourir: \u00abMais laissons-l\u00e0 ces questions morbides et revenons \u00e0 celle de mon d\u00e9c\u00e8s, d'ici deux ou trois jours si j'ai bonne m\u00e9moire. \u00c0 ce moment-l\u00e0 c'en sera fait des Murphy, Mercier, Molloy, Moran et autres Malone, \u00e0 moins que \u00e7a continue dans l'outre-tombe.\u00bb Mais il suffit de laisser un peu la plume \u00e0 Mallarm\u00e9 lui-m\u00eame pour qu'il retrouve une certaine sant\u00e9, voire un enjouement de bon aloi; voici comment sa lettre \u00e0 Villiers se conclut: \u00abPardonnez- moi donc mon silence, ancien sur votre envoi de \"Morgane\" ce magnifique d\u00e9veloppement de vous, que j'ai relu vingt fois, et futur sur les richesses que m'apportera le Journal \u2013 en faveur de mes maux.\u00bb (C, 367) Ce Journal tant attendu dans le \u00abfutur\u00bb ne devait pas fournir une imbattable martingale au po\u00e8te, mais la joie de se voir publi\u00e9.\n\nLa correspondance de Mallarm\u00e9 nous fait observer une bifurcation capitale entre un grandiose \u00abAvenir\u00bb dot\u00e9 d'une capitale et un \u00abfutur\u00bb moins impressionnant, plus r\u00e9alisable, mais tout aussi marqu\u00e9 par l'impr\u00e9visible. S'il fait souvent \u00e9tat d'un \u00abvague plan\u00bb conserv\u00e9 \u00abpour l'avenir, dans plusieurs ann\u00e9es\u00bb \u2013 c'est encore un programme de travail annon\u00e7ant, apr\u00e8s le \u00ablivre de \"Beaut\u00e9\"\u00bb, un trait\u00e9 de la \u00abLaideur\u00bb (C, 368-369) qui en serait la contrepartie \u2013, il avoue que la futurit\u00e9 de ces textes reste un \u00e9l\u00e9ment essentiel, constitutif: \u00abCela me vivifie. Si vous et le soleil pouviez faire le reste, par votre pareille chaleur amicale, vous sauveriez du n\u00e9ant de bien divines \u0152uvres, navr\u00e9es d'\u00eatre \u00e0 moiti\u00e9 plong\u00e9es dans le futur.\u00bb (C, 373)\n\nTout ceci rendrait mieux compte du lapsus calami de Marie Mallarm\u00e9 qui rempla\u00e7ait \u00abfutures\u00bb par \u00abf\u00fbtures\u00bb. En f\u00e9vrier 1869, Mallarm\u00e9 avait fait le v\u0153u de ne plus toucher une plume jusqu'\u00e0 P\u00e2ques, ce qu'il explique par l'entremise de sa femme, dont la graphie singuli\u00e8re a \u00e9t\u00e9 conserv\u00e9e: \u00abMa pens\u00e9e, occup\u00e9e par la plainitude l'Univers et distendu, perdait sa fonction normal: J'ai senti des sympt\u00f4mes tr\u00e8s inqui\u00e9tants causs\u00e9s par le seul acte d'\u00e9crire, et l'hyst\u00e9rie allait commencer a troubler ma parole.\u00bb ([sic]; C, 423) La convalescence que pressent Mallarm\u00e9, qui a vingt-sept ans \u00e0 ce moment, va impliquer un jeu constant avec le temps du futur: il lui faut attendre, encore et toujours, pour sortir des \u00abt\u00e9n\u00e8bres\u00bb de la \u00abgrand Nuit\u00bb ([sic]; C, 425) o\u00f9 il a \u00e9t\u00e9 plong\u00e9, que le soleil brille derechef sur un \u00abhomme nouveau\u00bb: \u00abLa premi\u00e8re phase de ma vie a \u00e9t\u00e9 finie. La conscience, exced\u00e9e d'ombres, se r\u00e9veille, lentement formant un homme nouveux, et doit, retrouver mon R\u00eave apr\u00e8s la cr\u00e9ation de ce dernier. Cela durera quelques ann\u00e9es pandant lesquelles j'ai \u00e0 revivre la vie de l'humanit\u00e9 depuis son enfance et prenant consience d'elle m\u00eame.\u00bb ([sic]; C, 425) Cette seconde et derni\u00e8re crise, la plus pathologique, nou\u00e9e autour du geste m\u00eame d'\u00e9crire, rel\u00e8gue le pass\u00e9 dans des ombres obscures, y compris un po\u00e8me en prose au titre et \u00e0 la pens\u00e9e aussi h\u00e9g\u00e9liens que \u00abLe Ph\u00e9nom\u00e8ne futur\u00bb, r\u00e9dig\u00e9 en 1864.\n\nL'idiomatique propre \u00e0 Mallarm\u00e9 compose avec ses rimes en -ur une \u00abNotion\u00bb capable de relier \u00abIgitur\u00bb, conclusion logique qui est aussi un personnage, et la notion de futur. Des textes plus tardifs comme \u00abLa D\u00e9claration foraine\u00bb, \u00e9crite vers 1887, r\u00e9\u00e9crit encore le th\u00e8me du \u00abPh\u00e9nom\u00e8ne futur\u00bb. \u00abIgitur\u00bb, ce \u00abbeau conte\u00bb extrait du vieux \u00abmal\u00bb (C, 440), comm\u00e9more la crise de la rencontre avec le N\u00e9ant, tandis que la correspondance prend acte de la d\u00e9cision d'accepter humblement de n'\u00eatre qu'un \u00ablitt\u00e9rateur\u00bb et non un philosophe h\u00e9g\u00e9lien ou un h\u00e9ros de l'Esprit absolu. Ceci se redouble d'un travail de fond sur la linguistique, nouvelle science sur laquelle l'\u0152uvre doit asseoir ses fondements. La crise a pourtant laiss\u00e9 \u00e0 Mallarm\u00e9 un profond scepticisme sur la possibilit\u00e9 d'un commencement radical; il a appris d'un tel bouleversement qui l'a amen\u00e9 aux portes de la folie et du mutisme que l'\u0152uvre \u00e0 venir ne pouvait \u00eatre pens\u00e9e que sur le mode inchoatif:\n\nCes heures critiques me permettent de revoir par \u00e9clairs ce qui fut mon r\u00eave de quatre ann\u00e9es, tant de fois compromis. Je le tiens, \u00e0 peu pr\u00e8s.\n\nMais commencer de suite, non. D'abord, il faut que je me donne le talent requis, et que ma chose, m\u00fbrie, immuable, devienne instinctive; presque ant\u00e9rieure, et non d'hier. (C, 509)\n\nLa distinction est cruciale: l'\u0152uvre \u00ab\u00e0 venir\u00bb ne peut \u00eatre \u00e9labor\u00e9e que si la r\u00e9serve d'\u00e9nergie linguistique, le \u00abtalent\u00bb accumul\u00e9, peut appara\u00eetre comme un vrai pass\u00e9, une ant\u00e9riorit\u00e9 ancr\u00e9e dans la personne de l'\u00e9crivain. Il faut en quelque sorte cr\u00e9er un r\u00e9flexe qui se d\u00e9guisera en archa\u00efsme avant de jeter les d\u00e9s et de diss\u00e9miner les lettres. Toute la probl\u00e9matique derridienne de la diss\u00e9mination se glisse d\u00e9j\u00e0 dans cette dramaturgie qui combine savoir et ruse, une ruse qui tente de jouer avec l'inconscient de la langue.\n\nLe futur incarnerait donc ce qui \u00abtarde\u00bb dans la promesse de l'\u0152uvre, un potentiel toujours recharg\u00e9 aux sources du pass\u00e9 qui se reconstitue au fur et \u00e0 mesure. Ce th\u00e8me d\u00e9borde l'id\u00e9e centrale chez Mallarm\u00e9 que le pr\u00e9sent n'existe pas. La non-existence du pr\u00e9sent, affirm\u00e9e en des termes cat\u00e9goriques, n'exclut pas un \u00ab\u00e9v\u00e9nement\u00bb myst\u00e9rieux, comme le confirme ce passage de \u00abL'action restreinte\u00bb, dans Quant au Livre:\n\nLe suicide ou abstention, ne rien faire, pourquoi? \u2013 Unique fois au monde, parce qu'en raison d'un \u00e9v\u00e9nement toujours que j'expliquerai, il n'est pas de Pr\u00e9sent, non \u2013 un pr\u00e9sent n'existe pas... Faute que se d\u00e9clare la Foule, faute \u2013 de tout. Mal inform\u00e9 celui qui se crierait son propre contemporain, d\u00e9sertant, usurpant, avec impudence \u00e9gale, quand du pass\u00e9 cessa et que tarde un futur ou que les deux se remm\u00ealent perplexement en vue de masquer l'\u00e9cart.\n\nSi le futur existe, il reste ouvert \u00e0 l'avenir, comme certains po\u00e8mes le sugg\u00e8rent, ainsi \u00abToast fun\u00e8bre\u00bb:\n\nCette foule hagarde! elle annonce: Nous sommes\n\nLa triste opacit\u00e9 de nos spectres futurs.\n\nOu encore, l'ouverture de l'\u00ab\u00c9ventail\u00bb de Madame Mallarm\u00e9:\n\nAvec comme pour langage\n\nRien qu'un battement aux cieux\n\nLe futur vers se d\u00e9gage\n\nDu logis tr\u00e8s pr\u00e9cieux\n\n\u00c9galement dans la conclusion du \u00abTombeau d'Edgar Poe\u00bb:\n\nCalme bloc ici-bas chu d'un d\u00e9sastre obscur\n\nQue ce granit du moins montre \u00e0 jamais sa borne\n\nAux noirs vols du Blasph\u00e8me \u00e9pars dans le futur.\n\nCe th\u00e8me revient sous une forme l\u00e9g\u00e8re et \u00e9rotique dans un \u00abSonnet\u00bb d\u00e9di\u00e9 \u00e0 M\u00e9ry Laurent:\n\nLe sais-tu, oui! pour moi voici des ans, voici\n\nToujours que ton sourire \u00e9blouissant prolonge\n\nLa m\u00eame rose avec son bel \u00e9t\u00e9 qui plonge\n\nDans autrefois et puis dans le futur aussi.\n\nCharm\u00e9 par la gr\u00e2ce sensuelle de sa ma\u00eetresse, Mallarm\u00e9 lui offre lyriquement et \u00e9rotiquement une premi\u00e8re version de la c\u00e9l\u00e8bre \u00e9pitaphe que Rilke avait fait inscrire sur sa pierre tombale: \u00abRose, \u00f4 pure contradiction, joie de n'\u00eatre le sommeil de personne sous tant de paupi\u00e8res.\u00bb\n\nS'il faut conclure, puisque j'ai commenc\u00e9 par une citation de Beckett, ce ne peut \u00eatre qu'avec lui que je terminerai. Beckett a su exprimer fortement le m\u00e9lange de d\u00e9sespoir et de confuses esp\u00e9rances que rec\u00e8le le concept du \u00abfutur\u00bb. Selon lui, nous sommes tous \u00abaccul\u00e9s \u00e0 des futurs\u00bb, emport\u00e9s de mani\u00e8re d\u00e9risoire en des guerres futiles, des pol\u00e9miques absurdes, des exils al\u00e9atoires, et m\u00eame le retour autrefois d\u00e9sir\u00e9 ne pourra mettre fin \u00e0 la nostalgie: \u00ab\u00c0 moi maintenant le d\u00e9part, la lutte et le retour peut-\u00eatre, \u00e0 ce vieillard qui est moi ce soir, plus vieux que je ne le serai jamais. Me voici accul\u00e9 \u00e0 des futurs.\u00bb\n\nAccul\u00e9, vraiment? Non, pas tout \u00e0 fait. Reste l'avenir et son babil joyeux, les Ah! et les Oh! jubilatoires de son enfant monstrueux.\n\n. Samuel Beckett, Fin de partie, Paris, Les \u00c9ditions de Minuit, 1957, p. 69.\n\n. Cf. J.-M. Rabat\u00e9, \u00abThe \"Mujic of the Footure\": Future, Ancient, Fugitive\u00bb, dans Futures of Jacques Derrida, op. cit., p. 179-200.\n\n. Cf. supra J. Derrida, \u00abNo apocalypse, not now (\u00e0 toute vitesse, sept missives, sept missiles)\u00bb, dans Psych\u00e9. Inventions de l'autre, op. cit., p. 363-370.\n\n. St\u00e9phane Mallarm\u00e9, Correspondance, Bertrand Marchal (\u00e9d.), Paris, Gallimard, coll. \u00abFolio\u00bb, 1995, p. 425. D\u00e9sormais abr\u00e9g\u00e9 en C, suivi du num\u00e9ro de la page. L'orthographe atypique de cette lettre a \u00e9t\u00e9 silencieusement normalis\u00e9e dans l'\u00e9dition de la Pl\u00e9iade de Bertrand Marchal; cf. Mallarm\u00e9, \u00abCorrespondance choisie\u00bb, dans \u0152uvres compl\u00e8tes, I, Bertrand Marchal (\u00e9d.), Paris, Gallimard, coll. \u00abBiblioth\u00e8que de la Pl\u00e9iade\u00bb, 1998, p. 742.\n\n. Je m'efforce de reproduire ici la typographie de l'\u00e9dition de Cosmopolis telle que l'on peut la trouver dans La premi\u00e8re \u00e9dition d'Un Coup de d\u00e9s par St\u00e9phane Mallarm\u00e9 pr\u00e9sent\u00e9e par David Mus, Dijon, \u00c9ditions du Tiroir, 1996. Cette page 7 correspond \u00e0 la page 425 de la Revue Internationale. Cf. aussi S. Mallarm\u00e9, \u0152uvres compl\u00e8tes, I, op. cit., p. 399.\n\n. J. Derrida, De la grammatologie, Paris, Les \u00c9ditions de Minuit, coll. \u00abCritique\u00bb, 1967, p. 14.\n\n. J. Derrida, Mal d'Archive. Une impression freudienne, Paris, Galil\u00e9e, coll. \u00abIncises\u00bb, 1995, p. 109.\n\n. W. Benjamin, \u00abTh\u00e8ses sur la philosophie de l'histoire\u00bb, tr. fr. Maurice de Gandillac, dans L'Homme, le langage et la culture, Paris, Deno\u00ebl\/Gonthier, 1974, p. 196.\n\n. Je renvoie \u00e0 l'excellente traduction anglaise du Vocabulaire europ\u00e9en des philosophies. Dictionnaire des intraduisibles, Barbara Cassin (dir.), Paris, \u00c9ditions du Seuil\/Dictionnaire Le Robert, 2004. Dans l'article \u00abPr\u00e9sent\u00bb, Pascal David insiste sur le fait que l'avenir inclut ce qui pourrait \u00eatre et que le futur nomme ce qui va \u00eatre. L'opposition entre le possible et le r\u00e9el ne me semble cependant pas \u00e9puiser ce couple s\u00e9mantique idiomatique. Cf. Dictionary of Untranslatables, Emily Apter, Jacques Lezra et Michael Wood (\u00e9ds), Princeton, Princeton University Press, 2014, p. 849-850.\n\n. Article \u00abFutur\u00bb, dans \u00c9mile Littr\u00e9, Le Littr\u00e9 (XM Littr\u00e9, v. 2), Dictionnaire de la langue fran\u00e7aise, texte int\u00e9gral en ligne, n. p.\n\n. Blaise Pascal, Pens\u00e9es, Michel Le Guern (\u00e9d.), Paris, Gallimard, coll. \u00abFolio\u00bb, 1977, vol. 1, p. 82.\n\n. Michel de Montaigne, Les essais, Jean Balsamo, Michel Magnien et Catherine Magnien-Simonin (\u00e9ds), Paris, Gallimard, coll. \u00abBiblioth\u00e8que de la Pl\u00e9iade\u00bb, 2007, p. 21.\n\n. Ibid., p. 108.\n\n. J. Joyce, \u00abLa Gr\u00e2ce\u00bb, dans Dublinois, tr. fr. J. Aubert, dans \u0152uvres, t. I, op. cit., p. 263. Les italiques sont dans le texte.\n\n. B. Pascal, Pens\u00e9es, op. cit., vol. II, p. 11.\n\n. Ibid., p. 12.\n\n. Ibid.\n\n. B. Pascal, \u00abTrait\u00e9 du triangle arithm\u00e9tique\u00bb, dans \u0152uvres compl\u00e8tes, Louis Lafuma (\u00e9d.), Paris, Seuil, 1963, p. 57.\n\n. Ibid., p. 58. Les italiques sont dans le texte.\n\n. Martin Heidegger, Sein und Zeit, T\u00fcbingen, Max Niemeyer, 1979, p. 4. La citation est en fran\u00e7ais dans le texte, tir\u00e9e de Blaise Pascal, Pens\u00e9es et opuscules, L\u00e9on Brunschvicg (\u00e9d.), Paris, Hachette, 1924, p. 169.\n\n. B. Pascal, Pens\u00e9es et opuscules, op. cit., p. 195.\n\n. Cf. Nicholas Rescher, Pascal's Wager: A Study of Practical Reasoning in Philosophical Theology, Notre Dame, University of Notre Dame Press, 1985.\n\n. B. Pascal, Pens\u00e9es, op. cit., vol. II, p. 13-15.\n\n. Cf. P. de Man, All\u00e9gories de la lecture. Le langage figur\u00e9 chez Rousseau, Nietzsche, Rilke et Proust, tr. fr. Thomas Trezise, Paris, Galil\u00e9e, coll. \u00abLa philosophie en effet\u00bb, 1989 [1979], p. 15.\n\n. B. Pascal, Pens\u00e9es, op. cit., vol. I, p. 76.\n\n. Ibid., vol. I, p. 157.\n\n. Ibid.\n\n. Les \u00e9diteurs des Essais dans la Pl\u00e9iade traduisent ainsi: \u00abLes bienfaits sont agr\u00e9ables dans la mesure o\u00f9 l'on estime pouvoir s'en revancher, mais quand ils sont par trop excessifs, c'est la haine, non la reconnaissance, qu'on \u00e9prouve en retour.\u00bb (Montaigne, Les Essais, op. cit., p. 985.)\n\n. Ibid.\n\n. Ibid.\n\n. B. Pascal, Pens\u00e9es, op. cit., vol. I, p. 81-82.\n\n. B. Pascal, Pens\u00e9es, op. cit., vol. I, p. 180.\n\n. Ibid.\n\n. Pierre Force, Le probl\u00e8me herm\u00e9neutique chez Pascal, Paris, Vrin, 1989, p. 99-101.\n\n. B. Pascal, Pens\u00e9es, op. cit., vol. II, p. 111.\n\n. B. Pascal, Pens\u00e9es, op. cit., vol. II, p. 40.\n\n. Ibid., vol. I, p. 216.\n\n. Cf. ibid., vol. I, p. 325.\n\n. Ibid., vol. I, p. 221.\n\n. B. Pascal, Pens\u00e9es, op. cit., vol. II, p. 92.\n\n. M. Heidegger, \u00abPourquoi des po\u00e8tes?\u00bb, dans Chemins qui ne m\u00e8nent nulle part, tr. fr. Wolfgang Brokmeier, Paris, Gallimard, 1962, p. 249. Traduction modifi\u00e9e. D\u00e9sormais abr\u00e9g\u00e9 en CH, suivi du num\u00e9ro de la page. M. Heidegger, Holzwege, Francfort-sur-le-Main, Vittorio Klostermann, 1950, p. 302. D\u00e9sormais abr\u00e9g\u00e9 en H, suivi du num\u00e9ro de la page.\n\n. Cit\u00e9 par M. Heidegger dans Chemins qui ne m\u00e8nent nulle part, op. cit., p. 250.\n\n. Rainer Maria Rilke, Les \u00c9l\u00e9gies de Duino. Les Sonnets \u00e0 Orph\u00e9e, tr. fr. Joseph-Fran\u00e7ois Angelloz, Paris, Garnier-Flammarion, 1992, p. 93. Voici la traduction d'Angelloz: \u00abVois, je vis. De quoi? Ni l'enfance, ni l'avenir\/ ne diminuent... Une existence surabondante\/ jaillit dans mon c\u0153ur.\u00bb\n\n. J. Derrida, Che cos'\u00e8 la poesia, \u00e9dition quadrilingue, Berlin, Brinkmann und Bose, 1990.\n\n. M. Heidegger, \u00abWhat Are Poets For?\u00bb, dans Poetry, Language, Thought, tr. angl. Albert Hofstadter, New York, Harper and Row, 1971, p. 142.\n\n. Roger Dragonetti, Un fant\u00f4me dans le kiosque. Mallarm\u00e9 et l'esth\u00e9tique du quotidien, Paris, Seuil, 1992.\n\n. Friedrich Nietzsche, Derni\u00e8res Lettres, tr. fr. Catherine Perret, Paris, Rivages, 1989, p. 56.\n\n. Dans Les mots anglais, \u00e0 propos de \u00abSingularit\u00e9s\u00bb linguistiques, Mallarm\u00e9 note que, en anglais, \u00abhazard ne repr\u00e9sente qu'un jeu de HASARD\u00bb (\u0152uvres compl\u00e8tes, II, op. cit., p. 1044) et ajoute plus loin que le mot \u00abHazard\u00bb, venu comme d'autres de l'arabe par le fran\u00e7ais, a \u00abgard\u00e9 l'orthographe intacte\u00bb. (Ibid., p. 1080.)\n\n. Cf. Yves Bonnefoy, \u00abL'Or du futile\u00bb, pr\u00e9face \u00e0 S. Mallarm\u00e9, Vers de circonstance, B. Marchal (\u00e9d.), Paris, Gallimard, 1996, p. 7-52.\n\n. Ibid., p. 37.\n\n. \u00abPourquoi? Parce que le futile, aussi utile soit-il, n'est pas n\u00e9cessairement ce qui s'impose avec \u00e9vidence, quand on est tout de m\u00eame un Mallarm\u00e9.\u00bb (Y. Bonnefoy, \u00abL'Or du futile\u00bb, dans S. Mallarm\u00e9, Vers de circonstance, op. cit., p. 38.)\n\n. Hannah Arendt, Condition de l'homme moderne, tr. fr. Georges Fradier, Paris, Sciences Humaines-Agora, 2002, p. 189.\n\n. Samuel Beckett, Malone meurt, Paris, UGE, coll. \u00ab10\/18\u00bb, 1970, p. 102.\n\n. S. Mallarm\u00e9, \u0152uvres compl\u00e8tes, II, op. cit., p. 217.\n\n. Ibid., I, op. cit., p. 27.\n\n. Ibid., p. 30.\n\n. Ibid., p. 38.\n\n. Ibid., p. 67.\n\n. S. Beckett, \u00abLe Calmant\u00bb, dans Nouvelles et Textes pour rien, Paris, Les \u00c9ditions de Minuit, 1955, p. 45.\n\n. Un bel exemple de l'image de l'enfant \u00e0 venir dans un futur de science-fiction, enfant merveilleux et monstrueux \u00e0 la fois, est donn\u00e9 dans le roman de Fran\u00e7ois Dominique, Dans la chambre d'Iselle, Paris, Verdier, coll. \u00abJaune\u00bb, 2015.\n\n# DU M\u00caME AUTEUR\n\nThink, Pig! Beckett at the Limit of the Human, New York, Fordham University Press, 2016.\n\nThe Pathos of Distance: Affects of the Moderns, New York, Bloomsbury, 2016.\n\nThe Cambridge Introduction to Literature and Psychoanalysis, Cambridge, Cambridge University Press, 2014.\n\nCrimes of the Future: Theory and its Global Reproduction, New York, Bloomsbury, 2014.\n\n\u00c9tant donn\u00e9s: 1) l'art, 2) Le crime. La modernit\u00e9 comme sc\u00e8ne du crime, Dijon, Les presses du R\u00e9el, 2010.\n\nThe Ethics of the Lie, trad. angl. Suzanne Verderber, New York, The Other Press, 2007.\n\n1913: The Cradle of Modernism, Oxford, Wiley-Blackwell, 2007. (Traduction chinoise: Shanghai, Shanghai Foreign Education Press, 2013).\n\nGiven: 1\u00ba Art, 2\u00ba Crime: Modernity, Murder and Mass Culture, Brighton, Sussex Academic Press, 2007.\n\nLacan literario. La experiencia de la letra, Mexico, Siglo XXI Editores, 2006.\n\nTout dire ou ne rien dire. Logiques du mensonge, Paris, Stock, 2005 (format Kindle).\n\nThe Future of Theory, Oxford, Wiley-Blackwell, 2002.\n\nJames Joyce and the Politics of Egoism, Cambridge, Cambridge University Press, 2001.\n\nJacques Lacan: Psychoanalysis and the Subject of Literature, Londres, Palgrave MacMillan, 2001.\n\nThe Ghosts of Modernity, Gainesville, University Press of Florida, 1996 (r\u00e9\u00e9d., 2010).\n\nLa p\u00e9nulti\u00e8me est morte. Spectrographies de la modernit\u00e9 (Mallarm\u00e9, Breton, Beckett et quelques autres), Seyssel, Champ Vallon, 1993.\n\nJames Joyce, Paris, Hachette, 1993.\n\nJoyce upon the Void: the Genesis of Doubt, Londres, Palgrave MacMillan, 1991.\n\nThomas Bernhard, Paris, Marval, 1991. (Traduction espagnole: Madrid, Debats, 1992.)\n\nJames Joyce: Authorized Reader, Baltimore, The Johns Hopkins University Press, 1991.\n\nMaurice Daranti\u00e8re. Les ann\u00e9es vingt, Dijon, Ulysse fin de si\u00e8cle, 1988\n\nLa beaut\u00e9 am\u00e8re. Fragments d'esth\u00e9tique: Barthes, Broch, Mishima, Rousseau, Seyssel, Champ Vallon, 1986.\n\nLanguage, Sexuality and Ideology in Ezra Pound's Cantos, Londres, Palgrave MacMillan, 1986.\n\nJames Joyce. Portrait de l'auteur en autre lecteur, Petit-R\u0153ulx, Cistre, 1984.\n\nDirection d'ouvrages collectifs et de num\u00e9ros de revues\n\n1922: Culture, Politics and Literature, Cambridge, Cambridge University Press, 2015.\n\nArt, or Listen to the Silence. Soun-Gui Kim on John Cage, Jacques Derrida, and Jean-Luc Nancy, avec Aaron Levy, Philadelphie, Slought, 2014.\n\nThe Cambridge Introduction to Literature and Psychoanalysis, Cambridge, Cambridge University Press, 2014.\n\nHandbook of Modernism Studies, Oxford, Wiley-Blackwell, 2013.\n\nH\u00e9l\u00e8ne Cixous\u2014Ex-Cities, avec Aaron Levy, Philadelphie, Slought, 2006.\n\nJacques Lacan, Paris, Bayard, 2005.\n\nWilliam Anastasi's Pataphysical Society, avec Aaron Levy, Philadelphie, Slought, 2005.\n\nPalgrave Advances to Joyce Studies, Londres, Palgrave MacMillan, 2004.\n\nInterfaces, \u00abArchitecture Against Death\u00bb (sur Arakawa et Madeline Gins), nos 21\/22, 2004.\n\nOn the Diagram: Marjorie Welish, avec Aaron Levy, Philadelphie, Slought, 2003.\n\nThe Cambridge Companion to Jacques Lacan, Cambridge, Cambridge University Press, 2003.\n\nLacan in America, New York, The Other Press, 2000.\n\nWriting the Image after Roland Barthes, Philadelphie, University of Pennsylvania Press, 1997.\n\nL'\u00e9thique du don. Jacques Derrida et la question du don, avec Michael Wetzel, Paris, M\u00e9tailli\u00e9, 1992. (Traduction en allemand: Ethik der Gabe, Berlin, Akademie Verlag, 1993.)\n\nJe rassemble les membres d'Osiris. Ezra Pound, Auch, Tristram, 1989.\n\nBeckett avant Beckett, Paris, Presses de l'\u00c9cole normale sup\u00e9rieure, 1985.\n\nTraductions et \u00e9ditions critiques\n\nPr\u00e9sentation et introduction de James Joyce, Works, Londres, Anthem Press, 2016.\n\nAppareil critique de Exils de James Joyce, traduction nouvelle de Jean-Michel Desprats, Paris, Gallimard, 2012.\n\nEau lourde, de Martin Amis, Paris, Gallimard, 2000.\n\nAngelus novus, de Beryl Schlossman, Dijon, Ulysse fin de si\u00e8cle, 1995.\n\nParticipation \u00e0 l'\u00e9dition d'Ulysse de James Joyce, t. I, Paris, Gallimard, coll. \u00abBiblioth\u00e8que de la Pl\u00e9iade\u00bb, 1995. \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n_the_ Australian   \nBeekeeping  \nManual\nThanks to Erica Siegel, wildlife photographer and native fauna enthusiast, for writing the section on native solitary bees in Chapter 14: Native Bees.\n\nFirst published 2015\n\nExisle Publishing Pty Ltd\n\n'Moonrising', Narone Creek Road, Wollombi,\n\nNSW 2325, Australia\n\nP.O. Box 60-490, Titirangi, Auckland 0642,\n\nNew Zealand\n\nwww.exislepublishing.com\n\nCopyright \u00a9 2015 in text: Robert Owen  \nCopyright \u00a9 2015 in photographs and illustrations: as credited on p. 335\n\nRobert Owen asserts the moral right to be identified as the author of this work.\n\nAll rights reserved. Except for short extracts for the purpose of review, no part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means, whether electronic, mechanical, photocopying, recording or otherwise, without prior written permission from the publisher.\n\nA CiP record for this book is available from the National Library of Australia.\n\nISBN 978 1 921966 88 0  \nePub ISBN 978 1 77559 247 1\n\nDesigned by Tracey Gibbs\n\nDisclaimer\n\nWhile this book is intended as a general information resource and all care has been taken in compiling the contents, neither the author nor the publisher and their distributors can be held responsible for any loss, claim or action that may arise from reliance on the information contained in this book.\n\nContents\n\nIntroduction\n\nChapter 1 Life history of the honey bee\n\n\\- Castes of the honey bee\n\n\\- The Western honey bee and its subspecies\n\n\\- Lifecycle of the honey bee\n\n\\- Mating and drone congregation areas\n\n\\- Life as a young adult bee\n\n\\- Collecting nectar and pollen\n\n\\- Communications and navigation\n\n\\- Honey bees as superorganisms\n\nChapter 2 Basic equipment and protective clothing\n\n\\- Equipment\n\n\\- Protective clothing\n\nChapter 3 Constructing hives\n\n\\- Constructing a hive\n\n\\- Other parts of a hive\n\nChapter 4 Frames and how to build them\n\n\\- Frames\n\n\\- Cleaning used frames\n\n\\- Attaching foundation to the frame\n\n\\- Using wax or plastic foundation\n\n\\- Using plastic sides for frames\n\nChapter 5 Your first bees\n\n\\- Obtaining your first bees\n\n\\- What kind of bees should I keep?\n\n\\- Should I buy an established hive or catch a swarm?\n\n\\- Swarms and swarm catching\n\n\\- Transferring an established colony\n\n\\- Moving hives\n\n\\- General maintenance and inspections during the first year\n\nChapter 6 Locating your bees\n\n\\- Locating hives on your property\n\n\\- Hive position in summer and winter\n\n\\- Urban beekeeping\n\n\\- Feeding bees water\n\nChapter 7 Spring and summer management\n\n\\- Inspections\n\n\\- Managing the build-up of bees in your hive to maximise honey production\n\n\\- Managing the honey flow\n\n\\- Merging colonies\n\n\\- Feeding syrup\n\n\\- Sugar candy\n\n\\- Feeding pollen or pollen substitute\n\n\\- Swarms\n\n\\- Providing hives for pollination\n\n\\- Collecting pollen\n\nChapter 8 Autumn and winter management\n\n\\- Autumn\n\n\\- The winter shutdown\n\n\\- Winter\n\nChapter 9 Extracting\n\n\\- Using a sieve to harvest honey\n\n\\- Use of an extractor to harvest honey\n\n\\- Choosing an extractor\n\n\\- Type of extractor to buy\n\n\\- Comb honey\n\n\\- Reuse of wax foundation\n\n\\- Colour of wax\n\nChapter 10 Preparing honey for sale and competitions\n\n\\- Preparing honey for sale\n\n\\- Preparing honey for competitions\n\nChapter 11 Rearing queens\n\n\\- The three reasons for a colony to produce a new queen\n\n\\- Hive splitting\n\n\\- Queen candy\n\nChapter 12 Sustainable beekeeping\n\n\\- Natural beekeeping hives\n\nChapter 13 The bee-friendly garden\n\n\\- Bee-friendly flowering plants\n\n\\- Further considerations for the bee-friendly garden\n\n\\- A garden is a work in progress\n\nChapter 14 Native bees\n\n\\- Social bees\n\n\\- Solitary bees\n\nChapter 15 Diseases of the honey bee\n\n\\- Diseases of brood\n\n\\- Good management practices to minimise disease in the colony\n\n\\- American Foul Brood (AFB)\n\n\\- European Foul Brood (EFB)\n\n\\- Sacbrood virus (SBV)\n\n\\- Chalkbrood\n\n\\- Adult bee diseases258\n\nChapter 16 Parasites of the honey bee\n\n\\- _Varroa_\n\n\\- _Tropilaelaps_ mites \u2014 the Asian mite\n\nChapter 17 Pests of the honey bee\n\n\\- Wax Moth\n\n\\- Small Hive Beetle (SHB)\n\n\\- Braula fly\n\nChapter 18 Other problems of the honey bee\n\n\\- Pesticide and fungicide poisoning\n\n\\- Chilled brood\n\n\\- Overheating of the hive\n\n\\- Damp hives\n\n\\- Cane toads\n\n\\- Ants\n\n\\- Mice\n\n\\- Colony Collapse Disorder (CCD)\n\nChapter 19 Other types of bees and wasps\n\n\\- The Asian honey bee _(Apis cerana)_\n\n\\- Bumblebees\n\n\\- Wasps\n\n\\- Asian hornets\n\n\\- Other races of _Apis mellifera_\n\n\\- Other species of Asian bees\n\nAppendix: Sizes of frames and boxes\n\nGlossary\n\nAcknowledgements and picture credits\n\nBibliography\n\nIndex\nIntroduction\n\nThe majority of beekeeping books currently for sale are either published in the United States or in the United Kingdom where beekeeping conditions are very different to those experienced in Australia. While there have been some excellent books previously published in Australia, I felt that the time was right for an alternative set of insights and experiences which I hoped would be of both interest and assistance to the beginner and the more experienced beekeeper. During the many years I have been involved in running Bob's Beekeeping Supplies, I have had the opportunity to speak with numerous beekeepers about their varied beekeeping experiences as well as listening and responding to their practical difficulties. This book is largely a culmination of those varied discussions. It is my hope that it will address and answer many of the questions being asked by other hobby beekeepers, particularly those who do not have access to either the combined experience of a club or an experienced mentor.\n\nA keen beekeeper will work to obtain information from many sources on how best to keep bees. These may include books, the Internet, YouTube, local beekeeping clubs and talking with other beekeepers. Glance through a number of different beekeeping books or have a look at a few online sites and you will notice that beekeeping practices around the world, although they differ to some degree in a number of particular areas, use many of the tools and techniques adopted and used in Australia. This reflects a common beekeeping heritage that began its evolution in Europe and the United States. Between 1851 and 1865 there were five inventions that changed traditional beekeeping management and practice from an inefficient backyard hobby to the modern more efficient practices that are used today. These inventions are:\n\n0.1: Drinking honey.\n\n0.2: Bee on a climbing rose.\n\n_Bee space_ \u2013 discovered by Reverend Lorenzo Lorraine Langstroth (the father of American beekeeping) in 1851.\n\n_The Langstroth hive_ \u2014 patented in 1852 by Reverend Lorenzo Lorraine Langstroth.\n\n_The smoker_ \u2013 invented by Moses Quinby in 1853 in the United States.\n\n_Comb foundation_ \u2013 invented by the German beekeeper J. Mehring in 1857.\n\n_The centrifugal extractor_ \u2013 invented in 1865 by Major D. Hruschka of Venice, Italy.\n\nThe Reverend Langstroth's invention made it possible for the first time to remove frames from a hive either to inspect or to extract the honey. Previously hives were often completely destroyed to obtain the honey collected by the occupying colony of bees.\n\nThe major benefits of the Langstroth hive are the ease and convenience it offers the beekeeper. This is perhaps not surprising as Langstroth himself saw bees as the god-given providers of sweet treats for mankind. The advantages of the Langstroth hive are:\n\n\u2022 The use of removable frames enables the beekeeper to readily inspect the colony for various reasons including checks for disease, the presence of the queen, brood pattern and numbers, swarm control and monitoring honey build-up.\n\n\u2022 The design enables a beekeeper to increase or decrease the size of the hive by adding or removing supers, thus increasing the area within the hive that bees can store honey.\n\n\u2022 It allows frames with drawn comb to be reused.\n\n\u2022 It enables the queen to be kept in a separate chamber or brood box away from stored honey.\n\nThere is an Australian view on how best to manage bees and many beekeepers in this country often disregard techniques and practices used successfully in other countries. You might have picked up beekeeping techniques along the way or taken advantage of the more formal education offered by local clubs. Whichever way your knowledge has been gained, I would strongly recommend that you look at all techniques used globally and select the ones that you believe are best for your own and your bees' situation. Do not become dogmatic about technique! In the early stages of beekeeping you will be bombarded with differing views on management and various hive manipulation techniques but try to absorb the information, and as your knowledge and experience grows, you can adopt or discard the information that suits you and your own hives.\n\nThis book, as well as describing the more traditional methods of keeping bees in Langstroth hives, also includes information on other types of hives including the natural beekeeping techniques that are becoming increasingly popular among hobby beekeepers. Natural beekeepers usually have strong philosophical opinions as to why their techniques are best for the bees in their care. There are many advantages and some disadvantages of natural beekeeping methods, both for bees and for the beekeeper. The chapter on natural beekeeping will, I hope, provide useful information for those wishing to join this growing movement.\n\nBeekeeping is a rewarding and addictive hobby. I have not yet met a beekeeper who does not find a hive and the activities of the bees that call it home, with all its variety and complexity, to be a topic of fascination. It's a good topic of discussion with other beekeepers and friends, as well as a source of continual learning. Recently beekeeping has become increasingly popular. This is due partly to increased publicity both in Australia and overseas of the increasing problems bees are facing due to environmental challenges and beekeeping practices. Another aspect is that the trend for more sustainable and lower-environmental-impact living is increasingly gaining a foothold. Many people are developing an interest in growing their own food or in sourcing organic produce untouched by chemicals and other artificial elements. The increased knowledge of food plants and their pollination requirements usually follows with some recognition of the value that the amazing honey bee adds to our lives. Not only does the bee produce the honey we so love to eat but it has unparalleled supremacy as a pollinator of the fruits and vegetables we enjoy on a daily basis. With a growing realisation of the importance of the bee it is a small step for someone to then become interested in keeping their own hives. If your interest in keeping bees is new, beware, once you have been initiated by your first sting, the love of bees is forever in your blood.\n\nFor me, writing this book has been a learning experience as I set out to try for myself most of the equipment and techniques explained in the following pages. I found that much of what other beekeepers had told me worked as they had explained, although equally some techniques did not work very well and I have not recommended their use in this book. Many common overseas practices that are not used in Australia also worked well and should be adopted by more beekeepers here; I have recommended these in the relevant parts of this book.\n\nSome may feel the information presented in the following pages is overly influenced by my own experiences of beekeeping in the south-eastern state of Victoria. Rest assured that the management and practical skills I describe apply equally well throughout the rest of the country but the timing may need some modification due to local climatic conditions.\n\nFinally, this book would not have been possible without the advice and input of many of my fellow beekeepers. The list of people who have generously contributed both time and information is provided at the end of the book. Any errors that have been made are my responsibility either for not fully understanding a topic or for not listening closely enough when techniques and practices were explained to me by those more knowledgeable on that subject than I. There is no 'one' or 'right' way of keeping bees and your way of doing things may differ to mine. Differences among beekeepers should be viewed as an opportunity to expand and develop knowledge and skills. Our hobby is fortunately one where knowledge is never complete and the bees will continue to teach us until we light our last smoker and pick up our last hive tool.\n\nIf any errors are noticed or you have suggestions for improvement please send them to:\n\nPracticalBeekeeping@owen.org.au\n\nBob Owen\n\nEltham, 2015 \n1.\n\nLife history of the honey bee\n\nThere are many books available on the life history, biology and anatomy of the honey bee. The web is also a mine of information for beekeepers wishing to expand their knowledge of this subject. The aim of this chapter is not to provide a detailed biology or life history of the honey bee, but to provide some basic information that can be used to better understand the content of this book. For more detailed information the reader is referred to the many excellent resources listed in the bibliography.\n\nCASTES OF THE HONEY BEE\n\nIn each bee colony there are three castes of bee.\n\n**A SINGLE FEMALE QUEEN**\n\nThere is usually only a single queen in each hive or colony and one of her main roles is to lay eggs. Once an egg has been laid by the queen she has no other role in the upbringing of the brood. The female nursery workers take over all responsibility for feeding and care of the young bee. Queen bees are larger than either female workers or male drones. Queens typically live between one and three years. Contrary to common belief, queens take no part in the decision making of the colony and all major decisions, such as how many drone or worker eggs are laid, or whether to swarm, are actually made collectively by the worker bees.\n\n1.1: A Caucasian bee, Apis mellifera caucasica. Notice the longer, slender abdomen of the queen compared to the female worker bees next to her. This allows the queen to deposit eggs at the bottom of empty brood cells.\n\n1.2: A female worker bee.\n\n1.3: Although this photo appears to show a bee covered in pollen, it is in fact a fl y of the order Diptera. The antennae of a fl y are characterised by an arista, a hair like structure which the photo shows coming off one of the antennal segments. The arista is used to detect changes in temperature and humidity.\n\n**TENS OF THOUSANDS OF FEMALE WORKERS**\n\n1.4: A male drone bee. Notice the larger eyes and the shorter, stubbier abdomen compared to the female worker bees next to him.\n\nA typical colony consists of at least 95 per cent workers and depending on the time of year between 20,000 and 60,000 workers may be present in a hive. Physically, workers are the smallest caste of honey bee. They also have a very limited ability to lay small numbers of unfertilised drone eggs if the colony becomes queenless. Workers perform all of the work in the hive, including rearing brood, cleaning cells, guarding the hive entrance and foraging. During the summer a typical worker lives between 30 and 50 days after emerging from her cell although this period is considerably longer in very cold areas where the bees are unable to leave the colony.\n\n**UP TO ABOUT 1000 MALE DRONES**\n\nThe only known role of a drone is to mate with queens from other colonies. Drones are usually intermediate in size between workers and mature queens and are easily differentiated from workers by their larger eyes and shorter, stubbier abdomen. Drones live about six weeks after emerging from their cells and are usually only found in colonies during the spring and summer.\n\nIt does not take long to be able to tell the difference between female workers and male drones, but queens are usually more difficult to find. They are the largest bee in the colony and can be recognised by their long, tapered abdomen which is used to deposit eggs at the bottom of empty brood cells. The main difficulty faced by the beekeeper when attempting to find a queen is that there is only one of them hiding among tens of thousands of other bees.\n\n1.5: A female worker, queen, male drone \u2014 three castes of honey bees.\n\nTHE WESTERN HONEY BEE AND ITS SUBSPECIES\n\nFirst, a note about the difference between the names Western honey bee and European honey bee. The Western honey bee includes all of the races (or subspecies) of _Apis mellifera_ and is found in Africa, Europe, the Middle East and regions much further east. European honey bees are the races of the Western honey bee, _Apis mellifera,_ found in Europe and parts of the Middle East. In Australia, the name European honey bee is used to denote the Italian, Caucasian, Carniolan and dark German races of bees.\n\nThe Western honey bee, _Apis mellifera,_ is one of twelve currently recognised species of honey bee found throughout the world. In spite of this it produces most of the commercial honey consumed globally. The reasons for its success are twofold. Firstly, after thousands of years of human harvesting and selection the European races of _Apis mellifera_ have been spread around the world and selectively bred for their docility and high honey yield. Secondly, the other eleven honey bee species are geographically isolated in parts of Asia and the Middle East. All of these eleven species have their ancestral home in South East Asia or India.\n\nThe European honey bee was first introduced into Australia in 1822. Since then it has spread to most parts of the country except for the more inhospitable desert regions. The European honey bee races found in Australia today are races or subspecies of _Apis mellifera._ The ancestors of the European honey bee spent most of the last 70,000 years of their existence living in their homelands of Europe, middle Europe, Eastern Europe and the foothills of the Ural Mountains near the Caspian Sea.\n\nThe traditional races of the European honey bee found in Australia are:\n\n\u2022 _Apis mellifera ligustica_ \u2013 Italian bees\n\n\u2022 _Apis mellifera caucasica_ \u2013 Caucasians\n\n\u2022 _Apis mellifera carnica_ \u2013 Carniolans\n\n\u2022 _Apis mellifera mellifera_ \u2013 often inaccurately called the dark German bee. It is in fact native to large areas of Europe from south-east France and Spain to Scandinavia. Although the European honey bees found in Australia are usually a mixture of all four subspecies of bees, feral bees are predominantly _Apis mellifera mellifera_ in their genetic origins.\n\n1.6: The Italian bee, Apis mellifera ligustica, is more yellow or straw-coloured than either Caucasians or Carniolan bees. Notice the dark Caucasian bee to the right of the queen that has drifted in from another hive.\n\n1.7: The Caucasian bee, Apis mellifera caucasica, from the Caucasus Mountains region of the former Soviet Union.\n\n1.8: The Carniolan bee, Apis mellifera carnica,from the region around Austria and the former Yugoslavia in south-eastern Europe.\n\nThe characteristics of these races of bees that are of interest to beekeepers are detailed in Chapter 5: Your First Bees.\n\nThe four races of _Apis mellifera_ found in Australia can interbreed with each other in order to produce viable offspring. Since all four races were introduced into Australia at different times since 1822 there has been extensive interbreeding among them. While the original four races of bee often have a different colour and possess different characteristics, the Australian honey bee is a mongrel mix of the four races. In South Australia, Kangaroo Island beekeepers claim the most pure Italian Ligurian bees globally. DNA and morphological studies of bees across Australia performed by the University of Sydney, Latrobe University in Melbourne and the University of Adelaide show that Kangaroo Island bees are not pure Italians and have significantly interbred with _Apis mellifera mellifera_ and other races.\n\n1.9: Geographic origins of the European honey bee.\n\nLIFECYCLE OF THE HONEY BEE\n\nFor a colony to survive there needs to be a healthy queen laying eggs that develop into adult worker and drone bees. Queens do not live forever and a new queen is also required when the colony swarms, taking with them the existing queen.\n\nThere are three situations in which a new queen is required \u2014 swarming, supersedure and emergency.\n\n**SWARMING**\n\n**PREPARING FOR SWARMING**\n\nTriggered by the increased flow of nectar during the spring and also by the lengthening hours of daylight, the queen starts to lay many more eggs, between 1000 and 2000 eggs per day, causing the number of bees in the colony to increase rapidly. Soon the hive becomes overcrowded and congestion is one of the major triggers for the colony to swarm (but not the only trigger). Although the first signs of swarming noticed by most beekeepers are queen cells appearing on brood frames, a colony's preparations for swarming actually start around a month before this. About three weeks before the colony starts building queen cells, the workers begin constructing drone cells into which male drone eggs are laid by the old queen. Drone cells are larger than their worker cell counterparts as they are required to hold the much bigger drone pupae and are typically located on the lower half of brood frames.\n\n1.10: The days that the drones go on their first mating flights and the queen goes on her mating flight are about the same. This means that the colony starts preparing drone cells about 25 days before the first queen cells can be seen. Once you first see drone cells inside a hive you know that the colony will start rearing queens in the next two to three weeks.\n\n1.11: Drone cells in a hive are a sign that the colony is planning to swarm, even before the first queen cells can be seen.\n\nA few weeks after the queen has started to lay more drone eggs the workers build about six to eight queen cells, usually at the bottom of the brood frames. Normal worker, drone or honey cells are almost horizontal on the frame; queen cells hang vertically downwards, are much larger and are peanut-shell shaped. Once an egg has been laid it will hatch after three days into a larva or grub. It is during the larval or grub stage that the workers select and feed any potential baby queen larvae large amounts of a high-quality protein-rich diet of a substance known as royal jelly. Royal jelly is produced by the hypopharyngeal glands of worker honey bees. The amount of royal jelly fed to larvae determines whether it becomes a worker or a queen. Worker larvae are fed royal jelly for the first three days of their life whereas larvae selected as potential queens are fed royal jelly throughout the entire larval stage.\n\n**SWARMING**\n\nIt takes about sixteen days after a queen egg has been laid for an adult queen to emerge unaided from her capped cell. Two or three days before the new queen emerges from her cell the older bees in the colony get ready to swarm. During this period the worker bees who are going to swarm gorge themselves with honey and mill around the outside of the hive before flying off with the old queen to a temporary landing place, say on a tree or on a building, usually less than a few hundred metres from the hive. When the swarm has settled at their temporary home, approximately 5 per cent of the scout bees start looking for a new permanent home for the swarm to move to. It is the older bees within the swarm that tend to become scout bees looking for a suitable new nest site. In Australia, about 90 per cent of all swarms that have not been captured by a beekeeper manage to find a new permanent home protected from bad weather. The remainder die, either shortly after swarming perhaps because they have been caught by heavy rainfall in an open area or because they do not find a suitable nest to protect themselves over winter.\n\nThe first of the queens to emerge from her cell will eat some honey, gain strength and will then attempt to locate the other queen cells, chew holes through the cell walls, and sting the remaining queens to death before they can emerge and challenge her. She is aided in her search for rival queens by the peculiar piping sound unhatched queens make shortly before they emerge. The new queen is not always successful in killing the remaining unhatched queens because if she was there would not be any after-swarms, headed by other virgin queens. In fact, after-swarms are frequently reported by beekeepers.\n\n1.12: Swarming and supersedure queen cells are located at the bottom of frames and have approximately the same shape, although supersedure queen cells are often stockier than swarming queen cells.\n\nThe newly emerged virgin queen will remain in the hive for about a week to ten days before her body has developed and strengthened sufficiently for her to go on her nuptial or first mating flight. Mating takes place at drone congregation areas (DCA) described later in this chapter.\n\nThe first swarm of the season, often called the prime swarm, may contain thousands to tens of thousands of worker bees and the old queen. Sometimes the colony from which the initial swarm originated may quickly swarm again and this is called an after-swarm. After-swarms \u2014 or secondary swarms, as they are sometimes called \u2014 are headed by a virgin queen and their departure will leave the colony even further depleted of workers. It occasionally happens that a colony may swarm a third time and this is referred to as a tertiary swarm and leaves behind very few workers in the original colony. Both secondary after-swarms and any tertiary swarms may contain insufficient numbers of workers to survive over the winter. The yearly swarming cycle of the colony will be repeated the following year as both new and old colonies increase in strength and then swarm again.\n\n**SUPERSEDURE**\n\nSupersedure occurs when an old queen is not perceived by the workers as being healthy or is not laying a sufficient number of eggs to keep the colony strong. Similar to the queen-rearing process used for swarming, with supersedure the workers encourage the queen to lay eggs in newly constructed queen cups and from one of these queen cells a new queen will emerge in sixteen days. The difference between supersedure and swarming is that with supersedure the old queen and the new queen both remain together in the colony for some period of time before the old queen dies, possibly of natural causes or due to physical eviction by workers from the hive. Outside of the hive, with no defences or ability to feed herself, she will very quickly die or be eaten. In the case of swarming the old queen permanently leaves the colony a few days before the new queen emerges from her cell.\n\n1.13: Emergency queen cells are usually found distributed around the centre of a brood frame among young brood.\n\n1.14: Drone pupae that have not yet hatched.\n\n1.15: Worker larvae ready to be capped. Capped worker cells are on the right.\n\nThe immediate cause of supersedure is probably diminished pheromone production by the older queen. The reason why the production of queen pheromone decreases is not clear but is more likely to occur with older queens than with younger queens. The workers detect the lowering level of queen pheromone and start the supersedure process. The process by which workers initiate both swarming and supersedure is the same: the workers build queen cells usually at the bottom of brood frames and the old queen lays eggs inside them. The remaining development of the supersedure queen is exactly the same as for a swarm queen.\n\n**EMERGENCY**\n\nThe production of queens by swarming and supersedure are events planned in advance by the workers and enable the queens to emerge in as healthy a state as possible. This process also helps to ensure that they produce strong, viable colonies as a result of swarming or supersedure.\n\nIn the case of emergency queens, however, these are formed when the queen dies suddenly and the normal swarming or supersedure process has not been started. In this situation the workers realise, after only a few hours, due to the lack of queen pheromone in the hive, that there is not a queen in the colony and they will choose some eggs or very recently hatched larvae to nurture into queens. The younger the larvae chosen to replace the missing queen, the healthier will be the new queen produced by this process. This is because older larvae will have less time to feed on royal jelly before their queen cell is capped.\n\nEmergency queen cells may be located anywhere on the frame although they are usually not found at the bottom of the frame. They are produced by widening pre-existing worker bee larvae cells. Emergency queen cells are different to swarm or supersedure queen cells in that the workers select pre-existing horizontal worker larvae cells and then modify them so that the larger queen cell droops down from them. Thus emergency queen cells can easily be recognised from swarm or supersedure queen cells. A common reason that an emergency queen needs to be raised is because the beekeeper has accidentally or deliberately killed or removed a viable queen from the colony.\n\n**DEVELOPMENT OF WORKER AND DRONE LARVAE**\n\nIn contrast to the larval life of the queen, worker and drone larvae are only initially fed high-quality royal jelly. After about three days they are fed a less nutritious diet of bee bread, which is a mixture of pollen, nectar and saliva produced by worker bees for consumption by larvae. It needs to be stressed that when the queen egg is laid there is no genetic, physical or biochemical difference between a female worker egg and a female queen egg. The only known reason for an egg to develop into a queen rather than into a worker is that the larvae selected to become queens are fed copious amounts of high-quality royal jelly during their entire larval feeding stage.\n\n1.16: Stages in the capping of brood cells.\n\n1.16a: Development of a female worker egg, through to larval and pupal stages, and finally emerging on the 21st day as an adult worker bee. The young bee remains inside the egg for approximately three days before hatching as a larva. The larva grows until, on about the ninth day, its cell is capped ready for the larva to change into a pupa.\n\n1.16b: Development of a male drone egg, through to larval and pupal stages, and finally emerging on the 24th day as an adult drone bee. The young bee remains inside the egg for approximately three days before hatching as a larva. The larva grows until, on about the tenth day, a day later than for a female worker larva, its cell is capped ready for the larva to change into a pupa.\n\n1.16c: The development of a queen, through to larval and pupal stages, and finally emerging on the 16th day as an adult queen. The young bee remains inside the egg for approximately three days before hatching as a larva. The larva is fed large amounts of high-quality royal jelly and grows until, on about the eighth day, a day sooner than for a female worker larva, its cell is capped ready for the larva to change into a pupa.\n\nMATING AND DRONE CONGREGATION AREAS (DCA)\n\nA queen will mate in mid-air, between 10 metres and 30 metres above the ground in areas called drone congregation areas or DCA. It is believed that queens and drones select a site to be a drone congregation area based on the visibility of key landmarks such as a large tree or open spaces.\n\nDrone congregation areas are spaced approximately every 2 kilometres apart and drones from many different colonies fly to an area for the opportunity to mate with a queen. Apart from a shorter, stubbier abdomen and absence of a stinger, the distinguishing feature of a drone is his large eyes. These have evolved to see a distant queen when she would otherwise be just a spot on the horizon. The drone's larger compound eyes also give him a wider field of vision. The competition among drones to mate is intense and there is literally a line-up of drones flying behind every queen in a fast-moving comet tail, waiting for the current drone that is mating to finish and fall off.\n\nWhen a drone starts mating, an endophallus (a penis-like structure that remains inverted inside the drone's abdomen prior to mating) is inserted into the queen's sting chamber, inside of which is the queen's vagina. The drone then falls backward causing the body fluids inside his abdomen to be pushed into his endophallus, causing it to further elongate and harden. Sperm is released into the queen and the drone falls further backwards causing the tip of his endophallus to tear off from his abdomen, where it remains temporarily inserted in the queen's sting chamber. With his endophallus and many of his internal organs missing or severely damaged, the drone falls to the ground and dies. The last drone's endophallus that is still attached to the queen when she returns to the hive is often referred to by beekeepers as 'the mating sign'.\n\n1.17: Drone and queen mating in mid-air. The male's endophallus, equivalent to a penis, snaps off during mating and the drone falls to the ground to die.\n\nIf the virgin queen does not mate with a sufficient number of drones on her first mating flight she will go on second and third mating flights to ensure a varied supply of drone sperm from multiple colonies. A typical queen mates with between ten and twenty drones during her nuptial flights. It is believed that a virgin queen will continue to go on mating flights and mate with drones until her oviduct is full with sperm.\n\nQueens do not mate inside their own hive, only at drone congregation areas away from the hive. The chances of a queen mating with a drone from her own colony are small because at any drone congregation area there will be hundreds of drones from other colonies. Also, studies indicate that drones and a virgin queen from the same colony probably go to different drone congregation areas to mate, further minimising the likelihood of inbreeding within the colony.\n\nHoney bees belong to the same order of insects as ants, wasps, bees and saw flies. Most of the insects in this order are unusual, although not unique, in that although female eggs are fertilised with stored drone sperm and have both a mother and a father, male eggs are not fertilised with sperm so have no father, only a grandfather.\n\nThe queen, using her long, thin abdomen and ovipositor (a modified stinger) places the egg perpendicular to the bottom of the cell. After about a day, as the egg develops, it will soften and fall over inside the cell. The outer membrane of the brood egg is a soft membrane, unlike the hard shell of a chicken egg, and the outer membrane will slowly dissolve over a couple of days allowing the larva, which is fully formed by day three inside the egg, to emerge. During the larval stage the nursery bees will feed larvae royal jelly, although feeding will stop once the brood cell has been capped. The larvae will eat this food and will grow over the five days before their cells are capped with a porous wax cover ready for the final stage of the brood's development into pupae. It is during the pupal stage that the brood's body finally develops into a form recognisable as a young bee. At the end of the twelve- to fourteen-day-long pupal stage the young female bee chews away the wax cap of the cell from the inside and pulls herself unaided out of the cell to emerge as a young adult bee.\n\nIf the queen has laid drone eggs in the larger-cell-sized drone comb constructed by the worker bees these larvae will be attended to in the same way by nursery bees. The difference is that whereas both queens and worker bees will have finished growing after five days drone larvae will take one further day before their cells can be capped. The drone also has a longer pupal stage: a worker will emerge from her cell at around 21 days after egg lay whereas a drone will not emerge until around day 24. It will be fed by nursery bees or will help itself to stored supplies and gain strength until it begins short orientation flights in preparation for its future role of mating with virgin queens in drone congregation areas.\n\n1.18: Stages in work roles of a worker.\n\nLIFE AS A YOUNG ADULT BEE\n\nVisual 1.18 shows the various stages that a worker bee transitions through from the time she first emerges from a cell to her final role as a forager collecting pollen, nectar and water. As a newly emerged adult the young bee needs to eat pollen, in the form of bee bread to obtain protein as her body still needs to grow and mature. This is stored in cells ready to be eaten. Bee bread is a mixture of pollen, honey, glandular juices and enzymes to promote the longevity of this food source. The five- to nine-day-old adult nursery bee will continue to eat bee bread to turn it into royal jelly. Royal jelly is made in the hypopharyngeal gland in her head and is fed to the developing larvae in the uncapped cells so that they too will mature and grow into adult bees.\n\nThe progress from one role to another as the worker matures is not well defined and varies both from bee to bee and on the particular needs of the hive. Perhaps the most dramatic change in role is from activities based solely within the hive to working outside the hive when she becomes a forager collecting food, water and propolis for the colony. Propolis, sometimes called bee glue, is a sticky resinous substance collected by foraging bees from resin-producing plants. The colony has many uses for propolis including blocking holes, cementing and strengthening cell bases, even embalming small animals that have died in the nest such as mice that are too large to be removed by the bees.\n\nThis dramatic shift in worker behaviour correlates with a change in protein levels within the body of worker bees. Of particular interest is a protein called vitellogenin, which makes up 30 to 50 per cent of the protein present in a bee's haemolymph (insect blood). Vitellogenin is used in the production of the royal jelly produced by workers for feeding to larvae. In addition to it being an ingredient in the food produced by nursery bees, vitellogenin has beneficial effects on the health of the bee, assisting in maintaining immune system functioning as well as in suppressing the production of juvenile hormone, a hormone that promotes the ageing process in insects. With the switch from nursing behaviour to foraging, workers greatly reduce their production of vitellogenin, leading to their ageing and eventual death. Older forager bees can return to nursing duties if required by the colony but do not fully return to the levels of vitellogenin production seen earlier in their lives for the reason that their pharyngeal glands would have reduced in size when they initially became foragers.\n\nAn adult worker bee lives for about 30 to 50 days during the spring and summer and may live considerably longer if born during the months preceding a cold winter. During the winter months in colder climates brood production may cease altogether with the adult bees huddling together on the comb in order to keep warm.\n\nThe drone has no elongated tongue or proboscis to suck nectar from flowers and with his almost hairless legs he has no useful apparatus to gather pollen. He is reliant on the worker bees in the hive to provide him with food and as he has no sting is unable to defend the colony during times of attack.\n\n1.19: Developmental differences between workers, queens and drones relating to DNA and feeding. Both the worker and the queen result from an egg fertilised with the sperm from a drone. The only development difference between a female worker and a queen is that the queen is fed far more high-quality royal jelly during its larval stage. The male drone only has its queen mother's DNA since its egg is not fertilised by male drone sperm. The food fed to the male drone larva is the same as fed to a female worker larva.\n\nAlthough there is ongoing research and some argument among beekeepers, the current wisdom is that the only function of a drone is to mate with a queen and that the male bee in the hive has no other role in support of the colony. It is not surprising then that when winter approaches and stores of honey and pollen must be preserved the workers evict the hapless drones and simply raise more the following spring when they are needed for mating.\n\nCOLLECTING NECTAR AND POLLEN\n\n**ORIENTATION FLIGHTS**\n\nBefore a worker bee can start foraging it must first learn to recognise the area around the hive so that it can use landmarks to safely return home. Since up to 2000 eggs a day are laid, a lot of potential foragers need to take their initial orientation flights every few days. The first orientation flights are short practice flights and enable a bee to memorise the location of its hive and any nearby landmarks to use for future reference. These flights can sometimes be recognised when there are hundreds of bees flying seemingly aimlessly outside a hive. Some people mistake these for robber bees but robber attacks are much more aggressive with a fast darting motion toward the hive entrance. Orientation flights are also often mistaken for swarming flights but careful observation will show that the bees are circling and moving near the hive whereas a swarm will, after a few minutes, leave the hive and move off. Drones must also orientate themselves to the position of the hive and begin taking short orientation flights when they are approximately eight days old.\n\n**FORAGING**\n\nForaging bees use their proboscis or mouth parts to suck nectar from flowers or water into a sac or honey stomach inside their body, known as a crop. When the crop is full of nectar or water the foraging workers return to the hive where the food is passed to specialised in-hive workers whose sole job is to take the nectar or water from the returning bee and to store it in empty cells. The process of mutual feeding by social insects or taking nectar or water from a returning forager bee is called trophallaxis _._ With trophallaxis, the returning forager brings up nectar from her crop into her mouth where it is taken by the in-hive worker. The regurgitated nectar is then stored in the crop of the in-hive worker where it is mixed with additional enzymes. The in-hive worker with the new supply of nectar will then take it to a honey cell for storage.\n\n1.20: Bees feeding each other. One bee contains a store of nectar or honey in its sac and is regurgitating it and feeding it to the other two bees. The scientific name for this process is trophallaxis.\n\nNectar has a high water content, often greater than 60 per cent and this needs to be reduced to about 17 per cent before the nectar can be called honey. When the honey has the correct water content the cell that it is placed in for storage is capped until it is needed for food by the colony. Evaporating water alone will not cause nectar to turn into honey and it is when the nectar is in the foraging bee's crop that it is mixed with enzymes to convert the sucrose in the nectar to fructose and glucose as well as causing other chemical changes to take place.\n\n1.21: The honey crop is the sac inside a worker's abdomen that is used to store nectar for transport from the flower back to the hive. When the sac is full of nectar its volume increases and takes up much more room inside the forager's body. The above two drawings show the honey crop both full and empty.\n\nIf you look inside a hive when there are many types of plants and trees flowering you will see a kaleidoscope of pollen colours inside the cells around the brood. This is because when a forager collects pollen it practises flower fidelity and only collects one type of pollen during a flight, flying from flower to flower of the same type of plant or tree. The foraging worker will become covered in pollen at the flower and uses her front and middle legs to clean her body of pollen and to store it on her specially adapted hairs on her back legs. This grooming is performed in the air shortly before she flies to the next flower or returns to the hive. While she is in the field the foraging bee mixes the collected pollen with honey stored in her crop for this purpose. This mixture forms a sticky paste that will attach itself more easily to her hind legs. It is easy to observe returning bees at the hive entrance with their two compressed pollen pellets attached to the outside of their rear legs and often you will also see excess loose pollen attached to the hair on their bodies.\n\nCOMMUNICATIONS AND NAVIGATION\n\n**THE DANCE LANGUAGE OF HONEY BEES**\n\nIn order for a complex colony of bees to survive they need to communicate with each other and to coordinate many of the colony's activities, particularly those relating to the collection of food. This enables a colony of bees to achieve much greater results than if each bee were to act on its own without regard to the activities of other bees. This coordination and communication between bees is at its greatest and most complex when foraging bees return and need to tell other bees within the colony of new food resources and the food source's direction and distance from the hive. These activities can be broken down into two separate but interrelated functions.\n\n1.22: Foragers store pollen on their rear legs to carry it back to the hive. Using the front two pairs of legs they brush pollen from their body, mix it with saliva and nectar or honey from their crop to make it sticky, and then attach it to their back legs.\n\n1.23: Pollen stored in a frame. Before storing pollen the bees mix it with honey and salivary juices containing enzymes so that it will store better and also become more nutritious.\n\n1.24: Different sized capped cells for workers and drones. Workers have smaller flat-capped cells while capped drone cells are larger and protrude above the surface of the comb.\n\n1. A returning forager needs to inform other bees within the colony of a new food source and its direction and distance. This is done using the waggle dance.\n\n2. Foraging bees leaving the colony must be able to navigate independently to the food source and then find their way back unaided to the hive.\n\nTHE WAGGLE DANCE\n\nIn order for a returning worker bee to communicate to other bees within the colony the location of new nectar, pollen, water or propolis resin sources, the returning bee performs a waggle dance. The waggle dance indicates the direction and distance of the new food source from the hive.\n\nMovement in the waggle dance consists of a bee running straight ahead for a short distance, up to about 1.5 centimetres, making a buzzing noise and energetically shaking vigorously, or waggling, her abdomen from side to side. At the end of the first straight run the bee turns in one direction to return to the start of the straight run. She then performs another straight run shaking her abdomen vigorously, and then at the end of the run turns in the other direction to return to the start of the straight run. This dance pattern draws out a figure of eight and, as a result, the waggle dance is often called a figure-of-eight dance. The waggle dancer periodically stops dancing to allow the watching workers to touch her with their antennae, probably to detect the scent of the food source.\n\n1.25: A forager bee doing the waggle dance inside the hive. The bee performing the dance is telling the watching foragers the direction and distance of the new food source. Periodically the watching foragers will stop the dancing bee and stroke her with their antennae. This is believed to provide the watching foragers with information about the scent of the new floral source that they can use to further refine their search for the food source.\n\nThe most significant aspect of the waggle dance is that it conveys to the watching forager bees both the direction and the distance to the food source from the hive. Direction to the food source is given by the angle off vertical that the returning worker aligns the straight waggle part of the semi-circle to. If the straight waggle part of the dance is performed at an angle of 40 degrees to the left of vertical, the direction of the food source is 40 degrees to the left of the direction of the sun from the hive.\n\nDistance to the food source is given by several means. There is the tempo of the abdominal waggle that decreases with greater distance to the food source from the hive. There is also the length in millimetres as well as length of time taken to perform the straight waggle portion of the dance which lengthens with increasing distance to the food source.\n\nThe accuracy of the dance is constrained by the speed and distance over which a dancing bee is able to pirouette back to her starting position during the return phase of the dance.\n\nIt used to be thought that returning forager bees used a simpler type of dance, the round dance, to tell other bees about a food source less than 100 metres from the hive. More recent studies have shown that all waggle dances follow a figure-of-eight pattern. Due to space constraints within the hive this pattern is sometimes difficult for both the bee to perform and for the beekeeper or scientist to identify.\n\n1.26: The orientation of the waggle dance to the vertical inside the hive tells the bees that are watching the direction relative to the sun of the new food source. If the dance is vertical, the food source is in the direction of the sun. If the dance is at an angle of, say, 40 degrees to the right of vertical, the food source is at an angle of 40 degrees to the right of the direction of the sun 1.25 from the outside of the hive.\n\n**NAVIGATION AND ORIENTATION DURING FLIGHT**\n\nAlthough the dance language provides information to forager bees about the direction and distance to food sources, it does not provide a mechanism for the forager bees to navigate during flight. Like all animals, the honey bee uses a variety of complex information to provide both navigation and orientation information during flight.\n\nHoney bees primarily use the sun and polarised light from the sun to provide direction information. The use of polarised light enables the bee to navigate by inferring the location of the sun when it is not visible (for example, by trees or partial cloud cover). Under full cloud they are unable to use polarised light and must navigate by landmarks. The most remarkable fact about the bee using the sun as her primary source of direction during flight is that she is able to compensate for the movement of the sun as time passes. Thus, if the sun has moved 10 degrees between the time a bee leaves a colony in search of food and the time she starts her return flight, she will make her own correction for the sun's new position in order to use it as a guide to return home. Geographic cues or landmarks are also used by bees as a secondary tool for navigation as well as the direction of the Earth's magnetic field.\n\nHONEY BEES AS SUPERORGANISMS\n\nBeekeepers often think of a hive as a single living entity and, when looking inside a hive, will see thousands of individual bees going about their daily specialised tasks in support of the colony. Each member of a honey bee colony works cooperatively with the other bees to ensure that the colony survives, with each individual bee working only for the good of the whole colony.\n\nBecause of this, honey bee colonies are considered to be superorganisms as the entire colony, rather than the bees individually, is viewed as the biological unit. Each individual within the colony is unable to survive in isolation to the colony as a whole. The queen and drones require the workers to keep them fed and rear the young, while the workers themselves are unable to reproduce and require the queen to produce the next generation.\n\nThe roles that individual bees perform that are essential for survival include nursery work, capping cells, storing food, guarding the colony entrance and foraging. Just as a colony would not survive without individual bees attending to their specialised tasks, individual bees would not survive for long outside of a colony that supports them. In this way each individual bee within the colony forms an unbreakable bond with the other members of the colony.\n\n**SUPERORGANISM DECISION MAKING**\n\nBecause superorganisms are made up of many individual members, the honey bee colony relies on self-organising behaviours rather than centralised control for the colony to function. These behaviours do the following:\n\n\u2022 Use simple, repeating feedback loops. For example, when a worker stings a beekeeper she also releases an alarm pheromone that recruits other workers to the attack.\n\n\u2022 Are used to gather information from multiple individuals.  \nFor example, a swarm scout returning to the swarm cluster with information on a possible new nest location needs to have the nest site validated by many other scouts before the swarm as a whole will move to the new location.\n\n1.27 A colony of honey bees is called a superorganism. Although made up of tens of thousands of individual bees, the bees work together for the survival of the colony.\n\n\u2022 Enable communication within the colony and ensure that information is not deliberately given to rival colonies; for example, the waggle dance is performed within the hive and information on new floral sources is only passed to members of the colony.\n\n\u2022 Use positive feedback cycles to initiate and coordinate actions by other members of the colony. For example, a returning forager uses the waggle dance to inform other foragers inside the hive of the location of a new food source. These newly recruited foragers will return from the food source and, if the new floral source is still providing nectar and pollen, will also perform the waggle dance to recruit many more foragers to visit the flowers.\n\n\u2022 Positive feedback enables the colony to make decisions collectively by involving many worker bees in the decision so that a reliable choice is made that benefits the colony as a whole. Examples include organised nest hunting by swarm scouts and the organisation of work activities within the hive.\n\n\u2022 Control foraging activities. For example, if in-hive bees did not immediately take nectar from a returning forager, foraging activities would be reduced. This would happen if the colony did not have sufficient empty cells in which to store incoming nectar.\n\nIn the honey bee colony it is not the instructions of an individual bee within the colony such as the queen, but rather the sheer number of bees, each with their own part to play in support of the colony, that enables the colony to survive, reproduce and the species to flourish in many parts of a hostile world.\n\nSUMMARY\n\n\u2022 Each colony of honey bees contains a single queen, between 20,000 and 60,000 female workers and up to 1000 male drones. All of the work and decision making within the colony is performed by the female workers. Each colony will swarm at least once each year, leaving behind about half of the old colony with a new queen. The other half of the colony will depart with the old queen to seek a new home and to establish a new colony.\n\n\u2022 The three most common races of bee kept by beekeepers in Australia are the Italian, Caucasian and Carniolan races, although the Italian bee is the race most frequently kept.\n\n\u2022 Worker bees live for about six weeks once they emerge from their capped cells as adults. The first three weeks as an adult are spent inside the hive performing duties such as looking after brood or cleaning. The last three weeks of their life are spent as forager bees searching for and collecting nectar, pollen, propolis and water. In comparison to worker bees, drone larvae take extra time to grow and to pupate into an adult. Both worker and drone bees take orientation flights prior to becoming foragers or going on mating flights.\n\n\u2022 Forager bees use a dance language to communicate with bees inside the hive the direction and distance to new food sources. The foraging bees use polarised light and landmarks to navigate to both the hive and floral sources.\n\n\u2022 A honey bee colony is regarded as an example of a superorganism as each bee in the colony works co-operatively for the good of the colony as a whole. The entity of the colony is paramount to the individuals in it. \n2.\n\nBasic equipment and protective clothing\n\n2.1: American design hive tools.\n\n2.2: Australian design hive tools. Two hybrid Australian\u2013American design tools are on the left.\n\n2.3: The chisel end of a hive tool can be used to remove burr comb like this as well as acting as a lever to prise open hives that have been stuck together with propolis.\n\nNew beekeepers usually have two initial questions:\n\n\u2022 What equipment and protective clothing do I need?\n\n\u2022 Where do I get my first hive of bees?\n\nThis chapter will give you an idea of the minimum amount of equipment needed to get the hobby beekeeper started. Chapter 5: Your First Bees will cover the second question.\n\nThe new beekeeper needs protective clothing, a hive tool, a smoker and a bee brush to be equipped and ready for their new hobby. The selection of this equipment is important as there are various designs available and, in the absence of good advice from a supplier or an experienced beekeeper, deciding which of a number of choices is right for your circumstances can be difficult.\n\nEQUIPMENT\n\n**HIVE TOOL**\n\nThere are two general designs of hive tool \u2014 the Australian and the American patterns. Both have their advantages although hobby beekeepers overwhelmingly prefer the Australian design, which is often called a J-tool for no better reason than it looks like a large metal J. The J hook at the end of the tool allows for easy removal of frames that are stuck in the hive either by honeycomb or by propolis. The wide crowbar-type hook at the end of an American hive tool is harder to use for this purpose. This is particularly true when the very first frame is removed from the hive, which when stuck fast can prove difficult to raise as the gap between the frames is too narrow for the beekeeper to get a good grip on the top of the frame. At the opposite end to the hook both types of hive tool have a sharp, chisel-like edge that is used to prise open lids, wedge apart frames and to loosen boxes adhered together by the bees. This edge is also useful to clean burr comb from frames, lids and boxes. Burr comb is the small pieces of wax honeycomb that worker bees build away from the central part of a frame, such as on the walls of the hive or underneath the lid.\n\nWhen choosing a hive tool check that the chisel edge is reasonably sharp and can easily be levered underneath a lid to prise it open. Some tools available on the market are too thick to work efficiently for this purpose. Many professional beekeepers prefer the American design because if they need to prise open dozens of hives every day, the flatter hook at the other end is more easily hit repeatedly with the palm of the hand without sustaining injury. It is common for a beekeeper to own two hive tools as these are inexpensive to purchase. An added advantage of owning two is that when one is mislaid, as it almost certainly will be, the beekeeper has another tool to use during hive inspections.\n\nA supplier will usually recommend the standard Australian J-tool to a new beekeeper; however, within the two general patterns of Australian and American hive tools there are many different styles and designs \u2014 some tools are longer while some are shorter. As the beekeeper gains more knowledge and skill it is worthwhile considering an additional design of tool.\n\nIf you suspect that a colony that you have just inspected is diseased, to reduce the probability of spreading the infection to other colonies, your hive tool can be cleaned by placing it inside a burning smoker. Alternatively the hive tool can be cleaned by leaving it in a jar of strong bleach or disinfectant.\n\n2.4: If you believe that a colony that you have just inspected is diseased, a quick and effective way to sterilise your hive tool is to leave it in a smouldering smoker.\n\n2.5: Some smokers use the older design with a grill insert in the smoker.\n\n2.6: Other smokers use the grate insert. Once the smoker has been lit it is much less likely to go out with this design of grate.\n\n**SMOKERS**\n\nSmokers are an essential part of the beekeeper's equipment. Filled with smouldering fuel to produce copious amounts of white smoke, the smoke is puffed over angry bees to calm them down. When bees smell smoke their instinctive fear of fire causes them to gorge on stored honey supplies. They are generally then more docile and have difficulty stinging, as they are unable to bend their overfull abdomens. On hot days when there is a shortage of food or because the bees were born with an aggressive streak, the beekeeper inspecting a hive is sometimes confronted by a mass of defensive bees. A few puffs of smoke will calm even the most agitated colony, allowing the beekeeper to inspect the hive in more comfortable conditions.\n\nIn times past the majority of smokers used by beekeepers were made of galvanised steel. These days, better quality smokers are made of stainless steel and various models are available to suit every beekeeper's pocket and preference. Smokers vary in size from small to large with each having its own advantages and also a few disadvantages. Many beginners turn to an experienced beekeeper for advice on which type and size of smoker to buy. Some will be told to buy the largest available as it will hold plenty of fuel and will not need topping up during hive inspections. There are also a number of beekeepers who swear by a small smoker that is a little larger than a soup can. Generally most suppliers will recommend a medium-sized smoker with dimensions approximately 25 centimetres to 27 centimetres high and 10 centimetres in diameter. This size is not too heavy to hold and has a reasonable fuel capacity to work a small number of hives. A commercial beekeeper with large numbers of hives to inspect may prefer a larger smoker with a greater fuel-holding capacity.\n\nFor the hobby beekeeper with only a few hives a medium-sized smoker is the most convenient one to use. When purchasing a smoker check that on the outside there is a protective cage or cover to prevent the hot cylinder touching you and causing burns. There are two types of lids on the majority of smokers sold. One is usually made of a single rolled piece of stainless steel which ends in a pointed spout \u2014 this is often referred to as an American style of smoker. The other commonly found shape is a dome with a welded spout. This style is generally referred to as a European, German or domed smoker. Keep in mind that the shape of the lid is purely cosmetic and that the other features of the smoker should also be considered.\n\nOne of those considerations is the design of the internal grill. This is used to distribute air and to prevent the fuel burning a hole through the base of the smoker. It's a simple, flat piece of perforated metal raised off the bottom of the unit. This style of grill is commonly found in the more traditional style of smoker, but the fuel tends to stop burning if the smoker bellows are not 'puffed' or depressed every minute or so. Recently gaining more popularity is the type of cylindrical insert usually seen in the European style of smoker. This essentially is a cylinder with a large number of perforated holes in it. A bolt centrally placed in the bottom holds the cylinder up and off the bottom of the smoker. This type of grill affords a lot more aeration to the fuel being burnt and as a result the beekeeper does not need to resort to pumping the bellows quite as often. This can be an advantage for the beginner whose hive inspections take longer thereby increasing the risk of either the smoker running out of fuel or going out because the beekeeper is concentrating on what is happening inside the hive and not on the smoker outside. There are advantages in a smoker that will continue smoking with minimal attendance as there is nothing worse than realising your smoker has gone out at a critical moment, for example, when you are covered with bees, which will require you to move away from the hive. There is also the time involved in relighting the smoker to consider.\n\n2.7: Lighting a smoker. Put crumpled newspaper into the smoker and light it using a barbecue lighter.\n\n2.8: When the paper is well alight keep puffing the bellows and slowly add dry pine needles until white smoke is pouring out.\n\n2.9: Puff the smoker until there is plenty of white (cool) smoke coming out.\n\n2.10: Putting dry gum leaves into a smoker to use as fuel instead of pine needles.\n\n2.11: Putting a roll of corrugated cardboard inside a smoker to use as fuel. Rolled inside the cardboard are some pine needles.\n\n2.12: Using a torch to light a smoker. This is a quick and easy way to light a smoker that contains any kind of fuel.\n\n2.13: Inspecting a hive. Step 1: When applying smoke to a hive, first puff some smoke into the entrance.\n\n2.14: Step 2: Next slightly raise the lid and puff smoke under the lid.\n\n2.15: Step 3: Wait about 30 seconds, remove the lid and hive mat then puff smoke over the top of the frames.\n\n2.16: A smoker needs to be carried safely inside a vehicle to avoid the hot metal causing a fire. I place the hot smoker in a metal bucket when I store it in my ute.\n\nGenerally the European smoker with the round cylinder type of grill is more expensive so this may be another consideration when deciding on which style to purchase. Keep in mind that a quality smoker will last many years with the only major ongoing repair being replacement of the smoker bellows from time to time. For this reason it is always worth asking if there are replacement bellows for the unit readily available before you buy the smoker.\n\nThere are many ways to light a smoker and every beekeeper has a personal preference for how to do this and for the fuel that is used. A barbecue lighter is an excellent way to provide the flame to light the smoker fuel as it has a long nose suitable for pushing deep into the fuel after it is placed in the smoker. First, some newspaper is either crumpled or shredded and loosely balled into the bottom of the smoker. Flame is applied to light the paper and then the bellows are gently puffed or depressed until the paper is alight. At this time the fuel of choice is added until it is well alight and smoking profusely. My personal preference is for dry pine needles, although paperbark, dry gum leaves, hessian sacking, sugar cane mulch or rolled up cardboard can also be used as fuel. Once the smoker is well alight add more fuel and press it down into the smoker, puffing the bellows all the time. Once this is complete and white smoke is pouring profusely from the smoker it is ready to use and the lid can be closed. A note of warning: please do not approach your bees while the smoker is igniting as bees require 'cool white smoke' rather than hot smoke.\n\nAn alternative to the barbecue lighter is to use a propane torch sold in hardware shops for brazing metal pieces together. These torches are easy to light with the click of a switch on the handle and their intense heat will very quickly get smoke pouring out of the smoker.\n\nIt is a good idea to keep a fireproof tin, bucket or empty ammunition box to place the smoker in after use if you are to travel some distance to and from your hives in a vehicle. Many a sad story has been told of bee suits and equipment catching alight in the back of vehicles due to carelessly carried burning smokers.\n\nA smoker is dangerous if left unattended and should be completely extinguished after use. Careful attention should be paid to where any hot embers or smoking fuel remains are discarded and water should be used to ensure the remains cannot reignite. It is a good idea to leave the embers to extinguish in the smoker and to check that they are completely cold prior to tipping them onto the ground. Smokers are not permitted on total fire ban days. Many beekeeping suppliers sell a liquid smoke that can be mixed with water and applied using a spray bottle for use on such days.\n\n2.17: Cleaning tar from a smoker lid. Step 1: Over time the inside of the lids of smokers get covered with a thick layer of tar. To remove the tar from the inside of the lid cover with methylated spirits.\n\n2.18: Step 2: Set fire to the methylated spirits and let it burn out, being careful that the bellows don't also catch alight.\n\n2.19: Step 3: Once the methylated spirits has burnt out the tar will have become brittle and can easily be removed using a hive tool.\n\n2.20 A well-loved smoker.\n\nOver time smoker lids get clogged up with heavy deposits of tar. The tar can reduce the size of the smoker nozzle making it less effective to use as well as making the lid difficult to close. To remove the tar from the lid, pour some methylated spirits over the inside of the lid and set it alight. Make sure that none of the methylated spirits is on the bellows or any wood or plastic parts of the unit. Once the methylated spirits has burnt out, the tar will have become brittle and can easily be broken off using your hive tool.\n\n2.21: Bee brushes are an essential part of a beekeepers toolbox. Most brushes are either made with nylon or horsehair bristles. Nylon is more durable although horsehair is gentler on the bees.\n\n**BEE BRUSHES**\n\nAlthough it may not be apparent to a new beekeeper, a bee brush is an essential and inexpensive item of equipment. A brush comes in handy when the beekeeper wants to remove bees from frames or off the rim of the hive box to replace the lid. Nylon or horsehair brushes are usually used. Nylon is easier to clean. Horsehair is softer but it is also prone to splaying out and may trap bees. After using the brush, rinse it in cold water to remove any honey or other residue and the brush should last for years. The brush needs to be thin to stop bees and debris getting caught in the hair, and the width or thickness of the brush should be about _1_ centimetre or less. Just like a hive tool a bee brush can transmit disease from one hive to another.\n\nPROTECTIVE CLOTHING\n\nEvery beekeeper needs to protect his\/her face as stings on the eyeballs, although extremely rare, may cause temporary blindness. Stings on other parts of the face, particularly the forehead, can cause significant swelling. With this in mind there are three types of protective clothing that can be used.\n\n\u2022 Veils that protect only the face.\n\n\u2022 A jacket with a built-in veil that offers good protection from the waist up.\n\n\u2022 Full-length overalls that offer complete protection for the whole body.\n\n**VEILS**\n\nVeils vary in design both in the way they protect the wearer's head and also in the way they are secured to the wearer's body. Some veils come without a sewn-in hat and these are worn over either a beekeeper's helmet or your hat of choice. The top of the veil is elasticised so that it fits over the crown of the hat. Over time the elastic can become loose and offer an entry point for bees. A veil with a hat sewn into it avoids this problem. Many hats have metal grills around the face while others are made of nylon or polyester netting. The bottom of the veil is made of fabric or netting, which will either have a long string attached for tying around the wearer's torso or, alternatively, an arrangement that fits over the shoulders and ties or is elasticised under the arms.\n\n2.22a and 2.22b: Veils are inexpensive and easy to put on. The disadvantage is that they offer less protection than a jacket or overalls since bees are able to get inside a veil more easily than with other types of protection. This is overcome with the design of veil in 2.22b with the pullover style top.\n\nWhile relatively inexpensive, veils have the significant disadvantage that it is often difficult to prevent bees gaining entry under the bottom of the veil and up into the face area. This is not generally a problem though, as the trapped bees are usually more concerned with getting out of the front of the veil to freedom. The psychological consequences, however, of having one or more bees inside a veil can be quite devastating. Again, the beekeeper will be required to weigh up the costs involved in starting with the cheaper veil and whether these compensate for the disadvantage is that they offer less protection than a jacket or extra vulnerability to stings. My opinion is overalls since bees are able to get inside a veil more easily than that veils do not provide the same level of with other types of protection. This is overcome with the design of veil in 2.22b with the pullover style top. protection as a jacket or overalls, although they are cheaper and many beekeepers use them, particularly professional beekeepers. It is critical for the beekeeper just starting out that a level of confidence is gained while working with bees and this will be boosted by the extra protection offered by a jacket or overalls.\n\nThe composition of the netting on veils used in all protective clothing is varied, so choose thicker nylon netting or metal gauze netting so that the veil will last longer. The mesh in the veil needs to be black since it is impossible for a beekeeper to see through a white mesh veil.\n\n**JACKETS**\n\nThe most popular option is the jacket with a built-in veil. The built-in veil provides good visibility and is usually open enough to provide plenty of air to the beekeeper. Some jackets come with a zipper up the front, like a normal jacket, or may be designed without a front zipper and pull over the wearer's head. If the jacket has a zipper up the front, check that the join where this reaches the veil does not have a gap that will allow bees to enter.\n\n2.23: This popular design of overalls uses a round hood supported by a round wire band and allows for plenty of ventilation and light. Note that there is no zipper on the pullover. The round metal band ensures that the protective netting cannot come in contact with the face.\n\n2.24: This style of jacket uses a fencing-style of hood.\n\nGood-quality jackets will have a Velcro patch where the jacket and attached veil zippers meet to ensure that bees cannot enter the jacket under the chin. Some manufacturers have gone further and designed the veil so that the zippers actually cross over each other in the front and are further sealed with a Velcro tab over the top. When selecting a jacket (or overalls), take particular care to check that the veil can be removed for laundering. To prolong the life of your veil it is generally unwise to launder it in a washing machine, as the abrasive action will destroy the netting inserts. Usually the veil will not need laundering as frequently as the jacket, and when required, this is better done by hand. Another important consideration is how easy it is to zip the veil closed when you are suited up. If there is only one zipper attaching the veil to the jacket it may be difficult to reach around the neck to grasp the zipper head when closing the hood. Always try on any jackets or overalls you are thinking of buying to make sure you can get in and out of the clothing easily.\n\nWhen wearing the jacket make sure that the elasticised bottom around the waist forms a seal against bees. The jacket should remain above the crotch because if it is pulled down further any gap formed between your legs will quickly be taken advantage of by bees in the vicinity. Try to buy garments with good quality, well- known branded zips. Jackets are about twice as expensive as veils, but the extra cost is well worth the added protection that they provide over a simple veil. This is particularly so for the beginner.\n\n2.25: The part where the zipper joins the hood and the jacket is a critical part of the design. To minimise the chances of bees entering the clothing make sure that the two ends of the zipper overlap and that there is also a Velcro cover over the zipper.\n\n2.26: Elastic bands are often attached to the wrist to place over the thumb of the beekeeper. The elastic bands stop the sleeves from riding up the arm.\n\nA good jacket or overall should have elastic loops sewn on both wrist cuffs. These hook over the thumbs to stop the sleeve moving up the arm, which can result in stings on the forearm or wrist if you are gloveless when working with your bees. The loops are also handy to prevent sleeves riding up when gloves are pulled on.\n\n**OVERALLS**\n\nFull-length overalls with a built-in veil are the most expensive option but do provide the best protection for the beekeeper. A big advantage of overalls is that they give you full protection and confidence when dealing with bees. The overalls should come with elasticised wrists and ankles to stop bees from crawling inside. Like the jacket, there should be elastic loops at the wrist to put over your thumbs to stop the sleeve moving up your arm. Again check the zippers and make sure that the fit is comfortable with plenty of room to move, particularly around the crotch area. The crotch to high shoulder point measurement is the key to a good fit in overalls, so don't rely solely on chest size, particularly if you buy online. There should be no restriction when the arms are raised over the head or when bending over with the arms placed near to the ground.\n\n2.27: Full-length overalls provide complete protection to the beekeeper and usually come with elasticised ankles as well as wrists.\n\n2.28: Protective clothing made out of aerated fabric is much cooler to wear on hot days. The fabric is made of three layers of material: an inner and outer shell made of nylon and a mesh material in the centre that provides thickness to the clothing as well as allowing for air flow.\n\nIn the summer beekeeping is hot work and it is best to get a very loose fitting jacket or pair of overalls that allow air to flow around the body and provide flexibility of movement. The material that the jacket or overalls are made of is important so try to buy a polyester\/cotton blend that will not shrink when washed. As a precaution, ask the supplier if the cloth in the overall or jacket is pre-shrunk. Generally the polyester content will ensure the longer life of the garment particularly at high-wear points such as the elbows or knees. It is possible to get pure cotton clothing, but again be aware of the possibility of shrinkage.\n\nA jacket or a pair of overalls purchased in the mid-price range should give the beekeeper many years of service. Keep in mind that the veil will usually need replacing during the life of the garment and you should check with the supplier if replacement veils are available. If not, you may need to occasionally repair a veil, as these are usually the first part of the garment to deteriorate. Veil repair is discussed later in this chapter.\n\nAERATED FABRIC\n\nThe standard garment made of either cotton or a polyester\/cotton mix with long sleeves and long legs in the case of the overalls does mean that beekeepers can become very hot even in mild weather. A recent innovation is the newer aerated garments that are now available. Even though they are much more expensive, protective clothing made of aerated fabrics keeps the beekeeper much cooler on hot days or during long hours of work. These garments are made of strong, thin outer and inner layers of fabric that are tear resistant and provide physical protection to a thicker rubberised inner layer. The rubberised inner layer is a loose weave net to allow air to circulate freely through the fabric. The inner and outer nylon layers on the surface are also net-like to allow air to circulate. The garments are see-though and due to their layers offer better protection against stings as well as being much cooler to wear than clothing made of conventional materials.\n\n**GLOVES**\n\nMost beekeeping gloves are made of cow or calf hide and have a long sleeve or gauntlet with an elasticised top to prevent bees crawling down into the glove. Gloves made of much softer goatskin may also be used. Although all of these gloves provide good protection against stings they have the disadvantage that the material is thick and finger dexterity or feeling is lost while wearing them. Some gloves come with an aerated wristband to keep the wrists cooler, but as the wristbands are frequently made of thinner material they can allow for the occasional sting.\n\n2.29: The conventional design of beekeeper's gloves includes the long gauntlet and a wrist vent that keeps the hands and wrist cool. The gauntlet of the glove is worn over the sleeve of the jacket or overall.\n\n2.30: If you use plastic gloves, wear a thin pair of cotton liners inside the gloves to stop your hands becoming sweaty and smelly.\n\nAll beekeepers work to a budget and many decide to use the thicker chemical-resistant work gloves they already have at home or those that can be bought cheaply from a hardware store. These gloves provide very good protection to the hands, but the lack of a gauntlet or long sleeve means that care needs to be taken to ensure that the wrist does not become uncovered during hive inspections. Also, since chemical gloves are impenetrable to sweat, hands become very smelly after even a short use \u2014 a smell that is difficult to wash off.\n\nIf plastic, rubber or nylon gloves are bought to handle your bees, buy a larger size and purchase some cotton gloves to go inside them since these will absorb sweat, are washable and will also ensure protection against stings to your hands.\n\n2.31: Nitrile gloves are inexpensive and provide a much better sense of touch than leather or plastic gloves. Although good for handling bees, working with queens and removing frames from hives, they are not suitable for lifting heavy hives as they may tear.\n\n2.32: If you do not wear a full pair of overalls a pair of thick socks will provide protection to the ankles. Bees congregate around the socks and, if given the opportunity, will climb up the leg before stinging you. Make sure that your jeans are tucked into a very thick pair of socks.\n\nAn alternative to each of the above is to use Nitrile gloves. These are the gloves used by mechanics to keep their hands clean when working on cars. Nitrile gloves have the advantage that they are strong, fit snugly over the hands, and have excellent feel and dexterity. Bees have trouble stinging through Nitrile gloves although they are not as sting-proof as cow or calf-hide gloves. Nitrile gloves should not be confused with the weaker latex dishwashing gloves found in many homes. Nitrile gloves can be found at the supermarket, hardware or car accessory store, although the cheapest place to buy them is online.\n\nMy preference for beekeeping clothing is a jacket, loose jeans, gloves and very thick socks to tuck my jeans into. Inevitably the bees will zero in on the socks and try to sting through them. Make sure that you always use thick socks and that your trousers are tucked in, even on brief inspections, otherwise the bees will seize the opportunity you present to them and sting your ankles.\n\nAn alternative way to protect your ankles is to wear gaiters available from any outdoor or ex-army shop. These are easy to put on and provide almost complete protection to your ankles. Alternatively, you can wear shin-length boots with trouser legs tucked in or gumboots with trouser legs over the boots and secured with an elastic band. Overalls will prevent most of the ankle vulnerability that can be experienced by the jacket-wearing beekeeper.\n\n**CLOTHING FOR CHILDREN**\n\nChildren are fascinated by bees and will often take an interest in the beekeeper's hobby. Overalls are recommended for children as they provide the most protection. Buy a little larger than is currently needed, as the child will quickly outgrow a suit that is a good fit when purchased. Always ensure that there is a good seal around the wrists and ankles. Long socks worn over overall legs with the addition of gumboots provide the best protection. Take particular care that the mesh in the veil cannot touch the child's face as a painful sting could turn the youngster off beekeeping for life. This can usually be achieved by encouraging the child to wear a baseball cap under the veil, particularly with the fencing style type of veil. Turn up the collar of the overall inside the veil as added protection. To make the overalls more attractive to the child, sew a small bee or other emblem onto the chest pocket of the overall, or even something totally non-bee related like a favourite cartoon character.\n\n2.33: Children love looking at bees so a properly fitting pair of overalls is essential.\n\n2.34: Beekeeping for all the family.\n\n2.35: Providing protection for children need not be expensive.\n\nGloves for children are available from some suppliers or alternatively kitchen gloves that are flocked inside can be used. Fit the gloves on the child over the overall sleeve. Buy some elastic and make a band to place over the glove so that there is a good seal preventing the possibility of any bees crawling inside. A further alternative is to buy a pair of child's gardening gloves from a discount store or hardware store and sew on a long gauntlet with an elasticised end.\n\nI cannot stress enough that children must be supervised around bees at all times. This is particularly true if a child is given a hot smoker to manage and keep alight.\n\n**REPAIRING A TORN VEIL**\n\nThere comes a time in the life of any protective clothing when the veil will tear and needs to be repaired. The best solution if the veil tears when the beekeeper is in the field well away from home is a little forethought, so always include some duct tape in your toolkit for emergency repairs both of the veil and the fabric of the clothing.\n\n2.36: I repair torn veils using a hot glue gun available from most hardware shops. The two edges of the torn gauze are held together and the glue is liberally applied. Hold together for a few seconds until the glue cools and hardens.\n\n2.37: Beekeepers should keep a roll of duct tape in their tool kit in case their protective clothing rips when they are out in the apiary.\n\nAt home there are more options available to repair a torn veil and my favourite is a hot glue gun available from any hardware store. When hot glue cools it is fairly opaque and usually does not detract too much from the beekeeper's field of vision. Silicon sealant can also work, but this is harder to use due to its longer drying time and it is less opaque than hot glue after it has cooled.\n\nSUMMARY\n\n\u2022 The three items of equipment that you will need as a beginner are a hive tool, a smoker and a bee brush. These come in various designs and it is worth talking with other beekeepers about their preferences or visiting a beekeeping supplies store to look at what is available.\n\n\u2022 There are three designs of protective clothing worn by beekeepers: the simple veil, a jacket with built-in veil, and a full overall with built- in veil. Gloves with long gauntlets are also needed to protect the hands from stings. These clothing items are critical for a beekeeper, so it is worth visiting a beekeeping supplies store to make sure that the clothing you buy is manufactured out of suitable fabric and is the correct size for you.\n3\n\nConstructing hives\n\n3.1: Making a hive.\n\n3.2: A wire mesh top cover placed under the lid allows visitors to view the bees without wearing protective clothing.\n\nAmong many new beekeepers there is initially some confusion about the design and choice of hive ware available. The most common type of hive used in Australia and the one most readily available through suppliers is the Langstroth. Named after its American inventor, the Reverend Lorenzo Langstroth, the hives are essentially a series of oblong boxes, stacked on top of one another and held together with a base and lid. Langstroth designed his hive with removable frames, which was a breakthrough in beekeeping in the mid-1800s but is now a legal requirement for all beekeepers in Australia to allow for ease of inspection for disease.\n\nWhile the boxes used in the Langstroth hives are identical, their use in the hive set-up determines the label by which they are referred to by beekeepers. The box placed on the hive base is called the brood box as this is where brood will be found, assuming, of course, that an excluder is used to restrict the movements of the queen and confine her to being able to lay eggs only in the bottom box. Any box above the excluder is called a super or honey super. It is exactly the same box in all respects, but defined and labelled by the role it plays in the hive set-up. The name super comes from the word 'superstructure', referring to the top-most parts of a building or structure.\n\nIf an excluder is not used to keep the queen in the bottom box, she will travel upward through the stack of available boxes and will lay eggs in the central frames in roughly an oval pattern. It is often for this reason that new beekeepers are encouraged to use excluders as, if nothing else, they know the queen and brood are in the bottom box and when it comes to taking honey they do not have to worry about avoiding brood frames throughout the honey supers. Of course, no theory is perfect and there are occasions when a smaller queen will slip through an excluder and go up into the top supers.\n\nTo further add to the confusion of the new beekeeper several sizes and depths of box are available from suppliers. Most hobby beekeepers use the full-size super and brood box. A description of each of the available sizes of boxes is given in the appendix.\n\nCONSTRUCTING A HIVE\n\nA complete beehive typically consists of:\n\n\u2022 a base to rest the hive on\n\n\u2022 a slotted bottom board, optional but increasingly used\n\n\u2022 one or two brood boxes sitting on the base\n\n\u2022 a queen excluder between the brood boxes and the supers\n\n\u2022 one or two supers in which to store honey\n\n\u2022 a hive mat used to stop worker bees building burr comb in the hive lid\n\n\u2022 a lid to cover the hive that will protect the colony from predators and bad weather.\n\nWhen I make up a hive in which to place a colony of bees I always place the queen in the bottom one or two boxes. The bottom of the hive is the natural place for the queen to lay eggs and for the nursery bees to raise brood. This arrangement allows the supers containing honey to be kept at the top of the hive where they can easily be inspected and the full frames of honey can be removed for extraction.\n\n3.3: Schematic of a hive.\n\n3.4: Hive bases come in many designs. This is a standard base with a solid bottom. The entrance size has been reduced although some designs leave the full front of the base open for returning foragers to enter and leave.\n\n3.5: This design of base utilises a longer entrance way so that guard bees can better protect the colony against robber bees from other colonies.\n\n**CONSTRUCTION OF THE PARTS OF A HIVE**\n\n**THE BASE**\n\nThe base of the hive protects the wooden hive parts from the damp by raising them above the ground. The base usually has built into it, around three of its top edges (two long sides and one short edge), a wooden riser that creates an entrance at the front of the base for the bees to enter and leave the hive through once a brood box is placed on top. Many commercially available bases have a riser attached on all four sides with a small entrance cut into the front riser for the bees to enter and exit the hive. In this type of base the width of the hive entrance is fixed and although it can be narrowed further by the beekeeper it cannot be widened should this be needed for extra ventilation. Risers can be obtained in thicknesses of between 10 millimetres and 50 millimetres depending on how much space the beekeeper wants below the brood box.\n\n3.6: A longer entrance way minimises the chances of robber bees or wasps entering the hive and stealing honey.\n\nThe flat bottom board or floor of the base is usually made of one of the following materials:\n\n\u2022 wood\n\n\u2022 plywood\n\n\u2022 Weathertex\n\n\u2022 galvanised steel\n\n\u2022 moulded plastic.\n\nThe flat bottom board usually rests on a set of cleats made of either hardwood or treated pine. In plastic versions the cleats may be moulded as part of the hive base. Base cleats keep the bottom board a few centimetres off the ground and help keep the hive dry and free of wood rot. Some beekeepers nail the base to the bottom of the brood box. I don't recommend this practice as it makes life difficult when it comes to maintaining, cleaning or inspecting the base board, which in this case can only be done once frames are removed from the attached box. A freestanding base makes good sense, particularly with the advent of Small Hive Beetle and other pests. When maintenance or cleaning is required on the base it can be removed as a single unit rather than requiring removal of both the box and the attached base, which will thoroughly disrupt the hive's resident colony.\n\nIf (and when) the _Varroa_ mite enters Australia (see Chapter 16: Parasites of the Honey Bee), a slightly different design of base board will be required to help manage this pest. To assist with the control of _Varroa_ mites overseas, the bottom boards are perforated by either using a stainless steel mesh or by using plastic slats. The aim of this type of vented bottom board is to allow _Varroa_ mites that have fallen off frames or bees to drop through the hive onto the ground where they will die outside the host colony. This effectively helps to break the _Varroa_ lifecycle and studies overseas have shown that the use of a vented bottom board effectively reduces _Varroa_ mite counts by about 30 per cent. These modified bottom boards are discussed in more detail in Chapter 16: Parasites of the Honey Bee.\n\n3.7: An American style base with a sloping landing strip attached to a separate stand.\n\n3.8: A slotted bottom board distributes air flow more evenly throughout the hive. This minimises draft through the brood box near the entrance and encourages the queen to lay eggs in that area. During the colder winter months the design of the board enables the bees to keep the brood box warmer. The wide slat at the front allows guard bees to better protect the entrance to the hive against intruders.\n\nAnother advantage of vented bottom boards, or aerated bottom boards as they are sometimes called, is that they allow a much greater flow of air through the hive. A US Department of Agriculture (USDA) study reported at the Apimondia beekeeping conference in Melbourne in 2007 stated that, with an increase in airflow past the colony, the queen lays far more eggs resulting in a much stronger and healthier colony. This result, however, was not substantiated in a more recent study of vented bottom boards conducted by Dr Doug Somerville of the NSW Department of Primary Industries. The hives that used vented bottom boards in Dr Somerville's study did, however, contain more honey than hives that did not use vented bottom boards.\n\nMany hobby beekeepers ask me about the size of the landing provided at the front of the hive entrance. Although bees seem to adapt well to feral nests that have only a hole in a tree or a wall as an entrance, most hives have a landing at the front for the bees to rest on before entering or leaving the hive. During hot summer evenings it is common to see thousands of bees outside and hanging off the front of the hive in an effort to prevent the colony from overheating by having too many bees on the inside. It is my preference to use a larger landing with a depth of about 5 centimetres. This is in contrast to some of the smaller landings that have a depth of around 1 centimetre.\n\nProfessional apiarists prefer to use narrow or no landings since they can more easily pack hives close together during transport to new sites either for pollination or to follow the flowering patterns of gum trees. In practice, landing strips are commonly provided by using an extended length of hive base board; that is, the base board extends beyond the length of the bottom box. For the hobbyist with time, more complicated slanted boards, often seen in North American beekeeping books, can be constructed.\n\n**SLOTTED BOTTOM BOARD**\n\nSome beekeepers, myself included, place a slotted bottom board (not to be confused with a vented bottom board which is part of the base) between the base and the first brood box. The slotted bottom board is a separate unit from the base board, but once placed on top of this, it becomes a permanent part of the hive set-up. The use of a slotted bottom board offers the following advantages and disadvantages.\n\n_Advantages:_\n\n\u2022 The wide slat at the front of the slotted bottom board reduces the ease with which robber bees and wasps can enter the hive and fly up among brood frames. This is because the longer entranceway allows the guard bees greater time in which to fight off intruders.\n\n\u2022 The use of slats distributes airflow throughout the entire brood box, not just at the front of the hive near the entrance. A queen is often reluctant to lay eggs near the entrance to the hive if there is a strong current of air coming in. The use of the slotted bottom boards encourages the queen to use the entire brood box for eggs.\n\n\u2022 During winter, workers use their bodies to reduce the amount of cold air entering the hive and they are able to straddle their bodies across the gaps between the slats to reduce the flow of cold air. This effectively allows the workers to keep the hive warmer and they require less honey over the winter to survive.\n\n_Disadvantages:_\n\n\u2022 The use of a slotted bottom board does reduce the amount of air circulating through the hive during hot days. I believe, though, that the benefits outweigh this disadvantage and by the start of spring, hives that use a slotted bottom board are far stronger than hives that do not use them\n\n\u2022 The use of a slotted bottom increases the overall cost of a hive.\n\n**BROOD BOXES AND SUPERS**\n\nAs mentioned earlier, there is no physical difference between a brood box and a super box and the same box can be used for both purposes.\n\nIn Australia the wood used to make hives is typically 22 millimetres thick. This is thicker than the commonly available 19-millimetre-thick wood found in hardware and building supply stores. There is no doubt that the thicker 22-millimetre wood provides better insulation for the colony both during the hot summer and during the cold winter. The thicker wood also makes for a stronger and more durable hive. Many years ago when the imperial system of measurement was in use, 7\/8-inch- thick wood was widely used in the construction industry and its use has continued to this day in the beekeeping industry with the conversion of 7\/8-inch wood to 22-millimetre-thick wood for hive-making. Generally suppliers will offer hive boxes in either radiata or hoop pine.\n\n**CONSTRUCTION OF BROOD BOXES AND SUPERS**\n\nThe corners of boxes are designed to fit together either using a rebated corner or by locked corners, often incorrectly called dovetail corners. For cost reasons rebated corners are the most frequently used corner for boxes, and when correctly assembled using glue and either nails or screws, make a very strong and durable corner.\n\nMany beekeepers believe that the locked corner is stronger than a rebated corner. Although this may have been true in the past, modern glues are so strong that a correctly assembled, rebated corner is just as robust as a locked corner. Perhaps the main advantage of a box assembled using locked corners is that the box holds itself together firmly while the beekeeper is inserting nails or screws into the corners.\n\nThe disadvantages of a locked corner is that more of the wood's end grain is exposed, making the box more likely to rot along exposed corners. Interestingly, a lot of anecdotal evidence after the severe bush fires experienced in Victoria during recent years indicates that the hives made with locked corners suffered more from ember attack than those hives made with rebated corners. Another disadvantage of locked corners, and one to be aware of, is that the interlocking corner needs to be assembled soon after manufacture. Because the short sides and the long sides of boxes can be made with wood cut from different trees and dried to slightly different standards, leaving locked corner sides unassembled for long periods may result in one side contracting a greater amount than the other, making them difficult to interlock with each other. The result is that the box walls may not fit and can easily split if forced together. To be fair, though, the walls of boxes made with rebated ends can also shrink differently although this is usually much less of a problem when it comes to assembly.\n\n3.9: The slotted bottom board fits between the base and the brood box.\n\n3.10: The interlocking corners of brood boxes or supers can be assembled with either 65mm galvanized nails or screws. If nails are used place five nails on one side of an edge and four nails on the other side. If 65mm long screws are used I only place them on one side of the edge due to their superior gripping power over nails.\n\n3.11: This rebated box is joined together using 65mm nails. Modern glues provide excellent adhesion to wood corners. There is no disadvantage in strength in using a well-glued rebated box rather than an interlocking edge box.\n\n3.12: Assembling a box with the aid of a clamp.\n\n3.13: To stop the edge of the box splitting when you insert screws, this handy drill available from most hardware stores will both drill and countersink a hole at the same time. If you are using nails, holes also need to be drilled, but not countersunk, to stop the wood from splitting.\n\nTo assemble the boxes I use either 65-millimetre-long cement-coated nails or 65-millimetre screws. My preference is for galvanised screws along with a good wood glue as this gives the box optimum strength. A hive is roughly the same temperature inside throughout the year while temperatures outside the box, depending on its location, will vary a great deal. This will place a great deal of stress on the wooden corners of the box and the extra time and effort put into careful construction will pay dividends in the long term. If I use nails I put them into both sides of a corner for strength; that is, where the corner of both the long side and short side of the box meet. On the other hand, when I use screws I find these provide a sufficiently strong corner for the box even if they are inserted only along the corner of the short end of the box.\n\n3.14: A full size super can weigh up to 35kg when full of honey. If the only way to hold the super is the grips cut into the wood these can be very awkward to pick up and carry. Adding cleats to a super makes lifting it easier, and they should be attached to all brood boxes and supers.\n\n3.15: Bees usually propolise boxes and lids together, making them difficult to prise apart. Planeing a bevel around the edge of the boxes and lid enables the hive tool to prise apart the box much more easily.\n\nThe preferred nails used to make boxes are usually cement- or glue-coated because of their superior adhesion to the wood when nailed in. 'Cement Coated' is the trade name for glue coated and the terms may be used interchangeably. As the nail is driven into the wood the friction causes the cement to melt and results in a much stronger attachment between the wood and the nail, creating a stronger corner.\n\nI use five screws at each corner of a full-size or WSP box (named after William S. Pender, the inventor) and three screws or nails on an Ideal box. If I use nails to secure the corners, in addition to the five nails used on the short corner of the box, I also insert another four nails into the other corner on the long side of the box for added strength.\n\nMost hive boxes come with pre-cut grips in the sides. The difficulty with pre-cut grips is that they are shallow and a full-sized super of honey is difficult to lift and hold using only your fingers. To overcome this I attach holding cleats made out of wood. Holding cleats can be placed on either the long or short side of the box. My preference is for the short side. The use of holding cleats makes lifting the super a lot easier and you are much less likely to strain your back. The cleats are basically a strip of wood the same length as the side of the box on which they are to be fixed. They are around 4 centimetres wide and 3 centimetres thick and allow for a strong grip when lifting a heavy box.\n\nThe last part of box construction is painting. I only ever use exterior latex house paint, usually from the discard pile at the local hardware or paint shop. These discarded tins are usually well-known brands that have been rejected due to unsatisfactory colour mixing. I believe that two coats of a good-quality exterior house paint provides sufficient protection for my hives. Some beekeepers prefer to apply three coats of paint and sometimes even a primer. Other beekeepers use oil-based paint and there are those beekeepers who prefer to varnish their hives.\n\nWhen buying paint do not skimp and buy cheaper brands as these frequently require three coats to provide a uniform, durable covering, whereas better quality paint will require only two coats to provide equal protection against the elements.\n\nHives are traditionally painted white as the idea is that this colour provides optimum reflection of the sun's rays and assists in keeping the colony cool during hot weather. These days, however, there is a move away from this practice, with many varying colours chosen by beekeepers. As a general rule, hives are not usually painted with black or dark paint, as this will absorb too much heat during the hot summer months. Keeping the colony cool during summer is more of a concern than keeping the same colony warm over a cold winter when a darker coloured paint would be beneficial.\n\nHave fun with your hives if you have the time and inclination \u2014 they can be an attractive garden feature in their own right. When my children were small I painted birds and other characters on my hives and these not only delighted them but many of our visitors. If you feel you do not have an artistic bent, buy some children's stencils and use these. Be aware though, that particularly in the more northerly parts of Australia hive wood may be treated with copper napthenate as a preservative. Copper napthenate is poisonous and hives treated with this chemical should not be handled by children.\n\nThe debate about whether it is necessary to paint the inside of the box as well as the outside rages back and forth with both sides holding strong views. I no longer paint the inside of my hives because I do not believe that it is either necessary for the protection of the wood or desirable to have paint next to bees or honey.\n\nOn some of my hives I paint a distinctive pattern at the front for that hive's returning foragers to home in on. The reason I do this is that if there are several identically painted hives placed next to each other returning foragers may get disoriented and drift into a neighbouring hive instead of their own.\n\n3.16: A creatively decorated hive.\n\n3.17: Horses painted on side of brood box.\n\n**THE QUEEN EXCLUDER**\n\nAlthough not all beekeepers use queen excluders, there are many good arguments both for and against their use in a hive, as explained below. Excluders are placed between the brood box and the honey super to reduce the likelihood that the queen will leave the brood box and move up into the supers to lay eggs in the honeycomb.\n\nExcluders are generally made of either galvanised steel or plastic, although some suppliers also offer a bamboo version. Any of these materials is suitable for use as an excluder. Plastic excluders are usually about half the price of metal excluders and many beekeepers choose them for this reason. The plastic in the excluder tends to become brittle over time, giving them a limited lifespan. Against this they have the real advantage that they provide no nooks and crannies for pests such as the Small Hive Beetle, _Aethina tumida,_ to hide in. Further they do not conduct either heat or cold inside the hive. If you choose a plastic excluder check that it is of the type that has rounded grill bars, as plastic excluders cut out of one flat sheet of plastic may have sharp grill edges that will damage worker bees' delicate wings. Metal excluders are generally much easier to clean and will outlast a plastic excluder by many years.\n\n3.18: A great hive decorated by one of our Tasmanian friends.\n\n3.19: Queen excluders are typically made of galvanised steel or plastic.\n\nBeekeepers often buy excluders without giving any consideration to the gap width between the grill bars. It is a simple fact that both queens and worker bees vary in size. Most excluders have a gap width of between 4.1 millimetres and 4.4 millimetres. This is a large variation in grill size and some workers will be unable to get through a 4.1-millimetre gap while a few queens can get through a 4.4-millimetre gap. Before buying an excluder check if your bees are large or small and try to buy an appropriate excluder for their size, which is usually one with a gap size of 4.3 millimetres or slightly greater. To decide if your bees are large or small you may need to look at bees from other people's hives to get a better idea of the comparison.\n\n**SHOULD I USE A QUEEN EXCLUDER?**\n\nMany beekeepers choose to use queen excluders, although in my experience roughly 10 per cent or less use no excluders at all. I have, over the years, kept hives in which I used queen excluders and hives in which I did not use them. My personal preference is to place an excluder in a hive. Nevertheless, there are advantages and disadvantages in their use. These are summarised overleaf.\n\n_Advantages:_\n\n\u2022 The use of an excluder keeps the queen below the honey supers so that frames of honey can be removed without any concern about brood being present.\n\n\u2022 Since the queen is unable to lay eggs in the super, the comb will remain white or light yellow without the unsightly dark wax caused by larvae skins, excreta and cocoons.\n\n\u2022 Keeping the queen in a restricted area of the hive, the brood box, makes her easier to find when needed.\n\n\u2022 Many workers will stay with the brood in the bottom brood box. This is an advantage when removing frames of honey since bees are more easily removed from honey frames than from frames containing brood.\n\n\u2022 Pollen is usually stored next to the brood so there will be little pollen to discolour the honey in the super.\n\n_Disadvantages:_\n\n\u2022 Airflow through the hive may be reduced and this can cause overheating on very hot days. Brood are unable to survive heat as well as they can cold.\n\n\u2022 Workers may experience difficulty getting through the grill of the excluder and be unable to store some of their honey in the super.\n\n\u2022 Excluders are expensive and easily damaged.\n\n\u2022 Some excluders may damage workers' wings as they struggle through the grill.\n\n\u2022 If the queen inadvertently gets through the excluder into the super and lays eggs, when the larger drones emerge they may get trapped above the excluder in the super.\n\n\u2022 Worker bees come in all sorts of sizes and a single gap width on a queen excluder may not be suitable for all the hives in your apiary. Queens also come in different sizes and the small virgin or newly mated queens may easily get through some excluders. The use of an excluder is for the sole benefit of the beekeeper and not the bee.\n\n\u2022 The main issue is that the excluder constricts the size of the brood nest and reduces the capacity for growth.\n\n\u2022 It may also encourage swarming.\n\n3.20: Plastic queen excluders are cheaper than metal excluders but are more likely to break during use.\n\n**THE HIVE MAT**\n\nThe hive mat rests on top of the frames in the uppermost box under the lid and discourages the workers from building burr comb inside the lid. Burr comb is a nuisance to the beekeeper as it makes the lid difficult to remove. Also, once the comb has been broken it makes a mess in and around the top of the hive, creating sticky, unpleasant inspection conditions. The hive mat can be made of a variety of materials although I usually use linoleum purchased as off-cuts from a floor-covering store and then cut to size. Not every beekeeper uses hive mats and many people, particularly professional beekeepers, are content to let the bees make burr comb in the lid.\n\n3.21: If an excluder is not used in a hive, the queen will lay brood in an oval pattern at the centre of the hive, leaving the frames on the outside to store honey.\n\nHive mats are cut so there is a 1-centimetre gap between the edges of the mat and the inside walls of the hive. This is to allow air to circulate through the hive and out of the vents in the lid. Circulation is important in keeping the colony cool during the summer and also assists in the evaporation of condensation during the winter.\n\n**LIDS**\n\nThe lid is a critical part of the hive since it protects colonies from heat, cold, inclement weather and from predators. Lids come in two designs: migratory lids and telescopic lids.\n\nThe migratory lid, named because of its use in migratory beekeeping, is the most common lid design used in Australia due to its lower cost, its utility when moving numbers of hives, and largely due to the fact that often it is the only design of lid that beekeeping supply stores sell.\n\nA migratory lid is designed so that the sides of the lid are flush with the sides of the hive box. This allows many hives to be packed closely together when they are transported on a truck or a trailer. The disadvantage of a migratory lid is that it can allow rain to run from the top of the lid and possibly enter the hive either under the lid or where the boxes meet. This may happen when the lid is accidently knocked off centre and will cause the colony to be exposed to the elements and predators.\n\nThe telescopic lid, although less well known in Australia, is designed to overlap the top hive box and those boxes underneath by approximately 2 centimetres, allowing water to run harmlessly from the hive lid to the ground. Further, by overhanging the top box they afford better protection against rain beating in under the hive lid. It is a pity that these lids are not more popular among hobby beekeepers as they provide better rain protection to hives than do the more commonly available migratory lids.\n\nThe top cover of either a migratory or telescopic lid is usually galvanised or Colorbond steel. Increasingly Weathertex, a long-lasting, manufactured, resin-infused hardwood, is used, as this offers protection without conducting heat or cold into the hive. Ordinary wood can also be painted and used. I tend to use a mixture of Weathertex and Colorbond both for practical and aesthetic reasons. Both come ready painted and during the hot Australian summers a well-insulated Weathertex lid offers the bees better protection from the heat. Weathertex comes with a 25-year written guarantee against deterioration during bad weather and does not require painting.\n\nLids that have a wood or plywood top provide good weather protection to the colony provided the wood or plywood is well painted. Another part of the lid design that I am often asked about is the necessity for vent holes in the rim. Some suppliers supply lid rims with the holes for vents pre-drilled, others do not. It is usual to see two vent holes in the front of the lid rim and two at the back of the lid rim covered by either a metal or plastic vent, which prevents robber bees entering the hive. Although the primary use of vent holes is to allow the colony to breathe when the entrance is closed for moving, the use of vents also allows moist air to escape from the hive during winter or during rainy or humid conditions. If moist air condenses on the inner roof it soon forms drops of water which will run down the inside walls of the hive. This dampness will encourage the development of _Nosema,_ Chalkbrood and other diseases in the colony.\n\n3.22: The steel corner angles used to join the telescopic sides of the wooden rim together also raise the rim a couple of millimetres above the super. This allows air to vent out of the super, keeping the hive dryer.\n\n3.23: The telescopic lid overhangs the sides of the hive and provides better protection to the colony against wet weather. A disadvantage of the telescopic lid is that it is much more difficult to pack together hives for transport. Underneath the telescopic lid in this photograph is a top feeder as well as insulation.\n\n3.24: The sides of a migratory lid are flush with the sides of the super, allowing more hives to be packed together for transport.\n\n3.25: Placing a hive mat above the top set of frames under the lids discourages the bees laying burr comb in the lid. The hive mat is cut so that there is about a 1cm gap between the mat and the sides of the box so that air can still circulate into the lid.\n\nMany migratory lids do not use air vents and the hive thus does not have the facility to allow damp air to escape. If the hive is kept over winter in dryer areas this may not be a problem, but I believe that a well-designed hive should have vents in its lid. Vent holes are often left out of lid rims during manufacture for cost reasons but they are easy enough to make with the correct drill bit. A hole diameter of between 15 millimetres and 25 millimetres should be used.\n\nIt is worth noting that some bees will consistently propolise lid vents if provided, but I am happy to give them that choice. Many experienced apiarists believe that if the vent is attached on the outside of the lid the bees are less likely to propolise it, although none of my inside vents has ever been closed by my bees. Beware though, that spiders are able to make nests in vent holes if the grill is attached to the inside and can bite fingers when the lid is removed.\n\n3.26: Airflow out of a hive with a telescopic lid.\n\n3.27: It is a legal requirement across Australia that all hives be marked with the registration number of the owner.\n\n3.28: Plastic hives are usually well designed and come in different colours for ease of identification.\n\nDepending on their design, telescopic lids use a different technique to allow for moist air to circulate out of the hive. In one common design the telescopic lid is raised about 2 or 3 millimetres above the top edge of the super. This small gap provides ample ventilation for a moist hive to vent damp air without being wide enough for the bees to exit or pests to enter the hive. The gap also prevents workers propolising the lid to the super, making it easier to remove.\n\nIn many respects this type of lid seems an improvement on the commonly used migratory lid and I believe that its use will grow in popularity in the coming years. Of course, the inventive practical beekeeper is already making design improvements to the standard migratory lid but the majority of beekeepers will buy their hive ware in a kit form from a beekeeping supply store. The popularity of new lid designs will be influenced by the willingness of suppliers to stock and explain their use and, of course, by beekeepers willing to pay a premium to try something different.\n\nWith the standard North American design for the telescopic lid, a crown board is needed to provide ventilation since this design does not include a 2- or 3-millimetre gap above the top edge of the super. Also, with the standard North American design a top board is needed since the bees are able to propolise the lid to the super and a hive tool is unable to reach under the lid to prise it open.\n\nIn Australia there is a legal requirement in each state and territory for each hive to be indelibly marked with the registration number of the owner. Most beekeepers paint their number on hives although some brand their number as well as marking their frames, as a further safeguard against theft.\n\n**PLASTIC HIVES**\n\nPlastic hives are relatively new to the beekeeping scene in Australia. There are now several types available that take eight or ten full-depth frames. Whether their popularity will grow to the point where they replace a high percentage of wooden hives remains to be seen. These hives are usually ready assembled, are convenient to use, will never need painting and will not rot. While they may be durable and convenient I have a philosophical objection to housing my bees in plastic. I may be old-fashioned and inflexible but that is choice I have made for my own apiary. I must now own up to the fact that I have minimal personal experience with the plastic hives available and much of what I have discovered has been from anecdotal evidence provided by the customers of my business. Some like them and, of course, others do not \u2014 just like everything else in life.\n\nMy only actual experience with a plastic hive is with one that was sent to me as a demonstration model. I did not house any bees in it but stored it outside until the sender arranged to collect it from me. We had one of Melbourne's 42\u00b0C days and the sides of the box buckled under the heat and the frames fell in. Interestingly the box returned to its normal shape during the evening so I was unable to photograph the distortion in the hive the next day.\n\nOTHER PARTS OF A HIVE\n\n**ADDITIONAL WEATHER PROTECTION ABOVE THE LID**\n\nI believe that it is beneficial to the bees in the hive to provide additional protection against bad weather above the lid. A piece of plywood or cement board can be used together with some pine to keep the additional cover raised off the existing hive lid. In addition to the above, during the very hot days of summer, to allow greater air circulation through the hive I will often slightly open the existing hive lid, moving the top cover further over to cover the gap in case of rain.\n\n3.29: Hot sun baking down on a metal hive lid can cause the colony to overheat. I cover my hives with this simple lid made of scrap plywood to provide additional protection against the hot Australian sun during the summer months.\n\n3.30: On hot days a lot of heat can enter the hive through the roof. To help minimise this, I sometimes place an Ideal box containing dry gum leaves between the top super and the lid. To stop the gum leaves falling into the hive I tack fly wire underneath the Ideal box. The box can also be filled with crumpled paper or other suitable insulating material.\n\n3.31: This lid rim has a layer of fly wire attached to it so that the beekeeper can remove the lid without the bees coming out to sting. This is a great idea for people who want to show visitors or children their bees without getting them dressed in protective clothing. These screened lids are also great for moving sealed colonies since they allow air in to the hive while stopping bees from escaping.\n\n3.32: Instead of a wire mesh top cover, a cover made of perspex can be placed under the lid to allow ease of viewing.\n\n**INSULATION UNDER LID**\n\nThis idea originated with the Warr\u00e9 hive. I sometimes provide additional insulation under the lid by filling an Ideal box with dry gum leaves or crumpled newspaper and placing it between the top super and the lid. The leaves are prevented from falling into the super by covering the bottom of the Ideal with wire netting of the type used in flywire screen doors.\n\nSUMMARY\n\nA hive consists of:\n\n\u2022 a base\n\n\u2022 one or more brood boxes\n\n\u2022 a queen excluder\n\n\u2022 one or more supers\n\n\u2022 a hive mat\n\n\u2022 a lid.\n\nAlthough they are called by different names, both a brood box and a super are identical. The use to which they are put in the hive set-up to either house the queen and brood or to store honey determines what they are called. Hives usually come in eight- or ten- frame sizes; they can hold either eight or ten frames per box. The height of the box can vary and the choice is usually made on the basis of weight since a full-size ten-frame super of capped honey can weigh as much as 40 kilograms. Ideal or WSP size supers are gaining in popularity since they are much lighter to lift when full of honey. Many beekeepers are choosing to use hives made of plastic rather than wood as they offer convenience, low maintenance and durability.\n4.\n\nFrames and how to build them\n\nFRAMES\n\n4.1: Frames are made of four pieces of wood: a top bar, a bottom bar and two sides. The slightly narrower bottom bar is used if beeswax foundation is to be inserted into the fame. The slightly thicker bottom bar with a groove cut in it is used if plastic foundation is used.\n\n4.2: The parts of a frame and the tools required to piece them together.\n\nThe frame is a key part of the hive and its design, assembly and ongoing maintenance is more critical than that of any other part of the hive. The frame is made up of four pieces of wood:\n\n\u2022 a top bar\n\n\u2022 two side bars\n\n\u2022 a bottom bar.\n\nBecause frames need to hold and support both brood comb and honeycomb, assembled frames need to provide support for the sheets of beeswax or plastic foundation that are inserted into them. To provide support to beeswax foundation sheets, a full-size frame usually includes four taut wires running horizontally between the side bars. Since the wire is taut, eyelets need to be inserted into the pre-drilled holes in the side bars to stop the wire cutting into the wood when the frame is holding a fully capped load of comb and honey. It is possible to use staples to do the same thing and several punched in to hold the wire on the side of the frame will take the strain off the wood.\n\nThe wooden side bars, the top bar and the bottom bar are usually held together using cement-coated nails with a size of approximately 1.4 millimetres x 35 millimetres.\n\nTwo nails are placed on either side of the top bar and hold it to the side bar. The bottom bar receives one nail on either side. Finally, a small tack is inserted on the top part of the side bar where it joins the top bar.\n\nThe reason for the extra nails on either side of the top bar is that it is the top bar of the frame that takes all of the strain. It will hold the weight of a full frame of comb and also be repeatedly levered out of the hive with a hive tool by the beekeeper. Again, staples can be used and some beekeepers find it easier to use them in preference to nails. Many commercially available pre-assembled frames are stapled. Using a good-quality glue such as PVA or Titebond III prior to nailing or stapling also helps to keep the frame together and add to its useful life.\n\n**FRAME WIRING BOARD**\n\nFrames can be wired by hand with a minimum of tools although I find that a frame wiring board as shown in Visual 4.12 makes the wiring much faster and much simpler than other methods.\n\n**FRAME-MAKING JIG**\n\nMost hobby beekeepers with a small number of hives need only assemble frames one at a time. For beekeepers with a larger number of hives who need to assemble many frames at the same time a frame-making jig will prove invaluable. A frame-making jig for full-sized frames can be made out of a modified WSP box. The use of a jig will significantly reduce not only the amount of time it takes to assemble frames, but will also make their assembly a much easier task.\n\n4.3: Step 1: Before joining the four pieces of wood together, first place a small amount of glue at each join.\n\n4.4: Step 2: Join the four pieces of wood together ready for nailing. You may need to gently tap the corners to get them to fit together snugly.\n\n4.5: Step 3: Next nail the top and bottom bar to the sides.\n\n4.6: Step 4: Since the joins on the top bar have the most stress on them, use two nails through the top and another tack through the side for added strength.\n\n4.7: Step 5: Placing a tack through the side of the top bar for added strength.\n\n4.8: Step 6: The bottom bar needs only one nail through each end since there is not much stress placed on these joins.\n\n4.9: Step 7: Two tacks are inserted into the sides of a side bar to attach the frame wire.\n\n4.10: Step 8: Eyelets are small and difficult to pick up. Using an eyelet tool makes placing eyelets into the side bars of the frames much easier.\n\n4.11: Step 9: Using an eyelet tool, insert eyelets into the side bars of the frames.\n\n4.12: Step 10: One of the tricky parts of attaching wire to frames is getting the wire tight. I use a frame wiring board to do this.\n\n4.13: Step 11: Threading the wire through the eyelets using the round door stops attached to the frame wiring board makes pulling the wire easier.\n\n4.14: Step 12: The tricky part of attaching wire to the frame is getting the wire tight. To do this, use the lever to push the side of the frame inwards.\n\n4.15: Step 13: Tie off the wire onto the two tacks, release the lever, and the side bar returning to its correct shape will tighten the wire.\n\n4.16: Step 14: Once the wire has been tied off, cut the ends and hammer in the tacks until they are flush with the side bars.\n\n4.17: Step 15: The wire has been tied off on the tacks and, once the lever has been released, the wire is taut.\n\n4.18: Some beekeepers attach the wire vertically.\n\n4.19: If you make lots of frames it is worth building a frame-making jig. Here I have used a modified full-size box as the jig. To make the jig cut a small amount of wood from the bottom of the box so that the frames rest on the floor when placed inside the jig. I have attached a Velcro strap across the top of the box to hold the frames in when the box is turned upside down to hammer nails into the bottom bars.\n\n4.20: Re-using old frames often requires that the groove in the top bar be cleaned. A cleaning tool is used to remove the wax from the groove.\n\n4.21: Before re-using an old frame the frame wire sometimes needs tightening. Here I am using a crimper to tighten the wire.\n\nCLEANING USED FRAMES\n\nFrames that contain black comb or those that have been damaged by Wax Moth need not be discarded but can be cleaned and reused. First remove the entire wax comb and then clean out the groove in the top bar. Next remove all of the old wax from the wire so that the new foundation will be easier to attach. If necessary tighten the frame wire using a crimper.\n\nATTACHING FOUNDATION TO THE FRAME\n\nAttaching the wax foundation inside the frame is one of the two tricky parts of making a frame; the other trick is getting the supporting wire taut. The best method of attaching wax foundation is to use a car battery charger to heat the wire so that the wax foundation melts into it. A 12-volt car battery charger, if used alone with stainless steel frame wire, would generate too much heat, may melt the wire and will damage wax sheets in an instant. It is better to use a resistor in series with one of the charger leads to reduce the current. I use a 2-ohm, 50-watt resistor to embed foundation onto stainless steel frame wire. This size resistor is unsuitable for use with galvanised or tinned wire, as the wire will not heat sufficiently for the foundation to melt into it. It is worth noting that many expensive car battery chargers cannot be used for embedding foundation as they often contain sensors that will shut off the charger if a car battery is not detected.\n\nStainless steel is the preferred material for frame wire since it does not corrode when it comes in contact with the natural acids contained in honey. Galvanised or tinned wire was previously used to make frames, but these two materials are no longer popular with beekeepers due to their corrosive properties. An important dimension when designing frames is the bee space. The bee space is about 10 millimetres wide and is the minimum gap that two bees can pass through at the same time when working on opposite comb. If the distance between the two combs is greater than 10 millimetres the worker bees will often fill it up with burr or bridging comb making it difficult and messy for the beekeeper to remove frames.\n\n4.22: Attaching and inserting wax foundation. Step 1: Place a sheet of wax foundation into the frame making sure that the wax fits snugly into the slot in the top bar.\n\n4.23: Step 2: Rest the foundation on a frame board with the wire resting on top of the foundation.\n\n4.24: Step 3:The frame-making board needs to be sufficiently deep that the sides of the frame do not touch the table. This will ensure that the wire is resting on the foundation.\n\n4.25: Step 4: Use an embedding tool connected to a car battery charger to attach the wax foundation to the wire. Connect a resistor in series with the charger otherwise too much current will flow and the wire will melt.\n\n4.26: Step 5: Alternatively, use an embedding tool to press the wire down into soft wax. This needs to be done on a warm day or indoors when the wax foundation is soft and wire can easily be pressed into it. The use of an embedder is not a hard technique to learn but it is not as effective as using a battery charger to attach wire to foundation.\n\nAlthough sheets of wax foundation come pre-cut for full size or other types of frames, I often use strips of wax or broken sheets of wax in my frames. The bees do not mind and soon build out the foundation to fill the frame.\n\n4.27: Instead of using a full sheet of foundation I often insert thin strips of foundation to provide a starter for the bees to build their own honeycomb. Using a thin starter strip is a good idea if you plan to eat the honeycomb since the comb will be much thinner and more delicate to eat than manufactured foundation.\n\n4.28: A thin strip of foundation being built down by the bees.\n\n4.29: A thin strip of foundation completely built out by the bees.\n\n4.30: Even broken pieces of foundation can be used to make up frames.\n\nUSING WAX OR PLASTIC FOUNDATION\n\nThere is a growing trend for professional beekeepers to use plastic foundation in frames rather than foundation made of beeswax. Although plastic foundation is stronger and less likely to break during honey extraction, bees will generally only draw or build comb on pre-waxed plastic foundation when there is a honey flow on or if they are being fed sugar. This is because bees prefer to build comb on natural beeswax and will only draw comb on plastic as a last resort. Once they have initially drawn out the plastic foundation with comb, however, they will keep using this comb without any ongoing resistance.\n\nIf you do decide to use plastic foundation the surface of the plastic sheet will need to be prepared by coating it with a thin layer of beeswax, otherwise the bees will not accept it. To coat the surface I first melt some beeswax in an old skillet or wide pan. Then, using a small paint roller, I dip the roller into the melted wax and run it over the foundation sheet, first on one side and then on the other.\n\nUSING PLASTIC SIDES FOR FRAMES\n\nThe majority of people use wooden frames but there are many suppliers of plastic frames. These are convenient to use and come in a variety of colours and designs to suit the beekeeper.\n\n4.31: Before placing plastic foundation into a hive it needs to be coated with wax. I melt the wax in a skillet and apply the wax using a paint roller.\n\n4.32: Flexing plastic foundation into a wooden frame. Frames made to hold plastic foundation have a groove cut into both the top bar and the bottom bar.\n\n4.33\/4.34\/4.35\/4.36: Plastic frames are sometimes used instead of wooden frames and come in many different designs and colours.\n\nSUMMARY\n\n\u2022 Frames are used in the hive for the workers to build brood comb and honeycomb on. Most frames are made of wood while the foundation that they hold is usually made of pure beeswax. Frames are more difficult to make than the bee hive and the beginner will need to learn the techniques used to both tighten the frame wire as well as attach the beeswax foundation.\n\n\u2022 There is a growing trend for beekeepers to use both frames and foundation made out of plastic. This is due to the superior durability and strength of plastic and, to some extent, the fact that frames made of wood can be time-consuming to wire and embed with foundation. In my experience, however, bees prefer foundation made of beeswax over foundation made of plastic.\n5.\n\nYour first bees\n\n5.1: Honey bee pollinating an almond flower in north-west Victoria.\n\nIn an ideal world every new beekeeper should either pair up with an experienced beekeeper or at least join a local beekeeping club. Beekeepers in clubs are generally friendly, helpful and eager to offer advice to new beekeepers. They are usually passionate about their hobby and proud of their hives. Even if you do not yet have your own hive, you may receive an invitation from a club member to inspect their hives, opening the supers and brood boxes under their guidance. Seize every opportunity and absorb the information given \u2014 it may contradict the book you have just finished reading but it will help you sort through any conflicting advice and information as you increase your own knowledge and practical experience. Clubs offer a lot to a new beekeeper as many have resources that can be borrowed including books, DVDs and extractors. Introductory workshops and courses on other aspects of beekeeping may be offered and specialist guest speakers are frequently invited to meetings. It is the easy access to a wide pool of varying experience that makes a club so valuable for the beginner.\n\nEven knowledgeable experienced beekeepers need specialised advice at times and it is common to see and hear them sharing information and offering assistance to each other at a bee club. If you do not have a club in your part of the country there are apiarist associations throughout Australia and these can often be a good starting point for information. Some beginners prefer to pay an experienced beekeeper to offer them some initial assistance and if this is within your budget it might be an alternative to a club if there is not one near to your local area.\n\nBees lead complex lives and a lot about them remains unknown. There are endless discussions even amongst professional beekeepers about the best way to manage their colonies under different conditions \u2014 and what works in colder Tasmania may not work in Queensland where tropical conditions make beekeeping very different.\n\nOBTAINING YOUR FIRST BEES\n\nIf you live near a large city that has a beekeeping club this will often be your starting point, as club members will provide advice and guidance on obtaining your first colony of bees. As a new beekeeper it is often more convenient to buy an established hive and to move this onto your property than it is to build a new hive and to arrange for either a swarm or nucleus of a queen and bees to be placed in it. Existing hives from a reputable supplier usually come with a guarantee against disease and the seller will often help to move and relocate the hive, advising you of a suitable aspect and location for bees on your property. Another advantage of buying an established hive is that you can ask the owner about the age of the queen, temperament of the bees, frequency of swarming and how much honey they produce. The answer to this latter question could be different once the colony is relocated to your area, which may not offer the same conditions for the bees.\n\nOn the other hand if you buy a hive from, say, an online site without any prior knowledge of beekeeping you will be unable to ask these questions and may be taken advantage of by an unscrupulous seller and sold a diseased hive, one riddled with Wax Moth or one so neglected that you may need to replace the frames and some of the hive ware. You will not have the skills to recognise the problems inside the hive and may be buying an expensive mistake.\n\nNumerous people, including myself, have purchased an existing hive and then started learning about the bees inside it. With the benefit of hindsight I do not recommend that you do this and feel that the common sense approach is to learn about bees before you undertake to keep them. It is unreasonable to expect a more experienced beekeeper to give you valuable time sorting out any problems when you yourself could not make the effort or take the time to acquire some initial knowledge. You have a responsibility to register and to manage your hives in such a way that they are not a nuisance to your neighbours. You cannot do this without the benefit of some prior understanding of the management of your bees. Even if you buy a hive from an established beekeeper you should ideally ask someone more experienced to inspect the colony inside the hive before accepting it to ensure that all is well with the bees. Of course this is not always practical, particularly if you know of no other beekeepers or live in a more isolated rural area. At the very least you should satisfy yourself that the hive has a steady stream of bees coming and going from the hive entrance. This may well mean one trip to the supplier to look at the hive during daylight and then another visit to collect the hive either in the evening or early morning. Do not buy hives that look obviously neglected, rotting or if the seller mentions they were Dad's but have been sitting in the paddock untouched for a few years. These types of hives require a lot of work to sort out and are not really the introduction a beginner should look for into the world of beekeeping.\n\nWHAT KIND OF BEES SHOULD I KEEP?\n\nThe three races of honey bee sold by queen rearers in Australia are the Italian, the Caucasian and the Carniolan bee. Beginner beekeepers often ask which is the best race of bees to keep and I have listed below the different characteristics of the three races.\n\n_Apis mellifera ligustica_ \u2014 Italian bees\n\n\u2022 A slightly smaller bee than other races, the main distinguishing feature being the yellowy and straw-coloured or light brown stripes across its abdomen. Italian bees tend to cling more tightly to the comb when the frame is shaken and are regarded as being good honey producers, less aggressive and easier to handle than other races. Italian bees build up numbers later in the spring than other races although they soon make up these numbers as spring progresses.\n\n_Apis mellifera caucasica_ \u2014 Caucasians\n\n\u2022 This race of bees originated in the Caucasus region north of the Black Sea and come in two types: the Mountain Grey strain from higher elevations and a yellowish strain from the lower elevations. The strain from the lower elevations tends to be more aggressive and stings freely while the Mountain Grey strain is gentle, industrious and survives winters well. They are good honey collectors although they tend to propolise hives more than other races.\n\n_Apis mellifera carnica_ \u2014 Carniolans\n\n\u2022 This bee closely resembles the German black bee both in appearance and habits. Carnies, as Carniolan bees are affectionately called, are docile although they have a reputation for swarming. They are, however, good collectors of honey.\n\nIn spite of the above discussion on the different races of queen bees sold in Australia, most bees found here are a mixture of Italian, Caucasian, Carniolan and dark German races. This is because even if the virgin queen is genetically pure, there is less control over the genetics of the drones she mates with, resulting in workers that are hybrids or cross-breeds.\n\nSHOULD I BUY AN ESTABLISHED HIVE OR CATCH A SWARM?\n\nThe advantage of buying an established colony is that you obtain a ready-made hive with an established colony of bees that can be placed immediately on your property without too much work. For the new beekeeper, the possibility of an established colony swarming during early spring is very real and may be a reason not to buy an established colony at this time. Although swarming is a natural part of the bee's lifecycle it takes experience to manage. If you are in the fortunate position of having an experienced beekeeper to offer assistance with your first hive swarming this will provide an excellent learning experience and allow you to hopefully approach the following spring swarming with some confidence.\n\nAn alternative method of obtaining bees is to first assemble your own hive and then catch a swarm to live inside it. If you are a member of a beekeeping club or know a local beekeeper they will often obtain a swarm for you. They may invite you to go with them to collect the swarm or alternatively you will provide an empty hive and the beekeeper will house the swarm inside it. Sometimes there is a small fee involved and in my own club there is also a prerequisite that a new beekeeper must have completed an introductory beekeeping course prior to being given a swarm. The advantage of a swarm is that it does not usually contain that many bees and these are less likely to swarm again during their first year. This allows time for the new beekeeper to gain knowledge and practical experience during the first year, ready for the swarming season the following year.\n\n5.2\/5.3\/5.4: Swarms often land in easy to access places such as on the side of a shed, on tree stumps near the ground or some swarms make a permanent nest under the eaves of houses. This nest has lived under the eaves of this house near Lake Boga, Victoria, for the last two years.\n\n5.5: The same permanent nest near Lake Boga when the bees moved away temporarily after part of the comb collapsed during very hot weather. The well-developed brood and honeycomb can be clearly seen.\n\nOne further option available is to buy frames of brood and adult bees. Many beekeepers will sell you four or more frames of brood and bees together with a viable queen to place in your hive to start a new colony. These may be placed in your own hive box or sold to you in a four- or five-frame nucleus box. If you obtain your colony in a nucleus box it is worth leaving it alone for a few weeks to allow the bees to orientate to your property.\n\nSWARMS AND SWARM CATCHING\n\nMost new or potential beekeepers, unsure of their skill level, are often reluctant to attempt to catch a swarm. If, however, a swarm is located in an easy-to-reach place, say on a low overhanging branch or on a fence post, then the task of collection is usually quite straightforward. Many swarm catchers become star attractions while removing swarms from public areas and can draw quite a crowd. Although members of the public should be kept at a safe distance it is a fascinating sight for them to see a mass of bees march into a box to join the rest of a collected swarm and queen.\n\n5.6: Swarms often make a home in possum boxes. This box containing a feral nest is ready to be taken down from the tree and the colony rehoused.\n\n5.7: Many swarms land in trees until they find a new home to move to.\n\nSwarms are generally very docile and even tempered since they are not protecting a colony containing brood or food; this is different to the aggressive temperament of nests of bees that are disturbed and immediately go on the attack to defend an established colony. Dressed in protective clothing, the swarm catcher only needs to brush or shake the swarm into a nucleus, brood box, cardboard box or bucket to collect them. Once the queen is in the chosen container the remaining bees will soon follow. It is worth mentioning that if a swarm has been in a location for several days and has started drawing comb then it is possible that they will become agitated when disturbed.\n\nSwarms are readily available in the swarming season, usually between early September and late December.\n\nMany swarms are located in difficult to reach places such as on high branches in trees or difficult to access parts of buildings. If the new swarm is located deep in shrubbery, a pair of secateurs will be a useful addition to the swarm collector's toolkit as these will efficiently remove obstructing smaller branches. A small saw is also useful to cut larger, thicker branches so that the branch and any attached swarm will fall or can be removed to a level more accessible to the swarm collector. If the swarm is located high in a tree, say up to about 5 metres, a plastic bucket attached to an extendable tent pole is useful. The bucket is raised until it is couple of centimetres below the swarm, then with an upward thrust the bucket is knocked into the branch directly where the swarm is located. The majority of the swarm will fall into the bucket, which is quickly lowered and the bees poured into a nucleus or similar box. The bucket is then raised again and the process repeated until most of the bees have been collected. Again, if the queen has been caught and is inside the holding box, the remaining bees will soon join her.\n\n5.8: A bucket attached to a long tent pole is sufficient to bring down many swarms from high up in trees or on the sides of buildings.\n\nIn most cases, even though swarms are not usually aggressive, the swarm catcher needs to use some ingenuity to find ways to access the colony and remove them. I would not allow the thought of possible difficulties to put you off trying to catch swarms that are in easily accessible locations. Catching, handling and managing swarms is a rewarding and educational experience and a useful and informative part of the beekeeper's art. In my experience, beekeepers who are prepared to collect swarms become better and more confident beekeepers as a result. Most feel a real sense of achievement plus they make an obvious contribution to the safety of their local community and save the swarm from possible extermination by other less 'bee-friendly' people or agencies.\n\nAn alternative way to attract a swarm, should one happen to be passing close by or if the bees are in a difficult to reach place, is to place a bait hive near the swarm's location and hope that scout bees will find the box, determine that it is a good home and return to the swarm to let them know of the bait hive's location. To improve the chances of this you can insert a vial of Nasonov scent inside the bait hive to attract the scouts. Nasonov scent is the pheromone from the Nasonov gland, a small gland located above the stinger of the worker bees. Nasonov scent is used by bees to mark their colony and is used as a homing signal so that returning foraging and scout bees can more easily find their own colony. A few beekeeping supply shops have a chemical Nasonov for sale and this can be inserted into a bait hive as a swarm attractant. If you are attempting to catch swarms from your own apiary or hive it is generally recommended that bait hives be placed approximately 100 to 250 metres away from the hive that is about to swarm.\n\n5.9: The Nasanov pheromone is the bees' natural homing signal. By placing a vial of Nasanov pheromone in an empty hive, preferably together with some dark foundation, it is often possible to catch your own swarm.\n\n5.10: Nucleus boxes are frequently used by beekeepers to catch swarms and move them to a permanent location. The nucleus entrance can be sealed with a rotating coloured disc.\n\n5.11: The difficulty when storing a swarm inside a nucleus and moving it long distances is that the colony may not have sufficient fresh air to breathe. At the side of my nucleus boxes I cut large holes and cover them with stainless steel meshing. This allows plenty of fresh air to enter the nucleus if I seal it for long periods\n\nIf a member of the public telephones to ask you to catch a swarm on their property you should consider charging a fee for this service. A lot of your valuable time and petrol will be used collecting swarms and many people who ask for swarms to be removed really do not appreciate the effort involved. Indeed, on occasion you may have to return to the property more than once.\n\nAfter the captured swarm of bees are in residence in their new home, you could add a frame of open or capped brood comb from another hive since the presence of this will make it much less likely that the swarm will leave. Another possibility if the swarm is to be housed in a nucleus box for a while is to consider the use of a nucleus entrance disc. This is a rotating, coloured disc-shaped piece of plastic that is essentially divided into four different entrances that can be selected by the beekeeper. One of these has a queen excluder, which enables the bees to come and go from the nucleus as the swarm is establishing itself, but will prevent the queen from leaving. Effectively, if she stays, there is an excellent chance that the bees will do likewise and the captured swarm will establish into a viable colony.\n\nIf the captured swarm is weak, either merge it with an existing colony or add some frames of brood to build up the number of workers.\n\nGenerally, as the spring swarming season approaches, the experienced beekeeper prepares for this by having a number of items ready and waiting should the opportunity arise to collect a swarm. These may include a nucleus box with undrawn or drawn frames, a strong cardboard box, gardening secateurs, a small saw, an Emlock fastener (named after its inventor, Mr E.M. Lock of Victoria), an old sheet, a sharp knife and a water spray bottle to calm the bees. A large, strong, flat piece of cardboard is useful to have if the bees are to be knocked from a branch and into a container or box. The cardboard can be placed under the area of the swarm before the collection commences. Any bees that fall to the ground will not get tangled in long grass or other debris (on the site) and can more easily access the nucleus or other collecting box, which should be placed at one edge of the cardboard.\n\nTRANSFERRING AN ESTABLISHED COLONY\n\nA more challenging task is to remove an established feral colony with brood and place it into a hive box. This is a messy task and more time consuming than catching a swarm. It is often referred to by beekeepers as 'a cut out', as the beekeeper will literally cut the comb from the feral nest to attach to some ready-prepared empty frames. Many feral swarms are located in possum boxes, walls of buildings, inside roof spaces or other difficult to reach places. The swarm collector needs to decide if it is practical to remove the colony, leave them there or kill them. If you are planning to move a feral colony from a possum box, bird box or elsewhere to a hive, make sure that you do this during the spring or summer when the colony has the best chances of surviving the move. Moving a feral nest during the autumn or winter is likely to result in the death of the colony.\n\nI will go through an example of a feral colony that has made its home in a possum box since this is a common situation, particularly in my local area.\n\n5.12: Rehousing a swarm from a possum box. Step 1: Place the possum box on a plastic sheet and remove the lid and one or two of the sides. This will allow you to gain access to the comb inside the box.\n\n5.13: Step 2: Once the comb is easy to reach, cut sheets of brood comb out of the possum box. A sharp knife with a long blade is best for this.\n\n5.14: Step 3: Next attach the sheets of brood comb to an empty frame using string. Concentrate on getting as much brood comb out of the possum box as possible.\n\n5.15: Step 4: Each sheet of comb is attached tidily to the frame.\n\n5.16: Step 5: Once you have a frame full of brood comb insert it into an empty full size hive. Repeat this until the hive is full of frames containing brood. When you have collected as much brood comb as possible, attach honeycomb to some empty frames and place them into the hive at the edges of the box.\n\n5.17: Step 6: The site is now tidied up and any remaining bees are brushed into the hive.\n\n5.18: Step 7: The rehoused swarm is now ready to be moved to its permanent location once the Emlock fastener has been attached to keep the hive in one piece and the entrance sealed.\n\nInitially the possum box will need to be removed from the tree and brought to ground level. There are various ways of doing this which I will not deal with here, although moving the colony from the possum box to the new hive needs to be done during the day when you can see clearly what is happening and are not rushed by oncoming dusk.\n\nPrepare your working surface with a large piece of plastic, which can be placed on either the ground or an outside table, and will make the clean-up of the resulting mess a lot easier. Once the possum box is on the ground or the table, carefully remove one or two of the walls of the box, taking care to minimise damage to the comb and not to overly excite the bees. Tools such as a screwdriver, hammer and saw will be required to take apart the box. I find a strong screwdriver more useful than a hive tool as it is easier to hammer the handle-end to force open joints in the wooden box using the screwdriver end. The possum box should be taken apart sufficiently that the hanging comb can be cut from the roof in slabs.\n\nBefore you start cutting out brood comb, depending on how the nest has been assembled by the bees, shake or brush as many bees as possible into a hive, making every effort to get the queen into the hive as well. Inside the hive there should be some frames of drawn comb so the queen can start laying eggs immediately and the foragers can quickly start bringing back nectar and pollen to store.\n\nLay some prepared wired frames (without foundation) horizontally on a sheet on the ground, then cut out a section of brood comb from the possum box colony and use string to attach the comb to one side of the wired frame. Select comb with brood and eggs over comb with honey and pollen. Continue this until you have removed the brood comb and then fill any remaining frames with any honeycomb, rubber banding in the same manner.\n\n5.19: Instead of using string to attach comb to frames an alternative method is to use elastic bands \u2014 the thick elastic bands used by the postman are ideal for this.\n\n5.20: If you plan to catch many swarms it will be useful if you make up some frames specially for holding cut out brood comb. Fence netting attached to frames that do not contain any frame wire make good comb holders. Attach the fence netting only to the top bar using cable ties or staples that can act as hinges.\n\nAn alternative to using string to attach brood comb to frames, particularly if you are regularly removing established nests, is to make up some frames with wire mesh on the sides as shown in the photograph.\n\nAttaching brood comb to a wired frame is an untidy business and it is unlikely that you will get the usual number of frames into the hive box or nucleus you are using, but this is not important. Any honeycomb left over can be placed in a sealed container and the honey extracted later as explained in method _1_ in Chapter Nine on extracting honey. Do make an effort, though, to cut out all comb containing larvae and eggs for your prepared wired frames. If at all possible leave the hive with the frames of cut-out comb and attendant bees next to where the possum box was for a day or two. This will allow any residual bees from the possum box to find their new home.\n\nAn alternative method of transferring brood comb from a possum box to a hive is to simply cut out sheets of comb and to rest them vertically inside a hive. To stop the brood comb touching other brood comb place thin strips of honeycomb between them so that the nursery bees will be able to feed the larvae. Although this is a simple technique if the hive is not to be moved, moving a re-housed feral colony using this method will probably result in the comb falling over and many of the bees being killed.\n\nIf all has gone to plan, inside the hive the queen and the workers will be busily establishing a home. This, however, is not the end of the hard work since in a few weeks you will need to tidy up all the old brood frames, replacing them with new frames and foundation. What you do next will depend on whether you placed your cut-out frames in a nucleus box or in an ordinary eight- or ten-frame hive box. If you used an ordinary hive box prepare a second box, inserting into it frames with new foundation and some used drawn comb. Remove from its base the box containing the previously cut-out frames, placing this gently to one side on a level, flat surface. Take the newly prepared second box and place it on the now available base. Next, remove each frame in turn from the old brood box and shake all of the bees into the new brood box sitting on the base. When you do this it is essential that you take care to transfer the queen to the new brood box otherwise you will need to repeat this process in another week or two. Once all the bees have been removed, place a queen excluder over the new brood box containing the queen and bees and carefully place or slide the old brood box on top. Over the next three weeks all of the eggs and brood in the top box containing the previously cut-out comb will hatch and you will be able to remove these untidy frames and clean them ready for reuse. With the queen established in the new bottom brood box, the workers will start drawing brood comb and the queen will recommence laying eggs.\n\nAfter a couple of weeks, if there is no sign of brood in the bottom box, but there are eggs and larvae in the top box this will indicate that the queen remains in the top box and you will need to repeat the process described above until the queen has been successfully transferred to the lower box. Once the queen has settled into the new bottom brood box and all of the brood has hatched in the top box, it is time to remove the frames in the top box and replace them with new frames and foundation, magically changing it into a super and providing room for the new colony to expand. As well as the old frames, the top box itself may be messy with remnant comb with the result that the whole box may need a good clean before it can be used again.\n\nIf, after a period of time, the newly hived colony is overly aggressive, not storing any quantity of honey or the queen is underperforming \u2014 that is, not laying many eggs \u2014 you will need to consider re-queening. Obtain a replacement queen from a reputable queen rearer and the performance of the hive should improve markedly after about three to four weeks.\n\nA newly caught swarm often needs some help to build up its strength quickly. The beekeeper can assist by feeding the colony with sugar syrup and\/or pollen substitute for a few weeks until the bees can stand on their own six legs and look after and sustain themselves. See Chapter Seven on spring and summer management for instructions on feeding syrup to colonies.\n\nWhen tidying up after catching a swarm or moving an established feral colony it is important to mask the smell of the pheromone that the bees have left behind. A lemon-smelling toilet freshener spray is often carried by swarm collectors for this purpose and this is applied liberally to the branches or other structure where the swarm had landed or the colony had made their home. If it is really difficult to get access to the area where the swarm or colony was, buy a bottle of almond oil essence and spray this where the bees were. Bees hate the smell and within a minute or two, depending on the accessibility of the colony, the bees will have disappeared looking for a less smelly place to live.\n\nMOVING HIVES\n\nIt is quite safe to move a hive inside a car if the hive entrance is well sealed and the lid, boxes and base are held together securely. The best way to do this is to use an Emlock fastener or to tie the hive together with ratchet straps of the type used for holding loads securely on the trays of utility vehicles. Duct tape may also be used to further seal the joins where the brood box and lid or base meets. Many new beekeepers feel safer moving a hive on a trailer rather than inside their car. The hive needs to be securely tied or strapped to the front of the trailer so it will not shift or move around during the journey. The gentle rocking of a carefully driven vehicle during transport of the colony has a calming effect on the hived bees and they are generally quite placid during the move.\n\nWhen I move a hive using my car or trailer I always carry a veil and gloves to deal with any emergency that may arise, such as bees escaping due to a box or lid becoming displaced.\n\n5.21: When moving a hive, the entrance may be sealed with a piece of metal.\n\n5.22: Make sure, though, that you seal the entrance after dark when all the bees are inside the hive or the front will soon be covered with returning bees, making it difficult to safely move.\n\nBees in an established hive should be prepared for moving after dusk when the foragers have returned home and all of the bees are inside the hive resting. Sealing the hive entrance can be done by inserting a damp sponge or rag, tacking a strip of wood over the entrance, using a perforated metal entrance closer or plenty of duct tape. Make sure though that there is not a long tail of sponge or rag hanging out of the entrance that could get caught and pulled out. Whatever method you choose, if you try to seal the hive during the day when many foragers are out, you will find that very soon there will be hundreds of returning bees attempting to get back inside the hive. This will make completing your job more difficult and rob the colony of bees who will be lost if the hive is moved without them. A collection of agitated bees will also make moving the hive more difficult.\n\nWhen choosing the entrance closure method, you should think about how long the colony is to be closed in and consideration should also be given to the temperature on the day of the move. If the hive is travelling for a short local trip in coolish weather the sponge or wooden seal can be used. If the hive is to travel some distance a ventilated entrance closure should be selected to allow the bees to keep air circulating throughout the hive. For any trips on very warm days or for long distances the wisest course is to use a ventilated travelling lid, which is a strong wire screen supported by a wooden surround. If the screened lid is used in full daylight, place a hive mat over the screen to limit the amount of bright sunlight and heat entering the hive. The base of the hive can also be modified to allow a better air circulation.\n\n5.23: These hives have ventilated lids allowing the colony to breathe when they are moved long distances.\n\n5.24: Emlocks are used to strap together a hive ready for moving. Note that soft foam has been pushed into the entrance to completely seal the hive. If the hive is only going to be closed for 20 minutes there should be sufficient airflow through the vents in the lid for the colony to breathe.\n\n5.25: Emlocks are assembled as shown. First attach the strap to the cradle by bending it and pushing it through one of the slots. Next the other end of the strap is wound around the handle.\n\nIt is better to wait until after dark before sealing the hive even if you plan to move it later that evening or very early the next day. As long as the day is not hot, it is safe to keep bees overnight and during the next morning in a well-ventilated and sealed hive. They must not, however, be left in the baking sun for even a short period of time. Ensure that they have fresh air to breathe and water to drink. Bees are living creatures; once they are confined in a hive it is impossible for them to regulate the inside temperature and they can very quickly overheat and die. Not only do you risk losing brood and bees but there is the added risk that the frames will melt down all over the colony.\n\nOnce the hive has arrived at its new home it should be moved into its permanent location before the entrance is opened. Prior to doing this, the site should have been selected and checked for suitability, with any brick bases or stands previously prepared. Preferably after dark or very early in the morning when the bees are still resting, move the hive quickly to the new location, remove the straps holding the hive together, and then remove whatever is being used to seal the entrance. If moved after dark and the entrance sealer removed, a handful of bees will slowly make their way out to check what is going on before returning inside the hive. Early the next morning you will see many bees flying around the hive taking orientation flights before leaving to look for nectar and pollen.\n\nMany beekeepers say that hives need to be moved further than 4 kilometres from their previous location otherwise the foragers will quickly return to their old home site. My experience is that I have moved hives very short distances, even 50 metres, without experiencing a great loss of bees. Some of the foragers may return to their old site although I would not let this deter you from moving a hive a short distance if it is necessary to do this, particularly if the circumstances surrounding the move are urgent.\n\nEmlocks, while not an essential item of equipment, are very useful when you need to strap a box together in order to move it. Straps for Emlocks can be made from polyester webbing, galvanised steel, stainless steel or Colorbond coated steel.\n\nPolyester webbing is the cheapest and also has the advantage that the flexibility of the strap makes it very easy to use. Perhaps the major disadvantage is that it cannot be left out for months in the sun and weather as the fabric will slowly disintegrate. For a hobby beekeeper who moves hives occasionally a polyester strapped Emlock is a good choice and when the strap deteriorates it is both easy and cheap to replace.\n\nGalvanised steel is a more robust material than polyester and can be left out in the sun for years without disintegrating. For this reason it is a strapping of choice in situations where hives are to be left outside for long periods.\n\nGalvanised and Colorbond steel are a lot stronger than polyester and the strap is unlikely to break during use. It does, however, have the disadvantage that it is not as flexible as polyester and is a little more difficult to use when strapping together a hive.\n\nProfessional beekeepers often use stainless steel strapping for their Emlocks, not because it is more rust proof than galvanised steel, but because it stretches and is less likely to break after constant use. Stainless steel strapping is slightly more expensive than galvanised steel, but is worth the extra cost due to its superior performance over a protracted period of time.\n\nAlthough I have never had an Emlock come apart on me, for peace of mind I always use two Emlocks to hold together a hive that I am moving. This extra insurance makes sense when you have 50,000 or more agitated and angry bees half a metre away from you inside a car.\n\nGENERAL MAINTENANCE AND INSPECTIONS DURING THE FIRST YEAR\n\nBetween spring and late summer, the peak period for honey flow, the beekeeper needs to regularly inspect the hive to check on the health of the colony, to manage swarming, and to see if honey needs to be extracted or a super added or removed. This is an ongoing procedure and is straightforward once it has been performed a few times.\n\nChapters Seven and Eight on spring and summer management and autumn and winter management provide detailed recommendations on how to approach the inspection of your hives. The instructions detailed in these chapters are suitable for beekeepers during their second and subsequent years of looking after bees. In the first year of your hobby, however, the new beekeeper will need to follow a slightly different course for inspections. This is because the beginner will not be familiar with typical colony behaviour or what to look for inside a hive. As a result, inspections during the first year should be aimed at gaining confidence and experience in opening the hive and learning about the colony of bees inside.\n\n5.26: Larvae and capped brood.\n\n5.27: A frame full of healthy capped brood.\n\n5.28: A new bee emerges from her cell. This is seen frequently inside a hive and is one of the delights to watch as a beekeeper.\n\n5.29: Queens are usually difficult to find. When you find a queen always try to mark her so that she will be easier to find during subsequent visits.\n\nConfidence is important and you will need to inspect the colony often so that you gain this quickly and feel safe from stings wearing your protective clothing. You will need to open your hive and look inside probably once a week for the first few months. (Later, the frequency with which you open your hive for inspection can be reduced to about once every three weeks during the spring and summer.) This weekly visit need not involve removing any of the frames during every inspection although gaining familiarity with this task will help in quickly developing your skills and confidence as a beekeeper.\n\n5.30: Queen cells in a hive are a sign that either the colony is about to swarm or they are replacing a poorly performing or missing queen.\n\n5.31: There needs to be plenty of pollen inside a hive if the colony is to raise healthy brood. Since a single floral source seldom provides the range of proteins that a colony needs to rear healthy brood, look for cells that contain a range of pollen colours since this indicates that the foragers are collecting food from several types of flora.\n\n5.32: The capped brood cells show an irregular or patchy distribution of brood; this indicates a diseased colony and you need to determine the cause of the disease and treat it accordingly.\n\n5.33: Castes of bees. From left to right: female worker, male drone, female queen.\n\nAnother aspect of gaining experience as a beekeeper is honing your powers of observation of the movements of the colony outside the hive. Watch the foragers come and go to see if they are bringing in pollen on their legs. Can you see worker bees feeding each other with nectar at the hive entrance? This is a process called trophallaxis. Observe the general look of the hive when things are going well, so that when they are not you will be able to see the difference even before you open the lid.\n\nSimilarly, the new beekeeper has to become familiar with the look and feel of the bees inside the hive. If anything changes later during the year it is then much easier to identify what is different and to seek advice.\n\nIf you bought the hive as a viable colony do you know the makeup of the hive you are inspecting? Is the colony in a brood box with an excluder between that and a honey super? Alternatively does the hive have two boxes but no excluder? We will assume for the purposes of explanation that you are inspecting a brood box, although in practice it will often be a honey super that will be the first box facing you when you remove the hive lid.\n\nAfter successfully lighting your smoker, before you approach the hive, make sure that the fuel is well alight and that the smoker is blowing a thick, cool white smoke. I can't stress this enough. Many new beekeepers do not ensure the smoker is well alight and smouldering plenty of cool smoke.\n\nBefore any attempt is made to open the hive, first puff smoke into the hive entrance. Then move to the rear or side of the hive and puff another couple of times under the lid as you gently slide it forward enough to give the smoker snout access. Wait about 15 seconds for the smoke to take effect, take off the lid and puff more smoke under the hive mat. This calms the bees and drives them down into the hive box. Remove the hive mat and quickly conduct the first visual check by looking down between the frames into the hive.\n\nIt is only necessary to use two or three puffs of smoke to subdue bees. Beginners often use excessive amounts of smoke, which the colony can take many days to recover from. If you are dealing with very aggressive bees, it may be necessary to wait for 2 or even 3 minutes after smoking before opening the hive.\n\nThen, starting with the second frame from one side, remove the frame and brush or shake most of the clinging bees back into the hive. Visually check both sides of the frame for eggs, larvae, brood patterns, capped or uncapped honey and pollen and make a mental note as your inspection proceeds if there is anything that concerns you and which you may need to discuss with an experienced beekeeper or do some research on. A nucleus box is a useful aid since you can temporarily store frames in it while inspecting your hive.\n\n5.34: Inspecting a hive. First remove the second frame in the hive as shown. The first frame next to the hive wall may be attached to the wood and become damaged if it is removed first.\n\n5.35: Next prise the first frame, next to the hive wall, away from the wood using the flat end of the hive tool. Once the frame has been detached from the hive wall it can be removed and inspected.\n\n5.36 American hive tools are designed differently to the 'J' style or Australian hive tool. The hook end is first used to separate the frame to be removed from its neighbouring frames and then it can easily be removed.\n\n5.37: When I remove frames from a hive I usually store the first frame on the ground resting against the brood box. With one of the frames from the hive removed, there is now plenty of room inside the hive to move, remove and inspect the remaining frames.\n\n5.38: An alternative method of storing frames during a hive inspection is to rest them on a frame holder attached to the super.\n\nI have assumed at this early stage that you are not familiar with disease in the hive so have not suggested a check for this. If, however, you have done your homework and feel you know what to look for this should also be part of your inspection.\n\nOnce you have removed and inspected the first frame you can either place it on the ground leaning against the hive, use a frame holder attached to the hive to temporally hold inspected frames, or place the frame in an empty super or nucleus box beside the hive. When you start it is a good practice to adopt the spare box method of storing your frames during hive inspections. It will keep your frames clean and safe. Frames leaning against the hive sometimes fall over or you can accidently put your foot through them when you are engrossed in your hive inspection.\n\nNext, use the flat end of the hive tool to prise the first frame (the second you are removing from the hive) away from the hive wall and remove it for inspection. Again, to give yourself plenty of room inside the super or brood box you are inspecting, make sure that after you remove this frame it is placed outside the hive leaning against the hive wall, hanging on the frame holder attached to the hive or in the spare super beside the other frame. The reason for starting your inspection with the second frame rather than the first frame next to the hive wall is because the first frame will often be stuck with honey or wax against the hive wall and pulling it out with another frame flush beside it may be difficult, resulting in a damaged frame, broken honeycomb and honey leaking out inside the hive.\n\nNext, gently prise the third frame away from the fourth frame and remove it for inspection. After inspecting the third frame, return it to the hive slightly away from where it was taken, allowing you plenty of room to inspect the fourth frame and so on. Once you have inspected all of the frames, return them to their original positions. As you gain in experience it will seldom be necessary to inspect every frame in a super or brood box since a lot of information can be obtained by looking down between the frames and by removing for inspection two or three frames randomly from either box.\n\nIf you are inspecting frames from the super all you are looking for is a full frame with about two-thirds of the cells capped with a wax layer, which will mean that the honey underneath the cappings is ready for extraction. If most or all frames in the super are almost fully capped, the frames are ready for extraction. If only one or two frames are full, you can remove these frames and extract them by hand or, if you own an extractor or want to wait until you have a greater number of full frames, you may choose to store them. If you choose to do this you will need to freeze frames to avoid any chance of Wax Moth larvae hatching and destroying the harvest. See Chapter Nine on extracting for details of how to do this. You will need to replace any full frames you remove with empty frames containing new foundation or empty comb. Alternatively, you can leave the capped frames inside the super for a week or two longer until the remaining frames are also capped and all the frames in the super can be removed and extracted at once.\n\nThe only times I fully inspect every frame in a brood box are when I am looking for the queen or I suspect that the colony is diseased and I need to determine if this is indeed the case and the extent of the disease. Also look for the queen when you remove frames from the brood box; she can be difficult to find, particularly for a beginner who has little experience spotting queens. When you find her it is well worth the effort of marking her thorax so that she will be easier to find when you need to. Bee supply stores sell a small round cage to hold the queen during this process and a marking pen to add a mark to her thorax. A hobby beekeeper with a few hives can use any colour pen but a larger apiarist may wish to go with the internationally recognised queen colours that signify at a glance the year the queen was introduced to the hive.\n\nWhile the hive is open, take the opportunity to clean the hive walls and frames of any burr comb and also remove any burr comb from inside the lid of the hive. Burr comb in the lid can be minimised by placing a hive mat above the top set of frames under the lid, making sure that there is sufficient gap around the mat for air to circulate. While the hive is open, also check for damaged frames that may need to be repaired or replaced.\n\nFor a detailed list of activities to be undertaken during each inspection in subsequent years see Chapters Seven and Eight on spring and summer management and autumn and winter management.\n\nSUMMARY\n\n\u2022 Bees can be purchased as part of an existing hive, caught when a colony has swarmed, or purchased from a beekeeper as a nucleus hive. Each of these methods is popular so you will need to choose which method of obtaining your first bees suits you best.\n\n\u2022 Once your colony is housed permanently on its site regularly inspect the hive both to gain confidence that your protective clothing will protect you from bee stings, and also to learn the ways and temperament of your colony. An important part of the beekeeper's work is to regularly inspect their hives to check that all is well with their bees and to remove any full frames of honey that need extracting. To become proficient and confident at these activities takes practice, although interest and enjoyment are also reasons for performing them regularly. \n6.\n\nLocating your bees\n\nLOCATING HIVES ON YOUR PROPERTY\n\n6.1: Try to position your hive so that the entrance faces north-east into the morning sun. This is important as an entrance that faces the morning sun is more likely to allow the hive to remain dry and will thus help minimise disease.\n\n6.2: When deciding on the optimum location for your hives there are several things that need to be taken into consideration: morning sun for the hive entrance, shade during the midday heat, a water source near by and easy access for the beekeeper.\n\nFor a hive to remain healthy and productive, its location and orientation is important. Place the hive so that the entrance faces a north-easterly direction into the rising morning sun as this helps to keep the hive dry. It is important that your bees leave the hive as early as possible to collect nectar and pollen, particularly if there is competition from other colonies of bees living in the area surrounding your hive. You want your bees to get the lion's share of any available nectar and pollen and an early start allows for maximum foraging opportunity. Just like other insects, bees are poikilothermic (this is often referred to as 'cold blooded') and the warming rays of the first sun in cooler weather will increase the ambient air temperature around the hive entrance, encouraging the bees to begin foraging activity.\n\nAs previously mentioned, prepare the site prior to moving the hive. If you propose to make a brick platform or place the hive on a stand, have this ready in position prior to moving the hive on to the site. If the hive is to be placed on a rural property, ensure that it is raised off the ground so that there is no risk of long grass and weeds blocking the entrance. This can happen very quickly in times of peak growth like early spring. Bees do not like wind and areas that tunnel wind or are very exposed should be avoided for hive placement.\n\nWhenever I move a hive to a new location I ensure that the hive floor is slanted slightly forward, allowing any water that has collected inside the hive to run out. If the hive were to slope slightly backwards water would pool at the rear of the base, causing the bees to endure wet damp conditions or to drown.\n\nWhile the hive entrance should face the morning sun, it should be covered by shade during the height of the mid-summer day, otherwise the hive will become too hot and the bees stressed. During the winter the opposite occurs and the hive should not be covered by shadow during the day since the bees will consume too much honey keeping the colony warm. Damp, shaded positions encourage mould and condensation in hives, neither of which should be encouraged for the health of your bees. Satisfying all requirements is not easy and the conditions you and your bees face will largely depend on which part of Australia you live in. Some hobby beekeepers find they need to keep hives in a different location in the summer than in the winter in order to keep the hive in optimum conditions for the bees.\n\n6.3: When you position your hive, ensure that it is sloping slightly towards the front to allow condensation to run out.\n\nOften, the biggest concern for the new beekeeper is how near to the house a hive of bees can be kept: will children be in danger, will pets be stung and will the bees bother neighbours? Many beekeepers, particularly in inner suburban situations, keep their bees only a few metres from house entrances, paths or from recreational areas on their property. In practice, I would not recommend this, although much depends on the temperament of the bees and how confident you and your family are around them. My hives were kept for many years only 10 metres from my back door without any incidents. If the bees in any of my hives became particularly aggressive I simply re-queened with a gentler queen and the problem was solved in less than six weeks. In my own situation my family were not frightened by the presence of bees and there were other areas on the property where my children could play away from the hives. To obtain a queen with a better temperament contact a queen rearer and ask how passive their queens are. Alternatively members of your local bee club or a reputable bee supply store will be able to recommend a queen rearer, usually from personal experience or from customer feedback.\n\nGenerally I have not found bees to be a problem with smaller animals and, over the years, I have spoken with many beekeepers who have also kept a variety of other domestic animals with no harmful incidents. To sound a precautionary note, when you introduce bees on to your property be aware that young puppies and kittens can suffer badly from bee stings, particularly if their breathing passages begin to swell, and a visit to the vet for an antihistamine injection may become necessary. Cats generally avoid a hive but older dogs might need to be stung before they, too, learn to stay away. Some beekeepers keep their hives in the same run as their hens and bantams and the two usually live contentedly side by side.\n\nProbably one of the few times that you need to make special arrangements for other animals is if you plan to keep hives in the same paddock as larger animals such as horses and cattle. Larger animals may see the hive as a rubbing post against which to ease an itch. In these instances, and in situations where there are other wild animals such as kangaroos or wombats with access to your property, it is better to fence off the hive with a small, inexpensive fence perhaps made out of star pickets and wire rather than risk the wellbeing of larger animals and your bees.\n\nI think it is fair to state that one of the biggest concerns for the urban hobby beekeeper can be immediate neighbours. Amazing as it may seem to the dedicated beekeeper, some people just do not want bees near their property. The majority of people will suffer an uncomfortable topical reaction to a sting and if a particularly vascular part of the body like the face is affected swelling can be severe. Nevertheless this should not be confused, as it often is, with a systemic allergic or anaphylactic reaction to a sting. Severe allergic reaction to bee venom occurs in roughly 3 per cent of the population, however local council health inspectors often joke that judging by the number of complaints raised by residents in their community they must have the most allergic population in the country. Allergy levels seem to increase in an area when some residents realise that even a not-so-near neighbour is keeping bees! Generally, the average person views the bee as a nasty insect that stings; in fact, many are unable to tell the difference between a bee and the more aggressive European wasp. European wasp and bee stings cause roughly the same amount of pain for a person, however wasps can repeatedly sting and are usually more of a nuisance since they are strongly attracted to barbecues, soft drink cans and food, while bees are usually found only on flowers or in the wrong place looking for a water source.\n\nAs a responsible beekeeper you have a duty of care to your neighbours and should do all that you can to ensure that your bees do not become a nuisance to others. This will involve you re-queening, if necessary for a calm strain of bee, and showing some consideration in the timing of your hive inspections; that is, if you live in close proximity to your neighbour don't open your hive when they are gathering for a barbecue or other outside function. Never forget the peace offering or the shameless bribe of the results of your bees' golden harvest.\n\nTo ensure that beekeeping does not overly inconvenience neighbours, each state and territory has developed its own Apiary Code of Practice. You can Google your relevant code of practice to access the information or call your local council. In Victoria, where I live, although the Apiary Code of Practice is a Victorian Government law, it is administered by local councils through their planning codes. Each code of practice will detail how best to keep bees so that they do not become a nuisance to neighbours or the public. If there is a complaint about your bees and you can show that you are following your state's code of practice, most apiary inspectors will support you in any dispute. With few exceptions most urban dwelling beekeepers can keep at least two hives on their property. To confirm the number of hives permitted in your own area you will need to check your state or territory's Apiary Code of Practice or contact your local council.\n\nAlthough regulations differ slightly around Australia most specify how close a hive can be placed to a neighbouring fence. This is usually around 3 metres from the adjoining fence line. Alternatively, there needs to be a solid, impenetrable barrier 2 metres or higher at the dividing line between the two properties. This barrier can be a solid fence or bushes, and dense trees also seem to suffice. Again the requirement in your own area will be specified in the applicable code of practice.\n\nIt is my experience, and I believe it is the experience of most beekeepers across Australia, that state government apiary inspectors are helpful and will work positively with beekeepers to minimise the impact of beekeeping on neighbours. Although this is also largely the case with local council inspectors, some councils or local authorities have passed laws restricting the keeping of bees within their jurisdiction. This can be a problem for hobby beekeepers, since the majority of bees in an area will come from other colonies, but if there are any incidents near to a beekeeper's property their bees will be blamed and rational discussion may not change a local council's view. Some urban beekeepers have removed their hives altogether as a result of council actions rather than face the costs of a court case. It is always worth checking your local council's beekeeping policies rather than assuming that because your state allows you to keep bees on your property, your local council will also allow this.\n\nHIVE POSITION IN SUMMER AND WINTER\n\nEven though the summer heat is unlikely to melt honeycomb, it can cause it to soften and sag inside the hive, particularly in areas of Australia that suffer from extreme summer temperatures. The build-up of heat inside the hive can be extreme, causing distress to the bees and resulting in poor conditions for them to raise brood. Winter cold I find easier to manage by insulating the hive with polystyrene sheets on the lid and by providing plenty of reserves of honey and pollen for bees to access when they are unable to leave the hive in search of food. I have also of late experimented with a wooden lid that telescopes over the sides of the existing hive lid but is supported above it, creating a large airspace between the two lids. This creates effective insulation in both summer and winter conditions while not impeding any airflow around the hive.\n\nFeral bees prefer to make nests high above the ground. To help make my hives feel more natural to the colony, if I can, I raise them off the ground about 30 centimetres. This has the advantage that a raised hive will:\n\n\u2022 allow better air circulation around and through the hive resulting in less condensation\n\n\u2022 improve air circulation which leads to healthier bees and a dryer hive; generally _Nosema_ or Chalkbrood will be reduced if circulation inside the hive is improved\n\n\u2022 slow grass growing around the hive and over its entrance during your absence, which can restrict or even prevent access for the bees.\n\nI try to raise the base of my hives roughly 30 to 40 centimetres off the ground; any more than this will make inspection and removing supers difficult.\n\nURBAN BEEKEEPING\n\nThere is a growing interest in beekeeping by people living or working in cities and this has resulted in many people, who live in very small townhouses or even apartments with balconies, realising that keeping bees is a hobby that they can participate in and enjoy. Even restaurateurs and other business people with access to roof space in inner cities are finding that keeping bees is a worthwhile activity. Many people believe that bees and cities do not mix, that there are insufficient flowers for food, or that pollution is too high to be good either for the bees or the resulting honey. This view is far from the truth and inner-city beekeeping often results in a lot more honey and healthier bees than rural beekeeping. Why is this?\n\nAlthough most people think that bees and rural areas go together, in fact, much of rural Australia is dry and supports bees only during certain times of the year. This is the reason that many professional beekeepers here transport their hives many hundreds of kilometres every few weeks to follow the flowering patterns of the gums or other nectar producing plants. If professional beekeepers kept their hives in a single rural location the amount of honey they extracted each year would more than halve compared to moving them around. Indeed, Australian professional beekeepers together with their North American counterparts are almost unique in practising migratory beekeeping by transporting their hives to follow flowering patterns. In other countries hives may be moved to take advantage of one floral source or another but not to the extent or for the distances they are transported both here and in the United States. For example, beekeepers from North Yorkshire and from parts of Scotland move their hives to take advantage of the seasonal flowering of heather.\n\n6.4: I raise my hives off the ground about 30cm wherever possible. This stops the entrance from getting blocked by long grass as well as allowing air to circulate under the hive keeping it cooler. Oily rags are tied around the legs to stop ants entering the hive.\n\nDuring dry months and particularly in times of protracted drought, rural areas may suffer water shortages with the result that many migratory beekeepers transport water considerable distances to their bees during these times.\n\nBees kept in urban areas are not faced with the same difficulties. Even in the largest Australian cities there is an abundance of well-kept parklands, botanic gardens, rivers and many homes with well watered gardens of exotic flowers and plants, many of which honey bees thrived upon for thousands of years, long before they were introduced into Australia.\n\nRural areas often contain large agricultural areas and a lot of these are based on monocultures, or growing only one type of plant in an area. In monocultural areas, bees may suffer compromised nutrition and a classic example of this in Australia is the vast almond orchards of north-eastern Victoria. Monocultural planting is not good for bees since it does not provide a sufficiently balanced or varied diet. Bees need a variety of nectar and pollen sources for optimum health. In contrast, urban areas offer a plethora of plant types for bees to feed from, both native bees and honey bees, so their nutrition is far better. Even pollution is not a problem as most urban hives are either kept on rooftops in the central business district or in quieter streets, well away from car and truck fumes. In fact, pesticide use in rural areas is generally far more of a concern than pollution in urban areas.\n\n6.5\/6.6\/6.7: Hives on rooftops.\n\n6.8: Hives on top of Federation Square in Melbourne.\n\n6.9: A hive in Hamilton Gardens, New Zealand.\n\n6.10: A beautiful location for a hive in Tasmania.\n\nMany hives are kept on rooftops in both large and smaller cities, particularly by businesses such as restaurants and hotels. The main consideration is safety and ease of access for the beekeeper to both place the hives on the roof and to inspect and remove frames for extraction. Wind may be strong on exposed rooftops and a sheltered location for hives will need to be chosen.\n\n6.11: Hives kept in the country.\n\nInterestingly, not only do urban bees collect more honey than their rural cousins, urban bees usually start to build up numbers in the spring sooner than country bees, giving them a head start when collecting nectar for honey production.\n\nFEEDING BEES WATER\n\nThere is a legal requirement that beekeepers must make water available to their bees. If there are no rivers or dams nearby, say within half a kilometre, an artificial source needs to be provided and one of the many ways to do this is to use a large chicken feeder. When I use a chicken feeder or other container to provide water for bees I make sure that there are sufficient stones, twigs, or floating objects in the water for the bees to stand on and access the water without drowning. I have heard tales of buckets of water being placed by beehives and quickly filling with drowned bees. It is not obvious to some people that bees cannot walk on water and must be provided with landing platforms. Bees are not fussy about the quality of water and seem to prefer either dirty water, say from a drainpipe, or swimming pool water, perhaps because it contains more minerals or salt. The problem with swimming pool water is that it is often your neighbour's pool that the bees like visiting and this can lead to strained neighbourly relations. To minimise this, ensure that there is plenty of water near hives for bees to drink, as once your bees fix on your own or a neighbour's pool as a source of water it is often difficult to persuade them away from it.\n\n6.12: Using a chicken feeder to provide water to bees.\n\n6.13: Bees drinking water off the ground.\n\nThe appetite and need a colony of bees has for water during a hot summer day can be astonishing. On hot days during a prolonged drought I regularly feed my bees 5 litres of water using a top feeder every two days. I find that mixing a little sugar with the water makes it much more palatable to the bees.\n\nSUMMARY\n\n\u2022 Decide beforehand where you are going to locate your hive and prepare the site in advance. As well as selecting the best location, this may include cutting the grass, generally tidying the site, and building a stand that the hive can be placed on.\n\n\u2022 In the interests of neighbourly relations take care when positioning your hive that it is not too close to an adjoining fence. The regulations of most states and territories will require a distance of around 3 metres in the absence of either a solid barrier or one provided by trees and shrubs.\n\n\u2022 The flight path of your bees should be considered and the hive positioned to avoid bee collisions with neighbours and people on public sidewalks. On your own property your hive should be placed away from children's recreational areas and commonly used access paths to clotheslines and outbuildings. If you follow your state's Apiary Code of Practice and there is a dispute with another person, the local apiary inspectors or council officials will usually be on your side in any dispute.\n\n\u2022 Urban beekeeping is gaining in popularity and, as surprising as it may seem, bees kept in urban or central business district areas fare better than rural or country bees. There is a legal requirement that water needs to be provided for your bees. If there is a natural pool or stream nearby this should be sufficient, otherwise you will have to provide a reliable water source. Bees can safely be kept around most animals and a gentle colony can be kept on a residential block with few concerns provided some care is taken with hive placement. It also helps if your immediate family accept and enjoy your hobby.\n7. Spring and summer management\n\nEven though the management of your hive will be discussed on a season-by-season basis in the following chapters, the situation in your particular hive will strongly be influenced by what happened in the previous season, or even earlier. The health of a colony during spring is dependent on its management during the previous autumn and winter. Also, any adverse effects on a colony during a season will have ramifications well into the future if not promptly managed by the beekeeper.\n\nAs a result, when reading these two chapters do not take each season in isolation, but take into consideration the health and productivity of the colony during the previous season as well as where you would like them to be in the following three to six months.\n\nThe climate in the various states and territories of Australia differs enormously and within these variations there are further localised climates that will affect the beekeeper's management of hives throughout the various seasons. This will mean that there are differing activities for beekeepers in various regions at any one time during the year and these will depend on the seasonal conditions of the particular area. Chapters Seven and Eight on spring and summer management and autumn and winter management will correspond to these seasons in each region of Australia, although the actual timing of these activities among beekeepers in various areas will vary.\n\nPreparing for the massive expansion in the number of bees during the early spring and then, six months later, preparing for the winter shutdown during the late autumn are two of the most important management activities that a beekeeper faces each year. Fortunately, both are straightforward and, once performed during the first year, should become second nature in the following years.\n\n7.1: Although almonds provide bees with a plentiful supply of pollen, they are a poor source of nectar and bees are unable to live off almond flowers alone.\n\nSeptember, October, November and December are the busiest months of the beekeeper's year. To minimise the amount of work that needs to be performed during this period make sure that all maintenance, building new hives and tidying of the apiary is performed during winter. Ensure that syrup feeders and any pollen\/pollen substitute stores are at hand, ready for use. If you plan to re-queen a number of hives, pre-order your queens well in advance as they are often in short supply during the spring. If your hives need to be moved, say to satisfy pollination contracts early in the spring, make sure that you have the logistics planned well in advance. Any delay in the move or the build-up of bees in the hive may make the contract invalid or mean that you miss a honey flow in a region to which you were planning to move your hives.\n\nTo understand hive management you need to understand both the lifecycle of the bee and of the colony. You will also need to understand the seasonal pattern of the flora of your area. While there is little that you can do about the floral sources growing in your area except to move your hives to a more suitable region during lean times, you should gain sufficient knowledge to see in advance that trees are not budding as they should and be forewarned of problems.\n\nThere is a lot that you can do to modify the natural yearly cycle of your bees and this cycle will need to be managed if you are to get the maximum honey harvest from your colonies.\n\nIn order to maximise honey production you will need to:\n\n\u2022 learn to recognise trees and plants and when they are in flower\n\n\u2022 find out when the main honey flow occurs in your area and which plants and trees provide this\n\n\u2022 manage your colonies so they build up a high number of foragers before the honey flow starts \u2014 the importance of high bee numbers prior to a honey flow cannot be overemphasised\n\n\u2022 manage your hives after the honey flow so that they overwinter well and can build up quickly again at the start of the following spring.\n\nIf you want to maximise the amount of honey that your bees produce do not allow bees to build up numbers on the honey flow. Rather, build up the colony beforehand so the hive has the maximum number of foragers going into a nectar flow and is able to take the best possible advantage of the good conditions. To do this you will need to know from observation during previous years when the honey flow starts in your area and to feed syrup to the bees about four weeks before the honey flow commences. For most of the southerly regions of Australia this is usually during mid-August, but will differ in other parts of the country. In areas with little variance in summer and winter temperatures, sufficient nectar sources are often available during the winter to keep colony numbers high.\n\nCorrect timing of the colony build-up ready for the honey flow is critical. It takes three weeks for an egg to emerge as an adult worker bee and then a further three weeks for the bee to become a forager. The six-week period from egg to forager needs careful planning if the number of foragers is to be at maximum strength at the start of the honey flow when the highest number of bees needs to be out collecting nectar and pollen.\n\nMost professional apiarists build up their colonies before spring by overwintering them in areas where there are winter-flowering plants rather than feeding with syrup. As an example, many beekeepers overwinter their colonies in the Mallee in north-east Victoria where there is sufficient nectar available to keep the colonies strong over winter.\n\nMake sure all of your hives are strong. A large number of bees divided between a few weak hives will produce a lot less honey than the same number of bees concentrated in a single hive. Two hives each containing 30,000 bees will produce a lot less honey than a single hive containing 60,000 bees. Larger hives mean that proportionally fewer bees are required for in-hive activities, allowing more bees to become foragers.\n\nINSPECTIONS\n\n**INSPECTING YOUR HIVES DURING EARLY SPRING**\n\nThe last winter inspection of your hive, or even just watching the coming and going of the bees to and from your hive, should prepare you for what to expect during your first spring inspection. Examine your hives in spring when the colony has stopped clustering and bees are out flying. Do not open your hives if the air temperature is below 18\u00b0C as this may kill many of the young brood and weaken the colony.\n\nChoose a warm day in late August or early September in which to open the hive and perform a thorough inspection of every box and frame, vital after the long winter months when you have been unable to do this.\n\nDuring this inspection you need to consider the following:\n\n\u2022 How strong is the colony?\n\n\u2022 Are eggs or brood present? If they are, you know that the queen is there even if you do not see her.\n\n\u2022 Is the brood pattern patchy or solid?\n\n\u2022 How much food is left in case the weather deteriorates and the bees are unable to forage?\n\n\u2022 Inspect for diseases such as American Foul Brood, European Foul Brood and Chalkbrood.\n\n**PROCEDURE FOR THE FIRST INSPECTION OF SPRING**\n\nRemove the lid from the hive and place it upside down on the ground next to the hive. Place the brood box on top of the upturned lid; if two brood boxes were used over winter place both boxes on the upturned lid.\n\nNow that you have complete access to the bottom board or hive base do the following:\n\n\u2022 Clean any debris off the bottom board and wash it down with a wet cloth.\n\n\u2022 If there is an entrance reducer, it doesn't need to be removed during the first inspection but can be removed a few weeks later.\n\nInspect and clean each brood box if two brood boxes were used over winter. Now is the time to reverse them by placing the previous top brood box onto the base. This is because over winter the colony may have moved to the upper brood box and will not be utilising the full volume of the two brood boxes. By reversing the position of the two brood boxes you will break up the winter cluster, giving the colony plenty of room for the queen to lay her eggs.\n\n7.2: If a two brood box hive is used over winter the colony often clusters in the top brood box. To encourage the colony to increase in numbers during early spring the position of the top and bottom brood boxes is reversed.\n\n7.3: If you have kept two brood boxes over winter, now is the time to reverse them.\n\nWhen inspecting the brood box be methodical and follow the steps outlined below.\n\n\u2022 With the first brood box on the base, remove each frame in turn and check it for brood, honey and pollen stores.\n\n\u2022 If brood is present, check that the brood pattern, both capped and uncapped, is solid. If the brood pattern is patchy this may mean disease, an infertile queen, or lack of food stores, and immediate action is required.\n\n\u2022 Check to see that the queen is present. If you are unable to locate her look for newly laid eggs or very young larvae to confirm her presence.\n\n\u2022 If you find a queen mark her as described in the chapter on rearing queens.\n\n\u2022 If you decide the colony needs re-queening it is usually too early to buy a new queen in September and this activity will probably need to be performed later in the season, say in October or November.\n\n\u2022 If you replace frames in a hive during early spring, preferably use frames with drawn comb kept over from the previous year so that the queen can start laying eggs in them immediately without the delay of the workers first needing to draw out the comb.\n\n\u2022 If there are empty frames between the brood and honey frames, place the frames containing honey next to the brood but be careful not to divide the brood by placing the frames of honey between the brood frames, effectively causing the brood to be divided into two parts.\n\n\u2022 If the weather has been poor check the hives weekly for food. If insufficient food is present start feeding immediately.\n\n_Adding a new super_\n\n\u2022 Now that you have inspected the brood area, if the colony is sufficiently strong you can add a super on top of the brood boxes and at this stage, if used, a queen excluder between them. A sign that a super needs to be added is either that the brood box is full of honey and brood or that when you take off the lid bees overflow the sides of the brood box. The super that is being added should also preferably only contain drawn comb since we are coming into the main honey flow period and the bees should not spend time drawing out new comb but should instead be storing nectar.\n\n\u2022 Often bees will not move up into a new super to store nectar. To overcome this, place some frames of brood in the super and the nursery bees will move up to look after the brood, encouraging other workers to follow.\n\n7.4: Bees overflowing from the top of a hive when it has been opened is a sign of congestion and that the colony may be preparing to swarm. If this is the case add another brood box or super to provide the colony with more space to grow.\n\n_Clean the hive mat and lid_\n\n\u2022 Clean the hive mat.\n\n\u2022 Remove and discard any burr comb from the inside of the lid.\n\n\u2022 If the inside of the lid looks damp additional ventilation is needed. When you place the lid on the brood box or super, raise it up slightly by placing some matchsticks, small twigs, stones or Popsicle sticks under the corners. This will allow damp air to escape from the hive, keeping the inside drier.\n\n7.5: Burr comb in a lid is untidy and makes the lid more difficult to remove. A hive mat will minimise bees building burr comb in the lid.\n\n7.6: Popsicle sticks under the lid provides additional ventilation for damp air to escape during early spring.\n\n7.7: Keep the area around your hives tidy and equipment in good condition.\n\n_Clean the area around the hives_\n\n\u2022 Control weeds and vegetation around the hive and next to the hive entrance. If you use a whipper snipper to cut the grass make sure that you wear protective clothing.\n\n\u2022 Bees require a lot of water at this time of year so make sure that it is available.\n\n\u2022 Remove any rubbish from around the hive and keep the apiary area clean and tidy.\n\nAn important aspect of hive inspection is swarm management. Many books and beekeepers say that to perform swarm prevention correctly the brood box needs to be inspected every week during the spring for queen cells. I find that a weekly inspection disturbs the colony too much and seriously interferes with their normal activities of nectar and pollen collection. The bees can become sufficiently agitated that you may not be able to go near the hive for several days without getting stung. Later in this chapter I will describe other methods of swarm management and if these are followed a detailed inspection of your hives need only be performed every three weeks.\n\n**INSPECTING HIVES DURING LATE SPRING AND SUMMER**\n\nThe second and subsequent stage of hive inspection of the season is a much-reduced version of the first stage of hive inspection explained on the previous pages. About every three weeks or less open the hive and check for the following:\n\n_Inspect the supers for honey_\n\n\u2022 Remove the lid and hive mat and inspect some of the frames in the super to determine how much honey is present. If most of the frames are two-thirds or more full of capped honey they can be removed ready for extraction. If the amount of capped honey is much less than this, determine how strong the colony is and how strong the honey flow is, to give you an idea of when to inspect next for capped honey ready to extract. See Chapter Nine on extracting for more details on removing frames for extraction.\n\n\u2022 Clean any burr comb from the box or frames.\n\n_Inspect and clean each brood box_\n\n\u2022 Every second or third inspection check in the brood boxes for a laying queen. If you are unable to locate her, look for newly laid eggs or very young larvae to confirm her presence.\n\n\u2022 If brood is present, check that the brood pattern, both capped and uncapped, is solid. If the brood pattern is patchy this may mean disease, an infertile queen or lack of food stores, and immediate action is required.\n\n\u2022 The above two checks should tell you if the colony needs re- queening.\n\n\u2022 If you replace frames in a brood box, preferably use frames with drawn comb.\n\n\u2022 If the weather has been bad, check the hives weekly for food. If insufficient food is present start feeding immediately.\n\n_Adding a new super_\n\n\u2022 If the colony is sufficiently strong and there is a good honey flow you can add another super on top of the existing super. The new super should contain some drawn comb so that the colony can start storing nectar in it immediately if necessary.\n\n\u2022 Workers often do not like moving up into a new super. To help the bees get used to the new super, try placing some frames of honey in it or even some capped brood.\n\n_Clean the lid_\n\n\u2022 Remove and discard any burr comb from the inside of the lid cavity.\n\n\u2022 If the inside of the lid looks damp at the start of spring additional hive ventilation is needed. When you replace the lid on the brood box or super, raise it up slightly by placing some matchsticks, small twigs, stones or Popsicle sticks under the corners. This will allow damp air to escape from the hive, keeping the inside drier.\n\n\u2022 Condensation inside a hive arises from bee metabolism used to keep the brood warm. If the weather is cold, ventilation is counter-productive if it makes it harder for the bees to thermo-regulate the brood. If the weather is extremely hot add additional ventilation by raising the lid slightly using matchsticks, small twigs, stones or Popsicle sticks under the corners. The lid should only be slightly raised for the duration of the heat wave.\n\n_Clean the area around the hives_\n\n\u2022 Control weeds and vegetation around the hive and, in particular, next to the hive entrance.\n\n\u2022 Bees require a lot of water during the summer; make sure that it is available.\n\n**FEDSS \u2014 FOOD, EGGS, DISEASE, SPACE, SWARMING**\n\nA useful mnemonic to remember when inspecting hives is FEDSS: Food, Eggs, Disease, Space, Swarming.\n\n\u2022 Food: are there sufficient stores of honey and pollen present?\n\n\u2022 Eggs: if you are unable to locate the queen, eggs or very young larvae in cells mean that she was present a day or two before.\n\n\u2022 Disease: check for the signs of disease.\n\n\u2022 Space: is there sufficient space in the hive for the queen to lay eggs or for the workers to store honey?\n\n\u2022 Swarming: during spring and early summer, if the colony is congested or queen cells are present, split the hive to make two colonies.\n\nMANAGING THE BUILD-UP OF BEES IN YOUR HIVE TO MAXIMISE HONEY PRODUCTION\n\nTo get the best possible honey production during a honey flow the number of bees in the hive needs to be at its maximum just before the flow. If the build-up is too slow and the number of foragers is still increasing at the start of the flow, the colony may miss much of the honey flow resulting in a poor harvest. If the colony strength is at its maximum too early the colony may swarm before the honey flow starts. Even if it does not swarm and the colony's population peaks well before the honey flow it will require extensive feeding to keep bee numbers up. At this point you may want to split strong hives and reunite them when needed to reduce the chances of swarming.\n\nIn the early spring, between roughly mid to late August and late September, monitor the number of bees in your hive. You can do this during a detailed inspection but as these are only performed roughly every three weeks a quick way to monitor congestion in your hive is to remove the lid and hive mat, and see how many bees spill over the top of the super. Also, when you remove the hive mat look down between the frames to see how many bees are in the top box. If the hive looks as if it is getting full of bees, add a second brood box or super to allow the colony to expand. As you increase the number of boxes, you can place the excluder either above the first or second box.\n\nThe trend these days among hobby beekeepers is to use two brood boxes and one or more supers for honey, so the excluder, if used, will be above the second box. If you are expanding the number of boxes in your hive from one to three, do not put the two additional boxes on at the same time but add the last box when the first two boxes are full of bees. Bees do not cope well with too much space and extra space also leaves the colony vulnerable to attack by pests such as Small Hive Beetle and Wax Moth. This is because there are insufficient bees in the available space to control these pests.\n\nWhen adding a third box to an existing two-box hive, if possible place the third box in the middle between the brood and existing super box, rather than on the top. Bees often do not like moving up into a new box, particularly if both the box and frames are new. Adding a new box sandwiched between the brood and an existing super will encourage them to move up to the familiar scent of the existing super and overcome any reluctance.\n\nMANAGING THE HONEY FLOW\n\nIt is not difficult to identify a honey flow using the following as a guide.\n\n\u2022 Bees are busy collecting nectar and are usually not aggressive \u2014 although on Messmate Stringybark the bees become aggressive defending and protecting their colonies!\n\n\u2022 White wax is being used to build new honeycomb.\n\n\u2022 A frame of honeycomb tipped on its side causes a shower of nectar to fall out.\n\n\u2022 The entrance of the hive is active with many bees coming and going.\n\nUnlike honey, nectar takes up a lot of room in the comb until most of the water has been evaporated. For this reason before the honey flow starts there needs to be sufficient space in the hive for the workers to store nectar. An empty super needs to be added before the flow if the colony is to collect and store as much honey as possible. It is best to use drawn comb in the new super as the bees will be able to store nectar in it immediately. If a new super with only foundation is added, the bees will expend time and energy drawing comb and not take full advantage of the honey flow and as a result not maximise the harvest for the colony and the beekeeper.\n\nIf foragers are bringing back a lot of nectar to the hive the colony may become honey-bound. This occurs when most of the available brood comb is filled with honey, leaving little if any space for the queen to lay eggs. As a result the queen may stop laying and the number of available foragers will decrease with the result that little new honey is being stored. A honey-bound hive needs to be managed quickly by removing frames full of honey and replacing them with empty frames of drawn comb.\n\n7.8: A frame that is two-thirds full of capped honey is ready to be extracted.\n\n7.9: If forager bees return to the hive with too much nectar the hive could become honey bound and the queen has little room to lay eggs. This results in a rapidly weakening colony.\n\nDuring a strong honey flow a super can be filled in one to two weeks, so add additional empty supers early to ensure that there is sufficient room for honey to be stored.\n\nIn order to maximise the amount of honey that bees can store in a super, place only seven frames in an eight- frame super or nine frames in a ten-frame super. This allows the bees to store more honey per comb and is easier for the beekeeper to extract.\n\nMERGING COLONIES\n\nIf you catch a small swarm or have a colony that has decreased in size significantly over winter and is not strong enough to provide much honey, the best option is to merge it with another colony in order to produce a single strong hive. This is a straightforward task using the newspaper method. If both colonies are in separate hives, decide which hive is the strongest and leave it on its base on the ground, remove the lid, place two sheets of newspaper over the hive and make some horizontal cuts in the newspaper using a sharp knife. The cuts should not allow the bees to immediately move through the slits since the two colonies will fight if they merge too quickly. Next, place the second hive, without the base, on top of the first hive, leaving only the lid on top of the second hive. The only barrier separating the two colonies is two sheets of newspaper. Over the next few days the bees in both the top and the bottom hives will slowly eat through the paper, getting used to each other's scent as they merge together. This means it is very unlikely that they will fight, but will calmly join to produce a much stronger and healthier colony.\n\nWhen merging two colonies it is preferable to kill the weaker queen before starting, but if I cannot locate her I just join them together and let them fight it out on the assumption that the stronger queen will win.\n\nA strong queen heads a colony that collected the most honey during the previous season. She also lays a solid brood pattern with plenty of brood being raised; that is, free from disease. A weak queen heads a colony that collected less than average quantities of honey during the previous season, does not lay a solid pattern of brood and may have a hive that is diseased. You may also prefer to eliminate the queen that produces the most aggressive workers.\n\n7.10: Two colonies of bees may be merged using the newspaper method. The lid of the first colony is removed and replaced with newspaper that has slits cut into it.\n\n7.11: Next, the brood box of the weaker colony is placed above the newspaper.\n\n7.12: After a few days the bees will have eaten through the newspaper and will have merged without fighting.\n\nFEEDING SYRUP\n\nAs part of managing hives for the spring build-up, many beekeepers feed their colony sugar syrup and\/or pollen substitute. There is a general consensus among beekeepers that bees should be fed a sugar\u2014water mixture at a 1:1 ratio during the spring and a stronger 2:1 mixture during autumn. Studies have shown that feeding bees the less concentrated 1:1 mixture in the spring encourages them to build up brood numbers and to go foraging. The stronger 2:1 mixture is better for encouraging the bees to convert the syrup directly to honey for use as food over winter.\n\nFeeding syrup to bees is straightforward and there are several methods that are widely used. These are:\n\n\u2022 frame feeders\n\n\u2022 front or Boardman feeders\n\n\u2022 top feeders\n\n\u2022 placing a large can of syrup above the top super\n\n\u2022 a Ziplock plastic sandwich bag.\n\nFrame feeders are useful for feeding syrup to weak colonies since the feeder can be placed immediately next to the brood frames where the in-hive bees can easily find the syrup and carry it back to feed the brood. Frame feeders are also good when raising queens since their small size enables them to fit easily into four- or five-frame nucleus boxes. For stronger colonies, some of the available smaller frame feeders are not very useful since they do not hold sufficient quantities of syrup, are often inconvenient to refill and require the colony to be disturbed more often than is good for them, particularly in cold or wet weather. To stop the bees drowning in the feeder I insert Gutter Guard as a ladder for them to walk up and down on. Pieces of cork, straw or small twigs can also be used for this as they will float on the surface of the syrup and provide landing pads for the bees as the level drops.\n\nFront or Boardman feeders are 0.5-litre to 2-litre reservoirs placed at the entrance to the hive with a small trough leading from the reservoir into the hive. Front feeders are easy to use, but have the disadvantage that they encourage robbing since the syrup is located at the hive entrance.\n\nTop feeders, sometimes referred to as Miller feeders after their inventor Dr C.C. Miller, are to my mind the most practical method to feed syrup to bees. The top feeder is essentially a very shallow wooden box; it has the same dimensions as a hive and sits under the hive lid. It consists of a large reservoir holding in excess of 5 litres of syrup. On one end of the feeder is a wide gap that allows the bees to crawl up from the hive and gain easy access to a small feeding reservoir. Top feeders have the advantage that they can hold a large amount of syrup and, if the feeder is designed carefully, the reservoir can be replenished without disturbing the bees or the beekeeper needing to wear protective clothing. To stop the bees drowning in the smaller feeding reservoir\n\nI again use Gutter Guard as a ladder for them to walk up and down on. Pieces of cork, Popsicle sticks, straw or small twigs can also be used for floats instead of Gutter Guard.\n\n7.13: A frame feeder is a popular method of feeding bees and a method that I use frequently.\n\n7.14: Feeding syrup to bees. A front feeder is convenient to use but increases the risk of robbing.\n\n7.15: Top feeders enable large quantities of syrup to be fed to a colony with minimal risk of robbing. I have had a 3.5 litre top feeder emptied every three days over about two weeks.\n\n7.16: Schematic diagram of a top feeder and a front feeder.\n\nAnother popular method of feeding syrup is to put an empty super above the top super or brood box. A large tin or plastic container is placed upside down, resting on some thin strips of supporting wood. Large coffee cans are ideal for this purpose. Punch some nail holes through the lid, about 1-millimetre in diameter, and the bees will drink the syrup through the holes. This is an inexpensive and convenient way to feed bees and is adopted by many larger beekeepers due to its ease of use. If the holes are kept about 1-millimetre in diameter a small quantity of syrup will flow out when the container is inverted but this will soon stop.\n\n7.17: A can of syrup inside an empty super above the top super. Two sticks are used to raise the bottom of the can above the frames.\n\n7.18: Ziplock bags of syrup with a slit across the top, resting on the top frames of the super, are an inexpensive way of feeding syrup to a colony.\n\n7.19: Inserting a slit in a Ziplock bag to allow the bees to drink the syrup.\n\n7.20: A block of candy in a baking tray can be used as an emergency food supply during the winter.\n\nZiplock sandwich bags are also an inexpensive and practical way to feed bees with syrup. Place the Ziplock bag on the hive mat or top bars of the uppermost frames and cut an opening in the top of the bag using a very sharp knife, razor blade or scissors.\n\nThere is only one rule for feeding bees syrup and that is to avoid any feeders placed outside the hive. The use of a central feeder placed away from the hive or away from a number of hives encourages robbing, feeds other beekeepers' hives or feral bees, is not hygienic and will lead to the spread of disease.\n\nSUGAR CANDY\n\nSyrup can be fed to bees during most of the year during the warmer months. During winter, however, particularly in colder regions like Tasmania and the higher mountainous regions of Victoria bees will not feed on syrup. During the winter months in an emergency I feed my bees with crystallised sugar candy, using the following formula.\n\n_Ingredients and equipment:_\n\n\u2022 2.5 kilograms of white sugar\n\n\u2022 0.5 litres of water\n\n\u2022 4-litre aluminium\/stainless steel saucepan\n\n\u2022 Wooden spoon\n\n\u2022 A bowl of cold water, big enough to sit the saucepan in or fill a sink with cold water\n\n\u2022 A shallow baking tray or lamington tin lined with baking paper\n\n\u2022 Plastic bags to store the slabs of sugar candy once it has crystallised in the baking tray.\n\n_Method:_\n\nPut the water into the saucepan on moderate heat and slowly add the sugar, stirring continuously. Bring it to the boil, then boil for 3 minutes. Remove the saucepan from the heat and stand in a bowl or a sink of cold water. While the mixture is cooling, keep stirring.\n\nWhen the solution starts to thicken pour it into the baking tray or tin, scraping as much as possible of the solution out of the saucepan with the wooden spoon before the solution starts to set or candy.\n\nWhen the slabs of sugar candy are cool they can be removed from the mould complete with any lining paper and stored in plastic bags until needed.\n\nTo use the candy I place it above the hive mat under the lid of the hive. The bees will find it here and eat it as they require. Candy is only an emergency feed if frames of capped honey are not available and cannot be used as a long-term feeding strategy.\n\nFEEDING POLLEN OR POLLEN SUBSTITUTE\n\nSyrup is fed to bees as a substitute for nectar so that they can build up stores of honey, albeit poor-quality honey. Honey is used to provide carbohydrates for the bees so that they have the energy to walk or fly, keep warm and to run their internal biological processes. This is only part of their needs, however, and for the colony to survive it also needs a plentiful supply of pollen. Pollen is used as a source of protein and minerals, and is essential if brood are to grow and develop. Nursery bees eat pollen in the form of bee bread and convert it in the hypopharyngeal gland in their head into royal jelly. The royal jelly is then fed to larvae and, in this way, they obtain the protein they need to grow into healthy bees.\n\n7.21: Brushing pollen into empty brood cells to feed protein to the colony. This method of providing pollen to a colony is more efficient than sprinkling pollen over the top of brood frames.\n\nPure pollen is more expensive than pollen substitute, which is typically made out of soy flour or yeast, although pollen is sometimes added to pollen substitute to increase the nutritional value of the food. Recently, irradiated pollen has become available from China and is growing in popularity among professional beekeepers.\n\nPreviously, the most common method of feeding pollen substitute to a colony was to mix the powder with sugar syrup to make a patty and to leave the patty above the hive mat for the bees to eat. This method, although convenient, has fallen out of favour since the patty attracts Small Hive Beetle and Wax Moth larvae that thrive in the protein-rich damp mixture. The preferred method today is to brush the dry pollen or pollen substitute powder directly into empty brood comb where it will last for a considerable amount of time. It is possible to place a teaspoonful of dry powder on the top of each brood frame and let some of the powder fall down over the bees. My experience with this method is that the powder, if not quickly consumed by the bees, will attract moisture and grow mildew. This will also happen to any powder that ends up on the base of the hive. As a result the base will need frequent cleaning so that it does not attract Small Hive Beetle and Wax Moth larvae.\n\nSWARMS\n\n**MINIMISING SWARMING BY COLONIES**\n\nWhile swarming is a natural part of the reproductive lifecycle of the honey bee, there are steps that can be taken by the beekeeper to minimise it. These include:\n\n\u2022 Ensuring that the hive is not congested with bees.\n\n\u2022 Alleviating congestion by the addition of a further brood box or super. If two brood boxes are already in use on the hive, swap the position of the two brood boxes. Take the top brood box and place this under the existing bottom box, as this will provide more room for the queen to lay eggs. The top brood box is usually more congested than the bottom box and swapping their positions will help to overcome this. If, however, both brood boxes are full and any queen cells are present you should artificially swarm or split the hive into two as described below.\n\n\u2022 Making sure that there is plenty of room for the bees to store honey and pollen. During a strong honey flow the bees will store nectar in brood cells and can cause the hive to become honey bound. If there is no room for nectar to be stored, either remove capped honey frames or add another super.\n\n\u2022 Regular re-queening with a young, healthy queen should minimise any swarming activity over the current season since queens seldom swarm until they are a year old. Another advantage of re-queening every year is that queens lay far more eggs during their first year, so the colony will be much stronger as a result.\n\n\u2022 Selecting queens that are of a lineage less likely to swarm.\n\nProfessional queen rearers raise queens that are selected for several attributes, one of which may be less of a tendency to swarm. Talk with several queen rearers about the traits that they select for and also with other beekeepers about their experience with commercially purchased queens.\n\n\u2022 Using artificial swarming techniques such as the Demaree (see p. 126) method to control swarming.\n\n\u2022 Providing sufficient ventilation to keep the hive cool during hot days and evenings.\n\nIf, despite your best efforts, you believe a hive is about to swarm you will probably be able to manage swarming using one of the techniques described below.\n\n**SPLITS**\n\nOne popular method of managing swarming is to fool the old queen into thinking that she has already swarmed and is living safely in a new hive. This is called the split method and is described here. Next to the hive that is about to swarm place another hive, preferably one containing a few frames of old brood comb. From the hive that is about to swarm, remove about half of the frames that have brood in them, very carefully inspect each frame that you have removed to see if there are any queen cells on them. Be aware that a careful inspection is called for since some queen cells are not easy to see. If there are queen cells present, place all of the frames with queen cells into the new brood box that you have prepared, leaving the old queen in the old hive with the remaining frames of brood and honey. The old queen will think that she has swarmed and will remain contentedly in the old box, while the new box containing the new queen cells will develop into a new colony with healthy bees. Of course it goes without saying that you must replace the frames removed from the old hive with new ones, either with foundation sheets or, even better, ready drawn comb if you have them.\n\nFor more detailed information on splitting a hive see the split method of rearing queens in Chapter Eleven.\n\n**DEMAREE METHOD**\n\nAnother popular and well-utilised method of managing swarming is the Demaree method, named after George Demaree who first published the details of the method in 1884.\n\nWhen using the Demaree method, the beekeeper first separates the queen from most of the brood by placing the majority of brood into a second brood box that the queen is unable to enter. The two brood boxes are separated by a super and two excluders. If you were to look at the set-up you would see a brood box with an excluder on top. Then a super is placed on top of this with a further excluder where the lid would normally sit. A second brood box is then placed on top of the excluder and a lid placed on top in the usual way. The result is a central honey super with two brood boxes, one on the top and one on the bottom. This results in very little congestion and adds room for the queen to lay eggs. In essence, the colony believes that it has already swarmed.\n\nHere are the basic steps:\n\n1. Lift the brood box from the base.\n\n2. Above the base place a new brood box filled with empty, drawn comb.\n\n3. Remove the centre two frames of drawn comb and set aside.\n\n4. Go back to the initial brood box and find the queen.\n\n5. Place the queen and two frames of sealed brood in the centre of the new brood box sitting on the base, leaving most of the worker bees in the top box.\n\n6. Place a queen excluder above this box.\n\n7. Above the queen excluder, place one or more empty honey supers and then the original brood box where you found the queen. Push the brood nest together in the centre and put the two empty drawn frames from step 3 on either end of the new brood box.\n\n8. Add your hive mat and lid.\n\n9. After one week go through the top brood box and remove any swarm cells.\n\n10. If necessary, the entire procedure may be repeated after nine or ten days if the hive continues to be congested.\n\nNow that you have the Demaree hive set-up this is what happens.\n\n\u2022 Most of the nursery bees stay with the brood in the top box and care for them.\n\n\u2022 A small number of nursery bees stay with the queen in the bottom brood box and look after the two frames of brood that were placed in it.\n\n\u2022 The older workers continue to forage for honey and pollen.\n\n7.22: Demaree Method. This two-box hive is congested and is likely to swarm.\n\n7.23\/7.24: The queen and one or two frames of brood are left in the bottom brood box while most of the bees and brood are placed in the top box. A third honey super is placed between the bottom brood box and the top brood box, with two queen excluders placed at the bottom and the top of the honey super.\n\n\u2022 The queen continues to lay eggs and has lots of room on new frames to do so.\n\nThis situation, while much like a hive that has already swarmed, has the major difference that both parts of the colony are in the same hive. You should therefore be aware of the following:\n\n\u2022 As soon as the queen scent decreases in the top box, the bees may try to raise a queen from young larvae.\n\n\u2022 You may need to destroy these cells or remove them to another nucleus hive.\n\n\u2022 After the brood hatches, the brood cells will be backfilled with honey.\n\n\u2022 In the end, the hive will not have swarmed, so it will contain lots of bees and lots of honey.\n\n\u2022 The growing hive may once again develop the urge to swarm, which is why a second Demaree is often needed.\n\nThe Demaree method can be a very effective tool to use in swarm control, but as you can see, it is quite labour intensive. It involves a lot of manipulation, provides good opportunities to lose or damage your queen and involves a lot of heavy lifting. On the other hand, not only does the Demaree method prevent swarming, but you can obtain some queen cells in the process. For the Demaree method to be successful you also need to be able to find the queen. Every time you inspect the brood box of a hive look for the queen and if you find her make sure that you mark her so she can be more easily found next time.\n\nOne more important point: when you set up the Demaree hive be sure to remove any swarm cells that are already present. Any cell not removed may hatch and cause a problem within the hive.\n\nPROVIDING HIVES FOR POLLINATION\n\nMany professional and hobby beekeepers provide hives to horticulturalists and others to pollinate their crops and trees. This is an extremely important aspect of today's apiary industry and will increase in importance in the future. Honey bees are by far the most important pollinators in the world, pollinating about 70 per cent of the types of food that we eat and 30 per cent of the actual food that we eat. Historically there has been a great reluctance by commercial crop growers to pay for the rental of hives, but this attitude is slowly changing. In the past growers have been able to rely on feral colonies of bees to pollinate their crops although if (or when) _Varroa_ or _Tropilaelaps_ enters Australia feral colonies will disappear and this benefit will quickly end.\n\nIt should be noted that providing hives for pollination purposes is a specialist activity as it requires a good level of knowledge and experience to have hives at optimum bee levels prior to placement, not to mention a lot of hard work for the beekeeper, both in preparing the colonies, delivering them and arranging to collect them again after a relatively short period. Providing hives for pollination may mean a reduced amount of honey being collected yearly and anyone wishing to offer this service needs to understand both the advantages of an attractive new source of revenue plus the disadvantages that providing this service may bring.\n\nThe preparation of colonies for pollination needs to be thought through and planned well in advance. A typical hive used for pollination consists of two brood boxes containing about a box and a half of bees. That is, the two brood boxes need to be almost full of bees for the hive to be acceptable to a crop grower. Also, since adult bees need to be about three weeks old before they start foraging for pollen there needs to be a high proportion of older adults in the colony. The colony will also need to have a high requirement for pollen and this is caused by the colony having a large number of uncapped brood that need feeding by the nursery bees.\n\nTo provide colonies that satisfy these requirements \u2014 large numbers of bees, many older adult bees that are foragers, and large numbers of uncapped brood that need feeding \u2014 takes careful planning and early preparation of the hive. If the pollination service is to be provided early in the spring, say for the almond pollination in August in north-east Victoria, preparation of the hives must start before winter and is likely to include supplementary feeding of both syrup and pollen substitute.\n\nHives destined to pollinate late-flowering crops may need to have frames of honey or capped brood removed to give the queen the room needed to continue to lay eggs.\n\nColonies to be used for pollination at any time of the year must have plenty of spare brood cells available so that the queen will not slow down her egg-laying rate. In addition, a congested hive may swarm which also needs careful management by the beekeeper to control.\n\nAnother unique feature of hives used for pollination is that the hives need to be moved often many hundreds of kilometres to the crops. Moving the hives is best done at night and the colony needs sufficient air to breathe. Most professional apiarists today shift their colonies 'open entrance', that is, they do not close the entrance and accept that about 10 per cent of their bees will be lost in transit. Professional beekeepers can save a lot of time by not closing the hive entrance and also know that the loss of about 10 per cent of their bees will be made up in about one week. Another popular method of moving hives long distances, particularly for hobby beekeepers, is to use a mesh cover instead of a lid to allow fresh air to circulate through the hive.\n\n7.25: Providing hives for almond pollination in north-east Victoria.\n\nHives should be placed near the crop to be pollinated when about 10 per cent of the crop is in flower. Earlier than this and there may not be enough nectar or pollen for them to live off and they may visit neighbouring crops owned by other people to collect food. Pollinating a nearby crop owned by an adjoining landowner may be an issue for the farmer paying for the pollination service. The hives should be removed from the crop when about 90 per cent to 95 per cent of the crop has stopped flowering. Many new beekeepers see pollination services as a quick way to make ready money but, as mentioned, providing healthy hives in optimum condition at the right time is a skilled and time-consuming occupation. Brokers who manage pollination contracts employ specialists to check that bee cover in hives is satisfactory, as the farmer does not want to pay for colonies that are weak or unable to work to capacity. In large monoculture crop areas where bees are brought in from great distances by various beekeepers, the chances of the transfer of disease and pests is greatly increased and these will need to be dealt with or controlled when the hives are removed.\n\nAnother important part of managing hives for pollination is to ensure that your bees are not killed by poisonous sprays applied either to the crop to be pollinated or to other nearby crops. This is best done by talking with the crop owner about any plans to spray near your hives and also by talking with other farmers or crop sprayers locally about their activities. If you are in doubt about the effect of the spray or its possible drift towards your hives it is best to remove them from areas of risk. It is seldom that a beekeeper providing pollination services will be fully or even partly compensated for lost hives caused by poisoning.\n\nCOLLECTING POLLEN\n\nHarvesting pollen is a good activity during the spring and summer when the flowers are blooming. Attaching a pollen trap to your hive is straightforward and you can collect a lot of pollen every day. In practice, since collecting pollen restricts how much pollen the bees are able to store and use to feed their own brood, I only use a pollen trap every other day. On the days that I am not collecting pollen I leave the entrance to the trap open so that the returning foragers can enter and leave the hive freely. Pollen absorbs water and will quickly grow mildew if it is not harvested every evening. Once I harvest the pollen I place it in a plastic bag and store it in the refrigerator or freezer.\n\nSUMMARY\n\n\u2022 Managing the colony to get it ready for spring and summer is an important activity that needs to be performed by the beekeeper. If you wish to have a strong colony of bees ready for the main nectar flow feed the bees a 1:1 sugar\u2014water syrup mixture from about late August onwards. The first hive inspection of spring is important, not only because it will be the first time that you have inspected the colony for a number of months, but also because you need to take the hive apart and clean a number of its components, particularly the base.\n\n\u2022 Hives should be inspected at least every three weeks during spring and summer. A good mnemonic to remember for what you need to look for when inspecting a hive is FEDSS \u2014 Food, Eggs, Disease, Space and Swarming.\n\n\u2022 Colonies are likely to swarm during spring or summer and managing this is a legal requirement for beekeepers, particularly in urban areas. Two of the most popular methods of managing swarming are the split hive and Demaree methods.\n\n\u2022 If you aim to provide hives for pollination make sure that the colonies are strong or the farmer may not accept your hives. Providing hives for pollination is a skilled occupation and needs careful planning. You may need to feed the colony a 1:1 sugar\u2014 water syrup well before the pollination contract to ensure that the queen lays to capacity and that the colony becomes strong. \n8.\n\nAutumn and winter management\n\nAUTUMN\n\n8.1: An experienced beekeeper shows beginners how to shut down a hive for winter.\n\n8.2: Cleaning a hive.\n\n8.3: Apiary ready for winter.\n\nAutumn is a critical time for hive management as the colony needs to be protected against robbing by other bees and also needs to be shut down ready for winter. If either of these two activities is performed poorly the colony may not survive or may enter spring with insufficient numbers of healthy workers available to forage for the nectar and pollen necessary to build up brood numbers.\n\n**FEEDING READY FOR WINTER**\n\nA colony needs between four and six full frames of honey to survive over winter. A colony in the cool temperate conditions of Victoria and Tasmania will need greater stores than one in the tropical north that does not need to use a lot of energy to keep warm. When starting your beekeeping activities it is better to err on the side of the bees and leave them at least six full frames. Many southern hobbyist beekeepers feel comfortable leaving a full super of eight frames for their bees over winter. Do not over rob your bees as you will end up with a weak colony going into winter and one that may not survive to see the spring. If sufficient numbers of full frames of honey are unavailable, either due to a poor season or because the beekeeper has robbed the hive of too much honey late in the season, the colony will need to be fed syrup that it can use to build up its stores. The previous chapter on spring and summer management explains the main methods used to feed sugar syrup and pollen\/pollen substitute to bees (see pp. 120\u20144). The use of sugar syrup is fine while temperatures are high enough so that the colony has not formed a winter cluster. If the colony runs out of honey during the winter either add a frame of stored honey, candy or sugar syrup close to the cluster so that the colony will be able to easily reach it without overly breaking the cluster. In some of the warmer areas of Australia winter temperatures may be sufficiently high that the colony does not form a cluster and there may even be forage available at some locations over the winter months.\n\nCandied sugar, though, is a last resort for the bees and cannot be relied upon to provide all the carbohydrates the bees require over a long winter since candied sugar is not that easy for the bees to consume. White sugar from the supermarket contains too little water for the bees to easily eat; candied sugar contains over half its volume as water, which the bees can more easily consume.\n\n**MINIMISING ROBBING**\n\nMost robbing of colonies occurs when nectar supplies are scarce and other honey bee colonies are desperately looking for food. In Victoria and Tasmania this typically occurs during the autumn. During this time the European wasp is often out in strength and these too can rob the hive of both honey, brood and adult bees. In other parts of the country, such as the Sydney region, one of the biggest honey flows is during the late autumn and thus robbing is less of a problem there at this time of year. To restrict robbing you should:\n\n8.4: Many beekeepers reduce the size of the entrance before winter to minimise the amount of cold air entering the hive.\n\n8.5: The entrance reducer inserted into the entrance of the hive.\n\n8.6: Bees feeding on pollen substitute.\n\n\u2022 Reduce the entrance of the hive to a width of 5 to 10 centimetres. Bees are better able to defend the smaller entrance from intruders.\n\n\u2022 Ensure that when you inspect the hive it is left open for the minimum amount of time.\n\n\u2022 Do not leave burr comb, frames of honey or spilt syrup near the hive to attract robbers.\n\nWhen you are near your hives, keep watching the entrance for unusual activity. Robber bees are very fast and aggressive in their attempts to enter the hive. If you notice this type of activity shut down the entrance to the hive completely for an hour and the robber bees may go away.\n\nNot long ago I inspected one of my hives on a Sunday and found that it was full of honey. It was too late in the day to extract so I planned to return the following weekend to remove the full frames. I returned the following Sunday and, to my astonishment, all the frames in the super were completely empty. Robber bees had found the colony and had stolen all of the honey, leaving an empty super.\n\nApart from mass robbing occasionally occurring, robbing attempts by individual intruders is an everyday occurrence and often just watching a hive entrance you will see bees pushing other bees away. This pushing away seldom involves stinging but the intruder soon gets the message and moves on. Potential robbers being pushed away from the hive often have different body markings and colours to the colony they are trying to penetrate.\n\nTHE WINTER SHUTDOWN\n\nThere are four reasons why, after the summer is over, a hive needs to be prepared for winter.\n\n\u2022 The hive probably consists of between three and four boxes and this is too many for the much smaller number of bees to keep warm over winter and to protect against pests such as Wax Moth and mice.\n\n\u2022 Depending on the region of Australia, the beekeeper needs to ensure that the colony has sufficient honey and pollen available to survive over winter. This, sadly, is often not the case and the main reason that previously healthy hives do not survive winter is that they run out of honey and are unable to keep the colony warm enough to survive. Bees around the Sydney area of New South Wales rarely run out of honey over winter because of Melaleuca trees and successful late autumn foraging.\n\n\u2022 Before the start of winter, pollen or pollen substitute may need to be provided by the beekeeper, so that if pollen supplies are short during early spring the workers can use the pollen supplement to make royal jelly to feed to the developing larvae.\n\n\u2022 Check on the strength and health of the colony and confirm that there is a queen present.\n\nThe climate in the region of Australia in which you live and the trees that are in flower during autumn will influence the date of your winter shutdown. Towards the end of April to late May you will need to take the following steps to prepare your hive for winter.\n\nOn a warm autumn day reduce the size of the hive down to one or two boxes. Assuming that you have a three-box hive, consisting of two brood boxes and one super, in order to reduce the number of brood boxes down to one or two; do the following:\n\n\u2022 First remove the lid from the hive and also place the top box, or super, at the side of the hive.\n\n\u2022 Next remove the queen excluder and place that at the side as well.\n\n\u2022 Place a plastic container that can hold the empty frames next to the hive.\n\n\u2022 Remove each full frame from the super and shake or brush the bees back into the hive.\n\n\u2022 Now place the frame, without any bees attached, into the plastic container and shut the lid \u2014 if you do not shut the lid, bees from the hive will smell the honey and within a couple of minutes the frame will again be covered with bees.\n\n\u2022 Take the second frame out of the super and repeat the procedure until all of the frames have been removed from the super and placed in the sealed plastic container.\n\n\u2022 Alternatively, a clearer board can be used to remove most of the bees from the super down to the brood boxes, as explained in the chapter on harvesting honey.\n\n\u2022 Take away both the empty super and the box of frames.\n\nI used to reduce my brood area down to one box over winter; more recently I only reduce it to two boxes. As long as there is not a queen excluder separating the two boxes the bees will be happy in either one or two boxes over winter.\n\nThe main exception to the above procedure occurs if you need to reduce the number of brood boxes, say if you have a three-box hive and do not use a queen excluder, and you do not want to kill any brood in the top brood box. Reducing the number of boxes in this situation is more time consuming and needs to be performed over a few weeks. Remove the top brood box and place it above a queen excluder making sure that all of the bees have first been shaken off into the bottom brood box. This will ensure that the queen is in the bottom brood box. When the brood box is placed above the queen excluder, worker or nursery bees will quickly move through the excluder to look after the brood left behind, leaving the queen below the excluder. Return to the hive after three weeks. All of the brood should have hatched in the top box and the box can now be removed as explained above.\n\nIf the frames are only half full of honey and you do not want to store them, use the following method to remove the honey from the frames before taking them away for storage. Remove each frame in turn, score the surface of the capped honeycomb with a scratcher and return the frame to the super. Repeat this process until all of the capped honeycomb in the super has been scored and honey is flowing freely from the cells. Carefully turn the super upside down and place it back onto the hive. Since the cells are now facing downwards all of the honey will flow out and will be quickly cleaned up by the bees. The bees will now store the honey in the lower parts of the hive since they will be unable to replace it in the downward-facing cells.\n\n8.7: Reducing size of hive for the winter. Both a super and the queen excluder have been removed from the threebrood box hive, leaving a two-brood box hive.\n\n8.8: I use an airtight plastic storage bin to store frames of honey over winter. Note that I have attached two pieces of wood at either side of the container to support the frames.\n\n8.9: Full frames of honey can be stored in polythene bags from the supermarket. Before placing the frames in the plastic container I freeze the frames in the freezer for two days to kill Wax Moth or other larvae.\n\nAfter a few days the frames will be clean and can be removed and stored. Turning a super upside down after first scoring the capped honeycomb is also a good way to remove hard or crystallised honey from a frame. The use of a queen excluder to stop the frames falling out when the super is turned upside down helps a lot.\n\n8.10: Removing half-full frames of honey. Step 1: If you have half-full frames of honey that need to be removed for the winter, one way of getting the bees to remove the honey for you is to score the surface of the capped and uncapped comb. Next, return the scored frames back to their super before placing the super upside down on the hive. This is also a good method for removing hard candied honey from frames.\n\n8.11: Step 2: Turn upside down the super of scored frames and place it upside down on the hive. A queen excluder is used to stop the frames from falling out when you are turning the super upside down.\n\n8.12: Step 3: The last step is to remove the queen excluder before placing a lid on the hive. Within a few days the bees will have removed all of the honey and stored it elsewhere in the hive. The empty frames can now be removed and stored for the winter.\n\n**CHECK THAT THE HIVE HAS SUFFICIENT FOOD TO LAST OVER WINTER**\n\nBefore placing the lid back on the hive, check that there is sufficient honey left in the brood boxes, about four to six full frames, for the bees to live on over winter. If not, and you have some full frames of honey taken previously from the same hive or some frames from other hives that are disease free, replace some of the emptier frames in the hive with full ones. Previously stored honey frames are best because honey contains minerals and enzymes that sugar syrup does not. If you do not have any full frames of honey available, follow one of the syrup-feeding procedures described earlier in Chapter Seven on spring and summer management. Before winter, if syrup needs to be fed to the colony so that they can quickly convert this into honey to store, use a ratio of 2:1 sugar\u2014water by weight for the syrup. The bees can much more easily convert this mixture to honey than the 1:1 ratio sugar\u2014water mixture used in the spring to encourage the queen to lay eggs.\n\nMost experienced beekeepers leave four or five full frames of honey in the hive over winter for the colony to feed on, and up to six frames if the hive is kept in a particularly cold location such as parts of Victoria or Tasmania. If too many frames of honey have been removed during the autumn, the colony may not have sufficient stores of honey to survive the winter. At the start of spring, finding no live bees in your hives and only dead ones with their heads pushed deep into cells as they desperately sought food is particularly distressing and a sign of very poor management.\n\nCHECK THE STRENGTH OF THE COLONY\n\nWhile you are inspecting the colony check how many bees are present. Going into winter with a weak colony will mean that the colony will not be able to build up strength by the time the first honey flow starts in early spring. If the colony is weak at the start of the winter and you are unable to feed it sufficiently so that it will increase in strength there is no point taking the weak hive through winter and then having to deal with it in the spring. Culling weak or diseased hives is best done during the autumn. It is the time to merge any weak hives with stronger colonies.\n\nMAKE SURE THAT THERE IS A LAYING QUEEN PRESENT\n\nCheck the hive for a laying queen. If you are unable to locate the queen look for eggs or young larvae and, if these are not present, you may need to either re-queen or merge the colony into another hive for the winter. If only capped brood is present, however, it may be impossible to introduce a new queen to save the colony.\n\nIf brood is present, look for an even brood pattern across a frame. Patchy brood may indicate a weak or failing queen or even disease.\n\nIf you decide that you need to re-queen you will have to do this earlier in the autumn. A young, vigorous queen will continue to lay later into the autumn and will also resume laying earlier in the spring ready for the first honey flow. Colonies headed by young queens are also much less likely to swarm during their first spring.\n\nEven if there is a queen present she may not be laying eggs at that moment. Check again in a couple of weeks to confirm if she is present or not.\n\nMAKE SURE THAT THE COLONY IS FREE FROM DISEASE\n\nCheck for signs of disease as this will need to be treated or managed if the colony is to survive and be strong, ready for the spring nectar flow. If the hive contains Small Hive Beetle you will need to introduce beetle traps or an approved miticide trap to kill the beetles.\n\nA strong foul smell may indicate that American Foul Brood or European Foul Brood is present. Treatment for these two diseases is discussed in Chapter Fifteen on brood diseases.\n\nOTHER THINGS TO CHECK FOR DURING AUTUMN MANAGEMENT AND PREPARING FOR WINTER\n\nChecking the hive during autumn is crucial to making sure your bees and the hive are healthy and will function well in spring. Some other tasks to do at this time are listed below.\n\n\u2022 While you are inspecting the hive check for any damaged parts of the hive that will need replacing or repairing over winter. Remove any damaged parts of the hive and ensure that the hive is in sound condition.\n\n\u2022 You may want to reduce the size of the entrance to about 5 to 10 centimetres to protect against robbing. Many beekeepers believe that this is useful to protect the colony against cold air although my experience is that a hive that has appropriate levels of honey in reserve does not need protecting against the cold. Also, if mice are a problem in your area consider adding a mouse excluder to the entrance to stop them entering the hive and making a cosy home for the winter, destroying comb in the process.\n\n\u2022 In many parts of the country warm winters may mean that the colony can remain active all year long and there may not be a need to close down the hive over winter.\n\n\u2022 Confirm that the hives are placed in a suitable location, free from drafts and facing the winter sun. Provide a wind break if necessary.Also, raise the hives 30 to 40 centimetres above the ground away from dampness; this will also allow fresh air to circulate.\n\n\u2022 Remove any frames that need replacing due to damage or dark or poorly formed comb.\n\n\u2022 If necessary, rearrange the frames so that brood is in the centre of the hive with stores of honey and pollen next to them.\n\n\u2022 Control weed and grass growth around the hive and keep the general area of the apiary clean of rubbish.\n\n\u2022 If large animals are present, either fence off hives or use Emlocks to keep them closed in case of a disturbance. Do not keep the Emlock strapping really tight as it may break, particularly after a lot of use or in cold weather.\n\n\u2022 Some beekeepers completely strip down the hive in the autumn, rather than in the spring, to replace damaged equipment and to clean the boxes, lid and base.\n\nWINTER\n\nDepending on the region of Australia that you live in, particularly in parts of Tasmania, the mountainous regions of Victoria, southern New South Wales and the Australian Capital Territory, you may not be able to open your hive between early May and late August. Much depends on the temperature since exposing the colony to cold may quickly harm the brood and severely reduce the strength of the hive coming into the spring honey flow.\n\nHoney bees do not hibernate over winter but, like penguins, cluster together to keep warm. The colder the air is around them, the tighter the cluster. The bees on the outside are closest to the cold and regularly change positions with bees deeper in the cluster in order to keep themselves warm. When the air temperature inside the hive drops to 18\u00b0 Celsius the bees start to form a loose cluster, and when the hive temperature has dropped to 14\u00b0 Celsius all of the bees are included in the cluster. It is important during the winter that this cluster is not broken during a hive inspection as many brood may die as a result. In all but the very coldest parts of Australia hives will carry brood throughout winter.\n\nThe adult bees generate heat by shivering, but there are limits to how much heat this can produce. As the temperature drops the cluster contracts into a tighter ball and it is this tighter ball that keeps the brood warm.\n\nThere may be drone cells left late in the autumn or even during the entire winter in the northerly, warmer regions of Australia. Drone cells are usually located along the bottom edge of a brood frame. If the temperature drops and the workers form a cluster around the brood, as the cluster grows smaller, fewer of the drones will be covered and so may die. This is another way of nature saying that male drones are dispensable when conditions are poor and that the focus of the colony is to maintain female brood since they will ultimately be needed to look after the colony.\n\n**WINTER MANAGEMENT**\n\nWinter management of the hive comes down to ensuring that there is sufficient honey available for the colony to survive on until spring. Since poor weather may make it impossible to open the hive to check how much honey is left, the best method of checking honey stores is to lift one end of the hive to see how heavy it is. A heavy hive has plenty of honey available for food while a light hive has little honey available for food and the colony needs feeding immediately. This can best be done by quickly taking the lid off the hive and replacing an empty frame with a frame full of honey stored since the previous autumn. Make sure though that the new frame of honey is placed next to the cluster since if the honey is too far away in the brood box the workers will not be able to break the cluster during very cold conditions to reach it and may starve to death anyway.\n\nIf a spare full frame of honey is not available, another way to feed emergency sugar to a starving colony is to place a block of candied sugar above the hive mat as described on p. 123.\n\nSUMMARY\n\n\u2022 Preparing your hive for winter is an important activity and needs to be performed by the beekeeper during autumn. If the hive includes a super and a queen excluder, both need to be removed leaving only one or two brood boxes for the colony to overwinter in. Winter survival of the colony is dependent on there being sufficient stores of honey present for the bees to feed on. In the colder regions of Australia six full frames of honey will need to be left for winter, and a smaller number in the warmer regions or where there is still a good honey flow during the colder months.\n\n\u2022 Robbing may become a problem during the autumn as there is generally little forage around and nearby hungry colonies may target your hives if they offer ample supplies of honey. One way to minimise robbing is to close down the size of the hive entrance to about 10 centimetres.\n\n\u2022 Once the temperature outside drops below 17o Celsius the hive should not be opened for long periods and brood frames should certainly not be removed for inspection.\n\n\u2022 Take the opportunity presented by the autumn closure of the hive to tidy up around the hive area, to make any repairs and to think about any new boxes and frames you will need to make over the quieter winter period.\n9. Extracting\n\nThe first extraction for a new beekeeper is always both exciting and daunting. Exciting because all the hard work, time and effort by the bees and beekeeper has finally come to fruition. Daunting, as extraction often appears to be a difficult task, which it is not, and the process raises the added concerns about how much honey will be extracted and its quality. My experience with new beekeepers indicates that they usually underestimate how much honey a hive will produce. The amount obviously varies according to the season, the available forage and, of course, the weather. In a good year roughly 20 to 30 kilograms per extraction is obtained and this may be repeated approximately two or three times each season. This amount of honey can be obtained from approximately ten or twelve full frames of honey. In favourable conditions the average hive will provide about 50 to 80 kilograms per season to the hobby beekeeper. This is a lot of honey.\n\nMany new beekeepers believe that the only way to extract honey is to use an extractor. While it is true that either a hand-cranked or electric extractor makes harvesting of honey easier, there are other inexpensive ways to do the job. I will begin with a discussion of the inexpensive labour-intensive ways to extract honey and will go on to discuss the use of an extractor at the end of the chapter.\n\nUSING A SIEVE TO HARVEST HONEY\n\nIf you do not have access to an extractor, the use of a sieve or strainer to drip and filter cut out comb is an inexpensive and simple way to harvest your honey. Begin by cutting the capped comb out of the frame in the area immediately under the first frame wire. Leave the comb above the wire, where it joins the top bar, untouched. This will provide the bees with a straight edge to redraw the comb when the frame is returned to the hive. Place the cut-out comb into a large sieve, chop it up with a knife and let the honey slowly drip through the mesh into a bucket or other container. As the honey needs to be liquid for this type of extraction to be successful it is better to choose a warm day. Alternatively leave the sieve and bucket in a warm kitchen to drain slowly. As the honey drains from the comb, periodically chop up the comb and compress it to allow more honey to escape. A quick way to speed up extraction using this method is to leave the sieve and bucket for a short period in filtered sun where the heat will quickly liquefy the honey and it will soon drain into the bucket. A plastic bag will need to be placed over the sieve to stop flies and dust contaminating the honey. Also keep watch for bees that will appear very quickly to rob the harvest if the sieve and honey are not completely sealed. It is important that you do not leave the honeycomb in the sun for too long or the honey will overheat and turn brown, losing much of its flavour and goodness.\n\n9.1: When using a kitchen sieve to extract honey, first cut the capped comb from the frames, place it in the sieve and cut it up further allowing the honey to flow through the sieve.\n\n9.2: Place a plastic bag over the sieve and pail to stop dust and bees landing on the comb.\n\nWhen removing sections of comb out of frames, cut half a frame length at a time between a pair of wires. Longer lengths are difficult to handle and are likely to fall outside the bucket or container holding the sieve and create a mess.\n\nA simple device to make and use when cutting comb out of frames is a length of wood 2.5 centimetres by 5 centimetres with a screw inserted in the centre of the long side. Choose a screw that is long enough to protrude through the 2.5-centimetre depth of the wood by at least 1 centimetre. The device is placed horizontally across a bucket with the screw point facing upwards. When the side of the frame is balanced on the screw it can be swivelled easily and the comb, as it is cut out, will fall into the bucket below. After this the cut-out comb can easily be moved from the bucket to the sieve without creating a mess.\n\n9.3: When you cut out sections of comb for placing in a sieve leave a strip of intact comb at the top for the bees to use as a starter to build new comb. This frame was left overnight in a hive and the colony had already started to build new comb from the starter strip.\n\n9.4: Using a scratcher to uncap honey comb is an easy and inexpensive way of preparing frames for the extractor.\n\nAs previously mentioned, make sure that when you cut out the honeycomb for extracting that you leave a strip of honeycomb above the top wire. The bees will use this as a template or starter to build new comb and you will not need to attach new foundation to the frame. Bees take longer to make their own foundation and also consume additional honey to make the wax. The benefit though is a more natural cell size and cleaner comb for the bees to use.\n\nA frame is generally ready to harvest when it is more than two-thirds full of capped honey. If in doubt turn the frame on its side and give it a shake \u2014 if any runny liquid falls from the frame it is probably not ready to harvest as the watery honey may ferment, spoiling the rest of the harvest.\n\nUSE OF AN EXTRACTOR TO HARVEST HONEY\n\nIf you have access to an extractor the uncapping process is different since you will leave most of the comb intact on the frame and remove only the surface capping that has been used to seal the ripe honey. To remove the capping the same wooden support and bucket described earlier is used. Again, a frame is ready to harvest when it is more than two-thirds full of capped honey.\n\n**REMOVING CAPPINGS**\n\n**METHOD 1**\n\nHolding a capped frame over an uncapping bucket, use a scratcher to score or break up the capping. Once the surface has been broken up, repeat the process on the other side of the frame. The frame should now be placed inside the extractor. Repeat the process on the remaining frames until the extractor is full of uncapped frames.\n\nA scratcher, although easy and flexible to use, has the disadvantage that it produces a lot of wax particles that will soon block the sieve used to filter the honey. As a result the sieve will need to be cleaned frequently during the extracting process.\n\n9.5: Using a knife to uncap honeycomb. This is slightly more difficult than using a scratcher but the advantage of it is that the capping comes off in sheets and does not block the honey filter.\n\n9.6: A convenient way to uncap frames or to remove sections of comb for filtering in a sieve is to use a wooden bar with a screw or nail pushed through it. The wooden bar sits on top of a plastic bucket.\n\n9.7: Before using knives to uncap frames of honey heat them in hot water so that they will more easily cut through the wax.\n\n**METHOD 2**\n\nAn alternative to using a scratcher is to use a long knife to cut off the capping. As before, the frame is rested on the screw and, with a downward sawing motion, the sharp edge of the knife slices the capping off the comb. Using a knife is slightly more difficult than using a scratcher, but has the advantage of peeling off the capping in a single sheet that does not block the filter with fine wax particles.\n\nSimple extracting knives are inexpensive and may need to be heated in hot water before use. Many beekeepers use two knives, one of which is heated in hot water while the other is being used to cut the capping. When the knife used for extracting cools down, it is returned to the hot water and the other knife removed, dried and used. On warm days, when I do most of my extracting, I have often used a knife without first heating it since I find the wax capping is sufficiently warm and pliable to be cut without the need to heat the knife. If you are heating a knife using water make sure you dry it quickly so that you do not contaminate either the frame you are uncapping or the cappings in the bucket below with water dripping off the knife.\n\n**METHOD 3**\n\nThe most convenient way to remove cappings is to purchase an electrically heated knife that will easily cut through the wax capping. Electric knives have a heating element and thermostat inside the blade. Some have adjustable thermostats, but most are factory set and will remain at the correct temperature for cutting through wax during uncapping. The knife is moved fairly quickly through the surface cappings in order to avoid scorching the honey contained in the uncapped cells.\n\nSteam-heated knives used to be popular and remain available at most beekeeping supply shops. Steam knives are cheaper than electric\n\nknives, but have the major disadvantage that they require a water boiler, which the\n\nbeekeeping supply store is unlikely to provide. Steam knives are also more cumbersome to use since there is a pair of hoses connected to the knife, the first to provide steam to heat the knife, the second to move the used steam away from the beekeeper. Some older beekeepers prefer to use a steam knife as they feel a steam knife retains its heat better than an electric knife, even though it is less convenient to use because of the need for an attached water boiler and steam pipes. If a beekeeper has simple plumbing skills a pressure cooker can be used to make a steam boiler. It is imperative when producing steam under any pressure that the boiler is not allowed to burn dry.\n\n**USING AN EXTRACTOR**\n\nOnce the capping has been scored or removed it is time to place the frames into an extractor to harvest the honey. The frames need to be inserted taking into account the direction of rotation of the cage. This is because the bee builds comb cells with the bottom of the cell pointing slightly downwards towards the back of the frame, and if the frame is rotated in the wrong direction the honey will pool in the bottom of the cell and not flow out under the force created by the rotating extractor.\n\nAs a general rule, when determining the number of turns that the frame should be rotated the 75\u2014150\u201475 rule applies. First, rotate one side of the frame 75 turns to remove half of the honey from one side. Turn the frame over and rotate the next side for 150 turns to remove all of the honey from this side. Now return the frame back to its initial position and rotate a further 75 turns. This process is used in order to prevent the frame 'blowing out'. If for example you start by rotating the first side of the full frame for 150 turns the weight of honey on the unspun side remains the same whereas the side that is being spun is becoming more fragile and loses substance and strength as the honey is thrown from the comb.\n\n9.9: Comb does not point out horizontally from the frame but slopes upwards at an angle of about 13 degrees. If this slope is not taken into account when inserting the frame into an extractor the honey will not be removed but will instead collect in the bottom of the cell.\n\n9.10: Worker bees build comb so that it is sloping slightly upwards. This minimises the chances of either honey or brood falling out.\n\n9.11: When inserting frames into an extractor, make sure that the orientation of the frame takes into account the direction of rotation of the extractor.\n\nThe unspun side of the frame, with the centrifugal force of the extractor and its weight, may push through the foundation, breaking the honeycomb and resulting in a lot of extra work for the beekeeper to repair or replace the wax foundation.\n\nSometimes, with thinner foundation, the comb will break during extraction. This is not a problem to the bees; just use your fingers to repair the comb so that there are no wide splits in it, return it to the hive and the bees will soon repair the damage leaving a flawless comb to store honey in.\n\nIf you have, say, only two frames left to extract and are using a three-frame extractor, you will need to keep an already extracted frame in the extractor cage so that the unit does not become unbalanced and wobble when rotating. When the two frames are empty of honey, you can remove all three frames and the extraction is over.\n\nAfter the frame has been removed from the extractor it is called a 'sticky' since it is covered in honey and sticky. If you do not plan to return the sticky frames to the hive after extraction, they cannot be stored covered with honey since this will attract bees, ants and other insects. The frames need to be returned to the hive for 24 hours immediately after extracting so that the bees can clean them, removing all the honey remnants, leaving the frames and comb spotless of honey and wax particles. After 24 hours you can remove the clean frames permanently from the hive for storage if required with little risk of attracting Wax Moth or Small Hive Beetle, unless the frames contain large amounts of protein from larvae skins or pollen, which Wax Moth larvae like to eat.\n\nOne point worth stressing is the necessity to uncap frames and to extract in a room, shed or garage that has doors that can be closed tight to keep out inquisitive bees. It only takes one or two bees to smell the honey and they will soon return to the hive to tell their sisters about the bountiful supply of delicious food that is currently waiting for them in your chosen extracting space. Very soon this space will be full of hundreds of bees all looking for the honey that you are extracting. Many years ago when I was new to both bees and extracting I set up my extractor in the garage, spun a few frames, and went inside for a cup of coffee, forgetting to shut the garage door. When I returned about twenty minutes later there were literally thousands of bees in my garage not only covering the extractor, frames and uncapping bucket, but flying around and covering just about everything else. Removing the bees was almost impossible and I had to patiently wait until dusk, when they returned to their hives. Only then could I continue extracting.\n\nIn addition to the above methods of honey extraction see Chapter Twelve on sustainable beekeeping on the use of a fruit press to extract honey.\n\nCHOOSING AN EXTRACTOR\n\nThere are a variety of designs of extractor on the market and the type of extractor that you buy will depend a lot on individual preference. There are, however, some general points that you need to consider when making your choice.\n\n\u2022 First, is the extractor made from food-grade stainless steel? This type of steel is non-magnetic, so a simple test to confirm if the steel is food grade is to touch the extractor with a magnet and see if the magnet is attracted to the metal. If it is not attracted you can be confident that the extractor is made from food-grade stainless steel. It is worth noting, however, that some parts of the extractor may not be food grade \u2014 for example, the metal gearing. This is not a problem as the gears will not usually come in contact with the honey being extracted.\n\n\u2022 Does the gate, at the bottom of the extractor to release the honey, leak when closed? This may be difficult to check, but look to see if the valve shuts tightly. In addition try opening and closing the gate several times to see if the bolt holding the moveable lid of the gate becomes loose. A further point to check is that the rubber or plastic O-ring sealing the gate entrance protrudes a reasonable distance above the body of the metal or plastic gate, say by 1-millimetre or more.\n\n\u2022 If you plan to extract from Ideal frames, make sure that they will be held in the extractor cage and not fall through to the bottom of the tank. If the extractor is in all other respects a good buy, this is not difficult to fix. Some stainless steel frame wire attached across the base of each frame-holding cage will generally prevent the frame from falling through the base of the cage and into the tank. If you are spending a lot of money on a new extractor and use large numbers of Ideal frames you may be better buying an extractor that has a cage designed to support both full depth and the Ideal size of frame.\n\n\u2022 Check that the frame, when it sits in the cage, does not touch the cross bar of the extractor when it is rotated, hindering rotation of the cages.\n\n\u2022 Consider whether the honey can escape freely from the tank through the honey gate without building up and touching the bottom of the cage. Honey that has built up and touches the bottom of the cage will stop the cage from being turned and put undue strain on the gearing mechanism.\n\n\u2022 Can the cage be easily removed for cleaning?\n\n\u2022 Ask if the spindle at the centre of the cage sits on a ball bearing at the bottom of the tank. If it does this will make turning the cage of the extractor much smoother and easier. You should also check if replacement ball bearings are available should you lose the one in the extractor while cleaning it.\n\n\u2022 What is the warranty period and are spare parts readily available?\n\n\u2022 Are the gears made of iron or nylon? Nylon gears are light but will wear out quickly and replacements may not be cheap or easy to obtain a few years after you purchased the extractor.\n\n\u2022 Many older second-hand extractors are made of galvanised steel. Galvanised steel is corroded by honey and is today not used to make extractors for reasons of food hygiene. Take this into account when offered an apparent second-hand bargain.\n\n\u2022 Some extractors have a plastic drum and are cheaper than extractors with stainless steel drums. I have used a plastic extractor several times and, although I would prefer to use a stainless steel extractor, plastic extractors are cheaper and are durable if handled with care.\n\n\u2022 The handle on the extractor may be mounted on top or on the side. Top-mounted handles may be less expensive than side-mounted handles, but the difference in cost is not significant. Try both types of handle and decide which you prefer before choosing one or the other.\n\nElectric extractors are much easier to use than manual extractors, but are significantly more expensive. The motor on the extractor needs to be powerful enough to turn a cage containing full frames without overheating. Electric extractors are a good option if you can justify the cost as they take all of the hard physical work out of extracting. Check to see if there is a switch to rotate the motor in either direction, and that the motor includes a variable speed control so that you can adjust the speed of the cage rotation.\n\nFor safety reasons the extractor needs to be earthed and confirmation obtained that it is made to a recognised international electrical safety standard such as for the European Union. Electric extractors made in China usually have motors that operate on 220V, not the Australian standard 240V. Watch out for this as the voltage difference may result in the motor overheating in use, particularly at higher speeds or with heavier frame loads. Transformers are available that convert the 240V power used in Australia to the 220V power that Chinese motors are designed to use, although these will add at least another $200 to the cost of the extractor.\n\n9.12: Electric extractors are much easier to use than manual extractors.\n\nTYPE OF EXTRACTOR TO BUY\n\nExtractors come in two types: tangential extractors and radial extractors.\n\n**TANGENTIAL EXTRACTORS**\n\nMost hobbyists use a tangential extractor due to their lower cost and also because they take up much less room. As the name suggests, the frames in a tangential extractor lie tangentially to the direction of rotation of the cage. Tangential extractors are compact, inexpensive and are easy to use and clean. They also keep their resale value and are a good investment, particularly if the cost is shared by a few beekeepers.\n\nTangential extractors can usually hold two, three or four frames at a time. My experience is that although a two-frame extractor sounds like a good deal due to its lower cost and compact size, they are not very practical since the frames are placed too close to the spindle, which is the axis of rotation of the cage. As a result, insufficient centrifugal force develops to effectively remove the honey from the centre of the frame. This may result in a frame that has had its honey removed on either side, but not in the centre. This is both frustrating and a waste of time.\n\n9.13: There are two designs of extractors available, tangential and radial. The tangential 9.13 extractor is more widely used by hobby beekeepers due to its lower price.\n\n**RADIAL EXTRACTORS**\n\nRadial extractors are much larger than tangential extractors and are used almost exclusively by professional apiarists. Radial extractors can hold a large number of frames ranging from, say, six up to 40 frames; some very large ones hold even more. Unlike the tangential extractor, frames are placed into a radial extractor in the same orientation as the spokes in a bicycle wheel. Apart from being able to hold many more frames, due to the orientation of the frame a radial extractor has the added advantage that the frames in it can be rotated at much higher speeds without damaging the wax foundation. The much higher speed of rotation also leads to higher centrifugal force, enabling the honey to be spun from each frame much more quickly.\n\n**FILTERS**\n\nSome extractors have a built-in filter and holding tank. If you only have a small number of hives these are the most convenient to use since the honey is extracted and filtered in a single operation and is ready to use immediately after extraction. Manual extractors of this type are useful for beekeepers with up to ten hives. The only disadvantage of this type of arrangement is that the extractor tank must be cleared of honey before the next extraction is undertaken. This may be a minor inconvenience or can be overcome by buying a second tank for the extractor at the time of the initial purchase.\n\nMany extractors do not have a built-in filter; instead, the honey, containing a lot of wax particles, pours out of a gate at the bottom of the extractor tank and into a pail resting on the floor beside the unit. In order to filter the honey, a sieve or other filter may be placed on top of the pail and as the honey flows from the extractor tank gate into the sieve it is filtered before passing into the pail. The sieve must be periodically checked and cleared of any build-up of wax or it will not filter efficiently. This arrangement has the advantage that when one bucket is full, another can be placed under the open honey gate and extracting can continue. This type of extractor is convenient for beekeepers with approximately six or more hives.\n\n**STORING FRAMES OF HONEY**\n\nAs I described in Chapter Eight on autumn and winter management, if you only have one or two frames of capped honey and do not plan to extract for some time, you need to store the capped frames. To do this you must first wrap them in a new, large garbage bag. Seal the bag with thick tape. Then place the frames in the freezer for two days to kill any Wax Moth or Small Hive Beetle larvae. Unless you are lucky enough to have plenty of freezer space this will usually involve freezing one frame at a time. Store the frames, still in their sealed garbage bags, inside an airtight plastic bin until you are ready to extract.\n\nCOMB HONEY\n\n**PREPARING FRAMES FOR COMB HONEY**\n\nPlastic 'sections' are available that are usually either square or round and, together with their plastic support, fit neatly into a frame. The bees draw comb and deposit honey within the section, which is later removed and sold. These sections of honey are expensive to produce and many colonies of bees will simply not draw them out satisfactorily. These days cut comb is the most common way honeycomb is sold, either in large square or round pieces cut from the frame or in small chunks of comb added to jars of liquid honey.\n\nBegin your attempts to obtain your own cut comb honey by using shallow or Ideal supers. Use very thin, new, pale beeswax foundation and just add a small starter strip to the top bar of the frame and let the bees make their own thin, perfectly white, honeycomb. The shallow super used for comb should be placed on the top of the hive, under the lid in order to avoid unnecessary bee traffic that will discolour the comb surface.\n\nColonies of bees used to produce cut comb honey should be on a good honey flow for the best results. The hive will need to be monitored and the honeycomb super removed once the frames are capped. Small Hive Beetle prevention is paramount and the frames should be carefully wrapped in plastic and placed in a freezer for at least 48 hours to kill any larvae or eggs. Once removed from the freezer they should be left in the plastic to come up to room temperature to avoid any condensation problems.\n\n**EXTRACTING COMB HONEY**\n\nAfter defrosting and ensuring the frame of honeycomb is at room temperature, place the frame on to a kitchen cutting board and carefully cut the rectangle of comb away from the wooden frame surround. Prior to doing this, inspect both sides of the frame and decide which is the more visually pleasing side to display in a container. This side should be placed face-up on your cutting board. Using a sharp knife, you can carefully cut squares of honey and place them into a plastic box or container. Your knife can be heated to make the process a little easier and improve the clean cut of the comb. An alternative way to cut out comb honey is to use a large pastry cutter available from most kitchen supply stores. These are often round in shape. It is also possible to buy a large, square, specialised comb cutter from some beekeeping supply stores.\n\nA piece of chunk comb honey is an attractive feature in a glass jar of liquid honey. Glass jars should be thoroughly sterilised before use. Once the chosen piece of cut comb has been removed, gently place it in the jar and slowly cover it with honey, ensuring that the honey is poured in gently enough so that air bubbles are not trapped spoiling the appearance of the product. If you have previously cut rounds of honeycomb for sale, the valuable leftover pieces can be used in this way.\n\nFor larger, square-cut or round-cut pieces of honeycomb I find that the least expensive and most useful containers are plastic takeaway food containers available in bulk from most discount or $2 type stores. If you require larger numbers of this type of container, enquire at some of your local takeaway restaurants where they obtain theirs and buy them directly from the same wholesaler.\n\nREUSE OF WAX FOUNDATION\n\nMost beekeepers will have wax left over after extracting and by the end of the year this can be a sizable amount. The question often asked is what can I do with the wax and how can I clean it? Cappings left after extracting or comb cut out of frames contain a lot of impurities: pollen and honey are present as are silk cocoons and other remains left by the pupae that need to be filtered out together with pieces of dead bee. The first stage in cleaning the wax is to melt it and then to filter it through an old sheet or piece of hessian sacking. The two most common methods used by hobbyists to clean wax are the saucepan method and the solar beeswax extractor.\n\n9.16: Chunks of honeycomb in jars of honey are an attractive way of displaying this product.\n\n9.17: Honeycomb stored in a takeaway food container.\n\n9.18: Preparing the wax. For small quantities of wax capping, melt the capping in a large saucepan bought from an op shop. The pan needs to contain about 3cm of water to stop the wax over-heating and catching fire.\n\n**SAUCEPAN METHOD**\n\nYou can use a large saucepan from your kitchen if it is no longer being used or find one at your local op shop. Choose a size that will comfortably fit all the wax that you have for melting. Fill this about a quarter full of water and raise the temperature of the water on a stove, slowly adding the cappings until they have all melted. The resulting melted cappings is messy and contains all sorts of unwanted material that will float on top. The beekeeper may need to negotiate the use of the family kitchen for this method and make sure any resultant mess or spilt cappings material is thoroughly cleaned up.\n\nIt is critical that the beekeeper remains close to the wax while it's being melted on the stove, to monitor its temperature. If the process is interrupted and the stove left unattended it should be turned off until the beekeeper can return and complete the operation.\n\nBefore melting the wax, prepare a large bucket by adding about 5 centimetres of water for the wax to float on while it hardens. Tie some cloth or hessian sacking around the top to act as a filter and set it up outside the house. Do make sure that there is a concave pocket in the cloth or sacking to catch the melted wax so that it will not overflow onto the ground beside the bucket. Next, very carefully pour the wax into the cloth on top of the bucket. The melted wax will flow through the cloth sieve leaving the slum-gum in the cloth pocket or depression in the material used. Slum-gum is the fibrous black mess left when cappings and comb are melted and filtered; this is particularly so for brood comb as it contains the silk cocoon, defecations and shed skin left behind by pupae as they develop and grow.\n\n9.19: Pour the molten wax into a large pail with hessian sacking tied over the top to filter out the bits of bee and cocoon, commonly called slum-gum.\n\n9.20: Once the wax has solidified in the pail, it can be removed and stored.\n\nWhen all of the wax has been filtered and the wax both inside the bucket and on top of the cloth has hardened, remove the cloth and hardened slum-gum and either dispose of it in the bin or in the compost. My experience is that although professional beekeepers have the equipment to extract further wax from slum-gum, this is very difficult for a hobby beekeeper and a lot of time can be wasted extracting very little additional wax.\n\nWhen the wax inside the bucket has hardened it can be removed. If you turn the block of set wax over you will notice some slum-gum has passed through the cloth and has congealed to the underside of the wax block. Cut this off with a hive tool or sharp knife and the remaining wax is usually sufficiently clean to swap for milled foundation from your local supplier.\n\nSince beeswax will be discoloured by pots and containers made of iron, brass, zinc and copper it is a good practice to melt wax only in containers made of aluminium or stainless steel.\n\nIf you plan to make candles or furniture polish from the wax, a further level of filtering, or rendering, will be required and the process will need to be repeated again. If the wax is to be used for cosmetic purposes to make face or hand cream, the process will need to be repeated yet again in order to get a higher quality, pure beeswax.\n\n**SOLAR BEESWAX EXTRACTORS**\n\nAnother way to melt the beeswax is to use a solar beeswax melter or extractor. This method has been used successfully over many years and has the advantage that the beekeeper need not stay watching the wax melt but can leave it in the melter\/extractor on a warm or hot day and let the sun do its work. A solar wax melter or extractor is usually a square or oblong box painted black or some other dark colour on the outside to attract heat, and white on the inside. Two sheets of glass or clear plastic are fitted to the lid allowing a space between the panes rather like double-glazing. This lid can be opened and shut by the beekeeper when a tray containing frames or wax pieces is inserted or removed.\n\nWooden supports are incorporated inside the box to raise the wax-holding tray. The tray should have a lip at one end so that the melted wax can pour into the container provided at the end of the tray. The melter box is placed in a sunny location and tilted at a slight vertical angle towards the sun.\n\nThe interior of the box acts like a greenhouse and heats up, melting the wax inside, which then runs off and is collected in a container placed inside at the bottom of the tilted box.\n\nCheck the solar extractor periodically to ensure that it has not run out of wax cappings. Larger solar wax extractors are useful for beekeepers that have space in the garden or have a lot of hives and thus a lot of wax to process. Smaller solar extractors that only hold one or two frames can also be used and are more convenient for the urban beekeeper confined to the average suburban block.\n\n9.21: An alternative way of processing beeswax is to melt and filter the wax in a large solar wax extractor.\n\n9.22: Smaller solar extractors can also be bought that are more convenient to use and store.\n\nAlthough Wax Moth would prefer to eat protein sources such as pollen, the moth can partly digest unclean honeycomb. It is important therefore that unfiltered beeswax be processed and cleansed as soon as possible after it has been removed from the frame. Neglect to do this and the moth may completely take over the unfiltered comb leaving a complete mess from which it will be difficult to extract any wax.\n\nCOLOUR OF WAX\n\nNew honeycomb made by workers is almost white in colour. If there is no brood present, such as in the case of a honey super above a queen excluder, the wax will only slowly yellow with age. The comb yellows in colour due to pollen and also to propolis.\n\nConversely honeycomb that contains brood quickly goes a chocolate brown colour and over three or four years turns almost black from pupal skins, silk cocoons and brood excrement. Although old dark wax does not look that attractive, it is in fact much less brittle than white wax since it contains a lot more oils. Also, bees appear to prefer the smell of darker foundation and a swarm is less likely to abscond from a hive containing dark foundation than from one containing very light foundation. I have previously mentioned that one of the main management tasks of the beekeeper is to regularly replace drawn frames with old chocolate brown or black comb with clean foundation. A few of the old frames, after replacement with new frames, could be kept to attract future swarms.\n\nFoundation comes in different thicknesses and is usually measured as a number of sheets per kilogram. Foundation typically comes in weights between about 11 sheets\/kilogram to 17 sheets\/kilogram; 11 sheets\/kilogram being the thicker and thus stronger foundation. Which to use will often be a personal choice guided both by price and what is available from your supplier. Thicker foundation is often easier to insert into the frame and due to its extra strength holds together better and is less likely to break when it is rotated in an extractor. Generally, the thicker the wax the better its strength and durability in use. If, however, you propose to cut the foundation out of the frame to extract by hand, use of the thinner product is usually fine as the bee will add its strength when both sides of the sheet are drawn out into comb.\n\nGood foundation should be pleasant to smell since the impurities will have been filtered out of it. As autumn progresses and sales of foundation drop, poorly filtered wax foundation that has been left to stand for some time will start to smell mouldy or earthy. Well-purified wax will continue to have a pleasant smell even after years of storage. Foundation, when left over winter, often forms a white bloom on its surface and many people believe that this is caused by a fungus. In fact, the white bloom occurs due to wax oils that have risen to the surface of the sheet and crystallised. Bloom melts at 39.8o Celsius, much lower than the melting point of beeswax itself, which is about 62o Celsius. Bloom can quickly be removed either by leaving the foundation in the sun for a few minutes or by gently heating the surface of the foundation sheet with a hair dryer. Bloom can also be partly removed by wiping the crystallised oils off the surface with a damp cloth. Winter bloom does not appear to be detrimental to bees or hinder them from drawing out the foundation sheet in the usual way.\n\nIf you like eating honeycomb or intend to market this product, it is better to use the thinnest foundation possible since the taste of thick foundation in the centre of a cut comb piece can be unpleasant. The best solution for the beekeeper who wishes to produce honeycomb is not to insert full-size sheets of foundation into frames at all but to put a 1- to 2-centimetre starter strip of foundation at the top of the frame and then leave the bees to draw their own thin, clean, white comb throughout the remainder of the frame. Honeycomb made by bees without foundation is by far the most pleasant to eat.\n\nSUMMARY\n\n\u2022 Extracting is an important part of beekeeping and can be performed inexpensively and easily. A simple and inexpensive method is to use a sieve to drain chopped up honeycomb. Beekeepers with more hives may prefer to buy or rent an extractor since these are convenient to use for, say, two or more hives. If you plan to use an extractor, the surface cappings of the honeycomb need to be removed either with a scratcher or with a hot knife. Wax cappings collected as a by-product of extracting can be melted down and sold back to wax foundation manufacturers, or melted down and filtered to make candles, hand cream or face cream.\n\n\u2022 Liquid honey is the most popular form of honey although many people also enjoy eating honeycomb. \n10.\n\nPreparing honey for sale and competitions\n\n10.1: Preparing comb honey for competitions.\n\nPREPARING HONEY FOR SALE\n\nColonies of bees usually produce much more honey than can be consumed by a family and selling the surplus honey to friends or at markets is a way to repay yourself some of the money you invested starting up as a beekeeper. Most first-year beekeepers do not harvest much honey, often because they became beekeepers after the main honey flow had finished, but also because their new colony of bees was not sufficiently populated with foragers to be very good at collecting nectar. During the second and subsequent years, as the bees learn how important it is to the beekeeper to produce lots of honey, they try harder and they often produce 50 to 80 kilograms per hive, sometimes a lot more in good floral areas during good years!\n\nIf you plan to give honey to family and friends the honey can be extracted and bottled in the family kitchen and, apart from general cleanliness, no particular set of hygiene standards will apply. If, however, you plan to sell honey commercially at a market or through a shop, the location where you intend to extract and bottle your honey comes under some very strict state and territory regulations enforced by your local council. These include monitoring and inspections for general hygiene and also a requirement that the working surfaces in the area must be cleanable and not porous. Porous surfaces enable the absorption of water and food, allowing bacteria to grow. While a glass working surface can be used this will not be as durable as surfaces made of stainless steel.\n\n10.2: First cut of honey.\n\nThe expense of setting up a suitable separate area to bottle and extract honey is one of the big problems facing hobby beekeepers wishing to sell honey commercially. The best way to find out about food preparation and handling requirements in your state is to attend a food- handling course run by the local council. Courses are usually inexpensive and typically last for one or two days or are held over three evenings during the week. Before anyone is given a licence to operate a honey processing kitchen it is a legal requirement to attend one of these courses. State regulations requiring the considerable expense of providing a separate working area often prevent hobby beekeepers selling honey commercially. If you feel that you have sufficient excess honey and a ready market of buyers it is worth contacting your local council to check what their requirements are since some of these rules are open to interpretation. Another obvious way to avoid the individual expense of setting up a separate area is to form a loose or formal cooperative undertaking with other individual or small producers which would enable the cost of hiring a suitable commercial kitchen to be shared.\n\n**LABELLING**\n\nAssuming that you comply with local council requirements and have the space to set up a separate extracting and bottling area, your next challenge will be to design and produce your own label. You will want the label to enhance the presentation of your honey and in addition to its visual appeal it must comply with food labelling regulations requiring that certain information is clearly displayed.\n\nThe requirements for labelling are complex and as a starting point I would suggest visiting a local supermarket, buying a jar of honey from a reputable supplier and following the type of information disclosed on their label as a guide for your own. The main points to note are that the label must show the major ingredient of a product (100 per cent pure Australian\n\nhoney) and it must include the contact details for the supplier. The weight of the contents should also be noted. Honey is a complex substance containing many trace elements, however, only the major nutritional components need be listed on the label. If you decide to use the nutritional information displayed on a commercial jar of honey as a guide do be aware that the composition of honey from different floral types varies significantly so the list of ingredients is only a guide for a typical blend of honey.\n\n10.3: This label listing nutritional information for honey is typical of one designed and used by a reputable honey producer.\n\nA small number people that I meet complain that imported or even some Australian bottled honey contains additives that are not listed on the label. The labelling laws for honey, as for all foods, are strict. If the label says 'pure honey' and does not list any further additives then you can be reasonably sure that it contains pure honey. I say reasonably as there have been cases where imported honey has been found to be adulterated with sugar syrup and other fillers. This is not a frequent occurrence and generally a jar labelled 100 per cent pure Australian honey is just that. If a person bottling and selling honey wishes to add a flavour or some other addition \u2014 say, extra pollen \u2014 this is permissible but the additive must be clearly included on the honey label.\n\n**PRESENTATION**\n\nLiquid honey purchased from shops and supermarkets is notable because of its clarity and lack of sediment. Supermarkets and other retailers do not want honey to become cloudy or to crystallise on their shelves as the uneducated consumer will interpret these as an indication that the honey is full of impurities or is old, and it will not sell. Consumers equally do not want their honey to become cloudy and start crystallising soon after they have purchased it. To achieve liquid, clear, sediment-free honey, commercial honey producers heat honey to make it more liquid allowing them to pass it through very fine filters to remove any pollen or particles of wax. This also dissolves any microscopic crystals of sugar that larger crystals may form around.\n\nThe advantage of home-produced honey is that it is not usually heated, preserving many nutrients and enzymes that are negatively affected by heat. Once consumers taste cold-extracted honey, they realise what they have been missing by opting for the over-filtered commercial product. Some people prefer to buy slightly cloudy honey as it is a sign that it contains pollen, which is believed to be highly nutritious. A well-presented jar of home-produced honey can usually be sold at a higher price than its commercial equivalent.\n\n10.4: Frame honey is commonly available at reputable restaurants during breakfast. Customers can cut off both honey and comb and spread it on toast or mix it with cereal.\n\nPREPARING HONEY FOR COMPETITIONS\n\nEvery beekeeper takes pride in both the health of their hives and the quality of the honey produced in those hives. One way to show off the quality of honey is to enter it into one of the many honey competitions that are held every year and to have it judged against honeys from other beekeepers. Although preparing liquid honey for entry into a competition is relatively easy and well within the skill level of even the novice beekeeper, many are reluctant to enter their honey because they are unfamiliar with what is required. This section will therefore cover what is involved in entering honey into a competition.\n\n**COMPETITION CLASSES**\n\nThere are a number of bee products that can be entered into most competitions. The focus will be specifically on the following:\n\n\u2022 liquid honey\n\n\u2022 granulated and creamed honey\n\n\u2022 chunk or cut honeycomb\n\n\u2022 frames of honey.\n\n**EXTRACTING HONEY AND PREPARATION FOR COMPETITION ENTRY**\n\nLiquid honey is the basic product of every hive and is typically the class that has the largest number of entries in a honey competition. Unfortunately, though, liquid honey is the most difficult class to prepare for and, due to the large number of entries, is the most difficult class to win a prize in. Do not be deterred as in both club competitions and even international events such as Apimondia, novice beekeepers have walked away with top prizes against stiff competition from experienced Australian and international beekeepers.\n\nIn order to prepare honey samples, the beekeeper could simply fill three or four bottles from the latest extraction. This method, although easy, is not setting the beekeeper up for success in the competition field. Instead, approximately six full frames that have been completely capped or are almost completely capped should be selected. This will ensure that the honey entered will have the maximum density.\n\nWhen selecting the frames, one way to ensure that all of the honey is from the same floral source or colour is to hold each frame against a bright light and to check that the colour of the honey across each frame, and between frames, is the same colour.\n\nTo remove the honey from the frames use either an extractor or a press to squeeze the wax comb until the honey runs out. Pressing the wax combs is preferable since fewer bubbles of air will enter the honey using this method. Once the honey has been extracted, it should be stored for two or three weeks at room temperature, preferably at around 37o Celsius, to allow air bubbles and other particles to rise to the top where they can be skimmed off. See visuals 12.13 and 12.14 in Chapter Twelve on sustainable beekeeping for using a fruit press to extract honey.\n\nSTRAINING THE HONEY\n\nOnce the honey has settled it is time to filter out any impurities. The liquid honey should be gently poured through a fine filter into another container. Fine nylon filters are available from beekeeping suppliers. Muslin cloth is also good at removing even very small particles and is widely used by beekeepers to filter their honey. To minimise the introduction of new air bubbles into the honey during filtering either of the following two methods are useful.\n\n**METHOD 1**\n\nTie the nylon or muslin cloth over the top of the container with a small dip in the top of the cloth like a saucer. Tip the container slightly over onto its side and gently pour the honey into the nylon or muslin. The filtered honey needs to flow gently down the side of the container to minimise the amount of new air bubbles that are introduced.\n\n**METHOD 2**\n\nSew the nylon or muslin cloth in a cone shape and tie the cone over the top of the container with the pointed end near the container bottom. Gently pour the honey into the cone, where it will pass through with minimal new air bubbles and filter out most of the residual wax particles.\n\nWhichever method is used, allow the filtered honey to remain in the container for about a week, again at room temperature or preferably at around 37o Celsius, to allow any residual air bubbles or fine particles to rise to the surface where they can again be skimmed off.\n\n10.5: A Nylex net tied over a pail with a hollow in the centre is a convenient way of filtering fine particles from honey. The filtered honey needs to flow gently down the side of the container to minimise the amount of new air bubbles that are introduced.\n\nA simple and effective technique for removing small particles of wax from the honey is to gently lay a strip of cling wrap across the surface. The wax particles will stick to the cling wrap allowing them to be removed when the plastic is lifted away.\n\n**POURING HONEY INTO A BOTTLE**\n\nContainers used in competitions are usually 375 gram or 500 gram glass jars. Each competition will have its own rules so the beekeeper should make enquiries or read the entry information to ensure they know what size and shape jars to use.\n\n10.6: A good way to fill jars with clear liquid honey and not get air bubbles in it is to let the honey settle in a pail that has a honey gate at the bottom. The jars can then be filled by letting honey flow through the gate.\n\nFor the best chance of winning a competition, good-quality jars should be purchased, not the cheaper variety that often have numerous fine air bubbles trapped in the glass of the jar. Cheaper glass jars often have a greenish tinge to them that will also detract from the visual appearance of your entry.\n\nBefore filling the jars with honey, wash each jar in soapy water, rinse well with fresh water and dry immediately with a lint-free towel. This will ensure that the jars are sparkling clean and do not have any lint on the inside or outside surfaces that will take marks off your entry. Usually the easiest way to fill a jar is simply to pour or ladle the honey into it. This, however, is not the best way to fill jars for competition entry since the top of the container will contain some air bubbles and other small particles, which will lower the quality of the beekeeper's entry.\n\nA better method is to allow the honey to settle in a container that has a wide valve or honey gate at the bottom. The jars can then be filled by letting honey flow through the gate. The honey at the bottom of the container will be cleaner than the honey at the top since, over a week, most of the air bubbles and other particles will have risen to the surface. Without a honey gate at the bottom, it will be almost impossible to retrieve clean bubble-free honey from the bottom of the container.\n\nIf a container with a honey gate is unavailable, an alternative method is to allow the honey to settle in a disposable plastic container, such as an ice-cream container. When the jars are ready to be filled a very sharp knife can be used to cut a small hole in the bottom of the plastic container allowing the honey to flow through the hole and into the jar.\n\nWhen filling the jars, care should be taken to ensure that they are slightly tilted and that the honey flows gently onto the side of the jar. This will further minimise the number of air bubbles introduced into the honey. Because air bubbles can be introduced into an otherwise excellent honey sample, at every step in the preparation of the final sample the beekeeper will need to constantly and carefully monitor the process, thinking of ways to remove impurities and eliminate new air bubbles. Completion of these tasks is well within the capabilities of even the novice beekeeper if care is taken during the filtering process.\n\nEach jar should be filled to the same height, slightly below the rim of the lid, so that the judge can see the top of the honey without opening the jar. Once the jars have been filled they can be stored in a dark place for a few weeks until the competition. Storing honey in the dark will minimise the risk of the colour of the honey changing before the competition day. When storing the jars, since judges also look at the quality of the jar that the honey is presented in, place some cardboard between the jars to stop them scratching each other.\n\nThe majority of competitions require two or three samples of the same honey to be submitted with every entry. If only two jars are required for an entry, three or four samples should be prepared and then, just before the competition, the best two samples should be selected for entry using the judging criteria explained a little later in this chapter. Again, this will maximise the chances of gaining a place or prize in the competition.\n\nIf the honey that you plan to exhibit becomes granulated it can be gently heated in a warm water bath or in warm air at a temperature no higher than 43\u00b0 Celsius. Heating the honey at temperatures higher than this will risk darkening its colour and losing some of its taste.\n\nEach entry class of honey will be restricted to a type of floral source, say, Yellow Gum. If it is necessary to blend honey from different hives to make up the required quantity the beekeeper should make sure that the honey from each hive comes from the same floral source and preferably from the same location.\n\nWith a good source of honey, the will to win and the patience to prepare and show honey to its best advantage, by following the steps above a prize is well within the grasp of even the newest beekeeper.\n\nMarking grades given in this book for all classes of hive products are used in the Sydney Royal Easter Show. Check with the competition organiser of the event that you wish to enter what their marking criteria are to ensure the best possibility of gaining high marks with your entry.\n\n10.7: If the honey that you plan to exhibit becomes granulated it can be gently warmed in a warm water bath or in warm air at a temperature no higher than 43\u00b0C.\n\n10.8: High marks will be given to the honey with the highest density. This can be estimated by inverting the jar and noting how quickly an air bubble rises to the surface.\n\nLIQUID HONEY\n\nEach competition organiser will publish in advance the judging and marking criteria for their particular competition.\n\nFlavour | 25  \n---|---  \nDensity | 25  \nColour | 25  \nAroma | 10  \nClearness | 10  \nBrightness | 5  \nTotal Points  | 100\n\nFLAVOUR\n\nThe honey submitted for judging needs to be palatable and free from any tang, harsh or biting taste due to fermentation, acidity or impurities from other floral types. As liquid honey is typically entered as part of a floral type \u2014 for example, Yellow Box \u2014 the entry needs to be typical of that class since the judge will be familiar with the different floral classes and will also have other honeys from that class to judge it against.\n\n10.9: A Pfund meter can be used to accurately estimate the colour of honey before entering it into a competition.\n\n10.10: The Pfund meter works by smearing a small amount of honey across the glass screen and closing a Perspex cover over it. The honey is viewed through the eyepiece against daylight and the colour of the honey is read out from a scale built into the meter.\n\n10.11: A jar of honey.\n\nDENSITY\n\nAll honeys vary in density and, for the same floral class, this characteristic varies both from location to location and also over time from flowers from the same location. The highest marks will be given to the honey with the highest density and this can be measured either by the use of a refractometer that measures the buoyancy of a glass rod at the surface of the honey, or by inverting the jar and noting how quickly an air bubble rises to the surface.\n\nCOLOUR\n\nMany honey competitions have separate classes for dark, medium and light honeys. The colour of honey can be measured with a Pfund honey grader. These graders are relatively inexpensive to buy online and are a good investment for the beekeeper planning to enter honey into many competitions. Dark and medium honey must come within a prescribed colour band, while the light honey class gives top marks to the lightest honey entered in the competition.\n\nAROMA\n\nEach type of honey has its own characteristic aroma and top marks are given to the entry with the most pleasing aroma, free from fermentation, the smell of being over-heated or any other offensive odour. Each floral class will have its own distinctive aroma \u2014 for example, Yellow Gum, Red Gum or Leatherwood from Tasmania \u2014 and this bouquet will be taken into account by the judge.\n\nCLEARNESS\n\nEach entry into a competition must be clear of visible impurities and air bubbles and have a sparkling appearance. The cleanliness of the jar will play an important part here and it is critical for the beekeeper to ensure that the jars chosen for use are clean and free from any surface marks such as lint, fingerprints or even air bubbles in the glass of the jar itself.\n\nBRIGHTNESS\n\nJust before the competition slightly heat the jars of honey to increase brightness. Slightly heating the honey will dissolve any sugar crystals and give added brilliance to the sample. Make sure, though, that there is sufficient time for the honey to cool to room temperature before it is judged.\n\nGRANULATED AND CREAMED HONEY\n\nThere are often separate classes for these two types of honey in competitions. The main differences between these prepared honeys are that granulated honey has a larger grain size than creamed honey and creamed honey also contains fine bubbles of air.\n\nGRANULATED HONEY\n\nHoney used to make granulated honey should be of the same high quality as used in liquid honey described above. The three major considerations that are used by judges to award points are the smoothness of the granulation, the flavour and the firmness of the set.\n\nEvenness of grain  | 30  \n---|---  \nFlavour | 30  \nFirmness | 30  \nColour | 10  \nTotal Points | 100\n\n10.12: Bees on honeycomb.\n\n10.13: Stirring starter into liquid honey to prepare candied honey.\n\nEVENNESS OF GRAIN\n\nCompetitions for granulated honey may include both fine- and coarse-grained classes. Fine-grained honey should be smooth with the grain almost indistinguishable to the tongue and with all grains an even size. Fine-grain granulated honey may be made with lucerne or clover. Honey from eucalypts gives a larger grain size.\n\nGRANULATION\n\nIn order to prepare granulated honey, a starter is added to initiate granulation, or crystallisation, of the liquid honey. This starter is coarse granulated honey that has either been prepared by the beekeeper or obtained elsewhere. When selecting a starter, it is important for the competition beekeeper to be aware that the consistency of the final granulated honey will mirror the granulation of the starter. For this reason the beekeeper should make sure that the evenness of grain of the starter is satisfactory before using it to prepare competition entry quality honey.\n\nAdd about 10 to 15 per cent of starter to the liquid honey that you wish to granulate, mix thoroughly without adding air bubbles to the mix until the entire mix is of the same colour and consistency. Leave this mix in a larger container to settle for 12 to 36 hours at 14\u00b0 Celsius. Before the granulated honey sets pour it slowly onto the inside surface of the show container making sure that air bubbles do not enter the mix. Leave the jar for about three weeks at 14\u00b0 Celsius for the granulated honey to set. When the granulated honey has become consistently firm throughout the container it is ready for competition entry.\n\nFLAVOUR\n\nThe flavour of the granulated honey will represent the floral source that the liquid honey comes from.\n\nFIRMNESS\n\nThe granulated honey mix needs to be firm but easy to spread. The starter used will strongly influence the firmness of the beekeeper's honey entry so this characteristic should be considered when selecting a suitable starter.\n\nCOLOUR\n\nThe colour should be between cream and off-white, but will be determined by the colour of the liquid honey used.\n\nCREAMED HONEY\n\nThe process used to prepare coarse granulated honey also applies to smooth creamed honey. The main differences are that the starter needs to be fine grained and that the colour of the final product needs to be white. The scoring scheme for both granulated and creamed honey is usually the same.\n\nWhen making creamed honey the starter should be added to the liquid honey in such a way as to incorporate air into the mixture, as this will give the creamed honey much of its white colour. Since creamed honey is very viscous, a household grade mixer is likely to overheat and burn out if used. If available, it is better to use a dough mixer and to beat the honey until the mix will not go any whiter.\n\nCHUNK HONEY\n\nChunk honey is a cut-out section of capped honeycomb that is placed in a jar of liquid honey.\n\nAppearance | 25  \n---|---  \nColour | 25  \nFlavour | 20  \nDensity | 20  \nClearness | 10  \nTotal Points  | 100\n\nThe beekeeper should select a frame containing capped honey with the capping smooth, white and pleasing to the eye. The wire should be removed from the frame, as this will make it easier to cut out a nice section of comb. Use a hot knife and cut out the selected section of comb, placing it gently onto a wire grill so that the honey at the edges, together with any pieces of wax, drains off.\n\nNext the chunk is placed into a jar ensuring that no force is used to pass the chunk through the opening. With the jar slightly tilted, high-quality liquid honey, prepared as previously described above, is gently poured onto the side of the jar making sure that\n\nno bubbles enter the honey. The liquid honey and comb need to be of the same floral type. The comb should rest in the honey with the top of the chunk resting just below the lid.\n\n10.14: To make a good presentation of cut comb honey, let the cut comb drain on a wire grill so that all of the honey is removed from the sides of the comb. Use tweezers to remove particles of wax comb for a smooth edge. This will make the display much more attractive to the judges.\n\nFRAMES OF HONEY\n\nAs well as chunk honey, there are often classes for frames of honey. A frame should be selected that is brood and pollen free and with surface capping of the honey that is white, smooth or regular, and pleasing to the eye. Since the frame that is used to store the honeycomb is also included in the competition make sure that the frames selected are well assembled and are of good quality.\n\nFullness | 30  \n---|---  \nColour of cappings | 30  \nEvenness | 20  \nGeneral appearance  | 20  \nTotal Points | 100\n\nThe quality of a frame of honey is very much dependent on the hive that it came from. When selecting hives from which to draw frames of honey for competition, look for colonies that store honey with a small amount of air under the cappings as this will give a whiter appearance than those frames that do not have air under the cappings. Once the beekeeper determines a frame of honey is of competition quality it should be removed immediately capping has finished so the bees do not get time to walk over it and discolour the surface. For this reason when selecting competition frames the beekeeper should keep away from colonies that produce a lot of propolis as this may discolour the wax and reduce its visual appeal. As part of the selection process the frame should be held up with a strong light behind it and a check made for impurities in the honey or wax such as pollen, propolis or brood as these will severely reduce the chances of winning a prize.\n\nThe judge will also look at the drawn frame. When selecting frames of honey make sure that the wooden frame is clean and does not contain propolis or other surface markings.\n\n10.15 Frame imperfections can lose points from an otherwise prize-winning frame of comb honey.\n\nFULLNESS\n\nJudges give the highest marks to frames that have been well drawn out.\n\nCOLOUR OF CAPPING\n\nCappings should be white with no differences in colour across the surface. Differences in colour may indicate different floral types stored across the frame and this will lose the competitor valuable marks.\n\nEVENNESS\n\nThe surface of the wax should be even with no depressions or rises.\n\nGENERAL APPEARANCE\n\nIt is important that both the capped honey and the frame are pleasing to look at. When setting up a super to prepare quality entries for competition the beekeeper should ensure that only new frames are used, as this will give an overall better appearance to the entry being judged. To ensure this, the beekeeper interested in entering competitions should inspect hives early in the season and any defective or dirty frames should be replaced with new frames. If the frame selected for competition entry has some minor blemishes these can be removed very carefully with fine sandpaper.\n\nIn order to protect the competition frame against damage, a frame holder may easily be made or purchased from a beekeeping supply store.\n\nSUMMARY\n\n\u2022 Honey prepared at home tastes far superior to honey purchased at a supermarket. If you have a surplus of honey it is usually straightforward to sell this to family, friends or co-workers. To do this there are no stringent legal or health rules apart from the common sense requirement of general cleanliness. If however you plan to sell honey commercially you will need to obtain certification for your honey extracting and bottling area from your local council.\n\n\u2022 Many beekeepers would like to enter their hive products in a competition and this is a lot easier than many novices believe. With a small amount of care and careful monitoring of the exhibit preparation process, even amateur beekeepers have succeeded in winning major prizes against stiff national and international competition.\n11. Rearing queens\n\nThere is no doubt that the success of the honey harvest each season depends on both the quality of the queen and the available flora. Without high-quality workers produced by a viable well-mated queen the overall health of the colony and resultant honey harvest will not be the best that you can get. Although the desire to harvest the maximum amount of honey is one of the main reasons for rearing and introducing a new queen, there are many others. These include:\n\n\u2022 learning new skills as a beekeeper\n\n\u2022 the satisfaction of rearing and introducing your own queens\n\n\u2022 selecting strains of queens that are better suited to the conditions in your area\n\n\u2022 to make money selling queens\n\n\u2022 the added prestige of being able to successfully raise queens rather than just keeping bees\n\n\u2022 to overcome the common difficulty of not being able to buy a new queen when you need one\n\n\u2022 to learn more about the science and genetics of bees\n\n\u2022 the pleasure in observing a fascinating natural process taking place within your own hives\n\n\u2022 engaging in a shared activity with other beekeepers, and\n\n\u2022 just for the heck of it.\n\n11.1:Workers covering the cage of a recently introduced queen.\n\n11.2:Marking a queen.\n\nThere are many well-proven methods used to rear queens, each with its own advantages and disadvantages. Some methods suit the hobby beekeeper with only a few hives while others are better suited to the serious or professional beekeeper with many hives and who needs to rear large numbers of queens each year. Queens can be reared almost anywhere in Australia, from Tasmania to Queensland and across to Western Australia. Although there are many commercial queen rearers in Queensland and New South Wales, there are also very successful and reputable rearers in the southerly states of Victoria and Tasmania.\n\nIn this chapter I will explain the hive splitting method for rearing queens. This is easy to execute and the technique has the added advantage of being a useful tool that can also be used to manage swarming.\n\nIf you plan to rear queens you may select any suitable hive, split the colony, and wait and see what the characteristics of the resulting queens are. Although this wait and see approach is easy to use, it will not provide you with the high quality queens that a more deliberate approach will allow.\n\nAssuming that you have several colonies that can be used to rear queens, select the colony that over the past season has displayed favourable characteristics that you would like to foster in your next generation colonies. These may include:\n\n\u2022 superior honey production\n\n\u2022 disease resistance\n\n\u2022 minimal aggression\n\n\u2022 good overwintering with strong build-up of brood in the spring, and\n\n\u2022 a reduced tendency to swarm.\n\nBy selecting a colony that manifests any of these characteristics you should be successful in producing high-quality queens. With care and practice, refining your queen-rearing skills over a number of seasons will ensure that the colonies in your apiary are headed by quality queens that possess the optimum characteristics for your area and needs.\n\nBefore discussing queen rearing in more detail, however, it is necessary to recap the reasons why a colony make a new queen under natural conditions and what is the optimum situation for this to be successful.\n\nTHE THREE REASONS FOR A COLONY TO PRODUCE A NEW QUEEN\n\nThe three reasons a colony may decide or need to raise a new queen are discussed in Chapter One on the life history of the honey bee. A sound understanding of these reasons is needed if you are to successfully rear queens. Without repeating the detail given earlier, the three reasons that a new queen may be raised by a colony are swarming, supersedure and emergency conditions.\n\nIn addition to what is going on inside the hive the second most important consideration in rearing queens is the nectar and pollen resources available to the colony. Rearing queens in any sort of number requires the production of large amounts of royal jelly and this in turn requires large numbers of young bees that have themselves been reared on a high-quality diet consisting of a variety of pollens and an adequate nectar flow.\n\nIn the event that floral conditions are not sufficient at the time that you wish to rear queens it is possible to supplement those conditions to a degree. Both nectar and pollen supplies can be augmented with substitutes from time to time. White sugar mixed in a 1:1 ratio with water to make sugar syrup provides a very close approximation to nectar and has an identical effect on brood rearing by the colony. There are many commercial irradiated pollen supplements available and these typically contain a blend of soy flour, yeasts and small quantities of amino acids and minerals. The other source of pollen is bee collected pollen that has been irradiated with gamma irradiation or collected from your own hives that you know are free of disease.\n\nWhen rearing queens the following steps are important. Firstly, the hives used to produce the queen cells must be strong, often much stronger than needed for collecting honey. This may entail merging two or more colonies, each of which would have been perfectly adequate if collecting honey, into a single very strong colony. During the process the size of the hive is also reduced to one or two boxes to ensure a crowded colony.\n\nWhen raising crowded colonies like this during the spring for queen rearing, great care must be taken to prevent swarming or all of your efforts will have been in vain. Regular inspection of the brood box is essential to identify and remove any swarm cells before they are capped.\n\nOnce you have decided which colony in your apiary possesses the optimum characteristics that you would like to propagate using new queens, you will need to manage that colony so that it is very strong (contains many worker bees), and is well fed, resulting in good larvae production and nursery bees producing a plentiful supply of royal jelly.\n\nHIVE SPLITTING\n\nHive splitting is the easiest of all methods for raising queens. The steps involved are easy to follow and can quickly be mastered by even an inexperienced beekeeper. The established queen in the brood box must be young and vigorous to minimise the risk of swarming. A young vigorous queen will also lay many more eggs to keep the colony strong while the introduced brood cells are developing in the top brood box.\n\nSTEP 1\n\nTake a strong two-brood box hive with young brood in both boxes and physically separate the two brood boxes into two separate hives. Assume that the queen is in the bottom brood box. I usually leave the bottom brood box in the same location. I will refer to this brood box as hive 1. Shake most of the bees from hive 1 into the second brood box, which I shall call hive 2. Hive 2 can either be placed next to hive 1 or much further away. When you shake most of the bees from hive 1 into hive 2, all of the foragers will quickly return to the original location, hive 1, balancing the number of bees in both boxes.\n\n11.3: The hive split method for rearing queens involves taking a strong, two-brood box hive and making two hives out of it. Leave the old queen in Hive 1 and ensure that there are plenty of very young larvae in Hive 2. The workers in Hive 2 should start making new queen cells overnight.\n\n11.4: Preparing a two-box hive to rear queens.\n\n11.5: Instead of splitting a hive, this beekeeper is placing some frames of young brood, capped honey and pollen into a nucleus. A plentiful supply of workers was included in the nucleus to rear new queens and to care for the existing brood.\n\n11.6: Workers raising new queens in a queenless hive.\n\n11.7: A nursery bee feeding royal jelly to a queen larva. A nursery bee entered the cell about every 30 seconds, stayed 2 or 3 seconds, then left.\n\nWhen preparing the two hives place the greater amount of sealed brood into hive 2; this will compensate for the drift of foraging bees back to the original hive entrance. The old queen remains in hive 1.\n\nNote that it does not matter which hive the queen is in. If you know which box the queen is in place at least one frame with eggs and less than one-day-old larvae in the box that does not contain the queen. If, on the other hand, you do not know which box the queen is in ensure that both boxes contain eggs and less than one-day-old larvae. If you have not previously found the queen it does not matter, as the workers will raise the new queens in whichever brood box is queenless.\n\n**STEP 2**\n\nWait three to five days and inspect whichever hive is queenless \u2014 or both hives if you do not know which of the two brood boxes contains the queen or, in other words, is 'queen-right'. The brood box that is queen-right should have new eggs and larvae while the brood box that does not have a queen will not contain any eggs or very young larvae and should show evidence of queen cells being made by the workers.\n\n**STEP 3**\n\nThe next step is to prepare a mating nucleus to take the ripe queen cells. A ripe queen cell is a capped cell out of which a queen will emerge in less than 24 hours. Each queen cell left in the split brood box will need its own mating nucleus and these need to be prepared in advance. Two or three 4- or 5-frame mating nucleus can be prepared from one strong eight- or ten-frame hive. Into each of the mating nucleus place two frames of brood, one frame of capped honey and one frame of pollen. Any remaining space can be filled with an empty frame of drawn comb. The old queen from the eight- or ten-frame hive can be placed into her own nucleus for future use.\n\n**STEP 4**\n\nTen days after the two brood boxes have been separated, about two days before the first queen emerges, it is time to move each queen cell to its own mating nucleus. Timing is important in removing each queen cell and placing them in their own nucleus to develop into mature queens. If you leave the capped queen cells together in the hive for too long the first queen to emerge will kill the other developing queens in their cells before they can be moved to their own nucleus.\n\n11.8: Cutting a ripe queen cell from a frame to transfer it to a mating nucleus.\n\n11.9: Attaching a ripe queen cell to a frame from the mating nucleus.\n\n11.10: Preparing mating nucleus for the queens to hatch in before going on their mating flights.\n\n11.11: A queen emerging from her cell unaided by workers.\n\n11.12:Once the queen has emerged from her cell, she wanders around unnoticed by other bees. It will take a few days before the new queen starts to release queen pheromone and the workers take any notice of her.\n\n11.13:Mating nucleus used by a queen rearer in northern Victoria.\n\nUsing a sharp modelling knife, cut out each capped queen cell, being very careful not to expose the resident queen, and then place it into its own mating nucleus. To do this, remove each capped queen cell from the brood box and press into the comb at the top of a frame of brood in the nucleus. Each queen cell removed needs to have its own mating nucleus if the maximum number of queens is to be reared during a hive split. Queen cells are fragile and vulnerable to extremes of temperature and drying out. Mating nucleus should ideally be close by and the transfer of cells should not be done in hot and dry conditions. Close the nucleus and leave it for a week. By this time the queens should have emerged from the cells although they may not have had the time to develop sufficiently to go on their mating flight.\n\n11.14: Queen cages are used to hold a new queen and her worker bees when they are to be placed in a new colony.\n\n11.15: A queen cage being used to insert a new queen into a queenless colony. Note that the entrance to the cage is facing upwards. This is to stop any dead bees inside the cage blocking the exit if it were to be facing downwards.\n\n**NEW QUEENS EMERGING FROM THE NUCLEUS HIVES**\n\nAbout one week after emerging as virgin queens from their capped cells the new queens will go on their mating flight.\n\nA few days after the mated queen has returned to the mating nucleus she will start laying eggs. You can leave the laying queen in the mating nucleus for several weeks while the colony builds up in numbers of workers. Once the mating nucleus is full of brood, honey and pollen, you can transfer the colony to its own eight-frame or ten- frame brood box and you now have a new hive to add to your apiary.\n\nAlternatively, if the aim of rearing queens was to re-queen existing colonies that need new, young, vigorous queens, find and remove the old queen before introducing the new queen using a queen cage. The entrance to the queen cage needs to be blocked with candy. It will take two to three days for the candy to be eaten allowing the queen to escape and take up her role in the colony. If the queen is introduced immediately into the colony the resident workers will not recognise her smell and will kill her. In the two or three days that it takes for the new queen to escape her cage, the resident workers will have become used to her queen pheromone and will usually allow her to become part of the colony without harming her.\n\nMake sure that the queen cage has very small gaps in it or the resident workers from the colony will enter the cage and kill the new queen. Alternatively, even if the gaps in the walls of the queen cage are too small for workers to enter, they may be sufficiently large for workers to bite and chew the new queen's feet through the gaps and thus kill her before she can leave the cage. Infrequently there is also the situation where workers will group and surround a cage, 'balling it' and smothering the queen inside.\n\nSo far I have only discussed the importance of selecting quality colonies from which to rear queens. Equally important, but much more difficult to manage, is ensuring that the virgin queen on her mating flight mates with quality drones from other colonies. Professional queen rearers often manage this by flooding the area around their queen-rearing apiary with colonies containing large numbers of quality drones. Although this practice is achievable by professional queen rearers it is not achievable by hobby beekeepers or small-scale queen rearers. Hobby beekeepers who rear queens can only hope that the virgin queen on her mating flight will mate with a sufficiently large number of drones to make the genetic diversity of her offspring strong, producing healthy, viable colonies.\n\nQUEEN CANDY\n\nIf you plan to send queens by mail to other beekeepers you will need to buy queen cages as well as make queen candy to block the entrance to the cage.\n\nQueen candy is different to the sugar candy used to feed colonies over winter in that queen candy is a fairly soft, putty-like mixture of honey and icing sugar while sugar candy is a solid block of white sugar. To make queen candy take a small amount of irradiated (disease-free) liquid honey and mix into it some finely granulated icing sugar. Keep adding and kneading in icing sugar until the resulting mixture has the consistency of putty. There are several types of queen cage available from suppliers and depending on the design you purchase the queen can be introduced in one of the following ways. For a two-entrance hole cage, begin by blocking one of the entrances to the queen cage with the queen candy. Next, place inside the cage the newly mated queen with about six worker bees to look after her until she is able to leave the cage. Once the queen and her escorts are inside the cage, the second entrance can usually be closed by a plastic plug. If your cage is of the type that has one entrance you will need to introduce the queen and her escorts before carefully inserting the candy plug.\n\nSUMMARY\n\n\u2022 Rearing queens is a valuable skill to learn as it will enhance your overall beekeeping knowledge and expertise. There are several methods of rearing queens and the easiest to master is the hive splitting method in which a single strong hive with brood in two boxes is separated into two independent hives. Properly managed, several new queens should emerge in one of the hives about fourteen days after the split.\n\n\u2022 High-quality queens can be reared by selecting a queen from your apiary that has, during the previous season, either produced the colony that stored the most honey, was the least aggressive, or was more disease resistant. \n12. Sustainable beekeeping\n\nThere are a growing number of beekeepers who practice what is loosely termed 'sustainable or natural beekeeping'. The philosophy of sustainable beekeeping differs from traditional hobby and commercial beekeeping in that practitioners devote a lot of thought and energy into providing what they believe are the optimum conditions to ensure their bees' existence is as close to that of a naturally occurring colony and as stress-free as possible. There is a view that bees 'know what they are doing' and should be left, with minimal interference, to get on with their business in the hive. The view is of a cooperative partnership between the beekeeper and the bees, with honey production being a secondary consideration.\n\n12.1: The extensive use of monocultures in agriculture has become a major problem for all species of bees. This photograph of a monoculture in Canada shows the difficulty bees have in getting a varied diet. Once the canola has stopped flowering there may not be any available food source for local bees.\n\nBeekeeping is seen ideally as a hobby or a small cottage industry and the commercial practice of many hives of bees foraging in a single area is seen as undesirable, both from the point of view of the bees and also the excessive competition created to many native pollinators in the area. Left to their own resources smaller numbers of bees would inhabit the same area and the number of hives would be regulated by the availability of food and water supplies. Many natural beekeepers believe that a large number of the pests, diseases and compromised quality of life of today's bees, including the well-publicised Colony Collapse Disorder (CCD) are largely due to, or certainly greatly contributed to, by current beekeeping practices. See p. 296 for more information about Colony Collapse Disorder.\n\nThere are various ways in which modern apiculture can cause stress and resultant poor health in bees. One of the more obvious practices causing nutritional stress is monoculture, where large areas of land are devoted to the production of a single crop. Notable examples are canola (rape and canola are the same thing) or almonds. Bees need a variety of pollens and nectars to provide the range of minerals and amino acids required to develop and raise healthy brood and to achieve innate and physical vigour as adults to perform the many duties demanded by the colony. Large areas planted with a single crop often do not allow bees to collect the variety of foods needed to sustain optimal health. Periods of monocultural exposure often supplemented by artificial feeding produce colonies more susceptible to disease, and both physical and nervous system distress.\n\nA fact sheet from the US Department of Agriculture on surveyed bee losses during the winter of 2012\u201313 records colony losses of around 30.5 per cent. A difference noted by the research scientist conducting the survey was the increase in colonies dwindling away rather than suffering classic CCD characteristics. Of interest is that beekeepers who took their bees to California for the almond pollination reported higher losses than beekeepers who did not take part in the pollination. Nearly 20 per cent of those beekeepers who moved their bees to pollinate Californian almonds lost 50 per cent or more of their colonies. The beekeepers participating in the survey managed around 600,000 colonies representing nearly 22 per cent of the US colony count of around 2.62 million.\n\nWhile poor nutrient uptake through an inadequate diet causes stress to bees, in many regions this is outside the control of the beekeeper. This is particularly so for those rural beekeepers who rely on the often fickle flowering of native trees and shrubs which are very dependent on climatic conditions. For the urban beekeeper, whose bees are able to forage from a wide range of native and exotic plantings throughout the local community, bee nutrition is much easier to manage. For the hobbyist with a few hives much can be done in the home garden to make it attractive and productive as a source of food for bees. Many beekeepers will also make an effort to locate hives near a wide range of flowering plants even if this is inconvenient due to distance and difficulty of site access.\n\n12.2: Urban gardens often have a wide range of exotic and native flowering plants that can provide bees with the range of pollens that they require. These gardens are often well watered and are not as subject to drought as country areas.\n\nA further major cause of stress to the bee is migratory beekeeping as practised in Australia and the United States. Commercial beekeepers, and some hobby beekeepers, move their hives every few weeks to a new location to follow the blooming pattern of local flora. As part of the practice of migratory beekeeping, hives can be loaded onto the back of a truck and shipped up to several hundred kilometres either overnight or during the day. Apart from the stress of being moved and the need to reorient themselves to a new location every few weeks, colonies are additionally stressed by being either confined to hives during their move or, in instances where hives are moved with the entrance open, some foraging bees will escape and not return, thus reducing the number of critical food gatherers in the hive.\n\nIn nature, if a bee colony takes up residence in, for example, a tree cavity, in the absence of human intervention or fire, nests may remain undisturbed for a number of years. Many of these feral colonies remain at the same location; they swarm regularly but will eventually abandon the nest, perhaps due to lack of space. If the colony is in a tree hollow, abandoning the nest may be caused by the tree decaying and becoming more open to the elements, ant invasion or a lack of food in the area. There are always exceptions and some very old trees with large hollows have supported viable colonies for a decade or more. If a suitable nest site is abandoned by its colony it is not unusual for the site to be re-occupied by later swarms.\n\nModern beekeeping practices since the invention and patent of the Langstroth hive, while a vast improvement on skeps and clay pots in which colonies were regularly destroyed in order to obtain honey, do nevertheless pose challenges to the wellbeing of the colony housed within it.\n\nWith conventional beekeeping, hobby beekeepers, particularly novice beekeepers, are encouraged to open their hives frequently so that they can sight the usual brood pattern, find the queen and generally get used to viewing a colony in good health so that later on there is some basis of comparison if the hive is opened and there is a change indicating a possible problem. During the swarming season again the beekeeper is encouraged every week to check for queen cells and, if present, to either destroy these or artificially swarm the hive by splitting it into two colonies. Natural beekeepers view over-frequent inspections as unnecessary and disruptive to bees, and many believe that much can be ascertained about the wellbeing or otherwise of a hive by careful repeated observation of the returning bees and the general outward appearance of colony behaviour outside the hive.\n\nWhile natural beekeepers regard the frequent manipulation of moveable frames as an unnecessary disruption to the life of bees, across Australia, Europe and North America there is a legal requirement for removable frames to be used so that apiary inspectors and beekeepers can easily check for the presence of disease. Indeed, the inability of earlier generations of beekeepers who used skeps, clay pots or other traditional hives to closely inspect their colonies was a major impediment to checking the health of their bees. In practical terms there is a trend among natural beekeepers to use top bar hives to house bees. To minimise disruption of the colony during inspections, some top bar hives have transparent acrylic sheets built into the walls. This allows the beekeeper to carry out visual inspections of the inside of the hive without disturbing the bees. The acrylic window or sheet will have a thick wooden door that can be closed between inspections so that the bees will not be disturbed by light continually entering the hive.\n\nThis is a major point of difference between traditional beekeepers and natural beekeepers who are strongly of the opinion that the hive should remain undisturbed as much as possible. There are a few reasons for this: firstly, the smoking and opening of a hive to remove frames is a significant disruption to the colony and the bees may take days to get over the intrusion before returning to their normal pattern of activities. Secondly, by opening the hive the beekeeper is altering both the nest scent and ambient temperature of the colony and this will take time and energy by the bees to return to normal. Also, brood exposed to extremes of heat or cold may either die or be subject to developmental problems during their growth and will not function properly as adult bees.\n\nHoney bees draw a variety of cell sizes when they make honeycomb, ranging from smaller cells at the top of the comb to lay worker eggs and to store honey, to larger cells at the bottom of the comb to raise drones. Natural beekeepers believe that the use of foundation embossed with a single cell size inhibits the bees from determining the optimum layout of the comb and hive, leading again to stress within the colony. By using milled foundation with smaller cell sizes suitable for worker bees and not drone bees, traditional apiarists ensure that there are a higher proportion of workers in the hive than would be the case in a feral or naturally occurring colony. Natural beekeepers argue that it is better to leave the selection of cell size to the colony. In this way the workers themselves will decide the optimum cell size and this size will vary across different parts of the nest and during different times of the year. Allowing the bees to make their own comb has a cost in reduced honey production (approximately 1 kilogram of wax takes 8 kilograms of honey to produce). But the reward comes in the form of healthier, more contented bees using clean, fresh comb, free from accumulated impurities and, of course, disease.\n\nAs previously mentioned, migratory beekeeping is practised by professional beekeepers in Australia and North America and although this practice is seen as maximising the amount of honey produced by each hive it causes significant stress to the bees. Moving many hives close together, particularly hives brought together from various parts of the country for pollination purposes, is a very effective way to transmit disease and pests widely and very quickly. Small Hive Beetle is a pest whose incursions have been aided by grouping both unaffected and affected hives in close proximity.\n\nTo use the pollination example again, if a colony of bees with the _Varroa_ mite was detected, say, at the Port of Melbourne, the Victorian Government's Department of Environment and Primary Industry and local beekeepers would mount a strong effort to trace and kill all colonies in the vicinity. If the same Varroa-infested colony was first detected in north-east Victoria during almond pollination in August, where thousands of hives have been trucked from across eastern Australia, there may be very little point in attempting to do anything about the mites. Any hives removed from the pollination site prior to detection would soon be distributed widely and the mite with them.\n\nAround one hundred years ago, diseases of honey bees were localised and bee populations globally were vulnerable to perhaps one or two diseases. Bees developed good resistance to these diseases which had been present in their particular locality in Europe for hundreds of years or even longer. Migratory beekeeping practices, while seen as an economic necessity for many beekeepers in the short term, may effectively have contributed to the creation of many long-term problems which will cause great economic loss, put strain on the industry and also severely compromise the bee it depends upon.\n\nSince the mid-twentieth century, when _Apis mellifera_ was transported across the globe on an unprecedented scale, bee diseases have spread exponentially into local bee populations that have little or no in-built resistance to them. The _Varroa_ mite was initially a minor pest of the Asian honey bee, _Apis cerana._ The two lived together for hundreds of years and _cerana_ developed a tolerance for the mite and the ability to keep it under control using effective grooming. When _Apis mellifera_ was introduced into Korea and Japan, _mellifera_ came into contact with _cerana_ bees carrying _Varroa_ mites and the mite soon jumped species from _cerana_ to _mellifera._ Commercial _mellifera_ beekeepers in Asia are believed to have used the Trans-Siberian Railway to ship infested _mellifera_ hives back to eastern Russia and from there into Europe, resulting in the disastrous situation we have today where _Varroa_ is considered the biggest problem of honey bees in the world and by far the largest cause of losses to beekeepers.\n\nNATURAL BEEKEEPING HIVES\n\nNatural beekeepers are not always fans of the Langstroth hive, although many commence their beekeeping life using them. Variants of the top bar hive are usually preferred for a more natural approach to beekeeping. The top bar hives usually favoured can be divided into two main designs: Warr\u00e9 vertical top bar hives and Kenyan and Tanzanian horizontal top bar hives.\n\nThe design of both of these types of hive will be explained shortly but essentially their use and appropriate management are believed by their adherents to minimise disruption to bees and to provide a more natural nest environment than the Langstroth hives.\n\nThere is no doubt that a thicker wooden wall for the hive, as recommended by Abb\u00e9 \u00c9imille Warr\u00e9 (see p. 194), provides better insulation for the colony and protects the bees from external disturbances. Langstoth hives purchased in Australia are generally made of wood that is 22 millimetres thick; this is a tradition from times when 7\/8-inch thick wood was readily available from saw mills and hardware shops. To obtain 22-millimetre-thick wood today to make your own hive is difficult since few mills cut to this size and you will need to buy large quantities of this wood to keep the cost down. Since this is beyond the resources or needs of most hobby beekeepers the next best option is to buy wood from the local hardware store, which is usually 19 millimetres thick.\n\n12.3: A skep hive next to a Warr\u00e9 hive. Skep hives, although the traditional symbol of beekeeping, are illegal to use in Australia since frames of brood cannot be removed to check for disease.\n\n12.4: The author's brightly painted Warr\u00e9 hive on the right.\n\n12.5: A Kenyan top bar hive.\n\n12.6: A Warr\u00e9 hive.\n\n**WARR\u00c9 HIVES**\n\nAbb\u00e9 Emile Warr\u00e9 (1867\u20131951), the French inventor of the Warr\u00e9 hive, believed that the optimum cross sectional area of a hive should be 300 millimetres x 300 millimetres square, 200 millimetres deep and that the hive should be made out wood preferably up to 35 millimetres thick for better insulation against heat and cold.\n\nOf course many beekeepers are of the opinion that their way of keeping bees is the only way and that all other methods are harmful and ineffective. Unless something has been lost in translation this attitude is apparent throughout Warr\u00e9's book on the construction and management of the Warr\u00e9 hive, often called the People's Hive. Warr\u00e9 has little tolerance for even the smallest change to the design of his hives or his management methods for keeping bees in them. In fact, bees of their own accord will make nests in a range of natural and artificial environments from hollow tree trunks, wall cavities, roof spaces, barbecues, and the odd possum or bird box. To state categorically that a bee prefers a box exactly 300 millimetres square and 200 millimetres deep is, to me, too dogmatic. In fact, many modern proponents of the Warr\u00e9 hive have devised their own, or adopted others adaptations to the original plans of construction but generally adhere to the management methods explained by Warr\u00e9. To be fair to Warr\u00e9, when he designed his hive he took into account what, in his view, was the optimum size of hive that the bees could manage taking into consideration the long cold winters experienced in Europe.\n\nA difference between the original design for the Warr\u00e9 hive and today's version is the removable frame. In the 1940s, when Warr\u00e9 specified the design, frames were fixed and the honey could only be extracted by breaking the comb. Warr\u00e9 was of the opinion that the modern removable frame was not a help in the early detection of disease but a major contributing cause. His view was that all that is needed to monitor the health of a colony is to closely watch the comings and goings of its resident bees to and from the hive entrance. Today, to comply with regulations for hive inspection for disease requiring moveable frames, a modern Warr\u00e9 hive frame includes removable top bars but not a bottom bar.\n\nWarr\u00e9 believed that using a bottom bar on frames restricted the ability of the bees to make natural honeycomb according to their needs. A thin starter strip of foundation is used underneath the top bar of the frame since the bees know intuitively the best shape and cell size for their colony. Interestingly, there have been recent variations on Warr\u00e9's original design with operators of these hives incorporating side bars on frames similar to a Langstroth hive. The side bars prevent the comb being attached to the side walls of the hive box where it may be severely damaged and break off when the frames are removed. In countries where the _Varroa_ mite is a problem there have also been modifications incorporated into the Warr\u00e9 hive base.\n\nThe design of the Warr\u00e9 hive also differs from a Langstroth hive in that instead of the top super box touching the lid, in the Warr\u00e9 hive a smaller buffering box is placed between the lid and first super box that provides additional insulation for the colony. This box is about 100 millimetres deep and is called a quilt as this term clearly explains its use. The quilt box contains a floor made of hessian cloth or aluminium flyscreen wire. The box is filled with dry leaves, straw, wood shavings or crumpled newspaper to provide insulation and the hessian or mesh wire attached to the bottom of the quilt box prevents the insulating material falling into the hive. The hessian or aluminium mesh floor very effectively allows damp air to circulate into the dry insulating material in the quilt box where it is absorbed, keeping the colony free from condensation. Since a lot of dampness can be absorbed by the insulating material, the beekeeper needs to check that this is not becoming mouldy and will need to periodically replace it with fresh dry insulation as required.\n\n12.7: A natural beekeeper holding a frame from his Warr\u00e9 hive.\n\n12.8: The quilt box on my Warr\u00e9 hive. My design is slightly different from that specified by Father Warr\u00e9 in that the quilt box is filled with crumpled newspaper or dry gum leaves. Flywire stops the leaves from falling into the hive.\n\nAnother major innovation of the hive is the design of the lid. Warr\u00e9 believed that the traditional flat metal lid was too disturbing for the bees since rain falling on it causes a drum-like noise inside the hive. Warr\u00e9 proposed two designs for lids, one of which is probably beyond those beekeepers without access to woodworking tools and facilities. The second, a much simpler design, can easily be made by most beekeepers with a little time and some basic woodworking tools. For all practical purposes the simpler design of lid for the Warr\u00e9 hive is effective and there is little to be gained by using the more sophisticated version of the lid apart from the inclination of some beekeepers to experiment with hives using both.\n\nA simpler lid again which would be effective in Australian conditions is to use the same flat roof design that is found in traditional Langstroth hive but to replace the metal cap with a thick wooden top that will not transmit the sound of rain so easily. The lid in this situation should be designed to be telescopic and not the migratory type of hive lid, as telescopic lids provide much better insulation and protection from the weather than do migratory hive lids. Wood to replace the metal cap can easily be sourced from scrap wood or from local building suppliers. If there is any difficulty sealing the joins of the lid the beekeeper can cover the lid with tar paper obtained again from a builder's supply outlet.\n\nAs with the Langstroth hive, a mat is placed on the top frames of a Warr\u00e9 hive, however the material used is hessian cloth soaked in flour and water and then ironed flat when dry.\n\nThe design of the base, apart from being nearly square, is the same as for the Langstroth hive. To ensure later ease of handling of the Warr\u00e9 hive lifting cleats need to be attached to the sides of each box.\n\n12.9: Part of the philosophy behind the Warr\u00e9 hive system is only to remove the top box since this contains honey. At the same time a new box is added at the bottom containing only frames. This process is called _nadiring._\n\n12.10: The top box is removed and a new bottom box is added.\n\n12.11: The nadiring process has been completed.\n\nManagement of the Warr\u00e9 hive is quite different to that of a Langstroth hive. When a new box is added to the Warr\u00e9 hive it is not placed on top of the existing hive boxes but is instead inserted at the bottom of the hive, a process more traditional beekeepers would call under-supering. This practice is referred to as nadiring in Warr\u00e9 terminology.\n\nIn the Warr\u00e9 hive set-up there is no queen excluder used and the queen is free to move throughout the hive. In their natural state, bees will build comb downwards as required, so it makes sense to provide additional space for the colony to expand at the bottom of the hive. This immediately raises the interesting question of how to lift a number of full boxes to slide a new empty box onto the hive base. Many books on Warr\u00e9 beekeeping show a rather sophisticated lifting system that seems on the face of it, or to me at least, to be totally impractical since it is beyond the capability of most beekeepers to either make or to obtain one of these lifts. The hobbyist with any carpentry skills or a bent for invention will relish facing and sorting out this problem, however, as it has not proved too much of an obstacle for an increasing number of beekeepers choosing this type of hive. Various websites provide many ideas and designs for Warr\u00e9 hive lifts and the interested reader should do a Google search for suitable plans.\n\nMany natural beekeepers who own a Warr\u00e9 hive perform under-supering, or nadiring, by removing all of the boxes off the base, adding an empty box with frames, and then returning the other boxes and placing them on top of the new bottom box. While this may be the only way to achieve the addition of a bottom box in the absence of a lifting device, it is in itself very disruptive to the colony of bees and goes against the principle of minimal disturbance.\n\nThe theory of the Warr\u00e9 hive is that since the queen will lay eggs in the bottom comb, and the workers will store honey in the top comb, Warr\u00e9 hive users need to extract honey by removing the top-most box from the hive \u2014 the box under the quilt. This box is taken away and the honey extracted by cutting out the comb and pressing it to squeeze out the honey. A common method of doing this is the use of a mechanical fruit press which can be procured either online or through many large specialist kitchen shops.\n\n**HORIZONTAL TOP BAR HIVES**\n\nA popular alternative to the Warr\u00e9 hive is the Kenyan top bar hive and its variant, the Tanzanian top bar hive. These two hives are often called horizontal top bar hives (HTBH), since extra room for a growing colony is obtained by allowing the colony to expand horizontally not vertically as it does in either a Warr\u00e9 or a Langstroth hive.\n\nIn the sections on both the horizontal top bar hive and Warr\u00e9 hives, I will use the terms frame and top bar interchangeably. (The term frame really applies to Langstroth hives since it contains a top bar, two side bars, as well as a bottom bar. Top bars in both Warr\u00e9 and horizontal top bar hives may contain side bars but do not use bottom bars.)\n\n12.12\/12.13: Using a fruit press to extract honey from a frame. This is a traditional method of extracting honey from honeycomb. Using a fruit press, a lot of the coarser wax particles are retained inside the press.\n\nThe advantages of the Kenyan and Tanzanian horizontal top bar hives over the Langstroth hive are:\n\n\u2022 The horizontal top bar hive is easy to make out of scrap wood or wood inexpensively purchased from a builders' supply store.\n\n\u2022 No heavy lifting is required when inspecting or managing the hive.\n\n\u2022 Frames can be easily removed for extraction.\n\n\u2022 The horizontal top bar hive can be built so that the top is at waist level, simplifying inspection of the hive.\n\n\u2022 Access to frames of brood or honey is straightforward, so they can conveniently be used as a training tool for new beekeepers. This is because both the brood chamber and the honey chamber (super) are at the same level and the beekeeper need not first remove the super to gain access to the brood chamber.\n\n\u2022 A single horizontal top bar hive can be used to house two separate colonies by the use of an internal partition, usually called a follower board, suitably positioned to keep the two colonies apart.\n\nThe disadvantages of the horizontal top bar hive are:\n\n\u2022 The hive does not use standardised parts like the Langstroth and a beekeeper may have difficulty purchasing a ready-made horizontal top bar hive or kit hive.\n\n\u2022 The hive needs to be placed on a level surface and, due to its large footprint, this may be more difficult than for a Langstroth or Warr\u00e9 hive.\n\n\u2022 Due to its larger size, moving a horizontal top bar hive may be more difficult than moving either a Langstroth or Warr\u00e9 hive. This can be overcome, however, by sliding the horizontal top bar hive onto a wheelbarrow which can be performed easily by one person.\n\n12.14: Making a hive out of a large log.\n\n12.15: A top bar hive on a rural property in Victoria.\n\nBefore discussing the management of a horizontal top bar hive the difference between Kenyan and Tanzanian top bar hives should be noted. The Kenyan horizontal top bar hive has sides that slope at about 30 degrees toward the base while the Tanzanian horizontal top bar hive has vertical sides. The remainder of the hive construction is the same except that the Tanzanian horizontal top bar hive does allow the use of frames with side bars if the beekeeper wishes to add them.\n\nThis is an advantage of the Tanzanian hive as frames similar to those used in Langstroth and Warr\u00e9 hives can be used. Side bars make a frame easier to remove as there will not generally be any comb attached to the sides of the hive. In addition side bars afford some strength and stability to the hanging comb, particularly in periods of very hot weather where melting from the top bar becomes a possibility.\n\nA new colony may be introduced into a horizontal top bar hive with any unneeded space blocked off by the use of internal follower boards. As the colony grows and requires more room, the follower board can be moved outwards and more top bars added to the area housing the colony.\n\nTo enable the bees to enter and exit the hive, around one to three 25-millimetre diameter holes are drilled, usually on one of the hive walls. These entrances can be blocked with corks when not required. Suitable corks can be obtained from stores supplying home winemakers or brewers. Alternatively, used leftover wine corks can be washed and aired to remove odour.\n\nAlthough the top bars provide a continuous surface over the top of the hive this surface contains cracks and is not weatherproof. To overcome this, a lid is placed over the entire horizontal top bar hive. This lid may contain extra insulation such as styrofoam or builder's blue house insulation sheeting to aid in further keeping the hive at a stable temperature.\n\nSimilar to the Warr\u00e9 hive, a horizontal top bar hive frame needs a thin strip of starter foundation attached to the top bar to guide the bees in drawing straight comb. This starter can be a thin strip of beeswax foundation inserted into a groove on the underside of the top bar frame held in place by pouring melted wax into a groove cut or routed for this purpose. An alternative method is to attach a triangular strip of wood to the underside of the top bar so that the bees will use the edge of the wood to guide their comb building. Popsicle sticks can also be glued into the routed grove to form a sharp edge to guide the bees in their comb making. If no guide is placed on the underside there is a risk that the bees will start drawing the comb at the edge of the top bar. This will result in an untidy, messy distribution of comb that will congest the hive and will prove difficult for the beekeeper to remove without damaging the honeycomb. This problem is true in particular with the Kenyan vertical top bar hive with its sloping sides, for as the bees build comb they attach it to the sides of the hive for strength. This can make removing frames difficult as there is the possibility that the entire comb can break off, leaving a mess for the beekeeper to clean up. To overcome this, a long sharp knife is needed to gently cut the comb from the hive walls before removing the top bar and, hopefully, the intact comb with it. If the comb is attached to the side walls this is usually only for the top 5 to 10 centimetres down the side of the box.\n\nOf the three types of hive discussed in this book \u2014 Langstroth, horizontal top bar hives and the Warr\u00e9 vertical top bar hive \u2014 I have had the practical experience of keeping my bees in primarily the Langstroth, which I admittedly prefer for my own convenience. I have, however, over the past two seasons run both a horizontal top bar hive and a Warr\u00e9 vertical top bar hive in order to educate myself on their management and to enable a more empirical assessment of the respective hives. It is also hoped that over a period of a few seasons as well as gaining some insight into the day-to-day practicalities of these types of hive I will observe any differences in bee populations hived in the different structures.\n\n12.16: Inspecting a small top bar hive.\n\n12.17: A strip of wax starter being built out by a colony of bees in a top bar hive.\n\n12.18: The use of Popsicle sticks to give the bees an edge to start building comb from is a popular method of making frames for both Warr\u00e9 and Kenyan top bar hives.\n\nFor the beekeeper interested in using one of the top bar hives there are three apparent disadvantages that may or may not prove of importance.\n\n\u2022 A horizontal top bar hive is not cheap to buy as a kit or ready assembled. Top bar hives, however, are inexpensive and easy to make, with plans readily available in books and off the Internet, for those with the necessary practical skills.\n\n12.19: Removing a frame from a Kenyan top bar hive is sometimes difficult since the bees attach comb to both the frame and the sides of the hive. The easiest way to remove the frame is to use a long knife to cut the comb from the sides of the hive.\n\n12.20: Step 1: Inspecting a frame from a Kenyan top bar hive. Since the foundation in a Kenyan top bar hive frame does not have any sides or wire to support it, if the frame were to be turned upside down the foundation would probably break off.\n\n12.21: Step 2: To inspect the opposite side of the frame first rotate the frame by 90 degrees so that the top bar is vertical.\n\n12.22: Step 3: Next twist the frame by 180 degrees so that you can inspect the other side of the foundation.\n\n\u2022 Demand for and resale value of a top bar hive is a lot less than for a Langstroth hive.\n\n\u2022 Both vertical top bar hives and horizontal top bar hives produce less honey than Langstroth hives. The reason for this is that since neither of the top bar hive designs uses queen excluders, the queen is free to lay eggs on the same frames as the workers store honey. This results in some frames of drawn comb holding both brood and honey. The honey is stored on the upper most part of the comb while the brood is on the lower part of the comb. This means that it is not possible to harvest the honey without killing many brood, which natural beekeepers would prefer not to do.\n\nBoth Warr\u00e9 and top bar hives are expensive and often difficult to purchase. As a result, many people apply natural beekeeping methods to the colony but buy a Langstroth hive in which to house their bees. This compromise is perfectly acceptable and is recognised as a legitimate approach to natural beekeeping.\n\nSUMMARY\n\n\u2022 Natural or sustainable beekeeping is gaining in popularity and there are now many proud owners of both Warr\u00e9 and Kenyan top bar hives. Natural beekeepers are often more concerned about the state of the environment and the plight of bees, in particular. The widespread planting of monocultures and the poor management practices that do not take into account the welfare of the bees are two of the concerns that natural beekeepers have about traditional beekeeping. Natural beekeepers view their relationship as a co-operative one with the bee and usually harvest less honey from a hive than the traditional Langstroth beekeeper. The honey taken is limited to the excess not required by the colony.\n\n\u2022 The top bar hives mentioned above are usually difficult to purchase as a kit and if available they are expensive. This will no doubt change as the popularity of top bar hives continues to increase. In the meantime, beekeepers who adhere to many natural beekeeping practices successfully use the Langstroth hive which is both readily available in kit form and at a relatively low price. \n13. The bee-friendly garden\n\nThe interdependent relationship between the honey bee and the flowering plant is one which has developed over millennia. The bee relies on the flowering plant for nectar and pollen without which it could not survive. Pollen, which is produced and found on the male organ of the flower, provides protein for the bee and is an essential component of the bee bread fed to developing larvae. Nectar, a sweet sugary solution, supplies carbohydrates to the bee and is produced in special glands called nectaries generally found in the deepest part of the flower. In some plants additional nectaries are found on petals, leaves and stems.\n\nTo gain access to a flower's nectary the bee will brush past the pollen-laden anthers or male part of the flower. Some of the pollen will adhere to the hairs on the bee and while it continues foraging the bee will inadvertently transfer some of the grains onto the sticky stigma, part of the female organs of the flower. A plant may be self-pollinating which means it will reproduce using its own pollen transferred onto its own stigma or it may be of a type that requires cross pollination from another plant of the same species. Due to the bee's flower fidelity, or habit of visiting one particular type of flower during a foraging trip, many plants of the same species will be successfully pollinated and will reproduce.\n\nBEE-FRIENDLY FLOWERING PLANTS\n\nAs a beekeeper it makes sense to populate the area around your hives with those plants that provide good supplies to your bees of the pollen and nectar essential for their wellbeing. There are myriad flowering plants available to suit the various climatic and gardening conditions found throughout a large continent like Australia. While advice can be sought from local garden centres and nurseries on the types of plant likely to thrive and flower in a certain area, for the beekeeper there are other additional considerations apart from personal preference that will influence plant choice. To earn a place in the bee-friendly garden a plant has to offer pollen, nectar or both and make these easily available to the visiting bee.\n\n13.1: A hive is part of the garden picture.\n\n**PLANT CHOICE**\n\nWhether you are renovating a small area of an existing garden or planning a new garden to maximise the forage available for your colonies of bees, there are certain initial considerations that will assist your final decision on what to plant. The following questions will guide you in your plant selection.\n\nDOES THE PLANT FLOWER? WHEN, HOW OFTEN AND FOR HOW LONG?\n\nWhile it is easy to quickly ascertain if a plant offers flowers, a further critical consideration before choosing a plant for the bee-friendly garden is the frequency of flowering. Find out at what time of the year a plant flowers, how often and for how long. Ascertain if it will repeat flower throughout the year or just the once. If it flowers once but has a lengthy flowering period it is worth considering. While for many plants flowering is restricted to the spring, during this time they produce copious amounts of pollen and\/or nectar just when the hive is building brood and bee numbers are rapidly increasing. In my garden several large old-fashioned single-flowered climbing roses keep the bees working to capacity for a number of weeks. Due to their size and voluminous flowering their contribution, particularly of pollen to the garden's resident colonies, is important.\n\nMany plants can be encouraged to repeat flower or sporadically flower for longer periods by judicious pruning and deadheading. Do a little research, check the plant label and see if some of the flowers you would like to grow will do this.\n\n13.2: Spring flowering climbing rose 'Pinkie' is a bee favourite in the author's garden.\n\n13.3: Bluebells, one of the many bee attracting spring bulbs.\n\n13.4: Summer flowering Crepe Myrtle.\n\n13.5: Sedum provides summer flowering for hot dry conditions.\n\nInitially limit yourself to a few good plant choices to start improving or developing your garden. Ensure that you select plant species that will offer a succession of flowers throughout the year and buy three or four plants of the same type, placing them together in blocks or informal groupings. As your knowledge grows and you become more confident with your plant choice you can add more of those species that do well in your particular area.\n\nTo create an environment that is more bee-friendly the aim is to choose a variety of plants that offer a succession of flowerings throughout the year. It is relatively easy to choose spring and early summer flowering species due to the large number of available plants that offer flowers during this period. Of equal importance are the late summer and autumn flowering plants that will sustain colonies after their initial build-up. Access to pollen and nectar is vital at this time as bees are building condition prior to winter and nutritious forage and continued storage of supplies is critical to the successful overwintering of colonies.\n\nIS THERE ACCESS TO NECTARIES?\n\nOften plants that succeed well in the garden and those most attractive to bees are the original species plant or old-fashioned cultivar that naturally evolved before we decided to improve on nature's design. The large double flowers so attractive to us with their complicated jumble of petals provide little or nothing in the way of pollen for the bee as they are often sterile; their sexual organs have disappeared or been modified to become extra petals. Nectaries, if they exist, are usually difficult or impossible for a bee to access.\n\nHighly bee attracting are simple or single flowers with an open, shallow cup-like face, as both the nectar and pollen that the bee seeks is readily accessible. Many single-flowering roses, poppies, spring blossom and various daisy species are good examples of this shape and type of flower.\n\nBees collect nectar from flowers using their proboscis, or long tongue, that measures around 5.7 to 6.7 millimetres. A flower with a nectar-producing part any deeper than this will not be within the reach of the average honey bee although will appeal to many smaller native foragers.\n\nA few plants with tubular or bell-shaped flowers may pose access problems for the bee as their nectaries are too deeply hidden at the base of a narrow tube of petals. For the most part, however, flowers with this shape are highly attractive to the bee and should be included in your plant choice. If you can look at the plant in the garden centre or plant nursery choose those tubular flowering plants that have a corolla tube (nectar-bearing part) wide enough for the bee to gain entry. Also a good choice are those tubular-shaped flowers that have overlapping petals which spread when the visiting bee forces them apart, often to completely disappear inside the flower intent on its nectar-seeking mission. Salvias, Grevilleas, Foxgloves, Penstemon and many fuchsia species (including some native types) have tubular-shaped flowers and are excellent bee-friendly choices if your garden conditions are suitable.\n\nPlants with a stem or stalk ending in a cluster of tiny individual flowers are particularly bee attracting. In addition to their pollen and nectar rewards the visiting bee is able to conserve energy during a foraging trip with many flowers readily available to it on the same stem. Lavender is an excellent example of this type of multi-floral flower as are _Achillea, Buddleja, Callistemon, Echium_ and the various _Hebe_ species.\n\nAnother similar flower type usually found at the end of a long slender stem and highly attractive to your bees is the round globe or semi-spherical shaped flower. These flowers are also composed of many individual tiny flowers and again provide the bee with energy efficient forage. _Echinop_ species, the globe artichoke _(Cynara scolymus_ ) and _Allium_ species such as garlic, onions, chives and the beautiful but underused flowering garden alliums are examples of this shape.\n\nIn Tasmania, the bumblebee is found alongside the honey bee and while it no doubt provides competition for forage, it also often enables the honey bee to gain access to those plants with long tubular flowers and hidden nectaries that would otherwise be inaccessible and unavailable to them. A bumblebee is able to pierce a hole in the outside petals of a flower near or over the location of the flower's nectary. It then inserts its tongue, or proboscis, into the opening to obtain nectar from the outside of the bloom. Once opened up like this, other pollinators, including the honey bee, will take advantage of the easy access offered to the flower's nectar.\n\nSIMPLE SINGLE FLOWERS\n\n13.6: Almond blossom.\n\n13.7: Poppy.\n\n13.8: Euryops is a winter-flowering daisy.\n\nTUBULAR SHAPED FLOWERS\n\n13.9: Salvia.\n\n13.10: A bee disappears inside a fuchsia flower.\n\nBELL SHAPED FLOWERS\n\n13.11: Blueberry flowers.\n\n13.12: Comfrey.\n\nMANY INDIVIDUAL FLOWERS ON ONE STEM\n\n13.13: Hebe.\n\n13.14: Californian Lilac.\n\n13.15: Buddleja davidii, commonly called butterfly bush.\n\n13.16: Garlic chive.\n\n13.17:Common chive.\n\nWILL THE PLANT THRIVE AND FLOWER IN THE AVAILABLE CONDITIONS?\n\nPersonal plant preference does to some extent become a secondary consideration in the bee-friendly garden. The primary focus of design and plantings is to offer year-round forage to the honey bee within the bounds of what will flower and survive in the conditions your property is able to provide.\n\nAs a starting point, survey your local area and look at the type of flowering plants that will survive and flourish \u2014 many do despite obvious neglect. Consider the overall climatic conditions of your area; frost tender plants will not flourish in very cold or snow prone parts of the country. Similarly, in hot dry arid areas heat-sensitive water-thirsty plants (often those with large fleshy leaves) will not thrive and you will waste a lot of time and energy trying to keep them alive.\n\nWalk around your own property in the morning and again in the afternoon. You may need to do this several times over a period of time. Consider the effect of sunlight and of shade as this will help you decide on the choice and eventual position of your plants. Work out how much sun the area to be planted receives throughout the day and how this is affected by the changing slant of the sun as it alters during the various seasons. If, for example, you are considering a sun-loving species of flowering plant, many of these will require four to six hours of good sun each day. If they receive filtered sunlight or are in shade for long periods they will not thrive and will become weak and succumb to both pests and disease.\n\nGive some thought to the aspect of your property and how this will affect your plantings. The area to be planted may face north but if there are tall mature trees, high fences or outbuildings these may affect the amount of sunlight reaching your plants for a large part of the day. Large trees will also provide both root and water competition for your establishing plants and inhibit their growth. If you live in an area prone to winter frosts bear in mind that buildings and dense shade can create 'frost pockets' for long periods in the winter and cause burning to sensitive plants from which they may not recover.\n\nIf your garden area is very exposed, for example on the top of a hill, a rise or near the coast and subject to salt-laden winds, these conditions should also be taken into account when you make your plant choices.\n\nWhile you cannot do much about the aspect and climate of your property you can greatly improve soil condition and enhance the survival prospects of your chosen plants. Soil is a living, breathing medium and its overall state is critical to whether plants will thrive and mature to the point of flowering and providing optimum nutrition for your bees. All soil from the opposite extremes of sandy to hard clay can be greatly improved with the addition of organic compost. A good supply of organic material will condition the soil in your garden, increase its water holding properties, provide a home for millions of soil essential micro-organisms and attract and and keep worms breeding and multiplying. Compost will also add the essential elements necessary to feed growing plants with the minerals and nutrients they require for healthy development and all-important flowering for your bees. A good layer of organic mulch will suppress weeds and provide a moisture-retentive protective blanket around your new plants.\n\n13.18: 'Cottage Garden Rose' with its abundant flowering is a bee favourite in spring.\n\n**COLOUR AND GROUP PLANTINGS**\n\nNature has endowed the bee with complicated compound eyes, or thousands of tiny lenses that are combined by the bee's brain into one picture. A group or several blocks or waves of plantings in bee-friendly colours will stand out to any passing bee and will be more obvious than a single specimen, unless of course the specimen is a large shrub or a tree covered in a multitude of flowers.\n\n13.19: A sight that is difficult for a passing bee to miss.\n\nBees are particularly drawn to the colours blue, purple, violet, white and yellow. During a foraging trip a bee tends to show an affiliation for one particular type of flower and will often ignore other nearby flowering plants that also provide pollen and nectar. Armed with this knowledge of bee behaviour you can plan a renovated or new garden by focusing on the colour and type of plant known to be favoured by the bee and then plant flowers of a single species in these colours in clumps or groups throughout your garden.\n\n13.20: What we see with our ordinary human vision.\n\nThe bee has the ability to see ultra-violet light, which humans cannot, and many of the marks or features it finds attractive in a flower are not visible to us. While a bee cannot see the colour red many flowers of this colour have other visible markings such as black spots and ultra-violet lines that act as a map to guide it to the nectary of the flower.\n\n13.21: What is seen by a bee with its ultraviolet vision.\n\nBy choosing groups of plants of differing sizes and height you will add structure and interest to your garden. In addition to being more visually appealing to you this will help your bees as different sizes and types of plant warm up in the sunshine at differing rates and as a result will release their nectar flow at different times of the day. Nectaries are not passive flower parts but are complicated active organs allowing many plants to regulate at what time of the day nectar will be secreted, in what concentration and to increase the flow in the presence of visiting pollinators.\n\nPlan your garden so that you have three or four different types of plant flowering at any one time to aid your bees' nutrition. Concentrate on those plants that like a sunny position rather than a shaded spot in the garden; the bee favours plants in sunny parts of the garden and will visit flowers in these areas and often ignore those in shade.\n\nFURTHER CONSIDERATIONS FOR THE BEE-FRIENDLY GARDEN\n\n**SPRAYS AND POWDERS**\n\nIf you use proprietary sprays and powders to control unwanted insects and other garden pests consider the effects that these will have on your own bees and all other beneficial insects and pollinators. Bees are very susceptible to pesticides, herbicides, insecticides, fungicides and their residues. You can decimate or at least severely compromise the health of the adult bees or brood in your hive with one careless application of spray.\n\nMany sprays available these days are systemic and they are absorbed into the living tissues and transportation systems of the plant. Any pollen, nectar or water droplets on the plant will be taken back to the hive by a foraging bee as nectar or pollen, stored and eventually fed to the developing brood. Topical sprays for either pests or disease do not discriminate between beneficial and pest insects and will harm or kill them all. Dusting powders such as derris dust, often labelled as a natural product, are also harmful to foraging insects and to your bees. You will perhaps decide to live with a few spots, chewed leaves and many perceived plant problems of a cosmetic nature rather than risk compromising your own colonies and those of other beekeepers.\n\n**NATIVE VERSUS EXOTIC**\n\nThe European honey bee has adapted well to the native flora of Australia and, in particular, to the copious nectar flow of many eucalypts. An increasing number of people prefer to plant native or indigenous species in their gardens and as the point of the bee-friendly garden is to provide flowering plants attractive to your bees for most of the year, with some research and the assistance of a specialist nursery you should be able to achieve a bee-friendly native garden with little difficulty in whatever region of Australia you live. For most of us a mixture of both native and introduced flowering plants will provide the largest choice of bee-attracting species for our gardens and ensure that our bees are easily able to access both pollen and nectar at various times throughout the year.\n\n13.22: Native hibiscus _Alyogyne huegelii._\n\n13.23: _Brachyscome multifida,_ commonly called rock daisy.\n\n13.24: _Hibbertia scandens,_ or snake vine.\n\n**BEE-FRIENDLY NATIVE FLOWERING SPECIES**\n\nBanksia _(Banksia_ spp.)\n\nBrachychiton _(Brachychiton_ spp.)\n\nBottle brush _(Callistemon_ spp.)\n\nGum tree _(Eucalyptus_ spp.)\n\nGrevillea _(Grevillea_ spp.)\n\nGuinea Flower _(Hibberta_ spp.)\n\nNative daisy bush ( _Olearia_ spp.)\n\nPaperbark, Honeymyrtle, Tea tree _(Melaleuca_ spp.) \u2014 note that some individual plants of the _Melaleuca_ spp. are commonly called tea tree; examples are _M. minutifolia_ and _M. alternifolia._\n\nSenna ( _Senna_ spp.)\n\nTea tree ( _Leptospermum_ spp.)\n\nDo your own research or ask a specialist native nursery or your local garden centre to suggest suitable plants within each species for your area and requirements.\n\n**INCORPORATING BEE-FRIENDLY PLANTS INTO THE GARDEN**\n\nRegardless of the size of your garden you can, depending on your inclination and time constraints, add a considerable amount of forage for your own and other visiting bees. While you may not wish to drastically alter a garden that you have nurtured and developed over many years it is always possible to find areas that you can add to or renovate with bee-attracting flowering plants.\n\nHERB GARDENS\n\nIf you are not really a gardener but can find a small amount of sunny space, consider planting a herb garden which will add to your own health and to the nutrition of visiting bees. A herb garden can be elaborately designed as an attractive addition to your property or be a simple patch of sunny fertile soil planted with a few herbs. Plant a flowering border or low hedge around the chosen area using thyme, garlic chives, sage or a low-growing species of lavender ( _L. stoechas_ ) and you will quickly add interest to your garden and extra flowers for your bees.\n\nA large number of the commonly used kitchen herbs such as rosemary, the various thymes, sage, marjoram, oregano, basil and parsley are very attractive to your bees. It will, of course, be necessary to allow your herbs to flower before they are useful to your colonies and this is not difficult if you limit your gathering to pinching out various areas of the plant for your own kitchen use. This will have the added effect of the plant growing further branched stems to support more flowers for your bees.\n\nHERBS FOR BEES AND YOU\n\nSweet basil ( _Ocimum basilicum_ )\n\nBorage ( _Borago officinalis_ )\n\nChives ( _Allium schoenoprasum_ )\n\nComfrey ( _Symphytum_ spp.)\n\nLavendar _(Lavendula_ spp.)\n\nLemon balm ( _Melissa officanalis_ )\n\nMarjoram ( _Oreganum marjorana_ )\n\nMint ( _Mentha_ spp.)\n\nOregano ( _Origanum vulgare_ )\n\nParsley ( _Petroselinum crispum_ )\n\nRosemary ( _Rosmarinus officinalis_ )\n\nSage _(Salvia officinalis)_\n\nThyme ( _Thyme_ spp.)\n\nMany herbs are vigorous. Comfrey needs space and is not a choice for a small garden. The mint family in particular should be planted with care and contained.\n\nVEGETABLE GARDEN\n\nIf you prefer the edible type of garden ensure that you leave several plants for your bees by allowing them to flower. Brassicas such as cauliflower, broccoli, cabbage and Brussels sprouts offer both pollen and nectar when allowed to flower, whereas many modern cultivated vegetables and self- pollinating peas and beans will offer nothing at all to your bees.\n\n13.25: Rosemary.\n\n13.26: Borage.\n\n13.27: Lavender.\n\nStrict soldier-like rows of vegetable plantings can be broken up by interspersing with groups of flowering plants which, in addition to providing nutrition for passing bees, may also serve to confuse and divert some of the pest species attracted to your vegetables.\n\nThere is a good deal of information available on both companion planting in the vegetable garden and planting to repel pest species. Further investigation of flowering species with these extra possibilities is a winning proposition for both your bees and you as a gardener.\n\nConsider planting a green manure crop, which will grow quickly, flower for your bees and before it sets seed can be dug to provide humus and minerals for your soil. Buckwheat is a plant that offers much; it is highly attractive to bees, produces edible seed for you to eat and can finally be dug in to fertilise and add humus to your soil as a green manure crop.\n\nFRUITING TREES AND PLANTS SUITABLE FOR BEES\n\nApple, crab apple ( _Malus_ spp.)\n\nAvocado ( _Persea americana_ )\n\nBlueberry ( _Vaccinium corymbosum_ )\n\nBlackcurrant ( _Ribes nigrum_ )\n\nBanana ( _Musa_ spp.)\n\nBroccoli, cabbage, cauliflower, brussels sprouts ( _Brassica_ spp.) \u2014 leave them to flower Cucumbers ( _Cucumis sativus_ )\n\nLemon, lime, orange, mandarin, cumquat ( _Citrus_ spp.) Macadamia _(Macadamia integrifolia_ var.)\n\nPassion fruit ( _Passiflora edulis_ )\n\nPersimmon ( _Diospyros_ spp.)\n\nPlum, cherry, peaches, nectarines and almonds _(Prunus_ spp.)\n\nPumpkin ( _Cucurbita maxima_ )\n\n13.28: Greek basil.\n\n13.29: Chilli plant.\n\n13.30: Leave a few vegetables to flower for your bees to forage in.\n\nRaspberry ( _Rubus idaeus_ )\n\nRedcurrant ( _Ribes rubrum_ )\n\nStrawberry ( _Fragaria x ananassa_ )\n\nTomato ( _Solanum lycopersicum_ )\n\n**VERTICAL FLOWERING SPACE**\n\nHEDGING\n\nInstead of the overdone non-flowering box hedge choose a low flowering plant that also responds to clipping if this is the look you prefer. You will have to restrain the clippers until after the flowering period of the particular plant has passed but your efforts will often result in a further flush of flowers for your bees. Taller hedging can be used to create windbreaks and soften existing fence lines. _Abelia, Callistemon, Ceanothus, Escallonia, Eriostemon, Westringia, Hebe_ and the rosemary leafed Grevilleas are all species that respond well to hedging and are good examples of bee-attracting shrubs.\n\n**BEE-ATTRACTING FLOWERING SPECIES FOR HEDGING**\n\nAbelia ( _Abelia_ spp.)\n\nEscallonia _(Escallonia_ spp.)\n\nGrevillea ( _Grevillea_ spp.) \u2014 fine rosemary leaved types are best Hebe ( _Hebe_ spp.) \u2014 numerous cultivars in various sizes and leaf types Lilly Pilly ( _Syzygium_ spp.)\n\nNative rosemary _(Westringia_ spp.)\n\nRugosa Rose (Rosa rugose)\n\nViburnum ( _Viburnum_ spp.) \u2014 deciduous and evergreen types available\n\n**CLIMBING PLANTS**\n\nFlowering shrubs or vines can be planted against existing fencing and walls to increase the vertical flowering space of your garden and the forage available to your bees. Pergolas and arbours add special interest to any property and are ideal supports for large flowering vines and climbers like the vigorous Wonga vine, _Hardenbergia_ , clematis and climbing rose species.\n\nSelection of climbing roses should concentrate on old-fashioned or more recent cultivars with either single or loosely formed blooms held in clusters on the plant. Avoid hybrid tea roses; their blooms are usually inaccessible to the bee until nearly spent and they do not produce sufficient numbers of flowers to be useful for forage. Consider the tough Rugosa roses. These are commonly used in other parts of the world for hedging and many have numerous large single flowers that are very attractive to bees. They are disease resistant and most have colourful hips for birds in the autumn. Unfortunately, they also have a number of thorns that may bother you \u2014 but not your bees.\n\n13.31: Hebe 'Lavender Blue' used as a loose hedge.\n\n13.32: Close up of Hebe 'Lavender Blue'.\n\n13.33: San Pedro cactus.\n\n13.34: Rugosa 'Frau Dagmar Hartopp'.\n\n13.35: Clematis 'Marie Boisselot'.\n\n13.36: Clematis 'Nellie Moser'.\n\nClimbing plants can grow to an enormous size so choose your species wisely and do not plant anything that will become too large and heavy for the structure supporting it.\n\nProvided with the right support, in temperate parts of the country large clematis are easy to grow. Several varieties can be planted close to each other to provide flowers from spring through to the early part of summer. Most will spot flower again in the autumn.\n\n**GROUND COVERS**\n\nSuitably chosen low-flowering groundcover plants can be used in your garden both as living mulch and as an extra source of forage for your bees. Some of the popular varieties such as the pink or white native _Myoporum parvifolium_ (Creeping Boobialla) and mauve or purple _Scaevola aemula_ (Fan flower) are easily grown. The exotic purple or white _Bacopa_ (Sutra) species and the mauve flowering _Convolvous mauritanicus_ are also attractive useful ground covers which when planted should be kept in check as they can become rampant in some areas. Prostrate ground-covering Grevilleas provide considerable pollen and nectar during the spring build-up of the hive and there are numerous original species and cultivars available to suit most garden environments.\n\n**GROUND COVER PLANTS SUITABLE FOR BEES**\n\nCreeping Boobialla _(Myoporum parvifolium_ spp.)\n\nCampanula ( _Campanula poscharskyana_ spp.)\n\nDead nettle ( _Lamium_ spp.)\n\nFan flower _(Scaevola aemula_ spp.)\n\nGrevillea ( _Grevillea_ spp.) \u2014 and also prostrate forms\n\nGuinea Flower ( _Hibberta_ spp.)\n\nPig face ( _Carpobrotus glaucescens_ spp.)\n\nSturt's Desert Pea _(Clianthus formosus_ spp.) \u2014 grafted varieties are less susceptible to root rot\n\nThyme ( _Thymus_ spp.)\n\n**GARDENING IN POTS**\n\nIf you live in an apartment or dense urban situation with or without your own colony of bees you can add to the flowering diversity of your area with potted flowers, herbs and some smaller flowering shrubs. The addition of several well-chosen bee-friendly flowering plants to your balcony or courtyard will offer food for visiting bees and will add interest and colour to your immediate surroundings. While you may feel your contribution is small you will add to the overall forage available for bees and other pollinators within your wider neighbourhood. Any flowering plants, even one or two, are better than none at all.\n\n**ENVIRONMENTAL WEEDS**\n\nIt is perhaps obvious that no one should plant and nurture anything that has been declared an environmental weed or is likely to become one. It is not always clear which plants fall into this pest category and to confuse matters further what is a weed in one part of Australia may not be in another. Your local council or plant nursery should be able to provide advice on what not to plant and many councils and local authorities supply residents with a list of plants considered environmental weeds and not safe to plant. Declared weed species may be native to Australia or introduced. Many environmental weeds are former garden plants or native species that have adapted too well to the conditions offered in a particular area, to the point that they threaten to take over large areas if left unchecked. Many of these species are very attractive to bees and in more rural areas are known to provide valuable forage during times of eucalypt and other flowering shortages. In the domestic situation the dandelion and many common flat-leaved lawn 'weeds' are sought out by the bee and you may decide to leave them where they are for this reason alone.\n\n13.38: Rose 'Crepescule'.\n\n13.39: Star Jasmine.\n\n13.40: Even one pot of bee-friendly flowers is better than none at all.\n\n**WATER**\n\nAs a general rule avoid flowering plants that like a lot of water. An increasing trend on plant labels is an indication of how much water a plant requires via a diagrammatic water drop system. A plant label with one drop indicates low water requirements. The more drops a label displays the greater the plant's water requirements for optimum growth. All plants will initially require some watering to establish their roots, but unless you live in a particularly dry or arid region, once established many can survive on natural rainfall alone and should be favoured choices.\n\nIf you live in an area prone to long dry spells consider concentrating on flowering shrubs and trees, as generally a shrub or tree is better able to sink its roots deep enough to survive short periods of hot weather and drought and will require less overall water and care. Nevertheless, there are many hardy perennials that will survive short dry periods and will both provide beauty to the garden and food and nutrition for bees and other foragers. Perennials are plants that do not need to be replanted annually; they have the added advantage that many species are easily propagated by taking cuttings from existing plant stocks or from plants in the gardens of others.\n\nIf you have the room consider adding a garden pond, even a small one, to aid your bees in their search for water. Ensure that you plant this with a variety of marginal and aquatic species that can be used by your bees as safe landing platforms to access the water while they drink.\n\nA GARDEN IS A WORK IN PROGRESS\n\nHowever you decide to improve your property and garden for your bees you will make mistakes or change your mind about some of your initial choices. A plant that does well in another beekeeper's garden may not thrive in yours. Any garden, including one being renovated or designed to be bee-kind, is never a static picture but an evolving one requiring ongoing labour, so if you are unhappy with a plant choice or its placement you can always change or improve on your efforts over the years. This becomes more feasible and an inexpensive task once you successfully take your own cuttings from those bee-friendly plants recommended by other beekeepers.\n\n13.41: Cotoneaster is an environmental weed in many parts of Australia but it is a magnet for bees in the spring and in autumn its red berries are loved by birds (who defecate its seeds ensuring its propagation).\n\nIf you have a large area to fill consider planting a few flowering 'sacrifice' plants to fill in the spaces until your chosen specimens gain enough bulk to provide useful forage. You may even decide that these 'fillers' provide more flowers for your bees than your chosen specimens and decide to prune and leave them.\n\nSow your own annuals, many of which can be purchased as seedlings quite inexpensively. If you would like to experiment with seeds start with sunflowers _(Helianthus annuus)_ or _Limnanthes douglasii,_ the poached egg plant. Both have larger seeds that are easy to handle and plant. The poached egg plant makes a beautiful spring annual border or potted display and is highly attractive to bees offering them both nectar and pollen.\n\nAvoid over hybridised plants (many will not produce viable seed) and limit your selection of annuals to the old-fashioned cottage garden plants such as alyssum, cosmos, single calendula, candytuft, nasturtiums, salvias and the native daisy _Brachyscombe iberidifolia._\n\nAnnual poppies are bee magnets with their copious amounts of pollen. Most varieties are easy to grow from seed but resent being moved so need to be sown where they are to flower. Although poppies readily self-seed it is a good idea to save a few seed pods either to share with other beekeepers or to cast around the garden the following spring to increase the numbers of flowers for your bees.\n\nTo be useful to your bees your plants must thrive and survive until the point of flowering. Even the most basic of composting efforts will improve the soil over time and improve the health and longevity of your chosen plants and as a result the nutrition of your bees.\n\nWhile your efforts may be aimed at providing additional nutrition for your own bees you will, by adding more floral content to your garden, add to the wider biodiversity of your local area and attract other beneficial insects and native pollinators to make homes in your garden. Australia has many native social and solitary bees that can be encouraged to take up residence along with your hives of European honey bees. As you learn more about the flowers that are attractive to your bees and whether they are good sources of pollen or nectar you may find your interest in flowering plants will grow and become an extension of your expertise as a beekeeper.\n\nHoney bee nutrition will become even more critical once the incursion of the _Varroa_ mite begins in Australia. The bee will have no initial resistance to the mite and will be affected by the many viruses it transmits. The hobby beekeeper will have an important role to play in both keeping and sustaining colonies of bees during the initial incursion. Providing additional forage from healthy plants during this period will benefit and support the bee, as it needs a wide variety of pollens to supply the complete source of the protein it needs and varied nectars to provide carbohydrates and energy.\n\nWhile a single property or a garden cannot provide all the nectar and pollen requirements of a colony it does add to the variety of plants and nutrition on offer and contributes to and increases the overall diversity of the available flora within a particular area for all visiting bees and foragers.\n\nOverleaf is a list of a range of plants that are bee-friendly and will bring bees into your garden.\n\n13.42: European honey bee and native Blue-banded bee foraging together.\n\n**BEE-ATTRACTING ANNUALS**\n\n---\n\nAlyssum _(Lobularia maritima)_ \u2014 a perennial that is usually grown as an annual\n\nCandytuft _(Iberis amara)_ \u2014 a perennial form (I. _sempervirens)_ is also available\n\nCosmos _(Cosmos bipinnatus)_\n\nForget Me Not _(Myositis alperstris)_\n\nLivingstone daisy _(Mesembryanthmum)_\n\nNasturtium _(Tropaeolum majus)_\n\nPoppies _(Papaver_ spp.) \u2014 adored by bees for their copious pollen\n\nPoached Egg Plant _(Limnanthes douglasii)_\n\nSunflower _(Helianthus annuus)_\n\nSwan River Daisy _(Brachyscombe iberidifolia)_\n\n**BEE-ATTRACTING PERENNIALS**\n\n---\n\nBacopa _(Sutera cordata)_\n\nBlue mist plant _(Caryopteris x Clandonensis)_\n\nCatmint ( _Nepeta_ spp.)\n\nClematis _(Clematis_ spp.) \u2014 both native and introduced Elephants Ears ( _Bergenia_ spp.)\n\nFan Flower _(Scaevola aemula)_\n\nHebe _(Hebe_ spp.)\n\nMichalmas daisy _(Aster x frikartii)_\n\nPentas _(Pentas lanceolata)_\n\nPenstemon ( _Penstemon_ spp.)\n\nPride of Maderia _(Echium candicans)_ \u2014 plant with care and remove dead flowerheads as this can be invasive Salvia ( _Salvia_ spp.) _\u2014_ annual and perennial types Scabiosa _(Scabiosa_ spp.) \u2014 especially S. _caucasica_ and S. _atropurpurea_ Sea Lavender ( _Limonium perezii_ )\n\nSedum ( _Sedum_ spp.)\n\nThrift _(Armeria_ spp.)\n\nValerian _(Centranthus ruber)_\n\nVerbascum _(Verbascumbombyciferum)_ \u2014 this is a bee magnet Wallflower ( _Erysimum_ cvrs)\n\nWinter rose ( _Helleborus_ spp.)\n\nYarrow ( _Achillea_ spp.)\n\n**BEE-FRIENDLY SHRUBS**\n\n---\n\nAbelia _(Abelia x grandiflora)_\n\nBanksia _(Banksia ericifolia),_ Heath Banksia, Banksia 'Giant Candles', Saw Banksia Buddleja _(Buddleja_ spp.) \u2014 _B. alternifolia, B. salviifolia, B. davidii_\n\nCalifornian Lilac _(Ceanothus_ spp.) \u2014 both tall and compact varieties Bottle Brush _(Callistemon_ spp.) \u2014 upright, weeping and dwarf varieties Escallonia ( _Escallonia_ spp.)\n\nGrevillea ( _Grevillea_ spp.) \u2014 various heights and forms including prostrate Hebe _(Hebe_ spp.) \u2014 miniature, small and tall varieties Hibiscus _(Hibiscus_ spp.)\n\nMexican Orange Blossom _(Choiysa ternata)_\n\nNative hibiscus _(Alyogyne huegelii)_\n\nNative Rosemary ( _Westringia_ spp.)\n\nNative Daisy Bushes _(Olearia_ spp.)\n\nRice Bush ( _Pimelea_ spp.)\n\nRock Rose _(Cistus_ spp.)\n\nRose ( _Rosa_ spp.) \u2014 with multiple single or loose-petalled blooms that open to show central cluster of stamens Sticky Hopbush _(Dodonaea viscosa)_\n\nThryptomene _(Thryptomene_ spp.)\n\n_Viburnum_ ( _Viburnum_ spp.)\n\nWax Flower ( _Philotheca myoporiodes_ )\n\n**FLOWERING TREES SUITABLE FOR BEES**\n\n---\n\nTrees should be chosen and planted with a view to their eventual size. Many eucalypt and fruiting trees are available grafted onto smaller or dwarf rootstock, making them a good choice in a large range of garden situations.\n\nBlueberry Ash _(Elaeocarpus reticulates)_\n\nJacaranda _(Jacaranda mimosifolia)_\n\nPin Cushion Hakea _(Hakea laurina)_\n\nSilky Oak _(Grevillea robusta)_\n\nSmall Leaved Lilly Pilly ( _Syzygium luehmannii_ )\n\nWater gum _(Tristaniopsis laurina)_\n14.\n\nNative bees\n\n14.1: Resin bee, Megachile mystacea, female.\n\n14.2: Female Leafcutter bee, Megachile pictiventris.\n\nThere are two general types of bee found in Australia: social bees and solitary bees. Social bees, like the European honey bee and the native social bees, form colonies consisting of thousands of individuals that work together to ensure the survival and wellbeing of the colony. Social native bees are stingless and are often referred to by the common name Stingless Bees. In this chapter, to ensure that there is no confusion between the European honey bee and native social (stingless) bees, both of which are 'social', I will refer to the native social bee using the name 'social bee'. Solitary bees, as their name suggests, lead solitary lives, independent of other bees. As solitary native bees have no large food stores in their nests to protect they are usually not aggressive but they are able to sting and should be treated with respect. While female solitary bees live in prepared nests, male solitary bees usually congregate or 'roost' at night in the open, clustered together.\n\nSOCIAL BEES\n\nIn Australia, there are 1600 described species of native bee, fifteen of which are social. These fifteen species can be classed into two genera (plural of genus): _Tetragonula_ (formerly _Trigona_ ) and _Austroplebeia_. Only four species have been 'domesticated' and kept in hives. These are:\n\n\u2022 _Tetragonula_\n\n\u2022 _Tetragonula carbonaria_\n\n\u2022 _Tetragonula hockingsi_\n\n\u2022 _Austroplebeia_\n\n\u2022 _Austroplebeia australis_\n\n\u2022 _Austroplebeia symei_\n\nThe natural range of Australia's social bees is mostly in the warmer northern regions of the country. There are, however, differences between the geographic regions inhabited by _Tetragonula_ and _Austroplebeia_. _Tetragonula_ prefer the wetter coastal regions while _Austroplebeia_ favour the drier inland regions. In order to survive in such different environments Australia's social bees have evolved a range of diverse behaviours.\n\nSocial bees have been an important part of indigenous culture for thousands of years with indigenous people using all of the nest contents, eating the honey, brood and pollen. In addition, the resin was used in crafts to waterproof baskets, and to attach tool and weapon heads to handles. The strong, tangy honey from social bees is called 'sugarbag' and commands a much higher price than honey made by the European honey bee. This difference in price is mainly due to the difficulty in collecting any marketable quantity of 'sugarbag' honey.\n\nAustralian social bees are small and black, measuring approximately 4 millimetres. _Austroplebeia_ species can be distinguished from _Tetragonula_ by their creamy coloured thoracic markings and nest architecture.\n\nAustralia's social bees have much smaller colony populations than honey bee colonies (10,000 bees vs. 50,000 bees), therefore they store proportionately smaller amounts of honey within their nests. Due to this fact, robbing a social bee nest may mean the death of the colony. This is particularly so in areas that are marginal for optimal foraging, such as the more southern regions of Australia. During a good nectar flow however, social bee colonies can store larger amounts of honey.\n\nFor _Tetragonula carbonaria_ to fly, the ambient temperature must be at least 18\u00b0 Celsius, while _Austroplebeia_ needs an ambient temperature of greater than 20\u00b0 Celsius.\n\nAs with the honey bee, there are three castes within a native social colony. These are a single queen, a large number of female workers and a small number of male drones. Unlike the multiple mating queens of the European honey bee, social bee queens are monandrous, mating with only a single drone.\n\n**_AUSTROPLEBEIA_**\n\nThere are nine species of _Austroplebeia_ listed in _The Zoological Catalogue of Australia_ and the two most commonly domesticated and studied are _Austroplebeia australis_ and _Austroplebeia symei._ Classification of the different _Austroplebeia_ species is based mainly on thoracic markings. _Austroplebeia_ bees are generally black with varying levels of cream or yellow markings on the rear edge of the thorax (scutellum), along the thoracic margins and on the face. _Austroplebeia_ are between 3.5 millimetres and 4.5 millimetres in length.\n\n14.3: Austroplebeia nest.\n\n14.4: Austroplebeia.\n\n14.5: Head of an Austroplebeia bee.\n\n14.6: Austroplebeia defending its nest against a much larger Apis mellifera.\n\n_Austroplebeia_ live in some of the most arid regions of Australia, where the climate extremes are harsh and food resources are often scarce. Species within the _Austroplebeia_ genus can be found throughout northern Australia, with _Austroplebeia symei_ having the widest distribution. _Austroplebeia symei_ can be found along the south-east coast of Queensland as well as in the northern parts of the Northern Territory. _Austroplebeia australis_ is found in Queensland, in New South Wales as far south as Kempsey and as far inland as Inverell.\n\n_Austroplebeia_ choose hollow trees to build nests. _Austroplebeia australis_ is typically found in hollows with a diameter of between 50 millimetres and 110 millimetres, and some nests have been recorded as being up to 6 metres in length. Brood populations may vary between 2000 and 13,000 and these variations may be due to cavity size, time of year and available floral resources.\n\nAll _Austroplebeia_ construct spherical brood cells and, with the single exception of _Austroplebeia cincta,_ make simple cell clusters. Open cells face outwards from the leading edge of the cluster. Clustered brood cells can be constructed to fit into the narrow, irregular cavities of the smaller trees or larger limbs favoured by _Austroplebeia_.\n\n**_TETRAGONULA_**\n\n_Tetragonula_ were reclassified from the genus _Trigona_ in 2012. Identification of species of the _Tetragonula_ genus is often difficult as some members may vary in size depending on their location. The largest, _Tetragonula hockingsi_ , measures approximately 4.5 millimetres in length, while the smallest, _Tetragonula clypearis,_ measures only 3.5 millimetres in length. As a result, the different nest architectures are an important aid in determining an identification of the species.\n\nThe distribution of _Tetragonula carbonaria_ is throughout northern and eastern Australia and extends from the Atherton Tablelands in Queensland to as far south as Bega in New South Wales, the most southerly geographic range of the social bees. _Tetragonula carbonaria_ chooses large cavities in which to build its nest. Apart from trees, _Tetragonula_ species also build nests in utility meter boxes, stone walls, beneath concrete paths and within door and wall cavities.\n\nAll of the _Tetragonula_ species in Australia build elongated, vertically oriented brood cells in regular, or nearly regular, structures. As previously mentioned there are some differences in the way that the species build their respective nests that can aid in their identification. A description of these differences is beyond the scope of this book and interested readers can consult the many excellent native bee resources listed in the bibliography.\n\nThe amount of honey that can be harvested from a _Tetragonula_ hive varies significantly. In Queensland and northern New South Wales, where bees can forage all year round, beekeepers are able to harvest 1 kilogram per hive per year. In the cooler regions further south, beekeepers should only harvest every two to three years, and it is recommended that hives kept in the Sydney basin or further south should not be harvested for honey at all.\n\n14.7: Tetragonula carbonaria nest.\n\n14.8: Tetragonula hockingsi nest.\n\n14.9\/14.10\/14.11: Tetragonula.\n\n**ALTERNATIVE POLLINATORS**\n\nPollination of crops using social bees is rare and commercial crops rely heavily on the European honey bee for pollination. Free pollination by feral honey bees is more widespread in Australia than in other regions of the world such as North America, Europe and New Zealand. Feral honey bees in these overseas regions have been almost wiped out by _Varroa_ infestations and the resulting spread of diseases by the mite. The reliability of feral honey bee pollination services in Australia is also expected to be drastically reduced if _Varroa_ is introduced into this country. One recent study counted 40 to 150 feral colonies per square kilometre of the European honey bee, _Apis mellifera,_ in the Wyperfeld National Park in north-east Victoria. As an example of how an introduced pest can affect feral colony numbers, between 2002 and 2006 it is estimated that more than 4500 colonies of European honey bees died out in Australia as a result of infestation with the Small Hive Beetle, _Aethina tumida._\n\nThe ongoing threat of disease to the European honey bee has caused many people to look for alternative pollinators including Australia's social bees. Interest in the use of social bees to pollinate macadamia trees increased when it was observed that the yield of macadamia nuts increased when they were planted next to uncleared native vegetation where there were significant numbers of native social bees. This is perhaps not surprising as both these native species of plant and insect would have evolved together over many millennia.\n\nIt has been estimated that Australian social bees have a flight range of 500 metres, although many beekeepers believe the range is more likely to be a kilometre. This is advantageous to the crop grower since any social bees brought in to pollinate crops are unlikely to forage further away to find other floral resources. Along with actual hive placement, the number of hives is important as fifteen to twenty native social bee hives per hectare are required in comparison with seven hives per hectare for the European honey bee. In addition to macadamia, social bees are also effective pollinators of lychee, avocados and watermelons.\n\nNative social bees seem to be relatively free of disease and at the point of writing there are no reports of any brood diseases having been seen. They do, however, suffer from predation, parasitism, and colony infestation. They face a similar range of threats from pests as do the European honey bee. As an example, the Small Hive Beetle, _Aethina tumida_ , can devastate both a weakened social bee colony as well as a European honey bee colony. Both _Tetragonula_ and _Austroplebeia,_ however, are more effective than the European honey bee at defending against Small Hive Beetle but can succumb if the colony is weak. Cane toads, _Rhinella marina_ (formerly _Bufo marinus),_ are also a threat to both European and social bee colonies.\n\nNest defence by native social bees is reported widely across the world. A variety of methods are used by native social bees to defend colonies including biting, resin daubing, chemical repellants, and locking onto the wings or body of intruders with their mandibles.\n\n_Tetragonula carbonaria_ exhibit a collective mechanism known as 'fighting swarms' during which bees from one hive attack or try to enter another hive. During a fighting swarm hundreds to thousands of these native bees are locked in aerial combat and fight to the death. These battles can last less than a day or extend over several days, restarting again each morning, and may result in thousands of dead and dying bees lying on the ground.\n\nMany people in northerly regions of Australia keep social bees in hives, usually as an aid to the bee's conservation, as the amount of honey that can be harvested is often small. In the southern parts of Australia where social bees are not naturally found, such as Victoria, Tasmania or South Australia, keeping _Tetragonula_ or _Austroplebeia_ bees in hives is not recommended due to the colder climatic conditions. Solitary bees, however, are found across Australia and there is much that an interested person can do to attract and encourage solitary bees to nest in their garden. Details on attracting and keeping native bees can be found in the bibliography.\n\nSOLITARY BEES\n\n_This section has been written by Erica Siegel, wildlife photographer and native fauna enthusiast._\n\nThere is a very large number of solitary bee species, ranging from minute bees, less than 3 millimetres long, to giants as long as 28 millimetres. Some of the more commonly seen and recognisable groups are described below.\n\n**BLUE-BANDED BEE \u2014 _AMEGILLA_**\n\nThere are about eleven species of Blue-banded bees in Australia. They are found in every state including Tasmania and are one of Australia's long-tongued bees within the _Apidae_ family and are known as buzz pollinators. These blue-banded bees range from 8 to 14 millimetres long; they have a sting but are not aggressive. They have thick red-brown hair on the thorax and glittering bands of iridescent, metallic blue or whitish hair across their black abdomens. Some, however, have green or orange bands. Males have five bands and females four. The colours are caused by microscopic tubes in each hair with the reflecting light causing the glittering colours.\n\nThey have yellow, cream or white markings on their faces and are often seen darting around the flowers of lavenders, and abelias and many other exotic and native flowers. The pollen is carried on their hind legs in sets of specialised hairs (termed scopae). The females build nests in shallow burrows in the ground, favouring river and dam banks, but they may also nest in mud bricks, soft sandstone banks or in soft mortar. Each female builds her own nest burrow but many females often nest together in the one place. The female Blue-banded bee constructs oval-shaped cells in her burrow, lining them with waterproof secretions. Before laying an egg in the cell she stocks it with a paste of pollen and nectar. After the egg is deposited she seals the cell with an earthen cap and when all the cells are filled and capped, she covers the nest with layers of soil to close it.\n\n14.12: Female Blue-banded bee, Amegilla cingulata.\n\n14.13: Blue-banded bee feeding on basil.\n\n14.14: Female Blue-banded bee, Amegilla cingulata.\n\n14.15: Male Blue-banded bees roosting, Amegilla cingulata.\n\nAccording to J.C. Cardale (Australian National Insect Collection, Canberra 1968), Blue-banded bees live around 40 days and about three generations of bees hatch during summer, depending on the location. Baby bees take about seven weeks to hatch. Those that do not hatch overwinter in their sealed cells as prepupae, emerging in the next spring.\n\nThe nests of female Blue-banded bees are stalked by the Neon Cuckoo bee _(Thyreus nitidulus)._\n\nMale Blue-banded bees roost in small groups out in the open at night, hanging on to twigs or stems with their mandibles and tucking their legs under their bodies. Recent research has shown that Blue-banded bees could be valuable pollinators of greenhouse tomatoes.\n\n14.16: Teddy Bear bee, Amegilla (Asaropoda) bombiformis.\n\n14.17: Teddy Bear bee, female.\n\n14.18: Male Teddy Bear bees roosting.\n\n**TEDDY BEAR BEE \u2014 _AMEGILLA (ASAROPODA)_**\n\nThere are about 25 species of _Amegilla (Asaropoda)_ recorded in Australia. Teddy Bear bees occur in all states of mainland Australia.\n\nMost species of these rotund, furry brown bees are 7 to 20 millimetres long and are covered in dense red-brown hair but some are a lighter buff colour. They are one of Australia's long-tongued bee groups and the females carry the pollen on special thick hair on their hind legs. The largest, the Dawson's Burrowing bee, _Amegilla (Asaropoda) dawsoni,_ is nearly 20 millimetres long and lives in Western Australia. It nests in groups of up to 10,000 in arid clay pans and mud flats. The males fight to their death to mate with virgin females when they emerge from their burrows as documented in the BBC TV series _Life._\n\nOther Teddy Bear bees build shallow 10-centimetre nest burrows in soft soil, creek banks, in weep holes of retaining walls, in gaps in sleepers surrounding garden beds and sometimes underneath houses. Many females may nest together in the one location but each female builds her own nest containing a cluster of 2-centimetre urn-shaped cells made of mud and lined with waterproof secretions. The cells all face upwards and the female stocks each cell with a paste of nectar and pollen on which she lays her egg. The cell is then sealed with a cap. Fully grown larvae may lie dormant until spring if winter is approaching.\n\nFemale Teddy Bear bees are stalked by the Domino Cuckoo bee, _Thyreus lugubris,_ which will lay her egg in unattended nests. The Cuckoo bee larvae will consume the food meant for the larvae of the Teddy Bear bee.\n\nTeddy Bear bees are known as buzz pollinators allowing them to pollinate flowers with enclosed pollen like _Solanum_ sp _._ , such as tomato, eggplant, chilli. They grasp the flowers with their front legs and use their flight muscles to vigorously vibrate the flower, dislodging the pollen.\n\nThey are also valuable pollinators of native wildflowers and forage on native flowers like _Senna clavigera, Hibbertia scandens, Hibbertia dentata, Dianella_ sp., _Melastoma affine,_ and exotic _Buddleja davidii, Cuphea_ sp.\n\nThe males roost together at night hanging onto stems or twigs with their mandibles. They sometimes roost near the roost of Blue-banded bees.\n\n**GREAT CARPENTER BEE, YELLOW AND BLACK CARPENTER BEE \u2014 _XYLOCOPA (KOPTORTOSOMA)_**\n\nGreat Carpenter bees are the largest native bees in Australia, ranging from 15 to 26 millimetres. According to measurements in Remko Ley's revision of the Carpenter bee genus, the males are either the same size or slightly bigger than the females.\n\nThere are six species of Great Carpenter bees in Australia and they are found in the warmer climate of northern Western Australia, northern New South Wales, Queensland and the Northern Territory. The females have a glossy black abdomen and bright yellow hair on the thorax. Males are covered uniformly with yellow, brown or olive green hair. Both males and females have black wings.\n\nThey are called Carpenter bees because they cut nest burrows with their jaws in soft timber such as the dead limbs of mango, frangipani and jacaranda trees. Other known nesting trees include _Ficus_ species, _Casuarina, Banksia, Lophostomon grandiflorus, Leptospermum_ species, dead flower stalks of Grass trees _(Xanthorrhorea_ sp.), Soursop _(Annona muricata),_ dead saplings of _Eucalyptus robusta,_ even old timbers in the backyard.\n\nThe nesting burrows may contain several tunnels and are partitioned into brood cells. The female lays her egg on a mixture of pollen and nectar then seals the cell with a disc of chewed wood particles before placing another egg on a paste of nectar and pollen, sealing it off till the tunnel is filled.\n\n14.19\/14.20: Great Carpenter bee, Xylocopa (Koptortosoma).\n\n14.21: Great Carpenter bee, female.\n\nNew generations of females often use old nesting burrows. Female Great Carpenter bees are known to share the burrow with their daughters creating a small community, even sometimes feeding their adult daughters, and are therefore referred to as 'para-social'.\n\nThe Great Carpenter bees are attracted to native _Melastome affine, Senna,_ foam bark ( _Jagera pseudorhus), Canavalia rosae_ and according to Dr Remko Lejis (Flinders University, SA) _Crotalaria, Wistaria (Callerya megasperma)_ and exotic _Cassia fistula._ They also visit other exotic plants like _Grewia occidentalis,_ Pigeon Pea _(Cajanus cajan),_ Cashew tree ( _Anacardium occidentale_ ), Leopard tree ( _Caesalpinia ferrea_ ), _Solanum_ sp., _Laburnums_ , _Jasmines_ , _Albizia_ sp., _Tipuana_ sp. (which is a declared weed).\n\nThey are able to buzz pollinate by grasping the flower with their legs and vigorously vibrating their thoracic muscles, thus agitating the flower via sound (sonication). Buzz pollination is important for wildflowers like _Melastoma affine_ or flowers where the pollen is trapped in narrow tubes.\n\nTo attract females, the males emit a pheromone smelling of flowers and pollen. The males are seldom seen as they establish their territories high up in the crown of trees. They detect intruding males by the pheromones they emit and defend their territories.\n\nSimilar species of Carpenter bees in other countries are good pollinators of passion fruit.\n\n14.22: Peacock Carpenter bee, Xylocopa (Lestis) bombylans.\n\n14.23: Peacock Carpenter bee, female.\n\n14.24: Peacock Carpenter bee, male in nest.\n\nPEACOCK CARPENTER BEE OR GREEN CARPENTER BEE \u2014 XYLOCOPA (LESTIS)\n\nXylocopa bombylans is found along the coast from Sydney to Cape York and _Xylocopa aeratus_ is found around Sydney, on the Dividing Range up to approximately the Stanthorpe area.\n\nThe female of these spectacular bees is a glossy metallic blue with purple tints and the male a metallic green with yellow tints. Peacock Carpenter bees are 17 to 20 millimetres long.\n\nThey cut 7 to 10-millimetre-wide nest burrows in the flower stalks of Grass trees _(Xanthorrhoea)_ or in the soft pithy dead timber of _Banksia, Tristania, Acacia, Leptosperum_ and _Casuarina_ , where they make rounded cells for their eggs. The cells are stocked with pollen and nectar formed into bee bread before the female lays a large egg on top. She then closes the cell with a plug of chewed wood particles. The nest burrow may be shared by males and females. The emerging young bees are fed in the nest for some time before they venture outside.\n\nPeacock Carpenter bees are able to buzz pollinate and are attracted to flowers needing buzz pollination such as _Hibbertia scandens_ and _Solanum jasminoides._ Land clearing has caused the loss of these stunning bees from Victoria and mainland areas of South Australia although they can still be found on Kangaroo Island in South Australia.\n\n14.25\/14.26\/14.27: Neon Cuckoo bee, Thyreus nitidulus.\n\n**CUCKOO BEE \u2014 _THYREUS_**\n\nThe Cuckoo bees are stunningly beautiful but these bees lead a life of deceit! They behave like cuckoo birds, laying their eggs in the nests of other bees. The Neon Cuckoo bee, _Thyreus nitidulus_ , with glittering metallic blue markings on a black body, lays its eggs in the nests of Blue-banded bees.\n\nThe Domino Cuckoo bee _(Thyreus lugubris)_ with white patches on a black body, lays its eggs in the nests of Teddy Bear bees.\n\nThere is also the Chequered Cuckoo bee _(Thyreus caeruleopunctatus)_ which has blue spots on a black body.\n\nAccording to overseas studies the cuticle of Cuckoo bees is thicker than normal for defence when fighting host bees. Their larvae have huge, sharp mandibles to kill the egg\/larvae of the host bee but sometimes they simply eat the provisions, resulting in a very stunted host bee emerging from the cell.\n\n**REED BEE \u2014 _EXONEURA_**\n\nThere are 69 species of Reed bee recorded in the Australian Faunal Directory. Reed bees can be found in all states of Australia with most found in coastal areas of Victoria, New South Wales and Queensland.\n\n14.28\/14.29: Reed bee, Exoneura.\n\nReed bees are slender bees measuring between 4 and 8 millimetres long. Most species have a red abdomen but some have a black abdomen. Many have white markings on their faces. Females have a sting but are not aggressive.\n\nThey nest inside dry, pithy twigs in plants such as raspberries and blackberries or in the dead fronds of tree ferns. Many nests can also be found in dead canes of the weed Lantana. They will also nest in small drilled holes in hardwood. They block the nest with their behind to stop ants or other predators from entering.\n\nThe females do not make cells or store pollen or honey but feed the larvae individually as they lie in the nesting cavity.\n\nReed bees feed on both native and exotic flowers including _Allysum, Acacia, Banksia, Eucalyptus, Grevillea,_ _Leptospermum_ and _Melaleuca._\n\n**LEAFCUTTER BEE \u2014 _MEGACHILE (EUTRICHARAEA)_**\n\nThere are believed to be about 27 species of Leafcutter bees from the genus _Megachile._ They are found in all states of Australia and can be found in both coastal and drier inland areas of Australia. The Leafcutter bees range in size from 6 to 15 millimetres. Most Leafcutter bees are black and have white or orange-gold stripes of hair on the abdomen. They carry pollen on special bristles underneath the abdomen. They can usually be distinguished from Resin bees by their relatively wide abdomen which tapers into a point, while resin bees have a more narrow and cylindrical abdomen. Leafcutters also alight on flowers with their wings spread, while Resin bees fold their wings.\n\nBee watchers often first discover these amazing bees when they notice rows of neat circular cuts on the edges of some leaves in their garden. Leafcutters use the discs of leaf as nest building material. They particularly like the soft leaves of roses, _Bauhinia, Desmodium_ and _Buddleja._\n\nThey use the cut leaves to make a tube as a nest for their eggs or line a hole in timber with it. Some species of Leafcutter bees nest in existing holes or crevices in timber or masonry. One species has been observed at the Australian Native Bee Research Centre (ANBRC) digging a shallow nest burrow in the soil of a pot plant.\n\n14.30: Leafcutter bee, female, Megachile.\n\n14.31: Leafcutter bee, female.\n\n14.32: Leafcutter bee, female, Megachile serricauda.\n\nEach nest is built by just one female. The leaves are cut in various shapes, round and elongated, to suit the construction of the cell for the egg. The cell is then stocked with a mixture of nectar and pollen in which the Leafcutter bee lays her egg. She cuts more circular leaves to close off the cell before constructing the next one until the hole or tube is filled. The hole is then plugged with rough leaf cuttings.\n\n14.33: Leafcutter bee leaving nest after it had fallen from its location.\n\nWhen the eggs hatch the tiny larva eats the provisions and when fully grown spins a silky cocoon and develops into a pupa then emerges as an adult bee. Immature bees may hibernate through winter and complete development into an adult the following spring.\n\nA minute wasp from the genus _Melittobia_ parasitises the Leafcutter bees and hundreds may be found in one cocoon eating the larva.\n\n**RESIN BEE \u2014 _MEGACHILE_**\n\nResin bees are found in all states of Australia. They carry pollen on scopal hairs underneath the abdomen.\n\n14.34: Resin bee, Megachile punctata.\n\nResin bees come in many colours and sizes. For example there are large black 14-millimetre bees with white tufts of hair, and small 8-millimetre black bees with bright orange abdomens. Resin bees nest in pre-existing holes made by wood-boring beetles or moths, in gaps in timber or stonework and will use artificial nests.\n\nThey are called Resin bees because they collect resins and gums to construct their nest and build partitions between their brood cells and to seal their nest holes. They roll the resin with their front legs into a ball before taking it to the nest. Plant material mixed with resin may also be used to close off the nest.\n\nThe female Resin bee mixes pollen and honey into a paste called bee bread on which she lays her egg as food for the developing larva. The eggs hatch into small white larvae, eat the food provided and then spin silken cocoons and develop into pupae before emerging as adult bees.\n\n14.35: Resin bee, _Megachile punctata._\n\n14.36: Resin bee, _Megachile mystacea,_ male.\n\nResin bees hibernate during winter and emerge in spring. The nests can be parasitised by wasps.\n\nBeekeepers sometimes notice Resin bees hanging around hives, trying to 'borrow' a little resin for their nests.\n\n**HOMALICTUS BEE \u2014 _HOMALICTUS_**\n\nThere are about 40 species of _Homalictus_ bees and they are found in all states of Australia, both along the coast and inland.\n\nAlthough very small, ranging from 5 millimetres to 8 millimetres long, the glittering _Homalictus_ bees come in a dazzling array of colours; 'golden blue', 'coppery red' and 'green tinged with purple, red or gold' are just a few of the colours listed by scientists. They carry pollen on unusually long, branched hairs under the abdomen.\n\n_Homalictus_ bees dig complex branching nests in the ground. Tiny oval brood cells are constructed in the nest shafts off the main shaft. Many females may live together in each nest, taking turns to guard the narrow nest entrance. One nest was found to be occupied by over 160 females!\n\n_Homalictus_ bees forage on a wide variety of flowers from many different genera such as mistletoe, _Angophera, Eucalyptus, Melaleuca, Tristania._\n\nThe female lays a single egg onto a ball of prepared food, then seals the brood cell up and starts constructing the next cell.\n\nMale _Homalictus_ bees roost at night in the open, clustered together like other solitary native male bees.\n\n14.37\/14.38: Homalictus, _Homalictus urbanus._\n\n14.39: Halictid bee, _Halictidae._\n\n14.40: _Homalictus urbanus._\n\n**LIPOTRICHES \u2014 _HALICTIDAE_**\n\nVarious species of _Lipotriches_ bees occur in all states of Australia according to records on the Atlas of Living Australia website. They live in woodlands, forests, deserts and urban areas.\n\n14.41: Lipotriches.\n\n14.42: Lipotriches sp.\n\n14.43: Lipotriches (Austronomia) phanerura.\n\n14.44: Female Lipotriches nest in burrows in the ground. At night, large numbers of males roost together in the open, clinging to grass or dry vegetation.\n\n14.45: Lipotriches bee.\n\n_Lipotriches_ bees measure about 10 millimetres and carry pollen on scopal hairs on the hind legs. Some species of male _Lipotriches_ have enlarged hind legs that may be used during courtship, according to overseas studies.\n\nMost _Lipotriches_ nest in burrows in the ground with each tunnel about 8 millimetres in width and widening a few centimetres below the surface to allow the guarding bee to retreat, allowing access to the entering bee. The main entrance is used by up to three females but each constructs her own side tunnel. The end of the tunnel contains a room where vertical brood cells are constructed. Pollen and nectar are deposited in each cell before an egg is laid. They guard the nest entrance, blocking it with their faces during the day and the tail at night. Some newly mated females spend the winter in their nest in semi hibernation.\n\nAt night large numbers of males cluster together on bare branches of trees, grass stems or other dry vegetation, the first arrivals attaching themselves by their mandibles and later arrivals clinging to the bodies of the first ones. Tarlton Rayment (A _Cluster of_\n\n_Bees,_ 1935) wrote that some clusters may contain thousands of male bees.\n\n**MASKED BEE \u2014 _AMPHYLAEUS, HYLAEUS_ AND _MEROGLOSSA_**\n\nMasked bees can be found throughout the world but the biggest variety is in Australia, with approximately 200 species making their home here. They can be found in every state and territory, from rainforest to deserts.\n\nThese slender black bees (most less than 10 millimetres long) are called 'masked bees' because they have pale markings on their faces. Many species also have a distinctive yellow spot on the thorax; some have a yellow collar.\n\nMasked bees have very little hair and carry pollen to their nests by swallowing it. The nests are usually in deserted mud wasp nests, pre-existing holes in wood and pithy stems of dead tree fern fronds. The females excavate 4 to 9-millimetre-wide burrows in the stems, then line them with waterproof cellophane-like secretions, according to scientists Dr Katja Hogendoorn and Dr Allan Spessa. Brood cells are then constructed and provisioned with pollen and nectar before the egg is laid. The cell is then enclosed with further waterproof cellophane-like material before being sealed.\n\nAdult Masked bees spend winter in their nest. In spring some bees will set up new nests and others will re-use the old nest. New nests are built by single bees while reused nests are sometimes used by a pair.\n\n14.46: Lipotriches.\n\n14.47\/14.48\/14.49: Masked bee.\n15. Diseases of the honey bee\n\nDISEASES OF BROOD\n\nThe key to the successful management of the health of a hive is for the beekeeper to inspect the colony and supporting frames closely during each hive opening and to look for the first signs of disease. Identified and treated early, it is often possible to minimise the effect of the disease or to eliminate it completely.\n\nMinimising and treating bee disease within a colony needs to be the primary aim of every beekeeper, both amateur and professional. Without a healthy colony honey production will not be at its maximum, wasting both time and money for the beekeeper and, more importantly, the prospect of disease being passed on to other healthy colonies nearby is very real.\n\nMany diseases have very subtle signs and new beekeepers in particular often have trouble recognising signs of a diseased hive. It is vital that the beginning beekeeper develop the habit of inspecting hives regularly so that experience is gained in looking at a healthy viable hive. Later on, if something changes within the hive and disease is suspected, it will then be much easier to recognising and identify any differences or changes, and seek advice from other beekeepers about the possible causes.\n\nAn important aspect of minimising the transmission of disease by beekeepers is to keep tools and equipment clean. If you suspect a hive that you have inspected is infected with American Foul Brood (AFB) or European Foul Brood (EFB), once a hive tool has been used inside the hive it should be sterilised before use on another hive. Sterilisation can be performed quickly by placing the exposed end of the hive tool inside the hot smoker when it is being carried or, alternatively, using a strong disinfectant such as household bleach in a container in which the hive tool or multiple tools can be kept between hive inspections.\n\n15.1: Diseased drone cells.\n\n15.2: Damage 15.2 caused by Greater Wax Moth.\n\nGOOD MANAGEMENT PRACTICES TO MINIMISE DISEASE IN THE COLONY\n\nAnother aspect of keeping colonies free of both brood and adult disease is good management practices by the beekeeper. These include:\n\n\u2022 ensuring appropriate nutrition\n\n\u2022 making sure that queen bees are young\n\n\u2022 keeping strong hives, possibly by merging with another clean colony if required\n\n\u2022 minimising dampness and condensation in the hive, particularly in colder areas during winter\n\n\u2022 practising comb rotation by replacing two or more old brood frames every year with new frames and foundation\n\n\u2022 trying not to move frames of honey or brood between hives since this may aid in the spread of disease\n\n\u2022 minimising stress on the colony by not unduly moving the hive or opening the hive too often.\n\nAMERICAN FOUL BROOD (AFB)\n\nAmerican Foul Brood, or AFB, caused by the bacterium _Paenibacillus larvae,_ has become one of the most serious bee diseases in Australia. AFB can cause extensive losses in both amateur and commercial apiaries although severe outbreaks can be minimised by good management practices, regular checks for the disease, and destroying or irradiating hives when the disease is confirmed.\n\nAmerican Foul Brood spores can remain dormant for over 50 years in a vegetative state on old equipment that was once used with infected colonies. Once larvae are more than three days old they are far less susceptible to infection by AFB. American Foul Brood is a notifiable disease and your state or territory's apiary inspectors need to be informed if you suspect an outbreak in any of your colonies.\n\n**SIGNS OF INFECTION**\n\nDuring the spring and autumn a thorough examination of brood frames for signs of the disease will need to be conducted. It should be noted that AFB can occur in hives at any time of the year. A close inspection of the frames may detect early infections, which often appear in only one or two larvae or pupae.\n\nTo the beginner beekeeper, American Foul Brood (AFB) and European Foul Brood (EFB) may appear similar and are often confused with each other. The following points may help to differentiate between them.\n\n\u2022 In heavily infected colonies the brood has a scattered, uneven pattern due to the intermingling of healthy cells with diseased cells.\n\n\u2022 Brood infected with AFB generally die after the cell has been capped. This is different to EFB, where brood usually die before the cell is capped.\n\n\u2022 AFB brood frames may show capped cells of dead brood with punctured and sunken cappings.\n\n\u2022 This peppered appearance of the brood usually allows AFB to be distinguished from EFB. In AFB the cappings are discoloured, while in EFB the cappings are not normally discoloured to any great extent.\n\n\u2022 Both AFB and EFB can result in brood under sealed cells exhibiting a sunken dark appearance with perforated cappings.\n\n\u2022 Brood affected with AFB will change colour from a healthy pearly white to a darker brown as the disease progresses.\n\n\u2022 Brown dead brood probed with a matchstick may show signs of a ropy consistency in infections of both AFB and EFB. If a cell infected with AFB is tested with the matchstick test the resulting ropiness may be 3 to 5 centimetres long, while the ropiness test on EFB affected brood may result in little or no stringing out of the cell contents.\n\n\u2022 Colonies infected with AFB may give off a putrid fish-like odour while colonies infected with EFB often give off a slightly sour or sometimes rotten faecal odour.\n\n\u2022 After about a month brood that have died from AFB dry out to a dark scale which adheres to the wall of the cell. Brood that have been killed by AFB lie on the bottom of the cell whereas those that have been killed by EFB may lie in a more unnatural sideways position as if they had been wriggling out of their normal position at the time of death.\n\n\u2022 The tongues of brood killed by AFB are often sticking out and sometimes attached to the roof of the cell. This is not the case with brood that have been killed by EFB.\n\n\u2022 The only accurate diagnosis is by laboratory analysis. There are, however, easy-to-use field diagnostic kits made by VITA available at a small price from beekeeping suppliers that provide reasonably accurate testing for AFB.\n\n15.3: The cappings on cells containing American Foul Brood (AFB) infected pupae are typically sunken and discoloured. Larvae infected with AFB usually die after the cell is capped although some strains are now appearing where the brood dies before the cell is capped.\n\n15.4: The remains of pupae killed by American Foul Brood (AFB) are ropey and can be pulled out using a matchstick, the aptly named matchstick test for AFB.\n\n15.5: Pupae that have recently died of AFB have a characteristic false tongue that attaches itself to the roof of the cell.\n\n15.6: Inexpensive kits can now be purchased to test for both American Foul Brood (AFB)and European Foul Brood (EFB).\n\n**SPREAD**\n\nThe control of AFB is very much in the hands of the beekeeper as the most common method of spread is via the use of contaminated equipment in healthy colonies. The use of infected hive tools, gloves or the interchange of hive parts within the apiary should be done with the greatest of care to avoid spreading any brood disease, not just AFB. Colonies may also become infected if bees are fed contaminated honey, rob honey from contaminated hives or feral colonies, or if infected comb is left on the ground within the apiary for other bees to feed off. Young larvae less than 53 hours after egg hatch are the most susceptible to AFB. Careful inspection and good hive and apiary management are the keys to preventing serious outbreaks of this disease. Transmission of AFB between feral colonies is rare and the disease mainly affects colonies belonging to professional and hobby beekeepers.\n\n**TREATMENT**\n\nFirst, the colony of bees infected with AFB needs to be killed with a strong insecticide to stop infected bees escaping or the infected comb being robbed. Contaminated materials such as the hive and frames are either burnt or sterilised using gamma irradiation. Both of these methods of sterilising AFB infected hives are very successful in minimising the incidence of the disease and are the only techniques approved by government apiary inspectors in each of the states. Fortunately, the incidence of AFB is much lower than other frequently occurring brood diseases such as European Foul Brood, Sacbrood or Chalkbrood disease.\n\nIf an infected hive is to be burnt this will need to be carried out in a pit in order to contain any wax and honey that is left over. The remaining ashes must be covered with 30 centimetres of soil to stop any residual bacteria from escaping.\n\nSterilisation of contaminated hives using gamma radiation is also frequently used to kill the bacteria although this option is only available in some of the larger capital cities in Australia. Before sending the infected hives for sterilisation any bees and honey need to be removed and destroyed using the techniques explained above. The extracted frames, boxes, hive covers, bottom boards and queen excluders are prepared for irradiation by wrapping them in a double layer of thick plastic garbage bags. After sterilisation hives are restocked with disease-free bees. For further details on the preparation of equipment for irradiation refer to: www.steritech.com.au\/content\/agriculture.\n\nIn Tasmania it is legal to treat mildly infected colonies with the antibiotic oxytetracycline hydrochloride (OTC) to control AFB. Severely infected colonies and hives must be destroyed as in all other states and territories of Australia.\n\nEUROPEAN FOUL BROOD (EFB)\n\nEuropean Foul Brood, or EFB, is caused by the bacterium _Melissococcus pluton_ and has become a serious bee disease in Eastern Australia. European Foul Brood will often cause extensive losses in both amateur and commercial apiaries and again outbreaks can be minimised by regular disease checks, early detection, and good management practices by the beekeeper.\n\n15.7: Larva infected with European Foul Brood (EFB) usually die before the cell is capped.\n\n15.8: Larvae infected with European Foul Brood (EFB).\n\n15.9: European Foul Brood scale with the characteristic gondola shape.\n\nEFB can remain dormant for over three years in a vegetative state on old equipment that once was used with infected colonies; it is highly contagious with all stages of larval development susceptible to infection. The disease is found throughout Australia.\n\nEFB is a notifiable disease and your state or territory's apiary inspectors need to be informed if you suspect an outbreak in any of your colonies.\n\n**SIGNS OF INFECTION**\n\nFortunately, EFB diseased colonies are usually easily recognised as being diseased although identification of the cause of the disease may be more difficult for the amateur due to its similarity to American Foul Brood. The signs associated with infection by EFB are very similar to those associated with AFB. As discussed above, the two main differences are:\n\n\u2022 Brood infected with AFB generally die after the cell has been capped. This is different to EFB where brood generally die before the cell is capped.\n\n\u2022 Brown dead brood probed with a matchstick may show signs of a ropy consistency in infections of both AFB and EFB. If a cell infected with AFB is tested with the matchstick test the resulting ropiness may be 3 to 5 centimetres long, while the ropiness test on EFB may only result in little or no stringing out of the cell contents.\n\nKeep in mind that diagnosis of EFB based solely on the signs described above is not always reliable. The only accurate diagnosis for EFB is by laboratory analysis, which beekeepers can access by submitting a comb sample or a smear of brood cell contents on a glass slide to a diagnostic laboratory. An EFB self-diagnostic kit is sold by VITA, and there is a similar kit for AFB diagnosis.\n\n**SPREAD**\n\nEuropean Foul Brood is highly contagious and the causes of spread are the same as for AFB. Infected colonies can survive for long periods with low spore counts of EFB without showing any serious signs of the disease. Sudden outbreaks of the disease can then occur due to a build-up of spore numbers; usually these outbreaks are caused by colony stress resulting from changed seasonal conditions, poor nutrition and moving bees. Young larvae less than 48 hours old are the most susceptible to EFB.\n\nEuropean Foul Brood enters the gut of the larvae and essentially competes with it, consuming food that is needed by the young brood. As a result, developing brood are left with insufficient food on which to survive and literally starve to death. Some experienced professional beekeepers say that EFB is likely to show its presence in a hive when there is a shortage of pollen coming in. If there is an abundance of pollen and large quantities of royal jelly are being fed to the larvae, there may be sufficient food for both the EFB and the larvae so that the disease will not show itself by harming the brood.\n\n**TREATMENT**\n\nThe only antibiotic recommended for the treatment of EFB is oxytetracycline hydrochloride (OTC), sometimes called Terramycin. Oxytetracycline hydrochloride is available only on prescription from a veterinarian or with an Order to Supply from an apiary officer of a state or territory government. Beekeepers have been known to treat infected colonies with too much OTC resulting in honey containing high residues of the antibiotic.\n\nWhile there may be times when antibiotic treatment would appear to be the only answer, its use is becoming increasingly less attractive due both to concerns that the honey may become contaminated and to the possibility that overuse will result in the development of antibiotic resistant strains of EFB bacteria. For the hobby beekeeper, the use of antibiotics has never really been a practical solution mainly due to the difficulty of obtaining the appropriate prescription. Alternative methods of treatment available to the hobbyist are described on p. 249.\n\nIf ample honey is stored in the hive a shortage of nectar should not be a problem, but good-quality pollen is another matter. Pollen is available either when it has been stored by the bees or when it is available from flowering plants. A good supply of pollen is essential in order to provide adequate protein levels that result in a well-balanced group of amino acids, important to reduce any nutritional imbalance that will stress the bees. A temporary lack of pollen can be overcome by feeding the bees either previously collected pollen or a pollen substitute.\n\nSACBROOD VIRUS (SBV)\n\nSacbrood, or SBV, is not common in Australia. The infection is caused by a virus and infected larvae typically die shortly after capping but before the change to the pupal stage of their development in the capped cell.\n\nSacbrood infected larvae form a plastic-like outer layer or skin that contains a watery fluid, giving the infected larvae a characteristic watery bloated look. Sacbrood infections are most common during the early spring. The colony at this time is very susceptible to both chill and nutritional stress, either of which can lead to further progression of the Sacbrood disease.\n\n15.10: Sacbrood virus.\n\n15.11: Although this larva looks as if it is infected with Sacbrood virus, the colony only tested positive for infection with European Foul Brood (EFB).\n\n**SIGNS OF INFECTION**\n\nSacbrood mainly affects female worker larvae although it may sometimes occur in male drones. The disease affects uncapped pupae as well as sealed brood that are seven to ten days old. Brood comb contaminated with Sacbrood show some degree of irregularity of brood pattern similar to EFB or AFB. Dead brood will be found scattered among healthy brood in the comb and their cappings may be discoloured, sunken, perforated or removed by the bees. Brood almost always die after the cell has been capped and generally not during the uncapped larval stage.\n\nDried brood that have died of SBV are located on the bottom of the cell, become brittle, and retain the upturned 'gondola' shape of the body. The dried scale can easily be removed without damaging the cell wall. Infected brood are typically odourless, though in advanced stages an occasional putrid smell may be detected.\n\n**SPREAD**\n\nIt is thought that nursery bees probably become infected with Sacbrood when cleaning out the cells of infected pupae and then pass the virus on to non-infected larvae when they feed larvae with brood food such as royal jelly from their hypopharyngeal glands. The virus breeds rapidly, leading to death shortly after capping. The virus may remain active for up to four weeks in the larval remains or in honey or pollen.\n\n**TREATMENT**\n\nThe incidence of SBV in most colonies is low because adult bees normally detect and quickly remove infected larvae. Good management practices by the beekeeper as detailed on p. 249 can help to alleviate Sacbrood.\n\nCHALKBROOD\n\nAlthough commonly found in hives, Chalkbrood is not usually a serious disease of honey bees and only very rarely kills an entire colony. If, however, the bees have a predisposition to the disease it can cause a gradual deterioration of the colony by killing a larger number of brood, thereby affecting adult bee numbers and resulting in a loss of honey production. Chalkbrood is caused by the fungus _Ascosphaera apis_ and there are signs that its incidence is increasing around the world, probably due in part to migratory beekeeping and to poor hygiene in the practices of some beekeepers. There has also been the suggestion that the importation of infected pollen and foundation from abroad may be an additional cause of the increase in incidence of the disease. The fungus can remain active on hives and other surfaces for up to fifteen years.\n\n**SIGNS OF INFECTION**\n\nChalkbrood can infect both unsealed larvae as well as sealed pupae. The main signs of the disease are as follows.\n\n\u2022 Infected brood are called mummies. When mummies are removed from their cell they appear to be solid, similar to lumps of chalk.\n\n\u2022 Partially chewed down mummies are often seen in open cells.\n\n\u2022 Mummies vary in colour from white to dark grey or black.\n\n\u2022 Many of the sealed cell caps may have a small pinprick-sized hole.\n\n\u2022 Dead brood will be dropped onto the hive floor by nursery bees and later moved outside the hive entrance by worker bees, leaving a litter of chalk-like mummies on the ground.\n\n\u2022 A hive heavily infected with Chalkbrood will show a patchy scattered brood pattern.\n\n\u2022 Chalkbrood is more prevalent during the spring since fungal growth is enhanced during cool, damp conditions, particularly in poorly ventilated hives.\n\n\u2022 Chalkbrood can be spread by workers, drones or queens.\n\n15.12: One of the signs of Chalkbrood infection is hard, off-white, chalk-like cadavers at the hive entrance.\n\n15.13: Pupae killed by Chalkbrood removed from cells.\n\n15.14: The tell-tale sign of Chalkbrood: white, hard cadavers peppered through the brood cells.\n\n**SPREAD**\n\nChalkbrood fungus, _Ascosphaera apis_ , can initially be passed to an uninfected colony in a number of ways, namely by forager bees returning with infected pollen or nectar, robber bees stealing honey from an infected colony or by the drift of bees from an infected colony to an uninfected colony. This latter scenario may arise when hives are moved with an open entrance or by the beekeeper not following hygienic management practices. The fungus is passed orally to uninfected bees through contaminated food or by worker bees cleaning the inside of an infected hive causing the fungus to be consumed. While adult bees are not susceptible to Chalkbrood disease they are the carriers and pass the fungus on to larvae that consume it, become infected and eventually die.\n\nYoung larvae between the ages of one to four days are the most susceptible to Chalkbrood disease. Once the Chalkbrood fungus has been consumed by the larvae and is inside its mid-gut, the fungus penetrates the gut wall and grows inside the body cavity of the larvae. As the infection develops the fungus eventually breaks out of the body cavity, penetrates the outer surface or skin, initially at the posterior or anal end but eventually breaking out over the body, covering the brood with a thick white layer of mycelium, similar to household mould on fruit and bread. At the end of each string of mould on the outside of a blackened corpse there will be a new infectious spore, with as many as 100 million to 1 billion active spores on the outside of each black corpse ready to be ingested by unsuspecting adult bees.\n\nHoney bee brood that have died with Chalkbrood infection are initially swollen in size, taking up the entire volume in the cell. Eventually the corpse dries out leaving the characteristic chalk-like pellet normally associated with the disease.\n\n**TREATMENT**\n\nThere are no chemical treatments for Chalkbrood although feeding sugar syrup to infected colonies has been shown to be beneficial in helping bees to fight this disease. Like so many other honey bee diseases, good management practices by the beekeeper lie at the frontline of control and treatment. These management practices are detailed on p. 249.\n\nADULT BEE DISEASES\n\n**NOSEMA**\n\n_Nosema_ is a fungus that infects the gut of the adult honey bee. Although it is rarely fatal for the colony, Nosemosis, as infection with _Nosema_ is called, can cause the colony harm, resulting in a significant loss in honey production and compromised bee health including the death of bees, particularly during the spring. _Nosema_ comes in two strains:\n\n\u2022 _Nosema apis_\n\n\u2022 _Nosema ceranae._\n\nBoth varieties of _Nosema_ are probably present at low levels in all honey bee colonies but rarely develop to become a significant infection unless conditions favour development of the disease.\n\nHistorically, it was _Nosema apis_ that infected honey bees but more recently _Nosema ceranae_ has jumped species from the Asian honey bee to the European honey bee. The more recent _Nosema ceranae_ fungal infection is the more lethal of the two and is believed to be causing Australian beekeepers the majority of the problems that are associated with _Nosema._ Infection with _Nosema_ is difficult for the beekeeper to diagnose and the only accurate test is to send a number of infected or suspect bees to a laboratory for analysis.\n\nGenerally _Nosema apis_ is more of a problem in regions with longer winters if bees are unable to leave the hive to defecate due to poor weather. _Nosema ceranae_ appears to be more prevalent in warmer climates.\n\nAlthough _Nosema apis_ has been known to infect honey bees for many years, the recently arrived _Nosema ceranae_ is proving to be far more harmful to bees and has the added ability to displace _Nosema apis_ within colonies. I will describe _Nosema ceranae_ as it is the most prevalent strain with the most impact on Australian bee populations.\n\n**SIGNS OF INFECTION**\n\nThere are no classic outward signs of Nosemosis and an infection may go undetected until the disease progresses sufficiently for the colony to be significantly affected. For this reason infection with _Nosema_ is often called the silent killer. The main signs to look for are:\n\n\u2022 a declining population of bees, particularly during the winter\n\n\u2022 bees that look shiny\n\n\u2022 a reduction in the egg-laying ability of the queen and her possible supersedure\n\n\u2022 general lethargy within the hive.\n\nUnlike _Nosema apis, Nosema ceranae_ typically does not cause diarrhoea and infection is more difficult for the beekeeper to detect and diagnose. Infection with _Nosema_ is more prevalent when nutrition is poor and weather conditions are wet and cold.\n\n**TREATMENT**\n\nIn order to minimise the effect of _Nosema_ on the colony, good management practices should be used as described on p. 249.\n\n_Nosema_ build-up is particularly strong during winter months when the bees are restricted from leaving the hive to defecate due to cold and poor weather. It is thus important for the beekeeper to ensure that hives are placed in warm, open areas with the entrance facing the morning sun. Also, raising the hive off the ground to enable good air circulation helps the bees fight off the disease. Many experienced apiarists believe that the best way to treat infection with _Nosema_ is to leave the hive alone for a few months while ensuring that the conditions described above are met.\n\nHives that have been contaminated by _Nosema_ may be thoroughly cleaned with glacial acetic acid. Spores of _Nosema ceranae_ are susceptible to death by freezing or by refrigeration for a week.\n\nThe more a colony with _Nosema_ is worked to produce honey or for pollination the worse the infection is likely to become.\n\n**SPREAD**\n\n_Nosema_ spores are spread by worker bees eating infected faeces; the faeces may be ingested when cleaning the hive (but not by cleaning brood cells since brood are rarely infected), or by coming into contact with bee faeces on flowers when collecting food or at water sources. Once inside the mid-gut the _Nosema_ spores infect the cells lining the stomach and multiply rapidly. Within six to ten days the infected cells become filled with new spores, the cell dies and bursts, with each bee releasing 30 to 50 million new spores in their faeces ready to infect other bees. Infection with _Nosema_ generally affects worker bees, probably because neither drones nor queens take part in the cleaning activities within the hive or foraging outside of it. Newly emerged bees are seldom infected by _Nosema,_ since they have not had the opportunity to come in contact with, and digest, faecal matter. Infection with _Nosema_ severely restricts the ability of bees to digest pollen, thus limiting their own growth and limiting their ability to make royal jelly and other foods. These foods are essential for feeding to:\n\n\u2022 developing larvae\n\n\u2022 the queen to support egg production\n\n\u2022 drones to ensure maturation of sperm\n\n\u2022 other adult workers.\n\n**DEFORMED WING VIRUS (DWV)**\n\nAlthough surveillance for Deformed Wing Virus, DWV, is ongoing, field studies by CSIRO during 2014 failed to detect Deformed Wing Virus in Australia. In spite of this, adult bees are sometimes observed in Australian hives that show the classic symptoms of Deformed Wing Virus \u2014 deformed, stubby wings and\/or misshapen bodies. These apparent signs of DWV are more likely to be the result of genetic birth defects or the larvae being too cold inside the brood cell.\n\n15.15: A bee infected with Deformed Wing Virus showing the classic signs of deformed wings and a misshapen body.\n\n15.16: Testing of bees from the colony that hosted this bee apparently infected with Deformed Wing Virus (DWV) did not show any signs of the virus. Similar signs to infection with Deformed Wing Virus can often have other causes such as brood chilled during cold weather.\n\nDeformed Wing Virus, though, is one of a few bee viruses easily recognisable due to its well-defined signs in infected bees. Typical signs of Deformed Wing Virus infection include:\n\n\u2022 stubby, useless wings\n\n\u2022 shortened, rounded abdomens\n\n\u2022 discolouring of the adult bee's body\n\n\u2022 paralysis.\n\nDeformed Wing Virus is known to be transferred by the _Varroa destructor_ mite. Studies of _Varroa destructor_ have showed that up to 100 per cent of the mites are carrying the virus. Since infestation with _Varroa_ mites is one of the main ways in which Deformed Wing Virus is passed to bees, ongoing monitoring for this virus is regarded as being a critical part of surveillance for any _Varroa_ incursion into Australia.\n\nSUMMARY\n\nMany of the diseases that currently infect honey bees are assisted in their spread by the beekeeper. As a result, good management practices are at the heart of any disease minimisation work.\n\nThe most common diseases of honey bees are:\n\n\u2022 American Foul Brood (AFB) \u2014 a bacterium with resistant, long-lived spores that kills capped brood\n\n\u2022 European Foul Brood (EFB) \u2014 a non-spore-producing bacterium that generally kills uncapped brood\n\n\u2022 Sacbrood \u2014 a virus that kills brood\n\n\u2022 Chalkbrood \u2014 a fungus that kills brood\n\n\u2022 _Nosema_ \u2014 a fungus that affects adult bees. Present in Australia in two forms: _Nosema apis_ and _Nosema ceranae_\n\n\u2022 Deformed Wing Virus (DWV) \u2014 a virus that causes adult bees to emerge with deformed wings and bodies.\n\nAll the above diseases except Deformed Wing Virus are found in Australia. American Foul Brood is the only disease for which the affected colony will need to be destroyed to minimise its spread.\n16. Parasites of the honey bee\n\nOnce _Varroa_ had been detected in honey bee colonies in Europe and North America there was a concerted effort globally to identify all of the mites that live with honey bees. To date, about 100 mites have been identified although only three (not including the pest, the Small Hive Beetle) have been found to cause economic loss to beekeepers. The three mites that cause economic loss are _Varroa destructor,_ Tracheal Mite _(Acarapis woodi)_ and the _Tropilaelaps_ mite. Both _Varroa destructor_ and to a lesser extent Tracheal Mites are causing severe economic loss to beekeepers across the world. _Tropilaelaps_ is restricted to Asia and is not yet a problem for beekeepers in Europe, North America, Australia or New Zealand. _Tropilaelaps_ has wiped out honey bees in many parts of New Guinea and is severely limiting European honey bee beekeeping in Asia.\n\n16.1: A _Varroa_ mite on the back of a worker bee.\n\n16.2: A _Varroa_ mite on the abdomen of a worker bee.\n\nBoth _Varroa_ and _Tropilaelaps_ are discussed below although Tracheal Mite, the least destructive of the three, is not discussed here. The reason for this is that the effect of the Tracheal Mite on honey bees is much less severe than either _Varroa_ or _Tropilaelaps._ Also Tracheal Mite is not as yet found in either Australia or New Zealand.\n\nVARROA\n\nThere are two species of _Varroa_ that infect the European honey bee. These are:\n\n\u2022 _Varroa destructor_\n\n\u2022 _Varroa jacobsoni._\n\n16.3: _Varroa_ mites infesting a colony.\n\nAlthough not yet in Australia, _Varroa_ and _Tropilaelaps_ are considered the two most serious threats to Australian beekeeping if they were to gain entry to this country. Considerable effort is being expended at both the government and apiarist association level to keep this mite out of Australia. Most experts believe however that it is only a matter of time before _Varroa_ enters the country since the mites are already endemic in every other country of the world including our closest neighbours New Zealand, the Solomon Islands, New Guinea and New Caledonia.\n\n16.4: _Varroa_ mites on a larva.\n\n16.5: Close up of Varroa mites photographed from the top and underneath.\n\n_Varroa destructor_ has long been a destructive parasite of the European honey bee, having jumped from its original host species, the Asian honey bee, _Apis cerana_ , probably in the 1940s but perhaps much earlier. More recently, _Varroa jacobsoni_ has been found to infest European honey bees in Papua New Guinea and this development is causing concern to beekeepers as the new parasitic mite adds additional strain to an already stressed species.\n\n**SIGNS OF INFESTATION**\n\n_Varroa destructor_ is large when compared to other mites and can easily be seen with the unaided eye. The width of the female mite is larger than her length, about 1.6 millimetres x 1.1 millimetres, with the female having a reddish colour to her body. Adult females of _Varroa destructor_ move into brood cells in order to reproduce, and newly emerged female adults may also be seen walking rapidly onthe surface of brood comb before entering another brood cell to reproduce. Individual mites are often seen attached to adult bees, clinging mostly to the bees' abdomen where they feed on haemolymph (blood).\n\nWhen a mature female _Varroa_ mite enters a brood cell she first lays a male mite egg and then about six female mite eggs. If the initial female mite is the only _Varroa_ to have entered the brood cell before it is capped, the male mite that she first lays is capable of fertilising his mother's daughter mites and expanding the number of viable _Varroa_ mites in the colony. If other female _Varroa_ mites enter the brood cell before it is capped the male mites can fertilise the daughter mites of the other _Varroa_ mothers and bring greater genetic diversity to the reproduction process.\n\nAdult _Varroa_ males and _Varroa_ male and female larvae are not seen outside of a brood cell as they are unable to survive without the wet and humid conditions found inside the capped cell. Immature mites are white in colour while adult females are reddish brown; adult males are smaller than females and, as previously noted, are not visible as their entire lifecycle is spent inside brood cells.\n\nA female _Varroa_ mite in a capped worker brood cell on average produces 1.4 to 1.5 mature daughter mites that emerge from the cell. The daughters remain on nurse bees for up to nine days before entering another brood cell to lay further _Varroa_ eggs. Honey bee worker brood emerge from their cells after about twelve days as pupae while drones emerge after about fifteen days from their pupal cells. Since a fertile _Varroa_ mother will lay several eggs inside a capped brood cell, the longer gestation period for male drones to emerge as adults means that about 2 to 2.5 mature _Varroa_ daughters will emerge from a male drone cell with the drone. It is the number of capped drone cells inside a colony that largely determines how many mature _Varroa_ adults will emerge and infect the hive.\n\nThe _Varroa_ mite attacks young brood in a capped cell by piercing the soft skin of the brood in order to feed on its haemolymph (blood). Damage to adult bees is usually restricted to heavy infestations when a substantial proportion of the next generation is parasitised. If only a single mite enters the brood cell, damage to the bee may not be visible although the adult bee will have a much shorter lifespan than unparasitised bees. If multiple _Varroa_ mites enter the brood cell, the brood larvae may die and any adult that emerges may be deformed with a shortened abdomen or deformed wings. Deformed wings are usually an indication of infection with Deformed Wing Virus but could also be caused by other factors such as chilled brood. In addition to physical signs, the behaviour of the adult bee may also be affected; for example, during orientation in flight or by experiencing difficulty in gathering food.\n\nA _Varroa_ infestation is a more complex parasitism than just the mites sucking brood haemolymph. During feeding the mite also injects viruses such as the Israeli Acute Paralysis Virus (IAPV) or Deformed Wing Virus (DWV) into parasitised brood. These and other viruses often have a significant impact on the bee's health, causing paralysis in adults, deformed wings, as well as severely reducing the lifespan of the infected bee.\n\n16.6: Sugar Shake Test 1. A cup full of bees is placed inside a container with a sieve built into the lid. Icing sugar is added to the bees and the container is gently rolled so that all of the bees become coated in icing sugar.\n\n16.7: Sugar Shake Test 2. Next the container is turned upside down over a bowl of water and gently shaken. Any mites will become dislodged from the bees and will fall into the water, leaving the bees still inside the container. The mites, if present, will float on the surface of the water and can easily be detected.\n\nThere are several ways to monitor for _Varroa_ although the most reliable but time-consuming method is to randomly open capped brood cells, particularly drone brood cells, and look for _Varroa_ mites. The brood needs to be removed from the cell with a forceps and the brood and cell inspected for the mite. About 100 to 200 cells must be inspected in this way before an accurate assessment can be made of the scale of any _Varroa_ mite infestation.\n\nAn alternative method is the Sugar Shake test. One hundred or so bees are poured into a jar or container with a metal gauze lid that has sufficiently fine mesh to enable the mites but not the bees to fall through the mesh. Once inside the jar the bees are covered with icing sugar and shaken to dislodge the mites. Then the jar is inverted so that the mites fall through the mesh into a bowl of water or onto a sheet of white paper. The beekeeper then inspects either the water surface or the sheet of paper for any mites that have fallen through.\n\nAnother method is to construct a tray that fits into the base of the hive. The tray is covered with mesh about 2 millimetres wide held about a centimetre above the floor of the tray. The tray is made of white or light material so that any _Varroa_ mites falling onto it can easily be seen by the beekeeper. Once the tray is placed inside the base, it should be inspected after one to three days for any signs of infestation. Often either a sticky board or gooey substance is placed below the mesh to trap fallen mites.\n\n16.8: _Varroa_ mites that have fallen to the bottom of a hive and become stuck on a sticky board on the base.\n\nThe control of _Varroa destructor_ is one of the most demanding and time-consuming tasks faced by beekeepers in countries where this mite is present. Not only do they have to manage the infestation but the learning curve to master the necessary skills to keep the mite under control can be quite a steep one.\n\n**SPREAD**\n\n_Varroa_ was discovered in Auckland, New Zealand, in April 2000. A subsequent survey found that the mite had already spread to large areas of the North Island and that eradication was not feasible. To slow the spread of the mite further south a line was drawn across the centre of the North Island across which it was not permitted to move hives. The line stayed in place from April 2000 to September 2003. At that time there was estimated to be 10,000 infested apiaries north of the line and 100 infested apiaries south of the line. It was decided that since there were already infested apiaries south of the line there was little point in keeping the quarantine zone. It has been estimated that the use of quarantine delayed the spread of _Varroa_ by about two years.\n\n16.9: A vented base board used in the detection and control of _Varroa_. The mesh is too small to allow bees to pass but _Varroa_ mites can easily drop through. Underneath the mesh can be placed a sticky board or the _Varroa_ can be allowed to fall through to the ground outside of the hive where they will die.\n\n16.10: Vented bases used to help control Varroa also provide ventilation to the hive during hot summer days. The different colour markers on this hive facilitates use of the base to hold three separate queen rearing nuclei, each with its own entrance marked with a different coloured design.\n\nIt was initially believed that the water barrier of Cook Strait that separates the North and South Islands of New Zealand would form a natural barrier to the spread of the mite. Even though Cook Strait is 22 kilometres wide at its narrowest point, _Varroa_ was found in Nelson at the top of the South Island in 2006. In 2008 all restrictions on moving bees and associated products were abandoned and the mite can now be found throughout much of the South and North Islands of the country.\n\nIn the United States, commercial movement of infested colonies for migratory beekeeping spread _Varroa_ throughout the country in five years. At the local level, the major cause of _Varroa_ spread is by other bees robbing collapsed colonies in search of honey. Other causes of the spread of the mite are by infested bees drifting to other hives or by foraging bees leaving mites on visited flowers, where they are picked up by other foragers. An infested queen used during re-queening or the merging together of two hives can also spread the mite. Beekeepers can help control the spread of _Varroa_ by ensuring that sound management principles are used that minimise its spread.\n\nIn the early stages of an Australian incursion the beekeeper will look to the government apiary inspector, beekeeping associations and beekeeping supply stores for advice and management information. The experiences of other countries in the control and management of _Varroa_ will become part of our own critical learning curve.\n\nJudging by overseas experience, once _Varroa_ reaches Australia and gains even a small foothold so that it cannot be eradicated, the spread of the mite throughout the country will be rapid. Experience in other _Varroa_ -affected countries also indicates that if a feral colony is infested with _Varroa_ it will only live about two to three years before the mite causes sufficient weakening for the colony to collapse and die. On the positive side, even though infestation by _Varroa_ is a serious and time-consuming problem, beekeepers with hives affected by the mite often report increased honey production with a smaller number of hives. The reasons given for this increased production vary from the ability and management skills of the individual beekeeper to other factors such as fewer feral colonies competing for nectar and pollen. The extra effort and time required for effective management control in each hive no doubt also results in beekeepers often reducing hive numbers to deal with the added workload.\n\n**TREATMENT**\n\nWhere _Varroa_ has been introduced into other countries, within a few years survival of the resident European honey bees has only been achieved through ongoing management by the beekeeper. Most colonies of bees have required significant and expensive levels of management to survive while feral colonies died out within two to three years. Larger numbers of feral colonies, however, have now returned to the United States and other countries where the mite is endemic. These colonies are still infested with _Varroa_ but the mites are not as harmful to the bees as they once were. In some cases the mite became less harmful to the colony; in other cases the colony became more resistant to infestation. Feral bees that have developed resistance to _Varroa_ are commonly called 'survivor stock'.\n\nSince _Varroa_ can be spread by the activities of the beekeeper, including the practice of migratory beekeeping, beekeepers will need to review the way that they manage their colonies, as this is one of the key tools in minimising the spread and the impact of the mite.\n\nOther management techniques that are being used overseas to minimise the growth of the mite in hives is to disrupt their breeding patterns by inserting drone brood comb, as it is the drone cells which the _Varroa_ mite prefers for reproduction. The drone comb is then removed, destroying the drone brood cells before the mite-infested drones are able to hatch out.\n\nMiticides are still the most popular way to control _Varroa_ overseas. The term miticide refers to any chemical that is used against mites such as Tracheal Mite, _Varroa_ or _Tropilaelaps._ The term insecticide is used to describe a chemical that is used to control insects such as the Small Hive Beetle. Many proprietary chemicals, such as Apithor, may be used as both a miticide and an insecticide.\n\nAlthough the use of chemicals to control _Varroa_ risks contaminating honey and wax with lethal pesticides, if used correctly by carefully following the instructions on the packet these risks can be minimised. It should be noted that medicating the hive should not be done unless it is absolutely essential. Applying chemicals to a hive as a protective measure without any signs of _Varroa_ is counterproductive since there will be toxic residues left in the hive.\n\nA further major problem with applying chemicals to control mites is that frequent applications will result in the mites becoming resistant to the chemical over a relatively short time. In order to minimise the possibility of _Varroa_ or other mites developing resistance to a particular chemical treatment the miticides used need to be rotated with the use of around three differently composed chemical treatments in order to delay the build-up of resistance to a single treatment. Treatment is usually effected around two to three times a year so the possibility of resistance build-up is an ongoing issue.\n\nThe chemicals used to treat _Varroa_ vary in their toxicity to both bees and to humans. The naturally occurring organic chemicals oxalic acid and formic acid are relatively safe to bees at lower concentrations but not to beekeepers and need to be handled with care. Chemicals such as Apiguard, Apistan and Apivar should not be used in hives if there are supers containing honey present. The resulting honey may become chemically contaminated and unfit for human consumption.\n\nAt the time of writing, _Varroa_ has not been detected in Australia. The chances are that this situation will not remain so for many more years and it is therefore imperative that if a beekeeper suspects an infestation that immediate contact is made with the government apiary inspector in order to have the suspect mites tested and confirmed. If an outbreak were to occur in Australia it is anticipated that the government will release a list of effective miticides that can be used inside the hive to treat the mite.\n\nApart from chemical miticides, many beekeepers in the United States and Europe are now working to develop Varroa-resistant or tolerant lines of bees. This has proven to be an effective, although currently underused, method of keeping _Varroa_ and other diseases in check and I expect to see much progress in this area in the years ahead. Using a genetically tolerant strain of queen has the major added advantage of keeping lethal chemicals away from the hive as the chemical solutions applied to _Varroa_ have now become part of the ever-increasing problem of hive contamination.\n\nWhen the mite is detected on our shores a properly formulated and managed integrated pest management plan (IPM) will need to be in place to help manage the incursion and spread. There is no doubt that the mite's arrival will kill tens of thousands of colonies and decimate the feral bee population. Overseas experience indicates that many commercial beekeepers' livelihoods will be severely affected, at least during the early years of upskilling required to control this parasite.\n\nTROPILAELAPS MITES \u2014 THE ASIAN MITE\n\n_Tropilaelaps_ mites are similar to _Varroa_ and are parasites of honey bee brood. _Tropilaelaps_ use the giant Asian honey bee, _Apis dorstata,_ as their natural host and four species have been documented _(Tropilaelaps clareae, Tropilaelaps koenigerum, Tropilaelaps thaii_ and _Tropilaelaps mercedesae)._ However, it has now been found that two of these, _Tropilaelaps clareae_ and _Tropilaelaps mercedesae_ , also infest the Western honey bee, _A. mellifera. Varroa_ has received enormous attention over the past three decades and, as a result, the importance of other parasitic mite species on bees has been largely ignored. Also, because at present _Tropilaelaps_ mites are only found in Asia, their threat has been largely overlooked in Australia. If _Tropilaelaps_ were to enter Australia, the United States or Europe, the mite's destructive power would be far greater than that of _Varroa_ due to its much shorter reproduction cycle. _Tropilaelaps clareae,_ which is frequently found infecting European honey bee colonies in Asia, has a wide distribution throughout Asia, and from Iran to Papua New Guinea, while other species of _Tropilaelaps_ have a much more restricted geographic distribution.\n\n16.11: _Tropilaelaps_ mites look very similar to _Varroa_ mites.\n\n16.12: _Varroa (L)_ and _Tropilaelaps (R)_ mites side by side.\n\n16.13: Workers removing excess drones from a colony.\n\nAdult _Tropilaelaps_ mites are light reddish-brown and, depending on the species, between 0.7 and 1.0 millimetres long and approximately 0.6 millimetres wide. Like _Varroa,_ they need to enter cells containing bee larvae shortly before capping in order to reproduce. Forty-eight hours after cell-capping the mite reproduces, laying three or four eggs, each of which will hatch within twelve hours. Five days after hatching the mite larvae will develop into adult mites.\n\n_Tropilaelaps_ mites depend on softer brood for food since their mouthparts cannot pierce the relatively tough body walls of adult bees. The developing mite feeds on the haemolymph (blood) of capped bee larvae, depriving it of the nutrients required for growth. At the same time as the young adult bee emerges from its cell, the new generation of adult mites (and their mother) leave the cell in search of fresh hosts to live off. Female _Tropilaelaps_ mites become pregnant while living in their initial cell and as pregnant females will die within two days of leaving the initial host bee they need to deposit their eggs quickly into a new cell if the eggs are to survive.\n\n**SIGNS OF INFESTATION**\n\n_Tropilaelaps_ is often mistaken for _Varroa_ although distinguishing between the two is relatively straightforward.\n\n\u2022 A _Varroa_ mite is larger, crab-shaped and wider than it is long.\n\n\u2022 _Tropilaelaps_ are about a third as wide as a _Varroa_ mite, or 1-millimetre long and 0.6 millimetres wide.\n\n\u2022 The _Tropilaelaps_ mite's body is elongated.\n\n\u2022 Adult _Tropilaelaps_ mites run rapidly over infested brood combs. Video shot by United States Department of Agriculture (USDA) researchers show the mites running around on the comb at a much faster rate than the _Varroa_ mite.\n\n\u2022 Unlike _Varroa, Tropilaelaps_ mites live in brood cells rather than on adult bees.\n\nInfection by _Tropilaelaps_ causes abnormal brood development and the death of both brood and adult bees that will cause colony strength to decline, resulting in absconding or the total collapse of the colony. With severe infestations, hives have a noticeable smell of decaying bee remains. Adult bees parasitised by _Tropilaelaps_ during development have reduced lifespans, lower body weight, and wing and leg deformities.\n\nEuropean honey bees, _Apis mellifera,_ lack the well-developed defences to _Tropilaelaps_ mites that the giant Asian bee, _Apis dorsata,_ has developed over thousands of years and are therefore more susceptible to the effects of infestation. Unlike _Apis mellifera, Apis dorsata_ can bite and injure _Tropilaelaps_ mites and it is also more efficient at grooming and removing the parasites.\n\nThe majority of _Tropilaelaps_ mites live and reproduce inside brood cells as they can achieve a longer lifecycle there, living only about three days on an adult honey bee outside of a pupal cell. Perhaps the most alarming characteristic of the mite is its reproduction rate and breeding cycle. One to four adult female mites are generally found in a cell reproducing at one time, however there are reports of mites being found in quantities over three times that many in just one cell. They can quickly outnumber any _Varroa_ mites in colonies, hatching in only twelve hours and reaching full maturity in just six days. Within 24 hours of emerging from a brood cell the mites enter another cell and begin reproducing, laying about four eggs per adult female at a time.\n\n**SPREAD**\n\nThe spread of _Tropilaelaps_ is similar to that of _Varroa. Tropilaelaps_ mites move easily and can readily transfer from one bee to another within the colony. To move between colonies they depend upon adult bees for transport through the natural processes of drifting, robbing and swarming. Within an apiary they can be spread through the distribution of infested combs and bees through poor beekeeping management practices. The rapid distribution of the mite to new areas is usually due to the beekeeper transporting colonies in search of new food or for pollination.\n\nThe lifecycle of _Tropilaelaps_ is similar to that of _Varroa destructor,_ but _Tropilaelaps_ develops much faster and the intervals between successive reproductive cycles are very short. As a result, if both types of mite are present in the same colony, _Tropilaelaps_ numbers can build up at a much faster rate than those of _Varroa._\n\n**TREATMENT**\n\nTreatment for _Tropilaelaps_ is expected to be similar to that of _Varroa,_ although since this mite has only been found in Asia and has not been detected in Australia, the United States or Europe, no approved treatment has been agreed upon. Similar to _Varroa_ mite detection in Australia it is expected that government apiary inspectors and beekeeping associations will issue directions for the control and management of _Tropilaelaps_ to assist the beekeeper.\n\nAn important difference between _Varroa_ and _Tropilaelaps_ is that a mature _Varroa_ mite can survive for many months outside of a brood cell. It does this by attaching itself to a bee and drinking haemolymph (blood) after biting into the host bee's abdomen. _Tropilaelaps_ mites are unable to survive in this way since their mouthparts are not sufficiently strong to bite into an adult bee's abdomen.\n\nOne way in which a _Tropilaelaps_ infestation can be controlled is to reduce the hive brood for several weeks. This can be done by either removing the queen from the hive for three weeks or by placing her inside a cage within the hive so that in either instance no more eggs are being laid.\n\nSUMMARY\n\n\u2022 The _Varroa_ mite is the most serious problem facing Western honey bee beekeepers today. Although not yet found in Australia, there is ongoing education of beekeepers about the mite as well as monitoring of colonies to detect any incursion. If (when) this mite were to become endemic in Australia, there would be a significant impact on both the honey and pollination industries in this country. Experience in other countries shows that the mite can be kept under control but only if significantly different management practices are mastered by the beekeeper. As previously mentioned it is expected that a range of chemical treatments will be approved and introduced to manage any incursion.\n\n\u2022 _Tropilaelaps_ , a mite currently only found on the Western honey bee in Asia, may cause even greater damage to the beekeeping industry than _Varroa_ if it were to escape out of Asia into Australia, North America or Europe. It is expected that if _Tropilaelaps_ were to be detected in Australia, management techniques to control the mite would be very similar to those used to control _Varroa._\n17.\n\nPests of the honey bee\n\n17.1: The Greater Wax Moth (GWM), Galleria mellonella.\n\n17.2: A Wax Moth larva.\n\n17.3: Cocoons in which the Wax Moth pupae live prior to emerging as fully grown moths.\n\nWAX MOTH\n\nWax Moth can cause severe damage to frames of honey and brood although infestation is usually limited to weaker hives that have insufficient numbers of bees to provide protective cover on frames to fight the moth.\n\nThere are two related, although physically different, types of Wax Moth in Australia: the Greater Wax Moth (GWM), _Galleria mellonella,_ and the Lesser Wax Moth (LWM), _Achroia grisella._ The Greater Wax Moth is the larger and more destructive of the two with a female adult length of about 20 millimetres and a wing span of between 30 millimetres and 41 millimetres. It is this pest that will be described in this chapter. The Greater Wax Moth is a brown-grey colour and may have a number of speckles on her wings. Infestation with Greater Wax Moth is often called _Galleriasis._ Larvae length when fully grown is up to 28 millimetres.\n\n**GREATER WAX MOTH**\n\nThe female Wax Moth enters colonies during the night when the defences of the hive are at their lowest. Once inside, the moth lays her eggs directly on comb or in other cracks in the hive that cannot easily be cleaned by the bees or seen by the beekeeper. Upon hatching, the larvae initially feed upon base-board debris before burrowing into pollen storage or honey cells. As the Greater Wax Moth larvae develop they burrow towards the centre of the colony where they are protected by the capped cells, leaving behind a trail of destruction in the form of a dense mass of silken webs and faecal debris. Unattended combs can be completely destroyed in as short a time as ten to fifteen days, and the larvae will also devour young brood if there is insufficient other food available. Between 50 and 150 eggs are laid in each batch by a single female and the maturation period for Wax Moth larvae, depending on temperature and the availability of food, typically takes eighteen to twenty days in tropical regions and a little longer in temperate zones of Australia. In very cold climates the eggs can remain semi-dormant for up to six months until warmer conditions arrive. The newly emerged larvae are a creamy white colour but as they reach maturity they turn grey. Infestation by the Greater Wax Moth, _Galleria mellonella,_ can result in the young brood being unable to leave their cells due to entanglement in the silken threads of the Wax Moth cocoons.\n\n17.4: A tangled mass of silk webbing is a later sign of infestation by the Greater Wax Moth. Black faeces are 17.4 visible in the silk matting.\n\nStudies overseas have shown that the newly hatched larvae of the Greater Wax Moth can travel more than 50 metres and are therefore capable of moving to neighbouring hives. The average colony copes well and will control a small number of larvae but if bee numbers are reduced for any reason Wax Moth larvae may take over and destroy the hive.\n\n17.5: Damage to a hive caused by Greater Wax Moth larvae.\n\nCareful identification of larvae by the beekeeper is necessary because Wax Moth larvae and Small Hive Beetle larvae are similar in appearance. Wax Moth larvae have three pairs of legs on the front of the body and have other uniform pairs of prolegs along the rest of the body. Small Hive Beetle larvae on the other hand only have three pairs of legs at the front of the body and no prolegs along the rest of the body. Wax Moth larvae are soft and fleshy, whereas the Small Hive Beetle larval body is rigid and hard. It is common for both types of pest to be present in the same hive. If larvae are found in the hive a simple way for the beekeeper to determine if Wax Moth is present rather than Small Hive Beetle is the presence of a mess of webbing in and on comb. This is a clear indicator of Wax Moth as Small Hive Beetle do not create either webbing or the characteristic Greater Wax Moth burrowing damage seen in the wooden parts of the hive or frame.\n\n**SIGNS OF INFESTATION**\n\nFor both types of Wax Moth the signs are the same: larvae crawling in the hive. Larvae burrowing through comb, particularly brood comb, are often the first signs of infestation in hives. Cocoons can also sometimes be found on the top of frames or in other parts of the hive. Cocoons are obvious due to their larger size and are often covered in very small black pellets of larval faeces.\n\nLater stages of infestation are very distinctive due to the tangled mass of silk webbing that covers the parts of the frame where larvae are burrowing. When the infestation is even further advanced, the comb in the frame falls apart, a very dense mass of impenetrable silk matting has been spun and larvae can be seen crawling over what is left of the comb.\n\nGreater Wax Moth larvae are responsible for the physical damage to hive boxes and frames that appears as gouges or holes in the wood. Often Wax Moth larvae can be seen crawling out of these holes when disturbed.\n\nThe adult Wax Moth is harmless and does not damage hives; it is only the larvae and pupae that cause serious problems for the beekeeper.\n\n**SPREAD**\n\nSince the main source of food for Wax Moths is found inside a hive or feral nest, the moth will fly from one colony to another in search of a new home or food.\n\n**TREATMENT**\n\nThere are no approved chemical treatments for the Greater Wax Moth within the hive so effective control comes down to the management practices of the beekeeper. Since the moth targets weaker colonies the beekeeper should ensure that the hive remains strong by merging weak colonies. A strong hive is a healthy hive so ensure that there is plenty of pollen or pollen substitute available for the bees and that there is a suitable reserve of capped honey during periods of drought. If the hive has too many boxes so that the distribution of bees throughout the hive is sparse, this will give the moth an opportunity to flourish in the exposed comb. It is therefore advisable to keep empty space within the hive to a minimum by limiting the number of supers added and by monitoring the space required by the bees. A healthy colony will cope well with a small infestation of Wax Moth larvae and the adult bees will quickly remove these. Like many pests and diseases affecting the honey bee, a weak colony will be most vulnerable to Wax Moth larvae build-up and may in time abscond or be completely annihilated.\n\nWax Moth also cause problems with frames containing used comb, unclean wax or used hive boxes stored in the open or in sheds. To minimise infestation, frames should be stored in airtight plastic containers, preferably in cool rooms. If the beekeeper plans to keep capped or partly capped frames of honey for any period of time any Wax Moth eggs or larvae can be killed by wrapping the frames individually in sealed plastic bags and freezing for two days. After that time the frames may be safely stored in airtight plastic boxes as all the larvae will have been killed.\n\nWax Moth will only settle and breed in dark locations and knowledge of this behaviour can be used to the beekeeper's advantage. Empty frames should be stored inside supers and stacked after placing sticks between each of the boxes to let in light. A lid is unnecessary, for as much light as possible should be allowed to enter through the top of the stacked boxes.\n\nSMALL HIVE BEETLE (SHB)\n\nThe Small Hive Beetle (SHB), _Aethina tumida,_ mainly affects weaker colonies as there are insufficient worker bees to remove or drive the beetles from the hive. The adult beetle lives in areas of the hive box inaccessible to bees; for example, under the hive mat or in places less often frequented by bees like the corners of lids and cracks in the wood of the box.\n\nThe larvae of Small Hive Beetle are often mistaken for Wax Moth larvae although there are significant differences between the two:\n\n\u2022 Small Hive Beetle larvae are smaller than Wax Moth larvae. Small Hive Beetle larvae are around 10.1 millimetres in length and the Lesser Wax Moth larvae are up to 13 millimetres in length.\n\n\u2022 Small Hive Beetle larvae have three pairs of legs at the front of their body with no prolegs, or smaller legs, along the rest of the body.\n\n\u2022 Small Hive Beetle larvae have a row of spines along their back together with two distinct spines protruding from the rear.\n\n17.6: The Small Hive Beetle.\n\n17.7: A close-up view of the Small Hive Beetle.\n\n17.8: Small Hive Beetle larva. Note the three pairs of legs at the front, the spines on top and the two distinct spines at the back.\n\n17.9: Small Hive Beetle larvae spoiling, or sliming, a frame of honey.\n\nSmall Hive Beetle is a relatively new pest to Australian beekeeping, first detected near to Sydney in October 2002. Even at this stage of detection the beetle had become well established around the greater Sydney area and its eradication was deemed impossible. Small Hive Beetle has spread widely since it was first discovered and can now be found in Queensland, New South Wales, Victoria, South Australia and Kununurra (north Western Australia). It was initially thought that the beetle would not move deep into Victoria as the colder climate would not suit its lifecycle, but like many pests it has proved very adaptable and can now be found in even the most southerly parts of the state. Over the next few years we will see beetle numbers rise and spread deeper into Australia. As Small Hive Beetle is still spreading throughout Australia, its full effect on the beekeeping industry has yet to be determined; however, indications of its impact to date in both New South Wales and Queensland show it will increase the workload of hobby and professional beekeepers as well as cause significant economic loss to the industry.\n\nLIFECYCLE OF THE SMALL HIVE BEETLE\n\nAdult female Small Hive Beetles will lay eggs directly onto food sources within the hive, such as pollen or brood combs. Alternatively, female Small Hive Beetles may deposit irregular masses of eggs in tiny crevices or cavities away from the bees. A female Small Hive Beetle may lay 1000 eggs in her lifetime, although some researchers suggest that the number of eggs produced in one female's lifetime might be upwards of 2000. The majority of these eggs hatch within three days.\n\nNewly hatched larvae immediately begin feeding on the available food including honey, pollen and bee brood, though they have a preference for bee brood. Maturation time for larvae is generally between ten and sixteen days. Once the larvae finish feeding, a wandering phase is initiated where male and female larvae leave the hive to find suitable soil in which to pupate. It is believed that the majority of larvae do this at night under cover of darkness.\n\nLarvae in the wandering stage may travel large distances from the hive to find suitable soil. Despite this, most larvae will pupate within 90 centimetres of the hive and will burrow down into the soil less than 10 centimetres in order to pupate. The period of time spent in the ground pupating can vary greatly depending on factors such as soil temperature, moisture and composition. Generally though, the majority of adult beetles emerge from the soil in approximately three to four weeks.\n\nUpon emerging from the ground as adults, the Small Hive Beetle flies in search of honey bee colonies and will probably identify potential host colonies by a range of smells emanating from the hive including honey and pollen. The beetle flies before or just after dark and, upon locating and entering the host colony, will seek out cracks and crevices where it can hide from the bees' often aggressive attempts to evict it.\n\nThe mating behaviour of the Small Hive Beetle is not fully understood although it is when the female adult enters the hive and lays her eggs that the lifecycle begins again.\n\nThe turnover rate from egg to adult emerging from the soil can be as little as four to six weeks; consequently, there may be as many as six generations in a twelve-month period under moderate climatic conditions.\n\nSmall Hive Beetles are a problem in hot, damp climates like Queensland and coastal New South Wales, but are also spreading into the colder regions of Victoria and South Australia.\n\nEven though there may not be any Small Hive Beetles in the drier areas of Australia, if discarded comb, slum-gum or residue from cleaning hives is left on the ground it may become sufficiently damp to host the beetle. Already the Small Hive Beetle is proving more adaptable to different climatic conditions than was initially thought and there is no reason to believe it will not continue to adapt and spread to other areas of the country.\n\n**SIGNS OF INFESTATION**\n\nDepending on how frequently hives are inspected, the following will give either a clear or fair indication that Small Hive Beetle is present.\n\n\u2022 Upon opening an infested hive, beetles are seen scurrying away out of sight, perhaps from on top of the hive mat.\n\n\u2022 For the less frequently inspected hive, noticing a slime oozing out of the hive entrance and a generally putrid smell similar to that of rotting oranges. The Small Hive Beetle itself is not the cause of this destruction but its larvae. As the larvae grow they burrow their way through the comb. They prefer comb that has contained brood or pollen. During the burrowing through honeycomb, a residue of yeast is left in the larvae excreta that quickly causes the honey to ferment, then to slime and fall to the base of the hive as a revolting slurry.\n\nEither of these situations is a sign of an established infestation of Small Hive Beetle that will need management. Other observable indications and checks for suspected Small Hive Beetle infestation include:\n\n\u2022 Small larvae burrowing through capped comb, both honeycomb and brood comb. Since the comb is capped the larvae may be difficult for the beekeeper to notice during a routine hive inspection.\n\n\u2022 Beetles inside non-capped areas of the comb. If unsure, tap the frame onto the lid and observe if any beetles fall out.\n\n\u2022 Inspect under the lid of the hive, on top and underneath the frames to see if there are any beetles there.\n\n\u2022 Pick up the bottom brood box and check if there are any larvae or beetles on the base board, particularly if the base board is dirty as the beetles may be hiding in the debris.\n\n\u2022 If the brood box is attached to the base board, remove all the frames from the brood box and very carefully inspect the base board from the top.\n\n\u2022 Check all the nooks and crannies in the hive where beetles may be hiding.\n\n\u2022 If plastic frames are used check that beetles are not hiding in the hollow ribs of the frames' sides.\n\n\u2022 Inspect metal queen excluders, particularly those with a solid strip edge, as the folded metal can provide a suitable hiding place.\n\n\u2022 Remove a super with frames and place it upside down on a lid. Wait for several minutes before returning the super to the hive and quickly inspect the lid for any beetles that may have congregated to hide from the light.\n\nOnce the beekeeper easily observes several beetles in a hive that is usually a good indication that there are many others and measures need to be taken to help the bees to keep the pest at controllable levels.\n\n**SPREAD**\n\nThe beetle is attracted to the smell of honey and flies from one hive to another. The migratory nature of commercial Australian beekeeping has ensured that the beetle can easily be shipped around the country, infesting new hives and areas wherever it is introduced. It is believed that the beetle can fly up to 10 kilometres and thus a single infested hive can infect other colonies for an area of over 300 square kilometres. Since feral hives also harbour the beetle, movement of commercial hives into new areas can inadvertently lead to infestation.\n\n**TREATMENT**\n\nAs previously noted, strong hives are reported to be less susceptible to Small Hive Beetle than weaker hives although this has been disputed by some experts, particularly in areas where the beetle has gained a good foothold. In order to obtain stronger colonies, merge weaker colonies or add a frame or two of capped brood to the weaker hive. With either of these two options make sure that you are not transferring any disease from an infected colony.\n\nSmall Hive Beetles are attracted to the smell of honey and use it to guide them to hives. As a consequence, to reduce the chance of the beetle being attracted to your hive by the smell, minimise the number of times you open the hive and keep the area around your hive clear of old honeycomb and debris that may act as an attractant. Small Hive Beetles will also fly with swarms so by catching a swarm you may unwittingly introduce the beetle into your apiary.\n\n17.10: To help manage Small Hive Beetle this base has been designed with trays of oil in the bottom. The beetles fall through the wire mesh into the oil and drown.\n\n17.11: Another design of base for trapping Small Hive Beetle. The removable box at the back of the base contains oil that the beetles fall into and drown.\n\n17.12: A trap filled with oil that is inserted between frames in both the brood box and the supers.\n\n17.13: Another method of managing Small Hive Beetles is to use a corrugated case filled with the miticide Apithor. This trap is called the Apithor Hive Beetle Harbourage.\n\n17.14: Filling a Small Hive Beetle trap with oil.\n\nThe beetles prefer to breed in a humid environment, so in order to keep the inside of hives a little drier, open the lid by a very small amount for a few hours to allow any moisture to escape, say by placing a twig under one side. Leaving the hive like this for longer periods will encourage robber bees, wasps or even more beetles to enter and this option can be time consuming so is really for the hobby beekeeper with time and a small number of hives.\n\nHive boxes should be kept in good condition; any cracks need to be repaired so that they cannot be used by the beetle to enter the hive and to hide within. Also ensure good hygienic practices in and around the hive by keeping the area clean and tidy and clearing any burr comb from inside the hive lid and removing waste material from the hive floor. This latter practice is made a lot easier if base boards are not fixed to brood or bottom boxes.\n\nMinimising the movement of the hive is also a good idea since by changing its location the beekeeper may be moving it close to a colony that already has the beetle and will unwittingly aid it in its spread.\n\nTo minimise the contamination of stored frames by the beetle, if possible keep them in a cool room below 10\u00b0 Celsius.\n\nBeekeepers should contact their state's apiary inspector to determine which chemical treatments are approved for use in the hive to manage infestation.\n\nApart from the above-mentioned general management practices, the following techniques have been successfully used to help colonies control the beetle.\n\n\u2022 Beetle traps can be placed inside the hive between frames. There are several types of beetle trap for sale through suppliers in Australia. A black-coloured plastic reusable model with a clip-off top is available as is a plastic disposable model with a black trapping surface and a deep clear reservoir. There is also a similar metal design available. The traps work on the principle that the bees chase the beetles into the slotted top of the trap where they drop into the reservoir and are killed either by drowning in a mixture of vegetable oil and a touch of apple cider vinegar or are smothered by garden lime or diatomaceous earth. Any kitchen vegetable oil can be placed in either trap and this option has the advantage that the oil mixture is non-toxic, does not have an unpleasant taste and will not contaminate honey if accidently spilled onto the frame. The disposable traps tend to have a deeper reservoir than the reusable models and have the additional advantage that the clear plastic reservoir makes it easy to do a dead beetle count and to check the level of oil.\n\n\u2022 Another popular method used to trap beetles is to use carpet underlay or felt-backed linoleum as a hive mat. The mat fibres trap the beetles and these can be disposed of during the next inspection of the hive by the beekeeper. A disadvantage of this method is that often the odd bee or two will also be caught in the felt.\n\n\u2022 Disposable kitchen wipes such as Chux Wipes and some types of carpet have also been shown to be effective in trapping Small Hive Beetles \u2014 success can often depend on the beetle numbers within a particular hive and how vigorously the bees pursue them.\n\n\u2022 Corflute, a type of plastic with a hollow, tubular core, can be inserted into the hive entrance or placed under the lid. The beetles will use the hollow tubes of the corflute to hide from the bees and the corflute can be removed by the beekeeper and destroyed. To ensure that the corflute can be conveniently inserted and removed from the hive entrance a piece of stiff wire is attached to it so that it can be pushed in or pulled out easily. An effective variation of corflute is a design by the New South Wales Department of Primary Industries and the Rural Industries Research and Development Corporation (RIRDC) that impregnates a hollow-fluted matrix with a miticide. The device is called the Apithor Hive Beetle Harbourage. The Apithor trap is pushed through the hive entrance and onto the base board of the hive. The beetles enter the hollow matrix to escape the bees and are killed by the miticide in the trap. This device has been shown to be effective in controlling beetles and the trap is small enough to be easily inserted and removed through the hive entrance. The miticide is only on the inside of the trap matrix where the mites hide so cannot be touched by the bees.\n\n\u2022 Another method of control is to use a specially constructed base that has a tray of vegetable oil, garden lime or diatomaceous earth with wire gauze above the tray so that the beetles can climb through the wire but the perforations are too small for the bees to enter. The beetles fall into the tray where they are killed and can be cleaned out by the beekeeper. For ease of use, the tray needs to be removable from the base without disturbing the rest of the hive. If the hive needs to be dismantled to remove the tray of dead beetles this will mitigate many of the advantages of the technique.\n\nThe design of traps for the Small Hive Beetle is limited only by the imagination of the beekeeper. Those with an inventive mind have successfully used empty CD cases, fishing tackle boxes and empty plastic screw containers as inexpensive, but effective, beetle traps. My own preference is for traps that contain an oil and vinegar mixture, as these seem to offer no adverse effects to bees or honey and are easy and inexpensive to install. I prefer the disposable traps with a deep clear reservoir as they are easy to refill, and I am able to monitor dead beetles with a quick visual check. The deep reservoir of the trap also makes it less likely that the contents will spill if the hive is moved. Although sold as disposable the traps can be used for long periods. Some bees have a tendency to propolise or wax up the top entry point of traps and you may need to keep an eye on this.\n\nBRAULA FLY\n\nThe scientific name for the Braula fly is _Braula coeca_ and is often incorrectly termed the Bee Louse. The fly is found in Tasmania but not in other parts of Australia.\n\nThe Braula fly is unusual in that it is wingless, flattened and lives in honey bee colonies by holding on tightly to the body hairs of adult bees, usually on the head, using a set of comb-like structures on its front legs. When hungry, the fly moves to the mouth of the bee and steals some of the food being fed to it by other bees. Adult Braula flies are reddish-brown in colour and are regularly mistaken for _Varroa_ mites.\n\nThe Braula fly does not damage or parasitise any stage of the honey bee lifecycle and is not usually considered a significant pest or threat to colonies. It is worth noting, however, that if Braula is living on a queen it may reduce the amount of available food, thereby resulting in an impaired ability to lay eggs.\n\n17.15: An inexpensive way to trap Small Hive Beetles is to lay a Chux cloth on top of the frames under the lid instead of a hive mat.\n\n17.16: A Braula fly.\n\n17.17: Braula flies on a bee's thorax.\n\nThe main economic impact of Braula fly occurs as a result of the fly laying eggs on capped honeycomb and the larval stage burrowing into the cell under the honey cappings. The sight of Braula larvae burrows in comb honey in particular can detract from its visual appeal for sale. Liquid honey is not affected by Braula, since most honey is extracted mechanically and will not be affected by the fly since it will be filtered out during the extraction process.\n\n**SIGNS OF INFESTATION**\n\nBraula flies are sufficiently large that they can be seen with the unaided eye on adult bees including the queen. Simple observation of adult bees, particularly the resident queen bee, may reveal the presence of adult Braula flies. Interestingly, tobacco smoke causes the Braula fly to fall off the bee so the useful addition of a few grams of tobacco to the smoker will cause the fly to fall to the floor of the hive. A sticky sheet of paper on the hive floor will catch any fallen Braula where they can easily be seen by the beekeeper.\n\n**TREATMENT**\n\nThe use of small amounts of tobacco in the bee smoker has been demonstrated as a very effective means of killing adult Braula flies. However, continued exposure to tobacco smoke will affect bees and may lead to some adult bees dying.\n\nComb honey should be stored in a freezer for two days after its removal from a beehive. This will ensure that all stages of the Braula fly larval lifecycle are killed. This will also serve to kill all larval stages of other beehive pests such as Wax Moth and Small Hive Beetle.\n\n**SPREAD**\n\nAdult Braula attach themselves to adult bees and can be spread by swarms, drifting bees, and queen bees. Overseas, Braula fly infestation is less common, possibly due to the chemical treatment of colonies for _Varroa_ which would also limit the incidence of Braula infestation.\n\nSUMMARY\n\nThere are several pests of the honey bee that, like Small Hive Beetle, can be very destructive of the colony, or, like Braula fly, are only minor irritants. The main pests are:\n\n\u2022 Wax Moth \u2014 which comes in two species, the Greater Wax Moth and the Lesser Wax Moth. The Greater Wax Moth is the more destructive of the two.\n\n\u2022 Small Hive Beetle \u2014 recently introduced into Australia, this beetle is causing serious economic harm to the beekeeping industry and is spreading further north and west and into the colder regions of Victoria and Tasmania.\n\n\u2022 Braula fly \u2014 found only in Tasmania, this is only a minor irritant to the honey bee and is not a cause for concern.\n\nCurrently the limited use of a Fipronil-based insecticide has been approved against Small Hive Beetle in the form of the Apithor Hive Beetle Harbourage. \n18. Other problems of the honey bee\n\nPESTICIDE AND FUNGICIDE POISONING\n\nA common cause of death or poor health in a colony is the ingestion of pesticides or fungicides by bees. Spraying flowers or orchards with pesticides or fungicides, both commercially and domestically, can lead to the death of a colony that collects nectar or pollen from that area. Typically a large ingestion of poison by the bees in a colony would be shown by many bees lying dead or dying outside of the hive. Although this is a worst-case scenario, it is a relatively common occurrence, particularly with large-scale crop spraying. Landowners or contractors who do not inform local beekeepers about planned spraying operations may be held responsible for the economic loss of the beekeeper's colonies.\n\n18.1: Bees killed by pesticides outside a hive.\n\n18.2: Bees showing the characteristic extended tongue or proboscis common with pesticide poisoning.\n\nA more common and perhaps more insidious scenario is the low-level absorption of pesticides or fungicides by bees at levels below those that are immediately obvious and evidenced by large-scale die off of bees. Sub-lethal poisoning \u2014 that is, a dose many orders of magnitude below a lethal dose \u2014 can result in poor brood laying by the queen, as well as insufficient collection of nectar or pollen by foragers. Also, as even low-level chemical poisoning weakens the immune system of bees this can result in a compromised immune system and susceptibility to higher levels of disease, for example, _Nosema_. A further demonstrated effect of sub-lethal doses of some pesticides is found in returning foragers unable to find their way back to the colony, with up to one-third failing to return to the hive for this reason.\n\nChemical pesticides may be applied topically to plants as a surface spray and in this situation bees absorb the chemical when they either crawl over the surface or collect nectar or pollen from affected plants. Other types of chemical insect control readily available are the systemic pesticides which, when applied, are absorbed by the plant into its cellular system and tissues. This can occur through the ground, by topical spraying or by direct incorporation into seeds. The plant, its pollen nectar and any excreted water droplets or resins carry the pesticide and unsuspecting bees forage and return to the hive with the chemical.\n\nIn the last ten or so years another class of insecticide chemicals has been developed and advanced: the neonicotinoids. These are systemic pesticides and were further developed as they were shown to have low levels of toxicity in mammals. Neonicotinoids are a class of neuro-active insecticides chemically similar and modelled on the natural insecticide nicotine. They work by blocking a specific neural pathway that is more abundant in insects than in mammals, resulting in selective or targeted toxicity of insects. Neonicotinoids work on the insect's central nervous system causing overstimulation, eventual paralysis and then death.\n\nOne of the available branded neonicotinoids, _Imidacloprid,_ has become the most widely used insecticide in the world and there is a concern that neonicotinoids may have some connection with the health problems honey bees are experiencing in the United States and parts of Europe. There is certainly fear among beekeepers that, at the very least, the use of this class of chemical may result in a culmination of many and varied problems for the honey bee.\n\nIn January 2013, the European Food Safety Authority stated that neonicotinoids pose an unacceptably high risk to bees and questioned the previous science relied upon by regulatory agencies to claim the safety of these pesticides. In response to this and the report of the Food Safety Authority in May 2013 the European Commission adopted a proposal to restrict the use of three pesticides belonging to the neonicotinoid family for a period of two years. The proposal restricts the use of these for seed treatment, soil application and also the foliar treatment of bee-attractant plants and cereal crops.\n\nThe Commission has stated that as new information becomes available it will review the conditions of approval of the three neonicotinoids and take into account any relevant scientific and technical developments. It is interesting to note that exceptions to the possibility of treating bee-attractant crops in closed greenhouses and open-air fields after flowering were contained in the proposal.\n\nAs with any controversy, it may be many years before there is sufficient data on the real effects of this class of pesticide on honey bee health and populations. Any low-level infection or agent can, when added to other environment stresses, be the tipping point to the compromised health and loss of viability of a colony and the fact that these insecticides were developed to target insects, of which the honey bee is one, will continue to cause beekeepers warranted concern.\n\nIn the United States large numbers of hives are routinely shipped around the country for pollination purposes, the bees spending long periods foraging in various monocultural crop situations with feeding supplemented by high fructose corn syrup. Given the ever-increasing number of diseases treated by numerous chemicals, plus the various mites including Tracheal Mite and _Varroa destructor,_ and pests like the Small Hive Beetle, plus the accumulating chemical build-up within hives, it is not too big a jump to understand concerns that the added burden of a neonicotinoid has pushed the bee to the point where its immune and other systems are severely compromised.\n\nFor the beekeeper, the best safeguard against accidental exposure of bees to lethal pesticides and fungicides is to develop an awareness of whether and where these chemicals are being used near hives. A diplomatic approach to a nearby user, who may be unaware of the affect the chemicals sprayed are having on nearby hives, may result if nothing else in their use during more bee-friendly times of the day or at least give the beekeeper an opportunity to either close or move hives during the period of spray drift.\n\nCHILLED BROOD\n\nA healthy colony containing a large number of workers is able to keep its brood at 34o Celsius to 35o Celsius. This is the optimum temperature for raising brood and this temperature needs to be well regulated if the colony is to remain healthy and to produce viable brood. If, for some reason, temperature regulation fails, the developing brood may become chilled to the point where development stops and death occurs.\n\nThere are several causes of chilled brood including:\n\n\u2022 smaller hives or nucleus that have insufficient numbers of workers to keep the colony warm\n\n\u2022 hives that are rapidly expanding during spring that contain large numbers of brood but do not yet have sufficient numbers of adult workers to keep the brood warm\n\n\u2022 hives that have lost large numbers of adult bees due to disease or pesticide poisoning\n\n\u2022 beekeepers opening hives in cold weather and allowing warm air to escape.\n\nThe best way to minimise chilled brood is to keep strong colonies and not to open hives in cold weather. Another technique is to reduce the size of the entrance during winter months to minimise the amount of cold air flowing through the colony. Fortunately in Australia there are few areas where it is sufficiently cold for long enough to cause real concerns about chilled brood.\n\nOVERHEATING OF THE HIVE\n\nA more common problem in Australia are hives overheating due to hot weather or to fire. Daytime temperatures in many inland parts of the country reach over 40 \u00b0Celsius during summer and this can prove detrimental, stressful and at times fatal, particularly if there is insufficient water nearby for the bees to cool the colony. To minimise the effect of heat, the following practices have proved to be effective.\n\n\u2022 Where possible, locate hives so that they are in shade during the hottest part of the day usually between 10 a.m. and 6 p.m.\n\n\u2022 Keep the entrance to the hive as open as possible.\n\n\u2022 Do not open the hive when the air temperature is above 33\u00b0 Celsius.\n\n\u2022 Ensure air vents at the hive lid are adequate and are unblocked to enable the fanning bees to keep cool air flowing through the hive.\n\n\u2022 Try using seven frames in an eight-frame super or brood box or nine frames in a ten-frame box, to allow for greater air circulation around the frames.\n\n\u2022 If possible, keep hives near larger bodies of water as this will both cool the air as well as provide drinking water for the colony. For the hobbyist with a few hives there are additional possibilities. The top of hives can be watered before it becomes too hot, taking care not to drown any bees. This and watering under and around the boxes frequently during extreme weather conditions will assist the bees' own efforts inside the hive where collected water is fanned with numerous wings in order to produce a cooling air-conditioned effect. Other hobbyist beekeepers place old beach umbrellas over their hives and water the top of these in order to offer some relief to bees in the hives below.\n\n\u2022 When transporting hives, make sure that there is a good airflow through the hive and that the entrance is not blocked off. Perforated hive-entrance closers can be used to prevent bees escaping while allowing air to pass in and out of the hive. The use of a screened travelling top is recommended as, again, it allows air to circulate and provides the bees with fresh air.\n\n\u2022 Paint hives with white or other light-coloured paint as darker colours attract and hold heat.\n\n\u2022 Provide insulating shade covers on top of the normal hive lid, particularly if this is metal.\n\nSince fire is a major concern to beekeepers in many areas, where possible keep hives away from long grass or vegetation that can burn. Reduce vegetation to a minimum around hives particularly during hot and dry periods.\n\nBees are affected by the heat and they can become very aggressive and difficult for the beekeeper to manage without good protective clothing and copious amounts of smoke. Obviously it is not a good idea to work hives in hot weather as this destroys the interior temperature the bees have fought so hard to maintain and will take a considerable time for them to re-regulate. If the beekeeper is aware a hot day is forecast and it is necessary to open the hive this should be done in the early morning when it is cooler or, if absolutely necessary, in the late evening when of course most of the bees will be inside and ready to object strongly to the interference. The best option is to leave hives alone until the temperature drops, hopefully after a few days. The beekeeper should always be aware that a total fire ban may be in place on very hot days and inspections will need to be done without the benefit of a smoker. Some beekeeping supply stores sell a liquid smoke that can be mixed with water and sprayed over the bees using a spray bottle. Although not as effective as smoke from a smoker, a spray or two of liquid smoke will help to calm the bees if the hive needs to be inspected on total fire ban days.\n\nDAMP HIVES\n\nThe classic sign of a damp hive is condensation on the inside of the lid. The condensation will pool into droplets which will drop down onto the bees in the hive. Dampness in hives contributes to or causes a lot of health problems in bees, including _Nosema_ and Sacbrood.\n\nIf dampness in hives is a problem try doing the following:\n\n\u2022 Add ventilation at the top of the hive either by providing air vents in the lid or by drilling 2.5-centimetre holes at the top of the uppermost super and covering with vents. Another technique is to place lollipop sticks, twigs or small stones under the lid rim, raising the lid by about 3 millimetres, thus providing greater airflow throughout the hive. A gap any larger may open the hive to predators or robbers. See also Chapter Seven on the procedure for the first inspection of spring.\n\n\u2022 Make sure that the base of the hive slopes forward so that any condensation flows out through the entrance and does not pool at the back of the base.\n\n\u2022 Keep the hive away from damp ground and preferably raise it off the ground about 30 centimetres.\n\n\u2022 Place the hive in a sunny but not a hot location.\n\nDampness mainly occurs during the winter when the bees respirate in order to keep the colony warm. Water is exhaled during respiration that then collects inside the hive in the form of damp lids, walls and bases. Adding additional ventilation during cold periods to allow the moisture to escape will be unhelpful, as this will cause cool air to enter the hive and the bees to respirate more intensely in order to to keep the colony warm.\n\nCANE TOADS\n\nThe cane toad, _Rhinella marina_ (formerly _Bufo marinus),_ an introduced species, can now be found widely over northern Australia, including New South Wales, Queensland and the eastern parts of the Northern Territory. Cane toads generally feed at night, spending the day hiding under logs, under hives or in small holes. Cane toads have become a major pest to the beekeeper as they sit outside a hive at night and devour many workers from around the hive entrance.\n\n18.3: A cane toad waiting at a hive entrance for exiting and returning bees.\n\n18.4: Raising hives above the ground about 60 centimetres will deter cane toads from attacking the foragers.\n\nAdult cane toads are usually heavily built and weigh an average of up to 1.8 kilograms. Their size may vary between 15 to 23 centimetres and their skin is warty. The colour on their back and sides varies from olive-brown or reddish-brown, grey, and yellow while their bellies are semi-yellow or semi-white with darker mottling. Their body is round and flat with prominent corneal crests and light mid-dorsal stripes. While their front feet are unwebbed, their back feet have tough, leathery webbing.\n\nThe easiest way for the beekeeper to minimise losses from cane toads is to raise the base of the hive about 60 centimetres above the ground. A variety of stands can be used for this purpose. Fixed stands are good if there is no plan to move the hive, but for the migratory beekeeper stands that can be easily assembled or taken down are best.\n\nANTS\n\nThere are many types of ant in Australia and they can be a significant pest by robbing honey, pollen, brood and even adult bees from the hive. Colonies that have been or are being robbed by ants may become aggressive and those that have been seriously and repeatedly attacked will become depleted and stressed with a resultant reduction in honey production.\n\n18.5: Coastal brown ants.\n\n18.6: Meat ants.\n\n18.7: Sugar ants.\n\n18.8: Green tree ants.\n\nAnts often become a problem for the beekeeper when sugar syrup is being fed to colonies and in this situation may invade the hive or syrup feeder in large numbers.\n\nThere are two small ants that cause problems to beekeepers in the northern parts of Australia: the small black ant and the coastal brown ant.\n\nThere are several larger ants that may also be a problem for the beekeeper and these include meat ants, sugar ants and green tree ants.\n\nAnts' nests can be destroyed by the application of chemical insecticides either in powder form or as a solution. Green ants, which make their nests high in trees, can be destroyed by applying insecticide around the base of the tree or by injecting insecticide directly into the nest. There are many insecticides approved for this purpose and the beekeeper should contact a local agricultural or beekeeping supplies store. Alternatively the nearest state apiary inspector will be able to provide the name of insecticides that have been approved for use in the local region.\n\nA common physical barrier for ants is to modify the cane toad stand mentioned on p. 103 and 293. The legs of the stand can be made impassable for ants by tying a cloth around them covered in sticky oil or grease. Insecticide applied carefully so as not to kill the bees may also be used but the insecticide needs to be reapplied more frequently than oil and may pose a risk to the hive of bees housed on the stand. Ants will be able to enter the hive even if it is fully protected by a physical barrier if long grass or other vegetation is allowed to touch the hive or parts of the stand above the physical barrier as they will use these as a bridge to gain access to the hive.\n\nMICE\n\nGenerally mice do not appear to be a large problem for the beekeeper in most parts of Australia. Bee equipment suppliers often sell mouse guards which attach to the front entrance of the hive, and from my many years of experience in my own beekeeping business, I find that customers in Victoria generally purchase entrance guards as a precautionary measure rather than to deal with specific mice problems. There is no doubt that the average beehive is an attractive proposition for a mouse looking for warm winter quarters and it is true that in some parts of the country mice may be a problem towards the end of autumn or during the winter. A mouse will eat comb and generally make a mess in a hive, dragging in straw and other nest material. Often the colony seems to ignore a live mouse in the hive even though the bees are more than capable of killing it. In instances where the colony does attack and sting to death an intruding mouse they are unable to remove it from the hive due to its large size. Not to be outdone the bees cover the dead mouse with antiseptic propolis to entomb the carcass and thereby prevent the spread of disease into the colony from the decaying body.\n\n18.9: Mouse guard to place over the hive entrance.\n\n18.10: A combined mouse guard and entrance closer with attaching brackets.\n\n18.11: Base with an internal mouse guard protecting the entrance.\n\nAs previously mentioned, specialised mice guards are available from some bee equipment suppliers to cover the entrance to hives. These are usually made of a galvanised metal and have a number of small holes to allow entry and exit from the hive by the bees but are too narrow to allow a mouse entry. If the beekeeper has or foresees that there may be a problem with mice entering hives and wishes to do something to narrow the hive entrance it is worth noting that a mouse is able to squeeze through a hole about the size of a pen, that is, approximately 6 millimetres in diameter to get at food or to make a home. Mice are able to do this because the bones in their head are relatively soft and additionally they have a slender, flexible body that can easily fit through a very small entrance. If mice are a problem, consider putting a mouse trap either outside or under the hive, suitably protected from the elements and from domestic and other wildlife. This may go part of the way to fixing any problems that may be caused to the colony by this inquisitive rodent.\n\nCOLONY COLLAPSE DISORDER (CCD)\n\nBefore discussing Colony Collapse Disorder it is important to note that a collapsed colony and Colony Collapse Disorder (CCD) are not the same thing. Colony Collapse Disorder is a very general disorder found in the United States, the causes of which are poorly understood, and it was regarded as being a major problem to American beekeepers. A collapsed colony, however, is more specific and can be caused by a variety of often-identifiable ailments affecting the bees in a particular colony.\n\nOnce Colony Collapse Disorder was first identified in the United States in November 2006, there was a concerted effort to identify a single cause for this large yearly loss of colonies, sometimes reaching as high as 30 per cent of all American hives lost during the winter months. The US Department of Agriculture (USDA), however, now acknowledges that CCD is not caused by a single pathogen or cause but by a range of issues some of which are man-made. These include:\n\n\u2022 Infestation by _Varroa,_ the most harmful affliction of the honey bee, is closely associated with high winter losses of colonies.\n\n\u2022 Multiple virus species have now been found in both bees and _Varroa_ mites, including Deformed Wing Virus (DWV), Israeli Acute Paralysis Virus (IAPV) and Black Queen Cell Virus (BQCV), and are believed to be significant factors in poor colony health.\n\n\u2022 _Varroa_ is known to increase the levels of viruses in bees, as it acts like a mosquito injecting viruses into the brood during feeding.\n\n\u2022 The bacterial disease European Foul Brood (EFB) is being detected more frequently in colonies and may be linked to colony loss.\n\n\u2022 Nutrition has a major impact on colony health and longevity. The increased use of monocultures, where a single crop is grown over very large areas and bees transported to the crop for the purpose of pollination \u2014 for example, almond pollination \u2014 is proving harmful to bees due to the lack of a balanced diet and the absence of any supplementary forage within the bees' flight range.\n\n\u2022 Gut microbes used by the honey bee to break down pollen, fight some gut diseases, digest food, transport nutrients from the gut into the body and for detoxifying chemicals are being harmed by the increasing use of pesticides, miticides and fungicides which the bees unintentionally digest.\n\n\u2022 Transporting hives sometimes many thousands of kilometres each season for pollination has been shown to be detrimental to bees' health.\n\n\u2022 High fructose corn syrup (HFCS), which is used widely in the United States as a food supplement, has been shown to be not as effective as honey in developing and keeping healthy the microbes within the gut of the bee.\n\n\u2022 The use by local councils and domestic home owners in the United States of herbicides to eliminate 'unsightly' broad leafed weeds on grass lands, parks and lawns is depriving bees of both valuable food and further adding toxins to their bodies.\n\n\u2022 Chemicals used within the hive to manage _Varroa,_ Small Hive Beetles and Tracheal Mite are proving more of a problem to bees than a solution to pest control.\n\n\u2022 Acute and sub-lethal effects of pesticides on bees have been increasingly documented and are a major concern to beekeepers. In some situations, two apparently harmless chemicals that are applied together prove to be significantly more harmful to bees than either chemical applied singly. These effects, together with the long-term cumulative build-up of apparently harmless chemicals, are not well understood and need to be further researched before we are able to discuss with confidence any harm they are or may cause to bees and other living things.\n\n\u2022 Poor management practices by the beekeeper.\n\nAlthough the above list represents what scientists today believe are the major contributing factors for a colony's collapse to be classified as being due to CCD, not all of the above need to be present for there to be irreparable harm to an individual colony and for the collapse to be classified as being due to CCD. Many of the contributing factors listed above come from issues that are unrelated to each other. As a result there will not be a single silver bullet that will eliminate CCD. The solutions to the problem will need to come from diverse areas and will involve many groups of people from industry, beekeepers, government and academic areas working together, and who could even today spend more time talking with each other in an effort to minimise the causes of large-scale bee deaths.\n\nSUMMARY\n\nThere are a number of general issues that may adversely affect your colony, although most can usually be easily dealt with by the beekeeper. Pesticide and fungicide poisoning is probably the issue that the beekeeper has the greatest difficulty managing. This is often because the beekeeper is unaware that chemicals are being used in the vicinity of hives and the people who apply them may not be aware of or give proper consideration to honey bee colonies in their area. Apart from pests and disease the more common general problems experienced by Australian beekeepers are:\n\n\u2022 pesticide and fungicide poisoning\n\n\u2022 chilled brood\n\n\u2022 overheating of the colony\n\n\u2022 damp hives\n\n\u2022 cane toads\n\n\u2022 ants\n\n\u2022 mice. \n19. Other types of bees and wasps\n\n19.1: The Asian honey bee (Apis cerana) workers on brood comb.\n\n19.2: Apis cerana worker.\n\n19.3: Apis cerana bees on a frame of brood comb in Vietnam.\n\n19.4: Hives of Apis cerana in Vietnam where the bee is commonly kept domestically for honey production.\n\nSo far I have discussed the European honey bee, _Apis mellifera,_ as well as some of the most important species of native bees found in Australia. There are also established populations of the invasive Asian honey bee _(Apis cerana),_ the Buff-tailed bumblebees _(Bombus terrestris),_ the solitary African Carder bee ( _Afranthidium_ ( _Immanthidium_ ) _repetitum_ ) and two species of wasp. Wasps are often mistaken for honey bees, but differ fundamentally in their appearance, behaviour and nesting biology. Also, globally there are other species of honey bee, as well as hornets that, if they were to enter Australia, could pose a significant biosecurity risk.\n\nTHE ASIAN HONEY BEE _(APIS CERANA)_\n\nGlobally there are twelve currently recognised species of honey bee. One of these, the Asian honey bee, _Apis cerana_ , is the most widely distributed honey bee in Asia, found natively from northern Russia and Japan, to Pakistan and Indonesia and everywhere in between. It is extensively used in Asian countries to produce honey, wax and also brood comb, which is eaten as a source of food. _Apis cerana_ has many common names in English including the Asian honey bee, Asiatic bee, Asian hive bee, Indian honey bee, Indian bee, Chinese bee, Eastern honey bee, Mee bee and fly bee.\n\nAcross Asia, _Apis cerana_ can be found at higher altitudes, at sea level, in tropical, subtropical and temperate regions, in deserts and in areas where there is high seasonal rainfall. The Asian honey bee has been extensively spread by human activity, most recently by missionaries to Papua New Guinea and the Solomon Islands in the 1970s, and the Cairns region of Australia in 2007. As a result of this widespread distribution, different races of the bee have very different temperaments and appearance, leading to the eight currently recognised subspecies of _Apis cerana._\n\n19.5: Hives of Apis mellifera also in Vietnam. Notice the white sacks kept on top of the hive that help keep the colony cool.\n\nAlthough _Apis cerana_ has its home across Asia, the particular race of _cerana_ found in Cairns in Queensland is _Apis cerana javana._ This race of _cerana_ is also found in the areas of Java, New Guinea and the Solomon Islands. _Apis cerana_ moved into the Solomon Islands in 2003 and quickly destroyed the limited local European honey bee industry. In May 2007, _Apis cerana_ was discovered in Cairns. The initial colony may have entered Australia hidden in the mast of a boat that had recently arrived from Papua New Guinea. By the time apiary inspectors became aware of the incursion the colony had swarmed and multiple colonies of the bee were located living in and around Cairns. Initial concern was due to the possibility that the bee would destroy the Australian beekeeping industry and a concerted effort to eradicate the bee was made by both Queensland's Department of Agriculture, Fisheries and Food (DAFF) and by Australian beekeepers. Despite these efforts _Apis cerana_ was declared endemic to Australia in 2011 when it was decided to be too difficult to eradicate. No more resources are currently being allocated to the eradication of _Apis cerana_ or efforts made to contain it within the Cairns region. At this point in time initial concerns about _Apis cerana_ have not been realised as the Asian honey bee found in Cairns does not appear to be significantly affecting local European honey bee colonies apart from the fact that they are both competing for the same floral resources.\n\nTo date, most honey bee research globally has been conducted on the European honey bee subspecies of _Apis mellifera,_ and much less research has been conducted on the other eleven species of honey bee. _Apis cerana_ is the best studied of the Asian honey bee species but there still remains a lot to be learnt about this species. This is particularly true for the race of _Apis cerana javana_ found in Cairns. The probable reason _Apis cerana javana_ has not been well studied is that it does not produce and store much surplus honey and is of very little economic importance in the areas where it can be found. As a result, the long-term impact of this race of bee in Australia is not fully understood. The effect of its increasing numbers on the European honey bee is only a best guess from limited knowledge of other races of _cerana_ in other parts of Asia, although a recent study conducted by the University of Sydney may give some indication of the future.\n\nThe study showed that in the Cairns area, about 33 per cent of European honey bee queens had mated with at least one _Apis cerana_ drone. A part of the study also compared European and _Apis cerana_ bees in China and found that 14 per cent of European honey bee queens had mated with at least one _Apis cerana_ drone. Conversely there were no cases found of an _Apis cerana_ queen mating with a European honey bee drone, possibly due to the large size of a European honey bee drone and the smaller size of an _Apis cerana_ queen. Although this study showed that _Apis cerana_ drones can and do mate with European honey bee queens, the resulting eggs were infertile and did not survive to adulthood. The effects of interspecies mating will therefore reduce the number of fertile eggs that a European honey bee queen will lay and must, as a result, produce weaker honey bee colonies.\n\nMorphologically, both _A. cerana_ and the European honey bee are very similar in appearance and only an experienced observer is able to tell the difference between bees of the two species. The main differences are outlined in the following table.\n\nApproximately 4 million years ago a common ancestor of the Western and Asian honey bees made the Near East and Africa their home and, due to the resulting geographical isolation between the bees in Asia and the Middle East and Africa, they diverged genetically and grew sufficiently apart to become two species.\n\nThe lifecycle of _A. cerana_ is almost identical to the European honey bee. The table below compares brood development times in days for _Apis cerana_ and the European honey bee, _Apis mellifera._\n\n_Apis cerana_ capped drone cells have a small pin-sized hole in them and European honey bee drone cells do not.\n\nLike the European honey bee, _Apis cerana_ make their nests in closed cavities like tree hollows, caves or between rocks or in the walls of houses. The tropical races of both the Western honey bee and _Apis cerana_ make less honey than non-tropical races and are also more prone to swarming than their sub-tropical cousins.\n\n_Apis cerana_ have far better grooming behaviour than European honey bees, making them better at brushing off _Varroa_ and other parasitic mites. Also, with _Apis cerana_ (their natural hosts), _Varroa_ infest drone cells, and therefore do not negatively affect the worker population of the colony. In contrast, in _Apis mellifera_ colonies _Varroa_ infest worker cells (due to these cells being of a similar size to _Apis cerana_ drone cells), and negatively affect the worker population.\n\nAcross Asia, _Apis cerana_ is kept in hives by local beekeepers to produce honey both to consume and to sell to augment income. _Apis cerana_ have been domesticated for honey production for many thousands of years and are often preferred over the European honey bee for domestication by poorer native people. This is because the set-up and on-going management costs for European honey bees are significantly higher than for _Apis cerana._ If a European honey bee colony were to die or abscond, replacing the colony would be much more expensive than for an _Apis cerana_ colony. A replacement _Apis cerana_ colony would be straightforward and almost free to obtain as the beekeeper need often only pay local village children a very small amount of money to find a nearby replacement feral colony.\n\nBUMBLEBEES\n\nBumblebees are social insects, like honey bees, with many species nesting in small holes made by rodents and other creatures in the ground. In Australia, one species of bumblebee, _Bombus terrestris,_ is found, and only in Tasmania. Bumblebees are easily distinguished from honey bees. They are much larger, are more round, have black and yellow bands on their abdomen and have more hair. Bumblebees make a loud buzzing sound when they fly, hence the name bumble, meaning a humming or droning sound, or a buzz. The buff-tailed bumblebee, _Bombus terrestris,_ was most likely introduced by accident into Tasmania and was found there in 1992 where it has been successfully living ever since.\n\n19.6: Worker of the bumblebee (B. terrestris) with the distinctive white patch at the rear.\n\n19.7: Commercial colonies of bumblebees being used to pollinate tomatoes inside a greenhouse in New Zealand.\n\nThere are about 250 species of bumblebees in the world. The majority of species are found in temperate climates in the northern hemisphere, with the centre of bumblebee diversity being Central Asia where most species fly between March and October. In the Southern Hemisphere, in Tasmania and in New Zealand where they are also found, bumblebees fly between September and June.\n\nBumblebees were deliberately introduced into New Zealand in 1885 to pollinate red clover. Up until that time red clover seeds were imported from England every year at a significant cost to the farmer. Although _Bombus terrestris_ is the dominant bumblebee in New Zealand and can be found all over the country, there are three other species found that are less well distributed. These are the large garden bumblebee ( _Bombus ruderatus_ ), the small garden bumblebee _(Bombus hortorum),_ and the short-haired bumblebee ( _Bombus subterraneus_ ),\n\nShort-haired bumblebees from New Zealand were being considered as a population source from which the species could be re-introduced into the United Kingdom after they died out there, but the bees were found to be too inbred to be suitable, and a Swedish population of the species was chosen instead.\n\nBumblebees, like honey bees, rely on collecting nectar and pollen from a variety of floral types to remain healthy and survive. Nectar and pollen are used as sources of carbohydrates and protein to raise brood, while nectar alone is used to provide energy to adult bees for flying or keeping the colony warm.\n\nBumblebees do not produce honey, but instead store small amounts of nectar. Whereas honey bee colonies produce honey to provide food during the colder winter months, bumblebee colonies die off annually with only newly mated queens overwintering to see the following summer. Due to this fundamental difference in lifecycle, they do not need to produce honey stores. A colony of _Bombus terrestris_ typically only contains about 400 workers during the peak nectar flow season in the spring, while a colony of honey bees contains about 50,000 workers.\n\nBumblebees live in habitats where there are flowering plants such as grassland, woodland edges, roadside verges, gardens and other undisturbed grass areas.\n\nOne of the noticeable points about _Bombus terrestris_ is the white patch at the bottom of the worker's abdomen. This is absent in the queen, who instead has a yellow band at the end of the abdomen.\n\nAgain, like the honey bee, there are three castes of bumblebee \u2013 queens, workers and drones. Typical characteristics of the castes of _Bombus terrestris are_ as follows.\n\n_Queen_\n\n\u2022 A queen is about 22 to 24 millimetres long.\n\n\u2022 She possesses a sting that she will use in defence if provoked.\n\n\u2022 The queen can sting multiple times, and does not lose her sting like a honey bee does, and therefore will not die.\n\n\u2022 Unlike honey bees, the bumblebee queen will construct comb for her nest, will forage for nectar and pollen during the early stages of nest construction and during times of extreme resource limitation, and will incubate her young.\n\n\u2022 The queen has a bald patch on the bottom of her abdomen so that she can efficiently transfer her body heat to brood cells.\n\n\u2022 A bumblebee queen will live for one year from autumn to autumn and the old queen will always die before the onset of winter.\n\n_Workers_\n\n\u2022 Workers are about 11 to 17 millimetres long.\n\n\u2022 They are able to sting multiple times as they do not lose their stinger during an attack. Bumblebees are usually less aggressive than honey bees.\n\n\u2022 The antennae of the worker are shorter than those of the drone and can be used to distinguish between the two castes. A worker has twelve segments on her antennae while a drone has thirteen segments.\n\n\u2022 The gender of a bumblebee can also be determined by counting the number of segments on its back: six segments for a worker and seven segments for a drone.\n\n\u2022 A worker lives for about eight to twelve weeks.\n\n_Drones_\n\n\u2022 Drones are about 14 to 16 millimetres long.\n\n\u2022 They do not possess a stinger.\n\n\u2022 The male abdomen is narrower and blunter at the end than that of the worker abdomen.\n\n\u2022 There are many more drones during the bumblebee mating season in late summer than at other times of the year.\n\n\u2022 Unlike honey bee drones, bumblebee drones do not die during the act of mating, and can therefore mate multiple times if given the opportunity.\n\n\u2022 Bumblebee drones forage on flowers just like male solitary bees, and unlike honey bee drones which must be fed in the hive by the female workers.\n\nBumblebee colonies last for one year and only the new queens survive past autumn. The new queens will have mated in the late summer before going into hibernation for the winter. The new queen will overwinter alone and will wake during a warm day in early spring to emerge from her resting place to search for a location for a new permanent nesting place for her soon to be established colony. The new nest is usually made of moss, grass and leaves in a hole in the ground, often hidden by long grass. In the United Kingdom, some _Bombus terrestris_ nests have been found in holes 2 metres deep. A favourite nesting place for _Bombus terrestris_ is under garden sheds.\n\nOnce the queen has selected the new nest site she will start building the nest by making a single thimble-shaped beeswax cell called a honey pot. The queen uses the honey pot to store honey that she has collected herself from flowers during the early spring. If a queen is seen flying with pollen attached to her hind legs then she has already started to build a nest, since this activity occurs only after she has selected a site for the new nest.\n\nThe queen stores pollen in a ball and lays about six eggs at a time in a clump in a depression she has built in the pollen. She then seals the pollen ball with wax and incubates it at between 30\u00b0 Celsius and 32\u00b0 Celsius to keep it warm. The queen feeds on honey from the honey pot, which is positioned within her reach so that she can continue to incubate the eggs without needing to move to obtain food.\n\nThe eggs hatch and the larvae feed on their bed of pollen. As they grow, the queen opens the wax covering and adds more pollen and nectar. She rests on the brood clump, incubating the larvae to speed their development. The larvae mature and each spins a cocoon of silk in which it pupates and completes development into an adult. After the new workers emerge their empty cocoons are used as storage pots for honey or pollen.\n\nSimilar to the European honey bee, the queen lays two kinds of egg: the fertilised worker egg and the unfertilised drone egg. The female worker eggs are fertilised with the sperm that she has stored from her mating flight the previous summer. Unlike the European honey bee, which mates with between ten and twenty drones, buff-tailed bumblebee queens are monandrous, mating with only a single drone.\n\nTypical development times for brood are:\n\n\u2022 eggs hatch to larvae after five days\n\n\u2022 larvae pupate after a further twelve days\n\n\u2022 pupae metamorphise into adult bumblebees after a further fourteen days.\n\nThe development time of a typical _Bombus terrestris_ bumblebee is about 30 days, depending on the temperature of the brood: a shorter period if the temperature is warmer, and longer if the temperature is cooler. The species of bumblebee will also affect the duration of the brood development cycle.\n\nThe first workers to emerge will remain in the nest for about a week before becoming foragers gathering nectar and pollen outside. Once the first set of brood has developed into foragers they take over from the queen the roles of collecting food and feeding the brood. At this time the queen stops venturing outside of the nest and devotes herself to laying and incubating eggs; these she lays in batches, not continuously like a honey bee queen.\n\nOnce nectar and pollen become readily available in the spring the _Bombus terrestris_ colony grows rapidly to about 400 adult workers. Depending on the supply of food, this size will remain stable until late summer when the old queen will lay drone eggs as well as raise new queens.\n\nThe male drones will leave the nest to find a queen and will not return to the colony. Drones sleep outside at night on flowers or in other sheltered places and spend their waking hours patrolling set routes in the hope of coming across a virgin queen. A new virgin queen will leave the nest in search of a mate. The exact sequence of events leading to mating is still unknown in _Bombus terrestris,_ but it is thought that queens are attracted to pheromone marks left by male bumblebees at regular intervals along their patrolled route.\n\nBumblebees are different to honey bees in that during the winter the entire colony dies except for the new queen who goes alone into hibernation. As noted previously, once spring comes the new queen will emerge on a warm day and look for a suitable site at which to build her new nest.\n\nBumblebees are excellent pollinators and there is a growing demand from some agriculturalists to import these creatures into mainland Australia for the greenhouse pollination of tomatoes. This move, however, is being resisted by others who believe that native solitary bees can perform this role as well, and that to introduce bumblebees into mainland Australia is importing another invasive species. There is a view held by some that bumblebees may preferentially pollinate non-native plant species, facilitating their spread. Furthermore, along with feral honey bees, bumblebees could impact native pollinators by competing for resources and affect cavity-nesting native animals by competing with them for nest hollows.\n\nThe bumblebee is able to forage from dawn until dusk and can fly up to 5 kilometres in search of food, pollinating large areas of plants. The temperature at which a bumblebee can fly is much lower than for a honey bee, as low as 5\u00b0 Celsius. For this reason bumblebees are more effective at pollinating plants in cold or wet regions where honey bees would experience difficulty remaining outdoors.\n\nMany species of bumblebees have longer tongues than honey bees and they can often collect nectar from more inaccessible places in the flower. Also, because bumblebees buzz (or shake) pollinate, that is they grasp the male stamen with their mouth pieces and shake it so that more pollen falls off, they are able to collect pollen from many plant species which honey bees are unable to pollinate.\n\nAs an example of buzz pollination, the male stamen of tomatoes only requires a small amount of vibration for pollen to fall off onto the female stigma of the flower. In nature this is performed by the wind and by insects, but in greenhouses where there is no wind and very few insects, bumblebees can be introduced to efficiently perform this role.\n\nBumblebees are good at pollinating many plants that are often grown in greenhouses, such as tomatoes, cucumbers, zucchinis, capsicum, strawberries, melon, avocados, and kiwi fruit. They readily adapt to confinement in large greenhouses, which honey bees do not, and their effectiveness as a pollinator makes them a useful ally to the horticulturist.\n\nWASPS\n\nBoth the European wasp, _Vespula germanica,_ and the Common or English wasp, _Vespula vulgaris,_ are found in mainland Australia and Tasmania. The two types of wasp are almost identical to look at and the two points of difference between them almost impossible to detect unless the wasps are stationary. Firstly the patterns on the face differ and secondly the European wasp also has two black spots on the top-most yellow band of the abdomen.\n\nAs the European wasp is the most prevalent of the two types most of this section will discuss this particular species. In practical terms the European wasp and the English wasp may be regarded as the same and the comments in this section about the European wasp will equally apply to the English wasp. These similar characteristics include lifecycle, nesting, venom, most body markings and aggressiveness.\n\nThe European wasp was first found in Tasmania in 1959 having arrived from New Zealand where it had established itself in 1945. By 1978 the European wasp had travelled to mainland Australia and was first found in Melbourne in Victoria.\n\nAt present, distribution of the European wasp appears to be restricted to the cool and wet climates of coastal southern Australia and it occurs throughout most of Victoria and Tasmania. In country New South Wales, nests have been located at Coonabarabran while several nests have been recorded in south-east Queensland. In South Australia, the European wasp is well established throughout the hills surrounding Adelaide and within Adelaide itself. In Western Australia, the European wasp has been recorded from Perth to Albany.\n\nAbout a year before the European wasp reached Tasmania, the English wasp, _Vespula vulgaris,_ was found in the eastern suburbs of Melbourne. The English wasp has not been as successful as the European wasp in acclimatising to Australia and has limited spread in the eastern parts of Melbourne and in Gippsland.\n\nStudies in Europe have shown that European wasps are similar to honey bees in that they are found in large communal nests headed by a queen and surrounded by many thousands of female workers. The size of the European worker wasp is between 1.2 centimetres and 1.6 centimetres long. Male drones are laid later in the season and only represent a small fraction of the wasps in the colony.\n\nThe entrance hole to a wasps' nest in the ground is typically about 25 to 30 millimetres wide and leads to a tube or tunnel about 1 metre long. The nests are usually built underground and the wasps are able to excavate dirt to enlarge the nest cavity \u2014 typically about the size of a football.\n\nLike the honey bee, wasps make cells to raise brood and to store food although cells are made of chewed wood fibres not wax. The colour of the wood fibre comb is greyish, reflecting the colour of the masticated wood fibres.\n\nEuropean wasps are similar to bumblebees in the construction of new colonies, although quite different to the honey bee. Wasp nests are started anew every spring from a single queen that has hibernated alone over the winter months. On a warm day in spring the queen wasp will emerge from her dormant state and will leave her winter's hiding place in search of nectar, honeydew or some other carbohydrate. Honeydew is the sugary secretion of aphids and is collected by wasps, native social bees, and honey bees as a form of food. Once the queen has fed she will look for a permanent home in which to build a nest and raise her new colony. Once the new nest site has been selected, the queen will build inside it about 30 brood cells and lay eggs in them. The eggs will turn into grub-like larvae and will develop into adult wasps. These wasps will then take over the role of collecting food, collecting wood fibres to build the comb, and raising the brood, leaving the queen with the sole responsibility of laying eggs. Wasps can often be seen scraping wood fence posts and chewing the wood fibres into a moist paste ready to be used to construct new cells in the nest.\n\n19.8: The European wasp, _V. germanica,_ in Tasmania. Notice the one large and two small black spots on its face, just above its mouth, which can be used to identify this species of wasp in Australia.\n\n19.9: A European wasp. Notice the pairs of black spots on its abdomen, which can also be used to identify this species of wasp in Australia.\n\n19.10: The Common or English wasp. Notice that there are not three spots on the face and that the black 'nose' is longer and more distinct than for a European wasp.\n\n19.11: A Common or English wasp, V. vulgaris, killing a honey bee.\n\n19.12: Nest of a European wasp. The nest is made of chewed wood and is similar to paper.\n\n19.13: English wasp queen beginning to build a nest and lay eggs.\n\nAs the summer progresses the colony will grow in size until, in the early autumn, a batch of new queens and male drones are laid, ready for mating later in the autumn. The male drones and the queens will leave the nest in search of mates; the drones will soon die and the mated queens will seek a suitable sheltered location in which to hibernate over the winter. Late in the autumn, larger, slow-flying queens may be seen looking for a suitable place in which to spend the winter months.\n\nOnce the queen has selected a home for the winter she will hold on to the structure of the cavity with her mouth; her legs, wings and antennae are all folded back closely against her body. In Europe during late autumn the original queen will die and the nest will slowly disintegrate before the onset of winter as the workers die out.\n\nWorkers of the European wasp eat meat as well as liking sweet foods such as nectar, honeydew and soft drinks. Wasps are also a pest of honey bees, robbing honey from hives as well as eating brood and cannibalising adult bees. An attack by wasps on a hive is distinctive in that a wasp, once it has caught an adult bee, will remove the bee's head, abdomen, legs and wings, and return to its own nest carrying only the thorax as this is the part of the bee that contains a lot of nutritious muscle. Workers raise brood on masticated, or chewed, meat protein from insect prey or scavenged meat or carrion.\n\nThe foraging range of a wasp rarely exceeds 1 kilometre and is usually about 500 metres. A wasp on a foraging trip typically has a zigzag flight path while it is looking for food, and will often fly close to the ground to look for opportunistic feeds there. On its return journey to the nest the wasp flies in a distinctive straight line and it is possible by looking at the return path of two wasps returning to the same nest to triangulate and make a good estimate of where the nest is located.\n\nStudies in New Zealand have shown that in one year about 4.9 per cent of honey bee colonies were seriously affected by wasps, with 1.9 per cent of the total colonies being completely destroyed. This can be compared with infection by American Foul Brood, which destroyed 0.44 per cent of hives during the same period.\n\nA wasp is more likely to attack humans than is a honey bee. A wasp can repeatedly sting its victim without being killed itself. Studies have shown that the pain inflicted by a wasp sting is the same intensity as a bee sting. Peak aggression of the wasp is during the autumn when colony size is at its maximum and food resources are short.\n\nDIFFERENCES IN COLONY LIFECYCLE BETWEEN EUROPE AND AUSTRALIA\n\nAlthough the above is true for wasp colonies in Europe, for reasons that are still unknown wasp colonies in warmer parts of both Australia and New Zealand show a very different lifecycle during the winter. In these countries the colony does not die out but survives, retaining up to 100 new queens over the winter. An obvious factor may be the shorter warmer winters experienced in Australia and New Zealand, some two months long compared to five months in much of Europe. With so many resident queens all surviving with the workers over the winter the colony comes out of winter extremely strong and may build up to 100,000 workers and 1 million cells in a nest; this is four times the maximum nest size of any wasp colony found in Europe. In Europe, a typical wasps' nest contains approximately 3000 to 4000 wasps and 12,000 to 13,000 cells.\n\nASIAN HORNETS\n\n19.14: The Asian hornet, V. velutina nigrithorax, is causing significant problems to beekeepers in France.\n\n19.15: Nest of an Asian hornet colony photographed during winter in France.\n\nAnother pest related to the honey bee, the Asian hornet, _Vespa velutina nigrithorax,_ although not found in Australia or New Zealand, is causing significant problems for beekeepers in France since its accidental introduction into that country (perhaps from China) in 2004. The Asian hornet is larger than a European honey bee with queens up to 30 millimetres and workers up to 25 millimetres in length. Asian hornets are easily recognised since their thorax is a velvety black or dark brown with distinctive yellowy-orange stripes. The head of the hornet is black with a yellowy-orange face.\n\nThe Asian hornet makes its nest in the open, often hanging from trees, using a papier-m\u00e2ch\u00e9 material made from chewed wood or plant material. The almost circular nest of the Asian hornet is easy to recognise as it is built in the open and is often about 1 metre in diameter.\n\nThe Asian hornet is a very aggressive predator of the European honey bee and can lead to the destruction of the entire colony. Asian hornets will position themselves about 30 centimetres from the entrance to a _mellifera_ hive and attack returning honey bees that are carrying pollen. The hornet will then take the honey bee to a nearby tree where it will remove the legs and wings and turn the remaining parts of the bee into a small meatball that it will take back to its own nest as food for larvae.\n\nIf the attack is severe, with multiple hornets attacking simultaneously, the effect on a _mellifera_ colony can be catastrophic. The attack could, over a short period of time, lead to the complete destruction of the hive or in lesser attacks the _mellifera_ colony could be so weakened so as to become vulnerable to other diseases or infestations by pests such as _Varroa_ or small hive beetle.\n\n19.16: Apis mellifera scutellata is native to east and parts of southern Africa. The accidental introduction of A. mellifera scutellata into Brazil and the subsequent interbreeding of this race with members of the European honey bee races led to the emergence of Africanized bees and their subsequent advance northwards to southern areas of the United States.\n\nOTHER RACES OF _APIS MELLIFERA_\n\nThe majority of this book is about the European honey bee, _Apis mellifera,_ but as mentioned in the section on the biology of the honey bee, other members of the species _Apis mellifera_ can also be found throughout Europe, most of Africa and much of the Middle East. Any of the ten distinct races of _Apis mellifera_ found in Africa would be classified as unwanted pests if they were to be found in Australia.\n\nSome African races of _Apis mellifera_ are very aggressive; for example, _Apis mellifera adansonii_ from Ghana or _Apis mellifera scutellata_ from central and southern east Africa. Although any of the ten races of honey bee found in Africa can technically be called African honey bees, it is for _Apis mellifera scutellata_ that this name is most commonly used.\n\nThe African honey bee, _Apis mellifera scutellata_ was introduced to Brazil in 1956. It escaped from a laboratory and slowly made its way north, eventually reaching the southern parts of the United States in 1990. Even though _Apis mellifera scutellata_ bees had crossed the Amazon basin and traversed the entire Central American isthmus spreading through Mexico, there was very little interbreeding with the small numbers of European honey bees that were present in these areas. Most African bees, by their nature, are usually aggressive and _Apis mellifera scutellata_ is no exception. The limited cross breeding of _Apis mellifera scutellata_ with the existing varieties of _Apis mellifera_ has given rise to the name Africanized bees, or, more popularly, Killer bees, to distinguish them from the purer races of African bees found in Africa. If either _Apis mellifera scutellata, Apis mellifera adansonii_ or most of the other races of African honey bees were to enter Australia, it is likely that the result would also be some very aggressive bees in Australia that would pose a significant threat to public health. Even today in Ghana _Apis mellifera adansonii_ is so aggressive that beekeepers only work these bees at night and change into their protective clothing before getting out of their vehicles.\n\n19.17: Distribution of _A. mellifera scutellata_ in Africa.\n\n19.18: Africanized honey bee (top left), next to a European honey bee (bottom right).\n\n19.19: Cape honey bee, _Apis mellifera capensis,_ on a flower.\n\n19.20: _A. mellifera capensis_ queen and her retinue.\n\nThe commonly called Cape bee, _Apis mellifera capensis_ , native to the Cape region of South Africa, would present a very different problem to beekeepers in Australia if it were to be found here. In southern Africa, around the Cape, _Apis mellifera capensis_ beekeepers were successfully keeping this race of honey bee for honey production. In 1990, in order to further exploit this race they took colonies of the Cape bee north to areas that were home to the African bee, _Apis mellifera scutellata._ This created a unique problem because, unlike other races of _Apis mellifera_ where workers can only lay unfertilised drone eggs, _Apis mellifera capensis_ workers can lay fully fertilised worker eggs that contain both the mother's DNA and the father's DNA, resulting in a viable worker being born without the _Apis mellifera capensis_ worker first mating with a drone. Once the Cape bee had been taken north into territory that was home to the African bee, Cape bee workers drifted into the hives of African bees where they promptly started to lay viable worker Cape bee brood. This led to a situation where inside an _Apis mellifera scutellata_ colony there was both an _Apis mellifera scutellata_ queen laying eggs and many Cape bee, _Apis mellifera capensis,_ workers laying eggs. In this situation the number of Cape bees inside the colony soon overtakes the number of African bees and the original colony ultimately absconds or collapses. This situation is currently causing significant disruption to beekeeping in the northern parts of South Africa and there does not appear to be a solution to the problem.\n\nOTHER SPECIES OF ASIAN BEES\n\nThere are believed to be twelve species of honey bee globally.\n\n_Cavity nesting bees_\n\n_1._ _Apis mellifera_ \u2014 European or Western honey bee\n\n_2._ _Apis cerana_ \u2014 Asian honey bee\n\n_3._ _Apis koschevnikovi_\n\n_4._ _Apis nuluensis_\n\n_5._ _Apis nigrocincta_\n\n_6._ _Apis indica_\n\n_Dwarf honey bees_\n\n_7._ _Apis andreniformis_\n\n_8._ _Apis florea_\n\n_Giant honey bees_\n\n_9._ _Apis dorsata_\n\n_10._ _Apis laboriosa_\n\n_11._ _Apis binghami_\n\n_12._ _Apis breviligula_\n\nAs mentioned previously, _Apis cerana_ is commonly called _'the_ Asian honey bee', while all of the above species collectively, excluding _Apis mellifera,_ are called Asian honey bees.\n\n19.21: A. florea worker.\n\n19.22: A. florea make their small nests in the open under small branches.\n\nDWARF SPECIES OF ASIAN HONEY BEES\n\nI mentioned earlier _Apis cerana,_ the Asian honey bee, which entered Australia through Cairns. The other ten species of Asian bee could also successfully establish themselves in Australia were they to land on our shores. One example would be the dwarf honey bee, _Apis florea._ The smaller Asian bees, such as _Apis florea,_ differ from both _Apis cerana_ and the European honey bee in that they form nests in the open, hanging from small branches often deep in bushes where they remain hidden. As a defensive strategy the smaller Asian honey bees abscond very quickly, often at the slightest physical disturbance.\n\nThe dwarf honey bee, _Apis florea_ , constructs a single comb nest in the open and does not collect or store much honey. The bee is gentle and, due to its small size, is unlikely to be recognised as a honey bee when visiting a flower. Due to its small size, gentle behaviour, and shyness, any incursion into Australia is likely to remain undetected for some time, allowing the bee to become well established before eradication can be attempted.\n\n_Apis florea_ has spread in recent years from its original home in south-east Asia and can now be found widely in east Africa, Israel, Jordan, Oman, and across the Saudi Arabian peninsula. It is not clear how much of this range expansion is due to human relocation, and how much is due to the natural spread of the species. Since the 1980s, _Apis florea_ has been spreading at a rapid pace throughout Sudan after being accidentally introduced into Khartoum, and is now found throughout 90 per cent of the country.\n\nGIANT SPECIES OF ASIAN HONEY BEES\n\nThe four giant honey bee species found in Asia are _Apis dorsata, Apis breviligula, Apis binghami_ and _Apis laboriosa._ Until recently, _Apis breviligula_ and _Apis binghami_ were considered to be subspecies of _Apis dorsata,_ but recent genetic studies suggest that they are a distinct species. The four giant honey bee species make their nests in the open, typically under cliffs, under large branches of trees or under the roofs of large buildings such as minarets. _Apis dorsata_ and _Apis laboriosa_ often nest in large aggregations of tens to hundreds of colonies, while _Apis breviligula_ and _Apis binghami_ tend to make single solitary nests. The giant honey bees, although successfully hunted for honey across Asia, India and Nepal, cannot be kept in hives and the collection of their honey represents a significant personal safety risk for the honey hunters who depend on them for their livelihood. Although this safety risk is partly due to the aggressiveness of the giant bees, part of the risk is also due to the heights that honey hunters need to climb in order to harvest comb. The honey produced by the various giant honey bees is considered by locals to be far superior in taste to that of other honey bee species, making the reward to be gained from the harvest worth the risks involved.\n\n19.23: A permanent _Apis dorsata_ nest in Asia. _Apis dorsata_ make their nests in the open on trees or buildings.\n\n19.24: _A. laboriosa_ worker taken in India in the Himalayas. _A. laboriosa_ are found at higher elevations in the Himalayas while _A. dorsata_ are much more widespread and are found at lower altitudes.\n\n19.25: _A. dorsata_ from the Philippines.\nAppendix:  \nSizes of frames and boxes\n\nInitially the choice of hive size offered to the new beekeeper is between a box that will hold eight frames, that is, an eight-frame box and one that will hold ten-frames, that is, a ten-frame box. More rarely there are beekeepers who use twelve-frame boxes, but these are no longer commonly seen or available from suppliers.\n\nIn Tasmania, Victoria and South Australia the majority of beekeepers use eight-frame hives, while those in New South Wales, Queensland, Western Australia, and the Territories favour ten-frame hives. I do not know why this is, but probably the answer can be found in the history of beekeeping and the culture that has grown up in the particular state. It is probably the same rationale as the initial choice of different gauge railway lines across parts of Australia.\n\nWhether you decide to use eight-frame or ten-frame hives you should, for practical reasons, stay with your initial choice and use the same size box for all future hives. Otherwise you will find that having a mix of both eight-frame and ten-frame hives will in the long run cause frustration when you want to rearrange the boxes in your apiary, as the various hive parts will not be interchangeable. I believe that if you have one eight-frame hive and one ten-frame hive, it is worth moving all of the bees, say, from the ten-frame hive into a new eight-frame hive and discarding the empty ten-frame hive. Either that or sell your ten-frame hive and use the money to replace it with a new eight-frame hive. This will save you a lot of inconvenience later. Having said that, of course, if you intend keeping a number of hives you may more easily use both sizes as you will have sufficient of each to manipulate boxes as you please.\n\nI should note here that not all hive and frame manufacturers use the same dimensions for the hive parts and I have sometimes experienced situations where a ten-frame box from one supplier will hold nine frames rather than the expected ten. This fact is not always readily apparent when you purchase a box flat packed and is not generally much of a problem as it is common practice to put nine frames into a ten-frame super or seven frames into an eight-frame super in order to allow the bees to draw the comb out more.\n\nAnother point to keep in mind when selecting the size of box is that an eight-frame box holds eight frames with room separating the frames that is usually sufficient for removing the frames without difficulty. The ten-frame box fits ten frames snugly, which can make separating the frames for removal difficult, particularly after winter when the frames have swelled due to moisture. When I use a ten-frame box my preference is to only insert nine frames for ease of removal, particularly when overwintering.\n\nWhy would you choose one size box over another? A lot depends on your age and the state of your back. A ten-frame super full of honey will weigh around 40 kilograms; on the other hand a ten-frame super offers you either one or two more frames of honey and a ten-frame brood box offers a queen extra laying space. For me the weight of the respective boxes is the deciding factor.\n\nWhat is called a 'full-depth' box containing a 'full-depth' frame is a standard used across Australia and can refer to either an eight- or ten-frame hive box. 'Full depth' refers to the depth (measured top to bottom) of the respective box and differentiates these from more shallow-depth boxes that are also available and used by beekeepers. Once you have made your choice of whether to use either an eight-frame or a ten-frame hive box, you will next have to consider the depth of the box. Your supplier or other beekeepers will no doubt offer advice, with most suggesting a full-depth box to sit on the hive base as a brood box. Which depth of box you will be recommended to use as a honey super will again largely depend on which state of Australia you happen to reside in.\n\nIn Australia there are currently six different depths of box in use. These are:\n\n\u2022 Jumbo or Deep:  | 280mm deep | \u2013 designed by Dadant in the United States for use as a brood chamber  \n---|---|---  \n\u2022 Full-depth: | 240mm deep |   \n\u2022 WSP: | 192mm deep |   \n\u2022 Manley: | 168mm deep |   \n\u2022 Ideal: | 144mm deep |   \n\u2022 Half: | 125mm deep |\n\nApart from the depth, the dimensions of the remaining sides of boxes are:\n\n\u2022 eight-frame:  | 51cm x 35cm  \n---|---  \n\u2022 ten-frame: | 51cm x 41cm\n\nThe box depths listed above are approximate because as previously noted not all hive manufacturers use exactly the same dimensions. They will be sufficiently similar, though, so that a full-size frame from one supplier will match the depth of a full-size frame from another supplier. Occasionally, though, a ten-frame box will only hold nine frames.\n\nProfessional beekeepers in the northern states of New South Wales and Queensland commonly use the shallower Manley or a WSP box, while in the southern states of Victoria and South Australia the WSP is more commonly used. Professional beekeepers often use the shallower WSP or Manley box for supers since back injury is the major cause of disability amongst beekeepers and choosing a lighter box helps alleviate ongoing suffering.\n\nThe shallowest depth box available is the Half-depth, although it is the slightly deeper Ideal depth that is most frequently used by hobbyists who want a lightweight box that can easily be lifted and carried without back strain. I believe that a major part of the reason for the choice of the Ideal box is due to the fact that many hobbyists are simply not aware of the existence of WSP or Manley boxes. This lack of awareness is often due to a limited range offered by suppliers, either because of the culture of the state, difficulty in obtaining supplies or simple lack of demand by customers.\n\nA.1: Relative size of the different designs of hive boxes. The height of the boxes is given in centimetres.\n\nAs a general rule, if you buy an existing hive and you suspect that the hive is home-made, check that the dimensions correspond to one of the standard depths listed above and are designed to hold either eight or ten frames. Many home-made hives are made to apparently random dimensions and finding additional suitable boxes or frames later may be impossible. Many hobby beekeepers with plenty of spare time make their own hive boxes, but usually buy standard frames to fit into them. If you are offered non-standard hives and frames by a seller, I would gently turn down the offer and look elsewhere for standard hives.\n\nThe typical beehive consists of one, two, three or more boxes called brood boxes or supers sitting on top of each other. Calling the bottom box a 'brood box' and the boxes on top of this 'supers' is confusing to many new beekeepers as they are in fact the same design of box with the same number of frames. The label 'brood box' is generally adopted for the bottom box, as it holds the brood.\n\nThe brood box at the bottom of the hive requires the largest depth box to enable the queen to lay as many eggs as possible during the spring honey flow. This is important since if I am to obtain the maximum number of workers in my hive to collect nectar, the queen needs to be able to lay the maximum number of eggs. To provide sufficient space for the queen to lay eggs I usually use two full-depth brood boxes unless I am building up a small colony from a swarm or a split. The much larger Jumbo or Deep brood box is better at supporting a large number of brood than a single full-depth brood box, but the large Jumbo box is not readily available for purchase in Australia. Hence two full-depth boxes stacked on top of each other provide similar space for the queen.\n\nJUMBO BOXES\n\nJumbo boxes are never used as supers due to their larger depth and their weight if they were to contain full frames of stored honey. Their use is restricted to housing brood. Since brood boxes are seldom moved by hobbyists, and a professional apiarist would probably use a mechanical lifter to move them, their use as a brood box is seldom a concern.\n\nFULL-DEPTH BOXES\n\nThe full-depth box is the standard depth across Australia for both eight-frame and ten-frame hives and is readily available at all beekeeping supply stores.\n\nWSP BOXES\n\nThe next box in depth is the WSP, named after its inventor W.S. Pender of New South Wales. These are available in both ten- and eight- frame versions.\n\nAs previously mentioned, a full-depth ten-frame super full of honey weighs about 40 kilograms and many beekeepers have strained their back from manually lifting them. The WSP, in a ten-frame size, has less depth and is therefore lighter to lift. The fact that it has less depth and therefore laying space means that the WSP box is seldom used as a brood box and is generally confined to use as a super. Ten-frame WSP boxes are widely used in New South Wales and Queensland although they are not often used in Victoria where the lighter, full-depth eight-frame super is more common.\n\nMANLEY BOXES\n\nNamed after Robert Manley (1888\u2013 1978), an English manufacturer of beekeeping equipment, this size box is frequently used in both Queensland and New South Wales and is also the most popular size of box used in the United States.\n\nIDEAL BOXES\n\nThe most common small-depth box used by hobbyists is the Ideal. The Ideal is just over half the depth of a full super and I find that many hobby beekeepers who want a smaller-depth super to reduce lifting weight choose this depth box. Personally, the disadvantage I find of an Ideal super is that the wooden sides and the top and bottom bars of the frames take up much of the space inside the box, leaving little space for bees to store honey. Another disadvantage I find when using smaller-depth supers is that I need three Ideal supers to store the same amount of honey as a single full-depth super. Ideal frames are also harder to make than full- depth frames due to the difficulty of tightening the supporting wire in the shallow frame. Ideal boxes are used extensively in Tasmania for both brood and honey supers.\n\nExtracting can also be more difficult from an Ideal frame, particularly when using an extractor with baskets designed to hold full-depth frames. On the plus side, Ideal frames without wire but with a small insert of starter comb are 'ideal' for drawing comb honey. Despite these perceived difficulties on my part many beekeepers choose to use this depth of box throughout their hive set up.\n\nHALF-DEPTH BOXES\n\nSimilar in depth to the Ideal, the half-depth box has the same advantages as the WSP or Ideal in reducing the weight of the super. Although I have never seen a half-depth box used in Victoria they are used in other parts of Australia.\n\nMy own preference when selecting a lighter super in which to store honey is to use a WSP, as I believe that this depth of box is the best compromise between the amount of honey stored and the ease of lifting the super. My own experience is that it is preferable to keep the depth of both the brood boxes and the supers the same. This is because when managing hives, frames from the brood box are often swapped with frames from the super and having different depth boxes makes hive management more difficult. \nGlossary\n\nACUTE BEE PARALYSIS VIRUS (ABV): A viral infection of the honey bee that causes paralysis of adult bees.\n\nABSCOND: This occurs when a colony of bees suddenly leaves a hive with few or no bees remaining behind. Absconding should not be confused with swarming; it is often caused by problems such as such as poor temperature\/ventilation within the hive, mite, pest, lack of food, or other intolerable problems.\n\nAMERICAN FOUL BROOD (AFB): A lethal infection of honey bees with the bacterium _Paenibacillus larvae._\n\nAFRICANIZED HONEY BEE: A term used to describe the South African honey bee _Apis mellifera scutellata_ or its hybrids, now found in both North and South America. Africanized honey bees are known for their volatile nature. The term Africanized honey bee is used to differentiate the type of bee found in the Americas from African honey bees, which although of the same race are only found in Africa.\n\nAFTER-SWARM: A swarm which leaves a colony with a virgin queen, shortly after the first (or prime) swarm has departed. The first after-swarm is also referred to as a secondary swarm while an after-swarm that leaves after the secondary swarm is also called a tertiary swarm.\n\nASIAN HONEY BEE: One of the twelve species of honey bee found globally, the Asian honey bee _(Apis cerana)_ is found across large areas of Asia as well as parts of Queensland.\n\nASIAN BEES: Of the twelve currently identified species of honey bees, eleven are only found in Asia and these are called by the generic name Asian bees. The twelfth type of bee, the European honey bee, _Apis mellifera,_ before being transported globally by humans, was only found in Africa, Europe and parts of the Middle East and is thus not an Asian bee. Note that Asian bee is a generic term for eleven species of honey bee found in Asia, while the term The Asian honey bee, AHB, usually applies to the single species, _Apis cerana._\n\nBAIT HIVE: An empty hive left to attract swarms of bees. The term sometimes refers to hives that have been left near ports to attract swarms that have reached Australia by ship and have eluded capture by quarantine. A bait hive may use the Nasonov pheromone to attract swarms.\n\nBEE SPACE: A space between two frames that is big enough to allow two bees to pass each other while working on opposite frames. A bee space is about 9.5mm wide, which is too small to encourage comb building, and too large to induce propolising activities.\n\nBLACK BEES: _Apis mellifera mellifera,_ sometimes called German bees.\n\nBRACE COMB: Pieces of seemingly random comb that connect hive parts together, such as between two frames or between an end frame and the wall of a hive. Brace Comb is a form of burr comb.\n\nBROOD: A general term to describe young bees that have not yet emerged from their cells as adults. Brood can be workers, drones or queens. The four stages of brood are egg, larva, pre-pupa, and pupa.\n\nBROOD CHAMBER: The area of the hive where the queen lays eggs and brood are reared; usually the lowermost parts of the hive.\n\nCAPPED BROOD: Brood in the pre-pupal or pupal stage inside a capped cell. Immature bees whose cells have been capped with a brown wax cover by other worker bees; inside the sealed cell, the non-feeding larvae are isolated and can spin cocoons prior to pupating.\n\nCAPPED HONEY: Cells full of ripe or mature honey, sealed or capped with beeswax.\n\nCAPPED LARVA: The stage of brood development when the larvae turns into pre-pupae and is ready to have its cell capped and to spin its silk cocoon, about the tenth day from the laying of the egg.\n\nCASTES: The three types of bee that comprise the adult population of a honey bee colony: the queen, workers and drones.\n\nCHRONIC PARALYSIS VIRUS (CPV): Sometimes called chronic bee paralysis virus, this is a disease mainly of adult bees.\n\nCELL CUP: The base of an artificial queen cell made of beeswax or plastic and used for rearing queen bees.\n\nCLEARER BOARD: A board used to clear bees from a super prior to removing the frames ready for extraction.\n\nCLIPPED QUEEN: Queen whose wing (or wings) has been clipped to stop her from swarming.\n\nCOLONY: A collection or family of bees living within a single social unit containing workers, drones and a queen.\n\nCOMB FOUNDATION: A thin sheet of beeswax with the base pattern of cells impressed on the sheet as a template for the bees to start making comb. Some foundation is also made of plastic.\n\nCOMB HONEY: Honey sold and eaten in the comb. Also referred to as cut comb honey.\n\nDEFORMED WING VIRUS (DWV): A virus of the honey bee, often associated with a _Varroa_ infestation, that causes newly emerged bees to have deformed wings.\n\nDISEASE RESISTENCE: The ability of an organism to avoid a particular disease; primarily due to genetic immunity or avoidance behavior.\n\nDOUBLE: A hive that consists of two boxes.\n\nDRIFT: The process in which bees join a different hive from their own, often due to loss of direction or when hives are placed too close together.\n\nDRONE CONGREGATION AREA (DCA): An area where many drones from surrounding colonies gather to mate with queens during their nuptial or mating flights. The location of drone congregation areas remains stable over many years.\n\nDYSENTERY: An abnormal condition of adult bees characterised by severe diarrhoea and usually caused by starvation, low-quality food, moist surroundings, or _Nosema apis_ infection. (See also _Nosema)_\n\nEMERGENCY QUEEN CELL: A queen cell in the middle of a brood frame that has been made quickly as a result of the queen bee dying suddenly.\n\nENTRANCE REDUCER: A device used to regulate the size of the bottom entrance. May be used to reduce hive entrance during winter months or during attacks from robber bees or wasps.\n\nESCAPE BOARD: A board having one or more one-way bee escapes in it. Used to empty a super of bees.\n\nESCORTS OR ESCORT BEES: Worker bees that are placed in a cage with a queen for shipping. Usually about four or five escort bees are included with the queen.\n\nEUROPEAN FOUL BROOD (EFB): An infectious disease of honey bee brood caused by the bacterium _Streptococcus pluton._\n\nEXTRACTION: Removal of honey from comb. Typically refers to the use of an extractor but also includes non-mechanical methods such as crush and strain.\n\nFAILING QUEEN: A queen that has reached the end of her fertile age and is not producing sufficient brood for the colony to survive. The colony is now ready for the supersedure of the queen.\n\nFERAL BEES: Bees that are not kept in hives by beekeepers.\n\nFERTILE QUEEN: A queen, inseminated instrumentally or mated with a drone, which can lay fertilised eggs.\n\nGUARDING: Bees stationed at the entrance of a hive to detect and ward off invaders and to examine entering bees. Guard bees are typically about three weeks old.\n\nHIVE STAND: A device that raises the bottom board off the ground and helps extend the life of the bottom board by keeping it dry.\n\nHONEY BEE RACE: Honey bee race is a classification of honey bees, in particular the European honey bee, _Apis mellifera,_ into different types. The most common races of _Apis mellifera_ found in Australia are Italian bees, Caucasian bees, Carniolan bees, and dark German bees.\n\nHONEY BOUND: A brood nest that is congested by cells\/comb filled with honey, reducing the amount of space that the queen has to lay eggs.\n\nHONEY FLOW: A time, usually in the spring and summer, when there are sufficient nectar-bearing plants blooming so that bees can store a surplus of honey.\n\nHONEY SUPERS: The hive bodies or boxes, usually above the brood box, used by the workers to store honey.\n\nHONEYDEW: The sweet secretion from aphids and scale insects. Since honeydew contains almost 90 per cent sugar; it is collected by bees and stored as honeydew honey.\n\nHOOP PINE: A high quality pine wood native to New South Wales and Queensland and used to make supers and other hive parts.\n\nHOPELESSLY QUEENLESS: A colony of bees that has been without a queen for several weeks and, as a result, is unable to make a new queen from young larvae.\n\nHYDROGEN PEROXIDE: Found in all honeys and provides most of its antibacterial properties.\n\nINBREEDING: Mating among related individuals that may cause genetic deformities in the bee.\n\nINFERTILE: A bee that is unable to produce a fertilised egg, typically applied to a laying worker.\n\nINTEGRATED PEST MANAGEMENT (IPM): A pest control method that uses a variety of complementary strategies including genetic, biological, cultural management, chemical management, as well as mechanical and physical devices such as screened bottom boards. IPM techniques are typically performed in three stages: prevention, observation, and intervention. It is an ecologically-friendly approach with the goal of significantly reducing or eliminating the use of pesticides while at the same time managing pest populations at an acceptable level.\n\nINTERLOCKING: A design of wooden hive in which the corners of the boxes interlock with each other.\n\nINTRODUCING: The process of introducing a new queen into an existing hive.\n\nINTRODUCING CAGE: A small wood, wire or plastic cage used to ship queens and also sometimes to release them into the colony. Also known as a queen cage.\n\nIRRADIATE: To irradiate a hive with gamma radiation to kill American Foul Brood or other diseases. There are few places across Australia that offer this. The facility is available in Melbourne, Sydney and Brisbane.\n\nISLE OF WIGHT DISEASE: _Acarapis woodi,_ a small mite that causes tracheal infections of the honey bee, is believed to have been the cause of the severe decline in the honey bee population on the Isle of Wight, off the south coast of England, in 1904.\n\nISRAELI ACUTE PARALYSIS VIRUS (IAPV): First identified in Israel but now infecting honey bees in many parts of the world, one of several viruses that causes paralysis of adult honey bees.\n\nITALIAN BEE: A common race of bees, _Apis mellifera ligustica,_ which originated in Italy. Bees have brown and yellow bands; usually gentle and productive, but tend to rob other hives. Italian bees are the most well known type of bee.\n\nKASHMIR BEE VIRUS (KBV): A naturally occurring virus infecting _Apis cerana,_ Kashmir Bee Virus jumped species to infect _Apis mellifera_ and is now a common infection of this bee.\n\nKILLER BEES: Africanized bee.\n\nLARVA: The second stage of development in the lifecycle of the bee. The three stages are egg, larva and pupa. The larva stage is often called a grub.\n\nLARVAE: The plural of larva.\n\nLIFTING CLEAT: A wooden cleat attached to the side of supers and brood boxes to make them easier to lift.\n\nLINE: A family or set of descendants bred from the same queen.\n\nMANIPULATIONS: The process by which frames in a hive are moved or replaced.\n\nMATED QUEEN: A queen that has gone on a mating flight or has been artificially inseminated with drone sperm.\n\nMATING FLIGHT: A short flight taken by a virgin queen during which she mates in the air with several drones from other colonies. Queens usually mate with ten to twenty drones on one or more mating flights.\n\nMIGRATORY BEEKEEPING: The practice of professional beekeepers and some hobby beekeepers to regularly move their hives to follow the flowering pattern of different types of trees.\n\nMIGRATORY LID: The lid of a hive that has the same outside dimensions as the super. Called a migratory lid since it is of a convenient size to allow beekeepers to transport their hives on the back of trucks. A lid that does not extend over the sides of the hive.\n\nMITICIDE: A chemical or biological agent which is applied to a colony to control parasitic mites such as _Varroa_ or Tracheal mite.\n\nNATIVE BEES: Bees that are native to Australia, for example, stingless bees. The European honey bee is not a native bee as it was introduced by Europeans in 1822.\n\nNATIVE POLLINATORS: Apart from native bees, other animals may also be native pollinators such as birds or other insects. (See Native Bees)\n\nNATURAL BEEKEEPING: A philosophy of keeping bees that puts the welfare of the colony ahead of the amount of honey collected during a season.\n\nNATURAL HONEY: Pure honey that has not been heated or finely filtered during processing prior to bottling.\n\nNECTAR FLOW: The time of the year when a tree or shrub produces a lot of nectar for bees to collect. The nectar flow of a particular tree often only lasts a few weeks, so professional beekeepers need to know what trees are flowering where and when and move their hives to follow the flowering trees.\n\nNEONICOTINOIDES: A class of insecticides chemically related to nicotine.\n\nNEST: A feral colony of bees that is not being managed by a beekeeper.\n\nNEWSPAPER METHOD: A technique to join together two unrelated colonies by providing a temporary newspaper barrier between them. The bees will chew their way through the newspaper over a few days and the two colonies should merge together without fighting.\n\n_NOSEMA:_ A disease caused by protozoan spore-forming parasites living in the gut of adult bees. The two types of nosema that infect adult bees are _Nosema apis_ and _Nosema cerana._ (See _Nosemosis)_\n\nNOSEMOSIS: To be infected with _Nosema._\n\nNUCLEUS HIVE: A colony of bees housed within a small brood box usually containing only four or five frames. Nucleus hives are often used to rear queens.\n\nNURSERY BEES: Young bees, three to ten days old, which feed and take care of developing larvae.\n\nOBSERVATION HIVE: A hive with walls made largely of glass or clear plastic to allow the observation of bees at work. Often used as a teaching aid during courses.\n\nORGANIC HONEY: Honey that has been made from nectar from flowers grown in organically certified regions.\n\nORIENTATION FLIGHTS: Short orienting flights taken by young bees, usually by large numbers at the same time and during the warm part of day, in order for them to learn to fly and to familiarise themselves with their surroundings.\n\nOVERCROWDING: A hive that has too many bees living in it for its size.\n\nOVERWINTERING: The survival of a colony over winter.\n\nOXYTETRACYCLINE (OTC): An antibiotic sold under the trade name Terramycin. Oxytetracycline is used to control American and European Foul Brood diseases.\n\nPARENT COLONY: The home colony from which swarms or splits originated.\n\nPARASITIC MITE SYNDROME (PMS): Signs of poor health in a colony of bees that has been infested with _Varroa_.\n\nPHEROMONE: Several kinds of scents produced by bees to establish a basic form of communication or to stimulate a response.\n\nPIPING: A series of sounds made by a queen, usually before she emerges from her cell.\n\nPLANTATION PINE: The most readily available plantation timbers in Australia are softwoods such as hoop pine, radiata pine and slash pine, the latter two being introduced species.\n\nPLASTIC FOUNDATION: Foundation placed inside a frame made entirely of plastic, used as an alternative to wax foundation. (See Plastic Frames)\n\nPLASTIC FRAMES: Frames that are constructed entirely of plastic, including the outer rim.\n\nPLAY FLIGHTS: Short flights taken in front and in the vicinity of the hive by young bees to acquaint themselves with the location of the hive. Play flights are sometimes mistaken for robbing or preparation for swarming. (See Orientation Flights)\n\nPOLLEN: The male reproductive cells of flowers. Pollen provides the protein in a young bee's diet and is frequently called bee bread when stored in cells in the colony. Pollen is an essential component of brood food. Honey is another essential component and provides the carbohydrate part of the bee's diet.\n\nPOLLEN BASKET: The area on the hind leg of bee adapted to carrying pellets of pollen or propolis back to the colony.\n\nPOLLEN SUBSTITUTE: A food material which is used as a substitute for naturally occurring pollen. Pollen substitute usually contains soy flour, brewers' yeast, powdered sugar, and other ingredients. Pollen substitute is used to stimulate brood rearing in periods of pollen shortage.\n\nPOLLEN TRAP: Device which forces the bees entering a hive to walk through a meshed screen, the pollen is brushed off the bees' legs by the screen and is collected from a collecting tray.\n\nPRIME SWARM: The first swarm to leave the parent colony, usually with the old queen. (See also After-swarm)\n\nPROPOLIS: A sticky resinous material collected from trees or other plants by bees; used to close holes and cover surfaces in the hive. Often referred to as bee glue. Propolis also has antimicrobial properties and is used by bees for this purpose.\n\nPROPOLISE: To fill with propolis, or bee glue; used by bees to strengthen comb and seal cracks.\n\nPROTEIN: Naturally occurring complex organic substances, such as pollen, and composed of amino acids. An essential food for brood to build body tissue before emerging as adult bees.\n\nPUPA: The final stage in the life of a developing baby bee after larva and before maturity. Pupae are only found inside sealed brood comb. Pupae is the plural of pupa.\n\nPYRETHRINS: A group of naturally occurring insecticides derived from the seed cases of the perennial plant _Chrysanthemum cinerariaefolium._\n\nQUEENLESS COLONY: A colony that does not have a queen.\n\nQUEEN-RIGHT: A colony of bees that contains a laying queen.\n\nRE-QUEEN: To introduce a new queen to a queenless hive. Usually to replace an old queen with a young one.\n\nROBBER BEES: Bees that rob other colonies of their honey. (See Robbing)\n\nROBBING: Bees that are stealing honey from other hives. This is a common problem particularly in autumn when nectar is not available in the field. The term also applies to bees that are cleaning out wet supers or cappings left uncovered by beekeepers.\n\nROPEY CHARACTERISTIC: A diagnostic test for American Foul Brood and European Foul Brood in which the decayed larvae or pupae form an elastic rope when drawn out with a matchstick.\n\nROYAL JELLY: Bee milk and worker jelly all refer to royal jelly, which is a pearly white, creamy substance produced by young worker bees to feed larvae. Royal jelly is a secretion of the hypopharyngeal glands located in the head of young worker honey bees.\n\nSACBROOD: A viral disease of larvae, usually nonfatal to the colony, which interferes with the moulting process; the dead larva resembles a bag of fluid.\n\nSCOUT BEES: Worker bees out searching for a new source of pollen, nectar, propolis, water, or a new home for a swarm of bees.\n\nSCREENED BOTTOM BOARD: A framed screen used instead of a solid base in order to improve ventilation through the hive. Also used as a means to control _Varroa,_ SHB and to allow debris to fall through to the outside of the hive.\n\nSEALED BROOD: See Capped Brood.\n\nSECTIONS: Small plastic or wooden boxes placed inside frames and used to produce comb honey.\n\nSLUM-GUM: The black, sticky, fibrous, oily or waxy material left when comb has been melted and filtered by a beekeeper.\n\nSMALL HIVE BEETLE (SHB): The Small Hive Beetle _(Aethina tumida)_ is a destructive pest of honey bee colonies, causing damage to comb, stored honey and pollen. If infestation is sufficiently heavy, this may cause the bees to abandon their hive. The beetles can also be a pest of stored combs, and honey (in the comb) awaiting extraction. Beetle larvae tunnel through combs of honey, feeding and defecating, causing discoloration and fermentation.\n\nSNOTTY BROOD: Sick brood with physical symptoms very similar to European Foul Brood (EFB). Snotty brood is also called 'Idiopathic Brood Disease Syndrome', which is a fancy way of saying that we haven't figured out what causes it yet and we are still looking.\n\nSOLAR WAX MELTER: A glass-covered box in which wax combs and cappings are melted by the sun's rays and wax is recovered in cake form.\n\nSPOTTED BROOD: An irregular brood pattern on the frame caused by disease or a failing queen.\n\nSPLIT: To divide a colony in two to increase the number of hives.\n\nSPRING BUILD-UP: The increase in the number of bees in a hive during very late winter and spring.\n\nSPRING DWINDLING: A decrease in the size of the colony population during spring instead of the significant population growth which should be experienced. Spring dwindling is not associated with swarming in which about half the colony leave the hive to found a new colony.\n\nSTICKY: A super frame that has had its honey removed and is now covered in honey and is sticky to touch.\n\n_STREPTOCOCCUS PLUTON:_ Bacterium that causes European Foul Brood (EFB).\n\nSUPER: A wooden box with frames containing foundation or drawn comb in which honey is to be produced. Named for its position above the brood nest, i.e. superstructure. The same size box is referred to as a hive body or brood box when it is situated below the honey supers and is to be used for brood rearing.\n\nSUPERING: Placing additional supers on a hive in order to collect excess honey and provide more room for workers in a crowded hive. (See Oversupering.)\n\nSUPERSEDURE: The replacement of a weak or old queen in a colony by a daughter queen. Shortly after the daughter queen begins to lay eggs, the mother queen disappears or is killed by the other bees.\n\nSWARM: A collection of bees containing at least one queen that has left its home colony to look for a new nest. Swarms are currently without a home site and are looking for a new one, often resting on a tree or other object.\n\nSWARM CELL: Queen cells usually found on the bottom of the combs before swarming.\n\nSWARMING: The natural method by which bee colonies propagate. When the hive or nest becomes congested with bees in the spring, about half the colony flies off with the old queen to find a new home, leaving a new queen behind to head the old colony.\n\nSWARMING SEASON: The time of year, usually between early spring and mid-summer, when colonies swarm.\n\nSYRUP: A mixture of sugar and water used to feed bees when nectar sources are scarce or the beekeeper wants the colony to make honey for the winter.\n\nTESTED QUEEN: A queen whose offspring have been tested to show that she has mated with a drone of her own race or has other qualities which would make her a good colony mother.\n\nTOP FEEDER: A syrup feeder that is placed at the top of the hive under the lid.\n\nTOP BAR HIVE (TBH): The top bar hive is a method to manage bees with removable combs which rely on top bars rather than frames for the combs. There are no standard dimensions for a TBH as there are for Langstroth hives. TBHs have some advantages for hobby beekeepers. (Also known as Kenyan top bar hive and Tanzanian top bar hive.)\n\nTRACHEAL MITE: A small mite, _Acarapis woodi,_ that infests the tracheas of the honey bee.\n\nTRIPLE: A hive consisting of three boxes. This configuration may be one brood box and two supers, or a single brood box and two supers.\n\nUNDERSUPERING: Placing an additional super beneath an existing super but above the brood chamber.\n\nUNITE: Combining two or more colonies to form a larger colony.\n\nUNRIPE: Honey in cells that have not been capped because the workers have not yet evaporated sufficient water to turn the nectar into honey. If unripe honey is extracted it can ferment.\n\nUNSEALED BROOD: Brood during the egg and larval stages that are not yet in sealed cells.\n\nUNSEALED HONEY: Honey in cells that have not yet been capped. (See also Unripe).\n\n_VARROA DESTRUCTOR:_ An external parasitic mite that attacks _Apis cerana_ and _Apis mellifera_ and develops in sealed brood cells. _Varroa destructor_ is closely related to _Varroa jacobsoni._\n\n_VARROA JACOBSONI:_ An external parasitic mite that attacks _Apis cerana_ and _Apis mellifera_ and develops in sealed brood cells. _Varroa jacobsoni_ is closely related to _Varroa destructor._\n\n_VARROA_ SENSITIVE HYGIENE (VSH): A genetically selected honey bee which appears to suppress _Varroa_ mite reproduction.\n\nVENT: A perforated piece of metal used to cover the air vent in the lid of a hive to stop robber bees or wasps entering.\n\nVENTILATED LID: A hive lid that has vents in the rim allowing air to circulate.\n\nVIRGIN QUEEN: A queen that has not yet left the colony to go on a mating flight and is thus not yet able to produce offspring.\n\nWASHBOARDING: Worker honey bees exhibit a group activity known as rocking or washboarding on the internal and external surfaces of the hive. This behaviour is believed to be associated with general cleaning activities but under what circumstances workers washboard is not known.\n\nWAX DIPPED: Hive parts that have been dipped in hot wax to protect the wood from the weather. Frames are not typically dipped in wax to protect them.\n\nWAX GLANDS: Eight glands located on the underside of a bee's abdomen from which wax is secreted.\n\nWAX MOTH: Usually refers to the Greater Wax Moth (GWM), _Galleria mellonella,_ whose larvae bore through and destroy honeycomb as they eat out its impurities. Wax Moth infested hives are covered in a thick, spider-web like material. The Lesser Wax Moth (LWM), _Achroia grisella,_ is the smaller of the two related species. Its larvae are not found in the large congregated numbers of the greater wax moth and are often solitary.\n\nWEATHERTEX: A range of long lasting processed wood fibres that have been rolled into sheets. Weathertex is frequently used to make hive bases and hive lids.\n\nWINTER CLUSTER: A closely packed colony of bees within a hive that forms a cluster to conserve heat when outside temperature falls below about 15\u00b0 Celsius.\n\nWINTER HARDINESS: The ability of some strains of honey bees to survive long winters by minimising their use of stored honey; the northern European black honey bee would be an example of this.\n\nWINTERING DOWN: Preparing a hive for the winter.\n\nWIRED FRAMES: Frames with wires holding sheets of foundation in place.\n\nWORKER: An unfertilised female bee that makes up the majority of a colony's population.\n\nWORKER JELLY: See Royal Jelly.\n\nWSP: A super or brood box with a depth of 192mm. Named after the inventor William Stanley Pender (1866-1931), a well-known New South Wales beekeeper. \nAcknowledgements and picture credits\n\nA book such as this could only have been written with the assistance of many people. In particular I would like to thank my wife Barbara for the many hours she spent working on the book, offering suggestions, correcting my grammar and generally tolerating my absence from many domestic duties over the three years that it took to complete the book. Barbara also wrote Chapter 13, 'The bee-friendly garden'. My two daughters, Laura and Sian, spent many hours helping me take the many photographs that were required.\n\nSpecial thanks go to Mary Ann, Daisy, Rosie, Lilly and Warren Higgs for letting me keep hives on their property, looking after them when I was busy with other work, taking many of the photographs in this book, assisting me in all manner of ways, and generally being very tolerant of my constant visits, which must have disrupted their lives considerably. Without the assistance and support of the Higgs family this book would not have been written.\n\nNative bees are outside my scope of knowledge and I am therefore grateful to Erica Siegel AAPS, AFIAP, wildlife photographer (www.ozbirds-wildlife.com) for writing the detailed section on native solitary bees in Chapter 14. Some of the information on solitary bees was obtained from Dr Anne Dollin, Australian Native Bee Research Centre (www.aussiebee.com.au).\n\nIdentification of many of the native bees in the photographs was undertaken by Dr Michael Batley, Australian Museum, New South Wales, who also reviewed the chapters on native bees and other species of bees and wasps. Grateful thanks are also due to Dr Megan Halcroft, (www.beesbusiness.com.au), for generously allowing me to use material from the chapter she wrote on Australian stingless bees in the book _Pot-Honey: A legacy of stingless bees_ (Springer, 2013). Dr Halcroft also reviewed the section on native social bees and offered many helpful suggestions.\n\nCathrine Hall worked with me for many hours rearing queens and showed me how to use the Jenter method of queen rearing. Cathrine Hall also reviewed Chapter 1, 'The life of the honey bee'. David Briggs, queen rearer, Glenrowan, generously gave up his time to show me how he reared high-quality Italian and Caucasian queens commercially as well as reviewing the chapter on rearing queens. Jody Gerdts from Bee Scientifics (www.beescientifics.com) spent a lot of time discussing her experiences rearing _Varroa_ -resistant queens while working at the University of Minnesota. Jody also reviewed Chapter 1, 'The life of the honey bee' as well as the chapter on rearing queens.\n\nThanks to Bob and Peter McDonald, Castlemaine, who lent me equipment to photograph as well as providing detailed reviews of the chapters on pests and diseases.\n\nIan Brown, who has 70 years' experience as a commercial and hobby beekeeper, read the manuscript in its entirety and provided many valuable comments. Ian Brown was my mentor when I started beekeeping and is still the person I turn to when I need advice.\n\nDr James Makinson, Queen Mary College, University of London, spent long hours reviewing and providing information for the chapters 'The life of the honey bee' and 'Other species of bees and wasps'.\n\nThe chapters on the pests and diseases of the honey bee were the most exacting to write, particularly since many of the maladies, such as _Varroa_ or _Tropilaelaps,_ have not yet reached Australia. Much of the information for these chapters was obtained after meeting with Dr Eric Mussen, University of California, Davis; and Prof. Ernesto Guzman and Prof. Gard Otis, both of the University of Guelph, Canada. Dr Eric Mussen also gave invaluable assistance by providing a very detailed peer review of all of the pest and eisease chapters. I would also like to thank Dr Mark Goodwin, Plant & Food Research, New Zealand, for taking the time to discuss with me the _Varroa_ incursion into New Zealand and Dr John Roberts, CSIRO, who provided a detailed review of the pests and diseases chapters.\n\nMartin O'Callaghan reviewed the chapter on sustainable beekeeping and allowed me to visit his home to take photographs of Warr\u00e9 and top-bar hives. Ron Rich reviewed the chapter 'Preparing honey for sale and competition'.\n\nThanks are also due to Prof. Ben Oldroyd, Bill Shay, Cindy Edwards, Howard Kirby, Anne Cliff, Roger Mitchell (UK), and Steve Pernal, Agriculture and Agri-Food Canada, for providing information and\/or reviews for different parts of the manuscript. George Bekier read the manuscript in its entirety and provided valuable comments.\n\nMany other people also assisted in various ways with the preparation of this book \u2014 far too many to mention individually \u2014 but I would like to thank each of them collectively for their time and insights.\n\nFinally an acknowledgement is due to the amazing bees mentioned throughout these pages. Without their influence this book would not have been written, and I hope the art and craft of beekeeping as described will assist readers in some small way towards an understanding their complex world. \n**PHOTOGRAPHS & DRAWINGS**\n\nMost of the photographs and drawings were taken or made by the author except for:\n\nAdrian Iodice: 12.3, 12.15a, 12.15b\n\nAlison Mellor: 14.23, 14.24\n\nBarbara Owen: 13.1, 13.2, 13.5, 13.6, 13.8\u201313.11, 13.15, 13.16, 13.20, 13.21, 13.24, 13.32\u201313.42, 13.44\u201313.46, 19.19, 19.21\n\nChris Lahy: 5.4, 5.5\n\nChris Luck: 19.14, 19.15\n\nCSIRO: 18.6\u201318.8, 19.9\n\nDenis Anderson, CSIRO: 16.11, 16.13, 17.6, 19.1, 19.12, 19.23\n\nDavid Briggs: 1.1, 1.6\n\nDaisy Higgs: 3.15, 4.3\u20134.26, 4.31, 4.32, 5.12\u20135.17, 8.10\u20138.12\n\nDave Reede: 12.1\n\nErnesto Guzman: 5.32, 15.3, 15.4, 15.15, 18.1\n\nErica Siegel: 13.15, 13.23, 13.24, 14.1, 14.2,\n\n14.9\u201314.23, 14.25\u201314.27, 14.30\u201314.38, 14.40, 14.42\u201314.45, 14.47\u201314.49\n\nGlenn Apiaries, USA: 19.3\u201319.5\n\nGeorge Vikentios: 13.43\n\nHamilton Gardens, New Zealand: 6.9\n\nHayley Carr: 6.6, 6.8\n\nIan Kimber: 17.1\n\nJason Critchley: 1.2, 1.15, 1.22, 1.27, 5.1, 5.2, 5.7, 5.28, 6.1, 6.13, 10.11, 10.12, 13.3, 13.6, 13.10\u201313.12, 13.16, 13.17, 13.25\u201313.28, 19.11\n\nJoe Horner: 1.7, 1.8\n\nJohn McLean, NZ: 1.14, 8.6\n\nJulie Workman: 19.19\n\nKate Nolan: 1.9, 1.16, 1.17, 1.19, 1.21, 1.25, 1.26, 3.21, 5.25, 6.2, 7.2, 19.17\n\nKen Walker, Museum Victoria: 14.5, 17.16\n\nLaura Matthews: 6.11\n\nLucy Wilks: 1.3, 1.23, 2.33, 9.9, 12.2, 13.4, 13.40, 15.16\n\nMegan Halcroft: 14.4, 14.7\n\nMarc Newman: 14.6\n\nNerina Lascelles: 3.16\n\nNick Monaghan: 14.39, 14.41\n\nBiobees, New Zealand: 19.6\n\nOleksandr Holovachov: 13.20, 13.21\n\nPaul Kozak, Government of Ontario, Canada: 15.8, 16.2, 16.4, 16.5, 16.8, 17.7\u201317.9\n\nPatrik Berger: 14.24\n\nPeter Boersch: 19.25\n\nPaul van Westendorp: 15.11\n\nR. Brito: 14.8\n\nRussell Zabel: 14.3, 18.3, 18.4\n\nTheresa Daley: 3.18, 5.3, 6.10\n\nTrevor Monson: 7.1, 7.25\n\nUSDA: 1.5, 15.12, 15.13, 16.1, 16.3, 16.11\n\nVanessa Kwiatkowski: 6.7\n\nWikimedia \u2014 Sarefo: 18.5; Charles Lam: 19.2; Vera Buhl: 19.6; JJ Harison: 19.8; Tim Evison: 19.10; Frank Hornig: 19.13; Gideon Pisanty: 19.21; L. Shyamal: 19.24\n\nWarwick Bone: 12.12, 12.13\n\nYeonsoon Bourke: 17.17\n\nZachary Huang: 5.33, 15.5, 15.7, 15.9, 15.10, 15.14, 16.12\nBibliography\n\nGENERAL\n\nBlackiston, H. 2015, _Beekeeping for Dummies,_ Wiley, Hoboken, New Jersey, USA\n\nBonney, Richard F. 1990, _Hive Management,_ Storey Publishing, Massachusetts, USA\n\nCramp, David. 2013, _A Practical Manual of Beekeeping,_ Spring Hill, Oxford, UK\n\nCliff, Ann. 2010, _Bee Book: Beekeeping basics, harvesting honey, beeswax, candles and other bee business,_ Aird Books, Melbourne, Australia\n\nFlottum, Kim. 2011, _Better Beekeeping,_ Quarry Books, Massachusetts, USA\n\nFlottum, Kim. 2010, _The Backyard Beekeeper: An absolute beginner's guide to keeping bees in your yard and garden,_ Quarry Books, Massachusetts, USA\n\nGoodman, Russell and Kaczynski, Peter. 2015, _Australian Beekeeping Guide,_ publication no. 14-098. Rural Industries & Research Development Corporation (RIDC), Canberra, Australia\n\nHughes, Craig. 2010, _Urban Beekeeping: A guide to keeping bees in the city,_ The Good Life Press, Preston, UK\n\nJohnson, Samantha and Johnson, Daniel. 2011, _The Beginner's Guide to Beekeeping,_ Voyageur Press, Minneapolis, Minnesota, USA\n\nMatheson, Andrew and Reid, Murray. 2011, _Practical Beekeeping in New Zealand,_ Exisle Publishing, Auckland, New Zealand\n\nNSW, Department of Industry, 2013, _Bee Agskills: A practical guide to farm skills,_ NSW Department of Primary Industries, publication no. B121, NSW, Australia\n\nSammataro, Diana and Avitabile, Alphonse. 2011, _The Beekeepers' Handbook,_ Comstock Publishing Associates, Ithaca, NY, USA\n\nWarhurst, Peter. 2013, _The Bee Book,_ self-published by Peter Warhurst, Queensland, Australia\n\nWoodward, David. 2010, _Queen Bee: Biology, rearing and breeding,_ Northern Bee Books, West Yorkshire, UK\n\nBEE HEALTH\n\nAston, David and Bucknall, Sally. 2010, _Keeping Healthy Honey Bees,_ Northern Bee Books, West Yorkshire, UK\n\nGoodwin, Mark. 1999, _Elimination of American Foulbrood Disease Without the Use of Drugs,_ National Beekeepers Association of New Zealand (NBANZ), Wellington, New Zealand\n\nGoodwin, Mark and Taylor, Michelle. 2007, _Control of Varroa,_ New Zealand Ministry of Agriculture and Forestry, Wellington, New Zealand\n\nPernal, Stephen and Clay, Heather. 2013, _Honey Bee Diseases and Pests,_ Canadian Association of Professional Apiculturists (CAPA), Beaverlodge, Alberta, Canada\n\nPlant Health Australia 2012, _Biosecurity Manual for the Honey Bee Industry,_ Plant Health Australia and Rural Industries Research & Development Corporation (RIRDC), Canberra, Australia\n\nSomerville, Doug. 2005, _Fat Bees, Skinny Bees: A manual on honey bee nutrition for bee keepers,_ publication no. 05\/054, Rural Industries Research & Development Corporation (RIRDC), Canberra, Australia\n\nSomerville, Doug. 2014, _Healthy Bees: Managing pests, diseases and other disorders of the honey bee,_ NSW Department of Primary Industries, NSW, Australia\n\nLIGURIAN BEES ON KANGAROO ISLAND, SOUTH AUSTRALIA\n\nKoulianos, S. and Crozier, RH. 1966, Mitochondrial DNA sequence provides further evidence that the honeybees of Kangaroo Island, Australia are of hybrid origin, _Apidologie,_ 27: 165\u201374\n\nGlatz, Richard V. 2014, Curious case of Kangaroo Island honeybee _Apis mellifera Linnaeus,_ 1758 _(Hymenoptera: Apidae)_ Sanctuary, _Australian Entomological Society Journal,_ DOI: 10.1111\/aen.12124\n\nOldroyd, BP, Sheppard, WS, Stelzer, JA. 1992, Genetic characterization of the bees of Kangaroo Island, South Australia, _Journal of Apicultural Research,_ 31 (3\/4: 141\u201348).\n\nSUSTAINABLE BEEKEEPING\n\nChandler, PJ. 2009, _The Barefoot Beekeeper,_ Biobees, Milton Keynes, UK\n\nConrad, Ross. 2013, _Natural Beekeeping,_ Chelsea Green Publishing, Vermont, USA\n\nHeaf, David. 2010, _The Bee-friendly Beekeeper: A sustainable approach,_ Northern Bee Books, West Yorkshire, UK\n\nMagnum, Wyatt. 2012, _Top-Bar Hive Beekeeping: Wisdom and pleasure combined,_ Stinging Drone Publications, Virginia, USA\n\nRowe, Tim. 2010, _The Rose Hive Method,_ Green Hat Books, UK\n\nTHE BEE-FRIENDLY GARDEN\n\nBlazey, Clive. 2012 _The Australian Vegetable Garden_ (2nd Edition), New Holland Publishers (Australia) Pty Ltd, Sydney, Australia\n\nBlazey, Clive. 2014, _There is No Excuse for Ugliness,_ The Diggers Club, Dromana, Victoria, Australia\n\nHemphill, John and Rosemary. 2007, _Herbs: Their cultivation and usage,_ Allen & Unwin, Sydney, Australia\n\nHooper, Ted and Taylor, Mike. 2006, _The Bee Friendly Garden,_ Alphabet and Image Ltd, Devon, UK\n\nLeech, Mark. 2012, _Bee Friendly: A planting guide for European honeybees and Australian native pollinators,_ Rural Industries Research & Development Corporation (RIRDC), publication no. 12\/014, Canberra, Australia\n\nLittle, Brenda. 2000, _Companion Planting in Australia,_ Spring Hill, Oxford, UK\n\nLittle, Maureen. 2011 _, The Bee Garden,_ Spring Hill, Oxford, UK\n\nMarshall, Tim. 2010, _Bug: The ultimate gardener's guide to organic pest control,_ ABC Books, HarperCollins, Sydney, Australia\n\nRouble, John. 1999, _Backyard Composting,_ Green Earth Books, UK\n\nUrquhart, Paul. 1999. _The New Native Garden: Designing with Australian Plants,_ New Holland Publishers (Australia) Pty Ltd, Sydney, Australia\n\nWoodward Penny. 2012, _Pest Repellent Plants_ (2nd edition), Hyland House, Carlton, Victoria, Australia\n\nWrigley, John and Fagg, Murray. 2002, _Starting Out With Natives,_ New Holland Publishers (Australia) Pty Ltd, Sydney, Australia\n\nXerces Society. 2011, _Attracting Native Pollinators,_ Storey Publishing, Massachusetts, USA\n\nNATIVE BEES\n\nDollin, Anne, Batley, Michael, Robinson, Martyn and Faulkner, Brian. 2007, _Native Bees of the Sydney Region, Australian Native Bee Research Centre_ , Richmond, NSW, Australia\n\nDollin, Anne. 2010, _Native Bees of Australia series,_ Australian Native Bee Research Centre, Richmond, NSW, Australia\n\n_Booklet 1 - Introduction To Australian Native Bees_\n\n_Booklet 2 - Nests of Australian Stingless Bees_\n\n_Booklet 3 - Behaviour of Australian Stingless Bees_\n\n_Booklet 4 - How to Recognise the Different Types of Australian Stingless Bees_\n\n_Booklet 5 - Keeping Australian Stingless Bees in a log or box_\n\n_Booklet 6 - Crop Pollination With Australian Stingless Bees_\n\n_Booklet 7 - Tips on Stingless Beekeeping by Australian Beekeepers_ (Vol. 1)\n\n_Booklet 8 - Tips on Stingless Beekeeping by Australian Beekeepers_ (Vol. 2)\n\n_Booklet 9 - Boxing and Splitting Hives_ \u2014 _A complete do it yourself guide for stingless bee keepers_\n\n_Booklet 10 - Tips on Stingless Beekeeping by Australian Beekeepers_ (Vol. 3)\n\nKlumpp, John. 2007, _Australian Stingless Beekeeping,_ Earthling Enterprises, Queensland, Australia\n\nNATURAL BEEKEEPING WEB RESOURCES\n\nwww.biobees.com\n\nwww.beekeepingnaturally.com.au\n\nBEEKEEPING WEB FORUMS\n\nwww.beesource.com\/forums\n\nwww.beemaster.com\n\nNATIVE BEE WEB RESOURCES\n\nwww.aussiebee.com.au\n\nwww.beesbusiness.com.au\n\nwww.ozbirds-wildlife.com\n\nwww.sugarbag.net\n\nwww.zabel.com.au\n**Index**\n\nItalicised page number indicate the presence of photos, diagrams, illustrations, or additional information in the captions.\n\nA\n\n_Aethina tumida see_ Small Hive Beetle\n\nAfrican honey bee \u2013\n\nafter-swarms\n\nair flow \u2013, _, _\n\nalmond flowers _\u2013_, _\u2013, _\n\nAmerican Foul Brood (AFB) \u2013, __\n\nanimals, proximity to hives\n\nannuals ,\n\nantibiotic residues\n\nants \u2013\n\nApiary Code of Practice\n\n_Apis binghami_\n\n_Apis breviligula_\n\n_Apis cerana see_ Asian honey bees\n\n_Apis dorsata_ \u2013, , __\n\n_Apis florea_ \u2013, __\n\n_Apis laboriosa_ , __\n\n_Apis mellifera_\n\nraces of \u2013\n\n_see also_ European honey bee\n\n_Apis mellifera adansonii_\n\n_Apis mellifera capensis_ \u2013, __\n\n_Apis mellifera carnica_ , ,\n\n_Apis mellifera caucasica\u2013, ,_ \u2013\n\n_Apis mellifera ligustica see_ Italian bee\n\n_Apis mellifera scutellata_ \u2013, _, _\n\nApithor Hive Beetle Harbourage\n\naroma, honey\n\n_Ascosphaera apis_ fungus \u2013\n\nAsian honey bee _\u2013_\n\nbehaviour\n\nbrood development\n\ncharacteristics \u2013\n\ncost\n\ndwarf species , __\n\ngiant species , __\n\nglobal distribution \u2013\n\nlifecycle\n\nAsian hornets , __\n\n_Austroplebia_ species \u2013, __\n\nB\n\nbase boards, vented\n\nbee behaviour, in cold temperatures \u2013\n\nbee bread\n\nbee brushes , __\n\nbee glue \u2013\n\nbee losses, US statistics\n\nbee-friendly gardens \u2013\n\nbeekeeping, undesirable practices \u2013\n\nbeeswax extractor, solar \u2013, __\n\nbeetle traps _,_ \u2013\n\nBlack Queen Cell Virus (BQCV)\n\nBlue banded bee _,_ \u2013, _, _\n\nBoardman feeder\n\n_Bombus terrestris see_ bumblebees\n\nbottling methods \u2013\n\nbottom boards\n\nAmerican-style __\n\nslotted __ ,\n\nvented \u2013\n\nboxes\n\ndepths\n\nrelative sizes __\n\nsizes \u2013\n\nBraula fly _(Braula coeca)_\n\nadult __\n\ndescribed \u2013\n\nmite on bee __\n\nbrightness, honey\n\nbrood\n\nchilled \u2013\n\ndiseases affecting \u2013, _,_ , __ , __\n\n_see also_ capping\n\nbrood boxes\n\nconstruction \u2013, __\n\ndescribed\n\npainting exteriors , _\u2013_\n\nreducing number\n\nreverse after winter _\u2013_\n\nspring inspection \u2013\n\nspring\/summer inspections \u2013\n\n_see also_ supers boxes\n\nbrood comb, remove from possum box \u2013, __\n\n_Bufo marinus_ \u2013, __\n\nbumblebees\n\ncharacteristics \u2013\n\ncolonies \u2013\n\ncommercial colonies __\n\ndistribution\n\nlifecycle , \u2013\n\nnectar collection method\n\nshort haired\n\nworker bees __ , \u2013\n\nburr comb __\n\ndifficulties with\n\ndiscouraging _, _\n\nremoving\n\nburrows, masked bees\n\nbuzz pollinators , , ,\n\nC\n\ncane toads \u2013, __\n\nCape honey bee \u2013, __\n\ncapping _, _\n\ndamaged by AFB __\n\ndrone cells __\n\nremoving \u2013, _, , _\n\nstages __\n\nworker cells __\n\nCarniolan bee , __ ,\n\nCaucasian bee _\u2013,_ , \u2013\n\ncavity nesting bees\n\ncell size, optimum \u2013\n\nChalkbrood \u2013, __\n\nChequered Cuckoo bee\n\nchildren's clothing , _,_\n\nchilled brood \u2013\n\nchunk honey \u2013\n\nclearness, honey\n\nclimbing plants , , , __\n\nclubs, membership benefits\n\nclusters, in cold temperatures\n\nCoastal Brown ants __\n\ncodes of practice\n\ncold temperatures\n\nbee behaviour \u2013\n\nclusters\n\ncolonies\n\n_Austroplebia_ species \u2013\n\nbuilding up\n\nbumblebees \u2013\n\nchecking strength of\n\nferal \u2013,\n\nmasked bees\n\nmerging \u2013, __\n\nwasps\n\n_see also_ nests\n\nColony Collapse Disorder (CCD) \u2013\n\ncolour\n\ngranulated honey\n\nhoney\n\ncomb honey\n\ncompetition preparation __\n\ndisplaying __\n\nextracting\n\nfreezer storage \u2013\n\nCommon wasp _see_ English wasp\n\ncompetition\n\nbottling methods \u2013\n\nchunk honey \u2013\n\nclasses within\n\nextraction and preparation \u2013\n\nframes of honey \u2013\n\ngranulated honey \u2013\n\nliquid honey \u2013\n\npreparing for\n\nstraining methods \u2013, __\n\ncongestion of hive __\n\ncontainer plants\n\ncopper napthenate\n\ncreamed honey, competition criteria\n\nD\n\nDawson's Burrowing bee\n\nDeformed Wing Virus (DWV) \u2013, __ , \u2013,\n\nDemaree method, swarm management \u2013, __\n\ndensity of honey _,_\n\ndevelopmental differences, queens, workers, drones __\n\nDiptera fly __\n\ndiseases\n\nadult bees \u2013, , \u2013,\n\nbrood _,_ \u2013, _, , _, __\n\ncheck for\n\ndrone cells , __\n\nglobal spread \u2013\n\nminimisation practices\n\nUnited States\n\ndomestic animals, near hives\n\nDomino Cuckoo bee\n\ndrone cells\n\nconstruction \u2013, __\n\ndiseased , __\n\nsign of swarming __\n\ndrone congregation areas (DCA) \u2013\n\ndrones __ , __ , __\n\nbumblebees\n\ncapped cells __\n\negg development __\n\nfeeding of eggs __\n\nlarvae development , \u2013, __\n\nlimited role \u2013\n\npupae __\n\nremoval of excess\n\nrole in hive\n\ndusting powders\n\ndwarf honey bees\n\nE\n\neggs\n\nfeeding differences __\n\nlaying and development\n\nelectric extractors , __\n\nemergency queen cells _,_\n\nEmlocks __ ,\n\nEnglish wasp\n\nAustralian distribution\n\nkilling a honey bee __\n\nmarkings __\n\nqueen nesting __\n\nentrance closure method \u2013\n\nenvironment of hives _see_ location of hives\n\nenvironmental weeds , __\n\nEuropean Foul Brood (EFB) \u2013, \u2013, __ , __ ,\n\nEuropean honey bee __\n\nbrood development\n\ncharacteristics \u2013\n\ngeographic origins, map __\n\nsubspecies in Australia\n\nEuropean wasp markings __\n\nmeateaters\n\nnest __\n\nstings\n\nexcluders __\n\nchoice of material __\n\nif not used __\n\nreasons for use \u2013\n\nuse of , \u2013\n\nexotic plants, with natives \u2013\n\nextraction\n\ncomb honey\n\nfor competition \u2013\n\nfirst experience\n\nframes ready for __\n\nby fruit press __\n\nkitchen sieve \u2013, _\u2013_\n\nextractors\n\nchoosing \u2013\n\nradial _,_\n\nsolar beeswax \u2013, __\n\nthree-frame _, _\n\ntypes\n\nusing \u2013\n\neyesight of bees \u2013\n\nF\n\nFEDSS (Food, Eggs, Disease, Space, Swarming) \u2013\n\nfeeders\n\nfront (Boardman)\n\nschematic diagram __\n\ntop \u2013\n\nuse of\n\nfeeding\n\neach other , __\n\nsyrup to hive \u2013, __\n\nfemale worker bees _see_ worker bees\n\nferal colonies \u2013,\n\n'fighting swarms'\n\nfilters\n\nflavour, competition criteria\n\nfloral source\n\ncompetition criteria\n\ninsufficient\n\none type per flight , __\n\nflowering trees\n\nflowers\n\nannuals\n\nbee-attracting , , , __ , __\n\nbell shaped __\n\nclimbing plants __\n\nground covers\n\nhedging plants\n\nherb gardens \u2013, , __\n\nlate summer\/autumn\n\nmulti-floral , _\u2013_\n\nnative plants _,_\n\nsimple single __\n\ntubular shaped , __\n\nvarious habits \u2013\n\nvegetable garden _\u2013_\n\nforaging \u2013\n\nformic acid, safety of\n\nfoundation\n\nattaching to frames \u2013, _\u2013_\n\ncharacteristics \u2013\n\nreuse methods \u2013, __\n\nusing strips __\n\nwax or plastic\n\nwhite bloom \u2013\n\nframe feeders _see_ feeders\n\nframe wiring board\n\nframe-making jig , __\n\nframes\n\nappearance for competition \u2013\n\ncleaning ,\n\ncompetition criteria , , __\n\nconstruction \u2013, , _\u2013_\n\nemptying for storage _\u2013_\n\ninspections , __\n\nplastic _\u2013_\n\npreparing \u2013\n\nready for extraction __\n\nsizes of \u2013\n\nstorage _,_ , _,_\n\nfruiting trees \u2013\n\nfull-depth boxes __ ,\n\nfungicides\n\nharm from\n\npoisoning \u2013\n\nG\n\ngaiters\n\n_Galleria mellonella see_ Greater Wax Moth\n\ngalvanised steel\n\ngarden conditions \u2013\n\ngardens\n\nbee-friendly \u2013\n\nwork in progress \u2013\n\nGhana, aggressive bees\n\ngiant honey bees\n\ngloves \u2013, , __\n\ngranulated honey\n\nadding starter __\n\ncompetition criteria\n\npreparation , __\n\nGreat Carpenter bee \u2013, __\n\nGreater Wax Moth\n\nadult __\n\ncocoons __\n\ndamage caused by , __\n\ninfestation signs _,_\n\nlarva __\n\nlarval hive damage __\n\ntreatment for \u2013\n\nGreen Carpenter bee\n\ngreen manure crop\n\nGreen tree ants _,_\n\ngreenhouse pollination\n\ngrevilleas, prostrate\n\nground covers\n\ngroup plantings \u2013\n\nH\n\nhalf-depth boxes __ ,\n\n_Halictidae_ bee __\n\nhedging plants, flowering species\n\nherb gardens \u2013, __\n\nherbicides ,\n\nhigh fructose corn syrup (HFCS)\n\nhive construction\n\nAmerican-style base , __\n\nthe base \u2013, _\u2013, _\n\nexplained \u2013, _\u2013_\n\nhive mats , __\n\nlids \u2013, __\n\nplastic _,_\n\nschematic _, _\n\n_see also_ brood boxes; supers boxes\n\nhive destruction, burning\n\nhive management\n\nautumn \u2013, \u2013\n\ndisease minimisation\n\nfirst year maintenance \u2013\n\ninventions \u2013\n\nto maximise production\n\nseasonal conditions \u2013\n\nWarr\u00e9 hives \u2013\n\nwinter\n\n_see also_ inspections\n\nhive mats , _, _\n\nhive purchase\n\nchoice of source \u2013\n\nestablished or new \u2013\n\nhive size, reducing for winter , __\n\nhive splitting\n\ndiagram __\n\nmanaging swarming \u2013\n\nfor raising queens \u2013\n\nhive tools __\n\nAmerican __\n\nexplained \u2013\n\nsterilising \u2013, _,_ \u2013\n\nhives\n\ncongested __\n\ndamp \u2013\n\nhoney-bound __\n\nmoving \u2013, _, _\n\noverheating \u2013\n\nraised height , __\n\nsealing methods \u2013\n\nwinter shutdown _\u2013,_ \u2013, __\n\n_see also_ weather protection\n\nhobby beekeepers, tips for\n\n_Homalictus urbanus_ , _, _\n\nhoney bees\n\ncastes \u2013, __\n\nas superorganisms , _,_\n\nthree races \u2013\n\nhoney crop __\n\nhoney production\n\nmanaging flow \u2013\n\nmaximising ,\n\nhoneycomb\n\nangle of slope __\n\n_see also_ comb honey\n\nhorizontal top bar hives\n\nadvantages \u2013\n\ndisadvantages\n\nexplained\n\n_see also_ top bar hives\n\nI\n\nIdeal boxes _,_ \u2013\n\n_Imidacloprid_ insecticide\n\ninner-city beekeeping \u2013, _,_ \u2013, __\n\ninspections\n\nearly spring \u2013\n\nfirst year hives \u2013\n\nframes , __\n\nfrequency \u2013\n\nlate spring\/summer \u2013\n\nmethod __\n\nspring\n\nswarm management \u2013\n\ntop bar hives\n\ninsulation _see_ weather protection interbreeding \u2013, , \u2013\n\ninventions, management practices \u2013\n\nIsraeli Acute Paralysis Virus (IAPV) \u2013,\n\nItalian bee\n\nbody colour __\n\ncharacteristics\n\nJ\n\njackets \u2013, __ , __\n\njumbo (deep) boxes __ ,\n\nK\n\nKenyan top bar hives __\n\nframe inspection __\n\nhorizontal\n\nvertical, frame removal\n\nKiller bees\n\nL\n\nlabelling containers \u2013\n\nLangstroth hives \u2013,\n\nLeafcutter bee _,_ , _,_ , __\n\n_Leioproctus_ species __\n\nLesser Wax Moth\n\ninfestation signs\n\nlarvae\n\nlids\n\nAustralian conditions\n\ncleaning ,\n\nconstruction __\n\nscreened __\n\ntypes \u2013\n\nventilated __\n\nWarr\u00e9 hives , __\n\nlifecycles\n\nAsian honey bee\n\nbumblebees \u2013\n\ngeographical differences\n\nhoney bees \u2013, __\n\nSmall Hive Beetle \u2013\n\n_Lipotriches_ species , __\n\nliquid honey marking criteria\n\nlocal authorities \u2013\n\nlocation of hives _, _\n\ncloseness to house\n\ncountry hives __\n\nimportant factors\n\nnear animals\n\norientation \u2013\n\nto prevent overheating \u2013\n\nproximity to neighbours \u2013\n\nregulations\n\nseasonal\n\nspring clean ,\n\nurban hives __\n\nM\n\nmacadamia trees, social bee pollination \u2013\n\nmanagement practices, inventions \u2013\n\nManley boxes __ ,\n\nmasked bees , __\n\nmatchstick test , _,_\n\nmating\n\ninterbreeding \u2013, , \u2013\n\nlack of need for\n\nprocess \u2013, __\n\nmating nucleus\n\npreparing for queen _,_ , __\n\nqueens emerging from \u2013\n\nMeat ants __\n\n_Megachile pictiventris,_ , __ , , __\n\nmice \u2013, __\n\nmigratory beekeeping __\n\nCCD factor\n\ndisease pest transmission \u2013\n\nreasons for , \u2013\n\nstress on bees\n\nUS diseases spread\n\n_Varroa_ spread \u2013\n\nmigratory lids\n\nMiller feeders \u2013\n\nmites \u2013\n\nmiticides ,\n\nmonocultures , \u2013, _\u2013_\n\nmouse guard __\n\nmummies\n\nN\n\nnadiring process ,\n\nNasonov scent \u2013, __\n\nnative plants, flowering \u2013,\n\nnatural beekeeping methods , \u2013, \u2013\n\nnavigation methods \u2013\n\nnectar\n\ncollection process \u2013,\n\nflower styles\n\nrole\n\ntimed release\n\nwater content\n\nnectaries\n\naccess to \u2013\n\nnectar release\n\nNeon Cuckoo bee , , __\n\nneonicotinoids\n\nnest defence, native social bee\n\nnests\n\nAsian honey bee\n\nAsian hornets , __\n\nbumblebees \u2013\n\ncavity nesting bees\n\ndwarf Asian honey bees __\n\ndwarf honey bees __\n\ngiant Asian honey bees _,_ , __\n\nGreat Carpenter bee \u2013\n\n_Lipotriches_ species\n\nmasked bees\n\nPeacock Carpenter bee\n\nTeddy Bear bee \u2013\n\nwasps ,\n\nnitrile gloves , __\n\n_Nosema ceranae_ fungus \u2013\n\nnovice beekeepers, tips for\n\nnucleus boxes\n\nmoving hives , __\n\npurchased colony\n\nnursery bees\n\nfeeding larvae\n\nfeeding queen larva __\n\nmaking royal jelly\n\nrole\n\nSacbrood infection \u2013\n\nnutritional information, labels\n\nO\n\norganic chemicals\n\norientation _see_ location of hives orientation flights , \u2013\n\noveralls _,_ \u2013, __\n\noxalic acid\n\noxytetracycline hydrochloride (OTC)\n\nP\n\nparasites \u2013\n\nPeacock Carpenter bee , __\n\nperennials ,\n\npesticides\n\nEuropean restrictions \u2013\n\nharm from\n\npoisoning \u2013\n\nsub-lethal effects\n\npests\n\nattacks by\n\nspread \u2013\n\nUnited States\n\nPfund meter __\n\npheromone\n\nto mask smell\n\nNasanov scent \u2013, __\n\nproduction by queen \u2013\n\nreleased with sting\n\nplant choice\n\nconsiderations \u2013, \u2013\n\nnatives _vs_ exotic \u2013\n\nplastic\n\nfoundation\n\nframes _\u2013_\n\nplastic hives _,_\n\npollen\n\ncollecting \u2013\n\ncollection\n\nfeeding \u2013, __\n\nrole\n\nstorage in hive __\n\nstorage on bee _, _\n\npollen substitute , __\n\npollination\n\nprocess\n\nproviding hives for \u2013\n\npollinators\n\nalternative \u2013\n\nbumblebees\n\nbuzz ,\n\npolyester webbing\n\npopsicle sticks , , __\n\npossum boxes, swarms in , , _\u2013,_ \u2013\n\npot gardens\n\nprime swarm\n\npropolis \u2013\n\nprotective clothing \u2013\n\nQ\n\nqueen cages , __\n\nqueen candy\n\nqueen cells\n\nemergency __\n\nlocation and shape , __\n\nsignificance __\n\ntransferring __\n\nqueen excluders _see_ excluders\n\nqueen rearing\n\ncolony choice \u2013\n\nhive preparation __\n\nhive splitting \u2013\n\nprofessionals\n\nreasons for\n\nqueens _, _\n\nautumn laying\n\nbumblebees , \u2013\n\ndeath of\n\ndevelopment stages __\n\nemerging \u2013, \u2013\n\nfeeding of eggs __\n\nlarval life\n\nlocation on transfer \u2013\n\nmarking , __\n\nmating process \u2013, __\n\nnew one required , \u2013\n\nphysical appearance\n\nrole in hive\n\nquilt box \u2013, __\n\nR\n\nradial extractors _,_\n\nReed bees\n\nregistration number, hive\n\nResin bee _,_ \u2013, _, _\n\n_Rhinella marina_ \u2013, __\n\nrisers\n\nrobber attacks\n\nrobbing, minimising \u2013\n\nroyal jelly ,\n\nS\n\nSacbrood Virus (SBV) \u2013, __\n\nsale\n\nlabelling for \u2013\n\npreparing for \u2013, _\u2013_\n\npresentation \u2013\n\nsaucepan method, melting wax _, _\n\nscout bees\n\nshort haired bumblebees\n\nshrubs\n\nskep hives __\n\nslum-gum\n\nSmall Hive Beetle\n\nadult __\n\nin Australia , \u2013\n\nbeetle trap __\n\ninfestation signs \u2013\n\nlarvae , _,_\n\nlifecycle \u2013\n\nspoiling honey frame __\n\nspread\n\ntrapping methods __\n\ntreatment for , \u2013\n\nsmokers\n\ncleaning off tar ,\n\nsafe transporting , __\n\ntypes __\n\nusing \u2013, _\u2013,_ , ,\n\nsocial bees \u2013\n\nsolar beeswax extractors \u2013, __\n\nsolitary bees , \u2013\n\nsplits _see_ hive splitting\n\nspraying\n\ncrops\n\ngarden pests\n\nstarter, adding\n\nsterilisation\n\nhive tools \u2013, \u2013\n\nhives\n\nstings\n\nreactions to \u2013\n\nwasps\n\nworker bees\n\nstraining methods \u2013, __\n\nSugar ants __\n\nsugar candy _,_\n\nSugar Shake Test , __\n\nsugarbag honey\n\nsuperorganisms _see_ honey bees\n\nsupers boxes\n\nadding\n\nadding new\n\nconstruction \u2013, , __\n\nexplained\n\n_see also_ brood boxes\n\nsupersedure \u2013\n\nswarm management\n\nDemaree method \u2013, __\n\nhive inspections \u2013\n\nsplitting hives \u2013\n\nswarm scouts ,\n\nswarming\n\npreparing for \u2013\n\nprocess \u2013\n\nswarms\n\nbuy or collect\n\ncapturing \u2013, _\u2013_\n\ncycle\n\nminimising \u2013\n\nrehousing \u2013, __\n\nremove from possum box _\u2013_\n\nsyrup, feeding \u2013, __\n\nsystemic pesticides\n\nT\n\ntangential extractors , __\n\nTanzanian horizontal top bar hive \u2013\n\nTeddy Bear bee \u2013, __\n\ntelescopic lids , \u2013\n\ntertiary swarms\n\n_Tetragonula_ species \u2013, , , ,\n\n_Thyreus_ species ,\n\ntools\n\nbee brushes , __\n\nfor capturing swarms\n\nexamples _\u2013_\n\nframe-making jig __\n\nhive tools __\n\nsterilising _\u2013_\n\nuncapping knife __\n\n_see also_ extractors; hive tools;\n\nsmokers\n\ntop bar hives\n\nbuying\n\ndisadvantages ,\n\ninspecting __\n\nnatural beekeeping\n\n_see also_ horizontal top bar hives\n\nTracheal Mite\n\ntrophallaxis , __\n\n_Tropilaelaps_ mites __\n\ndescribed \u2013\n\ninfestation signs \u2013\n\nspread\n\ntreatment for \u2013\n\nU\n\nunder-supering \u2013\n\nUnited States, pests and diseases ,\n\nurban beekeeping \u2013, _,_ \u2013, __\n\nV\n\n_Varroa_ mites _\u2013, , _\n\ncontrol measures \u2013,\n\ndescribed \u2013\n\nDWV spread\n\nimpact of ,\n\ninfestation signs \u2013\n\nspread \u2013\n\ntolerant lines of bees\n\ntreatment \u2013\n\nvegetable gardens \u2013\n\nveil repair \u2013, __\n\nveils \u2013, __\n\nvent holes \u2013\n\n_Vespa velutina nigrithorax_\n\n_Vespula germanica see_ European wasp\n\n_Vespula vulgaris see_ English wasp vision of bees \u2013\n\nvitellogenin\n\nW\n\nwaggle dance \u2013, _, ,_\n\nWarr\u00e9 hives , __\n\nbuying\n\ndesign \u2013\n\nnadiring process _,_\n\nwasps\n\neffect on bees\n\nidentification\n\nlifecycle, geographical differences\n\nstings\n\n_see also_ English wasp; European wasp\n\nwater\n\naccess for bees\n\navailability \u2013\n\nchicken feeder __\n\ngarden needs\n\nground water __\n\nwax\n\ncolour \u2013\n\nmelting \u2013, , __\n\nwax foundation _see_ foundation\n\nWax Moth\n\nweather protection\n\nabove lid , __\n\nfrom hot sun __\n\nunder lid\n\ntemperature control\n\ntypes of lids \u2013\n\nWeathertex lids\n\nWestern honey bee\n\nwhite bloom \u2013\n\nwinter\n\nbee behaviour \u2013\n\nfeeding hive \u2013\n\nhive management \u2013,\n\nhive shutdown _\u2013,_ \u2013, __\n\nworker bees _, , _\n\nbumblebees _,_ \u2013\n\ncapped cells __\n\ncapped larvae __\n\ndescribed\n\ndwarf Asian honey bees __\n\negg development __\n\nfeeding of eggs __\n\ngiant Asian honey bees __\n\nlarvae development , \u2013\n\nlife span\n\nwork role stages __\n\nWSP boxes _,_\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nCONTENTS\n\nINTRODUCTION\n\nVIEW FROM TOUFFOU\n\nCODETTA\n\nTHE DIGITAL REVOLUTION\n\nTHE SHORT MARCH\n\nTHE DIGITAL ECOSYSTEM\n\nTO BE OR NOT TO BE A MILLENNIAL\n\nTHE POST-MODERN BRAND\n\nCONTENT IS KING; BUT WHAT DOES IT MEAN?\n\nCREATIVITY IN THE DIGITAL AGE\n\n\"GIVE ME GOLD\"\n\nART OR SCIENCE?\n\nPERVASIVE CREATIVITY\n\nDATA: THE CURRENCY OF THE DIGITAL AGE\n\n\"ONLY CONNECT\"\n\nCREATIVE TECHNOLOGY: THE SWEET SPOT\n\nTHE THREE BATTLEGROUNDS\n\nPUTTING THE SOCIAL BACK INTO SOCIAL MEDIA\n\nTHE JOY OF MOBILITY\n\nCONTINUOUS COMMERCE\n\nDIGITAL TRANSFORMATIONS\n\nDIGITAL POLITICS\n\nDIGITAL GOVERNMENT\n\nDIGITAL TOURISM\n\nDIGITAL SOCIAL RESPONSIBILITY\n\nFIVE GIANTS OF ADVERTISING IN THE DIGITAL AGE\n\nBOB GREENBERG\n\nAKIRA KAGAMI\n\nMARTIN NISENHOLTZ\n\nMATIAS PALM-JENSEN\n\nCHUCK PORTER\n\nMY BRAIN HURTS\n\nTHE NEW SHAPE OF THE WORLD\n\nCULTURE, COURAGE, CLIENTS AND CASTANETS\n\nEPILOGUE\n\nENDNOTES\n\nINDEX\n\nACKNOWLEDGEMENTS\n\nPICTURE CREDITS\nINTRODUCTION\n\nVIEW FROM TOUFFOU\n\nUnlike authors who have to worry about why they are writing at all, my purpose is very narrow. The point of this book is to persuade people to read or re-read _Ogilvy on Advertising_ by David Ogilvy. It is still pure, pure gold. Yes, the cast has changed, the scenery is different, the plumbing is new, but the tragic and comic plots, sub-plots and counter-plots of this business remain persistently and defiantly unchanged. Of course, this irritates some people who really would rather all had changed completely.\n\nI am writing these words in Touffou, the home David retired to in South West France, in the room he used as his study. The desk on which _Ogilvy on Advertising_ was partly written is still here, although in a different room. The shelves in the study contain a range of books, which testify to his belief that the most productive people read the most widely. There's history, biography, architecture, travel \u2013 spines with titles that sum up a man's life. And, of course, there are the advertising books.\n\n_Touffou evolved over several centuries, but the original keep, built for defence, dates from the twelfth century._\n\nThe house nestles between some low wooded elevations and the banks of the River Vienne as it winds lazily through the countryside of Poitou. David settled here in 1973 with his third wife, Herta. The couple spent the next decades restoring Touffou, turning the grounds into a magnificent garden and creating a grand but friendly home.\n\n_David and Herta in San Francisco, 1984._\n\nDavid died here in 1999, and his ashes are scattered in the garden. Herta remains Ogilvy & Mather's materfamilias; and we continue to hold Board meetings, Executive Committee meetings, client meetings and workshops here.\n\nIn 2013, I called our Digital Council to Touffou. This was a group of young enthusiasts from around the world and from around the eco-system \u2013 mobile, customer relationship management (CRM), social, creative, technology. Our previous meetings had taken place in Palo Alto, CA, but there did not seem anything incongruous in talking about the future of communication in a medieval setting. In fact, it gave us something that we simply could not so easily get in California: perspective. Well into Ogilvy & Mather's own digital transformation, I wanted a discussion of a more fundamental kind. What is digital? Is it an evolution or a revolution? Is it so novel and specialized that we should treat it apart? Or it is something that needs to be baked into the heart of the business, an integrator in itself? We had guidance, in part, from a videotaped last testament that David left. He called it \"View from Touffou\". We still play it in training sessions. It makes a point about press advertising, but one that helped us answer the questions.\n\n_David's 'View from Touffou' video provides a posthumous take on digital. He would have viewed digital as a channel not as a discipline, one that cries out for rich content, and always in the service of selling._\n\nHis argument provided a flash of illumination, bringing into high relief the primacy of content over form. The meeting continued along a path divergent from the one being followed by so many others. It led us to see \"digital\" not as a \"discipline\" but rather as just a channel, a dramatic enhancer of \"traditional\" business, but not a parallel universe.\n1CODETTA\n\nA codetta is what ends a musical sequence.\n\n_Ogilvy on Advertising_ begins with a chapter called Overture. It is classic David Ogilvy. Plainspoken, forthright and resonant with his concise prose, it includes his famous line: \"I _hate_ rules.\"\n\nThe book is a simple yet demonstrative expression of David's breadth of knowledge, containing illustrative references to, among other things, eighteenth-century obstetrics and Horace. And it sparkles with his wit and good humour, ending with this note: \"If you think it is a lousy book, you should have seen it before my partner Joel Raphaelson did his best to delouse it. _Bless you, Joel_.\"\n\n_A codetta, or \"little tail\" in its native Italian, is a brief conclusion in music. It leads back into an exposition or recapitulation of the work before, or occasionally, to a section that develops the piece further. My codetta does the latter \u2013 building on David's book by adding some fresh notes on the nuances of the digital age._\n\nMy codetta rounds off what David began, though it will not be the last word.\n\n_Ogilvy on Advertising_ was written in the 1980s. Joel, David's literary amanuensis, recalls the speed with which it was produced: David posted a chapter to Joel every week. Joel was on sabbatical in Colorado, but his job was to edit it and make suggestions. Written partly in Touffou, at David's desk in his study, and partly in a chalet in Switzerland, it is a polemic.\n\nDavid did not think it was his best book, and he was right. _Confessions of an Advertising Man_ (1963) deserves that appellation. It is more literary: while many copywriters have found it difficult to extend short form into book form, David had a natural gift for book writing.\n\nBut _Ogilvy on Advertising_ was something else: a most elegant rant against what he believed to be a legion of misconceptions about our business; a primer for anyone interested in advertising; an expression of some very dogmatic views, skillfully excused as brevity; and a showcase for the work he admired most (including a sizeable chunk of his own oeuvre).\n\nWithin months of publication, _Ogilvy on Advertising_ was a storming success. It has become an advertising classic, remaining in print for over three decades, translated into multiple languages, and featured in legions of syllabi around the world. More than that, many people I meet, whether they're in the industry or not, tell me that the book is the first or only point of contact they have had with the agency he founded.\n\nI first met David in 1982, as a young Account Director in our London advertising agency. I was working in my small office there in the early evening. He happened to walk past, saw there was someone new inside, retraced his steps, came in and flopped in a chair. \"Who are you?\" Then, \"What are you doing?\" I was full of our recent win of the Guinness business, what wonderful work we were doing. He just looked at me and asked: \"But are they _gentlemen_?\" Some months later, their CEO, Ernest Saunders, \"fell like Lucifer\". David had a prescience that was often uncanny.\n\n_David produced the original manuscript for_ Ogilvy on Advertising _with characteristic efficiency, sharing a chapter every week with his friend and literary confidante, Joel Raphaelson._\n\n_This is the cover of David's original book, first published in 1983 \u2013 from which I've drawn inspiration for this version, and wisdom throughout my entire career._\n\nHe did not, however, predict the Digital Revolution, which has transformed what we think and do in so many ways, creating new concepts, new languages and new techniques. Nor, I sense (judging by his reaction to then the current phrase \"creative revolution\"), would he have liked the description very much.\n\n_David's appearance on_ Late Night with David Letterman _in 1983 for the launch of_ Ogilvy on Advertising _._\n\nBut it would be childish to criticize him for that, especially since he had spent much time since his retirement working on his \"first and last love\", direct marketing. When I started working in our Paris office, in 1994, to consolidate the IBM account in Europe, I discovered there were two offices: the \"posh\" advertising office just off the Avenue George V, and the \"down and dirty\" direct marketing office in the decidedly un-posh Rue Brunel. This was where my team was based, and also where David in retirement chose to have his office. It was a very deliberate gesture. He was cocking a snook at the self-indulgence, distaste for accountability and snobbishness of what was then seen to be the senior discipline. As a refugee from that discipline myself, I could feel some sympathy with the point.\n\nOne of the last working meetings David attended was a summit of our direct marketing leaders in the Ch\u00e2teau d'Esclimont, a turreted pile outside Paris. David was there as a treasured icon rather than as an active contributor, and played a passive role for most of the meeting, apparently peacefully somnolent at the rear of the room. Then the head of the Austrian office started a presentation of unprecedented complexity. It was a small triumph of process over content. After five minutes, the explosion happened.\n\n\"STOP!\" David bellowed. \"FOR GOD'S SAKE, STOP.\" And then slightly softer and more pained: \"I cannot understand or see the usefulness of what you are saying.\" And then, with acute pain: \"And from the land of _Mozart_!\" It was a mortifying and terrifying moment which no one in the room will easily forget, least of all that hapless Austrian colleague. He shouldn't feel bad about it if he's reading this now. But this is a clue for me about how David would respond to much of the verbiage, hype, redundancy, over-complication and ungrounded optimism that surrounds the Digital Revolution, observing and undermining the enormous gifts it brings.\n\n_A rather analogue picture from my early days in advertising. The last 30 years have witnessed a period of immense digital transformation, yet many of the principles drummed into me early on are as important as they ever were._\n\nIn his Overture, David stated that one of his objectives was to separate \"the eternal verities from the passing fads\".\n\nThe refrain of this Codetta, so many years on, is that these verities still exist, as true as ever. Conflict between what is significant and what is not is perennial. The digital world has merely spurred a flurry of new fads, but we are just reaching the time where some perspective is possible.\n\nThe time to reassert the verities has arrived.\n\nHALL OF FAME\n\nThe Digital Revolution has been punctuated by work that has, from time to time, helped to define it. I have put together six case studies, dotted through the book \u2013 a Hall of Fame \u2013 of those all-time greats, which have stood the test of time, and which still carry lessons today. Like all selections, it's very personal, and I am happy to be challenged!\n2THE DIGITAL REVOLUTION\n\nIn 1958 President Eisenhower established the Advanced Research Projects Agency (ARPA). The agency operated largely in the dark, but one of its undertakings, the creation of the ARPANET, was more or less an open secret. Its goal was simple but daunting: to create and ensure the survival of networked communications following the infrastructural disablement of a nuclear strike. The formation of ARPANET ultimately engaged the brightest minds in various fields, including electrical engineering, information architecture, mathematics, computer science, and even psychology, an ad hoc consortium of experts employed by the government (primarily the Department of Defense and NASA), private corporations and leading universities.\n\n_On 29 October 1969 the first message was transmitted over the ARPANET from 3420 Boelter Hall at UCLA's School of Engineering to Stanford Research Institute over 350 miles away. It crashed. The system managed a paltry \"Lo\" before stalling on the \"g\" of its intended command, \"Login\". We now send 200 billion emails and 50 billion text messages every day._\n\nThe first digital message transmitted over the network, on 29 October 1969, travelled from a Sigma 7 computer at the University of California, Los Angeles (UCLA) to a SDS 940 Host computer at the Stanford Research Institute (SRI), in Menlo Park, California. The system crashed mid-message, but the internet was born that day.\n\nOf course, it was just two nodes on one network. Within a year, that had grown to 14 nodes, then later to 100, and then to thousands. It started off as just one network but soon it linked up many others into a network of networks \u2013 an _internet_ \u2013 talking through a common language still in use today.\n\nAnd that was the primary motivation of the Digital Revolution: a military research program devised to ensure the survival of networked communications following the infrastructural disablement of a nuclear strike.\n\n_This is the origin of today's internet, scribbled as if on the back of an envelope. In December 1969 four nodes \u2013 the connection points between computers \u2013 were successfully linked for the first time._\n\n**Digital Comes of Age**\n\nThere has always been a yearning to designate someone as the inventor of the internet. History in this epoch is not so simple. Before, during and after the work of ARPANET, scores of individuals helped invent it. To each his own fame, but I believe \u2013 with the benefit of a reasonable amount of perspective \u2013 one can define 15 individuals who share a higher amount of responsibility than others for the Digital Revolution (see chart).\n\nWhat these men speak to is both the incremental way in which the internet evolved and that, after its first mission of national defence, the internet has been a technology looking for a commercial model, rather than one of commerce seeking a technological solution.\n\nANALOGUE VS DIGITAL PROCESS\n\n_The digital age hasn't called time on the analogue era, but it has offered an entirely different way of capturing information. Take the production process for developing a print advertisement. Whereas analogue technology represents information \u2013 like film negatives in a 35mm camera or the groove in a vinyl record \u2013 digital converts it into data, simple 0s and 1s. We may produce a final print ad that looks the same regardless of the method, but one is a representation while the other is a high-fidelity approximation._\n\nThe internet's now-ubiquitous force is transforming, among many other worlds, the world of advertising. In this arena, the challenges and opportunities presented by digital media are not unlike those faced by David Ogilvy \u2013 and Bill Bernbach, Rosser Reeves, Ted Bates and a whole legion of admen \u2013 upon the arrival of the previous great disruptor, television, in the 1950s.\n\nAs a transmitter, storer and generator of digital information, this internet with many diverse parts is what enables advertising in the digital age.\n\nOf course, since that first digital message in 1969, digital has not completely replaced analogue technology: a Rolex watch, for instance, where the hands represent passing time as an analogy, can still be more informative than its digital counterpart.\n\nThere is something profound in the philosophical as much as the practical impact of digital technology. It's down to the digits.\n\nAnalogue work is whole and of a piece, but digital is a binary approximation \u2013 albeit one of enormous fidelity \u2013 of the analogue world. Perhaps inevitably, a way of working which reduces everything to digits must be less holistic than one which does not. Here lies the real digital divide: the parts versus the whole. Which do you value most?\n\nThe answer is neither. The rate of digital advancement is so high that the two are easily confused. Amazing breakthroughs or the total package? You need both: but you need to be very clear what is, in fact, the end point you desire, and which is just a means of getting there.\n\nPerhaps, because of this uncertainty around its essence, the internet that has emerged is riven by conflict. It can best be described as the combination of frustrated idealism and strongly vested interests.\n\n_From its inception, the internet was a collaborative effort. These are the 15 people whose unique contributions democratized information technology and switched on the digital age._\n\n15 FOUNDERS OF THE INTERNET\n\nJoseph Carl Robnett Licklider\n\n**American (1915-1990).** Intellectual giant who earned undergraduate degrees in physics, maths, and psychology, and a PhD in psychoacoustics. Helped establish MIT Lincoln Laboratory in 1951. Two years later, became director of the Information Processing Techniques Office (IPTO). In 1962, wrote memo imagining an electronic commons open to all, \"the main and essential medium of informational interaction for governments, institutions, corporations, and individuals\". In other words, the internet.\n\nDonald Davies\n\n**British (1924-2000).** Co-inventor of packet-switching. Packet-switching methodology developed by ARPA incorporated ideas advanced by Americans Leonard Kleinrock, Paul Baran and Lawrence Roberts. Gordon Welchman coined the term \"packet\" as a unit of digital data, and co-invented packet-switching. Davies visited MIT in early 1960s, observing choke points in advanced time-sharing computer systems. Back in the UK, devised \"packet switching\", which divided computer messages into packets of code that could be tagged and sent independently, ultimately converging for reconstitution at a singular IP address. Packet-switching was central to the functioning of ARPANET.\n\nDouglas Engelbart\n\n**American (1925-2013).** Founder of Augmented Research Center (ARC) at non-profit Stanford Research Institute (SRI) in Menlo Park, CA, in 1966. Led team that created multilayered computer collaboration system NLS (oN-Line System). NLS broke ground by combining disparate computational elements, including hypertext links, graphical-user interface (GUI), mouse, information organized by relevance, screen windowing, and presentation programs (like PowerPoint).\n\nPaul Baran\n\n**Polish-born American (1926-).** Developed the concept of packets and packet-switching, but used different terminology. In 1960, joined RAND Corporation and led development of distributed array, or network, of packet-switching nodes. Later work expanded distributed communications idea to include OFDM (orthogonal frequency-division multiplexing), the notion that data could be transferred via several closely related channels. OFDM is the basis of digital broadband, DSL internet access, wireless networks and 4G mobile.\n\nRobert William \"Bob\" Taylor\n\n**American (1932-).** Director of IPTO in years leading to first ARPANET transmission. Recognized need for networked timesharing computer at Pentagon with those at UC-Berkeley and the System Development Corporation. This was ARPANET. In 1968, co-authored paper with Licklider laying out development of the internet. Headed Xerox PARC, which developed WYSIWYG word processing, the laser printer, and the Alto computer (on which the Mac was based).\n\nLawrence Roberts\n\n**American (1937-).** Neighbour and friend of Bob Taylor. Taylor recruited him to join ARPA's IPTO in 1966. While at ARPA, built system connecting large computers at MIT and UCLA.\"Two-node\" system paved way for additional connection of computers at the Universities of Utah and Stanford. Influenced by Licklider's writing, Roberts wrote code that enabled packet-switching. First such message transfer took place in October 1969 between computers at UCLA and Stanford.\n\nRobert Kahn\n\n**American (1938-).** Electrical engineer who invented TCP\/IP protocols with fellow UCLA alumnus Vint Cerf. Worked closely with Cerf after recruiting him to continue development of ARPANET, in 1973. Also with Cerf, served on Internet Engineering Task Force, getting ARPANET up and running. In 1972, Kahn demonstrated ARPANET's packet-switching technology at a public conference, considered a watershed event in development of the internet.\n\nBernard Marti\n\n**French (1943-).** Co-inventor of France's Minitel, a predecessor to the World Wide Web (WWW), launched locally in Brittany, France, in 1978, and nationwide in 1982 (ceased operating in 2012). Minitel was an integrated videotext online system that enabled ecommerce and had text chat, video chat and email. Minitel charged users for time spent online and took percentage of online purchases, establishing digital revenue models still in use.\n\nVinton \"Vint\" Cerf\n\n**American (1943-).** Original digital renaissance man. Co-created TCP\/IP protocols with Kahn. Led engineering of MCI Mail, the first internet-connected commercial email service in 1982\u201386. Worked on systems capable of concurrent transmission of data, information, voice and video over the internet. Now Google's Vice President and Chief Internet Evangelist.\n\nJon Postel\n\n**American (1943-1998).** Keeper of ARPANET RFCs, key to formation of ICANN (Internet Corporations for Assigned Names and Numbers). Part of UCLA contingent who formed ARPANET. Played key role in creating internet's structure. From 1969 until his death, wrote, edited and catalogued RFCs (Requests for Comment), the papers that shaped the the internet. Central figure in practical operation of internet. With Vint Cerf, in 1972, Postel devised \"socket number\" tracking system for internet domains. With colleague Paul Mockapetris, created DNS (Domain Name System).\n\nRobert Cailliau\n\n**Belgian (1947-).** Co-worker of WWW inventor Tim Berners-Lee at CERN, Switzerland. Information engineer and computer scientist switched from working on CERN's Large Hadron to lead organization's Office Computing Systems in Data Handling division. In 1989, Berners-Lee proposed hypertext system to give CERN researchers access to all of CERN's documents, in all forms. Berners-Lee created the system \u2013 World Wide Web \u2013 in Autumn 1990. Cailliau co-authored the WWW funding proposal, and later co-developed first web browser for the Mac OS, MacWWW.\n\nPaul Mockapetris\n\n**American (1948-).** Co-creator of DNS (Domain Naming System). Researcher at USC (with Postel) was part of the ARPANET team. Postel presented Mockapetris, inventor of SMTP (Simple Mail Transfer Protocol), with five proposals to improve domain service via host.text system. Mockapetris ignored proposals and co-wrote DNS with Postel. DNS distributed domain hosting across wide server network, providing redundancy to keep websites operating if any one server failed. Chief Scientist and Chairman of the Board of IP address infrastructure software provider Nominum since 1999. Has overseen DNS upgrades and security-software development, as well as creation of spam blacklist, which diverts messages from known malicious IP addresses into email spam folder.\n\nTim Berners-Lee\n\n**British (1955-).** With bachelor's degree in Physics from Oxford, started career as programmer, writing typesetting software for intelligent printers. At CERN in 1989, proposed hypertext project to facilitate sharing and updating of information among researchers. Prototype called ENQUIRE led to creation of World Wide Web. Took hypertext and connected it to the Transmission Control Protocol and domain name system to create the Web.\n\nJohn McCarthy\n\n**American (1955-2011).** Coined the term artificial intelligence in 1955. Implemented CTSS (Compatible Time-Sharing System) to allow use of computer resource (such as a networked machine of application) by many users through multi-programing and multi-tasking at the same time. During speech at MIT's centennial celebration, in 1961, suggested time-sharing could be sold like water, gas and electric utilities \u2013 the business model of cloud computing.\n\nMarc Andreessen\n\n**American (1971-)**. With fellow University of Illinois student Eric Bina, co-developed Mosaic web browser, the first web browser. A year later, introduced the Netscape search engine, which redefined the internet as potentially democratic, empowering, and enabling of digital transactions of every kind.\n\n_Airbnb is an almost entirely digital presence, but it still defaults to analogue communications in some areas, with billboards and print ads. Renting advertising space in areas with a large availability of rooms, or in the publications that sit within them, just goes to show that to \"belong anywhere\" means being everywhere._\n\nThe frustrated idealism stems from many of the founders, especially those close to the World Wide Web. They really did believe they were ushering in something that was free and equal, a virtual world that would be very much better and nobler than the real one. In many ways, they have been sadly disappointed, but that is another book. Suffice it to say that the internet is not free, and it is not equal. (They also thought it would be democratic: history has already shown that in the wrong hands it can be very undemocratic indeed. Look how it has been used to suborn elections or enable shadowy state actions; or how it can create an unreal bubble of misinformation. The Oxford Dictionaries declared 2016's International Word of the Year to be \"post-truth\", an idea brought to you by the internet.)\n\nAs the internet grew, the idea that it might just be better if it had a funding model \u2013 in other words, if the user paid \u2013 was ideologically unacceptable. When traditional media moved towards digital, they compounded the issue by giving their premium analogue content away for free. Only slowly have users' subscriptions been seen to be a necessary part of the model \u2013 and even now there are holdouts and deniers a-plenty.\n\nFor the advertising industry, these are very important points, as it has had to step in and fund much of the internet. No Google search you perform is really \"free\": it is paid for by advertising. How strange it is that it still seems free to so many people. In fact, Google makes an average $3.25 profit each year from every person on Earth. If you count only those of us who are connected, that figure jumps to $7.25. Google isn't free.\n\n\"...the internet that has emerged is riven by conflict. It can best be described as the combination of frustrated idealism and strongly vested interests.\"\n\nThe idealism of the commercial internet is a persistent gene, though, however misleading it may be. It has found full expression in the so-called Sharing Economy, a name that implies something essentially altruistic. I entirely fall on the side of those who recognize this as a piece of seductive re-framing that does not stand up to scrutiny. The transactions offered by Uber and Airbnb \u2013 enormously useful \u2013 are not about sharing: they are a rental agreement, that's all. And there are discounts to the real rental costs, which are absorbed by the unlevel playing fields in which they operate and which the user does not see. These platforms disintermediate, but they are not truly disruptive. If you really want to share a house\/room, I suggest couchsurfing.com.\n\nAs the internet has had to become more commercial, it has evolved into a set of highly vested media interests. It is these, much more than governments, which have given shape to the chaos. They have engaged in an orgy of wall building, and the gardens behind the walls have become chargeable properties. Entirely understandably, they have created a rhetoric of digital advertising that supports their agenda. It has a number of themes, including:\n\n\u2022It's all or nothing: either you are with us or against us. Old media are irretrievably dead: as of _now_.\n\n\u2022This is the world of new. Only new matters. This new is unlike any other previous new in every way possible.\n\nIt is the rhetoric of digital exclusivity and of digital exceptionalism. It is not so much philosophically driven as rooted in the hard need to attract advertising dollars; but it has become implicit in a raft of journalism and writing, and most books written on the Digital Revolution follow the theme.\n\n_Sometimes dismissive of traditional advertising, Google nonetheless has embraced non-internet channels to drive product interest, such as using television commercials in India and Pakistan to demonstrate the potential of search to bring people together._\n\nPersonally I could not agree more that much of \"old\" media will disappear or that there are many very novel aspects to the Digital Revolution. But the rhetoric does not do justice to the richness of the revolution. It creates a zero-sum game, with the \"digital Taliban\" (as John Hegarty says) on one side and latter-day Luddites on the other. But it is the zone in between that is fertile and rewarding.\n\nIt's also clear that if you are a client these days, the old rule of caveat emptor applies more than ever before. And that you need best advice to guide you through the rhetorical jungle with some sense of judicious, constructive scepticism.\n\n_The much-hyped Internet of Things can still work well in channels that don't plug in. Nest found comfort in print advertising and analogue billboards when it launched its high-tech home security cameras._\n3THE SHORT MARCH\n\nI first went to work in China in the late 1990s. I was still there at the turn of the millennium when I was much taken by a very curious statistic, which makes for one of the more recondite crossover charts in history. It was the time when the number of internet users in China surpassed for the first time the number of troglodytes, or cave dwellers. The latter were (and are) surprisingly numerous, around 30 million, but after 2001 the surging number of Chinese netizens soon left them behind.\n\nCAVE DWELLERS VS INTERNET USERS\n\n_At the turn of the millennium in China, online citizens began to outnumber those who inhabit caves for the first time \u2013 now that's progress. Internet usage has increased dramatically since._\n\nThis very Short March in China is a neat symbol for the force with which the Digital Revolution captured the world. It would be tiresome to repeat the statistics. But from the point of view of advertising, there will also be a critical crossover \u2013 the point when digital spending exceeds non-digital. It's coming soon.\n\n**Fragmentation**\n\nThe impact on our world has been explosive. The orderly chain whereby manufacturers stimulated demand by creating brands, which addressed a mass audience with one message at one time, which drove them to the stores, was shattered into fragments.\n\nYou can see the fragments everywhere: fragmented shareholders and stakeholders; audiences splintered into hundreds of mini-segments; more platforms and more touch points; supply chains and demand chains broken; more metrics and more things to say. And brands themselves fragmenting into hundreds of SKUs (Stock Keeping Units).\n\n_This is one of the most iconic charts of the digital age. Originally developed by the strategic advisory firm Luma Partners, it has been viewed over 6 million times and has spawned hundreds of lookalikes. There's never been a more graphic way of visualizing the cacophony of digital platforms, networks, apps and services._\n\nShattered advertising agencies, too. The full-service agency network has new competitors in pure play digital agencies, social media groups, content shops, branded entertainment producers, and activation agencies.\n\nNot only is there fragmentation, there is disintermediation as well. Amazon disintermediated the booksellers; Grindr, the gay bars; Uber, the taxis. Yes, they've all taken a hit, some more than others. But the story is always more complicated than it seems. And some of the disintermediators are less of a threat than they are a market expander. That seems to be the impact of Airbnb, for example.\n\n_I keep this dinosaur, sculpted by Sui Jianguo and given to me by colleague TB Song. One of our (great) clients, Tony Palmer of Kimberly-Clark, had described traditional agencies as dinosaurs, doomed to extinction. So, from 2008, it lived on my desk reminding me that this needn't be. Since then we became the world's largest digital agency network \u2013 and Kimberly-Clark is an \"at heart\" digital client._\n\nGoogle was seen (and certainly saw itself) as the disintermediator of advertising. It has been joined in that doubtless pleasant illusion by Facebook. Hiring distinguished creatives; building a sales force which talks to clients directly; hiring from the clients themselves to make their pitch even more compelling; offering a range of services usually provided by advertising agencies: none of these could be construed as friendly acts. And yet this is a medium \u2013 digital advertising \u2013 that receives some $6.9 billion and growing a year from the advertising business worldwide.\n\nI can remember, at various conferences over the past few years, all the polite denials from Google and Facebook speakers. \"Some mistake, surely?\"\n\nWell, at the time of writing, not one of the offerings proffered by the digital platforms competing with us has resulted in anything like disintermediation. The reality is that they have just not been able to demonstrate an ability to do what we do, which is to build brands on the principles of best advice. Meanwhile they do have a data reservoir, with all the pent-up leverage that brings, which we could never replicate. The truth is that we need each other.\n\nThe Digital Revolution is complex enough to require both them and us. We can and should try to work together. Around the world, I can see it starting to happen: as usual, the further from headquarters, the more there's real teamwork.\n\n_Orrin Klapp's writings on the gap between information and meaning, and the risks of social entropy from banality and noise, could hardly be more relevant to the Digital Revolution \u2013 despite being written before a single PC had hit the shelves._\n\n**Cacophony**\n\nThe unwanted child of fragmentation is cacophony. Noise, noise, everywhere. A few years ago I came across the writings of the now-forgotten Canadian sociologist Orrin Klapp, from the University at Western Ontario. They even smell slightly musty, as mine is a second-hand copy, but deserve to be displayed in neon now, because Klapp saw, with a clarity that still is scarce, exactly what would happen.\n\nComputers can speed the process of data, but give us little help in reading the meaning of the printout. Meaning has a reputation for arriving late \u2013 indeed the highest meaning, wisdom, is also slowest to arrive.... So society suffers a meaning gap, between input of factual information and the construction of common meaning. The paradox of the meaning gap generated by information overload is that ever more facts pile up, the credibility of which is unquestioned, but the overall meaning of the system is lacking and its rhetoric is rejected as hypocritical. The whole is less than the sum of the parts.\n\nThere is a series of phenomena which has created a new dimension to what was never a very easy challenge \u2013 getting people's attention: information overload, distraction, attention deficit, and so on. As journalist John Lorinc put it in a piece in _The Walrus_ :\n\nDigital communications have shown a striking capacity to subdue our attention into smaller and smaller increments; increasingly it seems as though the sheer glut of data itself has supplanted the hand of focused, reflective attention that make them useful in the first place.\n\nThere are two great truths all of us need to remember:\n\n1.We are designed to be selective.\n\n2.We are designed to make patterns.\n\nIt seems to me at least that the job of agencies assumes a simplicity we often forget amidst the cacophony of our overloaded, multi-tooled lives: we are the meaning creators.\n\nLorinc quotes a puzzled participant in a technology conference asking: \"If information is like the sea, what is seamanship?...We don't talk about 'human-wind interactions' \u2013 we talk about sailing.\"\n\nThe tension between humans and information is not another either\/or, a dichotomy which we observe but do not seek to influence: we need both sea and sailors, but it's the sailors who are vested with the task of deciding where to go and how.\n\nThis digital seamanship requires two things which are not generally deemed to be so important by quite a broad coalition of those which have led the Short March \u2013 from the new media, to the technology platforms, to the millions of applications.\n\nIt requires the ability to cohere, to bring together, to re-assemble the fragments, to make sense of them. And it requires some long-termism. The fragmentation has seen a descent into tactics as an end in themselves: any short-term program that can be measured suddenly gains respectability just because you know how many views or likes it gets. Someone has likened it to taking crack: instant hits are everything \u2013 and it is addictive.\n\nThere are voices of reason: bring on Melody Gambino, Director of Marketing at adtech firm Grapeshot.\n\nIt upsets me when I come across an all-too-prevalent mindset among the new wave of digital marketing technorati that glibly demeans the legacy of old-school Madison Avenue storytellers. Regardless of what some 'bro-geeks' who run ad tech companies might maintain, sacred advertising tomes like _Ogilvy on Advertising_ still do matter.4\n\nDavid Ogilvy, indeed, was slaying similar demons as far back as the 1950s. They don't go away: they just come back after a period of TV-imposed sobriety when mass media did develop some coherence, and they are now decked out in new, seductive clothes. In a speech to the 4A's (one of our premier industry trade groups) in 1955, he said,\n\nIt is my guess that 95 per cent of all campaigns now in circulation are being created without any real reference to long-term considerations.\n\nThey are being created ad hoc. Hence the oscillation. Hence the tacking. Hence the lack of any coherent personality from one season to another.\n\nHow tragically easy it is to stampede into change. But what golden rewards await the advertiser who has the brains to create a favourable brand image \u2013 and the stability to stick with it over a long period.\n\nThey still do.\n\nSTARBUCKS\n\nAGE OF INVENTION\n\nImagine yourself in a Seattle coffee shop in 1971, sipping a unique tasting coffee with a smooth, bold blend \u2013 a West Coast roast that makes regular coffee taste weak by comparison. Howard Schultz, who joined little-known Starbucks as Marketing Director a decade later, quickly saw potential in the brand's unique flavour to disrupt a lacklustre global coffee market. So he bought the company.\n\nAlongside taste, Schultz envisioned bringing the Italian coffee-house tradition to the US. He focused on transforming the experience. The server behind the counter became a Barista, someone who cared so much about \"customer intimacy\" he or she took the time to ask your name and write it on your cup. Schultz created an employee culture of caring, and Starbucks was consistently rated as one of the best places to work.\n\nThe Starbucks experience was immediately different, more exotic, a place where customers willingly adopted a new lingo to order their \"grande double skinny macchiato\". And it was underpinned by quality \u2013 full control over the supply chain, from growers to roasters to distribution, all to ensure exceptional flavour consistency. The result? Explosive growth across the US, and internationally, from the late 1990s.\n\nAGE OF APATHY\n\nA marketer by trade, Schultz has historically been dogmatic about not spending money on advertising; success had been achieved through word of mouth, and that was to continue.\n\nWhen Starbucks announced itself as \"the third place\", a destination between work and home open to everyone, the brand's efforts to refine experiential branding \u2013 at the exclusion of other marketing activity \u2013 became an obsession. Data showed that the average Starbucks customer already visited about six times a month, so resolutely focusing on in-store activities was, perhaps, a forgivable folly. But an ideological opposition to advertising caused the brand to suffer.\n\nInstead, local charity-giving programmes, continued service improvements and (failed) music distribution partnerships were weak substitutes. And seeking expansion through grocery retail, alongside increasing store footprint to 15,000 outlets in 50 countries, compromised the brand's local feel. In 2009, Starbucks began closing stores, letting go of one third of employees from its headquarters and 2,000 in total. The company's \"if you build it, they will come\" philosophy had failed. And the Starbucks experience became as vanilla as one of its lattes.\n\n_Founders Jerry Baldwin, Zev Siegl \u2013 an English and History teacher respectively \u2013 and Gordon Bowker \u2013 a writer \u2013 named Starbucks after a character in Moby Dick. An updated version of their original \"Siren\" logo, a woodcut depiction of a mermaid with two tails, still represents the brand today._\n\n_Howard Schultz famously said he does not believe in \"advertising\", but over the years his thirst has grown \u2013 starting with promotional ads, then partner ads, through to tactical ads that have a little more flavour, like this one._\n\n_Schultz's advertising awakening! \"Meet me at Starbucks\" was the first global brand campaign to bring the warmth of Starbucks to life in film. A valiant effort to capture the feeling of Starbucks outside of its locations, and timed (albeit a little late...) to fend off competitors both large and small._\n\n_The short march of digital, from the mid-1990s to the present day. The advent of the internet, access to broadband and adoption of mobile and has digitized global society. Digital advertising spend will surely overtake traditional by 2020._\n\n_Starbucks opened in 1971 as a single location in Seattle that sold only roasted beans. It took Howard Schultz joining to transform those beans into a global brand. He invented the idea of \"the third place\", a hub for friends, colleagues and communities to gather away from home or work. The formula worked._\n\nAGE OF COHERENCE\n\nWith Dunkin' Donuts and McDonald's upgrading their coffee offering in the US, copycat coffee chains expanding around the world, and the rise of boutique coffee shops on the corner of every hipster neighbourhood, Starbucks was forced to rethink.\n\nSchultz began with rebranding \u2013 a new logo emblazoned on every store and cup around the globe. But he knew that the product would no longer be enough. An earlier test of TV commercials, developed with Pepsi to advertise a range of ready-to-drink Starbucks beverages, had been both well received and, importantly, increased sales. And so, time for the unthinkable \u2013 advertising.\n\nIn 2014, Starbucks launched its first major advertising campaign \u2013 with an anthemic brand ad called \"Meet Me At Starbucks\". Shot in 59 stores across 28 countries over 24 hours, the ad went back to the brand's origins to project Starbucks as a place to connect, face to face, in the warmth, comfort and diversity of its coffee shops. The brand was still underweight on digital advertising and overweight on product (rather than masterbrand) advertising, but was learning fast.\n\nSchultz once said that \"Our stories are our billboard\". But those stories also seem to translate well as TV commercials, YouTube documentaries, print ads, and across a host of other channels too. Having developed a taste for advertising, Starbucks is now ranked by Fortune as one of the world's most admired companies.\n\nTIMELINE OF THE DIGITAL REVOLUTION\n\nEarly 1900s\n\nEnigma machines developed, with first models appearing after **WW1**. Looked like large portable typewriters in wooden boxes. First electromechanical computers, used to encode information to protect commercial, military and diplomatic communications.\n\n1940s\n\nTeam led by Alan Turing, English mathematician and pioneering computer scientist, develops a device to crack Nazi messages encoded using **Enigma** machines. Credited with creating the first \"universal machine\", a mathematical tool equivalent to a digital computer.\n\n1950s\n\nDevelopment of digital networks by government-funded, largely defence-related, projects in the US, UK and France. Early networks transfer \" **packets** \" of information, laying foundation for email.\n\n1960s\n\nComputer scientists and Pentagon employees create **COBOL** , or Common Business-Oriented Language, a computer code devised for utility companies to track usage but adopted quickly by many businesses. COBOL is still widely used today.\n\n1961\n\nIBM introduces the 1400 series, replacing cumbersome vacuum-tube technology with **transistors** , shrinking the size and cost of computers.\n\n1968\n\nIBM breaks new ground again with CICS (Customer Information Control System) **transaction-processing code** , replacing batched punch-card tallying. Companies use CICS to store customer information and conduct online transactions.\n\n1969\n\n**ARPANET** (Advanced Research Projects Agency Network) launches a data communication system funded by the US government, which forms the basis of the internet. Connects research centres at UCLA, Stanford, University of Utah, and University of California, Santa Barbara.\n\nARPANET uses IMPs (Interface Message Processors), a network of small computers similar to routers. Based on **packet switching** \u2013 disseminating one bundle of data to multiple IMPs at once, rather than circuit switching linear data transfer, like a telephone call. Later develops the TCP\/IP (Transmission Control Protocol\/Internet Protocol), enabling simultaneous multi-directional data communication among a network of computers \u2013 the basis of the internet.\n\n\"SENDING ELECTRONIC MAIL OVER THE ARPANET FOR COMMERCIAL PROFIT OR POLITICAL PURPOSES IS BOTH ANTI-SOCIAL AND ILLEGAL.\"\n\nAccording to a handbook published by MIT, whose AI Lab later hooked into the network.\n\n1971\n\nStudents at the Stanford Artificial Intelligence Laboratory and MIT arrange a **pot sale** via ARPANET. Is this the first online sale? The ARPANET users simply arranged a meeting over the network.\n\n\"THE SEMINAL ACT OF ECOMMERCE\"\n\nJohn Markoff, in his book _What the Dormouse Said_.\n\n1972\n\nThe term \"personal computer\" is coined. Refers to the Xerox Alto, whose **graphical-user interface** (GUI) provides inspiration years later for Apple Macintosh's and Microsoft Windows' operating systems.\n\n1974\n\nIBM Los Gatos Scientific Center develops a **portable computer prototype** called SCAMP (Special Computer APL Machine Portable).\n\nMotorola begins a run of firsts in the mobile market, including the 8000 series, aka the Brick Phone.\n\n\"JOEL, THIS IS MARTY. I'M CALLING YOU FROM A CELL PHONE, A REAL HANDHELD PORTABLE CELL PHONE...\"\n\nFirst words on a mobile phone call from Motorola researcher and executive Martin Cooper to chief competitor Joel S. Engel of Bell Labs, in New Jersey, 3 April 1973.\n\n1976\n\nFirst incidence of spam when marketing director of computer manufacturer Digital Equipment Corporation (DEC) sends message **promoting sales events** for its latest models to about 400 users of ARPANET. Draws sharp criticism from the connected community but generates sales among the target audience, in Southern California.\n\n1979\n\nEnglish inventor Michael Aldrich, working for Redifon Computers, creates R1800\/30 Compact Office System. The Times reports it enables users to \"place orders for goods, obtain information, and undertake programmed learning courses.\" Aldrich touts the \"place orders for goods\" part, and claims to have originated **ecommerce**.\n\n1981\n\nIBM rolls out first integrated-component personal-computer system, takes ownership of the acronym \" **PC** \". The 5150 computer includes a monitor, stand-alone keyboard, printer and paper stand. Also this year, Thomson Holidays UK installs first B2B online shopping systems.\n\n1982\n\nFrance Telecom introduces nationwide online-ordering system, **Minitel**. Ecommerce is becoming a thing.\n\n1984\n\nApple launches Macintosh with **$900,000 Super Bowl** commercial that reaches 46.4% of US households. Compuserve launches Electronic Mall in US and Canada, a major step in the development of B2C (business-to-consumer) ecommerce.\n\nMid-1980s\n\nIn an early form of **digital marketing** , ChannelNet (formerly SoftAd Group) places reader-response cards in magazines, sends respondents floppy discs with car-model information and test-drive offers.\n\n1992\n\n**SMS messaging** arrives on heels of 2G mobile network expansion and sharp increase in cell-phone use.\n\n1993\n\n**Clickable web ad** sold by Global Network Navigator (GNN) to law firm Heller, Ehrman, White, & McAuliffe, linking directly to firm's website. Questionable whether it was a \"clickable banner ad\" or just a link. Publishers such as Cond\u00e9 Nast and Time Inc. ramp up website development.\n\n1994\n\nFirst web magazine, **HotWired** , launches and becomes first site to provide ad clients with traffic reports, the first internet metrics. Coins the term \" **banner advertising** \", and creates modular spaces on web pages akin to newspaper and magazine pages. Posts the **first clickable ad** from AT&T.\n\nVibe magazine's website emerges as a favourite among advertisers. MCI Communications, Jim Beam, General Motors, and Timex pay Vibe $20,000 for homepage ad positioning. Spending for online ads already tens of millions of dollars. CompuServe, AOL and Netscape debut.\n\nUS **government promotes ecommerce** by moving web hosting from NSFNET to commercial network providers, including MCI and AT&T. Uptick in digital ad spending the following year due to the transfer of hosting from the government to business. Yahoo! and AltaVista **search engines** launch. Oregon-based Multi-Media Marketing Group (MMG) is founded, credited with coining the phrase **Search Engine Optimization** (SEO). SEO ranking becomes an important measure of the marketing power of a brand.\n\nFIRST CLICK-THROUGH BANNER AD\n\n\"HAVE YOU EVER CLICKED YOUR MOUSE RIGHT HERE? YOU WILL.\"\n\nAT&T was right: the click-through rate was 44%. Users who took the bait enjoyed a tour of seven museums around the world.\n\n1995\n\nNokia introduces first **smartphone** series, the Nokia 9000, a folding-notebook-style unit with internet access via WAP and a full QWERTY keyboard. DoubleClick founded, an early Application Service Provider (ASP) that serves ads (primarily banner ads). IPOs two years later as a Top 10 internet site.\n\n1996\n\nFirst **DVRs** introduced at Consumer Electronics Show, raising spectre of ad-skipping, the TV equivalent of ad blocking in digital media.\n\n1997\n\nNumber of people using the web, and search engines, reaches **70 million** , up from 16 million just two years earlier.\n\n1998\n\nTV ad revenue hits $8.3 billion. **Google** and MSN search engines launch. Google develops PageRank, measuring quality and strength of inbound links to determine relative value of sites. GoTo.com launches bidding for higher placement in search results.\n\n2000\n\n500% increase in the NASDAQ index shows technology companies performing at an all-time high. 10 March 2000: NASDAQ peaks at 5048 and the **tech market collapses**. Internet enters a new phase of information sharing, user-centred design, and collaboration.\n\nConsumers engage with brands in a more organic, personalized way. Marketing embraces the concept of \" **creating value for customers** \". Shift away from simply pushing advertising to customers online, towards delivering ads better suited to their lifestyle, personality, demographics, wants and needs.\n\n2001\n\n3G mobile connectivity debuts. Launched by Japan's NTT DOCOMO, in same year that the first **branded-content campaign** goes live. It is a series of short, dramatic video clips by A-list directors called \"The Hire\". BMW pays for the series and its cars are featured in each episode.\n\n2003\n\nSite-targeted advertising debuts. **Google AdWords** allows ad placement via keyword, domain name, topic, and demographics. Quickly becomes Google's main revenue source. Branded Content Marketing Association founded. Study shows consumers prefer native ads over traditional ones.\n\nPresident George W. Bush signs into law the Can-Spam Act (Controlling the Assault of Non-Solicited Pornography and Marketing Act).\n\n2005\n\nGoogle offers **personalized search** results informed by user's search history; launches branded version of Analytics.\n\n2007\n\n**Programmatic ad buying** debuts. Ad exchanges sell ads in inventory with multiple ad networks and are bought via real-time bidding on a per-impression basis. Subscribers to 3G networks worldwide reach **295 million** (just 9% the of global subscriber base). Music and video streaming booms.\n\n2009\n\nGoogle Instant offers real-time search results. **Ad.ly** in-stream advertising service pays Kim Kardashian $10,000 per tweet, a test of promoted tweets. Facebook self-service ad buying offers **targeted advertising** , allows targeting by **geo-location** and language. Amazon sales reach $25 billion.\n\n2010\n\nTwitter introduces **Promoted Trends and Promoted Tweets**. The first Promoted Trend is Disney's Toy Story 3. Virgin America, Starbucks and Bravo also pay for promotional placement.\n\n2011\n\nUS internet advertising revenue reaches **$7.68 billion** in the second quarter (24% increase from the second quarter of 2010). **Digital video** ads account for just 6% of all internet ad revenue in first half of 2011, but effectiveness leads to growth of in-stream advertising, such as TV advertising.\n\n**Ad-blocking** software debuts.\n\nMozilla announces that its Firefox web browser will include ad-blocking capability. Microsoft Internet Explorer, Apple Safari, and Google Chrome browsers follow suit. 42% of US households now own a DVR, with primary purpose of **skipping TV ads**.\n\n2014\n\nAd-bots create \" **fake traffic**\". Study finds that advertisers spend \"billions of dollars on online ads that real consumers never see\" because of automated ad counters, or ad bots. Bots create \"fake traffic\", which undermines reliability of publishers' audience metrics.\n\n2015\n\n**Real-time bidding** takes off, enabling real-time buying and selling of ads on a per-impression basis. Winning bids instantly display on publishers' sites. Auctions mimic **exchange mechanisms** used in financial markets. It's a $15 billion industry, projected to grow by 65% by 2020 (Business Insider).\n\nGlobal ad revenue reaches $17.08 billion (70% from mobile ads). **Amazon.com** accounts for more than half of all ecommerce growth, selling nearly 500 million SKU's in the US. Yahoo! confirms that it \"punishes\" users who employ ad blocking by holding back their personal email.\n\n2016\n\nAd spending on social media campaigns projected by eMarketer to reach $23.68 billion, a 33.5% increase over 2015. US Government formally accuse Russia of **state-sponsored hacking** intended to interfere with the outcome of the US 2016 election.\n\n2017\u20132019\n\nBy 2017, social media ad spend forecast to reach $36 billion, or 16% of all digital ad spending. Worldwide digital ad spending to grow from $226.7 billion to **$283 billion**, which translates to 35% and 39% of all media ad spending (eMarketer).\n4THE DIGITAL ECOSYSTEM\n\nKnitted together in the right way, the fragmented new media world gives us the sort of creative opportunity David Ogilvy could only have dreamed of. It's a dazzling ecosystem of opportunity if \u2013 and not because \u2013 you are sceptical in swallowing its own well-crafted hype.\n\nWhat success means in this ecosystem deserves some forensic probing. Who's doing really well, and who's not? Who's making money, and exactly how? It is strange, but in all the flood of journalistic commentary on the subject, it is impossible to find a simple, and systematic, set of comparisons to help answer these questions. Well, there's one specially constructed on the following pages.\n\nThere are two big and simple things one learns from doing an analysis like this:\n\n_Seemingly invincible social platforms have quickly succumbed to fitter successors. Six Degrees had a short life, and barely made it to the start of the twenty-first century. Next, Myspace and Friendster appeared, and then moved aside for Facebook, while Xanga gave way to YouTube. Orkut, until recently the most popular network in parts of the world, has largely disappeared from sight._\n\n\u2022For an advertiser, there's no one-stop shop. Most of these media complement each other in various ways. Which and how depends on our strategy, not theirs.\n\n\u2022It is a race of algorithms in which profitability is the prize, not mere scale. It's not won yet.\n\nThis is an ecosystem that is still evolving. My colleague Zach Newcombe, a partner at our global consulting network OgilvyRed, describes this as being a \"Cambrian\" moment: a point in time where the sudden arrival of maximum connectivity is spawning an explosion of new life forms. The question is, which are the fittest and will survive?\n\nIn 2011, Friendster failed because it missed the social interaction part of being a social network. It was like reading your r\u00e9sum\u00e9 at a cocktail party. There will be other Friendsters. The American social media life forms shown opposite may not have the best possible business model. Later on I will describe how life on the Chinese internet has evolved. It could have better evolutionary potential.\n\nBut what is obvious is that there is an emerging \"big three\" of digital life forms \u2013 Google, Facebook and Amazon \u2013 of which two have paid-for advertising very much at the heart of their algorithms, Google and Facebook. In the case of Google, it is not surprising. But it only took Facebook a few years to move from a quasi-altruistic social network to an aggressive media owner. I remember being criticized for saying early on that they were the biggest direct-marketing database in the world, but did not deign to recognize it. Now there is no compunction, and the combination of mobile access, accurate targeting, rate-card savviness and an increasing ability to generate a direct response makes it a highly attractive component of any media plan.\n\n_Facebook used to be all about its ads, confident in the allure of its huge audience to advertisers. Then marketers discovered the platform's more powerful function \u2013 social CRM. Now, Facebook is less of an advertising platform and more of a CRM vehicle, allowing brands to generate leads highly efficiently \u2013 for example, to launch new products like this campaign we did for Philips. Budgets are rightly being reallocated to capitalize on the effectiveness of this kind of social CRM._\n\nTogether this is beginning to look like a duopoly. Google and Facebook account for some $36 billion of the $69 billion digital media spend in the US. 52 per cent! When Theodore Roosevelt took on Standard Oil in 1906, it accounted for 70 per cent of the oil and kerosene market. Will our view of monopolistic trading change when Google and Facebook reach the same level? And will we have any Theodore Roosevelts around to slay the dragons?\n\nTHE DIGITAL ECOSYSTEM\n\nWelcome to the digital ecosystem \u2013 a land grab akin to the game of Risk, still to be played out.\n\nAdvertisers must understand the new terrain well and place the right strategic bets. Unlike the evolution of land masses, these are fluid territories. They change in size, splinter off and new ones still emerge.\n\nStart-ups appear with the potential to dominate the landscape, using algorithms as their armory. Smart though machine learning and AI may be, the biggest players now have such powerful network effects and sticky suite of features that it will take much more than it has before to conquer their territory.\n\n**The Death of Everything?**\n\nMeanwhile, what of \"traditional media\"? The pull from the duopoly has sucked the revenues out of it, especially in the press medium, and to some extent television. The \"digital exclusivity\" narrative has played this as the \"death of...\". The death of television, in particular, is given centre stage. The admirable journalist Bob Garfield has been a leading proponent \u2013 even having a chapter labelled \"The Death of Everything\" in his book _The Chaos Scenario_ (2009). I think even Bob would admit that the revolution has been rather less apocalyptic than his prophecies.\n\nIt is not a new claim. In 1996, Nicholas Negroponte, the author of _Being Digital_ , wrote that \"Television will disappear in less than ten years\". In the same year, the chairman of the BBC, Sir Christopher Bland, said: \"TV sets across the world will be jettisoned within a decade.\" More recently, it is the New York University academic Scott Galloway whose thesis has become somehow more ideological. He sees the death of TV as that of \"the advertising industrial complex\", the conspiracy of TV and advertisers which has been artificially sustained by big brands \u2013 in fact, the death of branding as we know it. This is also not a new angle. In 1994, Professor Ronald Rust and Richard Oliver prophesized exactly that in \"The Death of Advertising\", published in the _Journal of Advertising Research_.\n\nWhat has actually happened since then? It is informative to look at the table of measured media investment between 1999 and 2017 prepared by my colleague Adam Smith of GroupM. It shows that linear television (or non-time-shifted broadcast of cable viewing) has maintained a strong position in a market much expanded by digital. Its share of around 40 per cent is the same as it was in 1999. TV is not dead!\n\n_Graph showing that TV's percentage share of media investment over the last 16 years has barely changed, and remains at around 40 per cent._\n\nThe reason for this is that people's viewership of linear TV still holds up, despite the surge in digital media. Of course, the real story here is that digital has gained at the expense of the press medium. There was a death of classified ads, and direct response has shifted en masse to digital. Even the press medium is not completely dead. It can be resilient in places where strong content is justified by journalism of quality and consumed by an audience elite of sufficient scale or sufficient specialism. Then it still is a model sustainable by advertising \u2013 especially if circulation can be charged for. If any one of those parameters breaks down, the commercial model breaks down, too.\n\nSo TV is _not_ dead: though it is changing beyond recognition, it is more useful than ever before. \"Cord-cutting\", \"TV Everywhere\", and \"OTT\" (over-the-top) are roiling the TV landscape, but they are hardly fatal illnesses. In fact, they are heralding the age of greater TV consumption. The pay TV industry is adapting to new viewing patterns with multi-channel on-demand subscription services and consolidation. TV advertising, popularly thought to be a declining force, is thriving and growing fast in broadcast and digital. Television content is better than it ever has been before \u2013 more varied and of higher quality \u2013 but the audience is fragmented across different media. We don't know yet how to measure these audiences particularly well, and that (not cord-cutting) may be the biggest threat to TV of all.\n\nThere is no doubt that the \"TV is dead\" argument has led clients into a splurge of hypertargeting. My own belief (and it is only based on experience with real clients, nothing more) is that brands, having experimented, are now searching again for the fundamentals. What gives your brands the reach they need? What gives you the combination of greater reach and greater impact?\n\nIn the long term, where there is demand, supply usually configures itself to match it. It seems at least as likely that the media industry will conform to this and find new ways of delivering a total aggregated audience. Who knows, this might even pose an interesting challenge for the digital mass media: do they enter television or seek to slay it? My purpose in saying this is not to indulge in any wild prediction but simply to inveigh against the excesses of the blind anti-TV prejudice. TV is alive and well in developing markets, and alive in many developed markets.\n\n\"There is no such thing as digital marketing or digital advertising; there is just good marketing and good advertising.\"\n\n**6 reasons why TV is not dead**\n\n1.It offers the safest way to scale with net reach, which remains unobtainable in any other way.\n\n2.It does not suffer from ad fraud, which might taint up to a third of digital investment.\n\n3.Because TV content is consumed on devices other than TV, the TV audience is significantly underestimated. When the view numbers become \"real\", ad spend will increase.\n\n4.TV remains unquestionably the best vehicle for communicating emotion; brand platforms require emotion.\n\n5.There is a weight of econometric evidence which shows \u2013 conclusively \u2013 that cutting TV budgets damages sales.\n\n6.Online itself is sustaining TV: it has become the second largest spend category.\n\n**The Ghettoization of Digital**\n\nThis is a phrase used by my client on Dove for many years, one of the great contemporary marketers, Steve Miles. Yes, it is true: \"ghettoization\" is the unfortunate offspring of unthinking reverence. We revere, then we separate. We separate, then we erect silos to affirm the fact and protect the territory. And then we justify it by describing it as \"pure\" play.\n\nAs Steve says, there is no such thing as digital marketing or digital advertising; there is just good marketing and good advertising. As he puts it: \"If Dove is good at digital, then what makes it good there is exactly the same as what makes it good anywhere. It is the fundamentals of marketing.\"\n\nThe digital guru whom Nestl\u00e9 hired to digitalize its marketing, Pete Blackshaw, uses almost exactly the same words: \"The 'fundamentals' remain fundamental.\"\n\nSo much that is digital starts from the opposite view point. It is incidental not fundamental. It elevates the media platform above the brand platform. And, it is not scaleable.\n\nIt's in this light that I have my own take on the famous (or infamous) Super Bowl 2013 Oreo tweet (see below). It was the tweet that really registered for the first time the potential of being always-on \u2013 of exploiting a moment in time.\n\nAn Australian academic, Professor Mark Ritson, has done us all a service by dissecting the realities behind this tweet.\n\nBy dissecting the actual number of Oreo followers, the click-through rates, the amplification (re-tweets), Ritson shows that the tweet reached a mere 64,000 people \u2013 or 0.02 per cent of Oreo's customer base.\n\n_A social media team at the 2013 Super Bowl had the nerve, the agility and the creativity to take advantage of a blackout with this tweet, and then hyped it indefatigably. It provoked headlines such as \"How Oreo's Blackout Tweet Trumped Million-Dollar Super Bowl Ads.\"_\n\nIt's a point well made, and done in an amusingly irreverent way. But maybe its merit is more as an attack on the cult around the tweet, the silo triumphant, an easy dragon to slay, rather than what that tweet really betokened. So, ultimately, there's a bigger point. I've had the good fortune to run an advertising agency and a public relations agency. Those who hyped the Oreo tweet as an advertising breakthrough really did it a disservice. It was a brilliant piece of public relations, which spoke to a range of stakeholders, from staff to inventors. But it was \"ghettoized\" as a new formula of digital advertising, which it did not have the scale or replicability to be.\n\n_Our agency John Street in Toronto has always realized that when it comes to predicting the future of digital, there can be a humour deficit. Their antidote has been hilarious videos such as this one to satirize the ability of cats to generate huge views, and poke fun at the advertising industry in the process._\n\nIn the end, though, it made Oreo famous; it made Twitter famous; it made Mondelez (Oreo's parent company) famous; and it made the people who designed the tweet famous \u2013 and it even made the academic who \"umasked\" it famous.\n\nThe problem comes when new social media seeks to claim as a silo the credentials of the old. Quite simply, they don't work very impressively in a silo. They can work very impressively as part of a whole.\n\nWe have become victims of our desire to show ourselves to be successful. I plead guilty of signing off case histories where the measure is \"views\", as if views are all that matters. They do matter, because they show that what you do has got pull, that consumers enjoy it. But pull has to have conscience. It is easy to pull by pandering to the lowest common denominators of crowd culture: we would just put a cat in every video.\n\n\"It is easy to pull by pandering to the lowest common denominators of crowd culture: we would just put a cat in every video.\"\n\nThe ghettoization of digital leads to claims that are easy to attack. Branded content has been subjected to such attacks simply because it does not stack up the same viewing figures as online video creators do (see tables). How could it? It still has a branding or a selling purpose that necessarily constrains. No professional marketer is interested in views as a simple objective: the digital component is part of a gestalt. It often operates at the level of agenda setting, which is exactly what it does well. And we have a mountain of evidence that suggests it works best in concert with traditional media.\n\n**Rank** | **Video** | **Views All Videos (Millions)**  \n---|---|---  \n**Most Viewed Creators (vlog-style)** |  |   \n1 | Pew Die Pie | 13,411  \n2 | The Diamond Minecart | 8,196  \n3 | PopularMMOs | 6,605  \n4 | Smosh | 5,930  \n5 | Vanosgaming | 5,818\n\n_Look at the total views creators such as PewDiePie and Smosh get. Brands envy that, but those views are dependent on building \u2013 even pandering \u2013 to an audience built over years of daily video production. That's not something particularly useful for a brand._\n\n**Rank** | **Video** | **Views (Millions)**  \n---|---|---  \n**Most Viewed Brand Videos** |  |   \n1 | Akira: Shakira La La La | 561  \n2 | Android: Friends Furever | 201  \n3 | Dove: Real Beauty Sketches | 139  \n4 | Evian: Roller Babies | 133  \n5 | Metro Melbourne: Dumb Ways to Die | 114\n\n_These are rather more valuable numbers. These videos garnered these views because they were part of coordinated promotion campaigns that had commercial objectives._\n\n**Rank** | **Video** | **Views (Millions)**  \n---|---|---  \n**Most Viewed Videos** |  |   \n1 | Psy: Gangnam Style | 2,600  \n2 | Wiz Khalifa: See You Again | 2,000  \n3 | Mark Ronson\/Bruno Mars: Uptown Funk | 1,900  \n4 | Justin Bieber: Sorry | 1,800  \n5 | Taylor Swift: Blank Space | 1,800\n\n_Music videos routinely dominate the top views-per-video chart. This is because music is consumed via free media and streaming services. This doesn't represent the triumph of a few great artists so much as it does the total disruption of the music industry._\n\nOverclaims for social media give the (mainly academic) critics easy game to shoot out. It's damned as \"hoopla\"; or as \"a bullshit wagon\". Fair enough. But blanket criticism would be as misplaced as the exaggerated language of the ghettos and silos. Amplification and advocacy are real gains for any marketer. They happen not to be simplistically measurable in views.\n\nHere's the nub of the issue, for digital apologists and digital detractors alike. Both use the language of _advertising,_ and tend to think in advertising terms \u2013 reach, frequency, creative, consumers, advertising, brands and so on. It's hardly surprising: the title of this book also perpetuates the use of the term.\n\nBut advertising in the digital world is no longer _advertising_. The disciplines (and the very word speaks to a controlling desire to keep them as separate as possible) of advertising, public relations, direct marketing, sales enablement, and others, too, are truly collapsing; they no longer live apart. Much of advertising is public relations; much of public relations is direct marketing; much of direct marketing is advertising. Yet the debate is couched in advertising terms, while the real battle is about how well you integrate and the extent to which, by treating digital and traditional media as additive, you can be a better communicator.\n\nTHE DIGITAL SOCIAL CONTRACT\n\n_We have, perhaps unwittingly from the user's point of view, signed a new social contract in the digital age. Platforms, advertisers and users exchange value as a mix of data, dollars and engagement. More complex than the previous model \u2013 where media owners simply sold advertising or subscriptions \u2013 the new dynamic becomes more valuable to everyone as platforms grow and network effects kick in._\n\nBefore the Digital Revolution, the situation seemed quite clear: the media earned its revenue from a combination of what advertisers paid them for placing ads and consumers paid them on the cover price or subscription.\n\nNow there's a different type of relationship, a rather more subtle and less visible \"contract\" between the parties: the media owner, the advertiser, and the consumer.\n\n**The Age of And**\n\nI've been describing the digital world more as this _and_ that, rather than this _not_ that. This is no accident. When it comes to digital advertising we use the word \"and\" quite a lot. It's a word that has attracted attention before, especially by business theoreticians. Notable among these was Jim Collins in his book _Built to Last_ (2005). \"Embrace the genius of 'and'... a truly visionary company embraces both ends of a continuum, continuity and change.\"\n\nSince he wrote that, the zero-sum game has become more not less obvious \u2013 forcing artificial polarities on us.\n\nSo I would go a step further and say this should be the \"Age of And\".\n\nTo some extent, it is inevitable in all revolutions that there will be zealots versus resistors, taliban versus dinosaurs. The tensions are there: but perhaps we are getting to a point where we can see the benefit in recognizing that they are additive and complementary:\n\nanalogue _and_ digital\n\nintegration _and_ specialization\n\nslices _and_ scale\n\nmaths _and_ madness\n\nform _and_ content\n\nWe'll meet them all in this book.\n\nIn _Ogilvy on Advertising,_ David supplied a list of his favourite words. Let me add one of my own. There's an American word that describes a state of harmonious well-being, a word that somehow bubbled up from uncertain origins in the 1920s: copacetic. The answers lie in the harmony you find in-between \u2013 in the _copacetic consensus_.\n\nIt's worth singling out one \"zero sum\" here that has bedeviled our world since well before _Ogilvy on Advertising_ was published, and that is the slogan of Marshall McLuhan that the \"medium is the message\" (1964), which was given new digital life in a myriad of ways. I believe it to be wholly wrong. In this case the \"and\" is very subdued indeed: the message is the message \u2013 even though it can be influenced by the medium.\n\nDavid's surviving colleagues remember his advice: \"It's what you say, not how you say it.\" It's advice needed now much more than ever it was before.\n\n**The Coming War: YouTube and Facebook in Video**\n\nSome colleagues, a senior client and I recently spent a day meeting the two rivals, Google and Facebook, one after the other. It's a very special experience \u2013 like visiting Athens and Sparta in the same day.\n\nTo my colleague Rob Davis, who runs our video practice, it made concrete a general sense of impending battle.\n\nRob has started to distinguish the video networks between those that are \"ephemeral\" and those that are \"archival\".\n\nEphemeral networks achieve maximum impact in the moment. They are designed for interaction. Video viewing is just one feature the network's community uses. They have little search value, as they don't prioritize the archiving and storage of video. Facebook, Instagram, SnapChat and Periscope are all examples of ephemeral networks. This category is growing fast.\n\nArchival networks give a place to host and arrange content specifically for the video viewing experience. These networks are designed for watching video. They have tremendous search impact and serve as the go-to place for someone seeking video. YouTube, Vimeo and DailyMotion are among the remaining archival networks. Most others have ceased operation.\n\n\"The message is the message \u2013 and it can be influenced by the medium.\"\n\nBy late 2014, YouTube touted the delivery of 4 billion video views per day. No other platform came close. Facebook was delivering about 25 per cent as many videos. But by the time YouTube reached its 10th birthday in February 2015, something had changed. Facebook claimed it earned as many views as YouTube and would soon release data showing it had eclipsed YouTube's daily viewership.\n\nFacebook's dramatic increase in video views was driven by technological tweaks rather than a groundswell of viewership. They introduced auto-play videos, which would start playing as the user slowly scrolled over them, and began counting these automatic engagements as a \"view\", a sleight of hand that immediately doubled the daily view count. In contrast, YouTube only counted a view when the user clicked the play button. The new strategy essentially doubled views overnight.\n\n_One of the most enduring online video brands, The Young Turks network has become a media entity in its own right._\n\nWhat we saw during our visit to Google was an archival network struggling with the difficulties of maturity. During its rise, YouTube's influence defined the entire video marketplace. Unfortunately, YouTube also struggled with its identity. Every 18\u201324 months, the site would go through major changes: was it a content platform, a social network or a hub for interactive experiences? Clarity was achieved about two years ago when the ability to customize YouTube was eliminated. From that point on, it has been about delivering a standard video experience. The business model is based upon pre-roll advertising and a new ad-free subscription service, YouTube Red.\n\nWHAT DIGITAL ADDS AT ITS BEST\n\nIssue framing\n\nIntensity \u2013 leading to advocacy\n\nBuzz \u2013 word of mouth\n\nTopicality\n\nParticipation\n\nEngagement\n\nThe PewDiePies and Young Turks of the world form the basis of YouTube's cultural relevance, and drive the platform's traffic (along with music videos, which still make up a huge portion of the totals).\n\nUnlike YouTube, Facebook started without any focus on video. Initially, users were encouraged to embed videos from other networks into their Facebook posts. Facebook eventually added their own video service. Since then they have aggressively promoted distinct features in support of their version of viewability, which sometimes run counter to what we would recommend for YouTube or websites in general.\n\nFor instance, like most of its ephemeral network cousins, Facebook is a scrolling experience. Users thumb through, scrolling until something interesting catches their attention. We like to say content that works on Facebook \"stops the thumb\". Much of the advice we heard from Facebook during our visit was about thumb-stopping, but Facebook is also intent on delivering messages via the video asset even if the user does not choose to watch the video, creating a unique set of circumstances for brands and creatives.\n\nThumbnails (the static images which appear before a user plays a video) are what FullScreen calls \"the book covers of the online video world\". We know from experience outside of Facebook that video thumbnails should be attention-getting images that accurately portray the content within a video. Accepted best practice suggests never featuring products or logos within a thumbnail, as users are less likely to watch videos they think are overt advertising.\n\nHowever, Facebook's advice runs counter to this. They suggest thumbnails with product imagery and text. Why? The user who slows their scroll to read text or look at a product thumbnail will trigger the auto-play function, which counts as a view of the video.\n\nAs Facebook and YouTube continue to slug it out, other ephemeral networks are making inroads. Vine introduced the idea of a short \"loop\" video while Periscope and Facebook Live, the live streaming networks, caught the imagination of a public looking to broadcast. YouTube and Facebook have both recently instituted live capabilities.\n\nIn the long run, there must be a question about YouTube's ability to retain market share. Creators are in love with ephemeral. They are not giving up on YouTube, but it is becoming their library and search-target. The sharing and interactions are happening on other channels.\n\nWith Facebook expected to announce a full video library capability, they may be poised to take away YouTube's one main advantage.\n\n**5 lessons for leaders in the digital age**\n\nHowever fast-changing the digital ecosystem, the essence of leadership doesn't change one jot, though the levels of noise, obfuscations, hype and novelty do sometimes overwhelm us. These are the five things that seem important to me after 20 years of driving a digital agenda. I don't believe they're taught in any business school.\n\n1.Do keep asking the question \"but why?\" insistently, repeatedly, like a curious 5-year-old. \"Why\" is the only word that cuts through the fog, gets to the real problem and helps distinguish between the means and the end. Why? Why? Why? I even run training courses entitled \"The Power of Why\".\n\n2.Don't fetishize certainty. The ready availability of a slew of measures doesn't mean they are useful. Treat all KPIs (Key Performance Indicators) with caution, and the singular KPI with as much reserve as a plague spore. It could infect your whole organization with a distorted sense of priorities.\n\n3.Do programme yourself to total openness. You have a unique opportunity to break down internal silos. And you need the collaboration of a wider range of attitudes than ever before. Playing your cards close to your chest has become a loser's stratagem.\n\n4.Do avoid the divas when hiring. They abound, and they can easily dazzle, only then to disappoint. As one of my clients once said: \"Talent is a mere commodity, only perseverance differentiates.\" When I've repeated this to graduates, they tend to gasp, but it's never been truer than now.\n\n5.Do relish dualism. The tensions of the digital world can push you \u2013 or those around you \u2013 into zero-sum games. But success lies in managing the tension rather than clinging to the \"either\" or the \"or\". It's also more fun.\n\nDOVE\n\nTim Piper was a young Australian who had washed up in our Toronto office. As much a film director as a traditional writer, in 2005 he had written and directed a 30-second TV commercial called _Broken Escalato_ r for Becel Margarine. The escalator breaks down and the unfit passengers caught on it scream for help. He felt compelled to shoot a longer version. The Creative Director of OgilvyOne saw it, and told him about this \"thing called YouTube\". It was posted, and to everyone's surprise, got 50,000 views almost immediately.\n\nEvolution _was a viral hit \u2013 created by Tim Piper, his unwitting girlfriend, and our resourceful production team in Toronto. It was shot on a shoestring, and initiated a campaign that added $1.2 billion to Dove's brand value and increased revenue by $500 million._\n\nThe next year, the agency received an open brief on Dove to amplify the Campaign for Real Beauty. Tim presented a series of emotionally driven short-film ideas, including _Evolution_. This is a tribute to the entrepreneurial creative culture fostered by Creative Directors, Janet Kestin and Nancy Vonk, who empowered creators to present anything they were passionate about. The clients loved the idea of emotional short films, but preferred the other concepts to _Evolution_ , which seemed slightly off-brief. But Tim had a gut feeling that there was, as he puts it, \"some freak visual factor\", which would make people want to share this one.\n\nThere was no money allocated for the _Evolution_ video, so with the help of the director filming the approved films (Yael Staav) and agency producer Brenda Surminski, Tim was able to make Evolution off the back of another production, though he was sensible enough to let the client know. It was time for favours. He enlisted a local fashion photographer, make-up artist, post-production house, and even his girlfriend to be the talent. Tim wrote and art directed the spot with creative partner Mike Kirkland. The clients loved it.\n\nIt was posted with no distribution plan as such. But it just \"went\" \u2013 and became one of the early branded viral sensations of the internet. Some lessons are salutary: because there was no budget, it was never submitted to a market-research testing regime. The real learning was that if you can find something that creates visual resonance, that captures a brand's values in a concrete way, you've found a way of using video that people will not just notice but care about. You care, you share.\n\nBut let's go back to the very beginning. With David Ogilvy's help, Lever Brothers launched Dove in 1957. Differentiated from the outset as a beauty bar, it replaced the \"squeaky clean\" feel of typical soaps with moisturising cream, and featured accessible female celebrities of the era, such as Jean Shy and Pearline Watkins, in its ads. Dove has always been an authentic brand. By the early 2000s, in a crowded category fragmented by private label and online, our client asked us to transition Dove from a singular product to a portfolio of offerings.\n\nI've always believed in the notion that brands have Big IdeaLs\u2122 as well as big ideas. So our solution was to develop a cultural point of view that would be profitable while influencing society for the better. And if someone was to ask what Dove's was, I'd say: \"Dove believes the world would be a better place if women were allowed to feel good about themselves.\" It's a powerful organizing tool that relates back to the brand's best self. As Steve Miles, responsible for Dove globally, put it, \"Brands with purpose are not just socially beneficial, but are the pathway to superior growth\".\n\nThe beauty industry operates from the unquestioned assumption that beauty is something positive and pleasurable. So we were shocked to find out that a staggering 98 per cent of women did not feel happy with the way they looked. We envisioned a world where women believed that beauty is a source of confidence, not anxiety. And Dove was the brand to make that statement, with the launch of the Campaign for Real Beauty. It was to be a defining moment for Dove, and for the potential impact of a consumer brand's communications on culture.\n\nBut by the early 2010s, the Campaign For Real Beauty had become a victim of its own success. Real women were suddenly everywhere \u2013 the beauty industry was cleaning up its act, and teenage girls wanted to see real women in magazines. So we set to work re-evaluating our earlier insight, which while no longer disruptive, revealed a deeper layer. As 96 per cent of women still claimed to dislike the way they looked, the problem was no longer unrealistic beauty ideals. Our research found a new cause was women's internal monologue: 54 percent of women considered themselves their own worst critic and a third said their biggest anxiety was the \"pressure I put on myself to be beautiful\".\n\nSketches _is the most shared ad ever and one of the most watched videos of all time. It exposed women's overly critical assessment of their own beauty compared to the more generous appraisal of a stranger. Simply put, the film communicated that you are more beautiful than you think._\n\nDove understands that to keep mattering brands must continue to evolve, so over the next three years we kicked off each spring with an annual \"hero\" campaign, _Sketches_ in 2013 (see here), and its successors _Patches_ in 2014 and _Choose Beautiful_ in 2015. They earned 14 billion global impressions in the three years from 2013, and a media value of over $90 million from coverage in _Huffington Post_ to _Today_. Most of those impressions were digital. Today Dove is valued at over $5 billion, with more than 40 per cent of brand value attributed to advertising, according to WPP's BrandZ study. And Dove's own econometric modelling has eliminated price, confidence in the economy, distribution changes and media spend as the source of sales growth \u2013 proof that harnessing purpose in advertising can drive a brand into the heart of culture.\n\nOthers have tried to create a similar evolution for their client's brands as Tim started for Dove, but they so often miss the point. As Tim intuitively understood from his desk in Toronto, Dove doesn't prey on women's anxieties, it voices them. It turns them into a public issue that women unify against. Dove is an empathetic leader, an ecosystem for the digital age. Its voice is the internal voice of women everywhere, what women would like to say if society would let them. That millions of women have finally acknowledged their own beauty is perhaps the best success metric of them all.\n5TO BE OR NOT TO BE A MILLENNIAL\n\nIf you would believe some pundits, journalists, marketers and sociologists, Millennials represent a distinctive and unified tribe, the agents of an unprecedented transformation in the world. Well, I have been as guilty as any of them in bandying about the word \"Millennial\". The thing about labels is that we use them because they are easy.\n\nWe've been through Generation X and and now we have Generation Y, otherwise known as Millennials. Next to come are the Centennials. Oh, and to complete the collection, we should perhaps label older people as Generation S. Needless to say, each of these cohorts are different in certain key respects. I guess if one were to be even slightly cynical, that is hardly new or surprising. And various commentators have pointed out how much of the language used by Gen X to derogate Gen Y has in turn been used by Gen S to do the same to Gen X.\n\n_Labelling generations is useful only if it serves our understanding of each cohort. Millennials grew up amidst a Digital Revolution and have defined their lives differently as digital natives. Gen C, who are hot on their heels, were \"born digital\". We're yet to see the impact they will have on the world._\n\nBut what does make Millennials very different is that they are the generation which has coincided with the Digital Revolution. They are the first digital natives. Well, not even that really: at the upper end of the age spectrum, their experience might have been denominated by a Sony Walkman (the iteration that played CDs, not cassette tapes) or dial-up internet access. They were in at the primitive beginnings; and it's the Centennials \u2013 born after 2000 \u2013 who represent the real natives.\n\nIn fact, just as it is difficult to pin down Millennials in any useful way as one digital cohort, so it is also challenging to see them as homegrown in other respects. The first thing we tend to do in a presentation on Millennials is to deny that there is such a thing as a Millennial. We are, after all, talking about 38.1 per cent of the world's population \u2013 and there's little in common between Pakistan's 99 million Millennials and the US's 91 million. Confining ourselves to the US, it is quite difficult to talk of a homogeneous group when, for instance, income disparities between rich Millennials (most often the poster boys and girls for the tribe) and poor ones are so large.\n\nThe early characterizers of Millennials seem to me to have had some kind of perverted, vested interest in stigmatizing them. Of course, horror fascinates \u2013 and the picture painted was horrific. I call it the great \"narcissism calumny\": that this is a uniquely self-obsessed, me-loving, ruled-by-impulse generation. I just don't believe it. The academic research on which it was based has been widely criticized. And when Ogilvy & Mather did its own research, we found the exact opposite. They actually tend towards altruism \u2013 much more caring and giving to those less fortunate than their elders were.\n\n_Generation me? I simply don't agree. Our research suggests Millennials are more altruistic than their parents. They certainly seem to care more for people than things._\n\nA built-in sense of entitlement is another alleged hallmark of Millennials, but it just doesn't wash. In fact, compared to Gen X, they have a greater propensity for saving money and possess a sense of frugality, by, for instance, patiently recognizing that there is a \"point of readiness\" at which you can buy a home.\n\nMeanwhile, Centennials show a more extreme development of the same responses. They're much less inclined to see themselves as \"about fun\", or to do anything risky. And they are already and actually worried about the future, in particular, the environment.\n\nSo this is the American Millennial challenge: just being one coincides with a period in history when the good times are running out. And that does mean the emergence of distinct characteristics. They are much less likely to own homes, or to use credit cards, or to buy or drive cars, or to get married at the same age their parents did. They are much more likely to share. And to value diversity, and to believe that all lives matter \u2013 black, gay, white, straight. Thank goodness for Millennials.\n\nBut then there's a great danger of stepping into a second fallacy. It's the great millenarian fantasy that the social equivalent of a Second Coming is nigh. But most Millennials are not millenarians. Most do not believe that they will usher in a golden age where people don't have to work, where they trade in homegrown vegetables, and where they all live together in emancipated bliss. That dewy-eyed commentary is based on some exaggeration from quite minor trends. \"Normcore\", for instance \u2013 the fashion for wearing only commonplace items, bought from Walmart or LL Bean \u2013 does not seem to me likely to destroy the world's fashion industry. (And Normcore is hardly new: I have practised the extreme form of it \u2013 wearing second-hand clothes \u2013 with great satisfaction in the past without ever inciting a Second Coming.)\n\nOf course, Millennials don't have the same view of work, as anyone who employs them \u2013 and Ogilvy & Mather employs them in the thousands \u2013 knows. The old view was linear and compartmentalized. A Millennial is, in my experience, much more demanding \u2013 and rightly so \u2013 of how work fulfills his or her potential. That means more sideway shifts, more path changes, more of a link to personal interests, more opportunities to take a break and do something else. It may be difficult to manage, but who can deny that it is more civilized? And, again, we have to be careful. We are not about to witness the complete reform of organizational structures or the abolition of hierarchy. Rather, the system adapts \u2013 if it is intelligent enough.\n\nMILLENNIALS ARE HARD WORKING, ENTREPRENEURIAL AND NOT AFRAID OF RISK\n\n_Millennials are no longer faced with a simple career ladder \u2013 more like a game of snakes and ladders! They frequently make sideways shifts, take temporary breaks, and reach up and across to more technical challenges. When they fail, they dust themselves off and try another ladder. I admire both their effort and bravery; it must surely make for some interesting experiences along the way._\n\nMany of these views about Millennials, whether of merit or not, are influenced by perceptions of technology, and the impact it exerts on them. And here is the crux: a conflation of the digital revolution with the idea of generational change.\n\nBut are they one and the same?\n\nIn one sense, undoubtedly, yes. The great gift of the Digital Revolution is connectivity, and none have received it more than Millennials \u2013 or, to a larger extent, Centennials. The mobile phone is so important as the vehicle for digital access that lack or loss can produce identifiable anxiety syndrome. We all get it, but they get it worse.\n\nThe statistics I love most come from a 2016 AT&T online survey of 2,000 US citizens asking what they would sacrifice instead of giving up internet access for the rest of their life. Forty per cent said they would rather lose the sight in one eye, and 30 per cent responded that they would rather chop off one of their fingers. Bravo!\n\nConnectivity certainly becomes a cause of dependence, and spawns a series of behaviours, of which multitasking is the most ubiquitous. But, beyond that, we have to be careful not to assume that everyone exhibits the same attributes in response to the use of technology. I've not seen a shred of evidence to suggest they do.\n\nFor instance, demographic segmentation studies can often be soulless affairs. While they are capable of shedding light, they ought to be treated with caution, not least from giving one the false illusion that their \"segments\" are real people. That said, there are a few (surprisingly few) significant segmentation studies that have been conducted on the impact of digitalization. They all, of course, construct and label their segments differently. Take your choice between Techno-sploiters and Mouse Potatoes, Techno-gamers or Gadget Grabbers. But there are some conclusions that can be safely drawn from them:\n\n_For digital natives, the internet has truly become an extension of self. In this AT &T study from 2016, almost a third of the US digital native respondents surveyed claimed they'd prefer to chop off a finger than lose their phone._\n\n\u2022Humanity does not have a common set of attitudes toward technology.\n\n\u2022A decrease in technophobia, which is real, does not infer that techno indifference is also on the decline. In fact, the latter is rising.\n\n\u2022Even digital natives exhibit different attitudes dependent on the degree to which they are socially engaged. In other words, connectivity is not a constant; rather, it is a variable that manifests itself in different ways for different people. It sounds so obvious, but digital zealotry doesn't always put it that way. The reason, of course, is that in technology the \"leading edge\" is both an end and a means in itself. As a means of pushing boundaries, it is highly effective; as an end it has to be controlled, or it can control your thinking. \"It's leading edge or it just doesn't matter\" is a dangerous default option in digital advertising as much as it is in analogue advertising. Centennials themselves prove the point in perhaps a surprising way. They use social media differently than Millennials do. They don't use Facebook so much. They prefer instead small ecosystems such as Snapchat, which are more private.\n\nMILLENNIALS ARE MOST ENGAGED ONLINE, BUT THEY ALSO ENGAGE WITH ALL ADVERTISING FORMATS IN SELECTIVE WAYS\n\n_When it comes to engaging with a broad set of advertising formats, Millennials show remarkably similar tendencies to previous generations, albeit with more emphasis on the influence of friends, forums and familiar brands._\n\nAN AMERICAN DIGITAL NATIVE IS MORE LIKELY TO BE\n\n1. Ethically conscious\n\n2. Caring about self-image\n\n3. Culturally blended\n\n4. Committed single\n\n5. Mobile\n\n\u2022Centennials are more sceptical and wary. Two thirds prefer to interact with friends in person, as opposed to 15 per cent who would rather do so online. A majority prefer to make purchases in brick-and mortar shops rather than online.\n\n\u2022Research conducted by J. Walter Thompson shows that Millennial views about technology are much more nuanced than they are often portrayed. In fact, they fear being trapped by it.\n\n\u2022Millennials show a disturbing tendency not to follow the rules set by those who first tried to define them. For instance, they read. A study by Pew Research in 2015 showed a surprising pattern in book reading, with those aged 18\u201329 more likely than their elders to have read a book in the last 12 months. Now many of them are still students, but the point stands. Fully 80 per cent of the youngest group had read a book, compared to 71 per cent of those aged 30\u201349, 68 per cent of those 50\u201364 and only 69 per cent of those 65 and older.\n\n\u2022Millennials continue to engage with traditional advertising formats more than Gen S, in fact. Of course, they often do so in a different way.\n\n\u2022Millennials are \"meshers\". They use second devices to complement the content they're accessing on the first. And they're \"showroomers\", meaning, they research their purchases in the virtual universe, and then buy.\n\n\u2022As they buy, Millennials are waving a big yellow flag to all of us. They tend to be under-indexed in many of the big brand franchises. The reason for this is revealing: putting affordability aside, this so-called narcissistic generation actually searches for something beyond mere gratification. They look for authenticity, and for brands that behave well. The same is true for Centennials, but even more so.\n\n5 MYTHS ABOUT MILLENNIALS\n\n1. They're technophiliac.\n\n2. They're narcissistic.\n\n3. They don't read.\n\n4. They don't visit stores.\n\n5. They don't consume advertising.\n\nI'll get into that in greater depth later. For now, it's worth rubbing salt into what I hope is an open wound. Nothing exhibits more the danger of simplistic generalizations than the topic of Millennials. They are not what they are said to be. They are neither digital victims nor digital fanatics. As a result, they are both more complex \u2013 and more interesting.\n\n_Coca-Cola showed how readily Millennials will engage in television-like content with their #CokeTV Moments campaign. A series of shows, hosted by Millennials on a dedicated YouTube channel, bubbled with youthful candour as participants engaged with challenges and topics. It may be online, but CokeTV has all the hallmarks of teen-oriented television shows of old. Millennials are multi-channel thinkers and active participants, but they still like to lean back and watch too._\n\n_Millennials share something in common with every generation of young people \u2013 a reluctance to plan for the distant future. When it comes to retirement savings, however, starting early can have an outsized effect, something Prudential demonstrated in partnership with Harvard University psychologist, Dan Gilbert. In order to show how even tiny inputs can make a massive difference over time, Prudential and Gilbert erected a succession of dominos of increasing size and set them toppling. Even though they began with a normal sized domino, the last one to fall was 30 feet in height, setting a new world record in the process._\n6THE POST-MODERN BRAND\n\nFor as long as I can remember, brands have been in crisis. It used to be the threat of retailers' own labels, which, supposedly, would drive brands from the world. Now, in the digital age, they're apparently in mortal danger. Just scan the headlines: \"9 Iconic Brands That Could Be Dead\"; \"Is Brand Loyalty Dying a Slow and Painful Death?\"; \"Are Corporate Brands Dead?\" It's a pundits' paradise, and, as one put it rather more romantically, we must surely be seeing the \"Twilight of the Brands\".\n\n_The origin of brands is commonly attributed to the marking of livestock to denote ownership. We don't keep cattle at Ogilvy, but we've built on the legacy of this early form of branding, through trademarking to trust building and beyond._\n\nOf course, we can see the reason why brands might seem threatened. In a world of apparently perfect information and peer-group reviews, why do you need a brand itself to tell you it's good?\n\nBut don't write off brands. Far from dying, I believe they will survive in the digital age.\n\nAttempts are sometimes made to classify the history of branding in phases. It's risky to try to be too neat, as history textbooks are wont to be. But it is also tempting to try and label the great waves of branding \u2013 if for no other reason than to understand where brands have come from and where they might be headed next.\n\nIt all started with a red-hot iron \u2013 \"that's my steer\" \u2013 and then evolved into \"trademark branding\". The brand was a trademark of quality. Brands moved next into the era of hard facts and pointed argument: how to make our product different, at first with rational pitches. But then TV brought sound and moving pictures into consumers' homes, and we entered the era of brand image, of emotional branding. Each phase has tended to be accretive, overlaying and sometimes co-existing with the one that came before.\n\nAnd that's where _Ogilvy on Advertising_ stopped: at the high tide of the hard facts era of branding.\n\nThen the digital world started to kick in, and the idea of \"brand\", apparently defined in stone by gurus such as David Aaker or Philip Kotler, started to seem very old fashioned. We started to see the emergence of the post-modern brand.\n\n**Culture Vultures**\n\nThe word \"culture\" does not appear in _Ogilvy on Advertising_. But in the 1980s, brand owners started talking about culture as an asset that could be owned. The poster child for this was generally regarded to be Nike, which found a rich cultural seam in the narrative of competing through brute willpower, a battle of the individual against the world, the victory of inner strength over physical prowess. \"Just Do It\" preceded the internet, but upon the slogan's arrival in 1988 and thereafter, the addition of \"owned\" and \"earned\" to paid media became a powerful enabler of cultural cues. It created destinations for the brand to invite people into. As culture generated a shared system of meaning, it allowed brands to \"start a movement\". Before long, the phrase \"share of culture\" replaced \"share of market\" as an objective \u2013 and brands fought for their share like vultures fighting over meat.\n\nWAVES OF BRANDING\n\n_Branding has moved through a series of waves. In the original_ Ogilvy on Advertising _, David showed himself to be the inventor of emotional branding \u2013 he certainly had a big idea or two. Now, digital is forcing a return to the tangible. It's still grounded in the purpose and emotional promise of the preceding eras, but it's a new era of_ behavioural _branding \u2013 what we like to call \"Brands that DO\"._\n\nChampions for culture assailed the preceding era for its superficiality. The academic Douglas Holt, a pioneer of cultural branding, called it the \"commodity emotions trap\". Coupled with the bureaucracy of large companies, culture had tended, at its worst, to result in meaningless abstractions. Holt gives a hilarious account of one such, which all of us in the business must recognize:\n\nThey started out with the brand vision: this drink was to \"Enable a life lived absolutely, completely, and totally fulfilled\". The core proposition was \"A new thirst-quencher that empowers you to achieve far more than you ever thought\". But some managers disagreed; they did not think that this was quite right. So more research was commissioned and the brand vision changed. Now the brand would champion \"Up for Adventure\". And in the next iteration, they moved on to \"Refresh Your Day\". Then to \"Refueling Vitality\", \"Refreshment for an Active Lifestyle\", and \"Fuel for Life\". The company's management eventually settled on \"Refresh for Life\".\n\nAt its best, however, culture helped power brands to success. Holt, again, makes a crucial point: \"Nike's famed shoe innovations happened early on and do not coincide with the brand's takeoff. Nike succeeded with innovative cultural impressions, not innovative products.\" Nike launched its \"Word of Foot\" campaign featuring the personal stories of regular, everyday athletes. That set in motion over four decades of marketing dedicated to the idea that everyone is an athlete \u2013 not to mention the ubiquitous acceptance of athletic shoes as everyday footwear.\n\nThe culture vultures were certainly onto something. And nothing summed up a cultural aspiration more than a manifesto. Suddenly it seemed that we were awash with them; and manifesto writing became as much as an art form as the advertising that emerged from them.\n\nWe might call this progress. And \"progress\" was precisely the idea claimed by Johnnie Walker, one of the more successful examples of a brand undergoing the manifesto treatment.\n\nAt the turn of the twenty-first century, Johnnie Walker was stalling under market share pressure in both established and new territories. It would take a spirited combination of insight and iconography to ensure the brand's status into the next millennium.\n\nThis would be the role of the manifesto. In line with the changing values of twenty-first-century men, the brand came to understand that the notion of success was shifting from the material to the motivational. In essence, success was becoming defined not by what a man had, but what he might achieve and the direction in which he was heading. The brand's owner, Diageo, articulated the idea as \"Johnnie Walker inspires personal progress\", which was eloquently distilled down to the line, \"Keep Walking\".\n\n_Johnnie Walker's iconic Striding Man was flipped, literally, to point him in the direction of the future rather than the past._\n\nJohnnie Walker's manifesto cleverly reached into the brand's history to map its future. This trick is best illustrated by a new logo that heralded the brand's transformation. Initially sketched on a restaurant menu in the early 1900s, the brand's iconic Striding Man was flipped, literally, to point him in the direction of the future rather than the past.\n\n\"The bracing wind of transparency has blown through corporate corridors in a way that defies resistance.\"\n\nWhere the protagonists of culture have been less successful is turning their views into a system for working. Culture as something to tap into can be magical; culture turned into a \"model\" is a death trap. It either produces \"step processes\", which become pedestrian, or results in briefs of the \"please design me a provocative cultural expression as a counterpart to the rise of 'Trumpism'\": interesting clothes for a brand to borrow, perhaps, but not rich as a brief to a creative team. It's an ingredient of the brief, and it might be the intent of the work, but it's not a brief any creative would understand.\n\nIt is an annoying fact that explicitly cultural language is better for describing post-hoc what has happened that it is for provoking the happening in the first place.\n\n**Authenticity**\n\nHowever, the accelerating Digital Revolution started a new, fertile strand of thinking. It has brought with it an unprecedented pressure for transparency.\n\n_Toyota issued a mea culpa when it came unstuck due to a defective accelerator pedal, but the company took far too long to calm its critics._\n\nPut simply, there is no longer any hiding place. The bracing wind of transparency has blown through corporate corridors in a way that defies resistance. Yet there were still \u2013 and probably always will be \u2013 attempts at resistance. When Toyota first started to receive reports of accidents in the US in August 2009, and there were credible attributions of those accidents to unintended acceleration, their first reactions were slow. As the truth emerged, it seemed that it had to be dragged out. Now I do not believe there was any willful conspiracy not to be transparent; rather, the culture and politics of a conflicted organization just never put a premium on transparency as a value, and hundreds of small decisions added up to an overall behavioural trait. The learning came the hard way, and Toyota's brand suffered as a result.\n\nThe good news is that when it comes, transparency can heal. It has redemptive power. Toyota was able, after a while, to stimulate supporters' groups on Facebook in the US who felt the process of vilification had gone too far.\n\nHowever, how to avoid being there in the first place? Enter the Arthur W. Page Society of the US, who published a defining document of the digital age in 2007, \"The Authentic Enterprise\". As it says:\n\nIn such an environment, the corporation that wants to establish a distinctive brand and achieve long-term success must, more than ever before, be grounded in a sure sense of what defines it \u2013 why it exists, what it stands for and what differentiates it in a marketplace of customers, investors and workers. Those definitions \u2013 call them values, principles, beliefs, mission, purpose or value proposition \u2013 must dictate consistent behavior and actions.\n\nIn a word, authenticity will be the coin of the realm for successful corporations and for those who lead them.\n\n_IBM's proof of authenticity lies in its belief that the world's serious problems, in healthcare and security, cities and global commerce, can be resolved through humankind's unique ability to \"think\". This billboard dispays icons from IBM's Smarter Planet campaign, in which intelligent solutions contribute to smarter cities. And it shows how design can express a powerful idea in the digital age._\n\n_Dove re-establishes the relationship between women, their inner selves and their outward representation in media and advertising, exposing the rhetoric of the cosmetics industry. The poster's message? You don't have to conform to a stereotype to be beautiful._\n\n_More than any other brand that has tried to do so, Coca-Cola has occupied the territory of \"happiness\" by tapping into emotions that go so much deeper than the bubbles._\n\nIn place of the voice of authority, stakeholders in the digital world demand \"proof of authenticity\". Are you who you claim to be? And, who do you claim to be?\n\nOne of the co-authors of this white paper went on to become the IBM CMO, Jon Iwata. Jon is a remarkable client who helped reinvent traditional notions of marketing and public relations. He has an instinctive dislike of what he calls \"campaignery\", which has inspired his own work at IBM over three different interactions of the brand's public presentation of itself \u2013 in other words, its platform. A quotation from Abraham Lincoln is much in use at IBM: \"Character is the tree, reputation is the shadow.\" Is advertising in the digital age a player in the shadows, or a builder of character?\n\nIn working with clients such as IBM, Dove and Coca-Cola, we learned there was a simple way to capture their deeper values, so that IBM's proof of authenticity lay in its declaration that all the world's problems, from healthcare to security, and cities to global commerce, can be resolved through humankind's unique ability to \"think\". Dove's lay in re-establishing the relationship between women, their inner selves and their outward representation in media and advertising, thereby exposing the rhetoric of the cosmetics industry with the radical ideal that beauty is more than skin deep. And Coca-Cola's lay in acknowledging the common human need for authentic enjoyment, enhanced through our shared experiences, and expressed over the years with beautiful simplicity as \"open\", \"real\", \"happiness\", \"enjoy\" and \"feeling\". In each case the corporate values and the product values meshed together.\n\nWe have found it very useful to express this in terms of a Big IdeaL \u2013 one step on, if you like, from a Big Idea. What exactly do you stand for?\n\n**A Big IdeaL**\u2122\n\nA Big IdeaL is a type of positioning, but not all positionings are Big IdeaLs. A positioning can be based on a purely functional benefit: Brand X washes whiter, or is more refreshing. But a Big IdeaL is about having a worldview or a purpose that goes beyond a simple functional benefit, even though it should be supported by the functional aspects of the brand.\n\nAnd it's not a shallow emotional benefit: it touches culture. It's a belief system which drives everything that a brand does and helps it to attract widespread support. It's something to be voted on by consumers and stakeholders who have a bigger vote than ever before. For these reasons, at Ogilvy & Mather we have never written Big IdeaLs like \"normal\" positionings. We wanted to find a simple way of capturing a brand's Big IdeaL that would force the writer to express the brand's point of view on the world, or on life, or the country in which it operates.\n\nSo we developed a form of words which could work as a summary of a brand's Big IdeaL. In practice we find that completing this phrase forces the writer(s) to focus quite hard on what the brand's worldview really is. The phrase is deceptively simple because it could be completed in 30 seconds, but doing the necessary thinking to get it absolutely right could take months. The phrase is:\n\n\"It's a belief system which drives everything that a brand does and helps it to attract widespread support.\"\n\n**Brand X believes the world would be a better place if ..........................**\n\nTry completing the sentence for brands you are familiar with, brands that have momentum and a clear sense of identity. It might take a little while but the chances are you can come up with something that feels interesting, maybe even provocative, and quite specific to that brand.\n\nWe have found that the best Big IdeaLs seem to exist in the intersection between two realms of thought and experience.\n\nThey connect with the brand's \"best self\". This is not always exactly how the brand is right now, but how it is as its best. Clues to the brand's best self may be found in its heritage \u2013 those moments when the brand was most successful, or in the relationship that the brand's most loyal users have with it, or maybe in its visual identity. Bear in mind, however, that brands only exist in context: if the brand's greatest hour was in 1964, we need to reinterpret what made it great back then for a contemporary context. Big Ideals need to connect not just to a list of brand benefits, but to the little pieces of magic that comprise that brand's current or potential claim to greatness.\n\nBig Ideals also connect to a cultural tension. This is where we claim culture. \"Markets are conversations\", according to _The Cluetrain Manifesto_ and leading brands need to be interesting conversationalists. They are worth listening to when they have a valid point of view that resonates within culture. Coca-Cola's famously utopian \"Hilltop\" advertising was conceived against the backdrop of the bloody Vietnam War, when \"community\" was tensely pulled against notion brands.\n\nTogether these two things amount to what the brand believes in.\n\nThere are other ways to reach similar conclusions, and I have always been careful never to insist on this approach as a formula, or even necessarily to share it with clients even as it was guiding our work. It belongs to the kitchen not the front parlour.\n\n_Chipotle is the most striking example of how important it is in the digital age for a brand to actually do what it claims. Having set out its stall under the banner of \"food with integrity\" and won over mainstream fast-food customers, it suffered a sucession of embarrassing food-safety scandals. While the brand's ambition to serve fresh, locally sourced, high-quality ingredients is a noble one, the company's inability to deal with these crises has seriously undermined that promise._\n\nOthers in the advertising industry have been addicted to the word \"purpose\". It also works; although I feel sometimes it has led to an artificial leap into CSR (Corporate Social Responsibility). This brand has a purpose of X; therefore we need a CSR programme of Y to demonstrate it.\n\n\"The danger always is of being seen to buy authenticity.\"\n\nChipotle has recently been burned by this trend. Having set out its stall under the banner of \"food with integrity\", and won over mainstream fast-food customers in the process, the brand suffered a succession of embarrassing food-safety scandals, including a serious outbreak of _E. coli_ and norovirus. While the initial ambition of fresh, locally sourced, high-quality ingredients was a noble one, the company's inability to consistently deliver on its promise has undermined the integrity of its brand positioning. The punishment? A double-digit decline in stock-market value.\n\nThe danger, always, is of being seen to buy authenticity. It's not tradeable. One of the more ill-conceived attempts, and much maligned in the media, is that of Pepsi and the Kendall Jenner ad in 2017. The ad rather crassly traded on protest and tensions surrounding the police in America, two touchy subjects, and implied that all we needed to solve police violence, racial tension, and political anger was an ice-cold Pepsi. The backlash was immediate, multi-channel, and nearly universal. And it was deserved, too. Pepsi has used generational change to underpin its efforts to dethrone Coca-Cola, but greatly missed the mark this time.\n\n**From Authenticity to Belief**\n\nWe can convincingly demonstrate the power of _belief_ through research. In a study done at Ogilvy & Mather, we compared sets of brands and sorted them into two groups: those with a higher point-of-view rating and those with a lower one. In other words, those that had a belief about the world, or stood for something, and those that didn't so much. The research supported the notion that belief mattered, since consumers sorted them very clearly via their buying decisions.\n\n_American Express created OPEN forum, an online community sharing business advice between peers, leading entrepreneurs and industry bloggers. OPEN forum gives them greater exposure, business development tools, and an opportunity to build credibility for their business._\n\nSo we know that if a brand is seen to have a strong point of view, then its consideration, or its likelihood of being among a consumer's group of possible choices, is heightened. Brands with stronger points of view also rank higher in consumer perception. We created an algorithm and applied it on a larger scale to WPP's Millward Brown BrandZ database, one of the largest sets of brand data available anywhere. We found that the best-performing brands for point of view outperformed the lowest by 2.2 times in terms of their likelihood of future market share growth. Predictive methodology like this can't prove a relationship between ideals and business. Nor is there any business index I know that does so retrospectively. The macro evidence is prima facie; the micro evidence tends to require confidential company information, however convincing it may be. In other words, it could pay to believe.\n\nSo what makes this so powerful for the digital age? One very important thing: it provides a means by which a brand can organize itself amidst the digital chaos that surrounds it.\n\nARCHAEOLOGY OF MATTERING\n\n_\"Mattering\" requires more than a logo, a use case, or even a purpose. This cluster analysis of people's response to brand attributes reveals a new hierarchy for the digital age. It shows behavioural attributes rise to the top, far more important to people than the ones lower down in the sub-strata. Brands matter today because they behave in a way that helps people accomplish what they want to do. So, brands that are useful, practical and personalized are the ones that \"matter\" most._\n\nThis is the role of post-modern brands: to define their own space within the internet \u2013 their own ecosystem \u2013 which they populate with their own content. Brands act as editors, keeping what is good, junking what is bad; brands act as curators, exhibiting information in a way that is ordered and compelling. It is brands that re-assemble our attention, that provide a resort for those who are interested; it is brands that act as enablers of culture, watering holes for the herd; an enclave within the landscape of interruptions.\n\n**The Well-Behaved Brand**\n\nFor some while now \u2013 about the past five years \u2013 we at Ogilvy & Mather have been noticing something else. Brands may not be about to die but they are having to work harder to matter to people. If people are asked whether brands matter, more than half in developed countries and still more in developing countries say \"yes\". \"Mattering\" is quite a high bar, so these figures seem significant to me. However, our research shows that consumers are demanding much more from their brands: a logo, a reason why, an emotional promise, a cultural belief \u2013 all these may no longer be enough.\n\nIf you really get into what matters, an interesting picture emerges.\n\nThe things that matter group into those that are concerned with trademark, those that are functional, those that are emotional, those that are to do with belief \u2013 and then those that are to do with behaviour, what a brand does in order to matter to you. Think of it as a kind of \"archeology of mattering\" (see table). Today, the top layer is about doing: what does a brand offer by way of service? These are some of the things that matter to people.\n\nIn plain words, fewer empty promises, please. Show that you mean what you say by doing something. In the digital age brands that behave well have the edge.\n\nMany brands require significant transformation. In the past, putting a flag out, often with a new tagline or positioning, was the way to begin to make that happen. But in a world where few taglines are memorable or meaningful anymore, actions that demonstrate new points of view are a better way to drive change in perception.\n\n_It would be easy to assume that you don't reach business decision-makers through emotion. But nothing could be further from the truth. In a series of digital documentaries for Philips, we showcased the brand's innovation and leadership in infrastructure, healthcare and lighting in a more illuminating way. Our \"Breathless Choir\" film, which told how Philips' healthcare technology helps individuals with respiratory illness achieve fulfilling lives, was breathtaking. The jury at Cannes agreed \u2013 it won a Grand Prix award._\n\nThe Philips of today is a cutting-edge example of behaviour branding with an anthropological, design-led approach to innovation that engages the brand's key audiences in a deeper way and shows \u2013 rather than tells \u2013 how Philips thinks and acts as a business partner in healthcare and lighting.\n\nPHILIPS BREATHLESS CHOIR\n\nWhen Philips set about relaunching itself on the basis of \"Innovation + You\", it had a distinctive point of view \u2013 that innovation should be built not for its own sake, but around each of us. While other technology companies might focus on how they \"invent\" or \"think different\", the company founded in 1891 in Eindhoven in the Netherlands would be absolutely people-centric. And in doing so, it would shift from being seen as a consumer lightbulb brand, to a business-to-business company expert in lighting, healthcare and infrastructure technology.\n\nBut how to get there? Philips is a company that lives and breathes innovation. It tackles problems with an anthropological, design-led approach \u2013 but it's a culture that hasn't been exposed well. So, we helped Philips take a bold leap of faith, and use emotion to persuade even the most rational of business decision-makers.\n\nTogether we created \"Breathless Choir\", a film exploring how Philips' healthcare technology helps individuals with respiratory illness achieve fulfilling lives. And we did it with a level of emotion, authenticity and belief rarely evoked by B2B brands.\n\nWe enlisted the help of one of the foremost leaders of music \u2013 Gareth Malone \u2013 already well versed in assembling unlikely groups of people to transform their collective voices. Yet this would be his most difficult challenge \u2013 helping people with respiratory problems sing together, like Claire, a cystic fibrosis sufferer, or Lawrence, a 9\/11 first responder who lost a third of his lung function, and others with severe breathing issues.\n\nIt started very gently, some spoken words, a focus on the breath. And with patience, and practice, the group became one. Amazing! After a few days they started to believe in themselves. Finally, they put their belief to the test in a performance at the Apollo Theater in New York City \u2013 and the result was breathtaking. No matter who you are, the delight of seeing people exceed their limitations is powerful, and it changes your view of Philips.\n\nWe needed more than a 30-second spot to tell such a transformational story, and our choice of longer form content drove engagement and online sharing behaviour well above benchmarks across 20+ markets. The Economist, MSNBC, The Wall Street Journal and LinkedIn \u2013 the business media carried an unusually emotional film for their audience.\n\nAuthentic and unexpected, \"Breathless Choir\" showed how Philips innovates differently, how it solves problems around real people, how it thinks and acts as a technology partner, and how it holds a set of human beliefs that touch everything it does.\n\n**The Inclusive Brand**\n\nDavid Ogilvy did write: \"the consumer is not a moron, she is your wife\". It's disappointing, then, that 40 per cent of women typically do not recognize themselves in the advertising which is directed at them.\n\nIn the digital age, this is as little tenable as it is right. Sexism is an instantly shareable vice. As the activist Cindy Gallop puts it rather eloquently, \"Social media is a whole new way for us to do what we've been doing since the dawn of time: share the shit out of things.\"\n\nThe biggest advertiser in the world \u2013 and the biggest to address the problem \u2013 is Unilever. It had the courage to look into its own data and to confirm the stereotyping across its (and Ogilvy & Mather's) work: the really compelling thing about #Unstereotype is that the data showed that progressive portrayals of women were not just more enjoyable (how could they not be?) but more impactful. It's a business case, not just a moral case.\n\n_Pantene backed away for #notsorry, which encouraged women not to apologize when they've done nothing wrong in difficult social situations. \"Be strong and shine\", they said \u2013 but what on earth is the connection with shiny hair? Not surprisingly, there was no business lift to this campaign. It's borrowing an agenda: it's not rooting one in the brand._\n\nThe devil is in the details of how to do it. \"Femvertising\" has become a not very helpful catch-all phrase, and a veritable bandwagon of hashtags \u2013 #thisgirlcan, #inspireher, #girlscan, #likeagirl, #notsorry. It does not do the overall cause any good to assume that they are all equally well grounded. Pantene backed away from #notsorry, which encouraged women not to apologize when they've done nothing wrong in difficult social situations. \"Be strong and shine\", they said \u2013 but what on earth is the connection with shiny hair? Not surprisingly, there was no business lift to this campaign. It's borrowing an agenda: it's not rooting one in the brand.\n\nA shift seems to be taking place where the centre of the debate is moving from \"feminism\" to \"soft feminism\" to \"girl power\" to \"deeper fulfillment. The more it shifts, the more likely you are to look for work that gets deep into the real desires of women. When Under Armour says, \"I will what I want\", it reads that women are now surrounded by a lot of superficially \"empowering\" messages which in reality disempower, because they generalize the person. If sports products enable you to do what you want, they celebrate strength not weakness. Lesson number 1 in brand building: always go back to the product.\n\n_Always #LikeAGirl campaign was a hit with the judges at Cannes, and it garnered significant press and social attention. This is firmly in the girl power mode, but does it lack a connetction to the product, and a nuacnced expression of the needs of women?_\n\n_\"I will what I want\" \u2013 a truly empowering phrase that supports a deeper fulfillment of womanhood in a world that constantly seeks to undermine women._\n\nThere's a big issue, of course, at work here: and that is the mindset of the advertising (and marketing) business. This is the direct result of an inbuilt diversity issue. Put simply, there are not enough women in positions of management or in creative departments. And this reflects a broader issue in anything to do with STEM (Science, Technology, Engineering, Maths). So digital technology and creative advertising together don't make a diverse marriage: the opposite. It was depressing in writing this book that, for instance, the founders of the internet I selected, as well as the greats of Chapter 14, were all male. This, alas, was not unconscious bias.\n\n_Honey Maid Graham Crackers featured a gay family in \"This is Wholesome\", a moving TV commercial that evoked the parallels among wholesome families and wholesome foods. It aired in March 2014, at the height of the same-sex marriage debate in the US and provoked an outpouring of praise. And, sadly, derision. Rather than crumble in the face of virulent opposition, Honey Maid reaffirmed its values in \"Love\", which hit back at critics by showing artists spelling out the word_ love _using printouts of online hate mail and comments the brand received._\n\nIt won't improve in the advertising business until there is much closer parity between men and women in technology. And that will bring other benefits. As Shelley Zalis, the woman who did more than anyone to pioneer online market research, and who founded \"The Girls' Lounge\", a kind of pop-up workshop that combines activism with manicure, says, \"the social norm just becomes more nurturing, and nurturing companies are likely to be more successful\". Shelley popped up for us at one of our events at Cannes in 2016, and also emphasized that this whole debate is not just about women, it is about gender. It's about how men are portrayed also.\n\nWhen Guinness ran an ad featuring the Welsh rugby player, Gareth Thomas, who had come out in 2009, it rang a bell with me. Fourteen years earlier, Ogilvy & Mather lost the Guinness account in large part due to the uproar (especially among licensees) caused by the first gay Guinness commercial, in 1995 (certainly it was ahead of its time). That was a brave and lonely attempt to show gay normalcy by a brand known for its masculine qualities. A social revolution in 25 years has caused what seemed unacceptable to become not just normal but normal in the most apparently extreme redoubts of masculinity.\n\nSo one further communications impact of the digital world has been the opening of social silos. Brand owners have quickly realized that the targeting efficiency of the internet allows them to reach discrete groups, and, more importantly, to become accessible through relevant content to them. But then the lids have come off all together: slipstream is turning into mainstream.\n\n_Red Label tea, a staple in India and Pakistan, put a transgender band as ambassadors for its product, making a strong cultural statement in the process._\n\n_With the Proud Whopper, Burger King showed that the LGBQT community are the same on the inside as everyone else. How? Same burger, different wrapper._\n\nIt had become clear that this was not just about quick access to the commercial power of diverse audiences, but about the brands adapting in terms of their behaviour. Brands that do believe in diversity at their core.\n\n_A very high proportion of our staff are Millennials. So it's not only a matter of belief in what's right, but also a question of what's important to them that we support the LGBTQ community, and have been doing so proudly for a decade._\n\nLGBTQ mainstreaming has made this very clear. Tiffany & Co. is often perceived as epitomizing the traditional, but the brand challenged the norms of its category when it honoured tradition while celebrating modern love. The definition of timeless love has not changed, of course, but it has expanded, encompassing wider acceptance. Tiffany met this head on, and just two and a half years after same-sex marriage was legalized in New York, the brand included a gay couple in its advertising. It was a statement that this brand, a stalwart of the storied tradition of luxury gifting, endorsed the gay audience as mainstream.\n\nAnd when we launched Ogilvy Pride in 2008 it seemed entirely reasonable for it to identify with the LGBTQ constituency at a time when it was severely challenged. In fact, we know that almost 50 per cent of Millennials are more likely to support a brand after seeing a LGBTQ-themed ad.\n\nOur Ogilvy Pride network now operates in multiple markets, opening the eyes of staff and clients to the LGBTQ mainstream. And we count Stonewall as an Agency partner. I'm proud of that.\n\nBack in 2010, I founded an Islamic consulting unit within the Agency. We called it Ogilvy Noor, meaning \"light\". Its aim was very simple: to cast light on another constituency, grossly neglected by the West, both by many western companies selling in Islamic countries or in respect of the ignored (or sometimes shockingly stigmatized) Muslim constituency in the West. I keynoted the American Muslim Consumer Conference in 2010. It's fair to say that there's little one could have done to have appeared more out on a limb than this: but this is a constituency that is mainstream. Let's remind ourselves that some 23.2 per cent of the world's population adheres to Islam as of 2010, which will rise to 29.7 per cent by 2050 (Pew Research Center). And that the vast majority of Muslims living in the West see themselves unequivocally as mainstream.\n\nMuslim values are not un-modern: far from it. Shelina Janmohamed, our leader of Noor, is helping us understand the new Muslim Futurist: young; proud to be Muslim; trend and fashion conscious. But the vast majority feel that brands don't understand them. Being \"halal\" is not a question of narrow compliance to a formula; it's a matter of recognizing the deeper Islamic values and that recognition cannot be faked.\n\nLast, but not least, ethnicity is a denoter of the inclusive brand \u2013 especially in the US. Here's a country with multiple ethnic divides: Caucasian, Hispanic, Asian and African American. When I first moved to New York, I had a mentor, Jeff Bowman, an African American. Actually, I was supposed to mentor him, but it worked out the other way round. Jeff has gone on to make a career out of the Total Market thinking. This rejects the old American world where General Market agencies abandoned the segments and clients and used specialist, Black, or Hispanic, agencies to reach them \u2013 literally, as races apart. That made sense for a quite a long while, but changing demographic realities call this structure into question. These minorities, taken together, form the majority.\n\nJeff calls this the New Majority. By 2044, the majority of the population of the US will be composed of ethnic and racial minorities. The old General Market\/Multicultural Market segmentation no longer makes perfect sense when the General Market is itself multicultural. First, the multiethnic population is growing fast; second, the US population is increasingly cross-cultural in its tastes; and third, fragmented multicultural budgets are never large enough (even when demographically appropriate) to make the impact that a total market budget can.\n\nA total budget which is ethnically conscious and responsible is more likely to be effective than a small \"ethnic budget\" which is always vulnerable to being cut or to making false choices between various minorities. This means not going to the default position of only planning advertising to an audience of non-Hispanic whites.\n\nDigital media can become a way in which mainstream messages can touch and acknowledge ethnic context or product preferences. But they are being delivered as part of a holistic communications strategy and not from some parallel tactical universe.\n\nIf anyone doubts this, they should look for a moment at African American Millennials. Thanks to A.C. Nelson and its seminal report, _African American Millennials: Young, Connected and Black_ (2016), we have a clear view of them as tech savvy, articulate and at the leading edge of digital advancement \u2013 a good antidote to the old formula of \"will it play in Peoria?\"\n7CONTENT IS KING; BUT WHAT DOES IT MEAN?\n\n\"Content\" is one of the most used, reused, misused and abused words in the practice of marketing communications. It is associated, both consciously and subconsciously, with the phrase \"Content is King\", which is the title of an essay Bill Gates posted on the Microsoft website on 3 January 1996. \"Content is where I expect much of the real money will be made on the Internet, just as it was in broadcasting,\" Gates wrote.\n\nContent was not a word in David Ogilvy's lexicon, although he might have recognized \"contents\" very readily.\n\nWhen it would become a singular noun applied to media, it started to morph into many meanings. \"Branded content\" has become, perhaps, the most enduring of these.\n\n\"The vast majority of content produced on the internet remains unread, unwatched, unseen and unheard.\"\n\n**Branded Content**\n\nBranded content \u2013 communications contributed by brands within a vehicle whose primary purpose is to entertain rather than to sell \u2013 originally developed as a sub-discipline of the advertising business, with its own approaches, intellectual ethos and agencies. The role of the brand was essentially that of sponsor, whether it be of a whole property or as part of the entertainment narrative.\n\nThen we had the Digital Revolution. \"Branded content\" broke large with BMW's \"The Hire\", a series of eight short films produced for the internet and DVD in 2001 and 2002. Notable directors and actors were hired to make the 10-minute narrative segments, which showcased the performance benefits of BMW automobiles. This was a seminal example of content being enfranchised in a broader role.\n\nThe dramatic disintermediation of media opened up the means to create and distribute material to everyone. Absent a better term, the digital world glommed onto an amorphous, inoffensive one. Content started to be applied to anything that was produced and lived on the internet, from blogs to websites. It became short form and long form. It became client-sourced and crowd-sourced. And as it spread out in cyberspace \u2013 formless and fissiparous \u2013 we eventually realized we had to tame it. Brands and agencies evolved the concept of \"ecosystems of content\", which at least a brand could stake as its own digital territory, where the parts were actually linked, and where the sum of the parts created some value for the user.\n\n_Films got a head start on branded content with \"The Hire\", a series of short films featuring screen actor Clive Owen and shot by Hollywood directors such as Ang Lee. Distributed online and on DVD, each placed the car as the hero of a larger story._\n\nThe vast majority of content produced on the internet remains unread, unwatched, unseen and unheard. In 2014, Spotify released data indicating that only 80 per cent of the music-streaming service's songs has been listened to. That means 20 per cent of the music \u2013 some 4 million songs \u2013 had never been heard. A new service, Forgotify, sprang up to rectify what it deemed a \"musical travesty\", delivering playlists of \"neglected songs\". It has inspired other services as well. No Likes Yet supplied Instagram photos that hadn't received a single like, as the name suggests. If unseen video is more your thing, navigate over to Petit Tube where you can view the internet's most unpopular clips.\n\nAnd that's not even the ads. Once you factor in the number of lonely, unseen advertisements, the results grow more discouraging. Google says that 46 per cent of paid video ads are never viewed. For display ads, the number is even higher: 56 per cent of them go begging.\n\nThis has made content a rather dirty word \u2013 hardly kingly at all. In internal presentations, I use this image to stigmatize it \u2013 as so much digital landfill.\n\n_This is the image I use in presentations as a warning: avoid the landfill. Advertising must not become a dumping ground for content. As practitioners we must recognize content as something people chose to engage with, not discard._\n\nHow, then, to avoid the call of the dumpster?\n\nIt helps to have a sound definition. This is mine:\n\n**content**\n\nnoun\n\n_Communication so good you want to spend time with it or share it._\n\nThat's a high bar. It means that something worthy of being called content must so capture you that you choose to watch, read or listen to it. It must spur you to vouch for its value as you repost it on to your friends.\n\nThe parts agglomerated in these ways: what the brand _paid_ for, what it _owned_ , and what it _earned_ , known as P-O-E (see here). POE starts to define content in a more rigorous way, as something that is planned for.\n\nBut the question still remained, why consume the content?\n\nWhen we want clients to understand what content can be, we show a film clip from the early 1980s of David Ogilvy.\n\n\"It is content that we will produce in the future, not advertising, direct-mail pieces, or whatever.\"\n\nWhat, really?\n\nYes, really.\n\n\"It seems to me that [print] editors must know how to communicate [better] than we admen,\" he says. \"We admen have an unconscious belief that an ad has to look like an ad. Ad layouts signal to the reader, 'This is only an ad, skip it.' So always pretend you are an editor.\"\n\nThis is the mindset that is required for content online. In 2010, Ogilvy & Mather began hiring journalists, people who understood how to engage content beyond the confines of 30-second TV spots, not just in length but also in depth. Journalists expand investigation into a subject; the traditional creative team (a copywriter and an art director) compresses it. These are two very different things.\n\nAnd then we started looking for people with curatorial skills: collecting, re-presenting, showcasing, and acknowledging the source of material we did not originate ourselves \u2013 and doing it with pride. These skills are not common. They require subject matter speciality, even scholarship, combined with a generalist bent, as well as a broad network of other curators and experts to call on. Curators are, in the ponderous words of the American Alliance of Museums, \"information brokers who, through learned and creative interpretation, create meaningful experiences for people\".\n\nAll this betokens a shift in the business of agencies, which is still not recognized or even expected \u2013 namely, they are becoming publishers. It is _content_ that we will produce in the future, not advertising, direct-mail pieces, or whatever. As such, agencies will \u2013 or should \u2013 cease behaving like agents (who used to get paid by the media in a system invented in the mid-eighteenth century) and act more like the media itself. We have, if we want it, the chance to break the shackles of our past.\n\nAnd brand owners also need to think of themselves as publishers.\n\nPerhaps the first to do this overtly was Red Bull. When a beverage brand launches a global print magazine in a supposed post-print era, and then rapidly reaches a circulation of over 3 million monthly readers, it signals a readiness for consumers to accept the dominance of brand over product. It's part of a strategy that has redefined Red Bull as a publisher, specializing in categories favoured by aspirational young males, such as sports and music, and injecting the brand with an enviable dose of rocket fuel.\n\nThe brand is defiantly not engaging merely in \"content marketing\", but rather has teams in markets across the world, selling media to other brands, but scarcely mentioning Red Bull.\n\n_Logistics is a complicated but indispensable part of modern life. Through brand journalism we've helped educate UPS's business customers on the importance of logistics for future growth and profitability. Armed with more knowledge, they've been motivated to buy UPS services as a result._\n\n**Thinking About Content**\n\nHaving defined content, we can now start to attack it in a more structured way. To do that, we need to think about its properties.\n\nIn Ogilvy & Mather's experience, sometimes content is _magnetic_ , it attracts people; sometimes it is _immersive_ , you get embedded in it; sometimes it is _smart_ , enabling you in some way; and sometimes it is just downright _practical_.\n\nOf course, like all self-respecting business people, we have a natural tendency to like four-quadrant grids. Our content grid is shown overleaf, spanning from breadth (broad and mass) to personalization (personalized and individual) on one axis, and from utility (useful and informational) to entertainment (entertaining and emotive) value on the other.\n\nLike all such constructs, these quadrants are not hard and fast. There are overlaps and grey areas.\n\nInformation, for instance, can be highly entertaining; and many great pieces of content have provided it in documentary formats that entertain. Nonetheless, teasing out differences helps give a more thoughtful approach to this amorphous thing, content. But the differences are all more or less, rather than either\/or. Another helpful way of using this content matrix is to see it as a playground in which two great dividends can be looked for: creating experiences and creating engagement.\n\n_Red Bull has become something of a media giant (one that sells beverages). Record-breaking stunts, a winning Formula One team, global flugtag contests and box-cart races, music studios in major cities \u2013 and a magazine with one of the largest circulations among young males._\n\nThe digital world has put a premium on design: experimental design has become a critical pillar of the digital world.\n\nEnjoyment, however, comes through the power of stories.\n\nNeither designing experiences nor narrative enjoyment are remotely new concepts, but in the digital world they have both exploded in importance. They bring the \"content\" to content.\n\nTHE CONTENT MATRIX\n\n_The Content Matrix. Sometimes content is magnetic; sometimes it is immersive; sometimes it is smart; and sometimes it is practical._\n\n**Planning for Content**\n\nOne way of planning for a strategic approach to content would be to consider each of the four quadrants of our matrix.\n\n**1. Magnetic Content**\n\nHow many times have clients in the last few years said at the end of a brief, \"...and please will you make it viral.\" That's not something we like to hear. Not only do we mistrust \"virality\" itself \u2013 Ogilvy & Mather's own video practice has officially banned the word from all internal discussions \u2013 but the degree to which a piece of content is magnetic or not is often difficult to plan for or predict.\n\nAnd our understanding of the root causes \u2013 what makes for the magnetism \u2013 is only recently developing.\n\nThe magnetism of a piece of content is linked to arousal. Arousal is a state of heightened emotion, drawing people to share information or material. In short, it is what guides our fingers to the share icon.\n\nThe apogee of experience design \u2013 so far at least \u2013 may be The Museum of Feelings. Glade transformed more than 5,000 square feet (465 square metres) of Manhattan space into an immersive pop-up experience to appeal to its visitors' senses of sight, sound, touch and importantly smell. More reminiscent of an art installation than a marketing event, the space was designed to help people recognize how much our external environment affects our internal emotions. On the outside, the building would frequently change colour to reflect the emotional state of its surroundings, triggered by a sentiment analysis of New Yorkers' social media feeds. Inside, rooms were themed around five specific emotions: exhilaration, invigoration, joy, optimism and calm.\n\nIn the final room, visitors used polygraph-like machines to measure their biometrics, heart rate and skin salinity, which alongside external cues like building temperature and Twitter sentiment would combine to layer a corresponding colour over a user's selfie. This \"MoodLens\" transformed a simple selfie into a living work of art that was far more reflective of a visitor's emotions than a standard profile picture, laying bare an individual's genuine emotional wellbeing directly on social media.\n\n\"... the degree to which a piece of content is magnetic or not is often difficult to plan for or predict.\"\n\nWhen I first wrote a speech on this subject, my working title was \"Techniques of Arousal\". That seemed too redolent of that naughty old sexologist Dr Ruth Westheimer. But arousal is, in fact, a respectable academic study. It was recognized a long time ago that some television programming can create arousal very effectively, but the sharing impact of the internet has led to increasing study of the phenomena.\n\nYou can read the research of Taylor, Strutton, Thompson, Cushman, Earl, Binet, Field, Neba-Field, Riebe, Newstead, Mulkan, Berger. Certainly, you might have heard of Jonah Berger, Associate Professor of Marketing at the Wharton School at the University of Pennsylvania. He wrote the book called _Contagious_ (2013), which is useful.\n\nIn sum, all of these experts demonstrate that it is heightened emotions that drive people to share information.\n\nBerger has tended to emphasize the arousal of the subject (for instance, email sharing is boosted by the good feelings following physical exercise) as opposed to the emotional content of the message. He doesn't really study film. However, as practitioners, we know that content itself creates the context, and, therefore, the arousal. And other academics who have studied film confirm that social transmission is affected by emotional content.\n\n\u2022Arousal is caused most by _positive_ content.\n\n\u2022The emotion has to be _highly_ arousing.\n\n\u2022Arousal has a disinhibitory effect, which results in greater sharing.\n\nStimuli that provoke high arousal also trigger a high reminiscence effect: we remember them more. The more the emotion reflects the self-concept of the subject, the more likely sharing will take place.\n\nAnd, finally, our academics confirm for us that successfully designing high-arousal content is rather difficult. And they find something else to be critical: a sense of surprise, of the totally unexpected. As MacDonald and Ewing note in _Admap_ December 2012: \"The key emotional driver of sharing turned out to be a surprise.\"\n\nWhat happens then is that a ripple of creative joy starts to spread. In fact, what happens is physiological. The neuro-transmitter dopamine starts working; and the synapses in the brain start firing.\n\n_This is it \u2013 dopamine: the creative molecule. Food, sex, exercise \u2013 and even advertising \u2013 can trigger a release of dopamine from nerve cells (neurons) to other parts of the brain through the tiny space (synapse) in between. A spike of dopamine acts as a \"reward\", signalling pleasure, motivation, and most important of all, salience \u2013 alerting the brain to pay attention to whatever stimulus activated it._\n\nSo the key to understanding the internet is not technological; it's chemical.\n\nAnd the positive release which comes from arousal actually replicates the processes used in therapy. The social sharing of emotion produces immediate benefits, a therapeutic effect.\n\nWhat arouses us to share our happiness? We can discern a clue from one of the most successful promotions of all time. When Share a Coke was developed by our Sydney office, we had no idea that it would prove utterly magnetic and that it would generate billions of shares. Wherever it has run \u2013 and that's in most places around the world \u2013 it has helped reignite the relationship of young adults to one of the world's most iconic brands.\n\nShare a Coke aroused the sense that being socially connected is fundamental to our social and psychological well-being. Young adults tend to spend less and less time with family and more with peers, and these peer relationships become more intimate the more experiences are shared. We have always shared, but the internet allows us to share more. As futurist Stowe Boyd puts it, \"I am made greater by the sum of my connections; so are my connections.\"\n\n\"I do not like the word viral, and try to use it as little as possible.\"\n\nThat is one reason why I do not like the word viral, and try to use it as little as possible. I understand the analogy; but to me there is something malign about the word. When the research clearly tells us that internet sharing is the sharing of hedonic value, talking about making things viral somehow feels inappropriate.\n\nAnother word to be careful of is meme. Richard Dawkins, the ethnologist and evolutionary biologist, coined the term in his 1976 book _The Selfish Gene_. He described a meme as the cultural equivalent of a gene in biology. The latter self-replicates by a chemical process, while a meme undergoes the same process through cultural repetition and interpretation. Examples include a melody, line of poetry, fashion, and learned skills.\n\n_What arouses us to share our happiness? We can discern a clue from one of the most successful promotions of all time. When Share a Coke was developed by our Sydney office, we had no idea that it would prove utterly magnetic and that it would generate billions of shares. Wherever it has run\u2014and that's in most places around the world\u2014it has helped reignite the relationship of young adults to one of the world's most iconic brands._\n\nIronically, his idea became memetic itself. And now in the digital age it defines anything that spreads. But Richard's original meaning had much more to do with deeper \"transmitted code\", like the heroic codes of, say, Beowulf, than it does with bits of internet flotsam and jetsam, like bulldogs on skateboards.\n\nAnd my colleagues at Social@Ogilvy, Ogilvy's specialist social media practice, would argue against viral for very different reasons, believing that once there is the intrinsic surprise and charm in a piece of video \u2013 the preconditions \u2013 then social design kicks in, and through a whole series of interactions, optimizes the spread.\n\nIn internet terms, magnetic often means the wild and the wacky. This is the world of scary raptor reptiles, of carrot-eating contests, of mutant poodles. In fact, I once sat down and identified four types of humour that will \u2013 if done well \u2013 invariably spread.\n\n1. **Jocular** : and the joke is often against someone, as it is with the enduring genre of fail videos.\n\n2. **Pranky** : the \"lost\" bikini top that turns into shark's fin as a guy retrieves it.\n\n3. **Cute** : a computer mouse dangles on a cord that a cat holds in its mouth. \"They told me it was a mouse. They lied.\"\n\n4. **Absurd** : a bulldog riding a skateboard.\n\n_A skateboarding dog? An archetypal example of the absurd form of humour._\n\nThere is no doubt that humour like this can help create magnetic content. And Ogilvy & Mather have used all of these forms of humour to do so.\n\nHere's a prank.\n\nWe had to publicize a new car rental service: a flexible multiple subscription allowing one to use a car on demand no matter when or where.\n\nOur Paris office tricked a few unsuspecting members of the Parisian public into believing their cars had been crushed into blocks of scrap metal. We captured the ensuing drama, including the car owners' expressions of anger, with hidden cameras. Actors posing as police officers stepped in and offered a telephone help-line number. Those who took the bait and made the call were patched in to a national radio broadcast, during which many cursed profusely while receiving advice to rent instead of owning a car. More expletives were broadcast before Europcar was revealed as the prankster. (Of course, we didn't broadcast anyone without their permission.)\n\n_Our Paris office aided and abetted Europcar in a mischievous prank to publicize the company's new on-demand car rental service. How? We crushed people's cars. Well, we pretended to! Unsuspecting members of the public returned to the car park to find the remnants of what they believed to be their vehicles, with their reactions broadcast live on the radio. When they called the parking helpline and it was recommended they seize the opportunity to rent a car to get home, exasperation quickly turned to expletives. The prank not only generated laughs among listeners \u2013 and eventually the victims \u2013 it doubled subscriptions._\n\nThe message got through; subscriptions doubled.\n\nBut there is another story at work here. All this reminds me of the great nineteenth-century showman, P.T. Barnum, who understood exactly how to attract people to the circus. He drew in thousands by exhibiting a taxidermized monkey's torso attached to a fishtail. \"Roll up, roll up \u2013 see the mermaid!\" This is a crude form of arousal.\n\nThe art of arousal today has to aim higher. If we are going to entertain, we must do better. We have to tell stories. That bulldog riding a skateboard is a curiosity; it is not a story.\n\nA story requires structure, theme, mood, plot and characters rather than caricatures.\n\n\"The art of arousal today has to aim higher. If we are going to entertain, we must do better. We have to tell stories.\"\n\nDove's Real Beauty _Sketches_ film (2013) is a fine example of advertising as storytelling, and vice versa. It embraces the power of narrative, of viewer engagement, of appealing directly to a woman's sense of empathy. We invited women to sit for a forensic artist and asked them to describe themselves. Their replies exposed a skewed appraisal, one that emphasized their perceived physical flaws. Strangers were then invited to describe the women to the artist, and the appraisals were much more generous, framing the subjects' looks more holistically and using details other than purely physical.\n\nWhen the two sketches were revealed to each participant, the contrast between their negative self-perception and favourable external view could hardly have been more stark, surprising both to the women and the viewer. The emotional power of that reaction explains why the film has become one of the most watched pieces of advertising in the world.\n\nThe surprise here is shocking: the difference between how women see themselves, and how others see them. But it's also profoundly moving. It is dopamine rich.\n\n**2. Immersive Content**\n\nZen and the Art of Marketing Communications may seem a long way apart, but the Digital Revolution has brought them closer together than you might think.\n\nDigitalization had always carried with it the promise of a two-way communication. You, the individual, can be isolated. You can be invited to participate in the dialogue. And then you can join it. And, in joining, you can literally both find yourself and lose yourself.\n\nTo illustrate, I'll return to Dove. Imagine that you, a woman, were given the power to change the harmful female stereotypes that had become rote in digital media as much as in old media. We did not endear ourselves to Facebook when we created an application for Dove that did just that, using Facebook's own algorithm.\n\nWe invited women to expose the problem of negative advertising capitalizing on and directly targeting female insecurities using the \"Ad Makeover\" tool, a web application designed to subvert Facebook's own advertising marketplace and give women direct control over the ads. Typically used only by marketers, we connected the Facebook advertising auctions platform to a simple consumer interface, allowing women who visited the site to replace crude associations like \"Thunder thighs?\" with positive messages such as \"The perfect bum is the one you're sitting on\".\n\n_In \"Ad Makeover\", by turning Facebook's algorithm against those disreputable marketers who pray on women's insecurities, we empowered women to trade humiliation for admiration by replacing negative messages with ones of mutual support._\n\nOr consider the deep engagement UPS sought during the peak-shipping season. Parcel deliveries through the 2015 holiday season (from the day after Thanksgiving to Christmas Eve) accounted for about 60 per cent of all deliveries made that year. When our client UPS wanted to underpin its promise of a hassle-free holiday (after a difficult holiday season in 2014, due to atrocious weather) we guaranteed that we could deliver not just gifts but also wishes. Through the Wishes Delivered campaign, UPS invited package recipients to post a wish on UPS.com, or via social media using #WishesDelivered, and for each wish donated $1 to the Boys and Girls Clubs of America, The Salvation Army, and the Toys for Tots Literacy Program. One of many wishes fulfilled resulted in the delivery of a truckload of snow to school kids in Texas who had never seen the frozen white stuff.\n\nSuch programs create personal engagement at a very high level. Implicit in them is an element of play. Can I beat the system and change the Facebook algorithm? Can I make a wish that will be fulfilled against all apparent odds?\n\nBut what happens if the play becomes explicit? Enter the world of gaming, the point where content moves far beyond engaging and mildly immersive to being absorbing and completely immersive.\n\n_UPS used crowd-sourcing and a donation scheme to create a programme which solicited wishes for the holiday season. \"Wishes Delivered\" generated a cash contribution to charities for each approved wish. It was throughly engaging, but what really made it immersive was the stories. Stories like that of Carson, (the child in the middle), a young boy who formed a fast friendship with, and deep admiration of, Mr Eddie, his regular UPS driver. In fact, Carson had his own UPS outfit and enjoyed making his own packages. When the holidays came around, Mr Eddie arrived with more than just the usual deliveries. He had something far more special for Carson: his own child-sized truck and a day to be a real UPS driver, just like his hero._\n\nAcademic studies of video gaming define immersion as the state where gamers are so involved in the game that they become lost in it.\n\nThe evidence does, in fact, show that immersion exists on three distinct levels: a level of engagement, a period of adaptation and learning the controls; engrossment, where those controls become invisible; and total immersion. On this level you are cut off from reality to such an extent that the game is all and everything. In the words of one research study, it's \"a Zen-like state where your head just seems to know what to do, and your mind just seems to carry on with the story\".\n\nThis is what psychologists call \"spatial presence\". You create a model in your mind of the \"space\", but then move on to favour that \"space\" over real space, encouraged by cues that create a consistent sense of being \"there\". The environment has to be rich; that is to say, it has to dominate you with sensory stimulus. As Jamie Madigan writes: \"The more senses you assault and the more those senses work in tandem, the better. A bird flying overhead is good. Hearing it screech overhead is better\".\n\nIn its purest form, immersion requires skill sets which mesh ideation and technology together. The combination is not easily found.\n\nGames are important for commercial advertisers, not because a content program would be deficient without one, but because they contain ingredients and approaches which can be inspiring \u2013 and entrancing.\n\nAnd what successful games like these have in common is that they are all good stories. So let's not forget that a story in itself can entrance. Gaming psychologists call this \"flow\". It's when the interface between you and the story \u2013 game or novel \u2013 disappears. And it happens with full-blooded immersion. What the Digital Revolution has done is to make it possible for those stories to have an interactive component. At one extreme you can actually join up and become the storyteller.\n\n_With the launch of \"Farmed and Dangerous\", a satirical four-part comedy series premiering on online platform Hulu, Chipotle shunned product placement and chose instead to engineer its values directly into the show's DNA. Armed with an original script and a herd of exploding cows, the show was anointed \"one of the funniest things I have ever seen\" by media pundit Jim Cramer. Unfortunately a promotion is only ever as good as a product, and an_ E.coli _outbreak ended much of the good work._\n\nOr, you can participate as a viewer. Chipotle, again, did this technically very well, aggressively slaying agri-business in \"Farmed and Dangerous\".\n\nThe word not to forget in designing communication in this quadrant is \"entrancing\". Does the content reward you as it engages you? Otherwise it will become as quickly expendable as the vast mass of unconsumed content.\n\n**3. Smart Content**\n\nThere's been a lot of \"smart\" around in the last 20 years: smart cars, smart phones, smart carts, smart homes \u2013 and smart arses into the bargain. So, why trot it out again with content?\n\n_Participatory gaming is at the heart of the next-level brand experience our sister agency, David, created for Fanta. Rather than create a highly-scripted environment, Fanta offered its audience a full palette of tools and material to create gifs, videos, and beats to share with the world. Fanta expertly and helpfully inserted itself into a range of fun behaviours that its audience was already engaged with._\n\n_For Perrier, we invited people to enter_ Secret Place _and be a part of the best party ever. In a 90-minute feature film that worked like a video game, visitors have experienced 100,000 hours of play and have lived over 4 million lives._\n\nBecause at its best it signifies the point at which you can design your way into a better experience of life. This is not content which is there to entertain, but to help you, and in a very personal way.\n\nYou're a committed Amazon shopper, and you're looking for a home automation solution. The Amazon Echo is reasonably priced, highly interconnected with other technology, offers exclusive deals on Amazon, and voice recognition shopping.\n\nThat's smart.\n\nYou're an expectant mother in France. You're a little anxious and feeling insecure. You have so many things to keep track of and learn, and making sure that you (and baby) eat right, get enough rest and are progressing properly through pregnancy is a little overwhelming. Nestl\u00e9's Devenir Maman has you covered, keeping track of pregnancy milestones and helping you make the best choices for you and your baby.\n\n_Nestl\u00e9's Devenir Maman keeps track of a pregnant woman's milestones, deploying smart content to engage and inform._\n\nThat's smart.\n\nWhat's the psychology here?\n\nIt's all about being in control. Scientists now believe that control is a sum of different influences on personality, not just social learning but biological influences also. As one group of Dutch researchers has put it, \"feeling in control represents a blend of socially desirable traits that are all related to psychological well-being and successful performance.\" It is a direct enabler of self-regulation of behaviour, and that's exactly where content comes in as an enabler \u2013 a stimulus for, or a reinforcement of, that self-regulation.\n\nWhat is at work here is the power of design \u2013 integrating information and experience.\n\nAt an extreme, this leads to the phenomenon of \"self-quantification\". Individuals have never before been able to record the metrics that measure the routines of their life which affect their health and well-being. Now they can. Not just for the sake of losing weight or improving their fitness, but for a whole range of indicators, from sleep behaviour to work\/life patterns to mood changes.\n\nNot everyone, myself included, wants to be self-quantified, or \"body hacked\"! But the general principle of enablement, in a personalized way, is a very powerful one. It goes to the heart of the choices one makes as an individual, and how we can make better choices.\n\nDigital content is a precursor to making better choices. For instance, Nestl\u00e9 Milo paired an activity bracelet with a unique exercise and meal-planning app to help kids better understand the link between their diet and physical exercise. By combining challenges, video, avatars and real-time data, we appealed to their love of technology and games. So rather than force them outside to play, users were encouraged to take more steps each day in order to top the leaderboard among their friends, and follow tips and tricks to become \"champions\" in their chosen activity. They could see how much energy they burned in the process. Alongside, they could view their meal choices in terms of energy consumed, giving a quick indication of their progress towards a more balanced diet and exercise regime. By using more carrot and less stick, we gave kids an opportunity to actively enjoy being responsible with their health, and they jumped at the chance.\n\nIt's perhaps worthwhile to remember the second meaning of \"smart\": it is not just an adjective, it's an acronym \u2013 Specific, Measurable, Attainable, Realistic, Timely.\n\nThese are no longer goals for career-builders in business. The ordinary person has been enabled by the Digital Revolution to apply them to himself and herself, too.\n\n**4. Practical Content**\n\nOur final type of content is very practical. It sounds quite flat, but in fact it is empowering. It is content which helps elevate and enhance the collective experience of individuals, communities and businesses. This type of content puts resources in your hands.\n\n_Milo's activity bracelet and accompanying app combines exercises, meal planning tips and online games to keep kids healthier and more active offline._\n\nThe first resource is knowledge. Companies have always seen this as something to be shared. Some audiences have been catered for more than others. One thinks of motorists, for instance, for whom the Michelin Guides were created, or classics that have passed, such as the Shell Guides to the English countryside.\n\nIn the digital age, that knowledge can be distributed more accurately, in a more timely way, and also completely specifically to where you are.\n\nWhen Google wanted to showcase the capabilities of its new Chrome web browser, its Labs division reached out to Canadian Indie band Arcade Fire to help develop a new take on the music video for their single, \"We Used to Wait\".\n\nThe interactive short film _The Wilderness Downtown_ conceived by artist, entrepreneur and director Chris Milk, brought the user directly into the storyline. Unbeknownst to them, by inputting their name and the address of the home they grew up in, over the course of 5 minutes they would be taken back to their hometown. Guided by a teenage version of themselves, they would run through familiar streets and end up circling their childhood home to the sound of the band's atmospheric track.\n\nChrome has since grown to be the dominant browser, and Arcade Fire one of the biggest bands in the world.\n\n_In_ The Wilderness Downtown _the iconic indie band Arcade Fire's latest single \"We Used to Wait\" was chanted in the background as Google's Chrome browser quickly transported people to the neighbourhoods of their youth. A Google Labs experiment, it helped launch Chrome to become the world's most popular browser by combining a lightning fast experience with an incredible interactive video \u2013 an unusually immersive way to showcase a new product. Rarely is such emotional content so inherently practical too._\n\nUnder an editor's hand, knowledge brings a competitive edge. It expounds and justifies a point of view or positioning. But not just as a white paper, but rather as a dynamic program.\n\nOgilvy set up just such a program, the IBM Newsroom, to deliver rich, editorially driven content. Launched in 2014, the IBM Newsroom creates a wide range of properties: articles, videos, infographics, ebooks, SlideShares and more.\n\nWhen a brand delivers truth, it earns credibility, and with credibility comes consumer trust. As such, the Newsroom aims to deliver information its audience needs, not just content to promote the brand. In this way, Ogilvy helpd IBM join the conversation \u2013 not preach from the pulpit.\n\n_In 2012, Qualcomm microchips dominated their market, powering two thirds of smartphones. But the brand was not well known among consumers. Our agency was given a brief that paralleled the famous \"Intel Inside\" program of the 1990s \u2013 familiarize consumers with the underlying technology and establish a powerful ingredient brand. This time it was smartphones rather than PCs, and we called our solution Spark. Powered by a hub on the Qualcomm website, we re-imagined the future of communication and technology with stories of inventors changing the world. We hired industry veterans from_ USA Today _and_ PC Magazine _to steer an editorial team, who developed branded content. It all seemed to spark a lot of interest._\n\nQualcomm dominates the OEM (Original Equipment Manufacturer) market, powering about two-thirds of all smartphones, but the consumer side had been a low priority for them. Their task was to familiarize consumers with the underlying technology and establish Qualcomm as a powerful ingredient brand.\n\nSpark was the result. By hiring industry veterans from _USA Today_ and _PC Magazine_ to steer a team of editorial staff, developing branded content on sites such as Mashable, exploring syndication opportunities, and using tools such as Stumbleupon, Spark focused on generating and distributing highly practical, interesting and engaging content to early adopters and tech-savvy users \u2013 all of which was clearly attributable to Qualcomm.\n\n_Barneys New York zeroed in on the sort of journalism their audience wanted \u2013 and wanted from Barneys. The Window is an online lifestyle magazine-cum-ecommerce site that works well._\n\nNews is a resource that's freely available \u2013 but its over-availability creates an opportunity for curation. Let's aggregate and assemble the news that will appeal to our audience and distribute it to them as packaged content.\n\nThis is a space where success and failure are still being tested out. Big ambitions have bitten the dust.\n\nConsider GE, which sought to become a major news provider with properties such as \"Pressing\" (\"raising the national conversation, right, left, center\") and \"Mid-Market\" (\"the goal is, this is someone's Monday morning destination\"). Neither has survived. While its GE Reports site continues to produce excellent feature-style journalism that relates to its business, in these two they misread their audience. They aimed to fill what seemed to be a news void, but it was one that readers did not care to see filled \u2013 at least by GE.\n\nPractical content can become activist: it provides platforms for empowerment. It does not just provide contextual information but also a sense of inspiration within a community as well. This is exactly what our client American Express seeks to do with its OPEN Forum.\n\n_American Express developed OPEN Forum, an online community giving business owners access to advice from peers, leading entrepreneurs and industry bloggers. OPEN Forum gives them greater exposure, business development tools, and an opportunity to build credibility for their businesses._\n\nPractical content can also be incredibly useful, especially for an audience at the start of a steep learning curve, such as new mothers. What do contractions feel like? How accurate are ultrasounds? What music is best for my baby? These are the questions \u2013 among many, many others \u2013 that run through a woman's mind moments after finding out she's pregnant. And it's not just the next nine months that makes her anxious \u2013 it's the next few years!\n\n_Huggies' Mommy Answers is a go-to place for expectant mothers to ask questions freely and get feedback from professionals and peers. By targeting moms early \u2013 before they receive a care package provided at 90 per cent of hospitals by category leader Pampers \u2013 Mommy Answers creates the opportunity for Huggies to be a resource for new parents and build a lasting relationship with moms._\n\n_Ford invites customers to dream up their perfect Mustang \u2013 and build it too. Customers use its website or mobile apps to configure a bespoke vehicle._\n\nA final play, not as broad but still capable of scale within a defined audience of higher value customers, pushes empowerment into the physical space: the content becomes a utility. Yes, tools are also content. This is where digital design most influences experience.\n\nHow, for instance, to design a tool which enables you to configure a vehicle?\n\nOr to guide you through your menu choices?\n\nIn essence, digital enables companies to contextualize their products and services through information at greater scale than ever before.\n\nIt all starts with an idea. But an idea today has no wings unless it explores some of these digital dimensions. At its best it will play freely among them.\n\nWhen we created a program for Coke Zero in the US, the idea was that you wouldn't know it until you tried it. Armed with the research-driven knowledge that 85 per cent of Millennials had never tried the Coke variant, but knowing that half of them would become monthly drinkers once they did, we deployed digital technology in some rather impractical places to achieve some very practical results. A billboard that dispensed the drink, a TV commercial which viewers could Shazam for a sample, and other \"drinkable advertising\" got people trying and then buying Coke Zero.\n\n_The tool can be an object in itself. Ogilvy & Mather were asked to help Vittel publicize the case for regular hydration. Proving that smart technology doesn't always require the internet to power it, we put a twist on the everyday bottle top by integrating a simple mechanical timer into the cap itself. Turn the cap at the start of the day and at regular intervals a little flag would pop up to alert you to take a sip._\n\n_For Coke Zero we asked ourselves, why imagine the taste when you can try it for yourself? So we created the first drinkable advertising campaign, offering free samples through unexpected channels \u2013 including a twist on the classic billboard._\n\n**Organizing for the Content**\n\nSo does King Content have feet of clay? Not if it looks to what it can do at its best: to arouse, to entrance, to enable and to empower.\n\nBut there is one caveat. How to organize for it?\n\nAs it happens, that's not as easy as it sounds. Not only do we need to help brands and media partners work together in new ways, we ourselves in the advertising industry needed to find new structures. With anything new and complicated, it is easy to get caught up in the planning and strategy. Those elements are crucial, but when it comes to any kind of creative work, what matters in the end is the doing.\n\nI was once asked to paint a slogan on a slogan wall at one of our offices in China on the subject of strategy. Over-sated with several days of strategic discourse, I wrote the words \"superior execution is the highest form of strategy\". It was greeted with a certain surprise.\n\nBut never is that truer than in the case of content.\n\nAnd the battleground here is the Content Studio.\n\nOf course, they are not universally called that. There are kitchens, centres, newsrooms, hubs, shops, story labs \u2013 you name it.\n\nBut let's call them Content Studios. By the beginning of 2016, Ogilvy & Mather were operating some 126 of these for our clients around the world. And the demand is increasing exponentially.\n\nCONTENT STUDIO: SEVEN BASIC APPROACHES\n\nWhat do they do? There are seven basic approaches:\n\n_Content Studios need to do seven basic things well: foster community, develop an editorial lens, forge partnerships, be both curators and collaborators, act in real-time, be ready to manage crises and activate in a responsive, always-on manner._\n\nNot every studio performs the same functions. And, in some, particular functions are more important than others.\n\nBut in doing these things any Content Studio worth investing in has to justify itself in two ways.\n\n1. It has to have as its central guiding purpose the creation of a Content Calendar. It is this calendar which defines the whole rhythm of the studio. The inventor Robin Sloan borrowed a term from economics, and did us all a favour when he popularized the idea of \"stock and flow\". Stock is the bedrock, what you have to have all the time. But then:\n\nFlow is the feed. It's the posts and the tweets. It's the stream of daily and sub-daily updates that remind people you exist. It's the content you produce that's as interesting in two months (or two years) as it is today.\n\nIn fact, our social managers prefer to subdivide \"stock\" between \"hero\" content, planned annually and produced quarterly; and \"proactive content\", planned weekly and produced weekly. All the rest is real time.\n\nSo a perfect Content Studio's flow looks like the diagram opposite.\n\n2. It has to be dedicated to a second big purpose, which is to measure how effective all this is, and then not just measure but optimize.\n\nInitially it worried me that there were no tools available for assessing the effectiveness of different types of content. But then a brilliant young analyst in our New York team developed a system that can sort pieces of content by their performance against different metrics, including those that link directly to sales. She called it \"Pulse\".\n\nHOW THE CONTENT STUDIO WORKS\n\n_The workflow that lies behind the Content Studio, where the client's content strategy comes alive as a content calendar. It doesn't just happen: there has to be an engine that drives the development, distribution, and effectiveness. It's the same in principle whether you are Mastercard, with an audience of 30 million people a week across its digital and social properties, or a local restaurant chain, drumming up a customer base in the hundreds. The mantra is \"think, create, operate\"._\n\nAt the end of the day, it's keeping a finger on the pulse like this that holds King Content true to his promise.\n\nI do firmly believe in that promise. The concept of brands and agencies as publishers marks the first real change in our business since the invention of the commission system in the coffee houses of eighteenth-century London. It anticipates the final demise of that commission system, even if the industry is still at a very early stage of feeling comfortable with \"content\".\n\n**8 tips for content**\n\n1.Don't try to be a mainstream news provider.\n\n2.Don't compromise on quality: journalism is a craft not a commodity.\n\n3.Scale! Be willing to promote your content, to make sure it interrupts the interruptions. Don't be foolishly purist and expect you can just build it and they will come.\n\n4.Recognize that you need an engine to drive it.\n\n5.Don't forget why you are doing it. You are creating your own \"walled garden\" which people have to enjoy. Enjoyment has been proven to enhance not impede information processing.\n\n6.Make it sticky: link a lot; balance stock and flow; create series and sequences; build loops.\n\n7.Don't forget why you are doing it: who is the brand custodian and where is the brand conscience?\n\n8.Personalize. And use higher cost content as a reward for loyalty.\n\nCADBURY\n\nOne of the defining images of the digital age is a gorilla hammering drumsticks. How did that happen? Cadbury Dairy Milk's 2007\/08 marketing programme has assumed mythical status. The actual history is sometimes difficult to disentangle.\n\nIt began in a dire period for the brand: a salmonella scare leading to a recall of over a billion chocolate bars. And, generally, this was a brand which no longer enjoyed universal brand preference. It was highly polarized by age. Anyone over 35 considered it to be the best chocolate, and those under 35 did not.\n\nSo a brief went out with two components: to \"get the love back\", and at the same time, to remind younger consumers, in particular, that Cadbury chocolate is actually made with milk. There was a codicil given by the new Marketing Director, Phil Rumbol, who was insistent that the advertising should be as enjoyable to watch as it is to consume the product. What an exciting brief for an agency.\n\nUnfortunately, great briefs are never easy, especially if you sometimes know too much. The incumbent agency could not crack it. Phil gave the brief to Fallon, went to Australia on a business trip, and came back a week later to find there was a small raft of work to see.\n\n_Cadbury's famous_ Gorilla _commercial featured Phil Collins' track 'In the Air Tonight', which added an air of drama before the sticks came crashing down. He had been waiting for this moment, and it showed._\n\nThere was one campaign idea out of them which seemed to answer the second part of the brief: Glass and a Half Full productions, a flexible enough device for understanding milk content, and with that were four executions, among them _Gorilla_.\n\nIt was the very antithesis of linear, persuasive advertising, and the reactions up the line in Cadbury were of bemusement not amusement. Phil recalls it was something like: \"Let's get this right: You want to make an advert two times longer than a normal ad. It's got no chocolate in it, and it's got no message. Are you mad?\"\n\n_Like a disappointing movie sequel, a second film,_ Trucks _, didn't live up to expectations. It lacked the sheer joy and exuberance of the bar set by_ Gorilla _. Glass and a Half Full productions was looking half empty, not half full._\n\nWhat follows was a masterful exercise in stakeholder management. Phil persuaded his superiors at least to make the ad. Then it was shown to the senior management team, one of whom said: \"You are never showing that ad.\"\n\nPhil then requested, \"Can I ask for one last favour? Please take it home, watch it with your family over the weekend and see how they react.\" Of course, on the Monday the positive calls came in \u2013 yes, it had made them smile.\n\nThen came the meeting with Todd Stitzer, the CEO. Again, the blanket refusal. In this case, the marketing team answered the CEO with the standard defence: research. They did their research pre-tests. The results were not promising. It \"could be for any brand\", it \"didn't tell you anything new about the brand\". Of course, these were the answers to the wrong questions. The question which the \"old model\" research should have asked was \"does this, aspirationally, feel like something Cadbury Dairy Milk would do?\". By then the CMO was onside, and the 10 or 20 last pretests of Cadbury ads, when plotted from worst to best on the standard nine box matrix, had provided some permission to believe that this one was relatively \"airable\".\n\nAt the end of the day, an edit of respondents' reactions on video, moving from smiles to big, big smiles was what won the day. It was aired on 31 August 2007.\n\nThe impact was instantaneous. YouTube and Twitter were barely a year old, but the views and posts became an avalanche. With them came the spoofs \u2013 the tribute of the digital audience to something which engages them. So what was the idea?\n\nIt's been constructed in Cadbury culture as being about \"joy\". A latter-day Phil in Cadbury, Phil Chapman, was one of the most articulate explainers of this kind of joy: a world away from \"happiness\", something vital and visceral; in the case of _Gorilla_ something that even comes close to anger, letting feelings out of the system. Defining the idea is important. Cadbury, during the time between the two Phils took something of a wrong turn into a zone which was called joy (Joyville) but which was actually joyless. And Glass and a Half Full's immediate sequel to _Gorilla_, _Trucks_ is a laborious sequence of airport trucks \"fooling around\" where joy seems almost completely absent. Much, much later I remember Phil and his boss, Cadbury President Bharat Puri, saying of a South African script featuring a woman pregnant with triplets, as he signed off on it, \"with joy, you have to sniff it\". A lot has to do with charm, and craft plays a role here. The Italian Art Director Juan Cabral, who made _Gorilla_ , made it look wonderful \u2013 even adding the gold tooth which gives his scowl so much charm when the camera moves too close to him.\n\n_And with a bold and beautiful vibrato \u2013 the joy was back! Triplets evoked the essence of Gorilla, its indulgent irreverence, by way of three unborn babies singing acapella in their contented mother's womb._\n\nBack in 2007\/08, the sales decline went into reverse, and Cadbury posted a 5 per cent increase in revenue. Todd Stitzer stood up at an analyst's presentation in early 2008 to talk about the teamwork. He did it with a reasonably straight face.\n8CREATIVITY IN THE DIGITAL AGE\n\n**\" GIVE ME GOLD\"**\n\nDavid Ogilvy both managed his agency and acted as its creative director; and there have been others like him. But if you're a \"normal\" adman or woman working up the ladder by handling clients and running offices, you need a creative partner.\n\nI was lucky to have Tham Khai Meng as mine. A charming, quiet and understated Singaporean who was passionately obsessed by the quality of the work, he had led our Asian network, creatively predominant, for 11 years. So when I asked him over dinner in the unlikely venue of the Andaz restaurant in Lahore, kebabs in front of us, and the lowering mass of the Badshahi Masjid behind, to join me in New York, I had three fears.\n\n_The desk I shared with Khai at Ogilvy's global headquarters in Manhattan: a common work surface for a shared ambition \u2013 that we should strive for creative excellence and the utmost effectiveness._\n\nThe first was that I was deliberately flouting the conventional wisdom that we needed a big-ticket name. Or, worse, that an ethnic Chinese just would never pass muster in New York. And the third was that he would not come. But I knew that he was the one.\n\nHe came. We then shared a desk in New York, a completely alien notion there, but one which demonstrated that we could not be divided, and that our agenda was common.\n\nIn June 2008, we inherited a very average performance, winning some \u2013 but not many \u2013 awards at the Cannes Festival of Creativity. Ogilvy & Mather languished in the third division, and had done for years because, for many good reasons, award-winning creativity had not been an agency priority for some time. At one point Khai leaned towards my half of the table and said: \"I hate it. I can't work in a place that meekly accepts being slightly above average.\"\n\nSo we agreed to set out to become number one \u2013 \"Network of the Year\". We set five years as the time frame; we created a programme throughout the agency; we treated it like a combination of a military mission and a political campaign. The cry went out: \"Give me Gold\". It seemed crazy and impossible, but rather to our surprise, we achieved in it in two years.\n\nA central part of this has been the Cadre system, bringing together our best creative people once a year to talk about and improve the work. They're only invited if their offices are admitted to the Cadre; and that is calculated on points earned from awards won. We try to meet in interesting places. I have addressed the Cadre from a baroque pulpit in Cuzco, Peru, a tent in the Maasai Mara, Kenya, and a hotel saloon in Inverlochy, Scotland.\n\n_Developing advertising that challenges the creative convention is the work of artists, and a skill we've continually refined. When Khai declared, \"Give me gold!\", our creative teams across the world responded with work that set a higher bar._\n\nCreative awards are a dangerous thing in our business. They only represent excellence inasmuch as the juries who decide them can recognize it, or, in the politics that go with jury service, are prepared to reward it. David Ogilvy did not feel comfortable with them, although he was not alone in enjoying the lustre they brought.\n\nBut by the early part of this century, three things had changed since he received the Clio shown in this picture.\n\nFirst, the clients themselves had started attending Cannes in droves, attracted in a brilliantly successful strategy by the owners. It started with Procter & Gamble; and then the rest followed. For those clients, it became of a way of improving their own creativity, which in turn helped them achieve competitive advantage.\n\n_David was sometimes enamoured and often sceptical of advertising awards, and rightly so. In the years since his wins creative award shows have become big business._\n\nIndividual clients come and go, but collectively they changed Cannes from something self-indulgent into something more missionary, concerned with the value of creativity in business. The categories for entry expanded into effectiveness, technology and other disciplines beyond advertising. And, of course, there's nothing a client who attends Cannes likes more than to mount the stage in the Palais, or feel their choice of agency vindicated by its performance there.\n\nSecondly, the economic justification \u2013 to us at least \u2013 had become very clear. This is a business where success is directly related to the extent to which one's share of the best creative talent is disproportionately large, and Cannes is where you find them. I don't think there is any other business except for professional football where this is the case. Cannes provides a perfect recruitment mechanism. The stars in our business want to work in a company that takes creativity seriously, and where winning is seen as core to the purpose, not as an expendable luxury.\n\nThirdly, the arrival of the big holding companies \u2013 and the institution of a Holding Company of the Year award \u2013 meant the competition was featured in their annual reports. Performance was used by financial analysts hunting for any alchemical symptoms which might indicate underlying strength or weakness. All this happened at a time when bloggers were being critical of BDAs (Big Dumb Agencies), and the business model had to show itself to be as sexy as it was efficient.\n\nSo it was that when the 100th anniversary of David's birth had to be celebrated, we decided to take it (against the conventional wisdom) to Cannes. On 23 June 2011, people woke up to a gigantic red carpet spanning the entire length of the Croisette, which we subsequently cut up into small pieces and turned into David Ogilvy Red Carpet insoles, and used as a New Year's direct-mail piece sent to clients and friends. That year, we came second.\n\nBy mid-afternoon of 23 June 2012, the following year, it became clear that we were edging towards victory. A congratulatory text from my counterpart Andrew Robertson of BBDO, a typically generous gesture, confirmed it: their tally of points was at that time more accurate than ours. A few hours later, our teams invaded the stage in triumph. Since then, we've won four times more. Of course, we will not win forever, and rightly so.\n\n_The first time we de-camped to Cannes we proudly came second and cut up the red carpet as a celebratory direct mailshot. We won first place the following year._\n\nBut wanting to win, and working to win, has become a part of Ogilvy & Mather's culture. It taught me that my biggest battle was against our size. Big we were, and dumb we would be unless we _behaved_ as if we were small, as if every last prize mattered to us as an issue of life and death, because it reflected fundamentally on our worth as creators and as craftspeople.\n\nAnd we never pursued Cannes in isolation. It was always twinned with winning the Effies, which measure pure effectiveness.\n\n_Proof that agencies can be big without being dumb. The moment of victory is precious but fleeting; the greater value comes from the culture that winning creates._\n\nHowever, the acme of success must be to win both a Grand Prix at Cannes and a Grand Effie. This happened for the first time with our film _Evolution_ for Dove, in 2007 (see Hall of Fame case, pages 44\u201345).\n\n\"Big we were, and dumb we would be unless we behaved as if we were small.\"\n\n**ART OR SCIENCE?**\n\nDavid did not like the word \"creativity\". He described it in _Ogilvy on Advertising_ as \"hideous\". Rather, he saw himself in the business of inventing ideas.\n\nI think this repugnance for a word, which has, after all, become so widely accepted as a description of what entire industries do, came from a very visceral reaction to the over-statement of those who have argued \u2013 then and now \u2013 that advertising is exclusively an art and not a science. This precious and self-indulgent mindset led to work which, while pretty and interesting and entertaining, just does not sell.\n\nDavid was first and foremost a salesman. \"We sell or else\" was his motto; and, in origin at least, his salesmanship was founded on \"scientific\" principles, not least his admiration of the adman Claude Hopkins whose book _Scientific Advertising_ (1923) still has its admirers.\n\nThe former advertising planner Paul Feldwick has written an original and intelligent history of the theories of advertising, _The Anatomy of Humbug_ (2015). It is one of a kind, and it lays out the twists and turns of this debate over time. It has been an epic struggle between art and science.\n\nOn the side of art are Bill Bernbach and Charles Saatchi; and on the other side are the forces of science, where Claude Hopkins' mantle was inherited by Rosser Reeves and David Ogilvy.\n\nThis \u2013 the \"great debate\" of our business \u2013 is never ending but is now cast in a new light as we become more familiar with the digital age.\n\nThe sub-plot, though, has always been more subtle, and revolves around whether creativity is most effective as an appeal to reason or to emotion.\n\nThe apostles of reason argue that the fundamental purpose of advertising is to sell. Reeves' famous Anacin commercial is perhaps the best illustration of this.\n\n_In a unique take, Paul Feldwick turned his strategic planning sensibilities to the subject of how advertising works. Capturing a hundred years of ideas on the art versus science of advertising, the debate will surely run for at least another century._\n\nThose of emotion suggest that its role in consumer behaviour is paramount. Emotions account for brand preference, they argue, and trot out fMRI studies and likeability measures to prove the point.\n\nThen we saw over-emotional enthusiasts such as Kevin Roberts with his simple slogan of \"love marks\" ranged against the Australian academic Byron Sharp who believes that love is missing the point. We use our mammalian brain to decide, and it rewards simplicity and ease of remembering. It's an emotional decision but based on the simplest front-of-mind cues, nothing else. The fact that there are extremes in a debate is best adduced by the acerbic rhetoric as they slog about each other.\n\nThings start to become more interesting when we add in the very real dynamic of getting advertising approved. For many clients, approving an emotive piece of creativity can be a frightening proposition. It is difficult to judge. It requires an emotional response in return. Are you prepared to lay bare your soul in discussing it; how do you know it will work? And, of course, how can you defend it within your organization to those who are even more proof-demanding than you are?\n\nSo, a third level of debate opens up, the sub-sub-plot. How do you measure creativity?\n\n_Anacin pioneered competitive advertising, pitching itself to the headache sufferer as superior and different from the rest. In a US commercial from the 1940s, a confident spokesperson, talking straight to camera, directly compared the brand to Aspirin with Bufferin, which itself claimed to be twice as fast as Aspirin. To raise the stakes, he pronounced Anacin the \"fast, fast, fast\" remedy and likened it to a doctor's prescription \u2013 containing unique ingredients designed to remedy a headache without the typical side effects. To hammer the point home, an endorsement \u2013 that three out of four doctors recommend it. In fact, the main ingredient was just plain old Aspirin._\n\nOf course, the scientists lean towards quantitative testing; and not just post-testing (which even artists might find difficult to argue with), but pretesting. And, because no one will want to take the risk of spending millions of dollars in TV production before seeing those pre-test results, pre-testing with very inadequate stimulus material.\n\nHow can you hope to portray emotion in an animatic, or even a photomatic or stealomatic? They are crude proxies for film, but pre-tests thrive because they seem to provide certainty. I have known clients wait for a one-line email with the EI (a common testing measure) score in it, which would determine \"go\" or \"no go\".\n\nOf course, the artists (and the pre-test companies) point out the absurdity of this, and explain that a pre-test should be used diagnostically and not as surrogate decision-maker. But we all know that organizational politics mean that a red preview from Millward Brown renders a decision to air the copy politically very dangerous. And that is despite the fact that some highly successful pieces of TV advertising have run regardless of the test results, based on just an intuitive decision, of which the re-launch of Old Spice was a defining example of the digital age (see Hall of Fame case, pages 104\u2013105).\n\nIn the early 2000s, something funny happened: science came to the aid of art. The \"Ah-ha\" moment for me was in the unlikely venue of Changchun, in North Eastern China, in 2008. With temperatures of \u221220\u00b0C (\u22124\u00b0F), our planners were less than impressed by my choice of venue (chosen to emphasize all that we still needed to learn, because Changchun was the least well known of the planned cities of the twentieth century, as the sometime capital of the notorious Puppet Empire of Manchukuo). We had invited Dr Robert Heath, Professor of Advertising Theory at the University of Bath; and he gave a brilliant expos\u00e9 of the newly mined and startling implications of neuroscience for market research.\n\n\"For many clients, approving an emotive piece of creativity can be a frightening proposition.\"\n\nSimply put, neural scanning demonstrates that brand preference has a physical basis. Just look at the image below of the brain, in a blind and branded test of Coca-Cola. The glowing zones which are being stimulated by brand Coke relate to liking and memory.\n\nWhat Heath had succeeded in doing was to show how emotional content in advertising is often more powerful than rational messages; the latter can be easily filtered out by the brain, while the former can be processed even at low levels of attention, and then act as a kind of \"gatekeeper\" to the rational decision which all marketers yearn for. In other words, it may be easier to get positive test results or research validation for those rational campaigns that are solely product based, but the risk is that they will \"pass like ships in the night\". Logic persuades; but only emotions motivate. If people _want_ to use your brand, they will find a logical rationale themselves \u2013 but the wanting comes first.\n\n_A neural brain scan shows how the brain responds differently to the Coke brand compared to its taste. In the blind test (top), parts of the brain triggered by taste are active. But introduce the Coke brand (bottom), with its imagery and associations, and areas relating to memory and enjoyment are illuminated._\n\nOLD SPICE\n\nLike many others, my first tentative experience with aftershave was with Old Spice. The Beaujolais Nouveau of colognes, it was satisfying but shallow \u2013 a confection of nutmeg, star anise and citrus at the top of its perfume pyramid, with deeper, masculine notes below.\n\nLaunched in 1938, by the Shulton Company, it really had become grandpa's brand \u2013 the musk had become musty. It was fuddy-duddy and the very definition of uncool, a half-a-billion dollar business highly concentrated in a big and still-profitable footprint. Then the axe fell: the relaunch by Unilever of a demographically interesting range of men's toiletries which had previously just jogged along. It was all the more dangerous because Axe had launched in the US a few years earlier. Axe's weapon was the oldest known \u2013 sex \u2013 and the Axe effect depositioned Old Spice even more. The Axe user became more attractive to women, as simple as that.\n\n_The first attempt to refresh Old Spice looks a bit sour today. Although an important part of the journey, the Bruce Campbell campaign had the irreverence but not the style and demographic insight of the work that followed._\n\nBy 2006, things looked grim: Old Spice needed a better strategy and a change of agency. In this case it worked \u2013 after a while. Wieden & Kennedy Portland started by immersing themselves in the history of the brand, and its cultural rating in a way which would have won the plaudits of the cultural strategists. They spent two days combing through the archives to understand the history. The result? A strategic refocusing of Old Spice from \"old\" to \"experienced\". We're not your grandfather; we're your elder brother. Or, as it was expressed, \"experience is everything\". It doesn't appear in case histories, and not many people remember the Bruce Campbell commercial which launched it. He strides around a wood-panelled library. It wasn't his fault, but my goodness it seems laboured to me now. They were meant to communicate a sense of old-fashioned charming innuendo. And this reframing was aimed at teens and twenties consumers.\n\nBut in 2009 something else happened in the marketplace which caused alarm, and I was to observe it from the other side. It became apparent that Unilever was about to launch Dove Men+Care, the male extension of its female beauty brand. In fact, we were developing a positioning around men who were comfortable in their own skin. We had a Super Bowl film about to go into production, the apex of a socially designed program built around the microsite dovemencare.com, and aimed at evoking consumers' unsung moments of manhood.\n\nIn the meantime, the client structure changed at Old Spice. P&G had acquired Gillette in 2005, and had run it as a standalone unit. But in 2009, a new male grooming unit was formed in Cincinnati, incorporating both the Gillette and P&G brands. One of the Gillette clients assigned to it was Rishi Dhingra. In the frenzied period before the anticipated Dove launch, agencies were briefed afresh. Rishi recalls that he told them, \"We will give you the guardrails, but within that you have freedom\". That was a very un-P&G way of doing things, and indeed was the beginnings of an Old Spice exceptionalism which allowed it to be highly experimental, to use phrases like \"ridiculously masculine\" to complement the official strategy speak of \"help young men navigate to the stages of manhood\". Ideation took place at a very rapid pace \u2013 the agency was adamant that it would not give a predictable solution. Throughout the whole process, they pushed back against any effort to control, over-rationalize, and in Rishi they had an enlightened client who empathized with them. Out came Isaiah Amir Mustafa, an NFL player and small-time actor, but an inspirational piece of casting, to bring to life the idea: \"The man your man could smell like\". The first television commercial was beautifully shot by director Tom Kuntz. It was a leap of faith. It had to be \u2013 the agency had insisted on a pre-nuptial which broke the P&G rule that all advertising would be susceptible to pre-testing. As Rishi says, \"the true test was when the consumer reacted to it\", which they did in their millions. It's probably the feeling of attractive rebelliousness which makes Mustafa so effective. But also the strategic thought behind him, which was that 60 per cent of men's body wash is bought by women. By saying as he did \"look at your man, now back to me\", he simulated \u2013 and created \u2013 a conversation between men and women.\n\n_Old Spice's The Man Your Man Could Smell Like campaign featuring Isaiah Mustafa cleverly played on commonly felt notions of desire, pride, jealousy and masculinity \u2013 all in a uniquely quirky way \u2013 to take a swing at Axe. It worked._\n\nBy July 2010, with an eye still on Dove, the campaign changed into a more directly social gear. Mustafa filmed the so-called Response Campaign in two-and-a-half days, recording 186 personalized messages to Old Spice fans on Facebook, Twitter and YouTube. In 24 hours, with 6 million views, it became one of the most popular interactive campaigns in history, winning the online buzz war (capturing 76 per cent of it, in fact).\n\nAnd the upshot? Dove Men+Care still established itself in the contested male grooming market, but Old Spice transformed its brand equity and returned to volume growth in its home markets, then set about a programme of global expansion. By the time it reached India two years later, pre-nuptial had been all but forgotten. The empire had fought back and pretests were the order of the day again. But when it tested with young Indian men it was a top scoring ad. Rishi believes that they had seen the US versions on the internet and were predisposed to like them.\n\nAs the late Tim Broadbent, to whom we at Ogilvy & Mather owe so much of our own thinking on this subject, wrote:\n\nMostly what happens is that we want something, and then we come up with a rationale to justify why we want it. The justification is not the same as the motivation, although conventional market research can confuse the two.\n\nSo we learn about brands without being consciously aware that we are learning, which is why a narrow reliance on specifics such as the consumer recall of a selling point or a key visual contained in a pre-test misses the point. It is based on psychological models which long precede the digital age.\n\n_Visiting Changchun, one of the world's lesser known planned cities, with the late Tim Broadbent. Tim was a friend and ally in the pursuit of advertising effectiveness and the industry is indebted to his mastery of research \u2013 not least his warning that the stories we tell ourselves (and the market researchers) are not always to be believed._\n\nHowever, it is interesting to note that attempts by Heath and others to create a perfect pre-test have not met with great commercial success. Maybe the answer just lies in more intelligent use of the old studies?\n\nShortly after, the definitive proof emerged that emotive campaigns were twice as profitable as rational campaigns. It came to us courtesy of the study, \"Marketing in the Era of Accountability\" (2007). This should be on every student's digital bookshelf. Unfortunately, it languishes too much in the WARC archives, while the fruitless debate as to whether we are artists or scientists rages on.\n\nTime out, please.\n\nThe answer is that we are both.\n\nThe science of advertising works because of the art of advertising.\n\nBinet and Field show, definitively in a study of 880 campaigns, that it is _creative_ campaigns that outsell, on every business metric.\n\nOf course, these findings circle back beautifully to the issue of creative awards. In Cannes, as a jury you are inclined to award creative campaigns. In so doing, Tim pointed out, \"juries have been criticized \u2013 but their instincts are sound:...emotive campaigns are much more likely to strengthen the brand. They do better for the client. They deserve to win\". Not only that, but, in the digital age, their fame is amplified as a result of social buzz, creating a positive cycle of advocacy and excitement.\n\nIn a study of Cannes award winners, their campaigns were shown to be 11 times more effective than work that had not been awarded.\n\nBack to David Ogilvy. His first work in advertising was not to write an ad at all but to perform a consulting project. The report he generated is interesting for the pungency of its conclusions, so trenchant indeed that they resulted in Mather and Crowther being fired from the business (the famous work which later emerged came after a re-hiring). However uncomfortable the findings might have been to the client, they are rooted in observation, in evidence, in hypothesis \u2013 all the techniques which a little later led to David working for George Gallup as a researcher.\n\nBut, 25 years later, in 1955, it was the same Ogilvy who attacked the \"hard facts\" school of his friend and rival, Rosser Reeves, the founder of Ted Bates, in a famous speech in Chicago, and laid out a manifesto for brand image advertising. Suddenly he was joining the debate of what creativity is, and what it is for, from, apparently, the opposite end of the spectrum.\n\nOr course, in reality there was nothing contradictory in his evolved position. It simply reflects the enduring dualism about creativity, that it is neither purely an art form whose exact workings in the marketplace cannot be forensically proven nor is it a rigorous science susceptible to explanation only by rules and proof through numbers.\n\nAt the end of Paul Feldwick's book, he poses an (un-answered) question about how the Digital Revolution affects the sorts of debates that have characterized advertising's past.\n\nI think the effect is clear already. It is about the confirmation that dualism is not a \"benign conspiracy\", as Heath describes it, but is a far more substantive and holistic convergence of the two.\n\n\"This is creativity which pervades the world, not recognizing any barriers, divisions or silos.\"\n\nAt its root, digital media tends to be processed in a high-attention mode. So for agencies who do take an integrated approach, and who think of digital not as something apart but as just a part of advertising, the complementarity of artistic, emotional messaging with scientific, rational arguments starts to become real.\n\nDavid Ogilvy was, of course, at home with Direct Response \u2013 \"his first and last love\", as much as he was an inventor of \"brand image\". The digital age has a chance of demonstrating that there is no inconsistency between the two schools \u2013 unless there are other issues of politics, prejudices and preference which intervene.\n\nI find it a pleasing irony that it is a binary system that is leading to a communications world which is more unitary than ever before.\n\n**PERVASIVE CREATIVITY**\n\nThe \"great fragmentation\" both disfavours and favours creativity.\n\nAt its worst, it simply diminishes creative impact by allowing the parts \u2013 multifarious expressions in different media \u2013 to go their own way at the expense of the whole. Creative fragmentation follows on from industry fragmentation as a host of players and partners each struggle to find their own place in the sun.\n\n_David Ogilvy's \"The Theory and Practice of Selling the AGA Cooker\" is a masterpiece of direct marketing. Fifty years after its 1935 publication,_ Fortune _magazine dubbed it the best sales manual ever written. It shows a keen knowledge of product and customer as well as the barriers and drivers to sale. David notes that, \"The worst fault a salesman can commit is to be a bore,\" a peril he gives wide berth on every page of his entertaining manual. \"The good salesman combines the tenacity of a bulldog with the manners of a spaniel,\" David writes. \"If you have any charm, ooze it.\" Hunt this manual down online \u2013 it is easily found \u2013 and read it, even if you don't care in the slightest about cookery or appliances._\n\nBut at its best, it reaches out into the world in ways that previously would have scarcely been imagined.\n\nWhen I asked Khai to provide an adjective to describe the newness of this creative opportunity, he went silent for a while, and then obliged with \"pervasive\". This is creativity which pervades the world, not recognizing any barriers, divisions or silos.\n\nWhat I call \"packaged\" creativity, which delivers messages in neat bundles to interrupt your passive use of media, is essentially invasive. It seeks to intrude in defined spaces. It lobs carefully crafted missiles out into the world. But \"pervasive\" creativity spreads: it is essentially liquid. As Khai wrote:\n\nIf you want a metaphor for Pervasive Creativity, think of water; it is both vital for life and unstoppable when in full flood. It flows through cracks too small to see. It's always flowing, moving, exploring, and getting into everything, never getting stuck. Pervasive creativity means having one's antennae tuned to channel MUSE, looking for inspiration in everything.\n\nIn a sense, the digital age allows us to take mainstream something of the philosophy of graffiti. We can introduce messages in startling ways, just as an artist such as Banksy provokes by taking clean conventional messages in conventional contexts, and turning them on their head: \"sorry, the lifestyle you ordered is out of stock\".\n\nDon't pre-select your silo!\n\n_Banksy understands how to turn conventional formats, like a billboard, into an unlikely commentary on our lives. His lesson? Creativity doesn't care for pre-defined media plans._\n\nSo, in the digital world, the demands on creativity to be excellent are greater than when it was only invasive. Sure, we can reach people when they are waking up in the morning, but can we do it in a way that is appreciated but at the same time surprising?\n\nThe boundary between creative and gimmicky is a narrow one here. One might think that Oscar Mayer overstepped it when it started with the thought: \"What if smells could be digitized?\" Nine months of research and development later, it resulted in a dedicated mobile alarm-clock app that wakes you to the sound of sizzling bacon with an accompanying bacon-aroma-emitting smartphone device.\n\n_Do you really want to wake up every morning and smell the bacon? The difference between pervasive and invasive is a narrow one._\n\nThe premium, then, is on inspirational thinkers, who can transcend the \"bittiness\" of the digital world, and then jolt us into seeing things differently, but without descending into silliness. Sadly they are scarce.\n\nRaw: _pervasive creativity in Asia is a collection of striking everyday creativity, such as the type demonstrated by this man selling hand-decorated stuffed horses, found by street photographers along the roadsides and byways._\n\n_Coca-Cola showed that creativity can be just as fluid as a beverage with their Liquid And Linked concept of content excellence. The videos which articulate it were used internally, but are available on YouTube now._\n\nThe good news is that pervasive creativity recognizes a reality of the world that advertising agencies had rarely been interested in \u2013 that consumers can also be highly creative. Some years ago, we collected examples of this in Asia, and published them in a book called _Raw_ (2012).\n\nSo it's not now just a case that \"the consumer is not a moron, she is your wife\", but also of recognizing that she can be at least as creative as you are. Work which acknowledges this, and which allows that creativity to express itself, gains most from being pervasive. Now the consumer can even become a freelance copywriter.\n\nProbably the manufacturer who best articulated for themselves the notion of \"pervasive creativity\" was Coca-Cola. They labelled it \"liquid and linked\". The concept of link expressed the belief that a media neutral idea, if sufficiently inspirational, could pervade but at the same time cohere: pervasive, but also cohesive.\n\n**So What is an Idea?**\n\nThere is no doubt in my mind whatsoever that the digital age has hugely increased the importance of the idea in advertising, and yet the word is rarely given enough attention, let alone subjected to much analysis.\n\nIdeas that \"pass like ships in the night\" were always to be dreaded; and \"big\" ideas sought for. But what makes an idea \"big\"? Why is that more important than ever now? And, indeed, what is an idea?\n\nIt is curious how infrequently the last question is answered, precisely, granted that this is meant to be a business fuelled by ideas.\n\nOne of my treasured advertising books is a dog-haired volume called _Practical Advertising_ from 1909. It was published each year, in London, by our British antecedent, Mather & Crowther. It is their Mather in our name; and it was here that David Ogilvy came to work for his brother, Francis, in the 1930s.\n\nOne of the house ads in this book shows a client with a plump, well-fed face, in a winged collar, polka-dot tie and a pince-nez, scrutinizing a newspaper. The headline says, \"Looking Twice\". And the copy proceeds with a paragraph: \"How often do you have to look twice before finding your own announcement? How many casual readers, do you suppose, will be equally diligent?\"\n\nHow many indeed? And that, I suppose is a neat way of explaining why we need ideas. It's the ideas that make people notice things.\n\nBut have you ever been asked to define an idea? Please give yourself just 30 seconds \u2013 and pause without reading on.\n\nTry it now, right now. So what _is_ an idea? If you have pen and paper to hand, write your answer down.\n\nIt's not that easy, is it? And, strangely, not many advertising people have ever defined it very convincingly. Or, if they have, gone into print with it.\n\n_Did you look twice? A reminder from 1909 and our predecessor agency Mather & Crowther, London, that an advert is only valuable if it attracts your attention._\n\nSo we need to go to a philosopher to find the definition, not a blue-chip philosopher but a rambunctious, philandering, drunken, provocative and wannabe philosopher: Arthur Koestler. He was a brilliant writer: _Darkness at Noon_ (1940) is one of the best-ever written arguments against totalitarianism.\n\nIn his book on creativity, _The Act of Creation_ (1964), Koestler defines an idea as \"a bi-sociation of two previously unconnected thought matrices\". It may not be great philosophy, but it's a great definition \u2013 perhaps for the very reason that he was a creative writer himself.\n\nLet's paraphrase it as \"an unexpected combination of two previously unconnected things\".\n\nI've always found that it helps to keep in mind this definition of an idea when looking at work. Quite simply, it helps one pick the stronger ideas from the weaker.\n\n_Arthur Koestler \u2013 philosopher, provocateur and proponent of the best definition of creativity I've heard. His interpretation acts as my lens when evaluating work._\n\nThe more original, subtle, involving and intriguing the combination is, the more likely we are to notice it: the more it is a Big Idea.\n\nIdeas are precious things, and in the digital world it is clear that it is strong ideas alone that will defeat the noise, the fragmentation and the clutter.\n\nBut ideas have different forms.\n\nIn fact, there is something akin to a hierarchy of ideas:\n\n1.At the top, the _strategic_ idea, where an idea about how a business should be positioned, defines a platform for a company or a brand.\n\n2.Then there is the _campaign_ idea: what is it that ties together all the creative manifestations of the brand?\n\n3.And, finally, there are _executional_ ideas, smaller ideas within the campaign that provide its substance.\n\nStrategic ideas are longer term than campaign ideas. When I presented a new idea to Allianz, I was asked by the CEO how long I thought this idea would last. I said, \"at least ten years\". Campaigns are shorter: IBM wanted five years for its current platform.\n\nIn powerful strategic ideas there is usually a resolution of a tension of some kind; in campaigns the unexpected combinations are housed in a sequential framework; while executional ideas often draw their unexpectedness from how they are made or where they appear \u2013 like naughty children they can sometimes stray from the core idea, and need what can only be described as shepherding.\n\nIdeas are dependent on execution: and in the interplay between the two lies a second cause of \"bigness\".\n\nGive a weak idea to Alan Parker to film (if he would take it) and it will come out well. Give a strong idea to a hack director and it will come out poorly. I have always deployed \u2013 and trained my account people in \u2013 a simple matrix for assessing ideas (Ideas Vs. Execution opposite), on the basis that some process might be better than none. It is very simple, but can be a lethally effective career builder!\n\nIt's not just poor execution that can let down an idea, but it's also inconsistency in how the idea is executed. Quite rightly, one of the most awarded executions of the digital age is the Cannes Grand Prix Volvo film _The Epic Split_ (2013) made by Forsman & Bodenfors featuring Jean-Claude Van Damme (see here). Here is a strong idea: unexpected combination of _stability_ and _movement_. The stakes were high, as was the showmanship, and in this it pulled the same levers as the famous Krazy Glue TV commercial of the 1980s. Both defied our expectations. But what is generally forgotten is that this execution was just part of a larger campaign. The other executions \u2013 a hamster steering a truck, Volvo's CEO supported by the truck's hook \u2013 have been forgotten. They were not expressing the same idea at all, nor were these ideas. They were stunts, not surprises.\n\n\"Ideas are precious things, and in the digital world it is clear that it is strong ideas alone that will defeat the noise, the fragmentation and the clutter.\"\n\nThere are two new roles that advertising ideas play in the digital age, which I don't believe they have played before.\n\nFirst, they have become systems of management.\n\nAmidst the chaos, they offer a frame of reference for all a brand's activities; a principled compass against which activities can be assessed, accepted or rejected; and a visual and physical housing for how the brand interacts with its consumer. This is an editorial role.\n\nSecond, they are connectors to other ideas, in other disciplines, which are also undergoing redefinition. Brian Collins, my former colleague who ran our Brand Integration Group at Ogilvy & Mather and now heads his own successful design agency, refers to \"brand\" as the promise, and to \"design\" as the performance, evidence of a promise kept. More than developing an identity or creating an advertisement, brands are a tool to shape business \u2013 not something to build to, but something to build from.\n\nIDEA VS. EXECUTION\n\n_Here's a simple matrix to help evaluate big ideas. Plot the strength of an idea along the x-axis against the quality of execution on the y-axis. Try it yourself with this year's Super Bowl commercials. Advertisers pay upwards of $2 million for a 30-second slot, so you would expect them all to be in the top right. You might be surprised! A slick production and a celebrity cannot make up for the absence of a good idea. Only the ads in the top right quadrant deserve to lift the trophy in my view._\n\n_Outside of the haulage industry, most people pay little attention to trucks. Yet over 85 million viewers watched Jean-Claude Van Damme perform \"the most epic of splits\" to demonstrate the precision and stability of Volvo's Dynamic Steering. The campaign included other executions with lesser known people showcasing innovations, such as a lead Volvo technician confident enough to risk his life to explain the ground clearance accuracy of a new Volvo truck. The other ads didn't get as much traction \u2013 not because they lacked Van Damme's celebrity, but because the core idea was less compelling._\n\nFor Brian, brands live in four places \u2013 culture, environment, products, and the mind \u2013 and give us a blueprint to design systems that connect those spaces. The orientation is around the customer and their experience, so we put systems designers and engineers alongside the storytellers. And we use the brand to connect architecture, history, culture, product \u2013 so that when we get a brief from Hershey's to design a Times Square billboard, we instead give them a chocolate factory tourist destination. (Actually, we even generate revenue as our \"billboard\" became the highest grossing retail space in New York City \u2013 a real return on brand value).\n\nIn surveys, clients consistently express disappointment about the degree to which their presence is fragmented. Mark Addicks, CMO of General Mills, has complained that marketers \"are living in a world of chaos. They are desperate for order. They need a rulebook.\" Marketers, no longer able to keep track of customers across myriad touch points, find it increasingly difficult to get a single view of their customers.\n\nThe reason is that the digital world is intrinsically biased to the tactical, to ingenious small solutions, bubbling from the bottom up, but which do not really add up to anything.\n\nThe cure is simple. Find your own Big Idea. Then milk it mercilessly.\n\n**Telling Stories**\n\nIf 30 years ago you had asked someone what Beowulf (whom I have already mentioned once) had to do with advertising you would have got a funny look and not much of an answer. Since then it seems that everyone has become a storyteller.\n\nJust as ideas have become more important in the digital age, so, too, the power of storytelling has been re-discovered. It was always there, of course. But the internet puts the onus on the advertiser to attract users, and as advertising cannot rely solely on editorial content to bring them in, so it has become quite central.\n\nIt is stories that attract. And our stories have to be as good as, or better than, their stories.\n\n_Beowulf: an education in the art of storytelling._\n\nActually, the ingredients of a story have not changed much since Beowulf. There is a protagonist. The protagonist tries to achieve something. There is difficulty along the way. And then there is a resolution from which some lesson or meaning might be discerned. As my brilliant former client Javier Sanchez Lamelas of Coca-Cola says, \"storytelling allows you to say things you cannot otherwise say\".\n\nIn the digital age, our means to tell stories has expanded infinitely. Beowulf was recited; Oliver Twist was serialized in a weekly magazine. Our stories can be made available at any time, in any form, over any sequence. All they have to do is reward people.\n\nHow? Dopamine \u2013 the neurotransmitter we met in the last chapter that is released when we experience pleasure \u2013 makes us feel good. It is both a reward but it also anticipates reward, which is why we enjoy stories so much. We are just curious as to what is going to happen.\n\n_Humans are hardwired to tell and hear stories. For more than 100,000 years we have used story structure to process the outside world._\n\nMy colleague Khai has written memorably that, \"Man is a storytelling ape. He understands the world through story, and this is the way to move him.\"\n\nHe points to the critical importance of what Coleridge called the \"willing suspicion of disbelief\" when we enter a story world. We accept the rules of the story, and seeing those rules play out enhances our pleasure. We expect a good story to have a beginning, a middle and an end. Great storytelling in advertising always does. We recognize similar plots \u2013 in Beowulf's case, it is the Quest. We meet similar archetypes such as the Shadow, the enemy.\n\n_\"The Beauty Inside\" was a pioneering social film series from Intel and Toshiba, which not only challenged the notion of identity, but also the boundaries of audience participation. Beyond commenting or voting, contributors could take on the lead role of Alex by uploading footage of themselves \u2013 becoming part of a series with over 70 million viewers. The brands? The decades old \"Intel Inside\" idea got a new lease of life and Toshiba's Ultrabook was a prominent feature in Alex's unusual story._\n\n\"At its best, the art of storytelling is a foundation of advertising in the digital age.\"\n\nOne problem with storytelling in the digital age is that, at its extremes, it has become a fad. Every Tom, Dick or Harriet describes themself as a storyteller. As our friend Stefan Sagmeister pungently remarked at a festival dedicated to storytelling:\n\nI think all the storytellers are not storytellers. Recently, I read an interview with someone who designs roller coasters and he referred to himself as a \"storyteller\". \"No, F***head, you are not a storyteller, you're a roller coaster designer.\n\nNow some words of warning. It's easy for a pseudo-science to grow up around storytelling. Plots can become formulaic. I have heard market researchers trying to \"guide\" a story back to a formula, which there was no evidence that any consumer wanted. And archetype theory, while a helpful context for advertising, can also be dangerous. In recent years it has become popular to borrow from Joseph Campbell's model of archetypes and his seminal work, _The Hero with a Thousand Faces_ (1949). Campbell spent a lifetime studying the legends, myths and folktales of societies across the world. Regardless of the culture, he identified common stories repeated time and again, and his 12 archetypes reflect these familiar characters. Whether Henry Higgins in _Pygmalion_ , Gandalf in _Lord of the Rings_ , or Obi-Wan Kenobi from _Star Wars_ , the narrative of the Sage, who helps the hero along his journey, is all too familiar.\n\nBrands must be storytellers, and the most engaging brand stories are often the simplest \u2013 both new and familiar at the same time \u2013 but archetypes are rarely the inspiration. By framing a brand rigidly as one archetype versus another, the practitioner narrows the scope of storytelling opportunity a brand might explore. Equally, blending elements from different archetypes \u2013 another common trick \u2013 can undermine the value of the analysis altogether. At their most useful, archetypes are a convenient starting point from which to build a more meaningful discussion about a brand's story, behaviour and role in the world. At worst, they are a passing fad concocted by brand charlatans playing the Outlaw role themselves.\n\nHowever, they do appeal to clients who like rules, and who want the characters in their ads to look like archetypes. Too often I have heard clients say \"but \"X\" is not behaving like \"Y\": a sure recipe for bad advertising. There may be rules in storytelling, but the more they are visible the less the story will engage.\n\nAt its best, the art of storytelling is a foundation of advertising in the digital age. And the people who tell these stories need to borrow from the ways of working of the arch-storytellers in Hollywood. A story, which is episodic, is written in a very different way to the 30-second ad.\n\nTake for example \"The Beauty Inside\" (2012), a wonderful collaboration between Intel, Toshiba and the viewers who watched online over six, weekly episodes on Facebook. The story revolved around Alex, voiced by actor Topher Grace but played physically by a host of male and female actors, including some members of the audience. In a nod to Kafka, Alex wakes up every day as the same person but in a different body. He keeps a daily record of his unusual life via the video camera on his Toshiba Ultrabook, which he carries with him wherever he goes. This masterstroke meant _anyone_ in the audience could upload a piece to a camera (or webcam), which could then be interwoven into scenes within each episode. They are in the film.\n\nAnd that, after all, was exactly how Beowulf came into being: not the work of one copywriter and one visualizer, but of many minds, hearts and tongues over time.\n\nA Boy and his Atom _for IBM is evidence of just how innovative advertising can be. Armed with a small budget and IBM's Scanning Tunneling Electron Microscope, we used just a handful of atoms to make a unique short film. It was the first IBM video to earn a million views \u2013 literally overnight. Along with 30,000 YouTube likes, hundreds of articles, and a Cannes Lions award, it was shown at the TriBeCa film festival and features in the_ Guinness Book of World Records _as the world's smallest stop-motion film. A big success!_\n\nSometimes the biggest ideas are found in the smallest of places. As small as an atom.\n\nFrom awards alone, IBM should be recognized as one of the world's most innovative companies. Its employees have earned five Nobel Prizes, five National Medals of Technology, five Medals of Science, and a mere four Turing Awards. With a record-breaking count of 6,478 patents at the end of 2012, IBM was the highest recipient of US patents that year, maintaining 20 years of leadership in the patent space. But the public aren't interested in patents and awards.\n\nThey are, however, increasingly interested in science and technology. In 2012, there was a surge in interest in science, stimulated by news from CERN on discovering a fundamental particle supporting Higgs boson and NASA's _Seven Minutes of Terror_ video showcasing the Curiosity rover's landing. CNN and the _New York Times_ both commented on the public's burgeoning love affair with science, and on Facebook the popularity of IFLScience blog shares indicated the emergence of science as part of mainstream culture.\n\nIBM had a hidden gem that was ready-made for this audience: IBM can manipulate the atom. Few people know that IBM invented the Scanning Tunneling Electron Microscope, which enables the deliberate movement of single atoms, each a million times smaller than the width of human hair. Or that the company has lined up 12 atoms to create the world's smallest data storage module, a crucial future need, as big data gets even bigger and storage needs become paramount.\n\nRather than make another explainer video, we filmed _A Boy and his Atom_ , an animated short film comprised of 242 still images and created using IBMs electron microscope and 65 carbon monoxide two-atom molecules.\n\n**So, After All, What Does Make Work Great?**\n\nA Big Idea, evidently.\n\nAn engaging story.\n\nAn appeal to the emotions as well as to reason.\n\nAnd it seems to me there are other things.\n\nIn a word which I have usually resisted using, it must be \"edgy\". It's a word which best describes work after it's made, although it often appears in creative briefs as an aspiration. What then does it mean other than being the most general of aspirations?\n\n_What do you get if you cross a polling firm with two art critics? Art that Americans want but art lovers hate. I am grateful to have been recommended Komar and Melamid's book \u2013 proof that a scientific approach to art results in banality rather than a masterpiece._\n\nEdgy to me means the opposite of average. It means being prepared to take a risk. I remember Steve Harrison, at Ogilvy & Mather Direct in London in the early 1990s, writing a brilliant brochure on it. My client at SC Johnson, Salman Amin, once showed me a book he keeps in his office to remind him of the danger of \"average\". It is _Painting By Numbers, Komar and Melamid's Scientific Guide to Art_.\n\n_\"How to Carve an Elephant\" by Neil French and Tham Khai Meng \u2013 a quick guide to creativity that is even more relevant to the digital age than it was prior. Always strive for simplicity above all else._\n\n\"America's Most Wanted\" painting is what happens when one follows the average way. It carries the illusion of safety, but it is in fact the most dangerous way of all. In the digital age, it simply consigns you to oblivion.\n\nThe other great requirement is simplicity. I once asked Neil French, very much a pre-digital creative, to design an in-house poster to encourage better creative ideas. It was not the easiest of briefs, but what he did is shown above: \"How to Carve an Elephant\".\n\nThe digital age has created complexity in a myriad of ways. I have noticed that not a few self-pronounced \"digital\" creatives are incurably complicated.\n\nThe best way to be noticed and listened to is to be, at the root of whatever you are doing, beautifully simple.\n9DATA: THE CURRENCY OF THE DIGITAL AGE\n\nThe first recorded reference to big data was buried in a 1997 paper, but the idea had been making the rounds at Silicon Graphics (SGI) in the mid 1990s. While big data isn't so recent as we'd like to believe, the practical use of big data has emerged only recently as a proliferation of data sources has become available.\n\nWe give off a constant stream of what my former colleague Dimitri Maex calls \"digital exhaust\". Consider this: I like a particular type of red socks. I buy them, and that decision generates data about my location, my preferences, my habits and my finances. Now data of that kind has been available for quite some time, but assembling it was difficult. Big data isn't, therefore, a new kind of data. Rather, it is a proliferation of data sources, both structured and unstructured, and the means to use them:\n\n\u2022 **Structured data** \u2013 This is information stored with a high degree of organization. A database containing the inventory of a warehouse is structured data. So are the records of customers who bought a product at a certain time and by what payment means.\n\n\u2022 **Unstructured data** \u2013 This refers to information that isn't organized in a readily machine-readable format. A blog post is unstructured data. So is your email inbox or your Facebook feed.\n\nSince nearly every action today creates data, that data must be stored somewhere in order to be useful.\n\nEnter \"the Cloud\". The Cloud is nothing more than rented storage and computing power \u2013 much as early computers used timesharing to apportion processor cycles to individual users. But while that was the rationing of a scarce resource, the Cloud fosters the efficient and cost-effective distribution of an abundant one. Cloud servers, run by companies big and small all over the world, democratize the world of big data by reducing the cost of data storage and analysis. Companies can build and host their applications in a public cloud, and add both data storage capacity and computing power with a credit card. Businesses like Netflix and Uber were built in the cloud, scaling from start-up to dominance as their data assets increased exponentially.\n\nAbundant computing power means that algorithms can do more. A decade ago, a data scientist may have needed two days to analyze even a single stream of consumer data. He had to gather the data, clean the dataset, render it into structured data, and then run the algorithm. Confronted with such a burden, our data scientist was economical in his programing and in his choice of data. That can be done in hours \u2013 often less \u2013 today, but the principles remain: people still struggle with the fundamental questions of what data to analyze and why.\n\nThat's what's missing from those who have a vested interest in selling cloud-based products. They've romanced the Cloud to the point that it has started to seem to be an end in itself, rather than a means to something else. Cloud providers have engaged in vicious advertising wars, very much on the basis that \"my cloud is better than your cloud\".\n\nBut, of course, cloud technology has become a fungible commodity. Infrastructure as a Service (IaaS) and Software as a Service (SaaS) have both become commonplace. One has to look beyond to how and why they are used. For instance, the incorporation of SoftLayer into IBM in 2013 enabled it to compete with companies such as Amazon who were faster to the Cloud market. But how to differentiate the offering? That's where analytics came in. The distinguishing portions of IBM's offering are predictably rooted in the company's strengths in data analytics and business consulting.\n\n_Working on the IBM Cloud makes it easier to glean more insight and boosts the industry-specific business impact of technology for IBM Cloud customers. IBM has a suite of analytics tools \u2013 many best in class \u2013 to get more out of the big data that a company might store in the cloud; a business installing the cloud will more likely find a manufacturing expert to talk to at IBM than at any competitor._\n\nData is often presented as the currency of the digital age, or more aptly, as \"the new oil\" \u2013 a memorable phrase coined by data scientist Clive Humby back in 2006. So it is; but of course the role of data in the advertising business is hardly new. David Ogilvy loved direct response advertising for its rooting in data.\n\n\"For forty years,\" David wrote, \"I have been a voice crying in the wilderness, trying to get my fellow advertising professionals to take direct response seriously. Direct response was my first love. And later, my secret weapon.\"\n\n\"Just because you can collect data doesn't mean you should.\"\n\nDavid's love notwithstanding, data never did give us the ability in itself to find the nirvana of perfect solutions. I remember well the advice I was given when joining Ogilvy & Mather Direct in London as its Managing Director from the data-illiterate world of traditional advertising. I was desperately nervous that a client would one day ask me what response rate I would expect from a direct mailing: \"Just look them in the eye, and say, firmly, 4 per cent\" was the advice I was given. (It worked, by the way.)\n\nIn the pre-internet world, the cost of data dramatically fell as a result of innovations in computing. It nonetheless remained an asset constrained by the natural limitations of its available sources, which were mainly confined to the result of one's own campaigns, even though they gave us a degree of precision we had never had before.\n\nWhat the Digital Revolution has created is something radically different. Data has also become pervasive. This is what big data enthusiasts swoon over. With visibility into the entirety of the consumer journey, big data enthusiasts claim to be changing dramatically the lives of consumers. All that first party (or personal) data is going to lead us to a more customized, customer-centric world.\n\nActually, I don't much care for the term big data. As Dimitri says, it puts all the emphasis on the technology and not on the consumer experience. Big data has made things better for advertisers, but few consumers would say their lives have changed much as a result of it. We're focused on the data gathering and analysis instead of spending wisely on improving the consumer's life as a result of all that information.\n\nAgainst the tide of (excusable if disingenuous) enthusiasm, one has to table a series of big data cautions.\n\n**The Big Data Cautions**\n\n1.There is what I call the \"collection fallacy\": the more data you collect, the more valuable it will be. Just keep on collecting! I see no evidence at all that this is the case. The \"bigness\" of big data fuels this belief. Even though brute-force algorithms can churn through vast data sets in record time, that doesn't mean the analysis they produce will make the slightest bit of difference. Just because you can collect data doesn't mean you should. Instead, as Dimitri says, \"measure what you need to measure, not what you can measure. Only focus on the data that aligns with your goals.\" That's a decision you (and not your algorithm) will have to make.\n\n2.There is a \"utility deficit\". The data you have is simply not used. Near-limitless cloud storage capacity means that if any part of a company produces data, there is an excellent chance that data is stored somewhere. What's holding companies back? In my humble view, it is a lack of understanding of the fundamentals of data science.\n\n3.There is the \"siloization\" of big data. Silos are everywhere in business today. In agencies, the advertising team is separate from the direct marketers. The social media mavens don't talk to the PR team. Clients are the same way. One client, who shall remain nameless for obvious reasons, has one group of people decide on and assign creative work while another group is responsible for approving it! When people are siloed, they can't work together. When data is siloed, it can't work for you at all.\n\n4.There is a \"tolerance threshold\". Quite simply, big data seems a very natural plaything for Big Brother. Advertisers have pledged to regulate themselves, but individuals are worried about their privacy. Big data has made credit fraud detection much more effective, and it has made significant gains in fighting global terrorism. Those advances make our world safer, but thanks to data breaches, revelations of government surveillance, and awareness of the value of personal data, we're all feeling rather more insecure as well. The more the boosters of big data talk about how unfettered data flow will make our lives better, the more we recoil from the invasiveness we must sign up to for that utopian vision to come true.\n\nVALUE SPECTRUM\n\n_A value spectrum analysis can show which kinds of customers are, as Dimitri Maex puts it, potential pots of gold to be found, jackpots to be gambled at, acorns to be nourished, or lemons to be ignored._\n\nI think the time has come to think not big, but small. What is it within the enabling power of big data that is small enough to be beautiful?\n\nLet's call it the Really Useful Data Resolution. To follow through on it, you need to combine a wide-angle lens (to see the whole business landscape) with a narrow depth of field (to see precisely what the data can practically do). But the secret is to start problem first, not data first.\n\nDATA SETS COMPRISING A SINGLE ENTERPRISE POINT OF VIEW\n\n\u2022 Search intent modelling: specialist firms help marketers understand keyword clusters as well as provide insight into consumer thought processes.\n\n\u2022 Social listening: while less automated (for now) than search intent modelling, social listening can produce sentiment and semantic analysis that guides marketers. We did just that for a Las Vegas hotel. People raved about the view of a competing property's magnificent fountains. That data was fed to the creative, site development, PR, and search marketing teams, all of whom acted upon it to fill more rooms.\n\n\u2022 Primary research: asking consumers what they think may sound hoary and old fashioned, but the Digital Revolution has made the process much less expensive and vastly more responsive.\n\n\u2022 Results repository: it isn't a technological challenge to put all your data into one place, but it requires commitment. Performance and cost data is often stored in financial systems while marketing information is held elsewhere. By combining it, you can learn much more about what does and doesn't work \u2013 and why.\n\n_A word cloud as a pipe, which David Ogilvy would appreciate! The most useful data has strong visual appeal, even to show simple stats, such as the most frequently used terms in this book._\n\n_Data that works together can make instant improvements to business. Caesar's Palace sits across the street from a Las Vegas icon, the beautiful Bellagio fountains. Online reviews consistently raved about the great fountain view Caeser's Palace afforded. Rather than grumble about how much guests liked a competitor's signature feature, Caeser's Palace fed that data to the creative, site development, PR, and search marketing teams, all of whom acted upon it to fill more rooms._\n\n**Really Useful Data**\n\nData is not necessarily useful. Here are, in my experience, the seven pre-requisites of Really Useful Data.\n\n\"Customers expect brands to know them when they want to be known and to be anonymous when they don't.\"\n\n**1. Take the single enterprise point of view**\n\nCompanies have invested heavily in building \"single customer view\" data warehouses, storing everything the company knows about that customer. These valuable repositories of data help brands analyze the behaviour of an individual customer and target messaging to that person across multiple channels. But these days, we generate tons of relevant data that exist outside of corporate channels. My preferred supplier of red socks would be very interested indeed to learn that I've just liked a picture of green socks on my Facebook page. The solution is the single enterprise point of view which integrates company-owned data sets with consumer insight mined from platforms like Facebook, Tencent, Google, and their ilk (see here).\n\n_MetLife noticed that when people searched for life insurance, a particular CNN Money article showed up at the top of their search results time and time again. So we secured sponsorship of that article page and new applications flooded in. Even with complex technologies at our fingertips, often the simplest solutions are still the most effective._\n\n**2. Defeat platform impediments**\n\nIt is the tendency of platforms to present themselves as omniscient. They never are. The game used to be that you owned all the data yourself, but, because so many interactions are on platforms you do not control, marketers lose visibility into the lives of their customers. And that upsets marketers. Yes, powerful platforms can do amazing things. You can learn incredible things about consumers from Facebook; it's probably the world's largest database. Putting that insight into practice is no small task. Customers expect brands to know them when they want to be known and to be anonymous when they don't. Gathering and analyzing all the internal and external data requires \u2013 at this moment at least \u2013 an intensely complicated marketing technology stack, and no one piece of software can address it all.\n\nThe problem with platforms and marketing technology is that they, and the marketers who use them, see data analysis as a technology, not creative strategy. Marketers use a small fraction of the potential of their technology because changing the organization to make use of the insights they generate is hard, even if buying some new shiny toy is easy.\n\n**3. Distinguish between measurement and effectiveness**\n\nMy late colleague Tim Broadbent used to remind us of the old story of the surgeon who said the operation had been a success, but, unfortunately, the patient had died. That surgeon was a measurement culture man. He may have carried out the procedure in the approved manner, but measuring the connections of the process do not help if the end result goes wrong.\n\nData is dangerously prone to creating measurement cultures. But what we really need is effectiveness culture.\n\nA measurement culture is obsessed by process; while an effectiveness culture is obsessed by results. A measurement culture focuses on ticking the right boxes; while an effectiveness culture focuses on doing the right things. Measurement cultures look to the past \u2013 \"How well did we work?\" Effectiveness cultures look to the future \u2013 \"How can we do better next time?\"\n\n\"A measurement culture is obsessed by process; while an effectiveness culture is obsessed by results.\"\n\nTim gave us a real example. The client had both a problem and an opportunity. The problem was that customers who bought one of their products were unlikely to buy another one at a later date. In marketing this is called a low repurchase rate. The opportunity, though, was huge. We calculated that if the repurchase rate went up to the average of the major competitors, our client's revenues would increase by around 30 per cent.\n\nConsumer research showed that customers didn't repurchase because they found the product unreliable. They criticized the product's build quality. So we went to the manufacturing plant to interrogate the quality control. \"How do you test the products, and what are your standards?\"\n\nWe were told that product quality was improving each year. And there were demanding targets. This year's products had to be built to higher standards than last year's products, and last year's products were better than the year before, and so on. But, we asked, do you test the products against competitive products from rival companies? \"Oh no,\" they said, \"we test against our own internal benchmarks. That's the only scientifically valid comparison.\"\n\nNow, the trouble with this approach is that it doesn't work in the market. When you run a race, you can beat your own personal best time but still lose if another person runs faster. Yes, the company's management received measurements from the factory every year telling them product quality had improved. But other manufacturers were building to higher standards. And it's hardly surprising that consumers, faced with a choice, chose to buy products that didn't break or let them down so often. This client had a measurement culture, not an effectiveness culture. And it really hurt the business. Revenues were a third lower than they could have been.\n\n_Here is a useful illustration of the difference between measurement and effectiveness. What if you had a tennis racquet that could measure the power of your swing, the speed of your reactions, and the placement of your shots? We worked with Babolat to create a racquet that analyzed your swing data, and then helped you improve your swing._\n\nThere is no doubt that the digital age and big data have driven a worrying reliance on short-term measures. As Binet and Field have pointed out, the language is \"all about 'timely offers' delivered with zero wastage only to imminent purchases.\" As they ask: \"why bother with that slow-burn activity when sales can be switched on instantly with the latest Big Data tools?\" All the IPA's data suggests that this would be injurious to profitable brand building.\n\n**4. Re-discover econometric modelling**\n\nEconometric modelling was invented in the first quarter of the twentieth century by the Nobel Prize-winning economist Ragnar Frisch. As Dimitri writes in his excellent book _Sexy Little Numbers_ (2012): \"Econometrics is all about developing and applying _quantitative_ or _statistical_ methods to prove economic principles.\" Those techniques can be used in marketing and sales, helping forecast consumption and demand for products and brands.\n\nAt its best, modelling really can explain what is doing well in the market place and what is not. It makes the data work for you.\n\nEconometric modelling can do wonders. Consider this simple example: an econometric analysis of the impact of marketing on sales would help you forecast the impact a certain level of marketing spend would have on your sales. For some brands, there may be very little correlation because the most important variable is, perhaps, price, differentiation or distribution. But for another, often in the CPG (consumer packaged goods) category, the impact of marketing expenditure on sales is strong. How can you know how much you ought to spend? Econometric modelling helps find the exact level at which a brand gets the greatest possible return on its marketing investment.\n\nIt is strange that econometrics is not used more at the very point in history when the raw data it can feed upon has never been richer. We have seen that the best benchmark for studying effectiveness in the marketplace is the UK's IPA Awards. However, only in 15 per cent of case studies submitted has econometric modelling been used. What's the block?\n\nThere are three of them, in fact:\n\n1.The necessarily advanced statistical skills are rare.\n\n2.It is not always possible to capture the necessary data consistently.\n\n3.Marketers don't believe in it since it is often a black box.\n\n**5. Underdose with metrics**\n\nBig data needs to justify itself with a multiplicity of metrics. It overdoses on them. I believe that we gain most by rationing the numbers of metrics we define as critical to any particular mission. The reason why this does not often happen so much is very simple: it requires the painful thinking to be done upfront. What are we really trying to achieve and why? Agreed in this way, and then adhered to, you can build \"one version of the truth\", rather than getting mired in multiple, unresolved debates.\n\nFor some clients we've constructed a single Report Card with as few as five metrics which sensitively capture the health of the brand and its contribution to the business. For others, we build more comprehensive dashboards.\n\n_A good dashboard is used and shared. The best ones become forums for discussion about measurement. The design of the dashboard itself helps keep the marketing on target: deciding what data to show to decision-makers forces you to have a conversation about what data is important. And that leads you to align your metrics to your objectives._\n\n**6. Insights, insights, insights**\n\nReally Useful Data gives us insight. We thirst for more and more of them. Nothing pleases me more than seeing creative teams slavering for insights from data. There are three different types of insights from data:\n\n1.Observation: data can help show how something is doing.\n\n2.Improvement: data can help uncover the reasons why something is working or not working.\n\n3.Inspiration: data can spark an idea.\n\nCreative inspiration seems like magic, and data can inspire it in many ways. For me, it all boils down to this:\n\n**Prioritization:** Your data can show you whom to speak to and why.\n\n**Personalization:** Your data can point you towards certain categories of people, showing you why they behave in certain ways, and how you can reach them.\n\n**Precision:** Your data can help you generate the right message, delivered at the right time, to the right person, through the right medium.\n\n_Sometimes the data itself can be the idea \u2013 at least it was for a British Airways billboard campaign. We mounted an antenna that gathered flight data from passing passenger jets, instantly cross-referenced it with the flight information from Heathrow, and checked that the location and weather conditions made the airplane visible to people in sight of the billboard. That process, which took far less than a single second, triggered our billboard to interrupt its regular content in order to show a child pointing up at the passing plane._\n\n\"Made useful, it [big data] has the ability to take marketing and communications into whole new areas of creativity and precision.\"\n\nIBM WATSON\n\nThere is no better illustration of the potential data has to transform things \u2013 things well beyond marketing \u2013 than IBM Watson. Watson is IBM's cognitive computer. Cognitive computing is the remarkable result of machine learning algorithms churning through huge volumes of structured and unstructured data. We're only at the beginning of the cognitive age, but already we're seeing where it may take us. IBM's Watson is the world's most advanced cognitive system, built to analyze and extract knowledge from vast amounts of information with incredible speed and precision. Watson debuted to the public on the quiz show Jeopardy (above) in 2011, where it competed against \u2013 and handily beat \u2013 two of the best human players ever. But that was just a demonstration, a proof of concept. Soon after this victory, Watson was put to work, ploughing through reams of medical literature at a pace no human can match. It takes 160 hours a week to stay up to date on all the reading in a single medical field. This is no problem for Watson, and as a result, he's training with oncologists at Memorial Sloan-Kettering Cancer Center to help physicians anywhere make more informed decisions about patient care.\n\nWatson also helps companies serve their customers better. Today's consumers expect them to be highly responsive to their questions, comments, and complaints. Watson enables companies to tackle even tough questions quickly and accurately. Watson is helping financial services firms make money management recommendations. Watson even helped us write the T.V. commercials! Watson analyzed, for example, all of Bob Dylan's lyrics and then worked with our copywriters and art directors to create scripts for the commercial it would star in with Dylan. Cognitive computers working with human talent \u2013 that's the real promise of big data. Big data isn't a magical panacea, but when paired with human inspiration and a commitment to creativity, it can help us go to new places.\n\n**7. Optimal optimization**\n\nReally Useful Data becomes optimally useful when it is designed to optimize. How do you do that? With a closed-loop system.\n\nWe keep close tabs (via a dashboard) of what is happening out in the marketplace before our marketing tactics even start to run. We make sure they're in line with our objectives, set up a testing protocol, and initiate our tracking. Once the communications are live, we assess them in four time frames: real time, daily, weekly, and quarterly. We're looking for data such as the path consumers are following, the lifetime value of a customer, and brand preference indicators. That data tells us what is and isn't connecting, and we change things on the fly to do more of the former and (hopefully) less of the latter. To keep ourselves honest, we report our results on a weekly, monthly and quarterly cycle, using that data to inform our next effort.\n\nBig data _can_ be tamed. Made useful, it has the ability to take marketing and communications into whole new areas of creativity and precision. But it will never provide the nirvana its most extreme advocates suggest. It aids and supports the human thinking process. It short circuits hard work which otherwise could not be done. Through sheer volume it sees patterns we could not otherwise see.\n\nAt the end of the day, though, data is political, however much purists may not like to admit it. It can be ignored. It can be manipulated. It can be used to score points. Its impact can be disguised.\n\nThe organizations of the digital age are no different than those which preceded them. As long as functional specializations exist, it is likely that data will reflect these political agendas as much as it reflects any great objective truth enshrined in a _sanctum sanctorum_ of the temple of big data.\n10\"ONLY CONNECT\"\n\n\"Respondents admitted that they didn't completely understand today's marketing environment.\"\n\nSo was the understated conclusion of the research company Forrester in 2015 when they asked clients what the shift to digital meant for them.\n\nNo wonder.\n\nThe media landscapes have shifted tectonically.\n\nIn _Ogilvy on Advertising_ you can feel the old world of media. In fact, it was given scant attention. \"I have never worked in the media department of an agency,\" David wrote, \"but my observation of those who have been successful in this field leads me to think that they need an analytical mind, the ability to communicate numerical data in non-numerical formats, stability under pressure, and a taste for negotiation with the owners of media.\"\n\nAbout 20 years ago, a structural change started which was controversial. In fact, it began on 1 November 1997, when my counterpart Alan Fairnington of JWT Asia and I agreed to pool our Asian media departments and create the first true media independent, MindShare: a decision both advantageous and dear to the holding company that owned us, WPP, and not so endearing to our immediate bosses. But it was clear that, if the agencies were to compete effectively, they needed to have both the necessary economies of scale in software development and the aggregated muscle in buying power. It was the right thing to do.\n\nOf course, something was lost (as our immediate bosses feared), and that was the fruitful interface between creative people and media people. Who owned media planning \u2013 the creative agencies or the media agencies?\n\nIronically, the Digital Revolution has taken away much of the angst. We have observed the \"so-and-so is dead\" attitude that has accompanied it: the one claim which would be most accurate is \"media is dead\". That is to say the world of media in which many of us grew up. As Account Executives, there was one simple formula we learned: \"80 per cent coverage at 6 OTS\". That's what you'd say when a client asked you what weight of television campaign you would recommend. Reach (coverage) and frequency (Opportunities To See) were the sole media principles for managing a captive audience who had limited choices.\n\n_Minds of our own! Alan Fairnington of JWT and I went out on a limb to help create the first true media independent, MindShare. This is the press release from 1997, which made public our plan to unlock the combined media buying power of our agencies._\n\nWhat then followed was what the business technologist Steve Sammartino labelled the Great Fragmentation: the explosion of an orderly media universe into thousands upon thousands of programming options, and the parallel explosion of a homogeneous audience into one fragmented both by time and space \u2013 each fragment being, potentially, \"always on\".\n\nPAID, OWNED AND EARNED\n\n_Connections planning worships at the shrine of POE \u2013 paid, owned, earned. While media was previously bought and \"paid\" for via advertising, brands now \"earn\" it from PR, social mentions and partnerships, and \"own\" it via their digital and physical channels. The digital age didn't invent all these channels, but it did shepherd in a deeper level of sophistication in delivering campaigns._\n\n**Paid, Owned, Earned**\n\nMedia planning ceased to be central: \"connections planning\" emerged, in recognition that many points of connection with the digitized consumer could not just be paid for: rather, you could build and \"own\" your digital estate, and you could \"earn\" coverage. In Chapter 7 we briefly met the three unlikely letters which became the new paradigm: POE \u2013 Paid, Owned, Earned.\n\nPOE provided \u2013 and provides \u2013 a handy way of looking at the new media world, but it begs some important questions. How do you know where to connect? Or what to spend in making each connection? Or, indeed, how to connect?\n\nThere are many who place their faith blindly in full automation. But as my colleague Ben Richards, Ogilvy & Mather's Chief Strategy Officer, writes:\n\nPity the media operative today. Tasked into finding prospects for, say, Powerade, she checks the weather variable. She finds parts of the country where the temperature is 72\u00b0 or above. She sees that morning registers higher purchase rates than afternoon. She instructs the algorithm to look for people who are searching for sports activities on mobile devices \u2013 people who are clearly out and about. She instructs the algorithm to bid at 20 percent below the going rate. Her trade successfully completes. She feels she has won \u2013 for today. However, she worries about precisely where the ads will really appear, and ultimately she worries whether she will be replaced by that same algorithm tomorrow.\n\nJust as electronic trading has replaced the slamming phones, screaming brokers and ticker tape of the old trading floor, today's media buying appears to be hurtling towards full automation. Yet this is happening with little strategy and even less accountability. \"Connections\" is needed more than ever.\n\nFrom the start, intelligent CMOs started to realize that they needed to push their marketing organizations into digital media. Their spend patterns lagged their audiences' consumption patterns. So Keith Weed at Unilever, or Clive Sirkin at Kimberly-Clark, started to set quotas for their markets. At one point in Unilever, the \"target\" for digital spend was set at 20 per cent of analogue spend. At the time, it seemed ambitiously high, and somewhat arbitrary. Now the actual level is 24 per cent. Data is pushing this trend.\n\n\"It's amazing in retrospect that we simply didn't know much at all about the media behaviour of the consumers we ritually dedicated ourselves to.\"\n\nThe old media ecosystem delivered remarkably little data: very high-level segmentation of viewership confined to age and gender, often not ratified until 90 days later; and similarly generic circulation data for print readership audited twice a year. It's amazing in retrospect that we simply didn't know much at all about the media behaviour of the consumers we ritually dedicated ourselves to.\n\nNow, we also have that billowing \"digital exhaust\" of their media consumption, their use of devices and their non-media behaviours through a platform of first, second and third party data sources. Rich and in real time, their media profiles help us connect much, much better.\n\n**Deep Integration**\n\nBut something else is needed. How can we integrate everything around the consumer?\n\nIntegration has consistently emerged as the most intractable worry of the post-digital CMO in all surveys. And early practice did nothing more than to popularize what we call the \"matching luggage\" view of integration: if all your communication pieces in different media looked more or less the same, you're OK.\n\nThen, there was integration in a funnel, a child of the early 2000s: organizing around tasks such as creating awareness or gaining consideration, with channels generically deployed against each.\n\nEVOLUTION OF INTEGRATION\n\n_Behold the evolution of marketing mankind! Integration has its origins in Graphic Integration, where everything has a visual coherence but little more. After several stages of development, we are reaching to the holy grail of Dynamic Integration, in which every element of the business is working in harmony and managed in market and in real-time._\n\nToday, best practice sees integration as something organic, to be architected around the customer's experience, and ultimately, what we are moving towards \u2013 dynamic, with communications that move around people's real lives as they live them.\n\nDavid Ogilvy never believed in the disintegration that grew up in the post-war period as a symptom of disciplinary specialization \u2013 the \"disciplines\" of Public Relations, Direct Marketing, Sales, Promotion and so on. He also firmly rebutted the advertising-centric orthodoxies of his generation; Ogilvy & Mather brought, with his encouragement, these disciplines into our family very early on through a programme of acquisition. Still, too often, advertising strategies were developed in the advertising agency and handed over to the other disciplines for implementation. However seamlessly that was done, the integration was superficial, not deep.\n\nWhen I became CEO of Ogilvy in 2008, I felt that we had both a huge advantage and huge liability. We possessed all the disciplines, which other agencies didn't, and yet we did not feel deeply integrated. I hired Ben Richards from the connections agency Naked to find a solution. Naked was the parent of Communications Planning, but their definition of communications really stopped at advertising. Concepts of influence or loyalty were remote and alien to them. What Ben did was to create an operating system, Fusion\u2122, which was common across all the disciplines, which started with what the business problem was (rather than with the communication problem), and which, above all, was focused on the customer journey.\n\nUntil then, customer journeys had been a somewhat remote and academic construct, or something lodged exclusively in the Direct Marketing discipline. But now they could become a unifying concept, through which the real barriers and the potent drivers to engaging the consumer could be mapped.\n\nCUSTOMER JOURNEY OF AN ADVERTISING UNDERGRADUATE\n\n_Should you wish to devote your career \u2013 or at least some of it \u2013 to advertising, you might follow a path similar to this. Customer journeys help us identify the pleasures and pitfalls of a given experience, whether buying a new car, researching an interest, or making the bold step to study for a degree and get hired. By truly understanding people's engagement at each stage, we can identify where to help rather than hinder._\n\n\"Deep integration is digitally enabled in one very particular way: now, we can understand consumers' intentions.\"\n\nTwenty years ago, Ogilvy & Mather coined the phrase \"360\u00b0 Brand Stewardship\", which allowed us to do holistic integration. There was no doubt that the combination of in-depth specialisms provided much better programs. But we learnt there was something more. To get deep integration you must focus only on the _10\u00b0 which really matters_ , and do so in a way that is not just media neutral but also discipline neutral.\n\nDeep integration is digitally enabled in one very particular way: now, we can understand consumers' intentions. Search \u2013 in this case under its guise of Search Engine Marketing (SEM) as opposed to Search Engine Optimization (SEO) (see here) takes centre stage. I write \"search\" as we all do, as a one-word shorthand. Of course, what is meant is that we can understand _what consumers are searching for on the internet_.\n\nIt was the serial entrepreneur and journalist John Battelle who first characterized Google as the database of intent back in 2003: \"The Database of Intentions is simply this: The aggregate results of every search ever entered, every result list ever tendered, and every path taken as a result,\" he wrote. \"This information represents, in aggregate form, a place holder for the intentions of humankind \u2013 a massive database of desires, needs, wants, and likes that can be discovered, subpoenaed, archived, tracked, and exploited to all sorts of ends.\" Battelle has gone on to add social media and check-ins to this grouping.\n\nThe root of all is the not so humble keyword. We need to think of keywords as the common denominators of the digital age.\n\n_Here's an insight \u2013 weekly Google searches among new mothers almost double in frequency. So, we worked with baby food brand Gerber to help them learn about motherhood on their own terms, or, perhaps more accurately, their own search terms. Search intent data reveals much about a person's thoughts and intentions, and when you carefully sift it, you find a window into what they really want. For young mothers nursing a new baby, it highlights their curiosities and concerns \u2013 phrases like \"when to start baby cereal\" and \"how to establish healthy eating habits\". By understanding their knowledge gaps, we helped Gerber deliver precisely calibrated, highly-relevant videos to answer those needs in a supportive and engaging way. The idea was embraced by young mothers, and helped to get Gerber growing again._\n\n**How Keywords Work**\n\nGoogle is far from being the only database of intent. Baidu, Yandex and Yahoo! dominate in China, Russia and Japan respectively. Virtually every other portal \u2013 from Twitter to YouTube to Pinterest \u2013 is, de facto, a search engine.\n\nBy constructing a keyword universe we can learn about intent, not just from within Google.com but from a much broader territory. In fact, we should think of this as the largest focus group ever: a humongous source of insight into customers \u2013 who they are, what they like or don't like, and how they journey. But, in itself, its data is no more useful than any other. The magic only happens when we relate intent to content.\n\nWe made exactly this connection in order to revive Gerber, a well-known but out-dated baby food brand in the US that had suffered five years of declining sales. The data came from looking into search behaviours among Millennials, and our observations that weekly searches nearly double in frequency once a young woman becomes a mother. We then analyzed their queries in greater depth to better understand their questions, find common themes, and produce an entire video library to help provide highly relevant answers. We published video content that mapped directly their searches \u2013 queries about childhood nutrition such as \"when to start baby cereal\" and \"how to establish healthy eating habits\" \u2013 labelled with the exact same search terms they use to ensure easy discoverability on Google. And we used playlist and tagging features on YouTube to help people find the content even without paid advertising. By first mining data to truly understand new mothers and then building the go-to source for tailored advice on motherhood, the Gerber brand is growing again.\n\nSomething that I find irritating at worst and confusing at best is the way in which \"search\" is treated as if it is a distinct discipline, another industry vertical. This is a great misunderstanding. Search is fundamentally horizontal. It informs and enables all the communication disciplines \u2013 Advertising, Public Relations and Direct Marketing \u2013 and, because keywords, unlike P&L owners, don't recognize the distinctions between them, it acts as a major integrator for communications in the digital age.\n\nIt's also an area which suffers almost more than any other part of the digital landscape from acronym-itis, so please see the simple glossary opposite that has managed to keep me reasonably clear-headed about it.\n\n**How to Judge a Website: The Medium You Own**\n\nThe introduction that leads to the cocktail party conversation I most dread: \"This is Miles, he runs an advertising agency.\" I can see the follow-up coming from a mile away. \"I've got a website, but I'm not sure it's great. What do you think?\"\n\nMy question in response is: \"Well what do you want one _for_?\"\n\nThere are two types of answers. One is for the sheer numbers \u2013 the _audience_ \u2013 it brings me. The other is for how it explains and presents me.\n\nA GLOSSARY OF COMMON SEARCH TERMS\n\n**Annual Search Volume** Searches over a one-year period, based on the previous 12-month average. This is calculated by multiplying the Average Monthly Searches x 12.\n\n**Estimated Clicks** The estimated annual clicks for a particular group of domains, domain or URL. Estimated Clicks = Annual Searches x Click-Through Rate for the Page Rank. This is then summed up for aggregate views across Categories, Topics and Subtopics.\n\n**Opportunity Clicks** The estimated increase in annual clicks (by Category, Topic, Subtopic or Keyword). This involves a complex calculation that evaluates current client and competitor Page Rank as well as domains too well-entrenched to be dethroned to establish a target goal rank for each keyword. Opportunity Clicks = future clicks minus current Estimated Clicks.\n\n**Page Rank** Refers to the organic ranking position on Google. Page 1 = Position 1\u201310; Page 2 = Position 11\u201320. This data is pulled from a third-party tool that captures a snapshot of ranking positions at a given date and time.\n\n**Paid Search** An auction-based media channel that places advertisements at the top and bottom of the search results page. These ads vary from search query to search query and are chosen based on a search-engine's algorithm and the results of an auction between advertisers. A search-engine's algorithm examines many data points (over 100) to find the most relevant set of ads to show in response to someone typing a query, and uses the results of the auction to determine where those ads are located on the results page. When a user clicks an ad and goes to an advertiser's website, that advertiser is assessed a cost based on the amount of the various bids placed in the auction. Many other factors can affect the results of the auction, creating a rapidly evolving, highly complex market that acquires costs and solicits website traffic in real time.\n\n**Search Trend** Change in search volume based on the slope of the trend line from the Monthly Search Volume over a two-year period. Range includes Steep Drop, Drop, Flat, Growth and Steep Growth.\n\n**Search Seasonality** This metric utilizes Google Trends data over a 10-year period to determine the typical fluctuation from the norm during a particular month of the year.\n\n**Share of Search** The percentage of the available clicks that are likely going to a particular group of domains. This is a factor of the search volume for the Category, Topic or Sub Topic selected and the Estimated Clicks for the group of domains or domain.\n\n**Search Engine Marketing (SEM)** A form of digital marketing targeting performance in the search engines. SEM is comprised of paid search and organic search (also called SEO or Search Engine Optimization, see below). SEM really begins with an understanding of what your target audience is looking for and delivering accordingly. Sometimes you have to pay for visibility and other times you earn it by creating great content.\n\n**Search Engine Optimization (SEO)** Traditionally thought of as something that you do as you develop a site or after a site is developed. Success in SEO means that you have a pulse on what information\/products, etc., your customers or target are looking for, what format they want that information delivered, and that you are creating relevant content that surpasses other content across multiple platforms. SEO is typically called earned media because you can't buy your way to success, you have to earn visibility.\n\nLet's consider the numbers, first. And take, as an example, the website of a media property. Fortunately, I don't often receive direct pitches from media reps, but they go like this. There will be talk of \"delivering eyeballs\". It's the industry clich\u00e9 for the number of visitors a website attracts, also called \"uniques\", which is short for unique visitors and is also known as traffic. This is the first number you want to know, and the dominant traditional measurement of a website's performance. So you ask, \"What kind of traffic are you getting?\"\n\nSales rep: \"We're averaging 10 million uniques, and 60 million page views, up about 30 per cent in just the past four months.\"\n\nYou: \"Those numbers sound rather good.\"\n\nYou are not wrong. Anything in the tens of millions sounds pretty good. But what the rep just told you is that 10 million individual web users, excluding return visitors, view at least one page on the website per month, and that the numbers are trending upwards. Your reaction should be one of interest but not enthusiasm, considering that online newspaper giants like the _Washington Post_ and the _Guardian_ register more than 78 million and 120 million monthly average users (MAU), respectively.\n\nIn any case, the next question you pose to the rep might be: \"How many page views are we talking about?\"\n\nYou are asking about the average number of clicks per unique, which are also calculated on a monthly basis. Each click yields a new page view. When a viewer lands on a page and immediately clicks to navigate to another site, that's known as \"bouncing\". A high bounce rate is bad, needless to say. Web publishers don't like to talk about their bounce rate, which means, of course, that you should ask about it. Assuming a reader doesn't bounce, she sticks around, scrolling and clicking. Generally speaking, there's a correlation between the number of page views and a reader's level of engagement. More page views translate to more time spent on a site, and a greater likelihood that an ad will be effective. Ultimately, for your client's sake, you want a visitor to click on an ad. What happens next depends on the ad's creative content, and whether the user finds it compelling. A brief video advertisement may play in a pop-up window, or a reader may be invited to fill out a form, for example, to sign up for a newsletter or a special offer from your client.\n\nBut the moment a viewer clicks on an ad, she is deemed to be engaged by way of a click-through. One meaningful metric that doesn't involve click-through is an \"impression\". This describes when an ad simply appears, or is served, on a webpage. The presumption is that a reader saw the ad and afforded it some level of cognition. But when a click-through occurs, it triggers brand interaction and can be measured. So, in addition to asking about page views, you will also want to enquire about the click-through rate, or CTR. The number of clicks an ad generates, counted in thousands, is used, in part, to determine its cost. CPM, for cost per mile, is the total cost of an ad placement divided by 1,000.\n\nAs your meeting progresses, you can keep it on point by asking for more numbers, such as the conversion rate. After a reader is engaged, the next incremental goal is conversion, meaning the user takes action prompted by the ad. On a retail site, a conversion is a sale. Likewise, a political or cause-related ad achieves conversion when a user makes a donation or signs up to receive yet another inbox-clogging newsletter. If an ad is shared, that's also a conversion. If an ad goes viral, that's something of a different magnitude \u2013 one conversion that can lead to millions, like toppling dominoes. Related terms include \"funnels\", which are the steps that a user takes on the way to a conversion. A visitor who takes these steps is called an \"intended\". One editor for a top-tier online publishers says that his company is \"obsessed with not losing intendeds\", which is another way of saying, \"trying to increase the conversion rate\". So, you ask the rep, \"What's your conversion rate? Has an ad or any native content on your site gone viral?\"\n\nAnother telling metric is the amount of time a user spends on a site after the moment she arrives. Generally speaking, the longer a user lingers, the better. On most sites, time spent on-site means more page views and an increase in ad impressions and click-throughs. But the prevailing wisdom is that session time is a less important measure than the average time spent on a page. A good website holds the viewer's attention, and the best measure of her attention is how long a single page engages her.\n\n\"A website that builds traffic by word of mouth, or by its content being picked up and linked out from other sites, can be assumed to have more integrity and inherent interest than a site that pays its way to the No. 1 position in search results.\"\n\nSo far, so good. And, so many numbers. Aside from user demographics, which the rep ought to provide to you, you'll also want to know what drives traffic to the website. The vast majority of traffic is paid for. Just as advertisers pay Google AdWords for better and more frequent ad placements, websites pay to appear in the highest possible position in search results. Your question to the rep should be: \"How much of your traffic is paid, and how much is organic?\" Organic or incidental traffic measures the number of visitors to a site who arrive by means other than paid traffic. The relative worth of paid versus organic traffic is debatable, but on its face, the latter seems more valuable. A website that builds traffic by word of mouth, or by its content being picked up and linked out from other sites, can be assumed to have more integrity and inherent interest than a site that pays its way to the No. 1 position in search results.\n\nWe have a sense of how to judge our media site, but it appears we're on a roll \u2013 other party guests are joining us, each with a site of their own. There are many types of website, each with unique questions to ask and different metrics to measure. We might divide them into four groups.\n\n\u2022Media, directory and brochure sites (for example, the _Guardian_ , or a brand, say, Dove).\n\n\u2022Blogs and personal websites (for example, a Tumblr page or milesyoung.com).\n\n\u2022Ecommerce and marketplace sites (for example, Amazon and Airbnb).\n\n\u2022Social networks and sharing platforms (for example, Facebook and YouTube).\n\nBlogs and information sites would focus on page views and time on site, while ecommerce is more concerned with converting visits to sales. Social networks and sharing platforms, which tend to make their revenue from advertising, need their audiences to convert so they prioritize click-through rates. Business model is the real driver here; for example, platforms that rely on advertising, like YouTube, need highly engaged MAUs (Monthly Active Users), while subscription services, think Spotify, grow by converting eyeballs (or rather, earlobes) into paying customers. Fortunately our analytics software is set up to capture the right metrics based on the goals we've set.\n\nWe can move on to website optimization, the process of refining the site by A\/B testing alternative CTAs (Call To Actions), changing copy and imagery, editing contents or even completely overhauling the UI (User Interface) to improve our funnel metrics. And, of course, my inquisitor will ask about benchmarks \u2013 everyone always does. It is important to find and beat the norms to ensure we're getting the most out of our site. It's worth noting that category is still very important \u2013 low involvement categories offline are still low involvement online, although we can now try to engage people more often by measuring their browsing behaviour and re-targeting to bring them back.\n\n\"A good website doesn't draw attention to itself but rather serves as a hub, delivering the user to all of the components of your brand's digital ecosystem.\"\n\nSo much of it is numbers: how many impressions our site gets, the number of unique visitors, what proportion of them click through an ad or piece of content, how long is their average time on site and how many pages they view, what are the conversion rates at every level of the funnel, and so on. And then we look at these metrics through different lenses: segments, cohorts, time periods, devices, and so on.\n\nBut have you had enough of the numbers? I'm pretty sure I have. They are important, and should neither be ignored nor blithely accepted. But what's just as compelling is the second question: how does a website explain or represent me? And by \"me\", I mean anything that can be considered a brand, from an individual like you, to an automotive marque, to a digital retailer. The answer here is largely qualitative. It takes into account traditional design elements \u2013 font selection, legibility, colour palette, use of graphics, animation and video, density and placement of text, and so on. Most importantly how logical is the _architecture_ of the site? How is information organised and structured so the most interesting information surfaces the most easily? Ask for a web designer and then look for an architect!\n\nAt the end of the day, though, a website is only an advertisement. And, like any advertisement, it needs to have an idea within it. If you're lucky enough to have a brand idea already, it needs to be there not as \"matching luggage\", but in a way that brings all the elements to life. There is nothing more depressing than looking at a website that looks, feels and behaves as if it has come from a parallel universe \u2013 as if it is an idea-free zone.\n\nFinally, there is three-tiered chess game of interactive design. The best interactive design starts by facilitating intuitive navigation within the site itself. The less a visitor has to think about her next click, and the one after that, the better. The goal is a seamless experience that's enjoyable \u2013 you want a visitor to think well of your brand and return often to the site. That much is a given. But the design ideally should also steer the user to a conversion, and \u2013 here's the really difficult part \u2013 act more like a tool than a showpiece.\n\nA good website doesn't draw attention to itself but rather serves as a hub, delivering the user to all of the components of your brand's digital ecosystem. So the final question should be, \"What other digital channels are you using?\" The number of uniques and page views a site generates, which are provided by web analytics companies such as comScore or Nielsen Online, are only part of the calculus of a brand's digital audience. Social channels are of ever-increasing importance, as the balance of web access shifts away from laptops and desktops to mobile devices. A brand with a good digital strategy may use its website as a hub, but it must also deliberately build its presence and following via social channels. A dedicated YouTube channel, Facebook page, Twitter and Instagram feeds \u2013 all of these mechanisms grow awareness and reach. The sum of a brand's digital parts, including its website, should be greater than the whole.\n\n**Intimacy at Scale**\n\nWhen E.M. Forster, in his novel _Howards End_ , coined the quotable phrase \"only connect\", it was part of a longer sentence: \"Only connect the prose and the passion, and both will be exalted, and human love will be seen at its height.\"\n\nEnter the keymost of all keywords, I believe, for any vision of media in the digital world that really wants to connect intent and content, marketers and consumers, prose and passion: intimacy.\n\nIt's also probably the most difficult thing of all to strive for, because, like Ben Richards' threatened media operative, it seems as if at the end of the day everything might be replaced by a program: programmatic media trading automizes the negotiation of a price for rendering an exposure across different media. Programmatic advertising already accounts for some 20 per cent of all online advertising revenue, and it is still growing by upwards of 70 per cent per year. But as my famed colleague Rory Sutherland of OgilvyOne reminds us: \"Business completely fetishizes certainty. It takes big, complex problems, strips out all of the stuff that makes them complex, and solves them as if they're optimization problems.\" That's a good description of what programmatic advertising can do, optimizing itself into what Rory calls, \"very rapid Darwinian advertising\".\n\nOf course, there is much \"bad\" programmatic advertising. Rote buying results in BMW ads following you around the internet for six months after you bought a new Infiniti. Or it puts an un-skippable pre-roll ad for professional power tools in front of a middle-schoolgirl oriented school supply DIY video. And we also know that around 15 per cent of programmatic is just plain fake. But \"good\" programmatic allows us to buy audiences not just in context but by data; and more often than not in real time. We know that when an ad is delivered in real time to a person, they are twice as likely to interact with it.\n\nPROGRAMMATIC BUYING ECOSYSTEMS\n\n_Programmatic advertising is like a virtual factory. Money goes in at one end; and a delivered audience at a cost comes out at the other. The best programmatic advertising takes all of this technology and envelops the consumer in the most intimate of marketing experiences. The worst uses the same technology to undermine its own intent. Use it wisely._\n\nIf there is one secret to getting programmatic right, I believe it is a very obvious one, and that is to remember that technology is only the enabler. You start with as granular and as comprehensive a view as you can conceivably assemble of the individual (from any available source). Then you form a view of what creative messages will best engage him or her. Then you scale up. Shortcuts to scale will never be intimate.\n\nSearch has entered the picture again, not as a clarifier of intent, but more as a channel. Or, even better, your battlefield \u2013 the \"digital shelf\" your product sits on, with two zones, the \"organic\" zone, which is owned and earned, and the \"paid\" zone. The organic zone refers to your brand's appearance in search results due to search-engine optimization and the independent action of the search algorithms, or those results that appear as a result of the action of others \u2013 media placements, blog mentions, etc. The paid zone refers to those search results you have spent media money to generate.\n\nSo how do you win this battle?\n\n\u2022Your brand must be always-on with maximum visibility across desktop and mobile, all synchronized with your TV buy\n\n\u2022You must have a relentless focus on your Google Quality Score (QS)\n\n\u2022You must use insights from keyword modelling\n\n\u2022You must be present in Google Shopping, while taking care to manage the tension between your brand and the retailers that sell it.\n\n\"Good\" programmatic requires a reassertion of a fundamental principle of effective advertising: that you cannot separate the creative and the media processes. It can often seem that technology forces you to, but that way madness lies. Better, it can build them together, but only if intimacy is the starting point and scale the dividend \u2013 not the other way round.\n\nTHE GOOGLE QUALITY SCORE\n\nGoogle measures the quality of your website by the following measures:\n\n1. Click-through rate\n\n2. Landing page\n\n3. Historical performance\n\n4. Various relevancy factors\n\n5. Ad relevancy\n\n6. Keyword relevancy\n\n**Digital Video**\n\nThere's one piece of connective tissue which the digital world has seemed peculiarly slow to embrace, and that is digital video. Pre-rolls and interstitials have been disappointing to consumers. Porting over a TV commercial as an unskippable pre-roll add-on digital video doesn't do much for the brand and often frustrates the consumer, especially when the context of the ad and the video are wildly out of synch, such as a pickup truck commercial running before a teen-focused make-up tutorial. But how come \"good\" video is still such a relatively small percentage of online advertising spend \u2013 less than 20 per cent?\n\nI think it's down to two contradictory things: not joining the community and not collaborating with creators.\n\nDYNAMIC CREATIVE\n\n_For Weight Watchers, we cut manual optimization out of our diet and replaced it with technology that could crunch data in real time. Suddenly we could optimize ad creative in the moment \u2013 changing messaging, images and colour combinations on the fly \u2013 and immediately optimizing our results._\n\nThere is a failure to understand that in digital video you seek fans, you don't buy audiences. \"Fandom\" is moved by authentic passion; it isn't there to be \"hit\", but it's there to be conspired and collaborated with. If you want to enter into the mindset, then The Young Turks (TYT), an opinionated, youth-oriented online news network provides a good example. Fans prefer it to other news outlets and mob the presenters at conferences as if they were rock stars.\n\n_Qualcomm took a brave step in shifting its annual ad budget from media spend into the production of a Hollywood-style half-hour thriller released online. The short film,_ Lifeline _, resulted in advertising that people truly wanted to see. Directed by Academy Award-winner Armando Bo and starring Olivia Munn, Leehom Wang and Joan Chen,_ Lifeline _tells the story of a man's search for his missing girlfriend using the smartphone she has left behind. The capability of Qualcomm's Snapdragon processors were central to the story, with the hero using its enhanced security, advanced photo and video capture, superior connectivity and faster charging to find her. Set in Shanghai, with 70 per cent of the dialogue in Chinese (China is one of Qualcomm's main markets) the film has been viewed 20 million times, with another 100 million more including trailers and the behind-the-scenes video._\n\nI have been in many boardrooms in my career, and over the years they have not fundamentally changed. Nor have their marketing departments and their advertising agencies quite liberated themselves to be TYT. But there is something they all should recognize. It was something that Wendy Clark of Coca-Cola brilliantly encouraged: a 70%: 20%: 10% division of media investment. Seventy per cent is \"low risk, bread and butter\"; 20 per cent is \"innovative\"; but 10 per cent is spent on \"high-risk new ideas\". Unfortunately, it has not spread or stuck very much as a way of actually doing business. But if ever there was a case for the 10 per cent, it is digital video.\n\nIt would afford, for instance, experimental collaborations with creators. Our agency in New York actually identified the top 30 of them. They're not Boardroom material, but they have more consumer intimacy in the digital world than most influencers. Ryan Higa's launch of the Lenovo Yoga tablet was a seamless part of his ongoing fan interactions. As a result, his fans accepted, even looked forward to, a programme that was an obvious advertisement with the same sort of anticipation they displayed for his other videos.\n\n\"There is a failure to understand that in digital video you seek fans, you don't buy audiences.\"\n\nAnd it would sensitize you to the \"super fans\": a group that influences disproportionately: the 20 per cent of enthusiasts who drive 80 per cent of the engagement and conversation. Superfans help moderate comment streams and provide community leadership for the digital creator's audience. It is they who help shape the entertainment flow, and you reach them by becoming a part of it. Rhett and Link, for example, integrated Geico advertising into their video programming, but were it not for the brand name, you'd never know it was any different from their other videos. The brand reached Rhett and Link's superfans by the simple expedient of letting Rhett and Link be Rhett and Link. Something easier said than done for most brands.\n\nDoing these two things would allow you to have a much better chance of building a sustained audience as opposed to mere viewership. After all, audience is just the advertising world's way of describing community.\n\nBut there's also an alternative way of looking at digital video, not so much as a community-based audience but by learning from Hollywood. It requires client courage: the courage to commit, once in a while, to one big production. It means creating stories so compelling that they command attention. But then that attention, unlike the attention of a cinema audience, is scaled by a digitally informed distribution strategy.\n\n**Intimacy in Depth**\n\nHow intimate can you be? How deep can you go?\n\nOver the years the most thankless and frustrating presentations I have made were those to persuade clients to buy CRM (Customer Relationship Marketing) programs. I can now understand why.\n\nAt the very beginning of the digital era, Ogilvy & Mather perceived the value of creating not just one-off sales but long-term loyalty. We conducted tests which showed that those who received personalized communications did, indeed, purchase more than those who did not, and that they were prepared to give us information about themselves in return for useful information about our products. And we could calculate their \"lifetime value\" to us, and translate our data into measurable ROI (Return on Investment).\n\nBut we were seeing through a glass darkly. The medium was direct mail; the databases were primitive; the data was single source; the process was clunky and very expensive; and worst of all, it really required the clients to do a lot of hard work, too. Even when direct mail by post became email, something was missing.\n\nLike zealots convinced that what we believed was incontrovertibly true, we kept on making presentations to audiences who could not disagree but who did little \u2013 outside of certain easy-to-implement sectors, such as airlines where initially the relationship became as much a matter of points schemes as of deeper intimacy.\n\nCRM became a dirty acronym: and some companies have now banned it from their lexicons altogether because it brings the deeper principles into unnecessary disrepute.\n\nWhat CRM never achieved was to be really intimate \u2013 or truly engaging. Like many early adopters in the Digital Revolution, we were too obsessed with the tools and not enough with the experience the customer had. But \"data plus digital\" is a powerful combination which allows us to put the _customer's_ point of view at the centre of what we do, and not just the brand's.\n\nHi! I'm DAVE. They called me that because I suppose they wanted me to sound like a hip grandson of the great David. I'm not sure he'd like some of the words I stand for:\n\n**D** ata-inspired\n\n**A** lways-on\n\n**V** aluable\n\n**E** xperience\n\nBut he'd certainly sympathize with what I stand for \u2013 creating, in the digital age, engagement that sells.\n\n_Brian Fetherstonhaugh (the Chairman and CEO of OgilvyOne) and I introduce DAVE to the agency in one of my \"Miles' Minutes\" \u2013 my regular agency update that went out to 24,000 staff worldwide every few weeks._\n\nSo we invented DAVE.\n\nI don't know what David Ogilvy would have thought of DAVE, but it has certainly catapulted us into a new position as the world's premier Customer Engagement agency. It's a way of thinking as much as anything \u2013 and as it is creative at its core DAVE prompts us to do a number of very clever things:\n\n1.To define a customer ambition, which emerges from but is distinct from a business ambition: data gives us a very precise view of whom we are engaging, and exactly what value we want to unlock.\n\n2.To create personas of these customer segments, to spring them into life. There's no intimacy without knowing whom you are being intimate with.\n\n3.To map the customer journey: _all_ the relevant moments \u2013 touch points (and pain points) \u2013 along the persona's journey through life. It's a very thorough discovery, and needs to be painstaking. We've seen many slipshod ones. No pain, no gain.\n\n4.Create an engagement idea: where any other idea has to execute through unexpectedness, an engagement idea also has to provoke or invite the customer to take a role in the experience. Unlike many other ideas, it looks to the long term.\n\n5.Develop a blueprint for engagement. How do you bring it all together?\n\n6.Then turn that blueprint into an actual plan where the customer experience is mapped: how that idea feeds through each channel, impacts on each touch point and how intimate the experience will be.\n\nCUSTOMER-CENTRICITY\n\n_DAVE \u2013 or similar tools \u2013 finally provides a systematic path to customer centricity: the promised land of the digital age._\n\n**The Performance Brand**\n\nDAVE (and his other companions in the industry) represents the \"big\" leap we had always hoped for.\n\nBut can we leap further still? Is there such a thing as the performance brand? A brand where the ability to engage, and to do that programmatically, and then change the media and creative mix dynamically based on results as they emerge, is baked in?\n\nIt's coming.\n\nIt takes us into selling digital transformation itself to our clients, especially those who are marketing online.\n\nIt starts from straightforward programmatic buying, but adds in a hefty dose of digital analytics to super-optimize the results, and does so across all the channels, paid for and non-paid for. Then DAVE takes over the steering and it moves into customer engagement. Finally, it becomes completely omni-channel, relating behaviour in-store with behaviour offline.\n\nTo make this happen we need to work with 30 different technology partners; and create a marketing cloud in which to house the data.\n\nSo finally, everything connects. As the E.M. Forster quote about only connecting continues: \"Live in fragments no longer.\"\n\nIt does seem possible.\n11CREATIVE TECHNOLOGY: THE SWEET SPOT\n\nThere was a time in the pre-digital world when the phrase \"creative technology\" would have seemed like the most grotesque oxymoron. It's not necessarily a mainstream thought even now. Artists versus geeks, technophobes versus technophiles, \"right\" brain versus \"left\" brain... the clashes run on, but with, I think, decreasing intensity. For this is one of the biggest \"ands\" of the Age of And: the convergence between creativity and technology makes for a transformative sweet spot.\n\n**It Starts With Code**\n\nWe all use code even if we don't know our Java from our PHP, our C++ from our Ruby \u2013 even if we have never heard of Unix or Lisp.\n\nCode is everywhere. Code, as much as electricity itself, powers the electric grid that keeps our society operating. It also runs like an underground river through every system in our lives. It is all over our homes, governing our washing machines, playing our music and refrigerating our food. It runs our economy, often through fast but ancient (in computer terms) systems built on early languages such as Cobol and Fortran. It has long been in our cars, so much so that today's mechanics are as much computer engineers as they are grease monkeys, and it is even creeping into the humble light bulb.\n\nSimply put, code is a set of instructions for a computer, encoded in a language humans can make sense of and manipulate, that the computer can then translate into its own machine language. That language instructs the computer to configure its logic gates in such a way as to produce the result the coder desires.\n\nYou may very well come into advertising explicitly to avoid having to understand code, but just as code makes the world operate, it lies behind advertising, too.\n\nIf you really want to know how to code, find a few hours and read Paul Ford's magnificent essay, \"What is Code\" (2015).\n\nIt's too late for me to be a coder. But what I do appreciate is that the coder represents the foundation of advertising in the digital age. She or he assembles the foundations. There's good code and bad code, in both moral and functional terms. And there's creatively sensitive code such as the closely guarded algorithms used by high-frequency traders and search engines. In advertising, the programmer now defines whether you are good or bad at creative technology.\n\n_Code underpins the digital age, but it need not be as mysterious as it seems. Paul Ford's \"What is Code?\" is primer for anyone who wants to dig in to code themselves._\n\nWhat makes a great programmer that we would want at Ogilvy & Mather? There's the obvious fluency with the main languages, technology stacks, architectures or infrastructure, but our needs go beyond that. We need an elegant coder, one who can write for mobile and desktop, for low bandwidth and high. We need one with the social skills to work with clients, data scientists, account people and creatives alike. We need one sensitive to the importance of user experience, who is as allergic to kludgy interface as he or she is to kludgy code. Alternatively, you could have the most ham-fisted service-side engineer the world has ever seen so long as he is working with product and user-experience designers who have an instinctive flair for the customer's experience. And yet, even our most tech-oriented developers must build to enhance, rather than impede, creative application. But all this needs to be coordinated, and for that, we need a new animal.\n\n\"... the coder represents the foundation of advertising in the digital age.\"\n\nSo here comes another newbie, digitally spawned. We call them digital producers. In the tech industry, these people are often project managers, because all around, there are specialists who need to be coordinated, cajoled and encouraged. But the digital producer is not just a glorified digital project manager. He is the keeper of the vision, the link to the client, and often the talent scout for the project. Digital producers sit at the hub of the project, managing everything from user journeys to market research, from design to the engineering roadmap. To put a fine point on it, the digital producer is the leader of the project.\n\nThen there are the digital creatives: the people who dream up, write and subsequently make everything from banner ads to contextual, geo-located billboards, from company web pages to menu apps for quick-service restaurants. At one end of the scale, they handle InDesign or PhotoShop while coding a bit of HTML5 and CSS. At the other, they are artists, engineers, and architects, virtuosos of design and powerful coders. They're quick and nimble enough to get to a minimally viable product at Silicon Valley speeds while still upholding the brand's aesthetics and values.\n\nAnd finally, there's a very rare bird \u2013 the inventor. The person who gets technology so strategically and in such a creative way that he brings something radically new. It is she who asks if a tennis racquet itself can double as a coach; if virtual reality can give someone the experience of being at a hotel from half a world away; if kids can be urged to exercise more by linking a favourite drink to a fitness tracker and exercise program. These are people who can see that the purpose, stated or not, of one object is in fact the solution to another problem.\n\nThese four are not a hierarchy, and treating them as such seems to me to be utterly corrosive in a creative culture. Rather, they are a virtuous circle, each incrementally improving the other.\n\nTHE CREATIVE PROCESS FOR DIGITAL\n\n_The digital producer is the lead on digital projects, though less of a pilot and more an air traffic controller. Her role is to guide; to bring the various disciplines together in a disciplined way. In the digital age it has become the best way to create innovative work, such as our aeroplane-spotting billboard for British Airways (see alsohere)._\n\n**The Front End**\n\nThe front end where creativity and technology co-mingle has become known as UX \u2013 a deceptively simple acronym for User Experience. It describes what is a ferociously contested domain. As the internet developed, so did the need to make it easy to relate to. It was Don Norman who first gave us UX as a concept in 1998. As in other cases, what was originally meant morphed into something slightly different. Norman meant human-centred design. He writes, \"I invented the term because I thought Human Interface and usability were too narrow: I wanted to cover all aspects of the person's experience with a system, including industrial design, graphics, the interface, the physical interaction, and the manual.\" Now it means more like a place where information architecture meets research, strategy, content, psychology and visual design.\n\nIronically, for something that claims to make things simple, UX is never explained simply. It is impossible to find a chart that does so; indeed, they all look as if they have been designed by drunken locusts drawing in ink. Being as they are specialists in the user experience, UX professionals are in on the joke. I have a theory as to why that is. UX has evolved a collection of specialists who have never quite resolved their relationship with one another. Each specialism has its own view of what it is, and all have a vested interest in not defining it too clearly. And Ogilvy & Mather's charts, frankly, have not been much better. I briefed our people to design something clear and simple; and this is it:\n\nTHE PERFECT UX\n\n_This chart might not be perfect \u2013 as any UX expert knows, there is always room for improvement \u2013 but this attempt captures the UX process well._\n\nFrom this we can see that the UX designer sits at the intersection of customer needs, business needs, and the feasibility of meeting them on any given project. UX is as much a philosophy as it is a practice, and in many respects, it is everyone's job, particularly at an agency. That's more than just rhetoric: on some projects, the UX designer will do the work of a planner. On others, they'll be visual designers.\n\nA true jack-of-all-digital-trades, the agency UX designer needs to move easily among the following disciplines depending on the project and platform:\n\n\u2022Copywriting\n\n\u2022Information Design\n\n\u2022Sound Design\n\n\u2022Motion Design\n\n\u2022Graphic Design\n\n\u2022Interface Design\n\n\u2022Interaction Design\n\n\u2022Code\n\nWhy the staggering breadth? Because UX is a critical part of the fabric of advertising in the digital age. User experience is often the differentiator for a brand. Uber is cited as a great example of UX, and the real magic is the whole experience beyond just the elegant app. It supplies clear delivery of the kind of information waiting travellers want most, like fare and wait time, and marries that to more effective and consumer-friendly redistribution of capital and labour in the form of cars and drivers. Put that together and you transform a poor consumer experience \u2013 the opaque taxi market \u2013 and turn it into a great one. The total experience a consumer has with Uber is so well thought out, it's almost frictionless.\n\nAnd so we find a new, broader definition for the UX professional. If you book a flight through an airline's magnificent website, have a smooth check in, but then endure a terrible flight, then you've had a bad user experience. While she cannot influence air traffic control, the UX designer can address how the airline acts. In short, she is the person charged with enhancing every part of the interaction between a brand and its consumers. It is as essential as it is intangible.\n\n**The Back End**\n\nThe Back End is what makes anything happen. It's also the area where companies and agencies experience the most disappointment.\n\nBut before the disappointment, there's the fear.\n\nConsider the marketing technology business. It is skyrocketing. Taking the US alone, the number of vendors between 2014 and 2015 _doubled_.\n\n_Scott Brinker continually updates his famous chart as marketing technology solutions are invented at an astonishing rate. And yes, it's almost unreadable! A warning: technology is only useful with the skills and structures in place to make it sing. And never forget the big idea._\n\nAnd, of course, each vendor has a sales pitch. It is very interesting to consider the techniques technology vendors tend to use to demonstrate their products. They show simple \"flash\" demos to entice you, built to run smoothly in controlled conditions, on simple platforms. It all looks so easy. But it's not.\n\nAnd there's what Peter de Luca calls \"the 20% problem\". He should know: he's head of our marketing technology practice, and spends his whole life trying to remedy it. Peter says that you just need to ask a company \"what percentage of the marketing functionality of this marketing technology are you using?\" So very often the answer is 20 per cent, or something close to it.\n\nThe reason is primarily structural. There was no structure in place to support the licence when it was bought, or there are organizational gaps between the work groups or departments that should be benefiting from it. Incredibly, without really understanding this, disappointed clients often switch software vendors looking for the functionality they do not believe they have (even when they often do). They will still be disappointed.\n\nThe point is not about technology; it's about internal alignment.\n\nIf you think of the primary teams or departments in your client's organization that should be aligned, they are likely to be:\n\n\u2022Marketing Team: These are the planners and strategists, the people who think up and design the campaigns against specific business objectives.\n\n\u2022Operations Team: or the campaign execution team. These are the people that are actually working with the marketing technologies, or managing the vendors that do the heavy lifting.\n\n\u2022Analytics Team: people that manipulate data, find insights, and create reports for the campaigns.\n\nHOW NOT TO MAKE MARKETING AUTOMATION WORK FOR YOU\n\n\u2022 The **Marketing Team** starts off by submitting a simple ad-hoc campaign request lacking any real targeting criteria, test plan or cross-channel engagement strategy. This may be due to a lack of knowledge about the data that are available, past campaign successes, or the available functionality of the technology platform.\n\n\u2022 Then, with this simple request in hand, the **Operations Team** needs to use only the basic functionality of the platform to create an audience, build a simple email template, and launch the campaign.\n\n\u2022 The simplicity of this ad hoc campaign naturally leads to a lack of data to analyze, with limited insights and basic reports coming in from the **Analytics Team**.\n\nA simple request leads to a simple execution, which leads to limited insights. And that loop continues, leading to limited use of technology \u2013 only 20% of what's available.\n\n_Peter de Luca's take on making marketing automation work._\n\nHOW TO MAKE MARKETING AUTOMATION WORK FOR YOU\n\n\u2022 The **Marketing Team** is thinking strategically. They're being innovative in terms of channel engagement. They know what data exists. They're defining more complex campaigns to drive more personalization. They know what they can get out of their technology. In essence, they're pushing the Operations Team to do more interesting things.\n\n\u2022 In turn, when the **Operations Team** is being pushed with more strategic and innovative engagement strategies, they naturally become savvier with the technology platform. They're being pushed to use more of it. They're coordinating multiple channels in their journey, applying more complex business rules to drive more personalized experiences.\n\n\u2022 In turn, the **Analytics Team** has more data to analyze and more tests to evaluate. They're able to find more insights on what's worked and what hasn't worked. They're providing meaningful reports and dashboards to the Marketing Team in order to inspire them further.\n\nThe result is a positive loop that continues and continues.\n\nSo what happens when all is aligned perfectly, and the back end is on its best behaviour? For one thing, the brand's marketing technology investments are aligned to the sort of relationship it wants to have with customers. And for another, you can accomplish great things.\n\n_J\u00e4germeister teamed up with Shazam to become part of the entertainment experience for Millennials in a multi-pronged, automated campaign called \"Shazam the Stag\". Knowing that their audience craved moments that create social capital, the brand provided a digital ticket behind the scenes of the EDM (electronic dance music) scene \u2013 and it did so in the shopping aisle itself, turning every bottle, every sign, every shelf wobbler into a portal to a world of custom cocktails, bespoke playlists, and backstage access. Shoppers need only snap a picture of the J\u00e4germeister stag. Shazam and the back-end technology did the rest._\n\n**A Frontier How Far?**\n\nOf the waves of hype that have washed through the Digital Revolution, the most vigorous at the time of writing are those that promise the new frontier of Virtual Reality (VR) and Augmented Reality (AR); but a frontier how far away? And, indeed, how attainable?\n\nAs VR and AR are often used interchangeably and incorrectly, it behoves me to describe them clearly. In each case, technology is predicated around creating an enhanced experience.\n\n\u2022In VR, the experience takes place in any environment that is virtual \u2013 a movie, for instance, so that there is a simulation of physical presence.\n\n\u2022In AR, the experience takes place in a live, physical environment as a supplement added by digital assets.\n\nIn between there is a grey zone \u2013 mixed reality.\n\nPopular attention in each case is focused on the method of delivery: how you experience. But the innovation that has driven the experiences so far has been much more to do with the back end of the technology, where significant advances in individual fields have been assembled into a chain.\n\nThe first set of links is to do with capture: it's what enables us to see things in the round.\n\n_Since children are rarely enthusiastic about cleaning their teeth, Toothbrush Games uses augmented reality to encourage better dental hygiene habits. A two-minute game, featuring a live video of the user brushing their teeth alongside some fun characters, turns a chore into a challenge. The stroke of genius? It only works when the smartphone's microphone picks up the sound of brushing._\n\nThe second set of links is about process.\n\nSo far so good. And you would need a heart of pure flint not to be moved by some of the experience highs, which this incredible cocktail of innovation can produce.\n\nBut then, how do you view it? And it is in the area of display that the technology lags \u2013 and may do so for some time.\n\nAt each end of the spectrum you can view, on the one hand, with a self-assembly cardboard kit linked to your mobile phone or, on the other, with a clunky headset linked to a gaming PC. One is cheap; the other is very expensive. Let's also be honest: one is flimsy and primitive; the other is heavy and uncomfortable. There's a lot of work to be done before the chips can be miniaturized, de-thermalized, de-powered and then put into something resembling a hip pair of spectacles, which delivers enough speed and pixelation to equate to reality. And that's not even recognizing the biggest promise of all: the haptic promise.\n\nHaptics perception is created by the impact of sensory touch. Technologists have adopted this term to mean a touch-based interaction with a computer. Haptic controllers add touch to the mix, enabling you to manipulate digital objects in a virtual world. It requires a bi-directional interface between your brain and an interface that connects you to the world: at one level \"teaching\" you that you can feel virtual objects. While it's surprisingly convincing when done well \u2013 I've seen people flinch when they've dropped a purely virtual hammer on their imaginary hand! \u2013 any idea of mass-commercializing a hap-app in the reasonable future is haptic pie-in-the-sky. The front end has not caught up with the back end \u2013 and it could be rather a long time before it does.\n\n_Jim Beam makes a prized bourbon, The Devil's Cut, but you can't see what makes this spirit different from the outside. You have to know how it's made, and to illustrate that, Jim Beam worked with Bottle Rocket to create an immersive VR experience to dramatize the journey bourbon takes to become The Devil's Cut. Only in VR could you get right inside the bottle!_\n\nThere is, of course, an assumption made by all who seek to boost these technologies, which is that their effects are inevitably benign. But when the physical world becomes really indistinguishable from the digital, do we gain? Do we communicate better \u2013 or less? Do we take more out of life \u2013 or less? To be honest, at this stage \u2013 who knows? But a rational scepticism at least puts these questions on the table. The case for the \"metaverse\" sweeping all ahead of it is not yet proven.\n\nIt may seem that I'm \"down\" on VR\/AR. Not at all. It is very exciting and it will move out from its hard core of gamesters into the world of advertising.\n\nHere's what it offers:\n\n\u2022 **Perspective:** we can hyper-immerse as we tour through reality. Where we seek to simulate a product, we will be able to do so with an intensity never before managed. Imagine, for instance, wristwatch advertising. For years, designers have tried to make us believe we could see inside the workings in 2D. Now we'll be able to travel inside and around the mechanism of every last finely crafted Rolex or Piaget.\n\n\u2022 **Context:** providing extra meaning through explanation and references. Imagine opening and deploying a new food processor for the first time. You're led not just through the basics of how to assemble and operate it \u2013 but into making perfect mayonnaise, which does not curdle as each drop of oil melds into the emulsion, speeding up and slowing down with all the mentoring you might expect from Martha Stewart if she was standing beside you.\n\n**\u2022** **Intensity:** take any piece of emotive creativity, and it can be intensified a hundred-fold \u2013 at a cost, of course. Advertising has always appropriated the term \"anthem\" to describe its big set-pieces. \"Anthemic\" reality will make some corporate advertising seem very pallid. If you attach yourself to the values of the rainforest, how much more valuable is a hyper-immersive experience compared to a merely second-hand evocation of it.\n\nCAMERAS, HEADSETS AND HAPTICS\n\nCapturing the video needed for 360-degree video used to require a custom-built camera array. Now, the technology has matured to the point where prosumer 360-degree cameras are readily available. Animated VR, like standard computer animation, gets built in a virtual environment.\n\nHeadsets provide much of the VR experience by immersing the viewer's eyes and ears into an entirely virtual environment. Sensors detect head and body movement, giving the sense of movement through the created environment. More sophisticated setups use external positioning sensors to create a virtual room so that the viewer can move around to roam through the environment. Headsets range from expensive, high-tech models to cardboard units in which you slide a phone to act as the screen. VR headsets are becoming more sophisticated at a remarkable clip. While most are still tethered by wires, the next generation will be wireless, increasing the sense of reality. Future systems, much smaller and less obtrusive, will likely focus on mixed reality, enabling the real world to be a part of your chosen virtual environment.\n\nHaptics provide the feedback that make simulations feel more real. Anyone who has played a modern video game will be familiar with the \"buzz\" a controller can make to indicate an action. In VR, the experience is more sophisticated, but the idea is the same, but with a twist. The haptic controllers don't just give physical feedback; they give the user the means to interact with objects in the virtual environment, with physical feedback cues to fool the mind. That's now. Future devices will read hand and finger positions in space and give more nuanced feedback, leading to a much greater feeling of reality. The potential is there for much more than just entertainment, both good and bad. Take a real-life look at the book Ready Player One to get a feel for what a dystopian world based on virtual reality and haptics could be like.\n\n_VR, AR, and Mixed Reality depend on a suite of rapidly evolving technological innovations to create an experience. While many of these devices are relatively rare at this moment, consumer VR tech is on the cusp of becoming much less intrusive, and prices are dropping precipitously. Soon, there will come a point \u2013 still undefined \u2013 where the tech will be friendly enough, and the price low enough, that this tech jumps from early adopters to the mainstream._\n\nBut I do expect the practical uses of VR\/AR will be more educational than anything else. Walking around the Alamo with context delivered to you seems a no-brainer. Having your drive to work continuously informed by extraneous context might be rather trying.\n\nAnd this will also apply to advertising. So while the natural tendency is to see it developing in terms of glamorous consumer applications, its ability to supercharge demonstrability may make it as much a B2B technique, especially in high-value categories \u2013 choosing a new fleet, or selecting an architect, for instance.\n\nAnd in consumer goods, anything that lends itself to demonstration will be self-selecting: from the craftsmanship of luxury goods to the efficacy of personal-care products, to the performance of a new SUV to the hip design of a Mediterranean resort.\n\n_Amazon's Dash Button is a simple, single-function wireless node that connects to a WiFi network. It serves one purpose: once configured through your mobile, a push of the Dash Button will place an order for a specified product from Amazon. Out of dishwasher detergent? Press Dash to have a new supply shipped to your door. The innovative device does more than connect the Internet of Things, it connects the internet with brands. Imagine what else such wirelessly connected switches could do._\n\nPok\u00e9mon Go seems to be shepherding in an era of augmented reality, beginning with thousands of Pok\u00e9mon fanatics running around the city searching for computer-rendered creatures, but one that will logically expand to experiences beyond entertainment into fields such as travel, education and medicine. I confess; I am unlikely to ever go Pok\u00e9mon hunting. But I would love to take a historical tour of London or Beijing using an augmented reality travel app. As digital continues to augment reality, it creates all sorts of connections \u2013 to the past, present, future, and alternative versions of all three.\n\nVR and AR are the technologies of the moment, but as the Internet of Things (IoT) grows more ubiquitous, they may begin to merge with a growing orbit of connected devices. Trackers and smartwatches have already extended the functionality of our mobile devices and opened an extended channel for brands. But the same technologies that power front ends and back ends, and realities virtual and augmented, inspire different sorts of invention as well, further blurring the lines marking off advertising, product development, and marketing.\n\nAnd, here, of course, one must sound the familiar bugle call of warning. At the end of the day these are just techniques. They will only be as interesting, engaging and long lasting as the ideas that animate them \u2013 and the brands which pay for them.\n\nWHERE DID ALL THE POK\u00c9MON GO?\n\nIn the summer of 2016, a phenomenon was unleashed that captivated the world \u2013 for a while, at least.\n\nWhen little-known Niantic developed a side project \u2013 a game requiring players to hunt around their cities for monsters and hurl digital balls on their smartphone to catch them \u2013 it turned augmented reality mainstream. It launched on 6 July 2016, and the world went Pok\u00e9mon crazy.\n\nWhy the sudden hype? Firstly, an already powerful franchise in Pok\u00e9mon, which had been popular since the late 1990s. Secondly, an inherently viral platform where players became real-world, monster-chasing adverts \u2013 sparking immediate interest from confused bystanders. And thirdly, the app store effect quickly kicked in. Pok\u00e9mon\n\n_Pok\u00e9mon Go started life as an experiment, in partnership with Google's Maps and Earth divisions._\n\nGo was the most downloaded app ever in its first week of launch, achieving 7.2 million downloads compared to a paltry 2.2 million for Angry Birds 2. And it stuck at the top of the charts, passing 500 million downloads in little over a year. Free to download with in-app purchases, it drove a record-breaking month for Apple's App Store revenue. Pok\u00e9mon Go was making real money from AR monsters.\n\nBy mid-July, Pok\u00e9mon mania was in full swing. When a rare Pok\u00e9mon was discovered in New York's Central Park, thousands of players swarmed the area for a chance to collect points and climb the league table. Few of them spotted singer Justin Bieber among the crowd \u2013 all sights trained on the monsters on their screens, not real-life celebrities in their midst! Players were reportedly so distracted by the game that they were endangering themselves \u2013 one falling off a cliff, others from being mugged while roaming unknown streets looking for monsters. The app hit its peak of 28.5 million unique users within weeks. But then reality started to bite. By the end of summer player interest began to quickly drop off, settling at around 5 million unique visitors per month. This pattern \u2013 a burst of mass interest that quickly wanes \u2013 is the fad phenomenon writ large for the internet age. Popular mobile games like Candy Crush Saga and viral sensations like the ALS Ice Bucket challenge have followed a similar trajectory.\n\n_A familiar pattern. A dramatic hype-driven rise in unique visitors, followed by a period of decline until the natural level of truly interested users is found. Mobile games \u2013 like Candy Crush Saga and Clash of Clans \u2013 seem particularly prone to this \"faddish\" adoption._\n\n_Outdoor advertising \u2013 fitting for a game played in the real world, but surprisingly un-augmented \u2013 has been spotted in London, promoting new Pok\u00e9mon features. Even the most digital of platforms resorts to traditional methods of attracting attention and shoring up user numbers._\n\nBut falling user numbers hardly means a reversal of fortune for Pok\u00e9mon Go. A highly active user base of 5 million has netted an astonishing $1.7 billion in revenue since the game launched. And it's likely a preview of what's in store \u2013 AR experiences commanding extraordinary engagement levels and creating immense value from imagined worlds superimposed onto this one. As digital continues to augment reality, our senses will surely be expanded to encompass 5,000 years of civilization \u2013 finding ourselves seemingly able to slip through space and time, to move between worlds, and to catch a few monsters along the way.\n12THE THREE BATTLEGROUNDS: Social Media, Mobility, Continuous Commerce\n\nAs the digital age starts to \"settle\" there are three big battlegrounds where debate still persists and, in general, the future is not yet won. The first of these is social.\n\n**PUTTING THE SOCIAL BACK INTO SOCIAL MEDIA**\n\nIf ever there was a digital Eden, it was the early days of social media \u2013 days of innocence and hope, where friendship could be \"graphed\", and expanded in seemingly endless, experimental ways.\n\nAGE 18\u201334 DIGITAL AUDIENCE PENETRATION VS ENGAGEMENT OF LEADING SOCIAL NETWORKS\n\n_Facebook is the standout social medium in the United States. Almost every 18\u201334-year-old in the US engages with the monolithic platform for over 1,000 minutes \u2013 nearly 17 hours \u2013 every month according to comScore. But watch out for Instagram! It is becoming the platform of choice for younger Millennials._\n\nLooking back, those early views of social media \u2013 the attention given to the social graph of an individual and then to the collective spread of word of mouth \u2013 still seem prescient, but not now sufficient. In the absence of any more precise and subtle measures, the more fans (or followers) you had was quite simply, the better. \"Sharing\" or \"following\" suddenly become acceptable measures of brand advocacy. However, the sheer volume of connections started to become overwhelming. By 2015, the average American was spending 3.7 hours per day on a smartphone; the average Latin American whiled away 6.1 hours per day on social media.\n\nMarketing abhors a vacuum, and so brands rushed in. The distribution of social content by brands increased exponentially.\n\nBut something had gone wrong. The users themselves were not happy. In our research, we started to discover that almost 80 per cent of social media users felt the quality of online advertising was average to poor. Other research showed that 80 per cent of B2C (Business-to-Customer) content produced goes unnoticed by consumers.\n\nWhat had happened?\n\nIn the race for fans, fans had become audiences, and brands had started to hurt themselves by delivering messaging at the fans, the classic paradigm of analogue \"marketing\".\n\nThe content was often superficial. When Audi represented the idea of #PaidMyDues through a series of \"artistic\" images, they incited an onslaught of criticism. The fans insisted that the Instagram feed go back to images of cars. That was why they had followed Audi in the first place.\n\n_Authenticity matters on social platforms. When Audi started to over engineer its Instagram feed with #PaidMyDues, the audience demanded the brand get back to what it does best \u2013 cars._\n\n_Hashtags can be dangerous. The NYPD found itself in a spot of bother when the #MyNYPD was turned against it. It became the hashtag of choice to criticize the department's approach to fighting crime._\n\nAnd aimless connectivity can rebound horribly, as the New York Police Department discovered when they wandered up an irrelevant alley through the #MyNYPD hashtag.\n\nNo wonder the audience fought back: it had to.\n\nFor self-preservation people contracted their social graphs, retreating into smaller private networks, looking for walls with which to surround their gardens. The burgeoning growth of \"walled gardens\" \u2013 the more closed and intimate social platforms such as WhatsApp, WeChat or Facebook Messenger \u2013 has been the response. Meanwhile, the social platforms themselves started to tighten their filtering algorithms.\n\nBy 2014, we started to notice a drop in engagement levels on our clients' pages and launched a global analysis of more than 100 social programs. What we discovered was the decline of organic reach, in advance of this being recognized or admitted by vendors such as Facebook.\n\nFACEBOOK ZERO\n\n_As the opportunity for organic reach has declined on Facebook, brands have become increasingly willing to pay to reach the platform's audience instead._\n\nSimply put, this data demonstrated that the era of using social media as a free medium, which could be depended upon to inspire by word of mouth organically, was over, forever. The Earned in the triangle of Paid-Owned-Earned was about to undergo a major desertion of social media usage to the Paid.\n\nAt the same time, our understanding of what actually constituted a fan improved. There had been a strong streak of wish fulfilment in the industry: brand owners want to believe that because people are saying positive things about your brand they are brand advocates. This is not necessarily the case.\n\nCertainly there is no shortage of liking or of followers. Eighty-four per cent of people across 11 markets said, when we asked, that they had \"liked\" or followed a brand, product or service. More than half had actually interacted with a brand \u2013 although in a salutary finding which punctures a lot of the rhetoric about social marketing, 79 per cent never received a response back. Around 6 out of 10 people seem to be \"sharers\": they share their experiences of brands \u2013 both positive and negative \u2013 with others.\n\nFollowers are not the real prize. That lies in finding the true promoters of a brand. Only 2 in 10 merit this description. They are more active; they seek direct interaction with brands; they like associating with brands; and their networks of friends also talk about brands much more. They want news. They need feedback. And they are relentlessly discriminating. They are almost as willing as sharers to discuss terrible brand experiences online. The old direct-marketing adage of \"member get member\" very easily translates into member-lose-member.\n\nAlongside these sobering realities, much social marketing remains random, unmeasured and unused.\n\nFROM SOCIAL TO DEEP SOCIAL\n\n_Social moves from basic, through strategic, to transformative, when tactics give way to deep insight and engaging stories at every step of the customer journey._\n\nThe future lies in what we call \"deep social\". Social has, in a way, grown up. It is moving from a marketing and broadcast mindset to a publishing and tailored mindset. Shallow content gives way to more sophisticated insights, more interesting stories \u2013 and much more of the \"conversational\" promise that the early pioneers of social media craved.\n\nTHE ALS ICE BUCKET CHALLENGE\n\nIf you go to the Smithsonian's Museum of American History in Washington DC you'll find a very ordinary bucket. Why on earth is a bucket there? Well, it set off one of the most powerful viral marketing campaigns of the digital age.\n\nIn August 2014, celebrities, business moguls and millions of others voluntarily dumped a bucket of freezing cold water over their heads, donating to the little-known Amyotrophic Lateral Sclerosis (ALS) cause for the privilege. (Or, like US President Barack Obama and UK Prime Minister David Cameron, they had stumped up a larger amount to avoid their icy fate). The challenge's sudden popularity caught the ALS Association unaware; nobody internally had initiated the idea and the first sign of something unusual was at first a trickle, then a waterfall of new donations.\n\n_The bucket that started a movement, which Jeanette Senerchia previously used to mop her floors, now proudly stands in the Smithsonian's \"Giving for America\" exhibition._\n\nSo what was happening? Well, to find the source we need to visit a place called Pelham. It's a small town nestled in New York City's Westchester suburbs, only half an hour's train ride from Grand Central Station but a world away from bustling Manhattan. Pelham is a tight-knit community of fewer than 7,000 people, many of whom know each other well, or know someone who knows them. It is precisely this friendly, community atmosphere that created the conditions for the ALS Ice Bucket Challenge to flourish.\n\nOne of the town's residents is Jeanette Senerchia, a lovely, unassuming woman whose husband, Anthony, both suffer from ALS. It's a relatively unknown neurological disorder \u2013 also known as Lou Gehrig's Disease \u2013 for which there is currently no cure. The life expectancy of someone diagnosed with the disease is often a matter of years, as motor neurons controlling their muscle function deteriorate rapidly. Professor Stephen Hawking has famously lived with ALS for half a century, but he is more fortunate than most sufferers. ALS is a terminal illness.\n\nJeanette had, as she puts it, a \"very small\" foundation generating a little money for awareness of ALS, used for families who are affected. Her cousin-in-law, a pro golfer in Sarasota with whom she has a fun relationship, one day texted her and said, \"Look at your Facebook page\". On it was the original Ice Bucket Challenge. He'd come across the idea in his circles, but it had never really taken off or been associated with a specific cause. Then he thought of her and her \"very small\" foundation. Jeanette told me she texted back, \"You must be joking,\" but eventually she did it. Then she nominated her close friends in Pelham. They nominated their friends, who did the same. And so on \u2013 a ripple effect from a few local townspeople, to the whole community, to neighbouring towns, and finally, a tidal wave on Facebook.\n\nThe giant of social networks isn't a nebulous universe of strangers, but a web of communities where compelling content and ideas can quickly become a pandemic. Within weeks, Facebook's CEO, Mark Zuckerberg, had taken the ALS Ice Bucket Challenge (nominated by New Jersey Governor Chris Christie), alongside other celebrities such as Justin Timberlake, David Beckham and Justin Bieber, whose video gained over one million likes alone.\n\nLikes turned into donations **.** In just one month, the ALS Association in the US received almost $100 million, dwarfing the $2.7m donated over the same period the year before. Awareness soared as over 2 million videos and 4.5 million Twitter mentions saturated social and traditional media. Searches for information about ALS increased dramatically; the ALS Wikipedia page received 2.7 million views that August, compared to only 1.6 million in the preceding 12 months.\n\n_CEO, Mark Zuckerberg taking the ALS Ice Bucket Challenge. Nominated by New Jersey Governor, Chris Christie, in return he challenged Microsoft founder Bill Gates, Facebook COO Sheryl Sandberg and Netflix CEO Reed Hastings._\n\nYet some critics labelled the phenomenon \"slacktivism\", a nexus of entertainment and empathy, but ultimately a get-out from more serious involvement. While Jeanette had created a reason for the challenge, the charity risked being a footnote in the YouTube comments \u2013 more focus on the bucket than the cause, the LOLs rather than ALS. Others were concerned about a sudden redirection of donations to ALS that would otherwise go to a broader set of causes. Unsurprisingly, those involved with the disease defended the campaign, happy to finally be recipients of such large-scale awareness and unbridled support.\n\nThe critics are likely wrong.\n\nResearchers recently identified a new gene associated with the disease that will help uncover the triggers of ALS and lead to new treatments. The breakthrough was funded directly from the ALS Ice Bucket Challenge. Of $115 million raised over the eight weeks the campaign went viral, $77m was dedicated to research and $1 million \u2013 a drop in the bucket \u2013 went to the clever scientists at the University of Massachusetts Medical School who made the discovery.\n\nThere are few things more bracing than a bucket of ice dumped over one's head. A bigger shock, perhaps, is an unplanned campaign that traded fun, peer pressure and a dash of online narcissism for a scientific breakthrough, and all of it down to the intersection between small town familiarity and social media fame.\n\nWhere do the insights come from?\n\nFirst, from understanding that social is something that has to be seen across the whole customer journey. Taking a category at random \u2013 say, telephone providers in the UK \u2013 we can see that not one company is getting the basics of social presence right across an even abbreviated journey from awareness to loyalty.\n\nCOMPARISON OF UK TELEPHONE PROVIDERS ACROSS THE CUSTOMER JOURNEY\n\n_When it comes to social engagement, telephone providers in the UK have relative strengths. O2 is at least getting some things right, achieving broad awareness among UK social media users and converting reasonably well to consideration. But none get even the basics right across the whole customer journey._\n\nSecond, from understanding that while \"conversation\" is an easy word to use, it's a complex thing to understand. A conversation dissected and mapped will tell you what the key content interests, influential platforms and social behaviour of your fellow conversationalists are (see chart).\n\nThird, from understanding that some social conversationalists are more important than others. Journalists are, perhaps, an obvious example. Just consider the figures below.\n\nINFLUENCERS ARE IMPORTANT\n\n_Social media has extra clout when it influences the influencers. Journalists use social media to research, investigate, verify and publish stories._\n\nHOW THE MEANING OF MANSCAPING HAS CHANGED OVER TIME\n\n_Conversation is an easy word to say, more difficult to describe. Mapping ideas against search over time shows how the social dialogue evolves._\n\nBut anyone \u2013 not just journalists \u2013 can become influential, and they often do not identify as topic enthusiasts. It's no longer journalists who are influential. Again, analytics allows us to identify and rank the influencers that matter to you (see diagram).\n\nIt follows that none of this can be undertaken lightly. The \"community manager\" has changed beyond recognition, and the premium on his\/her skills is considerable. This is not a role to economize on! Their hi-touch is needed, not just to manage the original idea of a community as a group of people you somehow own, but to join in day by day, hour by hour, minute by minute, stimulating and facilitating them (see diagram).\n\nBut, more than that, they also have to have a pro-active publisher mind-set paired with the energy needed to help create a consistent brand experience.\n\nAt the end of the day, it is ideas that create conversations, and the stronger they are, the livelier the conversation and the more animated the community.\n\n_Here's a social CRM campaign for Philips Norelco shavers. First we identified precisely the right prospects on Facebook. Then we built awareness with engaging video ads of New York celebrity hairstylist Mark Bustos giving a confidence boost to homeless men by offering free haircuts. Lastly, we drove customers down the funnel with more product-focused content to boost sales._\n\n**Social CRM**\n\nBut it is the relatively new phenomenon of \"Social CRM\" that is most exciting for the future, particularly if you have ever worked in direct marketing.\n\nA few years ago, Ogilvy & Mather acquired a small company in Belgium. Social Lab is now one of the biggest assets we've got globally. They are pioneers in targeting consumers in social media, at the right time, with the things they are interested in. It's a no-brainer. Our targeted social content out-performs non-targeted content by a factor of 2.5.\n\nThey use \"lookalike\" modelling to target \"twins\" of people who know you already, and then leverage all the known interactions there are with them, engage and re-engage with them, test and re-test, never ignoring an action, tracking and re-targeting all the way, through from the initial prospecting to the final sale.\n\nTHE EVOLUTION OF THE COMMUNITY MANAGER\n\nDefinition of Community: not a fixed group of people you own and manage but flickering moments of opportunity based on shared passion\n\n_The role of community manager \u2013 equal parts publisher, customer support agent, brand guardian and social media maven \u2013 has changed considerably over the years._\n\nPeople often ask me, how do you measure the effects of social media?\n\nIn the past it was difficult to answer, as the \"intermediate\" measures (e.g. likes, shares, impressions) were just that \u2013 intermediate. Of course, one gold-plated ideal would be net promoter score \u2013 quantifying the value of an individual's recommendations to someone else to try a brand. AT&T developed some interesting work in this area; most notably for their \"At Summer Break\" program some years back. But systems like these rely on very heavy-duty econometric models, in this case created by management consultants. Unfortunately, they are just not affordable day to day; and as a result have not had wide-scale adoption.\n\nBut, social CRM does provide an answer. As long as there is a URL, and a tracking and attribution system to follow people around the internet, it is now possible to measure customers advancing along (or not) their Customer Journey.\n\nFor Facebook in particular, this might seem a long way from that idyllic Eden of their foundation myth. But it seems it might do a lot more like El Dorado as the cash registers ring with hard sales results.\n\n**THE JOY OF MOBILITY**\n\nThe heading is very carefully chosen. What's interesting is discussing what mobile enables, not what mobile is. But first, let's find out what mobile is not.\n\nIn September of 2000, I was asked to speak at the WAP Enhanced Services and GPRS Conference in London. I argued that the mobile phone would be a _sales promotion_ medium. This was a controversial point of view because \"advertising\" was busily claiming the phone as a new ad medium. Advertising would of course claim the Taj Mahal if impact (easy!) and scale (less easy!) could be demonstrated.\n\nI wasn't wholly right, but I was partly right. Mobile, because it gets one so close to the point of decision, is indeed especially useful in directing you to offers and deals in a specific location.\n\nIt is not an advertising medium of any significance, although the desire to think of it as such and use it as such is persistent. It is something much more than a device or a channel or, indeed, a medium.\n\nThe joy of mobile technology is that it offers mobility. It's about engagement with a whole new lifestyle, and it will be the premier platform of the digital age by far.\n\nEverything is now driven by an explosion of mobile connectivity. As my former colleague Martin Lange wrote in an internal note:\n\nTelecommunications providers have essentially become utility companies, providing access to something that people need... access to mobile devices worldwide is growing much faster than access to many of life's bare essentials.\n\nBy this, he means improved water, electricity and sanitation.\n\nAs more users connect, more connection and data points develop and the demand for more services increases. The only limitations are infrastructure, reach and bandwidth; and regulations.\n\nWhat is developing is fundamentally new: an engagement ecosystem. Mobility, born as it was out of ubiquitous connectivity, becomes a resource through which communications can be created \u2013 but only as a part of a larger view of what contributes service to the customer (see box).\n\nThis ecosystem in the West has been enabled largely by Apple, and in China by WeChat. Phones themselves are developing into a box of sensors which gather and deploy data.\n\nBut the software, the distribution model, the billing (including the connections to foreign exchange), and the link to the store all combine to make mobility seamless.\n\nCalvin Carter, Ogilvy & Mather's global guru in mobility, who runs our subsidiary called Bottle Rocket in Dallas, tells the story of how he founded the company the day after Steve Jobs opened the iPhone platform. Apple only charged 30 per cent commission for hosting an app on the website, and Calvin instantly saw an opportunity. He went home that night, sketched out five apps, and set about building them with some likeminded innovators. Four of the apps tanked. One, the app for National Public Radio in the US (see image), was a runaway success, and it launched Bottle Rocket.\n\n_Any parent knows that it is hard to get kids to drink water. Sugary, colourful drinks hold a powerful appeal, but kids need water to grow up healthy. Nestl\u00e9 wanted to provide a fun way to encourage kids to drink more water. And what better way than to use another means that children are drawn to \u2013 the smartphone. Nestl\u00e9 produced a children's book that lead families to an app where kids can meet their very own Tummyfish using a smartphone's camera. Managed by parents, the app uses daily data to make Tummyfish's behaviour mirror children's drinking choices \u2013 water makes her happy and playful and sugary drinks make her sad and inactive. Over time kids watch Tummyfish grow and unlock rewards for making healthy drinking choices, while parents can monitor their child's drinking habits and track behaviour change through daily reminders and statistics._\n\n_Burger King's most common ad unit is the 15-second commercial, but that's hardly enough time to tell consumers what makes their Whopper different from their competitors. But what, ask the creative team at Ogilvy's partner agency David, if a commercial could somehow trigger a digital experience? Voice-activated phones and devices provided the perfect vehicle. Many people keep their home devices or mobile phones near the TV or within audio range of their computers. Android phones and Google Home operate by continuously monitoring what is happening around them, listening for the triggering command, \"Okay Google\". Burger King took advantage of that with a commercial that intentionally activated the Google Home and Android voice recognition, with a spot that ended with the question: \"Ok Google, what is in the Whopper Burger?\" The question triggered thousands of devices to start reading all the list of ingredients from The Whopper's Wikipedia page._\n\nAn app can only ever be a \"bite-sized\" brand experience. Now there are millions of apps, and an extreme version of natural selection winnows out the non-performers. Why? Because inviting a brand onto your phone may be a small commitment, but it is also a significant one. And if it doesn't work, or feel right, then it is a particularly negative brand experience. Bite-sized can bite back.\n\nDone right, mobility can be transformative \u2013 indeed, disruptive. It is driving a change in bricks and mortar retail \u2013 or as Calvin calls it, \"Bricks and Mobile\". Your device is constantly with you. One study showed consumers pick up their phones 150\u2013200 times a day. It's the first thing you touch in the morning and last thing you set down at night. (And nearly two-thirds of Americans, at least, keep their phone close by overnight, too, or so says Gallup.)\n\nThe graph opposite shows how it impacts on the competing strategies of footwear retailers in the US, for instance.\n\nAnd it's giving brands opportunities to engage in various ways. Mobile services from retailers focus on more than just mobile commerce or tactics that drive consumers to their physical locations (such as coupons). They also use mobile to get people to the store, motivate them once they're there, satisfy them after the purchase, and keep them happy with customer service.\n\nTo make mobility happen, the mantra is \"mobile first\".\n\nOLD MOBILE\n\nMobile is a new, shiny toy with little measureable impact and thus given little priority.\n\nMobile is regarded as just another communications channel added to the media mix.\n\nTraditional metrics are applied from standard digital-marketing tactics.\n\nEvery company needs an app, regardless of value to the customer or the brand.\n\nMobile-specific contextual information is not exploited.\n\nNEW MOBILE\n\nAdvanced companies create engagement platforms that are mobile-first, if not mobile only.\n\nMobile is regarded as a service touch-point that can deliver competitive advantage through customer-centric utility or entertainment content.\n\nThere are mobile specific metrics as standards that can be integrated into measurement frameworks.\n\nInvestment is more focused, aligning the brand's value proposition with the customer need.\n\nSelf-learning systems such as Google Now analyze different kinds of external data to provide contextual value.\n\nMobile is a very tough medium to compete on. By its nature, people spend less time on a given page, are often multi-tasking and switch between applications more often than on the web. It's not easy to get consumers to watch a video, read an article or browse choices. Production on mobile is harder than web, which causes strains on the technology teams. So it is important to understand that it is always more difficult to go mobile first than just adapting non-mobile content to fit.\n\nBut when building a complete content strategy, it is a minimal requirement to have content optimized for mobile. Many people think that optimizing for mobile is simply a question of reformatting the content so it is visually optimized (scrolling down as opposed to having to move the screen horizontally and vertically). But this is only a small component of what mobile optimization really means. Most apps rely on data hosted somewhere beyond the mobile device itself. So what happens when the phone's connection drops? Developers call this the \"airplane mode\" problem. An app needs to function with an intermittent data stream. It also needs to bring in data from multiple sources, many of which are processor and bandwidth hogs. Surfacing data from legacy systems into mobile phone apps in a way that is bandwidth conservative and fast is no simple trick.\n\nTHE RETAIL MOBILE LANDSCAPE AT A GLANCE\n\n_Mobile innovation opens doors, especially in combination with an immersive shopping experience. The more successful footwear retail brands know this, comparing themselves to the best in the business \u2013 like Starbucks and Amazon \u2013 who use mobile not just to drive sales, but to nurture relationships._\n\n_NPR is the well-respected public radio service in the US, and many in the States depend on it for in-depth reporting, analysis, and entertainment. However, it's not always easy to tune in to. NPR stations are local operations of varying power, and they choose which of the network's offerings to broadcast and at what time. This makes tuning into a favorite show a potential challenge. Bottle Rocket's NPR One app solves that by providing on-demand listening for all the great NPR broadcasts no matter where you are._\n\nThere are challenges on the consumer side as well. Google and Nielsen did a study to find similarities and differences between mobile-content consumption and web-content consumption. They found that 40 per cent of all mobile searches were performed with local intent. And this is increasing.\n\nMobile localization has more profound impacts in the developing world. A mobile service called Tone provides connectivity to underserved regions. In Indonesia, Tone gives local fishermen a kit with a phone, educational material and carrier access with subsidized data. Eventually the service will pay for itself for carriers and users alike as fishermen learn more as a result of services like GPS, weather and fish maps.\n\nMobile first should always try to localize the content and make the content useful based on a variety of available data points. Think of space and time before simply reformatting any old available content. And align content with local context to increase its relevance.\n\nCalvin is a techie who likes pencils. His advice is always to start with a pencil, and a brutally clear understanding of what the small screen allows you to do. He calls it Lo-Fi techniques to achieve Hi-Fi solutions. Here's why:\n\n1.If you use computer design tools, the tool starts to impact what you do. But if it's your hand, some lead, some paper and you, then there are no rules.\n\n2.You're never going to get it right on the first try. So lo-fidelity tools like pencils and whiteboards enable you to easily let go of ideas that aren't good enough.\n\n3.You can work in public. Everyone from Bottle Rocket does their sketching on the walls now so that they can collaborate easily and with better interpersonal dynamics.\n\nThat done, scale becomes the Holy Grail. I have to tell you: the graveyard of small-scale apps is over-subscribed.\n\nWe have found there are four passports to scale in designing programs with mobility at their heart.\n\nThe first is to provide frictionless convenience. Mobile apps that make things more automatic, more seamless, and faster, help people get the things they need \u2013 without an intrusive upsell.\n\nThe second is to promise substantive gratification. Mobile can extend a product beyond its physical dimensions by integrating a real world and mobile experience.\n\nThe third is to hyper-personalize the program. Mobile devices are intimate companions, and like all such companions, we are willing to tell them a lot about ourselves.\n\nAnd the fourth is to give the customer an ultra-contextual experience. Phones know what time it is, where we are, what the weather is, and plenty of other contextual information. But, with a few exceptions like Google Now, brands tend to be parsimonious in using this data.\n\nAnd what of the future?\n\nWell, there are another billion people coming online by 2020, and they'll connect through mobile. Add to them some 25 to 50 billion new connected Internet of Things devices such as wearables, sensors, connected house applications, cars, infrastructure, and manufacturing equipment.\n\nThe current mobile infrastructure can't support the coming needs for data bandwidth. The services we're adding \u2013 such as video, virtual reality, connected homes and cars \u2013 require a massive increase in data.\n\n_Here's a tip: start with a pencil and let the technology follow. Lo-Fi first, Hi-Fi later._\n\nTechnology marches on, and 5G mobile promises to be the solution to these coming challenges, but 5G at global scale is still far in the future, and a hugely expensive proposition. You may faint at the size of your smartphone bill already, and that's without the investment in 5G infrastructure. For the foreseeable future, new services, content and applications will have to be created for current networks in order to scale them. And we need to find ways to make this affordable to everyone.\n\nGoogle and Facebook are already running initiatives to connect underserved communities, and while this serves their company-value propositions to connect people, don't forget that they are building commercially exploitable consumer segments to monetize.\n\nIf we're to do this right \u2013 and I hope we are \u2013 we'll need to find collaboration models among carriers, brand platforms and consumers that distribute the benefits of connectivity instead of making just another cheap way to put out more interruptive advertising.\n\n**CONTINUOUS COMMERCE**\n\nThe ecommerce leviathans created by the big players are predicated on a very clear business model. As Jeff Bezos of Amazon put it pithily:\n\nThere are two types of companies: those that work hard to charge customers more, and those that work hard to charge customers less. Both approaches can work. We are firmly in the second camp.\n\n_Chick-fil-A \u00ae is famous for its chicken sandwiches... and its queues. To allow customers to bypass the line, the chain built an app to let people order and pay for food instantly. Walk in, pick up, even sit down in the restaurant and eat \u2013 all without a hitch._\n\nOf course, the first camp works because many consumers are prepared to pay more for more \u2013 which includes intangible things. Perhaps this choice is also not so binary. There's a time for shopping at Amazon; and there's a time for shopping somewhere else. And brands that sell through Amazon understand that there isn't a point where they stop and Amazon starts, they know that Amazon is not a commercial silo completely apart, and that there is still a continuity of marketing, right along the distribution path.\n\n_Lego, a brand once strictly limited to analogue play, has greatly expanded its reach in recent years. Even before the Lego Movie delighted audiences worldwide, the Lego style of imaginative build-it-yourself fun had made its mark in console gaming. But integrating actual Lego play with the digital environment is another matter, one solved by Lego Mindstorms EV3. With them, a child can use a mobile device as a canvas to imagine what to build. The app translates that vision into a design which, when built, can be piloted via the mobile device as a remote-control robot._\n\nFIVE MOBILE USER EXPERIENCE MISTAKES BRANDS MAKE\n\n1. Brands often mistake mobile to be another digital channel when in reality it is fundamentally different.\n\n2. Brands are not geared for the speed of change associated with mobile \u2013 not just the technology but how users interact with it.\n\n3. Brands do not take full advantage of the analytics and insights that mobile can provide to build personal relationships with consumers.\n\n4. Brands do not recognize the personal nature of mobile and the level of relevance the right mobile tools can give users.\n\n5. Brands have not realized how much more decentralization is coming from mobile. Mobile experiences will evolve to consist of more micro interactions that will better serve the moment and context of the user.\n\n_Starwood Hotels took a different path, aggregating contextual data to improve travel. SPG Keyless is a mobile app that automatically checks you in and out of your hotel room, based on your booking time and geographical proximity. Bluetooth connectivity gives easy access to your room, as well as to elevators, pools and saunas. With push notifications to keep you up to date, guests can bypass check-in, check-out, and the front desk entirely! Most important of all, the app only works if customers sign up to the SPG loyalty programme, meaning a host of data is being collected to personalize the experience further._\n\nAmazon is more than just a retail juggernaut. It's becoming the shopper's main source for information. In fact, Amazon \u2013 not Google \u2013 is the leading tool mobile consumers use to compare prices, especially when they are \"showrooming\" (viewing a product in store while looking for a better deal online). A showrooming study has found shoppers opting for Amazon twice as often as Google. Marketers and sellers need to grapple with the implications of that.\n\nWhy ecommerce, anyway?\n\nDigitally enabled sales now account for about 8 per cent of all US retail sales, more than 14 per cent of UK sales, and about 12 per cent of Chinese sales. Those numbers are expected to grow in all three markets, as fast as nearly 20 per cent year-over-year in China. It has become simply unarguable that most retailers must do business in both the physical and digital worlds. Giving consumers different experiences in each is hardly to be recommended. They want a seamless experience. In fact, some 80 per cent of shoppers have said they are more likely to become loyal customers of a seamless cross-channel experience.\n\nSo the question is: if the digital and physical worlds are merged (which they are) and the consumer will reward a seamless union between them (which she will), why are we dividing up the world into ecommerce, mcommerce and traditional retail?\n\n\"...commerce is where you are at any given moment when you decide to search, shop, compare or buy.\"\n\nIt does not make sense.\n\nI'm a strong advocate of kicking the \"e\" in ecommerce into the dustbin of lost labels.\n\nCommerce is no longer sharply delineated in time or space \u2013 you're not in one mode while you are at home and another while getting to work, and in yet a third when you finally walk into a store, flip on your computer or turn on your phone (as if it's ever off anyway).\n\nNo, commerce is where you are at any given moment when you decide to search, shop, compare, or buy. Commerce is everywhere, all the time. It is continuous.\n\n_Amazon might be the giant of ecommerce, but it's a very nimble one. The company continuously innovates on its technology, media and logistics offerings to complement \u2013 and often supplement \u2013 its retail platform. It also outperforms Google as the authority for comparison shopping online._\n\nSo many ecommerce solutions don't answer consumers' needs. They only solve part of the problem. Some involve robust technological solutions but offer poor customer and brand experiences. Some offer beautiful imagery and storytelling but fail to deliver a reliable and scaleable technology infrastructure. Many commerce solutions are uni-channel, when, in fact, over 80 per cent of consumers' purchase journeys involve at least two channels.\n\nEven brands that have managed the transition from bricks and mortar to online well still get things wrong. Slow website load times, an absence of mobile optimization, indeed, anything other than a seamless integration of devices, has a big impact. Researchers at Aberdeen Group suggest that companies with a strong omni-channel presence retain 89 per cent of customers, compared to 33 per cent retention for those retailers with a weak omni-channel presence.\n\nMany fail because they begin and end at the transaction, embracing only part of the full commercial relationship with the customer \u2013 a relationship that begins well before the sale and continues well after the first transaction. In most relationships, the first transaction is not profitable; all the money is in the lifetime customer relationship.\n\n_Showrooming \u2013 where people try in-store but buy online \u2013 is very convenient for the customer but potentially challenging for bricks and mortar retailers._\n\nHOW TO MAKE THE MOST OF AMAZON\n\n**The basics:** Ensure that your product is represented on the site with accurate content and imagery that's optimized for Amazon's search engine, and that your inventory and distribution are set up to handle the channel's demands.\n\n**Mind your reviews:** Amazon's product reviews indicate consumer intent, user experience, and propensity to recommend. Compare comments on your products with those about your competitors. Check geographic and demographic data to inform segmentation strategies.\n\n**Pricing tips:** Amazon is the top source of information when a consumer is showrooming because of price and convenience. Amazon 1-Click turns a search into a sale with a screen tap. When you sell your product on Amazon, your competitors are both on- and offline; factor in this new consumer behaviour and exposure when pricing your product.\n\n**Understand the shopper:** Why does she use Amazon? What's important to her? How can you take advantage of Amazon's tools to meet her needs?\n\n**Drive traffic:** Learn your options, both on and off Amazon, for getting likely shoppers to your Amazon store. How can you fill your funnel with shoppers who are primed for the Amazon experience?\n\n**Check your numbers:** Challenge the data Amazon provides, and build your own set of KPIs. Distill both data sets into an easy-to-act-on dashboard.\n\n**Spend wisely:** Measure and adjust your tactics for driving traffic and conversion, and fine tune your marketing spend. Compare and contrast with your competitors' on-site initiatives.\n\n**Optimize your offer:** Now that you have some data, ask which of your products are doing well? How can you optimize products and packaging for the channel to do even better?\n\n**Use all available tools:** Use joint CRM programs and Amazon Vine to identify shoppers likely to favourably review your product, and incentivize them to do so.\n\nMy colleague Rory Sutherland reminds me of a speech David Ogilvy made to the Life Assurance Agency Management Association in 1965. David said:\n\nI myself have life assurance policies with three companies. Not one of them has even written me a letter suggesting that I buy more insurance from them. All they ever send me is premium notices. Bloody fools.\n\nWhat David was doing was proposing direct mail as part of an omni-channel tactic to meet a perennial opportunity \u2013 namely, that it is easier to cross-sell or upsell than it is to cold-sell. This is \"direct marketing\", and in many ways it anticipates the world of continuous commerce in its principles, though some of the disciplines that went into it, most notably testing, have tended to take an unfortunate leave of absence. As Rory puts it so well:\n\nThis form of advertising, though historically less celebrated, still holds great lessons for the digital age. It is based on continuous, responsive interactions, and incremental improvements rather than on fire-and-forget campaigns. It is stochastic rather than deterministic in its approach. And, at its best, it tries to operate according to the timescale of the consumer rather than at the behest of the advertiser. It is as much about target moments as target markets. And it aims as far as possible not to generalize but to disaggregate.\n\nThere are three pillars on which continuous commerce can be built:\n\n**Omni-channel**\n\nWhat does it mean? Very simply: continuity for the brand and the consumer across multiple channels, devices and locations to enable sales any time, anywhere. There are innumerable consumer entry points to commerce (social, web, mobile, kiosks, and more), and it is proven that the more opportunities you give a consumer to buy, the more she or he will. It could be argued that _all_ marketing communications elements should carry an opportunity to buy, be it a web banner, a kiosk, even a billboard.\n\nHere's how to do it \u2013 and do it very well indeed.\n\nAdidas is on the front foot with omni-channel, seamlessly integrating offline and online in both directions. Moving customers offline to online, the brand's \"endless aisle\" concept allows customers to browse additional stock online where not available in store. Meanwhile mobile app users get custom alerts on the launch of a new sneaker model, allowing them to check out different variations, before reserving online and collecting in store. Importantly this isn't an adhoc approach, the company is integrating stores globally into its bullish \u2013 \"impossible is nothing\" \u2013 omni-channel model.\n\nIncreasingly, it is the phone which is the hub for continuous commerce, the place where all the channels converge.\n\nIt follows that social media (supporting commerce, but not, please, \"social commerce\") offers a particularly acute portal to satisfying the psychology of shopping. As consumer psychologist Paul Marsden has pointed out, it can harness our human capacity for social learning. Social psychology teaches us that shoppers indulge in \"thin-slicing\" \u2013 screening out most information, and retaining thin slices of important information kindled by cues, along with a set of pre-programmed rules. These latter are the heuristics.\n\nFor instance, one heuristic concerns the benefit of following what others do. _They_ can't be wrong. This is all about social proof. Our Brazilian client \u2013 Magazine Luiza \u2013 is at the leading edge. Frederico, the CEO, knows the power of proof. He has launched social buying sites which perfectly match the Brazilian passion for social networking to the high penetration rate of direct sales. The concept was Magazine Voce (Your Store). It allows Facebook users to select up to 50 of their favourite items. If a friend makes a purchase based on one of these recommendations \u2013 showing proof \u2013 Magazine Luiza fulfills the order and pays the referring friend a commission.\n\nFor brands, continuous commerce says that a brand should not behave differently in a marketing mode to how it behaves in a sales mode. That often has not been the case, as different owners in the organization had responsibility for each. Those days of extreme silos must be numbered, though, in my experience they linger still to a surprising degree.\n\nA brand where they've been collapsed \u2013 and one where it is mission critical that this should be so \u2013 is Huggies. Do you know how many diapers in the US are sold through online retail? In 2017, it is estimated to be 22.4 per cent of all sales, according to Tabs Analytics. They found that ecommerce diaper sales grew even amidst a contraction in the whole category.\n\n**Relationship**\n\nThere's a word that's missing from hardcore ecommerce, and that's emotion \u2013 and for good reasons. After all, it's all about operating as an efficient mart. But I believe that emotion is an ingredient of continuous commerce.\n\nIf you think that's a trifle soft for this hard-nosed business, then consider this.\n\nWe have a partner \u2013 Motista \u2013 who has taken a big data approach to emotion. They've spent seven years looking at over a billion emotional data points for a sample of 1.2 million US customers from over hundreds of retailers. What they show, conclusively, is that emotional connection is the glue that holds omni-channel together. As Scott Magids, Motista's CEO, points out: \"we want to shop at retailers where we feel we belong with other people.\"\n\nThe \"what\" of commerce is easy to define, but it's the \"why\" that gets emotional: that sense of belonging but also sometimes of escape, or of indulgence, or as Magids says, reinforcement of the family.\n\n_The \"endless aisle\", a seamless journey where online and offline blend to offer customers an integrated retail experience, is omni-channel done correctly by Adidas._\n\nMotista's big data on emotion can shed illumination. To assess the financial impact of emotional connection and convergence, we segmented the landscape of retail customers into three groups:\n\n1.Satisfied customers who shop and buy in-store alone.\n\n2.Satisfied customers who shop and buy across the converged experience.\n\n3.Emotionally connected customers who shop and buy across omni-channel experiences.\n\nWe used these segments to answer two questions:\n\n1.What is the ROI of moving a satisfied in-store customer into omni-channel?\n\n2.What is the ROI of delivering emotional connection through the omni-channel experience?\n\nThe answer is \"a lot\".\n\nFor luxury retail, the annual spend for satisfied in-store customers goes from $637 per capita to $1,157 for satisfied omni-channel customers. That near-doubling of spend puts paid to the idea of ignoring omni-channel.\n\nGoing the extra mile to create emotional connection is even more attractive. Emotionally connected omni-channel luxury retail customers spend $1,640 per capita. The benefit of omni-channel emotional connection is a staggering 257 per cent increase in sales per customer.\n\n_Patagonia is a brand so committed to sustainability that it gives people reasons not to buy its own products! What might be damaging for others has become a competitive advantage \u2013 consumers make more informed choices, which influences product marketing decisions for the company and engenders a strong following among customers._\n\nLuxury retail is not an outlier. This pattern repeats itself in discount retail, the part of the retail segment that has the lowest rates of emotional connection so far.\n\nSometimes, it can be quite small things which trigger the emotion.\n\nTake Patagonia. \"Patagoniacs\" have a very personal relationship with the brand. They can smell phoniness, and are quick to call it out. On patagonia.com, the brand displays this authenticity from the moment customers hit the homepage, with stunning photography of athletes experiencing the natural world using Patagonia outdoor gear. Since people who live these adventures capture the vast majority of the photos, they are always accompanied with credits. But it is the wildly popular \"Notes from the Field\" essays within every print catalogue which pay the biggest natural dividend. While this may seem a relatively insignificant addition, it adds hugely to the authenticity of the brand, and demonstrates an attention to detail and transparency that is woven into the entire experience.\n\nThese authentic moments are not limited to great photography or writing. The digital channel continues brand storytelling in a consistent way. The Tin Shed e-zine, for example, is a dynamic microsite inspired by the tiny outbuilding in which rock climber and founder Yvon Chouinard started his piton company in the late 1960s. The shed's interior is photographed as it looks today, out behind corporate headquarters in Ventura, California. The multimedia interface places visitors in the middle of the room, and 360-degree technology allows them to turn and face each wall. Interactive objects and preview panels hint at the dozens of videos, images and articles from the latest dispatches sent by Patagonia ambassadors around the globe. While the primary goal is to engage and inspire visitors without distraction, they also can find links to share and can visit the product catalogue at any point.\n\n_Yvon Chouinard is the archetypal leader \u2013 a true outdoorsman who guides his company, employees and loyal customers with his moral compass. The values of Patagonia, like those of Chouinard, place environmental sustainability above all else._\n\nPatagonia has long been an industry leader when it comes to social and environmental responsibility. Corporate transparency is a significant part of that commitment, which leads to self-examination and improvement. In addition to supporting dozens of ongoing CSR initiatives, the company encourages employees to spend a month of every year volunteering for causes they believe in, and gives 1 per cent of sales to support environmental organizations around the world. So when designing the user experience for patagonia.com, we knew not to bury this content in some corner of the About Us page. It had to live front and centre, in the primary navigation.\n\nAnd when an important issue or piece of news needed to be broadcast, the brand was prepared to release the landing page entirely. In the Footprint Chronicles, visitors are invited to take an unprecedented tour of the environmental and social impacts inevitably involved in the garment-manufacturing process. And as a brand that seeks to do \"the least harm\", Patagonia launched the Common Threads Initiative with eBay, encouraging consumers to buy second-hand gear or sell gear they no longer use \u2013 thus extending a product's lifecycle as long as possible.\n\n_Patagonia's commitment doesn't waver even out in the wild. Its advertising simply restates the company's firmly held beliefs and weaves together passion for the outdoors, demand for high-performance gear, and concern for the environment \u2013 resulting in a sustainable and highly profitable loop._\n\n**Experience**\n\nThe experience of continuous commerce must delight at every touch point. Put bluntly, if I buy something online but want to wear it tonight, I expect to be able to pick it up at the store. If they don't have my size at the store, I expect it to be shipped by the sales clerk at no cost.\n\nOf course, I could expect more. I could expect advice before I chose. Hyper-personalized customer service can do that.\n\nHyper-personalized customer service is a pillar of Continuous Commerce. Based on my engagement loyalty, brand interaction and purchasing history, the offer can be personalized to me (versus a simple tiered set of criteria) and provided in a spectrum of ways.\n\nThe generality, however, is that digital channels have often lagged traditional ones in shopping experience. What research we have undertaken does show considerable dissatisfaction: some 23 per cent frustrated with websites (most of the time) and 25 per cent with mobile apps (most of the time).\n\n_On the high end of the continuous commerce spectrum is Kindle's Mayday button, which launches a video chat right in the app. A customer support person is immediately available to \"pilot\" a user through an experience by manipulating images on screen. Suddenly barriers between intent and action are removed. Imagine the possibilities! A clothing brand, for example, could provide you with an instant fashion consultant as you consider the sartorial requirements of your evening engagement._\n\nOften it is systemic issues which are at fault. For instance, as our partner Salman Amin from SC Johnson always reminds us, lots of the systems and processes underpinning an \"always-on\" commerce experience aren't actually always on, in the sense that they don't operate 24\/7\/365. This won't do.\n\nBrands need to face up to changing working practices and business processes, finding suppliers and services that can support uninterrupted service and creating workarounds for restrictions imposed by the IT system.\n\nThere are, however, some highly stimulating role models for beautifully delivered experience.\n\nThe more effective online shopping experiences (in terms of sales conversions) use multimedia in addition to static content. Video has become commonplace to showcase products. Tools that enable users to zoom in on details, change colours and further customize items are also frequently used. Live chat boxes pop up automatically, giving users the option to ask questions of a real person \u2013 not just to read bot boilerplate \u2013 to get a better feel for what they may buy. Luxury fashion brands such as Versace take the user experience to a new level, offering scheduled live feeds of runway shows with embedded digital mechanisms to purchase the products the user is viewing (or similar ones).\n\nStudies have shown all of these tactics drive up sales. Video is particularly effective. Research by EyeViewDigital.com has shown that 80 per cent of users click on video with fashion content, and that those who view the footage are 1.6 times more likely to make a purchase.\n\nFor the moment, it appears that \"too much of a good thing\" is impossible when it comes to multimedia online shopping experiences that are designed and executed well. With augmented and virtual reality moving into ecommerce, consumers are really in for a show, and online sales sites are bound to transform, and perhaps, find the tipping point that turns multiple digital devices into effective sales tools.\n13DIGITAL TRANSFORMATIONS\n\nThe Digital Revolution has upended almost every aspect of human experience: it influences how we relate to the governments who rule us and how we choose them; it influences how we spend our leisure time and where we choose to go; it influences our appreciation of social issues \u2013 and how they might be solved.\n\n_Ogilvy on Advertising_ gave the classic recipes for advertising tourism, politics, and good causes. Many of these have not changed much. But digital has transformed our ability to do all this in rather dramatic ways.\n\n_Simple posters like this that rail against corruption were supported by a novel variety of techniques, from holograms to animation._\n\n**DIGITAL POLITICS**\n\nDavid Ogilvy had a rule of no political advertising. Now and then \u2013 but not often \u2013 Ogilvy & Mather has broken this rule, only ever in an associated agency, and only on the basis that staff members are personally vested in it. We did so in the Indian General Election of 2014, through a subsidiary shop, and gave Piyush Pandey, Executive Chairman and Creative Director for Ogilvy & Mather India and South Asia, the space in which to create the work. In doing so, he gave creativity to a client who personally wanted it, Narendra Modi; but, more than that, it signalled how the whole genre is being influenced by the Digital Revolution.\n\n_India's Narendra Modi was an early social media user, and a first of his kind in politics. Dubbed \"King of social media\", his titles ultimately included Prime Minister. Despite the clamour surrounding President Donald Trump, Modi has 20 million more followers on Twitter._\n\nPolitical advertising has always been \"parent\u2013child\": the politician telling you what is good for you. Now the \"child\" can speak. The initial obstacle was to persuade the Bharatiya Janata party that the campaign should be centred around the individual. But once that was cleared, the way was opened for a Modi moment which was couched in the vernacular: using the language the voters used, and not the official, formal-sounding language of national television.\n\n_This anonymous Facebook page which commemorated the death of an activist was credited with starting the Egyptian Revolution of 2011. Today, two revolutions later, Facebook hosts views for and against both of them._\n\nModi himself had been an early social media convert, so the mood did not have to be invented. Piyush puts it this way: \"A man who's not supposed to be educated in the traditional, technological fashion adopted technology and digital. And that impacted the biggest movement that happened in our nation in many years.\" While Piyush and the associated agency did all of Modi's campaign advertising, when asked about digital, Piyush demures: \"Yes, I worked with him and his campaign, but the digital side he did on his own. And that's where I salute him.\" The Modi moment became a series of simple phrases for everyone, which then spread in social media, for those who had it, and later to those who did not have it \u2013 the icing on a very traditional media cake.\n\nCan social media be that cake? That was very much the view at the time of the Arab Spring. As one protester said: \"We use Facebook to schedule the protests, Twitter to coordinate, and YouTube to tell the world.\" That is a pretty good characterization of the role of social media. But it was not the \"social media revolution\" as much as a revolution where, at one (early) stage, social media galvanized action. In the later stages of these revolutions, social media has tended to fragment the majority, as each emerging faction quarrels with another. It is a tool: it serves multiple masters. The medium is not the agenda.\n\n_In 2012 our agency in Tunis created an event which was designed to be shared on social media, aimed at combatting apathy and dampening any desire to revert to dictatorships of the past. In this case, a large poster of the deposed dictator, Ben Ali, suddenly reappeared, disclosing a warning message when it was torn down._\n\nA very remarkable misunderstanding of the real power of digital social media is endemic to the standard treatments of Barack Obama's two presidential campaigns, which are often presented as the coming-of-age of new media and the displacing of traditional media. The real story is rather more complex.\n\n_The enthusiasm generated by the Obama campaign in 2008 is neatly symbolized by this barn in Ohio. An online activist requested, through the website, the correct Pantone colours and logo design, erected the necessary scaffolding and painted the whole barn in the Obama decal._\n\n_Like all great direct marketing, the Obama campaigns made much use of tried-and-tested techniques, such as member-get-member programmes, contests and A\/B testing. The famous \"I will be outspent\" letter underwent 17 versions in tests before it was sent to the broad list. And it raised $2.4 million in one day._\n\nIt began in 2004 with the campaign for the Democratic nomination of Howard Dean, when the tail end of the anti-war protest converged with the hangover of the dot.com bust. One of the young survivors of the latter was a charismatic whiplash of energy, Thomas Gensemer, who, along with Joe Rospars and other founders of the firm that became known as Blue State Digital, played around with email marketing and database management to help cohere an anti-war coalition which spanned young and old. The money flowed in to the Democratic campaign: and that, as Thomas says, \"sort of changed the way that we all saw the economics of campaigning\".\n\n_The Trump campaign, best known for its Tweets, used Facebook ad data analysis to target selectively and effectively \u2013 narrowing the fundraising gap and mobilizing support where it mattered, much more analagous to the Obama campaign._\n\nThey applied a similar strategy to Obama 2008, and insisted to David Plouffe, the campaign manager, that their role was central to the whole ecosystem of the campaign: the fundraising, the marketing and, most important of all, the field organization. He agreed. They started at the beginning of the campaign cycle, with an inspirational candidate, and harnessed the energy of Facebook and Twitter. But the hard work \u2013 the important work \u2013 was traditional database management: merging the donor database with the field-activist database with the voter database. As Thomas says, \"At that point you suddenly begin to look at people in multi-dimensions, whereas before they were handled incredibly transactionally.\"\n\nThe money flowed in again: $500 million online in 2008, $690 million in 2012. Where was it spent? On traditional TV, giving a huge share of voice advantage. This is a particularly poignant part because in the 2016 election the insurgent Donald Trump relied on a huge earned media advantage: \"authenticity\" fed out via Twitter.\n\nBut, more importantly, millions of people got vested in the campaign. The manager of a local field office in Cuyahoga, Ohio, could see that voter X who had given a few hundred dollars and knocked on some doors was a remarkably valuable target to create a whole experience around, with one aim \u2013 getting his or her labour. So the money financed the biggest field operation ever \u2013 tens of thousands of paid staffers, millions of volunteers, all singing from the same playbook \u2013 and socially enabled. (One technical note for non-American readers: the US \"Voter File\" and the long campaign cycles do make this recipe difficult to export in full \u2013 although it has been much demanded from all shades of the political spectrum in many countries.)\n\n_The Trump campaign unearthed a video of a 1996 speech by then First Lady Hillary Clinton. In the closing days of the campaign, this video was the centrepiece of a Facebook-first voter suppression campaign designed to suppress African American turnout._\n\n_The Clinton campaign spent time and effort on \"creative\" programs which talked to the young, and neglected the blue-collar voters she needed to win over._\n\n**The full story of the digital US Presidential campaign in 2016 is still emerging but by now some of the ingredients are clear:**\n\n\u2022Hillary Clinton had a massive fundraising advantage: over $1 billion compared to Trump's $650 million.\n\n\u2022Donald Trump's recipe of tweets and rallies earned a huge media advantage.\n\n\u2022Clinton's digital team focused on stylish and sophisticated Facebook videos aimed at young voters.\n\n\u2022Trump's digital director, Brad Parscale, ran a more traditional data-driven campaign, with considerable fundraising impact, and deploying sophisticated analytics, learning all the time through testing.\n\n\u2022This was accompanied by some highly targeted attempts at voter suppression of key Clinton supporters, African Americans for instance.\n\n\u2022Clinton eschewed traditional advertising at the end of the campaign and upweighted her Facebook presence. Investment in traditional media in critical swing states such as Pennsylvania and Wisconsin was discussed within her campaign, but never implemented.\n\n**DIGITAL GOVERNMENT**\n\nE-government started off with the simple \"technologizing of government\", but it took time for officials to understand that there then was a bigger game: redesigning governance \u2013 to put the user, the citizen, first.\n\nThe poster children for this are two small countries: Singapore and Estonia. In the case of Singapore, the intelligent island drove a \"digital transformation\" of its relationship with Singaporeans in a top-down, public service plan. By contrast, Estonia's was a decentralized program, with government as the catalyst and strong investment from the private, especially banking, sector. One ID and password for a citizen grants access. Both countries have easy-to-use interfaces, and high levels of user satisfaction. Estonia pushes the frontier a bit more: in just 20 minutes, you can register an EU-based company, for instance.\n\nBut small is easy; big is hard. And there has been no more poignant example of just how hard than the Healthcare.gov fiasco, when the US government set up a digital hub for Obamacare insurance \"marketplaces\". As someone remarked, it was the first time in history that a president had to stand in the Rose Garden and apologize for a broken website. Subsequent official enquiries revealed a series of reasons for the failure, all of which should give any of us the chills:\n\n\u2022A team convinced that they had to design the perfect solution, but who didn't leave enough time to implement it.\n\n\u2022Rigid procurement policies leading to the selection of the wrong type of contractor.\n\n\u2022Poor oversight as a result of divided leadership.\n\n\u2022Resistance to bad news and excessive \"path dependency\": we must land the plane or else.\n\nThere but for the grace of God...\n\nIn the aftermath, the US turned for help \u2013 surprisingly perhaps \u2013 to the UK's GDS (Government Digital Service), a nation that was a relative latecomer to e-government. This had been created as a result of a recommendation from Martha Lane Fox to create one body in the Cabinet Office (where Richard H. Thaler's \"Nudge Unit\" of behavioural scientists also sat \u2013 see Chapter 15), with the teeth to \"challenge any policy and practice that undermines good service design\".\n\nMy former colleague Russell Davies became its first Director of Strategy and describes it as \"what you'd get if you'd take a small government department and added a web development shop and a small creative agency.\"\n\nIts basic premise was that by making things more human, you make them more efficient. It's probably become the best example of how best to put user experience at the heart of government service, and it totally changed the basic process for something as simple as registering your vehicle or as complex as getting power of attorney. For 20 years, governments have basically put paper-based service online. But why not redesign the service?\n\n_If you go to the e-Estonia showroom, you'll get a vision of the future as interesting as any Apple Store. In fact for 100 Euros, you can become an e-Estonian; and enjoy the digital ID card just like a real Estonian._\n\nAt the centre is the website of GOV.UK, bare, clear and simple, removing everything that gets in the way of fast and easy access to information. Russell stopped a project that would have designed it with \"bells and whistles\", clever icons and gimmicks. The goal is not to be digital: it is to find the simplicity that digital can enable. A mantra became: \"the product is the service is the marketing\", and Russell, a former planner, became a kind of anti-planner, an un-marketer.\n\nOf course, this may be too purist a utopia. And governments do use all the devices of both analogue and digital communication in pursuit of their policy aims. But the impulse for better and better user experience \u2013 impelled by the \"Uberization\" and \"Amazonification\" of commercial transactions \u2013 is only going to grow.\n\n_In the UK, the GOV.UK website was labelled \"boring.com\" by a popular newspaper, but swept the award shows, including a D&AD Black Pencil. Russell Davies, Director of Strategy at the UK's Government Digital Service, had radically streamlined the website for GOV.UK to eliminate everything that got in the way of easy access to information. Its stripped-down simplicity echoes classic government information campaigns of the past \u2013 as does its use of one consistent, restrained font._\n\n_Meanwhile in the US, theHealthcare.gov website that launched in October 2013 got 20 million visits but could only effect 500,000 complete transactions, causing huge embarrassment. President Obama took the blame: \"Like any law, like any big product launch, there are going to be some glitches as this thing unfolds. That happens whenever you roll out a new program.\" In the aftermath of the Healthcare.gov debacle, the US turned to the UK's Government Digital Service for help. Less is more it seems._\n\n_Incredible !ndia created a distinctive identity for the nation; depicted familiar sites in an unfamiliar way; highlighted the scenic breadth of the nation; and linked people from all over the world to the welcoming embrace of India. As befits a nation both high-tech and rich in heritage, Incredible !ndia has always mixed traditional and digital media in its quest to boost India into the first rank of tourist destinations._\n\n**DIGITAL TOURISM**\n\nDavid Ogilvy devoted an entire chapter of _Ogilvy on Advertising_ to foreign travel. After all, he was known, as he put it, as \"the Grand Panjandrum of travel advertising\". The broadening and democratization of international travel took place in his lifetime: from just 25 million in 1950 to 435 million in 1990. Since then it has risen to over 1 billion per year.\n\nThe Digital Revolution since 1990 has facilitated the whole process of researching, buying and celebrating a holiday. The particular impact of communications is to enhance the whole experience \u2013 to bring the actual experience forward in the buying cycle, making it come as close to testing it as actually possible.\n\n_The State of Madhya Pradesh is huge, full of attractions, but was never a first-choice destination. Ogilvy used \"ombromanie\", the traditional art of creating images by hand shadows, to show that you can evoke sites and places without having to use photography. This hugely successful \"non-tourism\" campaign was backed by a rustic jingle sung by MP local actor, Raghubir Yadav._\n\nOne of the most successful tourism campaigns of the digital era has been \"Incredible !ndia\". Faced with a bundle of hostile perceptions \u2013 sweaty, dirty, nasty \u2013 this campaign turned them on their head. Through beautiful photography, it evoked an overwhelming experience: different, yes, but stimulating the senses not discomfiting them. In his autobiography, _Pandeymonium_ , Piyush Pandey writes that his mother used to tell him she liked to travel, to which he replied, \" _Muijhe ghumaane ka bahut shauk hai_ \" (I like to make people travel). The programme was always digital as well as analogue: creating buzz, distributing offers and offering a digital home for the individual programs of Indian states. Piyush advertised two of these as well.\n\nSocial media now provides us with the ability to pre-experience a destination. Who doesn't relate to the conflicting feelings of envy and joy by association that accompany a feed from a friend's Facebook holiday? You want to feel like you're there with them. We helped the city of Cape Town harness those sensations explicitly by inviting people (through extensive digital advertising) to send their Facebook profiles on a five-day trip to the Mother City in exchange for some positive feedback from friends and a chance to win the trip for real. Cape Town got increased awareness for its beauty and hidden gems as well as the benefit of an implicit peer recommendation.\n\nIt was surprisingly simple. Since Cape Town tourism didn't have buckets of money to spend on splashy TV commercials, they chose to hijack the place where everybody already goes to share their holidays: Facebook. Sign up, and you'd be able to virtually explore Cape Town in a five-day holiday. Users would get personalized content in their timelines that showed them all the different things they could see and do in one of Africa's most beautiful cities. Creating all the that content meant the Cape Town Tourism shot 150 POV videos, took 10,000 snapshots, and wrote 400 status updates. As a result, people got to visit Cape Town through the eyes of their Facebook profile... and all their friends did too.\n\nA country is the collective of its citizens. What if they were all asked to talk about their own country? That's what our amusing Random Swede campaign did for Sweden, a country where tourist revenues were reeling from recession. The campaign, which capitalized on Sweden's pride of place and historic commitment to freedom of expression, provided prospective tourists with direct access to 5,000 random Swedes who had signed up for the \"Dial-a Swede\" international telephone hotline. There is no better way to experience a country than by talking to its inhabitants, after all.\n\nOne hundred and ninety thousands calls came in from 186 different countries and 36,000 Swedes signed up as ambassadors. And what prospective tourists found out pleased them. They learned that Swedes are open and direct-talking people and are happy to answer any questions you may have about Sweden, that there is more to the country than its major cities, that Swedes love their country and would love for people to visit.\n\nIn _Ogilvy on Advertising_ , David asserts that the main purpose of advertising for good causes is to raise money for them. To do that, it has to tell people something about those causes. Nothing changed, except that we can do much more digitally. We can mobilize people behind things in a way that engages them personally, and that creates a range of behaviours \u2013 from simple support to more active endorsement to committed advocacy. The traditional divisions between public relations (issues management) and advertising (information promotion) collapse; the issue potentially becomes ownable by anyone.\n\n_Send Your Facebook Profile to Cape Town hoped to trigger consideration of the destination by potential travellers. Cape Town is far away for many people in the world, meaning that its beautiful sites remain unknown. People may have heard of the wonders of the Mother City, but nothing compares to seeing Table Mountain, the vineyards of Stellenbosch, or the V &A Waterfront for yourself. And if you can't make the journey, maybe your Facebook profile can, letting you see all the fun that could be had on a five-day excursion. The carefully constructed virtual tours did the job well. Bookings for Cape Town holidays in the year after the campaign jumped by 118 per cent._\n\n_How do you get people interested in visiting your country? Have them pick up the phone and dial a citizen directly! Your questions about Sweden, answered directly by a Swede \u2013 that's the way to find out about a place. Thirteen thousand calls were received in the first week of the campaign alone._\n\n_Another approach? Create the most appealing job posting in the world. Imagine the role of \"Island Caretaker of a UNESCO world heritage site\", which comes with a $100,000 salary and a job description including swimming, snorkelling, and making friends with locals. Tourism Queensland invited applicants to do just that \u2013 and over 1.4 million people from 200 countries submitted video applications online._\n\n**DIGITAL SOCIAL RESPONSIBILITY**\n\n\"Corporate\" social responsibility has been transformed in the digital age from a kind of high-handed philanthropy, providing financial support to \"good\" causes, to a deeper engagement with issues, which involves all stakeholders \u2013 from staff to customers to the supply and distribution chains. They no longer exist in silos and do not any more rely on glossy brochures for their information, but connect and interact digitally.\n\nPerhaps the most ambitious example of this I have been part of is the effort of Unilever to put environmental responsibility at the heart of everything it does. When Paul Polman became CEO he made the dramatic claim that he intended to double the group's revenues while halving its environmental impact. I'm not sure Paul ever did a lot of calculations before he formulated that equation, but with a gritty zeal and evangelical fervour he set out to show that it was actually possible. What it meant was an alignment behind the vision which involved everyone in the company. Some moved along faster than others, and I remember Paul saying that the challenge always lay in the middle of an organization. But soon it became very clear in Unilever that the purpose was to be rooted in the most basic product philosophy \u2013 and then fan out with everything the company did.\n\nWhether it was a case of buying sustainably sourced palm oil for ice cream or raising free-range chickens for eggs for mayonnaise, a demonstrable benefit could be shown, which, when accumulated over billions of consumption movements resulted in real, scaled impact.\n\nFarewell to the Forest _: an ancient tree speaks \u2013 \"Never in my 117 years did I imagine I'd come to this. Ironic as it may sound, it might be safer for me in the city than in the rainforest. You are the only living being that can help, and seeing your faces I know that you will.\" Unilever communicates that it is committed to making products sustainably without harming our forests._\n\nWhen Keith Weed briefed us on the communications eco-system which was to support the Unilever sustainable living plan, he was very clear that it should break precedents, signal a bright future, and create a specifically digital mobilization of all who touched the process \u2013 from the producers and growers to the end consumer. The result was an eco-system of content, from issue raising videos which generated millions of views and shares to in-depth editorial material.\n\n_Unilever's many brands took significant steps to create the bright future articulated by Paul Polman and Keith Weed. By 2016 the impact of those actions could be seen and measured. The sustained momentum of Unilever's sustainability initiative led the brand to prophesize that at last the end of the old world was upon us and showed how the CPG giant's brands were ushering in a brighter dawn._\n\nUnilever started to show that doing good in business is good for business; and the goodwill it generated in society at large played a part in its own defence when it became the object of an attempted hostile take-over bid in 2017.\n\nSaving the world is a big thing, but the same principles apply whatever the social task; you first raise the status and importance of the issue; you then identify your supporters; and then you mobilize them into action. It is the power of mobilization which distinguishes the digital contribution to social responsibility. At its most effective it can mobilize governments to change policy. One of David Ogilvy's favourite charities was the World Wildlife Fund, and we still work for them in different places. In Thailand, we created a programme against the ivory trade. It involved celebrities and ordinary individuals removing the letter of the alphabet which means \"elephant\" from their names. Very gradually, an army of 1.3million people was mobilized. The Thai Prime Minister felt impelled to call a press conference at the height of the campaign to declare his support for a crackdown on this appalling trade.\n\nDIGITAL SOCIAL RESPONSIBILITY: THREE RULES\n\n1.Be extremely sensitive to anything which remotely smells of \"green-washing\" \u2013 of insincerely stealing the clothes of an issue.\n\n2.Be prepared to shock the audience into understanding that your issue is important.\n\n3.Have a clear ask: what you want people to do, and, if they do it, how it will improve the issue.\n\nWhen same-sex marriage was being debated in France, our Paris office teamed up with Google+ and the association that \"owned\" the issue, Tous Unis pour l'Egalit\u00e9, to hold the first same-sex marriage in France, using Hangout, Google+'s video-conferencing facility. French couples connected to a mayor in Belgium (where same sex marriage is legal) who conducted the ceremony remotely. Millions were engaged: and the anomaly between the two countries poignantly highlighted.\n\n_A shopper's best friend? This ingenious outdoor campaign for Battersea Dogs and Cats Home, an animal shelter in London, is one of the best examples of digital ideas in action. Unsuspecting shoppers would take a leaflet, only to be followed around the shopping centre by an unashamedly cute rescue dog in search of a new home._\n\nMobilizing support is the prerequisite for raising funds. In _Ogilvy on Advertising_, David Ogilvy asserts that the main purpose of advertising for good causes is to raise the money for them. At the end of the day nothing changes, and \"digital giving\" \u2013 enabled by an online giving platform \u2013 will take an increasingly larger share of the donations bowl. As digital communications became more sophisticated, it is clear that aggressive, unbridled fundraising can be counter-productive. Data protection legislation in some countries makes it illegal, too. Brave charities, such as the UK's Royal National Lifeboat Institution, which have moved to opt-in only communications, have not faced disaster so far. But this highlights even more acutely the need for the issue itself to have the salience which can only come from creative communications; and also to treat the donor with maximum respect. The donor experience is another, perhaps higher, form of user experience.\n\n_A brand can be a catalyst for significant social change by staking out an unambiguous position on a cultural issue. But when you're Google, you can do more than just that: you can actively enable change. In 2013, same-sex marriage was illegal in France. In fact, it was the subject of intense cultural wrangling, and the outcome of the fight was still very much in doubt. Google saw a way to make it possible for same-sex couples to gain social and legal recognition of their love and commitment while promoting their brand-new Google Hangouts video-conferencing service. Since same-sex marriage was already legal in Belgium, French couples could stand \u2013 via Hangouts \u2013 in front of the mayor of Marchin, Belgium to exchange their vows while showing the world how technology and society can come together to change us all for the better._\n\n_At its very best, advertising can do more than just make people think or change behaviour on an individual basis. Sometimes, great social awareness advertising can change a society's views. When the letter for \"Chor Chang\" \u2013 which includes the Thai word for elephant \u2013 began to disappear from the names of celebrities, newspapers and companies, Thai citizens realized the damage being done by the ivory trade. In the wake of the campaign, Thailand changed its laws, destroying illegal ivory stockpiles, closing dealers, and wiping out the trade in elephant ivory._\n\n_The UK's Royal National Lifeboat Institution is the first charity to commit to opt-in fundraising. It's too early an experiment to read, but is evidently creating a different kind of donor base, smaller but more committed._\n\nNESCAF\u00c9\n\nThe digital case histories that tend to achieve profile in the trade press have intrinsic sex appeal. They are often small, sometimes interesting, but not always consequential. So for this entry into the Hall of Fame I have deliberately chosen a much more complex and much less apparently sexy example of digital transformation. It's got a flavour of empire strikes back.\n\nWhen Nestl\u00e9, the ultimate multinational, went about driving digital thinking into the hearts and minds of its global networks, it did so with earnestness, discipline and missionary zeal. It was not a question of letting a \"hundred flowers bloom\", but rather the opposite. They did bloom in the early days of the digital revolution: thousands of campaigns, in hundreds of markets, none scaled, none learning from each other, the sum of many parts in no way realized the promise of the cumulative investment.\n\nIn 2011 Nestl\u00e9 set itself the goal of being the No. 1 fast-moving consumer goods player in leveraging digital and social media to build brands and delight customers. It hired Pete Blackshaw, an early word-of-mouth pioneer to be its global head of digital marketing and social media. Pete established DAT, the Digital Acceleration Team, as a centre of excellence, a catalyst for change, and, most importantly in Pete's mind, a vehicle to promote rapid scaling. An outpost in Silicon Valley provides a point of external focus; it monitors and interacts with the interesting innovators there. But the whole remit of DAT is very practical, very down-to-earth. It reflects Pete's belief in fundamentals \u2013 \"fundamentals remain fundamental\". Digital is seen as something which should exist at the heart of Nestl\u00e9's approach to \"brand building the Nestl\u00e9 way\". It's not an adjunct, or a silo apart, or an execution facility: it's Mission Control.\n\nA good example of pioneering digital transformation at a brand level is Nescaf\u00e9. When Sean Murphy, the indefatigable global brand champion came to the coffee SBU, it was not in any way a cool brand, albeit a critical one in the portfolio, and one that Nestl\u00e9 was determined (in the word they chose carefully) to invigorate. As Sean puts it, \"the brand that had been waking up the world needed to wake itself up\". So it should reach out to Millennials \u2013 or they would pass it by. It must also look the part.\n\n_Three images which show the re-invention of Nescaf\u00e9 as a beverage with the kind of presence usually associated with soft drinks.  \nAn iconic red accent dramatizes wakefulness.  \nAn internet lounge in Tokyo.  \nSome of the series of simple digital conversation starters which helped bring the brand to life again._\n\nThe design agency, cba, transformed an untidy hodgepodge of different visual identities and replaced them with a stylish red accent to the word mark and a modern red mug: icons made for the digital age. It took more courage than you might imagine to jettison the baggage: but, once done, it immediately signified that Nescaf\u00e9 was competing in a very different world of all beverages, including soft drinks. The Nescaf\u00e9 creative platform consolidated around the idea of \"it all starts with a Nescaf\u00e9\".\n\nWe consume 6,000 cups of it a second, which means half a billion a day. But these consumption moments are very different from those of soft drinks or beer. They are often much more intimate \u2013 conversational moments in themselves.\n\nBut how to turn inspiration into consumption? That became the task of our office in Germany. They have created a flow of Facebook, Instagram and Snapchat posts, disseminated in multiple languages to Nestl\u00e9 markets around the world which give the brand a voice \u2013 and which encourage conversations. At the same time, Nescaf\u00e9 has become the first major brand to move its online presence to Tumblr, de-emphasizing traditional websites and increasing its engagement with a younger audience. From 2 million sleepy followers Nescaf\u00e9 now has 40 million much more engaged fans \u2013 who are increasingly becoming online buyers.\n14FIVE GIANTS OF ADVERTISING IN THE DIGITAL AGE: Greenberg, Kagami, Nisenholtz, Jensen and Porter\n\nIn _Ogilvy on Advertising_ , David chose six giants he considered to have been the most influential in modern advertising. They were Albert Lasker, Stanley Resor, Raymond Rubicam, Leo Burnett, Claude Hopkins and Bill Bernbach. I have undertaken the same exercise for this book, and gathered together this digital pantheon.\n\nLike David, I've used what might vulgarly be described as \"creative chops\" as the defining credential for entry, although my five demonstrate creative flair in very different ways. After that, they each represent an important aspect of the development of digital creativity and digital advertising, and represent it so well that I hope their choice will be at least understood. In a few cases their names may not be widely known, but I think that makes for a more interesting selection. Of course, I very much welcome alternative suggestions.\n\n_Bob Greenberg has made his name as a digital leader by embracing the same philosophy as his long-time client Nike. Whatever the digital challenge, Bob will \"Just Do It\", and typically before anybody else. Small wonder that his agency R\/GA has played a big part in some of the most signficant digital innovations._\n\n**BOB GREENBERG**\n\nWhen you sit with Bob Greenberg, you feel as if you are in the company of an Elizabethan magus. It is a matter both of appearance and aura. The appearance is extraordinary: clad always in difficult-to-define garments of black, the only contrast coming from three massive bracelets of steel, and topped by a black, brimless cap, from behind which two streams of grey hair cascade to his shoulders. This is surely some kind of digital Dr Dee of the twenty-first century, lodged incongruously in a huge cube dropped by Norman Foster into the middle of New York's Hudson Yards, occupying the whole 12th floor of a vintage Brutalist office building, the size of two football pitches. On the shelves behind him are ranged scores of carved stone Buddhas, all meticulously chosen (and scientifically vetted through scan electron microscopy before purchase) from just one \u2013 relatively obscure \u2013 dynastic period, the Northern Qi Dynasty, ad 550\u2013577. Now he's migrating to new territory, the Northern Wei, ad 386\u2013534. The auras of collector and collected do seem to converge a bit, and when Bob speaks it is in a low, measured, gentle and serene tone which suggests (and demonstrates) superior enlightenment.\n\nBob's cube is a long way from middle-class Chicago in the 1940s, where he grew up as one of three highly creative children.\n\nHe still feels like a Chicagoan in New York, not least because of his addiction \u2013 this is not too strong a word \u2013 to architecture. But he's not an establishment Chicagoan. Born dyslexic, he's always regarded himself as self-taught. The dyslexia is a trait he's proud to share with a group of highly original and disruptive business people, such as Steve Jobs, Richard Branson and Charles Schwab. He recalls without rancour seeing his sister just fly through pages of reading. Instead, he noticed that he saw the world differently. It's a habit he's made a remarkable career out of.\n\n\"His great contribution to the digital age is that he has seen it from the view of production.\"\n\nHis great contribution to the digital age is that he has seen it from the view of production. He is a producer: and he has elevated that viewpoint into a philosophy which is both coherent and inspiring, and which emphatically believes in technology and creativity as a partnership.\n\nIt all started by watching his slightly older brother Richard join the studio of Pablo Ferro. Pablo was the Cuban-born graphic designer who became the all-time great designer of titles. He added visual style to films. The Greenberg brothers entered a world that was quite dazzling \u2013 Ferro, Charles Eames and Saul Bass. They got the creative bug themselves, and decided to set up shop. They didn't tell anyone, but that shop was below where they were living in an apartment building at 130 East 38th, between Lexington and Park \u2013 a space just 20 feet wide by 70 feet deep. They built a studio, then brought in an animation camera. Before then the term graphic animation did not exist. Soon they were into new premises, and doing hundreds and hundreds of opening titles and special effects.\n\nBob started to develop technology: \"I did it because that's one of my things that I do,\" he says. He did it across film, video and computer graphics. By 1982, the brothers had bought one of the very first licences of Berkeley Unix \u2013 and developed their own software. They also used a lot of silicon graphics equipment.\n\nOne day in 1993, Bob took a trip out to see Jim Clark, one of the founders of Silicon Graphics, beleaguered at the time because PCs and Macs would make his machines irrelevant. As Bob relates:\n\nHe was always using the F word, and he said, \"What the f*** are you doing here?\" And I said, \"Well...\", and I couldn't remember why we made the appointment and why I was out there. But eventually it dawned on me that I wanted to talk to him about interactive advertising. So Jim said to me, \"That's very interesting. I want to show you this thing.\"\n\nThat thing was Mosaic, the first widely used web browser, which he had been working on with Marc Andreessen. Bob was impressed, and when he came back to New York, he determined to get out of visual effects and into the interactive boutique space.\n\nBob's company, R\/GA, now in its third incarnation, became a rock star of the digital age. It was never mainstream: it always had a distinct take on advertising from the point of view of technology. It was the Cartier of websites. No one piece of work symbolizes the work Bob did more than Nike+ (see Hall of Fame case, pages 214\u2013215).\n\nSometime around 2013, an unusual presentation box arrived at R\/GA. It had a blazon engraved on it, comprising Bob's beret, his glasses \u2013 and a FuelBand. Inside was a specially created pair of shoes. It was one of those rare moments of unsolicited client gratitude (and rare enough in our lives for Bob to confess that at first he thought it a new product sample). Thanks for what? Well, for creating a digital platform, which engages more than 40 million customers.\n\nIt felt very different when R\/GA first shoe-horned themselves into Nike's roster in a very conventional way, winning a small piece of digital business back in 2001. Since then they have always surprised. Nike never, ever, quite believed they had the ability to be the next \"big thing\": but they did. The FuelBand was the iconic device that created the whole idea of the connected athlete. Eventually, Nike was talked out of devices, probably by Apple as much as anyone. But an owned platform of over 60 million users remains, built on an unusual metric \u2013 Fuel \u2013 that's easy to use, that celebrates success, and that dives deep into just what that actually means for you. And with Nike, too, this digital landscape is tying itself into the physical world \u2013 the stores.\n\nNike+ took a brand into a new space. And Bob thinks in terms of spaces. Bauhaus founder, Walter Gropius, wrote that, \"The objective of all creative effort in the visual arts is to give form to space.\" Bob started to like the Bauhaus at the age of 16. R\/GA's previous office was a homage to Gropius. It is perhaps no surprise that Bob's holiday home is being conceived of as a homage to Mies van der Rohe's Farnsworth House. That steel-and-glass box is a perfect articulation of German modernism. Its orderly framework reconciles art and technology: a place for the human spirit to flourish in the age of mass production. This \u2013 Bob's fifth house \u2013 is being designed by Toshiko Mori, improbably perched on the rocks high above the Hudson Valley. Because it's glass, you can't hang artwork. So, more stone Buddhas.\n\nThe current Norman Foster office is much more than a space. It's a manifesto. It sets out to connect the physical space with the digital landscape. At the heart of this space is the tech stack, housed in its bunker. There are no crazy effusions of wiring here. Bob has directed them artfully into perfect order. Outside, if you feel calm, there's a reason. Bob chose techno lighting: it is fully programmable to match the circadian rhythms. If you want to configure a conference room to cerulean blue, you can. And as the days become longer the light becomes warmer. A St Bart's sunset? You can have it. And in the \"waffle\" ceiling the light is diverted so it does not shine down on you and tire you, while those same waffles are clad in noise-cancellation materials. There's a space reserved for gyrokinesis and gyrotonics. A calming environment like this might seem excessively cushy, but of course, it's cunning. \"It's a systematic way of supporting people to get to the most important thing which is retention\" \u2013 retention in a world where the digital duopoly of Google and Facebook are hiring by the thousands in New York.\n\nWith this connected technology, you can collaborate internally \u2013 or with London or Shanghai. The money goes into the screens, not the walls, which are blank \u2013 except for what Bob has collected. It's a luxury to be able to collect and have enough space to display what you collect. Not many people could place their motorcycle collection in their workspace: but here are the prized BMW's, including the pride of the pack, a recently reassembled Rizoma of 1973, together with a veritable stable of racing Ducatis. A small museum of past technologies \u2013 from the typewriter to the Mac \u2013 would do credit to any design museum. And then there's the art. Bob has one of the largest collection of L'Art Brut \u2013 outsider art, self-taught, vibrant pieces, often hung as a series, splashes of anarchism in this orderly environment. Of course, there's a curatorial system, mobile enabled.\n\nBob's frustration with architects is that they see technology as structure. He sees it as the foundation of a collaborative environment. He can't understand why R\/GA's architect, Norman Foster, wouldn't see his famous Comcast Building in Philadelphia as a digital landscape as well; or, for that matter, why Frank Gehry doesn't see the same opportunity in his Facebook building. Bob's space is designed to make people enjoy working together. And not to feel big. He once read a famous _Harvard Business Review_ article, which said that, after the size of 150 people, every creative organization loses something. Here, the space helps it fight back.\n\nBob's \"collaboration gene\" comes from his time in film production. It is intensely collaborative \u2013 much more so than traditional advertising. It might take 1,500\u20132,000 people to work on special effects for a big feature.\n\nThat collaborative mindset merges with the urge to innovate. He defines his business model as 60 per cent agency, 30 per cent consulting, and 10 per cent incubator, the last of these having hatched some 40 chickens already. I admire him most perhaps for his 30 per cent willingness to take on the accountancy practices and the management consultants. He's taking work away from them because they have no creative spirit. He's fearless in doing so, but it is his quietness which convinces. He's simply never seen the competition as the Wieden and Kennedys of this world: he's quite happy for them to fight for share among the traditional set. They just have not got digital.\n\nBob once stayed at Touffou, where he also slept in the library, thumbing through the advertising classics and reading the marginalia.\n\nHis advertising heroes are Leo Burnett, Bill Bernbach and David Ogilvy. He saw them all dealing with a very similar disruption to the disruption of the Digital Revolution. In their case, the disrupter was TV. Their responses to it, like his, were fundamentally creative: realizing ideas for the technology of the time.\n\nBob's reinvented himself three times so far; or, I should say, reincarnated himself \u2013 first as a production house, then as an interactive boutique and now as an agency for the connected world. He says he believes in a nine-year cycle of reinvention. The Buddha of Hudson Yards seems to have quite a few cycles left in him yet, despite the fact that he is ostensibly \"retired\".\n\nNIKE AND NIKE+\n\nIn talking to R\/GA's Bob Greenberg about Nike, it is understandable why he's such a champion of the famed sports brand. The swoosh logo is the most internationally recognized brand mark in the world. And \"Just Do It\" is one of the most well-known slogans. Rap artists have name-checked Nike in their music 687 times over the last decade, trumped only by Gucci and streets ahead of rival Adidas. To say that Nike has become a part of contemporary culture is an understatement; in the kingdom of sportswear and apparel, Nike reigns supreme.\n\nNike has always seemed one step ahead of the competition \u2013 from its launch of an innovative \"waffle\" shoe in 1974, to the release of \"Air\" technology in 1987, to a prescient Jordan sub-brand deal in the 1990s. The company has a knack for combining leading-edge technology with style and grace.\n\nIn the 2000s, it was the turn of the Nike+ FuelBand to lead the brand's evolution. Just as Apple's iPod was by no means the first mp3 player, Nike did not invent the fitness tracker. Nevertheless, FuelBand kicked off the billion-dollar wearables industry. Placing it on your wrist, it immediately felt different from anything else on the market, and it functioned differently too. This was far from a pedometer shoehorned into a bracelet: with its built-in micro USB and the later addition of Bluetooth, FuelBand combined smart technology with equally smart user insight to trounce its competitors. And it did so with an unusual metric, Fuel Points, a new currency for measuring not just steps \u2013 walking or running \u2013 but broader exercise workouts, such as yoga, cycling, weight lifting and cross training. Fuel Points are easy to use, celebrate success, and are uniquely meaningful data.\n\nFuelBand set the wearables market in motion, and helped Nike amass a tribe of 40 million customers, the Nike+ community, in the process. So much more than a glorified CRM database, the Nike+ community is a sports club where location matters less than common goals. Members join to track their progress; they stay to motivate each other and engage in friendly competition. It transforms the lonely art of jogging, with runners competing in races asynchronously across different time zones. Football players find local matches in which to participate. Skaters share trends. Ecommerce is a click away, with links to the latest performance enhancing Nike products. The Nike+ community does more, perhaps, than any of the company's other product offerings to achieve the brand ambition of inspiring the athlete in all of us, while at the same time providing a powerful direct-to-consumer channel for loyalists. The tech has changed over the years, from a sensor inserted in your shoe and synced with your iPod, to a Nike+ and TomTom co\u2011branded GPS running watch, to the latest incarnation of Apple's Watch \u2013 which now comes in an exclusive Nike+ version. But Nike has shown itself to be a solid team player, shifting technology partners to build its intellectual property, developing breakthrough solutions for its customers, and most importantly, growing the Nike+ community.\n\nThe partnership with Apple suggests the company's foray into consumer electronics is over. The business and tech press speculate that cost, quality issues or even a gentle push from Apple CEO, Tim Cook (who happens to be on Nike's board) forced a strategy shift. I think the reason is a simpler one. The company's instinct is to conduct itself in business as an athlete does in sport \u2013 to focus. In an era of the quantified self, \"body hacking\" and the connected athlete, hardware is a means to an end, not an end in itself. It's the community that counts.\n\n_What Apple did for portable music players with the iPod, Nike accomplished for wearables with the Nike+ FuelBand._\n\nNike+ isn't just valuable to users who keep track of their efforts, but also to Nike itself, which has been quietly transforming into an information-centric company. The Nike+ database gives exact information on how and where products are being used, allowing for better business decision-making around materials, suppliers and distribution channels. When a runner voluntarily inputs the model of their current training shoe, Nike has a precise way of measuring the life-cycle of its products down to the number of miles covered and over what terrain. And by opening the Nike+ dataset to the broader industry, Nike is putting itself and the Nike+ community at the centre of the entire sports business ecosystem. That's quite an achievement.\n\nBut what of the competition? Nike may consistently be running ahead of the pack, but there are other notable challengers. Adidas and, more recently, Under Armour continue to make strides within sporting sub\u2011categories such as football and American football respectively. They may play in a different league at the global brand level, but they have considerable fan bases. It all raises questions about a brand's potential for cultural dominance: if Nike really has won the cultural battle, how come Adidas hasn't died, and how is upstart Under Armour carving space for new entrants too? It stands to reason that sports brands, above most others, would engender tribes. Perhaps Nike needs other brands to build their own following; it relies on those who are loyal to three stripes to maintain allegiance to the swoosh.\n\nSo it is not a case of a single brand capturing the cultural high ground and the victor taking the spoils. Brands occupy different need states, they take variants of a cultural positioning \u2013 Nike is aspiration (Just Do It) while Adidas is confrontation (Impossible is Nothing), and Under Armour is self\u2011determinism (I Will). The tribes that gather and grow around these ideas fuel competition among brands and the category as a whole. The game is far from over.\n\nBut retirement is not really a very Buddhist concept.\n\n_Akira Kagami is the world's best-known Japanese ad man. Don't be fooled by his shirt and jacket, Kagami brings his brand of polite rebellion to every project. His work has spanned the whole of the digital revolution to date, championing creativity, continuous improvement, and a futurist's vision for what is on the horizon._\n\n**AKIRA KAGAMI**\n\nA \"gangling Japanese\", that's what Akira Kagami is. He is 6 feet tall. But when he was a junior at Dentsu in 1971, he was quite portly as well, at 242 pounds: his long-term boss, Akira Odagiri, remembers his shirt buttons were always in danger of popping off. He was known as the \"clever, talkative, chubby chap\".\n\nClever and talkative still: and he is also a very sensitive and modest person. He rose to the top of Dentsu, to be Skikkou Yakuin (\"corporate officer\"), but he did it with his calm personality, and his gold-plated client equity, not with the political elbows often necessary for ascending that particular ladder.\n\nKagami-san is the most famous Japanese adman in the world. It's as simple as that. He has been the public representative of Dentsu at festival after festival, but, also, of Japan. Understanding him is key to understanding the Japanese response to the digital age; as unique or as tasty as the Japanese take on, say, express trains or Wiener Schnitzel.\n\nKagami's life seems to have been about simultaneously conforming and not conforming: something, perhaps, that can only be understood in the context of Japan, where rules \u2013 written, spoken and implicit \u2013 abound.\n\nHis father was an itinerant salesman, always moving around. Kagami, born in Yamagata, in the north, did not settle in one place till his fourth grade. Since then he has lived in Tokyo. He first broke the rules when he declined to go to the _kokuritsu_ (\"national\" university), for which he was well qualified, in favour of Waseda, a private university. His father was furious. Kagami had to pay himself through by working on a construction site. He then discovered that translation was an easier gig: hence his excellent English skills.\n\nAt 19 he started writing; he's never stopped. After he left Waseda, he'd been freelancing for a TV station and thought he should work there. Restless and unorthodox, the station saw what they were dealing with. They thought he'd be bored, and pushed him to Dentsu. Admission was by examination: he got through and joined the Marketing Department.\n\nBut the Marketing Department thought and behaved by the rules. Odagiri-san, Dentsu's most creative leader, and his colleagues used to mock them: \"What's the point of a marketing department that can't come up with a hypothesis?\" It was the Creative Department that thought: and after a few years Kagami joined Odagiri and his brilliant cell of six creatives, the sole Waseda graduate in a University of Tokyo hive.\n\nHe read voraciously, sometimes four books a day. Under the mentorship of Odagiri he blossomed, and was encouraged to be courteously rebellious. The Japanese habit of mentorship is, to me, one of the most attractive traits of that society. Odagiri still mentors Kagami (and, indeed, mentors me!), although \"mentor\" in English carries narrower connotations than the Japanese word _onshi_ which feels much more like a life-guide.\n\nBut Kagami did conform to one rule. He stayed with Dentsu: for over 40 years.\n\nWhat was it like then for a rebellious young creative? For those in the West who are accustomed to a view of Dentsu as a bureaucratic behemoth, the answer will be very surprising: it was gloriously free. Complete freedom. Kagami just never appeared in the mornings. (\"That's really tough for me, punctuality.\")\n\n\"Kagami's life seems to have been about simultaneously conforming and not conforming: something, perhaps, that can be understood in the context of Japan, where rules \u2013 written, spoken and implicit \u2013 abound.\"\n\nIt's difficult to grasp the essence of Dentsu unless you've worked in Japan. The best way of thinking about it is as a state-within-a-state. It is certainly not a conventional advertising agency in a Western sense. They manage everything related to communication, from TV programmes to national events such as the Olympics. As a result of this, the comfortable and mutually supportive relationship between the government, the mass media and businesses was established. Dentsu fully supported the start-up of a private TV broadcasting system. Through this effort, Dentsu built a relationship with the media beyond just placing ads; and with the government, too. Twenty-nine per cent of the media bought in Japan goes through Dentsu. Who runs the Japanese advertising awards? Dentsu runs its own advertisement awards. Who runs the Japanese advertising association? Dentsu. Who runs the Japanese Film Production Association? Dentsu.\n\nDentsu's product is power as much as it is creativity. For years, opponents, or even defectors (and there have been some) have wishfully talked of a Big Bang in Japanese advertising, which would sweep away the \"pax Dentsua\": they have been utterly and serially wrong.\n\nAs Ogilvy & Mather started to become successful in Japan \u2013 in media terms \u2013 it became apparent around 20 years ago that we should acknowledge Dentsu's success. My own ex-Dentsu advisor arranged for me to make a courtesy visit there. It was important to show that we respected their role, and that we were happy to co-exist peacefully with their ecosystem. Polite conversation in formal armchairs it may have been \u2013 but a lifeline nonetheless.\n\nSince then, Dentsu has become \u2013 through an aggressive blitzkrieg of acquisitions \u2013 much more global. Kagami-san has been the creative face of that, himself working in and encouraging Dentsu's shift around the world, and especially in Asia. He does it with passion. It's not a duty he performs by rote. He marvels at the millennial, mobile mania of young Indonesia. He is fascinated \u2013 and also slightly scared \u2013 by China.\n\nHe tells the story of how he detected a typo in one of his ads in China, and panicked. Then he discovered that the Kanji (Chinese) character he had seen was different but the meaning was the same. His Chinese copywriter said, \"In Chinese you can find any Chinese character you wish to use so there is no concept of a typo!\" He sensed the exaggeration here, but at the same time felt both _Kyoi_ (wonder) and _Ky\u00f5i_ (threat) \u2013 _Ky\u00f5i_ is the homonym and has a radically different meaning \u2013 in what he saw as the bigness of this Chinese perspective.\n\nBut the heartland \u2013 and the way of doing business in Japan \u2013 remains fundamentally unchanged. And this explains Japan's unique take on the Digital Revolution. In one sense, it started here. When NTT's Docomo first launched I-mode, it seemed like it was just another teenage fad. But in less than a year, it started to be treated seriously by Japan Inc. One of the early adopters was Shiseido. In 2003, Shiseido-mode started sending out personalized messages containing advice on food and cosmetics during the critical days of a woman's menstrual cycle. It was anticipating the world of \"mobile first\" way before its richness could even be envisaged in the West.\n\nDentsu the media broker was initially anxious about the arrival of digital media. Would these minnows escape the great fishing net, and grow up to be whales?\n\n_A public service advertisement that reminds us of the scope of a child's imagination \u2013 and the limitations of our own. Consummate Kagami, this beautiful film depicts a child whose monotonous behaviour \u2013 constant scribbling on sheets of paper with black crayon \u2013 concerns both his parents and the doctors they call on. But the child has seen the whale first, long before the adults and onlookers, including the viewer. His \"problem\" is actually a source of wonder to us all. It is Japanese advertising at its best._\n\nCarefully and methodically, Dentsu closed the net: and \"digital\" never significantly escaped the ecosystem. So the Japanese difference is that TV has never ceded the high ground. It has absorbed the revolution, but has never been ideologically confronted by it, because in this market the special business relationship between TV as a medium and agencies as brokers has never been broken, and the commission system endures, which sustains it.\n\nKagami's view of digital is very much about improving the whole. It's not about creating another sect. Improvement is, after all, the Japanese way. Tonkatsu is an improved schnitzel; the Shinkansen an improved fast train. But the core idea remains a constant: it's just that it flows \u2013 naturally \u2013 through _all_ media in an integrated way. The Japanese system doesn't allow for disintegration. Integration is just assumed.\n\nThe digital work Kagami most admires outside of Dentsu is Uniqlock. Like Kagami's own work, it follows the Japanese principle of \" _ichi o kiite juu o shiru_ ,\" which means something like, \"When someone says 'one', you jump to ten\": you don't need to spell out all the numbers in between. So there is a very high premium on the implicit understanding in language, not the explicit.\n\nKagami, therefore, seems more tranquil, less engaged in the cut-and-thrust than many of his Western colleagues. He can escape into more important things \u2013 like sci-fi. When Odagiri first took Kagami to meet the client at Tokyo Fire and Marine Insurance, he was asked, \"Is this _the_ Akira Kagami?\" Odagiri didn't know his prot\u00e9g\u00e9 was already a well-known sci-fi writer, who had totally immersed himself in the genre since school. His current favourite writer is Greg Egan, whose works include _Permutation City_ and _Zendegi_.\n\nSci-fi does explain Kagami a lot: why, for instance, he was the first Japanese creative to embrace video games. But also why, when some people talked pretentiously about being part of a digital generation, he ceased to be so excited. It just became the norm. In the near-future of Iran, Zendegi is a virtual world which turns sour: there is a message here \u2013 the advice given to but spurned by the heroine, Nasim: \"If you want to make it human, make it whole.\"\n\n_Why pay for banner advertising when you can build your free media channel? Kagami cites the work planner and creative director Koichiro Tanaka did for Uniqlock \u2013 an unusual branded widget for Uniqlo, which combines the utility of a clock with synchronized music and live dance routines \u2013 as creativity at its best. The dancers, whose audition videos alone were viewed half a million times on YouTube, showcase the latest Uniqlo styles by changing their clothing dependent on the time and season. The campaign unlocked earned media from 27,000 bloggers in 76 countries, who each embedded the widget on their sites at zero cost to Uniqlo \u2013 but maximum impact on brand awareness. Sales ticked up nicely too._\n\nKagami's own famous novel, _Detective in an Uncertain World_ , has never been translated into English. It once again shows him rule-breaking. Much sci-fi has the unspoken rule that if you go back into the past you cannot change it. The present is, therefore, protected. But, what if, at the moment anyone makes a decision, there are then two different worlds \u2013 the world chosen and the world not chosen?\n\nFor Kagami, the present is not protected at all. He's busy re-choosing it.\n\nHe founded the Dentsu New School in 2011. It's about educating the educated in what's important. He hates the idea of teaching. To me it seems to be much less of a school and much more of an inspiration zone, where he has assembled brilliant outsiders to engage the students. They include Hiroshi Nakamira, an award-winning architect born in 1974, who established a new lifestyle concept in Japanese architecture, and Katsura Kaisi, the comedian, who performs Rakugo, a traditional Japanese comic art. And Ryo Shimizu, born in 1976, the Japanese gaming guru, who is a pioneer in mobile gaming.\n\nAll this is done for Dentsu.\n\nIf he could travel back in time to the Dentsu figure he admires most it would be Hideo Yoshida, who was President of Dentsu from June 1947 to January 1963. He died before they had a chance to meet. But it was Yoshida who laid down the Dentsu way, and core to it was that the business we are in is both science and art. Maybe some of that \"way\" has been lost over time: maybe the Dentsu of now is less able to understand a creative now than previously; maybe it's just become too mature.\n\nKagami is optimistic about the company: the rule-breaker remains loyal.\n\n_Relatively unknown to the advertising world today, Martin Nisenholtz has done more for digital transformation than most. Part-academic, part businessman, he set up Ogilvy's first \"interactive\" department in the 1990s, guiding the agency onto a more digital path. Students at Boston University, where he now resides as Professor of the Practice of Digital Communication, would struggle to find a more digitally enlightened mentor._\n\n**MARTIN NISENHOLTZ**\n\nMartin Nisenholtz must find it difficult to make up his mind whether he is the accidental academic or the accidental businessman. His life has moved between the two in a remarkably fruitful way. Quietly, but with conviction, he has been a major force in the digital era of communications. But no one's erected even a virtual statue to him; and yet maybe they should.\n\nIf you ask the average employee of Ogilvy & Mather today if they know who he is, or was, I am afraid the answer would be \"no\". Such is the fickleness of corporate memory! More than any other individual, he laid the foundation for the company's digital transformation. But, more importantly, he brought digital into the realm of big traditional agencies, or, as the inimitable George Parker described them, big dumb agencies (BDAs).\n\nWhile the debate (some would say war) between this model and the pure play model raged on for years, and while some BDAs chose not to speak the new language at all, it set the scene for many of us to change dramatically.\n\nI have to say that it was by no means certain that Ogilvy & Mather would become digital at heart, but we did so because of the work Martin Nisenholtz did back in the 1990s.\n\nToday he is Professor of the Practice of Digital Communication at Boston University. But if you imagine him confined to a professorial study, you'd be very wrong. Martin is a restless man. Unlike Bob Greenberg, he doesn't really like buildings at all. He worked in an iconic one for years \u2013 the Renzo Piano designed offices of the _New York Times._ Perhaps it was just the wrong building. Shimmering and stunning on the outside because of its curtain wall, on the inside that same wall gave you a sense of looking through bars. \"It felt like you were in a golden cage.\" Now he revels in being cage-less \u2013 and rootless. He is anywhere: and his students are as familiar with him in Starbucks as in the classroom.\n\nHe's passionate about those students. And about what he teaches. It's not a subject: he's teaching them to be ready for life \u2013 that \"they're not in Kansas anymore\"; or in Gansu, either (a third of his students are Chinese). He doesn't see any of the millennial stereotyping in them (although he hears them talk about their teenage siblings, who've known nothing other than a digital world, as \"different\"). They are thoughtful. They want substantive stuff. And that's what he gives them.\n\n\"Quietly, but with conviction, he has been a major force in the digital era of communications.\"\n\nHe makes them read books. Yes, books. And he requires them to write an essay each week. Because he defines the topic himself, they can't find the essay on the internet. Game up!\n\nLike many good teachers, Martin receives a reward by drinking in their enthusiasm. There's something narcissistic at play there: he's taken back to his own student life.\n\nIn that life, enthusiasm had to be created. It didn't come on plates in the Philadelphia of the 1970s. It was the Philadelphia of ruthless Mayor Rizzo; of permanent clashes between the radical organization MOVE and the police; of gross racial discrimination. His father was a letter carrier from the underprivileged, white Western suburbs. Martin went to a public high school in Springfield \u2013 and flourished. But only one kid from that school a year would make it to Harvard, and that year it was Martin's best friend. Martin ended up with the \"booby prize\" \u2013 Penn, the University of Pennsylvania.\n\nHis escape from this rather brutal landscape was photography. He just went around taking photographs. At one point he became a writer on the _Philadelphia Inquirer_. So he started to migrate, at Penn, inexorably into the world of media, studying for a PhD at the Annenberg School of Communication. And his enthusiasm for the world of media grew, and grew.\n\nAt the end of 1982, he received a call from New York, from NYU, to participate in the teletext trial. Martin never looked back to Philly after that, or the PhD. It was the big escape \u2013 physically and intellectually.\n\nIt's almost impossible to re-imagine those early days of teletext. It was sparse and linear. But a grant came from the National Endowment of the Arts: how to make of teletext more than what is on the surface. The unlikely base for the project was 144 Bleecker Street, a ramshackle building where no line ran straight at all. But it was in the Village. All Martin had to do was to walk down to the bars and find someone to help loosen up the \"squareness\" of teletext. The person he found was the artist Keith Haring. The potential of computer graphics was shown for the first time.\n\nMartin became Mr Teletext. The early NYU presentations seem antique, but it was these that attracted Ogilvy & Mather, looking for someone to head up a strange new project it had been given by Time Inc.: please help us make sense of our new Teletext initiative.\n\nSoon Martin had set up an interactive department called the Interactive Marketing Group. I believe Ogilvy & Mather was too timid at that time to put its name on it! But it did make the word \"interactive\" famous. It was the first such department in the world.\n\nIt was really tough for Martin: he was in his own digital wilderness. The media companies' interest waned. But he attracted new clients: General Foods; then American Express; then AT&T. A lot of what they were doing in the early days was taking PCs and putting them into public locations for promotional purposes, digital kiosks. Then a client, The Equitable, with a tiny new media team of two people, appointed the fledgling unit to help its sales force. Martin gave their agents the first PC-based software. It was on a 5\u00bc-inch floppy disc: it meant you could view interactions in real time in the customer's house. It was crude, but new and exciting and really revolutionary.\n\nIn fact, Martin has been an agent of _two_ advances.\n\nThe second was in the 2000s, when he became Head of Digital at the _New York Times_. He's the person who first chose RSS (Rich Site Summary) as the home for the newspaper's stories, which would then automatically alert readers to website updates. It is a standard that has become ubiquitous. He was also the person who set about registering readers for the first time.\n\nBut I think of Martin as someone who hasn't just fought on the barricades (from different sides, too) of the Digital Revolution, but as someone who epitomizes its most anguished inner conflict.\n\n_In the early days of digital interface design there were no experts to call on. Martin persuaded the artist Keith Haring to bring the blocky Teletext screen to life, which changed the face of computer graphics in an instant._\n\nHere he was in a storied newspaper brand, responsible for developing digital as quickly as humanly possible: the one metric \u2013 get as many digital readers as you can, to attract the advertisers in. There was a general consensus in the business that this migration was inevitable, and that it would be accompanied by some (but how much?) degradation of revenue. As Jeff Zucker famously expressed it in 2008, we must trade in our \"analogue dollars for digital pennies\". Then, a cataclysm: the great recession of 2010. The question finally hit home: can the model sustain giving the news \u2013 and premium news at that \u2013 away for free?\n\nThe debate was moderated in an exemplary way by _NYT_ 's Arthur Sulzberger. Here was a proprietor who realized that his great-great-grandfather had created a business by charging 2 cents a paper. But he was also very sensitive to the newsroom: and those staffers enjoyed the fun of being read digitally everywhere from Ulan Bator to Uruguay. He moderated a very sane and careful debate on the topic: charge or not? It was a series of discussions over lunch, on the 15th floor executive dining room of the Renzo Piano building, where a sculpture of a huge eagle surveys the masthead.\n\nMartin's going-in prejudice was not to charge. He was horrified at the idea of a simple paywall. It would kill off years of cultural adventure. But then the lunchers started to talk about the _FT_ : they had pioneered the \"metered model\": 10 free articles before the wall. It was a beautiful solution, which everyone converged on. In fact, the _NYT_ subscription model that subsequently emerged has been notably successful, both in absolute numbers and in relative resilience to the later challenges of the Facebook age. There was no zero-sum game here. You could win both ways, provided that you understand that journalism in the digital age is much more about creating interaction thank it ever was.\n\nJournalism has not gone away though. Martin remains frantic and passionate about the quality of digital advertising. He swears about it. West Philadelphia breaks into the nuanced language of the Boston academic!\n\nI can see Martin's point. There was a gale of insanity in the business. It did plaster the internet with stupid banners. David Weinberger and Doc Searls in \"New Clues\" posited that at one point in this revolution, direct marketing \"body snatched\" advertising. And so it did: the obsession with data pushed everything that Martin, or I, had learnt at Ogilvy into the trash heap \u2013 all that mattered was \"how many clicks?\"\n\n\"I think of Martin as someone who hasn't just fought on the barricades of the Digital Revolution, but as someone who epitomizes its most anguished inner conflict.\"\n\nThe message became the medium!\n\nThe advertisers Martin confronted \u2013 from the _NYT_ , as a media owner \u2013 were in rank denial. Of course, they aren't alone.\n\nA final note. Martin had just one interaction with David Ogilvy \u2013 at the same meeting I recalled in France here. Jerry Pickholz, the then Chairman of OgilvyOne, took Martin over and introduced him to David as a youngish fellow doing interesting things, and asked Martin to explain what he did. In two sentences, he did. David looked at him, confounded, and said, \"This is complete bullshit\", turned and walked away.\n\nMartin, the modest academic, entirely blames himself \u2013 not David. He'd used the language of tech \u2013 all jargon and obfuscation \u2013 and it was substantially longer than two minutes. He should have said that he had found a way of making word of mouth \u2013 manna to David \u2013 into something measurable. Or that we could turn Direct Response from a long and tedious process into something where all you had to do was press a button.\n\nMartin remains the \"holist\". He knows that culture in the deepest and broadest sense is what guides life, and provides the ultimate legacy of revolutions.\n\nHe has never written a book. And says he won't. But he should.\n\n_Matias Palm-Jensen, a jollier Swede than this picture might suggest, has been at the centre of digital since the early 1990s. His first award-winning campaign, for Swedish Post, foresaw the \"on-demand\" economy over a decade before Uber._\n\n**MATIAS PALM-JENSEN**\n\nMention the word Sweden to a multinational advertising network and it will send shivers down the corporate spine. In the past, I have felt them \u2013 full on. This is the market of doom: a global graveyard. Those of us who survive may have found an interesting niche with clients who value us \u2013 and tend to produce highly original work \u2013 but we are _not_ insiders.\n\nYet this is no backwater. Just add the word digital to Sweden and there is a dramatic re-frame. We all admire and revere its leading role in the Digital Revolution.\n\nIt is the inventor and incubator for some of the digital world's best ideas, from Skype to Spotify. It has exported digital talent all over the world, and kept a goodly clutch on them at home, too. It's become a hotbed of digital creativity.\n\nAnd, if there is to be a spokesman for this role, it would have to be Matias Palm-Jensen.\n\nOne day in 1992, he got a summons to a grand old house in the Gamla Stan, the medieval centre of Stockholm. He was a stocky, athletic, utterly self-confident young man in his early 30s, rather more saturnine than the average Swede. The room he was shown to was stately, furnished in a traditional English style. It was not unfamiliar to him and nor was his interlocutor. It was an unlikely looking room in which to foment, even a mite unwittingly, a digital revolution from, and yet that was exactly what was happening. Jan Stenbeck, for it was he who had called Matias, was a business mogul, who behaved \u2013 and looked \u2013 larger than life. A man of vast appetites, he used to go to Luxembourg to eat, and that's where his managers would fly in for meetings on a Saturday, then party with him on the Sunday. His favourite indulgence was a simple dish of mashed potatoes, egg yolk, cream and Russian caviar. This unlikely exterior belied a great visionary and innovator, and a nurturer of big ideas and big talents. Stenbeck had a simple request: \"Matias, I want you to do the biggest portal in Europe for the internet.\" He added that it was a \"meeting point that's coming on\" and \"we have to be the biggest\".\n\nMatias took space in the basement of Stenbeck's house. He remembered saying to himself, \"this is the future\", and set to work building a European portal from scratch, at that stage telephone enabled. It was called everyday.com.\n\nMeanwhile, something was astir in Sweden. It was on its way to becoming one of the most digital countries in the world \u2013 faster than any other. Outsiders marvelled. It had much to do with government sponsorship: an early push of computers into schools. On 5 February 1994, Premier Carl Bildt sent an email to President Bill Clinton: the first email contact between two heads of government. Bildt became excited; and two days later gave a speech \u2013 a very un-Swedish rhetorical speech \u2013 and one which did not even reflect official government policy. Its themes were mankind, technology and the future. He said: \"After the agricultural society and the industrial society we are facing the next big leap in human development \u2013 the information society.\" Bildt's moderate party and the Social Democrats ascended to power in the 1990s; the Social Democrats were later converts but by 1997 had drunk the Kool-Aid and introduced the \"home PC\" reforms for employees to lease a computer from the company they worked for deducted from their monthly pay. Penetration rocketed. In 1998, Carl Bildt's election manifesto contained the slogan \"broadband for everyone\". He lost the election but won the policy. Telia, aided by the housing cooperatives, soon created an internet with unprecedented penetration levels (92 per cent by 2010).\n\nMatias attributes some of this Swedish exceptionalism to a national bias towards doing: \"Sweden is a country of engineers.\" This may in part be due to a thinly spread population: the North in particular had a political impetus to be connected: the dark nights might help create legions of Nordic _otakus_ 1. The famous Swedish safety net, far from discouraging entrepreneurship, actually encouraged young entrepreneurs to take the risk, as, whatever happened, they would have a comfy landing.\n\nNot that Matias needed that. When, in 1996, he set up his own digital agency, Spiff, it soon became predicated on taking advantage of the internet boom. \"Everyone was doing homepages.\" They didn't. Spiff started with just four people: the other three came from a production agency, and were much younger than Matias. Spiff Industries did not create homepages. It broke the trend and made advertising in ways no other agencies imagined.\n\nThe first award they won was for the Swedish Post. For this campaign, they bought all banners on big Swedish media sites and then did daily callouts to Swedish influencers, such as Stefan Persson, CEO of H&M.\n\nFirst, they sent written letters to their targets that read: \"Go to the Internet today, and read the newspapers!\" Then, when their targets went online and visited the websites, they were greeted with digital banners directly addressing them: \"Hi, Stefan Persson, please order your lobster here!\" They simply had to click on the banner and choose, \"Yes, I want lobster\". Thirty minutes later, they had a deliveryman at their door, with nothing other than their request, fresh lobster.\n\nFor most people, this was a totally mind-shaking experience and suddenly a lot of \"important\" people understood what it was all about.\n\nSuccess fed success. Spiff continued to gain recognition but also grew in size. Soon they were managing 500 staff. It takes some calibre to handle that, although there's never been much self-doubt around Matias Palm-Jensen.\n\nHis mother was French, \"an intellectual, stupid, strange philosophy student in Paris\", who decided on a whim to go to Sweden with her best friend \u2013 \"not a very Parisian thing to do\". Of course she met a man. Franco-Swedish helps define Matias: Swedish he is, but with an even more exotic flavour than that might suggest. His grandparents had mixed with Mir\u00f3 and Dali. Pasolini came to stay \u2013 and also chose the basement.\n\n_Food delivery apps now seem ubiquitous, but Matias Palm-Jensen was already delivering lobster on-demand in the late 90s. His campaign for Swedish Post \u2013 which used a hack to give the appearance of personalized ads aimed at influential Swedes \u2013 was successful despite the rudimentary state of the web at the time. His agency developed campaigns that pushed at the edges of the internet, and people noticed._\n\nThe young Matias was very far from being the Swedish archetype, the engineer: he read and drew. Even today, if he wants to express something complex, he will draw it. But it was law and business that he studied at Uppsala University and Stockholm School of Economics, getting a degree in both. He went into government service; and became an expert in contracts before following his nose and straying off into creative consultancy.\n\nThat knowledge of contracts was to stand him in good stead. Spiff had become a hot property, and he sold it. Then there was the day when the other owners wanted to rename the agency to make it more stock-market friendly \u2013 to \"Technetologies\". Matias deliberated, and then gave them the name Drax, the evil business empire in _Goldfinger_. They simply thought it was a nice-sounding name, so on to the stock exchange it went. But when Matias needed people to service a client, the owners of Drax sent him untrained, unqualified heads from the North. That was it: he led his Spiff co-founders into a breakaway. All he lost was his stock; but, hell, at the end of the day, this was just a bubble.\n\nThe new name was rather warmer than Drax \u2013 as warm as you can get. What name would be the most unlikely you could find for a technology company? Something that says it is caring, and something you would look up to. Enter Farfar, the Swedish for \"grandpa\". And Farfar then became the hottest digital agency, housed at first, of course, underground, this time in a parking lot in Sk\u00e5negatan in the southern part of Stockholm (now called SOFO). But it set out to be the best in the world \u2013 to be Swedish but global.\n\nThe work was ravishing.\n\nFirst in line, was a program for Milko, which epitomized, ahead of its time, everything that digital offered. It gave people the unbelievable experience of editing for themselves.\n\nIt was viral in its market, before the word viral was stolen for digital. A Swedish dairy suddenly had a fan page in Brazil and a feature in _Wallpaper_.\n\nThe Vikings had arrived on the global advertising scene. Farfar received a Gold Lion for a campaign for Visit Sweden. This was celebrated with a big night out in Cannes. A sozzled Farfar creative buried the Lion (they are quite heavy, actually) in the sand on the beach in order to party a bit, came back to find the adjacent chairs had moved, and then after frantic digging couldn't find it. The missing lion was never recovered. This incident, however, proved to be great advertising, as Matias just turned it into a classic film.\n\n\"Even today, when he wants to express something complex, he will draw it.\"\n\nFarfar was premised on 80 per cent work and 20 per cent innovation. Fridays were for the latter, when they did whatever they thought was fun. But the business guy, the contract man, living inside Matias became restless again, kicking the traces. Farfar was sold in 2005 \u2013 to Aegis.\n\n_It may look basic by today's standards, but this early application for Milko gave users their first taste of controlling content by way of a simple editing feature._\n\nIt's difficult in retrospect to question the logic. Matias can still make a salesman's case. But culture is not easy to tabulate in contracts, and, not for the first time, a local agency felt the hostile embrace of a global acquirer. The media mindset of Aegis was very different. The philosopher in Matias quotes Upton Sinclair: \"It is difficult to get a man to understand something when his salary depends upon his not understanding it!\" Aegis did not understand Farfar. They soon bought Isobar, and Farfar now had an internal competitor.\n\nThe global Swede found that there was a jungle out there that was not so welcoming. He saw the bad side of Brazil when travelling there for an industry conference. He tweaked the whiskers of the jaguar. Following one of the giants of the Brazilian agency world onto a stage at a conference, he remembers half the audience getting up and leaving. He said to the remainder: \"Guys, old media just went out of the door.\" The giant, \"heard that, drinking his cocktail outside, and he came back, furious. He said: 'I want to ask you a question.' I declined: 'I have my speech to do.'\" The organizers improvised a debate at the end of his session, but the fury did not wane.\n\n_Most people treasure their Cannes Lions award, especially their first. A Farfar creative buried theirs in the sand. The story was used in a smart piece of advertising itself._\n\nAnd when his local Farfar started to feel the very un-grandfatherly punches of local protectionism more directly, our Swede beat a retreat. Not just from Brazil, but from Aegis.\n\nNick Brien scooped him up for McCann. A Swede on the board of McCann Worldgroup, Matias had an immediate impact. He called himself Nick's \"Surprise Officer\". He created a team whose role was always to make surprises. He did it deliberately because he didn't want to be an \"innovation officer\": he knew that they don't really make change. But you could possibly surprise someone into making change. Well, maybe you could have if Nick had stayed around, but he didn't. The game moved on. And so, again, did Matias.\n\nHas he found a home? He deserves it. When the digital world's symbolic Swede stopped travelling and started thinking about what it meant, he was struck by \u2013 above all \u2013 the transparency which the Digital Revolution had forced. Quite simply, it's much more difficult to lie. And his new baby is The Kind Collective. It's a collective because he's grown tired of hiring people: he just brings on the very best people when he needs them. It's not about apps or tech. It's about teaching people that they can be kind, even to their customers. It's about telling stories.\n\nThe driven businessman is certainly not doing this as charity: but it's no longer a world of contracts, of hiring and firing, of global adventures. The Franco-Swede has returned home, in more ways than one.\n\nThere's time to be passionate again: music, for instance. Matias laments that there has never been anyone in our industry who is exclusively responsible for the music. And it's usually the last thing to be thought of. He was always convinced that digital music was important: \"You can smell music.\" Right now he's working with Bj\u00f6rn Ulvaeus of ABBA, creating an interactive party experience.\n\nNot many digital creatives would get away with rewriting the lyrics of \"Mamma Mia\". But then not many would have the salesmanship to project it as a piece of change management.\n\n_Chuck Porter once met David Ogilvy, and resolved to model his success. Chuck's agency has been a bastion of creativity for 20 years, so he rightly sits among my list of influencers. David would approve._\n\n**CHUCK PORTER**\n\nThere was a moment in the mid-1970s, when a young freelance copywriter was nervously standing in a Madison Avenue office by a wall festooned with his concepts for a British Tourism Authority campaign, when David Ogilvy swept in, with his entourage, asking, \"Did you do these?\" Chuck Porter \u2013 for it was he \u2013 remembers that David looked \"kingly\"; and also that he said to himself: \"Someday I want to be like that.\"\n\nThe agency he built was the most striking of the new agencies that grew up in the Digital Revolution: a creative boutique that then transcended itself, and found both fame and scale.\n\nBut as Chuck says: \"I'm the most un-cool guy.\" A thoroughly nice 70-year-old, bursting with energy, who dotes on his Labrador and likes cooking, he has lived with his wife in the same Miami house for decades while the neighbourhood changed around them. He was, though, the \"master of the fireworks\" of the agency that bore his name, as it blazed through the crazy noughties, \"the agency that couldn't lose.\"\n\nThe niceness comes from Minnesota: but it's not the \"bad\" Minnesota nice of legend where you smile courteously at someone even if you're angry with them. Chuck does do angry when necessary: if, for instance, the writing is not up to scratch and the writer doesn't give a s**t, he'll yell at them. It's the _real_ nice of someone who doesn't seem to have an enemy, and who has a very controlled and un-jealous ego.\n\n_When MINI sought to enter the US market in 2002, Chuck's then small-ish agency was chosen to help the brand create buzz. He positioned a luxury small car \u2013 a lifestyle choice \u2013 to punch above its weight compared to similarly priced, larger alternatives. Striking executions \u2013 in print and outdoor \u2013 were a modern take on quintessential MINI. The campaign drummed up a lot of interest \u2013 and the brand sold almost 25,000 vehicles in the first year alone, and over half a million in the 10 years since launch._\n\nGrowing up in Minneapolis was not difficult. Everything works. It was a largely Scandinavian and friendly place. His parents owned restaurants; life was comfortable. You didn't notice the cold because you played hockey all winter. Jim Fallon (later founder of the agency bearing his name) and he were best friends at high school, and hung out all the time together. There were big brands in Minneapolis \u2013 General Mills, Pillsbury, 3M, Honeywell \u2013 and a marketing and advertising industry around them. Chuck remembers going to the house of a friend of his from Junior High; his dad was an art director at BBDO, and he would be doing layouts. It seemed a swell thing to do.\n\nHe started his first agency as an undergraduate, with some fellows from the advertising department. The father of one of them ran a company that made pre-stressed concrete panels: Chuck presented an ad for an architecture magazine featuring Mies van der Rohe and a \"terrible\" line about great architects. This first client asked what the heck it was. His blocks were cheap and quick, and had nothing to do with Mies van der Rohe. But they'd shot and set it already, so he ran it. Then he won a prize for a radio spot for the campus drugstore. He was hooked.\n\nAfter graduating, he did the dutiful thing and headed to law school. It never held him, although, like Matias Palm-Jensen, he did like the law. He lasted till the first Christmas, when he went to visit his brother, a Delta pilot, in Miami. He fell in love with the place immediately. He'd been looking at _Advertising Age_ and had read about two award-winning copywriters: he met with them, said he was a really good writer and asked them to give him a shot. They did. So he abandoned Law School, left his then-girlfriend, packed everything he had in a Chevy Camaro, and drove down. His first brief was a billboard for a bank installing cash dispensers. He wrote over 100 headlines: they thought around 99 were terrible, but alighted on one that read, \"Get out of Line\". It was put up by the airport, and he's never forgotten the thrill of seeing it. One of those copywriters, Rick Green, became his freelance partner for years.\n\nAs their family grew, it was Chuck's wife who suggested he get a proper job. He'd been 17 years a freelancer. One of the Miami agencies he had worked for was run by a man called Sam Crispin. Jamaica Tourist Board was a client, and one night in Montego Bay they were out drinking when Crispin handed him a legal pad and said, \"Write down what you want to come over here.\" Chuck wrote that he wanted his name on the door, and handed the pad back. Crispin said \"okay\": and so began the agency Crispin Porter, in 1988.\n\nCrispin was not an adman, but a successful businessman. After three years, he was content to sell out to the Midwesterner, who later bought his son's share. It was one of the few agencies in Miami. There was an exceptionally large freelance community \u2013 strong talent fleeing the winter in New York \u2013 but no one could afford to take them on full time. Chuck set out to build a proper agency. Crispin gave him the licence from the first day. So Chuck called the whole agency together and told them that the only thing he knew how to do was good work. He introduced one of his mantras: \"Try to do something really good today, and we'll let the future take care of itself.\"\n\nClients started to come: the local clients \u2013 the Florida Marlins, the _Miami Herald_ , the B2B arm of Del Monte. They won a Cannes Lion for one of the local clients, and then started attracting some out-of-state clients \u2013 from California, from Michigan.\n\nMeanwhile, Chuck had hired Crispin Porter's sixteenth employee. The agency's work didn't _look_ good. He had written some ads for a boat company to run in a boating magazine, but hated the designs. So he sent them to the owner of a design shop he had collaborated with in his freelance days and asked him to improve them. What came back was spectacular. Chuck called Bill Bogusky to say he had blown the boat out of the water. Bill said: \"It wasn't me. Alex did them.\" So, Chuck called his son Alex over and hired him. Alex Bogusky was then 20-something years old, a motor-cross junkie; but Chuck had known him since he was 10. There was a bond to build on: and one cannot help seeing a lot of Chuck's future role as that of an indulgent, energizing and patient parent.\n\nBogusky was \u2013 and is \u2013 one of the most brilliant creative people of his generation. It is the particular genius of Chuck that he enabled that genius \u2013 stormy and intense as it could be \u2013 and moulded it into a compelling partnership. By 1993, Bogusky was kicking the traces, so Chuck made him Creative Director instead of himself, \"because he was a better creative than me\".\n\nIconic work started coming out. Chuck had a very simple philosophy to \"research the hell out of strategies\", and then to let the work take care of itself. He has a pathological dislike of putting work into focus groups. The model first came to light with an anti-smoking campaign in Florida aimed at teens. It was called \"The Truth\". The research showed that a conventional message \u2013 \"smoking kills you\" \u2013 actually made rebellious kids want to smoke more. The Truth channelled this rebelliousness towards the behaviour of the tobacco companies, with students encouraged to make crank phone calls and provocative invasions of lobbies. Some of these were filmed, and turned into low budget TVCs. From the start The Truth website was the hub for the campaign. The budgets grew; the productions became glossier. But The Truth embodied Chuck's view that \"popular culture is currency\". You just need to capture it. And you needed someone with Bogusky's own rebelliousness to push the envelope, sometimes dangerously close to the edge.\n\n\"There was no rulebook, although when they came to write their own, its definition of advertising was 'anything that makes our clients famous'.\"\n\nSo began the glory years. Chuck's next mantra was \"Don't feed the gorilla.\" He was puzzled that the default strategy for challenged brands was always to bend over backwards trying to behave like a category leader. In fact, you need to do the exact opposite. When BMW's Mini Cooper launched in the US, the default instinct was to behave as with any other car launch, and so run a large outdoor campaign. However, they put together a 22-city pre-launch tour of Minis on top of SUVs with banners that asked: \"What are you doing for fun this weekend?\" Not only was this a celebration of the Mini's small size, but also made the statement that fun stuff goes on top of a car, whether it's your mountain bike, surfboard or camping equipment.\n\nWork like this did set part of the agenda for the digital age. It never started with television: Bogusky would refuse that brief. Chuck and he always thought in terms of \"fame\": what would this idea be if it were a press release? They did so because they had to. There wasn't enough money to do anything else.\n\nChuck always believed that they both benefited by not having ever worked in an advertising agency before. There was no rulebook, although when they came to write their own, its definition of advertising was: \"anything that makes our clients famous\".\n\nAs the business grew, Miami started to feel confining. It was difficult to attract staff there. The tipping point was when their chief recruiter got three refusals in a row from digital producers whose families did not buy Miami. They huddled, and decided to open a place that would be the exact opposite of Miami \u2013 and remain true to their inner challenger. They looked at Santa Fe (poor links), Portland and Boston (not different enough), and then, Boulder, Colorado. Both Chuck and Alex were skiers: they fell in love with it. And it certainly was the opposite of Miami: it's small, it's tightly integrated; everything's closed by 9.00 p.m. The only escape is nearly an hour away in Denver. And it's the \"most granola place you could find in the US\". At the agency's Christmas gathering, Chuck held up a big sign saying: \"Here's the good news: we're getting a new office.\" Then another sign went up: \"In Boulder, Colorado.\" Chuck asked anyone who wanted to make the move to come and see him, guessing that around 60\u201370 would vote yes. Two hundred and seventeen people came to see him that day to say they were in. And when, in 2006, Alex decided to move, a large part of the creative department followed him.\n\nBogusky's charisma started to flourish in an outsize way in the fresh Colorado air. Meanwhile, they had made a deal some years before with a small Canada holiday company, Multi Destinations Club (MDC), and Chuck became more involved in that. He had a very simple message for Miles Nadal, the CEO: \"You don't know anything about advertising, I do. If you let an agency do its creative best, it will make you money.\" At a high point, Crispin Porter + Bogusky was contributing 25 per cent of MDC's profits.\n\nChuck drove an expansive strategy for Crispin Porter + Bogusky. His first venture was to acquire Daddy, one of the new brilliant Swedish digital shops in Gothenburg. But he soon learnt that Europe is not the US: and that a challenger city strategy here was not appealing to clients. He needed to be in London. Other offices followed. And, for MDC, he assembled a kindergarten of Crispin feel-a-likes: Kirschenbaum, for instance, then 72andSunny. I think only a copywriter with legal training could have done it. And all the time he was protecting their own agency: buying it the resources and space it needed.\n\nMeanwhile, the Bogusky charisma continued to expand. It's interesting to reflect on how completely he captured (or maybe seduced is a better word) the trade press. Appearing on the covers of _Businessweek_ and _Fast Company_ , he had good looks, a whiff of danger, and overwhelming self-belief. The Agency of the Year awards flowed. The myth-making became more about Alex: the move to Boulder his own invention; the agency revelling in its image as a street shark. Clients also succumbed: Brandon Berger, my digital partner of recent years, recalls the relationship between the Boulder Managing Director, Jeff Hicks, and Alex: \"It was almost like Jeff was a drug dealer: bring Alex in, get them hooked on him, then take him away, and make them beg for more.\"\n\nThere was something of a cult around Bogusky. He chose his followers to be loyal to him. There was the \"Birthday Book\" he used during hiring interviews: where he would use the subject's birthdate and astrological profile as a means of testing which ones could cope with such an unusual inquisitorial line. He was ruthlessly demanding of his people \u2013 nothing wrong with that. But something about him could bug people, and allowed some escapees to become detractors. They created counter-myths: the sense that the Boulder office was some kind of advertising Jonestown. Of course, it wasn't. And, as someone has reminded me, \"Chuck was always there.\" It was a long rope, but there was a rope from the calm enabler to the brilliant firebrand.\n\n_Crispin & Porter in 1992 \u2013 when the right team got together to create magic over the next two decades. Chuck Porter sits front left, beside a cross-legged Alex Bogusky (front centre), whose name would later appear on the agency door._\n\nPerhaps, inevitably, a breaking point would come. Probably, Chuck was not surprised that it came. For a while, Bogusky had become disengaged from the business. Then they had both collaborated on a light-hearted book called _The 9-Inch Diet_ (2008). But Bogusky was moving well beyond that, to a deeper ideological disgust with the whole of the food supply chain. The storm finally broke when Chuck was at Cannes in June 2010. He started getting calls \u2013 from Miles Nadal of MDC, and from his clients. Only later was there a mail from Bogusky attaching a violent blog he'd written, asking Chuck what he thought about it. Chuck mailed him back and told him he had to decide: take this cause up and leave the agency \u2013 or stay. At 3.00 a.m. the call came: he resigned. Chuck has never evinced anything other than the total support he always showed him. He remains calm. Not \"Minnesota calm\", but real calm. And who can imagine that it was an easy experience for him?\n\nDoes Bogusky deserve that niche in the Hall of Fame? I like to think of it as another \"and\". He's somewhere there, unofficially. These two were joined at the hip, literally inseparable for years in anyone's mental landscape of the business. But it was Porter who noticed Bogusky, who chose Bogusky, who enfranchised Bogusky, who enabled Bogusky, who protected Bogusky. No Bogusky, no Porter. But, even more so: no Porter, no Bogusky.\n\n\"Crispin Porter + Bogusky at its best captured the opportunity of advertising in the digital age.\"\n\nAt the end of the day, Bogusky's heart was never in advertising. I think that shows in some of the work, where deep brand sensitivity is sometimes not there. Fame without brand can be a dangerous thing. In retrospect, the \"King\" of his Burger King work does feel like that. He himself said: \"My relationship with advertising was that I was not fond of it.\" It can sometimes show.\n\nCrispin Porter + Bogusky at its best captured the opportunity of advertising in the digital age. It believed in the internet because, at the beginning, the internet was cheap. And then it started to become more serious. In Boulder, Chuck and Alex acquired Selective, a technology company. They were early understanders of the fact that if you want to protect the work you also have to understand the coding. But they attracted a different type of talent. Jeff Benjamin, for instance, had made websites at Goodby Silverstein. Chuck didn't want to do websites. As he put it: \"digital was just another way to not do traditional advertising.\" But Benjamin made the ideas work.\n\nWhen the need came to remind people that Burger King makes chicken, the idea was as simple as \"what if we put a chicken\" on the World Wide Web \u2013 their claim was that it would do whatever you want. So: \"Can we do that\"? And, the answer from everyone was: \"Yes, we can give them 300 commands via the web that he can react to.\" And, so, enter the Subservient Chicken.\n\nMore central to a brand, perhaps, is the way the fusion between creativity and technology works for Domino's Pizza. This is a company that defines itself as a technology company that happens to make pizza. A beneficent client like this has allowed the agency to narrow the boundary between customer experience and connection. Anyone who has ordered a pizza and been glued to the countdown timer will know what I mean.\n\nCrispin Porter \\+ Bogusky today is no longer a challenger \u2013 and that's the joy of being a successful challenger. Seven years after the rupture of his Faustian pact with Bogusky (and I do think there was something at least mildly Faustian about it), Chuck Porter is a man at ease with himself. But he's still working hard \u2013 though now with an agency that's probably more at ease with itself and, in fact, a network more comfortable with building brands rather than just making fame. He has no hobbies, so nothing to retire to: it's the work. And he's proud as he sees that work reviving in the agency that has been his home.\n\nHe divides his time between Miami and Boulder. But when he needs to retreat close somewhere, he goes back to Minnesota. Not a pied \u00e0 terre in Manhattan or Paris for him, but a retreat he designed in Greenwood, MN, population 200. There is no nicer place for this nice man to re-fuel.\n\n_They say the early bird catches the worm, well this one caught the internet's imagination in one of the earliest interactive and viral campaigns of the digital age. Type in a command and Subservient Chicken would fulfil your request from a host of pre-determined moves \u2013 cleverly reinforcing the brand's \"have it your way\" message. (Anything outside of his repertoire would receive a humorous shake of the head)._\n\nAs someone who once worked with him once told me: \"I have never met anyone who doesn't love Chuck Porter.\"\n\n_Domino's Anyware has almost as many platforms connected as pizza toppings. Customers can order from their smart TV, smart watch, Ford car, Twitter and by text._\n\nThere are not too many people in advertising you can say that about.\n15MY BRAIN HURTS\n\nThe arrival of the digital age coincided with big leaps in neuroscience. In fact, MRI (Magnetic Resonance Imaging) is dependent on digitalized imaging, a technology of the 1970s \u2013 those pretty brain scans we became familiar with. Consequently, our knowledge of how the brain works is much greater than David Ogilvy's was. Without any doubt at all, it is knowledge he would have relished:\n\nThe quality of research will improve, and this will generate a bigger corpus of knowledge as to what works and what doesn't. Creative people will learn to exploit this knowledge, thereby improving their strike rate at the cash register.\n\nFrom the 1990s on, breakthroughs in academic neuroscience suddenly became populist, and the chief popularizers were the husband and wife team, Antonio and Hanna Damasio. They studied the case of the patient zero of neuroscience, Phineas Gage, a Vermont railroad foreman, who, in 1848, accidently set off an explosion that drove a steel rod into his skull from under his left eye through the top of his head. Miraculously, he survived. But Gage was not the same afterwards. From contemporary accounts, he seems to have lost all his charm \u2013 his emotional intelligence, or EQ. He turned from nice to nasty.\n\nAmazingly, the Damasios, from his surviving skull, re-built his brain with 3D software, and then were able to identify that the \"neural machinery\" which enables decision-making is informed by \"emotional\" parts of the brain as much as by \"rational\" parts. Those first parts had been removed by the rod, along with his charm.\n\nThis is the neural \"and\" underneath the \"Age of And\". It's the story that not every schoolchild knows, but should, because it explains that mankind is not purely a rational species.\n\nThe Damasios' academic research continued by using MRI to probe the workings of different parts of the brain: they identified very different roles for the limbic system, especially the almond-shaped amygdala, governing emotional inputs to decisions, and for the ventromedial prefrontal cortex, contributing intellectual reason.\n\nFace agnosia, along with the varied neuropsychological disturbances that may accompany it, can now be analyzed with experimental paradigms and correlated with neuro-anatomical loci of damage identified by neuroimaging methods.\n\n_Phineas Gage, a railroad foreman from Vermont in the United States, holding the rod that drove into his head. The unfortunate incident made him the first live case study for neuroscience._\n\nMy brain hurts!\n\nFortunately, Antonio Damasio in his popular book, _Descartes' Error_ (1994), demystifies it all. The Cartesian precept \u2013 \"I think, therefore I am\", was just plain wrong. \"I feel, therefore I am\" is more like it: or, in Damasio's words: \"we are not thinking machines, we are feeling machines that think.\"\n\nOf course, this is rather convincing proof of what one school of advertising has always believed. Rosser Reeves, turn in your grave.\n\nThe industry has implicitly absorbed these findings, although they are still insufficiently taught and explained. (A notable exception is my colleague, Chris Graves, who has read many academic articles on the subject and used them to distill afresh a scientific basis for Public Relations, which returns it to its roots as an applied social science. His brain does not hurt).\n\nUnfortunately, one of the besetting sins in our business has always been to grab on to the \"next new thing\", and treat it as a universal panacea. Thus was invented the pseudo-science of neuro-marketing.\n\nCorrelating the impact of marketing activity on the brain can be interesting \u2013 but as any statistician will tell you, correlation does not equal causation. Increased neural activity in the insula, for example, is linked to feelings of love and compassion. So if a brand triggers activity in this region this could be a positive sign. However, activity in this region is also linked to functions of memory, language, attention... and anger, disgust, and pain. We are a long way from a sophisticated enough understanding of the functioning of the brain to directly link neural activity to purchasing patterns.\n\nDecartes' Error _by Antonio Damasio, which explicates the role of feeling over thinking \u2013 that man is above all an emotional rather than rational animal._\n\nAs Dr Molly Crockett from the Department of Experimental Psychology at Oxford advises, we must avoid \"neuro-bunk\". She says: \"We haven't found a 'buy' button within the brain... yet.\"\n\nAs such, our focus should remain on the outcome of actual behaviour, rather than the intermediate measure of neural activity.\n\nThere are plenty of ways to use the foundations of neuroscience to improve marketing and advertising without building a dedicated superstructure above it. The opportunity, as my inimitable colleague and co-founder of Ogilvy Change, Rory Sutherland, who has done as much as anyone to evangelize its relevance, is to create the messages and cues which work on the \"adaptive unconscious\". For that we owe the ground-breaking research to the psychologists Amos Tversky and Daniel Kahneman, though it was Kahneman who, in this case, did the popularizing. He divided the brain into two parts and \"branded\" them \"System 1\" and \"System 2\". System 1 is the adaptive unconscious \u2013 instinctive, habitual, and often irrationally guided by biases and external circumstances. System 2 is more thought out, condensed and rational. When you are driving a car you might be in System 2; when you're in a food shop you are more likely in System 1.\n\nAt the Hay Festival in 2012, Kahneman was asked, \"Is there such a thing as a System 1 business?\" \"Advertising,\" he replied.\n\nAs Rory points out:\n\nMost good people are instinctively good psychologists. What we have lacked to date is a vocabulary and body of scientific work which we can use to explain and justify their own often counter-intuitive recommendations.\n\nWhat is now becoming apparent is that at the heart of most human decision-making lies something called the \"adaptive unconscious\". Far from being the rather pervy, dark part of the brain, much of what it does might be broadly characterized under \"common sense\". A collection of reliable heuristics and rules of thumb, which, over time, have become embedded in our inherited mental hardware.\n\nThese mental processes are the product of evolution. They operate automatically, effortlessly and to a great extent beyond our conscious awareness. They control the behaviour of people not by generating reasons but by generating feelings (approach\/withdraw\/trust\/punish\/anxiety\/arousal). They do not necessarily obey the rules of conventional logic, but they do have a logic all their own. And the metalogic of the unconscious is much more influential than previously thought: in fact, much as we want to believe the opposite is true, our feelings give rise to our beliefs, not the other way around. We do not rationalize and then form feelings \u2013 we feel and then we post-rationalize (or \"confabulate\"). We can no more directly and consciously control the workings of our adaptive unconscious than we can directly control our heart-rate.\n\nKAHNEMAN SYSTEM 1 AND SYSTEM 2\n\n_Kahneman's System 1 and System 2 model, where System 1 is our adaptive unconscious with its unthinking emotionality, and System 2 our rational side._\n\nThe brain was also being re-evaluated from a very different angle. What exactly was homo economicus, and how did his or her decisions impact as behaviour on the economy in which they lived? The debate had begun in 1955, the year when David Ogilvy gave his famous speech on branding in Chicago. Herbert A. Simon had published an article, \"A Behavioural Model of Rational Choice\" in the _Quarterly Journal_ , which for the first time questioned whether economic behaviour was only rational. By modelling the various alternatives open to close, the relationships between them which determined the \"pay-offs\", and the order of preference of those pay-offs, he concluded that real behaviour was largely unlikely ever to conform to the rules. How, for instance, can one ever be sure that a pay-off X will emerge from an outcome Y? What happens when alternatives are evaluated separately rather than all together before the process of choice starts?\n\nThe economist Richard H. Thaler adds:\n\nFollowing Simon's lead subsequent research has identified not only 150 or more heuristics or biases (mental shortcuts) that lead us to often make irrational (or erroneous) decisions, but also that the presentation of too many options can lead us to make no decision at all. Contrary to the received wisdom that abundant choice makes it more likely homo economicus will make the decision, the \"paradox of choice\" (as coined by Professor Barry Schwartz) means that biases are more likely to come to bear and we will make a decision based on much more limited criteria, or even delegate or defer the choice.\n\nThe question then comes: how do we organize the messages and the cues in such a way that they might better push us into a change of behaviour? Otherwise, this will all remain fascinating but useless.\n\n**The Nudge**\n\nWhile Herbert Simon's approach was largely falling on deaf ears in the financial world as it was so far removed from the economic orthodoxy of the day, there were a group of economists from the Chicago School that developed his work much further. The most famous of these are Richard H. Thaler and Cass R. Sunstein.\n\nTogether they built on Simon's work by taking the irrational operator and categorizing him as a \"human\", as opposed to the fictional \"econ\" of neoclassical economics. Their work led them to look at the heuristics and biases that influence all manner of economic decision-making and particularly to advocate \"libertarian paternalism\" \u2013 encouraging governments to structure processes and systems in such a way that the most socially beneficial choices are the easiest to make.\n\nBRAIN MAP\n\n_The MINDSPACE framework was developed by a UK government office with input from academics and behavioural scientists. Working through each letter of the mnemonic \u2013 M for messenger, I for incentives, and so on \u2013 can generate practical ideas that instigate real behavioural change. Our client, Kimberly-Clark, adopted the framework and added an extra letter to create MINSPACER. The all-important 'R' \u2013 for reciprocity \u2013 asks how people might act for others, not just for themselves._\n\nTogether, they popularized \u2013 in an eponymously titled book in 2008 \u2013 the idea of a \"nudge\", a behavioural intervention which pushes people in the right direction. Nudges at their best are optional; they should help people do what they want to do anyway; and they should be easy to avoid. \"Choice architecture\" is the system by means of which we structure nudges.\n\nNudging became immediately interesting to governments: how to nudge people into better behaviour \u2013 be it paying more taxes or drinking less. President Obama hired Sunstein; Prime Minister David Cameron hired Thaler. Ironically, though, the application of behavioural economics in its purer form has been far more popular in the UK than the US where it was \"invented\" \u2013 something to do with its simultaneous adaptation built by the Cabinet Office and by the IPA (Institute of Practitioners in Advertising), who have stimulated debate and encouraged good practices.\n\nTo be honest, not all the experiments have been useful. Some findings seem blindingly obvious. Isn't it to be expected that an organ-donor programme where you have to opt out will show better results than one where you opt in?\n\nAnd it has taken time for its opportunities to manifest themselves in strategic approaches which \"fit\" the usual ways we do strategy. One such is the \"Mindspace Framework\" \u2013 a process geared towards changing behaviour rather than changing minds and attitudes. It is that rare thing: an academically validated framework that integrates the findings of cognitive psychology, social psychology and behavioural economics \u2013 around a real-world problem.\n\nOriginally developed by the Institute for Government and the Cabinet Office in the UK to influence policy, it can be applied to any behavioural challenge as a checklist for generating practical ideas. It was borne out of a collaboration between a multi-disciplinary team of academics and behavioural science practitioners in London, who conducted a meta-analysis of the hundreds of heuristics, biases and principles available, and collected them into the simple mnemonic MINDSPACE. Each letter stands for a fundamental behaviour change principle: M for Messenger; I for Incentives; N for Norms; and so on. As such, it is a useful tool in the arsenal of organizations like Ogilvy Change for turning decades of research into practical application.\n\n\"The essence of a nudge is to change behaviour without coercion, to work with people's existing intentions.\"\n\nNeuroscience helps us persuade better; behavioural economics assists people to make choices. They emerged from different academic disciplines, but are mutually reinforcing. Of course, there are those who dislike consumerism in principle and may see them as dark perversions, preying on the unwary. But it is a great mistake to blame these methods for something much more fundamental you disagree with. Like any other tool in life \u2013 including a spade \u2013 they can be used for good or ill purposes. In the main, they have made advertising more responsible, not less. And in the era of transparency, they are easily audited. They are the bright secrets, not the dark secrets of the Digital Revolution (see following pages).\n\nNudges \u2013 whether they are big ones or small ones \u2013 are becoming an increasingly helpful way of making communications grounded in a business issue which might not otherwise have been clearly identified as such. For instance, you will sell more of your nappies if you frame yours as a positive opportunity for bonding time rather than simply a routine negative behaviour \u2013 and, for instance, put the hug into Huggies again.\n\nThe essence of a nudge is to change behaviour without coercion, to work with people's existing intentions. An unexpected opportunity was seen in food-processing plants around the world, whose hand-hygiene practices often fall short of agreed standards, not because workers don't want to wash or don't understand what to do, but because they simply haven't built effective habitual behaviours. For Kimberly-Clark Professional, who design and create innovative hand-washing solutions, this presented an opportunity to nudge. It took a collaborative research and development project with Ogilvy Change to invent a big nudge, which would subsequently be shown to reduce instances of dirty hands on the factory floor by 63 per cent. This nudge took inspiration from a study conducted in a UK hospital and involved the creation of a washable ink hand-stamp that, when applied before the communal wash process, would put a nasty looking bacteria stamp to the backs of workers' hands. The specially formulated ink was just sticky enough to require diligent washing to remove it. This made previously invisible germs \"visible\", created a familiar trigger to build a washing habit and made good hand-washing practices a social norm (because after all, each individual can now see that everyone is else washing, too). This nudge was big because it changed behaviour significantly, but also because it materially enhanced the hand-washing product itself, something that a typical communications approach can rarely claim to achieve.\n\n_Smart ideas can nudge people into changing behaviours. Our client, Kimberly-Clark, used a big nudge to improve factory hygiene levels by applying a washable stamp to workers' hands, which could only be removed by purposeful scrubbing. A small nudge can be equally clever, such as re-designing the language and layout of tax forms to encourage people to be more honest in what they report._\n\nAn example of a smaller nudge can be seen on application forms. This nudge was to simply move the signature box to the start of the form, rather than the traditional position at the end, because such a move primes honesty and acts as a commitment device for applicants. A study by a car insurance company found that when they ran a randomized controlled trial (RCT) on over 13,000 applications, there was a 10 per cent increase in the honesty of the mileage reported in the \"sign at the top\" forms, which equated to an average of US$97 on annual insurance premiums. Such a nudge is fascinating not just because it is low cost and scientifically validated, but because it highlights how there are \"unseen opportunities\" to change behaviour by designing the world for how people _really_ behave, rather than how we traditionally _think_ they think.\n\n12 BRIGHT SECRETS: FROM EARLY NEUROSCIENCE TO CUTTING EDGE BEHAVIOURAL ECONOMICS\n\n**HABIT FORMATION**\n\nBehaviour becomes habitual through repetition.\n\n**EDWARD THORNDIKE**\n\n(1874\u20131949)\n\nPsychologist\n\n1905 (Thorndike): The Law of Effect. Behaviours followed by pleasant consequences are more likely to be repeated than behaviours followed by unpleasant consequences.\n\n2010 (Lally et al.): Habit formation modelled in the real world with eating, drinking and activity behaviours revealed that automaticity takes anything from 18 to 254 days to achieve.\n\n**Ogilvy Change: Stamping Out Bad Hand Hygiene**\n\nThe use of hand hygiene safety stamps in food-processing factories encouraged habitual hand-washing through repetition.\n\n**SOCIAL NORMS**\n\nBehaviour is influenced by the need to be publicly compliant.\n\n**MUZAFER SHERIF**\n\n(1906\u20131988)\n\nSocial Psychologist\n\n1936 (Sherif): Social norms were first illustrated by the \"autokinetic effect\". Frames of reference were modified in the direction of group conformity.\n\nMore recent research shows that normative messages have the power to influence behaviour in a range of decision-making environments (e.g. energy conservation) without being detected.\n\n**Ogilvy Change: Direct Debit Leaflet Optimization**\n\nNormative messages used to encourage direct debit sign-ups for UK council tax payment.\n\n**CHUNKING**\n\nSeparating long strings of information into smaller \"chunks\" allows for better memory retention.\n\n**GEORGE A. MILLER**\n\n(1920\u20132012)\n\nCognitive Psychologist\n\n1956 Miller's Law: The average person can hold seven chunks (largest meaningful unit of information) +\/- two in their working memory.\n\n2001 (Cowan & Nelson): A review of capacity limits revealed four chunks to be the mental storage capacity for short-term memory.\n\n**Ogilvy Change: Medical Adherence**\n\nSeparating a 21-day course of medication into three manageable chunks increased compliance by 21%.\n\n**CONFIRMATION BIAS**\n\nPeople favour information that confirms rather than challenges their current beliefs.\n\n**PETER WASON**\n\n(1924\u20132003)\n\nCognitive Psychologist\n\n1960 (Wason): People focus on confirming their initial hypotheses rather than challenging them.\n\n1979 (Lord): People discount evidence that challenges their beliefs regarding capital punishment.\n\nMRI scans reveal that areas connected with reasoning are subordinated to those connected with emotion when beliefs are challenged.\n\n**People are Fragile: Vancouver Pedestrian Awareness campaign**\n\nUsing the confirmation bias to reduce jaywalking in Vancouver.\n\n**AMBIGUITY AVERSION**\n\nPeople prefer known risks over unknown risks.\n\n**DANIEL ELLSBERG**\n\n(b. 1931)\n\nEconomist, Activist\n\n1961 Ellsberg paradox: thought experiment showed people's overwhelming preference for gambling on known chances over unknown chances.\n\n2010 (Alary): In the real world, ambiguity aversion leads to an increased demand for insurance because people are averse to unknown events that will affect their lives.\n\n**Transport for London (TFL): Pedestrian Countdown at Traffic Signals**\n\nThe introduction of countdown timers between red and green pedestrian signals makes the \"risk\" of crossing the road known and increases feelings of safety.\n\n**IDENTIFIABLE VICTIM EFFECT**\n\nThe extent to which the vividness created by one human example creates superior impact.\n\n**THOMAS SCHELLING**\n\n(1921\u20132016)\n\nCognitive Psychologist\n\n1968 (Schelling): An identifiable victim will stimulate a more powerful emotional response than a statistical victim.\n\n2013 (Genevsky et al.): fMRI revealed that effect elicited by identifiable information reliably shifts preferences for giving.\n\n**No More Abuse: Memac Ogilvy and King Khalid Foundation campaign**\n\nUtilizing the Identifiable victim effect to raise awareness of domestic violence.\n\n**CONCRETENESS EFFECT**\n\nWords which are concrete (real and tangible) are processed faster and more accurately than abstract ones.\n\n**ALLAN PAIVIO**\n\n(1925\u20132016)\n\nPsychologist\n\n1971 (Paivio): The dual-coding theory. Concrete words activate both the visual and verbal systems in the brain.\n\n1983 (Schwanenflugel & Shoben): The context availability theory. Concrete words activate broader contextual verbal support allowing for faster processing.\n\n2000 (Jessen): fMRI suggests that both models combined explains the effect.\n\n**Nutrition Action Health Letter: Center for Science in the Public Interest**\n\nWe can more easily process the unhealthiness of popcorn from concrete descriptions (equivalent foods) than abstract ones (grams).\n\n**ANCHORING**\n\nIn decision making, people rely very heavily on the first piece of information received.\n\n**DANIEL KAHNEMAN**\n\n(b. 1934)\n\nPsychologist\n\n**AMOS TVERSKY**\n\n(b. 1937-1996)\n\nPsychologist\n\n**1** 974 (Kahneman and Tversky): In fast decision making 8x7x6x5x4x3x2x1 elicited four times higher estimates than 1x2x3x4x5x6x7x8 showing participants' heavy reliance on anchoring to the first few digits in the calculation.\n\n2006 (Ariely et al.): People placed 60\u2013120% higher bids in auctions when anchored to the last two digits of their social security number if those numbers were high compared to low.\n\n**Durex Cheaper campaign**\n\nAdvertisement anchors people to the high cost of baby products putting the cost of the Durex product into context.\n\n**HYPERBOLIC DISCOUNTING**\n\nA time inconsistent model of people's greater valuation of immediate rewards over distant rewards.\n\n**GEORGE AINSLIE**\n\n(b. 1944)\n\nPsychologist\n\n1974 (Ainsley): An overwhelming preference for sooner, smaller rewards over later, larger rewards was demonstrated with pigeons.\n\nMore recent studies show that people choose $50 today over $100 in a year. However, when given the same choice shifted in time by five years, they prefer to wait an additional year for the greater reward.\n\n**Jason Vale: Dieting campaign**\n\nDieting campaigns famously attract people by focusing on quick fixes with more immediate rewards.\n\n**FRAMING EFFECT**\n\nPeople react to choices differently depending on how they are presented.\n\n**DANIEL KAHNEMAN**\n\n(b. 1934)\n\nPsychologist\n\n**AMOS TVERSKY**\n\n(b. 1937-1996)\n\nPsychologist\n\n1979 (Tversky & Kahneman): \"Prospect theory\". People make decisions based on the potential value of losses and gains rather than the final outcome.\n\n1997 (Rothman & Salovey): Loss-framed messages are more persuasive for risky behaviours and gain-framed messages are more persuasive for low risk behaviours.\n\n**Ogilvy Change: Grubs campaign**\n\nThis campaign encouraged people to try grubs by activating people's fear of missing out: \"Don't miss your chance to try\".\n\n**NEURAL COUPLING**\n\nWhen we witness another's actions, our brain mirrors the neural response and we \"feel\" what we see.\n\n**GIACOMO RIZZOLATTI**\n\n(b. 1937)\n\nNeurophysiologist\n\n1980's (Rizzolatti): A monkey's motor neurons fire when watching and performing grasping actions. Monkey see = monkey do.\n\n1980 (Chong et al.): Neural responses will mirror neuron properties found in humans.\n\n2010 (Stephens et al.): Similar brain responses found when telling and hearing stories as though we are acting out the story ourselves.\n\n**Ogilvy Change and Nestl\u00e9: United for Healthier Kids in Mexico**\n\nCreated the Emmy-nominated reality-TV show \"Hermosa Esperanza\", which followed five families adopting lifestyle changes which encourage children to be healthier.\n\n**GOAL DILUTION**\n\nThe more goals there are, the less likely we believe one of them is likely to be fulfilled.\n\n**YING ZHANG**\n\n(b. unknown)\n\nAssistant Professor of Marketing\n\n2007 (Zhang et al): An increased number of goals (e.g. building muscles and losing weight) is perceived as more difficult to achieve than when those goals are combined into a single means (e.g. exercising).\n\n**Ogilvy Change and Public Health England: The 10 Minute Shake-Up campaign**.\n\nMade exercise seem more achievable by focusing on a simple burst of activity twice a day.\n\n_My \"12 Bright Secrets\" are a set of individuals who have each advanced an important area within neuroscience or behavioural economics \u2013 from \"habit formation\" to \"goal dilution\". Some of them are names you may have heard before, others are less well known but influential in their own right. As you'll see, the principle developed by each has been successfully applied in advertising to effect change._\n\nIt's clearly the case that choice architecture is made for the digital age. Not only do nudges help provide an integrating \"glove\" for digitally-led programs, but they are directly enabled by social media in particular, creating immediacy in helping a big decision, for instance, or turning otherwise private actions like voting into behaviours people can publicize, share and talk about.\n\nEven the most traditional actions can be treated with behavioural economics. In one of the largest ever randomly controlled trials, at the 2010 US Congressional Elections the Facebook news feeds of nearly 60 million people were modified. Salient information like the location of the nearest polling station, an \"I Voted\" button, and pictures of six friends who'd already done so were added to the News Feed. The results were conclusive with an additional 340,000 people going to the polls. The analysis showed that not only were people more likely to vote if they just saw other people they knew were doing the same, but even their more distant friends of friends were triggered as well. The seemingly inconsequential addition of a button and a picture can be enough to foster a behaviour as profound as deciding who governs the country.\n\nDavid Ogilvy is popularly quoted as saying: \"The trouble with market research is that people don't think what they feel, they don't say what they think and they don't do what they say.\"\n\nDecoding the human brain will never definitively solve those conundrums. But it has helped a bit. We can now apply a little more rigour to our understanding of why they may feel differently to what they say; and then to nudge them a little into actually doing something.\n\n_Social networks provide a boost to the polls. Peer pressure from \"I voted\" badges alongside pictures of Facebook friends encourages political involvement._\n16THE NEW SHAPE OF THE WORLD\n\n_Ogilvy on Advertising_ contained one chapter with a provocative title: \"Is America still top nation?\" Its influence is still huge, but \"top nation\" would be difficult to defend as a claim. The world is getting a different shape.\n\nIn September 1995, I arrived in Hong Kong as the Regional Director for Asia. In 1991 Ogilvy & Mather had opened its first mainland office, in Shanghai. Doing business on the mainland required a joint-venture contract with a domestic company, so we signed with the Shanghai Advertising Group. We have always benefited from having a first-mover advantage. Our first (and continuing) Chairman there was TB Song, a kind of Chinese David Ogilvy, endlessly curious and very literary (he would, before Ogilvy became too big, buy a copy of every new book he liked for each of our staff). TB was someone who seemed to define \"the Tao of advertising\", but nonetheless was a vigorously single-minded driver of growth. He had formed the first agency with foreign investors in Taiwan in 1985, later to become Ogilvy & Mather Taiwan. Over on the mainland, China still had only representative offices, which was the state of affairs until TB moved to Shanghai in 1991 to establish our new joint venture. So welcome was he in China that the Government gave him \"grace and favour\" lodging in the Summer Palace in our early days.\n\nAs I arrived, TB had just had a request for the hire of a significant executive turned down. This man was not cheap, but was going to be the catalyst of our next stage of growth. The refusal came from on high \u2013 from Farm Street, from Mayfair, and from Martin Sorrell himself of WPP.\n\n\"The mountains are high and the emperor is far away\" is a very useful Chinese proverb if you are an expatriate manager, but I never found it worked particularly well with Martin. I argued the case; Martin refused. I argued again; he refused again. And again. Finally, there was the response: \"China is a black hole. On your head be it.\" Martin will (I hope!) forgive me for saying this, but at the time he felt much more comfortable in India than in China (well, they played and talked about cricket). That changed very quickly; and he has been the person who, more than anyone else, saw the potential of Chinese advertising, and then relentlessly drove WPP's growth there. No city has ever been too far, no client ever too remote, for him to visit.\n\nBy 2017, China (including Hong Kong and Taiwan) accounted for over $1.6 billion of WPP's annual revenue; WPP is at least double the size of any of its competitors in the region. So much of that growth has been driven by the Digital Revolution. It has become a mature advertising market, which has enforced the Western theory of branding wholesale. I have always believed that there is something about the rich symbolism inherent in Chinese culture that has made it especially receptive to brands. It's very fertile territory. And, then, the concept of branding fitted perfectly into a government agenda committed to making state-owned enterprises modern customer-focused organizations.\n\nHaving said all that, the Chinese characters for brand   denote \"Ming Pai\" \u2013 _famous_ brand \u2013 and give a Chinese take on the concept, which has been rather more to do with fame than with uniqueness. This is epitomized by the infamous CCTV auctions, when all that matters is whether a brand gets the slot viewed by 130+ million people at once: instant fame. The Digital Revolution, however, as we'll see, changes that model.\n\n_We created the signature HelloMoto campaign for Motorola from China in 2004, then ran it widely in the US. It marks the beginning of a shift in seeing China as not just a huge target market, but an enormously creative one._\n\nWe quickly expanded beyond the global clients into the state-owned enterprises and private Chinese companies; and from the two big coastal cities, Shanghai and Beijing, into the provinces, where we believed the differential growth would be. For me, a proud moment came in March 2004, when the first Chinese-produced advertising, Motorola's HelloMoto campaign, ran in the US.\n\nIt was symbolic of the small beginnings of an impending shift of influence in the world. Asia, so long the importer of creativity, actually could export it. A one-way world could become two-way.\n\nAnd it was around this time that an iconic chart started circulating, which to me, a historian manqu\u00e9, was most seductive. It showed the global share of trade represented by China over time. Between ad 1 and the middle of the nineteenth century, it held steady above 25 per cent, number one in the world. In 1820, it reached a remarkable high-water mark of 33 per cent. Then it slid drastically to a low of 4.6 per cent in 1950.\n\nBut by 2017, China's contribution to the global gross domestic product stood at 14.84 per cent, not quite the record level it enjoyed in 1820 but a significant number \u2013 second only to that of the US and growing fast. It's back to the future, in a very big way.\n\nIn India, the advertising stimulus has been less digital, since, of all the markets in the world apart from Japan, it has remained addicted to the commission system. The earning of fees for digital work has therefore been artificially repressed. It has also been constrained by a complex and multi-layered wholesale and retail distribution system \u2013 the lack of a \"modern trade\" \u2013 which is what drives the expansion of advertising in the developing markets, as an efficient \"puller\" of demand. In other words, India's great advertising days are still to come. We have just seen a foretaste. It is _quantitatively_ under-playing.\n\nHowever, it has _qualitatively_ over-performed. An efflorescence of creativity has placed India among the most lively and exciting advertising markets in the world. Piyush Pandey, our Indian Chairman, who started life as a tea taster, is recognized as the prime mover. He demonstrates that the beliefs of David Ogilvy about what makes an idea \u2013 and then what makes an idea big \u2013 can have an Indian manifestation that owes nothing to the West, culturally, but draws from the deep, deep waters of Indianness; and Indianness is a multi-faceted and brilliant thing.\n\nAs a result of the personal charismas of Piyush and his colleagues in the industry, Indian advertising has developed a status in society difficult to find elsewhere in the world \u2013 film-star status, and very well deserved, too.\n\nIt is underpinned by something very important: Piyush has never forgotten where he came from. As he says, he does not wear India on his sleeve, but he does wear it on his heart. He avers he learnt as much from the carpenters who made his family's furniture as a child as he ever did formally. He combines great humility with a larger-than-life charisma. And he is generous to the last wicket, cricket being his abiding love, and team play such a natural analogy for producing great advertising.\n\nPiyush's advertising is culturally rooted. As a result, he has helped build definitively Indian brands \u2013 Pidilite, for instance \u2013 by telling powerful stories that resonate.\n\nPiyush's work is so popular, it's inherently viral. It spreads. But in his words: \"Let's all remember Paduyikiphed. If the story is not great or if the story is not human, no technology will save you.\"\n\nHe reminds us that it's not enough to be a multinational. You have to be multicultural \u2013 and multicultural even within India: to be Telugu or to be Tamil as much as to be Hindi. As Piyush once told me: \"Never forget that when a Rajasthani visits Kerala, he is stunned that the same coconut oil that he uses on his hair is used in cooking the fish he is eating.\"\n\n_India has become one of the most creative hubs in advertising, one where the industry still has a touch of Hollywood (or Bollywood) glamour to it \u2013 and it produces some of the best creative ideas in the world. This brilliant ad from Fevicol is a memorable example, where the product benefit is depicted by a common site across India \u2013 people clinging to the roof of a bus. Its inherent \"Indianness\" in both storytelling and tone is unlike anything elsewhere. And most importantly, the brand certainly gets stuck in your head!_\n\n**Velocity**\n\nI have no doubt whatsoever that these anecdotal characterizations of China and India do betoken a new shape of the world.\n\nBut, of course, it's broader than that.\n\nFor many years I, like many others, used in presentations, meetings, reviews and plans the concept of BRIC (Brazil, Russia, India, China), but I did so with increasing distaste. This was an acronym useful at a time in history, but also one which reflected the interests of those who coined it \u2013 in this case, Goldman Sachs. It was a financial markets' view of the world. There is nothing at all wrong with that. But it does not reflect the world of marketing and advertising, and of the masses of consumers (and businesses) who fuel it.\n\nThere is no shortage of demographic predictions about the future of the world, but it seemed to me that none of them really hone in on what makes advertising tick. So I asked a distinguished economist and demographer, Dr Surjit Bhalla, to go back to data purely from the point of view of where the consumer future lies. The results are startling.\n\n\"The global middle class is on the cusp of what can only be described as an explosion.\"\n\nThe global middle class is on the cusp of what can only be described as an explosion. By 2025, it will have increased by 35 per cent \u2013 rising to 4.6 billion people, almost 60 per cent of the world's population.\n\nBut the surprising thing is what the new shape of the world will look like.\n\nThere are 12 countries which are growing disproportionately: India, China, Pakistan, Indonesia, Bangladesh, Nigeria, Egypt, The Philippines, Vietnam, Brazil, Mexico, and Myanmar (and, one which will \u2013 politics permitting \u2013 quickly join the list, Iran). These are the V12 or Velocity 12 countries.\n\nIf you want a genuinely futurist index for charting bottom-line impact, this is it. It is composed of two primary indicators \u2013 a country's share of middle-class consumption in the world; and the projected rate of growth (or velocity) of this consumption, adjusted for Purchasing Power Parity.\n\nThis gives another angle to the concept of inequality. The world when measured by asset distribution is not equal; and grossly unfair concentrations of wealth exist, be it in 1 per cent or 5 per cent of the population. But when measured by income distribution, the world _is_ becoming more equal: the ground occupied by a middle class is becoming bigger and bigger.\n\nAt Ogilvy & Mather we like the word \"velocity\": it signifies more than just the economic growth of countries, which the expansion of middle-class consumers enables, but also the pace at which their daily life (increasingly digitally enabled) and their sentiments evolve.\n\nOf course, no projection will be perfectly fulfilled, and short-term bumps in the road there will be. Governments can accelerate or decelerate the velocity. The \"Lula middle-class\" of some 140 million people in Brazil grew up as a result of government policy in the administration of President Lula in the 2000s; as those policies have subsequently foundered, their purchasing ability has collapsed (and their sentiment has become dangerously embittered). But the cycle will turn \u2013 be it in two, five or 10 years.\n\nTHE NEW SHAPE OF THE WORLD:\n\nGROWTH IN MIDDLE-CLASS CONSUMERS 2015-25\n\n| **2015** | **2025**  \n---|---|---  \n**MEXICO** | **93** | **106**  \n**BRAZIL** | **140** | **159**  \n**NIGERIA** | **33** | **61**  \n**EGYPT** | **56** | **81**  \n**PAKISTAN** | **63** | **122**  \n**INDIA** | **431** | **828**  \n| **2015** | **2025**  \n---|---|---  \n**MYANMAR** | **13** | **26**  \n**INDONESIA** | **139** | **189**  \n**VIETNAM** | **30** | **49**  \n**BANGLADESH** | **22** | **59**  \n**CHINA** | **758** | **945**  \n**PHILIPPINES** | **25** | **43**\n\n  2025   NET GAIN (MILLIONS)\n\n_Source: Bhalla, Surjit S: Second Among Equals \u2013 The Middle Class Kingdoms of India and China; 2007; revised and data updated till 2025._\n\n_Cartographers, avert your eyes! If the world map reflected middle class population growth, it would look something like this. North America and Western Europe give way to a dominant Asia and South America._\n\nThere's one overwhelming thrust to this view of the world, and that's the shift to Asia; but within that a more dramatic shift to South Asia. It's not a shift that is yet viscerally recognized as something real by most multinational businesses. I set up our office in Pakistan in 2007. I can demonstrate there a strong consumer base; can show that it's much easier to grow a business to a particular fraction of an average Indian business than it would be to expand that Indian business by the same amount; can point to the exceptional local talent; can exhibit the success stories of a few multinationals who _have_ taken it seriously: and I still feel that I'm preaching in a wilderness of prejudice.\n\nThe fact is that we will be moving well beyond the Rise of China. In the next two decades, Chinese velocity will abate. India, Pakistan, Bangladesh, Indonesia and Vietnam will take up the slack. In particular, the next half-century will be India's: and yet in my view, most global marketers have not yet adjusted psychologically to this fact.\n\nWhat will this new world feel like?\n\nWell, very different to the world the West thinks of now.\n\nIt will be female, Muslim and urban.\n\nWomen are entering the labour force in the millions, spurred by better education. The Muslim middle class of the V12 will rise to 583 million people by 2030. And India will add 400 million, China 300 million and Nigeria 200 million new city dwellers by the middle of the next century.\n\nThose women, even in traditional societies, are starting to eclipse men in purchasing decisions; Muslims are looking for Halal values behind the products they buy. And the city dwellers will increasingly rely on digital connectivity as both citizens and consumers.\n\nThe digital age will have a very different global flavour. It's probably time to wake up to that fact.\n\n\"It's the V12 markets which will lead the world into the next big leap of internet growth.\"\n\nThe connectivity it brings is the dividend of this digital re-shaping of the world. Most commentators to date focus on the sheer scale of the advance of the internet.\n\nI remember visiting Facebook's Menlo Park HQ in its early days: conference rooms were named after countries representing the numbers of new users found: 75 million users \u2013 that's the Turkey room. It seemed a little overweening and, anyway, they soon ran out of rooms.\n\nIt's the V12 markets which will lead the world into the next big leap of internet growth. Of the unconnected globally, nearly half reside in China, India, Pakistan and Bangladesh. While the internet in India took more than a decade to move from 5.5 million to 100 million users, and another five years to reach 375 million users, it could reach 600 million by 2020. Six of the top 10 markets in internet growth are V12 markets. Low-tech markets like Bangladesh and Myanmar are emerging as \"straight to smartphone\" markets.\n\nThere are two big implications of this.\n\nThe first is that these populations want local content. Conceived in an English-speaking bubble, the mature internet now is non-English.\n\nAnd the second is that velocity is the sister of connectivity. Where penetration grows, economic growth roles accelerate disproportionately.\n\n**The Internet with Chinese Characteristics**\n\nThe exemplar of both is China: an internet that has gone its own way. I sometimes have difficulty in persuading my American clients that the digital ecosystem in China has become more sophisticated and more far-reaching than it is in the US. The \"Great Firewall\" acts as a perceptual screen, and is conflated into a general miasma of China detraction. Yet the reality is that, just as Deng Xiaoping's formula has worked for socialism, so it has worked for the internet: digitalization with Chinese characteristics.\n\nIn September of 1987, just a few subway stops to the west of Ogilvy's Beijing offices, at the Institute of High-Energy Physics, China sent its first international email. The famous subject header read: Crossing the Great Wall to Join the World\n\n\"I sometimes have difficulty in persuading my American clients that the digital ecosystem in China has become more sophisticated and more far-reaching than it is in the US.\"\n\nIronically, the first 30 years of China's internet has not been characterized by \"joining the world\". Far from it. The prevalent theme has instead been one of a domestic internet that has, in many ways, developed in parallel to the world's internet.\n\nAlthough the sender of that email was wrong about the first 30 years, he might be right about the next 30. Looking forward, I think we will see an outbound flow of internet innovation. Arguably, the non-Chinese internet will begin to look more and more like its Chinese counterpart. And as this happens \u2013 and it has already started \u2013 marketers will be looking to China to understand how to address challenges and opportunities faced years ago by their more experienced Chinese counterparts.\n\nEvery commentator has their own way of describing the scale of China's internet \u2013 a population of 688 million, at the end of 20153. To drive home the enormity of this number, some will tell you that that's twice as many Chinese internet users as there are Americans. Others might note that there are more Chinese internet users than there are people in Germany, Iran, Turkey, France, Thailand, the UK, Italy, Colombia, Spain and Canada combined. Wittier China watchers might tell you there are more Chinese internet users than there are teenagers in the world, or than there are cats or dogs alive today.\n\nWhat made it possible? And so quickly?\n\n\"Chinese leadership believed in the internet as the basis for a modern society.\"\n\nFirst, the government. Chinese leadership believed in the internet as the basis for a modern society. Early evidence of this includes the funding in 1988 of the country's first email system (via packet-switching), which provided a digital connection among governmental agencies and academic institutions in nine large cities, including Shanghai, Beijing and Guangzhou. In August 1993, Premier Li Peng approved a $3 million expenditure to begin building a national digital network providing access to public economic information; this was known as the Golden Bridge Project. A watershed event occurred in 1994, when China's internet was connected \u2013 over infrastructure provided by Sprint \u2013 to the global internet. Over the next several years, Chinese computer scientists and electrical engineers networked among their foreign counterparts, chiefly in the US and Japan, to accelerate the growth of China's internet. Today, only a small segment of the nation's population, 20 million rural people, have no broadband access, and efforts are underway to get them connected.\n\nAt the same time, the government protected the local platforms as they developed; their international rivals were either blocked altogether or severely constrained.\n\nMeanwhile the traditional media, largely government-owned, had gaps that the new media have been able decisively to fill. Dull and distrusted, Chinese internet users deserted state-controlled media and turned to each other for new ideas and fresh information: social media filled the gaps. Similarly, gaps in China's fragmented and creaky offline retail environment allowed sellers quickly to build distribution online at scale.\n\nIn the West, the predominant view of the Chinese internet is that it is fatally repressed by government censorship. Far from it. The controls that exist have done little to rein in the rampant usage of netizens. The few topics the government bans are of interest only to a minority. This is not to apologize for censorship, but simply to describe the reality \u2013 even though I know it is difficult to accept if you have not actually lived or worked in China. Far from feeling like an oppressed population with no freedom of speech, the vast majority of Chinese internet users behave online exactly as they would like to. We know: we watch them hourly.\n\n_It might look like an ordinary office, but it isn't. Our agency in Shanghai closely monitors real-time dashboards that provide instant metrics on activity across social platforms like WeChat, Weibo and Baidu._\n\n**Welcome to BAT!**\n\nBAT is the tri-partite world of the Chinese internet: Baidu, the biggest domestic search platform; Alibaba, China's number one ecommerce platform; and Tencent, the owner of China's biggest instant messaging apps.\n\nBAT is presided over by the \"three kings\", each one brilliant in his own way: Robin Li, the multi-billionaire founder of Baidu, the search engine with a phenomenal 80 per cent share of the Chinese market; Jack Ma, the self-made mogul who launched the ecommerce giant Alibaba in 1999; and Pony Ma, the instant-messaging pioneer who founded Tencent, which created WeChat, the largest instant messaging platform in the world.\n\nFollowing a theme of Chinese history, each king is building his kingdom to be as complete as possible.\n\nCan a winner take all? It's called the One Emperor Syndrome. So, they all now have a video site: Alibaba's TMall Box Office (TBO); Tencent's QQLive; and Baidu Video.\n\nAnd increasingly, they go to a client and offer a one-stop shop: so Alibaba will seek to bundle a client's logistics with its media and with its content. That doesn't just seek to disintermediate an advertising agency; it knocks it out of the game altogether. Fortunately for us, the individual offers are not, in the end, compelling \u2013 a case of generalization finding it difficult to trump specialization.\n\nInnovation is the driver of BAT. It started with Sina Weibo, a microblog platform, which began with a similar functionality to Twitter: 140-character text updates in reverse chronological order. Then, later on, it tended to innovate much more; Twitter has followed, with the addition of video, for instance.\n\nRobin Li, the graduate from New York State, chose the name Baidu for its meaning: \"search for one's own dream\", a reference to a Song Dynasty poem. His fierce battle with Google led to the latter's retreat from the Chinese market.\n\nJack Ma, 5 feet (1.52m) tall, lean and charismatic, could be China's richest man, by some counts. He was famously rejected for 30 jobs before becoming an intent entrepreneur. But he's also a self-confident face and voice of China.\n\nPony Ma founded Tencent in 1998. More private, even secretive, than the others, he at various times worked in the lowliest roles \u2013 including a janitor \u2013 to keep the company afloat. That's no longer necessary.\n\n_The three kings: Robin Li of Baidu, Jack Ma of Alibaba and Pony Ma of Tencent, who preside over China's largest search, ecommerce and messaging platforms respectively._\n\nBut it's WeChat that is most remarkable \u2013 and most revolutionary. A messaging, social and commerce app, it has fuelled China's use of QR codes \u2013 a technology that had fallen out of use almost completely in the West. Thanks to the platform, the scanning of QR codes is now ubiquitous in China: used by friends to exchange contacts, by marketers instead of links, and by merchants to facilitate payment. Ogilvy's staff in China have QR codes on their business cards.\n\n_This is a chart we show to our clients in China to explain the Chinese internet. BAT dominate, with each individual company having a near-monopoly on its primary business. For Baidu, that's search. It is China's answer to Google. For Alibaba, that's ecommerce. In fact, it's the world's largest ecommerce company (by number of sales, not revenue). Its pearls are Taobao and Tmall, which are the two primary ecommerce sites in China. For Tencent, that's social. The company owns both WeChat, the innovative and enormous mobile-based social app, and the QQ messenger service. Those are just the primary business lines. The BAT companies also dominate travel, e-payments, maps, games, entertainment, news, and security._\n\nThe killer advantage is the cross functionality. Without leaving WeChat, its hundreds of millions of users split the bill with friends, buy flowers, and even book doctor appointments. Facebook, Twitter and other Western platforms, although woefully behind in this regard, will be scrambling to provide similarly rich, multi-functional experiences over the next few years.\n\nAs WeChat continues to be the most used messaging app in China (impressive, since it launched a mere six years before this book went to press), advertisers on WhatsApp are struggling to make the platform work for them. WeChat itself is clear that it is morphing from a platform into a lifestyle brand: it does not want to be defined by anything that has gone before it. The average user uses WeChat 11 times a day.\n\nFor agencies, by the way, it has changed the way we relate with our Chinese clients. We often present work on WeChat, and client feedback more often than not will come on WeChat.\n\nThere is also much less time for \"over-thinking\". I find that Western advertisers operating in China often bemoan what they see as a lack of strategy: clients in China rush to execution without the due diligence that might be applied in the West. Yet rather than laziness and a lack of rigour, might this be a sane response to shorter attention spans and the fragmented media \u2013 trends we're likely to see more of in the West? Time will tell whether Chinese advertisers are right to think less, do more.\n\nThe export of China's social and digital innovation is a trend certain to continue. Chinese advertisers may even have reason to be smug, as their Western counterparts rush to make sense of the changes on the way. What can we then learn from Chinese marketers already succeeding in this world?\n\n**1. Social as a mindset**\n\nSuccessful Chinese brands have, on the whole, skipped the setting up of separate social, or even digital, departments within their organizations. In doing so they've avoided the silo structures that often stymie success in the West.\n\nSocial in China, more than anywhere else, has the power to make and break brands. In this most connected nation in the world, 91 per cent have at least one social media account. This massive population follows an average of eight brands via social, which gives these users enormous influence over the brands choices their peers make.\n\nVISIT BRITAIN\n\nWHAT'S IN A NAME?\n\nQUITE A LOT AS IT TURNS OUT...\n\nAmong young Chinese wealthy enough to travel, Britain has a reputation for being unwelcoming compared to other countries.\n\nSo, our agency helped VisitBritain \u2013 the government organization responsible for tourism to Great Britain \u2013 to get better at speaking their language. How? By letting Chinese citizens rename British landmarks.\n\nFor the first time in history, one country invited the citizens of another to come up with new names for famous places. The campaign kicked off with a call from the British Ambassador to China, alongside cinema and outdoor advertising, to submit names for over 100 iconic British attractions online.\n\nPeople immediately rose to the challenge. One participant suggested axing the name Sherwood Forest in favour of \"The Forest of Chivalrous Thieves\". Another renamed London's Shard the \"Tower to Pluck the Stars\", while the Lord Mayor's Gherkin was reduced to \"The Little Pickled Cucumber\".\n\nSo, we made the offer more enticing \u2013 come over and claim a landmark in person! Over 13,000 people did, including a die-hard Beatles fan who raced to rename Abbey Road. Each participant was rewarded by their suggested name appearing on Wikipedia, Baidu and Google Maps.\n\nAn unusual campaign, it unleashed online advocacy through a social idea and came to life in traditional channels too. Seen over 300 million times on Chinese social networks, it earned around \u00a515 million in free media. But the biggest success? Brits were seen as a little less standoffish, and Chinese tourism to Great Britain increased 27 per cent that year.\n\nYet despite its importance, social is not seen as a separate team, nor is it a separate line item, or a separate type of campaign. And never has it been.\n\nAdvertisers here know that to succeed in social \u2013 to get people talking about your brand in a positive way \u2013 you don't need to do something called \"social\". They know that the best social campaigns are not social campaigns. The best social campaigns, as measured by their ability to create positive word of mouth, are good TV commercials, good events, good PR campaigns and, even, good products.\n\nTo succeed in China has been to succeed in doing things outside of social, but with social in mind.\n\n**2. New interconnectivity between social and ecommerce**\n\nMore ecommerce promotions and exclusive products are built specifically with campaigns in mind. Brands such as Ray-Ban and Bulgari have created products specifically for launch via Chinese apps: Nice, a photo-sharing platform like Instagram, and Meipai, a social\/video platform akin to Vine.\n\n**3. Co-development, with consumers, of products using online communities**\n\nChinese marketers have been skilled in listening to the online conversation. In some cases, entire communities are engaged online to develop new iterations of products. Consider Xiaomi. The smartphone maker, claimed by some to be one of the fastest growing tech companies ever, owes its success partly to its online communities that provide feedback and suggestions to the operating system, which is updated weekly.\n\n**4. Is radio coming back to life?**\n\nChina consumes more podcasts per person than any other country. Twelve per cent of Chinese smartphone users claim to consume podcasts, compared with 5 per cent in the US and 4 per cent in the UK. Its music consumption is also growing faster. Marketers in China have been quick to respond to this, leading to a revival in the audio ad \u2013 a sleepy backwater for many years in Western advertising.\n\n**5. Bringing internet culture to the mass market**\n\nChina's internet culture spills into mainstream culture faster than in Western countries, and Chinese advertisers have been quick to exploit this. FatCat pet toys, a series of colourful, inexpensive (less than $2 apiece), plush toys for cats and dog, are available on Alibaba's AliExpress ecommerce site only to those who opt in to chat about the products. In other words, social media is the key that unlocks the privilege to buy the product. The mere act of engaging with others via a social platform is viewed as an inducement to buy a product.\n\n**Mobile Africa**\n\nIf China provides the first-era model of digital velocity, I think Equatorial Africa will provide the second. There's a mobile revolution at work \u2013 and one built around mobile money. The laying of cables along the coasts of East and West Africa has been accompanied by the arrival of affordable smartphones. In Kenya, they account for 50 per cent of mobile phone sales. But the innovation here has been mobile payments. Equatorial Africa has the majority of the world's mobile money accounts, linking the unbanked to a world of choice and security often lacking in cash-based operations.\n\nHow did it come about?\n\nA consortium of researchers unearthed that consumers in Uganda, Botswana and Ghana were trading mobile phone airtime as a proxy for currency. This caught the attention of a Kenyan mobile network operator, which supported the launch of M-Pesa \u2013 a mobile system for sending money \u2013 in 2007. Since then, it has grown in size, scope, and reach. M-Pesa can now be found in many African countries, India and Eastern Europe, and it accounts for fully 25 per cent of Kenya's GDP. M-Pesa has inspired another business called M-Kopa, which lets people in rural areas pay a small monthly fee wirelessly to have solar lighting and mobile-phone charging in their homes. Other, alternative payment forms are facilitating commerce in previously unbanked communities, ranging from Bitcoin in Afghanistan, to bKash in Bangladesh and Pay TM in India.\n\nThe mobile money culture has led to some exciting developments. Nigeria had a slower start with ecommerce. There are real barriers to wider adoption of online shopping: cost of delivery, concern about getting the right product and worry about the security of online payments. But enter Jumia, which builds in flexible, mobile-first payment structures and a pay-on-delivery model. It's now Nigeria's most trusted ecommerce brand.\n\nFinally chamas are credit unions popular in African economies. They offer consumers the means to create \"virtual stores\" for bulk purchase and distribution to individual consumers. Text messages or bulletins sent through messaging apps provide the access while mobile vouchers and mobile payments help smooth the transactions.\n\n**The End of Globalization?**\n\nFor marketers and communicators, \"right shaping\" to this new world is an existential challenge.\n\nOf course, it all depends on whether you are primarily a local brand or an international brand.\n\nLocal brands are no longer pallid, \"me-too\" feel-a-likes. They represent much more than half of our business, and they utilize all the tools and techniques of their global competitors. But they have one massive asset: their superior closeness to their customers. And in large part, this is enabled by their ability to think digital-first, whereas a global brand tends to think platform-first (because it is looking for efficiencies of scale) and digital second.\n\nWhenever I go to Manila, an irresistible craving comes over me. I yearn for a Jollibee's Peach and Mango Pie. Yes, I can recommend it as one of the most delectable desserts you could ever have, and I have over-indulged in it all over the archipelago. Jollibee leads Filipino fast food, above both McDonald's and KFC. It fills the \"need space\" usually occupied by McDonald's, which has been a formidable and aggressive attacker. When they created a campaign, \"What's in Joy's chicken?\", in which women named Joy confirmed that Chicken McDo was their chicken friend of choice; Jollibee hit back with a strong dose of Filipino pride, #ChickenJoyNation. Jollibee remains undeterred.\n\n_Chicken fights in the Philippines: when McDonald's relaunched its Chicken McDo, the local brand Jollibee with its Chickenjoy quickly asserted its position and responded._\n\nAn example of localism is Wardah, a personal care brand in Indonesia designed on Halal principles for progressive Muslims. It prospers in Indonesia's fertile social landscape through a digital engagement using hashtags such as #CantikHariDati (meaning Pretty Heart) and featuring users' Twitter and Instagram posts on its website, providing feedback for customers to see other real users and learn from them.\n\nMeanwhile, global brands no longer automatically enjoy the privilege of preference. Our research shows that in the V12 markets the proportion of consumers preferring local brands exclusively is twice as high as those preferring international. Of course, most do continue to buy both.\n\nBut, for the international brands, this presents a complex environment to navigate through. In fact, my client Alan Jope of Unilever says that, \"it takes marketing into the world of the Rubik cube\".\n\nYou have a local competitor that builds a millennial franchise in new channels in secondary cities using digital media. There's no off-the-cuff defence strategy that can be created in London or New York for that. So whereas the television age enabled one-size-fits-all global advertising with translation and minimal transculturation, the digital age has pushed the game the other way. To compete, you have to go digital (20 per cent of the global budget would be a modest number). And to go digital, you have to be, to some extent, local. That's not to say that there's no such thing as a \"global digital campaign\". There is. And it works by a process of hierarchy: a global strategy; a global creative platform; a global playbook of digital programs \u2013 but then there comes the point where every playbook fades and the highly localized, sensitive playground takes over. It's a very finely balanced art, because communications now meets politics. Can one be digitally impactful and still responsible to the strategies and creative definitions? Or is one impelled by local control of budgets and a sense of \"not invented here\" to indulge in a sense of localness which may not be responsible behaviour for the brand as a global whole? This will remain a perennial debate in the digital age.\n\n\"Rubik marketing\" is compounded by the multinationals' lack of stretch. It's the biggest change I have lived with over the last 10 years \u2013 the phenomenon of market drop off \u2013 where whole markets simply go \"dark\". Many of the dark markets are, ironically, the markets of the future.\n\nThe reasons for focus are unanswerable, but the consequences are unavoidable \u2013 it's like licensing competitive advantage in the long term. Enter the new Asian brands to fill the vacuum.\n\nAnd the case of Japan is a different type of example. In the 1970s, 1980s and 1990s, its value dramatically skewed the P&L of any multinational. Now, it is as if it has bowed, and politely left the world to which international companies market.\n\nThe Rubik syndrome isn't just something that affects Western brands. Back in October 2004, I organized a conference for Chinese companies that were contemplating a move into the rest of the world. It coincided with the Chinese government's statement of an objective to build Chinese brands. We held that conference at Touffou, and Herta Ogilvy recalls that the biggest fear of these early Chinese globalists was not the complexity of global markets but the presence of thirteenth-century ghosts in their bedrooms. More than 10 years on, Chinese brands represent the soft side of the hard power of the world's second largest economy.\n\nThat is not to say that they find it easy. In fact it is very tricky for them. The traditionally hierarchical structure of a Chinese company finds the fluid management of a global matrix unfamiliar. And, there is a \"manufacturing superiority complex\" baked into the culture of many enterprises, not to mention a (now changing) reluctance to configure products according to the tastes of foreign consumers as opposed to domestic ones. It works here in a market of 1.3 billion, so why change it?\n\n_Dabur is an Indian beauty brand with a rich local pedigree. It has a nuanced understanding of a strong cultural stream in Indian thought \u2013 Ayurveda \u2013 a system evolved over millennia relating to the continent's unique therapies and herbal compounds. This isn't just heritage positioning, it is a modern brand connecting well culturally through vivid, creative, rich stories about ingredients, responsive communications, and a utility portal to showcase beauty videos. Dabur uses these deep, local roots to protect against international competition._\n\nOf all the Chinese companies, the one that has committed the most seriously and successfully to branding in the digital age has been Huawei.\n\nA client meeting at Huawei's headquarters in Shenzhen is unlike any other. It's not the sheer scale of the multi-building campus (housing nearly 60,000 people, most of whom look like students); nor the Babylonian feel of the building where visitors come to be wowed by the Huawei experience; nor the frisson of realizing that all this has come about in just 25 years or so. Rather it is the sheer gutsy energy of the meetings, the sense that everything is impelled by a missionary, yet at the same time, very pragmatic, zeal.\n\n_Huawei's campus in Shenzhen, Guangdong, houses over 30,000 of its employees. The company will be one of the defining multinationals of the future._\n\nWhat is driving this? One of the most interesting business cultures to have emerged in the digital era, and the creation of Ren Zhengfei \u2013 who deserves a place in the global digital pantheon, which is as yet unfairly denied him in the West. He has a very clear sense of objective: amidst the deluge of data: \"Google will provide the water, and we will provide the pipelines\". He is driven as much by a sense of history and philosophy as by the future. (He has, for instance, encouraged Huawei's top 20 managers to study the rise and fall of the great nations from the sixteenth century, admires the British Glorious Revolution of 1688 and quotes the Duke of Wellington).\n\nPerhaps this is what gives him a uniquely Eastern-Western perspective of the \"Age of And\".\n\nHuawei culture officially lauds \"greyness\". The Chinese concept is _\"hui\"_   \u2013 which represents those shades of grey which lie between black and white. It implies the ability to be flexible, to compromise, to \"control the pace\", without losing the sense of direction. This does not imply softness: the wolf is the animal metaphor much used in Huawei, but also a distinctly Chinese \"animal\", the legendary \"Bei\"   This wolf had long front legs and short hind legs; the Bei has short front legs and long hind legs. \"The most effective organization is one that has Wolves and Bei cooperating closely\", Mr Ren says. And however vulpine and Bei-like the culture, all are encouraged to be restlessly self-critical, a Chinese sense of \"divine discontent\": \"people in ancient times even put aside a room for self reflection. Can't we learn to reflect on our failure?\" Such reflections have driven remarkable success. Huawei will be the first Chinese global brand \u2013 a global brand that happens to come from China.\n\nHuawei, Haier, Lenovo all continue their global advance. They will be joined by other \"new globals\", from India, Mexico and Indonesia in the first wave. And they will be part of a digital economy and a \"two-way\" world that was just inconceivable in 1982.\n\n_Mark my words, Huawei will be the first truly global brand originating from China. Just look at_ Shark Dancer _, an early experiment in content for them that beautifully emphasizes the power of touch to allow us to communicate. This is a brand on the attack._\n17CULTURE, COURAGE, CLIENTS AND CASTANETS\n\nIt's a sultry spring evening in Seville in the late 1980s. Fatigue and release, the end of another training programme. We're in the bar. And suddenly there is, from amidst our number, the strident sound of heels on the floor. It breaks the ennui as surely as a storm. The group opens out, and there's Jost van Nispen, no graceful Andalucian but one of the best direct marketers of his generation. A Dutchman, he moved to Spain and became besotted with it. And taught himself to dance. Nostrils flared, he looks very creditable. And then there's the sound of castanets.\n\nApplause. And somehow that memory \u2013 the clack-clack-clack of the castanets \u2013 sticks, one of hundreds of such moments in a career, trivial, transitory, fundamentally uninteresting except to those who were there \u2013 but accumulating like sediment to create a culture.\n\n_We are fortunate that David's opinions are inexhaustibly quotable. As an agency, we treasure his teachings in a little red book of quotations, which serves as a guide to our enduring company culture._\n\nIt helps if that culture rewards such memories, and treats them as significant. David Ogilvy created a culture like that \u2013 one of the strongest in the business. It shares a system of meaning, from his many aphorisms to his blind faith in the colour red.\n\nI'm often asked to define that culture.\n\nMy answer is that it is a humanistic culture, not a mechanistic one. It stemmed from David's interest in and liking for people (except very boring people, the \"dull dogs\" he advised agencies not to succor). In this he fell into the same camp as peers such as Bill Bernbach and Leo Burnett, but in a very different camp to Rosser Reeves, his one-time brother-in-law, who seems to me to have been fundamentally mechanistic in his approach to both life and business. When Ted Bates \u2013 his agency \u2013 was eventually sold (to WPP in 2003, and thus partly merged with Ogilvy & Mather), I remember some of the survivors there describing how completely void of culture it was. It was like they were walking out of a gulag and into a cathedral. Of all the founder cultures, David's has remained the most intense and vital. (Of course, I'm prejudiced.)\n\nMy predecessors and I have actively marketed it. It's powerful, because it combines a view about what good advertising is with some very basic rules of proper good behaviour and civilized etiquette.\n\nThere is no either\/or here. This is a big case of both\/and. The happier the agency, the better the work. Someone unkindly once said that joining Ogilvy & Mather was more like joining a cult than an agency. But, it tends to work. As other strong agency brands have found, taking the medicine which we prescribe to clients makes people want to join more and leave less.\n\nThere is no shortage of theories as to what makes an organization \u2013 collectively \u2013 courageous. Let me add mine: and it is quite possible that it is distorted by my experience in a creative services agency.\n\nI think it is directly related to the degree to which leadership can inculcate a day-to-day sense of being dissatisfied with where you are. The courage emanates not out of grand injunctions to be brave, but out of a sense of insecurity about where you are: that it's not good enough, that others are doing better, that something's missing, that you need another leap forward, that if you don't do something big, you'll... and so on. Courage is born of discontent.\n\nThe philosopher Jos\u00e9 Ortega y Gasset captured this sense of restlessness, a knowledge of what could be, coupled with a sense that we often fall short:\n\nThe essence of man is, discontent, divine discontent; a sort of love without a beloved, the ache we feel in a member we no longer have.\n\nHe wrote that in Argentina in the 1940s; and David used the same phrase, consciously or unconsciously, when he wrote: \"We have a habit of divine discontent with our performance. It is an antidote to smugness.\"\n\nIt's what leads you to tear up work and start again; to present something that hasn't been asked for; to break the conventions of a category; to ignore a research result if it doesn't feel right.\n\nI was once asked to assemble my ideal advertising agency from famous figures. I chose them because of their Divine Discontent. Enter BFME, the newest kid on the block (see here).\n\nDavid described the skills required to run an agency in _Ogilvy on Advertising_ as those of:\n\n... enthusiasts. They are intellectually honest. They have the guts to face tough decisions. They are resilient in adversity. Most of them are natural charmers. They are not bullies. They encourage communications upwards, and are good listeners.\n\n**The agency start up which never made the front cover of _Ad Age_ or _Campaign_...**\n\nThey remain.\n\nBut the digital age imposes some very specific requirements for someone who aspires to run an agency. Now, we would have to add three more qualities:\n\n1.You have to be a simplifier: because everything is so much more complex, it's not enough for managers just to preside as often they do \u2013 the law of complexity says that it, not they, will start driving outcomes. The ability to identify the simple core needs of a project or a mission does usually demand intellectual acuity. It's ironic, but dunces aren't great simplifiers. After that it requires a ruthless ability to weed out all that is not necessary \u2013 in thinking through it, in processing it and in delivering it. \"Is it really necessary?\" is the question the parrot on your shoulder has to ask each waking minute. Distilling that aim in the compelling language of lucid simplicity distinguishes the great simplifiers from the merely good ones.\n\n**The most brilliant agency ever? Designed for Divine Discontent, recruited from history. Here's whom I chose to run it:**\n\nSARAH BERNHARDT **(1844\u20131923)**\n\nThe \"divine Sarah\" of Oscar Wilde, discontent ran through every vein of this actress's body. She slept in coffins to better project herself into the mood of death. When she was dying, she chose to live on a 5th floor. One of her visitors asked why, and she said, \"so that when a man enters my bedroom he'll still be panting.\" That's the passion that attracts clients. She's the **Client Service Director**.\n\nCHRISTOPHER MARLOWE **(1564\u20131593)**\n\nThe dramatist, the best wordsmith money could buy, and great to go drinking with. Discontent killed him, unfortunately \u2013 he died in a tavern brawl. His heroes are somehow hipper than Shakespeare's, his more famous contemporary. His words terrify, as for instance, from the mouth of the dying Tamburlaine: \"Come, let us march against the powers of heaven\/And set black streamers in the firmament\/To signify the slaughter of the gods.\" More \"black streamers\", please. He's the **Creative Director**.\n\nSIGMUND FREUD **(1856\u20131939)**\n\nThe Viennese provocateur, mercilessly probing the unconscious for insights. He's the **Account Planner**. In Civilization and Its Discontents (1930) he sees love as the way of galvanizing creative endeavour, which helps us re-find our lost selves. \"The way of life which makes love the centre of everything, which looks for all satisfaction in loving and being loved.\" I have never seen great creativity come up out of hate. Hating consumers, hating clients: it's surprisingly common.\n\nCHARLES EAMES **(1907\u20131978)** \/\/ RAY EAMES **(1912\u20131988)**\n\nThe husband-and-wife team gave design a cultural and social importance that energized post-war America and shaped the twentieth century. What a team from hell to manage, but what divine discontent. From a converted garage in Venice, LA, Charles's mechanical mind and Ray's avant-garde leanings transformed the abstract into beautiful yet practical design. As Charles said: \"The details are not the details. They make the design.\" An early boss of mine used to repeat: \"Remember, God is in the details.\" At the time, I never knew why. Charles and Ray are the **UX team.**\n\n2.You have to be a collaborator, because nothing anymore is really apart: no man, but now no business, is an island. Some managers pretend theirs is, jealously guarding work they believe to be their proprietary skillsets. That \"proprietariness\" is a delusion. In the interconnected world, even if it did exist, it would be largely useless. I always identify two types: the crocodiles, who are snarkily concerned with how big is their slice of pie; and the lions, who grow the pie. If you have IP it will be enhanced and empowered by collaboration, not diminished. And if you are ambitious, you can harness the skills and assets of others. Profiting from but actually enjoying ambiguity is what makes you great. Certainty is the barbiturate of small minds.\n\n3.You have to be curious, because, if you aren't, you just won't attract the people you need to work for you. The best of them now are driven by curiosity. They are not excited by status or by money or by conventional badges of success, but they are insatiably curious \u2013 and their curiosity will drive the restlessness, the ferment, and the inventiveness which are in themselves intrinsically rewarding. Can an organization learn? If there is a tipping point at which curiosity at least corrects the natural tendency of organizations to conform, then it can. And, when it does, it will gather to itself a kind of differentiating charisma that kills the addiction to the comfortable, which will always drag it down.\n\n**Partners and Servants**\n\nOf course, all the above, however true I believe it to be, does not mean very much unless it is put in the context of why we are here at all. \"Wouldn't this be a wonderful business without clients?\" is the age-old question of the agency folk, in all agencies, at all levels.\n\nMaybe it would.\n\nBut it's probably more helpful to our survival to consider how we turn our clients into supporters rather than wishing them away.\n\nIt's easy to oscillate between contradictory views about what constitutes the ideal relationship of an agency with a client.\n\nClients get the advertising they deserve. I know some who are a malediction, and others who are an inspiration. Don't keep a dog and bark yourself. Any fool can write a bad advertisement, but it takes a genius to keep his hands off a good one.\n\nSince that era, the industry associations in both the UK and the US have worked hard to project the image of advertising as a profession, worthy of taking its place alongside accountancy or architecture. To some extent, they have succeeded, though such efforts have always run the risk of pushing too hard in the direction of its strategic role at the expense of its creative role, something to which the Digital Revolution has actually been a corrective.\n\nSuch efforts encourage our view that we are partners to clients. Our plea to them so often is for partnership. And it is certainly true that the best work invariably comes from relationships which are deep, long term and which have at their core the premise of partnership \u2013 a shared sense of responsibility for the results.\n\nOne danger of the partnership syndrome is that it creates blindness in agency people as to the real amount of time a client actually has to spend talking with us. We fail to see the pressures they face; the other priorities which pull them; the inconvenience which time wasted through a poor response can cause.\n\nThinking too much about partnership can lull us into a sense of security. We are a service business \u2013 and our partnerships are only as good as our last piece of work was liked (or, less importantly, worked). David intuitively recognized this. And is quite explicit about it.\n\nHe was quick to point out where processes were slowed down and produced bad advertising in the process. As he remarked in his book:\n\nOne of the biggest corporations in the world allows five levels to chew up its advertising. Each level has the power to veto, but only the Chief Executive Officer has the power of final approval. Don't strain your agency's output through more than two levels.\n\nI think the reason for this lies in the fact that while, for accountants, sets of accounts are, in the main, objectively assessed things, we have to sell, after all the strategic work is done, a product on which opinions will always contain some subjectivity. Cajoling, encouraging and assisting a client through the process of committing eventually to something subjective requires service skills with a higher-than-usual emotional component. Successful advertising people are hard-wired for that after years of experience. That is why it is challenging for clients successfully to cross over to the other side: there are few examples, and I would be hard put to name any who have been highly successful.\n\nQuite simply, it's difficult to turn from master to servant. So however much we may get close to realizing the aspiration of partnership, it behoves us to remember that we are partners _and_ servants, as I have tried to remind generations of trainees. At root, we are all waiters and waitresses really \u2013 however much we may not like to admit it.\n\nThere's one thing, though, which transforms the client-agency relationship, and that is actually liking your client. This sounds very trite, but, like many obvious things, it still deserves to be stated. The clients I have gone on to become friends with, we tend to do the best work for. They trust us and like us in return. Conversely, clients are quick to sense when we don't really like them. I've told my trainees that they have to think of a new client as being like a semi-trained and insecure dog: they sense hostility and they detect fear unconsciously.\n\nRespect for clients has, perhaps, diminished over time. My first client meetings were formal set pieces. When a client such as Cow & Gate came up from Wiltshire to see my agency in Lintas, London in the late 1970s, it was a big day out; rehearsed presentations followed by a sumptuous lunch in a private dining room until it was time for them to catch the early evening train home. The first time I organized a lunch like this, I was surprised to be asked what ice sculpture I wanted for the table centrepiece. I chose a dolphin playing with a ball.\n\nIn the digital age, it is so normal for ideas and programmes to be presented virtually, by video conference, or even, in China, on WeChat in text-based dialogues. But I believe that this does not displace the need to have personal contact \u2013 face-to-face, work into leisure \u2013 with clients. In fact, it makes it more imperative. And explains why many digital micro-shops fail to grow. They don't have the service gene.\n\nMore than ever, we need to re-personalize the de-personalized.\n\nSo, it's time to hear the castanets again. A shared culture of fun between client and agency is the strongest bond possible \u2013 assuming that the work is good. And agencies which appear to be having fun are \u2013 in my experience \u2013 invariably more attractive than those which do not.\n\nNow, sometimes I see the words \"have fun\" on a list of objectives presented by an office head. I know there is no fun to be had there. At least, I suppose, it is recognized as a need. But fun does not come organized from on high; it bubbles up from below. Like the flamenco, there is something spontaneous, zestful and uncontrollable about it.\n\n _Ol\u00e9! As\u00ed se baila!_ (That's how you dance!).\n18EPILOGUE\n\nDavid Ogilvy succumbed to the blandishments of the publisher of _Ogilvy on Advertising_ and wrote 13 predictions. In a similar spirit, here are mine.\n\n1.TV will continue to be the lynchpin medium, albeit transformed.\n\n2.The Indian ad market will be the most attractive in the world.\n\n3.The mobile handset will become as important to the whole of humanity as the bedtime pillow.\n\n4.\"Content\" will cease to be a dirty word and will increasingly distinguish brands that offer a service to customers from those which don't.\n\n5.WeChat will surpass Facebook.\n\n6.Virtual reality will find a niche, but will not take over the world.\n\n7.Top-notch writing skills will carry a huge premium as they decrease in supply.\n\n8.\"Pure play\" agencies will wither, or become main plays.\n\n9.Candidates for political office will continue to use dishonest advertising.\n\n10.New Asian multinationals will create the world's most dynamic brands.\n\n11.The debate as to whether our business is an art or a science will carry on forever, without resolution.\n\n12.A breakaway from Cannes will form, which is more purely creative.\n\n13.The word \"digital\" will ultimately disappear.\n\nENDNOTES\n\nCHAPTER 3: THE SHORT MARCH\n\n**** Orrin Edgar Klapp, _Opening and Closing_ , Cambridge University Press, 1978.\n\n**** John Lorinc, \"Driven to Distraction\", _The Walrus_ , 12 April 2007.\n\n**** Quoted in The Slant, Grapeshot Q&A, grapeshot.com, 28 May, 2015.\n\nCHAPTER 4: THE DIGITAL ECOSYSTEM\n\n**** Steve Miles, speech\n\n**** Pete Blackshaw, quoted in _The Internationalist_ , February 20, 2015.\n\n**** Holt, Douglas, _How Brands Become Icons_ , Harvard Business School Press, 2004.\n\n**** Professor Mark Ritson in _Marketing Week_ , 10 May 2012.\n\nCHAPTER 6: THE POST-MODERN BRAND\n\n**** Holt, Douglas, ibid\n\n**** Arthur W. Page Society, \"The Authentic Experience\", An Arthur W. Page Society Report.\n\nCHAPTER 7: CONTENT IS KING; BUT WHAT DOES IT MEAN?\n\n**** Charlene Jennett, Anna Cox, et al, \"Measuring and Defining the Experience of Immersion in Games\", _International Journal of Human-Computer Studie_ s, 66 (9): 641-666, September 2008.\n\n**** Jamie Madigan, \"The Psychology of Immersion in Video Games\", psychologyofgames.com, 27 July 2010.\n\n**** Robin Sloan, \"Stock and Flow\", Snarkmarket, 18 January 2010.\n\nCHAPTER 8: CREATIVITY IN THE DIGITAL AGE\n\n**** Tim Broadbent, _The Ogilvy & Mather Guide to Effectiveness_, 2012.\n\n**** Les Binet and Peter Field, \"Marketing in an Era of Effectiveness\", World\n\nAdvertising Research, 2007.\n\n**** To read David's great speech, plus a lot more of David's memorable writing, dip into a copy of _The Unpublished David Ogilvy_ , 1986.\n\n**** Tham Khai Meng in an all-company memo, 2009.\n\nCHAPTER 9: DATA: THE CURRENCY OF THE DIGITAL AGE\n\n**** Binet and Field, ibid.\n\nCHAPTER 10: \"ONLY CONNECT\"\n\n**** John Battelle, \"The Database of Intentions\", John Battelle's Search Blog, 2003.\n\nCHAPTER 11: CREATIVE TECHNOLOGY: THE SWEET SPOT\n\n**** Peter Merholz and Don Norman, \"Peter Merholz in Conversation with Don Norman about UX and Innovation\", Adaptive Path, December 2007.\n\nCHAPTER 12: THE THREE BATTLEGROUNDS\n\n**** Jeff Bezos's opening remarks in a letter on Amazon's homepage to introduce the Kindle.\n\nCHAPTER 14: FIVE GIANTS OF ADVERTISING IN THE DIGITAL AGE\n\n**** _Otaku_ is a peculiar Japanese term for those with obsessive interests.\n\n**** _Advertising Age_ , February 2014.\n\nCHAPTER 15: MY BRAIN HURTS\n\n**** David Ogilvy, _Ogilvy on Advertising_ , 1985, p.217.\n\n**** Antonio R. Damasio, D. Tranel and Hanna Damascio, \"Face Agnosia and the Neural Substrates of Memory\", _Annual Review of Neuroscience_ , 13(1), 89-109.\n\n**** Chris Graves, \"Brain, Behavior, Story\", Ogilvy Public Relations, 2014\n\n**** Sutherland, Rory, (2014, December) This Thing For Which We Have No Name, Edge Interview], retrieved from [www.edge.org\/conversation\/rory_sutherland-this-thing-for-which-we-have-no-name\n\n**** Richard H. Thaler, _Misbehaving: The Making of Behavioral Economics_ , W.W. Norton & Company, 2015.\n\n**** Dolan, P., Hallsworth, M., Halpern, D., King, D., Metcalfe, R., Vlaev, I. (2009) MINDSPACE: Influencing Behaviour Through Public Policy. London: Cabinet Office & Institute for Government.\n\n**** Lisa Shu, Francesca Gino, Max Bazerman, et al, \"When to Sign on the\n\nDotted Line? Signing First Makes Ethics Salient and Decreases Dishonest Self-Reports\", Working Paper 11-117, Harvard Business School, 2011.\n\nCHAPTER 16: THE NEW SHAPE OF THE WORLD\n\n**** CCTV's main evening news broadcast, Xinwen Lianbo, has a daily viewership of roughly 135 million, and is also one of the most expensive shows for ad buys, with its 2013 advertising slots selling for a record 5.4 million yen.\n\n**** Data from the World Bank, February 2017, retrieved from <http:\/\/databank.worldbank.org\/data\/download\/GDP.pdf>[  \n](http:\/\/databank.worldbank.org\/data\/download\/GDP.pdf)and  \nRobbie Gramer, \"Here's How the Global GDP is Divvied Up\", _Foreign Policy_ , February 24, 2017\n\n**3** According to CNNIC (The China Internet Network Information Center).\n\nCHAPTER 17: CULTURE, COURAGE, CLIENTS AND CASTANETS\n\n**** Jos\u00e9 Ortega y Gasset (1916), from his course lecture, \"Historical Reason\", Buenos Aires, 1940.\n\n**** David Ogilvy, _Ogilvy on Advertising_ , 1985, p.67.\n\n**3** Bond, R.M., Fariss, C.J., Jones, J.J., Kramer, A.D., Marlow, C., Settle, J.E., and Fowler, J.H. (2012). A 61-Million-Person Experiment in Social Influence and Political Mobilization, Nature, 489 (7415), 295-298.\n\n**Additional reference material:**\n\nJeffrey Bowman, _Reframe the Marketplace_ , Wiley, 2015.\n\nRichard Dawkins, _The Selfish Gene_ , Oxford University Press, 2016 (40th Anniversary Edition).\n\nPaul Feldwick, _The Anatomy of Humbug_ , Troubador Publishing, 2015\n\nPaul Ford, \"What is Code?\", _Businessweek_ , 11 July 2015.\n\n(Bloomberg.com\/graphics\/2015-paul-ford-what-is-code\/).\n\n\"Timeline of Computer History\", The Computer History Museum\n\nShelina Janmohamed, _Generation M_ , I.B. Tauris, 2016.\n\nArthur Koestler, _The Act of Creation_ , Last Century Media, 2014.\n\nRick Levine, et al, _The Cluetrain Manifesto_ , Basic Books, 2000.\n\nDimitri Maex, _Sexy Little Numbers_ , Crown Business, 2012.\n\nNicholas Negroponte, _Being Digital_ , Vintage, 1996.\n\nDavid Ogilvy, _Confessions of an Advertising Man_, Southbank Publishing, 2012\n\nDavid Ogilvy, _Ogilvy on Advertising_ , Prion Books, 1983.\n\nDavid Ogilvy, _The Unpublished David Ogilvy_ , Profile Books, 2014.\n\nPiyush Pandey, _Pandeymonium_ , Penguin Books, India, 2015.\n\nRosser Reeves, _Reality in Advertising_ , Widener Classics, 2015.\n\nKunal Sinha and David Mayo, _Raw: Pervasive Creativity in Asia_ , Clearview, 2014.\nINDEX\n\n(page numbers in italic type refer to illustrations and\n\ncaptions; the initials DO refer to David Ogilvy)\n\nA\n\nAaker, David\n\nA.C. Nelson\n\n_The Act of Creation_ (Koestler)\n\nad-blocking software\n\nAddicks, Mark\n\nAdidas , __\n\nAd.ly\n\nadvertising giants in digital age 210\u201335\n\nGreenberg 210\u201316, __\n\nKagami 216\u201320, __ , __\n\nNisenholtz 220\u20133, __\n\nPalm-Jensen 224\u20138, __\n\nPorter 228\u201335, __ , __\n\nAegis.\n\nAfrica, and mobile phones\n\nAGA Cooker __\n\nAge of And , , ,\n\nAinslie, George , __\n\nAirbnb __ ,\n\nAldrich, Michael\n\nAli, Ben __\n\nAlibaba , __ ,\n\nAllianz\n\nALS Ice Bucket Challenge 168\u20139, __ , __\n\nAltaVista, search engine of\n\nAlways __\n\nAmazon __ , __\n\nand disintermediation\n\nand ecommerce growth\n\nmaking most of\n\nas one of digital \"big three\"\n\nand price comparison , __\n\nrising sales of\n\nAmazon Echo\n\nAmerican Express __ , , __ ,\n\nAmin, Salman 188\u20139\n\namyotrophic lateral sclerosis (ALS) 168\u20139\n\nAnacin __\n\n_The Anatomy of Humbug_ (Feldwick) , __\n\nAndreessen, Marc , __ ,\n\nApple 174\u20136\n\nMacintosh launched by\n\nApple Safari, and ad-blocking software\n\nArab Spring , __\n\nArcade Fire , __\n\nUnder Armour , __\n\nArmour bra __\n\nARPANET __ , ,  ( _see also_ internet)\n\ncreation of and first message via\n\nArthur W. Page Society\n\nAT&T , , __ , ,\n\nAudi , __\n\nB\n\nBaidu\n\nBaldwin, Jerry __\n\nBangladesh:\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nand velocity\n\nBanksy , __\n\nBaran, Paul , __\n\nBarneys __\n\nBarnum, P.T.\n\nBass, Saul\n\nBAT , __\n\nBates, Ted ,\n\nBattelle, John\n\nBattersea Dogs and Cats Home __\n\nbattlegrounds 164\u201391, __\n\nand continuous commerce 180\u201391, __ , __\n\nmobile phones 174\u201380, __ , __ , __ , __\n\nmobility 174\u201380\n\nsocial media 164\u201373, __ , __ , __ , __ , __ , __ , __\n\n\"The Beauty Inside\" __ ,\n\nBeckham, David\n\n_Being Digital_ (Negroponte)\n\nBenjamin, Jeff\n\nBeowulf , , __ , ,\n\nBerger, Barndon\n\nBerger, Jonah 78\u20139\n\nBernbach, Bill , , ,\n\nBerners-Lee, Tim , __ ,\n\nBernhardt, Sarah\n\nBezos, Jeff\n\nBFME ,\n\nBhalla, Surjit\n\nBieber, Justin ,\n\nBig IdeaLs , 60\u20131, __\n\nBildt, Carl 224\u20135\n\nBlackshaw, Pete\n\nBland, Christopher\n\nBMW , , __ , , ,\n\nBo, Armando __\n\nBogusky, Alex 230\u20134, __\n\nBogusky, Bill\n\nBottle Rocket __, ,\n\nBowker, Gordon __\n\nBowman, Jeff\n\nbrain and behaviour 236\u201347, __ , __ , __ , __ , _244\u20136_ , __\n\nand nudging, _see main entry_\n\n\"Brand X believes . . .\"\n\nbranded content 72\u20135\n\nbrands ( _see also individual_ brands):\n\nand Big IdeaLs , 60\u20131, __\n\nand China 263\u20135, __\n\nand content, _see main entry_\n\ncrisis of\n\ndeath of, as we know them\n\nerrors of, concerning mobile users\n\nfour locations of\n\ninclusive\n\nand most-viewed videos\n\nand one message at a time\n\norigins of , __\n\nperformance\n\npost-modern 54\u201371\n\nas promise\n\nand Starbucks, _see main entry_\n\n360-degree stewardship of\n\nand transparency\n\nand videos\n\nwaves of __\n\nwell-behaved\n\nBrandZ ,\n\nBranson, Richard\n\nBrazil:\n\ndisproportionate growth in\n\nmiddle-class consumers in , __\u2013 _3_\n\nand social networking\n\nBRIC\n\nBrien, Nick\n\nBrinker, Scott __\n\nBroadbent, Tim , __ ,\n\n_Built to Last_ (Collins)\n\nBurger King __ , __ , __ ,\n\nBurnett, Leo , ,\n\nBush, George W.\n\nC\n\nCabral, Juan\n\nCadbury 96\u20137, __ , __\n\nCaesar's Palace __\n\nCailliau, Robert , __\n\nCameron, David ,\n\nCampaign for Real Beauty 44\u20135\n\nCampbell, Joseph\n\nCannes , 98\u2013101, __ , __,\n\nin author's predictions\n\nCannes Lions __ ,\n\nCarter, Calvin 174\u20136,\n\nCentennials , , ,\n\nCerf, Vinton \"Vint\" , __\n\nCERN\n\nChannelNet\n\n_The Chaos Scenario_ (Garfield)\n\nChapman, Phil\n\nChen, Joan __\n\nChick-fil-a __\n\nChicken McDo , __\n\nChina:\n\nand ad typo\n\nand brands 263\u20135, __\n\ncompanies' conference in\n\ndisproportionate growth in\n\nfirst international email from\n\nand Huawei 264\u20135, __ , __\n\nand internet 255\u201360, __ , __\n\ninternet __\n\ninternet users in , __\n\nmiddle-class consumers in __\u2013 _3_\n\nand QI codes\n\nand rich symbolism\n\nand social as a mindset 258\u20139\n\nand velocity\n\nand Visit Britain , __\n\nand WPP revenue 248\u20139\n\nChipotle __ , , __\n\n_Choose Beautiful_\n\nChouinard, Yvon , __\n\nClark, Jim\n\nClark, Wendy\n\nClinton, Bill\n\nClinton, Hillary , __\n\ncloud technology 120\u20131, __ , __\n\n_The Cluetrain Manifesto_\n\nCOBOL\n\nCoca-Cola __ , , __ , , , , __ , , __ , , __\n\ncode 150\u20132, __\n\nCoke Zero 92\u20133, __\n\nCollins, Brian 113\u201314\n\nCollins, Jim\n\nCompact\n\nCompuserve\n\nCond\u00e9 Nast\n\n_Confessions of an Advertising Man_ (Ogilvy)\n\n_Contagious_ (Berger)\n\ncontent 72\u201392\n\nin author's predictions\n\nbranded 72\u20135\n\nand Digital Revolution 72\u20133\n\neight tips for\n\nimmersive 82\u20135\n\n\". . . is King\"\n\nmagnetic 77\u201382\n\nmatrix of 77\u201392, __\n\nas misused word\n\norganizing for 93\u20135\n\npractical 88\u201392\n\nsmart 85\u20138\n\nthinking about 75\u20136\n\nContent Studios 93\u20135, __ , __\n\ncontinuous commerce:\n\nas battleground 180\u201391, __ , __\n\nand digitally enabled sales, growth in\n\nand emotion 186\u20137\n\nand experience\n\nomni-channel 185\u20136, __\n\nand relationship 186\u20139\n\nCook, Tim\n\nCorporate Social Responsibility (CSR)\n\nCow & Gate\n\ncreative technology 150\u201364\n\nand back end 154\u20137\n\nand code 150\u20132, __\n\nand front end 153\u20134\n\nand marketing automation , __\n\nand virtual, automated and mixed reality 157\u201361, __ , __ , __ , __\n\ncreativity:\n\nand art-vs-science debate 101\u20132\n\nand Digital Revolution 98\u2013119\n\nand ideas 110\u201314, __\n\nmeasuring\n\npervasive 107\u201310\n\nand stories 115\u201316\n\nin technology, _see_ creative technology\n\nword hated by DO\n\nCrispin Porter , , __ ,\n\nCrispin, Sam\n\nCrockett, Molly\n\ncustomer relationship management (CRM) , __ , , 172\u20133\n\nD\n\nDabur __\n\nDamasio, Antonio , 237\u20138\n\nDamasio, Hannah\n\n\"dark markets\"\n\ndata:\n\nbig, cautions concerning 122\u20133\n\nbig, and emotion 186\u20137\n\nbig, first reference to\n\nand cloud technology 120\u20131, __ , __\n\nas currency of digital age 120\u201331\n\nand econometric modelling 127\u20138\n\nand insights 129\u201331\n\nand measurement vs effectiveness 126\u20137, __\n\nand metrics, underdosing with , __\n\nas \"new oil\"\n\nand optimal optimization\n\npervasive nature of\n\nand platform impediments 125\u20136\n\nreally useful, prerequisites of 125\u201331, __ , __ , __\n\nand single-enterprise point of view ,\n\nstructured and unstructured\n\nvalue spectrum of __\n\nDAVE 148\u20139, __ , __\n\nDavies, Donald , __\n\nDavies, Russell , , __\n\nDavis, Rob\n\nDawkins, Richard 79\u201380\n\nde Luca, Peter , __\n\nDean, Howard\n\nDEC\n\nDentsu , ,\n\n_Descartes' Error_ (Damasio) , __\n\nDescartes, Ren\u00e9 237\u20138, __\n\nDhingra, Rishi\n\ndigital ecosystem _32\u20133_\n\nevolving nature of\n\nand traditional media\n\nDigital Revolution:\n\nadvertising giants in, _see_ advertising giants in digital age\n\nand analogue vs digital __\n\nand aroma\n\nand brands, _see main entry_\n\nbright secrets from 244\u20135\n\nand changes in advertising 39\u201340\n\nconnectivity as great gift of\n\nand content 72\u20133\n\nand creativity, _see main entry_\n\ndata as currency of 120\u201331\n\nand ghettoization ,\n\nand lessons for leaders\n\nprimary motivation of\n\nand social contract __\n\nand storytelling\n\ntimeline of 26\u20139\n\ntransformations brought about by, _see_ transformations in digital age\n\nand two-way communication\n\ndigital video 145\u20137\n\ndisintermediation\n\nDomestos __\n\nDomino's Pizza , __\n\nDoubleClick ,\n\nDove , 44\u20135, __ , __ , , __ , ,\n\nDove Men +Care\n\nDrax\n\nDunkin' Donuts\n\nE\n\nEames, Charles ,\n\nEames, Ray\n\nEgypt:\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nrevolution in __\n\nEisenhower, Dwight D.\n\nEllsberg, Daniel , __\n\neMarketer\n\nEngel, Joel S.\n\nEngelbart, Douglas , __\n\nEnigma machine , __\n\n_The Epic Split_ , __\n\nEquitable\n\nethnicity 70\u20131\n\nEuropcar , __\n\neveryday.com\n\n_Evolution_ , __\n\nEyeViewDigital.com __\n\nF\n\nFacebook , 82\u20133, __ , __ , __\n\nand ALS Ice Bucket Challenge\n\nin author's predictions\n\nand \"The Beauty Inside\"\n\nand Centennials\n\nchange in nature of\n\nand CRM __\n\nand digital-media spend\n\nand disintermediation\n\nas ephemeral network\n\ngrowing use of\n\nand IFLScience\n\nlaunch of\n\nand Nestl\u00e9\n\nand nudging __\n\nand Old Spice\n\nas one of digital \"big three\"\n\nand politics __ , , __\n\nrevenue of\n\nand tourism __\n\nand Toyota\n\nand underserved communities\n\nand video 40\u20131,\n\nFacebook Live\n\nFacebook Messenger\n\nFairnington, Alan\n\nFallon, Jim\n\nFanta __\n\nFarfar 226\u20138\n\nFatCat\n\nFeldwick, Paul 101\u20132,\n\nfeminism 66\u20138\n\nFetherstonhaugh, Brian __\n\n_Financial Times_ ( _FT_ )\n\nFirefox, and ad-blocking software\n\n5G\n\nFord __\n\nFord, Paul , __\n\nForgotify\n\nForster, E.M. ,\n\nFoster, Norman\n\nFrench, Neil\n\nFreud, Sigmund\n\nFriendster __ ,\n\nFrisch, Ragnar\n\nG\n\nGage, Phineas , __\n\nGallop, Cindy\n\nGalloway, Scott\n\nGambino, Melody\n\nGarfield, Bob\n\nGates, Bill\n\nGE\n\nGehry, Frank\n\nGeico\n\nGeneral Foods\n\nGeneration C\n\nGeneration S ,\n\nGeneration X ,\n\ngenerations:\n\nlabelling 46\u20137, __ , __\n\nMillennials 46\u201353, __ , __ , _52\u20133_\n\nGensemer, Thomas\n\nGerber , __\n\nGilbert, Dan\n\nGillette\n\nGoldman Sachs\n\nGoogle\n\nAdWords of ,\n\nas archival network 41\u20132\n\nand content consumption\n\ncost of, per person\n\nas database of intent\n\nand digital-media spend\n\nand disintermediation\n\nDoubleClick acquired by\n\nand Millennials' search behaviour , __\n\nand non-internet channels , __\n\nas one of digital \"big three\"\n\nand page ranking\n\n_PC Magazine_ 's verdict on\n\npersonalized search offered by\n\nrevenue of\n\nand same-sex marriage , __\n\nand underserved communities\n\nand video 40\u20131,\n\nGoogle Chrome:\n\nand ad-blocking software\n\nand Arcade Fire video , __\n\nGoogle Instant\n\nGoogle Now ,\n\nGoogle Quality Score\n\nGoTo.com\n\ngovernment, as transformation in digital age 197\u20139, __ , __\n\nGOV.UK , __\n\nGrace, Topher\n\nGrapeshot\n\nGraves, Chris\n\nGreen, Rick\n\nGreenberg, Bob 210\u201316, __\n\nGreenberg, Richard\n\nGrindr, and disintermediation\n\nGropius, Walter\n\nGuinness ,\n\nH\n\nHaier\n\nhaptics , , __\n\nHaring, Keith , __\n\nHarrison, Steve 118\u201319\n\nhashtags __\n\nHeath, Robert ,\n\nHegarty, John\n\n_The Hero with a Thousand Faces_ (Campbell)\n\nHershey\n\nHicks, Jeff\n\nHiga, Ryan\n\nHiggs boson\n\nHolt, Douglas 55\u20136\n\nHoney Maid __\n\nHopkins, Claude , ,\n\n_HotWired_ , launch of\n\n_Howards End_ (Forster)\n\nHuawei 264\u20135, __ , __\n\nHuggies __ ,\n\nHumby, Clive\n\nI\n\nIBM , __ , , __ , , , __ ,\n\nand CICS\n\nand cloud technology __\n\nand \"PC\"\n\nand SCAMP\n\nIBM Newsroom\n\nIBM Watson , __\n\nimmersive content 82\u20135 ( _see also_ content)\n\nIncredible India __ ,\n\nIndia:\n\nand Ayurveda __\n\ndisproportionate growth in\n\ngeneral election in\n\nand Indianness , __\n\nless digital advertising stimulus in\n\nmiddle-class consumers in __\u2013 _3_\n\nand non-internet channels\n\nand tourism\n\nand velocity\n\nIndonesia:\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nand velocity\n\nInstagram , , __ ,\n\nas ephemeral network\n\nin Philippines\n\nintegration:\n\ndeep 135\u20136,\n\nevolution of __\n\nIntel __ ,\n\ninternet ( _see also_ ARPANET; World Wide Web):\n\nadvertising pays for\n\nadvertising transformed by\n\nback-of-an-envelope beginnings of __\n\nbirth of\n\nbubble, bursting of __\n\nand China 255\u201360, __ , __\n\nChinese users of , __\n\nfounders of 14\u201315, __\u2013 _15_\n\nlooking for a commercial model\n\nmarch in progress of 26\u20139\n\nand media interests\n\nand memes 79\u201380\n\nand post-modern brands\n\nriven by conflict\n\nand targeting\n\nand \"viral\"\n\nwhat to give up instead of 48\u20139, __\n\nInternet Explorer, and ad-blocking software\n\nInternet of Things __ ,\n\nintimacy in depth 147\u20138\n\nIran, disproportionate growth in\n\nIslam 70\u20131\n\nIsobar\n\nIwata, Jon\n\nJ\n\nJ\u00e4germeister __\n\nJanmohamed, Shelina\n\nJim Beam __\n\nJobs, Steve ,\n\nJohn Street __\n\nJohnnie Walker 56\u20137, __\n\nJollibee\n\nJope, Alan\n\n_Journal of Advertising Research_\n\nK\n\nKagami, Akira 216\u201320, __ , __\n\nKahn, Robert , __\n\nKahneman, Daniel 238\u20139, __ , , __, , __\n\nKaisi, Katsura\n\nKestin, Janet __\n\nKFC\n\nKimberly-Clark 242\u20133, __\n\nKind Collective\n\nKindle __\n\nKirkland, Mike\n\nKlapp, Orrin , __\n\nKoestler, Arthur , __\n\nKotler, Philip\n\nKrazy Glue\n\nKuntz, Tom\n\nL\n\nLange, Martin\n\n_Late Night with David Letterman_\n\nLego __\n\nLenovo ,\n\nLever Brothers\n\nLGBTQ 68\u201370, __ , __ , __\n\nLi Peng\n\nLi, Robin , __\n\nlibertarian paternalism\n\nLicklider, Joseph Carl Robnett , __\n\n_Lifeline_\n\nLincoln, Abraham\n\nLinkedIn, launch of\n\nLorinc, John 22\u20133\n\nLuma Partners, chart developed by __\n\nM\n\nMa, Jack , __\n\nMa, Pony , __\n\nMcCann\n\nMcCarthy, John , __\n\nMcDonald's ,\n\nMCI\n\nMacintosh, launch of\n\nMcLuhan, Marshall\n\nMadigan, Jamie\n\nMaex, Dimitri , ,\n\nMagazine Luiza\n\nMagids, Scott\n\nmagnetic content 77\u201382 ( _see also_ content)\n\nMagnetic Resonance Imaging (MRI)\n\nmarketing automation , __\n\nMarlowe, Christopher\n\nMarsden, Paul\n\nMarti, Bernard , __\n\nMashable\n\nMather & Crowther , __\n\nmattering __ ,\n\nmedia ( _see also_ social media):\n\n\"dead\" claim concerning\n\nand Great Fragmentation\n\nnew, fragmented nature of ,\n\nold, and diminishing revenue\n\nold, DO's view of\n\nplanning, and \"connections planning\"\n\nshifting landscapes of\n\nfrom traditional to digital\n\nMeipai\n\nmemes 79\u201380\n\nMetLife __\n\nMexico:\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nMicrosoft\n\nand ad-blocking software\n\nmiddle-class consumers , _252\u20133_\n\nMiles, Steve ,\n\nMilk, Chris\n\nMilko 226\u20137, __\n\nMillennials 46\u201353, __ , __ , _52\u20133_\n\nand Coke\n\nand ethnicity\n\nsearch behaviour among\n\nMiller, George A. , __\n\nMindShare , __\n\nMINDSPACE __ ,\n\nMINI __\n\nMinitel\n\nM-Kopa\n\nMMG\n\nmobile phones:\n\nin Africa\n\nas battleground 174\u201380, __ , __ , __ , __\n\nand brands' errors\n\nenabling nature of\n\nfirst words on\n\nas new ad medium\n\nretail landscape __\n\nMockapetris, Paul , __\n\nModi, Narendra 192\u20133, __\n\nMotista 186\u20137\n\nMotorola , , __\n\nMozilla, and ad-blocking software\n\nM-Pesa\n\nMunn, Olivia __\n\nMuseum of American History 168\u20139, __ , __\n\nMuseum of Feelings , __\n\nMustafa, Isaiah Amir\n\nMyanmar:\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nMyspace __\n\nlaunch of\n\nN\n\nNadal, Miles ,\n\nNakamira, Hiroshi\n\nNaked\n\nNASA\n\nNASDAQ, increase in\n\nNational Public Radio , __\n\nNegroponte, Nicholas\n\nNescaf\u00e9 208\u20139, _208\u20139_\n\nNest __\n\nNestl\u00e9 , __ , 208\u20139\n\nNestl\u00e9 Milo __ ,\n\nNetflix\n\nneuroscience, _see_ brain and behaviour\n\n_New York Times_ ( _NYT_ ) , ,\n\nNewcombe, Zach\n\nNice\n\nNielson\n\nNigeria:\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nNike 54\u20135, , , 214\u201315, _214\u201315_\n\n_The 9-Inch Diet_ (Porter, Bogusky)\n\nNisenholtz, Martin 220\u20133, __\n\nNo Likes Yet\n\nNokia, first smartphone of\n\nNorman, Don __\n\nNSFNET\n\nNTT DOCOMO\n\nnudging , 240\u20137 ( _see also_ brain and behaviour)\n\nNYPD , __\n\nO\n\nObama, Barack , __ , , , __ ,\n\nOdagiri, Akira ,\n\n_Ogilvy on Advertising_ (Ogilvy) , , , , ,\n\nadvertising giants chosen by, _see_ advertising giants in digital age\n\nand brands 54\u20135\n\ncover of __\n\nfavourite words in\n\nfirst chapter of (\"Overture\") ,\n\nand foreign travel\n\nstorming success of\n\nand US's top-nation status\n\nwriting of\n\nOgilvy Change , , ,\n\nOgilvy, David __ , __ , __ ,\n\nadvertising giants chosen by, _see_ advertising giants in digital age\n\nand art-vs-science debate\n\nbook of quotations of __\n\nbranding speech of\n\ncentenary of , __\n\n\"creativity\" hated by\n\nculture created by\n\ndeath of\n\nand direct marketing\n\nand direct-response advertising ,\n\nand disintegration\n\nfilm clip of\n\nfirst advertising work of , __\n\non foreign travel\n\n4A speech of\n\non good causes , __ ,\n\nGreenberg favourite\n\n\"kingly\"\n\non Letterman show __\n\non life-assurance firms\n\non market research\n\non media\n\nand Nisenholtz\n\nand partnership syndrome\n\nand political advertising\n\non skills\n\nOgilvy, Francis 110\u201311\n\nOgilvy, Herta __ ,\n\nas Ogilvy & Mather's materfamilias\n\nOgilvy & Mather:\n\nand belief\n\nand Big IdeaLs 60\u20131, __\n\nand Cadre system\n\nand Cannes 98\u2013101, __ , __\n\nContent Studios of 93\u20135, __ , __\n\ndigital transformation of\n\nand Guinness\n\nJapan success of\n\nand political advertising\n\nShanghai office of\n\nSocial Lab acquired by\n\nsocial-media practice of\n\nand 360-degree stewardship\n\nand velocity\n\nOgilvy Noor 70\u20131\n\nOgilvy Pride , __ ( _see also_ LGBTQ)\n\nOld Spice , 104\u20135, __ , __\n\nOliver, Richard\n\nOreo 36\u20137, __\n\nOrkut __\n\nOrtega y Gasset, Jos\u00e9\n\nOscar Mayer , __\n\nOwen, Clive __\n\nP\n\npage ranking\n\npaid, owned, earned (POE) , , __ ,\n\nPaivio, Allan , __\n\nPakistan:\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nMillennials in\n\nand non-internet channels\n\nand velocity\n\nPalm-Jensen, Matias 224\u20138, __\n\nPalmer, Tony __\n\nPandey, Piyush , , , 249\u201350\n\nPantene __\n\nParker, Alan\n\nParker, George\n\nPatagonia 188\u20139, __ , __\n\n_Patches_\n\nPepsi ,\n\nPeriscope\n\nas ephemeral network\n\nPerrier __\n\nPetit Tube\n\nPewDiePie __ ,\n\nPhilippines:\n\nand chicken fights __\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nand Wardah\n\nPhilips __ , 64\u20135, __\n\nPickholz, Jerry\n\nPidilite\n\nPiper, Tim ,\n\nPlouffe, David\n\nPok\u00e9mon 162\u20133, __ , __\n\npolitics:\n\nin author's predictions\n\nas transformation in digital age 192\u20135, __ , __ , __ , __ , __\n\nPolman, Paul , __\n\nPorter, Chuck 228\u201335, __ , __\n\n\"post-truth\"\n\nPostel, Jon , __\n\n_Practical Advertising_ 110\u201311, __\n\npractical content 88\u201392 ( _see also_ content)\n\nand predicted death of television\n\nProcter & Gamble (P&G) ,\n\nprogrammatic advertising 143\u20135, __\n\nPuri, Bharat\n\nQ\n\nQI codes\n\nQualcomm 89\u201390, __ , __\n\nR\n\nRaphaelson, Joel:\n\nDO's letter to __\n\nand _Ogilvy on Advertising_\n\n_Raw_\n\nRay-Ban\n\nRed Bull\n\nRed Label __ , __\n\nReeves, Rosser , , ,\n\nRen Zhengfei ,\n\nResor, Stanley\n\nR\/GA\n\nRhett and Link\n\nRichards, Ben 134\u20135, ,\n\nRitson, Mark\n\nRizzolatti, Giacomo , __\n\nRoberts, Kevin\n\nRoberts, Lawrence , __\n\nRobertson, Andrew\n\nRoosevelt, Theodore\n\nRospars, Joe\n\nRoyal National Lifeboat Institution , __\n\nRubicam, Raymond\n\nRumbol, Phil 96\u20137\n\nRust, Ronald\n\nS\n\nSaatchi, Charles\n\nSagmeister, Stefan\n\nsame-sex marriage , , __ ( _see also_ LGBTQ)\n\nSammartino, Steve\n\nSanchez Lamelas, Javier\n\nSaunders, Ernest\n\nSchelling, Thomas , __\n\nSchultz, Howard , , __\n\nSchwab, Charles\n\nSchwartz, Barry\n\n_Scientific Advertising_ (Hopkins)\n\nsearch 137\u20138\n\nglossary of common terms for\n\nand keywords 137\u20138, __\n\nand Millennials' behaviour\n\nSearch Engine Marketing (SEM) ,\n\nSearch Engine Optimization (SEO) , ,\n\nSearls, Doc\n\nSelective\n\n_The Selfish Gene_ (Dawkins)\n\nSenerchia, Jeanette 168\u20139\n\nsexism\n\n_Sexy Little Numbers_ (Maex)\n\nsharing economy\n\nSharp, Byron\n\nShazam __\n\nSherif, Muzafer , __\n\nShimizu, Ryo\n\nShiseido.\n\nshopping online, first systems for\n\nshowrooming , __\n\nShy, Jean\n\nSiegl, Zev __\n\nSimon, Herbert A.\n\nSina Weibo\n\nSinclair, Upton\n\nSix Degrees __\n\n_Sketches_\n\nskipping ads ,\n\nSloan, Robin\n\nsmart content 85\u20138 ( _see also_ content)\n\nsmartphones, first\n\nSmith, Adam\n\nSmosh __\n\nSMS, arrival of\n\nSnapChat:\n\nand Centennials\n\nas ephemeral network\n\nsocial media ( _see also individual outlets_ ):\n\nad spending on\n\nand Arab Spring , __\n\nas archival networks\n\nas battleground 164\u201373, __ , __ , __ , __ , __ , __ , __\n\nas database of intent\n\nas ephemeral networks\n\noverclaims for\n\nsearch capabilities of\n\nand tourism __ , __\n\nsocial responsibility, as transformation in digital age 204\u20137, __ , __ , __\n\nSocial@Ogilvy\n\nSong, TB __ ,\n\nSorrell, Martin\n\nspam, first incidence of\n\nSpark\n\nSPG Keyless __\n\nSpiff ,\n\nSpotify\n\nSprint\n\nStaav, Yael __\n\nStandard Oil\n\nStarbucks 24\u20135, __\u2013 _5_\n\nStarwood Hotels __\n\nStenbeck, Jan\n\nStitzer, Tod ,\n\nStonewall\n\nStumbleupon\n\nSulzberger, Arthur\n\nSunstein, Cass R. 240\u20131\n\nSuper Bowl , __ , , __\n\nSurminski, Brenda __\n\nSutherland, Rory , , 238\u201340\n\nT\n\nTaylor, Robert William \"Bob\" , __\n\nTed Bates agency\n\nteletext 221\u20132, __\n\ntelevision:\n\nand changing digital ecosystem 34\u20135\n\nand DVR ,\n\nlinear viewership of\n\npredicted disappearance of\n\nTencent , , __\n\nThaler, Richard H. , 240\u20131\n\nTham Khai Meng , __ ,\n\nThomas, Gareth\n\nThompson Holidays\n\nThompson, J. Walter\n\nThorndyke, Edward , __\n\n3G:\n\ndebut of\n\ngrowing use of\n\nTiffany & Co.\n\nTimberlake, Justin\n\nTime Inc.\n\n_Tin Shed_ 188\u20139\n\nTone\n\nToothbrush Games __\n\nToshiba __ ,\n\ntourism:\n\nas transformation in digital age 200\u20133, __ , __ , __ , __\n\nand Visit Britain , __\n\nToyota\n\ntraditional media, and changing digital ecosystem\n\ntransformations in digital age 192\u2013209\n\nand government 197\u20139, __ , __\n\nand politics 192\u20135, __ , __ , __ , __ , __\n\nand social responsibility 204\u20137, __ , __ , __\n\nand tourism 200\u20133, __ , __ , __ , __\n\nTrump, Donald 195\u20136, __ , __\n\n\"The Truth\"\n\nTumblr\n\nTuring, Alan , __\n\nTV, rising revenue from\n\nTversky, Amos , , __ , , __\n\nTwitter\n\nand ALS Ice Bucket Challenge\n\nand Cadbury\n\nlaunch of\n\nand Oreo 36\u20137, __\n\nin Philippines\n\nand politics __ , , __\n\nand promoted trends and tweets\n\nrevenue of\n\n2G mobile network\n\nU\n\nUber ,\n\nand disintermediation\n\nUnilever , , 204\u20135, __\n\nUniqlock , __\n\nUPS __ , __ , __ , , __\n\nUX (user experience) 153\u20134, __\n\nV\n\nVan Damme, Jean-Claude , __\n\nvan Nispen, Jost\n\nvelocity ,\n\n_Vibe_\n\nVietnam:\n\ndisproportionate growth in\n\nmiddle-class consumers in __\u2013 _3_\n\nand velocity\n\n\"View from Touffou\", still played , __\n\nVine ,\n\nvirtual, automated and mixed reality 157\u201361, __ , __ , __ , __\n\nin author's predictions\n\nVisit Britain , __\n\nVittel __ , __\n\nVolvo , __\n\nVonk, Nancy __\n\nW\n\n_Walrus_\n\nWang, Leehom __\n\nWardah\n\nWason, Peter , __\n\nWatkins, Pearline\n\nweb advertising:\n\nand \"banner\" ads , __\n\nclickable, first , __\n\nwebsite, judging , 140\u20135\n\nWeChat , , 257\u20138, __\n\nin author's predictions\n\nWeed, Keith , __\n\nWeinberger, David\n\nWestheimer, Ruth\n\n\"What Is Code?\" (Ford) 150\u20131, __\n\nWhatsApp ,\n\nWikipedia\n\nWorld Wide Web ( _see also_ internet), and founders' idealism\n\nWorld Wildlife Fund __\n\nWPP , , , 248\u20139,\n\nX\n\nXanga __\n\nXerox Alto\n\nXiaomi\n\nY\n\nYahoo!, search engine of\n\nYing Zhang , __\n\nYoshida, Hideo\n\nYoung Turks __ , ,\n\nYouTube\n\nand ALS Ice Bucket Challenge\n\nas archival network 41\u20132\n\nand Cadbury\n\nand _Evolution_\n\nand live capabilities\n\nand Millennials , __\n\nand Millennials' search behaviour\n\nYouTube Red\n\nZ\n\nZalis, Shelley\n\nZhang, Ying , __\n\nZucker, Jeff\n\nZuckerberg, Mark , __\nACKNOWLEDGEMENTS\n\nThanks are due first to Lammy, who put up with the fractured holidays and weekends when I mostly wrote this book, and who provided sanity throughout.\n\nThen I must thank my predecessors at Ogilvy & Mather: Bill Philips, Graham Philips, Ken Roman, Charlotte Beers: without each of their contributions the brand would not be what it is. And special thanks go to Shelly Lazarus, my immediate predecessor, who guided and supported me in Asia and New York. My successor, John Seifert, has been an ardent supporter of the project. Herta Ogilvy provided encouragement, hospitality and friendship throughout my tenure: she is the direct line to David which all of us who have worked for the firm have felt and cherished. Joel Raphaelson generously gave his wisdom from \"the first time round\", and I crave his forgiveness for my latter day solecisms.\n\nAny wisdom in the book is collective much more than personal. How can I properly thank my colleagues at Ogilvy & Mather over the years? My creative partner Tham Khai Meng is the \"best of the best\": without him there would have been nothing. I was blessed to work with two inimitable strategists, each very different, Ben Richards and Colin Mitchell, as well as my Vice Chairman and Consulting Head, Carla Hendra. My brilliant colleagues throughout the organization have been indispensable collaborators. Thanks in particular to:\n\nBrian Fetherstonhaugh, Christopher Graves, Rory Sutherland, Calvin Carter, Piyush Pandey, the late Tim Broadbent, Kent Wertime, Thomas Crampton, Martin Lange, Pete Dyson, Mark Lainas, Madeleine Croucher, Julia Stainforth, Shelina Janmohamed, Andrew Barratt, Carlos Nunez, Paul Matheson, Dimitri Maex, Bjorn Stahl, Magnus Ivansson, Minoru Fujita, Maki Suzuki, Akira Odagiri, Mike McFadden, Jeremy Webb, Madeline Di Nonno, Jaime Prieto, Sean Muzzy, Mark Himmelsbach, Rajiv Rao, Maeve Countey, Mi hui Park, Beth Ann Ingrassia, Sabrina Allen, Robert Lear, Peter DeLuca and Liam Parker.\n\nA thank you to all the Ogilvy Discipline Heads, GBM leaders, Account and Creative teams that helped us gather all the material needed to put this publication together.\n\nI must also thank Steve Goldstein, Stacey Ryan-Cornelius and Lauren Crampsie for all their practical support; and the \"home team\" of Nikolaj Birjukow, Alexander Banon, Harley Saftler and John Vetrano. Joan Huber, Mariah MacCarthy, Laird Stiefvater and Ellen Pierce, Eleanor Tsang, Erin Clutcher, Lynn Whiston and Arlene Richardson all went beyond the call of duty.\n\nJeremy Katz and Samar Taher Khan helped produce the book at Ogilvy, and Samar assisted in a myriad of essential ways alongside me from beginning to end. Mark Dewings gave me completely invaluable help. Tuan Ching, Joohee Park and Jess Xuan contributed many design elements.\n\nMy deepest thanks go to them all.\n\nOutside of Ogilvy, I am indebted to my clients first and foremost. I have been blessed to know and work with them, and these are just a few who have helped push their \"partners\" into better and better work:\n\nJohn Hayes, Alison Bain, Tony Palmer, Clive Sirkin, Rishi Dhingra, Keith Weed, Marc Mathieu, Walter Susini, Steve Miles, Fernando Machado, Jon Iwata, Bharat Puri, Phil Chapman, Nikhil Rao, Dana Anderson, Salman Amin, Ann Mukherjee, Patrice Bula, Tom Buday, Quique Pendavis, Sean Murphy, Christine Owens, Maureen Healy, Peter Nota, Jeanette Senerchia, Javier Sanchez Lamelas and Rodolfo Echeverria.\n\nIt was Jonathan Goodman of Carlton Publishing who had the idea for this book, and then pushed and prodded until I finally wrote to him. I thank him, editors Alison Moss and Gemma Maclagan Ram, Art Director Russell Knowles and the whole team at Carlton for turning my untidy pile of pages and images into reality.\n\nFinally, there are others who have given freely of their time and knowledge: the \"famous five\" \u2013 Bob Greenberg, Akira Kagami, Martin Niesenholtz, Matias Palm-Jensen, Chuck Porter \u2013 and also Thomas Gensemer, Russell Davies, Surjit Bhalla, Phil Rumbold, Tim Piper, Adam Smith, Jeff Bowman, and Rob Norman.\nPICTURE CREDITS\n\nThe publishers would like to thank the following sources for their kind permission to reproduce the pictures in this book:\n\nHere \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9Jack Boss. This illustration appeared in \"Schenkerian-Schoenbergian Analysis and Hidden Repetition in the Opening Movement of Beethoven's Piano Sonata Op. here, No. 1,\" Music Theory Online 5\/1 (January 1999). It was authored by Jack Boss (jfboss@uoregon.edu), with whose written permission it is reprinted here; here \u00a9Joel Ralphson; here \u00a9Miles Young; here \u00a9SRI International, Image courtesy of SRI International; here \u00a9SRI International, Image courtesy of SRI International; here & here \u00a9Ogilvy & Mather; here \u00a9Airbnb; here \u00a9Google; here \u00a9Nest; here \u00a9Luma Partners LLC 2017; here credit unknown; here \u00a9Starbucks; here Chris Willson\/Alamy Stock Photo; here \u00a9Phillips; here \u00a9Facebook; here \u00a9Oreo; here \u00a9John St.; here Copyright by The Young Turks, LLC; here \u00a9Unilever; here \u00a9Ogilvy & Mather; here \u00a9Unilever; here \u00a9Time Magazine; here \u00a9AT&T; here \u00a9Coca-Cola; here \u00a9Prudential; here \u00a9Prudential; here \u00a9Johnnie Walker; here \u00a9Toyota; here \u00a9IBM; here \u00a9 Unilever; here \u00a9Coca-Cola; here \u00a9Chiptole; here \u00a9American Express; here \u00a9Phillips; here \u00a9P&G; here \u00a9P&G; here \u00a9Under Armour; here \u00a9Honey Maid; here \u00a9 Unilever; here \u00a9Burger King; here \u00a9Ogilvy & Mather; here \u00a9BMW; here \u00a9UPS; here \u00a9Red Bull here \u00a9 SC Johnson; here \u00a9Coca-Cola; here \u00a9Europcar; here \u00a9 Unilever; here \u00a9UPS; here \u00a9Chipotle Mexican Grill; here \u00a9 Coca-Cola; here \u00a9 Nestl\u00e9; here \u00a9 Nestl\u00e9; here \u00a9Nestl\u00e9; here \u00a9Google; here \u00a9Qualcomm; here \u00a9Barneys New York; here \u00a9American Express; here \u00a9 Kimberly-Clark; here \u00a9Ford Motor Company; here \u00a9Nestl\u00e9; here \u00a9Coca-Cola; here \u00a9Mondelez; here \u00a9Mondelez; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9Dr. Robert Heath; here \u00a9P&G; here \u00a9P&G; here \u00a9Ogilvy & Mather; here \u00a9P&G; here \u00a9Oscar Mayer \u2013 Kraft Foods; here \u00a9Steffen Billhardt Photography; \u00a9Thomas Billhardt Photography); here \u00a9Coca-Cola; here \u00a9Getty Images; here \u00a9AB Volvo\/Forsman & Bodenfors; here \u00a9AB Volvo\/Forsman & Bodenfors; here \u00a9Ogilvy & Mather; here \u00a9Intel Corporation; here \u00a9IBM; here Robert Lachman\/Los Angeles Times via Getty Images; here \u00a9Neil French; \u00a9Tham Khai Meng; \u00a9Ogilvy & Mather; here \u00a9IBM; here \u00a9CNN Money; \u00a9Metropolitan Life Insurance Company; here \u00a9Babolat; here \u00a9British Airways; here \u00a9IBM; here \u00a9Nestl\u00e9; here \u00a9Qualcomm; here \u00a9Ogilvy & Mather; here \u00a9Bloomberg Business Week; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here Produced by Scott Brinker(@cheifmartec.com) here \u00a9Ogilvy & Mather; here \u00a9Mast-J\u00e4germeister; here \u00a9National German Centre for Health Information; here \u00a9Monica Wisnlewska\/Shutterstock.com; here \u00a9Bottle Rocket; here \u00a9Amazon; here \u00a9Niantic; \u00a9Nintendo; here \u00a9Niantic; \u00a9Nintendo here \u00a9ComScore; here \u00a9Niantic; \u00a9Nintendo here \u00a9comScore; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here (background) Olena Yakobchuk\/Shutterstock.com; here \u00a9Philips; here \u00a9Nestl\u00e9; here \u00a9Burger King; here \u00a9NPR; here \u00a9Bottle Rocket; here \u00a9CFA Properties, Inc.; here \u00a9The Lego Group; here \u00a9Starwood; here \u00a9Amazon; here \u00a9marketoonist.com here \u00a9Adidas AG; here \u00a9Patagonia; here \u00a9Patagonia; here \u00a9Patagonia; here \u00a9Amazon; here \u00a9Piyush Pandi; hereb \u00a9Piyush Pandi; here \u00a9Engagement Citoyen; here \u00a9Ministry of Tourism, India; here \u00a9Ministry of Tourism, India; here \u00a9Cape Town Tourism; here \u00a9The Swedish Tourism Association; here \u00a9Tourism Queensland; 204 \u00a9Unilever 205 \u00a9Unilever here \u00a9Battersea Dogs and Cats Home; here \u00a9Google; here \u00a9WWF \u2013 Thailand; here \u00a9Royal National Lifeboat Institution; here \u00a9Ogilvy & Mather; here \u00a9Nestl\u00e9; here \u00a9Nestl\u00e9; 210 \u00a9Bob Greenburg; here \u00a9Ogilvy & Mather; here \u00a9Nike; here \u00a9Akira Kagami; here \u00a9Akira, here \u00a9Uniqlo, here \u00a9Martin Nisenholtz; here Photo by Martha Stewart; here \u00a9Matias Palm-Jensen; here \\- here \u00a9Swedish Post; \u00a9Matias Palm-Jensen; here \u00a9Milko; \u00a9Matias Palm-Jensen; here \u00a9Cannes Lions; \u00a9Matias Palm-Jensen; here Chuck Porter; here \u00a9Mini; here \u00a9CP&B; here \u00a9Burger King; here \u00a9Dominos Pizza; here \u00a9Kimberly-Clark; here \u00a9Kimberly-Clark; here \u00a9Ogilvy & Mather; here \u00a9John T. Miller; here \u00a9Ogilvy & Mather; here \u00a9Armorer Wason; here \u00a9Ogilvy & Mather; here \u00a9Chris Graves; here \u00a9Durex; here \u00a9JasonVale; here \u00a9Grubs; here \u00a9Nestl\u00e9 here \u00a9Public Health England; here Reproduced with Permission of Motorola Mobility LLC, \u00a92004, Motorola Mobility LLC.; here \u00a9Pedilite Industries; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9Ogilvy & Mather; here \u00a9BAT\/Ogilvy & Mather; here \u00a9Visit Britain; here \u00a9Jollibee Foods Corporation; here \u00a9McDonalds; here \u00a9Dabur; here \u00a9Huawei Technologies Co. Ltd. here \u00a9Huawei Technologies Co. Ltd.\n\nEvery effort has been made to acknowledge correctly and contact the source and\/or copyright holder of each picture and Carlton Books Limited apologizes for any unintentional errors or omissions, which will be corrected in future editions of this book.\nBloomsbury USA\n\nAn imprint of Bloomsbury Publishing Plc\n\n1385 Broadway | 50 Bedford Square  \n---|---  \nNew York | London  \nNY 10018 | WC1B 3DP  \nUSA | UK\n\nwww.bloomsbury.com\n\nThis electronic edition published in 2018 by Bloomsbury Publishing Plc\n\nBLOOMSBURY and the Diana logo are trademarks of Bloomsbury Publishing Plc\n\nFirst published in Great Britain by Goodman 2017\n\nFirst U.S. edition published by Bloomsbury 2018\n\nText \u00a9 Ogilvy & Mather, 2017\n\nDesign \u00a9 Carlton Books Ltd., 2017\n\nAll rights reserved  \nYou may not copy, distribute, transmit, reproduce or otherwise make available this publication (or any part of it) in any form, or by any means (including without limitation electronic, digital, optical, mechanical, photocopying, printing, recording or otherwise), without the prior written permission of the publisher. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.\n\nNo responsibility for loss caused to any individual or organization acting on or refraining from action as a result of the material in this publication can be accepted by Bloomsbury or the author.\n\nISBN: 978-1-6355-7146-2 (HB)   \nISBN: 978-1-6355-7147-9 (eBook)\n\nLibrary of Congress Cataloging-in-Publication Data is available.\n\nTo find out more about our authors and their books please visit www.bloomsbury.com where you will find extracts, author interviews and details of forthcoming events, and to be the first to hear about latest releases and special offers, sign up for our newsletter.\n\nBloomsbury books may be purchased for business or promotional use. For information on bulk purchases please contact Macmillan Corporate and Premium Sales Department at specialmarkets@macmillan.com.\n\n#\n\n  1. Cover\n  2. Title Page\n  3. CONTENTS\n  4. INTRODUCTION\n    1. VIEW FROM TOUFFOU\n  5. 1 CODETTA\n  6. 2 THE DIGITAL REVOLUTION\n  7. 3 THE SHORT MARCH\n  8. 4 THE DIGITAL ECOSYSTEM\n  9. 5 TO BE OR NOT TO BE A MILLENNIAL\n  10. 6 THE POST-MODERN BRAND\n  11. 7 CONTENT IS KING; BUT WHAT DOES IT MEAN?\n  12. 8 CREATIVITY IN THE DIGITAL AGE\n    1. \"GIVE ME GOLD\"\n    2. ART OR SCIENCE?\n    3. PERVASIVE CREATIVITY\n  13. 9 DATA: THE CURRENCY OF THE DIGITAL AGE\n  14. 10 \"ONLY CONNECT\"\n  15. 11 CREATIVE TECHNOLOGY: THE SWEET SPOT\n  16. 12 THE THREE BATTLEGROUNDS\n    1. PUTTING THE SOCIAL BACK INTO SOCIAL MEDIA\n    2. THE JOY OF MOBILITY\n    3. CONTINUOUS COMMERCE\n  17. 13 DIGITAL TRANSFORMATIONS\n    1. DIGITAL POLITICS\n    2. DIGITAL GOVERNMENT\n    3. DIGITAL TOURISM\n    4. DIGITAL SOCIAL RESPONSIBILITY\n  18. 14 FIVE GIANTS OF ADVERTISING IN THE DIGITAL AGE\n    1. BOB GREENBERG\n    2. AKIRA KAGAMI\n    3. MARTIN NISENHOLTZ\n    4. MATIAS PALM-JENSEN\n    5. CHUCK PORTER\n  19. 15 MY BRAIN HURTS\n  20. 16 THE NEW SHAPE OF THE WORLD\n  21. 17 CULTURE, COURAGE, CLIENTS AND CASTANETS\n  22. 18 EPILOGUE\n  23. ENDNOTES\n  24. INDEX\n  25. PICTURE CREDITS\n  26. ACKNOWLEDGEMENTS\n  27. eCopyright\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nReclams   \nLateinisches Zitaten-Lexikon\n\nVon  \nMuriel Kasper\n\nReclam\nF\u00fcr Raban und Maura\n\nAlle Rechte vorbehalten\n\n\u00a9 2014 Philipp Reclam jun. GmbH & Co. KG, Stuttgart\n\nUmschlaggestaltung: Stefan Schmid Design, Stuttgart\n\nGesamtherstellung: Reclam, Ditzingen\n\nMade in Germany 2014\n\nRECLAM, UNIVERSAL-BIBLIOTHEK und RECLAMS UNIVERSAL-BIBLIOTHEK sind eingetragene Marken der Philipp Reclam jun. GmbH & Co. KG, Stuttgart\n\nISBN: 978-3-15-960488-6\n\nISBN der Buchausgabe: 978-3-15-019196-5\n\nwww.reclam.de\nEinleitung\n\nSo ein paar grundgelehrte Zitate zieren den ganzen Menschen.\n\nHeinrich Heine, _Reisebilder_ , Zweiter Teil\n\nEs mag seltsam anmuten, da\u00df eine Sammlung lateinischer Zitate mit einem Spruch eines deutschen Dichters beginnen soll \u2013 zumal einem so ironischen. Kaum ein anderes Zitat jedoch gibt die Zwiesp\u00e4ltigkeit eines solchen Unterfangens in der heutigen Zeit so treffend wieder.\n\nAuch wenn im Internet Kommunikationen in lateinischer Sprache stattfinden, so ist generell eher eine r\u00fcckl\u00e4ufige Tendenz in der ernsthaften Auseinandersetzung mit der Sprache und Kultur Roms zu verzeichnen \u2013 eine Tendenz, die bei der zunehmenden Relevanz anderer Wissensbereiche verst\u00e4ndlich ist, die Liebhaber der Reicht\u00fcmer jener Kultur jedoch eine enorme geistige Verarmung der kommenden Zeiten voraussehen l\u00e4\u00dft.\n\nDieser Tendenz steht die oft mi\u00dfbrauchte Aura der Gelehrsamkeit gegen\u00fcber, die in den Augen vieler die lateinische Sprache auch heute noch umgibt: Manch einer bedient sich solcher Zitate, um mit ein paar souver\u00e4n dahergesagten lateinischen Worten seine vermeintlich \u00bbtiefe Gelahrtheit anzubringen\u00ab (wieder Heine, siehe oben). \u00bbDelirant, isti Romani!\u00ab \u2013 \u00bbDie spinnen, die R\u00f6mer!\u00ab mag sich da der mit unverst\u00e4ndlichem Latein bombardierte Gespr\u00e4chspartner sagen.\n\nSowohl dem Verschwinden einer ganzen Kultur als auch dem seichten, allein auf Prestige bedachten Wissen m\u00f6chte dieses B\u00fcchlein entgegenwirken. Es soll Neugierde wecken, zu weiterer Lekt\u00fcre anregen und vor allem die Bed\u00fcrfnisse der Benutzer erf\u00fcllen, die ein bestimmtes Zitat suchen: durch knappe, exakte Hinweise auf den genauen Wortlaut eines Zitats, dessen Fundort und, wenn n\u00f6tig, durch Erl\u00e4uterungen zum literarischen Kontext und kulturellen Hintergrund. Leser, die mit der Sprache Roms bislang keinen Kontakt hatten, sind hier ebenso angesprochen wie Kenner der lateinischen Literatur, da neben den Informationen zur Situierung der Zitate h\u00e4ufig Parallelstellen aufgef\u00fchrt werden. Die streng alphabetische Anordnung der Eintr\u00e4ge erlaubt es, auch ohne Kenntnis des Lateinischen ein im Wortlaut vorliegendes Zitat ausfindig zu machen. Wer lediglich rudiment\u00e4re Sprachkenntnisse besitzt, dem soll durch die textnahen (f\u00fcr deutsche Ohren daher bisweilen vielleicht etwas widerspenstigen) \u00dcbersetzungen der lateinischen Originale wieder auf die Spr\u00fcnge geholfen und das Zitat so auch sprachlich n\u00e4hergebracht werden. Die \u00bbHolprigkeit\u00ab mancher \u00dcbersetzung wird, wo dies m\u00f6glich ist, durch analoge sprichw\u00f6rtliche Wendungen des Deutschen wettgemacht, die in vielen F\u00e4llen Karl Simrocks Sammlung deutscher Sprichw\u00f6rter entnommen sind. Bei der Suche nach dem Ausspruch eines bestimmten Urhebers leistet ein Quellenregister, das bei den wichtigsten Schriftstellern nach Einzelwerken oder Werkgruppen aufgegliedert ist, eine Hilfestellung. Lesern, die ein Zitat f\u00fcr einen bestimmten Anla\u00df suchen oder sich nicht mehr an den genauen Wortlaut erinnern, mag ein knappes Schlagwortregister dienlich sein.\n\nDie ausgew\u00e4hlten Zitate beschr\u00e4nken sich nicht auf antike Quellen, sondern illustrieren das Fortleben der lateinischen Sprache \u00fcber das Mittelalter hinaus bis in die Neuzeit: Queen Elizabeth II. hat mit ihrem Ausspruch aus dem Jahre 1992 hier in zeitlicher Hinsicht das letzte Wort. Au\u00dfer literarischen Zitaten fanden auch zahlreiche Wendungen aus dem kirchlichen und juristischen Bereich sowie aus Studentenliedern neben Ausspr\u00fcchen historischer Pers\u00f6nlichkeiten Eingang. Frappant ist bereits nach kurzem Schm\u00f6kern die gedankliche N\u00e4he vieler Zitate zu heutigen Lebenssituationen; h\u00e4ufig zeigt sich auch eine Parallelit\u00e4t zu den sprachlichen Motiven des Deutschen (z. B. \u00bbUna hirundo non facit ver\u00ab \u2013 \u00bbEine Schwalbe macht noch keinen Fr\u00fchling\u00ab). Befremdlich hingegen sind etwa die zahlreichen misogynen Eintr\u00e4ge, die, auch wenn bereits das Zeitalter der \u203apolitical correctness\u2039 eingel\u00e4utet wurde, hier aus zwei Gr\u00fcnden nicht fehlen sollen: Zum einen w\u00e4re die Sammlung ohne sie nicht vollst\u00e4ndig und w\u00fcrde das Bild des \u00fcberlieferten Gedankenguts verf\u00e4lschen. Zum andern bieten Hintergrundinformationen zu solchen Zitaten mitunter die Mittel, ewig Gestrige zu entwaffnen, wurden doch manche Ausspr\u00fcche erst im Laufe ihres Gebrauchs frauenfeindlich gedeutet, w\u00e4hrend sie im Kontext des Originals anders intendiert waren.\n\nUm einen effektiven Gebrauch dieses Buchs zu gew\u00e4hrleisten, sind noch einige Hinweise vonn\u00f6ten:\n\nWie erw\u00e4hnt, sind die Zitate streng alphabetisch geordnet. Bei Prosa-Zitaten k\u00f6nnen einleitende Partikel (z. B. \u00bbet\u00ab, \u00bbnam\u00ab) wegfallen, auch wenn sie dem Original voranstehen. Ausschlaggebend f\u00fcr eine solche Auslassung ist die jeweils g\u00e4ngige Zitierweise \u2013 ein Vorgehen, das sicherlich anfechtbar ist. Verse hingegen werden unter Beachtung der Satzgrenzen vollst\u00e4ndig zitiert. Sind Ausspr\u00fcche in mehreren Formen \u00fcberliefert, so wird dies durch Verweise vermerkt (siehe z. B. \u00bb\u203adisiecti membra poetae\u2039, h\u00e4ufig zitiert als \u203adisiecta membra poetae\u2039\u00ab).\n\nBei den Quellenangaben wurde stets versucht, einen w\u00f6rtlichen Beleg f\u00fcr die \u00fcberlieferte Wendung zu finden, was nicht immer m\u00f6glich war. Ist der Eintrag gegen\u00fcber dem Original nur geringf\u00fcgig syntaktisch ver\u00e4ndert, so wird dies durch Zuf\u00fcgung von \u00bbnach\u00ab (z. B. \u00bbnach Cicero\u00ab) verdeutlicht. Eine Ausnahme bilden allerdings vollst\u00e4ndige S\u00e4tze, die im Original in eine l\u00e4ngere Periode eingebettet sind; sie werden als in sich geschlossene S\u00e4tze zitiert. Sind Zitat und Quelle nur inhaltlich miteinander verwandt, wird der Fundort mit \u00bbvgl.\u00ab eingeleitet. Wendungen, die bereits in der Antike als sprichw\u00f6rtlich galten, werden als solche gekennzeichnet. Ein besonderes Anliegen dieses B\u00fcchleins liegt darin, m\u00f6glichst die jeweils \u00e4lteste Quelle anzugeben; darauf folgen bei besonders bekannten Zitaten in chronologischer Reihenfolge, wenn diese feststellbar ist, Beispiele f\u00fcr eine Wiederaufnahme oder Weiterentwicklung des jeweiligen Zitats. Sie stellen jedoch nur eine geringe Auswahl dar, da der Rahmen dieses Buches es nicht erlaubt, die komplette Geschichte eines Motivs oder Gedankengangs vorzulegen. So wird man vergeblich nach Verweisen auf die Verwendung der Zitate in den einzelnen Nationalliteraturen suchen.\n\nSchlecht \u00fcberlieferte Textstellen sowie _versus suspecti_ werden nicht als solche markiert, sofern sie in den gel\u00e4ufigen Textausgaben enthalten sind. Generell richten sich die Stellenangaben nach den Z\u00e4hlungen der Standardausgaben (vornehmlich Teubner, Oxford University Press und Loeb). Dies gilt auch f\u00fcr Fragmentsammlungen; da hier die Z\u00e4hlungen stark divergieren, findet der jeweils konsultierte Editor Erw\u00e4hnung. In den seltenen F\u00e4llen auff\u00e4lliger Archaismen wurden Zitate orthographisch vereinheitlicht; die Interpunktion geht einen Mittelweg zwischen syntaktischer und semantischer Motivation, \u00e4hnlich den Gepflogenheiten englischsprachiger Editoren. Bei Bibelstellen folgt die Z\u00e4hlung der Vulgata.\n\nF\u00fcr zahlreiche lateinische Zitate lie\u00dfen sich griechische Quellen angeben. Hier sollen sich jedoch die Angaben auf F\u00e4lle beschr\u00e4nken, in denen das Lateinische eine mehr oder weniger w\u00f6rtliche \u00dcbersetzung einer griechischen Textstelle darstellt, der lateinische Autor selbst auf eine griechische Fassung verweist oder die griechische Urheberschaft so offensichtlich ist, da\u00df sie nicht unerw\u00e4hnt bleiben durfte. Die Titel der griechischen Werke sind in deutscher oder lateinischer Sprache angegeben, die Namen der Autoren nach der jeweils gel\u00e4ufigen Form standardisiert.\n\nNun wird sich der Leser fragen, nach welchen Kriterien hier entschieden wurde, was ein Zitat sei und in welcher Form es aufgenommen werde, vielmehr noch: wieso \u00fcberhaupt ein neues Zitatenlexikon einer toten oder zumindest f\u00fcr tot erkl\u00e4rten Sprache publiziert werden m\u00fcsse.\n\nZun\u00e4chst zur Auswahl: Das vorliegende B\u00fcchlein ist nat\u00fcrlich den Sammlungen verpflichtet, die im Anschlu\u00df an dieses Vorwort aufgelistet werden. Die \u00dcberlegung, da\u00df manche Zitate nur noch als solche gelten, weil ein Lexikon-Verfasser sie vom vorausgehenden \u00fcbernahm, verwehrte jedoch einigen Zitaten die Aufnahme. Statt dessen avancierte der tats\u00e4chliche Gebrauch der Wendungen zum ausschlaggebenden Kriterium \u2013 mit der Einschr\u00e4nkung, da\u00df in Zweifelsf\u00e4llen auch pers\u00f6nliche Vorlieben zum Tragen kamen. Andere Eintragungen beruhen auf der eigenen Sammlung h\u00e4ufig zitierter Wendungen. Der Wortlaut richtet sich generell nach dem Original; wenn Original und zitierte Version stark voneinander abweichen, wird dies erw\u00e4hnt.\n\nKeine Ber\u00fccksichtigung fanden sprichw\u00f6rtliche Wendungen, die nicht auf eine konkrete Quelle zur\u00fcckgehen, Floskeln und Versatzst\u00fccke, die gelegentlich in Fremdw\u00f6rterlexika und recht umfassend bei Bartels erl\u00e4utert werden. Nach \u00bbDe gustibus non est disputandum\u00ab etwa wird man vergeblich suchen. Nur in seltenen F\u00e4llen wird mit dem K\u00fcrzel \u00bbHW\u00ab auf Hans Walthers _Proverbia sententiaeque Latinitatis medii aevi_ verwiesen. \u00bbHWS\u00ab bezeichnet die Weiterbearbeitung durch Schmidt. Aussagen historischer Pers\u00f6nlichkeiten entbehren gelegentlich einer konkreten Quelle. Wenn sie jedoch in mehreren Geschichtswerken belegt werden, fehlen sie auch in diesem Buch nicht. Am\u00fcsanten _Maccaroni_ -Spr\u00fcchen, Hybriden aus lateinischen und deutschen W\u00f6rtern, wie Zoozmann sie bisweilen darbietet, wurde hier keine Aufmerksamkeit zuteil.\n\nDoch wozu ein weiteres lateinisches Zitatenlexikon? Zugegeben: Die Zahl der erst in den letzten Jahren in lateinischer Sprache ge\u00e4u\u00dferten Worte, denen Fl\u00fcgel verliehen wurden, ist beklagenswert gering. Von der Aufnahme aktuell hinzugekommener Zitate kann also nur in wenigen F\u00e4llen die Rede sein. Verbesserungsbed\u00fcrftig schien in den meisten bisherigen Werken jedoch vor allem die Quellenforschung: Bisweilen werden die Quellen nur vage angegeben oder fehlen g\u00e4nzlich, obwohl sie eindeutig zu lokalisieren sind; nicht selten finden sich leider auch falsche Angaben. Auch das vorliegende B\u00fcchlein stellt in dieser Hinsicht nur einen kleinen Fortschritt dar, keineswegs eine endg\u00fcltige Erkenntnis. Daneben wurde im Vergleich zu einigen Vorg\u00e4ngern vermieden, Zitate nach Vertauschung der Wortreihenfolge mehrfach unter verschiedenen Buchstaben des Alphabets erscheinen zu lassen, da dies den Eindruck von der Gr\u00f6\u00dfe des \u00fcberlieferten Zitatenschatzes f\u00e4lscht und denjenigen, der mehrere Buchstaben hintereinander liest, verwirrt. Neu ist in dieser Sammlung vor allem die kompakte Form, sowohl was das Format betrifft als auch in Hinblick auf die L\u00e4nge der Erkl\u00e4rungen, ohne da\u00df dadurch die Zahl der Eintr\u00e4ge oder die Verst\u00e4ndlichkeit beeintr\u00e4chtigt w\u00fcrde. Sicherlich wird ein jeder Leser einige Zitate vermissen oder auch zur einen oder anderen Quelle eine Erg\u00e4nzung bieten k\u00f6nnen; entsprechende Hinweise nehme ich dankbar entgegen.\n\nNach diesen spr\u00f6den Pr\u00e4liminarien ist es mir eine Freude, Dr. Thorsten F\u00f6gen und Dr. Antje Sch\u00e4fer meinen Dank f\u00fcr ihre zahlreichen Hinweise und Ratschl\u00e4ge w\u00e4hrend der gesamten Arbeitsphase auszusprechen. Mein besonderer Dank gilt ebenso Daniela B\u00e4stlein, Tanja Diemer-Benedict und Dr. Christoph Benedict, Dr. Sigrid Eckart, Dr. Dr. Sotera Fornaro, Dr. Wilfried Henning, Dr. Stephan Kriesel, Matthias M\u00fcller, Dr. Eleonore Pieh, Kerstin Stange, Dr. Thorsten Umland, Prof. Dr. Hans-Joachim Zimmermann und meinen Lehrern in W\u00fcrzburg und in Heidelberg. Daneben soll die Bibliothek des Instituts f\u00fcr Klassische Philologie der Universit\u00e4t Heidelberg nicht unerw\u00e4hnt bleiben; ohne die dortigen hervorragenden Arbeitsbedingungen w\u00e4re dieses Werk wesentlich erschwert worden.\n\nNun bleibt nur noch der Wunsch an den Leser, diese Sammlung m\u00f6ge ihm Freude bereiten und ihn weniger zu unreflektiertem Zitieren lateinischer Worte als vielmehr zu eigenen Gedanken inspirieren, denn:\n\n**Nemo gloriari nisi suo debet.**\n\nNiemand soll sich einer Sache r\u00fchmen au\u00dfer der eigenen.\n\nSeneca, _Epistulae morales_ 41,7.\n\n_Anmerkung zur vierten Auflage_\n\nAn dieser Stelle m\u00f6chte ich den zahlreichen Lesern danken, die durch ihre Hinweise wesentlich zur Verbesserung der vorhergehenden Auflagen beigetragen haben, besonders Bert F\u00fcssenich, Prof. Dr. Mathias Schmoeckel, Andrea-Eva Smolka, Prof. Dr. Rodney Thomson, Prof. Dr. Andrew Watson und Friedemann Weitz.\n\nM. K. \nBenutzte und zitierte Literatur\n\nBartels, Klaus: Veni, Vidi, Vici. Gefl\u00fcgelte Worte aus dem Griechischen und Lateinischen. Z\u00fcrich\/M\u00fcnchen 21992.\n\nBayer, Karl: Expressis verbis. Z\u00fcrich\/M\u00fcnchen 1996.\n\n\u2013 Nota bene! Das lateinische Zitatenlexikon. Z\u00fcrich\/M\u00fcnchen 31999.\n\nB\u00fcchmann, Georg: Gefl\u00fcgelte Worte. Der Zitatenschatz des deutschen Volkes. Berlin \/ Frankfurt a. M. \/ Wien 411998.\n\nBury, Ernst: In medias res. Lexikon lateinischer Zitate und Wendungen. Berlin 1999. (Digitale Bibliothek. 27.)\n\nCato, Otto: Lateinische Zitate, Kernspr\u00fcche und Redensarten. Hildesheim 1981. Neu bearb. von Hugo Birnbaum. Ebd. 51991.\n\nDuden. Bd. 12: Zitate und Ausspr\u00fcche. Bearb. von Werner Scholze-Stubenrecht [u. a.]. Mannheim\/Leipzig\/Wien\/Z\u00fcrich 1993.\n\nEichholz, Karl: Lateinische Citate mit deutscher \u00dcbersetzung. Lateinische Spr\u00fcche, W\u00f6rter und Spr\u00fcchw\u00f6rter. Hamburg 21900.\n\nFinzi, Giuseppe: __ Dizionario di Citazioni Latine ed Italiane. Palermo 21979.\n\nHelfer, Christian: Crater Dictorum. Saarbr\u00fccken 21995.\n\nKudla, Hubertus: Lexikon der lateinischen Zitate. M\u00fcnchen 1999.\n\nLiebs, Detlef: Lateinische Rechtsregeln und Rechtssprichw\u00f6rter. M\u00fcnchen 61998.\n\nOtto, August\/H\u00e4ussler, Reinhard: Die Sprichw\u00f6rter und sprichw\u00f6rtlichen Redensarten der R\u00f6mer. Leipzig 1890. Nachdr. Hildesheim 1988.\n\nReichert, Heinrich: Unverg\u00e4ngliche lateinische Spruchweisheit. Hamburg 81997.\n\nSellner, Alfred: Latein im Alltag. Wiesbaden 261998.\n\nSepp, Bernhard: Varia. Eine Sammlung von lateinischen und deutschen Versen, Spr\u00fcchen und Redensarten. Augsburg 81894.\n\nSimrock, Karl: Die deutschen Sprichw\u00f6rter. Frankfurt a. M. 1846. Neuausg. Stuttgart 1988. 1995.\n\nTosi, Renzo: __ Dizionario delle sentenze Latine e Greche. Mailand 111996.\n\nWalther, Hans: Carmina medii aevi posterioris Latina. Bd. 2: Proverbia sententiaeque Latinitatis medii aevi. Lateinische Sprichw\u00f6rter und Sentenzen des Mittelalters in alphabetischer Anordnung. Ges. und hrsg. von Hans Walther. 5 Tle. G\u00f6ttingen 1963\u201367. [Zit. als: HW.]  \nBd. 2. N. F.: Proverbia sententiaeque Latinitatis medii aevi ac recentioris aevi. Lateinische Sprichw\u00f6rter und Sentenzen des Mittelalters und der fr\u00fchen Neuzeit in alphabetischer Anordnung. Aus dem Nachla\u00df von Hans Walther hrsg. von Paul G. Schmidt. 3 Tle. Ebd. 1982\u201386. [Zit. als: HWS.]\n\nWerner, Jakob \/ Flury, Peter: Lateinische Sprichw\u00f6rter und Sinnspr\u00fcche des Mittelalters. Heidelberg 21966.\n\nZoozmann, Richard: __ Zitaten- und Sentenzenschatz der Weltliteratur. \u00dcberarb. von Otto A. Kielmeyer. Reinbek b. Hamburg 1997. \nA\n\n**ab alio amentatas hastas torquere**\n\nSpeere verschie\u00dfen, die ein anderer mit Schwungriemen versehen hat\n\n(d. h.: sich auf eine Autorit\u00e4t berufen; vgl.: B\u00fcchsenspanner)\n\nNach Cicero, _De oratore_ 1,242. Vgl. **hastas iacere...**\n\n**Ab alio exspectes, alteri quod feceris.**\n\nErwarte vom anderen, was du (selbst) ihm getan.\n\nPublilius Syrus, _Sententiae_ A 2; ebenso Seneca ( _Epistulae morales_ 94,43), der den Spruch als bekannte Regel bezeichnet. Vgl. Fragment Fab. pall. ex inc. inc. fab. 82 Ribbeck; Matth\u00e4us 7,12, und Lukas 6,31. Vgl. **Ne quid exspectes...**\n\n**Abducet praedam, qui occurrit prior.**\n\nDie Beute wird mit sich fortf\u00fchren, wer sich als erster auf sie st\u00fcrzt.\n\n(Vgl.: Wer zuerst kommt, mahlt zuerst.)\n\nNach Plautus, _Pseudolus_ 1198 f.\n\n**Ab hac regula mihi non licet transversum digitum discedere.**\n\nVon dieser Richtschnur darf ich keinen Finger breit abweichen.\n\nNach Cicero ( _Academici libri priores_ 2,58), der die Wendung bereits als Sprichwort bezeichnet.\n\n**ab hoc et ab hac et ab illa**\n\nvon diesem und von dieser und von jener\n\nEpigramm von Friedrich Taubmann, _Taubmanniana_ , Abt. 4. Angesichts schwatzender Waschfrauen verk\u00fcndet er: \u00bbQuando conveniunt ancilla, Sibilla, Camilla sermonem faciunt, & ab hoc, & ab hac, & ab illa.\u00ab (\u00bbWenn die Magd, Sibilla und Camilla zusammenkommen, schw\u00e4tzen sie von diesem und von dieser und von jener.\u00ab)\n\n**Abi atque abstine manum!**\n\nGeh weg und nimm deine Hand weg!\n\nPlautus, _Casina_ 229; vgl. Terenz, _Adelphoe_ 781.\n\n**ab igne ignem**\n\nvom Feuer Feuer\n\nCicero, _De officiis_ 1,52. \u2013 Es galt als Pflicht, jedem Bed\u00fcrftigen Feuer vom eigenen Herd zu gew\u00e4hren.\n\n**Abi in malam crucem!**\n\nGeh zur Kreuzigung!\n\n(Vgl.: Geh zum Henker!)\n\nNach Plautus, _Mostellaria_ 850; vgl. _Poenulus_ 295; Terenz, _Andria_ 317.\n\n**Abiit, excessit, evasit, erupit.**\n\nEr ging weg, entschwand, entkam, st\u00fcrzte davon.\n\nCicero ( _In L. Catilinam_ 2,1) \u00fcber Catilina.\n\n**ab imis unguibus usque ad verticem summum**\n\nvon den Zehen bis oben zum Scheitel\n\n(vgl.: vom Scheitel bis zur Sohle)\n\nCicero, _Pro Q. Roscio comoedo_ 20; vgl. Petron, _Satyrica_ 102,13: \u00bba capillis usque ad ungues\u00ab (\u00bbvon den Haaren bis zu den Zehen\u00ab) und Apuleius, _Metamorphoses_ 3,21.\n\n**ab imo pectore**\n\naus tiefster Brust\n\nNach (z. B.) Lukrez, _De rerum natura_ 3,57; Catull, _Carmina_ 64,198; Vergil, _Aeneis_ 1,371 und 1,485; Ovid, _Metamorphoses_ 2655 f.\n\n**Ab initio nullum semper nullum.**\n\nAnfangs nichtig, immer nichtig.\n\nVgl. _Digesta_ 50,17,29.\n\n**Ab Iove principium, musae: Iovis omnia plena.**\n\nVon Jupiter her (nehmet) den Anfang, Musen: Alles ist von Jupiter erf\u00fcllt.\n\nVergil, _Bucolica_ 3,60; vgl. auch _Aeneis_ 7,219.\n\n**A bove maiore discat arare minor.**\n\nVom gr\u00f6\u00dferen Ochsen lerne der kleinere pfl\u00fcgen.\n\n(Vgl.: Wie die Alten sungen, so zwitschern die Jungen.)\n\nHW: 04.\n\n**ab ovo**\n\nvom Ei an\n\nHoraz ( _Ars poetica_ 147) spielt hier auf das Zwillingsei der Leda an, aus dem Helena geschl\u00fcpft war. Er lobt Homer, da er den Trojanischen Krieg __ eben nicht mit der Geburt Helenas, sondern \u00bbin medias res\u00ab (148), d. h. mitten im Geschehen begonnen habe.\n\n**ab ovo \/ usque ad mala**\n\nvom Ei bis zu den \u00c4pfeln\n\n(d. h.: vom Anfang bis zum Ende)\n\nHoraz, _Sermones_ 1,3,6 f. \u2013 Das r\u00f6mische Mahl wurde h\u00e4ufig mit einem Ei begonnen und mit einem Apfel beendet. Vgl. **ab ovo.**\n\n**Absens heres non erit.**\n\nWer nicht zur Stelle ist, wird nicht erben.\n\nHWS: 34367.\n\n**absentem qui rodit amicum**\n\nwer einen abwesenden Freund herabsetzt\n\nHoraz, _Sermones_ , 1,4,81.\n\n**Absit invidia verbo!**\n\nMi\u00dfgunst sei dem Worte fern!\n\n(D. h.: mit Verlaub, mit Respekt)\n\nLivius, _Ab urbe condita_ 9,19,15; vgl. 36,7,7. Oft auch zitiert als: \u00bbAbsit invidia dicto.\u00ab (\u00bbMi\u00dfgunst sei dem Gesagten fern.\u00ab)\n\n**Absoluta sententia expositore non indiget.**\n\nVollkommene Worte bed\u00fcrfen keines Interpreten.\n\nVgl. _Digesta_ 32,25,1.\n\n**ab urbe condita (a.u.c.)**\n\nseit Gr\u00fcndung der Stadt (d. h.: Roms)\n\nVarro ( _De gente populi Romani_ ) hatte das Jahr 753 als Zeitpunkt der Gr\u00fcndung Roms errechnet. Livius benannte seine 142 B\u00fccher umfassende Geschichte Roms bzw. des R\u00f6mischen Reiches entsprechend _Ab urbe condita_.\n\n**Abusus non tollit usum.**\n\nMi\u00dfbrauch hebt ein Gebrauchsrecht nicht auf.\n\nVgl. Cicero, _Topica_ 17.\n\n**A cane non magno saepe tenetur aper.**\n\nEin kleiner Hund f\u00e4ngt oft einen wilden Eber.\n\nOvid, _Remedia amoris_ 422.\n\n**a capillis usque ad ungues** siehe **ab imis unguibus...**\n\n**accepto damno ianuam claudere**\n\ndie T\u00fcre schlie\u00dfen, nachdem Schaden erlitten wurde\n\n(vgl.:... wenn das Kind bereits im Brunnen liegt)\n\nHWS: 34389.\n\n**Accidit in puncto, quod non speratur in anno.**\n\nEs ereignet sich in einem Augenblick, was man sich in einem ganzen Jahr nicht erhofft.\n\n(Vgl.: Im Augenblick kann sich begeben, was man nie gedacht im Leben.)\n\nLaut Julius Wilhelm Zincgref ( _Teutscher Nation Apophthegmata_ 1, S. 78) pflegte Kaiser Ferdinand I. diese Worte zu \u00e4u\u00dfern, \u00bbwann etwas unversehens zugieng\u00ab. Vgl. HW: 242.\n\n**Accipe quam primum; brevis est occasio lucri!**\n\nGreif schleunigst zu; kurz nur bietet sich die Gelegenheit, zu Geld zu kommen.\n\nMartial, _Epigrammata_ 8,9,3.\n\n**Accipere quam facere praestat iniuriam.**\n\nUnrecht erleiden ist besser als Unrecht tun.\n\nCicero, _Tusculanae disputationes_ 5,56. Vgl. **Patiare potius...**\n\n**accipitri columbas credere**\n\neinem Habicht Tauben anvertrauen\n\n(vgl.: den Bock zum G\u00e4rtner machen)\n\nNach Ovid, _Ars amatoria_ 2,363.\n\n**Accusare nemo se debet nisi coram deo.**\n\nNiemand mu\u00df sich selbst bezichtigen \u2013 au\u00dfer vor Gott.\n\nLiebs A 15.\n\n**Acheronta movebo** siehe **Flectere si nequeo superos...**\n\n**Acheruntis pabulum**\n\nFutter f\u00fcr den Acheron\n\nPlautus, _Casina_ 157. \u2013 Der Acheron ist einer der vier Str\u00f6me der Unterwelt.\n\n**A Corydon, Corydon, quae te dementia cepit!**\n\nAch Corydon, Corydon, welch Wahnsinn hat dich ergriffen!\n\nVergil, _Bucolica_ 2,69.\n\n**acta agere**\n\nErledigtes erledigen\n\n(vgl.: leeres Stroh dreschen)\n\nNach (z. B.) Plautus, _Cistellaria_ 703 und _Pseudolus_ 260, sowie Terenz, _Phormio_ 419, der die Wendung bereits als gel\u00e4ufig bezeichnet. Vgl. Cicero, _De amicitia_ 85 und _Ad Atticum_ 9,18,3.\n\n**actus fidei**\n\nGlaubensakt\n\nDie Quelle dieses Ausdrucks ist unbekannt. Er wurde im Portugiesischen zu _auto da f\u00e9_ , womit die Urteilsverk\u00fcndigung und -vollstreckung der spanischen und portugiesischen Ketzergerichte bezeichnet wurde.\n\n**Acu tetigisti** siehe **Tetigisti acu**\n\n**ad absurdum**\n\nbis zur Sinnwidrigkeit\n\nDie Wendung ist so in der Antike nicht belegt. Vgl. z. B. Cicero, _Tusculanae disputationes_ 1,61: \u00bbAbsurdum id quidem.\u00ab (\u00bbDas ist Unsinn.\u00ab)\n\n**adamanta movere**\n\nStahl (d. h.: ein hartes Herz) bewegen\n\nNach Ovid, _Amores_ 3,7,57 f. und _Ars amatoria_ 1,659.\n\n**ad bestias**\n\nzu den wilden Tieren\n\n(d. h.: zum Kampf mit wilden Tieren verurteilt werden)\n\n(z. B.) _Digesta_ 28,1,8,4.\n\n**Adde fidem dictis auxiliumque refer!**\n\nHalte dein Versprechen und erwidere die Hilfeleistung!\n\nOvid, _Heroides_ 12,194.\n\n**Adde, quod insidiae sacris a vatibus absunt.**\n\nDazu kommt, da\u00df den heiligen Dichtern Hinterh\u00e4ltigkeit fernliegt.\n\nOvid, _Ars amatoria_ 3,539.\n\n**Additus ab insolente Gallo ponderi gladius.**\n\nVom dreisten Gallier wurde zum Gewicht ein Schwert hinzugelegt.\n\nLivius, _Ab urbe condita_ 5,48,9. \u2013 Im Jahre 390 v. Chr. warf der Senonenf\u00fcrst Brennus nach der Einnahme des Kapitols beim Abwiegen des L\u00f6segelds aus \u00dcberheblichkeit sein Schwert auf die Waagschale und tat den ber\u00fchmten Ausspruch: **Vae victis!** (\u00bbWehe den Besiegten!\u00ab).\n\n**adhuc flagranti crimine**\n\nbei noch brennendem Verbrechen\n\n(d. h.: auf frischer Tat)\n\nNach _Codex Iustinianus_ 9,13,1,1. \u2013 Die Wendung wurde sp\u00e4ter abgek\u00fcrzt zu _in flagranti_ (\u203aauf frischer Tat\u2039).\n\n**Adhuc sub iudice lis est** siehe **Grammatici certant...**\n\n**Adhuc tua messis in herba est** siehe **Sed nimium...**\n\n**Ad impossibilia nemo obligatur** siehe **Impossibilium nulla obligatio est**\n\n**ad incitas redigere**\n\nin h\u00f6chste Verlegenheit bringen\n\nNach Plautus, _Poenulus_ 907. \u2013 Zu erg\u00e4nzen ist _calces_ (\u203aSpielsteine\u2039), so da\u00df die Wendung w\u00f6rtlich bedeutet \u203adie Spielsteine unbeweglich machen\u2039, d. h. jemanden schachmatt setzen.\n\n**ad infinitum usque**\n\nweiter bis ins Unendliche\n\n(z. B.) Quintilian, _Institutio oratoria_ 11,2,41.\n\n**Ad ingenium redis.**\n\nDu kehrst zu deiner Veranlagung zur\u00fcck.\n\n(D. h.: Du wirst wieder du selbst.)\n\nTerenz, _Hecyra_ 113.\n\n**Aditum nocendi perfido praestat fides.**\n\nRedlichkeit erm\u00f6glicht einem Treulosen Zugang zu sch\u00e4dlichem Handeln.\n\nSeneca, _Oedipus_ 686. \u2013 Jemand, der sich durch vermeintliche Redlichkeit Vertrauen erheischt hat, kann, dieses Vertrauen ausnutzend, um so gr\u00f6\u00dferen Schaden anrichten.\n\n**Adiuvat in bello pacatae ramus olivae.**\n\nIm Krieg ist der Zweig des friedlichen \u00d6lbaumes von Nutzen.\n\nDer in der Verbannung befindliche Ovid ( _Epistulae ex Ponto_ 1,1,31) erw\u00e4hnt den Friedenskaiser Augustus in seinen Gedichten und erhofft sich dadurch eine Verbesserung seiner Situation.\n\n**Adiuvat in duris aliquos praesentia rebus.**\n\nIm Unheil n\u00fctzt manchen pers\u00f6nliches Erscheinen.\n\nOvid, _Epistulae ex Ponto_ 2,7,53.\n\n**ad Kalendas Graecas (soluturi)**\n\nan den griechischen Kalenden (werden sie bezahlen)\n\nNach Sueton ( _Vita divi Augusti_ 87,1) war dies eine Redewendung des Kaisers Augustus. \u2013 Im r\u00f6mischen Kalender bezeichnete man den Monatsersten als \u203aKalenden\u2039. Die Griechen hingegen z\u00e4hlten die Tage des Monats ohne besondere Bezeichnung und hatten folglich keine Kalenden. So ist die Wendung also vergleichbar mit dem deutschen \u00bbSankt Nimmerleinstag\u00ab.\n\n**Ad maiorem Dei gloriam (vicit pietas).**\n\nZum gr\u00f6\u00dferen Ruhme Gottes (siegte die Fr\u00f6mmigkeit).\n\nPapst Gregor I., _Dialogi_ 1,2,6. \u2013 Die Wendung wurde durch Ignatius von Loyola zum Wahlspruch des 1534 gegr\u00fcndeten Jesuitenordens.\n\n**Ad medicam dubius confugit aeger opem.**\n\nWer bedenklich krank ist, sucht \u00e4rztlichen Beistand.\n\nOvid ( _Epistulae ex Ponto_ 3,4,8) bittet seinen Freund Rufinus, seine Dichtung wohlwollend aufzunehmen: So wie ein Kranker einen Arzt, so brauche er, der durch die Verbannung geschw\u00e4chte Dichter, einen geneigten Leser. Vgl. **Parcendum est animo...**\n\n**admodum tenui filo suspensum esse**\n\nan einem ziemlich d\u00fcnnen Faden h\u00e4ngen\n\n(vgl.: an einem seidenen Faden h\u00e4ngen)\n\nNach Valerius Maximus, _Facta et dicta memorabilia_ 6,4,1.\n\n**ad nauseam usque**\n\nbis zum Erbrechen, bis zum \u00dcberdru\u00df\n\nHWS: 34444a.\n\n**Adnuit et totum nutu tremefecit Olympum.**\n\n(Jupiter) nickte gew\u00e4hrend und lie\u00df durch sein Nicken den ganzen Olymp erbeben.\n\nVergil, _Aeneis_ 9,106.\n\n**Ad nummum convenit.**\n\nEs stimmt auf den Pfennig.\n\nCicero, _Ad Atticum_ 5,21,12.\n\n**ad oculos**\n\nvor Augen\n\nCicero, _Partitiones oratoriae_ 20. \u2013 \u00bbad oculos demonstrieren\u00ab bedeutet \u203ajemandem etwas vor Augen f\u00fchren\u2039.\n\n**Adolescentia deferbuit.**\n\nDie Jugend hat sich ausgetobt.\n\nNach Cicero, _Pro M. Caelio_ 43.\n\n**Ad omnia alia aetate sapimus rectius.**\n\nIn allen anderen Angelegenheiten werden wir mit dem Alter weiser.\n\nTerenz, _Adelphoe_ 832 ff.: \u00bbAd omnia alia aetate sapimus rectius. \/ Solum unum hoc vitium adfert senectus hominibus: \/ adtentiores sumus ad rem omnes quam sat est: \/ quod illos sat aetas acuet.\u00ab (\u00bbIn allen anderen Angelegenheiten werden wir mit dem Alter weiser. Nur dieses eine Laster bringt das Alter den Menschen: Wir streben alle \u00fcber die Geb\u00fchr nach Besitz: Diesbez\u00fcglich wird das Alter jene gen\u00fcgend anspornen.\u00ab)\n\n**Adora, quod incendisti!**\n\nBete an, was du einge\u00e4schert hast.\n\nNach Gregor von Tours ( _Historia Francorum_ 2,31) war der Hl. Remigius, Bischof von Reims, von Clodovechs Frau zur Taufe ihres Mannes gerufen worden. W\u00e4hrend der Zeremonie soll er folgende Worte gesprochen haben: \u00bbMitis depone colla, Sigamber; adora, quod incendisti, incende, quod adorasti.\u00ab (\u00bbBeuge gelassen deinen Nacken, Sicamber; bete an, was du verbrannt hast, verbrenne, was du angebetet hast.\u00ab)\n\n**Ad praesens ova cras pullis sunt meliora.**\n\nHeute Eier (zu haben) ist besser als morgen Hennen.\n\n(Vgl.: Der Spatz in der Hand ist besser als die Taube auf dem Dach.)\n\nRabelais, _Gargantua et Pantagruel_ 3,42.\n\n**Ad summam laetitiam cumulus accedit.**\n\n(Dies) kommt als Gipfel zur \u00fcbergro\u00dfen Freude (noch) hinzu.\n\nNach Cicero, _Ad Atticum_ 4,21.\n\n**ad usum Delphini**\n\nf\u00fcr den Gebrauch des Dauphin\n\n_Dauphin_ war seit dem 12. Jahrhundert der Beiname der Grafen von Albon, dann Titel der Grafen von Vienne. Ihr Land, die Dauphin\u00e9, eine historische Provinz im S\u00fcdosten Frankreichs, fiel 1349 an die franz\u00f6sische Krone. Es wurde dem jeweiligen Thronfolger, von nun an auch _Dauphin_ genannt, als Apanage bestimmt. \u2013 K\u00f6nig Ludwig XIV. hatte Jacques Bossuet und Pierre Daniel Huet damit beauftragt, f\u00fcr seinen Sohn, den _Dauphin_ , eine von anst\u00f6\u00dfigen Stellen gereinigte Lekt\u00fcre klassischer Werke zu erstellen. Heute wird diese Bezeichnung auch ironisch f\u00fcr zensierte oder f\u00fcr den Schulgebrauch zurechtgemachte Textausgaben verwendet.\n\n**Adversae res admonent religionum.**\n\nUngl\u00fcck erinnert an die Religion.\n\n(Vgl.: Not lehrt beten.)\n\nNach Livius, _Ab urbe condita_ 5,51,9.\n\n**adverso flumine**\n\ngegen den Strom\n\n(z. B.) Caesar, _De bello Gallico_ 7,60,3; 7,61,3 und _De bello civili_ 3,30,4. \u2013 Die Wendung ist bei Caesar w\u00f6rtlich gemeint. Sp\u00e4ter erst wurden \u00e4hnliche Wendungen bildlich gebraucht, z. B. bei Juvenal, _Saturae_ 4,89 f.\n\n**adversum stimulum calces (iactare)**\n\nwider den Stachel l\u00f6cken\n\n(d. h.: vergeblich Widerstand leisten)\n\nTerenz, _Phormio_ 78; vgl. Apostelgeschichte 26,14. \u2013 Dieses Bild findet sich bereits in der griechischen Literatur bei Aischylos ( _Agamemnon_ 1624) und Euripides ( _Die Bacchantinnen_ 795).\n\n**Adversus necessitatem ne dii quidem resistunt.**\n\nGegen (den Zwang der) Notwendigkeit leisten nicht einmal die G\u00f6tter Widerstand.\n\nNach Livius, _Ab urbe condita_ 9,4,16.\n\n**advocatus diaboli**\n\nAnwalt des Teufels\n\nVolkst\u00fcmliche Bezeichnung des amtlichen Vertreters der Gegenargumente bei Selig- und Heiligsprechungsprozessen (auch _promotor fidei \u203a_ F\u00f6rderer des Glaubens\u2039), dessen Widersacher der _advocatus Dei_ \u203ader Anwalt Gottes\u2039 oder auch _postulator_ \u203aForderer (der Heiligsprechung)\u2039 war.\n\n**Aegroto dum anima est, spes est.**\n\nSolange einem Kranken Atem innewohnt, gibt es noch Hoffnung.\n\nCicero ( _Ad Atticum_ 9,11,3) erw\u00e4hnt dieses Sprichwort, um seine eigene Situation zu schildern: Solange Pompeius noch in Italien war, hegte Cicero die Hoffnung, da\u00df ein B\u00fcrgerkrieg noch zu vermeiden sei. Vgl. **Dum spiro, spero.**\n\n**A.E.I.O.U.**\n\n(En!) Amor Electis Iniuriis Ordinat Ultor. (Sic Fredericus ego rex mea iura rego.)\n\n(Siehe!) Liebe, R\u00e4cher des Unrechts, waltet \u00fcber den Auserw\u00e4hlten. (So f\u00fchre ich, K\u00f6nig Friedrich, mein Recht aus.)\n\nDer Legende zufolge soll Friedrich der Sch\u00f6ne (Friedrich III.) diese Auslegung der Abk\u00fcrzung an einem Schrank angebracht haben, nachdem ein Widersacher die Abk\u00fcrzung an die Burgwand geschrieben und ihre Bedeutung als \u00bb **A** ller **e** rst **i** st **\u00d6** sterreich **V** erderben\u00ab erkl\u00e4rt hatte. (Die Buchstaben \u203aU\u2039 und \u203aV\u2039 wurden gleich geschrieben.) In einer anderen Version wird das Akronym mit Albrecht III. in Verbindung gebracht:\n\nAlbertus Electus Imperator Optamus Vivat \u2013 Der zum Kaiser gew\u00e4hlte Albrecht lebe hoch.\n\nWeitere Auslegungen sind:\n\nArchidux Electus Imperator Optime Vivat \u2013 Der zum Kaiser gew\u00e4hlte Erzherzog lebe bestens hoch.\n\nAustriae Est Imperare Orbi Universi \u2013 Es ist Aufgabe \u00d6sterreichs, den ganzen Erdkreis zu beherrschen.\n\nAustria Erit In Orbe Ultima \u2013 \u00d6sterreich wird bestehen bzw. sich ausdehnen bis ans Ende der Welt.\n\nAustria Et Imperium Optima Unita \u2013 \u00d6sterreich und das Reich sind bestens vereint.\n\nAller Ehren ist \u00d6sterreich voll.\n\nAlles Erdreich ist \u00d6sterreich untertan.\n\n**Aequalitas non parit bellum.**\n\nKr\u00e4ftegleichheit bringt keinen Krieg hervor.\n\nHWS: 34523.\n\n**Aequam memento rebus in arduis \/ servare mentem.**\n\nDenke daran, in widrigen Zeiten ein ruhiges Herz zu bewahren.\n\nHoraz ( _Carmina_ 2,3,1 ff.) warnt in epikureischer Gesinnung ebenso vor \u00fcberm\u00e4\u00dfiger Freude, die die Seelenruhe st\u00f6ren k\u00f6nnte: \u00bb... non secus in bonis \/ ab insolenti temperatam \/ laetitia, moriture Delli.\u00ab (\u00bb... und es [das Herz] in guten Zeiten ebenso von \u00fcberm\u00fctiger Freude fernzuhalten, sterblicher Dellius.\u00ab)\n\n**Aequat omnis cinis.**\n\nDie Asche macht alle gleich.\n\n(Vgl.: Der Tod macht alle gleich; er fri\u00dft Arm und Reich.)\n\nSeneca, _Epistulae morales_ 91,16.\n\n**Aequitas numquam contravenit legi.**\n\nBilligkeit l\u00e4uft niemals dem Gesetz zuwider.\n\nLiebs A 53.\n\n**Aequo animo audienda sunt imperitorum convicia.**\n\nDie Schm\u00e4hungen der Einf\u00e4ltigen mu\u00df man sich mit Gleichmut anh\u00f6ren.\n\n(Vgl.: Was von mir ein Esel spricht, das acht' ich nicht.)\n\nSeneca, _Epistulae morales_ 76,4.\n\n**Aequo animo poenam, qui meruere, ferant.**\n\nWer die Strafe verdient hat, soll sie gleichm\u00fctig ertragen.\n\nNach Ovid, _Amores_ 2,7,12.\n\n**Aequum atque iniquum regis imperium feras.**\n\nErtrage den Befehl des K\u00f6nigs, ob gerecht oder ungerecht.\n\nSeneca, _Medea_ 195.\n\n**Aequum est \/ peccatis veniam poscentem reddere rursus.**\n\nEs ist billig, da\u00df, wer um Nachsicht f\u00fcr Verfehlungen bittet, sie seinerseits gew\u00e4hrleistet.\n\nHoraz, _Sermones_ 1,3,74 f.\n\n**aere perennius** siehe **Exegi monumentum...**\n\n**Aeris alieni comes miseria.**\n\nElend ist der Begleiter von Schulden.\n\n(Vgl.: Borgen macht Sorgen.)\n\nNach Plinius d. \u00c4. ( _Naturalis historia_ 7,119) ist dies einer der drei weisen Spr\u00fcche Chilons, die zu Delphi in Gold niedergeschrieben waren.\n\n**Aetas cinaedum celat, aetas indicat.**\n\nDie Zeit verh\u00fcllt einen L\u00fcstling, die Zeit enth\u00fcllt ihn.\n\nPublilius Syrus, _Sententiae_ A 24.\n\n**Aetas volat.**\n\nDie Zeit fliegt dahin.\n\nNach Cicero, _Tusculanae disputationes_ 1,76.\n\n**Aetate fruere! Mobili cursu fugit.**\n\nGenie\u00dfe das Leben! In schnellem Lauf flieht es dahin.\n\nSeneca, _Phaedra_ 446.\n\n**Aethiopem lavare**\n\neinen \u00c4thiopier waschen\n\n(Vgl.: Einen Mohren kann man nicht wei\u00df waschen. Und: Schwarz geboren hat's Waschen verloren.)\n\nJeremia 13,23; vgl. Hieronymus ( _In Sophoniam_ 2,12 und _Adversus Rufinum_ 2,23), der ein \u00e4hnliches (jud\u00e4isches) Sprichwort erw\u00e4hnt. Wie aus Zenobius (Paroemiographi Graeci) 1,46 ersichtlich, geht die Wendung auf ein griechisches Sprichwort zur\u00fcck (Aesop, _Fabulae_ 274 Hausrath).\n\n**Affirmantis probatio.**\n\nDer Beweis (obliegt) dem, der eine Behauptung aufstellt.\n\nVgl. _Digesta_ 22,3,2.\n\n**Afflavit deus, et dissipati sunt.**\n\nGott blies, und sie wurden zersprengt.\n\nInschrift auf einer Gedenkm\u00fcnze zur Erinnerung an den Untergang der spanischen Armada im Jahre 1588.\n\n**Age, libertate Decembri utere!**\n\nWohlan, genie\u00dfe die Dezemberfreiheit!\n\nNach Horaz, _Sermones_ 2,7,4 f. \u2013 Im Dezember fanden die Saturnalien, eine Art Fastnacht, statt, w\u00e4hrend deren sich z. B. Sklaven als Herren auff\u00fchren durften. Vgl. **Non semper Saturnalia erunt.**\n\n**Agnosco veteris vestigia flammae.**\n\nIch erkenne die Spuren der alten Flamme wieder.\n\nVergil, _Aeneis_ 4,23. \u2013 Dido entdeckt im Gespr\u00e4ch mit Anna die Liebe zu ihrem verstorbenen Mann Sychaeus wieder.\n\n**Agunt opus suum fata.**\n\nDie Schicksalsg\u00f6ttinnen sind am Werke.\n\nSeneca, _Ad Marciam de consolatione_ 21,6.\n\n**A Iove percussus non leve vulnus habet.**\n\nWer von Jupiter getroffen ist, tr\u00e4gt keine leichte Wunde davon.\n\nOvid, _Epistulae ex Ponto_ 1,7,50.\n\n**A lasso rixa quaeritur.**\n\nWer m\u00fcde ist, sucht Streit.\n\nDer Spruch wird schon von Seneca ( _De ira_ 3,9,5) als altes Sprichwort bezeichnet.\n\n**albae gallinae filius**\n\nder Sohn einer wei\u00dfen Henne\n\nNach Juvenal, _Saturae_ 13,141. \u2013 Der Bezug dieser Wendung ist ungekl\u00e4rt. Eventuell besteht eine Verbindung zu den seit Livia am Kaiserhof gehaltenen wei\u00dfen H\u00fchnern. Aus dem Kontext geht hervor, da\u00df sich Calvinus f\u00fcr etwas Besonderes h\u00e4lt, wom\u00f6glich wie sich ein kaiserliches wei\u00dfes Huhn von der Schar gew\u00f6hnlicher H\u00fchner abhebt.\n\n**Albus an ater sit, nescio.**\n\nOb er wei\u00df oder schwarz ist, wei\u00df ich nicht.\n\n(D. h.: Er ist mir gleichg\u00fcltig.)\n\nNach Catull, _Carmina_ 93,2.\n\n**Alea iacta est.**\n\nDer W\u00fcrfel ist geworfen.\n\nLaut Sueton ( _Vita divi Iulii_ 32) tat Caesar diesen Ausspruch (\u00bbiacta alea est\u00ab nach Menander, \u00fcberliefert bei Athenaios 13,599e) im Jahre 49 v. Chr., als er den Rubicon \u00fcberschritt und somit als Ausdruck seines Machtanspruchs gegen\u00fcber Rom trotz Verbots die Grenzen seines Amtsbereichs mit dem Heer verlie\u00df. Plutarch ( _Pompeius_ 60) zufolge bediente sich Caesar des griechischen Idioms, das den Imperativ (lat. _esto)_ statt des Indikativs verwendet. Vgl **. Caesar citra...**\n\n**Aliam aetatem alia decent.**\n\nF\u00fcr unterschiedliche Lebensalter ziemen sich jeweils andere Dinge.\n\n(Vgl.: Alles zu seiner Zeit. Und: Andere Zeiten, andere Sitten.)\n\nNach Plautus, _Mercator_ 984: \u00bbItidem ut tempus anni, aetatem aliam aliud factum condecet.\u00ab (\u00bbWie f\u00fcr unterschiedliche Jahreszeiten, so ziemt sich auch f\u00fcr unterschiedliche Lebensalter eine andere Handlungsweise.\u00ab)\n\n**Aliena nobis, nostra plus aliis placent.**\n\nFremdes gef\u00e4llt uns, Unseres mehr den anderen.\n\nPublilius Syrus, _Sententiae_ A 28. Vgl. Horaz, _Epistulae_ 1,14,11: \u00bbCui placet alterius, sua nimirum est odio sors.\u00ab (\u00bbWem das Los eines anderen gef\u00e4llt, dem ist das eigene nat\u00fcrlich verha\u00dft.\u00ab) \u00c4hnlich Seneca, _De ira_ 3,31,1.\n\n**Aliena vitia in oculis habemus, a tergo nostra sunt.**\n\nDie fremden Fehler haben wir vor Augen, die eigenen liegen hinter unserem R\u00fccken.\n\n(vgl.: den Balken im eigenen Auge nicht sehen)\n\nSeneca, _De ira_ 2,28,8; vgl. Matth\u00e4us 7,3 und Lukas 6,42.\n\n**alienis gloriari bonis**\n\nsich fremder G\u00fcter r\u00fchmen\n\n(vgl.: sich mit fremden Federn schm\u00fccken)\n\nNach Phaedrus ( _Fabulae_ 1,3,1), der in Anlehnung an \u00c4sop am Beispiel der Dohle, die sich mit Pfauenfedern schm\u00fcckt, die Eitelkeit der Menschen aufzeigt. Vgl. Horaz, _Epistulae_ 1,3,18 ff., und Hieronymus, _Praefatio ad Paulinianum_. Beide sprechen, \u00e4hnlich der deutschen Wendung, von fremden _colores_ (\u203aFarben\u2039, d. h. bunten Federn).\n\n**Alii sementem faciunt, alii metent.**\n\nDie einen s\u00e4en, die anderen werden ernten.\n\nHWS: 34620.\n\n**Aliis inserviendo consumor.**\n\nIm Dienst f\u00fcr andere verzehre ich mich.\n\n(z. B.) Gabriel Rollenhagen, _Nucleus Emblematum selectissimorum_ , Embleme 31. Er vergleicht einen Herrn, der sich um seine Untergebenen verdient macht, mit einer Kerze, die anderen Licht spendet und dabei selbst vergeht. Bismarck hat die Wendung in einem Brief vom 6. November 1852 an Leopold von Gerlach zu \u00bbPatriae inserviendo consumor\u00ab (\u00bbIm Dienst f\u00fcr das Vaterland verzehre ich mich\u00ab) umgem\u00fcnzt.\n\n**Aliis leporem excitavi.**\n\nF\u00fcr andere habe ich einen Rammler aufgescheucht.\n\nBei Petron ( _Satyrica_ 131,7) wird diese schl\u00fcpfrige Wendung als Sprichwort bezeichnet.\n\n**Aliis ne feceris, quod tibi fieri non vis.**\n\nTu anderen nicht an, was du nicht erleiden willst.\n\n(Vgl.: Was du nicht willst, da\u00df man dir tu, das f\u00fcg auch keinem andern zu.)\n\nNach _Historia Augusta_ , \u00bbVita Alexandri Severi\u00ab 51,8; vgl. Matth\u00e4us 7,12 und Lukas 6,31. Vgl. **Quod tibi...**\n\n**Aliis si licet , tibi non licet.**\n\nAuch wenn es anderen erlaubt sein mag: Dir ist es nicht erlaubt.\n\nTerenz, _Heauton timorumenos_ 797. Vgl. **Quod licet Iovi...**\n\n**a limine**\n\nvon der Schwelle\n\nNach Catull, _Carmina_ 68,4. \u2013 Der Dichter antwortet hier auf den Brief seines Freundes Allius, der ihn u. a. gebeten hatte, ihn \u00bbvon der Schwelle des Todes zur\u00fcckzuholen\u00ab. Heute wird die Wendung \u00bbeinen Antrag _a limine_ ablehnen\u00ab in der Bedeutung von \u203aeinen Antrag gleich abweisen, ihn gar nicht erst behandeln\u2039 verwendet.\n\n**Aliud legunt pueri, aliud viri, aliud senes.**\n\nAnderes lesen sich Knaben, anderes M\u00e4nner, anderes Greise heraus.\n\nHW: 792a.\n\n**Aliud sceptrum, aliud plectrum.**\n\nDas Szepter ist eine Sache, die Laute eine andere.\n\n(D. h.: Jeder versteht sich auf etwas anderes.)\n\nHW: 792b.\n\n**alius et idem**\n\nanders und doch derselbe\n\nNach Horaz, _Carmen saeculare_ 10.\n\n**Alter alterius auxilio eget.**\n\nDer eine bedarf der Hilfe des anderen.\n\nNach Sallust _, Bellum Catilinarium_ 1,7.\n\n**alter ego** siehe **Amicus est tamquam alter ego**\n\n**Alter frenis eget, alter calcaribus.**\n\nDer eine bedarf der Z\u00fcgel, der andere der Sporen.\n\nLaut Cicero ( _Ad Atticum_ 6,1,12) hat Isokrates so die Historiker Ephorus und Theopompus beschrieben.\n\n**Alterius non sit, qui suus esse potest.**\n\nEinem anderen geh\u00f6re nicht, wer sich selbst geh\u00f6ren kann.\n\nNach Cicero, _De re publica_ 3,37.\n\n**Alternant spesque timorque vicem.**\n\nHoffnung und Furcht wechseln sich st\u00e4ndig ab.\n\nOvid, _Heroides_ 6,38.\n\n**Alter rixatur de lana saepe caprina.**\n\nDer andere streitet oft um die Ziegenwolle.\n\nHoraz, _Epistulae_ 1,18,15. \u2013 Die Frage, ob man bei dem Fell einer Ziege von \u203aHaaren\u2039 oder von \u203aWolle\u2039 sprechen m\u00fcsse, stand sprichw\u00f6rtlich f\u00fcr g\u00e4nzlich nichtige Angelegenheiten.\n\n**Altissima flumina minimo sono labuntur.**\n\nDie gr\u00f6\u00dften Fl\u00fcsse gleiten mit dem geringsten Ger\u00e4usch dahin.\n\nNach Curtius Rufus, _Historia Alexandri Magni_ 7,4,13. \u2013 Kobares erw\u00e4hnt hier ein Sprichwort der Baktrianer: \u00bb... canem timidum vehementius latrare quam mordere altissimaque flumina minimo sono labi\u00ab (\u00bb... da\u00df ein \u00e4ngstlicher Hund heftiger belle als bei\u00dfe und da\u00df die tiefsten Fl\u00fcsse mit dem geringsten Ger\u00e4usch dahinglitten.\u00ab)\n\n**altum silentium**\n\ntiefes Schweigen\n\nNach Vergil, _Aeneis_ 10,63.\n\n**amabilis \/ insania**\n\nliebensw\u00fcrdiger Wahnsinn\n\nHoraz, _Carmina_ 3,4,5 f.\n\n**amantes amentes**\n\nVerliebte, Verwirrte\n\n(Vgl.: Minne verkehrt die Sinne.)\n\nTitel eines Lustspiels von Gabriel Rollenhagen, wohl nach Plautus, _Mercator_ 82: \u00bbAmens amansque ut animum offirmo meum.\u00ab (\u00bbLiebend und rasend, wie ich war, fa\u00dfte ich mir ein Herz.\u00ab) Vgl. **Inceptio est amentium, non amantium** und **Nemo in amore videt.**\n\n**Amanti nihil difficile** siehe **Nihil difficile amanti**\n\n**Amantium irae amoris integratio est.**\n\nDer Zorn der Liebenden ist Erneuerung der Liebe.\n\n(Vgl.: Liebeszorn ist neuer Liebeszunder.)\n\nTerenz, _Andria_ 555.\n\n**Amare et sapere vix deo conceditur.**\n\nZu lieben und (dabei) den Verstand zu bewahren ist selbst einem Gott kaum m\u00f6glich.\n\nPublilius Syrus, _Sententiae_ A 22.\n\n**amare tamquam oculos**\n\nwie die (eigenen) Augen lieben\n\n(vgl.: wie seinen Augapfel h\u00fcten)\n\nNach Plautus, _Miles gloriosus_ 984.\n\n**Ambages narras.**\n\nDu sprichst in R\u00e4tseln.\n\nNach Terenz, _Heauton timorumenos_ 318 f.\n\n**Amens incurrit in columnas.**\n\nEin Verr\u00fcckter rennt gegen S\u00e4ulen.\n\nNach Cicero, _Pro M. Aemilio Scauro_ 45m.\n\n**Ames iudicio, non amore iudices.**\n\nM\u00f6gest du mit Urteil lieben, aber nicht nach der Liebe urteilen.\n\nPseudo-Seneca, _De moribus_ 48.\n\n**Amici, diem perdidi!**\n\nFreunde, ich habe einen Tag verloren!\n\nSueton ( _Vita divi Titi_ 8,1) \u00fcber den Kaiser Titus: \u00bbAtque etiam recordatus quondam super cenam, quod nihil cuiquam toto die praestitisset, memorabilem illam meritoque laudatam vocem edidit: Amici, diem perdidi.\u00ab (\u00bbUnd als er sich einmal beim Abendessen daran erinnerte, da\u00df er w\u00e4hrend des ganzen Tages niemandem irgendwie dienlich gewesen war, tat er den bekannten, oft gepriesenen Ausspruch: \u203aFreunde, ich habe einen Tag verloren!\u2039\u00ab) Vgl. Ausonius, _Gratiarum actio ad Gratianum imperatorem pro consulatu_ 16.\n\n**Amicitiae immortales, mortales inimicitiae debent esse.**\n\nFreundschaften sollen unsterblich, Feindschaften sterblich sein.\n\nLivius ( _Ab urbe condita_ 40,46,12) f\u00fchrt dies bereits als Sprichwort an.\n\n**amico amicus**\n\ndem Freund (wie) ein Freund (gesinnt)\n\nTerenz, _Phormio_ 562: \u00bbSolus est homo amico amicus.\u00ab (\u00bbNur der Mensch ist Freund dem Freund.\u00ab) Vgl. Plautus, _Miles gloriosus_ 660.\n\n**Amicum esse unum animum in duobus corporibus.**\n\nEin Freund sei eine Seele in zwei K\u00f6rpern.\n\nNach Aristoteles, _Nikomachische Ethik_ 9,4,1166a 31.\n\n**Amicus certus in re incerta cernitur.**\n\nEin sicherer Freund l\u00e4\u00dft sich in unsicheren Umst\u00e4nden erkennen.\n\n(Vgl.: Freunde in der Not gehen zehn auf ein Lot.)\n\nCicero ( _De amicitia_ 64) zitiert hier Ennius, _Hectoris lytra_ Fragment 216 Warmington.\n\n**Amicus esse mihi coepi.**\n\nIch habe angefangen, mir selbst ein Freund zu sein.\n\nSeneca ( _Epistulae morales_ 6,7) zitiert hier Hekaton mit der Erkl\u00e4rung, wer sich selbst ein Freund geworden sei, habe gro\u00dfe Fortschritte gemacht: Er werde niemals allein und stets allen ein Freund sein.\n\n**Amicus est tamquam alter ego.**\n\nEin Freund ist gleichsam ein zweites Ich.\n\nNach Ambrosius, _De spiritu sancto_ 2,13,154. \u2013 Die Bezeichnung eines guten Freundes als zweites Ich geht laut Diogenes Laertius ( _Vitae philosophorum: Zenon_ 7,23) auf Zenon zur\u00fcck. Bei Aristoteles ( _Nikomachische Ethik_ 9,4,1166a 31) findet sich \u00bballos autos\u00ab (\u00bbein zweiter eben solcher\u00ab). \u00c4hnlich bei Cicero ( _De amicitia_ 80): \u00bb(verus amicus) est enim is, qui est tamquam alter idem.\u00ab (\u00bb[Ein wahrer Freund] ist n\u00e4mlich der, der gleichsam ein zweiter solcher [wie man selbst] ist.\u00ab)\n\n**Amicus Plato, sed magis amica veritas.**\n\nPlaton ist mir ein Freund, doch mehr Freund ist mir die Wahrheit.\n\nNach Aristoteles, _Nikomachische Ethik_ 1,4,1096a 16.\n\n**Amittit merito proprium, qui alienum adpetit.**\n\nEs verliert zu Recht Eigenes, wer Fremdes anstrebt.\n\nPhaedrus, _Fabulae_ 1,4,1.\n\n**amor ac deliciae generis humani**\n\nLiebe und Liebling (d. h.: Wonne) des Menschengeschlechts\n\nSueton ( _Vita divi Titi_ 1,1) bezeichnet so den Kaiser Titus.\n\n**Amor electi...** siehe **A.E.I.O.U.**\n\n**Amore, more, ore, re \/ iunguntur amicitiae.**\n\nDurch Liebe, Sitte, Tat und Mund geschlossen wird der Freundschaftsbund.\n\nHW: 999.\n\n**amor fati**\n\nLiebe zum Schicksal\n\nNietzsche, _Ecce homo_ , \u00bbWarum ich so klug bin\u00ab, 10.\n\n**amor sceleratus habendi**\n\nverbrecherische Habsucht\n\nOvid, _Metamorphoses_ 1,131.\n\n**Amor vincit omnia** siehe **Omnia vincit Amor...**\n\n**Anathema sit!**\n\nEr soll verflucht sein!\n\nGalaterbrief 1,8 f.; vgl. R\u00f6merbrief 9,3 und Korintherbriefe 1,12,3 und 1,16,22.\n\n**Anceps malum urget.**\n\nEin doppeltes \u00dcbel droht.\n\n(vgl.: zwischen zwei Feuer geraten)\n\nNach Livius, _Ab urbe condita_ 3,28,9.\n\n**An ego totiens de eadem re audiam?**\n\nSoll ich mir so oft (noch) dasselbe anh\u00f6ren?\n\n(vgl.: ewig das alte Lied, immer dieselbe Leier h\u00f6ren)\n\nTerenz, _Adelphoe_ 128.\n\n**Anguilla est, elabitur.**\n\nEr ist (wie) ein Aal; er entschl\u00fcpft.\n\nPlautus, _Pseudolus_ 747.\n\n**anima candida**\n\neine lautere Seele\n\nNach Horaz, _Sermones_ 1,5,41.\n\n**Anima est amica amanti.**\n\nDie Freundin ist des Liebenden Lebenskraft.\n\nPlautus, _Bacchides_ 193.\n\n**Animam debet.**\n\nEr schuldet seine Seele.\n\n(Vgl.: mit Leib und Seele verschuldet sein)\n\nTerenz, _Phormio_ 661.\n\n**anima naturaliter Christiana**\n\ndie von Natur aus christliche Seele (des Menschen)\n\nNach Tertullian, _Apologeticum_ 17,6.\n\n**animi sub vulpe latentes**\n\nunter einem Fuchspelz versteckte Gesinnung\n\nHoraz, _Ars poetica_ 437.\n\n**animula vagula blandula**\n\nschweifendes, schmeichelndes Seelchen\n\nHadrian, Fragment 3,1 Baehrens.\n\n**Animum rege, qui nisi paret, \/ imperat** siehe **Ira furor brevis est...**\n\n**Animus aequus est optimum aerumnae condimentum.**\n\nBei Tr\u00fcbsal ist Gleichmut die beste W\u00fcrze.\n\nPlautus, _Rudens_ 402.\n\n**Animus est in patinis** siehe **Iam dudum animus...**\n\n**Animus meminisse horret.**\n\nDie Seele erschaudert bei der Erinnerung.\n\nVergil, _Aeneis_ 2,12. \u2013 Aeneas erschaudert, als ihn Dido vom Untergang Trojas berichten l\u00e4\u00dft. Plinius d. J. ( _Epistulae_ 6,20,1) zitiert diese Stelle zu Beginn seines Briefes \u00fcber den Vesuv-Ausbruch des Jahres 79 n. Chr.\n\n**An nescis longas regibus esse manus?**\n\nWei\u00dft du etwa nicht, da\u00df die H\u00e4nde der K\u00f6nige weit reichen? (Vgl.: Gro\u00dfe Herren haben lange H\u00e4nde.)\n\nOvid, _Heroides_ 17,166. Vgl. auch Seneca, _Epistulae morales_ 82,5.\n\n**An nescis, mi fili, quantilla prudentia mundus regatur** (bzw. **regatur orbis** ) **?**\n\nWei\u00dft du etwa nicht, mein Sohn, mit wie geringem Verstand die Welt regiert wird?\n\nDiese Worte, oft auch als \u00bbVidebis, mi fili, quam parva sapientia regatur mundus?\u00ab zitiert, soll Axel Oxenstierna zu seinem Sohn gesagt haben, als dieser z\u00f6gerte, die Aufgabe des schwedischen Gesandten in M\u00fcnster anzunehmen. Evtl. Ausspruch von Papst Julius III. gegen\u00fcber einem portugiesischen M\u00f6nch (vgl. B\u00fcchmann).\n\n**annus horribilis**\n\nein schreckliches Jahr\n\nK\u00f6nigin Elisabeth II. von England bezeichnete so r\u00fcckblickend das Jahr 1992, das von zahlreichen Krisen innerhalb der k\u00f6niglichen Familie gepr\u00e4gt war. Im Gegensatz zu \u00bbAnnus Mirabilis\u00ab, dem Titel eines Historiengedichtes von John Dryden.\n\n**ante antidotum quam venenum**\n\ndas Gegenmittel vor dem Gift (geben)\n\nHieronymus (z. B. in _Adversus Rufinum_ 2,34) verwendet dieses schon in der Antike bekannte Sprichwort hier ironisch: Er fragt Rufinus, warum er denn das Vorwort zu den B\u00fcchern, die er anklagt, nicht kenne. Rufinus erf\u00fclle wohl das Sprichwort, indem er sich schon gegen diese B\u00fccher wappne, bevor er sie \u00fcberhaupt l\u00e4se.\n\n**Ante mortem nemo beatus** siehe **Dicique beatus...**\n\n**aperta transire**\n\nan Offensichtlichem vorbeigehen\n\n(vgl.: den Wald vor lauter B\u00e4umen nicht sehen)\n\nNach Seneca, _Epistulae morales_ 68,4.\n\n**Apes debemus imitari.**\n\nWir m\u00fcssen es den Bienen gleichtun.\n\n(vgl.: flei\u00dfige Bienen)\n\nSeneca ( _Epistulae morales_ 84,3) bezeichnet diese Aufforderung bereits als Sprichwort. Vgl. Horaz, _Carmina_ 4,2,27 ff.\n\n**apices iuris**\n\njuristische Spitzfindigkeiten\n\n_Digesta_ 17,1,29,4.\n\n**Apparet id etiam caeco.**\n\nDas sieht auch ein Blinder.\n\nNach Livius, _Ab urbe condita_ 32,34,3.\n\n**Appello a papa male informato ad papam melius informandum.**\n\nIch appelliere vom schlecht informierten Papst an den gr\u00fcndlicher zu informierenden Papst.\n\nMartin Luther zugeschrieben.\n\n**Apta mihi vis est, vi tristia nubila pello.**\n\nMir ziemt Gewalt, mit Gewalt vertreib ich das d\u00fcstere Gew\u00f6lk.\n\nOvid, _Metamorphoses_ 6,690. \u2013 Der Windgott Boreas droht mit Gewalt, nachdem die Athener ablehnend auf seine Werbung um die geliebte Oreithya, Tochter des athenischen K\u00f6nigs Erechtheus, reagiert haben.\n\n**Apud paucos post rem manet gratia.**\n\nBei wenigen bleibt nach (Erhalt) der Sache Dankbarkeit.\n\nSeneca, _De beneficiis_ 1,12,2.\n\n**Aqua et panis est vita canis.**\n\nWasser und Brot, das ist ein Hundeleben.\n\nHW: 1234a.\n\n**Aqua haeret.**\n\nDas Wasser stockt.\n\nCicero ( _De officiis_ 3,117) bezeichnet diese Worte als Sprichwort. \u2013 Gemeint ist vermutlich das Wasser in einer Wasseruhr: Wenn es stehen bleibt, kann man nicht mehr sehen, wieviel Uhr es ist, d. h. man wei\u00df nicht mehr, woran man ist. Mitunter wird auch die \u00dcbersetzung \u00bbDa hapert es\u00ab geboten.\n\n**Aquam a pumice nunc postulas.**\n\nDu verlangst nun Wasser von einem Bimsstein (d. h.: Geld von einem Armen).\n\nPlautus, _Persa_ 41.\n\n**Aquila non captat muscas.** Auch: **Aquila non aucupatur.**\n\nEin Adler f\u00e4ngt keine Fliegen.\n\nErasmus, _Adagia_ 3,2,65. Vgl. **Leones non sunt...**\n\n**arare bove et asino**\n\nmit einem Ochsen und einem Esel pfl\u00fcgen\n\n(d. h.: ungeschickt vorgehen)\n\nNach Deuteronomium 22,10, und Hieronymus, _Epistulae_ 123,5.\n\n**Arare ma(ve)lim quam sic amare!**\n\nIch wollte lieber pfl\u00fcgen als so lieben!\n\nPlautus, _Mercator_ 356.\n\n**arbiter elegantiae**\n\nSchiedsrichter \u00fcber die Eleganz\n\nNach Tacitus ( _Annales_ 16,18,2) Beiname Petrons, der aus dessen Funktion am Hofe Neros resultierte.\n\n**Arbores serit agricola, quae alteri saeculo prosint** siehe **Serit arbores...**\n\n**arcem facere e cloaca**\n\naus einer Kloake eine Burg machen\n\n(vgl.: aus einer M\u00fccke einen Elefanten machen)\n\nCicero, _Pro Cn. Plancio_ 95. Vgl. **E rivo flumina magna facis**.\n\n**Arcus nimium tensus rumpitur.**\n\nEin allzu stark gespannter Bogen bricht entzwei.\n\nPhaedrus ( _Fabulae_ 3,14,10) warnt: \u00bbCito rumpes arcum, semper si tensum habueris.\u00ab (\u00bbSchnell wirst du den Bogen brechen, wenn du ihn immer gespannt h\u00e4ltst.\u00ab)\n\n**Arduum videtur res gestae scribere.**\n\nEs erscheint schwierig, Geschichte niederzuschreiben.\n\nSallust, _Bellum Catilinarium_ 3,2.\n\n**Arma amens capio, nec sat rationis in armis.**\n\nVon Sinnen ergreife ich die Waffen, doch auch in den Waffen ist nicht gen\u00fcgend Rat.\n\nVergil, _Aeneis_ 2,314.\n\n**Arma nisi ut summoveas.**\n\nWaffen nur, um Waffen abzuwehren.\n\nHWS: 899.\n\n**Armaque in armatos sumere iura sinunt.**\n\nDas Recht l\u00e4\u00dft zu, Waffen gegen Bewaffnete zu ergreifen.\n\nOvid, _Ars amatoria_ 3,492.\n\n**Arma virumque cano (Troiae qui primus ab oris...)**\n\nIch singe von den Waffenk\u00e4mpfen und dem Mann (der als erster von den K\u00fcsten Trojas...)\n\nVergil, _Aeneis_ 1,1.\n\n**Arrectae sunt horrore comae.**\n\nDie Haare sind aufgerichtet vor Entsetzen (d. h.: sie stehen zu Berge).\n\nNach Vergil, _Aeneis_ 4,280. Vgl. **Obstipui steteruntque...**\n\n**Arrige aures, Pamphile!**\n\nSpitze die Ohren, Pamphilus!\n\nTerenz, _Andria_ 933; vgl. Vergil, _Aeneis_ 1,153.\n\n**Ars longa, vita brevis** siehe **Vita brevis...**\n\n**Ars prima regni est posse et invidiam pati.**\n\nDie wichtigste Kunst der K\u00f6nigsherrschaft: auch Neid ertragen zu k\u00f6nnen.\n\nSeneca, _Hercules furens_ 353.\n\n**Artem non odit, nisi qui non novit.**\n\nDie Kunst ha\u00dft nur, wer sie nicht kennt.\n\nHW: 1476.\n\n**asinus ad lyram**\n\n(wie) ein Esel beim Lyraspiel\n\n(d. h.: k\u00fcnstlerisch unbegabt)\n\nVarro soll Gellius zufolge ( _Noctes Atticae_ 3,16,13) in griechischer Sprache in einer Satire beschlossen haben, k\u00fcnstlerisch unbegabte S\u00f6hne zu enterben. Vgl. Phaedrus, _Fabulae_ Appendix Perottina 14, wo die Unf\u00e4higkeit eines Esels, die Lyra zu spielen, dargestellt wird.\n\n**Asinus asino et sus sui pulcher.**\n\nEin Esel erscheint einem Esel sch\u00f6n und ein Schwein einem Schwein.\n\n(Vgl.: Gleich und gleich gesellt sich gern.)\n\nHW: 1541.\n\n**Asinus asinum fricat.**\n\nEin Esel reibt sich (gern) an einem Esel.\n\n(Vgl.: Eine Hand w\u00e4scht die andere.)\n\nVgl. _Mutuum muli scabunt_ (\u00bbMaultiere reiben einander\u00ab), Titel einer Satire Varros (Fragment 295 B\u00fccheler); ebenso Ausonius, _Technopaegnion_ 4. Vgl. **Manus manum lavat.**\n\n**asinus in tegulis**\n\nein Esel auf dem Dach\n\n(d. h.: eine unwahrscheinliche Begebenheit)\n\nPetron, _Satyrica_ 63,2.\n\n**Aspera perpessu fiunt iucunda relatu.**\n\nWas hart zu ertragen war, wird beim Erz\u00e4hlen angenehm.\n\n_Monosticha Catonis_ 32.\n\n**Asperitas odium saevaque bella movet.**\n\nRoheit f\u00fchrt zu Ha\u00df und grausamem Krieg.\n\nOvid, _Ars amatoria_ 2,146.\n\n**Astutam vapido servas in pectore vulpem.**\n\nIn deiner verdorbenen Brust bewahrst du einen schlauen Fuchs.\n\nPersius, _Saturae_ 5,117.\n\n**At pulchrum est digito monstrari et dicier: Hic est!**\n\nDoch es ist sch\u00f6n, wenn die Leute mit dem Finger auf einen zeigen und sagen: Das ist er!\n\nAntwort eines Freundes, den Persius ( _Saturae_ 1,26 f.) fragt: \u00bbO mores, usque adeone \/ scire tuum nihil est, nisi te scire hoc sciat alter?\u00ab (\u00bbO Sitten! Bedeutet also dein Wissen nichts, wenn der andere nicht wei\u00df, da\u00df du wei\u00dft?\u00ab) \u2013 Es galt als Zeichen der Anerkennung, wenn andere mit dem Finger auf jemanden zeigten. Vgl. **Monstror digito praetereuntium.**\n\n**Atqui vivere, Lucili, militare est.**\n\nLeben freilich, Lucilius, bedeutet k\u00e4mpfen.\n\nSeneca, _Epistulae morales_ 96,5. Vgl. **Militia est...**\n\n**Atrocitati mansuetudo est remedium.**\n\nGegen Wildheit ist Sanftmut ein Heilmittel.\n\nPhaedrus, _Fabulae_ Appendix Perottina 12,15.\n\n**At suave est ex magno tollere acervo.**\n\nAber es ist angenehm, von einem gro\u00dfen Haufen nehmen zu k\u00f6nnen.\n\nHoraz ( _Sermones_ 1,1,51) lehnt f\u00fcr materiellen Reichtum pl\u00e4dierende Argumente wie das obige ab.\n\n**At vindicta bonum vita iucundius ipsa.**\n\nAber Rache ist ein erg\u00f6tzlicheres Gut als das Leben selbst.\n\nJuvenal, _Saturae_ 13,180.\n\n**a. u. c.** siehe **ab urbe condita**\n\n**Audacter calumniare, semper aliquid haeret.**\n\nVerleumde frech! Irgend etwas bleibt immer h\u00e4ngen.\n\nFrancis Bacon, _De dignitate et augmentis scientiarum_ 8,2, Parabola 34, nach Plutarch, _Moralia_ 65d. Dieser erkl\u00e4rt, da\u00df, auch wenn man sich von Verleumdungen wieder erhole, doch immer eine Narbe bleibe.\n\n**Audax omnia perpeti \/ gens humana ruit per vetitum nefas.**\n\nTollk\u00fchn alles zu ertragen, st\u00fcrzt sich das Menschengeschlecht auf verbotene Taten.\n\nHoraz, _Carmina_ 1,3,25 f.\n\n**Audendo magnus tegitur timor.**\n\nDurch Wagemut wird gro\u00dfe Furcht vertuscht.\n\nLukan, _De bello civili_ 4,702.\n\n**Audentes fortuna iuvat.**\n\nDen Wagemutigen hilft das Gl\u00fcck.\n\n(Vgl.: Frisch gewagt ist halb gewonnen.)\n\nVergil, _Aeneis_ 10,284. \u2013 Seneca ( _Epistulae morales_ 94,28) z. B. erw\u00e4hnt die Wendung als bekanntes Sprichwort. Vgl. Ovid, _Ars amatoria_ 1,608: \u00bbAudentem Forsque Venusque iuvat.\u00ab (\u00bbDem Mutigen helfen Gl\u00fcck und Venus.\u00ab) Ebenso Livius, _Ab urbe condita_ 8,29,5. Daneben findet sich h\u00e4ufig: \u00bbFortes Fortuna (ad)iuvat\u00ab (\u00bbDen Tapferen hilft das Gl\u00fcck\u00ab), so z. B. bei Terenz ( _Phormio_ 203); bei den Elegikern hingegen findet sich in R\u00fcckgriff auf Ovid meist: \u00bbAudentes Venus iuvat.\u00ab (\u00bbDen Wagemutigen hilft Venus.\u00ab)\n\n**Audiatur et altera pars!**\n\nEs m\u00f6ge auch die Gegenpartei angeh\u00f6rt werden!\n\nHW: 1723. Diese Rechtsregel findet sich in \u00e4hnlicher Formulierung bereits z. B. bei Euripides ( _Die Herakliden_ 179 f.) und Demosthenes ( _Rede gegen_ _Timokratus_ 149\u2013151). Bei Seneca ( _Medea_ 199 f.) hei\u00dft es: \u00bbQui statuit aliquid parte inaudita altera, \/ aequum licet statuerit, haud aequus fuit.\u00ab (\u00bbWer etwas festlegt, ohne vorher die andere Partei geh\u00f6rt zu haben, handelt ungerecht, selbst wenn er eine gerechte Entscheidung gef\u00e4llt hat.\u00ab) Vgl. Augustinus, _De duabus animabus_ 14,22: \u00bbAudi alteram partem!\u00ab (\u00bbH\u00f6re die andere Partei an!\u00ab) \u00c4hnlich _Digesta_ 48,17,1.\n\n**Aufer te hinc!**\n\nSchaff dich weg von hier!\n\nTerenz, _Phormio_ 559.\n\n**Augiae cloacas purgare**\n\ndie Augiasst\u00e4lle s\u00e4ubern\n\nNach Seneca ( _Apocolocyntosis_ 7,5), der hier auf eine der zw\u00f6lf Taten des Hercules anspielt: die S\u00e4uberung der St\u00e4lle des Augias, dessen riesige Rinder- und Schafherden das Land in weiten Teilen so durch ihren Dung verschmutzt hatten, da\u00df es nicht mehr bebaut werden konnte.\n\n**Augusto felicior, Traiano melior!**\n\nGl\u00fccklicher als Augustus, besser als Trajan!\n\nEutropius ( _Breviarium ab urbe condita_ 8,5) zufolge wurde mit diesem Ausruf des Kaisers Hadrian gedacht. Er wird hier mit Augustus und seinem Adoptiv-Vater Trajan verglichen.\n\n**aura popularis**\n\nder Hauch des Volkes\n\n(d. h.: die stets wechselnde Gunst des Volkes)\n\n(z. B.) Cicero, _De haruspicum responsis_ 43, und Vergil, _Aeneis_ 6,816; vgl. Horaz, _Carmina_ 3,2,20; Livius, _Ab urbe condita_ 3,33,7.\n\n**aurea mediocritas**\n\nder goldene Mittelweg\n\nNach Horaz, _Carmina_ 2,10,5. \u2013 Was gemeint ist, erl\u00e4utert z. B. Cicero genauer in seiner Beschreibung des Ideals der Peripatetiker ( _De officiis_ 1,89): \u00bbmediocritatem illam..., quae est inter nimium et parum\u00ab (\u00bbjene Mitte..., die zwischen dem Zuviel und dem Zuwenig liegt\u00ab).\n\n**Aurea sunt vere nunc saecula: plurimus auro**\n\nVenit honos, auro conciliatur amor.\n\nGolden sind wahrhaftig nun die Zeiten: Das h\u00f6chste Amt wird f\u00fcr Gold verkauft, mit Gold verschafft man sich Liebe.\n\nOvid, _Ars amatoria_ 2,277 f.\n\n**aurem sibi pervellere**\n\nsich selbst das Ohr langziehen\n\nNach Seneca, _De beneficiis_ 4,36,1 und 5,7,6 sowie _De vita beata_ 10,3. Mitunter findet sich die Erkl\u00e4rung, die Wendung bedeute \u203aseinem Ged\u00e4chtnis auf die Spr\u00fcnge helfen\u2039, denn nach Plinius d. \u00c4. ( _Naturalis historia_ 11,251) ist das Ohr der Sitz des Ged\u00e4chtnisses. Man ber\u00fchre es, um sich an etwas zu erinnern.\n\n**Auribus teneo lupum.**\n\nIch halte einen Wolf bei den Ohren.\n\nTerenz ( _Phormio_ 506) und Varro ( _De lingua Latina_ 7,31) bezeichnen dieses Bild als bekannte Redewendung. Vgl. Sueton, _Vita Tiberii_ 25,1.\n\n**aurificis statera examinare**\n\nauf der Goldwaage pr\u00fcfen\n\n(vgl.: auf die Goldwaage legen)\n\nNach Cicero, _De oratore_ 2,159.\n\n**auri sacra fames** siehe **Quid non mortalia...**\n\n**Auro loquente omnis oratio inanis est.**\n\nWenn das Gold spricht, ist jede Rede sinnlos.\n\n(Vgl.: Redet Geld, so schweigt die Welt.)\n\nHW: 1807b.\n\n**Aurora Musis amica.**\n\nDie Morgenr\u00f6te ist die Freundin der Musen.\n\n(Vgl.: Morgenstund' hat Gold im Mund.)\n\nHW: 1815a.\n\n**Austria(e)...** siehe **A.E.I.O.U.**\n\n**Aut bibat aut abeat!**\n\nEr trinke oder verschwinde!\n\n(Vgl.: Sauf oder lauf!)\n\nCicero ( _Tusculanae disputationes_ 5,118) bezeichnet diesen Spruch als ungeschriebenes Gesetz bei griechischen Gastm\u00e4hlern.\n\n**Aut Caesar aut nihil!**\n\nEntweder Caesar oder nichts!\n\nInschrift, die Cesare Borgia auf einer B\u00fcste von C. Iulius Caesar anbringen lie\u00df. Die Wendung wurde sein eigener Wahlspruch. Sueton ( _Vita Caligulae_ 37,1) berichtet, bereits Caligula soll bei einem ausschweifenden Gelage ge\u00e4u\u00dfert haben: \u00bb... aut frugi hominem esse opportere... aut Caesarem\u00ab (\u00bb... sparsam solle der Mensch sein... oder ein Caesar\u00ab).\n\n**Aut prodesse volunt aut delectare poetae.**\n\nDie Dichter wollen entweder n\u00fctzen oder erfreuen.\n\nHoraz ( _Ars poetica_ 333) erkl\u00e4rt, der beste Dichter sei der, der den Leser zugleich zu erfreuen und zu belehren verm\u00f6ge (343): \u00bbOmne tulit punctum, qui miscuit utile dulci.\u00ab (\u00bbJeden Punkt [d. h.: alle Stimmen] erhielt, wer das N\u00fctzliche mit dem Angenehmen mischte.\u00ab) Bei der Stimmenabgabe stach man mit einem Stab einen Punkt in ein Wachst\u00e4felchen ein.\n\n**Aut regem aut fatuum nasci oportet.**\n\nMan mu\u00df entweder als K\u00f6nig oder als Narr geboren werden.\n\nSeneca ( _Apocolocyntosis_ 1,1) bezieht diese bereits in der Antike als Sprichwort bekannte Wendung auf den Kaiser Claudius. Auf ihn treffe sie in doppelter Hinsicht zu: F\u00fcr seine Zeitgenossen sei er sowohl ein K\u00f6nig als auch ein Dummkopf.\n\n**Avaritia omnia vitia habet.**\n\nHabsucht birgt alle Laster in sich.\n\n(Vgl.: Geiz ist die Wurzel allen \u00dcbels.)\n\nGellius ( _Noctes Atticae_ 11,2,2) f\u00fchrt diese Worte als Ausspruch Catos d. \u00c4. ( _Carmen de moribus_ Fragment 1 Jordan) an.\n\n**Avaritia prima scelerum mater.**\n\nHabsucht ist die Urmutter aller Verbrechen.\n\nNach Claudian, _De consulatu Stilichonis_ 2,111; vgl. Timotheusbriefe 1,6,10.\n\n**(H)ave , imperator, morituri te salutant!**\n\nHeil dir, Kaiser, die Todgeweihten gr\u00fc\u00dfen dich!\n\nSueton ( _Vita divi Claudii_ 21,6) berichtet, da\u00df die Gladiatoren, die auf dem Fuciner See eine Seeschlacht darstellen sollten, Claudius mit diesem Ausruf begr\u00fc\u00dft h\u00e4tten. Der Kaiser soll geantwortet haben: \u00bbAut non.\u00ab (\u00bbOder nicht.\u00ab) Darauf h\u00e4tten die Gladiatoren den Kampf verweigert. Claudius' \u00c4u\u00dferung h\u00e4tten sie n\u00e4mlich so verstanden, da\u00df er ihnen vergebe und sie nicht k\u00e4mpfen m\u00fc\u00dften. Erst nach einigen Bitten und Drohungen seitens Claudius soll dann die Schlacht stattgefunden haben. \nB\n\n**Barba non facit philosophum.**\n\nEin Bart macht (noch lange) keinen Philosophen.\n\nNach Gellius ( _Noctes Atticae_ 9,2,4 f.) kam einst ein Bettler zum ehemaligen Konsul Herodes Atticus und bat ihn um Geld f\u00fcr Brot. Auf die Frage, wer er denn \u00fcberhaupt sei, antwortete der Bettler ungehalten, es sei doch offensichtlich, da\u00df er ein Philosoph sei. Darauf erwiderte Herodes: \u00bbVideo barbam et pallium, philosophum nondum video.\u00ab (\u00bbIch sehe einen Bart und einen griechischen Umhang, einen Philosophen sehe ich noch nicht.\u00ab) Vgl. **Cucullus non facit...** und **Una hirundo...**\n\n**Barbarus hic ego sum, qui non intellegor ulli.**\n\nHier bin ich ein Barbar, der ich von niemandem verstanden werde.\n\nOvid ( _Tristia_ 5,10,37) \u00fcber seine Lage in Tomi am Schwarzen Meer, wohin er verbannt worden war.\n\n**Beati possidentes!** siehe **Non possidentem multa...**\n\n**Beatus ille, qui procul negotiis.**\n\nGl\u00fccklich ist, wer fern von Gesch\u00e4ften.\n\nHoraz, _Epodi_ 2,1. \u2013 Vgl. Joseph Freiherr von Eichendorffs \u00bbWanderlied der Prager Studenten\u00ab ( _Aus dem Leben eines Taugenichts_ , Kap. 9): \u00bbBeatus ille homo, \/ qui sedet in sua domo \/ et sedet post fornacem \/ et habet bonam pacem.\u00ab (\u00bbGl\u00fccklich jener Mensch, der in seinem Hause sitzt und hinter dem Ofen sitzt und wohl seinen Frieden hat.\u00ab)\n\n**Bella gerant alii, Protesilaus amet!**\n\nAndere m\u00f6gen Kriege f\u00fchren, Protesilaus soll lieben!\n\nOvid, _Heroides_ 13,84; vgl. _Heroides_ 17,254.\n\n**Bella gerant alii, tu felix Austria nube!**\n\n**Nam quae Mars aliis, dat tibi regna Venus.**\n\nKriege m\u00f6gen die anderen f\u00fchren, du, gl\u00fcckliches \u00d6sterreich, heirate!\n\nDenn die K\u00f6nigreiche, die der Kriegsgott Mars anderen gibt, gibt dir die Liebesg\u00f6ttin Venus!\n\nDieses Epigramm \u00fcber die Heiratspolitik Kaiser Maximilians I. wurde lange Matthias I. Corvinus zugeschrieben, findet sich aber einigen Quellen zufolge bereits auf einem Siegel des Herzogs Rudolf IV. aus dem Jahre 1363.\n\n**bellaque matribus \/ detestata**\n\nund die von den M\u00fcttern verfluchten Kriege\n\nHoraz, _Carmina_ 1,1,24 f.\n\n**bellum omnium contra omnes**\n\nKrieg aller gegen alle\n\nThomas Hobbes ( _Leviathan_ 1,14 und _De cive_ 1,12) beschreibt so den Zustand der Menschheit vor der Festlegung einer staatlichen Rechtsordnung. Vgl. bereits Platon, _Gesetze_ 1,625e, 626a, und Lucilius, Fragment 1145\u201351 Warmington.\n\n**Bellum omnium pater.**\n\nDer Krieg ist der Vater aller Dinge.\n\nErasmus ( _Adagia_ 3,5,36) f\u00fchrt diese Wendung auf Lukian ( _De ratione conscribendae historiae_ 2) zur\u00fcck. Die Formulierung findet sich freilich schon bei Heraklit (Diels-Kranz, Fragmente der Vorsokratiker 53): \u00bb\u03c0\u03cc\u03bb\u03b5\u03bc\u03bf\u03c2 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u03bc\u03b5\u03bd \u03c0\u03b1\u03c4\u03ae\u03c1 \u2013 polemos panton men pater\u00ab.\n\n**Bellum se ipsum alet.**\n\nDer Krieg wird sich selbst n\u00e4hren.\n\nLaut Livius ( _Ab urbe condita_ 34,9,12) ein Ausspruch Catos d. \u00c4. (Fragment Dicta memorabilia 74 Jordan).\n\n**Bene facta in luce se conlocari volunt.**\n\nGute Taten wollen ins Licht gesetzt werden.\n\nCicero, _Tusculanae disputationes_ 2,64.\n\n**Bene ferre magnam \/ disce fortunam!**\n\nLerne, gro\u00dfes Gl\u00fcck gut zu ertragen!\n\nHoraz, _Carmina_ 3,27,74 f.\n\n**Bene qui latuit, bene vixit.**\n\nGut hat gelebt, wer im Verborgenen gelebt.\n\nOvid, _Tristia_ 3,4,25. \u2013 Die Sentenz geht auf Epikur zur\u00fcck (Fragment 551 Usener: \u00bbLebe im Verborgenen\u00ab). Vgl. **Nec vixit...**\n\n**bis ad eundem**\n\nzweimal an denselben (Stein sto\u00dfen)\n\nCicero ( _Ad familiares_ 10,12,2) weist auf die sprichw\u00f6rtliche Verwendung dieser Ellipse hin.\n\n**Bis dat, qui cito dat** siehe **Inopi beneficium bis dat...**\n\n**Bis vincit, qui se vincit in victoria.**\n\nZweimal siegt, wer sich im Sieg besiegt.\n\nPublilius Syrus, _Sententiae_ B 21.\n\n**bolum e faucibus eripere**\n\neinen guten Fang aus dem Rachen rei\u00dfen\n\n(d. h.: jemandem die Beute wegschnappen)\n\nNach Terenz, _Heauton timorumenos_ 673.\n\n**bona fide**\n\nmit gutem Glauben\n\n(d. h.: auf Treu und Glauben)\n\n(z. B.) _Digesta_ 18,1,27.\n\n**Bona nemini hora est, quin alicui sit mala.**\n\nF\u00fcr niemanden ist eine (bestimmte) Stunde gut, die nicht (gleichzeitig) f\u00fcr einen anderen schlecht ist.\n\n(Vgl.: Des einen Freud ist des anderen Leid.)\n\nErasmus, _Adagia_ 4,5,7.\n\n**Boni pastoris est tondere pecus, non deglubere.**\n\nDie Aufgabe eines guten Hirten besteht darin, das Vieh zu scheren und nicht zu schinden.\n\nNach Sueton, _Vita Tiberii_ 32,2.\n\n**Bonis necesse est noceat, qui parcit malis.**\n\nDen Guten schadet zwangsl\u00e4ufig, wer die Schlechten schont.\n\nPseudo-Seneca, _De moribus_ 114.\n\n**Bono nulla cuiusquam boni invidia est.**\n\nDer Gute hegt keine Mi\u00dfgunst gegen irgend etwas Gutes.\n\nSeneca, _Epistulae morales_ 65,10.\n\n**Bonos corrumpunt mores congressus mali.**\n\nSchlechter Umgang verdirbt gute Sitten.\n\nTertullian ( _Ad uxorem_ 1,8); vgl. Korintherbriefe 1,15,33; wohl nach Menander Fragment 187\/218 K\u00f6rte-Thierfelder bzw. Euripides Fragment 1024 Nauck.\n\n**Bonus animus in mala re dimidium est mali.**\n\nGute Gesinnung in einer schlechten Sache ist halbes \u00dcbel.\n\nPlautus, _Pseudolus_ 452.\n\n**Bonus esse quam videri maluit** siehe **Esse quam videri...**\n\n**Bonus vir semper tiro** siehe **Semper homo bonus tiro est**\n\n**Bos lassus fortius figit pedem.**\n\nEin ersch\u00f6pfter Ochse setzt den Fu\u00df fester auf.\n\nHieronymus ( _Epistulae_ 102,2) f\u00fchrt die Wendung als Sprichwort an.\n\n**bracatae et Transalpinae nationes**\n\ndie hosentragenden V\u00f6lker von jenseits der Alpen\n\nCicero ( _Ad familiares_ 9,21,2) bezeichnet so die Gallier. Sowohl in Griechenland als auch im R\u00f6mischen Reich galten Hosen als barbarisch.\n\n**brevi manu**\n\nkurzerhand\n\n_Digesta_ 23,3,43,1.\n\n**Brevis a natura vita nobis data est.**\n\nEin kurzes Leben ist uns von der Natur gegeben.\n\nCicero, _Orationes Philippicae_ 14,32; vgl. auch _Tusculanae disputationes_ 3,69, wo Theophrast bedauert, da\u00df den Hirschen, die dies nicht zu sch\u00e4tzen w\u00fc\u00dften, ein so langes Leben gegeben werde, den Menschen hingegen ein kurzes.\n\n**Brevis esse laboro, \/ obscurus fio.**\n\nIch bem\u00fche mich, kurz zu sein, (und) werde unverst\u00e4ndlich.\n\nHoraz, _Ars poetica_ 25 f.\nC\n\n**caelum ac terras miscere**\n\nHimmel und Erde vermischen\n\n(d. h.: gro\u00dfen L\u00e4rm machen)\n\nNach Livius, _Ab urbe condita_ 4,3,6.\n\n**Caelum, non animum mutant, qui trans mare currunt.**\n\nDas Klima, (doch) nicht die Gesinnung \u00e4ndern die, die \u00fcbers Meer fahren.\n\n(Vgl.: Reisen wechseln das Gestirn, aber weder Kopf noch Hirn.)\n\nHoraz, _Epistulae_ 1,11,27; vgl. Seneca, _Epistulae morales_ 28,1.\n\n**Caesar citra Rubiconem.**\n\nCaesar ist diesseits des Rubicon.\n\n(D. h.: Er hat die Entscheidung getroffen, den Flu\u00df Rubicon und somit auch das Gesetz zu \u00fcbertreten.)\n\nNach Cicero, _Orationes Philippicae_ 6,5 und 7,26 (allerdings \u00fcber Marcus Antonius). Vgl. **Alea iacta est**.\n\n**Caesar non supra grammaticos.**\n\nDer Kaiser (steht) nicht \u00fcber den Grammatikern.\n\nKaiser Sigismund soll, als er auf dem Konzil von Konstanz vom Erzbischof von Piacenza darauf aufmerksam gemacht wurde, das Wort \u203aschisma\u2039 sei nicht m\u00e4nnlichen Geschlechts, erkl\u00e4rt haben: \u00bbEgo sum rex Romanus et supra grammaticam.\u00ab (\u00bbIch bin r\u00f6mischer K\u00f6nig und stehe \u00fcber der Grammatik.\u00ab) Daraufhin sei er von jenem mit genanntem Ausspruch zurechtgewiesen worden. In der Antike wird von \u00e4hnlichen Anekdoten berichtet, so bei Sueton, _De grammaticis_ 22, und Cassius Dio, _R\u00f6mische Geschichte_ 57,17.\n\n**calcem impingere**\n\neinen Fu\u00dftritt versetzen\n\n(d. h.: anfangen)\n\nNach Petron, _Satyrica_ 46,5.\n\n**canis a non canendo**\n\nDer Hund (hei\u00dft _canis_ ), weil er nicht singt (lat. _non canit_ ).\n\nNach Varro ( _De lingua Latina_ 7,32), der ironische Beispiele f\u00fcr unzutreffende Volksetymologien angibt, die h\u00e4ufig W\u00f6rter gleichen Klanges miteinander in Verbindung bringen. Vgl. **lucus a non lucendo**.\n\n**Canis timidus vehementius latrat quam mordet** siehe **Altissima flumina...**\n\n**Cantabit vacuus coram latrone viator.**\n\nSingen wird der mittellose Wanderer, wenn er vor einem R\u00e4uber steht.\n\nJuvenal, _Saturae_ 10,22.\n\n**Cantatores amant humores.**\n\nS\u00e4nger lieben Feuchtes.\n\nHW: 2313.\n\n**Cantilenam eandem canis.**\n\nDu singst dasselbe Lied.\n\n(d. h.: die alte Leier)\n\nTerenz, _Phormio_ 495.\n\n**Capita aut navia!**\n\nK\u00f6pfe oder Schiff! (r\u00f6misches Spiel; vgl.: Kopf oder Zahl!)\n\nMacrobius ( _Saturnalia_ 1,7,22) l\u00e4\u00dft Praetextatus berichten, Janus habe als erster M\u00fcnzen gepr\u00e4gt. Saturn, den er hochsch\u00e4tzte, habe er verewigt, indem er auf die eine Seite einer jeden M\u00fcnze dessen Antlitz, auf die andere das Schiff, auf dem Saturn zu ihm gekommen war, pr\u00e4gte.\n\n**captatio benevolentiae**\n\ndas Haschen nach Wohlwollen\n\nDiese Nominalf\u00fcgung ist in der Antike nicht belegt. Vgl. Cicero, _De inventione_ 1,51,21, und Boethius, _Commentaria in Topica_ _Ciceronis_ 1. \u2013 Gemeint ist die Taktik, zu Beginn einer Rede das Wohlwollen der Zuh\u00f6rer zu gewinnen.\n\n**Caput eius agitur.**\n\nEs geht um seinen Kopf.\n\nNach Cicero, _De amicitia_ 61.\n\n**caput (sinistra manu) perfricare**\n\nsich (mit der linken Hand) am Kopf kratzen\n\n(d. h.: sich nachdenklich zeigen)\n\nNach Cicero, _In L. Calpurnium Pisonem_ 61.\n\n**Caput prurit.**\n\nDer Kopf juckt (mir).\n\nPlautus, _Bacchides_ 1193. Vgl. **Dorsus totus prurit.**\n\n**Carpe diem!** siehe **Dum loquimur...**\n\n**Casus magister alius et paene numerosior.**\n\nEin weiterer Lehrmeister ist der Zufall und fast der h\u00e4ufigere.\n\nPlinius d. \u00c4. ( _Naturalis historia_ 17,101) sieht zwei Lehrmeister des Menschengeschlechts: die Natur und in fast noch mehr F\u00e4llen den Zufall.\n\n**Cato esse quam videri bonus malebat** siehe **Esse quam videri...**\n\n**Causa aliqua subest.**\n\nDahinter steckt irgendein Grund.\n\nNach Cicero, _De finibus_ 5,29.\n\n**Cave canem!**\n\nH\u00fcte dich vor dem Hunde!\n\nVarro, _Eumenides_ Fragment 143 B\u00fccheler, und Petron, _Satyrica_ 29,1. Die Warnung findet sich in zahlreichen Fu\u00dfbodenmosaiken vor antiken Haust\u00fcren.\n\n**Cedant arma togae, concedat laurea laudi!**\n\nWeichen m\u00f6gen die Waffen der Toga, der Lorbeer der l\u00f6blichen Tat!\n\nCicero zitiert sich in _De officiis_ 1,77 selbst aus _De consulatu suo_ Fragment 16 Taglia, ebenso in der Rede _In L. Calpurnium Pisonem_ 73 f.; vgl. auch _Orationes Philippicae_ 2,20. \u2013 Die Toga galt als Merkmal des B\u00fcrgers im Gegensatz zu den Waffen des Legion\u00e4rs. W\u00e4hrend Feldherrn lorbeertragend in Triumphz\u00fcgen gefeiert wurden, hielt man auf Politiker Lobreden.\n\n**Cedo maiori.**\n\nIch weiche dem Gr\u00f6\u00dferen (d. h.: dem St\u00e4rkeren).\n\nMartial, _Epigrammata_ , Spectaculorum liber 32: \u00bbCedere maiori virtutis fama secunda est. \/ Illa gravis palma est, quam minor hostis habet.\u00ab (\u00bbDem St\u00e4rkeren zu weichen l\u00e4\u00dft dem Tapferen einen zweiten Ruhm zuteil werden. Unangenehm ist n\u00e4mlich der Sieg, den der [eigentlich] unterlegene Feind einstreicht.\u00ab) Vgl. **Maiori concede...**\n\n**Celerius quam asparagi cocuntur.**\n\nDas ist schneller als Spargel gekocht.\n\nSueton ( _Vita divi Augusti_ 87,1) berichtet, Augustus habe mit dieser Wendung besonders schnell erledigte Unterfangen beschrieben.\n\n**Centum sunt causae, cur ego semper amem.**\n\nHundert Gr\u00fcnde gibt es, weshalb ich immer verliebt bin.\n\nOvid, _Amores_ 2,4,10.\n\n**Certa mittimus, dum incerta petimus.**\n\nSicheres lassen wir fahren, w\u00e4hrend wir Unsicheres anstreben.\n\nPlautus, _Pseudolus_ 685.\n\n**Cessante causa cessat effectus.**\n\nWenn die Ursache wegf\u00e4llt, entf\u00e4llt auch die Wirkung.\n\nDamasus Bohemus, _Burchardica, sive regulae canonicae_ 114, und Thomas von Aquin, _Summa theologiae_ 1,96,3,3.\n\n**Cetera mitte loqui!**\n\nLa\u00df es, vom \u00fcbrigen zu sprechen!\n\n(D. h.: Sprich nicht weiter!)\n\nHoraz, _Epodi_ 13,7.\n\n**Cetera res expediet** siehe **Tantummodo incepto...**\n\n**Ceterum censeo Carthaginem esse delendam.**\n\nIm \u00fcbrigen beantrage ich, da\u00df Karthago zerst\u00f6rt werden soll.\n\nCato d. \u00c4. zugeschriebener Ausspruch, der in \u00e4hnlicher Form z. B. von Plutarch, _Cato maior_ 27,2, und Florus, _Epitomae de Tito Livio_ 2,15,4 f., belegt wird. Cato soll vor dem Dritten Punischen Krieg diese Forderung hartn\u00e4ckig in jeder Senatssitzung gestellt haben.\n\n**Charta non erubescit** siehe **Epistula non erubescit**\n\n**Christianos ad leones!**\n\nDie Christen vor die L\u00f6wen!\n\nTertullian ( _Apologeticum_ 40,2) berichtet, man habe die Christen als Ursache f\u00fcr jedwedes Unheil angesehen und stets gerufen: \u00bbChristianos ad leonem!\u00ab (\u00bbDie Christen vor den L\u00f6wen!\u00ab) Er f\u00e4hrt fort: \u00bbTantos ad unum?\u00ab (\u00bbSo viele vor einen?\u00ab)\n\n**Cibi condimentum fames est.**\n\nDer Hunger ist die W\u00fcrze der Speise.\n\n(Vgl.: Hunger ist der beste Koch.)\n\nNach Cicero, der diesen Spruch in _De finibus_ 2,90 als von Sokrates stammend anf\u00fchrt. H\u00e4ufig auch zitiert als: \u00bbFames optimum condimentum.\u00ab (\u00bbHunger ist das beste Gew\u00fcrz.\u00ab)\n\n**Cibus omnis in illo \/ causa cibi est.**\n\nBei ihm ist jedes Essen Grund zu neuem Essen.\n\nOvid, _Metamorphoses_ 8,841 f. Es geht um Erysichthon, der f\u00fcr seine Freveltaten mit unstillbarem Hunger bestraft worden war.\n\n**circulus vitiosus**\n\nein fehlerhafter Kreis\n\nDie Wendung bezeichnet einen Trugschlu\u00df, bei dem ein noch zu f\u00fchrender Beweis selbst Pr\u00e4misse ist. Eine solche unsinnige Argumentationsweise legt Aristoteles ( _Analytica priora_ 2,5,57b 18 ff.) ausf\u00fchrlich dar. Heute ist mit dieser Wendung meist ein \u00bbTeufelskreis\u00ab gemeint.\n\n**Cita mors ruit.**\n\nSchnell st\u00fcrzt der Tod herein.\n\nNach Horaz, _Sermones_ 1,1,8: **Horae momento cita mors venit...**\n\n**citius , altius, fortius**\n\nschneller, h\u00f6her, weiter (eigtl.: st\u00e4rker)\n\nDas von Pierre de Coubertin ( _Olympische Erinnerungen_ 25) gepr\u00e4gte Motto der neuzeitlichen Olympischen Spiele nach einem Ausspruch des P\u00e8re Henri Didon, Prior des Coll\u00e8ge d'Arcueil.\n\n**Cito enim arescit lacrima, praesertim in alienis malis.**\n\nSchnell trocknet die Tr\u00e4ne, zumal wenn sie sich auf fremdes Ungl\u00fcck bezieht.\n\nCicero ( _Partitiones oratoriae_ 57) r\u00e4t dazu, in Gerichtsprozessen das harte Schicksal seines Mandanten darzustellen, doch nicht allzu ausf\u00fchrlich, da man nur begrenzt Mitleid f\u00fcr andere aufbringe. Gegen zu gro\u00dfe Genauigkeit spreche noch ein weiterer Punkt: \u00bbMinuta est enim omnis diligentia.\u00ab (\u00bbKleinlich ist jede [Form von] Genauigkeit.\u00ab)\n\n**Cito ignominia fit superbi gloria.**\n\nDer Ruhm eines Hochm\u00fctigen wird rasch zur Schande.\n\n(Vgl.: Hochmut kommt vor dem Fall.)\n\nPublilius Syrus, _Sententiae_ C 10.\n\n**Civis Romanus sum!**\n\nIch bin (doch) ein r\u00f6mischer B\u00fcrger!\n\nCicero, _Actio secunda in C. Verrem_ 5,147 und 5,162. \u2013 Laut L. Suettius sollen dies zu Unrecht eingekerkerte R\u00f6mer gerufen haben. Vgl. **Et facere et pati...**\n\n**classici auctores**\n\ndie klassischen Autoren\n\nNach Gellius, _Noctes Atticae_ 19,8,15. \u2013 Urspr\u00fcnglich bezeichnet \u00bbclassicus\u00ab B\u00fcrger, die zur obersten Steuerklasse geh\u00f6ren, bald aber allgemein \u203aErstklassiges\u2039.\n\n**Claude os, aperi oculos!**\n\nHalte den Mund und mach die Augen auf!\n\nHWS: 35536.\n\n**clavum clavo eicere**\n\neinen Nagel mit einem anderen heraustreiben\n\n(vgl.: den Teufel mit dem Belzebub austreiben)\n\nNach Cicero, _Tusculanae disputationes_ 4,75.\n\n**Clericus clericum non decimat.**\n\nEin Geistlicher nimmt von einem anderen Geistlichen keinen Zehnten.\n\n(Vgl. auch pejorativ: Eine Kr\u00e4he hackt der anderen kein Auge aus.)\n\nVgl. _Corpus iuris canonici_ , De decimis 3,2; **Cornix cornici...**\n\n**C + M + B (Christus mansionem benedicat.)**\n\nChristus segne dieses Haus.\n\nDie Abk\u00fcrzung wird bisweilen auch als \u00bbCaspar, Melchior und Balthasar\u00ab aufgel\u00f6st.\n\n**Coactus volui.**\n\nIch wollte, (wenn auch) unter Zwang.\n\n_Digesta_ 4,2,21,5.\n\n**Cogitatione finge!**\n\nStell dir vor!\n\nNach Cicero, _Pro T. Annio Milone_ 79.\n\n**Cogitationis poenam nemo patitur.**\n\nNiemand mu\u00df Strafe f\u00fcr seine Gedanken erleiden.\n\n(Vgl.: Die Gedanken sind frei.)\n\n_Digesta_ 48,19,18.\n\n**Cogito, ergo sum.**\n\nIch denke, also bin ich.\n\nRen\u00e9 Descartes, _Principia philosophiae_ 1,7 und 1,10; in franz\u00f6sischer Sprache bereits in _Discours de la m\u00e9thode_ 4.\n\n**coincidentia oppositorum**\n\nZusammenfall der Gegens\u00e4tze\n\nNikolaus von Kues f\u00fchrt diese Wendung in _De coniecturis_ (76,12 und \u00f6fter) als Gottesbeweis (bes. 82) an: Gott vereinige Widerspr\u00fcchliches, so wie eine Kreislinie bei unendlich gro\u00dfem Radius zu ihrem Gegenteil, zu einer Geraden wird.\n\n**colludere cum altero**\n\nmit einem anderen zusammenspielen\n\n(d. h.: unter einer Decke stecken)\n\nNach Cicero, _Actio secunda in Verrem_ 2,58.\n\n**Comes facundus in via pro vehiculo est.**\n\nAuf einer Reise ersetzt ein beredter Begleiter den Reisewagen.\n\n(Vgl.: Gef\u00e4hrte munter k\u00fcrzt die Meilen.)\n\nPublilius Syrus, _Sententiae_ C 17.\n\n**Communia sunt amicorum inter se omnia.**\n\nUnter Freunden ist alles gemeinsam.\n\nTerenz ( _Adelphoe_ 804) bezeichnet dies als bekannte Redewendung. Vgl. z. B. Cicero ( _De officiis_ 1,51), der sie als Sprichwort der Griechen (z. B. Platon, _Phaedrus_ 279c) anf\u00fchrt.\n\n**Compedes, quas ipse fecit, ipse gestabit faber.**\n\nDie Fesseln, die er selbst hergestellt, wird der Schmied selbst tragen.\n\n(vgl.: sich in der eigenen Schlinge fangen)\n\nNach Ausonius, _Bissula_ 1,6.\n\n**Compelle intrare, ut impleatur domus mea.**\n\nN\u00f6tige sie hereinzukommen, auf da\u00df mein Haus voll werde.\n\nLukas 14,23.\n\n**Compesce mentem!**\n\nB\u00e4ndige deinen Zorn!\n\nHoraz, _Carmina_ 1,16,22.\n\n**Componitur orbis \/ regis ad exemplum.**\n\nDie Welt formt sich nach dem Beispiel des Herrschers.\n\nClaudian, _De quarto consulatu Honorii Augusti_ 299 f. Vgl. **Ignavus servos...** und **Plane qualis...**\n\n**compressis manibus sedere**\n\nmit zusammengepre\u00dften H\u00e4nden dasitzen\n\n(vgl.: die H\u00e4nde in den Scho\u00df legen)\n\nNach Livius ( _Ab urbe condita_ 7,13,7), der dies als bekannte Redewendung bezeichnet.\n\n**Conclamatum est.**\n\nEs ist laut beklagt worden.\n\n(D. h.: Alles ist aus.)\n\nTerenz, _Eunuchus_ 348. \u2013 Bei Begr\u00e4bnissen wurde der Tote laut beweint.\n\n**concordia discors** siehe **rerum concordia discors**\n\n**Concordia domi, foris pax!**\n\nEintracht zu Hause, drau\u00dfen Friede!\n\nInschrift am Holstener Tor zu L\u00fcbeck.\n\n**Concordia parvae res crescunt, discordia maxumae dilabuntur.**\n\nDurch Eintracht wachsen (auch) kleine Staaten, durch Zwietracht zerfallen (sogar) die gr\u00f6\u00dften.\n\nLaut Sallust ( _Bellum Jugurthinum_ 10,6) \u00e4u\u00dferte Micipsa, K\u00f6nig von Numidien, diese Worte auf dem Sterbebett. Seneca ( _Epistulae morales_ 94,46) zitiert sie als von Marcus Agrippa stammend.\n\n**condicio sine qua non**\n\neine Bedingung, ohne die nicht...\n\n(d. h.: eine unerl\u00e4\u00dfliche Bedingung)\n\nVgl. z. B. Cicero, _Topica_ 61: \u00bbHoc igitur sine quo non fit, ab eo in quo certe fit, diligenter est separandum.\u00ab (\u00bbDas also, ohne das sich das Ergebnis nicht einstellt, mu\u00df sorgf\u00e4ltig von dem getrennt werden, wodurch das Ergebnis sicher eintritt.\u00ab)\n\n**congregare cum leonibus vulpes**\n\nF\u00fcchse mit L\u00f6wen vereinigen\n\nMartial ( _Epigrammata_ 10,100,3) wirft einem anderen Dichter Plagiatentum vor, doch sei dies leicht zu durchschauen: Ein Gedicht Martials in dem Buch dieses Dichters falle so aus der Reihe wie ein Fuchs unter L\u00f6wen.\n\n**Conscia mens recti famae mendacia ridet.**\n\nEine Gesinnung, die sich des Rechten bewu\u00dft ist, lacht \u00fcber die L\u00fcgen des Ger\u00fcchts.\n\n(Vgl.: Ein gutes Gewissen ist ein sanftes Ruhekissen.)\n\nNach Ovid, _Fasti_ 4,311.\n\n**Conscientia mille testes.**\n\nEin gutes Gewissen (ist soviel Wert wie) tausend Zeugen.\n\nQuintilian ( _Institutio oratoria_ 5,11,41) bezeichnet die Wendung als Sprichwort.\n\n**Consuetudo est quasi altera natura.**\n\nDie Gewohnheit ist gleichsam eine zweite Natur.\n\nNach Cicero, _De finibus_ 5,74. \u2013 Sp\u00e4ter wird die Stelle h\u00e4ufig als \u00bbnatura secunda\u00ab zitiert. Vgl. Ovid, _Ars amatoria_ 2,345.\n\n**Consummatum est.**\n\nEs ist vollbracht.\n\nLaut Johannes (19,30) die letzten Worte Christi.\n\n**Contenti estote.**\n\nSeid zufrieden.\n\nFriedrich Schiller, _Wallenstein_ ; Wallensteins Lager, 8. Auftritt, V. 589.\n\n**Contenti simus hoc Catone!**\n\nSeien wir mit diesem Cato zufrieden!\n\nSueton ( _Vita divi Augusti_ 87,1) berichtet, Augustus habe mit dieser Wendung dazu aufgefordert, sich mit den jeweiligen Umst\u00e4nden abzufinden. \u2013 Cato d. \u00c4. (Cato Censorius) galt als besonders sittenstrenger R\u00f6mer, der stets die gegenw\u00e4rtigen Zust\u00e4nde beklagte.\n\n**Contra quis ferat arma deos?**\n\nWer wollte die Waffen gegen die G\u00f6tter richten?\n\nTibull, _Elegiae_ 1,6,30.\n\n**Contraria contrariis curantur.**\n\nGegens\u00e4tzliches wird durch Gegens\u00e4tzliches geheilt.\n\n(vgl.: den Teufel mit Weihwasser vertreiben)\n\nDer Gedanke wird Hippokrates zugeschrieben. Vgl. HWS, z. B. 35737.\n\n**Contra vim mortis non est medicamen in hortis.**\n\nGegen die Macht des Todes w\u00e4chst kein Heilmittel in den G\u00e4rten.\n\n(Vgl.: Wider des Todes Kraft hilft kein Kr\u00e4utersaft.)\n\n_Regimen sanitatis Salerni_ , De salvia 2.\n\n**Conturbasti mihi \/ rationes omnes.**\n\nDu hast mir die ganzen Rechnungen durcheinandergebracht.\n\n(vgl.: einen Strich durch die Rechnung machen)\n\nNach Terenz, _Eunuchus_ 868 f.\n\n**convenienter naturae vivere**\n\nim Einklang mit der Natur leben\n\n(z. B.) Cicero ( _De officiis_ 3,13), der dies als das h\u00f6chste Gut der Stoiker anf\u00fchrt. Vgl. **secundum naturam vivere**.\n\n**Conversa subito fortuna est.**\n\nGewendet hat sich pl\u00f6tzlich das Gl\u00fcck.\n\nCornelius Nepos, _Atticus_ 10,1.\n\n**corio suo ludere**\n\num seine Haut spielen\n\n(d. h.: seine Haut zu Markte tragen)\n\nNach Martial, _Epigrammata_ 3,16,4.\n\n**Cornix cornici numquam oculos effodit.**\n\nEine Kr\u00e4he hackt einer anderen niemals die Augen aus.\n\nNach Macrobius, _Saturnalia_ 7,5,2. Vgl. **Clericus clericum non decimat**.\n\n**Coronemus nos rosis antequam marcescant!**\n\nLasset mit Rosen uns bekr\u00e4nzen, bevor sie verbl\u00fchen!\n\nWeisheit 2,8.\n\n**corpus delicti**\n\nGegenstand (Werkzeug) eines Verbrechens\n\nProsper Farinacius, _Variae quaestiones_ (z. B.) 1,1,1.\n\n**corvus albus**\n\nein wei\u00dfer Rabe\n\n(vgl.: ein seltener Vogel)\n\nNach Juvenal, _Saturae_ 7,202. Vgl. **rara avis.**\n\n**crabrones irritare**\n\nHornissen aufreizen\n\n(vgl.: in ein Wespennest stechen)\n\nNach Plautus, _Amphitruo_ 707.\n\n**crambe repetita**\n\nmit aufgew\u00e4rmtem Kohl\n\n(d. h.: die alte Leier; kalter Kaffee)\n\nJuvenal, _Saturae_ 7,154.\n\n**Cras amet, qui numquam amavit; quique amavit, cras amet!**\n\nEs liebe morgen, wer noch nie geliebt hat; wer (bereits) geliebt hat, liebe morgen!\n\nRefrain des _Pervigilium Veneris_.\n\n**Cras legam.**\n\nMorgen werd' ich's lesen.\n\nNach Sueton ( _Vita divi Iulii_ 81,4) soll Caesar so geantwortet haben, als man ihm am Tage seines Todes eine warnende Nachricht zustecken wollte.\n\n**crassa Minerva**\n\nmit fetter Minerva\n\n(d. h.: unverbildeten Verstandes, mit gesundem Menschenverstand)\n\nNach Horaz, _Sermones_ 2,2,3. \u2013 Die r\u00f6mische G\u00f6ttin Minerva war die Patronin des Handwerks, der K\u00fcnste, der Lehrer und \u00c4rzte. Vgl. **invita Minerva** und **pingui Minerva**.\n\n**Cras vives ? Hodie iam vivere, Postume, serum est.**\n\nMorgen wirst du leben? Heute zu leben, Postumus, ist bereits zu sp\u00e4t.\n\nMartial, _Epigrammata_ 5,58,7. Der Name Postumus ist wohl eine Anspielung auf Horaz: **Eheu fugaces...** Vgl. auch **Non est, crede...**\n\n**Credat Iudaeus Apella, \/ non ego.**\n\nDas soll der Jude Apella glauben, nicht ich.\n\nHoraz ( _Sermones_ 1,5,100 f.) dr\u00fcckt seine Ungl\u00e4ubigkeit gegen\u00fcber einem Naturwunder aus. Apella war ein h\u00e4ufig bei Freigelassenen anzutreffender Name und bezeichnet hier vermutlich keine tats\u00e4chliche Person.\n\n**Crede experto!**\n\nGlaube einem Erfahrenen!\n\nSilius Italicus, _Punica_ 7,395. Vgl. **Experto credite!**\n\n**Crede mihi, sacra populi lingua est.**\n\nGlaube mir, die Rede des Volkes ist heilig.\n\nSeneca Rhetor, _Controversiae_ 1,1,10. \u2013 Vgl. Hesiod, _Werke und Tage_ 763 f.\n\n**Credo, quia absurdum.**\n\nIch glaube es, weil es widersinnig ist.\n\nHW: 3707a. Vgl. Tertullian ( _De carne Christi_ 5,4), der \u00fcber den Tod Christi sagt: \u00bbProrsus credibile est, quia ineptum est... Certum est, quia impossibile est.\u00ab (\u00bbDas mu\u00df man geradezu glauben, weil es unsinnig ist.... Es ist sicher, weil es unm\u00f6glich ist.\u00ab)\n\n**Credo, ut intellegam.**\n\nIch glaube, um zu verstehen.\n\nAnselm von Canterbury, _Proslogion_ 1.\n\n**Credula res amor est.**\n\nEin leichtgl\u00e4ubig Ding ist die Liebe.\n\nOvid, _Metamorphoses_ 7,826.\n\n**Crescentem sequitur cura pecuniam.**\n\nDem wachsenden Geld folgt die Sorge.\n\nHoraz, _Carmina_ 3,16,17.\n\n**Crescit amor nummi, quantum ipsa pecunia crevit.**\n\nDie Liebe zum Geld w\u00e4chst in gleichem Ma\u00dfe wie das Geld selbst.\n\nJuvenal, _Saturae_ 14,139.\n\n**Crescit animus, quotiens coepti magnitudinem attendit.**\n\nMein Geist w\u00e4chst, sooft er die Gr\u00f6\u00dfe des Unterfangens beachtet.\n\nSeneca, _Naturales quaestiones_ 3, praefatio 4.\n\n**Crimine ab uno \/ disce omnes.**\n\nAus dem einen Verbrechen begreife alle.\n\nVergil, _Aeneis_ 2,65 f.: \u00bbAccipe nunc Danaum insidias et crimine ab uno \/ disce omnes.\u00ab (\u00bbVernimm nun die List der Danaer und aus dem einen Verbrechen begreife alle\u00ab, d. h.: das verbrecherische Wesen des gesamten Volkes.)\n\n**Cucullus non facit monachum.**\n\nEine Kapuze macht keinen M\u00f6nch.\n\nHWS: 35960. Vgl. **Barba non facit...** und **Una hirundo...**\n\n**Cui bono?**\n\nWem n\u00fctzt es?\n\n(z. B.) Cicero, _Pro S. Roscio Amerino_ 84 und 86; _Pro T. Annio Milone_ 32; _Orationes Philippicae_ 2,35. Cicero zufolge stammt das Wort von Lucius Cassius Longinus Ravilla.\n\n**Cui dolet, meminit.**\n\nWer den Schmerz hat, hat auch die Erinnerung.\n\n(Vgl.: Gebranntes Kind scheut das Feuer.)\n\nCicero, _Pro L. Murena_ 42: \u00bbCui placet, obliviscitur, cui dolet, meminit.\u00ab (\u00bbWem es angenehm ist, der vergi\u00dft es, wer den Schmerz hat, dem bleibt die Erinnerung.\u00ab)\n\n**Cui nolis saepe irasci, irascaris semel.**\n\nWem du nicht oft z\u00fcrnen willst, dem z\u00fcrne einmal.\n\nPublilius Syrus, _Sententiae_ C 28.\n\n**Cuius regio, eius religio.**\n\nWessen Land, dessen Religion.\n\n(Vgl.: Wes Brot ich e\u00df', des Lied ich sing'.)\n\nNach Joachim Stephani, _Institutiones iuris canonici_ 1,7, S. 52, wohl in Anlehnung an eine \u00e4hnliche F\u00fcgung im Augsburger Religionsfrieden von 1555.\n\n**Cuiusvis hominis est errare, nullius nisi insipientis perseverare in errore.**\n\nJeder Mensch kann sich irren, doch nur ein Narr verharrt im Irrtum.\n\n(Vgl.: Irren ist menschlich.)\n\nCicero ( _Orationes Philippicae_ 12,5) f\u00e4hrt fort: \u00bbPosteriores enim cogitationes, ut aiunt, sapientiores solent esse.\u00ab (\u00bbDie sp\u00e4teren \u00dcberlegungen pflegen, wie man sagt, weiser zu sein.\u00ab) Vgl. **Errasse humanum est...**\n\n**Cuivis potest accidere, quod cuiquam potest.**\n\nWas irgendeinem passieren kann, kann jedem passieren.\n\nPublilius Syrus, _Sententiae_ C 34.\n\n**Cultus magnificus addit hominibus auctoritatem.**\n\nGro\u00dfartiges Auftreten vergr\u00f6\u00dfert das Ansehen der Leute.\n\n(Vgl.: Kleider machen Leute.)\n\nNach Quintilian ( _Institutio oratoria_ 8, prooemium 20) ist die Wendung ein Ausspruch eines griechischen Dichters. Vgl. **Vestis virum reddit**.\n\n**cum basi sua metiri**\n\nmit dem Sockel zusammen messen\n\n(d. h.: \u00fcbersch\u00e4tzen)\n\nNach Seneca, _Epistulae morales_ 76,31.\n\n**cum grano salis**\n\nmit einem Korn Salz\n\n(d. h.: nicht ganz ernst zu nehmen)\n\nNach Plinius d. \u00c4. ( _Naturalis historia_ 23,149) fand Pompeius im Notizbuch von Mithridates ein Rezept f\u00fcr ein Mittel, das einen ganzen Tag lang gegen jegliches Gift im voraus Sicherheit bieten sollte. Das Rezept endet mit den Worten \u00bbunter Beif\u00fcgung eines K\u00f6rnchen Salzes\u00ab (\u00bbaddito salis grano\u00ab).\n\n**Cum insanientibus furere necesse est.**\n\nMit den Verr\u00fcckten mu\u00df man rasen.\n\n(vgl.: mit den W\u00f6lfen heulen)\n\nNach Petron, _Satyrica_ 3,2.\n\n**Cum libentissime edis, tum auferatur cena!**\n\nWenn du mit dem gr\u00f6\u00dften Vergn\u00fcgen i\u00dft, soll das Mahl abgetragen werden.\n\n(d. h.: aufh\u00f6ren, wenn es am sch\u00f6nsten ist)\n\nBei Gellius ( _Noctes Atticae_ 15,8,2) spricht Favonius\/Favorinus (s. Malcovati, Orat. Rom. Frag., Bd.1, S. 203 f.) jedoch von einer dem entgegengesetzten Handlungsdevise: \u00bbPraefecti popinae atque luxuriae negant cenam lautam esse, nisi cum lubentissime edis, tum auferatur et alia esca melior atque amplior succenturietur.\u00ab (\u00bbDie, die sich auf E\u00dfkunst und Ausschweifungen verstehen, bestreiten, da\u00df ein Gastmahl elegant sei, wenn nicht dann abgetragen werde, wenn man mit dem gr\u00f6\u00dften Vergn\u00fcgen i\u00dft, und statt dessen eine bessere und reichhaltigere Speise aufgetragen werde.\u00ab)\n\n**cum mula pepererit**\n\nwenn eine Mauleselin geworfen hat\n\nSueton, _Vita Galbae_ 4,2. \u2013 Solche unm\u00f6glichen Naturerscheinungen (Adynata) werden h\u00e4ufig zur Beschreibung unerf\u00fcllbarer Bedingungen herangezogen. \u00c4hnlich im Deutschen: \u00bbWenn die Fl\u00fcsse aufw\u00e4rts flie\u00dfen...\u00ab. Vgl. Properz, _Elegiae_ 1,15,29.\n\n**cum ratione insanire**\n\nmit Vernunft unvern\u00fcnftig sein\n\n(vgl.: sehenden Auges blind)\n\nNach Terenz, _Eunuchus_ 63.\n\n**Cum sale panis \/ latrantem stomachum bene leniet.**\n\nBrot mit Salz wird einen knurrenden Magen gut beruhigen.\n\nHoraz, _Sermones_ 2,2,17 f.\n\n**Cum tacent clamant.**\n\nIndem sie schweigen, schreien sie.\n\n(vgl.: beredtes Schweigen)\n\nCicero, _In L. Catilinam_ 1,21; vgl. auch _Pro P. Sestio_ 40; Terenz, _Eunuchus_ 476; Ovid, _Ars amatoria_ 1,574.\n\n**Cuncta fluunt, omnisque vagans formatur imago;**\n\nIpsa quoque assiduo labuntur tempora motu.\n\nAlles flie\u00dft, jede Erscheinung bildet sich wechselnd neu. Selbst die Zeit gleitet in steter Bewegung dahin.\n\nOvid ( _Metamorphoses_ 15,178 f.) l\u00e4\u00dft Pythagoras diese Worte sprechen. Vgl. die nachtr\u00e4gliche Formulierung der Lehre Heraklits: \u00bbpanta rhei.\u00ab (\u00bbAlles flie\u00dft.\u00ab) Vgl. **Nihil est toto...**\n\n**Cunctando restituit rem** siehe **Unus homo nobis...**\n\n**Cuneus cuneum trudit** siehe **Malo arboris nodo...**\n\n**curriculum vitae**\n\nLebenslauf, Lebensbahn\n\n(z. B.) Cicero, _Academici libri posteriores_ 1,44 und _Pro C. Rabirio_ 10,30.\n\n**Currit ferox aetas.**\n\nUnaufhaltsam enteilt die Zeit.\n\nNach Horaz, _Carmina_ 2,5,13 f.\n\n**curvo dinoscere rectum**\n\nGerades von Krummem (d. h.: B\u00f6ses von Gutem) unterscheiden\n\nHoraz, _Epistulae_ 2,2,44.\n\n**cutem curare**\n\ndie Haut pflegen\n\n(d. h.: es sich wohl ergehen lassen)\n\nHoraz ( _Epistulae_ 1,4,15 f.) spielt in ironischer Selbstbezichtigung auf die unzutreffende Interpretation der Philosophie Epikurs als reines Luststreben an: \u00bbMe pinguem et nitidum bene curata cute vises \/ cum ridere voles, Epicuri de grege porcum.\u00ab (\u00bbDu wirst mich fett und gl\u00e4nzend in wohlgepflegter Haut antreffen, wenn du (einmal) lachen willst, ein Schweinchen aus der Herde Epikurs.\u00ab) Vgl. **in cute curanda...**\nD\n\n**Da locum melioribus!**\n\nMach Besseren Platz!\n\nTerenz, _Phormio_ 522.\n\n**Damna damnis continuantur.**\n\nSchaden reiht sich an Schaden.\n\n(Vgl.: Ein Ungl\u00fcck kommt selten allein.)\n\nNach Tacitus, _Agricola_ 41,3.\n\n**Damnant, quod non intellegunt.**\n\n(Die meisten) mi\u00dfbilligen, was sie nicht verstehen.\n\n(Vgl.: Was der Bauer nicht kennt, das fri\u00dft er nicht.)\n\nNach Quintilian, _Institutio oratoria_ 10,1,26.\n\n**Danaum \/ fatale munus**\n\ndas verh\u00e4ngnisvolle Geschenk der Danaer\n\nNach Seneca, _Agamemnon_ 627 f. \u2013 Gemeint ist das h\u00f6lzerne Pferd, das die Danaer (Griechen) nach ihrem scheinbaren Abzug von Troja zur\u00fcckgelassen hatten. Trotz der Warnungen des Priesters Laokoon vor diesem \u00bbDanaergeschenk\u00ab schafften die Trojaner das Pferd in die Stadt, und aus seinem Bauch kam eine Schar griechischer Krieger hervor, die die Stadt zu Fall brachten. Vgl. **donum exitiale Minervae** , **Latet error...** und **Quidquid id est, timeo** **Danaos...**\n\n**dantur opes nullis nunc nisi divitibus** siehe **Semper pauper eris...**\n\n**Da pignus!**\n\nGib ein Pfand!\n\n(D. h.: Wollen wir wetten!)\n\nPlautus, _Epidicus_ 699.\n\n**Da requiem , requietus ager bene credita reddit.**\n\nG\u00f6nne (ihr) Ruhe; ein ausgeruhter Acker gibt das Anvertraute reichlich zur\u00fcck.\n\nOvid ( _Ars amatoria_ 2,351 f.) bedient sich mehrerer recht schl\u00fcpfriger Naturgleichnisse, wenn er dem Liebhaber r\u00e4t, sich f\u00fcr eine Weile von der Geliebten zu trennen, nachdem sie sich bereits an ihn gew\u00f6hnt hat: \u00bbTerraque caelestes arida sorbet aquas.\u00ab (\u00bbUnd ausged\u00f6rrte Erde saugt den Regen auf.\u00ab)\n\n**Da spatium vitae, multos da, Iuppiter, annos!**\n\nGib (lange) Lebensdauer, gib, Jupiter, viele Jahre!\n\nJuvenal, _Saturae_ 10,188. Er f\u00fchrt dieses Begehren im folgenden (190 f.) als Beispiel f\u00fcr die unsinnigen W\u00fcnsche seiner Mitmenschen an: \u00bbSed quam continuis et quantis longa senectus \/ plena malis!\u00ab (\u00bbDoch wie voll ist das lange Alter von andauernden und gro\u00dfen \u00dcbeln!\u00ab)\n\n**Dat census honores. \/ Census amicitias: pauper ubique iacet.**\n\nVerm\u00f6gen bringt Ehren. Verm\u00f6gen bringt Freundschaften: Der Arme (jedoch) liegt in allen Angelegenheiten darnieder (d. h.: er ist machtlos).\n\nOvid, _Fasti_ 1,217 f. \u2013 Aus einer Rede des Gottes Janus \u00fcber die Herrschaft des Geldes. Vgl. auch _Amores_ 3,8,55.\n\n**Dat legem natura tibi, non accipit ipsa.**\n\nDie Natur gibt dir ein Gesetz, sie selbst nimmt keines (von dir) an.\n\n_Dicta Catonis_ , Appendix 7.\n\n**Dat, quodcumque libet...** siehe **Nempe dat id...**\n\n**Dat veniam corvis, vexat censura columbas.**\n\nMit Raben hat die Zensur Nachsicht, den Tauben setzt sie zu.\n\n(Vgl.: Die Gro\u00dfen l\u00e4\u00dft man laufen, die Kleinen m\u00fcssen h\u00e4ngen.)\n\nJuvenal ( _Saturae_ 2,63) l\u00e4\u00dft Laronia die Vorw\u00fcrfe gegen das weibliche Geschlecht verteidigen, indem sie die \u00dcbeltaten der M\u00e4nner aufz\u00e4hlt. Ihrer Meinung nach verzeihen M\u00e4nner die Fehler ihrer Geschlechtsgenossen allzu leicht und blicken statt dessen eher auf die belanglosen Vergehen der Frauen.\n\n**Da vacuae menti, quo teneatur, opus!**\n\nGib deinem leeren Geist eine Aufgabe, die ihn packe.\n\nOvid, _Remedia amoris_ 150.\n\n**Davos sum, non Oedipus.**\n\nDavus bin ich, nicht Oedipus.\n\nTerenz, _Andria_ 194. \u2013 Oedipus hatte ein von der Sphinx gestelltes R\u00e4tsel gel\u00f6st. Mit dieser Anspielung erkl\u00e4rt der Sklave Davus zwar sein Unverm\u00f6gen, eine bestimmte Frage zu beantworten, bekundet jedoch gleichzeitig seine literarische Bildung.\n\n**Debemur morti nos nostraque.**\n\nUns selbst und unser Hab und Gut schulden wir dem Tode.\n\nHoraz, _Ars poetica_ 63.\n\n**decantata fabula**\n\neine heruntergeleierte Geschichte\n\n(vgl.: die alte Leier)\n\nNach Cicero, _Ad Atticum_ 13,40; vgl. Seneca, _Epistulae morales_ 24,6.\n\n**Decet imperatorem stantem mori.**\n\nEs ziemt sich f\u00fcr einen Kaiser, im Stehen zu sterben.\n\nNach Sueton ( _Vita divi Vespasiani_ 24) hat der todkranke Vespasian, diese Worte murmelnd, noch einmal versucht sich aufzurichten, starb jedoch in den Armen derer, die ihm zu Hilfe kamen.\n\n**Decet verecundum esse adulescentem.**\n\nEs ziemt sich f\u00fcr einen jungen Mann, bescheiden zu sein.\n\nPlautus, _Asinaria_ 833.\n\n**decies repetita placebit** siehe **ut pictura poesis**\n\n**Decipimur specie recti.**\n\nWir werden vom Schein des Rechten get\u00e4uscht.\n\nHoraz, _Ars poetica_ 25.\n\n**Dediscit animus sero, quod didicit diu.**\n\nSp\u00e4t (erst) vergi\u00dft das Ged\u00e4chtnis, was es \u00fcber lange Zeit hinweg gelernt.\n\nSeneca, _Troades_ 633.\n\n**de facto**\n\ntats\u00e4chlich, auf Grundlage einer Tat\n\n_Digesta_ 8,5,2,3; vgl. **de iure**.\n\n**Deficiente PECU deficit omne NIA.**\n\nMangelt es an BAR, so auch an jeglicher SCHAFT.\n\nHW: 5305 und Rabelais, _Gargantua et Pantagruel_ 3,41, als Parodie auf den Stil Ennius', der gerne zusammengeh\u00f6rende W\u00f6rter trennte. Vgl. **disiecti membra poetae.**\n\n**Deficit omne, quod nascitur.**\n\nAlles, was entsteht, erlischt (wieder).\n\nQuintilian, _Institutio oratoria_ 5,10,79.\n\n**Deforme etiam est de se ipsum praedicare, falsa praesertim.**\n\nEs ist ungeb\u00fchrlich, sich selbst zu r\u00fchmen, zumal wegen etwas, was gar nicht stimmt.\n\n(Vgl.: Eigenlob stinkt.)\n\nCicero, _De officiis_ 1,137.\n\n**De homine enim dicitur, cui necesse est mori.**\n\nVom Menschen n\u00e4mlich ist die Rede, der zwangsl\u00e4ufig sterben mu\u00df.\n\nCicero ( _De fato_ 17) f\u00fchrt die Aussage \u00bbScipio wird sterben\u00ab als seltenes Beispiel f\u00fcr eine Vorhersage an, die, obwohl sie sich auf die Zukunft bezieht, absolut sicher ist, da jeder Mensch einmal sterben mu\u00df. Die Wendung wird bisweilen auch verk\u00fcrzt zitiert als: \u00bbHomini necesse est mori.\u00ab\n\n**de integro**\n\nvon neuem\n\n(z. B.) Cicero ( _De divinatione_ 1,131), der Pacuvius ( _Chryses_ Fragment 6 Argenio) zitiert.\n\n**de iure**\n\nvon Rechts wegen\n\n_Digesta_ 8,5,2,3; vgl. **de facto**.\n\n**Deliciae populi, quae fuerant domini.**\n\nDes Volkes Freude ist, was des Herrn war.\n\nMartial, _Epigrammata_ , Spectaculorum liber 2,12.\n\n**Deligere oportet, quem velis diligere.**\n\nDu solltest dir so einen ausw\u00e4hlen, den du gerne lieben w\u00fcrdest.\n\nDer Auctor ad Herennium (4,29) f\u00fchrt diese Sentenz nicht aus inhaltlichen Gr\u00fcnden, sondern als Beispiel f\u00fcr \u00e4hnlich klingende, aber nicht verwandte W\u00f6rter (Paronomasie, lat. _ad_ \\- oder _agnominatio_ ) an.\n\n**Delirant, isti Romani!**\n\nDie spinnen, die R\u00f6mer!\n\nGoscinny und Uderzo, _Asterix_.\n\n**De me ego facio coniecturam.**\n\nIch ziehe von mir aus(gehend) einen Schlu\u00df.\n\n(vgl.: von sich auf andere schlie\u00dfen)\n\nTerenz, _Heauton timorumenos_ 574.\n\n**Deme supercilio nubem.**\n\nNimm die Wolke von deiner Braue.\n\n(D. h.: Zeige eine freundlichere Miene.)\n\nHoraz, _Epistulae_ 1,18,94.\n\n**De mortuis nil nisi bene!**\n\n\u00dcber Tote (soll) nur in guter Weise (gesprochen werden)!\n\nNach Demosthenes ( _Rede gegen Leptines_ 104) stammen diese Worte von Solon. Diogenes Laertius ( _Vitae philosophorum: Chilon_ 1,70) spricht die Anweisung Chilon zu.\n\n**De nihilo nihil** siehe **Nihil ex nihilo exsistere...**\n\n**Denique non omnes eadem mirantur amantque.**\n\nSchlie\u00dflich bewundern und lieben nicht alle das gleiche.\n\nHoraz, _Epistulae_ 2,2,58.\n\n**Denique \/ nullum est iam dictum, quod non dictum sit prius.**\n\nEs gibt schlie\u00dflich kein Wort mehr, das nicht fr\u00fcher schon gesagt worden w\u00e4re.\n\nTerenz ( _Eunuchus_ 40) wendet sich gegen den Vorwurf, seine St\u00fccke seien nur Plagiate griechischer Werke.\n\n**de omni re scibili et quibusdam aliis**\n\n\u00fcber alles Wi\u00dfbare und manches andere\n\nNach Giovanni Pico della Mirandola ( _Conclusiones_ , De mathematicis secundum opinionem propriam 11), der behauptete, vermittels Zahlen zur Entdeckung und zum Verst\u00e4ndnis alles Wi\u00dfbaren gelangen zu k\u00f6nnen. Oft auch zitiert als \u00bbde omnibus rebus et quibusdam aliis\u00ab (\u00bb\u00fcber alle Dinge und manches andere\u00ab).\n\n**Deos absentis testis memoras.**\n\nDu f\u00fchrst abwesende G\u00f6tter als Zeugen an.\n\nPlautus, _Mercator_ 627.\n\n**Deo volente, nobis viventibus.**\n\nWenn Gott will und wir leben.\n\nDie sogenannte Condicio Iacobea, der \u00bbJakobeische Vorbehalt\u00ab, nach Jakobusbrief 4,15.\n\n**De profundis (clamavi ad te, Domine).**\n\nAus der Tiefe (habe ich zu dir gerufen, Herr).\n\nPsalm 129,1.\n\n**De scurra multo facilius dives quam pater familias fieri potest.**\n\nAus einem Narr kann viel eher ein Reicher als ein (rechter) Familienvater werden.\n\nNach Cicero ( _Pro P. Quinctio_ 55), der die Wendung als Sprichwort bezeichnet.\n\n**Desinant \/ maledicere, malefacta ne noscant sua!**\n\nSie sollten aufh\u00f6ren, \u00dcbles zu reden, damit man ihnen nicht ihre eigenen \u00dcbeltaten vorh\u00e4lt.\n\nTerenz, _Andria_ 22 f.\n\n**Desunt inopiae multa, avaritiae omnia** siehe **Inopiae desunt multa...**\n\n**de te fabula narratur** siehe **Mutato nomine de te...**\n\n**de tripode dictum**\n\nvom Dreifu\u00df aus gesprochen\n\n(d. h.: r\u00e4tselhaft wie ein Orakel)\n\nHWS: 36122. \u2013 Im Tempel des Apollon-Heiligtums zu Delphi befand sich ein Erdspalt, aus dem bet\u00e4ubende D\u00e4mpfe hervorstr\u00f6mten. Die Seherin Pythia sa\u00df auf einem Dreifu\u00df \u00fcber diesem Spalt und gab, von den D\u00e4mpfen berauscht, Orakel. Vgl. **Eris mihi magnus Apollo**.\n\n**deus ex machina**\n\nder Gott aus der Maschine\n\nDie lateinische Quelle der Wendung ist unbekannt. \u2013 Platon ( _Cratylus_ 425d) l\u00e4\u00dft Sokrates im Gespr\u00e4ch mit Hermogenes erkl\u00e4ren, die Urspr\u00fcnge der W\u00f6rter, besonders der Namen von Dingen, seien ungekl\u00e4rt. Man k\u00f6nne nat\u00fcrlich sagen, sie seien von den G\u00f6ttern gegeben. Dieser R\u00fcckgriff auf die G\u00f6tter gleiche dem Vorgehen vieler Trag\u00f6diendichter, die in verfahrenen Situationen einen Gott auf der B\u00fchne erscheinen lassen, indem sie einen Schauspieler mit Hilfe einer Maschine auf die B\u00fchne herablassen. \u2013 Heute wird mit dieser Wendung oft ein \u203aunverhoffter Retter\u2039 bezeichnet.\n\n**Deus nobis haec otia fecit.**\n\nEin Gott (Augustus) hat uns diese Mu\u00dfe beschert.\n\nVergil, _Bucolica_ 1,6.\n\n**deus sive natura**\n\nGott oder Natur\n\nSpinoza ( _Ethica_ 4, praefatio) spicht von \u00bbinfinitum Ens, quod Deum, seu Naturam appellamus\u00ab (ein \u00bbunendlich Seiendes, das wir Gott oder Natur nennen\u00ab).\n\n**Deus sum , si hoc ita est.**\n\nIch bin ein Gott, wenn das stimmt.\n\nTerenz, _Hecyra_ 843.\n\n**dextro tempore**\n\nzur rechten Zeit\n\nHoraz, _Sermones_ 2,1,18.\n\n**Di bene vortant!**\n\nDie G\u00f6tter m\u00f6gen es zum Guten wenden!\n\n(D. h.: Gottes Segen!)\n\nNach Plautus, _Trinummus_ 502.\n\n**Dicenda tacenda locutus.**\n\nEr hat \u00fcber Dinge, die man sagen darf, und \u00fcber solche, die man verschweigen sollte, gesprochen.\n\nHoraz, _Epistulae_ 1,7,72.\n\n**Diceres aliquid.**\n\nDu k\u00f6nntest etwas sagen.\n\n(D. h.: In deinen Worten k\u00f6nnte eine wichtige Bedeutung liegen, wenn...)\n\nCicero, _Tusculanae disputationes_ 3,35.\n\n**Dices hic porcos coctos ambulare.**\n\nDu wirst sagen, da\u00df hier gebratene Ferkel herumlaufen.\n\n(vgl.: die gebratenen Tauben des Schlaraffenlandes)\n\nPetron, _Satyrica_ 45,4: \u00bbTu si aliubi fueris, dices hic porcos coctos ambulare.\u00ab (\u00bbWenn du [erst einmal] woanders gewesen bist, wirst du sagen, da\u00df hier gebratene Ferkel herumlaufen.\u00ab)\n\n**Dic, hospes , Spartae nos te hic vidisse iacentis,**\n\n**Dum sanctis patriae legibus obsequimur.**\n\nWanderer, kommst du nach Sparta, verk\u00fcndige dorten, du habest \/ uns hier liegen gesehn, wie das Gesetz es befahl.\n\nCicero, _Tusculanae disputationes_ 1,101 (\u00fcbers. von Friedrich Schiller, \u00bbDer Spaziergang\u00ab, V. 97 f.), nach Simonides (Fragment 119 Edmonds) bzw. Herodot, _Geschichte_ 7,228,2. \u2013 Grabepigramm f\u00fcr die 300 Spartaner, die im Jahre 480 v. Chr. ihr Leben gelassen hatten, als sie bei den Thermopylen versuchten, das Perserheer aufzuhalten.\n\n**Dicique beatus \/ ante obitum nemo supremaque funera debet.**\n\nUnd keiner darf vor seinem Tod und seinem letzten Geleit gl\u00fccklich genannt werden.\n\n(Vgl.: Man soll den Tag nicht vor dem Abend loben.)\n\nOvid ( _Metamorphoses_ 3,136 f.) \u00fcber Cadmus. Nach Herodot ( _Geschichte_ 1,32,7) soll dies Solon zu Kr\u00f6sus gesagt haben. H\u00e4ufig auch zitiert als \u00bbAnte mortem nemo beatus\u00ab.\n\n**dicis causa**\n\nzum Schein, der Form wegen\n\n(z. B.) Cicero, _Actio secunda in C. Verrem_ 4,53.\n\n**Dico ego, tu dicis, sed denique dixit et ille,**\n\n**Dictaque post toties non nisi dicta vides.**\n\nIch sage es, du sagst es, aber schlie\u00dflich sagt es auch jener, und nach so viel Gesagtem siehst du nur noch Gesagtes (erg.: und nicht mehr die Ph\u00e4nomene selbst).\n\nJohann Wolfgang Goethe, _Farbenlehre_. \u00dcberschriftsvers zu seiner Enth\u00fcllung der Theorie Newtons.\n\n**Dictum, factum.**\n\nWort (oder) Tat\n\nTerenz, _Andria_ 381 f.: \u00bbDictum factum invenerit \/ aliquam causam, quam ob rem eiciat oppido.\u00ab (\u00bbOb Wort oder Tat, er wird schon einen Grund finden, um sie aus der Stadt zu vertreiben.\u00ab) \u2013 Oft wird diese Wendung mit \u00bbGesagt, getan\u00ab \u00fcbersetzt.\n\n**Dictum sapienti sat est.**\n\nF\u00fcr den Verst\u00e4ndigen ist genug gesagt worden.\n\nPlautus, _Persa_ 615; vgl. Terenz, _Phormio_ 541.\n\n**diem perdidi** siehe **Amici, diem perdidi!**\n\n**dies ater**\n\nein schwarzer Tag\n\nAls \u00bbschwarz\u00ab wurden die Tage bezeichnet, die auf die Kalenden, Nonen und Iden (erster, f\u00fcnfter bzw. siebter und dreizehnter bzw. f\u00fcnfzehnter Tag des Monats) folgten. Laut Gellius ( _Noctes Atticae_ 5,17,1 f.) erkl\u00e4rte Verrius Flaccus in _De verborum significatu_ den Ursprung dieser Bezeichnung folgenderma\u00dfen: Die R\u00f6mer waren am 18. Juli 390 v. Chr. an der Allia vernichtend geschlagen worden, nachdem sie am Vortag, dem Tag nach den Iden, geopfert hatten. Daraufhin seien Opfergaben an den Kalenden, Nonen und Iden verboten worden. Vgl. Macrobius, _Saturnalia_ 1,16,22 f.\n\n**Dies diem docet.**\n\nEin Tag lehrt den anderen.\n\nVgl. Publilius Syrus, _Sententiae_ D 1: **Discipulus est prioris...**\n\n**Dies irae, dies illa \/ solvet saeclum in favilla.**\n\nDer Tag des Zorns, jener (letzte) Tag, wird die Zeitlichkeit in Asche aufl\u00f6sen.\n\nAnfangsverse eines Hymnus des katholischen Traueramtes (12.\u201313. Jh.), mitunter dem Franziskaner Thomas von Celano zugeschrieben. Vgl. Zefanja 1,15.\n\n**differre ac procrastinare**\n\nverschieben und vertagen\n\n(vgl.: auf die lange Bank schieben)\n\nNach Cicero, _Pro S. Roscio Amerino_ 26.\n\n**Difficile est saturam non scribere.**\n\nEs ist schwierig, keine Satire zu schreiben.\n\nJuvenal, _Saturae_ 1,30.\n\n**Difficilis , facilis, iucundus, acerbus es idem,**\n\n**Nec tecum possum vivere nec sine te.**\n\nUnzug\u00e4nglich und umg\u00e4nglich, gef\u00e4llig und abweisend bist du zugleich: Ich kann nicht mit dir leben noch ohne dich.\n\nMartial, _Epigrammata_ 12,46.\n\n**Difficilis optimi perfectio.**\n\nSchwierig ist die Vollendung des Besten.\n\nCicero, _Brutus_ 137.\n\n**Dignum laude virum Musa vetat mori.**\n\nEinen Mann, der des Lobes w\u00fcrdig ist, l\u00e4\u00dft die Muse (der Dichtkunst) nicht sterben.\n\nHoraz ( _Carmina_ 4,8,28) glaubt, da\u00df Dichter durch ihre Verse verewigt w\u00fcrden. Vgl. **Quem referent...**\n\n**Diligitur nemo, nisi cui Fortuna secunda est.**\n\nGeliebt wird nur, wem die Gl\u00fccksg\u00f6ttin Fortuna hold ist.\n\nOvid, _Epistulae ex Ponto_ 2,3,23.\n\n**Di meliora!**\n\n(M\u00f6gen) die G\u00f6tter (es) zum Besseren (wenden)!\n\n(D. h.: Gott bewahre!)\n\nCicero ( _De senectute_ 47) berichtet, der alte Sophokles habe so auf die Frage, ob er noch Geschlechtsverkehr mit Frauen habe, geantwortet. \u2013 Vgl. Seneca, _Epistulae morales_ 47,8.\n\n**Dimidium facti est coepisse.**\n\nDie H\u00e4lfte der Tat besteht darin, angefangen zu haben.\n\n(Vgl.: Frisch begonnen ist halb gewonnen.)\n\nAusonius, _Epigrammata_ 81,1; vgl. Seneca, _Epistulae morales_ 34,3, und **Dimidium facti, qui coepit, habet...** \u2013 Der Gedanke geht bereits auf Platon ( _Gesetze_ 6,753e) zur\u00fcck.\n\n**Dimidium facti, qui coepit, habet: Sapere aude, \/ incipe!**\n\nDie H\u00e4lfte der Tat hat, wer begonnen hat. Wage es (einmal), Vernunft zu \u00fcben, fang nur erst an!\n\n(Vgl.: Frisch begonnen ist halb gewonnen.)\n\nHoraz, _Epistulae_ 1,2,40 f., zitiert bei Kant, _Beantwortung der Frage: Was ist Aufkl\u00e4rung?_ Vgl. **Dimidium facti est coepisse.**\n\n**Dimidium plus toto.**\n\nDie H\u00e4lfte ist mehr als das Ganze.\n\nNach Hesiod ( _Werke und Tage_ 40 f.) hat man um so weniger Scherereien, je weniger Geld man besitzt. Der lateinische Ursprung der Wendung ist unbekannt.\n\n**Di nos quasi pilas homines habent.**\n\nDie G\u00f6tter handhaben uns Menschen gleichsam wie Spielb\u00e4lle.\n\nPlautus, _Captivi_ 22.\n\n**dira necessitas**\n\ndie furchtbare Notwendigkeit\n\nHoraz, _Carmina_ 3,24,6.\n\n**Diruit, aedificat, mutat quadrata rotundis** siehe **Quod petiit, spernit, repetit...**\n\n**Dis aliter visum.**\n\nDie G\u00f6tter haben es anders beschlossen.\n\nVergil, _Aeneis_ 2,428.\n\n**Disce aliquid! nam cum subito fortuna recessit,**\n\n**Ars remanet vitamque hominis non deserit umquam.**\n\nErlerne etwas! Denn wenn pl\u00f6tzlich das Gl\u00fcck weicht, so bleibt doch die Kunst und l\u00e4\u00dft den Menschen nie im Stich.\n\n_Disticha Catonis_ 4,19.\n\n**Disce bonas artes, moneo, Romana iuventus!**\n\nLerne, mahne ich, r\u00f6mische Jugend, die edlen K\u00fcnste!\n\nOvid, _Ars amatoria_ 1,459.\n\n**Discipulus est prioris posterior dies.**\n\nSch\u00fcler des Vortages ist der folgende Tag.\n\n(Vgl.: Der heutige Tag ist des gestrigen J\u00fcnger.)\n\nPublilius Syrus, _Sententiae_ D 1. Vgl. die vermutlich daraus entwickelte Wendung **Dies diem docet**.\n\n**Discite iustitiam, moniti, et non temnere divos!**\n\nSeid gewarnt: Lernet Gerechtigkeit und verachtet nicht die G\u00f6tter!\n\nVergil, _Aeneis_ 6,620.\n\n**Discrepant facta cum dictis.**\n\nTaten und Worte stimmen nicht \u00fcberein.\n\nNach Cicero, _De finibus_ 2,96.\n\n**disiecti membra poetae**\n\ndie Glieder des zerst\u00fcckelten Dichters\n\nOft zitiert als \u00bbdisiecta membra poetae\u00ab (\u00bbdie zerst\u00fcckelten Glieder des Dichters\u00ab). \u2013 Horaz ( _Sermones_ 1,4,62) grenzt sich hier von Ennius ab: Dessen Verse enthielten noch poetische Elemente (eben die \u00bbGlieder des zerst\u00fcckelten Dichters\u00ab in Anspielung an den von den M\u00e4naden zerrissenen sagenhaften S\u00e4nger Orpheus), selbst wenn man sie aus dem Versma\u00df l\u00f6se. Horaz' eigene Satirenverse seien jedoch, vom Versma\u00df gel\u00f6st, nichts als reine Umgangssprache. Vgl. **Deficiente...**\n\n**Dives aut iniquus aut iniqui heres.**\n\nDer Reiche ist entweder ein Schelm oder eines Schelmen Erbe.\n\nHieronymus, _Epistulae_ 120,1.\n\n**Dives erit, magno quae dormit tertia lecto.**\n\nReich sein wird die, die als Dritte im gro\u00dfen Bette schl\u00e4ft.\n\nJuvenal, _Saturae_ 2,60. \u2013 Gemeint ist eine Ehefrau, die gegen finanzielle Entsch\u00e4digung die Homosexualit\u00e4t ihres Mannes deckt und somit \u00bbDritte im Bett\u00ab ist. Vgl. Seneca Rhetor, _Controversiae_ 2,1,35 f., und Apuleius, _Metamorphoses_ 9,27 ff.\n\n**Dives est, cui tanta possessio est, ut nihil optet amplius.**\n\nReich ist, wer solch gro\u00dfen Besitz hat, da\u00df er nichts weiter w\u00fcnscht.\n\nNach Cicero, _Paradoxa Stoicorum_ 42.\n\n**Dives qui fieri volt, \/ et cito volt fieri.**\n\nWer reich werden will, will es auch schnell werden.\n\nJuvenal, _Saturae_ 14,177 f.\n\n**Dives qui sapiens est.**\n\nReich ist, wer weise ist.\n\nHoraz ( _Sermones_ 1,3,124) stellt diesen stoischen Lehrsatz allerdings in Frage.\n\n**Divide et impera!**\n\nTeile und herrsche!\n\nDiese in der Antike nicht belegten Worte illustrieren das au\u00dfenpolitische Vorgehen der R\u00f6mer: Die eroberten St\u00e4mme erhielten nach getrennten Verhandlungen unterschiedliche Vorrechte. Prosper M\u00e9rim\u00e9e ( _Chronique du r\u00e8gne de Charles IX._ , pr\u00e9face) schreibt diesen Ausspruch, allerdings in franz\u00f6sischer Sprache (\u00bbdiviser pour r\u00e9gner\u00ab), Louis XI. von Frankreich zu. Die lateinische Fassung wird bei Traiano Boccalini ( _La bilancia politica_ , Bd. 1, S. 136, und Bd. 2, S. 225) als gel\u00e4ufige Maxime angef\u00fchrt.\n\n**Dividuum fac!**\n\nLa\u00df die H\u00e4lfte (dir gen\u00fcgen)!\n\nTerenz ( _Adelphoe_ 241) mahnt zur Bescheidenheit, denn wer mit der H\u00e4lfte nicht zufrieden sei, verliere wom\u00f6glich das Ganze.\n\n**Divisum sic breve fiet opus.**\n\nDerart geteilt wird die Arbeit k\u00fcrzer werden.\n\nMartial, _Epigrammata_ 4,82,8.\n\n**Divitiis potietur heres.**\n\nDeines Reichtums wird sich ein Erbe bem\u00e4chtigen.\n\nHoraz, _Carmina_ 2,3,20.\n\n**Dixerat ille aliquid magnum.**\n\nJener hatte etwas Gro\u00dfes (d. h.: Prahlerisches) gesagt.\n\nVergil, _Aeneis_ 10,547.\n\n**Dixi et salvavi animam meam.**\n\nIch habe gesprochen und meine Seele gerettet.\n\nNach Ezechiel (3,19) leistet der Prophet das Seine, indem er das Volk warnt. Wer nicht zuh\u00f6re, trage selber Schuld.\n\n**Docendo discimus** siehe **Homines dum docent, discunt**\n\n**docta dicta**\n\ngelehrte Worte\n\nNach Lukrez, _De rerum natura_ 2,987.\n\n**Doctrina est fructus dulcis radicis amarae.**\n\nGelehrsamkeit ist die s\u00fc\u00dfe Frucht einer bitteren Wurzel.\n\n_Monosticha Catonis_ 40.\n\n**Doctum doces.**\n\nDu informierst einen (bereits) Informierten.\n\nPlautus, _Poenulus_ 880.\n\n**Dolo malo haec fiunt omnia.**\n\nAll das geschieht mit Arglist.\n\nNach Terenz, _Eunuchus_ 515.\n\n**Dolus an virtus, quis in hoste requirat?**\n\nList oder Tugend, wer wollte beim Feind danach fragen?\n\nVergil, _Aeneis_ 2,390. \u2013 Coroebus ruft die Trojaner dazu auf, mit List gegen die Griechen vorzugehen, denn gegen\u00fcber einem Feind seien alle Mittel recht.\n\n**domi leones, foras vulpes**\n\nzu Hause L\u00f6wen, drau\u00dfen Schlauf\u00fcchse\n\n(Vgl.: Daheim ist _ein_ Mann zwei.)\n\nPetron, _Satyrica_ 44,4,14; vgl. Horaz, _Sermones_ 2,3,186, und **in pace leones...**\n\n**Domine, dirige nos!**\n\nHerr, lenke uns!\n\nWappeninschrift der City of London.\n\n**Dona praesentis cape laetus horae ac \/ linque severa!**\n\nGenie\u00dfe froh die Gaben der gegenw\u00e4rtigen Stunde und la\u00df das Unangenehme beiseite!\n\nHoraz, _Carmina_ 3,8,27 f.\n\n**Donec eris sospes, multos numerabis amicos;**\n\n**Tempora si fuerint nubila, solus eris.**\n\nSolange du gl\u00fccklich bist, wirst du viele Freunde haben; in d\u00fcsteren Zeiten wirst du allein sein.\n\nOvid, _Tristia_ 1,9,5 f. Vgl. **Horrea formicae tendunt...**\n\n**donum exitiale Minervae**\n\ndas unheilvolle Geschenk Minervas\n\nVergil ( _Aeneis_ 2,31) bezeichnet so das Trojanische Pferd. Vgl. **Danaum fatale munus** , **Latet error...** und **Quidquid id est, timeo Danaos...**\n\n**Dorsus totus prurit.**\n\nDer ganze R\u00fccken juckt (mir).\n\n(D. h.: Ich f\u00fcrchte Schl\u00e4ge. Mir juckt das Fell.)\n\nPlautus, _Miles gloriosus_ 397. Vgl. **Caput prurit.**\n\n**Dos est magna parentium \/ virtus.**\n\nEine gro\u00dfe Mitgift ist die Tugend der Eltern.\n\nHoraz, _Carmina_ 3,24,21 f.\n\n**Dote imperium vendidi.**\n\nF\u00fcr die Mitgift habe ich meine Eigenm\u00e4chtigkeit verkauft.\n\n(Vgl.: Wer nach Geld heiratet, verliert seine Freiheit.)\n\nPlautus, _Asinaria_ 87.\n\n**Do, ut des.**\n\nIch gebe (dir), damit du (mir) gibst.\n\nAltr\u00f6mische Rechtsformel, die bei Vertragsabschl\u00fcssen oder Tauschgesch\u00e4ften gebraucht wurde. \u2013 Vgl. _Digesta_ , 19,5,5, prologus.\n\n**duabus sellis sedere**\n\nauf zwei St\u00fchlen sitzen\n\n(vgl.: auf zwei Hochzeiten gleichzeitig tanzen)\n\nNach Seneca Rhetor ( _Controversiae_ 7,3,9) hat Laberius seinem Freund Cicero mit diesen Worten scherzhaft vorgeworfen, da\u00df er gleichzeitig Caesar und Pompeius schmeichle.\n\n**Ducunt volentem fata, nolentem trahunt.**\n\nDen Willigen f\u00fchrt das Schicksal, den Unwilligen rei\u00dft es mit.\n\nSeneca ( _Epistulae morales_ 107,11) zitiert hier in lateinischer Sprache Kleanthes (Fragment 527 von Arnim).\n\n**Dulce est desipere in loco** siehe **Misce stultitiam...**\n\n**Dulce et decorum est pro patria mori.**\n\nS\u00fc\u00df und ehrenvoll ist's, f\u00fcr das Vaterland zu sterben.\n\nHoraz ( _Carmina_ 3,2,13 f.) f\u00e4hrt fort: \u00bbMors et fugacem persequitur virum.\u00ab (\u00bbDer Tod verfolgt auch einen fliehenden Mann\u00ab; d. h.: Man mu\u00df kein Held sein, um zu sterben.)\n\n**Dulce mihi furere est.**\n\nEs ist mir eine Wonne, vor Freude zu rasen.\n\nHoraz, _Carmina_ 2,7,28.\n\n**Dum excusare credit, accusat.**\n\nW\u00e4hrend er zu entschuldigen glaubt, klagt er an.\n\nNach Hieronymus, _Epistulae_ 4,3.\n\n**Dum fata sinunt, \/ vivite laeti!**\n\nSolange das Schicksal es erlaubt, lebt froh!\n\nSeneca, _Hercules furens_ 177 f.\n\n**Dum ferrum candet, cudere quemque decet.**\n\nMan mu\u00df das Eisen schlagen, solange es gl\u00fcht.\n\n(d. h.: das Eisen schmieden, solange es hei\u00df ist)\n\nHW: 6519.\n\n**Dum loquimur, fugerit invida \/ aetas. Carpe diem!**\n\nW\u00e4hrend wir sprechen, flieht die mi\u00dfg\u00fcnstige Zeit. Genie\u00dfe den Tag!\n\nHoraz ( _Carmina_ 1,11,7 f.) mahnt dazu, in der Gegenwart zu leben und nicht auf die Zukunft zu bauen. So f\u00e4hrt er fort: \u00bb... quam minimum credula postero.\u00ab (\u00bbVertraue dich m\u00f6glichst wenig dem n\u00e4chsten Tag an!\u00ab) \u2013 Vgl. Ovid, _Amores_ 1,11,15. Vgl. **Tu ne quaesieris...**\n\n**Dummodo sit dives, barbarus ipse placet.**\n\nWenn er nur reich ist, gef\u00e4llt selbst ein Barbar.\n\nOvid, _Ars amatoria_ 2,276.\n\n**Dum moliuntur , dum conantur, annus est** siehe **Nosti mores mulierum...**\n\n**Dum res et aetas et sororum \/ fila trium patiuntur atra.**\n\nSolange die Verh\u00e4ltnisse und die Zeit und die schwarzen F\u00e4den der drei Schwestern es erlauben.\n\nHoraz, _Carmina_ 2,3,15 f. \u2013 Die drei Schwestern sind die Parzen, die Schicksalsg\u00f6ttinnen, die die Lebensf\u00e4den der Menschen spinnen, doch irgendwann abschneiden.\n\n**Dum Roma deliberat, Saguntum perit.**\n\nW\u00e4hrend Rom (noch) \u00fcberlegt, geht Sagunt (bereits) unter.\n\nNach Livius, _Ab urbe condita_ 21,7,1. \u2013 Hannibal eroberte im Jahre 219 v. Chr. die mit Rom verb\u00fcndete Stadt Sagunt, bevor die R\u00f6mer \u00fcberhaupt einen Entschlu\u00df zur Hilfeleistung gefa\u00dft hatten.\n\n**Dum spiro, spero.**\n\nSolange ich (noch) atme, hoffe ich.\n\nVgl. Cicero, _Ad Atticum_ 9,11,3: **Aegroto dum...** und Seneca, _Epistulae morales_ 70,6. \u2013 Im Gegensatz zu Cicero distanziert sich Seneca von einer solchen Haltung.\n\n**Dum vitant stulti vitia, in contraria currunt.**\n\nW\u00e4hrend Dumme Fehler zu vermeiden suchen, laufen sie ins Gegenteil (d. h.: sie erreichen das Gegenteil ihrer Bem\u00fchungen und begehen neue Fehler).\n\nHoraz, _Sermones_ 1,2,24.\n\n**Duo cum faciunt idem, non est idem.**\n\nAuch wenn zwei dasselbe tun, ist es nicht dasselbe.\n\nNach Terenz, _Adelphoe_ 823 ff. Vgl. **Ille crucem sceleris pretium tulit...**\n\n**duo parietes de eadem fidelia dealbare**\n\nzwei W\u00e4nde aus demselben T\u00fcnchgef\u00e4\u00df wei\u00dfen\n\n(vgl.: zwei Fliegen mit einer Klappe schlagen)\n\nCicero, _Ad familiares_ 7,29,2; vgl. **in saltu uno...**\n\n**Dura lex , sed lex.**\n\nEin hartes Gesetz, aber ein Gesetz.\n\nVgl. _Digesta_ 40,9,12.\n\n**Durate et vosmet rebus servate secundis!**\n\nBleibt hart und erhaltet euch f\u00fcr g\u00fcnstige Zeiten!\n\nVergil, _Aeneis_ 1,207.\n\n**Durius in terris nihil est, quod vivat, amante.**\n\nNichts Beharrlicheres lebt auf der Welt als ein Liebender.\n\nProperz, _Elegiae_ 2,17,9.\n\n**Durum est contendere cum victore.**\n\nEs ist hart, mit einem Sieger zu k\u00e4mpfen.\n\nNach Horaz, _Sermones_ 1,9,42 f.\n\n**Durum! Sed levius fit patientia,**\n\n**Quidquid corrigere est nefas.**\n\nHart (mag das Schicksal sein)! Aber leichter wird durch Geduld, was zu \u00e4ndern nicht erlaubt ist.\n\nHoraz, _Carmina_ 1,24,19 f.\n\n**Durum telum necessitas.**\n\nEine harte Waffe ist Notwendigkeit.\n\n(Vgl.: Mu\u00df ist eine harte Nu\u00df.)\n\nVgl. Livius, _Ab urbe condita_ 4,28,5. Vgl. **Necessitas ultimum...**\nE\n\n**Ea est natura hominum.**\n\nDas ist die Natur der Menschen.\n\nVgl. Cicero, _De finibus_ 5,66.\n\n**eandem incudem diem noctemque tundere**\n\nTag und Nacht auf denselben Ambo\u00df schlagen\n\n(d. h.: fortw\u00e4hrend dasselbe tun)\n\nNach Cicero, _De oratore_ 2,162.\n\n**Ecce homo!**\n\nSiehe: (Welch) ein Mensch!\n\nNach Johannes (19,5) rief Pontius Pilatus diese Worte aus, als er den dornengekr\u00f6nten Jesus dem Volk zeigte; auch Titel eines Werkes von Nietzsche.\n\n**Ecquid is homo habet aceti in pectore?**\n\nHat dieser Mensch denn etwa Essig in der Brust?\n\n(D. h.: Grollt er?)\n\nPlautus, _Pseudolus_ 739.\n\n**Edepol, qui amat, si eget, misera adficitur, aere, aerumna.**\n\nBei Gott! Wer liebt, ist mit \u00fcblem Kummer behaftet, wenn es ihm (gleichzeitig) an Geld mangelt.\n\nPlautus, _Curculio_ 142.\n\n**Edere oportet, ut vivas, non vivere, ut edas** siehe **Non ut edam...**\n\nEdite , bibite!\n\nE\u00dft, trinkt!\n\nNach Strabo ( _Geographie_ 14,5,9) und Arrian ( _Anabasis_ 2,5,4) sollen diese Worte auf dem Grab des mythischen Assyrerk\u00f6nigs Sardanapalus gestanden haben. Vgl. den Refrain eines Studentenliedes von Christian Wilhelm Kindleben (1748\u20131785): \u00bbEdite, bibite, collegiales!\u00ab. H\u00e4ufig findet sich auch: \u00bbEde, bibe, lude! Post mortem nulla voluptas!\u00ab (\u00bbI\u00df, trink, spiel! Nach dem Tod gibt es kein Vergn\u00fcgen mehr.\u00ab)\n\n**E duobus malis minus eligendum est.**\n\nVon zwei \u00dcbeln ist das kleinere zu w\u00e4hlen.\n\nNach Cicero ( _De officiis_ 3,3), der dies als bekannte, von den Gelehrten \u00fcberlieferte Regel bezeichnet. Er f\u00fchrt weiter aus, da\u00df man, wenn es denn m\u00f6glich sei, selbst aus dem Schlechten noch Gutes ziehen m\u00fcsse. \u2013 Der Gedanke findet sich bereits bei Aristoteles ( _Nikomachische Ethik_ 5,2,1129b 8). Vgl. **minima de malis**.\n\n**Effugit mortem, quisquis contempserit; timidissimum quemque consequitur.**\n\nDem Tod entkommt, wer ihn verachtet; gerade die \u00c4ngstlichsten ereilt er.\n\nCurtius Rufus, _Historia Alexandri Magni_ 4,14,25.\n\n**e flamma petere cibum**\n\nsein Essen aus dem Feuer holen\n\nNach Terenz, _Eunuchus_ 491; vgl. Catull, _Carmina_ 59,3. \u2013 Bei Leichenfeiern wurden mit den Toten auch Speisen verbrannt. Diese aus dem Feuer zu stehlen, galt als besonders frevelhaft.\n\n**Ego in portu navigo.**\n\nIch segele (bereits) im Hafen.\n\n(D. h.: Ich habe meine Sch\u00e4fchen ins trockene gebracht.)\n\nTerenz, _Andria_ 480.\n\n**Ego pretium ob stultitiam fero.**\n\nIch erhalte den Lohn meiner Torheit.\n\nTerenz, _Andria_ 610.\n\n**Ego spem pretio non emo.**\n\nIch kaufe Hoffnung nicht f\u00fcr Geld.\n\nTerenz, _Adelphoe_ 219.\n\n**Ego te intus et in cute novi.**\n\nDich kenne ich innen und auf der Haut (d. h.: in- und auswendig).\n\nPersius, _Saturae_ 3,30.\n\n**Eheu fugaces, Postume, Postume, \/ labuntur anni!**\n\nAch, Postumus, Postumus, schnell entgleiten die fl\u00fcchtigen Jahre!\n\nHoraz, _Carmina_ 2,14,1 f.\n\n**Ei mihi, quod nullis amor est sanabilis herbis!**\n\nWeh mir, da\u00df Liebe durch kein Kraut heilbar ist!\n\nOvid, _Metamorphoses_ 1,523.\n\n**Eis nunc praemium est, qui recta prava faciunt.**\n\nHeutzutage gibt es Belohnungen f\u00fcr die, die Gutes schlecht machen.\n\nTerenz, _Phormio_ 771.\n\n**Eiusdem est farinae.**\n\nEr ist aus demselben Mehl (gebacken wie wir).\n\n(vgl.: aus demselben Holz geschnitzt sein)\n\nNach Persius, _Saturae_ 5,115.\n\n**Elephanti corio circum tectus est.**\n\nMit einer Elefantenhaut ist er ringsum bedeckt.\n\n(Vgl.: Er hat ein dickes Fell.)\n\nPlautus, _Miles gloriosus_ 235.\n\n**Emas, non quod opus est, sed quod necesse est.**\n\nKaufe nicht, was du brauchst, sondern was n\u00f6tig ist.\n\nSeneca ( _Epistulae morales_ 94,27) zitiert hier wie im folgenden Cato d. \u00c4. ( _De agricultura_ Fragment 10 Jordan): \u00bbQuod non opus est, asse carum est.\u00ab (\u00bbWas man nicht braucht, ist selbst f\u00fcr ein Ass zu teuer.\u00ab)\n\n**Emere melius est, cui imperes.**\n\nDu solltest dir besser jemanden kaufen, dem du Vorschriften machen kannst.\n\nPlautus, _Trinummus_ 1061.\n\n**Epicuri de grege porcus** siehe **cutem curare**\n\n**Epistula enim non erubescit.**\n\nEin Brief err\u00f6tet n\u00e4mlich nicht.\n\nCicero, _Ad familiares_ 5,13,1. \u2013 In Anlehnung an die deutsche Wendung _Papier ist geduldig_ oft auch zitiert als \u00bbCharta non erubescit\u00ab. Vgl. Ambrosius, _De virginibus_ 1,1,1, und **Fiscus non erubescit**.\n\n**equus Troianus**\n\nein Trojanisches Pferd\n\nCicero ( _Actio secunda in C. Verrem_ 4,52) bezeichnet Verres, der die von ihm verwalteten Sikuler aufs \u00fcbelste geschr\u00f6pft hatte, als ebenso unheilvoll wie das Trojanische Pferd. Vgl. Cicero, _Pro L. Murena_ 78, und **Danaum fatale munus**.\n\n**Ergo bibamus!**\n\nLa\u00dft uns also trinken!\n\nHermann Hagen, _Carmina clericorum_ , Canticum potatorium 1 ff.: \u00bbErgo bibamus \/ ne sitiamus \/ vas repleamus.\u00ab (\u00bbLa\u00dft uns also trinken, damit wir nicht d\u00fcrsten, la\u00dft uns das Glas f\u00fcllen.\u00ab) \u00bbErgo bibamus\u00ab ist auch der Refrain eines Trinkliedes (\u00bbHier sind wir versammelt zu l\u00f6blichem Tun\u00ab) von Johann Wolfgang Goethe und war laut Francesco da Buti ( _Commento sopra la divina comedia_ , Anmerkung zu Purgatorio 24) der Lieblingsspruch von Papst Martin IV.\n\n**Ergo vivamus, dum licet esse bene.**\n\nLa\u00dft uns also leben, solange uns erlaubt ist, wohlauf zu sein.\n\nPetron, _Satyrica_ 34,10.\n\n**Eripe te morae!**\n\nEntrei\u00dfe dich dem Zaudern!\n\n(D. h.: Handele endlich!)\n\nHoraz, _Carmina_ 3,29,5.\n\n**Eripitur persona, manet res.**\n\nDie Maske wird abgerissen, die Sache selbst bleibt (in ihrem Wesen).\n\nLukrez ( _De rerum natura_ 3,58) meint, das wahre Wesen der Menschen sei erst in widrigen Umst\u00e4nden zu erkennen: In bedenklichen Situationen g\u00e4ben sie jegliche Vernunft auf und fr\u00f6nten irrationaler Gottesfurcht.\n\n**Eripuit coelo fulmen sceptrumque tyrannis.**\n\nEr entri\u00df dem Himmel den Blitz, das Szepter den Tyrannen.\n\nAnne Robert Turgot (1727\u20131781) in einem Epigramm auf einer B\u00fcste Benjamin Franklins.\n\n**Eripuitque Iovi fulmen viresque tonandi.**\n\nUnd sie entri\u00df Jupiter den Blitz und die Donnerkraft.\n\nManilius ( _Astronomica_ 1,104) preist so die Vernunft, die die Menschen von religi\u00f6ser Furcht befreite.\n\n**Eris mihi magnus Apollo.**\n\nDu wirst mir ein gro\u00dfer Apoll sein.\n\nVergil, _Bucolica_ 3,104. \u2013 Der Hirte Damoetas stellt seinem Freund Menalcas ein schwieriges R\u00e4tsel und verk\u00fcndet, sein Freund werde f\u00fcr ihn wie Apoll sein, falls er die Aufgabe zu l\u00f6sen verm\u00f6ge. \u2013 Apoll war der Herr zahlreicher Orakel. Vgl. **de tripode dictum**.\n\n**Eritis sicut dii scientes bonum et malum.**\n\nIhr werdet wie G\u00f6tter um Gut und B\u00f6se wissen.\n\n1. Mose 3,5. \u2013 Versprechen der Schlange an Eva.\n\n**E rivo flumina magna facis.**\n\nAus einem B\u00e4chlein machst du riesige Str\u00f6me.\n\n(vgl.: aus einer M\u00fccke einen Elefanten machen)\n\nOvid, _Epistulae ex Ponto_ 2,5,22; vgl. **arcem facere e cloaca**.\n\n**Errare humanum est** siehe **Errasse humanum est...**\n\nErrare... malo cum Platone... quam cum istis vera sentire.\n\nIch m\u00f6chte lieber mit Platon irren, als mit jenen die Wahrheit erkennen.\n\nCicero ( _Tusculanae disputationes_ 1,39) distanziert sich hier von den Pythagor\u00e4ern.\n\n**Errasse humanum est et confiteri errorem prudentis.**\n\nIrren (eigtl.: geirrt zu haben) ist menschlich und einen Irrtum einzugestehen Kennzeichen eines Weisen.\n\nHieronymus, _Epistulae_ 57,12. \u2013 Der Gedanke findet sich schon bei Theognis ( _Elegien_ 327 f.) und bei Sophokles ( _Antigone_ 1023 f.). Vgl. auch (z. B.) Terenz, _Adelphoe_ 579; Seneca Rhetor, _Controversiae_ 4,3, und **Cuiusvis hominis est errare...**\n\n**Errat et in nulla sede moratur amor.**\n\nDie Liebe schweift umher und verweilt an keinem Wohnsitz.\n\nOvid, _Ars amatoria_ 3,436.\n\n**Esse quam videri bonus malebat.**\n\nEr wollte lieber gut sein als scheinen.\n\nLaut Sallust ( _Bellum Catilinarium_ 54,6) war dies eine Maxime Catos d. J.\n\n**Est deus in nobis, agitante calescimus illo.**\n\nEs wohnt ein Gott in uns; regt er sich, so ergl\u00fchen wir.\n\nOvid, _Fasti_ 6,5; vgl. _Ars amatoria_ 3,549 und _Epistulae ex Ponto_ 3,4,93 und 4,2,25 f.\n\n**Est enim proprium stultitiae aliorum vitia cernere, oblivisci suorum.**\n\nEs zeugt n\u00e4mlich von Dummheit, die Fehler anderer wahrzunehmen, die eigenen (jedoch) zu vergessen.\n\n(vgl.: den Splitter im eigenen Auge nicht sehen)\n\nCicero, _Tusculanae disputationes_ 3,73.\n\n**Est et fideli tuta silentio \/ merces.**\n\nAuch f\u00fcr treues Schweigen gibt es sicheren Lohn.\n\n(Vgl.: Reden ist Silber, Schweigen ist Gold.)\n\nHoraz, _Carmina_ 3,2,25 f.\n\n**Est modus in rebus, sunt certi denique fines.**\n\nEs liegt ein Ma\u00df in den Dingen, kurz, es gibt bestimmte Grenzen.\n\nHoraz, _Sermones_ 1,1,106.\n\n**Est nobis voluisse satis.**\n\nGewollt zu haben reicht uns (schon).\n\n(vgl.: Man mu\u00df den Willen f\u00fcr das Werk nehmen.)\n\nTibull, _Elegiae_ 4,1,7. Vgl. **Quod si...** und **Ut desint...**\n\n**Esto , quod esse videris!**\n\nSei, was du zu sein scheinst!\n\nVermutlich abgeleitet von folgendem ironischen Rat Juvenals ( _Saturae_ 5,113): \u00bbEsto, ut nunc multi, dives tibi, pauper amicis.\u00ab (\u00bbSei also, wie jetzt viele, reich f\u00fcr dich, arm f\u00fcr die Freunde.\u00ab)\n\n**Est quaedam flere voluptas.**\n\nWeinen ist eine gewisse Lust.\n\nOvid, _Tristia_ 4,3,37.\n\n**Est rerum omnium magister usus.**\n\nErfahrung ist in allen Dingen Lehrmeisterin.\n\nCaesar, _De bello civili_ 2,8,3.\n\n**E tenui casa saepe vir magnus exit.**\n\nUnd aus einer \u00e4rmlichen H\u00fctte kommt oft ein gro\u00dfer Mann.\n\nNach Seneca, _Epistulae morales_ 66,3.\n\n**Et facere et pati fortia Romanum est.**\n\nTapfer zu handeln wie auch tapfer zu leiden ist Kennzeichen eines R\u00f6mers.\n\nLivius, _Ab urbe condita_ 2,12,9. \u2013 Der Sage nach hatte Mucius Cordus im Jahre 508 n. Chr. versucht, den Etruskerk\u00f6nig Porsenna, der Rom belagerte, zu t\u00f6ten. Als die Etrusker ihn jedoch fa\u00dften, soll er seine r\u00f6mische Identit\u00e4t betont (\u00bbRomanus sum civis\u00ab \u2013 \u00bbIch bin ein r\u00f6mischer B\u00fcrger\u00ab) und zum Beweis seines Mutes seine rechte Hand in ein Feuer gehalten haben. M\u00f6glicherweise erhielt er daher seinen Beinamen Scaevola (\u203aLinkhand\u2039). Von solchem Heldenmut beeindruckt, soll Porsenna die Belagerung aufgehoben haben. \u2013 Eine \u00e4hnliche Tat berichtet Plutarch ( _Agesilaus_ 2) \u00fcber Agesilaus. Vgl. **Civis Romanus sum.**\n\n**et genus et formam regina pecunia donat** siehe **Scilicet uxorem cum dote...**\n\n**Et hoc, quod senectus vocatur, paucissimorum est circuitus annorum.**\n\nAuch das, was man das hohe Alter nennt, ist nur eine Spanne sehr weniger Jahre.\n\nSeneca, _Ad Marciam de consolatione_ 11,5.\n\n**Etiam capillus unus habet umbram suam.**\n\nAuch ein einziges Haar wirft seinen Schatten.\n\nPublilius Syrus, _Sententiae_ E 15.\n\n**Etiam innocentes cogit mentiri dolor.**\n\nSchmerz zwingt sogar Unschuldige zu l\u00fcgen.\n\nPublilius Syrus, _Sententiae_ E 1.\n\n**Etiam periere ruinae.**\n\nSelbst die Ruinen gingen unter.\n\nLukan ( _De bello civili_ 9,969) \u00fcber den Untergang Trojas.\n\n**Etiam qui faciunt, oderunt iniuriam.**\n\nAuch die, die Unrecht tun, hassen es.\n\nPublilius Syrus, _Sententiae_ E 10.\n\n**Et in Arcadia ego.**\n\nAuch in Arkadien bin ich (erg.: der Tod).\n\nSpruch auf einem Gem\u00e4lde von Giovanni Francesco Barbieri (il Guercino). \u2013 Arkadien wird in der Neuzeit als Inbegriff einer idyllischen Landschaft verstanden. Im Gegensatz zur genannten grammatisch korrekten \u00dcbersetzung und zur vermutlichen Intention Guercinos wird mit dieser Wendung heute nicht mehr die Allmacht des Todes, der selbst in Arkadien anzutreffen ist, impliziert. Vielmehr wird das Zitat im Sinne einer Kontemplation \u00fcber eine gl\u00fcckliche Vergangenheit interpretiert (\u00bbAuch ich war in Arkadien\u00ab). Ma\u00dfgeblichen Einflu\u00df auf diese Entwicklung hatten zwei Gem\u00e4lde Poussins, die weniger vor dem Tode warnen, als da\u00df sie Erinnerungen an eine Hirtenidylle wachrufen.\n\n**Et mihi forsan, quod tibi negarit, \/ porriget hora.**\n\nUnd vielleicht wird mir die Stunde gew\u00e4hren, was sie dir versagte.\n\nHoraz, _Carmina_ 2,16,31 f.\n\n**Et mihi si non vis parcere, parce meis!**\n\nUnd wenn du mich nicht schonen willst, so schone die Meinen!\n\nOvid, _Heroides_ 4,162.\n\n**Et nimium vixisse diu nocet.**\n\nAuch allzulang gelebt zu haben schadet.\n\nOvid, _Metamorphoses_ 6,38.\n\n**Et non facere facere est.**\n\nAuch Nichtstun (d. h.: Unterlassen) ist Tun.\n\nVgl. _Digesta_ 50,17,121.\n\n**Et oleum et operam perdidi.**\n\n\u00d6l und M\u00fche habe ich verschwendet.\n\n(D. h.: Vergeblich habe ich alle Hebel in Bewegung gesetzt.)\n\nBei Plautus ( _Poenulus_ 332) tut eine Sklavin mit diesen Worten kund, sie habe sich vergeblich mit kosmetischen \u00d6len herausgeputzt. Cicero ( _Ad Atticum_ 13,38,1) hingegen weist mit der Wendung \u00bboleo et opera\u00ab (\u00bbmit \u00d6l und M\u00fche\u00ab) darauf hin, da\u00df er nachts gearbeitet und somit viel Lampen\u00f6l verbraucht habe. Vgl. **Oleum perdit et impensas**.\n\n**Et piscatorem piscis amare potest?**\n\nUnd kann der Fisch einen Fischer lieben?\n\nMartial ( _Epigrammata_ 6,63,6) warnt so Marianus, der einer Person nachstellt, die nur anstrebt, ihn zu beerben. Vgl. **Munera misit...**\n\n**Et pius est patriae facta referre labor.**\n\nEs ist auch ein frommes Unterfangen, die Geschichte des Vaterlandes zu erz\u00e4hlen.\n\nOvid, _Tristia_ 2,322.\n\n**Et quod temptabam scribere, versus erat.**\n\nUnd was ich zu schreiben versuchte \u2013 es war ein Vers.\n\nOvid ( _Tristia_ 4,10,26) erkl\u00e4rt recht selbstbewu\u00dft, er k\u00f6nne gar nicht umhin, sich in Versen auszudr\u00fccken.\n\n**... et quorum pars magna fui.**\n\n... und worin ich eine gro\u00dfe Rolle spielte.\n\nVergil, _Aeneis_ 2,6. \u2013 Aeneas berichtet vom traurigen Schicksal Trojas. Er selbst habe dabei am eigenen Leibe viel \u00dcbles erfahren m\u00fcssen.\n\n**Et res non semper, spes mihi semper adest.**\n\nUnd (wenn) die (erhoffte) Sache nicht immer bei mir ist, so ist immer die Hoffnung (darauf) bei mir.\n\nOvid, _Heroides_ 18,178.\n\n**Et semel emissum volat irrevocabile verbum.**\n\nUnd einmal ausgesprochen, fliegt das Wort unwiderruflich davon.\n\nHoraz, _Epistulae_ 1,18,71. Vgl. **Nescit vox...**\n\n**Et senis amplexus culta puella fugit.**\n\nUnd ein kultiviertes M\u00e4dchen meidet die Umarmungen eines Alten.\n\nTibull, _Elegiae_ 1,9,74.\n\n**Et sit humus cineri non onerosa tuo!**\n\nUnd m\u00f6ge die Erde deiner Asche keine Last sein!\n\nOvid ( _Amores_ 3,9,68) in seinem Klagelied \u00fcber den Tod Tibulls.\n\n**Et sol crescentis decedens duplicat umbras.**\n\nUnd die weichende Sonne verdoppelt die wachsenden Schatten.\n\nVergil, _Bucolica_ 2,67.\n\n**Et tenuit nostras numerosus Horatius aures.**\n\nUnd unsere Ohren hielt der rhythmenreiche Horaz (in seinem Bann).\n\nOvid, _Tristia_ 4,10,49.\n\n**Et tu, Brute?**\n\nAuch du, Brutus?\n\nLaut Sueton ( _Vita divi Iulii_ 82,2) richtete Caesar bei seiner Ermordung diese Frage in griechischer Sprache (\u039a\u03b1\u1f76 \u03c3\u1f7a \u03c4\u03ad\u03ba\u03bd\u03bf\u03bd? \u2013 \u00bbAuch du, Kind?\u00ab) an Brutus, der zusammen mit den anderen M\u00f6rdern auf ihn einstach. \u2013 Vgl. Cassius Dio, _R\u00f6mische Geschichte_ 44,19,5.\n\n**Euge, poeta!**\n\nBravo, Dichter!\n\nPersius, _Saturae_ 1,75.\n\n**Euhoe! Recenti mens trepidat metu.**\n\nAch! Von frischer Furcht zittert (noch) das Herz.\n\nHoraz, _Carmina_ 2,19,5.\n\n**Eunt via sua fata.**\n\nDas Schicksal geht seinen Weg.\n\nSeneca, _Ad Marciam de consolatione_ 21,6.\n\n**Eventus docebit.**\n\nDer Ausgang (eines Unterfangens) wird es lehren.\n\n(Vgl.: Erfahrung macht klug.)\n\nLivius ( _Ab urbe condita_ 22,39,10): \u00bbNec eventus modo hoc docet \u2013 stultorum iste magister est...\u00ab (\u00bbDies lehrt nicht nur der Erfolg \u2013 er ist ein Lehrmeister der Toren...\u00ab); daher auch \u00bbeventus magister stultorum\u00ab (\u00bbdas Ergebnis ist der Lehrmeister der Toren\u00ab).\n\n**eventus magister stultorum** siehe **Eventus docebit**\n\n**ex alieno incommodo suam petere occasionem**\n\nim Ungl\u00fcck anderer die Gelegenheit (zum eigenen Vorteil) suchen\n\nNach Livius, _Ab urbe condita_ 4,58,2.\n\n**exanclare omnes labores**\n\nalle M\u00fchen bis zum Ende durchstehen\n\nNach Cicero, _Tusculanae disputationes_ 1,118.\n\n**ex animi nostri sententia**\n\naus unserer Herzens\u00fcberzeugung\n\nLaut Quintilian ( _Institutio oratoria_ 8,5,1) eine typische Schwurformel.\n\n**Ex avaritia omnia scelera ac maleficia gignuntur.**\n\nAus der Habsucht entspringen alle Verbrechen und \u00dcbeltaten.\n\n(Vgl.: Habsucht ist die Wurzel allen \u00dcbels.)\n\nNach Cicero, _Pro S. Roscio Amerino_ 75.\n\n**Excitat auditor studium.**\n\nDer Zuh\u00f6rer regt den Eifer (des Vortragenden) an.\n\nOvid, _Epistulae ex Ponto_ 4,2,35 f.: \u00bbExcitat auditor studium, laudataque virtus \/ crescit, et immensum gloria calcar habet.\u00ab (\u00bbDer Zuh\u00f6rer regt den Eifer an, und die Tugend w\u00e4chst, wenn sie gelobt wird, und [die Aussicht auf] Ruhm ist ein gewaltiger Ansporn.\u00ab)\n\n**Excludat iurgia finis!**\n\nDer (genaue) Zeitpunkt soll Streitigkeiten ausschlie\u00dfen.\n\nHoraz ( _Epistulae_ 2,1,38) sucht ironisch nach der Anzahl der Jahre, die ein Dichter tot sein mu\u00df, um als Dichter von Rang zu gelten.\n\n**Exegi monumentum aere perennius.**\n\nIch habe (mir) ein Denkmal geschaffen, dauerhafter als Erz.\n\nHoraz, _Carmina_ 3,30,1. \u2013 F\u00fcnf Verse weiter verk\u00fcndet Horaz selbstbewu\u00dft, er habe durch seine Dichtung Unsterblichkeit erlangt: \u00bbNon omnis moriar.\u00ab (\u00bbIch werde nicht g\u00e4nzlich sterben.\u00ab)\n\n**exemplum facere**\n\nein Exempel statuieren\n\nNach Terenz, _Eunuchus_ 948.\n\n**Exercitatio artem parat.**\n\n\u00dcbung f\u00fchrt zur Kunst.\n\n(Vgl.: \u00dcbung macht den Meister.)\n\nNach Tacitus, _Germania_ 24,1.\n\n**ex eventu iudicium facere, non ex consilio**\n\nsich sein Urteil aufgrund des Ergebnisses, nicht der Absicht bilden\n\nNach Cicero, _Ad familiares_ 1,8,5.\n\n**Ex factis, non ex dictis amici pensandi.**\n\nNach Taten, nicht nach ihren Worten soll man Freunde beurteilen.\n\nNach Livius, _Ab urbe condita_ 34,49,7.\n\n**Ex igne ut fumus, sic fama ex crimine surgit.**\n\nWie aus dem Feuer Rauch, so entsteht aus der Anklage Leumund.\n\n_Monosticha Catonis_ 14.\n\n**Ex incomprehensibili parvitate harenae funis effici non potest.**\n\nAus der unfa\u00dfbaren Kleinheit der Sandk\u00f6rner kann kein Strick gedreht werden.\n\nColumella ( _De re rustica_ 10, praefatio 4) bezeichnet diese Worte als bekanntes griechisches Sprichwort; vgl. Macarius Hieromonachus (Paroemiographi Graeci) 3,97. Er erl\u00e4utert mit dieser Wendung, da\u00df er in seinem Buch \u00fcber so viele Einzelheiten berichten werde, da\u00df kein geschlossenes Ganzes entstehen k\u00f6nne.\n\n**Exitus acta probat.**\n\nDer Ausgang pr\u00fcft die Tat.\n\nOvid, _Heroides_ 2,85.\n\n**Exitus in dubio est.**\n\nDer Ausgang ist zweifelhaft.\n\nOvid, _Metamorphoses_ 12,522.\n\n**Ex magna coena stomacho fit magna poena.**\n\n**Ut sis nocte levis, sit tibi coena brevis.**\n\nReichliches Essen zur Nacht beschert dem Bauch gro\u00dfe Pein:\n\nDamit du nachts leicht schl\u00e4fst, sei deine Nachtmahlzeit klein!\n\n_Regimen sanitatis Salerni_ , De coena.\n\n**Ex malis multis malum, quod minimum est, id minime est malum.**\n\nUnter vielen \u00dcbeln ist das, welches das geringste ist, am wenigsten ein \u00dcbel.\n\nPlautus, _Stichus_ 120.\n\n**Ex minimis seminibus nascuntur ingentia.**\n\nAus winzigen Keimen entsteht Riesiges.\n\nNach Seneca, _De providentia_ 1,2.\n\n**Ex nihilo nihil** siehe **Nihil ex nihilo exsistere...**\n\n**Exoriare aliquis nostris ex ossibus ultor!**\n\nM\u00f6ge sich aus unseren Gebeinen irgendein R\u00e4cher erheben!\n\nVergil, _Aeneis_ 4,625. \u2013 Dido verflucht Aeneas, der sie verlassen hat, und ruft die Karthager mit den obigen Worten zur Rache auf. Die R\u00f6mer sahen diese Rache sp\u00e4ter in Hannibal eingel\u00f6st.\n\n**Ex oriente lux.**\n\nAus dem Osten (kommt) das Licht.\n\nVielleicht nach Matth\u00e4us 2,2. \u2013 Die Wendung bezog sich vermutlich zun\u00e4chst nur auf die Sonne und wurde sp\u00e4ter auf Christentum und Kultur \u00fcbertragen.\n\n**Experti scire debemus.**\n\n(Das) m\u00fcssen wir aus Erfahrung wissen.\n\nCicero, _Pro T. Annio Milone_ 69.\n\n**Experto credite!**\n\nGlaubet dem, der es aus Erfahrung wei\u00df!\n\nVergil, _Aeneis_ 11,283. \u2013 Der Argiver Diomedes, der selbst die Tapferkeit des Aeneas im Kampf erprobt hat, r\u00e4t seinen Landsleuten zum B\u00fcndnis mit den Trojanern. \u2013 Ebenso Ovid, _Ars amatoria_ 3,511, vgl. _Fasti_ 5,674, und Cicero, _Topica_ 74, sowie **Crede experto!**\n\n**Expertus metuit.**\n\nDer Erfahrene ist furchtsam.\n\n(Vgl.: Gebranntes Kind scheut das Feuer.)\n\nHoraz, _Epistulae_ 1,18,87.\n\n**Ex quovis ligno non fit Mercurius** siehe **Non enim ex omni ligno...**\n\n**ex re fabellae**\n\nzum Thema passende Geschichten\n\nNach Horaz, _Sermones_ 2,6,78.\n\n**ex tempore**\n\naus dem Stegreif\n\n(z. B.) Cicero, _De divinatione_ 1,72.\n\n**Extra ecclesiam nulla salus** siehe **Salus extra ecclesiam...**\n\n**extrema omnia experiri**\n\nauch das \u00c4u\u00dferste versuchen\n\nSallust, _Bellum Catilinarium_ 26,5.\n\n**extremis digitis attingere**\n\n(nur) mit den Fingerspitzen anfassen\n\n(d. h.: sich einer Sache nur kurz zuwenden)\n\nCicero ( _Pro M. Caelio_ 28) f\u00fchrt dies als gebr\u00e4uchliche Redewendung an.\n\n**ex ungue leonem (pingere)**\n\nnach der Pranke den L\u00f6wen (malen)\n\n(d. h.: aus einem Teil das Ganze ersehen)\n\nPlutarch ( _Moralia_ 410c) f\u00fchrt den entsprechenden griechischen Ausdruck auf den Dichter Alkaios zur\u00fcck (Fragment Inc. lib. 118 Lobel), w\u00e4hrend Lukian ( _Hermotimus_ 54) diese F\u00e4higkeit dem Bildhauer Phidias zuschreibt.\n\n**Ex uno puteo similior numquam potis**\n\n**Aqua aquae sumi, quam haec est atque ista hospita.**\n\nAus einem Brunnen kann niemals Wasser gesch\u00f6pft werden, das diesem Wasser \u00e4hnlicher w\u00e4re, als diese da jener Fremden \u00e4hnlich ist.\n\n(Vgl.: Sie gleichen einander wie ein Ei dem anderen.)\n\nPlautus, _Miles gloriosus_ 551 f. Vgl. **Neque aqua...**\nF\n\n**Faba ista in me cudetur.**\n\nDiese Bohne wird auf mir gestampft werden.\n\n(D. h.: Diese Suppe werde ich ausl\u00f6ffeln m\u00fcssen.)\n\nNach Terenz, _Eunuchus_ 381.\n\n**Faber est suae quisque fortunae.**\n\nJeder ist seines Gl\u00fcckes Schmied.\n\nNach Sallust ( _Ad Caesarem senem de re publica oratio_ 1,2), der diesen Ausspruch auf Appius Claudius Caecus zur\u00fcckf\u00fchrt. Vgl. **Fortunam sibi...**\n\n**Fabulae!**\n\nGeschw\u00e4tz!\n\nTerenz, _Heauton timorumenos_ 336.\n\n**Fabula quanta fui!**\n\nZu welchem Gespr\u00e4chsstoff bin ich geworden!\n\nHoraz, _Epodi_ 11,8.\n\n**Facere docet philosophia, non dicere.**\n\nZu handeln lehrt die Philosophie, nicht zu reden.\n\nSeneca, _Epistulae morales_ 20,2.\n\n**Facetiae omnium sermonum condimenta.**\n\nWitzeleien sind die W\u00fcrze aller Gespr\u00e4che.\n\nVgl. Cicero, _De amicitia_ 66.\n\n**facies Hippocratica**\n\nein Hippokratisches Gesicht\n\nHippokrates ( _Prognostikon_ 2) beschreibt besonders treffend das vom bevorstehenden Tod gekennzeichnete Gesicht eines Sterbenden.\n\n**Facies non omnibus una \/ nec diversa tamen.**\n\nKein Gesicht unter ihnen allen ist gleich und dennoch keines verschieden.\n\nOvid ( _Metamorphoses_ 2,13 f.) beschreibt so die Gesichter der Nereiden (f\u00fcnfzig Meeresg\u00f6ttinnen).\n\n**Facile omnes, cum valemus, recta consilia aegrotis damus.**\n\nLeicht geben wir alle, solange wir gesund sind, Kranken gute Ratschl\u00e4ge.\n\nTerenz, _Andria_ 309.\n\n**Facilis descensus Averno.**\n\nLeicht ist der Abstieg zur Unterwelt.\n\nVergil, _Aeneis_ 6,126 ff.: \u00bbFacilis descensus Averno: \/... \/ sed revocare gradum superasque evadere ad auras, \/ hoc opus, hic labor est.\u00ab (\u00bbLeicht ist der Abstieg zur Unterwelt..., aber den Schritt umzukehren und in die Oberwelt zu entkommen, das bedeutet M\u00fche und Not.\u00ab)\n\n**Facilitate nil homini melius neque clementia.**\n\nNichts steht dem Menschen besser als Leutseligkeit und Milde.\n\nTerenz, _Adelphoe_ 861.\n\n**Facis de necessitate virtutem.**\n\nDu machst aus der Not eine Tugend.\n\nHieronymus, _Adversus Rufinum_ 3,2; vgl. Pseudo-Quintilian, _Declamationes_ 4,10.\n\n**Facis farinam.**\n\nDu machst (alles zu) Mehl.\n\nMartial, _Epigrammata_ 8,16,5.\n\n**Facit indignatio versum.**\n\nDie Entr\u00fcstung macht den Vers.\n\nJuvenal, _Saturae_ 1,79: \u00bbSi natura negat, facit indignatio versum.\u00ab (\u00bbWenn [die Begabung] versagt, macht die Entr\u00fcstung den Vers.\u00ab)\n\n**Faciunt favos et vespae.**\n\nAuch Wespen bauen (Honig-)Waben.\n\nTertullian, _Adversus Marcionem_ 4,5.\n\n**Facta infecta fieri nequeunt.**\n\nGeschehenes kann nicht ungeschehen gemacht werden.\n\nNach Terenz, _Phormio_ 1034. Vgl. **Factum est...**\n\n**Fac tantum cupias, sponte disertus eris.**\n\nHandle so, wie du willst! Von selbst wirst du beredt sein.\n\nOvid, _Ars amatoria_ 1,610.\n\n**Factum abiit, monumenta manent.**\n\nDie Tat ist vergangen, die Denkm\u00e4ler bleiben.\n\nOvid, _Fasti_ 4,709.\n\n**Factum est illud; fieri infectum non potest.**\n\nJenes ist geschehen; man kann es nicht ungeschehen machen.\n\nPlautus, _Aulularia_ 741. Vgl. **Facta infecta...**\n\n**Faenum habet in cornu: longe fuge!**\n\nEr hat Stroh am Horn: Nichts wie weg!\n\n(D. h.: Er ist wie ein wilder Stier.)\n\nHoraz, _Sermones_ 1,4,34. \u2013 F\u00fcr diese Wendung werden zwei Interpretationen angef\u00fchrt: zum einen, da\u00df gef\u00e4hrliche Tiere durch Stroh an den H\u00f6rnern gekennzeichnet wurden, zum anderen (und wahrscheinlicher), da\u00df es sich um einen st\u00f6\u00dfigen Stier handelt, der gerade noch im Stroh herumw\u00fctete.\n\n**Fallaces enim sunt rerum species.**\n\nDenn tr\u00fcgerisch ist der Schein der Dinge.\n\nSeneca, _De beneficiis_ 4,34,1.\n\n**Fallacia \/ alia aliam trudit.**\n\nEin Betrug treibt den n\u00e4chsten hervor.\n\nTerenz, _Andria_ 778 f.\n\n**Fallit enim vitium specie virtutis.**\n\nDas Laster n\u00e4mlich t\u00e4uscht durch den Anschein der Tugend.\n\n(Vgl.: Laster stehlen der Tugend die Kleidung.)\n\nJuvenal, _Saturae_ 14,109.\n\n**Fallitur augurio spes bona saepe suo.**\n\nHoffnung auf Gutes betr\u00fcgt sich oft selbst durch ihr Vorgef\u00fchl.\n\n(Vgl.: Hoffen und Harren macht manchen zum Narren.)\n\nOvid, _Heroides_ 17,234.\n\n**Fama crescit eundo** siehe **Mobilitate viget...**\n\n**famam extendere factis**\n\nseinen Ruhm durch Taten vergr\u00f6\u00dfern\n\nVergil, _Aeneis_ 10,468; vgl. 6,806.\n\n**Fama nihil est celerius.**\n\nNichts ist schneller als das Ger\u00fccht.\n\nNach Livius, _Ab urbe condita_ 24,21,5; vgl. Vergil, _Aeneis_ 4,174.\n\n**Famaque post cineres maior venit.**\n\nUnd wenn wir zu Asche geworden, w\u00e4chst der Ruhm.\n\nOvid, _Epistulae ex Ponto_ 4,16,3.\n\n**Fama volat.**\n\nDas Ger\u00fccht fliegt (eilends).\n\nVergil, _Aeneis_ 3,121; 7,392 und 8,554.\n\n**Fames optimum condimentum** siehe **Cibi condimentum fames est**\n\n**Familiaritas librorum parit insaniam.**\n\nDer Umgang mit B\u00fcchern f\u00fchrt zum Wahnsinn.\n\nErasmus, _Colloquia familiaria_. _Abbatis et Eruditae_.\n\n**Fas est et ab hoste doceri.**\n\nEs ist auch erlaubt, sich vom Feinde belehren zu lassen.\n\nOvid, _Metamorphoses_ 4,428.\n\n**Fata obstant.**\n\nDas Schicksal stellt sich dagegen.\n\nVergil, _Aeneis_ 4,440.\n\n**Fata viam invenient.**\n\nDas Schicksal wird seinen Weg finden.\n\nVergil, _Aeneis_ 3,395.\n\n**Fatetur facinus is, qui iudicium fugit.**\n\nSein Verbrechen gesteht, wer das Gericht flieht.\n\nPublilius Syrus, _Sententiae_ F 9.\n\n**Faucibus teneor** bzw. **premor.**\n\nMan bedr\u00e4ngt mich bei der Kehle.\n\nPlautus, _Casina_ 943; vgl. Cicero, _Pro A. Cluentio Habito_ 84.\n\n**Favete linguis!**\n\nBeg\u00fcnstigt (die Kulthandlungen) mit euren Zungen (erg.: indem ihr sie stillhaltet)!\n\nCicero, _De divinatione_ 2,83, und Horaz, _Carmina_ 3,1,2. \u2013 Bei offiziellen religi\u00f6sen Anl\u00e4ssen gebot ein Herold mit diesem Ruf Schweigen, um eine St\u00f6rung der Feier durch ein unbedachtes omin\u00f6ses Wort auszuschlie\u00dfen. Seneca ( _De vita beata_ 26,7) erkl\u00e4rt, diese Wendung leite sich nicht von \u203aGunst\u2039 ( _favor_ ) her, sondern befehle Schweigen, damit die Opferhandlung nach den Kultbr\u00e4uchen vollzogen werden k\u00f6nne, ohne da\u00df ein ungutes Wort sie unterbreche. Vgl. Cicero, _De divinatione_ 1,102; Ovid, _Fasti_ 2,654; Plinius d. \u00c4., _Naturalis historia_ 28,11.\n\n**fax et turbo sequentis saeculi**\n\nBrandfackel und Wirbelsturm des folgenden Jahrhunderts\n\nFlorus ( _Epitomae de Tito Livio_ 2,14) \u00fcber Antonius.\n\n**Felix, heu, nimium felix!**\n\nGl\u00fcckliche, ach, allzu G\u00fcckliche!\n\nVergil, _Aeneis_ 4,657. \u2013 Dido klagt, sie w\u00e4re gl\u00fccklich gewesen, h\u00e4tte sie Aeneas niemals zu Gesicht bekommen.\n\n**Felix, qui potuit rerum cognoscere causas.**\n\nGl\u00fccklich, wer die Urgr\u00fcnde der Welt zu erkennen vermochte.\n\nVergil ( _Georgica_ 2,490 ff.) vergleicht sich mit dem epikureischen Wissenschaftler Lukrez, dessen Verdienst darin liege, da\u00df er die Menschen durch seine Atomlehre von der Angst vor den G\u00f6ttern und dem Tode befreit habe. Anschlie\u00dfend preist er jedoch auch die Kenntnis der G\u00f6tter, die f\u00fcr das l\u00e4ndliche Leben von Belang sind (493): \u00bbFortunatus et ille deos qui novit agrestis.\u00ab (\u00bbGl\u00fccklich ist auch jener, der die l\u00e4ndlichen G\u00f6tter kennt.\u00ab)\n\n**Felix, qui quod amat , defendere fortiter audet.**\n\nGl\u00fccklich, wer, was er liebt, tapfer zu verteidigen wagt.\n\nOvid, _Amores_ 2,5,9.\n\n**Feras, non culpes, quod mutari non potest.**\n\nErtrage (und) mi\u00dfbillige nicht, was nicht zu \u00e4ndern ist.\n\nPublilius Syrus, _Sententiae_ F 11.\n\n**Fere fit malum malo aptissimum.**\n\nB\u00f6ses pa\u00dft f\u00fcr gew\u00f6hnlich am besten zu B\u00f6sem.\n\nLivius, _Ab urbe condita_ 1,46,7.\n\n**Feriuntque summos \/ fulgura montis.**\n\nUnd Blitze schlagen auf den Berggipfeln ein.\n\nHoraz, _Carmina_ 2,10,11 f.; vgl. Seneca, _Agamemnon_ 96.\n\n**Ferreus adsiduo consumitur anulus usu.**\n\nEin eiserner Ring wird durch st\u00e4ndigen Gebrauch abgenutzt.\n\nOvid, _Ars amatoria_ 1,473. Vgl. **Gutta cavat lapidem...** , **Percussu crebro...** und **Stillicidi casus...**\n\n**Ferroque nocentius aurum.**\n\nUnd Gold ist sch\u00e4dlicher als Eisen.\n\nOvid, _Metamorphoses_ 1,141.\n\n**Ferrum tuum in igne est.**\n\nDein Eisen ist im Feuer.\n\n(D. h.: Dein Schicksal steht nun auf dem Spiel.)\n\nNach Seneca, _Apocolocyntosis_ 9,6.\n\n**Fervet olla, vivit amicitia** siehe **Sociorum olla malle fervet**\n\n**Fervet opus .**\n\nDie Arbeit gl\u00fcht (d. h.: sie wird mit Feuereifer betrieben).\n\nVergil, _Aeneis_ 1,436.\n\n**Festina lente!**\n\nEile mit Weile!\n\nNach Sueton ( _Vita divi Augusti_ 25,4) hat Augustus diese Worte oft in griechischer Sprache ge\u00e4u\u00dfert; vgl. Euripides, _Die ph\u00f6nizischen_ _M\u00e4dchen_ 599 und Gellius ( _Noctes Atticae_ 10,11,5). Vgl. **Sat celeriter...**\n\n**Fiat iustitia et pereat mundus!**\n\nEs geschehe Gerechtigkeit, und wenn die Welt dabei untergeht!\n\nMarino Sanuto, _I diarii_ 33. \u2013 Laut Johannes Manlius ( _Loci communes_ , decalogus 8) und Julius Wilhelm Zincgref ( _Teutscher Nation Apophthegmata_ 1, S. 78) war dies die Devise Kaiser Ferdinands I. \u2013 Liebs zufolge ist \u00bbmundus\u00ab hier mit \u00bbdie Gro\u00dfen dieser Welt\u00ab zu \u00fcbersetzen. Der Ausspruch w\u00e4re dann entgegen dem heutigen Gebrauch kein Beispiel von Rechtsfanatismus, sondern dr\u00fcckte die standesunabh\u00e4ngige Gleichheit aller vor dem Gesetz aus.\n\n**Fiat lux!**\n\nEs werde Licht!\n\n1. Mose 1,3.\n\n**Fidem non servanti fides servanda non est.**\n\nDem gegen\u00fcber, der sein Wort nicht h\u00e4lt, braucht man sein eigenes Wort nicht zu halten.\n\nDamasus Bohemus, _Burchardica, sive regulae canonicae_ 61.\n\n**fides spectata**\n\nerwiesene Treue\n\nNach Cicero, _Divinatio in Q. Caecilium_ 11.\n\n**fidus Achates**\n\nder getreue Achates\n\nAchates, ein Gef\u00e4hrte des Aeneas, dessen Treue bei Vergil mehrfach erw\u00e4hnt wird (z. B. _Aeneis_ 6,158), wurde zum sprichw\u00f6rtlich \u203atreuen Begleiter\u2039.\n\n**finis Poloniae**\n\ndas Ende Polens\n\nWohl legend\u00e4rer Ausruf Tadeusz Kosciuszkos nach der Niederschlagung des von ihm initiierten Aufstandes 1794 nach der zweiten polnischen Teilung.\n\n**Firma valent per se.**\n\nStarkes gedeiht von selbst.\n\nOvid, _Epistulae ex Ponto_ 3,4,7.\n\n**Fiscus non erubescit.**\n\nDer Fiskus err\u00f6tet nicht.\n\n(D. h.: Er nimmt auch Steuern aus anr\u00fcchigen Einnahmen an.)\n\nIn Anlehnung an Cicero, _Ad familiares_ 5,13,1: **Epistula enim non erubescit** ; **Lucri bonus est odor ex re quaelibet** und **Non olet**.\n\n**Fistula dulce canit, volucrem dum decipit auceps.**\n\nDie Fl\u00f6te t\u00f6nt s\u00fc\u00df, w\u00e4hrend der Vogelf\u00e4nger den Vogel t\u00e4uscht.\n\nWarnung in den _Disticha Catonis_ (1,27 B) vor allzu schmeichelhaften Menschen.\n\n**Fit via vi.**\n\nEs entsteht ein Weg durch Gewalt.\n\nVergil, _Aeneis_ 2,494, zitiert z. B. von Seneca, _Epistulae morales_ 37,3.\n\n**Flamma fumo est proxima.**\n\nDie Flamme ist dem Rauch am n\u00e4chsten.\n\n(D. h.: Wo Rauch ist, da ist auch Feuer.)\n\nPlautus, _Curculio_ 53.\n\n**Flectere si nequeo superos, Acheronta movebo.**\n\nKann ich die Himmlischen nicht beugen, so werde ich Acheron (d. h.: die Unterwelt) in Bewegung setzen.\n\n(vgl.: alle Hebel in Bewegung setzen)\n\nVergil, _Aeneis_ 7,312. Es spricht die G\u00f6ttermutter Juno.\n\n**Floccum non interduim.**\n\nIch g\u00e4be keine Faser daf\u00fcr.\n\n(vgl.: keinen Pfifferling darum geben)\n\nPlautus, _Trinummus_ 994.\n\n**Fluctuat nec mergitur.**\n\nEs schwankt, aber geht nicht unter.\n\nWappendevise der Stadt Paris. \u2013 Auf dem Wappen ist ein Schiff abgebildet, das die Stadt symbolisiert.\n\n**foedum inceptu, foedum exitu**\n\nh\u00e4\u00dflich am Anfang, h\u00e4\u00dflich am Ende\n\nLivius ( _Ab urbe condita_ 1, praefatio 10) sieht den Vorteil von Geschichtsstudien darin, da\u00df sie Beispiele f\u00fcr jegliche Form menschlicher Erfahrung darb\u00f6ten. Man k\u00f6nne f\u00fcr sich entscheiden, welche davon man nachahmen und welche man, da sie erwiesenerma\u00dfen sowohl in ihrem Beginnen als auch in ihrem Ausgang schlecht seien, vermeiden wolle.\n\n**Fontes ipsi sitiunt.**\n\nDie Quellen selbst d\u00fcrsten.\n\nCicero ( _Ad Quintum fratrem_ 3,1,11) schreibt an seinen Bruder, zur Zeit sei gerade auch seine eigene schriftstellerische Ader vertrocknet.\n\n**foris sapere, sibi non posse auxiliarier**\n\ndrau\u00dfen weise sein, (aber) sich selbst nicht zu helfen wissen\n\nNach Terenz, _Heauton timorumenos_ 923.\n\n**Forma bonum fragile est.**\n\nSch\u00f6nheit ist ein zerbrechliches Gut.\n\nOvid, _Ars amatoria_ 2,113. Vgl. **Ingenii dotes...**\n\n**Formosa facies muta commendatio est.**\n\nGutes Aussehen ist eine stumme Empfehlung.\n\nPublilius Syrus, _Sententiae_ F 4.\n\n**Forsan et haec olim meminisse iuvabit.**\n\nVielleicht wird es einst angenehm sein, sich auch daran zu erinnern.\n\nVergil, _Aeneis_ 1,203.\n\n**Fortasse non poterit hoc sic abire.**\n\nVielleicht wird sich dies nicht ohne weiteres erledigen lassen k\u00f6nnen.\n\nNach Cicero, _De finibus_ 5,7.\n\n**Fortes fortuna adiuvat** siehe **Audentes fortuna iuvat**\n\n**Fortior in fulva novus est luctator arena.**\n\nSt\u00e4rker ist ein neuer Ringer in der mit gelbem Sand gef\u00fcllten Arena.\n\nOvid ( _Tristia_ 4,6,31) erl\u00e4utert, da\u00df man seine Gef\u00fchle stets auffrischen m\u00fcsse, denn nur wer sich an seinen Zustand noch nicht gew\u00f6hnt habe, werde in der Lage sein, heftig gegen ihn anzuk\u00e4mpfen.\n\n**Fortiter in re, suaviter in modo!**\n\nEnergisch in der Sache (selbst), sanft in der Vorgehensweise!\n\nNach Claudio A(c)quaviva, _Industriae ad curandos animae morbos_ 2,4.\n\n**Fortuna caeca est** siehe **Non enim solum...**\n\n**Fortunae filius, in manu illius plumbum aurum fiebat.**\n\nEin Gl\u00fcckskind, in seiner Hand wurde Blei zu Gold.\n\nPetron, _Satyrica_ 43,7; vgl. Horaz, _Sermones_ 2,6,49.\n\n**Fortuna fortes metuit, ignavos premit.**\n\nDas Schicksal f\u00fcrchtet die Tapferen, die Feigen bedr\u00e4ngt es.\n\nSeneca, _Medea_ 159.\n\n**Fortunam sibi quisque parat.**\n\nSein Gl\u00fcck schafft sich jeder selbst.\n\n(Vgl.: Jeder ist seines Gl\u00fcckes Schmied.)\n\nNach Plautus, _Trinummus_ 363: **Nam sapiens...** Vgl. **Faber est...**\n\n**Fortuna multis dat nimium, satis nulli.**\n\nDas Gl\u00fcck gibt vielen zuviel, aber keinem genug.\n\nMartial, _Epigrammata_ 12,10,2. Vgl. **Nec quisquam...**\n\n**Fortuna non mutat genus.**\n\nDas Gl\u00fcck \u00e4ndert nicht die Art (d. h. die Herkunft des Menschen).\n\nHoraz, _Epodi_ 4,6.\n\n**Fortuna obesse nulli contenta est semel.**\n\nDas Schicksal gibt sich nicht damit zufrieden, jemandem (nur) einmal zu schaden.\n\n(Vgl.: Ein Ungl\u00fcck kommt selten allein.)\n\nPublilius Syrus, _Sententiae_ F 18.\n\n**Fortuna opes auferre, non animum potest.**\n\nDas Schicksal kann Reicht\u00fcmer, doch nicht den Geist rauben.\n\nSeneca, _Medea_ 176.\n\n**Fortunatus et ille, deos qui novit agrestes** siehe **Felix, qui potuit...**\n\n**Fortuna, ut saepe alias, virtutem est secuta.**\n\nDas Gl\u00fcck folgte, wie auch sonst oft, der Tapferkeit.\n\nLivius, _Ab urbe condita_ 4,37,7.\n\n**Fortuna vitrea est: tum, cum splendet, frangitur.**\n\nGl\u00fcck ist wie Glas: Wenn es gl\u00e4nzt, bricht es.\n\n(Vgl.: Gl\u00fcck und Glas, wie leicht bricht das!)\n\nPublilius Syrus, _Sententiae_ F 24.\n\n**Fraus est accipere, quod non possis reddere.**\n\nEs ist Betrug anzunehmen, was du nicht wiedergeben kannst.\n\nPublilius Syrus, _Sententiae_ F 7.\n\n**Frenos imponit linguae conscientia.**\n\nDas Schuldbewu\u00dftsein legt der Zunge Z\u00fcgel an.\n\nPublilius Syrus, _Sententiae_ F 31.\n\n**frondem in silvis non cernere**\n\ndas Laub in den W\u00e4ldern nicht wahrnehmen\n\n(vgl.: den Wald vor lauter B\u00e4umen nicht sehen)\n\nOvid, _Tristia_ 5,4,9 f.: \u00bbNec frondem in silvis, nec aperto mollia prato \/ gramina, nec pleno flumine cernit aquam.\u00ab (\u00bbEr sieht nicht im Walde das Laub und nicht im offenen Gefilde die biegsamen Halme noch das Wasser im tiefen Flu\u00df.\u00ab)\n\n**Fronte capillata, post est occasio calva.**\n\nVorne hat die Gelegenheit einen Haarschopf, hinten ist sie kahl.\n\n(vgl.: die Gelegenheit beim Schopfe packen)\n\n_Disticha Catonis_ 26 B; vgl. Phaedrus, _Fabulae_ 5,8,2.\n\n**Fugiendo in media fata ruitur.**\n\nDurch Flucht ger\u00e4t man (oft) mitten ins Verderben.\n\nLivius, _Ab urbe condita_ 8,24,4.\n\n**Fugit hora** siehe **Vive memor leti!...**\n\n**Fugit inreparabile tempus** siehe **Sed fugit interea...**\n\n**Fuimus Troes, fuit Ilium et ingens \/ gloria Teucrorum.**\n\nTrojaner waren wir (einst), Ilion war (einst) und ebenso der gewaltige Ruhm der Teukrer.\n\nVergil, _Aeneis_ 2,325 f. \u2013 Klage des Priesters Panthus beim Anblick des brennenden Troja. \u2013 Ilion ist eine weitere Bezeichnung f\u00fcr Troja; Teuker war der erste trojanische K\u00f6nig.\n\n**Fumantem nasum vivi ursi ne temptaveris!**\n\nWage dich nicht an die dampfende Nase eines lebenden B\u00e4ren heran!\n\nNach Martial, _Epigrammata_ 6,64,28.\n\n**fumos vanos vendere**\n\nleeren Rauch verkaufen\n\nNach Martial, _Epigrammata_ 4,5,7. \u2013 Als \u00bbRauch\u00ab bezeichnet Martial angebliche Informationen \u00fcber den Kaiser, die einige Gestalten am Hofe verkauften.\n\n**Fumus ignem.**\n\nDer Rauch (zeigt) das Feuer (an).\n\n(Vgl.: Wo Rauch ist, da ist auch Feuer.)\n\nNach Cicero, _Partitiones oratoriae_ 34.\n\n**fundamenta tamquam in aqua ponere**\n\ndie Fundamente gleichsam aufs Wasser setzen\n\n(vgl.: auf Sand bauen)\n\nNach Cicero, _De finibus_ 2,72.\n\n**Fundamentum autem est iustitiae fides.**\n\nGrundlage der Gerechtigkeit jedoch ist Pflichttreue.\n\nCicero, _De officiis_ 1,23.\n\n**Fundum alienum arat, incultum familiarem deserit.**\n\nEr pfl\u00fcgt fremdes Land, das eigene l\u00e4\u00dft er unbebaut.\n\nPlautus, _Asinaria_ 874. \u2013 Die Wendung wird oft auf Ehebrecher \u00fcbertragen.\n\n**Furor fit laesa saepius patientia.**\n\nZur Wut wird (eine) allzu oft beleidigte Geduld.\n\nPublilius Syrus, _Sententiae_ F 13.\n\n**Furorne caecus vos rapit?**\n\nRei\u00dft euch blinde Wut dahin?\n\nNach Horaz, _Epodi_ 7,13.\n\n**furor Teutonicus**\n\nteutonisches W\u00fcten\n\nNach Lukan ( _De bello civili_ 1,255 f.), der so die Germaneneinf\u00e4lle des Jahres 113 v. Chr. beschreibt.\nG\n\n**Gallia est omnis divisa in partes tres...**\n\nDas Gesamtgebiet Gallien ist in drei Teile gegliedert (von denen die Belgier einen bewohnen, einen anderen die Aquitaner, den dritten die V\u00f6lker, die in ihrer eigenen Sprache Kelten, in der unsrigen Gallier genannt werden).\n\nCaesar, _De bello Gallico_ 1,1,1.\n\n**Gallina scripsit.**\n\nEin Huhn hat das geschrieben.\n\n(vgl.: Kr\u00e4henf\u00fc\u00dfe)\n\nPlautus, _Pseudolus_ 30.\n\n**Gallus in suo sterquilino plurimum potest.**\n\nAuf seinem eigenen Misthaufen ist der Hahn der M\u00e4chtigste.\n\n(Vgl.: Der Hahn ist K\u00f6nig auf seinem Miste.)\n\nNach Seneca ( _Apocolocyntosis_ 7,3), der den Kaiser Claudius verspottet: Er habe zwar in Rom Autorit\u00e4t gehabt, im Himmel sei er jedoch machtlos.\n\n**Gaudeamus igitur, iuvenes dum sumus!**\n\nFreuen wir uns also, solange wir noch jung sind!\n\nVers aus dem Studentenlied \u00bbDe brevitate vitae\u00ab, dessen Neufassung von Christian Wilhelm Kindleben stammt (der eigentliche Urheber ist unbekannt); _Gaudeamus_ ist au\u00dferdem der Titel einer Sammlung von Studentenliedern von Josef Victor Scheffel.\n\n**Gaudii comes maeror.**\n\nDer Freude Begleiterin ist die Trauer.\n\nHW: 10256a.\n\n**Geminat peccatum, quem delicti non pudet.**\n\nDie Schuld verdoppelt, wer sich seines Vergehens nicht sch\u00e4mt.\n\nPublilius Syrus, _Sententiae_ G 11. Vgl. **Quem paenitet pecasse...** und **Si paenitet, haud nocet error**.\n\n**genium mero curare**\n\nseinen Schutzgeist mit Wein umsorgen\n\n(d. h.: sich g\u00fctlich tun am Weine)\n\nNach Horaz, _Carmina_ 3,17,14 f. \u2013 Der _genius_ ist der Schutzgeist des Mannes. Umsorgt man ihn mit Wein, tut man sich selbst etwas Gutes.\n\n**genius loci**\n\nder Schutzgeist des Ortes\n\nNach (z. B.) Vergil, _Aeneis_ 5,95.\n\n**gens humana ruit per vetitum nefas** siehe **Audax omnia perpeti...**\n\n**genus irritabile vatum**\n\ndas reizbare Geschlecht der Dichter\n\nHoraz, _Epistulae_ 2,2,102.\n\n**genus mutabile mulierum** siehe **Heia age...**\n\n**Gladiator in arena capit consilium.**\n\nDer Gladiator fa\u00dft seinen Entschlu\u00df (erst) in der Arena.\n\nSeneca ( _Epistulae morales_ 22,1) bezeichnet diese Wendung bereits als Sprichwort: Ein Gladiator habe zwar schon vor dem Kampf eine Strategie, der richtige Zeitpunkt f\u00fcr die einzelnen Schritte lasse sich aber erst im Kampf an der Gestik und Mimik seines Gegners erkennen.\n\n**gloriari alienis bonis** siehe **alienis gloriari bonis**\n\n**Gloria virtutem tamquam umbra sequitur.**\n\nDer Ruhm folgt der Tugend gleichsam wie ein Schatten.\n\nNach Cicero, _Tusculanae disputationes_ 1,109; vgl. Seneca, _Epistulae morales_ 79,13.\n\n**Gradus ad Parnassum**\n\nStufen zum Parna\u00df (Musenberg)\n\nTitel einer Materialiensammlung von Paul Aler, in der Versatzst\u00fccke f\u00fcr das Dichten in lateinischer Sprache dargeboten werden, wie etwa typische Wortverbindungen, Synonyme etc. \u2013 Am Parna\u00df liegt das Apollon-Heiligtum von Delphi. Der Berg wurde als Lieblingssitz Apolls und der Musen angesehen.\n\n**Graeca non leguntur.**\n\nGriechisches wird nicht gelesen.\n\nIm Mittelalter wurden griechische Passagen der Gesetzessammlung _Corpus iuris civilis_ , f\u00fcr die keine \u00dcbersetzungen vorlagen, \u00fcbergangen, da die damaligen Rechtskundigen des Griechischen nicht mehr m\u00e4chtig waren. Vgl. im Gegensatz dazu Cicero, _Pro Archia_ _poeta_ 23.\n\n**Graecia capta ferum victorem cepit et artis \/ intulit agresti Latio.**\n\nGriechenland, das erobert worden war, hat selbst den rauhen Sieger erobert und im b\u00e4uerlichen Latium (Italien) die K\u00fcnste eingef\u00fchrt.\n\nHoraz, _Epistulae_ 2,1,156 f.\n\n**Gram. loquitur, Dia. vera docet, Rhe. verba colorat, Mu. canit, Ar. numerat, Geo. ponderat, As. docet\/colit astra.**\n\nDie Grammatik redet (richtig), die Dialektik lehrt das Wahre, die Rhetorik schm\u00fcckt die Worte, die Musik singt, die Arithmetik z\u00e4hlt, die Geometrie w\u00e4gt ab, die Astronomie lehrt\/k\u00fcmmert sich um (den Lauf der) Gestirne.\n\nHW: 10354. Merkspruch auf die Sieben Freien K\u00fcnste (\u00bbartes liberales\u00ab), von denen die ersten drei das Trivium, die folgenden vier das Quadrivium bilden.\n\n**Grammatici certant, et adhuc sub iudice lis est.**\n\nDar\u00fcber streiten die Grammatiker, und noch ist der Streitfall vor dem Richter.\n\n(Vgl.: Dar\u00fcber streiten sich die Gelehrten.)\n\nHoraz ( _Ars poetica_ 78) erkl\u00e4rt, da\u00df zwar Homer als erster Epiker feststehe, doch wer die erste Elegie verfa\u00dft habe, sei noch nicht erwiesen.\n\n**Grata superveniet, quae non sperabitur, hora** siehe **Inter spem curamque...**\n\n**Gratia gratiam parit.**\n\nFreundlichkeit f\u00fchrt zu Freundlichkeit.\n\n(Vgl.: Wie es in den Wald hineinschallt, so schallt es hinaus.)\n\nNach Sophokles, _Aias_ 522.\n\n**Grave ipsius conscientiae pondus.**\n\nSchwer ist die Last des Gewissens selbst.\n\nCicero, _De natura deorum_ 3,85.\n\n**Gravissimum est imperium consuetudinis.**\n\nDie schlimmste Herrschaft ist die der Gewohnheit.\n\n(vgl.: die Macht der Gewohnheit)\n\nPublilius Syrus, _Sententiae_ G 8.\n\n**Gutta cavat lapidem, consumitur anulus usu.**\n\nDer Tropfen h\u00f6hlt den Stein; ein Ring wird durch den Gebrauch abgenutzt.\n\nOvid ( _Epistulae ex Ponto_ 4,10,5) spielt hier wohl auf Properz, _Elegiae_ 2,25,16 an. Vgl. Lukrez, _De rerum natura_ 4,1286 f., und Tibull 1,4,18; **Ferreus adsiduo consumitur...** , **Percussu crebro...** und **Stillicidi casus...**\nH\n\n**Habeas corpus!**\n\nDu sollst deinen K\u00f6rper haben!\n\nDer Habeas-Corpus-Erla\u00df wird bereits im 14. Jahrhundert in England erw\u00e4hnt. 1679 wurde er englisches Staatsgrundgesetz, sp\u00e4ter auch in die amerikanische Verfassung aufgenommen. Er verf\u00fcgt, da\u00df niemand ohne richterlichen Haftbefehl verhaftet oder ohne gerichtliche Untersuchung in Haft gehalten werden darf.\n\n**Habeas tibi!**\n\nDu sollst deinen Willen haben (erg.: auch wenn er zu nichts Gutem f\u00fchrt).\n\n(z. B.) Plautus, _Stichus_ 615; vgl. 1. Mose 38,23, und Sueton, _Vita divi Iulii_ 1,3.\n\n**Habemus confitentem reum.**\n\nWir haben einen gest\u00e4ndigen Angeklagten.\n\nNach Cicero, _Pro. Q. Ligario_ 2.\n\n**Habemus papam.**\n\nWir haben einen Papst.\n\nPatricius ( _Ritus ecclesiastici SS_. _Romanae ecclesiae_ 1,1,4) erl\u00e4utert, da\u00df mit folgendem Ruf die Neuwahl eines Papstes verk\u00fcndet werde: \u00bbAnnuncio vobis gaudium magnum: Papam habemus.\u00ab (\u00bbIch verk\u00fcnde euch eine gro\u00dfe Freude: Wir haben einen Papst.\u00ab)\n\n**Habent insidias hominis blanditiae mali.**\n\nSchmeicheleien eines B\u00f6sen bergen Fallen.\n\n(Vgl.: Einem Schmeichler und Wolf ist nicht zu trauen.)\n\nPhaedrus, _Fabulae_ 1,19,1.\n\n**Habent sua fata libelli.**\n\nB\u00fcchlein haben (auch) ihr Schicksal.\n\nTerentianus Maurus, _De litteris_ 1286: \u00bbPro captu lectoris habent sua fata libelli.\u00ab (\u00bbJe nach der Aufnahme durch den Leser haben die B\u00fcchlein ihr Schicksal.\u00ab)\n\n**Habet profecto in ventre confidentiam.**\n\nEr hat wahrlich Selbstvertrauen im Bauch.\n\nPlautus, _Captivi_ 812.\n\n**Habet salem.**\n\nEr hat Salz (d. h.: einen scharfen Verstand und Witz).\n\nTerenz, _Eunuchus_ 400.\n\n**Habet silices pectus eius.**\n\nSeine Brust birgt Kieselsteine.\n\n(D. h.: Er hat ein Herz aus Stein.)\n\nNach Ovid, _Tristia_ 3,11,4.\n\n**Habet suum venenum blanda oratio.**\n\nSchmeichelnde Rede hat ihr eigenes Gift.\n\nPublilius Syrus, _Sententiae_ H 12.\n\n**habuisse et nihil habere**\n\ngehabt zu haben und nichts zu haben\n\nPlautus, _Rudens_ 1321: \u00bbMiserum istuc verbum et pessimum est, habuisse, et nihil habere.\u00ab (\u00bbDas ist ein j\u00e4mmerliches Wort und das allerschlimmste: \u203agehabt zu haben\u2039 [im Lateinischen eben ein Wort: _habuisse_ ], wenn man nichts hat.\u00ab)\n\n**Hac urget lupus, hac canis.**\n\nHier lauert der Wolf, dort der Hund.\n\nHoraz ( _Sermones_ 2,2,64) f\u00fchrt die Wendung als Sprichwort an. Vgl. **Incidit in Scyllam...** und **inter canem et lupum.**\n\n**Haec a te non multum abludit imago.**\n\nDieses Bild ist dir nicht allzu un\u00e4hnlich.\n\nHoraz, _Sermones_ 2,3,320.\n\n**Haec ornamenta sunt mea.**\n\nDiese sind meine Zierde.\n\nLaut Valerius Maximus ( _Facta et dicta memorabilia_ 4,4) zeigte einst eine reiche Freundin der Gracchenmutter Cornelia ihren Schmuck. Statt ihrerseits Geschmeide hervorzuholen, verwies Cornelia mit den obigen Worten auf ihre Kinder.\n\n**Haec res et iungit, iunctos et servat amicos.**\n\nDies verbindet und h\u00e4lt Freunde verbunden.\n\nHoraz ( _Sermones_ 1,3,54) beschreibt schmeichlerische Nachsicht unter Freunden: Ein schielender Freund etwa \u203ablinzele nur verliebt\u2039; ein allzu grober werde \u203akraftvoll\u2039 genannt.\n\n**Haeres in medullis.**\n\nDu h\u00e4ngst (mir) im Mark.\n\n(D. h.: Du bist in meinem Herzen.)\n\nCicero, _Ad familiares_ 15,16,2.\n\n**Haeret lateri letalis harundo.**\n\nEs haftet in der Seite der t\u00f6dliche Pfeil.\n\nVergil, _Aeneis_ 4,73.\n\n**Ha, sine me fati non meminisse mei!**\n\nAch, la\u00df mich nicht an mein Schicksal erinnern!\n\nOvid, _Tristia_ 4,4,40.\n\n**Hanc veniam petimusque damusque vicissim.**\n\nWir bitten um diese Nachsicht und gew\u00e4hren sie unsererseits.\n\nHoraz ( _Ars poetica_ 11) bittet um das Zugest\u00e4ndnis dichterischer Freiheit, die jedoch nicht so weit gehen d\u00fcrfe, da\u00df der K\u00fcnstler Unvereinbares kombiniere. Er vergleicht den Dichter mit dem Maler, der auch nicht den Kopf einer sch\u00f6nen Frau auf den K\u00f6rper eines h\u00e4\u00dflichen schwarzen Fisches setzen d\u00fcrfe.\n\n**Hannibal ad portas!**\n\nHannibal ist vor den Toren!\n\nOft f\u00e4lschlich als \u00bbHannibal ante portas\u00ab zitiert. \u2013 Cicero ( _De finibus_ 4,22 und _Orationes Philippicae_ 1,11) erw\u00e4hnt Hannibals Vorsto\u00df bis an die Tore Roms im Jahre 211 stellvertretend f\u00fcr h\u00f6chst gef\u00e4hrliche Situationen. Bei Livius ( _Ab urbe condita_ 23,16,2) wird die Wendung im eigentlichen historischen Sinne verwendet. Vgl. **Vincere scis...**\n\n**Has litteras gallina scripsit** siehe **Gallina scripsit**\n\n**hastas abicere**\n\ndie Lanzen wegwerfen\n\n(vgl.: die Flinte ins Korn werfen)\n\nNach Cicero, _Pro L. Murena_ 45.\n\n**hastas iacere, quas alius ministrat**\n\nLanzen schleudern, die ein anderer herreicht\n\n(vgl.: B\u00fcchsenspanner)\n\nVgl. Cicero, _Topica_ 65. Vgl. **ab alio amentatas...**\n\n**Haud aequum facit, \/ qui, quod didicit, id dediscit.**\n\nNicht richtig handelt, wer, was er gelernt, (wieder) verlernt.\n\nPlautus, _Amphitruo_ 687 f.\n\n**Haud est nocens, quicumque non sponte est nocens.**\n\nEs s\u00fcndigt nicht, wer nicht absichtlich s\u00fcndigt.\n\nSeneca, _Hercules Oetaeus_ 886.\n\n**Haud est virile terga fortunae dare.**\n\nEs ist nicht m\u00e4nnlich, dem Schicksal den R\u00fccken zuzukehren.\n\nSeneca, _Oedipus_ 86.\n\n**Haud stulte sapis.**\n\nDu bist nicht dumm in deiner Weisheit.\n\n(D. h.: Du bist nicht auf den Kopf gefallen.)\n\nTerenz, _Heauton timorumenos_ 323.\n\n**Haustus aquae mihi nectar erit.**\n\nEin Schluck Wasser wird Nektar f\u00fcr mich sein.\n\nOvid, _Metamorphoses_ 6,356.\n\n**Hectora quis nosset, felix si Troia fuisset?**\n\n**Publica virtutis per mala facta via est.**\n\nWer w\u00fcrde Hektor kennen, wenn Troja gl\u00fccklich gewesen w\u00e4re? Der Weg seiner Tapferkeit wurde erst durch sein Ungl\u00fcck bekannt.\n\nOvid, _Tristia_ 4,3,75 f.\n\n**Heia age , rumpe moras! Varium et mutabile semper \/ femina.**\n\nWohlan, brich das Zaudern! Ein wankelm\u00fctiges und ver\u00e4nderliches Wesen ist stets die Frau.\n\nVergil, _Aeneis_ 4,569 f. \u2013 Aeneas wird im Traum davor gewarnt, l\u00e4nger bei Dido zu verweilen.\n\n**Heredis fletus sub persona risus est.**\n\nDas Weinen des Erbens erweist sich unter der Maske als Lachen.\n\n(Vgl.: Des Erben Weinen ist heimlich Lachen.)\n\nPublilius Syrus, _Sententiae_ H 19.\n\n**Hereditatem inhiat quasi esuriens lupus.**\n\nEr giert nach dem Erbe wie ein verhungernder Wolf.\n\nPlautus, _Stichus_ 605.\n\n**Heu, melior quanto sors tua sorte mea!**\n\nAch, um wieviel besser ist dein Los als meines!\n\nOvid, _Amores_ 1,6,46.\n\n**Heu miseri, qui bella gerunt!**\n\nAch, die Elenden, die Kriege f\u00fchren!\n\nLukan, _De bello civili_ 4,382.\n\n**Heu, nihil invitis fas quemquam fidere divis!**\n\nAch, es ist niemandem erlaubt, auf die unwilligen G\u00f6tter zu vertrauen!\n\nVergil, _Aeneis_ 2,402.\n\n**Heu, patior telis vulnera facta meis!**\n\nAch, ich leide an den Wunden meiner eigenen Geschosse!\n\nOvid, _Heroides_ 2,48. \u2013 Phyllis gr\u00e4mt sich, ihrem Mann Demophoon selbst das Boot f\u00fcr seine Abfahrt bereitgestellt zu haben, da sie jetzt seine Untreue f\u00fcrchtet.\n\n**Heu pietas , heu prisca fides!**\n\nAch Fr\u00f6mmigkeit, ach altehrw\u00fcrdige Treue!\n\nVergil, _Aeneis_ 6,878. \u2013 Anchises gedenkt mit diesen Worten des tugendhaften Marcellus.\n\n**Hic finis fandi.**\n\nDas war das Ende des Redens.\n\n(D. h.: Das waren die letzten Worte von Jupiters Rede.)\n\nVergil, _Aeneis_ 10,116.\n\n**Hic locus est, partis ubi se via findit in ambas.**\n\nHier ist die Stelle, wo sich der Weg in zwei Richtungen trennt.\n\nVergil, _Aeneis_ 6,540. \u2013 Am Scheideweg zwischen Elysium und Tartarus mahnt die Sibylle, die Aeneas durch die Unterwelt leitet, den Helden weiterzugehen.\n\n**Hic niger est, hunc tu, Romane, caveto!**\n\nDieser ist schwarz (in der Seele), den, R\u00f6mer, meide!\n\nHoraz ( _Sermones_ 1,4,85) warnt so vor untreuen Freunden, Schw\u00e4tzern und anderen \u00fcblen Gesellen.\n\n**Hic Rhodus, hic salta!**\n\nHier ist Rhodus, hier springe!\n\nNach Aesop, _Fabulae_ 33 Hausrath: Ein F\u00fcnfk\u00e4mpfer r\u00fchmt sich, zu Rhodos einen gro\u00dfen Sprung gemacht zu haben. Er wird mit diesen Worten aufgefordert, den Sprung zum Beweis an Ort und Stelle zu wiederholen.\n\n**Hic solet eventus facta nefanda sequi.**\n\nEin solches Resultat pflegt ruchlosen Taten zu folgen.\n\nOvid, _Heroides_ 14,16.\n\n**Hic ver adsiduum atque alienis mensibus aestas.**\n\nHier ist ewig Fr\u00fchling und auch in ungew\u00f6hnlichen Monaten (d. h.: im Winter) Sommer.\n\nVergil, _Georgica_ 2,149.\n\n**Hinc illae lacrumae !**\n\nDaher diese Tr\u00e4nen!\n\nTerenz, _Andria_ 126. \u2013 Bei der Bestattungsfeier f\u00fcr die Nachbarin Chrysis begegnet der alte Simo deren \u00e4u\u00dferst ansehnlicher Schwester Glycerium; nun erkennt er, da\u00df die Tr\u00e4nen seines Sohnes Pamphilus nicht aus dessen vermeintlicher Trauer um Chrysis herr\u00fchren, sondern vielmehr der Liebe zu ihrer sch\u00f6nen Schwester Glycerium entspringen. \u2013 Bei Cicero ( _Pro Caelio_ 61) und Horaz ( _Epistulae_ 1,19,41) wird die Wendung bereits im heutigen Sinne stellvertretend f\u00fcr Aha-Erlebnisse jeglicher Art verwendet. Juvenal ( _Saturae_ 1,168) erweitert die Wendung: \u00bbInde irae et lacrimae\u00ab (\u00bbDaher der Zorn und die Tr\u00e4nen\u00ab).\n\n**Hinc omne principium, huc refer exitum!**\n\nVon hier nimm den Anfang, hierauf bezieh das Ende!\n\nHoraz, _Carmina_ 3,6,6.\n\n**hircum olere**\n\nnach Bocksgeruch stinken; stinken wie ein Bock\n\n(z. B.) nach Horaz, _Sermones_ 1,2,27 und 1,4,92.\n\n**Hirundines aestivo tempore praesto sunt, frigore pulsae recedunt... Item falsi amici.**\n\nBei Sommerwetter sind die Schwalben zugegen; von der K\u00e4lte vertrieben, ziehen sie sich zur\u00fcck. So auch (handeln) falsche Freunde.\n\nDer Auctor ad Herennium (4,61) f\u00fchrt diese Regel nicht ihres Inhalts wegen an, sondern als Beispiel f\u00fcr einen Vergleich, dessen einmal aufgenommene Metaphorik zum gesamten Sachverhalt pa\u00dft und bis zum Ende durchgehalten wird. Vgl. Plautus, _Stichus_ 520\u2013522.\n\n**Hoc age!**\n\nTu das!\n\nHoraz, _Epistulae_ 1,18,88.\n\n**Hoc decet, hoc leges duxque pudorque iubent.**\n\nSo geh\u00f6rt es sich, so befehlen es die Gesetze, der Kaiser und das Schamgef\u00fchl.\n\nOvid, _Ars amatoria_ 3,614.\n\n**Hoc erat in votis: modus agri non ita magnus.**\n\nDas war einer meiner W\u00fcnsche: ein St\u00fcck Land, nicht so gro\u00df.\n\nHoraz ( _Sermones_ 2,6,1) dankt seinem G\u00f6nner Maecenas f\u00fcr das Landgut, das er von ihm erhielt. Vgl. **Immoritur studiis...**\n\n**Hoc feci , dum licuit, intermisi, quoad non licuit.**\n\nDas tat ich, solange es erlaubt war, und unterlie\u00df es, solange es nicht erlaubt war.\n\nCicero, _Orationes Philippicae_ 3,33.\n\n**Hoc habet!**\n\nDas hat's!\n\n(D. h.: Das sitzt.)\n\nPlautus, _Mostellaria_ 715. \u2013 Mit diesem Ausspruch verk\u00fcndeten die Gladiatoren, da\u00df sie einen Gegner t\u00f6dlich getroffen haben.\n\n**Hoc Herculi, Iovis satu edito, potuit fortasse contingere, nobis non item.**\n\nDas k\u00f6nnte vielleicht einem Hercules, aus Jupiters Saat hervorgekommen, gelingen, uns (aber) nicht in gleicher Weise.\n\nCicero ( _De officiis_ 1,118) spielt hier auf Hercules am Scheidewege an.\n\n**Hoc nobis vitium maximum est, cum amamus, tum perimus.**\n\nDas ist unser gr\u00f6\u00dfter Fehler: Wenn wir lieben, gehen wir daran zugrunde.\n\nPlautus ( _Truculentus_ 190) beschreibt so die Eigenart von Liebenden, s\u00e4mtliche Schmeicheleien begierig in sich aufzusaugen, ohne deren Wahrheitsgehalt zu hinterfragen.\n\n**Hoc opus, hoc studium parvi properemus et ampli.**\n\nDiesem Werk, dieser Besch\u00e4ftigung wollen wir uns eilends zuwenden, wir Nichtigen und Wichtigen.\n\nHoraz, _Epistulae_ 1,3,28. \u2013 Gemeint ist die Philosophie.\n\n**Hoc plus verere, quod licet tantum tibi.**\n\nF\u00fcrchte du viel mehr, da\u00df dir soviel erlaubt.\n\nSeneca ( _Octavia_ 450) warnt hier Nero vor \u00dcbermut. Dieser hatte, nachdem er Claudius zum Gott geweiht hatte, behauptet, er m\u00fcsse sich nicht um G\u00f6tter k\u00fcmmern: Er schaffe sie selbst.\n\n**Hoc sensus ostendit, animus credidit.**\n\nDer Sinn zeigte es an; der Geist glaubte es.\n\nSeneca, _Epistulae morales_ 117,13.\n\n**Hoc signo vinces.**\n\nUnter diesem Zeichen wirst du siegen.\n\nNach Eusebius ( _Vita Constantini_ 1,28), \u00fcber die Kreuzeserscheinung vor der Schlacht an der Milvischen Br\u00fccke. Andere lateinische \u00dcbersetzungen des griechischen Originals sind \u00bbHoc signo victor eris\u00ab, \u00bbIn hoc signo vinces\u00ab und \u00bbHac vince\u00ab.\n\n**Hoc volo, sic iubeo! Sit pro ratione voluntas!**\n\nDas will ich, so befehle ich! Statt einer Begr\u00fcndung gelte mein Wille!\n\nJuvenal ( _Saturae_ 6,223) f\u00fchrt in seiner \u00bbWeibersatire\u00ab diesen Befehl als typisch f\u00fcr herrische Frauen an. Im obigen Fall fordert eine Ehefrau von ihrem Gatten die sofortige Hinrichtung eines Sklaven ohne weitere Untersuchung seiner Schuld.\n\n**Hominem experiri multa paupertas iubet.**\n\nArmut l\u00e4\u00dft den Menschen vieles versuchen.\n\n(Vgl.: Not macht erfinderisch.)\n\nPublilius Syrus, _Sententiae_ H 8.\n\n**hominem involutum aestimare**\n\neinen eingewickelten Menschen einsch\u00e4tzen\n\n(vgl.: die Katze im Sack kaufen)\n\nNach Seneca, _Epistulae morales_ 80,9.\n\n**Hominem te esse memento!** siehe **Respice post te...**\n\n**Homines amplius oculis quam auribus credunt... longum iter est per praecepta, breve et efficax per exempla.**\n\nDie Menschen glauben eher ihren Augen als ihren Ohren. \u00dcber Vorschriften f\u00fchrt ein langer Weg, ein kurzer und wirkungsvoller \u00fcber Beispiele.\n\nSeneca, _Epistulae morales_ 6,5.\n\n**Homines dum docent , discunt.**\n\nW\u00e4hrend die Menschen lehren, lernen sie.\n\nSeneca ( _Epistulae morales_ 7,8) r\u00e4t dazu, sich sowohl mit Menschen zu umgeben, die einen zu bessern verm\u00f6gen, als auch selbst zu versuchen, andere zu bessern. \u2013 Oft zitiert als: \u00bbDocendo discimus.\u00ab (\u00bbIndem wir lehren, lernen wir.\u00ab)\n\n**Homines quo plura habent, eo cupiunt ampliora.**\n\nJe mehr die Menschen haben, desto mehr w\u00fcnschen sie.\n\nVgl. Iustinus ( _Historiae Philippicae_ 6,1,1), der diese Eigenschaft besonders den Spartanern zuschreibt.\n\n**Homines sumus, non dii** siehe **Nemo nostrum...**\n\n**Homini necesse est mori** siehe **De homine enim...**\n\n**Hominum causa omne ius constitutum est.**\n\nAlles Recht ist um der Menschen willen gesetzt.\n\nNach _Digesta_ 1,5,2.\n\n**Homo bulla.**\n\nDer Mensch ist eine Luftblase (d. h.: er zerplatzt so leicht wie eine Seifenblase).\n\nVarro ( _De re rustica_ 1,1,1) bezeichnet die Wendung als Sprichwort, das die Verg\u00e4nglichkeit des Menschen verdeutliche; vgl. Petron, _Satyrica_ 42,3, und Persius, _Saturae_ 2,10.\n\n**Homo doctus in se semper divitias habet.**\n\nDer Gelehrte hat an sich selbst immer einen Schatz.\n\nPhaedrus, _Fabulae_ 4,23,1. Vgl. **Omnia mecum...**\n\n**Homo homini lupus est.**\n\nDer Mensch ist dem Menschen ein Wolf.\n\nBei Plautus ( _Asinaria_ 495) begr\u00fcndet so ein Herr seine Weigerung, einem Sklaven Geld zu leihen: \u00bbLupus est homo homini, non homo, quom qualis sit, non novit.\u00ab (\u00bbDer Mensch ist dem Menschen ein Wolf, kein Mensch, wenn er nicht wei\u00df, welcher Art [sein Gegen\u00fcber] ist.\u00ab) \u2013 Maxime von Thomas Hobbes in seinem _Leviathan_.\n\n**Homo ludens**\n\nder spielende Mensch\n\nTitel eines 1938 erschienenen Buches des niederl\u00e4ndischen Kulturhistorikers Johan Huizinga, der das Spiel als Grundkategorie menschlichen Verhaltens und menschlicher Entwicklung und somit als kulturbildenden Faktor betrachtet.\n\n**homo mensura** siehe **Omnium rerum homo mensura est**\n\n**homo novus**\n\nein neuer Mann\n\n(d. h.: ein Aufsteiger)\n\n(z. B.) Cicero, _De re publica_ 1,1 und _De officiis_ 1,138. \u2013 Mit diesem Terminus wurden r\u00f6mische B\u00fcrger aus nichtadeligen Familien bezeichnet, denen dennoch der Aufstieg in kurulische \u00c4mter und in die Nobilit\u00e4t gelungen war.\n\n**Homo proponit, sed deus disponit.**\n\nDer Mensch schl\u00e4gt vor, aber Gott ordnet an.\n\n(Vgl.: Der Mensch denkt und Gott lenkt.)\n\nThomas a Kempis, _Imitatio Christi_ 1,19,9; vgl. Spr\u00fcche 16,9.\n\n**Homo semper aliud, fortuna aliud cogitat.**\n\nDer Mensch denkt immer anders als das Schicksal.\n\nPublilius Syrus, _Sententiae_ H 14.\n\n**Homo sum, humani nil a me alienum puto.**\n\nIch bin ein Mensch, nichts Menschliches, glaube ich, ist mir fremd.\n\nTerenz, _Heauton timorumenos_ 77. \u2013 Dieser Ausdruck des Verst\u00e4ndnisses f\u00fcr menschliche Schw\u00e4chen wird bereits in der Antike zitiert, so z. B. bei Cicero ( _De officiis_ 1,30) und bei Seneca ( _Epistulae morales_ 95,53).\n\n**Homo totiens moritur, quotiens amittit suos.**\n\nDer Mensch stirbt, sooft er Nahestehende verliert.\n\nPublilius Syrus, _Sententiae_ H 13.\n\n**Homunculi quanti sunt, cum recogito!**\n\nWie klein die Menschlein sind, wenn ich's recht bedenke!\n\nPlautus, _Captivi_ 51.\n\n**Honesta mors turpi vita potior.**\n\nEin ehrlicher Tod (ist) besser als ein sch\u00e4ndliches Leben.\n\nTacitus, _Agricola_ 33,6.\n\n**Honeste parcas improbo, ut parcas probo.**\n\nZu Recht darf man einen Schuft schonen, um (gleichzeitig auch) einen Anst\u00e4ndigen zu schonen.\n\nPublilius Syrus, _Sententiae_ H 22.\n\n**Honos alit artes.**\n\nEhre n\u00e4hrt die K\u00fcnste.\n\nCicero, _Tusculanae disputationes_ 1,4.\n\n**Honos est praemium virtutis.**\n\nEhre ist der Lohn der Tugend.\n\nNach Cicero, _Brutus_ 281.\n\n**Honos habet onus.**\n\nEhre hat B\u00fcrde.\n\n(Vgl.: Ehren beschweren; W\u00fcrden sind B\u00fcrden.)\n\nVgl. Varro ( _De lingua latina_ 5,73), der _honos_ (\u203aEhre\u2039) von _onus_ (\u203aLast\u2039) ableitet, denn alles Ehrenvolle sei immer auch eine gro\u00dfe Last. Vgl. auch HW: 11126.\n\n**Horae \/ momento cita mors venit aut victoria laeta.**\n\nIm Verlauf einer Stunde kommt der rasche Tod oder der frohe Sieg. (D. h.: Alles ist offen.)\n\nHoraz, _Sermones_ 1,1,7 f.\n\n**Horrea formicae tendunt ad inania numquam.**\n\nIn leere Speicher begeben sich Ameisen nie.\n\nOvid ( _Tristia_ 1,9,9) stellt in seiner Einsamkeit einen niederschmetternden Vergleich auf: So wenig wie sich Ameisen in leere Speicher begeben, so wenig zieht es einstmalige Freunde zu dem, der alles verloren hat. Vgl.: **Donec eris sospes...**\n\n**Horresco referens.**\n\nIch schaudere beim Erz\u00e4hlen.\n\nVergil, _Aeneis_ 2,204.\n\n**horribile dictu, horribile visu**\n\nschrecklich zu sagen, schrecklich anzusehen\n\nDie Wendung ist antik nicht belegt. Vgl. z. B. Vergil, _Aeneis_ 3,621, und HWS: 37287.\n\n**horror vacui**\n\nder Schrecken der Leere\n\nRabelais, _Gargantua et Pantagruel_ 1,5: \u00bbNatura abhorret vacuum.\u00ab (\u00bbDie Natur schreckt vor der Leere zur\u00fcck.\u00ab) \u2013 In der Antike diente die Vorstellung einer \u00bbScheu\u00ab der Natur vor dem Leeren als Erkl\u00e4rungsansatz f\u00fcr Saugwirkungen.\n\n**Hos ego versiculos feci, tulit alter honores.**\n\nIch habe diese Verslein gedichtet, ein anderer die Ehre davongetragen.\n\nVergil (Fragment 257 B B\u00fccheler-Riese) vergleicht sich anschlie\u00dfend mit den Bienen, die auch arbeiteten, doch nicht f\u00fcr sich, sondern f\u00fcr den Imker: \u00bbSic vos non vobis.\u00ab (\u00bbSo [arbeitet auch] ihr, doch nicht f\u00fcr euch.\u00ab)\n\n**Hostis est uxor, invita quae ad virum nuptum datur.**\n\n(Wie) ein Feind ist eine Gattin, die gegen ihren Willen dem Mann gegeben wird.\n\nPlautus, _Stichus_ 140.\n\n**Huic \/ quantam fenestram ad nequitiem patefeceris!**\n\nWelch weites Fenster hast du ihm zu seiner Schlechtigkeit ge\u00f6ffnet!\n\n(vgl.: T\u00fcr und Tor \u00f6ffnen)\n\nTerenz, _Heauton timorumenos_ 480 f.\n\n**Humanas actiones non ridere, non lugere neque detestari, sed intellegere curavi.**\n\nIch habe mich bem\u00fcht, die Handlungen der Menschen nicht zu bel\u00e4cheln noch zu beweinen oder abzulehnen, sondern sie zu verstehen.\n\nNach Spinoza, _Tractatus politicus_ 1,4.\n\n**Humanum amare est, humanum autem ignoscere est.**\n\nEs ist menschlich zu lieben, aber es ist auch menschlich zu verzeihen.\n\nPlautus, _Mercator_ 319.\n\n**Humiles laborant, ubi potentes dissident.**\n\nDer kleine Mann leidet, wo M\u00e4chtige sich streiten.\n\nPhaedrus, _Fabulae_ 1,30,1.\n\n**Hunc hominem decet auro expendi.**\n\nDiesen Menschen sollte man mit Gold aufwiegen.\n\nPlautus, _Bacchides_ 640.\nI\n\n**Iacet omnis virtus, fama nisi late patet.**\n\nAlle Tugend liegt darnieder, wenn nicht der Ruf sie weithin bekannt macht.\n\nPublilius Syrus, _Sententiae_ I 36.\n\n**Iam dudum animus est in patinis.**\n\nSchon lange ist mein Geist in den Sch\u00fcsseln.\n\nTerenz, _Eunuchus_ 816. \u2013 Antwort des Sklaven Sanga auf die Aufforderung, sich, wie es sich f\u00fcr einen Soldaten zur rechten Zeit zieme, wieder auf Heim und Herd einzustellen. Heute meist verwendet als: \u00bbIch bin nicht bei der Sache.\u00ab\n\n**Iam finis erat.**\n\nSchon war das Ende da.\n\nVergil ( _Aeneis_ 1,223) \u00fcber das Ende des ersten Mahles der Aeneaden auf libyschem Grund.\n\n**Iam labor in fine est.**\n\nSchon ist die M\u00fche beendet.\n\nOvid, _Metamorphoses_ 13,373.\n\n**Iam proximus ardet \/ Ucalegon!**\n\nSchon brennt der Nachbar Ukalegon!\n\nVergil, _Aeneis_ 2,311 f. \u2013 Brandszenerie w\u00e4hrend der Eroberung Trojas: Aeneas schwebt in h\u00f6chster Gefahr, da bereits das Haus seines Nachbarn brennt.\n\n**Iamque non pugna, sed caedes erat.**\n\nSchon war es kein Kampf mehr, sondern ein Blutbad.\n\nCurtius Rufus, _Historia Alexandri Magni_ 4,15,32.\n\n**Iamque quiescebant voces hominumque canumque.**\n\nSchon schwiegen die Stimmen der Menschen und der Hunde.\n\nOvid ( _Tristia_ 1,3,27) beschreibt so eindringlich die Stimmung in Rom w\u00e4hrend seiner letzten Nacht vor der Abreise in die Verbannung.\n\n**Iam scies patrem tuum mercedes perdidisse.**\n\nDu wirst schon merken, da\u00df dein Vater das (Lehr)geld umsonst ausgegeben hat.\n\nPetron ( _Satyrica_ 58,8) spricht von verschwendetem Lehrgeld, wenn jemand nicht das leistet, was man von ihm erwarten k\u00f6nnte.\n\n**Iam seges est, ubi Troia fuit.**\n\nSchon sind Felder, wo einst Troja war.\n\nOvid, _Heroides_ 1,53.\n\n**Iam tempus illi fecit aerumnas leves.**\n\nSchon hat ihm die Zeit den Kummer erleichtert.\n\n(Vgl.: Zeit heilt Wunden.)\n\nSeneca, _Thyestes_ 305.\n\n**Ibi fas, ubi proxima merces.**\n\nDort (sehen sie) das Rechte, wo sofortige Belohnung (lockt).\n\nLukan ( _De bello civili_ 10,408) tadelt die K\u00e4uflichkeit der Soldaten.\n\n**Id demum est homini turpe, quod meruit pati.**\n\nGerade das ist f\u00fcr den Menschen eine Schmach, was zu erleiden er verdient hat.\n\nBei Phaedrus ( _Fabulae_ 3,11,7) wehrt sich so ein Eunuch gegen den Vorwurf, kein Geschlecht zu besitzen. Das liege schlie\u00dflich nicht in seiner Verantwortung, sondern sei vom Schicksal bestimmt.\n\n**Idem velle atque idem nolle, ea demum firma amicitia est.**\n\nDasselbe zu wollen und dasselbe nicht zu wollen, gerade darin liegt best\u00e4ndige Freundschaft.\n\nSallust, _Bellum Catilinarium_ 20,4.\n\n**Id enim maxime quemque decet, quod est cuiusque maxime suum.**\n\nDas n\u00e4mlich ziemt sich f\u00fcr jeden am meisten, was seinem Wesen am meisten entspricht.\n\nCicero, _De officiis_ 1,113.\n\n**Id facere laus est, quod decet, non quod licet.**\n\nEs ist l\u00f6blich zu tun, was sich ziehmt, und nicht, was erlaubt ist.\n\nSeneca, _Octavia_ 454.\n\n**Ieiunus raro stomachus vulgaria temnit.**\n\nEin selten n\u00fcchterner Magen verachtet Gew\u00f6hnliches.\n\nHoraz, _Sermones_ 2,2,38.\n\n**Ignavis precibus fortuna repugnat.**\n\nUnt\u00e4tigen (blo\u00dfen) Gebeten widersteht das Gl\u00fcck.\n\nOvid, _Metamorphoses_ 8,73.\n\n**Ignavis semper feriae sunt.**\n\nFaule haben st\u00e4ndig Ferien.\n\nNach Theokrit, _Idyll_ 15,26.\n\n**Ignavum scelus est tantum fuga.**\n\nEin feiges Verbrechen ist die blo\u00dfe Flucht.\n\nLukan, _De bello civili_ 9,283.\n\n**Ignavus servos rector facit esse protervos.**\n\nEin fauler Aufseher macht freche Sklaven.\n\n(Vgl.: Wie der Herr, so's G'scherr.)\n\nHW: 11379. Vgl. **Componitur orbis...** und **Plane qualis...**\n\n**ignem gladio scrutare**\n\nmit dem Schwerte in der Glut herumw\u00fchlen\n\nHoraz, _Sermones_ 2,3,276. \u2013 Bei Hieronymus ( _Adversus Rufinum_ 3,39) findet sich eine \u00e4hnliche Warnung: \u00bbIgnem gladio ne fodias!\u00ab (\u00bbStochere nicht mit dem Schwert im Feuer herum!\u00ab) Beide Stellen gehen vermutlich auf eine pythagor\u00e4ische Warnung zur\u00fcck, unbeherrschte Menschen nicht zu Gewalttaten anzuregen.\n\n**ignem igne incitare**\n\nFeuer mit Feuer anfachen\n\nSeneca ( _De ira_ 2,20,2) r\u00e4t hier mit Platon ( _Gesetze_ 2,666a) davon ab, feurige Temperamente noch durch den Genu\u00df von Wein anzustacheln.\n\n**Ignis aurum probat, miseria fortes viros.**\n\nFeuer pr\u00fcft Gold, Not tapfere M\u00e4nner.\n\nSeneca, _De providentia_ 5,10.\n\n**Ignis quo clarior fulsit, citius extinguitur.**\n\nJe heller das Feuer scheint, desto leichter ist es gel\u00f6scht.\n\nSeneca, _Ad Marciam de consolatione_ 23,4.\n\n**Ignoramus et ignorabimus.**\n\nWir wissen es nicht und werden es nicht wissen.\n\nNach Emil Du Bois-Reymond ( _\u00dcber die Grenzen des Naturerkennens_ ; Vortrag vom 14. August 1872): Der Naturforscher sei nicht nur gewohnt, \u00bbgegen\u00fcber den R\u00e4thseln der K\u00f6rperwelt\u00ab zu verk\u00fcnden: \u00bbIgnoramus.\u00ab (\u00bbWir wissen es nicht.\u00ab) Vielmehr m\u00fcsse er \u00bbgegen\u00fcber dem R\u00e4thsel aber, was Kraft und Materie seien... ein f\u00fcr allemal zu dem viel schwerer abzugebenden Wahrspruch sich entschlie\u00dfen: \u203aIgnorabimus.\u2039 (\u203aWir werden es nicht wissen.\u2039)\u00ab. _Ignoramus_ war bereits Titel einer 1615 erschienenen Kom\u00f6die von George Ruggle.\n\n**Ignorantia facti excusat, ignorantia iuris nocet.**\n\nUnkenntnis von Tatsachen ist eine (g\u00fcltige) Entschuldigung, Unkenntnis des Rechts (aber) schadet.\n\nNach _Digesta_ 22,6,9, praefatio.\n\n**Ignoratio futurorum malorum utilior est quam scientia.**\n\nDie Unkenntnis zuk\u00fcnftiger \u00dcbel ist n\u00fctzlicher als das Wissen darum.\n\nCicero, _De divinatione_ 2,23.\n\n**Ignoscas aliis multa, tibi nihil!**\n\nAnderen magst du viel verzeihen, dir nichts!\n\nAusonius, _Septem sapientium sententiae_ 18.\n\n**Ignosce fatenti!**\n\nVerzeihe dem Gest\u00e4ndigen!\n\n(Vgl.: Wo Reue ist, da ist auch Gnade.)\n\nTibull, _Elegiae_ 1,6,29.\n\n**Ignoti nulla cupido** siehe **Quod latet...**\n\n**Iliacos intra muros peccatur et extra.**\n\nInnerhalb und au\u00dferhalb der Mauern Ilions (Trojas) wird ges\u00fcndigt.\n\nHoraz, _Epistulae_ 1,2,16. \u2013 W\u00e4hrend der Belagerung Trojas kommt es bei beiden Parteien zum Streit, z. B. zwischen Antenor und Paris sowie zwischen Achilles und Agamemnon.\n\n**Illa tamen nunc me composuisse piget.**\n\nDennoch verdrie\u00dft es mich nun, es gedichtet zu haben.\n\nOvid, _Tristia_ 5,1,8.\n\n**Illa tamquam cycnea fuit divini hominis vox et oratio.**\n\nDiese Rede war auch gleichsam der Schwanengesang eines erhabenen Mannes.\n\nCicero ( _De oratore_ 3,6) berichtet so von Crassus' letzter Rede.\n\n**Ille crucem sceleris pretium tulit, hic diadema.**\n\nDieser erhielt als Lohn f\u00fcr sein Verbrechen das Kreuz, jener eine Krone.\n\n(D. h.: Die Menschen werden nach unterschiedlichen Ma\u00dfst\u00e4ben gerichtet.)\n\nJuvenal, _Saturae_ 13,105. Vgl. **Duo cum faciunt idem, non est idem**.\n\n**Ille potest auctore suam defendere vocem.**\n\nEr kann, was er sagt, verteidigen durch die Autorit\u00e4t (dessen, der es ihm eingab).\n\nOvid, _Metamorphoses_ 13,218.\n\n**Ille potest vacuo furari litore harenas,**\n\nUxorem stulti siquis amare potest.\n\nJener kann Sand vom verlassenen Strand stehlen, der die Frau eines Dummen lieben kann.\n\n(D. h.: So leicht ist es, einen Dummen zu betr\u00fcgen.)\n\nOvid, _Amores_ , 2,19,45 f.\n\n**Ille terrarum mihi praeter omnis \/ angulus ridet.**\n\nJener Winkel l\u00e4chelt mir vor allen (anderen) auf der Erde zu.\n\nHoraz ( _Carmina_ 2,6,13 f.) \u00fcber Tarent.\n\n**Illicita amantur, excidit quidquid licet.**\n\nWir lieben, was nicht erlaubt ist; was erlaubt ist, f\u00e4llt aus (unserer Gunst).\n\nSeneca, _Hercules Oetaeus_ 357.\n\n**Illi robur et aes triplex \/ circa pectus erat.**\n\nJener (der erste Seefahrer) hatte Eichenholz und dreifaches Erz um seine Brust.\n\nHoraz ( _Carmina_ 1,3,9 f.) w\u00fcnscht dem nach Griechenland aufbrechenden Vergil eine gute Reise und erinnert an die Tollk\u00fchnheit der ersten Seefahrer.\n\n**Illo neminem reperire potest sui similiorem.**\n\nEr kann keinen finden, der ihm \u00e4hnlicher w\u00e4re als jener.\n\n(Vgl.: Er gleicht ihm wie ein Ei dem anderen.)\n\nNach Cicero, _Orationes Philippicae_ 3,31.\n\n**illotis manibus**\n\nmit ungewaschenen H\u00e4nden\n\nPlautus, _Poenulus_ 316.\n\n**illuc, unde negant redire quemquem**\n\ndorthin, von wo, wie man sagt, niemand zur\u00fcckkehrt\n\nCatull, _Carmina_ 3,12.\n\n**illud amicitiae quondam venerabile nomen**\n\njenes vorzeiten verehrungsw\u00fcrdige Wort \u00bbFreundschaft\u00ab\n\nOvid, _Epistulae ex Ponto_ 2,3,19.\n\n**illud iners quidem, iucundum tamen nil agere, nihil esse**\n\ndieses zwar tr\u00e4ge, aber doch so angenehme Nichtstun, Nichtssein\n\nPlinius d. J., _Epistulae_ 8,9,1.\n\n**Illum, quo laesa est, horret adire locum.**\n\n(Mein Kahn) schaudert davor zur\u00fcck, sich dem Orte zu n\u00e4hern, an dem er (bereits) Schaden erlitt.\n\nOvid, _Tristia_ 1,1,86.\n\n**Ima permutat levis hora summis.**\n\nNiedrigstes vertauscht die unstete Zeit mit H\u00f6chstem.\n\nSeneca, _Thyestes_ 598.\n\n**Immedicabile cura \/ ense recidendum est, ne pars sincera trahatur.**\n\nDas Unheilbare mu\u00df sorgf\u00e4ltig mit dem Schwert zur\u00fcckgeschnitten werden, damit nicht das Unversehrte angesteckt werde.\n\nOvid, _Metamorphoses_ 1,190 f.\n\n**Immensum gloria calcar habet** siehe **Excitat auditor...**\n\n**Immoritur studiis et amore senescit habendi.**\n\nEr vergeht vor Eifer und altert vor Besitzgier.\n\nHoraz ( _Epistulae_ 1,7,85) betont hier seine innere Unabh\u00e4ngigkeit von seinem G\u00f6nner Maecenas: Wer reich beschenkt werde, vergehe f\u00fcr gew\u00f6hnlich irgendwann vor Eifer, noch reicher zu werden. Vgl. **Hoc erat in votis...**\n\n**Impavidum ferient ruinae** siehe **Si fractus...**\n\n**Imperare sibi maximum imperium est.**\n\nSich selbst zu beherrschen ist die gr\u00f6\u00dfte Herrschaft.\n\nSeneca, _Epistulae morales_ 113,30. Vgl. **Totum igitur...**\n\n**Imperat aut servit collecta pecunia cuique.**\n\nEs gebietet oder dient das angesammelte Geld einem jeden.\n\nHoraz, _Epistulae_ 1,10,47.\n\n**Imperia sic excelsa fortunae obiacent.**\n\nSo liegen hohe Reiche dem Schicksal ausgesetzt da.\n\nSeneca, _Oedipus_ 11.\n\n**Imperitia confidentiam, eruditio timorem creat.**\n\nUnerfahrenheit f\u00fchrt zu Selbstvertrauen, Bildung zu Scheu.\n\nHieronymus, _Epistulae_ 73,10.\n\n**imperium et libertas et fortunae**\n\nReich und Freiheit und Reicht\u00fcmer\n\nNach Cicero ( _In L. Catilinam_ 4,19), f\u00fcr den dies die Grundwerte des r\u00f6mischen Staates sind.\n\n**Impius infando miles corrumpitur auro.**\n\nDurch ruchloses Gold wird der frevelhafte Soldat verdorben.\n\nMartial, _Epigrammata_ 5,69,5.\n\n**Imponit finem sapiens et rebus honestis.**\n\nEine Beschr\u00e4nkung legt der Weise auch schicklichen Dingen auf. (Vgl.: Alles in Ma\u00dfen!)\n\nJuvenal, _Saturae_ 6,444.\n\n**Impossibilium nulla obligatio est.**\n\nEs besteht keine Verpflichtung zu Unm\u00f6glichem.\n\n(D. h.: Niemand wird gezwungen, etwas Unm\u00f6gliches zu tun.)\n\n_Digesta_ 50,17,185. \u2013 Diese Rechtsregel wird auch in leicht ver\u00e4nderter Form zitiert: \u00bbAd impossibilia nemo obligatur\u00ab (\u00bbZu Unm\u00f6glichem wird niemand verpflichtet\u00ab); \u00bbLex cogit neminem ad impossibilia\u00ab (\u00bbDas Gesetz zwingt niemanden zu Unm\u00f6glichem\u00ab) und \u00bbUltra posse nemo obligatur\u00ab (\u00bbNiemand wird verpflichtet, mehr zu k\u00f6nnen\u00ab).\n\n**Improbe Amor, quid non mortalia pectora cogis!**\n\nUners\u00e4ttliche Liebe, wozu treibst du nicht die sterblichen Herzen!\n\nVergil ( _Aeneis_ 4,412) bekundet so seine Teilnahme an Didos Schicksal, als sie sich vor dem scheidenden Aeneas dem\u00fctigt. Vgl. **Quid non mortalia pectora...**\n\n**Impune quaelibet facere, id est regem esse.**\n\nUngestraft zu tun, was beliebt, hei\u00dft K\u00f6nig sein.\n\nSallust, _Bellum Iugurthinum_ 31,26. \u2013 Das Wort \u203aK\u00f6nig\u2039 hatte f\u00fcr die R\u00f6mer eine \u00e4u\u00dferst negative Konnotation und grenzte an den Begriff des Tyrannen.\n\n**in aere aedificare**\n\nin der Luft bauen\n\n(vgl.: Luftschl\u00f6sser bauen)\n\nNach Augustinus, _Sermones_ 2,6,7.\n\n**in aere piscari, in mari venari**\n\nin der Luft fischen, auf dem Wasser jagen\n\n(d. h.: schier Unm\u00f6gliches tun)\n\nNach Plautus, _Asinaria_ 99 f.\n\n**In alio pediculum vides, in te ricinum non vides.**\n\nBeim andern siehst du eine winzige Laus, bei dir nicht den Holzbock.\n\n(vgl.: den Balken im eigenen Auge nicht sehen)\n\nPetron, _Satyrica_ 57,7.\n\n**In amore forma plus valet quam auctoritas.**\n\nIn der Liebe gilt die Sch\u00f6nheit mehr als das Ansehen.\n\nPublilius Syrus, _Sententiae_ I 39.\n\n**In angustiis amici apparent.**\n\nIn der Bedr\u00e4ngnis zeigen sich (wahre) Freunde.\n\nPetron, _Satyrica_ 61,9.\n\n**inanis quaedam profluentia loquendi**\n\neine Art leerer Redeflu\u00df\n\nCicero, _Partitiones oratoriae_ 81.\n\n**In aqua scribis.**\n\nDu schreibst auf Wasser.\n\n(Vgl.: Du redest gegen eine Wand.)\n\nNach Catull ( _Carmina_ 70,3 f.), der dazu r\u00e4t, den von einer Frau zu ihrem Geliebten gesprochenen Worten keine Aufmerksamkeit zu schenken: \u00bbSed mulier cupido quod dicit amanti, \/ in vento et rapida scribere oportet aqua.\u00ab (\u00bbDoch was eine Frau dem begierigen Liebsten sagt, soll man in den Wind und ins rei\u00dfende Wasser schreiben.\u00ab)\n\n**In arena aedificas.**\n\nDu baust auf Sand.\n\nVgl. Matth\u00e4us 7,26.\n\n**In audaces non est audacia tuta.**\n\nGegen K\u00fchne ist K\u00fchnheit kein sicheres Hilfsmittel.\n\nOvid, _Metamorphoses_ 10,544.\n\n**... in aurem utramvis otiose ut dormias**\n\n... damit du auf welchem Ohr du willst in Ruhe schlafen kannst\n\nTerenz, _Heauton timorumenos_ 342.\n\n**In causa facili cuivis licet esse diserto.**\n\nIn einer einfachen Sache kann jeder beredt sein.\n\nOvid, _Tristia_ 3,11,21.\n\n**Incende, quod adorasti** siehe **Adora, quod incendisti!**\n\n**Inceptio est amentium, non amantium.**\n\nDas ist ein Unterfangen Wahnsinniger, nicht Liebender.\n\nTerenz, _Andria_ 218. Vgl. **amantes amentes**.\n\n**incerti fallax fiducia Martis**\n\nfalsches Vertrauen auf unstetes Kriegsgl\u00fcck\n\nSilius Italicus, _Punica_ 6,333.\n\n**Incertus animus scire cum cupiat, timet.**\n\nDer unsichere Geist f\u00fcrchtet, obwohl er zu wissen begehrt.\n\nSeneca, _Oedipus_ 209.\n\n**Incertus exitus et anceps fortuna belli.**\n\nUngewi\u00df sind Kriegsausgang und Kriegsgl\u00fcck.\n\nCicero, _Pro M. Marcello_ 15.\n\n**Incertus pectora versat amor.**\n\nUnstet wendet Amor (dein) Herz hin und her.\n\nOvid, _Ars amatoria_ 3,718.\n\n**Incidit in foveam, quam fecit.**\n\nEr f\u00e4llt in die Grube, die er gegraben hat.\n\n(Vgl.: Wer anderen eine Grube gr\u00e4bt, f\u00e4llt selbst hinein.)\n\nPsalm 7,16. Vgl. **Qui fodit...**\n\n**Incidit in Scyllam, qui vult vitare Charybdin.**\n\nEs ger\u00e4t in die (F\u00e4nge der) Skylla, wer Charybdis meiden will.\n\nWalter von Ch\u00e2tillon ( _Alexandre\u00efs_ 5,301) greift hier auf die antike Vorstellung zur\u00fcck, auf dem \u00f6stlichen Ufer der Meerenge von Messina sitze das Ungeheuer Skylla, w\u00e4hrend sich ihm gegen\u00fcber auf der anderen Seite der Strudel Charybdis befinde. Vgl. Homer, _Odyssee_ 12,85 ff.; **Hac urget lupus...** und **inter canem et lupum**.\n\n**Incipe pollicitis addere facta tuis!**\n\nBeginne deinen Versprechungen Taten hinzuzuf\u00fcgen!\n\n(Vgl.: Versprechen will ein Halten haben.)\n\nOvid, _Amores_ 2,16,48.\n\n**incorrupta fides nudaque veritas**\n\nunbescholtene Treue und ungeschminkte Wahrheit\n\nHoraz ( _Carmina_ 1,24,7) beweint den Tod des Quin(c)tilius Varus, dem niemand je an Treue und Redlichkeit gleichkommen werde. Vgl. **Multis ille...**\n\n**incredibile dictu**\n\nunglaublich zu sagen\n\n(z. B.) Cicero, _Orationes Philippicae_ 2,106.\n\n**Increscunt animi, virescit vulnere virtus.**\n\nIhr Mut w\u00e4chst; die Tugend wird gest\u00e4rkt durch die Wunde.\n\nGellius ( _Noctes Atticae_ 18,11,4) zitiert hier Furius Fragment 3 Baehrens.\n\n**In culpa est animus, qui se non effugit unquam.**\n\nSchuldig macht sich eine Geisteshaltung, die sich selbst nie entflieht.\n\nHoraz ( _Epistulae_ 1,14,13) warnt davor, immer andere zu beneiden und sein eigenes Schicksal zu hassen.\n\n**Incustoditum captat ovile lupus.**\n\nDer Wolf trachtet nach einem unbewachten Schafstall.\n\nOvid, _Tristia_ 1,6,10.\n\n**in cute curanda plus aequo operata iuventus**\n\ndie Jugend, die mehr als recht damit besch\u00e4ftigt ist, die eigene Haut zu pflegen (d. h.: es sich wohl ergehen zu lassen)\n\nHoraz, _Epistulae_ 1,2,29.\n\n**Indecorant bene nata culpae.**\n\nSchuld verschandelt (auch) Wohlgeborene.\n\nHoraz, _Carmina_ 4,4,36.\n\n**Inde datae leges, ne firmior omnia posset.**\n\nDaher wurden Gesetze geschaffen, damit der St\u00e4rkere nicht alles verm\u00f6ge.\n\nOvid, _Fasti_ 3,279.\n\n**Indicium mores nobilitatis habent.**\n\nGute Sitten sind Zeichen edler Gesinnung.\n\nOvid, _Epistulae ex Ponto_ 3,2,104.\n\n**In digitis hodie percoquam, quod ceperit.**\n\nIch werde heute zwischen den Fingern gar kochen, was er f\u00e4ngt.\n\n(D. h.: Er wird nichts fangen.)\n\nPlautus, _Rudens_ 902.\n\n**Indigna digna habenda sunt, erus quae facit.**\n\nUnehrenhafte Taten m\u00fcssen f\u00fcr ehrenhaft gehalten werden, wenn ein Herr sie ausf\u00fchrt.\n\nPlautus, _Captivi_ 200.\n\n**Indignere licet: iuvat inconcessa voluptas.**\n\nAuch wenn es dich emp\u00f6rt: Das unerlaubte Vergn\u00fcgen macht Spa\u00df.\n\nOvid, _Amores_ 3,4,31.\n\n**Indocti discant et ament meminisse periti.**\n\nLaien m\u00f6gen (hier) lernen und Kenner sich der Erinnerung freuen.\n\nBei Alexander Pope ( _Essay on Criticism_ 739 ff.) finden sich diese Worte in englischer Sprache. J. F. H\u00e9nault stellt sie seinem _Nouvel Abr\u00e9g\u00e9_ _chronologique de l'histoire de France_ in lateinischer \u00dcbersetzung voran.\n\n**In dubiis tutum est inopem simulare tiranno.**\n\nIn der Gefahr ist es f\u00fcr einen Tyrannen sicher, einen Armen zu spielen.\n\nLukan, _De bello civili_ 8,241.\n\n**in dubio pro reo**\n\nim Zweifel f\u00fcr den **Angeklagten**\n\nW\u00f6rtlich findet sich diese Wendung erstmals bei Aegidius Bossius, _Tractatus varii_ , Titulus de Favoribus defensionis 2. Vgl. _Digesta_ 42,1,38; 44,7,47; 48,19,5, praefatio, und 50,17,56.\n\n**In eadem es navi.**\n\nDu bist im selben Boot.\n\n(Vgl.: Wir sitzen alle im selben Boot.)\n\nCicero, _Ad familiares_ 2,5,1.\n\n**In eadem re utilitas et turpitudo esse non potest.**\n\nEs kann nicht in derselben Sache Nutzen und Schande zugleich liegen.\n\nCicero, _De officiis_ 3,35.\n\n**in eandem copulam conicere**\n\nmit demselben Strick binden\n\n(vgl.: in einen Topf werfen; unter einen Hut bringen)\n\nNach Seneca, _De vita beata_ 7,1.\n\n**In eo Catonis materies atque indoles est.**\n\nIn ihm ist der Stoff und die Veranlagung f\u00fcr einen (neuen) Cato.\n\nNach Cicero, _Actio secunda in C. Verrem_ 3,160. \u2013 Cato d. \u00c4., der sich gegen jeglichen griechischen Einflu\u00df wehrte, galt als R\u00f6mer par excellence.\n\n**In eodem luteo haesitas.**\n\nDu steckst (noch) im selben Dreck.\n\nTerenz, _Phormio_ 780.\n\n**ineptiis suis plaudere**\n\nseinen eigenen Albernheiten Beifall klatschen\n\nNach Tacitus, _Dialogus_ 32,7.\n\n**Inesse quin etiam sanctum aliquid et providum putant.**\n\nSie (die Germanen) glauben sogar, da\u00df (den Frauen) etwas Heiliges und Voraussehendes innewohne.\n\nTacitus, _Germania_ 8,2.\n\n**Infandum, regina, iubes renovare dolorem.**\n\nUnsagbaren Schmerz, K\u00f6nigin, hei\u00dft du erneuern.\n\nVergil, _Aeneis_ 2,3. \u2013 Aeneas' Antwort auf Didos Bitte, vom Untergang Trojas und seinen Irrfahrten zu berichten.\n\n**Infantes perhibet et stultos dicere verum.**\n\nMan sagt, da\u00df Kinder und Narren die Wahrheit sprechen.\n\nVgl. Cicero, _Topica_ 75.\n\n**Infectum nemo reddet, \/ quod fugiens semel hora vexit.**\n\nNiemand wird ungeschehen machen, was die fliehende Stunde einmal fortgetragen hat.\n\nNach Horaz, _Carmina_ 3,29,47 f.\n\n**Infirmi est animi exiguique voluptas \/ ultio** siehe **Minuti \/ semper...**\n\n**in flagranti** siehe **adhuc flagranti crimine**\n\n**In flammam flammas, in mare fundis aquas.**\n\nIn die Flamme gie\u00dft du **Flammen, ins Meer Wasser.**\n\nOvid, _Amores_ 3,2,34. Vgl. _Amores_ 2,10,13 f.: \u00bbQuid folia arboribus, quid pleno sidera caelo, \/ in freta collectas alta quid addis aquas?\u00ab (\u00bbWas f\u00fcgst du den B\u00e4umen Bl\u00e4tter, dem vollen Himmel Sterne, dem Meer gesammeltes Wasser hinzu?\u00ab)\n\n**Ingenii dotes corporis adde bonis!**\n\nF\u00fcge geistige Gaben der Sch\u00f6nheit des K\u00f6rpers hinzu!\n\nOvid, _Ars amatoria_ 2,112. Vgl. **Forma bonum...**\n\n**Ingeniis stimulos subdere fama solet.**\n\nDen Geist pflegt Ruhm anzustacheln.\n\nOvid, _Tristia_ 5,1,76.\n\n**Ingenium est omnium \/ hominum ab labore proclive ad lubidinem.**\n\nDer Sinn aller Menschen (wendet sich) rasch von der M\u00fche zur Lust.\n\nTerenz, _Andria_ 77 f.\n\n**Ingenium mala saepe movent.**\n\nOft weckt Not Talent.\n\n(Vgl.: Not macht erfinderisch.)\n\nOvid, _Ars amatoria_ 2,43. Vgl. **Paupertas artes omnes perdocet**.\n\n**Ingenium probitas artemque modestia vincit.**\n\nIhre Redlichkeit und Rechtschaffenheit \u00fcbertreffen ihr Talent und ihre Geschicklichkeit.\n\nStatius, _Silvae_ 3,5,67.\n\n**Ingenium res \/ adversae nudare solent, celare secundae.**\n\nTalent pflegen widrige Umst\u00e4nde zu enth\u00fcllen, w\u00e4hrend Gl\u00fcck es verbirgt.\n\nHoraz, _Sermones_ 2,8,73 f.\n\n**Ingens telum necessitas.**\n\nEine ungeheure Waffe ist die Notwendigkeit.\n\nLivius, _Ab urbe condita_ 4,28,5. Vgl. **Necessitas ultimum...**\n\n**Ingenuum nasci facile est.**\n\nAls freier Mann geboren zu werden ist leicht.\n\nNach Petron, _Satyrica_ 57,11.\n\n**Ingrata misero vita ducenda est in hoc.**\n\nElender, daf\u00fcr mu\u00dft du ein unangenehmes Leben f\u00fchren.\n\nHoraz, _Epodi_ 17,63.\n\n**Ingrato homine nihil impensius est.**\n\nNichts ist kostspieliger, als ein undankbarer Mensch (zu sein).\n\nPlautus, _Bacchides_ 394.\n\n**Ingratus unus omnibus miseris nocet.**\n\nEin einziger Undankbarer schadet allen Armen.\n\nPublilius Syrus, _Sententiae_ I 14.\n\n**In hac re omni pede standum est.**\n\nIn dieser Angelegenheit mu\u00df man auf dem ganzen Fu\u00df stehen (d. h. man kann sie nicht aus dem Stegreif erledigen).\n\nNach Quintilian ( _Institutio oratoria_ 12,9,18), der diese Worte als Redewendung aus dem b\u00e4uerlichen Sprachgebrauch bezeichnet. Gemeint ist die F\u00e4higkeit, stets f\u00fcr jede Art von Rede bereit zu sein.\n\n**in horam vivere**\n\nin die Stunde hinein (d. h.: in den Tag hinein, von der Hand in den Mund) leben\n\nNach Cicero, _Orationes Philippicae_ 5,25.\n\n**Inicietque manum formae damnosa senectus,**\n\nQuae strepitus passu non faciente venit.\n\nEs legt die Hand auf die Sch\u00f6nheit das verderbliche Alter, das mit lautlosem Schritt daherkommt.\n\nOvid, _Tristia_ 3,7,35 f.\n\n**in integrum restituere**\n\nin seine Rechte einsetzen\n\nNach Cicero, _Pro A. Cluentio Habito_ 98.\n\n**In ipso adhuc lacte matris tenerum cor meum pie praebiberat.**\n\nMit der Muttermilch selbst hatte mein zartes Herz ihn fromm schon getrunken.\n\nAugustinus ( _Confessiones_ 3,4,8) berichtet, der Name des Heilands sei ihm schon von klein auf vertraut gewesen.\n\n**Iniqua numquam regna perpetuo manent.**\n\nUngerechte Herrschaft ist niemals dauerhaft.\n\n(Vgl.: Unrecht Gut gedeiht nicht.)\n\nSeneca, _Medea_ 196.\n\n**Iniquissimam pacem iustissimo bello anteferro.**\n\nIch ziehe den ungerechtesten Frieden dem gerechtesten Krieg vor.\n\nNach Cicero, _Ad familiares_ 6,6,5.\n\n**Iniquitati proxima est severitas.**\n\nStrenge ist der Ungerechtigkeit sehr nahe.\n\nPseudo-Seneca, _De moribus_ 95.\n\n**Iniuriam aures facilius quam oculi ferunt.**\n\nUnrecht ertragen die Ohren leichter als die Augen.\n\nPublilius Syrus, _Sententiae_ I 35.\n\n**Iniuriam qui facturus est, iam facit.**\n\nWer im Begriff ist, Unrecht zu tun, tut bereits Unrecht.\n\nSeneca, _De ira_ 1,3,1.\n\n**Iniuriarum remedium est oblivio.**\n\nEin Heilmittel gegen Unrecht liegt im Vergessen.\n\nPublilius Syrus, _Sententiae_ I 21, und Seneca, _Epistulae morales_ 94,28.\n\n**Iniusta ab iustis impetrari non decet.**\n\nEs ziemt sich nicht, Unrechtes von Gerechten zu erlangen.\n\nPlautus, _Amphitruo_ 35.\n\n**In laqueos auceps decideratque suos.**\n\nDer Vogelf\u00e4nger war in sein eigenes Netz geraten.\n\n(vgl.: sich im eigenen Strick verfangen)\n\nOvid, _Remedia amoris_ 502; vgl. auch _Ars amatoria_ 1,646.\n\n**in limine**\n\nauf der Schwelle\n\n(d. h.: schon zu Beginn)\n\n(z. B.) Vergil, _Aeneis_ 11,423.\n\n**In magnis et voluisse sat est** siehe **Quod si deficiant...**\n\n**in malam crucem** siehe **Abi in malam crucem!**\n\n**in manu illius plumbum aurum fit** siehe **fortunae filius...**\n\n**in medias res** siehe **ab ovo**\n\n**In medio luto est.**\n\nEr sitzt mitten im Dreck.\n\n(vgl.: in der Tinte sitzen)\n\nPlautus, _Pseudolus_ 984.\n\n**In melle sunt linguae sitae vostrae atque orationes,**\n\n**Facta atque corda in felle sunt sita atque acerbo aceto.**\n\nVoll Honig sind eure Zungen und Worte, eure Taten und Herzen voller Galle und bitteren Essigs.\n\nPlautus, _Truculentus_ 178 f.\n\n**In me omnis spes mihi est.**\n\nIch setze meine ganze Hoffnung auf mich selbst.\n\nTerenz, _Phormio_ 139.\n\n**In necessariis unitas, in dubiis libertas, in omnibus caritas. In den notwendigen Dingen Einigkeit, in den umstrittenen Freiheit, in allem Liebe.**\n\nNach Friedrich L\u00fccke ( _\u00dcber das Alter, den Verfasser, die urspr\u00fcngliche Form und den wahren Sinn des kirchlichen Friedensspruches..._ ) wird der Spruch meist f\u00e4lschlich Augustinus zugesprochen. Er stamme in Wahrheit in etwas anderer Form von Rupertus Meldenius. Vgl. HW: 11894a.\n\n**in nuce**\n\nin der Nu\u00df, im Kern\n\nBei Plinius d. \u00c4. ( _Naturalis historia_ 7,85) berichtet Cicero von einer Pergamenthandschrift der _Ilias_ im Miniaturformat, die in einer Nu\u00dfschale Platz gefunden habe. Sp\u00e4ter gewinnt die Wendung die Bedeutung \u203ain Kurzfassung\u2039 bzw. \u203aim Kern\u2039.\n\n**In omni certamine, qui opulentior est, etiamsi accipit iniuriam, tamen, quia plus potest, facere videtur.**\n\nIn jedem Wettstreit scheint der, der m\u00e4chtiger ist, auch wenn er Unrecht hinnehmen mu\u00df, dennoch, weil er mehr Macht hat, Unrecht zu tun.\n\nSallust, _Bellum Iugurthinum_ 10,7, und Quintilian, _Institutio oratoria_ 8,5,4.\n\n**Inopiae desunt multa, avaritiae omnia.**\n\nDer Armut mangelt es an vielem, dem Geiz an allem.\n\n(Vgl.: Dem Armen geht viel ab, dem Geizigen alles.)\n\nPublilius Syrus, _Sententiae_ I 7.\n\n**Inopi beneficium bis dat, qui dat celeriter.**\n\nDoppelt spendet dem Armen, wer schnell gibt.\n\nPublilius Syrus, _Sententiae_ I 6.\n\n**Inops, potentem dum vult imitari, perit.**\n\nWenn der Arme es dem Reichen gleichtun will, so geht er zugrunde.\n\nPhaedrus, _Fabulae_ 1,24,1.\n\n**in os laudare** siehe **Vereor coram...**\n\n**in otio inconcusso iacere**\n\nin unersch\u00fctterter **Mu\u00dfe daliegen**\n\n(vgl.: auf der faulen Haut liegen)\n\nSeneca ( _Epistulae morales_ 67,14) warnt davor, in Mu\u00dfe dazuliegen, denn sie sei nur Windstille. Erst die nach den Pr\u00fcfungen des Charakters eintretende Ruhe sei echte Ruhe.\n\n**in pace leones et in proelio cervi**\n\nim Frieden L\u00f6wen und im Kampf Hirsche\n\nNach Tertullian, _De corona militis_ 1,5. \u2013 Oft findet sich die Gegen\u00fcberstellung von L\u00f6wen und F\u00fcchsen (z. B. Horaz, _Sermones_ 2,3,186, und Petron, _Satyrica_ 44: **domi leones...** ) oder von L\u00f6wen und Hasen (z. B. Sidonius Apollinaris, _Epistulae_ 5,7,5: \u00bbIn praetoriis leones, in castris lepores\u00ab \u2013 \u00bbIm Hauptquartier [sind sie noch] L\u00f6wen, im Kriegsdienst Hasen\u00ab).\n\n**in partibus infidelium**\n\nin den Gebieten der Ungl\u00e4ubigen\n\nAb dem 7. Jahrhundert ein Zusatz zum Titel von Bisch\u00f6fen, deren Dioz\u00f6sen durch Sarazenen eingenommen wurden und die dennoch weiter im Amt blieben (seit der Abschaffung dieser Bezeichnung durch Papst Leo XIII. im Jahre 1882 \u00bbTitularbisch\u00f6fe\u00ab).\n\n**In patria natus non est propheta vocatus.**\n\nWer im Vaterland geboren ist, wird nicht \u00bbProphet\u00ab genannt.\n\n(Vgl.: Der Prophet gilt nichts im eigenen Land.)\n\nHW: 11918. Vgl. Markus 6,4.\n\n**in pertusum dolium dicta ingerere**\n\nWorte in ein l\u00f6chriges Fa\u00df f\u00fcllen\n\n(d. h.: tauben Ohren predigen)\n\nNach Plautus, _Pseudolus_ 369.\n\n**In plerisque rebus mediocritas optima est.**\n\nIn den meisten Dingen ist der Mittelweg am besten.\n\nCicero, _De officiis_ 1,130.\n\n**in portum navigare**\n\nin den Hafen einlaufen\n\n(d. h.: in Sicherheit gelangen)\n\nNach Terenz, _Andria_ 480: **Ego in portu navigo.**\n\n**In praeteritum subvenire ne di quidem possunt.**\n\nBei Vergangenem k\u00f6nnen nicht einmal die G\u00f6tter Hilfe leisten.\n\nPlinius d. J., _Panegyricus_ 40,3.\n\n**In praetoriis leones, in castris lepores** siehe **in pace leones...**\n\n**In principio erat verbum.**\n\nAm Anfang war das Wort.\n\nJohannes 1,1.\n\n**In prolem transcurrit gratia patrum.**\n\nDie Dankbarkeit gegen\u00fcber dem Vater geht auf die Kinder \u00fcber.\n\nClaudian, _De consulatu Stilichonis_ 2,51.\n\n**Inquietum est cor nostrum, donec requiescat in te.**\n\nUnruhig ist unser Herz, bis es Ruhe findet in dir.\n\nAugustinus, _Confessiones_ 1,1,1.\n\n**In quod me conieci malum!**\n\nIn welch \u00fcble Lage habe ich mich da gebracht!\n\nTerenz, _Hecyra_ 132.\n\n**In rebus dubiis plurimi est audacia.**\n\nIn gef\u00e4hrlicher Lage vermag K\u00fchnheit am meisten.\n\nPublilius Syrus, _Sententiae_ I 30.\n\n**In re mala, animo si bono utare, adiuvat.**\n\nIn \u00fcblen Umst\u00e4nden hilft es, guten Mut zu bewahren.\n\nPlautus, _Captivi_ 202.\n\n**in saltu uno duos apros capere**\n\nin einem Waldtal zwei Eber fangen\n\n(vgl.: zwei Fliegen mit einer Klappe schlagen)\n\nNach Plautus, _Casina_ 476. Vgl. **duo parietes...**\n\n**Insanus medio flumine quaeris aquam.**\n\nIn deiner Raserei verlangst du mitten im Strom nach Wasser.\n\nProperz, _Elegiae_ 1,9,16.\n\n**Insatiabilis est humanus animus.**\n\nUners\u00e4ttlich ist der Menschengeist.\n\nNach Livius, _Ab urbe condita_ 4,13,4.\n\n**In scirpo nodum quaeris.**\n\nDu suchst den Knoten in der Binse.\n\n(D. h.: Du suchst Schwierigkeiten bzw. Fehler, wo keine sind.)\n\nPlautus, _Menaechmi_ 247.\n\n**In se magna ruunt: laetis hunc numina rebus \/ crescendi posuere modum.**\n\nGro\u00dfes f\u00e4llt in sich selbst zusammen: Diese Beschr\u00e4nkung des Wachstums hat der g\u00f6ttliche Wille dem Erfolg auferlegt.\n\nLukan, _De bello civili_ 1,81 f.\n\n**In servitutem cadere de regno grave est.**\n\nVon der K\u00f6nigsherrschaft in die Sklaverei hinabzufallen ist hart.\n\nSeneca, _Phoenissae_ 598.\n\n**Insidias et campus habet.**\n\nAuch das Feld birgt T\u00fccken.\n\nMartial ( _Epigrammata_ 12,14,5) warnt vor scheinbar ungef\u00e4hrlichen Situationen.\n\n**In silvam ne ligna feras!**\n\nTrag nicht Holz in den Wald!\n\n(vgl.: Eulen nach Athen tragen)\n\nNach Horaz, _Sermones_ 1,10,34. Vgl. **noctuam Athenas deportare...** und **ululam Athenas portare**.\n\n**In silvis l\u00e9pores, in verbis quaere lep\u00f3res!**\n\nIn den W\u00e4ldern suche Hasen, in den Worten Anmut!\n\nHW: 12053a.\n\n**in sinu atque in deliciis venenatam viperam habere**\n\neine giftige Schlange wie einen Liebling am Busen halten\n\n(vgl.: eine Schlange am Busen n\u00e4hren)\n\nNach Cicero, _De haruspicum responsis_ 50.\n\n**Insperata accidunt magis saepe quam quae speres.**\n\nUnerhofftes geschieht \u00f6fter als das, was man hofft.\n\n(Vgl.: Unverhofft kommt oft.)\n\nPlautus, _Mostellaria_ 197.\n\n**inspicere tamquam in speculum in vitas omnium**\n\ngleichsam wie in einen Spiegel auf das Leben aller schauen\n\nTerenz, _Adelphoe_ 415.\n\n**in stomacho ridere**\n\ntrotz des \u00c4rgers lachen\n\n(d. h. eine gute Miene zum b\u00f6sen Spiel machen)\n\nNach Cicero, _Ad familiares_ 2,16,7.\n\n**In tacito cohibe gaudia clausa sinu.**\n\nHalte in verschwiegener Brust die Freude verschlossen.\n\nProperz ( _Elegiae_ 2,25,30) warnt vor vorzeitiger Liebesfreude.\n\n**integer vitae scelerisque purus**\n\nvom Leben nicht verdorben und rein von Schuld\n\nHoraz ( _Carmina_ 1,22,1) preist sein eigenes unbescholtenes und gewaltfreies Leben.\n\n**In teneris consuescere multum est.**\n\nWichtig ist die Gew\u00f6hnung im zarten Alter.\n\nVergil, _Georgica_ 2,272.\n\n**Intentant omnia mortem.**\n\nAlles deutet drohend auf den Tod hin.\n\nVergil, _Aeneis_ 1,91.\n\n**inter canem et lupum**\n\nzwischen Wolf und Hund\n\nNach Plautus, _Casina_ 971, und Horaz, _Sermones_ 2,2,64: **Hac urget lupus...** und **Incidit in Scyllam...**\n\n**Interdum lacrimae pondera vocis habent.**\n\nBisweilen haben Tr\u00e4nen das Gewicht von gesprochenen **Worten.**\n\nOvid, _Epistulae ex Ponto_ 3,1,158.\n\n**Interdum vulgus rectum videt, est ubi peccat.**\n\nBisweilen sieht die breite Masse das Richtige, manchmal (aber) macht sie Fehler.\n\nHoraz, _Epistulae_ 2,1,63.\n\n**Interea fiet aliquid.**\n\nIn der Zwischenzeit wird irgend etwas geschehen.\n\nTerenz, _Andria_ 314. \u2013 Charinus dr\u00fcckt so seine Hoffnung aus, die Hochzeit zwischen seiner Geliebten, Philumena, und Pamphilus noch verhindern zu k\u00f6nnen.\n\n**Interim velim a sole mihi non obstes!**\n\nSteh mir zun\u00e4chst nicht in der Sonne!\n\nValerius Maximus, _Facta et dicta memorabilia_ 4,3, ext. 4. \u2013 Als Alexander der Gro\u00dfe zu Diogenes aus Sinope kam und ihn fragte, ob er etwas f\u00fcr ihn tun k\u00f6nne, sagte dieser lediglich: \u00bbMox... de ceteris, interim velim a sole mihi non obstes.\u00ab (\u00bbDazu sp\u00e4ter, im Moment w\u00fcrde ich mir nur w\u00fcnschen, da\u00df du mir nicht in der Sonne stehst.\u00ab) Vgl. **Nunc quidem paululum a sole**.\n\n**inter iocos et seria**\n\nzwischen Scherz und Ernst\n\nVgl. Cicero, _De finibus_ 2,85, und Sallust, _Bellum Iugurthinum_ 96,2.\n\n**inter os atque offam**\n\nzwischen Mund und Bissen\n\n(Vgl.: Man soll den Tag nicht vor dem Abend loben.)\n\nGellius ( _Noctes Atticae_ 13,18,1) zitiert Cato d. \u00c4. ( _De aedilibus vitio creatis_ Fragment 65,1 Jordan), der vor vorschneller Freude warnt, denn selbst \u00bbzwischen Mund und Bissen\u00ab k\u00f6nne noch viel dazwischenkommen.\n\n**Inter sacrum saxumque sto nec, quid faciam, scio.**\n\nZwischen Altar und Stein stehe ich und wei\u00df nicht, was ich tun soll.\n\nPlautus, _Captivi_ 617, vgl. _Casina_ 970.\n\n**Inter spem curamque, timores inter et iras,**\n\n**Omnem crede diem tibi diluxisse supremum.**\n\nIn Hoffnung und Sorgen, in Furcht und Zorn, glaube immer, da\u00df dir der letzte Tag geschienen (d. h.: das letzte St\u00fcndlein geschlagen) habe.\n\nHoraz ( _Epistulae_ 1,4,12 ff.) f\u00e4hrt optimistisch fort: \u00bbGrata superveniet, quae non sperabitur hora.\u00ab (\u00bbAngenehm wird die Stunde \u00fcberraschen, die man nicht erhoffte.\u00ab)\n\n**In tirannos!**\n\nWider die Tyrannen!\n\nMotto der Titelvignette zur zweiten Auflage von Friedrich Schillers Drama _Die R\u00e4uber_ von 1782.\n\n**Intra \/ fortunam debet quisque manere suam.**\n\nJeder sollte die Grenzen seiner Bestimmung einhalten.\n\nOvid, _Tristia_ 3,4,25 f.\n\n**Introite, nam et heic dii sunt!**\n\nTretet ein, denn auch hier sind G\u00f6tter!\n\nGotthold Ephraim Lessing (Motto zu _Nathan der Weise_ ), der die Wendung f\u00e4lschlicherweise Gellius zuschreibt. Eigtl. nach Aristoteles ( _De partibus animalium_ 1,645a 21), der Heraklit zitiert.\n\n**In tumultu festinatio quoque tarda est.**\n\nIn der Verwirrung ist sogar Eile langsam.\n\nCurtius Rufus, _Historia Alexandri Magni_ 9,9,12.\n\n**Intus et in cute te novi** siehe **Ego te intus et in cute novi**\n\n**In unoquoque virorum bonorum \u203a(quis deus incertum est) habitat deus\u2039.**\n\nIn jedem guten Menschen \u00bbwohnt ein Gott, (doch es ist unsicher, was f\u00fcr einer)\u00ab.\n\nSeneca ( _Epistulae morales_ 41,2) zitiert hier einen Teilvers von Vergil ( _Aeneis_ 8,352).\n\n**in usum Delphini** siehe **ad usum Delphini**\n\n**Invenit arma furor.**\n\nDas W\u00fcten fand Waffen.\n\nLukan, _De bello civili_ 3,671.\n\n**Invenit interdum caeca columba pisum.**\n\nMitunter findet auch eine blinde Taube eine Erbse.\n\n(Vgl.: Auch ein blindes Huhn findet einmal ein Korn.)\n\nHW: 12732.\n\n**Invenitque protinus patella operculum.**\n\nUnd es fand sogleich die Sch\u00fcssel ihren Deckel.\n\n(Vgl.: Jeder Topf hat seinen Deckel.)\n\nHieronymus, _Epistulae_ 127,9.\n\n**Invia virtuti nulla est via.**\n\nDer Tugend ist kein Weg ungangbar.\n\nOvid, _Metamorphoses_ 14,113.\n\n**Invictumque virum vincit dolor.**\n\nUnd den unbesiegten Mann besiegt der Schmerz.\n\nOvid, _Metamorphoses_ 13,386. \u2013 Aiax ist wutentbrannt, als er im Streit um die Waffen des Achilles Odysseus unterliegt. Durch seinen anschlie\u00dfenden Selbstmord beweist er, da\u00df \u00bbniemand au\u00dfer Aiax Aiax \u00fcberwinden kann\u00ab (390).\n\n**Invidia gloriae comes.**\n\nDer Neid ist der Begleiter des Ruhms.\n\nCornelius Nepos, _Chabrias_ 3,3.\n\n**Invidus alterius macrescit rebus opimis.**\n\nDer Neider magert ab, wenn sein N\u00e4chster im Fetten sitzt.\n\nHoraz, _Epistulae_ 1,2,57.\n\n**Invidus, iracundus , iners, vinosus, amator**\n\n**Nemo adeo ferus est, ut non mitescere possit.**\n\nOb neidisch, hitzig, tr\u00e4ge, trunks\u00fcchtig, l\u00fcstern \u2013 niemand ist so wild, da\u00df er nicht friedlich werden k\u00f6nnte.\n\nHoraz, _Epistulae_ 1,1,38 f.\n\n**In vino veritas.**\n\nIm Wein (liegt) Wahrheit.\n\nNach Plinius d. \u00c4. ( _Naturalis historia_ 14,141), der die Wendung als Sprichwort bezeichnet. Die \u00e4lteste nachweisbare Quelle ist wohl Alkaios Fragment 128 Lobel. \u2013 Tacitus ( _Germania_ 22) berichtet, die Germanen berieten sich stets beim Weine, da man sich betrunken nicht verstellen k\u00f6nne. Entscheidungen tr\u00e4fen sie jedoch erst am n\u00e4chsten Tag in N\u00fcchternheit. \u2013 Die Wendung wird oft scherzhaft abge\u00e4ndert zu \u00bbIn vino feritas\u00ab (\u00bbIm Wein liegt Leidenschaft\u00ab).\n\n**Invisurus aliquis facilius quam imitaturus.**\n\nJemand wird es leichter tadeln als nachmachen.\n\nNach Plinius d. \u00c4. ( _Naturalis historia_ 35,63), der berichtet, der Maler Zeuxis sei mit seinem Bildnis eines Athleten so zufrieden gewesen, da\u00df er darunter geschrieben habe, es sei leichter, ihn zu tadeln, als ihn nachzumachen.\n\n**invita Minerva**\n\ngegen den Willen Minervas\n\nCicero, _De officiis_ 1,110. \u2013 Gemeint ist \u203anicht in Sch\u00f6pferlaune\u2039, da Minerva als Patronin der Handwerker, K\u00fcnstler, Lehrer und \u00c4rzte galt. In _Ad familiares_ 12,25,1 verweist Cicero auf das Zusammentreffen seines Verhandlungserfolgs mit dem Senatsbeschlu\u00df, die MinervaStatue wiederaufzurichten. Vgl. **crassa Minerva** und **pingui Minerva**.\n\n**Invitat culpam, qui peccatum praeterit.**\n\nWer ein Vergehen \u00fcbergeht, l\u00e4dt zu (neuen) Fehltritten ein.\n\nPublilius Syrus, _Sententiae_ I 9.\n\n**In vitium ducit culpae fuga.**\n\nZu (neuer) Verfehlung f\u00fchrt die Flucht vor dem Fehler.\n\nHoraz, _Ars poetica_ 31.\n\n**Iota unum aut unus apex non praeteribit a lege.**\n\nEs wird kein Jota und kein Akzent des Gesetzes \u00fcbergangen werden.\n\n(D. h.: Auch nicht das Geringste wird ge\u00e4ndert werden.)\n\nMatth\u00e4us 5,18.\n\n**Iovis omnia plena** siehe **Ab Iove principium...**\n\n**Ipsa olera olla legit.**\n\nDer Topf sucht sich sein Gem\u00fcse selbst.\n\nCatull ( _Carmina_ 94,2) legt dar, wie gew\u00f6hnlich Ehebruch sei: Schlie\u00dflich suche einander, was zueinander geh\u00f6re.\n\n**Ipsa quidem virtus sibimet pulcherrima merces.**\n\nDie Tugend ist sich selbst der sch\u00f6nste Lohn.\n\nSilius Italicus, _Punica_ 13,663.\n\n**Ipse dixit.**\n\nEr selbst hat es gesagt.\n\nCicero ( _De natura deorum_ 1,10) berichtet, die Anh\u00e4nger des Pythagoras h\u00e4tten oftmals keine rationale Begr\u00fcndung f\u00fcr ihre Behauptungen geben k\u00f6nnen, sondern sich schlichtweg darauf berufen, da\u00df ihre Worte von Pythagoras selbst stammten.\n\n**Ipse mihi asciam in crus impegi.**\n\nIch habe mir selbst die Axt ins Schienbein geschlagen.\n\n(vgl.: sich ins eigene Fleisch schneiden)\n\nPetron, _Satyrica_ 74,16. Vgl. **sponte asceam cruribus suis illidere.**\n\n**Ipse se prodit.**\n\nEr verr\u00e4t sich selbst.\n\nSeneca ( _Epistulae morales_ 10,2) erkl\u00e4rt, f\u00fcr einen Toren sei sogar der einzige Vorteil der Einsamkeit, n\u00e4mlich keinen Denunzianten f\u00fcrchten zu m\u00fcssen, verloren, denn der Dumme verrate sich selbst.\n\n**I! Quid stas, lapis!**\n\nGeh! Was stehst du herum, du Stein?\n\nTerenz, _Heauton timorumenos_ 831.\n\n**Ira concitum pectus doma!**\n\nBez\u00e4hme die ha\u00dferregte Brust!\n\nSeneca, _Medea_ 506.\n\n**Iracundia fortitudinis quasi cos est.**\n\nZorn ist gleichsam der Tugend Wetzstein.\n\nNach Cicero ( _Academici libri priores_ 2,135), der diese Worte als eine Erkenntnis der Alten Akademie bezeichnet.\n\n**Iracundiam qui vincit, hostem superat maximum.**\n\nWer den J\u00e4hzorn besiegt, \u00fcberwindet den st\u00e4rksten Feind.\n\n(Vgl.: Wer den Zorn bezwingt, hat einen gro\u00dfen Feind besiegt.)\n\nPublilius Syrus, _Sententiae_ I 22.\n\n**Ira furor brevis est: animum rege, qui nisi paret, \/ imperat.**\n\nZorn ist ein kurzer Wahnsinn. Beherrsche deine Leidenschaften, denn wenn sie (dir) nicht gehorchen, befehlen sie (dir).\n\nHoraz, _Epistulae_ 1,2,62 f.\n\n**Ira initium insaniae.**\n\nDer Zorn ist der Beginn des Wahnsinns.\n\n(Vgl.: Der Zornige hat alle Sinne bei sich bis auf f\u00fcnf.)\n\nCicero ( _Tusculanae disputationes_ 4,52) zitiert hier Ennius (Fragment Varia 32 Warmington).\n\n**ire tendere de fumo ad flammam**\n\naus dem Rauch in die Flamme (d. h.: zum Wesentlichen) streben\n\nAmmianus Marcellinus ( _Res gestae_ 14,11,12) bezeichnet die Wendung bereits als bekanntes Sprichwort.\n\n**Irus et est subito, qui modo Croesus erat** siehe **Nempe dat id...**\n\n**Is cadet ante senem, qui sapit ante diem.**\n\nDer wird vor dem Greis untergehen, der vor seiner Zeit weise wird.\n\n(Vgl.: Kluge Kinder werden selten alt.)\n\nHW: 12953.\n\n**Istuc est sapere : non quod ante pedes modo est \/ videre, sed etiam illa, quae futura sunt, \/ prospicere.**\n\nDas ist Klugheit: nicht nur zu sehen, was vor den F\u00fc\u00dfen liegt, sondern auch zu beachten, was kommen wird.\n\nTerenz, _Adelphoe_ 386 ff.\n\n**Ita conparatam esse hominum naturam omnium, \/ aliena ut melius videant et diiudicent \/ quam sua!**\n\nDa\u00df das Wesen aller Menschen so geartet ist, da\u00df sie Fremdes besser sehen und beurteilen als Eigenes!\n\nTerenz, _Heauton timorumenos_ 503 ff.\n\n**Ita fugias ne praeter casam!**\n\nFliehe nicht so an der H\u00fctte vorbei!\n\nTerenz ( _Phormio_ 768) bezeichnet diese Wendung bereits als sprichw\u00f6rtlich.\n\n**Ita vita est hominum, quasi quom ludas tesseris.**\n\nSo ist das Leben der Menschen: als ob man mit W\u00fcrfeln spielte.\n\nTerenz, _Adelphoe_ 739.\n\n**Iter ad superos gloria pandet.**\n\nRuhm \u00f6ffnet den Weg zu den \u00dcberirdischen.\n\nSeneca, _Hercules Oetaeus_ 1988.\n\n**Iubet igitur nos Pythius Apollo noscere nosmet ipsos.**\n\nDer pythische (d. h.: delphische) Apoll befiehlt uns also, uns selbst zu erkennen.\n\nCicero, _De finibus_ 5,44. \u2013 Am Apollontempel von Delphi findet sich die einem der sieben Weisen zugeschriebene griechische Inschrift \u00bbgnothi seauton\u00ab (\u00bbErkenne dich selbst\u00ab). Vgl. **Nosce te ipsum**.\n\n**Iucundi acti labores.**\n\nAngenehm sind (bereits) geleistete Anstrengungen.\n\n(Vgl.: Nach getaner Arbeit ist gut ruh'n.)\n\nCicero ( _De finibus_ 2,105) bezeichnet diese Wendung bereits als Sprichwort und f\u00fcgt einen passenden Vers aus dem verlorenen St\u00fcck _Andromeda_ des Euripides hinzu (Fragment 131 Nauck): \u00bbSuavis laborum est praeteritorum memoria.\u00ab (\u00bbS\u00fc\u00df ist die Erinnerung an vergangene M\u00fchen.\u00ab) Vgl. Cicero, _Ad familiares_ 5,12,4.\n\n**Iucundum et carum sterilis facit uxor amicum.**\n\nAngenehm und liebenswert macht den Freund eine unfruchtbare Gattin (erg.: weil man ihn zu beerben hofft).\n\nJuvenal, _Saturae_ 5,140.\n\n**Iudex damnatur, cum nocens absolvitur.**\n\nDer Richter wird verurteilt, wenn ein Schuldiger freigesprochen wird.\n\nPublilius Syrus, _Sententiae_ I 28.\n\n**Iudice Fortuna cadat alea.**\n\nMit Fortuna als Richterin falle der W\u00fcrfel.\n\nPetron, _Satyrica_ 122,176.\n\n**iungere vulpes et mulgere hircos**\n\nF\u00fcchse ins Joch spannen und B\u00f6cke melken\n\nVergil ( _Bucolica_ 3,91) greift auf eine sprichw\u00f6rtliche Wendung zur\u00fcck, um auszudr\u00fccken, da\u00df jemand etwas g\u00e4nzlich Unsinniges tut.\n\n**Iupiter pluvius**\n\nder regenreiche Jupiter\n\nJohann Wolfgang Goethe, \u00bbWandrers Sturmlied\u00ab, V. 75, nach Tibull, _Elegiae_ 1,7,26.\n\n**Iuppiter est, quodcumque vides, quodcumque moveris.**\n\nJupiter ist, was auch immer du siehst, durch was auch immer du bewegt wirst.\n\nLukan, _De bello civili_ 9,580.\n\n**iurare in verba magistri**\n\nauf die Worte des Meisters schw\u00f6ren\n\nHoraz ( _Epistulae_ 1,1,14 f.) lehnt Kritiklosigkeit gegen\u00fcber Leitfiguren ab und bekundet seine philosophische Unabh\u00e4ngigkeit: \u00bbQuo me cumque rapit tempestas, deferor hospes.\u00ab (\u00bbWohin auch immer mich der Sturm treibt, dorthin begebe ich mich als Gast.\u00ab) Vgl. im Gegensatz dazu Goethe, _Faust I_ , V. 1988 f.: \u00bbAm besten ist's auch hier, wenn Ihr nur Einen h\u00f6rt \/ und auf des Meisters Worte schw\u00f6rt.\u00ab\n\n**Iuravi lingua, mentem iniuratam gero.**\n\nIch habe mit der Zunge geschworen, mein Geist bleibt unvereidigt.\n\nCicero ( _De officiis_ 3,108) zitiert hier Euripides ( _Hippolytos_ 612).\n\n**Ius civile neque inflecti gratia neque perfringi potentia neque adulterari pecunia debet.**\n\nDas \u00bbIus Civile\u00ab (\u00bbB\u00fcrgerrecht\u00ab) darf weder durch Gunsterweisung gebeugt noch durch Macht gebrochen noch durch Geldzuwendungen verf\u00e4lscht werden.\n\nNach Cicero, _Pro A. Caecina_ 73.\n\n**Ius est in armis, opprimit leges timor.**\n\nDas Recht liegt in den Waffen, Furcht unterdr\u00fcckt die Gesetze.\n\nSeneca, _Hercules furens_ 253.\n\n**Ius summum saepe summa est malitia.**\n\nDas strengste Recht ist oft die gr\u00f6\u00dfte Schikane.\n\nTerenz ( _Heauton timorumenos_ 796) bezeichnet die Wendung als Sprichwort. Vgl. **Summum ius...**\n\n**Iustitia erga deum religio dicitur.**\n\nRechtsempfinden gegen\u00fcber den G\u00f6ttern nennt man Religion.\n\nNach Cicero, _Partitiones oratoriae_ 78.\n\n**iustus et tenax propositi vir**\n\nein gerechter und, was seine Vors\u00e4tze betrifft, hartn\u00e4ckiger Mann\n\nNach Horaz, _Carmina_ 3,3,1.\n\n**iuvat inconcessa voluptas** siehe **Indignere licet...**\n\n**Iuvat ipse labor.**\n\nDie M\u00fche an sich tut wohl.\n\nMartial, _Epigrammata_ 1,107,7 f.: \u00bbIn steriles nolunt campos iuga ferre iuvenci: \/ Pingue solum lassat, sed iuvat ipse labor.\u00ab (\u00bbD\u00fcrre Felder pfl\u00fcgen junge Stiere ungern: Fetter Boden macht zwar m\u00fcde, aber die M\u00fche an sich tut wohl.\u00ab)\n\n**Iuvat o meminisse beati \/ temporis.**\n\nEs ist erfreulich, sich einer gl\u00fccklichen Zeit zu erinnern.\n\nOvid, _Metamorphoses_ 7,797 f.\n\n**Iuvenile vitium est regere non posse impetum.**\n\nEs ist ein (typischer) Fehler der Jugend, sein Ungest\u00fcm nicht b\u00e4ndigen zu k\u00f6nnen.\n\nSeneca, _Troades_ 250. \u2013 Agamemnon warnt vor ma\u00dfloser Ausnutzung von Macht, denn (259): \u00bbModerata durant.\u00ab (\u00bbGez\u00fcgelte [Macht] ist von Dauer.\u00ab)\nL\n\n**Labitur occulte fallitque volatilis aetas.**\n\nDie fl\u00fcchtige Jugend gleitet heimlich dahin und t\u00e4uscht uns.\n\nOvid, _Amores_ 1,8,49.\n\n**Labor omnia vicit \/ improbus.**\n\n\u00dcberm\u00e4\u00dfige Arbeit hat alles \u00fcberwunden.\n\nVergil ( _Georgica_ 1,145 f.) beschreibt den Wechsel vom Goldenen Zeitalter unter Saturn zu den folgenden Zeitaltern unter Jupiter: Von nun an mu\u00dften die Menschen zwar arbeiten, doch f\u00fchrte die Arbeit auch zur Entstehung der Handwerke und K\u00fcnste.\n\n**Laetitia iuvenem, frons decet tristis senem.**\n\nFreude steht einem jungen Mann, ein trauriges Antlitz einem Alten.\n\nSeneca, _Phaedra_ 453.\n\n**Laetitia vana evadit.**\n\nDie Freude entschwindet leer.\n\nLivius, _Ab urbe condita_ 23,12,12.\n\n**Laetus sorte tua vives sapienter.**\n\nIn Freude \u00fcber dein Los wirst du weise leben.\n\nHoraz, _Epistulae_ 1,10,44.\n\n**Lapidem, \/ non hominem putas.**\n\nDu h\u00e4ltst (mich) wohl f\u00fcr einen Stein, nicht f\u00fcr einen Menschen.\n\nTerenz, _Hecyra_ 214 f.\n\n**lapsus linguae**\n\nein Ausrutscher der Zunge\n\n(d. h.: ein Versprecher)\n\nNach Sirach 20,18.\n\n**Laterem lavem.**\n\nIch d\u00fcrfe wohl einen Ziegelstein waschen.\n\n(d. h.: etwas vollkommen Sinnloses tun)\n\nTerenz, _Phormio_ 186.\n\n**latere tecto abscedere**\n\nmit gedeckter Flanke abziehen\n\n(d. h.: mit heiler Haut davonkommen)\n\nTerenz, _Heauton timorumenos_ 672.\n\n**Latet anguis in herba.**\n\nEs lauert eine Schlange im Grase.\n\nVergil, _Bucolica_ 3,93.\n\n**Latet error; equo ne credite, Teucri!**\n\nEs verbirgt sich (darin) ein Betrug; mi\u00dftraut dem Pferd, Teukrer!\n\nVergil, _Aeneis_ 2,48. \u2013 Laokoon warnt die Trojaner vor dem h\u00f6lzernen Pferd der Griechen. Vgl. **Danaum fatale munus** , **donum exitiale Minervae** und **Quidquid id est, timeo Danaos...**\n\n**Latet et lupus ore cruento \/ sub vellere molli.**\n\nAuch der Wolf verbirgt sich mit bluttriefendem Maul unter einem weichen (Schaf-)Fell.\n\nPrudenz, _Psychomachia_ 791 f.\n\n**Laudamus veteres, sed nostris utimur annis.**\n\nWir loben die alten Zeiten, doch wir leben (gut und gern) in den unsrigen.\n\nOvid, _Fasti_ 1,225.\n\n**Laudant illa, sed ista legunt.**\n\nSie loben jenes, aber dieses lesen sie.\n\nMartial, _Epigrammata_ 4,49,10.\n\n**Laudato ingentia rura, \/ exiguum colito!**\n\nLobe die riesigen Felder, doch bestelle ein kleines!\n\nVergil, _Georgica_ 2,412 f.\n\n**laudator temporis acti**\n\nein Lobredner vergangener Zeit\n\nHoraz, _Ars poetica_ 173. \u2013 Mit dieser Wendung verspottet man heute die \u00bbewig Gestrigen\u00ab.\n\n**Laudatur ab his, culpatur ab illis.**\n\nVon diesen wird er gelobt, von jenen getadelt.\n\nHoraz, _Sermones_ 1,2,11.\n\n**laudat venalis, qui volt extrudere merces** siehe **Multa fidem promissa levant**\n\n**Laus alit artes.**\n\nLob n\u00e4hrt die Kunst.\n\nSeneca ( _Epistulae morales_ 102,16) zitiert einen ungenannten \u00bbalten Dichter\u00ab; siehe Fragment poet. Rom. inc. p. 137,1 Baehrens.\n\n**Legem enim brevem esse oportet, quo facilius ab imperitis teneatur.**\n\nEin Gesetz mu\u00df n\u00e4mlich kurz sein, damit es um so leichter von Unkundigen behalten werde.\n\nSeneca ( _Epistulae morales_ 94,38) zitiert hier Posidonius, um sich anschlie\u00dfend von ihm zu distanzieren.\n\n**Leges bello siluere coactae.**\n\nDie Gesetze wurden durch den Krieg gezwungen zu schweigen.\n\nLukan, _De bello civili_ 1,277.\n\n**Legi, intellexi, condemnavi.**\n\nIch las, begriff, verdammte.\n\nNach Sozomenus, _Historia ecclesiastica_ 5,18. \u2013 Mit diesen Worten reagierte Julian Apostata auf die Schrift \u00bb\u00dcber die Wahrheit\u00ab von Apollinaris von Laodicaea. Die Bisch\u00f6fe antworteten ihrerseits: \u00bbDu hast es gelesen, aber nicht verstanden. Wenn du es verstanden h\u00e4ttest, h\u00e4ttest du es nicht verurteilt.\u00ab\n\n**Legis virtus haec est: imperare, vetare, permittere, punire.**\n\nDie Eigenschaft des Gesetzes ist zu befehlen, zu verbieten, zu erlauben und zu bestrafen.\n\n_Digesta_ 1,3,7.\n\n**Legum denique idcirco omnes servi sumus, ut liberi esse possimus.**\n\nWir sind schlie\u00dflich alle deswegen Diener der Gesetze, um frei sein zu k\u00f6nnen.\n\nCicero, _Pro A. Cluentio Habito_ 146.\n\n**Lenis alit flammas, grandior aura necat.**\n\nEin leichter Wind n\u00e4hrt die Flammen, ein st\u00e4rkerer l\u00f6scht sie.\n\nOvid, _Remedia amoris_ 808. Vgl. **Nutritur vento...**\n\n**Lenit albescens animos capillus.**\n\nMein ergrauendes Haar d\u00e4mpft meine Leidenschaften.\n\nHoraz, _Carmina_ 3,14,25.\n\n**Lentescunt tempore curae.**\n\nMit der Zeit lassen die Sorgen nach.\n\nOvid, _Ars amatoria_ 2,357.\n\n**Leones \/ non sunt papilionibus molesti.**\n\nL\u00f6wen sind Schmetterlingen nicht l\u00e4stig.\n\nMartial, _Epigrammata_ 12,61,5 f. Vgl. **Aquila non captat muscas**.\n\n**leonina societas** siehe **Numquam est fidelis...**\n\n**Lepus tute es, pulpamentum quaeris?**\n\nSelbst ein Hase, suchst du nach Wildbret?\n\nTerenz, _Eunuchus_ 426.\n\n**Leti mille repente viae.**\n\nIn den Tod f\u00fchren unvermutet tausend Wege.\n\nTibull, _Elegiae_ 1,3,50.\n\n**Letum non omnia finit.**\n\nDer Tod beendet nicht alles (erg.: denn es gibt noch die Manen, die Geister der Verstorbenen).\n\nProperz, _Elegiae_ 4,7,1.\n\n**Leve est miserias ferre, perferre est grave.**\n\nEs ist leicht, Ungl\u00fcck zu ertragen, es zu \u00fcberstehen ist schwer.\n\nSeneca, _Thyestes_ 307.\n\n**levi defungi poena**\n\nmit einer leichten Strafe davonkommen\n\nNach Livius, _Ab urbe condita_ 29,21,6.\n\n**levis notae macula adspergi**\n\nmit dem Flecken einer leichten R\u00fcge versehen werden\n\n(d. h.: mit einer leichten R\u00fcge davonkommen)\n\nNach _Codex Iustinianus_ 3,28,27.\n\n**Levissimus quisque et futuri improvidus spe vana tumens.**\n\nGerade die Leichtsinnigen und die, die sich nicht um die Zukunft k\u00fcmmern, schwellen an vor eitler Hoffnung.\n\nTacitus, _Historiae_ 1,88,3.\n\n**leviter malo perstringi**\n\nleicht vom Unheil gestreift werden\n\n(vgl.: mit einem blauen Auge davonkommen)\n\nNach Seneca, _Naturales quaestiones_ 6,1,2.\n\n**Lex cogit neminem ad impossibilia** siehe **Impossibilium nulla...**\n\n**Lex non promulgata non est rata.**\n\nEin unver\u00f6ffentlichtes Gesetz ist nicht rechtskr\u00e4ftig.\n\nVgl. _Digesta_ 50,16,131,1.\n\n**Lex posterior derogat priori.**\n\nEin sp\u00e4teres Gesetz hebt ein fr\u00fcheres auf.\n\n_Digesta_ 1,4,4.\n\n**Lex prospicit , non respicit.**\n\nEin Gesetz blickt voraus, nicht zur\u00fcck (d. h:. es gilt nicht r\u00fcckwirkend).\n\nHW: 13711.\n\n**Lex universa est, quae iubet nasci et mori.**\n\nEs gibt ein allgemeing\u00fcltiges Gesetz, das Entstehen und Vergehen befiehlt.\n\nPublilius Syrus, _Sententiae_ L 5.\n\n**Libenter homines id, quod volunt, credunt.**\n\nGern glauben die Menschen das, was sie wollen.\n\nCaesar, _De bello Gallico_ 3,18,6.\n\n**Liberae sunt enim nostrae cogitationes.**\n\nFrei sind n\u00e4mlich unsere Gedanken.\n\nCicero, _Pro T. Annio Milone_ 79.\n\n**Libera fortuna mors est; capit omnia tellus, \/ quae genuit.**\n\nLosgel\u00f6st vom Schicksal ist der Tod; die Erde nimmt alles (wieder) auf, was sie hervorgebracht hat.\n\nLukan, _De bello civili_ 7,818 f.\n\n**Libertas: potestas vivendi, ut velis** siehe **Quod est enim libertas...**\n\n**Libido effrenata effrenatam appetentiam efficit.**\n\nUngez\u00fcgelte Lust f\u00fchrt zu ungez\u00fcgelter Begierde.\n\nNach Cicero, _Tusculanae disputationes_ 4,15.\n\n**licentia poetica** siehe **poetica licentia**\n\n**Lingua fuit damno.**\n\nSch\u00e4dlich war die Zunge (d. h.: das Gerede).\n\nOvid, _Metamorphoses_ 2,540.\n\n**Lingua haeret metu.**\n\nDie Zunge stockt (mir) vor Angst.\n\nTerenz, _Eunuchus_ 977.\n\n**Lingua mali pars pessima servi.**\n\nDie Zunge ist der schlimmste Teil eines schlechten Sklaven.\n\nJuvenal, _Saturae_ 9,121.\n\n**Lingua, sile! Non est ultra narrabile quicquam.**\n\nSchweig, Zunge! Es gibt da nichts weiter zu erz\u00e4hlen.\n\nOvid, _Epistulae ex Ponto_ 2,2,59.\n\n**Linque severa!**\n\nLa\u00df den Ernst beiseite!\n\nHoraz, _Carmina_ 3,8,28.\n\n**lippis notum et tonsoribus**\n\nden Trief\u00e4ugigen und den Barbieren bekannt (d. h. allen, die Zeit f\u00fcr Klatsch haben)\n\nHoraz, _Sermones_ 1,7,3.\n\n**Lis litem parit.**\n\nStreit bringt Streit hervor.\n\nHW: 13871a.\n\n**Lis verbis minimis interdum maxima crescit.**\n\nEin furchtbarer Streit entsteht bisweilen aus den geringsten Worten.\n\n_Disticha Catonis_ 2,11,2.\n\n**lite pendente**\n\nbei schwebendem Verfahren\n\n_Codex Iustinianus_ 8,36(37),2.\n\n**litigare cum ventis**\n\nmit den Winden streiten\n\n(d. h.: etwas Sinnloses tun)\n\nNach Petron, _Satyrica_ 83,7.\n\n**Litore quot conchae, tot sunt in amore dolores.**\n\nSoviel Muscheln am Strand, soviel Schmerzen sind in der Liebe.\n\nOvid, _Ars amatoria_ 2,519.\n\n**Litura tamen exstet!**\n\nDie Tilgung soll wenigstens (sichtbar) bleiben!\n\nSueton ( _Vita divi Claudii_ 16,1) berichtet, Claudius habe, immer wenn er einen Vermerk aus der Zensur streichen lie\u00df, darauf geachtet, da\u00df man wenigstens die Streichung erkenne.\n\n**Litus ama!... Altum alii teneant.**\n\nLiebe den Strand! Auf das hohe Meer m\u00f6gen andere fahren.\n\nVergil, _Aeneis_ 5,163 f. Ratschlag an einen Steuermann.\n\n**litus arare**\n\nden Strand pfl\u00fcgen\n\n(d. h.: sich vergeblich abm\u00fchen)\n\nOvid, _Tristia_ 5,4,48; vgl. _Heroides_ 5,116: \u00bbNon profecturis litora bubus aras.\u00ab (\u00bbDu pfl\u00fcgst den Strand mit Rindern, die nicht vorankommen.\u00ab)\n\n**Loca quae nimium grata fuere, cave!**\n\nH\u00fcte dich vor Orten, die dir allzu lieb waren.\n\nOvid, _Remedia amoris_ 738.\n\n**Longius aut propius mors sua quemque manet.**\n\nFr\u00fcher oder sp\u00e4ter erwartet jeden der Tod.\n\nProperz, _Elegiae_ 2,28,58.\n\n**longo, sed proximus intervallo**\n\nin weitem Abstand, aber (doch) der n\u00e4chste\n\nVergil, _Aeneis_ 5,320.\n\n**longum iter est per praecepta** siehe **Homines amplius...**\n\n**Lucet . Eamus, \/ quo ducit gula!**\n\nEs wird hell. Gehen wir, wohin uns die Kehle f\u00fchrt!\n\nHoraz ( _Epistulae_ 1,6,56 f.) fordert ironisch dazu auf, sich der Prasserei hinzugeben, falls essen der Weg zum Gl\u00fcck sei. So f\u00e4hrt er fort: \u00bbPiscemur, venemur!\u00ab (\u00bbLa\u00dft uns also fischen, jagen!\u00ab)\n\n**Lucri bonus est odor ex re \/ qualibet.**\n\nWoher der Gewinn auch stammen mag, sein Geruch ist gut.\n\n(Vgl.: Geld stinkt nicht.)\n\nJuvenal ( _Saturae_ 14,204 f.) spottet \u00fcber den Stellenwert, den der Besitz bei den Menschen einnimmt: \u00bbHoc discunt omnes ante alpha et beta puellae.\u00ab (\u00bbDies lernen alle M\u00e4dchen noch vor dem Alphabet.\u00ab) Vgl. **Non olet**.\n\n**lucus a non lucendo**\n\nEinen Hain (nennt man) _lucus_ , weil in ihm kein Licht leuchtet (lat. _non lucet_ ).\n\nQuintilian ( _Institutio oratoria_ 1,6,34) h\u00e4lt zahlreiche Volksetymologien f\u00fcr fraglich, die ein Wort aus dessen Gegenteil ableiten. Auch Varro verspottet solche R\u00fcckschl\u00fcsse; vgl. **canis a non canendo**.\n\n**ludibrio habere**\n\nzum Gesp\u00f6tt machen\n\nTerenz, _Hecyra_ 149.\n\n**Ludit in humanis divina potentia rebus.**\n\nIn menschlichen Angelegenheiten spielt eine g\u00f6ttliche Macht mit.\n\nOvid, _Epistulae ex Ponto_ 4,3,49.\n\n**Ludunt formosae, casta est, quam nemo rogavit.**\n\nDie Sch\u00f6nen verlustieren sich; z\u00fcchtig ist die, die niemand (je) fragte.\n\nOvid, _Amores_ 1,8,43.\n\n**lupo agnum eripere**\n\ndem Wolf ein Lamm entrei\u00dfen\n\n(d. h.: etwas nahezu Unm\u00f6gliches tun)\n\nPlautus, _Poenulus_ 776.\n\n**Lupo ovem commisisti.**\n\nDu hast einem Wolf ein Schaf anvertraut.\n\nTerenz, _Eunuchus_ 832. Vgl. **O praeclarum custodem ovium lupum**.\n\n**lupum sub ovis pelle celare**\n\nden Wolf unter dem Schafspelz verbergen\n\nLaktanz zugeschrieben; vgl. Matth\u00e4us 7,15.\n\n**Lupus in fabula.**\n\n(Da ist) der Wolf (wie) in der Fabel.\n\n(Vgl.: Wenn man den Teufel nennt, kommt er gerennt.)\n\nTerenz, _Adelphoe_ 537, und Cicero, _Ad Atticum_ 13,33,1. Vgl. Plautus, _Stichus_ 577.\n\n**Lupus non curat numerum.**\n\nDer Wolf k\u00fcmmert sich nicht um die Zahl.\n\n(D. h.: Er fri\u00dft auch die mit einer Zahl gekennzeichneten Schafe.)\n\nNach Vergil, _Bucolica_ 7,51 f.\n\n**lux mundi**\n\ndas Licht der Welt\n\nMatth\u00e4us 5,14.\n\n**Luxuriae quicquid dederis, perfluet.**\n\nAlles, was du den Ausschweifungen hingabst, wird dir verloren sein.\n\nPhaedrus, _Fabulae_ Appendix Perottina 7,12.\n\n**Luxuriant animi rebus plerumque secundis.**\n\nIn gl\u00fccklichen Zeiten werden die Herzen meist \u00fcberm\u00fctig.\n\n(Vgl.: Wo Gl\u00fcck aufgeht, da geht Demut unter.)\n\nOvid, _Ars amatoria_ 2,437.\nM\n\n**Macte \/ virtute (esto)!**\n\nSei deiner Tugend wegen gepriesen!\n\n(z. B.) Horaz, _Sermones_ 1,2,31 f.; vgl. Vergil, _Aeneis_ 9,641 f. \u2013 W\u00f6rtlich: (Sei) verherrlicht wegen deiner Tugend!\n\n**Magis illa iuvant, quae pluris emuntur.**\n\nMehr erfreut jenes, das man teurer ersteht.\n\nJuvenal, _Saturae_ 11,16.\n\n**Magistra vitae philosophia.**\n\nDie Philosophie ist die Lehrmeisterin des Lebens.\n\nCicero, _Tusculanae disputationes_ 2,16.\n\n**Magistrum memet ipsum habeo.**\n\nIch habe mich selbst zum Lehrer.\n\nHieronymus ( _De viris illustribus_ , praefatio) bezeichnet diese Worte als bekannte Wendung.\n\n**Magis unde cadas, \/ quam quo refert.**\n\nWichtiger ist, von wo man f\u00e4llt, als wohin.\n\nSeneca, _Thyestes_ 925 f.\n\n**Magnae periclo sunt opes obnoxiae.**\n\nGro\u00dfe Reicht\u00fcmer sind Gefahren ausgesetzt.\n\nPhaedrus, _Fabulae_ 2,7,14.\n\n**Magna inter molles concordia.**\n\nGro\u00dfe \u00dcbereinstimmung herrscht unter den Woll\u00fcstigen.\n\nJuvenal, _Saturae_ 2,47.\n\n**magnas inter opes inops**\n\ninmitten gro\u00dfer Reicht\u00fcmer arm\n\nHoraz, _Carmina_ 3,16,28.\n\n**Magna tamen spes est in bonitate dei.**\n\nDennoch besteht gro\u00dfe Hoffnung auf die G\u00fcte des Gottes.\n\nOvid, _Epistulae ex Ponto_ 1,6,46.\n\n**Magna voce sonas manuque tota.**\n\nDu l\u00e4\u00dft es mit lauter Stimme und der ganzen Hand (d. h.: eifrig gestikulierend) erklingen.\n\nMartial, _Epigrammata_ 6,19,8.\n\n**Magni animi magnis honoribus fiunt.**\n\nEdle Gesinnung kommt von hohen \u00c4mtern.\n\nNach Livius, _Ab urbe condita_ 4,35,9.\n\n**Magni constant regum amicis bona consilia.**\n\nGute Ratschl\u00e4ge kommen Freunden von K\u00f6nigen teuer zu stehen.\n\nNach Seneca, _De ira_ 3,14,6. \u2013 Der K\u00f6nig Kambyses war von seinem Freund Praexaspes dazu angehalten worden, weniger Wein zu trinken. Cambyses jedoch behauptete, auch nach vielen Bechern Wein noch eine ruhige Hand zu haben, und bewies dies, indem er in betrunkenem Zustand Praexaspes' Sohn mitten ins Herz scho\u00df.\n\n**Magnifica verba mors prope admota excutit.**\n\nPrahlerische Worte vertreibt der nah heranger\u00fcckte Tod.\n\nSeneca, _Troades_ 575.\n\n**Magnificum est laudari a laudato viro.**\n\nEs ist r\u00fchmlich, von einem gelobten Mann gelobt zu werden.\n\nNach z. B. Cicero ( _Tusculanae disputationes_ 4,67), der Naevius zitiert ( _Hector profiscens_ Fragment 17 Warmington). Vgl. Seneca, _Epistulae morales_ 102,16.\n\n**Magni fures parvum ducunt.**\n\nDie gro\u00dfen Diebe f\u00fchren den kleinen ab.\n\nNach Diogenes Laertius ( _Vitae philosophorum: Diogenes_ 6,45), der berichtet, der Philosoph Diogenes habe diese Bemerkung gemacht, als er sah, wie Priester einen Unterbeamten, der eine Schale aus einem Tempel entwendet hatte, abf\u00fchrten.\n\n**magni nominis umbra**\n\nein Schatten seines gro\u00dfen Namens\n\n(vgl.: nur ein Schatten seiner selbst sein)\n\nLukan ( _De bello civili_ 1,135) bezeichnet so Pompeius.\n\n**Magno se iudice quisque tuetur.**\n\nJeder sch\u00fctzt sich durch (Berufung auf) eine hohe Autorit\u00e4t.\n\nLukan ( _De bello civili_ 1,127) \u00fcber Caesar und Pompeius.\n\n**Magnum iter ascendo, sed dat mihi gloria vires.**\n\nEinen langen Weg steige ich hinauf, doch (die Hoffnung auf) Ruhm gibt mir Kraft.\n\nProperz, _Elegiae_ 4,10,3.\n\n**Magnum timoris remedium clementia est.**\n\nEin wirksames Heilmittel gegen Angst ist Milde.\n\nSeneca, _Octavia_ 442.\n\n**Magnum vectigal est parsimonia** siehe **Non intellegunt homines...**\n\n**Magnus sibi ipse non facit finem dolor.**\n\nGro\u00dfer Schmerz setzt sich selbst keine Grenze.\n\nSeneca, _Troades_ 786.\n\n**Maiora cupimus, quo maiora venerunt.**\n\nWir begehren um so mehr, je mehr uns zugefallen ist.\n\nSeneca, _De beneficiis_ 2,27,3.\n\n**Maior e longinquo reverentia.**\n\nGr\u00f6\u00dfer ist die Verehrung aus der Ferne.\n\nTacitus, _Annales_ 1,47,2.\n\n**Maiores maiora sonant.**\n\nGr\u00f6\u00dferes t\u00f6nen Gr\u00f6\u00dfere.\n\nMartial, _Epigrammata_ 9, prologus 7.\n\n**Maioresque cadunt altis de montibus umbrae.**\n\nUnd l\u00e4ngere Schatten fallen von den hohen Bergen.\n\nVergil, _Bucolica_ 1,84.\n\n**Maior famae sitis est quam \/ virtutis.**\n\nGr\u00f6\u00dfer ist der Hunger nach Ruhm als der nach Tugend.\n\nJuvenal, _Saturae_ 10,140 f.\n\n**Maiori concede, minorem ne contempseris!**\n\nWeiche dem Gr\u00f6\u00dferen, verachte nicht den Kleineren!\n\n_Disticha Catonis_ , prologus 10 und 47. Diese beiden getrennt \u00fcberlieferten Mahnungen werden h\u00e4ufig zusammen zitiert. Vgl. **Cedo maiori.**\n\n**Maior post otia virtus.**\n\nGr\u00f6\u00dfer ist die (Bereitschaft zur) Tugendhaftigkeit nach der Mu\u00dfe.\n\n(Vgl.: Rast soll vor der Tat gehen.)\n\nStatius, _Silvae_ 4,4,34.\n\n**Maiorque videtur \/ et melior vicina seges.**\n\nGr\u00f6\u00dfer und besser erscheint des Nachbarn Saat.\n\n(Vgl.: Des Nachbarn Braten ist stets feister.)\n\nJuvenal, _Saturae_ 14,142 f.\n\n**Maior sum, quam cui possit fortuna nocere.**\n\nIch bin zu gro\u00df, als da\u00df mir das Schicksal schaden k\u00f6nnte.\n\nOvid, _Metamorphoses_ 6,195. \u2013 Worte der Niobe kurz vor ihrem Untergang.\n\n**Maius erat nostris viribus istud onus.**\n\nZu gro\u00df war f\u00fcr unsere Kr\u00e4fte diese Last.\n\nOvid, _Tristia_ 4,10,36.\n\n**Maius opus mores composuisse suos.**\n\nEine gr\u00f6\u00dfere Aufgabe ist es, sein Temperament zu b\u00e4ndigen.\n\nOvid, _Ars amatoria_ 3,370.\n\n**Mala causa silenda est.**\n\nEin \u00fcbler Beweggrund mu\u00df verschwiegen werden.\n\nOvid, _Epistulae ex Ponto_ , 3,1,147.\n\n**Malae naturae numquam doctore indigent.**\n\nB\u00f6se Gesinnung bedarf niemals eines Lehrers.\n\nPublilius Syrus, _Sententiae_ M 20.\n\n**Mala est inopia, ex copia quae nascitur.**\n\nSch\u00e4ndlich ist die Armut, die aus Reichtum hervorgeht.\n\nPublilius Syrus, _Sententiae_ M 69.\n\n**Mala mens, malus animus.**\n\n\u00dcbler Geist, \u00fcbler Sinn.\n\nTerenz, _Andria_ 164.\n\n**Malam rem, cum velis honestare, improbes.**\n\nWenn du etwas \u00dcbles ehren willst, verdamme es.\n\nPublilius Syrus, _Sententiae_ M 53.\n\n**Male creditis hosti!**\n\nZu eurem Verderben glaubt ihr dem Feind!\n\nOvid, _Fasti_ 2,225.\n\n**Male facere qui vult, numquam non causam invenit.**\n\nWer \u00dcbles tun will, findet immer einen Grund daf\u00fcr.\n\nPublilius Syrus, _Sententiae_ M 28.\n\n**Male parta male dilabuntur.**\n\nAuf schlimme Weise Erworbenes zerrinnt auf schlimme Weise.\n\n(Vgl.: \u00dcbel gewonnen, \u00fcbel zerronnen.)\n\nCicero, _Orationes Philippicae_ 2,65, zitiert Naevius Fragment Ex trag.? 38 Warmington. Vgl. Plautus, _Poenulus_ 844.\n\n**Male secum agit aeger, medicum qui heredem facit.**\n\nSchlecht geht der Kranke mit sich selbst um, der den Arzt als Erben einsetzt.\n\nPublilius Syrus, _Sententiae_ M 24.\n\n**Malignus comes quamvis candido et simplici rubiginem suam adfricuit.**\n\nEin b\u00f6sartiger Begleiter steckt auch einen noch so makellosen und ehrlichen Menschen mit seiner Verkommenheit an.\n\n(Vgl.: Schlechte Gesellschaft verdirbt gute Sitten.)\n\nSeneca, _Epistulae morales_ 7,7.\n\n**Malis consiliis pares adepti eventus.**\n\nDurch schlechte Pl\u00e4ne haben sie ein ebensolches (d. h.: ein schlechtes) Ergebnis erlangt.\n\nLivius, _Ab urbe condita_ 6,8,8; vgl. auch 6,10,8.\n\n**Mali sunt homines, qui bonis dicunt male.**\n\nSchlecht sind die Menschen, die schlecht von Guten sprechen.\n\nPlautus, _Bacchides_ 118.\n\n**Mallem nescisse futura.**\n\nIch wollte, ich w\u00fc\u00dfte nicht um die Zukunft.\n\nOvid, _Metamorphoses_ 2,660. \u2013 Ocyro\u00eb versagt die Stimme, als sie weiter von der Zukunft des Aesculap-Kindes sowie der ihres Vaters Chiron k\u00fcnden will. Daraufhin verwandelt sie sich in eine Stute.\n\n**Malleus manubrio sapientior.**\n\nDer Hammer ist gescheiter als der Stiel.\n\n(Vgl.: Das Ei will kl\u00fcger sein als die Henne.)\n\nHWS: 38090. Vgl. Plautus, _Epidicus_ 524 f.\n\n**Malo arboris nodo malus cuneus requirendus est.**\n\nF\u00fcr einen argen Knoten im Baum mu\u00df man einen argen Keil suchen.\n\nHieronymus ( _Epistulae_ 69,5) beschreibt die Wendung als bekanntes Sprichwort. Es wird oft zitiert als \u00bbCuneus cuneum trudit.\u00ab (\u00bbEin Keil treibt den anderen.\u00ab)\n\n**Malo benefacere tantundem est periclum, \/ quantum bono malefacere.**\n\nEinem Schlechten Gutes zu tun ist ebenso gef\u00e4hrlich wie einem Guten Schlechtes zu tun.\n\nPlautus, _Poenulus_ 633 f.\n\n**Malo, quam bene olere, nil olere.**\n\nIch will lieber nach nichts riechen, als lieblich zu riechen.\n\nMartial ( _Epigrammata_ 6,55,5) ist ein allzu lieblicher Geruch stets verd\u00e4chtig. Vgl. **Non bene olet...**\n\n**Malum alienum cave gaudium facias tuum!**\n\nH\u00fcte dich, anderer \u00dcbel zu deiner Freude zu machen!\n\nCaecilius Balbus, _Sententiae_ 110.\n\n**Malum consilium consultori est pessimum.**\n\nEin schlechter Ratschlag ist f\u00fcr den am schlechtesten, der ihn erteilt (erg.: denn er wird daf\u00fcr bestraft werden).\n\nBei Varro ( _De re rustica_ 3,2,1) wird die Wendung bereits als Sprichwort bezeichnet. Vgl. Gellius ( _Noctes Atticae_ 4,5, nach Hesiod, _Werke und Tage_ 266), der folgende Anekdote zu berichten wei\u00df: Etrurische Seher, die von den R\u00f6mern um eine Weisung gebeten worden waren, h\u00e4tten absichtlich einen falschen Rat erteilt. Als dies aufgedeckt wurde, seien sie get\u00f6tet worden, und das Volk h\u00e4tte obige Worte in den Stra\u00dfen gesungen.\n\n**malum discordiae**\n\nder Apfel der Zwietracht\n\nIustinus, _Historiae Philippicae_ 12,15,11. \u2013 Gemeint ist der Zankapfel der Eris mit der Aufschrift \u00bbDer Sch\u00f6nsten\u00ab, den sie auf der Hochzeit von Peleus und Thetis unter die G\u00e4ste warf (Hygin, _Fabeln_ 92). So l\u00f6ste sie einen Streit unter den G\u00f6ttinnen Hera, Aphrodite und Athene aus, die Paris als Schiedsrichter w\u00e4hlten. Aphrodite, die Paris Helena als Belohnung versprochen hatte und dadurch letztlich den Trojanischen Krieg ausl\u00f6ste, ging als Siegerin aus diesem Streit hervor.\n\n**malum necessarium**\n\nein notwendiges \u00dcbel\n\n_Historia Augusta_ , \u00bbVita Alexandri Severi\u00ab 46,5.\n\n**Malum nullum est sine aliquo bono.**\n\nKein \u00dcbel ist g\u00e4nzlich ohne Gutes.\n\n(Vgl.: Kein \u00dcbel ist so gro\u00df, es hat ein Gl\u00fcck im Scho\u00df.)\n\nNach Plinius d. \u00c4., _Naturalis historia_ 27,9.\n\n**Malum panem tibi tenerum et siligineum fames reddet.**\n\nSchlechtes Brot wird dir der Hunger wie zartes Weizenbrot erscheinen lassen.\n\nNach Seneca, _Epistulae morales_ 123,2.\n\n**Manat rumor sine capite.**\n\nEs zirkuliert ein Ger\u00fccht ohne Kopf (d. h.: dessen Urheber man nicht ausfindig machen kann).\n\nCicero, _Pro Cn. Plancio_ 23.\n\n**Maneat nostros ea cura nepotes.**\n\nDiese Sorge erwarte unsere Enkel.\n\nVergil, _Aeneis_ 3,505. \u2013 Gemeint ist die Einigung von Griechenland und Italien, der alten und der neuen Heimat des Aeneas.\n\n**Manet alta mente repostum.**\n\nEs bleibt in tiefster Seele verborgen.\n\nVergil, _Aeneis_ 1,26. \u2013 Junos Groll ob des Paris-Urteils ist tief in ihrer Seele verankert.\n\n**manibus pedibusque obnixe omnia facere**\n\nsich mit H\u00e4nden und F\u00fc\u00dfen allem widersetzen\n\nNach Terenz, _Andria_ 161 f.\n\n**Manum de tabula!**\n\nDie Hand vom Bild!\n\nCicero, _Ad familiares_ 7,25,1; ebenso Petron, _Satyrica_ 76,9, und Plinius d. \u00c4., _Naturalis historia_ 35,80. \u2013 Plinius zufolge behauptete der Maler Apelles, er sei in allem genauso gut wie der ebenso ber\u00fchmte Maler Protogenes oder jener sei sogar besser; aber eines habe er, Apelles, Protegenes voraus: Er wisse zur rechten Zeit die Hand von der Maltafel zu nehmen (\u00bbquod manum de tabula sciret tollere\u00ab). Cicero und Petron benutzen die Wendung in der Bedeutung \u203azur rechten Zeit aufh\u00f6ren\u2039.\n\n**Manus manum lavat.**\n\nEine Hand w\u00e4scht die andere.\n\nSeneca, _Apocolocyntosis_ 9,6, und Petron, _Satyrica_ 45,13. Vgl. **Asinus asinum fricat.**\n\n**Marcet sine adversario virtus.**\n\nDie Tatkraft erlahmt ohne Hindernisse (d. h.: wenn sie nicht herausgefordert wird).\n\nSeneca, _De providentia_ 2,4.\n\n**Mare quidem commune certo est omnibus.**\n\nDas Meer jedoch geh\u00f6rt sicher allen gemeinsam.\n\nPlautus, _Rudens_ 975.\n\n**maria montisque polliceri**\n\nMeere und Berge versprechen\n\n(d. h.: das Blaue vom Himmel versprechen)\n\nSallust, _Bellum Catilinarium_ 23,3.\n\n**Mars dubius, nec certa Venus** siehe **Militat omnis amans**\n\n**Marte nostro**\n\nmit unserem eigenen Kriegsgott\n\n(d. h.: auf eigene Faust)\n\nCicero ( _De officiis_ 3,34) bezeichnet diese F\u00fcgung als sprichw\u00f6rtlich. Vgl. **suo Marte**.\n\n**massa perditionis, unde electi sumus**\n\ndie Masse der Verdammnis, aus der wir auserw\u00e4hlt sind\n\nAugustinus ( _Sermones_ 26,12,13) nimmt hier Bezug auf R\u00f6merbrief 9,21.\n\n**Materiam superabat opus.**\n\nDas Werk \u00fcbertraf das Material.\n\n(D. h.: Noch bewundernswerter als das Material war die Ausf\u00fchrung des Werks.)\n\nOvid, _Metamorphoses_ 2,5 \u2013 Beschreibung der _regia Solis_ , des Sonnenpalastes.\n\n**Mater timidi flere non solet.**\n\nDie Mutter eines \u00c4ngstlichen pflegt nicht zu weinen.\n\nNach Cornelius Nepos, _Thrasybulus_ 2,3. Er bezeichnet die Wendung als altes Sprichwort, das besage, da\u00df man sich um einen Vorsichtigen und \u00c4ngstlichen weniger sorgen m\u00fcsse als um einen \u00dcberm\u00fctigen.\n\n**Matres omnes filiis \/ in peccato adiutrices.**\n\nAlle M\u00fctter sind ihren S\u00f6hnen Helferinnen bei Verfehlungen.\n\nTerenz, _Heauton timorumenos_ 991 f.\n\n**Matura, dum libido manet!**\n\nBeeil dich, solange die Lust anh\u00e4lt!\n\n(vgl.: das Eisen schmieden, solange es noch hei\u00df ist)\n\nNach Terenz, _Phormio_ 716.\n\n**Mature fias senex, si diu velis esse senex!**\n\nWerde fr\u00fch alt, wenn du lange alt sein willst!\n\nNach Cicero ( _De senectute_ 32), der sich von dieser Aufforderung, die er als altbekanntes Sprichwort bezeichnet, distanziert.\n\n**Maxima debetur puero reverentia.**\n\nH\u00f6chste Ehrfurcht schuldet man dem Knaben.\n\nJuvenal ( _Saturae_ 14,47) mahnt Eltern, sich vor den Augen ihrer Kinder zu beherrschen, da sie schlechte Beispiele schnell nachahmten.\n\n**Maxima de nihilo nascitur historia.**\n\nAus dem Nichts entsteht eine sehr gro\u00dfe Geschichte.\n\nProperz, _Elegiae_ 2,1,16.\n\n**Maxima quaeque domus servis est plena superbis.**\n\nGerade die gr\u00f6\u00dften H\u00e4user sind voll mit hochm\u00fctigen Sklaven.\n\nJuvenal, _Saturae_ 5,66.\n\n**Maximum indicium est malae mentis fluctatio.**\n\nDas sicherste Anzeichen f\u00fcr eine schlechte Geistesverfassung ist Unstetigkeit.\n\nSeneca, _Epistulae morales_ 120,20.\n\n**Maximum remedium irae dilatio est.**\n\nDas beste Mittel gegen Zorn ist der Aufschub (erg.: des Zornausbruchs).\n\n(Vgl.: Harren ist des Zornes Arznei.)\n\nSeneca, _De ira_ 3,12,4.\n\n**Mea culpa , mea culpa, mea maxima culpa.**\n\nDurch meine Schuld, durch meine Schuld, durch meine \u00fcbergro\u00dfe Schuld.\n\nAus dem \u00bbConfiteor _\u00ab_ der katholischen Messe.\n\n**mea parvitas** bzw. **mea tenuitas**\n\nmeine Wenigkeit\n\nValerius Maximus, _Dicta et facta memorabilia_ , prologus, bzw. Gellius, _Noctes Atticae_ 12,1,24.\n\n**Mea \/ virtute me involvo.**\n\nIch h\u00fclle mich in meine eigene Tugend.\n\nHoraz, _Carmina_ 3,29,54 f.\n\n**Media in vita in morte sumus.**\n\nMitten im Leben sind wir im Tod.\n\nAnfangsvers einer einzeln \u00fcberlieferten Strophe eines mittelalterlichen Kirchenliedes, das einigen Quellen zufolge vom St. Gallener M\u00f6nch Notker Balbulus stammt und seit dem 11. Jahrhundert belegt ist.\n\n**medias in res** siehe **ab ovo**\n\n**Medicamen calamitatis est aequanimitas.**\n\nIm Ungl\u00fcck ist Gleichmut die (rechte) Medizin.\n\nPublilius Syrus, _Sententiae_ M 34.\n\n**Medice, cura te ipsum!**\n\nArzt, heile dich selbst!\n\nLukas 4,23; vgl. Sirach 18,20.\n\n**Medicina vinci fata non possunt.**\n\nDas Schicksal kann nicht durch Medizin besiegt werden.\n\nNach Pseudo-Quintilian, _Declamationes_ 268,10.\n\n**Medicus curat, natura sanat.**\n\nDer Arzt kuriert, die Natur heilt.\n\nHW: 14564e.\n\n**Medicus enim nihil aliud est quam animi consolatio.**\n\nEin Arzt ist ja nichts anderes als Seelentr\u00f6stung.\n\nPetron, _Satyrica_ 42,5.\n\n**Mediocritas est inter nimium et parum** siehe **aurea mediocritas**\n\n**Medio de fonte leporum \/ surgit amari aliquid.**\n\nMitten aus dem Quell der Annehmlichkeiten steigt etwas Bitteres empor.\n\nLukrez ( _De rerum natura_ 4,1133 f.) \u00fcber die Nichtigkeit materieller Reicht\u00fcmer, deren Genu\u00df oft etwa durch Gewissensbisse getr\u00fcbt werde.\n\n**Medio tutissimus ibis.**\n\nIn der Mitte wirst du am sichersten gehen.\n\nOvid, _Metamorphoses_ 2,137. \u2013 Der Sonnengott r\u00e4t so seinem Sohn Phaeton, die Mitte \u2013 hier im konkreten Sinne gemeint \u2013 der Bahn einzuhalten. Das Zitat wird heute meist \u00fcbertragen gebraucht. Vgl. **Parce, puer, stimulis...**\n\n**Me duce carpe viam!**\n\nUnter meiner F\u00fchrung nimm den Weg!\n\nOvid, _Metamorphoses_ 8,208. \u2013 Daedalus mahnt Icarus, ihm genau zu folgen.\n\n**Melior est condicio possidentis.**\n\nBesser ist die Lage dessen, der Besitz hat.\n\n(Vgl.: Der Habich ist besser als der H\u00e4ttich.)\n\nNach _Digesta_ 20,1,10. \u2013 Laut Liebs besagt diese Rechtsregel, da\u00df eine Sache, um die sich zwei streiten, im Zweifelsfalle eher dem zugesprochen wird, der sie bereits in Besitz hatte, als dem, der sie f\u00fcr sich reklamiert. Meist wird die Wendung jedoch in einem allgemeineren Sinne verstanden: Konkreter Besitz ist besser als potentieller Besitz.\n\n**Melior mihi dextera lingua est.**\n\nMir dient meine Rechte besser als die Zunge.\n\nOvid, _Metamorphoses_ 9,29. \u2013 Hercules und der Flu\u00dfgott Achelous warben beide um Deianira. Nachdem Achelous seinen Gegner mit Worten geschm\u00e4ht hatte, setzte dieser sich mit Taten zu Wehr.\n\n**Melior tutiorque est certa pax quam sperata victoria.**\n\nBesser und sicherer ist gesicherter Frieden als erhoffter Sieg.\n\nLivius, _Ab urbe condita_ 30,30,19.\n\n**Melius est unum malum pati quam multa.**\n\nEs ist besser, ein \u00dcbel zu erleiden als viele.\n\n_Historia Augusta_ , \u00bbVita Alexandri Severi\u00ab 65,5. \u2013 Gemeint ist hier: Ein schlechter Herrscher, der sich mit guten Menschen umgibt, ist ein geringeres \u00dcbel als ein guter Herrscher, der sich mit schlechten Menschen umgibt.\n\n**Melius homines exemplis docentur.**\n\nBesser werden die Menschen durch Beispiele belehrt.\n\nPlinius d. J. ( _Panegyricus_ 45,6) f\u00e4hrt fort: \u00bbQuae in primis hoc in se boni habent, quod adprobant, quae praecipiunt fieri posse.\u00ab (\u00bbDiese haben vor allem den Vorzug, da\u00df sie den Beweis liefern, da\u00df das, was sie uns vorschreiben, auch ausgef\u00fchrt werden kann.\u00ab)\n\n**Me, me, adsum, qui feci!**\n\nMich, mich, ich bin es, der es getan hat!\n\nVergil, _Aeneis_ 9,427. \u2013 Nisus will Euryalus retten, indem er die Aufmerksamkeit der gegnerischen Rutuler auf sich lenkt.\n\n**Memento mori!** siehe **Respice post te...**\n\n**Memento omnia mihi et in omnes licere!**\n\nBedenke, da\u00df mir alles sogar gegen\u00fcber allen erlaubt ist.\n\nNach Sueton ( _Vita Caligulae_ 29,1) antwortete so Caligula auf eine Ermahnung seiner Gro\u00dfmutter Antonia. Vgl. **Si libet...**\n\n**Memento semper finis, et quia perditum non redit tempus.**\n\nDenke immer an das Ende, zumal auch da die verlorene Zeit nicht zur\u00fcckkehrt.\n\nThomas a Kempis, _Imitatio Christi_ 1,25,43.\n\n**Me miseram , quod amor non est medicabilis herbis!**\n\nIch Elende! Da\u00df die Liebe nicht durch Kr\u00e4uter heilbar ist!\n\nOvid, _Heroides_ 5,149.\n\n**Memoria minuitur, nisi eam exerceas.**\n\nDas Ged\u00e4chtnis l\u00e4\u00dft nach, wenn man es nicht \u00fcbt.\n\nNach Cicero, _De senectute_ 21.\n\n**Mendacem memorem esse oportet.**\n\nEin L\u00fcgner mu\u00df ein gutes Ged\u00e4chtnis haben.\n\nNach Quintilian ( _Institutio oratoria_ 4,2,91) eine g\u00e4ngige Wendung.\n\n**Mendaci homini ne verum quidem dicenti credere solemus.**\n\nEinem L\u00fcgner pflegen wir nicht einmal dann zu glauben, wenn er die Wahrheit spricht.\n\n(Vgl.: Wer einmal l\u00fcgt, dem glaubt man nicht, und wenn er auch die Wahrheit spricht.)\n\nNach Cicero, _De divinatione_ 2,146; vgl. Phaedrus, _Fabulae_ 1,10,1 f.: **Quicumque turpi...**\n\n**Mens agitat molem.**\n\nDer Geist bewegt die Materie.\n\nVergil, _Aeneis_ 6,727. \u2013 Der tote Anchises, der von seinem Sohn Aeneas in der Unterwelt besucht wird, steht ihm Rede und Antwort auf die Frage nach dem Ursprung und dem Schicksal der Seelen.\n\n**Mens est, quae diros sentiat ictus.**\n\nDer Geist ist es, der die grausamen Stiche sp\u00fcrt.\n\nOvid, _Metamorphoses_ 4,499.\n\n**Mensque magis gracili corpore nostra valet.**\n\nUnd mein Geist ist st\u00e4rker als mein zarter K\u00f6rper.\n\nOvid, _Epistulae ex Ponto_ 1,5,52.\n\n**Mensque pati durum sustinet aegra nihil.**\n\nUnd mein kranker Geist ist nicht in der Lage, etwas Hartes zu ertragen.\n\nOvid, _Epistulae ex Ponto_ 1,5,18.\n\n**mens sana in corpore sano**\n\nein gesunder Geist in einem gesunden K\u00f6rper\n\nJuvenal ( _Saturae_ 10,356) verspottet in seiner zehnten Satire die sinnlosen W\u00fcnsche, die die Menschen stets den G\u00f6ttern antr\u00fcgen. Wenn man um irgend etwas bitten solle, dann darum, da\u00df sich in einem gesunden K\u00f6rper ein gesunder Geist befinde (\u00bbOrandum est, ut sit mens sana in corpore sano\u00ab). Meist jedoch wird die Wendung als Aufforderung zu einer gesunden Lebensweise aufgefa\u00dft: In einem gesunden K\u00f6rper wohne n\u00e4mlich auch ein gesunder Geist.\n\n**Mens semper, quod timet, esse putat.**\n\nDer Verstand glaubt stets, da\u00df wahr sei, was er f\u00fcrchtet.\n\nOvid, _Ars amatoria_ 3,720.\n\n**mens sibi conscia recti**\n\nein sich des Rechten bewu\u00dfter Sinn\n\n(d. h.: Rechtsempfinden)\n\nVergil, _Aeneis_ 1,604.\n\n**mente atque sensu absentissimus**\n\nin Geist und Sinn v\u00f6llig abwesend\n\nAugustinus, _Confessiones_ 4,4,8.\n\n**Mentem praecipitat animus.**\n\nDas Herz st\u00fcrzt den Verstand um.\n\nNach Quintilian ( _Institutio oratoria_ 6,2,6), der aufzeigt, da\u00df ein objektives Urteil nicht mehr m\u00f6glich sei, sobald Gef\u00fchle im Spiel seien. Das Herz schreibe den Augen vor, was sie sehen sollten.\n\n**Meo sum pauper in aere.**\n\nIn meinem eigenen Geld bin ich arm.\n\n(D. h.: Ich habe zwar keine Reicht\u00fcmer, aber auch keine Schulden.)\n\nHoraz, _Epistulae_ 2,2,12.\n\n**Me quoque pectoris \/ temptavit in dulci iuventa \/ fervor.**\n\nAuch mich hat in der s\u00fc\u00dfen Jugend die Herzensglut ergriffen.\n\nHoraz, _Carmina_ 1,16,22 ff.\n\n**Meretrix tantisper blanditur, dum illud, quod rapiat, videt.**\n\nEine Dirne schmeichelt nur, solange sie vor Augen hat, was sie an sich rei\u00dfen k\u00f6nnte.\n\nPlautus, _Menaechmi_ 193.\n\n**Metasque dati pervenit ad aevi.**\n\nUnd er gelangte an das Ende der ihm beschiedenen Lebenszeit.\n\nVergil, _Aeneis_ 10,472, zitiert bei Seneca, _Ad Marciam de consolatione_ 21,5.\n\n**Metiri se quemque suo modulo ac pede verum est.**\n\nEs ist wahr, da\u00df sich jeder nach seinem eigenen Ma\u00dfstab mi\u00dft.\n\nHoraz, _Epistulae_ 1,7,98.\n\n**Metus hos, regni spes excitat illos.**\n\nDie einen bewegt die Furcht vor der Tyrannei, die anderen die Hoffnung auf sie.\n\nLukan, _De bello civili_ 7,386.\n\n**Metus improbos compescit, non clementia.**\n\nFurcht, nicht G\u00fcte b\u00e4ndigt Frevler.\n\nPublilius Syrus, _Sententiae_ M 49.\n\n**Meum est propositum in taberna mori.**\n\nMein Vorsatz ist es, in einer Kneipe zu sterben.\n\nArchipoeta, 10,12,1.\n\n**Meus mihi, suus cuique est carus.**\n\nMeiner (hier: mein Sohn) ist mir teuer, (wie) jedem der seine.\n\nPlautus, _Captivi_ 400.\n\n**Mihi crede, verum gaudium res severa est** siehe **Verum gaudium...**\n\n**Mihi cura futuri.**\n\nIch sorge mich um die Zukunft.\n\nOvid, _Metamorphoses_ 13,363.\n\n**Mihi enim liber esse non videtur, qui non aliquando nihil agit.**\n\nMir scheint n\u00e4mlich nicht frei zu sein, wer nicht auch bisweilen nichts tut.\n\nCicero ( _De oratore_ 2,24) l\u00e4\u00dft Crassus erkl\u00e4ren, er habe Scaevola gegen\u00fcber einmal die obigen Worte ge\u00e4u\u00dfert, und f\u00fcgt hinzu: \u00bbHoc ipsum nihil agere et plane cessare delectat.\u00ab (\u00bbEben dieses Nichtstun und g\u00e4nzliche Rasten ist angenehm.\u00ab)\n\n**Mihi heri et tibi hodie!**\n\nGestern mir und heute dir!\n\nSirach 38,23: \u00bbMemor esto iudicii mei; sic enim erit et tuum: mihi heri et tibi hodie.\u00ab (\u00bbGedenke des Richterspruchs \u00fcber mich, denn so wird auch der deine sein: Mich traf er gestern und dich trifft er heute.\u00ab)\n\n**Mihi istic nec seritur nec metitur.**\n\nF\u00fcr mich wird dort weder ges\u00e4t noch geerntet.\n\n(D. h.: Es hat f\u00fcr mich keinerlei Belang.)\n\nPlautus, _Epidicus_ 265.\n\n**Mihi res, non me rebus subiungere conor.**\n\nIch versuche, mir die Dinge und nicht mich den Dingen zu unterwerfen.\n\nHoraz, _Epistulae_ 1,1,19.\n\n**Militat omnis amans.**\n\nJeder Liebende leistet Kriegsdienst.\n\nOvid, _Amores_ 1,9,1 und 2. \u2013 Den Vergleich von Liebe und Kriegswesen f\u00fchrt der Dichter in Vers 29 weiter: \u00bbMars dubius, nec certa Venus.\u00ab (\u00bbDas Kriegsgl\u00fcck schwankt, noch ist das Liebesgl\u00fcck sicher.\u00ab) Vgl. **Militiae species amor est**.\n\n**Militiae species amor est.**\n\nDie Liebe ist eine Art Kriegsdienst.\n\nOvid, _Ars amatoria_ 2,233; vgl. **Militat omnis amans**.\n\n**Militia est vita hominis super terram.**\n\nEin Kriegsdienst ist das Leben der Menschen auf der Erde.\n\nHiob 7,1. Vgl. **Atqui vivere...**\n\n**Mille animos excipe mille modis!**\n\nErhasche tausend Herzen auf tausend Weisen.\n\nOvid, _Ars amatoria_ 1,756.\n\n**Mille hominum species et rerum discolor usus.**\n\n(Es gibt) tausend Arten von Menschen, und unterschiedlich ist ihre Lebensf\u00fchrung.\n\nPersius, _Saturae_ 5,52.\n\n**Mille ioci Veneris.**\n\nTausendf\u00e4ltig (sind) die Spiele der Venus.\n\nOvid, _Ars amatoria_ 3,787.\n\n**Mille modis leti miseros mors una fatigat.**\n\nEin einziger Tod qu\u00e4lt die Elenden mit zahlreichen Todesformen.\n\nStatius, _Thebais_ 9,280.\n\n**Mille modos inter leti mors una timori est, \/ qua coepere mori.**\n\nUnter den tausend Formen des Todes f\u00fcrchten die Menschen die eine, durch die sie (erst) anfangen zu sterben (d. h.: den langsamen Tod).\n\nLukan, _De bello civili_ 3,689 f.\n\n**minima de malis**\n\ndie geringsten unter den \u00dcbeln (erg.: soll man sich aussuchen)\n\nCicero ( _De officiis_ 3,105) bezeichnet die Wendung als altes Sprichwort. Vgl. **E duobus...**\n\n**minima maxima**\n\ndas Geringste (wie) das Gr\u00f6\u00dfte\n\nCicero, _Ad Quintum fratrem_ 3,1,10.\n\n**Minima non curat praetor.**\n\nUm g\u00e4nzlich nichtige Belange k\u00fcmmert sich der Richter nicht.\n\nNach _Digesta_ 4,1,4; vgl. Cicero ( _De natura deorum_ 2,167; 3,86; 3,90 und 3,93), bei dem es mehrfach hei\u00dft, die G\u00f6tter k\u00fcmmerten sich nicht um Geringf\u00fcgigkeiten.\n\n**Minimum decet libere, cui multum licet.**\n\nWem viel erlaubt ist, dem ziemt es, sich wenig zu erlauben.\n\nSeneca, _Troades_ 336.\n\n**Minimum eripit fortuna, cui minimum dedit.**\n\nAm wenigsten entrei\u00dft das Gl\u00fcck, wem es am wenigsten gegeben hat.\n\nPublilius Syrus, _Sententiae_ M 44.\n\n**Minimum est, quod amantibus obstat.**\n\nEs ist (nur) etwas ganz Geringes, was die Liebenden hindert.\n\nOvid, _Metamorphoses_ 3,453. \u2013 Narzi\u00df, der in sein eigenes Spiegelbild verliebt ist, klagt dar\u00fcber, das Objekt seiner Begierde im Wasser wahrzunehmen, ohne es ber\u00fchren zu k\u00f6nnen. Dabei stehe nur die d\u00fcnne Wasseroberfl\u00e4che zwischen ihm und dem Angebeteten.\n\n**Minuentur atrae \/ carmine curae.**\n\nDie d\u00fcsteren Sorgen werden durch ein Lied verringert werden.\n\nHoraz, _Carmina_ 4,11,35 f.\n\n**Minuet vindicta dolorem.**\n\nDeine Rache wird den Schmerz verringern.\n\nOvid, _Amores_ 1,7,63.\n\n**Minus saepe pecces, si scias, quod nescias.**\n\nWeniger h\u00e4ufig wirst du s\u00fcndigen, wenn du wei\u00dft, was du nicht wei\u00dft.\n\nPublilius Syrus, _Sententiae_ M 67.\n\n**Minuta enim est omnis diligentia** siehe **Cito enim...**\n\n**Minuti \/ semper et infirmi est animi exiguique voluptas \/ ultio.**\n\nEs ist stets Zeichen eines kleinlichen und schw\u00e4chlichen und beschr\u00e4nkten Geistes, an Rache Vergn\u00fcgen zu finden.\n\nJuvenal, _Saturae_ 13,189 ff. Vgl. **Vindicta \/ nemo...**\n\n**Miror, quod non ridet haruspex, haruspicem cum videt.**\n\nIch wundere mich, da\u00df ein Opferschauer nicht lacht, wenn er einen Opferschauer sieht.\n\nNach Cicero ( _De divinatione_ 2,51), der diesen Ausspruch Cato d. \u00c4. (Fragment Dicta memorabilia 64 Jordan) zuschreibt und ihm zustimmt.\n\n**Misce stultitiam consiliis brevem!**\n\n**Dulce est desipere in loco.**\n\nMische auch ein wenig Torheit unter deine Besonnenheit. S\u00fc\u00df ist's, zur rechten Zeit zu tollen.\n\nHoraz, _Carmina_ 4,12,27 f. Die Wendung geht wohl auf Menanders _Polumenoi_ (\u00bbDie zum Kauf Angebotenen\u00ab) Fragment 354 K\u00f6rte-Thierfelder zur\u00fcck. Vgl. Seneca ( _De tranquilitate animi_ 17,10), der einen ungenannten griechischen Dichter zitiert: \u00bbAliquando et insanire iucundum est.\u00ab (\u00bbBisweilen ist es auch angenehm zu tollen.\u00ab) Zus\u00e4tzlich behauptet Seneca mit Aristoteles: \u00bb **Nullum magnum ingenium sine mixtura dementiae fuit.** \u00ab Vgl. auch **Saepe enim audivi...**\n\n**miscuit utile dulci** siehe **Aut prodesse volunt...**\n\n**Misera est magni custodia census!**\n\nEin Elend ist die Bewachung eines gro\u00dfen Verm\u00f6gens!\n\nJuvenal, _Saturae_ 14,304.\n\n**Misera mens incredula est.**\n\nDer leidende Geist ist ungl\u00e4ubig.\n\n(Vgl.: Gebranntes Kind scheut das Feuer.)\n\nSeneca, _Hercules Oetaeus_ 1979.\n\n**Misera mors sapienti non potest accidere.**\n\nEin elender Tod kann einem Weisen nicht widerfahren.\n\nNach Cicero, _In L. Catilinam_ 4,3.\n\n**misera plebs**\n\ndas armselige Volk\n\nNach Horaz, _Sermones_ 1,8,10.\n\n**Miser est homo, qui amat.**\n\nElend ist der Mensch, der liebt.\n\nPlautus, _Asinaria_ 616; vgl. _Persa_ 179.\n\n**Miseret te aliorum, tui nec miseret nec pudet.**\n\nDu erbarmst dich anderer, nicht aber deiner selbst und sch\u00e4mst dich nicht.\n\nPlautus, _Trinummus_ 431.\n\n**Miser ex potente fiat, ex misero potens.**\n\nAus einem M\u00e4chtigen werde ein Elender, aus einem Elenden ein M\u00e4chtiger.\n\nSeneca, _Thyestes_ 35.\n\n**Miserias lenit quies.**\n\nRuhe lindert Leiden.\n\nSeneca, _Medea_ 559.\n\n**Miserum est aliorum incumbere famae.**\n\nEs ist erb\u00e4rmlich, sich auf den Ruhm anderer zu st\u00fctzen.\n\n(vgl.: sich mit fremden Federn schm\u00fccken)\n\nJuvenal, _Saturae_ 8,76.\n\n**Miserum te iudico, quod numquam fuisti miser.**\n\nIch halte dich f\u00fcr elend, da du niemals elend warst.\n\nSeneca, _De providentia_ 4,3.\n\n**Mitius inveni quam te genus omne ferarum.**\n\nJede Art wilder Tiere habe ich sanfter gefunden als dich.\n\nOvid, _Heroides_ 10,1.\n\n**Mitte leves spes et certamina divitiarum.**\n\nLa\u00df die eitlen Hoffnungen und die K\u00e4mpfe um Reicht\u00fcmer.\n\nHoraz, _Epistulae_ 1,5,8.\n\n**Mobilis et varia est ferme natura malorum.**\n\nUnstet und ver\u00e4nderlich ist in der Regel das Wesen der B\u00f6sen.\n\nJuvenal, _Saturae_ 13,236.\n\n**Mobilitate viget viresque acquirit eundo.**\n\n(Das Ger\u00fccht) ist von gro\u00dfer Beweglichkeit und gewinnt im Lauf an Kr\u00e4ften.\n\nVergil, _Aeneis_ 4,175. Auch zitiert als: \u00bbFama crescit eundo.\u00ab (\u00bbDas Ger\u00fccht w\u00e4chst im Lauf.\u00ab)\n\n**mobilium turba Quiritium**\n\ndie Masse der wankelm\u00fctigen Quiriten\n\nHoraz, _Carmina_ 1,1,7. \u2013 _Quirites_ bezeichnet das r\u00f6mische Volk. Das Wort h\u00e4ngt eventuell mit dem Quirinus, einer Ortsgottheit des Quirinals (eines der sieben H\u00fcgel Roms) zusammen.\n\n**Moderata durant** siehe **Iuvenile vitium est regere...**\n\n**Modica voluptas laxat animos et temperat.**\n\nMa\u00dfvolles Vergn\u00fcgen schw\u00e4cht die Leidenschaften und m\u00e4\u00dfigt sie.\n\nSeneca, _De ira_ 2,20,3.\n\n**Modice et modeste melius est vitam vivere.**\n\nEs ist besser, ma\u00dfvoll und bescheiden zu leben.\n\nPlautus, _Persa_ 346.\n\n**Modo sic, modo sic, inquit rusticus, varium porcum perdiderat.**\n\n\u00bbMal so, mal so (geht das Leben)\u00ab, sprach der Bauer; er hatte ein geflecktes Schwein verloren.\n\nPetron, _Satyrica_ 45,2.\n\n**Modus omnibus rebus, soror, optimum est habitu.**\n\nIn allen Situationen, Schwester, ist es am besten, Ma\u00df zu halten.\n\nPlautus, _Poenulus_ 238.\n\n**Mollia sunt parvis prata terenda rotis.**\n\nKleine R\u00e4der m\u00fcssen auf weichem Gras rollen.\n\nProperz ( _Elegiae_ 3,3,18) spielt in den vorausgehenden Versen mit dem Gedanken, sich dem Epos zuzuwenden. Phoebus verweist ihn jedoch sofort auf sein bisheriges Metier, die Liebesdichtung (oft als \u00bbversus mollis\u00ab bezeichnet), zur\u00fcck.\n\n**molli bracchio**\n\nmit zartem Arm\n\n(vgl.: jemanden wie ein rohes Ei behandeln)\n\nCicero, _Ad Atticum_ 2,1,6.\n\n**Moniti meliora sequamur.**\n\n(Da wir) gewarnt (sind), wollen wir besseren (Ratschl\u00e4gen) folgen.\n\nVergil, _Aeneis_ 3,188.\n\n**monitoribus asper**\n\nMahnern unzug\u00e4nglich\n\nHoraz, _Ars poetica_ 163.\n\n**Monstror digito praetereuntium.**\n\nDie Vorbeigehenden zeigen mit dem Finger auf mich.\n\nHoraz, _Carmina_ 4,3,22. \u2013 Im antiken Rom war es durchaus salonf\u00e4hig, mit dem Finger auf eine bekannte Person zu zeigen. Eine solche Aufmerksamkeit galt als Kompliment. Vgl. **At pulchrum est digito...**\n\n**monstrum horrendum, informe, ingens, cui lumen ademptum**\n\nein schreckliches, ungestaltes, riesiges Ungeheuer, des Augenlichts beraubt\n\nVergil, _Aeneis_ 3,658. \u2013 Beschreibung des Kyklopen Polyphem, den Odysseus geblendet hatte.\n\n**montes auri pollicens**\n\nBerge von Gold versprechend\n\nTerenz, _Phormio_ 68; vgl. Plautus, _Stichus_ 25 und _Miles gloriosus_ 1065: \u00bbargenti montes\u00ab (\u00bbBerge von Silber\u00ab).\n\n**monumentum aere perennius** siehe **Exegi monumentum...**\n\n**Mora mera est monerier.**\n\nEs ist reine Zeitverschwendung, daran erinnert zu werden.\n\nPlautus ( _Captivi_ 396) f\u00fcgt der Alliteration noch ein Element hinzu: \u00bbQuae memini, mora mera est monerier.\u00ab (\u00bbEs ist reine Zeitverschwendung, an etwas erinnert zu werden, an das ich mich erinnere.\u00ab)\n\n**Mora semper amantes \/ incitat, exiguum si modo tempus habet.**\n\nEin Verzug stachelt Liebende immer an, sofern er nur von kurzer Dauer ist.\n\nOvid, _Ars amatoria_ 3,473 f.\n\n**Moribus et forma conciliandus amor.**\n\nDurch guten Charakter und durch Sch\u00f6nheit mu\u00df Liebe gewonnen werden.\n\nOvid, _Heroides_ 6,94.\n\n**Mori est felicis, antequam mortem invoces.**\n\nGl\u00fccklich ist, wer stirbt, bevor er den Tod herbeiruft.\n\nPublilius Syrus, _Sententiae_ M 5.\n\n**Morior: Desinam mori posse.**\n\nIch sterbe: Ich werde (also damit) aufh\u00f6ren, sterben zu k\u00f6nnen.\n\n(Vgl.: Tod ist des Todes Ausgang.)\n\nNach Seneca ( _Epistulae morales_ 24,17), der versucht, mit diesem Argument seinem Adressaten Lucilius die Angst vor dem Tode zu nehmen.\n\n**Mori volenti desse mors numquam potest.**\n\nDen, der zu sterben w\u00fcnscht, kann der Tod niemals im Stich lassen.\n\nSeneca, _Phaedra_ 878.\n\n**Mors certa , hora incerta.**\n\nDer Tod ist gewi\u00df, die Stunde ungewi\u00df.\n\nVgl. Plautus, _Captivi_ 732; Cicero, _De senectute_ 74; Matth\u00e4us 25,13, und Seneca, _Epistulae morales_ 99,9.\n\n**Mors etiam saxis nominibusque venit.**\n\nDer Tod ereilt sogar Steine und Namen.\n\nAusonius ( _Epitaphia_ 32,10) erkl\u00e4rt, da\u00df auch Ruhm nicht unsterblich sei, denn selbst Monumente zerfielen.\n\n**Mors innocentem sola fortunae eripit.**\n\nNur der Tod entrei\u00dft den Unschuldigen dem Schicksal.\n\n(D. h.: Nur der Tod sch\u00fctzt davor, vom Schicksal ereilt zu werden.)\n\nSeneca, _Oedipus_ 934.\n\n**Mors ipsa refugit \/ saepe virum.**\n\nSelbst der Tod floh oft vor dem Mann.\n\nLukan, _De bello civili_ 2,75 f. \u2013 Gemeint ist Marius: Einst sei ein Feind, der ihn erschlagen wollte, pl\u00f6tzlich erstarrt und habe das Schwert fallengelassen.\n\n**Mors laborum ac miseriarum quies est.**\n\nDer Tod ist Ruhe von M\u00fchen und Elend.\n\n(Vgl.: Der Tod ist das Ende aller Not.)\n\nNach Cicero, _In L. Catilinam_ 4,7.\n\n**Mors mihi munus erit.**\n\nDer Tod wird ein Geschenk f\u00fcr mich sein.\n\nOvid, _Metamorphoses_ 9,181. \u2013 Hercules wird von Schmerzen zerrissen, da er sich das mit dem Blut des Nessus getr\u00e4nkte Hemd angezogen hat. Daraufhin setzt er seinem Leben ein Ende, indem er sich selbst auf dem Berg Oeta verbrennt.\n\n**Mors misera non est, aditus ad mortem est miser.**\n\nDer Tod ist kein Elend: Der Weg zum Tod ist elend.\n\nFragment Ex inc. fabulis 109\/203 Ribbeck, zitiert bei Quintilian ( _Institutio oratoria_ 8,5,6).\n\n**Mors nobis tempus habetur iners.**\n\nUnt\u00e4tig verbrachte Zeit ist f\u00fcr mich (wie) der Tod.\n\nOvid, _Epistulae ex Ponto_ 1,5,44.\n\n**Mors omni aetati est communis.**\n\nJedem Alter ist der Tod gemein.\n\nNach Cicero, _De senectute_ 68.\n\n**Morsque minus poenae quam mora mortis habet.**\n\nUnd der Tod ist (mir) eine geringere Strafe als die Verz\u00f6gerung des Todes.\n\nOvid, _Heroides_ 10,82.\n\n**Mors sola fatetur, \/ quantula sint hominum corpuscula.**\n\nDer Tod allein zeigt, wie nichtig die K\u00f6rperchen der Menschen sind.\n\nJuvenal, _Saturae_ 10,172 f.\n\n**Mors ultima linea rerum.**\n\nDer Tod ist die \u00e4u\u00dferste Grenze alles Irdischen.\n\nHoraz, _Epistulae_ 1,16,79.\n\n**Mortalia facta peribunt.**\n\nDie Taten der Sterblichen werden vergehen.\n\nHoraz, _Ars poetica_ 68.\n\n**Mortalis nemo est, quem non attingat dolor \/ morbusque.**\n\nEs gibt keinen Sterblichen, den nicht Schmerz und Krankheit ber\u00fchrten.\n\nCicero ( _Tusculanae disputationes_ 3,59) zitiert hier aus _Hypsipyle_ , einem heute verlorenen St\u00fcck von Euripides (Fragment 757 Nauck).\n\n**Morte carent animae.**\n\nDie Seelen sind frei vom Tod (d. h.: unsterblich).\n\nOvid, _Metamorphoses_ 15,158.\n\n**Mortem effugere nemo potest.**\n\nDem Tod entrinnen kann niemand.\n\nNach Cicero, _Orationes Philippicae_ 8,29.\n\n**Mortem misericors saepe pro vita dabit.**\n\nDer Barmherzige wird oft den Tod statt des Lebens geben.\n\n(D. h.: Bisweilen ist es barmherziger, jemanden zu t\u00f6ten, als ihn leben zu lassen.)\n\nSeneca, _Troades_ 329.\n\n**mulgere hircum** siehe **iungere vulpes...**\n\n**Muliebrium vitiorum fundamentum avaritia est.**\n\nAller weiblichen Laster Grund ist die Habsucht.\n\nSeneca Rhetor, _Controversiae_ Exc. 2,7.\n\n**Mulieres in ecclesiis taceant.**\n\nDie Frauen sollen in den Gemeinden schweigen.\n\nKorintherbriefe 1,14,34. \u2013 Meist zitiert als \u00bbMulier taceat in ecclesia\u00ab. (\u00bbDie Frau schweige in der Gemeinde.\u00ab)\n\n**Mulieres sunt ferme ut pueri levi sententia.**\n\nFrauen sind, fast wie kleine Kinder, von leichtfertigem Urteil.\n\nTerenz, _Hecyra_ 312.\n\n**Mulier recte olet, ubi nihil olet.**\n\nEine Frau riecht gut, wenn sie nicht riecht.\n\nPlautus, _Mostellaria_ 273.\n\n**Multa ante temptes, quam virum invenias bonum.**\n\nVieles wirst du versuchen (m\u00fcssen), bis du einen Guten findest.\n\nPublilius Syrus, _Sententiae_ M 63.\n\n**Multa docet fames.**\n\nVieles lehrt der Hunger.\n\nNach Seneca, _Epistulae morales_ 15,7.\n\n**Multa eveniunt homini, quae vult, quae non vult.**\n\nViel passiert einem Menschen, ob er es will oder nicht.\n\nPlautus, _Trinummus_ 361.\n\n**Multa ferunt anni venientes commoda secum,**\n\n**Multa recedentes adimunt.**\n\nViele Annehmlichkeiten bringen die voranschreitenden Jahre mit sich, viele nehmen sie wieder weg, wenn sie weichen.\n\nHoraz, _Ars poetica_ 175 f. Vgl. **Multa senem...**\n\n**Multa fidem promissa levant.**\n\nViele Versprechungen schm\u00e4lern das Vertrauen.\n\nHoraz ( _Epistulae_ 2,2,10 f.) f\u00e4hrt fort: \u00bb... ubi plenius aequo \/ laudat venalis, qui vult extrudere merces.\u00ab (\u00bb... wenn man die Waren, die man ver\u00e4u\u00dfern will, \u00fcber Geb\u00fchr lobt.\u00ab)\n\n**Multa ignoscendo vir potens potentior.**\n\nVieles zu verzeihen macht einen starken Mann st\u00e4rker.\n\nPublilius Syrus, _Sententiae_ M 42.\n\n**Multa loquens et cuncta silens non ambo placemus.**\n\n(Ich) der ich viel sprach, und (du), der du alles verschwiegst, wir mi\u00dffielen beide.\n\nAusonius, _Epistulae_ 29,47.\n\n**Multa mulierum sunt vitia.**\n\nZahlreich sind die Laster der Frauen.\n\nPlautus, _Poenulus_ 1203.\n\n**Multa petentibus \/ desunt multa.**\n\nDenen, die vieles verlangen, mangelt es an vielem.\n\nHoraz, _Carmina_ 3,16,42 f.\n\n**Multaque, dum fiunt turpia, facta placent.**\n\nUnd vieles, was h\u00e4\u00dflich ist, w\u00e4hrend es geschieht, gef\u00e4llt uns, wenn es geschehen ist.\n\nOvid, _Ars amatoria_ 3,218.\n\n**Multa senem circumveniunt incommoda.**\n\nViele Unannehmlichkeiten umgeben einen alten Mann.\n\nHoraz, _Ars poetica_ 169. Vgl. **Multa ferunt...**\n\n**Multa tulit fecitque puer, sudavit et alsit** siehe **Qui studet optatam cursu...**\n\n**Multi ad fatum \/ venere suum, dum fata timent.**\n\nViele haben ihr Schicksal erf\u00fcllt, w\u00e4hrend sie es f\u00fcrchteten.\n\nSeneca, _Oedipus_ 994 f.\n\n**Multis damnosa fuere \/gaudia.**\n\nVielen war die Freude sch\u00e4dlich.\n\nClaudian ( _De quarto consulatu honorii Augusti_ 334 f.) warnt davor, sich vorzeitig sicher zu w\u00e4hnen.\n\n**Multis eget, qui multa habet.**\n\nVieler Dinge bedarf, wer vieles hat.\n\nNach Gellius, _Noctes Atticae_ 9,8,1.\n\n**Multis \/ et sua mortifera est facundia.**\n\nVielen bringt auch ihre Beredsamkeit den Tod.\n\nJuvenal, _Saturae_ 10,9 f.\n\n**Multis ille bonis flebilis occidit,**\n\n**Nulli flebilior quam tibi, Vergili.**\n\nVon vielen Guten beweint, starb er, von keinem mehr beweint als von dir, Vergil.\n\nHoraz ( _Carmina_ 1,24,9 f.) anl\u00e4\u00dflich des Todes von Quintilius Varus, der mit Horaz und Vergil eng befreundet war. Vgl. **incorrupta fides...**\n\n**Multis utile bellum.**\n\nVielen n\u00fctzt der Krieg.\n\nLukan, _De bello civili_ 1,182.\n\n**Multo facilius divitem quam patrem familias fieri** siehe **De scurra multo facilius...**\n\n**Multorum calamitate vir moritur bonus.**\n\nDer Tod eines Guten ist das Ungl\u00fcck vieler.\n\nPublilius Syrus, _Sententiae_ M 48.\n\n**Multos, qui sperant, decepit spes.**\n\nViele Hoffende t\u00e4uschte die Hoffnung.\n\nNach Plautus, _Rudens_ 401.\n\n**Multos timere debet, quem multi timent.**\n\nViele mu\u00df f\u00fcrchten, wen viele f\u00fcrchten.\n\nPublilius Syrus, _Sententiae_ M 30. Vgl. Seneca ( _De ira_ 2,11,3), der Laberius (Fragment 126 Ribbeck) zitiert.\n\n**multum, non multa**\n\nviel, nicht vielerlei\n\n(Vgl.: Weniger ist mehr.)\n\nNach Plinius d. J. ( _Epistulae_ 7,9,15), der berichtet: \u00bbAiunt enim multum legendum esse, non multa.\u00ab (\u00bbEs hei\u00dft n\u00e4mlich, man solle viel lesen, nicht vielerlei.\u00ab) \u2013 Vgl. Quintilian, _Institutio oratoria_ 10,1,59.\n\n**Munditiis capimur.**\n\nVon Sauberkeit lassen wir uns gewinnen.\n\nOvid ( _Ars amatoria_ 3,133) mahnt die Frauen, von allzu aufwendiger Haartracht Abstand zu nehmen: M\u00e4nnern gefalle vielmehr schlichte Sauberkeit.\n\n**Mundus vult decipi.**\n\nDie Welt will get\u00e4uscht werden.\n\nBei Sebastian Brant ( _Das Narrenschiff_ ) findet sich der Spruch in deutscher Sprache, in lateinischer Sprache dann bei Sebastian Franck, _Paradoxa_ 238. Sp\u00e4ter wurde der Zusatz \u00bbErgo decipiatur!\u00ab (\u00bbAlso m\u00f6ge sie get\u00e4uscht werden!\u00ab) beigef\u00fcgt.\n\n**Munera , crede mihi, capiunt hominesque deosque.**\n\nGeschenke, glaube mir, erobern Menschen wie G\u00f6tter.\n\nOvid, _Ars amatoria_ 3,653.\n\n**Munera misit in hamo.**\n\nEr schickt Geschenke mit einem Haken.\n\nNach Martial, _Epigrammata_ 6,63,5, und Plinius d. J., _Panegyricus_ 43,5. Vgl. **Et piscatorem...**\n\n**Munificus facito vivas, non parcus, amicis.**\n\nLebe freigiebig deinen Freunden gegen\u00fcber, nicht knickrig.\n\n_Disticha Catonis_ 3,9,2.\n\n**Musice aetatem agit.**\n\nEr f\u00fchrt ein luxuri\u00f6ses Leben.\n\n(D. h.: Er lebt wie der Vogel im Hanfsamen.)\n\nNach Plautus, _Mostellaria_ 729.\n\n**Mus miser est, antro qui solum clauditur uno.**\n\nDie Maus ist arm dran, die in einer einzigen H\u00f6hle eingesperrt ist.\n\nCrotus Rubianus, _Epistulae Virorum Obscurorum_ (Magister Bernhardus Plumilegus) 1,3.\n\n**Mus sapiens aetatem non cubili umquam committit suam.**\n\nDie Maus, ein weises Tier, vertraut ihr Leben niemals (nur) einem einzigen Schlupfloch an.\n\nNach Plautus, _Truculentus_ 868 f.\n\n**Mutare quod non possis, ut natum est, feras!**\n\nWas du wohl nicht \u00e4ndern kannst, ertrage, wie es ist.\n\nPublilius Syrus, _Sententiae_ M 62.\n\n**Mutato nomine de te \/ fabula narratur.**\n\nDie Geschichte handelt von dir; nur der Name ist ge\u00e4ndert.\n\nHoraz, _Sermones_ 1,1,69 f.\n\n**Mutat via longa puellas.**\n\nEine lange Reise ver\u00e4ndert die M\u00e4dchen.\n\n(Vgl.: Aus den Augen, aus dem Sinn.)\n\nProperz, _Elegiae_ 1,12,11.\nN\n\n**Nabis sine cortice.**\n\nDu wirst ohne Korkg\u00fcrtel schwimmen.\n\nHoraz ( _Sermones_ 1,4,120) l\u00e4\u00dft seinen Vater erkl\u00e4ren, der Sohn werde seinen eigenen Weg ohne Besch\u00fctzer oder Gew\u00e4hrsmann gehen k\u00f6nnen, wenn er einmal eine gewisse Reife erlangt habe.\n\n**Nam et ipsa scientia potestas est.**\n\nDenn schon das Wissen an sich ist Macht.\n\nFrancis Bacon, _Meditationes sacrae_ 11 (\u00bbde haeresibus\u00ab).\n\n**Nam fuit ante Helenam cunnus taeterrima belli \/ causa.**\n\nDenn (schon) vor Helena war das weibliche Geschlecht der sch\u00e4ndlichste Kriegsgrund.\n\nHoraz, _Sermones_ 1,3,107 f. \u2013 Herodot beginnt seine Historien mit einer Aufz\u00e4hlung von Kriegen, deren Ursache der Raub einer Frau gewesen sei (so Io, Medea und Helena). Vgl. **Nulla fere causa est...**\n\n**Nam melius duo defendunt retinacula navem.**\n\nDenn besser sichern zwei Taue das Schiff.\n\n(Vgl.: Doppelt h\u00e4lt besser.)\n\nProperz, _Elegiae_ 2,22,41.\n\n**Nam mora dat vires.**\n\nDie Weile bringt Kraft.\n\n(Vgl.: Gut' Ding will Weile haben.)\n\nOvid, _Remedia amoris_ 83.\n\n**Nam non conveniens omnibus omnis erit.**\n\nDenn nicht jede (Farbe) wird zu allen passen.\n\nOvid ( _Ars amatoria_ 3,188) r\u00e4t den Frauen, die Farbe der Kleidung genau auf ihren Typ abzustimmen.\n\n**Nam quem ferret, si parentem non ferret suum?**\n\nDenn wen k\u00f6nnte er ertragen, wenn er nicht seinen eigenen Vater ertr\u00fcge?\n\nTerenz, _Heauton timorumenos_ 202. \u2013 Chremes erkl\u00e4rt, da\u00df V\u00e4ter zwar oft streng erschienen, doch nur das Beste f\u00fcr ihre S\u00f6hne wollten. Daher k\u00f6nne man sie leicht ertragen.\n\n**Nam qui amat, cui odio ipsust, eum bis facere stulte duco.**\n\nWer den liebt, der ihn ha\u00dft, der handelt, glaube ich, doppelt t\u00f6richt.\n\nTerenz, _Hecyra_ 343.\n\n**Nam risu inepto res ineptior nulla est.**\n\nDenn nichts ist alberner als albernes Lachen.\n\nCatull, _Carmina_ 39,16.\n\n**Nam sapiens quidem, pol ipsus fingit fortunam sibi.**\n\nDenn, bei Gott, der Weise formt sich sein Schicksal selbst.\n\n(Vgl.: Jeder ist seines Gl\u00fcckes Schmied.)\n\nPlautus, _Trinummus_ 363.\n\n**Nam sera numquam est ad bonos mores via.**\n\nDenn der Weg zu einem guten Charakter ist niemals sp\u00e4t.\n\n(D. h.: Es ist nie zu sp\u00e4t, sich zu bessern.)\n\nSeneca, _Agamemnon_ 242.\n\n**Nam si hic mali est quicquam, em illic est huic rei caput.**\n\nWenn n\u00e4mlich hier etwas B\u00f6ses ist, dann ist er dort der Kopf der Sache (d. h.: dann steckt er dort dahinter).\n\nTerenz, _Andria_ 458.\n\n**Nam tua res agitur, paries cum proximus ardet,**\n\n**Et neglecta solent incendia sumere vires.**\n\nDenn es geht um dich, wenn die Nachbarwand brennt, und unbeachtete Feuer pflegen m\u00e4chtig zu werden.\n\nHoraz, _Epistulae_ 1,18,84 f.\n\n**Nam uxor contenta est, quae bona est, uno viro.**\n\nDenn eine Frau, die eine gute Frau ist, ist mit einem Mann zufrieden.\n\nPlautus, _Mercator_ 824. \u2013 Diese Aussage ist jedoch, anders als sie meist verstanden wird, keine Zurechtweisung untreuer Frauen. Sie pl\u00e4diert vielmehr daf\u00fcr, bei Mann und Frau dieselben moralischen Ma\u00dfst\u00e4be anzuwenden. So f\u00e4hrt Syra fort: \u00bbQui minus vir una uxore contentus siet?\u00ab (\u00bbWarum sollte ein Mann weniger zufrieden mit einer einzigen Frau sein?\u00ab)\n\n**Nam vita morti propior fit cotidie.**\n\nDenn das Leben kommt dem Tod t\u00e4glich n\u00e4her.\n\nPhaedrus, _Fabulae_ 3, epilogus 10.\n\n**Nam vitiis nemo sine nascitur; optimus ille est, \/ qui minimis urguetur.**\n\nDenn kein Mensch wird ohne Fehler geboren; der beste ist der, den die geringsten bedr\u00e4ngen.\n\nHoraz, _Sermones_ 1,3,68 f.\n\n**Nascentes morimur finisque ab origine pendet.**\n\nWir sterben bereits, wenn wir geboren werden, und das Ende h\u00e4ngt am Beginn.\n\nManilius, _Astronomica_ 4,16.\n\n**Nascitur exiguus, sed opes acquirit eundo.**\n\n(Ein Strom) ist klein, wo er entsteht, in (seinem) Verlauf aber nimmt er an St\u00e4rke zu.\n\nOvid ( _Ars amatoria_ 2,343) r\u00e4t hier, der Liebe Zeit zur Gew\u00f6hnung zu geben, damit sie schlie\u00dflich stark werde.\n\n**Naso suspendis adunco \/ ignotos.**\n\nDu h\u00e4ngst Unbekannte **an der Hakennase auf.**\n\n(D. h.: Du r\u00fcmpfst die Nase \u00fcber Unbekannte.)\n\nHoraz, _Sermones_ 1,6,5 f.\n\n**Natura cupiditatem ingenuit homini veri videndi.**\n\nDie Natur hat dem Menschen die Begierde eingepflanzt, die Wahrheit zu sehen.\n\nCicero, _De finibus_ 2,46.\n\n**Natura dedit usuram vitae tamquam pecuniae nulla praestituta die.**\n\nDie Natur hat (uns) den Nie\u00dfbrauch des Lebens gew\u00e4hrt wie den eines Darlehns, f\u00fcr das (vorher) kein R\u00fcckzahlungstermin festgelegt wurde.\n\nNach Cicero, _Tusculanae disputationes_ 1,93.\n\n**Natura duce errari nullo pacto potest.**\n\nUnter der F\u00fchrung der Natur kann man auf keine Weise irren.\n\nNach Cicero, _De legibus_ 1,20.\n\n**naturae convenienter vivere** siehe **convenienter naturae vivere**\n\n**Natura enim duce utendum est.**\n\nMan mu\u00df n\u00e4mlich die Natur als F\u00fchrerin w\u00e4hlen.\n\nSeneca, _De vita beata_ 8,1.\n\n**Natura est paucis contenta.**\n\nDie Natur ist (bereits) mit wenigem zufrieden.\n\nHW: 15924. Vgl. Seneca, _Epistulae morales_ 16,9: \u00bbNaturalia desideria finita sunt.\u00ab (\u00bbDie nat\u00fcrlichen Begierden sind begrenzt.\u00ab)\n\n**Naturalia non sunt turpia.**\n\nNat\u00fcrliche Dinge sind nicht schimpflich.\n\nNach Euripides, _Hypsipyle_ Fragment 757,9 Nauck, \u00fcberliefert von Stobaios, _Anthologie_ 3,29,56. \u2013 Der Gedanke geht auf den Kyniker Diogenes zur\u00fcck, der laut Diogenes Laertius ( _Vitae philosophorum_ : _Diogenes_ 6,69) allen nat\u00fcrlichen Bed\u00fcrfnissen in der \u00d6ffentlichkeit nachzukommen pflegte.\n\n**Naturam expelles furca, tamen usque recurret.**\n\nAuch wenn du die Natur mit der Gabel austreiben wirst, wird sie dennoch zur\u00fcckgelaufen kommen.\n\nHoraz, _Epistulae_ 1,10,24.\n\n**Naturam quidem mutare difficile est.**\n\nEs ist n\u00e4mlich schwer, sein Wesen zu \u00e4ndern.\n\nSeneca, _De ira_ 2,20,2.\n\n**Natura non facit saltus.**\n\nDie Natur macht keine Spr\u00fcnge.\n\nIn franz\u00f6sischer Sprache bereits bei Leibniz, _Nouveaux essais sur l'entendement humain_ , Pr\u00e9face: \u00bbLa nature ne fait jamais des sauts.\u00ab Lateinisch von Carl von Linn\u00e9, _Philosophia botanica_ 77. \u2013 Die Vorstellung einer stufenlosen, flie\u00dfenden Abfolge aller phylo- wie ontogenetischen Vorg\u00e4nge stammt von Aristoteles (vgl. _Historia animalium_ 8,1,588b 4 ff. und _De partibus animalium_ 4,5,681a 12 ff.). Vgl. dagegen Maximus Tyrius ( _Philosophumena_ 9,4a), der auch sprunghafte Ver\u00e4nderungen in nat\u00fcrlichen Abl\u00e4ufen kennt.\n\n**naufragia ex terra intueri**\n\nvom Land aus Schiffbr\u00fcche betrachten\n\nCicero ( _Ad Atticum_ 2,7,4) erkl\u00e4rt, er wolle von nun an die politischen Katastrophen als Unbeteiligter betrachten und selbst keine Politik mehr betreiben. \u2013 Vgl. **Suave mari magno...**\n\n**Naufragium rerum est mulier male fida marito.**\n\nDen Ruin bedeutet eine Frau, die ihrem Ehemann untreu ist.\n\n_Monosticha Catonis_ 6.\n\n**Navigare necesse est.**\n\nSeefahrt ist notwendig.\n\nNach Plutarch ( _Pompeius_ 50,1) soll Pompeius eine Sturmwarnung mit den Worten abgetan haben: \u00bbNavigare necesse est, vivere non necesse est.\u00ab (\u00bb[Jetzt loszu]segeln ist n\u00f6tig, zu leben ist nicht n\u00f6tig.\u00ab) \u2013 Der Spruch ist am Haus der Seefahrt in Bremen zu lesen.\n\n**Ne bis in idem!**\n\nNicht zweimal im gleichen (Proze\u00df)!\n\nNach Demosthenes, _Rede gegen Leptines_ 147; vgl. Quintilian, _Institutio oratoria_ 7,6,4. \u2013 Laut Liebs N 6 findet sich die w\u00f6rtliche Version in einer Glosse zu Gratian, _Decreta_ 2,2,14,1.\n\n**nec caput nec pedes**\n\nweder Kopf noch Fu\u00df (haben)\n\nCicero, _Ad familiares_ 7,31,2.\n\n**Nec cito credideris!**\n\nGlaube nicht schnell!\n\nOvid ( _Ars amatoria_ 3,685) warnt davor, Nachrichten \u00fcber angebliche Seitenspr\u00fcnge einer geliebten Person allzu schnell zu glauben. Als Beispiel f\u00fchrt er Procris an, deren unberechtigte Eifersucht ihren Tod verursachte.\n\n**Nec cito desisto nec temere incipio.**\n\nIch h\u00f6re weder schnell auf (zu lieben) noch beginne ich \u00fcberst\u00fcrzt.\n\nProperz ( _Elegiae_ 2,20,36) beteuert seine ewige Treue.\n\n**Nec cor nec caput habet.**\n\nEr hat weder Herz noch Kopf.\n\nSeneca ( _Apocolocyntosis_ 8,1) \u00fcber den verstorbenen Kaiser Claudius.\n\n**Nec domo dominus, sed domino domus honestanda est.**\n\nNicht das Haus mu\u00df den Herrn, sondern der Herr das Haus ehren.\n\nCicero ( _De officiis_ 1,139) stellt die Tugendhaftigkeit des Hausherrn jeglichem Zierat des Hauses voran.\n\n**Necessitas ante rationem est.**\n\nDie Notwendigkeit kommt vor der Vernunft.\n\nCurtius Rufus, _Historia Alexandri Magni_ 7,7,10.\n\n**Necessitas est lex temporis.**\n\nNotwendigkeit ist das Gesetz der Stunde.\n\nSeneca Rhetor, _Controversiae_ 4,4.\n\n**Necessitas ultimum ac maximum telum est.**\n\nDie Notwendigkeit ist die letzte und die st\u00e4rkste Waffe.\n\nNach Livius, _Ab urbe condita_ 4,28,5. Vgl. **Durum telum...** und **Ingens telum...**\n\n**Necessitatem ferre , non flere addecet.**\n\nEs ziemt sich, die Notwendigkeit zu ertragen, nicht sie zu beweinen.\n\nPublilius Syrus, _Sententiae_ N 58.\n\n**Necessitatem ne dii quidem superant.**\n\nDie Notwendigkeit \u00fcberwinden nicht einmal die G\u00f6tter.\n\nNach Simonides, Fragment 19 Edmonds.\n\n**Necessitati est parendum.**\n\nDer Notwendigkeit mu\u00df man gehorchen.\n\n(Vgl.: Not kennt kein Gebot.)\n\nNach Cicero, _De officiis_ 2,74.\n\n**Nec est ullum magnum malum praeter culpam.**\n\nUnd es gibt kein gro\u00dfes \u00dcbel au\u00dfer der Schuld.\n\nNach Cicero, _Ad familiares_ 6,4,2.\n\n**Nec facile est aequa commoda mente pati.**\n\nUnd es ist nicht leicht, Angenehmes gleichm\u00fctig (d. h.: ohne \u00fcberm\u00fctig zu werden) zu ertragen.\n\nOvid, _Ars amatoria_ 2,438.\n\n**Nec iuga taurus amat; quae tamen odit, habet** siehe **Odero, si potero...**\n\n**Nec lusisse pudet, sed non incidere ludum.**\n\nUnd man sch\u00e4mt sich nicht, herumget\u00e4ndelt zu haben, aber (es w\u00e4re besch\u00e4mend,) der T\u00e4ndelei kein Ende zu setzen.\n\nHoraz, _Epistulae_ 1,14,36.\n\n**Nec manet ulla sui similis res.**\n\nUnd kein Ding bleibt sich selbst \u00e4hnlich.\n\nLukrez, _De rerum natura_ 5,830.\n\n**nec mora nec requies**\n\nweder Rast noch Ruhe\n\nVergil, _Aeneis_ 5,458.\n\n**Nec mortem effugere quisquam nec amorem potest.**\n\nWeder dem Tod noch der Liebe kann irgendein Mensch entgehen.\n\nPublilius Syrus, _Sententiae_ N 57.\n\n**Nec muta profecto reperta ulla est aut hodie aut ullo in saeclo mulier.**\n\nUnd in der Tat hat sich weder jetzt noch in irgendeinem Zeitalter je eine stumme Frau finden lassen.\n\nNach Plautus ( _Aulularia_ 125 f.), der die Wendung bereits als Sprichwort bezeichnet.\n\n**Nec possum tecum vivere nec sine te** siehe **Difficilis, facilis...**\n\n**Nec puero gladium!**\n\nUnd dem Knaben kein Schwert!\n\nAugustinus ( _Epistulae_ 104,2,7) bezeichnet diese Mahnung als sprichw\u00f6rtlich.\n\n**Nec quisquam adhuc inventus est, cui, quod haberet, esset satis.**\n\nUnd es ist noch keiner gefunden worden, dem, was er hat, genug w\u00e4re.\n\nNach Cicero, _Paradoxa Stoicorum_ 52. Vgl. **Fortuna multis...**\n\n**Nec quod fuimusve sumusve, \/ cras erimus.**\n\nUnd was wir waren oder sind, werden wir morgen nicht mehr sein.\n\nOvid, _Metamorphoses_ 15,215 f.\n\n**Nec scire fas est omnia.**\n\nEs ist auch nicht recht, alles zu wissen.\n\nHoraz, _Carmina_ 4,4,22.\n\n**Nec semper feriet, quodcumque minabitur arcus.**\n\nNicht immer wird der Bogen treffen, worauf er drohend gerichtet.\n\nHoraz, _Ars poetica_ 350.\n\n**Nec si non obstatur , propterea etiam permittitur.**\n\nUnd was nicht verhindert wird, ist deswegen nicht auch erlaubt.\n\nCicero, _Orationes Philippicae_ 13,14.\n\n**Nec te fata tenent post funera?**\n\nUnd h\u00e4lt dich das Schicksal nach dem Tode nicht fest?\n\nSeneca, _Hercules Oetaeus_ 1953. \u2013 Ausruf Alkmenes, als ihr der Geist ihres toten Sohnes Hercules erscheint.\n\n**Nec tellus eadem parit omnia.**\n\nUnd ein und derselbe Boden bringt nicht alles hervor.\n\nOvid ( _Ars amatoria_ 1,757) erinnert die M\u00e4nner daran, da\u00df alle Frauen sehr unterschiedlich seien und man die jeweils entsprechende Verf\u00fchrungsmethode w\u00e4hlen m\u00fcsse.\n\n**nec verbis solum, sed etiam verberibus castigare**\n\nnicht nur mit Worten, sondern auch mit Schl\u00e4gen z\u00fcchtigen\n\nNach Cicero, _Tusculanae disputationes_ 3,64; vgl. **Non opus est verbis...**\n\n**Nec vixit male, qui natus moriensque fefellit.**\n\nUnd wer unbemerkt \/ sich in die Welt hinein- und wieder \/ hinausgeschlichen, hat nicht schlimm gelebt.\n\nHoraz ( _Epistulae_ 1,17,10; \u00fcbers. von Christoph Martin Wieland) untermauert hier nochmals seinen Lebensentwurf nach der epikureischen Weisung \u00bbLebe im Verborgenen!\u00ab Vgl. **Bene qui...**\n\n**Ne discere cessa!**\n\nH\u00f6re nicht auf zu lernen!\n\n_Disticha Catonis_ 3,1.\n\n**Nefas propter vitam vivendi perdere causas!**\n\nEs ist sch\u00e4ndlich, um des Lebens willen den Sinn des Lebens zu verlieren.\n\nNach Juvenal, _Saturae_ 8,83 f.\n\n**neglecta solent incendia sumere vires** siehe **Nam tua res agitur...**\n\n**Ne illius modi iam nobis magna civium**\n\n**Penuria est, homo antiqua virtute ac fide!**\n\nWahrhaftig haben wir jetzt schon gro\u00dfen Mangel an derartigen B\u00fcrgern, an einem Mann von alter Tugend und Treue!\n\nTerenz, _Adelphoe_ 441 f.\n\n**Ne Iuppiter quidem omnibus placet.**\n\nNicht einmal Jupiter gef\u00e4llt allen.\n\n(Vgl.: Allen Leuten recht getan ist eine Kunst, die niemand kann.)\n\nErasmus ( _Adagia_ 2,7,55) schreibt diese Worte Theognis ( _Elegien_ 25) zu.\n\n**Nemo adeo ferus est, ut non mitescere possit** siehe **Invidus, iracundus, iners...**\n\n**Nemo autem regere potest nisi qui et regi.**\n\nNiemand jedoch kann herrschen, der nicht auch beherrscht werden kann.\n\nSeneca, _De ira_ 2,15,4.\n\n**Nemo contra deum nisi deus ipse.**\n\nKeiner (kann) gegen Gott (sein) au\u00dfer Gott selbst.\n\nJohann Wolfgang Goethe, _Dichtung und Wahrheit_ , 20. Buch.\n\n**Nemo dignitati perditae parcit.**\n\nNiemand schont verlorene W\u00fcrde.\n\n(Vgl.: Ehre hin, alles hin.)\n\nSeneca, _De clementia_ 1,22,1.\n\n**Nemo enim est tam senex, qui se annum non putet posse vivere.**\n\nDenn niemand ist so alt, da\u00df er nicht glaubte, noch ein Jahr leben zu k\u00f6nnen.\n\nCicero, _De senectute_ 24; vgl. Seneca, _Epistulae morales_ 12,6.\n\n**Nemo enim potest personam diu ferre.**\n\nNiemand n\u00e4mlich kann auf Dauer eine Maske tragen.\n\nSeneca, _De clementia_ 1,1,6.\n\n**Nemo errat uni sibi, sed dementiam spargit in proximos.**\n\nKeiner irrt f\u00fcr sich allein, sondern er verbreitet seine Verr\u00fccktheit unter seinen N\u00e4chsten.\n\nSeneca, _Epistulae morales_ 94,54.\n\n**Nemo in amore videt.**\n\nNiemand sieht in der Liebe.\n\n(Vgl.: Liebe macht blind.)\n\nProperz, _Elegiae_ 2,14,18. Vgl. **amantes, amentes** und **Inceptio est . . .**\n\n**Nemo ita pauper vivit, quam pauper natus est.**\n\nNiemand lebt so arm, wie er geboren wurde.\n\nHW: 16361.\n\n**Nemo malus felix, minime corruptor.**\n\nKein Schurke ist gl\u00fccklich, am wenigsten ein Verf\u00fchrer.\n\nJuvenal, _Saturae_ 4,8.\n\n**Nemo me impune lacessit.**\n\nNiemand reizt mich ungestraft.\n\nUrspr\u00fcnglich Inschrift im Wappen des \u00bbOrder of Thistle\u00ab und Randinschrift der in Lleantrisant gepr\u00e4gten, Schottland repr\u00e4sentierenden Pfundst\u00fccke, auf deren R\u00fcckseite eine Distel abgebildet ist.\n\n**Nemo mortalium omnibus horis sapit.**\n\nKeiner unter den Sterblichen ist zu jeder Stunde weise.\n\nPlinius d. \u00c4. ( _Naturalis historia_ 7,13) bezeichnet die Wendung als Sprichwort.\n\n**Nemo nascitur sapiens, sed fit.**\n\nNiemand wird als Weiser geboren, sondern er wird es (erst).\n\nNach Seneca, _De ira_ 2,10,6.\n\n**Nemo nostrum non peccat; homines sumus, non dii.**\n\nKeiner von uns, der nicht s\u00fcndigte; wir sind Menschen, nicht G\u00f6tter.\n\nNach Petron, _Satyrica_ 75,1.\n\n**Nemo potentes aggredi tutus potest.**\n\nNiemand kann die M\u00e4chtigen ungef\u00e4hrdet angreifen.\n\nSeneca, _Medea_ 430.\n\n**Nemo prudens punit, quia peccatum est, sed ne peccetur.**\n\nKein besonnener Mensch straft, weil gefehlt worden ist, sondern damit nicht (wieder) gefehlt werde.\n\nSeneca ( _De ira_ 1,19,7) zitiert hier Platon ( _Gesetze_ 11,934a).\n\n**Nemo rex non ex servis est oriundus, nemo servus non ex regibus.**\n\nKein K\u00f6nig stammt nicht (auch) von Sklaven ab, kein Sklave nicht (auch) von K\u00f6nigen.\n\nSeneca ( _Epistulae morales_ 44,4) zitiert Platon, _Theaetetus_ 175a.\n\n**Nemo sapiens concupivit.**\n\nKein Weiser hat es (d. h.: das Geld) begehrt.\n\nSallust ( _Bellum Catilinarium_ 11,3) wettert gegen Geiz: \u00bbAvaritia pecuniae studium habet, quam nemo sapiens concupivit.\u00ab (\u00bbGeiz hat in sich das Streben nach Geld, das doch kein Weiser [je] begehrte.\u00ab)\n\n**Nemo silens placuit, multi brevitate loquendi.**\n\nNiemand gefiel durch Schweigen, viele (aber) durch die K\u00fcrze ihres Ausdrucks.\n\nAusonius, _Epistulae_ 24,44.\n\n**Nemo solus satis sapit.**\n\nNiemand ist f\u00fcr sich allein weise genug.\n\nPlautus, _Miles gloriosus_ 885.\n\n**Nemo tam divos habuit faventes,**\n\n**Crastinum ut posset sibi polliceri.**\n\nNiemand hatte ihm so geneigte G\u00f6tter, da\u00df er sich den morgigen Tag h\u00e4tte versprechen k\u00f6nnen.\n\n(D. h.: Niemand stand bisher so in der Gunst der G\u00f6tter, da\u00df er sich des folgenden Tages sicher sein konnte.)\n\nSeneca, _Thyestes_ 619 f.\n\n**Nempe dat id, cuicumque libet Fortuna rapitque,**\n\n**Irus et est subito, qui modo Croesus erat.**\n\nGibt doch die Gl\u00fccksg\u00f6ttin und raubt es, wem immer sie will, und Irus ist pl\u00f6tzlich, wer eben noch Kr\u00f6sus war.\n\nOvid, _Tristia_ 3,7,42 f. \u2013 Irus war ein Bettler am Hofe des Odysseus, Kr\u00f6sus der unerme\u00dflich reiche K\u00f6nig der Lyder.\n\n**ne pilo quidem minus**\n\num kein Haar weniger\n\nCicero, _Ad Quintum fratrem_ 2,16,5.\n\n**Neque aqua aquae nec lacte est lactis, crede mihi, usquam similius \/ quam hic tui est tuque huius autem.**\n\nWeder ist Wasser jemals Wasser noch Milch der Milch, glaube mir, \u00e4hnlicher als dieser dir und du diesem.\n\nPlautus, _Menaechmi_ 1089 f. Vgl. **Ex uno puteo...**\n\n**Neque enim mihi cornea fibra est.**\n\nUnd mein Inneres ist n\u00e4mlich nicht aus Horn.\n\n(D. h.: Ich habe sehr wohl Gef\u00fchle.)\n\nPersius, _Saturae_ 1,47. \u2013 Catull ( _Carmina_ 23,12) und Plinius d. \u00c4. ( _Naturalis historia_ 31,102) verbinden Horn mit Trockenheit. Sidonius Apollinaris ( _Epistulae_ 4,1,4) spricht von den \u00bbcorda cornea\u00ab (\u00bbhornigen Herzen\u00ab) ungehobelter barbarischer V\u00f6lker.\n\n**neque fictum neque pictum neque scriptum**\n\nweder plastisch dargestellt noch gemalt noch geschrieben\n\n(d. h.: weder in der Bildhauerei noch in der Malerei noch in der Dichtung dargestellt)\n\nNach Plautus, _Asinaria_ 174.\n\n**Neque habet plus sapientiae quam lapis.**\n\nUnd er hat nicht mehr Verstand als ein Stein.\n\nPlautus, _Miles gloriosus_ 236.\n\n**Neque id ad vivum reseco.**\n\nUnd ich schneide es nicht bis aufs lebendige Fleisch zur\u00fcck.\n\n(D. h.: Ich will darauf nicht genauer eingehen.)\n\nCicero, _De amicitia_ 18.\n\n**Neque imbellem feroces \/ progenerant aquilae columbam. Auch bringen wilde Adler keine friedliche Taube hervor.**\n\nHoraz, _Carmina_ 4,4,31 f.\n\n**Neque in tenero stat tibi corde silex.**\n\nUnd du hast keinen Kiesel im weichen Herzen.\n\n(vgl.: ein Herz von Stein haben)\n\nTibull, _Elegiae_ 1,1,64.\n\n**Neque semper arcum \/ tendit Apollo.**\n\nAuch Apoll spannt seinen Bogen nicht immer (d. h.: auch er gibt sich bisweilen der Mu\u00dfe hin).\n\nHoraz, _Carmina_ 2,10,19 f. Vgl. **Quandoque bonus dormitat Homerus**.\n\n**Ne quid exspectes amicos, quod tute agere possies.**\n\nErwarte nicht von Freunden, was du selbst tun k\u00f6nntest.\n\nGellius ( _Noctes Atticae_ 2,29,20) zitiert Ennius (Fragment Saturae ex libris inc. 58 Vahlen). Vgl. **Ab alio exspectes...**\n\n**Ne quid nimis !**\n\nNichts allzu sehr!\n\n(Vgl.: Nichts im \u00dcberma\u00df!)\n\nNach Platon ( _Protagoras_ 343b) eine der von den Sieben Weisen stammenden griechischen Inschriften auf dem Apollontempel zu Delphi (meist Chilon oder Solon zugeschrieben). Lateinisch zuerst bei Terenz ( _Andria_ 61). Ausonius ( _Ludus septem sapientium_ 155), Augustinus ( _De beata vita_ 32) u. a. zitieren sie nach Terenz. Der Platon-Text wird mitunter auch als \u00bbNi(hi)l nimis\u00ab \u00fcbersetzt, z. B. bei Seneca, _Epistulae morales_ 94,43. Vgl. **Vitiosum est ubique...**\n\n**Nervi belli pecunia infinita.**\n\nDie Haupttriebfeder des Krieges sind unbegrenzte Geldmittel.\n\nNach Cicero, _Orationes Philippicae_ 5,5; vgl. auch _De imperio Cn. Pompei_ 7,17, und Diogenes Laertius, _Vitae philosophorum: Bion_ 4,48. Ihm zufolge sah Bion von Borysthenes Reichtum als \u00bbden Nerv des Erfolges\u00ab an. Hieraus entstand \u00bbnervus rerum\u00ab (\u00bbder Nerv der Dinge\u00ab) als Metapher f\u00fcr Geld. Vgl. **Vectigalia nervi sunt rei publicae**.\n\n**nescia mens hominum fati sortisque futurae**\n\nMenschengeist, nicht kundig des Schicksals und des k\u00fcnftigen Loses\n\nVergil, _Aeneis_ 10,501.\n\n**Nescis quid vesper serus vehat.**\n\nDu wei\u00dft nicht, was der sp\u00e4te Abend bringt.\n\nTitel einer Satire Varros (Fragment 333 B\u00fccheler), der auf ein bekanntes r\u00f6misches Sprichwort anspielt (siehe **Praesenti credere fortunae...** ), das gemahnt, den Tag nicht vor dem Abend zu loben. Diese Anspielung ist jedoch ironisch (etwa im Sinne von: \u00bbMan kann nie wissen, was das Abendessen bringen wird!\u00ab), da Varro in seiner Satire lediglich darlegt, wie man ein Abendessen so planen k\u00f6nne, da\u00df es gelinge.\n\n**Nescit vox missa reverti.**\n\nDas gesprochene Wort kann nicht zur\u00fcckkehren.\n\nHoraz, _Ars poetica_ 390. Vgl. **Et semel...**\n\n**Ne supra crepidam sutor!**\n\nDer Schuster (urteile) nicht \u00fcber die Sandale hinaus!\n\n(Vgl.: Schuster, bleib bei deinem Leisten!)\n\nLaut Plinius d. \u00c4. ( _Naturalis historia_ 35,85) hatte ein Schuster zun\u00e4chst die Darstellung einer Sandale auf einem Bild des ber\u00fchmten Malers Apelles getadelt, da eine \u00d6se fehle. Als der Maler dies korrigierte, wurde der Schuster so \u00fcberm\u00fctig, da\u00df er auch die Darstellung eines Beines tadeln zu m\u00fcssen meinte, woraufhin er von dem Maler in seine Schranken verwiesen wurde. Die so entstandene Wendung war bereits in der Antike sehr gel\u00e4ufig. Vgl. **Quam quisque...**\n\n**Ne tempora perde precando!**\n\nVergeude keine Zeit mit Bitten!\n\nOvid, _Metamorphoses_ 11,286. \u2013 Ce\u00ffx, der K\u00f6nig von Trachis, gebietet Peleus, der ihn um Schutz ersucht, keine weitere Zeit durch Bitten zu verschwenden, da ihm ohnehin alles gew\u00e4hrt werde.\n\n**ne vitiosa quidem nuce emere**\n\nnicht einmal f\u00fcr eine hohle Nu\u00df kaufen\n\nNach Plautus, _Miles gloriosus_ 316.\n\n**Nihil ab illo vacat; opus suum ipse implet.**\n\nNichts ist frei von jenem (Gott); er selbst erf\u00fcllt sein Werk.\n\nSeneca, _De beneficiis_ 4,8,2.\n\n**Nihil ad rem.**\n\n(Das tut) nichts zur Sache.\n\nCicero, _Tusculanae disputationes_ 5,63.\n\n**Nihil adsuetudine maius.**\n\nNichts ist m\u00e4chtiger als die Gewohnheit.\n\nOvid ( _Ars amatoria_ 2,345) r\u00e4t, zu Beginn einer Liebschaft keine M\u00fchen zu scheuen, damit sich die Geliebte an die Annehmlichkeiten gew\u00f6hne und dann bei Abwesenheit des Mannes um so heftiger entflamme.\n\n**Nihil agendo homines male agere discunt.**\n\nDurch Nichtstun lernen die Menschen, \u00dcbles zu tun.\n\n(Vgl.: M\u00fc\u00dfiggang ist aller Laster Anfang.)\n\nColumella ( _De re rustica_ 11,1,26) schreibt diese Worte Cato d. \u00c4. zu.\n\n**Nihil agere delectat** siehe **Mihi enim liber esse...**\n\n**Nihil aliud est ebrietas quam voluntaria insania.**\n\nTrunkenheit ist nichts anderes als freiwillige Tollheit.\n\nSeneca, _Epistulae morales_ 83,18. Der Gedanke geht vermutlich auf Cato d. \u00c4. (Fragment Dicta memorabilia 79 Jordan) zur\u00fcck.\n\n**Nihil difficile amanti.**\n\nNichts ist schwer f\u00fcr einen Liebenden.\n\nCicero, _Orator_ 33.\n\n**Nihil dulcius est ultione.**\n\nNichts ist s\u00fc\u00dfer als Rache.\n\nPseudo-Quintilian, _Declamationes_ 381.\n\n**Nihil enim est ei veritatis luce dulcius.**\n\nNichts ist ihm (dem menschlichen Geist) n\u00e4mlich s\u00fc\u00dfer als das Licht der Wahrheit.\n\nCicero, _Academici libri priores_ 2,31.\n\n**Nihil enim honestum esse potest, quod iustitia vacat.**\n\nNichts kann n\u00e4mlich ehrenhaft sein, was der Gerechtigkeit entbehrt.\n\nCicero, _De officiis_ 1,62.\n\n**Nihil enim, quod ad ultimum sui perventurum est finem, non et mature et celeriter incipit.**\n\nNichts n\u00e4mlich, was zu seinem letzten Ziel gelangen will, beginnt nicht auch fr\u00fchzeitig und schnell.\n\nValerius Maximus, _Facta et dicta memorabilia_ 8,7, ext. 2.\n\n**Nihil eripit Fortuna, nisi quod dedit.**\n\nNichts entrei\u00dft das Gl\u00fcck, was es nicht (zuvor) gegeben hat.\n\n(Vgl.: Das Gl\u00fcck ist ein Geber und ein Nehmer.)\n\nSeneca, _De constantia sapientis_ 5,4.\n\n**Nihil est ab omni \/ parte beatum.**\n\nNichts ist in jeder Hinsicht gl\u00fccklich.\n\nHoraz, _Carmina_ 2,16,27 f.\n\n**Nihil est annis velocius.**\n\nNichts ist schneller als die Jahre.\n\nOvid, _Metamorphoses_ 10,520.\n\n**Nihil est enim simul et inventum et perfectum.**\n\nDenn nichts ist, wenn es erfunden wird, zugleich auch vollendet.\n\nCicero, _Brutus_ 71.\n\n**Nihil est enim tam miserabile, quam ex beato miser.**\n\nNichts ist n\u00e4mlich so erb\u00e4rmlich, wie wenn aus einem Gl\u00fccklichen ein Elender wird.\n\nCicero, _Partitiones oratoriae_ 57.\n\n**Nihil est enim virtute amabilius.**\n\nEs gibt n\u00e4mlich nichts Liebenswerteres als die Tugend.\n\nCicero, _De amicitia_ 28.\n\n**Nihil est in intellectu, quod non sit prius in sensu.**\n\nNichts ist im Verstand, was nicht zuvor in der Wahrnehmung w\u00e4re.\n\nNach Thomas von Aquin, _Quaestiones disputatae de veritate_ 2,3,19; vgl. Leibniz, _Nouveaux essais sur l'entendement_ _humain_ , Pr\u00e9face.\n\n**Nihil est miserius quam animus hominis conscius.**\n\nEs gibt nichts Elenderes als einen Menschen mit schlechtem Gewissen.\n\nPlautus, _Mostellaria_ 544.\n\n**Nihil est perpetuum datum** siehe **Ut sunt humana...**\n\n**Nihil est, quod deus efficere non possit.**\n\nEs gibt nichts, was der Gott nicht bewirken k\u00f6nnte.\n\nNach Cicero, _De natura deorum_ 3,92.\n\n**Nihil est toto, quod perstet, in orbe, \/ cuncta fluunt.**\n\nNichts gibt es auf der ganzen Welt, das Bestand h\u00e4tte. Alles flie\u00dft.\n\nOvid, _Metamorphoses_ 15,177 f. (in Anlehnung an Heraklit). Vgl. **Cuncta fluunt...**\n\n**Nihil ex nihilo exsistere vera sententia est.**\n\nEs ist ein wahrer Spruch, da\u00df nichts aus Nichts entsteht.\n\nBoethius, _Consolatio philosophiae_ 5,1, prologus. \u2013 Der Gedanke findet sich bereits bei Aristoteles ( _Physik_ 1,187a 28\u201329), der sich auf Anaxagoras beruft, ebenso an zahlreichen Stellen bei Lukrez (z. B. _De rerum natura_ 1,150), der auf die Physik Epikurs zur\u00fcckgreift, sowie bei Persius, _Saturae_ 3,83 f.\n\n**Nihil homini amico est opportuno amicius.**\n\nNichts ist einem Menschen willkommener als ein Freund zur rechten Zeit.\n\nPlautus, _Epidicus_ 425.\n\n**Nihil inimicius quam sibi ipse.**\n\nNichts ist ihm sch\u00e4dlicher als er sich selbst.\n\nCicero, _Ad Atticum_ 10,12a,3.\n\n**Nihil natura portionibus parit.**\n\nNichts bringt die Natur st\u00fcckweise hervor.\n\nPlinius d. \u00c4., _Naturalis historia_ 17,177.\n\n**Nihil nimis!** siehe **Ne quid nimis!**\n\n**Nihil novi sub sole** siehe **Nihil sub sole...**\n\n**Nihilo sepse plus quam alterum diligat!**\n\nEr liebe sich selbst um nichts mehr als den N\u00e4chsten!\n\nCicero ( _De legibus_ 1,34) bezeichnet diese Worte als eine Leitlinie der Pythagor\u00e4er. Er berichtet von ihnen weiter, da\u00df sie ihren Besitz untereinander teilten und einen Freund als ein zweites Ich ansahen. Vgl. Porphyrius, _Das Leben des_ _Pythagoras_ 33.\n\n**nihil praeter suum negotium agere**\n\nnichts tun, was \u00fcber die eigenen Angelegenheiten hinausgeht\n\nCicero, _De officiis_ 1,125.\n\n**Nihil, quo stat loco, stabit; omnia sternet abducetque secum vetustas.**\n\nNichts wird da stehen, wo es (jetzt) steht; das Alter wird alles niederwerfen und mit sich rei\u00dfen.\n\nSeneca, _Ad Marciam de consolatione_ 26,6.\n\n**Nihil se scire dicit nisi id ipsum.**\n\nEr sagt, er wisse nichts au\u00dfer diesem Umstand (da\u00df er nichts wei\u00df).\n\nNach Cicero ( _Academici libri posteriores_ 1,16), der \u00fcber die Gespr\u00e4chstaktik des Sokrates berichtet, dieser habe nichts selbst behauptet, sondern stets nur andere widerlegt und betont, da\u00df er selbst au\u00dfer seiner eigenen Unwissenheit nichts wisse.\n\n**Nihil sub sole novum.**\n\nNichts Neues unter der Sonne.\n\nNach Kohelet 1,9 f. \u2013 Oft als \u00bbNihil novi sub sole\u00ab zitiert.\n\n**Nihil tam difficile est, quin quaerendo investigari possiet.**\n\nNichts ist so schwierig, da\u00df es nicht durch Suchen ergr\u00fcndet werden k\u00f6nnte.\n\nTerenz, _Heauton timorumenos_ 675.\n\n**Nihil tam firmum est, cui periculum non sit, etiam ab invalido.**\n\nNichts ist so fest, da\u00df ihm nicht sogar von Schwachem Gefahr drohte.\n\nCurtius Rufus, _Historia Alexandri Magni_ 7,8,15.\n\n**Nihil tam munitum, quod non expugnari pecunia possit.**\n\nNichts ist so sicher gesch\u00fctzt, da\u00df es nicht mit Geld erobert werden k\u00f6nnte.\n\nCicero, _Actio prima in C. Verrem_ 4.\n\n**Nihil virtuti invium.**\n\nDer Tugend ist nichts unzug\u00e4nglich.\n\nNach Tacitus, _Agricola_ 27,1.\n\n**ni(hi)l admirari**\n\nnichts anstaunen\n\n(z. B.) Cicero ( _Tusculanae disputationes_ 3,30), der berichtet, Anaxagoras habe, als ihm der Tod seines Sohnes gemeldet wurde, geantwortet: \u00bbSciebam me genuisse mortalem.\u00ab (\u00bbIch wu\u00dfte, da\u00df ich einen Sterblichen hervorgebracht habe.\u00ab) Hieran ankn\u00fcpfend legt Cicero dar, da\u00df Weisheit darin bestehe, sich geistig auf alle Zwischenf\u00e4lle einzustellen und sich \u00fcber nichts zu wundern. \u2013 Ebenso Horaz, _Epistulae_ 1,6,1; vgl. Seneca, _Epistulae morales_ 8,5.\n\n**nil conscire sibi, nulla pallescere culpa**\n\nsich nichts (\u00dcblem) bewu\u00dft sein, vor keiner Schuld erblassen m\u00fcssen\n\nHoraz, _Epistulae_ 1,1,61.\n\n**Nil est dictu facilius.**\n\nNichts ist leichter zu sagen.\n\nTerenz, _Phormio_ 300.\n\n**Nil homini certum est.**\n\nNichts ist dem Menschen sicher.\n\nOvid, _Tristia_ 5,5,27.\n\n**Nil melius muliere bona, nil quam mala peius.**\n\nEs gibt nichts Besseres als eine gute Frau, nichts Schlimmeres als eine schlechte.\n\nPetrus Abelardus, _Monita ad Astralabium_ 175.\n\n**Nil mortalibus ardui est.**\n\nNichts ist den Sterblichen (zu) steil (d. h.: [zu] schwer zu erreichen).\n\nHoraz, _Carmina_ 1,3,37.\n\n**Nil nocere.**\n\n(Dem Patienten wenigstens) in keiner Hinsicht schaden.\n\nNach Hippokrates ( _Epidemien_ 1,11), der zwei Grunds\u00e4tze \u00e4rztlicher Behandlung anf\u00fchrt: zu helfen \u2013 oder wenigstens nicht zu schaden.\n\n**Nil non permittit mulier sibi, turpe putat nil.**\n\nAlles erlaubt sich die Frau, nichts h\u00e4lt sie f\u00fcr unschicklich.\n\nJuvenal ( _Saturae_ 6,457) spottet \u00fcber Frauen, die glauben, sich aufgrund ihres Reichtums alles erlauben zu k\u00f6nnen.\n\n**Nil prodest, quod non laedere possit idem.**\n\nNichts ist n\u00fctzlich, was nicht auch ebenso schaden k\u00f6nnte.\n\nOvid, _Tristia_ 2,266.\n\n**Nil sine magno \/ vita labore dedit mortalibus.**\n\nNichts, das frei von gro\u00dfer M\u00fche w\u00e4re, hat das Leben den Sterblichen gegeben.\n\nHoraz, _Sermones_ 1,9,59 f.\n\n**Nimia familiaritas parit contemptum.**\n\nAllzu gro\u00dfe Vertraulichkeit f\u00fchrt zu Verachtung.\n\nJohannes Klimakos ( _Scala Paradisi_ 8; lat \u00dcbers. vermutl. aus 15. Jh., ed. von Antonio Cerruti 1875) bezeichnet diese Wendung bereits als Sprichwort. Die _Scala_ wird mitunter Augustinus, dem Hl. Bernhard und Guigo Carthusiensis zugeschrieben.\n\n**Nimias delicias facis.**\n\nDu treibst zuviel der Scherze.\n\nPlautus, _Casina_ 528.\n\n**Nimium altercando veritas amittitur.**\n\nDurch allzu langen Streit verliert man die Wahrheit (aus dem Blick).\n\nPublilius Syrus, _Sententiae_ N 40.\n\n**Nisi caste, saltem caute!**\n\nWenn schon nicht keusch, dann wenigstens vorsichtig!\n\nHW: 16940a.\n\n**... nisi tute tibi obvius obstes.**\n\n... wenn du dir nicht selbst im Wege stehst.\n\nLukrez, _De rerum naturae_ 4,1150.\n\n**Nitimur in vetitum semper cupimusque negata.**\n\nWir streben immer nach dem Verbotenen und begehren, was uns versagt ist.\n\n(Vgl.: In verbotenen Teichen fischt man gern.)\n\nOvid, _Amores_ 3,4,17.\n\n**Nitor in adversum.**\n\nIch stemme mich dagegen.\n\n(D. h.: Ich biete ihm die Stirn.)\n\nOvid, _Metamorphoses_ 2,72.\n\n**Nobilis equus umbra quoque virgae regitur.**\n\nEin edles Pferd l\u00e4\u00dft sich schon durch den Schatten der Gerte lenken.\n\nCurtius Rufus, _Historia Alexandri Magni_ 7,4,18.\n\n**Nobilitas sola est atque unica virtus.**\n\nAdel liegt einzig und allein in der Tugend.\n\n(Vgl.: Tugend macht edel, aber Adel macht nicht Tugend.)\n\nJuvenal, _Saturae_ 8,20.\n\n**Nocere facile est, prodesse difficile.**\n\nSchaden ist leicht, n\u00fctzen schwierig.\n\nQuintilian ( _Institutio oratoria_ 8,5,6) f\u00fchrt die obige Wendung als Beispiel f\u00fcr eine recht gew\u00f6hnliche Ausdrucksweise an. Derselbe Inhalt werde von Ovid ( _Medea_ Fragment 1 Lenz) in wesentlich eindrucksvollere Worte gefa\u00dft: \u00bbServare potui; perdere an possim, rogas?\u00ab (\u00bbIch vermochte zu retten: Da fragst du, ob ich Verderben bringen k\u00f6nnte?\u00ab)\n\n**Nocet empta dolore voluptas** siehe **Sperne voluptates...**\n\n**noctuam Athenas deportare**\n\neine Eule nach Athen tragen\n\nNach Cicero ( _Ad familiares_ 6,3,4), der in griechischer Sprache Aristophanes ( _V\u00f6gel_ 301) zitiert. Vgl. **In silvam ne ligna feras!** und die andere \u00dcbersetzung des griechischen Originals **ululam Athenas portare**.\n\n**nolens volens**\n\nnicht wollend wollend\n\n(d. h.: wohl oder \u00fcbel)\n\nVgl. Cicero, _De natura deorum_ 1,17; Petron, _Satyrica_ 71; Martial, _Epigrammata_ 8,44,16.\n\n**Noli \/ barbam vellere mortuo leoni!**\n\nZupfe nicht einem toten L\u00f6wen den Bart!\n\nMartial ( _Epigrammata_ 10,90,9 f.) gemahnt die alte Ligeia mit diesen Worten, nicht ihre Scham zu zupfen; das zieme sich nur f\u00fcr junge Frauen. Sie hingegen sei nicht nur alt, sondern fast schon tot.\n\n**Noli equi dentes inspicere donati!**\n\nNimm nicht die Z\u00e4hne eines geschenkten Pferdes in Augenschein!\n\n(Vgl.: Einem geschenkten Gaul schaut man nicht ins Maul.)\n\nHieronymus ( _Commentarius in Ephesios_ , praefatio) bezeichnet diese Regel als bekanntes Sprichwort.\n\n**Noli me tangere!**\n\nR\u00fchr mich nicht an!\n\nLaut Johannes (20,17) sprach so der auferstandene Jesus zu Maria Magdalena. Daher die Bezeichnung des Waldspringkrauts als \u203aKr\u00e4utlein R\u00fchr-mich-nicht-an\u2039.\n\n**Noli turbare circulos meos!**\n\nSt\u00f6re meine Kreise nicht!\n\nNach Valerius Maximus, _Facta et dicta memorabilia_ 8,7, ext. 7. \u2013 Der Mathematiker Archimedes hatte bei der Eroberung von Syrakus durch die R\u00f6mer im Jahre 212 v. Chr. laut Befehl des Marcellus verschont bleiben sollen. Als Soldaten in sein Haus eindrangen, war er so in seine in den Sand gezeichneten geometrischen Formen vertieft, da\u00df er auf die Frage eines Soldaten nicht antwortete, wer er sei. Er soll Valerius zufolge nur \u00bbNoli, obsecro, istum disturbare\u00ab (\u00bbIch bitte dich, bringe diesen [Sand] nicht in Unordung\u00ab) geantwortet haben, worauf ihn ein Soldat t\u00f6tete.\n\n**Nomen amicitia est, nomen inane fides.**\n\nEin Wort (nur) ist Freundschaft, ein leeres Wort Treue.\n\nOvid ( _Ars amatoria_ 1,740) warnt davor, die Geliebte in der Gegenwart von Freunden allzu sehr zu preisen, sonst mache ein Freund sie dem Liebenden wom\u00f6glich abspenstig.\n\n**nomen atque omen**\n\nName und Vorbedeutung\n\nBei Plautus ( _Persa_ 625) gibt eine Prostituierte ihren Namen mit _Lucris_ an, was an _lucrum_ (\u203aGewinn\u2039) denken l\u00e4\u00dft. Heute wird die Wendung meist zitiert als \u00bbNomen est omen\u00ab (\u00bbDer Name ist [gleichzeitig] auch ein Anzeichen [des Charakters]\u00ab).\n\n**Nomina sunt odiosa.**\n\nNamen sind verha\u00dft.\n\n(D. h.: Es ist oft schon schlimm, nur den Namen zu nennen.)\n\nNach Ovid ( _Heroides_ 13,54), der Laodamia in ihrem Brief an Protesilaos die Namen ihr verha\u00dfter Orte aufz\u00e4hlen l\u00e4\u00dft mit der Bemerkung: \u00bbNomina sunt ipso paene timenda sono.\u00ab (\u00bbDie Namen sind fast schon in ihrem Klang furchtbar.\u00ab) \u2013 Vgl. Cicero, _Pro S. Roscio Amerino_ 47.\n\n**Non Archimedes melius potuit discribere.**\n\nArchimedes h\u00e4tte das nicht besser berechnen k\u00f6nnen.\n\nCicero, _Pro A. Cluentio Habito_ 87. \u2013 Cicero selbst hat das Grab des Mathematikers Archimedes in Syracus entdeckt, als er 75 v. Chr. Qu\u00e4stor in Sizilien war ( _Tusculanae disputationes_ 5,64 ff.).\n\n**Non avis aucupibus monstrat, qua parte petatur.**\n\nNicht zeigt der Vogel den Vogelf\u00e4ngern, von welcher Seite aus er erhascht werden kann.\n\nOvid, _Ars amatoria_ 3,669.\n\n**Non bene olet, qui bene semper olet.**\n\nNicht gut riecht, wer immer gut riecht.\n\nMartial ( _Epigrammata_ 2,12,4) h\u00e4lt allzu Gepflegte f\u00fcr verd\u00e4chtig, denn wer immer dufte, stehe nicht in gutem Geruch. Vgl. **Malo quam...**\n\n**Non cuivis homini contingit adire Corinthum** siehe **Principibus placuisse viris...**\n\n**Nondum omnium dierum sol occidit.**\n\nNoch ist nicht die Sonne aller Tage untergegangen.\n\n(Vgl.: Noch ist nicht aller Tage Abend.)\n\nNach Livius, _Ab urbe condita_ 39,26,9.\n\n**Non enim ex omni ligno debet Mercurius exsculpi.**\n\nNicht aus jedem Holz n\u00e4mlich soll eine Merkurstatue geschnitzt werden.\n\nNach Apuleius ( _Apologia_ 43,4), der mit diesem Ausspruch Pythagoras' verdeutlichen will, da\u00df man nicht jedweden Knaben als Medium f\u00fcr Prophezeiungen heranziehen d\u00fcrfe. H\u00e4ufig auch zitiert als \u00bbEx quovis ligno non fit Mercurius\u00ab (\u00bbNicht aus jedem Holz entsteht ein Merkur\u00ab).\n\n**Non enim numero haec iudicantur, sed pondere.**\n\nDies wird n\u00e4mlich nicht nach der Zahl, sondern nach dem Gewicht beurteilt.\n\nCicero ( _De officiis_ 2,79) erl\u00e4utert, in gewissen F\u00e4llen z\u00e4hle nicht Quantit\u00e4t, sondern Qualit\u00e4t. Genauso z\u00e4hle bei Rechtsstreitigkeiten nicht, welche Partei mehr Personen umfasse, sondern welche Partei tats\u00e4chlich recht habe.\n\n**Non enim possumus , quae vidimus et audivimus, non loqui.**\n\nDenn wir k\u00f6nnen nicht, was wir gesehen und geh\u00f6rt haben, nicht verk\u00fcnden.\n\n(D. h.: Wir m\u00fcssen es unbedingt verk\u00fcndigen.)\n\nApostelgeschichte 4,20.\n\n**Non enim solum ipsa Fortuna caeca est.**\n\nDenn das Gl\u00fcck ist nicht nur selbst blind.\n\nCicero, _De amicitia_ 54: \u00bbNon enim solum ipsa Fortuna caeca est, sed eos etiam plerumque efficit caecos, quos complexa est.\u00ab (\u00bbDenn das Gl\u00fcck ist nicht nur selbst blind, es macht auch meist diejenigen blind, die es in seine Arme schlie\u00dft.\u00ab) \u2013 Aus Cicero, _Orationes Philippicae_ 13,10 ist ersichtlich, da\u00df die blinde Gl\u00fccksg\u00f6ttin in der Antike ein gel\u00e4ufiges Bild war.\n\n**Non est ad astra mollis e terris via.**\n\nDer Weg zu den Sternen von der Erde aus ist nicht bequem.\n\nNach dieser Formulierung von Seneca ( _Hercules furens_ 437) bildete sich das bekannte Sprichwort \u00bbPer aspera ad astra\u00ab (\u00bbDurch rauhe Gefilde [f\u00fchrt der Weg] zu den Sternen\u00ab). Hesiod ( _Werke und Tage_ 289 f.), Xenophon ( _Memorabilien_ 2,1,21 ff.) und Silius Italicus ( _Punica_ 4,603 f.) unterstreichen alle, da\u00df der Weg zur Tugend hart sei. Der Hl. Benedikt \u00fcbertrug dieses Bild auf seine Ordensregel. So solle jedem neu aufgenommenen Bruder \u00bballes Harte und Strenge auf dem Wege, der zu Gott f\u00fchrt\u00ab, vorausgesagt werden: \u00bbPraedicentur ei omnia dura et aspera, per quae itur ad Deum\u00ab ( _Regulae Benedicti_ 58,8). Vgl. **Sic itur ad astra**.\n\n**Non est beatus, esse se qui non putat.**\n\nNicht gl\u00fccklich ist, wer nicht gl\u00fccklich zu sein glaubt.\n\nPublilius Syrus, _Sententiae_ N 61.\n\n**Non est, crede mihi, sapientis dicere: vivam.**\n\n**Sera nimis vita est crastina: vive hodie!**\n\nGlaube mir, es zeugt nicht von Weisheit zu sagen: Ich werde leben. Allzu sp\u00e4t ist es, morgen zu leben: Lebe heute!\n\nMartial, _Epigrammata_ 1,15,11 f. Vgl. **Cras vives?**\n\n**Non est de pastu ovium quaestio, sed de lana.**\n\nHier ist nicht die Frage nach dem Weiden der Schafe gestellt, sondern nach der Wolle.\n\nPapst Pius II. zugeschrieben.\n\n**Non est enim consilium in vulgo.**\n\nDie Menge zeigt n\u00e4mlich keine Besonnenheit.\n\nCicero, _Pro Cn. Plancio_ 9.\n\n**Non est viri timere sudorem.**\n\nEs ist nicht mannhaft, den Schwei\u00df zu f\u00fcrchten.\n\nSeneca, _Epistulae morales_ 31,7.\n\n**Non est vivere, sed valere vita est.**\n\nLeben hei\u00dft nicht am Leben zu sein, sondern gesund zu sein.\n\nMartial, _Epigrammata_ 6,70,15.\n\n**Non fit sine periclo facinus magnum nec memorabile.**\n\nOhne Risiko geschieht keine gro\u00dfe und denkw\u00fcrdige Tat.\n\nTerenz, _Heauton timorumenos_ 314.\n\n**Non honor est, sed onus.**\n\nNicht Ehre ist's, sondern Last.\n\n(Vgl.: Ehren beschweren; W\u00fcrden sind B\u00fcrden.)\n\nOvid, _Heroides_ 9,31.\n\n**Non ignara mali miseris succurrere disco.**\n\nDa Unheil ich (selbst) erfahren, lerne ich, den Elenden zu Hilfe zu kommen.\n\nVergil, _Aeneis_ 1,630. \u2013 Mit diesen Worten nimmt Dido, der selbst gro\u00dfes Leid widerfahren war, Aeneas und seine Gef\u00e4hrten auf.\n\n**Non intellegunt homines, quam magnum vectigal sit parsimonia.**\n\nDie Menschen verstehen nicht, welch **gro\u00dfe Einnahmequelle in der Sparsamkeit liegt.**\n\nCicero, _Paradoxa Stoicorum_ 49.\n\n**non invita Minerva** siehe **invita Minerva**\n\n**Non liquet (N. L.)**\n\nEs ist nicht klar.\n\nMit dieser Formel wurde nach Abschlu\u00df der Beweisaufnahme gefordert, statt der Urteilsf\u00e4llung den Strafproze\u00df wiederaufzunehmen, wenn die Beweislage noch nicht eindeutig war. \u2013 Vgl. Cicero, _Pro A. Cluentio Habito_ 76, und Gellius, _Noctes_ _Atticae_ 14,2,25.\n\n**Non me pudet fateri nescire, quod nesciam.**\n\nIch sch\u00e4me mich nicht zuzugeben, da\u00df ich nicht wei\u00df, was ich nicht wei\u00df.\n\nNach Cicero ( _Tusculanae disputationes_ 1,60), der seine Unf\u00e4higkeit bekennt zu erkl\u00e4ren, woraus die Seele bestehe.\n\n**Non missura cutem, nisi plena cruoris hirudo.**\n\nDer Blutegel wird nicht von der Haut lassen, ehe er sich nicht mit Blut vollgesogen hat.\n\nHoraz ( _Ars poetica_ 476) vergleicht verr\u00fcckte Dichter mit Blutegeln, da sie sich gleichsam an einen Zuh\u00f6rer klammern und ihn mit ihren Werken \u203azu Tode lesen\u2039.\n\n**Non nobis solum nati sumus.**\n\nWir sind nicht nur f\u00fcr uns geboren.\n\nCicero ( _De officiis_ 1,2; vgl. _De finibus_ 2,45) zitiert Platon (Brief IX, _An Archytas_ 358 A).\n\n**Non nostrum inter vos tantas componere lites.**\n\nEs ist nicht unsere Sache, so gro\u00dfe Streitigkeiten zwischen euch zu schlichten.\n\nVergil, _Bucolica_ 3,108.\n\n**Non oculi tacuere tui.**\n\nDeine Augen haben nicht geschwiegen.\n\nOvid, _Amores_ 2,5,17.\n\n**Non olet.**\n\nEs stinkt nicht.\n\n(Vgl.: Am Gelde riecht man es nicht, womit es verdient ist.)\n\nW\u00f6rtlich bei Cicero ( _Orator_ 154), der von verschiedenen idiomatischen Ph\u00e4nomenen der lateinischen Sprache spricht, die nicht \u00bbdanach r\u00f6chen, woher sie kommen\u00ab, d. h. deren Herkunft unklar sei. \u2013 Heute bezeichnet man mit dieser Wendung meist Sachverhalte, denen nichts Anr\u00fcchiges innewohnt. Diese Bedeutung entstand infolge einer von Sueton ( _Vita divi_ _Vespasiani_ 23,3) berichteten Begebenheit: Vespasian ging in seinem Geldhunger so weit, selbst den Urin mit einer Steuer (vielleicht f\u00fcr F\u00e4rber und Walker) zu belegen. Von seinem Sohn deshalb getadelt, hielt er ihm eine Handvoll M\u00fcnzen unter die Nase. Als dieser sagte, sie r\u00f6chen nicht, entgegnete ihm Vespasian: \u00bbDabei stammt es (das Geld) doch vom Urin.\u00ab Vgl. **Fiscus non erubescit** und **Lucri bonus est odor ex re quaelibet**.\n\n**Non omne, quod licet, honestum est.**\n\nNicht alles, was erlaubt ist, ist auch ehrenhaft.\n\n_Digesta_ 50,17,144.\n\n**Non omnia possumus omnes.**\n\nWir k\u00f6nnen nicht alle alles.\n\nLucilius, _Saturae_ 5,26 (Fragment 246 Warmington), und Vergil, _Bucolica_ 8,63. Vgl. Macrobius, _Saturnalia_ 5,16,7 und 6,1,35.\n\n**Non omnibus dormio.**\n\nIch schlafe nicht f\u00fcr alle.\n\nCicero ( _Ad familiares_ 7,24,1) zitiert hier Lucilius (Fragment 251 Warmington). Ihm zufolge soll ein gewisser Cipius sich oft schlafend gestellt haben, um seiner Frau die M\u00f6glichkeit zu geben, einflu\u00dfreichen G\u00e4sten ganz besondere Aufmerksamkeit zu schenken. Als jedoch einmal ein Sklave meinte, unbemerkt Wein stehlen zu k\u00f6nnen, soll Cipius die obigen Worte ausgerufen haben.\n\n**Non omnis moriar** siehe **Exegi monumentum...**\n\n**Non opus est verbis, sed fustibus.**\n\nEs bedarf keiner Worte, sondern eines Schlagstocks.\n\nCicero ( _In L. Calpurnium Pisonem_ 73) \u00e4u\u00dfert sich ungehalten \u00fcber die Dummheit seines Kritikers, der eher mit Schl\u00e4gen als mit Worten belehrt werden m\u00fc\u00dfte. Vgl. **nec verbis solum, sed etiam...**\n\n**Non ovum tam simile ovo.**\n\nKein Ei ist einem anderen so \u00e4hnlich.\n\nQuintilian ( _Institutio oratoria_ 5,11,30) f\u00fchrt diese Wendung als Beispiel f\u00fcr einen Vergleich an, der besonders gro\u00dfe \u00c4hnlichkeit ausdr\u00fcckt.\n\n**Non plus ultra!**\n\nNicht dar\u00fcber hinaus!\n\nNach Pindar ( _Nemeische Oden_ 3,21 ff.), der von den S\u00e4ulen den Hercules spricht, \u00fcber die hinauszusegeln sehr schwierig sei. Einigen Quellen zufolge soll Hercules selbst diese Worte auf die S\u00e4ulen geschrieben haben. Heute wird die Wendung f\u00fcr Un\u00fcbertreffliches, f\u00fcr Dinge, \u00fcber die nichts hinausgeht, verwendet.\n\n**Non posse videtur muscam excitare.**\n\nEr scheint keine M\u00fccke aufscheuchen zu k\u00f6nnen.\n\n(vgl.: keiner Fliege etwas zuleide tun k\u00f6nnen)\n\nSeneca, _Apocolocyntosis_ 10,3.\n\n**Non possidentem multa vocaveris \/ recte beatum.**\n\nNicht den, der viel besitzt, nennt man mit Recht gl\u00fccklich.\n\nHoraz ( _Carmina_ 4,9,45 ff.) f\u00e4hrt fort: \u00bbRectius occupat \/ nomen beati, qui deorum \/ muneribus sapienter uti \/ duramque callet pauperiem pati \/ peiusque leto flagitium timet.\u00ab (\u00bbMit gr\u00f6\u00dferem Recht hei\u00dft sich der gl\u00fccklich, der die Gaben der G\u00f6tter weise zu nutzen und die harte Armut zu ertragen versteht und Frevel mehr scheut als den Tod.\u00ab) \u2013 Das Zitat wird h\u00e4ufig so abgewandelt, da\u00df es den gegenteiligen Sinn ergibt: \u00bbBeati possidentes!\u00ab (\u00bbGl\u00fccklich die Besitzenden!\u00ab)\n\n**Non possumus .**\n\nWir k\u00f6nnen nicht.\n\nAntwort des Papstes Clemens VII. (wohl in Anspielung auf Apostelgeschichte 4,20), der von Heinrich VIII. um die Scheidung von Katharina von Aragonien gebeten worden war. Heinrich, der noch von Papst Leo X. als \u00bbfidei defensor\u00ab (\u00bbVerteidiger des Glaubens\u00ab) ausgezeichnet worden war, verf\u00fcgte daraufhin die Abspaltung der englischen Kirche vom katholischen Glauben.\n\n**Non profecturis litora bubus aras** siehe **litus arare**\n\n**Non puto illum capillos liberos habere.**\n\nIch glaube nicht, da\u00df jener unverschuldete Haare hat.\n\n(Vgl.: Er hat mehr Schulden als Haare auf dem Kopf.)\n\nPetron, _Satyrica_ 38,12.\n\n**Non quam diu, sed quam bene acta sit, refert.**\n\nNicht wie lange, sondern wie gut du (dein Leben) gef\u00fchrt hast, z\u00e4hlt.\n\nSeneca ( _Epistulae morales_ 77,20) erl\u00e4utert, wie bei Theaterst\u00fccken, so z\u00e4hle auch im Leben nicht, wie lange, sondern wie gut es (aus)gef\u00fchrt wurde.\n\n**Non revocare potes, qui periere dies** siehe **Senescimus; effugit aetas...**\n\n**Non scholae, sed vitae discimus** siehe **Non vitae, sed scholae discimus**\n\n**Non semper idem floribus est honor \/ vernis.**\n\nNicht immer tragen die Fr\u00fchlingsblumen dieselbe Zierde (d. h.: auch sie verbl\u00fchen).\n\nHoraz, _Carmina_ 2,11,9 f.\n\n**Non semper Saturnalia erunt.**\n\nEs werden nicht immer Saturnalien sein.\n\n(Vgl.: Es ist nicht alle Tage Sonntag.)\n\nSeneca, _Apocolocyntosis_ 12,2. \u2013 Am Ende des Monats Dezember wurde das unserem Karneval \u00e4hnliche Saturnalienfest gefeiert. Seneca vergleicht in seiner Satire den Gerichtsbetrieb unter Claudius mit eben diesem Fest. Vgl. **Age, libertate Decembri utere!**\n\n**Non, si male nunc, et olim \/ sic erit.**\n\nAuch wenn es (uns) jetzt schlecht geht, es wird nicht (immer) so sein.\n\nHoraz, _Carmina_ 2,10,17 f.\n\n**Non sum ego, qui fueram.**\n\nIch bin nicht der, der ich gewesen war.\n\nOvid ( _Tristia_ 3,11,25) beklagt, er sei durch das Exil g\u00e4nzlich gebrochen.\n\n**Non sum, qualis eram.**\n\nIch bin nicht der, der ich war.\n\nHoraz ( _Carmina_ 4,1,3) bittet die Liebesg\u00f6ttin Venus, ihn zu verschonen, da er nicht so sei wie als junger Mann.\n\n**Non sunt facienda mala, ut eveniant bona.**\n\nEs darf nichts B\u00f6ses getan werden, damit Gutes daraus hervorgehe.\n\nDamasus Bohemus, _Burchardica, sive regulae canonicae_ 107.\n\n**Non sus Minervam!**\n\nDie Sau (soll) nicht Minerva (belehren)!\n\nCicero ( _Academici libri posteriores_ 1,18) bezeichnet die Wendung als Sprichwort: Es sei l\u00e4cherlich, Minerva (die G\u00f6ttin der \u00c4rzte, K\u00fcnstler, Lehrer und Handwerker) belehren zu wollen.\n\n**non tamquam transfuga, sed tamquam explorator**\n\nnicht als \u00dcberl\u00e4ufer, sondern als Spion\n\nSeneca ( _Epistulae morales_ 2,5) h\u00e4lt sich nicht f\u00fcr einen \u00dcberl\u00e4ufer zur Philosophie Epikurs, wenn er bisweilen bei ihm Anleihen macht, sondern vielmehr f\u00fcr einen stoischen Spion.\n\n**Non tantum autem, qui rapuit, verum is quoque qui recepit, tenetur.**\n\nNicht nur, wer geraubt hat, sondern auch wer (die Ware) in Empfang genommen hat, wird \u00fcberf\u00fchrt.\n\n(Vgl.: Der Hehler ist nicht besser als der Stehler.)\n\nNach _Digesta_ 47,9,3,3.\n\n**Non temptanda , quae effici omnino non possint.**\n\nMan soll nicht versuchen, was \u00fcberhaupt nicht bewerkstelligt werden kann.\n\nQuintilian ( _Institutio oratoria_ 4,5,17) bezeichnet diese Wendung als griechisches Sprichwort.\n\n**Nonumque prematur in annum.**\n\nBis ins neunte Jahr hinein soll es zur\u00fcckgehalten werden.\n\nHoraz ( _Ars poetica_ 388) r\u00e4t, ein Gedicht erst nach neun Jahren Bearbeitung herauszugeben. Die Dauer von neun Jahren ist eine Anspielung auf den neoterischen Dichter C. Helvius Cinna, der neun Jahre lang an seinem Werk _Smyrna_ gefeilt hatte (vgl. hierzu Catull, _Carmina_ 95).\n\n**Non ut diu vivamus, curandum est, sed ut satis.**\n\nSorgen mu\u00df man sich darum, nicht lange, sondern genug zu leben.\n\nSeneca ( _Epistulae morales_ 93,2) wettert gegen \u00bbunm\u00e4\u00dfige Sucht nach Gelehrsamkeit\u00ab und pl\u00e4diert f\u00fcr einen gesunden Menschenverstand. \u2013 Diese Worte werden sehr unterschiedlich \u00fcbersetzt. So findet sich neben der oben gebotenen Version auch z. B. \u00bb... sondern gen\u00fcgend lange zu leben\u00ab, aber auch \u00bb... sondern tatenreich\u00ab oder \u00bb... sondern anst\u00e4ndig zu leben\u00ab.\n\n**Non ut edam, vivo, sed ut vivam edo.**\n\nNicht um zu essen lebe ich, sondern um zu leben esse ich.\n\nQuintilian ( _Institutio oratoria_ 9,3,85) f\u00fchrt die Wendung als Beispiel f\u00fcr eine Antithese an. Vgl. Auctor ad Herennium 4,39.\n\n**Non vitae, sed scholae discimus.**\n\nNicht f\u00fcr das Leben, sondern f\u00fcr die Schule lernen wir.\n\nSeneca ( _Epistulae morales_ 106,12) wettert gegen \u00bbunm\u00e4\u00dfige Sucht nach Gelehrsamkeit\u00ab und pl\u00e4diert f\u00fcr einen gesunden Menschenverstand. \u2013 Die Wendung wird oft im umgekehrten Sinne zitiert: \u00bbNon scholae, sed vitae discimus.\u00ab\n\n**Nosce te (ipsum)!**\n\nErkenne dich (selbst)!\n\nNach Platon ( _Protagoras_ 343b) eine der von den Sieben Weisen stammenden griechischen Inschriften auf dem delphischen Apollontempel. Lateinisch z. B. bei Cicero ( _Tusculanae disputationes_ 1,52; 5,70 sowie _De finibus_ 5,44: **Iubet igitur nos...** ) und Seneca ( _Epistulae morales_ 94,28).\n\n**Nos iam fabula sumus.**\n\nWir sind schon zum Tagesgespr\u00e4ch geworden.\n\nBei Terenz ( _Hecyra_ 620) ist diese Wendung allerdings anders zu verstehen: Der Sprecher meint hier, er habe schon die typischen Rollen eines Alten aus dem gesamten B\u00fchnenrepertoire \u00fcbernommen und sei somit selbst schon zum Theaterst\u00fcck (\u00bbfabula\u00ab) geworden.\n\n**Nos numerus sumus et fruges consumere nati.**\n\nWir sind eine (blo\u00dfe) Zahl und dazu geboren, Fr\u00fcchte zu essen.\n\nHoraz, _Epistulae_ 1,2,27.\n\n**Nos quoque floruimus, sed flos erat ille caducus.**\n\nAuch wir bl\u00fchten (einst), doch jene Bl\u00fcte war verg\u00e4nglich.\n\nOvid, _Tristia_ 5,8,19.\n\n**Nosse velint omnes, mercedem solvere nemo.**\n\nWissen m\u00f6chten sie alle, Lohn zahlen keiner.\n\nJuvenal ( _Saturae_ 7,157) \u00fcber Rhetoriksch\u00fcler, die, wenn es ans Zahlen geht, behaupten, sie h\u00e4tten noch nichts gelernt.\n\n**Nosti mores mulierum: \/ dum moliuntur, dum conantur, annus est.**\n\nDu kennst die Gepflogenheiten der Frauen: W\u00e4hrend sie etwas planen und versuchen, ist das Jahr herum.\n\nTerenz, _Heauton timorumenos_ 239 f. Vgl. **Novi ingenium...**\n\n**nostri farrago libelli**\n\ndas bunte Gemisch unseres B\u00fcchleins\n\nJuvenal, _Saturae_ 1,86.\n\n**Nova nunc religio in te istaec incessit.**\n\nEine neue (d. h.: bisher ungewohnte) Fr\u00f6mmigkeit hat dich nun erfa\u00dft.\n\nTerenz, _Andria_ 730.\n\n**Novi ingenium mulierum: \/ nolunt, ubi velis; ubi nolis, cupiunt ultro.**\n\nIch kenne das Wesen der Frauen: Wo du willst, wollen sie nicht; wo du nicht willst, begehren sie sogar.\n\nTerenz, _Eunuchus_ 812 f. Vgl. **Nosti mores...**\n\n**novum et ad hunc diem inauditum**\n\nneu und bis zum heutigen Tag ungeh\u00f6rt\n\nNach Cicero, _Pro Q. Ligario_ 1,1. \u2013 Im Lateinischen schwingt gleichzeitig die Bedeutung \u203aunerh\u00f6rt\u2039 mit.\n\n**nuda veritas** siehe **incorrupta fides nudaque veritas**\n\n**Nudo detrahere vestimenta me iubes.**\n\nDu befiehlst mir, einem Nackten die Kleider auszuziehen.\n\nPlautus, _Asinaria_ 92.\n\n**nugae canorae**\n\nwohlt\u00f6nende Nichtigkeiten\n\nNach Horaz, _Ars poetica_ 322.\n\n**Nulla dies maerore caret.**\n\nKein Tag ist frei von Kummer.\n\n(Vgl.: Jeder Tag hat seine Plag!)\n\nSeneca, _Troades_ 77.\n\n**Nulla dies sine linea.**\n\nKein Tag ohne eine Linie.\n\nPlinius d. \u00c4. ( _Naturalis historia_ 35,84) berichtet, der Maler Apelles habe keinen Tag vergehen lassen, ohne eine Linie zu zeichnen. Daraus habe sich ein Sprichwort entwickelt, das mit der oben genannten Wendung \u00fcbereinstimmen d\u00fcrfte (vgl. Paroemiographi Graeci: Apostolius 16,44c).\n\n**Nulla fere causa est, in qua non femina litem \/ moverit.**\n\nEs gibt in der Regel keinen Proze\u00df, in dem nicht eine Frau den Streit verursacht h\u00e4tte.\n\nJuvenal, _Saturae_ 6,242 f. \u2013 Schon Herodot beginnt seine Historien mit einer Aufz\u00e4hlung der ersten Kriege, die alle durch den Raub einer Frau hervorgerufen worden seien (z. B. Io, Medea und Helena). Vgl. **Nam fuit ante Helenam...**\n\n**Nulla iniuria est, quae in volentem fiat.**\n\nKein Unrecht ist, was einem nach reinem Willen widerf\u00e4hrt.\n\n_Digesta_ 47,10,1,5.\n\n**Nulla nisi ardua virtus.**\n\nTugend ist stets beschwerlich.\n\n(D. h.: Man erreicht sie nur durch gro\u00dfe M\u00fchen.)\n\nOvid, _Ars amatoria_ 2,537.\n\n**Nulla poena sine lege.**\n\nKeine Strafe ohne Gesetz.\n\nAnselm von Feuerbach, _Lehrbuch des gemeinen in Deutschland g\u00fcltigen peinlichen Rechts_ \u00a7 20,1; vgl. _Digesta_ 50,16,131.\n\n**Nulla potentia longa est.**\n\nKeine Machtstellung ist von Dauer.\n\nOvid, _Metamorphoses_ 2,416.\n\n**Nulla praeterea lex, quae puniat inscitiam capitalem, nullum exemplum vindictae.**\n\nEs gibt au\u00dferdem kein Gesetz, das gemeingef\u00e4hrliche Unwissenheit bestraft, kein Beispiel daf\u00fcr, da\u00df sie geahndet worden w\u00e4re.\n\nPlinius d. \u00c4., _Naturalis historia_ 29,18.\n\n**Nulla salus bello; pacem te poscimus omnes.**\n\nIm Krieg liegt kein Heil; wir alle fordern Frieden von dir.\n\nVergil, _Aeneis_ 11,362.\n\n**Nulla tam facilis res, quin difficilis siet, \/ quam invitus facias.**\n\nKeine Sache ist so leicht, da\u00df sie nicht schwer w\u00fcrde, wenn du sie widerwillig ausf\u00fchrst.\n\nTerenz, _Heauton timorumenos_ 805 f.\n\n**Nullum magnum ingenium sine mixtura dementiae fuit.**\n\nKein gro\u00dfes Genie war ohne Beimischung von Verr\u00fccktheit.\n\n(Vgl.: Genie und Wahnsinn wohnen nah beieinander.)\n\nSeneca ( _De tranquilitate animi_ 17,10) schreibt diese Feststellung Aristoteles ( _Problemata_ 30,953a) zu. Der Gedanke findet sich jedoch bereits bei Platon (z. B. _Phaedrus_ 245a). Vgl. **Misce stultitiam...** und **Saepe enim audivi...**\n\n**nullum quidem pilum boni viri habere**\n\nkein einziges Haar eines ehrenwerten Mannes (an sich) haben\n\nNach Cicero ( _Pro Q. Roscio comoedo_ 20), der behauptet, Fannius sei immer glatt rasiert, damit an ihm auch wirklich kein ehrenwertes Haar sei.\n\n**Nullus agenti dies longus est.**\n\nEinem T\u00e4tigen ist kein Tag lang.\n\nSeneca, _Epistulae morales_ 122,3.\n\n**Nullus dolor est, quem non longinquitas temporis minuat ac molliat.**\n\nEs gibt keinen Schmerz, den nicht lange Zeitdauer verringerte und milderte.\n\n(Vgl.: Zeit heilt Wunden.)\n\nCicero, _Ad familiares_ 4,5,6.\n\n**Nullus est liber tam malus, ut non aliqua parte prosit.**\n\nKein Buch ist so schlecht, da\u00df es nicht an irgendeiner Stelle n\u00fctzte.\n\nPlinius d. J. ( _Epistulae_ 3,5,10) berichtet, obiger Ausspruch stamme von seinem Onkel Plinius d. \u00c4., der jedes Buch, das er las, exzerpierte. Er glaubte n\u00e4mlich, da\u00df selbst in schlechten B\u00fcchern verwertbare Informationen zu finden seien. So behauptet er im Vorwort seiner _Naturalis Historia_ (17), etwa zweitausend Schriftrollen f\u00fcr die Erstellung seines Werkes gelesen zu haben.\n\n**Numerantur enim sententiae, non ponderantur.**\n\nDie Stimmen werden n\u00e4mlich gez\u00e4hlt und nicht gewogen.\n\nPlinius d. J., _Epistulae_ 2,12,5.\n\n**Numquam accedo, quin abs te abeam doctior.**\n\nNiemals besuche ich dich, ohne da\u00df ich dich nicht gescheiter verlie\u00dfe.\n\nTerenz, _Eunuchus_ 791.\n\n**Numquam aliud natura, aliud sapientia dicit.**\n\nNiemals sagt die Natur das eine, die Weisheit das andere.\n\n(D. h.: Natur und Weisheit stimmen stets \u00fcberein.)\n\nJuvenal, _Saturae_ 14,321.\n\n**Numquam est fidelis cum potente societas.**\n\nNiemals ist das B\u00fcndnis mit einem M\u00e4chtigen verl\u00e4\u00dflich.\n\nPhaedrus, _Fabulae_ 1,5,1, nach Aesop 155 Hausrath. \u2013 Aus der Fabel wird deutlich, da\u00df derjenige, der gemeinsam mit einem L\u00f6wen jagt, immer den k\u00fcrzeren ziehen wird, da L\u00f6wen nach der Jagd die ganze Beute f\u00fcr sich beanspruchen (vgl. dt. \u00bbL\u00f6wenanteil\u00ab). Hieraus entwickelte sich die Wendung \u00bbleonina societas\u00ab (\u00bbL\u00f6wengesellschaft\u00ab; z. B. _Digesta_ 17,2,29,2).\n\n**Numquam nimis dicitur, quod numquam satis discitur.**\n\nNie wird zuviel gesagt, was nie genug gelernt wird.\n\nSeneca, _Epistulae morales_ 27,9.\n\n**Numquam potest non esse virtuti locus.**\n\nNiemals kann nicht der rechte Ort f\u00fcr Tugend gegeben sein.\n\n(D. h.: Tugend ist immer recht am Platz.)\n\nSeneca, _Medea_ 161.\n\n**Numquam Stygias fertur ad umbras \/ inclita virtus.**\n\nNiemals wird ber\u00fchmte Tugend zu den stygischen Schatten hinabgetragen.\n\n(D. h.: Helden bleiben durch ihren Ruhm am Leben.)\n\nSeneca, _Hercules Oetaeus_ 1983 f. \u2013 Die Styx ist ein Unterweltsflu\u00df, das \u00bbWasser des Grauens\u00ab.\n\n**Numquam virtutis molle documentum est.**\n\nTugend zu beweisen ist niemals einfach.\n\nSeneca, _De providentia_ 4,12.\n\n**Nunc adbibe puro \/ pectore verba, puer, nunc te melioribus offer!**\n\nJetzt, da du (noch) ein Knabe bist, trinke reinen Herzens diese Worte, jetzt \u00f6ffne dich den Besseren!\n\nHoraz ( _Epistulae_ 1,2,67 ff.) f\u00fchrt das Bild weiter aus: \u00bbQuo semel est imbuta recens servabit odorem \/ testa diu.\u00ab (\u00bbEin frisches Tongef\u00e4\u00df wird lange den Geruch, mit dem es einmal getr\u00e4nkt, behalten.\u00ab) Auch Quintilian ( _Institutio oratoria_ 1,1,5) betont mit diesem Bild, da\u00df in der Jugend Gelerntes besonders stark pr\u00e4gt.\n\n**Nunc animis opus, Aenea, nunc pectore firmo.**\n\nJetzt ist Mut, Aeneas, jetzt ist ein starkes Herz vonn\u00f6ten.\n\nVergil, _Aeneis_ 6,261.\n\n**Nunc est bibendum, nunc pede libero \/ pulsanda tellus.**\n\nJetzt hei\u00dft es trinken, jetzt mit freiem Fu\u00df die Erde stampfen (d. h.: tanzen).\n\nHoraz ( _Carmina_ 1,37,1 f.) spielt mit diesen Anfangsworten der Ode anl\u00e4\u00dflich des Selbstmordes von Kleopatra auf die Worte des Alkaios nach dem Tode des Tyrannen Myrsilos von Mytilene an (Fragment Inc. libr. 8 Lobel).\n\n**Nunc hic dies aliam vitam adfert, alios mores postulat.**\n\nDieser heutige Tag bringt ein anderes Leben mit sich, er fordert andere Sitten.\n\nTerenz, _Andria_ 189, zitiert bei Cicero, _Ad familiares_ 12,25,5.\n\n**Nunc quidem paululum a sole.**\n\nNun (geh mir) wenigstens ein bi\u00dfchen aus der Sonne.\n\nCicero ( _Tusculanae disputationes_ 5,92) zufolge soll Diogenes aus Sinope diese Worte Alexander dem Gro\u00dfen entgegengebracht haben, der ihn gefragt hatte, ob er etwas f\u00fcr ihn, Diogenes, tun k\u00f6nne. Vgl. Plutarch, _Alexander_ 14 und **Interim velim a sole mihi non obstes**.\n\n**Nunc vino pellite curas** siehe **O fortes peioraque passi...**\n\n**Nusquam est, qui ubique est.**\n\nNirgends ist, wer \u00fcberall ist.\n\n(Vgl.: Jedermanns Gesell ist niemands Freund.)\n\nSeneca ( _Epistulae morales_ 2,2) r\u00e4t davon ab, sich zu vielen Philosophen zuzuwenden, da man sich so verzettele.\n\n**nutrimentum spiritus**\n\nNahrung f\u00fcr den Geist\n\nInschrift der fr\u00fcheren K\u00f6niglichen Bibliothek in Berlin. \u2013 In seinem Buch \u00fcber alt\u00e4gyptische Baudenkm\u00e4ler schreibt der Abb\u00e9 Jean Terrasson, die Inschrift der Bibliothek von Memphis sei \u00bbLa Nourriture de l'\u00e2me\u00ab gewesen.\n\n**Nutritur vento, vento restinguitur ignis.**\n\nFeuer wird durch Wind gesch\u00fcrt, durch Wind wieder gel\u00f6scht.\n\nOvid, _Remedia amoris_ 807. Vgl. **Lenis alit...**\nO\n\n**Obsequium amicos, veritas odium parit.**\n\nGef\u00e4lligkeit schafft Freunde, Wahrheit Ha\u00df.\n\n(Vgl.: Willfahren macht Freunde, Wahrheit Feinde.)\n\nTerenz, _Andria_ 68; vgl. Quintilian ( _Institutio oratoria_ 8,5,4), der diese Stelle zitiert.\n\n**Obstipui steteruntque comae et vox faucibus haesit.**\n\nIch erstarrte, und die Haare standen mir zu Berge, und die Stimme stockte mir in der Kehle.\n\nVergil, _Aeneis_ 2,774 und 3,48.\n\n**Obstipui tacitus sustinuique gradum.**\n\nErstaunt verstummte ich und hielt meine Schritte an.\n\nOvid, _Fasti_ 6,398.\n\n**Occasionem oblatam tenete!**\n\nHaltet die sich bietende Gelegenheit fest!\n\n(vgl.: die Gelegenheit beim Schopfe packen)\n\nCicero, _Orationes Philippicae_ 3,34.\n\n**Occide, verbera, ure!**\n\nT\u00f6te, schlag zu, brenne!\n\nSeneca ( _Epistulae morales_ 7,5) zufolge hechelte das Volk oft mit \u00e4hnlichen Worten nach stets neuen Grausamkeiten bei den Gladiatorenk\u00e4mpfen.\n\n**Occidis fabulans.**\n\nDu bringst (mich) noch schwatzend (d. h.: durch dein Schwatzen) um.\n\nPlautus, _Menaechmi_ 922.\n\n**Occultum visus decurrere piscis ad hamum.**\n\nEr wurde gesehen, wie er wie ein Fisch auf den versteckten Haken zuschwamm.\n\nHoraz, _Epistulae_ 1,7,74.\n\n**O cives, cives, quaerenda pecunia primum est, \/ virtus post nummos.**\n\nO B\u00fcrger, B\u00fcrger, zun\u00e4chst mu\u00df man nach dem Geld trachten, die Tugend kommt nach den M\u00fcnzen.\n\nHoraz, _Epistulae_ 1,1,53 f. \u2013 Der Spruch soll am Medius Ianus, einem Durchgang am Forum, wo die Bankiers anzutreffen waren, gestanden haben.\n\n**Oculi sunt in amore duces.**\n\nDie Augen sind die Lenker in der Liebe.\n\n(Vgl.: Die Augen sind der Liebe Pforten.)\n\nProperz, _Elegiae_ 2,15,12.\n\n**O curas hominum! O quantum est in rebus inane!**\n\nO Sorgen der Menschen! O wieviel Nichtiges ist in der Welt!\n\nPersius, _Saturae_ 1,1, nach Lucilius, _Saturae_ 1,2 (Fragment 2 Warmington); wohl auch Anspielung auf Lukrez (z. B. _De rerum natura_ 1,330). Wie selbstironisch dieser Beginn der Satiren des Persius ist, wird durch die Frage im n\u00e4chsten Vers deutlich: \u00bbQuis leget haec?\u00ab (\u00bbWer wird [all] das lesen?\u00ab)\n\n**Oderint, dum metuant!**\n\nSie m\u00f6gen (mich ruhig) hassen, solange sie (mich) f\u00fcrchten!\n\nSueton ( _Vita Caligulae_ 30,1) schreibt diesen Ausspruch dem Kaiser Caligula zu. Er findet sich bereits bei Accius ( _Atreus_ Fragment 4 D'Ant\u00f2) und wird in der lateinischen Literatur h\u00e4ufig zitiert.\n\n**Oderint, dum probent !**\n\nSie m\u00f6gen (mich ruhig) hassen, solange sie (mich eigentlich) guthei\u00dfen.\n\nTiberius soll Sueton zufolge ( _Vita Tiberii_ 59,2) Schm\u00e4hverse seiner Gegner so aufgefa\u00dft haben, als w\u00e4ren sie nicht aus sachlicher \u00dcberzeugung, sondern aus Unmut \u00fcber seine Ma\u00dfnahmen verfa\u00dft.\n\n**Odero , si potero; si non, invitus amabo.**\n\nIch werde (sie) hassen, wenn ich es kann; wenn nicht, werde ich (sie) widerwillig lieben.\n\nOvid, _Amores_ 3,11,35. \u2013 Meist wird auch die folgende Zeile zitiert: \u00bbNec iuga taurus amat; quae tamen odit, habet.\u00ab (\u00bbUnd der Stier liebt nicht das Joch; dennoch tr\u00e4gt er, was er ha\u00dft.\u00ab)\n\n**Oderunt hilarem tristes tristemque iocosi.**\n\nDie Traurigen hassen den Fr\u00f6hlichen, die Lustigen den Traurigen.\n\nHoraz, _Epistulae_ 1,18,89.\n\n**Oderunt peccare boni virtutis amore.**\n\nAus Liebe zur Tugend hassen es die Guten zu s\u00fcndigen.\n\nHoraz, _Epistulae_ 1,16,52.\n\n**Odia qui nimium timet, \/ regnare nescit.**\n\nWer Ha\u00df allzu sehr f\u00fcrchtet, versteht nicht zu herrschen.\n\nSeneca, _Oedipus_ 703 f. Vgl. **Regna custodit metus.**\n\n**Odi et amo.**\n\nIch hasse und liebe.\n\nCatull, _Carmina_ 85,1.\n\n**Odi profanum vulgus et arceo.**\n\nIch hasse das gew\u00f6hnliche Volk und halte es (mir) fern.\n\nHoraz, _Carmina_ 3,1,1.\n\n**odium humani generis**\n\nHa\u00df auf das Menschengeschlecht\n\nNach Tacitus ( _Annales_ 15,44,4) wurde unter Nero den Christen eine feindliche Gesinnung gegen\u00fcber der ganzen Menschheit nachgesagt.\n\n**O dulce nomen libertatis.**\n\nO s\u00fc\u00dfes Wort: Freiheit!\n\nCicero, _Actio secunda in C. Verrem_ 5,163.\n\n**O fallacem hominum spem!**\n\nO tr\u00fcgerische Hoffnung der Menschen!\n\nCicero, _De oratore_ 3,7.\n\n**O faustum et felicem diem!**\n\nO gl\u00fccklicher und gesegneter Tag!\n\nTerenz, _Andria_ 956.\n\n**O fortes peioraque passi**\n\n**Mecum saepe viri, nunc vino pellite curas!**\n\nO tapfere M\u00e4nner, die ihr mit mir schon oft Schlimmeres ertragen habt, nun vertreibt mit Wein die Sorgen!\n\nHoraz, _Carmina_ 1,7,30 f.\n\n**O fortunate adulescens, qui tuae virtutis Homerum praeconem inveneris!**\n\nO gl\u00fccklicher J\u00fcngling, der du Homer als Preiser deiner Tapferkeit gefunden hast!\n\nLaut Cicero ( _Pro Archia poeta_ 24) Ausruf Alexanders des Gro\u00dfen am Grabe Achills in Sigeum.\n\n**O fortuna, ut numquam perpetuo es bona.**\n\nO Schicksal, da\u00df du niemals in einem fort g\u00fcnstig bist.\n\nTerenz, _Hecyra_ 406.\n\n**Ohe, \/ iam satis est!**\n\nHalt, jetzt ist es genug!\n\nHoraz, _Sermones_ 1,5,12 f.\n\n**O ierum, ierum, ierum, o quae mutatio rerum!**\n\nO weh, welch Wandel der Welt!\n\nRefrain eines Studentenliedes, das erstmals am 9. August 1825 im \u00bbFreim\u00fctigen\u00ab (Berlin) erschien. Der Verfasser ist unbekannt.\n\n**O imitatores , servum pecus!**\n\nO Nachahmer, Sklavenherde!\n\nHoraz, _Epistulae_ 1,19,19.\n\n**oleum addere camino**\n\n\u00d6l ins Feuer gie\u00dfen\n\nNach Horaz, _Sermones_ 2,3,321.\n\n**Oleum perdit et impensas.**\n\nEr verschwendet \u00d6l und Aufwand.\n\nVgl. Hieronymus, _Epistulae_ 57,12; vgl. **Et oleum et operam perdidi**.\n\n**O magnam stultitiam timoris, id ipsum, quod verearis, ita cavere, ut, cum vitare fortasse potueris, ultro accersas et attrahas!**\n\nO diese grenzenlos t\u00f6richte Angst, sich gerade vor dem, was man bef\u00fcrchtet, so zu h\u00fcten, da\u00df man es, obwohl man es vielleicht h\u00e4tte vermeiden k\u00f6nnen, noch obendrein heranholt und herbeizieht!\n\nCicero, _Epistulae ad M. Brutum_ 25,4.\n\n**O matre pulchra filia pulchrior!**\n\nO sch\u00f6ner Mutter sch\u00f6nere Tochter!\n\nHoraz, _Carmina_ 1,16,1. \u2013 Vermutlich ist wie in der folgenden Ode Horazens Geliebte Tyndaris gemeint.\n\n**O mihi praeteritos referat si Iuppiter annos!**\n\nO g\u00e4be mir Jupiter die vergangenen Jahre zur\u00fcck!\n\nVergil, _Aeneis_ 8,560. \u2013 Abschiedsworte des greisen Euander an seinen Sohn Pallas. Jener mu\u00df mit Aeneas in den Krieg gegen Turnus ziehen, aus dem er nie zur\u00fcckkehren wird. Vgl. Nestors Worte bei Homer, _Ilias_ 7,132 und 11,670.\n\n**Omina sunt aliquid.**\n\nAn Vorzeichen ist etwas (Wahres).\n\nOvid, _Amores_ 1,12,3.\n\n**Omitte ista!**\n\nLa\u00df es!\n\nCicero, _Academici libri posteriores_ 1,2.\n\n**Omne animal se ipsum diligit.**\n\nJedes Lebewesen liebt sich selbst.\n\nCicero, _De finibus_ 5,24.\n\n**Omne animi vitium tanto conspectius in se**\n\n**Crimen habet, quanto maior qui peccat habetur.**\n\nJedes Laster der Seele tr\u00e4gt um so sichtlichere Schuld in sich, je angesehener der ist, der sich verfehlt.\n\nJuvenal, _Saturae_ 8,140 f.\n\n**Omne in amore malum, si patiare, leve est.**\n\nJedes \u00dcbel in der Liebe ist leicht, wenn du es ertragen willst.\n\nProperz, _Elegiae_ 2,5,16.\n\n**Omne in praecipiti vitium stetit.**\n\nJedes Laster stand am Scheitelpunkt.\n\nF\u00fcr Juvenal ( _Saturae_ 1,149) hat das Laster in Rom seinen H\u00f6hepunkt erreicht; jetzt droht der Abgrund.\n\n**Omnes enim immemorem beneficii oderunt.**\n\nAlle hassen n\u00e4mlich den, der sich an eine Wohltat nicht erinnert.\n\nCicero, _De officiis_ 2,63.\n\n**Omnes eodem cogimur.**\n\nWir alle werden zum selben Ort getrieben (d. h.: zum Hades, in die Unterwelt).\n\nHoraz, _Carmina_ 2,3,25.\n\n**Omnes homines aequales sunt.**\n\nAlle Menschen sind gleich.\n\n_Digesta_ 50,17,32.\n\n**omnes inlacrimabiles \/ urgentur ignotique longa \/ nocte** siehe **Vixere fortes...**\n\n**Omnes muti eloquerentur prius hoc quam ego.**\n\nAlle Stummen spr\u00e4chen es eher aus als ich.\n\nNach Plautus, _Persa_ 240.\n\n**Omnes sibi malunt melius esse quam alteri.**\n\nAlle m\u00f6chten lieber, da\u00df es ihnen selbst besser geht als ihrem N\u00e4chsten.\n\n(Vgl.: Das Hemd ist mir n\u00e4her als der Rock.)\n\nTerenz ( _Andria_ 427) bezeichnet diese Wendung als altes Sprichwort.\n\n**Omnes una manet nox** siehe **Sed omnes una...**\n\n**Omne supervacuum pleno de pectore manat.**\n\nAlles \u00dcberfl\u00fcssige entflie\u00dft dem vollen Herzen.\n\nHoraz ( _Ars poetica_ 337) r\u00e4t den Dichtern zu K\u00fcrze, da \u00dcberfl\u00fcssiges dem Ged\u00e4chtnis des Lesers schnell entfalle.\n\n**Omne tulit punctum, qui miscuit utile dulci** siehe **Aut prodesse volunt...**\n\n**Omnia conando docilis sollertia vincit.**\n\nIndem sie alles versucht, siegt die gelehrige Erfindungsgabe.\n\nManilius, _Astronomica_ 1,95.\n\n**Omnia cum amico delibera, sed de ipso prius.**\n\nBerate dich mit deinem Freund in allem, aber zuvor berate dich \u00fcber ihn selbst!\n\nSeneca, _Epistulae morales_ 3,2.\n\n**Omnia fert aetas, animum quoque.**\n\nAlles nimmt (uns) das Alter, sogar das Ged\u00e4chtnis.\n\nVergil, _Bucolica_ 9,51.\n\n**Omnia, Lucili , aliena sunt, tempus tantum nostrum est.**\n\nAlles ist fremdes Gut, nur die Zeit ist unser Eigen, Lucilius.\n\nSeneca, _Epistulae morales_ 1,3.\n\n**Omnia mecum porto mea.**\n\nIch trage alles Meinige bei mir.\n\nCicero ( _Paradoxa Stoicorum_ 8) berichtet, Bias aus Pri\u00ebne, der seinen gr\u00f6\u00dften Besitz in seinen geistigen F\u00e4higkeiten sah, habe so geantwortet, als er bei der Einnahme seiner Vaterstadt dazu aufgefordert wurde, sein Hab und Gut zusammenzupacken. Die Wendung l\u00e4\u00dft sich auf Menander, Einzelverse Fragment 569 J\u00e4kel, zur\u00fcckf\u00fchren. Vgl. z. B. Seneca ( _Epistulae morales_ 9,18 f.), der eine \u00e4hnliche Aussage Stilpon von Megara zuschreibt; \u00e4hnlich _De constantia sapientis_ 5,6. Phaedrus ( _Fabulae_ 4,23,14 und 26 f.) wiederum l\u00e4\u00dft Simonides \u00e4hnliche Worte sprechen. Vgl. **Homo doctus...**\n\n**Omnia mors aequat.**\n\nDer Tod macht alle(s) gleich.\n\n(Vgl.: Ob arm, ob reich, der Tod macht alle gleich.)\n\nClaudian, _De raptu Proserpinae_ 2,302.\n\n**Omnia mors poscit; lex est, non poena, perire.**\n\nAlles fordert der Tod; Gesetz ist es (und) keine Strafe zu sterben.\n\nSeneca, _Epigrammata_ 1,7 (\u00bbDe tranquilitate temporis\u00ab).\n\n**Omnia munda mundis.**\n\nDen Reinen ist alles rein.\n\nTitusbrief 1,15. \u2013 Diese Worte werden heute meist als Verspottung von \u00fcberm\u00e4\u00dfiger Gutgl\u00e4ubigkeit und Naivit\u00e4t aufgefa\u00dft.\n\n**Omnia mutantur, nihil interit.**\n\nAlles wandelt sich, nichts vergeht.\n\nOvid, _Metamorphoses_ 15,165.\n\n**Omnia orta occidunt.**\n\nAlles Entstandene geht (wieder) unter.\n\nSallust, _Bellum Iugurthinum_ 2,3.\n\n**Omnia praeclara rara.**\n\nAlles Herrliche ist selten.\n\nCicero, _De amicitia_ 79.\n\n**Omnia prius experiri quam armis sapientem decet.**\n\nF\u00fcr einen Weisen ziemt es sich, alle (Mittel) eher als Waffen zu versuchen.\n\nTerenz, _Eunuchus_ 789.\n\n**Omnia quadrata currunt.**\n\nAlles l\u00e4uft wie wohlgef\u00fcgt.\n\n(vgl.: wie am Schn\u00fcrchen laufen)\n\nPetron, _Satyrica_ 43,7.\n\n**Omnia sunt hominum tenui pendentia filo.**\n\nAlle menschlichen Belange h\u00e4ngen an einem d\u00fcnnen Faden.\n\nOvid, _Epistulae ex Ponto_ 4,3,35.\n\n**Omnia vertuntur.**\n\nAlles wandelt sich.\n\nProperz, _Elegiae_ 2,8,7.\n\n**Omnia videre, multa praetermittere, pauca monere.**\n\nAlles sehen, vieles unbeanstandet lassen, weniges anmahnen.\n\nPapst Johannes XXIII. zugeschrieben.\n\n**Omnia vincit Amor, et nos cedamus Amori!**\n\nAlles besiegt Amor; auch wir wollen Amor weichen!\n\nDiese von Vergil ( _Bucolica_ 10,69) dem Elegiker Gallus in den Mund gelegten Worte bezeichnet bereits Macrobius ( _Saturnalia_ 5,16,7) als sprichw\u00f6rtlich.\n\n**omni ex parte in suo genere perfectum**\n\nin seiner Art in jeder Hinsicht vollkommen\n\nCicero, _De amicitia_ 79.\n\n**Omnis ars naturae imitatio est.**\n\nJede Kunst ist eine Nachahmung der Natur.\n\nSeneca, _Epistulae morales_ 65,3.\n\n**Omnis habet sua dona dies.**\n\nJeder Tag hat seine Geschenke.\n\nMartial ( _Epigrammata_ 8,78,7) will nicht, wie oft verstanden, ausdr\u00fccken, da\u00df jeder Tag sein Gutes habe, sondern er beschreibt die protzigen Spiele des Stella, bei denen jeden Tag Geschenke unter dem Volk verteilt wurden.\n\n**Omnis homo mendax.**\n\nJeder Mensch ist ein L\u00fcgner.\n\nPsalm 115,11; vgl. R\u00f6merbrief 3,4.\n\n**Omnis in modo est virtus.**\n\nDie Tugend liegt ganz im (rechten) Ma\u00df.\n\nSeneca, _Epistulae morales_ 66,9.\n\n**Omnis res est iam in vado.**\n\nDie ganze Sache ist schon im seichten Wasser.\n\n(D. h.: Wir haben unsere Sch\u00e4fchen schon ins trockene gebracht.)\n\nTerenz, _Andria_ 845.\n\n**omnium consensu**\n\nin \u00dcbereinstimmung aller\n\n(z. B.) Caesar, _De bello Gallico_ 7,4,6.\n\n**Omnium enim rerum principia parva sunt.**\n\nAller Dinge Anfang ist klein.\n\nCicero, _De finibus_ 5,58.\n\n**omnium officiorum observantissimus**\n\nalle Pflichten genauestens beachtend\n\nPlinius d. J., _Epistulae_ 7,30,1.\n\n**Omnium rerum homo mensura est.**\n\nDer Mensch ist das Ma\u00df aller Dinge.\n\nDieser Ausspruch des Protagoras, der seine Schrift \u00bb\u00dcber die Wahrheit, oder: Niederringende Reden\u00ab er\u00f6ffnet, ist mehrmals in griechischer Sprache \u00fcberliefert, so z. B. von Sextus Empiricus ( _Adversus mathematicos_ 7,60; vgl. Diels-Kranz, Fragmente der Vorsokratiker 80). Platon ( _Theaetetus_ 152a) erkl\u00e4rt die Aussage im Sinne eines Relativismus: F\u00fcr jeden Menschen seien die Dinge gerade so, wie sie ihm zu sein schienen.\n\n**Omnium rerum mors est extremum.**\n\nDer Tod ist das Ende aller Dinge.\n\nNach Cicero, _Ad familiares_ 6,21,1.\n\n**O passi graviora, dabit deus his quoque finem.**\n\nO ihr, die ihr (schon) Schlimmeres ertragen habt, ein Gott wird auch diese (Qualen) beenden.\n\nVergil, _Aeneis_ 1,199.\n\n**O praeclarum custodem ovium lupum!**\n\nO welch pr\u00e4chtiger Schafhirte: ein Wolf!\n\n(Vgl.: Wo der Wolf weidet die Herd, sind die Sch\u00e4flein bald verzehrt.)\n\nCicero ( _Orationes Philippicae_ 3,27) bezeichnet diese Worte als bekannte Wendung. Vgl. **Lupo ovem commisisti**.\n\n**Optimum est non nasci.**\n\nAm besten ist, (gar) nicht (erst) geboren zu werden.\n\nNach Sophokles, _Oedipus Coloneus_ 1224 f.\n\n**O, quam difficile est crimen non prodere vultu!**\n\nO wie schwierig ist es, das Verbrechen nicht durch seine Miene zu verraten!\n\nOvid, _Metamorphoses_ 2,447. \u2013 Die von Jupiter geschw\u00e4ngerte Kallisto kann kaum die Fassung bewahren, als Diana, die von dieser Tat nichts wei\u00df, sie anh\u00e4lt, sich wieder unter ihre jungfr\u00e4ulichen Begleiterinnen zu begeben.\n\n**O quam miserum est nescire mori!**\n\nWie elend es ist, nicht sterben zu k\u00f6nnen!\n\nSeneca, _Agamemnon_ 611.\n\n**O quantum est in rebus inane!** siehe **O curas hominum...**\n\n**O quid solutis est beatius curis!**\n\nO was ist gl\u00fcckseliger, als frei von Sorgen zu sein!\n\nCatull, _Carmina_ 31,7.\n\n**Ora et labora. Deus adest sine mora.**\n\nBete und arbeite! Gott ist da ohne Verzug.\n\nWahlspruch der Benediktiner. \u2013 Die vorliegende Pr\u00e4gung entstand wohl erst im Mittelalter. In der _Regula Benedicti_ (48 und 57) wird jedoch bereits der hohe Stellenwert der Arbeit betont.\n\n**Orator est vir bonus, dicendi peritus.**\n\nEin Redner ist ein t\u00fcchtiger, im Reden erfahrener Mann.\n\nNach Cato, Fragment _De Rhetorica?_ 14 Jordan, zitiert bei Seneca Rhetor, _Controversiae_ 1, praefatio 9.\n\n**O saeculum, o litterae! Iuvat vivere!**\n\nO Jahrhundert, o Wissenschaft! Es ist eine Freude zu leben!\n\nUlrich von Hutten in einem Brief an Willibald Pirckheimer vom 15. Oktober 1518.\n\n**O sancta simplicitas** siehe **sancta simplicitas**\n\n**Oscula qui sumpsit, si non et cetera sumit,**\n\nHaec quoque, quae data sunt, perdere dignus erit.\n\nWer K\u00fcsse sich nahm und das \u00fcbrige nicht nimmt, verdient es auch, was ihm gegeben wurde, zu verlieren.\n\nOvid, _Ars amatoria_ 1,669 f.\n\n**Ossa ac pellis totus est.**\n\nEr ist ganz Knochen und Haut.\n\nPlautus, _Aulularia_ 564.\n\n**O tempora , o mores!**\n\nO Zeiten, o Sitten!\n\nCicero, _In L. Catilinam_ 1,2. \u2013 Cicero gebraucht die Wendung an mehreren Stellen und wird von Seneca Rhetor ( _Suasoriae_ 6,3) zitiert. Vgl. z. B. Martial, _Epigrammata_ 9,70,1 und 5.\n\n**Otia me somnusque iuvant.**\n\nMu\u00dfe und Schlaf erfreuen mich.\n\nMartial ( _Epigrammata_ 12,68,5) will dem hektischen Leben abschw\u00f6ren.\n\n**O Tite, tute, Tati tibi tanta, tyranne, tulisti!**\n\nO Titus Tatius, du Tyrann, du hast dir dieses so gro\u00dfe (Ungl\u00fcck) selbst eingebracht!\n\nEnnius, _Annales_ Fragment 109 Warmington; zitiert z. B. von Priscian, _Institutiones Grammaticae_ 12,23.\n\n**otium cum dignitate**\n\nMu\u00dfe mit W\u00fcrde\n\nCicero ( _Pro P. Sestio_ 98) will nun keine \u00f6ffentlichen \u00c4mter mehr auf sich nehmen. Allerdings hat er bereits wegen seiner fr\u00fcheren Dienste Ansehen erworben. In _De oratore_ 1,1 und _Ad familiares_ 1,9,21 bezieht er die Wendung nicht auf sich als Privatmann, sondern auf den Staat, dem er einen ehrenvollen Frieden w\u00fcnscht.\n\n**Otium sine litteris mors est et hominis vivi sepultura.**\n\nMu\u00dfe ohne geistige Besch\u00e4ftigung bedeutet den Tod und die Bestattung eines lebendigen Menschen.\n\nSeneca, _Epistulae morales_ 82,3.\n\n**O vita , misero longa, felici brevis!**\n\nO Leben, lang f\u00fcr den Elenden, kurz f\u00fcr den Gl\u00fccklichen!\n\nPublilius Syrus, _Sententiae_ O 3.\nP\n\n**pacem te poscimus omnes** siehe **Nulla salus bello...**\n\n**Pacta sunt servanda.**\n\nVertr\u00e4ge m\u00fcssen eingehalten werden.\n\nNach Papst Gregor IX., _Decretales_ 1,35,1. Vgl. _Digesta_ 2,14,1 und 2,14,7,7.\n\n**Paete, non dolet.**\n\nPaetus, es tut nicht weh.\n\nNach Plinius d. J. ( _Epistulae_ 3,16,6) hat Arria, als sie und ihr Mann Paetus im Begriff waren, gemeinsam Selbstmord zu begehen, sich als erste den Dolch in die Brust gesto\u00dfen, ihn herausgezogen und ihrem Mann mit den obigen Worten gereicht. \u2013 Martial ( _Epigrammata_ 1,13) berichtet dasselbe, doch soll Arria gesagt haben: \u00bbGlaube mir, die Wunde, die ich mir zugef\u00fcgt habe, schmerzt nicht, aber die, die du dir zuf\u00fcgen wirst, Paetus, schmerzt.\u00ab\n\n**Palam in eum tela iaciuntur, clam subministrantur.**\n\n\u00d6ffentlich werden Geschosse auf ihn geschleudert, heimlich werden sie nachgereicht.\n\nCicero ( _Pro M. Caelio_ 20) verdeutlicht hier, da\u00df seiner Meinung nach hinter den Ankl\u00e4gern des Caelius andere Drahtzieher stecken.\n\n**Palleat omnis amans, hic est color aptus amanti.**\n\nEs erbleiche jeder Liebende, das ist die zu einem Liebenden passende Farbe.\n\nOvid, _Ars amatoria_ 1,729.\n\n**Pallida mors aequo pulsat pede pauperum tabernas \/ regumque turris.**\n\nDer bleiche Tod klopft mit gerechtem Fu\u00df an \u00e4rmliche H\u00fctten wie an hohe K\u00f6nigspal\u00e4ste.\n\n(Vgl.: Ob arm, ob reich, der Tod macht alle gleich.)\n\nHoraz, _Carmina_ 1,4,13 f.\n\n**panem et circenses**\n\nBrot und Spiele\n\nJuvenal, _Saturae_ 10,81. \u2013 Mit Lebensmittelspenden und aufwendigen Spielen hielten die Kaiser sich das Volk geneigt.\n\n**Parant amicos lautae res, dubiae probant.**\n\nEine stattliche Verm\u00f6genslage schafft Freunde, eine bedenkliche pr\u00fcft sie.\n\n(Vgl.: Gl\u00fcck macht Freunde, aber Not bew\u00e4hrt sie.)\n\nPseudo-Seneca, _De moribus_ 51.\n\n**par bene comparatum**\n\nein gut zusammengestelltes Paar\n\nNach Iustinus, _Historiae Philippicae_ 6,2.\n\n**Parce gaudere oportet et sensim queri.**\n\nEs ziemt sich, sich ma\u00dfvoll zu freuen und in aller Stille zu klagen.\n\nPhaedrus, _Fabulae_ 4,18,9.\n\n**Parcendum est animo miserabile vulnus habenti.**\n\nEine Seele, die an einer elenden Wunde leidet, mu\u00df geschont werden.\n\nOvid ( _Epistulae ex Ponto_ 1,5,23), im Exil befindlich, bittet seinen Freund Maximus um Nachsicht f\u00fcr seine angeblich nachlassenden Dichterf\u00e4higkeiten. Vgl. **Ad medicam dubius confugit aeger opem**.\n\n**Parce, puer, stimulis et fortius utere loris!**\n\nSei zur\u00fcckhaltend mit dem Treibstachel, Knabe, und benutze mehr die Z\u00fcgel!\n\nOvid, _Metamorphoses_ 2,127. \u2013 Der Sonnengott Phoebus warnt seinen Sohn Pha\u00ebton davor, die Pferde, die den Sonnenwagen ziehen, allzu schnell laufen zu lassen. Vgl. **Medio tutissimus ibis**.\n\n**parcere subiectis et debellare superbos**\n\nUnterworfene schonen und Hochm\u00fctige niederschlagen\n\nVergil, _Aeneis_ 6,853. \u2013 Anchises prophezeit seinem Sohn Aeneas, der ihn in der Unterwelt besucht, die k\u00fcnftige Maxime Roms.\n\n**Parcite paucarum diffundere crimen in omnes.**\n\nH\u00fctet euch, die Schuld weniger auf alle zu \u00fcbertragen.\n\nOvid, _Ars amatoria_ 3,9.\n\n**Pares cum paribus facillime congregantur.**\n\nGleiche gesellen sich aufs leichteste zu Gleichen.\n\n(Vgl.: Gleich und gleich gesellt sich gern.)\n\nNach Cicero ( _De senectute_ 7), der die Wendung als altes Sprichwort bezeichnet. Vgl. Homer, _Odyssee_ 17,218, und Platon, z. B. _Symposion_ 195 b.\n\n**Paritur pax bello.**\n\nFrieden erh\u00e4lt man durch Krieg.\n\n(Vgl.: Wer Friede haben will, mu\u00df zum Kriege r\u00fcsten.)\n\nCornelius Nepos, _Epaminondas_ 5,4. Vgl. **Qui desiderat...**\n\n**par negotiis neque supra**\n\nseinen Aufgaben gewachsen, aber nicht mehr\n\nTacitus ( _Annales_ 6,39) \u00fcber Poppaeus Sabinus, der nur durch seine Freundschaft zu einflu\u00dfreichen M\u00e4nnern an \u00c4mter gelangt war, sich aber nicht durch besondere F\u00e4higkeiten auszeichnete.\n\n**par nobile fratrum**\n\nein edles Br\u00fcderpaar\n\nHoraz ( _Sermones_ 2,3,243) bezeichnet so ironisch die ber\u00fchmt-ber\u00fcchtigten S\u00f6hne des Arrius.\n\n**Par pari datum hostimentum est.**\n\nZur Vergeltung ist Gleiches f\u00fcr Gleiches gegeben worden.\n\nPlautus, _Asinaria_ 172.\n\n**par pari respondere**\n\nGleiches mit Gleichem beantworten\n\n(hier: zur\u00fcckzahlen, was man schuldet)\n\nNach Cicero, _Ad Atticum_ 16,7,6. Vgl. Terenz, _Adelphoe_ 73.\n\n**Pars sanitatis velle sanari fuit.**\n\nEin Teil der Heilung war (schon immer) geheilt werden zu wollen.\n\nSeneca, _Phaedra_ 249.\n\n**parta tueri**\n\nErworbenes wahren\n\nOvid, Ars _amatoria_ 2,13.\n\n**Parturient montes, nascetur ridiculus mus.**\n\nKrei\u00dfen werden die Berge, geboren wird eine l\u00e4cherliche Maus.\n\nHoraz ( _Ars poetica_ 139) warnt mit diesen Worten davor, Werke allzu prahlerisch anzuk\u00fcndigen. Die Wendung geht wohl auf ein griechisches Sprichwort zur\u00fcck; vgl. Diogenianus 8,75, au\u00dferdem Phaedrus, _Fabulae_ 4,24. Plutarch ( _Agesilaus_ 36,5) berichtet, die \u00c4gypter h\u00e4tten Agesilaus mit den entsprechenden Worten wegen seines Kleinwuchses verlacht. Vgl. Athenaios, _Deipnosophistae_ 14,616d.\n\n**Parum est autem luxuriae, quod naturae satis est.**\n\nDer Schwelgerei jedoch ist das (zu)wenig, was f\u00fcr die Natur (bereits) genug ist.\n\nSeneca, _De vita beata_ 13,1.\n\n**Parva leves capiunt animos.**\n\nKleinigkeiten erobern leichtsinnige Gem\u00fcter.\n\nOvid, _Ars amatoria_ 1,159.\n\n**Parvum parva decent.**\n\nF\u00fcr Kleine ziemt sich Kleines.\n\nHoraz, _Epistulae_ 1,7,44.\n\n**Pascitur in vivis livor.**\n\nNeid weidet sich an den Lebenden.\n\nOvid ( _Amores_ 1,15,39) ist sich sicher, da\u00df, auch wenn er zu Lebzeiten verkannt wird, ihm nach seinem Tod der verdiente Ruhm zuteil werde.\n\n**Passibus ambiguis fortuna volubilis errat.**\n\nSchwankenden Schritts irrt das wandelbare Schicksal umher.\n\nOvid, _Tristia_ 5,8,15.\n\n**pater historiae**\n\nVater der Geschichtsschreibung\n\nNach Cicero ( _De legibus_ 1,5), der Herodot so bezeichnet.\n\n**pater patriae**\n\nVater des Vaterlandes\n\nCicero ( _In L. Calpurnium Pisonem_ 6) erl\u00e4utert, Catulus habe ihn vor versammeltem Senat mit dieser Ehrenbezeichnung versehen; vgl. Seneca, _De clementia_ 1,14,2. Auch von Caesar (vgl. Livius, _Periochae_ 116) und Augustus (Augustus, _Res gestae_ 35; vgl. Sueton, _Vita divi Augusti_ 58) ist \u00fcberliefert, da\u00df sie diesen Ehrennamen erhalten haben. Tiberius (vgl. Tacitus, _Annales_ 1,72,1) soll ihn zur\u00fcckgewiesen haben. Vgl. **Quod bonum...**\n\n**Pater, peccavi in caelum et coram te.**\n\nVater, ich habe ges\u00fcndigt, gegen den Himmel und vor dir.\n\nLukas 15,18 und 15,21 \u2013 Worte des verlorenen Sohnes bei seiner R\u00fcckkehr.\n\n**Patet omnibus veritas; nondum est occupata.**\n\nDie Wahrheit steht allen offen; sie ist noch nicht (von jemandem f\u00fcr sich) eingenommen worden.\n\nSeneca, _Epistulae morales_ 33,11.\n\n**Patiare potius ipse quam facias scelus.**\n\nErleide eher ein Verbrechen, als selbst eines zu begehen.\n\n(Vgl.: Besser Unrecht leiden als Unrecht tun.)\n\nSeneca, _Phoenissae_ 494. \u2013 Iocasta versucht, den Bruderkampf ihrer S\u00f6hne zu verhindern, und warnt ihren Sohn Polynices vor unlauterem Verhalten. Vgl. **Accipe quam...**\n\n**Patior sine crimine poenam.**\n\nIch erleide Strafe, ohne ein Verbrechen (begangen zu haben).\n\nOvid, _Metamorphoses_ 9,372. \u2013 Dryope war in einen Baum verwandelt worden, als sie Lotusbl\u00fcten gepfl\u00fcckt und dabei unwissentlich die in eine Blume verwandelte Nymphe Lotis verletzt hatte.\n\n**Patriae solum omnibus carum est.**\n\nDer Boden des Vaterlandes ist allen teuer.\n\n(Vgl.: Heimat mein, was kann besser sein?)\n\nNach Cicero, _In L. Catilinam_ 4,16.\n\n**Patria est , ubicumque est bene.**\n\nDas Vaterland ist, wo auch immer man sich wohl f\u00fchlt.\n\nCicero ( _Tusculanae disputationes_ 5,108) zitiert vermutlich aus dem _Teucer_ des Pacuvius (Fragment ex inc. inc. fabulis Ribbeck 92, wahrscheinlich nach Aristophanes, _Plutos_ 1151). Oft auch zitiert als \u00bbUbi bene, ibi patria\u00ab. Vgl. auch HW, z. B. 20842a.\n\n**Patria est, ubicumque vir fortis sedem sibi elegerit.**\n\nDas Vaterland ist, wo auch immer sich ein tapferer Mann seinen Wohnsitz erw\u00e4hlt hat.\n\nCurtius Rufus ( _Historia Alexandri Magni_ 6,4,13) berichtet von diesen Worten Nabarzanes', des Generals von Darius III.\n\n**Pauca accipe contra!**\n\nNimm ein wenig dagegen an!\n\n(D. h.: La\u00df mich kurz darauf erwidern!)\n\nHoraz, _Sermones_ 1,4,38.\n\n**Pauci ex multis sunt amici, homini qui certi sient.**\n\nEs gibt nur wenige unter den vielen Freunden, die einem Menschen (wirklich) sicher sind.\n\n(Vgl.: Viel Freunde und wenig Nothelfer.)\n\nPlautus, _Pseudolus_ 390.\n\n**Pauci, quod sinit alter, amant.**\n\nWenige lieben, was der andere gew\u00e4hrt.\n\nOvid, _Amores_ 3,4,26.\n\n**Paucis temeritas est bono, multis malo.**\n\nF\u00fcr wenige ist Leichtsinn gut, f\u00fcr viele schlecht.\n\nPhaedrus, _Fabulae_ 5,4,12.\n\n**Paulum sepultae distat inertiae \/ celata virtus.**\n\nWenig nur unterscheidet sich verborgene Tugend von begrabener Unt\u00e4tigkeit.\n\nHoraz, _Carmina_ 4,9,29 f.\n\n**Pauperis est numerare pecus.**\n\nDes Armen Sache ist es, sein Vieh zu z\u00e4hlen.\n\nOvid ( _Metamorphoses_ 13,824) l\u00e4\u00dft Polyphem vor seiner Angebeteten Galatea mit seinen ungeheuer gro\u00dfen Herden prahlen: Er habe soviel Vieh, da\u00df er es nicht einmal mehr z\u00e4hlen k\u00f6nne.\n\n**Paupertas artes omnes perdocet.**\n\nArmut lehrt alle K\u00fcnste.\n\n(Vgl.: Not macht erfinderisch.)\n\nNach Plautus, _Stichus_ 178. Vgl. **Ingenium mala saepe movent**.\n\n**Paupertas impulit audax, \/ ut versus facerem.**\n\nVerwegene Armut trieb mich an, Verse zu schmieden.\n\nHoraz, _Epistulae_ 2,2,51 f.\n\n**pauper ubique iacet** siehe **Dat census honores...**\n\n**Pax Cererem nutrit, pacis alumna Ceres.**\n\nDer Friede n\u00e4hrt Ceres, Z\u00f6gling des Friedens ist Ceres.\n\nOvid, _Fasti_ 1,704. \u2013 Ceres steht metonymisch f\u00fcr die Landwirtschaft.\n\n**Pax est tranquilla libertas.**\n\nFrieden ist ungest\u00f6rte Freiheit.\n\nCicero, _Orationes Philippicae_ 2,113.\n\n**Pax hominibus bonae voluntatis!**\n\nFriede den Menschen, die guten Willens sind!\n\nLukas 2,14.\n\n**Pax optima rerum.**\n\nFrieden ist die beste Sache.\n\nSilius Italicus ( _Punica_ 11,592) f\u00e4hrt fort (593 f.): \u00bbPax una triumphis \/ innumeris potior.\u00ab (\u00bbEin Frieden[sschlu\u00df] ist besser als unz\u00e4hlige Triumphz\u00fcge.\u00ab)\n\n**Pax vobiscum .**\n\nDer Friede sei mit euch.\n\nNach Lukas (24,36) und Johannes (20,19), die beide berichten, Jesus habe seine J\u00fcnger bei seiner Erscheinung nach der Kreuzigung mit den Worten \u00bbPax vobis\u00ab (\u00bbFriede sei euch\u00ab) begr\u00fc\u00dft.\n\n**Pecca fortiter!**\n\nS\u00fcndige tapfer!\n\nLuther an Melanchthon am 1. August 1521: \u00bbEsto peccator et pecca fortiter, sed fortius fide et gaude in Christo, qui victor est peccati, mortis et mundi.\u00ab (\u00bbSei ein S\u00fcnder und s\u00fcndige tapfer, aber tapferer noch glaube und freue dich in Christus, der Sieger ist \u00fcber S\u00fcnde, Tod und Welt.\u00ab)\n\n**Peccare pauci nolunt, nulli nesciunt.**\n\nWenige (nur) wollen nicht s\u00fcndigen, keiner (aber) w\u00fc\u00dfte nicht (wie man s\u00fcndigt).\n\nPublilius Syrus, _Sententiae_ P 35.\n\n**Pectus est enim, quod facit disertos.**\n\nEs ist n\u00e4mlich das Herz, das beredt macht.\n\nQuintilian ( _Institutio oratoria_ 10,7,15) r\u00e4t dem Redner, sich den Inhalt der Rede bildhaft vor Augen zu stellen und das, was man sagt, selbst zu empfinden, da man nur \u00fcberzeuge, wenn man aus dem Herzen spreche.\n\n**Pecuniam in loco neglegere maxumum interdum est lucrum.**\n\nAm rechten Ort das Geld nicht zu beachten bringt bisweilen den gr\u00f6\u00dften Gewinn.\n\nTerenz, _Adelphoe_ 216.\n\n**pecuniam per gulam ventremque transmittere**\n\ndas Geld durch die Gurgel und durch den Bauch jagen\n\nNach Pseudo-Quintilian, _Declamationes_ 260.\n\n**Pedibus timor addidit alas.**\n\nDen F\u00fc\u00dfen verlieh die Furcht Fl\u00fcgel.\n\nVergil, _Aeneis_ 8,224.\n\n**Peiora iuvenes facile praecepta audiunt.**\n\nAuf schlechtere Vorschriften h\u00f6rt die Jugend gern.\n\nSeneca, _Thyestes_ 309. \u2013 Der Diener f\u00fchrt noch weiter aus (311): \u00bbSaepe in magistrum scelera redierunt sua.\u00ab (\u00bbOft sind Verfehlungen wieder auf den Lehrer zur\u00fcckgekommen.\u00ab)\n\n**Peior est bello timor ipse belli.**\n\nSchlimmer als Krieg ist die eigentliche Angst vor dem Krieg.\n\nSeneca, _Thyestes_ 572.\n\n**pendere narrantis ab ore**\n\nam Mund des Erz\u00e4hlers h\u00e4ngen\n\n(vgl.: jemandem an den Lippen h\u00e4ngen)\n\nNach Vergil ( _Aeneis_ 4,79), der beschreibt, wie Dido wieder und wieder Aeneas' Bericht vom Untergang Trojas zu h\u00f6ren begehrt.\n\n**Per aspera ad astra!** siehe **Non est ad astra...**\n\n**Percussu crebro saxa cavantur aquis.**\n\nDurch h\u00e4ufiges Tropfen werden Felsen vom Wasser geh\u00f6hlt.\n\nOvid, _Epistulae ex Ponto_ 2,7,40. Vgl. **Ferreus adsiduo consumitur...** , **Gutta cavat lapidem**... und **Stillicidi casus...**\n\n**Perdere est dignus bona, \/ quis nescit uti.**\n\nG\u00fcter zu verlieren verdient, wer sie nicht zu benutzen wei\u00df.\n\nSeneca, _Phaedra_ 442 f.\n\n**Perdere est patriam grave, \/ gravius timere.**\n\nEs ist hart, das Vaterland zu verlieren, h\u00e4rter, es zu f\u00fcrchten.\n\nSeneca, _Troades_ 912 f.\n\n**Perdere posse sat est, si quem iuvet ipsa potestas.**\n\nVernichten zu k\u00f6nnen ist genug f\u00fcr den, den Macht (um ihrer) selbst (willen) erfreut.\n\nOvid, _Heroides_ 12,75.\n\n**per fas et nefas**\n\nmit Recht und Unrecht\n\n(d. h.: mit allen Mitteln)\n\n(z. B.) nach Lukan, _De bello civili_ 5,313.\n\n**Perfer et obdura! Dolor hic tibi proderit olim.**\n\nHalte durch und sei hart! Dieser Schmerz wird dir einst n\u00fctzen.\n\nOvid, _Amores_ 3,11,7; vgl. auch _Ars amatoria_ 2,178, und Horaz, _Sermones_ 2,5,39. Bereits Catull ( _Carmina_ 8,11) sucht sich mit \u00bbperfer, obdura\u00ab vom Liebesschmerz, verursacht durch Lesbia, zu befreien.\n\n**Pergo ad alios, venio ad alios, deinde ad alios: una res.**\n\nIch gehe weiter zu den einen, komme zu anderen, schlie\u00dflich (wieder) zu anderen: immer dasselbe.\n\nPlautus ( _Captivi_ 488) l\u00e4\u00dft den Parasiten Ergasilus dar\u00fcber klagen, da\u00df ihn alle zum Narren halten.\n\n**Periculum in mora.**\n\nGefahr (liegt) im Verzug.\n\nDer preu\u00dfische Kriegsminister Albrecht von Roon in einem Telegramm an Bismarck am 18. September 1862: \u00bbPericulum in mora. D\u00e9p\u00eachez-vous.\u00ab Wohl nach Livius, _Ab urbe condita_ 38,25,13.\n\n**Periere mores, ius, decus, pietas, fides.**\n\nVergangen sind Sitten, Recht, Ehre, Fr\u00f6mmigkeit und Pflichtgef\u00fchl.\n\nSeneca, _Agamemnon_ 112.\n\n**Periit ius fasque bonumque.**\n\nVergangen sind menschliches und g\u00f6ttliches Recht und die Gerechtigkeit.\n\nStatius, _Thebais_ 1,154.\n\n**... perinde ac si cadaver essent.**\n\n... als ob sie ein Leichnam w\u00e4ren.\n\nIgnatius von Loyola, _Conclusiones_ 6,1,6. Daher der \u00bbKadavergehorsam\u00ab.\n\n**Per me equidem sint omnia protinus alba!**\n\nVon mir aus sei k\u00fcnftig alles wei\u00df!\n\n(D. h.: Ich will ein Auge zudr\u00fccken, f\u00fcnf gerade sein lassen.)\n\nPersius, _Saturae_ 1,110.\n\n**Per me isti pedibus trahantur!**\n\nVon mir aus sollen diese bei den F\u00fc\u00dfen gezogen werden!\n\n(D. h.: Der Teufel soll sie holen!)\n\nCicero, _Ad Atticum_ 4,18,2.\n\n**Permitte divis cetera!**\n\n\u00dcberlasse den G\u00f6ttern das \u00fcbrige!\n\nHoraz, _Carmina_ 1,9,9.\n\n**per mutuum colloqium et consolationem fratrum**\n\ndurch das wechselseitige Gespr\u00e4ch und die Tr\u00f6stung der Br\u00fcder untereinander\n\nLuther (\u00bbSchmalkaldische Artikel\u00ab, Dritter Teil, Vom Evangelio) f\u00fchrt Beispiele auf, die zeigen, da\u00df Gott \u00bbvberschwenglich reich jnn seiner Gnade\u00ab sei.\n\n**perpetuum mobile**\n\nunaufh\u00f6rlich Bewegliches (d. h.: ohne angetrieben zu werden)\n\nCaspar Schott, (z. B.) _Technica curiosa_ 10,1: \u00bbmobile perpetuum physico-mechanicum\u00ab. Die Idee des Perpetuum Mobile ist wohl schon \u00e4lter.\n\n**Persaepe evenit, ut utilitas cum honestate certet.**\n\nSehr oft geschieht es, da\u00df Nutzen und Anstand im Widerstreit liegen.\n\nCicero, _Partitiones oratoriae_ 89.\n\n**Per scelera semper sceleribus tutum est iter.**\n\nDie sichere Bahn f\u00fcr Verbrechen ist stets (die) \u00fcber (neue) Verbrechen.\n\nSeneca, _Agamemnon_ 115. Vgl. **Scelere velandum...**\n\n**per varios casus, per tot discrimina rerum**\n\ndurch mancherlei Ungl\u00fcck, durch so viele Gefahren\n\nVergil, _Aeneis_ 1,204.\n\n**Pessima sit, nulli non sua forma placet.**\n\nAuch wenn sie die Allerh\u00e4\u00dflichste ist: Keiner Frau gef\u00e4llt ihr \u00c4u\u00dferes nicht.\n\nOvid, _Ars amatoria_ 1,614.\n\n**Pessimum inimicorum genus laudantes.**\n\nDie schlimmste Art von Feinden: Lobredner.\n\nTacitus, _Agricola_ 41,1.\n\n**Pestis in amicitia pecuniae cupiditas.**\n\nUnheilstifter in der Freundschaft ist Geldgier.\n\nNach Cicero, _De amicitia_ 34.\n\n**Philosophia ancilla theologiae.**\n\nDie Philosophie ist die Dienerin der Theologie.\n\nWohl in Anlehnung an Petrus Damiani, _De divina omnipotentia_ 5,621.\n\n**pia desideria**\n\nfromme W\u00fcnsche\n\nTitel einer Schrift (1627) des katholischen Theologen Herman Hugo. Vgl. auch die Schrift \u00bbPia Desideria oder: Herzliches Verlangen nach gottgef\u00e4lliger Besserung der wahren evangelischen Kirchen\u00ab (1675) des lutherischen Theologen Philipp Jakob Spener. Heute bezeichnet man mit dieser Wendung meist unerf\u00fcllbare W\u00fcnsche.\n\n**pia fraus**\n\nfrommer Betrug\n\nNach Ovid ( _Metamorphoses_ 9,711), der so die Tat der Telethusa bezeichnet, die ihre neugeborene Tochter als Sohn ausgab, um sie vor der durch den Vater gebotenen T\u00f6tung zu bewahren.\n\n**Pictoribus atque poetis \/ quidlibet audendi semper fuit aequa potestas.**\n\nMalern wie Dichtern war es stets gleicherma\u00dfen erlaubt zu wagen, was auch immer (ihnen) beliebte.\n\nHoraz ( _Ars poetica_ 9 f.) f\u00fchrt diese Worte als Behauptung Pisos an. Darauf antwortet er selbst, seiner Meinung nach seien der dichterischen Freiheit Schranken auferlegt: Alle Darstellungen m\u00fc\u00dften im Rahmen des Nat\u00fcrlichen bleiben. Vgl. **ut pictura poesis**.\n\n**Pietas fundamentum est omnium virtutum.**\n\nFr\u00f6mmigkeit (gegen\u00fcber den Eltern) ist das Fundament aller Tugend.\n\nCicero, _Pro Cn. Plancio_ 29.\n\n**pingui Minerva**\n\nmit fetter Minerva\n\n(d. h.: unverbildeten Verstandes; mit gesundem Menschenverstand)\n\nNach Cicero ( _De amicitia_ 19), der diese Wendung als sprichw\u00f6rtlich bezeichnet. \u2013 Die r\u00f6mische G\u00f6ttin Minerva war die Patronin des Handwerks, der K\u00fcnste, der Lehrer und \u00c4rzte. Vgl. **crassa Minerva** und **invita Minerva**.\n\n**Pinguis venter non gignit sensum tenuem.**\n\nEin fetter Bauch bringt keine zarte Empfindung hervor.\n\nHieronymus, _Epistulae_ 52,11, nach Apostolius (Paroemiographi Graeci) 5,22. Vgl. die andere \u00dcbersetzung des griechischen Sprichwortes: **Plenus venter non studet libenter**.\n\n**pinnas incidere**\n\ndie Fl\u00fcgel stutzen\n\nNach Cicero, _Ad Atticum_ 4,2,5.\n\n**Pisces natare oportet.**\n\nFische m\u00fcssen schwimmen.\n\nPetron, _Satyrica_ 39,9. \u2013 Trimalchio fordert mit diesen Worten seine G\u00e4ste zum Trinken auf.\n\n**Placeat homini, quicquid deo placuit.**\n\nM\u00f6ge dem Menschen gefallen, was dem Gott gefiel.\n\nSeneca, _Epistulae morales_ 74,20.\n\n**Plane qualis dominus, talis et servus.**\n\nWirklich: Wie der Herr, so auch der Sklave.\n\n(Vgl.: Wie der Herr, so's G'scherr.)\n\nPetron, _Satyrica_ 58. Vgl. Cicero ( _Ad familiares_ 1,9,12), der Platon zitiert ( _Gesetze_ 4,711c: \u00bbwie die Herrscher, so die Untertanen\u00ab); Cicero, _Ad Atticum_ 5,11,5, in Anspielung auf Platon, _Staat_ 8,563c (\u00bbwie die Herrin, so die H\u00fcndin\u00ab); im umgekehrten Sinne bei Hieronymus, _Epistulae_ 7,5 (\u00bbwie die Gelenkten, so der Lenker\u00ab). Die heute h\u00e4ufig zitierte Wendung \u00bbQualis rex, talis grex\u00ab \u2013 \u00bbwie der K\u00f6nig, so die Herde\u00ab stammt aus dem Mittelalter (HW: 23250 und HWS: 39840a 17a). \u2013 Vgl. **Componitur orbis...** und **Ignavus servos...**\n\n**Plaudite , cives!**\n\nKlatscht Beifall, B\u00fcrger!\n\nDer \u00fcbliche Abschlu\u00df einer Kom\u00f6die lautete \u00bbPlaudite!\u00ab (\u00bbApplaudiert!\u00ab). Sueton ( _Vita divi Augusti_ 99,1) zufolge soll der sterbende Augustus in griechischer Sprache seine Freunde um Applaus gebeten haben, da er seine Rolle in der Kom\u00f6die des Lebens gut gespielt habe.\n\n**Plenus venter facile de ieiuniis disputat.**\n\nEin voller Bauch diskutiert leicht \u00fcber das Fasten.\n\n(Vgl.: Voller Bauch lobt das Fasten.)\n\nHieronymus, _Epistulae_ 58,2; vgl. auch _Epistulae_ 52,11.\n\n**Plenus venter non studet libenter.**\n\nEin voller Bauch studiert nicht gern.\n\nDie Wendung geht wohl auf ein bei Apostolius ( Paroemiographi Graeci, 5,22) genanntes griechisches Sprichwort zur\u00fcck. Eine Quelle f\u00fcr die uns \u00fcberlieferte F\u00fcgung ist nicht belegt; vermutlich stammt sie aus dem Mittelalter. Vgl. die andere \u00dcbersetzung des griechischen Sprichwortes: **Pinguis venter non gignit sensum tenuem**.\n\n**Plerumque gratae divitibus vices.**\n\nMeist sind den Reichen Abwechslungen willkommen.\n\nHoraz, _Carmina_ 3,29,13.\n\n**Ploratur lacrimis amissa pecunia veris!**\n\nMit echten Tr\u00e4nen wird verlorenes Geld beweint.\n\nJuvenal, _Saturae_ 13,134.\n\n**Pluraque sunt semper deteriora bonis.**\n\nEs gibt immer mehr Schlechtere als Gute.\n\nBei Ovid ( _Ars amatoria_ 3,256) stehen diese Worte in folgendem Zusammenhang: W\u00e4hrend er seine Ratschl\u00e4ge in Sachen Liebe an alle richtet, kommen sie letztlich eher den H\u00e4\u00dflichen zugute; die Sch\u00f6nen wenden sich seltener an den Liebesdichter.\n\n**Plurima mortis imago.**\n\nSehr vielgestaltig ist das Erscheinungsbild des Todes.\n\nVergil, _Aeneis_ 2,369.\n\n**Pluris est oculatus testis unus quam auriti decem.**\n\nEin Augenzeuge ist mehr wert als zehn Ohrenzeugen.\n\nPlautus, _Truculentus_ 489.\n\n**Plus apud me tamen vera ratio valebit quam vulgi opinio!**\n\nMehr wird bei mir jedoch echte Vernunft gelten als die Meinung des Volkes.\n\nCicero, _Paradoxa Stoicorum_ 8.\n\n**plusculum se invitasse**\n\nsich etwas mehr g\u00fctlich getan haben\n\n(d. h.: einen sitzen haben)\n\nNach Plautus, _Amphitruo_ 283.\n\n**Plus docet quam scit.**\n\nEr lehrt mehr, als er wei\u00df.\n\nPetron, _Satyrica_ 46,6.\n\n**Plus dolet, quam necesse est, qui ante dolet, quam necesse est.**\n\nMehr als n\u00f6tig leidet, wer leidet, bevor es n\u00f6tig ist.\n\nSeneca, _Epistulae morales_ 98,8.\n\n**Plus exempla quam peccata nocent.**\n\nSchlechte Beispiele schaden mehr als (tats\u00e4chliche) S\u00fcnden.\n\nNach Cicero ( _De legibus_ 3,32), der hier vor allem von den Adligen spricht, die als offizielle Personen in h\u00f6herem Ma\u00dfe als andere das Volk zur Nachahmung anstifteten und daher Vorbildcharakter haben sollten.\n\n**Plus oportet scire servum quam loqui.**\n\nEin Sklave sollte mehr wissen als sagen.\n\nPlautus, _Miles gloriosus_ 477; vgl. auch _Epidicus_ 60.\n\n**plus quam possis audere**\n\nmehr wagen, als man kann\n\nNach Quintilian, _Institutio oratoria_ 1,1,32.\n\n**Plus significas quam loqueris.**\n\nDu deutest mehr an, als du aussprichst.\n\nSeneca, _Epistulae morales_ 59,5.\n\n**Plus sonat quam valet.**\n\nEs t\u00f6nt mehr, als es vermag.\n\nSeneca, _Epistulae morales_ 40,5.\n\n**Plus tamen timor quam ira celeritatis habet.**\n\nAngst ist dennoch von gr\u00f6\u00dferer Schnelligkeit als Zorn.\n\nLivius ( _Ab urbe condita_ 6,32,10) zufolge war die Flucht der ver\u00e4ngstigten Latiner und Volsker zu schnell, als da\u00df das erz\u00fcrnte r\u00f6mische Heer sie h\u00e4tte einholen k\u00f6nnen.\n\n**Plus valet humanis viribus ira dei.**\n\nMehr als menschliche Kr\u00e4fte vermag der Zorn eines Gottes.\n\nOvid, _Tristia_ 5,12,14.\n\n**plusve minusve**\n\nmehr oder weniger\n\n(z. B.) Ovid, _Fasti_ 5,110, wo die Wendung in einem anderen Sinn verwendet wird als heute gel\u00e4ufig: Ovid nimmt sich vor, keine der Musen \u00bbmehr oder weniger\u00ab zu preisen als eine andere.\n\n**Poena potest demi, culpa perennis erit.**\n\nDie Strafe kann genommen werden, die Schuld (jedoch) wird ewig bleiben.\n\nOvid, _Epistulae ex Ponto_ 1,1,64.\n\n**Poena reversura est in caput ista tuum.**\n\nDie Strafe wird auf dein eigenes Haupt zur\u00fcckfallen.\n\nOvid, _Ars amatoria_ 1,340.\n\n**Poenas garrulus ipse dabit.**\n\nDer Schw\u00e4tzer selbst wird bestraft werden.\n\nOvid ( _Amores_ 2,2,60) warnt einen Sklaven davor, einen Ehebruch seiner Herrin anzuzeigen: Oft glaube der Ehemann n\u00e4mlich seiner Frau und verurteile den Sklaven als Schw\u00e4tzer und Verleumder.\n\n**Poeta nascitur, orator fit.**\n\nEin Dichter wird geboren, ein Redner wird (erst durch \u00dcbung).\n\nNach Florus Fragment 419,2 Baehrens.\n\n**poetica licentia**\n\ndichterische Freiheit\n\n(z. B.) nach Seneca, _Naturales quaestiones_ 2,44,1; vgl. Ovid, _Amores_ 3,12,41.\n\n**pollices premere**\n\ndie Daumen dr\u00fccken\n\nPlinius d. \u00c4. ( _Naturalis historia_ 28,25) zufolge gemahnt ein altes Sprichwort, demjenigen, dem man Gl\u00fcck w\u00fcnscht, die Daumen zu dr\u00fccken.\n\n**Pollicitis dives quilibet esse potest.**\n\nIn Versprechungen kann jeder reich sein.\n\n(Vgl.: Er verspricht goldene Berge und ist keinen Heller wert.)\n\nOvid ( _Ars amatoria_ 1,444) r\u00e4t, der Geliebten viele Versprechungen zu machen, selbst wenn man sie nicht erf\u00fcllen kann. Einmal geweckte Hoffnung halte lange an.\n\n**Pone irae frena modumque!**\n\nLege dem Zorn Z\u00fcgel und Ma\u00df an!\n\nJuvenal, _Saturae_ 8,88.\n\n**Pone modum laetis!**\n\nSetze deinem Erfolg eine Grenze!\n\n(D. h.: Werde nicht \u00fcberm\u00fctig!)\n\nStatius, _Thebais_ 2,406.\n\n**Poscimur.**\n\nWir sind gefordert.\n\nHoraz ( _Carmina_ 1,32,1) wird aufgefordert, eines seiner Gedichte vorzulesen.\n\n**Possum multa tibi veterum praecepta referre.**\n\nIch k\u00f6nnte dir von vielen Vorschriften der Alten berichten.\n\nVergil, _Georgica_ 1,176.\n\n**Possum nil ego sobrius.**\n\nN\u00fcchtern vermag ich nichts.\n\n(Vgl.: Wein ist der Poeten Heiliger Geist.)\n\nMartial ( _Epigrammata_ 11,6,12) verk\u00fcndet, da\u00df er, wenn er trinke, gleichsam f\u00fcnfzehn Dichter in sich versp\u00fcre.\n\n**Post amicitiam credendum est, ante amicitiam iudicandum.**\n\nNach der Freundschaft mu\u00df man vertrauen, vor der Freundschaft urteilen.\n\nSeneca ( _Epistulae morales_ 3,2) r\u00e4t, man solle sich seine Freunde sorgf\u00e4ltig aussuchen; betrachte man sie aber einmal als Freunde, m\u00fcsse man ihnen auch vertrauen.\n\n**Post certas hiemes uret Achaicus**\n\n**Ignis Iliacas** (oft auch: **Pergameas** ) **domos.**\n\nNach einer bestimmten Anzahl von Wintern (d. h.: in absehbarer Zeit) wird der Ach\u00e4er Feuer Ilions H\u00e4user verbrennen.\n\nHoraz, _Carmina_ 1,15,35 f. \u2013 Nereus sagt Paris den Untergang Trojas durch den von den Griechen gestifteten Brand voraus.\n\n**Post coenam stabis aut passus mille meabis.**\n\nNach dem Essen sollst du ruh'n (eigentlich: stehen) oder tausend Schritte tun.\n\nInterpolierter Vers aus dem _Regimen sanitatis Salerni_ 1,212 De Renzi.\n\n**Post coitum omne animal triste.**\n\nNach dem Koitus ist jedes Gesch\u00f6pf betr\u00fcbt.\n\nVermutlich nach Pseudo-Petron (Fragment 4,10,1 Baehrens).\n\n**Post equitem sedet atra cura.**\n\nHinter dem Reiter sitzt die d\u00fcstere Sorge.\n\nHoraz, _Carmina_ 3,1,40.\n\n**Posteriores enim cogitationes, ut aiunt, sapientiores solent esse** siehe **Cuiusvis hominis est errare...**\n\n**Post malam segetem serendum est.**\n\nNach einer schlechten Saat mu\u00df man (erneut) s\u00e4en (d. h.: den Mut nicht verlieren).\n\nSeneca, _Epistulae morales_ 81,1.\n\n**Post mortem in morte nihil est, quod metuam, mali.**\n\nNach dem Tod gibt es nichts \u00dcbles im Tode, was ich f\u00fcrchten k\u00f6nnte.\n\nPlautus, _Captivi_ 741.\n\n**Post mortem nulla voluptas** siehe **Edite, bibite!**\n\n**Post nubila Phoebus.**\n\nNach den Wolken (kommt) die Sonne.\n\nNach Alanus ab Insulis, _Doctrinale minus_ 1,33.\n\n**Potentissimus est, qui se habet in potestate.**\n\nAm st\u00e4rksten ist, wer sich selbst in der Gewalt hat.\n\nNach Seneca ( _Epistulae morales_ 90,34), der den Ausspruch Demokrit zuschreibt.\n\n**Potest taurum tollere, qui vitulum sustulerit.**\n\nEinen Stier kann hochheben, wer ein Kalb hochgehoben hat.\n\nNach Petron, _Satyrica_ 25,6, wo Quartilla behauptet, sich nicht daran erinnern zu k\u00f6nnen, jemals Jungfrau gewesen zu sein: Schon als junges M\u00e4dchen habe sie sich mit kleinen Jungen verlustiert, sp\u00e4ter dann gr\u00f6\u00dferen M\u00e4nnern zugewandt. Solch eine Handlungsweise sei die Grundlage f\u00fcr obige Wendung, die sie als altes Sprichwort bezeichnet.\n\n**Potior est, qui prior ad dandum est.**\n\nMehr vermag, wer fr\u00fcher da ist zum Bezahlen.\n\n(Vgl.: Wer zuerst kommt, mahlt zuerst.)\n\nNach Terenz, _Phormio_ 533.\n\n**potius amicum quam dictum perdere**\n\nlieber einen Freund verlieren, als einen Witz (unterdr\u00fccken)\n\nQuintilian ( _Institutio oratoria_ 6,3,28) bezeichnet die Wendung als gel\u00e4ufig. Vgl. Horaz, _Sermones_ 1,4,34 f.\n\n**Potius sero quam numquam.**\n\nLieber sp\u00e4t als nie.\n\nNach Livius, _Ab urbe condita_ 4,2,11.\n\n**Praeceptor Germaniae**\n\nLehrer Deutschlands\n\nEhrentitel Philipp Melanchthons.\n\n**praedicare de Romulo et Remo**\n\nvon Romulus und Remus erz\u00e4hlen\n\n(d. h.: bei Adam und Eva beginnen)\n\nNach Cicero, _De legibus_ 1,8.\n\n**Praemium incertum petis, \/ certum scelus.**\n\nDu strebst einen unsicheren Gewinn, (doch) eine sichere Schuld an.\n\nSeneca, _Phoenissae_ 632 f.\n\n**praesens absens**\n\nanwesend (und doch) abwesend\n\nTerenz, _Eunuchus_ 192. \u2013 Phaedria bittet die Kurtisane Thais, auch wenn sie physisch bei ihrem Soldaten sei, in ihrem Herzen doch stets bei ihm, Phaedria, zu weilen. Vgl. Terenz, _Adelphoe_ 73.\n\n**Praesenti credere fortunae non decet.**\n\nEs ziemt sich nicht, auf das gegenw\u00e4rtige Gl\u00fcck zu vertrauen!\n\nNach Livius ( _Ab urbe condita_ 45,8,6), der fortf\u00e4hrt: \u00bb... cum quid vesper ferat, incertum sit.\u00ab (\u00bb... da, was der Abend bringen mag, unsicher ist.\u00ab) Vgl. **Nescis quid vesper...**\n\n**Praestat amicitia propinquitati.**\n\nFreundschaft ist besser als Verwandtschaft.\n\n(Vgl.: Ein guter Freund ist mehr wert als hundert Verwandte.)\n\nCicero ( _De amicitia_ 19) erkl\u00e4rt, von Freundschaft k\u00f6nne nicht die Rede sein, sobald der gute Wille fehle, w\u00e4hrend Verwandtschaft auch ohne diesen bestehen bleibe.\n\n**Praeterita mutare non possumus.**\n\nVergangenes k\u00f6nnen wir nicht \u00e4ndern.\n\nCicero, _In L. Calpurnium Pisonem_ 59.\n\n**praeter propter**\n\nn\u00e4herungsweise\n\nGellius, _Noctes Atticae_ 19,10,4.\n\n**Praeter speciem stultus est.**\n\nEr ist d\u00fcmmer, als er aussieht.\n\nPlautus, _Mostellaria_ 965.\n\n**Praetervehitur aures.**\n\nEs geht an den Ohren vorbei.\n\n(D. h.: Es geht zum einen Ohr hinein und zum andern wieder hinaus.)\n\nNach Cicero, _Pro L. Cornelio Balbo_ 4.\n\n**premere argumentum**\n\nein Argument betonen\n\n(d. h.: auf einem Argument beharren)\n\nNach Cicero, _Tusculanae disputationes_ 1,88.\n\n**Pretio parata vincitur pretio fides.**\n\nTreue, die durch Bestechung zustande kam, wird durch Bestechung (wieder) aufgel\u00f6st.\n\nSeneca, _Agamemnon_ 287.\n\n**Pridie caveat, ne faciat, quod pigeat postridie!**\n\nAm Vortag m\u00f6ge sie sich h\u00fcten, etwas zu tun, dessen sie sich am n\u00e4chsten Tag sch\u00e4men k\u00f6nnte.\n\nPlautus, _Stichus_ 122.\n\n**Prima et maxima peccantium est poena peccasse.**\n\nDie erste und die gr\u00f6\u00dfte Strafe f\u00fcr einen S\u00fcnder ist, ges\u00fcndigt zu haben.\n\nNach Seneca, _Epistulae morales_ 97,14.\n\n**Primum vivere , deinde philosophari.**\n\nZuerst leben, dann philosophieren.\n\n(D. h.: Zuerst ein t\u00e4tiges Leben f\u00fchren, dann philosophieren.)\n\nDie Quelle des Zitats ist nicht nachgewiesen. Der Vorrang des praktischen Lebens \u00fcber die theoretische Philosophie wird jedoch bereits von Cicero betont ( _Ad Marcum filium_ Fragment 4 Watt; \u00fcberliefert von Laktanz, _Divinae institutiones_ 3,14,17).\n\n**Primus in orbe deos fecit timor.**\n\nFurcht schuf zuerst die G\u00f6tter auf Erden.\n\n(Vgl.: Wer sich f\u00fcrchtet, der lauf' in die Kirche.)\n\nPetron, Fragment 27,1 Ernout, und Statius, _Thebais_ 3,661.\n\n**Principibus placuisse viris non ultima laus est.**\n\nNon cuivis homini contingit adire Corinthum.\n\nDen besten M\u00e4nnern gefallen zu haben ist nicht das geringste Lob. Es gelingt nicht jedem Menschen, nach Korinth zu fahren.\n\nHoraz, _Epistulae_ 1,17,35 f. \u2013 Korinth gilt nach einem griechischen Sprichwort als St\u00e4tte des Luxus. Hier jedoch ist wohl gemeint, da\u00df nicht jeder vollkommene Tugend erlangen kann.\n\n**Principiis consentiant exitus.**\n\nDen Anf\u00e4ngen m\u00f6ge der Ausgang entsprechen.\n\n(Vgl.: Wie der Anfang, so das Ende.)\n\nCicero, _Ad familiares_ 11,5,3.\n\n**Principiis obsta!**\n\nWehre den Anf\u00e4ngen!\n\nOvid ( _Remedia amoris_ 91) f\u00e4hrt fort: \u00bbSero medicina paratur.\u00ab (\u00bbZu sp\u00e4t wird die Medizin zubereitet.\u00ab) Vgl. die Anspielung bei Seneca, _Epistulae morales_ 72,11.\n\n**Priusquam incipias, consulto et, ubi consulueris, mature facto opus est.**\n\nBevor man beginnt, bedarf es der \u00dcberlegung, und sobald man \u00fcberlegt hat, rechtzeitiger Ausf\u00fchrung.\n\nSallust, _Bellum Catilinarium_ 1,6.\n\n**Probavit non rem publicam suam esse, sed se rei publicae.**\n\n(Der Herrscher) hat bewiesen, da\u00df der Staat nicht ihm, sondern er dem Staat geh\u00f6re.\n\nSeneca ( _De clementia_ 1,19,7) fordert ein solches Verhalten von einem guten Herrscher.\n\n**Probitas laudatur et alget.**\n\nRedlichkeit wird gelobt und mu\u00df (doch) frieren.\n\n(Vgl.: Redlichkeit lobt jedermann, jedoch l\u00e4\u00dft man sie betteln gahn.)\n\nJuvenal, _Saturae_ 1,74.\n\n**Procellae quanto plus habent virium, tanto minus temporis.**\n\nJe mehr Kraft St\u00fcrme haben, desto weniger Zeit.\n\n(D. h.: Starke St\u00fcrme ziehen schnell vor\u00fcber.)\n\nSeneca, _Naturales quaestiones_ 7,9,3.\n\n**procul negotiis** siehe **Beatus ille, qui procul negotiis**\n\n**Procul omnis esto \/ clamor et ira.**\n\nFern sei (uns) alles Geschrei und Z\u00fcrnen!\n\nHoraz, _Carmina_ 3,8,15 f.\n\n**Procul, o procul este profani!**\n\nFern, o ferne stehet, Unheilige!\n\nVergil ( _Aeneis_ 6,258). \u2013 Die Sibylle warnt die Gef\u00e4hrten des Aeneas, nicht mit ihm in die Unterwelt hinabzusteigen.\n\n**prodenda , quia prodita**\n\nzu \u00dcberlieferndes, da \u00fcberliefert\n\n(D. h.: Man mu\u00df \u00fcberliefern, was \u00fcberliefert wurde.)\n\nNach Herodot, _Geschichte_ 7,152,3. Vgl. die andere \u00dcbersetzung: **Relata refero.**\n\n**Prodigus et stultus donat, quae spernit et odit.**\n\nEin Verschwender und Tor gibt weg, was er verachtet und ha\u00dft.\n\nHoraz, _Epistulae_ 1,7,20.\n\n**pro domo**\n\nf\u00fcr das eigene Haus\n\n(d. h.: im eigenen Interesse)\n\nCicero hielt im Jahre 57 v. Chr. nach seiner R\u00fcckkehr aus dem Exil die Rede _Pro domo sua ad pontifices_ (heute meist zitiert als _De domo..._ ), um die R\u00fcckgabe seiner Grundst\u00fccke zu erwirken, die Clodius eingenommen hatte. Heute fa\u00dft man die Wendung im Sinne eines Pl\u00e4doyers in eigener Sache auf.\n\n**Profecto deliramus interdum senes.**\n\nIn der Tat spinnen wir Alten bisweilen.\n\nPlautus, _Epidicus_ 392.\n\n**Profecto enim vita vigilia est.**\n\nIn der Tat n\u00e4mlich ist das Leben ein Wachdienst.\n\n(D. h.: Ein echtes Leben ist nur ein wachsames Leben.)\n\nPlinius d. \u00c4., _Naturalis historia_ , praefatio 18.\n\n**Pro lucri pallida tabes!**\n\nO bleiche Pest der Geldsucht!\n\nLukan, _De bello civili_ 4,96.\n\n**Prona est timoris semper in peius fides** siehe **Quid nimis miseri...**\n\n**Propone ante oculos deum!**\n\nF\u00fchre (dir) Gott vor Augen!\n\nCicero ( _De natura deorum_ 1,114) verwendet diese Worte in ganz anderem Sinn: Der epikureischen Auffassung, die G\u00f6tter k\u00fcmmerten sich nicht um die Menschen, sondern bef\u00e4nden sich in st\u00e4ndiger Kontemplation ihres eigenen Gl\u00fccks, setzt er entgegen: \u00bbConprehende igitur animo et propone ante oculos deum nihil aliud in omni aeternitate nisi \u203aMihi pulchre est\u2039 et \u203aEgo beatus sum\u2039 cogitantem.\u00ab (\u00bbMan stelle sich also im Geiste und bildlich einen Gott vor, der auf alle Ewigkeit nichts anderes denkt als \u203aMir ist wohl\u2039 und \u203aIch bin gl\u00fccklich\u2039.\u00ab)\n\n**Proque sua causa quisque disertus erat.**\n\nUnd f\u00fcr seine eigene Sache war ein jeder beredt.\n\nOvid ( _Fasti_ 4,112) spricht hier von Liebenden, die f\u00fcr ihre jeweilige Angebetete singen.\n\n**Prospera animos efferunt.**\n\nGl\u00fcckliche Umst\u00e4nde macht die Herzen \u00fcberm\u00fctig.\n\nSeneca, _Agamemnon_ 252.\n\n**Prospera omnes sibi vindicant, adversa uni imputantur.**\n\nErfolge nehmen alle f\u00fcr sich in Anspruch, Mi\u00dferfolge werden einem einzigen zugeschrieben.\n\nTacitus, _Agricola_ 27,1.\n\n**Pro superi! quantum mortalia pectora caecae \/ noctis habent!**\n\nO G\u00f6tter! Welch schwarze Nacht wohnt in den Herzen der Sterblichen!\n\nOvid, _Metamorphoses_ 6,472 f.\n\n**Proximus sum egomet mihi.**\n\nIch bin mir selbst der N\u00e4chste.\n\nTerenz, _Andria_ 636.\n\n**Prudens futuri temporis exitum**\n\nCaliginosa nocte premit deus.\n\nWeise h\u00fcllt der Gott den Ausgang (d. h.: die Ereignisse) der kommenden Zeit in d\u00fcstere Nacht ein.\n\nHoraz, _Carmina_ 3,29,29 f.\n\n**Prudens interrogatio quasi dimidium sapientiae.**\n\nEine kluge Frage ist gleichsam die H\u00e4lfte der Weisheit.\n\nFrancis Bacon zugeschrieben.\n\n**Pueri inter sese quam pro levibus noxiis iras gerunt!**\n\nAus was f\u00fcr nichtigen Anl\u00e4ssen Knaben einander z\u00fcrnen!\n\nTerenz, _Hecyra_ 310.\n\n**Pueri puerilia tractant** siehe **Sunt pueri pueri, pueri...**\n\n**Pugna nocet, citharae noxque Venusque iuvant.**\n\nKampf schadet, Zitherspiel, Nacht und Liebe erfreu'n.\n\nOvid, _Heroides_ 3,116.\n\n**Pugna suum finem, cum iacet hostis, habet.**\n\nDer Kampf hat sein Ende, wenn der Feind darniederliegt.\n\nOvid ( _Tristia_ 3,5,34) erl\u00e4utert, da\u00df Milde nur bei den Gro\u00dfen zu finden sei, so wie der edle L\u00f6we den Kampf schnell zu Ende f\u00fchre, w\u00e4hrend der Wolf und der B\u00e4r den Tod lange hinausz\u00f6gerten.\n\n**pugnos edere**\n\nF\u00e4uste essen\n\n(d. h.: Pr\u00fcgel bekommen)\n\nNach Plautus, _Amphitruo_ 309.\n\n**Pulchrum eminere est inter illustres viros.**\n\nEs ist sch\u00f6n, unter angesehenen M\u00e4nnern hervorzustechen.\n\nSeneca, _Octavia_ 472.\n\n**Pulchrum est vitam donare minori.**\n\nEs ist r\u00fchmlich, einem Geringeren das Leben zu schenken.\n\nStatius, _Thebais_ 6,816.\n\n**Pulsate, et aperietur vobis.**\n\nKlopft, und es wird euch ge\u00f6ffnet.\n\nMatth\u00e4us 7,7; Lukas 11,9.\n\n**Pulvis es, et in pulverem reverteris.**\n\nStaub bist du und zum Staub wirst du zur\u00fcckkehren.\n\n1. Mose 3,19.\n\n**Pulvis et umbra sumus.**\n\nStaub und Schatten sind wir.\n\nHoraz ( _Carmina_ 4,7,16) erkl\u00e4rt, da\u00df alles von Natur aus verg\u00e4nglich sei.\n\n**punctum saliens**\n\nder springende Punkt\n\nNach Aristoteles, _Historia animalium_ 6,3,561a 9 ff. Die lateinische Pr\u00e4gung findet sich mit Bezug auf Aristoteles bei Ulisse (Ulyssis) Aldrovandi ( _Ornithologia_ 14,1, Unterkapitel \u00bbSalacitas. Coitus. Partus.\u00ab). Die Wendung bezeichnete urspr\u00fcnglich den in einem Ei wahrnehmbaren zuckenden Punkt, das Herz. Heute meint sie den in einer Frage entscheidenden Punkt.\n\n**Punica fides**\n\npunische Treue\n\nNach Sallust, _Bellum Iugurthinum_ 108,3. \u2013 Seit den Kriegen mit Hannibal sagte man den Puniern Treulosigkeit nach.\n\n**Pupillus est avarus aetatis brevis.**\n\nEin M\u00fcndel ist ein Geizhals f\u00fcr eine kurze Zeitspanne.\n\nCaecilius Balbus, _Sententiae_ 153. \u2013 M\u00fcndel durften nicht selbst\u00e4ndig Geschenke machen und waren somit zwangsl\u00e4ufig geizig.\n\n**Putent se in alium orbem terrarum delatos.**\n\nSie k\u00f6nnten glauben, sie w\u00e4ren in eine andere Welt verschlagen.\n\nPetron, _Satyrica_ 1,2.\nQ\n\n**Quadripedante putrem sonitu quatit ungula campum.**\n\nDer Huf l\u00e4\u00dft mit vierf\u00fc\u00dfigem Donnern das staubige Feld erbeben.\n\nVergil, _Aeneis_ 8,596, in Anlehnung an Homer und Ennius. \u2013 Aeneas reitet mit seinen Mannen in den Kampf gegen Turnus.\n\n**Quaecumque ex merito spes venit, aequa venit.**\n\nHoffnung, welcher Art auch immer, besteht zu Recht, wenn sie auf Verdienst fu\u00dft.\n\nOvid, _Heroides_ 2,62.\n\n**quaedam quaestio subdifficilis, num quando amici novi... veteribus sint anteponendi**\n\n(Es besteht) eine gewisse wenig schwierige Frage, ob neue Freunde stets den alten vorzuziehen seien\n\nCicero ( _De amicitia_ 67) zu einem grotesken Vergleich zwischen Pferden und Freunden: Bei Reittieren seien stets die jungen und frischen vorzuziehen. Wahre Freundschaften hingegen bed\u00fcrften einiger Zeit: \u00bbVerum illud est, quod dicitur, multos modios salis simul edendos esse, ut amicitiae munus expletum sit.\u00ab (\u00bbJenes Sprichwort ist wahr, da\u00df man viele Scheffel Salz zusammen essen mu\u00df, um die Bedingungen einer Freundschaft zu erf\u00fcllen.\u00ab)\n\n**Quae dant, quaeque negant, gaudent tamen esse rogatae.**\n\nOb sie sich hingeben oder sich verweigern, sie freuen sich dennoch, gefragt worden zu sein.\n\nOvid ( _Ars amatoria_ 1,345) erkl\u00e4rt, fast alle Frauen seien jederzeit zu Willen, und selbst wenn sie sich verweigerten, schmeichelten ihnen Ann\u00e4herungsversuche.\n\n**Quae enim domus tam stabilis, quae tam firma civitas est, quae non odiis et discidiis funditus possit everti?**\n\nWelches Haus n\u00e4mlich ist so stabil, welcher Staat so gefestigt, da\u00df er nicht g\u00e4nzlich zerst\u00f6rt werden k\u00f6nnte durch Ha\u00df und Zerr\u00fcttung?\n\nCicero ( _De amicitia_ 23) z\u00e4hlt die Vorteile wahrer Freundschaft auf und bietet zur Abschreckung obiges Beispiel von den Auswirkungen von Ha\u00df und Zerr\u00fcttung.\n\n**Quae fecisse iuvat, facta referre pudet.**\n\nMan sch\u00e4mt sich (oft) der Dinge, die angenehm waren, als man sie tat, wenn man (sp\u00e4ter) von ihnen erz\u00e4hlt.\n\nOvid, _Heroides_ 19,64.\n\n**Quae fuerant vitia, mores sunt.**\n\nWas (einst) Fehler waren, sind (jetzt) Sitten.\n\nSeneca, _Epistulae morales_ 39,6.\n\n**Quae fugiunt, celeri carpite poma manu!**\n\nFr\u00fcchte, die dahinschwinden, pfl\u00fccket mit rascher Hand!\n\nOvid, _Ars amatoria_ 3,576.\n\n**Quae fuit durum pati, \/ meminisse dulce est.**\n\nEs ist angenehm, sich an das zu erinnern, was hart zu ertragen war.\n\nSeneca, _Hercules furens_ 656 f.\n\n**Quaelibet eloquio fit bona causa tuo.**\n\nDurch deine Beredsamkeit wird jedes Anliegen gut.\n\nOvid, _Tristia_ 1,9,46.\n\n**Quaelibet officio causa sit apta tuo.**\n\nJeglicher Anla\u00df sei dir f\u00fcr eine Aufmerksamkeit recht.\n\nOvid ( _Ars amatoria_ 1,152) r\u00e4t dem Liebenden, keine Gelegenheit f\u00fcr Ann\u00e4herungsversuche auszulassen. So soll er etwa selbst dann Staub von dem Gewand der Angebeteten wischen, wenn dort keiner ist.\n\n**Quae medicamenta non sanant, ferrum sanat;**\n\nQuae ferrum non sanat, ignis sanat;\n\nQuae vero ignis non sanat, insanabilia reputari oportet.\n\nWas Medikamente nicht heilen, heilt das Schwert; was das Schwert nicht heilt, heilt das Feuer; was aber das Feuer nicht heilt, mu\u00df als unheilbar angesehen werden.\n\nNach Hippokrates, _Aphorismen_ 7,87. \u2013 Die lateinische Version der ersten beiden Zeilen findet sich als Motto zu Friedrich Schillers _R\u00e4ubern_.\n\n**Quae modo pugnarunt, iungunt sua rostra columbae.**\n\nTauben, die gerade erst miteinander fochten, vereinigen (nun) ihre Schn\u00e4bel.\n\nOvid, _Ars amatoria_ 2,465.\n\n**Quae nocent, docent.**\n\nWas schadet, belehrt.\n\n(Vgl.: Aus Schaden wird man klug.)\n\nMartin Luther zugeschrieben; vgl. Heuseler, _Luthers Sprichw\u00f6rter_ 292.\n\n**Quae res bene vortat!**\n\nM\u00f6ge sich die Sache zum Guten wenden!\n\nPlautus, _Trinummus_ 500.\n\n**Quae venit ex tuto, minus est accepta voluptas.**\n\nLust, die aus der Sicherheit daherkommt (d. h.: die uns ohne Gefahr in den Scho\u00df f\u00e4llt), ist weniger willkommen.\n\n(Vgl.: Je h\u00f6her die Trauben h\u00e4ngen, desto s\u00fc\u00dfer sind sie.)\n\nOvid, _Ars amatoria_ 3,603.\n\n**Qualis artifex pereo!**\n\nWelch ein K\u00fcnstler geht an mir verloren!\n\nSueton ( _Vita Neronis_ 49,1) zufolge sprach Nero diese Worte kurz vor seinem Selbstmord.\n\n**Qualis rex, talis grex** siehe **Plane qualis dominus...**\n\n**Qualis sit animus, ipse animus nesciet.**\n\nWie die Seele beschaffen ist, wird die Seele selbst nicht wissen.\n\nCicero, _Tusculanae disputationes_ 1,53.\n\n**Qualis vir , talis oratio.**\n\nWie der Mann, so die Rede.\n\nNach Pseudo-Seneca, _De moribus_ 73. Vgl. **Talis hominibus...**\n\n**Quamlibet infirmas adiuvat ira manus.**\n\nZorn hilft noch so schwachen H\u00e4nden (d. h.: er macht sie st\u00e4rker).\n\nOvid, _Amores_ 1,7,66.\n\n**Quam miser est, qui fert asperiora fide!**\n\nWie elend ist, wer H\u00e4rteres erleidet, als man glaubt.\n\nOvid, _Epistulae ex Ponto_ 4,10,36.\n\n**Quam non est facilis virtus! Quam vero difficilis eius diuturna simulatio!**\n\nWie ist doch die Tugend nicht leicht! Wie schwer aber ist (erst) ihr st\u00e4ndiges Vorheucheln!\n\nCicero, _Ad Atticum_ 7,1,6.\n\n**Quamquam ridentem dicere verum, \/ quid vetat?**\n\nDoch was verbietet, l\u00e4chelnd die Wahrheit zu sagen?\n\nHoraz, _Sermones_ 1,1,24 f.\n\n**Quam quisque norit artem, in hac se exerceat.**\n\nJeder \u00fcbe sich in der Kunst, die er erlernt hat!\n\n(Vgl.: Schuster, bleib bei deinem Leisten!)\n\nCicero ( _Tusculanae disputationes_ 1,41) bezeichnet die Wendung als griechisches Sprichwort. Vgl. **Ne supra...**\n\n**Quam veterrimus, tam homini optimus est amicus.**\n\nJe \u00e4lter ein Freund ist, um so besser ist er f\u00fcr den Menschen.\n\nPlautus, _Truculentus_ 173.\n\n**Quamvis sint sub aqua, sub aqua maledicere temptant.**\n\nAuch wenn sie unter Wasser sind, so versuchen sie, unter Wasser zu schimpfen.\n\nOvid, _Metamorphoses_ 6,376. \u2013 Die lykischen Bauern hatten Latona (Leto), die mit ihren Zwillingen von Iuno (Hera) vertrieben worden war, das Wasser verweigert. Latona verwandelte sie daraufhin in Fr\u00f6sche, doch selbst in ihrer neuen Gestalt h\u00f6rten sie nicht auf, sie zu schm\u00e4hen.\n\n**Quandoque bonus dormitat Homerus.**\n\nMitunter schl\u00e4ft (auch) der gute Homer.\n\nHoraz, _Ars poetica_ 359; vgl. Quintilian, _Institutio oratoria_ 10,1,24, und **Neque semper arcum tendit Apollo**.\n\n**Quando tumet venter, produntur facta latenter.**\n\nWenn der Bauch anschwillt, werden heimliche Taten ans Licht gef\u00fchrt.\n\n(Vgl.: Wenn der Bauch schwillt, sieht man, wo es gilt.)\n\nHW: 23562.\n\n**Quanti est sapere!**\n\nWieviel ist's doch wert, weise zu sein!\n\nTerenz, _Eunuchus_ 791.\n\n**Quantum animis erroris inest!**\n\nWieviel Irrtum ist in den Herzen!\n\nOvid, _Fasti_ 2,789.\n\n**Quantum est adhibere hominem amicum, ubi quid geras!**\n\nWie gut ist es, einen Freund heranzuziehen, wenn man etwas unternimmt!\n\nPlautus, _Persa_ 595.\n\n**Quantum mutatus ab illo!**\n\nWie sehr hat er sich im Vergleich zu jenem (der er fr\u00fcher war) ver\u00e4ndert!\n\nVergil, _Aeneis_ 2,274. \u2013 Der Geist des gefallenen Hektor erscheint Aeneas und mahnt ihn, aus Troja zu fliehen. Hektor zeigt sich dabei nicht wie sonst in der Pose eines strahlenden Siegers, sondern tr\u00e4nen\u00fcberstr\u00f6mt.\n\n**Quantum oculis, animo tam procul ibit amor.**\n\nWie fern die Liebe aus den Augen (schwindet), so fern wird sie aus dem Herzen schwinden.\n\n(Vgl.: Aus den Augen, aus dem Sinn.)\n\nProperz, _Elegiae_ 3,21,10.\n\n**Quantum quisque ferat, respiciendus erit.**\n\nEin jeder wird danach zu ber\u00fccksichtigen sein, wieviel er bringt.\n\nOvid, _Amores_ 1,8,38. \u2013 Dipsas r\u00e4t einem jungen M\u00e4dchen dazu, sich mehrere Liebhaber zu nehmen und diese nach ihrem Verm\u00f6gen zu behandeln.\n\n**Quare alium pro me, qui queat ista pati!**\n\nDaher (suche) einen anderen statt meiner, der diese Dinge ertragen kann!\n\nOvid, _Amores_ 3,11,28.\n\n**Quare verbis parcam? Gratuita sunt.**\n\nWarum soll ich an Worten sparen? Sie sind (doch) umsonst.\n\nSeneca ( _Epistulae morales_ 29,2) warnt vor einer solchen Einstellung. Vielmehr solle man seine philosophischen Erkenntnisse gezielt an willige Zuh\u00f6rer richten.\n\n**quasi aranearum fila**\n\ngleichsam wie Spinnweben\n\nHieronymus ( _Epistulae_ 125,6) \u00fcber liederliche Frauen, die ihre Laster nicht mehr vertuschen, sondern den Schleier, der ihre Taten verbergen k\u00f6nnte, \u203agleichsam wie Spinnweben\u2039 zerrei\u00dfen.\n\n**Quasi aurum igni, sic benevolentia fidelis periculo aliquo perspici potest.**\n\nGleichsam wie Gold im Feuer, so wird treues Wohlwollen in Gefahren erkennbar.\n\nNach Cicero, _Ad familiares_ 9,16,2.\n\n**Quasi piscis, itidem est amator lenae: nequam est nisi recens.**\n\nF\u00fcr die Kupplerin ist der Liebhaber wie ein Fisch: Er ist nichts wert, wenn er nicht frisch ist.\n\nPlautus, _Asinaria_ 178.\n\n**quasi umbra sequi**\n\nwie ein Schatten folgen\n\nNach Plautus, _Casina_ 92.\n\n**Qua vicit , victos protegit ille manu.**\n\nMit der Hand, mit der er siegte, sch\u00fctzt jener die Besiegten.\n\nOvid ( _Amores_ 1,2,52) bittet den Liebesgott Cupido, der ihn doch schon besiegt habe, ihn nun nicht weiter mit Liebespein zu qu\u00e4len: Auch Caesar sei schlie\u00dflich Besiegten gegen\u00fcber gn\u00e4dig.\n\n**Quemcumque dederit exitum casus, feram!**\n\nWas auch immer f\u00fcr ein Ende mir das Schicksal bestimmt hat, ich werde es ertragen!\n\nSeneca, _Phaedra_ 138.\n\n**Quemcumque miserum videris, hominem scias.**\n\nWen auch immer du elend sehen wirst: Wisse, da\u00df er ein Mensch ist.\n\nSeneca, _Hercules furens_ 463. \u2013 Auf diese Behauptung des Lycus entgegnet Amphitruo: \u00bbQuemcumque fortem videris, miserum neges.\u00ab (\u00bbWen auch immer du tapfer sehen wirst: Nenne ihn nicht einen Elenden.\u00ab)\n\n**Quem di diligunt, \/ adulescens moritur.**\n\nJung stirbt, wen die G\u00f6tter lieben.\n\n(Vgl.: Wen Gott liebt, den holt er zu sich.)\n\nPlautus ( _Bacchides_ 816 f.) zitiert hier Menander ( _Der doppelte Betr\u00fcger_ Fragment 111 K\u00f6rte-Thierfelder).\n\n**Quem metuunt, oderunt: quem quisque odit, perisse expetit.**\n\nWen sie f\u00fcrchten, den hassen sie. Wen man ha\u00dft, dessen Tod w\u00fcnscht man sich.\n\nCicero ( _De officiis_ 2,23) zitiert hier Ennius (Fragment Ex tragoediis inc. 410 Warmington). Vgl. Ovid, _Amores_ 2,2,10.\n\n**Quem paenitet peccasse, paene est innocens.**\n\nWen es reut, gefehlt zu haben, der ist fast unschuldig.\n\nSeneca, _Agamemnon_ 243. Vgl. **Geminat peccatum, quem...** und **Si paenitet, haud nocet error**.\n\n**Quem referent Musae, vivet.**\n\nVon wem die Musen berichten werden, der wird leben.\n\nTibull, _Elegiae_ 1,4,65. Vgl. **Dignum laude...**\n\n**Quem saepe transit casus, aliquando invenit.**\n\nWen das Ungl\u00fcck oft \u00fcbergeht, den findet es eines Tages (doch).\n\nSeneca, _Hercules furens_ 328.\n\n**Quem sua culpa premit, deceptus omitte tueri!**\n\nWenn jemanden (wirklich) Schuld belastet, so sch\u00fctze du, der du get\u00e4uscht worden bist, ihn nicht.\n\nHoraz ( _Epistulae_ 1,18,79) gemahnt, jemanden, der sich als schuldig erwiesen hat, nicht l\u00e4nger zu unterst\u00fctzen, auch wenn man ihn vorher f\u00e4lschlicherweise f\u00fcr unschuldig hielt.\n\n**Qui alteri exitium parat, \/ eum scire oportet sibi paratam pestem.**\n\nWer einem anderen den Untergang bereitet, der mu\u00df wissen, da\u00df auch f\u00fcr ihn Verderben bereitsteht.\n\n(Vgl.: Wer anderen eine Grube gr\u00e4bt, f\u00e4llt selbst hinein.)\n\nCicero ( _Tusculanae disputationes_ 2,39) zitiert Ennius, Fragment Ex trag. inc. _Medea_ 271 Warmington.\n\n**Qui amant, ipsi sibi somnia fingunt.**\n\nLiebende denken sich ihre Tr\u00e4ume selbst aus.\n\nVergil, _Bucolica_ 8,108.\n\n**Qui amat periculum, in illo peribit.**\n\nWer die Gefahr liebt, wird darin umkommen.\n\nSirach 3,27.\n\n**quia me vestigia terrent**\n\nweil mich die Fu\u00dfstapfen abschrecken\n\nHoraz ( _Epistulae_ 1,1,74, nach Aesop, _Fabulae_ 147 Hausrath) begr\u00fcndet seine Weigerung, sich den Ansichten des gemeinen Volkes anzuschlie\u00dfen, wie folgt: Die obigen Worte habe einst ein Fuchs auf die Frage geantwortet, warum er einen kranken L\u00f6wen nicht in dessen H\u00f6hle besuche: Die Spuren schreckten ihn ab, da sie alle in die H\u00f6hle f\u00fchrten, keine jedoch hinaus; vgl. (dt.) \u00bbsich in die H\u00f6hle des L\u00f6wen wagen\u00ab.\n\n**Quia qui alterum incusat probri, sumpse enitere oportet.**\n\nDenn wer einen anderen ob einer \u00dcbeltat anklagt, mu\u00df selbst (vor Tugend) strahlen.\n\nPlautus, _Truculentus_ 160.\n\n**Qui asinum non potest, stratum caedit.**\n\nWer den Esel nicht pr\u00fcgeln kann, pr\u00fcgelt den Sattel.\n\n(Vgl.: Man schl\u00e4gt den Sack und meint den Esel.)\n\nPetron, _Satyrica_ 45,8.\n\n**Qui autem non proficit, deficit.**\n\nWer jedoch nicht vorangeht, f\u00e4llt zur\u00fcck.\n\n(Vgl.: Wer rastet, der rostet.)\n\nPapst Leo I. (der Gro\u00dfe), _Sermones_ 59,8.\n\n**Qui bene amat, bene castigat.**\n\nWer richtig liebt, der z\u00fcchtigt richtig.\n\n(Vgl.: Je lieber das Kind, desto sch\u00e4rfer die Rute.)\n\nVgl. z. B. Sirach 30,1 und Hebr\u00e4erbrief 12,6.\n\n**Qui bene imperat, paruerit aliquando necesse est.**\n\nWer gut herrscht, mu\u00df einst gehorcht haben.\n\nCicero ( _De legibus_ 3,5) f\u00e4hrt fort: \u00bb... et qui modeste paret, videtur, qui aliquando imperet, dignus esse.\u00ab (\u00bb... und wer in Bescheidenheit gehorcht, erscheint w\u00fcrdig, selbst einmal zu befehlen.\u00ab)\n\n**Qui confitetur, proxime innocentiam est.**\n\nWer gesteht, ist der Redlichkeit sehr nahe.\n\nPseudo-Seneca, _De moribus_ 94.\n\n**Quicquid agis, prudenter agas, et respice finem!**\n\nWas auch immer du tust, handle klug und bedenke das Ende!\n\n_Gesta Romanorum_ 103.\n\n**Quicquid discis , tibi discis.**\n\nWas auch immer du lernst, du lernst es f\u00fcr dich.\n\nPetron, _Satyrica_ 46.\n\n**Quicquid init ortus, finem timet: ibimus omnes.**\n\nAlles, was einen Anfang genommen hat, f\u00fcrchtet das Ende: Wir werden alle untergehen.\n\nStatius, _Silvae_ 2,1,218.\n\n**Quicquid loquitur, sal merum est.**\n\nWas auch immer er sagt, ist reines Salz (d. h.: es hat feinen Witz).\n\nAfranius, _Compitalia_ Fragment 3 Ribbeck.\n\n**Quicumque turpi fraude semel innotuit,**\n\n**Etiam si verum dicit, amittit fidem.**\n\nWer einmal durch einen sch\u00e4ndlichen Betrug bekannt geworden ist, verliert die Glaubhaftigkeit, auch wenn er die Wahrheit spricht.\n\n(Vgl.: Wer einmal l\u00fcgt, dem glaubt man nicht, und wenn er auch die Wahrheit spricht.)\n\nPhaedrus, _Fabulae_ 1,10,1 f. Vgl. **Mendaci homini...**\n\n**Quid crastina volveret aetas, \/ scire nefas homini.**\n\nWas die k\u00fcnftige Zeit bringen w\u00fcrde, war dem Menschen zu wissen verboten.\n\nStatius, _Thebais_ 3,562 f.\n\n**Quid deceat, non videt ullus amans.**\n\nKein Liebender achtet darauf, was sich ziemt.\n\nOvid, _Heroides_ 4,154.\n\n**Quid dem, quid non dem? Renuis tu, quod iubet alter.**\n\nWas soll ich geben, was nicht? Du weist zur\u00fcck, was der andere befiehlt.\n\nHoraz ( _Epistulae_ 2,2,63) weigert sich zu versuchen, mit seiner Dichtung allen gerecht zu werden.\n\n**Quid de quoque viro et cui dicas, saepe videto!**\n\n\u00dcberlege oft, was du \u00fcber jemanden sagst und wem du es sagst.\n\nHoraz, _Epistulae_ 1,18,68.\n\n**Qui dedit beneficium, taceat. Narret, qui accepit.**\n\nWer (jemandem) eine Wohltat erwiesen hat, schweige. Es erz\u00e4hle der davon, dem sie erwiesen wurde.\n\nSeneca, _De beneficiis_ 2,11,2.\n\n**Qui desiderat pacem, praeparet bellum.**\n\nWer sich nach Frieden sehnt, der bereite Krieg.\n\n(Vgl.: Wer den Frieden haben will, mu\u00df zum Kriege r\u00fcsten.)\n\nVegetius, _Epitoma rei militaris_ 3, praefatio. Vgl. Cicero, _Orationes Philippicae_ 7,19; Publilius Syrus, _Sententiae_ P 16; Livius, _Ab urbe condita_ 6,18,7. Oft auch zitiert als \u00bbSi vis pacem, para bellum!\u00ab (\u00bbWenn du Frieden willst, bereite den Krieg!\u00ab) Vgl. auch **Paritur pax bello.**\n\n**Quid est autem turpius quam senex vivere incipiens?**\n\nWas ist sch\u00e4ndlicher als ein Greis, der zu leben beginnt?\n\nSeneca ( _Epistulae morales_ 13,17) gemahnt, nicht alle Hoffnungen auf die Zukunft zu richten.\n\n**Quid est enim libertas? Potestas vivendi, ut velis.**\n\nWas ist denn Freiheit? Die M\u00f6glichkeit so zu leben, wie du willst.\n\nCicero, _Paradoxa Stoicorum_ 34.\n\n**Quid est pietas nisi voluntas grata in parentes?**\n\nWas ist kindliche Liebe anderes als dankbare Gesinnung gegen\u00fcber den Eltern?\n\nCicero, _Pro Cn. Plancio_ 80.\n\n**Quid est sapientia? Semper idem velle atque idem nolle.**\n\nWas ist Weisheit? Immer dasselbe zu wollen und dasselbe abzulehnen.\n\nSeneca, _Epistulae morales_ 20,5, vgl. 109,16.\n\n**Quid est suavius quam bene rem gerere \/ bono publico?**\n\nWas ist angenehmer, als etwas gut und zum allgemeinen Wohl auszuf\u00fchren?\n\nPlautus, _Captivi_ 498 f.\n\n**Quid est veritas ?**\n\nWas ist Wahrheit?\n\nLaut Johannes (18,38) richtete Pontius Pilatus diese Frage an Jesus, der verk\u00fcndet hatte, er sei auf die Welt gekommen, um der Wahrheit ein Zeugnis zu leisten.\n\n**Quid faciant pauci contra tot milia fortes?**\n\nWas sollen wenige Tapfere gegen so viele Tausende (Gegner) verm\u00f6gen?\n\nOvid ( _Fasti_ 2,229) \u00fcber die Familie der Fabii, die im Jahre 479 v. Chr. angeboten hatten, allein den Krieg gegen Veii zu f\u00fchren, und 477 geschlagen wurden.\n\n**Quid faciemus nos?**\n\nWas werden wir tun?\n\nFriedrich Schiller, _Wallenstein_ , Wallensteins Lager, V. 549.\n\n**Quid facies, facies Veneris si veneris ante?**\n\n**Ne sedeas? sed eas, ne pereas per eas.**\n\nWas wirst du tun, wenn du vor das Antlitz der Venus trittst? Wirst du dich etwa setzen? Nein, du sollst weggehen, damit du dadurch nicht vergehst.\n\nWortspiel im Vorwort der _Bi\u00e9vriana ou Jeux de mots_ des M. de Bi\u00e8vre.\n\n**Quid harenae semina mandas?**\n\nWas streust du Samen in den Sand?\n\n(d. h.: etwas Unsinniges tun)\n\nOvid, _Heroides_ 5,115.\n\n**Quid hic statis tota die otiosi?**\n\nWas steht ihr hier den ganzen Tag m\u00fc\u00dfig herum?\n\nMatth\u00e4us 20,6.\n\n**Quid id ad me?**\n\nWas geht mich das an?\n\nPlautus, _Menaechmi_ 722.\n\n**qui digito scalpunt uno caput**\n\ndie sich mit (nur) einem Finger am Haupte kratzen\n\nJuvenal ( _Saturae_ 9,133) beschreibt so Gecken, die um keinen Preis ihre modische Frisur in Unordnung bringen wollen.\n\n**Quid iuvat extinctos ferrum demittere in artus?**\n\nWas n\u00fctzt es, das Schwert in leblose Glieder zu sto\u00dfen?\n\nOvid, _Epistulae ex Ponto_ 4,16,51.\n\n**Quid leges sine moribus?**\n\nWas (sollen) Gesetze ohne Moral?\n\nHoraz, _Carmina_ 3,24,35.\n\n**Quid me alta silentia cogis \/ rumpere?**\n\nWas zwingst du mich, das tiefe Schweigen zu brechen?\n\nVergil, _Aeneis_ 10,63 f.\n\n**Quid moror exemplis, quorum me turba fatigat?**\n\nWas halte ich mich mit Beispielen auf, deren Menge mich erm\u00fcdet?\n\n(D. h.: Es gibt so viele Beispiele hierf\u00fcr. Warum sollte ich sie alle anf\u00fchren?)\n\nOvid, _Remedia amoris_ 461.\n\n**Quid nimis miseri volunt, \/ hoc facile credunt.**\n\nWas Ungl\u00fcckliche allzu sehr w\u00fcnschen, das glauben sie leicht.\n\nSeneca ( _Hercules furens_ 313 ff.) f\u00e4hrt fort (316): \u00bbProna est timoris semper in peius fides.\u00ab (\u00bbDer Glaube der Angst ist immer Schlimmerem zugeneigt\u00ab; d. h.: man f\u00fcrchtet meist das Schlimmste.) Vgl. Caesar, _De bello Gallico_ 3,18,6.\n\n**Quid non Amor improbus audet!**\n\nWas wagt der freche Amor nicht!\n\nOvid, _Fasti_ 2,331.\n\n**Quid non cogit amor?**\n\nWozu zwingt nicht die Liebe (die Menschen)?\n\nMartial, _Epigrammata_ 5,48,1.\n\n**Quid non ebrietas dissignat? Operta recludit.**\n\nWas zeigt Trunkenheit nicht an? Sie deckt Verborgenes auf.\n\nHoraz, _Epistulae_ 1,5,16.\n\n**Quid non, longa dies, quid non consumitis, anni?**\n\nWas, langer Tag, was, Jahre, verzehrt ihr nicht?\n\nMartial, _Epigrammata_ 9,49,9.\n\n**Quid non mortalia pectora cogis, \/ auri sacra fames!**\n\nWozu zwingst du nicht die Herzen der Sterblichen, sch\u00e4ndliche Goldgier!\n\n(Vgl.: Goldhunger, wen machst du nicht zum Unger!)\n\nVergil, _Aeneis_ 3,56 f. Vgl. **Improbe Amor...**\n\n**Quid non sentit amor?**\n\nWas merkt nicht die Liebe?\n\nOvid, _Metamorphoses_ 4,68.\n\n**Quid novi ex Africa?**\n\nWas (gibt es) Neues aus Afrika?\n\nNach einem griechischen Sprichwort (vgl. Aristoteles, _De generatione animalium_ 2,7,746b 7 f., und Plinius d. \u00c4., _Naturalis historia_ 8,42), das von einer unendlichen Artenvielfalt in Libyen spricht. Aristoteles bietet als Erkl\u00e4rung f\u00fcr dieses Ph\u00e4nomen, alle Tiere k\u00e4men an den Wasserl\u00f6chern zusammen und pflanzten sich dort beliebig miteinander fort.\n\n**Quid nunc stupes tamquam hircus in ervilia?**\n\nWas stierst du jetzt wie ein Ziegenbock in einem Feld von Kichererbsen?\n\n(vgl.: jemanden anstarren wie die Kuh den Kaiser)\n\nPetron, _Satyrica_ 57,11.\n\n**Quid opus est verbis?**\n\nWas bedarf es der Worte?\n\nTerenz, _Andria_ 165.\n\n**Quid opus nota noscere?**\n\nWas ist es n\u00f6tig, Bekanntes zu erfahren?\n\nPlautus, _Miles gloriosus_ 636.\n\n**Quid peius muliere aut audacius?**\n\nWas gibt es Schlechteres oder Verwegeneres als eine Frau?\n\nPlautus, _Miles gloriosus_ 307.\n\n**Quidquid amor iussit, non est contemnere tutum.**\n\nZu mi\u00dfachten, was auch immer die Liebe befahl, ist kein sicheres Unterfangen.\n\nOvid, _Heroides_ 4,11.\n\n**Quidquid delirant reges, plectuntur Achivi.**\n\nWas immer die K\u00f6nige T\u00f6richtes begehen: Die Achiver werden geschlagen (d. h.: sie m\u00fcssen es ausbaden).\n\nHoraz, _Epistulae_ 1,2,14. \u2013 Infolge des Streits zwischen dem Peliden Achilles und dem Atriden Agamemnon kam gro\u00dfes Unheil \u00fcber das Heer der Achiver (Griechen).\n\n**Quidquid erit, superanda omnis fortuna ferendo est.**\n\nWas auch immer es sei, man mu\u00df jeden Schicksalsschlag \u00fcberwinden, indem man ihn ertr\u00e4gt.\n\nVergil, _Aeneis_ 5,710.\n\n**Quidquid excessit modum, \/ pendet instabili loco.**\n\nWas auch immer das Ma\u00df \u00fcberschreitet, h\u00e4ngt an einem unsicheren Ort.\n\nSeneca, _Oedipus_ 909 f.\n\n**Quidquid id est, timeo Danaos et dona ferentis.**\n\nWas es auch sei, ich f\u00fcrchte die Danaer, und wenn sie Geschenke bringen.\n\nVergil, _Aeneis_ 2,49. \u2013 Laokoon warnt die Trojaner vor dem h\u00f6lzernen Pferd. Vgl. **Danaum fatale munus** , **donum exitiale Minervae** und **Latet error...**\n\n**Quidquid in altum \/ fortuna tulit, ruitura levat.**\n\nWas auch immer das Schicksal in die H\u00f6he trug, erhebt es, um es wieder zu zerst\u00f6ren.\n\nSeneca, _Agamemnon_ 100 f.\n\n**quidquid in buccam venerit**\n\nwas auch immer (einem) in den Mund kam\n\nNach Cicero, _Ad Atticum_ 1,12,4.\n\n**Quidquid praeter spem eveniat, omne id deputare esse in lucro.**\n\nWas entgegen der Hoffnung geschehen mag, das sei als Gewinn anzurechnen.\n\nTerenz, _Phormio_ 246, zitiert bei Cicero, _Tusculanae disputationes_ 3,30.\n\n**Quidquid sub terra est, in apricum proferet aetas.**\n\nWas auch immer unter der Erde ist, wird die Zeit ans Licht bringen.\n\nHoraz, _Epistulae_ 1,6,24.\n\n**Quidquid venerit obviam, loquitur.**\n\nWas ihm gerade \u00fcber den Weg lief, davon redet er.\n\nNach Martial, _Epigrammata_ 11,6,7.\n\n**Quid Romae faciam? Mentiri nescio.**\n\nWas soll ich in Rom tun? Ich wei\u00df nicht zu l\u00fcgen.\n\nJuvenal ( _Saturae_ 3,41) l\u00e4\u00dft einen Freund den Sittenverfall in Rom beklagen.\n\n**Quid, si nunc caelum ruat?**\n\nWas, wenn der Himmel nun einst\u00fcrzte?\n\nLaut Terenz ( _Heauton timorumenos_ 719) waren diese Worte bereits in der Antike eine stehende Wendung, mit der man Pessimisten verspottete.\n\n**Quid sit futurum cras, fuge quaerere!**\n\nH\u00fcte dich, danach zu fragen, was morgen geschehen wird!\n\nHoraz, _Carmina_ 1,9,13.\n\n**Quid tibi vis?**\n\nWas willst du f\u00fcr dich?\n\n(D. h.: Was hast du vor, was planst du?)\n\nTerenz, _Eunuchus_ 559.\n\n**Quid verbis opus est?**\n\nWas bedarf es der Worte?\n\nTerenz, _Andria_ 99 (72).\n\n**Quid vesper ferat, incertum est** siehe **Praesenti credere fortunae...**\n\n**Quid vesper serus vehat... sol tibi signa dabit.**\n\nWas der sp\u00e4te Abend bringt, k\u00fcndet dir die Sonne durch Zeichen an.\n\nVergil, _Georgica_ 1,461 ff.\n\n**Quid vici prosunt aut horrea?**\n\nWas helfen Anwesen und Kornspeicher?\n\nHoraz ( _Epistulae_ 2,2,177) h\u00e4lt Reichtum f\u00fcr unwichtig, da der Tod nicht bestechlich sei: Arm wie Reich raffe er gleicherma\u00dfen dahin.\n\n**Qui e nuce nuculeum esse vult, frangit nucem.**\n\nWer aus der Nu\u00df den Kern essen will, der zerbricht die Nu\u00df.\n\nPlautus ( _Curculio_ 55) f\u00e4hrt unmi\u00dfverst\u00e4ndlich fort: Ebenso m\u00fcsse, wer sexuellen Verkehr mit Frauen haben wolle, sich den Weg mit K\u00fcssen bahnen.\n\n**Quieta non movere!**\n\nEinmal zur Ruhe Gekommenes soll man nicht aufr\u00fchren!\n\nNach einem griechischen Sprichwort in der Sammlung des Macarius Hieromonachus (Paroemiographi Graeci 5,98). Vgl. Sallust, _Bellum Catilinarium_ 21,1. In der obigen lateinischen Version verwendet von Bismarck in einer Rede vom 14. April 1891.\n\n**Qui fodit foveam, incidet in eam.**\n\nWer einen Graben gr\u00e4bt, wird hineinfallen.\n\n(Vgl.: Wer anderen eine Grube gr\u00e4bt, f\u00e4llt selbst hinein.)\n\nSpr\u00fcche 26,27 und Kohelet 10,8. Vgl. **Incidit in foveam...**\n\n**Qui genus iactat suum, \/ aliena laudat.**\n\nWer mit seiner Herkunft prahlt, lobt Fremdes.\n\nSeneca, _Hercules furens_ 340 f.\n\n**Qui habet aures audiendi, audiat!**\n\nWer Ohren hat zu h\u00f6ren, der h\u00f6re!\n\nMatth\u00e4us 11,15.\n\n**Qui invident, egent.**\n\nDie neiden, leiden Mangel.\n\nPlautus, _Truculentus_ 746.\n\n**Qui ipse sibi sapiens prodesse non quit, nequiquam sapit.**\n\nEin Weiser, der sich selbst nicht helfen kann, ist umsonst weise.\n\nEnnius, _Medea_ Fragment 271 Warmington.\n\n**Qui modeste paret, videtur, qui aliquando imperet, dignus esse** siehe **Qui bene imperat...**\n\n**Qui monet, quasi adiuvat.**\n\nWer ermahnt, hilft gleichsam.\n\nPlautus, _Curculio_ 460.\n\n**Qui multum habet, plus cupit.**\n\nWer viel hat, verlangt nach mehr.\n\nSeneca, _Epistulae morales_ 119,6.\n\n**Qui nescit dissimulare, nescit regnare.**\n\nWer nicht zu t\u00e4uschen wei\u00df, wei\u00df nicht zu herrschen.\n\nLudwig XI. von Frankreich zugeschrieben. \u00c4hnlich HW: 24329.\n\n**Qui nil potest sperare, desperet nihil.**\n\nWer nichts hoffen kann, soll an nichts verzweifeln.\n\nSeneca, _Medea_ 163. Vgl. **Una salus...**\n\n**Qui nolet fieri desidiosus, amet!**\n\nWer nicht tr\u00e4ge werden will, der liebe!\n\nOvid, _Amores_ 1,9,46.\n\n**Qui non est hodie, cras minus aptus erit.**\n\nWer es heute nicht ist, der wird morgen (noch) weniger geeignet sein.\n\nOvid ( _Remedia amoris_ 94) warnt davor, die Heilung von der Liebe zu sp\u00e4t in Angriff zu nehmen, denn auch Medizin k\u00f6nne man zu sp\u00e4t einsetzen.\n\n**Qui non vetat peccare, cum possit, iubet.**\n\nWer nicht zu s\u00fcndigen verbietet, obwohl er es k\u00f6nnte, befiehlt es (zu s\u00fcndigen).\n\n(Vgl.: Selbst schuldig ist der Tat, wer nicht straft die Missetat.)\n\nSeneca, _Troades_ 291.\n\n**Quintili Vare, legiones redde!**\n\nQuintilius Varus, gib (mir meine) Legionen zur\u00fcck!\n\nSueton ( _Vita divi Augusti_ 23,2) zufolge soll Augustus noch Monate nach der verlorenen Schlacht im Teutoburger Wald (9 n. Chr.) diese Worte oft ausgerufen und seinen Kopf gegen eine T\u00fcr geschlagen haben.\n\n**Qui pavet vanos metus, \/ veros meretur.**\n\nWer aus falscher Furcht zittert, der verdient richtige.\n\nSeneca, _Oedipus_ 700 f.\n\n**Qui per virtutem perit, at non interit.**\n\nWer tugendhaft stirbt, vergeht jedoch nicht (erg.: denn es bleibt sein Ruhm).\n\nPlautus, _Captivi_ 690.\n\n**Qui poterit sanum fingere, sanus erit.**\n\nWer einen Gesunden mimen kann, wird gesund werden.\n\nOvid, _Remedia amoris_ 504.\n\n**Qui, quae vult, dicit, ea quae non vult, audiet** siehe **Si mihi perget...**\n\n**Quique aliis cavit, non cavet ipse sibi.**\n\nEr, der f\u00fcr andere sorgte, sorgt nicht f\u00fcr sich selbst.\n\nIn seiner Aufz\u00e4hlung von Pl\u00e4tzen, die sich f\u00fcr die Liebe eignen, nennt Ovid ( _Ars amatoria_ 1,84) auch das Forum: Dort seien viele Rechtsgelehrte, die sich um die Belange anderer k\u00fcmmerten. Was sie selbst betrifft, seien sie jedoch so sorglos, da\u00df sie dort von der Liebe \u00fcberrascht w\u00fcrden.\n\n**Qui sapit , in tacito gaudeat ipse sinu.**\n\nWer weise ist, freue sich im schweigsamen Herzen (d. h.: in aller Stille).\n\nTibull, _Elegiae_ 4,13,8.\n\n**Quis custodiet ipsos \/ custodes?**\n\nWer wird die W\u00e4chter selbst bewachen?\n\nJuvenal ( _Saturae_ 6,347 f.) emp\u00f6rt sich \u00fcber den Sittenverfall unter den r\u00f6mischen Frauen, die nun schon so l\u00fcstern seien, da\u00df sie sogar den von ihrem Ehemann bestellten W\u00e4chter verf\u00fchrten.\n\n**Qui se ipse laudat, cito derisorem inveniet.**\n\nWer sich selbst lobt, wird schnell einen Sp\u00f6tter finden.\n\nPublilius Syrus, _Sententiae_ Q 45.\n\n**Quis est enim, qui totum diem iaculans non aliquando colliniet?**\n\nWen g\u00e4be es denn, der den ganzen Tag herumsch\u00f6sse und nicht irgendwann tr\u00e4fe?\n\nCicero, _De divinatione_ 2,121.\n\n**Quis fallere possit amantem?**\n\nWer k\u00f6nnte eine Liebende t\u00e4uschen?\n\nVergil, _Aeneis_ 4,296.\n\n**Quis famulus amantior domini?**\n\nWelcher Diener liebt seinen Herrn mehr?\n\nColumella ( _De re rustica_ 7,12,1) preist die Treue der Hunde.\n\n**Quis furor, o cives, quae tanta licentia ferri?**\n\nWas f\u00fcr ein W\u00fcten, B\u00fcrger, welch ungeheure Z\u00fcgellosigkeit der Waffen!\n\nLukan, _De bello civili_ 1,8.\n\n**Qui sibi semitam non sapiunt, alteri monstrant viam.**\n\nDie, die ihren eigenen Weg nicht kennen, zeigen anderen die Richtung.\n\nCicero, _De divinatione_ 1,132.\n\n**Quis legem det amantibus? \/ Maior lex amor est sibi.**\n\nWer wird Liebenden Gesetze auflegen? Ein h\u00f6heres Gesetz ist die Liebe sich selbst.\n\nBoethius, _Consolatio Philosophiae_ 3, Metrum 12,47 f.\n\n**Quis leget haec?** siehe **O curas hominum...**\n\n**Quis minor est autem quam tacuisse labor?**\n\nWas aber ist weniger m\u00fchevoll als zu schweigen?\n\nOvid, _Amores_ 2,2,28.\n\n**Quis non odit sordidos, varios, leves, futiles?**\n\nWer ha\u00dft nicht die Widerlichen, die Wankelm\u00fctigen, die Leichtsinnigen, die Taugenichtse?\n\nCicero, _De finibus_ 3,38.\n\n**Quisquis amas, loca sola nocent, loca sola caveto!**\n\nWer auch immer du bist, der liebt: Einsame Orte sind sch\u00e4dlich, h\u00fcte dich vor einsamen Orten!\n\nOvid, _Remedia amoris_ 579.\n\n**Quisquis magna dedit, voluit sibi magna remitti.**\n\nWer gro\u00dfe Geschenke gab, wollte, da\u00df ihm wiederum gro\u00dfe gemacht w\u00fcrden.\n\nMartial, _Epigrammata_ 5,59,3.\n\n**Quisquis peccat inops, minor est reus.**\n\nWer als Mittelloser s\u00fcndigt, ist weniger schuldig.\n\nPetron, _Satyrica_ 133,3,10.\n\n**Quisquis plus iusto non sapit, ille sapit.**\n\nWer nicht \u00fcber die Ma\u00dfen weise ist, der ist weise.\n\nMartial, _Epigrammata_ 14,210,2.\n\n**Quis Solem fallere possit?**\n\nWer k\u00f6nnte Sol t\u00e4uschen?\n\nOvid, _Ars amatoria_ 2,573. \u2013 Venus und Mars hatten ihr Liebeslager gut versteckt, doch der Sonnengott sah die beiden und berichtete Venus' Gatten Vulcanus davon.\n\n**Qui studet optatam cursu contingere metam,**\n\n**Multa tulit fecitque puer, sudavit et alsit.**\n\nWer im Wettlauf dem ersehnten Zielstein zustrebt, hat als junger Mensch viel ertragen und geleistet, hat geschwitzt und gefroren.\n\nHoraz ( _Ars poetica_ 412 f.) erkl\u00e4rt, weder nat\u00fcrliche Begabung noch \u00dcbung allein reichten aus f\u00fcr einen guten Dichter. Vielmehr m\u00fcsse zur Begabung die \u00dcbung hinzukommen, so wie auch ein guter L\u00e4ufer bereits als Kind trainiere.\n\n**Quis tulerit Gracchos de seditione querentes?**\n\nWer ertr\u00fcge es, wenn sich die Gracchen \u00fcber Aufruhr beklagten?\n\nJuvenal, _Saturae_ 2,24. \u2013 Tiberius und Gaius Sempronius Gracchus hatten als Volkstribunen in den Jahren 133 bis 121 v. Chr. durch ihre volksfreundlichen Gesetzesvorschl\u00e4ge die sogenannten Gracchischen Unruhen bewirkt.\n\n**Qui tacet, consentire videtur.**\n\nWer schweigt, scheint beizustimmen.\n\nPapst Bonifaz VIII., _Liber sextus decretalium_ 5,12, de regulis iuris 43.\n\n**Qui tacet, non utique fatetur, sed tamen verum est eum non negare.**\n\nWer schweigt, gesteht zwar nicht, aber dennoch ist es wahr, da\u00df er nicht abstreitet.\n\n_Digesta_ 50,17,142 (Paulus, _Ad edictum_ 54). Vgl. Cicero, _De inventione_ 1,32,54; Seneca Rhetor, _Controversiae_ 10,2,6.\n\n**Qui tarde fecit, diu noluit.**\n\nWer sp\u00e4t handelt, hat lange nicht gewollt.\n\nSeneca ( _De beneficiis_ 1,1,8) ermahnt dazu, mit Wohltaten nicht zu z\u00f6gern, denn solches Verhalten zeige nur, da\u00df man eigentlich nicht wohlt\u00e4tig sein wolle. F\u00fcr den Empfangenden sei aber gerade der Wille des anderen wichtig.\n\n**Qui terret , plus ipse timet.**\n\nWer Schrecken verbreitet, hat selbst mehr Furcht.\n\nClaudian, _De quarto consulatu honorii Augusti_ 290.\n\n**Qui timide rogat, \/ docet negare.**\n\nWer \u00e4ngstlich fragt, lehrt abzulehnen.\n\nSeneca, _Phaedra_ 593 f.\n\n**Qui uti scit, ei bona; illi, qui non utitur recte, mala.**\n\nWer die Dinge zu benutzen wei\u00df, dem sind sie G\u00fcter, dem, der sie nicht richtig benutzt, \u00dcbel.\n\nTerenz, _Heauton timorumenos_ 196.\n\n**Quivis beatus versa rota fortunae ante vesperum potest esse miserrimus.**\n\nJeder Gl\u00fcckliche kann, wenn sich das Rad des Schicksals dreht, vor dem Abend zum Allerelendsten werden.\n\nAmmianus Marcellinus, _Res gestae_ 26,8,13.\n\n**Qui vult amari, languida regnat manu.**\n\nWer geliebt werden will, regiert mit nachsichtiger Hand.\n\nSeneca, _Phoenissae_ 659.\n\n**Quocumque aspexi, nihil est nisi mortis imago.**\n\nWohin auch immer ich blicke, ist nur ein Bild des Todes.\n\nOvid, _Tristia_ 1,11,23.\n\n**Quod boni est, id tacitus taceas tute tecum et gaudeas.**\n\nBehalte dein Gl\u00fcck stillschweigend f\u00fcr dich und erfreue dich daran.\n\nPlautus, _Epidicus_ 651.\n\n**Quod bonum faustum felix fortunatumque sit!**\n\nWas gut, g\u00fcnstig, gl\u00fccklich und gedeihlich sei!\n\nNach Cicero ( _De divinatione_ 1,102), der erl\u00e4utert, dies sei ein typischer Spruch der an Omen glaubenden Vorfahren der R\u00f6mer. In \u00e4hnlicher F\u00fcgung findet sich die Wendung z. B. bei Plautus ( _Trinummus_ 41). Livius erl\u00e4utert zum einen ( _Ab urbe condita_ 1,17,10), der Interrex habe \u00e4hnliche Worte ausgerufen, als das Volk selbst einen K\u00f6nig als Nachfolger des Romulus w\u00e4hlen sollte. Zum anderen f\u00fchrt er an, die Decemviri h\u00e4tten das Volk auf diese Weise aufgefordert, die Zw\u00f6lftafelgesetze zu lesen und zu \u00fcberdenken. Sueton ( _Vita divi Augusti_ 58) zufolge sei Augustus \u00e4hnliches gew\u00fcnscht worden, als man ihn als **pater patriae** begr\u00fc\u00dfte.\n\n**Quod caret alterna requie, durabile non est.**\n\nWas keine zeitweilige Pause kennt, ist nicht dauerhaft.\n\n(Vgl.: Was nicht rastet und nicht ruht, tut in die L\u00e4nge nicht gut.)\n\nOvid, _Heroides_ 4,89.\n\n**Quodcumque libuit facere victori, licet.**\n\nWas auch immer ihm gef\u00e4llt, ist dem Sieger erlaubt zu tun.\n\nSeneca, _Troades_ 335.\n\n**Quod cupio mecum est; inopem me copia fecit.**\n\nWas ich begehre, habe ich; die F\u00fclle machte mich arm.\n\nOvid, _Metamorphoses_ 3,466. \u2013 Narzi\u00df verzweifelt an seiner Selbstverliebtheit: Gerade die Tatsache, da\u00df er in Besitz dessen ist, was er liebt, verwehrt ihm eine Zusammenkunft mit der geliebten Person.\n\n**Quod dare non possis, verbis promittere noli,**\n\nNe sis ventosus, dum vir bonus esse videris.\n\nWas du nicht geben kannst, das versprich auch nicht mit Worten, damit du nicht ein Windbeutel seist, w\u00e4hrend du ein guter Mann scheinen willst.\n\n_Disticha Catonis_ 1,25.\n\n**Quod decet, honestum est, et quod honestum est, decet.**\n\nWas sich ziemt, ist anst\u00e4ndig, und was anst\u00e4ndig ist, ziemt sich.\n\nCicero, _De officiis_ 1,94.\n\n**Quod differtur, non aufertur.**\n\nWas aufgeschoben wird, wird nicht aufgehoben.\n\nArnobius, _Commentarii in Psalmos_ 36; vgl. z. B. Properz, _Elegiae_ 2,3,8, und Plinius d. J., _Epistulae_ 8,11,3.\n\n**Quod dubites , ne feceris!**\n\nWor\u00fcber du Zweifel hegst, das la\u00df sein!\n\nPlinius d. J. ( _Epistulae_ 1,18,5) nennt diese Worte eine alte Regel der Weisen.\n\n**Quod erat demonstrandum.**\n\nWas zu beweisen war.\n\nNach Euklid, z. B. _Elementorum liber_ 3,4. Die lateinische Version ist im Mittelalter belegt, siehe HW: 25788e.\n\n**Quod ergo intellego, id etiam credo; at non omne quod credo, etiam intellego.**\n\nWas ich also verstehe, das glaube ich auch; aber nicht alles, was ich glaube, verstehe ich auch.\n\nAugustinus, _De magistro_ 37,2.\n\n**Quod fors feret, feremus aequo animo.**\n\nWas das Schicksal bringen wird, werden wir gleichm\u00fctig ertragen.\n\nTerenz, _Phormio_ 138.\n\n**Quod hodie non est, cras erit; sic vita truditur.**\n\nWas heute nicht ist, wird morgen sein; so geht das Leben.\n\nPetron, _Satyrica_ 45,2.\n\n**Quod latet, ignotum est: ignoti nulla cupido.**\n\nWas verborgen ist, ist unbekannt; nach Unbekanntem (gibt es) kein Verlangen.\n\n(Vgl.: Was ich nicht wei\u00df, macht mich nicht hei\u00df.)\n\nOvid ( _Ars amatoria_ 3,397) r\u00e4t den Frauen, ihre Sch\u00f6nheit zu zeigen.\n\n**Quod licet, ingratum est, quod non licet, acrius urit.**\n\nWas erlaubt ist, ist reizlos, was nicht erlaubt ist, brennt heftiger.\n\n(Vgl.: In verbotenen Teichen fischt man gern.)\n\nOvid, _Amores_ 2,19,3.\n\n**Quod licet Iovi , non licet bovi.**\n\nWas Jupiter erlaubt ist, ist dem Ochsen (noch lange) nicht erlaubt.\n\nHW: 2587. Diese Wendung ist in der Antike nicht belegt. Sie nimmt ihren Ursprung vermutlich von dem Terenzischen **Aliis si licet, tibi non licet**.\n\n**Quod mihi praedicas, vitium id tibi est.**\n\nDer Fehler, den du mir anh\u00e4ngst, ist deiner.\n\nPlautus, _Amphitruo_ 402.\n\n**Quod non est in actis, non in mundo.**\n\nWas nicht in den Akten ist, ist nicht in der Welt (d. h.: es existiert nicht f\u00fcr den Richter).\n\nHW: 25928a.\n\n**Quod non exspectes, ex transverso fit.**\n\nWas man nicht erwartet, kommt unverhofft.\n\n(Vgl.: Unverhofft kommt oft.)\n\nPetron, _Satyrica_ 55,3.\n\n**Quod non fecerunt barbari, fecerunt Barberini.**\n\nWas die Barbaren nicht getan haben, taten die Barberini.\n\nCarlo Castelli zugeschriebener Ausspruch \u00fcber die Pl\u00fcnderung des Pantheons durch den Barberini-Papst Urban VIII.\n\n**Quod non potest, vult posse, qui nimium potest.**\n\nWas er nicht kann, will der k\u00f6nnen, der zuviel kann.\n\nSeneca, _Phaedra_ 215.\n\n**Quod non vetat lex, hoc vetat fieri pudor.**\n\nWas das Gesetz nicht verbietet, das verbietet der Anstand.\n\nSeneca, _Troades_ 334.\n\n**Quod omnes tangit, debet ab omnibus approbari.**\n\nWas alle ber\u00fchrt, mu\u00df auch von allen gebilligt werden.\n\nPapst Bonifaz VIII., _Liber sextus decretalium_ 5,12, de regulis iuris 29, nach _Codex Iustinianus_ 5,59,5,2.\n\n**Quod periit , periit.**\n\nWas untergegangen ist, ist untergegangen.\n\n(Vgl.: Vorbei ist vorbei.)\n\nPlautus, _Cistellaria_ 703.\n\n**Quod petiit , spernit, repetit, quod nuper omisit.**\n\nWas (mein Sinn) erstrebte, verachtet er; er will wiederhaben, was er neulich weggab.\n\nHoraz ( _Epistulae_ 1,1,98 ff.) malt seine schwankende Geisteshaltung zwei Verse weiter aus: \u00bbDiruit, aedificat, mutat quadrata rotundis.\u00ab (\u00bbEr [mein Geist] rei\u00dft ab, baut, vertauscht das Eckige mit dem Runden.\u00ab)\n\n**Quod posse fieri non putes, metuas tamen.**\n\nDas, wovon du nicht glaubst, da\u00df es geschehen k\u00f6nne, f\u00fcrchtest du dennoch.\n\nSeneca, _Oedipus_ 26.\n\n**Quod pudeat socios, prudens celare memento!**\n\nWas deine Gef\u00e4hrten besch\u00e4men k\u00f6nnte, das \u2013 denke daran \u2013 halte weise geheim!\n\n_Disticha Catonis_ 2,7,1.\n\n**Quod pudet, facilius fertur quam illud, quod piget.**\n\nLeichter ertr\u00e4gt man das, was einen besch\u00e4mt, als das, was einen \u00e4rgert.\n\nPlautus, _Pseudolus_ 281.\n\n**Quodque fuit vitium, desinit esse mora.**\n\nWas ein Mangel war, h\u00f6rt auf, ein Hinderungsgrund zu sein.\n\nOvid ( _Ars amatoria_ 2,654) erkl\u00e4rt, da\u00df man sich an k\u00f6rperliche M\u00e4ngel des Partners schnell gew\u00f6hne.\n\n**Quodque parum novit, nemo docere potest.**\n\nNiemand kann lehren, wovon er zu wenig versteht.\n\nOvid, _Tristia_ 2,348.\n\n**Quod quisque fecit, patitur, auctorem scelus \/ repetit.**\n\nEin jeder leidet unter dem, was er getan; das Verbrechen kommt wieder auf seinen Urheber zur\u00fcck.\n\nSeneca, _Hercules furens_ 735 f.\n\n**Quod ratio non quit, saepe sanavit mora.**\n\nWas der Verstand nicht kann, heilte oft die Zeit.\n\nSeneca, _Agamemnon_ 130.\n\n**Quod saepe fieri non potest, fiat diu. \/ Mors eligatur longa.**\n\nWas nicht oft geschehen kann, geschehe lang. W\u00e4hle einen langen Tod.\n\nSeneca, _Oedipus_ 948 f. \u2013 Oedipus' Worte, bevor er sich die Augen ausrei\u00dft.\n\n**Quod satis est, cui contingit, nihil amplius optet!**\n\nNichts weiter w\u00fcnsche, wer erhielt, was genug ist!\n\nHoraz, _Epistulae_ 1,2,46.\n\n**Quod scripsi, scripsi.**\n\nWas ich geschrieben habe, habe ich geschrieben.\n\nLaut Johannes (19,22) Antwort des Pontius Pilatus auf die Aufforderung der j\u00fcdischen Priester, nicht \u00bbrex Iudaeorum\u00ab (\u00bbK\u00f6nig der Juden\u00ab) auf das Kreuz zu schreiben.\n\n**Quod sequitur, fugio, quod fugit, usque sequor.**\n\nWas mir folgt, das will ich nicht, was sich mir entzieht, das verfolge ich.\n\nOvid, _Amores_ 2,19,36.\n\n**Quod si deficiant vires, audacia certe**\n\n**Laus erit: In magnis et voluisse sat est.**\n\nAuch wenn die Kr\u00e4fte fehlen, wird die K\u00fchnheit sicher Lob bringen. Bei gro\u00dfen Unterfangen ist es genug, den Willen gehabt zu haben.\n\n(Vgl.: Hier gilt der Wille f\u00fcr das Werk.)\n\nProperz, _Elegiae_ 2,10,5 f. Vgl. **Est nobis...** und **Ut desint vires...**\n\n**Quod sis , esse velis, nihilque malis!**\n\nWas du bist, das wolle sein, und nichts wolle lieber (sein)!\n\nMartial, _Epigrammata_ 10,47,12.\n\n**Quod supra nos, nihil ad nos!**\n\nWas \u00fcber uns ist, ist nichts f\u00fcr uns!\n\n(D. h.: Die Handlungen der G\u00f6tter betreffen uns nicht.)\n\nMinucius Felix ( _Octavius_ 13,1) zitiert hier Sokrates. Tertullian ( _Ad nationes_ 2,4,15) schreibt den Ausspruch Epikur zu.\n\n**Quod tibi fieri non vis, alteri ne feceris.**\n\nWovon du nicht willst, da\u00df es dir geschehe, das tu deinem N\u00e4chsten nicht an.\n\n(Vgl.: Was du nicht willst, da\u00df man dir tu, das f\u00fcg auch keinem andern zu.)\n\n_Historia Augusta,_ \u00bbVita Alexandri Severi\u00ab 51,8; vgl. Tobias 4,16. Vgl. **Aliis ne feceris...**\n\n**Quod verum, simplex, sincerumque est, id est naturae hominis aptissimum.**\n\nWas wahr, einfach und aufrichtig ist, ist der Natur des Menschen am angemessensten.\n\nNach Cicero, _De officiis_ 1,13.\n\n**Quod vides perisse, perditum ducas.**\n\nWas, wie du siehst, untergegangen ist, das halte f\u00fcr verloren.\n\nCatull, _Carmina_ 8,2.\n\n**Quo fata trahunt retrahuntque sequamur.**\n\nWohin auch das Schicksal uns zieht und wieder wegzieht, dorthin wollen wir folgen.\n\nVergil, _Aeneis_ 5,709.\n\n**Quo me cumque rapit tempestas, deferor hospes** siehe **iurare in verba magistri**\n\n**Quo mihi fortunam, si non conceditur uti?**\n\nWozu (n\u00fctzt) mir das Gl\u00fcck, wenn es mir nicht gestattet wird, davon Gebrauch zu machen?\n\nHoraz, _Epistulae_ 1,5,12.\n\n**Quoniam non potest id fieri, quod vis, \/ id velis, quod possit.**\n\nDa nun nicht geschehen kann, was du willst, wolle, was geschehen kann.\n\nTerenz, _Andria_ 305 f.\n\n**Quo non penetras, livor improbe?**\n\nWohin dringst du nicht, sch\u00e4ndlicher Neid?\n\nPseudo-Quintilian, _Declamationes_ 13,4.\n\n**Quo plura habent, eo ampliora cupientes.**\n\nJe mehr sie haben, desto mehr begehren sie.\n\nIustinus, _Historiae Philippicae_ 6,1,1.\n\n**Quo plura possis, plura patienter feras.**\n\nJe mehr du vermagst, desto mehr ertrage geduldig.\n\nSeneca, _Troades_ 254.\n\n**Quo quisque est maior, magis est placabilis irae.**\n\nJe gr\u00f6\u00dfer jemand ist, desto eher kann sein Zorn gelindert werden.\n\nOvid, _Tristia_ 3,5,31.\n\n**Quos amor verus tenuit, tenebit.**\n\nWen wahre Liebe (zusammen)hielt, die wird sie (zusammen)halten.\n\n(Vgl.: Wo Liebe, da ist Treue.)\n\nSeneca, _Thyestes_ 551.\n\n**Quos cogit metus \/ laudare, eosdem reddit inimicos metus.**\n\nDie, welche Furcht zwang (sich gegenseitig) zu loben, die wird Furcht auch wiederum zu Feinden machen.\n\nSeneca, _Thyestes_ 207 f.\n\n**Quos deus perdere vult, dementat prius.**\n\nDie, die Gott verderben will, die schl\u00e4gt er zuerst mit Wahnsinn.\n\nIn griechischer Sprache in den Scholien zu Sophokles, _Antigone_ 620.\n\n**Quos ego...!**\n\nEuch (werd') ich...!\n\nVergil, _Aeneis_ 1,135. \u2013 Der Meergott Neptun droht so den vier Winden, die von Aeolus freigelassen worden waren und ohne sein Gehei\u00df die Flotte des Aeneas versprengt hatten.\n\n**Quot caelum stellas, tot habet tua Roma puellas.**\n\nSo viele Sterne der Himmel, so viele M\u00e4dchen hat dein Rom.\n\nOvid, _Ars amatoria_ 1,59.\n\n**Quot homines, tot sententiae.**\n\nSo viele Menschen, so viele Meinungen.\n\nTerenz, _Phormio_ 454, und Cicero, _De finibus_ 1,15. Vgl. Horaz, _Sermones_ 2,1,27 f.; Ovid, _Ars amatoria_ 1,759.\n\n**Quousque tandem abutere, Catilina, patientia nostra?**\n\nWie lange schlie\u00dflich willst du noch, Catilina, unsere Geduld mi\u00dfbrauchen?\n\nCicero, _In L. Catilinam_ 1,1. Vgl. Sallust, _Bellum Catilinarium_ 20,9, und Livius, _Ab urbe condita_ 6,18,5. \u2013 Der ber\u00fchmte Auftakt der Anklagerede Ciceros gegen den Verschw\u00f6rer Catilina, der auch Cicero selbst ermorden lassen wollte.\n\n**(Domine,) quo vadis ?**\n\n(Herr,) wohin gehst du?\n\nPapst Linus, _Martyrium beatri Petri apostoli_ 6. \u2013 Als Nero Rom angez\u00fcndet hatte, floh Petrus aus der Stadt. Er traf Jesus und fragte ihn, wohin er gehe. Dieser antwortete: \u00bbRomam venio iterum crucifigi.\u00ab (\u00bbIch gehe nach Rom, um ein zweites Mal gekreuzigt zu werden.\u00ab)\nR\n\n**Rana in paludem ex aureo resilit throno.**\n\nEin Frosch h\u00fcpft (selbst) von einem g\u00fcldenen Thron wieder in den Sumpf zur\u00fcck.\n\nHW: 26241.\n\n**Rapienda rebus in malis praeceps via est.**\n\nIn der Bedr\u00e4ngnis mu\u00df man eilends den schnellsten Weg einschlagen.\n\nSeneca, _Agamemnon_ 154.\n\n**Rapimur, quo cuncta feruntur.**\n\nWir werden (im Strudel der Ereignisse) dorthin mitgerissen, wohin alles gezogen wird.\n\nLukan, _De bello civili_ 8,522.\n\n**Rapit omnia casus.**\n\nEs rei\u00dft mit sich fort alles der Zufall.\n\nLukan, _De bello civili_ 7,487.\n\n**Rapuit vires \/ pondusque malis, casus animo \/ qui tulit aequo.**\n\nDen \u00dcbeln raubte die Kraft und das Gewicht, wer das Schicksal gleichm\u00fctig ertrug.\n\nSeneca, _Hercules Oetaeus_ 231 ff.\n\n**rara avis**\n\nein seltener Vogel\n\nHoraz, _Sermones_ 2,2,26; vgl. Persius, _Saturae_ 1,46, und Juvenal ( _Saturae_ 6,165). Vgl. **corvus albus**.\n\n**Rara est adeo concordia formae \/ atque pudicitiae.**\n\nSo selten ist der Einklang von Sch\u00f6nheit und Schamhaftigkeit.\n\nJuvenal, _Saturae_ 10,297 f.\n\n**Rara in tenui facundia panno.**\n\nSelten steckt Beredsamkeit in einem sch\u00e4bigen Mantel.\n\nJuvenal ( _Saturae_ 7,145) kritisiert, da\u00df wahre Beredsamkeit bei den Zuh\u00f6rern eine geringere Rolle spiele als das materielle Verm\u00f6gen des Redners.\n\n**Rara iuvant.**\n\nSeltenes gef\u00e4llt.\n\n(Vgl.: Selten ist angenehm.)\n\nMartial, _Epigrammata_ 4,29,3.\n\n**Rari quippe boni.**\n\nSelten freilich sind gute Menschen.\n\nJuvenal, _Saturae_ 13,26.\n\n**Raro antecedentem scelestum \/ deseruit pede poena claudo.**\n\nSelten lie\u00df die Strafe hinkenden Fu\u00dfes vom vorausgehenden S\u00fcnder ab.\n\n(D. h.: Meist holt die Strafe den S\u00fcnder ein.)\n\nHoraz, _Carmina_ 3,2,31 f.\n\n**Rarum est felix idemque senex.**\n\nSelten ist derselbe Mensch gl\u00fccklich und alt.\n\nSeneca, _Hercules Oetaeus_ 643.\n\n**Ratio et consilium propriae ducis artes.**\n\nBerechnung und Klugheit sind charakteristische Eigenschaften eines Feldherrn.\n\nNach Tacitus, _Historiae_ 3,20,2.\n\n**Ratio fatum vincere nulla valet.**\n\nKeine Berechnung kann das Schicksal besiegen.\n\nOvid, _Tristia_ 3,6,18.\n\n**Rationi nulla resistunt \/ claustra.**\n\nDer Vernunft widersetzen sich keine Hindernisse.\n\nManilius, _Astronomica_ 1,541 f.\n\n**Ratio quasi quaedam lux lumenque vitae.**\n\nDie Vernunft ist gleichsam Licht und Leuchte des Lebens.\n\nNach Cicero, _Academici libri priores_ 2,26.\n\n**Rebus in angustis facile est contemnere vitam.**\n\nIn der Bedr\u00e4ngnis ist es leicht, das Leben zu verachten.\n\nMartial, _Epigrammata_ 11,56,15.\n\n**Receptores non minus delinquunt quam adgressores.**\n\nDie Empf\u00e4nger (von Diebesgut) verfehlen sich nicht weniger als die R\u00e4uber.\n\n(Vgl.: Der Hehler ist nicht besser als der Stehler.)\n\n_Digesta_ , 47,9,3,3.\n\n**redire in viam**\n\nauf den Weg zur\u00fcckkommen\n\n(vgl.: wieder auf den rechten Weg finden)\n\nNach Terenz, _Andria_ 190.\n\n**reges et regum vita praecurrere amicos**\n\nwas das Leben betrifft, K\u00f6nige und Freunde von K\u00f6nigen \u00fcberholen\n\nHoraz ( _Epistulae_ 1,10,33) erkl\u00e4rt, ein gl\u00fcckliches und ausgef\u00fclltes Leben hinge nicht vom Besitz ab.\n\n**Regia, crede mihi, res est succurrere lapsis.**\n\nEs ist, glaube mir, etwas K\u00f6nigliches, Gefallenen zu helfen.\n\nDer im Exil befindliche Ovid ( _Epistulae ex Ponto_ 2,9,11) wendet sich mit diesen Worten an den thrakischen K\u00f6nig.\n\n**Regi frenis nequit \/ et ira.**\n\nAuch Zorn kann nicht mit Z\u00fcgeln geb\u00e4ndigt werden.\n\nSeneca, _Troades_ 279 f.\n\n**Regna custodit metus.**\n\nFurcht sch\u00fctzt die K\u00f6nigreiche.\n\n(D. h.: Der K\u00f6nig ist sicher, solange er seine Untertanen einsch\u00fcchtert.)\n\nSeneca, _Oedipus_ 704. Vgl. **Odia qui...**\n\n**Regnare nolo, liber ut non sim mihi.**\n\nIch m\u00f6chte nicht herrschen, sofern ich mich (dann) nicht frei f\u00fchle.\n\n(D. h.: Selbst K\u00f6nig wollte ich nicht sein, wenn es auf Kosten meiner Entscheidungsfreiheit w\u00e4re.)\n\nPhaedrus, _Fabulae_ 3,7,27. \u2013 Ein Wolf beneidete einen Hund, weil dieser immer wohlgen\u00e4hrt war. Als er dann jedoch erfuhr, da\u00df der Hund den Befehlen seines Herren gehorchte, zog er es vor, ein armer, doch freier Wolf zu bleiben.\n\n**Relata refero.**\n\nIch berichte (nur mir) Berichtetes.\n\nNach Herodot, _Geschichte_ 7,152,3. Vgl. die andere \u00dcbersetzung des griechischen Originals: **prodenda, quia prodita**.\n\n**relicta non bene parmula**\n\nwobei ich meinen kleinen Schild sch\u00e4ndlicherweise liegen lie\u00df\n\nHoraz ( _Carmina_ 2,7,10) \u00fcber die Schlacht von Philippi.\n\n**Religentem esse oportet, religiosus ne fuas.**\n\nEs ziemt sich, gottesf\u00fcrchtig zu sein, damit man nicht abergl\u00e4ubisch werde.\n\nGellius ( _Noctes Atticae_ 4,9,1) zitiert Nigidius Figulus ( _Commentarii Grammatici_ Fragment 4 Legrand), der die Worte einem alten Dichter zuschreibt, den er nicht namentlich nennt; vgl. Fragment Tragic. Rom. Ex. inc. inc. fab. 81 Ribbeck.\n\n**Rem facias , rem, \/ si possis, recte, si non, quocumque modo rem!**\n\nSchaffe dir ein Verm\u00f6gen auf rechte Weise, wenn du kannst, wenn nicht, dann auf welche Weise auch immer.\n\nSolche Leits\u00e4tze lehnt Horaz ( _Epistulae_ 1,1,65 f.) ab. F\u00fcr ihn ist pers\u00f6nliche Integrit\u00e4t wichtiger als Besitz.\n\n**rem fidemque perdere**\n\nVerm\u00f6gen und Vertrauen verlieren\n\nPlautus, _Curculio_ 504.\n\n**Rem tene, verba sequentur!**\n\nHalte an der Sache fest, die Worte werden folgen!\n\nCato d. \u00c4. ( _De rhetorica?_ Fragment 15 Jordan) unterstreicht hier seine rhetorikfeindliche Haltung. Der Inhalt ist ihm wichtiger als die sprachliche Form.\n\n**Repetitio est mater studiorum.**\n\nWiederholung ist die Mutter der Studien.\n\nHWS: 40753, vermutlich nach Horaz, _Ars poetica_ 365.\n\n**Requiescat in pace (R. I. P.)**\n\nEr\/Sie m\u00f6ge in Frieden ruhen.\n\nNach Psalm 4,9. Dort meint \u203aruhen\u2039 jedoch tats\u00e4chlich den Schlaf und nicht den Tod.\n\n**rerum concordia discors**\n\nzwietr\u00e4chtige Eintracht der Dinge\n\nHoraz, _Epistulae_ 1,12,19; vgl. Ovid, _Metamorphoses_ 1,433, und Lukan, _De bello civili_ 1,98.\n\n**Rerum omnium custos est memoria.**\n\nAller Dinge Hort ist das Ged\u00e4chtnis.\n\nCicero, _Partitiones oratoriae_ 3.\n\n**Res ad triarios redit.**\n\nDie Sache kommt auf die Triarier zur\u00fcck.\n\n(D. h.: Es ist zum \u00c4u\u00dfersten gekommen.)\n\nLivius ( _Ab urbe condita_ 8,8,11) bezeichnet die Wendung als bekanntes Sprichwort. \u2013 Die Triarier, die erfahrensten Soldaten, waren das dritte Treffen im Heer und kamen erst zum Einsatz, wenn die ersten beiden Treffen zur\u00fcckgeworfen worden waren, die Lage also bedenklich wurde.\n\n**Res, aetas , usus semper aliquid apportat novi.**\n\nDie Umst\u00e4nde, die Zeit und die Erfahrung bringen immer etwas Neues mit sich.\n\n(D. h.: Man kann das Leben nie ganz vorausplanen.)\n\nNach Terenz, _Adelphoe_ 856.\n\n**Res age, tutus eris.**\n\nSei gesch\u00e4ftig, (und) du wirst sicher sein.\n\nOvid ( _Remedia amoris_ 144) erkl\u00e4rt, so wie eine Pflanze Wasser brauche, um zu wachsen, so bed\u00fcrfe die Liebe der Unt\u00e4tigkeit. Daher k\u00f6nne man sich durch eifrige Arbeit vor der Liebe sch\u00fctzen.\n\n**Res amicos invenit.**\n\nBesitz findet Freunde.\n\n(D. h.: Wer reich ist, hat viele Freunde.)\n\nPlautus, _Stichus_ 522.\n\n**reservatio mentalis**\n\ninnerer Vorbehalt\n\nHermann Busenbaum _Medulla theologiae moralis_ 3,2,2,4. \u2013 Gemeint ist eine Eidesformulierung, die bewu\u00dft mehrere Deutungen zul\u00e4\u00dft und sich somit nicht festlegt. F\u00fcr die Jesuiten galt ein solcher innerer, nicht ausgesprochener Vorbehalt nicht als Meineid. Vgl. dagegen Luther ( _An den christlichen Adel deutscher Nation_ , Weimarer Ausgabe I,6, S. 425), der eine solche Eidesformulierung ablehnt. Ebenso Blaise Pascal, _Les provinciales_ 9.\n\n**Res est imperiosa timor.**\n\nFurcht ist ein gebieterisch' Ding.\n\nMartial, _Epigrammata_ 11,58,8.\n\n**Res est ingeniosa dare.**\n\nDas Geben erfordert Verstand.\n\nOvid, _Amores_ 1,8,62.\n\n**Res est solliciti plena timoris amor.**\n\nLiebe ist eine Angelegenheit voll unruhiger Furcht.\n\nOvid, _Heroides_ 1,12.\n\n**Res immoderata cupido est.**\n\nBegierde ist etwas Grenzenloses.\n\nOvid, _Epistulae ex Ponto_ 4,15,31.\n\n**Res in cardine est.**\n\nDie Sache ist am Wendepunkt.\n\n(D. h.: Die Entscheidung steht bevor.)\n\nServius ( _In Aeneidem_ 1,672) bezeichnet die Wendung als bekanntes Sprichwort.\n\n**res iudicata**\n\nein bereits entschiedener Fall\n\nNach Cicero, _Partitiones oratoriae_ 126.\n\n**Res loquitur ipsa.**\n\nDie Sache spricht (f\u00fcr sich) selbst.\n\nCicero, _Pro T. Annio Milone_ 53; vgl. Terenz, _Eunuchus_ 705.\n\n**Respice post te, hominem te esse memento!**\n\nBlicke hinter dich; bedenke, da\u00df du ein Mensch bist!\n\nTertullian ( _Apologeticum_ 33,4) berichtet, da\u00df derselbe Staatssklave, der die prunkvolle Krone bei Triumphz\u00fcgen \u00fcber das Haupt des Triumphators hielt, diesem zu den Beifallskundgebungen des Volkes die obigen Worte ins Ohr zu fl\u00fcstern hatte. Oft wird auch \u00bbMemento mori\u00ab zitiert, eine Wendung, deren eigentliche Quelle im dunkeln liegt. Sie verdeutlicht aber einen Grundgedanken der delphischen Theologie und des delphischen Menschenbildes und ist heute Wahlspruch des Kart\u00e4userordens. Mitunter findet sich die Angabe, diese Wendung gehe auf ein alemannisches Gedicht aus der Zeit um 1090 zur\u00fcck, das von einem Dichter Noker, wom\u00f6glich dem Abt des Klosters Zwiefalten, verfa\u00dft worden sei.\n\n**Respiciet deus \/ bene cogitata.**\n\nDer Gott wird Wohl\u00fcberlegtes beachten.\n\nSeneca, _Thyestes_ 489 f.\n\n**Responsum date!**\n\nGebt Antwort!\n\nHoraz, _Epodi_ 7,14.\n\n**Res similis fictae, sed quid mihi fingere prodest?**\n\nDie Sache ist (zwar) einer erdichteten \u00e4hnlich, doch was n\u00fctzt mir das Erdichten?\n\nOvid, _Metamorphoses_ 13,935.\n\n**Restat iter caeli; caelo temptabimus ire!**\n\nEs bleibt der Weg durch den Himmel; durch den Himmel zu gehen werden wir versuchen!\n\nOvid, _Ars amatoria_ 2,37. \u2013 Daedalus sieht f\u00fcr sich und seinen Sohn nur die M\u00f6glichkeit, durch die L\u00fcfte zu entkommen, da K\u00f6nig Minos von Kreta alle anderen Wege versperrt hat.\n\n**Res timida est omnis miser.**\n\nJeder Elende ist ein furchtsames Ding.\n\nOvid, _Epistulae ex Ponto_ 2,7,37.\n\n**Retinent mores, quod perdidit aetas.**\n\nDie Gewohnheit bewahrt, was das Alter verdorben hat.\n\nClaudian ( _In Eutropium_ 1,97) spricht hier von einer gealterten Prostituierten, die, obwohl ihr Alter es verbietet, immer noch aus Gewohnheit das Bordell aufsucht.\n\n**retrorsum \/ vela dare**\n\ndie Segel wenden\n\n(d. h.: zur\u00fcckkehren)\n\nHoraz, _Carmina_ 1,34,3 f.\n\n**Rex eris, si recte facies.**\n\nEin K\u00f6nig wirst du sein, wenn du richtig handelst.\n\nNach Horaz ( _Epistulae_ 1,1,59 f.), der diese Wendung als Kindervers anf\u00fchrt. In den Scholien findet sich die komplette Version: \u00bbRex erit qui recte faciet; qui non faciet, non erit.\u00ab (\u00bbEin K\u00f6nig wird sein, wer richtig handeln wird; wer nicht [richtig] handeln wird, wird es nicht sein.\u00ab)\n\n**Rex est , qui metuit nihil; \/ rex est, qui cupiet nihil.**\n\nEin K\u00f6nig ist, wer nichts f\u00fcrchtete; ein K\u00f6nig ist, wer nichts begehren wird.\n\nSeneca, _Thyestes_ 388 f.\n\n**Rex ipse sine aculeo.**\n\nDer (Bienen)k\u00f6nig selbst ist ohne Stachel.\n\nSeneca ( _De clementia_ 1,19,3) erl\u00e4utert anhand des Beispiels der stachellosen Bienenk\u00f6nigin (die in der Antike f\u00fcr ein m\u00e4nnliches Tier gehalten wurde), da\u00df auch in der Natur Milde ein Merkmal der Herrscher sei.\n\n**Rex numquam moritur.**\n\nDer K\u00f6nig stirbt nie.\n\nRechtsinstitutionen bleiben erhalten, auch wenn der entsprechende Amtsinhaber oder W\u00fcrdentr\u00e4ger stirbt. Vermutlich nach Goffredus Tranensis, _Summa super titulis Decretalium, De officio et potestate iudicis delegati_ 18. Vgl. Ernst Kantorowicz, _The King's Two Bodies_ 7,3.\n\n**Rex regnat, sed non gubernat.**\n\nDer K\u00f6nig herrscht, aber regiert nicht.\n\nJan Zamoyski soll K\u00f6nig Sigismund III. (1566-1632) zugerufen haben: \u00bbRegna, sed non impera!\u00ab (\u00bbRegiere, aber herrsche nicht (wie ein Despot)!\u00ab) Adolphe Thiers formulierte sp\u00e4ter die obige Regel in franz\u00f6sischer Sprache ( _National_ 18. 01.1830). Gemeint war nun, ein K\u00f6nig solle zwar die h\u00f6chste Autorit\u00e4t sein, aber keine politische und administrative Macht haben.\n\n**ridentem dicere verum** siehe **Quamquam ridentem...**\n\n**Ride, si sapis, o puella, ride!**\n\nLach, wenn du weise bist, M\u00e4dchen, lache!\n\nMartial ( _Epigrammata_ 2,41,1) zitiert diesen Vers eines heute unbekannten Dichters und warnt dann sarkastisch vor dessen Anwendung: Maximina, die nur noch drei pechschwarze Z\u00e4hne habe, solle tunlichst ihren Mund geschlossen halten.\n\n**Ridet argento domus.**\n\nEs strahlt vor Silber das (ganze) Haus.\n\nHoraz, _Carmina_ 4,11,6.\n\n**Ridetur , chorda qui semper oberrat eadem.**\n\nAusgelacht wird, wer immer bei derselben Saite danebengreift.\n\nHoraz, _Ars poetica_ 356.\n\n**Ridiculum acri \/ fortius et melius magnas plerumque secat res.**\n\nScherz zerlegt gro\u00dfe Fragen meist auf forschere und treffendere Weise als bei\u00dfender Scharfsinn.\n\nHoraz, _Sermones_ 1,10,14 f.\n\n**Risum teneatis, amici?**\n\nK\u00f6nnt ihr das Lachen zur\u00fcckhalten, Freunde?\n\nHoraz, _Ars poetica_ 5.\n\n**Rivalem possum non ego ferre Iovem.**\n\nEinen Nebenbuhler kann ich nicht dulden, (und) sei es Jupiter (selbst).\n\nProperz, _Elegiae_ 2,34,18.\n\n**Roma aeterna**\n\ndas ewige Rom\n\nNach Tibull, _Elegiae_ 2,5,23.\n\n**Romae Tibur amem ventosus, Tibure Romam.**\n\nIn Rom w\u00e4re ich Wankelm\u00fctiger lieber in Tibur, in Tibur lieber in Rom.\n\nHoraz, _Epistulae_ 1,8,12.\n\n**Roma locuta, causa finita.**\n\nRom hat gesprochen, der Fall ist beendet.\n\nNach Augustinus, _Sermones_ 131,10. \u2013 Die zweigliedrige F\u00fcgung findet sich in franz\u00f6sischer Sprache in dem Gedicht \u00bbPhilotanus\u00ab des Abb\u00e9 de Gr\u00e9court: \u00bbRome a parl\u00e9, l'affaire est termin\u00e9e.\u00ab\n\n**Romam cuncta undique atrocia atque pudenda confluunt celebranturque.**\n\nIn Rom str\u00f6mt alles Schreckliche und Sch\u00e4ndliche von \u00fcberall her zusammen und wird (auch noch) gefeiert.\n\nNach Tacitus, _Annales_ 15,44,4.\n\n**Romanus sum civis** siehe **Et facere et pati...**\n\n**Rosa de spinis floret.**\n\nDie Rose bl\u00fcht am Dornenstrauch.\n\nNach Hieronymus ( _Vita Hilarionis_ 2), der die Wendung als sprichw\u00f6rtlich bezeichnet.\n\n**rudis indigestaque moles**\n\neine rohe, ungeordnete Masse\n\nOvid ( _Metamorphoses_ 1,7) beschreibt den Zustand der Welt zu Beginn ihrer Existenz.\n\n**Ruendi \/ in ferrum mens prona viris.**\n\nDiesen M\u00e4nnern steht der Sinn danach, sich in das Schwert zu st\u00fcrzen.\n\nLukan, _De bello civili_ 1,460 f.\n\n**Ruere in servitium.**\n\nSie st\u00fcrzten sich in die Knechtschaft.\n\nTacitus, _Annales_ 1,7.\n\n**Rumor in ambiguo est.**\n\nDas allgemeine Gerede ist unentschieden.\n\nOvid ( _Metamorphoses_ 3,253) beschreibt, wie unterschiedlich das Volk die Rachetat der Diana beurteilte. \u2013 Diana war vom J\u00e4ger Actaeon beim Bade erblickt worden. Aus Rache lie\u00df sie ihn, in einen Hirsch verwandelt, von seinen eigenen Hunden zerfleischen.\n\n**Rumpe moras!** siehe **Heia age...**\n\n**Rumpitur invidia.**\n\nEr zerplatzt vor Neid.\n\nMartial, _Epigrammata_ 9,97,1 et passim.\nS\n\n**sacrificium intellectus**\n\nOpfer des Verstandes\n\nNach Korintherbriefe 2,10,5. \u2013 Gemeint sind Entscheidungen, bei denen man seinen Verstand dem Glauben hintansetzt, die Verstandeserkenntnis also opfert.\n\n**Sacrilegia minuta puniuntur, magna in triumphis feruntur.**\n\nKleine Verbrechen werden bestraft, gro\u00dfe in Triumphz\u00fcgen gefeiert.\n\n(Vgl.: Die Kleinen h\u00e4ngt man, die Gro\u00dfen l\u00e4\u00dft man laufen.)\n\nSeneca, _Epistulae morales_ 87,23.\n\n**Saepe enim audivi poetam bonum neminem sine inflammatione animorum existere posse et sine quodam afflatu quasi furoris.**\n\nOft habe ich geh\u00f6rt, kein guter Dichter k\u00f6nne leben, ohne da\u00df seine Leidenschaft entflammt sei, und ohne einen gewissen Anflug von Wahnsinn.\n\nNach Cicero ( _De oratore_ 2,194), der den Gedanken Demokrit und Platon (z. B. _Menon_ 99c) zuschreibt. Vgl. **Misce stultitiam...** und **Nullum magnum ingenium...**\n\n**Saepe est etiam sub palliolo sordido sapientia.**\n\nOft steckt auch unter einem schmutzigen Gewand Weisheit.\n\nCicero ( _Tusculanae disputationes_ 3,56) zitiert hier Statius, Fragment Ex inc. fab. 255 Warmington.\n\n**Saepe in coniugiis fit noxia, si nimia est, dos.**\n\nOft wird in der Ehe eine allzugro\u00dfe Mitgift zum Unheil.\n\n(Vgl.: Ist das Geld die Braut, so taugt die Ehe selten was.)\n\nAusonius, _Technopaegnion_ 7,1; vgl. Juvenal, _Saturae_ 6,460.\n\n**Saepe in magistrum scelera redierunt sua** siehe **Peiora iuvenes...**\n\n**Saepe refert animus lusus gravitate carentes.**\n\nOft erinnert sich mein Herz an die Ausgelassenheit, die des Ernstes entbehrte.\n\nOvid ( _Epistulae ex Ponto_ 1,9,9) trauert um den Tod seines Freundes Celsus.\n\n**Saepe solent auro multa subesse mala.**\n\nOft pflegen im Gold viele \u00dcbel zu stecken.\n\nTibull, _Elegiae_ 1,9,18.\n\n**Saepe stilum vertas!**\n\nWende oft den Griffel!\n\n(D. h.: Tilge Geschriebenes und feile am Text!)\n\nHoraz, _Sermones_ 1,10,72. \u2013 Wie sich an modernen Bleistiften oben ein Radiergummi befindet, so hatten die Schreibgriffel einen platten Teil zum Gl\u00e4tten des Wachses auf den Schreibtafeln, in das Buchstaben eingeritzt worden waren.\n\n**Saepe tacens vocem verbaque vultus habet.**\n\nOft hat eine schweigende Miene Stimme und Worte.\n\n(vgl.: beredtes Schweigen)\n\nOvid, _Ars amatoria_ 1,574.\n\n**Saepe tepent alii iuvenes, ego semper amavi.**\n\nAndere junge M\u00e4nner sind oft erkaltet (in der Liebe); ich habe immer geliebt.\n\nOvid, _Remedia amoris_ 7.\n\n**Saepe tulit lassis sucus amarus opem.**\n\nOft hat bitterer Saft den Ermatteten Hilfe gebracht.\n\n(Vgl.: Bitter dem Mund, dem Herzen gesund.)\n\nOvid, _Amores_ 3,11,8.\n\n**Saepe vindicta obfuit.**\n\nOft hat Rache geschadet.\n\nSeneca, _Hercules furens_ 1187.\n\n**Saepe viri fallunt.**\n\nOft betr\u00fcgen die M\u00e4nner.\n\nOvid ( _Ars amatoria_ 3,31) h\u00e4lt Frauen in puncto Liebe f\u00fcr ehrlicher als M\u00e4nner.\n\n**Saevaque diducto stipite flamma perit.**\n\nDie wilde Flamme erlischt, wenn das Holzscheit geteilt wird.\n\nOvid ( _Remedia amoris_ 446) r\u00e4t, wer sich von Liebesqualen befreien wolle, solle sich mindestens zwei Geliebte nehmen, denn so werde die Leidenschaft halbiert.\n\n**Saevis inter se convenit ursis.**\n\n(Sogar) die wilden B\u00e4ren kommen miteinander \u00fcberein.\n\nJuvenal, _Saturae_ 15,164.\n\n**Saevit amor ferri et scelerata insania belli, \/ ira super.**\n\nEs w\u00fctet die Gier nach dem Schwerte und des Krieges sch\u00e4ndlicher Wahnsinn, Zorn noch dazu.\n\nVergil, _Aeneis_ 7,461 f.\n\n**sal Atticus**\n\nattisches Salz\n\n(d. h.: feiner Witz)\n\n(z. B.) nach Cicero, _Ad familiares_ 9,15,2.\n\n**salivam movere**\n\nden Speichel in Bewegung setzen\n\n(d. h.: den Mund w\u00e4\u00dfrig machen)\n\nNach Seneca, _Epistulae morales_ 79,7.\n\n**Salus extra ecclesiam non est.**\n\nAu\u00dferhalb der Kirche gibt es kein Heil.\n\nCyprianus, _Epistulae_ 73,21,2.\n\n**Salus populi suprema lex esto!**\n\nDas \u00f6ffentliche Wohl soll das oberste Gesetz sein!\n\nNach Ciceros Auffassung ( _De legibus_ 3,8) sollte das \u00f6ffentliche Wohl das h\u00f6chste Anliegen der Pr\u00e4toren, Richter und Konsuln sein.\n\n**sancta simplicitas**\n\nheilige Einfachheit\n\nHieronymus ( _Epistulae_ 57,12) verteidigt die Schlichtheit der Sprache der J\u00fcnger, da ihr etwas Heiliges innewohne. Im ironischen Sinne, die Einfalt eines Gl\u00e4ubigen beklagend, soll Johannes Hus diese Worte auf dem Scheiterhaufen ausgerufen haben, als ein Bauer noch weiteres Holz herantrug. Vgl. Julius Wilhelm Zincgref, _Teutscher Nation Apophthegmata_ 3, S. 383; Eusebius, _Kirchengeschichte_ 10,3.\n\n**Sapere aude; \/ incipe!** siehe **Dimidium facti, qui coepit,...**\n\n**Sapienter idem \/ contrahes vento nimium secundo \/ turgida vela.**\n\nWeise wirst du wieder bei zu g\u00fcnstigem Wind die bauschigen Segel raffen.\n\nHoraz, _Carmina_ 2,10,22 ff.\n\n**sapienter vitam instituere**\n\nsein Leben weise einrichten\n\n(z. B.) nach Terenz, _Andria_ 67.\n\n**Sapienti aetas condimentum, sapiens aetati cibus est.**\n\nAlter ist W\u00fcrze f\u00fcr den Weisen, der Weise ist die Nahrung des Alters.\n\nPlautus ( _Trinummus_ 368) l\u00e4\u00dft Philto erkl\u00e4ren, da\u00df nicht das hohe Alter allein, sondern der Charakter Weisheit bedinge.\n\n**sapientia prima \/ stultitia caruisse** siehe **Virtus est vitium fugere...**\n\n**Sapientia vino obumbratur.**\n\nWeisheit wird vom Wein umschattet.\n\nNach Plinius d. \u00c4. ( _Naturalis historia_ 23,41) ist dies eine sprichw\u00f6rtliche Wendung.\n\n**Sapientissimus est, cui, quod opus sit, ipsi venit in mentem.**\n\nDer Weiseste ist der, dem selbst einf\u00e4llt, was er braucht.\n\nNach Cicero ( _Pro A. Cluentio Habito_ 84), der diese Worte als gebr\u00e4uchliche Wendung bezeichnet.\n\n**sapientum octavus**\n\nder achte der Weisen\n\nHoraz ( _Sermones_ 2,3,296) l\u00e4\u00dft hier den Sklaven Damasippus Stertinius als achten unter die Sieben Weisen (Bias, Chilon, Kleobulus, Periander, Pittakos, Solon und Thales) einreihen.\n\n**Sapit nimio plus quam Thales.**\n\nEr ist gar sehr viel weiser als Thales.\n\nNach Plautus, _Bacchides_ 122. \u2013 Thales aus Milet war einer der Sieben Weisen.\n\n**Sat celeriter fit, quidquid fit satis bene.**\n\nSchnell genug wird ausgef\u00fchrt, was gut genug ausgef\u00fchrt wird.\n\nNach Sueton ( _Vita divi Augusti_ 25,4) soll Augustus diese Worte oft im Munde gef\u00fchrt haben. Vgl. **Festina lente!** \u00c4hnlich Cato d. \u00c4. (Fragment Dicta memorabilia 80 Jordan): \u00bbSat cito, si sat bene.\u00ab \u2013 \u00bbSchnell genug (wird etwas getan), wenn (es) gut genug (getan wird).\u00ab\n\n**satis eloquentiae, sapientiae parum**\n\ngen\u00fcgend Beredsamkeit, zu wenig Weisheit\n\nSallust, _Bellum Catilinarium_ 5,5.\n\n**satis superque**\n\ngenug und dar\u00fcber hinaus\n\n(d. h.: mehr als genug)\n\n(z. B.) Cicero, _De natura deorum_ 2,2.\n\n**Satis vixi; invictus enim morior.**\n\nIch habe genug gelebt; ich sterbe n\u00e4mlich unbesiegt.\n\nCornelius Nepos ( _Epaminondas_ 9,4) schreibt diese Worte Epaminondas zu.\n\n**Satura quidem tota nostra est.**\n\nDie Satire freilich geh\u00f6rt ganz uns.\n\nQuintilian ( _Institutio oratoria_ 10,1,93) betont, da\u00df die Satire eine rein r\u00f6mische Gattung ist, die auf kein griechisches Vorbild zur\u00fcckgeht.\n\n**Saxa loquentur.**\n\nDie Felsen werden sprechen.\n\nLukan, _De bello civili_ 6,618. \u2013 Die thessalische Hexe Eric(h)tho verk\u00fcndet, da\u00df man die Zukunft aus der Erde, dem \u00c4ther, dem Chaos, dem Meer, den Feldern und aus den Felsen des Rhodope-Gebirges erfahren k\u00f6nne. Vgl. Lukas 19,40.\n\n**scapham scapham dicere**\n\neinen Kahn einen Kahn nennen\n\n(vgl.: das Kind beim Namen nennen)\n\nNach Aristophanes, Fragment Inc. fab. 927 Kassel-Austin. Vgl. z. B. Plutarch, _Moralia_ 178b; Lukian, _Iuppiter Tragoedus_ 32, und Julian, _Orationes_ 7,208a.\n\n**Scelere velandum est scelus.**\n\nEin Verbrechen mu\u00df durch ein Verbrechen vertuscht werden.\n\nSeneca, _Phaedra_ 721. Vgl. **Per scelera...**\n\n**Scholae sunt humanitatis officinae, efficiendo nimirum, ut homines vere homines fiant.**\n\nSchulen sind Produktionsst\u00e4tten der Menschlichkeit, sofern sie bewirken, da\u00df aus Menschen wirklich Menschen werden.\n\nJohann Amos Comenius, _Didactica Magna_ 10,3.\n\n**Scientia nobilitat.**\n\nWissenschaft adelt.\n\nHW: 27590 f. \u2013 Der Spruch soll sich auch in einer Glosse zum _Codex Iustinianus_ 2,6,7 befinden.\n\n**Scilicet uxorem cum dote, fidemque et amicos,**\n\nEt genus et formam regina pecunia donat.\n\nNat\u00fcrlich werden eine Frau mit Mitgift, Ansehen und Freunde, edle Abstammung und Sch\u00f6nheit durch die K\u00f6nigin Geld gew\u00e4hrleistet.\n\nHoraz, _Epistulae_ 1,6,36 f.\n\n**Scinditur incertum studia in contraria vulgus.**\n\nDas unsichere Volk ist nach gegenteiligen Bestrebungen gespalten.\n\nVergil, _Aeneis_ 2,39. \u2013 Das Volk teilt sich beim Anblick des Trojanischen Pferdes in zwei Lager, die zu gegenteiligen Ma\u00dfnahmen greifen wollen.\n\n**Scio : tu coactus tua voluntate es.**\n\nIch wei\u00df: Du bist mit deiner Einwilligung gezwungen worden.\n\nTerenz, _Andria_ 658.\n\n**Scire tuum nihil est, nisi te scire hoc sciat alter** siehe **At pulchrum est digito monstrari...**\n\n**Scitum est periclum ex aliis facere, tibi quod ex usu siet.**\n\nEs ist weise, aus anderen (d. h.: aus den Erfahrungen anderer) eine Lehre zu ziehen, die dir n\u00fctze.\n\nTerenz, _Heauton timorumenos_ 210, vgl. 221.\n\n**Scribendi recte sapere est et principium et fons.**\n\nWeise zu sein ist Anfang und Quelle richtigen Schreibens.\n\nHoraz, _Ars poetica_ 309.\n\n**Scribimus indocti doctique poemata passim.**\n\nWir alle, Ungelehrte wie Gelehrte, schreiben allerorten Gedichte.\n\nHoraz ( _Epistulae_ 2,1,116 f.) erkl\u00e4rt, da\u00df in praktischen Berufen der Schuster bei seinen Leisten bleibe (\u00bbTractant fabrilia fabri\u00ab \u2013 \u00bbDie Handwerker gehen mit Werkzeug um\u00ab), doch zu dichten ma\u00dfe sich jeder an.\n\n**scrupulum alicui inicere**\n\njemandem ein Steinchen hineinwerfen\n\n(d. h.: jemandem einen Floh ins Ohr setzen)\n\nNach Terenz, _Adelphoe_ 228.\n\n**sectores collorum**\n\nHalsabschneider\n\nCicero, _Pro S. Roscio Amerino_ 80.\n\n**Secunda non habent umquam modum.**\n\nGl\u00fcck hat niemals ein Ma\u00df.\n\nSeneca, _Oedipus_ 694.\n\n**Secundas fortunas decent superbiae.**\n\nStolz ziemt sich f\u00fcr das Gl\u00fcck (d. h.: f\u00fcr die, denen das Gl\u00fcck hold ist).\n\nPlautus, _Stichus_ 300.\n\n**secundum naturam vivere**\n\nmit der Natur im Einklang leben\n\n(d. h.: naturgem\u00e4\u00df leben)\n\n(z. B.) Cicero, _De finibus_ 2,34. \u2013 Eine solche Lebensf\u00fchrung galt der Alten Akademie und dem Peripatos als h\u00f6chstes Gut. Vgl. **convenienter naturae vivere**.\n\n**Sed dux malorum femina, haec scelerum artifex.**\n\nAnf\u00fchrerin bei den \u00dcbeltaten aber ist die Frau; in Verbrechen ist sie K\u00fcnstlerin.\n\nSeneca, _Phaedra_ 559.\n\n**Sed fugit interea, fugit irreparabile tempus.**\n\nDoch es entflieht indes, es entflieht die unwiederbringliche Zeit.\n\nVergil, _Georgica_ 3,284.\n\n**Sed nimium properas, et adhuc tua messis in herba est.**\n\nAber du bist \u00fcbereilig, und deine Ernte ist noch auf dem Halm.\n\n(vgl.: \u00fcber ungelegte Eier sprechen)\n\nOvid, _Heroides_ 17,263.\n\n**Sed non sum credulus illis.**\n\nAber ich glaube ihnen nicht so leicht.\n\nVergil, _Bucolica_ 9,34.\n\n**Sed nunc non erat his locus.**\n\nDoch hier war nicht der geeignete Ort daf\u00fcr.\n\nHoraz, _Ars poetica_ 19.\n\n**Sed omnes una manet nox \/ et calcanda semel via leti.**\n\nDoch alle erwartet die eine Nacht und der nur einmal zu betretende Weg des Todes.\n\nHoraz, _Carmina_ 1,28,15 f.\n\n**sed piger ad poenas princeps, ad praemia velox**\n\nein F\u00fcrst aber, der langsam beim Strafen und schnell beim Belohnen ist\n\nDer im Exil befindliche Ovid ( _Epistulae ex Ponto_ 1,2,121) bezeichnet so Augustus, bei dem Ovids Freund Fabius Maximus ein gutes Wort f\u00fcr den verbannten Dichter einlegen soll.\n\n**Sed taciti fecere tamen convicia vultus.**\n\nAber die schweigsamen Mienen machten dennoch Vorw\u00fcrfe.\n\n(vgl.: beredtes Schweigen)\n\nOvid, _Amores_ 1,7,21.\n\n**Se ipse amans sine rivali.**\n\nWer sich selbst liebt, hat keinen Rivalen.\n\nNach Cicero, _Ad Quintum fratrem_ 3,6,4; vgl. Horaz, _Ars poetica_ 444.\n\n**Semel emissum volat irrevocabile verbum** siehe **Et semel emissum...**\n\n**Semen est sanguis Christianorum.**\n\nEin Same ist das Blut der Christen.\n\nTertullian ( _Apologeticum_ 50,13) erkl\u00e4rt, da\u00df die Christen gerade wegen der M\u00e4rtyrer neue Anh\u00e4nger bek\u00e4men.\n\n**semper aliquid haeret** siehe **Audacter calumniare...**\n\n**Semper avarus eget.**\n\nEin Habgieriger leidet immer Mangel.\n\nHoraz, _Epistulae_ 1,2,56.\n\n**Semper ego auditor tantum?**\n\nSoll ich immer nur Zuh\u00f6rer sein?\n\nJuvenal ( _Saturae_ 1,1) will sich mit seinem Werk endlich f\u00fcr die Unmengen schlechter Literatur r\u00e4chen, denen er bislang als Zuh\u00f6rer ausgesetzt war.\n\n**Semper homo bonus tiro est.**\n\nEin guter Mensch ist immer ein Anf\u00e4nger.\n\nMartial, _Epigrammata_ 12,51,2.\n\n**Semper honos nomenque tuum laudesque manebunt.**\n\nImmer werden deine Ehre, dein Name und dein Ruhm bleiben.\n\nVergil, _Aeneis_ 1,609. \u2013 Aeneas preist Dido, die ihn freundlich aufgenommen hat.\n\n**semper idem**\n\nimmer derselbe\n\n(z. B.) Cicero, _Tusculanae disputationes_ 3,31. Er zitiert Xanthippe, die berichtet, ihr Mann Sokrates habe immer dieselbe Miene zur Schau getragen. Das Motto der K\u00f6nigin Elisabeth I. von England war \u00bbSemper eadem\u00ab (\u00bbImmer dieselbe\u00ab).\n\n**Semper inops, quicumque cupit.**\n\nImmer arm ist, wer begehrt.\n\nClaudian, _In Rufinum_ 1,200.\n\n**Semper nocuit differre paratis** siehe **Tolle moras...**\n\n**Semper pauper eris, si pauper es, Aemiliane;  \nDantur opes nulli nunc nisi divitibus.**\n\nImmer wirst du arm sein, wenn du arm bist, Aemilianus; heutzutage werden Reicht\u00fcmer nur Reichen gegeben.\n\nMartial, _Epigrammata_ 5,81.\n\n**Semper sursum!**\n\nImmer aufw\u00e4rts!\n\nNach Platon, _Staat_ 10,621c.\n\n**Semper tibi proximus esto.**\n\nSei dir selbst immer der N\u00e4chste.\n\n_Disticha Catonis_ 1,40,2.\n\n**Senectus enim insanabilis morbus est.**\n\nDas Alter ist n\u00e4mlich eine unheilbare Krankheit.\n\nSeneca, _Epistulae morales_ 108,28.\n\n**Senectus ipsa est morbus.**\n\nDas Greisenalter an sich ist eine Krankheit.\n\nTerenz, _Phormio_ 575.\n\n**Senescimus ; effugit aetas: \/ Utere rene tuo!**\n\nWir werden alt, die Zeit flieht dahin: N\u00fctze deine Lenden!\n\nAusonius ( _Epigrammata_ 24,1 f.) wendet sich hier an die alternde Galla, die er einst mit den obigen Worten vor ihrer Keuschheit warnte. Nun wird er noch eindringlicher: \u00bbNon revocare potes, qui periere dies.\u00ab (\u00bbMan kann nicht die Tage zur\u00fcckrufen, die vergangen sind.\u00ab) Obwohl Gallas Sch\u00f6nheit bereits verbl\u00fcht ist, wirbt er weiter um sie: \u00bbDa, fruar, et si non quod volo, quod volui.\u00ab (\u00bbGib [dich mir hin]; ich werde genie\u00dfen \u2013 wenn auch nicht das, was ich will, so doch, was ich einst wollte.\u00ab)\n\n**Sentit amans sua damna fere.**\n\nDer Liebende ist sich in der Regel seines eigenen Ruins bewu\u00dft.\n\nOvid, _Tristia_ 4,1,33.\n\n**Sequitur superbos ultor a tergo deus.**\n\nDie Hochm\u00fctigen verfolgt im R\u00fccken ein r\u00e4chender Gott.\n\n(Vgl.: Hochmut kommt vor dem Fall.)\n\nSeneca, _Hercules furens_ 385.\n\n**Sequitur vara vibiam.**\n\nDas Querholz folgt dem Balken.\n\n(D. h.: Ein \u00dcbel folgt dem anderen; beide \u00dcbel passen perfekt ineinander wie zwei Balken.)\n\nAusonius ( _Technopaegnion_ 4) bezeichnet diese Wendung als altes Sprichwort.\n\n**Sera parsimonia in fundo est.**\n\nAm Grund kommt die Sparsamkeit zu sp\u00e4t.\n\nSeneca ( _Epistulae morales_ 1,5, nach Hesiod, _Werke und Tage_ 368 f.) bezeichnet diese Worte als altes Sprichwort. Gemeint ist: Man darf nicht erst zu sparen anfangen, wenn man bereits auf dem Boden eines Gef\u00e4\u00dfes angelangt ist; dort findet sich nur noch der Bodensatz.\n\n**Sera tamen tacitis poena venit pedibus.**\n\nSp\u00e4t kommt die Strafe auf leisen Sohlen (doch sie kommt).\n\nTibull, _Elegiae_ 1,9,4.\n\n**Seris venit usus ab annis.**\n\nMit den sp\u00e4ten Jahren kommt die Erfahrung.\n\n(Vgl.: Erfahren kommt mit den Jahren.)\n\nOvid, _Metamorphoses_ 6,29.\n\n**Serit \/ arbores, quae alteri saeclo prosint.**\n\nEr pflanzt B\u00e4ume, die in sp\u00e4teren Jahren n\u00fctzen werden.\n\nCicero ( _De senectute_ 24) zitiert Statius, _Synephebi_ Fragment 200 Warmington.\n\n**Serius aut citius sedem properamus in unam.**\n\nLangsamer oder schneller eilen wir alle zu dem einen Ruhesitz (d. h.: in den Tod).\n\nOvid, _Metamorphoses_ 10,33.\n\n**Sermo datur cunctis, animi sapientia paucis.**\n\nSprache wird allen gegeben, Weisheit des Herzens wenigen.\n\n_Disticha Catonis_ 1,10,2.\n\n**Sero clipeum post vulnera sumo.**\n\nZu sp\u00e4t ergreife ich, schon verwundet, den Schild.\n\nOvid, _Tristia_ 1,3,35.\n\n**Sero recusat ferre, quod subiit, iugum.**\n\nZu sp\u00e4t weigert er sich, das Joch zu tragen, unter das er (bereits) getreten ist.\n\nSeneca, _Phaedra_ 135.\n\n**Sero sapiunt.**\n\nSp\u00e4t kommen sie zu Verstand.\n\nFestus (Fragment 510,11 Lindsay) berichtet von diesem trojanischen Sprichwort aus der verlorenen Trag\u00f6die _Equus Troianus_ , deren Urheber Naevius oder Livius Andronicus ist (vgl. Warmington Bd. 2, S. 623). Zitiert bei Cicero, _Ad familiares_ 7,16,1. \u2013 Gemeint sind die Trojaner, die zu sp\u00e4t beschlossen, Helena zur\u00fcckzugeben.\n\n**Sero te amavi , pulchritudo tam antiqua et tam nova, sero te amavi.**\n\nSp\u00e4t hab' ich dich geliebt, Sch\u00f6nheit, so alt und doch so neu, sp\u00e4t hab' ich dich geliebt!\n\nAugustinus ( _Confessiones_ 10,27,38) \u00fcber seine Bekehrung zum christlichen Glauben.\n\n**Serpet hoc malum, mihi credite, longius quam putatis.**\n\nDieses \u00dcbel, glaubt mir, wird schleichend um sich greifen \u2013 weiter, als ihr denkt.\n\nCicero, _Pro C. Rabirio Postumo_ 15.\n\n**Serum est cavendi tempus in mediis malis.**\n\nZu sp\u00e4t ist der Zeitpunkt, wenn man sich (erst) mitten in den \u00dcbeln vorsieht.\n\nSeneca, _Thyestes_ 487.\n\n**Serva me, servabo te.**\n\nRette mich, ich werde dich retten.\n\n(Vgl.: Eine Hand w\u00e4scht die andere.)\n\nPetron, _Satyrica_ 44,3.\n\n**Servare cives maior est virtus patriae patri.**\n\nSeine B\u00fcrger zu retten ist ein gr\u00f6\u00dferes Verdienst f\u00fcr einen Vater des Vaterlandes.\n\nSeneca ( _Octavia_ 444) l\u00e4\u00dft hier die Figur Seneca gegen\u00fcber Nero die Meinung vertreten, da\u00df es f\u00fcr einen Landesvater eine gr\u00f6\u00dfere Tat sei, eigene B\u00fcrger zu retten, als Feinde zu zerst\u00f6ren.\n\n**Servat multos fortuna nocentes \/ Et tantum miseris irasci numina possunt.**\n\nDas Schicksal rettet viele Schuldige, und die G\u00f6tter k\u00f6nnen (bisweilen) nur den Ungl\u00fccklichen z\u00fcrnen.\n\nLukan, _De bello civili_ 3,448 f.\n\n**Serviet aeternum, qui parvo nesciet uti.**\n\nEwig Sklave sein wird, wer nicht verstehen wird, von wenigem zu leben.\n\nHoraz, _Epistulae_ 1,10,41.\n\n**Sese omnes amant.**\n\nAlle lieben sich selbst.\n\nPlautus, _Captivi_ 477.\n\n**sesquipedalia verba**\n\nsechsf\u00fc\u00dfige (d. h.: ellenlange) W\u00f6rter\n\nHoraz, _Ars poetica_ 97.\n\n**Si alteram talem victoriam reportavero, mea erit pernicies.**\n\nWenn ich noch solch einen Sieg erringe, wird er mein Untergang sein.\n\nNach Plutarch ( _Moralia_ 184c) Worte des Pyrrhus nach der Schlacht von Heraclia 280 v. Chr. oder nach der Schlacht von Ausculum im darauffolgenden Jahr. Vgl. dt. \u00bbPyrrhussieg\u00ab.\n\n**Si bene te novi...**\n\nWenn ich dich recht kenne...\n\nHoraz ( _Epistulae_ 1,18,1) beginnt so einen an Lollius gerichteten Brief.\n\n**Sibi famam diminuit, qui se indignis comparat.**\n\nSeinen guten Ruf verringert, wer sich mit Unw\u00fcrdigen vergleicht.\n\nPhaedrus, _Fabulae_ Appendix Zander 10,11.\n\n**Sibi quisque peccat.**\n\nEin jeder s\u00fcndigt f\u00fcr sich.\n\n(D. h.: Jeder mu\u00df selbst daf\u00fcr geradestehen.)\n\nPetron, _Satyrica_ 45,10.\n\n**Sibi quisque ruri metit.**\n\nJeder erntet f\u00fcr sich auf seinem Landgut.\n\nPlautus, _Mostellaria_ 799.\n\n**Si cadere necesse est, occurrendum discrimini.**\n\nWenn man (schon) fallen mu\u00df, mu\u00df man der Gefahr entgegenlaufen.\n\nNach Tacitus, _Historiae_ 1,33,2.\n\n**Sic erat in fatis.**\n\nSo stand es im Buch des Schicksals.\n\nOvid, _Fasti_ 1,481.\n\n**Sic itur ad astra.**\n\nSo steigt man zu den Sternen.\n\nVergil, _Aeneis_ 9,641; aufgegriffen bei Seneca, _Epistulae morales_ 48,11 und 73,15 sowie **Non est ad astra mollis e terris via**.\n\n**Sic me servavit Apollo.**\n\nSo hat mich Apoll gerettet.\n\nHoraz ( _Sermones_ 1,9,78) in Anlehnung an Lucilius ( _Saturae_ Fragment 267 Warmington), der wiederum Homer ( _Ilias_ 20,443) zitiert. \u2013 Apoll, Gott der Dichtkunst, rettet hier seinen Sch\u00fctzling Horaz vor einem geschw\u00e4tzigen Schmeichler.\n\n**Sic saepe ingenia calamitate intercidunt.**\n\nSo gehen oft gro\u00dfe Talente durch Ungl\u00fcck zugrunde.\n\nPhaedrus, _Fabulae_ Appendix Perottina 14,7.\n\n**Sic transit gloria mundi.**\n\nSo vergeht die Herrlichkeit der Welt.\n\nPatricius, _Ritus ecclesiastici SS. Romanae ecclesiae_ 1,2,3. \u2013 Wenn der neugew\u00e4hlte Papst aus der Kapelle des Heiligen Gregor heraustritt, wird dreimal unter Ausruf dieser Worte ein B\u00fcndel Werg verbrannt. Vgl. Matth\u00e4us 24,35, und Thomas a Kempis, _Imitatio Christi_ 1,3,30.\n\n**Sic vive cum hominibus, tamquam deus videat.**\n\nLebe so mit deinen Mitmenschen, als ob der Gott es s\u00e4he.\n\nSeneca, _Epistulae morales_ 10,5.\n\n**Sic vos, non vobis** siehe **Hos ego versiculos feci...**\n\nSi deus pro nobis, quis contra nos?\n\nWenn Gott f\u00fcr uns ist, wer kann wider uns sein?\n\nR\u00f6merbrief 8,31.\n\n**si digito quem attigisset...**\n\nwenn er ihn (auch nur) mit dem Finger ber\u00fchrt h\u00e4tte...\n\n(erg\u00e4nze: h\u00e4tte ihn das schon Kopf und Kragen gekostet)\n\nCicero, _Tusculanae disputationes_ 5,55.\n\n**Si foret in terris, rideret Democritus.**\n\nWenn Demokrit (noch) auf der Erde w\u00e4re, w\u00fcrde er lachen.\n\n(D. h.: Er h\u00e4tte etwas zu lachen.)\n\nHoraz, _Epistulae_ 2,1,194. \u2013 Demokrit von Abdera trug den Beinamen \u00bbder lachende Philosoph\u00ab, da er angeblich immer \u00fcber die Torheiten der Menschen lachen mu\u00dfte. Ihm wird Heraklit, der \u00bbweinende Philosoph\u00ab, gegen\u00fcbergestellt.\n\n**Si fractus inlabatur orbis, \/ impavidum ferient ruinae.**\n\nWenn zerborsten die Welt einst\u00fcrzte, tr\u00e4fen die Tr\u00fcmmer einen Unerschrockenen.\n\nHoraz ( _Carmina_ 3,3,7 f.) glorifiziert so die R\u00f6mer, die auch in \u00e4u\u00dferster Not noch ihren Mut bewahrten.\n\n**signatus septem sigillis**\n\ndas mit sieben Siegeln versiegelte (Buch)\n\nNach Apokalypse 5,1.\n\n**Si iudicas, cognosce; si regnas, iube!**\n\nWenn du urteilst, untersuche; wenn du herrschst, befiehl!\n\nSeneca, _Medea_ 194.\n\n**Silent enim leges inter arma.**\n\nInmitten der Waffen schweigen n\u00e4mlich die Gesetze.\n\nCicero, _Pro T. Annio Milone_ 11; vgl. Quintilian, _Institutio oratoria_ 5,14,17 und **Leges bello siluere coactae.**\n\n**Si libet, licet.**\n\nWenn es beliebt, ist es erlaubt.\n\n(Vgl.: Erlaubt ist, was gef\u00e4llt.)\n\nDer _Historia Augusta_ , \u00bbVita Antonini Caracallae\u00ab 10,2, zufolge soll Julia, die Stiefmutter und sp\u00e4tere Frau Caracallas, ihn daran erinnert haben, da\u00df er der Kaiser sei und selbst die Gesetze schaffe. Daher sei ihm, was immer er w\u00fcnsche, erlaubt. Vgl. **Memento omnia...**\n\n**Simia quam similis, turpissima bestia, nobis!**\n\nWie \u00e4hnlich ist uns der Affe, dieses \u00e4u\u00dferst scheu\u00dfliche Tier!\n\nEnnius (Fragment Sat. Ex. libr. inc. 23 Warmington), zitiert bei Cicero, _De natura deorum_ 1,97.\n\n**Si mihi perget, quae vult, dicere, ea, quae non vult, audiet.**\n\nFalls er mir weiter sagt, was er will, wird er (zu) h\u00f6ren (bekommen), was er nicht (h\u00f6ren) will.\n\nTerenz, _Andria_ 920.\n\n**Similem habent lactucam labra comedente asino carduos.**\n\nSeine Lippen halten Lattich, wie wenn ein Esel Disteln fri\u00dft.\n\nLucilius (Fragment catal. Warmington, Bd. 1, S. 422) zufolge soll Crassus Agelastus (\u00bbder Nicht-Lachende\u00ab) nur einmal in seinem Leben gelacht haben, n\u00e4mlich als er dieses Sprichwort h\u00f6rte.\n\n**simplex munditiis**\n\neinfach, was den Schmuck betrifft\n\n(d. h.: von schlichter Eleganz)\n\nHoraz, _Carmina_ 1,5,5.\n\n**Simulare certe est hominis.**\n\nT\u00e4uschung ist sicherlich typisch f\u00fcr Menschen.\n\nTerenz, _Adelphoe_ 734.\n\n**Simul flare sorbereque haud factu facile est.**\n\nBlasen und zugleich schlucken ist nicht leicht zu bewerkstelligen.\n\n(vgl.: auf zwei Hochzeiten gleichzeitig tanzen)\n\nBei Plautus ( _Mostellaria_ 791) wird diese Wendung als sprichw\u00f6rtlich bezeichnet.\n\n**Sincerum est nisi vas, quodcumque infundis, acescit.**\n\nWenn das Gef\u00e4\u00df nicht sauber ist, wird sauer, was auch immer du hineingie\u00dft.\n\nHoraz ( _Epistulae_ 1,2,54) erkl\u00e4rt mittels dieses Gleichnisses, da\u00df bei innerer Unausgeglichenheit auch ein gro\u00dfes Verm\u00f6gen keine Zufriedenheit schaffe.\n\n**Sine amicitia vita est nulla.**\n\nOhne Freundschaft gibt es kein Leben.\n\nNach Cicero, _De amicitia_ 86.\n\n**Sine Cerere et Libero friget Venus.**\n\nOhne Ceres und Liber friert die Venus.\n\nTerenz ( _Eunuchus_ 732) bezeichnet die Wendung als Sprichwort: zitiert bei Cicero, _De natura deorum_ 2,60. \u2013 Ceres steht f\u00fcr Ackerbau und Nahrung, Liber f\u00fcr Wein.\n\n**sine ira et studio**\n\nohne Zorn und Eifer\n\n(d. h.: objektiv)\n\nTacitus, _Annales_ 1,1,3; vgl. auch _Historiae_ 1,1, und Cicero, _Pro M. Marcello_ 29.\n\n**Sine pennis volare haud facile est.**\n\nOhne Fl\u00fcgel zu fliegen ist nicht leicht.\n\nPlautus ( _Poenulus_ 871) l\u00e4\u00dft Syncerastus verdeutlichen, da\u00df Anstrengung nichts bewirkt, wenn die n\u00f6tigen Mittel fehlen.\n\n**Sine philosophia nemo intrepide potest vivere.**\n\nOhne die Philosophie kann niemand unverzagt leben.\n\nNach Seneca, _Epistulae morales_ 16,3.\n\n**sine sudore et sanguine**\n\nohne Schwei\u00df und Blut (zu vergie\u00dfen)\n\nEnnius, _Achilles_ Fragment 22 Warmington, zitiert bei Cicero, _De officiis_ 1,61.\n\n**Si nihil infesti durus vidisset Ulixes,**\n\nPenelope felix, sed sine laude foret.\n\nWenn der leidgepr\u00fcfte Odysseus auf keine Gefahren gesto\u00dfen w\u00e4re, w\u00e4re Penelope zwar gl\u00fccklich gewesen, doch ohne Ruhm.\n\nOvid, _Tristia_ 5,5 (6),51 f. \u2013 Odysseus war wegen des Trojanischen Krieges und der anschlie\u00dfenden Irrfahrt zwanzig Jahre lang von seiner Gattin Penelope getrennt.\n\n**Si non proficient artes, veniemus ad arma.**\n\nWenn die K\u00fcnste nichts n\u00fctzen, werden wir zu den Waffen \u00fcbergehen.\n\nOvid, _Heroides_ 20,47. \u2013 Acontius will zur Not Cydippes Liebe mit Gewalt erlangen.\n\n**Sint Maecenates, non deerunt, Flacce, Marones.**\n\nWenn es Leute wie Maecenas gibt, dann, Flaccus, werden auch Leute wie Maro nicht fehlen.\n\nMartial, _Epigrammata_ 8,55,5. \u2013 Mit Maro ist Publius Vergilius Maro (Vergil) gemeint; Maecenas war sein heute sprichw\u00f6rtlich gewordener G\u00f6nner, sein M\u00e4zen.\n\n**Sint, ut sunt, aut non sint!**\n\nSie sollen sein, wie sie sind, oder nicht sein!\n\nLorenzo Ricci, Ordensgeneral der Jesuiten, soll Papst Clemens XIII. gegen\u00fcber diese Worte bez\u00fcglich einer m\u00f6glichen \u00c4nderung der Ordensverfassung ge\u00e4u\u00dfert haben.\n\n**Si paenitet, haud nocet error.**\n\nWenn man reuig ist, schadet die Verfehlung nicht.\n\nPrudenz, _Psychomachia_ 396. Vgl. **Geminat peccatum, quem...** und **Quem paenitet pecasse...**\n\n**... si parva licet componere magnis.**\n\n... wenn es erlaubt ist, Kleines mit Gro\u00dfem zu vergleichen.\n\nVergil ( _Georgica_ 4,176) vergleicht die Arbeit der Kyklopen mit der der Bienen. \u2013 Vgl. Vergil, _Bucolica_ 1,23, und Ovid, _Metamorphoses_ 5,416 f.\n\n**Si qua venit sero, magna ruina venit.**\n\nKommt Unheil sp\u00e4t, so kommt es vernichtend.\n\nProperz, _Elegiae_ 2,25,28.\n\n**Si qua voles apte nubere, nube pari.**\n\nWenn du passend (d. h.: gl\u00fccklich) heiraten willst, dann heirate einen Ebenb\u00fcrtigen.\n\nOvid, _Heroides_ 9,32.\n\n**Si quid novisti rectius istis, \/ candidus inperti; si nil, his utere mecum.**\n\nWenn du etwas Besseres wei\u00dft als jene (Leitlinien), dann teile du Redlicher sie (mir) mit; wenn nicht, so folge diesen (bisherigen Leitlinien) zusammen mit mir.\n\nHoraz, _Epistulae_ 1,6,67 f.\n\n**Si quid te vetat bene vivere, bene mori non vetat.**\n\nWenn dir etwas verbietet, gut zu leben, so verbietet es dir (doch) nicht, gut zu sterben.\n\nSeneca, _Epistulae morales_ 17,5.\n\n**Si rota defuerit, tu pede carpe viam.**\n\nWenn dir ein Rad fehlt, mache den Weg zu Fu\u00df.\n\nOvid ( _Ars amatoria_ 2,230) weist darauf hin, da\u00df man eine Geliebte unter keinen Umst\u00e4nden warten lassen d\u00fcrfe.\n\n**Si sapis, \/ quod scis, nescis.**\n\nWenn du weise bist, wei\u00dft du nicht, was du wei\u00dft.\n\n(D. h.: Wer weise ist, stelle sich in heiklen Angelegenheiten besser unwissend.)\n\nTerenz, _Eunuchus_ 721 f. Vgl. **Tu nescies...**\n\n**Siste furibundum impetum!**\n\nGebiete dem w\u00fctenden Angriff Einhalt!\n\nSeneca, _Phaedra_ 263.\n\n**Si tacuisses, philosophus mansisses.**\n\nWenn du geschwiegen h\u00e4ttest, w\u00e4rest du ein Philosoph geblieben.\n\nNach Boethius, _Consolatio philosophiae_ 2,7. Ein angeblicher Philosoph war einst von einem Gespr\u00e4chsgegner so aus der Fassung gebracht worden, da\u00df er sich eben durch seinen mangelnden Gleichmut als falscher Philosoph entpuppte. Vgl. Hiob 13,5 und Spr\u00fcche 17,28.\n\n**Sit caeca futuri \/ mens hominum fati: liceat sperare timenti.**\n\nDer menschliche Geist mag f\u00fcr das kommende Schicksal blind sein: Es sei ihm wenigstens erlaubt zu hoffen, wenn er schon f\u00fcrchten mu\u00df.\n\nLukan, _De bello civili_ 2,14 f.\n\n**Sit procul omne nefas; ut ameris, amabilis esto!**\n\nFern sei alles Ungeh\u00f6rige; damit du geliebt wirst, sei liebensw\u00fcrdig!\n\nOvid, _Ars amatoria_ 2,107. Vgl. **Si vis amari...** und **Ut ameris...**\n\n**Sit tibi credibilis sermo consuetaque verba!**\n\nDeine Rede sei glaubw\u00fcrdig, deine Worte die \u00fcblichen!\n\nOvid ( _Ars amatoria_ 1,467) r\u00e4t jungen M\u00e4nnern dazu, in Liebesbriefen eine schmeichelnde, doch nicht zu ausgefallene Sprache zu verwenden.\n\n**Sit tua cura sequi, me duce tutus eris.**\n\nDeine Sorge sei (nur) zu folgen; unter meiner F\u00fchrung wirst du sicher sein.\n\nOvid, _Ars amatoria_ 2,58. \u2013 Daedalus ermahnt so seinen Sohn Icarus, ihm zu folgen und weder der Sonne noch dem Meer zu nahe zu kommen.\n\n**Si vir es, i!**\n\nWenn du ein Mann bist, geh!\n\nOvid, _Fasti_ 6,594. \u2013 Tullia fordert so ihren Mann Superbus auf, ihren Vater zu t\u00f6ten und die K\u00f6nigsherrschaft an sich zu rei\u00dfen.\n\n**Si vis amari, ama!**\n\nWenn du geliebt werden willst, liebe!\n\nSeneca ( _Epistulae morales_ 9,6) zitiert hier Hekaton (Fragment 27 Fowler). Vgl. **Sit procul...** und **Ut ameris...**\n\n**Si vis me flere, dolendum est \/ primum ipsi tibi.**\n\nWenn du willst, da\u00df ich weine, mu\u00dft du selbst zuerst Schmerz empfinden.\n\nHoraz ( _Ars poetica_ 102 f.) erl\u00e4utert, wahre Dichtung setze voraus, da\u00df der Dichter das Dargestellte selbst empfinde.\n\n**Si vis pacem, para bellum!** siehe **Qui desiderat pacem...**\n\n**societas leonina** siehe **Numquam est fidelis...**\n\n**Societatis vinculum est ratio et oratio.**\n\nDas Band der Gesellschaft sind Vernunft und Sprache.\n\nNach Cicero, _De officiis_ 1,50.\n\n**Sociorum olla malle fervet.**\n\nDer Topf von Freunden kocht schlecht.\n\n(D. h.: Da ist nichts zu holen.)\n\nPetron, _Satyrica_ 38,13. \u2013 Nach einem griechischen Sprichwort oft zitiert als \u00bbFervet olla, vivit amicitia\u00ab (\u00bbSiedet der Topf, so bl\u00fcht die Freundschaft\u00ab).\n\n**socordia atque desidia bonum otium conterere**\n\nin Sorglosigkeit und Unt\u00e4tigkeit die gute Mu\u00dfe vergeuden\n\nSallust, _Bellum Catilinarium_ 4,1.\n\n**Sola deos aequat clementia nobis.**\n\nNur Milde macht uns den G\u00f6ttern ebenb\u00fcrtig.\n\nClaudian, _De quarto consulatu honorii Augusti_ 277.\n\n**sola fide**\n\nallein durch den Glauben\n\nNach R\u00f6merbrief 1,3,28.\n\n**Solamen miseris socios habuisse malorum.**\n\nEin Trost f\u00fcr die Elenden ist es, im Ungl\u00fcck Gef\u00e4hrten zu haben.\n\nSpinoza, _Ethik_ 4,57, nach Aesop, _Fabulae_ 191 Chambry.\n\n**Solent mendaces luere poenas maleficii.**\n\nL\u00fcgner pflegen f\u00fcr ihre Missetat Strafe zu b\u00fc\u00dfen.\n\nPhaedrus, _Fabulae_ 1,17,1.\n\n**Solet a despectis par referri gratia.**\n\nOft wird (uns) von denen, die wir verachtet haben, gleicher Dank gezollt.\n\n(Vgl.: Wie es in den Wald hineinschallt, so schallt es hinaus.)\n\nPhaedrus, _Fabulae_ 3,2,1.\n\n**Soli igitur hoc contingit sapienti, ut nihil faciat invitus.**\n\nDem Weisen allein also gelingt es, da\u00df er nichts gegen seinen Willen tut.\n\nCicero, _Paradoxa Stoicorum_ 34.\n\n**Solitudinem faciunt , pacem apellant.**\n\nSie schaffen eine W\u00fcste und nennen das Frieden.\n\nTacitus, _Agricola_ 30,4. \u2013 Der Barbar Calgacus \u00e4u\u00dfert sich so \u00fcber die R\u00f6mer.\n\n**Sollicitae mentes speque metuque pavent.**\n\nAufgeregte Gem\u00fcter zittern vor Hoffnung und vor Furcht.\n\nOvid, _Fasti_ 3,362.\n\n**Sol omnibus lucet.**\n\nDie Sonne leuchtet f\u00fcr alle.\n\nPetron, _Satyrica_ 100,1. \u2013 Encolpius versucht so, seine Eifersucht zu d\u00e4mpfen: Alle wirklichen G\u00fcter auf der Erde geh\u00f6rten der Allgemeinheit; daher m\u00fcsse man auch die eigene Geliebte mit anderen teilen.\n\n**Solve senescentem mature sanus equum.**\n\nSei weise und l\u00f6se das alternde Pferd rechtzeitig (von den Z\u00fcgeln).\n\n(D. h.: H\u00f6re zur rechten Zeit auf.)\n\nHoraz, _Epistulae_ 1,1,8.\n\n**Somnus est imago mortis.**\n\nDer Schlaf ist ein Abbild des Todes.\n\nNach Cicero, _Tusculanae disputationes_ 1,92.\n\n**Sorex suo perit indicio.**\n\nDie Spitzmaus kommt dadurch um, da\u00df sie sich selbst anzeigt (indem sie quiekt).\n\nNach Terenz, _Eunuchus_ 1024. Vgl. Ovid, _Heroides_ 11,72.\n\n**Sors tua mortalis: non est mortale, quod optas.**\n\nDein Los ist das eines Sterblichen: Was du begehrst, ist (jedoch) nichts Sterbliches.\n\nOvid, _Metamorphoses_ 2,56.\n\n**spargere voces \/ in vulgum ambiguas**\n\nzweideutige Reden im Volke verbreiten\n\nVergil, _Aeneis_ 2,98 f.\n\n**Spatio brevi \/ spem longam reseces!**\n\nDa die (Lebens)zeit kurz ist, begrenze deine lange Hoffnung!\n\nHoraz, _Carmina_ 1,11,6 f.\n\n**Spectatum veniunt, veniunt, spectentur ut ipsae.**\n\nSie kommen zum Zuschauen, sie kommen (aber auch), um selbst gesehen zu werden.\n\nOvid, _Ars amatoria_ 1,99; vgl. Aelian ( _Bunte Geschichten_ 7,10), der berichtet, Sokrates habe mit entsprechenden Worten seine Frau Xanthippe zurechtgewiesen, die einer Prozession nicht in ihrem alten Mantel zusehen wollte.\n\n**spemque metumque inter dubii**\n\nschwankend zwischen Hoffnung und Furcht\n\nVergil, _Aeneis_ 1,218.\n\n**Spem retine! Spes una hominem nec morte relinquit.**\n\nBewahre die Hoffnung! Die Hoffnung allein verl\u00e4\u00dft den Menschen auch nicht im Tode.\n\n_Disticha Catonis_ 2,25,2.\n\n**Spem teneo, salutem amisi.**\n\nIch halte (noch) an der Hoffnung fest; mein Heil habe ich (bereits) verloren.\n\nPlautus, _Mercator_ 592.\n\n**Sperat quidem animus, quo eveniat, dis in manu est.**\n\nDer (menschliche) Geist hegt zwar Hoffnungen, worauf die Sache aber hinausl\u00e4uft, liegt in der Hand der G\u00f6tter.\n\n(Vgl.: Der Mensch denkt, Gott lenkt.)\n\nPlautus, _Bacchides_ 144.\n\n**Sperne voluptates, nocet empta dolore voluptas.**\n\nVerachte die Vergn\u00fcgungen; es schadet das mit Schmerz erkaufte Vergn\u00fcgen.\n\nHoraz, _Epistulae_ 1,2,55.\n\n**Spes est, quae pascat amorem.**\n\nDie Hoffnung ist es, die die Liebe n\u00e4hrt.\n\nOvid, _Metamorphoses_ 9,749.\n\n**Spiritus magnos fuga!**\n\nMeide den Gro\u00dfmut!\n\nSeneca, _Troades_ 505.\n\n**sponte asceam cruribus suis illidere**\n\nsich absichtlich eine Axt ins Bein schlagen\n\nNach Apuleius, _Metamorphoses_ 3,22,6. Vgl. **Ipse mihi asciam...**\n\n**spumans ex ore scelus, anhelans ex infimo pectore crudelitatem**\n\naus dem Mund Verbrechen hervorsch\u00e4umend, aus der tiefsten Brust Grausamkeit aushauchend (vgl.: Gift und Galle speien)\n\nAuctor ad Herennium 4,68.\n\n**stans pede in uno**\n\nauf einem Fu\u00df stehend\n\nHoraz ( _Sermones_ 1,4,10) \u00fcber den Schnelldichter Lucilius, der sich zum Dichten nicht einmal hinsetzt.\n\n**stante pede**\n\nstehenden Fu\u00dfes, sofort\n\nHWS: 43072b\n\n**Stat nulla diu mortalibus usquam \/ Fortuna titubante fides.**\n\nKeine Treue w\u00e4hrt den Sterblichen je lange, wenn das Gl\u00fcck schwankt.\n\nSilius Italicus, _Punica_ 11,3 f.\n\n**Stat sua cuique dies. Breve et inreperabile tempus \/ omnibus est vitae.**\n\nEinem jedem steht der (Todes)tag fest. Kurz und unwiederbringlich ist die Lebenszeit aller.\n\nVergil, _Aeneis_ 10,467 f.\n\n**statua taciturnius**\n\nschweigsamer als eine Statue\n\nHoraz ( _Epistulae_ 2,2,83) bezeichnet so den typischen Weisen, der Abgeschiedenheit liebt und wegen seiner Schweigsamkeit oft f\u00fcr verr\u00fcckt gehalten wird.\n\n**Stertit noctes et dies.**\n\nEr schnarcht Tag und Nacht.\n\nTerenz, _Eunuchus_ 1079.\n\n**Stillicidi casus lapidem cavat.**\n\nDer Fall des Tropfens aus der Dachrinne h\u00f6hlt den Stein.\n\nLukrez, _De rerum natura_ 1,313. Vgl. **Ferreus adsiduo consumitur...** , **Gutta cavat lapidem...** und **Percussu crebro...**\n\n**Stimulos dedit aemula virtus.**\n\nSporen verlieh die wetteifernde T\u00fcchtigkeit.\n\nLukan, _De bello civili_ 1,120,3.\n\n**Stomachor omnia.**\n\nIch rege mich \u00fcber alles auf.\n\nCicero, _Ad Atticum_ 14,21.\n\n**strenua inertia**\n\ngesch\u00e4ftiges Nichtstun\n\nNach Horaz, _Epistulae_ 1,11,28.\n\n**Studia adulescentiam alunt, senectutem oblectant.**\n\nDie Wissenschaft n\u00e4hrt die Jugend und erg\u00f6tzt das Alter.\n\nCicero, _Pro Archia poeta_ 7,16.\n\n**Stultitia est, pater, venatum ducere invitas canes.**\n\nEs ist eine Dummheit, Vater, unwillige H\u00fcndinnen zur Jagd zu f\u00fchren.\n\n(vgl.: die Hunde zur Jagd tragen)\n\nPlautus, _Stichus_ 139.\n\n**Stultitiam patiuntur opes.**\n\nReichtum kann Dummheit erdulden.\n\n(D. h.: Wer reich ist, kann es sich leisten, dumm zu sein.)\n\nHoraz, _Epistulae_ 1,18,29.\n\n**Stultorum eventus magister est** siehe **Eventus docebit**\n\n**Stultorum infinitus est numerus.**\n\nDie Zahl der Dummen ist unendlich.\n\nKohelet 1,15.\n\n**Stultorum plena sunt omnia.**\n\nDer Dummen voll ist alles.\n\nCicero, _Ad familiares_ 9,22,4.\n\n**Stultus es, qui facta infecta facere verbis postules.**\n\nDumm ist, wer glaubt, durch Worte Geschehenes ungeschehen machen zu k\u00f6nnen.\n\nPlautus, _Truculentus_ 730.\n\n**sua confessione indui et iugulari**\n\nsich im eigenen Gest\u00e4ndnis verwickeln und erdrosseln\n\nNach Cicero, _Actio secunda in C. Verrem_ 5,166.\n\n**Sua cuique deus fit dira cupido.**\n\nDie eigene schreckliche Begierde wird einem jedem zum Gott.\n\n(D. h.: Die eigene Begierde bestimmt das Leben.)\n\nVergil, _Aeneis_ 9,185.\n\n**Suae quemque fortunae maxime paenitet.**\n\nJeden verdrie\u00dft das eigene Schicksal am meisten.\n\nNach Cicero, _Ad familiares_ 6,1,1.\n\n**Suam quisque homo rem meminit.**\n\nJeder erinnert sich an seine eigenen Angelegenheiten.\n\n(D. h.: Jeder ist sich selbst der N\u00e4chste.)\n\nPlautus, _Mercator_ 1011.\n\n**Sua quemque premit terroris imago.**\n\nJeden bedr\u00e4ngt sein eigenes Schreckensbild.\n\n(D. h.: Jeder hat andere Alptr\u00e4ume.)\n\nLukan, _De bello civili_ 7,773.\n\n**Sua quisque exempla debet aequo animo pati.**\n\nEin jeder ertrage seine Vorbilder mit Gleichmut.\n\n(D. h.: Man mu\u00df erdulden, so behandelt zu werden, wie man andere zuvor behandelt hat.)\n\nPhaedrus, _Fabulae_ 1,26,12.\n\n**Suave mari magno turbantibus aequora ventis**\n\nE terra magnum alterius spectare laborem.\n\nEs ist angenehm, wenn auf hoher See die Winde das Meer aufw\u00fchlen, vom Land aus die gro\u00dfe Bedr\u00e4ngnis des anderen zu sehen.\n\nLukrez, _De rerum natura_ 2,1 f. Vgl. **naufragia ex terra intueri**.\n\n**Suavis laborum est praeteritorum memoria** siehe **Iucundi acti labores**.\n\n**Subiere minores \/ saepe casas superi.**\n\nG\u00f6tter haben schon oft geringere H\u00fctten betreten.\n\nOvid, _Metamorphoses_ 5,282 f. \u2013 Mit vermeintlicher Bescheidenheit versucht Pyreneus, der K\u00f6nig von Daulis, die Musen bei einem Unwetter in sein Haus zu locken, um sie zu vergewaltigen.\n\n**Sublimi feriam sidera vertice.**\n\nMit erhabenem Scheitel werde ich die Sterne streifen.\n\nHoraz ( _Carmina_ 1,1,36) \u00fcber seinen Dichterruhm.\n\n**Sub noctem cura recursat.**\n\nW\u00e4hrend der Nacht kehrt die Sorge zur\u00fcck.\n\nVergil, _Aeneis_ 1,662.\n\n**Sub omni lapide scorpius dormit.**\n\nUnter jedem Stein schl\u00e4ft ein Skorpion.\n\nNach Macarius Hieromonachus (Paroemiographi Graeci) 8,69.\n\n**sub reservatione Iacobea** siehe **Deo volente...**\n\n**sub specie aeternitatis**\n\nunter dem Blickwinkel der Ewigkeit\n\nSpinoza, _Ethik_ 5,29 f. \u2013 Gemeint ist die Betrachtung verg\u00e4nglicher Dinge als Erscheinungen der einzigen wahren Substanz.\n\n**Successore novo vincitur omnis amor.**\n\nJede Liebe wird von einer neuen folgenden besiegt.\n\nOvid, _Remedia amoris_ 462.\n\n**Successus ad perniciem multos devocat.**\n\nErfolg zieht viele ins Verderben hinab.\n\nPhaedrus, _Fabulae_ 3,5,1.\n\n**Successus improborum plures allicit.**\n\nDer Erfolg von Schuften lockt (noch) mehr (Schufte) an.\n\nPhaedrus, _Fabulae_ 2,3,7.\n\n**sucum ac sanguinem amittere**\n\nSaft und Blut verlieren\n\n(d. h.: keine Energie mehr haben)\n\nNach Cicero, _Ad Atticum_ 4,18,2.\n\n**sui generis**\n\nvon eigener Art\n\n(d. h.: besonders)\n\n(z. B.) Plinius d. \u00c4., _Naturalis Historia_ 23,134.\n\n**Suis fortuna cuique fingitur moribus.**\n\nDurch den eigenen Charakter bestimmt sich einem jeden das Schicksal.\n\nNach Cicero ( _Paradoxa Stoicorum_ 34), der diese Worte einem \u00bbweisen Dichter\u00ab zuschreibt. Vgl. Cornelius Nepos, _Atticus_ 11,6: \u00bbSui cuique mores fingunt fortunam hominibus.\u00ab (\u00bbDer eigene Charakter formt den Menschen das Schicksal.\u00ab)\n\n**Sum apud te primus.**\n\nIch bin bei dir der erste.\n\n(vgl.: bei jemandem die Hauptrolle spielen)\n\nTerenz, _Eunuchus_ 90.\n\n**summa summarum**\n\nalles in allem\n\nPlautus, _Truculentus_ 25.\n\n**Summisque negatum \/ stare diu.**\n\nUnd den M\u00e4chtigen ist es verwehrt, lange (in ihrer hohen Position) zu verweilen.\n\nLukan, _De bello civili_ 1,70 f.\n\n**Summum ius summa iniuria.**\n\nDas h\u00f6chste Recht ist das h\u00f6chste Unrecht.\n\n(D. h.: Man soll das Recht nicht auf die Spitze treiben.)\n\nCicero ( _De officiis_ 1,33, nach Terenz, _Heauton timorumenos_ 796: **Ius summum...** ) bezeichnet die Wendung als altes Sprichwort. Vgl. z. B. Columella, _De re rustica_ 1,7,2.\n\n**Sunt animis etiam sua nubila.**\n\nAuch der Verstand hat (bisweilen) seine Wolken.\n\n(D. h.: Er ist getr\u00fcbt.)\n\nPrudenz, _Hamartigenia_ 89.\n\n**sunt certi denique fines** siehe **Est modus in rebus...**\n\n**Sunt Iovis omnia plena** siehe **Ab Iove principium...**\n\n**Sunt lacrimae rerum.**\n\nEs gibt Tr\u00e4nen der Welt.\n\nVergil, _Aeneis_ 1,462. \u2013 Aeneas, der im Schutze eines Nebels unerkannt durch Karthago wandert, sieht an einem Tempel eine Darstellung des Trojanischen Krieges und ist ergriffen von dem Gedanken, da\u00df menschliche Geschicke allerorten auf Mitgef\u00fchl sto\u00dfen.\n\n**Sunt pueri pueri, pueri puerilia tractant.**\n\nKinder sind Kinder; Kinder treiben Kindereien.\n\nHW: 30797b nach Korintherbriefe 1,13,11, bzw. Horaz, _Epistulae_ 2,1,116: **Scribimus indocti...**\n\n**Sunt superis sua iura.**\n\nDie G\u00f6tter haben ihre eigenen Gesetze.\n\nOvid, _Metamorphoses_ 9,500.\n\n**Sunt verba et voces.**\n\nEs gibt Worte und Stimmen (d. h.: Zuspr\u00fcche und Zauberformeln, die Abhilfe leisten k\u00f6nnen).\n\nHoraz ( _Epistulae_ 1,1,34) r\u00e4t, auch wenn man sein Ziel nicht ganz erreichen k\u00f6nne, dennoch auf alle Hilfsmittel zur\u00fcckzugreifen, um diesem Ziel n\u00e4herzukommen.\n\n**suo Marte**\n\naus eigener Kraft\n\n(z. B.) Cicero, _Orationes Philippicae_ 2,95. Vgl. **Marte nostro**.\n\n**Superflua non nocent.**\n\n\u00dcberfl\u00fcssiges schadet nicht.\n\nAugustinus, _De civitate Dei_ 4,27, und _Codex Iustinianus_ 6,23,27.\n\n**Surdas iamdudum aures pulso.**\n\nSchon lange schlage ich an taube Ohren.\n\n(vgl.: tauben Ohren predigen)\n\nCurtius Rufus, _Historia Alexandri Magni_ 9,2,30.\n\n**surdis auribus dicta**\n\ntauben Ohren Gesagtes\n\nLivius, _Ab urbe condita_ 3,70,7. Vgl. **vana surdis...**\n\n**surdo narrare fabulam**\n\neinem Tauben eine Geschichte erz\u00e4hlen\n\nNach Terenz, _Heauton timorumenos_ 222.\n\n**Sursum corda, gratias agimus.**\n\nHimmelw\u00e4rts die Herzen, wir sagen Dank.\n\nEinleitungsworte der Pr\u00e4fation der katholischen Messe. Nach den Klageliedern des Jeremia 3,41. Ebenso Patricius, _Ritus ecclesiastici SS. Romanae ecclesiae_ 1,2,2.\n\n**Suspecta ne sint, longa colloquia amputa!**\n\nK\u00fcrze die lange Rede, damit sie nicht verd\u00e4chtig wirke!\n\nSeneca, _Medea_ 530.\n\n**Sustine et abstine!**\n\nHalte durch und halte dich fern!\n\n(Vgl.: Leide und meide!)\n\nGellius ( _Noctes Atticae_ 17,19,6) zitiert hier Epiktet in griechischer Sprache. Die lateinische Version findet sich bei HW: 30937a.\n\n**suum cuique (tribuere)**\n\neinem jeden das Seine (zollen)\n\nCicero ( _De legibus_ 1,19) bezeichnet diese Worte als Grundsatz der griechischen Gesetze. Gellius ( _Noctes Atticae_ 13,24,1) berichtet von folgendem Ausspruch Catos d. \u00c4. (Fragment Inc. orat. rel. 10 Jordan): \u00bbSuum cuique per me uti atque frui licet.\u00ab (\u00bbVon mir aus sei es einem jeden erlaubt, was er hat, zu nutzen und es zu genie\u00dfen.\u00ab) Vgl. _Digesta_ 1,1,10. \u2013 Der Spruch wurde von den Nationalsozialisten, die ihn in Buchenwald anbrachten, in seiner Intention mi\u00dfbraucht.\n\n**Suum cuique pulchrum est.**\n\nJedem ist das Seine sch\u00f6n.\n\nCicero, _Tusculanae disputationes_ 5,63; vgl. auch _Ad Atticum_ 14,20,3, und **Te tua...**\n\n**Suum quemque decet.**\n\nF\u00fcr jeden ziemt sich das Seine.\n\nPlautus, _Stichus_ 693.\n\n**Suus est mos cuique genti.**\n\nJedes Volk hat seine eigene Sitte.\n\n(Vgl.: Andre L\u00e4nder, andre Sitten.)\n\nPrudenz, _Contra orationem Symmachi_ 2,89.\n\n**Suus rex reginae placet.**\n\nDer eigene K\u00f6nig gef\u00e4llt der K\u00f6nigin.\n\nPlautus, _Stichus_ 133.\nT\n\n**tabula rasa**\n\neine gegl\u00e4ttete Tafel\n\n(vgl.: ein unbeschriebenes Blatt)\n\nDie R\u00f6mer schrieben Notizen auf Wachst\u00e4felchen, die man nach Gebrauch wieder glattstreichen konnte. Den Vergleich des menschlichen Geistes mit einer solchen Tafel stellt bereits Sokrates an (Platon, _Theaetetus_ 191c). Vgl. Aristoteles, _De anima_ 3,430a 1. W\u00f6rtlich findet sich die Wendung bei Albertus Magnus ( _De anima_ 3,2,17), der so die kindliche Seele bezeichnet, da sie noch v\u00f6llig rein und unbeeinflu\u00dft sei. Ebenso Thomas von Aquin, _Summa theologiae_ 1,79,2.\n\n**Tacent: satis laudant.**\n\nSie schweigen: (Damit) loben sie gen\u00fcgend.\n\nTerenz, _Eunuchus_ 476.\n\n**Tacere multis discitur vitae malis.**\n\nSchweigen lernt man aus des Lebens zahlreichen \u00dcbeln.\n\nSeneca, _Thyestes_ 319.\n\n**Talis hominibus fuit oratio, qualis vita.**\n\nWie ihr Leben, so war der Menschen Rede.\n\nSeneca ( _Epistulae morales_ 114,1) bezeichnet die Wendung als griechisches Sprichwort. Vgl. Diogenes Laertius ( _Vitae philosophorum: Chilon_ 1,58), der \u00e4hnliche Worte Solon zuschreibt, und Cicero ( _Tusculanae disputationes_ 5,47), der Sokrates als Quelle anf\u00fchrt. Vgl. **Qualis vir...**\n\n**Tam deest avaro, quod habet, quam quod non habet.**\n\nDer Geizige leidet Mangel an dem, was er hat, wie an dem, was er nicht hat.\n\nPublilius Syrus, _Sententiae_ T 3; zitiert bei Quintilian, _Institutio oratoria_ 8,5,6.\n\n**Tamdiu discendum est, quamdiu nescias.**\n\nMan mu\u00df so lange lernen, wie man (etwas) nicht wei\u00df.\n\nSeneca ( _Epistulae morales_ 76,3) f\u00e4hrt fort, wenn man einem antiken Sprichwort Glauben schenke, m\u00fcsse man sogar so lange lernen, wie man lebe (\u00bb... quamdiu vivas\u00ab).\n\n**Tam felix utinam quam pectore candidus essem!**\n\nW\u00e4re ich doch so gl\u00fccklich, wie ich reinen Herzens bin!\n\nOvid, _Epistulae ex Ponto_ 4,14,43.\n\n**tamquam truncus stipesque**\n\nwie ein Baumstumpf und Holzklotz\n\n(d. h.: nur Staffage)\n\nCicero ( _In L. Calpurnium Pisonem_ 19) spricht hier von einem Konsul, der keineswegs einen richtigen Politiker darstelle, sondern gerade einmal f\u00e4hig sei, wie ein Pfahl dazustehen, der ein Schild mit der Aufschrift \u00bbKonsul\u00ab trage.\n\n**Tantae molis erat Romanam condere gentem.**\n\nSo gro\u00dfe M\u00fche bedeutete es, das r\u00f6mische Volk zu begr\u00fcnden.\n\nVergil ( _Aeneis_ 1,33) wirft zu Beginn seines Epos einen Blick auf das Resultat: Rom, das neue Troja.\n\n**Tantaene animis caelestibus irae?**\n\nSteckt in himmlischen Wesen solcher Zorn?\n\nVergil, _Aeneis_ 1,11.\n\n**Tantummodo incepto opus est, cetera res expediet.**\n\nEs bedarf nur eines Anfangs, das \u00fcbrige wird sich erledigen.\n\nSallust, _Bellum Catilinarium_ 20,10.\n\n**Tantum ne noceas , dum vis prodesse, videto!**\n\nSieh nur zu, da\u00df du nicht schadest, w\u00e4hrend du zu helfen w\u00fcnschst.\n\nOvid, _Tristia_ 1,1,101.\n\n**Tantum religio potuit suadere malorum.**\n\nSo viel Unheil vermochte der Aberglaube anzuraten.\n\nLukrez ( _De rerum natura_ 1,101) verteidigt die Philosophie gegen die Religion. Als Beispiel f\u00fchrt er die Opferung Iphigenies an, die zeige, zu welch unsinnigen Taten religi\u00f6ser Glaube f\u00fchren k\u00f6nne.\n\n**Tarda venit dictis difficilisque fides.**\n\nSp\u00e4t und widerwillig kam der Glaube an das Gesagte.\n\nOvid, _Fasti_ 3,350.\n\n**Tardo amico nihil est quicquam iniquius.**\n\nEs gibt nichts Ung\u00fcnstigeres als einen tr\u00e4gen Freund.\n\nPlautus, _Poenulus_ 504.\n\n**Te, Fortuna, sequor.**\n\nDir, Schicksal, folge ich.\n\nLukan ( _De bello civili_ 1,226) zufolge verk\u00fcndete Caesar nach der \u00dcberschreitung des Rubikon, er lasse nun Frieden und Gesetze hinter sich und folge dem Schicksal.\n\n**Te hominem esse memento!** siehe **Respice post te...**\n\n**Telum ira facit.**\n\nDer Zorn schafft (ihm) eine Waffe.\n\nVergil, _Aeneis_ 7,508.\n\n**Tempora labuntur, tacitisque senescimus annis**\n\nEt fugiunt freno non remorante dies.\n\nDie Zeit schwindet dahin und wir altern durch unmerkliche Jahre, und die Tage fliehen, da kein Z\u00fcgel ihnen Einhalt gebietet.\n\nOvid, _Fasti_ 6,771 f.\n\n**Tempora mutantur et nos mutamur in illis.**\n\nDie Zeiten \u00e4ndern sich, und wir \u00e4ndern uns mit ihnen.\n\nVgl. Corippus, _Iohannis_ 7,91. Matthias Borbonius (in: Jan Gruter, _Delitiae poetarum Germanorum_ ) schreibt diese Worte in der Version \u00bbOmnia mutantur\u00ab Kaiser Lothar I. (795\u2013855) zu. Die obige Version soll in den _Proverbialia dicteria_ des Andreas Gartner zu finden sein.\n\n**Tempore qui longo steterit, male curret.**\n\nWer lange Zeit stand, wird schlecht laufen.\n\n(Vgl.: Wer rastet, der rostet.)\n\nOvid, _Tristia_ 5,12,25.\n\n**tempore uno \/ duobus locis simul esse**\n\nzu einem Zeitpunkt an zwei Orten zugleich sein\n\n(vgl.: auf zwei Hochzeiten gleichzeitig tanzen)\n\nNach Plautus, _Amphitruo_ 567 f.\n\n**Tempori aptari decet.**\n\nMan mu\u00df sich der Zeit anpassen.\n\nSeneca, _Medea_ 175.\n\n**Tempori cedere semper sapientis est habitum.**\n\nDen zeitlichen Umst\u00e4nden nachzugeben hat immer als weise gegolten.\n\nNach Cicero, _Ad familiares_ 4,9,2.\n\n**Temporis ars medicina fere est.**\n\nDie Kunst der Zeit (d. h.: der Wahl des richtigen Zeitpunkts) ist fast eine Medizin.\n\nOvid, _Remedia amoris_ 131.\n\n**tempus edax rerum**\n\ndie alles verschlingende Zeit\n\n(vgl.: der Zahn der Zeit)\n\nOvid, _Metamorphoses_ 15,234; vgl. _Epistulae ex Ponto_ 4,10,7.\n\n**Tempus est etiam maiora conari.**\n\nEs ist (die richtige) Zeit, auch Gr\u00f6\u00dferes zu versuchen.\n\nLivius, _Ab urbe condita_ 6,18,13.\n\n**Tempus in agrorum cultu consumere dulce est!**\n\nS\u00fc\u00df ist's, die Zeit auf die Bearbeitung der \u00c4cker zu verwenden.\n\nOvid, _Epistulae ex Ponto_ 2,7,69. \u2013 Die Zeit, in der man sich dem Ackerbau widmen kann, steht hier metonymisch f\u00fcr den Frieden.\n\n**Tendis iners docto retia nota mihi.**\n\nUngeschickt spannst du mir, der ich erfahren bin, bekannte Netze.\n\nProperz, _Elegiae_ 2,32,20.\n\n**Tendit in ardua virtus.**\n\nTugend strebt Schwieriges an.\n\nOvid, _Epistulae ex Ponto_ 2,2,111.\n\n**Teneas tuis te!**\n\nHalte dich an deine Angelegenheiten!\n\nNach Horaz, _Sermones_ 2,3,324.\n\n**tenero lascivior haedo**\n\nmunterer als ein zartes B\u00f6ckchen\n\nOvid, _Metamorphoses_ 13,791. \u2013 Der h\u00e4\u00dfliche Polyphem bezeichnet so die liebliche Galatea.\n\n**Ter bibe, vel toties ternos, sic mystica lex est.**\n\nDreimal trinke oder ebensooft je drei (d. h.: neunmal) \u2013 so lautet das mystische Gesetz.\n\nAusonius, _Griphus ternarii numeri_ 1.\n\n**terra incognita**\n\nein unbekanntes Land\n\nVielleicht nach Tacitus ( _Agricola_ 10,4), der von bislang unbekannten Inseln spricht.\n\n**Terra mortalium semper ancilla.**\n\nDie Erde ist stets den Menschen dienlich.\n\nNach Plinius d. \u00c4., _Naturalis historia_ 2,155.\n\n**Terraque securae sit super ossa levis.**\n\nDie Erde m\u00f6ge dir, Ruhende, leicht auf den Gebeinen wiegen.\n\nTibull, _Elegiae_ 2,4,50.\n\n**Tertium non datur.**\n\nEin Drittes wird nicht gegeben.\n\n(D. h.: Eine dritte M\u00f6glichkeit ist ausgeschlossen.)\n\nVgl. z. B. Cicero, _Ad familiares_ 9,22,1, und Publilius Syrus A 6.\n\n**Tertius e caelo cecidit Cato!**\n\nEin dritter Cato ist vom Himmel gefallen!\n\nJuvenal, _Saturae_ 2,40. \u2013 Anspielung auf die beiden gestrenge r\u00f6mische Moral verk\u00f6rpernden Porcier: M. Porcius Cato Censorius und seinen Urenkel M. Porcius Cato Uticensis.\n\n**Te semper anteit saeva necessitas.**\n\nVor dir l\u00e4uft immer die grimmige Notwendigkeit her.\n\nHoraz, _Carmina_ 1,35,17.\n\n**Te spectem, suprema mihi cum venerit hora,**\n\n**Te teneam moriens deficiente manu.**\n\nSchau'n will ich dich, wenn die letzte Stunde mir schl\u00e4gt, sterbend dich halten mit schon ermattender Hand.\n\nTibull, _Elegiae_ 1,1,59 f.\n\n**Testis unus, testis nullus.**\n\nEin Zeuge ist kein Zeuge.\n\nVgl. _Codex Iustinianus_ 4,20,9, und **Uni testi nec Catoni creditum est**.\n\n**Tetigisti acu.**\n\nDu hast es mit der Nadel(spitze) ber\u00fchrt.\n\n(Vgl.: den Nagel auf den Kopf treffen)\n\nPlautus, _Rudens_ 1306.\n\n**Te tua , me delectant mea.**\n\nDich erfreut das Deine, mich das Meine.\n\nCicero ( _Tusculanae disputationes_ 5,63) erkl\u00e4rt hier, Dichter seien stets von ihrem eigenen Werk \u00fcberzeugt. Vgl. **Suum cuique...**\n\n**Tibi aras , tibi occas, tibi seris, tibi idem metis.**\n\nF\u00fcr dich pfl\u00fcgst du, f\u00fcr dich eggst du, f\u00fcr dich s\u00e4st du, und f\u00fcr dich erntest du ebenfalls.\n\nPlautus, _Mercator_ 71.\n\n**Tibi ut opus est facto, face!**\n\nHandle, wie du handeln mu\u00dft.\n\nCicero ( _De finibus_ 5,29) nach Terenz, _Heauton timorumenos_ 80.\n\n**Timeo lectorem unius libri.**\n\nIch f\u00fcrchte den, der (nur) ein einziges Buch gelesen hat.\n\nThomas von Aquin zugesprochen.\n\n**Timeri autem tam domi molestum est quam foris.**\n\nGef\u00fcrchtet zu werden jedoch ist unangenehm, zu Hause wie drau\u00dfen.\n\nSeneca ( _Epistulae morales_ 105,4) warnt davor, sich bei anderen unbeliebt zu machen, denn wer gef\u00fcrchtet werde, m\u00fcsse auch selbst f\u00fcrchten.\n\n**Timet timentes.**\n\nEr f\u00fcrchtet die, die f\u00fcrchten.\n\nSeneca, _Oedipus_ 706. \u2013 Kreon erkl\u00e4rt, da\u00df Tyrannen meist ihre Untertanen genauso f\u00fcrchteten, wie sie von diesen gef\u00fcrchtet w\u00fcrden. So f\u00e4hrt er fort: \u00bbMetus in auctorem redit.\u00ab (\u00bbDie Furcht kommt auf ihren Urheber zur\u00fcck.\u00ab) Vgl. _Epistulae morales_ 105,4.\n\n**Timidi est optare necem.**\n\nDen Tod (nur) zu w\u00fcnschen ist feige.\n\nOvid ( _Metamorphoses_ 4,115) l\u00e4\u00dft Pyramus mit diesen Worten Selbstmord begehen.\n\n**Tolle , lege!**\n\nHeb es auf und lies!\n\nAugustinus ( _Confessiones_ 8,12,29) berichtet, er habe einst ein Kind diese Worte rufen geh\u00f6rt und sich dazu aufgefordert gef\u00fchlt, sofort zur Bibel zu greifen. Die Lekt\u00fcre der zuf\u00e4llig aufgeschlagenen Stelle (R\u00f6merbrief 13,13) habe seine religi\u00f6se Umkehr eingeleitet.\n\n**Tolle moras ! Semper nocuit differe paratis.**\n\nBeende das Zaudern! Immer hat Aufschub denen, die bereitstanden, geschadet.\n\nLukan, _De bello civili_ 1,281. \u2013 Curios Rat an Caesar, der z\u00f6gernd am Rubicon stand. Vgl. **Alea iacta est**.\n\n**Tolluntur in altum, \/ ut lapsu graviore ruant.**\n\nSie werden in die H\u00f6he gehoben, um in heftigerem Fall zusammenzust\u00fcrzen.\n\nClaudian ( _In Rufinum_ 1,22 f.) erkl\u00e4rt, er wolle sich nie wieder \u00fcber den Aufstieg eines schlechten Menschen beschweren, da er seit Rufinus wisse, da\u00df ein B\u00f6sewicht um so tiefer falle.\n\n**tondere usque ad vivam cutem**\n\nbis auf die lebende Haut scheren\n\nNach Plautus, _Bacchides_ 242.\n\n**Tota erras via.**\n\nAuf dem ganzen Weg irrst du.\n\n(D. h.: Du bist auf dem Holzweg.)\n\nTerenz, _Eunuchus_ 245.\n\n**Totaque res vacillat et claudicat.**\n\nUnd die ganze Sache schwankt und hinkt.\n\nCicero, _De natura deorum_ 1,107.\n\n**totidem hostes quot servi**\n\nso viele Sklaven, so viele Feinde\n\nSeneca ( _Epistulae morales_ 47,5) kritisiert diese bereits in der Antike als Sprichwort bekannte Wendung: Sklaven seien nicht von sich aus Feinde ihrer Herren, sondern w\u00fcrden erst durch schlechte Behandlung dazu gemacht.\n\n**Totum igitur in eo est, ut tibi imperes.**\n\nAlles beruht also darauf, da\u00df du dir selbst gebietest.\n\nCicero, _Tusculanae disputationes_ 2,53. Vgl. **Imperare sibi...**\n\n**Totus homuncio nil est.**\n\nDas ganze elende Menschlein ist ein Nichts.\n\nPetron, _Satyrica_ 34.\n\n**Tractant fabrilia fabri** siehe **Scribimus indocti...**\n\n**Trahit sua quemque voluptas.**\n\nJeden rei\u00dft seine Leidenschaft hin.\n\nVergil, _Bucolica_ 2,65.\n\n**Tranquillas etiam naufragus horret aquas.**\n\nDer Schiffbr\u00fcchige f\u00fcrchtet auch ruhige Gew\u00e4sser.\n\n(Vgl.: Gebranntes Kind scheut das Feuer.)\n\nOvid ( _Epistulae ex Ponto_ 2,7,8) entschuldigt sich bei seinem Freund Atticus f\u00fcr seine \u00fcberm\u00e4\u00dfge Verzagtheit, die in seinen bisherigen Erfahrungen begr\u00fcndet sei.\n\n**Tres faciunt collegium.**\n\nDrei bilden ein Kollegium.\n\nNach _Digesta_ , 50,16,85. \u2013 Hier bezieht sich die Wendung auf einen Verein im juristischen Sinne. Heute wird mit diesen Worten konstatiert, da\u00df gen\u00fcgend Personen beisammen sind, um eine Lehrveranstaltung stattfinden zu lassen (\u00bbTres faciunt colloquium\u00ab).\n\n**Tria verba non potest iungere.**\n\nDrei Worte kann er nicht aneinanderreihen.\n\n(D. h.: Er bringt keine drei zusammenh\u00e4ngenden Worte heraus.)\n\nSeneca, _Epistulae morales_ 40,9.\n\n**Tristis eris, si solus eris.**\n\nTraurig wirst du sein, wenn du allein sein wirst.\n\nOvid ( _Remedia amoris_ 583) r\u00e4t, sich vor Liebeskummer zu sch\u00fctzen, indem man Gesellschaft suche.\n\n**Truditur dies die.**\n\nDer (eine) Tag wird vom (n\u00e4chsten) Tag weggesto\u00dfen.\n\n(D. h.: Dem vorausgehenden Tag ist der kommende schon auf den Fersen.)\n\nHoraz, _Carmina_ 2,18,15.\n\n**Tua quod nil refert, ne cures!**\n\nSorge dich nicht um Dinge, die dich nichts angehen.\n\nPlautus, _Stichus_ 320.\n\n**Tua quod nil refert, percontari desinas!**\n\nH\u00f6re auf zu erforschen, was dich nichts angeht!\n\nTerenz, _Hecyra_ 810.\n\n**Tu dic, mecum quo pignore certes!**\n\nSag, um welches Pfand du mit mir wetten willst!\n\nVergil, _Bucolica_ 3,31.\n\n**Tu, homo, adigis me ad insaniam.**\n\nDu, Mensch, treibst mich (noch) zum Wahnsinn!\n\nTerenz, _Adelphoe_ 111.\n\n**Tu mihi sola places.**\n\nDu allein gef\u00e4llst mir.\n\nOvid, _Ars amatoria_ 1,42.\n\n**Tunc omnia iure tenebis, \/ cum poteris rex esse tui.**\n\nDu wirst mit Recht alles in deiner Macht haben, wenn du K\u00f6nig \u00fcber dich selbst sein kannst.\n\nClaudian, _De quarto consulatu honorii Augusti_ 261 f.\n\n**Tu ne cede malis, sed contra audentior ito!**\n\nWeiche dem \u00dcbel nicht, sondern geh ihm k\u00fchner noch entgegen!\n\nVergil, _Aeneis_ 6,95. \u2013 Die Sibylle von Cumae fl\u00f6\u00dft Aeneas Mut ein, nachdem sie ihm zun\u00e4chst furchtbare Kriege und einen \u00bbanderen Achill\u00ab vorhergesagt hat.\n\n**Tu ne quaesieris , scire nefas, quem mihi, quem tibi finem di dederint.**\n\nForsche du dem nicht nach \u2013 es zu wissen ist verwehrt \u2013, welches Ende mir, welches dir die G\u00f6tter vorbestimmt haben.\n\nHoraz ( _Carmina_ 1,11,1 f.) zieht aus der obigen Erkenntnis folgenden Schlu\u00df: \u00bb **carpe diem** \u00ab (\u00bbpfl\u00fccke, d. h. n\u00fctze den Tag\u00ab).\n\n**Tu nescies , quod scis, si sapies.**\n\nDu wirst nicht wissen, was du wei\u00dft, wenn du weise bist.\n\nNach Terenz, _Heauton timorumenos_ 748. \u2013 Ein Sklave warnt seinen Kollegen, er solle sich besser dumm und unwissend stellen. Vgl. **Si sapis...**\n\n**Tunica propior palliost.**\n\nDie Tunika ist mir n\u00e4her als der Mantel.\n\n(Vgl.: Das Hemd ist mir n\u00e4her als der Rock.)\n\nPlautus, _Trinummus_ 1154.\n\n**Tu quod cavere possis, stultum admittere est.**\n\nEs ist dumm zuzulassen, wovor du dich h\u00fcten k\u00f6nntest.\n\nTerenz, _Eunuchus_ 761.\n\n**Turdus ipse sibi malum cacat.**\n\nDie Drossel kackt ihr eigenes Ungl\u00fcck.\n\nHW: 31933c. \u2013 Aus dem Kot der Drossel wurde Leim gefertigt, mit dem man V\u00f6gel fing.\n\n**Turpe est aliud loqui, aliud sentire.**\n\nEs ist sch\u00e4ndlich, etwas anderes zu sagen, als man denkt.\n\nSeneca ( _Epistulae morales_ 24,19) f\u00e4hrt fort: \u00bbQuanto turpius aliud scribere, aliud sentire.\u00ab (\u00bbWieviel sch\u00e4ndlicher, etwas anderes zu schreiben, als man denkt.\u00ab)\n\n**Tu si animum vicisti, potius quam animus te, est, quod gaudeas.**\n\nWenn du deine Leidenschaften eher als sie dich besiegt hast, hast du einen Grund, dich zu freuen.\n\nPlautus, _Trinummus_ 310.\n\n**Tu si hic sis, aliter sentias.**\n\nW\u00e4rest du hier (an meiner Stelle), d\u00e4chtest du anders.\n\nTerenz, _Andria_ 310.\n\n**Tute hoc intristi: Tibi omne est exedendum.**\n\nDu hast dir das eingebrockt: Du mu\u00dft das alles aufessen.\n\n(vgl.: die Suppe ausl\u00f6ffeln, die man sich eingebrockt hat)\n\nTerenz, _Phormio_ 318; vgl. Ausonius, _Bissula_ , praefatio 1,5.\n\n**Tutemet mirabere.**\n\nDu wirst dich selbst wundern.\n\nTerenz, _Heauton timorumenos_ 374.\nU\n\n**Ubi amici, ibidem opes.**\n\nWo Freunde, da Verm\u00f6gen.\n\nPlautus ( _Truculentus_ 885) bezeichnet diese Wendung bereits als Sprichwort. Vgl. Quintilian ( _Institutio oratoria_ 5,11,41), der die Wendung als stehenden Ausdruck bezeichnet.\n\n**Ubi bene, ibi patria** siehe **Patria est, ubicumque est bene.**\n\n**Ubi defecerit ratio, ibi est fidei aedificatio.**\n\nWo der Verstand versagt, dort beginnt die Erbauung durch den Glauben.\n\nAugustinus, _Sermones_ 247,2.\n\n**Ubi erit victoriae spes, \/ si offenditur Deus?**\n\nWo wird Hoffnung auf Sieg bestehen, wenn man Gott angreift?\n\nFriedrich Schiller, _Wallenstein_ , Wallensteins Lager, V. 533 f.\n\n**Ubi leonis pellis deficit, vulpina est insuenda.**\n\nWo es am L\u00f6wenfell mangelt, mu\u00df man sich den Fuchspelz \u00fcberziehen.\n\n(D. h.: Wo einem die Kr\u00e4fte fehlen, mu\u00df man schlau sein.)\n\nNach Phaedrus, _Fabulae_ Appendix Perottina 25.\n\n**Ubi semel quis peieraverit, ei credi postea, etiamsi per plures deos iuret, non oportet.**\n\nDem, der einmal einen Meineid geleistet hat, darf man, auch wenn er bei mehreren G\u00f6ttern schw\u00f6rt, nicht glauben.\n\n(Vgl.: Wer einmal l\u00fcgt, dem glaubt man nicht, und wenn er auch die Wahrheit spricht.)\n\nCicero, _Pro C. Rabirio Postumo_ 36.\n\n**Ubi sum , ibi non sum; \/ ubi non sum, ibi est animus.**\n\nWo ich bin, dort bin ich nicht; wo ich nicht bin, dort ist meine Seele.\n\nPlautus, _Cistellaria_ 211 f. \u2013 Alcesimarchus ist vor lauter Liebestaumel g\u00e4nzlich verwirrt.\n\n**Ubi terrarum sumus?**\n\nWo in aller Welt sind wir?\n\nCicero, _Pro C. Rabirio Postumo_ 37.\n\n**udi tamquam mures**\n\nna\u00df wie M\u00e4use\n\n(vgl.: pudelna\u00df)\n\nPetron, _Satyrica_ 44, 18.\n\n**ulcus tangere**\n\neine Wunde ber\u00fchren\n\n(vgl.: den Finger in die Wunde legen)\n\nTerenz, _Phormio_ 690.\n\n**ultima ratio regis**\n\ndie letzte Methode des K\u00f6nigs\n\nKanoneninschrift z. B. der preu\u00dfischen Gesch\u00fctze seit 1742. Vielleicht nach Pedro Calder\u00f3n de la Barca ( _En la vida todo es verdad y todo mentira_ , V. 1188), der Waffen als \u00bb\u00faltima ra\u00e7\u00f3n de reyes\u00ab bezeichnet.\n\n**ultima Thule**\n\ndas \u00e4u\u00dferste Thule\n\n(d. h.: der \u00e4u\u00dferste Norden)\n\nVergil, _Georgica_ 1,30. \u2013 Der Seefahrer Pytheas von Massilia hatte um das Jahr 325 v. Chr. auf einer Erkundungsfahrt im \u00e4u\u00dfersten Norden eine Insel mit dem Namen Thule entdeckt. Vgl. Pytheas, Fragmente 6a, 6c, 7a und 11b Mette.\n\n**Ultra posse nemo obligatur** siehe **Impossibilium nulla...**\n\n**ululam Athenas portare**\n\neine Eule nach Athen tragen\n\nNach Cicero ( _Ad familiares_ 6,3,4), der in griechischer Sprache Aristophanes ( _V\u00f6gel_ 301) zitiert. Vgl. **In silvam ne ligna feras!** und die andere \u00dcbersetzung des griechischen Originals **noctuam Athenas deportare**.\n\n**Umbra et imaginibus utimur.**\n\nWir verwenden Schatten und Abbilder.\n\nCicero ( _De officiis_ 3,69) erkl\u00e4rt, da\u00df man kein wirkliches Abbild des wahren Gesetzes und der echten Gerechtigkeit habe, sondern dem Menschen nur eine Art blo\u00dfer Skizze vorliege.\n\n**Umbram suam metuit.**\n\nEr f\u00fcrchtet seinen eigenen Schatten.\n\nQ. Cicero, _Commentariolum Petitionis_ 9.\n\n**una atque eadem voce**\n\nmit ein und derselben Stimme\n\nCicero, _De officiis_ 2,42.\n\n**Una domus non alit duos canes.**\n\nEin Haus ern\u00e4hrt nicht zwei Hunde.\n\nNach Aristophanes, _Wespen_ 927.\n\n**Una hirundo non facit ver.**\n\nEine Schwalbe macht keinen Fr\u00fchling.\n\nNach (z. B.) Gregor von Nazianz, _Orationes_ 39,14, nach Aristoteles, _Nikomachische_ _Ethik_ 1,6,1098a 16. Vgl. **Barba non facit...** und **Cucullus non facit...**\n\n**Una salus victis nullam sperare salutem.**\n\nDie einzige Rettung der Besiegten ist, auf keine Rettung zu hoffen.\n\nVergil, _Aeneis_ 2,354. Vgl. **Qui nil...**\n\n**Unde fames homini vetitorum tanta ciborum est?**\n\nWoher kommt dem Menschen so gro\u00dfer Hunger nach verbotenen Speisen?\n\nOvid, _Metamorphoses_ 15,138.\n\n**Unde gentium?**\n\nWoher in aller Welt?\n\nPlautus, _Asinaria_ 90 und nach _Epidicus_ 483.\n\n**Unde habeas, quaerit nemo; sed oportet habere.**\n\nWoher man hat, fragt niemand; aber haben mu\u00df man.\n\nJuvenal ( _Saturae_ 14,207) bezeichnet diese Worte als gel\u00e4ufige Maxime.\n\n**Undique enim ad inferos tantundem viae est.**\n\nVon \u00fcberallher ist n\u00e4mlich der Weg in die Unterwelt gleich weit.\n\nCicero ( _Tusculanae disputationes_ 1,104) zufolge lehnte der sterbende Anaxagoras mit diesen Worten ab, vor seinem Tod noch in seine Geburtsstadt gebracht zu werden.\n\n**unguibus ac pedibus repugnare**\n\nsich mit Krallen und F\u00fc\u00dfen wehren\n\n(vgl.: sich mit H\u00e4nden und F\u00fc\u00dfen wehren)\n\nNach Lukrez, _De rerum natura_ 5,1037.\n\n**unguibus et dentibus**\n\nmit den N\u00e4geln und den Z\u00e4hnen\n\n(vgl.: mit H\u00e4nden und F\u00fc\u00dfen)\n\nAugustinus, _Contra Faustum Manichaeum_ 22,73; vgl. Lukian, _Mortuorum dialogi_ 11,4.\n\n**Uni testi nec Catoni creditum est.**\n\nEinem einzigen Zeugen glaubt man nicht, selbst wenn es Cato w\u00e4re.\n\n(Vgl.: Ein Zeuge ist kein Zeuge.)\n\nHieronymus, _Adversus Rufinum_ 2,24; vgl. Deuteronomium 17,6, und Matth\u00e4us 18,16. Vgl. **Testis unus, testis nullus**.\n\n**Unum cum noris, omnes noris.**\n\nKennst du einen, kennst du alle.\n\nTerenz, _Phormio_ 265.\n\n**Unum pro multis dabitur caput.**\n\nEin Haupt wird f\u00fcr viele gegeben werden.\n\n(D. h.: Einer wird f\u00fcr viele sterben.)\n\nVergil, _Aeneis_ 5,815. \u2013 Palinurus wird zur Rettung seiner Gef\u00e4hrten sein Leben lassen.\n\n**unus et idem**\n\nein und derselbe\n\n(z. B.) Horaz, _Epistulae_ 2,2,200.\n\n**Unus homo nobis cunctando restituit rem.**\n\nEin einziger Mann hat uns (allen) durch sein Zaudern den Staat wiederhergestellt.\n\nEnnius ( _Annales_ 12,1 Fragment 360 Warmington) beschreibt so die von Quintus Fabius Maximus Verrucosus im Zweiten Punischen Krieg angewandte Verz\u00f6gerungstaktik. Vgl. auch Vergil, _Aeneis_ 6,846 (\u203aHeroenschau\u2039).\n\n**unus \/ multorum**\n\neiner von vielen (Unaufgekl\u00e4rten)\n\nHoraz, _Sermones_ 1,9,71 f.\n\n**Urbem venalem et mature perituram, si emptorem invenerit!**\n\nK\u00e4ufliche Stadt, die bald zugrunde gehen wird, wenn sie (nur) einen K\u00e4ufer findet!\n\nSallust, _Bellum Iugurthinum_ 35,10. \u2013 Worte Iugurthas, als er auf einen Beschlu\u00df des Senates Rom verlassen mu\u00df.\n\n**urbi et orbi**\n\nf\u00fcr die Stadt (Rom) und f\u00fcr den (Welt)Kreis\n\n(d. h.: f\u00fcr die ganze Welt)\n\nDer Segen des Papstes nach Patricius, _Ritus ecclesiastici SS. Romanae ecclesiae_ 1,1,4. Diese Kollokation findet sich in der Antike in anderem Zusammenhang bei Cicero, _In L. Catilinam_ 1,9; Ovid, _Ars amatoria_ 1,174 und _Fasti_ 2,683.\n\n**Urceatim pluit.**\n\nEs regnet kr\u00fcgeweise.\n\n(Vgl.: Es gie\u00dft wie aus K\u00fcbeln.)\n\nNach Petron, _Satyrica_ 44,18.\n\n**Urget diem nox et dies noctem.**\n\nDie Nacht folgt dr\u00e4ngend dem Tag und der Tag der Nacht.\n\nHoraz, _Epodi_ 17,25.\n\n**Urit miserum gloria pectus.**\n\nEs verbrennt seine elende Brust (das Verlangen nach) Ruhm.\n\nSeneca, _Hercules Oetaeus_ 620.\n\n**Urticae proxima saepe rosa est.**\n\nDer Brennessel ganz nah ist oft die Rose.\n\nOvid ( _Remedia amoris_ 46) erl\u00e4utert, Gegens\u00e4tzliches liege oft nah beieinander. So lehre er selbst sowohl die Liebeskunst als auch die Heilung von der Liebe.\n\n**Usque adeone mori miserum est?**\n\nIst denn das Sterben gar so arg?\n\nVergil, _Aeneis_ 12,646.\n\n**Usus magister est optimus.**\n\nErfahrung ist die beste Lehrerin.\n\nCicero, _Pro C. Rabirio Postumo_ 9.\n\n**Usus tyrannus.**\n\nDie Gewohnheit ist ein Tyrann.\n\n(vgl.: die Macht der Gewohnheit)\n\nNach Herodot, _Geschichte_ 3,38 und 7,104.\n\n**... ut aliquid fecisse videamur**\n\n... damit es so aussieht, als h\u00e4tten wir etwas unternommen\n\nNach Laktanz, _Institutiones divinae_ 7,4.\n\n**Ut ameris , ama!**\n\nUm geliebt zu werden, liebe!\n\nMartial, _Epigrammata_ 6,11,10. Vgl. **Sit procul...** und **Si vis amari...**\n\n**Ut ameris, amabilis esto!** siehe **Sit procul omne nefas...**\n\n**Ut desint vires, tamen est laudanda voluntas.**\n\nWenn auch die Kr\u00e4fte fehlen, mu\u00df dennoch der (gute) Wille gelobt werden.\n\n(Vgl.: Hier gilt der Wille f\u00fcr das Werk.)\n\nOvid, _Epistulae ex Ponto_ 3,4,79. Vgl. **Est nobis...** und **Quod si deficiant vires...**\n\n**Utendum est aetate, cito pede labitur aetas.**\n\nMan mu\u00df die Zeit nutzen: Schnellen Fu\u00dfes vergeht die Zeit.\n\nOvid, _Ars amatoria_ 3,65.\n\n**Ut fragilis glacies interit ira mora.**\n\nWie zerbrechliches Eis vergeht Zorn auf Dauer.\n\nOvid, _Ars amatoria_ 1,374.\n\n**Ut homo est, ita morem geras.**\n\nWie der Mensch ist, so mu\u00df man sich verhalten.\n\n(vgl.: mit den W\u00f6lfen heulen)\n\nTerenz, _Adelphoe_ 431.\n\n**Utile opus manuum vario sermone levemus.**\n\nLa\u00dft uns das n\u00fctzliche Werk der H\u00e4nde durch buntes Gespr\u00e4ch erleichtern.\n\nOvid, _Metamorphoses_ 4,39.\n\n**Utilis interdum est ipsis iniuria passis.**\n\nBisweilen ist Unrecht selbst f\u00fcr die n\u00fctzlich, die es erlitten haben.\n\nOvid, _Heroides_ 17,187.\n\n**uti possidetis**\n\nwie ihr besitzt\n\n(d. h.: nach eurem augenblicklichen Besitzstand)\n\n_Digesta_ 43,17.\n\n**ut locuples moriaris, egentis vivere fato**\n\num als Reicher zu sterben, das Schicksal eines Armen durchleben\n\nJuvenal ( _Saturae_ 14,137) fragt sich hier nach dem Sinn von \u00fcberm\u00e4\u00dfiger Sparsamkeit.\n\n**ut nil supra**\n\nda\u00df nichts dar\u00fcber (geht)\n\nTerenz, _Andria_ 120.\n\n**ut pictura poesis**\n\nwie ein Gem\u00e4lde, so auch ein Gedicht\n\nHoraz ( _Ars poetica_ 361) vergleicht Gedichte mit Bildern: Manche beeindruckten bei n\u00e4herer Betrachtung, andere eher aus der Ferne; manche liebten das Dunkel, andere wiederum wollten im Lichte stehen; manche gefielen bei einmaligem Hinsehen, andere, nachdem sich die Prozedur zehnmal wiederholt habe (\u00bbdecies repetita placebit\u00ab). Vgl. **Pictoribus atque poetis...**\n\n**Ut quimus, quando, ut volumus, non licet.**\n\n(Wir handeln) wie wir k\u00f6nnen, wenn es nicht erlaubt ist (zu handeln), wie wir wollen.\n\nNach Terenz ( _Andria_ 805), der die Wendung als sprichw\u00f6rtlich bezeichnet.\n\n**Ut quisque fortuna utitur, \/ ita praecellet.**\n\nWie ein jeder sein Gl\u00fcck nutzt, so wird er sich auszeichnen.\n\n(Vgl.: Jeder ist seines Gl\u00fcckes Schmied.)\n\nPlautus, _Pseudolus_ 679 f.\n\n**Ut rem servare suave est!**\n\nWie angenehm es ist, sein Verm\u00f6gen zu bewahren!\n\nPlautus, _Truculentus_ 342.\n\n**Ut saepe summa ingenia in occulto latent!**\n\nWie oft h\u00f6chste Begabung im Verborgenen liegt!\n\nPlautus, _Captivi_ 165.\n\n**Ut sementem feceris, ita metes.**\n\nWie du ges\u00e4t hast, so wirst du ernten.\n\nCicero, _De oratore_ 2,261.\n\n**Ut sis nocte levis, sit tibi coena brevis** siehe **Ex magna coena...**\n\n**ut solet accipiter trepidas urgere columbas**\n\nwie der Habicht die zitternden Tauben zu bedr\u00e4ngen pflegt\n\nOvid ( _Metamorphoses_ 5,606) l\u00e4\u00dft die von Diana in ein Gew\u00e4sser verwandelte Arethusa so ihre Flucht vor dem Flu\u00dfgott Alpheus beschreiben.\n\n**Ut sunt humana, nihil est perpetuum datum.**\n\nWie die menschlichen Dinge (so) sind: Nichts ist (uns) auf ewig gegeben.\n\nPlautus, _Cistellaria_ 194.\n\n**Ut tute es, item omnes censes esse?**\n\nGlaubst du, da\u00df alle so sind, wie du selbst bist?\n\nPlautus, _Rudens_ 1099.\n\n**Ut valeas, multa dolenda feres.**\n\nUm zu genesen, mu\u00dft du viel Schmerzhaftes ertragen.\n\nOvid, _Remedia amoris_ 226.\n\n**Uxorem fato credat adesse suo.**\n\nEr mag glauben, seine Frau k\u00e4me zu seinem Gl\u00fcck noch hinzu.\n\nOvid, _Remedia amoris_ 566. Oft auch zitiert: \u00bbUxorem fato credat obesse suo.\u00ab (\u00bbEr mag glauben, seine Frau stehe seinem Gl\u00fcck im Wege.\u00ab)\n\n**Uxori nubere nolo meae.**\n\nMeine Frau will ich nicht zum Manne nehmen.\n\nMartial ( _Epigrammata_ 8,12,2) begr\u00fcndet hier seine Weigerung, eine reiche Frau zu heiraten: Dann sei er seiner Frau unterlegen und n\u00e4hme gleichsam die Stellung der Frau in der Ehe ein, die seiner Meinung nach untergeordnet zu sein hat.\n\n**Uxorium se praebet.**\n\nEr zeigt sich seiner Gattin sklavisch ergeben.\n\n(vgl.: Pantoffelheld)\n\nNach Macrobius, _Saturnalia_ 7,3,19.\nV\n\n**Vacare culpa magnum est solacium.**\n\nFrei von Schuld zu sein ist ein starker Trost.\n\nCicero, _Ad familiares_ 7,3,4.\n\n**Vade mecum.**\n\nGehe mit mir.\n\nPantagruel (Rabelais, _Gargantua et Pantagruel_ 2,28) bezeichnet so eine Feldflasche, die er stets bei sich tr\u00e4gt. Heute wird die Wendung meist f\u00fcr B\u00fccher, die einen ein Leben lang begleiten, gebraucht, vielleicht in Anlehnung an eine _Vade mecum_ genannte Epigrammsammlung von Peter Lotichius.\n\n**Vae victis!**\n\nWehe den Besiegten!\n\nPlautus, _Pseudolus_ 1317. H\u00e4ufiger zitiert nach: Livius, _Ab urbe condita_ 5,48,9: **Additus ab insolente Gallo...**\n\n**Valere malo quam dives esse.**\n\nIch m\u00f6chte lieber gesund als reich sein.\n\nNach Cicero ( _De officiis_ 2,88), der diese \u00dcberlegung als Beispiel f\u00fcr die Abw\u00e4gung von physischen und materiellen Vorteilen heranzieht.\n\n**Valetudo decrescit, accrescit labor.**\n\nDas Wohlbefinden nimmt ab, die Pein nimmt zu.\n\nPlautus, _Curculio_ 219.\n\n**vallis lacrymarum**\n\nTal der Tr\u00e4nen\n\n(vgl.: Jammertal)\n\nNach Psalm 83,7.\n\n**vana surdis auribus canere**\n\ntauben Ohren Vergebliches vorsingen\n\n(vgl.: tauben Ohren predigen)\n\nLivius, _Ab urbe condita_ 40,8,10. Vgl. **surdis auribus dicta**.\n\n**Vanescitque absens et novus intrat amor.**\n\nSo verbla\u00dft der Abwesende, und eine neue Liebe tritt ein.\n\n(Vgl.: Aus den Augen, aus dem Sinn.)\n\nOvid ( _Ars amatoria_ 2,358) warnt so vor allzu langer Abwesenheit des Geliebten.\n\n**Vanitas vanitatum!**\n\nEitelkeit der Eitelkeiten!\n\nKohelet 1,2 und 12,8; vgl. Thomas a Kempis, _Imitatio Christi_ 1,1,11.\n\n**Variatio delectat.**\n\nAbwechslung erfreut.\n\nIn der Antike findet sich statt \u00bbvariatio\u00ab stets \u00bbvarietas\u00ab, so z. B. bei Cicero, _De natura deorum_ 1,22, und bei Phaedrus, _Fabulae_ 2, prologus 10.\n\n**Varium et mutabile semper \/ femina** siehe **Heia age, rumpe moras...**\n\n**vas electionis**\n\nauserw\u00e4hltes Gef\u00e4\u00df\n\nApostelgeschichte 9,15. \u2013 Gemeint ist Paulus.\n\n**Vectigalia nervi sunt rei publicae.**\n\nDie Steuern sind die Nerven des Staates.\n\nNach Cicero, _Pro lege Manilia_ 17. Vgl. **Nervi belli pecunia infinita**.\n\n**Velle parum est; cupias, ut re potiaris, oportet.**\n\nZu wollen ist zu wenig: Du mu\u00dft begehren, dich der Sache zu bem\u00e4chtigen.\n\nOvid, _Epistulae ex Ponto_ 3,1,35.\n\n**Velle suum cuique est.**\n\nJeder hat seinen (ganz) eigenen Willen.\n\n(Vgl.: Des Menschen Wille ist sein Himmelreich.)\n\nPersius, _Saturae_ 5,53.\n\n**velut inter ignes \/ luna minores**\n\nwie der Mond unter kleineren Feuern (Sternen)\n\nHoraz ( _Carmina_ 1,12,48 f.) preist das Julische Geschlecht, das unter allen anderen hervorgl\u00e4nze.\n\n**Venalis populus, venalis curia patrum, \/ est favor in pretio.**\n\nK\u00e4uflich ist das Volk, k\u00e4uflich die Kurie der V\u00e4ter, ihre Unterst\u00fctzung h\u00e4ngt vom Preis ab (den man ihnen daf\u00fcr zahlt).\n\nPetron, _Satyrica_ 119,41 f.\n\n**Venenum in auro bibitur.**\n\nGift trinkt man aus Gold (d. h.: aus einem goldenen Becher).\n\nSeneca, _Thyestes_ 453.\n\n**Venia sit dicto.**\n\nDas Wort sei gestattet.\n\n(d. h.: mit Verlaub)\n\nPlinius d. J., _Epistulae_ 5,6,46.\n\n**Venienti occurrite morbo!**\n\nLaufe der kommenden Krankheit entgegen (d. h.: gebiete ihr Einhalt, bevor sie g\u00e4nzlich um sich greift).\n\nPersius, _Saturae_ 3,64.\n\n**Veniet quondam felicior aetas.**\n\nEs wird einst ein gl\u00fccklicheres Zeitalter kommen.\n\nSilius Italicus, _Punica_ 11,123.\n\n**Venimus ad summum fortunae.**\n\nWir sind zum Gipfel des Gl\u00fccks gekommen.\n\nHoraz, _Epistulae_ 2,1,32.\n\n**Venit amor gravius quo serius.**\n\nJe sp\u00e4ter die Liebe kommt, um so heftiger.\n\nOvid, _Heroides_ 4,19.\n\n**Venit post multos una serena dies.**\n\nNach den vielen (Tagen) kam (schlie\u00dflich) ein heiterer Tag.\n\nTibull, _Elegiae_ 3,6,32.\n\n**Veni, vidi, Deus vicit!**\n\nIch kam, sah, Gott siegte!\n\nKaiser Karl V. nach der Gefangennahme des Kurf\u00fcrsten Johann Friedrich von Sachsen und der Einnahme Wittenbergs; vgl. Julius Wilhelm Zincgref, _Teutscher Nation Apophthegmata_ 1, S. 71.\n\n**Veni, vidi, vici.**\n\nIch kam, sah, siegte.\n\nSueton ( _Vita divi Iuli_ 37,2) berichtet, da\u00df beim Triumphzug Caesars anl\u00e4\u00dflich des Sieges \u00fcber Pharnaces von Pontus (47 v. Chr.) eine Tafel mit diesen Worten gezeigt wurde, die nicht die Ereignisse dieses Krieges, sondern die Geschwindigkeit, mit der er beendet wurde, dokumentieren sollte.\n\n**Venter non habet aures.**\n\nEin Bauch hat keine Ohren.\n\nVgl. Seneca, _Epistulae morales_ 21,11: \u00bbVenter praecepta non audit; poscit, appellat.\u00ab (\u00bbEin Bauch h\u00f6rt nicht auf Vorhaltungen; er fordert, ruft.\u00ab) Plutarch ( _Cato maior_ 8,1) berichtet, Cato habe, als er versuchte, das Volk von einer Forderung nach Korn abzuhalten, gesagt, es sei schwierig, mit einem Bauch zu diskutieren, da er keine Ohren habe.\n\n**ventis remis**\n\nmit Winden (und) Rudern\n\n(vgl.: volle Kraft voraus)\n\nCicero, _Ad familiares_ 12,25,3.\n\n**Ventum ad supremum est.**\n\nEs ist zum \u00c4u\u00dfersten gekommen.\n\nVergil, _Aeneis_ 12,803.\n\n**Ventum seminabunt et turbinem metent.**\n\nSie werden Wind s\u00e4en und Sturm ernten.\n\nHosea 8,7.\n\n**Verae amicitiae sempiternae sunt.**\n\nWahre Freundschaften sind von ewiger Dauer.\n\nCicero, _De amicitia_ 32.\n\n**Vera redit facies, dum simulata perit.**\n\nDas wahre Aussehen kehrt zur\u00fcck, w\u00e4hrend das vorget\u00e4uschte verschwindet.\n\nPetron, _Satyrica_ 80,9.\n\n**verba dare**\n\n(blo\u00dfe) Worte geben\n\n(d. h.: jemanden t\u00e4uschen)\n\n(z. B.) Terenz, _Andria_ 211.\n\n**Verba dat in ventos.**\n\nEr spricht seine Worte in den Wind.\n\nOvid, _Amores_ 1,6,42. Hier allerdings \u00fcbergibt der Schlaf Worte dem Wind: Der Schlafende nimmt nicht wahr, was man zu ihm spricht.\n\n**Verba dat omnis amor.**\n\nJede Liebe verleiht (sch\u00f6ne) Worte.\n\nOvid, _Remedia amoris_ 95.\n\n**Verba puellarum foliis leviora caducis.**\n\nDie Worte der M\u00e4dchen sind leichter als herabfallende Bl\u00e4tter.\n\nOvid, _Amores_ 2,16,45.\n\n**verba sine penu et pecunia**\n\nWorte ohne Nahrung und Geld\n\n(d. h.: Worte, doch keine materiellen Leistungen)\n\nPlautus, _Captivi_ 472.\n\n**Verba volant , scripta manent.**\n\nDie Worte fliegen hinfort, die Schrift bleibt.\n\nHW: 33093a.\n\n**verborum paupertas, immo egestas**\n\neine Armut, ja D\u00fcrftigkeit der Worte\n\nNach Seneca, _Epistulae morales_ 58,1.\n\n**verbo tenus**\n\nw\u00f6rtlich, bis aufs Wort\n\n(z. B.) Cicero, _De legibus_ 3,14.\n\n**Vere calor redit ossibus.**\n\nIm Fr\u00fchjahr kehrt die W\u00e4rme in die Knochen zur\u00fcck.\n\nVergil, _Georgica_ 3,272.\n\n**Verecundari neminem apud mensam decet.**\n\nBei Tische ziemt es sich f\u00fcr niemanden, zimperlich zu tun.\n\nPlautus, _Trinummus_ 478. \u2013 Gemeint ist: Wenn es ums blo\u00dfe \u00dcberleben geht, darf man keine ideologischen \u00dcberlegungen anstellen, sondern mu\u00df nehmen, was kommt.\n\n**Vereor coram in os te laudare amplius.**\n\nIch scheue mich, dich in deiner Gegenwart weiter zu loben.\n\nTerenz, _Adelphoe_ 269.\n\n**Vereri perdidit.**\n\nEr hat seine Ehrfurcht verloren.\n\nPlautus, _Bacchides_ 158.\n\n**Veris magna paratur \/ fama bonis.**\n\nGro\u00dfen Ruhm erlangt man durch wahre Verdienste.\n\nLukan, _De bello civili_ 9,593 f.\n\n**Veritas numquam perit.**\n\nDie Wahrheit geht niemals unter (d. h.: sie kommt letztlich immer zum Vorschein).\n\nSeneca, _Troades_ 614.\n\n**Veritas odit moras.**\n\nWahrheit ha\u00dft Verz\u00f6gerung.\n\nSeneca, _Oedipus_ 850.\n\n**Veritas odium parit** siehe **Obsequium amicos...**\n\n**Veritatem dies aperit.**\n\nDer Tag offenbart die Wahrheit.\n\nSeneca ( _De ira_ 2,22,3) warnt vor vorschnellen Schl\u00fcssen: Oft scheine ein Sachverhalt auf den ersten Blick klar, doch k\u00e4me die Wahrheit erst nach einiger Zeit zum Vorschein.\n\n**Veritatis simplex oratio est.**\n\nDer Wahrheit Rede ist einfach.\n\nSeneca ( _Epistulae morales_ 49,12) zitiert hier Euripides, _Die ph\u00f6nizischen M\u00e4dchen_ 469.\n\n**vero distinguere falsum**\n\nvom Wahren das Falsche unterscheiden\n\n(z. B.) Horaz, _Epistulae_ 1,10,29.\n\n**Versate diu, quid ferre recusent, \/ quid valeant umeri!**\n\n\u00dcberlegt lange, was eure Schultern zu tragen sich weigern und was sie tragen k\u00f6nnen.\n\nHoraz ( _Ars poetica_ 39 f.) mahnt die Dichter, sich nur an Themen heranzuwagen, die ihre F\u00e4higkeiten nicht \u00fcbersteigen.\n\n**Versatur celeri Fors levis orbe rotae.**\n\nDas leichte Rad des Schicksals dreht sich in schnellen Kreisen.\n\nTibull, _Elegiae_ 1,5,70.\n\n**Verum gaudium res severa est.**\n\nWahre Freude ist eine ernste Sache.\n\nSeneca ( _Epistulae morales_ 23,4) grenzt hier oberfl\u00e4chliche Freude, die oft durch L\u00e4cheln zum Ausdruck komme, von wahrer Freude ab: Sie entstehe dadurch, da\u00df man die stoischen Forderungen zur Erlangung der Gem\u00fctsruhe erf\u00fclle, indem man z. B. keine Furcht vor Tod und Schmerzen empfinde.\n\n**Verum mihi videtur, quod est.**\n\nWahr scheint mir (nur) das zu sein, was ist.\n\nNach Augustinus, _Soliloquia_ 2,8.\n\n**Verus amicus est tamquam alter idem** siehe **Amicus est tamquam alter ego**\n\n**Verus amor nullum novit habere modum.**\n\nWahre Liebe kennt kein Ma\u00df.\n\nProperz, _Elegiae_ 2,15,30.\n\n**Vestigia terrent** siehe **quia me vestigia terrent**\n\n**Vestis virum reddit (facit).**\n\nDas Kleid macht den Mann.\n\n(Vgl.: Kleider machen Leute.)\n\nErasmus, _Adagia_ 3,1,60, mit Verweis auf Quintilian, _Institutio oratoria_ 8, prooemium 20 ( **Cultus magnificus...** ), und Homer, _Odyssee_ 6,29 f.\n\n**Vexilla regis prodeunt...**\n\nDes K\u00f6nigs Banner r\u00fccken vor...\n\nVenantius Fortunatus, _De passione domini hymnus_ 1. Diese Worte zitierte Bismarck in seiner \u00bbPolenrede\u00ab vom 28. Januar 1886.\n\n**Via hostibus, qua fugiant, munienda est.**\n\nDen Feinden soll man einen Weg bahnen, auf dem sie fliehen k\u00f6nnen.\n\nNach Vegetius ( _Epitoma rei militaris_ 3,21), der diese Worte als eine von Scipio aufgestellte Milit\u00e4rregel bezeichnet.\n\n**Via trita est tutissima.**\n\nDer ausgetretene Weg ist der sicherste.\n\nLiebs V 21.\n\n**Vicina sunt vitia virtutibus.**\n\nBenachbart sind die Laster den Tugenden.\n\nHieronymus, _Dialogus contra Luciferianos_ 14.\n\n**Vicit et superos amor.**\n\nDie Liebe hat selbst G\u00f6tter (schon) besiegt.\n\nSeneca, _Hercules Oetaeus_ 473.\n\n**victima nil miserantis Orci**\n\nein Opfer des erbarmungslosen Orcus\n\nHoraz, _Carmina_ 2,3,24. \u2013 Orcus ist der Herr des Totenreichs.\n\n**Victi vincimus.**\n\nObwohl besiegt, siegen wir.\n\n(D. h.: Der Spie\u00df wird sich umdrehen.)\n\nPlautus, _Casina_ 510.\n\n**Victoria pax, non pactione parienda.**\n\nDurch einen Sieg, nicht durch einen Pakt soll man den Frieden schaffen.\n\nCicero, _Ad familiares_ 10,6,1.\n\n**Victrix causa diis placuit, sed victa Catoni.**\n\nDie siegreiche Sache gefiel den G\u00f6ttern, die besiegte dem Cato (Uticensis).\n\nLukan, _De bello civili_ 1,128. \u2013 Aus dem B\u00fcrgerkrieg ging Caesar nicht nur als Sieger \u00fcber Pompeius hervor, sondern auch \u00fcber den j\u00fcngeren Cato, den Verfechter der alten republikanischen Staatsidee. Da Cato Caesars Alleinherrschaft nicht ertragen konnte, beging er nach der Schlacht bei Thapsus im Jahre 46 v. Chr. Selbstmord.\n\n**Victrix fortunae sapientia.**\n\nSiegerin \u00fcber das Schicksal ist die Weisheit.\n\nJuvenal, _Saturae_ 13,20.\n\n**Videant consules, ne quid res publica detrimenti capiat.**\n\nM\u00f6gen die Konsuln Sorge tragen, da\u00df der Staat keinen Schaden nehme.\n\nNach (z. B.) Cicero, _In L. Catilinam_ 1,4, und Caesar, _De bello civili_ 1,53 und 1,7,5. \u2013 Formel, mit der der Senat den Notstand erkl\u00e4rte.\n\n**Video meliora proboque, \/ deteriora sequor.**\n\nIch sehe das Bessere und billige es, dem Schlechteren folge ich.\n\nOvid, _Metamorphoses_ 7,20 f.\n\n**Videre nostra mala non possumus.**\n\nAlii simul delinquunt, censores sumus.\n\nWir k\u00f6nnen unsere eigenen Fehler nicht sehen; sobald aber andere sich verfehlen, sind wir (ihre) Richter.\n\n(vgl.: den Balken im eigenen Auge nicht sehen)\n\nPhaedrus, _Fabulae_ 4,10,4 f.\n\n**Videte, quaeso, quid potest pecunia!**\n\nSeht, so bitte ich, was das Geld vermag!\n\nPlautus, _Stichus_ 410.\n\n**Vigilavit iustitiae oculus sempiternus.**\n\nEs wachte das ewige Auge der Gerechtigkeit.\n\nAmmianus Marcellinus, _Res gestae_ 28,6,25.\n\n**Vilius argentum est auro, virtutibus aurum.**\n\nGeringer als Gold ist Silber, geringer als die Tugenden das Gold.\n\nHoraz, _Epistulae_ 1,1,52.\n\n**vim vi depellere**\n\nGewalt mit Gewalt vertreiben\n\nNach Cicero, _Pro P. Sestio_ 39; vgl. Damasus Bohemus, _Burchardica, sive_ _regulae canonicae_ 72; _Digesta_ 4,2,12,1 und 43,16,1,27.\n\n**Vina dabant animos.**\n\nDer Wein gab (ihnen) Mut.\n\nOvid, _Metamorphoses_ 12,242.\n\n**Vina quies sequitur.**\n\nAuf den Wein folgt die Ruhe.\n\nOvid, _Fasti_ 3,305.\n\n**Vince animos iramque tuam, qui ceteras vincis!**\n\nBesiege deinen Unmut und deinen Zorn, du, der du (alles) \u00fcbrige besiegst!\n\nOvid, _Heroides_ 3,85.\n\n**Vince omnem, miles, virtute laborem!**\n\n\u00dcberwinde, Soldat, jede M\u00fchsal durch deine Tugend!\n\nSilius Italicus ( _Punica_ 12,511) zufolge ermahnte Hannibal seine Soldaten in Capua mit diesen Worten.\n\n**Vincere scis, Hannibal, victoria uti nescis.**\n\nDu verstehst zu siegen, Hannibal, aber den Sieg zu nutzen verstehst du nicht.\n\nLivius, _Ab urbe condita_ 22,51,4, zitiert bei Florus, _Epitomae de Tito Livio_ 1,22,19, und Ammianus Marcellinus, _Res gestae_ 18,5,6. \u2013 Maharbal soll mit diesen Worten Hannibal gedr\u00e4ngt haben, nach seinem Sieg \u00fcber die R\u00f6mer bei Cannae (216 v. Chr.) sofort gegen Rom zu marschieren. Vgl. **Hannibal ad portas!**\n\n**Vincere vult animos, non satiare Venus.**\n\nBesiegen, nicht befriedigen will Venus die Herzen.\n\nAusonius, _Epigrammata_ 56,2.\n\n**Vincit sanctos dira libido.**\n\nDie schreckliche Lust besiegt (selbst) Tugendhafte.\n\nSeneca, _Phaedra_ 981.\n\n**Vincula da linguae, vel tibi vincla dabit.**\n\nLege deiner Zunge Fesseln an, oder sie wird dir Fesseln anlegen.\n\nHW: 33446.\n\n**Vincuntur molli pectora dura prece.**\n\nBesiegt werden harte Herzen durch weiche Bitten.\n\nTibull, _Elegiae_ 3,4,76.\n\n**Vindicta \/ nemo magis gaudet quam femina.**\n\nNiemand erfreut sich mehr an Rache als eine Frau.\n\nJuvenal, _Saturae_ 13,191 f. Vgl. **Minuti \/ semper...**\n\n**Vino diffugiunt mordaces sollicitudines.**\n\nBeim Wein verfl\u00fcchtigen sich die bei\u00dfenden Sorgen.\n\nNach Horaz, _Carmina_ 1,18,4.\n\n**Vino forma perit, vino corrumpitur aetas.**\n\nWein richtet die Sch\u00f6nheit zugrunde; durch Wein wird die Bl\u00fcte des Lebens verdorben.\n\nProperz, _Elegiae_ 2,33,33.\n\n**Vinum lac Veneris.**\n\nWein ist die Milch der Venus.\n\nHW: 33476.\n\n**Vinum laetificat cor hominis.**\n\nWein erfreut des Menschen Herz.\n\nPsalm 103,15.\n\n**viperam sub ala nutricare**\n\neine Schlange unter der Achsel n\u00e4hren\n\n(vgl.: eine Schlange am Busen n\u00e4hren)\n\nNach Petron, _Satyrica_ 77,2 (nach Aesop 62 Hausrath); vgl. Cicero, _De haruspicum responsis_ 50.\n\n**viribus, ingenio, specie, virtute, loco, re**\n\nwas Kr\u00e4fte, Begabung, Aussehen, Tugend, Position, Besitz betrifft\n\nHoraz ( _Epistulae_ 2,2,203) behauptet, in bezug auf alle Aspekte des Lebens eine Mittelposition einzunehmen: Nie sei er der erste noch jemals der letzte.\n\n**viribus unitis**\n\nmit vereinten Kr\u00e4ften\n\nWahlspruch des \u00f6sterreichischen Kaisers Franz Joseph I., gepr\u00e4gt von Joseph Ritter von Bergmann.\n\n**vir omnium horarum**\n\nein Mann f\u00fcr alle Stunden\n\nQuintilian ( _Institutio oratoria_ 6,3,110) zufolge wurde von Asinius Pollio berichtet, da\u00df er zu Spa\u00df wie Ernst f\u00e4hig, d. h. ein Mann f\u00fcr jede Gelegenheit gewesen sei.\n\n**Virtus est medium vitiorum.**\n\nTugend ist die (rechte) Mitte zwischen den Fehlern.\n\nHoraz, _Epistulae_ 1,18,9.\n\n**Virtus est vitium fugere et sapientia prima \/ stultitia caruisse.**\n\nDie Tugend liegt darin, das Laster zu fliehen, und von Dummheit frei zu sein ist der Beginn der Weisheit.\n\nHoraz, _Epistulae_ 1,1,41 f.\n\n**Virtus et summa potestas \/ non coeunt.**\n\nTugend und absolute Macht passen nicht zusammen.\n\nLukan, _De bello civili_ 8,494 f.\n\n**Virtus in astra tendit, in mortem timor.**\n\nTugend strebt nach den Sternen, Furcht nach dem Tod.\n\nSeneca, _Hercules Oetaeus_ 1971.\n\n**Virtus mihi \/ in astra et ipsos fecit ad superos iter.**\n\nTugend hat mir den Weg zu den Sternen und selbst zu den G\u00f6ttern ge\u00f6ffnet.\n\nSeneca, _Hercules Oetaeus_ 1942 f.\n\n**Virtus omnia in sese habet.**\n\nTugend birgt alles in sich.\n\nPlautus, _Amphitruo_ 652.\n\n**Virtus post nummos!** siehe **O cives, cives...**\n\nVirtus, recludens immeritis mori\n\nCaelum, negata temptat iter via.\n\nTugend, die den Himmel denen \u00f6ffnet, die nicht zu sterben verdienen, versucht sich an einem Pfad, der (anderen) verwehrt ist.\n\nHoraz, _Carmina_ 3,2,21 f.\n\n**Virtute decet, non sanguine niti.**\n\nAuf Tugend, nicht auf Blut mu\u00df man sich st\u00fctzen.\n\nClaudian, _De quarto consulatu honorii Augusti_ 220.\n\n**Virtutem incolumem odimus.**\n\nWir hassen lebendige Tugend.\n\nHoraz ( _Carmina_ 3,24,31) erkl\u00e4rt, die meisten Menschen beneideten tugendhafte Menschen. Erst wenn verdiente Menschen st\u00fcrben, w\u00fcnsche man sich deren Tugend zur\u00fcck.\n\n**Virtutis est domare, quae cuncti pavent.**\n\nTapferkeit liegt darin zu bezwingen, was alle f\u00fcrchten.\n\nSeneca, _Hercules furens_ 435.\n\n**vis comica**\n\nkomische Kraft\n\n(d. h.: Kraft der Komik)\n\nDiese Kollokation beruht auf einem Mi\u00dfverst\u00e4ndnis: Caesar hatte in einem Gedicht Terenz' Komik (\u00bbcomica virtus\u00ab, in: Sueton, _Terentius_ 5) gelobt. Das Adjektiv \u00bbcomica\u00ab ist dabei von einigen Lesern f\u00e4lschlicherweise auf die in der n\u00e4chsten Zeile folgende \u00bbvis\u00ab bezogen worden. Inhaltlich besteht jedoch kein Widerspruch.\n\n**Vis consilii expers mole ruit sua.**\n\nKraft ohne \u00dcberlegung bricht durch ihre eigene Wucht zusammen.\n\n(Vgl. Friedrich Schiller, \u00bbDas Lied von der Glocke\u00ab, V. 350: \u00bbWo rohe Kr\u00e4fte sinnlos walten...\u00ab)\n\nHoraz, _Carmina_ 3,4,65.\n\n**Viscus merus vestra est blanditia.**\n\nPurer Vogelleim ist eure Schmeichelei.\n\n(vgl.: jemandem auf den Leim gehen)\n\nPlautus, _Bacchides_ 50.\n\n**Vis nulla cruentam castrat avaritiam.**\n\nKeine Gewalteinwirkung schw\u00e4cht grausamen Geiz.\n\nClaudian, _In Eutropium_ 1,192.\n\n**Vita brevis est, ars longa.**\n\nDas Leben ist kurz, die Kunst lang.\n\nNach Seneca ( _De brevitate vitae_ 1,1), der Hippokrates, _Aphorismen_ 1,1, zitiert.\n\n**Vitae summa brevis spem nos vetat incohare longam.**\n\nDie kurze Spanne des Lebens verbietet uns, lange Hoffnungen zu hegen.\n\nHoraz, _Carmina_ 1,4,15.\n\n**vitam impendere vero**\n\nsein Leben der Wahrheit widmen\n\nJuvenal, _Saturae_ 4,91.\n\n**Vitam regit fortuna, non sapientia.**\n\nGl\u00fcck regiert das Leben, nicht Weisheit.\n\nCicero ( _Tusculanae disputationes_ 5,25) zufolge war Theophrast wegen dieser in seinem _Callisthenes_ gelobten Behauptung oft kritisiert worden. Vgl. Plutarch, _De fortuna_ 97c.\n\n**Vitaque mancipio nulli datur, omnibus usu.**\n\nDas Leben wird niemandem als Eigentum, allen (aber) zum Gebrauch gegeben.\n\nLukrez, _De rerum natura_ 3,971.\n\n**Vitavi denique culpam, \/ non laudem merui.**\n\nIch habe nur gar die Schuld gemieden, (doch) kein Lob verdient.\n\nHoraz, _Ars poetica_ 267 f.\n\n**vita vigilia est** siehe **Profecto enim vita...**\n\n**Vitia nobis sub virtutum nomine obrepunt.**\n\nLaster beschleichen uns unter dem Decknamen der Tugend.\n\nSeneca, _Epistulae morales_ 45,7.\n\n**Vitiosum est ubique, quod nimium est.**\n\nFehlerhaft ist alles, was zuviel ist.\n\n(Vgl.: Nichts im \u00dcberma\u00df!)\n\nSeneca, _De tranquilitate animi_ 9,6. Vgl. **Ne quid nimis.**\n\n**Vivat Bacchus!**\n\nBacchus lebe!\n\nWolfgang Amadeus Mozart, _Die Entf\u00fchrung aus dem Serail_ , 2. Aufzug, 7. Auftritt, Duett zwischen Pedrillo und Osmin (Libretto von Gottlieb Stephanie und Christoph Friedrich Bretzner).\n\n**viva voce**\n\nmit lebender Stimme\n\n(d. h.: m\u00fcndlich, nicht schriftlich)\n\n(z. B.) nach Seneca Rhetor, _Controversiae_ , praefatio 11.\n\n**Vive memor leti! Fugit hora, hoc quod loquor inde est.**\n\nLebe eingedenk des Todes! Die Zeit flieht dahin; was ich sage, ist schon hinweg.\n\nPersius, _Saturae_ 5,153.\n\n**Vive memor quam sis aevi brevis!**\n\nLebe eingedenk der K\u00fcrze deiner Lebenszeit!\n\nHoraz, _Sermones_ 2,6,97.\n\n**Vivere est militare** siehe **Atqui vivere...**\n\n**Vivere si recte nescis, decede peritis!**\n\nWenn du nicht recht zu leben wei\u00dft, weiche den Erfahrenen (d. h.: denen, die zu leben verstehen)!\n\nHoraz, _Epistulae_ 2,2,213.\n\n**Vive vale!**\n\nLebe (und) lebe wohl!\n\n(z. B.) Horaz, _Epistulae_ 1,6,67.\n\n**Vivimus, ut numquam sensu careamus amaro.**\n\nWir leben derart, da\u00df wir nie einer bitteren Empfindung entbehren.\n\nOvid, _Epistulae ex Ponto_ 1,2,37.\n\n**Vivitur parvo bene.**\n\nEr lebt von wenigem gut.\n\nHoraz, _Carmina_ 2,16,13.\n\n**Vivos voco, mortuos plango, fulgura frango.**\n\nDie Lebenden rufe ich, die Toten beklage ich, die Blitze breche ich.\n\nFriedrich Schiller, \u00bbDas Lied von der Glocke\u00ab, Motto.\n\n**Vix ea nostra voco.**\n\nKaum nenne ich dies das unsere.\n\nOvid, _Metamorphoses_ 13,141. \u2013 Odysseus erkl\u00e4rt hier, da\u00df man nichts, was man nicht durch eigenes Verdienst erlangt habe \u2013 als Beispiel nennt er edle Abstammung \u2013 wirklich sein eigen nennen k\u00f6nne.\n\n**Vixere fortes ante Agamemnona \/ multi.**\n\nViele Tapfere lebten (bereits) vor Agamemnon.\n\nHoraz ( _Carmina_ 4,9,25 ff.) zufolge sind Helden wie z. B. Hector nicht allein aufgrund ihrer Taten ber\u00fchmt, sondern vor allem, weil sie von einem Dichter verewigt wurden. Die nicht von einem Dichter \u00bbbeweinten\u00ab Helden jedoch umgebe die tiefe Nacht der Unbekanntheit: \u00bbSed omnes inlacrimabiles \/ urgentur ignotique longa \/ nocte.\u00ab (\u00bbAber alle Unbeweinten und Unbekannten werden von einer langen Nacht bedr\u00e4ngt.\u00ab)\n\n**Vixit, dum vixit, bene.**\n\nEr hat, solange er lebte, gut gelebt.\n\nTerenz, _Hecyra_ 461.\n\n**Vix ossibus haerent.**\n\nSie h\u00e4ngen kaum in ihren Knochen.\n\n(vgl.: nur noch aus Haut und Knochen bestehen)\n\nVergil, _Bucolica_ 3,102.\n\n**Vocat labor ultimus omnes.**\n\nDie \u00e4u\u00dferste Not ruft alle.\n\nVergil, _Aeneis_ 11,476.\n\n**Vocatus atque non vocatus audit.**\n\nGerufen oder ungerufen gehorcht er.\n\nHoraz ( _Carmina_ 2,18,40) spricht hier von Orcus, dem Herrscher der Unterwelt, der jeden von den M\u00fchen des Lebens befreit, ob man ihn ruft oder nicht.\n\n**volatilis aetas**\n\ndie fl\u00fcchtige Zeit\n\nOvid, _Metamorphoses_ 10,519.\n\n**Volenti non fit iniuria** siehe **Nulla iniuria est...**\n\n**volgivaga Venus**\n\n\u00fcberall umherschweifende Liebe\n\nNach Lukrez, _De rerum natura_ 4,1071.\n\n**volubilis fortuna**\n\nwandelbares Gl\u00fcck\n\nNach Valerius Maximus, _Dicta et facta memorabilia_ 7,1,1.\n\n**Voluptatem maeror comes consequitur.**\n\nAuf das Vergn\u00fcgen folgt wie ein Begleiter die Trauer.\n\n(Vgl.: Ohne Verdru\u00df ist kein Genu\u00df.)\n\nNach Plautus, _Amphitruo_ 635.\n\n**volventibus annis**\n\nw\u00e4hrend die Jahre sich dahinw\u00e4lzen\n\n(d. h.: im Laufe der Jahre)\n\nVergil, _Aeneis_ 1,234.\n\n**voraces impensae**\n\ngefr\u00e4\u00dfige Unkosten\n\nNach Valerius Maximus, _Facta et dicta memorabilia_ 7,1,2.\n\n**Vos exemplaria Graeca \/ nocturna versate manu, versate diurna.**\n\nW\u00e4lzet die griechischen Vorbilder bei Nacht (und) bei Tag.\n\nHoraz, _Ars poetica_ 268 f.\n\n**vox clamantis in deserto**\n\ndie Stimme des Rufenden in der W\u00fcste\n\nJesaja 40,3 und Matth\u00e4us (3,3) \u00fcber Johannes den T\u00e4ufer.\n\n**Vox faucibus haesit** siehe **Obstipui steteruntque...**\n\n**Vox populi vox dei.**\n\nDie Stimme des Volkes (ist) die Stimme Gottes.\n\nPetrus Blesensis, _Epistulae_ 15, nach Jesaja 66,6; vgl. Seneca Rhetor, _Controversiae_ 1,1,10.\n\n**Vulcanum naribus efflant.**\n\nFeuer blasen sie aus den N\u00fcstern.\n\nOvid ( _Metamorphoses_ 7,104) beschreibt so die Stiere, mit denen Iason k\u00e4mpfen mu\u00dfte.\n\n**Vulgare amici nomen, sed rara est fides.**\n\nVerbreitet ist die Bezeichnung \u00bbFreund\u00ab, doch selten ist die Treue.\n\nPhaedrus, _Fabulae_ 3,9,1.\n\n**Vulgi opinio mutari vix potest.**\n\nDie Meinung der breiten Menge l\u00e4\u00dft sich kaum \u00e4ndern.\n\nCicero, _Topica_ 73.\n\n**Vulgus amicitias utilitate probat.**\n\nDie breite Menge mi\u00dft Freundschaften an ihrem Nutzen.\n\nOvid, _Epistulae ex Ponto_ 2,3,8.\n\n**Vulpes non iterum capitur laqueo.**\n\nEin Fuchs l\u00e4\u00dft sich nicht zweimal mit einer Schlinge fangen.\n\nNach einem griechischen Sprichwort, z. B. Zenobius (Paroemiographi Graeci) 1,67; vgl. Horaz, _Sermones_ 2,7,70 f.\n\n**Vulpes pilum mutat, non mores.**\n\nEin Fuchs wechselt den Balg, nicht den Charakter.\n\nSueton ( _Vita divi Vespasiani_ 16,3) zufolge Ausruf eines Ochsentreibers, der Vespasian nach dessen Thronbesteigung um seine Freiheit gebeten hatte und daf\u00fcr zahlen sollte.\n\n**Vultus indicat mores.**\n\nDie Miene zeigt den Charakter an.\n\nNach Cicero, _De legibus_ 1,27.\n\n**Vultus loquitur , quodcumque tegis.**\n\nDeine Miene spricht aus, was auch immer du verheimlichst.\n\nSeneca, _Hercules Oetaeus_ 705.\nRegister lateinischer Schlagw\u00f6rter und Namen\n\n**a** bire\n\nabsurdum\n\nabusus\n\naccipiter\n\nAchaicus\n\nAchates\n\nAcheron\n\nAchivi\n\nacta, actiones, agere\n\nadamas\n\nadolescens, adolescentia\n\nadorare\n\naeger, aegrotum\n\nAemilianus\n\nAeneas\n\naequalis, aequus\n\naequalitas\n\naequa mens, aequanimitas, aequus animus\n\naequare\n\naequitas\n\naequor\n\naes (alienum)\n\naetas, aevum\n\naeternitas, aeternus\n\nAethiops\n\nAfrica\n\nAgamemnon\n\nalea\n\nalienus\n\nalter\n\namabilis, amare, amator, amor\n\nambages\n\namens\n\namicitia\n\namicus\n\nanima, animus\n\nanimal\n\nannus\n\nApella\n\naper\n\napis\n\nApollo\n\naqua\n\naquila\n\narare\n\narbor\n\nArcadia\n\nArchimedes\n\narcus\n\n(h)arena\n\narma\n\nars, artifex\n\narx\n\nasinus\n\nasparagus\n\nasper, asperitas\n\nastrum\n\nater\n\nAthenae\n\nAtticus\n\nauctor, auctoritas\n\naudacia, audax, audere\n\naudire, auditor\n\n(se) auferre\n\nAugias\n\nAugustus\n\naureus, aurum\n\nauris\n\nAustria\n\nauxiliari, auxilium\n\navaritia, avarus\n\nAvernus\n\navis\n\n**B** acchus\n\nbarba\n\nbarbarus\n\nBarberini (Papst Urban VIII.)\n\nbeatus\n\nbellum, debellare, imbellem\n\nbeneficium\n\nbenevolentia\n\nbestia\n\nbibere\n\nblandiri, blanditia, blandulus, blandus\n\nbos\n\nbrevis, brevitas\n\nBrutus\n\nbucca\n\n**c** adaver\n\ncaecus\n\ncaelum\/coelum\n\nCaesar\n\ncanere, cantare\n\ncanis\n\ncapitalis, caput\n\ncarmen\n\nCarthago\n\ncasa\n\ncasus\n\nCatilina\n\nCato\n\n(abs-, de-, dis-, con-, re-)cedere\n\nc(o)ena\n\ncensor, censura, census\n\nCeres\n\n(in)certus\n\nCharybdis\n\nchorda\n\nChristianus\n\nChristus\n\ncibus\n\ncinis\n\ncirculus\n\ncitus\n\ncivis, civitas, civile\n\nclementia\n\nclericus\n\ncogitare, cogitatio\n\ncollegium\n\ncollum\n\ncolumba\n\ncomes, comitari\n\nconcordia\n\nconiugium\n\nconscientia, conscire, conscius\n\nconsensus\n\nconsilium\n\nconsolatio\n\nconsul, consulere, consultor\n\ncontrarius\n\ncor\n\nCorinthus\n\ncorium\n\ncorpus, corpusculum\n\ncorvus\n\nCorydon\n\ncrabro\n\ncrambe\n\ncrimen\n\nCroesus\n\ncrus\n\ncrux\n\nculpa, culpare\n\ncupere, concupiscere, cupiditas, cupido\n\ncura, curare\n\ncustodia, custos, custodire\n\ncutis\n\n**d** amnum, damnosus\n\nDanai\n\nDavos\n\nDecember\n\ndelirare\n\nDelphinus\n\ndementia, dementare\n\nDemocritus\n\ndesipere\n\ndetrimentum\n\ndeus\n\ndextera\n\ndiabolus\n\ndies\n\ndigitus\n\ndimidium\n\ndiscere\n\ndiscordia, discors\n\ndives, divitiae\n\ndividuum\n\ndivinus, divus\n\n(per)docere, docilis, doctor, doctus\n\ndolere, dolor\n\ndolus\n\ndomus\n\ndos\n\ndubitare, dubium\n\n**e** brietas\n\necclesia\n\neloqui, eloquentia, eloquium\n\nEpicurus\n\nepistula\n\neques, equus\n\n(ob)errare, error\n\nexemplar, exemplum\n\nexspectare\n\n**f** aber\n\nfabella, fabula, fabulari\n\nfacies\n\nfacinus, factum\n\nfacundia, facundus\n\nfallere, fallax\n\nfama\n\nfames\n\nfarina\n\nfatum\n\nfauces\n\nfelix\n\nfemina\n\nfera, ferus\n\nferreus, ferrum\n\nfidelis, fidere, fides, fidus\n\nfilia, filius\n\nfilum\n\nfingere\n\nfinire, finis, infinitus\n\nFlaccus\n\nflamma\n\nflebilis, flere, fletus\n\nflorere, flos\n\nflumen, fluvius\n\nforma, formare, formosus, informis, deformis\n\nfortuna\n\nfors\n\nfrenum, effrenatus\n\nfur, furari\n\nfurere, furor, furibundus\n\n**G** allia, Gallus\n\ngallina, gallus\n\ngaudere, gaudium\n\ngens\n\nGermania\n\ngladius\n\ngloria, gloriari\n\nGracchi\n\nGraecia, Graecus\n\ngratia\n\n**h** aerere\n\nHannibal\n\nhasta\n\nHector\n\nherba\n\nHercules\n\nheres, hereditas\n\nHippocrates\n\nhircus\n\nhirundo\n\nhistoria\n\nHomerus\n\nhonestus, honestare, honestas, honos\n\nhora\n\nHoratius Flaccus\n\nhorreum\n\nhorror, horribilis, horrescere\n\nhostis\n\n**i** gnavus\n\nignis\n\nignorare\n\nignoscere\n\nIliacus, Ilium\n\nimago\n\nimperare, imperium, imperator, imperiosa\n\nimpetus\n\ninceptio, incipere\n\niners, inertia\n\ningeniosus, ingenium\n\ninimicitia, inimicus\n\niniquitas, iniquus\n\niniuria\n\ninitium\n\ninnocens, innocentia\n\ninopia, inops\n\ninsania, insanire, insanus\n\nintellegere\n\ninvidere, invidia, invidus\n\niocosus, iocus\n\nira, iracundia, iracundus, irasci\n\nIrus\n\niter\n\niucundus\n\niudex, iudicare, iudicium\n\nIuppiter\n\nius, iustitia, iustus\n\niuvenile, iuvenis, iuventa, iuventus\n\n**K** alendae\n\n**l** aborare, labor\n\nlacrima\n\nlaetificare, laetitia, laetus\n\nlapis\n\nlatere, latenter\n\nLatium\n\nlaudare, laus\n\nlector, legere\n\nleo\n\nlepus\n\nletalis, letum\n\nlex\n\nlibellum, liber\n\nliber, libertas\n\nLiber\n\nlibido\n\nlimen\n\nlingua\n\nlis, litigare\n\nlivor\n\nlocus\n\nlorum\n\nLucilius (Freund Senecas d. J.)\n\nLucilius (Dichter)\n\nlucere, lumen, lux\n\nlucrum\n\nludere\n\nluna\n\nlupus\n\nluxuria, luxuriare\n\n**M** aecenas\n\nmagister, magistra\n\nmalum (Apfel)\n\nmalum\n\nmanus\n\nmare\n\nMars\n\nmassa\n\nmater\n\nmedicabilis, medicamen, medicamentum, medicus\n\nmediocritas, medium\n\nmeminisse, memor, memoria\n\nmendacium, mendax\n\nmens, mentalis\n\nmerces\n\nMercurius\n\nmeretrix\n\nmetere, messis\n\nmetus, metuere\n\nmiles, militare, militia\n\nMinerva\n\nmiser, miserabilis, miseria, misericors\n\nmobilis, mobilitas\n\nmodus\n\nmoles, molestus\n\n(de)monstrare\n\nmonumentum\n\nmora, morari\n\nmorbus\n\nmori, mors, mortalis, mortifer\n\nmos\n\nmulier\n\nmundus\n\nmus\n\nMusa\n\n**n** asci\n\nnatura, naturalis\n\nnaufragium, naufragus\n\nnausea\n\nnavigare, navigium, navis\n\nnecessarius, necesse, necessitas\n\nnegotium\n\nnervus\n\nnihil, nil\n\nnocere\n\nnocturnus, nox\n\nnubere\n\nnubes, nubilum\n\nnummus\n\nnux\n\n**o** ccasio\n\noculus\n\nodiosus, odisse, odium\n\nodor\n\nOedipus\n\nolere\n\noleum\n\nolla\n\nOlympus\n\nopera\n\nops\n\nopus\n\noratio, orator\n\norbis\n\nOrcus\n\nos, oris\n\nos, ossis\n\nosculum\n\notiosus, otium\n\novile, ovis\n\novum\n\n**p** abulum\n\npacatus\n\nPaetus\n\npalle(sc)ere, pallidus\n\npalli(ol)um\n\nPamphilus\n\npanis\n\npapa\n\npapilio\n\nparere\n\nparies\n\nparmula\n\nParnassus\n\npars\n\nparsimonia\n\npasci, pastor, pastus\n\npassus\n\npater\n\npatientia, pati\n\npatina\n\npatria\n\npauper, paupertas\n\npax\n\npeccare, peccatum\n\npectus\n\npecunia\n\npecus\n\nPenelope\n\npenna\n\nPergameus\n\npericulum\n\npersona\n\npes\n\npestis\n\nphilosophari, philosophia, philosophus\n\nPhoebus\n\npietas, pius\n\npignus\n\npiscis, piscari\n\nPlato\n\npoema, poesis, poeta, poeticus\n\npoena, punire\n\nPolonia\n\npopularis, populus\n\nporcus\n\npossidere, possessio\n\nPostumus\n\npotestas\n\npraeda\n\npraedicare\n\npraesentia\n\nprincipium\n\nProtesilaus\n\npudet, pudicitia, pudor\n\npuella, puer\n\npugna, (re)pugnare\n\nPunicus\n\nPythius\n\n(re) **q** uies, (re)quiescere\n\nQuirites\n\n**r** ectus\n\nregere\n\nregina, rex\n\nregnare, regnum\n\nreligens, religio, religiosus\n\nremedium\n\nremus\n\nRemus\n\nren\n\nres adversae\/angustae\/arduae\/durae\/ incertae\/malae\n\nres laetae\/lautae\/secundae\n\nres publica\n\nrete\n\nreus\n\nreverentia\n\nRhodus\n\nrixa, rixari\n\nRoma\n\nRomanus\n\nRomulus\n\nrosa\n\nRubico\n\nruina\n\nrumor\n\n**s** aeculum\n\nSaguntum\n\nsal\n\nsalus\n\nsanare, sanitas, sanus\n\nsanctus\n\nsanguis\n\nsapere, sapientia\n\n(in)satiabilis, satiare, sat(is)\n\nSaturnalia\n\nsaxum\n\nsceleratus, scelus\n\nsceptrum\n\nschola\n\nscientia\n\nscorpius\n\nscribere\n\nScylla\n\nseges\n\nsenectus, senescere, senex\n\nsententia\n\nsermo\n\nservire, servitium, servitudo, servus\n\nsidus\n\nsigillum\n\nsilentium, silere\n\nsilva\n\nsitire, sitis\n\nsocietas, socius\n\nsol\n\nsolacium, solamen\n\nsomnus\n\nSparta\n\nspecies\n\nsperare, spes\n\nspirare, spiritus\n\nstilum\n\nstimulus\n\nstomachari, stomachus\n\nstudere, studium\n\nstultitia, stultus\n\nStyx\n\nsuave\n\nsuccessus\n\nsuperbia, superbus\n\n**t** acere, taciturnus, tacitus\n\ntellus\n\ntelum\n\ntempus\n\nterra\n\ntestis\n\nTeucer\n\nTeutonicus\n\nThales\n\nthronus\n\nThule\n\nTibur\n\ntimere, timidus, timor\n\ntiro\n\nTraianus\n\ntriumphus\n\nTroianus, Tros, Troia\n\ntruncus\n\ntyrannus\/tirannus\n\n**U** calegon\n\nulcisci, ultio, ultor\n\nUlixes\n\numbra, obumbrare\n\nunguis\n\nurbs\n\nursus\n\nutile, utilitas\n\nuxor\n\n**v** anitas, vanus\n\nvariatio, varius\n\n(Quintilius) Varus\n\nvas\n\nvates\n\nvelum\n\nvenenum, venenatus\n\nvenia\n\nventer\n\nventus, ventosus\n\nVenus\n\nverbum\n\nVergilius Maro\n\nveritas, ver(ecund)us\n\nversus\n\nvespa\n\nvesper\n\nvestigium\n\nvestimentum, vestis\n\nvia\n\nvictor(ia), vincere\n\nvideri\n\nvindicare, vindicta\n\nvinosus, vinum\n\nvipera\n\nvir, virile\n\nvirtus\n\nvis\n\nvita, vivere\n\nvitium\n\nvocare, vox\n\nvoluntarius, voluntas\n\nvoluptas\n\nvotum\n\nvulgari, vulgaris, vulgus\n\nvulnus\n\nvulpes\n\nvultus\nRegister deutscher Schlagw\u00f6rter\n\nDie Verweise beziehen sich auf das lateinische Schlagwortregister.\n\n**A** bend _s_. vesper\n\n(Abend)essen _s_. c(o)ena\n\nAbwechslung _s_. variatio\n\nAdler _s_. aquila\n\nAlter, alter Mann _s_. senectus, senex\n\nAnf\u00e4nger _s_. tiro\n\nAnfang, Beginn _s_. initium, principium\n\nAngeklagter _s_. reus\n\nAngriff, Schwung _s_. impetus\n\nAntlitz _s_. vultus\n\nAnwesenheit _s._ praesentia\n\nApfel _s._ malum\n\nApoll _s_. Apollo\n\nArbeit, M\u00fche _s_. labor\n\nArznei, Arzt _s_. medicamen, medicamentum, medicus\n\nAsche _s_. cinis\n\nAuge _s_. oculus\n\nAusschweifungen _s_. luxuria\n\nAussehen, Antlitz _s_. facies\n\nAutor, Autorit\u00e4t _s_. auctor, auctoritas\n\n**B** \u00e4r _s_. ursus\n\nBarbar _s_. barbarus\n\nBauch _s_. venter\n\nBaum _s_. arbor\n\nBaumstumpf _s_. truncus\n\nBefehl _s._ imperium\n\nBegabung, Veranlagung _s_. ingenium\n\nBegierde, Verlangen _s_. cupiditas (cupere)\n\nBegleiter, Kamerad _s_. comes\n\nBeispiel _s_. exemplum\n\nBeredsamkeit _s_. eloquentia, eloquium, facundia\n\nBesitz, Reichtum _s_. possessio\n\nBeute _s_. praeda\n\nBewacher, Bewachung _s_. custodia, custos\n\nBeweglichkeit _s_. mobilitas\n\nBiene _s_. apis\n\nBild _s_. imago\n\nBitte, Gebet _s_. votum\n\nBlume, Bl\u00fcte _s_. flos\n\nBlut _s_. sanguis\n\nBogen _s_. arcus\n\nBrief _s_. epistula\n\nBrot _s_. panis\n\nBrust _s_. pectus\n\nBuch, B\u00fcchlein _s_. libellum, liber\n\nB\u00fcrger, B\u00fcrgerschaft _s_. civis, civitas\n\nBurg _s_. arx\n\n**C** hristen _s_. Christianus\n\n**D** ank _s_. gratia\n\nDenkmal _s_. monumentum\n\nDezember _s._ December\n\nDichter _s_. poeta, vates\n\nDieb _s_. fur\n\nDiener _s_. servus\n\nDirne _s_. meretrix\n\nDurst _s_. sitis\n\nDummheit _s_. stultitia\n\n**E** he _s_. coniugium\n\nEhre _s_. honos\n\nEhrerweisung _s_. reverentia\n\nEi _s_. ovum\n\nEifer, Studium _s_. studium\n\nEintracht _s_. concordia\n\nEisen _s_. aes, ferrum\n\nEitelkeit _s_. vanitas\n\nEnde, Grenze _s_. finis\n\nErbe _s_. heres\n\nErde _s_. tellus, terra\n\nErfolg _s_. successus\n\nErinnerung _s_. memoria\n\nErnte _s_. messis\n\nEsel _s_. asinus\n\nEssen, Speise _s._ c(o)ena, cibus\n\nEwigkeit _s_. aeternitas\n\n**F** aden _s_. filum\n\nFehler, Laster _s_. vitium\n\nFeind, Feindschaft _s_. hostis, inimicitia, inimicus\n\nFels, Stein _s_. saxum\n\nFeuer _s_. ignis\n\nFinger _s_. digitus\n\nFisch _s_. piscis\n\nFlamme _s_. flamma\n\nFl\u00fcgel, Feder _s_. penna\n\nFlu\u00df, Strom _s_. flumen, fluvius\n\nFrau, Ehefrau _s_. femina, mulier, uxor\n\nFreiheit _s_. libertas\n\nFreude, Vergn\u00fcgen _s_. gaudium, laetitia\n\nFreund _s_. amicus\n\nFreundschaft _s_. amicitia\n\nFrieden _s_. pax\n\nFr\u00f6mmigkeit _s_. pietas\n\nFuchs _s_. vulpes\n\nFurcht, Angst _s_. metus, timor\n\nFu\u00df _s_. pes\n\n**G** allien, Gallier _s_. Gallia, Gallus\n\nGedicht, Lied _s_. carmen, poema, poesis\n\nGeduld _s_. patientia\n\nGef\u00e4hrte _s_. socius\n\nGef\u00e4\u00df _s_. vas\n\nGefahr _s_. periculum\n\nGeist _s_. mens, spiritus\n\nGeistlicher _s_. clericus\n\nGeld _s_. aes, pecunia\n\nGelegenheit _s_. occasio\n\nGelehrter _s_. doctus (docere)\n\nGemeinde _s_. ecclesia\n\nGerechtigkeit _s_. iustitia\n\nGeruch _s_. odor\n\nGer\u00fccht _s_. fama, rumor\n\nGesch\u00e4ft _s_. negotium\n\nGeschichte _s_. fabula, historia\n\nGeschlecht _s_. gens\n\nGescho\u00df _s_. telum\n\nGesellschaft _s_. societas\n\nGesetz _s_. lex\n\nGesundheit _s_. salus, sanitas\n\nGewalt, Kraft _s_. vis\n\nGewinn, Profit _s_. lucrum\n\nGewissen, Bewu\u00dftsein _s_. conscientia\n\nGier, Habsucht _s_. avaritia\n\nGift _s_. venenum\n\nGlaube, Treue, Pflicht _s_. fides\n\nGleichmut _s_. aequa mens, aequanimitas, aequus animus\n\nGl\u00fcck, g\u00fcnstige Umst\u00e4nde _s_. res laetae\/lautae\/secundae\n\nGl\u00fcck, Schicksal _s_. fortuna\n\nGold _s_. aurum\n\nGott _s_. deus\n\nGras, Kr\u00e4uter, Halm _s_. herba\n\nGriechisch _s._ Achaicus, Achivi, Graecus\n\nGriffel _s_. stilum\n\n**H** abicht _s_. accipiter\n\nH\u00e4lfte _s_. dimidium, dividuum\n\nHahn, Huhn _s._ gallus, gallina\n\nHals _s_. collum\n\nHand _s_. dextera, manus\n\nHase, Rammler _s_. lepus\n\nHa\u00df _s_. odium\n\nHaus _s_. casa, domus\n\nHaut _s_. corium, cutis\n\nHeil _s_. salus\n\nHeilmittel _s_. cura, remedium\n\nHeirat _s_. nubere\n\nHerrschaft, Befehl _s_. imperium, regnum\n\nHerrscher, K\u00f6nig, K\u00f6nigin _s_. imperator, rex, regina\n\nHerz _s_. cor\n\nHilfe _s_. auxilium\n\nHimmel _s_. caelum\n\nHirte _s_. pastor\n\nHoffnung _s_. spes\n\nHornisse _s_. crabro\n\nHund _s_. canis\n\nHunger _s_. fames\n\n**I** rrtum, Verfehlung _s_. error\n\n**J** ahr _s_. annus\n\nJugend, junger Mann _s_. adolescentia, adolescens, iuventa, iuventus, iuvenis\n\n**K** adaver _s_. cadaver\n\nKampf _s_. pugna\n\nKind, Knabe, M\u00e4dchen _s._ puella, puer\n\nKleidung _s_. vestimenta, vestis\n\nKnochen _s_. os, ossis\n\nK\u00f6nig, K\u00f6nigin _s_. rex, regina\n\nK\u00f6rper _s_. corpus, corpusculum\n\nKohl _s_. crambe\n\nKollegium _s_. collegium\n\nKonsul _s_. consul\n\nKopf, Haupt _s_. caput\n\nKraft _s_. vis\n\nKrankheit _s_. morbus\n\nKraut _s._ herba\n\nKreis _s_. circulus\n\nKreuz, Keuzigung _s_. crux\n\nKrieg _s_. bellum\n\nKriegsdienst _s_. militia\n\nKriegsgl\u00fcck, -gott _s._ Mars\n\nK\u00fchnheit _s_. audacia\n\nKunst, K\u00fcnstler _s_. ars, artifex\n\nKu\u00df _s_. osculum\n\n**L** eben _s_. vita\n\nLebensalter, Zeit _s_. aetas, aevus\n\nLehrer(in) _s_. magister, magistra\n\nLeser _s_. lector\n\nLicht _s_. lumen, lux\n\nLiebender _s_. amare\n\nList _s_. dolus\n\nLob _s_. laus\n\nL\u00f6we _s_. leo\n\nLohn, Preis _s_. merces\n\nL\u00fcge, L\u00fcgner _s_. mendacium, mendax\n\nLust _s_. libido, voluptas\n\n**M** acht, Befugnis _s_. potestas\n\nM\u00e4dchen _s_. puella\n\nM\u00e4zen _s_. Maecenas\n\nMagen, Bauch _s_. stomachus\n\nMantel _s_. palli(ol)um\n\nMaske _s_. persona\n\nMa\u00df, Vorgehensweise _s_. modus\n\nMasse, Last _s_. massa, moles\n\nMaus _s_. mus\n\nMeer _s_. mare\n\nMehl _s_. farina\n\nMeinung, Worte _s_. sententia\n\nMilde _s_. clementia\n\nMi\u00dfbrauch _s_. abusus\n\nMitgift _s_. dos\n\nMitte, Mittelma\u00df _s_. medium, mediocritas\n\nMond _s_. luna\n\nM\u00fcnze, Pfennig _s_. nummus\n\nMund, Gesicht _s_. bucca; os, oris\n\nMuse _s_. Musa\n\nMu\u00dfe _s_. otium\n\nMutter _s_. mater\n\n**N** acht _s_. nox\n\nNahrung, Speise _s_. cibus\n\nNatur, Wesen _s_. natura\n\nNeid, Mi\u00dfgunst _s_. invidia, livor\n\nNerv _s_. nervus\n\nNetz _s_. rete\n\nNichts _s_. nihil, nil\n\nNotwendigkeit _s_. necessitas\n\nNu\u00df _s_. nux\n\nNutzen _s_. utilitas\n\n**O** chse _s_. bos\n\nOdysseus _s_. Ulixes\n\n\u00d6l _s_. oleum\n\n\u00d6sterreich _s_. Austria\n\nOhr _s_. auris\n\nOrt _s_. locus\n\n**P** apst _s_. papa\n\nPause, Ruhe _s_. (re)quies\n\nPest, Unheil _s_. pestis\n\nPfand _s_. pignus\n\nPferd _s_. equus\n\nPhilosoph, Philosophie _s_. philosophus, philosophia\n\nP\u00f6bel _s_. vulgus\n\n**R** abe _s_. corvus\n\nRache _s_. ultio, vindicta\n\nRachen, Schlund _s._ fauces\n\nR\u00e4tsel _s_. ambages\n\nRat(schlag) _s_. consilium\n\nRecht _s_. ius\n\nRede, Geschw\u00e4tz _s_. oratio, sermo\n\nReichtum _s_. divitiae, ops\n\nReiter _s_. eques\n\nReligion _s_. religio\n\nRichter, Zensor _s_. censor, iudex\n\nRose _s_. rosa\n\nRuder _s_. remus\n\nRuhe _s._ (re)quies\n\nRuhm _s_. gloria\n\nRuinen, Tr\u00fcmmer _s_. ruina\n\n**S** aat _s_. seges\n\nSaite _s_. chorda\n\nSalz _s_. sal\n\nSand _s_. (h)arena\n\nSchaden _s_. damnum, detrimentum\n\nSchaf _s_. ovis\n\nScham(haftigkeit) _s_. pudicitia, pudor\n\nSchatten _s_. umbra\n\nSchein, Aussehen _s_. species\n\nScherz, Spiel _s_. iocus\n\nSchicksal _s_. fatum, fors\n\nSchienbein _s_. crus\n\nSchiff _s_. navigium, navis\n\nSchiffbruch, Schiffbr\u00fcchiger _s_. naufragium, naufragus\n\n(kleiner) Schild _s_. parmula\n\nSchlaf _s_. somnus\n\nSchlange _s_. vipera\n\nSchmeichelei _s_. blanditia\n\nSchmerz _s_. dolor\n\nSchmied _s_. faber\n\nSch\u00f6nheit _s_. forma\n\nSchrecken, Entsetzen _s_. horror\n\nSchuld _s_. culpa\n\nSchulden _s_. aes alienum\n\nSchule _s_. schola\n\nSchwalbe _s_. hirundo\n\nSchwein _s_. porcus\n\nSchwelle _s_. limen\n\nSchwert _s_. gladius\n\nSeele, Geist _s_. anima, animus\n\nSegel _s_. velum\n\nSieg, Sieger _s_. victor, victoria\n\nSiegel _s_. sigillum\n\nSitte _s_. mos\n\nSklave, Sklaverei _s_. servus, servitium\n\nSkorpion _s_. scorpius\n\nSoldat _s_. miles\n\nSonne _s_. sol\n\nSorge _s_. cura\n\nSpargel _s_. asparagus\n\nSparsamkeit _s_. parsimonia\n\nSpeer, Lanze _s._ hasta\n\nSpiel _s_. ludere\n\nSpur _s_. vestigium\n\nStaat _s_. res publica\n\nStachel _s_. stimulus\n\nStadt _s_. urbs\n\nStahl _s_. adamas\n\nStein _s_. lapis\n\nStern _s_. astrum, sidus\n\nStille _s_. silentium\n\nStimme, Wort _s_. vox\n\nStolz _s_. superbia\n\nStrafe, Pein _s_. poena\n\nStreit _s_. lis, rixa\n\nStunde _s_. hora\n\nS\u00fcnde _s_. peccatum\n\nSzepter _s_. sceptrum\n\n**T** ag _s_. dies\n\nTat, Faktum _s_. acta, actio, facinus, factum\n\nTaube _s_. columba\n\nTeufel _s_. diabolus\n\nThron _s_. thronus\n\nTier, Bestie _s_. animal, bestia, fera\n\nTod _s_. letum (letalis), mors\n\nTopf _s_. olla\n\nTr\u00e4gheit _s_. inertia\n\nTr\u00e4ne _s_. lacrima\n\nTreue _s_. fides\n\nTriumph _s_. triumphus\n\nTroja, trojanisch _s_. Iliacus, Pergameus, Teucer, Troia, Troianus\n\nTrost _s_. consolatio, solacium, solamen\n\nTrunkenheit _s_. ebrietas\n\nTugend _s_. virtus\n\nTyrann _s_. tyrannus\n\n**\u00dc** bel _s_. malum\n\n\u00dcbereinstimmung _s_. consensus\n\nUngerechtigkeit _s_. iniquitas\n\nUngl\u00fcck, Not _s_. miseria\n\nUnheil, Not _s_. res adversae\/angustae\/arduae\/durae\/incertae\/malae\n\nUnrecht _s_. iniuria\n\nUnschuld _s_. innocentia\n\nUnterwelt _s_. Acheron, Avernus\n\nUnwissenheit _s_. ignorare\n\nUrteil _s_. iudicium\n\n**V** ater _s_. pater\n\nVaterland _s_. patria\n\nVerbrechen, Schuld _s_. crimen, scelus\n\nVergn\u00fcgen _s_. gaudium\n\nVers _s_. versus\n\nVerstand _s_. intellegere\n\nVerzeihung, Erlaubnis _s_. venia\n\nVieh _s_. pecus\n\nVogel _s_. avis\n\nVolk _s_. populus (popularis)\n\nVorstellung, Gedanke _s_. cogitatio\n\n**W** ache, W\u00e4chter _s_. custodia, custos\n\nWaffen _s_. arma\n\nWahnsinn _s_. dementia, furor, insania\n\nWahrheit _s_. veritas\n\nWald _s_. silva\n\nWasser _s_. aqua\n\nWeg _s_. via\n\nWein _s_. vinum\n\nWeinen _s_. fletus\n\nWeisheit _s_. sapientia\n\nWelt _s_. mundus, orbis\n\nWerk _s_. opus\n\nWespe _s_. vespa\n\nWille _s_. voluntas\n\nWind _s_. ventus\n\nWissenschaft _s_. scientia\n\nWohltat _s_. beneficium\n\nWohlwollen _s_. benevolentia\n\nWolf _s_. lupus\n\nWolke _s._ nubes, nubilum\n\nWort _s_. sententia, verbum\n\nW\u00fcrfel _s_. alea\n\nWunde _s_. vulnus\n\nWut _s_. furor\n\n**Z** ehe, Kralle _s_. unguis\n\nZeit _s_. tempus\n\nZeitalter _s_. aetas, aevum\n\nZeuge _s_. testis\n\nZiegenbock _s_. hircus\n\nZ\u00f6gern _s_. mora\n\nZorn _s_. ira, iracundia\n\nZ\u00fcgel _s_. frenum, lorum\n\nZufall, Schicksal _s_. casus\n\nZuh\u00f6rer _s_. auditor\n\nZunge _s_. lingua\n\nZweifel _s_. dubium\n\nZwietracht _s_. discordia\nRegister der Autoren und Quellen\n\n**A** ccius (170 \u2013 um 86 v. Chr.)\n\nAelian (um 170 \u2013 um 235 n. Chr.)\n\nAesop (?)\n\nAfranius (2. Jh. v. Chr.)\n\nAischylos (um 525 \u2013 um 456 v. Chr.)\n\nAlanus ab Insulis (1115\/28 \u2013 um 1202)\n\nAlbertus Magnus (um 1200 \u2013 1280)\n\nAlbrecht III. (um 1349 \u2013 1395)\n\nAldrovandi, Ulisse (1522\u20131605)\n\nAler, Paul (1656\u20131727)\n\nAlkaios (620\u2013580)\n\nAmbrosius (339\u2013397)\n\nAmmianus Marcellinus (330 \u2013 um 395 n. Chr.)\n\nAnselm von Canterbury (1033\u20131109)\n\nApokalypse\n\nApollinaris von Laodicaea (um 310 \u2013390 n. Chr.)\n\nApostelgeschichte\n\nApostolius (um 1420 \u2013 1453)\n\nApuleius (um 125 \u2013 um 180 n. Chr.)\n\nA(c)quaviva, Claudio (1543\u20131615)\n\nArchipoeta (um 1130 \u2013 1140)\n\nAristophanes (um 257 \u2013 um 180 v. Chr.)\n\nAristoteles (384 \u2013 322 v. Chr.)\n\nArnobius (2. H\u00e4lfte 3. Jh. \u2013 1. Drittel 4. Jh.)\n\nArrian (um 95 \u2013 175 n. Chr.)\n\nAthenaios (um 200 n. Chr.)\n\nAuctor ad Herennium (um 87 \u2013 82 v. Chr.)\n\nAugustinus (354 \u2013 430 n. Chr.)\n\nAugustus (63 v. Chr. \u2013 14 n. Chr.)\n\nAusonius (um 310 \u2013 um 393 n. Chr.)\n\n**B** acon, Francis (1561\u20131626)\n\nBenedikt, Hl. (um 480 \u2013 um 547 n. Chr.)\n\nBergmann, Joseph Ritter von (1796\u20131872)\n\nBismarck, Otto von (1815\u20131898)\n\nBoccalini, Traiano (1556\u20131613)\n\nBoethius (um 480 \u2013 524 n. Chr.)\n\nBonifaz VIII., Papst (um 1235 \u2013 1303)\n\nBorbonius, Matthias (1566\u20131629)\n\nBorgia, Cesare (1475\u20131507)\n\nBossius, Aegidius (Egidio Bossi; 1487\u20131546)\n\nBrant, Sebastian (um 1457 \u2013 1521)\n\nBusenbaum, Hermann (1600\u20131668)\n\n**C** aecilius Balbus\n\nCaesar (um 100 \u2013 44 v. Chr.)\n\nCalder\u00f3n de la Barca, Pedro (1600\u20131681)\n\nCassius Dio (um 155 \u2013 um 235 n. Chr.)\n\nCastelli, Carlo (gest. 1666)\n\nCato d. \u00c4. (234 \u2013 149 v. Chr.; und Pseudo-Cato)\n\n_Dicta_\n\n_Disticha_\n\n_Monosticha_\n\n_Sonstige_\n\nCatull (um 84 \u2013 um 54 v. Chr.)\n\n(Marcus Tullius) Cicero (106 \u2013 43 v. Chr.; und Pseudo-Cicero)\n\n_Academici_ _libri_\n\n_Ad Atticum_\n\n_Ad familiares_\n\n_De amicitia_\n\n_De divinatione_\n\n_De finibus_\n\n_De legibus_\n\n_De natura deorum_\n\n_De officiis_\n\n_De oratore_\n\n_De re publica_\n\n_De senectute_\n\n_In L. Calpurnium Pisonem_\n\n_In L. Catilinam_\n\n_In C. Verrem_\n\n_Orationes Philippicae_\n\n_Orator_\n\n_Paradoxa Stoicorum_\n\n_Partitiones oratoriae_\n\n_Pro T. Annio Milone_\n\n_Pro A. Cluentio Habito_\n\n_Topica_\n\n_Tusculanae disputationes_\n\nReden (weitere)\n\nSonstige\n\n(Quintus Tullius) Cicero (um 102 \u2013 43 v. Chr.)\n\nClaudian (um 373 \u2013 404 n. Chr.)\n\nClemens VII., Gegenpapst (1342\u20131394)\n\n_Codex Iustinianus_ (Teil des _Corpus iuris civilis_ ; entst. 528 \u2013 534 n. Chr.)\n\nColumella (1. Jh. n. Chr.)\n\nComenius, Johann Amos (1592\u20131670)\n\nCorippus (6. Jh. n. Chr.)\n\n_Corpus iuris canonici_ (12.\u201315. Jh.)\n\nCrotus Rubianus (Johannes J\u00e4ger; um 1480 \u2013 1545)\n\nCurtius Rufus (1. Jh. n. Chr.?)\n\nCyprianus (um 200 \u2013 258 n. Chr.)\n\n**D** a Buti, Francesco (um 1324 \u2013 1406)\n\nDamasus Bohemus (lehrte um 1210 \u2013 1227)\n\nde Bi\u00e8vre, Marquis (1747\u20131789)\n\nde Coubertin, Pierre (1863\u20131937)\n\nde Gr\u00e9court, Abb\u00e9 Jean-Joseph-Baptiste (1683\u20131743)\n\nDemokrit (um 470 \u2013 um 380 v. Chr.)\n\nDemosthenes (384 \u2013 322 v. Chr.)\n\nDescartes, Ren\u00e9 (1596\u20131650)\n\nDeuteronomium\n\n_Digesta_ (Teil des _Corpus iuris civilis_ ; 533 n. Chr.)\n\nDiogenes Laertius (2.\/3. Jh. n. Chr.)\n\nDiogenianus (1. H\u00e4lfte 2. Jh. n. Chr.)\n\nDryden, John (1631\u20131700)\n\nDu Bois-Reymond, Emil (1818\u20131896)\n\n**E** ichendorff, Joseph Freiherr von (1788\u20131857)\n\nElisabeth I. (1533\u20131603)\n\nElisabeth II. (geb. 1926)\n\nEnnius (239 \u2013 169 v. Chr.)\n\nEpiktet (um 50 \u2013 um 138 n. Chr.)\n\nEpikur (341 \u2013 270 v. Chr.)\n\nErasmus (um 1466 \u2013 1536)\n\nEuklid (um 300 v. Chr.)\n\nEuripides (um 485 \u2013 um 407 v. Chr.)\n\nEusebius (um 260 \u2013 339 n. Chr.)\n\nEutropius (4. Jh. n. Chr.)\n\nEzechiel\n\n**F** arinacius, Prosper (Daniel P. Farinacci; 1554\u20131618)\n\nFavonius (1. Jh. v. Chr.) \/ Favorinus (80\/90 \u2013 Mitte 2. Jh. n. Chr.)\n\n(Sextus Pompeius) Festus (2. Jh. n. Chr.)\n\nFeuerbach, (Paul Johann) Anselm Ritter von (1775\u20131833)\n\nFlorus (2. Jh. n. Chr.)\n\nFranck, Sebastian (1499 \u2013 um 1542)\n\nFriedrich III. (der Sch\u00f6ne; 1289\u20131330)\n\nFurius (1. Jh. v. Chr.)\n\n**G** alaterbrief\n\nGartner, Andreas (16. Jh.?)\n\nGellius (um 130 \u2013 um 170 n. Chr.)\n\n_Gesta Romanorum_ (13.\/14. Jh.)\n\nGoethe, Johann Wolfgang (1746\u20131832)\n\nGoffredus Tranensis (gest. 1245)\n\nGoscinny, Ren\u00e9 (1926\u20131977), und Uderzo, Albert (geb. 1927)\n\nGratian (Ende 11. Jh. \u2013 um 1160)\n\nGregor I., Papst (um 540 \u2013 601 n. Chr.)\n\nGregor IX., Papst (um 1170 \u2013 1241)\n\nGregor von Nazianz (330 \u2013 390 n. Chr.)\n\nGregor von Tours (um 538 \u2013 594 n. Chr.)\n\nGruter, Jan (1560\u20131627)\n\nil Guercino (Giovanni Francesco Barbieri; 1591\u20131666)\n\n**H** adrian (76 \u2013 138 n. Chr.)\n\nHagen, Hermann (1884\u20131898)\n\nHebr\u00e4erbrief\n\nHekaton (2. Jh. v. Chr.)\n\nH\u00e9nault, Jean-Fran\u00e7ois (1685\u20131770) 145\n\nHeraklit (um 550 \u2013 um 480 v. Chr.)\n\nHerodot (um 490 \u2013 430 v. Chr.)\n\nHesiod (um 700 v. Chr.)\n\nHeuseler, J. A. (19. Jh.?)\n\nHieronymus (um 347 \u2013 um 419 n. Chr.)\n\nHiob\n\nHippokrates (um 460 \u2013 um 370 v. Chr.)\n\n_Historia Augusta_ (vermutlich Ende 4. Jh. n. Chr.)\n\nHobbes, Thomas (1588\u20131679)\n\nHomer (vermutlich 8. Jh. v. Chr.)\n\nHoraz (65 \u2013 8 v. Chr.)\n\n_Ars poetica_\n\n_Carmen saeculare_\n\n_Carmina_\n\n_Epistulae_\n\n_Epodi_\n\n_Sermones_\n\nHosea\n\nHugo, Hermann (1588\u20131629)\n\nHuizinga, Johan (1872\u20131945)\n\nHutten, Ulrich von (1488\u20131523)\n\nHygin (um 60 v. Chr. \u2013 10 n. Chr.) 182\n\n**I** gnatius von Loyola (1491\u20131556)\n\nIustinus (2.\/3. Jh. n. Chr.)\n\n**J** akobusbrief\n\nJeremia\n\nJesaja\n\nJohannes\n\nJohannes XXIII., Papst (1881\u20131963)\n\nJohannes Klimakos (um 579 \u2013 um 639 n. Chr.)\n\nJulian (331 \u2013 363 n. Chr.)\n\nJulius III., Papst (1487\u20131555)\n\nJuvenal (58\/67 \u2013 um 127 n. Chr.)\n\n**K** antorowicz, Ernst (1895\u20131963)\n\nKarl V., Kaiser (1519\u20131556)\n\nKindleben, Christian Wilhelm (1748\u20131785)\n\nKleanthes (um 331 \u2013 233 v. Chr.)\n\nKohelet\n\nKorintherbriefe\n\nKosciuszko, Tadeusz (1746\u20131817)\n\n**L** aberius (106\/115 \u2013 43 v. Chr.)\n\nLaktanz (250\/260 \u2013 317 n. Chr.)\n\nLeibniz, Gottfried Wilhelm von (1646\u20131716)\n\nLeo I. (der Gro\u00dfe), Papst (gest. 460)\n\nLessing, Gotthold Ephraim (1729\u20131781)\n\nLiebs, Detlef (geb. 1936)\n\nLinn\u00e9, Carl von (1707\u20131778)\n\nLinus, Papst (1. Jh. n. Chr.)\n\n(Titus) Livius (59 v. Chr. \u2013 17 n. Chr.)\n\nLivius Andronicus (um 284 \u2013 um 204 v. Chr.)\n\nLothar I., Kaiser (795\u2013855)\n\nLotichius, Peter (1528\u20131560)\n\nLucilius (um 180 \u2013 um 101 v. Chr.) 47, 235, 247, 342, 344\n\nLudwig XI. (1786\u20131868)\n\nLudwig XIV. (1638\u20131715)\n\nL\u00fccke, Gottfried Christian Friedrich (1791\u20131855)\n\nLukan (39 \u2013 65 n. Chr.)\n\nLukas\n\nLukian (um 120 \u2013 Ende 2. Jh. n. Chr.)\n\nLukrez (um 99 \u2013 um 55 v. Chr.)\n\nLuther, Martin (1483\u20131546)\n\n**M** acarius Hieromonachus (um 300 \u2013 um 390 n. Chr.)\n\nMacrobius (um 400 n. Chr.)\n\nManilius (1. H\u00e4lfte 1. Jh. n. Chr.)\n\nManlius, Johannes (vermutlich 15. Jh.)\n\nMarkus\n\nMartial (um 40 \u2013 um 103 n. 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Chr.)\n\n_Pervigilium Veneris_ (3.\/4. Jh. n. Chr.)\n\nPetron (gest. 66 n. Chr.; und Pseudo-Petron)\n\nPetrus Abelardus (1079\u20131142)\n\nPetrus Blesensis (um 1135 \u2013 1204)\n\nPetrus Damiani (1007\u20131072)\n\nPhaedrus (um 50 n. Chr.)\n\nPico della Mirandola, Giovanni (1463\u20131494)\n\nPindar (522\/518 \u2013 um 446 v. Chr.) 236\n\nPius II., Papst (1405\u20131464)\n\nPlaton (um 428 \u2013 um 348 v. Chr.)\n\nPlautus (um 250 \u2013 um 184 v. Chr.)\n\n_Amphitruo_\n\n_Asinaria_\n\n_Aulularia_\n\n_Bacchides_\n\n_Captivi_\n\n_Casina_\n\n_Cistellaria_\n\n_Curculio_\n\n_Epidicus_\n\n_Menaechmi_\n\n_Mercator_\n\n_Miles gloriosus_\n\n_Mostellaria_\n\n_Persa_\n\n_Poenulus_\n\n_Pseudolus_\n\n_Rudens_\n\n_Stichus_\n\n_Trinummus_\n\n_Truculentus_\n\nPlinius d. \u00c4. (um 23 \u2013 79 n. Chr.)\n\nPlinius d. J. (um 61 \u2013 um 113 n. Chr.)\n\nPlutarch (um 46 \u2013 um 125 n. Chr.)\n\nPope, Alexander (1688\u20131744)\n\nPorphyrius (4. Jh. n. Chr.)\n\nPosidonius (135 \u2013 151 n. Chr.)\n\nPriscian (5.\/6. Jh. n. Chr.)\n\nProperz (um 50 \u2013 16 v. Chr.)\n\nProtagoras (um 480 \u2013 421 v. Chr.)\n\nPrudenz (348 \u2013 um 405 n. Chr.)\n\nPsalmen\n\nPublilius Syrus (1. Jh. v. Chr.)\n\nPythagoras (um 570 \u2013 um 480 v. Chr.)\n\nPytheas (2. H\u00e4lfte 4. Jh. v. Chr.)\n\n**Q** uintilian (um 35 \u2013 um 100 n. Chr.; und Pseudo\u2013Quintilian)\n\n**R** abelais, Fran\u00e7ois (um 1494 \u2013 1553)\n\n_Regimen sanitatis Salerni_ (11. Jh.; vermutlich von Johannes von Mailand)\n\nRicci, Lorenzo (1703\u20131775)\n\nR\u00f6merbrief\n\nRollenhagen, Gabriel (1583\u20131619)\n\nRoon, Albrecht von (1803\u20131879)\n\nRudolf IV., Herzog (1339\u20131365)\n\nRuggle, George (1575\u20131622)\n\n**S** allust (86 \u2013 35 v. Chr.)\n\nSanuto, Marino (1466\u20131535)\n\nSchiller, Friedrich (1759\u20131805)\n\nSchott, Caspar (1608\u20131666)\n\nSeneca (d. J.; 4 v. Chr. \u2013 65 n. 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Chr.)\n\n**T** acitus (um 55 \u2013 um 120 n. Chr.)\n\nTaubmann, Friedrich (1565\u20131613)\n\nTerentianus Maurus (2.\/3. Jh. n. Chr.)\n\nTerenz (195\/185 \u2013 159 v. Chr.)\n\n_Adelphoe_\n\n_Andria_\n\n_Eunuchus_\n\n_Heauton timorumenos_\n\n_Hecyra_\n\n_Phormio_\n\nTerrasson, Abb\u00e9 Jean (1670\u20131750)\n\nTertullian (um 160 \u2013 um 220 n. Chr.)\n\nTheognis (6.\/5. Jh. v. Chr.)\n\nTheokrit (um 310 \u2013 um 250 v. Chr.)\n\nThiers, Adolphe (1797\u20131877)\n\nThomas a Kempis (um 1379 \u2013 1471)\n\nThomas von Aquin (1225\u20131274)\n\nThomas von Celano (gest. um 1260)\n\nTibull (um 50 \u2013 um 17 v. Chr.)\n\nTimotheusbriefe\n\nTitusbrief\n\nTobias\n\nTurgot, Anne Robert (1727\u20131781) 91\n\n**U** derzo, Albert _s._ Goscinny, Ren\u00e9\n\n**V** alerius Maximus (1. H\u00e4lfte 1. Jh. n. Chr.)\n\nVarro (116 \u2013 27 v. Chr.)\n\nVegetius (Ende 4. Jh. n. Chr.)\n\nVenantius Fortunatus (um 530 \u2013 um 600 n. Chr.)\n\nVergil (70 \u2013 19 v. Chr.)\n\n_Aeneis_\n\n_Bucolica_\n\n_Georgica_\n\nFragmente\n\nVerrius Flaccus (gest. 14 n. Chr.)\n\n**W** alter von Ch\u00e2tillon (um 1135 \u2013 um 1200)\n\nWalther, Hans (1884\u20131971; zit. als: HW und HWS)\n\nWeisheit\n\n**X** enophon (um 430 \u2013 354 v. Chr.) 232\n\n**Z** amoyski, Jan (1542\u20131605)\n\nZefanja\n\nZenobius (1.\/2. Jh. n. Chr.)\n\nZincgref, Julius Wilhelm (1591\u20131635)\nHinweise zur E-Book-Ausgabe\n\nDie E-Books des Reclam-Verlags verwenden entsprechend der jeweiligen Buchausgabe Sperrungen zur Hervorhebung von Textpassagen. Diese Textauszeichnung wird nicht von allen Readern unterst\u00fctzt.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nPraise for  \nOVERDELIVER\n\n_\"Nobody has paid closer attention to the world of direct response advertising than Brian Kurtz. His new book\u2014his opus magnum\u2014brings that world to you, in great detail. This is a must-have book for anyone in the business . . . and anyone seriously interested in how people decide what, when, and how to buy.\"_\n\n\u2014 **Bill Bonner** , founder, The Agora\n\n_\"In_ Overdeliver _, Brian chronicles almost four decades of hardcore experience that comes with building an iconic $100 million direct marketing company (and then a multimillion-dollar business on his own)\u2014and then committing to teaching all he has learned, condensing it down into something you can absorb in an afternoon. Brian Kurtz is a different kind of author. He's forgotten more about marketing and sales than most newcomers will ever discover.\"_\n\n\u2014 **Frank Kern**\n\n_\"I've known Brian Kurtz for over 20 years, and he is a direct marketing and direct mail genius. . . . In_ Overdeliver _, Brian shares how you can use offers, copy, multichannel marketing, continuity and a lot more to be a better marketer, grow your business, make more money and make a big impact in your market. If you want to learn from one of the greatest marketing minds alive, read this book.\"_\n\n\u2014 **Joe Polish** , founder, Genius Network and Genius Recovery\n\n_\"Filled with more useful information and history than any other book I'm familiar with. . . . You should dramatically improve your results after reading this book.\"_\n\n\u2014 **Joe Sugarman** , direct marketing legend\n\n_\"Brian Kurtz was not only one of the wunderkinds who practically invented DM publishing in the 1980s, but he was also one of the elder statesmen of the industry who guided its transition from paper to electronics in the 2000s. [In this book he] assimilates the best insights and ideas of the industry's best and brightest business leaders._ Overdeliver _is a must-read.\"_\n\n\u2014 **Mark Ford** , chief growth strategist, The Agora, and author of two dozen books, including the bestseller _Ready Fire Aim_\n\n_\"Brian . . . delivers a career manual and strategy book that covers every major principle that will affect your bottom line. This book should be on every marketer's shelf and its case studies should be burned into every marketer's muscle memory.\"_\n\n\u2014 **Perry Marshall** , author, _80\/20 Sales & Marketing, Ultimate Guide to Google AdWords_, and _Ultimate Guide to Facebook Advertising_\n\n_\"There's a very small group of true direct response wizards out there . . . [and Brian is] one of the guys who can change your career and business in a heartbeat with just a casually tossed off piece of advice. . . . This book is your invitation to join that elite conversation. To finally learn the mind-sets, tactics, techniques, and habits of the best in the biz. Do NOT read this book. Instead, devour it. Insert it into your brain and make it part of your DNA. It's that tight.\"_\n\n\u2014 **John Carlton** , copywriting know-it-all, grizzled consultant, and author of _The Entrepreneur's Guide to Getting Your Shit Together_\n\n_\"Brian Kurtz's_ Overdeliver _refreshes and reminds of the elemental and critical forces that drive response while providing us with all new ways of creating new opportunities in direct marketing. . . . Destined to be one of the classics.\"_\n\n\u2014 **Greg Renker** , co-founder, Guthy-Renker\n\n_\"_ Overdeliver _is the bridge that connects the timeless fundamentals of direct response marketing to the state-of-the-art marketing strategies, tactics, and channels of today. . . . Brian Kurtz is a living legend . . . nobody but him could have written this book. I consider it to be among the must-read direct marketing books that should be on the bookshelf of every marketer, every copywriter, and every entrepreneur.\"_\n\n\u2014 **Ryan Levesque** , #1 national best-selling author of _Choose_ and _Ask_\n\n_\"Brian Kurtz knows real marketing better than almost anyone. This book is a must-read for marketers of all generations. From tried and true modern marketing tactics to pearls of 'old school' wisdom,_ Overdeliver _provides a roadmap for jump-starting any organization's marketing efforts.\"_\n\n\u2014 **Ryan Deiss** , founder and CEO, Digital Marketer\n\n_\"People claim to be marketing experts when their actual results are never tracked or measured. That is exactly what_ Overdeliver _is_ not _about. Brian has almost 40 years of obsessing, learning and mastering direct response marketing and now he's sharing those secrets in this brilliant book. Read it and implement as fast as humanly possible.\"_\n\n\u2014 **Dean Graziosi** , _New York Times_ best-selling author of _Millionaire Success Habits_ and serial entrepreneur\n\n_\"Whatever you are paying for this book, you're paying too little. Marketing is the lifeblood of your business and this book should be required reading for everyone.\"_\n\n\u2014 **Ryan Lee** , founder of REWIND\n\n_\"Why do successful entrepreneurs pay Brian Kurtz $20,000 or more per year to be in his Titans Mastermind Group? You'll see for yourself when you read this book.\"_\n\n\u2014 **Steve Dworman** , infomercial writer\/producer and author of _$12 Billion of Inside Marketing Secrets Discovered Through DRTV_\n\n_\"Brian has become one of the industry's leading mentors\u2014sharing his knowledge of direct marketing and well-honed strategies with some of the fastest-growing companies in our space. Any event he puts on or book he writes will always be at the top of my list\u2014and it should be at the top of yours too.\"_\n\n\u2014 **Kim Krause Schwalm** , A-list copywriter and copy mentor\n\n_\"If you want to succeed in marketing and DON'T buy this book, you'd better pray your competition can't read.\"_\n\n\u2014 **Arthur P. Johnson** , copywriter, product developer, and infomercial star\n\n_\"Got a low-priced product that you're having trouble selling due to high media costs? Brian reveals how he sold $2 billion worth of products, $39 at a time. Do you think Facebook invented look-alike audiences? Look inside this book and you'll find out how Brian and his team were creating regression models (better than look-alikes) back when Zuck was still in diapers! Do yourself a favor and read this book.\"_\n\n\u2014 **Parris Lampropoulos** , marketing strategist and copywriter\n\n_\"For years, I've given to students of marketing a list of 14 must-read books\u2014I've now added Brian's book to a new list of 15._ Overdeliver _is the best direct marketing book since his and my mentor Dick Benson's 1987_ Secrets of Successful Direct Mail. _\"_\n\n\u2014 **Richard A. Viguerie** , pioneer of political direct mail and chairman of American Target Advertising, Inc.\n\n_\"If the word 'marketing' makes you cringe, this book is for you. And if the word 'marketing' makes you salivate, this book is for you._ Overdeliver _is loaded with time-tested strategies and invaluable ideas . . . a marketing balm to the sore business soul.\"_\n\n\u2014 **Victoria Labalme** , performing artist and performance strategist and member, Speaker Hall of Fame\n\n_\"Over the years, I've paid Brian tens of thousands of dollars for his money-making, business-building, life-changing advice and insights. Now they can be yours for the price of this book.\"_\n\n\u2014 **David Deutsch** , A-list copywriter and author of _Think Inside the Box!_\n\n_\"If you want to make a lot of money in direct mail, e-commerce, TV, radio, consulting, yard sales, or lemonade,_ Overdeliver _is the roadmap that can make you relentlessly richer. How can I be sure? Brian is directly responsible for generating millions of dollars in income for my business. He can do the same for you . . . all you have to do is listen.\"_\n\n\u2014 **Eric Betuel** , copywriter, president of Imagination Inc.\n\n_\"I just read a new 'classic' for direct response marketers! I learned\u2014and relearned\u2014so much. Overdeliver is a MUST READ for newbies\u2014as well as experienced marketers and copywriters.\"_\n\n\u2014 **Carline Anglade-Cole** , copywriter and consultant, Cole Marketing Solutions\n\nALSO BY BRIAN KURTZ\n\n_The Advertising Solution_ with Craig Simpson\n\n\u25a0\n\nCopyright \u00a9 2019 by Brian Kurtz\n\n**Published in the United States by:** Hay House, Inc.: www.hayhouse.com\u00ae \u2022 **Published in Australia by:** Hay House Australia Pty. Ltd.: www.hayhouse.com.au \u2022 **Published in the United Kingdom by:** Hay House UK, Ltd.: www.hayhouse.co.uk \u2022 **Published in India by:** Hay House Publishers India: www.hayhouse.co.in\n\n_Indexer:_ Joan Shapiro\n\n_Cover design:_ Scott Breidenthal\n\n_Interior design:_ Nick C. Welch\n\nAll rights reserved. No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the form of a phonographic recording; nor may it be stored in a retrieval system, transmitted, or otherwise be copied for public or private use\u2014other than for \"fair use\" as brief quotations embodied in articles and reviews\u2014without prior written permission of the publisher.\n\n**Library of Congress Cataloging-in-Publication Data**\n\nNames: Kurtz, Brian, author.\n\nTitle: Overdeliver : build a business for a lifetime playing the long game in direct response marketing \/ Brian Kurtz.\n\nDescription: 1st Edition. | Carlsbad : Hay House Inc., 2019.\n\nIdentifiers: LCCN 2018059646| ISBN 9781401956752 (hardback) | ISBN 9781401956769 (ebook)\n\nSubjects: LCSH: Electronic commerce. | Marketing. | BISAC: BUSINESS & ECONOMICS \/ Marketing \/ Direct. | BUSINESS & ECONOMICS \/ E-Commerce \/ Internet Marketing. | BUSINESS & ECONOMICS \/ Marketing \/ General.\n\nClassification: LCC HF5548.32 .K867 2019 | DDC 658.8\/72--dc23 LC record available at <https:\/\/lccn.loc.gov\/2018059646>\n\n**Hardcover ISBN:** 978-1-4019-5675-2\n\n**E-book ISBN:** 978-1-4019-5676-9\n\n**E-audio ISBN:** 978-1-4019-5691-2\n\n10 9 8 7 6 5 4 3 2 1\n\n1st edition, April 2019\n\nPrinted in the United States of America\n_No one can do anything valuable in life  \nwithout a loving and supportive family, and  \nwithout Robin, Alex, and Madeline, I'd be lost._\nCONTENTS\n\n_Foreword by Jay Abraham_\n\n_Introduction_\n\n**CHAPTER 1:** Overdeliver\n\n**CHAPTER 2:** Original Source\n\n**CHAPTER 3:** How Paying Postage Made Me a Better Marketer\n\n**CHAPTER 4:** List Building & RFM\n\n**CHAPTER 5:** Offers\n\n**CHAPTER 6:** Creative & Copy\n\n**CHAPTER 7:** Multichannel Marketing\n\n**CHAPTER 8:** Customer Service & Fulfillment\n\n**CHAPTER 9:** Continuity & LTV\n\n**CHAPTER 10:** Playing the Long Game\n\n_Further Reading_\n\n_Index_\n\n_Acknowledgments_\n\n_About the Author_\nFOREWORD\n\nI rarely endorse books, let alone write forewords. Why? Well, I don't want to sound brash, but I don't want my reputation for preeminence being associated with mediocre material. But writing a foreword for this book is an exception I'm ecstatic to make. I am putting the full force of my reputation and endorsement on the line for _Overdeliver_ by my dear friend Brian Kurtz.\n\nThis book you are about to devour (and believe me, it's like that famous potato chip commercial\u2014you won't be able to read just one chapter) is going to explode the market of mastery. What daily workouts are to your muscles or what nutrients are to your mind and body, this book will be to building businesses that last a lifetime.\n\nSome time ago, Brian co-wrote a book about the greats of direct response advertising. It was very informative, but I felt it didn't scratch the surface of his knowledge, brilliance, and teaching ability. Brian possesses a fascinating and ever-evolving way of thinking and a truly original mind.\n\nSo I challenged Brian with an audacious reward if he could blow the roof off everything else that had been written to date about direct response marketing. I challenged him to painstakingly create a hybrid book that would encompass the countless real-life case studies from his own career as well as his world-class colleagues' success stories.\n\nI wanted to see a unique integration of his vast, incredibly effective knowledge along with a clear identification of how readers of these success stories could apply these enduring, immutable principles (which no one talks about) to their marketing. If he could interweave these stories, examples, and case studies into a one-of-a-kind format\u2014where every page contained a story, an analogy, or an action step\u2014not only would I write one of the most compelling forewords ever, I would personally back the success of the book.\n\nI knew unreservedly that he could do it, so I felt confident making perhaps the most outrageous guarantee ever offered in order to induce you to read and keep rereading this book until the message becomes ingrained permanently in your mind and guides all you do.\n\nBrian, of course, outdid himself, so now I'm on the hook to make good on my promise. And so I personally guarantee you\u2014everyone who purchases this book and reads it all the way through, with a notepad and pen to identify the most relevant ideas and concepts that apply directly to them\u2014that I will personally write a check for the purchase price if Brian's ideas fail to propel a meaningful improvement in not one but _at least three_ facets of your business.\n\nDescribing _Overdeliver_ as a mere marketing book is a disservice both to Brian and to you as the reader. I don't want this groundbreaking compendium to be misperceived as something light or general or a regurgitation of anything written before it. Brian's approach, experience, and strategies for transforming theoretical material into actionable systems are stunningly unique.\n\nIn direct response marketing, a key action is to tell the prospect what they're going to learn, tell them how to learn it, and then tell them what they just learned\u2014but to do it in different enough ways so that the power of the message cannot be missed or denied. Brian does this to perfection throughout the book. I found myself reveling in the way he shares a lifetime of lessons and a world of insights, perspectives, and updated and modified implementations in this fast-moving, nontheoretical world in such a way that we cannot help but see the tremendous value of each lesson.\n\nYou will be unimaginably changed as a strategist, a marketplace psychologist, and a consequential thinker after reading this book, and even better, become a far more masterful marketer than the person you were before. This is no mere brain dump. Brian has achieved a profound knowledge transfer by adroitly intertwining stories with principles, principles with applications, applications with implications of how it all directly applies to you. He ties together the principles of direct marketing in a cohesive way that will improve your business\u2014no matter the type, size, or marketing methods used\u2014and in so doing, he succeeds in achieving something rare indeed.\n\nThis book will be a true catalyst for your growth\u2014with Brian acting as your measurable marketing alchemist, transforming knowledge into fuel that can power and propel enormous growth and prosperity for years to come. But the economic rewards you'll achieve are equaled only by the powerhouse of newfound understanding you will gain from this fresh new slant on direct response marketing.\n\nIt's probably best now to provide some teasers (or what Brian refers to as \"fascinations\" in his chapter on creative and copy) to stimulate your desire and forge your perception of what this book is and what it isn't. Here are a few of my favorite insights from this book:\n\n  * Why almost every marketer has a limited understanding of all that marketing really entails\n  * Where great ideas come from and how to generate more breakthrough concepts in your industry than all your competitors combined\n  * Why today's marketer almost always underperforms the capabilities of their marketplace and which activities they could pursue to precisely identify where the biggest underperforming opportunities lie\n  * A better formula for multiplying results by meeting prospects where their response buttons are red hot and always making it easier for them to say yes than no to your offers\n  * Where the maximum money is in your business and industry and how to mine the richest veins in a consistent, sustainable way\n  * A creativity switch for turning on (and up) your creative abilities\u2014and those of your copywriters and creative partners\n  * Understanding markets: identifying who's got the most money to spend in your market and why\n  * Changing the game you play, making new rules your competition can't compete with or win at\n\nHere's a factoid you might find interesting: One of my good friends followed the basic lessons Brian shares here and\u2014taking only $100 but measuring and maximizing every element\u2014turned it into over a billion dollars in sales. This is no exaggeration and should give you some food for thought regarding just how valuable this book could be to you!\n\nThere's a famous book both Brian and I love, _Reality in Advertising_ , which Brian talks about in more depth in Chapter 2. It sets out some of the most important realizations and understanding and underrecognized truths about advertising ever discovered. I'd say that what _Reality in Advertising_ is to that market, _Overdeliver_ is to all forms of business marketing today. In summary, no one in any stage of marketing should fail to read Brian Kurtz's _Overdeliver_ at least three times. It's a new classic.\n\nI'm proud, thrilled, and thoroughly impressed at the outsize contribution Brian will make to your marketing mastery and business success in these pages. I feel privileged to have been able to write this foreword, and I'm excited for the impact this book is about to have on your marketing and business life. I could say much more\u2014but you get that I love this book and the way it's written and presented. But even beyond that, I love the enormity of usable maxims it constantly and beautifully provides.\n\nI hope these words encourage you to read _Overdeliver_ carefully and often.\n\n\u2014 Jay Abraham\nINTRODUCTION\n\n_Those who did it have an obligation to teach it;  \nthose who teach gain many riches._\n\nOn my first day as a direct marketer, a crazed movie buff with a revolver shot the president of the United States. Against the odds, President Reagan survived and saw his popularity rate rise to over 70 percent as a result. But I was also quite lucky that day. While it was far from a full day of work, with everyone glued to the TV set watching the news unfold, it was the day I started working for Marty Edelston and his scrappy little company, Boardroom Reports.\n\nOver the next three decades, Boardroom went from being a fairly unknown publisher to an iconic brand and direct marketer, and I'm proud to say I had a direct hand in that exponential growth. I went from being a junior list manager to executive vice president of the company, working across the most competitive niches with the best copywriters and direct marketers the world has ever seen. Marty saw to it that my education was absolutely immersive and, as one of the titans of the industry, he set an incredible standard to live up to every day.\n\nA few weeks before that pivotal day early in 1981, a headhunter called to ask me if I would like to interview for a job at this up-and-coming newsletter company called Boardroom Reports as an in-house list manager. Like I knew what that meant.\n\nHaving graduated with an English degree, I had been working at my first job with a company in the theater industry (mainly because they were the only ones who would hire me). I liked the sound of this cool young company. Marty seemed so entrepreneurial and creative that I decided to see where things would go, even though I had no idea what a list manager did. So I accepted the role, thinking that I would move over to editorial as soon as I had the chance, and started sponging up everything I could about lists in the meantime. The job felt like a good fit, and soon I was joking that I had been voted \"most likely to become a list manager\" in my high school yearbook.\n\nThe role was essentially to be a salesman. My job was to make sure everyone in the direct mail community was renting (and mailing to) the Boardroom lists, which were some of the most responsive lists in the industry.\n\nA year after getting started, I went into Marty's office to apply for a junior editorial role that had opened up. Despite having taken to lists like a natural, enjoying myself, and starting to excel beyond people who had been at it much longer, I thought I \"should\" be in editorial, since it was closer to what I'd studied.\n\nBut Marty took a hard look at me and said, \"I think you have a nose for marketing. I think that you should stay on the list side of the business and learn everything you can about direct marketing.\"\n\nLittle did I know at the time that following Marty's advice that day would lead to a career I never could have imagined. Like any great direct marketer, Marty understood the critical importance of lists. He was one of the best direct marketers in the business, and here he was telling me that he saw a career for me there. I had tremendous respect for Marty, so if he said I was good, then I was going to listen. I stayed put, and over the coming months, I learned what a list manager did and I learned all about what list brokers did as well. It was the list brokers who represented the mailers (i.e., marketers), and because they bought all the media (lists), they were among the most powerful and wealthy folks in the direct mail business. Many of them were making as much money as the stock brokers enjoying the boom on Wall Street at the same time.\n\nLearning that a list broker was not a stock broker was an important early lesson. Every list manager was obsessed with getting their lists into the hands of the right brokers, and there's where I found an edge, because my obsession was a little different. The most important thing I did was to make time to talk and meet with other list managers\u2014my \"competitors\"\u2014and not just the list brokers. I felt like I could never get enough time talking with the other list managers. They all had something to teach me, and every time I picked up a bit of new information from one of them, selling Boardroom's lists got easier for me.\n\nOn the surface, these encounters took away from selling time with the list brokers (since it was the brokers, not the other managers, who were responsible for my livelihood). But I was playing a different game, and I was willing to give up some short-term income to spend time sizing up how our competitors were selling the lists in their portfolios. What I discovered changed everything for us at Boardroom and has stood me in good stead for the rest of my career\u2014the lesson is what I now call competition as coexistence.\n\nOver the next few years, I earned the reputation of being the best list manager in the country. It all started with talking to list managers more than list brokers to build my career.\n\nYears later, as I adapted my list knowledge to all areas of marketing, I became Marty's partner in every aspect of the business. I had the freedom to think of new ideas daily, with phenomenal resources at my fingertips: the best copywriters, consultants, database experts, list mavens, and numbers gurus, and a database of customers and former customers that fluctuated between five and ten million names. Those names\u2014real people\u2014would become the cornerstone of our success.\n\nMarty used to say that we were in the business of saving lives, that as a result of what we were selling, people were saved from financial ruin and health disasters. In the end, our success all came down to becoming experts on our lists, building deep empathy with our audience, and knowing the market better than anyone else\u2014and doing all this long before we ever tried to figure out the best offers and the best copy. We learned that to create a company that was truly transformational, the list (again, we're talking about real people here) has to be an obsession.\n\n* * *\n\nAs time went on and Boardroom became more and more successful, Marty and I talked a lot about the obligation we had to teach what we knew . . . so much so that our financial advisors would beg us to do a little less teaching and a little more selling. But everything we were doing was so powerful that we knew we had a unique opportunity to share the abundance with other people who were willing to do the work.\n\nEugene (Gene) M. Schwartz, who I'll be talking a lot about in this book, was one of the greatest copywriters and Renaissance men who ever lived, and he was a key figure in this resolution to teach as much as we could. His death inspired us to republish his lost classic, _Breakthrough Advertising_ , for a new generation.\n\nMarty was not a sentimental guy. A few days after Gene's funeral, instead of waxing poetic, Marty focused on the lessons Gene had left us. He wanted to bring those lessons to the marketers and copywriters coming up at the time, as quickly as possible, so we got to work on the new edition of Gene's book right away. Marty was all business, always driving forward.\n\nSo when Marty died in 2013, I knew I had to continue holding up my promise to teach everything we'd learned over the 32 years we worked together, from list selection and running campaigns to maximizing relationships with customers and competitors. This book is how I fulfill that commitment. I accept that some young marketers might look at me like I'm some dinosaur from a bygone era; but while I playfully include \"T. rex\" in my e-mail address, that doesn't mean this book is going to be a stroll down memory lane. Instead, it's a way to share everything I've learned over a long career in direct marketing and to help the next generation of marketers understand that the principles of direct response are both eternal and essential.\n\nI would like this book to serve as the bridge between those principles and the state-of-the-art multichannel marketing world of today. I believe it's my responsibility to pass on the fundamental lessons of direct response marketing to today's marketers and entrepreneurs who are hungry to learn them and apply them. Other people have already written about many of the marketing concepts you'll find in this book: product development and market selection, recency-frequency-monetary (RFM) scores, list segmentation, lifetime value (LTV), continuity programs (i.e., membership programs with a renewal component). I'm not the first nor will I be the last to write and talk about any of that.\n\nWhat I _am_ first at is putting it all under the umbrella of my experience and combining the ideas in light of everything I've worked on throughout my career. I have a unique position among marketers, as my journey took me from building a large direct marketing company offline to moving online and bringing some of the biggest players in the industry along with me.\n\nI've been in some form of the direct response business for nearly 40 years, seeing the same principles work across all channels, offline and online. I've been able to take a lot of previously disconnected ideas and connect them in ways that make 1 + 1 = 64. (This is a concept that originally came from Gene Schwartz\u2014that when you get the right combination of ideas and experience, growth becomes exponential instead of staying linear.)\n\nMaybe you're new to copywriting, marketing, and entrepreneurship and wanting to improve your skills; maybe you're a seasoned marketer who is looking for a few new ideas, confident that you'll be able to have a huge impact with what you find. Either way, you've probably run into quite a few entrepreneurs who are uncomfortable with the idea that marketing is critical to every kind of business. You might have even met a few people who view marketing as a necessary evil (this could even have been you!).\n\nI often say that marketing isn't everything . . . it's the only thing. That might sound like I only care about selling and making money, but it's just the opposite. When marketers understand what they're doing at the deepest level and act with integrity and purpose, their work becomes a vehicle for bringing positive change into the lives of the people they serve.\n\nBut let me be clear: It doesn't matter if your product or service saves lives, transforms businesses, heals relationships, or just makes you outrageously wealthy. It cannot and will not make an ounce of difference to anyone until you decide to aggressively sell it (within the boundaries you determine are right for your business). Having a direct marketing focus was the key to that growth as we moved from direct mail and print to TV and the Internet. It's all about getting in front of the people who need you most. This book is going to teach you how to sell aggressively without ever losing sight of the people you are selling to or compromising the respect and care they deserve.\n\nThis book is about _direct marketing_ , which is _measurable marketing_ in any medium. While direct mail was the medium I cut my teeth on (and I'm proud of it), I cringe when marketers think that direct marketing is only about direct mail.\n\nThis book is _not_ about general advertising, or what we call the Mad Men model: advertising that builds brand awareness, and maybe evokes an emotional response, but has no way for the potential customer to actually _respond_ to the ad or for the marketer to track that response. With most general advertising, you can't measure how effective the ad was because there's no way to measure whether the customer takes an action as a result of the ad. Direct marketing, on the other hand, is all about measurability and therefore accountability. It doesn't matter what medium we are talking about, and it doesn't matter if we are talking about marketing offline (like direct mail) or online (like e-mail).\n\nAll the advertising you pay for, regardless of the medium, should be measurable, which is why direct marketing is so powerful. It's the only way to get a specific return on your investment\u2014every time you run a campaign, there has to be some way to measure it. Whether it's including a phone number, an e-mail address, or a web address to respond to, there must be some way to measure the response to the ad. Having some kind of response device with every marketing message you send in any medium\u2014one that enables you to measure effectiveness and profitability\u2014is the only way to know if your marketing is really working for the people in your target audience.\n\nBefore we dive into the lessons and obsessions I gathered over a long career in direct marketing, here are nine fundamentals for you to keep in mind as you work through the book (whether you are directly involved in marketing products and services or are an entrepreneur who needs to hire marketing talent).\n\n1. Be Involved in All Your Marketing Efforts, Even If You Outsource Everything\n\nMost people with a background in direct response probably won't need more encouragement on this one\u2014we're all data junkies who love getting our hands dirty on every kind of campaign. But if your role also demands that you focus on other things outside of marketing (product development, finances, hiring, etc.), make sure you never let go of the marketing reins completely. Keep an eye on everything that everyone is doing, both inside and outside your organization.\n\nThe best people I ever hired internally were the people in operations, analytics, and finance who understood and recognized the value of direct marketing. Being on the same page meant we could move so much faster than our competitors. When everyone on your team is on the same page, you don't get slowed down by red tape. It might sound like a big ask to be hands-on with marketing when it's not your core competency or to only hire people who love marketing, no matter their job title. But remember that every marketing message is a representation of you and your brand. Look after your marketing like you would your own child\u2014it connects you to your customers, and that relationship must be protected at all costs.\n\n2. Marketing Is Not Evil\n\nI totally understand mission-based businesses and the need to be \"elegant\" or \"classy\" with all marketing and sales messages. But marketing your product, _especially_ when it is mission based, is about serving your customers, not taking from them.\n\nAlways remember this important premise, drilled into me by one of my mentors, Jay Abraham (still one of the world's most influential thought leaders on business and marketing, and author of the foreword to this book): it is irresponsible to bring your product or service to the marketplace without the same passion with which you created it.\n\nThere's an ethical line each of us won't cross (which comes down to your personal perspective), and that's important to define clearly. But there's nothing to be ashamed of in bringing your life's work to as many people as possible if you are doing it in a way that's congruent with your ethics and overall philosophy. To do this may require copy platforms and creative that you consider aggressive; but as long as you are working in the spirit of making a big impact, sell hard. The more people you reach, the better.\n\n3. No One Spends Enough Time on Lists\n\nYou're going to hear that no one spends enough time on lists a few times throughout this book. Take it to heart. Dick Benson (another mentor and the smartest man I ever met on the science and strategies of direct mail) said this to me the day I met him, circa 1985, when we were researching and comparing lists. I never forgot it, particularly since he later revised the comment to say that the team I led at Boardroom was the only exception.\n\nI can't emphasize enough that you must pay close attention to your customer list, your prospect list, and how you approach list building at a very detailed level. The real live people who make up those lists are truly the lifeblood of any business driven by direct response (and even many businesses that wouldn't identify themselves as being in direct response).\n\nIgnoring your lists is like ignoring your family. It's a prescription for disaster. Making sure you talk to different segments of your list based on their relationship with you is critical to maximize success, and we'll talk about this at length in the following chapters.\n\n4. Customers Refund Transactions . . . Not Relationships\n\nE-mail is probably the most prevalent sales and communication medium in the world today, and it is also the most effective and efficient way to market any product or service when used properly. But just because e-mail is cheap doesn't mean it should be used indiscriminately or without careful thought about every message you send. The care that you take with your e-mail list (what I would call your online family) might be the most important thing you do as a marketer, both now and in the future.\n\nIf someone recommends that you mail every day, don't take it as gospel . . . take it under advisement. But here's better advice: mail when you have something really worthwhile to say. Blogging every day with purpose is effective; throwing out random thoughts or sending pictures of last night's dinner might be counterproductive. You wouldn't want your own family to start ignoring you because you were sending stuff they didn't care about. Think of your online family the same way. If you are just throwing out whatever you can think of every morning, including unrelated or insignificant content, you run the risk of burning out your list, causing your online family to ignore you when you really want them to pay closer attention.\n\nAlways look to build relationships for a lifetime, and deliver huge amounts of valuable content before you ever try to sell something. Because e-mail is cost-efficient, this is so much easier than in the past, when we only had offline techniques. Understanding lifetime value (which has its own chapter later on) gives all marketers real leverage and the maximum chance for success in multiple channels. It's this focus on the long game that you need to develop in your customer relationships.\n\n5. Credibility & Transparency Trump All\n\nRespect your customers. If they're following you, don't insult them with anything but transparency. Showing proof of your approach through testimonials, case histories, and professional endorsements is about building your case, not bragging. And when you make a mistake or want to show vulnerability, people always appreciate transparency.\n\nIsn't the cover-up always worse than the crime? There are countless stories in the world of marketing where a business confessed to something they were not proud of or failed at and came back stronger than ever. But once there's a cover-up, you're doomed.\n\n6. Always Think _Direct Marketing_\n\nRemember\u2014direct marketing is _not_ just direct mail. It's measurable marketing in any channel. Measurable and accountable advertising is always what you want, and don't let anyone sell you a brand and image advertising \"campaign\" without being able to show a clear return on your investment (ROI).\n\nI've been saying for years that the Internet is the ultimate direct marketing medium . . . and while that won't change anytime soon, _all marketing is direct marketing_ in my world. Expecting an acceptable ROI must always be standard operating procedure. Just because a lot of stuff is cheap to do online doesn't mean you should allow sloppy campaigns out the door or skimp on requiring tangible results.\n\n7. Use Your Personal Brand in Your Marketing When Applicable\n\nMany marketing superstars understand their personal brand deeply and always keep it totally in sync with their products and services. They're confident talking about their achievements and know that their information is valuable to their customers. However, many entrepreneurs and business owners are squeamish about putting themselves out there for fear of sounding boastful or disingenuous. In response to that, I will quote baseball pitcher Dizzy Dean (and I think John Wayne and others have used this one too): \"It ain't bragging if you done it.\"\n\n8. Advertising Opportunities Are Now Infinite\n\nThere are endless ways to market your products and services today. None of us can master them all, because there's simply too much to know. This is why now is the best time ever to be a marketer . . . but this is also what makes it dangerous.\n\nHaving spent my entire career buying \u00e0 la carte from the best experts across multiple marketing niches, I can assure you this approach will pay for itself. I tell entrepreneurs to run\u2014not walk\u2014away from anyone who claims they can be your one-stop shop for all marketing, creative, and media buying.\n\nRemember that when you have more choice across channels, you also see more specialization and therefore more experts. I don't think I have to convince you that experts are much better to work with than generalists\u2014and certainly better than generalists posing as experts. In today's supercharged, multichannel and technology-rich marketing environment, no one can be an expert on everything. Make sure you're always searching for the expert who is the leading specialist in the channel you want to use.\n\n9. SingleChannelMarketingIsSoBoring.com\n\nFeel free to go to this website because it just redirects to my website, www.briankurtz.net. I really _do_ own this URL, just to prove a point.\n\nNo business should rely on a single marketing channel, because you never know when the landscape might change. If Facebook shuts down your ad account, or Google changes its search algorithm, or your e-mail provider slaps you for an infringement, you must have other channels dialed in to keep your business solvent.\n\nThis is not about one channel being better than another channel. That is, it's not an \"or\" but an \"and.\" Not only is buying your media on a single channel boring, it's also dangerous. And the danger is easily avoided when you've committed to working with the best of the best across the different marketing disciplines, as I mentioned in the last point. You would expect the people who manage your money to diversify your financial risk, and you should expect the people who manage your marketing to do the same.\n\nRemember, you have to meet your customers where they are, and there's not a single industry where buyers use only one channel. Meet them everywhere they spend time. And you can only serve your customers if your business stays afloat, so diversify your marketing to ensure you can provide for them over the long haul.\n\n* * *\n\nWith all the tools, case studies, and information at our disposal, today's marketers have more opportunity than ever to leave an indelible mark on the world. While I can't guarantee what the market will do or which platform will take off next, if you approach direct response marketing with a spirit of service and healthy competition, your business and marketing can be a force that changes countless lives for the better\u2014and pays off for you too.\n\nThe nine fundamentals listed above were originally written for entrepreneurs and business owners who are _not_ marketers but who I believe must embrace these principles to build their businesses for a lifetime. This book will dive deeply into all these principles, giving you valuable understanding whether or not you are an experienced marketer.\nCHAPTER 1\n\nOVERDELIVER\n\n_Everything in business (and life) is not a revenue event,  \nbut everything is a relationship event._\n\nDuring my almost four decades in direct response marketing, I was personally responsible for the circulation of close to two billion pieces of physical mail. Thankfully, because I saw some incredible opportunities to grow with the company, stamp-licking duties fell to other people. That said, my role came with some huge responsibilities and a lot of pressure. As I spent more time with Marty Edelston, the founder of Boardroom\u2014and with the world-class marketers his company attracted\u2014it became obvious that I would have to continually level up my game to keep pace with them.\n\nThe concepts you'll learn in this chapter came from that journey of self-improvement and striving for a new standard in my own behavior. I believe learning how to overdeliver is at the core of real success in direct marketing (and dare I say, in all aspects of your life).\n\nOverdelivering to anyone who is an existing customer, a previous customer, or a potential customer is obvious and something you have certainly read elsewhere; but you can take the concept to a new level if you also overdeliver on your marketing messages. I will explain throughout this book why all these forms of overdelivering are about playing the long game (i.e., building a business that lasts a lifetime) and building a business you love.\n\nTHE FOUR PILLARS OF BEING EXTRAORDINARY\n\nWhen I first started thinking about writing this book, I knew it needed to focus on the fundamental truths of direct response marketing. I also knew that without the right mind-set and attitude, people would never reap the full benefit of these strategies. But the last thing I wanted to do was write a self-help book. A lot of personal development advice gives me hives. But if you want to be extraordinary, you're going to have to do some work on yourself. I've had some great models in every part of my life, but in this area, much of what I know about becoming extraordinary came from Marty Edelston.\n\nAt 2:00 A.M. on the day of Marty's funeral, I was sitting at my kitchen table, fighting through my tears to construct a fitting eulogy. I was grateful, and terrified, that his family had invited me to speak. I wanted my words to be extra special for the man who had changed my life. Marty always would tell me that he was just an ordinary guy who was able to do extraordinary things. To his close friends while growing up and to his many relatives, he was simply \"Mickey from Newark, New Jersey,\" trying to do the best he could do every day. But under that everyman exterior, excellence was his mantra. He had stationery that read, \"Good, better, best. Never let it rest until the good is better and the better best.\"\n\nAs I sat there in my kitchen, just hours before hundreds of people were going to pay tribute to him, four pillars emerged that illustrated why Marty was an extraordinary entrepreneur, marketer, and benefactor:\n\n  1. He outworked everyone.\n  2. He had insatiable curiosity.\n  3. He always surrounded himself with people smarter than he was in areas where he lacked knowledge or expertise.\n  4. He focused on helping others first. Saving lives was his ultimate mission.\n\nNot a bad way to live. I've come to believe that these are the pillars that enable you to constantly overdeliver in your business (and your life), and that they will make you extraordinary too if you choose to follow the same path Marty did.\n\n1. Outwork Everyone\n\nNothing worth having comes easy. Without exception, all the copywriters, marketers, and entrepreneurs I've seen make it have this attitude in common. They know there is no EASY button to get the life and business they want. They have an unwavering commitment to their craft, and they are willing to do whatever it takes to be the best.\n\nBecoming the best at something is not just a question of working the most hours. It's what you _do_ with those hours and your efforts. Most people in any given field will do what it takes to maintain a baseline of skill, but real, differentiating success comes from deliberate practice, seeking critical feedback, and getting coaching from people who are ahead of you. These extra elements will determine whether you'll be able to outwork and outperform the other people in your industry.\n\nAnd when you experience success, the first thing you should do (after a short celebration) is to try to beat the control. If you're not a copywriter or not familiar with the term, I should tell you that the control is the promotion that's currently performing best in your marketing, and your task is always to beat it. What I'm talking about here is not just for copywriters\u2014it's for all of us. Celebrate your successes, and then get right back to the grindstone to try to improve on what you just did to keep working toward the next level.\n\n2. Possess Insatiable Curiosity\n\nWe are all drawn to people who have insatiable curiosity about life. When you meet someone who reads widely and deeply, you always know you've met someone extraordinary. The aforementioned Gene Schwartz, one of the greatest copywriters who has ever lived (and someone who will come up a lot more in this book), was one of those people. He read tabloids like the _National Enquirer_ , attended all the popular films, and read all the pulp novels that came out in order to understand what people in America truly cared about. He also read scholarly journals and business books and was a patron of the arts. He was probably the most curious man I ever met, always questioning what motivated people unlike himself to behave a certain way. He wanted to know where the world was going\u2014and why. His deep interest in popular culture and what everyday Americans were paying attention to made him one of the most insightful and successful marketers in history.\n\nMarty and Gene were two peas in a pod. They talked endlessly about what they were reading and what it all meant. Other people read deeply in single topics or specific disciplines, one at a time, to learn every single thing about that area. Going deep, instead of going wide, is a characteristic of some of the other best marketers and copywriters who have ever lived. My good friend and one of those great copywriters, Parris Lampropoulos, immersed himself so deeply in a subject that he would eventually know more than the leading experts in those fields. Parris became one of the best health copywriters in the world by immersing himself wherever his curiosity led him about the topics he was working on.\n\nWhen I was diagnosed with stage-two prostate cancer back in 2008, I called Parris immediately after speaking to my doctor\u2014and before speaking to anyone else\u2014because he knew _everything_ there was to know in the field. He had written about alternative cancer treatments for years and kept up with the latest breakthroughs in care and treatment in even more depth than many doctors I knew. _That's_ insatiable curiosity.\n\nWhen you work in an established organization, the odds are high that you are surrounded by superstars who have near-infinite knowledge in areas you don't know squat about. If you're working on a team, take full advantage of that at every chance you get. It will serve you well if you ever go out on your own.\n\nWhether your focus is narrow or wide, curiosity is irresistible and immediately sets you apart from your competition. It makes you an authority and attracts people to you because they can sense that you understand them\u2014their fears, hopes, and needs\u2014better than anyone else.\n\n3. Surround Yourself with Smart People\n\nAs the saying goes, if you're the smartest person in the room, you're in the wrong room.\n\nEarly in my career, being the \"list expert\" made it easy to show off my knowledge in that one area. But there were plenty of other smart marketers working alongside me, and many of them had been at it much longer than I had. So I tried to shut up and stay humble and draw on their expertise to make my own work stronger.\n\nEven the legendary Dick Benson, one of the greatest direct response marketers of all time and possibly the most knowledgeable man about direct mail who ever lived, knew this. One day, after I'd asked him how he knew so much about direct mail, he held his thumb and forefinger about an inch apart in front of me and said, \"I know everything about this much.\" And it was true. Part of Dick's great success came from utilizing other people's expertise when he needed to go wide and sticking to what he was good at by continuously going deep.\n\nIf you can learn to identify (and then use) other people's expertise, you'll build momentum so much faster than if you try to learn everything yourself. Curiosity is very powerful, but no one can know everything, and the sooner you accept that, the better. You have to learn to identify and ask for the help you need, because you must be able to delegate the tasks you don't like to do or are not especially good at. You will never succeed at the highest level as an entrepreneur if you don't focus on doing the things that are your Unique Ability\u00ae, to use a phrase from the world's top coach for entrepreneurs, Dan Sullivan.\n\nThat brings us to some questions: Who do you hang out with in your professional life, and why? How do you stay accountable to honing your craft? Are you willing to \"pay to play\" in order to get access to big opportunities, mentors, and connections with the most successful people in your field?\n\nOne of the most important career decisions you will ever make is which communities you align yourself with in the pursuit of excellence. Being an apprentice to mentors who can show you the way and then surrounding yourself with peers with equal or more talent is the key. David Deutsch is one example of this. When I met him, he was an apprentice to the copywriting giant Jim Rutz. He put himself in a humble position to learn everything he could. Since then, David has become one of the best copywriters in the world and is now himself one of the top coaches for up-and-coming copywriters. Other A-list players consider him a peer, and through this network he gets (and gives) support, ideas, and constant motivation to improve his craft.\n\nThere are quite a few copywriters working today who paid to spend time with Gary Halbert to see if they had the chops to become successful copywriters. When I made a joke to one of them that they paid to be Gary's butler, the response was that being a butler would have been a lot easier. Gary worked them hard and taught them everything he knew, and they have the scars (and success) to prove it. Of course, there were no physical beatings involved, but Gary demanded that if you were in his world, helping him with his work, you were there to learn, absorb, and adapt. It's like being an unpaid intern in today's work environment, but there were instances where the \"butler\" paid Gary tens of thousands of dollars (which was nonrefundable) just to make sure they were committed to the \"copywriting boot camp\" Gary put them through.\n\nThis is why I am a member of six mastermind groups today and run two more of my own, because I see that as the only way to ensure that I'm never the smartest person in the room. For those not familiar with the term _mastermind_ , it is a concept that can be traced to Napoleon Hill as early as the late 1920s. In Hill's own words, a \"master mind\" is \"coordination of knowledge and effort between two or more people who work towards a definite purpose in a spirit of harmony. . . . No two minds ever come together without, thereby, creating a third invisible, intangible force which may be likened to a third mind.\" Masterminding is the best way to live most powerfully following this third pillar of being extraordinary.\n\n4. Help Others First\n\nWhen you default to generosity and always helping others, it's easy to outwork everyone. It's easy to stay curious and engaged. It's easy to find the right mentors and peers (though often your mentors choose you, once they've seen what you're made of). An attitude of giving freely releases you from needing to \"get something\" from everyone around you.\n\nWhen you come from a place of generosity with no expectation of getting something in return, you display your passion and integrity to the people who matter . . . and you also get noticed. Passion never goes out of style, and the people in your life (including your customers and contemporaries) will be drawn to you. This is not about being a pushover or ignoring your own priorities. It's about recognizing that a rising tide lifts all boats, and the more you contribute to the success of others, the more success you will see for yourself.\n\nWith your work\u2014whatever it is\u2014always think about helping others first. Helping the people you serve (both your customers and associates) get what they want puts generosity first.\n\nMarty used to say, \"You only go through life once, so you might as well be the world's best.\" He knew that being the best was not just about knowing or doing the most but also about producing the best work possible, according to his high standards of integrity, and publishing books and products that could actually be lifesaving for the people who bought them. His focus on creating books and newsletters in the health niche was driven by this desire to change and even save lives\u2014he saw an opportunity to educate our customers about the things that really affected them the most (and as you'll see shortly, that altruistic desire is not in conflict with making a lot of money).\n\nThose are the four pillars to becoming extraordinary: outworking everyone, having insatiable curiosity, surrounding yourself with people smarter than yourself, and always thinking of how you can change the lives of others. I learned these from Marty Edelston under the umbrella of what it means to truly overdeliver in all aspects of your life.\n\nTHE CONTROL IS YOUR ENEMY\n\nThere is nothing more exhilarating for a copywriter or marketer than getting a new control\u2014a campaign that sets a new standard of performance, converts first-time buyers into second-time buyers or renewals, pays out by whatever criteria is necessary, brings customers back as multibuyers over and over, and, simply put, beats the pants off the previous control.\n\nWhen I got started, a good control package in direct mail might run for months, even years, before a new package could knock it off its perch. These days, you can have a new control online in hours (if you have the traffic and the ability to make statistically significant tests).\n\nBut no matter how long a control lasts, it's your enemy. I learned this from Gary Bencivenga, one of the true legends of the direct marketing world. He believed, deeply, that the control is there to get beat and that the best marketers always have the discipline to keep improving their work even when they're beating everyone else. Though Gary is one of the nicest guys in the world\u2014charming, smart, and a joy to be around\u2014if you were trying to beat one of his controls, he was a ferocious competitor. In his prime, no one wanted to even _try_ to beat one of his controls (and most copywriters couldn't do it if they did try).\n\nThe thing that made Gary so formidable was that before anyone else tried to beat him, he would try to beat himself, regardless of how successful his current control was. This is how a copywriter can overdeliver for their client. And then how you can overdeliver to your audience. If Gary didn't beat himself the first time, he kept at it longer than any other copywriter would have. He would write package after package until he couldn't find a single thing to improve. And if he was trying to beat _your_ control, you had better watch out. With an attitude like that, it's not surprising that he rarely lost.\n\nThere's a difference between tweaking a winner to make it a bigger winner and creating a brand-new promotion to get a new (even bigger) winner. Both are very important, and testing should never stop. The day you get a new control is the day to start thinking about how to beat it. That's advice for copywriters and marketers alike. Discipline doesn't stop when you submit the package, mail the campaign, run the ad, or create an online launch.\n\nOf course, sometimes it's time to move on to a new product, but everything can always be improved on, or changed completely, and a philosophy of always trying to beat the control (i.e., \"the control is your enemy\") is a foundation of direct marketing.\n\nTHE POWER OF 100:0\n\nWe all know that no matter how close we are to certain people, it's easy to fall out of touch. And if some of those people are particularly complacent, it can be even more challenging. Often if you don't reach out, you won't hear from them. They're not proactive enough to make a first move, so you have to lead the way if you want to stay in touch.\n\nSome of my old buddies who fall into this category like to joke that they are always sitting by the phone waiting for my call. Earlier in my life, I wondered why it always had to be me that made the extra effort to keep in touch and why they weren't as motivated to maintain the relationship. But I learned that my disappointment was wasted energy. It wasn't that they didn't love me. They just didn't have it in them to work that hard with everything else going on in their lives. But when we did connect, it was always a blast. These were lifelong, deep connections, so eventually I accepted that I would be the one to be much more proactive about keeping these relationships meaningful. These days, we joke a lot about how we're beating the odds when we actually manage to get together.\n\nConsider eliminating the phrase \"I'll meet you halfway\" in dealing with people in your life, difficult or not, acquaintances or close friends, whether you are in a tough negotiation or just disagreeing on something. If I had expected all my relationships to be a perfect give-and-take, if I had waited for these guys to meet me halfway, I would have missed out on years of friendship with some of the most important people in my life. It's easy to feel hurt or to take it personally when you are making a disproportionate effort. But when you bring 100 percent to any relationship, the payoff of having those people on board is almost always worth the extra effort.\n\nIt's the same in business. Many of your customers are going to be like my buddies. Customers might not be knocking down your door, but when you make the effort to reach out with no reservations or ulterior motives, you'll get great results because they value hearing from you. You can let the imbalance annoy you, own you, or make you quit . . . or you can accept it as reality and make the most of it. World-famous speaker and author Sean Stephenson, the man whose mission it is to rid the world of insecurity (having been wheelchair-bound his entire life), says, \"I love everyone because as soon as I don't love you, you own me.\" I would add that \"loving everyone\" in the context of not expecting the same percentage of love in return will ensure that you never feel beholden to anyone else; and you won't think anyone else owes you anything either.\n\nI learned the value of this early on, and I stuck to it, because I knew it would pay dividends. Everyone I have ever worked for throughout my career has considered me a wimp for not holding out for every dollar on a contract and not walking away when demands from the other side of a negotiation became unreasonable. I also took a lot of flak for giving away outrageous amounts of product to our customers when they complained and bending over backward to make difficult customers happy. But I slept well every night over these past four decades in business, knowing that I was overdelivering for my customers, vendors, and partners, and feeling secure in my integrity and the integrity of our company.\n\nThis concept is called 100:0, which I learned at a personal development conference (so you can see I don't _always_ get hives at the idea of personal development). I've put my own spin on it, and I have to tell you, it takes away so much stress. You give 100 percent of your effort at all times, even if the other side is giving nothing. All that stress of keeping score, of tracking how much they gave you compared to what you gave them\u2014it's just not worth it. I encourage you to let it go. It's just wasted energy.\n\nThe aforementioned Dan Sullivan promotes a concept I love around building an abundance mind-set as opposed to a scarcity mind-set. He shares that the scarcity mind-set is characterized by envy, guilt, and anger, which keeps you small and limits your ability to grow. The abundance mind-set, the mind-set of always giving 100 percent, is characterized by gratefulness, creativity, and cooperation. This enables you to access opportunity and ingenuity and to exponentially grow your capabilities.\n\nGivers, Takers & Matchers\n\nAdam Grant, the author of one of the books that has been most impactful for me, _Give and Take_ , developed the concept of givers, takers, and matchers after studying more than 30,000 people in multiple industries all around the world. He found that we all fall into one of these three behavioral categories: about 25 percent of us are givers, 19 percent are takers, and 56 percent are matchers in our interactions with other people.\n\nGivers are the people who are constantly helping the folks around them. They're always sharing their time, effort, money, and attention with others\u2014even when it drains them or takes up more time and resources than they can afford. Takers are self-serving in their interactions with other people. They're always on the lookout for what you can do for them, and they don't care what they could do for you. Matchers strike a balance right in the middle. They'll do something for you if you do something for them. It's the \"I'll scratch your back if you scratch mine\" approach.\n\nEarly in his book, Grant asks which group you think will be the _least_ successful in their buisness and personal endeavors\u2014givers, takers, or matchers\u2014and reveals that it's the givers. But as you read on, he also asks which group you think will be the _most_ successful . . . turns out it's also the givers. The important lesson is _how_ you give, not _how much_ you give.\n\nOne more thing I want to share from this book is what Grant refers to as the five-minute favor. This is all about looking for ways to have a big impact in a short period of time. It might be making a business introduction, throwing a surprise bonus into the delivery for a loyal customer, or spending a few minutes sharing some feedback or experience with someone in your business that is going to accelerate their ability to perform. It only takes a few moments, and it has a big effect.\n\nMarty was definitely a giver, and while he didn't have Grant's system in place, he knew how to give effectively, and he led the company with a spirit of generosity. Together, we built a culture of really overdelivering for our customers and employees, and it was this obsession with giving people more than they ever would have expected that took us from humble beginnings to becoming a direct marketing giant.\n\nMarty would often say to me, \"So Brian, who did you give your right arm to today?\" He didn't ask me this to be mean or to shame me for my overenthusiastic giving. He was asking out of care and concern. He didn't want me to get taken advantage of, which is a risk of being a giver and of overdelivering. But it's a risk that's worth taking, because overdelivery sets you apart and keeps people coming back to you.\n\nThis overall philosophy of overdelivery gave us\u2014and it will give you\u2014incredible momentum, because customers always know when you are in their corner. In my career, I didn't always get it right, and I'll talk about a few of my biggest mistakes in the coming chapters. But the customers I dealt with in the past and the ones I deal with today always know one thing: they can count on me (and whoever is on my team) to be proactive and reliable.\n\nMy advice is to always have something incredible to offer. And while occasionally people might take advantage of you and be disappointed nonetheless (there are dangers in setting expectations so high when you overdeliver), net-net, the pros outweigh the cons, and it is the most powerful thing you can do to set yourself apart from the competition.\n\nKEY TAKEAWAYS\n\nObsessing about giving your customers more than they could ever have dreamed\u2014being relentless in your pursuit of excellence and building an abundance mind-set\u2014will put you on an unstoppable path, no matter where your business has come from, where it is right now, or where it's going in the future.\n\nIn the spirit of overdelivery (and to make it as easy as possible for you to refer back to this book later on), you'll find a recap section at the end of each chapter. These will include the key points covered\u2014a CliffsNotes version that will jog your memory or give you clear talking points if you want to explain anything you find useful to the people you connect with.\n\n  * Not everything we do in marketing (and in life) is a revenue event . . . but everything we do is a relationship event.\n  * \"Overdelivery\" (while not a word) is a way of life. When you go above and beyond what you promise to your prospects, customers, and partners, you weave integrity, loyalty, and growth into the fabric of your life and business.\n  * There are four pillars to becoming extraordinary in any pursuit: \n    1. Outwork everyone.\n    2. Possess insatiable curiosity.\n    3. Surround yourself with smart people.\n    4. Help others first.\n  * The control is your enemy. When you get a NEW winner with your marketing or promotion, your customers benefit more and you also gain more customers\u2014but that is the time to start trying to beat that control and not rest on your laurels.\n  * Give 100 percent of your effort in all your relationships\u2014personally, professionally, with friends and family, with colleagues, and with your audience and customers\u2014even when it seems like the other party is making no effort at all. The best results in any relationship come from consistent, generous effort that demands nothing in return.\n  * Remember that people are either givers, takers, or matchers. When you choose to be a giver, you do run the risk of people taking advantage of you, but you also set yourself up to truly overdeliver. This gives you a competitive advantage, makes your customers love you, and will help you find your work more fulfilling, all the while helping you actively cultivate an abundance mind-set that will bring even more riches into your own life.\n  * Perhaps the best summary of all that I've said comes from Unitarian minister L. P. Jacks: \"A master in the art of living draws no sharp distinction between his work and his play, his labor and his leisure, his mind and his body, his education and his recreation. He hardly knows which is which. He simply pursues his vision of excellence in whatever he is doing and leaves others to determine whether he is working or playing. To himself he always seems to be doing both.\"\n\nCHAPTER 2\n\nORIGINAL SOURCE\n\n_Learn the rules like a pro so you can break them like an artist._\n\nThis chapter's opening quote is often attributed to Pablo Picasso . . . and the Dalai Lama. . . and _Life's Little Instruction Book_ by H. Jackson Brown, Jr. It's ironic, when we're talking about the importance of original source, that we can't attribute the quote to its original source (i.e., the person who said it).\n\nEvery lesson I learned throughout my career at Boardroom, and everything I've done since (including running high-end mastermind groups and consulting with companies to help them implement direct response principles in their marketing), has been based on original source material from the greatest marketers who ever lived. My first book, _The Advertising Solution_ , profiled six of those greatest marketers, and their fingerprints are all over this book too. Believe it or not, all the marketing you've ever done is probably based on something they pioneered, even if you've never heard of them before.\n\nWhen you recognize and understand the fundamentals of any field, you're better equipped to innovate and break the rules of the particular game you're playing. Marketing is no different. If you don't know the fundamentals, you might accidentally break the rules and get lucky, but you can't rely on getting lucky.\n\nBuilding a business that will keep its customers for a lifetime requires much more than the occasional successful fluke. And when you have a deep, intrinsic appreciation of the principles and strategies that were developed over the years by the titans of direct marketing, you get to stand on their shoulders. Taking what they learned (and proved over and over again), you build on foundations they laid, and what you create will be much more solid because these principles are timeless. They are still 100 percent relevant to today's marketplace.\n\nThis might be obvious to you, but just in case it's not, having a deep understanding of the fundamentals of direct response marketing from the pioneers who first implemented them will enable you to achieve better results with any and all marketing programs you work on today and into the future. You will be able to serve your customers in powerful, profitable ways without having to repeat the painful learning curves and costly errors that these entrepreneurs and marketers suffered before you.\n\nRosser Reeves, one of the greatest ad men who ever lived (he developed the idea of the unique selling proposition (USP) and coined the unforgettable M&M's \"melt in your mouth, not in your hand\" slogan), wrote about why it's so important to understand the fundamentals in the introduction to his landmark book _Reality in Advertising_ :\n\nThis book attempts to formulate certain theories about advertising\u2014many of them quite new, and all of them based on twenty years of the most intensive research.\n\nBefore the ink is dry, some man will sit down and write a campaign that ignores every word contained here. What is more, this campaign will work.\n\nHowever, this does not invalidate this book.\n\nThe croupier at a roulette wheel knows that at some moment a player may violate all the rules of probability. If the wheel spins long enough, some number (say, thirteen) will come up twenty times in a row.\n\nBut roulette wheels, in the long run, do not lose money on exceptions, nor do advertising agencies or their clients, in the long run, make money on them. Agencies and clients, like actuaries, must invoke the laws of probability and determine (out of hundreds of campaigns) how they can make these laws work to their benefit.\n\nAs you will see, when the laws of probability are observed, it is possible to add a heavy percentage to every advertising dollar. Nor will this procedure rule out genius or fail to give genius full room to exercise its rich and golden talents.\n\n_Reality in Advertising_ was written in 1961, which makes Rosser Reeves one of those original sources of innovation and wisdom. When _The Advertising Solution_ was published, I did a long interview with marketing icon Perry Marshall, and unexpectedly, we got into the weeds on original source as a core concept, not just talking about the legends who came before us but about what they taught us for today. While the six legends I profiled are considered the fathers of modern advertising, in reality they were all direct marketers (i.e., they understood the importance of measuring everything), even if they were trapped in the bodies of what we thought were general advertisers (those who only focus on image advertising). Perry's insights on Rosser Reeves, and how he thinks about learning from the best, are worth sharing here too, since Perry agrees that all marketers today owe a huge debt of gratitude to those who came before us:\n\n[Reeves] was the first person to define the unique selling proposition [USP]. If you go read his book, it's relatively short, but it has so many dense ideas and says so much in such a short space of time that you really need to know. . . . It's like eating a bunch of pecan pie. You feel it's a bit much; you can't eat giant chunks of it; you need to absorb it.\n\nThis is not the way most people want to consume information, but if you want to be a master, you'll seek out the original source and you'll really be a student of that original guy, not a student of a student of a student of the guy.\n\nTo become a world-class marketer, to master the art and science of building businesses for a lifetime, you must first and foremost be a student\u2014a student of human psychology and of the core principles of direct response marketing.\n\nCORE PRINCIPLE #1: MARKETING IS ABOUT PSYCHOLOGY\n\nYou need to understand what _moves_ your market\u2014that is, what makes them buy. And to do that, you need to know, love, and respect your market. This is one of the first principles of marketing, and it comes down to us from the legend that was Gene Schwartz.\n\nI have never met anyone who read more than Gene. He knew it wasn't enough to simply study your client's products or your own products, or to only study the audience you are trying to reach with those products. Studying those things is indeed critical to success, but he went way beyond studying the project or task at hand. What separated Gene from mortal men was that he understood modern culture and society at its core, by reading everything\u2014scholarly journals, business books, fiction, and nonfiction, anything he could get his hands on\u2014with the purpose of knowing his audience and what they wanted.\n\nIn Gene's own words, \"You cannot lose touch with the people of this country no matter how successful or potent you are; if you don't spend at least two hours a week finding out where your market is today, you are finished!\" Of course, Gene spent way more than two hours a week on this, and while he was looking at what his market was interested in today, he was also exploring where other markets might be tomorrow. He knew that he had to care about the things that the people in his audience cared about. He knew that if he looked down on them, assumed that he was smarter than them, or jumped to conclusions about their problems, fears, hopes, and aspirations, he could kiss his success good-bye.\n\nAs David Ogilvy famously said, \"The customer is not a moron. She's your wife.\" He knew that talking down to customers, patronizing them, or doubting their ability to make sound decisions would just hurt a campaign. If you're selling good products to the right market segment, the customer will know if you have their best interests at heart. But they'll see right through you just as fast if you think you're better than them. That's why you always have to assume that your customers are at least as smart as you are. Regard them as you would the most important people in your life\u2014people who deserve your respect, love, and attention.\n\nMany people think that direct marketing is only about the nuts and bolts, such as copywriting, media buying, and offer creation. But more important, it is about human behavior. We need to immerse ourselves in understanding why people do the things they do at the deepest level so we can learn to predict what our customers want and need and how they will act as consumers.\n\nGene knew that understanding human behavior is the key to everything. This is why his industry-defining book, _Breakthrough Advertising_ , was (and continues to be) so powerful. Gene said over and over, publicly and in private, \"The greatest mistake marketers make is trying to create demand.\"\n\nMany marketers working online with state-of-the-art techniques feel that they are in fact creating demand for new products; but I maintain that since marketing is about psychology, they're really just tapping into existing desires. You're better off letting the market tell you what they demand.\n\nThis was one of the most fascinating things about Gene Schwartz (and many of the other great copywriters). He was never about trying to create mass desire; he saw his job as channeling and directing the desires already present in his audience. And when one of the best marketers who ever lived lets us in on his secret to selling so effectively, we must pay attention.\n\nEverything Gene was talking about in 1966 still applies today, without exception, which is why there was a 500-person waiting list for _Breakthrough Advertising_ when I republished it in early 2017. Many of the marketers and copywriters who were on that waiting list had been born years (and even decades) after the book was first published. The best marketers get that the fundamentals of human behavior have not changed since 1966, and they're smart enough to take the lesson without needing to work it all out for themselves.\n\nCORE PRINCIPLE #2: THERE'S NOTHING WRONG WITH SHORTCUTS\n\nThe power of going to the original source is, in part, that you gain access to a wealth of knowledge you could never hope to gather by yourself. Even if you limited yourself to the six legends alluded to earlier\u2014Hopkins, Collier, Caples, Ogilvy, Schwartz, and Halbert\u2014you would have hundreds of years of accumulated practice, experience, and testing to learn from. In some marketing circles, the prevailing thought is that you have to work things out yourself. Everyone wants to test every little thing on their own customers, even when they don't have the numbers to make these tests statistically significant (we'll talk about why that's so important in Chapter 5).\n\nThere's nothing wrong with wanting accurate data, but there's no point risking your shirt to learn firsthand the key lessons already figured out by earlier marketers. Throughout my career, I nearly lost everything (several times), and it's my hope that the lessons in this book will save you from the same potential disasters. Shortcutting your learning like this is not laziness. It's not cheating, and it's not unethical\u2014though I understand that in a culture obsessed with \"doing the work,\" it can be hard to stomach using someone else's information.\n\nIt might seem hypocritical for me to say that you should outwork everyone while using someone else's hard-earned lessons. But using existing information and then applying your own genius to it just allows you to work more effectively. Don't try to reinvent the wheel. Every great success story is built on preexisting technology and information. Nothing happens in a vacuum, especially not where human psychology and market behavior are concerned.\n\nWhen it comes to leveraging the psychology of your customers, you first need to understand mass desire, states of awareness, and market sophistication (all concepts from Schwartz in _Breakthrough Advertising_ ). Maximizing the response from your customers then requires you to understand all the principles of direct marketing that we will explore: list segmentation, recency, frequency, monetary scores, and copywriting. To make your marketing (and business) profitable over the long term, you then need to understand the cost of acquisition, lifetime value, continuity, and renewability of all your marketing programs. All that will be discussed in detail in this book.\n\nWhat you _don't_ need is to build all this knowledge from scratch. There are exceptional books written on many of these topics (see the reading list at the end of this book), and I'm hoping that the one you're reading now will begin to fill in any gaps in your arsenal. Using original source material and the work of early direct response marketers can fuel the fire you've been stoking, so work smart and take advantage of all the accumulated knowledge out there.\n\nCORE PRINCIPLE #3: USE WHAT'S ALREADY WORKING\n\nIn his prime, Bill Jayme was the most sought-after copywriter in the world. He had a two-year waitlist of clients who wanted him to write a direct mail package for them. Every major magazine publisher and nonprofit was on that list, along with a slew of other mailers in every category at the time. Bill charged top dollar, and he had an incredible track record of success in a wide range of topics, using many different approaches and copy platforms. He was regarded as both a poet and a marketing expert, and he is still famous for some of the most memorable and successful promotions in the history of direct mail.\n\nMy favorite Jayme headline was an outer envelope for the magazine _Psychology Today_ ; they used it as their control for years:\n\n_Do you close the bathroom door even when you're the only one at home?_\n\nNo need to answer . . . but I bet you want to read the rest of the package now. Here is another Jayme gem:\n\n_How much do you tip the waitress when you're planning to steal the ashtray?_\n\nCreativity like this made us (and so many others) huge Jayme fans . . . and we always had a prominent place on the Bill Jayme waiting list. He wrote some wonderful packages for Boardroom, including the one that launched our retirement newsletter, _Bottom Line\/Tomorrow_ , which led with this headline:\n\n_How to Succeed in Business Retirement without Really Trying_\n\nAfter every winning assignment, Marty sent Bill some celebratory gift. That was Marty's style\u2014to treat the people who helped grow his business like royalty. (It's just as important to treat your partners as well as you treat your customers.) After Marty sent Bill one of these gifts, Bill sent Marty a small white envelope with these provocative words penned across the front:\n\n_Deeply and Irrevocably Personal . . ._\n\nEnclosed was a funny, witty thank-you note like no other from Bill. The guy could write eloquently about anything to anyone. (I still have two personal notes from him framed and hanging on my wall.)\n\nMarty took Bill's envelope, marched into our art department, and found the artist who was mocking up a direct mail piece for our niche book _The Encyclopedia of Estate Planning._ Marty told him that we had an envelope test for him to execute . . . and the plain, standard outer envelope with the words \"Deeply and Irrevocably Personal\" in simple typewriter font became the control for that book for years. Though it had never been intended as creative material, this subject copy on the outer envelope went on to sell millions of dollars' worth of a book about estate planning (one of those topics everyone _needs_ to know about, but no one _wants_ to know about).\n\nThe package, written by our secret weapon copywriter Mel Martin, was already doing well before the Jayme envelope. However, the new text drastically lifted response, and the estate planning book became an even bigger winner despite being on a narrow topic most people would really rather not think about. We sold more copies of this book on a niche topic than many New York Times bestsellers once we started using that envelope. This is a prime example of what I call \"stealing smart,\" which comes from a quote I use often: \"Stealing is a felony; stealing smart is an art.\"\n\nBill's \"copy\" captured Marty's attention, and he knew it would capture our audience too. The lesson is that great copy, headlines, and subject lines can come from anywhere; don't assume it has to be a formal assignment for your copywriter. Never stop searching for ideas while watching the news, reading a book, or looking through your mail (online or offline). Make a virtue of being aware of what conversations are happening in people's heads. These little gems can turn into headlines and subject lines\u2014and they are everywhere. You have to be ready to pounce on them at a moment's notice.\n\nBill was famous for saying that the outer envelope was like the \"hot pants on the hooker\"; and I am sure he would say the same today about anything that could capture the attention of your prospects, whether on an envelope or in the subject line of an e-mail. Like all of the most successful copywriters and marketers past and present, read everything you can and always be on the lookout for ways to get into the minds of your prospects, to collect the language and perspectives that would draw people irresistibly into your offers.\n\nWhen Marty realized we would be able to mail millions of names because of this new Jayme envelope, he sent Bill a check for the \"new copy.\" Bill tried to return the check, which was for thousands of dollars, obviously surprised that Marty felt the need to pay him for using a line from a personal letter. (Though I think he was secretly pretty pleased that even his off-the-cuff work was getting such a good response.) Of course, Marty wouldn't have any of that and insisted that Bill keep the money.\n\nCORE PRINCIPLE #4: SWIPE FILES ARE YOUR SECRET WEAPON\n\nSwipe files are one of the most powerful tools in any marketer's toolbox. (And if you're not a copywriter, a swipe file is a collection of proven copy and campaigns that you study and refer to in order to internalize what works for a given market and as inspiration for models that will work for your own campaigns.) Some people feel like it's stealing to study and internalize the work of other people, but really it's just being smart. Again, if there's a shortcut in front of you that will save you time and money or make your existing work drastically better, use it. The swipe files of the pioneers of direct marketing and copywriting are some of the most powerful pieces of original source material available anywhere.\n\nFor example, Jim Rutz was one of the greatest copywriters of all time (not to mention one of the most colorful characters this business has ever seen). Every copywriter, working or aspiring, owes Jim a debt of gratitude, as he was a pioneer in creating the win-win royalty arrangements for creative talent that many of us still use today. Simply put, he was one of the first copywriters to ensure that when a copywriter wrote a blockbuster package that made millions for their client, they would be paid royalties based on how many names were mailed\u2014the more successful the package was for the client, the more money the copywriter made. While this is standard operating procedure for many of today's most successful copywriters, offline and online, Jim broke new ground when he brought this approach to his clients.\n\nJim left us with some of the greatest promotional copy ever written, a compensation model that every copywriter today should thank Jim for . . . direct mail formats that he innovated . . . and dozens of A-list copywriters who couldn't copy him but were his prized students.\n\nIf you are a marketer of any product or service, studying the work of this one-of-a-kind talent will be equivalent to a graduate course in direct response. There's a lot more about Jim that I'd like to share here to illustrate the impact a good swipe file can have.\n\nI don't mean to be disrespectful when I say that organized religion owes its success, at least partially, to great copy. No matter how many times I brag about Boardroom's huge sales numbers selling _The Book of Inside Information_ or _The World's Greatest Treasury of Health Secrets_ in the 1980s, 1990s, and 2000s, they will never be the all-time bestsellers on the planet. We couldn't even come close to _Atlas Shrugged_ , much less the Bible. I bring this up because Jim Rutz understood the power of religion. He actually organized his own church\u2014one that was true to the very early origins of Christianity, when people met in each other's homes in small groups. He was _very_ serious about it and had quite a following.\n\nJim used his incredible talent as a copywriter to build his Open Church movement with story and messaging that was as persuasive as the best magalogs he wrote for the largest marketers in the world. You may not have ever heard the term _magalog_ ; Jim was one of the first to innovate and use this format. Magalogs are self-mailers that look like magazines or catalogs (adding to their perceived value and shelf life), but they are actually promotions. These worked extremely well (and still do)\u2014and they were a breakthrough in long-form direct mail for decades after Jim and a handful of copywriters perfected them in the 1970s and '80s.\n\nReading Jim's copy within state-of-the-art formats and then watching his promotions perform in the marketplace was like a religious experience for me (pun intended). His strong faith made him unique among all the great copywriters. He had faith in God, of course, but he also had faith that the products he was selling could help people, and he never doubted the intelligence of his readers.\n\nI used to kid him that I thought he had a two-person copywriting agency in his house\u2014himself and God\u2014and it just wasn't fair to have a partner with that much power. I wasn't completely joking, though\u2014Jim could beat anyone.\n\nIt was in his early days as an A-list copywriter that Jim started the practice of not charging an up-front fee. Instead, as I mentioned, he collected royalties based on the number of pieces the client was able to mail. He was confident he could write the package \"for free\" because he knew he would write a winner and beat all comers\u2014and then collect more than any up-front fee he could have charged. We are talking about a royalty of a nickel or more _per piece mailed_. That may not seem like much, but when your client is mailing in the millions, it adds up. Jim's winning packages became controls for some of the biggest mailers in the country, and he got a nickel for each one the mail carrier delivered. For example, if one huge client mailed around 20 million Jim Rutz pieces, and they were not his only client, let's do the math. Just from that one client, 20 million pieces at $.05 each is a cool million. Then, if you figure that he had dozens of clients, many of whom mailed as many of his control packages in similar numbers, it was a pretty nice living. Over time, he again started charging an up-front fee, since the waiting list for his services was so long and he had to go to the highest bidder (gotta love America, which he did). Later, he routinely charged $50,000 as an up-front fee for those who wanted to hire him, and even up to $60,000 or $75,000 on occasion.\n\nAll that is to say that Jim Rutz _really_ knew what he was doing. But I only discovered what made him so exceptional at writing winning copy after we had already been working together for several years.\n\nSince he had worked out how to build a religious movement from the ground up, I had always assumed that writing copy to sell books and newsletter subscriptions came easily to him. Then one day, I visited him at his home in Colorado Springs and was surprised to see that he had no furniture in his living room. Instead, there were just neat piles of successful direct mail from every significant direct marketer at that time. He gave me a tour around the room, naming each company as he pointed to each stack: \"Here's the Boardroom pile . . . the Phillips pile . . . the Agora pile . . . the Rodale pile.\" These were the top publishers and direct marketers at that time, and Jim wrote for all of them. He studied every package in all the categories he wrote for in incredible detail, until he knew the material back to front.\n\nHe immersed himself relentlessly in the copy that worked, absorbing the language, structure, rhythm, and hooks that made people pay attention and pull out their wallets. He studied all his competition, all the controls, and that made him one of the best copywriters that ever lived. Put a different way, Jim was a swipe file junkie . . . which most of the great copywriters are.\n\nA few years ago, I hosted an epic event called Titans of Direct Response. It brought together the best copywriters and direct marketers who ever lived (I'll tell you more about that throughout this book). During a panel of the best of the best talking about copy, it became clear that almost every one of them was obsessed with having a comprehensive swipe file. What's more, all of them still spent significant time immersed in other people's copy, soaking up what the market was responding to.\n\nI encourage you to follow their lead. You can't beat a swipe file for deep insight and inspiration. Building up a robust swipe file like Jim did is the closest you will ever get to a personalized education in copywriting from the best minds in the business.\n\nMy personal commitment to swipe files is significant. For example, at the Titans of Direct Response event, all attendees were given over a thousand pages of the best packages in Boardroom's history, plus an additional swipe file from one of the greatest marketers and copywriters of all time, Dan Kennedy (also a speaker at the event). My commitment continues with the Bill Jayme Collection I've put together, which is over two hundred pieces of his best work. I'm also working on a similar collection for Jim Rutz's best packages. Simply put, swipe files are gold, and they create an opportunity for marketers to see what worked in the past and how those winning ideas and platforms can translate into new breakthroughs in the present and the future. It's a resource that most marketers don't capitalize on.\n\nCORE PRINCIPLE #5: THAT NEW THING MIGHT NOT BE SO NEW\n\nOne of the most important by-products of learning about the foundational principles of any industry is that it keeps our egos in check. It's pretty easy to let yourself get carried away with your own genius if you think you're the first person to come up with some groundbreaking new idea. But if you know that you picked up all the salient points from concepts developed decades ago (before spreadsheets and predictive data modelling), you're going to be able to put it to use much more effectively because you will have so many examples to work from.\n\nCase in point: I was on a podcast not long ago where the interviewer decided to take a deep dive into the topic of regression modelling. This is the technique used to statistically predict response rates before mailing a list of names (using direct mail) on large databases and individual lists, sometimes modelling millions of names at a time with predictive results. It's a technique I used all throughout my career (with the help of a topnotch statistician, since I was an English major in college), and a technique that has been around a very long time . . . yes, even before the Internet.\n\nAnd it still works.\n\nI was surprised the interviewer decided to go down this path in our discussion\u2014we were getting a bit granular, and I was afraid we would lose the audience. But it was an interesting topic and something so few people in the online world ever ask about, so I went with the flow . . . and then he posted the interview on Facebook. One of the responses was both flattering and telling:\n\nWow! This interview with Brian Kurtz is fantastic. In it he talks about this incredible modelling technique he used earlier in his career. And I thought Facebook invented lookalike models. Mind blown!\n\nI'm not sharing this to give anyone a hard time who has never heard of regression modelling . . . but it illustrates an important point. That is, having a deep understanding of the modelling techniques that came before (including look-alike models, which are far less predictable) can only make you better as you buy ads from models created on Facebook. In this example, knowing the original source approach\u2014i.e., modelling on established lists\u2014will enable you to ask different, more probing questions about your campaigns. This will also give you ideas and insights on how you can make the most of each one as well as create brand-new ideas as hard data comes in.\n\nThe reason Facebook has enjoyed such colossal success with techniques like this is simple: they know it's irrelevant whether or not they invented the technique . . . what matters is that they understand how incredibly powerful this proven technique is when applied to their platform.\n\nStuff like this happens every day with my own clients and with some of the greatest online marketers I know and love. We regularly see examples where having a historical perspective on the fundamentals of direct marketing leads to new insights (and higher levels of success) using a current, state-of-the-art technique. This is the reason I always tell people not to ignore books written before the Internet. Imagine missing out on the insights of Claude Hopkins's _Scientific Advertising_ just because you thought something written in 1923 couldn't be relevant today. It doesn't matter that Gene Schwartz wrote _Breakthrough Advertising_ in 1966\u2014his insights into human psychology and the copywriting methods inspired by his work are as powerful today as the day he first published them.\n\nI regularly say that this is absolutely the best time in the history of the world to be a marketer . . . and having as much knowledge and information as possible on what has worked in the past will only make you bigger and stronger when using the effective tools and methodologies being developed every day. Many of these tools rely on advanced technology, and many on advanced logic, but _all_ of them rely on the direct marketing principles of the past.\n\nI urge you to become a devoted student of the original source material of direct response marketing. Not only will it give you an incredible edge over your competition, but it will help you build a business for a lifetime and create relationships with your customers and partners that are profound and profitable.\n\nKEY TAKEAWAYS\n\n  * When you learn the rules like a pro, you'll be able to break them like an artist. Deeply understanding the rules, principles, and methods developed by all the marketers who have come before you will allow you to innovate and experiment effectively.\n  * You develop that deep understanding of the rules of marketing by studying the original source material produced by the legends of the marketing industry.\n  * There are five key principles at play in direct response marketing: \n    1. Marketing is about psychology. What moves your market?\n    2. There's nothing wrong with shortcuts. If someone else has already done the research, use it. Don't waste time doing all the research yourself.\n    3. Use what's already working. There is material for great copy and promotions everywhere, so keep your eyes open. But if something is already working, let it run. Don't reinvent the wheel.\n    4. Swipe files are your secret weapon. Study control packages (winners from other marketers) to internalize the rules of powerful copy and for insight and inspiration. This is the closest thing you will get to a master's degree in direct response.\n    5. That new thing might not be so new. Materials and techniques developed in the 1920s and the 1960s are still completely relevant and effective\u2014and they provide the foundation for every technology and tactic we are using today.\n\nCHAPTER 3\n\nHOW PAYING POSTAGE MADE ME A BETTER MARKETER\n\n_I'm a direct mail guy, and proud of it._\n\nAs I've said before, being responsible for close to two billion pieces of mail in my career is something I am extremely proud of\u2014and the lessons I learned are worth sharing with you even if you have no intention of ever using direct mail in your marketing mix. And when I say direct mail, I do mean those paper envelopes that slide into the mailbox in front of your house . . . or catalogs . . . or even the aforementioned magalogs.\n\nDirect mail was risky then, and it is now. It's expensive, and it takes a lot of heavy lifting to get a campaign off the ground. At the height of Boardroom's success, I believe we were paying as much as some of the largest direct marketers in the country for postage. I was part of a group of direct marketers that lobbied for postal reform and lower postal rates in Washington, D.C., and I loved that I got to be in the same room with companies like National Geographic and the American Association of Retired Persons (AARP). They had no idea what this little company from Connecticut was all about, but our postage bill got me in the room.\n\nBut that wasn't the most important thing about paying a lot of postage. The most important thing was to be 100 percent sure that you had done everything you could to make a campaign a winner, or the costs of postage, printing, and production could kill you.\n\nIn contrast, online marketers today have it a little easier. Don't get me wrong\u2014I'm often blown away by the expertise and speed of execution many online marketers bring to their businesses. But I think you'd agree that having to pay all those extra costs made me a bit more careful with my marketing spending. It required me to be much more precise with list selection, offers, and creative, and that extra care and patience paid off . . . then and now.\n\nMost online marketers today don't have to deal with being that precise. While I think they _should_ be that precise, the economics don't require it. When a rumor floated around the Web a few years ago that e-mail service providers were going to start charging \"postage,\" a small, sadistic part of me wondered if that might force more online marketers to take much better care of their campaigns and audiences. Of course, it all came to nothing (which was probably for the best), but it did mean that e-mail costs kept going down. In turn, this meant that marketers were able to continue being sloppy, abusing their lists (i.e., customers), and wasting precious opportunities to build sustainably profitable businesses.\n\nThat's not to say that _everybody_ is lazy or sloppy, but you can see those traits all over the Internet. Lazy list selection, incongruent offers, and crappy creative means that a company's best customers often slip through the cracks, never to return. It's the kiss of death for any business.\n\nDON'T BE SLOPPY\n\nOver time, Boardroom built a database of millions of people who bought something at some point. But size isn't really the point. You could have a list of thousands of names or hundreds of names . . . maybe only dozens of names. What really matters is looking at that audience and thinking about when they bought, how much they bought, and how much they spent while buying. These values are known as the recency, frequency, and monetary (RFM) scores, something I lived by when I began working in direct mail in the 1980s. And it's as relevant today, offline and online. (We'll dive deeper into RFM in the next chapter.)\n\nRFM is the foundation of basic segmentation, which we were doing with large computers over millions of names back in the 1980s, and it's still the best way to segment today, whether you have a large list or not. No matter what size your list is, they're all real people. They join your list because they know, like, and\/or trust you, and you have the privilege of continuing to sell to them as long as you don't get lazy. That's the way I see it.\n\nAs with many direct mail and direct marketing companies in the 1980s and 1990s, we were mailing millions of names, and we could calculate the profit and loss down to the tiniest amount. Our business was sustainably profitable for nearly three decades because of the way we selected our lists, the way we developed our offers, and the way we executed our creative. Because we always knew our key numbers, we could calculate what we could afford to spend to acquire a new customer and then what it would take for that customer to eventually become profitable. Knowing these numbers was critical because at that time in direct mail, with long-form formats, the average cost to get a mailing out the door could be as much as $500 (or more) per thousand pieces mailed. Since we were doing extravagant, expensive, long-form direct mail (including those 32-page magalogs that looked like magazines but were really promotions) and printing lots of different formats, we had to think carefully before we sent anything out. Even though e-mail is cheap and might only cost 50\u00a2 (or even less) per thousand to e-mail, 50\u00a2-per-thousand marketing still deserves $500-per-thousand thinking.\n\nLet me emphasize this: If you don't know your numbers, you're not really using direct marketing. And if you're not using direct marketing, then you really don't know how you're doing.\n\nMarketers today often think e-mail is free. But there is always a cost. Gordon Grossman, who built _Reader's Digest_ into a publishing giant in the 1960s (and virtually invented sweepstakes as a marketing tool), used to say that if you aren't careful with your list segment selection, or if you send a lackluster offer, you will lose customers. He stressed that every communication with your list needed to sell something or accomplish something. He also said you should ask yourself this before planning any promotion or communication: \"If it doesn't lose an order or cause a rejection, it doesn't matter.\"\n\nEvery communication you make with anyone in your offline or online family will be perceived as either positive or negative. Therefore, you can lose orders and encourage rejections with sloppy marketing, even when it's cheap. If you have a responsive audience, you can still make money with a poor offer, but there may well be attrition from the list because of an offer that's not congruent or not carefully targeted. There is a real risk that the next time you send an offer\u2014even if it's much better\u2014you will have lost the trust of your audience. If they are no longer sure you are committed to excellence because your previous offer to them lacked congruence, they will be less likely to respond positively with the new offer\u2014and going forward as well.\n\nSome online marketers have figured this out. They've carefully built a tribe (what I like to call an online family) rather than just a list, and they know that they have to provide a huge amount of useful content before they try to sell something. They segment their lists down to a very detailed level, creating tailored offers for each of the different segments, and then they test different creative across each of the offers.\n\nOn the other hand, there are marketers out there who are just sending \"the next offer\" (compelling or not) to their list on a regular basis. Not to mention sending offers from partners or affiliates that also may not be compelling (or congruent). (Much more about this dangerous practice later on.)\n\nTRACK THE METRICS THAT MATTER\n\nOne day, pre-Internet, Marty and I were looking at the company financials. We were growing like crazy and mailing very aggressively for our largest and most popular newsletter, _Bottom Line\/Personal._ The acquisitions were good (i.e., we were getting a lot of new subscribers), we already had around 800,000 paid subscribers on file as indicated on the \"data card\" that the industry used to select the list for their mailings. We were looking for patterns in the data that would help us direct our efforts for the next few years. We were tracking our RFM as well as our customer acquisition costs, but we needed a better way to predict our revenues into the future.\n\nThen Marty glanced up from a stack of papers with an inspired look on his face and said to me, \"We're _not_ in the business of bringing on new subscribers to _Bottom Line_ \u2014we're in the business of _Bottom Line_ renewals.\"\n\nBy extrapolating this analysis over a period of years, we realized we could lose money on a new customer for a number of years and still turn a profit. That number of years is different for every company.\n\nIf you picked up our financial reports and just looked at the numbers for the first year of a customer's relationship with us, it would have looked like we were losing our shirts. But the key was calculating how much those first-year subscribers would contribute over the entire time they were subscribed. Renewals were the biggest part of that total, but having multiple products to sell them (in our case, other newsletters and many books on similar subjects) was truly the key to maximizing the lifetime value (LTV) of a new subscriber. If one metric matters most above all others, it is lifetime value\u2014and doing everything you can to know that number as accurately as possible should be your top priority.\n\nI believe most online marketers today understand the value of a new customer and also understand the direct marketing rule of thumb that says it's always better (and more profitable) to keep an existing customer than acquire a new one. Being laser focused on renewability and offering as many synergistic products and services as possible to that new customer is where the focus should be rather than just adding new customers.\n\nLifetime value analysis is not easy to do, but when you do it, it's a game changer for the business. Dick Benson called this the bogey, old military slang for an unidentified aircraft\u2014but he repurposed it to mean the unidentified costs and long-term profit associated when acquiring a new customer. As a consultant to Boardroom and so many other publishers and direct marketers over his career, one of his primary goals was to help us identify and calculate our bogey.\n\nDick expanded his definition of a bogey to essentially mean what a marketer was willing to lose on an initial order to make it back (and more) in subsequent years. We needed to know how much we could afford to pay for a customer based on what they would spend with us over time. When you think about this for your own business, consider how much you spent to get a new customer, how much you make in year one from that customer, and how much it's going to take for you to get that money back in year two or maybe longer, depending on your cash flow needs and how much you can invest in a new customer.\n\nI recommend finding somebody to work with who can give you an accurate calculation, which in turn will enable you to buy more media that you didn't think you could afford. While I'm not the expert in doing the calculation\u2014I relied on far more skilled analysts and statisticians to get those numbers\u2014the main thing here is to make sure you're always focused on renewability and building a stable of new and exciting products or services.\n\nKeep in mind that calculating LTV should also be channel specific. A new customer originating from direct mail is different from one originating from e-mail, and that customer will be different from one originating from Facebook. It's important to make these calculations based on the original channel the customer came from because two things are most likely happening:\n\n  1. You're missing out on big opportunities with channels that don't look profitable on the first sale.\n  2. You're probably spending way too much on channels that do look profitable on the first sale but are really losers over time.\n\nTracking every new customer from their first interaction or purchase by their original channel is the most important first step you can take in building a sustainable marketing strategy. In the prehistoric age of direct mail only, it was about tracking the first sale based on the offer they were sent; in the online world, you can track interactions before the purchase too, which allows you to predict the timeline from prospect to customer by channel, along with the channel-specific LTV.\n\nTHE IMPACT OF GREAT PRODUCTS ON LTV\n\nReid Tracy, CEO of Hay House, when once asked the secret to selling a lot of books replied, \"Write a good book.\"\n\nWhat does that have to do with LTV? Everything.\n\nWhether it's a book, a course, a newsletter\u2014anything\u2014the way to keep customers engaged and with you for the longest amount of time possible is to give them good stuff.\n\nIn the world of newsletter and magazine publishing, there is a universal belief: \"Marketers sell subscriptions; editors sell renewals.\" In making the case that renewals and continuous business with your best customers is the key for long-term growth (through increased LTV), this premise is applicable to all businesses, not just publishing.\n\nThe quality of your products or services is directly correlated with increasing your renewal rates and LTV. Look what happens when quality is low: You've heard the stories of 50 percent return rates on launches online, and if you lose half your customers immediately, you've got no chance for long-term success. So . . . \"write a good book\" every time to protect your LTV.\n\nOnce we started seeing this pattern in our data, we were able to really focus on getting every customer to renew their subscription to the newsletter, rather than putting all our attention on the first sale. To me, there's nothing more important than the _second_ sale, and that's why having multiple products\u2014and high-quality products\u2014is key. So many people get sucked into the importance of that first sale, not realizing that the real upside for any business is in repeat purchases. But when you take a long view, you get to make fully informed decisions about which lists and products you should double down on.\n\nArmed with our bogey and this longer view, we were able to prune our customer acquisition channels down so that we were only working with people who really wanted the information we were selling and would pay for years on end to keep soaking it up. (We'll get deep into the details of how this happened in Chapter 9.) Not only did we track the lifetime value of the original channel and the renewal rates at different time periods, but we also tracked the performance of the promotions we sent. The next chapter will explore this further, with examples of how different kinds of promotions bring on different kinds of new customers and once in the fold, understanding that not all new customers are created equal. Follow-up promotions need to be consistent with the promotions that brought them in in the first place.\n\nTracking the performance of the different packages, the new customers acquired from those packages, and how we made the second sale (and beyond) to those customers taught us a lot about the type of customer that will convert based on _how_ you're selling. This is subtle but critical. Pay close attention to this as you test different copy, formats, and offers. It's one of those direct mail lessons that doesn't often get applied online . . . but it should.\n\nAn overriding principle is that things change. You have to calculate your bogeys regularly, because your business and the channels you're working with will evolve.\n\nYou also need to calculate everything on the basis of your business model. You need to know when you have your money back. If you're not a mature company, focus on the sources that have shorter bogeys. The return on your investment doesn't have to be immediate, but it may need to be faster based on your cash flow and overhead requirements. When you're established and have stable cash flow, you can go with the longer bogeys and tolerate more risk.\n\nFOLLOW THE ANECDOTAL EVIDENCE\n\nThe most logical person I ever met was a man named Adolph Auerbacher. His name may not be familiar to you if you're under a certain age, but he was one of the most influential people in both magazine publishing and direct marketing. Adolph was a senior executive at Meredith Corporation when it was one of the largest magazine publishers in the world. He was the brains behind the tremendous growth of magazine brands such as _Better Homes and Gardens_ and _Ladies' Home Journal._ If you haven't heard of those titles, go look them up. They were incredibly successful and case histories worth learning from.\n\nAdolph spoke softly and gave sage advice in very few words. He was always on point, and something he said to me in the early 1980s has stuck with me for over 30 years: \"Follow the anecdotal evidence.\"\n\nWhat he meant was to look at what was happening out in the world, think critically about any current trends and what had happened previously in the market, and then measure it against our experiences, strengths, and weaknesses to find a good way to capitalize on the opportunities in front of us. No one _said_ that Boardroom should go into infomercials (28-minute TV spots), but the anecdotal evidence told us that since we had a lot of information to share, we should take advantage of this new long-form medium. Long-form copy had drastically outperformed short-form copy for selling our books and information in print, so it made sense that a 28-minute TV spot would be more effective than a 2-minute spot. Following the anecdotal evidence (and a bout of insomnia) eventually led to a franchise that created sales well over $200 million employing a trifecta of media: direct mail, direct response television, and online display advertising.\n\nThe story begins in 1988, when every magazine seemed to be selling subscriptions on TV, using one-minute or two-minute ads. The _Sports Illustrated_ team was pranking people in shoe stores with their sneaker phone campaign and turning the recordings into ads: They would mount the sneaker phone on the wall with all the other shoes, and it looked so much like an actual sneaker that customers would be startled when the shoe they were considering started ringing. The phone came free with a subscription to _Sports Illustrated_ , and between the sneaker phone and the football phone that had preceded it, the magazine sold millions of dollars' worth of subscriptions on TV.\n\nI was jealous\u2014TV was one medium that we had not conquered. Unlike direct mail, it didn't have postage and printing costs, but there were still other huge up-front costs in producing spots that made it a risky medium . . . but one that was high risk with high reward.\n\nIn 1988, our most popular consumer newsletter was the aforementioned _Bottom Line\/Personal (BL\/P)_ , and our best-selling book was called _The Book of Inside Information_ ( _BII_ ), a compilation of Bottom Line's greatest hits (the best articles that had previously appeared in the newsletters). We had an incredibly loyal readership, and the information was very valuable, so we figured that if TV worked for _Sports Illustrated_ , it could work for us.\n\nBut our short-form TV ads bombed spectacularly. Even with spots as \"long\" as two minutes\u2014about four times as long as most other TV ads at the time\u2014there was no way to adequately sell products like ours, covering complex subjects with an unknown brand behind them. That is, our product line was basically unknown (except to the \"lucky millions\" in our direct mail universe). The TV market was very broad and hard to segment (this was 1988, remember), and viewers needed a huge amount of education before they would be confident enough to pick up the phone and place an order for products like ours.\n\nIn the late 1980s, an \"old media\" was brought back to life after a hiatus due to Ronald Reagan deregulating it. TV infomercials were long form (28 minutes and 30 seconds). I started following the work of Guthy-Renker, who created and produced huge winners with this format, many of them with content that had some affinity to ours. The Guthy-Renker show \"Personal Power\" (with Tony Robbins) inspired me the most. By 1991, over 100 million Americans (in 200 media markets) had seen the ad\u2014which is astounding given the infancy of the infomercial industry at the time. I keenly observed that 28 minutes and 30 seconds was a lot longer than 2 minutes and would give us enough time to tell a story that would educate people and make them want to buy. So Marty and I met with Bill Guthy and Greg Renker to tell them about our amazing Bottom Line franchise.\n\nWe knew we could help millions of consumers with life-changing information, and we wanted to explore creating a program for this brand that would reach beyond our existing customers. We wanted Guthy-Renker to do for us what they did with \"Personal Power.\" (If this program was before your time, look it up on YouTube. It sold like crazy, and it's an incredible early direct-marketing case study, suitable for the swipe file I know you are building since reading Chapter 2.) Our logic seemed solid: \" _Sports Illustrated_ sells with simple envelope mailings in direct mail and complements that with 2-minute TV spots. We sell with long 12-page letters, 16- or 32-page magalogs, and even 64-page digests (bookalogs) in direct mail, so long-form direct mail should translate to around 28 minutes to sell via TV!\"\n\nBrilliant, right? Nope. We couldn't get it off the ground.\n\nOur first attempt at a TV infomercial went absolutely nowhere. Not because of Guthy-Renker or because we didn't understand the audience, but where Tony Robbins had a whole suite of congruent products\u2014tapes, CDs, and workbooks\u2014we only had the one book, a newsletter, and other disparate pieces that didn't make sense to sell together.\n\nIt was clear we hadn't collected _all_ the evidence on what made infomercials work and didn't understand at the time how to use our assets to make the medium successful for us, despite the logic that long-form TV would be more effective than short-form TV. Even though I knew there was a huge opportunity for us with infomercials, the prospect of spending the time and money to figure out how to make such an expensive medium work for us soon collapsed under its own weight. So we went back to doing what we knew best.\n\nFast-forward a decade or so. I was sitting up at 2:00 A.M. just after New Year's Eve with a nasty bout of indigestion (I am so glad to have given up sugar now). I flipped on the TV to see an incredible pitchman, Kevin Trudeau, sell a single book for 28 minutes and 30 seconds, at $29.95, without a recognizable company, without a known brand, and frankly, without much credibility. If you've never heard of Trudeau, look him up as a cautionary tale about what happens when you prioritize fast money over truly caring for your customers. I won't comment further on his business practices, how he sold, or what he delivered. The point here is that he was using the medium that had been taunting me all these years, and he was clearly having a lot of success as evidenced by how often his shows were running\u2014direct marketers don't keep buying expensive media if it's not paying out. This is another rule of thumb I learned from doing direct mail: If you saw the same control mailing piece in your mailbox over a period of months (sometimes years), you knew it was a successful control and \"swipe worthy.\" Connecting that rule to TV, when someone pays for pricey TV time (like they might pay for postage), they are not simply washing, rinsing, and repeating without reading results and determining profitability by some measure. What really bugged (and eventually inspired) me was that he was selling a product that did not compare with or meet the standards of all our legitimate, high-quality material and experts. I knew I needed to reengineer what he was doing to achieve similar success.\n\nAdolph's advice popped into my brain, and once again I knew it was time to follow the anecdotal evidence. I called the number on the screen to order Kevin's book and was on the phone for what felt like _hours_. The rep on the other end of line tried to sell me every version of the book\u2014CD, digital, one wrapped in a Snuggie (just kidding on that last one)\u2014followed by a Walmart discount card, a travel club, and a host of other offers. There were upsells and downsells, cross-sells and discounts . . . and now I saw the model that would work for us. We wouldn't sell _everything_ on the phone, but we could sell something else to raise the average order value far above the price listed in the infomercial.\n\nThis is a good example of a direct marketing principle that I love\u2014that is, every company can ratchet up or ratchet down the aggression of the sales message and the offer as long as they are measuring everything every step of the way. So while we were following the Trudeau model, we only sold things that were congruent with our brand and that we felt comfortable selling to a new audience. We followed the anecdotal evidence on what was already working and stayed within the boundaries we were comfortable with.\n\nThen I had my lightbulb moment. We already had a book that could be the lead offer: _The World's Greatest Treasury of Health Secrets_. I reached out to infomercial producer and direct marketing genius Steve Dworman to help us execute this idea. Steve had his own lightbulb moment when he realized that the doctors we worked with, who had material in the book, could all be interviewed for maximum credibility. This would help the audience know, like, and trust us, and would lead them to understand the offer.\n\nFor the first show, we chose well-known radio personality Barry Farber as the host. He interviewed one of our copywriters, Arthur Johnson, who (like all great copywriters) had insatiable curiosity in all things health. Johnson also served as the \"medical editor\" for the show, representing the doctors and the material they shared. We ran the rest of the campaign just like we did with direct mail\u2014multiple entry points and plenty of back-end offers, upsells, and subscriptions\u2014and everything came together in a perfect storm. That first show did incredibly well.\n\nThat gave us the confidence to up our game, so we decided to really aim high for our next show. We brought on Hugh Downs, a well-respected newsman, as the new interviewer. Hugh again interviewed Arthur Johnson, our copywriter\/medical editor. This show outperformed the previous one, as we added many doctor guests to talk about their own article in the book. And we kept pushing. The show's next iteration was live at the Hugh Downs School of Communications at Arizona State. While the previous shows had doctors on a panel, this one was also attended by doctors\u2014including a Nobel Prize winner\u2014who contributed through audience participation, adding to the gravitas and credibility.\n\nThese progressive iterations inspired us to create new variations of health books with more variety and cutting edge health information than previous ones. _Treasury of Health Secrets_ became _Ultimate Healing_ , with a new format and refreshed content. We kept striving for new heights once we had the formula figured out, and it worked for quite a while. The moral of the story is that although we realized in 1989, when infomercials first became popular, that long-form TV would be much better for our products than short-form TV, it took until 2004 to find the right approach, over 15 years after we made the intitial connection. Trusting the logic and the anecdotal evidence was step one; figuing out the model took some time.\n\nAnd don't forget\u2014once we found the model, we didn't just imitate it (i.e., follow Trudeau); we adapted it to our own image.\n\nTHE VALUE OF DISCIPLINE\n\nIt took us a long time to make television work for Boardroom's brand. I chewed on it for close to two decades, coming back to it over and over, every time I got new information or saw someone else having great success with the medium. Fortunately, the wait was worth it, and the lessons were profound. When we entered the world of infomercials, we were told that the success rate for new shows was something like 1 in 15. Some experts even told us it was more along the lines of 1 in 20. But lucky us! We had a blockbuster success right out of the gate with our first show and proceeded to have even bigger winners in our subsequent shows. We beat the odds by a wide margin, with four out of our first five shows becoming blockbusters.\n\nHowever, on further review, maybe we weren't so lucky after all. Once we figured things out, we should have been extremely careful with TV after taking such a long time to reach success. But with our huge hits up front, we got sloppy. We started planning all kinds of shows in new categories and taking bigger risks with unproven products and formats. Not only that, but we took all these risks when a recession was looming.\n\nNot everyone agreed that a financial crisis was about to hit, but we should have factored in the possibility as we planned our aggressive strategy, particularly since we knew TV was one of the riskiest mediums. Unfortunately, we had been reading our own press clippings and assumed we could do no wrong. And I was by far the biggest proponent of our aggressive approach to getting another big win; thus, I was also the biggest culprit when results were not good. This was about my own hubris and it was my call, not anyone else's.\n\nLet me be clear that there is no fault in trying to build on momentum when you're achieving huge success in a particular channel, but you do need to stay grounded and objective. Plenty of people with years of experience in TV advertising told me to be careful. They told me that TV was more volatile than other channels and that there were a lot of ups and downs with it . . . but I became too overconfident to hear them. Folks internally told me the same thing. Lots of people were gushing about how incredible we were and how we were an exception to the rules of infomercial success . . . and I bought it. While we _were_ seeing amazing results, and had come up with a unique formula, we just kept churning out ideas without doing our research or holding the campaigns to the kind of standards we did with our direct mail.\n\nComing straight off super-successful campaigns that defied the odds, I thought I had the Midas touch . . . and I then proceeded to produce a series of shows that were not winners. While the excursion into TV was the most successful multichannel campaign in my career, we could have made many millions more if I had been more disciplined with the medium.\n\nWe learned a lot of valuable lessons from those shows that did not work. The biggest lesson is that if you don't approach every direct response campaign with discipline and objective standards, you're dead in the water. Having the privilege of working with the best marketers in the world at Boardroom (inside and outside) made it almost worthwhile to have some shows not be as successful or campaigns that fell short so that I (we) could learn from them and come out even better the next time.\n\nDiscipline is what made the direct mail generation of marketers so successful. Every single campaign sent through the United States Postal Service had to be meticulously researched and planned so that nothing was wasted. Every test had to mean something. Every test needed to light the path to a potential breakthrough and a new control package.\n\nDirect mail marketers are always testing new approaches against the control to find the new winner. The best marketers online today think about testing the same way\u2014that's why they are the best marketers online today. They know that selling so easily and cheaply online is an opportunity that should be used and not abused, given that it's unlikely you'll have to pay postage anytime soon. I remember an urban legend in the early days of the Internet that postage was going to be assessed for every e-mail\u2014it was a made-up story\u2014but I have to admit that creating a tougher barrier to entry for anyone and everyone to use e-mail kind of intrigued me as a marketer who knew how to pay postage and make money. My fantasy was that this would perhaps weed out the imposters. But as we know, this did not happen. But when they truly understand that there is a huge cost to sending marketing messages that are inappropriate, not targeted, and lack discipline, maybe they will see the light. Their sloppiness hurts all of us.\n\nSimply put, the best marketers in any medium know that waste sucks. Discipline isn't just something for the folks paying postage. It benefits every marketer, no matter whether you're using e-mail, TV, radio, affiliates, or pay-per-click advertising. The fact that you don't have to pay for postage to send your online marketing messages is not a license to beat your list into submission until they buy.\n\nIn the spirit of online marketers learning from the discipline of direct mail, here are nine best practices that can be applied to all media.\n\n1. Use Content Strategically\n\nEverything you send doesn't have to _sell_ something, but everything you send must _achieve_ something. When you are marketing\u2014in any medium\u2014while everything is not a revenue event, everything is a relationship event. Familiarize yourself with what different types of strategic content look like and always be sure that it fits with the content and products you eventually sell. Another way to express this is that the ability to give away your best content for free has created its own marketing medium\u2014and it should be completely consistent with the content you sell.\n\n2. Deploy the Best Resources for Your Copy & Creative\n\nSurround yourself with successful marketers who understand direct response and copy. And remember that developing breakthrough creative is not just about employing the newest techniques, although those are important. It's much more about paying a lot of attention to your messaging. In addition, it's imperative to develop a clear strategy first and then work with the best creative talent you can find to execute that strategy. Creative and copy is not an afterthought. Spend time and money on hiring the best. Just because it's inexpensive to launch a promotion online, it is not an excuse to be lazy with your copy and creative. Get those heavy hitters in your circle to sign up for all your messaging. Listen carefully to what they tell you about how your copy looks when it gets delivered and where you should be tweaking. If possible, hire those people to help you.\n\n3. Find Out What's Missing\n\nFind some secret shoppers to buy and experience your products; also, use focus groups to get information you cannot get anywhere else. Dan Kennedy calls this \"marketing by walking around.\" Find people to buy, bend, and break your offerings without giving away who they are; and constantly talk to, and hang out with, people who represent your ideal audience. These aren't experts in direct response and advanced copy. They're the type of people who could potentially be your best prospects and customers. Several world-class copywriters use an innovative technique when writing copy to a particular audience. They pay a panel of people who seem like their target customer to read their work before they send it to the client. This gives them real-world opinions and reactions before the copy actually gets in front of a \"live\" audience.\n\nPerry Marshall has his own version of how to \"learn what you can't know.\" His idea is to get on Fiverr and hire people to read the material, then ask what they liked and hated, what they wanted more of and less of, whether or not they would pay for the product in a store, and why. Paying for this kind of feedback will give you accurate, useful insights that will help you improve your material. Listening to your customers, or people just like them, is always going to be the most powerful tool you can use to shape your products and marketing, no matter what you're selling or how you're selling it.\n\n4. Sweat the Small Stuff\n\nInvest the time and effort to agonize over every word in your copy. Even though copy has less impact on your campaign performance than list selection and the offer itself, why let it go out at any standard below perfection?\n\n5. Look for Consequential Thinking in Your Messaging\n\nI learned the term \"consequential thinking\" from Marty. It means putting yourself in the prospect's shoes and seeing how you would react to the elements of the copy. Does it take you through a sequence that makes sense? In direct mail, this is a science. How the mailing piece is received has a huge impact, and every element matters\u2014the placement of the address, the order the recipient sees the pieces in the envelope . . . Online, of course, you have many more ways to guide your prospect through a sequence. Navigation and site order or page design play an important role here, and you should think about how your audience goes through your landing page (e.g., sales pages, opt-in pages, first screen on a home page, etc.). Does each page make sense based on what the audience has seen immediately before? Are you guiding them in a logical, comfortable way through your sequence?\n\nAlways ask, \"Who is the audience that this will most appeal to?\" Conversely, think about who your copy could possibly alienate. If your copy does have the potential to alienate, consider whether those people are a good fit to become your customers. If not, don't worry about it\u2014it's okay to scare off the people who will never buy from you anyway.\n\n6. What's the Logic Line?\n\nThis is another one I learned from Marty, and it takes the idea of consequential thinking to a more strategic level. Is there a \"logic line\" that flows through your marketing? Does each part of the story follow from what came before it? Is there a thread that runs through your argument, tying everything together in a logical, believable way? The logic line is closely related to being consistent with your brand identity. It might be helpful to think of it that way too.\n\nThe purpose of each sentence should be to make sure you can move the reader to the next sentence. You need a logic line within each marketing message you send, but you also need a logic line for your business as a whole. Is each message congruent with your marketing message overall? Will it resonate with what you've sent in the past, or does it contradict earlier messages? For folks working with affiliates, this is especially important. The affiliates you partner with are a reflection of you, so they need to meet your logic line too. If there is a contradiction in the message, you need to decide if you want to move in a different direction or if you want to rein a particular piece in (or not allow a particular affiliate into the mix) to better fit the overarching message of your business.\n\n7. Do You Care?\n\nNo matter how hard-hitting the copy might be, it's crucial to be empathetic. Is there some element of care and concern for your end user? Do you actually care about the problem your offer is supposed to solve for your audience, and does the message you are sending communicate respect and care for those seeking help? People see straight through fake interest, so if you don't care about the customer or the outcome they get, don't send the message.\n\n8. Give Them a Reason to Care about You\n\nAudiences are basically selfish. It's not their job to care about your business or what you do. To that end, create your marketing assuming that nobody cares what you have to say. Instead, write in such a way that you _give_ them a reason to care. No matter how much you believe your product, service, or message is a \"need to have,\" always assume your customers think of it as a \"nice to have.\" Your job is to convince your audience to shift their belief from one to the other.\n\n9. Understand the Basic Rules of Language\n\nYou don't have to be obsessed with correct grammar or perfect punctuation\u2014enjoyable content and copy often uses informal language. But if you are going to violate the rules and standards of a language, you really need to understand what you're doing. Any nonstandard usage must be in line with how your audience uses that language.\n\nGene Schwartz always said, \"Grammar is overrated.\" I believe he was saying that communicating with your audience in their own style always trumps perfect grammar and usage\u2014but _make sure you get it right_. Messing with the structures people understand can really trip them up if they don't get it on an intuitive level. If you're not sure, stick with simple language so that everyone in your audience will understand it immediately.\n\nWHAT IF YOU HAD TO PAY POSTAGE?\n\nAs a marketer, the stakes are high. Maybe you don't have to pay for postage, but you depend on the opinion and respect of your audience. That puts your reputation and authority at stake every time you communicate with them. It's hard to measure the cost of every off-brand or incongruent message, even if the e-mail you use to send it is virtually free. I can assure you that you will find the hidden cost in the long run when your customers don't trust you and your prospects are turned off before they ever buy anything. If you can't be bothered to approach your online family with discipline and care in your communication, it's better not to contact them at all. Save yourself the hidden opportunity and goodwill costs until you have something really stellar to offer or something meaningful to say.\n\nKEY TAKEAWAYS\n\n  * Direct mail (and other offline) success stories have left a lot of clues for all of us to learn from. Your numbers don't lie, and they will enable you to identify campaigns that aren't working (and should be killed) as clearly as they will show which campaigns are winners (and should be expanded).\n  * Take calculated risks and build off previous successes before you reinvent the wheel. Follow the anecdotal evidence and remember that consistency and discipline are _everything_ in direct response marketing.\n  * A windfall of profit does not mean you have unlimited funds for testing. Pretend you have to pay drastically more than you actually pay if you feel yourself getting sloppy.\n  * Be disciplined. Don't waste any opportunity to build rapport or a deeper relationship when communicating with your potential or existing audience, and don't piss them off either.\n  * When campaigns are expensive, it forces you to be 100 percent sure that you have done everything you can to make the campaign a winner. If you act as if your campaigns are extremely expensive to send, that will ensure that you only send campaigns that have been crafted to be as effective and profitable as possible.\n  * Track the metrics that matter: the lifetime value (LTV) of customers, channels, and campaigns and your \"bogey\" (what you are willing to lose on an initial order) are critical numbers to measure over time.\n  * Don't read your own press clippings; listen when the experts give you advice; and don't let hubris or any unproven beliefs take you off course.\n  * There are nine best practices gathered from direct mail that can be applied to any kind of direct response marketing: \n    1. Use content strategically.\n    2. Deploy the best resources (e.g., copywriters and designers) for your copy and creative.\n    3. Find out what's missing from your campaigns through constant questioning, testing, and research.\n    4. Sweat the small stuff.\n    5. Use consequential thinking (i.e., how will your prospect consume what you send?).\n    6. What's the logic line? Can the prospect follow your logic throughout the promotion?\n    7. Do you care about your customers?\n    8. Give those customers and potential customers a reason to care about you.\n    9. Understand the basic rules of language. And know the language your customers and potential customers use.\n\nCHAPTER 4\n\nLIST BUILDING & RFM\n\n_\"No mailer [or marketer] spends enough time on lists.\"_\n\n\u2014 DICK BENSON\n\nWhen you really think about it, the only thing that matters is who you're communicating with\u2014and how you're communicating with them. It's always about the list. I've said this so many times that it's becoming a running joke about me, both in the marketing world and in my own house. My daughter gives me a hard time about my solution to nearly every problem being \"It's always about the list!\"\n\nWe joke around, but it's true: every business lives and dies by their list. I often worry that the word _list_ objectifies what we're talking about, so I want to say this again because it's worth repeating: the list is real people. I'll be using the word \"list\" throughout this chapter, but please understand that every time you see the word _list_ , I'm talking about people, your online family.\n\nAnd when I say \"the list\" here, I mean anyone in your audience. The term came from direct mail, where there would be a huge database of names to pull from, but it also applies to your e-mail list, your social media followers, prospects who see your ads in other media\u2014anyone that you are building a relationship with and promoting to.\n\nI focus relentlessly on this because the majority of my career was spent in the highly competitive world of business-to-consumer (B2C) marketing. There it's critical to create a powerful connection with the people on your list. Your advantage is how you communicate with them. Over time we had millions of names on our list, buyers and nonbuyers (and having the records of who _hadn't_ bought was just as important in how we sold to and communicated with each segment).\n\nAfter going out on my own, I still concentrated on building my list, but now in the business-to-business (B2B) world. My list at the time of writing is at around nine thousand names. I think about these people the same way I thought about those nine million people I obsessed about at my old job. No matter how big your list is, it's your most valuable asset.\n\nIn this chapter, I'm going to share the strategies and tactics I used then and still use now for myself and for clients to build a responsive database, regardless of size.\n\nWHEN 41 PERCENT IS A MAJORITY\n\nA direct marketing rule of thumb is the formula 40\/40\/20. This says that 40 percent of the success of any direct response campaign is based on the targeting, segmentation, and quality of the list; 40 percent of the success is based on how relevant and irresistible the offer is; and 20 percent of the success is based on the quality of the creative material and the messaging.\n\nBut the list is the most important part. Over time, as I've worked with clients, this has become so apparent that I've changed the formula to 41\/39\/20, with the list taking the lead with 41 percent, followed by 39 percent for the offer and 20 percent for the creative.\n\nIf you have a perfectly targeted list, you are well on your way to success. I've seen many situations where the list selection or the audience the marketer was trying to reach was off. Then\u2014even though the business ran the best creative from the best copywriters, all with a compelling offer\u2014the campaigns got virtually zero response.\n\nIf you have a carefully selected list, populated with people who are perfectly suited to what you're offering, you can throw average creative at them (red box with a flashing arrow, anyone?) with a fairly unremarkable offer, and you'll still make some money.\n\nI am not _recommending_ average creative or unremarkable offers. If you are an online marketer using great affiliates (i.e., people with lists that are compatible with your offer) who endorse your product enthusiastically, even while using very basic creative and offers, many campaigns will still make a lot of money just because their lists are such a fit for your offer. But you can make a lot _more_ money when you combine perfect list selection with an irresistible offer and some world-class creative. That's when you start building a business for a lifetime and when things really start exploding, whether you're offline or online.\n\nI recently saw someone online saying the 40\/40\/20 rule is dead and touting a new \"25\/25\/25\/25\" rule. Their theory: a campaign's success is based on 25 percent list, 25 percent offer, 25 percent creative, and 25 percent technology. This just doesn't make sense to me. I respect anyone who wants to debunk an old theory, but this one doesn't land. I believe wholeheartedly that technology makes us better marketers, but to quote advertising legend Bill Bernbach: \"Never adapt your idea to the technique, adapt the technique to the idea.\"\n\nYou want to come up with the perfect offer to make to the perfect list with the perfect creative. But don't get attached to the medium, or to any one technology. The medium or technology you use is not what matters most\u2014the list is what matters most. Maybe Facebook will be the right platform, maybe display advertising will be.\n\nMaybe space advertising in print magazines or newspapers would be better, or even direct mail. It could be TV or radio or working with affiliates. These are all good channels\u2014all of them work\u2014but what is critical is knowing which medium is going to be the right place to find your audience and engage with them most deeply. Technology is an important tool however you market, but it is not a building block like the list, the offer, or the creative.\n\nA CLASSIC APPROACH TO LIST BUILDING\n\nWhen Marty first had the idea for his flagship newsletter, _Boardroom Reports,_ he really wanted to launch it as a magazine. Marty came up in the golden age of print\u2014he was a very successful space advertising salesman for magazines and was also fascinated with direct mail. Launching a magazine that could compete with the big, established titles at the time would have required an astronomical amount of money. So he decided to build up the idea as a newsletter instead. That meant that there would be no advertising (i.e., no revenue from outside advertisers). Launching _Boardroom Reports_ as a newsletter would keep production costs down, because unlike a magazine, a newsletter could print on less expensive paper and there was no need for extravagant design. Newsletters were first and foremost about the content.\n\nBut Marty saw an opportunity to market his newsletter like most companies were marketing magazines at the time\u2014sending what were called bill-me-later offers, which meant that he was willing to give away free trial issues before getting payment. At the time most newsletters were cash-with-order (the new subscriber had to send cash or credit card up front with their order, like we do online today). But because Marty wanted to grow his newsletter larger and more quickly, he used bill-me-later offers instead.\n\nA lot of publications were sending two or three free issues (or what we call \"grace issues\") before they sent the reader a bill for the subscription. Because Marty couldn't hold out that long for cash flow, he went with a lower risk model: he would send the newsletter to potential customers with _one_ free trial issue and a bill-me-later order form. While he carried some risk, he was committed to quality content that he knew people would pay for, and he was confident that enough people would pay to get _Boardroom Reports_ once they sampled it. This model allowed us to get to critical mass much faster than requiring cash up front.\n\nWe were able to extend credit to prospects like this because of our deeply targeted list selection, _based on their previous buying behavior_ (we mailed lists of customers who had previously paid for bill-me-later offers from other publishers). We also knew that the quality of the content we were producing was groundbreaking and that the offers we were making were congruent and valuable too. The recipients, we thought, would immediately see the value of paying up. Later on we were able to take advantage of new credit-screening technology to cross-reference prospects with credit reports to ensure that only people with good credit ratings would get the bill-me-later offer, removing anyone who had bad debt and was unlikely to pay up.\n\nThis was risky because the model was mostly used by publications (e.g., magazines) who had another revenue stream: advertising. Some newsletter marketers thought this was insane (and still do), but we took the risk, and it paid off massively. Marketers I meet today who sell digital products online have a hard time wrapping their heads around this idea\u2014and for good reason. But this could be game-changing for anyone who can crack the code on a true bill-me-later offer using today's technology. It would help to build a larger list, more quickly and profitably, and open the way to all kinds of potential promotions. I leave it to you to think about all the possibilities. Some online marketers have experimented (and some have had success) giving away their product without a payment up front\u2014but this online model for a bill-me-later usually still includes taking a credit card number which they charge 30 days later (or sometime in the near future). It's what I call a hard-bill-me, and it's a worthwhile test. But I am sure you can see that being able to do a bill-me-later offer without asking for any payment information up front would increase response considerably (i.e. totally risk free and hassle-free for the buyer . . . no need to think about the charge on the credit card that will be coming at the time of the order). The key to making a true bill-me-later offer work is to make sure the offer only goes to more creditworthy people (doable with technology) and that your product is everything you promise in the sales copy, so there is no remorse (i.e. little or no bad debt).\n\nBack to the launch of _Boardroom Reports._ Marty knew that the list selection would be good and the offer would be irresistible, so now all we needed was world-class copy. Who else but Gene Schwartz could write that launch package? The package he wrote was, of course, incredible, and I'll share its details in Chapter 6.\n\nMarty couldn't mail as much as he would have liked at the start, since he had to manage the cash flow so carefully, but he proved this list-building model. Advanced list selection and segmentation, an offer where we could give away great content up front for free, with copy and creative written by one of the best copywriters in the world . . . that's a prescription for success in any medium, past or present.\n\nOf course, this only works if your content is actually good. If your product sucks, no one will pay for it\u2014even if you _give it away_ to targeted lists with a great sales letter. A lot of marketers miss this when they don't pay attention to the quality of their product, which leads to the 40\/40\/20 rule (or the 41\/39\/20 rule) falling apart.\n\nTargeted list segmentation is the most important thing you can do with any marketing program . . . and then adding a great offer and the best creative is the formula for success. But if you are not delivering on what you promise, no execution of any kind of percentage rule will help you.\n\nHOW TO FIND THE RIGHT 41 PERCENT\n\nMost marketers agree that their list is their most important asset. This was true for the direct mail industry, and I believe that rule will continue to hold. If you think that the idea of mailing other people's lists (i.e., affiliate marketing) was invented by online marketers, let's talk about where it originated in the direct mail world.\n\nDirect mail companies regularly used their lists to access other relevant lists through exchanges, and they frequently rented other people's lists. The primary tool of the list industry for marketers using direct mail was the data card. Every list that was available in direct mail had a data card with (supposedly) all the pertinent information any mailer or broker would need to make a decision on whether to mail the list or not. Assuming that the data cards were complete and accurate, list managers (the people who represented the lists in the marketplace) would simply mail data cards in big stacks to list brokers (the people who worked with the marketers on list selection).\n\nData cards and the dance that list managers and brokers created around them was how the list business worked. Looking at how the most successful direct mail companies worked within this system to find the most responsive lists for their products or services is instructive for anyone marketing today in any medium.\n\nMost list managers were lazy and did not give all the compelling reasons for list brokers to recommend their lists to clients. They usually just threw numbers at the brokers, expecting the brokers to make sense of the data in a way that would allow them to do something interesting or innovative with the list on behalf of their client. Even when list managers met with list brokers and presented the data in person, they would only talk about the lists, but not about the offers and creative that had gone into building those lists . . . which really was the key to whether a list would work or not. Not only did that mediocre approach undercut a list manager's opportunity to build great partnerships, it also turned a specialty product\u2014a unique list\u2014into a commodity.\n\nThe reason I got so good at selling lists was that I saw this hole in the system. When I presented to list brokers, I not only presented the data cards on Boardroom's lists, but I also brought the promotions that got the names onto the lists in the first place. I went way beyond giving the broker a pile of statistics to digest\u2014I profiled the psychology of the buyers and subscribers on that list based on the offers and creative they had responded to.\n\nBeing able to explain how and why the names on a particular list got there was far more valuable than being able to rattle off a bunch of demographic data. I could also explain the segmentation we had used to create our list\u2014that is, the promotions our best customers had responded to before. I could make connections between the copywriter we used for our winning promotions and the copywriter (or approach) the client was using, highlighting where there was even more synergy.\n\nAll this presented a powerful picture of how our lists would perform based on data that could never be discovered on a data card. Digging deep into the psychology of the lists you promote (for example, when choosing affiliates and joint venture partners) is critical to success. Again, your list is the single most valuable asset in your business. Marketing to lists that will bring in new customers with high lifetime value and intense interest and loyalty toward your company is always the business you are in as a direct marketer.\n\nLIST BUILDING TODAY\n\nThe marketers getting the best results online today are constantly giving away their best stuff, and their audiences love them for it. Giving away high-quality content to get people to opt in to an e-mail list or some kind of offer is the best way to build trust and authority, even in saturated markets.\n\nList building for the long term goes way beyond finding your ideal customers\u2014although that is a starting point. It's also about quality over quantity\u2014and an ideal customer is someone who will stay with you beyond the first sale and hopefully for a long time into the future. But they have to be a good fit in the first place. Your responsibility is then to continue to deliver quality to them with those subsequent offers and the products and services you sell them. The material in your promotions should be super valuable, and the offers you make should be congruent with the content you gave away to get people on the list in the first place. If you sell a product that looks like the material you gave away, and you consistently add more value as you move your audience toward a sale, they'll pay for whatever is coming next.\n\nI know this sounds really simple\u2014because it is. We're a little spoiled online today, because we can distribute content (promotion and product) so cheaply. The danger is that people often send out whatever idea comes to them without really thinking through the congruency issues or the opportunity and goodwill costs that this offer could have on the future responsiveness of their list. Congruency is the most important thing you can build into your business\u2014it's the glue that makes your audience stick with you. This goes back to the discipline I talked about in Chapter 3. If you acted as if you had to produce a physical product, pay for shipping, and cover the costs of failed payments, you would be much more careful about what you expose your list to. And it pains me to think about online marketers I've met who seem fine with 50 percent return rates (or more) when they promote what they say are their best products. Congruency has to be off when return rates are high.\n\nOf course, no matter what your model is, you have to be working in a market where there is enough opportunity for you to make the economics work in your favor. Since that is a choice that's made long before you even start building a list, let's dig into that here.\n\nTABLE SELECTION (OR, WHAT MARKETERS CAN LEARN FROM POKER PLAYERS)\n\nIn some businesses, the likelihood of success or failure is determined before you ever start building a list or sending a campaign. This is what poker players call \"table selection.\" For our purposes, this means how you decide which industry to go into in order to make the economics of direct response work for you. It's about identifying big, motivated markets where your expertise and positioning will give you an edge on your competition. For example, a lot of direct mail businesses made fortunes because they selected the finance and health markets, which are both huge evergreen markets with a wide variety of niches to be developed. I'm a stickler for going deep as opposed to going wide, and choosing markets where we can go really deep and become a \"category of one\" leads to the biggest opportunities. For example, being a leader in general health and wellness is a tough place to start, but coming up with a unique diet for a specific ailment or disease, or developing a supplement that does one thing or solves one health issue with unique ingredients not found anywhere else seems like a much better place to start. Your list selection, offers, and copy can go a mile deep rather than a mile wide when you are a category of one . . . and you can capture a bigger slice of a small market first rather than trying to capture a large slice of a huge market. I'm not saying you shouldn't reach for the stars or want to go wide with your vision\u2014but going narrow to wide is almost always the better path for a disciplined direct marketer.\n\nWhile _Boardroom Reports_ was Marty's first product and his passion, he didn't go into it because it was an obvious opportunity where no one else was doing anything but he had an angle to make it a category of one. He had a real frustration that while the big magazines such as _Forbes, Fortune, Businessweek_ , and _The Wall Street Journal_ would tell you what was happening at the big companies, they really didn't tell you how to run a business. He saw a void in the marketplace. _Boardroom Reports_ was a passion project for him, a way to scratch his own itch, _but_ he also thought it could be a big market, one relatively untapped and with a lot of potential. He knew he could go deep into that subject, even if he didn't know at the time where it would take the company.\n\nFortunately, as time went on, Marty stayed open to what the market wanted. _Boardroom Reports_ was by then very successful, but this audience of executives eventually started telling him that there were other topics _they_ were more passionate about, specifically health, personal finance, and issues affecting them as consumers. The audience told him where to focus, and in 1980, a year before I got to Boardroom, the company launched _Bottom Line\/Personal_. This essentially appealed to the consumer side of the same executive who subscribed to _Boardroom Reports._\n\n_BL\/P_ focused on things like health, personal finance, auto, taxes, home improvement, personal development . . . practical and useful nonbusiness information that an affluent executive would want to have at their fingertips in order to create more abundance in all aspects of their life. For example, subscribers showed increased interest in articles such as how to prevent a heart attack at their desk (i.e., how to live another day and make more money in their business). The health angle was an obvious addition, and that market was really growing at the time, so it's not surprising that we had lots of success right out of the gate with the new publication. _BL\/P_ grew so rapidly that it soon eclipsed _Boardroom Reports_ in circulation and overall responsiveness. That led us to create other offers in such consumer-interest areas, while abandoning the idea of more books in business-related categories.\n\nClearly this new consumer focus engaged a much wider audience, and it was also one that was easier to reach through direct mail, since there were many more responsive lists available to us on the consumer side. Furthermore, as we got into consumer publishing in a bigger way, we realized that the category of health was huge by itself, despite looking initially like a niche that would fit within _BL\/P_. That led us to the creation of a new line of health books. Once again the audience told us how to segment.\n\nWe went on to have big wins in the finance and health categories, and in time in another market that our audience told us they were very interested in as well\u2014taxes. Without listening carefully to our audience, we never would have figured out this intense interest in taxes, which was less obvious than such categories as health and personal finance. To meet this obsession, our third newsletter became _Tax Hotline_ ; and with the success of all our health books pointing the way, launching a health newsletter became the obvious next publication. After these, we launched _Bottom Line\/Tomorrow_ , focusing on another area of intense interest among our customers\u2014retirement.\n\nMaking the transition from business to consumer interest, then adding taxes, health, and retirement as additional categories to cover, we kept going where our customers led us. And we did it with newsletters _and_ books. We watched our competition carefully at the time, and we saw the most successful companies also build their lists in the niche categories that their audiences wanted the most information about. Choosing the right market is sometimes about letting the market choose you.\n\nCOMPILED LISTS VERSUS RESPONSE LISTS\n\nIn the direct mail world, all lists fall into two categories: compiled lists and response lists. Simply put, compiled lists are about quantity, while response lists are about quality. Compiled lists are about who the prospects are or where they live; response lists are about what they've done and how they did it.\n\nA compiled list is based solely on demographic information, collated from census data. You can think of a compiled list as being like a phone book. A list compiler starts with names and addresses from public sources, collects data, and layers information onto the names. For example, you might buy or rent a list of names from a compiler with standard industrial classification (SIC) codes appended, which indicate which industry someone works in, allowing you to target people who work in that industry. Lists are not always completely accurate as much of the data compilers use may be \"implied\" from census data and geography . . . but it is always better than mailing names from the phone book. Speaking of geography, another big data point on compiled lists is zip code: where the people on the list live, from the state level all the way down to their neighborhood. Retail stores use compiled lists to attract people who live within a certain radius of their locations, since this kind of data is the most useful to them. A compiled list will tell you any number of facts about those names, but it has nothing to do with how those people _respond_ to anything.\n\nPut another way, the single biggest thing missing from a compiled list is response. That's not to say that people on a compiled list won't respond to an offer; it means that they are not being selected based on how they have behaved or have responded to an offer in the past. In the world of direct marketing that I grew up in\u2014whether the offers were for newsletters, books, magazines, charities, or consumer goods\u2014previous response to other offers was far more important in getting people to respond to our offers than any demographic data.\n\nResponse lists also have all kinds of categories. There are inquiries, free subscribers, active buyers, ex-buyers\u2014and response lists are considered more valuable than compiled lists because there are three key elements at play, which I think you could have guessed by now: recency, frequency, and monetary (RFM), which we are about to take a deep dive into. For now, suffice it to say that if someone has recently interacted with you, has interacted with you frequently, or has spent money with you, they're far more likely to respond positively to an offer than they would if you came at them cold (as you often do with compiled lists).\n\nUnderstanding compiled lists versus response lists, as we defined them in direct mail, has a huge application for people in the online marketing world today. List selection\u2014and list building\u2014begins and ends with response. It's all about tracking the behavior of every person on your list, from hitting a LIKE button to buying a $2,000 product.\n\nHere's a simple example from the affiliate space: If you want to do an affiliate deal with someone and they will only promote your offer to the people on their list who inquired but didn't buy, or who bought over a year ago, that list is unlikely to be as valuable as the list of people who are _currently_ buying from the affiliate. While a 50:50 split is the norm for an affiliate deal, you could make the case that certain segments of someone's list might be worth less than a 50 percent commission. You'll have to set your own parameters for the affiliate deals you make, but I hope you see that how we value names in direct mail is relevant to e-mail names too. . . . When valuing direct mail names, a list of people who inquired about an offer or who bought a year ago always costs less than a list of people who are current, active buyers. An e-mail list of similar \"nonbuyers\" should cost less too; the affiliate partner should give you more than a 50\/50 split, as an example.\n\nThat brings us to the foundational direct marketing concept of recency, frequency, monetary (RFM). While we may not talk about it much today in the online world, it is relevant to everything we do in marketing and list development.\n\nRECENCY, FREQUENCY, MONETARY (RFM)\n\nI've been shocked by how many people practicing direct response marketing today are not aware of RFM. Analyzing the RFM data on your customer database might be the most critical, high-leverage thing you can do with your list. RFM is a pretty simple concept, accessible to anyone, and much of it can be done by observation and \"tallying\" (or tagging) your prospects and customers. But many people aren't using this method, even when they could be getting huge gains from it.\n\nThat said, RFM can be both simple and sophisticated. We used it on multimillion-name databases, and you may need a statistician to help calculate RFM formulas and models for you if you have a large list. Regardless, understanding this bedrock concept should get you thinking in different ways about the markets you promote to and the people who are already in your internal database.\n\nR Is for \"Recency\"\n\n\"Recency\" says that a prospect or customer who has interacted with you recently is more responsive (and immediately more valuable) than someone who responded less recently. When I started in the business, I remember that this didn't make sense to me. I thought that if someone just bought from you, they would have less money to buy something else right away. That seemed like a logical line of thought, but actually humans behave very differently.\n\nWhen someone is in buying mode, _don't get in their way_. In that moment, they are looking for help with something, and you can present a sequence of high-quality solutions that can solve their problem. Make the appropriate offers early and often. The best online marketers today understand this concept of recency and striking while the iron is hot. Some do get a bit overzealous\u2014and like anything else, selling should be done strategically and thoughtfully. It's not about selling them everything under the sun immediately because they just made a purchase.\n\nWhen you start understanding the data on _when_ people buy, _how_ they buy, and _why_ they buy, you can start segmenting names and predicting the best times to make them offers\u2014and the best sequence for those offers. And the products don't all have to be yours. If you don't have anything else to offer, sell affiliate products.\n\nI remember an old-time legacy publisher who would not rent their mailing list of book buyers to other mailers offering books in direct mail. They foolishly thought that every customer had a set budget to spend on books per year, and they wanted their customers to spend that budget only with them. It's no surprise that this company is no longer in business, since they failed to understand that all boats rise when we keep the folks on our list active and happy with all kinds of relevant offerings, even if it means recommending other people's products occasionally. Making an appropriate affiliate offer to your list can increase the responsiveness of your list if you do it strategically and in a way that capitalizes on the list's recency.\n\nFortunately, most marketers know the importance and value of recency. When you buy something online today, it can be a surprise when you _don't_ get a cross-sell or upsell immediately after your purchase. If you cross-sell and upsell aggressively, right after the new customer just bought something, you already understand recency quite well.\n\nIn direct mail the segments of recent buyers are called hotlines, and list owners still charge a premium for those names. Over a few decades of mailing millions of names and almost always selecting hotlines from lists, I can tell you with certainty that choosing this segment is usually the make-or-break factor on whether a list pays out.\n\nBut marketers cannot live on recency alone.\n\nF Is for \"Frequency\"\n\n\"Frequency\" pushes us to combine the most recent buyers with the most frequent buyers. These customers, called multibuyers, are always among the most valuable people on any list. But they don't have to _buy_ frequently to be super valuable\u2014they can also inquire or respond frequently and still be valuable. For example, if someone on my list buys a direct marketing book and then buys a second direct marketing book, they become a VIP on the basis of their purchase frequency. I'm confident they'll buy a third and fourth book sometime in the future. But I also had someone correspond with me\u2014without buying anything\u2014to ask very sophisticated questions based on information I had provided in blog posts. That frequency of communication led him to join one of my high-level mastermind groups. Because we had this frequent contact, he became hugely valuable in the long run. Frequency comes in all shapes and sizes. It's easy to recognize the value of frequent buyers; frequent contacts require you to pay more attention.\n\nRecency and frequency give you a one-two punch that will enable you to segment any list, no matter what size, in order to focus on the people who will be your best customers for subsequent products or offerings. Now, if you're an experienced marketer, this might sound basic to you. _Of course_ someone who bought from you multiple times is a better customer than someone who bought from you once (or never), right?\n\nBut let me add creative and copy into the mix here: Are you communicating with your multibuyers (or your \"multiqueries\") using tailored language, creative, and offers, based on their purchasing and interaction history with you? Or are you sending the same message to the multibuyer who bought their third product from you today as you are to the one-time buyer from six months ago? The first group is \"family\" . . . the second group are \"guests.\" And to keep multibuyers coming back, you need to target each group differently. (The same is true of frequent contacts too.)\n\nThere are even cases where frequency trumps recency. Let's say you had a customer who bought multiple products and then disappeared six months ago. If you don't contact them again with messaging that speaks to them like family (\"We want you back!\") rather than messaging that speaks to them as a guest (like any other new prospect), you could be missing a huge opportunity to reengage them with your offers. And based on the direct marketing rule of thumb that previous buyers are your most valuable prospects, this would still apply if they stopped buying a year ago or even more. I remember mailing to expired subscribers, many of whom had subscribed multiple times to a newsletter\u2014but not for five to ten years. We were able to revive many of them with the right offer and messaging. Direct mail also has an advantage here, since we can enlist the help of the United States Postal Service's extensive change of address file (COA), allowing us to update previous buyers with new addresses fairly easily. This is an extreme example of when frequency can trump recency and the point is to not look at anything in isolation when looking at real people (i.e., your list). And there is a further lesson for those of you who work exclusively online: Keeping track of as many previous buyers as possible with an accurate e-mail address could be money in the bank for a very long time.\n\nYou might know this instinctively, but I have consulted with too many marketers and heard too many horror stories from folks who use one-size-fits-all copy to all segments of their audience. Something as simple as \"we want you back\" to a segment of frequent former customers (who may not be recent) can increase your response rate significantly.\n\nFrequency analysis is critical and should be done on any list of any size. The more frequently someone has bought from you in the past, the more frequently they are likely to buy from you in the future. If you can get someone who has bought multiple products, and they also exist within a hotline of recent buyers, now you've got the cream of the crop. And if the frequency has led to them spending a large amount of money with you, knowing that information could enable you to spend more on reselling that frequent buyer than others who have not spent as much.\n\nThat's why there is a third leg to the RFM stool that we need to be aware of, since the more your customer spends with you, the more you can invest in them for even more sales in the future.\n\nM Is for \"Monetary\"\n\nWe round out this RFM formula by making sure you know the total amount of money every person on your list has spent with you. Creating tiers based on different spending levels will give you far more information about how much customers might spend with you in the future. However, the amount of money spent by each customer in isolation can be deceptive. Combining it with recency and frequency is the most powerful way to segment your list.\n\nIn my previous world of $39 newsletters, for example, monetary value was the least important element of the RFM selection criteria, except when the customer had also bought recently and frequently. That is, one $39 purchase alone is not very powerful, but the customer becomes more valuable as they buy more over time.\n\nIf you are selling a high-ticket item, one purchase could be a lot more significant in determining the future buying behavior of that customer. If a buyer has spent a high dollar amount in the past, they're likely to do so again in the future. As with recency and frequency, how we talk to people who have spent a lot of money with us, as opposed to people who have spent less money with us, can once again make our messaging more powerful to each segment.\n\nRFM IS AS IMPORTANT ONLINE AS IT IS OFFLINE\n\nRFM is a universal language for all marketers, and it is a cornerstone for list selection and segmentation. It is still the best way to analyze past behavior and how that past behavior will dictate future behavior. Again and again, over decades of use across multiple industries, it's obvious that these factors are worth paying extra for in the external lists we select, and it should guide how we communicate and market to the lists we own as well.\n\nWhen you're working online, it's even more important to be slicing and dicing your list through an RFM formula according to how people are buying and behaving. Remember, make sure you don't have one-size-fits-all promotions. Talk to your customers based on their relationship with you, calculated through RFM. This approach can be a game changer and can turn up some unexpected results. Let me share a quick story to bring this home.\n\nI have a friend and client who has something like 18 different related products, and she does a great job of cross-selling and upselling those products to existing customers who have bought some (but not all) of them. But when I saw a breakdown of her buyer list, the majority of buyers had only bought one product, and most of the others had bought two or three at most. It seemed that with some RFM segmenting tweaks, there would be huge potential to get more of those one- to three-time buyers to buy _much_ more, given all the related offerings available. She immediately got on board with this when I noted that there was one person on the list who had bought all 18 products (and no, it was not a relative!) . . . and I asked the question, \"When was the last time you invited this eighteen-time buyer to dinner?\"\n\nOf course, I was being a little sarcastic . . . but I was also trying to make an important point. It's easy to forget that lists are people too. We can lose sight of that when we spend all day sitting behind our computers and spreadsheets. What would my client have discovered if she had in fact taken that customer out for a meal? Imagine all the insight she would have gleaned\u2014the buying motivation, the questions and doubts they had overcome, the language they used to describe the incredible value they were obviously getting from her products. So from that question, we started surveying the most recent, frequent, and high-dollar-spending customers, finding out why they bought multiple products and also in what order they bought the products, looking for trends in buying behavior. That led to a logical (and much more successful) contact strategy. We started offering products to previous buyers in a sequence that made more sense based on what the multibuyers had told us about their purchasing motivation and also by tracking their buying behavior over time rather than simply making random offerings. It was all based on RFM. We paired this with fresh copy and creative that spoke to why their recent purchase led perfectly to the next purchase, customizing the copy to different segments.\n\nI have used this concept of contact strategy in the past too. I recall the exercise of going through the highest-value customers, figuring out the next best product to offer them, and creating a logical path for new subscribers and buyers. We looked at the transaction data on every customer we'd ever had, and after we tracked the purchase history on many of them, we would map out the most appropriate pathways. The path forward was based on how they initially got on the list and what promotions made them buy most often a second time. And then we would try to determine what the second, third, and fourth purchases would most likely be on the basis of each unique entry point to our funnel (though we weren't calling it a funnel back then).\n\nWe were trying to predict what their next purchase would be, and we would then automatically send them an inexpensive promotion for it. A simple and obvious example was when we discovered that the buyers of a health book were most inclined to subscribe to a health newsletter as their next purchase. Once we knew that, rather than wait until the next promotion went out for the health newsletter, we implemented a new contact strategy that every time a new customer bought a health book, we would send them a low-cost piece of direct mail offering them a free trial to the newsletter. The original book was sold as a bill-me-later offer, so when we received payment for the book, it would automatically trigger the bill-me-later offer for the newsletter, since the customer had proven they were interested _and_ creditworthy. They became paid subscribers to the newsletter at a very high rate with a much lower promotion cost. Having been recent book buyers\u2014specifically, creditworthy book buyers who were intensely interested in health\u2014made them more likely to be frequent buyers.\n\nGOING DEEPER: REGRESSION MODELLING & DEMOGRAPHIC SEGMENTATION\n\nThe key to RFM is being able to collect, understand, and use all the transaction data on your own list, and when appropriate, outside lists.\n\nEvery time someone on our database did something\u2014bought a product, ordered a product and didn't pay, returned something, made an inquiry\u2014we noted that action on their individual customer record, making what I call a tick on the file. Every name on our database had these ticks on them, showing everything each person did, so that our statistician could access all that information when running a model on the list each time we wanted to do a new promotion. These ticks enabled us to build models to find out who would be most likely to respond to our offers. It was almost always based first and foremost on RFM. If there was a combination of a customer being a recent buyer, buying something more than once, and spending a lot of money, those features would always rise to the top.\n\nOur modelling technique, called regression modelling, starts with mailing a cross section of a large list (say 50,000 or 100,000 names out of a list of 2 million) to collect real, paid responses to a particular promotion. Because the selection on the cross section segment was not precise, we would lose money on this up-front mailing, but its purpose was research. (And while mailing this many names makes this technique sound like it's only for large mailers or marketers, there are ways to \"tick your file\" no matter how small your list is. The segments you create for promoting from that data might be slender, but I know of no better way to predict future response than by tracking, analyzing, and acting on past response and behavior.)\n\nThe statistician would then analyze the buying patterns (based on all the common ticks) across all the people who responded to the test mailing and then rank the remaining names on the larger list we hadn't mailed yet based on their commonalities with the names that responded. Then it was time to narrow the list selection. We would build a gains chart, which allows you to divide the whole list into deciles (10 percent segments) or demi-deciles (5 percent segments) in order to predict what the response rate would be in the top deciles or demi-deciles. That would determine how we would approach the wider list to make the mailing profitable. It gave us a clear cutoff point because a particular decile or demi-decile would show where profitability was likely to drop off.\n\nWhile we were taking the names that responded and finding more names that \"looked like\" them from the bigger list, this was far from a look-alike model. This was much more sophisticated because it was all based on response and buying behavior (what we call transaction data)\u2014not just demographics or somebody hitting a LIKE button.\n\nI hope I didn't lose you with all that detail, because there is a reason I wanted to share this widely used and successful technique for modelling millions of names at a time. My intent in going so deep is to show how the same principles are at play when you create a model of any kind on any list based on RFM\u2014and you don't need a huge computer or a statistician to do it. However, let me be clear that I was not doing this analysis myself. This _is_ sophisticated stuff, and we hired a Ph.D. in statistics to handle this (although for smaller lists, you can probably simulate the same kind of analysis yourself). But it was all based on RFM at its core.\n\nOnce we had these models, we might consider overlaying additional demographic data, such as household income, but none of that data was ever as powerful as the transaction data.\n\nA lot of marketers confuse the significance of RFM and demographic data and segment their lists based on demographics as much as on behavior. But always keep in mind that demographics are _not_ reliable indicators of purchasing behavior, and in most campaigns, the demographic data did not have a material impact on who we would mail. Sometimes it would add a few names into the top deciles or demi-deciles, but ultimately our name selection was based on when they bought, what they bought, how often they bought, and how much they had spent.\n\nThis is critical for online marketers to understand because a lot of the segmentation tools available rely far too heavily on demographic data. Sometimes you have to do a bit of the heavy lifting yourself to get the RFM information you really need. For example, it's much more valuable to know that someone bought a course on copywriting than to know they live in a city where a lot of copywriters live. Knowing demographic data on the people we promote to (things like race, ethnicity, gender, age, education, profession, occupation, income level, and marital status) or getting access to psychographic data about them (attitudes, aspirations, and other psychological criteria) are not meaningless. Both can be valuable, but it's always going to be RFM that carries the day.\n\nUnderstanding the dynamics and data of the lists you are working with is absolutely critical in direct response marketing, regardless of your channel of choice. The concepts of RFM, contact strategy, and regression modelling are critical in caring for your customers, delivering the value they need, and in so doing, building a business for a lifetime.\n\nKEY TAKEAWAYS\n\n  * Your list is the most important asset in your business\u2014but never forget that your list is made up of real people. Think of them like family members who deserve your love and respect.\n  * Follow the 40\/40\/20 rule, which says 40 percent of a campaign's success relies on the list selection, 40 percent on the offer, and 20 percent on the creative. Or if you really want to dial it in, make that 41 percent list\u2014so that the list has the clear priority\u201439 percent offer, and 20 percent creative.\n  * Pick niches where you can go really deep instead of going shallow across a wide range of niches. Just as importantly, listen to your audience when they tell you they want to go deeper on a particular subject (whether they tell you explicitly or you can infer it from their buying behavior).\n  * There are two types of lists you can buy or create: compiled lists and response lists. While they are terms that originated in the world of direct mail, it's useful to look at the distinction whenever you look at lists. Compiled lists are built solely from demographic and possibly psychograhic data; response lists are built based on actual responses people have given to previous offers. Response lists are generally much more targeted than compiled lists, since those people have already indicated a specific interest in what you are offering.\n  * Analyzing the RFM data of your customer database might be the most high-leverage thing you can do with that list of names. RFM can be applied to tiny lists just as well as huge lists (though when you have a huge list, you might need to bring on a statistician to help you manage it all). Regardless of the list's size, RFM should always be used to determine the types of campaigns being sent to different segments.\n\nCHAPTER 5\n\nOFFERS\n\n_The right offer should be so attractive that  \nonly a lunatic would say \"no.\"_\n\n\u2014 CLAUDE HOPKINS\n\nIf I ruffled any feathers in the last chapter by stating that the list is more important than the offer, rest assured that the importance of the offer will not be shortchanged in any way. That is what we will explore in this chapter. And if you want to stick with 40\/40\/20 rather than 41\/39\/20, I'm fine with that too.\n\nPlenty of other books will give you lists of different offers you can make or the exact process to follow to develop good offers. This chapter is about the _strategy_ of offers\u2014how to think about creating them in order to make them as effective as possible. Whether you are selling information, physical products, or experiences, your offer needs to appeal so deeply to your audience that it's irresistible to them. This is why it's so critical to get your list targeting right\u2014once you know _who_ you are marketing to, it's much easier to work out what you should be offering them. Whether 41\/39 or 40\/40, list and offer must be in sync all the time.\n\nSTART WITH NICHE\n\nThe power of choosing a niche (sometimes referred to as a vertical) should never be underestimated when bringing products and services to the marketplace. Finding niches that fit with your superpowers and then going a mile deep rather than a mile wide is often the key to success . . . and it is way more satisfying to be a master of one thing than trying to be a master of everything. I also believe that owning a niche is easier, more practical, and enables you to make more impact in the area you are most proficient in than attempting to be all things to all people.\n\nSometimes you need someone to point out when you are going too wide with your idea for a new product or service. Sometimes the niche is obvious to you without much help from others. Regardless, the rewards for carving out a niche are always worth it. I want to share two stories that illustrate different approaches to niche:\n\n  1. Being far too ambitious with an initial offer\n  2. The \"horizontal vertical\" offer: how starting narrow and expanding gradually can create an incredible windfall\n\nWHY SHOULD I LISTEN TO YOU?\n\nThis first story is about a driven entrepreneur who presented his big idea to a mastermind group we were in together in order to get some advice and feedback. Let's call him John. After exposing his offer-to-be to the harsh light of day and fielding some tough questions, John walked away with a much bigger idea\u2014despite the revised idea being of interest to a much smaller audience. Because the new, smaller audience we identified was a much better fit for his business and mission, his list would be much hungrier for what he had to offer.\n\nJohn's initial idea was to create an information product (and eventual coaching program) for men going through a midlife crisis. That's a _big_ goal, and I was just one of many skeptics in the group regarding whether he could really pull it off. While I wasn't trying to be nasty or obnoxious, I had to hit him with an uncomfortable question: \"Why are you qualified to teach and coach men going through a midlife crisis?\" The implied question there was, \"Why should I listen to you?\" Be advised: This is a question all customers ask, whether consciously or not, and it's why establishing authority and credibility is so critical in any promotion. You have to be able to answer that question with integrity and experience, or no offer you can make will strike the right chord.\n\nJohn's answer actually told me he might just be the guy to get this thing done. He had gone through a terrible time late in his career when he spiraled into an awful depression, went through a divorce, lost contact with his kids, went bankrupt, thankfully survived a suicide attempt\u2014and came out the other side stronger and better than ever.\n\nObviously he had some lessons to share. But with _all_ the men who go through a crisis? Millions of guys of all ages go through crises; I didn't think there would be a one-size-fits-all solution to such a big problem. Other skeptics in the room started throwing out more questions:\n\n  * What would be the entry point to his funnel (i.e. his initial offer)?\n  * What were the \"hot buttons\" that would get someone's interest?\n  * What did he know about different segments of the market?\n\nEventually, frustrated with how broad his focus was, I asked another question: \"What did you do for a living at the time of your breakdown?\" He answered that he was an attorney. Now I thought we were on to something. It immediately became clear that he could create his product or program specifically for lawyers going through midlife crises rather than all men. Wouldn't the relatability lead to more powerful promotion? Couldn't it also be more personalized? Even if what he delivered to self-described recovering attorneys was exactly what he had in mind for every man (no matter what they did for a living before their crisis), owning a bigger slice of a targeted market where he had deep understanding and empathy seemed like a much better path.\n\nHe could always break into new markets later on if he wanted to, but I had a feeling that once he dominated a niche where he could be a standout leader, he might be less interested in going wider. We concluded that it's much more effective to focus on a tight niche and avoid competing with other gurus who were trying to be all things to all people. This is one of the biggest mistakes most marketers make: they start too broad. It's better to start narrow, and widen your focus over time, than to start wide and try to gradually hone it down.\n\nSometimes it's obvious how to narrow down your market, like we were able to do with our lawyer friend here. And sometimes you need to poll the marketplace and find out where the need is and how you can uniquely fill it. In other cases, you can just talk to your best customers to see how deep you can go.\n\nIf you have a solution that can appeal to \"anyone in any category,\" chances are that the marketing message is so broad that you will struggle to get new clients. Here's a question, in a shorthand version, to ask in order to narrow down the market: in your current business, which clients have cut you the biggest checks? Or, who has spent the most money with you over some specified period of time? You'll probably find that your biggest clients\u2014those who pay you the most money\u2014are in similar categories, and that knowledge will enable you to build a strategy that goes deep to appeal to prospects in that same category. It's almost like a miniature regression model: find the clients who have the most recency, frequency, and monetary value so you can go and look for more of those.\n\nIt's far better to create your best offer for people you understand the best and who understand you _or_ to create offers for people that \"look like\" people who have loved your offers in the past.\n\nA NICHE OF 30 MILLION\n\nMy second story relates to how I spent a good part of my career selling useful (and often lifesaving) health information to consumers. Although we had gotten started selling general health books that were more encyclopedic, when we niched down, we were able to tap into one of the most powerful niches in health.\n\nOur successes early on were always thanks to careful research, since we knew there was a real and present danger that if we went too wide all the time, we would look like our competition and risk commoditizing our content. Of course, the temptation to stay broad and appeal to everyone was tantalizing. This story is about trusting your gut and collective experience in addition to using research appropriately . . . and it's about being able to break the rules of direct marketing because you know what the rules are in the first place. It's important to recognize when current circumstances are different, and then seize opportunities when they are staring you in the face. This epitomizes that quote, \"Learn the rules like a pro so you can break them like an artist.\"\n\nOver the early years of selling general health information, we started using a process that would be deployed before we launched any new niche book through direct mail: the Q test. Here _Q_ stands for questionnaire, though I have also heard these referred to as concept tests or survey tests. Whatever you call them, it is a scientific way to predict tangible results using quantitative research. I learned this technique from direct marketers who had been using it for many years\u2014even decades\u2014before I discovered it, and before I had the good sense to use this kind of research too.\n\nFirst step: write a blurb for a new book. We always made it exciting, but we also made sure it was a description that we could deliver on. That is, the book we would ultimately create or buy the rights to needed to match the blurb closely enough that the test would be reliable.\n\nSecond step: put together at least four to six new book blurbs to test against each other, sometimes doing as many as eight in a single Q test.\n\nThird step: select a universe from our previous book buyers. We were looking for a cross section from our internal database. Then we would mail approximately 2,000 buyers in three waves. The participants in the survey were some of our best customers, based on our RFM analysis. This was all done with physical mail, since we were also looking for direct mail responsiveness, but these days Q tests are done online just as easily.\n\nThe Three Waves of Our Q Tests\n\n**Wave 1:** A postcard told the folks a survey was coming and we wanted their help.\n\n**Wave 2:** The survey itself was sent, usually with a $1 or $2 bill attached, which encouraged them to participate so we could continue to bring them the best books in the future. In the survey, after each blurb, there were only two possible responses for the respondent to check:\n\n  Would Order.\n\n  Would Not Order.\n\n(There are many Q test formats that can have up to four responses: would definitely order, would probably order, might order, would definitely not order. It's much more effective to have only two options.)\n\n**Wave 3:** A follow-up letter in an envelope reminding them to fill out the survey.\n\nAll waves were mailed first class, which, although costly, guarantees deliverability, and that is worth it. If you're Q testing by e-mail, make sure you take deliverability and open rates into account so that you get enough responses to read the results accurately.\n\nUsing this methodology, we received what was close to the industry standard for response rates when Q testing\u2014between 40 percent and 50 percent. Yes, that's _40 percent to 50 percent_ , not .04 percent to .05 percent, or .004 percent to .005 percent. And after the mailing was complete, marketers would be armed with 800 to 1,000 responses (which is about the number you should look for when Q testing, offline or online) that showed the comparative interest between titles. Extrapolating that data, using some benchmarking from the past, meant we never launched a book unsuccessfully once we started Q testing. It's expensive and time consuming . . . and I know of no better technique to determine the best products and offers for your core audience.\n\nOf course, we eventually moved a lot of our surveys online, but it is harder to send a dollar bill through a phone line. If you use this methodology online I suggest you use other incentives, like free reports or discount codes for future purchases. But I recommend that you still e-mail in three waves. There were very few things that would compel us _not_ to do a Q test on a new potential book title. But like I said earlier, there are always exceptions to the rules.\n\nIn one case, the exception was the epidemic of diabetes and prediabetes in the United States, an ailment that affected upward of 30 million people at the time, and a book that landed on my desk unexpectedly at exactly the right moment. We were handed an opportunity (sugar-free, of course) on a silver platter from a world-class copywriter who had given up writing to build a company producing exercise and nutrition programs for type 2 diabetics and prediabetics.\n\n_The 30-Day Diabetes Cure_ was a wonderful book and program that we felt could help hundreds of thousands of people\u2014 _and_ it came with the outline for a direct mail promotion package from a seasoned copywriter we knew and trusted, who was available to write it immediately. We also knew that diabetes was a topic of intense interest among our customers for general health books, and many diabetes-related headlines and lead materials were performing well in current promotions for many of those books.\n\nHowever, despite being such a hot topic, we knew that if we Q tested a niche title on diabetes to a list of buyers who had only bought general health books in the past, there was a very good chance it would not get an acceptable response to make the cut and be developed as our next project. So now it was time to break the rules. I suggested that we should bypass putting this one into a Q test and just launch it.\n\nI was willing to take this risk because our window of opportunity could close by the time we got our results. Not only could we lose the copywriter's availability, but being a niche product, the Q test results wouldn't necessarily give us an accurate picture of the market's attitude to the topic even though it was one of the scariest health trends in the market at the time, and consumers needed lots of information. This was a situation where understanding the history of the niche might have been more powerful than anything we could have learned from additional research. This was an offer we had to make immediately.\n\nI have made many errors over the decades I've been doing this marketing stuff, and jumping into this proejct without Q testing was risky. Fortunately it paid off. We jumped on the opportunity by getting into the mail quickly, and we never looked back. Our friend and world-class copywriter gave us a gift with his diabetes book and eating program, and it proved to be one of the best launches ever in the category. The topic is still going strong for health marketers everywhere because, unfortunately, the epidemic has only gotten worse. We were doing our part to reverse that trend.\n\nLooking back on this incredible success, I've often wondered if I would have broken the Q test rule had I not known how important this topic was to our audience. I doubt it. But isn't that one of the beauties of experience?\n\nDirect marketing success comes from accumulated wisdom, perpetual curiosity, unbridled optimism . . . and, of course, being a slave to your numbers and measuring everything. We were lucky on that launch, but it was luck created by experience and having played by the rules enough to know when to break them. That's how the best offers are created.\n\nI encourage you to go with your gut once in a while when you have strong anecdotal evidence or information you trust about a marketplace or topic. Even though direct marketing is such a numbers-driven business (which is what I love most about it), there is still room for logic, instinct, and courage. Horizontal verticals (i.e., big topics in niche areas) like the diabetes epidemic don't come along often. But I believe that if you are always thinking niche first, these opportunities have a much better chance of showing up. Even if you never get to have a horizontal vertical in your career, being number one in any niche\u2014especially if it's a niche you are passionate about\u2014will almost always beat being one of many in a crowded, noisy marketplace that also has many imposters.\n\nIn addition, with direct response marketing, where everything is measurable and containing your costs is always part of the equation (since everything has to eventually pay out), there is no such thing as a \"failure.\" It's either a breakthrough or it's education. You win or you learn.\n\nLET YOUR MARKET SHAPE YOUR OFFERS\n\nOne of the biggest offer breakthroughs in my career can be traced to a new branding strategy. As a serial direct marketer who relies on measurability all the time, talking about branding gives me hives, but while this next story has a branding component to it, it also teaches us to never throw advertising, marketing, or promotion dollars at any offer that doesn't have an expected, measurable return.\n\nTo review: Boardroom Inc. was launched in 1972 with _Boardroom Reports_ , which taught small business owners how to run their businesses better. It was a publication filled with practical, useful information, and subscriptions were sold almost exclusively through direct mail.\n\nIn 1980, the company launched _Bottom Line\/Personal_ , which, as I mentioned earlier, essentially appealed to the consumer side of the same executive who subscribed to _Boardroom Reports_ , covering health, personal finance, personal development, and so on. As you read earlier, the company exploded (in a good way) and _really_ got busy: health, tax, and retirement newsletters followed, as did dozens of books in those categories too . . . and eventually we had a product line where a majority of our offers focused on health, personal finance, taxes, and consumer information.\n\nBusiness information had taken a back seat. So what happened to poor _Boardroom Reports_? Well, the core audience interested in business information\u2014while enthusiastic for the newsletter\u2014had shrunk quite a bit due to attrition and lack of available outside lists. And of course, our house (internal) list was not responding all that well to the business-related newsletter with the new focus on consumer-related editorial. That is, attracting new subscribers to _Boardroom Reports_ was hard, while new customer acquisition in our new consumer categories was easier and more lucrative. But it was difficult for the company to let go of its firstborn, and _Boardroom Reports_ chugged along for a while, albeit marginally.\n\nThen we had an innovative idea for a test, although it's sort of obvious in hindsight. Here was the logic: The database of existing customers (after the explosion of _BL\/P_ ) had huge identification and affinity with the Bottom Line brand. So instead of giving up on the consumers who still might want business information, we met them where they identified with us the most. _Boardroom Reports_ was renamed _Bottom Line\/Business_ and was redesigned to look like an affiliated publication with _BL\/P_. We then created promotions that looked and sounded more like what our customers were used to seeing for our rapidly growing consumer publications.\n\nThe rebranding created a huge lift in response from the house audience to the newly renamed _Bottom Line\/Business_ and breathed new life into our original publication from customers we had paid to acquire a long time ago and from new subscribers to our consumer publications too. It allowed us to create a whole new line of products, which in turn kept the newsletter alive (and profitable) for a few more years. When we eventually folded _Bottom Line\/Business_ , it was done by choice and not as much by necessity\u2014always a better way to go. At that point we realized that the subject area was an outlier; we had gotten so much better at marketing in the consumer categories and at that time they deserved 100 percent of our editorial focus.\n\nBut we learned valuable lessons we could take forward from this transition from _Boardroom Reports_ to _Bottom Line\/Business_ :\n\n  * Our first health newsletter, _Health Confidential_ , had a huge growth spurt when it was rebranded as _Bottom Line\/Health._\n  * Boardroom's tax newsletter, _Tax Hotline_ , got some new life when it went wider as _Bottom Line\/Wealth._\n  * _Bottom Line\/Tomorrow_ (the retirement newsletter) and _Bottom Line's Natural Healing with Dr. Mark Stengler_ were both launched with titles under the Bottom Line umbrella. We didn't even bother trying anything _but_ Bottom Line branding at the outset of any new launches after seeing the power of the umbrella brand. We were fast learners, I guess.\n  * All the books we published became publications of Bottom Line Books rather than Boardroom Books.\n\nThis all might sound a bit simple now. But the fact that Boardroom was never a household name made it less obvious that the company was sitting on a killer brand with a pretty large audience and that this audience was just waiting for us to ask them what kind of products they wanted to see.\n\nI'd also like to quote ice hockey legend Wayne Gretzky in this context: \"Skate to where the puck is going to be, not where it has been.\" We didn't change the entire branding and offer strategy of our company on a whim or because some brand consultant said it would be cool, but because our audience told us that's where we should go. And who are we to argue with the folks who pay the rent and the electric bill?\n\nCHOOSING THE RIGHT CAMPAIGN STRUCTURE FOR YOUR OFFER (OR, HOW YOU SELL PREDICTS HOW THEY WILL RESPOND)\n\nHow you present your offer to your lists, internally and externally, will often determine the kind of response you get. Not just the number of responses, but the kind of people that end up becoming your new customers. For example, if you sell your offer with a sweepstakes, the people who respond are likely to be impulsive and might not have a high lifetime value, being more interested in winning millions of dollars than actually investing in your products or services. Or they might not pay as much for a particular offer and they might only do free trials (and are therefore more likely to be tire kickers). While they may still be profitable, they will have a lower lifetime value than other segments. But if you sell with long-form copy, online or offline, lifetime value is likely to increase, because the customers are going to be more engaged with the offer from the outset. That's what I mean when I say that how you sell predicts how the list will respond.\n\nRegardless of the offers you make and how you sell them, you have to track how the list buys and how (and if) they buy again later. I've seen this play out many times, whether I was renting the Boardroom list to other mailers or selling our own products to our house list. I was always very conscious of those different buyer profiles (i.e., what offers they had previously responded to) to segment the lists and ensure the right people were sent the most appropriate offer with the most appropriate type of campaign.\n\nThe Survey Package versus the Bookalog\n\nOnce upon a time, back in the dark ages called the 1990s, one of the greatest copywriters of all time, Gary Bencivenga, wrote a lucrative front-end promotion for _BL\/P_ that we called the survey package. By \"front end\" I mean it was to attract new customers who were mostly unfamiliar with the newsletter and the brand. The survey package was a fairly simple letter, in a standard envelope, with a gift certificate attached to the letter. The gift certificate was also the order device. The letter began:\n\nI need your honest opinion . . . and I'm willing to reward you handsomely for it. That's why I've enclosed the above gift certificate, which entitles you to two very valuable gifts in exchange for your opinion.\n\nThe goal was to have prospects send in the \"survey\" with the gift certificate to get a free trial subscription to the newsletter. It was a generous offer: six free issues and a premium book before they had to pay anything, a true \"try it before you buy it,\" or what you now know as the bill-me-later approach.\n\nGetting people to subscribe with a survey felt like a gimmick and not the best way to sell such high-quality information. Despite that, the campaign was incredibly successful. The survey package performed like a sweepstakes, in that people felt they would be missing out on something from us if they didn't participate.\n\nAs you might expect, since the trial subscribers came into the newsletter without a lot of selling or extensive background on how great _BL\/P_ really was, the number of people who actually paid us the $29.95 subscription price after the free trial period was lower than previous packages. That's what I meant when I said the survey was more of a gimmick rather than the most powerful way to sell such important consumer information. We had a great hook that led to an impulsive response, but the promotion did not reflect what the product was all about. Impulse gets people to respond . . . but it doesn't necessarily get them to stick around.\n\nThe bottom line (pun intended) was that the survey package was, net-net, a huge success since it added many new paying subscribers, despite the low pay-up rate. However, the low pay-up rate ultimately became a problem, and it didn't feel great that many of the trial subscribers were unresponsive to the material.\n\nI really didn't like the survey package. There was nothing illegal or immoral about it, but it always felt like a less than admirable way to sell the best consumer newsletter in America. A rule of thumb to being a world-class direct marketer\u2014at least in my playbook\u2014is to believe that the control is your enemy. It may be the current winner, but it's there to be beaten too. Whether I liked it or not, the survey package would have to remain the control until it was beaten on the battlefield of statistical significance . . . so I was pretty motivated to create a new package to beat it. Lucky for me, I was working with the best copywriter on the planet at the time.\n\nGary Bencivenga also saw the flaws with selling this offer via the survey package, despite the success it had up front. He is the one who did the deep thinking (and heavy lifting) to then beat his own control. Gary wrote a masterpiece to test against the survey package: a 64-page bookalog (which looks like a digest magazine) titled _The Little Black Book of Secrets._ There was no survey or quick, impulsive gift certificate order device featured in those 64 pages. And, in fact, it wasn't all selling copy either.\n\nThere's an expression we used before longer formats became more prevalent in direct mail: selling the sizzle but not the steak. This meant that we didn't want to give away the \"meat\" of the product in our attempts to sell it. The intrigue and mystery was part of the selling process. However, longer direct mail at that time was trending toward giving away some steak along with the sizzle, and Gary's bookalog did that very well. You might even say that promotions like that bookalog were a precursor to the content marketing we see online today.\n\nSo what were the front-end results of the bookalog against the survey package? _Much_ lower: it produced approximately half the number of new trial subscribers on the front end. But the back-end results were just the opposite. The percentage of people who actually paid for the newsletter after taking a trial subscription from the bookalog was close to double that of the survey package . . . and they paid _long_ before they got all of their six free issues, which I think is a tribute to the \"steak\" included in the bookalog.\n\nPut another way, the bookalog got lower numbers initially but more quality engagement in the long run, and that led to much higher profits when we analyzed the subscribers from the bookalog offer after the first sale. Tracking those subscribers who initially came in from the bookalog into their second year renewal (what we called conversions) showed us the true miracle. These subscribers were engaged in such a way from the initial promotion that their expectations were met in year one of their subscription . . . and then long into the future. They were worth so much more long term than the survey subscribers.\n\nThe LTV of subscribers who came in from the bookalog over those who originally came in from the survey package was a real game changer. The bookalog became the new control, and it inspired many packages like it in the future. We also sold all sorts of books to our subscriber list, and the response rate to all our book offers (also using longer promotional copy) was significantly higher among the subscribers who came in from the bookalog over the survey package subscribers. That is, new long-form copy promotions did better with people who came in from previous long-form copy promotions.\n\nAnd there was one more way the subscribers from the bookalog added to the company's profit: when we rented our lists to the outside world (equivalent to folks using their list for revenue share or affiliate offers online today), outside mailers saw higher response rates when _they_ mailed names derived from the bookalog. Quality over quantity indeed\u2014how we sold (bookalog versus survey) had a statistically significant impact on how our audience responded to our offers.\n\nThe Paranoid Package versus the Happy-Go-Lucky-Package\n\nThis second example of \"how you promote is how they will respond\" is also an example of how you promote creating new list universes if you pay close attention.\n\nJust after the dinosaurs became extinct, around 1986, we tested what we called a \"paranoid package\" for _BL\/P_ with the headline _\"_ The World Crisis of 1986. _\"_ The control at the time was a package that sold a wonderful premium book, _How to Do Everything Right_ \u2014a promotion that was much more upbeat and optimistic. (I guess anything seems upbeat compared to a world crisis.)\n\nOn the surface, the test results were clear: _How to Do Everything Right_ defended its position as the control by a wide margin when we looked at the mailing in its entirety. It looked like paranoia could be put to the side until the next world crisis. However, something strange happened on the way to burying that test: When we looked at how each package did on a list-by-list basis, we found that there were certain names who were less interested in the mainstream offers and fell more in the opportunity-seeker category. These people responded to the world crisis package at a much higher rate than they did to the control.\n\nTo cut to the chase, we discovered that we had two list universes\u2014two different sets of names with two distinct interests. So we ended up with _two_ controls\u2014one upbeat and one paranoid\u2014each mailing to the list universe most suited to the messaging.\n\nKey lesson: if we could create multiple control offers for different audiences in the most expensive medium, direct mail\u2014with extra printing, sorting, and postage costs\u2014shouldn't you be thinking about multiple controls to various list segments in your online business all the time? While one-size-fits-all creative _can_ be profitable, it is not the strategy for long-term business building. Just about everyone on your list will be on other people's lists too. But they come to you to meet a _particular_ interest or need, so the makeup of your list is always going to be one of a kind. And sometimes selling in a different style can help you understand the makeup of your list(s) even more effectively. There are no unique names, only unique lists.\n\nIn the words of Jay Abraham, this is getting everything you can out of all you've got. Here are the main lessons from these examples:\n\n  * Read your numbers, listen to your audience, and look for the hidden offer opportunities\u2014and then act on the opportunities that present themselves.\n  * Don't underestimate the impact of how you promote on the kind of customers you will attract.\n  * Make sure you can calculate the lifetime value of every new customer based on the promotion they came in on. The initial sale is only one metric\u2014and LTV, not the first sale, may be the most important metric when deciding on your most profitable acquisition vehicle.\n  * If you think these concepts are only true for direct mail, think again. All these principles apply on any channel you decide to use.\n\nSOME OF THE BEST OFFERS AREN'T ORIGINAL\n\nDuring my first decade at Boardroom, creating offers and products\u2014mainly for books and newsletters\u2014was relatively easy. We were publishing some of the most useful newsletters in the world at the time, including _Boardroom Reports_ , _Bottom Line\/Personal_ , _Tax Hotline_ , and _Health Confidential._ We created best-selling books by taking the \"greatest hits\" of our newsletters and putting them into huge volumes, indexed and categorized, and sold millions of copies of those \"new\" books.\n\nI never lost sleep over not being on the _New York Times_ bestseller list . . . we were much happier selling millions of books and helping millions of consumers without needing to get involved with bookstores or anything that resembled trade publishing. We had a wonderful formula for creating these books and then had the best copywriters at the time (including Gene Schwartz, Gary Bencivenga, Jim Rutz, Mel Martin, Clayton Makepeace, and others) work their magic on each one.\n\nWe priced our books higher than anything similar you would find in a bookstore; we also created higher value than anything we could have done in a traditional bookselling environment. It worked like crazy for years. But then we were faced with a moment of truth in the late 1980s: we started running out of content doing these greatest-hits volumes.\n\nOur bestsellers were often over 500 pages. They were encyclopedic, and we joked that we sold books by the pound. These kind of books sold much better than our niche titles (although we had a lot of success with many big niches, like the horizontal vertical we discussed before\u2014the diabetes book).\n\nSimply put, books by the pound was what our core customers wanted . . . but it was getting tougher and tougher to deliver those kinds of books without duplicating content. _The Book of Inside Information_ , or what I called _Bottom Line_ 's Greatest Hits, sold vast numbers of hardcover copies over many years, at $30 a book. We had offshoots of that book, including _The Big Black Book_ and _The Book of Secrets_ , and they all sold well.\n\nThere was also our book _Healing Unlimited_ , which sold over two million copies; our big tax books; and even a title on estate planning, which sold hundreds of thousands of copies (as you recall, this was partially thanks to Bill Jayme). Even \"vertical\" books, including the estate planning book and the diabetes book, became a little more \"horizontal\" when direct mail targeting was involved.\n\nBut how were we going to expand this formula? We had a hungry database waiting for more big books and nothing to sell them. Enter Gordon Grossman, the man who was the architect of _Reader's Digest_ in the 1960s and 1970s, and who I hired as a consultant in the 1990s. He looked at what we were doing and said, \"Brian, what makes you think that all the content for your books has to be your own stuff?\"\n\nThat simple question changed everything for us. I realized at that moment that I could buy our new books instead of having to make them all. Gordon had been involved in the practice of buying trade books and converting them to direct mail books at _Reader's Digest_ and also for his other big consulting client, Rodale Press. I put this lightbulb idea, invented by others, into action\u2014and added my own twist.\n\nIt started with a trip to Barnes and Noble when I asked to borrow a hand truck. Thus armed, I trundled around the store to every category that fit with the interests of the millions of names in our database. I got a good workout running all over the store, piling the hand truck higher and higher with books. I visited every category that made sense, including health, fitness, finance, investing, taxes, personal development, food and nutrition, retirement, and so on. There were so many amazing books in those categories, most of which had an inch of dust on them because no one was picking them up. I knew many of them just needed a marketing makeover.\n\nI also realized that if any of those dusty volumes had sold 10,000 copies in their lifetime, they would have been considered bestsellers by the trade at that time. Although most of them had not hit anywhere near that level, I was confident we could change that with our direct mail formula. Thinking ahead, I knew I could propose a windfall opportunity to some old-school trade publishers, who knew we were better equipped to take their books from the store to the mail. My master plan was to approach those trade publishers and secure the direct marketing rights, then blow their minds with how many books we could sell with direct mail when the title was selling virtually nothing in bookstores.\n\nFive or six trips to my car later, after purchasing 40 or 50 books and weighing down my 1985 Toyota Camry until my tailpipe was almost dragging on the ground, I had the candidates to rival our current direct mail bestsellers.\n\nThe next step was to go through the books to see which ones had the most useful information from the most credible sources. Many books got nixed by the editors: \"We can't put our name on _that!_ \" Others got nixed by the copywriters: \"There's not enough juicy information for compelling copy!\" But the books that got through this gauntlet were put into the new product pipeline.\n\nWe also had to come up with an amazing pitch to the trade publishers for the direct marketing rights (which took some doing), and we had to create a system to reposition any appropriate trade books to direct mail. Now, this was before we had started doing Q tests, so fortunately our approach worked well.\n\nThe books we added to our direct mail lineup were often in paperback in the bookstore already, selling for less than $20. But our direct mail version of the book would always be hardcover (which has a higher perceived value), and we would add bonuses and premiums to make the new version drastically more appealing than the one in the bookstore. This repositioning made the offer for direct mail unrecognizable as what it had been before. It was this \"apples to oranges\" comparison of offers that also got all the trade publishers to give us the rights to sell the books, and justifiably so. They saw the book as a single title on the shelf at a bookstore; we saw the book as part of a treasury of information. Our price points, which started at $30, eventually rose to $40 plus shipping and handling. All the while the same book was still sitting on bookstore shelves at half the price and with none of the attention.\n\nThat's how we created a whole book division with content we didn't have to produce. I didn't invent the idea of buying the rights to books to sell in other channels. But without these direct mail offers, these books would have died on the vine (or shelf) without helping an enormous number of people who needed that information. The way we did it enabled us to both bring fantastic content to the world and have multiple outside titles that sold in the hundreds of thousands. A few even sold more than a million copies.\n\nThere are a few lessons here for any marketer, offline or online:\n\n  1. Look everywhere and anywhere for undersold or underutilized assets or products right under your nose that could be of interest to your audience. I'm not talking about doing another affiliate mailing\u2014I'm talking about buying the rights to those products and making them your own. This same principle applies to digital products as well, not just books and physical products.\n  2. Know the power of your name and your brand to your audience and look for products that you would be proud to put your name on.\n  3. Nothing gets sold without a great sales message, regardless of your medium, but the offer has to have legs of its own. Think about whether it has real credibility before you fall in love with your new product idea. Have seasoned copywriters look at everything you are considering so you can get an objective view on it before you try to get them to write anything.\n\nAnd of course, the offer has to be unique and compelling.\n\nASSESSING YOUR ASSETS\n\nThe first thing I do when I take on a new client (or onboard someone into one of my mastermind groups) is to have a conversation with them that begins with, \"Where were you? Where are you now? And where would you like to go?\" Since the people I like working with the most are always up to something bigger and better, the \"where would you like to go\" part always seems to include something about \"exponential growth\" or at least \"significant\" growth. And no one answers the question of where they want to go with \"staying where I am now.\"\n\nBut it's the next set of questions and the inquiry that follows that is most interesting. Almost everyone I've worked with assumes that creating huge growth involves inventing something new or doing something completely different than what they have done in the past, with completely new people, assets, and resources. And that might be the case. But I maintain that it is never the starting point.\n\nThis is where I would start: Your next big idea might be right under your nose . . . or in a drawer somewhere . . . or tucked away in something called a file cabinet . . . or simply sitting on your hard drive. (This idea is summed up in the phrase \"getting everything you can out of all you've got,\" which happens to be the title of one of my favorite books by one of my mentors, Jay Abraham.) Another way to put this: When brainstorming (with yourself or others) about that next big idea, first think deeply about assets you already own and control before thinking about trying to attain new assets.\n\nI think the biggest reason I am a \"mastermind junkie\" is that I love to surround myself all the time with people smarter than me and to have others I trust and admire tell me things I can't see (and will never see) about my life and my business. Having others making you aware of your overlooked assets from the outside looking in\u2014and pushing you to go deep with what you have already rather than coveting something new and shiny\u2014is another reason to surround yourself with smart people who will tell you the truth.\n\nHaving sat on many \"hot seats\" in masterminds over the years, and participated in the audience for hundreds more, helping myself and others seek the wisdom of the room, it's often shocking how obvious the answer becomes when someone else asks the right probing questions. A problem or opportunity that seemed so daunting at the outset usually becomes clearer and manageable to solve when someone else forces you to focus on resources that are right at your fingertips.\n\nAgain, the first step is to go deep on \"assessing your assets.\" Start by asking, _What are all the things I have access to or own that have already been created, written, developed, or curated\u2014and probably with a sunk cost?_\n\nOf course we should all think big and look beyond where we have been and where we are presently, and I am by no means suggesting that exponential growth is only the result of rehashing, recycling, and never buying something brand new. But don't ignore everything you have built, created, and achieved just because you see it as the \"past.\" In fact it is all that material in all forms that can propel you even further in the future.\n\nBy assessing your assets in this way, you will create the most innovative offers. I guarantee that there will be more than enough time for you to explore the exciting (and unknown)\u2014but looking at what is already there is where I recommend you begin. I know this might sound simple. Hopefully it will be helpful to you if we focus on three areas and some questions you can ask yourself regarding the inventory you own before you go shopping outside.\n\n1. Lists\n\nHow many active or previous customers do you have on a list somewhere (anywhere)? How many people can you reach who know you, your products, or your service who have not bought from you in the past but still might take your call (that is, open an e-mail from you because it's from you)?\n\nYour first step is to take inventory of all the lists you already own. And that includes groups of people not in some sort of automated database (your Facebook friends, your LinkedIn contacts, and your e-mail contacts all count). I can't tell you how many times someone tells me about their next big idea and the first thing they do is talk about extravagant marketing plans that include promoting to _huge_ audiences who have no idea who they are and know nothing about their history or track record.\n\nI know you are well aware that \"warm traffic\" is more responsive than \"cold traffic\" . . . but you would be amazed how many people go after the shiny object (such as cold traffic from the millions on Facebook) before approaching the \"boring\" previous buyers and prospects who might know and love them already. I don't even like using the term \"traffic\" when looking at potential audiences, since traffic in a literal sense is unpredictable\u2014and the last thing we want in direct response marketing is unpredictability if we have some ability to make the media we buy predictable.\n\nBelieve it or not, there are many marketers today that won't even test offers and concepts to their Facebook friends (for free) before they buy costly ads on Facebook. The reason most people justify going outside before looking inside is usually a version of this (which is a real quote from a somewhat experienced marketer launching a new product with a new offer): \"I have 3,000 people who know me and love me (and some have even sent me a check) . . . but I want this new product to be _big_!\"\n\nI maintain that you will get bigger faster (and prove any concept with more confidence) by going a mile deep with those 3,000 first rather than going a mile wide in another medium. Whether you sell business-to-consumer like I did (selling millions of subscriptions and books $39 at a time) or whether you sell business-to-business at much higher prices, the philosophy is the same. Mining your \"house list\" first in as many ways as you can think of (different offers to different segments, special copy based on interest areas, etc.) will become the foundation of your business and fuel the creation of your most successful offers. And after you prove the new idea with those folks, there will be more than enough time (with a lot less risk) to go after the millions of new people who can't wait to hear from you too.\n\nBut take care of those who love you the most first. And by \"love\" I mean anyone you might have touched in the past, whether they bought from you or not. Also, always look at the economics of being big in a smaller, niche market where you are already a hero (and that will probably spend more with you) versus being small in a larger, broad market that has never heard of you. It's worth doing the math.\n\n2. Content\n\nWhat content do you own or have the ability to use on an unlimited basis? What products do you own or have access to that have relevance (or once had relevance)?\n\nThis is a resource we all take for granted. It's so easy to think that once it's been printed or sent in an e-mail, you can't use it again. The idea of \"repurposing content\" has nothing to do with selling or distributing the same stuff over and over again to people who can't remember if they bought it or consumed it. It has everything to do with the fact that good material is always worth repeating . . . and updating . . . and thereby transforming previously used information into new information. Simply put, before creating anything new from scratch, see what you have already that can be repurposed powerfully.\n\n3. Promotions\n\nWhat previous winners do you have in your swipe file that have been retired? What previous _almost_ winners or winners in a different medium are currently sitting in that same swipe file? And what losers do you have that surprised you when they lost, but you never went back to try to turn them around?\n\nEverything you have done in the past with your marketing efforts is an archive of educational materials, not just sales letters, order pages, and copy platforms that are \"one and done.\" Furthermore, anything that has worked well in the past for you or someone else should never die. Even anything that was a total dog (i.e., didn't work well at all) for you or someone else is a building block for future promotions. While I had very little luck in my career taking a huge winner from years past and just reusing it after simply giving it a rest, I have had winner after winner by repeating copy platforms, headline structures, and some previously used promotional copy with new and restructured offers.\n\nI've talked about the power of swipe files and why every successful copywriter and marketer has elaborate systems to catalog anything and everything they can build upon for their own future promotions. Remember, you can look at winners from others and \"steal smart\" (i.e., adapt winning formulas to whatever you might be working on yourself). I say this often, but it is worth repeating here: \"Stealing is a felony; stealing smart is an art.\"\n\nI know you have probably been taught the lesson of \"being grateful for what you have\" and its kissing cousin, \"you don't know what you have until it's gone.\" The beauty of marketing today is that nothing you have done (or do) will ever be gone. Plus, we have more media choices than ever before, more content available than ever before, and access to more successful promotions to study in an instant than ever before.\n\nThere is a dark side to the fact that nothing you've done in the past is ever gone. Just ask the kid who put a picture of himself on Facebook at a frat party five years ago and hopes the person interviewing him for a job today doesn't see it. On the other hand, every person you have ever had a previous relationship with, everything you have ever written, and every sales approach you have ever used is also around forever\u2014and it's all far more usable (and less embarrassing) than pictures from a frat party.\n\nHOW TO USE PREMIUMS (OR, WHEN OVERDELIVERY & BEST OFFER COLLIDE)\n\nBonuses and premiums have always been among the most important tools in the direct response playbook and should be considered with every offer. They can turn a good offer into a great one and can be the difference between mediocre response and massive response. Many pioneers in the direct response business (magazines, for example) always emphasized premiums, such as free reports, books, and even hard goods (remember the sneaker phone from _Sports Illustrated_?).\n\nOne of the top copywriters I ever worked with, a master at maximizing the effectiveness of premiums, is Eric Betuel. He had been writing control after control, and we wanted him to tackle a hot new product for us called _The Bottom Line Yearbook._ The book was hot because it was made up of the best annual content created from multiple sources . . . and it was a continuity, updated every year and sold as a kind of subscription, which would add a lot to the profitability long term if a new offer could beat the control offer.\n\nThe control offer Eric was trying to beat gave away a bonus report with the book, and he suggested increasing that to four bonus reports that were related to some of the most important topics covered in the book. Four bonuses instead of one increased response to the offer significantly. So we upped our game and started offering 50 free reports. Once 50 became the control, of course we tested giving away 100 free reports. And while the increased _number_ of bonuses obviously created higher perceived value, we got the most bang for our buck by choosing report headlines that covered the most salient interest areas, creating many more entry points for our audience.\n\nGoing from 1 premium to 4 to 50 to 100 actually didn't cost us a lot more money the way we produced the reports, and the value of this overdelivery (an irresistible offer addressing the hottest topics) made a huge difference to the offer's profitability. One thing we did to keep the costs down while being able to increase the number of premiums from 4 to 100 was by binding all the special reports into one volume for delivery. But the direct mail piece displayed the titles of all 100 reports individually; that way the customer could see exactly how much value they were getting by being able to read every topic of every report. Each report was only a few pages long, but the consumer advice was deeply impactful on a wide variety of topics.\n\nThis is a great example of getting the most out of a premium-driven offer. You can see the value added here\u2014by offering 100 bonus reports, you get 100 extra entry points, any one of which could be the key reason someone will buy the book. A reader might buy the whole package on the basis of just one of these bonus reports. And our revenue and profit for _The Bottom Line Yearbook_ rose significantly as a result of massively overdelivering high-quality information to our customers inside an irresistible and super-compelling offer. And very little was brand-new content, just great content from the archives that had been updated.\n\nTHE SIGNIFICANCE OF STATISTICAL SIGNIFICANCE\n\nOne of my favorite marketing cartoons (I have collected many over the years) shows three marketing people in weighted sacks at the bottom of a polluted river.\n\nOne says (from his sack), \"I could sell this!\" That's curiosity (and the importance of dreaming) about what makes a great offer.\n\nOne says, \"It could work!\" That's optimism that the offer is going to be great for the customers.\n\nThe last one says, \"Numbers. I need numbers!\" That's measuring how the offer performs to get real data.\n\nI use those three geniuses to create my direct marketing offer success formula. Most companies that sell with direct response have plenty of curiosity and optimism, but without the final part of this formula, measurement, they spend much more time strategizing than implementing. Of course, you always have to assess the downside risks of doing faster tests (and more tests); but when there is a chance for a big winner, get it out there quickly.\n\nRemember: curiosity + optimism + measurement = direct marketing offer success. What that really means is that you have to be disciplined with your testing. Test the first segment that you know _should_ work for this offer. If they don't respond, kill the product. Don't even go to the next segments for testing\u2014 _you don't have a business with that product if your most targeted customers are not interested._ As Gordon Grossman always said, \"First find out if you have a business.\" This is the first essential step.\n\nIf you have a brand-new product you're excited about, should you test 15 variations of it to see which one will perform best? No. You test whatever you need to test to prove that you have a business with that product first. Then get a critical mass of engagement from paying customers . . . and _then_ you can test your variations.\n\nThis is why you should never walk out of a product development meeting without a clear idea of the tests you want to make\u2014your only initial focus should be to find out if you have a business for that product. Most online marketers are pretty impulsive about testing many things at the outset, since they can roll out products very quickly. And on the surface, the cost of multiple tests (even in a launch) is not financially prohibitive. So why do they need to think through testing a new product so intently? For one thing, there is a massive hidden cost to rolling out products that aren't robust enough to stand up to basic testing. There might not be a monetary cost to you in the short term, but if you present an inappropriate offer to a list, those people are less likely to respond in the future, and you won't be able to track why their responsiveness declined. Put another way, you might have a success in terms of overall profit but the nonbuyers probably won't tell you why they didn't buy\u2014and if the offer was a disconnect, you might end up losing those folks forever. You will have no chance of reviving them with your next offer because they will feel betrayed. And if you get in the habit of disconnected offers, this pattern will continue. There's opportunity cost, goodwill cost, time cost\u2014you accumulate huge losses over time when you don't test properly.\n\nDick Benson was a stickler for proper testing methodology. I think if Dick were alive today, he would be excited that technology has enabled us to test more frequently, faster, and much more inexpensively than we ever could have dreamed in the days before the Internet. But with all that excitement, I think he would have identified a new problem. I can even hear him now: \"Do you and all your clients understand statistical significance enough to accurately identify the winners of the tests you are doing?\"\n\nA quick primer on statistical significance: You must have enough data (and the right data) to take the results of a test and then roll a product out to a much larger universe of names with similar success and confidence. This requires large amounts of data so that your results are replicable.\n\nA while back I was consulting with a marketer who didn't understand statistical significance. We were starting a test on a $97 offer, and just as we were getting the first results, he turned to me and said, \"We've got a winner with test A over test B! Test A has 12 orders and test B has 8. This is great!\"\n\nOn a $97 offer, I can _assure_ you that 12 orders over 8 orders does not mean you have a statistically valid winner. Statistical significance requires many more orders. (If you don't have a list that allows for thousands of responses, you're probably better off focusing on finding more of the right people for your list and relying on the anecdotal evidence they give you rather than running tests that cannot reach true statistical significance.)\n\nBut not only do you need to test for statistical significance, _you also need to test the right things_. I like to combine Dick Benson's obsession with testing with another maxim from Gordon Grossman: \"Don't make tiny tests.\" You need to test the things that can give you significant lifts in response, revenue, or profit (or a combination of all those), not just the things that make you feel good.\n\nHere's a tip: Whenever you are in a brainstorming meeting discussing things to test, after you get all the possibilities written on a whiteboard, posted all over the room on sticky notes, or written in crayon on the wall, _look to test things that could give you at least a 30 percent lift in your key metric_. For example, changing the color of your guarantee box is unlikely to give you a 30 percent lift in front-end conversions, but testing the headline on that page might. Depending on your product, offer, and price, you could look for a little less than a 30 percent lift . . . but this is not a bad benchmark to hopefully help you make fewer \"tiny tests.\"\n\nTHE POWER OF PYRAMIDING\n\nThe concept of pyramiding will prevent you from making big mistakes when rolling out a new offer, whether you think you have a big winner or not. When you test a small quantity and your results say it's a winner, you only roll out to an additional select set of names (that look like the names you initially tested) to prove the first result. If the results hold up there, you \"pyramid\" a little more, promoting to a larger set of people who look like the second group. Regardless of price point, you need a certain level of response that will be a good indicator of how a bigger portion of the list will perform.\n\nFor example, the rule of thumb in direct mail (if you could afford it) was to mail around 5,000 names from a particular list in the hope that you would get around 100 responses to the offer you were testing. In most cases with a winning result on 5,000 names from a specific list with a specific selection on that list, you would pyramid to 25,000, and if the 25,000 validated the initial test result, you might pyramid to 100,000.\n\nThere are many factors that can affect results as you go from smaller quantities to larger quantities, so the guiding principle is to have discipline and be cautious but continue to expand.\n\nThe most important thing to know about pyramiding is that at each step, as you move up the pyramid, the test needs to pay for itself. You don't move on to bigger quantities and take on more risk until you've consistently proven the method and the profitability.\n\nOFFERS ARE CREATIVE\n\nEven though the _next_ chapter is about creative and copy, don't underestimate how much creativity is involved in coming up with your most compelling offers. In this chapter I didn't want to give you tactics on building offers, or to create a checklist of things to test, or give you lots of copy advice around offers. Many of the titles in the bibliography at the back of this book (see Further Reading) have lists of potential offers and tests you can consider that are based on decades of solid results and case histories. Here I want to lay out philosophies around offer testing that can be applied to many things you might be working on in your business today.\n\nHaving said that, offers and creative (copy) go hand in hand\u2014the offers you make must be consistent with the messaging you use.\n\nI remember a wonderful copywriter who wrote some very successful controls for me but didn't spend any significant time constructing the offers that went with his very well-written sales letters, headlines, brochures, and outer envelopes. Looking back on this now, I think we missed much bigger opportunities for success with this talented writer by not pushing him to be more holistic with the package, forcing him to think through the offers that went with his copy more deeply. Whether we could have done better or not with that copywriter, the lesson is that offers _are_ creative . . . and this is a perfect segue into the next chapter. But before we leave the subject of offers, a reminder: Coming up with the best offers for your products and services is something that's personal and specific to your business; and after you create your most irresistible offers, make sure you test them and build on them using the bedrock principles we've covered here in Chapter 5.\n\nKEY TAKEAWAYS\n\nMaking your best offers is another area in direct response that has a lot of moving parts. This chapter ranged far and wide, and I hope you will have a sense of everything that goes into delivering world-class offers to your carefully selected lists. Here's what you need to keep \"top of mind\":\n\n  * Choose niches where you can go a mile deep on your offers rather than a mile wide.\n  * When you choose your niche, make sure it fits with your \"superpowers\" and that there is ample reason for people in your audience to listen to you. Why are you uniquely positioned to give them what they need? How can you use that position to create innovative, powerful offers that will make them feel like you've been reading their mind?\n  * Survey your audience when developing your offers. Using the Q test methodology, you can collect hard evidence on the offers your ideal customers will be willing to pay for.\n  * Sometimes your audience will help you develop additional offerings over time. If your audience's interests evolve, evolve with them. Maximize the offers you have that are working, but don't stay so attached to them that you never innovate to meet the market where it's going.\n  * How you sell is how they will respond, that is, the structure of your campaigns and offers can have a huge impact on how your audience responds to them. Make sure the copy and creative package are congruent with the type of offer you are making and with the psychology of the ideal customer for that offer.\n  * Your offers don't all have to be original. Look at the assets you already have that could be repurposed, or look at buying the relevant assets if that makes sense for you. If you have an established brand, putting your name on an existing, proven asset can create some easy wins.\n  * Use bonuses and premiums to overdeliver even more on your best offers (and they should _all_ be world-class).\n  * Make sure you test all your offers, provided you have the volume to generate statistically significant results. Remember, don't make \"tiny tests.\" Once you have the results, you can roll out the offer using the pyramid structure throughout your list to validate the test results without risking your shirt.\n\nCHAPTER 6\n\nCREATIVE & COPY\n\n_\"It's not the copywriter's job to create desire.\"_\n\n\u2014 GENE SCHWARTZ\n\nUnder the 41\/39\/20 rule, I hope it's clear that I am _not_ suggesting that creative and copy (the design and the words you use in your marketing) are half as important as list or offer. In fact, creative and copy are just as important for effective campaigns as list selection and tailored offers. However, what I _am_ suggesting is that you must have everything regarding your targeting and list selection buttoned up with your best offer _first_ in order to make sure you don't waste precious creative dollars on expensive copywriters and designers.\n\nIt's a tragedy that so many marketers outsource their creative to amateurs when it's so vitally important. But many marketers don't pay as much attention to creative as they should because when you have a perfectly targeted list and an irresistible offer, sending mediocre creative can still create a profitable promotion. For example, an online marketer with a perfect affiliate partner (\"perfect\" meaning having an audience suited to the offer) will probably generate a decent number of orders and sales regardless of the messaging.\n\nBut even if it saves you money, there is no reason to coast along with mediocre messaging, since great creative will significantly increase your list's responsiveness to your offer and your overall success. When you can add killer copy to an irresistible offer and the tightest list selection you can manage, you've got the direct marketing formula that will create a business for a lifetime instead of creating a series of one-hit wonders.\n\nCopywriting legend John Carlton made the brilliant observation that the next big things in marketing may not be in the areas of whizbang technology but rather in the nuts and bolts that career direct marketers are especially good at: building responsive lists, creating the right offers, and developing the right copy approaches.\n\nThat's not to say that technology won't continue to move at the speed of light. But when it comes to making sales, no one can argue that the right list\u2014combined with the right offer and strong creative\u2014still wins, every time. The marketers who spend their time building up these big three (and less time figuring out the next big quick win or ninja technique) will be the winners for the long haul. Great copy and creative approaches create sustainable, profitable businesses; ninja techniques create single revenue events. Now, this chapter is not about how to write copy. I am not a copywriter. There are many fantastic books available about how to write powerful copy (and you can go to my reading list at the back of this book for my recommendations), but this book is not one of them. That said, I know good copy when I see it. That's because I had the privilege of working with some of the best copywriters who have ever lived, which taught me a lot about what to look for in copywriters and the work they produce.\n\nWhether you write copy yourself or you aspire to write copy, or even if you have no intention of ever writing copy, I believe that understanding seven key characteristics present in every great copywriter will give you a powerful perspective that will enable you to create the most successful campaigns possible. Interestingly, these seven characteristics have also been present in the makeup of every great _marketer_ I have ever worked with as well\u2014which tells me that to be a great copywriter is also to be a great marketer (and number five below talks about where the worlds of copywriters and marketers collide).\n\nTHE SEVEN CHARACTERISTICS OF WORLD-CLASS COPYWRITERS\n\n\"Do you know any good copywriters?\"\n\nThis might be the single most asked question I get regularly\u2014and I mean multiple times a week for decades. I guess I must look like a headhunter for copywriters. The question always frustrates me because I can never give someone a quick answer. I believe it is a disservice if I do. There are so many variables, and as you will read in this chapter, it's not \"one size fits all\" when it comes to matching a copywriter with a marketer. Copywriting is a specialty, not a commodity, and finding someone who is suited for your offer, products, or company is never a shot in the dark.\n\nOf course I know good copywriters. I worked with\u2014and still work with\u2014the best of the best. But to find the copywriters who will have the most impact for your business, I believe it's critical to understand what makes the best of the best tick.\n\nIf you are hiring or looking for the best copywriters, or aspire to be one yourself, success leaves clues. What follows are the key factors to look for in a candidate's work and behavior, or your own. If you're a copywriter yourself, be prepared to internalize these ideas and cultivate these seven qualities so that you meet the standard when you're being hired by shrewd marketers who know what to look for; and if you are one of those shrewd marketers, go through the process of finding the right fit for your business and see if your best candidate(s) have what it takes _before_ hiring.\n\n1. Hunger\n\nSpending thousands of hours researching and learning the craft of writing copy will make you one of the best in your field. Gary Halbert talked about how difficuilt it is to become a great copywriter as a type of \"hard labor,\" and he suggested to writers working for him: \"Do one hour's worth of road work every morning, right after you get up, for six days a week.\" Like road work, writing copy is hard labor and not everyone is cut out for it. You just sweat a little differently.\n\nMarty Edelston's first pillar to becoming extraordinary was outworking everyone\u2014and it's not only about working more hours. It's about showing dedication to the craft. If you accept that real expertise is developed over thousands of hours of deliberate practice, if you study and practice writing great copy without complaining and you keep a focused mind, you'll be well on your way to becoming one of the best in your field.\n\nEvery great copywriter is willing to do the work. They know this isn't easy, but their hunger drives them forward. Here's another maxim from Gary Halbert on this theme: \"Get yourself a collection of good ads and DM pieces, and copy them in your own handwriting.\" Many of the greatest active copywriters who are now coaching young copywriters use this technique (and it has no resemblance to writing \"I refuse to write crappy copy\" 100 times on the blackboard).\n\nStudies from around the world have found that handwriting (instead of typing) has a profound impact on our brains. It taps into what's called visual-motor memory, and it increases our ability to quickly recall information and execute commands and encourages higher order, abstract thinking (the free flow of creative ideas). Of course, copying out successful promotion packages from the past by hand requires patience and a particular mind-set\u2014the hunger I am talking about to become the best. It's hard, it takes a long time, it's repetitive . . . but it has worked for the best copywriters who have ever lived. Who are we to argue?\n\nShortly after I republished and began selling Gene Schwartz's lost classic _Breakthrough Advertising_ , a young copywriter who knew about this technique sent me a fat package in the mail that contained dozens of handwritten pages\u2014the complete first three chapters of _Breakthrough Advertising._ Now, in every copy of that book, I enclose a letter welcoming and congratulating the buyer on purchasing what I think is the most important book ever written on copy, creative, and, dare I say, human behavior. I suggest that they read the first three chapters several times before moving on to the rest of the book.\n\nWhile this book wasn't an ad or a direct mail piece, one young copywriter was listening, and he was clearly hungry . . . or maybe he was just trying to get on my good side (and he did). That was an unexpected package, but I will tell you that this copywriter got on my radar for the future\u2014not because he showed any particular talent by copying three chapters from a book word for word, but because he copied three chapters from a book that will become his bible for a lifetime.\n\nHunger also frees you from the fear of a lousy first draft. Gary Halbert is so quotable when it comes to the drive it takes to become a great copywriter: \"I don't exactly know what I am going to write about today, so what I am going to do is just keep putting words down on paper until I start to get some direction.\" Beware of the copywriter who complains about writer's block. The best writers I have worked with over almost 40 years all follow Gary's advice and understand that it won't be perfect on the first pass. And more importantly, they know they have to just keep writing to get each piece where it needs to be. (Hunger goes hand in hand with tons of patience.)\n\n2. Curiosity\n\nWhen it comes to being a great copywriter or marketer, nothing is more important than insatiable curiosity. As I mentioned earlier, a glowing example of this is Gene Schwartz, who read everything. His prolific consumption of media and information of every kind came from a need to know what made people tick\u2014knowledge he wanted in order to be able to write to them at the deepest level. Remember his warning not to lose touch with the people in your audience, and his advice to spend two hours a week (at least) finding out where the market is.\n\nArthur Johnson, the star of our infomercials, became so immersed in every area of health and medicine he wrote about that he went from being a copywriter to being a medical writer and editor. Copywriter Parris Lampropoulos was almost as knowledgeable as my urologist on alternative prostate cancer treatments because of the hard work and insatiable curiosity he had for that topic and all the other health topics he ever got involved with.\n\nInsatiable curiosity is also tied closely to being an expert researcher. The best direct marketing companies today who hire copywriters to work full time in their organizations sometimes don't have them write any significant copy for six months (or longer) while they deeply research the topics they will be writing about in the years ahead.\n\nAs David Ogilvy warned: \"Advertising people who ignore research are as dangerous as generals who ignore decodes of enemy signals.\" Research (such as Q testing) is important to everyone in the marketing process, but especially to the copywriter. When assessing the level of curiosity of copywriters, it's a red flag when they show little interest in research. If they are not obsessed with knowing everything they can about their audience before writing a word of copy, they are copywriters to avoid.\n\nThe most prolific copywriters I have ever known and worked with are as interested in the \"41 percent\" as the marketers are\u2014they know that understanding their audience (i.e., the list) at every level is the key to writing the most powerful copy. If a copywriter I'm considering hiring doesn't ask me for a \"list history\" (every list mailed for the product they are writing for\u2014with the results\u2014good and bad), I immediately write them off. The best of the best always come back to me to find out if I have any of the promotions that were used to create the lists that were being mailed most successfully for the product they are working on.\n\n3. Smarts\n\nCopywriting is seen as a very lonely profession; you really can't write copy by committee. On the other hand, every copywriter knows that if they don't have people around them to help them hone their craft and ideas, they will never get their copy to a world-class level. The best of the best understand this, and that's what makes them the best of the best.\n\nA question I ask every copywriter\u2014and a question I would ask myself if I wrote copy\u2014is \"Who do you hang out with professionally, and why?\" I then ask, \"How do you stay accountable to your craft, constantly improving your skills and insights, even when everybody is telling you how great you already are?\" Even the most talented and confident copywriters understand how important it is to have constructive critics available to them 24\/7. The communities you align yourself with may be the most critical career decision you will ever make . . . and there is not one copywriter or marketer I know who goes it alone.\n\nCopywriting is a profession, like so many others, that lends itself to apprenticeship. Sometimes (though rarely) this will happen organically; other times, you will need to pay to play by joining private masterminds, attending high-leverage events, and paying for coaching from the greats who have come before you.\n\nFor example, many young copywriters paid Gary Halbert a huge (and nonrefundable) sum of money to work with him for a short time (but long enough for him to figure out if they had the right stuff to become a great copywriter). And from that group, successful and well-known copywriters are still working today, long after Gary Halbert's death, all trained by the master.\n\nThe sacrifice they made to get this education was partly due to hunger, partly to curiosity, and all about what I call \"smarts\"\u2014they became great because they were accountable to someone smarter than themselves. They paid significantly in both time and money, and it set them apart from the pack for good.\n\nBut leave it to Halbert to be a contrarian on his own philosophy (he was quite a character): \"A support system is like a garden, and you always need to be on the lookout for weeds to pull.\" I guess you have to be smart about who you get your smarts from too.\n\n4. Passion\n\nIt's so much easier to write about something that you have passion for than something you care little about and are only doing for a payday. Almost all the greatest copywriters began their careers writing about material they knew a lot about and loved writing about as well.\n\nAlthough not a copywriter, filmmaker Steven Spielberg is one of the greatest creative talents of all time. His three most successful early films\u2014 _Jaws, Close Encounters of the Third Kind_ , and _E.T.\u2014_ on the surface simply look like blockbuster films about terror, science fiction, and special effects. What they are really about is family and loss, and they were expressions of Spielberg's childhood.\n\nI bring this up because when you look at the early work of many great copywriters, it's often driven by something from their personal experience that made them go a mile deep before they went a mile wide. (When I speak publicly about this characteristic, the visual I use is an upside-down funnel: narrow to wide.) That's one expression of passion\u2014copywriters starting their careers writing about something they already know and care about. Or at least, writing about things they _want_ to know more about and taking a deep dive to discover everything there is to know.\n\nWhen you look at the careers of the most successful copywriters, it looks like they could write about anything, any time. But it doesn't start that way. Whenever I interview a new copywriter, I really try to understand their journey and how they got to my office with an interest in writing about this kind of offer.\n\nThe deeper they've gone into subjects they were curious about or had some connection to in their past, the more confident I get that they will take those same deep dives in the present . . . and that they care enough about the topic to write me a winner.\n\nThere's another angle on passion: we should want to hire copywriters who think in terms of impact before there's ever a conversation about money (and we should hang out with other marketers who think that too). Don't get me wrong, making money is important\u2014but impact with passion goes hand in hand with profit. We have a responsibility as marketers to teach and reach aggressively, and it is actually irresponsible to not sell hard when we know we've got the goods to change lives. That's the passion I want in my copywriters.\n\nPeople who do not consider themselves marketers might say that we're just trying to justify our existence with this philosophy. But I implore you, whether you are a marketer or not, to look at your list, your customers, your community, your tribe, and your potential prospects as your most important asset. These assets will enable you to achieve everything you dream of creating over the course of your professional life.\n\nYour copywriters must be in step with you here\u2014passionate about the project, clear on how much impact it can have, and congruent with everything you believe. Of course you should aim to make a lot of money together. But if you both come from the perspective of making a difference, I guarantee you will leave a huge mark on the world\u2014and build a more profitable business for the long term too.\n\nSometimes your passion won't be directly related to the subject matter of what you're writing about or marketing, but being passionate is always powerful, and the energy from your passion will often drive curiosity in other areas. For example, even though Gene Schwartz didn't write a lot of direct response copy about art, he was both one of the greatest copywriters who ever lived _and_ one of the greatest modern art collectors and experts of his time. Art was his passion, and he used the energy and perspectives it gave him to take deep dives into all kinds of niches with his writing.\n\nIt's interesting to note that Gene's obituary only had one paragraph about his direct marketing and copywriting success; the rest was all about his impact on modern art. He may not have written any control packages for art products, but his passion showed up in everything he did. This quote hints at how his passion for art translated into his genius for copy: \"The arts not only imbue our sense of sight, balance, movement, touch and hearing, they also lift our logical minds\u2014the traditional focus of modern education\u2014into the reaches of possibility, invention, and genius.\"\n\n5. Understanding Direct Marketing Principles\n\nThe most successful copywriters are far more than just writers. They are true advisors and partners in every element of the client's marketing and business growth.\n\nSo how do you become an advisor? Simply put, you have to understand the direct marketing principles to which your copy is going to be applied. In writing my first book about six legends of advertising, all of whom wrote copy, I was astounded at how obsessed all of them were with lists and offers, and not just with the poetry they wrote.\n\nRobert Collier, the father of the sales letter, said: \"Study your reader first, your product second.\" The sales letter guy was thinking about lists. I love that. And while they were all obsessed with lists, they were obsessed with offers too. John Caples said: \"Test everything. Doubt everything. Be interested in theories, but don't spend a large sum of money on a theory without spending a little money to test it first.\"\n\nEvery copywriter who has had a long career, or expects to have a long career, needs to know almost as much about lists and offers as they do about writing copy. At a minimum, they need to understand RFM and LTV. That's why if you can understand even the basics of direct marketing, you will have a huge advantage over other copywriters competing for the same assignments. And if you're a marketer, having a copywriter who can truly partner with you on all your marketing initiatives will also be a huge advantage. The best in the business know this and are constantly reading, studying, and learning, both from books and from experts.\n\n6. Humility\n\nSince everything we do in direct response marketing is measurable, there is very little room for arrogance and hubris. The most successful copywriters will tell you all about their winners, but they'll tell you even more about their losers. That's not to say that we don't want our copywriters to have supreme confidence in their work. But if we're working with partners who are always hungry, curious, smart, passionate, and well educated, then hopefully confidence will come naturally to them despite many failures.\n\nBefore I left Boardroom, a young copywriter wrote to me and said: \"I will be the best copywriter who has ever walked in the door at Boardroom. We should talk about an assignment.\" (I didn't want to be outright rude, so I put the blame on myself: \"Shame on me\u2014I've been doing this for over 30 years, and I've never heard of you, so I'm glad you brought this to my attention.\")\n\nThis guy's approach is exactly how not to do it. The hubris put me off immediately, especially since the quality of his work didn't back up his claim . . . and he apparently had no idea that some of the greatest copywriters of all time had walked through the doors at Boardroom previously. He didn't even do research on _that_ . . . so you've probably guessed that he never got hired.\n\nOn the other side of the spectrum, in 2014 America's best living copywriter, Gary Bencivenga, and America's top TV infomercial producer, Greg Renker, both spoke at the Titans of Direct Response event. They were headline speakers, yet both thought that they should pay for their admission to the event because of the other people who were speaking. They came on my invitation\u2014as keynote presenters\u2014but saw themselves as humble students.\n\nOf course, I didn't take their money . . . and I learned a valuable lesson about humility. These two giants would have been justified in throwing their weight around, making demands and expecting constant attention, but instead they behaved nothing like the prima donnas they could have been. If Greg and Gary can be so humble after such incredible careers, the rest of us can keep our pride and egos in check too.\n\n7. How They Share Their Success\n\nWhen hiring a copywriter, the first thing clients usually ask for is a portfolio of work. The portfolio is always important. But I purposely put this as the last of the seven characteristics to show that you won't get to a portfolio of success without prioritizing the other six characteristics. Clearly, the more you write (as a copywriter) or the more you test (as a marketer), the more lessons you will learn on the field of play, and the larger your portfolio will become. More proof is, of course, always better. Build up a portfolio you are proud of, and don't be scared to show it. But show who you are first and what you've done second. Hunger, curiosity, smarts, passion, direct marketing knowledge, and humility come first.\n\nBased on the characteristics we've just discussed, it's obvious that a great copywriter is a secret weapon that should be deployed in every part of your business.\n\nThere were years when our highest-paid \"employees\" never showed up at the office. Gary, like so many other copywriters with a similar mind-set, was probably at home, not resting on his laurels. While he was collecting royalties for blockbuster winners he was also always thinking about how to beat them with even bigger winners. When people asked if paying huge royalties bothered us, Marty and I would simply say: \"No! If our copywriters are making that much, can you imagine how much _we_ are making?\" This philosophy enabled us to mail millions of pieces, because Gary and all our top gun copywriters wrote packages that beat all comers by a wide margin, and then they followed them up by going back to the drawing board every time to make the next one even better.\n\nFeeling bad because a copywriter makes too much money is shortsighted, because it shows you're not thinking about the growth of your business. I maintain that is scarcity thinking. A mediocre copywriter will hold you back. Despite all I said previously about being your own best copywriter, good copywriters have a unique talent, and it's one that should never be shortchanged.\n\nDon't believe what you read about the toughest job for a copywriter being how long it takes them to count their money while hanging out at their beach house in the south of France. The best copywriters are relentlessly disciplined, and we can all take many lessons from them as we study how they work and what makes them tick; and if they are able to buy that ch\u00e2teau, it's because they deserve it.\n\nThroughout the rest of this chapter, I'm going to share how you can best leverage copywriters and other creative talent across all your marketing to make the most of these incredible assets.\n\nTHE COPYWRITER IN THE COAL MINE\n\nThis expression refers to the old practice of sending canaries down into coal mines with miners. Any dangerous gasses would kill the birds before they would do any serious harm to the miners, giving them an early warning system and a chance to exit the tunnels before things got dicey.\n\nNow, I don't want you to think my thesis is that we should actually sacrifice our copywriters for the sake of saving _other_ humans (yes, copywriters are people too), but the best copywriters are always ahead of the curve. With their insatiable curiosity and need to research everything at the deepest level before putting pen to paper, they are in the best position to heed the warnings of what is happening in the marketplace and what will (and won't) move people to take action.\n\nWe can stop the metaphor here. No one needs to die (canary _or_ copywriter), but I want to talk about some examples of why I think the top copywriters are the folks to watch (and lean on heavily) if you want to know the best route through your marketing tunnel.\n\nBoardroom's four top copywriters, responsible for over _600 million pieces_ of successful direct mail since 1995, are a Mount Rushmore of copywriters, and they are heroes to me: Eric Betuel, David Deutsch, Arthur Johnson, and Parris Lampropoulos. They consistently wrote winning promotions, and the company could never have made it through 40 years without their talent and persistence.\n\nMarty was not the expert in any one area we wrote about, but he was the most inquisitive man on the planet. And he had a strong sense of justice that was central to what he wanted to achieve at Boardroom\u2014he was a true bloodhound for our audience. He was always looking out for the little guy. However, Marty wasn't the only bloodhound in this partnership. The copywriters were equally aggressive in how they approached inside information and helping consumers get the edge whenever possible. These four writers (and others before them) were able to write in his voice, perfecting the bloodhound approach and tracking down vital knowledge for the average American. It was amazing to behold.\n\nThe bloodhound publisher only hired bloodhound copywriters. And even if you are not like Marty, you should hire like Marty. That is, even if you are not a bloodhound yourself, hire copywriters who understand this concept. The best copywriters plunge into one coal mine after another\u2014often uncharted or controversial territory\u2014and let themselves be completely immersed: they read, research, study, and then read and research and study some more. And sometimes they even do their research over dinner.\n\nYou should also hire copywriters who understand your vision and philosophy at a very deep level well before they write one word for you.\n\nOne of our trademarks for gathering information and content for our newsletters was bringing our experts together on a regular basis (more about this in Chapter 10). We would assemble experts from all the disciplines we covered to meet and debate to see what sparks might fly when they got together in a moderated discussion. Those sparks always turned into the best story ideas, and the best story ideas often turned into the best promotion copy.\n\nSo when Marty had a brainstorm to bring together experts from one discipline\u2014taxes\u2014he was smart enough to invite not only our tax editors, to get story ideas, but also one of our copywriters, Parris Lampropoulos, who was working on our tax newsletter promotion at the time. What came out of that meeting were dozens of great story ideas . . . and a blockbuster control package for our newsletter _Tax Hotline._\n\nParris loved the content that came out of the discussions at the meeting, but he went even deeper\u2014he saw something in the format and the idea behind the meeting as well. He set up his promotion around a \"secret meeting of the country's top tax experts\" spilling the beans on things they would not normally talk about in public and the reader having access to this inside information. This approach would convince readers that _Tax Hotline_ was the most valuable publication of its kind. It wasn't just the content we got from the meeting that led to this winning package, but also the intrigue of who was in the room and what they knew that made this a blockbuster. Only a copywriter could have thought of that.\n\nDON'T LEAVE YOUR COPYWRITERS ALONE IN THE COAL MINE\n\nIn the past, I have readily admitted that I am a copywriter wannabe. Despite the fact that this is never going to be my gift, I have come to believe that at some point, we will all be our own best copywriters. Put a different way, we all need to be a bigger part of the creative process for our campaigns whether we do the actual writing or not.\n\nThis refers back to what I said about passion\u2014and not only the passion of the copywriter for the topic they are writing about, but also the passion of the marketer or entrepreneur for the product or service they are offering. Marketers and entrepreneurs make the mistake all the time of giving a copywriter an assignment without immersing themselves in the project right at the beginning.\n\nI encourage you to hire the best copywriters you can find to push the envelope (pun intended) on your behalf\u2014but it's not just about their writing talent. It's also about their gut instinct on what will make your audience respond and what will differentiate you from your competition.\n\nAlways allow your copywriters to probe your editors, your gurus, your experts (and you) to make sure there is not more material in those incredible brains that could create some of the most exciting and breakthrough articles or concepts. These can lead to more compelling promotion copy, and yes, more sales of your product or service. In the right hands, that information\u2014even if it seems controversial, crazy, or weird\u2014can become cutting-edge, transformative copy. So keep an open mind, spill your guts (to a seasoned professional), and search high and low for your next best idea.\n\nI'm not recommending that you ever be irresponsible or push the envelope for the sake of pushing the envelope. But it reminds me of the classic Henry Kissinger story (which I will tell in my own words): A speech writer for Kissinger sent in his first draft of an upcoming speech, and Kissinger sent it back to him with a question written in the margin: \"Is this the best you can do?\" The speech writer went back to work. He sent in another draft, only to get the same response. This back-and-forth happened several more times until the speechwriter said: \"This is the best I can do! I can't possibly improve one more word.\" To which Kissinger then replied: \"In that case, now I will read it.\"\n\nA-list copywriters would never dream of showing their early versions to a client. They only bring them the most complete version, after immersing themselves deep in the coal mine, getting down and dirty to turn up every piece of information they need to deliver the goods for that promotion. I knew that those copywriters had not only honed the copy on their own with multiple versions and approaches, but they had also bounced it off trusted colleagues for ideas, tweaks, and opinions (which I discussed earlier in this chapter under #3 of the seven characteristics all A-list writers possess). When you hire the best, you get a bonus\u2014the wisdom and insight of the writers and thinkers _they_ think are the best as well.\n\nAnd don't forget: Before they write, while they are writing, and after they deliver a draft to you, don't leave them down there in the coal mine on their own. Get in there with them and make sure they have all the tools they need every step of the way so they can work their magic, but never forget your vision and passion too.\n\nTHE ART & SCIENCE OF COPY\n\nMost people think one of two things about copy: that it's some mysterious art form, a panacea for every campaign problem, or that it's an extremely technical science\u2014a plug-and-play system that works according to a formula. The truth is that it's somewhere between the two. The linchpin that brings both elements together is the list\u2014and how the copywriter thinks about it.\n\nI can't emphasize enough that the best copywriters I ever worked with (and I worked with nearly all of them) were obsessed with the list and the audience first. I know I am repeating myself, but I want to make sure you are obsessed with this relationship between copy and list too. Gene Schwartz, Gary Halbert, Gary Bencivenga\u2014and every other super successful copywriter I have ever met and worked with\u2014all of them wanted every scrap of information they could get on the list (or audience, or in today's jargon, avatar) before they ever wrote a word.\n\nSo what does that tell you?\n\nIt tells you that any copywriter worth their salt will be obsessed with the audience they're writing for\u2014and they will be chomping at the bit to get into the research. I want to repeat this again too: my general rule of thumb is that if a copywriter doesn't ask me for a list history in our initial interview, I don't hire them. (While list history was documented list by list in the world of direct mail, compiling a list history in any medium you work in is doable, and you should consider it a requirement.)\n\nIn addition, any copywriter who does not do a deep-dive interview with their client about the new project and its target audience is a copywriter you don't want to hire. This might sound like I'm back to saying the list is more important than anything else . . . and while the list _is_ the most important thing, that's not my point here. What I'm really saying now is that to write your best copy, and to get copy that's going to move your business forward consistently, you have to understand how your copy relates to the list.\n\nASKING \"WHAT'S NOT HERE?\"\n\nOne of the greatest lessons I learned from working with leading copywriters is to never be satisfied with a product as is\u2014to always ask the question, What's not here that should be?\n\nWhenever we handed Gene Schwartz a book and asked him to write a winning promotion for it, he would start by asking us what was _not_ in the book. That's often what would inspire him to write the most compelling copy he could. One of my favorite Gene Schwartz-isms is, \"Copy is not written. Copy is assembled.\" Gene had a voracious appetite to gather (assemble) everything in plain sight, and also those things that were not so obvious\u2014which would then maximize the impact of what he eventually wrote. He wanted to know everything: what we had cut, what we had ignored, and what we thought was too out there to be included. He made an assumption that what we had was good but not great (yet) and that we had an opportunity at the start of each project to build marketing capital _into_ the product as it was being created.\n\nGene's view was that even if the book was well done, there was huge upside (and no downside) in pushing our editors and marketers to find the material that no one else was talking about . . . which would then enable him to write much more compelling copy and teasers. Let me expand on this with some examples that came from copywriters who were dissatisfied with their assignments as given. Each time, we turned their dissatisfaction into unbridled enthusiasm by letting them ask a few simple questions of their clients with the purpose of uncovering the most impactful copy platforms.\n\nMy favorite is the story of Gene interviewing Marty about the new publication Marty wanted to launch in 1972, to be titled _Boardroom Reports._ Marty had a vision for a newsletter that would teach business owners and entrepreneurs what they really needed to know about running their businesses. He wanted to include only the most important information from leading experts, helpful and actionable information he felt the business magazines were missing. The goal of _Boardroom Reports_ was to be useful and practical from cover to cover. Marty asked Gene to write the promotion to launch his big idea and gave him the premise above\u2014but Gene was too smart to simply run off and start writing. He grilled Marty at length, digging deep into why Marty would put his life savings into launching this idea. They spent many hours together over many days before Gene wrote a single word of copy. The result was the headline that launched everything for Boardroom:\n\nNOW!  \nREAD 300 BUSINESS MAGAZINES IN 30 MINUTES!\n\nAnd get the guts of every one of their most valuable ideas\u2014in super condensed form you just can't forget!\n\nLooking back, knowing that Boardroom eventually became one of the most successful direct marketers of newsletters in the decades that followed, we could assume it was Gene's brilliant copy that started it all. And of course Marty gave him all the credit.\n\nHowever, talking with Gene some years later, I asked him about this headline (which was just the beginning of an incredibly engaging sales letter). He said to me: \"You can give me all the credit you want but I didn't write it. Marty did.\" Gene's genius was not just in the writing . . . it was also in how he extracted what he needed from Marty's brain. According to Gene, all the best material in that sales letter was in Marty's head already and just needed to be brought up to the surface.\n\nGene's great ability was in how he questioned Marty and then created the story around his answers. He knew that Marty (his client, who was really his partner in this promotion) was also an experienced expert who was passionate about the topic and who was constantly researching how he could deliver a better product. Gene knew that gold (i.e., good copy) was just waiting to be mined. (Of course, there _was_ genius in Gene's writing too . . . even if he was too humble to admit it.)\n\nAfter leaving Boardroom, I had a similar experience with a client: I wasn't writing their copy from scratch, but we were brainstorming how to create a copy approach on a landing page that would get prospects to order (which was not happening with their current strategy). During a long interview process with the entrepreneur, all I did was ask questions around two themes:\n\n  1. \"Why should I listen to you?\"\n  2. \"What's in it for me if I buy your product?\"\n\nWhat poured out of her was passion and copy that seemed to write itself.\n\nIt was her story of persistence and grit, how she got to where she is today, why she wants her tribe to have the same success, and why the product will be a game changer for their businesses. In the end, we had copy that captured her readers' attention at a whole new level. And even though she needed someone with copy experience to bring it up to the surface, she\u2014not the copywriter\u2014is the one who \"wrote\" it.\n\nWhen I think about this process, perfected by the world's best copywriters, it reminds me of the services available today for thought leaders to write books quickly and efficiently. Even those who say they could never find the time (or have the discipline) to write a book can all \"talk a book\"\u2014and a transcribed interview capturing what is in their brain becomes a book super fast. When I see so many folks being able to \"write\" books this way, I smile and think of the interview processes employed by the best copywriters of all time. After all, a great promotion is sort of like an authored book for many entrepreneurs and gurus selling their vision to a wide audience.\n\nYOUR BEST STUFF MIGHT BE ON THE CUTTING ROOM FLOOR\n\nSomething you should always keep in mind is that your best material (and copy approach) might not be in the spec assignment but rather, as they say in film editing, what is on the cutting room floor. It might also be in a wastebasket or deleted file somewhere, waiting to be snatched back before it is lost forever.\n\nI recall a time when I was working on an alternative health newsletter written by a naturopath (who was our guru for the newsletter), and we needed a new control badly. I approached one of our A-list copywriters and asked him if he would take on the assignment. When he started digging into the past issues of the newsletter and previous promotions, he was underwhelmed: \"There's nothing new or groundbreaking here . . . and this is an area of health that should be all about new and groundbreaking. I don't think I can take this assignment. Based on this material, what I will write will be boring and not likely to be very successful.\"\n\nAfter some begging and pleading on my part, the copywriter consented to dive in some more. He interviewed the guru, all the editors, and all the marketers who were working on the newsletter at the time.\n\nWhat he found out was startling: There were all sorts of stories and ideas that never got past an initial idea meeting. The editors had vetoed lots of material and left it on the cutting room floor because it seemed too controversial. And I didn't blame them\u2014they were only being super conscientious. However, that discovery became an incredible opportunity: could \"controversial\" become \"new and groundbreaking\"?\n\nRather than lament the fact that we couldn't cover the controversial material, we went back to the editors to see which of these potential stories could be researched further and resurrected. We knew there must be _something_ there that we could share with our audience without taking any big risks. Our readers were hungry for brand-new health information they had never heard before, and we knew we could give them something useful and responsible without crossing the line into sharing incomplete or undocumented information.\n\nWell, we didn't revive _all_ those discarded ideas, stories, and research. But we were able to save a surprising amount, and this brought what we thought was dead material back to life, much of it becoming some of the most exciting articles the newsletter had ever published. And our copywriter had many new entry points into the promotion with leads, headlines, and stories\u2014not risky, but super exciting. The result was a new control that beat the old control by a wide margin.\n\nI'd like to leave you with two important takeaways from these stories:\n\n  1. While I always say that you should never leave your copy and creative to amateurs and to only hire the best copywriters and creative talent, I also think you are often your own best copywriter\u2014that is, you are the best source of the most compelling information you can deliver. With the right copywriter guiding you, you might actually write your next great promotion simply by talking and thinking about your \"big why.\"\n  2. Never assume that there isn't more to your product, service, or message. The more you can differentiate from what's out there already, the better. Search everywhere to find some of those differentiators that might have been dismissed before the proper research or hard work had been done.\n\nTALK TO THEM LIKE YOU KNOW THEM . . . ESPECIALLY IF YOU DO\n\nIn 2016, Donald Trump won the U.S. presidential election. While political advertising is often boring and even irritating, the one thing we can say about the 2016 campaign is that it was a marketing case study for the ages. This is not a political book, and I won't tell you who I voted for; nonetheless, marketers can learn _a lot_ from Trump's campaign about matching message to list.\n\nGene Schwartz also taught about this relationship between message and list. He said, \"There is your audience. There is the language. There are the words that they use.\"\n\nWhen writing my first book, I was surprised that all six men I was profiling were supremely interested in knowing their audience\u2014despite all of them being known more as copywriters than anything else. When you read their work, and especially their quotes about audience and list, their writing sometimes seems almost secondary, a consequence of having gotten to know the people who would read it. Remember Gene Schwartz's warning: you cannot lose touch with the people of this country, no matter how successful or potent you are.\n\nGene and all the copywriters at his level knew that their best shot at connecting with almost any market was to know as much about them as possible, and the only way to do that was to read everything they were reading, to understand everything they were experiencing in modern culture, and truly live in their shoes.\n\nIt should be noted that the copywriters I knew and worked with rarely had much in common with or lived a similar lifestyle to the audiences they wrote for. . . but they did everything they could to be in their heads when they wrote to them. For example, Gene lived in New York City, in a penthouse on Park Avenue, and had one of the most impressive modern art collections in the country. He was a very sophisticated man with an incredible intellect, but that never stopped him from being a student of all types of people.\n\nThere is another secret all the greats focused on intensely: they learned how the audience communicated with each other. Every successful copywriter uses language that is in sync with the market they are writing for . . . and it takes a lot of time and effort to make sure they get every word and every sentence perfect.\n\nDid Donald Trump spend the kind of time and effort researching his market and honing his language that Schwartz and Halbert did? Probably not. But he figured out the right words and the right tone to garner over 60 million votes, despite never having run for any office in his life.\n\nI love what David Ogilvy said, to drive the point home even further on language: \"Never use jargon words like reconceptualize, demassification, attitudinally, judgmentally; they are hallmarks of a pretentious ass.\" There are probably times (maybe in some academic journals) when you could get away with words like that\u2014but only if your readers use those words too. As marketers and copywriters, use words and phrases that are unfamiliar to your reader at your peril. It only takes one pretentious word to ruin a sales letter.\n\nI do the following exercise with clients to remove every word in their promotions that can't answer these two questions in the affirmative:\n\n  1. Would they know what this word or phrase means?\n  2. Would your customers use that word or phrase while talking with each other?\n\nIt seems like a pretty simple litmus test, yet if you read copy through this lens, you may be surprised by how little some copywriters and marketers seem to really know about their audience.\n\nIn short, know the language of your audience. Sometimes it will be simple, other times complex. Sometimes it will be clinical, other times emotional. Know your audience, know their language, know the words they use.\n\nSome ideas to ensure you do this well:\n\n  * Hang out in online forums where your audience spends time to see how they talk and what words they use.\n  * Read reviews on Amazon for books that your audience is likely reading and see how they write about them.\n  * Create customer panels (paid groups of people who are typical of your audience and the kind of people you are trying to find more of). You can try out headlines, phrases, and copy platforms on them to see how they respond and react.\n\nSomeone in the Trump campaign knew something about this . . . or maybe they just got lucky. But I'd say it's more than luck. We know that when audience and language are aligned, big things happen.\n\nTHERE'S NO ONE-SIZE-FITS-ALL CREATIVE\n\nTailoring your creative and promotions to your different list segments is critical. It seems nuts that anyone would try to come up with one-size-fits-all creative, given how inexpensive e-mail and most online marketing is. We had different control packages for different segments in direct mail, where it was very expensive to do, so there's no excuse not to tailor your creative online to specific list segments.\n\nThere's never any reason to talk to everyone on your list the same way. You want to talk to former customers differently than you talk to current customers, who are both very different from prospects who have never bought, who are then also different from names acquired through joint ventures . . . and those are only the obvious segments.\n\nFirst and foremost, you must segment your list on the basis of source and behavior, which is much of what we talked about previously regarding RFM. The next step is to create messaging for each segment that makes sense on the basis of source and behavior, using language and creative appropriate to each segment. The goal is always to maximize the response rate of each segment rather than trying to increase the overall response rate with one-size-fits-all creative.\n\nThat said, it's important not to segment to the point of diminishing returns. Keep your focus on building segments that have meaningful differences and will yield meaningful results as they respond. You might only have a couple of variations. But let your audience tell you how many segments make sense.\n\nTo make the economics work in direct mail, we couldn't afford too many segments with different promotions because we had the barrier of paying postage and printing\u2014every time we had different creative for different list segments, that added a lot of costs. Regardless, we would still often have two or three controls for each product, which would be sent to different segments of our existing and potential audience, based on all the data we had collected on them.\n\nWhile I believe no direct marketer should ever \"go cheap\" with copy and creative offline or online, taking out the cost of postage and printing should be an incentive for online marketers to get even _more_ creative with their creative (to different list segments). Split testing in direct mail had additional costs because when you mailed smaller quantites of a particular piece of creative, postage and printing were more expensive too. Online you can split test much more inexpensively and efficiently, but of course, everything still needs to get measured and everything still needs to be meaningful (remember, no tiny tests).\n\nUNDERSTANDING SEGMENT PSYCHOLOGY\n\nWhenever you are building a promotion, you can run through lots of numbers\u2014the total number of names on a list, the new names on a list, the multibuyers\u2014but often, the more interesting piece of information is what type of content brought each group of names onto the list.\n\nFor example, we had entire segments on the Boardroom list that came in through fascination pieces, which were very curiosity-driven and were focused on the people we affectionately called mail-order junkies. These fascinations were the handiwork of the best copywriter you never heard of: Mel Martin. While Gene pioneered the use of interesting bullet points in copy, Mel turned fascinations into a true art form.\n\nMy friend and direct marketing historian Denny Hatch described fascinations like this:\n\nFascinations. Teasers. Taking an old-fashioned teaser\u2014usually found on an envelope\u2014and stuffing an entire mailing full of them, nakedly appealing to the emotions that scare people and drive them to action.\n\nMel had so much success with fascinations that eventually they became a staple technique of copywriters everywhere and have even become commonplace in today's copy.\n\nBy knowing that a name (also a person!) had subscribed based on a deep need to satisfy some intellectual curiosity or to find answers to questions that had been plaguing him or her, we also knew that he or she would likely respond well to other fascination campaigns.\n\nThis also showed that when there is synergy between lists and offers, you can feel confident that their responses will be similar, if, for example, you are doing an affiliate deal with a complementary list and you know that they have used a similar creative approach to what you used to convert your own customers. More clues, such as if price points are in the same range, if bonuses and premiums are prevalent, and so on, tell you where synergies lie in the offers and the products and how they are promoted.\n\nAn obvious example is that if you're a guru in a particular field and an affiliate has done well with guru-driven offers in a similar space, then that will probably be a high-performing list for you to promote to. On the other hand, you may have affiliate partners who are not guru driven, and therefore talking to their lists will require a different approach and\/or language.\n\nAll these elements\u2014the source, the type of promotion, the piece of creative that got the name on the list\u2014are subtleties that are rarely touched on when most people are doing list research. Yes, it's more demanding and requires more legwork than just throwing a one-size-fits-all campaign together, but that's why there is such an amazing opportunity in thinking this way: you basically have the strategy to yourself.\n\nHere are some questions to help you dig down into the psychology of each list segment:\n\n  * What type of content brought someone onto the list?\n  * What was the tactic that got the person to opt in?\n  * What's the psychology of the buyer?\n  * What triggered the decision to buy?\n  * Who wrote the copy?\n  * Has that writer produced other packages or promotions that perform well with this list or lists like it? (And can you use that writer again for similar offers to similar lists?)\n  * How long was the copy?\n  * What was the format\u2014long form, short form, free trial, sweepstakes?\n  * What was the conversion behavior? Did the person download something, opt in, mail something, make a purchase?\n\nNote that there's a direct correlation between how much prospects engage with the initial promotion and how long they will stay with you. For example, if someone joins with long copy, is highly engaged, and buys promptly, you want to track what kind of promotion he or she came in on and how that differs from the people who came in on short copy. Usually people who buy with short copy are more likely to have made a more impulsive purchase and won't be as wedded to the product, because they haven't read as much about it and haven't engaged as much with your promotion piece.\n\nIn all of this, you are looking for the underlying motivation of the people on your lists. And the more you understand, the more you can segment intelligently and the more you can create precise copy and messaging to those segments.\n\nKEY TAKEAWAYS\n\nWhile this chapter didn't talk about how to write great copy (see the recommended reading at back of this book for that information), you now understand why the final 20 percent of the 41\/39\/20 rule is so critical. Your copy and creative shape how your list experiences your offers and are how you connect with them in a meaningful (and profitable) way.\n\n  * Don't outsource your copy and creative to amateurs. These are critical elements in getting your targeted lists to respond positively to your curated offers.\n  * World-class copywriters share seven characteristics: \n    1. Hunger (their dedication to the craft)\n    2. Curiosity (their obsession with learning about the audience and its history)\n    3. Smarts (the way they surround themselves with people who know what they don't)\n    4. Passion (their deep interest in the material they are writing about)\n    5. Understanding direct marketing principles (to become true advisors and partners on their clients' campaigns and businesses)\n    6. Humility (they'll tell you about their winners, but even more about their losers)\n    7. How they share their success (their generosity)\n  * Copywriters are like canaries in the coal mines: they know what's happening in the market long before anyone else does.\n  * Support your copywriters any way you can. Don't leave them sweating in dirty coal mines if you already have the expertise they need to produce great creative.\n  * Copy is both an art and a science, and sometimes the best material for a campaign will have been left on the cutting room floor before the copywriter even arrived on the scene (so make sure you give them a chance to resurrect it).\n  * There's no such thing as one-size-fits-all creative. You need to communicate with your audience according to the relationship you already have with them and based on their previous behaviors.\n\nCHAPTER 7\n\nMULTICHANNEL MARKETING\n\n_\"Never adapt your technique to the idea; adapt  \nyour idea to the technique.\"_\n\n\u2014 BILL BERNBACH\n\nMany online businesses (and plenty of offline businesses too) default to marketing on one channel, whether out of convenience or because they know one channel above all others. This might be the most dangerous thing you can do in marketing, regardless of why you are doing it. This chapter is about why you should market across different channels and how to measure how each of those channels is performing for you.\n\nI learned from Dan Kennedy a long time ago that the number to avoid in business is 1 (of anything)\u2014and this lesson needs to be top of mind as we focus on the importance of marketing in multiple channels. This is not just a nice idea\u2014it is the key to survival and, therefore, to being able to overdeliver today . . . and on into the future. It's understandable why many marketers will default to one big channel\u2014many of the largest platforms have millions of users with multiple advertising opportunities and advanced reporting technology. Most businesses will find an active part of their audience on any given platform. But just because it's easy to use one huge channel doesn't mean it's your only option. It doesn't even mean it's your _best_ option.\n\nSize _might_ matter\u2014or it might not. Many marketers make the mistake of thinking that if they are not able to promote on the largest platforms (the ones with the most people on them), they will inevitably have trouble scaling their business. I am all for marketing to the biggest universes available, but if those universes are unresponsive to your offer, their size really doesn't matter. In fact, when we look at lifetime value (LTV), we see that it's often smaller, more targeted lists and audiences that are the key to long-term growth. Fewer orders at higher dollar amounts can often be much more profitable than more orders at lower dollar amounts. And you might achieve higher LTV in the process, making that profit even bigger in subsequent years with additional purchases.\n\nWhen we looked at the differences between compiled lists and response lists in direct mail in Chapter 4, we saw firsthand how quality almost always beats quantity. This is a universal principle to keep in mind as we explore all the marketing channels that are available. For some businesses, the most popular channel at any given time, even if it's the biggest, might actually be a _bad_ option. This may be particularly true if you're looking for quality over quantity (if, for example, you're selling a high-ticket item).\n\nI often hear marketers lamenting that they can't convert prospects with similar profiles to the people who have become customers in the past. But those marketers are missing the point. Don't be fooled by what you think customers _should_ do . . . it's all about what they _have done_ in the past. Previous response is the best predictor of future response\u2014which has nothing to do with whether the channel they come from is large or small. There are so many ways to build your list, so many channels to choose from, and putting all your eggs in one basket is never a good idea.\n\nMy good friend and marketing thought leader Michael Fishman says, \"Never define yourself by a channel.\" I didn't call myself a direct mail marketer in the 1980s, and I recommend you don't call yourself an Internet marketer today. Instead, look at yourself as a direct marketer who chooses to market through various channels, offline and online, whatever is best for your products, services, and ideas, always thinking about measurability and profitability. You're not a social media marketer, you're not an e-mail marketer, you're not a direct mail marketer . . . you're a direct response marketer whose most effective channel is social media or e-mail or direct mail. Defining yourself by the channel you use most could prevent you from being open-minded about effective channels that you have not considered.\n\nThis is why diversity is key. As my financial planner once told me: \"While you may not make a killing with my investment philosophy, you will never get killed.\" Of course I want you to make a killing with all your marketing efforts, but I want you to do it with discipline and a mind-set that always emphasizes multiple channels so you don't get killed. We've all heard the horror stories of people running their businesses exclusively on one channel or platform and then getting slapped (even shut down) by a change in a policy or an algorithm.\n\nIf, for example, you run anything that you get paid for on someone else's platform and you do not have complete control, it's always good to have a plan B. I heard one particular story of an entire online business vanishing _literally overnight_ when the e-commerce platform the business was built on (the one and only platform the business was using) woke up one morning (assuming e-commerce platforms sleep) and decided that the online business advertising on their platform wasn't compliant with their rules. This online business had been running smoothly (and making millions) for years exclusively on this one huge e-commerce platform; after being cut off, this business had to start all over again, whether the reasons they got shut down were valid or not. Needless to say, this business now markets on multiple channels.\n\nI'm not saying this happens every day, but again, diversification is key. It's critical to be aware all the time of what you do control and what you don't control as you are building your business. Given how passionate you are about your product or service, and how hard you've worked on it, don't risk it all by using just one channel. Your product deserves to live a long life on different platforms.\n\nYou will also find much greater reach, with a more robust and varied audience, when you don't restrict yourself to a single channel. The trick is in knowing _which_ of those opportunities to focus on, because while you need to be diversified, you also can't chase every shiny \"game-changing media opportunity.\" There is a delicate balance between exploring every potential channel and stretching yourself so thin you can't deliver effectively on any of them.\n\nThat's why meeting your customers where they want to be met is far more important than being on the hottest, most cutting-edge channel. Which, of course, goes back to knowing your list and testing everything with measurable results.\n\nFINDING THE CHANNEL OF CHOICE FOR YOUR CUSTOMERS\n\nVery few people\u2014including the people who make up your lists\u2014go to one place for all their information. For example, when it comes to news and current events, no one is _just_ using Facebook or Twitter or CNN or Reuters or those things called newspapers. Everyone uses multiple sources to gather information, educate themselves, and seek entertainment. No matter what you're selling or who you're selling to, it's up to you to do the research on what approaches will be most effective. Remember the wise advice of Bill Bernbach that opened this chapter\u2014adapt your idea to the technique, not the technique to your idea.\n\nAt the time of this writing, social media seems to be where everybody is hanging out the most. People want to keep up with their friends and to keep an eye on the various groups they've joined and the pages they've liked. Maybe they also use this medium to get their news and to find out about cool things to do or buy. Now, many of those same people will also check their e-mail and, believe it or not, walk out to the end of their driveway to check what's in the mailbox too (and that one is the least crowded inbox you could send to today). In addition, many of these people might read newspapers, magazines, and books.\n\nAnd that's not all. They also watch TV, listen to the radio and podcasts, go to the movies, use apps on their phones . . . you get the idea. If you're trying to reach your future customers in only one place, I hope I've made the case that you might want to expand your thinking.\n\nKey Questions (& Research to Consider) to Determine the Channels of Choice for Your Core Market\n\n  * What is the biggest pain or problem your core market is facing? Where do they talk about it? (And can you \"eavesdrop\" there or, even better, advertise there?)\n  * Where are your competitors advertising? And, from what you can tell, where are they receiving the highest engagement from their audience (which is also your audience)?\n  * On your competitors' best channels, can you find out anything about how those people got there in the first place?\n  * Where else does your audience get their information before they buy, and where else do they do their research?\n  * What's the demographic profile of the audience? How old are they? With what platforms are they most familiar and comfortable?\n  * What's the psychographic profile of the audience? What other things are they interested in, and how do they self-identify?\n  * How do they pass their time? How do they inform, educate, and entertain themselves?\n\nDeveloping a profile like this for each customer segment can highlight the opportunities open to you across multiple channels.\n\nWhether it's through launching ad campaigns on a different platform, building partnerships with complementary businesses, or moving into physical marketing (direct mail isn't dead!), building out this deep understanding is what will give you leverage.\n\nNow, there is a lot to think about when you're diversifying, because the opportunities are, dare I say, infinite\u2014and getting overwhelmed is a real threat here. It reminds me of a simple yet profound quote I heard from an investment banker addressing the increased value of companies that have diversified their media, and those that offer best in class and unique offerings in one medium: \"Advertising opportunities are now infinite.\" Marketers are at risk of paralysis by analysis because of all the options in front of them. Don't let yourself fall into that trap. Do the deep dive on where and how your customers spend their time and attention, and then ignore everything else.\n\nTHE DANGERS OF ONE-STOP SHOPPING\n\nIn the interest of keeping yourself sane and using your energy in your unique zone of genius, hire the best people working on each of your chosen channels to ensure your business makes the most of each one. I've said before that this is the best time ever to be a marketer, but with all the new opportunities also comes additional risk\u2014and not just from being overwhelmed. With greater specialization in every channel comes more opportunity for charlatans and imposters too. So while we want to dive in and test every new thing we hear about, we also need to be very careful.\n\nDick Benson once told me his simple strategy for mitigating these risks: \"Never buy everything in one place.\" Back in the 1980s, with far fewer media choices, I still bought media and creative (and all marketing services) \u00e0 la carte . . . and it's even more important to buy that way today when the choices are infinite. After spending my entire career knowing exactly where to find the experts I needed in every aspect of direct marketing, the fact that the choices are now more vast and more specialized than ever got me thinking even more about the dangers of one-stop shopping. I wouldn't do it then, and I certainly wouldn't do it now.\n\nI don't want to make any advertising or marketing agency folks feel bad, but even in the prehistoric era of \"finite\" advertising opportunities, I never met an agency of any kind that I liked very much, because there was no way they could be that good at everything they said they offered. It wasn't about the people, just the model. Often there was one thing they specialized in that I could trust them to do, but there was no way I would put all my eggs in a basket labeled \"full-service direct marketing agency.\"\n\nI went to a specific source to rent lists for direct mail, to another source who knew more about space ads in newspapers than anyone else, to another who just placed inserts in outgoing packages from catalogers (what we called PIPs or package insert programs), and to yet another to buy broadcast (radio and TV). Even within something like broadcast, there were multiple specialists: remnant radio ads were different from other kinds of radio buys, and short-form TV was different from long-form infomercials. Buying broadcast could include four different vendors . . . or more. And don't get me started about buying copy and creative in one place or as part of a full-service model.\n\nWhen it came to hiring the best copywriters, it was painfully obvious that any copywriter who was working in an agency (when they could be making 10 times their salary as a successful freelancer if they were really good) was not at the top of my list as a prospect to write for me. (Of course, if you found Gary Bencivenga while he was still a pup at the ad agency Ogilvy & Mather, that's a win.) As I hope became clear in Chapter 6, copywriting is not a commodity, it is a specialty, and it must be bought from specialists who are not claiming to be able to write everything for everybody.\n\n\u00c0 la carte buying was, and still should be, a rule of thumb for everything in marketing, not just media and creative. Database building, sophisticated modelling and list segmentation, purchasing, production, printing\u2014every role requires experts. Some are more commoditized than others (e.g., printing) and some are super specialized (e.g., proprietary software to build a customer profile) . . . but why not always seek out best in class? Even the most commoditized functions are often differentiated by things like superior customer service or personalized solutions inside a basic offering.\n\nAlthough this chapter is specifically about multichannel marketing, I want to stress the importance of choosing specialists over generalists. This is how you will maximize each channel\u2014by bringing in an expert for each one. That was true when there were far fewer channels, and it's even more important today when marketing opportunities and technology are both confusing and dizzying. Anyone who tells you they can handle all your marketing needs in their \"one-stop shop\" is someone you should run away from fairly quickly.\n\nAnd as technology gets even more sophisticated and makes us more efficient\u2014and we see the arrival of new channels (i.e., new ways to reach people with our marketing messages), there will be even more specialization . . . which leads to even more niche areas of expertise. With so many people getting so smart in tighter and tighter niches, it would be a shame not to make the most of these categories of one\u2014they've spent so much time becoming the world's best in their \"thing,\" you should want them on your team.\n\nI believe multichannel marketing is the \"make or break\" in terms of how you sell and market in a world that is as noisy as ever . . . and that demands excellence as much as ever too. It is the core premise I start with when I build the makeup and content for my mastermind groups and any consulting I do today. It's not _just_ about being multichannel . . . I want you to be multichannel with the best experts working for you on every one of the channels you choose.\n\nBegin with intelligent outsourcing, and then buy \u00e0 la carte from specialists. Dick Benson, the man who mentored me in a world where advertising opportunities were still finite, drilled this into my head so I could share it with you today: \"Do what you do best and buy everything else around the corner . . . and never settle for second best.\"\n\nMULTICHANNEL METRICS\n\nIn case you haven't picked up on this yet, direct response marketing is all about measurement. John Wanamaker, considered a pioneer in advertising and marketing in the late 19th century and early 20th once said, \"Half the money I spend on advertising is wasted; the trouble is, I don't know which half.\" If Wanamaker had the measurement tools and discipline we have at our disposal today, he could have made back some of that \"wasted money.\" It's all about putting out a campaign, being able to track a specific response from it, and then determining your next move(s). You never want to be stuck wondering what's working.\n\nThis is why I have very little tolerance for general advertising or marketing that is not accountable and measurable, complete with metrics that indicate whether each campaign was actually successful. And success is not only in the eye of the beholder either. It also needs to hold up in the eye of other key players, such as your accountant and your analyst. It's up to you to figure out whether the media you're using is working for you\u2014and that's what we want to explore together now.\n\nFor a campaign to be considered successful, or for any medium to be used continually, it needs to make money\u2014enough money to justify its ongoing existence in your media mix. There's no way to know if a campaign or channel meets that criteria unless you are tracking some key metrics. Once you establish those key metrics for every campaign and every channel, it's time to scale.\n\nLifetime Value (LTV)\n\nLifetime value is, above every other metric, the one number I urge you to focus on most. The lifetime value of a customer is the dollar amount you can expect to sell to a single customer over the course of your entire relationship with them.\n\nLet's talk about LTV with a real-life and fairly simple case history. I remember sitting in the audience at a mastermind meeting a few years ago, before everyone had made _funnel_ the most important marketing term ever. It was also before Facebook had figured out how robust their advertising model could be, so it was still super cheap to run Facebook ads.\n\nOne of the young, dynamic marketers in my group diagrammed on a flip chart how he sold a $10 report on Facebook and then created a funnel to convert that $10 buyer into a raving fan (and multibuyer) over time\u2014making Facebook a key component in his marketing media mix, way ahead of the curve. His competitors, offering a similar $10 product, would usually receive very few sales early on (not understanding the quantity versus quality argument we talked about earlier). When the initial revenue from those sales did not equal the total advertising spend for the Facebook ads, they would jump ship to find even cheaper media. Their inability to figure out a way to make this channel work simply eliminated them as competition, due to their failure to understand lifetime value.\n\nSo what was the difference? My guy figured out that with these low-ticket buyers, the key was to determine who would buy a second product related to the first (for a little higher price), and then create a third and a fourth offer in an intelligent ascension model with price increases that made sense.\n\nHe tested dozens of subsequent offers and price points, again showing he was a student of direct marketing. And while he never came close to getting his money back on the _first_ sale (or even the second), through all his testing, he created a series of offers that eventually led to an average sale of over $70 from a new $10 buyer.\n\nBecause he tracked the average lifetime value of the new buyers well past their second order, he was able to make back enough money to pay for the advertising within a reasonable period of time and without making cash flow difficult. He was also able to buy a lot more media from Facebook with confidence that it would be profitable in an acceptable period of time for him. (Remember what I mentioned previously when we talked about bogeys\u2014it's up to _you_ to determine what is acceptable, and you can't do that without a firm command of your numbers and understanding LTV.)\n\nMore importantly, our young gun was building a _business_ , not simply looking for a way to be a one-hit wonder, making quick cash on a single sale. While many of his competitors were lamenting that they couldn't make Facebook advertising work selling a $10 product, he was buying the ad space for his $10 product that could have been _their_ ad space for _their_ $10 product. He was continuing successfully while his competitors looked elsewhere for media that \"worked.\"\n\nHis biggest competitive advantage was that he understood direct marketing principles while others who were advertising on Facebook at the time did not. But it wasn't just that\u2014this guy understood that the first purchase is just the tip of the iceberg and that it's usually not the first sale that makes your money back, it's about all the sales you make throughout the lifetime of that customer.\n\nLet me give you another example of creating serious upside by doing an accurate lifetime value calculation. A while ago, I started consulting with a company that was doing well selling almost exclusively through referrals. However, when I asked them about their existing customer base\u2014people who paid between $10,000 and $20,000 to be in a specialized program\u2014they told me that the average person stayed with them for three years.\n\nI did some quick LTV math and showed them that they could project $30,000 to $60,000 of revenue for each new client. I told them it was fine to continue with their referrals strategy\u2014it was clearly working\u2014but I encouraged them to not make it a stand-alone strategy. With such a robust lifetime value, we started talking about and diving deeper into other channels that might not be as profitable initially but would be a way to bring in more $10,000 to $20,000 clients. And they realized they had some significant opportunities to grow.\n\nThe chances are that other channels might not yield new clients who stick around for three years; however, even if \"inferior\" media yielded new clients who didn't last that long, my guess is that they would still be very profitable. Of course, we would test and measure everything to confirm my hunch. But we were able to agree that referrals was not _the_ marketing strategy but just one of many (once we understood the real and potential LTV of new clients).\n\nThe Real Cost to Acquire a New Customer\n\nTo keep your business afloat and each marketing channel profitable, you have to understand how much you need to spend to acquire a profitable customer on that channel. Think of it this way: whenever you spend money on a marketing channel, you are \"buying\" a new customer. When you calculate the price tag for each customer, you must include the cost of producing each unit you sell and ship to them if it's physical product, including materials, product development, storage, shipment, staff costs, taxes, and\u2014if you really want to go deep and account for everything\u2014depreciation and amortization of any debts. Failure to account for these costs of acquisition can kill your business slowly and painfully, and it will be difficult to identify the problem. The business will gradually bleed out because what you earn from your customers will not cover all your costs.\n\nA successful entrepreneur once told me that the most important person in his organization, an organization that prided itself on being a world-class marketer, was his analyst, or \"numbers guy.\" Without that person dissecting response rates, cost of acquisition, and lifetime value, he knew he could never grow. (And \"dissecting\" includes making recommendations on how to expand in each media channel intelligently.)\n\nIn my past life, mostly in the world of direct mail and physical product, accounting for every penny in terms of media, postage, printing, and cost of goods sold was one of the most important things we did. And while you might not be worried about these things if you only deliver digital products, I guarantee there are all sorts of costs that need to be loaded up before you determine whether a specific media channel is working.\n\nAttribution\n\nBeing a great multichannel marketer can cause some headaches, and attribution is a big one. But if those headaches lead to increased profits, I always prefer to deposit more money in the bank and just take an extra aspirin.\n\nAttribution is the way you track a customer's journey through their experience with you from the first time they interact with you until the point that they put down money to become a customer. This allows you to see the channel each customer comes from, the path they followed that influenced them to convert, and how each channel performs over time. Most businesses will have multiple touchpoints along the customer journey, often across different channels, so the variables in any campaign can become quite complex if you are trying to optimize across each touchpoint. Attribution ensures you're able to simplify that complex web of interactions into a list of steps to be addressed.\n\nThe real value in having clear attribution is that you can track the performance of each channel over time, including the performance when you introduce a new campaign or control. It can also show you the lifetime value of a customer based on the channel they came from\u2014for example, you might find that customers from your e-mail list have a higher lifetime value than customers that come through affiliate offers. This lets you see where to spend more (or less) money and can highlight new opportunities for each channel. The extent that you can attribute every order or sale that you ever make to a particular channel will help you maximize the profitability of that channel.\n\nIn direct mail, we knew where just about every order was coming from by individual list. That enabled us to easily attribute orders to specific sources, and we could make media decisions with the utmost confidence. Even in an era when technology was less advanced than it is today, we always had a key code on each piece of return mail that told us which campaign it came from and what list the response came from too.\n\nBut as we expanded into new channels, such as our infomercial television program, the lines became more blurry. For example, how do we attribute the order of someone who saw our infomercial and decided not to call in an order from the 800 number on the screen but then received our direct mail piece for the same product three weeks later and ordered? This person would now be a direct mail buyer, not a TV buyer, but would they have bought had they not seen the infomercial? And my problems around attribution were not nearly as complicated as a $1.8 billion business (Guthy-Renker) in many more channels than I was playing in. When I interviewed Greg Renker at the Titans of Direct Response event, he shared his experience.\n\nAt the time, Guthy-Renker was running ads on TV, sending direct mail, and hosting kiosks in malls, as well as creating e-mail funnels, online display advertising, and more. Everything was working together to bring in orders, so trying to identify where an order really originated was very difficult, even if they had added an identifier indicating where every order came from (such as a specific 800 number, key code on a mailing label, or source code on a coupon). Even an enormous company struggled with this (although since that time, a few tools have been created that make at least the online part of attribution much easier).\n\nRegardless, you need to at least _try_ to attribute each order to a channel so that you can assess the effectiveness of your campaigns on each channel. And it's okay if you have to split the attribution\u2014some channel (or combination of channels) need to get credit for the order, even if it's just a piece of the order. Put procedures in place across every channel with a way to track response to media. If the customer calls or sends an e-mail or signs up in person, they should get a specific type of order number. If they see a commercial on TV, an 800 number should be set up for that specific commercial.\n\nOf course, you can't account for everything. If the customer saw an infomercial and was sold on the product, forgot to call in an order, and then remembered when they saw the kiosk at the mall, should the infomercial get the credit or the kiosk? You can train kiosk staff to ask customers where they first heard about the product, and so give you a chance to split the attribution, but this is far from a precise science. However, it's the best you can do, and some data here is better than none.\n\nDo everything you can to get that attribution so you can stay within your channels with confidence, knowing they are paying out. Remember that this is the foundation of direct marketing, and everything has to be measurable\u2014it's not enough just to have a good feeling. Far too often, attribution is the metric that gets missed. In fact, I speak to marketers all the time who have never even thought about attribution . . . but without it, all the other data they collect is flawed.\n\n\"Conversion\" Defined & Why It's Your Best Friend\n\nWhen I understood the power of conversions, I began to understand the true power of direct marketing. In the subscription world, the conversion was the first renewal, not the initial purchase. We started with a free trial offer to our newsletters and got a certain amount of those folks to pay us for a one-year subscription. And of those folks, those who \"converted\" to a second year (i.e., made their second purchase) became the gold in our business.\n\nToday in marketing, the term \"conversion\" is thrown around too freely. It's often confused with acquisition\u2014that is, when a marketer goes into a new channel and gets someone to pay them once, they consider it a conversion. But I don't want to get caught up in semantics, and if a conversion for you is your first sale, so be it. However, if that's the case, it's time to track the conversions on your conversions.\n\nI can't emphasize enough how important it is to record all the activity of every new customer as soon as they enter your orbit. When you acquire a new customer, it's exhilarating. I've never met a marketer who doesn't get a thrill when someone sends them real money for a product or service they believe in. But all new customers are not alike . . . some will never buy a second product from you, some will buy anything you ever offer, and some will end up somewhere in between.\n\nKnowing where every one of those new customers came from and then tracking their behavior over time will enable you to determine the sources that your best new customers came from, all based on conversion. And then track carefully the renewal (i.e., next order, or \"conversion of conversion\") and so on. In addition, as a marketer you will have a lot more fun by interacting and making attractive offers to the converted (the people who have bought from you more than once) than the nonconverted, although I know it sometimes feels more exciting to get that first sale.\n\nLifetime value is the key calculation to determine what media to buy and how much of it to buy. Your focus should always be on how long a customer stays with you and how much they spend with you. And that you are calculating _all_ costs associated with acquiring and keeping (servicing) that customer. Remember, pay more attention to the people who already love you (existing customers, including conversions) versus those you are trying to attract for the first time (new customers, or acquisitions). When it comes to multichannel marketing and diversifying your portfolio, there is nothing more important than knowing all these numbers and having them at your fingertips all the time.\n\nONLINE TO OFFLINE TO ONLINE (O to O to O)\n\nDespite all the challenges around attribution and calculating an accurate LTV for every new customer, the opportunities available to us in combining online and offline media are too rich to ignore. Online to offline to online (or offline to online to offline, or any combination that fits for your audience) is a concept that I believe not enough people are talking about.\n\nTo define O to O to O, let me state it this way: Always be aware of the life cycle of a new customer entering your world. They can start in either O (online or offline), then be directed to the alternate, and then pulled back to the original source (O) where they started.\n\nThe ability to combine today's technology with the fundamentals of direct marketing and direct response learned over decades of testing is exciting. For example, bringing a highly qualified buyer or prospect from a direct mail piece or a print ad into an online funnel will certainly create customers with higher lifetime value.\n\nMany marketers who work online have told me that their customers originating in a variety of offline media have a higher LTV than those who first engaged with them online. And the marketers who have figured out that _starting_ online doesn't mean they have to be _exclusively_ online have had some incredible results by going offline.\n\nFor example, using direct mail on the back end of a winning offer from e-mail or the Web can create more intimacy and value with those customers, which is often a winning formula. Great examples of this can be found with numerous health offers geared toward an older audience who love direct mail but can be engaged initially in an online channel. They may then buy in the future from either channel, or both. Again, this is all about meeting the customer where they want to be met. And don't forget about direct response television, space advertising, or inbound and outbound telemarketing to weave in and out of different kinds of offers, whether they start online or offline.\n\nThe breakthrough really comes when you get consumers to transfer from one medium to another, regardless of whether the medium is online or offline. You then have command of all the different media, and you are in a better position to monetize in a variety of different ways. Many marketers who you would probably classify as \"online\"\u2014that is, they send traffic to a landing page\u2014offer a free copy of their book (plus shipping). They send that book all over the country and even overseas, probably losing money on that kind of an offer and even the first sale. But shipping a book means they now have a physical mailing address.\n\nThese savvy direct marketers initiate using an online offer to get into the direct mail business. Having the physical addresses of their customers means that they can make offers in the mail, where fewer competitors are probably making their offers. Those physical offers (in this case a real book) are so valuable that customers are often motivated to move through to other products and services offered by the marketer, many much higher priced. The marketer has the choice to meet the customer in the medium they are most comfortable with.\n\nHaving spent most of my career marketing to an \"older demographic\" (folks 55-plus in age and even 65-plus and 75-plus), I have heard hundreds of times from marketers and nonmarketers alike, \"You better start marketing to a younger audience since your audience will die off soon.\" While I didn't ignore them, I looked at it a little differently (in the spirit of meeting customers where they want and need to be met): I love marketing to an older demographic\u2014they have the two most important things a younger demographic doesn't have: time and money. I don't want my customer base to get younger . . . I just want to be prepared for the next group of 55-year-olds when they are ready for my products and services.\n\nRemember the case histories I told you about estate planning and diabetes? While those are ostensibly topics for \"older people,\" the 35-year-olds today will need the same information on those topics when they are 55 (and dare I say, they should be thinking about \"pre-estate planning\" and prediabetes long before that). But regardless, when your audience is ready for the information (whatever you might be selling), my opinion is that you then deliver it in the format and media they want to receive it. That's another take on multichannel and \"O to O to O.\"\n\nThinking about how online and offline can work together, you can see how multichannel marketers work at an extremely high level. They know their initial costs and they know how much they are willing to invest to bring a long-term customer on board. And they can do it online or offline, or both, using techniques and strategies that truly give them an unfair advantage. They may even make their first offer on television or at a live event. No matter what, their initial sale (whether it's a book, a course, or something else) becomes the gateway into their ascension program, and they have the ability to use multiple channels and platforms, online and offline, as part of that program.\n\nMultichannel marketing is really the wave of the future\u2014it engages your customers in all the places that they spend their time (online and offline)\u2014and protects your business against unpredictable events in the business environment. It makes you a more robust marketer and opens up all kinds of opportunities that you just can't access when you're only focused on a single channel.\n\nKEY TAKEAWAYS\n\nMultichannel marketing is a critical element in maximizing your business growth, protecting yourself from unexpected downturns, and getting the kind of payoff you really want to see for yourself and your customers.\n\n  * As Dan Kennedy says, the most dangerous number in business is 1. If you are only marketing on one channel, you are risking the long-term viability of your business.\n  * Diversifying the channels you use for your marketing\u2014and the media mix you deploy across them\u2014protects your downside. It guards against being wiped out by an unfavorable change in a law or algorithm by always having more than one way to get profitable customers.\n  * The biggest channels are not necessarily the most responsive or profitable\u2014a smaller universe with a higher LTV is always preferable to a larger universe with a lower LTV.\n  * Your audience does not use just one channel to get all their information. Work out their chosen mix of channels, and market across all of them as appropriate.\n  * Avoid \"one-stop shopping\" for your marketing at all costs. There are so many different areas of expertise in direct marketing that no one can master them all; buy \u00e0 la carte to ensure you are working with the best providers in every area.\n  * Key metrics to track across all parts of your multichannel marketing strategy include: \n    * LTV\n    * Cost to acquire customers (including cost of goods sold)\n    * Attribution\n    * Conversion (not to be confused with acquisition)\n  * Take your customers from online channels to offline channels and back again to maximize their engagement and LTV with you.\n\nCHAPTER 8\n\nCUSTOMER SERVICE & FULFILLMENT\n\n_\"Marketing by walking around is a requirement, not a choice\"_\n\nIn the early 1980s, baseball writer Thomas Boswell wrote a book called _How Life Imitates the World Series._ As a lifelong, diehard baseball fan, I loved this title. Baseball, like direct marketing, is a big part of my life, and I often look for ways where my two passions collide. It happens surprisingly often.\n\nI am not just a casual fan of the game (or of this direct marketing thing either). I was a pretty good catcher in Little League\u2014a legend in my own mind at 12 years old. But in youth baseball, when you turn 13, you go from a miniature field to a much bigger one\u2014the same size field major league players play on.\n\nBeing a fat kid (\"husky\" to my family), that big field was a problem for me. I was quite slow, and navigating all that real estate just to get on base was a problem. And while I could still catch, the distances to throw the ball were too far for me as well. But I still loved the game and wanted to be part of it even if my body type (and ability) was standing in the way of my future baseball career.\n\nHow could I get on the field competently? That's when I became an umpire, often the most hated man on the field, but in my mind, I could be the savior, creating order out of chaos, especially with little leaguers. I knew they needed me, even if _they_ didn't know it. I loved the game, I knew the rules, I had a loud voice, and being a little husky and slow was not an issue. In fact, it seemed like all the umpires I ever saw on TV were fat.\n\nBut umpiring gave me much more than just a hobby (inside a passion) and, as it turned out, a way to avoid obesity\u2014it is also related to my passion for direct marketing in more ways than I ever would have imagined.\n\nHOW BASEBALL IMITATES DIRECT MARKETING\n\nBack to why I love the title, _How Life Imitates the World Series._ I became an umpire at 16 years old and have done it my entire life: Little League, high school level, men's and women's softball. My dream is to umpire at the Little League World Series. You probably also have passions in your life that, on the surface, have no relationship to your \"work\" . . . however, ask anyone about their hobbies, interests, exercise routines, etc., and they will tell you how the things they do outside of work give them more focus when they are back at their desk. I maintain that\u2014as it said on the back of Marty Edelston's business card\u2014\"the master in the art of living makes little distinction between his work and his play.\" But for this part of the discussion, let's assume there is a distinction. At least a little one.\n\nAs with all the activities in your life, the concentration required to do something well is demanding\u2014whether it's being a lawyer, an accountant, or a dentist, or if it's mountain biking, skiing, or yoga\u2014and in all cases and situations, distractions will lead to failure. In the case of your livelihood, you could lose a client, have a downturn in business, or even get fired from a job; in sports, it could lead to physical injury.\n\nWith umpiring, while you could get hit in the head with a baseball if you are daydreaming, the real injuries come when you get yelled at for missing a call. I know you might get a correction during an imperfect downward-facing dog in a yoga class, but missing an obvious call in a baseball game will have a coach loudly arguing with you\u2014in your face and possibly kicking dirt at you\u2014while the parent of the kid who was the victim of your awful call will be screaming from the stands about what a despicable person you are.\n\nMaybe I should take up yoga.\n\nA copywriter friend once asked me, \"Why would _anyone_ want to be an umpire?\"\n\nThere had to be something motivating me to be abused so much, and I came up with three answers. The first two I have already touched on:\n\n  1. I love the focus required to do it well.\n  2. I love creating order from chaos by applying established principles (in this case to the game of baseball).\n  3. I love that underneath it all, umpiring is about customer service and fulfillment.\n\nThis chapter is about #3, and when I say fulfillment here, I am not talking about my personal fulfillment but rather fulfillment for the customer. No one comes to a baseball game to see the umpire, just as no one goes to any sporting event to see the referee or official of that game. The best an umpire can get at the end of the game is \"Nice game, ump.\" But the moment an umpire makes a mistake, all hell breaks loose.\n\nIf you have ever worked in customer service, fulfillment, or any part of a business that is all about making the behind-the-scenes stuff seamless, I think you might be getting the connection. Most customers don't interact with a company just to experience their customer service and fulfillment. They are buying, receiving, and enjoying their product (sort of like playing in a baseball game on a beautiful spring day), but if something is not right with the experience, it will be time to get satisfaction for the wrong that has been inflicted upon them.\n\nNo one wants to feel wronged\u2014and if you don't make it right for them (quickly), I guarantee that the lifetime value of this customer will decrease significantly, maybe even to nothing, or less than nothing. This is the marketing after the marketing\u2014and when we screw up on delivering for our customers, it is absolutely the worst thing we can do in business. As I've said in other ways throughout this book already, it's much easier to keep a customer happy (and increase their lifetime value) than to get a new customer . . . although it does feel sexier to get new customers all the time.\n\nSaving a loyal customer after they feel you have not fulfilled your promise (at any point in the relationship) is your most important sales function. You do this by making world-class customer service your highest priority. While I worried about the title of this book being _Overdeliver_ , worried that potential readers might think it is only about delivering impeccable customer service, I decided to go with it because it is such a core principle of marketing. And now you know that overdelivering is also much more than this.\n\nThe fact that so many marketers today put up with 30 percent to 50 percent return rates on an initial sale (I guess because keeping 50 percent to 70 percent of new customers is good enough) makes me cringe. And even if accepting that return rate on the initial sale is the prevailing philosophy, these marketers still don't spend enough time caring for and nurturing the folks who _don't_ return the product initially, losing them later by not being laser focused on doing everything to keep these customers happy and coming back for more.\n\nHaving a total commitment to the highest quality (and integrity) on the initial sale is the most important first step . . . but that same mind-set needs to be in place with all customers throughout the lifetime of their relationship with you. And when you are thinking about customer service and fulfillment, it's even more important to have this maxim front and center all the time, one that I made up and one that I try to live by (and which you have seen at the opening to Chapter 1): \"Everything in business (and in life) is not a revenue event, but everything is a relationship event.\"\n\nHere's another way I like to put this as it pertains to customer service: Lifetime value increases the longer someone loves you and wants to buy from you in the future like they have in the past, but this requires you to take care of them like they are family, even without any exchange of funds.\n\nUnfortunately, too many entrepreneurs and business owners treat their customer service and fulfillment as afterthoughts, and not just the function but the people they hire for those jobs too. These shortsighted executives act as though the actual real-world interaction with real-life people who buy from them doesn't matter once a sale is made. This is a big mistake. Alas, the fulfillment manager or customer service rep, on their _best_ day, will get the feedback equivalent of \"Nice game, ump.\" If there are no problems, these folks remain invisible. But when they blow a call (i.e., make a mistake) they become the least popular employees in the company.\n\nIt takes a certain kind of personality to umpire . . . or to be a fulfillment manager or customer service representative. Taking pride in giving exquisite service\u2014and being willing to do it without being noticed\u2014is extremely rare. These are the people who overdeliver at the highest level. Pay close attention to people you hire for these roles, and pay them well too. They are your first line of defense in protecting against leakage of your existing and potential customers, and they are your last line of defense against customers leaving you forever.\n\nThe wonderful telephone operator who solves a customer's problem behind the scenes and saves an order or avoids a cancellation feels like they are part of sales and revenue creation (or what we can call revenue saving). And the delighted customer, who receives a positive outcome, will have their day made and might just become a customer for life. The folks I know who are the best at customer service receive satisfaction in saving the day, whether it's noticed by one or by many. But while they may not be looking for positive reinforcement, go out of your way to applaud them. Give them accolades for their excellent skill at keeping customers; they deserve this just as much as the sales rep or copywriter deserves praise for figuring out creative ways to add new customers.\n\nMake sure you look after every part of your customer service and fulfillment, because the only way to stay in business for the long haul is to focus relentlessly on keeping your customers happy. And that includes keeping your employees happy too. Even if things go wrong, if you are committed to making things right and saving every customer with every interaction, no matter how negative or potentially damaging, you will build resilience into your business\u2014because, as we know, all businesses thrive on repeat purchases.\n\nYou can even base your entire differentiation strategy as a company on how you treat your customers. Look at such well-known retail businesses as Nordstrom, Stew Leonard's, Ritz Carlton\u2014they are differentiated in extremely competitive industries due to their unrivalled customer service. I love tracking what companies have done to create superior levels of service for their best customers, especially with commoditized products. Premium services such as Amazon Prime and the American Express Black Card overdeliver and create customer experiences like no one else.\n\nCreating what is called a barrier to switch is critical and will be discussed later in this chapter. For now, I will just say that your product or service is only a commodity if you let it be a commodity. Now let's look at some examples that prove why customer service and fulfillment are marketing functions (although I hope I have made that case already).\n\nIT'S EASIER TO KEEP A CUSTOMER THAN TO GET A NEW ONE\n\nI don't mind repeating this tenet of direct marketing: it's easier to keep a customer than to get a new one. It really is that important. And I learned it by staying late at the office to field calls from angry customers.\n\nMy theory is that if you can overdeliver and transform a customer's bad experience into something positive, almost making it a game of sorts, you turn them into a loyal fan for life. The lessons you can learn about human behavior by going deep with one unhappy customer can teach you lessons you can apply to multiple customers with the same (or similar) issues. Never assume you are making only one mistake in the eyes of each angry customer; rather, assume that you are making many mistakes all the time and that learning what it will take to save each customer is research you can take to the bank.\n\nAs Claude Hopkins said in terms of attracting new customers, \"We cannot go after thousands of men until we learn how to win one.\" This applies to _all_ customers, including those we have gone after and won already. And about them, I like to say we cannot understand why thousands of customers stay with us until we learn why one would consider leaving us.\n\nNo problem, issue, or complaint is too small. All complaints are relevant, even if you think the customer is being unreasonable or trying to take advantage of you. And there's always an opportunity to learn about your audience and to spot the holes in your process that are leaving you exposed.\n\nI used to stay late at the office to answer the phone. The only people who called after 8:00 P.M. were angry customers, and I heard every complaint under the sun\u2014from missing issues of a newsletter to irritation that a book had not arrived yet. There was one person whose dog ate pages 17 and 18 of the special bonus that contained the most important secret regarding treatment of their type 2 diabetes, and I recall a sad story about the postman who left a much anticipated book in the snow.\n\nEverything our customer service folks hear needs to be heard and documented and shared with everyone on our marketing teams. Even things that on the surface seem to be beyond our control. Listening and interacting with your customers will give you far deeper insights than any spreadsheet or report on the makeup of your database. This is what I mean by \"marketing by walking around.\" It could also be \"marketing by listening more to your customers,\" or \"marketing by talking more to your customers\" or even \"marketing by eavesdropping on your customers\"\u2014but only in the most ethical way and always with the goal to become more valuable to them.\n\nAs an owner, there are so many things that are not in your direct line of vision. Yes, you should read reports from customer service and monitor complaints regularly, but there is nothing like talking to someone who gave you their money (and their trust) when they might want you to give them their money back. Issues that look like minutia on the customer service side can end up having big implications for how you make decisions in the future.\n\nI learned this by accident, staying late. But that experience led me to making it mandatory for marketing personnel to sit in on customer service calls on a regular basis. The valuable firsthand lessons you learn by paying attention to the bad news (which you will hate hearing at the time) will very often lead to making positive changes in your business.\n\nOne obvious thing I learned through my experience taking calls late at night was that what starts as confrontational can quickly be diffused through generosity of speech and spirit (i.e., not arguing but just listening and acknowledging their position). This can be followed up with giving away bonuses and unexpected extras as a thank-you for bringing the issue to your attention.\n\nAs you may recall from the example of going from 4 bonuses to 100, I believe this generosity mind-set that was part of both the front-end marketing and the back-end marketing enabled us to create more premiums and bonuses in our acquisition packages and then always encouraged us to give as much as possible later on if there was any dissatisfaction. This cut down on many kinds of complaints. Customers were always getting more than they ever would have expected up front. If they were dissatisfied later on, they were far less confrontational and their problems were easier to resolve. And we still overdelivered on the back end too.\n\nWhenever I picked up the phone and heard an angry customer, I was ready to play the game that every direct marketer must master: to overdeliver and give the customer more than they would ever expect. It's an opportunity for you to discover the unanswered questions, to see their unexpected perspectives, and to shine a light on the unmet fears and desires of your audience.\n\nThe key then is to react quickly and to just keep saying yes until the customer is satisfied. This is how you learn what holes need to be plugged for _all_ your customers, some of whom are too timid to make the call and are suffering in silence. Always assume that there are many more people with the same problem as the person who was bold enough to make an issue of it. There will always be a silent group waiting for a solution. This exercise can give you insight you cannot easily get anywhere else. And if your evening automated attendant doesn't allow you to talk to real people, or if all your customer service is outsourced, make it a practice to spend some time listening in to what your customers are telling the folks you have entrusted to be on your front lines.\n\nAll this ties in to that critical direct marketing rule of thumb: it is always easier to keep a customer than to get a new one. Remember the adage that marketers sell subscriptions, but editors sell renewals. Remember that this applies way beyond subscription offers\u2014this lesson is universal. Another way to express this concept beyond subscription marketing: Marketers sell the first order; product developers, content creators, _and customer service and fulfillment people_ sell renewals.\n\nIf you don't truly deliver the product or service you sell in the initial promotion, you can resell later until you are blue in the face and you won't get a renewal or repeat order. And even if you deliver what you promised on the front end, remember that the relationship is just beginning. Make sure the customer loves that first product, and continue offering the same quality when you sell them subsequent products. In between all that selling, however, the communication channels have to stay open all the time in order to satisfy every customer who, for example, doesn't get the bonus you promised or receives one of your amazing products three weeks later than you said you would deliver it. It's all connected. And of course, be ready to answer the phone when they call and need something more.\n\nTHE SECRETS OF SECRET SHOPPERS\n\nAs I mentioned earlier, many insightful entrepreneurs hire secret shoppers (people on the payroll _outside_ the company to go through every aspect of their sales and marketing operation) to find out where there might be a broken link or a hole in the operation that no one would ever see without this anonymous monitored buying. One brilliant entrepreneur had his secret shopper answer one question\u2014and one question only\u2014at every step of the process as they experienced the organization's sales funnel: \"How does that make me feel?\" This secret shopper bought everything, returned a lot, complained a lot, got on the phone with customer service people, sent e-mails\u2014basically everything they could possibly do to stress the system as much as possible. Whenever the answer to the \"how did they feel\" question was anything short of thrilled (or at the least very satisfied), there were tweaks to be made in the system.\n\nAs soon as the customer feels like you're not interested in them, they are gone. They will take their money and go elsewhere. And in a world where people spend time sharing pictures of their dinner on social media, they certainly will make their displeasure known to many more people than just your customer service rep if things go wrong. Technology and the ability to share a lot more with a lot more people has quickly made impeccable customer service even more of a requirement for every business today. In our interconnected world, bad news travels faster than ever before. I don't have to emphasize how damaging bad word-of-mouth can be\u2014it obviously brings your LTV way down, it makes the job tougher for your front-end\/acquisition marketers and copywriters to sell to new customers, and it compromises your ability to deliver on what you've promised, assuming the bad news about the customer experience you deliver becomes more widespread.\n\nOne secret shopper story I heard reminds me to add that you must put your ego on the back burner to get information you can use and apply. This entrepreneur heard that after his expert sales people ascended customers at a live event into a high-priced coaching program (at $15,000 a year)\u2014after they had only bought much lower-priced products and services previously\u2014it was often four full days after the event before someone from his company would contact this new VIP customer.\n\nHe asked his sales staff, \"If someone paid you $15,000, would you ignore them for four days? Or would you make them your highest priority?\" This entrepreneur immediately saw that shortening the time between the sale and the onboarding, consultation, and simply getting the client started would immediately take away potential buyer's remorse (especially since they just spent so much money) and likely increase their LTV too.\n\nWhile that might at first feel like overdelivering, this kind of thing is just the right thing to do, particularly since these are the customers you dream about. And if they start coming into your business, you must nurture them proactively and promptly. No matter how high up the food chain you are at your company\u2014including being the owner\u2014the responsibility of keeping your customers happy is ultimately yours.\n\nCONTROL THE DISTRACTIONS YOU CAN CONTROL (WITH YOUR CONTROL)\n\nThere's another important area of customer service that has less to do with solving problems, fielding complaints, or satisfying customers. It's about being present with your audience\u2014being conscious of what your prospects and customers might be going through in other aspects of their lives.\n\nUnfortunately, many game-changing entrepreneurs and inventors have their best stuff ignored because of world events and things beyond their control that decrease interest in their offer. Major distractions in the marketplace can negatively affect the results you see for the products and services you are promoting. Some are predictable and can be avoided. Some are less predictable, but even in those cases, there may be ways to make lemonade from lemons. And some are just plain bummers all the way around, things that are unpredictable and unavoidable, leading to terrible results beyond your control.\n\nLet's explore all three in the context of customer service.\n\nPredictable & Avoidable Market Distractions\n\nI heard from many people in late 2016 that their marketing results (i.e., response, revenue, and profit) just before and after that year's U.S. presidential election were lower than expected. While people were probably paying far more attention to this election than most, the point is that we need to recognize anything we know is coming down the pike that can be distracting to our potential audience.\n\nThese marketers should have known better\u2014 _every_ presidential election is worth avoiding as a launch date, mail date, or promotion period. And I'm sure there are parallel events in countries outside the U.S. that could lead to distraction in the marketplace and diminished results. During my days doing large direct mail programs, we always avoided the time immediately before and after the vote, and this is a good practice no matter which channels you use.\n\nIn direct mail, it was a lot harder to predict when the mail would land (which we called the in-home date). You couldn't always rely on the mailman to deliver at the same pace. Today, although it's even easier to predict in-home dates with most forms of online media, we never mailed or advertised near any major event due to what we called the distraction factor. We always wanted to reduce risk whenever we could.\n\nA presidential election is one distraction you can work around . . . unlike war, horrific world events, weather disasters, or just unexpected news. I recommend that you make planning for predictable and avoidable events part of your standard operating procedure. The example of the 2016 election is an obvious case, but there are many others (and this is not just an American phenomenon). In my years of planning promotions in various media\u2014direct mail, print, inserts, TV, radio, e-mail\u2014we mapped out our mailing and promotion schedule at the beginning of each year and took as many planned major world events into account as we could, such as the Olympics, The World Cup, elections, and so on.\n\nIn the 1980s and 1990s, we had mailings going out every week. We never had fewer than 40 mailings a year, all of which were over a million pieces\u2014sometimes over five million pieces. There was even one year when we had over 52 mailings and had to double up a couple of weeks (while not mailing the same names more than once on the same mail date). Note: In the world of direct mail, if you rented names for a one-time mailing but names appeared on more than one list, you were allowed to mail duplicates on a future date using the same offer. And there were some weeks we simply avoided in every calendar year, since we kept accurate records on periods when responsiveness was very low for certain kinds of offers. Simply put, so many mailings with not enough mail dates.\n\nNo matter how often you send out promotions, online or offline, I recommend that you map out your promotion schedule at the beginning of the year and work around any events that are predictable distractions, as well as dates when your potential audience will be uninterested in hearing from you, such as a tax newsletter arriving in late April, right after the filing deadline.\n\nPredictable distractions and potential irrelevance are both marketing issues and customer service issues. It's marketing since you don't want your campaigns to bomb because of events and circumstances that are in your power to plan around. There is also an important yet overlooked aspect of customer service at work here: if your offer is ignored because you didn't plan for potential distractions or seasonal relevance, you could lose a customer forever, or never get one in the first place, simply because they felt disrespected or annoyed.\n\nLess Predictable, but Still Avoidable\n\nDo you remember Operation Desert Shield in 1990 and 1991? In August of 1990, Iraq's invasion of Kuwait took everyone by surprise. If you look back at that very tense period, America had troops in Saudi Arabia, literally waiting for their marching orders, while everyone at home in the U.S. was anxiously watching the news, waiting to see how the situation would unfold.\n\nNot that anyone cared (nor should they have), but during the troop buildup, we had a huge quantity of direct mail ready to go out. They were in mail bags, sitting at a \"lettershop\" (a warehouse where the mailing was being staged), and I remember meeting with my staff and with key consultants every day regarding what to do. We wanted to time the mailing as best we could without diminishing the responsiveness and recency of the lists we had rented\u2014that is, we tried to find the sweet spot of making the mailing successful while avoiding having the promotion hit mailboxes just as our troops invaded Kuwait (a big distraction).\n\nWe had recency issues, since the names were getting older by the day (remember the R from RFM?), and that was colliding with a severe distraction factor. We wanted to be sensitive to what our potential customers were going through at the time. We waited a while during the troop buildup in Saudi Arabia and eventually timed the mailing when the distraction slowed down, early in 1991. We traded off some responsiveness (because the names aged a bit) to avoid mailing into total distraction.\n\nWe generated less revenue on that promotion than we had planned, but the waiting period also gave us time to revise our budgets, since we figured we would probably not do as well once we held the mail for so many weeks. But the mailing was far from a disaster, and thankfully the United States avoided a war at that time as well.\n\nWhile everyone likes to make as much money as possible, giving up a little bit of profit by giving up a little recency and not marketing into a period where your audience is distracted (assuming you can wait a little while until the distraction is diminished) is a trade-off that is worth it. You won't completely write off the promotion\u2014and you won't compromise the goodwill of your audience either.\n\nA quick side note on how situations like this became an advertising opportunity for some: I heard from industry sources during Desert Shield that CNN was able to increase their advertising rates significantly, because the number of eyeballs looking at the TV matters, and at times like these, that number goes up. One medium's distraction is another medium's attraction.\n\nThere are always ways to try to use a distraction to your advantage, of course. This was something we experienced firsthand once we got into TV advertising, although this one is hard to recommend broadly. In different periods of the mid-2000s, I believe we were spending as much as any marketer advertising with infomercials at that time. At times we experimented with timing some ad buys week to week, trying to calculate when we could get the most eyeballs on a particular promotion based on the news cycle. It was kind of like day trading, which gave me a lot of anxiety, and like I said, I don't recommend it. Not only did the risk outweigh the reward, it also felt a bit exploitative.\n\nWhen something unpredictable happens, you have to take care of yourself and your business, and at the same time, you can look for ways to help others too. This hit home for me in September of 1989 when the devastation of Hurricane Hugo led to the deaths of at least 60 people. Hugo was a category-five storm and was responsible for an estimated $10 billion in damage, mostly in the Southeastern United States and the Caribbean. It was one of the worst storms ever to hit the United States.\n\nIt seemed meaningless to send the millions of pieces of mail we had ready to go out that week as we watched homes in South Carolina swept away and lives destroyed. I remember watching a news report where there was a mailbox floating down a street and feeling so impacted by it . . . it emphasized to me that despite being my livelihood, direct mail was pretty insignificant at that moment.\n\nBut we had a business to run and our team's livelihoods to protect as well. So after a restless night, I called the mailing facility where we had that huge campaign ready to go out and had them resort the mail. We had time before it went out, so I asked them to remove all the mailing pieces set to go out within a 100 mile radius around Hugo's path as it came up the southeastern coast of the U.S. (it was easy to omit by zip codes). My logic was that if someone's mailbox was floating down the street, the odds that they would want to buy a copy of _The Book of Inside Information_ were pretty low. To my mind, it was wasteful and disrespectful to the folks who were suffering to try to send them mail, and it seemed ridiculous that the postal workers in the affected region would also have to handle and store all that extra mail. Selling a book was not a high priority for me in light of the awful situation, and buying it was not relevant to them\u2014staying safe and working out how to rebuild their lives was going to be the only focus for those people.\n\nAfter the decision not to mail them, I decided it wasn't enough, and that we could do more. We reached out to every charitable fundraiser who had used our lists over the years. The subscriber and buyer lists we owned were some of the most responsive names for fundraisers, and I told every one of them (including the American Red Cross, Save the Children, Habitat for Humanity, and others) that they could have as many of our \"best names\" as they wanted\u2014for free\u2014for any fundraising effort related to Hurricane Hugo.\n\nAt the time, our lists sold for over $100 per thousand names, so this was a big donation\u2014and we made an additional monetary donation to each of those charities who took free names from us (as long as our money would be targeted to Hugo) to help with their mailing costs. We ended up shipping hundreds of thousands of names to a variety of fundraisers and donated a significant amount of cash too. Those mailing efforts raised much needed funds for people who needed it most; later we made this standard procedure for any disaster where direct mail (and our responsive lists) could be used to help.\n\nAll this is to say that even when things go wrong\u2014and you have some warning\u2014you can protect yourself more than you think, while making a difference and turning a disaster into a positive experience for both your company and your customers.\n\nBummers You Just Can't Predict\n\nOn Labor Day weekend in the United States, response rates for many kinds of space advertising are traditionally very high, and in 1997, we had scheduled a large newspaper advertising campaign in anticipation. There was no way we could have predicted that Princess Diana would die tragically, in highly publicized circumstances, just days before. People were devastated by the news and glued to their TV sets. Everyone's Labor Day campaigns tanked that year. There was no time to pull the advertising like we did with the natural and (manmade) disasters where we had some warning, which meant that our newspaper campaign received less than 50 percent of the responses expected on a million-dollar media buy. Sometimes you just have to be prepared to take a hit when everything is out of your control. This is another reason to have your media buys diversified and to never be reliant on just one medium for the lion's share of your revenue and profit. Multichannel marketing is not just a neat sounding term, it should be your mantra.\n\nBut I believe situations like this made us even more sensitive to what we could control and what we couldn't. We were already pretty good at planning for distractions where we had some control, as indicated by our actions around Desert Shield and Hurricane Hugo. And we were always on our guard and encouraged others to do the same, and also to be extremely proactive in how we all approach our customers in volatile times. It's much better to be paranoid and proactively try to control as much you can with your campaigns than to leave things to chance and let the chips fall where they may\u2014especially in situations where people are just not as interested in what you have to offer as you are.\n\nMarty, my partner on all these adventures in marketing, once had T-shirts made up quoting the venture capitalist Frederick Adler: \"Paranoia is not a psychosis . . . it's survival.\"\n\nTHE BARRIER TO SWITCH\n\nLet's talk about one final element in building out customer service and fulfillment as functions of your marketing strategy: creating a barrier to switch. The barrier to switch is one of the most important things you can build into your business. It's about making it almost impossible for your customers to leave you because you give them such great service or a unique experience that would be hard to duplicate if they moved to a competitor.\n\nMaking it difficult (even painful) for your customers to leave you sounds like the antithesis of loving your audience and providing the highest level of customer service. But keeping customers for life is all about your deepest commitment to your audience and has nothing to do with holding them captive. If you have the better solution, I maintain that it is your duty (Jay Abraham might say it's your moral responsibility) to keep them with you, rather than have them leave you for an inferior competitor.\n\nI hinted at this earlier when I gave examples of commodity businesses that make customer service their differentiator. Lifetime relationships and customers who never want to leave you happen when you create a shared history and a shared vision of your future together. It doesn't matter whether your business sells $5 widgets or $20,000 masterminds . . . there is always a way to create a deep connection and loyalty with your customers. And yes, the loyalty has to go both ways\u2014when customers know that you will go to any length for them, they'll be yours for life.\n\nIn my own life, three brands have created a barrier to switch that keep me going back to them again and again. At one end of the cost scale, there's a barrier to switch from my dry cleaner. They gave me a beautiful garment bag with my name embossed on it, along with a barcode that contains all the details they need to never lose my pants and how I like my shirts done. All I have to do is bring my dirty clothes in the garment bag, then they scan the code and they instantly know who I am, how I want everything done, and when I want to pick them up. I don't have to fill anything out, and when I return to pick up my clothes, they are neatly hung in the garment bag and my credit card has already been charged. Sure, I can go to the next dry cleaner who has embossed bags, but as long as my current dry cleaner has exquisite customer service, and the quality is as good as any other dry cleaner, a barrier to switch has been established. They created differentiation inside of a commoditized service.\n\nAt a higher point on the cost scale, Ritz-Carlton records my preferences from every stay, so my experience with them gets better every time I book one of their hotels. When I get to the hotel in San Francisco, they have arranged my stay with all the little details passed on from what I wanted during my stay with them in Chicago. They know what kind of room I want, the type of food I order, the requests I'm likely to make. This makes Ritz-Carlton my first preference every time I travel (and no, they're not comping me for the endorsement). I don't want to stay anywhere else, because the other hotels don't know me, and I don't want to go back to being just another name getting another average experience in another impersonal hotel. (Loyalty points with other hotels I stay at also create a barrier to switch . . . but when The Ritz knew the kind of pillows I preferred as I checked in, it put them in a different category).\n\nIn the case of my car, Infiniti has also created a barrier to switch, again through differentiated customer service. I'm not a big car guy, and since I only buy a new car every eight or nine years, I'm not a great customer for the dealer either. At the time of writing this book, my 2008 Infiniti\u2014which is a really nice car\u2014might be due for a replacement any day. But I also might have it until 2028. However, it's going to be really hard for me to buy any car other than an Infiniti because of the barrier to switch they have built in, even with a terrible customer like me. I have lifetime oil changes and services, I get first access to all their new models whenever I want, I will get a discount as a previous owner, and the dealer knows me by name when I walk onto the lot, as do the service people when I go for those free oil changes. They still treat me like a king _even though I buy so rarely_ , because they know it's going to make it very hard for me to go buy an Acura or a Lexus. They are focused on the LTV of their customer, always thinking about the long-term, cumulative monetary value as much as high frequency purchasing. And although they are still waiting for me to buy my one car a decade, my wife bought an Infiniti during the period they were waiting on me. They can add that second sale to my LTV, I guess.\n\nCreating a barrier to switch is about knowing what your customers want on a tactical level. The more you know the specific details they love, the easier it will be for you to create that barrier, and your entire business can be built around increasing lifetime value. For me, the barrier to switch is just another spin on customer service and fulfillment being critical marketing functions.\n\nKEY TAKEAWAYS\n\nYour customers are the most important people in your business. Keeping them happy\u2014serving them, creating world-class experiences, overdelivering on what you promise, and always innovating for their benefit\u2014is your primary role.\n\n  * Customer service and fulfilment are critical marketing functions. It's the \"marketing after the marketing\" that shows your customers that you do really care about them.\n  * It's much easier to keep existing customers and to increase their LTV\u2014through overdelivering on customer service and fulfilment\u2014than to acquire new customers.\n  * People who can deliver exquisite customer service and fulfilment are extremely rare, so if you find someone who excels at this, do everything you can to make them a permanent part of your team. They should be compensated, developed, and involved just as well as the other important players in your organization.\n  * Customer service and fulfilment can provide the basis for your entire differentiation strategy (as it has for Ritz-Carlton, Amex Black, and Amazon Prime\u2014all premium experiences that create unshakeable customer loyalty).\n  * Listening to your customers and actually interacting with them will give you deeper, more actionable insights on improving your company than any spreadsheet or report could ever do.\n  * Use secret shoppers to help you identify where you can improve your customer experience\u2014check your ego at the door, listen hard to what they find, plug any gaping holes immediately, and then get to work fine-tuning.\n  * Cultivating a spirit of generosity and overdelivery throughout your company makes it far easier to meet the demands of customer service and fulfilment.\n  * Don't forget that customer service and fulfillment extends to when, where, and how your customers receive your marketing campaigns. Always consider their context: plan your campaigns so that you don't send them into predictable and avoidable distractions, and do everything you can to mitigate situations that are less predictable (or that are completely out of left field).\n  * Always create a barrier to switch if you possibly can. Make it extremely difficult for your customers to leave you by giving them powerful, positive experiences every time they interact with your company.\n\nCHAPTER 9\n\nCONTINUITY & LTV\n\n_\"No direct marketing business can survive without repeat business.\"_\n\n\u2014 BOB STONE\n\nSome years ago, I attended a presentation given by a well-known marketer who mostly works online. He opened with a slide that said, \"A product is not a business.\"\n\nI can't tell you how much that warmed my heart. It made me realize that there were top marketers today who understand that it is not the first order that will be the key to their success, but rather that they need to develop a suite of products to ensure a foundation of future success. Copywriting legend John Carlton gave me another spin on this (from the creative side) when he said, _\"_ A promotion is not a business.\" Put those two quotes together and you actually have your direct marketing philosophy.\n\nAs you are figuring out your first product to offer to a select marketplace, think\u2014even at that early stage\u2014about products two, three, and four that you can offer to those buyers. Don't be fooled into thinking that when your first promotion does extremely well that your work is done or that you can ride on that successful promotion for any significant length of time.\n\nOnce you have a first-time buyer, what would they expect from you as the next offer? I love the idea of creating new products, but I like creating a franchise or suite of products rather than one breakthrough product even more.\n\nI have engaged in many lengthy discussions over the years, whether in my own business or consulting on other businesses, about what the audience we were selling to would want next, coupled with what we would be proud to put our name on. Sometimes the conclusion was to _make_ the product, and sometimes it was to _buy_ (i.e., partner on) the product. But in all cases, this line of thinking is the key to creating raving fans that will stick with you for a very long time.\n\nWe've established that list selection is the most important element of a successful direct marketing campaign\u2014and when you combine careful segmentation with powerful offers and compelling promotions, you create an intense focus on maximizing the lifetime value of every new customer you bring into the fold. But as you can see here, we've added the element of always thinking about multiple products and offers that are natural extensions for your audience\u2014before you even launch the first winning product or promotion.\n\nTarget the right people, have multiple offerings from the outset, and never rely on one winning promotion for a significant length of time. This \"business plan,\" while simple, is not easy. One of my favorite authors on the subject of personal finance and money is my friend David Bach. David's books have a consistent theme of \"finishing rich,\" which I love, considering so much marketing advice we receive on a regular basis is about getting rich quick.\n\nIn this chapter, I want to make sure you have a keen understanding of LTV as the key metric in your business above all others, because it's this number that will give you the best possible chance of finishing rich. We have talked quite a bit already about LTV, because it is such an important principle and it influences every part of direct response\u2014but we're going to go really deep in this chapter. The reason it's so important is that it helps you take the long view of any channel you test.\n\nMany people jump to conclusions about particular channels when they don't understand LTV. They give up on a channel that they might think doesn't pay out, while their better informed competitors, as a result of _their_ understanding of LTV, stay in the same channel with acceptable returns and an ability to scale. Three things can happen if you don't track the lifetime value of the customers that come through a particular channel:\n\n  1. You give up on the channel too soon.\n  2. You stick with the channel too long.\n  3. You experience a fluke and misattribute your success.\n\nIn the first instance, because you don't have command of your numbers, on the surface it looks like you're spending way too much to justify acquiring customers on that channel. This is what happened to all those guys competing with my friend advertising on Facebook with their $10 offers. They couldn't stomach not breaking even on the first purchase\u2014because they weren't thinking about a second, third, or fourth purchase. This is why my friend ended up with the channel all to himself and was making back nearly _eight times_ his ad spend on Facebook while everyone else in the space thought the channel didn't work for their offer.\n\nIn the second instance, it looks like you're making good money from each customer that comes through the given channel, but in actual fact you're losing money on them in the long run. For example, let's say you spend $15 to acquire a customer through ads on a big platform and make $30 on their first purchase\u2014you've made $15 per sale (assuming you're incorporating your cost of goods into your acquisition cost). So on the surface, this platform looks like a winner. But when you start looking at the LTV of these new customers, you might discover that they never buy again . . . or buy infrequently . . . and that your $15 of net profit is short-lived and a onetime revenue event with no chance to add additional profit in the future.\n\nYou get this false positive because you didn't track the lifetime value of the new customer. And then you end up distracted from looking at other channels where you might add many more repeat purchases (leading to a higher LTV) that would justify the acquisition cost. You are not building for the future.\n\nNote that original list selection and segmentation is critical here too. That is, if you are prospecting for new customers who are less qualified to buy repeat products in the first place, you are setting yourself up to be a one-hit wonder rather than a lifetime value hero.\n\nIn the third instance\u2014and this happens a lot \u2014you stumble onto a winner but neglect to calculate the LTV of any of your channels, which, if calculated, would identify the most profitable ones and lead to greater expansion (with confidence) in those channels. That is, you get something right, but you can't explain it. You misattribute your success to your own genius and never learn the underlying principle. This means you can't reproduce the success but keep expecting you will be able to. And you won't invest as much in a valuable channel since you don't know all your numbers. There may also be additional profits that you could attribute to initial buyers once they are intertwined in your multichannel marketing strategy, and if you can calculate that into their lifetime value, you may be able to go back and \"lose more\" on the first sale. (We'll dive deeper on that in a moment.)\n\nNot recognizing the full lifetime value of every new buyer across every channel they buy from is the most dangerous of the three scenarios. In the first scenario, you'll abandon a profitable channel and jump to something else that seems to work better. In the second, you'll be running in place\u2014and hopefully that won't lead to anything fatal. In the third scenario, however, you'll assume that you're doing everything right and applying the same thinking to every channel\u2014not realizing that the expected value of a new channel may be inaccurate.\n\nUnderstanding LTV allows you to simply sidestep these problems. The power of direct response is that it's marketing by numbers. Yes, you have to know which numbers to track (and you have to actually _do_ the tracking), but once you have them, you'll be able to model out the trajectory of the business for the months and years ahead. It's a superpower that saves you the uncertainty and volatility that can so easily kill a business.\n\nLTV is the linchpin in those calculations. When you know the LTV of a set of customers, you can then develop \"the bogey\"\u2014a simple calculation that answers the question, \"How much are you willing to _lose_ on your initial order to make it all back (and more) in subsequent years?\"\n\nTHE BOGEY MAN\n\nUnder the supervision of my mentor Dick Benson and a team of best-in-class direct marketers, Boardroom's flagship consumer newsletter, _Bottom Line\/Personal_ , reached a million paid subscribers at its height. This was quite a feat, since the newsletter took no advertising and had to exist (and profit) based solely on subscription revenue (and a little bit from list rental income). Getting to a million subscribers profitably was even more daunting because the subscription revenue was based on a very low price point (between $30 and $40 per year).\n\nEnter Dick Benson, the smartest man who ever lived when it came to direct mail. (By the way, Benson's 31 \"Rules of Thumb\" in his classic book _Secrets of Successful Direct Mail_ still apply, at least conceptually, to everything we do online today.) Dick told us that the way to grow _BL\/P_ was to establish the bogey, and I can safely say that calculating this number and basing all our marketing decisions on it was the single biggest reason Boardroom became as large as it did.\n\nWe calculated on a list-by-list basis how much we were willing to lose on the initial subscription. That calculation took into account the average renewal rates of those initial subscribers, list rental income we could derive from those subscribers for as long as they were on file as active, and what the average new subscriber would buy from us after the first sale (such as subscriptions to other newsletters and books in similar subject areas).\n\nDespite the fact that _BL\/P_ was by far the most profitable product in the company's stable, it was certainly not stand-alone. Lifetime value is always about the interaction of multiple channels, multiple products, and every way a new customer contributes to your bottom line.\n\nThe end result was that we got rich slowly. That is, the money in our bank accounts steadily increased and over time made us less concerned about monthly cash flow. This slow and steady process started with a one-year bogey (making a new subscription profitable a year after we sold it); then we moved to a two-year bogey, and eventually a three-year bogey. Who would have believed when we launched the publication that we would be willing and able to wait until year three to break even on a new subscriber?\n\nThe day Marty and I realized we were in the business of _renewals_ was the day I believe we really understood LTV. We looked at all the sources of revenue for the company: subscription income, book sales, list rental. And as we studied it all, the _BL\/P_ renewals was by far the most significant number and accounted for the highest margins (since it was always cheaper to renew a subscriber than to get a new one). The cross-selling and upselling associated with the newsletter was the whipped cream on the sundae, and it was important too. The renewal revenue (plus the related revenue streams) allowed us to be much more aggressive with our marketing for new subscribers, given what we could lose in the first years of a customer's relationship with us.\n\nKnowing that renewals and multiple product sales rule the day, the emphasis on editorial excellence takes on even more significance. We were committed to making sure that subscribers to any of our newsletters would renew at 50 percent to 70 percent after the first year. We wanted them to be so satisfied that they would trust us implicitly when we offered them additional editorial products (other newsletters and books). This strategy\u2014getting renewals as high as possible and creating subsequent products of only the highest quality\u2014enabled us to mail more names at an upfront loss to get new subscribers for a long-term profit.\n\nWe eventually reached a million paid subscribers. It took a long time . . . but it was worth it.\n\nYOU CAN SPEND MORE MONEY ON YOUR UP-FRONT MARKETING THAN YOU THINK\n\nEarlier I mentioned a client who offered a high-priced coaching program which cost $20,000 per year. They weren't spending much money on acquisition marketing to new, outside lists, and I wondered why, with such a high price point. Even a one-year buyer was worth a lot to them.\n\nAnd when I asked what their average renewal rate was for the program, I was blown away. They said it was three times\u2014that is, their best customers were $60,000 buyers, which _screamed_ to me that we could devote a lot more dollars to marketing. I suggested all kinds of high-priced acquisition efforts (expensive three-dimensional direct mail, radio, TV, live events . . . ). They were doing some of this already, but the real progress started happening once we understood the lifetime value of a new customer and, with that number at our fingertips, realized we could invest a lot more in our marketing. (Yes, LTV calculations can be made\u2014and must be made\u2014whether your offer is priced at $10, $30, or $20,000.)\n\nThis situation taught me to ask this as my first question when entering a company for a consultation of any kind: What is the lifetime value of a new customer worth, and are you willing to lose money short term to make a lot more money long term? It's an updated version of the question Dick Benson asked Marty and me when he knew that developing a bogey was critical to our future success.\n\nIf the marketer can't invest more in first-time buyers because of financial limitations, that's one thing, but if they _won't_ do it because they don't want to play a long game and are only focused on short-term revenue and profit, that will be the biggest impediment to their growth. One of the main things to remember with LTV is that every dollar you spend on marketing and media does not need to be made back immediately. I understand that marketing isn't free, but if you intend to stick around for a long time, and you believe in what you are doing at the deepest level, you will embrace this notion of getting rich slowly.\n\nDan Sullivan, the world's top coach for entrepreneurs, says this when he talks about abundance: \"Never spend less when you can spend more.\" He was not talking about spending more on marketing specifically, but the principle still applies, particularly if we factor in a proper understanding of LTV. Spend as much as you can on up-front marketing as long as you know exactly when you will make back your investment because you have calculated lifetime value of every new customer or client.\n\nWe tried to embody this as much as possible. Depending on the product, we would extend our bogey out two or three years. On _BL\/P_ , we were willing to be in the negative until the second annual renewal (i.e., the third year) because we knew those customers were so valuable in the long run. On other products, we would want to break even on the first conversion, because we knew those lists were more impulsive and less likely to become loyal repeat buyers.\n\nWhen we started the book division, we had to be really disciplined. We could have had a bogey there as well, even though the books were based on a one-shot model\u2014rather than a renewal model\u2014but people who bought a book from us would often buy other books and newsletters, so we would see an increased lifetime value even in these one-time buyers. Without the renewal stream, we were more cautious\u2014however, that didn't mean we weren't calculating lifetime value.\n\nAll lifetime values are not created equal. You must look at the potential revenue streams by product and channel and calculate accordingly. Being less aggressive with our book marketing than we were with our newsletter marketing was difficult because we had seen such incredible results by investing deeper in media with a higher lifetime value. But with books, we stuck to our guns\u2014we had to make money on the first sale. This prevented us from mailing a lot more names for the books, but having the books make at least some money up front allowed us to have a much more aggressive bogey for _BL\/P_ and never got us into trouble from a cash-flow perspective.\n\nSimply put, we stayed super aggressive without being greedy . . . and we stayed committed to the lifetime value calculation that we knew would be most profitable across all products. In today's marketplace, you might have certain \"hero products\" that are worth extending the bogey on because they are the most profitable based on lifetime value.\n\nBut to make the model work, you have to make sure there are other products in your business where the cash-flow cycle is tighter but just as predictable. This keeps you solvent\u2014you want to make sure you're not extending out so far on every product that you run into cash-flow problems. For example, if we had extended a three-year bogey on all our products, we would not have been able to scale, and we probably would have gone broke. Once you have millions in the bank, it is much easier, but that doesn't mean you can ever afford to get sloppy. You can't afford to be complacent just because you've got a lot of money in the bank.\n\nRich Schefren, a marketing thought leader once told me, simply and eloquently: \"You don't have a real business until you can profitably buy customers consistently from cold sources\" (not JVs, affiliates, etc.). What he meant was once you can spend real money on media that pays out and you are not reliant on media that is some kind of \"revenue share\" (with affiliates), you are taking that leap from a \"product\" or a \"promotion\" and into a business. In the context of this chapter, it applies in that if you can buy customers consistently _at a loss initially_ , knowing exactly when they become profitable, you have a real (and _big_ ) business for the long run.\n\nDirect marketing is all about the numbers\u2014not just the numbers coming in, but the numbers going out too. Good money management is _critical_ if you're going to scale your business because you have to be able to service the increasing volume of customers, continue producing and fulfilling your product at a high standard, and often start working with consultants and partners that come in to handle areas outside your competency. To do all that, you always want to put some money aside that you're not going to reinvest, which becomes your cushion. This rainy day fund protects your assets and allows you to take a few risks too.\n\nPROTECTING YOUR LTV WITH CONGRUENT MARKETING\n\nEverything you do in your business needs to be congruent. While we've spent this chapter so far talking about lifetime value as a number, staying in business for a lifetime combines knowing your numbers with knowing who you want to be in the world and what you want your business to represent. That means building a culture into your business that reflects who you are and what you value most.\n\nIf you have a strong foundation that is congruent with your purpose, it's easy to attract the right audience and to develop offers and creative that are congruent and appropriate for them. And not surprisingly, when you focus on congruence, the number that you calculated in lifetime value per customer tends to increase exponentially. Whether you make the subsequent products yourself or buy them from someone else in some kind of partnership, whatever you put your name (brand) on has to be congruent\u2014to you as a company and to your customers.\n\nCreative is also an area where congruence can be at risk: what might work best to bring in new customers may be inconsistent with how you really want to sell. I have seen many thought leaders struggle when they begin marketing aggressively for the first time. Unfortunately, they start with the premise that marketing is evil\u2014and then my job is to show them how to sell congruently yet still remain aggressive. It's a sweet spot that we can all get to, and it's different for everyone. But I maintain that being as aggressive as possible, while being as congruent as possible, is the formula for the highest lifetime value.\n\nSince we're talking about marketing, and money is always involved, things can get tricky when your desire to be congruent collides with paying the electric bill. If you have a good reputation and a solid bank of goodwill with your customers, and therefore with your prospects too, it's easier to recover from a \"congruence fail,\" but you can't count on it. I made one of these congruence fails, and thankfully survived, learning a valuable lesson that I refer to these days as \"the moose on the table.\"\n\nTHE MOOSE ON THE TABLE\n\nI once received a card from a business associate that gave this sage advice about meetings: \"None of us is as dumb as all of us.\" Have you ever made decisions in your life, business or personal, where you look back and wonder why no one suggested you do the opposite . . . or simply do nothing at all? It's critical for someone to always call out \"the moose on the table\" . . . the thing no one wants to say, the thing everyone is afraid to suggest. (In my mind, the phrase conjures up an image of a dead, decaying moose right in the middle of a meeting room table . . . while everyone talks around the carcass, refusing to acknowledge that it's even there.)\n\nThat's how I felt after I made the second biggest mistake of my career (the biggest, as you recall, was reading our press clippings and assuming our quick and unexpected success in long-form infomercials was something easily reproduced). No one likes sharing their failures, but as Marty always said, the only things worth talking about are the things you can't talk about. So here's how I nearly tanked our LTV with an incongruent offer.\n\nIn the early 1990s, list rental was lucrative for the company. But we didn't accept advertising, and we were aggressive marketers, reinvesting most of our profits back into the business in order to mail as many millions of names as we could to grow circulation and to create long-term revenue. Direct mail was (and still is) very expensive because it scales and because it works. But that expense can be very stressful to manage.\n\nOne day, we were sent a request from a mailer to use our lists for an offer that, simply put, we would never have thought to expose our list to. Not surprisingly, this mailer often had trouble getting list owners to give them names for list rental. For me as the list manager, this seemed like an easy rejection: it wasn't something I felt comfortable approving for our customers. However, before I rejected the order, I turned the envelope upside down to see if there was anything else in it, and out came a check from the list broker who sent the request . . . for $100,000.\n\nThat was _a lot_ of money for a list rental, since it was basically for a one-time use of the list. It was roughly the same amount we would receive for over two weeks in list rental income at the time, across _all_ mailers using our list.\n\nIn direct mail, the recipient of the mailing rarely knew which list was used to rent their name. I went to a couple of my mentors at the time outside the company to get their opinion. The general feeling was that there probably wouldn't be any harm in it.\n\nLooking back I should never have asked anyone who didn't have skin in the game with our customers for their advice. I should have followed my gut and sent the check back with a rejection.\n\nI knew that was the right thing to do . . . but I didn't do it.\n\nEven though none of our customers traced the mailing to us, one of our most trusted exchange partners (a huge publisher we exchanged millions of names with) got a complaint from one of _their_ subscribers. At first, they couldn't figure out where this rogue campaign had come from . . . but eventually they realized that this person was subscribed to their publication _and_ ours. Soon enough, all the evidence pointed to me. I had to confess to our exchange partner to renting names to that campaign. I would never do anything like this again, and it was a lesson for a lifetime. Simply put, your LTV on all channels will go to zero if you gamble with the respect of the most important people to your business\u2014your customers.\n\nYou might read this story and think you would never do something like that. And I hope that's the case. But it's hard to know ahead of time what lines you might cross if you need cash and it seems like your business depends on you making a distasteful deal. Before you make your final decision, think long and hard about congruency and how being incongruent can affect the lifetime value of your customers . . . and of your business.\n\nThese are the critical lessons I took from this situation:\n\n  * Never make decisions when the need for money can take precedence over your core principles and what you know is right.\n  * Never forget that as marketers we are all connected. This is more true today than ever, since we mail for other people under our own names (as affiliates). But even on channels where you can hide for the most part (exchanging names in direct mail was more anonymous), we found out the hard way that everything comes to light eventually. Every action you take can have repercussions that you can't predict, given how interwoven marketing communities are and how much overlap many lists have.\n  * Always consult with multiple people with a variety of viewpoints who will tell you when there might be a moose on the table.\n\nOf course, whether an offer is questionable is often in the eyes of the beholder . . . but always think about the perspective of your core customers before you send something that could be incongruent. I've seen offers sent to lists online today that are so disconnected to the audience that it's obvious the mailing was done for a quick cash injection or fulfilling a commitment to mail ( reciprocation) or some other reason that has little to do with serving the customers.\n\nI know of one online guru who had nearly half his list unsubscribe\u2014a list that took years to build based on trust and integrity\u2014because he sent an offer that had no congruence for his audience. In fact, it was a subject they had disdain for. Not only did he not make any money on the mailing, but he lost half his customers in the process. Ouch.\n\nIf you are playing a long game, there is no reciprocation for another marketer that should ever compromise how you want to treat your best customers. Or prospects. Or even suspects (people you eventually want to turn into prospects and then customers). All in all, you can't create value for a lifetime without a total commitment to delivering exquisite products, value, integrity, congruency, and consistency.\n\nLifetime value is way more than a number.\n\nKEY TAKEAWAYS\n\nWe've talked about LTV a lot in this book, and I hope you see that I've beat that drum so relentlessly because it's such a fundamental part of building a thriving business.\n\n  * A product is not a business. A promotion is not a business either.\n  * LTV is much more than just a number. Keeping LTV top of mind all the time will save you from making damaging decisions, show you major growth opportunities, and give you the best chance of \"finishing rich.\"\n  * LTV allows you to accurately predict your bogey\u2014how much you can afford to lose on your initial sales or cost of acquisition in order to make it back (and more) over time.\n  * Knowing your LTV and bogey allows you to spend aggressively (and strategically) on marketing because you know exactly when you will make your money back.\n  * Renewals and multiple product orders rule the day, which means that you must develop editorial excellence throughout your company. Remember, marketers sell subscriptions, but editors sell renewals; or, more broadly, product sellers sell subscriptions but product producers sell renewals.\n  * Be aggressive but disciplined in how you spend on customer acquisition. Don't extend your bogey out so far that you run into cash-flow problems. If you can, vary the bogey across your different product lines to ensure that you have regular cash flow regardless of the different product sales cycles.\n  * Staying in business for a lifetime requires knowing your numbers and doing everything you can to maintain a congruent message and approach that connects deeply with your customers.\n  * Be incongruent at your own risk. And when the decision comes down to needing the money, make sure there's no moose on the table.\n\nCHAPTER 10\n\nPLAYING THE LONG GAME\n\n_\"Life is long.\"_\n\n\u2014 MARTIN EDELSTON\n\nI hate the word _networking_. And that is despite having a reputation as someone who connects the dots as well as anyone once I know the superpowers of the people I interact with.\n\nI guess networking and making valuable connections is one of my own superpowers\u2014but I still hate the term. My distaste for the word (and the idea behind it) stems from my late 20s, when I was featured in a trade magazine in one of those \"30 under 30\" profile pieces, which on the one hand I was very proud of, though on the other hand, I was a bit embarrassed. The profile dubbed me a \"strategic schmoozer,\" which to me implied that I was more fluff than substance . . . or that I had some secret agenda when I met people or added them to my network. Because of that article, I was concerned that people might perceive me as disingenuous. I believe it is the reason why I am so uninterested in superficial networking today\u2014and why my take on this important area has evolved into something much more.\n\nI'm glad I picked up on this before I was all grown up. I realized that it was not enough to simply _tell_ everyone about my motivation to help as many people as possible by connecting them to others who could help them. I needed to _show_ everyone that I was much more than a strategic schmoozer. I became intensely driven to create synergistic partnerships and relationships . . . but I needed to come up with different ways to do it than simply accumulating people and being the life of the party.\n\nAfter that article, I thought more deeply about this concept. I remember attending a multiday personal development seminar and when I arrived, they asked us to define who we were\u2014and I defined my core motivations as \"Know everyone. Do the right thing.\"\n\nAt the end of the seminar, after lots of discussion and hard work on myself, they asked me to answer the same question, phrasing it, \"I am the possibility of ___________.\" The word that filled in the blank for me was, and still is, \"contribution.\" I wanted to be a strategic _contributor_ first, with the connection flowing naturally from there. Networking for me had now become contributing in order to connect, and a much longer (and more meaningful) game had begun.\n\nLEARNING FROM GIANTS\n\nThis chapter is a collection of lessons accumulated over my career, both during my time at Boardroom and since. I've been privileged to know and learn from some of the best minds covering three generations of direct marketers; and while there is strategic and tactical brilliance at play, it's the intangibles that have made all of them so remarkable.\n\nAnd I didn't ask these folks to write blurbs for this book simply to sell more books. I approached them because they have been the biggest influencers throughout my career. In every case, the relationship began with how I could first contribute to them . . . and then letting things take their course. Having Jay Abraham write the foreword to this book is indicative of how much I stand on the shoulders of giants and how important it is for me to recognize as many of them as I can. This is directly related to Adam Grant's \"givers, takers, and matchers\" concept . . . and now I can share examples with you of how this played out for me over almost 40 years in direct marketing and how I believe you can apply these principles to your life too.\n\nI didn't want this to be a personal development book, not because the world doesn't need more personal development, but again, because there are better people than me to write those books. However, it would be a disservice to you, your career, and your business if we didn't take a deep dive into these intangibles. They will help make overdelivery, humility, and curiosity a way of life, and they will make you a better direct marketer too.\n\nTHE REAL VITAMIN C (IT'S NOT IN YOUR OJ)\n\nNot only do I hate the word \"networking,\" but I also think it's one of the most overused terms in the business world today. And given the omnipresence of social media, and that these powerful connection tools are often in the hands of amateurs, the idea of creating a network by pinging anyone and everyone with a smartphone makes me hate the term more than ever.\n\nAccepting a request from every stranger who sends you one on Facebook or LinkedIn is far from the gold standard. In fact, I wouldn't really even call that networking; it's just screensucking (as ADHD expert Ned Hallowell calls it) . . . with little or no chance for true connection. Clicking a button is never going to build a genuine relationship with anyone.\n\nI would like to share some observations and techniques I use to establish real connection with people, even in a world where relationship building has been reduced to clicking on a face (Facebook) or a one-sentence introduction supplied by someone else (LinkedIn).\n\nTo truly connect with people, you must contribute to them first with no expectation of a return (even though you'll probably get a return if you're investing in high-quality people). Ned Hallowell, who I just referred to as one of the world's leading experts on ADHD, was also a frequent contributor to the Boardroom newsletters and a brilliant author of close to 20 books (my favorites are _Crazy Busy_ and _Driven to Distraction_ ). He sent me a beautiful note some years ago in response to a blog post I wrote about the notion of \"contributing to connect\" replacing the notion of networking:\n\nYou are a master of what this world needs most, and what has been the cornerstone of all my books: the power of connection. I call it the other Vitamin C, Vitamin Connect, and we live in a world where most people suffer from a deficiency of it. Glued to their electronics, they get massive doses of electronic connection, which in no way imparts the life-enhancing, indeed life-saving, power of the true Vitamin Connect . . . and the beauty of it is that it is free, and infinite in supply.\n\nI know it sounds old-fashioned, but face-to-face connection trumps all. The fastest way to become deficient in Vitamin C(onnect) is to stare at your computer all day, screensucking and pretending that you have real relationships with people on the other end of an algorithm.\n\nWhile there is obviously lots of utility\u2014and convenience\u2014in connecting electronically, that doesn't mean you should be taking shortcuts if your first interaction with someone is online. For example, I send dozens of personal e-mails every week to folks who send me LinkedIn requests\u2014that is, before I simply accept the invitation, I look at their profile and shared connections and acknowledge what kind of overlap we have in our respective worlds.\n\nI also encourage them to look at some of my writing and to opt into my list (in addition to scoping out my profile) for the purpose of staying in touch and starting a real relationship. I am so far away from thinking about selling them anything at this point\u2014or trying to figure out what the benefit might be. However, I know that the odds of creating any kind of mutual benefit are much lower if I simply hit ACCEPT.\n\nIt's also what happens after this that matters. If they opt in to my list and they e-mail me, even in response to an automated e-mail blast or blog post, I answer the e-mail one-on-one; and of course, if there is ever any chance to meet them face-to-face sometime in the future, I would look for that opportunity when it makes sense.\n\nWhatever you do, do something that creates a _real_ connection with that person\u2014even at the earliest stages of what could be a lifelong relationship. This philosophy needs to be consistent everywhere in your life in how you contribute to connect. I guarantee that you will find this kind of networking far more rewarding than just hitting ACCEPT without finding out more about the people you come into contact with.\n\nIt's sort of ironic that the guy who mailed close to two billion pieces of direct mail (not exactly an intimate form of communication) is talking to you about how to create a personal relationship with everyone you come in contact with . . . but I really mean it.\n\nBeing a mass marketer has not stopped me\u2014in fact, it has only fueled me\u2014to connect more deeply whenever possible and appropriate. That's why I loved answering customer service calls; it's why I return every e-mail that comes into my inbox; and it's why I meet as many folks as I can if I think I can contribute to them in some way.\n\nI understand that sometimes you can't be that personal. But here is what I want to stress: never connect at a superficial level when connecting at a deeper level will only take a little more time and effort. It's worth it. Start as you want to finish. Develop a deeper relationship in the long term by avoiding shallow communication in the short term. And always put something special (and personal) into all your communications.\n\nIf you start with what you can contribute to the other person first, you are guaranteed to inject the maximum amount of \"Vitamin C\" into your bloodstream. It's so simple, yet so powerful. Another way to dose up on \"Vitamin C\" is differentiation: do something different to stick out\u2014and not just for the sake of sticking out. Stick out because it really matters to create something extra special with every person you interact with.\n\nI speak often to recent college graduates about opportunities in direct marketing. I ask them how many of them send an e-mail after they interview for a job or meet someone new who might be able to help them with their career. Just about everyone raises their hand. Then I ask, \"How many of you actually send a handwritten card or a formal business letter as follow-up?\" After I explain how you actually do that using the old artifacts of pen and paper, maybe a third of the room raise their hands. Sometimes I'm happy to see any hands at all. Then I suggest a different approach with my final question: \"And how many of you look for areas of common interest with the person interviewing you\u2014an author, a magazine, a subject area\u2014and then use that in your follow-up, beyond just mentioning something in an e-mail? For example, sending a handwritten note or letter and maybe an article about something you discussed?\"\n\nOf course, no one raises their hand for that one.\n\nAt that point, I always whisper loudly into my microphone to really emphasize the point: \"If you do something like that, no one else will be doing it, and that's a good thing. Even if you don't get the job, you will be remembered, which is always better than being forgotten.\"\n\nThis applies to anyone in business looking to find common ground and to connect beyond a superficial level. Whether you're at the beginning of your career or established in your career, contributing to connect should always be part of your playbook.\n\nMarty always sent what he called \"nice notes\" all year round rather than just sending the important people in his world a card during holidays and milestones with everyone else. He believed that you should connect with people as the mood hits you, as things come across your desk that would create connection.\n\nThese are just a couple of tactics you can put your own stamp on. The ingenuity you use in how you contribute\u2014and then connect\u2014needs to be congruent with who you are and what you are comfortable with\u2014but don't just do what everyone else does.\n\nBottom line: get your daily dose of Vitamin C(onnect) and avoid being a strategic schmoozer at all costs. When you invest intentionally in relationships with remarkable people, the return you get far outweighs any financial investment you could make. But I'm not naive. Being in business and creating relationships is also about making money. Contributing to connect might actually be your biggest moneymaker in the long run too (since many successful people want to partner with people they know, like, and trust).\n\nHowever, this is a chapter about playing the long game and creating not just a network but rather \"relationship capital\"\u2014the account that will have the biggest impact on your life and business, because it's worth a lot more than cash.\n\nTHE COMPOUND INTEREST ON RELATIONSHIP CAPITAL\n\nMy love for contributing and connecting has turned me into a mastermind junkie and a people junkie. I can never get enough great content from great people, and most of the other world-class marketers I know are the same way. It's impossible to calculate the value of being a lifelong student. The friendships and relationships I've developed this way mean the world to me and have been incredibly powerful forces in every part of my life. (Not to mention all the frequent flier miles I've accumulated.)\n\nMoving from strategic schmoozer to \"contributing to connect\" and eventually learning how to invest deeply in relationship capital has been a lifelong journey. It started with that uncomfortable article, which motivated me to do a lot of work on myself. When it started having a big impact on my life, I kept at it. And it culminated in a huge way in 2014 with the Titans of Direct Response event . . . and shortly after the event and retiring from Boardroom, I started my blog. One of my first posts was about how I got all those huge names to speak at the Titans event (and, even more impressive, got some of them to come out of retirement and\/or seclusion). From Dan Kennedy and Jay Abraham to Gary Bencivenga and Greg Renker, the biggest living legends in direct marketing graced the event . . . and I can admit to you here that making those requests was painless (and gratifying).\n\nNow, I could simply chalk up their willingness to appear to the fact that I'm a nice guy and that I always try to be appreciative and respectful. But it was more than that. The event was a tribute to Marty, who had died the year before. His stature in the industry as a legendary contributor didn't hurt in getting folks to step up, and at the time, I actually thought it was mostly that aspect that made the event such an easy sell to all these powerful speakers.\n\nBut since then, I have brought together more amazing direct response marketers and copywriters in my two mastermind groups, along with guests to regularly speak to those groups. These guests are all titans in their specialties, so I knew there was something more here. It's the relationship capital I had built with all of them.\n\nYou may find this hard to believe, but relationship capital is far more valuable than any asset in your financial portfolio. It has nothing to do with getting rich quick. It travels well, and it is accepted everywhere. This kind of capital is forever, and what you can \"buy\" with it doesn't take up storage space or require heavy management fees from a financial planner (although it does need to be watched very closely).\n\nRELATIONSHIP CAPITAL IS AN INVESTMENT FOR A LIFETIME\n\nSo how does the special currency of relationship capital get deposited into your personal account and accumulate compound interest over time? Well, we can start with why all those icons in direct response marketing said yes so quickly to speak at Titans, why they continue to say yes, and why it was easy and natural for me to make the request of each of them. As I said, politeness and my connection to Marty has a lot to do with it . . . but building up my personal account had been a long time in the making.\n\nI know how it _didn't_ happen. It had nothing to do with trying to build the largest LinkedIn or Facebook account . . . and it didn't happen by contacting people only when I wanted or needed something. It also didn't happen by accident\u2014it happened by being committed to consistent communication and always thinking about how I could contribute to the other person first. But even more important, it also happened because I truly care about these people and have consistently let them know how much I appreciate them, sincerely, and never just as a way to ask them for something.\n\nWithout playing a long game of contributing first and always caring, any \"new deposits\" into your account might as well be Monopoly money. There's no \"interest\" accumulated when you are simply playing a numbers game with your contacts, only going a mile wide and not thinking about how you can go a mile deep.\n\nSomeone said to me a while ago that it is far more gratifying and powerful to be interested than interesting. That doesn't mean you should sit on your butt all day not creating stuff that others find interesting . . . but try to notice when your conversations are all about you. At that point, turn the conversations around to focus on the people around you and the people you want to connect with.\n\nI've been accused many times of writing too much about myself instead of focusing exclusively on _you_ , and I accept that I'm not perfect at this. And there are certainly a lot of I's in this book. It takes a lot of practice and a lot of humility to get the balance right. If this is a new concept for you, it can be hard to even make a start, given the conversations going on in our heads all day about the most important person in our lives: ourselves. Those are the voices we hear all day telling us how interesting we are\u2014the \"legends in our own minds\" thing. That's why I recommend that you have people around you all the time who will make you accountable (and will tell you to shut up when it's all about you).\n\nIt's key to have true friends and mentors who keep you accountable so you don't always assume your next idea is your best idea ever. Making requests for accountability from your closest friends, advisers, and coaches regarding how you are showing up for people is also critically important\u2014that is, getting some checks and balances on when you are talking too much about your own interests. I think you will agree that your closest friends should always tell you the truth, with love never leaving the room.\n\nThere are two things that must be eliminated when dealing with the folks around you in order to see the maximum yield on your relationship capital: complacency and arrogance. If you are too complacent (not engaging or being interested in the other person) or if you are arrogant (being a bit too \"interesting\" and making it all about you), you need an honest intervention from someone who cares about you. You can't invest in relationship capital if either of those things is getting in the way of how you interact with everyone around you. These two things will stop you in your tracks from creating a bigger future for yourself. And since I can't trust myself to be 100 percent aware all the time if either of those two things are creeping in, I surround myself with people who have permission to let me know immediately and loudly if I am becoming complacent or arrogant. I chose those people because I know they will tell me what I need to hear and that they will communicate it only with love and respect.\n\nHere's the quick prescription for building your relationship capital account, which automatically includes compound interest:\n\n  * It's always about what you can contribute first to the other person, not what you can get from them. Find out what that person cares about and invest in that.\n  * Be appreciative and respectful always . . . I guess that's pretty obvious, but good manners never go out of style.\n  * Don't come out of nowhere with your communications to anyone. Be consistent, and don't presume to make an \"ask\" of someone, no matter how small, if you haven't invested in the relationship first.\n  * Have people around you to let you know when you are being complacent . . . firmly, directly, and in a spirit of love and respect.\n  * Have people around you that will let you know when you are being arrogant . . . firmly, directly, and in a spirit of love and respect (though a kick in the head might also be in order here).\n  * Do your homework when entering into any new \"investment\" (i.e., relationship). You probably ask a ton of questions before investing your money; you should ask just as many (or more) when bringing another person into your world.\n\nA high-value relationship capital account doesn't happen easily or without a lot of time and effort. It must be valued as a privilege and not simply an asset. Cherish it and nurture it as you make additional deposits every day.\n\nTHE POWER OF INTENTIONAL DINNERS\n\nMarty loved bringing together experts for meals . . . he created \"expert lunches\" early in his career to gather the best content for his publications and to make himself smarter. Those lunches evolved into something much more, what he eventually called The Boardroom Dinners. In many industry circles\u2014including direct marketing, publishing, media, health, finance\u2014these became legendary.\n\nWhile eating dinner seems like a fairly straightforward endeavor, creating dinners over decades that were magical events involved a lot of planning and thinking. Buying dinner for a group of people is easy; making dinner an experience people will never forget takes a little work. Since I would encourage everyone to bring some variation of intentional dinners into their lives, I will share today what makes creating these gatherings so magical . . . and how profound they can be for you, your career, and everyone you feed.\n\nThe first dinner took place in Marty's New York City apartment, overlooking Central Park, catered by world-class chefs. Because of the size of his apartment, the dinners were limited to a maximum of 14 people (which is a fine number to make them work well). Marty would host a dinner each month, and it was considered a privilege when anyone in the Boardroom family of writers, sources, clients, and partners received an invitation.\n\nAs the company became more mature and the word spread about these special dinners, they only got bigger and more profound. Guests told their friends about the amazing conversations, connections, and projects that were coming out of the dinners, and soon people were reaching out to us for invitations. After Marty gave up the apartment, he began holding the dinners at the world-famous Four Seasons restaurant in New York City, where a seat at one of his \"salons\" became one of the most coveted invites across multiple industries.\n\nRick Frishman, an author and friend who attended many of these dinners, profiled them in his book _Networking Magic_ , calling them \"networking events that pay homage to a master networker and to his amazing passion for the best\" and that \"it is an honor to be invited.\" Another regular attendee, marketing consultant Ken Glickman, said that they were \"evenings that create a tremendous amount of enthusiasm and positive energy.\"\n\nKeep in mind, though, that all these extraordinary outcomes won't happen by accident. The more you plan in advance for success, the more success you will have. As you think about creating your own version of intentional dinners, remember the lessons we've talked about in this book regarding the need to love your customers (giving them more than they would ever expect), how to approach your partners (always making them feel like teammates), and the importance of being interested and invested in making people feel special. Dinners like these are a perfect way for you to express these core beliefs.\n\nBe all about intention. Leave nothing to chance. Make your intentional dinner a platform and an individual expression of your own insatiable curiosity and that of your guests. Bring your brilliance, and the brilliance of those you most respect, forward in a bold way. By inviting other people who are as voracious and curious in self-education as yourself, you've created another way to \"expand your library.\"\n\nMaking the Invite List\n\nIn the early days, Marty hosted the dinners by himself; later on, I had the privilege of co-hosting over 150 dinners with him. If you were an expert who appeared in any of our publications, you were immediately added to the guest list for the future. If you were an expert, author, speaker, or thought leader we met at a conference, or if you were someone we heard in the media, or someone fascinating we read about, your name would also be added to our invite list.\n\nAssembling experts in every industry and vocation, and having them share with each other in their area of expertise, guaranteed that everybody left the room smarter after dinner. Marty's simple summation about the dinners was \"There is no better way to spend an evening.\"\n\nYou can do the same thing, adding potential dinner guests as you go (in your current circles), and then adding guests in circles you constantly build as you make deposits into your relationship capital account. Plan them out in advance and give guests a choice of dates, especially if you have a lot of people in your world who live out of town.\n\nDinner Logistics: Intentionality Matters\n\nMake sure you get biographies or CVs for all the guests in advance. The night before each dinner, read each person's biography carefully, and on the basis of common interest areas and connections you can plainly see (or sometimes just infer), create a seating chart to maximize the experience for every attendee. Seat people near each other who have clear synergies or have the potential to create opportunities together. Sometimes you might just think certain people would find each other fascinating.\n\nI follow this procedure whenever I host a dinner at one of my Titans Mastermind meetings, or any time I host any kind of dinner, which I do often. While the events would probably work even if the seating were left to chance\u2014since all the attendees are incredibly interesting people doing incredibly interesting things\u2014the chances of creating bigger opportunities for guests is exponentially improved with this kind of planning.\n\nThe proof is that these dinners create more lasting friendships and profitable projects than any other kind of dinner I've ever attended. As Rich Frishman puts it:\n\nSince these kinds of dinners place a higher premium on the quality of the guests than on the areas of their expertise, focusing on a diverse range of businesses, professions, and disciplines gathering around the table is the priority. Guests in the past have been spellbound by information disclosed by experts on everything from terrorism to sex therapy. Medical researchers have explained the latest breakthroughs in their fields and a former White House staffer from the Kennedy White House has shared inside stories about the JFK era.\n\nSometimes randomness can even lead to fireworks, and I recall Marty and I being a little devilish when setting up the table when we did it together. There was the dinner when we had America's leading anti-tobacco physician scheduled to sit next to a well-known and very successful (but chain-smoking) publishing executive . . . and since those were the days when you could smoke cigarettes indoors, it was not surprising that neither guest was all that happy. As I remember, the executive left before dinner and the doctor ended up with an empty seat next to him. Oh well. Another time we sat a well-known physician\u2014very traditional and set in his ways\u2014next to a naturopathic physician. The naturopath promptly began making a case that more physicians needed to learn more about diet and nutrition in medical school, instead of simply learning to dispense prescription drugs for illness as the first line of defense. No one left in a huff that night . . . but the conversation was a lot spicier than the food.\n\nOnce you assemble your guests at their assigned seats, explain the purpose of the dinner\u2014to bring together some of the most fascinating people you could find to share unique insights and ideas, have compelling conversations, and enjoy some excellent food at the same time. Or whatever your vision might be.\n\nPrepare an index card from each guest's bio or C.V. and make it your job to introduce each of your special guests, with the intention of having everyone at the table be _interested_ in each person rather than allowing each guest to share their own bio, which would only make them _interesting_. And while this is subtle, reading their bios for them rather than having them introduce themselves is far more powerful. It's always better to have others brag about you than to do it yourself.\n\nAfter reading all the bios, tell everybody that it is time to enjoy their dinner, enjoy their neighbor, and if you want to steal from my playbook, remind them that they are all strategically placed at the table so they will have nothing in common with their dinner mates (jokingly emphasizing how strategically you have actually placed them).\n\nThen let them know that after dinner, you will play \"pass the mic,\" with each guest sharing something in their area of expertise for a few minutes. For those who tell you they are not really experts in anything (which is never true), hold your thumb and forefinger an inch apart and remind them of the Dick Benson premise that \"Everyone knows everything about this much.\" (See below for more on this under \"Everyone Belongs.\")\n\nIf you like, have a few questions prepared for specific attendees if there is a hot topic you would like to discuss with the table.\n\nDinner Follow-Up\n\nAfter the dinner conversation is completed and before the guests leave, promise everyone that they will get a list with the contact information of all the attendees. In addition, if any of your guests have a book, an article, or material they want to share with the group, invite them to send that to you so you can include those extras in the \"goodie bag\" that becomes the follow-up package from the dinner (including the contact info). A bulky package beats a follow-up e-mail every time.\n\nEveryone Belongs\n\nDon't allow anyone to play small at your dinners. As you create bigger and better dinners, someone is bound to look around and wonder if they really belong. I say shut that down immediately, having fallen into the same trap myself. If this does happen, explain why they belong as much as anyone and that you are very excited that they are there, and that you can't wait to hear what they were going to share. If they don't believe you, emphasize to them that everybody is an expert in something, and when you share your expertise and your passion in a way that is totally giving, extraordinary things happen for everyone around you. And that's precisely the reason why dinners like these are so profound in their simplicity.\n\nA small but important point: it's critical that there is no outside noise and that you have complete privacy. And one other crucial detail: the dinner must be at one long table so everyone can see everyone else at all times. Always close the table, with the hosts (or host and a VIP) at the head and tail. It gives the room a feeling of much more community when it's closed. I don't recommend more than 30 at one of these events, and I have found that 15 to 20 is an ideal number.\n\nOne of the most incredible experiences of my career was when I had 67 people at a VIP dinner at the Titans of Direct Response event\u2014yes, at one long table. At the other end of the table from me was marketing icon and Titans keynote speaker Dan Kennedy. I would have needed a bullhorn to get his attention, but, thank goodness, we had a microphone system when we played \"pass the mic.\" We didn't get to everyone around the table that night, but we did teach everyone how to set up an intentional dinner as a tribute to Marty, who invented them.\n\nFrishman sums it up perfectly:\n\nStart with the best people you can reach. Invite your most interesting, enjoyable, entertaining friends and contacts. Invite people you've heard about but don't know. Invite people whom you've wanted to meet. Select guests who are experts in fields that interest you and in areas that you know nothing about. It doesn't have to be all talk. If you know musicians, poets, or entertainers, then ask them to your gathering to liven up and diversify the mix.\n\nWork on a scale that you can afford. Although good food certainly helps, great people should be your top priority. So when you're starting out, think first about the quality of your guests, about attracting the very best people. And if the food is a way to attract heavy hitters, do whatever you can, within your means, to get them to the table. Then get them talking.\n\nPrepare and ask questions that will make your guests expound. Once they're talking, sit back and let the magic work.\n\nYou'll be amazed at what can happen when you add intentionality and structure to something as simple as a dinner.\n\nA SIMPLER (BUT EQUALLY POWERFUL) VERSION OF INTENTIONAL DINNERS\n\nEven if the idea of organizing one of these dinners seems a bit overwhelming or you are not comfortable doing something quite that large, there is another format I love (since I never want you to waste an opportunity to learn and grow with every meal . . . and I am not talking about your waistline).\n\nTwo amazing entrepreneurs I am lucky enough to consider great friends, Michael Fishman and Ramit Sethi, invented their own version of an intentional dinner format that I will call \"The Six-Person Anytime\/Anywhere Dinner.\" This works best when you have a planning partner (as Michael and Ramit had in each other). Each person invites two superstars in their world that the partner has not met before or doesn't know well. You end up with six people in a single conversation for the entire dinner. It's okay in this format for each person to do their own intro (with prompts from the hosts). My favorite prompts are:\n\n  * What is your superpower?\n  * What is your biggest success?\n  * What is your biggest failure?\n  * If you could have dinner with anyone in history (besides the other five people at this table!) who would it be, and why?\n\nOf course, you will have many more questions of your own as people start talking.\n\nImperative for The Six-Person Anytime\/Anywhere Dinner is a round table and a quiet spot in a restaurant (if you can't do it in a private room). These smaller dinners are so easy to do, and you can do them as often as you like. For example, whenever I attend a large event or seminar and I did not have time to plan anything in advance, I always look for five other people that I know I can learn from and create a Six-Person Dinner on the fly.\n\nI think you can see that something as simple yet profound as these intentional dinners can be a pretty hefty deposit into your relationship capital account on a regular, consistent basis.\n\nI'll spend the rest of this chapter discussing five additional concepts that will contribute to investing in your own relationship capital. I think each of these can be adapted with all you do in business\u2014and in your personal life, too.\n\n1. EMBRACE COMPETITION AS COEXISTENCE\n\nWhen we approach business with a spirit of generosity and humility, everyone we come into contact with has to up their game. It's good for us individually, and it's good for business. This is one of the reasons I have a portrait of legendary New York Yankees pitcher Mariano Rivera in my office, even though I'm a die-hard Mets fan. (Those of you who are not from New York may not know that Yankees fans and Mets fans are super competitive with each other and it's understood that you must choose whether to be a fan of one or the other but never both.)\n\nWhile I never really root for the Yankees, Rivera is still my favorite player of all time because he embodied this philosophy of competitive coexistence. He is the best closer the game has ever seen. (A closer is the pitcher responsible for getting the last three batters out when your team is ahead\u2014and they are the hardest three outs, since getting them means you win and not getting them means you lose.) What is fascinating about Rivera is that he was the best at this high-pressure job in the history of the game, yet he only had one pitch (most pitchers have an arsenal of different pitches, like a fastball, a curveball, a changeup, etc.). The one pitch Rivera had\u2014called a cut fastball\u2014was lethal. It looked good enough to hit until the last second, then it dropped down drastically as a batter tried to hit it. The result was usually Rivera 1, batter 0. Much of the time, he was unhittable.\n\nWhat is extraordinary, however, is that he was an open book about his superpower of throwing this unhittable pitch with every player and coach that came to him for help\u2014whether they were on his team or not. Instead of guarding his incredible ability and refusing to share it, he openly shared how to throw it to anyone who asked. He knew that his role wasn't just to win games, but to make everyone, and the sport, better. A rising tide lifts all boats. While I can't prove this, I also think he knew that no one could really throw the pitch as well as he did (even with personal instruction from him); and if for some reason they did learn how to do it better, he'd have a worthy competitor to surpass instead of just being the best with no real competition.\n\nIt's the same in business. If you share your secret sauce, and you're generous with your time and insights, everyone in your industry will have to raise their standards. Sharing your best stuff freely will inevitably help your customers trust you more and make your competitors respect you more. And an added bonus is that your ego will be kept in check\u2014always a good thing.\n\nCopywriting great Gary Bencivenga was Rivera-like in terms of teaching his most valuable writing secrets. And since he still had a formidable track record of getting winners against all the top copywriters he ever competed with, I know that embracing this concept only made him better.\n\nI don't mean to depress you with another take on this, but whether you like it or not, a hundred years from now (no matter how young you are today), we will all be in the ground with lots of dirt on top of us. Taking your best work and best ideas to the grave might be okay for you, but it's not for me. I only say this to put competition into perspective. Facing down your own mortality has a way of helping you stay calm and let go of having to constantly keep what you think are secrets to yourself.\n\n2. DON'T MAKE YOUR LAWYER RICH\n\nSome of my best friends are attorneys. I also know many recovering attorneys who I like a lot too. However, even though we all need legal advice from time to time, I will submit to you that there are better ways to do business than living lawsuit to lawsuit and always thinking \"adversarial\" rather than \"peaceful coexistence.\"\n\nThe story that follows proves that it's a lot harder to build relationship capital when you immediately think \"problem first\" rather than \"potential opportunity first.\" And it's also a lot harder if you are one of those people who will die on a sword over \"the principle of the thing.\" If you tend to hold on to things at all costs to prove a point, I encourage you to read on, and if you don't, read on anyway, because I think this story will reinforce how we should always try to turn lemons into lemonade as our _first_ course of action.\n\nWhen I hosted the Titans of Direct Response event in 2014, one of the first things I needed to do was to create a logo\u2014so I went to an online design site and held a contest. I ended up with something I loved, with colors and a font that fit the magnitude of the event, and a titan's shield that was powerful yet subtle.\n\nUsing the event as a springboard, I branded everything I have done since under Titans: Titans Mastermind, Titans Master Class, Titans DVDs, and Titans Marketing (which I was even able to trademark). After launching Titans Marketing about a year after the 2014 event, a friend e-mailed me to say that there was an up-and-coming online marketing company calling themselves Titans of Marketing.\n\nDespite having a trademark, I don't _own_ the word \"titan,\" and the term gets around: it's been used by NFL teams past and present, by various gurus in other industries, and even by a company that places advertising at bus stops in New York City. But Titans of Marketing? That sounded too close for comfort. So I went to their website, and while they were not competing directly with me, what was shocking was their logo\u2014the colors and font were identical to mine, but instead of the shield, there was a Spartan helmet. What was also strange was that the helmet logo was from a design given to me during my own design contest, which I rejected in favor of the shield. What to do?\n\nMy attorney suggested we send a cease and desist letter, which I considered . . . but not for very long. Others suggested I sue them outright (and not all those folks were lawyers, mind you). Still others told me to forget about them, saying that anyone who would show such little imagination and ingenuity would not stand the test of time and they would be out of business before we knew it.\n\nI decided not to take any legal action, which was easy, based on all the lessons I had learned up until that point regarding the waste of time and money a lawsuit entails. After 30-plus years of watching people rip off promotions, copy, and titles (and more) from right under our noses and the noses of friends and colleagues alike, I knew that chasing every little infraction by taking some kind of legal action was only a distraction and would not be a good use of my energy or time.\n\nAnd of course, pursuing stuff like this is a great a way to add to the billable hours of your friendly attorney, often with little or no return on that money. When legal situations arose throughout my career, I did a simple calculation in my mind of not only the money going out to an attorney but the opportunity cost and money lost by not staying focused on marketing. I remember a comment from Marty once when we were dealing with a legal situation that involved a rip-off: \"I don't want lawyers running our business. I just want reasonable and intelligent lawyers helping us do the right thing. And suing will never be our first option.\"\n\nFor a while I decided to just forget about those \"other Titans,\" until one day I discovered that one of the partners in that business had sent me a friend request on Facebook (which I hadn't gotten around to accepting yet)\u2014and he had even liked a post of mine.\n\nI decided to reach out to him directly. No lawyer required. I privately messaged him on Facebook, asking him why he would be so lazy with the naming of his company, and especially with the development of his typeface and logo. I wrote the note assuming he had heard of my epic event, and that he wanted to cash in on the tsunami that was the Titans brand in direct marketing. Well, he clearly was not aware of my press clippings . . . he didn't even know who I was or what I was talking about. I got a good lesson in getting over myself. But that's not the biggest lesson from this story.\n\nOnce we connected and I showed him what I was talking about\u2014my logo and branding next to his\u2014he was shocked. Together we figured out what happened, and both of us learned two very valuable lessons.\n\nLesson #1 (Micro)\n\nWhen you go to a public site to get work from designers who are not getting paid top dollar, don't assume they are being 100 percent original. It became clear to us that that someone took a good design that failed to win a previous contest (mine) and figured they could recycle it for another contest without changing very much. Of course, \"stealing is a felony; stealing smart is an art.\" I guess this example was neither a felony nor art. It was just opportunistic on the designer's part . . . and the Titans of Marketing were simply caught in the crossfire. I didn't need to accuse anyone of anything.\n\nLesson #2 (Macro)\n\nJumping to conclusions\u2014and jumping to legal action without all the facts\u2014is a waste of time and money. When I discovered the truth, I felt there was now lemonade to be made from lemons. I scheduled a call to learn more about these competitive Titans, their business, their overall mission, and their future plans as coexisting Titans.\n\nWhat I found were two young, ambitious entrepreneurs\u2014full of enthusiasm for marketing and the direct response business we all love\u2014actually, the kind of marketers I regularly seek to attract to my tribe. On the call, they volunteered to change their name before I even asked, saying they would be going in a completely new direction with their branding and their logo. They seemed inspired to raise the bar higher for their business after the call, and I was inspired to help them.\n\nI offered my consulting services at no charge to create their new brand, and I began asking more about what they offered to their clients and how we might even work together. It turned out that their skill set had tremendous value to various clients and mastermind members I was serving at the time. And while I didn't ask them for anything specific in return, they offered to sell, as an affiliate partner, the $2,000 product from the Titans of Direct Response event I had created (a package of DVDs, interviews, and swipe files that were sold to those who were unable to attend). And of course I offered them an affiliate commission for every one they sold.\n\nThese guys went from being the jerks ripping me off to my newest affiliate partner. This felt much better than paying legal fees, and I'm sure they would agree that the last thing they would have wanted was the stress and cost of legal action from me.\n\nAnd the story goes on . . . I spoke at a large online marketing conference some years later, and one of the partners from the former Titans of Marketing was sitting in the front row while I was on stage. After the event, he was one of the first people to post on Facebook saying nice things about me (and no, it had nothing to do with the affiliate commissions I had paid him in the past). He went from potential adversary to friend for life.\n\nThis is not an isolated incident. I have had many experiences over my career that speak to this theme. Many don't involve legal issues or lawyers, but they all work on the premise that assuming the worst of people as your first reaction is _not_ a winning formula for a peaceful life. Or for a relationship capital account you can access at any point in your career.\n\nI even had a dose of inspiration from my mentor on this topic on one of the saddest days of my life. Marty was on his deathbed, and I couldn't believe it when he asked what I was working on. The man was unstoppable, right to the end. I told him about a few tricky deals that were going on at the company, and through a difficult breath he just said, \"Be fair.\"\n\nObviously Marty had taught me that lesson over the previous 30 years too, and it stays with me every day. Don't always think in terms of getting the advantage, whether it's your partners, employees, consultants, or competitors\u2014they all deserve respect and fair treatment, even on the days when it's the last thing you feel like giving them. Keep litigious thinking out of all you do. Of course, sometimes you may actually need a lawyer . . . but above all, be fair.\n\n3. REMEMBER THAT SOMEONE IS ALWAYS WATCHING\n\nOn January 1, 2000, my clock and computer continued to work (despite the fear that the world would end with the coming of the new millennium) . . . and later that month, I traveled to Los Angeles for Jay Abraham's \"all superstar event\" dedicated to some new marketing medium called the Internet. Jay\u2014along with fellow direct response rock stars Audri and Jim Lanford\u2014created an event like no other at the time. They called it the Billion Dollar Internet Strategy Setting Summit. Not a big promise at all, right?\n\nDespite the hyperbole, it sounded interesting. And after all, it was Jay. Only someone with his experience could legitimately put a billion-dollar claim on an unproven medium . . . and he delivered a massive amount of value. I learned a ton at the event and made some lifelong friends there as well, many of whom became important people that I've followed and learned from over the years, including Audri and Jim Lanford, Paul Hartunian, Dana Blankenhorn, Jim Sterne, Corey Rudl, and of course, Jay Abraham. And unbeknownst to me, there were people watching me back then who I wasn't even aware of\u2014future superstars who became life changing for me much later.\n\nThirteen years later, a LinkedIn request hit my inbox out of the blue from a guy by the name of Jeff Walker. I had heard of Jeff and admired him, but I had no idea that he was at that event back in 2000. If you don't know Jeff by name, you may know his Product Launch Formula process\u2014which has been responsible for close to a billion dollars' worth of online sales of products and services to date (I guess it proves that Jay knew what he was talking about when he named the seminar).\n\nJeff is a marketing genius and one of the real good guys in this business. He's got a heart of gold to go along with being a marketing badass (and to me, \"badass\" is at the top of the marketing food chain). But if not for the Abraham-Lanford event (and my philosophy of responding to LinkedIn requests), Jeff would have remained just one more amazing person I looked up to in this fantastic industry.\n\nMy response to the LinkedIn request from Jeff was \"Is this _the_ Jeff Walker?\" And to my surprise, I received a response back from Jeff saying he had followed my career for many years and he was a fan. Who knew? I surely didn't. I had known of him for a long time, but at the time of his LinkedIn request, I can't say we had ever truly connected . . . so when I asked him how he knew me, he told me this story.\n\nHe had scraped and clawed his way to Los Angeles to attend Jay's event in 2000 (with no idea how he was going to afford it) to learn for himself how to create a billion-dollar Internet strategy. He said he watched me moderate a roundtable discussion on something at that event and had been impressed. He couldn't believe how much I was sharing without any expectation of anything in return, since I was a paying attendee and not technically a speaker. I had no recollection of this at all . . . but it wasn't simply dumb luck that I was on my best behavior when Jeff was observing me. Consistency, congruency, integrity, being yourself at all times, always showing up with your best, no matter the situation\u2014those qualities win every time. And as this crazy situation shows, you never know who's watching and what opportunities might turn up for you when you are playing a long game.\n\nThis reconnection to Jeff on LinkedIn led to an invitation to his Platinum Plus mastermind group in January of 2013, where I met and interacted with 30 of the most successful folks in Jeff's tribe. I fell in love with all of them. They all just gave and gave . . . and still give and give. Jeff attracts the best of the best marketers with the biggest hearts. In addition, despite most of them being a lot younger than me, they knew as much (or more) about direct response marketing (and how it drives all marketing, regardless of channel) as I did. I probably had 20 more years of experience than most of them, but even so, I was by no means the smartest person in the room\u2014so I knew I was in the right room.\n\nMy experience with Jeff's mastermind made me join the group as a paying member, and I have never looked back. Being a member has been equivalent to studying for my Ph.D. in online marketing (and entrepreneurship). I should have my degree in a few decades. Not only have many of the members of that mastermind been guest speakers at my own mastermind group, I have done a considerable amount of business with others\u2014and it's where I got to know Reid Tracy at a much deeper level, which led to this book being published by Hay House. All that from just being a nice guy and sharing at a roundtable, not knowing everyone who was there but knowing that everyone who was there mattered.\n\nAnd I'm so glad that the world didn't end on December 31, 1999.\n\n4. SEND CHRISTMAS CARDS IN JULY\n\nWhen the doorbell rang early one December morning, I was reluctant to answer. But I could see a guy through the window wearing a brown suit (not one of my favorite fashion colors), and when I made out the UPS logo on his shirt and saw his matching brown truck in my driveway, I ran to open the door. Whatever he had for me was going to be better than any e-mail or PDF I was reading or downloading. Packages are welcome on my doorstep all year round\u2014how about for you?\n\nThen later in the day, I waited in a very long line at the post office behind a guy with a shoebox full of Christmas cards stamped and ready to go. I thought it was kind of early for that, being the week after Thanksgiving, but I admired his preparation. And it got me thinking.\n\nIf you are one of those people who sends cards to everyone important in your life sometime before Christmas, do you also stay in touch all year round? Kudos to you if you do that, since I believe that true contribution and connection with the most important people in your life is not a one-time-a-year thing; and I also believe it's much more powerful when we go beyond just sending cards.\n\nI am not anti-Christmas or anti\u2013New Year, so please don't call me a grinch, but I would like you to consider a different take on all of this. Thinking you have deepened your relationships with everyone in your address book simply by sending a card might be worth reconsidering. It's not the worst thing you can do, but there are additional meaningful things you may want to consider.\n\nHow many of the cards that you receive during the holidays are memorable? Do you save them? Do they enhance your life beyond that moment? The beauty of thinking about everyone in your life all year round and not having to look for an excuse (or holiday) to connect is that it truly builds relationship capital. And I will add that the lumpier the packages you send, the better\u2014but anything is better than nothing. E-mail is nice . . . a physical card is better . . . and something that barely fits in the mailbox is best.\n\nHere's a quick story to illustrate why \"lumpy direct mail\" is also not just for folks over 50. I was with a client a few years ago and everyone around the table was younger than me (this happens more often these days . . . which I love). I asked the youngest person at the table (she was in her mid-20s), \"Do you own a mailbox?\" Thank goodness she said yes, because I was afraid my next question was going to be \"Do you know what a mailbox is?\"\n\nThen I asked, \"When you go to your mailbox, if there is something in it that is thicker, bulkier, and not in a traditional envelope (and addressed by hand), would you open that first when you get back inside?\" She said she actually opens stuff like that _before_ she gets inside, way before checking her other mail . . . and way, way before she checks her e-mail or phone. Remember, your least crowded inbox is not in your computer.\n\nThis focus group of one told me it's possible that sending three-dimensional packages to the people we love most might not have a minimum age requirement. I have since checked this out with other 20-somethings and my research has proved the theory that they also like big packages in their mailboxes. Lumpy packages have stuck out in crowded mailboxes for decades. They stick out even more now in less crowded mailboxes. They're fun for all ages in all seasons. And they are especially powerful when personalized.\n\nIn addition, assuming that your customer doesn't live in an area where parcels are routinely stolen, your package and the message will actually get delivered. With impact. And keep in mind that the United States Postal Service is often kinder and gentler than whoever is responsible for delivering your e-mail, and they will not label your physical mail as \"spam\"\u2014it will almost always get delivered.\n\nAs you have learned, list segmentation rules . . . whether it's a huge subscriber list or buyers list or a list of business VIPs or just friends and family. Sending thoughtful gifts and small, targeted mailings is just another form of segmented, personalized direct mail that anyone can use effectively.\n\nThe Power of Regifting (To Get You Started)\n\nBelieve it or not, you can make a huge impact without paying anything _except_ the postage . . . and if you think you look like a cheapskate when you regift, think again. Let me give you some examples that might inspire you to think of ways to deepen your contributions to the VIPs in your life without ever going to a store or logging onto Amazon. Hopefully you will see in the examples below that it is always the thought that counts and that the gift can simply be part of the guaranteed delivery system of the thought. Think \"presence\" over \"presents\" . . . but do both with your gifting and regifting.\n\nReading is fundamental.\n\nEvery book you receive as a gift that you own already is thoughtfulness ready for a mini direct mail campaign. When I receive a duplicate book, I immediately think, _Who in my life would this help, enhance, or create value for?_ It's the note with the book that shows how much you care and how much you were thinking about them . . . and if you need to admit it's a \"regift,\" feel free.\n\nSugar is sweet but not for me.\n\nI often receive a lot of food that I don't eat or can't eat (e.g., sweets). I recently received something delicious from an exotic place, which was very thoughtful\u2014I still sent a thank-you note. Unfortunately, the delicacies in question were not on my diet. But I remembered a good friend of mine had just vacationed in the same exotic place, so I sent them to him with a special note saying how much he meant to me.\n\nPromote the heroes in your life.\n\nWhen I was hosting one of my first mastermind meetings, I presented two (clean and new) T-shirts I'd received for supporting a wonderful volunteer organization to two of the members who sat on the hot seat during the meeting. They were asking for feedback and advice from the group, specifically about how they could add more charitable giving into their business models. These \"regifts\" were presented in public and in person (I saved on postage too!) . . . and I included personal notes so I could tell each of them how much I admired their missions and commitment to giving.\n\nThe endorphin rush.\n\nWalking to the table in my office where I pile up my gift packages every day for different people in my life may not register as exercise on my Fitbit, but every package makes someone else bigger and stronger, with care and intention. And there is no bigger achievement for me each day than figuring out how I can touch someone with something special just for them. The reason I needed to hire an assistant who could come to my house once a week (no virtual assistant in the Philippines for me) was precisely to take those packages to the post office. I love this kind of \"direct mail\"\u2014but I don't love standing in line at the post office.\n\nNothing is a throwaway.\n\nThis is not about \"one person's garbage is another person's treasure\". . . and of course, giving your unwanted goods and clothes to charity is always a good thing to do. But at least once a day, if I have not created a package for someone, I walk around my office or check my bookshelves or go to my storage closet and see what reminds me of who. As the great Jay Abraham taught me (and what this last chapter is all about), \"Your relationship capital is the most valuable asset in your portfolio.\" And one huge way to achieve \"compound interest\" on that account is to touch everyone in your life in the most meaningful ways and do it whenever the mood hits you\u2014or whenever you've got some \"junk\" lying around.\n\n5. STAY HUMBLE, EVEN WHEN YOU THINK YOU'VE GOT IT MADE\n\nWhen Gary Bencivenga (the aforementioned Mariano Rivera of copywriting) retired, he hosted an event called the Bencivenga 100. It was a two-day farewell tour for the world's top copywriter, sharing everything he had learned along the way. It was $5,000 per seat and attended by a who's who of direct marketing and copywriting. It was truly incredible\u2014one of those events that, years later, everyone says they went to, even if that's not quite true. It's sort of like those epic sports events that everyone says they went to, which makes you wonder how 500,000 people could fit into a 50,000-seat stadium.\n\nThe first day of the Bencivenga 100 was on May 20. I remember because it was my birthday. I hate my birthday but I love baseball, and it was my lucky day: my beloved Mets were playing the hated Yankees . . . and I had a ticket. I was sitting in the upper deck directly behind home plate, and for the first time in my life, after attending hundreds of live baseball games, I caught a foul ball, hit by the best Mets player at the time, Mike Piazza. I was thrilled. But at the same time, I categorize things like souvenir baseballs the same way I categorize mugs and cute little boxes from faraway lands. Put simply, they are just dust catchers. So . . . what to do with the baseball?\n\nI came up with a killer idea. The next day, at day two of the Bencivenga 100, I took the baseball through the crowd during breaks and got every copywriter I knew to sign it. It was an all-star team of copywriters who were attending, including Gary Halbert, John Carlton, Parris Lampropoulos, David Deutsch, Jim Punkre, Richard Armstrong, Clayton Makepeace . . . and I know if I keep going, I will just insult everyone I don't name. Suffice it to say that anyone who was anyone in direct response copywriting was at the event, and I got all of them to sign the baseball. I was then privileged to have the opportunity to present the baseball to Gary after the afternoon break in front of the entire crowd.\n\nBut here's the hard lesson I learned from it later on:\n\nA few years after the event, a copywriter I had never heard of sent me a LinkedIn request. Per my LinkedIn procedure, I checked out his profile and sent a personal e-mail with my usual goal of creating potential synergies right at the beginning of our relationship. He sent me a beautiful response, telling me how much he admired Boardroom, and that he had followed my career for many years. He then said he was actually at the Bencivenga 100 event (and he wasn't bluffing), and that when I was walking around the room on day two collecting those autographs for Gary's baseball, \"you made eye contact with me, sized me up, and then just kept on walking.\"\n\nI will never forget how that hit me right in the gut . . . I must have seemed like such a jerk. Fortunately for me, he wasn't bitter or angry. He admitted that he was a rookie in the business at the time, and that getting to that event had been a launching pad for his career. But it made me think about all the people we meet (or go out of our way _not_ to meet) every day. We make judgments that some people are more \"useful\" to us than others.\n\nBut it's important for all of us to realize that everyone we come into contact with has a story, a contribution, and the ability to add value to our lives and the lives of others. And this is even truer as we attain more fame and fortune ourselves. It's so easy to get lost in the echo chamber of success, of people fawning over you and nodding along to everything you say like you can do no wrong. Keeping both feet firmly on the ground is critical to maintaining your good reputation and to thinking clearly. That's why I'm a broken record on having people around you who will tell you the honest truth and save you from your own ego.\n\nI thought a lot about that copywriter at the Bencivenga 100. Surrounded by hall of famers, he chose not to be intimidated, seeing the opportunity right in front of him and how far he had come, instead of how far he still had to go. That's the kind of person I want around me . . . and I could have had him (and many others like him) as part of my universe much earlier if I had been more attentive and more humble and had asked more questions along the way. Well, better late than never. It's never too late to make up for a lost contribution.\n\nIf you're ever on the receiving end of this kind of interaction, let me add this from Nelson Mandela's 1994 inaugural speech, a quote that I believe originally came from Marianne Williamson: \"Our deepest fear is not that we are inadequate; our deepest fear is that we are powerful beyond measure.\" Have no fear . . . you are not inadequate.\n\nAnd don't think that even today I am never in a situation like that young copywriter was at the Bencivenga 100. I never hold it against someone for not noticing me, because it's up to me to get noticed. Some people (like me at that event) can just be a little slow to notice your greatness. Keep holding up your end of the bargain (i.e., being powerful beyond measure), and put in the hard work, because remember, you never know when someone else has their eye on you.\n\nKEY TAKEAWAYS\n\nSuccess in business is not just about the numbers. It's also the result of surrounding yourself with remarkable people who lift you up\u2014and being that kind of person for others. The more proactively you build your connections with other great people, the more growth you will see, both personally and professionally.\n\n  * Be a strategic contributor, and remember that personal connection in business trumps just about every other strategy for professional growth.\n  * Don't connect superficially, or simply \"network\"\u2014always put your best effort into your interactions with people, whether they are new relationships or longstanding ones.\n  * There is a compound interest that develops on your relationship capital. The value of your relationships increases over time, in terms of the fulfillment you get from them as well as the potential for personal and business growth and opportunity. Relationship capital is an investment for a lifetime.\n  * Be proactive in building interesting relationships: intentional dinners are a fun and effective way of doing this. When creating your intentional dinners, make your invite list carefully, plan the logistics and seating thoroughly, and make sure there is a way for guests to connect and follow up with each other after the event.\n  * Beyond intentional dinners, there are five simple ways to invest in your relationship capital: \n    1. Embrace competition as coexistence.\n    2. Don't make your lawyer rich.\n    3. Remember that someone is always watching.\n    4. Send Christmas cards in July.\n    5. Stay humble, even when you think you've got it made.\n\n\u25a0 \u25a0 \u25a0\n\nAs we come to the end of this book, I'm reminded of a wonderful proverb: \"We make a living by what we get, but we make a life by what we give.\" As an entrepreneur and marketer, you have an incredible opportunity to contribute your unique genius to the world.\n\nRemember, your list, the people on the receiving end of your business, is the most important asset you will ever have.\n\n  * Treat those people with the care and respect they deserve.\n  * Make them great offers that will truly serve them.\n  * Invest in the world-class creative that will communicate those offers in ways that will resonate with them\u2014as individuals\u2014as much as possible.\n  * Run your campaigns by the numbers, measuring everything.\n  * Be committed to the fundamentals of direct response, never being afraid to dig deep into the original source material, where it all came from\u2014it will only make you a better marketer in the future.\n  * Always keep RFM and LTV top of mind.\n  * Protect your business by diversifying and using multiple channels; invest heavily in your customer support and fulfillment; and always overdeliver.\n  * And finally, be in the game for the long run and play fair. You'll sleep better, and you will be richer for it in every way.\n\nFURTHER READING\n\nDIRECT MARKETING (ONLINE AND OFFLINE)\n\nBrunson, Russell. _Expert Secrets: The Underground Playbook for Creating a Mass Movement of People Who Will Pay for Your Advice_. New York: Morgan James Publishing, 2017.\n\nBrunson, Russell. _DotCom Secrets: The Underground Playbook for Growing Your Company Online_. New York: Morgan James Publishing, 2015.\n\nCaples, John. _Tested Advertising Methods. 4th ed_. Englewood Cliffs, NJ: Prentice-Hall Career & Personal Development, 1982.\n\nDworman, Steven. _$12 Billion of Inside Marketing Secrets Discovered Through Direct Response Television Sales: And how you can profitably apply them to your business._ Beverly Hills, CA: SDE, Inc., 2004.\n\nGodin, Seth. _Permission Marketing: Turning Strangers into Friends and Friends into Customers._ New York: Simon & Schuster, 1999.\n\nHatch, Denny. _Method Marketing: How to Make a Fortune by Getting Inside the Heads of Your Customers._ Chicago: Bonus Books Inc., 1999.\n\nHatch, Denny, and Don Jackson. _2,239 Tested Secrets for Direct Marketing Success: The Pros Tell You Their Time-Proven Secrets._ Chicago: NTC Business Books, 1998.\n\nLevesque, Ryan. _Ask. The counterintuitive online formula to discover exactly what your customers want to buy...create a mass of raving fans...and take any business to the next level._ Nashville: Dunham Books, 2015.\n\nLibey, Donald R., and Christopher Pickering. _Libey and Pickering on RFM and Beyond: Wisdom, Analytics and Tactics for Improving Multichannel Profitability._ Expanded 2nd edition. West Des Moines, IA: Merit Direct Press, 2005.\n\nMarshall, Perry. _80\/20 Sales and Marketing: The Definitive Guide to Working Less and Making More._ Irvine, CA: Entrepreneur Press, 2013.\n\nPolish, Joe. _Joe's Marketing Book._ Tempe, AZ: privately printed, 2016.\n\nReeves, Rosser. _Reality in Advertising_. New York: Alfred A. Knopf, 1961.\n\nRies, Al, and Jack Trout. _The 22 Immutable Laws of Marketing: Violate Them at Your Own Risk!_ New York: HarperCollins Publishers, 1993.\n\nSimpson, Craig, and Brian Kurtz. _The Advertising Solution: Influence Prospects, Multiply Sales, and Promote Your Brand._ Irvine, CA: Entrepreneur Press, 2016.\n\nKennedy, Dan S. _The Ultimate Marketing Plan: Target Your Audience! Get Out Your Message! Build Your Brand!_ Avon, MA: Adams Business, 2011.\n\nSkrob, Robert. _Retention Point: The Single Biggest Secret to Membership and Subscription Growth for Associations, SAAS, Publishers, Digital Access, Subscription ... Membership and Subscription-Based Businesses._ Tallahassee, FL: Membership Services, Inc., 2018.\n\nStone, Bob, and Ron Jacobs. _Successful Direct Marketing Methods: Interactive, Database, and Customer Marketing for the Multichannel Communications Age_ , 8th ed. New York: The McGraw-Hill Companies, Inc., 2008.\n\nThrockmorton, Joan. _Winning Direct Response Advertising: How to Recognize It, Evaluate It, Inspire It, Create It._ Englewood Cliffs, NJ: Prentice-Hall Inc., 1986.\n\nWalker, Jeff. _Launch: An Internet Millionaire_ ' _s Secret Formula to Sell Almost Anything Online, Build A Business You Love, And Live The Life Of Your Dreams_. New York: Morgan James Publishing, 2014.\n\nCOPYWRITING AND CREATIVE\n\nAccountable Advertising, Inc. \"Gary Bencivenga's Marketing Bullets,\" Marketing Bullets Archives. 2018. <https:\/\/marketingbullets.com\/archive>.\n\nBird, Drayton. _How to Write Sales Letters That Sell: Learn the Secrets of Successful Direct Mail_ , 2nd ed. London: Kogan Page, 2002.\n\nBly, Robert. _The Copywriter's Handbook: A Step-by-Step Guide to Writing Copy That Sells_. New York: Dodd, Mead & Company, 1985.\n\nCarlton, John. _Kick-Ass Copywriting Secrets of a Marketing Rebel._ Reno, NV: Carlton Ink, 2002.\n\nCarlton, John. _The Entrepreneur's Guide To Getting Your Shit Together_. Scotts Valley, CA: CreateSpace Independent Publishing Platform, 2013.\n\nCollier, Robert. _The Robert Collier Letter Book._ East Setauket, NY: Robert Collier Publications, 1937 (revised 1950, 1978).\n\nDeutsch, David, _Think inside the Box._ Delray Beach, FL: Early to Rise, 2006.\n\nEdwards, Ray. _How to Write Copy That Sells: The Step-By-Step System for More Sales, to More Customers, More Often._ New York: Morgan James Publishing, 2016.\n\nFord, Mark Morgan, and Will Newman. _Persuasion: The Subtle Art of Getting What You Want._ Delray Beach, FL: American Writers & Artists, Inc., 2014.\n\nForde, John, and Michael Masterson. _Great Leads: The Six Easiest Ways to Start Any Sales Message_. Delray Beach, FL: American Writers & Artists, Inc., 2011.\n\nHalbert, Bond. _The Halbert Copywriting Method Part III: The Simple, Fast & Easy Editing Formula That Forces Buyers to Read Every Word Of Your Ads!_ Scotts Valley, CA: CreateSpace Independent Publishing Platform, 2016.\n\nHalbert, Gary. _The Boron Letters._ Los Angeles: Bond Halbert Publishing, 2013.\n\nHalbert, Gary. \"The Gary Halbert Letter,\" The Gary Halbert Letter. 2003. www.thegaryhalbertletter.com.\n\nHandley, Ann. _Everybody Writes: Your Go-To Guide to Creating Ridiculously Good Content_. Hoboken, NJ: John Wiley & Sons, Inc., 2014.\n\nHanly, Laura. _Content That Converts: How to Build a Profitable and Predictable B2B Content Marketing Strategy._ Lisbon: Hanly Creative, 2017.\n\nHatch, Denny. _Write Everything Right!: Let the world's highest-paid writers show you the secrets of making readers love your: e-mails, letters, memos, blog, ... website and yes, especially your r\u00e9sum\u00e9!_ Philadelphia: Direct Marketing IQ, 2014.\n\nHopkins, Claude. _My Life in Advertising._ Chicago: NTC Business Books, 1966.\n\nHopkins, Claude. _Scientific Advertising._ Chicago: NTC Business Books, 1966.\n\nKennedy, Dan. _The Ultimate Sales Letter: Attract New Customers. Boost Your Sales._ 4th ed. Avon, MA: Adams Business, 2011.\n\nMakepeace, Clayton. \"The Total Package: Business-Building Secrets for Growth-Obsessed Companies,\" Makepeace Total Package. www.makepeacetotalpackage.com.\n\nMiller, Donald. _Building a StoryBrand: Clarify Your Message So Customers Will Listen_. Nashville: HarperCollins Leadership, 2017.\n\nNicholas, Ted. _Magic Words That Bring You Riches._ London: Ted Nicholas, Chesham House, 1995.\n\nOgilvy, David. _Confessions of an Advertising Man._ Harpenden, Herts, UK: Southbank Publishing, 1963.\n\nOgilvy, David. _Ogilvy on Advertising._ New York: Random House, 1983.\n\nPressfield, Steven. _The War of Art: Break Through the Blocks and Win Your Inner Creative Battles._ New York: Black Irish Entertainment LLC, 2002. Paperback edition.\n\nSchwab, Victor O. _How to Write a Good Advertisement: A Short Course in Copywriting_. New York: Harper & Row, 1962.\n\nSchwartz, Eugene, M. _Breakthrough Advertising._ Westport, CT: Titans Marketing, 2017. Original copyright 1966 by Barbara Schwartz. _(Author's note: This classic could easily be in the Direct Marketing section or the Market Psychology section, since this is the one book that transcends everything we do in marketing. I put it in with Copywriting and Creative as a tribute to Gene Schwartz, possibly the greatest copywriter of all time.)_\n\nSchwartz, Eugene, M. _The Brilliance Breakthrough: How to Talk and Write So That People Will Never Forget You._ Westport, CT: Titans Marketing, 2017. Original copyright 1994 by Eugene M. Schwartz.\n\nSugarman, Joe. _Advertising Secrets of the Written Word: The Ultimate Resource on How to Write Powerful Advertising Copy from One of America's Top Copywriters and Mail Order Entrepreneurs._ Las Vegas: DelStar Publishing, Inc., 1998.\n\nSugarman, Joe. _The Adweek Copywriting Handbook: The Ultimate Guide to Writing Powerful Advertising and Marketing Copy from One of America's Top Copywriters_. Hoboken, NJ: John Wiley & Sons, Inc., 2007.\n\nSugarman, Joe. _Triggers: 30 Sales Tools You Can Use to Control the Mind of Your Prospect to Motivate, Influence and Persuade._ Las Vegas, NV: DelStar Books, 2014.\n\nZinsser, William. _On Writing Well: The Classic Guide to Writing Nonfiction_. 7th ed., revised and updated. New York: HarperCollins Publishers, 2006. First published 1976.\n\nBUSINESS GROWTH\n\nAbraham, Jay. _Getting Everything You Can Out of All You've Got: 21 Ways You Can Out-Think, Out-Perform, and Out-Earn the Competition._ New York: Truman Talley Books \/ St. Martin's Press, 2000.\n\nAgugliaro, Mike. _Nine Pillars 2.0: The Game-Changing Formula to Create Massive Transformation in Your Business and Life._ New Jersey: privately printed, 2017.\n\nBettger, Frank. _How I Raised Myself from Failure to Success in Selling._ New York: Prentice Hall Press, 1947.\n\nChowdhury, Subir. _The Ice Cream Maker: An Inspiring Tale About Making Quality the Key Ingredient in Everything You Do._ New York: Currency \/ Doubleday, 2005.\n\nEdelston, Martin, and Marion Buhagiar. _\"I\" Power: The Secrets of Great Business in Bad Times_. Fort Lee, NJ: Barricade Books, 1992.\n\nFields, Jonathan. _Uncertainty: Turning Fear and Doubt into Fuel for Brilliance_. New York: Penguin Group, 2011.\n\nFrishman, Rick, and Jill Lublin. _Networking Magic: How to Find Connections that Transform Your Life._ 2nd edition, updated. New York: Morgan James Publishing, 2014.\n\nGladwell, Malcolm. _Outliers: The Story of Success._ New York: Hachette Book Group, 2008.\n\nHarnish, Verne. _Mastering the Rockefeller Habits: What You Must Do to Increase the Value of Your Growing Firm._ New York: SelectBooks, Inc., 2006. Paperback edition.\n\nHarnish, Verne. _Scaling Up: How a Few Companies Make It...and Why the Rest Don't._ Mastering the Rockefeller habits 2.0. Ashburn, VA: Gazelles Inc., 2014.\n\nKeller, Gary W., and Jay Papasan. _The ONE Thing: The Surprisingly Simple Truth Behind Extraordinary Results._ Austin, TX: Bard Press; 2013.\n\nKennedy, Dan. _No B.S. Sales Success in the New Economy_. Irvine, CA: Entrepreneur Press, 2010.\n\nKim, W. Chan, and Renee Mauborgne. _Blue Ocean Strategy: How to Create Uncontested Market Space and Make the Competition Irrelevant,_ Expanded edition. Boston: Harvard Business Review Press, 2015.\n\nKoenigs, Mike. _Publish And Profit: A 5-Step System For Attracting Paying Coaching and Consulting Clients, Traffic and Leads, Product Sales and Speaking Engagements_. San Diego: privately printed, 2012.\n\nLee, Ryan. _Passion to Profits_. Connecticut: privately printed, 2011.\n\nLong, David. _Built to Lead: 7 Management R.E.W.A.R.D.S Principles for Becoming a Top 10% Manager._ Las Vegas: Next Century Publishing, 2014.\n\nMasterson, Michael. _Ready, Fire, Aim: Zero to $100 Million in No Time Flat._ Hoboken, NJ: John Wiley & Sons, Inc., 2008.\n\nMcCarthy, Ken. _The System Club Letters: 57 Big Ideas to Transform Your Business and Your Life._ Tivoli, NY: privately printed, 2013.\n\nMcKeown, Greg. _Essentialism: The Disciplined Pursuit of Less_. New York: Crown Business, 2014.\n\nNewport, Cal. _Deep Work: Rules for Focused Success in a Distracted World_. New York: Grand Central Publishing, 2016.\n\nSullivan, Dan. _The Dan Sullivan Question_. Toronto and Chicago: The Strategic Coach, 2009.\n\nTribby, MaryEllen. _Reinventing the Entrepreneur: Turning Your Dream Business into a Reality._ Hoboken, NJ: John Wiley & Sons, Inc., 2013.\n\nVaynerchuk, Gary. _Jab, Jab, Jab, Right Hook: How to Tell Your Story in a Noisy Social World._ New York: HarperCollins Publishers, 2013.\n\nWickman, Gino, and Mark C. Winters. _Rocket Fuel: The One Essential Combination That Will Get You More of What You Want from Your Business._ Dallas: BenBella Books, Inc., 2015.\n\nDIRECT MAIL\n\nBenson, Richard. _Secrets of Successful Direct Mail._ New York: The McGraw-Hill Companies, Inc., 1987.  \n _(Author's note: See page 251 for a sample.)_\n\nBurnett, Ed. _Complete Direct Mail List Handbook: Everything You Need to Know About Lists and How to Use Them for Greater Profit._ Englewood Cliffs, NJ: Prentice Hall, 1988.\n\nGrossman, Gordon. _Confessions of a Direct Mail Guy_. Westport, CT: Titans Marketing (original copyright Gordon W. Grossman), 2006.\n\nHarper, Rose. _Mailing List Strategies: A Guide to Direct Mail Success._ New York: The McGraw-Hill Companies, Inc., 1986.\n\nHatch, Denison. _Million Dollar Mailings: The Art and Science of Creating MoneyMaking Direct Mail, Including Secrets of Using Direct Mail to Make Money on the Internet._ Chicago: Bonus Books, 2001. Originally published in 1992 by Denison Hatch.\n\nHodgson, Richard S., _Direct Mail and Mail Order Handbook._ 3rd edition. Chicago: The Dartnell Corporation, 1988.\n\nKennedy, Dan, and Craig Simpson. _The Direct Mail Solution: A Business Owner's Guide to Building a Lead-Generating, Sales-Driving, Money-Making Direct-Mail Campaign._ Irvine, CA: Entrepreneur Press, 2014.\n\nSchwartz, Eugene M. _Mail Order: How to Get Your Share of the Hidden Profits That Exist in Your Business._ Westport, CT: Titans Marketing, 1982. (Original copyright Boardroom Reports, Inc.)\n\nMARKET PSYCHOLOGY AND INSPIRATION\n\nAnderson, Hans Christian. _The Emperor's New Clothes._ Boston: Houghton Mifflin Company, 1949.\n\nAndrews, Andy. _How Do You Kill 11 Million People?: Why the Truth Matters More Than You Think._ Nashville: Thomas Nelson, 2011.\n\nAriely, Dan. _Predictably Irrational, Revised and Expanded Edition: The Hidden Forces That Shape Our Decisions._ New York: Harper Perennial, 2010. Originally published in 2008 by HarperCollins.\n\nBach, David. _Debt Free For Life: The Finish Rich Plan for Financial Freedom._ New York: Crown Business, 2010.\n\nBach, David. _The Automatic Millionaire, Expanded and Updated: A Powerful One-Step Plan to Live and Finish Rich._ New York: Crown Business, 2016. Originally published in 2003 by Broadway Books.\n\nBurchard, Brendon. _High Performance Habits: How Extraordinary People Become That Way._ Carlsbad, CA: Hay House, Inc., 2017.\n\nBurg, Bob, and John David Mann. _The Go-Giver, Expanded Edition: A Little Story About a Powerful Business Idea._ New York: Penguin Random House LLC, 2015. Originally published in 2007 by Penguin Random House LLC.\n\nCairns, Julie Ann. _The Abundance Code: How to Bust the 7 Money Myths for a Rich Life Now._ Carlsbad, CA: Hay House, Inc., 2015.\n\nCarnegie, Dale. _How to Win Friends & Influence People._ New York: Pocket Books, 1936.\n\nCarty, Thomas. _Backwards, in High Heels: Faith Whittlesey, Reagan's Madam Ambassador in Switzerland and the West Wing._ Oxford: Casemate Publishers, 2012\n\nCialdini, Robert. _Influence: The Psychology of Persuasion, Revised Edition._ New York: William Morrow and Company, 1993. Originally published in 1984 by William Morrow and Company.\n\nCialdini, Robert. _Pre-Suasion: A Revolutionary Way to Influence and Persuade._ New York. Simon & Schuster Paperbacks, 2016.\n\nCoelho, Paulo. _The Alchemist._ New York: HarperCollins Publishers, 1993.\n\nGrant, Adam. _Give and Take: Why Helping Others Drives Our Success._ New York: Penguin Group, 2013.\n\nGraziosi, Dean. _Millionaire Success Habits: The Gateway to Health & Prosperity._ Expanded edition. Carlsbad, CA: Hay House, Inc., 2019.\n\nHallowell, Edward M., and John J. Ratey. _Driven to Distraction: Recognizing and Coping with Attention Deficit Disorder from Childhood through Adulthood._ New York: Anchor, 2011. Originally published in 1994 by Pantheon Books.\n\nHallowell, Edward M. _Success Strategies for the Crazy Busy._ Audiobook edition. Chicago: Nightingale-Conant, 2014.\n\nHemphill, Barbara. _LESS CLUTTER More Life: A Life's Teachings._ Pasadena, CA: True Roses, Inc., 2014.\n\nHogshead, Sally. _Fascinate, Revised and Updated: How to Make Your Brand Impossible to Resist_. New York: HarperCollins, 2016.\n\nHyman, Mark. _The Blood Sugar Solution: The UltraHealthy Program for Losing Weight, Preventing Disease, and Feeling Great Now!_ New York: Little, Brown and Company, 2012.\n\nIkonn, Alex, and UJ Ramdas. _The Five-Minute Journal: A Happier You in 5 Minutes a Day._ Toronto: Intelligent Change, 2016.\n\nKeppe, Norberto. _The Origin of Illness: Psychological, Physical and Social_. Englewood Cliffs, NJ: Campbell Hall Press, 2000.\n\nKlaff, Oren. _Pitch Anything: An Innovative Method for Presenting, Persuading, and Winning the Deal._ New York: The McGraw-Hill Companies, 2011.\n\nMaltz, Maxwell. _Psycho-Cybernetics: Updated and Expanded._ New York: Perigee, 2015.\n\nMeisel, Ari, and Nick Sonnenberg. _Idea to Execution: How to Optimize, Automate, and Outsource Everything in Your Business._ Austin, TX: Lioncrest Publishing, 2016.\n\nMeisel, Ari. _The Art of Less Doing: One Entrepreneur's Formula for a Beautiful Life._ Austin, TX: Lioncrest Publishing, 2016.\n\nPausch, Randy. _The Last Lecture._ London: Hodder & Stoughton, 2008.\n\nRand, Ayn. _The Fountainhead._ With an afterword by Leonard Peikoff. New York: The Bobbs-Merrill Company, 1943. Renewed copyright in 1971 by Ayn Rand.\n\nRivera, Mariano, and Wayne Coffey. _The Closer._ New York: Back Bay Books \/ Little, Brown and Company, 2014.\n\nSethi, Ramit. _I Will Teach You To Be Rich._ New York: Workman Publishing, 2009.\n\nSilver, Yanik. _Evolved Enterprise: An Illustrated Guide to Re-think, Re-image & Re-invent Your Business to Deliver Meaningful Impact & Event Greater Profits._ Washington, D.C.: Ideapress Publishing, 2015.\n\nSilverstein, Shel. _The Giving Tree._ New York: Harper & Row, 1964.\n\nSeuss, Dr. _Oh, The Places You'll Go!._ New York: Random House, 1960.\n\nStephenson, Sean. _Get Off Your \"But\": How to End Self-Sabotage and Stand Up for Yourself._ San Francisco: Jossey-Bass, 2009.\n\nVoss, Chris. _Never Split the Difference: Negotiating As If Your Life Depended on It._ New York: HarperCollins, 2016.\n\nWeintraub, Jerry. _When I Stop Talking, You'll Know I'm Dead: Useful Stories from a Persuasive Man._ New York: Twelve, 2010.\n\nBENSON'S RULES OF THUMB\n\n**Author's note:** I thought it would be extremely useful and insightful to include Dick Benson's 31 \"Rules of Thumb\" that open his classic book, _Secrets of Successful Direct Mail,_ as an addendum to the reading list I have created for _Overdeliver._ Please remember that Dick wrote these rules pre-Internet, yet every one of them has an application to online media.\n\n_\"In defense of these prejudices: I have a lot of scar tissue backing up these principles. I offer them to you with this qualifier: They work for me.\"_\n\n\u2014 DICK BENSON\n\n  1. A two-time buyer is twice as likely to buy as a one-time buyer. Most of the experts I know who issue catalogs, handle circulation for publications, or raise funds by mail know this to be true.\n  2. The same product sold at different prices will result in the same net income per thousand mailed.\n  3. Sweepstakes will improve results by 50 percent or more.\n  4. A credit or bill-me offer will improve results by 50 percent or more.\n  5. Tokens or stickers always improve results.\n  6. Memberships renew better than plain subscriptions by 10 percent or more.\n  7. \"Department store\" pricing always pays except for membership offers.\n  8. You can never sell two things at once.\n  9. Self-mailers almost never work.\n  10. The more believable a special offer, the more likely its success.\n  11. The addition of installment payments for an item over $15 will increase results by 15 percent.\n  12. Dollar for dollar, premiums are better incentives than cash discounts.\n  13. Adding elements to a mailing package, even though obviously adding cost, is more likely to pay out than cheapening the package.\n  14. For magazines a \"soft\" offer (\"Try a complimentary copy at our risk\") is better than a hard offer (cash or \"bill me\").\n  15. A Yes-No option will increase orders.\n  16. \"FREE\" is a magic word.\n  17. Two premiums are frequently better than one.\n  18. Long copy is better than short copy.\n  19. Personalized letters work better to house lists (those who have bought or subscribed before) than to \"cold\" lists.\n  20. Brochures and letters should stand alone and each of them should contain all the information.\n  21. Direct mail should be scrupulously honest.\n  22. Subscriptions sold at half-price for at least eight months will convert at renewal time just as strongly as subscriptions sold for a full year at full price.\n  23. Lists are the most important ingredient to the success of a promotional mailing.\n  24. The offer is the second most important ingredient of direct mail.\n  25. Letters should look and feel like letters.\n  26. An exclusive reduced price to a house list will more than pay its way.\n  27. To predict final results from a promotion, you can assume you will always receive as many more orders as you've received in the past week. This projection will generally be valid beginning with the second week's orders and continuing thereafter.\n  28. A follow-up mailing dropped two weeks after the first mailing will pull 50 percent of the original response.\n  29. An incentive to pay cash when you offer both cash and credit options reduces net response.\n  30. Test-mailing packages are best when they come from independent creative sources.\n  31. Offers of subscriptions using two terms (i.e., 8 months, 16 months) will pull more money . . . but 10 percent fewer orders.\n\n_Nothing works all the time, but ignore any of these rules at your own peril._\nINDEX\n\nA\n\nAbraham, Jay, xxiv, , \u2013, ,\n\nAbundance mind-set\n\ncustomer service and generosity,\n\n100:0 concept and,\n\nAccountability,\n\nAdvertising, direct marketing vs., xxii\n\n_Advertising Solution, The_ (Kurtz),\n\nAffiliate deals,\n\n\u00c0 la carte buying of marketing services, \u2013\n\nAnecdotal evidence, following, \u2013\n\nArrogance, avoiding,\n\nAssets, assessing, \u2013\n\nAttribution, \u2013\n\nAudience. _See_ List management\n\nAuerbacher, Adolph, , \u2013\n\nB\n\nBach, David,\n\nBarrier to switch, , \u2013\n\nBehavior. _See also_ Customers\n\n_Breakthrough Advertising_ (Schwartz) on, \u2013,\n\nhuman behavior and effect on marketing, \u2013\n\ntracking of, \u2013\n\ntransaction data, \u2013\n\nBencivenga, Gary\n\nBencivenga 100, \u2013\n\nbookalog by, \u2013,\n\non controls, \u2013\n\nexpertise shared by,\n\nhumility of, \u2013\n\nBenson, Dick\n\non \"bogey,\" 38\n\non expertise,\n\non list management importance, xxiv\u2013xxv\n\nmultichannel marketing and, ,\n\nreliance on smart people by,\n\n\"Rules of Thumb,\" \u2013\n\n_Secrets of Successful Direct Mail_ ,\n\non testing methodology,\n\nBest practices\n\ncare and concern for customers,\n\nconsequential thinking in messaging,\n\ncopy and creative resources, \u2013\n\ncopywriting perfection,\n\ncustomer feedback,\n\ncustomers' care for your business,\n\ndiscipline of, \u2013\n\nlanguage usage,\n\nlogic line, \u2013\n\nstrategic use of content,\n\nBetuel, Eric,\n\nBillion Dollar Internet Strategy Setting Summit (2000),\n\nBill-me-later offers, \u2013,\n\n_BL\/P_. _See Bottom Line\/Personal_ (Boardroom)\n\nBoardroom Dinners (intentional dinners) model, \u2013\n\nBoardroom Inc. _See also Bottom Line\/Personal_ (Boardroom)\n\n_The Book of Inside Information_ ,\n\n_The Bottom Line Yearbook_ ,\n\n_The Encyclopedia of Estate Planning_ ,\n\ngrowth of, xvii\n\n_Healing Unlimited_ ,\n\n_Health Confidential_ ,\n\n_How to Do Everything Right_ , \u2013\n\nlaunch of,\n\nlegacy of copywriters of,\n\n_The Little Black Book of Secrets_ ,\n\n_Tax Hotline_ ,\n\n_The 30-Day Diabetes Cure_ , \u2013\n\ntrade books repositioned as direct mail by, \u2013\n\n_Boardroom Reports_ (Boardroom)\n\nlaunch of, \u2013, \u2013\n\nniche markets of, \u2013\n\nrebranding of, \u2013\n\n\"Bogey,\" 38\n\nBookalogs, \u2013,\n\n_Book of Inside Information, The_ (Boardroom), \u2013,\n\nBoswell, Thomas, \u2013\n\n_Bottom Line\/Personal_ (Boardroom)\n\n_Boardroom Reports_ rebranded as,\n\ncustomer focus of, \u2013\n\ninfomercials by, \u2013\n\nLTV of customers of, \u2013, \u2013\n\noffers and, , \u2013,\n\n_Bottom Line\/Tomorrow_ (Boardroom),\n\n_Bottom Line Yearbook, The_ (Boardroom),\n\nBranding strategy\n\naggression vs. brand congruency,\n\nfundamentals of personal brand, xxvii\n\noffers and, \u2013\n\n_Breakthrough Advertising_ (Schwartz)\n\nhandwritten letter sent with, \u2013\n\non human behavior, \u2013,\n\nrepublishing of, by Kurtz, xx, \u2013\n\nC\n\nCampaign. _See_ Offers\n\nCaples, John,\n\nCards, for relationship capital, , \u2013\n\nCarlton, John, ,\n\nChange, expecting,\n\nChange of address files (COA), \u2013\n\nCharacteristics of world-class copywriters\n\ncuriosity, \u2013\n\nhumility, \u2013\n\nhunger, \u2013\n\nimportance of copywriting and, \u2013 ( _See also_ Copywriting)\n\npassion, \u2013\n\nportfolio of work and sharing success, \u2013\n\nsmarts, \u2013\n\nunderstanding of direct marketing principles,\n\nCollier, Robert,\n\nCompensation, royalties as,\n\nCompetition\n\nas coexistence, xix, \u2013\n\nstudying controls of competitors,\n\nCompiled lists, \u2013\n\nComplacency, avoiding,\n\nConcept tests (Q tests), \u2013\n\nCongruent marketing, \u2013\n\nConnection (\"Vitamin C\"), \u2013\n\nConsequential thinking in messaging,\n\nContact strategy, \u2013\n\nContent. _See also_ Copywriting\n\nassessing assets for offers, \u2013\n\nstrategic use of,\n\nControls. _See also_ Measurability\n\nbeating, , , , ,\n\ncreativity of, \u2013,\n\ndefined,\n\ngaining new controls, \u2013\n\nstudying controls of competitors,\n\nConversion, \u2013\n\nCopywriting, \u2013. _See also_ Creative\n\ncharacteristics of world-class copy writers, \u2013\n\ncopy and creative resources for, \u2013\n\ncopywriters as key to marketing, \u2013\n\ncreativity of, \u2013\n\nimportance of, \u2013\n\nlanguage usage, , \u2013\n\nperfection of,\n\nrelationship between message and list for, \u2013\n\nresearch for, \u2013\n\nrevisiting discarded ideas for, \u2013\n\nsegment psychology for, \u2013\n\nsupport for, \u2013\n\ntailoring, \u2013\n\nCore Principles\n\ncreativity (#3), \u2013\n\nmarketing as psychology (#1), \u2013\n\noriginal source material and prior success (#5), \u2013\n\nshortcuts (#2), \u2013\n\nswipe files (#4), \u2013\n\nCreative\n\ncongruent marketing and,\n\ncreativity (Core Principle #3), \u2013\n\ncreativity of offers, \u2013\n\ndefined,\n\ntailoring of \u2013\n\nCuriosity, of copywriters, \u2013\n\nCustomers. _See also_ Customer service and fulfillment; Lifetime value (LTV); Multichannel marketing; Relationship capital\n\nbuying behavior of, \u2013, , \u2013, \u2013\n\nfeedback by, , \u2013\n\nmultibuyers, \u2013\n\nrenewals by, \u2013\n\nCustomer service and fulfillment, \u2013\n\nbarrier to switch and, , \u2013\n\nbaseball analogy to, \u2013\n\nfor customer retention, \u2013\n\ndistractions from, \u2013\n\n\"marketing by walking around,\" \u2013\n\nsecret shoppers' insight for, \u2013\n\nD\n\nDatabase. _See_ List management\n\nData cards, \u2013\n\nDecision-making, \"moose on the table\" and,\n\nDeutsch, Dan,\n\nDifferentiation, \u2013\n\nDirect mail\n\ncreative copywriting for, \u2013\n\nexpense of postage for, \u2013,\n\nlist management for, online marketing,\n\n\"lumpy direct mail,\" \u2013\n\nonline marketing vs., \u2013\n\nswipe files for, \u2013\n\nDirect marketing, \u2013\n\nbest practices for, \u2013\n\ndefined, xxii\n\ndirect mail vs., xxvi\n\ndiscipline for, \u2013\n\nfollowing marketing trends and, \u2013\n\nfundamental lessons of, xxiii\u2013xxviii\n\nlifetime value (LTV) and, \u2013\n\nlist management for direct mail vs. online marketing and,\n\nmeasurability of, xxii\u2013xxiii\n\nunderstanding of, by copywriters,  ( _See also_ Copywriting)\n\nDiscarded ideas, revisiting, \u2013\n\nDiscipline\n\nfollowing best practices for, \u2013\n\nvalue of, \u2013\n\nDistractions, dealing with, \u2013\n\nDowns, Hugh,\n\nDworman, Steve,\n\nE\n\nEdelston, Marty. _See also_ Boardroom Inc.\n\non avoiding legal problems,\n\nBoardroom Dinners (intentional dinners) of, \u2013\n\nBoardroom Reports growth and, xvii\n\nBoardroom Reports launch by, \u2013\n\ncelebratory gifts by,\n\ncharacterization of,\n\non consequential thinking,\n\non copywriting,\n\ncuriosity of,\n\nFour Pillars of Being Extraordinary of, \u2013, \u2013\n\nas \"giver,\" \u2013\n\nlegacy of, ,\n\non logic line, \u2013\n\n\"nice notes\" by,\n\non teaching for success, xx\n\nE-mail\n\ncost of,\n\nfor customer relationship, xxv\u2013xxvi\n\nEmployees, happiness of,\n\n_Encyclopedia of Estate Planning, The_ (Boardroom Reports),\n\nEnd users. _See_ Customers\n\nEthics, xxiv\n\nExpertise\n\nBoardroom Dinners (intentional\n\ndinners) model for, \u2013\n\nas fundamental lesson of direct marketing, xxvii\n\nreliance on smart people for, \u2013\n\nsharing, with competition,\n\nF\n\nFacebook, \u2013\n\nFarber, Barry, \u2013\n\nFinancial issues\n\ncompensation with royalties,\n\ndiscipline and, \u2013\n\nof mailing, \u2013\n\nmarketing trends and, \u2013\n\nmeasuring LTV, \u2013\n\nmonetary, defined, \u2013 ( _See also_ Recency, frequency, and monetary (RFM) scores)\n\nrecency, frequency, and monetary (RFM) scores, \u2013\n\ntracking performance, \u2013\n\nFishman, Michael,\n\nFive-minute favor concept,\n\n41\/39\/20 formula, \u2013,\n\nFour Pillars of Being Extraordinary\n\nEdelston and, \u2013\n\nhelping others first (Pillar 4), \u2013\n\noutworking everyone else (Pillar 1),\n\npossessing insatiable curiosity (Pillar 2), \u2013\n\nsurrounding yourself with smart people (Pillar 3), \u2013\n\nFrequency, defined, \u2013. _See also_ Recency, frequency, and monetary (RFM) scores\n\nFrishman, Rick, , ,\n\nFundamental lessons of direct\n\nmarketing\n\ncredibility and transparency, xxvi\n\nof direct marketing measurability, xxvi\n\nethics, xxiv\n\nexpertise, xxvii\n\ninvolvement, xxiii\n\nrelationship management vs. customer importance, xxv\u2013xxvi\n\ntime needed for list management, xxiv\u2013xxv\n\nusing multiple channels for marketing, xxviii\n\nG\n\nGifts, for relationship capital, \u2013\n\n_Give and Take_ (Grant),\n\nGivers, Takers & Matchers concept, \u2013,\n\nGlickman, Ken,\n\nGrace issues,\n\nGrammar, language usage and,\n\nGrant, Adam,\n\nGrossman, Gordon, , \u2013, ,\n\nGuthy, Bill,\n\nGuthy-Renker, , \u2013\n\nH\n\nHalbert, Gary, , \u2013, , \u2013\n\nHallowell, Ned, \u2013\n\nHandwriting, impact of,\n\nHay House,\n\n_Healing Unlimited_ (Boardroom),\n\n_Health Confidential_ (Boardroom),\n\nHelping others first (Pillar 4), \u2013\n\nHiring practices, , , \u2013\n\nHopkins, Claude, ,\n\nHorizontal vertical niche, defined,\n\n_How Life Imitates the World Series_ (Boswell), \u2013\n\n_How to Do Everything Right_ (Boardroom), \u2013\n\nHumility\n\nof copywriters, \u2013\n\nfor relationship capital, \u2013\n\nHunger, of copywriters, \u2013\n\nHurricane Hugo, effect on marketing, \u2013\n\nI\n\nInfiniti, \u2013\n\nInfomercials\n\nadvent of, \u2013\n\ndiscipline needed for, \u2013\n\nIntentional dinners (Boardroom Dinners), \u2013\n\nJ\n\nJayme, Bill, \u2013\n\nJohnson, Arthur, \u2013, \u2013\n\nK\n\nKennedy, Dan, , , ,\n\nKissinger, Henry,\n\nKurtz, Brian. _See also_ Boardroom Inc.\n\nblog of,\n\n_Breakthrough Advertising_ (Schwartz), republished version, \u2013\n\nbriankurtz.me, xxviii\n\nL\n\nLampropoulos, Parris, , ,\n\nLearning from others, \u2013\n\nLegal problems, avoiding, \u2013\n\nLifetime value (LTV), \u2013\n\ncongruent marketing for, \u2013\n\nimportance of, \u2013\n\nas key metric, \u2013\n\nmultichannel marketing and, \u2013\n\noffers and,\n\nrenewals and profit, \u2013\n\ntargeting customers and, \u2013\n\nup-front marketing expenses and, \u2013\n\nLinkedIn, \u2013,\n\nList management\n\nassessing assets for offers, \u2013\n\ncompiled lists vs. response lists, \u2013\n\ncongruent marketing and, \u2013\n\ndevoting time to, xxiv\u2013xxv\n\nfor direct mail vs. online marketing,\n\nlist, defined, \u2013\n\nlist as donation, example, \u2013\n\nlist building and, \u2013\n\nlist history and,\n\nlist managers and list brokers, xvii\u2013xx\n\nrelationship between message and list for copywriting, \u2013\n\nsegmentation and list selection, \u2013\n\ntime needed for list management, xxiv\u2013xxv\n\n\"tribe,\"\n\nunderstanding market for offers, \u2013 ( _See also_ Offers)\n\n_Little Black Book of Secrets, The_ (Boardroom),\n\nLogic line, \u2013\n\nLTV. _See_ Lifetime value (LTV)\n\n\"Lumpy direct mail,\" \u2013\n\nM\n\nMagalogs, , \u2013\n\nMarketing as psychology (Core Principle #1), \u2013\n\n\"Marketing by walking around,\" \u2013\n\nMarketing channels. _See_ Multichannel marketing\n\nMarshall, Perry, ,\n\nMartin, Mel,\n\nMeasurability\n\nof copywriting success, \u2013\n\nimportance of, for direct marketing, xxii\u2013xxiii\n\nLTV as key metric, \u2013\n\nLTV calculations and up-front marketing expenses, \u2013\n\nof multichannel marketing, \u2013\n\nregression modelling for, \u2013\n\nof statistical significance, \u2013\n\ntracking performance, \u2013\n\nMeredith Corporation,\n\nMessaging\n\naggression vs. brand congruency,\n\nconsequential thinking in messaging,\n\nrelationship between message and list for copywriting, \u2013\n\nMonetary, defined, \u2013\n\n\"Moose on the table,\"\n\nMultibuyers, \u2013\n\nMultichannel marketing, \u2013\n\nattribution and, \u2013\n\nconversion and, \u2013\n\nas fundamental lesson of direct marketing, xxviii\n\nimportance of using multiple channels, \u2013\n\nLTV of customers and, \u2013\n\nmeasuring, \u2013\n\none-stop shopping dangers, \u2013\n\nonline to offline to online (O to O to O) for, \u2013\n\nresearching channel of choice for customers, \u2013\n\ntargeting for market and,\n\ntelevision for, \u2013\n\nN\n\nNetworking, \u2013. _See also_ Relationship capital\n\n_Networking Magic_ (Frishman),\n\n\"Nice notes,\"\n\nNiche marketing, \u2013,\n\nO\n\nOffers, \u2013\n\nassessing assets for, \u2013\n\nbill-me-later offers, \u2013,\n\nbranding strategy and, \u2013\n\ncampaign structure and, \u2013\n\ncreativity of, \u2013\n\nfinding niches for,\n\n41\/39\/20 formula and, \u2013,\n\nmeasuring statistical significance of, \u2013\n\npremiums with, \u2013\n\npyramiding, \u2013\n\nQ tests and, \u2013\n\nrepositioning older products as, \u2013\n\nunderstanding market for, \u2013\n\nOgilvy, David, \u2013, ,\n\n1 + 1 = 64 concept, xx\n\n100:0 concept, \u2013\n\nOnline marketing. _See also_ Financial issues\n\ndirect mail vs., \u2013\n\nlist management for, direct mail vs.,\n\nrecency, frequency, and monetary (RFM) scores for, \u2013\n\ntesting for,\n\nOnline to offline to online (O to O to O), \u2013\n\nOpen Church movement,\n\nOperation Desert Shield, effect on marketing, \u2013\n\nOriginal source material, \u2013\n\nbasing work on fundamentals of, \u2013\n\ncreativity (Core Principle #3), \u2013\n\nmarketing as psychology (Core Principle #1), \u2013\n\nprior success of (Core Principle #5), \u2013\n\nresearching, \u2013\n\nshortcuts (Core Principle #2), \u2013\n\nswipe files (Core Principle #4), \u2013\n\nOutworking everyone else (Pillar 1),\n\nOverdeliver, \u2013\n\ncontrols and, \u2013\n\nFour Pillars of Being Extraordinary for, \u2013\n\nGivers, Takers & Matchers concept for overdelivering, \u2013\n\n100:0 concept for overdelivering, \u2013\n\nrelationship management vs. customer importance, xxv\u2013xxvi, \u2013\n\nP\n\n\"Paranoid package\" example, \u2013\n\nPassion, of copywriters, \u2013\n\n_Personal Power_ (Robbins),\n\nPlatinum Plus, \u2013\n\nPortfolio of copywriting work,\n\nPossessing insatiable curiosity (Pillar 2), \u2013\n\nPremiums as offers, \u2013\n\nPromotions, assessing assets for offers, \u2013\n\nPyramiding, \u2013\n\nQ\n\nQ tests, \u2013\n\nR\n\n_Reader's Digest_ , , \u2013\n\n_Reality in Advertising_ (Reeves), \u2013\n\nRecency, frequency, and monetary (RFM) scores, \u2013\n\nbill-me-later offers and, \u2013\n\ncompiled lists vs. response lists, \u2013\n\ndefined, \u2013\n\nfrequency, defined, \u2013\n\nindustry niche (\"table selection\") and, \u2013\n\nlist building and, \u2013\n\nlist importance and, \u2013\n\nmonetary, defined, \u2013\n\nfor online marketing, \u2013\n\nrecency, defined, \u2013\n\nrecency issues and effect on customer service, \u2013\n\nregression modelling and demographic segmentation, \u2013\n\nfor segmentation, \u2013\n\ntargeting and, \u2013, \u2013\n\nReconnection with prior contacts, \u2013, \u2013\n\nReeves, Rosser, ,\n\nRegression modelling, \u2013, \u2013\n\nRelationship capital, \u2013\n\navoiding legal problems and, \u2013\n\nbenefits of, \u2013\n\ncards and gifts for, , \u2013\n\ncustomer importance vs., xxv\u2013xxvi\n\nembracing competition as coexistence, \u2013\n\nhumility for, \u2013\n\nintentional dinners (Boardroom Dinners) model for, \u2013\n\nas investment in future, \u2013\n\nlearning from others, \u2013\n\nmaking connection with (\"Vitamin C\"), \u2013\n\nnetworking and, \u2013\n\nreconnection with prior contacts and, \u2013, \u2013\n\nReligion, power of, \u2013\n\nRenker, Greg, , \u2013, \u2013\n\nRepositioning, \u2013\n\nResearch\n\nby copywriters, \u2013\n\nrelationship capital and, \u2013\n\nresearching customers' channel of choice, \u2013\n\nResponse lists, \u2013\n\nResponse rate, to Q Testing, \u2013\n\nRetention of customers, \u2013\n\nReturn rates, by customers,\n\nRFM. _See_ Recency, frequency, and monetary (RFM) scores\n\nRobbins, Tony,\n\nRodale Press,\n\nRoyalties,\n\n\"Rules of Thumb\" (Benson), \u2013\n\nRutz, Jim, , \u2013\n\nS\n\nScarcity mind-set vs. abundance mind-set,\n\nSchefren, Rich,\n\nSchwartz, Eugene (Gene) M. Boardroom Reports launch and,\n\n_Breakthrough Advertising_ , xx, , \u2013\n\ncopywriting and, \u2013\n\ncuriosity of,\n\non language usage, , \u2013\n\non marketing as psychology, , \u2013\n\n1 + 1 = 64 concept of, xx\n\n_Scientific Advertising_ (Hopkins),\n\nSecret shoppers, \u2013\n\n_Secrets of Successful Direct Mail_ (Benson),\n\nSegmentation\n\nLTV and list selection, \u2013\n\nrecency, frequency, and monetary (RFM) scores, \u2013, \u2013\n\nsegment psychology needed for copywriting, \u2013\n\ntargeted list segmentation,\n\n( _See also_ Target marketing)\n\nSethi, Ramit,\n\nShortcuts (Core Principle #2), \u2013\n\nSingle channel marketing, dangers of, \u2013\n\nSmarts, of copywriters, \u2013\n\nSocial media\n\nFacebook, \u2013\n\nLinkedIn, \u2013,\n\nmultichannel marketing and,\n\nSpielberg, Steven,\n\nStatistical significance, \u2013\n\nStrategic use of content,\n\nSuccess, repeating (Core Principle #5), \u2013\n\nSuccess, shared by copywriters, \u2013\n\nSullivan, Dan, , ,\n\nSurrounding yourself with smart people (Pillar 3), \u2013\n\nSurvey package promotion example, \u2013,\n\nSurvey tests (Q tests), \u2013\n\nSwipe files (Core Principle #4)\n\nassessing assets for offers, \u2013\n\noverview, \u2013\n\nT\n\n\"Table selection,\" \u2013\n\nTarget marketing. _See also_ List management\n\nmailing thoughtful cards\/gifts to subscribers or acquaintances, \u2013\n\nrecency, frequency, and monetary (RFM) scores and, \u2013, \u2013\n\ntargeted list segmentation,\n\n_Tax Hotline_ (Boardroom),\n\nTelevision infomercials, \u2013\n\nThird party offers,\n\n_30-Day Diabetes Cure, The_ , \u2013\n\nTitans of Direct Response (event), , \u2013, ,\n\nTracy, Reid,\n\nTrade books, repositioned as direct mail, \u2013\n\nTransaction data, \u2013\n\n\"Tribe,\"\n\nTrudeau, Kevin,\n\nTrump, Donald, \u2013\n\nU\n\n_Ultimate Healing_ (Boardroom),\n\nUnique selling proposition (USP), ,\n\nU.S. Postal Service\n\nchange of address files (COA), \u2013\n\nzip codes for compiled lists,\n\nV\n\nVertical niche, defined,\n\n\"Vitamin C\" (connection), \u2013\n\nW\n\nWalker, Jeff, \u2013\n\nWanamaker, John,\n\nWilliamson, Marianne,\n\n_World's Greatest Treasury of Health Secrets, The_ (Boardroom),\n\nZ\n\nZip codes, \nACKNOWLEDGMENTS\n\nWe all know we can't do anything in life without a loving and supportive family, and in my case, I've been so lucky to have Robin, Alex, and Madeline as my cheering section. In addition, my sister Ellen and my supportive parents have been there from the start (even though my parents never really understood what I did for a living\u2014my Mom still doesn't\u2014and she thinks I should get an M.B.A.!).\n\nAnd there are many more people I would like to thank and acknowledge. I stand on the shoulders of so many giants from so many phases of my career.\n\nSomeone suggested that this book could be titled _All My Mentors Are Dead._ Sounds a little maudlin, I know, but my philosophy when I was in my 20s and 30s was to have mentors in their 60s and 70s, since they had all the wisdom and I had none. Unfortunately, that means I've already lost way too many of the people who taught me the most.\n\nLeading that list is Marty Edelston, who is all over this book and all over my life. I've been blessed to have had someone who was both a business mentor and almost like a second father to me.\n\nGene Schwartz, who comes up time and again not only in this book but in everything I do, is a giant of a man to be looked up to, and not just because he was so tall. I also want to thank his wife, Barbara Schwartz, who trusted me to be the shepherd of her beloved husband's work. Without her (and her trusted adviser Steve Scherer), the world would not have complete access to Gene's brilliance as they do now, specifically _Breakthrough Advertising_ and _The Brilliance Breakthrough_. And a shout-out to my friend and direct marketer Jim DiCola for partnering with me to bring _The Brilliance Breakthrough: How to Talk and Write So People Will Never Forget You_ back from the dead (no more $4,000 copies on Amazon\u2014it is now totally accessible and affordable to everyone in marketing and copywriting). Having that Gene Schwartz lost classic available in addition to _Breakthrough Advertising_ is a blessing. Other greats who are no longer with us and who I miss terribly:\n\nDick Benson, the curmudgeon's curmudgeon to many, but to me he was always \"Timid Timothy,\" as he liked to be called. No one could understand that one . . . but I did.\n\nRodney Friedman, Benson's partner in some of the most successful direct marketing franchises ever, who left us way too soon.\n\nLee Epstein, who taught me not just how to give, but to give without an expectation of a return.\n\nBill Jayme, whose creative contributions were always matched by his huge heart.\n\nAdolph Auerbacher, the man with the best judgment ever.\n\nJim Rutz, one of the liveliest minds ever to write copy.\n\nFred Catona, \"Mr. Direct Response Radio.\"\n\nScott Haines, one of Gary Halbert's prized students, also left us way too soon. As did Audri and Jim Lanford, masters of innovation.\n\nBob Stone, the man who wrote \"The Bible\" of Direct Marketing.\n\nOther mentors and friends, mostly retired but not forgotten, include Gordon Grossman, who is quoted a lot in this book and who taught me so much about direct marketing fundamentals and databases; and Denny Hatch, who taught me how to \"write everything right\" (the title of one of three of his books that are listed in my reading list).\n\nCopywriters have been a big part of my life. While I'm done calling myself a copywriter wannabe, I also know how much I owe to all the great copywriters I have ever worked with wishing I could be as prolific as they are. That's why I will always be a copywriter groupie. So many copywriters made me look so good over so many years (and they were always super generous, saying that I helped _them_ look good because I knew something about lists). But my skill was nothing compared to theirs. I want to acknowledge the original Mount Rushmore of Boardroom copywriters\u2014in addition to Gene Schwartz, there was the late Mel Martin, the best copywriter no one ever heard of; the aforementioned Jim Rutz; and of course, Gary Bencivenga, the \"Mariano Rivera of copywriting,\" considered the top copywriter alive today.\n\nThen there's the more modern Mount Rushmore\u2014the copywriters I worked with most in the second half of my career at Boardroom: Eric Betuel, David Deutsch, Arthur Johnson, and Parris Lampropoulos. Four geniuses, and four special friends as well.\n\nBefore I go on to talk about all of the other copywriters I have learned so much from\u2014as well as marketers and thought leaders of all kinds who influenced me the most\u2014I have to call out a \"moose on the table\": I am well aware that to this point, I have not listed any women in my professional acknowledgments . . . yet.\n\nDuring the early part of my career, there were far fewer women teaching and mentoring\u2014and thank goodness there are so many more today.\n\nI received a lot of flak at the \"Titans of Direct Response\" event in 2014 for not having any female speakers, and I sporadically receive e-mails in response to my blog posts when I talk about my key mentors from my past (who are mostly male), asking, \"Why no women?\"\n\nAs far as that event, because it was a tribute to Boardroom and Marty Edelston, the sad truth is that when Boardroom was founded in 1972 (and in the years that followed), the builders, consultants, and copywriters from the outside were mostly men. I know some women even boycotted the event, which hurt. I did my best to explain, and I have had to live with this sad fact despite the event being a huge success.\n\nNote: I can tell you, however, that many of the key folks internally at Boardroom during my tenure were women, and most of the senior staff while I was there were female as well. Two extraordinary women, Joan Throckmorton and Rose Harper, helped build and shape Boardroom from the outside, even in the early years, Joan as a copywriter and Rose as a list broker. Since neither one was alive in 2014, I distributed their out-of-print classic books to the most prolific and dynamic women in direct response marketing who were in the audience at \"Titans.\" Their books are included in my reading list despite being out of print. I've got my copies of their books in a prominent spot on my bookshelf.\n\nI do my best to pay it forward (and I will continue to do so) to the women (and men) who have shaped my world and the world of so many other direct response marketers . . . and you will see many amazing women in the rest of these acknowledgments. The change has done us good.\n\nThere were many other copywriters (and designers) that made me and everyone else at Boardroom look good for many years, writing control after control. In addition to every copywriter I've mentioned already, there was Marcella Allison, Richard Armstrong, Sol Blumenfeld, Belinda Brewster, Carline Anglade-Cole, Henry Cowen, Rob Davis, Lee Euler, Pat Garrard, Lori Haller, Bill Jayme, Barnaby Kalan, Kent Komae, Clayton Makepeace, Bernie Mazel, Don Pierce, Jim Punkre, Michele Raes, Heikki Ratalahti, Kim Krause Schwalm, Alice Taus, and David Wise. My apologies to anyone I did not mention.\n\nOther copywriters and \"teachers of writing copy\" I also learned so much from (although I don't recall any of these folks writing a control for Boardroom\u2014which is not my only criteria for greatness!) include Jon Benson, Drayton Bird, Bill Bonner, John Carlton, Craig Clemens, Doug D'Anna, Ray Edwards, Ed Elliott, Mark Ford, John Forde, Marie Forleo, \"Doberman Dan\" Gallapoo, David Garfinkel, Justin Goff, Bond Halbert, Kevin Halbert, Don Hauptman, \"Big Jason\" Henderson, Daniel Levis, Ryan Markish, Sam Markowitz, Scott Martin, Ted Nicholas, Caleb O'Dowd, Mike Palmer, Traian Sava, Joe Schriefer, Ben Settle, Richard Stanton-Jones, Amit Suneja, Mike Ward, and Irving Wunderman.\n\nI also want to thank \"copywriter trainers\" and coaches Katie Yeakle, Denise Ford, Kevin Rogers, Kira Hug, Rob Marsh, Abbey Woodcock, K. C. Baney, Ramit Sethi, Joanna Wiebe, and anyone else who takes on this noble mission\u2014for all they do to train the next generation of A-list copywriters and for letting me into their respective worlds to teach and be part of that mission.\n\nAnd even though I didn't know all six legends from my first book, they live on in this book too\u2014in addition to Gene Schwartz, there's John Caples, Robert Collier, Gary Halbert, Claude Hopkins, and David Ogilvy. Of course, I owe an additional debt of gratitude to my co-author of _The Advertising Solution_ , Craig Simpson, who could have done the book on his own but brought me in as a partner, which gave me the confidence to write this book. And thanks to Lawrence Bernstein for helping us create a swipe file for the ages (www.TheLegendsBook.com) from those six legends as a companion resource to the book. That was also an inspiration for the creation of www.OverdeliverBook.com, which I hope you will all access as a complement to Overdeliver.\n\nThanks to all the people who participated in the _Breakthrough Advertising_ interviews (not otherwise mentioned in my acknowledgments), which enabled so many additional eternal truths to pass into the next generation, and many of those truths are covered in this book: Bob Bly, Harlan Kilstein, Milt Pierce, Yanik Silver, Joe Vitale.\n\nSince we now can all agree that \"lists are 41 percent,\" and the most important part of every marketing campaign (with apologies to the copywriters!), I will always be indebted to the two men who taught me the list business\u2014Dave Florence and Mike Manzari. And I am also indebted to Richard Viguerie, who was one of my first \"professors\" of direct mail along with Dick Benson.\n\nI think fondly on those days as a \"list manager\" and learning the business from Dave, Mike, and Richard, which led me into a world of direct marketing (mostly direct mail and print and eventually online) in the 1980s, 1990s, and early 2000s where I made so many lifelong friends to this day (although we have lost a few along the way). All are (or were) teachers and students of the craft with me every step: Marie Adolphe, Susan Allyn, Tom Amoriello, Karen Arbegast, Simon Aronin, Bill Baird, John Baldwin, Terri Bartlett, Stacy Parks Berver, Jock Bickert, Arthur Blumenfield, Stuart Boysen, JoAnna Brandi, Annette Brodsky, Nora Brophy, Mike Brostoff, Ed Burnett, Cindy Butehorn, Jimmy Calano, Tom Callahan, Brian Carnahan, Jeff Carneal, Cary Castle, Nancy Cathey, Frank Cawood, Sandy Clark, Bob Cohn, Kevin Cordray, Pat Corpora, Tom Corry, Dan Danielson, Bill Denhard, JoAnne Dunn, Mal Dunn, Suzanne Eastman, Robert Englander, Larry Epstein, Barbara Erlandson, John Finn, Kevin Finn, Ed Fones, Carrie French, Joe Furgiuele Hank Garcia, Lois Geller, Jonah Gitlitz, David Gitow, Dick Goldsmith, Deb Goldstein, Dan Gonzalez, Jerry Gould, Fran Green, Denise Greiner, Judy Haselhoef, Peggy Hatch, John Hendrickson, Gary Hennerberg, Gail Henry, Leon Henry, Stuart Hotchkiss, Beth Hurwitz, Charles Inlander, Karen Isenberg, Stuart Jordan, Barbara Kaplowitz, Joel Katz, Graeme Keeping, Dee Kendall, Bob Keppel, Dan Kern, Jackie Kern, Beth Dent Ketzner, Bob King, Liz Kislik, Dave Klein, Jeff Kobil, Ed Krug, Cyndi Lee, Don Libey, Tim Litle, Marc Liu, Michael Loeb, Lorna Lowery, Coleen Malc, Carole Mandel, Lon Mandel, Harriet Maneval, Harvey Markovitz, Bob Marty, Angelo Matera,Linda McAleer, Jeff Meltzer, Steve Millard, Jon Mulford, Joel Nadel, Ted Nicholas, Andrea Nierenberg, Hal Oringer, Charles Pace, John Pahmer, Chris Paradysz, Pierre Passavant, Dave Penzell, Ben Perez, Reed Phillips, Tom Phillips, Jerry Pont, Rick Popowitz, Jim Prendergast, Kathi Ramsdell, Steve Rapella, Donn Rappaport, Tom Reynolds, Tom Rocco, Marji Ross, Mac Ross, Nat Ross, Hank Rossi, Nick Sarantakos, Bruce Seide, Harold Shain, Joe Shain, Mike Shapiro, Ed Sheehan, David Shepard, Christine Slusarek, Laurie Spar, John Squires, Ralph Stevens, John Suhler, Bob Teufel, Jenny Thompson Rich Vergara, Frank Ward, Wallace Ward, Sue Webb, Martin Weiss, Ann Wixon, Chip Wood, Andy Yoelin, Alan Zamchick, Dawn Zier, Caroline Zimmerman.\n\nA special acknowledgment to Steve Dworman, who taught me the direct response television business and, together with the amazing team at Boardroom, created one of the biggest direct marketing success stories of multichannel marketing\u2014ever.\n\nI'd like to acknowledge my accountability group\u2014who I like to call my \"$25 group\"\u2014the cheapest mastermind around to join, but one that is as rich as any with information and influence\u2014my brothers Michael Fishman, Jim Kwik, and Ryan Lee. And thanks to the unwavering support from two \"brothers in arms,\" guys who always have my back and know this business as well as anyone I have ever met, Alan Kraft and Jim Kabakow.\n\nThanks to the three people I go to when I need the ultimate in guidance and coaching: Joe Polish, Dan Sullivan, and Jeff Walker.\n\nAnd there are a whole host of marketing geniuses, thought leaders, entrepreneurs, and pioneers I follow to this day, and will follow to the day I die, beginning with the keynote speakers from the Titans of Direct Response event not yet mentioned elsewhere: Dan Kennedy, Perry Marshall, Joe Sugarman, Ken McCarthy, and Greg Renker.\n\nAnd so many others who have taught me so much and believed in me over the years, including Mike Agugliaro, Kristen Arnold, Lou Aronica, David Bach, Ellyn Bader, Eric Bakey, Allen Baler, Erin Baler, Craig Ballantyne, Joe Barton, Bari Baumgardner, Lee Bellinger, Susan Berkley, Dan Benamoz, Larry Benet, Dave Berg, Bob Beverley, Josh Bezoni, Paul Bigham, Daniela Birkelbach, Dana Blankenhorn, Chandler Bolt, Chrissy Borchardt, Joshua Boswell, Patrick Bove, Shelley Brander, Rae Brent, Ross Bridgeford, Justin Brooke, Dr. Joy Browne, Russell Brunson, Shaun Buck, Dr. Ruth Buczynski, Erin Burch, Todd Brown, Brendon Burchard, Bob Burg, Michael Cage, Julie Cairns, Jesse Cannone, Mike Capuzzi, Adee Cazayoux, Alex Charfen, Joseph Choi, Rich Christiansen, Robert Cialdini, Lisa Cini, Ian Clark, Mike Cline, Travis Cody, Melinda Cohan, Mike Colella, Paul Colligan, Chad Collins, John Corcoran, Jeff Corriveau, Dave Coyle, JV Crum III, Cedric Crumbly, Fernando Cruz, Phil Cuzzi, Stan Dahl, Stephen Dean, Eelco de Boer, Dave Dee, Ryan Deiss, Gina DeLong, Rachelle Diaz, Mike Disney, Eric Douay, Hugh Downs, Andy Drish, John Lee Dumas, Mary Jane Dykeman, Bo Eason, Dawn Eason, Yuri Elkaim, Ryan Ellefsen, Roxanne Emmerich, Al Evans, Marc Evans, Marty Fahncke, Michelle Falzon, Summer Felix, Ilan Ferdman, Jonathan Fields, Jim Fitzgerald, Jason Fladlian, Joe Foley, Denise Ford, Nick Fosberg, Tim Francis, Dr. Stephen Franson, Katrin Fr\u00f6\u00dfler, Jason Friedman, Rick Frishman, Dylan Frost, Roy Furr, Jayson Gaignard, Thaddeus Gala, John Gallagher, Jason Garber, Randy Garn, Susan Garrett, Arthur Gelber, Kevin Gianni, Ben Glass, Elaine Glass, Ken Glickman, JB Glossinger, Gary Goldstein, David Gonzalez, Denise Gosnell, Bill Gottlieb, Bill Gough, Helmut Graf, Carla Graubard, Steve Gray, Dean Graziosi, Seth Greene, Craig Gross, Aharon Pinchas Grundman, David (Jay) Guigue, Garrett Gunderson, Audrey Hagen, Dr. Ned Hallowell, Derek Halpern, Will Hamilton, Rick Harmon, Verne Harnish, Paul Hartunian, Clay Hebert, Barbara Hemphill, Cameron Herold, Doug Hill, Tracy Hinken, Jess Holland, Shel Horowitz, Martin Howey, Jeremey Hunsicker, Don Hutcheson, Khoa Huynh, Dr. Mark Hyman, Danny Iny, Dean Jackson, Richard James, Debra Jason, Joline Johannes, Jeff Johnson, Shane Johnston, Darcy Juarez, Attila Pongor-Juhasz, Dr. Joel Kahn, Mark Kaplan, Danny Katz, Noah Katz, Brian Kay, Jesse Kay, David Kekich, Frank Kern, Erik Kerr, Rachel Kersten, Sidney Kess, Gail Kingsbury, Mastin Kipp, Esther Pinky Kiss, Mindie Kniss, Mike Koenigs, Jason Korman, Steve Krein, Ken Krell, Dr. Judy Kuriansky, Dan Kuschell, Nicholas Kusmich, Victoria Labalme, Vishen Lakhiani, Tim Larkin, Glen Larson, Sage Lavine, Carey Leader, Ryan Levesque, Senator Joseph Lieberman, Justin Livingston, Steve Loflin, David Long, Pauline Longdon, Michael Lovitch, Rick Lugash, Margaret Lynch, Jeff Madoff, Alice March, Frank Maguire, Michael Maidens, Wendy Makepeace, Alex Mandossian, Steven Mandurano, Brad Martineau, Paul Martinez, Clate Mask, Harry Massey, Kathy Mauck, Dane Maxwell, Shannon McCaffery, Andrew McCall, Richard McFarland, Ryan McGrath, Greg McKeown, Amy Dow McLaren, Stu McLaren, Ari Meisel, Blue Melnick, Ed Mendlowitz, Dan Meredith, Mike Michalowicz, Christian Mickelsen, Sheila Mielcarek, James Mielnik, Donald Miller, Eunice Miller, Richard Miller, Mike Monahan, Jeff Moore, Brian Moran, Tom Morkes, Ben Morris, Marisa Murgatroyd, Jeff Murphy, Nick Nanton, Praveen Narra, Dr. Michael Neal, Ken Neumann, Sebastien Night, Myles Norin, Bill O'Hanlon, Ed O'Keefe, Dr. Tom Orent, Nick Ortner, Eben Pagan, Dr. Steven Palter, Kathleen Patten, David Paull, Tim Paulson, Molly Pearson, Peter Pearson, Sarah Petty, David Phelps, Kim Walsh-Phillips, Greg Poulos, Annie Hyman Pratt, UJ Ramdas, Phil Randazzo, Hallvard Reiso, Ken Rendell, Norman Rentrop, Eric Rhoads, Mike Rhodes, Lee Richter, Buck Rizvi, Susan RoAne, Ocean Robbins, Robin Robins, Erico Rocha, Dan Roitman, Olivier Roland, Greg Rollett, Zac Romero, Anna Rossi, Lisa Rossi, Richard Rossi, Christopher Ruddy, Ginger Rutz, Myra Salzer, Rich Schefren, Patricia Schramm, Congressman Christopher Shays, Ron Sheetz, Farukh Shroff, Ben Simkin, Sonia Simone, Steve Sims, Steve Sipress, Ryan Skelly, Robert Skrob, Babs Smith, Dr. Michael Solomon, Frank Sonnenberg, Nick Sonnenberg, Paulette Sopoci, Selena Soo, Robert Spector, Jonathan Sprinkles, Casey Stanton, Dr. Mark Stengler, Dr. Sean Stephenson, Sarah Anne Stewart, Marc Stockman, Hillary Styles, Neil Sutton, Ed Swain, Jaime Tardy, Ricardo Teixeira, Rich Thau, David Thompson, Kevin Thompson, Susan Peirce Thompson, Mark Timm, MaryEllen Tribby, Christie Turley, Josh Turner, Zim Ugochukwu, Ron Urbach, Daryl Urbanski, Erin Verbeck, JJ Virgin, Chris Vos, Jon Walker, Shannon Waller, James Wedmore, Dr. Jeremy Weisz, Joel Weldon, Barbara Weltman, Ambassador Faith Whittlesey, Kyle Widner, David Will, Ethan Willis, Ann Wilson, Mark Winters, Stephanie Winston, Dave Woodward, Ian Wyatt, Keith Yackey, Pamela Yellen, Melissa Young, Rob Zadotti. And trying to list everyone means I missed someone . . . thank you to anyone I should have mentioned and didn't.\n\nI'd like to thank everyone I worked with at Boardroom over my 34 years. Any time this book makes it seem like I did anything great by myself, I did not. We hired the best, which made Boardroom an icon in the direct marketing world and a company that attracted the brightest minds in the business. Everyone there did, and still does, incredible work and makes Boardroom the most amazing direct marketing think tank that has ever existed.\n\nThanks to the people I work with now in Titans Marketing\u2014my assistant, Carla Zwaan, who runs so much of the business day to day, my marketing partner, Chris Mason, and the wonderful Mary Grace \"Macy\" Mahait who does so much behind the scenes (and on the other side of the world) in the Philippinnes. And I rarely make a move in the business without consulting my \"consigliere\" Peter Hoppenfeld, who is always there for me too, as are my financial advisors Barry Gothelf and Tom Henske and my IT support genius, Paul Schietinger.\n\nThank you to all my Titans Mastermind and Titans Master Class members mentioned and not mentioned by name above, past, present, and future, as I continue my mission to reach more people with what I have to teach\u2014they make it all worthwhile, and they inspired me to write this book as much as anyone.\n\nI also want to thank the early readers of the manuscript: Jay Abraham, Jim DiCola, Perry Marshall, Rob Hanly, and Robert Skrob.\n\nAnd a special shout-out to my \"Uncle Jay\" (Jay Abraham) for writing the foreword and being a guiding light and motivator to me for more than 25 years, and he remains that today.\n\nA big thank-you and acknowledgment to my developmental editor on this book, Laura Gale. I could not have gotten here without her\u2014she had incredible patience when I didn't do my homework and kept me on the straight and narrow to get this book finished.\n\nThank you to the folks at Morgan James\u2014particularly Karen Anderson and David Hancock\u2014for their support.\n\nFinally, I want to thank Reid Tracy, Patty Gift, Anne Barthel, Mary Norris, Rachel Shields Ebersole, Marlene Robinson, Stacey Smith, Tricia Breidenthal, Lindsay McGinty, Lauren Rosenthal, Catherine Veloskey, Celeste Phillips, Steve Morris, and the rest of the amazing team at Hay House for believing in me and this book.\nABOUT THE AUTHOR\n\nBrian Kurtz has been a serial direct marketer for almost 40 years and never met a medium he didn't like . . . and while he's had much success, he also must admit that trying to sell subscriptions and books on the back of ATM receipts and under yogurt lids were only a \"good idea at the time.\"\n\nAs a key business builder at Boardroom Inc. with founder Martin Edelston, Brian worked with, and was mentored by, many of the top marketers and copywriters who have ever lived.\n\n_Overdeliver_ is Brian's second book. His first book, co-authored with Craig Simpson, was _The Advertising Solution_ (2016), which profiled and taught core marketing fundamentals from six of the greatest advertising legends ever (www.TheLegendsBook.com).\n\nToday Brian consults and works directly with bleeding-edge direct response marketing companies and entrepreneurs in a wide variety of categories and in all channels through his two mastermind groups, Titans Mastermind and Titans Master Class (and soon to be a third, Titans Xcelerator). He also publishes books and materials for direct response marketers (including owning the exclusive rights to the landmark work by Eugene Schwartz, _Breakthrough Advertising_ , with Gene's wife, Barbara).\n\nHe writes and speaks regularly at marketing and copywriting events. Find out more about Brian at briankurtz.net and access free resources accompanying this book at www.OverdeliverBook.com.\n\nThe only thing Brian loves more than direct marketing is his family.\n\nAnd umpiring baseball is right up there with direct marketing.\nWWW.OVERDELIVERBOOK.COM\n\nThank you for purchasing _Overdeliver_!\n\nConsistent with the title, I want to continue to \"overdeliver\" to you for many years to come.\n\nIf you go to www.OverdeliverBook.com, you will be treated to priceless and hard-to-find resources from many of the greatest direct-response marketers and copywriters of all time, most of whom have been profiled in this book.\n\nThese are free resources to help you build your business for a lifetime playing the long game.\n\nWhat I have put together so far (and I will be adding even more to www.OverdeliverBook.com regularly):\n\n  * Rare and classic videos from many of the \"Titans of Direct Response,\" many of which I spoke about in _Overdeliver_\n  * Tips and secrets from top copywriters not available anywhere else\n  * \"The lost chapters\" from _Overdeliver_ . . . research and writing I have done since this book was completed\n  * PDFs and access to books that are either out-of-print or will only be available on this site\n  * Swipe files to study and adapt from all marketing channels to get your messages noticed and remembered, whether you work online, offline or both\n  * A compilation of the best headlines of all time . . . over 10,000 of them\n  * And much more as I find best-in-class information and best practices that will help you build your business in the present . . . and into the future.\n\n**www.OverdeliverBook.com**\n\nIn the meantime, keep overdelivering for anyone you serve . . . and go forth and multiply!\n**Hay House Titles of Related Interest**\n\n_YOU CAN HEAL YOUR LIFE, the movie,_ starring Louise Hay & Friends  \n(available as a 1-DVD program, an expanded 2-DVD set,  \nand an online streaming video)  \nLearn more at www.hayhouse.com\/louise-movie\n\n_THE SHIFT, the movie,_  \nstarring Dr. Wayne W. Dyer  \n(available as a 1-DVD program, an expanded 2-DVD set,  \nand an online streaming video)  \nLearn more at www.hayhouse.com\/the-shift-movie\n\n\u25a0 \u25a0 \u25a0\n\n_HIGH PERFORMANCE HABITS: How Extraordinary People  \nBecome That Way,_ by Brendon Burchard\n\n_MILLIONAIRE SUCCESS HABITS: The Gateway to Wealth & Prosperity,_  \nby Dean Graziosi\n\n_OPPORTUNITY: How to Win in Business & Create a Life You Love,_  \nby Eben Pagan\n\n_WOMEN ROCKING BUSINESS: The Ultimate Step-by-Step Guidebook to  \nCreate a Thriving Life Doing Work You Love,_ by Sage Lavine\n\nAll of the above are available at your local bookstore,  \nor may be ordered by contacting Hay House (see next page).\n\n\u25a0 \u25a0 \u25a0\nWe hope you enjoyed this Hay House book. If you'd like to receive our online catalog featuring additional information on Hay House books and products, or if you'd like to find out more about the Hay Foundation, please contact:\n\nHay House, Inc., P.O. Box 5100, Carlsbad, CA 92018-5100  \n(760) 431-7695 or (800) 654-5126  \n(760) 431-6948 (fax) or (800) 650-5115 (fax)  \nwww.hayhouse.com\u00ae \u2022 www.hayfoundation.org\n\n**_Published in Australia by:_**  \nHay House Australia Pty. Ltd., 18\/36 Ralph St., Alexandria NSW 2015  \n_Phone:_ 612-9669-4299 \u2022 _Fax:_ 612-9669-4144 \u2022 www.hayhouse.com.au\n\n**_Published in the United Kingdom by:_**  \nHay House UK, Ltd., Astley House, 33 Notting Hill Gate, London W11 3JQ  \n _Phone:_ 44-20-3675-2450 \u2022 _Fax:_ 44-20-3675-2451 \u2022 www.hayhouse.co.uk\n\n**_Published in India by:_** Hay House Publishers India,  \nMuskaan Complex, Plot No. 3, B-2, Vasant Kunj, New Delhi 110 070  \n _Phone:_ 91-11-4176-1620 \u2022 _Fax:_ 91-11-4176-1630 \u2022 www.hayhouse.co.in\n\n**Access New Knowledge.  \nAnytime. Anywhere.**\n\nLearn and evolve at your own pace  \nwith the world's leading experts.\n\nwww.hayhouseU.com\n\n# TABLE OF CONTENTS\n\n  1. Cover Page\n  2. Title Page\n  3. Copyright Page\n  4. Table of Contents\n  5. Foreword by Jay Abraham\n  6. Introduction\n  7. CHAPTER 1: Overdeliver\n  8. CHAPTER 2: Original Source\n  9. CHAPTER 3: How Paying Postage Made Me a Better Marketer\n  10. CHAPTER 4: List Building & RFM\n  11. CHAPTER 5: Offers\n  12. CHAPTER 6: Creative & Copy\n  13. CHAPTER 7: Multichannel Marketing\n  14. CHAPTER 8: Customer Service & Fulfillment\n  15. CHAPTER 9: Continuity & LTV\n  16. CHAPTER 10: Playing the Long Game\n  17. Further Reading\n  18. Index\n  19. Acknowledgments\n  20. About the Author\n\n## LIST OF PAGES\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. vii\n  8. viii\n  9. ix\n  10. xi\n  11. xiii\n  12. xiv\n  13. xv\n  14. xvi\n  15. xvii\n  16. xix\n  17. xx\n  18. xxi\n  19. xxii\n  20. xxiii\n  21. xxiv\n  22. xxv\n  23. xxvi\n  24. xxvii\n  25. xxviii\n  26. xxix\n  27. xxx\n  28. xxxi\n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\norbitbooks.net\n\norbitshortfiction.com\n\n**Begin Reading**\n\nTable of Contents\n\nA Preview of _Promise of Blood_\n\nNewsletters\n\nCopyright Page\n\nIn accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher constitute unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\n_To my sister-in-law, Nancy Watt_\n\n# Chapter One\n\n# Albinkirk\u2014Ser John Crayford\n\nThe Captain of Albinkirk forced himself to stop staring out his narrow, glazed window and do some work.\n\nHe was jealous. Jealous of a boy a third of his age, commanding a pretty company of lances. Riding about. While he sat in a town so safe it was dull, growing old.\n\n_Don't be a fool_ , he told himself. _All those deeds of arms make wonderful stories, but the doing is cold, wet and terrifying. Remember?_\n\nHe sighed. His hands remembered everything\u2014the blows, the nights on the ground, the freezing cold, the gauntlets that didn't quite fit. His hands pained him all the time, awake or asleep.\n\nThe Captain of Albinkirk, Ser John Crayford, had not started his life as a gentleman. It was a rank he'd achieved through pure talent.\n\nFor violence.\n\nAnd as a reward, he sat in this rich town with a garrison a third the size that it was supposed to be on paper. A garrison of hirelings who bossed the weak, abused the women, and took money from the tradesmen. A garrison that had too much cash, because the posting came with the right to invest in fur caravans from the north. Albinkirk furs were the marvel of ten countries. All you had to do to get them was ride north or west into the Wild. And then come back alive.\n\nThe captain had a window that looked north-west.\n\nHe tore his eyes away from it. Again.\n\nAnd put pen to paper. Carefully, laboriously, he wrote:\n\n_My Lord,_\n\n_A Company of Adventure\u2014well ordered, and bearing a pass signed by the constable\u2014passed the bridge yesterday morning; near to forty lances, each lance composed of a knight, a squire, a valet and an archer. They were very well armed and armoured in the latest Eastern manner\u2014steel everywhere. Their captain was polite but reserved; very young, refused to give his name; styled himself The Red Knight. His banner displayed three lacs d'amour in gold on a field sable. He declared that they were, for the most part, your Grace's subjects, lately come from the wars in Galle. As his pass was good, I saw no reason to keep him._\n\nSer John snorted, remembering the scene. No one had thought to warn him that a small army was coming his way from the east. He'd been summoned to the gate early in the morning. Dressed in a stained cote of fustian and old hose, he'd tried to face down the cocky young pup in his glorious scarlet and gold, mounted on a war horse the size of a barn. He hadn't enough real soldiers to arrest any of them. The damned boy had _Great Noble_ written all over him, and the Captain of Albinkirk thanked God that the whelp had paid the toll with good grace and had good paper, as any incident between them would have gone badly. For him.\n\nHe realised he was looking at the mountains. He tore his eyes away. Again.\n\n_He also had a letter from the Abbess at Lissen Carak. She had sent to me last autumn for fifty good men, and I had to refuse her\u2014your Grace knows I am short enough of men as it is. I suppose she has offered her contract to sell-swords in the absence of local men._\n\n_I am, as your Grace is aware, almost one hundred men under strength; I have but four proper men-at-arms, and many of my archers are not all they should be. I respectfully request that your Grace either replace me, or provide the necessary funds to increase the garrison to its proper place._\n\n_I am your Grace's humblest and most respectful servant, John Crayford_\n\nThe Master of the Guild of Furriers had invited him to dinner. Ser John leaned back and decided to call it a day, leaving the letter lying on his desk.\n\n# Lissen Carak\u2014The Red Knight\n\n\"Sweet Jesu,\" Michael called from the other side of the wall. It was as high as a man's shoulder, created by generations of peasants hauling stones out of fields. Built against the wall was a two-storey stone house with outbuildings\u2014a rich manor farm. Michael stood in the yard, peering through the house's shattered main door. \"Sweet Jesu,\" the squire said again. \"They're all dead, Captain.\"\n\nHis war horse gave the captain the height to see over the wall to where his men were rolling the bodies over, stripping them of valuables as they sought for survivors. Their new employer would not approve, but the captain thought the looting might help her understand what she was choosing to employ. In his experience, it was usually best that the prospective employer understand what he\u2014or she\u2014was buying. From the first.\n\nThe captain's squire vaulted over the stone wall that separated the walled garden from the road and took a rag from Toby, the captain's page. Sticky mud, from the endless spring rain, covered his thigh-high buckled boots. He produced a rag from his purse to cover his agitation and began to clean his boots. Michael was fussy and dressed for fashion. His scarlet company surcoat was embroidered with gold stars; the heavy wool worth more than an archer's armour. He was well born and could afford it, so it was his business.\n\nIt was the captain's business that the lad's hands were shaking.\n\n\"When you feel ready to present yourself,\" the captain said lightly, but Michael froze at his words, then made himself finish his task with the rag before tossing it back to Toby.\n\n\"Apologies, m'lord,\" he said with a quick glance over his shoulder. \"It was something out of the Wild, lord. Stake my soul on it.\"\n\n\"Not much of a stake,\" the captain said, holding Michael's eye. He winked, as much to amuse the onlookers of his household as to steady his squire, who was pale enough to write on. Then he looked around.\n\nThe rain was light\u2014just enough to weigh down the captain's heavy scarlet cloak without soaking it through. Beyond the walled steading stretched fields of dark, newly planted earth, as shining and black in the rain as the captain's horse. The upper fields toward the hills were rich with new greenery and dotted with sheep. Good earth and fertile soil promised rich crops, as far as the eye could see on both sides of the river. This land was tamed, covered in a neat geometric pattern of hedgerows and high stone walls separating tilled plots, or neatly scattered sheep and cattle, with the river to ship them down to the cities in the south. Crops and animals whose riches had paid for the fortress nunnery\u2014Lissen Carak\u2014that capped the high ridge to the south, visible from here as a crenelated line of pale stone. Grey, grey, grey from the sky to the ground. Pale grey, dark grey, black.\n\nBeyond the sheep, to the north, rose the Adnacrags\u2014two hundred leagues of dense mountains that lowered over the fields, their tops lost in the clouds.\n\nThe captain laughed at his own thoughts.\n\nThe dozen soldiers nearest him looked; every head turned, each wearing matching expressions of fear.\n\nThe captain rubbed the pointed beard at his chin, shaking off the water. \"Jacques?\" he asked his valet.\n\nThe older man sat quietly on a war horse. He was better armed than most of the valets; wearing his scarlet surcote with long, hanging sleeves over an Eastern breastplate, and with a fine sword four feet long to the tip. He, too, combed the water out of his pointed beard while he thought.\n\n\"M'lord?\" he asked.\n\n\"How did the Wild make it here?\" The captain asked. Even with a gloved hand keeping the water from his eyes, he couldn't _see_ the edge of the Wild\u2014there wasn't a stand of trees large enough to hide a deer within a mile. Two miles. Far off to the north, many leagues beyond the rainy horizon and the mountains, was the Wall. Past the Wall was the Wild. True, the Wall was breached in many places and the Wild ran right down into the country. The Adnacrags had never been cleared. But here\u2014\n\nHere, wealth and power held the Wild at bay. _Should have held the Wild at bay._\n\n\"The usual way,\" Jacques said quietly. \"Some fool must have invited them in.\"\n\nThe captain chuckled. \"Well,\" he said, giving his valet a crooked smile, \"I don't suppose they'd call us if they didn't have a problem. And we need the work.\"\n\n\"It ripped them apart,\" Michael said.\n\nHe was new to the trade and well-born, but the captain appreciated how quickly he had recovered his poise. At the same time, Michael needed to learn.\n\n\"Apart,\" Michael repeated, licking his lips. His eyes were elsewhere. \"It _ate_ her. Them.\"\n\n_Mostly recovered,_ the captain thought to himself. He nodded to his squire and gave his destrier, Grendel, a little rein so he backed a few steps and turned. The big horse could smell blood and something else he didn't like. He didn't like most things, even at the best of times, but this was spooking him and the captain could feel his mount's tension. Given that Grendel wore a chamfron over his face with a spike a foot long, the horse's annoyance could quickly translate into mayhem.\n\nHe motioned to Toby, who was now sitting well to the side and away from the isolated steading-house and eating, which is what Toby tended to do whenever left to himself. The captain turned to face his standard bearer and his two marshals where they sat their own fidgeting horses in the rain, waiting for his commands.\n\n\"I'll leave Sauce and Bad Tom. They'll stay on their guard until we send them a relief,\" he said. The discovery of the killings in the steading had interrupted their muddy trek to the fortress. They'd been riding since the second hour after midnight, after a cold camp and equally cold supper. No one looked happy.\n\n\"Go and get me the master of the hunt,\" he added, turning back to his squire. When he was answered only with silence, he looked around. \"Michael?\" he asked quietly.\n\n\"M'lord?\" The young man was looking at the door to the steading. It was oak, bound in iron, and it had been broken in two places, the iron hinges inside the door had bent where they'd been forced off their pins. Trios of parallel grooves had ripped along the grain of the wood\u2014in one spot, the talons had ripped through a decorative iron whorl, a clean cut.\n\n\"Do you need a minute, lad?\" the captain asked. Jacques had seen to his own mount and was now standing at Grendel's big head, eyeing the spike warily.\n\n\"No\u2014no, m'lord.\" His squire was still stunned, staring at the door and what lay beyond it.\n\n\"Then don't stand on ceremony, I beg.\" The captain dismounted, thinking that he had used the term lad quite naturally. Despite the fact that he and Michael were less than five years apart.\n\n\"M'lord?\" Michael asked, unclear what he'd just been told to do.\n\n\"Move your arse, boy. Get me the huntsman. Now.\" The captain handed his horse to the valet. Jacques was not really a valet. He was really the captain's man and, as such, he had his own servant\u2014Toby. A recent addition. A scrawny thing with large eyes and quick hands, completely enveloped in his red wool cote, which was many sizes too big.\n\nToby took the horse and gazed at his captain with hero-worship, a big winter apple forgotten in his hand.\n\nThe captain liked a little hero-worship. \"He's spooked. Don't give him any free rein or there'll be trouble,\" the captain said gruffly. He paused. \"You might give him your apple core though,\" he said, and the boy smiled.\n\nThe captain went into the steading by the splintered door. Closer up, he could see that the darker brown was not a finish. It was blood.\n\nBehind him, his destrier gave a snort that sounded remarkably like human derision\u2014though whether it was for the page or his master was impossible to tell.\n\nThe woman just inside the threshold had been a nun before she was ripped open from neck to cervix. Her long, dark hair, unbound from the confines of her wimple, framed the horror of her missing face. She lay in a broad pool of her own blood that ran down into the gaps between the boards. There were tooth marks on her skull\u2014the skin just forward of one ear had been shredded, as if something had gnawed at her face for some time, flensing it from the bone. One arm had been ripped clear of her body, the skin and muscle neatly eaten away so that only shreds remained, bones and tendons still hanging together... and then it had been replaced by the corpse. The white hand with the silver IHS ring and the cross was untouched.\n\nThe captain looked at her for a long time.\n\nJust beyond the red ruin of the nun was a single clear footprint in the blood and ordure, which was already brown and sticky in the moist, cool air. Some of the blood had begun to leech into the pine floor boards, smooth from years of bare feet walking them. The leeched blood blurred the edge of the print, but the outline was clear\u2014it was the size of a war horse's hoof or bigger, with three toes.\n\nThe captain heard his huntsman come up and dismount outside. He didn't turn, absorbed in the parallel exercises of withholding the need to vomit and committing the scene to memory. There was a second, smudged print further into the room, where the creature had pivoted its weight to pass under the low arch to the main room beyond. It had dug a furrow in the pine with its talons. And a matching furrow in the base board that ran up into the wattle and plaster. A dew claw.\n\n\"Why'd this one die here when the rest died in the garden?\" he asked.\n\nGelfred stepped carefully past the body. Like most gentlemen, he carried a short staff\u2014really just a stick shod in silver, like a mountebank's wand. Or a wizard's. He used it first to point and then to pry something shiny out of the floorboards.\n\n\"Very good,\" said the captain.\n\n\"She died for them,\" Gelfred said. A silver cross set with pearls dangled from his stick. \"She tried to stop it. She gave the others time to escape.\"\n\n\"If only it had worked,\" said the captain. He pointed at the prints.\n\nGelfred crouched by the nearer print, laid his stick along it, and made a clucking sound with his tongue.\n\n\"Well, well,\" he said. His nonchalance was a little too studied. And his face was pale.\n\nThe captain couldn't blame the man. In a brief lifetime replete with dead bodies, the captain had seldom seen one so horrible. Part of his conscious mind wandered off a little, wondering if her femininity, the beauty of her hair, contributed to the utter horror of her destruction. Was it like desecration? A deliberate sacrilege?\n\nAnd another, harder part of his mind walked a different path. The monster had placed that arm _just so._ The tooth marks that framed the bloody sockets that had been her eyes. He could imagine, far too well.\n\nIt had been done to leave terror. It was almost _artistic._\n\nHe tasted salt in his mouth and turned away. \"Don't act tough on my account, Gelfred,\" he said. He spat on the floor, trying to get rid of the taste before he made a spectacle of himself.\n\n\"Never seen worse, and that's a fact,\" Gelfred said. He took a long, slow breath. \"God shouldn't allow this!\" he said bitterly.\n\n\"Gelfred,\" the captain said, with a bitter smile. \"God doesn't give a fuck.\"\n\nTheir eyes met. Gelfred looked away. \"I will know what there is to know,\" he said, looking grim. He didn't like the captain's blasphemy\u2014his face said as much. Especially not when he was about to work with God's power.\n\nGelfred touched his stick to the middle of the print, and there was a moment of _change_ , as if their eyes had adjusted to a new light source, or stronger sunlight.\n\n\"Pater noster qui es in caelus,\" Gelfred intoned in plainchant.\n\nThe captain left him to it.\n\nIn the garden, Ser Thomas's squire and half a dozen archers had stripped the bodies of valuables\u2014and collected all the body parts strewn across the enclosure, reassembled as far as possible, and laid them out, wrapped in cloaks. The two men were almost green, and the smell of vomit almost covered the smell of blood and ordure. A third archer was wiping his hands on a linen shirt.\n\nSer Thomas\u2014Bad Tom to every man in the company\u2014was six foot six inches of dark hair, heavy brow and bad attitude. He had a temper and was always the wrong man to cross. He was watching his men attentively, an amulet out and in his hand. He turned at the rattle of the captain's hardened steel sabatons on the stone path and gave him a sketchy salute. \"Reckon the young 'uns earned their pay today, Captain.\"\n\nSince they weren't paid unless they had a contract, it wasn't saying much.\n\nThe captain merely grunted. There were six corpses in the garden.\n\nBad Tom raised an eyebrow and passed something to him.\n\nThe captain looked at it, and pursed his lips. Tucked the chain into the purse at his waist, and slapped Bad Tom on his paulder-clad shoulder. \"Stay here and stay awake,\" he said. \"You can have Sauce and Gelding, too.\"\n\nBad Tom shrugged. He licked his lips. \"Me an' Sauce don't always see eye to eye.\"\n\nThe captain smiled inwardly to see this giant of a man\u2014feared throughout the company\u2014admit that he and a woman didn't \"see eye to eye.\"\n\nShe came over the wall to join them.\n\nSauce had won her name as a whore, giving too much lip to customers. She was tall, and in the rain her red hair was toned to dark brown. Freckles gave her an innocence that was a lie. She had made herself a name. That said all that needed to be said.\n\n\"Tom fucked it up already?\" she asked.\n\nTom glared.\n\nThe captain took a breath. \"Play nicely, children. I need my best on guard here, frosty and awake.\"\n\n\"It won't come back,\" she said.\n\nThe captain shook his head. \"Stay awake anyway. Just for me.\"\n\nBad Tom smiled and blew a kiss at Sauce. \"Just for _you_ ,\" he said.\n\nHer hand went to her riding sword and with a flick it was in her hand.\n\nThe captain cleared his throat.\n\n\"He treats me like a whore. I _am not._ \" She held the sword steady at his face, and Bad Tom didn't move.\n\n\"Say you are sorry, Tom.\" The captain sounded as if it was all a jest.\n\n\"Didn't say one bad thing. Not one! Just a tease!\" Tom said. Spittle flew from his lips.\n\n\"You meant to cause harm. She took it as harm. You know the rules, Tom.\" The captain's voice had changed, now. He spoke so softly that Tom had to lean forward to hear him.\n\n\"Sorry,\" Tom muttered like a schoolboy. \"Bitch.\"\n\nSauce smiled. The tip of her riding sword pressed into the man's thick forehead just over an eye.\n\n\"Fuck you!\" Tom growled.\n\nThe captain leaned forward. \"Neither one of you wants this. It's clear you are both _posturing._ Climb down or take the consequences. Tom, Sauce wants to be treated as your peer. Sauce, Tom is top beast and you put his back up at every opportunity. If you want to be part of this company then you have to accept your place in it.\"\n\nHe raised his gloved hand. \"On the count of three, you will both back away, Sauce will sheathe her weapon, Tom will bow to her and apologise, and Sauce will return his apology. Or you can both collect your kit, walk away and kill each other. But not as my people. Understand? Three. Two. One.\"\n\nSauce stepped back, saluted with her blade and sheathed it. Without looking or fumbling.\n\nTom let a moment go by. Pure insolence. But then something happened in his face, and he bowed\u2014a good bow, so that his right knee touched the mud. \"Humbly crave your pardon,\" he said in a loud, clear voice.\n\nSauce smiled. It wasn't a pretty smile, but it did transform her face, despite the missing teeth in the middle. \"And I yours, ser knight,\" she replied. \"I regret my... attitude.\"\n\nShe obviously shocked Tom. In the big man's world of dominance and submission, she was beyond him. The captain could read him like a book. And he thought _Sauce deserves something for that. She's a good man._\n\nGelfred appeared at his elbow. Had probably been waiting for the drama to end.\n\nThe captain felt the wrongness of it before he saw what his huntsman carried. Like a housewife returning from pilgrimage and smelling something dead under her floor\u2014it was like that, only stronger and wronger.\n\n\"I rolled her over. This was in her back,\" Gelfred said. He had the thing wrapped in his rosary.\n\nThe captain swallowed bile, again. _I love this job,_ he reminded himself.\n\nTo the eye, it looked like a stick\u2014two fingers thick at the butt, sharpened to a needlepoint now clotted with blood and dark. Thorns sprouted from the whole haft, but it was fletched. An arrow. Or rather, an obscene parody of an arrow, whittled from...\n\n\"Witch Bane,\" Gelfred said.\n\nThe captain made himself take it without flinching. There were some secrets he would pay the price to preserve. He flashed on the last Witch-Bane arrow he'd seen\u2014and pushed past it.\n\nHe held it a moment. \"So?\" he said, with epic unconcern.\n\n\"She was shot in the back\u2014with the Witch Bane\u2014while she was alive.\" Gelfred's eyes narrowed. \"And then the monster ripped her face off.\"\n\nThe captain nodded and handed his huntsman the shaft. The moment it left his hand he felt lighter, and the places where the thorns had pricked his chamois gloves felt like rashes of poison ivy on his thumb and fingers\u2014if poison ivy caused an itchy numbness, a leaden pollution.\n\n\"Interesting,\" the captain said.\n\nSauce was watching him.\n\n_Damn women and their superior powers of observation,_ he thought.\n\nHer smile forced him to smile in return. The squires and valets in the garden began to breathe again and the captain was sure they'd stay awake, now. Given that there was a murderer on the loose who had monster-allies in the Wild.\n\nHe got back to his horse. Jehannes, his marshal, came up on his bridle hand side and cleared his throat. \"That woman's trouble,\" he said.\n\n\"Tom's trouble too,\" the captain replied.\n\n\"No other company would have had her.\" Jehannes spat.\n\nThe captain looked at his marshal. \"Now Jehannes,\" he said. \"Be serious. Who would have Tom? He's killed more of his own comrades than Judas Iscariot.\"\n\nJehannes looked away. \"I don't trust her,\" he said.\n\nThe captain nodded. \"I know. Let's get moving.\" He considered vaulting into the saddle and decided that he was too tired and the show would be wasted on Jehannes, anyway. \"You dislike her because she's a woman,\" he said, and put his left foot into the stirrup.\n\nGrendel was tall enough that he had to bend his left knee as far as the articulation in his leg harness would allow. The horse snorted again. Toby held onto the reins.\n\nHe leaped up, his right leg powering him into the saddle, pushing his six feet of height and fifty pounds of mail and plate. Got his knee over the high ridge of the war-saddle and was in his seat.\n\n\"Yes,\" Jehannes said, and backed his horse into his place in the column.\n\nThe captain saw Michael watching Jehannes go. The younger man turned and raised an eyebrow at the captain.\n\n\"Something to say, young Michael?\" the captain asked.\n\n\"What was the stick? M'lord?\" Michael was different from the rest\u2014well born. Almost an apprentice, instead of a hireling. As the captain's squire, he had special privileges. He could ask questions, and all the rest of the company would sit very still and listen to the answer.\n\nThe captain looked at him for a moment. Considering. He shrugged\u2014no mean feat in plate armour.\n\n\"Witch Bane,\" he said. \"A Witch-Bane _arrow._ The nun had _power._ \" He made a face. \"Until someone shot the Witch Bane into her back.\"\n\n\"A nun?\" Michael asked. \"A nun who could work _power_?\" He paused. \"Who shot her? By Jesu, m'lord, you mean the Wild has _allies_?\"\n\n\"All in a day's work, lad. It's all in a day's work.\" His visual memory, too well trained, ran through the items like the rooms in his memory palace\u2014the splintered door, the faceless corpse, the arm, the Witch-Bane arrow. He examined the path from the garden door to the front door.\n\n\"Wait on me,\" he said.\n\nHe walked Grendel around the farmyard, following the stone wall to the garden. He stood in his stirrups to peer over the wall, and aligned the open garden door with the splintered front door. He looked over his shoulder several times.\n\n\"Wilful!\" he called.\n\nHis archer appeared. \"What now?\" he muttered.\n\nThe captain pointed at the two doors. \"How far away could you stand and still put an arrow into someone at the _front_ door.\"\n\n\"What, shooting through the house?\" asked Wilful Murder.\n\nThe captain nodded.\n\nWilful shook his head. \"Not that far,\" he admitted. \"Any loft at all and the shaft strikes the door jamb.\" He caught a louse on his collar and killed it between his nails. His eyes met the captain's. \"He'd have to be close.\"\n\nThe captain nodded. \"Gelfred?\" he called.\n\nThe huntsman was outside the front door, casting with his wand over a large reptilian print in the road. \"M'lord?\"\n\n\"See if you and Wilful can find any tracks out the back. Wilful will show you where a bowman might have stood.\"\n\n\"It's always fucking me\u2014get Long Paw to do it,\" Wilful muttered.\n\nThe captain's mild glance rested for a moment on his archer and the man cringed.\n\nThe captain turned his horse and sighed. \"Catch us up as soon as you have the tracks,\" he said. He waved at Jehannes. \"Let's go to the fortress and meet the lady Abbess.\" He touched his spurs ever so lightly to Grendel's sides, and the stallion snorted and deigned to move forward into the rain.\n\nThe rest of the ride along the banks of the Cohocton was uneventful, and the company halted by the fortified bridge overshadowed by the rock-girt ridge and the grey walls of the fortress convent atop it, high above them. Linen tents rose like dirty white flowers from the muddy field, and the officer's pavilions came off the wagons. Teams of archers dug cook pits and latrines, and valets and the many camp followers\u2014craftsmen and sutlers, runaway serfs, prostitutes, servants, and free men and women desperate to gain a place\u2014assembled the heavy wooden hoardings that served the camp as temporary walls and towers. The drovers, an essential part of any company, filled the gaps with the heavy wagons. Horse lines were staked out. Guards were set.\n\nThe Abbess's door ward had pointedly refused to allow the mercenaries through her gate. The mercenaries had expected nothing else, and even now hardened professionals were gauging the height of the walls and the likelihood of climbing them. Two veteran archers\u2014Kanny, the barracks room lawyer of the company, and Scrant, who never stopped eating\u2014stood by the camp's newly-constructed wooden gate and speculated on the likelihood of getting some in the nun's dormitory.\n\nIt made the captain smile as he rode by, collecting their salutes, on the steep gravel road that led up the ridge from the fortified town at the base, up along the switchbacks and finally up through the fortress gate-house into the courtyard beyond. Behind him, his banner bearer, marshals and six of his best lances dismounted to a quiet command and stood by their horses. His squire held his high-crested bassinet, and his valet bore his sword of war. It was an impressive show and it made good advertising\u2014ideal, as he could see heads at every window and door that opened into the courtyard.\n\nA tall nun in a slate-grey habit\u2014the captain suppressed his reflexive flash on the corpse in the doorway of the steading\u2014reached to take the reins of his horse. A second nun beckoned with her hand. Neither spoke.\n\nThe captain was pleased to see Michael dismount elegantly despite the rain, and take Grendel's head, without physically pushing the nun out of the way.\n\nHe smiled at the nuns and followed them across the courtyard towards the most ornate door, heavy with scroll-worked iron hinges and elaborate wooden panels. To the north, a dormitory building rose beyond a trio of low sheds that probably served as workshops\u2014smithy, dye house and carding house, or so his nose told him. To the south stood a chapel\u2014far too fragile and beautiful for this martial setting\u2014and next to it, by cosmic irony, a long, low, slate-roofed stable.\n\nBetween the chapel's carved oak doors stood a man. He had a black habit with a silk rope around the waist, was tall and thin to the point of caricature, and his hands were covered in old scars.\n\nThe captain didn't like his eyes, which were blue and flat. The man was nervous, and wouldn't meet his eye\u2014and he was clearly angry.\n\nFlicking his eyes away from the priest, the captain reviewed the riches of the abbey with the eye of a money-lender sizing up a potential client. The abbey's income was shown in the cobbled courtyard, the neat flint and granite of the stables with a decorative stripe of glazed brick, the copper on the roof and the lead gutters gushing water into a cistern. The courtyard was thirty paces across\u2014as big as that of any castle he'd lived in as a boy. The walls rose sheer\u2014the outer curtain at his back, the central monastery before him, with towers at each corner, all wet stone and wet lead, rain slicked cobbles; the priest's faded black cassock, and the nun's undyed surcoat.\n\n_All shades of grey_ , he thought to himself, and smiled as he climbed the steps to the massive monastery door, which was opened by another silent nun. She led him down the hall\u2014a great hall lit by stained glass windows high in the walls. The Abbess was enthroned like a queen in a great chair on a dais at the north end of the hall, in a gown whose grey had just enough colour to appear a pale, pale lavender in the multi-faceted light. She had the look of a woman who had once been very beautiful indeed\u2014even in middle age her beauty was right there, resting in more than her face. Her wimple and the high collar of her gown revealed little enough of her. But her bearing was more than noble, or haughty. Her bearing was commanding, confident in a way that only the great of the land were confident. The captain noted that her nuns obeyed her with an eagerness born of either fear or the pleasure of service.\n\nThe captain wondered which it was.\n\n\"You took long enough to reach us,\" she said, by way of greeting. Then she snapped her fingers and beckoned at a pair of servants to bring a tray. \"We are servants of God here\u2014don't you think you might have managed to strip your armour before you came to my hall?\" the Abbess asked. She glanced around, caught a novice's eye, raised an eyebrow. \"Fetch the captain a stool,\" she said. \"Not a covered one. A solid one.\"\n\n\"I wear armour every day,\" the captain said. \"It comes with my profession.\" The great hall was as big as the courtyard outside, with high windows of stained glass set near the roof, and massive wooden beams so old that age and soot had turned them black. The walls were whitewashed over fine plaster, and held niches containing images of saints and two rich books\u2014clearly on display to overawe visitors. Their voices echoed in the room, which was colder than the wet courtyard outside. There was no fire in the central hearth.\n\nThe Abbess's people brought her wine, and she sipped it as they placed a small table at the captain's elbow. He was three feet beneath her. \"Perhaps your armour is unnecessary in a nunnery?\" she asked.\n\nHe raised an eyebrow. \"I see a fortress,\" he said. \"It happens that there are nuns in it.\"\n\nShe nodded. \"If I chose to order you taken by my men, would your armour save you?\" she asked.\n\nThe novice who brought his stool was pretty and she was careful of him, moving with the deliberation of a swordsman or a dancer. He turned his head to catch her eye and felt the tug of her power, saw that she was not merely pretty. She set the heavy stool down against the back of his knees. Quite deliberately, the captain touched her arm gently and caused her to turn to him. He turned to face her, putting his back to the Abbess.\n\n\"Thank you,\" he said, looking her in the eye with a calculated smile. She was tall and young and graceful, with wide-set almond-shaped eyes and a long nose. Not pretty; she was arresting.\n\nShe blushed. The flush travelled like fire down her neck and into her heavy wool gown.\n\nHe turned back to the Abbess, his goal accomplished. Wondering why the Abbess had placed such a desirable novice within his reach, unless she meant to. \"If I chose to storm your abbey, would your piety save you?\" he asked.\n\nShe blazed with anger. \"How dare you turn your back on me?\" she asked. \"And leave the room, Amicia. The captain has bitten you with his eyes.\"\n\nHe was smiling. He thought her anger feigned.\n\nShe met his eyes and narrowed her own\u2014and then folded her hands together, almost as if she intended to pray.\n\n\"Honestly, Captain, I have prayed and prayed over what to do here. Bringing you to fight the Wild is like buying a wolf to shepherd sheep.\" She looked him in the eye. \"I know what you are,\" she said.\n\n\"Do you really?\" he asked. \"All the better, lady Abbess. Shall we to business, then? Now the pleasantries are done?\"\n\n\"But what shall I call you?\" she asked. \"You are a well-born man, for all your snide airs. My chamberlain\u2014\"\n\n\"Didn't have a nice name for me, did he, my lady Abbess?\" He nodded. \"You may call me Captain. It is all the name I need.\" He nodded graciously. \"I do not like the name your chamberlain used. Bourc. I call myself the Red Knight.\"\n\n\"Many men are called bourc,\" she said. \"To be born out of wedlock is\u2014\"\n\n\"To be cursed by God before you are born. Eh, lady Abbess?\" He tried to stop the anger that rose on his cheeks like a blush. \"So very fair. So _just_.\"\n\nShe scowled at him for a moment, annoyed with him the way older people are often annoyed with the young, when the young posture too much.\n\nHe understood her in a glance.\n\n\"Too dark? Should I add a touch of heroism?\" he asked with a certain air.\n\nShe eyed him. \"If you wrap yourself in darkness,\" she said, \"you risk merely appearing dull. But you have the wit to know it. There's hope for you, boy, if you know that. Now to business. I'm not rich\u2014\"\n\n\"I have never met anyone who would admit being rich,\" he agreed. \"Or to getting enough sleep.\"\n\n\"More wine for the captain,\" snapped the Abbess to the sister who had guarded the door. \"But I can pay you. We are afflicted by something from the Wild. It has destroyed two of my farms this year, and one last year. At first\u2014at first, we all hoped that they were isolated incidents.\" She met his eye squarely. \"It is not possible to believe that any more.\"\n\n\"Three farms this year,\" said the captain. He fished in his purse, hesitated over the chain with the leaf amulet, then fetched forth a cross inlaid with pearls instead.\n\n\"Oh, by the wounds of Christ!\" swore the Abbess. \"Oh, Blessed Virgin protect and cherish her. Sister Hawisia! Is she\u2014\"\n\n\"She is dead,\" the captain said. \"And six more corpses in the garden. Your good sister died trying to protect them.\"\n\n\"Her faith was very strong,\" the Abbess said. She was dry eyed, but her voice trembled. \"You needn't mock her.\"\n\nThe captain frowned. \"I never mock courage, lady Abbess. To face such a thing without weapons\u2014\"\n\n\"Her faith was a weapon against evil, Captain.\" The Abbess leaned forward.\n\n\"Strong enough to stop a creature from the Wild? No, it was not,\" said the captain quietly. \"I won't comment on evil.\"\n\nThe Abbess stood sharply. \"You are some sort of atheist, are you, Captain?\"\n\nThe captain frowned again. \"There is nothing productive for us in theological debate, my lady Abbess. Your lands have attracted a malignant entity\u2014an enemy of Man. They seldom hunt alone, especially not this far from the Wild. You wish me to rid you of them. I can. And I will. In exchange, you will pay me. That is all that matters between us.\"\n\nThe Abbess sat again, her movements violent, angry. The captain sensed that she was off balance\u2014that the death of the nun had struck her personally. She was, after all, the commander of a company of nuns.\n\n\"I am not convinced that engaging you is the right decision,\" she said.\n\nThe captain nodded. \"It may not be, lady Abbess. But you sent for me, and I am here.\" Without intending to, he had lowered his voice, and spoke softly.\n\n\"Is that a threat?\" she asked.\n\nInstead of answering, the captain reached into his purse again and withdrew the broken chain holding a small leaf made of green enamel on bronze.\n\nThe Abbess recoiled as if from a snake.\n\n\"My men found this,\" he said.\n\nThe Abbess turned her head away.\n\n\"You have a traitor,\" he said. And rose. \"Sister Hawisia had an arrow in her back. While she faced something terrible, something very, very terrible.\" He nodded. \"I will go to walk the walls. You need time to think if you want us. Or not.\"\n\n\"You will poison us,\" she said. \"You and your kind do not bring peace.\"\n\nHe nodded. \"We bring you no peace, but a company of swords, my lady.\" He grinned at his own misquote of scripture. \"We don't make the violence. We merely deal with it as it comes to us.\"\n\n\"The devil can quote scripture,\" she said.\n\n\"No doubt he had his hand in writing it,\" the captain shot back.\n\nShe bit back a counter\u2014he watched her face change as she decided not to rise to his provocation. And he felt a vague twinge of remorse for goading her, an ache like the pain in his wrist from making too many practice cuts the day before. And, like the pain in his wrist, he was unaccustomed to remorse.\n\n\"I could say it is a little late to think of peace now.\" He sneered briefly and then put his sneer away. \"My men are here, and they haven't had a good meal or a paid job in some weeks. I offer this, not as a threat, but as a useful piece of data as you reason through the puzzle. I also think that the creature you have to deal with is far worse than you have imagined. In fact, I'll go so far as to say it's far worse than I had imagined. It is big, powerful, and angry, and very intelligent. And more likely two than one.\"\n\nShe winced.\n\n\"Allow me a few minutes to think,\" she said.\n\nHe nodded, bowed, set his riding sword at his waist, and walked back into the courtyard.\n\nHis men stood like statues, their scarlet surcoats livid against their grey surroundings. The horses fretted\u2014but only a little\u2014and the men less.\n\n\"Be easy,\" he said.\n\nThey all took breath together. Stretched arms tired from bearing armour, or hips bruised from mail and cuirass.\n\nMichael was the boldest. \"Are we in?\" he asked.\n\nThe captain didn't meet his eye because he'd noticed an open window across the courtyard, and seen the face framed in it. \"Not yet, my honey. We are not in yet.\" He blew a kiss at the window.\n\nThe face vanished.\n\nSer Milus, his primus pilus and standard bearer, grunted. \"Bad for business,\" he said. And then, as an afterthought, \"m'lord.\"\n\nThe captain flicked him a glance and looked back to the dormitory windows.\n\n\"There's more virgins watching us right now,\" Michael opined, \"Then have parted their legs for me in all my life.\"\n\nJehannes, the senior marshal, nodded seriously. \"Does that mean one, young Michael? Or two?\"\n\nGuillaume Longsword, the junior marshal, barked his odd laugh, like the seals of the northern bays. \"The second one said she was a virgin,\" he mock-whined. \"At least, that's what she told me!\"\n\nComing through the visor of his helmet, his voice took on an ethereal quality that hung in the air for a moment. Men do not look on horror and forget it. They merely put it away. Memories of the steading were still too close to the surface, and the junior marshal's voice had summoned them, somehow.\n\nNo one laughed. Or rather, most of them laughed, and all of it was forced.\n\nThe captain shrugged. \"I have chosen to give our prospective employer some time to consider her situation,\" he said.\n\nMilus barked a laugh. \"Stewing in her juice to raise the price, is that it?\" he asked. He nodded at the door of the chapel. \"Yon has no liking for us.\"\n\nThe priest continued to stand in his doorway.\n\n\"Think he's a dimwit? Or is he the pimp?\" Ser Milus asked. And stared at the priest. \"Be my guest, cully. Stare all ye like.\"\n\nThe soldiers chuckled, and the priest went into the chapel.\n\nMichael flinched at the cruelty in the standard bearer's tone, then stepped forward. \"What is your will, m'lord?\"\n\n\"Oh,\" the captain said, \"I'm off hunting.\" He stepped away quickly, with a wry smile, walked a few steps toward the smithy, concentrated... and vanished.\n\nMichael looked confused. \"Where is he?\" he asked.\n\nMilus shrugged, shifting the weight of his hauberk. \"How does he do that?\" he asked Jehannes.\n\nTwenty paces away, the captain walked into the dormitory wing as if it was his right to do so. Michael leaned as if to call out but Jehannes put his gauntleted hand over Michael's mouth.\n\n\"There goes our contract,\" Hugo said. His dark eyes crossed with the standard bearer's, and he shrugged, despite the weight of the maille on his shoulders. \"I told you he was too young.\"\n\nJehannes eased his hand off the squire's face. \"He has his little ways, the Bourc.\" He gave the other men a minute shake of his head. \"Let him be. If he lands us this contract\u2014\"\n\nHugo snorted, and looked up at the window.\n\nThe captain reached into the palace in his head.\n\n_A vaulted room, twelve sided, with high, arched, stained glass windows, each one bearing a different image set at even intervals between columns of aged marble that supported a groined roof. Under each window was a sign of the zodiac, painted in brilliant blue on gold leaf, and then a band of beaten bronze as wide as a man's arm, and finally, at eye level, a series of niches between the columns, each holding a statue; eleven statues of white marble, and one iron-bound door under the sign of Ares._\n\n_In the exact centre of the room stood a twelfth statue\u2014Prudentia, his childhood tutor. Despite her solid white marble skin, she smiled warmly as he approached her._\n\n_\"Clementia, Pisces, Eustachios,\" he said in the palace of his memory, and his tutor's veined white hands moved to point at one sign and then another._\n\n_And the room moved._\n\n_The windows rotated silently above the signs of the zodiac, and the statues below the band of bronze rotated in the opposite direction until his three chosen signs were aligned opposite to the iron-bound door. And he smiled at Prudentia, walked across the tiles of the twelve-sided room and unlatched the door._\n\n_He opened it on a verdant garden of rich summer green\u2014the dream memoryof the perfect summer day. It was not always thus, on the far side of the door. A rich breeze blew in. It was not always this strong, his green power, and he deflected some with the power of his will, batting it into a ball and shoving it like a handful of summer leaves into a hempen bag he imagined into being and hung from Prudentia's outstretched arm. Against a rainy day. The insistent green breeze stirred through his hair and then reached the aligned signs on the opposite wall and_\u2014\n\nHe moved away from the horses without urgency, secure in the knowledge that Michael would be distracted as he moved\u2014and so would the watcher in the window.\n\nThe captain's favourite phantasms depended on misdirection more than aethereal force. He preferred to add to their efficacy with physical efficiency\u2014he walked quietly, and didn't allow his cloak to flap.\n\nAt the door to the dormitory he reached into his memory palace and\n\n_leaned into the vaulted room. \"Same again, Pru,\" he said._\n\n_Again the sigils moved as the marble statue pointed to the signs, already aligned above the door. He opened it again, allowed the green breeze to power his working, and let the door close._\n\nHe walked into the dormitory building. There were a dozen nuns, all big, capable women, sitting in the good light of the clerestory windows, and most of them were sewing.\n\nHe walked past them without a swirl of his scarlet cloak, his whole will focused on his belief that _his presence there was perfectly normal_ and started up the stairs. No heads turned, but one older nun stopped peering at her embroidery and glanced at the stairwell, raised an eyebrow, and then went back to her work. He heard a murmur from behind him.\n\n_Not entirely fooled then,_ he thought. _Who are these women?_\n\nHis sabatons made too much noise and he had to walk carefully, because power\u2014at least, the sort of power he liked to wield\u2014was of limited use. The stairs wound their way up and up, turning as tightly as they would in any other fortress, to foul his sword arm if he was an attacker.\n\n_Which I am, of a sort_ , he thought. The gallery was immediately above the hall. Even on a day this grey, it was full of light. Three grey-clad novices leaned on the casemates of the windows, watching the men in the yard. Giggling.\n\nAt the edge of his power, he was surprised to find traces of _their_ power.\n\nHe stepped into the gallery, and his sabaton made a distinct metallic scratch against the wooden floor\u2014a clarion sound in a world of barefoot women. He didn't try to strain credulity by _willing_ himself to seem normal, here.\n\nThe three heads snapped around. Two of the girls turned and ran. The third novice hesitated for a fatal moment\u2014looking. Wondering.\n\nHe had her hand. \"Amicia?\" he said into her eyes, and then put his mouth over hers. Put an armoured leg inside her thighs and trapped her\u2014turned her over his thigh as easily as throwing a child in a wrestling match, and she was in his arms. He rested his back plate against the ledge of the cloister and held her. Gently. Firmly.\n\nShe wriggled, catching her falling sleeve against the flange that protected his elbow. But her eyes were locked on his\u2014and huge. She opened her lips. More there than simple fear or refusal. He licked her teeth. Ran a finger under her chin.\n\nHer mouth opened under his\u2014delicious.\n\nHe kissed her, or perhaps she kissed him. It was not brief. She relaxed into him\u2014itself a pleasing warmth, even through the hardened steel of his arm harness and breastplate.\n\nKisses end.\n\n\"Don't take the vows,\" he said. \"You do not belong here.\" He meant to sound teasing, but even in his own head his voice dripped with unintended mockery.\n\nHe stood straight and set her on the ground, to show that he was no rapist. She blushed red from her chin to her forehead, again. Even the backs of her hands were red. She cast her eyes down, and then shifted her weight\u2014he watched such things. She leaned forward\u2014\n\nAnd slammed a hand into his right ear. Taking him completely by surprise. He reeled, his back hit the wall with a metallic _thud_ , and he caught himself\u2014\n\n\u2014and turned to chase her down.\n\nBut she wasn't running. She stood her ground. \"How dare you judge me?\" she said.\n\nHe rubbed his ear. \"You mistake me,\" he said. \"I meant no hard judgment. You wanted to be kissed. It is in your eyes.\"\n\nAs a line, it had certainly worked before. In this case, he felt it to be true. Despite the sharp pain in his ear.\n\nShe pursed her lips\u2014full, very lovely lips. \"We are all of us sinners, messire. I struggle with my body every day. That gives you no right to it.\"\n\nThere was a secret smile to the corner of her mouth\u2014really, no smile at all, but something\u2014\n\nShe turned and walked away down the gallery, leaving him alone.\n\nHe descended the stairs, rubbing his ear, wondering how much of the exchange had been witnessed by his men. Reputations can take months to build and be lost in a few heartbeats and his was too new to weather a loss of respect. But he calculated that the grey sky and the angle of the gallery windows should have protected him.\n\n\"That was quick,\" said Michael, admiringly, as he emerged. The captain was careful not to do anything as gross as tuck his braes into his hose. Because, had he taken her right there against the cloister wall, he would still have re-dressed meticulously before emerging.\n\n_Why didn't I?_ He asked himself. _She was willing enough._\n\n_She liked me._\n\n_She hit me very hard._\n\nHe smiled at Michael. \"It took as long as it took,\" he said. As he spoke, the heavy iron-bound door opened and a mature nun beckoned to the captain.\n\n\"The devil himself watches over you,\" Hugo muttered.\n\nThe captain shook his head. \"The devil doesn't give a fuck, either,\" he said, and went to deal with the Abbess.\n\nHe knew as soon he crossed the threshold that she'd elected to take them on. If she'd decided _not_ to take them on, she wouldn't have seen him again. Murder in the courtyard might have been closer to the mark.\n\nExcept that all the soldiers she had couldn't kill the eight of them in the courtyard. And she knew it. If she had eight good men, she'd never have sent for him to begin with.\n\nIt was like Euclidean geometry. And the captain could never understand why other people couldn't see all the angles.\n\nHe rubbed at the stinging in his ear, bowed deeply to the Abbess, and mustered up a smile.\n\nShe nodded. \"I have to take you as you are,\" she said. \"So I will use a long spoon. Tell me your rates?\"\n\nHe nodded. \"May I sit?\" he asked. When she extended a reasonably gracious hand, he picked up the horn wine cup that had obviously been placed for him. \"I drink to your eyes, ma belle.\"\n\nShe held his gaze with her own and smiled. \"Flatterer.\"\n\n\"Yes,\" he said, taking a sip of wine and continuing to meet her stare over the rim like a proper courtier. \"Yes, but no.\"\n\n\"My beauty is long gone, with the years,\" she said.\n\n\"Your body remembers your beauty so well that I can still see it,\" he said.\n\nShe nodded. \"That was a beautiful compliment,\" she admitted. Then she laughed. \"Who boxed your ear?\" she asked.\n\nHe stiffened. \"It is an old\u2014\"\n\n\"Nonsense! I educate children. I know a boxed ear when I see one.\" She narrowed her eyes. \"A nun.\"\n\n\"I do not kiss and tell,\" he said.\n\n\"You are not as bad as you would have me believe, messire,\" she replied.\n\nThey gazed at each other for a few breaths.\n\n\"Sixteen double leopards a month for every lance. I have thirty-one lances today\u2014you may muster them and count them yourself. Each lance consists of at least a knight, his squire, and a valet; usually a pair of archers. All mounted, all with horses to feed. Double pay for my corporals. Forty pounds a month for my officers\u2014there are three\u2014and a hundred pounds for me. Each month.\" He smiled lazily. \"My men are very well disciplined. And worth every farthing.\"\n\n\"And if you kill my monster tonight?\" she asked.\n\n\"Then you have a bargain, lady Abbess\u2014only one month's pay.\" He sipped his wine.\n\n\"How do you tally these months?\" she asked.\n\n\"Ah! There's none sharper than you, even in the streets of Harndon, lady. Full months by the lunar calendar.\" He smiled. \"So the next one starts in just two weeks. The Merry month of May.\"\n\n\"Jesu, Lord of the Heavens and Saviour of Man. You are not cheap.\" She shook her head.\n\n\"My people are very, very good at this. We have worked on the Continent for many years, and now we are back in Alba. Where you need us. You needed us a year ago. I may be a hard man, lady, but let us agree that no more Sister Hawisias need die? Yes?\" He leaned forward to seal the deal, the wine cup between his hands, and suddenly the weight of his armour made him tired and his back hurt.\n\n\"I'm sure Satan is charming if you get to know him,\" she said quietly. \"And I'm sure that if you aren't paid, your interest in the Sister Hawisia's of this world will vanish like snow in strong sunshine.\" She gave him a thin-lipped smile. \"Unless you can kiss them\u2014and even then, I doubt you stay with them long. Or they with you.\"\n\nHe frowned.\n\n\"For every steading damaged by your men, I will deduct the price of a lance,\" she said. \"For every man of mine injured in a brawl, for every woman who complains to me of your men, the price of a corporal. If a single one of my sisters is injured\u2014or violated\u2014by your Satan's spawn, even so much as a lewd hand laid to her or an unseemly comment made, I will deduct _your_ fee. Do you agree? Since,\" she said with icy contempt, \"Your men are so well disciplined?\"\n\n_She really does like me,_ he thought. _Despite all._ He was more used to people who disliked him. And he wondered if she would give him Amicia. She'd certainly put the beautiful novice where he could see her. How calculating was the old witch? She seemed the type who would try to lure him with more than coin\u2014but he'd already pricked her with his comment about Sister Hawisia.\n\n\"What's the traitor worth?\" he asked.\n\nShe shook her head. \"I do not believe in your traitor,\" she said, pointing on the enamel leaf on a wooden platter by her side. \"You carry this foul thing with you to trick fools. And I am not a fool.\"\n\nHe shrugged. \"My lady, you are allowing your dislike for my kind to cloud your judgment. Consider: what could make me to lie to you about such a thing? How many people should have been at that steading?\" he asked.\n\nShe met his eye\u2014she had no trouble with that, which pleased him. \"There should have been seven confreres to work the fields,\" she allowed.\n\n\"We found your good sister and six other corpses,\" the captain countered. \"It is all straightforward enough, lady Abbess.\" He sipped more wine. \"One is missing when none could have escaped. None.\" He paused. \"Some of your sheep have grown teeth. And no longer wish to be part of your flock.\" He had a sudden thought. \"What was Sister Hawisia doing there? She was a nun of the convent, not a labourer?\"\n\nShe took a sharp breath. \"Very well. If you can prove there is a traitor\u2014or traitors\u2014there will be reward. You must trust that I will be fair.\"\n\n\"Then you must understand: my men will behave badly\u2014it is months since they were paid, and longer since they've been anywhere they might spend what they don't have. The writ of my discipline does not run to stopping tavern brawls or lewd remarks.\" He tried to look serious, though his heart was all but singing with the joy of work and gold to pay the company. \"You must trust that I will do my best to keep them to order.\"\n\n\"Perhaps you'll have to lead by example?\" she said. \"Or get the task done quickly and move on to greener pastures?\" she asked sweetly. \"I understand the whores are quite comely south of the river. In the Albin.\"\n\nHe thought of the value of this contract\u2014she hadn't quibbled at his inflated prices.\n\n\"I'll decide which seems more attractive when I've seen the colour of your money,\" he said.\n\n\"Money?\" she asked.\n\n\"Payment due a month in advance, lady Abbess. We _never_ fight for free.\"\n\n# Lorica\u2014A Golden Bear\n\nThe bear was huge. All of the people in the market said so.\n\nThe bear sat in its chains, legs fully extended like an exhausted dancer, head down. It had leg manacles, one on each leg, and the chains had been wrought cunningly so that the manacles were connected by running links that limited the beast's movement.\n\nBoth of its hind paws were matted with blood\u2014the manacles were also lined in small spikes.\n\n\"See the bear! See the bear!\"\n\nThe bear keeper was a big man, fat as a lord, with legs like tree trunks and arms like hams. His two boys were small and fast and looked as if they might have a second profession in crime.\n\n\"A golden bear of the Wild! Today only!\" he bellowed, and his boys roamed through the market, shouting \"Come and see the bear! The golden bear!\"\n\nThe market was full, as market can only be at the first breath of spring when every farmer and petty-merchant has been cooped up in a croft or a town house all winter. Every goodwife had new-made baskets to sell. Careful farmers had sound winter apples and carefully hoarded grain on offer. There were new linens\u2014shirts and caps. A knife grinder did a brisk trade, and a dozen other tradesmen and women shouted their wares\u2014fresh oysters from the coast, lambs for sale, tanned leather.\n\nThere were close on five hundred people in the market, and more coming in every hour.\n\nA taproom boy from the inn rolled two small casks up, one at a time, placed a pair of boards across them and started serving cider and ale. He set up under the old oak that marked the centre of the market field, a stone's throw from the bear master.\n\nMen began to drink.\n\nA wagoner brought his little daughter to see the bear. It was female, with two cubs. They were beautiful, with their gold-tipped blond fur, but their mother smelled of rot and dung. Her eyes were wild, and when his daughter touched one of the cubs the fearsome thing opened its jaws, and his daughter started at the wicked profusion of teeth. The growing crowd froze and then people shrank back.\n\nThe bear raised a paw, stretching the chains\u2014\n\nShe stood her ground. \"Poor bear!\" she said to her father.\n\nThe bear's paw was well short of touching the girl. And the pain of moving against the spiked manacles overcame the bear's anger. It fell back on all fours, and then sat again, looking almost human in its despair.\n\n\"Shh!\" he said. \"Hush, child. It's a creature of the Wild. A servant of the enemy.\" Truth to tell, his voice lacked conviction.\n\n\"The cubs are wonderful.\" The daughter got down on her haunches.\n\nThey had ropes on them, but no more.\n\nA priest\u2014a very worldly priest in expensive blue wool, wearing a magnificent and heavy dagger\u2014leaned down. He put his fist before one of the cubs' muzzles and the little bear bit him. He didn't snatch his hand back. He turned to the girl. \"The Wild is often beautiful, daughter. But that beauty is Satan's snare for the unwary. Look at him. Look at him!\"\n\nThe little cub was straining at his rope to bite the priest again. As he rose smoothly to his feet and kicked the cub, he turned to the bear master.\n\n\"It is very like heresy, keeping a creature of the Wild for money,\" he said.\n\n\"For which I have a licence from the Bishop of Lorica!\" sputtered the bear master.\n\n\"The bishop of Lorica would sell a licence to Satan to keep a brothel,\" said the priest with a hand on the dagger in his belt.\n\nThe wagoner took hold of his daughter but she wriggled free. \"Pater, the bear is in pain,\" she said.\n\n\"Yes,\" he said. He was a thoughtful man. But his eyes were on the priest.\n\nAnd the priest's eyes were on him.\n\n\"Is it right for us to hurt any creature?\" his daughter asked. \"Didn't God make the Wild, just as he made us?\"\n\nThe priest smiled and it was as terrible as the bear's teeth. \"Your daughter has some very interesting notions,\" he said. \"I wonder where she gets them?\"\n\n\"I don't want any trouble,\" the wagoner said. \"She's just a child.\"\n\nThe priest stepped closer, but just then the bear master, eager to get a show, began to shout. He had quite a crowd\u2014at least a hundred people, and there were more wandering up every minute. There were half a dozen of the earl's soldiers as well, their jupons open in the early heat, flirting with the farmers' daughters. They pushed in eagerly, hoping to see blood.\n\nThe wagoner pulled his daughter back, and let the soldiers pass between him and the priest.\n\nThe bear master kicked the bear and pulled on the chain. One of his boys began to play a quick, staccato tune on a tin whistle.\n\nThe crowd began to chant, \"Dance! Dance! Dance, bear, dance!\"\n\nThe bear just sat. When the bear master's tugging on the chains caused her pain, she raised her head and roared her defiance.\n\nThe crowd shuffled back, muttering in disappointment, except for the priest.\n\nOne of the soldiers shook his head. \"This is crap,\" he said. \"Let's put some dogs on it.\"\n\nThe idea was instantly popular with his mates, but not at all with the bear master. \"That's my bear,\" he insisted.\n\n\"Let me see your pass for the fair,\" said the sergeant. \"Give it here.\"\n\nThe man looked at the ground, silenced, for all his size. \"Which I ain't got one.\"\n\n\"Then I can take your bear, mate. I can take your bear and your boys.\" The sergeant smiled. \"I ain't a cruel man,\" he said, his tone indicating that this statement was untrue. \"We'll put some dogs on your bear, fair as fair. You'll collect the silver. We'll have some betting.\"\n\n\"This is a gold bear,\" said the bear master. He was going pale under his red, wine-fed nose. \"A gold bear!\"\n\n\"You mean you spent some silver on putting a bit of gilt on her fur,\" said another soldier. \"Pretty for the crowd.\"\n\nThe bear master shrugged. \"Bring your dogs,\" he said.\n\nIt turned out that many of the men in the crowd had dogs they fancied against a bear.\n\nThe wagoner slipped back another step, but the priest grabbed his arm. \"You stay right here,\" he said. \"And your little witch of a daughter.\"\n\nThe man's grip was like steel, and the light in his eyes was fanatical. The wagoner allowed himself, reluctantly, to be pulled back into the circle around the bear.\n\nDogs were being brought. There were mastiffs\u2014great dogs the size of small ponies\u2014and big hounds, and some mongrels that had replaced size with sheer ferocity. Some of the dogs sat quietly while others growled relentlessly at the bear.\n\nThe bear raised its head and growled too\u2014once.\n\nAll the dogs backed away a step.\n\nMen began to place bets.\n\nThe bear master and his boys worked the crowd. If he was hesitant to see his bear in a fight, he wasn't hesitant about accepting the sheer quantity of silver suddenly crossing his palm. Even the smallest farmer would wager on a bear baiting. And when the bear was a creature of the Wild\u2014\n\nWell it was almost a religious duty to bet against it.\n\nThe odds against the bear went up and up.\n\nSo did the number of dogs, and they were becoming unmanageable as the pack grew. Thirty angry dogs can hate each other as thoroughly as they hate a bear.\n\nThe priest stepped out of the ring. \"Look at this creature of Evil!\" he said. \"The very embodiment of the enemy. Look at its fangs and teeth, designed by the Unmaker to kill men. And look at these dogs men have bred\u2014animals reduced to lawful obedience by patient generations of men. No one dog can bring down this monster alone, but does anyone doubt that many of them can? And is this lesson lost on any man here? The bear\u2014look at it\u2014is mighty. But man is more puissant by far.\"\n\nThe bear didn't raise its head.\n\nThe priest kicked it.\n\nIt stared at the ground.\n\n\"It won't even fight!\" said one of the guards.\n\n\"I want my money back!\" shouted a wheelwright.\n\nThe priest smiled his terrible smile. He grabbed the rope around one of the little cubs, hauled the creature into the air by the scruff of the neck, and tossed it in among the dogs.\n\nThe bear leaped to its feet.\n\nThe priest laughed. \"Now it will fight,\" it said.\n\nThe bear strained against its manacles as the mastiffs ripped the screaming cub to shreds. It sounded like a human child, terrified and afraid, and then it was gone\u2014savaged and eaten by a dozen mongrels. Eaten alive.\n\nThe wagoner had his hands over his daughter's eyes.\n\nThe priest whirled on him, eyes afire. \"Show her!\" he shrieked. \"Show her what happens when evil is defeated!\" He took a step towards the wagoner\u2014\n\nAnd the bear moved. She moved faster than a man would have thought possible.\n\nShe had his head in one paw and his dagger in the other before his body, pumping blood across the crowd, hit the dirt. Then she whirled\u2014suddenly nothing but teeth and claws\u2014and sank the heavy steel dagger into the ground _through_ the links of her chain.\n\nThe links popped.\n\nA woman screamed.\n\nShe killed as many of them as she could catch, until her claws were glutted with blood, and her limbs ached. They screamed, and hampered each other, and her paws struck them hard like rams in a siege, and every man and woman she touched, she killed.\n\nIf she could have she would have killed every human in the world. Her cub was dead. _Her cub was dead._\n\nShe killed and killed, but they ran in all directions.\n\nWhen she couldn't catch any more, she went back and tore at their corpses\u2014found a few still alive and made sure they died in fear.\n\n_Her cub was dead._\n\nShe had no time to mourn. Before they could bring their powerful bows and their deadly, steel-clad soldiers, she picked up her remaining cub, ignored the pain and the fatigue and all the fear and panic she felt to be so deep in the tame horror of human lands, and fled. Behind her, in the town, alarm bells rang.\n\nShe ran.\n\n# Lorica\u2014Ser Mark Wishart\n\nOnly one knight came, and his squire. They rode up to the gates at a gallop, summoned from their Commandery, to find the gates closed, the towers manned, and men with crossbows on the walls.\n\n\"A creature of the Wild!\" shouted the panicked men on the wall before they refused to open the gates for him\u2014even though they'd summoned him. Even though he was the Prior of the Order of Saint Thomas. A paladin, no less.\n\nThe knight rode slowly around the town until he came to the market field.\n\nHe dismounted. His squire watched the fields as if a horde of boglins might appear at any moment.\n\nThe knight opened his visor, and walked slowly across the field. There were a few corpses at the edge, by the dry ditch that marked the legal edge of the field. The bodies lay thicker as he grew closer to the Market Oak. Thicker and thicker. He could hear the flies. Smell the opened bowels, warm in the sun.\n\nIt smelled like a battlefield.\n\nHe knelt for a moment, and prayed. He was, after all, a priest, as well as a knight. Then he rose slowly and walked back to his squire, spurs catching awkwardly on the clothes of the dead.\n\n\"What\u2014what was it?\" asked his squire. The boy was green.\n\n\"I don't know,\" said the knight. He took off his helmet and handed it to his squire.\n\nThen he walked back into the field of death.\n\nHe made a quick count. Breathed as shallowly as he could.\n\nThe dogs were mostly in one place. He drew his sword, four feet of mirror-polished steel, and used it as a pry-bar to roll the corpse of a man with legs like tree trunks and arms like hams off the pile of dogs.\n\nHe knelt and took off a gauntlet, and picked up what looked like a scrap of wool.\n\nLet out a breath.\n\nHe held out his sword, and called on God for aid, and gathered the divine golden power, and then made a small working.\n\n\"Fools,\" he said aloud.\n\nHis working showed him where the priest had died, too. He found the man's head, but left it where it lay. Found his dagger, and placed a _phantasm_ on it.\n\n\"You arrogant idiot,\" he said to the head.\n\nHe pulled the wagoner's body off the mangled corpse of his daughter. Turned aside and threw up, and then knelt and prayed. And wept.\n\nAnd finally, stumbled to his feet and walked back to where his squire waited, the worry plain on his face.\n\n\"It was a golden bear,\" he said.\n\n\"Good Christ!\" said the squire. \"Here? Three hundred leagues from the wall?\"\n\n\"Don't blaspheme, lad. They brought it here captive. They baited it with dogs. It had cubs, and they threw one to the dogs.\" He shrugged.\n\nHis squire crossed himself.\n\n\"I need you to ride to Harndon and report to the king,\" the knight said. \"I'll track the bear.\"\n\nThe squire nodded. \"I can be in the city by nightfall, my lord.\"\n\n\"I know. Go now. It's one bear, and men brought it here. I'll stem these fools' panic\u2014although I ought to leave them to wallow in it. Tell the king that the Bishop of Jarsay is short a vicar. His headless corpse is over there. Knowing the man, I have to assume this was his fault, and the kindest thing I can say is that he got what he deserved.\"\n\nHis squire paled. \"Surely, my lord, now it is you who blaspheme.\"\n\nSer Mark spat. He could still taste his own vomit. He took a flask of wine from the leather bag behind his saddle and drank off a third of it.\n\n\"How long have you been my squire?\" he asked.\n\nThe young man smiled. \"Two years, my lord.\"\n\n\"How often have we faced the Wild together?\" he asked.\n\nThe young man raised his eyebrows. \"A dozen times.\"\n\n\"How many times has the Wild attacked men out of pure evil?\" the knight asked. \"If a man prods a hornet's nest with a pitchfork and gets stung, does that make the hornets evil?\"\n\nHis squire sighed. \"It's not what they teach in the schools,\" he said.\n\nThe knight took another pull at his flask of wine. The shaking in his hands was stopping. \"It's a mother, and she still has a cub. There's the track. I'll follow her.\"\n\n\"A golden bear?\" the squire asked. \"Alone?\"\n\n\"I didn't say I'd fight her in the lists, lad. I'll follow her. You tell the king.\" The man leaped into his saddle with an acrobatic skill which was one of the many things that made his squire look at him with hero-worship. \"I'll send a phantasm to the Commandery if I've time and power. Now go.\"\n\n\"Yes, my lord.\" The squire turned his horse and was off, straight to a gallop as he'd been taught by the Order.\n\nSer Mark leaned down from his tall horse and looked at the tracks, and then laid a hand on his war horse's neck. \"No need to hurry, Bess,\" he said.\n\nHe followed the track easily. The golden bear had made for the nearest woods, as any creature of the Wild would. He didn't bother to follow the spoor exactly, but merely trotted along, checking the ground from time to time. He was too warm in full harness, but the alarm had caught him in the tiltyard, fully armed.\n\nThe wine sang in his veins. He wanted to drain the rest of it.\n\nThe dead child\u2014\n\nThe scraps of the dead cub\u2014\n\nHis own knight\u2014when he was learning his catechism and serving his caravans as a squire\u2014had always said _War kills the innocent first._\n\nWhere the stubble of last year's wheat ran up into a tangle of weeds, he saw the hole the bear had made in the hedge. He pulled up.\n\nHe didn't have a lance, and a lance was the best way to face a bear.\n\nHe drew his war sword, but he didn't push Bess though the gap in the hedge.\n\nHe rode along the lane, entered the field carefully through the gate, and rode back along the hedge at a canter.\n\nTracks.\n\nBut no bear.\n\nHe felt a little foolish to have drawn his sword, but he didn't feel any inclination to put it away. The fresh tracks were less than an hour old, and the bear's paw print was the size of a pewter plate from the Commandery's kitchens.\n\nSuddenly, there was crashing in the woods to his left.\n\nHe tightened the reins, and turned his horse. She was beautifully trained, pivoting on her front feet to keep her head pointed at the threat.\n\nThen he backed her, step by step.\n\n_Crash._\n\n_Rustle._\n\nHe saw a flash of movement, turned his head and saw a jay leap into the air, flicked his eyes back\u2014\n\nNothing.\n\n\"Blessed Virgin, stand with me,\" he said aloud. Then he rose an inch in his war saddle and just _touched_ his spurs to Bess's sides, and she walked forward.\n\nHe turned her head and started to ride around the wood. It couldn't be that big.\n\n_Rustle._\n\n_Rustle._\n\n_Crack._\n\n_Crash._\n\nIt was _right there._\n\nHe gave the horse more spur, and they accelerated to a canter. The great horse made the earth shake.\n\n# Near Lorica\u2014A Golden Bear\n\nShe was being hunted. She could smell the horse, hear its shod hooves moving on the spring earth, and she could _feel_ its pride and its faith in the killer on its back.\n\nAfter months of degradation and slavery, torture and humiliation she would happily have turned and fought the steel-clad war man. Glory for her if she defeated him, and a better death than she had imagined in a long time. But her cub mewed at her. The cub\u2014it was all for the cub. She had been captured because they could not run and she would not leave them, and she had endured for them.\n\nShe only had one left.\n\nShe was the smaller of the two, and the gold of her fur was brighter, and she was on the edge of exhaustion, suffering from dehydration and panic. She had lost the power of speech and could only mew like a dumb animal. Her mother feared she might have lost it for life.\n\nBut she had to try. The very blood in her veins cried out that she had to try to save her young.\n\nShe picked the cub up in her teeth the way a cat carried a kitten, and ran again, ignoring the pain in her paws.\n\n# Lorica\u2014Ser Mark Wishart\n\nThe knight cantered around the western edge of the woods and saw the river stretching away in a broad curve. He saw the shambing golden creature in the late sunlight, gleaming like a heraldic beast on a city shield. The bear was running flat out. And so very beautiful, Wild. Feral.\n\n\"Oh, Bess,\" he said. For a moment he considered just letting the bear go.\n\nBut that was not what he had vowed.\n\nHis charger's ears pricked forward. He raised his sword, Bess rumbled into a gallop and he slammed his visor closed.\n\nBess was faster than the bear. Not much faster, but the great female was hampered by her cub and he could see that her rear paws were mangled and bloody.\n\nHe began to run her down as the ground started to slope down towards the broad river. It was wide here, near the sea, and it smelled of brine at the turn of the tide. He set himself in his saddle and raised his sword\u2014\n\nSuddenly, the bear released her cub to tumble deep into some low bushes, and turned like a great cat pouncing\u2014going from prey to predator in the beat of a human heart.\n\nShe rose on her haunches as he struck at her\u2014and she was faster than any creature he'd ever faced. She swung with all her weight in one great claw-raking blow, striking at his horse, even as his blow cut through the meat of her right forepaw and into her chest\u2014cut deep.\n\nBess was already dead beneath him.\n\nHe went backwards over his high crupper, as he'd been taught to. He hit hard, rolled, and came to his feet. He'd lost his sword\u2014and lost sight of the bear. He found the dagger at his waist and drew it even as he whirled. Too slow.\n\nShe hit him. The blow caught him in the side, and threw him off his feet, but his breastplate held the blow and the claws didn't rake him. By luck he rolled over his sword, and got to his feet with it in his fist. Something in his right leg was badly injured\u2014maybe broken.\n\nThe bear was bleeding.\n\nThe cub mewed.\n\nThe mother looked at the cub. Looked at him. Then she ran, picked the cub up in her mouth and ran for the river. He watched until she was gone\u2014she jumped into the icy water and swam rapidly away.\n\nHe stood with his shoulders slumped, until his breathing began to steady. Then he walked to his dead horse, found his unbroken flask, and drank all the rest of the contents.\n\nHe said a prayer for a horse he had loved.\n\nAnd he waited to be found.\n\n# West of Lissen Carak\u2014Thorn\n\nA two hundred leagues north-west, Thorn sat under a great holm-oak that had endured a millennium. The tree rose, both high and round, and its progeny filled the gap between the hills closing down from the north and the ever deeper Cohocton River to the south.\n\nThorn sat cross-legged on the ground. He no longer resembled the man he had once been; he was almost as tall as a barn, when he stood up to his full height, and his skin, where it showed through layers of moss and leather, seemed to be of smooth grey stone. A staff\u2014the product of a single, straight ash tree riven by lightning in its twentieth year\u2014lay across his lap. His gnarled fingers, as long as the tines of a hay fork, made eldritch sigils of pale green fire as he reached out into the Wild for his coven of spies.\n\nHe found the youngest and most aggressive of the Qwethnethogs; the strong people of the deep Wild that men called daemons. _Tunxis._ Young, angry, and easy to manipulate.\n\nHe exerted his will, and Tunxis came. He was careful about the manner of his summons; Tunxis had more powerful relatives who would resent Thorn using the younger daemon for his own ends.\n\nTunxis emerged from the oaks to the east at a run, his long, heavily muscled legs beautiful at the fullness of his stride, his body leaning far forward, balanced by the heavy armoured tail that characterized his kind. His chest looked deceptively human, if an unlikely shade of blue-green, and his arms and shoulders were also very man-like. His face had an angelic beauty\u2014large, deep eyes slanted slightly, open and innocent, with a ridge of bone between them that rose into the elegant helmet crest that differentiated the male and female among them. His beak was polished to a mirror-brightness and inlaid with lapis lazuli and gold to mark his social rank, and he wore a sword that few mere human men could even lift.\n\nHe was angry\u2014but Tunxis was at the age when young males are always angry.\n\n\"Why do you summon me?\" he shrieked.\n\nThorn nodded. \"Because I need you,\" he answered.\n\nTunxis clacked his beak in contempt. \"Perhaps I do not need you. Or your games.\"\n\n\"It was my games that allowed you to kill the witch.\" Thorn didn't smile. He had lost the ability to, but he smiled inwardly, because Tunxis was so young.\n\nThe beak clacked again. \"She was nothing.\" Clacked again, in deep satisfaction. \" _You_ wanted her dead. And she was too young. You offered me a banquet and gave me a scrap. A _nothing._ \"\n\nThorn handled his staff. \"She is certainly nothing now.\" His _friend_ had asked for the death. Layers of treason. Layers of favours asked, and owed. The Wild. His attention threatened to slip away from the daemon. It had probably been a mistake to let Tunxis kill in the valley.\n\n\"My cousin says there are armed men riding in the valley. In _our valley._ \" Tunxis slurred the words, as all his people did when moved by great emotion.\n\nThorn leaned forward, suddenly very interested. \"Mogan saw them?\" he asked.\n\n\"Smelled them. Watched them. Counted their horses.\" Tunxis moved his eyebrows the way daemons did. It was like a smile, but it caused the beak to close\u2014something like the satisfaction of a good meal.\n\nThorn had had many years in which to study the daemons. They were his closest allies, his not-trusted lieutenants. \"How many?\" Thorn asked patiently.\n\n\"Many,\" Tunxis said, already bored. \"I will find them and kill them.\"\n\n\"You will _not_.\" Thorn leaned forward and slowly, carefully, rose to his feet, his heavy head brushing against the middling branches of the ancient oak. \"Where has she found soldiers?\" he asked out loud. One of the hazards of living alone in the Wild was that you voiced things aloud. He was growing used to talking to himself aloud. It didn't trouble him as it had at first.\n\n\"They came from the east,\" Tunxis said. \"I will hunt them and kill them.\"\n\nThorn sighed. \"No. You will find them and watch them. You will watch them from afar. We will learn their strengths and weaknesses. Chances are they will pass away south over the bridge, or join the lady as a garrison. It is no concern of ours.\"\n\n\"No concern of _yours,_ Turncoat. Our land. Our valley. Our hills. Our fortress. Our power. Because you are weak\u2014\" Tunxis's beak made three distinct clacks.\n\nThorn rolled his hand over, long thin fingers flashing, and the daemon fell flat on the ground as if all his sinews had been cut.\n\nThorn's voice became the hiss of a serpent.\n\n\"I am _weak_? The soldiers are _many_? They came from the _east_? You are a fool and a child, Tunxis. I could rip your soul from your body and eat it, and you couldn't lift a claw to stop me. Even now you cannot move, cannot summon power. You are like a hatchling in the rushing water as the salmon comes to take him. Yes? And you tell me 'many' like a lord throwing crumbs to peasants. Many?\" he leaned down over the prone daemon and thrust his heavy staff into the creature's stomach. \" _How many exactly, you little fool?_ \"\n\n\"I don't know,\" Tunxis managed.\n\n\"From the east, the south-east? From Harndon and the king? From over the mountains? Do you know?\" he hissed.\n\n\"No,\" Tunxis said, cringing.\n\n\"Tunxis, I like to be polite. To act like\u2014\" He sought for a concept that could link him to the alien intelligence. \"To act like we are allies. Who share common goals.\"\n\n\"You treat us like servants! We serve no master!\" spat the daemon. \"We are not like your _men_ , who lie and lie and say these pretty things. We are Qwethnethogs!\"\n\nThorn pushed his staff deeper into the young daemon's gut. \"Sometimes I tire of the Wild and the endless struggle. _I am trying to help you and your people reclaim your valley. Your goal is my goal._ So I am not going to eat you. However tempting that might be just now.\" He withdrew the staff.\n\n\"My cousin says I should never trust you. That whatever body you wear, you are just another man.\" Tunxis sat up, rolled to his feet with a pure and fluid grace.\n\n\"Whatever I am, without me you have no chance against the forces of the Rock. You will _never_ reclaim your place.\"\n\n\"Men are weak,\" Tunxis spat.\n\n\"Men have defeated your kind again and again. They burn the woods. They cut the trees. They build farms and bridges and they raise armies and your kind _lose._ \" He realised that he was trying to negotiate with a child. \"Tunxis,\" he said, laying hold of the young creature's essence. \"Do my bidding. Go, and watch the men, and come back and tell me.\"\n\nBut Tunxis had a power of his own, and Thorn watched much of his compulsion roll off the creature. And when he let go his hold, the daemon turned and sprinted for the trees.\n\nAnd only then did Thorn recall that he'd summoned the boy for another reason entirely, and that made him feel tired and old. But he exerted himself again, summoning one of the Abnethog this time, that men called wyverns.\n\nThe Abnethog were more biddable. Less fractious. Just as aggressive. But lacking a direct ability to manipulate the power, they tended to avoid open conflict with the magi.\n\nSidhi landed neatly in the clearing in front of the holm oak, although the aerial gymnastics required taxed his skills.\n\n\"I come,\" he said.\n\nThorn nodded. \"I thank you. I need you to look in the lower valley to the east,\" he said. \"There are men there, now. Armed men. Possibly very dangerous.\"\n\n\"What man is dangerous to me?\" asked the wyvern. Indeed, Sidhi stood eye to eye with Thorn, and when he unfolded his wings their span was extraordinary. Even Thorn felt a twinge of real fear when the Abnethog were angry.\n\nThorn nodded. \"They have bows. And other weapons that could hurt you badly.\"\n\nSidhi made a noise in his throat. \"Then why should I do this thing?\" he asked.\n\n\"I made the eyes of your brood clear when they clouded over in the winter. I gave you the rock-that-warms for your mate's nest.\" Thorn made a motion intended to convey that he would continue to heal sick wyverns.\n\nSidhi unfolded his wings. \"I was going to hunt,\" he said. \"I am hungry. And being summoned by you is like being called a dog.\" The wings spread farther and farther. \"But it may be that I will choose to hunt to the east, and it may be that I will see your enemies.\"\n\n\"Your enemies as well,\" Thorn said wearily. _Why are they all so childish?_\n\nThe wyvern threw back its head, and screamed, and the wings beat\u2014a moment of chaos, and it was in the air, the trees all around it shedding leaves in the storm of air. A night of hard rain wouldn't have ripped so many leaves from the trees.\n\nAnd then Thorn reached out with his power\u2014gently, hesitantly, a little like a man rising from bed on a dark night to find his way down unfamiliar stairs. He reached out to the east\u2014farther, and a little farther, until he found what he always found.\n\nHer. The lady on the Rock.\n\nHe probed the walls like a man running his tongue over a bad tooth. She was there, enshrined in her power. And with her was something else entirely. He couldn't read it\u2014the fortress carried its own power, its own ancient sigils which worked against him.\n\nHe sighed. It was raining. He sat in the rain, and tried to enjoy the rise of spring around him.\n\n_Tunxis killed the nun, and now the lady has more soldiers._ He had set something in motion, and he wasn't sure why.\n\nAnd he wondered if he had made a mistake.\n\n# Chapter Two\n\n# Harndon Palace\u2014The Queen\n\nDesiderata lay on the couch of her solar chewing new cherries and savouring the change in the air. Because\u2014 _at last_ \u2014spring had come. Her favourite season. After Lent would come Easter and then Whitsunday, and the season of picnics, of frolics by the river, or eating fresh fruit, wearing flowers, walking barefoot...\n\n... and tournaments.\n\nShe sighed at the thought of tournaments. Behind her, Diota, her nurse, made a face. She could see the old woman's disapproval in the mirror.\n\n\"What? Now you frown if I sigh?\" she asked.\n\nDiota straightened her back, putting a fist into it like a pregnant woman. Her free hand fingered the rich paternoster at her neck. \"You sound like a whore pleasing a customer, mistress\u2014if you'll pardon the crudity of an old woman\u2014\"\n\n\" _Who's known you all these years_ ,\" the Queen completed the sentence. Indeed, she'd had Diota since she was weaned. \"Do I? And what do you know about the sounds whores make, nurse?\"\n\n\"Now, my lady!\" Diota came forward, waggling a finger. Coming around the screen, she stopped as if she'd hit an invisible barrier. \"Oh! By the Sweet Lord\u2014put some clothes on, girl! You'll catch your death! It's not spring yet, morsel!\"\n\nThe Queen laughed. She was naked in the new sunlight, her tawny skin flecked with the imperfections of the glass in her solar's window, lying on the pale brown profusion of her hair. She _drew_ something from the sunlight falling on her skin\u2014something that made her glow from within.\n\nDesiderata rose and stood at the mirror\u2014the longest mirror in the Demesne, made just for her, so that she could examine herself from the high arches of her feet, up her long legs, past her hips and thighs and the deep recess of her navel to her breasts, her upright shoulders, her long and tapered neck, deep cut chin, a mouth made for kissing, long nose and wide grey eyes with lashes so long that sometimes she could lick them.\n\nShe frowned. \"Have you seen the new lady in waiting? Emmota?\" she said.\n\nHer nurse chuckled alongside her. \"She's a child.\"\n\n\"A fine figure. Her waist is thin as a rail.\" The Queen looked at herself with careful scrutiny.\n\nDiota smacked her hip. \"Get dressed, you hussy!\" she laughed. \"You're looking for compliments. She's nothing to you, Miss. A child. No breasts.\" She laughed. \"Every man says you are the beauty of the world,\" she added.\n\nThe Queen continued to look in the mirror. \"I am. But for how long?\" She put her hands up over her head, arching her back as her chest rose.\n\nHer nurse slapped her playfully. \"Do you want the king to find you thus?\"\n\nDesiderata smiled at her woman. \"I could say yes. I want him to find me _just_ like this,\" she said. And then, her voice coloured with power, she said, \"Or I could say I am as much myself, and as much the Queen, naked, as I am clothed.\"\n\nHer nurse took a step away.\n\n\"But I won't say any such thing. Bring me something nice. The brown wool gown that goes with my hair. And my golden belt.\"\n\n\"Yes, my lady.\" Diota nodded and frowned. \"Shall I send some of your ladies to dress you?\"\n\nThe Queen smiled and stretched, her eyes still on the mirror. \"Send me my ladies,\" she said, and subsided back onto the couch in the solar.\n\n# Lissen Carak\u2014The Red Knight\n\nAt Ser Hugo's insistence, the master archers had set up butts in the fields along the river.\n\nMen grumbled, because they'd been ordered to curry their horses before turning in, and then, before the horses were cared for, they were ordered to shoot. They had ridden hard, for many long days, and there wasn't a man or woman without dark circles under their eyes.\n\nBent, the eldest, an easterner, and Wilful Murder, fresh back from failing to find a murderer's tracks with the huntsman, ordered the younger men to unload the butts, stuffed with old cloth or woven from straw, from the wagons.\n\n\"Which it isn't my turn,\" whined Kanny. \"An' why are you always picking on us?\" His words might have appeared braver, if he hadn't waited until Bent was far away before saying them.\n\nGeslin was the youngest man in the company, just fourteen, with a thin frame that suggested he'd never got much food as a boy, climbed one of the tall wagons and silently seized a target and tossed it down to Gadgee, an odd looking man with a swarthy face and foreign features.\n\nGadgee caught the target with a grunt, and started toward the distant field. \"Shut up and do some work,\" he said.\n\nKanny spat. And moved very slowly towards a wagon that didn't have any targets in it. \"I'll just look\u2014\"\n\nBad Tom's archer, Cuddy, appeared out of nowhere and shoved him ungently towards the wagon where Geslin was readying a second target. \"Shut up and do some work,\" he said.\n\nHe was slow enough that by the time he had his target propped up and ready for use, all nine of the other butts were ready as well. And there were forty archers standing a hundred paces distant, examining their spare strings and muttering about the damp.\n\nCuddy strung his bow with an economy of motion that belied long practice, and he opened the string that held the arrows he had in his quiver.\n\n\"Shall I open the dance?\" he said.\n\nHe nocked, and loosed.\n\nA few paces to his right, Wilful Murder, who fancied himself as good an archer as any man alive, drew and loosed a second later, contorting his body to pull the great war bow.\n\nBent put his horn to his lips and blew. \"Cease!\" he roared. He turned to Cuddy. \"Kanny's still down range!\" he shouted at the master archer.\n\nCuddy grinned. \"I know _just_ where he is,\" he said. \"So does Wilful.\"\n\nThe two snickered as Kanny came from behind the central target, running as fast as his long, skinny legs would carry him.\n\nThe archers roared with laughter.\n\nKanny was spitting with rage and fear. \"You bastard!\" he shouted at Cuddy.\n\n\"I told you to work faster,\" Cuddy said mildly.\n\n\"I'll tell the captain!\" Kanny said.\n\nBent nodded. \"You do that.\" He motioned. \"Off you go.\"\n\nKanny grew pale.\n\nBehind him, the other archers walked up to their places, and began to loose.\n\nThe captain was late to the drill. He looked tired, and he moved slowly, and he leaned on the tall stone wall surrounding the sheepfold that Ser Hugh had converted to a tiltyard and watched the men-at-arms at practice.\n\nDespite fatigue and the weight of plate and mail Ser George Brewes was on the balls of his feet, bouncing from guard to guard. Opposite him, his \"companion\" in the language of the tiltyard, was the debonair Robert Lyliard, whose careful fighting style was the very opposite of the ostentatious display of his arms and clothes.\n\nBrewes stalked Lyliard like a high-stepping panther, his pole-arm going from guard to guard\u2014low, axe-head forward and right leg advanced, in the Boar's Tooth; sweeping through a heavy up-cut to rest on his right shoulder like a woodcutter in the Woman's Guard.\n\nFrancis Atcourt, thick waisted and careful, faced Tomas Durrem. Both were old soldiers, unknighted men-at-arms who had been in harness for decades. They circled and circled, taking no chances. The captain thought he might fall asleep watching them.\n\nBad Tom came and rested on the same wall, except that his head projected clear above the captain's head. And even above the plume on his hat.\n\n\"Care to have a go?\" Tom asked with a grin.\n\nNo one liked to spar with Tom. He hurt people. The captain knew that despite all the plate armour and padding and mail and careful weapon's control, tiltyard contests were dangerous and men were down from duty all the time with broken fingers and other injuries. And that was without the sudden flares of anger men could get when something hurt, or became personal. When the tiltyard became the duelling ground.\n\nThe problem was that there was no substitute for the tiltyard, when it came to being ready for the real thing. He'd learned that in the east.\n\nHe looked at Tom. The man had a reputation. And he had dressed Tom down in public a day before.\n\n\"What's your preference, Ser Thomas?\" he asked.\n\n\"Longsword,\" Bad Tom said. He put a hand on the wall and vaulted it, landed on the balls of his feet, whirled and drew his sword. It was his war sword\u2014four feet six inches of heavy metal. Eastern made, with a pattern in the blade. Men said it was magicked.\n\nThe captain walked along the wall with no little trepidation. He went into the sheepfold through the gate, and Michael brought him a tilt helmet with solid mesh over the face and a heavy aventail.\n\nMichael handed him his own war sword. It was five inches shorter than Bad Tom's, plain iron hilted with a half-wired grip and a heavy wheel of iron for a pommel.\n\nAs Michael buckled his visor, John of Reigate, Bad Tom's squire, put his helmet over his head.\n\nTom grinned while his faceplate was fastened. \"Most loons mislike a little to-do wi' me,\" he said. When Tom was excited, his hillman accent overwhelmed his Gothic.\n\nThe captain rolled his head to test his helmet, rotated his right arm to test his range of motion.\n\nMen-at-arms were pausing, all over the sheepfold.\n\n\"The more fool they,\" the captain said.\n\nHe'd watched Tom fight. Tom liked to hit hard\u2014to use his godlike strength to smash through men's guards.\n\nHis father's master-at-arms, Hywel Writhe, used to say _For good swordsmen, it's not enough to win. They need to win their own way. Learn a man's way, and he becomes predictable._\n\nTom rose from the milking stool he'd sat on to be armed and flicked his sword back and forth. Unlike many big men, Tom was as fast as the tomcat that gave him his name.\n\nThe captain didn't strike a guard at all. He held his sword in one hand, the point actually trailing on the grass.\n\nTom whirled his blade up to the high Woman's Guard, ready to cleave his captain in two.\n\n\"Garde!\" he roared. The call echoed off the walls of the sheep fold and then from the high walls of the fortress above them.\n\nThe captain stepped, moved one foot off line, and suddenly he had his sword in two hands. Still trailing out behind him.\n\nTom stepped off-line, circling to the captain's left.\n\nThe captain stepped in, his sword rising to make a flat cut at Tom's head.\n\nTom slapped the sword down\u2014a _rabatter_ cut with both wrists, meant to pound an opponent's blade into the ground.\n\nThe captain powered in, his back foot following the front foot forward. He let the force of Tom's blow to his blade rotate it, his wrist the pivot\u2014sideways and then _under_ Tom's blade.\n\nHe caught the point of his own blade in his left hand, and tapped it against Tom's visor. His two handed grip and his stance put Tom's life utterly in his hands.\n\n\"One,\" he said.\n\nTom laughed. \"Brawly feckit!\" he called.\n\nHe stepped back and saluted. The captain returned the salute and sidestepped, because Tom came for him immediately.\n\nTom stepped, then swept forward with a heavy downward cut.\n\nThe captain stopped it, rolling the blade well off to the side, but as fast as he could bring his point back on line, Tom was inside his reach\u2014\n\nAnd he was face down in sheep dip. His hips hurt, and now his neck hurt.\n\nBut to complain was not the spirit of the thing.\n\n\"Well struck,\" he said, doing his best to bounce to his feet.\n\nTom laughed his wild laugh again. \"Mine, I think,\" he said.\n\nThe captain had to laugh.\n\n\"I was planning to chew on your toes,\" he said, and drew a laugh from the onlookers.\n\nHe saluted, Tom saluted, and they were on their guards again.\n\nBut they'd both shown their mettle, and now they circled\u2014Tom looking for a way to force the action close, and the captain trying to keep him off with short jabs. Once, by thrusting with his whole sword held at the pommel, he scored on Tom's right hand, and the other man flicked a short salute, as if to say \"that wasn't much.\" And indeed, Ser Hugo stepped between them.\n\n\"I don't allow such trick blows, my lord,\" Hugh said. \"It'd be a foolish thing to do in a melee.\"\n\nThe captain had to acknowledge the truth of that assertion. He had been taught the Long Point with the advice _never use this unless you are desperate. Even then_ \u2014\n\nThe captain's breath was coming in great gasps, while Tom seemed to be moving fluidly around the impromptu ring. Breathing well and easily. Of course, given his advantages in reach and size, he could control most aspects of the fight, and the captain was mostly running away to keep his distance.\n\nThe last five days of worry and stress sat as heavily on his shoulders as the weight of his tournament helm. And Tom was very good. There was really little shame in losing to him. So the captain decided he'd rather go down as a lion than a very tired lamb. And besides, it would be funny.\n\nSo\u2014between one retreat and the next blow\u2014he swayed his hips, rotated his feet so that his weight was back, and let go the sword's hilt with his left hand. Eastern swordsmen called it \"The Guard of One Hand.\"\n\nTom swept in with another of his endless, heavy, sweeping blows. Any normal man would have exhausted himself with them. Not Tom. This one came from his right shoulder.\n\nThis time, the captain tried for a _rebatter_ defence\u2014his sword sweeping up, one handed, coming slightly behind Tom's but cutting as fast as a falcon strikes its prey. He caught Tom's sword and drove it faster along its intended path as he stepped slightly off-line and _forward_ , surprising his companion. His free left hand shot out, and he punched Tom's right wrist, and then his left hand was between the big man's hands, and Tom's aggressive pursuit of his elusive opponent carried him forward\u2014the captain's left hand went deeper, and he achieved the arm lock, and twisted, in complete possession of the man's sword and shoulder\u2014\n\nAnd nothing happened. Tom was _not_ rotated. In fact, Tom's rush turned into a swing, and the captain found himself swinging off Tom's elbow and the giant turned to the left, and again, and the captain couldn't let go without tumbling to the ground.\n\nHis master-at-arms had never covered this situation.\n\nTom whirled him again, trying to shake him off. They were at a nasty impasse. The captain had Tom's sword bound tight, and his elbow and shoulder in a lock too. But Tom had the captain's feet off the ground.\n\nThe captain had his blade free\u2014mostly free. He hooked his pommel into Tom's locked arms, hoping it would give him the leverage to, well, to do what should have happened in the first place. The captain's sense of how combat and the universe worked had received a serious jar.\n\nBut even with both hands\u2014\n\nTom whirled him again, like a terrier breaking a rat's neck.\n\nUsing every sinew of his not inconsiderable muscles, the captain pried his pommel between Tom's arms and levered the blade over Tom's head and grabbed the other side, letting his whole weight go onto the blade.\n\nIn effect, he fell, blade first, on Tom's neck.\n\nThey both went down.\n\nThe captain lay in the sheep muck, with his eyes full of stars. And his breath coming like a blacksmith's bellows.\n\nSomething under him was moving.\n\nHe rolled over, and found that he was lying entangled with the giant hillman, and the man was laughing.\n\n\"You're mad as a gengrit!\" Tom said. He rose out of the muck and smothered the captain in an embrace.\n\nSome of the other men-at-arms were applauding.\n\nSome were laughing.\n\nMichael looked like he was going to cry. But that was only because he had to clean the captain's armour, and the captain was awash in sheep dip.\n\nWhen his helmet was off, he began to feel the new strain in his left side and the pain in his shoulder. Tom was right next to him.\n\n\"You're a loon,\" Tom said. He grinned. \"A loon.\"\n\nWith his helmet off, he could still only just breathe.\n\nChrys Foliack, another of the men-at-arms who had hitherto kept his distance from the captain, came and offered his hand. He grinned at Tom. \"It's like fighting a mountain, ain't it?\" he asked.\n\nThe captain shook his head. \"I've never\u2014\"\n\nFoliack was a big man, handsome and red-headed and obviously well-born. \"I liked the arm lock,\" he said. \"Will you teach it?\"\n\nThe captain looked around. \"Not just this minute,\" he said.\n\nThat got a laugh.\n\n# Harndon Palace\u2014The King\n\nThe king was in armour, having just trounced a number of his gentlemen on the tilt field, when his constable, Alexander, Lord Glendower\u2014an older man with a scar that ran from his right eyebrow, all the way across his face, cleaving his nose from right to left so deeply as to make most men he met wince\u2014and then down across his face to his mouth, so that his beard had a ripple in it where the scar had healed badly, and he always looked as if he was sneering\u2014approached with a red-haired giant at his back.\n\nGlendower's scar couldn't have suited a man worse as he was, as far as the king was concerned, the best of companions, a man little given to sneering and much to straight talk unlaced by flattery or temper. His patience with his soldiers was legendary.\n\n\"My lord, I think you know Ranald Lachlan, who has served you two years as a man-at arms.\" He bowed, and extended an arm to the red-bearded man, who was obviously a hillman\u2014red hair, facial scarring, piercing blue eyes like steel daggers, and two ells of height unhidden by the hardened steel plate armour and red livery of the Royal Guard.\n\nRanald bowed deeply.\n\nThe king reached out and clasped his hand. \"I'm losing you,\" he said warmly. \"The sight of your great axe always made me feel safe,\" he laughed.\n\nRanald bowed again. \"I promised Lord Glendower and Sir Ricard two years when I signed my mark,\" the hillman said. \"I'm needed at home, for the spring drive.\"\n\nSir Ricard Fitzroy, so indicated, was the captain of the guard.\n\n\"Your brother is the Drover, I know,\" the king said. \"It's a troubled spring, Ranald. Alba will be safer if your axe is guarding beeves in the hills, rather than guarding the king, safe in Harndon. Eh?\"\n\nRanald shrugged, embarrassed. \"There'll be fighting, I ha'e na' doot,\" he admitted. Then he grinned. \"I have no doubt, my lord.\"\n\nThe king nodded. \"When the drive is over?\" he asked.\n\n\"Oh, I have reason to come back,\" he said with a grin. \"My lord. With your leave. But my brother needs me, and there are things\u2014\"\n\nEvery man present knew that the things Ranald Lachlan wanted involved the Queen's secretary, Lady Almspend\u2014not an heiress, precisely. But a pretty maid with a fair inheritance. A high mark for a King's Guardsman, commonly born.\n\nThe king leaned close. \"Come back, Ranald. She'll wait.\"\n\n\"I pray she does,\" he whispered.\n\nThe king turned to his constable. \"See that this man's surcoat and kit are well stored; I grant him leave, but I do not grant him quittance from my service.\"\n\n\"My lord!\" the man replied.\n\nThe king grinned. \"Now get going. And come back with some tales to tell.\"\n\nRanald bowed again, as ceremony demanded, and walked from the king's presence to the guardroom, where he embraced a dozen close friends, drank a farewell cup of wine, and handed the Steward his kit\u2014his maille hauberk and his good cote of plates beautifully covered in the royal scarlet; his two scarlet cotes with matching hoods, for wear at court, and his hose of scarlet cloth. His tall boots of scarlet leather, and his sword belt of scarlet trimmed in bronze.\n\nHe had on a doublet of fustian, dark hose of a muddy brown, and over his arm was his three-quarter's tweed cloak.\n\nThe Steward, Radolf, listed his kit on his inventory and nodded. \"Nicely kept, messire. And your badge...\" The king's badge was a white heart with a golden collar, and the badges were cunningly fashioned of silver and bronze and enamel. \"The king expressly stated you was to keep yours, as on leave and not quit the guard.\" He handed the badge back.\n\nRanald was touched. He took the brooch and pinned his cloak with it. The badge made his tweed look shabby and old.\n\nThen he walked out of the fortress and down into the city of Harndon, without a backwards glance. Two years, war and peril, missions secret and diplomatic, and the love of his life.\n\nA hillman had other loyalties.\n\nDown into the town that grew along the river's curves. From the height of the fortress, the town was dominated by the bridge over the Albin, the last bridge before the broad and winding river reached the sea thirty leagues farther south. On the far side of the bridge, to the north, lay Bridgetown\u2014part and not part of the great city of Harndon. But on this side, along the river, the city ran from the king's fortress around the curve, with wharves and peers at the riverside, merchants' houses, streets of craftsmen in houses built tall and thin to save land.\n\nHe walked down the ramp, leading his two horses past the sentries\u2014men he knew. More hand clasps.\n\nHe walked along Flood Street, past the great convent of St. Thomas and the streets of the Mercers and Goldsmiths, and down the steep lanes past the Founders and the Blacksmiths, to the place where Blade Lane crossed with Armour Street, at the sign of the broken circle.\n\nThe counter was only as wide as two broad-built men standing side by side, but Ranald looked around, because the Broken Circle made the finest weapons and armour in the Demesne, and there were always things there to be seen. Beautiful things\u2014even to a hillman. Today was better than many days\u2014a dozen simple helmets stood on the counter, all crisp and fine, with high points and umbers to shade the eye, the white work fine and neat, the finish almost mirror bright, the metal blue-white, like fine silver.\n\nAnd these were simple archer's helmets.\n\nThere was an apprentice behind the counter, a likely young man with arms like the statues of the ancient men and legs to match. He grinned and bobbed his head and went silently through the curtain behind him to fetch his master.\n\nTad Pyel was the master weapon smith of the land. The first Alban to make the hardened steel. He was a tall man with a pleasant round face and twenty loyal apprentices to show that the mild disposition was not just in his face. He emerged, wiping his hands on his apron.\n\n\"Master Ranald,\" he said. \"Here for your axe, I have no doubt.\"\n\n\"There was some talk of a cote, of maille as well,\" Ranald added.\n\n\"Oh,\" Tad nodded absently to his apprentice. \"Oh, as to that\u2014-Continental stuff. Not my make. But yes, we have it ready for you.\"\n\nEdward, the apprentice, was shifting a wicker basket from the back, and Ranald opened the lid and looked at the river of gleaming mail, every ring riveted with a wedge so small that most of the rings looked as if they'd been forged entire. It was as fine as the hauberk he'd worn as a King's Man.\n\n\"This for thirty leopards?\" Ranald asked.\n\n\"Continental stuff,\" the master replied. He didn't actually sniff, but the sniff was there. Then the older man smiled, and held out a heavy pole with the ends wrapped in sacking. \"This would cut it as a sharp knife cuts an apple.\"\n\nRanald took it in his hands, and was filled with as sweet a feeling as the moment that a man discovers he is in love\u2014that the object of his affection returns his feelings.\n\nEdward cut the lashings on the sacking, revealing a sharp steel spike on one end, ferruled in heavy bronze, balancing an axe blade at the other end\u2014a narrow crescent of bright steel, as long as a man's forearm, ending in a wicked point and armed with a vicious back-hook. All balanced like a fine sword, hafted in oak, with steel lappets to guard against sword cuts.\n\nIt was a hillman's axe\u2014but incomparably finer, made by a master and not by a travelling smith at a fair.\n\nRanald couldn't help himself, and he whirled it between his hands, the blade cutting the air and the tip _not quite_ brushing the plaster of the low room.\n\nEdward flattened himself against the wall, and the master nodded, satisfied.\n\n\"The one you brought me was a fine enough weapon,\" the master said. \"Country made, but a well-made piece. But the finish,\" he winced. And shrugged. \"And I thought that the balance could be improved.\"\n\nThe spike in the butt of the haft was as long as a knight's dagger, wickedly sharp and three-sided.\n\nRanald just smiled in appreciation.\n\nThe master added two scabbards\u2014a sheath of wood covered in fine red leather for the axe, and another to match for the spike.\n\nRanald counted down a hundred silver leopards\u2014a sizeable portion of two years' pay. He looked admiringly at the helmets on the counter.\n\n\"They're spoken for,\" the master said, catching his eye. \"And none of them would fit your noggin, I'm thinking. Come back in winter when my work is slow, and I'll make you a helmet you could wear to fight a dragon.\"\n\nThe air seemed to chill.\n\n\"Naming calls,\" Edward said, crossing himself.\n\n\"Don't know what made me say that,\" said the master. He shook his head. \"But I'd make you a helmet.\"\n\nRanald carried his new maille out to his pack horse, who was not as fond of it as he was, resenting the weight and the re-packing of the panniers it necessitated. He came back for the axe, and put it lovingly into the straps on his riding horse, close to hand. No one watching doubted that he'd handle it a dozen more times before he was clear of the suburbs. Or that he'd stop and use it on the first bush he found growing by the road.\n\n\"You ride today, then,\" the master said.\n\nRanald nodded. \"I'm needed in the north,\" he said. \"My brother sent for me.\"\n\nThe weapon smith nodded. \"Send him my respects, then, and the sele of the day on you.\"\n\nThe hillman embraced the cutler, stepped through the door, and walked his horses back up the old river bank.\n\nHe stopped in the chapel of Saint Thomas, and knelt to pray, his eyes down. Above him, the saint was martyred by soldiers\u2014knights in the Royal Livery. The scene made him uncomfortable.\n\nHe bought a pie from a ragged little girl by the Bridge Gate, and then he was away.\n\n# Harndon City\u2014Edward\n\n\"There goes a fearsome man,\" said the master to his apprentice. \"I've known a few. And yet as gentle as a lady. A better knight than many who wear spurs.\"\n\nEdward was too smitten with hero-worship to comment.\n\n\"And where's our daring mercer?\" asked the master.\n\n\"Late, your worship,\" said his apprentice.\n\nTad shook his head. \"That boy would be late to his own funeral,\" he said, but his voice suggested he had nothing but praise for the mercer. \"Pack the helmets in straw and take them round to Master Random's house, will you, Ned?\"\n\nNo matter how kind your master is, there's no apprentice who doesn't relish a trip beyond the Ward. \"May I have a penny to buy baskets?\"\n\nMaster Thaddeus put coins in his hand. \"Wish I'd made him a helmet,\" he said. \"Where'd the thought of a dragon come from?\"\n\n# Harndon Palace\u2014the Queen\n\nDesiderata sat primly on an ivory stool in the great hall, its stucco walls lined with the trophies taken by a thousand brave knights\u2014the heads of creatures greater and smaller, and a very young dragon's head, fully the size of a horse, filling the northern wall beneath the stained glass window like a boat hull protruding from the sea. To her, it never quite looked the same way twice, that dragon\u2014but it was huge.\n\nShe sat peeling a winter apple with a silver knife. Her hair was a halo of brown and red and gold around her\u2014a carefully planned effect, as she sat in the pool of light thrown by the king's beloved rose window. Her ladies sat around her, skirts spread like pressed flowers on the clean checkerboard marble floor, and a dozen of the younger knights\u2014the very ones who should have been tilting in the tiltyard, or crossing swords with the masters\u2014lounged against the walls. One, the eldest of them by half a dozen years and some fighting, was called \"Hard Hands\" for his well-known feat of killing a creature of the Wild with a single blow of his fist. It was a story he often told.\n\nThe Queen disliked men who boasted. She made it her business to know who was worthy and who was not\u2014indeed, she viewed it as her sacred role. She loved to find the shy ones\u2014the brave men who told no one of their deeds. She thought less of the braggarts. Especially when they sat in _her_ hall and flirted with _her_ ladies. She had just determined to punish the man when the king came in.\n\nHe was plainly dressed, in arming clothes, he smelled like horses and armour and sweat, and she wrapped herself around him and his smell as if they were newly wed. He smiled down into her face and kissed her nose.\n\n\"I love it when you do that,\" he said.\n\n\"You should practise your tilting more often, then,\" she said, holding his arm. Behind the king, Ser Driant stood rubbing his neck, and behind him, Ser Alan, and the constable, Lord Glendower. She laughed. \"Did you defeat these poor knights?\"\n\n\"Defeat?\" asked Driant. He laughed ruefully. \"I was crushed like a bug in an avalanche, my lady. His Grace has a new horse that's bigger than a dragon.\"\n\nSer Alan shrugged. \"I was unhorsed, yes, lady.\" He looked at Ser Driant and frowned. \"I think it rude to suggest the king's _horse_ rolled you on the sand,\" he said.\n\nDriant laughed again. He was not a man who stayed downcast for long. \"There's a great deal of me to hit the ground,\" he said, \"and that ground is still frozen.\" He rubbed his neck again, peering past the Queen to her ladies sitting with their knights. \"And you lads\u2014where were you when the blows were being dealt and received?\"\n\nHard Hands nodded appreciatively. \"Right here in the warm hall, basking in the beauty of the Queen and all these fair flowers,\" he said. \"What man goes voluntarily to fight on frozen ground?\"\n\nThe king frowned. \"A man preparing for war?\" he asked quietly.\n\nHard Hands looked about him for support. He'd mistaken the bantering tone of conversation for permission to banter with the king.\n\nThe Queen smiled to see him humbled so swiftly.\n\n\"Out beyond the walls are creatures who would crack your armour to eat what lies within\u2014or to drink your soul,\" said the king, and his voice rang through the hall as he walked beneath the rows of heads. \"And alone of these fair flowers, Ser knight, _you_ know the truth of what I say. You have faced the Wild.\" The king was not the tallest man in the room or the handsomest. But when he spoke like this, no other man could compare.\n\nHard Hands looked at the floor and bit his lip in frustration. \"I sought only to entertain, Sire. I beg your pardon.\"\n\n\"Seek my pardon in the Wild,\" the king said. \"Bring me three heads and I will be content to watch you flirt with the Queen's ladies. Bring me five heads and you may flirt with the Queen.\"\n\n_If you dare_ , she thought.\n\nThe king grinned, stopped by the younger man and clapped a hand on his shoulder. Hard Hands stiffened.\n\nHe did not want to leave the court. It was plain to see.\n\nThe king put his lips close to Hard Hand's ear, but the Queen heard his words. She always did.\n\n\"Three heads,\" the king whispered through the smile on his lips. \"Or you will stay in your castle and be branded faithless and craven.\"\n\nThe Queen watched the effect on her ladies and held her peace. Hard Hands was quite a popular man. Lady Mary, who was known as \"Hard Heart\" had been heard to say that perhaps his hands were not so very hard, after all. Seated nearest to the Queen, she pursed her lips and set her mouth, determined not to show the Queen her hurt. Behind this vignette, the king waved to his squires and set off up the main stairs to his arming room.\n\nWhen the king was gone, Desiderata sat back down on her stool and picked up her sewing\u2014an arming shirt for the king. Her ladies gathered round. They felt her desire and closed themselves against the younger knights, who looked to Hard Hands for leadership. Or had. Now they were disconsolate at losing their leader. They left with the sort of loud demonstration that young men make when socially disadvantaged, and the Queen laughed.\n\nHard Hands stopped in the arch of the main door and looked back. He met her eye, and his anger carried clearly across the sun beams that separated them.\n\n\"I will come back!\" he shouted.\n\nThe other young men looked afraid at his outburst, and pushed him out the door.\n\n\"Perhaps,\" purred the Queen. She smiled, much like a cat with a tiny piece of tail sticking out between its teeth.\n\nThe ladies knew that smile. They were silent, and the wisest hung their heads in real, or well-feigned, contrition, but she saw through all of them.\n\n\"Mary,\" said the Queen gently. \"Did you let Hard Hands into your bed?\"\n\nMary, sometimes called Hard Heart, met her eye. \"Yes, my lady.\"\n\nThe Queen nodded. \"Was he worthy?\" she asked. \"Answer me true.\"\n\nMary bit her lip. \"Not today, my lady.\"\n\n\"Perhaps not ever\u2014eh? Listen, all of you,\" she said, and she bent her head to her ladies. \"Emmota\u2014you are latest amongst us. By what signs do you know a knight worthy to be your lover?\"\n\nEmmota was not yet fully grown to her womanhood\u2014fourteen years old. Her face was narrow without being pinched and a clear intelligence shone in her eyes. She was nothing next to the Queen, and yet, the Queen admitted to herself, the girl had something.\n\nBut in this instant, her wits deserted her, and she blushed and said nothing.\n\nThe Queen smiled at her, as she was always tender for the lost and the confounded. \"Listen, my dear,\" she said softly. \"Love only those worthy of your love. Love those who love themselves, and love all around them. Love the best\u2014the best in arms, the first in the hall, the finest harpist, and the best chess player. Love no man for what he owns, but only for what he does.\"\n\nShe smiled at all of them. And then pounced. \"Are you pregnant, Mary?\"\n\nMary shook her head. \"I did not allow him that liberty, my lady.\"\n\nThe Queen reached out and took Mary's hand. \"Well done. Ladies, remember\u2014we award our love to those who deserve us. And our bodies are an even greater prize than our love\u2014especially to the young.\" She looked at each in turn. \"Who does not yearn for the strong yet tender embrace? Who does not sigh for skin soft as fine leather over muscles as hard as wood? But get with child\u2014\" she locked eyes with Mary, \"\u2014and they _will_ call you a whore. And you may _die_ , bearing that bastard. Or worse, perhaps; find yourself living meanly, rearing his bastard child, while he rides to glory.\" She looked at the window. \"If you are not locked away in a convent.\"\n\nEmmota raised her head. \"But what of love?\" she asked.\n\n\"Make your love a reward, not a raw emotion,\" the Queen said. \"Any two rutting animals feel the emotion, child. Here, we are only interested in what is _best._ Rutting is not best. Do you understand?\"\n\nThe girl swallowed carefully. \"Yes, I think so,\" she said. \"But then\u2014why would we ever lie with any man?\"\n\nThe Queen laughed aloud. \"Artemis come to earth! Why, because it is for the love of us that they face terror, girl! Do you think it is some light thing to ride out into the Wild? To sleep with the Wild, eat with it, live with it? To face it and fight it and kill it?\" The Queen leaned down until her nose almost touched the sharp point of Emmota's nose. \"Do you think they do it for the good of humanity, my dear? Perhaps the older ones\u2014the thoughtful ones. They face the dangers for us all because they have seen the alternative.\" She shook her head. \"But the young ones face the foe for just one thing\u2014to be deemed worthy of you, my dear. And you control _them_. When you let a knight into your lap you reward him for his courage. His prowess. His _worth._ You must judge that it has been earned. Yes? You understand?\"\n\nEmmota gazed into the eyes of her Queen with worship. \"I understand,\" she said.\n\n\"The Old Men\u2014the Archaics of long ago\u2014they asked 'Who shall guard the guardians?' \" The Queen looked around. \"We shall, ladies. We choose the best of them. We may also choose to punish the worst. Hard Hands was not deserving, and the king found him out. We should have known first\u2014should we not? Did none of you suspect he was merely a braggart? Did none of you wonder where his prowess lay, that he made no show or trial of it?\"\n\nMary burst into tears. \"I protest, madame.\"\n\nThe Queen gave her a small embrace. \"I relent. He is a good man-at-arms. Let him go prove it to the king. And prove himself worthy of you.\"\n\nMary curtsied.\n\nThe Queen nodded, and rose to her feet. \"I go to attend the king. Think of this. It is our duty. Love\u2014our love\u2014is no light thing. It is be the crown of glory, available to the best and only the best. It should be hard won. Think on it.\"\n\nShe listened to them she went up the stairs\u2014broad marble stairs of that the Old Men had wrought. They didn't giggle, which pleased her.\n\nThe king was in the Arming Room, with two squires\u2014Simon and Oggbert, as like as two peas in a pod, with matching freckles and matching pimples. He was down to his shirt and his hose and his braes. His leg harnesses still lay on the floor having been removed, and each squire held a vambrace, wiping them down with chamois.\n\nShe smiled radiantly at them. \"Begone,\" she said.\n\nThey fled, as adolescent boys do when faced with beautiful women.\n\nThe king sat back on his bench. \"Ah! I see I have won your esteem!\" he grinned, and for a moment he was twenty years younger.\n\nShe knelt and undid a garter. \"You are the king. You, and you alone, need never _win_ my esteem.\"\n\nHe watched her unbuckle the other garter. She buckled the two of them together and placed his leg harnesses together on a table behind her, and then, without hurry, she sat in his lap and put her arms around his neck and kissed him until she felt him stir.\n\nAnd then she rose to her feet and unlaced her gown. She did it methodically, carefully, without taking her eyes off him.\n\nHe watched her the way a wolf watches a lamb.\n\nThe gown fell away leaving her kirtle\u2014a sheath of tight silk from ankle to neck.\n\nThe king rose. \"Anyone might come in here,\" he said into her hair.\n\nShe laughed. \"What care I?\"\n\n\"On your head be it, lady,\" he said, and produced a knife. He pressed the flat of the point against the skin of her neck and kissed her, and then cut the lace of her kirtle from neck to waist, the knife so sharp that the laces seemed to fall away, and his cut so careful that the blade never touched her skin through the linen shift beneath it.\n\nShe laughed into his kiss. \"I love it when you do that,\" she said. \"You owe me a lace. A silk one.\" Her long fingers took the knife from him. She stepped back and cut the straps of her shift at her shoulders and it fell away and she stabbed the knife into the top of the table so that it stuck.\n\nHe rid himself of his shirt and braes with more effort and far less elegance, and she laughed at him. And then they were together.\n\nWhen they were done, she lay on his chest. Some of his hair was grey. She played with it.\n\n\"I am old,\" he said.\n\nShe wriggled atop him. \"Not so very old,\" she said.\n\n\"I owe you more than a kirtle lace of silk,\" he said.\n\n\"Really?\" she asked, and rose above him. \"Never mind the shift, love\u2014Mary will replace the straps in an hour.\"\n\n\"I was not being so literal. I owe you my life. I owe you\u2014my continued interest in this endless hell that is kingship.\" He grunted.\n\nShe looked down. \"Endless hell\u2014but you love it. You _love_ it.\"\n\nThe king pulled her down and hid his face in her hair. \"Not as much as I love you.\"\n\n\"What is it?\" she asked, playing with his beard. \"You are plagued by something...?\"\n\nHe sighed. \"One of my favourite men left me today. Ranald Lachlan. Because he has to make himself a fortune in order to wed your Lady Almspend.\"\n\nShe smiled. \" _He_ is a worthy man, and he will prove himself or die trying.\"\n\nThe king sighed. \"Yes,\" he said. \"But by God, woman, I was tempted to give him a bag of gold and a knighthood to keep him by me.\"\n\n\"And you would have deprived him of the glory of earning his way,\" she said.\n\nHe shrugged and said, \"It is good that one of us is an idealist.\"\n\n\"If you are in a giving mood,\" she said, \"might we have a tournament?\"\n\nThe king was a strong man with a fighter's muscles, and he sat up despite her weight on his chest. \"A tournament. By God, lady\u2014is that what this was in aid of?\"\n\nShe grinned at him. \"Was it so bad?\"\n\nHe shook his head. \"I should be very afraid, were you to decide to do something I didn't fancy, inside that pretty head. Yes, of course we can have a tournament. But the wrong men always win, and the town's a riot for a week, and the castle's a mess, you, my dear, are a mess, and I have to arrest men whose only crime was to drink too much. All that, for your whim?\" He laughed.\n\nDesiderata laughed, throwing her head back, and she read his desire in his eyes. \"Yes!\" she said. \"All that for my whim.\"\n\nHe laughed with her. And then frowned. \"And there are the rumours from the north,\" he said.\n\n\"Rumours?\" she asked. Knowing full well what they were\u2014war and worse in the northlands, and incursions from the Wild. It was her business to know them.\n\nThe king shrugged. \"Never mind, love. We shall have a tournament, but it may have to wait until after the spring campaign.\"\n\nShe clapped her hands. Spring was, at last, upon them.\n\n# Chapter Three\n\n# Lissen Carak\u2014The Red Knight\n\nAmy's Hob managed to get his nag to a gallop for long enough to reach the Captain in good time. The company was stretched out along the road in march order\u2014no wagons, no baggage, no followers. Those were in camp with a dozen lances as guards.\n\n\"Lord\u2014Gelfred says he's found its earth. Away in the forest. Trail and hole.\" Amy's Hob was a little man with a nose that had been broken as often as he'd been outlawed.\n\nThe scout held up his hunting horn, and in it was a clod of excrement.\n\n_The fewmets_ , thought the captain, wrinkling his nose in distaste. Gelfred's revenge for his impiety\u2014sometimes close adherence to the laws of venery could constitute their own revenge. He gave the scout a sharp nod. \"I'll just take Gelfred's word for it, shall I?\" he said. He stood in his stirrups and bellowed \"Armour up, people!\"\n\nWord moved down the column faster than a galloping horse. Men and women laced their arming caps and donned their helmets\u2014tall bassinets, practical kettle hats, or sturdy barbutes. Soldiers always rode out armed from head to foot\u2014but only a novice or an overeager squire rode in his helmet or gauntlets. Most knights didn't don their helmets until they were in the face of the enemy.\n\nMichael brought the captain's high-peaked helmet and held it high over his head to slide the mail aventail, the cape that protected the neck and depended from the lower rim of the helmet, over his shoulders. Then he seated the helmet firmly on the padded arming cap, visor pinned up.\n\nThe captain motioned for his squire to pause and reached up to pull the ends of his moustache clear of the mail. He was very proud of his moustache. It did a great deal to hide his age\u2014or lack of it.\n\nThen Michael adjusted the fall of the aventail over his breastplate, checked the buckles under his arms, and pushed the gauntlets on to his master's hands, one at a time, while the captain watched the road to the north.\n\n\"How far up the road?\" he asked Hob.\n\n\"A little farther. We'll cross the burn and then follow it west into the trees.\"\n\nHe had the second gauntlet on, and Michael unbuckled the captain's riding sword and took his long war sword from Toby, who was standing between them on foot, holding it out, a look of excitement on his plain face and a biscuit in his free hand.\n\nMichael handed the shorter riding sword down to Toby, and girded him with the sword of war. Three and a half pounds of sharp steel, almost four feet long.\n\nThe weight always affected the captain\u2014that weight at his side meant business.\n\nHe looked back, standing a little in his stirrups, feeling the increased weight of his armour.\n\nThe column had tightened up.\n\n\"How far?\" he asked Hob.\n\n\"A league. Less. Not an hour's walk.\" Hob shrugged. His hands were shaking.\n\n\"Standard front, then. At my word\u2014Walk!\" called the captain. He turned to his squire. \"Whistle. Not the trumpet.\"\n\nMichael understood. He had a silver whistle around his neck. Carlus, the giant trumpeter and company armourer, shrugged and fell back.\n\nThe column shifted forward, into a walk, the horses suddenly eager, ears pricked forward and heads up. The chargers quivered with excitement\u2014the lighter ronceys ridden by the archers caught the bug from the bigger horses. Along the column, the less able riders struggled to control their mounts.\n\nUp a long hill they went, and then back down\u2014to a burn running fast with the water of two days' rain. Hob led them west into the trees.\n\nNow that they were at the edge of the Wild, the captain had time to note that the trees were still nearly bare. Buds showed here and there, but the north country was not yet in spring, and snow lay in the lee of the larger rocks.\n\nHe could see a long way in these woods.\n\nAnd that meant other things could see him, especially when he was resplendent in mirror-white armour, scarlet and gilt.\n\nHe led them on for another third of a league, the column snaking along behind him, two abreast, easily negotiating the sparse undergrowth. The trees were enormous, their branches thick and long, but stretching out high above even Bad Tom's head.\n\nBut when an inner sense said that he was courting disaster\u2014 _imagine that taloned monster in among this column before we were off our horses and ready_ \u2014he raised his right fist to signal a halt and then spread his arms\u2014always good exercise, in armour\u2014and waved them downwards, once. _Dismount._\n\nHe dismounted carefully, to Grendel's disgust. Grendel liked a fight. Liked to feel the hot squirt of blood in his mouth.\n\n_Not this time_ , the captain thought, and patted his destrier's shoulder.\n\nToby came and took his head.\n\n\"Don't go wandering off, young Toby,\" the captain said cheerfully. \"All officers.\"\n\nMichael, already off his horse and collected, blew a whistle blast. Then handed the captain a short spear with a blade as long as a grown man's arm at one end and a sharp spike at the other.\n\nJehannes and Hugo and Milus walked up, their armour almost silent.\n\n\"Gelfred has the beast under observation. Less than a league away. I want a spread line, heavy on the wings, light in the loins, and every man-at-arms with an archer tight to his back.\" The captain glanced about.\n\n\"The usual, then,\" said Jehannes. His tone suggested that the captain should have said as much.\n\n\"The usual. Fill the thing full of arrows and get this done.\" This was not the right moment to spar with Jehannes, who was his best officer, and disapproved of him nonetheless. He looked around for inspiration.\n\n\"Thick woods,\" Jehannes said. \"Not good for the archers.\"\n\nThe captain raised his hand. \"Don't forget that Gelfred and two of our huntsmen are out there,\" he said. \"Don't let's shoot them full of arrows, too.\"\n\nThe rear two-thirds of the column came forward in an orderly mob and rolled out to the north and south, forming a rough crescent two hundred ells long, in three rough ranks\u2014knights in the front rank, squires in the middle, both men covering an archer to the rear. Some of the archers carried six-foot bows of a single stave, and some carried heavy crossbows, and a few carried eastern horn bows.\n\nThe captain looked at his skirmish line and nodded. His men really were good. He could see Sauce, off to the north, and Bad Tom beyond her. What else could they do? Be outlaws? He gave them purpose.\n\n_I like them_ , he thought. _All of them. Even Shortnose and Wilful Murder._\n\nHe grinned, and wondered who he would be, if he had not found this.\n\n\"Let's get this done,\" he said aloud. Michael blew two sharp blasts, and they were moving.\n\nHe'd counted two hundred paces when Gelfred appeared off to his left. He waved both arms, and the captain lifted a fist, and the line shuffled to a halt. A single shaft, released by a nervous archer, rattled through the underbrush and missed the huntsman by an ell. Gelfred glared.\n\nMilus spat. \"Get his name,\" he growled. \"Fucking new fuck.\"\n\nGelfred ran to the captain. \"It's big,\" he said. \"But not, I think, our quarry. It is\u2014I don't know how to describe it. It's different. It's bigger.\" He shrugged. \"I may be wrong.\"\n\nThe captain weighed this. Looked into the endless trees. Stands of evergreen and alder stood denser than the big, older oaks and ashes.\n\nHe could feel it. It knew they were there.\n\n\"It's going to charge us,\" the captain said. He spoke as flatly as he could, so as not to panic his men. \"Stand ready,\" he called. _To hell with silence_.\n\nBehind him, Michael's breathing grew louder.\n\nGelfred spanned his crossbow. He wasn't wearing armour. Once he had a bolt on the stock, he stepped into line behind Michael.\n\nThe captain reached up and lowered his visor, and it fell across his face with a loud _snap._\n\nAnd then his vision was narrowed to the two long slits in his faceplate, and the tiny breathing holes that also gave him his only warning of any motion coming from below. His own breath came back into his mouth, warmer than the air. The inside of the helmet was close, and he could taste his own fear.\n\nThrough the slits, the woods went on and on, although they seemed darker and stiller than before.\n\nEven the breeze had died.\n\nSilence.\n\nNo bird song.\n\nNo insect noise.\n\nMichael's breathing inside his dog-faced Thuruvian helmet sounded like the bellows in a forge running full-out at a fair. _His first time,_ the captain thought to himself.\n\nThe line was shuffling a little. Men changed their stances\u2014the veterans all had heavy spears, or pole-axes, and they shifted their weight uneasily. The crossbowmen tried to aim. The longbowmen waited for a target before they drew. No man could hold a hundred-pound weight bow for long at the full draw.\n\nThe captain could feel their fear. He was sweating into his armingcote. When he shifted, cold air came in under his arms and his groin, but the hot sweat ran down his back. His hands were cold.\n\nAnd he could feel the tension from his adversary.\n\n_Does it have nerves too? Fear? Does it think?_\n\nNo birds sang.\n\nNothing moved.\n\nThe captain wondered if anyone was breathing.\n\n\"Wyvern!\" shouted Bad Tom.\n\nIt exploded from the trees in front of the captain\u2014taller than a war horse, the long, narrow head full of back-curved teeth, scales so dark that they appeared black, so polished they seemed to be oiled.\n\nIt was fast. The damned things always were.\n\nIts wave of terror was a palpable thing, expanding like a soap bubble around it\u2014the full impact of it struck the captain and washed over him to freeze Michael where he stood.\n\nGelfred raised his crossbow and shot.\n\nHis bolt hit something and the creature opened its maw and screeched until the woods and their ears alike rang with its anger.\n\nThe captain had time to take his guard, spear high, hands crossed, weight back on his right hip. His hands were shaking, and the heavy spearhead seemed to vibrate like a living thing.\n\nIt was coming right for him.\n\n_They always do._\n\nHe had a long heartbeat to look into its golden-yellow eyes, flecked with brown\u2014the slitted black pupil, the sense of its _alienness._\n\nOther archers loosed. Most missed\u2014taking panicked shots at ranges far closer than they had expected. But not all did.\n\nIt ran forward over the last few yards, its two powerful, taloned legs throwing up clods of earth as it charged the thin line of men, head low and forward, snout pointed at the captain's chest. Wings half open, beating the air for balance.\n\nGelfred was already spanning his crossbow, confident that his captain would keep him alive for another few heartbeats.\n\nThe captain shifted his weight and uncrossed his hands\u2014launching the hardest, fastest swing in his repertoire. Cutting like an axe, the spearhead slammed into the wyvern's neck, into the soft skin just under the jaw, the cut timed so that the point stopped against the creature's jawbone... and its charge rammed it onto the point, pushing it deeper and then through the neck.\n\nHe had less than a heartbeat to savour the accuracy of his cut. Then the captain was knocked flat by a blow from its snout, his spear lodged deep in the thing's throat. Blood sprayed, and the fanged head forced itself down the shaft of his spear\u2014past the cross guard, ripping itself open\u2014to reach him. Its hate was palpable\u2014it grew in his vision, its blood lashed him like a rain of acid, and its eyes\u2014\n\nThe captain was frozen, his hands still on the shaft, as the jaws came for him.\n\nAfraid.\n\nBut his spearhead had wide lugs at the base, for just such moments as this and the wyvern's head caught on them, just out of reach. He had a precious moment\u2014recovered his wits, put his head down, breaking the gaze\u2014\n\n\u2014as in one last gout of blood, it broke the shaft, jaws open and lunged\u2014\n\nThe hardened steel of his helmet took the bite. He was surrounded by the smell of the thing\u2014carrion, cold damp earth, hot sulphur, all at once. It thrashed, hampered by the broken spear in its gullet, trying to force its jaws wider and close on his head. He could hear its back-curved teeth scrape, ear-piercingly, over his helmet.\n\nIt gave a growl to make his helmet vibrate, tried to lift him and he could feel the muscles in his neck pull. He roared with pain and held hard to the projecting stump of the shaft as the only support he had. He could hear the battle cries\u2014loud, or shrill, depending on the man. He could hear the meaty sounds of strikes\u2014he could _feel_ them\u2014as men's weapons rained on the wyvern.\n\nBut the creature still had him. It tried to twist his head to break his neck, but its bite couldn't penetrate the helmet for a firmer grip. Its breath was all around him, suffocating him.\n\nHe got his feet beneath him and tried to control his panic as the wyvern lifted him clean from the ground. He got his right hand on his heavy rondel dagger\u2014a spike of steel with a grip. With a scream of fear and rage, he slammed it blindly into the thing's head.\n\nIt spat him free and he dropped like a stone to the frozen ground. His dagger spun away, but he rolled, and got to his feet.\n\nDrew his sword.\n\nCut. All before the wave of pain could strike him\u2014he cut low to high off the draw, left to right across his body and into the joint behind the beast's leg.\n\nIt whirled and before he could react, the tusked snout punched him off his feet. Too fast to dodge. Then threw back its head and screamed.\n\nBad Tom buried his pole-axe in its other shoulder.\n\nIt reared away. A mistake. With two wounded limbs, it stumbled.\n\nThe captain got his feet under him, ignored the fire in his neck and back, and stood, powering straight forward, coming at it from the side this time. It turned to flatten Bad Tom, and Jehannes, suddenly in front of it, hit it on the breastbone with a war hammer. Its face was feathered with barbs and arrows. There were more in the sinuous neck. Even as it turned and took another wound, in the moment that the head was motionless it lost an eye to a long shaft, and its body thrashed\u2014a squire was crushed by a flick of the wyvern's tail, his back breaking and armour folding under the weight of the blow.\n\nHugo crushed its ribs with a mighty, two-handed overhead blow. George Brewes stabbed it with a spear in the side and left the weapon there while he drew his sword. Lyliard cut overhand into the back of its other leg; Foliack hammered it with repeated strokes.\n\nBut it remained focused on the captain. It swatted at him with a leg, lost its balance, roared, and turned on Hugo who had just hit it again. It closed its jaws on the marshal's head, and his helmet didn't hold, The bite crushed his skull, killing him instantly. Sauce stepped over his headless corpse and planted her spear in its jaw, but it flung her away with a flick of the neck.\n\nThe captain leaped forward again and his sword licked out. This time, his cut took one of the thing's wings clean off its body, as easy as a practice cut on a sapling. As the head turned and struck at him the captain stood his ground, ready to thrust for the remaining eye\u2014but the head collapsed to the earth a yard from him, almost like a giant dog laying his head down at his master's feet, and the baleful eye tracked him.\n\nHe thrust.\n\nIt whipped its head up, away from the point of the sword, reared, remaining wing spread wide and thrashing the men under it, a ragged banner of the Wild\u2014\n\n\u2014and died, a dozen bolts and arrows catching it all together.\n\nIt fell across Hugo's corpse.\n\nThe men-at-arms didn't stop hacking at it for a long time. Jehannes severed the head, Bad Tom took one leg off at the haunch, and two squires got the other leg at the knee. Sauce rammed her long rondel into every joint, over and over. Archers continued to loose bolts and arrows into the prone mound of its corpse.\n\nThey were all covered in blood\u2014thick, brown-green blood like the slime from the entrails of a butchered animal, hot to the touch, so corrosive that it could damage good armour if not cleaned off immediately.\n\n\"Michael?\" the captain said. His head _felt_ as if it had been pulled from his body.\n\nThe young man struggled to get his maille aventail over his head, failed, and threw up inside his helmet. But there was wyvern blood on his spear, and more on his sword.\n\nGelfred spanned his crossbow one more time, eyes fixed on the dead creature. Men were hugging, laughing, weeping, vomiting, or falling to their knees to pray, others merely gazed blank-eyed at the creature. The wyvern.\n\nAlready, it looked smaller.\n\nThe captain stumbled away from it, caught himself, mentally and physically. His arming cote was soaked. He went instantly from fight-hot to cold. When he stooped to retrieve his dagger, he had a moment's vertigo, and the pain from his neck muscles was so intense he wondered if he would black out.\n\nJehannes came up. He looked\u2014old. \"Six dead. Sweet William has his back broken and asks for you.\"\n\nThe captain walked the few feet to where Sweet William, an older squire in a battered harness, lay crumpled where the tail and hindquarters had smashed him flat and crushed his breastplate. Somehow, he was alive.\n\n\"We got it, aye?\" he said thickly. \"Was bra'ly done? Aye?\"\n\nThe captain knelt in the mire by the dying man's head. \"Bravely done, William.\"\n\n\"God be praised,\" Sweet William said. \"It all hurts. Get it done, eh? Captain?\"\n\nThe captain bent down to kiss his forehead, and put the blade of his rondel into an eye as he did, and held the man's head until the last spasm passed, before laying his head slowly in the mire.\n\nHe was slow getting back to his feet.\n\nJehannes was looking to where Hugo's corpse lay under the beast's head. He shook his head. Looked up, and met the captain's eye. \"But we got it.\"\n\nGelfred was intoning plainchant over the severed head. There was a brief flare of light. And then he turned, disgust written plain on his face. He spat. \"Wrong one,\" he said.\n\nJehannes spat. \"Jesu shits,\" he said. \"There's another one?\"\n\n# North of Harndon\u2014Ranald Lachlan\n\nRanald rode north with three horses\u2014a heavy horse not much smaller than a destrier and two hackneys, the smallest not much better than a pony. He needed to make good time.\n\nBecause he needed to make good time he went hard all day and slept wherever he ended. He passed the pleasant magnificence of Lorica and her three big inns with regret, but it was just after midday and he had sun left in the sky.\n\nHe didn't have to camp, exactly. As the last rays of the sun slanted across the fields and the river to the west, he turned down a lane and rode over damp manured fields to a small stand of trees on a ridge overlooking the road. As he approached in the last light, he smelled smoke, and then he saw the fire.\n\nHe pulled up his horses well clear of the small camp, and called out, \"Hullo!\"\n\nHe hadn't seen anyone by the fire, and it was dark under the trees. But as soon as he called a man stepped from the shadows, almost by his horse's head. Ranald put his hand on his sword hilt.\n\n\"Be easy, stranger,\" said a man. An old man.\n\nRanald relaxed, and his horse calmed.\n\n\"I'd share my food with a man who'd share his fire,\" Ranald said.\n\nThe man grunted. \"I've plenty of food. And I came up here to get away from men, not spend the night prattling.\" The old fellow laughed. \"But bad cess on it\u2014come and share my fire.\"\n\nRanald dismounted. \"Ranald Lachlan,\" he said.\n\nThe old man grinned, his teeth white and surprisingly even in the last light. \"Harold,\" he said. \"Folk around here call me Harold the Forester, though its years since I was the forester.\" He slipped into the trees, leading Ranald's packhorse.\n\nThey ate rabbit\u2014the old man had three of them, and Ranald wasn't so rude as to ask what warren they'd been born to. Ranald still had wine\u2014good red wine from Galle, and the old man drank a full cup.\n\n\"Here's to you, my good ser,\" he said in a fair mockery of a gentleman's accent. \"I had many a bellyful of this red stuff when I was younger.\"\n\nRanald lay back on his cloak. The world suddenly seemed very good to him, but he remained troubled that there were leaves piled up for two men to sleep, and that there were two blanket rolls on the edge of the fire circle, for one man. \"You were a soldier, I suspect,\" he said.\n\n\"Chevin year, we was all soldiers, young hillman,\" Harold said. He shrugged. \"But aye. I was an archer, and then a master archer. And then forester, and now\u2014just old.\" He sat back against a tree. \"It's cold for old bones. If you gave me your flask, I'd add my cider and heat it.\"\n\nRanald handed over his flask without demur.\n\nThe man had a small copper pot. Like many older veterans Ranald had known, his equipment was beautifully kept, and he found it without effort, even in the dark\u2014each thing was where it belonged. He stirred his fire, a small thing now the rabbit was cooked, made from pine cones and twigs, and yet he had the drink hot in no time.\n\nRanald had one hand on his knife. He took the horn cup that was offered him, and while he could see the man's hands, he said \"There was another man here.\"\n\nHarold didn't flinch. \"Aye,\" he said.\n\n\"On the run?\" Ranald asked.\n\n\"Mayhap,\" said Harold. \"Or just a serf who oughtn't to be out in the greenwood. And you with your Royal Guardsman's badge.\"\n\nRanald was ready to move. \"I want no trouble. And I offer none,\" he said.\n\nHarold relaxed visibly. \"Well, he won't come back. But I'll see to it that the feeling is mutual. Have some more.\"\n\nRanald lay under his cloak without taking off his boots and laid his dirk by his side. Whatever he thought about the old man, there were plenty of men who would cut another's throat for three good horses. And he went to sleep.\n\n# Harndon\u2014Edward\n\nThaddeus Pyel finished mixing the powder\u2014saltpeter and charcoal and a little sulphur. Three to two to one, according to the alchemist who made the mixture for the king.\n\nHis apprentices were all around him, bringing him tools as he demanded them\u2014a bronze pestle for grinding charcoal fine, spoons of various sizes to measure with.\n\nHe mixed the three together, carried the mixture outside into the yard, and touched a burning wick to them.\n\nThe mixture sputtered and burned, with a sulphurous smoke.\n\n\"Like Satan cutting a fart,\" muttered his son Diccon.\n\nMaster Pyel went back into his shop and mixed more. He varied the quantities carefully, but the result was always the same\u2014a sputtering flame.\n\nThe boys were used to the master's little ways. He had his notions, and sometimes they worked, and other times they didn't. So they muttered in disappointment but not in surprise. It was a beautiful evening, and they went up on the workshop roof and drank small beer. Young Edward, the shop boy and an apprentice coming up on his journeyman qualification, stared at the rising moon and tried to imagine _exactly_ what the burning powder did.\n\nIn all his imagings it was something to do with a weapon, because at the sign of the broken circle, that's what they did. They made weapons.\n\n# Albinkirk\u2014Ser John Crayford\n\nSer John was taking exercise. Age and weight had not prevented him from swinging his sword at his pell\u2014or at the other four men-at-arms who were still willing enough to join him.\n\nSince the young sprig had ridden through with his beautifully armed company, the Captain of Albinkirk had been at the pell three times. His back hurt. His wrists hurt. His hands _burned._\n\nMaster Clarkson, his youngest and best man-at-arms, backed out of range and raised his sword. \"Well cut, Ser John,\" he said.\n\nSer John grinned, but only inside his visor where it wouldn't show. Just in that moment, all younger men were the enemy.\n\n\"Ser John, there's a pair of farmers to see you.\" It was the duty sergeant. Tom Lickspittle, Ser John called him, if only inside his visor. The man couldn't seem to do anything well except curry favour.\n\n\"I'll see them when I'm done here, Sergeant.\" Ser John was trying to control his breathing.\n\n\"I think you'll want to\u2014to see them now.\" _That was new._ Lickspittle Tom never questioned orders. The man gulped. \"My lord.\"\n\n_That makes this some sort of emergency._\n\nSer John walked over to his latest squire, young Harold, and got his visor lifted and his helmet removed. He was suddenly ashamed of his armour\u2014brown on many surfaces, or at least the mail was. His cote armour was covered in what had once been good velvet. How long ago had that been?\n\n\"Clean that mail,\" he said to Harold. The boy winced, which suited Ser John's mood well. \"Clean the helmet, and find me an armourer. I want this recovered in new cloth.\"\n\n\"Yes, Ser John.\" The boy didn't meet his eye. Lugging armour around the Lower Town would be no easy task.\n\nSer John got his gauntlets off and walked across the courtyard to the guard room. There were two men\u2014prosperous men; wool cotes, proper hose; one in all the greys of local wool, one in a dark red cote.\n\n\"Gentlemen?\" he asked. \"Pardon my armour.\"\n\nThe man in the dark red cote stood forward. \"Ser John? I'm Will Flodden and this is my cousin John. We have farms on the Lissen Carak road.\"\n\nSer John relaxed. This was not a complaint about one of his garrison soldiers.\n\n\"Go on,\" he said, cheerfully.\n\n\"I kilt an irk, m'lord,\" said the one called John. His voice shook when he said it.\n\nSer John had been a number of places. He knew men, and he knew the Wild. \"Really?\" he said. He doubted it, instinctively.\n\n\"Aye,\" said the farmer. He was defensive, and he looked at his cousin for support. \"There was tre of 'em. Crossing my fields.\" He hugged himself. \"An' one loosed at me. I ran for ta' house, an' picked up me latchet and let fly. An' tey ran.\"\n\nSer John sat a little too suddenly. Age and armour were not a good mix.\n\nWill Flodden sighed. \"Just show it to him.\" He seemed impatient\u2014a farmer who wanted to get back to his farm.\n\nBefore he even undid the string securing the sack, Ser John knew what he was going to see. But it all seemed to take a long time. The string unwinding, the upending of the sack. The thing in the sack had stuck to the coarse fabric.\n\nFor as long as it took, he could tell himself that the man was wrong. He'd killed an animal. A boar with an odd head, or some such.\n\nBut twenty years before Ser John had stood his ground with thousands of other men against a charge of ten thousand irks. He remembered it too damned well.\n\n\"Jesus wept. Christe and the Virgin stand with us,\" he said.\n\nIt was an irk, its handsome head somehow smaller and made ghastly having been severed from its sinuous body.\n\n\"Where, exactly?\" he demanded. And turning, he ignored Tom Lickspittle, who was a useless tit in a crisis. \"Clarkson! Sound the alarm and get me the mayor.\"\n\n# Lissen Carak\u2014The Red Knight\n\nPatience had never been the captain's greatest virtue, and he paced the great hall of the convent, up and back, up and back, his anger ebbing and flowing as he gained and lost control of himself. He suspected that the Abbess was keeping him waiting on purpose; he understood her motives, he read her desire to humble him and keep him off guard; and despite knowing that he was angry, and thus off guard.\n\nGradually, frustration gave way to boredom.\n\nHe had time to note that the stained glass of the windows in the clerestory had missing panels\u2014some replaced in clear glass, and some in horn, and one in weathered bronze. The bright sunlight outside, the first true sign of spring, made the rich reds and blues of the glass glow, but the missing panes were cast into sharp contrast\u2014the horn was too dull, the clear glass too bright, the metal almost black and sinister.\n\nHe stared at the window depicting the convent's patron saint, Thomas, and his martyrdom, for some time.\n\nAnd then boredom and annoyance broke his meditation and he began to pace again.\n\nHis second bout of boredom was lightened by the arrival of two nuns in the grey habit of the order, but they had their kirtles on, open at the neck and with their sleeves rolled up. Both had heavy gloves on, tanned faces, and they bore an eagle on a perch between them.\n\nAn _eagle._\n\nBoth of them bowed politely to the captain and left him with the bird.\n\nThe captain waited until they were clear of the hall and then walked over to the bird, a dark golden brown with the dusting of lighter colour that marked a fully mature bird.\n\n\"Maybe a little too fully mature, eh, old boy?\" he said to the bird, who turned his hooded head to the sound of the voice, opened his beak, and said \" _Raawwk!_ \" in a voice loud enough to command armies.\n\nThe bird's jesses were absolutely plain where the captain, who had been brought up with rich and valuable birds, would have expected to see embroidery and gold leaf. This was a Ferlander Eagle, a bird worth\u2014\n\n\u2014worth the whole value of the captain's white harness, which was worth quite a bit.\n\nThe eagle was the size of his entire upper body, larger than any bird his father\u2014the captain sneered internally at the thought of the man\u2014had ever owned.\n\n_\"Raaaawk!\"_ the bird screamed.\n\nThe captain crossed his arms. Only a fool released someone else's bird\u2014especially when that bird was big enough to eat the fool\u2014but his fingers itched to handle it, to feel its weight on his fist. Could he even fly such a bird?\n\n_Is this another of her little games?_\n\nAfter another interval of waiting, he couldn't stand it any more. He pulled on his chamois gloves and brushed the back of his hand against the talons of the bird's feet. It stepped obligingly onto his wrist and it weighed as much as a pole-axe. More. His arm sank, and it was an effort to raise the bird back to eye level and place it back on its perch.\n\nWhen it had one foot secure on the deerskin-padded perch, it turned its hooded head to him, as if seeing him clearly, and closed its left foot, sinking three talons into his left arm.\n\nEven as he gasped, it stepped up onto its perch and turned to face him.\n\n_\"Rawwwwwwwk,\"_ it said with obvious satisfaction.\n\nBlood dripped over his gauntlet cuff.\n\nHe looked at the bird. \"Bastard,\" he said. And he went back to pacing, albeit he now cradled his left arm in his right for twenty trips up and down the hall.\n\nHis third bout of boredom was broken by the books. He'd given them only a cursory glance on his first visit, and had dismissed them. They displayed the usual remarkable craftsmanship, superb calligraphy, painted scenes, gilt work everywhere. Worse, both volumes were collections of the _Lives of the Saints_ , a subject in which the captain had no interest whatsoever. But boredom drove him to look at them.\n\nThe leftmost work, beneath the window of Saint Maurice, was well-executed, the paintings of Saint Katherine vivid and rich. He chuckled to wonder what lovely model had stood in a monk's mind, or perhaps a nun's, as the artist lovingly re-created the contours of flesh. Saint Katherine's face did not show torment, but a kind of rapture\u2014\n\nHe laughed and passed to the second book, pondering the lives of the devout.\n\nWhat struck him first was the poor quality of the Archaic. The art was beautiful\u2014the title page had a capital where the artist was presented, sitting on a high stool, working away with a gilding brush. The work was so precise that the reader could see that the artist was working on the very title page, presented again in microcosm.\n\nThe captain breathed deeply in appreciation of the work, and the humour of it. And then he began to read.\n\nHe turned the page. He imagined what his beloved Prudentia would have said about the barbaric nature of the writer's Archaic. He could all but see the old nun wagging her finger in his mother's solar.\n\nShook his head.\n\nThe door to the Abbess's private apartments opened and the priest, hurried past, hands clasped together and face set. He looked furious.\n\nBehind him, the Abbess gave a low laugh, almost a snort. \"I thought you'd find our book,\" she said. She looked at him fondly. \"And my _Parcival._ \" She indicated the bird.\n\n\"I can't see how such a brutally bad transcription merited the quality of artist,\" he said, turning another page. \"If that's your bird, you are braver than I thought.\"\n\n\"Am I?\" she asked. \"I've had him for many years.\" She looked fondly at the bird, who bated on his perch. \"Can you not see why the book is so well wrought?\" she asked with a smile that told him that there was a secret to it. \"You do know that we have a library, Captain? I believe that our hospitality might extend as far as allowing you to use it. We have more than fifty volumes.\"\n\nHe bowed. \"Would I shock you if I said that the _Lives of the Saints_ held little interest for me?\"\n\nShe shrugged. \"Posture away, little atheist. My gentle Jesu loves you all the same.\" She gave him a wry smile. \"I am sorry\u2014I would love to spar with you all morning, but I have a crisis in my house. May we to business?\" She waved him to a stool. \"Still in armour,\" she said.\n\n\"We are still on the hunt,\" he said, crossing his legs.\n\n\"But you killed the monster. Don't think we are not grateful. In fact, I regret taking the tone I did, especially as you lost a man of great worth, and since you were so very effective.\" She shrugged. \"And you have done your work before the new month\u2014and before my fair opens.\"\n\nHe made a sour face. \"My lady, I would like to deserve your esteem, and few things would give me greater pleasure than to hear you apologise.\" He shrugged. \"But I am not here to spar, either. Unworthily, I assumed you kept me cooling my heels to teach me humility.\"\n\nShe looked at her hands. \"You could use some, young man, but unfortunately, I have other issues before me this day or I would be happy to teach you some manners. Now, why do you say you do not deserve my regard?\"\n\n\"We have killed a monster,\" he acknowledged. \"But not the one that killed Sister Hawisia.\"\n\nShe jutted out her jaw\u2014a tic he hadn't seen before. \"I must assume that you have ways to know this. You must pardon me if I am sceptical. We have _two_ monsters? I remember your saying the enemy seldom hunts alone this far from the Wild\u2014but surely, Captain, you know that we are not as far from the Wild as we once were.\"\n\nHe wished for a chair with a back. He wished that Hugo were alive, and he hadn't been saddled with internal issues of discipline that should have been Hugo's. \"May I have a glass of wine?\" he asked.\n\nThe Abbess had a stick, and she thumped it on the floor. Amicia entered, eyes downcast. The Abbess smiled at her. \"Wine for the captain, dear. And do not raise your eyes, if you please. Good girl.\"\n\nAmicia slipped out the door again.\n\n\"My huntsman is a Hermetic,\" he said. \"With a licence from the Bishop of Lorica.\"\n\nShe waved a hand. \"The orthodoxy of Hermeticism is beyond my poor intellect. Do you know, when I was a girl, we were forbidden to use High Archaic for any learning beyond the _Lives of the Saints_. I was punished by my chaplain as a girl for reading some words on a tomb in my father's castle.\" She sighed. \"You read the Archaic, then,\" she said.\n\n\"High and low,\" he answered.\n\n\"I thought as much... and there cannot be so many knights in the Demesne who can read High Archaic.\" She made a motion with her head, as if shaking off fatigue. Amicia returned, brought the captain wine and backed away from him without ever raising her eyes\u2014a very graceful performance.\n\nShe wore that curious expression again. The one he couldn't read\u2014it held both anger and amusement, patience and frustration, all in one corner of her mouth.\n\nThe Abbess had taken Parcival the eagle on her wrist, and she was stroking his plumage and cooing at him. While the arm of her throne-like chair helped support the great raptor, the captain was impressed by her strength. _She must be sixty_ , he thought.\n\nThere was something about the Abbess\u2014the Abbess and Amicia. It was not a similarity of breeding\u2014two more different women could not be imagined, the older woman with an elfin beauty and slim bones, the younger taller, heavier boned, with strong hands and broad shoulders.\n\nHe was still staring at Amicia when the Abbess's staff thumped the floor.\n\nThe word _hermetic_ rolled around the captain's busy brain, and curled itself in the corner of Amicia's mouth. But the staff took his attention.\n\n\"Assuming I believe you\u2014what does your huntsman say?\" the Abbess demanded.\n\nThe captain sighed. \"That we got the wrong one. My lady, no one but a great Magus or a mountebank can tell us _why_ the enemy acts as they do. Perhaps one of them is calling to others for reinforcements. Perhaps you have a nest of them. But Gelfred assures me that the signs left by Sister Hawisia's killer are _not_ the same as those of the beast we slew and my men\u2014all of them\u2014are exhausted. It will take them a day to recover. They've lost a gallant leader, a man they all respected, so I am sorry, but we will not be very aggressive for a few days.\" He shrugged.\n\nShe looked at him for a long time, and finally crossed her hands on the top of her staff and laid her long chin on them. \"You think I do not understand,\" she said. She shrugged. \"I do. I do not believe you seek to cheat me.\"\n\nHe didn't know what to make of that.\n\n\"Let me tell you my immediate concerns,\" she said. \"My fair opens in a week. The first week of the fair is merely local produce and prizes. Then the Harndon merchants come upriver in the second week to buy our surplus grain and our wool. But in the second and third weeks of the fair, the drovers come down from the moors. That is when the business is done, and that's when I need my bridge and my people to be safe. You know why there is a fortress here?\" she asked.\n\nHe smiled. \"Of course,\" he said. \"The fortress is merely to guarantee the bridge.\"\n\n\"Yes,\" she admitted. \"And I have been lax in letting my garrison drop\u2014but if you will pardon an old woman's honesty, soldiers and nuns are not natural friends. Yet these attacks\u2014I hold this land by knight service and garrison service, and I do not have enough men. The king will send a knight to dispense justice at the fair and I dread his discovering how my penny-pinching ways have put these lands at risk.\"\n\n\"You need me for more than just monster-hunting,\" he said.\n\n\"I do. I would like to purchase your contract for the summer, and I wonder if you have a dozen men-at-arms\u2014archers, even\u2014who could stay when you go. Perhaps men you'd otherwise pension off, or men who've been wounded.\" She shrugged. \"I don't even know how to _find_ a new garrison. Albinkirk used to be a fine town\u2014and a place where such men could be found\u2014but not anymore.\" She took a deep breath.\n\nHe nodded. \"I will consider it. I will not pretend, since we are being honest with each other, that my company does not need a steady contract. I would like to recruit, too. I need men.\" He thought a moment. \"Would you want women?\"\n\n\"Women?\" asked the Abbess.\n\n\"I have women\u2014archers, men-at-arms.\" He smiled at her chagrin. \"It's not so uncommon as it once was. It is almost accepted over the sea, on the Continent.\"\n\nShe shook her head. \"I think not. What kind of women would they be? Slatterns and whores taught to fight? Scarcely fit companions for women of religion.\"\n\n\"You have a good point, my lady. I'm sure they are far less fit as companions then the sort of _men_ who are attracted to a mercenary company.\" He leaned back, stretching his legs to ease the pressure on his lower back.\n\nTheir eyes met, sharp as two blades crossed.\n\nShe shrugged. \"We are not adversaries. Rest if you must. Consider my offer. Do you need a service for the dead?\"\n\nFor the first time, he allowed himself to feel warmth for the lady Abbess. \"That would be greatly appreciated.\"\n\n\"Not all your men reject God as you do?\" she said.\n\n\"Much the opposite.\" He rose to his feet. \"Soldiers are as inclined to nostalgic irrationality as any other group\u2014perhaps more so.\" He winced. \"I'm sorry my lady, that was rude, in response to your very kind offer. We have no chaplain. Ser Hugo was a gentleman of good family who died in his faith, whatever you may think of me. A service for the dead would be very kind of you, and would probably do much to keep my people in order\u2014ahem.\" He shook his head. \"I appreciate your offer.\"\n\n\"You are really quite sweet in your well-mannered confusion,\" she said, also rising. \"We will get along well enough, Ser Captain. And you will, I hope, forgive me if I counter your blatant lack of respect for my religion with an attempt to convert you to it. Whatever has been done to you, it was not Jesu who did it, but the hand of man.\"\n\nHe bowed. \"That's just where you are wrong, my lady.\" He reached for her hand, which she offered, to kiss\u2014but the imp in him could not be stopped, and so he turned it over and kissed her palm like a lover.\n\n\"Such a little boy,\" she said, but she was clearly both pleased and amused. \"A rather wicked boy. Service tonight, I think, in the chapel.\"\n\n\"You will allow my company into the fortress?\" he asked.\n\n\"Since I intend to employ you as my garrison,\" she replied, \"I will, in time, have to trust you inside the walls.\"\n\n\"This is a sharp change of direction, my lady Abbess,\" he said.\n\nShe nodded and swept towards the inner door to the convent. \"Yes it is,\" she said. She gave him a very straight-backed courtesy. \"I know things, now.\"\n\nHe stopped her with his hand. \"You said the Wild is closer now. I've been away. Closer how?\"\n\nShe released a breath. \"We have twenty farms which we have taken from the trees. There are more families here than when I was first a novice\u2014more families. And yet. When I was young, nobles hunted the Wild in the mountains\u2014expeditions into the Adacrags were a knight errant's dream. The convent used to host them in our guest house.\" She glanced out the window. \"The border with the Wild used to be fifty leagues or more to the north and the west of us, and while the forest was deep, trustworthy men lived there.\" She met his eyes. \"Now, my fortress is the border, as it was in my grandfather's time.\"\n\nHe shook his head. \"The wall is two hundred leagues north of here. And as far west.\"\n\nShe shrugged. \"The Wild is not. The king was going to push the Wild back to the wall,\" she said wearily. \"But I gather his young wife takes all his time.\"\n\nHe smiled. And changed the subject. \"Tell me what the book is?\" he asked.\n\nShe smiled. \"You will enjoy puzzling it out for yourself,\" she said. \"I wouldn't want to deny you that pleasure.\"\n\n\"You are a wicked old woman,\" he said.\n\n\"Ah,\" she smiled. \"You are beginning to know me, messire.\" She smiled, all flirtation, and then paused. \"Captain, I have decided to tell you something,\" she said. She wasn't hestitant. She was merely careful. \"About Sister Hawisia.\"\n\nHe didn't move.\n\n\"She told me that we had a traitor in our midst. And that she would unmask him. I was supposed to be at the farm that day. She insisted on going in my place.\" The Abbess looked away. \"I'm afraid that monster was meant for me.\"\n\n\"Or your brave sister unmasked the traitor and he killed her for it. Or he already knew she intended to unmask him, and set a trap.\" The captain hadn't shaved in days, and he scratched absently under his chin. \"Who knows of your movements and decisions, my lady?\"\n\nShe sat back. Her staff smacked against the floor in real agitation. Their eyes met.\n\n\"I am on your side,\" he said.\n\nShe was fighting tears. \"They are _my people,_ \" she said. She bit her lip and gave her head a shake that moved the fine linen folds of her wimple. \"Bah\u2014I am not a schoolgirl. I will have to think, and perhaps look at my notes. Sister Miram is my vicar, and I trust her absolutely. Father Henry attends me at most hours. Sister Miram has access to everything in the fortress and knows most of my thoughts. John le Bailli is my factor in the villages and the king's officer for the Senechally. I will arrange that you meet them all.\"\n\n\"And Amicia,\" the captain said quietly.\n\n\"Yes. She attends me at most hours.\" The Abbess's eyes locked with the captain's. \"She and Hawisia were not friends.\"\n\n\"Why not?\" asked the knight.\n\n\"Hawisia was gently born, nobly born. She had great power.\" The Abbess looked out the window, and her bird bated slightly at her movement.\n\n\"Put him back on his perch, please?\" she asked.\n\nThe knight collected the great bird on his fist and transferred his great weight to the perch. \"Surely he is a royal bird?\"\n\n\"I had a royal friend, once,\" said the Abbess, with a curl of her lips.\n\n\"And Amicia is not gently born?\" the Red Knight prodded.\n\nThe Abbess met his glance and rose. \"I will leave you to make such enquiries yourself,\" she said. \"I find that I am uninterested in gossiping about my people.\"\n\n\"I have angered you,\" said the knight.\n\n\"Messire, creatures of the Wild are killing my people, one of them is a traitor and I have to hire sell-swords to protect me. Today _everything_ angers me.\"\n\nShe opened the door, and he had a glimpse of Amicia, and then the door closed behind her.\n\nGiven an unexpected moment of freedom, he walked to the book. It stood under a window of Saint John the Baptist so he began to turn the pages, looking for the saint's story.\n\nThe Archaic was painful, stilted, ill-phrased, as if a schoolgirl had translated the Archaic to Gothic and then back, making grievous errors in both directions.\n\nThe calligraphy was inhuman in its perfection. In ten pages, he could not find a pen error. Who would labour so over such a bad book?\n\nThe secret of the book merged in his mind with the secret that hid in the corner of Alicia's downturned mouth, and he began to look more carefully at the lavish illuminations.\n\nFacing the tale of Saint Paternus was a complex illustration of the saint himself, in robes of red, white and gold. His robes were richly embellished, and in one hand he held a cross.\n\nIn the other hand he held an alembic instead of an orb, and inside the alembic were minute figures of a man and a woman...\n\nThe captain looked back to the Archaic, trying to find the trace of a reference\u2014was it heresy?\n\nHe stood up, releasing the vellum cover. _Heresy is none of my concern_ , he thought. Besides\u2014whatever that smug old woman was, she was no secret heretic. He walked slowly across the hall, his sabatons clinking faintly as he walked, and his mind still on the problem of the book. _She was right, damn her_ , he thought.\n\n# Heading North\u2014A Golden Bear\n\nThe mother bear swam until she could no longer swim, and then she lay up all day, cold to the bone and weary from blood loss and despair. Her cub sniffed at her and demanded food, and she forced herself to move to find some. She killed a sheep in a field and they fed on it; then she found a line of bee-hives at the edge of another field and they ate their way through the whole colony, eight hives, until both bears were sticky and drunk on sugar. She licked raw honey into the wounds the sword had made. Men were born without talons, but the claws they forged for themselves were deadlier than anything the Wild might give them.\n\nShe sang for her daughter, and called her name.\n\nAnd her cub mewed like an animal.\n\nWhen Lily was stronger, they went north again. That night, she smelled the pus in her wounds. She licked it and it tasted bad.\n\nShe tried to think of happier days\u2014of her mate, Russet, and her mother's den in the distant mountains. But her slavery had gone on too long, and something was dimmed in her.\n\nShe wondered if her wound was mortal. If the warrior man had poisoned his claw.\n\nThey lay up another day and she caught fish, no kind she recognised, but something that tasted a little of salt. She knew that the great Ocean was salted, perhaps the river had a spring run of sea fish.\n\nThey were easy to catch, even for a wounded bear.\n\nThere were more hives at a field edge, whose outraged human guard lofted arrows at them from his stone croft. None of them struck home, and they slipped away.\n\nShe had no idea where she was, but her spirit said to go north. And the river flowed from her home, she could taste the icy spring run off. So she kept moving north.\n\n# The Great North Road\u2014Gerald Random\n\nGerald Random, Merchant Adventurer of Harndon, looked back along the line of his wagons with the satisfaction of the captain for his company, or the Abbot for his monks. He'd mustered twenty-two wagons of his own, all in his livery colours, red and white, their man-high wheels carefully painted with red rims and white spokes; the sides of every wagon white with red trim, and scenes from the Passion of the Christ decorating every side panel, all the work of his very talented brother-in-law. It was good advertising, good religion, and it guaranteed that his carters would always form his convoy in order\u2014every man, whether they could not read or figure or not, knew that the God Jesu was scourged by knights in their guard room, and _then_ had to carry his own cross to Golgotha.\n\nHe had sixty good men, mostly drapers' and weavers' men, but a pair of journeymen goldsmiths and a dozen cutlers, and some bladesmiths and blacksmiths, a handful of mercers and grocers too, all armed and well armoured like the prosperous men they were. And he had ten professional soldiers he'd engaged himself, acting as his own captain\u2014good men, every one of them with a King's Warrant that he had borne arms in the king's service.\n\nGerald Random had such a warrant himself. He'd served in the north, fighting the Wild. And now he was leading a rich convoy to the great market fair of Lissen Carak as the commander, the principle investor, and the owner of most of the wagons.\n\nHis should be the largest convoy on the road and the best display at the fair.\n\nHis wife Angela laid a long white hand on his arm. \"You find your wagons more beautiful than you find me,\" she said. He wished that she might say it with more humour, but at least there _was_ humour.\n\nHe kissed her. \"I've yet time to prove otherwise, my lady,\" he said.\n\n\"The future Lord Mayor does not take his wife for a ride in the bed-carriage while his great northern convoy awaits his pleasure!\" she said. She rubbed his arm through his heavy wool doublet. \"Dinna' fash yourself, husband. I'll be well enough.\"\n\nGuilbert, the oldest and most reliable-looking of the hireling swordsmen, approached with a mixture of deference and swagger. He nodded\u2014a compromise between a bow and a failure to recognise authority. Random took it to mean something like _I have served great lords and the king, and while you are my commander, you are not one of them._\n\nRandom nodded.\n\n\"Now that I see the whole convoy,\" Guilbert nodded at it. \"I'd like six more men.\"\n\nRandom looked back over the wagons\u2014his own, and those of the goldsmiths, the cutlers, the two other drapers, and the foreign merchant, Master Haddan, with his tiny two wheel cart and his strange adult apprentice, Adle. Forty-four wagons in all.\n\n\"Even with the cutlers' men?\" he asked. He kept his wife there by taking her hand when she made to slip away.\n\nGuilbert shrugged. \"They're fair men, no doubt,\" he said.\n\nWages for six more men\u2014Warrant men\u2014would cost him roughly the whole profit on one wagon. And the sad fact was that he couldn't really pass any part of the cost on to the other merchants, who had already paid\u2014and paid well\u2014to be in his convoy.\n\nMoreover, he had served in the north. He knew the risks. And they were high\u2014higher every year, although no one seemed to want to discuss such stuff.\n\nHe looked at his wife, contemplating allowing the man two more soldiers.\n\nHe loved his wife. And the worry on her face was worth spending more than the value of a cart to alleviate. And what would the profit be, should his convoy be taken or scattered?\n\n\"Do you have a friend? Someone you can engage at short notice?\" he asked.\n\nGuilbert grinned. It was the first time that the merchant had seen the mercenary smile, and it was a surprisingly human, pleasant smile.\n\n\"Aye,\" the man said. \"Down on his luck. I'd esteem it a favour. And he's a good man\u2014my word on it.\"\n\n\"Let's have all six. Eight, if we can get them. I have a worry, so let's be safe. Money is not all there is,\" he said, looking into his wife's eyes, and she breathed out pure relief. Some dark omen had been averted.\n\nHe hugged her for a long time while apprentices and journeymen kept their distance, and when Guilbert said he needed an hour by the clock to get his new men into armour\u2014meaning they'd pawned theirs and needed to redeem it\u2014Random took his wife by the hand and took her upstairs. Because there were so _many_ things that were more important than money.\n\nBut the sun was still in the middle of the spring sky when forty-five wagons, two hundred and ten men, eighteen soldiers, and one merchant captain started north for the fair. He knew that he was the ninth convoy on the great road north\u2014the longest to assemble, and consequently, the last that would reach Lissen Carak's great supply of grain. But he had the goods and the wagons to buy _so much_ grain that he didn't think he'd be the loser, and he had a secret\u2014a trade secret\u2014that might make him the greatest profit in the history of the city.\n\nIt was a risk. But surprisingly for a man of money, as the lords called his kind, Gerald Random loved risk as other men loved money, or swords, or women, and he set his sword at his hip, his dagger on the other, with a round steel buckler that would not have disgraced a nobleman, and smiled. Win or lose, this was the moment he loved. Starting out. The dice cast, the adventure beginning.\n\nHe raised his arm, and he heard the sounds of men responding. He sent a pair of the mercenaries forward, and then he let his arm fall. \"Let's go!\" he called.\n\nWhips cracked, and animals leaned into their loads, and men waved goodbye to sweethearts and wives and children and brats and angry creditors, and the great convoy rolled away with creaking wheels and jingling harness and the smell of new paint.\n\nAnd Angela Random knelt before her icon of the Virgin and wept, the tears as hot as her passion of an hour before.\n\n# Lissen Carak\u2014The Red Knight\n\nSeven men had died fighting the wyvern. The corpses were wrapped in plain white shrouds because that was the rule of the Order of Saint Thomas, and they gave off a sickly sweet smell\u2014corruption and zealous use of sweet herbs, and bitter myrrh burned in the censors that hung in the front of the chapel.\n\nThe whole fighting strength of his company stood in the nave, shifting uneasily as if facing an unexpected enemy. They wore no armour, bore no weapons, and some were very ill-dressed; not a few wore their arming cotes with mail voiders because they had no other jacket, and at least one man was bare-legged and ashamed. The captain was plainly dressed in black hose and a short black jupon that fitted so tightly that he couldn't bend over\u2014his last decent garment from the Continent. His only nod to his status was the heavy belt of linked gold and bronze plaques around his hips.\n\nTheir apparent penury contrasted with the opulence of the chapel\u2014even with the shrines and crosses swathed in purple for Lent, or perhaps the more so _because_ the purple of Lent was so rich. Except that nearer to hand, the captain could see the edge of a reliquary peaking out from beneath its silken shroud, the gilt old and crazed, the wood broken. Tallow, not wax, burned in every sconce except the altar candelabra, and the smell of burning fat was sharp against the sweet and the bitter.\n\nThe captain noted that Sauce wore a kirtle and a gown. He hadn't seen her dressed as a woman since her first days with the company. The gown was fine, a foreign velvet of ruddy amber, somewhat faded except for one diamond shaped patch on her right breast.\n\n_Where her whore's badge was sewn_ , he thought. He glared at the crucified figure over the altar, his pleasant, detached mood destroyed. _If there is a god, how can he allow so much fucking misery and deserve my thanks for it?_ The captain snorted.\n\nAround him the company sank to their knees as the chaplain, Father Henry, raised the consecrated host. The captain kept his eyes on the priest, and watched him throughout the ritual that elevated the bread to the sacred body of Christ\u2014even surrounded by his mourning company, the captain had to sneer at the foolishness of it. He wondered if the stick-thin priest believed a word of what he was saying\u2014wondered idly if the man was driven insane by the loneliness of living in a world of women, or if he was consumed by lust instead. Many of the sisters were quite comely, and as a soldier, the captain knew that comeliness was in the eye of the beholder and directly proportionate to the length of time since one's last leman. Speaking of which\u2014\n\nHe happened to catch Amicia's eye just then. He wasn't looking at her\u2014he was very consciously not looking at her, not wanting to appear weak, smitten, foolish, domineering, vain...\n\nHe had a long list of things he was trying not to appear.\n\nHer sharp glance said, _Don't be so rude\u2014Kneel,_ so clearly he almost felt he had heard the words said aloud.\n\nHe knelt. She had a point\u2014good manners had more value than pious mouthing. If that was her point. If she had, indeed, even looked at him.\n\nMichael stirred next to him, risked a glance at him. The captain could see that his squire was smiling.\n\nBeyond him, Ser Milus was trying to hide a smile as well.\n\n_They want me to believe. Because my disbelief threatens their belief, and they need solace._\n\nThe service rolled on as the sun flared its last, nearly horizontal beams throwing brilliant coloured light from the stained glass across the white linen shrouds of the dead.\n\n_Dies ir\u00e6! dies illa_\n\n_Solvet s\u00e6clum in favilla:_\n\n_Teste David cum Sibylla!_\n\nThe coloured light grew\u2014and every soldier gasped as the blaze of glory swept across the bodies.\n\n_Tuba, mirum spargens sonum_\n\n_Per sepulchra regionum,_\n\n_Coget omnes ante thronum._\n\n_It's a trick of the light, you superstitious fools!_ He wanted to shout it aloud. And at the same time, he felt the awe just as they did\u2014felt the increase in his pulse. _They hold the service at this hour to take advantage of the sun and those windows_ , he thought. _Although it would be very difficult to time the whole service to arrange it_ , he admitted to himself. _And the sun cannot be at the right angle very often._\n\nEven the priest had stumbled in the service.\n\nMichael was weeping. He was scarcely alone. Sauce was weeping and so was Bad Tom. He was saying \"Deo gratias\" over and over again through his tears, his rough voice a counterpoint to Sauce's.\n\nWhen it was all done, the knights of the company bore the corpses on litters made from spears, out of the chapel and back down the hill to bury them in the consecrated ground by the shrine at the bridge.\n\nSer Milus came and put his hand on the captain's shoulder\u2014a rare familiarity\u2014and nodded. His eyes were red.\n\n\"I know that cost you,\" he said. \"Thanks.\"\n\nJehannes grunted. Nodded. Wiped his eyes on the back of his heavy firze cote sleeve. Spat. Finally met his eye too. \"Thanks,\" he said.\n\nThe captain just shook his head. \"We still have to bury them,\" he said. \"They remain dead.\"\n\nThe procession left by the chapel's main door, led by the priest, but the Abbess was the focal point, now in severe and expensive black with a glittering crucifix of black onyx and white gold. She nodded to him and he gave her a courtly bow in return. The perfection of the Abbess's black habit with its eight-pointed cross contrasted with the brown-black of the priest's voluminous cassock over his cadaver-thin body. And the captain could smell the tang of the man's sweat as he passed. He was none too clean, and his smell was spectacular when compared with the women.\n\nThe nuns came out behind their Abbess. Virtually all of the cloister had come to the service, and there were more than sixty nuns, uniform in their slate grey habits with the eight pointed cross of their order. Behind them came the novices\u2014another sixty women in paler grey, some of a more worldly cut, showing their figures, and others less.\n\nThey wore grey and it was twilight, but the captain had no difficulty picking out Amicia. He turned his head away in time to see an archer known as Low Sym make a gesture and give a whistle.\n\nThe captain suddenly felt his sense of the world restored. He smiled.\n\n\"Take that man's name,\" he said to Jehannes... \"Ten lashes, disrespect.\"\n\n\"Aye, milord.\" Marshal Jehannes had his hand on the man's collar before the captain had taken another breath. Low Sym\u2014nineteen, and no woman's friend\u2014didn't even thrash. He knew a fair cop when he felt one.\n\n\"Which I was\u2014\" he began, and saw the captain's face. \"Aye, Captain.\"\n\nBut the captain's eyes rested on Amicia. And his thoughts went elsewhere.\n\nThe night passed in relaxation, and to soldiers, relaxation meant wine.\n\nAmy's Hob was still abed, and Daud the Red was fletching new arrows for the company and admitted to being \"poorly,\" company slang for a hangover so bad it threatened combat effectiveness. Such a hangover would be punishable most days\u2014the day after they buried seven men was not one of them.\n\nThe camp had its own portable tavern run by the Grand Sutler, a merchant who paid the company a hefty fee to ride along with his wagons and skim their profits when they had some to share. He, in turn, bought wine and ale from the fortress's stores, and from the town at the foot of Lissen Carak\u2014four streets of neat stone cottages and shop fronts nestled inside the lower walls and called \"The Lower Town.\" But the Lower Town was open to the company as well, and its tavern, hereabouts known as the Sunne in Splendour, was serving both in its great common room and out in the yard. The inn was doing a brisk business, selling a year's worth of ale in a few hours. Craftsmen were locking up their children.\n\nThat was not the captain's problem. The captain's problem was that Gelfred was planning to venture back to the tree line alone, while the captain had no intention of risking his most valuable asset without protection. And no protection was available.\n\nGelfred stood in the light rain outside his tent, swathed in a three-quarter length cloak, thigh high boots and a heavy wool cap. He tapped his stick impatiently against his boot.\n\n\"If this rain keeps up, we'll never find the thing again,\" he said.\n\n\"Give me a quarter hour to find us some guards,\" the captain snapped.\n\n\"A quarter hour we may not have,\" Gelfred said.\n\nThe captain wandered through the camp, unarmoured and already feeling ill-at-ease with his decision to dress for comfort. But he, too, had drunk too much and too late the night before. His head hurt, and when he looked into the eyes of his soldiers, he knew that he was in better shape than most. Most were still drinking.\n\nHe'd paid them. It improved his popularity and his authority, but it gave them the wherewithal to be drunk.\n\nSo they were.\n\nJehannes was sitting in the door of his pavilion.\n\n\"Hung over?\" the captain asked.\n\nJehannes shook his head. \"Still drunk,\" he answered. Raised a horn cup. \"Want some?\"\n\nThe captain mimed a shudder. \"No. I need four sober soldiers\u2014-preferably men-at-arms.\"\n\nJehannes shook his head again.\n\nThe captain felt the warmth go from his heart to his cheeks. \"If they are drunk on guard, I'll have their heads,\" he growled.\n\nJehannes stood up. \"Best you don't go check, then.\"\n\nThe captain met his eye. \"Really? That bad?\" he said it mildly enough, but his anger came through.\n\n\"You don't want them to think you don't give a shit, do you, _Captain_?\" Jehannes had no trouble holding his eye, although the marshal's were bloodshot and red. \"This is not the moment to play at discipline, eh?\"\n\nThe captain sat on an offered stool. \"If something comes out of the Wild right now, we're all dead.\"\n\nJehannes shrugged. \"So?\" he asked.\n\n\"We're better than this,\" the captain said.\n\n\"Like fuck we are,\" Jehannes said, and took a deep drink. \"What are you playing at? _Ser?_ \" He laughed grimly. \"You've taken a company of broken men and made something of them\u2014and now you want them to act like the Legion of Angels?\"\n\nThe captain sighed. \"I'd settle for the Infernal Legion. I'm not particular.\" He got to his feet. \"But I will have discipline.\"\n\nJehannes made a rude noise. \"Have some discipline tomorrow,\" he said. \"Don't ask for it today. Show some humanity, lad. Let them be sad. Let them fucking _mourn._ \"\n\n\"We mourned yesterday. We went to church, for god's sake. Murderers and rapists, crying for Jesus. If I hadn't seen it, I'd have laughed to hear about it.\" Just for a moment, the captain looked very young indeed\u2014and confused, annoyed. \"We're in a battle. We can't take a break to mourn.\"\n\nJehannes drank more wine. \"Can you fight every day?\" he asked.\n\nThe captain considered. \"Yes,\" he said.\n\n\"You ought to be locked up, then. We can't. Give it a rest, _Captain._ \"\n\nThe captain got to his feet. \"You are now the constable. I'll need another marshal to replace Hugo. Shall I promote Milus?\"\n\nJehannes narrowed his eyes. \"Ask me that tomorrow,\" he said. \"If you ask me again today, I swear by Saint Maurice I'll beat you to a fucking pulp. Is that clear enough?\"\n\nThe captain turned on his heel and walked away, before he did something he'd regret. He went to Jacques\u2014as he always did, when he'd reached bottom.\n\nBut his old valet\u2014the last of his family retainers\u2014was drunk. Even the boy Toby was curled up on the floor of the captain's pavilion, with a piece of rug pulled over him and a leg of chicken in one hand.\n\nHe looked at them for a long time, thought about having a temper-tantrum, and decided that no one was sober enough to bother. He tried to arm himself and found that he couldn't get beyond his chain hauberk. He put a padded cote over it, and took his gauntlets.\n\nGelfred had the horses.\n\nAnd that is how the captain came to be riding with his huntsman, alone, on the road that ran along the river, sore back, pulled neck muscles, and all.\n\n# North of Harndon\u2014Ranald Lachlan\n\nRanald was up with the dawn. The old man was gone, but had left him a deer's liver fried in winter onions\u2014a veritable feast. He said a prayer for the old man and another when he found the man had thrown a blanket over his riding horse. He cleared the camp and had mounted up before the sun was above the eastern mountains.\n\nIt was a ride he'd done with the king a hundred times. Following the highway north along the Albin, except that where the great river wound like an endless snake, the highway ran as straight as the terrain would allow, deviating only for big hills and rich manors, and crossing the Albin seven times via the seven great stone bridges between Harndon and Albinkirk. Lorica was the first bridge. Cheylas was the second\u2014a pretty town with red-tile roofs and round chimney-stacks and fine brick houses. He ate a big meal at the sign of the Irk's Head, and was out the door before the ale could tempt him to stay the night. He changed to his big hackney and rode north again, crossing Cheylas Bridge while the sun was high in the sky and making for Third Bridge as fast as his horse would go.\n\nHe crossed Third Bridge as darkness was falling. The Bridge Keeper didn't take guests\u2014a matter of law\u2014but directed him politely to a manor farm on the west bank. \"Less than a league,\" the retired soldier said.\n\nRanald was pleased to find the man's directions were spot on, because the night was dark and cold, for spring. In the North, the Aurora played in the sky, and there was a feel to it that Ranald didn't like.\n\nBampton Manor was rich beyond a hillman's ideas of rich\u2014but Ranald was used to how rich the southland was. They gave him a bed and a slice of game pie, and a cup of good red wine, and in the morning, the gentleman who owned the farms smiled at his offer of repayment.\n\n\"You are a King's Guardsman?\" the young man asked. \"I am\u2014I would like to be a man-at-arms. I have my own harness.\" He blushed.\n\nRanald didn't laugh. \"You'd like to serve the king?\" he asked.\n\nThe young man nodded. \"Hawthor Veney,\" he said holding out his hand.\n\nHis housekeeper bustled up with a bag. \"Which I packed you a lunch,\" she said. \"Good for a ploughman, good for a knight, I says.\"\n\nRanald bowed to her. \"Your servant, ma'am. I'm no knight\u2014just a servant of the king, going home to see my family.\"\n\n\"Hillman?\" she said, and sniffed. It was a good sniff\u2014it suggested that hillmen themselves were not always good people, but that she'd already decided in his favour on the matter.\n\nHe bowed again. To young Hawthor, he said, \"Do you _practise_ arms, messire?\"\n\nHawthor beamed, and the older housekeeper cackled. \"It's all he does. Doesn't plough, doesn't reap, won't even attend the haying. Doesn't chase servant girls, doesn't drink.\" She shook her head.\n\n\"Goodwife Evans!\" Hawthor said with the annoyance of a master for an unservile servant.\n\nShe sniffed again\u2014another sniff entirely.\n\nRanald nodded. \"Would you care to measure your sword against mine, young ser?\"\n\nIn a matter of minutes they were armed and padded in jupons and gauntlets and helmets, standing in the farmhouse yard with a dozen labourers for an audience.\n\nRanald liked to fight with an axe, but service in the King's Guard required knowledge of the courtly sword. Four feet of steel. The boy\u2014Ranald didn't think of himself as old, but found that Hawthor made him feel old with every comment he made\u2014had a pair of training weapons, not too well balanced, probably made by a local man, a little heavy. But they were perfectly serviceable.\n\nRanald waited patiently in a garde. Mostly, he was interested in seeing how the boy came at him\u2014a man's character was visible in his swordsmanship.\n\nThe boy stood his ground. He put his sword on his shoulder, and came forward in a position that fencing masters called \"The Garde of the Woman.\" His stance was too open and he didn't seem to understand that he needed to cock the sword back as far as he could. _The sort of little error you would spot up when arms are your profession,_ thought Ranald. But he liked how patient the boy was.\n\nThe boy closed with assurance, and launched his attack without a false preamble\u2014no bobbing or weaving or wasting effort.\n\nRanald cut into the boy's attack and knocked his blade to the ground.\n\nThe boy didn't wait for the whole move, but back-stepped.\n\nRanald's sword licked out and caught him in the side of the head despite his retreat.\n\n\"Oh!\" Hawthor said. \"Well struck.\"\n\nThe rest was much the same. Hawthor was a competent lad, for a young man without a master-at-arms to teach him. He knew lots of wrestling and very few subtleties, but he was bold and careful, a superb combination for a man so young.\n\nRanald paused to get out of his heavy jupon and to write the boy a note. \"Take this to Lord Glendower with my compliments. You may be asked to serve a year with the pages. Where are your parents?\"\n\nHawthor shrugged. \"Dead, messire.\"\n\n\"Well, if the goodwife can spare you,\" he said. And he was still smiling as he headed for the Fourth Bridge, at Kingstown.\n\n# North of Harndon\u2014Harold Redmede\n\nHarold Redmede looked down at the sleeping hillman with a smile. He packed his gear silently, left the hillman the better part of a venison liver, picked up his brother's gear as well, and humped it all to the stream.\n\nHe found his brother asleep under a hollow log with his threadbare cloak all about him. Sat and whittled, listening to the Wild, until his brother woke on his own.\n\n\"He was harmless,\" Harold said.\n\n\"He was a king's man, and thus a threat to every free man,\" said Bill.\n\nHarold shrugged. \"I've been a king's man,\" he said. It was an old argument, and not one likely to be resolved. \"Here, have some venison and the cider I saved for you. I brought you fish hooks, twenty good heads for arrows and sixty shafts. Don't shoot any of my friends.\"\n\n\"An aristo is an aristo,\" Bill said.\n\nHarold shook his head. \"Bollocks to you, Bill Redmede,\" he said. \"There's right bastards in the nobles and right bastards in the commons, too.\"\n\n\"Difference is that a right bastard commoner, you can break his head with your staff.\" Bill took a piece of his brother's bread as it was sliced off with a sharp knife.\n\n\"Cheese?\" Harold asked.\n\n\"Only cheese I'll see this year.\" Bill sat back against a tree trunk. \"I've a mind to go put a knife in your guest.\"\n\nHarold shook his head. \"No you won't. First, I drank with him, and that's that. Second, he's wearing mail and sleeping with a dirk in his fist, and I don't think you're going to off a hillman in his sleep, brother o' mine.\"\n\n\"Fair enough. Sometimes I have to remember that we must be fair in our actions, while the enemy is foul.\"\n\n\"I could still find you a place here,\" Harold said.\n\nBill shook his head. \"I know you mean well, brother. But I am what I am. I'm a Jack. I'm down here recruiting new blood. It's going to be a big year for us.\" He winked. \"I'll say no more. But the day is coming.\"\n\n\"You and your day,\" Harold muttered. \"Listen, _William._ You think I don't know you have five young boys hid in the beeches north of here? I even know whose boys they are. Recruits? They're fifteen or sixteen winters! And you have an _irk_ for a guide.\"\n\nBill shrugged. \"Needs must when the de'il drives,\" he said.\n\nHarold sat back. \"I know irks is folks,\" he said, waving a hand. \"I've met 'em in the woods. Listened to 'em play their harps. Traded to 'em.\" He leaned forward. \"But I'm a forester. They kill other folks. Bill. If you're on their side you're with the Wild, not with men.\"\n\n\"If the Wild makes me free, mayhap I'm with the Wild.\" Bill ate more bread. \"We have allies again, Harold. Come with me. We can change the world.\" He grimaced to himself. \"I'd love to have a good man at my back, brother. We've some right hard cases, I'll admit to you.\" He leaned forward. \"One's a priest, and he's the worst of the lot. You think I'm hard?\"\n\nHarold laughed. \"I'm too fucking old, brother. I'm fifteen winters older than you. And if it comes to that\u2014\" He shrugged. \"I'll be with my lord.\"\n\nBill shook his head. \"How can you be so blind? They oppress us! They take our land, take our animals, grind us\u2014\"\n\n\"Save it for the boys, Bill. I have six foot of yew and a true shaft for any as tried to grind me. But that won't make me betray my lord. Who, I may add, fed this village himself when other villages starved.\"\n\n\"Farmers are often good to their cattle, aye,\" Bill said.\n\nThey looked at each other. And then both grinned at the same time.\n\n\"That's it for this year, then?\" Harold asked.\n\nBill laughed. \"That's it. Here, give me your hand. I'm off with my little boys for the greenwood and the Wild. Mayhap you'll hear of us.\" He got up, and his long cloak shone for a moment, a dirty white.\n\nHarold embraced him. \"I saw bear prints by the river; a big female and a cub.\" He shrugged. \"Rare down here. Watch out for her.\"\n\nBill looked thoughtful.\n\n\"Stay safe, you fool,\" he said, and swatted him. \"Don't end up eaten by irks and bears.\"\n\n\"Next year,\" Bill said, and was gone.\n\n# Lissen Carak\u2014The Red Knight\n\nGelfred led them west along the river for miles, on a road that became increasingly narrow and ill-defined, until they had passed the point where they fought the wyvern and the road disappeared entirely. There were no longer any fields; the last peasant's cot was miles behind them, and the captain could not even smell smoke on the cool spring breeze, which instead carried an icy hint of old snow. The Abbess had not been exaggerating. Man had lost this land to the Wild.\n\nFrom time to time Gelfred dismounted in patches of sunlight and drew his short, silver-tipped wand from his belt. He would take his rosary from his belt and say his beads, one prayer at a time, eyes flicking nervously to his captain, who sat impassively on his horse. Each time, he would lay the shrivelled, thorned stick of Witch Bane on the ground at his feet, and each time it pointed, straining like a dog on a leash.\n\nEach time, they rode on.\n\n\"You use the power of grammerie to track the beasts?\" the captain said, breaking the frosty silence. They were riding single file along a well-defined track, the old leaves deeply trodden. It was easy enough to follow, but the road was gone. And by almost any measure, they were in the Wild.\n\n\"With God's aid,\" Gelfred said, and looked at him, waiting for the retort. \"But my grammerie found us the wrong beast. So now I'm looking for the man. Or men.\"\n\nThe captain made a face, but refused to rise to the comment about God. \"Do you sense their power directly?\" he asked. \"Or are you following the same spore a dog would follow?\"\n\nGelfred gave his captain a long look. \"I'd like your permission to buy some dogs,\" he said. \"Good dogs. Alhaunts and bloodhounds and a courser or two. I'm your Master of the Hunt. If that is true then I would like to have money, dogs, and some servants who are not scouts and soldiers.\" He spoke quietly, and his eyes didn't rest on the captain. They were always roaming the Wild.\n\nSo were the captain's.\n\n\"How much are we talking about?\" the captain asked. \"I love dogs. Let's have dogs!\" He smiled. \"I'd like a falcon.\"\n\nGelfred's head snapped around, and his horse gave a start. \"You would?\"\n\nThe captain laughed aloud. It was a sound of genuine amusement, and it rang like a trumpet through the woods.\n\n\"You think you are fighting for Satan, don't you, Gelfred?\" He shook his head.\n\nBut when he turned to look at his huntsman, the man was down off his horse, pointing off into the woods.\n\n\"Holy Saint Eustace! All praise for this sign!\" he said.\n\nThe captain peered off through the bare branches and caught the flash of white. He turned his horse\u2014no easy feat on the narrow track between old trees\u2014and he gasped.\n\nThe old stag was not as white as snow\u2014that much was obvious, because he had a patch of snow at his feet. He was the colour of good wool, a warm white, and there were signs of a long winter on his hide\u2014but he was white, and his rack of antlers made him a hart; a noble beast of sixteen tines, almost as tall at the shoulder as a horse. Old and noble and, to Gelfred, a sign from God.\n\nThe stag eyed them warily.\n\nTo the captain he was, palpably, a creature of the Wild. His noble head was redolent with power\u2014thick ropes of power that seemed, in the unreal realm of phantasm, to bind the great animal to the ground, the trees, the world in a spider web of power.\n\nThe captain blinked.\n\nThe animal turned and walked away, his hooves ringing on the frozen ground. He turned and looked back, pawed the old snow, and then sprang over a downed evergreen and was gone.\n\nGelfred was on his knees.\n\nThe captain rode carefully through the trees, watching the branches overhead and the ground, trying to summon his ability to see in the phantasm and struggling with it as he always did when his heart was beating fast.\n\nIt had left tracks. The captain found that reassuring. He found the spot where the beast had stood, and he followed the prints to the place where it had turned and pawed the snow.\n\nHis riding horse shied, and the captain patted her neck and crooned at her. \"You don't like that beast, do you, my honey?\" he said.\n\nGelfred came up, leading his horse. \"What did you see?\" he asked. He sounded almost angry.\n\n\"A white hart. With a cross on his head. I saw what you saw.\" The captain shrugged.\n\nGelfred shook his head. \"But why did _you_ see it?\n\nThe captain laughed. \"Ah, Gelfred\u2014are you so very holy? Shall I pass word of your vow of chastity on to the maids of Lonny? I seem to remember one young lass with black hair\u2014\"\n\n\"Why must you mock holy things?\" Gelfred asked.\n\n\"I'm mocking you. Not holy things.\" He pointed a gloved hand at the place where the stag had pawed the snow. \"Run your wand over that.\"\n\nGelfred looked up at him. \"I beg your pardon. I am a sinful man. I should not give myself airs. Perhaps my sins are so black that there is nothing between us.\"\n\nThe captain's trumpet laugh rang out again. \"Perhaps I'm not nearly as bad as you think, Gelfred. Personally, I don't think God gives a fuck either way\u2014but I do sometimes wonder if She has a wicked sense of humour and I should lighten up.\"\n\nGelfred writhed.\n\nThe captain shook his head. \"Gelfred, I'm still mocking you. I have problems with God. But you are a good man doing his best and I apologise for my needling. Now\u2014be a good fellow and pass your wand over the snow.\"\n\nGelfred knelt in the snow.\n\nThe captain winced at how cold his knees must be, even through his thigh high boots.\n\nGelfred spoke four prayers aloud\u2014three Pater Nosters and an Ave Maria. Then he put his beads back in his belt. He raised his face to the captain. \"I accept your apology,\" he said. He took the wand from his belt, raised it, and it snapped upright as if it had been struck by a sword.\n\nGelfred dug with his gloved hands. He didn't need to dig far.\n\nThere was a man's corpse. He had died slowly, from an arrow in the thigh that had severed an artery\u2014that much they could reasoned from the blood that soaked his braes and hose into a frozen scarlet mass.\n\nAll of his garments were undyed wool, off white, well made. He wore a quiver that was full of good arrows with hardened steel heads\u2014the captain drew them one by one and tested the heads against his vambrace.\n\nGelfred shook his head. The arrows alone were worth a small fortune.\n\nThe dead man's belt pouch had a hundred leopards or more in gold and silver, a fine dagger with a bronze and bone hilt and a set of eating tools set into the scabbard, and his hood and cloak were matching undyed wool.\n\nGelfred opened his cloak and took out a chain with an enamelled leaf.\n\n\"Good Christ,\" he said, and sat back.\n\nThe captain was searching the snow using his sword as a rake, combing up old branches under the scant snow cover.\n\nHe found the bow after a minute. If was a fine war bow, heavy, sleek, and powerful\u2014not yet ruined by the exposure to the snow.\n\nGelfred found the arrow that had killed him after assiduous casting, using his power profligately, casting it wider and wider. He had the body, had the blood, had the quiver. The connections were strong enough that it was only a matter of time, unless the arrow was a very long way away.\n\nIn fact, the arrow was near the road where they had left it, almost on their trail, buried in six inches of snow. Blood was still frozen to the ground where the arrow had been torn from the wound.\n\nThe arrow was virtually identical to the fifteen in the quiver.\n\n\"Mmm,\" said the captain.\n\nThey took turns watching the woods while the other stripped the corpse of clothes, chain, boots, belt, knife\u2014of everything.\n\n\"Why didn't something eat him?\" Gelfred asked.\n\n\"Enough power here to frighten any animal,\" the captain said. \"Why didn't the man who killed him strip his corpse and take the arrows? And the knife?\" He shook his head. \"I confess, Gelfred\u2014this is\u2014\" he snorted.\n\nGelfred didn't raise his eyes. \"There's plenty of folk live in the Wild.\"\n\n\"I know _that_.\" The captain raised an eyebrow. \"I'm from the north, Gelfred. I used to see Outwallers every day, across the river. There's whole villages of them.\" He shook his head. \"We raided them, sometimes. And other times, we traded with them.\"\n\nGelfred shrugged. \"He isn't an Outwaller.\" He looked at the captain as if he expected trouble. \"He's one of those men and women who want to bring down the lords. They say we'll\u2014that is, that they'll be free.\" His voice was detached and curiously non-committal.\n\nThe captain made a face. \"He's a Jack, isn't he? The bow? The leaf brooch? I've heard the songs.\" He shook his head at his huntsman. \"I know there's folk who want to burn the castles. If I were born a serf, I'd be out there with my pitchfork, right now. But Jacks? Men dedicated to fighting for the Wild? Who would fund them? How do they recruit? It makes no sense.\" He shrugged. \"To be honest, I'd always assumed the Jacks were made up by the lords to justify their own atrocities. Shows what a little youthful cynicism will get you.\"\n\nGelfred shrugged. \"There are always rumours.\" His eyes slipped away from the captain's.\n\n\"You're not some sort of secret rebel, Gelfred?\" The captain forced the other man to meet his eyes.\n\nGelfred shrugged. \"Does it brand me a traitor to say that sometimes the whole sick wheel of the world makes me want to kill?\" He dropped his eyes, and the anger went out of him. \"I don't. But I understand the outlaws and the outwallers.\"\n\nThe captain smiled. \"There. At last, you and I have something in common.\" He rolled the frozen corpse and used the dead man's sharp knife to slit his hose up the back. He cut the waistband of the man's linen braes, stiff with frozen blood, and took them as well. He got a sack from his heavy leather male that sat behind his saddle, and filled it with the dead man's belongings.\n\nHe tossed the purse to Gelfred. \"Get us some dogs,\" he said.\n\nNaked, the dead man didn't look like a soldier in the army of evil. The thought made the captain purse his lips. He leaned over the corpse\u2014as white as the snow around it\u2014and rolled it over again.\n\nThe death wound went in under the arm, straight to the heart, and had been delivered with a slim bladed knife. The captain took his time, looking at it.\n\n\"His killer came and finished him. And was so panicked, they didn't know their man was already dead.\"\n\n\"Already dead?\" Gelfred asked.\n\n\"Not much blood. Look at his cote. There's the entry\u2014there's the blood. But not much.\" The captain crouched on his heels. \"This is a puzzle. What do you see, Gelfred?\"\n\n\"His kit is better than ours,\" Gelfred said.\n\n\"Satan pays well,\" the captain shrugged. \"Or perhaps he merely pays on time.\" He looked around. \"This is not what we came for. Let's go back to the trail and look for the monster.\" He paused. \"Gelfred, how can you conjure with Witch's Bane?\"\n\nGelfred walked a few paces. \"I've heard it can't be done,\" he said with a shrug. \"But it can. It's like mucking out a stall\u2014you just try not to get the shit on you.\"\n\nThe captain looked at his huntsman with a whole new appreciation. Sparring about religion had defined their relationship in the weeks since the captain had engaged him.\n\n\"You are potent,\" the captain said.\n\nGelfred shook his head eyes on the trees. \"I feel that we've disturbed a balance,\" he said, ignoring the compliment.\n\nThe captain led his horse to a downed tree. He could vault into the saddle, but he felt sore in every limb, and his neck hurt where the wyvern had tried to snap it, and he was still more than a little hung over, and he used the downed tree to mount.\n\n\"All the more reason to keep moving,\" he said. \"We're not in the Jack-hunting business, Gelfred. We kill monsters.\"\n\nGelfred shrugged. \"My lord\u2014\" he began. He looked away. \"You have power of your own. Yes?\"\n\nThe captain felt a little frisson run down his back. _Run? Hide? Lie?_\n\n\"Yes,\" he said. \"A little.\"\n\n\"Hmm,\" Gelfred said, noncommittally. \"So. Now that I have eliminated the... the Jack from my casting, I can concentrate on the other creature.\" He paused. \"They were bound together. At least,\" he looked scared. \"At least, that's how it seemed to me.\"\n\nThe captain looked at his huntsman. \"Why do you think someone killed the Jack, Gelfred?\"\n\nGelfred shook his head.\n\n\"A Jack helps a monster kill a nun. Then, another man kills him.\" The captain shivered. The chainmail under his arming cote did a wonderful job of conducting the cold straight to his chest.\n\nGelfred didn't meet his eye.\n\n\"Not money. Not weapons.\" The captain began to look around. \"I think we're being watched.\"\n\nGelfred nodded.\n\n\"How long had the Jack been dead?\" the captain asked.\n\n\"Two days.\" Gelfred was sure, as only the righteous can be sure.\n\nThe captain stroked his beard. \"Makes no sense,\" he said.\n\nThey rode back to the track, and Gelfred hesitated before facing west. And then they began to ride.\n\n\"The stag was a sign from God,\" Gelfred said. \"And that means the Jacks are but tools of Satan.\"\n\nThe captain looked at his huntsman with the kind of look fathers usually have for young children.\n\nWhich, the captain thought, was odd, since Gelfred was ten years his senior.\n\n\"The stag was a creature of the Wild, every bit as much as the wyvern, and it chose to manifest itself as it did because it opposes whomever aids the Jacks.\" The captain shrugged. \"Or so I suspect.\" He met his huntsman's eye. \"We need to ask ourselves why a creature of the Wild helped us find the body.\"\n\n\"So you are an Atheist!\" Gelfred asked. Or rather, accused.\n\nThe captain was watching the woods. \"Not at all, Gelfred. Not at all.\"\n\nThe trail narrowed abruptly, killing their conversation. Gelfred took the lead. He looked back at the captain, as if encouraging him to go on, but the captain pointed over his shoulder and they rode on in silence.\n\nAfter a few minutes, Gelfred raised a hand, slipped from the saddle, and performed his ritual.\n\nThe stick in his hand snapped in two.\n\n\"Holy Saint Eustace,\" he said. \"Captain\u2014it is right here with us.\" His voice trembled.\n\nThe captain backed his horse a few steps to get clear of the huntsman's horse and then took a heavy spear from its bucket at his stirrup.\n\nGelfred had his crossbow to hand, and began to span it, his eyes wide.\n\nThe captain listened, and tried to see in the phantasm.\n\nHe couldn't see it, but he could feel it. And he knew, with sudden weariness, that it could feel him too.\n\nHe turned his horse slowly.\n\nThey were at the top of a bank\u2014the ground sloped sharply to the west, down to a swollen stream. He could see where the track crossed the stream.\n\nOn the eastern slope, towards the fortress, the ground fell away more slowly and then rose dramatically up the ridge they had just descended, and the captain realised that the ridge was littered in boulders\u2014rocks big enough to hide a wagon, some so large that trees grew from the top of them.\n\n\"I think I may have been rash,\" the captain began.\n\nHe heard the sharp click as Gelfred's string locked into the trigger mechanism on his bow.\n\nHe was looking at an enormous boulder the size of a wealthy farmer's house. Steam rose over it, like smoke from a cottage fire.\n\n\"It's right there.\" He didn't turn his head.\n\n\"Bless us, Holy Virgin, now and in the hour of our deaths. Amen.\" Gelfred crossed himself.\n\nThe captain took a deep breath and released it softly, fighting his nerves. The ground between them and the rock was tangled with scrubby spruce, downed trees, and snow. Miserable terrain for his horse to cross in a fight. And he wasn't on Grendel\u2014he was on a riding horse that had never seen combat.\n\nNot wearing armour.\n\n_I'm an idiot_ , he thought.\n\n\"Gelfred,\" he said, without turning his head. \"Is there more than one? What is downslope?\"\n\nGelfred's voice was calm, and the captain felt a spurt of affection for the huntsman. \"I believe there is another.\" Gelfred spat. \"This is my fault.\"\n\n\"Is this our killer?\" the captain asked. He was quite proud of his conversational tone. If he was going down, he would die like a gentleman. That pleased him.\n\nGelfred was also a brave man. \"The one upslope is the killer,\" he said. \"By the wounds of Christ, Captain\u2014what _are_ they?\"\n\n\"Stick close,\" the captain said. \"You're the huntsman, Gelfred. What are they?\"\n\nHe began to ride forward, down the trail to the west. He passed Gelfred, who came in so tight behind him that the captain could feel the warmth of his horse. Down the steep slope to the stream, and he could no longer see the boulders, but he could hear movement\u2014crashing movement.\n\nAcross the stream in a single leap of his horse. He could feel her terror.\n\nHe could feel his own.\n\nHe rode five yards, holding his mount down to a trot by sheer force of will and knee. She wanted to bolt. Ten yards. He heard Gelfred splash across the stream instead of leaping it and he turned his horse. She didn't want to turn.\n\nHe put his spur into her right side.\n\nShe turned.\n\nGelfred's eyes were as wide as his horse's.\n\n\"Behind me,\" the captain said.\n\nHe was facing their back trail. He backed his horse again, judging the distance.\n\n\"I'm dismounting,\" Gelfred said.\n\n\"Shut up.\" The captain fought for enough mental control to enter the room in his head. Closed his eyes\u2014forced them closed against the crashing sound from the top of the ridge to the east.\n\n_Prudentia?_\n\n_She stood in the centre of the room, her eyes wide, and he ran to her, took her outstretched hand and pointed it over his shoulder._\n\n_\"Katherine, Ares, Socrates!\" he called. He ran to the door, grasped the handle, and turned the key while the room spun around him._\n\n_The lock clicked open and the door crashed back against his leg, throwing him from his feet so that he fell heavily on the marble floor. The breeze was an icy green wind, and on the other side of the door_ \u2014\n\n_It was caught on his shoulder where he had fallen, and the gale was sliding him along the floor as it forced the door open._\n\n_He wondered what would happen if the door crashed back against its hinges._\n\n_He wondered whether he could die in the small, round room._\n\n_Had to assume he could._\n\n_I rule here! he said aloud. He put a knee under himself, as he would if he was wrestling with a big man. Used the key for leverage. Pushed the door with his shoulder._\n\n_For a long set of heartbeats, it was like pushing a cart in mud. And then he felt the shift\u2014minute\u2014but the tiny victory lit his power like a mountebank's flare and he slammed the door closed as his net of power wove itself like a giant spider's web across the stream._\n\nThe horse was fighting him, and the thing was halfway down the hill, coming straight down the track, its bulk breaking branches on either side of the trail while its taloned feet gouged clods of earth out of the ground.\n\nHis mind shied away from looking at its head.\n\nHe couched his lance, timing his charge.\n\nHorses are complex animals, delicate, fractious and sometimes very difficult. His fine riding horse was spirited and nervous on the best of days, and was now terrified, wanting only to flee.\n\nGelfred's crossbow loosed with a flat crack and the bolt caught the thing under its long snout and it shrieked. It slowed.\n\nThirty yards. The length of the tiltyard in his father's castle. Because this had to be just right.\n\nThe _adversarius_ \u2014the captain had never seen one, but had to assume that this was the fabled enemy of man\u2014lengthened its loping stride to leap the brook.\n\n_A daemon._\n\nThe captain rammed his spurs into his mount. Sometimes, horses are simple. His riding horse exploded forward.\n\nThe adversarius leaped again at the edge of the stream, its hooked beak already reaching for his face, arms spread wide.\n\nIt seemed to slow as it crossed the water\u2014vestigial wings a blur of angry motion, maned head with a helmet crest of bone curving above it, spraying spittle as the thing tried to snap at the fine web of Power he had cast over the near bank. It would only last a moment\u2014already the daemon was blowing through the mild restraint the way a big child, angry and frightened, tears through spider web.\n\nHe tracked the thing's right eye with his lance tip like it was an opponent's crest; the brass ring; the upper left corner of the shield on the quintain. Held in place like an insect pinned to a page, it tried to rear back just as his spear point glanced off the ocular ridge and plunged into the soft tissue of the eye, the strong steel of the long spear head breaking the bone above and below the eye socket, driving the point deeper and deeper, the whole weight of the man and horse behind it.\n\nHis lance shaft snapped.\n\nThe creature's legs spasmed and its talons tore into his horse's forequarters, raking flesh and tendon from bone, flaying the poor animal while it screamed. The captain flew back over its rump on force of the impact, with no brace against his back from a tilting saddle. The horse reared and the talons eviscerated it, its guts spilling onto the road in a great gout.\n\nThe daemon got its feet on the ground and its forearms shredded the last of his web of power\u2014\n\nIt turned from the ruin of his horse and he saw the damage he'd done, the angry orange of its remaining eye\u2014no slit, no pupil. Nothing but fire. It _saw_ him.\n\nThe terror of its presence pounded him like a hammer of spirit\u2014for a moment, the terror was so pure in him that he had no self. He was only fear.\n\nIt came for him then, rising up fast on its haunches\u2014and, like a puppet with the strings cut, it collapsed atop the corpse of his riding horse.\n\nHe gagged, clamped down on the vomit, failed, and heaved everything in his stomach down the front of his jupon. When he was done he was sobbing in the backwash of the terror.\n\nAs soon as he had any control over himself, he said \"Ware! There's another!\"\n\nGelfred approached him slowly, holding a cup in one hand and a cocked and loaded crossbow balanced carefully on his arms.\n\n\"It's been a long time.\" He shook his head. \"I've prayed the whole rosary, waiting for you to recover.\" He was shaking. \"I don't think the other one is coming.\"\n\nThe captain spat out the taste of vomit. \"Good\" he said. He wanted to say something witty. Nothing came. \"Good.\" He took the cup. \"How\u2014long did I kneel here?\"\n\n\"Too long,\" Gelfred said. \"We need to ride.\"\n\nThe captain's hands shook so hard he spilled the wine.\n\nGelfred put his arms around him.\n\nThe captain stood in that unwanted embrace and shook. Then he washed in the creek. He felt violated. And different. He was suddenly afraid of everything. He didn't feel at all like a man who had faced a daemon, the greatest adversary to the rule of man, in single combat, and the adoration in Gelfred's eyes made him sick.\n\n_Tomorrow, I will no doubt be insufferable_ , he thought.\n\nGelfred cut the head from the daemon.\n\nHe threw up again, a stream of bile, and wondered if he could ever face a creature of the Wild again. His bones felt like jelly. There was something in the pit of his gut\u2014something that had gone.\n\nHe knew exactly what this felt like: like being beaten by his brothers. Beaten and humiliated. He knew that feeling well. They'd been younger than him. They'd hated him. He'd made their lives a misery, when he learned that\u2014\n\nHe spat.\n\nSome things are best left unexamined. He held the line at that memory, and felt his fear recede a little, like the first sign the tide is in ebb.\n\nIt would pass, then.\n\nGelfred couldn't get the horse to bear the head. The captain didn't have enough concentration to conjure anything to help them with it. So they tied a rope to the head and dragged it.\n\nIt would be a long walk back to camp. And after an hour, something behind them began to howl, and the captain felt the hairs on the back of his neck rise.\n\n# Lissen Carak\u2014Mogan\n\nMogan watched her cousin's killer as he mounted slowly and rode up the road.\n\nMogan was a hunter, not a berserker. Her cousin's death terrified her, and until she had understood it she was not going down to face the men on the road. Instead, she edged cautiously from rock to rock, keeping well out of their line of sight, and she watched them with her superb eyes, made for spotting the movement of prey a mile across the plains to the west.\n\nWhen they were well clear of the scene of the fight, she trotted down the ridge.\n\nTunxis lay in a pitiful heap, his once mighty frame hunched and flattened by death, and there were already birds on the corpse.\n\n_They cut his head off._\n\nIt was horrendous. Mogan threw back her head and howled her rage and sorrow.\n\nAfter her third howl, her brother came. He had four hunters with him, all armed with heavy war-axes or swords.\n\nThurkan looked at his nest-mate's corpse and shook his great head. \"Barbarians,\" he spat.\n\nMogan rubbed her shoulder against his. \"One man killed him. I chose not to try him. He killed our cousin so easily.\"\n\nThurkan nodded. \"Some of their warriors are terrifying, little sister. And you had no weapon to open his armour.\"\n\n\"He had no armour,\" Mogan said. \"But he had Power. Our Power.\"\n\nThurkan paused, sniffing the air. Then he walked to the edge of the stream, and back, several times, while his nestmates stood perfectly still.\n\n\"Powerful,\" Thurkan said. He paused and licked his shoulder where a mosquito had penetrated his armoured flesh. Insects. How he hated them. He batted helplessly with a taloned forefoot at the cloud that was gathering around his head. Then he bent over his cousin's form, raised his talons, and turned his cousin's corpse to ash in a flash of emerald light.\n\nLater, as they ran through the forest, Thurkan mused to his sister. \"This is not as Thorn thinks it to be,\" he said.\n\nMogan raised her talons to indicate her complete lack of interest in Thorn. \"You seek to dominate him, and he seeks to dominate you, but as he is not of our kind your efforts are wasted,\" she said scathingly.\n\nThurkan took a hundred running steps before he answered. \"I don't think so, little sister. I think he is the rising power of the Wild, and we must cleave to him. For now. But in this matter he is blind. This fortress. The Rock. Here we are, masters of the woods from the mountains to the river\u2014and he would have us leave our winnings to assault this one place. And now the Rock has a defender\u2014one who is also a power of the Wild.\" Thurkan ran on. \"I think Thorn is making an error.\"\n\n\"You seek his throat, and his power,\" Mogan said. \"And it is we who wish to return to the Rock.\"\n\n\"Not if the cost is too high. I am not Tunxis.\" Thurkan leapt a log.\n\n\"How can the Rock have a defender who is one of us? And we not know him?\" Mogan asked.\n\n\"I don't know,\" Thurkan admitted. \"But I will find out.\"\n\n# Chapter Four\n\n# South of Lorica\u2014Ser Gawin\n\nGawin Murien of Strathnith, known to his peers as Hard Hands, rode north along the Albin River in his armour, a knight bent upon errantry. And the further north he rode, the deeper his anger grew.\n\nAdam, the elder of his two squires, whistled, bowed from his saddle to every passing woman, and looked at the world with whole-hearted approval. _He_ was not sorry to be leaving the court of Harndon. Far from it. Far from the great hall, far from the rounds of dancing and cards and hunts and flirtation, squires lived in barracks under the absolute domination of the oldest and toughest. Younger men got little food and much work, and no chance of glory. Adam was the squire of a named and belted knight, and on errantry, he expected to have a chance to win his own place in song. At Harndon, all he got was black eyes and bad food.\n\nToma, the younger squire, rode with his head down. Adam could make nothing of him, beyond his mumbled answers and his clumsy work. He seemed young for his age and deeper in misery than a boy should be.\n\nGawin wanted to do something for him, but he was having a hard time seeing through his own anger.\n\nIt wasn't _fair._\n\nThe words were meaningless. His oaf of a father had beaten any notion of fairness from him from birth. Gawin knew that the world gave you nothing but struggle. That you had to make your own luck. And a thousand more such aphorisms all with the same general message, but, by God and all the saints, Gawin had done his time, faced his monster and killed the literally damned thing in single combat with his gauntlets after his sword broke. He remembered it vividly, just as he remembered going to fight the damned thing out of sheer guilt.\n\n_I killed my brother._\n\nIt still made him sick.\n\nHe didn't want to have to face the foe again, not for all the pretty ladies in court and not for all the lands he stood to inherit. He was no coward. He'd _done it._ In front of his father and fifty other men. There probably weren't fifty knights in all Alba\u2014from one end of the Demesne to the other\u2014who had bested a daemon in single combat. He certainly hadn't wanted to.\n\nBut he had. And that should have been that.\n\nBut of course, the king hated him, as he hated all his brothers, hated his mother, loathed his father.\n\n_Fuck the king. I'll ride home to Pater._\n\nStrathnith was one of the greatest fortresses in the Demesne. It was a citadel of the Wall, and the Muriens had held it for generations. The Nith was a mighty river\u2014almost an inland sea\u2014that defined the ultimate border between the Demesne and the Wild. His father ruled the fortress and the thousands of men and women who paid their taxes and depended on it for protection. He thought about the great hall; the ancient rooms, some built by the Archaics. The sounds of the Wild carrying across the broad river.\n\nThe constant bickering, the drunken accusations. The family fighting.\n\n\"Good Christ, I might as well go find a cursed monster and kill it,\" he said aloud. Going home meant returning to a life of constant warfare\u2014in the field against the daemons, and in the hall against his father. And his brothers.\n\n_I killed my brother._\n\n\"They can have it,\" he said.\n\nHe'd been sent south, the young hero, to win a bride at court. To raise the family in the estimation of the king.\n\nAnother of his father's brilliant plans.\n\nHe had fallen in love, but not with a woman. Rather, he'd fallen in love with women. And the court. Music. Card games. Dice. Good wine and wit. Dancing.\n\nStrathnith wasn't going to offer _any_ of those things. He couldn't stop himself from thinking about it. In retrospect, maybe his loathsome brother had a point.\n\n_His mother_ \u2014\n\nHe banished the thought.\n\n\"Lorica, m'lord,\" Adam sang out. \"Shall I find us an inn?\"\n\nThe idea of an inn helped douse his moment of self-doubt. Inns\u2014good ones\u2014were like miniature courts. A little rougher, a little more home-spun. Gawin smiled.\n\n\"The best one,\" he said.\n\nAdam grinned, touched his spurs to his horse, and rode off into the setting sun. Drink. And maybe a girl. He thought fleetingly of Lady Mary, who so obviously loved him. A beautiful body, and, he had to admit, a fine wit. And the daughter of the Count. She was a fine catch.\n\nHe shrugged.\n\nThe sign of the Two Lions was an old inn built on the foundation of an Archaic cavalry barracks, and it looked like a fortress; it had its own curtain wall, separate from Lorica's town wall, and it had a tower in the north-east corner where any soldier could see the original gate had been. Built against the tower was a massive building of white plaster and heavy black beams, with a hipped thatch roof with expensive copper sheathing around the chimneys; glass windows opened onto the porches that ran all along the front and sunlit side, and four massive chimneys, all new masonry, rose out of the roof.\n\nIt was like a piece of the Palace of Harndon brought into the countryside. Lorica was an important town, and the Two Lions was an important inn.\n\nAdam appeared to hold his horse. \"A king's knight is very welcome here,\" he said through his grin. Adam liked to serve a great man\u2014it rubbed off. Especially forty leagues north of the city.\n\nA prosperous man, razor thin, wearing a fine woollen hood lined in silk and edged with silver crosses and a fur band, swept off the last and bowed to the ground. \"Edard Blodget, m'lord. At your service. I won't call my inn humble\u2014it's the best inn on the Highway. But I do like to see the king's knights.\"\n\nGawin was startled to see a commoner so well dressed and so frankly spoken\u2014startled, but not displeased. He returned the bow, all the way to the ground. \"Ser Gawin Murien,\" he said. \"Knighthood doesn't necessarily make a man rich, Master Blodget. May I enquire\u2014\"\n\nMaster Blodget gave a tight lipped smile. \"Your own room for a silver leopard. Share with your squires for two cats more.\" He raised an eyebrow. \"I can make it cheaper, m'lord, but it will be in a common room.\"\n\nGawin mentally reviewed his purse. He had a good memory, well-trained, and so he could almost literally see the contents\u2014four silver leopards and a dozen heavy copper cats. And gleaming among them, a pair of rose nobles, solid gold, worth twenty leopards apiece. Not a fortune, by any means, but enough that he needn't stint on his first night on the road, or on the second.\n\n\"Adam will take care of it, then. I would prefer we were all in a room. With a window, if that's not too much to ask?\"\n\n\"Clean linen, window, well-water, and stabling for three horses. The pack horse will cost another half a cat.\" Blodget shrugged, as if such petty amounts were beneath him, which they probably were. The Two Lions was at least a third of the size of the massive fortress of Strathnith, and was probably worth\u2014Gawin tried to do the mathematics in his head\u2014wished for his tutor\u2014and finally arrived at a figure that had to be recklessly wrong.\n\n\"I'm flattered you came to greet me in person,\" Gawin said with another bow.\n\nBlodget grinned from ear to ear.\n\n_Another thing I learned at court\u2014men like to be flattered just as much as ladies,_ Gawin thought.\n\n\"I have a group of singers tonight, m'lord\u2014on their way to court, or so they hope. Will you join us for dinner in the common room? It ain't a great hall\u2014but it's not bad. And we'd be honoured to have you sit with us.\"\n\n_Of course, I'm as fond of flattery as the next man._\n\n\"We will join you for dinner and music,\" he said with a slight bow.\n\n\"Evensong at Saint Eustachios. You'll hear the bell,\" the innkeeper said. \"Dinner follows the service directly.\"\n\n# Harndon City\u2014Edward\n\nMaster Pyle appeared in the yard after evensong and asked for a volunteer.\n\nEdward had a girl, but she worked, too. She would have to understand, because a chance to work with the master was every apprentice's dream.\n\nThe master mixed the powder differently this time. Edward didn't see how. But he moved a heavy iron pitch-bowl for repousse work into the yard, and cleared a lot of old rubbish\u2014bits of ruined projects and soft wood for making temporary moulds and hordles\u2014out of the way, in case they should catch fire. It wasn't skilled work, but he was still working for the master.\n\nThe smoke was thicker this time, and the flame burned whiter.\n\nMaster Pyle looked at it, fanning his face to get the evil smelling smoke to clear. He had a bit of a smile.\n\n\"Well,\" he said. He looked at Edward. \"Are you ready for your examination, young man?\"\n\nEdward took a deep breath. \"Yes,\" he said. And hoped that didn't sound too cocky.\n\nBut Master Pyle nodded. \"I agree.\" He looked around the yard. \"Clear all this up, will you?\"\n\nThat night in the loft, the apprentices whispered. The older boys knew when the master was making progress. They could tell just by the way he held his head. And because rewards suddenly emerged from the master's purse, and boys got new work, and apprentices were suddenly tested to be journeymen. Lise, the eldest female cutler, had gone to the masters the week before. She'd passed.\n\nAnd so Edward Chevins, senior apprentice and sometime shop boy, found himself up for journeyman. It was so sudden it made his head spin, and before the next morning was old enough to drink his beer, the Guild Hall had checked his papers, the Guild Masters examined him, his nerves were wracked, his hands shook\u2014and he was left to sweat, alone, in a richly decorated room fit to entertain a king. It was plenty to overawe a seventeen-year-old blade smith.\n\nEdward was a tall, gangly young man with sandy red hair and too many freckles. Standing under the stained glass of Saint Nicholas, he could think of twenty better answers he might have given to the question: \"How do you achieve a bright, constant blue on a blade with a heavy forte and a needle point?\"\n\nHe groaned. The other four boys who'd been tested with him looked at him with a mixture of sympathy and hope. It was too easy to believe that someone else's failure raised one's own hopes of success.\n\nAn hour later, the masters came into the hall. They all looked a little red in the face, as if they'd been drinking.\n\nMaster Pyle came and put a ring on his finger\u2014a ring of fine steel. \"You're made, boy,\" he said. \"Well done.\"\n\n# Lorica\u2014Ser Gawin\n\nGawin was awakened from his nap by the sound of men shouting in the courtyard. Angry voices have a timbre to them\u2014especially when men mean violence.\n\nAdam was at his bed. He had a heavy knife in his hand. \"I don't know who they are, m'lord. Men from overseas. Knights. But\u2014\" Squires didn't speak ill of knights. It was never a good idea. So Adam shrugged.\n\nGawin rolled off his bed, wearing only his braes. He pulled a shirt over his head, and with Toma's help got his legs into his hose and his torso laced into his pourpoint and his hose tied on.\n\nDown in the courtyard one voice sounded clear above the others. Accented, but powerful controlled, elegant. The words ended with a long, clear laugh that sounded like bells.\n\nGawin went to his window and threw it open.\n\nThere were a dozen armoured men in the courtyard. At least three were true knights, and wore armour as good as Gawin's own. Their men-at-arms were nearly as well armoured. It was possible they were _all_ knights.\n\nThey all wore the same badge\u2014a rose, gules, on a field d'or.\n\nNot anyone he knew.\n\nThe leader with the magnificent laugh had silver-gilt hair and fine features\u2014in armour, he looked like a statue of Saint George. He was _beautiful._\n\nGawin felt ill-dressed and somewhat doltish in comparison.\n\nMaster Blodget stood in front of this saint with his hands on his hips.\n\n\"But,\" the knight had a smile on his face, \"But that is the room I want, Master Innkeeper!\"\n\nBlodget shook his head. \"There's a gentleman in that room\u2014a belted king's knight, in that room. First come, first served, m'lord. Fair is fair.\"\n\nThe knight shook his head. \"Throw him out, then.\"\n\nToma had his master's doublet and helped him into it. While Adam did the laces, Toma fetched his riding sword.\n\n\"Follow me,\" Gawin snapped at the scared boy, and sprang down the stairs. He went through the common room\u2014empty, because every man in the inn was in the courtyard watching the fun.\n\nHe stepped through the door and the knight turned to look at him. He smiled.\n\n\"Perhaps I don't wish to leave my room,\" Gawin called. He hated that his voice wavered. There was nothing to fear, here\u2014just a misunderstanding, but the kind wherein a knight had to make a good show.\n\n\"You?\" he asked. His tone of disbelief wasn't mocking\u2014it was genuine. \"You are a king's knight? Ah\u2014Gaston, they need us here!\"\n\nCloser up, the men in the courtyard were huge. The smallest of them was a head taller than Gawin, and he was not a small man.\n\n\"I have that honour,\" Gawin said. He tried to find something wittier to say, but he was more interested in defusing the tension than in scoring points.\n\nThe one called Gaston laughed. The rest laughed too.\n\nThe beautiful knight leaned down from his saddle. \"Have your man clear your things from that corner room,\" he said. And then, in a particularly annoying tone, he added, \"I would esteem it a favour.\"\n\nGawin found that he was angry.\n\n\"No,\" he said.\n\n\"That was ill-said, and not courteous,\" the knight answered him with a frown. \"I shall have it. Why make this difficult? If you are a man of honour then you may cede it to me with a good grace, knowing I am a better man than you.\" He shrugged. \"Or fight me. That would be honour too.\" He nodded to himself. \"But to stand here and tell me I can't have it; that makes me angry.\"\n\nGawin spat. \"Then let us fight, ser knight. Give me your name and style, and I will name the weapons and the place. The king has announced a tournament in a two months, perhaps\u2014\" Even as he spoke, the man was dismounting.\n\nHe gave his reins to Gaston and turned, drawing his sword\u2014a four-foot long war sword. \"Then fight.\"\n\nGawin squeaked. He wasn't proud of the squeak, but he was unarmoured and had only his riding sword\u2014a good blade, but a single handed weapon whose only real purpose was to mark your status in life and keep riff-raff at arm's length.\n\n\"Garde!\" the man called.\n\nGawin reached out and drew his sword from the scabbard Toma held, and brought it up in a counter cut that just stopped his opponent's first heavy overhand blow. Gawin had time to bless his superb Master at Arms\u2014and then the giant cut at him again and he slipped to the side, allowing the heavier sword to slide off his own like rain off a roof.\n\nThe bigger man stepped in as quick as a cat and struck him in the face with one gauntleted hand, knocking him to the ground. Only a turn of his head saved him from spitting teeth. But he was a knight of the king\u2014he rolled with the impact, spat blood, and came to his feet with a hard cut at his opponent's groin.\n\nA single-handed sword has advantages in a fight with a heavier sword. It is quicker, even if the wielder is smaller.\n\nGawin funnelled his anger into his sword and cut\u2014three times, on three different lines, trying to awe the giant with a flurry of blows. The sword rang off the mirrored finish of his opponent's armoured wrist on the third cut. It was a fight ending blow.\n\nIf his opponent wasn't covered in steel.\n\nThe giant attacked, drove him back two steps, and then Toma screamed. The boy had been unprepared for a fight and stood frozen, but now tried to turn and run he'd became entangled with his master's defensive flurry. Gawin almost fell, and the bigger man's long sword licked out, caught his, and drove his thrust deep into Toma.\n\nHe kicked Gawin in the groin when he turned to look at Toma, whose head was cut nearly in half by the blade. Gawin fell, retching with the pain, and the big knight showed no mercy; knelt on his back, and pushed his nose into the mud in the courtyard. He stripped the sword from Gawin's hand.\n\n\"Yield,\" he said.\n\nNortherners were reputedly stubborn and vengeful. Gawin, in that moment, swore to kill this man, whomever he might be, if it cost him his life and his honour to do it.\n\n\"Fuck yourself,\" he said through the mud and blood in his mouth.\n\nThe man laughed. \"By the law of arms, you are my prisoner, and I will take you to your king to show him how very much he needs me.\"\n\n\"Coward!\" Gawin roared. Even as part of his mind suggested that slumping in pretended swoon might be the wiser course.\n\nA gauntleted hand rolled him over and pulled him up. \"Get your things out of my room,\" he said. \"I will pretend I did _not_ hear you say such a thing to me.\"\n\nGawin spat blood. \"If you think you can take me to the king and _not_ be bound for murder\u2014\"\n\nThe blond man sniffed. \"You killed your own squire,\" he said. He allowed himself just the slightest smile at the words and, for the first time, Gawin was afraid of him. \"And calling a man who has bested you in a test of arms 'coward' is poor manners.\"\n\nGawin wanted to speak like a hero, but rage, sorrow, fear, and pain spat his words out \"You killed Toma! You are no knight! Attacking an unarmoured man? With a war sword? In an _inn_?\"\n\nThe other man frowned. He leaned close.\n\n\"I should strip you and have you raped by the grooms. How dare you call me\u2014me!\u2014an unfit knight? Little man, I am Jean de Vrailly, I am the greatest knight in the world, and the only law I recognise is the law of Chivalry. Yield to me, or I will slay you where you stand.\"\n\nGawin looked into that beautiful face\u2014unmarred by anger, rage, or any other emotion\u2014and he wanted to spit in it. His father would have.\n\n_I want to live._\n\n\"I yield,\" he said, and hated himself.\n\n\"All these Alban knights are worthless,\" de Vrailly laughed. \"We will rule here.\"\n\nAnd then they all dismounted, leaving Gawin alone in the courtyard with the body of his squire. The boy was quite dead.\n\n_I killed him_ , Gawin thought. _Sweet Christ._\n\nBut it wasn't over yet, because Adam was a brave man, and he died alone in the doorway of their corner room.\n\nOne of the foreigners threw all his kit through the window after he heard his squire die. They laughed.\n\nGawin knelt on the stones by Toma and, after an hour, when the bells rang for evensong, the innkeeper came to him.\n\n\"I've sent for the sheriff and the lord,\" he said. \"I'm so sorry, m'lord.\"\n\nGawin couldn't think of anything to say.\n\n_I killed my brother._\n\n_I killed Toma._\n\n_I have been defeated and yielded._\n\n_I should have died._\n\nWhy had he yielded? Death would have been better than this. Even the innkeeper _pitied_ him.\n\n# Lorica\u2014de Vrailly\n\nGaston was wiping the blood from his blade, fastidiously examining the last four inches where he'd hacked repeatedly into the young squire's guard, battering his defences until he was overwhelmed and then dead. His blade had taken some damage in the process and would need a good cutler to restore the edge.\n\nDe Vrailly drank wine from a silver cup while his squires removed his armour.\n\n\"He cut you, the man in the courtyard,\" Gaston said, looking up from his task. \"Don't try to hide it. He cut you.\"\n\nDe Vrailly shrugged. \"He was swinging wildly. It is nothing.\"\n\n\"He got through your guard.\" Gaston sniffed. \"They aren't really so bad, these Albans. Perhaps we will have some real fights.\" He looked at his cousin. \"He hit you hard,\" he pointed out, because de Vrailly was rubbing his wrist for the third time in as many minutes.\n\n\"Bah! They have little skill at arms.\" De Vrailly drank more wine. \"All they do is make war on the Wild. They have forgotten how to fight other men.\" He shrugged. \"I will change that, and make them better at defeating the Wild as I do. I will make them harder, better men.\" He nodded to himself.\n\n\"Your angel has said this?\" Gaston asked, with obvious interest. His cousin's encounter with an angel had benefited the whole family, but it was still a matter that puzzled him.\n\n\"My angel has commanded it. I am but heaven's tool, cousin.\" De Vrailly said it without the least irony.\n\nGaston took a deep breath, looking for his great cousin to show a little humour, and found none. \"You called yourself the best knight in the world,\" he said, trying to raise a smile.\n\nDe Vrailly shrugged as Johan, his older squire, unlaced his left rerebrace and began to remove the arm harness over the wound on his wrist. \"I am the greatest knight in the world,\" he said. \"My angel chose me _because_ I am the first lance in the East. I have won six battles; I have fought in twelve passages of arms and never been wounded; I have killed men in every list in which I've fought; in the melee at Tours\u2014\"\n\nGaston rolled his eyes. \"Very well, you are the best knight in the world. Now tell me why we've come to Alba, besides bullying the locals.\"\n\n\"Their king will proclaim a tournament,\" de Vrailly said. \"I will win it, and emerge as the King's Champion.\" He nodded, \"and then I will be the king, to all intents and purposes.\"\n\n\"The angel has said this?\" Gaston asked.\n\n\"You question my angel, cousin?\" De Vrailly frowned.\n\nGaston rose and sheathed his sword. \"No, I merely choose not to believe everything I'm told\u2014by you or any other man.\"\n\nDe Vrailly's beautiful eyes narrowed. \"Are you calling me a liar?\"\n\nGaston smiled a crooked smile. \"If we continue like this we will fight. And while you may be the best knight in the world, I believe I have bloodied your knuckles more than once\u2014eh?\"\n\nTheir eyes crossed, and Gaston saw the glitter in de Vrailly's. Gaston held his gaze. Few men could do it. Gaston had the benefit of a lifetime of practice.\n\nDe Vrailly shrugged. \"You couldn't have asked this before we left home?\" he asked.\n\nGaston wrinkled his nose. \"When you say fight, I fight. Yes? You say: gather your knights, we go to conquer Alba. I say: lovely, we shall all be rich and powerful. Yes?\"\n\n\"Yes!\" de Vrailly said, through his smile.\n\n\"But when you tell me that an Angel of God is giving you very specific military and political advice\u2014\" Gaston shrugged.\n\n\"We are to meet the Earl of Towbray in the morning. He will engage us in his mesne. He desires what my angel desires.\" For the first time, de Vrailly seemed to hesitate.\n\nHe pounced. \"Cousin\u2014 _what does your angel desire?_ \"\n\nDe Vrailly drank more wine, put the cup down on the sideboard, and shrugged out of his right arm harness as his younger squire opened the vambrace. \"Who can know what an angel desires?\" he said quietly. \"But the Wild here must be _destroyed._ That's what the king's father intended. You know they burned swathes of the wood between the towns to do it? They waited for windy days and set fires. The old king's knights fought four great battles against the Wild\u2014and what I would give to have been part of that. The creatures of the Wild came forth to do battle\u2014great armies of them!\" His eyes shone.\n\nGaston raised an eyebrow.\n\n\"The old king was victorious in the main, but eventually, he sent to the East for more knights. His losses were fearsome.\" De Vrailly looked as if he could see it happening. \"His son\u2014now the king\u2014has fought well to hold what his father gained, but he takes no new land from the Wild. My angel will change that. We will throw the Wild back beyond the wall. I have seen it.\"\n\nGaston released a long-held breath. \"Cousin, just how fearsome were these losses?\"\n\n\"Oh, heavy, I suppose. At the Battle of Chevin, King Hawthor is said to have lost fifty thousand men.\" De Vrailly shrugged.\n\nGaston shook his head. \"Numbers that large make my head ache. That's the population of a large city. Have they replaced their losses?\"\n\n\"By the good Saviour, no! If they had, do you think we could challenge for the rulership of this land with three hundred lances?\"\n\nGaston spat. \"Good Christ\u2014\"\n\n\"Do not blaspheme!\"\n\n\"Your angel wants us to take this realm with three hundred lances so that he can launch a war against the Wild?\" Gaston stepped close to his cousin. \"Should I slap you to wake you up?\"\n\nDe Vrailly rose to his feet. With a gesture, he dismissed his squires. \"It is not seemly that you question me on these matters, cousin. It is enough that you summoned your knights and now you follow me. Obey me. That is all you need to know.\"\n\nGaston made a face like a man who has discovered a bad smell. \"I have always followed you,\" he said.\n\nDe Vrailly nodded his head.\n\n\"I have also saved you from a number of mistakes,\" Gaston added.\n\n\"Gaston,\" de Vrailly's voice suddenly softened. \"Let us not disagree. I am advised by heaven. Do not be jealous!\"\n\n\"Then I should like to meet your angel,\" Gaston said.\n\nDe Vrailly narrowed his eyes. \"Perhaps,\" he said, \"perhaps my angel is only for me. After all\u2014I alone am the greatest knight.\"\n\nGaston sighed and moved to the window where he looked down at the lone figure kneeling on the smooth stones of the courtyard. The bodies had been taken, laid out and wrapped in linen ready for burial, but still the Alban knight knelt in the courtyard.\n\n\"What do you plan to do with that man?\" Gaston asked.\n\n\"Take him to court to prove my prowess. Then I'll ransom him.\"\n\nGaston nodded. \"We should offer him a cup of wine.\"\n\nDe Vrailly shook his head. \"He does penance for his weakness\u2014for the sin of pride, in daring to face me, and for his failure as a man-at-arms. He should kneel there in shame for the rest of his life.\"\n\nGaston looked at his cousin, his face half turned away. He fingered his short beard. Whatever he might have said was interrupted by a knock on the door. Johan put his head in.\n\n\"An officer of the town, _monsieur._ To see you.\"\n\n\"Send him away.\"\n\nAfter a pause in which Gaston poured himself wine, Johan reappeared. \"He says he must insist. He is _not_ a knight. Merely a well-born man. He is not in armour. He says he is the sheriff.\"\n\n\"So? Send him away.\"\n\nGaston put a hand on his cousin's shoulder. \"Their sheriff's are king's officers, are they not? Ask him what he wants.\"\n\nJohan could be heard speaking, and then shouting, and then the door slammed open. Gaston drew his sword, as did de Vrailly. Their gentlemen poured in from adjoining rooms, some still fully armed.\n\n\"You are Jean de Vrailly?\" asked the newcomer, who didn't seem to care that he was surrounded by armed foreigners who topped him by a head or more. He was in doublet and hose, with high boots and a long sword belted at his waist. He was fiftyish and running to fat, and only the fur on his hood, his bearing and the sword at his hip suggested he was a man of any consequence. But he glowered.\n\n\"I am,\" de Vrailly answered.\n\n\"I arrest you in the name of the king for the murder of\u2014\"\n\nThe sheriff was knocked unconscious with a single blow from Raymond St. David, who let the body fall to the floor. \"Bah,\" he said.\n\n\"They are soft,\" de Vrailly said. \"Did he bring men-at-arms?\"\n\n\"Not one,\" Raymond said. He grinned. \"He came alone!\"\n\n\"What kind of a country is this?\" Gaston asked. \"Are they all insane?\"\n\nIn the morning, Gaston's retainers collected the dull-eyed Alban knight from the courtyard and packed him onto a cart with his armour; his horses were tethered behind. He tried to engage the Alban in conversation and was repelled by the man's look of hatred.\n\n\"Destriers,\" his cousin commanded. There was a lot of grumbling at the order\u2014no knight liked to ride his war horse when the occasion didn't demand it. A good war horse, fully trained, was worth the value of several suits of armour\u2014and a single pulled muscle, a strain, a cut, or a bad shoe was an expensive injury.\n\n\"We must impress the earl.\"\n\nDe Vrailly's household knights formed up in the inn's great courtyard while the lesser men-at-arms prepared in the field outside the town. They had almost a thousand spears, as well as three hundred lances. Gaston had already been out the gate, seen to the lesser men, and was back.\n\nThe innkeeper\u2014a surly, sharp faced fellow\u2014came out and spoke to the Alban knight on the cart.\n\nDe Vrailly grinned at him, and Gaston knew there would be trouble.\n\n\"You!\" de Vrailly shouted. His clear voice rang across the courtyard. \"I take issue with your measure of hospitality, Ser Innkeeper! Your service was poor, the wine bad, and you attempted to interfere in a gentleman's private matter. What have you to say for yourself?\"\n\nThe rat-faced innkeeper put his hands on his hips. Gaston shook his head. He was actually going to _discuss it_ with a knight.\n\n\"I\u2014!\" he began, and one of de Vrailly's squires, already mounted, reached out and kicked him. The kick caught him in the side of the head and he fell without a sound.\n\nThe other squires laughed and looked to de Vrailly, who dropped a small purse on the unconscious man. \"Here's _money_ , innkeeper.\" He laughed. \"We will teach these people to behave like civilized people and not animals. Burn the inn!\"\n\nBefore the last wagon of their small army had pulled out onto the road, a column of smoke was rising over the town of Lorica, and high into the sky.\n\nAn hour later, Gaston was at his cousin's side when they met the Earl of Towbray and his retinue where the Lorica road crossed the North road. The man had fifty lances\u2014a large force for Alba. The earl was fully armoured and wore his helmet. He sent a herald who invited _The Captal de Vrailly and all those who attend him_ to ride forward and meet the earl under the shade of a large oak that grew alone at the crossroads.\n\nGaston smiled at the earl's caution. \"Here is a man who understands how the world works,\" he said.\n\n\"He grew up among us,\" de Vrailly agreed. \"Let us ride to meet him. He has six lances with him\u2014we shall take the same.\"\n\nThe earl raised his visor when they met. \"Jean de Vrailly, Sieur de Ruth?\" he asked.\n\nDe Vrailly nodded. \"You do not remember me,\" he said. \"I was quite young when you toured the east. This is my cousin Gaston, Lord of Eu.\"\n\nTowbray clasped hands with each in turn, gauntleted hand to gauntleted hand. His knights watched them impassively, visors closed and weapons to hand.\n\n\"Did you have trouble in Lorica?\" the earl asked, pointing at the column of smoke on the horizon.\n\nDe Vrailly shook his head. \"No trouble,\" he said. \"I taught some lessons that needed to be learned. These people have forgotten what a sword is, and forgotten the respect due to the men of the sword. A poor knight challenged me\u2014I defeated him, of course. I will take him to Harndon and ransom him, after I display him to the king.\"\n\n\"We burned the inn,\" Gaston interrupted. He thought it had been a foolish piece of bravado, and he was finding his cousin tiresome.\n\nThe earl glared at de Vrailly. \"Which inn?\" he asked.\n\nDe Vrailly glared back. \"I do not like to be questioned in that tone, my lord.\"\n\n\"The sign of two lions. You know it?\" Gaston leaned past his cousin.\n\n\"You burned the Two Lions?\" The earl demanded. \"It has stood there forever. Its foundations are Archaic.\"\n\n\"And I imagine they are still there for some other peasant to build his sty upon.\" De Vrailly frowned. \"They scurried like rats to put out the fire, and I did nothing to stop them. But I was offended. A lesson needed to be made.\"\n\nThe earl shook his head. \"You have brought so _many_ men. I see three hundred knights\u2014yes? In all of Alba there might be four thousand knights.\"\n\n\"You wanted a strong force. And you wanted me,\" de Vrailly said. \"I am here. We have common cause\u2014and I have your letter. You said to bring all the force I could muster. Here it is.\"\n\n\"I forget how rich the East is, my friend. Three hundred lances?\" The earl shook his head. \"I can pay them, for now, but after the spring campaign we may have to come to another arrangement.\"\n\nDe Vrailly looked at his cousin. \"Indeed. Come spring we will have another arrangement.\"\n\nThe earl was distracted by the cart in the middle of the column.\n\n\"Good Christ,\" he said suddenly. \"You don't mean that Ser Gawin Murien is your prisoner? Are you _insane_?\"\n\nDe Vrailly pulled his horse around so hard Gaston saw blood on the bit.\n\n\"You will not speak to me that way, my lord!\" De Vrailly insisted.\n\nThe earl rode down the column, heedless of his men-at-arms' struggle to stay with him. He rode up to the wagon.\n\nGaston watched his cousin carefully. \"You will not kill this earl just because he annoys you,\" he said quietly.\n\n\"He said I was insane,\" de Vrailly countered, mouth tight and eyes glittering. \"We can destroy his fifty knights with a morning's work.\"\n\n\"You will end with a kingdom of corpses,\" Gaston said. \"If the old king really lost fifty _thousand_ men in one battle a generation ago, this kingdom must be almost empty. You cannot kill everyone you dislike.\"\n\nThe earl had the Alban knight out of the cart and on horseback before he rode back, his visor closed and locked and his knights formed closely behind him.\n\n\"Messire,\" he said, \"I have lived in the East, and I know how this misunderstanding has sprung up. But in Alba, messire, we do not keep to _The Rule of War_ at all times. In fact, we have something we call the _The Rule of Law._ Ser Gawin is the son of one of the realm's most powerful lords\u2014a man who is my ally\u2014and Ser Gawin acted as any Alban would. He was not required to be in his armour at that hour\u2014not here, and not when taking his ease at an inn. He is not in a state of war with you, messire. By our law, you attacked him perfidiously and you can be called to law for it.\"\n\nDe Vrailly made a face. \"Then your law is something that excuses weakness and devalues strength. He chose to fight and was beaten. God spoke on the matter and no more need be said.\"\n\nThe earl's eyes were just visible inside his visor and Gaston had his hand on his sword; while the earl was speaking reasonably, his hand was on the pommel of an axe at his saddle bow. His knights all had the posture\u2014the small leaning forward, the steadying hand on a horse's neck\u2014of men on the edge of violence. They were one step away from a disaster of blood. He could sense it.\n\n\"You will apologise to him for the barbaric deaths of his squires, or our agreement is at an end.\" The earl's voice was firm, and his hand was steady on his axe. \"Listen to me, messire. You _cannot_ take this man to court. The king has only to hear his story and you will be arrested.\"\n\n\"There are not enough men-at-arms in this country to take me,\" de Vrailly said.\n\nThe earl's retainers drew their swords.\n\nGaston raised his empty, armoured hands and interposed his horse between the two men. \"Gentlemen! There was a misunderstanding. Ever it has been so, when East meets West. My cousin was within his rights as a knight and a seigneur. And you say this Ser Gawin was also within his rights. Must we, who have come so far to serve you, my lord Earl\u2014must we all pay for this misunderstanding? As it pleases God, we are all men of good understanding and good will. For my part, I will apologise to the young knight.\" Gaston glared at his cousin.\n\nThe beautiful face showed understanding. \"Ah, very well,\" he said. \"He is the son of your ally? Then I will apologise. Although, by the good God! He needs training in arms.\"\n\nGawin Murien had recovered enough of his wits to pack his armour onto one horse and mount another. Then he followed the earl through the column, the way a child follows his mother.\n\nThe earl raised his visor. \"Gawin!\" he called out. \"Lad, the foreign knights\u2014they come from different customs. The Lord de Vrailly will apologise to you.\"\n\nThe Alban was seen to nod.\n\nDe Vrailly halted his horse well out of arm's reach, while Gaston rode closer. \"Ser Knight,\" he said, \"for my part, I greatly regret the deaths of your squires.\"\n\nThe Alban knight nodded again. \"Very courteous of you,\" he said. His voice was flat.\n\n\"And for mine,\" de Vrailly said, \"I forgive your ransom, as the earl insists that by your law of arms, I may have encountered you unfairly.\" The last word was drawn from him as if by a fish hook.\n\nMurien looked a less-than-heroic figure in his stained cote-hardie and his hose ruined by a night of kneeling in the courtyard. He didn't glitter. In fact, he hadn't even put his knight's belt back on, and his sword still lay on the bed of the wagon.\n\nHe nodded again. \"I hear you,\" he said.\n\nHe turned his horse, and rode away.\n\nGaston watched him go, and wondered if it would have been better for everyone if his cousin had killed him in the yard.\n\n# Chapter Five\n\n# Harndon Palace\u2014Harmodius\n\nHarmodius Magus sat in a tower room entirely surrounded by books, and watched the play of the sun on the dust motes, as it shone through the high, clear glass windows. It was April\u2014the season of rain but also the season of the first serious, warm sun\u2014when the sunlight finally has its own colour, its own richness. Today, the sky was blue and a cat might be warm in a patch of sunshine.\n\nHarmodius had three cats.\n\n\"Miltiades!\" he hissed, and an old grey cat glanced at him with weary insolence.\n\nThe man's gold-shod stick licked out and prodded the cat, whose latest sleeping spot threatened the meticulously drawn pale blue chalk lines covering the dark slate floor. The cat shifted by the width of its tail and shot the Magus a disdainful look.\n\n\"I feed you, you wretch,\" Harmodius muttered.\n\nThe light continued to pour through the high windows, and to creep down the whitewashed wall, revealing calculations in chalk, silver or lead pencil, charcoal, even scratched out in dirt. The Magus used whatever came to hand when he felt the urge to write.\n\nAnd still the light crept down the wall.\n\nIn the halls below, the Magus could sense men and women\u2014 _a servant bringing a tray of cold venison to his tower's door; a gentleman and lady engaged in a ferocious tryst that burned like a small fire almost directly under his feet\u2014where was that? It must be awfully public\u2014and the Queen, whoburned like the very sun._ He smiled when he brushed over her warmth. Oftimes, he watched others to pass the time. It was the only form of phantasm he still cast regularly.\n\n_Why is that?_ he wondered, idly.\n\nBut this morning, well. This morning his Queen had asked him\u2014-challenged him\u2014to do something.\n\n_Do something wonderful, Magus!_ she had said, clapping her hands together.\n\nHarmodius waited until the sun crossed a chalk line he had drawn, and then raised his eyes to a particular set of figures. Nodded. Sipped some cold tea that had a film of dust\u2014what was that dust? Oh\u2014he had been grinding bone for oil colour. He had bone dust in his tea. That wasn't entirely disgusting.\n\nAll three cats raised their heads and pricked their ears.\n\nThe light intensified and struck a small mirror with the image of Ares and Taurus entwined on the ivory back\u2014and then shot across the floor in a focused beam.\n\n\"Fiat lux!\" roared the Magus.\n\nThe beam intensified, drawing in the light around it until the cats were cast in shadow while the beam sparkled like a line of lightning\u2014passing above the chalk designs, through a lens, and into a bubble of gold atop his staff. Unnoticed by him, it struck a little off centre and a tiny fragment of the white beam slid past the staff to dance along the far wall, partially reflected by the golden globe, partly refracted by the mass of energy seething inside the staff itself. The intense light flicked up sharply, licked at gilding of a triptych that adorned the sideboard, struck a glass of wine abandoned hours before. And, still focused, it passed across the east wall, its rapid passage burning away a dozen or more characters of a spell written in an invisible, arcane ink, hidden under the paint.\n\nThe older cat started, and hissed.\n\nThe Magus felt suddenly light headed, as if at the onset of a fever or a strong head-cold. But his mind was abruptly clear and sharp, and the staff was giving off the unmistakable aura of an artefact filling with power. He saw the rogue light fragment and deftly moved the mirror and the focus to touch the staff perfectly.\n\nHe clapped his hands in triumph.\n\nThe cats looked around, startled, as if they had never seen his room before\u2014and then went back to sleep.\n\nHarmodius looked around the room. \"What in the name of the triad just happened?\" he asked.\n\nHe didn't need rest. Even after casting a powerful _phantasm_ like the last, the feeling of the _helios_ in his staff made him tingle with anticipation. He'd promised himself that he'd wait a day... perhaps two days... but the temptation was strong.\n\n\"Bah,\" he said aloud, and the cats flicked their ears. He hadn't felt so alive in many years.\n\nHe took a heavy flax mop and scrubbed the floor, eliminating every trace of the complex chalk pattern that had decorated it like an elaborate Southern rug. Then, despite his age and his heavy robes, he was down on his knees with a square of white linen, scrubbing even the cracks between the slate slabs until there was not a trace of pale blue chalk. However eager he was, he was also fastidious about this\u2014that no trace of one phantasm should linger while he performed another. Experience had taught him that lesson well.\n\nThen he went to a side table and opened the drawer, wherein was laid a box of ebony bound in silver. The Magus loved beautiful things\u2014and when the consequences of bad conjuring were soul-destruction and death, the presence of beautiful things help reassure and steady him.\n\nInside the box lay a nested set of instruments made of bronze\u2014a compass, a pair of calipers, a ruler with no markings; a pencil which held silver suspended in alum and clay and wax, blessed by a priest.\n\nHe wrapped a string around the pencil, measuring the length against the ruler, and began to pray. \"O, Hermes Trismegistus,\" he began, and continued in High Archaic, purifying himself, clearing his thoughts, invoking God and his son and the prophet of the magi while another part of his mind calculated the precise length of string he would need.\n\n\"I should not do this today,\" he told the fattest cat. The big feline didn't seem to care.\n\nHe knelt on the floor, not to pray, but to draw. Putting a sliver of wood into a slot in the slate, he used the string, shaking with the tension in his hands, to guide his hand through a perfect circle, and into the circle, with the help of the ruler and a sword, he inscribed a pentagram. He wrote his invocation to God and to Hermes Trismegistus in High Archaic around the outside, and only the clamour of the cats for their noonday feast kept him from attempting his work right there and then.\n\n\"All three of you are man's best practice for dealing with demons,\" he said as he fed them fresh salmon, new caught in the River Albin and sold in the market.\n\nThey ignored him and ate, and then rubbed against him with loud protests of eternal love.\n\nBut his words gave him pause, and he unlocked the heavy oak door to the tower chamber and walked down one hundred and twenty-two steps to his sitting room where Mastiff, the Queen's man, sat reading in an armchair. The man leapt to his feet when the Magus appeared.\n\nThe Magus raised an eyebrow and the man bowed. But Harmodius was in a hurry\u2014a hurry of passion\u2014and little incivilities would have to wait. \"Be so kind as to hurry and beg the Queen's indulgence: would she do me the kindness to pay me a visit?\" he asked, and handed the man a plain copper coin\u2014a sign between them. \"And ask my laundress to pay me a visit.\" He handed over a handful of small silver change. Some of the coins were as small as sequins.\n\nMastiff took the coins and bowed. He was used to the Magus and his odd ways, so he hurried off as if his life depended on the journey.\n\nThe Magus poured himself a cup of wine and drank it off, stared out the window, and tried to convince himself to let it go for a day. Who would care?\n\nBut he felt ten years younger, and when he thought of what he was about to prove he shook his head, and his hand trembled on the cup.\n\nHe heard her light step in the hall, and he rose and bowed deeply when she entered.\n\n\"La,\" she said, and her presence seemed to fill the room. \"I was just saying to my Mary\u2014I'm bored!\" She laughed, and her laugh rose to the high rafters.\n\n\"I need you, your Grace,\" he said with a deep bow.\n\nShe smiled at him, and the warmth of it left him more light headed still. Afterwards, he could never decide whether lust played a part in what he felt for her; the feeling was strong, possessive, awesome, and dangerous.\n\n\"I am determined to work a summoning, your Grace, and would have you by me to steady my hand. I hope it will be wonderful.\" He bowed over hers.\n\n\"My dear old man, she looked at him tenderly. He felt in her regard a flaw\u2014she pitied him. \"I honour your efforts, but don't tax yourself to impress me!\"\n\nHe refused to be annoyed. \"Your Grace, I have made such summonings many times. They are always fraught with peril, and like swimming, only a fool does such a thing alone.\" In his mind's eye, he imagined swimming with her, and he swallowed heavily.\n\n\"I doubt that I can do anything to support a mighty practitioner such as you\u2014I, who only feel the sun's rays on my skin, and you, who feel his power in your very soul.\" But she went to the base of the long staircase eagerly and led him to the top, her feet lighter on the treads than his by half a century. And yet he was not breathing hard when they reached the top.\n\nShe kicked off her red shoes on the landing and entered his chamber carefully, barefoot, avoiding the precise markings on the floor. She paused to look at them. \"Master, I have never seen you work something so\u2014daring!\" she said, and this time her admiration was unfeigned.\n\nShe went to stand in the sun which now covered the east wall instead of the west. She stood there studying the equations and lines of poetry, and then she began to scratch the ears of the old fat cat.\n\nHe purred a moment, sank his fangs into her palm, and mewed when she swatted him with her other hand.\n\nHarmodius shook his head and poured honey on the punctures the cat had left. \"I've never known him to bite before,\" he said.\n\nShe shrugged with an impish smile and licked the honey.\n\nHe, too, removed his shoes.\n\nHe went to his wall of writing and pushed his nose close, reading two lines written in silver pencil. Then, taking up a small ebony wand, he wrote the two lines in the air, and left letters of bright fire behind\u2014thinner than the thinnest hair, and yet perfectly visible from where either of them stood.\n\n\"Oh!\" said the Queen.\n\nHe smiled at her. He had the briefest temptation to kiss her and another desire, equal but virtually opposite, to back out of the whole thing.\n\nShe reminded him of\u2014\n\n\"Bah,\" he said. \"Are you ready, your Grace?\"\n\nShe smiled and nodded.\n\n_\"Kaleo se, CHARUN,\"_ the Magus said, and the light over the pentagram paled.\n\nThe Queen took a step to the right, and stood in the full beam of the sun from the high windows, and the old cat rubbed against her bare leg.\n\nShadow began to fill the pentagram. The Magus took up his staff, and held the hollow golden end like a spear point between himself and the inscribed sign on the floor.\n\n\"Who calls me?\" came a whisper from the fissure in light that flickered like a butterfly above the pentagram.\n\n\" _KALEO_ ,\" Harmodius insisted.\n\nCharun manifested beneath the shadow. The Magus felt his ears pop, and the sun seemed dimmed.\n\n\"Ahhh,\" he hissed.\n\n\"Power for knowledge,\" Harmodius said.\n\nThe shadows were drawn into a creature that was like a man, except he was taller than the highest bookcase, naked, a deep white veined in blue like old marble, with tough, leathery wings that swept majestically from well above his head to the floor in a perfect arc that any artist would have admired.\n\nThe smell he brought with him was alien\u2014like the smell of lye soap being burned. Neither clean nor foul. And his eyes were a perfect, black blank. He carried a sword as tall as a man and wickedly barbed, and his head held both alien horror and angelic beauty in one\u2014an ebony-black beak inlaid with gold; huge, almond shaped eyes, deep and endless blue like twin sapphires, and a bony crest filled with hair, like the decoration on an Archaic helmet.\n\n\"Power for knowledge,\" Harmodius said again.\n\nThe demon's blank eyes regarded him. Who knew what they thought? They seldom spoke, and they didn't often understand what a magus asked.\n\nAnd then, as swiftly as an eagle seizes a rabbit, the sword shot out and cut the circle.\n\nHarmodius's eyes narrowed, but he had not lived as long as he had by giving in to panic. \" _Sol et scutum Dominus Deus_ ,\" he said.\n\nThe second strike of the sword licked out through the circle but rang off the shield that had formed over the demon. The creature looked at the shield, glowing a bubbly purple shot with white, and began to prod it with the sword. Sparks began to cascade down the sides of the shield, shaped like a bright bell of colour suspended over the daemon. Smoke began to rise from the floor.\n\nHarmodius struck his staff against the edge of the circle where the sword had cut his pattern. \" _Sol et scutum Dominus Deus!_ \" he roared.\n\nThe rift in the circle closed, and the creature reared back and hissed.\n\nThe Queen leaned in towards it, and Harmodius felt a pang of pure terror that she would unwittingly cross the circle. But he could not say anything to her. To do so would be to betray the energy of his summoning\u2014his entire will was bent on the creature that had manifested, and the circle, the pentagram, and the shield.\n\nHe was, he realised, juggling too many balls.\n\nHe considered letting the shield go\u2014right up until the demon breathed fire.\n\nIt blossomed like a flower, flowing to cover the entire surface of the shield, and the room was suddenly hot. The fire could not pass the shield\u2014but the heat from it could, and the deamon's heat changed the contest of wills utterly. Even as he began to consider the possibility that he might be defeated, Harmodius's mind viewed this fact with fascination. Despite the shield, he could smell the creature, and he could feel the heat.\n\nAs suddenly as they had appeared, the flames retreated from the edges of the circle and fled back into the creature's mouth. The heat dropped perceptibly.\n\nDesiderata leaned in until her nose touched the unsolid surface of the shield. And she _laughed._\n\nThe demon turned to her, head cocked, for all the world like a puppy. And then he laughed back.\n\nShe curtsied, and then began to dance.\n\nThe demon watched her, rapt, and so did the Magus.\n\nShe expressed herself in her hips, and in the rise of her hands above her head as she danced a mere dozen steps\u2014a dance of spring, na\u00efve and unflawed by practice.\n\nThe creature inside the bubble of power shook his head. \" _Eyah!_ \"\n\nHe took a step towards her, and his head touched the edge of the pentagram, and he howled with rage and swept his sword across the sigil, cutting a gouge in the slate floor that broke the circle.\n\nShe extended a foot and crossed her toes over the break, and it was healed.\n\nHarmodius breathed again. Quick as a terrier after a rat, he struck his staff through the shield and poured the power he had collected from his phantasm into the demon.\n\nIt whirled from the Queen to face the Magus, sword poised\u2014but took no action. Its mighty chest rose and fell. Its aspect changed, suddenly\u2014it rose into the air, glowing white, an angel with wings of a swan, and then it fell to the slate floor and its writhing changed to the hideous controlled motion of a millipede larger than a horse, cramped in the confines of the shield. Harmodius raised his wand, joy surging though his heart\u2014the pure joy of having truly tested a theory and found it to contain more gold than dross.\n\nHarmodius's took his staff from the circle and spat \" _Ithi!_ \"\n\nThe pentagram was empty.\n\nHarmodius was too proud to slump. But he went to the Queen's side and threw his arms around her with a familiarity he never knew he dared.\n\nShe kissed him tenderly.\n\n\"You are an old fool,\" she said. \"But a brilliant, brave old fool, Harmodius.\" Her smile was warm and congratulatory. \"I had no idea\u2014I've never seen you do anything like it.\"\n\n\"Oh,\" he said, into the smell of her neck\u2014and a galaxy of new learning occurred to him in that moment. But he backed away and bowed. \"I owe you my life,\" he said. \"What are you?\"\n\nShe laughed, and her laugh threatened to mock all evil straight out of fashion. \"What am I?\" she asked. She shook her head. \"You dearest old fool.\"\n\n\"Still wise enough to worship at _your_ feet, your Grace.\" He bowed very low.\n\n\"You are like a boy who attacks a hornet's nest to see what will come out. And yet I smell the triumph of the small boy on you, Harmodius. What have we learned today?\" She subsided suddenly into a chair, ignoring the scrolls that covered it. \"And where did this sudden burst of daring come from? You are a byword for caution in this court.\" She smiled, and for a moment, she was not a na\u00efve young girl, but an ancient and very knowing queen. \"Some say you have no power, and are a sort of Royal Mountebank.\" Her eyes flicked to the pentagram. \"Apparently, they are wrong.\"\n\nHe followed the wave of her hand and hurried to pour her wine. \"I cannot say for certain sure what we learned today,\" he said carefully. Already his careful manner was reasserting itself. But he _knew_ he was right.\n\n\"Talk to me as if I was a student\u2014a stupid squire bent on acquiring the rudiments of hermeticism,\" she said. She sipped his wine and her look of contentment and the flinging back of her head told him that she, too had known a moment of terror. She was mortal. He was not always sure of that. \"Because I can use power, I think you assume that I know how it functions. That we have the same knowledge. But nothing can be further from the truth. The sun touches me, and I feel God's touch, and sometimes, with his help, I can work a miracle.\" She smiled.\n\nHe thought that her self-assurance could, if unchecked, make her more terrifying than any monster.\n\n\"Very well, your Grace. You know there are two schools of power\u2014two sources for the working of any phantasm.\" He laid his staff carefully in a corner and then knelt to wipe the pentagram from the floor.\n\n\"White and black,\" she said.\n\nHe glared at her.\n\nShe shrugged with a smile. \"You are so easy, my Magus. There is the power of the sun, pure as light, unfettered, un-beholden\u2014the very sign of the pleasure of God in all creation. And there is the power of the Wild\u2014for which, every iota must be exchanged with one of the creatures that possess it, and each bargain sealed in blood.\"\n\nHarmodius rolled his eyes. \"Sealed. Bargained for. Blood does not really enter into it.\" He nodded. \"But the power is there\u2014it rises from the very ground\u2014from grass, from the trees, from the creatures that live among the trees.\"\n\nShe smiled. \"Yes. I can feel it, although it is no friend to me.\"\n\n\"Really?\" he asked, cursing himself for a fool. Why had he _not_ asked the Queen earlier? A safer experiment sprang to his mind. But what was done was done. \"You can feel the power of the Wild?\"\n\n\"Yes!\" she said. \"Stronger and weaker\u2014even in those poor dead things that decorate the hall.\"\n\nHe shook his head at his own foolishness\u2014his hubris.\n\n\"Do you sense any power of the Wild in this room?\" he asked.\n\nShe nodded. \"The green lamp is an artefact of the Wild, is it not? A faery lamp?\"\n\nHe nodded. \"Can you take any of the power it pours forth and use it, your Grace?\"\n\nShe shuddered. \"Why would you even ask such a thing? Now I think you dull, Magus.\"\n\n_Hah_ , he thought. _Not so hubristic as all that._\n\n\"And yet I conjured a powerful demon of the abyss\u2014did I not?\" he asked her.\n\nShe smiled. \"Not one of the greatest, perhaps. But yes.\"\n\n\"Allied to the Wild\u2014would you say?\" he asked.\n\n\" _God is the sun and the power of the sun\u2014and Satan dwells in the power of the Wild._ \" She sang the lines like a schoolgirl. \" _Daemons must use the power of the Wild. When Satan broke with God and led his legions to hell, then was magic broken into two powers, the green and the gold. Gold for the servants of God. Green for the servants of Satan._ \"\n\nHe nodded. Sighed. \"Yes,\" he said. \"But of course, it is more complicated than that.\"\n\n\"Oh, no,\" she said, showing that glacial self-assurance again. \"I think men often seek to overcomplicate things. The nuns taught me this. Are you saying that they lied?\"\n\n\"I just fed a demon with the power of the sun. I conjured him with the power of the sun.\" Harmodius laughed.\n\n\"But\u2014no, you banished him!\" Her silver laugh rang out. \"You tease me, Magus!\"\n\nHe shook his head. \"I banished him after feeding him enough power to make him _grow_ ,\" the Magus said. \"Pure Helios, which I drew myself using my instruments\u2014lacking your Grace's special abilities.\" _Whatever they may be._\n\nShe gazed at him, eyes level, devoid of artifice or flirtation, mockery or subtle magnetism or even her usual humour.\n\n\"And this means?\" she asked, her voice a whisper.\n\n\"Ask me again, your Grace, after I conjure him back a week hence. Tell me you will stand at my side that day\u2014I'm beholden to you, but with you\u2014\"\n\n\"What do you seek, Magus? Is this within the circlet of what the church will countenance?\" She spoke slowly, carefully.\n\nHe drew a breath. Released it. _Sod the church_ , he thought. And aloud he said, \"Yes, your Grace.\" _No, your Grace. Perhaps not. But they're not scientists. They're interested in preserving the status quo._\n\nThe Queen gave him a beautiful smile. \"I am just a young girl,\" she said. \"Shouldn't we ask a bishop?\"\n\nHarmodius narrowed his eyes. \"Of course, your Majesty,\" he said.\n\n# The North Road\u2014Gerald Random\n\nRandom's convoy moved fast, by the standards of convoys\u2014six to ten leagues a day, stopping each night at the edge of a town and camping in pre-arranged fields, with fodder delivered to their camp along with hot bread and new-butchered meat. Men were happy to work for him because he was a meticulous planner and the food was good.\n\nBut they had a hundred leagues to go, just to make Albinkirk, and another forty leagues east after that, to the fair, and he was later than he wanted to be. Albinfleurs\u2014little yellow balls of sweet-scented, fuzzy petals that grew only on the cliff edges of the great river\u2014were blooming in the hayfields that lined the roads; and when they were on his favourite sections of road\u2014the cliff-edge road along the very edge of the Albin which ran sixty feet or more below them in the vale\u2014the Albinfleurs were like stripes of yellow below him and layers of yellow on the cliffs nearly a mile distant on the other side. It was years since he'd left late enough to see the Albinfleurs. They didn't grow in the north.\n\nBut after three glorious days of solid travel, they came to Lorica, and the Two Lions. His usual stop and supplier of bread and forage was a smoking hulk. It took him a day to establish a new supplier and get the material he needed, and the story of how the inn had been burned and the sheriff beaten by foreigners angered him. But the innkeeper had sent to the king, and stood in his yard with a bandaged head watching workmen with a crane lift the charred rooftrees off the main building.\n\nHe used one of his precious mercenaries to send a message about the killings back to the Guild Master in Harndon. Harndoners didn't usually concern themselves with the doings of the lesser towns. But this was business, friendship, and basic patriotism all in one package.\n\nThe following day not one but two of his wagons broke spokes on their wheels\u2014one so badly that the wooden wheel split and the iron tyre popped off the wheel. That meant finding a smith and wheelwright and forced him to go _back_ to Lorica, where he had to stay in an inferior inn while his convoy crept north without him. He had to do it himself\u2014the men in Lorica knew him but none of his hirelings, not Judson the draper, nor any of the other investors.\n\nIn the morning, the two wagons were ready to move, and he grudgingly paid the agreed fee for making two apprentice wheelwrights and a journeyman work by rush light through the night. Plus an extra silver leopard to the blacksmith for getting the tyre on before matins.\n\nHe finished his small beer and mounted his horse, and the smaller train was on the road as soon as he'd taken the Eucharist from a friar who said Mass at a roadside shrine. That roadside Mass was full of broken men and women\u2014wastrels, a pair of vagabonds, and a troop of travelling players. Random had never been troubled by the poor. He gave them alms.\n\nBut the broken men worried him, for both his convoy and his purse. There were four of them, although they didn't seem to be together. Random had never been robbed by men he'd just attended Mass with, but he didn't take any chances, either. He mounted, exchanged meaningful glances with his drovers, and the carts moved on.\n\nOne of the broken men followed them on the road. He had a good horse and armour in a wicker basket, and he seemed listless. Random looked back at him from time to time.\n\nEventually, the man caught them up. But he hadn't put on his armour and he didn't even seem to know they were there. He rode up, slowly catching and then overtaking them.\n\nHarndoners traditionally called the men they'd attended Mass with that day Brother or Sister, and so Random nodded to the stranger.\n\n\"The Peace of God to you, Brother,\" he said, a little pointedly.\n\nThe man looked startled to be addressed.\n\nIn that moment, Random realised he wasn't a broken man at all but a dirty gentleman. The differences were clear in his quality\u2014the man had a superb leather-covered jupon worth a good twenty leopards, even covered in dirt. Hip boots with gold spurs. Even if they were silver gilt, they were worth a hundred leopards by weight.\n\nThe man sighed. \"And to you, messire.\"\n\nHe rode on.\n\nRandom hadn't come to relative riches in the cut-throat world of Harndon's shippers and guilds without having some willingness to grab at Fortuna's hairs. \"You're a knight,\" he said.\n\nThe man didn't rein in, but he turned his head and, feeling the weight shift, his horse stopped.\n\nThe man turned to look at him, and the silence was painful.\n\n_What have we here?_ Random wondered.\n\nFinally, the young man\u2014under his despair, the man was younger than Random by a generation\u2014nodded.\n\n\"I am a knight,\" the young man said, as if confessing a sin.\n\n\"I need men,\" Random said. \"I have a convoy on the road and if you wear spurs of gold, I'd be honoured to have you. My convoy is fifty good wagons headed north to the fair, and there's no dishonour in it. I fear only bandits and the Wild.\"\n\nThe man shook his head minutely and turned away, and his horse ambled on, a good war horse which was over-burdened with man and armour, the weight ill-distributed and ruining the horse's posture.\n\n\"Are you sure?\" Random asked. It never hurt to try.\n\nThe knight kept riding.\n\nRandom let his drovers stop for lunch, and then they pushed on\u2014into the evening and even a little after dark.\n\nIn the morning, they rose and were moving on before the sun was a finger above the river which curved, snake-like, to the east. Later in the morning they descended into the vale and crossed the Great Bridge, the edge of the Inner Counties. He had a fine meal at the Crouching Cat with his drovers, who were honoured by his willingness to join them and pleased to eat so good a meal.\n\nAfter lunch they crossed Great Bridge, twenty-six spans built by the Archaics and painstakingly maintained. And then climbed the far bank for an hour, with the drovers leading the horses. They crested the far bank, and Random saw the knight again, kneeling at a roadside chapel, tears cutting deep channels in the road-dust on his face.\n\nHe nodded to him, and rode on.\n\nBy evening he caught up the rest of the convoy, already in camp, and he was welcomed back by the men he'd left. His drovers regaled their peers with the minutiae of their days, and Guilbert saluted and told him how the column had proceeded, and Judson was resentful that he was back so soon.\n\nBusiness as usual.\n\nA little after dark, one of the goldsmith boys came to his wagon and saluted like a soldier. \"Messire?\" he asked. \"There's a knight asking for ye.\" The boy had a crossbow on his shoulder, and was obviously puffed with pride at being on watch, on convoy, and in such an important role. _Henry Lastifer_ , the name floated up from the merchant's storehouse of ready knowledge.\n\nRandom followed the boy to the fire. Guilbert was there, and Old Bob, another of the men-at-arms.\n\nAnd the young knight from the road, of course. He was sitting, drinking wine. He rose hurriedly.\n\n\"May I change my mind?\" he blurted.\n\nRandom smiled. \"Of course. Welcome aboard, Ser Knight.\"\n\nGuilbert smiled broadly. \"M'lord, is more like. But he's the king's mark. And that's a sword.\" He turned to the knight. \"Your name, m'lord?\"\n\nThe young man waited so long it was obvious he was going to lie. \"Ser Tristan?\" he said, wistfully.\n\n\"Fair enough,\" Guilbert said. \"Come wi' me, and we'll see to it you have a place to sleep.\"\n\n\"Mind you,\" said Random. \"You work for Guilbert and then for me. Understand?\"\n\n\"Of course,\" said the young man.\n\n_What am I getting myself into?_ Random thought. But he felt satisfied with the man, broken or not. King's knights were trained to a high level\u2014especially trained to fight the Wild. Even if the young man was a little addled... well, no doubt he was in love. The gentry were addicted to love.\n\nHe slept well.\n\n# North of Lorica\u2014Bill Redmede\n\nBill Redmede led his untrained young men up the trail. Their irk stayed well ahead, moving like smoke through the thick trees. He tended to return to the column from the most unexpected directions, even for a veteran woodsman like Bill.\n\nThe lads were all afraid of him.\n\nBill rather liked the quiet creature, which spoke only when it had something to say. Irks had something about them. It was hard to pin down, but they had some kind of nobility\n\n\"Right files watch the right side of the trail,\" Bill said, automatically. \"Left files watch the _left_ side.\" Three days on the trail and all he did was mother them.\n\n\"I need a break,\" whined the biggest and strongest of them. \"Christ on the Cross, Bill! We're not boglins!\"\n\n\"If you was, we'd move faster,\" Redmede said. \"Didn't you boys do any _work_ on the farm?\"\n\nIt was worse when they made camp. He had to explain how to raise a shelter. He had to stop them from cutting their twine, and teach them how to make a fire. A _small_ fire. How to be warm, how to be dry. Where to take a piss.\n\nTwo of them sang while they worked, until he walked up and knocked one to the ground with a blow of his fist.\n\n\"If the king catches you because you are _singing_ , you will hang on a gibbet until the crows pick your bones clean and then the king's fucking sorcerer will grind your bones to make the colours for his paints,\" Bill said.\n\nThe angry silence of wronged young men struck him from all sides.\n\n\"If you fail, you will die,\" he said. \"This is not a summer lark.\"\n\n\"I want to go home,\" said the biggest man. \"You're worse than an aristo.\" He looked around. \"And you can't stop all of us.\"\n\nThe irk materialised out of the dusk. He looked curiously at the big man. Then he turned to Bill. \"Come,\" he said in his odd voice.\n\nBill nodded to them, the debate now unimportant. \"Don't go anywhere,\" he said, and followed the irk.\n\nThey crossed a marsh, over a low ridge, and then down to a dense copse of spruce.\n\nThe irk turned and made a motion with its head. \"Bear,\" it said. \"A friend. Be kind, Man.\"\n\nNear the centre of the spruce was a great golden bear. It lay with its head in its paws, as if it was resting. A beautiful cub stood licking its face.\n\nAs Bill come up, the bear stirred. It raised its head and hissed.\n\nBill stepped back, but the irk steadied him, and spoke in a sibilant whisper.\n\nThe bear rolled a little, and Bill could see it had a deep wound in its side, full of pus\u2014pus was dryed on either side of the wound, and it stank.\n\nThe irk squatted down in a way a man could not have done. Its ear drooped\u2014this was sadness, which Bill had never seen in an irk.\n\n\"The bear dies,\" the irk said.\n\nBill knew the irk was right.\n\n\"The bear asks\u2014can we save her cub?\" The irk turned and Bill realised how seldom the elfin creature had met his eyes, because in that moment, the irk's gaze locked with his, and he all but fell into the forest man's regard. His eyes were huge, and deep like pools\u2014\n\n\"I don't know a thing about bears,\" Bill said. He squatted by the big mother bear. \"But I'm a friend of any creature of the Wild, and I give you my word that if I can get your cub to other golden bears, I will.\"\n\nThe bear spat something, in obvious pain.\n\nThe irk spoke\u2014or rather, sang. The line became a stanza, full of liquid rhymes.\n\nThe bear coughed.\n\nThe irk turned. \"The cub\u2014her mother named her for the yellow flower.\"\n\n\"Daisy?\"\n\nThe irk made a face.\n\n\"Daffodil? Crocus? I don't know my flowers.\"\n\n\"In water.\" The irk was frustrated.\n\n\"Lily?\"\n\nThe irk nodded.\n\nSo he reached out a hand to the cub, and the cub bit him.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain was so tired and so drained by the fear that it was all he could do to push one boot in front of the other as the trail became a track and the track became a road.\n\nNothing troubled them but the coming darkness, their exhaustion, and the cold. It was late in the day and increasingly clear that they would have to camp in the woods. The same woods which had produced a daemon and a wyvern.\n\n\"Why didn't it kill us?\" the captain asked. _Two daemons._\n\nGelfred shook his head. \"You killed that first one. Pretty. Damn. Fast.\" His eyes were always moving. They had reached the main road, and Gelfred pulled up on his horse's reins. \"We could ride double,\" he said.\n\n\"You'll lame that horse,\" the captain snapped.\n\n\"You _cast a spell._ \" Gelfred wasn't accusatory. He sounded more as if he was in pain.\n\n\"Yes,\" the captain admitted. \"I do, from time to time.\"\n\nGelfred shook his head. He prayed aloud, and they rode on until a drizzle began and the light began to fade.\n\n\"We'll have to stand watches,\" the captain said. \"We are very vulnerable.\" He could barely think. While Gelfred curried the poor beast, he gathered firewood and started a fire. He did everything wrong. He gathered bigger wood and had no axe to cut it; then he gathered kindling and broke it into ever smaller and better sorted piles. He knelt in his shallow fire-pit and used his flint and steel, shaving sparks onto charred cloth until he had an ember.\n\nThen he realised that he hadn't built a nest of tow and bark to catch the ember.\n\nHe had to start again.\n\n_We're a pair of fools._\n\nHe could _feel_ that the woods were full of enemies. Or allies. It was the curse of his youth.\n\n_What exactly have I stumbled into?_ he asked himself.\n\nHe made a little bird's nest of dry tow and birchbark shreds, and made sparks again, his right hand holding the steel and moving precisely to strike the flint in his left hand. He got a spark, lit the char\u2014\n\nDropped it into the tow and bark\u2014\n\nAnd blew.\n\nThe fire caught.\n\nHe dropped twigs on the blaze until it was steady, and then built a cabin of dry wood, carefully split with his hunting knife. He was very proud of his fire when he'd finished, and he thought that if the Wild took him here, at least he'd started the damned fire first.\n\nGelfred came and warmed his hands. Then he wound his crossbow. \"Sleep, Captain,\" he said. \"You first.\"\n\nThe captain wanted to talk\u2014he wanted to think, but his body was making its own demands.\n\nBut before he could go to sleep he heard Gelfred move, and he was out of his blankets with his sword in his fist.\n\nGelfred's eyes were big in the firelight. \"I just wanted to move the head,\" he said. \"It\u2014it's hard to have it there. And the horse hates it.\"\n\nThe captain helped to move the head. He stood there, in the dark, freezing cold.\n\nThere was something very close. Something powerful.\n\nPerhaps building the fire had been a mistake, like coming out into the woods with just one other man.\n\n_Prudentia? Pru?_\n\n_Dear boy._\n\n_Pru, can I pull the Cloak over this little camp? Or will I just make a disturbance in casting?_\n\n_Cast quietly, as I have taught you._\n\n_He touched her marble hand, chose his wards and gardes, and opened the great iron door to his palace. Outside was a green darkness\u2014thicker and greener than he liked._\n\n_But he took carefully from the green, and closed the door._\n\nHe staggered with the effort.\n\nSuddenly he couldn't stay upright. He fell to his knees by the daemon's head.\n\nThe darkness was thick.\n\nThe head still had something of its aura of fear about it. He knelt by it\u2014knees wet in the damp, cold leaves, and the cold helped to steady him.\n\n\"M'lord?\" Gelfred asked, and he was obviously terrified. \"M'lord!\"\n\nThe captain worked on breathing for a moment.\n\n\"What?\" he whispered.\n\n\"The stars went out,\" Gelfred said.\n\n\"I cast a little\u2014concealment over us,\" the captain said. He shook his head. \"Perhaps I mis-cast.\"\n\nGelfred made a noise.\n\n\"Let's get away from this thing,\" the captain said, and he got to his feet, and together the two men stumbled over tree roots to their tiny fire.\n\nThe horse was showing the whites of its eyes.\n\n\"I have to sleep,\" he said.\n\nGelfred made a motion in the dark. The captain took it for acceptance.\n\nHe slept from the moment his head went down, despite the fear, to the moment Gelfred woke him with a hand on his shoulder.\n\nHe heard the hooves.\n\nOr talons.\n\nWhatever it was, he couldn't see the thing making the noise. Or anything else.\n\nThe fire was out and the night was too dark to see anything. But something very large was moving\u2014just an arm's length away. Maybe two.\n\nGelfred was right there, and the captain put a hand on his shoulder to steady them both.\n\n_Skerunch._\n\n_Snap._\n\n_Tick._\n\nAnd then it was past them, moving down the hill to the road.\n\nAfter an aeon, Gelfred said \"It didn't see us or smell us.\"\n\nThe captain said _Thanks, Pru._\n\n\"My turn to watch,\" he said.\n\nGelfred was snoring in ten minutes, secure in his lord in a way the captain could not be in himself.\n\nThe captain stared into the darkness, and it became his friend more than his foe. He watched, and as he watched, he felt his heartbeat settle, felt his pains fade. He made an excursion into his palace of memory\u2014reviewing sword cuts, castings, wards, lines of poetry.\n\nBeyond the bubble of his will the night passed slowly. But it did pass.\n\nEventually, the faintest light coloured the eastern sky, and he woke Gelfred as gently as he could. He lowered his ward when they were both awake and armed, but there was nothing waiting for them, and they found the horse, and the head.\n\nJust around the clearing where they'd slept, a pair of deep tracks\u2014cloven, with talons and a dew claw\u2014pierced the forest leaf mold.\n\nGelfred started. The captain watched as he followed the tracks\u2014\n\n\"Are we borrowing trouble, Gelfred?\" he asked, following a few paces behind.\n\nGelfred looked back and pointed at the ground in front of him. When the captain joined him, he saw multiple tracks\u2014perhaps three sets, or even four.\n\n\"What you fought yesterday. Four sets of prints. Here's one moving more slowly. Here's two moving fast\u2014here they pause. Sniffing.\" He shrugged. \"That's what I see.\"\n\nCuriosity\u2014the kind that gets cats killed\u2014pulled the two of them forward. In ten more steps, there were eight or ten sets of tracks, and then, in another ten steps\u2014\n\n\"Sweet Son of Man and all the angels!\" Gelfred said.\n\nThe captain shook his head. \"Amen,\" he added. \"Amen.\"\n\nThey stood on a bank over a gully wide enough for a pair of wagons and a little deeper than the height of a man on a horse. It ran from west to east. The base was clear of undergrowth, like a\u2014a road.\n\nThe whole gully was a mass of churned earth and tracks.\n\n\"It's an army!\" Gelfred said.\n\n\"Let's move,\" said the captain. He turned and ran back to their clearing and settled his gear on the poor horse.\n\nThen they were moving.\n\nFor a while, every shadow held a daemon\u2014until they passed it. The captain didn't feel recovered; he was cold, hungry, and afraid even to make tea. The horse was lame from the cold and from being insufficiently cared for on a cold, damp spring night, and they rode her anyway.\n\nIt turned out they didn't have to go very far, which probably saved her life. The camp's sentries must have been alert, because a mile from the bridge, they were met by Jehannes leading six lances in full armour.\n\nJehannes's eyes were still bloodshot, but his voice was steady.\n\n\"What in the name of Satan were you doing?\" Jehannes demanded.\n\n\"Scouting,\" the captain admitted. He managed to shrug, as if it was a matter of little moment. He was very proud of that shrug.\n\nJehannes looked at him with the look that fathers save for children they intend to punish later\u2014and then he caught sight of the head being dragged in the mud. He rode back to look at it. Bent over it.\n\nHis wide and troubled eyes told the captain that he had been right.\n\nJehannes turned his horse with a brutal jerk of the reins.\n\n\"I'll alert the camp. Tom, give the captain your horse. M'lord, we need to inform the Abbess.\" Jehannes's tone had changed. It wasn't respectful, merely professional. This was now a professional matter.\n\nThe captain shook his head. \"Give me Wilful's horse. Tom, stay at my back.\"\n\nWilful Murder dismounted with his usual ill grace and muttered something about how he was always the one who got screwed.\n\nThe captain ignored him, got a leg over the archer's roncey with a minimum of effort, and set off at a fast trot, Wilful holding onto another man's stirrup leather and running full out, and then they stretched to a racing gallop across the last furlongs, with Wilful seeming to run alongside in ten league boots.\n\nThe guard had already turned out at the camp gate\u2014a dozen archers and three men-at-arms, all in their kit and ready to fight. For the first time since he'd set his spear under his arm the day before, the captain's heart rose a fraction.\n\nThe head dragged in the dirt behind Gelfred's horse left a wake of rumour and staring.\n\nThe captain pulled up before his pavilion and dropped from the saddle. He considered bathing, considered washing the clots of ordure from his hair. But he wasn't positive he had the time.\n\nHe settled for a drink of water.\n\nJehannes, who had paused to speak to the Officer of the Watch, rode up, tall and deadly on his war horse.\n\nTwo archers\u2014Long Sam and No Head, were ramming the head down on a stake.\n\nThe captain nodded at them. \"Outside the main gate,\" he said. \"Where every cottager can see it.\"\n\nJehannes looked at it for too long.\n\n\"Double the guard, put a quarter of the men-at-arms into harness round the clock as a quarter-guard, and draft a plan to clear the villages around the fortress,\" the captain said. He was having trouble with words\u2014he couldn't remember being so tired. \"The woods are full\u2014 _full_ of the Wild. They have amassed an army out there. We could be attacked any moment.\" He seized an open inkwell on his camp table and scrawled a long note. He signed it in big capitals\u2014good, educated writing.\n\n_The Red Knight, Captain._\n\n\"Get two archers provisioned and mounted as fast as you can\u2014a pair of good horses apiece, and on the road. Send them to the king, at Harndon.\"\n\n\"Good Chryste,\" said Jehannes.\n\n\"We'll talk when I've seen the Abbess,\" the captain called, and Toby brought up his second riding horse, Mercy. He mounted, collected Bad Tom with a glance, and rode up the steep slope to the fortress.\n\nThe gate was open.\n\nThat was about to change.\n\nHe threw himself from Mercy and tossed the reins to Tom, who dismounted with a great deal less haste. The captain ran up the steps to the hall and pounded on the door. The priest was watching from his chapel door, as he always watched.\n\nAn elderly sister opened it and bowed.\n\n\"I need to see the lady Abbess as soon as may be,\" the captain said.\n\nThe nun flinched, hid her eyes and closed the door.\n\nHe was tempted to pound on it with his fists again, but chose not to.\n\n\"You and Gelfred killed that thing?\" Bad Tom asked. He sounded jealous.\n\nThe captain shook his head. \"Later,\" he said.\n\nBad Tom shrugged. \"Must have been something to see,\" he said wistfully.\n\n\"You're\u2014listen, not now, eh? Tom?\" The captain caught himself watching the windows in the dormitory.\n\n\"I'd ha' gone wi' you, Captain,\" Tom said. \"All I'm saying. Think of me next time.\"\n\n\"Christ on the cross, Tom,\" the captain swore. It was his first blasphemous oath in a long time, so naturally, he uttered it just as the frightened, elderly nun opened the heavy door.\n\nHer look suggested she had heard a few oaths in her day. She inclined her head slightly to indicate that he should follow her so he climbed the steps and crossed the hall in her wake, to the doorway he'd never passed through but from whence wine had been served, and stools brought.\n\nShe led him down a corridor lined with doors and up a tightly winding stair with a central pillar of richly carved stone, to an elegant blue door. She knocked, opened the door and bowed.\n\nThe captain passed her, returning her bow. He wasn't too tired for courtesy, it appeared. His mind seemed to be coming back to him and he found that he was sorry to have blasphemed in the hearing of the nun.\n\nIt was like the feeling returning to an arm he'd slept on\u2014the gradual retreat of numbness, the pins and needles of returning awareness, except that it was emotion returning, not his senses.\n\nThe Abbess was sitting on a low chair with an embroidery frame. Her west window caught the mid-day rays of the spring sun. Her scene showed a hart surrounded by dogs, a spear already in his breast. Bright silk-floss blood flowed down his flank.\n\n\"I saw you come in. You lost your horse,\" she said. \"You stink of phantasm.\"\n\n\"You are in great peril,\" he replied. \"I know how that sounds. But I mean it, just the same. This is not a matter of a few isolated creatures. I believe that some force of the Wild seeks to take this fortress and the river crossing. If they cannot take it by stealth and subterfuge, they will come by direct assault. And the attack could come at any hour. They have massed, in large numbers, in your woodlands.\"\n\nShe considered him carefully. \"I assume this isn't a dramatic way of increasing your fee?\" she asked. Her smile was subtle, betraying fear and humour in the same look. \"No?\" she asked, with a catch in her voice.\n\n\"My huntsman and I followed the spore\u2014the Hermetical spore\u2014of the daemon that murdered Hawisia,\" he said.\n\nShe waved him to a stool, and he found a cup of wine sitting on the side table. He drank it\u2014the moment the cup touched his lips, he found that he was tilting it back, feeling the acid fire rush down his gullet. He put the cup back down, a little too hard, and the horn made a click on the wood that caused the Abbess to turn.\n\n\"It is bad?\" she asked.\n\n\"We found a man's corpse first. He was dressed as a soldier\u2014as a _Jack._ \" He took a deep breath. \"Do you remember the Jacks, Abbess?\"\n\nHer eyes wandered far from him, off into another time. \"Of course,\" she said. \"My lover died fighting them,\" she said. \"Ah, there's a reason for penance. My lover. Lovers.\" She smiled. \"My old secrets have no value here. I know the Jacks. The secret servants of the Enemy. The old king exterminated them.\" She raised her eyes to his. \"You found one. Or at least you showed me a leaf.\"\n\n\"Dead. Looked as if he had been killed, quite recently, by one of his own.\" The captain found a flagon of wine and poured a second cup. \"I'm going to wager that he died a few hours after Sister Hawisia. Killed by another of his kind, as if that makes sense.\" He shook his head. \"Then we went west, still following the spore.\" He sat down again, a little too hard.\n\nShe watched him.\n\n\"Then we found the creature.\" He stared at her. \"An adversarius. You know what they are?\" he asked.\n\n\"Every person of my generation knows what they are.\" She covered her eyes with her hand for a moment. \"Daemons. The Wardens of the Wild.\"\n\nHe let another long breath go. \"I thought they had been exaggerated.\" He looked out the window. \"At any rate, there were two of them. I can only assume that the Jacks and the daemons are working together. If they are, this cannot be a random incident\u2014I believe they're the harbingers of an attack, testing your strength, and I assume that your fortress is the target. It certainly has immense strategic value. I need to ask you to let my troops in, close the gates, place yourself in a posture of defence, and victual the fortress\u2014call in your people, of course. And send word to the king.\"\n\nShe looked at him for a long time. \"If you planned to take my fortress yourself...\" she said. And left it there.\n\n\"My lady, I agree that it would be a brilliant stratagem. I even agree that I might try something like. I have fought in the East\u2014we did such things there.\" He shrugged. \"This is my country, my lady. And if you doubt me\u2014and you have every reason to doubt me\u2014you have only to look at what my archers are putting up outside the gates of our camp.\"\n\nShe looked out the window.\n\n\"You could tell me that there's an angel of the Lord outside the gates of your camp, telling your archers that I'm the most beautiful woman since Helen, and I couldn't see it well enough to believe you,\" she said. \"But\u2014I have seen you. I can smell the power on you. And\u2014now I understand other things I have seen.\"\n\n\"You are an astrologer,\" he said. _I am slow_ , he thought.\n\n\"Yes. And you are very difficult to read, as if\u2014as if you have some protection from my art.\" She smiled. \"But I am no novice, and God has given me the power to look at souls. Yours is rather curious\u2014as I expect you know.\"\n\n\"Oh, God has been very good to me,\" he said.\n\n\"You mock and are bitter, but we face a crisis, and I am not your spiritual mother.\" Her voice changed, becoming sharper, and yet deeper. \"Although I would be, if you would let me in. You need His spirit.\" She turned away. \"You are armoured in darkness. But it is a false armour, and will betray you.\"\n\n\"So people tell me,\" he said. \"Yet it's served me well so far. Answer me this, Abbess. Who else was at that manor?\"\n\nThe Abbess shrugged. \"Later...\"\n\nThe captain looked at her for a long time. \"Who _else_ was there?\"\n\nShe shook her head. \"Later. It is not the issue now, when I have a crisis of my tenure. I will not fail. I will hold this place.\"\n\nHe nodded. \"So you will put this fortress in a posture of defence?\" he asked.\n\nShe nodded. \"This minute.\" She raised a hand bell and rang it.\n\nThe elderly nun came immediately.\n\n\"Fetch the gate warder and the sergeant at arms. And ring the alarm,\" the Abbess ordered in a firm voice. She went to the mantel on her fireplace, and opened a small box of ivory carved in the Cross of the Order of Saint Thomas. In it was a slip of milk-white birch bark.\n\n\"You're sure about this?\" she whispered.\n\n\"I am,\" he said.\n\n\"I need to share your assurance,\" she said.\n\nHe sat back. \"I could not make this up. You say you smell the power of the phantasm on me\u2014\"\n\n\"I believe that you have met and defeated another monster. It is possible that you found a dead Jack.\" She shrugged. \"It is possible I have a traitor inside my walls. But once I cast this summoning, the Master of My Order will come with all his knights. He will probably demand that the king raise an army.\"\n\n\"That's is just about what is required here,\" said the Red Knight.\n\n\"I cannot have them come to my aid for nothing,\" she said.\n\nThe Red Knight sat back. His back hurt, and his neck hurt, and he felt the dull anger of complete fatigue. He bit back a retort, and then another.\n\n\"What will satisfy you?\" he asked.\n\nShe shrugged. \"I believe you. But I must be sure.\"\n\nHe nodded. Irrationally angry.\n\n\"Fine,\" he said. He rose, and bowed.\n\nShe reached for his hand.\n\nHe stepped back. \"No time like the present,\" he spat.\n\n\"Captain!\" she said. \"You are not a small child.\"\n\nHe nodded, held onto his anger, and stalked out.\n\n\"What did she say?\" Tom asked.\n\n\"She wants us to find their army, not just the signs of it,\" the captain said.\n\nTom grinned. \"That will be a mighty feat of arms,\" he said.\n\nSer Milus had the banner, and the rest of his entourage was ready to mount. But the sergeant at arms stood in the gate with only the postern open. They would have to walk their horses out the gate. Even while cursing this delay, the captain commended the old witch. She took his warning seriously.\n\n\"Captain!\"\n\nHe turned to see Amicia running barefoot across the courtyard.\n\n\"Let's go,\" Tom grunted. \"I'll get a convoy together.\"\n\n\"Twenty lances,\" the captain said.\n\n\"Aye,\" said Tom. He winked as he left.\n\nAmicia reached him. He felt her through the aether as she came up. He could smell her, an earthy, female smell, clean and bright, like a new sword. Like a taste of the Wild.\n\n\"The Abbess sends this,\" she said levelly. She held out a small scroll. \"She says she will take immediate steps, so you are not to think yourself ignored.\"\n\nHe took the scroll from her hand.\n\n\"Thank you,\" he said. He managed a smile. \"I am tired and difficult.\"\n\n\"You have fought for your life,\" she said. Her eyes held his. \"There is no fatigue like that of fear and war.\"\n\nHe might have denied it. Knights don't admit to fear. But her gentle voice held an absolute certainty. It was healing. It was forgiving.\n\nIt was admiring.\n\nHe realised that he had been holding her hand the whole time. She flushed, but did not snatch it away.\n\n\"Lady, your words are a tonic to a tired man.\" He bowed and kissed her hand. It _was_ a tonic. That or she had cast a spell on him unnoticed.\n\nShe laughed. \"I am no lady, but a simple novice of this house,\" she said.\n\nHe tore himself away from her, or they might have stood far too long in the courtyard, with the first sun of the spring resting on them.\n\nHe read the scroll as he rode down the gravel path from the main gate to the Lower Town. Much of the path was walled, and some of it paved, making a fortified road, itself a defence.\n\nSomeone had put a great deal of money into this fortress.\n\nHe cantered through the town. His shoulder didn't hurt at all. But his right hand tingled for another reason entirely, and he laughed aloud.\n\n# Harndon Palace\u2014Desiderata\n\nDesiderata led her knights and ladies out into the spring.\n\nIt was early days yet, and even the heartiest of her bold young friends would not slip into the river naked today. But it was warm enough to ride fast, and to lay a picnic out on blankets.\n\nLady Mary directed the laying out of the food. Spontaneity, with Desiderata, often involved careful preparation and a great deal of work. Usually by Lady Mary.\n\nLady Rebecca Almspend, the Queen's bookish secretary, sat behind her, ticking items off as they were unpacked. They were old allies and childhood friends.\n\nRebecca kicked off her shoes. \"It _is_ spring,\" she said.\n\nMary smiled at her. \"When a young man's fancy turns to war,\" she said.\n\n\"Too true. They've left us for the first foe in the field, and that is enough to turn any girl's head.\" Rebecca frowned. \"I think he'll offer for me. I thought he might before he left.\"\n\nMary pursed her lips, looking at the two stone jars of marmalade\u2014the Queen's favourite. She could eat a great deal of marmalade. \"Did we really bring just two jars?\"\n\n\"Honestly, Mary, the stuff costs the earth\u2014oranges from the south? White sugar from the Islands?\" Rebecca tossed her head. \"She'll have no teeth when she's thirty.\"\n\n\"No one would notice,\" Mary said.\n\n\"Mary!\" Rebecca was appalled to find her friend weeping. She slipped off her stump, and threw her arms around Mary. She was widely known as _sensible_ , which seemed to mean that all of them could cry on her shoulders. In this case, she stood with her stylus in one hand and her wax tablet in the other, clutched to her friend's back, feeling a little foolish.\n\n\"He left without so much as a good-bye!\" Mary said, fiercely. \"Your hillman loves you, Becca! He'll come back for you, or die in the attempt. Murien only loves himself, and I was a fool\u2014\"\n\n\"There, there,\" Rebecca muttered. Over by the willows that lined the river, there was laughter\u2014the flash of the queen's hair.\n\n\"Look, she has her hair down,\" Mary said.\n\nThey both laughed. The Queen tended to let her hair down out of its coif at the least excuse.\n\nRebecca smiled. \"If I had her hair, I'd let it down too.\"\n\nMary nodded. She stepped back from their embrace and wiped her eyes. \"I think we're ready. Tell the servants to start laying plates.\" She looked around at the trees, the angle of the sun. It was beautiful\u2014as spring-like as could be imagined, like a scene in an illustrated manuscript.\n\nAt her word, Mastiff, the Queen's man, stepped out from behind a tree and bowed. He snapped his fingers, and a dozen men and women moved with the precision of dancers to lay out the meal. They were done in the time it would take a man to run to the river.\n\nMary touched Mastiff's elbow. \"You work miracles, as always, ser,\" she said.\n\nHe bowed, obviously pleased. \"You are too kind, my lady,\" he said. He and his team melted back into the trees, and Mary summoned the Queen and her friends to lunch.\n\nThe Queen was barefoot, lightly clad in green with her hair free down her back and her arms bare in the new sun. Some of the young men were fully clad, but two of them, both knights, wore simple homespun tunics and no leggings, like peasants or working men. The Queen seemed to be favouring them\u2014and the short tunics and bare legs did show off their muscles to good advantage.\n\nWhen they sat on the new grass to eat, though, they had to fold their legs very carefully. This made Mary smile, and meet Rebecca's eye who grinned and looked away.\n\nLady Emmota, the youngest of the Queen's lady's, had her hair down as well, and when the Queen sat, Emmota sat next to her and the Queen pulled the girl to lie down with her head in the Queen's lap. The Queen stroked her hair. The young girl gazed at her with adoration.\n\nMost of the young knights were unable to eat.\n\n\"Where is my lord?\" asked the Queen.\n\nLady Mary curtsied. \"And it please you, he is hunting, and said he might join us for lunch if the hart allowed him.\"\n\nThe Queen smiled. \"I am second mistress to Artemis,\" she said.\n\nEmmota smiled up at her. \"Let him have his blood,\" she said.\n\nTheir eyes met.\n\nLater, while the young men fenced with their swords and bucklers, the women danced. They wove wreathes of flowers, and danced in rings, and sang old songs that were not favoured by the church. As the sun began to sink, they were flushed, and warm, down to their kirtles, and now all of them were barefoot in the grass, and the knights were calling for wine.\n\nThe Queen laughed. \"Messires,\" she said, \"none of my ladies will get a green back for the quality of your fencing, however much we ladies may be affected by the rising sap of spring.\"\n\nThe women all laughed. Some of the men looked crestfallen. A few\u2014the best of them\u2014laughed at themselves and their fellows alike, but none of them answered her.\n\nRebecca put a hand on Mary's bare arm. \"I miss him too,\" she said. \"Gawin would have given her a witty answer.\"\n\nMary laughed. \"I love her\u2014and she's right to speak. Emmota will fall into the first strong arms that will have her. It's all the light and the warmth and the bare legs.\" At a motion from the Queen, she walked over and offered a hand to draw the Queen to her feet. The Queen kissed her lady.\n\n\"You arrange everything so well, Mary.\" She took her hands. \"I hope you had a pleasant day as well.\"\n\n\"I am easily pleased,\" Mary said, and the two women smiled at each other, as if enjoying a private joke.\n\nRiding back, they rode three abreast, with the Queen flanked by Lady Mary and Lady Rebecca. Behind them, Emmota rode between two young knights, her head back, laughing.\n\n\"Emmota is vulnerable,\" Mary said carefully.\n\nThe Queen smiled. \"Yes. Let us break up these laughs and long glances. It is far too early in the season.\"\n\nShe straightened her back and gave her horse a check, turned in her saddle like a commander in a tapestry.\n\n\"Gentles! Let us race to the Gates of Harndon!\"\n\nSer Augustus, one of the young men in a peasant's smock, laughed aloud. \"What is the forfeit?\" he called.\n\n\"A kiss!\" called the Queen, and she gathered her horse under her.\n\nOne of the squires blew a horn, and they were away into the fading spring light in a riot of colour and noise, the last of the sun on brilliant greens and blues and brght scarlet, gold and silver.\n\nBut the Queen's kiss was never in danger. Her southern mare seemed to scarcely touch the road as she skimmed along, and the Queen was the first horsewoman in her court\u2014back straight, shoulders square, hips relaxed, and the two of them seemed like a single creature as they led the excited pack of young courtiers along the road, over the bridge, and up the long hill, recently lined with fine houses, to the gates of the city.\n\nThe Queen touched her crop to them, first of all the pack by two lengths, and Lady Rebecca was second, flushed and delighted at her own prowess.\n\n\"Becca!\" cried the Queen in delight. As the others rode up, she kissed her secretary. \"You are riding more for your hillman?\"\n\n\"Yes,\" she said modestly.\n\nThe Queen beamed at her.\n\n\"Are you the Queen, or has some wild hussy stolen the Queen's horse?\" said a voice from inside the gate, and Diota emerged. \"Put your hair up, _my lady._ And put some decent clothes on.\"\n\nThe Queen rolled her eyes.\n\n# Lissen Carak\u2014The Red Knight\n\nThe Red Knight drank off a cup of wine from the saddle. He handed the cup down to Toby.\n\n\"Listen up, messires,\" he said. \"Gelfred\u2014we have to assume their camp is between us and Albinkirk.\"\n\nGelfred looked around. \"Because we didn't come across it last night, you mean?\"\n\nThe captain nodded. \"Exactly. Let's look at this for a moment. The farm that was hit was east of the fortress.\"\n\nSer Jehannes shrugged. \"You found the dead Jack west of here, though. And it stands to reason he was returning to camp.\"\n\nThe captain looked at him for a moment, and then shook his head. \"Damn,\" he said. \"I hadn't thought of that.\"\n\nBad Tom leaned in. \"Can't be south. They can't be across the river.\"\n\n\"West and north, I'm thinking,\" said Gelfred. \"I'm sensing there's a high ridge that way, that runs parallel to the ridge that the fortress is on.\"\n\n\"This could take days,\" Ser Jehannes said.\n\nThe captain seemed to glow with vitality, an impossible feat for a man who had fought two monsters in three days.\n\n\"Messires,\" he said, \"This is what we do. All the men-at-arms in the centre, in one group. Pages will ride ahead, ten horse lengths between men. We will stop whenever I whistle, and dismount. And _listen._ The archers will follow well to the rear, also in a long skirmish line. In the event of a fight, the archers will close on the battle and the men-at-arms will remain under my command. Because we are _not_ going out to fight. We are going out to find evidence of a force of the Wild mustering. The only occasion to fight will be to rescue one of our scouting parties.\" His voice was clipped, professional, and had the self-assurance of a prince. Even Jehannes had to admit his plan was correct.\n\n\"Gelfred, when we locate their camp, we will make a brief demonstration.\" He grinned. \"To occupy their attention.\" He winked at Cuddy, who nodded.\n\n\"I'm thinking you mean an archery demonstration,\" he said.\n\nThe captain nodded and continued. \"You and your men will conceal yourselves nearby and report what happens when we leave. We will withdraw due east, and come down into the Vale of the Cohocton. If there is pursuit, they will have the sun in their eyes.\" The captain looked at Cuddy. \"If we are pursued\u2014\"\n\n\"I dismount the lads and ambush your pursuers. If I ain't been hit myself.\" He nodded. \"I know the game.\"\n\nThe captain clapped his armoured shoulder. \"Everyone see it?\"\n\nHis squire, Michael, was pale. \"We're going out into the woods, looking for an army of creatures of the Wild?\" he asked.\n\nThe Red Knight smiled. \"That's right,\" he said.\n\nAs their leader turned his war horse and raised his baton to give an order, Jehannes turned to Tom. \"He's drunk.\"\n\n\"Nah. He's a loon, like I am. He wants a fight. Give him his head.\" Tom grinned.\n\n\"He's drunk!\" Jehannes repeated.\n\nSer Milus shook his head. \"Only on love,\" he said.\n\nJehannes spat. \"Worse and worse.\"\n\nThey rode west first, and the road was very familiar. As soon as they reached the edge of the woods, the pages split off, riding ahead, their skirmish line widening and widening to the north. The men-at-arms turned into the woods behind them in a compact mass, and then came the archers. Gelfred rode with the captain, and his scouts were nowhere to be seen.\n\nAfter enough time to terrify most of the pages, who rode in fear of imminent ambush by unimaginable monsters, the captain's whistle rang out.\n\nEvery man reined in his horse and slipped to the ground.\n\nThey were still for a long time.\n\nThe captain's whistle sounded again, two long blasts.\n\nThey mounted and rode forward. It was late afternoon. The sky had patches of blue, and a man could be warm from the sun, the weight of his harness, and his nerves.\n\nOr cold, from the same causes.\n\nMen tire quickly when they are scared. A patrol in hostile terrain is the most tiring thing a soldier can do short of violence. The captain blew his whistle each time he had completed a silent count to fifteen hundred. Stopping gave his men a rest.\n\nThe sun began to slant more, and the light grew redder. The sky to the west was clear.\n\nThey began to climb Gelfred's ridge, and the tension began to grow.\n\nAbout halfway up the ridge, the captain's whistle sounded, and the company dismounted.\n\nThe captain motioned to Michael, who stood at his shoulder.\n\n\"Whistle: horseholders.\"\n\nMichael nodded. He took off his right gauntlet, picked up the silver whistle on the cord around his neck, and blew three long and three short notes. After a pause, he blew the same call again.\n\nAll around them, men-at-arms handed their horses to squires. Behind them, at the base of the hill, every sixth archer took the horses of his mates and led them to the rear.\n\nThe captain watched it all, wondering if the pages, who he couldn't see, were also obeying.\n\nHe could _feel_ the enemy. He could smell the green of the Wild. He listened, and he could almost hear them. Idly, he wondered why Amicia smelled like the Wild.\n\nThere was a distant trumpeting noise, like the belling of a hart.\n\n\"Jehannes, you have the men-at-arms. I'm going to take command of the pages. Michael, on me.\" He handed his reins to Toby and started up the hill. His harness was almost silent, and he moved fast enough to leave Jehannes's protests behind.\n\nBad Tom stepped out and followed him.\n\nThe hill was steep, and the pages were two hundred paces further up the ridge. He breathed in relief when he saw them\u2014too clumped up, but all dismounted, and he passed a boy of fifteen with six horses headed down the hill.\n\nClimbing a steep ridge in armour reminded him of just how little sleep he'd had since the first fight, against the wyvern, but through his fatigue he could still feel the place on his fingers where Amicia had touched him.\n\nMichael and Tom had trouble keeping up with him.\n\nHe reached the pages. Jacques had them spreading out already. He smiled at the captain.\n\n\"Nice job,\" he whispered.\n\n\"We're going to the top, I take it?\" asked Jacques.\n\nThe captain looked right and left. \"Yes,\" he said. He motioned to Michael, who gave one whistle blast.\n\nThe pages were lightly armed. They weren't woodsmen, but they slipped up the hill like ghosts, at a pace that left the captain breathless. The hill steepened and steepened as they climbed, until the very top was almost sheer, and the pages hauled themselves up from tree to tree.\n\nThere was a scream, a wicked hiss of arrows, a boy of no more than sixteen roared, \"For God and Saint George!\" and there was the unmistakable sound of steel on steel.\n\nAn arrow, nearly spent, rang off the captain's helmet.\n\nSuddenly, he had the spirit to run to the crest of the hill. The trees were dense, and branches reached for him, but a man in armour can run through a thicket of thorns and not take a scratch. He grabbed a slim oak, pulled with all his strength, and found himself at the top.\n\nThere was a small hollow, with a fire hidden by the bulk of the hill, and a dozen men.\n\nNot men.\n\nIrks.\n\nLike men, but thinner and faster, with brown-green skin like bark, almond eyes and pointed teeth like wolves. Even as the captain stopped in surprise, an arrow rang off his breastplate and a dozen pages burst from the trees to the right of the irks around the fire and charged.\n\nThe captain lowered his head and ran at the irks, too.\n\nThey loosed arrows and fled away north, and the pages gave chase.\n\nThe captain stopped and opened his visor. Michael appeared at his side, sword out, buckler on his left hand. He could smell woodsmoke, lots of woodsmoke.\n\n\"We've found them!\" Michael said.\n\n\"No. A dozen irks is not an army of darkness,\" said the captain. He looked at the sky.\n\nTom came up behind him.\n\n\"Tom? We have an hour of good light. The pages are running down their sentries.\" He looked at the veteran man-at-arms. He shrugged. \"I don't really know all that much about fighting the Wild,\" he admitted. \"My instinct is to keep going forward.\"\n\nTom nodded. \"It's the Wild,\" he said. \"They never have a reserve. Yon won't have anything like a quarter guard.\" He shrugged.\n\nThe captain knew the decision they made now was pivotal. Any losses out here didn't bear thinking about. Caution would dictate\u2014\n\nHe thought of her touch on his hand. Her admiration.\n\nHe turned to Michael. \"Tell the archers to prepare an ambush half a league back. men-at-arms to guard the horses at the base of the ridge. This is the pages only. Understand?\"\n\nMichael nodded. \"I want to come with you.\"\n\n\"No. Give me your whistle. Now move! Tom, with me.\"\n\nThey ran down the northern side of the ridge, toward the sound of screams and fighting.\n\nLater, the captain admitted that he'd let the pages get too far ahead of him. The deep woods and fading light made it almost impossible to maintain communications.\n\nHe ran down the ridge with Tom beside him, crashing recklessly through thickets. He all but fell into a steep-sided vale; a small stream that cut deeply into the side of the ridge. It was easier to go east, so he followed it, passing three corpses\u2014all irks.\n\nAt the base of the ridge, with his breath coming in great shuddering gasps, there was a shallow stream and, on the far side, a path. And along the path\u2014\n\nTents. But no pages.\n\nThere were fifty men, most of them stringing bows.\n\nThe captain stopped. He'd made enough noise coming down the ridge to catch their attention but with the sun at his back, despite his armour, they were easier for him to see than he was for them.\n\nTom and Jacques and a dozen pages who had followed them down the hill slipped in behind old trees. There were screams off to the west\u2014screams and something else.\n\n\"Fucking Jacks,\" Jacques said.\n\nThe men across the stream turned, almost as one. A small horde of boglins and irks bolted down the path from the west. It was odd to see the monsters of myth running.\n\nThe Jacks began to shuffle.\n\nSeveral of them drew their great bows and shot west.\n\nThe captain looked around. \"Follow me,\" he said. \"Make a _lot_ of noise.\"\n\nThey all looked at him.\n\n\"One. Two. Three.\" He broke cover, and bellowed \"THE RED KNIGHT!\"\n\nThe effect was electric. The captain was south of and slightly behind the line of Jacks, and they had to look over their shoulders to see him. Immediately, men began to flee with the boglins and the irks.\n\nThe pages behind him roared his battlecry, and Bad Tom roared his\u2014\"Lachlan for Aa!\"\n\nThere are different types of soldier. Some men are trained to stand under fire, waiting for their turn to inflict death. Others are like hunters, slipping from cover to cover.\n\nThe Jacks were not of a mind to stand and fight. It wasn't their way. One arrow, launched from a mighty bow, slammed into the captain's scarlet surcote, punched through it, and left a dent a finger deep and bruised him like a kick from a mule. And then the Jacks were gone.\n\nThe captain grabbed Bad Tom by the shoulder. \"Stop!\" he roared.\n\nTom's eyes were wild. \"I have nae' wet my sword!\" he shouted.\n\nThe captain kept a hand on him, like a man calming a favourite dog. He blew the recall on the whistle\u2014three long blasts, and then three more, and then three more.\n\nThe pages stopped. Many wiped their swords on dead things, and all of them drank from their water bottles.\n\nFrom the east came a long scream. It was an alien sound, and it sobered them.\n\n\"Up and over the ridge. Straight back the way we came, tight and orderly. Now.\" The captain pointed his sword up the ridge. \"Stay by the stream!\" he called.\n\nNow there was a baying and roaring in the woods to the east. Roaring, infernal screams, and something else, something that was huge and terrible and fell, and as tall as the trees.\n\nHe turned to run up the ridge.\n\nTom was still at his shoulder. \"I have nae killed a one!\" he said. \"Just let me kill one!\"\n\nSuiting action to word, Tom turned as a gout of green fire smashed into the ground, not two horselengths from Tom's outstretched sword. It exploded with a roar and suddenly the very stones seemed to be on fire.\n\nTom smiled and raised his sword.\n\n\"Tom!\" the captain screamed. \"This is not the time!\"\n\nBoglins and irks were crossing the stream at the foot of the ridge, led by a golden bear, as tall as a war horse and shining gold like the sun. When it roared, its voice filled the woods like a storm wind.\n\n\"What the _fuck_ is that?\" asked Tom. \"By god, I want a cut at that!\"\n\nThe captain pulled hard at the hillman's arm. \"With me!\" he ordered, and ran.\n\nGrudgingly, Tom turned and followed him.\n\nThey made the top of the ridge. The bear was not charging them, it seemed content to lead the boglins and the irks. But behind them came something far worse. And much larger.\n\nThe pages had waited for the captain a little way down the ridge, in itself an act of fine discipline and bravery. But as soon as he caught them up, they turned and ran for the base and their horses.\n\nThe captain could barely move his steel clad feet, and never had leg armour seemed so pointless, so heavy, as it did when the first of the enemy began to crest the hill behind him. They were close.\n\n# West of Lissen Carak\u2014Thorn\n\nThorn's initial reaction to the assault on his camp was panic. It took him long minutes to recover from the shock and when he did, the sheer effrontery of it filled him with an irrational rage. As he reached out through his creatures, he was shocked to find how pitiful and few were his human attackers. A few dozen of them, and they had sent his Jacks running down the path, broken fifty irks, and killed an outpost of boglins who were caught napping after a feed.\n\nHe stopped the rout by killing the first irk to pass him, in spectacular style. The creature exploded in green fire, raining burning flesh on the others, and the Magus raised his hoary arms and the rout stopped.\n\n\"You fools!\" he roared at them. \"There are fewer than fifty of them!\" He wished he had his daemons but they were already scouting Albinkirk. His wyverns were close, but not close enough. He poured his will into two of the golden bears and sent his forces up the ridge after the raiders. His Wild creatures would be far more nimble in the woods then mere men. The bears were faster than horses on their home ground.\n\nOne of the boglin chiefs stood at his side, his milk-white chiton all but glowing in the setting sun.\n\n\"Tell your people that they will feast. Anything they catch is for their own.\"\n\nExrech saluted with a sword. He released a cloud of vapour\u2014part power and part scent. And then he was away, racing loose-limbed up the ridge with boglins following like a brown tide at his heels.\n\n# West of Lissen Carak\u2014The Red Knight\n\nThe captain tried to be the last man, shoving his flagging pages along before him by force of will, but the weaker among them were used up. One, a little plumper than he ought to have been, stopped to breathe hard.\n\nThe enemy were fifty paces away. Closer with every heartbeat.\n\n\"Run!\" roared Tom.\n\nThe boy threw up, looked behind him and froze.\n\nA boglin paused and shot him with an arrow.\n\nHe screamed and fell, kicking, into his own vomit.\n\nTom heaved the writhing boy over his shoulders and ran. His sword licked out\u2014caught an irk in the top of the knee, and the thing screamed and fell, clutching at the wound.\n\nThe captain paused\u2014they were trying to surround him. He punched at the nearest and impaled him, took two cuts on his leg armour, and suddenly it _had_ been worth it to wear the stuff all afternoon.\n\nThere were, in moments, _hundreds_ of boglins. They seemed to boil up out of the ground in terrifying numbers. They moved like ants and covered the forest floor as fast as he could back away. Their armoured heads rose above his knight's belt.\n\nBehind him, he heard a trumpet call and Cuddy's voice, as clear as on parade, called \"Nock! And Loose!\"\n\nThe captain was still on his feet, but there was a sharp pain in his left thigh where a boglin was trying to sink its jaws into his flesh, and his legs were all but immobilized by the press of creatures when something reached for his soul through the aether.\n\nHe panicked.\n\nHe couldn't see. The brown boglins were everywhere, clamping onto him, and he wasn't fighting anymore, he was just trying to keep his feet, and the pressure of the phantasm was bearing down harder and harder on his soul.\n\nThen, even through his helmet and his fear, he could hear the hiss of the warbow arrows, like the fall of vicious sleet.\n\nThe arrows hit.\n\nThree of them hit him.\n\n# West of Lissen Carak\u2014Thorn\n\nThorn paused at the top of the ridge to watch the last moments of the raiding party. The boglins weren't as fast as the irks, but the irks were running the enemy down. The tide of boglins would finish the fight.\n\nAny fight.\n\nHe prepared a casting, gathering the raw force of nature to him through a web of half-rational portals and paths.\n\nAt the base of the ridge, one of the fleeing raiders paused.\n\nThorn reached out for him, grasped him and felt his will slip off the man like claws around a stone.\n\nAnd then fifty enemy archers stood up from concealment, and began to fill the air with wood and iron.\n\n# West of Lissen Carak\u2014The Red Knight\n\nThe captain was hit more than a dozen times more. Every strike was like being kicked by a mule. Most fell on his helmet, but one ripped across his inner thigh, cutting through his hose and his braes. He was blind with pain, dazed by the repeated impacts.\n\nBut he was armed cap \u00e0 pied in hardened steel armour, and the boglins trying to kill him were not.\n\nWhen every one of Cuddy's archers had loosed six shafts, the V-shaped space between the arms of the ambush was silent. Nothing was left alive.\n\nCuddy ordered his men forward to collect their shafts as the captain raised his visor, aware that there was still something\u2014\n\nAt the top of the hill, the figure of horror stepped out where they could all see him, and raised his arms\u2014\n\nHe still functioned through the panic because he'd been afraid so damned often he was used to it now.\n\n_The captain touched Prudentia's hand. Above his head, the three great levels of his palace spun like gaming wheels._\n\n_\"Don't open the door!\" Prudentia said. \"He's right there!\"_\n\n_Faced with imminent immolation, the captain opened the door._\n\n_There was an entity of the Wild. Right outside the door to his mind._\n\n_He made a long, sharp dagger of his will and punched it into the entity, leaning out through the door to do so._\n\n_Prudentia caught him._\n\n_The door slammed shut._\n\n_\"You're insane,\" she said._\n\nIn the world the great figure stumbled. It didn't fall, but the intensity of its gathered power stumbled with it. And dissipated.\n\n\"To horse!\" he captain roared. Behind the monstrous figure on the ridge he could see thrashing tentacles approaching and fresh hordes of monsters.\n\nThe massive thing, like two twin trees, reared up and a flash of green fire covered the hillside. It fell shorter than it might have, or more men might have died, but archers were reduced to bones\u2014a page burned green like a hideous barn-lamp for three heartbeats before vanishing\u2014and dozens of wounded creatures on the ground were immolated as well.\n\nBehind him, men were mounting\u2014pages and archers hurried horses to their riders. This was their most practised movement: escape.\n\nBut the captain's sense of the enemy was that he'd get one more gout of fire in.\n\nHe got a leg over Grendel's saddle and\n\n_Passed back into the palace._\n\n_\"Shield, Pru!\" he called. He pulled raw power from the sack hanging on her arm as the sigils turned above them\u2014Xenophon, St. George, Ares._\n\n_The first spell any magister learned. The measure of an adept's power._\n\nHe made a buckler, small and nimble, and threw it far forward, into his adversary's face.\n\nBehind him, the corporals ordered men into motion, but they needed no urging, and the company moved away, down the hill.\n\nThe captain turned Grendel and rode, running as fast as the heavy horse would allow\u2014\n\nThe two-horned thing in the woods reached out with his staff\u2014\n\nThe captain's shield\u2014his very strongest, smallest, neatest casting\u2014-vanished like a moth in a forge fire.\n\nThe captain felt his shield go\u2014felt it vanish\u2014had a taste of the sheer power of his adversary\u2014but training told.\n\nQuick as a cat pouncing, the captain spun his horse to face the foe and\n\n_reached in and cast again\u2014a wider arc to cover horse and rider_\n\nThe green fire ran across the ground like a rising tide, immolating everything that lay in its path\u2014scarring trees, reaping grass and flowers, boiling squirrels in their own skin. It struck the air in front of Grendel's chamfron\u2014\n\nIt was like watching a sand-castle give way under the power of the waves.\n\nHis second shield was weaker, but the green fire had crossed hundreds of paces of ground and its puissance was ebbing\u2014and still it eroded the shield\u2014slowly, and then more quickly as Grendel half-reared in panic, alone in a sea of incandescent green.\n\n_He put everything he had\u2014every shred of stored power_\n\nHe could smell burning leather, and he could see\u2014trees. Upright and black.\n\nGrendel screamed and bolted.\n\nAll he wanted to do was sleep, but Cuddy needed reassurance. \"You was in full harness\u2014\" said the Master Archer.\n\n\"It was the right decision,\" the captain agreed.\n\n\"I can't believe we hit you so many times,\" Cuddy said, shaking his head. Even as he spoke, Carlus, the armourer and company trumpeter, was working with heat and main strength to get the dents out of the captain's beautiful helmet.\n\n\"I'll be more careful to whom I give extra work details in future,\" the captain agreed.\n\nCuddy left the tent, still muttering.\n\nMichael got his captain out of the rest of his armour. The breast plate was badly dented in two places. The arm harnesses were untouched.\n\n\"Wipe my blade first,\" muttered the captain. \"Boglins; I've heard their blood is caustic.\"\n\n\"Boglins,\" Michael said. He shook his head. \"Irks. Magic.\" He took a deep breath. \"Did we win?\"\n\n\"Ask me that in a month, young Michael. How many did we lose?\"\n\n\"Six pages. And three archers, in the retreat when yon _thing_ began to rain fire on us.\" Michael shrugged.\n\nTheir retreat had become a rout. Most of the men had ridden back to camp almost blind with terror, as more and more monsters crested the ridge and entered the field, following that fire-raining figure of terror.\n\n\"Well.\" The captain allowed his eyes to close for a moment and then jolted awake. \"Son of a bitch. I have to tell the Abbess.\"\n\n\"They might attack us again, any moment,\" Michael said.\n\nThe captain gave him a hard look. \"Whatever they are, they aren't so different from us. They know fear. They do not want to die. We hurt them today.\" _They hurt us, too. I was too rash. Damn it all._\n\n\"So now what happens?\" Michael asked.\n\n\"We scurry into the fortress. And that thing comes and lays siege to us.\" The captain got slowly to his feet. For a moment, absent the weight of his harness, he felt as if he could fly. Then the fatigue settled again like an old and evil friend.\n\n\"Attend me,\" he said.\n\nThe Abbess received him immediately.\n\n\"It seems you were correct. Your men look badly beaten.\" She averted her eyes. \"That was unworthy,\" she allowed.\n\nHe managed a smile. \"My lady, you should see the state they're in.\"\n\nShe laughed. \"Is that cockiness or truth?\"\n\n\"I think we killed a hundred boglin and fifty irks. Perhaps even a few Jacks. And we kicked the hornet's nest.\" He frowned. \"I saw their leader\u2014a great horned creature. Like a living tree, but malevolent.\" He shrugged, trying to forget his panic. Tried to keep his voice light. \"It was huge.\"\n\nShe nodded.\n\nHe put that nod away for future consideration. Even in his fatigue, he caught that she knew something.\n\nShe went to the mantel of her chimney and picked up her curious ivory box. This time, she opened it and took the slip of bark between her hands. It turned jet-black. He felt her casting. Then she threw it in the fire.\n\n\"What shall I do now?\" the captain asked. He was too tired to think.\n\nShe pursed her lips. \"You tell me, Captain,\" she said. \"You are in command.\"\n\n# Lissen Carak\u2014Father Henry\n\nFather Henry watched the mercenary come down the steps of the Great Hall with the Abbess on his arm, and his skin crawled to watch that spawn of Satan touch her. The man was young and pretty, for all his bruises and the dark circles under his eyes, and he had an air about him that Father Henry knew in his soul was all pretence; the sham of concern and the worm of falsity.\n\nThe big mercenary barked a laugh. And then the sergeant at arms and the master warder both appeared from the donjon tower.\n\nFather Henry knew his duty\u2014knew that he could not allow major decisions to be made without him. He walked forward to join them.\n\nThe Abbess gave him a look that he suspected was meant to drive him away, but he schooled his face to hide his feelings and bowed to the loathsome killer and his minion.\n\nThe master warder rolled his eyes. \"Nothing for you here, Father,\" he said.\n\nThe old soldier had never liked him, had never made a confession.\n\nThe mercenary returned his bow pleasantly enough, but the Abbess didn't introduce him or let any one else do so. She indicated the mercenary. \"The captain is now the Commander of this fortress. I expect all of you to give him your ready obedience.\"\n\nThe master warder nodded and the sergeant at arms, who commanded the tiny garrison, merely bowed. A possible ally, then.\n\n\"My lady!\" Father Henry rallied his arguments. His thoughts were a riot of confused images and conflicting motives, but they were united by the knowledge that _this man must not be given command of the fortress._ \"My lady! This man is an apostate, an unrepentant sinner, a bastard child of an unknown mother by his own admission.\"\n\nThe mercenary now looked at him with reptilian hate.\n\n_Good._\n\n\"I've never suggested my mother was unknown,\" he said with mild condescension.\n\n\"You cannot allow this piece of _scum_ into our fortress,\" the priest said. He was too vehement, he could see them closing their minds against him. \"As your spiritual adviser\u2014\"\n\n\"Father, let us continue this conversation at a more seemly time and place,\" the Abbess said.\n\nOh, how he hated her tone. She spoke to him\u2014him, a man, a _priest_ \u2014as if he was an errant child and just for a moment the quality of his rage must have shown through, because all of them\u2014except the mercenary\u2014took a step back.\n\nThe mercenary, on the other hand, looked at him as if seeing him for the first time, and gave a sharp nod.\n\n\"I feel you are making a grave error, my lady,\" the priest began again, but she turned on him with a speed that belied her years and put her hand on the pectoral cross he wore.\n\n\"I understand that you disagree with my decision, Father Henry. Now please _desist_.\" Her tone of ice froze him in place.\n\n\"I will not stop while the power of the Lord\u2014\"\n\n\" _Me Dikeou!_ \" she hissed at him.\n\n_The bitch was using arcane powers on him._ And he found himself unable to speak. It was as if his tongue had gone to sleep. He couldn't even form a word in his mind.\n\nHe staggered back, scarred hands over his mouth, all of his suspicions confirmed and all of his petty errors transformed into acts of courage. She had used witchcraft against him. She was a witch\u2014an ally of Satan. Whereas he\u2014\n\nShe turned to him. \"This is an emergency, Father, and you were warned. Return to your chapel and do penance for your disobedience.\"\n\nHe fled.\n\n# North of Lissen Carak\u2014Thorn\n\nThorn strode east as fast as his long legs would carry him, a swarm of faeries around his head like insects, feeding on the power that clung to him like moss to stone. \"We continue,\" he said to the daemon at his side.\n\nThe daemon surveyed the wreckage of tents and the scatter of corpses. \"How many did you lose?\" he asked. His crest moved with agitation.\n\n\"Lose? Only a handful. The boglins are young and unprepared for war.\" The great figure shook like a tree in the wind.\n\n\"You took a wound yourself,\" Thurkan said.\n\nThorn stopped. \"Is this one of your dominance games? One of them distracted me. He had a little magic and I was slow to respond. It will not happen again. Their attack had no real affect on us.\"\n\nThe great figure turned and shambled east. Around him, irks and boglins and men packed their belongings and prepared to march.\n\nThurkan loped alongside, easily keeping pace with the giant sorcerer. \"Why?\" he asked. \"Why Albinkirk?\"\n\nThorn stopped. He despised being questioned, especially by a troublemaker like Thurkan, who saw himself\u2014a mere daemon\u2014as his peer. He longed to say, \"Because I will it so.\"\n\nBut this was not the moment.\n\n\"Power summons power,\" he said.\n\nThurkan's head-crest trembled in agreement. \"So?\" he asked.\n\n\"The irks and boglin hordes are restless. They have come here\u2014was that at your bidding, daemon?\" Thorn leaned at the waist. \"Well?\"\n\n\"Violence summons violence,\" Thurkan said. \"Men killed creatures of the Wild. A golden bear was enslaved by men. It cannot be borne. My cousin was murdered; so was a wyvern. We are the guardians. We must act.\"\n\nThorn paused, and pointed his staff. They were passing to the north of the great fortress; it was just visible from here, high on its ridge to the south.\n\n\"We will never take the Rock with the force we have,\" Thorn said. \"I might act to destroy it, or I might not. This is not my fight. But we are allies, and I will help you.\"\n\n\"By leading us away from that which we wish to reclaim?\" snapped the daemon.\n\n\"By unleashing the Wild against a worthy goal. An _attainable_ goal. We will strike a blow that will rock the kingdoms of man, and that will send a signal throughout the Wild. Many, many more will come to us. Is this not so?\"\n\nThurkan nodded slow agreement. \"If we burn Albinkirk, many will know it and many will come.\"\n\n\"And then,\" said Thorn, \"we will have the force and the time to act against the Rock, while the men worry over smoking ruins.\"\n\n\"And you will be many times more powerful than you are now,\" Thurkan said suspiciously.\n\n\"When you and yours can again drink from the spring of the Rock, and mate in the tunnels beneath the Rock, you will thank me,\" Thorn said.\n\nTogether, they began to walk east.\n\n# Chapter Six\n\n# Prynwrithe\u2014Ser Mark Wishart\n\nTwo hundred leagues and more south of the Cohocton, well west of Harndon, the Priory of Pynwrithe was a beautiful castle rising from a spur of solid rock, a hundred years old, with high battlements, four slim towers with arched windows, topped in copper-gilt roofs, and a high arched gate that made some visitors exclaim that the whole must have been built by the Faery.\n\nSer Mark Wishart, the Prior, knew better. It had been built by a rich thug, who had given it to the church to save his soul.\n\nIt was a very comfortable place to live. A dream for a soldier who had lived most of his life having to sleep on the cold hard ground. The Prior was standing in his shirt, in front of a roaring fire, with a piece of bark in his hand\u2014a small piece of birch bark, which had just turned almost perfectly black. He turned it over and over in his hands, and winced at the pain in his shoulder. The she-bear had hurt him badly.\n\nIt was a chilly morning, and from the glazed glass window, he could see that there had been a frost\u2014but a mild one. Spring was in the air. Flowers, crops, new life.\n\nHe sighed.\n\nDean\u2014his new servant boy\u2014appeared with a cup of small beer and his clean mantle. \"My lord?\" he said, an evocative question, for two words.\n\nThe boy was far too intelligent to spend his life pouring hippocras for old men.\n\n\"Hose, braes, double, and a cote, lad,\" the Prior said. \"Summon the marshal and my squire.\"\n\nThomas Clapton, the Marshal of the Order of Saint Thomas of Acon, was in his solar before the Prior had his hose laced to his doublet\u2014-something he could not get used to allowing a servant to do.\n\n\"My lord,\" the marshal said, formally.\n\n\"What's our fighting strength, right now?\" the Prior asked.\n\n\"In the priory?\" the marshal asked. \"I can find you sixteen knights fit to ride this morning. In the Demesne? Perhaps fifty, if I give you the old men and boys.\"\n\nThe Prior lifted the birchbark and his marshal went pale.\n\n\"And if we make knights of all our squires who are ready?\" Ser Mark asked.\n\nThe marshal nodded. \"Then perhaps seventy.\" He rubbed his beard.\n\n\"Do it,\" said Ser Mark. \"This isn't some minor incursion. She would never call us unless it was war.\"\n\n# Harndon Palace\u2014Harmodious\n\nHarmodius cursed his age and peered into the silver mirror, looking for any redeeming features and finding none. His bushy black and white eyebrows did not recommend him as a lover and nor did his head; he was bald on top with shoulder length white hair, the ruined skin of age and slightly stooped shoulders.\n\nHe shook his head, more at the foolishness of desiring the Queen than at his reflection. He admitted to himself that he was happy enough with his appearance, and with its reality.\n\n\"Hah!\" he said to the mirror.\n\nMiltiades rubbed against him, and Harmodius looked down at the old cat.\n\n\"The ancients tell us that memory is to reality as a seal in wax is to the seal itself,\" he said.\n\nThe cat looked up at him with aged disinterest.\n\n\"Well?\" he asked Miltiades. \"So is my memory of the image of myself in the mirror a new level of removal? It's the image of an image of reality?\" He chuckled, pleased with the conceit, and another came to him.\n\n\"What if you could perform a spell that altered what we saw between the eye and the brain?\" he asked the cat. \"How would the brain perceive it? Would it be reality, or an image, or an image of an image?\"\n\nHe glanced back at the mirror. Pursed his lips again and began to climb the stairs. The cat followed him, his heavy, four-foot gait an accusation and a complaint about overweight infirmity.\n\n\"Fine,\" Harmodius said, and turned to scoop Miltiades up, putting a hand in the middle of his back at the pain. \"Perhaps I could exercise more,\" he said aloud. \"I was a passable swordsman in my youth.\"\n\nThe cat's grey whiskers twitched a reproach.\n\n\"Yes, my youth was quite some time ago,\" he said.\n\nSwords, for example, had changed shape since then. And weight.\n\nHe sighed.\n\nAt the top of the stairs he unlocked the door to his sanctum and reset the light wards he'd left on the place. There was very little to guard here, or rather, his books and many artefacts were supremely valuable, but it was the king that guarded them, not the lock and the wards. If he ever lost the king's confidence\u2014\n\nIt didn't bear thinking about.\n\nWanting Desiderata must be the common denominator of the entire court, he thought and laughed, mostly at himself, before going to the north wall where shelves of Archaic scrolls, many of them gleaned from daring raids into the necropoli of the distant southlands, waited for him like pigeons in a cote. _I used to be a very daring man._\n\nHe deposited Miltiades on the ground and the cat walked heavily to the centre of the room and sat in the sun.\n\nHe began to read on the origins of human memory. He picked up a day-old glass of water and drank from it, tasting some of yesterday's flames and a little chalk, and said \"Hmm,\" a dozen times as he read.\n\n\"Hmmm,\" he said again, and carefully re-rolled the scroll before sliding it back into the bone tube that protected it. The scroll itself was priceless\u2014one of perhaps three surviving scrolls of the Archaic Aristotle, and he always meant to have it copied but never did. He was tempted, sometimes, to order the destruction of the other two, both held in the king's library.\n\nHe sighed at his own infantile pride.\n\nThe cat stretched out in the sun and went to sleep.\n\nThe other two cats appeared. He didn't know where they had been\u2014and suddenly wasn't sure he could remember when he'd adopted them, or where they had sprung from at all.\n\nBut he had found the passage that he remembered, about an organ in the tissue of the brain that transmitted the images from the eye for the mind.\n\n\"Hmm,\" he said to himself with a smile, and reached down to pat the old cat who bit his hand savagely.\n\nHe jerked his bloody hand back and cursed.\n\nMiltiades got up, walked a few steps and settled again. Glared at him.\n\n\"I need a corpse. Perhaps a dozen of them,\" he said, flexing his fingers and imagining the dissection. His master had been quite enamored of dissection... and it had not ended well.\n\nIt had led him to make a stand with the Wild at the Field of Chevin. The old memory hurt, and Harmodius had an odd thought\u2014he thought _when did I last think about the fight at Chevin?_\n\nIt poured into his mind like an avalanche, and he staggered and sat under the impact of the memories\u2014the strange array of the enemy, with Jacks on the flanks and all their monstrous creatures in the centre, so that the kingdom's knighthood was raked with arrows as they rode forward through the waves of terror to face the creatures of the Wild.\n\nHis hands shook.\n\nAnd his master had stood with them. And thrown carefully considered workings designed to baffle and deceive, that had led the king's archers to loose their shafts into their own knights, and to fight each other\u2014\n\n_And so I attacked him_. Harmodius didn't treasure the memory, or that of the king begging him to do _something._ The suspicion of the barons, each assuming he would betray them and join the Wild as well.\n\nHis master's eyes when they locked wills.\n\n_He cast, and I cast._ Harmodius shook his head. _Why did he join our enemy? Why? Why? Why? What did he learn when he began to dissect the old corpses?_\n\n_Why have I not thought on this before?_\n\nShrugged. \"My hubris differs from his hubris,\" he said to his cats. \"But I pray to God that he may yet see the light.\" At least enough to reduce him to a small mound of ash, he continued in his head. A really powerful light. Like a lightning bolt.\n\nSome things were best not said aloud, and naming could most definitely call. He had triumphed over his master, but no corpse had ever been found, and Harmodius knew in his bones that his mentor was still out there. Still part of the Wild.\n\n_Enough of this,_ he thought, and reached for another scroll on memory. He scanned it rapidly, took a heavy tome of grammerie down from a high shelf, referred to it, and then began to write quickly.\n\nHe paused and tapped his fingers rapidly on an old beaker while trying to think who could provide him with fresh corpses for his work. No one in the capital. The town was too small, the court too full of intrigue and gossip.\n\n\"Who would feed you if I took a trip?\" he asked. Because, already, his pulse was racing. He hadn't left his tower in\u2014he couldn't remember when he'd last left Harndon.\n\n\"Gracious Divinity, have I been here since the battle?\" he asked Miltiades.\n\nThe cat glared at him.\n\nThe Magus narrowed his eyes suddenly. He couldn't remember this cat as a kitten, or where the cat had come from. There was something out of step in his memories.\n\n_Christ_ , he thought, and sat in a chair. He could remember picking the kitten out of the dung heap by the stables, intending to dissect it. But he hadn't.\n\nHow had he lost that memory?\n\nWas it even a true memory?\n\nA spear of pure fear lunged through his soul. The beaker crashed to the floor, and all the cats jumped.\n\n_I have been ensorcelled._\n\nHe drew power quickly, in a whispered prayer, and performed a small and subtle working with it. Indeed, it was so subtle it scarcely required power.\n\nThe tip of his staff glowed a delicate shade of violet, and he began to move it around the room.\n\nThe violet remained steady for some time until, as he paused with the staff held up, to look at his own chalk marks on one wall, the tip flared pink and then a deep, angry red.\n\nHe waved it again.\n\nRed.\n\nHe leaned closer to the wall. He moved the tip of the staff back and forth in ever smaller arcs, and then he muttered a second casting, speaking stiffly the way a man does when he fears he's forgotten his lines in a play.\n\nA line of runes was suddenly picked out in angry fire-red. Wild runes, concealed under the paint on the wall.\n\nAcross the middle was a scorch mark that had erased a third of the writing.\n\n\"By the divine Christ and Hermes saint of Magisters,\" he said. He staggered back, and sat, a little too suddenly. A cat squalked and twitched its tail out from beneath him.\n\nSomeone had placed a binding spell on the walls of his sanctum. A binding laid on him.\n\nOn a hunch, he placed his staff where he had positioned it yesterday, to power it. He sighted along the line from his crystal to the head of the staff\u2014\n\n\"Pure luck.\" he said. \"Or the will of God.\"\n\nHe stood in thought. Then he took a deep breath. Sniffed the air.\n\nHe gathered power slowly and carefully, using a device he had in the corner, using an ancient mirror he had on a side table, using in the final instance a vial filled with shining white fluid.\n\n_In the palace of his mind, on a black and white tiled floor like an infinite chessboard, pieces moved\u2014like chess pieces and yet not like. There were pawns and rooks and knights, but also nuns and trees and ploughs and catapults and wyverns. He slowly resolved them into a pattern, each piece positioned on a tile of its own._\n\n_He poured his gathered power slowly out on the altar in the centre of the floor._\n\nWith the casting hovering, potent with a will to locate but still un-realised, in his mind, he climbed the twenty steps from his sanctum to the very top of his tower. He opened the door and stepped out onto a wooden hoarding, like a massive balcony, that ran all the way around the top of the tower. The spring sun was bright and the air was clear but the breeze was cold.\n\nHe saw the sea to the south-east. Due south, Jarsey spread like a storybook picture of farms and castles, rolling away for leagues. He raised his arms and released his phantasm.\n\nInstantly, he felt the power behind him, in the north.\n\nNo surprise there.\n\nHe walked slowly around the hoardings, his staff thumping hollowly on the wooden planks. His eyes stayed on the horizon. He looked due west, and there was, to his great enhanced vision, a faint haze of green off to the west along the horizon. Just as it ought to be, where the Wild held sway. But the border was farther than a man could ride in five days on a good horse, and the tinge of green stemmed from the great woods beyond the mountains. A threat\u2014but one that was always there.\n\nHe walked around the tower.\n\nLong before he reached the northernmost point, he saw the bright green flare. His spell was potent and he used it carefully, tuning his vision to get every scrap of knowledge from his altered sight.\n\nThere it was.\n\nHe refined the casting, so that instead of a complex web of lenses bouncing light, he reduced his effort to a single shining green strand, thinner than a strand of a spider web, running from the north directly to his tower. He had no doubt it ran to the very runes on his wall.\n\n_Damn._\n\n\"Was I fantasising about the Queen a moment ago?\" he asked the wind. \"What a fool I have been.\"\n\nHe didn't sever the strand. But he let go most of the Aethersight that had allowed him to see the threats displayed, and he reduced that, too, until he could just see the glimmer of his thread. Now his great phantasm took almost no golden light to power it.\n\nHe strode down into the tower with sudden purpose, and carefully shut the door behind him.\n\nHe picked up his staff, took the first wands to come under his hand and a heavy dagger with a purse, and went back out of his library, leaving the door wide open. He went down one hundred and twenty-two steps to the floor below, picked up a heavy cloak and a hat and fought the urge to pause there. He walked through the open door and shut it behind him, aware that all three cats were watching him from the top of the stairs.\n\nHe longed for an ally and, at the same time, doubted everything.\n\nBut he had to trust someone. He chose his Queen, stopped at the writing desk beyond the door and wrote.\n\n_Urgent business calls me to the north. Please tell the king that I have the gravest fears that I have been manipulated by an ancient enemy. Be on your guard._\n\n_I remain your Majesty's least humble servant, Harmodius_\n\nHe walked rapidly to the head of the twisting stairs and started down them, cursing his long staff and making as much haste as he could. He was trying to remember when he had last come down the stairs. Had it been yesterday?\n\nHe cast a very minor working ahead of him, now afraid that there might be spells to prevent his departure, but he could see nothing. That didn't help. If his fears were correct, his eyes might betray him, or be a tool of the enemy. Did his vision in the aether function in the same manner as natural sight?\n\n_Richard Plangere used to ask us, \"What is this natural of which you speak\" and we'd all be silent._\n\n_Richard Plangere, the spell on my wall stinks of you._\n\nCaught up in his thoughts, Harmodius almost missed a step. His foot slipped, and for a moment he hung at the edge of a forty-foot fall to the cobblestones below, and the only enemy he was fighting was old age and memory. He got the rest of the way down the stairs with nothing worse than a pain in his side from walking too fast.\n\nHis tower opened on the main courtyard, fifty paces on a side and lined with the working buildings of the king's government, although there were more of those down along the west wall as well, where there high windows looked down on the mighty river.\n\nHe walked to the stable. Men and women bowed deeply at his approach. His actions were scarcely secret, and he wondered briefly if he would have been better served leaving in the dark of night. Anyone could be an informant. Equally, he feared to go back to his chambers.\n\n_What am I afraid of?_\n\n_Have I lost my mind?_\n\nHe built a mental compartment around his chambers and all his associated thoughts and fears and closed the door on them. _I may be at the edge of madness, or I may have just discovered a terrible secret,_ he thought.\n\nThere were two grooms in the stable, working quickly and efficiently to unsaddle a dozen royal horses in hunting tack. They stopped when they saw the Magus.\n\nHe tried a smile. \"I need a horse,\" he said. \"A good one, for a journey.\"\n\nBoth of them looked at him as if he was insane.\n\nThen they looked at each other.\n\nFinally, the older nodded. \"Whatever you like, m'lord,\" he said. \"I can gi' you a courser\u2014a fine big mare callit Ginger. If it please you?\"\n\nHarmodius nodded, and before he could grow any more afraid a big bay was led out, a light saddle on her back. Harmodius looked up at that saddle with an old man's despair, but the younger groom had anticipated his look and moved to help, bringing him a stool.\n\nHarmodius stepped up on the stool and forced his leg up over the horse's back.\n\nThe ground seemed a long way down.\n\n\"Thank you, lad,\" Harmodius said. The boys handed him up his staff, two wands, and his purse, dagger, and cloak. The elder boy showed him how to stow it all behind the saddle.\n\n\"See that this note makes it to the Queen. Deliver it in person. This is my ring, so you may reach her\u2014every guard in the palace should know it. Do you understand me, boy?\" he asked, and realised that he was a figure of terrible fear to these two boys. He tried on a smile. \"You'll get a reward.\"\n\nThe younger smiled bravely. \"I'll take it, Master.\"\n\n\"See you do.\" He nodded.\n\nAnd then they were gone, and he was riding.\n\nHe rode through the gate without so much as a nod from the two Royal Guardsmen who stood there, either scanning the approach or sound asleep. The brims of their ornate helmets hid their eyes.\n\nHis horse's hooves rang hollowly against the drawbridge. The palace and its surrounding castle was merely the citadel in an extensive series of works\u2014three rings of walls and two other castles\u2014that towered above the ancient city of Harndon. Twice in Alba's history the entire Demesne had been reduced to the people that could huddle inside these walls.\n\nWhen the Wild came.\n\nHe rode down the slope of the castle mound into High Street\u2014the main street of the city of Harndon, that ran from gate to gate until it became the High Road and passed through the countryside, out to the town of Bridge where it crossed the mighty river, in the first of seven bridges. The river ran like a great snake from the north to the south of Alba, while the road cut straight across it.\n\nHere the road was a steep street lined with magnificent white-walled houses, each as tall and turreted as small castles. They were adorned in gilt and black iron with red or blue doors, tile or copper roofs, marble statuary painted and unpainted, and windows, clear or stained, high or wide. Each house was a palace and had its own character.\n\n_I used to dine here. And here. How long have I been under?_\n\nThe pressure in his chest eased as Harmodius rode down the hill, looking at the palaces of courtiers and great knights and wondering how it was that he had never visited any of them.\n\nHe rode through the Inner Gate without glancing at the guards. It was chilly in the wind, and he struggled with his cloak as he rode through Middle Town, and peered out into the High Cheaping, the city's principal market. The Cheaping was a market square two or three times the size of the courtyard of the castle, and packed with stalls and the bustle of commerce. He watched it as he passed, and then he was into the lower town, the Cheaping in local dialect, crossing Flood Street at the Bridge Gate, and his heart began to beat faster. He saw no threat\u2014but he expected one.\n\nThe men at the Bridge Gate had all of their attention on a magnificent retinue of knights and armoured men-at-arms entering the city. Harmodius looked at it from under his hood, trying to make out the blazon and guess whom the lord might be\u2014not anyone he had ever seen at court. A tall man, heavy with muscle.\n\nThe guards clearly wanted no part of making the decision to let the giant and his men into the town. Nor did they have any attention to spare for solitary old men riding out.\n\nThe knight commanding the retinue did, though, and turned to watch him as he rode by. His glance sharpened\u2014and then the Lieutenant of the Lower Gate appeared, armoured head to toe and holding not a wax tablet and stylus but a pole-axe, with four more knights at his back. The foreigner stiffened, and Harmodius rode past him while he was distracted.\n\nThrough the gate, down the slope past the lesser merchants who were only allowed to display their wares outside the walls\u2014in the Ditch, as men liked to call it. He rode past the mountebanks, the players, and the workmen building bleachers and barriers for the Whitsunday Play.\n\nHe pursed his lips and touched his heels to the horse's flanks, and the mare, delighted to be out in the spring and bored by the pace, sprang forward.\n\nHarmodius cantered along beside the market and continued past the outer ring of homes, the poorest still associated with the city, and past the first fields, each surrounded by a ring of rocks and old, painstakingly cleared tree stumps. The soil here was not the best. He cantered along the road for a further half a mile, pleased with his horse but still in the grip of fear, and came to the bridge.\n\nStill no one challenged him.\n\nHe crossed the first great span, stopped, spat into the river, and worked two powerful spells while he was safe in the bright sunshine at the centre of the bridge. Hermeticism functioned best in sunlight; while most workings of the Wild couldn't cross running water without enormous effort or the water's Hermetic permission. There was no power on earth that could take him in bright sunlight, in the middle span of flowing fresh water.\n\nAnd if there was such a power, he had no chance against it anyway.\n\nThen he went the rest of the way across and took the road north.\n\n# The Behnburg Road, East of Albinkirk\u2014Robert Guissarme\n\nRobert Guissarme was tall and cadaverously thin despite his intake of mutton and ale. Men said that his appetite for food was only exceeded by his appetite for gold. He called his company of men a _Company of Adventure_ , like the best Eastern mercenaries, and he dressed well in leather and good wool, or in bright armour made by the best Eastern smiths.\n\nNo one knew much of his birth. He claimed to be the bastard son of a great nobleman, whom he was careful never to name\u2014but he was known from time to time to lay a finger to his nose when a great man passed him on the road.\n\nHis sergeants feared him. He was quick to anger, quick to punish, and as he was the best man-at-arms of his company none of them wanted to cross him. Especially not right now; he was sitting fully armed on his charger, in deep fog, looking at a pair of peddlers who had passed them the night before, and who now stood in the middle of the road. They had been carefully butchered, flayed, and then set on posts in the road so that their heads seemed locked in endless screams of abject agony.\n\nSince yesterday, he had pushed his convoy north-west along the bad road that connected Albinkirk to the east\u2014to the Hills, and then over the mountains to Morea, and the land of the Emperor. He'd started his convoy in Theva, the city of slavers, and had pushed his men so hard that their horses began to fail. As for the long chain of slaves that was their principal cargo\u2014he no longer cared much whether they lived or died. They had been entrusted to him in Theva; a long line of broken men and women\u2014some pretty, some ugly, and all with the blank despair of the utterly beaten human being. He'd been told that they were a valuable consignment, being skilled slaves\u2014cooks, menservants, housemaids, nurses, and whores.\n\nHis company had treated them well enough on the long trip west. Well enough, despite the frowns of the Emperor's Knight\u2014a pompous bastard too proud to share his meals with a mere _mercenary_. After Albinkirk the man would no longer be his problem.\n\nBut when they passed Behnburg, the last town before Albinkirk, and found the town's garrison and population huddled within their walls in fear of un-named terrors, he'd started to hurry west, leaving the rest of the spring flood of merchants to hurry along in his wake. A dozen with wagons and good horses had paid him in gold to stay with his convoy.\n\nHe'd only taken the job transporting slaves to pay his passage\u2014rumour had it that the fortress convent at Lissen Carak was offering payment in gold for monster-hunting work, and Guissarme needed the work. Or his company did.\n\nOr perhaps they could manage a little longer. He sat his charger, at eye level with the corpses who had been killed, he now saw, by the act of their impaling.\n\nHe'd heard of impalement. Never seen it before. He couldn't tear his eyes away.\n\nHe was still gazing at them, rapt, when the arrows began to fall.\n\nThe first hit his horse. The second struck his breastplate with enough force to unseat him and sprang away and then he was falling. Men were screaming around him, and he could hear his corporals shouting for order. Something struck him in the groin and he felt a hot, rapidly spreading damp. Heard the sound of hooves\u2014heavy horses moving fast, although with an odd rhythm. He couldn't see well.\n\nHe tried to raise his head, and _something_ crouched over him, coming for his face\u2014\n\n# The Behnburg Road, East of Albinkirk\u2014Peter\n\nPeter watched the arrows fly from the woods that lined the road with a sort of hopeless, helpless anger.\n\nIt was so _obvious_ an ambush. He couldn't believe anyone had walked into it.\n\nChained by the yoke around his neck to the women front and back, he couldn't run.\n\nHe didn't have the words, but he tried all the same.\n\n\"Fall down!\" he shouted. \"Down!\"\n\nBut the panic was already coming. The terror\u2014he'd never felt such terror. It came directly behind the arrows, and washed over him like dirty water leaving fear behind. The two women to whom he was bound ran in different directions, stumbled, and fell together, taking him to the ground with them.\n\nThe arrows continued to fall on the soldiers, who mostly died. Only a small knot of them were still fighting.\n\nSomething\u2014he couldn't see very well in the late morning ground fog\u2014something came out of the fog moving as fast as a knight on a horse and slammed into the column. Men and horses screamed anew, and the terror increased to the point where his two companions simply curled into balls.\n\nPeter lay still and tried to make his head work. Watched the creatures coming at the column. They were daemons. He had heard of them in his home, and here they were, and they were feeding on the corpses. Or perhaps the living.\n\nA wyvern fell from the sky on the blonde woman ahead of him, its beaked head ripping at her guts. The woman behind him shrieked and got to her knees, arms extended, and a gout of pure green passed inches over Peter's head and slammed into the thing, which gave off an overpowering smell of burning soap.\n\nIt pivoted on its hips like a dancer, the action ripping the screaming woman under its forefoot in two and snapping the chain that connected the slaves. The end of the chain whipped around the creature's leg.\n\nThe wyvern unwound the chain fastidiously, using a talon, and the woman at Peter's back cast again, two handfuls of raw spirit shot out with an hysterical scream. The wyvern screamed back as it was hit, hundreds of times as loud, snapped its wings open and flung itself on the woman.\n\nPeter rolled beneath it, the newly snapped end of the chain running through his yoke, which caught on a tree root and wrenched his neck. Free, he was on his feet and running into the fog.\n\nA flash, and he was thrown flat. Silence\u2014he got to his feet and ran on, and only after a hundred panicked steps did he realise he was deaf and the shirt on his back was charred.\n\nHe ran on.\n\nHis mouth was so dry he could not swallow, and his thighs and calves burned as if they, too, had been burnt. But he ran until he crossed a deep stream, and there he drank his fill and lay gasping until he passed out.\n\n# Albinkirk\u2014Ser Alcaeus\n\nSer Alcaeus rode up to Albinkirk on a blown horse, with his destrier trotting along behind him. He'd lost his squire and his page in the fighting but his valet, a boy too young to swing a sword to any effect, had somehow survived with the pack horse.\n\nAlcaeus pounded on the town's west gate with his sword hilt. A pair of scared looking guards opened the main gate the width of one horse to let him in.\n\n\"There is an army of the Wild out there,\" Alcaeus gasped. \"Take me to your captain.\"\n\nThe captain of the town was an old man\u2014at least as fighters went\u2014grey bearded and tending to fat. But he was booted and spurred, wearing a hauberk of good iron rings and a belt that showed his paunch to unfortunate effect.\n\n\"Ser John Crayford,\" he said, holding out a hand.\n\nSer Alcaeus thought it unlikely that the man had ever been knighted. And he wondered how such an ill-favoured lout had come to command such an important post.\n\n\"I was with a convoy of fifty wagons on the Behnburg Road,\" Alcaeus said. He sat suddenly. He hadn't intended to sit, but his legs went out from under him.\n\n\"The Wild,\" he said. He tried to sound sane and rational and like a man whose word could be trusted. \"Daemons attacked us. With irks. A hundred, at least.\" He found that he was having trouble breathing.\n\nIt was difficult even telling it.\n\n\"Oh, my God,\" he said.\n\nSer John put a hand on his shoulder. The man seemed bigger somehow. \"How far, messire?\" he said.\n\n\"Five leagues.\" Alcaeus took a deep breath. \"Maybe less. East of here.\"\n\n\"By the Virgin!\" the Captain of Albinkirk swore. \"East, you say?\"\n\n\"You believe me?\" Alcaeus said.\n\n\"Oh, yes,\" said the captain. \"But east? They went _around_ the town?\" He shook his head.\n\nAlcaeus heard boots on the steps outside. He raised his head and saw the same man who'd let him into the city, with a pair of lower-class men.\n\n\"They say there's boglins in the fields, Ser John.\" The sergeant shrugged. \"That's what they say.\"\n\n\"My daughter!\" the younger man shouted. It was more like a shriek than a shout. \"You have to save her.\"\n\nSer John shook his head. \"I'm not taking a man out that gate. Steady, man.\" He poured the man a cup of wine.\n\n\"My _daughter_ !\" the man said in anguish.\n\nSer John shook his head. \"I'm sorry for your loss,\" he said, not unkindly. He turned to the sergeants. \"Sound the alarm. Bar the gates. And get me the mayor and tell him I'm imposing martial law. _No one is to leave this town._ \"\n\n# East of Albinkirk\u2014Peter\n\nPeter woke at a jerk of his heavy yoke. It was a hand-carved wooden collar with a pair of chains that ran down to his hands, allowing some movement, and a heavy staple for attaching him to other slaves, and he'd slept in it.\n\nTwo Moreans, easterners with scrips and heavy backpacks, wearing hoods and the air of men recently released from fear, stood over him.\n\n\"One survived then,\" the taller one said, and spat.\n\nThe shorter one shook his head. \"Hardly a fair return on the loss of our cart,\" he said. \"But a slave's a slave. Get up, boy.\"\n\nPeter lay in abject misery for a moment. So, naturally, they kicked him.\n\nThen they made him carry their packs, and the three of them started west along a trail through the woods.\n\nHis despair didn't last long. He had been unlucky\u2014or perhaps he had been lucky. They fed him; he cooked their meagre food and they let him have some bread and a little of the pea soup he'd made them. Neither of them were big men, or strong, and he thought he could probably kill them both, if only the yoke came off his shoulders.\n\nBut he couldn't get it off. It had been his constant companion for a month of walking over snow and ice, sleeping with the cold and hellish thing while the soldiers raped the women to either side of him and waiting to see if they would take a turn on him.\n\nHe bruised his wrists again and again trying get free of the thing. He daydreamed of using it as a weapon to crush these puny men.\n\n\"You're a good cook, boy,\" the taller man said, wiping his mouth.\n\nThe thin man frowned. \"I want to know what happened back there,\" he said, after drinking watered wine from his canteen.\n\nThe thicker man shrugged. \"Bandits? Cruel bastards, no doubt. I never saw a thing\u2014I just heard the fighting and\u2014well, you ran, too.\"\n\nThe thinner man shook his head. \"The screams,\" he said, and his voice shook.\n\nThey sat and glowered at each other, and Peter looked at them and wondered how they managed to survive at all.\n\n\"We should go back for our cart,\" said the thinner man.\n\n\"You must've had a bump on the head,\" the fatter one said. \"Want to be a slave? Like him?\" he gestured at Peter.\n\nPeter hunched by the fire and wondered if lighting it had been a good idea, and wondered how these two could be so foolish. At home, they had had daemons. These idiots must know of them too.\n\nBut the night passed\u2014a night in which he never slept, and the two fools slumbered after tying his yoke to a tree. They snored, and Peter lay awake, waiting for a hideous death that never came.\n\nIn the morning, the easterners rose, pissed, drank the tea he'd made, ate his bannock and started west.\n\n\"Where'd you learn to cook, boy?\" the thicker man asked him.\n\nHe shrugged.\n\n\"Now that's a saleable skill,\" the man said.\n\n# The Toll Gate\u2014Hector Lachlan\n\nDrovers hated tolls. There was no way to love them. When you have to drive a huge herd of beasts\u2014mostly cattle, but small farmers put in parcels of sheep, and even goats as well\u2014representing other men's fortunes, across mountain, fen, fell, swamp and plain, through war and pestilence, tolls are the very incarnation of evil.\n\nHector Lachlan had a simple rule.\n\nHe didn't pay tolls.\n\nHis herd numbered in the hundreds, and he had as many men as a southern lord had in an army; men who wore burnies of shining rings and carried heavy swords and great axes slung from their shoulders. They looked more like the cream of a mercenary army than what they were. Drovers.\n\n\"I didn't mean to cross you, Lachlan!\" the local lordling pleaded. He had that tone, the one Hector hated the most\u2014wheedling bluster, he called it, when a man who had pretended he was cock of the north started begging for his life.\n\nHector hadn't even drawn the great sword that sat across his hip and rump. He merely leaned his forearm on the hilt. He stroked his moustache idly and ran a hand through his hair, looked back down the long, muddy train of cattle and sheep that extended behind him, as far as the eye might see on the mountain track.\n\n\"Just pay me the toll. I'll\u2014see to it you ha' the coins back soon enough.\" The other man was tall, well-built, and wearing a chain hauberk worth a fortune, every link riveted closed, strong as stone.\n\nHe was afraid of Hector Lachlan.\n\nBut not afraid enough to let the long convoy of beasts past. He had to be seen to try and collect the toll. It was the way, in the hills, and his own fear would make him angry.\n\nSure enough, even as Hector had the thought, he saw the man's face change.\n\n\"Be damned to you, then. Pay the damned toll or\u2014\"\n\nHector drew his sword. He wasn't hurried by his adversary's anger, fear, or the fifty armed men at his back. He drew the long sword at his own pace, and allowed the heavy pommel to rotate the sword in his hand, so that the point aimed unwaveringly at the other man's face.\n\nAnd punched the needle sharp point through the other man's forehead with all the effort of a shoemaker punching a hole in leather. The armoured man crumpled, his eyes rolling up. Already dead.\n\nHector sighed.\n\nThe dead man's retinue stood rooted to the ground in shock\u2014a shock that would last a few more heartbeats.\n\n\"Stop!\" Hector said. It was a delicate art\u2014to command without threatening them and provoking the very reaction he sought to avoid.\n\nThe body crashed to the ground, the dead man's heels thrashing momentarily.\n\n\"None of ye need to die,\" he said. There was a thread of the dead man's blood on the tip of his sword. \"He was a fool to demand a toll of me, and every man here knows it. Let his tanist take command, and let us hear no more about it.\" Lachlan got the words out, and for a moment the men he was facing teetered on the knife-edge of doubt and greed and fear and loyalty\u2014not to the dead man but to the code that required them to avenge him.\n\nThe code won.\n\nLachlan heard the grunt that signified their refusal, and he had both hands on his sword, swinging a heavy overhand blow at the nearest man. He had a sword in his hand, but was too slow to save his own life; the heavy swing batted his parry aside and cut through his skull from left eyebrow to right jaw, so that the top of his head spun away, cleanly severed.\n\nHector's own men started to come forward, abandoning their places with his herd. Which meant that when this was over, with all the attending noise, violence, blood and ordure, a day would be lost while they collected all the beasts who ran off into the glens and valleys.\n\nSomeone\u2014some ancient philosopher Lachlan couldn't remember from the days when a priest came to teach him letters\u2014had said that the hillmen would conquer the world, if only they would ever stop fighting among themselves.\n\nHe pondered that as he killed his third man of the day, as his retinue charged with a shout, and as the doomed men of the toll gate tried to make a stand and were cut down.\n\n# Lissen Carak\u2014The Red Knight\n\nThe camp below the Abbey vanished as quickly as it had appeared, the tents folded and packed into the wagons, the wagons double-teamed and hauled up the steep slope into the fortress.\n\nThe first chore that faced all of them was billeting the company. Captain and Abbess walked quickly through the dormitory, the great hall, the chapel, the stables, and the storehouses, adding, dividing, and allocating.\n\n\"I will need to bring all my people in, of course,\" the Abbess said.\n\nThe captain bit his lip and looked at the courtyard. \"Eventually, we may have to re-erect our tents here,\" he said. \"Will you use the Great Hall?\"\n\n\"Of course. It's being stripped even now,\" she said. She shrugged. \"It is Lent\u2014all of our valuables are put away already.\"\n\nOne of the company's great wagons was just crossing the threshold of the main gate. Its top just fitted under the lintel.\n\n\"Show me your stores and all your storage places,\" he said.\n\nShe led him from cellar to cellar, from store room to the long, winding, airless steps that led deep into the heart of the living rock under their feet, to where a fresh spring burbled away into a pool the size of a farm pond. She was slower coming back up the winding steps than she had been going down.\n\nHe waited with her when she stopped to rest.\n\n\"Is there an exit? Down there?\" he asked.\n\nShe nodded. \"Of course\u2014who would hollow out this mountain and not make one? But I haven't the strength to show you.\" They reemerged through the secret door behind the chapel altar, and the Abbess was immediately surrounded by grey-clad sisters, each demanding her attention\u2014matters of altar care, of flowers for the next service, of complaints about the rain of blasphemous oaths falling from the walls, now fully manned.\n\n\"All you cock suckers get your fucking arses in armour or I'll chew off the top of your sodding skulls and fuck your brains,\" Bad Tom was dressing down a dozen men-at-arms just going onto the wall. His tone was conversational and yet it fell into a moment of silence and was carried everywhere inside the fortress.\n\nAn older sister stared at her Abbess in mute appeal.\n\n\"Your sisters are silent,\" the captain said.\n\nThe Abbess nodded. \"All are allowed to speak on Sundays. Novices and seniors may speak when they are moved to\u2014which is seldom for seniors and often for novices.\" She made a gesture with her hands. \"I am their ambassador to the world.\" She pointed at the cowled figure who followed her. \"This is Sister Miram, my chancellor and my vicar. She is also allowed to speak.\"\n\nThe captain bowed to Sister Miram, who inclined her head slightly.\n\nThe Abbess nodded. \"But she prefers not to.\"\n\n_Whereas you_ \u2014the captain thought that perhaps she liked to speak more than she let on, and liked to talk to him, to have an adult to spar with. Yet he did not doubt her piety. To the captain, piety came in three brands\u2014false piety, hypocritical piety, and hard won, deep and genuine piety. He fancied that he could tell them apart.\n\nAt the far end of the chapel stood Father Henry. He looked harried\u2014hadn't bathed or shaved, the captain suspected. He looked at the Abbess. \"Your priest is in a bad way,\" he said.\n\nHe knew that she had cast a phantasm on him last night. She'd done it expertly, and so revealed she was more than a mere mathematical astrologer. She was a magus. She'd probably known the instant he cast his glamour in her yard, and on her sisters.\n\nAnd she was not the only magus. There were wheels within the wheels that powered this situation. He looked at Sister Miram, his sense of power reaching tentatively towards her, like a third hand.\n\n_Aha_. It was as if Sister Miram had slapped that hand.\n\nThe Abbess was looking at the priest. \"He's in love with me,\" she said dismissively. \"My final lover. Gentle Jesu, might you not have sent me someone handsome and gentle?\" She turned and smiled wryly. \"I suspect he was sent me as a penance. And a reminder of what\u2014of what I was.\" She shrugged. \"The Knights of our Order didn't send us a priest last winter, so I took him from a local parish. He seemed interesting. Instead, I find he's\u2014\" She paused. \"Why am I telling you this, messire?\"\n\n\"As your captain, it is my duty to know,\" he said.\n\nShe considered him. \"He's a typical ignorant parish priest\u2014can scarcely read Archaic, knows the Bible only from memory, and thinks women are less than the dirt on his bare feet.\" She shook her head. \"And yet he came here, and he is drawn to me.\"\n\nThe captain smiled at her, took her right hand between his and kissed it. \"Perhaps _I_ am your last lover,\" he said.\n\nAs he did it, he saw the priest squirm. _Oh, my, what fun._ The man was loathsome, but his piety was probably genuine too.\n\n\"Should I box your ears for that? I understand that's the fashion,\" the Abbess said. \"Please desist, Captain.\"\n\nHe retreated as if she'd struck him. Sister Miram was frowning.\n\nTo regain his composure, he summoned Jehannes and Milus. \"Get the drovers to dismantle the wagons. Put the hardware in the cellars\u2014Abbess, we're going to need some guides.\"\n\nThe Abbess sent for the old garrison\u2014eight non-noble men-at-arms hired at the Great Fair a dozen years before. They were led by Michael Ranulfson, a grizzled giant with gentle manners, the sergeant at arms the captain had met briefly the night before.\n\n\"You know that I've placed the captain in charge of our defence,\" she said. \"His men need help moving in, and guides to the storerooms. Michael\u2014I trust them.\"\n\nMichael bowed his head respectfully, but his eyes said _on your head be it._\n\n\"How are you set for hoardings?\" the captain asked. \"Do you have pre-cut lumber?\"\n\nThe old sergeant at arms nodded. \"Aye. Hoardings, portable towers, a pair of trebuchets, some smaller engines.\" He rolled his head on his neck, as if trying to rid himself of a stiffness. \"When you are in garrison, you may as like do a good job of work.\"\n\nThe captain nodded. \"Thanks, Ser Michael.\"\n\n\"I'm no knight,\" Michael said. \"My da was a skinner.\"\n\nThe captain ignored his statement to look at Jehannes. \"As soon as the lads are unpacked, give this man fifty archers and all the riff-raff and get the hoardings up while the men-at-arms stand to.\"\n\nJehannes nodded, obviously in full agreement.\n\n\"Store the dismantled wagons wherever the hoardings are now,\" the captain said. \"And then we'll start on patrols to fetch in the peasants. Gentlemen, this place is going to be packed as tight as a cask of new-salted mackerel. I want to say this in front of the Abbess. There will be no rape and no theft by our men. Death penalty on both. My lady, I can't do much about casual blasphemy, but an effort will be made\u2014you understand me, gentlemen? Make an effort.\"\n\nShe nodded. \"It is Lent,\" she said.\n\nJehannes nodded. \"I gave up wine,\" he said, and then stared at the floor.\n\n\"Jesu does not care what you give up, but rather, what you give him,\" Sister Miram replied, and Jehannes smiled shyly at her.\n\nShe returned his smile.\n\nThe captain released a heavy sigh. \"Ladies, you may well cure all of our souls yet, but it must wait until the hoardings are up and all your people are safe. Michael, you are in charge of them. I recommend that my men live in the towers and galleries\u2014if we have time, we'll build them beds.\"\n\n\"My people will go four to a room,\" the Abbess said. \"I can take the older girls and single women from the farms into the dormitory, and all the men and their families will go in the hall. Overflow into the stables.\"\n\nMichael nodded. \"Yes, my lady,\" he replied. He turned to the captain. \"I'm at your orders.\" He looked back and forth. \"Will we hold the Lower Town?\"\n\nThe captain stepped up onto the gate wall and looked down at the four streets of the town, a hundred feet below.\n\n\"For a little while,\" he said.\n\n# Albinkirk\u2014Ser Alcaeus\n\nSer Alcaeus passed a bad night and drank too much wine in the morning. The man whose daughter had been abducted sat in the garrison barracks and wept, and demanded that the garrison send out a sortie to her rescue.\n\nThe mayor agreed with him, and hot words were exchanged.\n\nAlcaeus didn't want any part of it. They were too alien\u2014the commoners were both too servile and too free, and Ser John was no knight. Even the churches were wrong. Mass was said in low Archaic.\n\nIt was disorienting. Worse than the convoy of slaves had been, because he could ignore them.\n\nMid-morning, as he finished his ablutions\u2014he, the Emperor's cousin, washing without so much as a servant or slave to help him\u2014he heard the mayor's shrill voice in the guardroom, demanding that Ser John come out.\n\nAlcaeus dressed. He had spare shirts because the boy had saved his packhorse, and he'd see the page richly rewarded for it.\n\n\"Come out of your hole, you doddering old coward!\" shrieked the mayor.\n\nAlcaeus was trying to lace his cuffs by himself. He _had_ done his own in the past, but not since he became a man. He had to press his right hand against the stone of the castle wall and pin the knot in place.\n\n\"Master Mayor?\" he heard. It was Ser John, his voice calm enough.\n\n\"I demand that you gather all the useless mouths you call your garrison and go out and find this man's daughter. And open the gates\u2014the grain convoys are on their way. This town needs money, though I'm sure you've been too _drunk_ to notice.\" The mayor sounded like a fishwife\u2014a particularly nasty one.\n\n\"No,\" said the captain. \"Was that all?\"\n\nAlcaeus couldn't, in that moment, decide exactly what he thought of the knight. Over-cautious? But memories of yesterday's ambush were still burned onto the backs of his eyelids.\n\nHe reached for his boots\u2014uncleaned, of course. He pulled them on, and fought with all the buckles, his head suddenly full of irks and boglins and worse things. The road. The confusion.\n\nHe had been trained to fight the Wild. Until yesterday, he'd only fought other men\u2014usually one to one, with knives, at court.\n\nThe images in his head made him shudder.\n\n\"I order you!\" the mayor screamed.\n\n\"You can't order me, Master Mayor. I have declared martial law, and I, not you, am the power here.\" Ser John sounded apologetic rather than dismissive.\n\n\"I represent the _people_ of the town. The burgesses, the merchants, and the artisans!\" The mayor's voice sank to a hiss. \"You don't seem to understand\u2014\"\n\n\"I understand that I represent the king. And you do not.\" Ser John's voice remained level.\n\nAlcaeus had made his decision. He was going to go support the low-born knight. It didn't matter what the two men were debating\u2014it was their manners. Ser John was knightly. He might even survive at court.\n\nAlcaeus tested his feet in his boots, and took his heavy dagger and put it in his belt. He never left his rooms without a dagger. Then he went out into the hall\u2014a hall crowded with garrison soldiers listening to the argument in the main room below. He ran light-footed down the stairs.\n\nHe'd missed an exchange. When he entered, the mayor, red-faced, thin and tall and blond as an angel, was silent, his mouth working.\n\nSer Alcaeus went and stood behind the old knight. He noted that the mayor wore a rich doublet of dark blue velvet trimmed in sable, and a cap to match, embroidered with irks and rabbits. He smiled\u2014his own silk doublet was worth about fifty times the value of the mayor's.\n\nThe irks in the mayor's cap were ironic, to say the least.\n\n\"This is Ser Alcaeus,\" Ser John said. \"The Emperor's ambassador to our king. Yesterday his convoy was attacked by _hundreds_ of Wild creatures.\"\n\nThe mayor shot a venomous glance at him. \"So you say. Go do your _fucking_ job, sell-sword. Aren't you even a little humiliated to think that this man's daughter is the plaything of monsters while you sit and drink wine?\"\n\nThe man\u2014who stood behind the mayor with a dozen other men\u2014gave a sob and sank to a wooden bench, his fist in his mouth.\n\n\"His daughter has been dead since yesterday and I won't risk men to look for her corpse,\" Ser John said with casual brutality. \"I want all the woman and children moved to the castle immediately, with victuals.\"\n\nThe mayor spat. \"I forbid it. Do you want to panic the town?\"\n\nSer John shrugged. \"Yes,\" he said. \"In my professional opinion\u2014\"\n\n\"You _have_ no professional opinion. You were a sell-sword\u2014what? Forty years ago? And then a drinking crony of the king's. Very professional!\" The mayor was beside himself.\n\nAlcaeus realised the man was afraid. Terrified. And that terror made him belligerent. It was a revelation. Alcaeus was not, strictly speaking, a _young_ man. He was twenty-nine, and he thought he knew how the world worked.\n\nYesterday had been a shock. And now today was a shock too. He watched the fool mayor, and watched Ser John, and understood something of their quality.\n\n\"Messire mayor?\" he asked in his stilted Gothic. \"Please\u2014I am a stranger here. But the Wild is real. What I saw was real.\"\n\nThe mayor turned and looked at him. \"And who in God's name are you?\" he asked.\n\n\"Alcaeus Comnena, cousin to the Emperor Manual, may his name be praised, the drawn sword of Christ, the Warrior of the Dawn.\" Alcaeus bowed. His cousin was too old to draw a sword but his titles rolled off the tongue, and he was annoyed by the mayor.\n\nThe mayor was, for all his belligerence and terror, a merchant and an educated man. \"From Morea?\" he asked.\n\nAlcaeus thought of telling this barbarian what he thought of their casual use of Morea for the Empire. But he didn't bother. \"Yes,\" he shot back.\n\nThe mayor drew a breath. \"Then if you are a true knight, _you_ will go and rescue this man's daughter.\"\n\nAlcaeus shook his head. \"No. Ser John is correct. You must call in your out-farmers and move the people into the castle.\"\n\nThe mayor shook his fist. \"The _convoys_ are coming. If we close the gates, this town will _die_!\" He paused. \"For the love of God! There's _money_ involved.\"\n\nSer John shrugged. \"I hope the money helps when the boglins come,\" he said.\n\nAs if on cue, an alarm bell sounded.\n\nAfter the mayor pounded out of the castle, Alcaeus went out on the wall and saw two farms burning. Ser John joined him. \"I told him to bring the people in last night,\" he muttered. \"Fucking idiot. Thanks for trying.\"\n\nAlcaeus watched the plumes of smoke rise and his stomach did flips. Suddenly, again, he was seeing those the irks under his horse. He had once, single-handed, fought off four assassins who were going for his mother. Irks were much, much worse. He tasted bile.\n\nHe thought of lying down.\n\nInstead, he drank wine. After a cup, he felt strong enough to visit his page, who was recovering from terror in the resilient way teenagers so. He left his page to cuddle with a servant girl and walked wearily back to the guard room, where there was an open cask of wine.\n\nHe was on his fourth when Ser John's fist closed around his cup. \"I take it you are a belted knight,\" Ser John said. \"I saw your sword, and you've used it. Eh?\"\n\nSer Alcaeus got up from his chair. \"You dared draw my sword?\" he asked. At the Emperor's court touching a man's sword was an offence.\n\nThe old man grinned mirthlessly. \"Listen, messire. This town is about to be attacked. I never thought to see it in my lifetime. I gather you had a bad day yesterday. Fine. Now I need you to stop draining my stock of wine and get your armour on. They'll go for the walls in about an hour, unless I miss my guess.\" He looked around the empty garrison room. \"If we fight like fucking heroes and every man does everything he can, we might just make it\u2014I'm still trying to get that fool to send the women into the castle. This is the Wild, Ser Knight. I gather you've tasted their mettle. Well\u2014here they come again.\"\n\nSer Alcaeus thought that this was a far, far cry from being a useful functionary at his uncle's court. And he wondered if his true duty, given the message he had in his wallet, was to gather his page and ride south before the roads closed.\n\nBut there was something about the old man. And besides, the day before he'd run like a coward, even if he'd had the blood of three of the things on his sword first.\n\n\"I'll arm,\" he said.\n\n\"Good,\" Ser John said. \"I'll help, and then I'll give you a wall to command.\"\n\n# Abbington-on-the-Carak\u2014Mag the Seamstress\n\nOld Mag the seamstress sat in the good, warm sunshine on her doorstep, her back braced against the oak of her door frame, as she had sat for almost forty years of such mornings. She sat and sewed.\n\nMag wasn't a proud woman, but she had a certain place, and she knew it. Women came to her for advice on childbirth and savings, on drunken husbands, on whether or not to let a certain man visit on a certain night. Mag knew things.\n\nMost of all, she knew how to sew.\n\nShe liked to work early, when the first full light of the sun struck her work. The best time was immediately after Matins. If she managed to get straight to her work\u2014and in forty years of being a lay sister, helping with the altar service in her village church, of tending to her husband and two children she had missed the good early morning work hours all too often.\n\nBut when she got to it\u2014when cooking, altar service, sick infants, aches and pains, and the will of the Almighty all let her be\u2014why, she could do a day's work by the time the bells rang for Nones in the fortress convent two leagues to the west.\n\nAnd this morning was one of those wonderful mornings. She'd been the lay server at church, which always left her with a special feeling, and she had laid flowers on her husband's grave, kissed her daughter in her own door yard and was now home in the first warm light, her basket by her side.\n\nShe was making a cap, a fine linen coif of the sort that a gentleman wore to keep his hair neat. It wasn't a difficult object and would take her only a day or two to make, but there were knights up at the fortress who used such caps at a great rate, as she had reason to know. A well-worked cap that fit just so was worth half a silver penny. And silver pennies were not to be sneered at, for a fifty-three-year-old widow.\n\nMag had good eyes, and she pricked the fine linen\u2014her daughter's linen, no less\u2014with precision, her fine stitches as straight as a sword blade, sixteen to the inch, as good or better as any Harndon tailor's work.\n\nShe put the needle into the fine cloth and pulled the thread carefully through, feeling the fine wax on the thread, feeling the tension of the fine cloth, and aware that she pulled more than the thread with each stitch\u2014every one gathered a little sun. Before long, her line of stitches _sparkled_ , if she looked at it just so.\n\nGood work made her happy. Mag liked to examine the fine clothes that came through to Lis the laundress. The knights in the fortress had some beautiful things\u2014usually ill-kept but well made. And many less well-made clothes, too. Mag had plans to sell them clothes, repairs, darning\u2014\n\nMag smiled at the world as she stitched. The sisters were, in the main, good landlords, and much better than most feudal lords. But the knights and their men brought a little colour to life. Mag didn't mind hearing a man say fuck, as long as he brought a little of the outside world to Abbington-on-the-Carak.\n\nShe heard the horses, and her eyes flicked up from her work. She saw dust rising well off to the west. At this hour, it could never bode well.\n\nShe snorted and put her work in her basket, carefully sheathing her best needle\u2014Harndon work, there was no local man who could make such\u2014in a horn needle case. No crisis was so great that Mag needed to lose a needle. They were harder and harder to get.\n\nMore dust. Mag knew the road. She guessed there were ten horses or more.\n\n\"Johne! Our Johne!\" she called. The Bailli was her gossip, and occasionally more. He was also an early riser, and Mag could see him pruning his apple trees.\n\nShe stood and pointed west. After a long moment, he raised an arm and jumped down from the tree.\n\nHe dusted his hands and spoke to a boy, and heartbeats later that boy was racing for the church. Johne jumped the low stone wall that separated his property from Mag's and bowed.\n\n\"You have good eyes, m'ame.\" He didn't smirk or make any obvious gesture, which she appreciated. Widowhood brought all sorts of unwelcome offers\u2014and some welcome ones. He was clean, neat, and polite, which had become her minimum conditions for accepting even the most tenuous of male approaches.\n\nShe enjoyed watching a man of her own age who could still jump a stone wall.\n\n\"You seem unconcerned,\" she murmured.\n\n\"To the contrary,\" he said quietly. \"If I were a widowed seamstress I would pack all my best things and be prepared to move into the fortress.\" He gave her half a smile, another bow, and sprang back over the wall. \"There's been trouble,\" he added.\n\nMag didn't ask foolish questions. Before the horses rode into their little town square, shaded by an ancient oak, she had two baskets packed, one of work and one of items for sale. She filled her husband's travelling pack with spare shifts and clothes, and took her heaviest cloak and a lighter cloak\u2014for wearing and sleeping, too. She stripped her bed, took the bolster and rolled the blankets and linens tightly around it to make a bundle.\n\n\"Listen up!\" called a loud voice\u2014a _very_ loud voice\u2014from the village square.\n\nLike all her neighbours she opened the upper half of her front door and leaned through it.\n\nThere were half a dozen men-at-arms in the square, all mounted on big horses and wearing well-polished armour and scarlet surcotes. With them were as many archers, all in less armour with bows strapped across their backs, and as many valets.\n\n\"The lady Abbess has ordered that the good people of Abbington be mustered and removed into the fortress immediately!\" the man bellowed. He was tall\u2014huge, really, with arms the size of most men's legs, mounted on a horse the size of a small house.\n\nJohne the Bailli, walked across the square to the big man-at-arms, who leaned down to him, and the two spoke\u2014both of them gesturing rapidly. Mag went back to her packing. Out the back she scattered feed for her chickens. If she wasn't here for a week, they'd manage, longer, and they'd all be taken by something. She had no cow\u2014Johne gave her milk\u2014but she had her husband's donkeys.\n\n_My donkeys,_ she reminded herself.\n\nShe'd never packed a donkey before.\n\nSomeone was banging at her open door. She shook her head at the donkeys, who looked back at her with weary resignation.\n\nThe big man-at-arms stood on her stoop. He nodded. \"The Bailli said you'd be ready to move first,\" he said. \"I'm Thomas.\" His bow was sketchy, but it was there.\n\nHe looked like trouble from head to foot.\n\nShe grinned at him, because her husband had looked like trouble, too. \"I'd be more ready if I knew how to pack a donkey,\" she said.\n\nHe scratched under his beard. \"Would a valet help? I want people moving in an hour. And the Bailli said that if people saw you packed, they'd move faster.\" He shrugged.\n\nOff to the right, a woman screamed.\n\nThomas spat. \"Fucking archers,\" he snarled, and started back out the door.\n\n\"Send me a valet!\" she shouted after him.\n\nShe got a produce basket down from the shed and began to fill it with perishable food, and then preserves. She had sausage, pickles, jam, that was itself valuable\u2014\n\n\"Good wife?\" asked a polite voice from the doorway. The man was middle-aged, and looked as hard as rock and as sound as an old apple. Behind him was a skinny boy of twelve.\n\n\"I'm Jaques, the captain's valet. This is my squire, Toby. He can pack a mule\u2014I reckon donkeys ain't much different.\" The man took his hat off and bowed.\n\nMag curtsied back. \"The sele of the day to you, ser.\"\n\nJacques raised an eyebrow. \"The thing of it is, ma'am\u2014we're also to take all your food.\"\n\nShe laughed. \"I've been trying to pack it\u2014\" Then his meaning sunk in. \"You mean to take my food for the garrison.\"\n\nHe nodded. \"For everyone. Yes.\" He shrugged. \"I'd rather you made it easy. But we will take it.\"\n\nJohne came to the door. He had a breast and back plate on and nodded to Jacques. To Mag, he said, \"Give them everything. They are from the Abbess, we have to assume she will repay us.\" He shrugged. \"Do you still have Ben's crossbow? His arming jack?\"\n\n\"And his sword and dagger,\" Mag said. She opened her cupboard, where she kept her most valuable things\u2014her pewter plates, her silver cup, her mother's gold ring, and her husband's dagger and sword.\n\nToby looked around shyly, and said to Jacques, \"This is a rich place, eh, master?\"\n\nJacques smiled grimly and gave the boy a kick. \"Sorry, ma'am. We has some bad habits from the Continent, but we won't take your things.\"\n\n_But you would under other circumstances, and anything else you fancied_ , she thought.\n\nJohne took her by her shoulders. It was a familiar, comfortable thing, and yet a little too possessive for her taste, even in a crisis.\n\n\"I have a locking box,\" he said. \"There's room in it for your cup and ring. And any silver you have.\" He looked into her eyes. \"Mag, we may never come back. This is war\u2014war with the Wild. When it's done, we may not have homes to return to.\"\n\n\"Gentle Jesu!\" she let slip. Took a shuddering breath, and nodded. \"Very well.\" She scooped up the cup and ring, tipped over a brick in her fireplace and took out all her silver\u2014forty-one pennies\u2014and handed it all to the bailli. She saved out one penny, and she gave it to Jacques.\n\n\"This much again if my donkeys make it to the fortress,\" she said primly.\n\nHe looked at it for a moment. Bit it. And flipped it to the boy. \"You heard the lady,\" he said. He nodded to her. \"I'm the captain's valet, ma'am. A piece of gold is more my price. But Tom told me to see to you, and you are seen to.\" He gave her a quick salute and was out her door, headed for Simon Carter's house.\n\nShe looked at the boy. He didn't seem very different from any other boy she knew. \"You can load a donkey?\" she asked.\n\nHe nodded very seriously. \"Do you\u2014\" he looked around. He was as skinny as a scarecrow and gawky the way only growing boys can be. \"Do you have any food?\" he asked.\n\nShe laughed. \"You'll be taking it all anyways, won't you, my dear?\" she asked. \"Have some mince pie.\"\n\nToby ate the mince pie with a determination that made her smile. While she watched him, still packing her hampers, he ate the piece he was given and then filched a second as he headed for her donkey.\n\nA pair of archers appeared next. They lacked something that Ser Thomas and Jacques the valet had both possessed. They looked dangerous.\n\n\"What have we here?\" asked the first one through the door. \"Where's the husband, then, my beauty?\" His voice was flat, and so were his eyes.\n\nThe second man had no teeth and too much smile. His haubergeon was not well kept, and he seemed like a half-wit.\n\n\"Mind your own business,\" she said, her voice as sharp as steel.\n\nDead-eyes didn't even pause. He reached out, grabbed her arm, and when she fought him he swept her legs out from under her and shoved her to the floor. His face didn't change expression.\n\n\"House's protected,\" said the skinny boy said from the kitchen. \"Best mind yourself, Wilful.\"\n\nThe dead-eyed archer spat. \"Fuck me,\" he said. \"I want to go back to the Continent. If I wanted to be a nurse-maid\u2014\"\n\nMag was so stunned she couldn't react.\n\nThe archer leaned down and stuck his hand in the front of her cote-hardie. Gave her breast a squeeze. \"Later,\" he said.\n\nShe shrieked, and punched him in the crotch.\n\nHe stumbled back, and the other one grabbed her hair, as if this was a practised routine\u2014\n\nThere was a sharp _crack_ and she fell backwards, because the archer had released her. He was kneeling on the floor with blood pouring out of his face. Thomas was standing over him, a stick in his hand.\n\n\"I tol' 'em that this house was protected!\" the thin boy shouted.\n\n\"Did you?\" The big man said. He eyed the two archers.\n\n\"We was gentle as lambs!\" said the one with dead eyes.\n\n\"Fucking archers. Piss off and get on with it,\" the big man said, and offered her a hand up.\n\nThe two archers got to their feet and went out the back to collect her chickens and her sheep and all the grain from her shed, all the roots in her cellar. They were methodical, and when she followed them into the shed, the dead-eyed one gave her a look that struck fear into her. He meant her harm.\n\nBut soon enough the boy had her donkeys rigged and loaded, and she put her husband's pack on her back, her two baskets in her arms, and went out into the square.\n\nFrom where she stood, her house looked perfectly normal.\n\nShe tried to imagine it burned. An empty basement yawning at the sun. She could see the place where she rested her back when she sewed, rubbed shiny with use, and she wondered if she would ever find such a well-lit spot.\n\nThe Carters were next to be ready\u2014they were, after all, a family of carters with two heavy carts of their own and draught animals, and six boys and men to do the lifting. The bailli's housekeeper was next, with his rugs\u2014Mag had lain on one of those rugs, and she blushed at the thought. She was still mulling over her instinctive use of his name\u2014his Christian name\u2014\n\nThe Lanthorns were the last, their four sluttish daughters sullen, and Goodwife Lanthorn, in her usual despair, wandered the village's column of animals, begging for space for her bag and a basket of linen. Lis the laundress was surrounded by soldiers, who competed to carry her goods. But she knew many of them by name, having washed their linens, and she was both safely middle-aged _and_ comely, an ideal combination in the soldiers' eyes.\n\nAt last the Lanthorns were packed\u2014all four daughters eyeing the soldiers\u2014and the column began to move.\n\nThree hours after the men-at-arms rode into Abbington, the town was empty.\n\n# Albinkirk\u2014Ser Alcaeus\n\nSer John gave him a company of crossbowmen\u2014members of the town's guilds, all of them a little too shiny in their guild colours. Blue and red predominated, from the furriers, the leading guild of Albinkirk. He might have laughed to think that he, cousin to the Emperor, was commanding a band of common-born crossbowmen. It would have amused him, but...\n\nThey came at sunset, out of the setting sun.\n\nThe fields looked as if they were crawling with insects and then, without a shout or a signal the irks changed direction and were coming up the walls. Ser Alcaeus had never seen anything like it, and it made his skin crawl.\n\nThere were daemons among them, a dozen or more, fast, lithe, elegant and deadly. And they simply ran up the walls.\n\nHis crossbowmen loosed and loosed into the horde coming at them, and he did his best to walk up and down behind them on the crenellations, murmuring words of encouragement and praising their steadiness. He knew how to command, he'd just never done it before.\n\nThe first wave almost took the wall. A daemon came right over and started killing guildsmen. It was nothing but luck that its great sword bounced off a journeyman armourer's breastplate and the man's mates got their bolts into the lethal thing. It still took four more men down while it died, but the sight of the dead daemon stiffened the guildsmen's spines.\n\nThey staved off the second wave. The daemons had grown careful and led from the back. Alcaeus tried to get his crossbowmen to snipe at them, but there was never a moment when they could do anything but fight the most present danger.\n\nA guild captain came to him where he was standing, leaning heavily on his pole-axe because he knew enough not to waste energy in armour. The man saluted.\n\n\"M'lord,\" he said. \"We're almost out of bolts. Every lad brings twenty.\"\n\nSer Alcaeus blinked. \"Where do you get more?\"\n\n\"I was hoping you would know,\" said the guild officer.\n\nSer Alcaeus sent a runner, but he already knew the answer.\n\nThe third wave got over the walls behind them, they heard it go. The sounds of fighting changed, there was sudden shrieking and his men started to look over their shoulders.\n\nHe wished he had his squire\u2014a veteran of fifty battles. But the man had died protecting him in the ambush and so he had no one to ask for advice.\n\nSer Alcaeus set his jaw and prepared to die well.\n\nHe walked along the wall again as the shadows lengthened. His section was about a hundred paces, end to end\u2014Albinkirk was a _big_ town, even to Ser Alcaeus who hailed from the biggest city in the world.\n\nHe stopped when he saw three of his men looking back at the town.\n\n\"Eyes front,\" he snapped.\n\n\"A house on fire!\" some idiot said.\n\nMore men turned and, just like that, he lost them. They turned, and then there was a daemon on the wall, killing them. It moved like fluid, passing through men, round them, with two axes flashing in its taloned hands\u2014even as Alcaeus watched, one of the daemon's taloned feet licked out to eviscerate a fifteen-year-old who'd had no breastplate.\n\nAlcaeus charged. He felt the fear that it generated\u2014but in Morea knights trained for this very thing, and he knew the fear. He ran through it, blade ready\u2014\n\nIt hit him. It was faster by far, and an axe slammed into his arm. He was well-trained and caught much of the blow. His small fortune in plate armour ate the rest, and then he was swinging.\n\nIt had to pivot to face him. The twitch of its hips took a heartbeat, and he swung his pole-axe up from the garde of the boar, like a boy swinging a pitchfork at haying, but with twice the speed.\n\nSer Alcaeus was as shocked as the daemon when his axe caught the other creature's axe-hand and smashed it. Ichor sprayed and its axe fell. It slashed at him with the left, turned and kicked him with a taloned foot. All four talons bit through his breastplate and knocked him flat, but none reached him through his mail and padded arming cote.\n\nA crossbow struck the daemon. Not a bolt but the bow itself, swung by a terrified guildsman.\n\nThe daemon bounded onto the wall, scattering defenders, and jumped.\n\nAlcaeus got to his feet. He still had his pole-axe.\n\nHe was proud of himself for two breaths, and then he realised that the town behind him was afire, and there were two more daemons on the wall with him, and irk arrows were suddenly everywhere. Worse, they were coming from the town.\n\nHe had a dozen men by him, including the stunned looking man who'd hit the daemon with his crossbow. The rest of his fools were leaving the wall, running for their houses.\n\nHe shook his head and cursed. They were surrounded, half his men gone, and it was growing dark rapidly.\n\nHe made his decision. \"Follow me!\" he called, and ran along the wall. He was headed for the castle, which towered over the western end of town by the river gate. It had its own defensive walls.\n\nThe whole town was falling. It was the only place to make a stand.\n\nWhen he paused to breathe, Albinkirk was afire from south to north, and a sea of Wild creatures were running through the streets. He knew the difference between the irks\u2014elfin and gnarled and satanic in the firelight\u2014and the boglins, with their leather midsections and their oddly-jointed arms. He'd studied pictures. He'd trained for this, but it was like a nightmare. He was running _again_ with the half dozen of his crossbowmen who stuck with him. The rest ran off into the town despite his admonitions. One died at their feet, ripped to pieces by boglins and consumed by something worse.\n\nHe could see the river, and the castle, but the next section of wall was crowded with enemies. The streets below were worse.\n\nBut at the edge of the firelight, he could see a company of soldiers with spears still holding one street, a crowd of panicked refugees behind them pressing on the castle gates.\n\nUnbeckoned, a thought whispered into his head.\n\n_Time to earn your spurs._\n\n\"Let me go first,\" he said to his crossbowmen. \"I will charge. You will follow me and kill anything that gets past me. You understand?\"\n\nHe longed, just for a second, for wine and his lyre, and for the feeling of a woman's breast under his hand.\n\nHe raised his pole-axe.\n\n_\"Kyrie Eleison!\"_ he sang, and charged.\n\nThere were perhaps sixty boglins on the wall. It was too dark to count, and he wasn't that interested.\n\nHe smashed into them, taking them by surprise. The first one died, and after that nothing went right. His pole-axe fouled in the boglin; his blow had caught the thing in an armpit, and it fell off the wall taking his precious weapon with it.\n\nHe was instantly surrounded.\n\nHe got a dagger unsheathed with a practised _flick_ \u2014because a bastard cousin of the Emperor does not survive long at court without being able to use a dagger expertly, in or out of armour\u2014and then they piled on him and he was all but buried standing up.\n\nHis right arm began stabbing largely of its own accord.\n\nA tremendous blow knocked him forward, and he stumbled a few steps smashing pieces of boglin beneath his feet\u2014suddenly panicked that he would fall off the wall. Panic powered his limbs, he spun and felt his steel-clad back slam into the crenellations. Suddenly his arms were free, and the thing trying to open his visor was the top priority, and then it was gone too and he was clear.\n\nHis right arm was slick with green-brown blood. He took up the low guard\u2014All Gates are Iron\u2014with his dagger back over his right hip, left fist by his left hip, looking over his left shoulder.\n\nA boglin threw a spear at him.\n\nHe blocked it with his left hand, and stumbled forward into them. His breath was coming in great bursts, but his brain was clear, and he rammed the point of his heavy dagger into the first one, right through its head, and ripped it out again. His armoured fist snapped out in a punch and smashed the noseless face of a second.\n\nThe next two boglins were folded over their midriffs, shot with bolts. He stepped past them, his dagger switching hands with a dexterity his uncle's master of arms would have approved of, he was drawing his sword right-handed as he advanced.\n\nThe boglins began to back away.\n\nHe charged them.\n\nThey had their own gallantry. One creature gave its life to trip him, and died on his dagger as he fell. He rolled on a shoulder, but then there was nothing under his feet\u2014\n\nHe hit a tiled roof, slid, hit a stone lintel with his armoured shoulder, flipped...\n\nAnd landed in the street, on his feet. He still had both sword and dagger and took the time to thank God for it.\n\nAbove him, on the wall, the boglins were staring at him. \"Follow me!\" he shouted to his men. He hadn't meant to come down to the street\u2014but from here he could see irks coming along the wall from behind his archers.\n\nTwo made the jump. The rest froze, and died where they stood.\n\nThe three of them ran for the castle, which was lit up as if it was a royal palace ready for a great event. Albinkirk was ablaze, and the streets were carpeted with dead citizens and their servants and slaves.\n\nIt was a massacre.\n\nHe ran as well as he could in sabatons. His two surviving archers ran at his heels, and they killed the only two enemies they found, and then they were in the open street in front of the castle's main gate.\n\nThe spearmen were still holding the street.\n\nThe gate was still shut.\n\nAnd the three of them were on the wrong side of the fighting.\n\nHe flipped up his visor. He no longer cared that he might die; he had to breathe. He stood there for as long as it took for his breathing to slow\u2014bent double, he was easy meat for any boglin or irk who wanted him.\n\n\"Messire!\" shouted the panicked crossbowmen.\n\nHe ignored them.\n\nIt seemed like eternity, but he got his head back up after he vomited on the cobbles. There was a half-eaten young boy at his feet, his body cast aside after his legs had been gnawed to the bone.\n\nAcross the square, the spearmen were barely holding. There were fifteen of them, or perhaps fewer, and they were holding back a hundred irks and boglins. The Wild creatures weren't particularly enthusiastic\u2014they wanted to loot, not fight. But they kept pressing in.\n\nAlcaeus pointed across the small square. \"I'm going into that,\" he said to the crossbowmen. \"I intend to cut my way through to the spearmen. Die here or die with me\u2014it's all one to me.\" He looked at the two scared boys. \"What are your names?\" he asked.\n\n\"James,\" said the thin one.\n\n\"Mat,\" said the better accoutered one. He had a breastplate.\n\n\"Span, then. And let's do this thing,\" he said.\n\nHe knew that he didn't want to do it\u2014and he knew that if he didn't make himself go then he'd die right here, probably still trying to catch his breath.\n\n\"Holy Saint Maurice, stand with me and these two young men,\" he said. And then, to the boys, \"Walk right behind me. When I say to loose, kill the creatures closest to me.\" He began to walk around the edge of the square.\n\nOff to the right a pack of irks were fighting over bales of furs. He ignored them.\n\nA daemon loped into an alley, chasing a screaming, naked man, and he ignored it, too. He kept walking, hoarding his strength, sabatons making a grim metallic sound on the bloody stones.\n\nHe didn't look back. He just kept going, under a tree hanging over a house wall, and then along a stone bench on which, in happier days, drunks had no doubt passed out.\n\nWhen he was ten paces from the back of the enemy mob, he shrugged. He wanted to pray, but nothing came to his mind but the sight of a beautiful courtesan in Thrake.\n\n\"Loose,\" he said.\n\nTwo bolts snapped into the mass of Wild flesh, and he followed them in, his sword and dagger flashing.\n\nThe lowest caste of boglins had no armour, but just their soft leather carapaces, and he cut them open, slammed them to earth, and crushed them with his fists. One.\n\nTwo.\n\nThree.\n\nFour.\n\nFive.\n\nHe couldn't breathe. He couldn't see. He had no more to give\u2014\n\n\u2014but he struck blindly, and something caught his dagger hand and threw him to the ground.\n\nHe rolled to his feet, because he was a knight, as an irk\u2014one of the deadly ones\u2014slammed a spear into his midriff. He went backwards and suddenly there were men all around him\u2014\n\n_Men!_\n\nHe was in among the spearmen. It put power into his limbs, and he got up again, his sword rising and falling.\n\nHe could see the thin crossbowman, James, still standing. The boy had flattened some of the things with his crossbow, and now had his side sword in his hand.\n\nThe creatures, panicked by even this very small attack from their rear, were flinching away from them both.\n\nSer Alcaeus gathered himself. One more time.\n\nHe tottered forward, and swung\u2014one.\n\nTwo.\n\nThree times. In those swings two boglins went down. The big irk flinched, turned, and hopped back.\n\nThe two hellish things feeding on the older boy died on James's sword, and then abruptly the square cleared.\n\nBehind them huddled two hundred shocked survivors.\n\nThe men on the castle walls finally opened the gate. Or perhaps were ordered to, now it was safer, and people flooded through, utterly panicked. More died, trampled by others, than at the Wild's hands\u2014the crush of women panicked beyond the capacity for anything but herd animal flight.\n\nThe spearmen backed up after them, step by step.\n\nStep.\n\nBy.\n\nStep.\n\nIn the shadowed streets beyond the square a pair of daemons rallied their own panicked forces, and added irk archers\u2014good ones. Using the light of the burning town, the irks began to loose long shots across the square. Their elfin bows were light but deadly.\n\nSer Alcaeus couldn't cover them all. He was almost immune to their hits but the shafts hurt when they struck his helmet or his greaves, and he was already beyond normal pain, beyond normal fatigue. He looked to the right and left and found that he had reached the gate. The guards were trying to close it; he was trying to back in. But the crush of injured men and trampled corpses underfoot was jamming them open as the enemy made their charge.\n\nHe was able to get his sword arm up in time; he managed to cover himself against a daemon's heavy sword, and then old Ser John was there. He had a mace. It had a five-foot handle.\n\nHe used it well.\n\nHe stepped out past Ser Alcaeus, bouncing on the balls of his feet as if eager for the contest, and his mace moved like a piston. The daemons flinched back from his strike. A boglin died. Another daemon took a blow in the torso and staggered and the mace hit its foot, shattering the bone. It screamed as it went down.\n\nIt wasn't glorious work but Alcaeus bent and grabbed the corpse of a trampled woman and threw it out into the darkness.\n\nThe gate moved.\n\nHe got his hands under a dead boglin's skull and threw the corpse into its fellows.\n\nThe gate moved another hand's breath.\n\n\"Ser John!\" he shrieked. His voice was hoarse and cracked.\n\nThe old knight bounced, cut, and suddenly bounded back.\n\nAlcaeus stumbled after him.\n\nThe gates slammed shut. Terrified sergeants slammed the timbers home into the sockets that held them, and blows rained on the outside surface of the gate from the creatures outside. One irk, braver or craftier than the rest, ran up the gate and got a leg over before one of Ser John's archer's spiked it to the wooden hoardings with a clothyard shaft. The professionals on the wall held\u2014the wave failed, and died.\n\nSer John fell to his knees. \"Too gods-damned old for this,\" he said, staring at the courtyard full of refugees.\n\nBut the gate held. The wall held.\n\nAlcaeus tottered to a pillar in the colonnade and tried to open his faceplate, but he couldn't raise his arms. He hit his head on the colonnade. He couldn't breathe.\n\nStrange hands flipped the catches of his visor and lifted it. Air flooded him. Sweet, wonderful air, tainted only with the harsh screams of people too maddened by fear to do anything else.\n\nIt was James the crossbowman. \"I've got it,\" he said. \"Just stand still.\" The boy hauled the helmet right off his head.\n\nHe pulled off the gauntlets. And Alcaeus slumped to the ground, his back against the colonnade.\n\nSer John appeared in front of him. \"I need you on the walls.\"\n\nAlcaeus groaned.\n\nThe boy stood in front of him. \"Let him breathe! He saved everyone!\"\n\nSer John snorted. \"They ain't saved until they's saved, boy. Ser knight? To the walls.\"\n\nAlcaeus reached out a hand.\n\nSer John caught it, and pulled him to his feet.\n\n# Harndon City\u2014Edward\n\nMaster Pyel's first commission for him was the dullest project he could imagine. It was something he could have done when he was fourteen.\n\nHe was to take twenty iron bars and make staves like barrel staves, then forge-weld them into a single column with bands to keep the staves together. Bands every handspan. Inside diameter to be a constant diameter of one inch.\n\nDull.\n\nStill, he was smart enough to know that Master Pyle wouldn't have given him the work if it didn't matter. He was careful with his measurements, and he decided to construct a mandrel to keep the _insides_ of the staves equidistant while he forge-welded them. That took time, too. He planished the mandrel and then polished it endlessly.\n\nHe had a moment of deep satisfaction when another journeyman, Lionel, grinned. \"You know,\" he drawled, obviously relishing the moment. \"You can order an apprentice to do that.\"\n\n_I'm a fool_ , he thought, happily. He left Ben the shoemaker's boy to use pumice on his lovely mandrel while he went out into the evening to fence with his mates and show Anne his ring. Better, to show Anne's parents. Apprentices didn't marry\u2014but a journeyman was a person of consequence. He was a man.\n\nThe next morning, he had the mandrel ready, thanked the apprentice like a good master, and then whipped the forge welds into shape, smoothing both inside and outside. It turned out to be more finicky than he had expected, and took him all day.\n\nMaster Pyle looked at the result and slammed it against the oak tree in the yard. The welds held. He smiled. \"You made a mandrel,\" he said.\n\n\"Had to,\" replied Edward.\n\nMaster Pyle made a face. \"My design is flawed,\" he said. \"How're your casting skills?\"\n\nEdward shrugged. \"Not that good, Master,\" he admitted.\n\nThe next morning when the sun rose, he was down by the river, casting bells with the Foibles\u2014rivals, but friends.\n\n# Lissen Carak\u2014The Red Knight\n\nHundreds of leagues to the north, the same sun rose on a fortress which was complete in every warlike respect\u2014high wooden hoardings crowned the turrets and curtain walls, and a major engine of war stood atop every tower: the donjon tower bore the weight of a trebuchet, and smaller mangonels and ballistae decorated the smaller towers.\n\nAside from a dozen men on duty, the garrison, who had laboured two days and nights by torchlight, lay asleep in heaps of straw. The dormitory was full of local people and so was the hall and the stable.\n\nSauce awakened the captain because there was movement down by the river. He had placed a garrison of ten archers, three men-at-arms and a pair of knights in the tower at the bridge under Ser Milus's command the evening before. They had their own food and a mirror with which to signal, and this morning they were apparently flashing away merrily.\n\nSer Jehannes had gone with them, as a mere man-at-arms. He had gone without comment and left no note. The captain awoke to find it still on his mind.\n\n\"Damn him,\" he said, staring at the newly whitewashed plaster over his head. Jehannes had always disliked him because he was young and well-born.\n\nAs far as the captain saw it, Messire Jehannes could have both his birth and his youth. He lay on his bed, his breath steaming in the air, and found himself growing angrier and angrier.\n\n\"Damn who?\" asked Sauce. She flashed him a smile that was probably meant to be winsome. She was an attractive woman, but the missing front teeth and the scar on her face tended to made her winsome look slightly savage.\n\nSauce and the captain went backaways. The captain considered confiding in her\u2014but he was the captain, now. Everyone's captain.\n\nHe got his feet on the cold stone instead. \"Never mind. Call Toby for me, will you?\"\n\nShe leered. \"I'm sure I could dress you, mesself.\"\n\n\"Maybe you could and maybe you can't, but neither will get me moving fast enough.\" He stood up, naked and she swatted at him with her gloves and went out calling for Toby.\n\nToby and Michael arrived together, Toby with clothes, Michael, sleepy to the point of clumsiness, with a cup of steaming wine.\n\nThe captain armed himself in the ruddy light of the new sun, Michael fumbling with buckles and laces so that it seemed to take twice as long as arming usually did and he almost regretted sending Sauce away. But he ran lightly down the steps to the great courtyard and patted Grendel's nose when he was led out. He took the tall bassinet on his head, pulled steel gauntlets over his hands, and vaulted into Grendel's war-saddle. He was giving his men a good example\u2014he was also riding out of a fortress into the unknown.\n\nIt occurred to him as he ducked his head to pass through the narrow postern\u2014he had ordered that the main gate be shut for the duration\u2014that if nothing attacked them, he was going to look a ripe fool. Followed by the image of a taloned foot ripping the guts out of his riding horse, which made his stomach lurch and his throat go cold.\n\nHe rode down the steep road, leaning well back into the comforting buttress of his war-saddle, with Wilful Murder, Sauce, Michael Rankin and Gelfred all fully armed at his back. At the base of the hill he turned away from the bridge and rode west\u2014not onto the narrow track he'd followed and fought the daemon, but around the base of the fortress.\n\nHe rode slowly around it, looking up so hard that his neck hurt, examining his hoardings from their attackers' perspective. The fortress was a hundred feet above him, huge, imposing and very, very far away.\n\nAfter he passed the donjon the first trebuchet released. He heard the crack of the wood base of the counter-weight striking its restraint and saw the rock pause at the height of its arc. Then it fell with a crash well to the west.\n\nThe captain turned to Wilful Murder. \"Go and put an orange stake on it, Will. They won't loose again.\"\n\n\"It's always me,\" Wilful grumbled and did as he was told.\n\nThe rest continued to ride around the base of the fortress. Two other engines released, and both times the captain sent Wilful off to mark the fall of shot.\n\n\"Tough nut,\" Sauce said, suddenly.\n\n\"Some of our enemy have wings,\" the captain replied and he nodded heavily, because he was in full harness and couldn't really shrug well. \"But yes. With our company on the walls and all the defences up we should be able to hold until we starve.\" He looked beyond her. \"We'll lose the Lower Town first, then Bridge Castle.\" He shrugged. \"But the\u2014the king will come first.\"\n\nWith that, he leaned his weight forward and led them at a slow, lumbering canter across the fields to the Bridge Castle.\n\nMilus met him, also fully armoured, at the tower gate. Behind him, on the bridge, were a dozen heavy wagons laden with goods and fifty or more men and women all pale as parchment. Merchants.\n\n\"Come for the fair,\" Milus said. He made a face. \"They say there's five convoys behind them.\"\n\nThe captain turned and looked at Michael, who grimaced. \"We don't even have all the farmers in,\" he said. \"Fifty, you say? And their wagons?\"\n\n\"And I'll bet they don't have any food,\" the captain said. \"I'll guess they have carts full of cloth and luxury goods, because they've come to _buy_ grain.\" He looked around. \"How many more mouths can you take, Milus?\"\n\nThe older knight narrowed his eyes. \"I can take all of 'em,\" he admitted. \"And thirty more like 'em. But I'll need more grain, more salt meat, more of everything to do it. Except water. We've plenty of that, out of the river.\"\n\nBack up the hill he went to report to the Abbess. A heavy military wagon was raised from the cellars and reassembled, then loaded to heaping with food and provender, and hand-hauled down the steep slope, teams of men on gate winches letting it down a few feet a time. The captain disarmed, handing his harness to his squire. His hips were screaming, and once it was finally off he felt light enough to fly away.\n\nEven as they increased the supplies to the lower fort, more merchants arrived. Some were angry at the interruption of trade, and some were clearly already terrified. The captain went back down the hill and wasted the morning trying to calm them. He finally told them to send a deputation up the hill to the Abbess.\n\nThen he made the climb back up to the fortress to hide in his Commandery, a small cell with a door directly onto the courtyard and a pair of arched windows separated by a fluted column. Open, the windows let in a spring breeze carrying the scent of wildflowers and jasmine, and he could see fifteen leagues to the east over the low hills.\n\nToday, instead of turning to the parchment scrolls full of accounts that awaited him, he unbuckled his sword and hung it on the man-high bronze candelabra and leaned his elbows on the sill of the leftmost window.\n\nBooted footsteps announced Michael. \"Your armour,\" the young man said quietly.\n\nThe captain turned to see two archers with a heavy wicker basket, and his valet with an armload of dressed lumber. While he watched, the archers argued about which pre-cut peg went in which hole and the valet stared off into space while idly providing the correct piece, even when the archers asked for the wrong thing. Before the sun had moved the width of a finger, they had assembled a rack for the captain's armour, man sized, a little taller than the man himself, and Michael dressed the heavy wooden form carefully. A good arming rack could speed a man into his harness by precious minutes. And with every inch of the fortress convent crammed to capacity and past it with soldiers and refugeees, his office was his sleeping room.\n\nWhen the archers and the valet went back out, the noise vanished and the captain returned to his window.\n\n\"Will that be all, ser?\" Michael asked.\n\n\"Well done, Michael,\" the captain said.\n\nThe younger man jumped as if he'd been bitten. \"I\u2014that is\u2014\" he laughed. \"Your valet, Jacques, did most of it.\"\n\n\"The more credit to you that you give him credit,\" the captain observed.\n\nEmboldened, Michael came forward and, very slowly, leaned into the right hand window. His stealthy progress was not unlike that of a convent cat the captain had observed that morning, which had been intent on stealing a piece of cheese. He smiled. It took Michael as long to rest in the window as it had for the three men to build the armour rack. \"We're fully provisioned,\" Michael said carefully.\n\n\"Hmm. No commander facing a siege ever admits being 'fully provisioned,' \" the captain said.\n\n\"So now we wait?\" Michael asked.\n\n\"Are you a squire or an apprentice captain?\" the captain asked.\n\nMichael stood up straight. \"My pardon, ser.\"\n\nHe grinned wickedly. \"I don't mind an intelligent question, and especially not when it helps me think. I do have to think, young Michael. Plans don't just come full-blown into my head. Next we're going to use a powerful magic, something potent, grave and dire. The Archaics used it well and often. All the histories describe it, and yet no romance of chivalry ever mentions it.\"\n\nMichael pulled a face that told the captain he'd wit to tell when he was being baited.\n\n\"What spell?\" he asked.\n\n\"No spell,\" the captain advised. \"But it's a kind of magic nonetheless. We're provisioned and armed, we've repaired our fortifications, and the enemy are not yet at the gates. So what shall we do?\"\n\n\"Compel the rest of the peasants into the walls?\" Michael asked.\n\n\"No. That's done.\"\n\n\"Build outworks?\"\n\n\"We lack the force to man them, so no.\" The captain paused. \"Not so bad, though.\"\n\nMichael's frustration was obvious. \"Summon a tame daemon?\" he asked.\n\nThe captain scratched his pointed beard. \"No,\" he said. \"Although if I knew how to I might.\"\n\nMichael shrugged.\n\n\"Two words,\" the captain encouraged him.\n\nMichael shook his head. \"Higher walls?\" he asked, knowledge of his own inadequacy making him sound petulant.\n\n\"No.\"\n\n\"More arrows?\"\n\n\"Not bad, but no.\"\n\n\"Find allies?\" Michael asked.\n\nThe captain was silent a moment at that, looking east. \"We have already summoned our allies, but that's not bad at all,\" he said. \"A very useful thought, and one that I may pursue.\" He looked at the fashionably green-clad scion of the aristocracy and added. \"But no.\"\n\n\"Damn,\" Michael said. \"Can I give up?\"\n\n\"As squire, or as apprentice captain?\" the captain asked. \"You started this, not me.\" The captain picked up the short baton of office that he almost never carried. It had belonged to the previous captain, and had some history and authority to it\u2014enough that the captain suspected it might have a touch of phantasm about it. \"You have thirty-one lances, give or take; sixteen elderly but competent sergeants and one well-constructed, if elderly, fortress on good ground. You must defend a ford, a bridge, a constant flow of terrified merchants and a vulnerable Lower Town with inadequate walls. Tell me your plan. If it's good enough, I'll claim it's my own and use it. There are stupid answers but there's no right answer. If your answer is good, you live and make a little money. If your answer is bad, you fail and die and just for extra points, a lot of harmless people, some actual nuns and a bunch of farmers will die with you.\" The captain had an odd look in his eyes. \"Let's hear it.\"\n\nMichael had sprouted enough hairs on his chin that it might honestly be called a beard and he played with them for a while. \"All in our current situation? Fully provisioned and so on?\"\n\nThe captain nodded.\n\n\"Send messengers for aid. Enlist allies from local lords. Button up the fortress, tell the merchants to go hang themselves, and prepare for the enemy.\" Michael looked out over the woods to the east while he thought on.\n\n\"Messengers sent. Allies cost money and our profit on this is slim as it is. We were in pretty desperate straits before we got this job. And those merchants represent a source of cash to us. I leave aside the morality of the thing. We can make them pay for protection and split the money with the abbess. Fair is fair\u2014it's her fortress and our steel.\" The captain's gaze was out the window, on the distant woods.\n\nThe sun moved in the sky.\n\n\"I give up,\" Michael admitted. \"Unless it's something very simple like more rocks for siege engines, or more water.\"\n\n\"I think I'm glad that you can't find it, lad, because you have a brain and your family has a lot of war craft. And if you don't see it, perhaps _they_ won't see it either.\" The captain pointed out the window.\n\n\"They? The Wild?\" Michael asked quietly.\n\nThe captain scratched at his beard again. \"Active patrolling, Michael. Active patrolling. Starting in about six hours, I'm putting our lances out in fast-moving patrols. In all directions, but mostly east. I want to be familiar with the terrain, to relocate our foe, and then I'm going to ambush, harass, irritate, and annoy him and his minions until they go elsewhere looking for easier prey. If they choose to come here and lay siege to us I intend to have them leave a trail of blood\u2014or whatever they have for blood\u2014through that forest.\"\n\nMichael was looking at his hands, which were trembling. \"You intend to go out into the Wild?\" he asked, incredulous. \"Again?\"\n\n\"If the initiative is in the woods, I'll seize it in the woods,\" said the captain. \"You think the enemy are ten feet tall and made of adamantine. I think they have a corps of men as servants, archers and woodsmen, who have so little war-craft that I can see the smoke of their dinner fires from here.\" The captain put a hand on his squire. \"And ask yourself\u2014why is the main body of our enemy to the _east_ ?\" He looked out. \"Gelfred is out there right now,\" he said quietly.\n\nMichael whistled. \"Blessed Saint George. Have they passed us by?\"\n\nThe captain smiled. \"Well guessed, young Michael. Our enemy has bypassed us\u2014a tribute to our preparations and our little raid. But there's a reason you don't bypass a fortress, and I'm about to teach him. Unless,\" he smiled, and just for a moment, he showed his youth. \"Unless it's all a fucking trap.\"\n\nMichael swallowed.\n\n\"Anyway, his human allies are right there as well\u2014to the east. Don't point. I suspect that some of the birds are spies.\" The captain turned away.\n\n\"Then they can see everything we do!\" Michael said.\n\n\"Everything,\" the captain said with evident satisfaction. \"Go to the refectory, find some parchment, write me a list of all of your notions for the defence of this position, and then go polish something.\" He smiled. \"But first, get me some wine.\"\n\n\"I was afraid,\" the squire blurted. \"In the fight with the wyvern\u2014I was so afraid I could barely move.\" He breathed heavily. \"I can't stop thinking about it.\"\n\nThe captain nodded. \"I know,\" he said.\n\n\"But it will get better, won't it? I mean\u2014I'll get used to it. Won't I?\" he asked.\n\n\"No.\" The captain shook his head. \"Never. You never get used to it. You shake, vomit, foul your braes, piss yourself, whatever you do, every fucking time. What you get used to is the power of the fear, the onset of the terror. You learn you can face it. Now get me some wine, drink a couple of cups yourself, and get back to work.\"\n\n\"Yes, m'lord.\"\n\nThere was a constant flow of men and materiel up and down the hill, from the top of the fortress to Bridge Castle. The war engines on the towers lofted practice rounds into the fields, and trusted corporals took patrols out into the farmland\u2014careful, wary patrols on fast horses. The closest farmers had responded well enough to the alarm bells and yesterday's summons, and Abbington, the biggest of the hamlets, was clear, but the more distant had only sent children to ask for more information, and none of them had brought in any of their precious grain unless the soldiers had brought it themselves. The patrols either went to fetch in the timid or led out farmers who had believed it was merely a drill.\n\nAnd the more prosperous yeoman had other questions.\n\n\"Who is going to pay for our grain?\" demanded a strong middle-aged man with an archer's forearms and a handsome head of brown hair. \"This is my treasure, ser knight\u2014my precious store. What we scrimp and save up over the winter turns to silver when the merchants come in the spring. Who's paying for it now?\"\n\nThe captain directed all such questions, firmly and quietly, to the Abbess.\n\nAs the sun set on the third full day the cellars were bursting with grain. A further hundredweight lay at the foot of the track that ran up the hill to the fortress where a cart had broken loose and smashed to pieces, and now every wagon up or down ran with ropes attached to the gate winches\u2014and the main gate had stood open all day.\n\nThe hundredweight of grain had the curious effect of dragging birds out of the sky to eat the free bounty. Archers, led by Gelfred, netted them.\n\nThe fortress was so packed with people that there were men and women planning to sleep on the stone flags on straw despite the briskness of the evening. Torches burned all around the courtyard and a bonfire burned in the centre, the flickering orange light reflecting off the towers, the donjon and the sparkling dormitory windows. Chickens\u2014hundreds of chickens\u2014ran about the courtyard and the rocks on the ridge below the gate. Pigs rooted in the convent garbage at the base of the cliff, nigh on two hundred of them. The convent sheepfold, hard against the eastern walls, was also full to bursting and in the last light a man standing in the Abbess's solar could see the glitter of a dozen men-at-arms and as many archers, bringing in another thousand sheep from the eastern farms.\n\nThe captain stood in the Abbess's solar and watched patrols, the sheep, and the formal closing of the gate. He followed Bent's craggy form as the big archer changed the watch in the donjon, marching the off-going watch around the whole circuit as he collected them and put fresh men in their places. It was an impressive and efficient ceremony, and it had the right effect on the villagers, most of whom had never seen so many armed men in their lives.\n\nThe captain sighed. \"In an hour's time a virgin will have been deflowered and a husbandman will have lost his farm at dice,\" he said.\n\n\"You have a virgin in mind?\" the Abbess asked.\n\n\"Oh, I'm quite above such earthy concerns.\" The captain continued to watch, and he was smiling.\n\n\"Because you are worried, you mean. You must be worried that nothing has come at us yet,\" the Abbess said.\n\nThe captain pursed his lips and shook his head. \"I'd rather be a ripe fool, the laughing stock of every soldier in Alba,\" he said, \"then face a siege by those things. I don't know where they are yet or why they let us get everyone under cover. In my dark moments I think our walls are already undermined, or they have a legion of traitors inside the walls\u2014\" He raised a hand, making a warding-off motion. \"But in truth, I can only hope they know as little of us as we know of them. The day before yesterday we were easy meat. Today, if sheer fear doesn't break us, we could hold for a year.\" He glanced at her worried face.\n\nShe shrugged. \"How old are you, Captain?\"\n\nHe was clearly uncomfortable with the question.\n\n\"How many sieges have you seen?\" she asked. \"How many Wild creatures have you faced in combat?\" She turned towards him and stepped forward, boring in on her target. \"I'm a knight's daughter, Captain. I know these are not polite questions, but by God I feel I deserve to know the answer.\"\n\nHe leaned against the wall. Scratched under his chin for a moment, staring off into space. \"I've killed more men than I have monsters. I've stood one siege and, to be fair, we broke it in the second month. I'm\u2014\" He turned his head and met her eye. \"I'm twenty.\"\n\nShe made a sound between a satisfied hrmmf and a snort.\n\n\"But your divination told you that.\" He straightened from the wall. \"I'm young, but I've seen five years of unending war. And my father\u2014\" He paused, and the pause became a silence.\n\n\"Your father?\" she asked quietly.\n\n\"Is a famous soldier,\" he finished, his voice very quiet.\n\n\"I've entrusted my defences to a child,\" the Abbess said, but she pursed her lips in self-mockery.\n\n\"A child with a first rate company of lances. And there is, truly, no better sell-sword captain in Alba. I know what I'm doing. I've seen it and done it before, and I've studied it, unlike the rest of my breed. I've studied them all\u2014Maurikos and Leo and Nikephoros Phokas, even Vegetius. And if I may say so, it's too late to change your mind now.\"\n\n\"I know,\" she said. \"I'm afraid.\" She drank her wine and, quite spontaneously, she took his hand. \"I'm fifty,\" she admitted. \"I've never withstood a siege, myself.\" She let his hand go and bit her lip. \"Are you afraid?\"\n\nHe took her hand again and kissed it. \"Always. Of everything. My mother made me a coward. She taught me, very carefully, to fear _everything._ Starting with her. See? You are become my confessor.\" He smiled crookedly. \"I am the world's expert at overcoming fear. Cowardice is the best school for courage, I find.\"\n\nShe had to smile. \"Such a wit. Vade retro!\"\n\nHe nodded. \"I'm too tired to get out of the chair.\"\n\nTheir laughter and light conversation lasted through the rest of her wine, and his. Finally she said, after looking out the window, \"And what do you fear most?\"\n\n\"I fear failure,\" he said. He laughed at his own words. \"But alone of the people in this fortress, I have no fear of the Wild whatsoever.\"\n\n\"Are you posturing?\" she asked.\n\nHe stared into her fire for a little. \"No,\" he said with a sigh. \"I need to go look at the watch. I have tried something reckless tonight. I need to make sure my people are ready for it. You know that your enemy is using animals to watch us\u2014yes?\"\n\n\"Yes,\" she said, very quietly.\n\n\"Do you know anything else, my lady? Anything that would help your very young captain save your walls?\" He leaned toward her.\n\nShe looked away. \"No,\" she said.\n\nHe put his wine cup on the oak sideboard with a click. \"I told you the truth.\"\n\n\"Let us have a few moments to marshal our forces,\" she said with a wan smile. \"Go see to your watch. My few tawdry secrets are not in any way germane to our siege.\"\n\nHe bowed, and she waved him away, so he went out into the stairwell. It was dark.\n\nHer door closed, and he began to feel his way down the stone steps when a hand closed on his.\n\nHe knew her in a moment, and pulled the hand to his lips\u2014faster than she could take it away. He heard her sigh.\n\nThat moment he considered crushing her against the stone wall. But it occurred to him that she must be there by the Abbess's commission, and it would be rude, to say the least, to attack the novice outside the Abbess's door. Or something like that went through his head\u2014before her lips came down on his and her hands pushed against his shoulders.\n\nHis heart pounded. His mind went blank.\n\nHe could feel her power, now. As their bodies moved together\u2014her tongue probing his\u2014they were _generating_ power.\n\nShe broke their kiss and stepped away\u2014a sudden absence of warmth in the dark\u2014and said \"Now we are even.\" She took his hand. \"Come.\"\n\nShe led him down the dark stone steps. Across the hall\u2014the bonfires in the courtyard made the stained glass figures flicker and wriggle as if they were animated, and fitful rainbows played across the hall floor. After the complete darkness of the Abbess's solar stairwell, the hall seemed well enough lit.\n\nShe was taking him to the books. Halfway across the hall they kissed again. No one could have said which of them initiated it. But when his hand moved across her bodice, she stepped away.\n\n\"No,\" she said. \"I want to show you this, and I am not your whore.\"\n\nBut she kept his hand. Led him to the book. \"Have you seen it?\" she asked.\n\n\"Yes.\"\n\n\"Did you understand?\" she asked, flipping the pages.\n\n\"No,\" he admitted. There is nothing a young man enjoys less than telling the object of his affection how little he knows.\n\nHer not-quite-a-smile played somewhere in the corner of her mouth. \"You are one of us, are you not? I can feel you.\"\n\nHis eyes were on hers, but when she looked at the book he looked too. Looked at the alembic in St. Pancreas's hand. And followed the saint's pointing finger to a diagram, lower on the page\u2014a tree.\n\nHe flipped to another page, where another saint pointed\u2014this time to a cloud.\n\n\"Is this a test?\" he asked.\n\nShe smiled. \"Yes.\n\n\"Then I guess the book is a code. The shapes that the saints point to indicate the shape of a template that, when covering the text, will indicate what the reader should read.\" He ran his finger over the text across from St. Eustachios. \"It is a grimoire.\"\n\n\"A fantastically detailed, internally coded, referential grimoire,\" she said, and then bit her tongue which he found, just at that moment, intensely erotic. He reached to kiss her, but she made the dismissive motion women make when boys are tiresome. \"Come,\" she said.\n\nHe followed her across the hall. He was conscious, at a remove, that he had a watch to oversee; a siege to command. But her hand in his held such promise. It was smooth, but rough. The hand of a woman who worked hard. But still smooth; like the surface of good armour.\n\nShe dropped his hand the moment she opened the courtyard door, and they were in the light again.\n\nHe wanted to say something to her\u2014but he had no idea _what_ he wanted to say.\n\nShe turned and looked back at him. \"I have one more thing to show you,\" she said.\n\nEven as she spoke, she pulled a cowl of _not-seeing_ around herself.\n\nHe was being tested in another way.\n\nHe reached into the palace of his memory and did the same. He was there for long enough to see Prudentia looking at him with ferocious disapproval, and that the green spring outside his iron door was building up into a storm of epic proportions.\n\nAnd then they slipped across the courtyard. They were scarcely invisible\u2014one of the Lanthorn girls, spinning in a reel with a young archer, saw the captain clearly because she was dancing and she deftly avoided him as she whirled.\n\nBut he was not interrupted as he passed.\n\nShe stopped at the iron-bound dormitory door and he manipulated his phantasm so that it linked to hers. It was a very intimate thing to do\u2014something he had never done with anyone but Prudentia, and which the sight of her had reminded him of.\n\n_She used to say that the mind was a temple, an inn, a garden, and an outhouse, and that casting with another magus partook of worship, intimate conversation, sex, and defecation._\n\nBut as his power reached to hers, hers accepted it, and they were linked.\n\nHe winced.\n\nShe winced as well.\n\nAnd then they were in the dormitory, standing in a small hall where, on his former visit, older nuns had sat to read or to do needlework. There was light here. Most of the nuns were out in the yard, but two still sat quietly.\n\n\"Look at them,\" Amicia said. \"Look.\"\n\nHe didn't have to look too hard. Tendrils of power played about them.\n\n\"All of you have the power?\" he asked.\n\n\"Every one of us,\" she said. \"Come.\"\n\n\"When will I see you again?\" he managed, as she led him along the northern curtain behind the stable block. An apple tree grew there, in a stone box set into the wall. There was a bench around it.\n\nAmicia settled onto the bench.\n\nHe was too befuddled to seek to kiss her, so he simply sat.\n\n\"All of you are witches?\" he asked.\n\n\"That's an ugly word for _you_ to use, man-witch,\" she said. \"Sorcerer. Warlock.\" She looked out over the wall.\n\nFar to the east he saw the barest smudge of orange, and it instantly recalled him to his duty. \"I must go,\" he said. He wanted to impress her\u2014he wanted not to seem to need to impress her. \"I've sent people to do something I should have done myself,\" he blurted.\n\nShe didn't seem to pay him any heed. \"I thought that you needed to know what the stakes were,\" she said. \"I don't think _she_ is going to tell you. This is a place of power. And the Masters of our Order have filled it with women of power, and with powerful artefacts. Now it shines like a beacon.\n\nHe felt blind and foolish at her words. But Prudentia's rules\u2014on the use of power, on using the sight of power\u2014which were wisdom in a world that distrusted the magi, had deprived him of this insight.\n\n\"That, or she meant me to tell you,\" Amicia added. Her head slumped for the first time that evening.\n\n\"Or she expected me to reason it out for myself,\" he said bitterly. He felt the time flowing away as if he had an hourglass in his hand\u2014he felt tonight's raid slipping west into the trees, and he felt the lack of alertness on his watch, and sensed a thousand forgotten details, like a tendril of power attached to his soldiers that was pulling him from her side. And the glow far in the east\u2014what was that?\n\nAnd then he felt her, and it was like a chain that tethered him to the bench.\n\n\"I must go,\" he said again. But youth, and his hand, betrayed him, and he was again in her arms or she in his.\n\n\"I do not want this,\" she said as she kissed him again.\n\nSo he broke free. Broke the binding between them with a thought, and stepped away. \"Do you often come here?\" he asked, his voice hoarse. \"To the tree?\"\n\nShe nodded, barely perceptible in the odd light.\n\n\"I might write to you,\" he said. \"I want to see you again.\"\n\nShe smiled. \"I imagine you'll see me every day,\" she said. \"I don't want this. I don't need it. You don't know me. We should walk away.\"\n\n\"If I strike you now we can end as we started,\" he said. \"With a kiss and a blow. But you want me as I want you. We are _bonded._ \"\n\nShe shook her head. \"That sort of thing is for children. Listen, _Captain._ I have been a wife. I know how a man feels between my legs. Ah! You wince. The novice is not a virgin. Shall I go on? I lived across the wall. I was an Outwaller. No, look!\" She peeled back the collar of her gown, and her shoulder was covered in tattoos.\n\nBathed in the distant firelight her shoulder gleamed, and all he felt was desire.\n\n\"I was taken young, and grew to womanhood among them. I had a husband\u2014a warrior, and we might have grown old together, he as war chief and I the shaman. Until the Knights of the Order came. They killed him, they took me, and here I am. And I do not need rescuing. I _live_ in the world of spirit. I have come to love Jesus. Every time I kiss you, I hurtle backwards in my life to another place. I cannot be with you. I will not be a mercenary's whore. I sacrificed myself this evening so that you could see what you are so obviously blind to\u2014because you are so very afraid of your power.\" She turned her head. \"Now go.\"\n\nThe lines of power to his soldiers were taut as cables. He was ignoring his duty. It was like a broken bone\u2014a scream of pain. But he couldn't let what was between them rest.\n\n\"You wanted me as much as ever I wanted you, from the moment your eyes met mine. Don't be a hypocrite. You sacrificed yourself this evening? Rather, you craved this evening and built yourself a reason to let yourself have it.\" Even as he spoke the words, he cursed himself for a fool. It was not what he wanted to say.\n\n\"You have no idea what I do or do not want,\" she said. \"You have no idea the life I have led.\"\n\nHe took a half-step away\u2014the sort of half-step a swordsman takes when he changes from defence to attack. \"I grew up with five brothers who hated me, a father who ignored and despised me, and a doting mother who wanted to make me a tool of her revenge,\" he hissed. \"I grew up across the river from your Outwaller villages. When I looked out of my tower I saw you Outwallers in the land of freedom. You had a husband who loved you? I had a succession of sweethearts placed in my bed by my mother to spy on me. You would have been an Outwaller shaman? I was being trained to lead armies of the Wild to crush Alba and rid the earth of the king. So that my mother could feel avenged. Knights of the Order came for you? My brothers ganged up to beat me, to please my supposed father. It was _good fun._ \" He found that his voice was rising and spittle flew from his mouth.\n\nSo much for self-control; he had said too much. Far too much. He felt sick.\n\nBut he was not done. \"But _fuck that._ I am not the Antichrist, even if God himself decrees I should be. I will be what I will, not what anyone else wills, as can you. Be what you choose. You love Jesus?\" he asked, and something black passed into his mind. \"What has he done for you? Love me instead.\"\n\n\"I will not,\" she said, quite calmly.\n\nHe didn't will himself to walk away. He didn't feel a thing\u2014he didn't feel an urge to reply. It was like being cut with a very sharp sword, and watching your arm fall to the ground.\n\nThe next he knew, he was standing in the guard box over the gate.\n\nBent, the duty archer, stood with his arms crossed. When he saw the captain he twirled his moustaches. \"You've sent out a sortie,\" he said. \"Or somewhat similar. I can't find Bad Tom or half the men-at-arms for duty.\"\n\n\"It's about to happen,\" the captain said, mastering himself. \"Tell the watch to be alert. Tell them\u2014\"\n\nHe looked up. But the stars were silent and cold.\n\n\"Tell them to be alert,\" he said, at a loss. \"I have to attend the Abbess.\"\n\nHe got himself to the jakes and threw up. Wiped his chin on an old handkerchief and threw it after his puke, which would have scandalized a laundress. And then he pulled himself up straight, nodded, as if to an invisible companion, and walked back into the hall.\n\nThe Abbess was waiting for him.\n\n\"You met with my handmaiden,\" she said.\n\nHis armour was adamantine. He smiled. \"A merry meeting,\" he said.\n\n\"And you saw to your guards,\" she said.\n\n\"Not enough,\" he said. \"Lady, there are too many secrets here. I do not know what the stakes are. And perhaps I am simply too young for this.\" He shrugged. \"But we have two foes\u2014the enemy outside and the enemy within. I wish you would tell me what you know.\"\n\n\"If I told you everything I knew you would scourge me with whips of fire,\" said the Abbess. \"It is a passage in the Bible on which I often ponder.\" She rose from her throne and crossed the hall to the book. \"You have solved this riddle?\" she asked.\n\n\"Using the enormous hints provided,\" he answered.\n\n\"It was not my place to tell you,\" she said. \"When our kind swear oaths, those oaths bind our power.\"\n\nHe nodded.\n\n\"You are as tense as a bowstring,\" she said. \"Is that the effect of Amicia?\"\n\n\"I have played a trump card tonight,\" he admitted. \"And I let my tryst interfere with duty. Things are not done as I would like on an evening when I have taken a gamble that now seems reckless.\" He paused, and said what he had boiling inside him. \"I do not enjoy being toyed with.\"\n\nThe Abbess picked up her onyx rosary and adjusted her wimple. She shrugged. \"No one does,\" she said dismissively. \"I don't deal in the imagery of gambling,\" she said. \"But perhaps we can do some good, and by our presence prevent the dicing and the deflowering you were worried about,\" she said. \"Let's walk among our people, Captain.\"\n\nThey walked out, and she put a hand on his arm, very much the lady, and a veiled sister came and carried her train, which was longer and more ornate than any other sister's in the convent. Indeed, the captain suspected that her habit was far from the rule as laid down for sisters of Saint Thomas. She was a rich and powerful woman who had somehow turned to this life.\n\nWhen they entered the courtyard all conversation stopped. A ring of dancers moved to the sound of a pair of pipes and a psalter played by none other than the captain's squire. The musicians continued to play, and the dancers paused, but the Abbess gave them a firm nod of approval and the dancing continued.\n\n\"When will they come at us?\" the Abbess asked quietly.\n\n\"Never, if I have my way,\" the captain said pleasantly.\n\n\"It's better to make your money without fighting?\" she asked.\n\n\"Always,\" he said, bowing deeply to Amicia, who stood watching the dancers. She nodded coolly in return. But he had armed himself against her and he continued without a pause. \"But I also like to win. And winning requires some effort.\"\n\n\"Which you will make eventually?\" she said. But she smiled. \"We spar so naturally I might have to do some penance for flirtation.\"\n\n\"You have a gift for it that must have won you many admirers,\" he said gallantly.\n\nShe struck the back of his hand with her fan. \"Back in the ancient times when I was young, you mean?\"\n\n\"Like all beautiful women, you seek to make an insult of my flattery,\" he returned.\n\n\"Stand here. Everyone can see us here.\" She nodded to Father Henry, who was standing hesitantly between the chapel and the steps to the Great Hall.\n\nThe captain thought that the man was a-boil with hostility. A year ago, the captain, in one of his first acts on taking command, had executed a murderer in the company\u2014an archer who had started to kill his comrades for their loot. Torn had been a non-descript man, an outlaw. The captain eyed the priest. He had something of the same look. It wasn't really a look. A feel. A smell.\n\n\"Father Henry, I don't believe that you've been properly introduced to the captain.\" She smiled, and her eyes flashed\u2014a glimpse of the woman she had been, who knew that a flash of her eyes would restore any admirer to obedience. A predator who liked to play with her food.\n\nFather Henry offered a long hand to shake. It was moist and cold. \"The Bourc, his men call him,\" he said. \"Do you have a name you prefer?\"\n\nThe captain was so used to dealing with petty hostility that it took a moment to register. He turned his full attention on the priest.\n\nThe Abbess shook her head and pushed the priest by the elbow. \"Never mind. I will speak to you later. Begone, ser. You are dismissed.\"\n\n\"I am a priest of God,\" he said. \"I go where I will, and have no master here.\"\n\n\"You haven't met Bad Tom,\" the captain said.\n\n\"You have a familiar look about you,\" Father Henry added. \"Do I know your parents?\"\n\n\"I'm a bastard, which you've already found cause to mention,\" the captain said. \"Twice, man of God.\"\n\nThe priest withstood his glare. But his eyes were as full of movement as a man dancing on coals. After too long a pause, the priest turned on his heel and walked away.\n\n\"You go to great lengths to hide your heredity,\" the Abbess noted.\n\n\"Do you know why?\" the captain asked.\n\nThe Abbess shook her head.\n\n\"Good,\" said the captain. His eyes were on the priest's back. \"Where did he come from? What do you know about him?\"\n\nThe dance had finished, and men were bowing, women dipping deep courtesies. Michael had just noted that his lord had witnessed his troubadour skills and flushed deep red in the torchlight, and the Abbess cleared her throat.\n\n\"I told you. I took him from the parish,\" the Abbess murmured. \"He has no breeding.\"\n\nThe sky to the east lit up, as if from a flash of lightning, but the flash lasted too long and burned too red, for as long as it took a man to say a Pater Noster.\n\n\"Alarm!\" roared the captain. \"Gate open, all crossbows armed, get the machines loaded. Move!\"\n\nSauce had been watching the dancers. She paused, confusion written on her face. \"Gate _open_?\" she asked.\n\n\"Gate open. Get a sortie ready to ride, you'll be leading it.\" The captain pushed her towards her helmet.\n\nMost of his men were already moving, but if he hadn't been beguiled by the evening's revelations, they'd have been in their armour already.\n\nAlready, a dozen men-at-arms stood by their destriers in the torchlit gateway, their squires and valets scrambling to arm them. Archers scrambled from the courtyard onto the catwalks around the curtain walls, some even bare-arsed in the light of the courtyard fires, their hose down and their shirttails dangling.\n\nThere was a second flash of fire to the east, half as long as the last.\n\nThe captain was grinning. \"I hope you didn't need olive oil for anything really important,\" he said, and squeezed her arm in a very familiar way. \"May I take my leave? I should be back with you before the next bell.\"\n\nShe eyed him in the fire-lit darkness. \"This is your doing, and not the enemy's?\" she asked.\n\nHe shrugged. \"I hope so,\" he said. Then he leaned close. \"Hellenic fire. In their camp. Or so I hope.\"\n\n# North of Harndon\u2014Harmodius\n\nDissection is one of those skills a man never really forgets. Harmodius had exhumed the corpse himself\u2014not much of a risk, given the haste with which it had been buried.\n\nHe was only interested in the brain, anyway. Which was as well, as the thorax was badly damaged and the central body cavity was largely empty. Something had eaten the guts.\n\nHarmodius was above such feelings as nausea. Or so he kept telling himself. A steady spring rain fell on his back, darkness was falling, and he was in the midst of the northern wilderness, but the body was there for the taking and it was, after all, what had started him on this mad-cap chase. That, and the firm and magnetic draw of the power. Power like a beacon.\n\nHe took out a hunting trousseau\u2014a pair of very heavy knives and half a dozen smaller, very sharp ones\u2014and quickly and accurately flensed the skin from the dead man's skull, folded the flaps back, took a trepan from his pack, and lifted a piece of skull the size of a triple leopard of solid silver.\n\nThe light was failing, but it was still clear the brain-matter was rotting.\n\nHarmodius took an eating knife from his purse, pulled out the sharp pick, and dug around carefully. He used the tip of the knife to cut away small portions of the rotting material\u2014\n\nHe spat out a mouthful of salty saliva. \"I will not vomit,\" he declared aloud. Dug again.\n\nIt was too damned dark. He pulled a candle from his increasingly upset horse's pack, and lit it with sorcery. There was no breeze at all, and the candle hissed in the light rain. He lit two more, squandering the beeswax.\n\nHe trepanned again, but it was no use. The brain-matter was too rotten. Or his theory was entirely wrong. Or rather, Aristotle's theory was entirely wrong.\n\nThe Magus left the body where it was, lying half exhumed in the rain. He washed his hands in the creek at the base of the hill, repacked his knives, extinguished the candles, and reloaded his horse, who now shied at every noise. He reached out, and felt the power gathering the north.\n\n_Jesu Christe_\n\nThe Magus paused, one booted foot already in a stirrup. There was something\u2014\n\nThe creature gave itself away with a growl, and his mare bolted. Harmodius managed to get a hand on the saddle-bow and clung on for a furlong until the frightened animal turned. Harmodius rode the turn, using the force of her movement to help get his leg over the saddle at last. The moon was new and distant, the rain was covering the stars, and the night was dark. He prayed, rapidly and incoherently, that his mare stayed on the road.\n\nHe got his right foot in the stirrup and the reins in his left hand, and he pulled on them. Ginjer did _not_ obey.\n\nHe reined in sharply, and felt for the baton he used as a riding whip. It took him what seemed hours to locate it in his belt, and hours more, apparently, to press it to her neck, a trick he'd learned from a knight.\n\nHe cast a simple thought, a phantasm that allowed him to see in the dark.\n\nWhat he saw froze his blood. The horse balked, and he almost went over the cantle.\n\n\"Sweet Jesu!\" he said aloud.\n\nSomething was standing in the road, waiting for him.\n\nOff to his left, the north-west, the sky erupted\u2014a long orange flash. Its dim light further illuminated the familiar\u2014too familiar\u2014shape of the creature standing on the road.\n\nIt tossed the corpse aside and loped at him. But he had time to feel the convulsion in the northern power first. To ponder, for an instant, the fact that the long flash of orange beyond the horizon had reached him long before the convulsion of power, a matter of great interest to a hermeticist. He had never fully investigated the effects of distance on power\u2014\n\nIt was a form of panic, to spin off thought after thought, none of them connected to the monster on the road ahead of him, or the wave of terror that it emitted like a fist of fear.\n\n\" _Adeveniat regnum tuum_ ,\" the Magus spat.\n\nA lance of fire sprang from his riding stick to the winged creature, whose head was bathed in flame for as long as man might draw a deep breath. The liquid parts of the creature's head vaporized and its skull exploded, lit by the intense flame of the lance of fire.\n\nThe fire went out, apart for some pale blue flames that licked at the creature's neck for a few heartbeats before sizzling out.\n\nSilence fell, in which the creature's tail lashed the ground\u2014 _thump, thump, thump_ \u2014and then was still.\n\nThe silence went on, and on. The night smelled of singed hair and burnt soap.\n\nThe Magus drew a deep breath. Raised his riding crop and blew gently on the silver rune set in the gold cap. He smiled to himself, despite the fatigue that settled on his shoulders like a haubergeon of mail, and allowed himself a single \"heh.\"\n\nHe watched the northern horizon as the fire flickered there again, then dismounted and walked through the darkness to the creature's side and muttered \"Fiat lux.\" His light was blue and pale, but it sufficed.\n\nHe made a clucking sound, reached out into the night with his senses, recoiled from what he found there and ran for his horse.\n\n# East of Lissen Carak\u2014Peter\n\nPeter lay in a state of angry exhaustion and watched the pale fire flicker in the distant west. He had to tear his eyes away from it and watch the darkness to be sure that the whole thing wasn't just his imagination. But it was true\u2014above the endless trees, somewhere to the west, there was a great fire. So great, it reflected from the cliff face above him in long flashes of light.\n\nHis two \"masters\" slept through it.\n\nHe struggled with his yoke again, surrendered again, and fell asleep.\n\nAwoke to the smaller man kneeling beside him.\n\n\"Cook,\" he said. \"Wake up. Something is out here with us,\" he added. There was fear in his voice.\n\n\"What the fuck are you doing?\" asked the other Morean.\n\n\"I'm letting him out of this yoke,\" said the smaller man. \"I'm not going to run and leave him to die. Jesu\u2014I'm a better man than that.\"\n\n\"He's a pagan, or a heretic, or some such filth. Leave him.\" The first man was loading the mule as fast as he could. It was dark, but not true dark\u2014the first pale light of morning. And something heavy was moving in the bush.\n\n\"I am a Christian man,\" Peter said.\n\n\"See?\" said the smaller man. He fumbled with the chains. Grunted.\n\n\"Come on!\" shouted his friend.\n\nThe shorter man pulled again, slammed the yoke against a rock, and scrambled to his feet. \"Sorry,\" he said. \"We don't have the key.\" And he followed his mate into the woods, leaving Peter lying on the ground.\n\nHe lay there and waited to die.\n\nBut no one came for him, and you can only be so terrified for so long.\n\nHe got to his feet and stumbled over a stump he'd made himself the night before. The axe handle bruised his shin. The idiots had left their axe.\n\nHe plucked it from the stump. He went through the camp, over the broken ground in the near dark\u2014camp was too strong a word for a place where three men had built a fire the size of a rabbit and lain on the bare ground. But by the fire he found an earthenware cup, still intact, and a tinderbox with both char cloth, flint and a steel.\n\nPeter knelt on the ground and prayed to God. He managed a bittersweet thanks, and then he put the cup and the tinderbox into the front of his shirt, tied them in place, and made his way to the road, just a few horse lengths to the north. It was the main road from the eastern seaports to the Albin Plains. He knew that much.\n\nTo the east lay civilization and safety\u2014and slavery.\n\nTo the west lay the Albin River and the Wild. Peter had seen the Wild, red in tooth and claw. And it had not enslaved him. So he shouldered the axe and headed west.\n\n# Harndon Palace\u2013Desiderata\n\nShe read the note with ill-concealed irritation. \"He gave this to you when?\" she asked the terrified boy.\n\n\"Yesterday, r'Grace,\" he mumbled. \"Which\u2014er\u2014cook sent me to Cheapside and me mum was sick\u2014\"\n\nShe looked at him. She was annoyed\u2014she loved the useless old Magus the way she loved her magnificent Eastern riding horse, and his recent display of _real_ power made him even more exciting.\n\n\"An he took a horse\u2014a fine horse\u2014r'Grace. Had leather bags\u2014had hisn staff.\" The boy's desire to please was palpable, and she relented.\n\nShe turned to Lady Almspend and motioned at her waist. \"Give the boy a leopard for his pains and send Mastiff to the Magus's rooms in the tower. I would like a full report.\" She made a face. \"Sir Richard?\"\n\nSir Richard Fitzroy was the old king's bastard son, a handsome man, a fine knight, and a reliable messenger. He doted on the Queen, and the Queen appreciated his stability.\n\nHe was attending her, obviously courting Lady Almspend now that his low-born rival was gone.\n\nShe beckonned to him. \"Sir Richard\u2014I need a private word with the king,\" she said.\n\n\"Consider it done,\" he allowed, and bowed himself out.\n\n# East of Albinkirk\u2014Gerald Random\n\nGerald Random woke to hear Guilbert Blackhead rapping for entrance to his tent\u2014knocking on the tent's cross-pole with his sword hilt. Random was on his feet in an instant, dagger in hand, and he was awake in another.\n\n\"What is it?\" he asked, fumbling for the hooks and eyes that would open the flap.\n\n\"No idea. But you had better see it.\" Guilbert's urgency was carried fluently.\n\nRandom was out of the tent in another few heartbeats.\n\nThey were camped in a narrow meadow on the banks of the Albin, and the great river was in full flood, running fast and deep and almost silent, the black water sullen in the damp night air. They'd been hit by rain squalls again and again all day, and men and animals were still as wet and as sullen as the water.\n\nFar off, north-east, the first crags of the mountains should have been visible, but low clouds drifted right over them, obscuring them for minutes at a time and then clearing just as rapidly, keeping the grass and the trees full of water.\n\nAs the next low cloud passed by, the Adnacrag Mountains loomed even in the darkness. Random thought that they might make the fortress town of Albinkirk in four more days. It was not the distance but the condition of the road at this time of year which delayed them. The river road, with its stone bridges and deep stone foundations built by the Archaics, was the only one a sane man would travel with heavy wagons. Every other road was fetlock deep in mud. But all the same, it was not easy.\n\nThere was an orange glow to the north.\n\n\"Just watch,\" Guilbert said.\n\nAfter six days on the road Random had the warrant-man's measure\u2014careful, cautious, and thorough. Perhaps not the man for a deed of daring, but just the sort of man to work a convoy. The guard posts were always manned and constantly checked.\n\nWhatever he was trying to show the merchant, it was important.\n\nRandom watched a flicker\u2014was it more than that? North-west, towards the fair. Perhaps\u2014but they were too far for the fair to be visible. It was fifty leagues away or more\u2014they were not yet to Albinkirk.\n\n\"There!\" said the mercenary.\n\nThere was, just for a moment, a pinpoint of light that burned like a star above the glow of Albinkirk.\n\nRandom shrugged. \"That's all?\" he asked.\n\nGuilbert nodded, clearly unhappy about it.\n\n\"I'm for bed, then,\" Random said. \"Wake me if we're attacked,\" he added. He wished, later, that he hadn't been quite so snappish.\n\n# Lissen Carak\u2014Mag the Seamstress\n\nMag the seamstress sat on a barrel, staying out of the way. The day had passed well enough\u2014she'd helped Lis wash shirts and been paid in solid coin for her work; had remembered her skills at avoiding pinching fingers, or delivering a slap where it was needed. The mercenaries were like nothing she'd ever seen\u2014aggressive beyond anything a town of peasants had to offer.\n\nShe knew that, had the circumstances differed, they'd have killed her sheep, taken her chickens, her silver, and probably raped and killed her as well. These were hard men\u2014bad men.\n\nBut they shared their wine and danced in the evening, and she had a hard time seeing them for what they probably were. Thieves and murderers. Because the Abbess said the Wild was going to attack them, and these men were all they had as defenders, and Mag thought...\n\nWhatever she thought, she must have drifted off after the flashes in the sky. And suddenly they came out of the darkness in blackened armour, led by Thomas, who she now knew was Ser Thomas, riding hard on a destrier covered in sweat; six men-at-arms, twenty archers and some armed valets, all galloping up the twisting road and through the gate almost at her feet.\n\nBad Tom was the first off his horse, and he bent his knee to the captain. \"Just as you said,\" he panted. \"We _fucked_ 'em.\" He rose stiffly.\n\nThe captain embraced the bigger man. \"Go get your harness off and get a drink,\" he said. \"With my thanks, Tom. Well done.\"\n\n\"And who's gonna take the lamp-black off my mail?\" complained one of the archers\u2014the one with dead eyes. He looked up, and his terrifying eyes found her unerringly with their promise of violence.\n\nHe grinned at her. The other men called him Will, and she'd learned it stood for Wilful Murder, of which he had apparently been convicted.\n\nShe flinched.\n\n\"How was it?\" asked the captain.\n\nThomas laughed his huge laugh. \"Gorgeous, Cap'n!\" he said, and swung down.\n\nThe other men laughed, a little wildly, as Mag knew men she knew that Thomas was really laughing, and the others had endured something sharp and horrible.\n\nThey'd survived it, and triumphed.\n\nThe captain embraced the big man again, and shook his hand. He went among the archers, helping them dismount and giving each his hand, and Mag saw the Abbess was right next to him and that she was _blessing_ them.\n\nShe clapped her hands and just managed not to laugh.\n\n# Harndon Palace\u2014Desiderata\n\nAs evening fell, Desiderata watched the foreign knight with the pleasure of a connoisseur for a true artist. He was tall\u2014a head taller than every other man in the great hall\u2014and he moved with a grace that God only bestowed on women and exceptional athletes. His face was like that of a saint\u2014bright gold hair and sculpted features that were not _quite_ too fine for a man. His red jupon fitted to perfection, his white hose were silk, not wool, and the wide belt of gold plaques on his slim hips was a mute testament to riches, privilege, and bodily power.\n\nHe bowed deeply before the king, sinking all the way to one knee with graceful courtesy.\n\n\"My lord King, may I present the noble Jean de Vrailley, Captal de Ruth, and his cousin Gaston D'Albret, Sieur D'Eu.\" The herald proceeded to name their coats of arms and their heraldic achievements.\n\nDesiderata already knew the foreign knight's achievements.\n\nShe watched his eyes, and he watched the king.\n\nThe king scratched his beard. \"It is a long way from the _Grand Pays_ ,\" he said. \"Is all of Galle at peace, that you can bring so many knights to my lands?\" He said the words easily, and yet his eyes were hard and his face blank.\n\nDe Vrailly remained on one knee. \"An angel commanded me to come and serve you,\" he said.\n\nHis sponsor, the Earl of Towbray, turned sharply.\n\nDesiderata extended her sense\u2014her warmth, as she thought of it\u2014towards him, and the foreign knight burned like the sun.\n\nShe inhaled, as if to inhale his warmth, and the king glanced at her.\n\n\"An angel of God?\" the king asked. He leaned forward.\n\n\"Is there another kind?\" de Vrailly asked.\n\nDesiderata had never heard a man speak with such simple arrogance. It _hurt_ her, like a physical blemish on a beautiful flower. And yet, like many blemishes, it had its own fascination.\n\nThe king nodded. \"How do you intend to serve me, Ser Knight?\" he asked.\n\n\"By fighting,\" de Vrailly said. \"By making unrelenting war upon your enemies. The Wild. Or any men who oppose you.\"\n\nThe king scratched his beard.\n\n\"An angel of God told you to come and kill my enemies?\" he asked. Desiderata thought the knight spoke with irony but she couldn't be sure. De Vrailly blinded her in some strange way. He filled the room.\n\nShe closed her eyes and she could still sense him.\n\n\"Yes,\" he said.\n\nThe king shook his head. \"Then who am I to deny you,\" he said. \"And yet I sense that you, in turn, desire something of me?\"\n\nDe Vrailly laughed, and the sweet musical sound of it filled the room. \"Of course! I would be your heir in exchange, and this kingdom shall, after you, be my own.\"\n\nThe earl staggered as if he had been struck.\n\nThe king shook his head. \"Then, angel or no, I think it would be best if you went back to Galle,\" he said. \"My wife will bear me an heir of my body, or I will appoint my own choice.\"\n\n\"Of course!\" de Vrailly said. \"But of course, my king!\" He nodded, his eyes shining. \"But I will prove myself and become your choice. I will serve you, and you will see that there is no one like me.\"\n\n\"And you know this because an angel told you.\"\n\n\"Yes,\" said Jean de Vrailly. \"And I offer to prove it on the body of any man you send against me, on horse or foot, with any weapon you care to name.\"\n\nHis challenge, delivered in his sweet angelic voice from the bended knee of the suppliant, had all the authority of a decree. Men flinched from it.\n\nThe king nodded, as if satisfied.\n\n\"Then I look forward to placing my lance against yours,\" he said. \"But not as a challenge to your angel. Merely for the pleasure of the thing.\"\n\nDesiderata saw the perfect knight exchange a glance with his cousin. And she had no idea what thought they shared, but they were pleased. Pleased with themselves, and perhaps pleased with the king. It warmed her, so she smiled.\n\nGaston, the Sieur D'Eu, smiled back at her, but the golden de Vrailly, never took his eyes from the king. \"I should love to match lances with you, sire,\" he said.\n\n\"Well, not tonight. It's too dark. Perhaps tomorrow.\" The king looked at the Earl of Towbray and nodded. \"I thank you for bringing me this splendid man. I hope I have the revenue to keep him and his army!\"\n\nThe earl chewed his moustache for a moment, and then shrugged. \"My pleasure, your Majesty,\" he replied.\n\n# Lissen Carak\u2014The Red Knight\n\n\"God be with you,\" the Abbess said quietly, laying her hands on Wilful Murder's head, and he flinched.\n\nShe caught the captain's eye as the narrow gateway began to clear.\n\n\"Any pursuit?\" he asked Ser George Brewes, the rear file leader\u2014a man ready to be a corporal. One of Jehannes's cronies, not one of Tom's. Still waiting in the gate, aware it was open, eyes on the darkness outside.\n\nBrewes shrugged. \"How would I know?\" he asked. But he relented. \"I wouldn't think it.\" He shook his head. \"We lit ten farms' worth of the woods, and sent the fire downwind right at their camp.\"\n\n\"How many Jacks?\" the captain asked.\n\n\"At least a hundred. Maybe thrice that\u2014there's no proper counting in the dark, ser.\" Brewes shrugged. \"M'lord,\" he added, as an afterthought.\n\nA pair of valets and an archer came up and began to winch the main gate shut.\n\n\"Ware!\" shouted a voice from the highest tower, the one over the nuns' dormitory, and the captain heard the unmistakable sound of a crossbow snapping off a shot.\n\nSomething passed over the moon.\n\nThankfully every man was on the walls and alert, or it might have been worse when the wyvern came down into the courtyard on wings a dozen ells wide, and its claws wreaked ruin among the unarmoured dancers and singers and merry-makers, but before the screams started it sprouted a dozen bolts, and it raised its head and screeched a long cry of anger and pain, and leapt back into the air.\n\nThe captain saw Michael, unarmoured, hurdle a pair of corpses and draw his heavy dagger, flinging himself at the wyvern's back as it lifted into the air. Its tail flicked\u2014and slammed full force into the squire's hip. Michael screamed in pain and was thrown a horse's length to the stone.\n\nThe Red Knight didn't waste the time provided by his squire. He was down from the gatehouse, sword in gloved hand, before Michael's scream had echoed off the stable walls and the chapel.\n\nThe wyvern whirled to finish the squire, and Bad Tom stepped between the monster and its prey. The big man had a long, heavy spear in hand, and he attacked, thrusting for the thing's head. It was fast\u2014but its sinuous neck served the creature as a man's torso serves a man, and when it flicked its head to avoid the spear, it could neither strike nor rise into the air until it had its balance back.\n\nBad Tom stepped in closer, shortened his grip on the spear and struck hard, thrusting the spear brutally into the thing's chest where the neck met its underbelly.\n\nLong shafts began to feather the thing's wings and abdomen.\n\nIt screamed and leaped into the air, wings beating hard, slamming its tail at Tom, but the big man jumped high and cleared the lashing tail by a fraction. But he missed the flicker of a wing in the dark, and the wingtip creased his backplate and slammed him to the ground.\n\nThe archers on the walls loosed shaft after shaft. Wilful Murder stood a horse's length away, drawing shafts from the quiver at his hip and loosing carefully\u2014aiming for any vulnerable part.\n\nThe bonfire in the courtyard illuminated their target, and the wickedly forged arrowheads cut into the beast's hide like chisels through wood as the sparks from the courtyard fires rose like fireflies in the weakening wingbeats.\n\nThe captain was behind and above it when it leaped for the air, and he leaped too. He hit its neck and his sword whipped around its throat. His left hand grabbed the sword at the other side and he let himself drop, his sword become a vicious fulcrum, dragging the wyvern's head down. It lost height and crashed on the steps of the chapel, his sword deep in the soft underside the neck, its jaws unable to reach him, the wyvern injuring itself as its head slammed into the steps again and again in fury and panic.\n\nA lone crossbowman ran along the parapet, leaped down to the courtyard, stumbled, righted himself, and loosed his heavy weapon into the wyvern's head from a distance of a few feet. The power of the bolt snapped its head back, and the captain rolled to his left, loosed his left hand, and got to his feet, his heavy blade already lashing out for the neck\u2014again, and again, and then, when the head came up, he caught the blade in his left hand again, and slashed down into the creature's head, his blade sliding down its armoured scales to slice softer flesh. He made ten strokes in as many heart beats, and the head suddenly snapped back, the whole beast rolled like a man and the brave crossbowman died when the mighty claws took him round the waist and tore him in half.\n\n\"There's another!\" shouted Tom, off to his left.\n\nThe tip of the thrashing tail caught his right ankle and ripped his feet from under him, and the captain cursed that he was not in armour.\n\nHe hit his head on a chapel step and lost an instant.\n\nThe wyvern reared over him.\n\nA woman\u2014the seamstress\u2014appeared out of the darkness on his right, and threw a barrel at the monster\u2014clipped the thing's head, and it lost its balance, and one of his engineers loosed a scorpion into it.\n\nThe power of the scorpion shaft was so great that it took the creature's neck and punched it _through_ the chapel doors so hard that where the creature's head smashed into the stone the lintel cracked. He heard its neck break. The shaft did a hundred leopards' damage inside the chapel, the wyvern's death struggles did a hundred more, and a river of gore spoiled the sacred carpet on the marble floor.\n\nThe captain got to his feet and found that he'd kept his sword. His chamois gloves were ruined and his left hand was bleeding where he'd grabbed the blade too high, above the area left dull for such purposes. He'd twisted his ankle, and he had to blink rapidly bring the world, spinning around him, back into focus.\n\nThe thing twitched, and he buried his point in the eye he could reach.\n\nThe courtyard fire glimmered on the belly of the second wyvern.\n\nForty archers threw shaft after shaft, so that the fortress seemed to have a new column of sparks rising into the fire-lit monster, and something happened\u2014not suddenly, like the strike of the siege shaft, but gradually the wyvern's wings tore, holed, it lost lift and screamed in fear as the men below brought it down and it realised there was no escape from the deadly upwards rain of steel. It slipped lower and lower, wings beating more frantically, turned sharply and suddenly one mighty wing failed. It plummeted to the hillside and crashing down with such weight and speed that the captain felt the steps shake under his boots.\n\n\"Sortie!\" the captain shouted. He meant to shout, but it came out as more of a croak... although it was understood, and his eight armoured knights had the gate open and were away down the road, led by Sauce.\n\nAs the courtyard stilled it showed twenty dead people\u2014dead or terribly maimed. A girl of fifteen or so screamed and screamed, and the woman who had thrown the barrel bent and gathered her into her arms.\n\nA child tried to drag himself by his arms, because he had no legs.\n\nNuns were suddenly pouring from their dormitory\u2014ten, twenty, fifty women, surrounding the injured and the dead in a storm of grey wool and clean linen, spreading out to access the scale of the dead, injured and traumatised. The captain slumped against a wall, his right leg a torrent of pain, and wished he could just slide into unconsciousness.\n\nShe screamed again and again. His eyes flickered to her but only after a long look did he see that most of the left side of her upper torso was _gone._ He couldn't believe she was alive, or screaming. The woman who had saved his life was covered in her blood\u2014shiny with it, trying to help her\u2014and there was _nothing_ to be done.\n\nHe wished the screaming woman would just die.\n\nA pair of nuns wrapped her tight in a sheet, round and round, and the sheet turned red as fast as they could wrap another layer, and still she screamed, becoming one voice amongst a chorus of anguish that filled the night.\n\nHe staggered up and stumbled to Michael, who lay crumpled against the chapel.\n\nThe boy was alive.\n\nHe looked around for Amicia. She had been standing right there\u2014there, where the woman screamed. But she was gone. He shouted for a sister\u2014for anyone\u2014and four responded. They ran their hands carefully over him before lifting him away from Michael.\n\nMen were shouting now. Even over the screams, their shouts were triumphant, but he ignored them and dragged himself over to Tom.\n\nTom was sitting against the stable. \"Backplate took it,\" he said with a grin. \"Christ, I thought I was done.\" He pointed at the sword. \"Nice trick, that.\"\n\n\"Half-sword versus wyvern,\" the captain said. \"A standard move. All the best masters teach it.\" He stripped away the ruin of his left glove and wrapped it tight around his cut. \"I just need more practice.\"\n\nTom chuckled. \"Sauce just killed t'other, I'll wager,\" he said, pointing at the cheering archers.\n\nSure enough, the next moments brought the mounted sortie back through the main gate, dragging the head of the second wyvern. Brought to earth by fifty arrows, it had died on their lance tips without injuring a single human.\n\nTom nodded. \"That was well done, Captain.\"\n\nThe captain shrugged. \"We were ready, we laid our trap, you burned their camp and surprised them, and they _still_ killed our people.\" He shook his head. \"I wasn't ready enough. I was caught lollygagging.\"\n\nTom shrugged back. \"They killed a lot of people.\" He raised an eyebrow. \"But not many of _our_ people.\"\n\n\"You're a hard bastard, Tom Mac Lachlan.\"\n\nBad Tom shrugged, obviously taking it as a compliment, then something caught his eye in the chapel. He wrinkled his nose as if he'd smelled something bad.\n\n\"What?\" asked the captain.\n\n\"Ever notice how they're always smaller when they're dead?\" Tom asked. \"It's just the fear that makes 'em seem so big.\"\n\nThe captain nodded. He was looking at the wyvern too, and he had to admit that it _was_ smaller than it had seemed in the fight. And it looked different. Paler. A mass of wounds and cuts and barbs.\n\nAlmost pitiful.\n\nTom smiled and started to get to his feet, and the Abbess was there.\n\nHe expected anger or recriminations from her, but she merely extended a hand and took his.\n\n\"Let us heal your people,\" she said.\n\nThe captain nodded, still pressing his glove tight around his hand. There was a lot of blood. She got an odd look on her face, just before he fainted in her arms.\n\n# Albinkirk\u2014Ser Alcaeus\n\nDeep in the marches of the next night, the enemy attacked the castle of Albinkirk.\n\nSer Alcaeus had passed beyond fatigue. He was in a world lived one heartbeat at a time, and events passed him in a series of illuminated flashes, as if lightning was playing on all of them.\n\nThere were some assaults on the walls of the castle, but unlike the low stone curtain walls of the town the castle walls were too high and too well maintained for the flood of Wild creatures to climb. The handful of beasts who made it to the top were killed.\n\nBut every attack cost him a little more.\n\nOne flash was a fight with an irk\u2014a tall, thin, beautiful creature with a hooked nose like a raptor's beak and chain armour as fine as fish scales that turned his sword again and again. And when, by dint of desperate strength, he knocked it to the stones, and its helmet spun away, the irk's eyes begged for mercy. Like a man's.\n\nAlcaeus would remember that. Even as his dagger terminated it he registered that it, too, had humanity.\n\n... and what followed was worse.\n\nBecause something came.\n\nIt was huge and foreboding, out in the horrifying fire-lit ruins of the town. It strode forward with a hideous shambling gait, and it was as tall as the city wall or taller.\n\nIt was _alive._\n\nAnd now it raised its staff\u2014the size of a mounted knight's heavy lance, or bigger\u2014and a line of white-green fire struck the castle wall. The stone deflected in it a wash of white-green fire for as long as the terrified men on the wall might have counted to ten.\n\nAnd then there was a rending _crack_ and the wall breached, about ten paces to the left of the gate. The whole wall moved. Men fell\u2014chunks of flint fell to crush the creatures below.\n\nThen the monster raised its arms and seemed to call the stars down from the heavens, and as they began to plummet, Alcaeus fought not to fall on his face and hide.\n\nThe stars screamed down from the clear sky, falling to earth with an eerie, unearthly wail, and struck. One struck out in the fields, killing a wave of boglins. One struck in the centre of the town, and the cloud of fire reached into the heavens. The whole castle moved, and a cloud of dust reached like a fist into the heavens.\n\nThe third struck the castle wall mere feet from the great crack, and an enormous piece of masonry and stone fell outward with a crash.\n\nAlcaeus ran for the breach, and found himself with another armoured man\u2014Cartwright, he thought, or the Galle, Benois. The breach was narrow\u2014two men wide.\n\nThey filled it with their bodies.\n\nAnd the enemy came for them.\n\nAt some point, Benois fell. He was stunned, and Alcaeus tried to cover him, but the enemy reached a hundred hands and talons for his feet, sank claws into his flesh and dragged him to the edge of the wall, inch by inch. He screamed, unmanned with horror, and tried to rise. Boglin weapons cut him in the soft places not covered by armour, peeled his plate away.\n\nThey were eating him alive.\n\nAlcaeus struck and struck again, powered by desperate fear, and he straddled the screaming man's body and cut and cut.\n\nIt wasn't enough. And then Benois grabbed at _his_ ankles.\n\nHe ripped himself clear, and leaped back into the uncertain footing of the breach, and Benois was gone, a pile of hellspawn feeding on him, his armour torn open\u2014\n\nAlcaeus made himself breathe.\n\nSuddenly Ser John was there with his mace. The five foot weapon moved like a goodwife's broom on a new spring morn, and he shattered first the boglins around them, and then Benois's skull.\n\nThere was a flash of light to the east\u2014a distant _whump_ of displaced air. A column of flame leaped up perhaps a league away. Perhaps two.\n\nThen another\u2014even greater.\n\nThe creatures of the Wild faltered, looked over their shoulders, and the fury of their assault rapidly abated.\n\n# Albinkirk\u2014Thorn\n\nIn an instant, Thorn knew that something had gone wrong.\n\nHe'd drained himself by calling even the smallest stones from the heavens. It was a showy, inaccurate and inefficient working, but it had spectacular results when it worked. And he loved to cast it, the way a strong man loves to show his strength.\n\nThe daemons were impressed, and that alone was worth the fatigue. Better, the town was utterly destroyed and it had been far, far easier than even he had hoped.\n\n_I have grown so strong,_ he thought. What he had planned as a mere diversion had become a triumph. She would hear of it and cower in fear.\n\n_Perhaps taking the Rock is worth doing after all. Perhaps I will refashion myself as a warlord._\n\nBut the twin pillars of fire behind him came from his camp\u2014the camp where his greatest allies, the irks and the boglins, stored their food and their belongings and their slaves and their loot. And it was afire.\n\nHe had left his most trusted troops had been left to guard it.\n\nHe turned with his army and strode for it.\n\nWithout his willing it, the bulk of his Wild creatures turned and followed him. They had no discipline, and they went like a shoal of fish\u2014\n\n# Albinkirk\u2014Ser Alcaeus\n\nAlcaeus watched them go, slumped against the wall. The Gallish man-at-arms looked like a butchered animal, his bones stripped. The boglins had feasted on him.\n\nThe sun was rising, and the lower town was an abattoir of horrors. In the main square irks had taken the time to carefully flay a man and hang him on a cross. He was still alive.\n\nJames the crossbowman stepped into the breach. He took a long look, raised his weapon and shot the crucified man. It was a good shot, given the range. The man's screaming, skinless head dropped, and he was silent.\n\nSer John was slumped against the other wall. James helped the old man get his visor up. He winked.\n\nHe _winked._\n\nIn that moment the old knight became a hero, in Ser Alcaeus's estimation.\n\nAlcaeus had to smile back, despite so many things. The loss of Benois hurt. The feel of the man's hands on his ankles\u2014\n\n\"I need you to ride to the king,\" Ser John said. \"Right now, while whatever miracle this respite may be lasts.\"\n\nAlcaeus must have agreed with him, because an hour later he was on his best horse, unarmoured, and galloping south. It was a desperate gamble.\n\nHe was too tired to care.\n\n# Chapter Seven\n\n# South of Albinkirk\u2014Master Random\n\n\"Gates of Albinkirk are broken, ser,\" Guilbert reported. He shrugged. \"There's fires burning in the town and it looks like a fucking fist, beg your pardon, punched the cathedral. King's banner still flies over the castle but none answered my hail.\"\n\nJohn Judson, worshipful draper, and St. Paul Silver, a goldsmith, drew their horses closer to Random where he sat with Old Bob, Guilbert's friend and the last man he'd hired, a bald, ruddy skinned drunkard whose voice and carriage suggested that the spurs on his heels were actually his.\n\nOld Bob was the oldest man in the company, and had a much-broken nose, bumpy with knots of erupting flesh. His straggly salt and pepper hair erupted from a narrow zone around his ears and was always dirty, but the man's eyes were deep and intelligent and a little disturbing, even to a man as experienced as Gerald Random.\n\nHe wore good armour, and he wore it all the time.\n\n\"That's what the peasants said yesterday,\" Old Bob noted calmly.\n\nRandom looked at the other merchant venturers. \"Albinkirk in ruins?\" he asked. \"I've fought up here, friends. The border is a hundred leagues farther north, and even then\u2014the Wild is west and north of us, not here.\"\n\n\"Something did this, all the same,\" Judson said. The corners of his lips were white, the lips themselves drawn tight. \"I say we go back.\"\n\nPaul Silver wore high boots like a gentleman. Goldsmiths were often better dressed than their customers. It was the way of the world. But Silver had also served the king and wore a heavy sword, an expensive weapon meticulously kept ready for battle. \"We'd be fools to ignore that something is going on,\" he agreed. \"But bad as this is I'm not sure it means a convoy of nearly fifty wagons should turn back.\"\n\nAlbinkirk rose on a high hill at the next great bend in the river. Ships would make it this far north, later in the summer when the floods were done and all the ice was out of the mountains\u2014when the run off wasn't carrying whole pine trees, big enough to stove in a round ship, down mountainsides and out into the great river. Albinkirk was the northernmost town that could be reached by ship, and yet the southernmost at the edge of the Great Forest that covered the mountains. Once it had hosted the Great Fair, but poor management and rapacious tolls had forced it to move further east, to the convent at Lissen Carak.\n\nToday Albinkirk was a corpse, red-tile roofs looking grey and old, or fire-blacked, in the distance, and the spire of the cathedral gone.\n\n\"What's happened to the cathedral?\" Random asked.\n\nOld Bob made a face. \"They was attacked by dragons.\" He shrugged. \"Or Satan himself.\"\n\nRandom took a deep breath. This was the sort of moment for which he lived. The great decision. The gamble.\n\n\"We could leave the road. Turn east on this side of the river and use the bridge at Lissen Carak,\" he found himself saying. \"Keep the river between us and Albinkirk.\"\n\n\"A river won't stop wyverns,\" Old Bob said.\n\n\"Not much choice anyway,\" Guilbert said. \"The gates are shut, so we can't exactly take the High Road.\"\n\n\"They should _want_ us in that town,\" Random said.\n\nJudson watched him, and his face held something Random didn't recognise\u2014horror? Fear? Curiosity?\n\nBut finally the man worked up his courage and spoke his mind. \"I'll be taking my wagons back south,\" he said carefully.\n\nRandom nodded. Judson had the second largest contingent\u2014eight wagons, a sixth of the total.\n\n\"I reckon I'll take my share of the sell-swords, too,\" Judson said.\n\nRandom thought for a fraction of a heart beat and shook his head. \"How do you reckon that, messire?\"\n\nJudson shrugged, but his eyes were angry. \"I paid for eight wagons to join your convoy,\" he said. \"I reckon that's a quarter the cost of the sell-swords, so I'll take four of them. Six would be better.\"\n\nRandom nodded. \"I see,\" he said. \"No, and you know it doesn't work that way. You joined my convoy for a fee. If you leave it\u2014that's on you. You didn't purchase a share, you purchased a place.\"\n\n\"You think the King's Court will see it that way?\" Judson asked. Fear had made him bold. \"I'll be back in a few days, telling my story.\" He shrugged and looked away. \"Give me my half-dozen swords, and I'll say nothing.\" Judson looked at Paul, and then he leaned forward. \"You want to be Lord Mayor, Random? Start playing the game.\"\n\nRandom looked at him, and then shook his head. \"No. I won't quarrel with you, and I won't give you a sword, much less six. Go your own way. It should be safe enough.\"\n\n\"You'd send me back without a single man?\" Judson demanded.\n\n\"I'm not sending you back. You're going. Your decision.\" Random looked at Guibert and Old Bob. \"Unless one of you has cold feet too?\"\n\nOld Bob scratched something unspeakable on his nose. \"Going on ain't going to be good,\" he said. \"But I don't need to go back.\"\n\nGuilbert looked at the older man. \"What's not going to be good?\" he said. \"What in hell's name are you on about?\"\n\n\"Wyverns,\" said the old man. \"Daemons, irks and boglins.\" He grinned, and he looked truly horrific. \"The Wild lies ahead.\"\n\n# Harndon Palace\u2014Desiderata\n\nThe lists were pristine\u2014the gravel carefully tended and unmarked, the barriers crisp and white with new lime, like a farmer's dooryard fence except for the fancy red posts at either end, each topped with a brightly polished brass globe the size of a man's fist.\n\nThe stands were virtually empty. The Queen sat in her seat, her ladies around her, and her young knights in the lower tiers of the seats, tossing early flowers to their favourite ladies.\n\nThere were professional spectators\u2014a dozen men-at-arms, most from the castle garrison of archers. Word had spread quickly, and the rumour was that the king had been challenged by this foreign knight and meant to show him a thing or two.\n\nDesiderata watched her husband sitting quietly by the little wooden shack where he had armed. He was drinking water. His hair was long and well kept, but even at this distance, she could see the grey in the dark brown.\n\nAt the other end of the lists, his opponent's hair was an unmarked gold, the gold of sunset, of polished brass, of ripe wheat.\n\nSer Jean finished whatever preparations he was making and had a quiet word with his cousin, while his squire held the biggest war horse the Queen had ever seen\u2014a beautiful creature, tall and elegant of carriage, its gleaming black hide unrelieved by marks of any kind, with a red saddle and blue furniture all pointed in red and gold. Ser Jean's arms, a golden swan on a field of red and blue, decorated the peak of his helm, the tight, padded surcote over his coat armour, the heavy drapery over the rump of his horse, and the odd little shield, curved like the prow of a war-galley, that sat on his left shoulder.\n\nIt was a warm day. Perhaps the first truly warm day of spring, and the Queen bathed in the sun like a lioness and gave forth a glow of her own that bathed her ladies and even the knights on the seats below her.\n\nToday the foreign knight glanced up at her quite frequently, as was her due.\n\nShe looked back at the king. By comparison, he looked small and just a little dingy. His squires were the best in the land, but he _liked_ his old red arming cote and his many-times-repaired plate, fashioned in the mountains far across the ocean when hardened steel was a new thing and carefully repaired by his armourer ever since. He _liked_ his old red saddle with the silver buckles, and if they left traces of black tarnish on the leather, it was still a fine saddle. Where the foreign knight was new and shone from top to toe, her king was older\u2014worn.\n\nHis war horse was smaller too\u2014Father Jerome, the king called him, and he was veteran of fifty great jousts and a dozen real fights. The king had other, younger, bigger horses, but when he went to fight for real he rode Father Jerome.\n\nThe herald and the master of the list called them to action. It was friendly play\u2014the spears were bated. Desiderata saw Gaston, the foreign knight's friend, say something to the king, pointing to his neck, with a bow.\n\nThe king smiled and turned away.\n\n\"He's not wearing his gorget under his aventail,\" Ser Driant said in her ear. \"Young Gaston asks why, and requests that the king wear it.\" He nodded in approval. \"Very proper. His man wishes to fight hard, and he doesn't want to be accused of injuring the king. I'd feel the same way myself, if the king had challenged me.\"\n\n\"The king did _not_ challenge him,\" Desiderata said.\n\nSer Driant gave her a queer look. \"That's not what I heard,\" he said. \"Still, I suspect the king will tip him in the sand and that will be that.\"\n\n\"Men say that man is the best knight in the world,\" the Queen replied, a little coldly.\n\nSer Driant laughed. \"Men say such things about any pretty knight,\" he said. He looked at Ser Jean, mounting with a vault and taking his lance. \"Mind you, the man is the size of a giant.\"\n\nThe Queen felt a mounting unease, such as she had never experienced watching men fight. This was her place\u2014her role. It was her duty to be impartial as the armoured figures crashed together, and to judge the best of them. To forget that one was her lover and king and the other an ambitious foreigner who had all but accused her of being barren.\n\nShe should judge only their worthiness.\n\nBut as the two men manoeuvred their horses on either side of the barrier, she felt a band tighten around her heart. He had forgotten to ask for her favour, and she almost lifted the scarf she held in her hand.\n\nShe couldn't remember the last time she'd watched men fight without bestowing her favour on one, or perhaps both.\n\nSer Jean wore a foreign type of helmet, a round-faced bascinet with a low, round brow and a heavy dog-faced visor with the Cross of Christ in brass and gold.\n\nThe king wore the high-peaked helmet more typical of Albans, with a pointy visor that men called pig-faced but which always reminded the Queen of a bird\u2014a mighty falcon.\n\nEven as she watched, he flicked the visor down and it closed with a click audible across the field.\n\nThere was a stir up in the castle yard\u2014soldiers craning their necks, others moving to the walls, while still others jammed up at the gate beyond which were sounds of shouting and galloping horses.\n\nThe Queen did not often pray. But as she watched the king she put her right hand to the rosary around her neck and prayed to the Queen of Heaven, asking her for grace\u2014\n\nTwo horses flashed past the gate, galloping along the cobbled road to the lists down by the moat yard, their riders shouting and horseshoes striking sparks that leapt even in the sunlight.\n\nShe could feel the gathering of powers in the tiltyard, exactly as she had been able to feel the first gathering of Harmodius's not inconsiderable power, but this was power of another order\u2014like bright white light on a dark day.\n\nThe foreign knight touched his spurs to his horse.\n\nThe king spurred Father Jerome, almost in the same instant. In another time, she would have applauded.\n\nThe two messengers were racing along the moat road, neck and neck, as king and knight charged each other\u2014\n\n\u2014and Ser Jean's horse shied beneath him as a great horse fly sunk its sting far into the black horse's unprotected nose, where the soft lips emerged from beneath the chamfron.\n\nThe war horse balked, missed a stride, and half-reared, half-turned from the barrier. Ser Jean fought for his seat, tried to force his mount's head back to the barrier, but he was hopelessly out of line and now too slow to strike with real force. He raised his lance and then cast it aside as his pained horse reared again.\n\nThe king came on at full tilt, back straight, Father Jerome perfectly collected under him, lance aimed like a swift arrow from some ancient god's bow. A foot short of Jean de Vrailly's prow-shaped shield his lance tip swept up, plucked the swan from his helmet, and then the king thundered by, his lance dropping again to strike the brass globe on the last post of the lists. He struck it squarely, so hard that it ripped from its post and flew through the air to bounce and roll past Ser Gaston, past the two messengers thundering up the rise to the lists, and into the moat.\n\nThe Queen applauded... and yet felt that the king\u2014she tried to keep the thought in check\u2014that he might have voided his lance and passed his opponent without taking his crest. It would have been a generous act, and such things were done, between friends, when a knight was obviously struggling with his horse.\n\nDe Vrailly rode back toward his own end, back straight, horse now firmly under control.\n\nA dozen royal archers ran to get between the king and the two riders, who were bearing down on him with intent, shouting but their words indistinct. They both held scrolls, the colourful ribbons dangling.\n\nThe archers parted to let them through when the king opened his faceplate and beckoned to the messengers. He was grinning like a small boy in his victory.\n\nThe Queen wasn't sure whether this was the outcome of her prayer or not, and so she prayed again as the messengers reached the king, dismounting to kneel at his feet even as his squires began to take his armour.\n\nAt the same end of the lists, only a few feet away, Jean de Vrailly dismounted. His cousin spoke sharply to him, and the tall knight ignored the smaller man, and drew his sword\u2014almost too fast to follow.\n\nHis cousin slapped him\u2014hard\u2014on the elbow of his sword arm, and the foreign knight fumbled his sword\u2014the only clumsy movement she'd ever seen him make. He turned on his cousin, who stood his ground.\n\nThe Queen knew unbridled rage when she saw it, and she held her breath, a little shocked to see the Galle so out of control\u2014but even as she watched, the man mastered himself. She saw him incline his head very slightly to his cousin, as if acknowledging a hit in the lists.\n\nHe turned and spoke to one of his squires.\n\nThe man collected the mighty horse's reins and began to strip its barding with the help of a pair of pages.\n\nShe lost the action for a moment while she tried to take in what she had seen.\n\nSuddenly the king was by her side.\n\n\"He's very angry,\" the king said, while bowing over her hand. He sounded content with his opponent's anger. \"Listen, sweeting. The fortress at Lissen Carak is under attack by the Wild. Or so both these two messengers say.\"\n\nShe sat up. \"Tell me!\" she demanded.\n\nSer Gaston came up, approaching the king with the deference that his cousin never seemed to show even when kneeling.\n\n\"Your Grace\u2014\" he began.\n\nThe king raised a hand. \"Not now. The joust is over for the day, my lord, and I thank your cousin for the sport. I will be riding north with all my knights as soon as I can gather them. One of my castles, and not the least of them, is under attack.\"\n\nSer Gaston bowed. \"My cousin requests that he might ride one course against you.\" He bowed. \"And he wishes your Grace to know that he honours your Grace's horsemanship\u2014he sends you his war horse, in hopes your Grace might school him as well as your own is schooled.\"\n\nThe king smiled like a boy who's been well-praised by a parent. \"Indeed, I love a horse,\" he said. \"I do not claim the good knight's horse and arms, you understand\u2014but if he offers.\" The king licked his lips.\n\nSer Gaston nodded to where the squire was leading the now-unarmed horse. \"He is yours, your Grace. And he asks that he be allowed to take another horse and have another course with your Grace.\"\n\nThe king's face closed like the visor of his helmet had clicked down. \"He has ridden one,\" the king said. \"If he wishes another chance to prove himself, he may gather his knights and ride with me to the north.\" The king seemed on the verge of saying more, and then he steadied himself. But he allowed himself a small, kingly smile and said, \"And tell him that I'll be happy to loan him a horse.\"\n\nBut Gaston bowed. \"We will ride with you, your Grace.\"\n\nBut the king had already dismissed him, and turned to the Queen.\n\n\"It's bad,\" he said. \"If the writer of this letter knows his business, it's very bad. Jacks. Daemons. Wyverns. The might of the Wild has joined against us.\"\n\nAt the names, all of her ladies crossed themselves.\n\nThe Queen rose to her feet. \"Let us help these worthy gentlemen,\" she said to her ladies. She rose and kissed the king's face. \"You will need carts, provender, supplies, canteens and water casks. I have the lists to hand. You gather your knights, and I'll have the rest ready to follow you before noon.\" In a moment, the winds of war\u2014actual war, with all it implied about glory and honour and high deeds\u2014blew away her fancy for the foreign knight.\n\nAnd her lover was the _king._ Going to war with the Wild.\n\nHe looked into her eyes with adoration. \"Bless you!\" he whispered. And her king turned, and shouted for his constable. And the Earl of Towbray, who was ready to hand.\n\nTowbray had the grace to give the king a wry smile. \"How convenient that I have all my armed strength to hand, your Grace. And that you have summoned your knights to a tournament.\"\n\nThe king usually had no time for Towbray, but just for a moment they shared something. The king clapped the other man's shoulder. \"If only I _had_ planned it,\" he said.\n\nTowbray nodded. \"My knights are at your service.\"\n\nThe king shook his head. \"That's the trouble with you, Towbray. Just when I've found reason to despise you, there you are doing something to help. And unfortunately a year hence, you'll do something to spoil it again.\"\n\nTowbray bowed. \"I am what I am, your Grace. In this case, your Grace's servant.\"\n\nHe glanced at the Queen.\n\nShe didn't see his look, already busy with a list of long-bodied wagons available in the town of Harndon.\n\nBut the king followed Towbray's glance, and his lip curled.\n\nTowbray had been watching the king, too. It was easy to dismiss him\u2014he didn't seem to have any finer feelings, or to have any purpose beyond the tilt field and his wife's bed.\n\nAnd yet here was the Wild, launching an attack, and the king _just happened_ to have already summoned his host. That kind of luck seemed to happen to the king all the time.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain woke in the Abbey infirmary, his head on a feather pillow, his hands\u2014the left heavily wrapped in bandages\u2014laid neatly on a white wool blanket atop a fine linen sheet. The sun shone through the narrow window well over his head and the shaft of light lit Bad Tom, snoring in the opposite bed. A young boy lay with his face to the wall in the next bed, and an older man with his whole head wrapped in linen opposite him.\n\nHe lay still for a moment, oddly happy, and then it all came back to him in a rush. He shook his head, cursed God, sat and got his feet on the floor.\n\nHis movement caused the duty sister to raise her head. He hadn't noticed her. She smiled.\n\nAmicia.\n\n\"Aren't you afraid to be alone with me?\" he asked.\n\nHer composure was palpable, like armour. \"No,\" she answered. \"I am not afraid of you, sweet. Should I be?\" She rose to her feet. \"Besides, Tom is only just asleep and old Harold\u2014who has leprosy\u2014sleeps very lightly. I trust you not to disturb them.\"\n\nThe captain winced at the word trust. He leaned towards her\u2014she smelled of olive oil and incense and soap\u2014and had to fight the urge to put his hands on her hips, her waist\u2014\n\nShe cocked her head a little to one side. \"Don't even think it!\" she said, sharply, but without raising her voice.\n\nHis cheeks burned. \"But you like me!\" he said. It seemed to him one of the stupidest things he'd ever said. He gathered himself, his dignity, his role as the captain. \"Tell me why you always fend me off?\" he asked, his voice controlled, light hearted, and false. \"You didn't fend last night.\"\n\nShe met his look, and hers was serious, even severe. \"Tell me why you curse God on rising?\" she asked.\n\nThe silence between them lasted a long time, during which he even considered telling her.\n\nShe took his left hand and started to unwrap the bandage. That hurt. A little later, Tom opened one eye. The captain did not particularly enjoy watching him admire her hips and her breasts as she moved with her back and side to him.\n\nHe winked at the captain.\n\nThe captain did not wink back.\n\nAfter she'd put a oregano poultice on his hand and wrapped it in linen, she nodded. \"Try not to seize the sharp bits when fighting grim beasts in future, messire,\" she said.\n\nHe smiled, she smiled, their silence forgotten, and he left feeling as light as air. It lasted all the way down the steeply turning stairs, until he saw the twenty-three tight-wrapped white bundles under an awning in the otherwise empty courtyard.\n\nIn the aftermath of the battle, the Abbess had ordered all of her people to stay indoors. No one would sleep in the open air, no matter how balmy and spring-laden it was. Services were held in a side chapel\u2014the main chapel was now sleeping quarters.\n\nHe passed under the arch to his Commandery, and found Michael, who was busy writing, with Ser Adrian, the company's professional clerk. Michael rose stiffly and bowed. Adrian kept writing.\n\nThe captain couldn't help but smile at his squire, who was obviously alive and _not_ one of the bundles in the courtyard. His face asked the question.\n\n\"Two broken ribs. Worse than when I tried to ride my father's destrier,\" Michael said ruefully.\n\n\"In a business where we take daring and courage for granted, yours was a brave act,\" the captain said, and Michael glowed. \"Stupid,\" the captain continued, putting a hand on the young man's shoulder, \"and a little pointless. But brave.\"\n\nMichael continued to beam with happiness.\n\nThe captain sighed and went to his table, which was stacked high with scrolls and tubes. He found the updated roster. It was due the first of every month, and tomorrow was the first of May.\n\n_Why had he even considered telling her why he cursed God?_\n\nPeople were often stupid, but he wasn't used to being one of them.\n\nHe read through the roster. Thirty-one lances\u2014thirty, because Hugo was dead and that broke his lance. He needed a good man-at-arms\u2014not that there seemed to be any to be found in this near wilderness. There must be local knights\u2014younger sons eager for glory, or for a little cash, or with a pregnancy to avoid.\n\nThe whole stack of paperwork made him tired. But he still needed men, and then there was the Wild to consider as well.\n\n\"I need to talk to Bad Tom when he's well enough. And to the archers from last night. Who was most senior?\" he asked.\n\nMichael took a deep breath. The captain knew he was testing the bounds of the pain against the inside of the bandage with that breath\u2014knew this from having broken so many ribs himself.\n\n\"Long Paw was the senior man. He's awake\u2014I saw him eating.\" Michael rose to his feet.\n\nThe captain held up a hand. \"I'll see him with Tom. If he can leave the infirmary.\" His hand was throbbing. He initialled the muster roll. \"Get them, please.\"\n\nMichael paused, and the captain swallowed a sigh of irritation. \"Yes?\"\n\n\"What\u2014what happened last night?\" Michael shrugged. \"I mean, all the men feel we won a great victory, but I don't even know what we did. Beyond killing the wyverns,\" he said, with the casual dismissiveness of youth.\n\nThe captain felt like yelling, _We killed two wyverns, you useless fop._ But he understood the boy's attitude, albeit unspoken.\n\nThe captain sat carefully in a low backed folding chair made of a series of arches linked at the base\u2014it was a beautiful chair with a red velvet cushion which welcomed him, and he leaned back. \"Are you the apprentice captain asking? Or my squire?\"\n\nMichael raised an eyebrow. \"I'm the apprentice captain,\" he said.\n\nThe captain allowed the younger man a small smile. \"Good. Tell me what you _think_ we did.\"\n\nMichael snorted. \"Saw that coming. Very well. All day we sent out patrols to gather in farmers. I didn't realise it at the time, but more patrols went out than came back.\"\n\nThe captain nodded. \"Good. Yes. We're being watched, all the time. But the creatures watching us aren't very bright. Do you have any of the power?\"\n\nMichael shrugged. \"I studied it but I can't hold all the images in my mind. All the phantasms.\"\n\n\"If you capture a beast and bend it to your will, you can look through its eyes\u2014it's a potent phantasm but it is wasteful. Because you must first overcome the will of another creature\u2014a massive effort, there\u2014and then _direct_ that effort. And in this case you must do so over distance.\"\n\nMichael listened, utterly fascinated. Even Ser Adrian had stopped writing.\n\nThe captain glanced at him, and the clerk shook his head and started to get to his feet. \"Sorry,\" he mumbled. \"No one ever talks about this stuff.\"\n\nThe captain relented. \"Stay. It is part of our lives and our way of war. We use scouts because we don't have a magus to use birds. Even if we did I'd rather use scouts. They can observe and report, can make judgments as to numbers, can tell if they see the same three horses every day. A bird can't make those judgments, and the magus's perceptions of whatever the bird sees is filtered through\u2014something.\" The captain sagged. \"I don't know what, but I imagine it as a pipe that's too small for all the information to get through, as if everything is seen through water or fog.\"\n\nMichael nodded.\n\n\"The Wild has no scouts, so I guessed that our enemy was using animals as spies. We have trapped a lot of birds, and then I misled him.\" The captain crossed his hands behind his head.\n\n\"And with cook fires. You told me so.\" Michael leaned forward.\n\n\"Gelfred isn't down at the Bridge Castle, not much anyway. He's out in the woods, watching their camps. He has been since we realised the bulk of the Wild army had gone around us. Want to talk about brave? I sent patrols out with a weapon\u2014something the Moreans make. Olive oil, ground oil, whale oil will do\u2014bitumen, if you can get it, plus sulphur and saltpeter. There's dozens of mixtures and any artificer knows them. It makes sticky fire.\"\n\nMichael nodded. The clerk crossed himself.\n\n\"Even the creatures of the Wild sleep. Even the adversarius is just a creature. And when they gather to attack men\u2014well, it stands to reason that they must have a camp. Do they talk? Do they gather at campfires? Play cards? Fight amongst themselves?\" The captain looked out of the window. \"Have you ever thought, Michael, that we are locked in a war without mercy against an enemy we don't understand at all?\"\n\n\"So you've watched them, and attacked their camp,\" Michael said with satisfaction. \"And we hit them hard.\" Now Michael was smiling.\n\n\"Yes and no. Perhaps we didn't touch them,\" the captain said. \"Perhaps Bad Tom and Wilful Murder put some fire on some meaningless tents, and then they followed our boys back and hit us harder\u2014killing twenty-three people for the loss of just two wyverns,\" the captain said.\n\nMichael's smile froze. \"But\u2014\"\n\n\"I want you to see that victory and defeat are a question of perception, unless you are dead. You know every man and woman in the company\u2014in this fortress\u2014feels we won a great victory. We fired the enemy's camps, and then we killed a pair of his most fearsome monsters in ours.\" The captain got to his feet as Michael nodded.\n\n\"And because of this perception, everyone will fight harder and longer, and be braver, despite my fucking mistake to allow civilians into the courtyard which cost us twenty-three lives. Despite that, we're _winning._ \" The captain's eyes locked on Michael's. \"Do you see?\"\n\nMichael shook his head. \"It wasn't your fault\u2014\"\n\n\"It was my fault,\" the captain said. \"It's not my moral burden\u2014I didn't kill them. But I could have kept them alive if I hadn't been distracted that evening. And keeping them alive is my duty.\" He stood up straight and picked up the baton of the command. \"Best know this, if you want to be a captain. You have to be able to look reality in the eye. I fucked their lives away. I can't go to pieces about it, but neither can I forget it. That's my job. Understand?\"\n\nMichael nodded and gulped.\n\nThe captain made a face. \"Excellent. Here endeth the lesson about victory. Now, if it is not too much trouble, I'd like Long Paw and Bad Tom, please.\"\n\nMichael stood and saluted. \"Immediately!\"\n\n\"Harumf,\" said the captain.\n\nLong Paw was fifty, his once red hair mostly grey and a mere tonsure around a bald pate, with an enormous moustache and long sideburns so that he had more hair on his face than on his head. His arms were unnaturally long and despite his status as an archer and not a man-at-arms, he was reputed the company's best swordsman. The rumour was he had once been a monk.\n\nHe clasped hands with the captain and grinned. \"That was a little too exciting.\"\n\nBad Tom came in after him, a head taller than either the captain or the archer, his iron grey hair curiously at odds with his pointed black beard. His forehead had a weight of bone that made his head look like the prow of a ship, and no one would call him a handsome man. He looked scary, even in broad daylight, dressed in nothing but a shirt and an infirmary blanket. He clasped hands with the captain and the archer, grinned at Ser Adrian, and settled every inch of his gigantic frame into one of the arched chairs.\n\n\"Good plan,\" he said to the captain. \"I had fun.\"\n\nMichael slipped in. No one had invited him, but his face suggested that no one had told him he couldn't come, either.\n\n\"Get us all a cup of wine,\" the captain said, which indicated that he was welcome enough.\n\nWhen five horn cups were on five chair arms, and when Ser Adrian had his lead poised to write, Tom tasted his wine, leaned back and said, \"We hit 'em hard. Not much to say\u2014worst part was getting there. The lads was fair skittish, and every shadow had a boglin or an irk in it, and I thought once I was going to have to cut Tippit in half to shut him the fuck up. So I leaned over him\u2014\"\n\nLong Paw grinned. \"Leaned over him with that giant dagger in his fist!\"\n\n\"And Tippit pissed himself,\" Bad Tom said with evident satisfaction. \"Call him Pishit from now on.\"\n\n\"Tom,\" Long Paw cautioned.\n\nTom shrugged. \"If he can't cut it he should go weave blankets or cut purses. He's a piss poor archer and one day he's going to get a man killed. Anyway, we rode most of the way there, and we moved fast, 'cause you said\u2014\" Bad Tom paused, obviously at a loss for the words.\n\n\"Your only stealth will be speed.\" _One of Hywel's many aphorisms._\n\n\"That's what you said,\" Tom agreed. \"So we didn't sweat it too much, but went for them. If they had sentries, we never saw 'em, and then we were in among their fires. I slit a lot of sleeping cattle,\" he said, with a horrible smile. \"Stupid fucks, asleep with a killer among them.\"\n\nRemorse was not in Tom's lexicon. The captain winced. The big man looked at Long Paw. \"I got busy. You tell it.\"\n\nLong Paw raised an eyebrow. \"All the archers had an alchemical on our backs. I threw mine in a fire\u2014to start the ball, so to speak.\" He nodded. \"They were _spectacular_. If that's the word.\" Long Paw was obviously proud of it.\n\nTom nodded. \"Made us plenty of light,\" he said, and the words, combined with his look, were horrible enough that Long Paw looked away from him.\n\n\"We didn't see no tents. But there was men sleeping on the ground, critters too. And beasts\u2014horses, cattle, sheep. And wagons, dozens of them. They've been hitting the fair convoys, or I'm a Galle.\"\n\nThe captain nodded.\n\n\"We burned it all, killed the animals, and then any critter we come across too.\"\n\n\"What critters? Boglins? Irks? Tell me,\" the captain asked, and the words just hung there, between them.\n\nTom made a face. \"Little ones. Boglins and irks mostly. You know. Nightmares and daemons pursued us. Fucking daemons are fast. I fought a golden bear, sword to its axe and claws.\" He blew his nose into his hand and flicked the contents out the window. \"But I didn't get to fight a daemon,\" he said regretfully.\n\nThe captain wondered if, in the entire world, there was another man who could regret not having met a creature that projected terror.\n\nBad Tom was not like other men.\n\n\"How many? Total? What are we still up against?\" the captain asked.\n\nLong Paw shrugged. \"Dark and fire, Cap'n. My word ain't worth shit\u2014but I say we killed maybe fifty men and more creatures.\" He shrugged. \"And all we really did was kick the ant hill.\"\n\nTom gave Long Paw a look of appreciation. \"What he said,\" Tom admitted. \"We kicked the ant hill. But we kicked it hard.\"\n\nMichael sputtered. \"You two killed fifty _Jacks_?\" he asked.\n\nTom looked at him as if he'd discovered a bad smell. \"We had help, younker. And it weren't all _Jacks._ I killed I don't know how many\u2014five? Ten\u2014before I realised they was all yoked together. Poor fucks.\"\n\nMichael made a choking sound. \"Captives?\" he managed.\n\nTom shrugged. \"Got to think so.\"\n\nMichael's outrage showed, and the captain raised a hand. Pointed at the door. \"More wine,\" he said. \"And take your time.\"\n\nLong Paw shook his head as the young man slammed out. \"Not for me, Captain. It'll send me to sleep.\"\n\n\"I'm done, anyway,\" the captain said. \"Better result than I thought. Thanks.\"\n\nLong Paw clasped his hand again. \"One for the books, Cap'n.\"\n\nThe clerk looked at his pencil scrawl. \"I'll just copy this out for fair,\" he said, exchanging a parting look with Long Paw and heading for the door himself.\n\nHis departure left the captain alone with Bad Tom, who stretched his naked legs out beneath his blanket and took a long drink of his wine.\n\n\"That Michael's too soft for this life,\" Tom said. \"He tries, and he ain't worthless, but you should let him go.\"\n\n\"He doesn't have anywhere _to_ go,\" the captain said.\n\nTom nodded. \"I'd wondered.\" He took another sip and grinned. \"That girl\u2014the nun?\"\n\nThe captain looked blank.\n\nTom wasn't fooled for a moment. \"Don't give me that. Asking you why you curse God. Listen, you want my advice\u2014\"\n\n\"I don't,\" the captain said.\n\n\"Get a knee between her legs and keep it there 'til you're inside her. You want her\u2014she wants you. I'm not saying rape her.\" Tom said this with a professional authority that was more horrible than his admission of killing the captives. \"I'm just saying that if you get it done, you can have a warm bed as long as you're here.\" He shrugged. \"A warm bed and a soft shoulder. Good things for a man in command. None of the lads will blame you.\" His unspoken thought came through, too. _Some of the lads might see you in a better light for it._\n\nTom nodded at the captain, and the captain felt a black rage boil up inside him. He worked on it\u2014trying to shape it, trying to plug it. But it was like the brew they'd sent against the enemy\u2014oily black, and when it hit fire\u2014\n\nBad Tom took a deep breath and stepped back. \"Beg your pardon, Captain,\" he said. He said it with as much assurance as he'd suggested everything else. \"Overstepped, I expect.\"\n\nThe captain swallowed bile. \"Are my eyes glowing?\" he asked.\n\n\"Little bit,\" Tom said. \"You know what's wrong with you, Captain?\"\n\nThe captain leaned on the table, the burst of rage dying away and leaving fatigue and a headache of Archaic proportions. \"Many things.\"\n\n\"You're a freak, just like me. You ain't like them. Me\u2014I take what I want and let the rest go. You want them to love you.\" Tom shook his head. \"They don't love the likes of us, Captain. Even when I kill their enemies, they don't love me. Eh? You know what a sin-eater is?\"\n\nThat came out of nowhere. \"I've heard the name.\"\n\n\"We have 'em up in the hills,. Usually some poor wee bastard with one eye or no hands or some other freak. When a man dies, or sometimes a woman, we put a piece of bread soaked in wine\u2014they used to soak it in blood\u2014on the corpse. Goes on the stomach and the heart. And the poor wee man comes and eats the bread, and takes all the dead's sin on them. So the dead un goes off to heaven, and the poor wee man goes to hell.\" Tom was far away, in memory. The captain had never seen him that way before. It was odd, and a little scary, to be intimate with Bad Tom.\n\n\"We're sin-eaters, every one of us,\" Tom said. \"You and me, sure\u2014but Long Paw an' Wilful Murder and Ser Hugo and Ser Milus and all the rest. Sauce too. Even that boy. We eat their sin. We kill their enemies, and then they send us away.\"\n\nThe captain had a flash of the daemon eviscerating his horse. _We eat their sin._ Somehow, the words hit him like a thunderclap, and he sat back. When he was done with the thought\u2014which cascaded away like a waterfall, taking his thoughts in every direction\u2014he realised the shadows had changed. His wine was long gone, Bad Tom was gone, his legs were stiff, and his hand hurt.\n\nMichael was standing in the doorway with a cup of wine in his hand.\n\nThe captain dredged a smile out of his reverie, shrugged and took the wine.\n\nHe drank.\n\n\"Jacques went down to Bridge Castle with grain and came back with a message for you from Messire Gelfred,\" Michael said. \"He says it's urgent he speak to you.\"\n\n\"Then I'll have to put my harness on,\" the captain said. He sounded whiney, even to himself. \"Let's get it done.\"\n\n# The Albinkirk Road\u2014Ser Gawin\n\nHe had lost track of time.\n\nHe wasn't sure what he was any more.\n\nGawin rode through another spring day, surrounded by carpets of wildflowers that flowed like morning fog beneath his horse, rolling away in clumps and hummocks, a thousand perfect flowers in every glance, blue and purple, white and yellow. In the distance, all was a carpet of yellow green from the haze of sun on the mountainsides that were coming closer every day, their peaks woven like a tapestry in and out of the stands of trees that grew thicker and closer every mile.\n\nHe'd never had the least interest in flowers before.\n\n\"Ser knight?\" asked the boy with the crossbow.\n\nHe looked at the boy, and the boy flinched. Gawin sighed.\n\n\"Ye weren't moving,\" the boy said.\n\nGawin pressed his spurs to his horse's side, and shifted his weight, and his destrier moved off. His once-handsome dark leather bridle was stained with the death of fifty thousand flowers, because Archangel ate every flower he could reach as soon as he'd figured out that the once fierce hands on the reins weren't likely to stop him eating. That's what his misery meant to his war horse\u2014more flowers to eat.\n\n_I am a coward and a bad knight._ Gawin looked back at a life of malfeasance and tried to see where he'd gone wrong, and again and again he came back to a single moment\u2014torturing his older brother. The five of them ganging up on Gabriel. Beating him. The pleasure of it\u2014his screams\u2014\n\n_Is that where it started?_ he asked himself.\n\n\"Ser knight?\" the boy asked again.\n\nThe horse's head was down, and they'd stopped again.\n\n\"Coming,\" Gawin muttered. Behind him, the convoy he was not guarding rolled north, and Gawin could see the Great Bend ahead, where the road turned to head west.\n\nWest towards the enemy. West, where his father's castle waited full of his mother's hate and his brother's fear.\n\n_Why am I going west?_\n\n\"Ser knight?\" the boy asked. This time, there was fear in his voice. \"What's that?\"\n\nGawin shook himself out of his waking dream. The goldsmith's boy\u2014Adrian? Allan? Henry?\u2014was backing away from a clump of trees just to his left.\n\n\"There's something there,\" the boy said.\n\nGawin sighed. The Wild was _not_ here. His horse stood among wildflowers, and last year this field had been ploughed.\n\nThen he saw the sickly-pale arm, light brown, shiny like a cockroach, holding a stone-tipped javelin. He saw it and it saw him in the same moment, and he leaned to the left with the habit of hard training and ripped his long sword from its scabbard.\n\nThe boglin threw its weapon.\n\nGawin cut the shaft out of the air.\n\nThe boglin gave a thin scream of anger, balked of its prey, and the goldsmith's boy shot it. His crossbow loosed with a snap and the bolt went home into the creature with a slurpy thud and came straight out the other side in a spray of gore, leaving the small horror to flop bonelessly on the wildflowers for as long as a trout might take to die, making much the same gasping motions with its toothless mouth, and then its eyes filmed over and it was gone.\n\n\"They always have gold,\" the goldsmith's boy said, taking a step towards it.\n\n\"Step back, young master, and load that latch again.\" Gawin was shocked at his voice\u2014calm, commanding. Alive.\n\nThe boy obeyed.\n\nGawin backed Archangel slowly, watching the nearest woods.\n\n\"Run for the wagons, boy. Sound the alarm.\"\n\nThere was more movement, more javelin heads, a flash of that hideous cockroach brown, and the boy turned and ran.\n\nGawin slammed his visor down.\n\nHe wasn't in full armour. Most of it was in a goldsmith's wagon, wrapped in tallow and coarse sacking in two wicker baskets because he had no squires to keep it. And because wearing it might have meant something.\n\nSo he was wearing his stained jupon, his boots, his beautiful steel gauntlets and his bassinet, riding a horse worth more than three of the wagons full of fine wools he was protecting. He backed Archangel faster, sawing the reins back and forth as his destrier all but trotted backwards.\n\nThe first javelin came out of the woods, high. He had his sword in his right hand, all the way down by his left side, the position his father's master at arms had taught him. He could hear the man saying \"Cut up, mind! Not into your own horse, ye daft thing!\"\n\nHe cut up, severing the weapon's haft and breaking its flight.\n\nBehind him, he heard the boy yelling \"To arms! To arms!\"\n\nHe risked a long glance back at the convoy. It was hard to focus through the piercing of his visor, hard to pick up distant movement, but he thought he could see Old Bob directing men in all directions.\n\nHe turned back to see the air full of javelins, and he cut\u2014up, down, up again as fast as thought. A javelin haft caught him in the side of his head and rang his helmet like a bell, even with his padded arming cap. He smelt his own blood.\n\nTurned his horse's head\u2014because once they'd all thrown, he had a moment to get around, and get away.\n\nTwo of them were running for him. They were fast, moving like insects\u2014so low to the ground that they were a danger to horses' legs. Archangel reared, pivoted on his hind legs, and a powerful forefoot shot out like a boxer's punch.\n\nGawin flicked his sword out along his fingers, lengthening his grip until he was holding only the disc-shaped pommel and, in the same motion, made a wrist cut down and back.\n\nArchangel's boglin popped like a ripe melon, its chest and neck caving in with a dull thud and a fine spray of ichor. Gawin's screeched as the cold iron pierced its hide\u2014iron was poison to its kind, and it screamed its hate as its tiny soul rose from its corpse like a minute thundercloud that dissipated on the first whiff of breeze.\n\nAll at once they were away, the big horse galloping easily over the wildflowers. Gawin had trouble breathing. His visor seemed to cut all the air from his lungs and his chest was tight.\n\nAs he rode, he could see there were other knots of the things\u2014perhaps four or five groups of them spread across the flowers like shit stains on a pretty dress, and suddenly he was filled with a fey energy, a will to do a great deed and die in the accomplishment.\n\n_I am a knight_ , he thought fiercely.\n\nGawin sat up in the saddle, holding his long, sharp sword with new purpose, and he turned Archangel and raised it at the boglins. Something dead within him rekindled as the sun lit the blade like a torch.\n\nHe felt the touch of something divine, and he saluted as if riding in a tournament.\n\n\"Blessed Saint George,\" he prayed, \"let me die as I wish I had lived.\"\n\nHe put his spurs to Archangel\u2014gently, a nudge rather than a rake\u2014and the great horse thundered forward.\n\nThe boglins scattered. Javelins flew past and then he was among them, through them, using his knees to turn Archangel in a long curve toward the next clump, who were already running for the trees.\n\nGawin had no plans to survive so he thundered after them, slaying any that stood or were merely too slow to escape, leaning far out from the saddle\u2014\n\nSomething called from inside the wood\u2014a wail that froze his blood.\n\nIt was out of the woods and at him in heartbeats.\n\nArchangel was ready, pivoted his whole great bulk as Gawin's weight shifted so that the war horse moved like his own feet in a fight, and the huge enemy\u2014scent of burned hair and soap and old ashes\u2014shot past. One taloned arm stretched out like an angry cat's paw, reaching for Archangel's neck, but the war horse was fast, and some steel-shod forefoot smashed the taloned hand with lethal precision.\n\nThe thing screamed, its left talon hanging limply, the bones broken. It rose on its hind feet, raised its right claw, and fire shot from its outstretched talons\u2014a beam of fire that caught his body where the mail aventail of his helmet hung over his padded jupon. It had no pressure, no impact, and Gawin ducked his head, putting the peak of his helmet into the flame by instinct rather than training. His left eye flared with pain even as the first cold knife of agony pierced his left shoulder. His body, with no guidance from his mind, cut down blindly with the sword.\n\nHis blow was weak and badly directed\u2014the edge of the blade didn't even bite into the thing's hide\u2014but the sword's weight fell on its brow ridge, and it stumbled.\n\nArchangel shouldered it. Gawin almost lost his seat, his back and rump crashing into the high-backed saddle as his war horse made its own fighting decisions and leaped forward, bearing down the monster again with weight and momentum, so that the creature's stumble was more off balance and the horse landed two more blows with its steel-shod forefeet, forcing the creature onto all fours. It roared with pain as it put weight on the broken limb.\n\nAnd then the grass was full of boglins thrusting their stone-tipped spears at him, and some of them scoring hits. The deerskin of his padded jupon turned a few and the damp sheep's wool stuffing turned others, but at least one punched straight through and into his skin. Unthinking, he touched his spurs to Archangel and the great horse responded with a mighty leap forward, and then they were running free.\n\nGawin turned him in a wide circle. He couldn't see from his left eye, and the pain in his side was so great that he could scarcely feel it\u2014or anything else.\n\n_I want that thing,_ he thought. _Let them take that head back to Harndon and show it to the king, and I will be content._\n\nHe got Archangel around. The horse had at least two wounds\u2014both from javelins. But like his rider he was trained to fight hurt, and went at his prey with all the spirit he could have asked.\n\nBut the monster was running\u2014weight forward, low to the ground, only three legs working, a dozen boglins gathered tight around it in the strong sunlight, as they fled into the trees.\n\nGawin reined up\u2014surprised at himself. Death lay waiting in those trees. But it was one thing to fight to the death out here under the sun, and another to follow the Wild into the waiting trees and die alone\u2014and for nothing. He reined up, and looked at the litter of broken boglins, and his view of them suddenly narrowed\u2014he tasted salt in his mouth, and copper, and\u2014\n\n# Lorica\u2014Ser Gaston\n\nLorica again.\n\nGaston spat the foreign name as he watched the grey stone walls approach. He flicked a look at his cousin, who was riding serenely at his side.\n\n\"We are going to be arrested,\" Gaston said.\n\nJean made a face. \"For what?\" he asked. He laughed, and at the silvery peel of his laughter, other men smiled all down the column. Their contingent was third; first the king's household, then the Earl of Towbray's, and then theirs. They had more knights than the king and the earl together.\n\n\"We killed the two squires. I locked the sheriff in a shed. You burned the inn.\" Gaston winced as he said the last. Ten days in Alba and he was beginning to appreciate just how poor their behaviour had been.\n\nJean shrugged. \"No one of worth was involved except the knight,\" he said. His voice rode the edge of a sneer. \"And he has chosen not to take exception. He has shown especial wisdom in this, I think.\"\n\n\"Nonetheless, the king will learn exactly what happened in the next hour or so,\" Gaston said.\n\nJean de Vrailly gave his cousin a sad smile. \"My friend, you have much to learn of the workings of the world. If we were in the least danger, my angel would have told me. And it seems to me that our knights make up the best part of this column\u2014bigger, better men in superb armour on fine horses. We can always fight. And if we fight, we will win.\" De Vrailly shrugged again. \"You see? Simple.\"\n\nGaston considered taking his own men and riding away.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain rode through the postern gate of the Bridge Castle with no one but Michael, also mailed and armed. They'd ridden out of the upper postern of the fortress with a minimum of fuss\u2014two men-at-arms on detail. But the captain rode fast and hard down the ridge because the sky was full of crows to the west. He noted there wasn't a bird to be seen over the fortress or the castle.\n\nHe dismounted in the Bridge Castle courtyard, where big merchant wagons were parked hub-to-hub leaving just room for a sortie to form up. As the captain looked around he realised that all the wagons were occupied. The merchants were living in them. No wonder Ser Milus said he had room. Over by the main tower, dogs whined and barked\u2014four brace of good hounds. He stopped and let them smell him. Dogs made him smile with their enthusiastic approval. All dogs liked him.\n\nCleg, Ser Milus's valet, came and led him into the main tower, where the garrison had their quarters on the ground floor\u2014plenty of paliasses of new straw, with six local women and another half-dozen company trulls sitting on the floor and sewing. They were making mattresses\u2014there were twenty ells of striped sacking already measured and cut, as the captain had seen done in a dozen countries. Clean sacks made good mattresses while dirty linen spread disease\u2014any soldier knew it.\n\nThe women rose to their feet and curtsied.\n\nThe captain bowed. \"Don't let me disturb you, ladies.\"\n\nSer Milus took his hand and a pair of archers\u2014older, steady men, Jack Kaves and Smoke, pushed the merchants away. Three of them were waving scrolls.\n\n\"I protest!\" the taller man called. \"My dogs\u2014\"\n\n\"I'll take you to law for this!\" called a stout man.\n\nThe captain ignored them and went up a set of tight steps to the uppermost floor, where tents had been used to partition the tower into sleeping quarters for officers.\n\nSer Jehannes nodded curtly to the captain. He nodded back.\n\n\"Ready to move back up the hill?\" the captain said.\n\nJehannes nodded. \"Do I owe an apology?\"\n\nThe captain lowered his voice. \"I pissed you off, and you sulked about it. I need you. I need you at the fortress, giving orders, kicking arses and taking names.\"\n\nJehannes nodded. \"I'll go back up with you.\" He looked over to Gelfred, and indicated the huntsman with a nod. \"It's bad.\"\n\n\"No one ever summons me for good news.\" The captain was relieved that he hadn't lost his most senior man forever, and clapped the man on the back hoping it was the right gesture. \"I'm sorry,\" he said.\n\nJehannes paused. \"I am also sorry,\" he said. \"I am differently made to you, and I lack your certainty.\" He shrugged. \"How is Bent doing?\"\n\n\"Very well indeed.\" Bent was the archer in Ser Jehannes's lance\u2014and also the most senior archer in the fortress.\n\n\"I'll send you Ser Brutus,\" the captain said to Milus, who grinned.\n\n\"You mean you're trading me the best knight in the company for a kid with an archer he can't control?\" He laughed. \"Never mind\u2014Jehannes outranked me and never did any work anyway.\"\n\nThe captain thought\u2014not for the first time\u2014how sensitive his mercenaries were. Jehannes had chosen to go to the castle garrison as a mere man-at-arms rather than go to the fortress with the captain, because he was angry. And everyone knew it, because there was no privacy, in a camp or in a garrison. And now that he and the captain had made it up everyone was very gentle about it. The teasing would start later. The captain thought it remarkable that such men had so much tact, but they did.\n\nGelfred was waiting, and from his expression, he was about to explode.\n\nThe captain went into his \"room\" and sat at the low camp table on a leather stool. Gelfred beckoned to the other two officers, and both came in. Jehannes paused in the doorway and spoke to someone just outside the tent flap wall. \"Clear this floor,\" he said.\n\nThey heard men grumbling, and then Marcus, Jehannes's squire, said in his guttural accent, \"All clear, sers.\"\n\nGelfred looked around. \"Not sure where to start.\"\n\n\"How about the beginning? And with a cup of wine?\" The captain tried to be light hearted, but the others looked too serious.\n\n\"The merchants came in\u2014two of them had animals.\" Gelfred shrugged. \"I'm telling this badly. Two of them had a dozen good falcons and some dogs. I took the liberty of securing them. Aye?\"\n\nA dozen good falcons and some hunting dogs would be worth a fortune. No wonder the merchants were so incensed.\n\n\"Go on,\" the captain said.\n\n\"Today is the first morning I've been here.\" Gelfred cleared his throat. \"I've been in the woods.\"\n\n\"You did a beautiful job,\" the captain said. \"Tom hit their camp just right\u2014didn't even see a guard.\"\n\nGelfred smiled at the praise. \"Thanks. Anyway. Starting this morning, I\u2014\" He looked at Ser Milus. \"I started flying the hawks against the birds\u2014those that watch the castle.\" He shrugged. \"I know this sounds lame\u2014\"\n\n\"Not at all,\" said the captain.\n\nGelfred breathed a sigh of relief. \"I was afraid you'd think me mad. Will you trust that I can see\u2014I can _see_ \u2014that some of the animals are servants of the Enemy?\" He whispered the last part.\n\nThe captain nodded. \"Yes. I believe it. Go on.\"\n\nJehannes shook his head. \"It sounds blasphemous to me,\" he said.\n\nGelfred put his hands on his hips in exasperation. \"I have a licence from the _Bishop_ ,\" he said.\n\nThe captain shrugged. \"Get on with it, Gelfred.\"\n\nGelfred brought out a game bag. It was stiff with blood, but then game bags generally were.\n\nHe extracted a dove\u2014a very large specimen indeed\u2014laid it on the camp table, and stretched out its wings.\n\n\"The gyrfalcon took it down about two hours ago,\" he said. \"No other bird we have is big enough.\"\n\nThe captain was staring at the message tube on the bird's leg.\n\nGelfred nodded. \"It came _out_ of the abbey, Captain,\" he said.\n\nMilus handed him a tiny scroll no larger than his smallest finger. \"Low Archaic,\" he said. \"Has to limit the suspects.\"\n\nThe captain ran his eyes over the writing. Neat, precise, and utterly damning\u2014a list of knights, men-at-arms and archers; numbers, stores and defences. But no description. Nothing with which to catch a spy.\n\n\"Limit the suspects in a convent?\" the captain said bitterly. \"A hundred women, _every one of whom can read and write low Archaic._ \" And use power.\n\nOne of whom he knew was an Outwaller.\n\nGelfred nodded. \"We have a traitor,\" he said, and the captain's heart sank.\n\nThe captain leaned his head on his hand. \"This is why you needed to meet me here,\" he said.\n\nGelfred nodded. \"The traitor isn't here,\" he said. \"The traitor is in the fortress.\"\n\nThe captain nodded for a while, the way a man will when he's just heard bad news and can't really take it all in. \"Someone killed the Jack in the woods,\" he said. His eyes met Gelfred's. \"Someone stabbed Sister Hawisia in the back.\"\n\nGelfred nodded. \"Yes, my lord. Those are my thoughts, as well.\"\n\n\"Someone co-operated with a daemon to murder a nun.\" The captain scratched under his beard. \"Even by my standards, that's pretty bad.\"\n\nNo one smiled.\n\nThe captain got to his feet. \"I'd like to have you hunt our traitor down, but I need you out in the woods,\" he said. \"And it is going to get worse and worse out there.\"\n\nGelfred smiled. \"I like it.\" He looked around. \"Better than in here, anyway.\"\n\n# Lorica\u2014Ser Gaston\n\nOutside the town, a deputation of ten wagons full of forage, four local knights, and the town's sheriff waited under the Royal Oak. The king rode up and embraced the sheriff, and the king's constable accepted the four young knights and swore them to their duty. The quartermaster took charge of the wagons.\n\nThe sheriff was midway through telling the king of the burning of the Two Lions when he turned white, then red.\n\n\"But that is the man!\" he said. \"Your Grace! _That_ is the man who ordered the inn burned!\" He pointed at de Vrailly.\n\nDe Vrailly shrugged. \"Do I know you, ser?\" he asked, and rode to the king, the sheriff, and the other member's of the Royal Household gathered under the great tree.\n\nThe sheriff sputtered. \"You\u2014Your Grace, _this_ is the miscreant who ordered the inn burned! Who allowed the innkeeper to be beaten, a loyal man and a good\u2014\"\n\nDe Vrailly shook his head mildly. \"You call _me_ a miscreant?\"\n\nThe king put his hand on de Vrailly's bridle. \"Hold hard, my lord. I must hear this accusation.\" The king glared at his sheriff. \"However baseless it is.\"\n\n\"Baseless?\" the sheriff shouted.\n\nDe Vrailly smiled. \"Your Grace it is true. My squires kicked the worthless _paysant_ and burned his inn as a lesson for his insolence.\" He raised his left eyebrow just a hair\u2014his beautiful nostrils flared, and his lips thinned.\n\nThe king took a deep breath. Gaston watched him very carefully. He had already loosened the sword at his hip in its sheath. Not even de Vrailly would get away this time. The king's justice could not be seen as weak in front of his own people, his vassals and his officers.\n\n_De Vrailly is insane,_ Gaston thought to himself.\n\n\"Ser knight, you must explain yourself,\" the king said.\n\nDe Vrailly raised both eyebrows. \"I am a lord, and I have the High Justice, the Middle Justice, and the Low Justice right here in my scabbard. I need no man's leave to take a life. I have burned more peasants' cots than a boy has pulled the wings off flies.\" De Vrailly shook his head. \"Take my word, your Grace: the man received due payment for his foolishness. Let us hear no more about it.\"\n\nThe sheriff put his hands on the pommel of his saddle as if to steady himself. \"I have never heard the like of this. Listen, your Grace\u2014this pompous foreigner, this so-called knight, also killed two squires of Ser Gawin Murien, and then, when I approached him, had me beaten. I was thrown into a shed, tied and bound. When I was rescued, I found the inn burned.\"\n\nGaston pushed his horse into the angry group. \"Your words in no way prove my lord's guilt,\" he insisted. \"You did not witness any of these things, yet now pronounce them truth.\"\n\n\"You were the one who hit me!\" the sheriff said.\n\nGaston had to restrain himself from shrugging. _You are an ineffective, useless man and a shame on your king\u2014and you were in my way._ But he smiled, glanced at the king, and offered his hand. \"For that I apologise. My cousin and I were newly landed, and failed to understand the laws of these parts.\"\n\nThe king was firmly in the cleft stick of conflicting emotions, goals and needs\u2014his indecision showed clearly on his face. He needed Jean de Vrailly's three hundred knights and he needed to be seen to give justice. Gaston willed the sheriff to take his hand and clasp it. He _willed_ it, and so did the king.\n\n\"Messire, my cousin and I have joined the king to ride against the Wild.\" Gaston's voice was low, urgent and yet soothing. \"I _beg_ your forgiveness before we go into battle.\"\n\nGaston prayed that the king wasn't looking at his cousin, whose expression at the word _beg_ would have curdled milk.\n\nThe sheriff sniffed.\n\nThe king's shoulders began to relax.\n\nAlmost as if against his will, the sheriff of Lorica took Gaston's hand and clasped it. He left his glove on, which was rude enough, and he didn't meet Gaston's eye.\n\nBut the king seized the moment. \"You will pay reparations to the town and to the innkeeper,\" he said. \"The sum to be the full value of the inn and all of its goods and chattels. The sheriff will investigate the value and send a writ.\" The king turned in his saddle to address the Captal de Ruth. \"You, who have announced your willingness to serve me, will first serve my sentence on this: your wages and those of all of your knights will be paid, in lieu of fine, to the innkeeper and to the town until the value set by the sheriff has been discharged.\"\n\nJean de Vrailly sat on his horse, his beautiful face still and peaceful. Only Gaston knew he was considering killing the king.\n\n\"We\u2014\" he began, and the king turned in his saddle, showing some of the flexibility he had showed jousting.\n\n\"Let the captal speak for himself,\" the king said. \"You are glib in your cousin's defence, my lord. But I must hear him speak his acceptance for himself.\"\n\nGaston thought, _He is very good at this. He has understood my cousin better than most men, and he has found a way to punish him while keeping him close and using his prowess against his enemies. Jean and his angel will not dominate this king in an afternoon._ Outwardly, he bowed.\n\nAnd glared at Jean.\n\nJean bowed as well. \"I came to fight your enemies, your Grace,\" he said in his charming accent. \"At my own expense. This _ordinance_ makes little difference to me.\"\n\nGaston winced.\n\nThe king looked around him, gathering eyes, gathering the opinions men cast with their body language, in subtle facial expressions, in the fretting of their horses. He pushed his tongue against his teeth\u2014which Gaston had already come to read as a tic of frustration.\n\n\"That is not sufficient,\" the king said.\n\nDe Vrailly shrugged. \"You wish me to say that I accept your law and your writ?\" he said, and contempt dripped from every word.\n\n_Here we go_ , Gaston thought.\n\nThe Earl of Towbray pushed his horse between the king and the captal. \"This,\" he said, \"is my fault.\"\n\nBoth the king and de Vrailly looked at him as if he'd come between them in the lists.\n\n\"I invited the Captal to Alba to serve me, and I failed\u2014even after a youth spent fighting on the Continent\u2014to understand how he would see us.\" The earl shrugged. \"I will bear the cost, for my mistake.\"\n\nDe Vrailly had the good grace to appear surprised. \"But\u2014no!\" he said suddenly. \"But I insist! I must bear it.\"\n\nThe king was looking at the Earl of Towbray the way a man might look at a rare flower suddenly discovered on a dung hill.\n\nGaston remembered to breathe.\n\nAnd in moments men were chattering with relief, the convoy was forming up, and Gaston could ride to his cousin's side.\n\n\"This is _not_ what the angel told me would happen,\" he said.\n\nGaston raised an eyebrow.\n\nDe Vrailly shrugged. \"But it will suffice. It irks me, cousin, to hear you crawl to a creature like that _sheriff._ You must avoid such things, lest they form a habit.\"\n\nGaston sat still for a moment, and then leaned forward. \"It irks me, cousin, to hear you put on airs before the King of Alba. But I assume you cannot help yourself.\" He turned, and rode back to his own retinue, and left Jean to ride by himself.\n\n# West of Lissen Carak\u2014Thorn\n\nThorn was dimly conscious of his body while he sat beneath the giant holm-oak and reached out over the sea of trees. He was aware of himself at the centre; of the fear and anger from the Jacks; the restive arrogance of the qwethnethogs; the mourning of the winged abnethogs; the distant presence that heralded the arrival of the Sossag people from the north, across the wall. He was aware of every tree past its tenth season; of the large patches of iris flowers; of the wild asparagus growing by the river where a man had built a cottage a century before; of the cattle that his raiders had taken to feed the Jacks; of the tuft-eared lynx that was both terrified and angered to have his army camped in its territory, and of the thousand other presences rolling away to the limits of his _kenning._\n\nHe sympathised with the lynx. Unknowable, powerful creatures with filthy thoughts and polluted bodies, dirty with fear and hate, had come to his woods and fired his camp, terrifying his allies, destroying his trees and making him seem weak. The greater qwethnethogs would question whether he was worthy of service\u2014the very strongest might even waste themselves and their energy on a challenge for mastery.\n\nIt was difficult for a Power of the Wild to have trusted lieutenants. But he would continue to attempt such relationships, for the good of the Wild and their cause.\n\nHe rose from beneath his tree and walked into camp, scattering lesser creatures and frightening the Jacks. He walked west, to the handful of golden bears who had allied themselves with him and had made huts of brush and leaves. He nodded to Blueberry, a huge bear with blue eyes.\n\nThe bear rose on his haunches. \"Thorn,\" he said. The bears were afraid of nothing, not even him.\n\n\"Blueberry,\" Thorn said. \"I wish to recruit more of your people. Let me have the child and I will take it to the ice caves.\"\n\nBlueberry thought for a moment. \"Yes,\" he said. \"Better food, and females. Well thought.\" Sunset, the largest of the bears, brought the cub. She was small enough for Thorn to carry easily and mewed at him when he took her. He stroked her fur, and she bit at him, tasted his odd flesh, and sneezed.\n\nHe left the bears without another word and started north. When he stretched his legs, he moved faster than a galloping horse, and he could travel that fast for as long as he wished. He cradled the little bear and moved faster still.\n\nBefore the sun had dipped a finger's breadth, he was too far from his camp to hear the thoughts of his allies, or to smell the fires of the men who had chosen to serve him. He crossed a series of beaver meadows, enjoying their health, feeling the trout in the streams and the otters in the banks, and he crossed a big stream flowing south from the Adnacrags. There he turned and followed the banks north, into the mountains. Leagues flew by. Thorn drew power from the hills, valleys, water, and the trees. He drew more than power.\n\nHe drew inspiration.\n\nWar was not his choice. It had been an accident. But if he had to make war now, he needed to remind himself _why_. He would make war for this. For the wilderness. To keep it clean.\n\nAnd, of course, for himself. He was growing more powerful with every creature which chose to come and follow him.\n\nThe stream began to climb, faster and faster\u2014up a great ridge, and then down, and then up again. He was in the foothills now, and his passage was like a strong wind in the trees. Deer looked up startled. Afraid.\n\nBirds fled.\n\nHe knew the valley he wanted\u2014the valley of the creek that the Sossag called the Black, that flowed from the ice caves under the mountains. It was a special place, almost as imbued with power as the Rock.\n\nThe bears ruled it.\n\nHe climbed a steep path, almost a road, from the stream to the top of the ridge, and waited. He was fifty leagues from his army. He set the bear on the ground and waited.\n\nThe sun began to set behind him and he let his mind wander, wondered if the enemy would try to raid his camp again. It occurred to him, now that he was far from the problem, that the enemy captain must have someone watching his camp. _Of course._ How else would he have known where to attack. _He must be using animals as spies._\n\nIt was suprising how much clarity he could achieve when he was not bombarded with the chaos of other creatures.\n\n\"Thorn.\"\n\nThe speaker was old, a bear who had lived more then a century. He was called Flint, and he was acknowledged as a Power. He stood almost as tall as Thorn, and while he had white at his ears and muzzle, his body was strong and firm as a new apple in fall.\n\n\"Flint.\"\n\nThe old bear reached out and the little bear ran to him.\n\n\"Her mother was enslaved and tortured by men,\" Thorn said. \"To be fair, she was then rescued by other men, and brought to Blueberry at my camp.\"\n\n\"Men,\" said Flint. Thorn could feel the old bear's anger, and his power.\n\n\"I have burned Albinkirk,\" Thorn said, and realised what a pointless boast it was. Flint would know.\n\n\"With stars from the sky,\" Flint said. His deep voice was like the sound of a rasp cutting into hardwood.\n\n\"I have come to ask\u2014\" Faced with Flint, it was suddenly difficult to explain. Bears were well known for their complete contempt for organisation. For government. Rules. War. Bears would kill when roused. But war repelled them.\n\n\"Do not ask,\" Flint said.\n\n\"What I do\u2014\" Thorn began.\n\n\"Has nothing to do with bears,\" Flint said. He nodded. \"This is the cub of Sunbeam, of the Clan of the Long Dam. Sunbeam's brother will no doubt come and avenge her.\" The old bear said this with obvious sadness. \"As will his friends.\" Flint picked up the cub. \"They are young, and understand nothing. I am old. I see you, Thorn. I know you.\" He turned his back and walked away.\n\nAll at once Thorn wanted to chase down the old bear and sit at his feet. Learn. Or protest his\u2014not his innocence, but his intentions.\n\nBut another part of him wanted to turn the old bear to ash.\n\nIt was a long walk back to camp.\n\n# Lissen Carak\u2014Sister Miram\n\nSister Miram was missing her favourite linen cap, and she took the moments between study of High Archaic and Nones to visit the laundry. She raced down the steps of the north tower\u2014for a large woman, she was very fast\u2014but then a flash of intuition made her pause at the door to the laundry. Six sisters laboured away, their hands and faces red, stripped to their shifts in the heat of the room. A dozen local girls worked with them.\n\nLis Wainwright was also stripped to her shift. Forty years had not ruined her figure. Miram might have smiled, but she didn't. Beyond Lis were younger girls. Miram knew them all\u2014had taught them. The Carters and the Lanthorns. The Lanthorn girls were simpering. There wasn't usually a lot of simpering in the laundry.\n\nA hundred nuns and novices generated a great deal of laundry. The addition of four hundred farmers, their families, and two hundred professional soldiers, forced the laundry to boil linen day and night. The drying lines were stretched at every hour, and even senior sisters like Sister Miram received their linens slightly damp and badly ironed. Or left them missing things like caps.\n\nShe looked around for Sister Mary, whose week it was to run the laundry, and heard a man's voice. It was a cultured voice, singing.\n\nShe listened intently. Singing a Gallish romance.\n\nShe couldn't see him, but she could see the four Lanthorn girls in their shifts, giggling, preening and showing a great deal of leg and shoulder.\n\nMiram's eyes narrowed. The Lanthorn girls were what they were, but they didn't need some smooth-talking _gentleman_ to encourage them on their road to hell. Miram strode across the damp floor and there he was, leaning in the laundry door. He had a lute, and he was not alone.\n\n\"Your name, messire?\" she asked. She had pounced so swiftly that he was locked in indecision\u2014keep playing, or flee?\n\n\"Lyliard, ma soeur,\" he said sweetly.\n\n\"You are a knight, messire?\" she asked.\n\nHe bowed.\n\n\"None of these four unmarried maidens is of noble birth, messire. And while it may suit you to bed them, their pregnancies and their unwed lives will weigh heavily on my convent, my sisters, and your soul.\" She smiled. \"I hope we understand each other.\"\n\nLyliard looked as if he'd been hit by a wyvern. \"Ma soeur!\"\n\n\"You look like a squire,\" Sister Miram said to the young man at Lyliard's elbow. He also had a lute and while he lacked Lyliard's dash and polish Miram's opinion was he'd get there in time. And he was handsome, in a raffish, muscular way.\n\n\"John of Reigate, sister.\" He was young enough to drop his eyes and look like a schoolboy caught out in a lark. Which he was. She had to remember that they killed for a living, but they were still people.\n\nThe third man was the handsomest. He had polish _and_ good looks. And he blushed.\n\n\"And you are the captain's squire,\" she said.\n\nHe shrugged. \"Unfair. My fame proceeds me.\"\n\n\"Don't ape your master,\" Miram said. \"The three of you, gently born, should be ashamed of yourselves. Now go.\"\n\nLyliard looked abashed. \"Listen, sister, we merely crave some female company. We are not bad men.\"\n\nShe sniffed. \"Do you mean you would pay for what you take?\" She looked at all three of them. \"You seduce innocents instead of committing out and out rape? Is that supposed to impress me?\"\n\nThe captain's squire sniffed. His left hand patted the bandage around his waist. \"You really have no idea who or what we are. What we face.\"\n\nMiram caught his eye and stepped very close, close as a lover. Almost nose to nose. His eyes were blue, and she had once been a woman to enjoy handsome men.\n\nHers were a deep, old green.\n\n\"I know, young squire,\" she said. _\"I know exactly_ what you face.\" She didn't blink and he couldn't tear his gaze away from her. \"Save your posturing for whores, _boy._ Now go and say twenty Pater Nosters, mean them, and think about what it might mean to be a knight.\"\n\nMichael would have liked to have stood his ground, but the moment her regard dropped away, he stumbled a step.\n\nShe smiled at the three of them, and they backed away from the door.\n\nSister Miram went back into the laundry, where the Lanthorn girls were huddling, terrified, and trying to cover their bare legs.\n\nSister Mary came in, carrying a huge basket. \"Miram!\" she called out. \"What's amiss?\"\n\n\"The usual,\" Miram said. And started searching for her missing cap.\n\n# North of Lissen Carak\u2014Thorn\n\nThorn felt bitten by the old bear's disdain. His walk back was full of thoughts about how the men in the Rock had, apparently, inflicted two defeats on him. He had to face the hard truth; to the irks and boglins and even to the daemons, these little fiery pinpricks were defeats.\n\nHe didn't really think that either of his lieutenents would challenge him, and he reached out more and more to the east as he walked, until he could feel the intense _wrongness_ of the invaders. They were not like the peasants, the nuns, and the shepherds in the fortress. They smelt of violence.\n\nHe had always hated their kind, even when he walked among them as a man.\n\nAlso in the fortress, surrounded by all that cold stone worked by man, the enchantments an aeon old and proof against all but his strongest enchantment, he could feel the Abbess, a sun of power, with her nuns a star field behind her.\n\nHe flinched away from her.\n\nAnd the tendrils of his questing power saw another, darker sun\u2014the beacon that the daemons had seen\u2014that Thurkan, the sharpest of the daemons, had seen and avoided. The shielded one, who had resisted, however briefly, his workings on the battlefield.\n\n_The bears hadn't refused him, precisely. But nor were they helping him with any force but a few angry warriors bent on revenge._ He drew deep breath of clean air and turned north, back into the mountains, and lengthened his stride until he was all but running, his giant body now moving faster than the fastest horse. He could get where he wanted to with a phantasm, but he was suddenly wary of using too much power. Power attracted other power, and in the Wild, that could spell a quick end\u2014all too often, something bigger than you arrived unexpectedly. And ate you.\n\nEven as he ran the forest highways, Thorn contemplated eating Turkan.\n\n# Lissen Carak\u2014Kaitlin\n\nThe four Lanthorn girls were quick to recover from Sister Miram, and the afternoon found them coring winter apples behind the kitchens. There were no sisters and no novices.\n\nThe eldest Lanthorn girl was Elissa. She was dark haired, as tall as a man, thin, with long legs and very little figure and a nose like a hawk. Despite this men found her irresistible, mostly because she smiled a great deal and was selective in her use of the family's principle weapon: a sharp tongue.\n\nMary was the second daughter. She was the very opposite of her elder sister; short, but not squat, with a full figure, guinea gold hair, a narrow waist and a snub nose. She thought herself a great beauty and was always puzzled when boys preferred Elissa.\n\nFran was brown haired, full-lipped and full hipped. She had her mother's looks, her father's brains and sense of honesty, and she seldom cared whether boys noticed her or not.\n\nAnd Kaitlin was the youngest: just fifteen. She was not as tall as Elissa, not as full-figured as Mary, nor yet as witty, or as cutting, as Fran. She had pale brown hair that framed a heart-shaped face, and she appeared to be the quietest and most respectable of the Lanthorns.\n\n\"Bitch,\" Fran said, tossing a core aside. \"She thinks we're going to be good little girls with pig shit on our feet for the rest of our lives.\"\n\nElissa looked around carefully. \"We have to play this right,\" she said thoughtfully. She ate a slice of apple, deftly taking a knife from beneath her kirtle, cutting a slice, wiping the knife on her apron and putting back in her sheath faster than most people could follow. She looked down her long nose at Fran. \"I hearby convene a meeting of the 'Marry a Noble' club.\"\n\n\"Silly kids' nonsense,\" Mary scoffed. She was eighteen. \"No one around here is going to marry any of us.\" She flicked her eyes around the circle. \"Maybe Kaitlin,\" she admitted.\n\nFran tossed an apple core viciously into the sty behind them. \"If _some people_ would stop making the beast with two backs with every farm boy in every blessed hay stack\u2014\"\n\nElissa's smile didn't even thin. \"Ahh, Fran, you'll go a virgin to your wedding, won't ya?\" She snorted.\n\nFran's next apple core hit Elissa in the nose and she hissed.\n\nMary shrugged. \"Scarcely matters if I bed 'em or don't,\" she said, \"seeing they say I did, and folks believe 'em.\"\n\nThe others nodded.\n\nElissa shrugged. \"Listen, the men-at-arms don't talk to the farmers. They don't know _shit_ about our lives. And even the archers\u2014\" She shrugged. \"The archers have more money than any _farm boy_ in this place. The men-at-arms\u2014\"\n\n\"They ain't all gents,\" Mary said. \"I wouldn't touch that Bad Tom if I had armour on.\"\n\nFran shrugged. \"I rather like him.\"\n\n\"You're dumber than I thought then. Aren't you supposed to be the smartest, fastest one? He gives me the creeps.\" Mary shivered.\n\nElissa raised a hand for silence. \"That's as may be. What I'm saying is that we\u2014\" She looked around. \"We have something. Of value.\" She smiled. The smile lit her face and turned her from a square jawed young harridan into a very attractive woman. Mary turned and saw that Elissa's smile was for a middle-aged squire just walking past the kitchen with a pail of ash. Off to polish armour somewhere.\n\nElissa folded up her smile and put it away. \"There's sixty men-at-arms,\" she said. \"Sixty chances one of them might marry one of us.\"\n\nMary snorted.\n\nBut Fran leaned forward, the apple in her hand forgotten. \"You might have something there,\" she said.\n\nElissa and Fran weren't usually allies. But Elissa met her look and both smiled.\n\n\"So we don't,\" Elissa said. \"We just don't. That's all you have to do, girls. _Don't_. Let's see what we're offered.\"\n\nMary wasn't so sure. \"So what. We don't bed them? What else do we do? You're planning to learn to shoot a bow? Go to Mag and take up fine sewing?\"\n\nElissa shook her head.\n\n\"Lis won't stop opening her legs for any likely lad,\" Mary said.\n\n\"Lis can do as she likes. She's old and we're not.\" Fran looked around. \"Captain's not bad looking.\"\n\nElissa made a crude noise. \"He's doing one of the nuns.\"\n\n\"He ain't!\" said Kaitlin. She'd been silent thus far, but some things couldn't be allowed to pass.\n\n\"Oh, you're an expert, are ya?\" asked Mary.\n\n\"I clean his room,\" Kaitlin said. She blushed. \"Sometimes.\"\n\nElissa looked at her. \"You, young maiden, are a dark horse.\"\n\n\"I ain't!\" Kaitlin said, prepared for their mockery.\n\n\"You go right in his room?\" Elissa asked.\n\n\"Almost every day.\" Kaitlin looked around. \"What?\"\n\nElissa shrugged. \"One of us could be in his bed.\"\n\nKaitlin put a hand to her mouth. Mary spat. Fran, frankly, looked as if she was considering it.\n\n\"Too desperate,\" Fran pronounced. \"He's scary, too.\"\n\n\"Creepy,\" said Mary.\n\n\"His squire's pretty as a picture,\" Elissa said.\n\nKaitlin blushed. Luckily the rest weren't watching.\n\n# North-west of Lissen Carak\u2014Thorn\n\nThorn needed to know more. He needed his friend in the Rock to be less coy. Thorn summoned birds from the air even as he ran through the woods in the failing light. Now he was climbing ridges. The descent on the north side was never as steep as the ascent had been, and he was going higher and higher into the mountains. The trees thinned, and he moved faster as the land opened up.\n\nA pair of ravens descended to his fists as if they were hawks to a knight. He spoke to them, planted messages in their wise heads, and sent them to the fortress. No one ever suspected ravens. They rose above him and then soared away to the south-east, and he turned and saw how very high he had come.\n\nHe looked out over the wilderness. At his feet\u2014far, far below\u2014was the chain of beaver ponds like miniature lakes sparkling in the last of the sun. The stream that connected them was a thread of silver, visible here and there in the warp and weft of trees.\n\nHe turned and climbed higher. The trail was steeper now, and he was not so fast. He had to use his long, powerful arms to pull himself from tree to tree. The stream began to descend in a series of waterfalls at his side.\n\nFinally, he pulled himself over a slick rock and raised himself by main force to the top, his arms spread wide, grunting with effort as they lifted the full weight of his giant body. At his feet was a pool, deep and black, and a waterfall dropped a hundred feet into it. The spray coated him in moments. He stooped and drank deep of the magic pool.\n\nA head broke the surface, just an arm's length away, and he started.\n\n_Who drinks in my pool?_\n\nThe words appeared in his mind without a sound being spoken.\n\n\"I am called Thorn,\" he said.\n\nThe creature rose from the pool, black water flowing from him. As he moved up the side of the pool he grew and grew. His skin was jet-black and shone like obsidian.\n\nHe moved fast yet appeared to be perfectly still; the transitions were difficult to catch, movement always seemed to happen at the corner of Thorn's eye. And when the creature fully emerged, he was a quarter taller than the sorcerer.\n\nA shining black stone golem, with no face, no eyes, no mouth.\n\n_I do not know you._\n\n\"I know a little of you,\" Thorn said. \"I know that I need allies. Your kind are said to be fearsome warriors.\"\n\n_I can feel your power. It is considerable._\n\n\"I can see your speed and strength. They, too, are considerable.\" Thorn nodded.\n\n_Enough talk. What do you WANT?_\n\nThe mind shout almost brought Thorn to his knees. \"I want a dozen of your kind as my guards. As soldiers.\"\n\nThe smooth monster threw back his head and laughed, and suddenly there was a mouth after all, with cruel teeth. The stone of his face\u2014if it was stone\u2014seemed to flow like water. _We serve no one._\n\nThorn would have smiled if he still had the ability to. Instead, he simply cast his binding. Simultaneously, he shielded his mind from the shout that was sure to follow.\n\nThe troll stiffened. He screamed, and his teeth clashed like rocks in a flooded stream, and his smooth arms grew hands and talons that reached for Thorn.\n\nThe sorcerer didn't stir. The net of his will settled in sparkling green strands over the creature and tightened, and that quickly it was over.\n\n_I will slay you and all your kind in ways too horrible for your mind to encompass._\n\nThorn turned. \"No you will not,\" he said. \"Now, obey. We have more of your kind to find, and a long night ahead of us.\"\n\nThe troll thrashed in his binding like a wolf in a cage. He screamed, his bell-like voice ringing across the wilderness.\n\nThorn shook his head minutely. \"Obey,\" he said again, and pushed a little more of his will into the binding.\n\nThe monster resisted, showing\u2014or growing\u2014wicked black in a black mouth. His whole body stretched for Thorn.\n\nTo Thorn, it was like arm wrestling with a child. A strong child\u2014but a child nonetheless. He slammed his will down on the troll's, and it crumbled.\n\nThat was the way of the Wild.\n\nThe other trolls weren't hard to find, and the second was considerably easier to _press_ than the first had been... but the seventh was much harder than the sixth, and by the time the sun had set he had a tail of mighty trolls and that sense a man gets when he has lifted so much weight that he can no longer lift his arms.\n\nHe sat in a narrow gully, and listened to the wind while his blank-faced trolls crouched all around him.\n\nAfter some time, as the sun began to slip beneath the rim of the world and he felt better, he reached out a tendril of his power toward the dark sun in the distant fortress.\n\nAnd he recoiled from what he found, because\u2014\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain was leaning on the wall, the curtain wall that covered the outer gate. He'd walked here almost without volition, because the confines of the Commandery were suddenly too close and airless.\n\nHe'd written her a note. Because he was not fifteen he had written one, not ten of them, and he'd placed it in the crotch of the old apple tree. And then, after cursing himself for waiting and hoping she might appear by some sympathetic magic, he'd walked to the wall for some air.\n\nThe stars burned in the distant heavens, and there were fires in the Bridge Castle courtyard below him. The Lower Town at the foot of the ridge was empty\u2014a skeleton guard held it and no more. And there was no light.\n\nHe looked out at the darkness\u2014the Wild was as dark as the sea.\n\nSomething was looking for him. At first it was a prickle in his hair, and then a presentiment of doom, and then, suddenly, he'd never felt so vulnerable in all his life, and he crouched on the battlement fighting a particularly awful childhood memory.\n\nWhen it didn't relent, didn't let up, he took a deep breath and forced himself to his feet. He turned and made himself walk, despite the crushing fear, up the steps set into the wall to the first tower. The second step was so hard he had to use his hands on the fourth and fifth\u2014by the eighth he was crawling. He pushed, made a sword of his will, and pushed through. The feeling relaxed like the grip of an unwelcome suitor as soon as he entered the stone structure.\n\nBent leaped to his feet, a deck of painted cards in his hand. \"Captain!\" he shouted, and a dozen archers leaped to their feet and snapped their salutes.\n\nThe captain glanced around. \"At ease,\" he said. \"Who's on the walls?\"\n\n\"Acrobat,\" Bent answered. \"Half-Arse on the main curtain, Ser Guillam Longsword and Snot commanding the towers with the engines. Watch changes in a glass.\"\n\n\"Double up,\" the captain ordered. He wanted to apologise\u2014 _Sorry, boys, I have a creepy feeling, so I'm costing a lot of you a night's sleep._ But he'd learned not to apologise when he gave an unpopular order, much less over-explain it. And the successful raid had given him credit in the hard currency of leadership\u2014no commander is ever much better than his last performance.\n\nBent grimaced, but he started lacing up his embroidered leather jack. Like many of the other veterans, Bent wore his fortune on his body\u2014a subtle brag, a statement of his worth, a willingness to see that fortune taken by his killer. The dark-skinned man looked around, and like true soldiers his fellow gamblers avoided his eye.\n\n\"Hetty, Crank, Larkin, with me. Hetty, if you don't want the duty, don't be so obvious about sneaking to the jakes.\" Bent glared at the youngest man in the tower room and then turned back to the captain. \"That sufficient, m'lord?\"\n\nThe captain didn't know Bent very well\u2014he was Ser Jehannes's man\u2014but he was impressed that his most senior archer would take the trick on the wall himself. \"Carry on,\" he said coldly, and walked across the room surveying the piles of coins on the tables, and the dice and cards, as he did. He was pretty sure Ser Hugo would never have allowed such overt gambling. So he scratched his beard and beckoned to Bent.\n\nThe archer came up like a dog expecting a kick.\n\nThe captain pointed at the money on the main table. He didn't say a thing.\n\nBent raised an eyebrow and opened his mouth.\n\n\"Save it,\" the captain said. \"Remind me of the company rule on gambling.\"\n\nBent made a face. \"Total value of the game not to exceed a day's pay for the lowest man,\" he recited.\n\nTwo rose nobles gleamed up at the captain, with more than a dozen silver leopards and a pile of copper cats by them. The captain put his hand over the pile. \"Must be mine then,\" he said, \"I'm the only man in the company who makes this kind of money every day.\"\n\nBent swallowed but his eyes narrowed in anger.\n\nThe captain lifted his hand, leaving the pile untouched. He locked eyes with the archer and smiled. \"You get me, Bent?\"\n\nThe archer all but sighed with relief. \"Aye, Captain.\"\n\nThe captain nodded. \"Good night, Bent,\" he said, and touched the man's shoulder, to say, _And over is over, unless you dick up._ He'd learned from experts, and he wanted to believe he was doing the captain thing well.\n\nHe walked out onto the wall, and there it was again\u2014not the fear, but the feeling he was being watched. Scrutinized. He was ready for it this time, and he reached into the round room, and\u2014\n\n_\u2014there was Prudentia._\n\n_\"He is looking for you,\" she said. \"His name is Thorn. A Power of the Wild. Do you remember how to avoid being found?\"_\n\n_He stopped to kiss her hand._\n\n_\"How do you know it is this Thorn?\" he asked._\n\n_\"He has a signature, and he has cast many times tonight, gathering allies. If you would pay attention to the Aethereal, instead of dabbling\u2014\"_\n\n_He smiled. \"I'm not interested. Too much like hard work.\"_\n\n_The door was open a crack. He often left it that way to give himself fast access to power, and tonight he could feel that searching presence through the crack in the door\u2014more powerfully, if anything, than he had felt it on the wall._\n\n_Of course._\n\n_He continued past Prudentia and pushed the door firmly shut. The heavy iron latch fell into place with a comforting click._\n\n# North-west of Lissen Carak\u2014Thorn\n\n\u2014the dark sun went out like a torch thrown into a pool.\n\nHe was disoriented, at first. The dark sun had dimmed and strengthened, dimmed and strengthened, and years of patient growth of power had taught him not to read too much into the fluctuations in power wrought by distance, weather, old phantasms that lingered like ghosts of their former powers, or animals who used power the way bats used sound. There were thousands of natural factors that occluded power the way other factors might affect sound.\n\nIn fact, he thought that the use of power and the movement of sound might usefully be studied together. The thought pleased him, and he spun off a part of himself to contemplate the movement of sound over distance as an allegory\u2014or even as a direct expression\u2014of power. Meanwhile, he sat and breathed in the night air and maintained, almost without effort, the chains of power that bound the trolls, and a third part of him looked for the dark sun with increasing frustration.\n\nA fourth aspect considered his next move.\n\nThe conflict at the Rock had now forced a gathering of resources and allies that involved risks and challenges he had not anticipated. If he continued gathering, he would soon reach a level that would at least appear to challenge his peers\u2014already the mighty Wyrm of the Green Hills was awake to him, raising, as it were, a scaled green eyebrow at his speedy accumulation of power and lesser creatures, men and resources. The old bear in the mountains did not love him either. And at some future point the trolls, scarcely people and more like cruel animals as they were\u2014would come to resent his chains and find a way to throw them off.\n\nSo might one feudal lord be alarmed\u2014or at least deeply curious\u2014when a neighbour called in his vassals and raised an army.\n\nThe allegory occurred to the fourth self, because the fourth self had once been a man\u2014a man capable of raising armies.\n\nBefore he'd learned the truth.\n\nAnd at another level, the fortress was obviously not going to crumple at his command. Albinkirk's outer wall had fallen so easily that he'd allowed himself to be seduced by the easy victory\u2014but the citadel itself, full of terrified humans, was not yet under his claws and the easy conquests were over.\n\nAnd whatever the dark sun was, it was powerful and dangerous, and the men of violence who surrounded it were deadly enemies who he would not underestimate again. Neither could he accept their pollution of his land, attack of his camp, or the preceeding endless cycle of challenge and counter challenge that had led him to grant a favour and confront the fortress more directly.\n\nAnd where, exactly, was his professed friend in the Rock?\n\nEnough.\n\nHe had made his choices, and they led to making war. Now he had to marshal his assets without affronting his peers, rip the fortress from the face of the world\u2014a warning and a tale for all his foes\u2014and grant the Rock to the Wild.\n\nAnd all the while he contemplated this next move, that part of him which was enjoying the cool of night continued to avoid the golden light cast by the Abbess, as if the mere admission of her existence would be a defeat.\n\nTwenty leagues to the south a hundred of his creatures stirred and rumbled and slept in the cold darkness, and two hundred men huddled close to their fires and posted too many sentries, and over the mountain to the north, hundreds of Sossag warriors woke and made their fires and prepared to come to his cause. And west, and north, creatures woke in their burrows, their caves, their holes and hides and homes\u2014more irks, more boglins, and mightier creatures\u2014a clan of daemons, a moiety of golden bears. And because power called to power, they were coming to him.\n\nThe trolls would counter the knights. The Sossag would give him more reliable scouts. The irks and boglins were his foot soldiers. By morning, he would have a force to deal with anything that humankind could offer. Then he would close his claws around the fortress.\n\nOf course, there was irony in his trust of men, rather than creatures of the Wild, to fight other men.\n\nWith this decision his selves collapsed, one by one, back into the body under the tree, and that body stretched, sighed, and was almost like a man's.\n\nAlmost.\n\n# Lissen Carak\u2014Kaitlin Lanthorn\n\nKaitlin sighed, and half rolled against the figure beside her. She sighed again, and wondered why her sister had to take up so much of the bed... and then she suddenly knew where she was, and she made a noise in her throat. The man next to her turned and put a hand on her breast, and she smiled. And then moaned a little.\n\nHe licked her under her chin and kissed her, his tongue dabbing away at the corner of her mouth like a questing thing, and she laughed and threw her arms around him. She was not a slut like her sisters, and she'd never had a man in her bed before in her life, but she was not going to be bound by their sordid plans or their poor taste. She was in love.\n\nHer lover ran his tongue across the base of her ear lobe while one lazy finger traced the line of her nipple. She laughed, and he laughed.\n\n\"I love you,\" Kaitlin Lanthorn said into the darkness. She had never said the words before, not even when he'd first had her maidenhead.\n\n\"And I love you, Kaitlin,\" Michael said, and put his mouth over hers.\n\n# Chapter Eight\n\n# North of Albinkirk\u2014Peter\n\nPeter the cook's first thought was _I am still a free man._\n\nHe had walked for two days along the road east, and he hadn't seen another man. Yesterday noon he'd smelt smoke, and seen the fortress rising to the south\u2014a fortress he had to assume was the town of Albinkirk, although his appreciation of the landscape of western Alba was virtually non-existent and he had only the comments of his captors and the mercenaries to go by.\n\nThe two traders must have travelled some way to the east, then. And he was virtually back where he'd begun.\n\nOr he was walking in circles.\n\nBut the sight of Albinkirk, five leagues or so distant, oppressed him. It was, after all, the place they'd been taking him to be sold. So he turned up the first decent path leading north, into the mountains, and he followed it even though setting his feet to it required an act of courage and marked his first decision.\n\nHe was not going back.\n\nHe'd rid himself of his yoke. It was easier than he'd expected\u2014given time and some large rocks, he'd simply bashed it to flinders.\n\nLike all slaves, and many other men, he had heard that there were people who lived at peace with the Wild. Even at home, there were those who did more\u2014\n\nBest not to think of those people. Those who sold their souls.\n\nHe didn't want to think about it. But he went north, the axe over his shoulder, and he walked until darkness fell, passing a dozen abandoned farmsteads and stealing enough food that he couldn't easily carry more. He found a good bow, although it had no arrows and no quiver. It was odd, going into the abandoned cabins along the trail\u2014in some, the steaders had carefully folded everything away; chests full of blankets, plate rails full of green glazed plates from over the eastern mountains, Morean plates and cups and a little pewter. He didn't bother to steal any of these, except a good horn cup he found on a chimney mantel.\n\nIn other houses, there was still food on the table, the meat rotting, the bread stale. The first time he found a meal on the table he ate it, and later he burped and burped until he threw it up.\n\nBy the twelfth cabin, he'd stopped being careful.\n\nHe went into the barn, and there was a sow. She'd been left because she was heavy, gravid, and the farmer was too soft hearted\u2014or just too pragmatic\u2014to try and drive her to Albinkirk in her condition.\n\nHe was wondering if he was hard-hearted enough to butcher her when he heard the barn's main door give a squeak.\n\nHe saw the Wild creature enter. It was naked, bright red, its parody of hair a shocking tongue of flame. It had an arrow on its bow, the iron tip winked with steel malevolence, and it was pointed at Peter's chest.\n\nPeter nodded. His throat had closed. He fought down his gorge, and the shaking of his arms, and managed to say, \"Hello.\"\n\nThe red thing wrinkled its lips as if he smelled something bad and Peter's perspective shifted. _It_ was actually a man in red paint, with his hair full of some sort of red mud.\n\nPeter turned slightly to face the man. He held up his empty hands. \"I will _not_ be a slave,\" he said.\n\nThe red man raised his head and literally looked down his nose at Peter, who shivered. The arrow, at full draw, didn't waver.\n\n\"Ti natack onah!\" the red man said in a tone full of authority. A human voice.\n\n\"I don't understand,\" Peter said. His voice trembled. The red man was obviously a war-leader of some sort, which meant that there were others around. Whatever kind of men they were, they were not what Peter had expected. They raised his hopes, and dashed them.\n\n\"Ti natack onah!\" the man said again, with increasing insistence. \"TI NATACK ONAH.\"\n\nPeter put his hands up in the air. \"I surrender!\" he said.\n\nThe red man loosed his arrow.\n\nIt passed Peter, missing him by the width of his arm, and Peter felt his bowels flip over. He crouched, his knees going out from under him, and he put his arms around himself, cursing his own weakness. _And so quickly I am a slave again._\n\nBehind him there was a scream.\n\nThe red man had put an arrow into the sow's head, and she twitched a few times and was dead.\n\nSuddenly the barn was full of painted men\u2014red, red and black, black with white handprints, black with a skull face. They were terrifying and they moved with a liquid, muscular grace that was worse than all his imaginings of the creatures of the Wild. As he watched, they butchered the sow and her unborn piglets, and he was pulled from the barn\u2014roughly, but without malice\u2014and the red man lit a torch from a very ordinary looking fire-kit and set fire to the shingles of the barn.\n\nIt lit off like an alchemical display, despite weeks of rain.\n\nMore of the warriors came, and then more\u2014perhaps fifty arrived within the hour. They passed through the cabin, and when the roof fell into the raging inferno of the barn, they gathered half-burned boards into a smaller fire, and then another and another, until they had a fire that ran the length of the small cabin, and then staked the unborn piglets on green alder and some iron stakes they found in the cabin and roasted them. Other men found the underground storage cellar, and pushed dried corn into the coals, and apples\u2014hundred of apples.\n\nBy now there were a hundred painted men and women, and darkness was falling. Most had a bow and arrows; a few had a long knife, or a sword, or even, in one case, a pair of swords. A few had long falls of hair in brilliant colours, but most had a single stripe of hair atop their heads, and another across their genitals. They looked odd to him, but it was only after his brain began to grow accustomed to them that he realised that they had no fat on their frames.\n\nNo fat.\n\nLike slaves.\n\nNo one was watching him. He was no threat, but he was also no use. He had a dozen opportunities to run, and he went as far as the edge of the clearing, where the farmer had been hacking down trees older than his own grandfather to make room for his crops. Then he stopped, lay on the low branch of an apple tree, and watched.\n\nBefore the ruddy sun was gone from the sky, he stripped off his hose and his braes\u2014cheap, dirty, torn cloth\u2014and walked back among them in his shirt. A few of them had shirts of deerskin or linen, and he hoped that he was making a statement.\n\nHe still had the wallet hanging over his shoulder, and the axe.\n\nAnd the bow.\n\nHe came and stood near the barn-fire, feeling the warmth, and his stomach did somersaults as he smelled the burning pig-flesh.\n\nOne of the painted people had set fire to the cabin, and there was laughter. Another painted man had burned himself stealing pinches of pig's flesh from the sow's carcass, and all the warriors around him were laughing like daemons.\n\nIf there was a signal he didn't hear it. But suddenly, they all fell on the piglets as if a dining bell had sounded, and ate. It was like watching animals eat. There were few sounds besides those of chewing, ripping flesh from bone, punctuated by the spitting out of burned bits and cartilage and the continuing sound of laughter.\n\nIf it hadn't been for the laughter, it might have been nightmarish. But the laughter was warm and human, and Peter found he had stepped closer and closer to the fires, drawn by the smell of food and the sound of laughter.\n\nThe red man was close to him. Suddenly, their eyes met, and the red man gave him the flash of a grin\u2014almost a grimace\u2014and waved a rib at him.\n\n\"Dodeck?\" he asked. \"Gaerleon?\"\n\nThe other warriors close by turned and looked at Peter.\n\nOne man\u2014taller than most, painted an oily black, with oily black hair and a single slash of red across his face\u2014turned and grinned. \"You want to eat?\" he said. \"Skadai asks you.\"\n\nPeter took another step forward. He was intensely conscious of his legs and neck and face, naked and very different from theirs.\n\nThe red man\u2014Skadai\u2014waved to him. \"Eat!\" he said.\n\nAnother warrior laughed and said something in the alien tongue, and Skadai laughed. And the black warrior laughed too.\n\n\"Was it your pig?\" the black warrior asked.\n\nPeter shook his head. \"No,\" he said. \"I was just passing through.\"\n\nThe black warrior appeared to translate this to his friends, and handed him a portion of pig.\n\nHe ate it. He ate too fast, burning his hands on the skin and his tongue on the meat and fat.\n\nThe black warrior handed him a gourd that proved to be full of wine. Peter drank, sputtered and then handed it back. The burns on his hands suddenly hurt.\n\nThey were all watching him.\n\n\"I was a slave,\" he said suddenly. As if they could understand. \"I won't be a slave again. I'd rather be dead. I won't be a slave for you.\" He took a deep breath. \"But short of slavery, I'd like to join you.\"\n\nThe black warrior nodded. \"I was a slave, too,\" he said. He smiled wryly. \"Well\u2014of sorts.\"\n\nIn the morning, they were up with the first tendrils of dawn, moving down the narrow road that Peter had climbed the day before. They moved in complete silence, their only communication via whistles and birdcalls. Peter attached himself to the black warrior, who called himself Ota Qwan. Ota Qwan followed Skadai, who seemed, as far as Peter could tell, to be the captain. Not that he issued any orders.\n\nNo one spoke to Peter, but then, no one spoke much anyway, so he focused on trying to move the way they did. He was not a noisy man in the woods, and no one cautioned him\u2014he followed Ota Qwan as best he could, across an alder swamp, up a low ridge dotted with stands of birch, and then west along a deer trail through beech highlands, with a lake stretching away to the right of the long ridge and the great river stretched out to the left.\n\nThey moved across the face of the wilderness, sometimes following trails and sometimes following the flow of the terrain, and their paths gradually made sense to Peter\u2014they were following a relatively straight course west, and avoiding the river. He had no sense of how many of them there were, even when they made camp, because that night, they simply stopped and lay down, all nestled tightly in a tangle of bodies and limbs. No one had a blanket, few had shirts, and the night was cold. Peter found he despised the touch of another against him, front and back, but his revulsion was quickly forgotten in sleep.\n\nIn the rainy grey of not-quite-morning, he offered some very stale bread from his sack to Ota Qwan, who accepted it with gratitude, took a small bite, and handed the crust on. Men watched it eagerly, but no one protested when the bread ran out before the mouths did. Peter hadn't even had a bite. He had expected it to be handed back. But he shrugged.\n\nOn his second night with the painted people he couldn't sleep at all. It rained softly, and the sensation of wet flesh\u2014paint, grit, and a man's naked thigh against his own\u2014made him get up shiver alone. Eventually he crept back into the pile of bodies, disgusted but almost frozen.\n\nThe next day was agony. The whole group moved faster, running the length of a grass meadow curiously criss-crossed by an arterial profusion of canals the width of a man's outstretched arm. The painted people leaped them with ease, but Peter fell into several, and always received a hand out and a belly laugh for his troubles.\n\nThe painted people wore supple, thin leather shoes, often the same colour as their paint so that he hadn't noticed them at first. His cheap slave shoes were falling to pieces, and the great meadow was littered with sharp sticks pointing up from the ground. He hurt his feet a dozen times, and again, he was helped along and laughed at.\n\nHe was limping badly, exhausted, utterly unaware of his surroundings, so when Ota Qwan stopped Peter all but walked over him.\n\nJust the length of a horse in front of them stood a creature straight from nightmare\u2014a beautiful monster as tall as a plough horse, and as heavy, with a crested head like a helmeted angel, a raptor's beak and blank eyes grey, the colour of new-wrought iron. It had wings\u2014small, but heart-breakingly beautiful.\n\nPeter couldn't even look at it, because for the third time in as many days he was terrified beyond his ability to think.\n\nOta Qwan put a reassuring hand on his shoulder.\n\nSkadai raised a hand. \"Lambo!\" he said.\n\nThe monster grunted, and raised a taloned claw-hand.\n\nPeter had time to note that its left claw was wrapped in linen, the way an injured man's hand would be wrapped in bandages.\n\nThen the monster grunted again\u2014if it spoke, the tones were too deep for Peter to understand\u2014and then it was gone into the underbrush. Skadai turned and raised his bow. \"Gots onah!\" he shouted.\n\nThere was an answering roar from all around them, and Peter was staggered to discover that there were dozens\u2014perhaps hundreds\u2014of painted warriors around him.\n\nHe grabbed at Ota Qwan. \"What\u2014what was that?\" he asked.\n\nOta Qwan gave him a wry smile. \"That was what men call an adversarius,\" he said. \"A warden of the Wild.\" He eyed Peter for a moment. \"A daemon, little man. Still want to be one of us?\"\n\nPeter took a breath but it was hard. His throat was closed again.\n\nOta Qwan put an arm around his shoulder. \"Tonight we'll be in a regular camp. Maybe we can talk. You must have questions. I know a little.\" He shrugged. \"I love living with the Sossag. I am one. I would never go back, not even to be a belted earl.\" The black painted man shrugged. \"But it ain't for everyone. And the Sossag are Free People. If you don't want to continue with them, well, just walk away. The Wild might kill you, but the Sossag won't.\"\n\n\"Free People?\" Peter asked. He'd heard it said before.\n\n\"You have a lot to learn.\" Ota Qwan smacked his shoulder. \"Move now. Talk later.\"\n\n# Dormling\u2014Hector Lachlan\n\nHector Lachlan walked into the courtyard of the great inn at Dormling like a prince coming into his kingdom, and men came out to stare, even applaud. The Keeper came in person, and shook his hand.\n\n\"How many head?\" he asked.\n\nLachlan grinned. \"Two thousand, six hundred and eleven,\" he said. \"Mind you, master, that includes the goats, and I'm not so very fond of goats.\"\n\nThe Keeper of Dormling\u2014a title as noble and powerful as any in the south, for all it belonged to a big bald man in an apron\u2014clapped Lachlan on the back. \"We've expected you a ten-day. Your cousin's here to join you. He says it's bad to the south.\" He added, \"We were afraid you might be broke, or dead.\"\n\nLachlan accepted the cup of wine that the Keeper's own daughter pressed into his hand. He raised it to her. \"I drink to you, lass,\" he said.\n\nShe blushed.\n\nHector turned back to the Keeper. \"The Hills are empty,\" he said, \"which trouble in the south explains. How far south? Is it the king?\"\n\nThe Keeper shook his head.\n\n\"Your cousin told me that Albinkirk was afire,\" he said. \"But come in and sit, and bring your men. The pens are ready, even for twenty-six hundred and eleven beasts. And I'm eager to buy\u2014if I serve you a steak tonight, Hector Lachlan, you'll have to sell me the cow first. I'm that short.\"\n\nThe staff of the Dormling Inn descended on the drovers like an avenging army, carrying trays of leather tankards of strong ale and mountains of soft bread and sharp cheese. By the time the youngest, dustiest drover far in the rear had been served his welcome cup and his bread, their lord was clean of the mud of the trail, sitting in a room as fine as most lord's halls, looking at a new tapestry from the East and smiling at the back of a local woman\u2014a grown woman with a mind of her own, as he'd just discovered. He rubbed his bicep where she'd pinched him hard as a land crab and laughed.\n\n\"Cawnor tried to levy a toll on me,\" he went on.\n\nThe Keeper and the rest of the audience shook their heads.\n\nThe drover shrugged. \"So we opened the road. I doubt there's enough of his men left alive to hold their fort, should someone decide to take it from them now.\" Drovers never sought to hold land. Drovers drove.\n\nHis cousin Ranald pushed through the crowd.\n\n\"You're taller!\" Hector said, and crushed him in an embrace. Then he sank back into his chair and took a long draught of ale. \"Albinkirk afire? That's ill news. What of the fair?\"\n\nRanald shook his head. \"I was moving fast. I kept moving. I was already on the east side when I reached Fifth Bridge, so I stayed there and rode cross country.\" He shrugged. \"I didn't see a thing.\"\n\nThe Keeper shrugged. \"If I'd thought you'd be here today, I'd have held the two bloody peddlers,\" he said. \"They claimed they'd been part of a merchant convoy headed west from Theva. Lost all their goods and slaves.\"\n\nHector nodded. \"Like enough.\" This was the time of year for the great convoys.\n\n\"A pair of Moreans. Said it was an ambush. Whole convoy destroyed.\" The Keeper shrugged. \"My sons say there was a good sized Theva convoy a ten-day back as well, on the south road, so they didn't pass by here.\" He shrugged again. \"I have no faith in Moreans, but they had no reason to lie either.\"\n\n\"Ambushed by what?\" Hector asked.\n\n\"They couldn't agree,\" the landlord said.\n\n\"They said the Wild,\" said a bold young farmer, a frequent customer, and a suitor for one of the Keeper's daughters. \"Or, leastways, the younger one said the Wild.\"\n\nThe Keeper shrugged again. \"That's true. Some of them said it was the Wild.\"\n\nHector nodded slowly. \"I certainly didn't see an animal bigger than a dog the whole trip here,\" he said. He shook his head in weary disgust. \"The Wild is set against Albinkirk? Where's the _king_ ? His people eat my cattle too.\"\n\nThe Keeper sighed. \"I don't know, and that's a fact,\" he said. \"I've put two of my sons and a dozen men on fast horses out to get ye news. We'll see what they have. Folk have spied Outwallers in the woods. Sassogs. I think that if they were really there, they'd have been seen and et alive\u2014but then I'm a suspicious bastard.\"\n\nHector took a deep breath. \"So it's war then.\"\n\nThe Keeper looked away. \"I hope not.\"\n\nHector took another pull of ale. \"Hope in one hand and shit in the other and see which one smells most. How long until you hear from your fast horsemen?\"\n\n\"Tomorrow,\" the Keeper said.\n\n\"Assuming the Outwallers don't eat them.\" Hector kicked his sword out in front of him to make room for his legs and sat back, tipping his chair against the wall. \"By the five wounds of Christ, Keeper. This will be an adventure to remember then\u2014taking the drove into an army of the Wild. Not even my da did the like.\"\n\n\"Waste of courage and arrows, though, if the fair ain't happening,\" the Keeper said. \"Lissen Carak may be so many burned cots and splintered stones when you get there.\"\n\nHector thumped his chair back down. \"Truth in what you say,\" he said. \"And no use pondering it until I know more.\" He looked around at the dozen men in the room. \"But I have a real harper and a dozen other players in my tail\u2014and unless Dormling's fallen on hard times, I wager a golden noble to a copper cat we can have us some fine music and dancing to rival the fairies tonight. So enough talk of war. Let's have wine and music.\"\n\nIn the far doorway, the tall serving woman tapped her foot and nodded approvingly.\n\nThe Keeper's youngest daughter clapped her hands. \"Now that's why you're the Prince of Drovers,\" she said approvingly. \"To Hector, Prince of the Green Hills!\"\n\nHector Lachlan frowned. \"The Green Hills have no lord but the Wyrm of Erch,\" he said. \"The dragon will have no rival, and can hear all that's said by men, so let's not be naming me to the lordship of any hills\u2014eh, Keeper?\"\n\nThe Keeper took a long pull of his own ale and put an arm around his daughter's shoulders, and said \"Honey, you know never to speak so. The Wyrm is no friend of man\u2014but he's no foe to us, as long as we stay clear of him and keep the sheepfolds where he commands. Eh?\"\n\nShe burst into tears and fled the room with every eye on her, and then the moment passed and the woman in the doorway clapped her hands. \"Bother the Wyrm!\" she said boldly. \"I want the harper!'\n\n# Harndon Palace\u2014Desiderata\n\nDesiderata lay back on the daybed in her solar, wearing only a long shift of sheer linen and silk hose with red leather garters. Her nurse clucked disapprovingly at her mistress's d\u00e9shabill\u00e9 and began the herculean task of gathering up her shoes.\n\nDesiderata had a scroll, a day book, and a lead pencil encased in silver with her, and she was writing furiously. \"Why don't they build all the cart wheels to the same size?\" she asked.\n\nDiota made a face. \"Because wheelwrights don't share their measures, mistress.\"\n\nDesiderata sat up. \"Really?\"\n\nDiota clucked, looking for a second damask slipper. She found it under the daybed. \"Every wheelwright builds to their own set of sizes\u2014usually given 'em by their father or grandfather. Some build cartbeds to the width of the narrowest bridge\u2014I grew up in the mountain country, and the Bridge of Orchids was the narrowest lane in the baillie. No carter would build a cart wider that that, and no wheelwright\u2014\"\n\nDesiderata made an impatient noise. \"I take your point. But military carts\u2014\" She shook her head. \"There are no military carts. We have vassals who give cart service. They hold their cottages and farms in exchange for providing a cart and a driver. Can you imagine anything clumsier? And when their cart breaks down, it's the king's problem.\" She chewed on the silver pencil. \"He needs a professional train. Carts built for war, with carters paid a wage.\" She scribbled furiously.\n\n\"I imagine it costs too much, my lady,\" Diota said.\n\nDesiderata shook her head. \"You know what it costs to repair the wheels on one cart? War does not need to be this expensive.\"\n\n\"You make me laugh, my lady,\" Diota said. She had found both of the red calfskin slippers\u2014a miracle in itself\u2014and she was putting the whole collection all on shoe forms to keep them stretched.\n\nDesiderata gave her nurse the sort of smile that squires at court fought to gain. \"I make you laugh, my sweet?\"\n\n\"You are the Queen of Beauty, with your head full of romance and starshine, and now you're organising his supply train.\" Diota shook her head.\n\n\"Without forage and fodder, a knight and his horse are worthless,\" Desiderata said. \"If we want them to win glory, they must be fed.\" She laughed. \"You think my head is full of starshine, nurse\u2014look inside a young man's head. I wager that half of the young louts who try to look down my dress and fight to kiss my hand will ride off to do great deeds without even a nose bag for their chargers. Without an oiled rag to touch up their sword blades. Without a sharpening stone or a fire kit.\" She tossed her head to move her mane of hair. \"I've watched knights my _whole life._ Half of them are good fighters\u2014fewer than a tenth make even marginally competent soldiers.\"\n\nDiota made a face. \"Men. What more need be said?\"\n\nDesiderata laughed. She picked up a second scroll. \"I'm moving forward with the plans for the tournament. The king will have almost the full muster of knighthood together anyway, so I'll move the date by a month\u2014the fourth Sunday in Pentecost isn't a bad time for a big show. The planting will be done and only the haying will be in.\"\n\n\"Fourth Sunday\u2014Lorica Cattle Fair,\" said Diota.\n\nDesiderata sighed. \"Of course.\" She made a face. \"Drat.\"\n\n\"Have your tournament at Lorica instead.\"\n\n\"Hmmm,\" Desiderata mused. \"Very good for the town\u2014good for our relations there, as they'll rake in a profit. And I understand my husband had to make some concession there.\"\n\n\"Because your perfect knight burned the Two Lions,\" Diota spat. \"Foreign fuck!\"\n\n\"Nurse!\" Desiderata swung a pillow with great accuracy, catching her nurse in the back of the head with a soft tassel.\n\n\"He's a lout in armour.\"\n\n\"He's reputed to be the best knight in the world,\" breathed the Queen. \"You cannot judge him by the standards\u2014\"\n\n\"By the Good Christ,\" Diota said. \"If he's the best knight in the world, then he should _embody_ the standards.\"\n\nThey glared at each other. But Diota knew her duty. She smiled. \"I'm sure he is a great knight, my lady.\"\n\nThe Queen shook her head. \"I confess he lacks something,\" she admitted.\n\nDiota made a noise.\n\n\"Thank you, nurse. That will be quite enough. Despite your ill-mannered grumbling, I take your point\u2014no doubt the king does need something nice done for Lorica. Holding the tournament there\u2014if the timing is right, if the army is returning that road, and if the town fathers are in favour\u2014yes, it would do very nicely. And I would get my tournament.\" She rang a silver bell, and the door to the solar opened to admit her secretary, Lady Almspend, one of the few university-taught women in Alba.\n\n\"Two letters, if you please, Becca.\"\n\nLady Almspend curtsied, sat at the writing table, and produced a silver pen and ink from her purse.\n\n\"To the Mayor and Sheriff of Lorica, the Queen of Alba sends greeting\u2014\"\n\nShe dictated quickly and fluently, pausing while her secretary filled in titles and appropriate courtesies with equal fluency. It was the habit of kings and queens to employ scholars of repute as secretaries, as most of the nobility couldn't be bothered to learn the task and employed others to do any actual writing. But Rebecca Almspend managed to write fine poetry and research the works of the troubadours of the last two centuries, and still found time to do her job thoroughly.\n\n\"To his Alban Majesty, from your devoted, loving wife\u2014\"\n\nLady Almspend gave her an arch look.\n\n\"Oh, say what I mean, not what I say,\" Desiderata pouted.\n\n\"You Grace will forgive me if I suggest that sometimes your performance as a wilful beauty overshadows your obvious intelligence,\" Lady Almspend said.\n\nDesiderata let the nails of her right hand pass lightly down the back of her secretary's arm. \"Let my letter be coy, and let him gather how very brilliant I am by looking at the design of his new war carts,\" she said. \"Telling him how very clever I am will only cause him distress. Men, my dear Becca, are like that, and you will never attract a lover, not even a bespectacled merchant prince who adores your head for long columns of figures, if you wear wimples that hide your face and seek to prove to every lover that you are the smarter of the pair.\" The Queen knew perfectly well that her intellectual secretary had attracted the devotion of the strongest and most virile of the King's Guardsmen\u2014it had been something of a wonder at the court. Even the Queen was curious how it had happened.\n\nLady Almspend was perfectly still, and the Queen knew she was biting back a hot remark.\n\nThe Queen kissed her. \"Be at peace, Becca. In some ways, I am far more learned than you.\" She laughed. \"And I am the Queen.\"\n\nEven the staid Lady Almspend had to laugh at the truth of this. \"You are the Queen.\"\n\nLater, when giving justice, the Queen summoned two of the king's squires, and sent them with the letters\u2014one was delighted to go to the army, if only for a day or two, and the other, rather more dejected, riding to a merchant town to deliver a letter to a retired knight.\n\nThe Queen allowed them both to kiss her hand.\n\n# North of Harndon\u2014Harmodius\n\nHarmodius was on his second night without sleep. He tried not to think about how easily he'd done such things forty years before. Tonight, riding very slowly down the road on an exhausted horse, he could only hope to keep his hands on the saddle, hope that the horse didn't stumble, throw a shoe, or simply collapse beneath him.\n\nHe'd drained every reserve of energy. He'd set wards, thrown bolts, and built phantasmal dissuasions with the abandon of a much younger man. All his carefully hoarded powers were gone.\n\nIn a way, it had been marvellous.\n\nYoung magi have energy and old ones have skill. Somewhere in the continuum between young and old lies a practitioner's greatest moment. Harmodius had assumed his had been twenty years ago, and yet last night he'd thrown a curtain of fire five furlongs long\u2014and swept it ahead of his galloping horse like a daemonic plough blade.\n\n\"Heh,\" he said aloud.\n\nAn hour after he'd extinguished the fiery blade, he'd met a foreigner on an exhausted horse, who had watched him with wary eyes.\n\nHarmodius had reined in. \"What news?\" he asked.\n\n\"Albinkirk,\" the man breathed. He had a Morean accent. \"Only the castle holds. I must tell the king. The Wild has struck.\"\n\nHarmodius had stroked his beard. \"Dismount a moment, and allow me to send the king a message as well?\" he said. \"I'm the King's Magus,\" he added.\n\n\"Ser Alcaeus Comnena,\" the dark-visaged man replied. He swung his leg over the horse's rump.\n\nHarmodius had given him some sweet wine. He was pleased to see the foreign knight attend his horse\u2014rubbing the gelding down, checking his legs.\n\n\"How's the road?\" asked the knight.\n\nHarmodius permitted himself a moment of glowing satisfaction. \"I think you'll find it clear,\" he said. \"Alcaeus? You're the Emperor's cousin.\"\n\n\"I am,\" said the man.\n\n\"Strange meeting you here,\" Harmodius said. \"I've read some of your letters.\"\n\n\"I'm blushing, and you can't see it. You must be Lord Harmodius, and I've read everything you written about birds.\" He laughed, a little wildly. \"You're the only barba\u2014only foreigner whose High Archaic is ever read aloud at court.\"\n\nHarmodius had a werelight going, and was writing furiously. \"Yes?\" he asked absently.\n\n\"Although you haven't written a thing in five years, now? Ten?\" The younger knight had shaken his head. \"I am sorry, my lord. I had thought you dead.\"\n\n\"You weren't far wrong. Here\u2014deliver this to the king. I'm going north. Tell me\u2014did you see any Hermetics fighting against Albinkirk?\"\n\nSer Alcaeus had nodded. \"Something enormous came against the walls. It pulled the very stars from the sky, and threw them at the castle.\"\n\nThey had clasped hands.\n\n\"I long to meet you under more auspicious circumstances,\" Ser Alcaeus had said.\n\n\"And I you, ser.\"\n\nAnd with that, each had ridden off\u2014one north, and one south.\n\n_Who can pull stars from the sky and hurl them at castle walls?_ Harmodius asked himself, and worried that there was only one answer.\n\nThe last light of day had shown him smoke rising over Albinkirk, and if the town was gone then he was bereft of a plan.\n\nHis original impulse was all but gone. The evidence of the road, and Ser Alcaeus, was that a marauding army of the Wild had come down on the north of Alba, and he was afraid\u2014to the core of his chilled and weary bones\u2014that all the work of the old King Hawthor was undone. Worse, whatever had cast the ensorcellment on him was out there. With that army.\n\nAnd yet he hadn't pointed his horse's head back south. When he came to the road that turned west, into the woods, and saw fresh wagon ruts on it he turned his horse's head that way and followed them.\n\nPart of that was pragmatic. He'd fought three bands of the Wild to win through this much of the road to Albinkirk. He wasn't ready to fight a fourth.\n\nTwo hours later, somewhere in the darkness, a horse gave a long snort and then a soft whicker, and his horse answered.\n\nHarmodius sat up.\n\nHe let his horse stumble forward. The horses would find each other quicker than he could, and they rode on for long minutes. He stared with unaided eyes at the darkness that pressed in on the road like a living thing.\n\nThe other horse whinnied.\n\nHis horse gave a call, almost a mule's bray, in return.\n\n\"Halt! You on the road\u2014halt and dismount, or you'll have enough crossbow bolts in you to play a porcupine in a show.\" The voice was loud, shrill, and sounded very young, which made the speaker dangerous. Harmodius slid from his horse, knowing in his bones that he was unlikely to be able to remount. His knees hurt. His calves hurt. \"I'm off,\" he said.\n\nA bull's-eye lantern opened its baleful eye in front of him, the powerful oil lamp all but blinding him.\n\n\"Who are you, then?\" asked the annoying young voice.\n\n\"I'm the fucking King of Alba,\" Harmodius snapped. \"I'm an old man on a done horse and I'd love to share your fire, and if I was a horde of boglins you'd already be dead.\"\n\nThere were chortles from the darkness.\n\n\"There you are, Adrian. Put that weapon down, Henry. If he's riding a horse, he ain't a creature of the Wild. Eh? Did you think of that, boy? What's your name, old man?\" The new voice was authoritative without being noble. The bland accent of court was completely absent.\n\n\"I'm Harmodius Silva, the King's Magus.\" He walked forward into the lantern light, and his horse followed him, as eager for rest and food as his rider. \"And that's not a tall tale,\" he added.\n\n\"Sounds pretty tall,\" said the new voice. \"Come to the fire and have a cup of wine. Adrian, back to your duty, boy. Young Henry, if you point that weapon at me again I'll break your nose.\"\n\nThe man was in armour and had a heavy axe across his arms, but he stripped off a chain mitten and clasped Harmodius's hand. \"They call me Old Bob,\" he said. \"Man-at-arms to the great and near great,\" he laughed. \"You really Lord Silva?\"\n\n\"I truly am,\" said Harmodius. \"Do you really have a safe camp and wine? I'll pay a silver leopard to have a boy see to my horse.\"\n\nThe man-at-arms laughed. \"Long night?\"\n\n\"Three long nights. By the blood of Christ and his resurrection\u2014I've been fighting for three days.\"\n\nThey emerged into the circle of light from a big fire, and over the fire towered a heavy trestle that held the chains of three heavy cauldrons\u2014and a pair of lanterns hung from the cross bar. It was the strongest light he'd seen since sundown. By the candlelight he could see a dozen men crouched over something on the ground, and the tall wheel of a heavy wagon. And beyond that, another.\n\n\"You've reached Master Random's convoy,\" the man-at-arms said. \"Fifty wagons, or near enough, all the guilds of Harndon represented.\"\n\nHarmodius nodded. He'd never heard of Master Random, but then, as had become increasingly clear over the last three days, he'd been lost to the world for ten years or more.\n\n\"You're safe enough here,\" Old Bob said. \"Boglins ambushed us today,\" he said, and shrugged, clearly unhappy about it.\n\n\"You took losses?\" Harmodius was anxious to ask about the numbers and strength of the opposition, but his desire for information was at war with his fatigue.\n\n\"The young knight,\" Old Bob motioned with his great axe at the group of men gathered around something on the ground. \"He was badly wounded fighting a daemon out of the Wild.\"\n\nHarmodius sighed. \"Make way,\" he said.\n\nThey had a candle, and the horse leech was cleaning the man's wounds with vinegar. The young knight had lost a great deal of blood and, stripped naked, he looked pale and vulnerable. The new spring flies were feasting on him.\n\nHarmodius cast almost without thinking, putting a small banishment on the flies.\n\nFatigue, which had seemed like a coat of mail, suddenly clamped like a vice around his heart. But he knelt by the wounded man, and Old Bob held the lantern high.\n\nJust for a moment the wounded knight looked like the king.\n\nHarmodius bent closer, examining the wounds. Three punctures, some slashes\u2014nothing so deadly as to kill a healthy man, until he saw the burns. In the ruddy candle light, the man's eye looked like a pit of red.\n\n\"Sweet Jesu,\" the Magus said.\n\nThe odd pieces of dirt he'd seen on the man's shoulder was no dirt at all\u2014his chain links had burned right into his shoulder. The burns weren't even red, they were black.\n\n\"He faced down an adversarius,\" a man said. \"A daemon of hell. Even when it threw fire at him.\"\n\nHarmodius felt his eyes closing. He didn't have the power to save this brave soul which was frustrating, especially as he only needed a little organic power to stabilise the burns. It took much skill, much shaping of the phantasm, but little power.\n\nThe concept that the power needed was organic gave him a thought, though.\n\nHe touched his own reserves\u2014the items he'd carefully enchanted over the years\u2014drenched in sun power, imbued, impregnated, imprinted with the rich golden light of the Sacred Sun. All were cold and empty. And so was his greatest reservoir\u2014his own skin. Empty, cold, tired.\n\nAnd yet, by the logic of the experiment in his tower\u2014\n\n\"Everyone stand away,\" he ordered. He lacked the energy to explain; either this was going to work, or it was not. \"I'm exhausted,\" he said to the old soldier. \"You know what that means?\"\n\n\"Means ye can't heal. Eh?\" said Old Bob.\n\n\"Just so. I'm going to try tapping a local source. If I fail, nothing will happen. If I succeed\u2014\" Harmodius rubbed his eyes. \"By Hermes and all the saints, if I succeed, I think something will happen.\"\n\nOld Bob snorted. \"Are you always that clear?\" He held out a beaker. \"Drink first. Good red wine.\"\n\nHarmodius waved it away. The other men backed away, or fled to the fire\u2014no one fancied watching a wizard work except Old Bob, who looked on with a cat's wary curiosity.\n\nHarmodius reached out into the darkness until he found a pool of the green power that he knew had to be there. It wasn't far away. He didn't enquire what it was. He simply seized its power\u2014\n\n\u2014and the night exploded with shrieking.\n\nOne does not work with the ultimate powers of the universe for many years without developing a concentration bordering on utter ruthlessness. Harmodius focused on the power, which was difficult to lay hold of, difficult to _seize,_ and it tasted wrong, somehow. That wrongness would instantly have repelled him, had he not the scientific assurance of his earlier experiment that a creature of the Wild could interact with the Hermetic. The reverse had to be true.\n\nThe shrieking went on, and the men around him moved in a disciplined kind of panic\u2014seizing weapons, calming horses. Harmodius was aware of them, but not enough to break the iron chain of his connection to the distant green source of power, which he took like an infant sucking greedily at a breast.\n\nRuthlessly.\n\nAnd then it was in him, filling him with its odd, bitter, wintergreen tang, and there was far, far more than he needed for his small enchantment. But he worked with it, first a complex binding, then two simultaneous phantasms run by dividing himself into two working halves, as his master had taught him so long ago. And there was so much power that he could divide himself once again, to leave an awareness to watch the darkness. Taking the green energy, as he had, seemed to have kicked a beehive.\n\nA village witch could funnel power independently through each hand. Harmodius could use each finger as a channel, and could use other foci on his body\u2014rings and the like\u2014as reserves, or clamps.\n\nHe used many of them now.\n\nHis first use of it was to look into the burns. They were worse than he'd thought by the firelight\u2014the blackened skin was charred, and in some cases the damage ran all the way through the skin layer to the fat and muscle.\n\nThese were lethal burns.\n\nIndeed, the man was slipping away even as Harmodius worked to block the pain and heal the greatest wounds.\n\nHealing burns was the most difficult of all forms of healing and Harmodius, for all his power, was no healer. He juggled tendrils of power for a dozen heartbeats, attempting to rebuild charred tissue and in the process only charring more. The control required was incredible and in his frustration and fatigue, he let slip a greater packet of the green power than he intended. It was rolling through him in waves and he passed it straight into the young man's shoulder.\n\nHarmodius had heard of healing miracles, but he had never witnessed one before. Under his hand, a patch of skin the size of a bronze sequin healed. The angry marks that pulsed in Harmodius's enhanced vision simply faded and were gone.\n\nIt was incredible.\n\nHarmodius had no idea what he had done, but he was an empirical magus and so he reached for more power, drew it from its source like a man trying to haul in a great ocean fish with a light rod, and then pumped it through his hands into the swaths of burned flesh...\n\n... and they healed.\n\nA section of the knight's neck the size of the palm of a hand closed over and healed.\n\nHe reached for more power, seized it, struggled with the source and overcame it by main force of will, and then hauled with all his trained might, ripping the green power into his soul and then passing it down his hands into the knight, whose eyes suddenly opened with a great cry.\n\nHarmodius stumbled back.\n\nThe screams from the woodlands stopped.\n\n\"Why did you kill me?\" the young knight asked plaintively. \"I was so beautiful!\"\n\nHe slumped and his eyes closed.\n\nHarmodius reached out and touched him. He was asleep, and the skin on his neck, chest, back, and shoulders was flaking away, the blackness and scabs simply falling away from the new flesh underneath.\n\nNew, pale flesh.\n\nWith scales.\n\nHarmodius flinched, trying to understand what he had done.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain woke still tired. He got up, called for Toby, and stumbled to his wash basin.\n\nToby came in, chewing on a biscuit, and began to lay out his clothes. He moved warily, and the captain assumed from his averted head that something was wrong. Whatever it was, the captain would have to work it out for himself.\n\n\"What news, Toby?\" the captain asked.\n\n\"Boglins in the fields,\" the boy said, and went back to chewing.\n\n\"Where's Michael?\" the captain asked, when no one came to help him point his hose.\n\nToby looked away. \"At chapel, I reckon.\"\n\n\"Only if Jesu came and visited Michael in person in the night,\" the captain said. Mornings made him nasty. Toby wasn't to blame, but the boy idolized the squire and he wasn't going to rat him out.\n\nThe captain pointed his own hose, and took an old arming doublet and began to lace it up. He didn't call for Michael until he was ready to lace the cuffs. When the young man still wasn't there, he nodded to Toby. \"I'm going to go find him,\" he said.\n\nToby looked terrified. \"I'll go, master!\"\n\nThe captain felt annoyed. \"We can go together,\" he said, and his long legs took him out of the solar and down the hall to the Commandery where Michael slept.\n\nToby tried to beat him to the door, but a combination of shorter legs and deference kept him a stride behind.\n\nThe captain flung the heavy oak door open.\n\nMichael leaped from his bedroll, a long dagger in his right fist. He was naked. So was the beautiful young girl he put behind him.\n\n\"Michael?\" the captain said to the dagger.\n\nMichael blushed. The blush started just above his groin, ran in splotches over his chest and up his neck to his face. \"Oh my God\u2014my lord, I'm so sorry\u2014\"\n\nThe captain looked at the girl. Her blush was even brighter.\n\n\"That's my laundry maid, I believe,\" he said. He raised an eyebrow. \"Perhaps maid is the wrong word, given the circumstances.\"\n\nShe hid her head.\n\n\"Get dressed. Michael. It's full light, and when that poor young woman walks down the steps to the courtyard, every person in the fortress will know where she's been; either with you, with me, or with Toby. Perhaps with all three. Toby at least has the virtue of being her own age.\"\n\nMichael was trying to put his dagger away.\n\n\"I love her!\" he said hotly.\n\n\"Wonderful. That love is about to bring down a mountain of consequences that may end in your no longer being in my employ.\" The captain was angry.\n\n\"At least she's not a nun!\" Michael said.\n\nThat stopped the captain. And filled him with black rage; in a moment, he went from a distant, weary amusement to the flat desire to kill. He was struggling not to draw a weapon. Or use his fists. Or his power.\n\nMichael took a step back and Toby placed himself between the captain and the squire.\n\nHeavy, strong arms suddenly encircled the captain from behind. He thrashed, angry beyond sense, but he couldn't break the grip. He tried to plant his feet and headbutt his adversary, but the man lifted him straight off the floor.\n\n\"Whoa!\" said Bad Tom. \"Whoa there!\"\n\n\"His eyes are glowing!\" Michael said, and his voice was trembling, Kaitlin Lanthorn cowering in the corner.\n\nTom spun the captain and slapped him clear across the face.\n\nThere was a pause. The captain's power hung in the air\u2014palpable even to non-talents. Kaitlin Lanthorn saw it as a cloud of golden green around his head.\n\n\"Let go of me, Tom,\" said the captain.\n\nTom put his feet on the ground. \"What was that about?\"\n\n\"My idiot squire deflowered a local virgin, for sport.\" The captain took a deep breath.\n\n\"I love her!\" Michael shouted. Fear made his voice high and whiney.\n\n\"Like enough,\" Tom said. \"I love all the women I fuck, too.\" He grinned. \"She's just one of the Lanthorn sluts. No damage done.\"\n\nKaitlin burst into tears.\n\nThe captain shook his head. \"The Abbess\u2014\" he began.\n\nTom nodded. \"Aye. She won't take it well.\" He looked at Michael. \"I won't ask you what you were thinkin', 'cause I can guess it well enough.\"\n\n\"Get him out of my sight,\" the captain said. \"Toby, get the girl dressed and get her... I don't know. Can you get her out of here without everyone seeing?\"\n\nToby nodded soberly. \"Aye,\" he said, eager to help. Toby didn't like it when his heroes were angry, especially not with each other.\n\nThe captain had a splitting headache, and he wasn't even into the day yet.\n\n\"What are you doing here, anyway?\" he asked Tom.\n\n\"Sauce has a patrol out and there's the remnants of a convoy in the Bridge Castle,\" Tom said. \"Bad news.\"\n\nSauce reported an hour later, handing a child down off the saddlebow of her war horse and saluting her captain crisply.\n\n\"Twenty-three wagons. All burned. Sixty corpses found, not yet ripe, and not much of a fight.\" She shrugged. \"Slightly chewed.\" She lowered her voice, as there were dozens of people in earshot, all looking for news. \"Many eaten down to sinew and bone, Captain.\"\n\nThe captain fingered his beard, looked at the desperate people surrounding his horse, and knew that any morale won by his raids on the enemy camp was now dissipated in a fresh wave of terror.\n\n\"Back to your work,\" the captain called.\n\n\"We ain't got no work!\" a man shouted, and the crowd in the courtyard rumbled angrily.\n\nThe captain had mounted in anticipation of taking out a patrol. He was restless and depressed himself, and craved action\u2014anything to distract him.\n\nBut he was the captain. He nodded to Gelfred. \"Go north, and move fast. You know what we want.\"\n\nHe swung one spurred foot over Grendel's back and slid from the saddle. \"Wilful Murder, Sauce, on me. The rest of you\u2014well done. Get some rest.\"\n\nHe led them inside. Michael dismounted too, looking as furious as the captain felt having lost an opportunity to substitute honest fear for nagging terror. He clearly knew that he now had no opportunity to expiate his sin. But he took his own destrier and the captain's and headed for the stable without untoward comment.\n\nSister Miram\u2014the heaviest and thus most easily identified of the sisters\u2014was passing through the courtyard with a basket of sweet bread for the children. The captain caught her eye, and waved.\n\n\"The Abbess will want to hear this,\" he said to her. She put a biscuit in his hand with a look that might have curdled milk\u2014a look of blanket disapproval.\n\nThere was a slip of vellum underneath it.\n\n_Meet me tonight_\n\nA bolt of lightning shot through him.\n\nThe Abbess arrived while he was still standing in his solar. He'd just stripped off his gauntlets and placed them on the sideboard, his helmet was still on his head. Sauce took it from him, and he turned to find the Abbess, hands clasped loosely in front of her, wimple starched and perfect, eyes bright.\n\nThe captain had to smile, but she did not return it.\n\nHe sighed. \"We've lost another convoy coming to the fair\u2014six leagues to the west, on the Albinkirk road. More than sixty dead. The survivors are panicking your people, and they aren't helping mine much.\" He sighed. \"In among them are refugees from Albinkirk, which, I am sorry to report, has fallen to the Wild.\"\n\nTo Sauce, he said, \"In future, no matter how badly off they are, take new refugees to Ser Milus. Let him keep their ravings contained.\"\n\nSauce nodded. \"I should have thought\u2014\" she said wearily.\n\nThe captain cut her off. \"No, I should have thought of it, Sauce.\"\n\nWilful Murder shook his head. \"It's worse than you think, Captain. You're not from around here, eh?\"\n\nThe captain gave the archer a long look, and Wilful quailed.\n\n\"Sorry, ser,\" he said.\n\n\"It happens that I know the mountains to the north well enough,\" the captain said quietly.\n\nWilful was not so easily put down though. He produced something from his purse and put it on the table.\n\nThe Abbess turned as white as parchment when she saw it.\n\nThe captain raised an eyebrow.\n\n\"Abenacki,\" he said.\n\n\"Or Quost, or most likely Sassog.\" Wilful nodded respectfully. \"So you _are_ from around here.\"\n\n\"How many?\" the captain asked.\n\nWilful shook his head. \"At least one. What kind of question is that?\" The feather he had placed on the table\u2014a heron feather\u2014was decorated with elaborate quillwork from a porcupine, the quills dyed bright red and carefully woven up the stem of the feather.\n\nWilful looked around, and then, like a conjuror, produced a second item, very like the first in look\u2014a small pouch, decorated with complex leather braids. When his audience looked blank, he grinned his broken-toothed grin. \"Irks. Five feet of muscle and all of it mean. They make amazing stuff. Fey folk, my mother used to call them.\" He looked at the Abbess. \"They like to eat women.\"\n\n\"That's enough, Wilful.\"\n\n\"Just saying. And there was tracks.\" He shrugged.\n\n\"Nicely done, Wilful. Now give me some quiet.\" The captain pushed his chin towards the door.\n\nWilful might have been surly, but he found a silver leopard pushed across the table to him, too. He bit it, grinned and left.\n\nThe captain glanced at the Abbess as soon as they were alone. \"What's going on here?\" he asked in his pleasant but professional voice. \"This isn't the random violence of the Wild, an isolated incident, a murder, a couple of creatures come over the wall on a rampage. This is a war. Daemons, wyverns, irks and now the Outwallers. All we seem to lack is a few boglins, a goblin or two, and then maybe the Dragon will enter the field too. Abbess, if you know anything, I think this is the time to tell me.\"\n\nShe met his gaze. \"I can make some educated guesses,\" she said. Her lips curled down. \"I gather that the youngest Lanthorn girl spent the night here?\" she said archly.\n\n\"Yes she did. I raped her repeatedly and threw her naked into the courtyard in the morning,\" the captain said. His annoyance showed. \"Damn it, this matters.\"\n\n\"And Kaitlin Lanthorn doesn't? My Jesu says she matters as much as you do, ser knight. As much as I do. Perhaps more. And spare me your posturing, boy. I know why you're so touchy. She spent the night with your squire. I know. I have just spent a few minutes with the girl. We spoke about this.\" She looked at him. \"Will he marry her?\"\n\n\"You can't be serious,\" the captain said. \"He's the son of a great lord. He may be on the outs with his family just now, but they'll forgive him soon enough. His kind doesn't marry farm sluts.\"\n\n\"She was a virgin a few days ago,\" the Abbess said. \"Calling her a whore doesn't make her one. Nor does it make you stand any better in my sight.\"\n\n\"Fine,\" said the captain. \"She's a fine upstanding lass with impeccable morals and my nasty squire got her to bed. I'll see to it that he pays for it\u2014both morally and financially. Now can we please talk about the true threat here?\"\n\n\"Maybe we already are. So far, no creature of the Wild has done so much harm as your men have done,\" the Abbess said.\n\n\"Untrue, my lady. I swear on my word: I will see to it justice is done for this young woman. I confess that she looked quite unsluttish this morning, and very young. I am embarrassed my squire has acted in such a way.\"\n\n\"Like master, like man,\" the Abbess said.\n\nThe captain clenched his fists. He mastered himself, unclenched them, and steepled his hands instead.\n\n\"I think you are avoiding the topic. Sister Hawisia was murdered. Her murder was planned. Perhaps she was the target\u2014perhaps you were. The daemon that did the killing had inside help. The men who helped the daemon then fell out among themselves and one killed the other, burying his body on the west road. Shortly after, we arrived. We found a wyvern and killed it. Gelfred and I found a pair of daemons; one died and the other escaped. We scouted and found an army forming under a powerful sorcerer. As of this morning, the woods around us are full of enemies and the road to Albinkirk is cut. Albinkirk has fallen to the Wild, and I put it to you, my lady, that you know more than you are telling me. What is really going on here?\"\n\nShe turned her head away. \"I know nothing,\" she said, in a tone that merely showed that she was a poor liar.\n\n\"You cut down the sacred grove? Your farmers are raping dryads? By all you hold holy, my lady Abbess, if you do not help me understand this, we're all going to die here. This is a full invasion, the first that has been seen since your youth. Where have they come from? Has the north fallen? Why has the Wild come here in such strength? I grew up with the Wall. I've been to Outwaller villages, eaten their food. There are far more than we admit\u2013tens of thousands. If they have come to support the Wild directly, we will be swept away in the sea of foes. So what _exactly_ is happening here?\"\n\nThe Abbess took a breath as if to steady herself, succeeded, and raised an eyebrow. \"Really, Captain, I have no more idea than you. The actions of the savages are beyond me. And the Wild is just a name we give to an amalgam of evil, is it not? Is it not sufficient that we are holy, and seek to preserve ourselves, our God, and our way of life? And they seek to take that from us?\"\n\nThe captain met her gaze and shook his head. \"You know more than that. The Wild is not so simple.\"\n\n\"It hates us,\" the Abbess said.\n\n\"That's no reason to mass against you now,\" replied the captain.\n\n\"There's burned trees and new fields out east toward Albinkirk,\" Sauce said.\n\nThe Abbess turned, as if to reprimand the woman, but shrugged. \"We have to expand as our people expand. More peasants to feed required more fields.\"\n\nThe captain looked at Sauce. \"How many burned trees? I don't remember them.\"\n\n\"They're not right along the road. I don't know\u2014ask Gelfred.\"\n\n\"They go all the way to Albinkirk,\" the Abbess admitted. \"We agreed to burn the forest between us and bring in more farmers. What of it? It was the old king's policy, and we need that land.\"\n\nThe captain nodded. \"It was the old king's policy, and it led to the Battle of Chevin.\" He rubbed his beard. \"I hope that one of my messengers made it to the king, because right now we're in a whole heap of shit.\"\n\nMichael came in with cups of wine. He flushed very red when he saw the Abbess.\n\nThe captain glanced at him. \"All officers, Michael. Get Ser Milus from the Bridge Castle too.\"\n\nMichael sighed, served the wine, and left again.\n\nThe Abbess pursed her lips. \"You wouldn't abandon us,\" she said, but it was more a question than a statement.\n\nThe captain was looking through his window to the west. \"No, my lady, I wouldn't. But you must have known there would be a response.\"\n\nShe shook her head, anger warring with frustration. \"By Saint Thomas and Saint Maurice, Captain, you task me too heavily! I did no more than was my right, even my duty. The Wild was beaten\u2014or so I'm told by both the sheriff and the king. Why should I not expand my holdings at the cost of some old trees? And when the killing started\u2014Captain, understand that I had no idea that the killings were connected, not until\u2014\"\n\nThe captain leaned forward. \"Let me tell you what I think,\" he said. \"Hawisia unmasked a traitor, and died for it.\"\n\nThe Abbess nodded. \"It is possible. She asked to go to the outholdings when, ordinarily, I would have gone.\"\n\n\"She was your chancellor? The post Sister Miram holds now?\" he asked.\n\nShe shook her head. \"No. She had more power than the other sisters, but she was too young to hold an office.\"\n\n\"And she was widely disliked,\" Sauce said.\n\nThe Abbess flinched, but she didn't deny it.\n\nThe captain had his head in his hands. \"Never mind. We're here now and so are they. It's my guess that the Jacks, or the daemons, or both, were going to kill _you_ and seize the Abbey in a coup de main; Hawisia ruined it all somehow, either by confronting the traitor or by taking your place. We may never know.\" He shook his head.\n\nThe Abbess looked at her hands. \"I loved her,\" she said.\n\nThe Red Knight knelt by her and put his hands on hers. \"I swear I will do my best to hold this fortress and save you. But, my lady, I still feel you know something more. There is something personal about all this, and you still have a traitor within your walls.\" When she didn't answer him, he got up from his knee. She kissed his cheek, and he smiled. He handed her a cup of wine.\n\n\"Not your usual contract, ser knight,\" she said.\n\n\"Damn it, my lady, this _is_ my usual contract: it's a war between rival barons, except that this time the rival baron can't be negotiated with or turned from his path or simply murdered, and they are all good ways of avoiding a knock-down fight. But in every other respect you and the Wild are feuding border lords. You've taken a piece of his land, and in turn he's raiding you and threatening your home.\"\n\nAs the captain spoke, his officers trickled in\u2014Bad Tom, Ser Milus, Ser Jehannes, Wilful Murder, and Bent. The others were either asleep or on patrol.\n\nThe Abbess was brought a chair.\n\n\"Park wherever you can,\" the captain said. \"I'll try and make this brief. I'd say we're almost surrounded, and our enemy hasn't bothered to build trench lines and trebuchets. Yet. But he's got enough force to close the woods and every road around us. He's got Outwallers\u2014who are those men and women who live in the Wild, for you godless foreigners.\" The captain gave Ser Jehannes a mirthless smile. \"I'm guessing he has a hundred or more Outwallers, a thousand irks, and perhaps fifty to a hundred other creatures of the types we've already seen\u2014wyverns, daemons and the like.\" He shrugged. \"I'm guessing our enemy is a potent magus.\"\n\nBad Tom whistled. \"Lucky we didn't get ourselves killed trying for their camp then.\"\n\nThe captain nodded. \"When you move fast and plan well, you deserve a little luck,\" he said. \"But yes, I'd say that getting away with that raid seized our luck with both hands.\"\n\n\"So now what?\" Sauce asked.\n\n\"First, Jehannes, you are now the constable. Ser Milus, you are now marshal. Tom, you are now first lance. Sauce, you are now a corporal. In one sweep, I'm short three knights. Milus, are there any likely lads in your refugees? The merchants?\"\n\nMilus scratched under his chin. \"Archers? Hell, yes. Men-at-arms? Not a one. But I'll tell you what there is down in my little kingdom\u2014there's two wagon loads of armour in barrels, and some nice swords, and a dozen heavy arbalests. All for sale at the fair, of course.\"\n\n\"Better than what we have?\" the captain asked.\n\n\"White plate\u2014the new hardened breastplates.\" Ser Milus licked his lips. \"The swords are good, the spearheads better. The arbalests as heavy as anything we have.\"\n\nThe Abbess smiled. \"Those were for me, anyway.\"\n\nThe captain nodded. \"Take it all. Tell the owners we'll give them chits for it and settle up at the end if we're still alive. How heavy are these arbalests?\"\n\n\"Bolts a forearm in length and thick as a child's wrist,\" Ser Milus said.\n\n\"Put them on frames. Two for you and the rest up here for me.\" The captain looked at the Abbess. \"I want to build an outwork.\"\n\n\"Anything you like,\" she said.\n\n\"I want to put all your farmers and all the refugees to work and I want your help seeing that I get no insolence from them. I need them to work quickly and be quiet.\" The captain took out a scroll of parchment and unrolled it.\n\n\"My squire is a gifted young man, and he drew this,\" he said. Michael flushed uncontrollably. \"We want a deep V-shape of walls on both sides and ditches outside the walls; built three hundred paces from Bridge Castle, where the road from the Lower Town starts up the hill. It will allow us to send soldiers and supplies freely back and forth from the Lower Town to Bridge Castle. Put boards all along the bottom so men can walk quickly, without being seen, and put three bridges over it, so our sorties can move easily about the fields. See this cutaway? A nice hollow space under the boards. Good place for a little surprise.\" He grinned and most of the soldiers grinned back.\n\n\"We'll _also_ put a wall along the Gate Road, running all the way to the top. We should have done it in the first place, anyway. Towers here and here, on earth bastions.\" He rubbed his beard. \"First, we put in covered positions for these new frame crossbows\u2014here and here\u2014so that if they attack while we're building, it's all a trap and they lose a couple of their own for nothing. Last, we improve the path from the postern gate to the Lower Town.\"\n\nAll the soldiers nodded.\n\nExcept Tom. Tom spat. \"We don't have the fucking men to hold all that wall,\" he said. \"Much less in both directions.\"\n\n\"No we don't. But building it will keep the peasants quiet and busy, and when our enemy attacks we're going to make him pay for it, and then let him have it.\"\n\nTom grinned. \"Of course we are.\"\n\nThe captain turned to the others. \"I'm assuming that our enemy doesn't have a lot of experience in fighting men,\" he said. \"But even if he does, we won't have lost much with these distractions.\"\n\nThe Abbess looked pained. Her eyes had a hunted look, and she turned away. \"He is a man. Or he was, once.\"\n\nThe captain winced. \"We face a man?\"\n\nThe Abbess nodded. \"I have felt the brush of his thought. He has some small reason to\u2014to fear me.\"\n\nThe captain looked at her, gazing as intently as a lover into her flecked brown and blue eyes, and she held his gaze as easily as he held hers.\n\n\"It is none of your affair,\" she said primly.\n\n\"You are not telling us things that would be of value to us,\" the captain said.\n\n\"You, on the other hand, are the very soul of openness,\" she replied.\n\n\"Get a room,\" Tom muttered under his breath.\n\nThe captain looked at Ser Milus. \"We cut the patrols down to two a day, and we launch them at my whim. Our sole remaining interest is getting any more convoys in here safely, or in turning them away. Albinkirk is gone. Sauce\u2014how far did you go today?\"\n\nShe shrugged. \"Eight leagues?\"\n\nThe captain nodded. \"Tomorrow\u2014no, tomorrow we won't send anything. Not a man. Tomorrow we dig. The day after, we send four patrols, in all directions except west. The day after that, I'll take half the company west along the road, as fast as we can go. We'll aim for twenty leagues, pick up merchants or convoys we can, and get a look at Albinkirk. Then back here, all with enough force to kill whatever opposes us.\"\n\nTom nodded. \"Aye, but against a hundred Outwallers, in an ambush, we'll just be dead. And that's without a couple of daemons and maybe a pair of wyverns and a hundred irks to eat our bodies afterwards. Eh?\"\n\nThe captain wrinkled his lips. \"If we surrender the initiative and hunker down here we're all dead too,\" he said. \"Unless the king comes with his army to relieve us.\"\n\nThe Abbess agreed.\n\n\"For all I know the Wall fortresses have already fallen,\" the captain said. His eyes narrowed, as if the subject had particular interest to him. \"Whatever the case, we cannot count on any help from the outside, nor can we hope that this is an isolated incident. We have to behave as if we have an unending supply of men and materiel, and we have to try to keep the road east open. We need to lure our enemy into some battles of our choosing.\" He looked around at his officers. \"Everyone understand?\" He looked at the Abbess. \"We have to be ready to destroy the bridge.\"\n\nShe nodded. \"There's a phantasm to do it. I have it. It is regularly maintained: when a certain key is turned in the gate lock, the bridge will fall into the river.\"\n\nThe officers nodded their approval.\n\nThe captain stood. \"Very well. Ser Milus, Ser Jehannes, you are in charge of my construction project. Tom, Sauce, you will lead the patrols. Bent, get the arbalest frames up, and placed in those four covered positions,\" he smiled, \"where Michael marked them. Bent, take charge of running the rotations inside the fortress too. Don't worry about who is a man-at-arms, who's a valet, who's an archer. Just get the numbers right.\"\n\nThey all nodded.\n\n\"You planning to take a nap?\" Bad Tom asked.\n\nThe captain smiled at the Abbess. \"My lady and I are going to raise a nice fog,\" he said. \"She is a very potent magus.\"\n\nHe had the pleasure of watching her eyes widen in surprise.\n\nJehannes paused. \"And you? Captain?\"\n\n\"I am a modestly talented magus.\" He nodded to his new constable. \"Ah, Michael. Please don't go anywhere.\"\n\nThe other officers rattled out, Michael stood uncomfortably by the door, and after a moment it was just the three of them.\n\n\"What do you have to say for yourself?\" the Abbess asked.\n\nMichael writhed. \"I love her,\" he said.\n\nShe smiled, to his shock.\n\n\"That is the best answer you could have given, under the circumstances. Will you wed her?\" she asked.\n\nThe captain made a noise.\n\nMichael stood straight. \"Yes.\"\n\n\"What a dashing young fool you are, to be sure,\" said the Abbess. \"Who's son are you?\"\n\nMichael's lips tightened, so the Abbess beckoned to him, and he came to her side. She leaned forward, touched his forehead, and there was a magnificent burst of colour and sparkling shards, as if a sunlit mirror had shattered.\n\n\"Towbray's son,\" she said, and laughed. \"I knew your father. You have twice the looks and twice the grace he ever did. Is he still a weak man who changes sides with every twist of the wind?\"\n\nMichael stood his ground. \"Yes, he is,\" he said.\n\nThe Abbess nodded. \"Captain, I will take no action until our war is resolved. But what I say now, I say as a woman who has lived at court with the great. And as an astrologer. This boy could do much worse than Kaitlin Lanthorn.\"\n\nMichael looked at his captain, whom he feared more than ten abbesses. \"I love her, my lord,\" he said.\n\nThe captain thought of the note in his gauntlet, and of what the Abbess had just said\u2014he'd felt the power of her words, which had bordered on prophecy.\n\n\"Very well,\" he said. \"All the best romances bloom in the midst of a good siege. Michael, you are not so much forgiven as pardoned for this. Your pardon does not include further tumblings of said girl in my solar. Understood?\"\n\nThe Abbess looked long and hard at the squire. \"Will you marry her?\" she asked.\n\n\"Yes,\" said the squire, defiantly, bowed and left the room.\n\nThe captain looked at the Abbess and grinned. \"And the sisters will go with her? They'll liven up castle life, I have no doubt.\"\n\nShe shrugged. \"He should marry her. I can feel it.\"\n\nThe captain sighed. And sighed again when he realised that there was no one to help him disarm.\n\n\"Shall we go and make fog?\" he asked.\n\nShe extended her hand. \"Nothing would please me more.\"\n\n# Lissen Carak\u2014Bad Tom\n\nBad Tom stared at the captain's steel-clad back, slim as a blade, as he squired the Abbess down the corridor to the steps. Jehannes made as if to pass him, and Tom put his arm up and blocked him.\n\nThey glared at each other, but if they had had fangs they'd have been showing.\n\n\"Give it a rest,\" Tom said.\n\n\"I don't like taking orders from a boy,\" Jehannes said. \"He's a boy. An inexperienced boy. He's hardly older than his squire. That _gifted young man._ \" He spat.\n\n\"Give it a rest, I said.\" Tom spoke with the kind of finality that starts fights, or sometimes ends them. \"You were never going to be captain. You haven't the brains, you haven't the hard currency, and most of all, you haven't the birth for it. He has all three.\"\n\n\"I hear the boy almost lost the castle because he can't keep his hands off some nun. He was off billing and cooing while you were out with the sortie. That's what I hear.\" Jehannes leaned back and crossed his arms.\n\n\"You know what makes me piss myself laughing when I watch you?\" Tom leaned forward until his nose was almost touching the older man's nose. \"When he issues his orders, you just fucking obey like the trained dog you are. And that's why you hate him. Because he's born to it. He's not new at this, he's the bastard of some great man, he grew up in one of the big houses, with the best tutors, the best weapons masters, the best books, and five hundred servants. He gives orders better than I do, because it's never occurred to him that anyone would disobey. And you don't. You just _obey_. And later, you hate him for it.\"\n\n\"He's not one of us. When he has what he wants, he'll go.\" Jehannes looked around.\n\nTom leaned back, shifted until his shoulders fitted neatly along a line of stone. \"That's where you are wrong, Jehan. He _is_ one of us. He is a broken man, a lost soul, whatever crap you want to call us. He has everything to prove, and he values us. He\u2014\" Tom spat. \"I like him,\" he said. And shrugged. \"He's a loon. He'll fight anyone, anytime.\"\n\nJehannes rubbed his chin. \"I hear you.\"\n\n\"All I ask,\" said Tom. He didn't do anything obvious, but a subtle shift of his hips cleared the corridor. Jehannes stood straight, and then, quick as thought, his rondel dagger was in his hand\u2014poised at shoulder height.\n\n\"Not planning to use it,\" he said. \"But don't threaten me, Tom Lachlan. Save it for the archers.\"\n\nThe knight turned and walked away, sheathing his dagger easily.\n\nTom watched him go with a slight smile on his lips.\n\n\"Catch all that, young Michael?\" he said, levering his giant form upright.\n\nMichael blushed.\n\n\"Not for his ears\u2014hear me? Men talk. Sometimes with their bodies, sometimes like old fishwives. Not his business.\" He looked at Michael, who was not quite cowering in the doorframe.\n\nBut Michael was afraid, yes, but also determined. \"I'm his squire.\"\n\nTom rubbed his chin. \"So you get to decide some things. If you hear two archers talking about stealing from a third, would you peach?\"\n\nMichael managed to meet his eyes. \"Yes.\"\n\n\"Good. And talking about raping a nun?\" he asked.\n\nMichael held his eye. \"Yes.\"\n\n\"Good. And talking about how much they hate him?\"\n\nMichael paused. \"I see what you mean.\"\n\n\"He's not their friend, he's their captain. He's pretty good at it, and he's better every day. But what he don't know won't hurt him. Get me?\" Tom leaned in close.\n\n\"Yes.\" Michael didn't back away. He tried to stand tall.\n\nTom nodded. \"You've guts, young Michael. Try not to get dead. We might make a man-at-arms out of you yet.\" He grinned. \"Nice, that little chit of yours. Best act quickly if you want to keep her for yourself.\"\n\nOut in the yard, a dozen archers and a pair of squires were gathered around a girl, and they were all furiously peeling carrots.\n\n# Lissen Carak\u2014Father Henry\n\nThe priest watched the sell-swords come out of their leader's room. The captain, the source of infection. _She_ came out first, and the Bastard was holding her hand like they were lovers. Perhaps they were\u2014if he was an imp of Satan then pleasuring an old whore would be just his mark. Aristocrats. Birds of a feather.\n\nBile flooded his throat, and his hands shook a little to think that he had\u2014he had\u2014\n\nHe ducked his head to avoid looking at them, and went back to his sermon. But it was a long time before his hands were steady enough to scrape the old parchment as clean and thin as he needed it to be.\n\nAnd when the biggest of the sell-swords came down the steps, he caught the priest's eye and smiled.\n\nHenry felt fear go through him like a wave of cold and dirty water. What did the man know?\n\nHe got up from his work table as soon as the giant walked off, and he slunk across the chapel to the prie-dieu in the chapel. Reached under the altar cloth to make sure it was still there. His war-bow. His arrows.\n\nHe sagged with relief, and hurried back to his work table, imagining one of his shafts in the giant's groin. Listening to him scream.\n\n# Dormling\u2014Hector Lachlan\n\nThe fast horsemen hadn't learned enough to change Hector's mind. He looked at the rough sketch of the country and shook his head. \"If I go east, I'd as well take my beasts over the mountains to Theva,\" he said. \"And I don't intend to do that. I have customers in Harndon and Harnford waiting for their cattle. West of the mountains, there's no way to pass a few thousand head except the road.\"\n\nThe Keeper had spent the night dancing and drinking his own ale as well as some nasty foreign spirits and his head was pounding. \"So wait here and send a message to the king,\" he said.\n\nLachlan shook his head. \"Sod that. I'll be away in the first light. What can you give me, Keeper? How many men?\"\n\nThe Keeper grimaced. \"Perhaps twenty helmets.\"\n\n\"Twenty? You have a hundred swords here, wasting your money and standing about idle.\"\n\nThe Keeper shook his head in turn. \"The Wild's coming,\" he said. \"I can't just drift away like some. I have to hold this place.\"\n\n\"You can hold this place with thirty men. Give me the rest.\"\n\n\"Maybe thirty like you\u2014thirty heroes. Normal men? I need sixty.\"\n\n\"So now you'll give me forty? That's better. Forty brings me near a century strong\u2014enough to watch both ends of the herd and still leave a sting in my tail.\" Lachlan looked over his sketch. \"When we come down out of the hills, it'll be worse\u2014I'll want horses. So I'll take fifty of your swords and two hundred head of horses.\"\n\nThe Keeper laughed. \"Will you now?\" he asked.\n\n\"Yes. For a third of my total profit,\" Lachlan said.\n\nThe Keeper's eyes widened. \"A _third?_ \"\n\n\"Of the profit. In silver, payable when I'm standing on your doorstep on my road home.\" Lachlan was smiling as if he knew the punch line to a secret joke.\n\n\"And nothing if you're dead,\" said the Keeper.\n\n\"I confess, paying my debts won't matter so much to me if I'm dead,\" Lachlan answered.\n\nThe Keeper pondered a while. The tall serving woman came in, and the Keeper was surprised to see that nothing but the blandest of smiles passed between them. He'd been _sure_ the dark-haired woman was the drover's type.\n\n\"I need your trade, and you're a well-known man,\" the Keeper said. \"But you're trying to take all my horses and half my fighting strength on a wild-haired adventure with little profit and a great deal of death.\" He rubbed his head. \"Tell me why I should help you?\"\n\nLachlan kicked his sword blade around his chair and sat back. \"If I told you I was going to make the greatest profit in my family's history by getting this herd through to the south\u2014\" he said.\n\nThe Keeper nodded. \"Sure, but\u2014\" The drover's cheerful arrogance annoyed him.\n\n\"If I said that success would leave the king in my debt, and open new markets for my beef,\" Lachlan said.\n\n\"Perhaps,\" the Keeper said.\n\n\"If I said I lay with your youngest in the night, and she carries my son in her womb, and that I'll do this for her bride price and be your family?\" Lachlan said.\n\nThe Keeper sat up, rage crossing his face.\n\n\"Don't you lose your temper with me, Will Tollins. She came to me free, and I'll wed her and be happy on it.\" But Lachlan put his hand on the hilt of his sword just in case.\n\nThe Keeper held his eye. And they sat there, diamond cut diamond, for a long time.\n\nAnd then the Keeper smiled. \"Welcome to my family.\"\n\nLachlan held out a great hand, and the other man took it.\n\n\"Forty-five swords, and all the horses I can raise by tomorrow, and I get half your profits\u2014a quarter for my own and quarter for Sarah's bride price. And you wed her today.\" He had the drover's rough hand in his own, and he felt no falsehood in it.\n\nHector Lachlan, the Prince of the Drovers, withdrew his hand, spat and held it out, and Will Tollins, the Keeper of Dormling, took it, and the Inn of Dormling rang to a second night of revelry.\n\nOn the next morning, Lachlan led his tail and all his herds out onto the road into the watery sunlight. Every man had a shirt of shining rings, every ring riveted closed on a forge, every hauberk fitted to its wearer over a heavy cote of elk hide quilted full of sheep's wool, and every man had a heavy bow or a crossbow, a sword at least four feet high to the cross, and some had axes as tall as they themselves were, over their shoulders. Every man had a tall helmet with a brim that kept the rain off your face and a sharp point at the top, and a cloak of furze, and long leather boots like leather hose that went to the hip. The Keeper's men wore hoods of scarlet with black lines woven in, so that they made a check of red and black, and Lachlan's men had a more complex weave of red and blue and grey that made a colour that seemed to change with the trees and the rain. And Sarah Lachlan stood in her father's door-yard with a wreath of spring jasmine in her hair, kissing her man again and again while the flame of her hair lit the morning like a second sun. Her former suitor, the farmer, stood in the watery sun with a great axe on his shoulder, determined to go and die rather than face life without her.\n\nLachlan put an arm around him. \"There's other girls, lad,\" he said.\n\nHector Lachlan took his great green ivory horn and put it to his lips and blew, and the deep note sounded up the vale and down it to the Cohocton, over leagues of ground. Deer raised their heads to listen, and bears paused in their orgy of spring eating, and beavers, surveying the dams broken by the spring rains, looked up from calculations. And other things\u2014scaled and taloned, chiton brown or green\u2014raised their heads and wondered. The horn call rang from hillside to cliff edge.\n\n\"I am Hector Lachlan, Drover of the Green Hills, and today I set forth to drive my herd to Harndon!\" Hector shouted. \"Death to any who oppose me, and long life to those who aid me.\" He sounded the horn again, and kissed his new wife, and took a charm from around his neck and gave it to her.\n\n\"Wish me luck, love,\" he said.\n\nShe kissed him, and gave him not one tear. But she looked at her father defiantly and was shocked to find him smiling back.\n\nThen Hector crushed her to his burnie, and then he was walking out the gate. \"Let's go!\" he shouted, and the drive lumbered into motion.\n\n# West of Albinkirk\u2014Gerald Random\n\nGerald Random scratched his head for the tenth time that morning under his linen coif and wished that he could stop his convoy for long enough to wash his hair.\n\nThe Magus rode by his side, all but asleep in the saddle. Random couldn't help looking at him with the same proprietary air a man might look at a beautiful woman who's suddenly consented to sleep with him. Having the Magus by his side was incredible luck, like a tale of errantry come to life.\n\nThe convoy was down a wagon this morning\u2014the farrier's cart had been parked badly, had sunk in the rain and both oxen's throats had to be slit. The farrier himself was dry eyed, as his tools had been distributed among forty other carts and he'd been promised a place on the return. Altogether it was a small loss, but the whole column was exhausted, and Random was, for the first time, seriously considering turning around and going back south. The total loss of the convoy was a risk he could not really afford\u2014his financial losses if the convoy failed but the stock survived would merely set him back ten years. But if it was destroyed, he'd be ruined.\n\n_And dead, you fool_ , he thought to himself. _Dead men never become lord mayor, nor sheriff._\n\nSet against that, he'd brought them through an ambush and one straight-up fight and last night most of them had snatched a little sleep. He was reasonably sure that, with the Magus at his side, they could cut their way to the fair at Lissen Carack.\n\nBut what if they arrived to find no fair? The farther north-west they went, the less likely it seemed that there was a fair at Lissen Carack. Or even a convent.\n\nOn the other hand, going back seemed both craven _and_ dangerous. And the old Magus had been very clear: he was going to Lissen Carack, not back down the river to the king.\n\nHe scratched his itching head again.\n\nHe was seven leagues west of Albinkirk, if his notions of the road were accurate. About two days travel to the fords of the Cohocton, and another full day at oxen speed up the north side to the Convent.\n\nThe sun rose fully and the sky was truly blue for the first time in three days. Men's clothes dried, they warmed up, and the chatter of a well-ordered company began to spread. Men ate stale bread and drank a little wine, or small beer if they had it fresh, or hard cider if not, and the column rolled briskly along.\n\nThe soldiers were twitchy\u2014Old Bob had a dozen mounted men spread a hundred horse-lengths wide in the trees ahead of the wagons, and the rest covered the rear in a tight knot ready to charge in any direction under Guilbert.\n\nThey didn't stop to eat a noonday meal, but rolled on.\n\nWhen the sun was well down in the sky, Old Bob rode back to report that they were coming to one of the turn fields, the cleared fields maintained specifically for the fair convoys to camp.\n\n\"Looks like hell,\" he said. \"But it has fresh water and it's clear enough.\n\nIn fact, raspberry prickers had grown up over most of the field, and while one small convoy seemed to have made camp there a few days before, they had stayed to the edge by the road and cut no brush.\n\nGuilbert sent his men out into the raspberry canes in armour, to cut armloads of the stuff with their swords, and he had the archers lash them in bundles on the sturdy Xs of a pair of fascine horses, cradles made of heavy logs where armloads of brush and prickers could be wrapped tight. In the last three hours of daylight, while the boys cooked and brought in water and the older men saw to the animals and circled the wagons, the soldiers build a rampart of raspberry cane bundles.\n\nAnd then, the evening being dry, they set fire to the rest of the field. The canes went up like dried wood, burning into the edge of the trees in a few minutes.\n\nThe Magus came awake to watch the sparks rise into the clear night air.\n\n\"That was extremely foolish,\" he said.\n\nRandom was eating a garlic sausage. \"Why?\" he asked. \"Clear field for archery. No cover for the little boglins and spider irks.\"\n\n\"Fire calls strong as naming,\" the old Magus said. \"Fire is the Wild's bane.\" He glared at the merchant, and his glare held weight.\n\nRandom had been glared at all his life. \"Convoy's safer with a clear field around,\" he said, like an angry boy.\n\n\"Not if six wyverns come, you idiot. Not if a dozen golden bears decide you've intruded\u2014not if even a pair of daemon wardens decide you've broken the Forest Law. Then your clear field will not save you.\" But he looked resigned. \"And irks have nothing to do with spiders. Irks are Fae. Now\u2014where's my patient?\"\n\n\"The young knight? Sound asleep. He wakes up, talks to himself, and goes back to sleep.\"\n\n\"Best thing for him,\" Harmodius said. He walked around the circle of wagons, found his man, and looked him over.\n\nHarmodius put the blanket back after a long look, and then the younger man's eyes opened.\n\n\"You might have just let me live,\" he said. He looked pained. \"Sweet Jesu\u2014I mean you might have let me die.\"\n\n\"No one _ever_ thanks me,\" the Magus agreed.\n\n\"I'm Gawin Murien,\" he said. Groaned. \"What have you done to me?\"\n\n\"I know who you are,\" the Magus said. \"Now they can call you Hard Neck.\"\n\nNeither man laughed.\n\n\"I don't really know what I did to you. I'll work it out over the next few days. Don't worry about it.\"\n\n\"You mean, don't worry that I'm gradually turning into some loathsome God-cursed enemy of man who will try to slay and eat all my friends?\" Gawin asked. His voice strove for calm, but there was panic in it.\n\n\"You have a vivid imagination,\" Harmodius said.\n\n\"So I've always been told.\" Gawin looked at his own upper left arm and recoiled in horror. \"Good Christ, I have scales. It wasn't a dream!\" His voice rose suddenly, and his eyes narrowed. \"By Saint George\u2014my lord, must I ask you to kill me?\" His eyes went far away. _\"I was so beautiful_ ,\" he said, in another voice.\n\nHarmodius made a face. \"So very dramatic. I seized the power to heal you from something of the Wild.\" He shrugged. \"I wasn't really fully in control of the power, but never mind that. Without it, you'd have died. And whatever you may feel about it right now, death is not better!\"\n\nThe young knight rolled away, closing his eyes. \"Like you would know. Go away and let me sleep. Oh, Blessed Virgin, am I doomed to be a monster?\"\n\n\"I very much doubt it,\" Harmodius said, but he knew that his own slight doubt was not very reassuring.\n\n\"Please leave me alone,\" the knight said.\n\n\"Very well. But I'll be back to check on you.\" Harmodius reached out with a tendril of power and it was his turn to recoil at what he saw. Gawin saw his reaction.\n\n\"What's happening to me?\"\n\nHarmodius shook his head. \"Nothing,\" he lied.\n\nAn hour after full dark, the enemy struck. There was a whistle of arrows from the darkness, and two of the guildsmen on guard fell\u2014one silently, the other with the panicked screams of a man in pain.\n\nGuilbert had the wagons manned and alert in a hundred heartbeats. Which was as well, because a wave of boglins, announced by a sinister rustling, exploded into the north face of the wagon-fort.\n\nBut Guilbert was an old campaigner, and his dozen archers shot fire-arrows into the piles of cane and brush left around the old clearing, and most of them caught. And then, by the flickering light of spring bonfires, the guildsmen and the soldiers killed. Having negotiated the raspberry cane walls, the boglins were almost incapable of climbing the tall wagons after, and they died in dozens trying.\n\nBut the red arrows arching like vicious dragonflies over the fires began to annoy the defenders. The arrows lacked the potency to penetrate good mail, and their flint heads shattered easily, but they sunk deep in exposed flesh, and men who took them, even as a scratch on the hand, became fevered in an hour.\n\nHarmodius went from man to man, pulling the poison by grammerie. He'd had a day to gather power and rest, and he was full of sunlight, his aids charged and ready except for the two wands, whose charging required greater time, attention, and investment.\n\nWhen the fires burned down, he cast a powerful phantasm of light on a tree way out at the edge of the raspberry thicket. He repeated the spell six times, all the way around the wagon-fort, to back-light their attackers and blind their archers. But the Hermetic cost was immense, and he was shouting his power to the world.\n\nAs his sixth light casting began to fade, and the deadly, wasp-like arrows began to come in again, Harmodius felt the presence of an enemy. A practitioner.\n\nAnother magus.\n\nThere was a moment's warning\u2014possibly as the other one raised a defensive ward.\n\nHarmodius raised his own. And then, like a man fighting with a sword and buckler, he pushed his ward across the open space between himself and the other source of power. If his ward was held close to his body, it could only cover _him._ Held close to the other magus, the ward could cover the whole convoy.\n\nA simple exercise in mathematica that most practitioners never learned.\n\nIt cost a fraction more energy to maintain the ward over there than here.\n\nEnergy exploded against his ward and was deflected. Irks and boglins died under the onslaught of phantasms which should have been supporting them.\n\nHarmodius smiled wickedly. Evidently whoever was out there had a great deal of raw power and very, very little training.\n\nIn his youth, Harmodius had been an accomplished swordsman. And the practice of hermetical combat had many close analogues in swordsmanship. Harmodius had always meant to write a treatise on the subject.\n\nAs his adversary prepared another attack, Harmodius dashed through the labyrinthine palace of his memory, stacking wards and gardes in a sequence he'd practised but never used.\n\nHis opponent's next attack came with more force\u2014a titanic, angry upwelling of power that came as a lurid green stripe across the night.\n\nHis first ward was voided. The enemy had moved off line, realising the strength of his forward defence.\n\nHis second ward, however, caught the attack and subtly displaced it\u2014and the third ward reflected it down yet another line\u2014right back into the caster, who was struck squarely by his own phantasm.\n\nHis wards flared a deep blue-green\u2014and Harmodius struck. In the tempo of the opponent's own attacks, he launched a line of bright, angelic white\u2014a line like a lance that connected his index finger and the enemy's wards. It cost Harmodius almost no power, but the enemy, having over-committed to warding in the wrong place, now used his reserve ward to block...\n\n... nothing. The light beam was just that. Light. There was no force behind it.\n\nLike a fencing master going for an elegant, killing thrust, Harmodius drew power for his attack, and launched it, all in a tenth of a beat of a panicked guildsman's heart. And as the blow went in\u2014over one ward, under a second, and through the weakened energies of the third\u2014he felt his enemy collapse. Felt him experience the despair of defeat.\n\nAnd without intending to, he reached out and seized something\u2014just as he had taken power to save the young knight. But this time, he took the essence of the enemy sorcerer much faster and more thoroughly.\n\nHis opponent's power was extinguished like a candle.\n\nHarmodius took a deep breath, and realised that he was now more powerful than he had been when he started the night.\n\nHe cast a seventh light without any opposition.\n\nThe irks faded into the brush, and the rest of the night passed as slowly as he'd ever known, but with no further attacks.\n\n# West of Albinkirk\u2014Gerald Random\n\nA horse length from the Magus, Random stood with Old Bob. The last exchange of phantasm happened incredibly quickly. Random had watched it.\n\nIn the distance, something screamed.\n\nA cruel smile spread across Harmodius's lips.\n\nRandom glanced at Old Bob, who was looking at him. \"That was\u2014\"\n\nOld Bob shook his head. \"Legendary,\" he said.\n\nIn the morning, the convoy confronted the truth\u2014the broken bodies of a hundred boglins lay among the wagons. No man could deny what they had fought. Several vomited. Every man crossed himself and prayed.\n\nRandom approached the Magus who sat, cross-legged in the open, greeting the rising sun with his arms across his lap.\n\n\"May I interrupt?\" he asked.\n\n\"I'd rather you didn't,\" grumbled the Magus.\n\n\"My apologies,\" Random said. \"But I need some information.\"\n\nThe Magus snapped his eyes open. \"If you do not let me do this, I will have fewer arrows to my bow when they come again,\" he said.\n\nRandom bowed. \"I think we should turn and go back.\"\n\nThe Magus frowned. \"Do as you must, merchant. Leave me be!\"\n\nRandom shook his head. \"Why shouldn't I turn back?\"\n\nHarmodius's voice was savage. \"How do I know, you money-grubbing louse? Do as you like! Just _leave me alone!\"_\n\nOld Bob was already mounted and Guilbert stood by his horse with a short, oddly curved bow across his saddle. Today was his turn to ride point.\n\nOld Bob gestured tp Messire Random. \"Well?\" he asked.\n\n\"We press on for Lissen Carak,\" Random said.\n\nOld Bob rolled his eyes. \"What the fuck for?\"\n\nRandom looked back at the Magus. And shrugged. \"He made me angry,\" Random said, with simple honesty.\n\nOld Bob looked at the pile of dead boglins. \"You fought 'em before?\" he asked.\n\nRandom nodded.\n\n\"Take every man to the pile and make him look at them. Careful like. In daylight. Make every man touch one. Make every man see where they're weak.\" He shrugged. \"It helps.\"\n\nRandom hadn't thought of any of those things. So he ordered it done, and stood there while Old Bob hauled a corpse off the pile.\n\nThe guildsmen flinched as he slammed the body down.\n\n\"Don't be afraid, lads,\" Old Bob said. \"It's dead.\"\n\n\"Fucking _bug_ ,\" said one of the cutlers.\n\n\"Not bugs. More like\u2014\" Old Bob shrugged. \"Get the Magus to tell you what they're like. But look. They have hard parts and soft parts. Hard on the chest. Soft as cheese here under the arms.\" To demonstrate, he took his arming dagger and thrust it into the muddy brown skin, which extended like membrane from the soft-hard chitinous armour of the torso, under the arm with no effort at all. Green-black ichor covered his blade, but none leaked out.\n\n\"A thrust is always deadly,\" Old Bob said. He struck, and his heavy dagger blade punched through the thing's tough shell, and a putrid odour filled the air. One of the salters vomited.\n\nOld Bob walked over and kicked him. \"Do that when you're fighting and you're fucking dead. Hear me? Look at it. _Look at it!_ \" He looked around at the startled apprentices. \"Everyone touch it. Take one off the pile for yourself, and try it with your sword. _Do it._ \"\n\nAs Guilbert rode to the head of the column, he muttered to Harold Redlegs\u2014loud enough to be heard\u2014\"Because the old Magus made him mad? That makes all sorts of sense.\"\n\nIt didn't make any sense to Random, either.\n\nBut an hour down the road, Harmodius rode up next to the merchant and bowed in the saddle.\n\n\"My pardon if I was brusque,\" he said. \"Sunrise is a very important moment.\"\n\nRandom laughed. \"Brusque, is it?\" he asked. But then he laughed. It was a beautiful day, the woods were green, and he was commanding the biggest convoy of his life.\n\nRiding to war beside a living legend.\n\nHe laughed again, and the old Magus laughed, too.\n\nThirty wagons behind, Old Bob heard their laughter and rolled his eyes.\n\n# North of Albinkirk\u2014Peter\n\nThe Sossag People had gathered almost their full fighting strength and brought it south across the wall. Ota Qwan said so, ten times a day, and the second full day in camp, he saw the whole fighting strength of the Sossag gathered in one place, the great clearing a mile south of camp. He stopped counting men when he reached several hundred, but there had to be a thousand painted warriors and another few hundred unpainted men and women. He'd learned that to be painted was to declare a willingness to die. Unpainted men might fight\u2014or not, if they had other immediate interests, like a new wife or new children.\n\nPeter had also learned that the Sossag had little interest in cooking. He had tried to win a place through efforts with a copper pot and a skillet, but his beef stew with stolen wine was eaten noisily and quickly by the band with which he travelled, the Six River Sossag who also called themselves the \"Assegatossag\" or \"Those who follow where the Squash Rots\" as Ota Qwan explained.\n\nThey ate it, and went about their business. No one thanked him or told him what a fine meal it had been.\n\nOta Qwan laughed. \"It's food!\" he said. \"The Sossag don't eat that well, and we all know what it is like to grow hungry. Your meal was excellent in that there was enough for everyone.\"\n\nPeter shook his head.\n\nOta Qwan nodded. \"Before I was Ota Qwan, I understood what it was like to cook, to eat well, to _dine._ \" He laughed. \"Now, I understand many things, and none of them involve fine wines or crunchy bread.\"\n\nPeter hung his head a little and Ota Qwan slapped his back.\n\n\"You will earn a place. Everyone says you work hard. That is all the People expect of a newcomer.\"\n\nPeter nodded.\n\nBut that night he made a number of new friends. Dinner had been a simple soup with some seasonings and deer meat that Ota Qwan had contributed, and one of the reptilian monsters had come, sniffed the carcass of the deer, and made its strange crying noises until Skadai came at a run.\n\nPeter had been afraid, but it had left them in peace, the deer meat had gone into the soup, and all was well.\n\nWhen the soup was served, two of the boglins came out of the woods. They were slim without being tall\u2014when they stood erect, they were only the height of a tall child, and their heads were more like insects than men, with the skin stretched tight over light bones perched on bodies with a bulbous armoured torso and four very mammalian appendages. Their legs were thin and heavily muscled, the arms whipcord-tight. They were hideous, and just watching one move was the stuff of nightmare. They didn't mix much with Sossag, although Peter had seen Skadai speak to a group of them.\n\nThey also seemed to come in three types\u2014the commonest were red-brown and moved very quickly; the second type were clearly warriors, with a more heavily armoured carapace and paler, almost silver, skin. The warriors were almost as tall as a man, and every appendage had a spike. The Sossag used the Albin name for those ones\u2014 _wights._\n\nAnd finally, there was what seemed to be a leader class\u2014long and thin, like great mantis creatures. The Sossag called them _priests_.\n\nThese two creatures were both lowly workers, each carrying a bow and a spear, naked except for a quiver and a canteen. Peter tried not to watch the liquid sliding of plate on plate in their lower abdomens. It was disturbing.\n\nThey stopped by his fire. Both rotated their heads in unison, their strange lobe-shaped eyes seeing the fire and the man together.\n\n\"Guk fud?\" said the nearer of the two. His voice was scratchy, almost a screech.\n\nPeter tried to get past his fear. \"I don't understand,\" he said.\n\n\"U guk fud?\" said the other one. \"Gud fud?\"\n\nThe first one shook its head and abdomen\u2014an alien display, but Peter understood it was showing impatience. \"Me try fud,\" he screeched.\n\nPeter still didn't understand their shrieks, but the pointing claw-hands seemed to indicate his stew pot.\n\nNone of the Sossag were rising to help him. As usual, they had eaten to satiety and now lay on the ground, virtually unmoving, although every one of them was watching him. Ota Qwan was smiling\u2014a hard, cruel smile.\n\nPeter bent, turning his back on the creatures, and poured stew into a bowl. He added a little wild oregano and handed it to the nearer of the two monsters.\n\nHe took it, and Peter watched him sniff the bowl. He wished he hadn't. Watching the thing's not-entirely-inhuman nose split open to reveal a cavernous hole in the face with spiky hairs\u2014\n\nThe thing made a loud scratching noise with two of its arms and poured the whole bowl straight into the hole in its face.\n\nThrew back its head at an unnatural angle and screamed.\n\nThen it held out the bowl for more.\n\nPeter scooped two more bowls, put oregano on both, and handed one to each boglin.\n\nThe entire process was repeated.\n\nThe smaller of the two boglins opened and closed its beak-mouth three or four times, emitting a chemical reek that caught at the back of Peter's throat.\n\n\"Fud gud!\" it chirped.\n\nLong, agile tongues of a shocking pink-purple emerged from their mouths, and they licked the bowls clean.\n\nThey emitted a long scratching noise together, and raced off, running lightly on the ground, bent half over.\n\nPeter stood by his fire with two empty food bowls. He was shaking a little.\n\nSkadai came. \"You have been honoured,\" he said. \"They seldom notice us.\" He looked like a man with more to say, but then he pursed his lips, patted Peter on the shoulder, smiled and loped off, as he always did.\n\nPeter was still trying to decide what to make of the incident when the woman came and put a hand on the small of his back.\n\nThat hand on his back was a palpable thing\u2014another means of communication, a thing he hadn't expected, and it conveyed a wealth of information to him\u2014so much, in fact, that an hour later he was between her legs... and moments after that another man kicked him in the head.\n\nSuch a blow might have killed, but the painted man was barefoot and Peter had a little warning. And despite being a former slave and a cook, Peter had been bred to war, so as the kick turned his head, as he ripped himself free of the dark-haired woman's embrace, he was already moving, calculating, reaching for the knife he wore around his neck.\n\nThe painted man expected him to be easy prey. He screamed, in rage or feigned rage, and attacked again. Peter had rolled on his back with the force of the kick, and he had the knife in his hand, and when the painted man\u2014his red and black and white mixed in blotchy patches that looked like a skin sickness\u2014jumped at him, Peter killed him as easily as such a thing could ever happen. He rammed his blade deep into the man's belly and then rolled him over as he screamed in shock and desperation, his wild eyes suddenly wide with the despair of agony leading inevitably to death.\n\nPeter ripped the knife up his abdomen, spilling his guts and covering himself in the man's gore.\n\nThen, full of his own terror, he plunged the knife into the man's eyes, one and two.\n\nBy then, the blotchy man was dead.\n\nPeter lay there for a moment. Every one of the last hundred heartbeats was open to him like a long book, carefully read, and the remnants of his erection reminded him that he had passed from one extreme of life to the other in that time.\n\nHe tried to get to his feet but his knees were shaking and there were men all around him.\n\nAll Sossag men.\n\nSkadai held out a hand and hauled him to his feet with a firmness that seemed threatening. But was not.\n\nThen Ota Qwan was there, with a steadying hand.\n\n\"Open your mouth,\" he said.\n\nPeter opened it, and Skadai stuck a bloody finger into his mouth and began to chant. Ota Qwan grabbed his arm tightly. \"This is important,\" he said. \"Listen: Skadai says, 'Take your foe, Gruntag, into your body.' \" Ota Qwan squeezed again. \"Skadai says, 'Now you and Gruntag are one. What you were, he is. What he was, you are.'?\"\n\nPeter wanted to retch at the taste of coppery, warm blood inside his mouth.\n\n\"I say, don't make a habit of killing Sossag,\" Ota Qwan said.\n\n\"He attacked me!\" Peter squeaked.\n\n\"You were fucking his woman, who was only using you to be rid of an inferior man. She avoided the shame of sending him from her blankets by arranging for you to kill him. Understand?\" Ota Qwan turned to a group of painted men and said something, and they all laughed.\n\nPeter spat. \"What's so funny?\"\n\nOta Qwan shook his head. \"Our humour. Yours later, but not now, I think.\"\n\n\"Tell me,\" Peter said.\n\n\"They asked how you were. I said you weren't sure whether the dick or the knife went in more smoothly.\" Ota Qwan's eyes were a bright blue, and the man was amused. \"You are now a man, and a Sossag. Killing your own should not be a habit, but you must know by now what the Wild is.\"\n\nPeter spat again. \"It's every hand raised against every other hand,\" he said. He had trained to kill all his young life, and his first failure to kill another man had made him a slave. But this sudden success felt more like rape than victory. He was covered in blood and worse, and yet these men were congratulating him. \"There is no law.\"\n\nOta Qwan shook his head. \"Don't be foolish,\" he said. \"There are many laws. But the greatest of them is that the strongest is the strongest. And every creature, weak or strong, makes a good meal.\" He laughed. \"It's no different at the king's court. But here, it's fair and honest, at least in that no one lies. Skadai is faster and deadlier than I will ever be. I will never challenge him. But another man might\u2014or a woman\u2014and the matrons would name a form of challenge, and the challenger would face Skadai. Or perhaps simply attack him\u2014but that sort of victory does not always result in the killer gaining the power and prestige he seeks. Am I making sense?\"\n\n\"Too much sense,\" Peter said. \"I want to wash.\" Peter wanted free of this alien man and his paint and his aura of violence.\n\n\"I tell you this because now other warriors see you as a man and you may be challenged. Or simply killed. Up until now, I have protected you.\" Ota Qwan shrugged.\n\n\"Why kill me?\" Peter asked.\n\nOta Qwan shrugged. \"To raise the number of men they've killed? Or to claim Senegral, your woman?\" He laughed. \"Grundag died easily because he thought you were a slave. He wasn't much of a man, but he was a fighter, and his very stupidity made men afraid of him. They are not afraid of you\u2014although the way you opened him and cut out his eyes may make some men afraid. But many men want Senegral, and she doesn't like to say no.\"\n\nPeter had made it to the stream, and despite the cold and the sharp rocks, he threw himself into the low pool where the men washed their cups, heedless of the layer of water-swollen grains where a hundred wooden bowls had been washed after dinner. Heedless of leeches. Wanting only to get the sticky blood and intestinal matter off his hands and his belly and his groin.\n\nFrom the water, he said, \"Perhaps I should just kill her.\"\n\nOta Qwan laughed. \"An elegant solution, except that her brothers and sisters would then surely kill you.\"\n\nThe water woke his brain and froze his skin. He put his head under the water and come up floundering, feet aching from trying to balance on sharp rocks. \"What can I do?\" he asked.\n\n\"Paint!\" Ota Qwan said. \"As a warrior on a mission, you are exempt from such treatment. Unless you provoke it, of course. But men are not as swift as other animals, as deadly in a fight, as well-taloned, or as long limbed. Eh? But in a pack, we are the deadliest animals in the Wild, and when we paint, we are a pack. Do you understand?\"\n\nPeter shook his head. \"No,\" he said. \"But I will paint. And that commits me to make war against people I do not know to gain a little peace at home.\" He laughed. His laughter was strange and wild and a little crazed. \"But they enslaved me, so they can take the consequences.\"\n\nOta Qwan nodded. \"I knew you would make one of us from the moment I met you,\" he said. \"Don't disdain us. We do as other people do, we just don't call it by pretty names. We make war now to support Thorn, but also so that all the other killers and all the other predators will see our strength and leave us in peace. Will fear us. So we can go home and grow squash. It is not all war and knives in the dark.\"\n\nPeter sighed. \"I hope not.\"\n\nOta Qwan made a noise. \"You need to paint soon, I think. And have a name. But I will let someone else name you.\"\n\nHe gave Peter a hand out of the stream, and then took him to a fire, where he removed the horde of leeches stuck to the former cook. On another day, the leeches would have appalled him, but Peter bore their removal with hardly a glance, earning a respectful grunt from an older man.\n\nThen Ota Qwan spoke, and all the men and several women stiffened, paid close attention, and went to their blanket rolls, returning with pretty round boxes of pottery and wood\u2014some covered in remarkable designs made with coloured hair or quills, and some made of gold or silver.\n\nEvery little vessel held paint\u2014red, black, white, yellow, or blue.\n\n\"May I paint you?\" Ota Qwan asked.\n\nPeter smiled. \"Of course,\" he said. He was exhausted and almost asleep.\n\nThree men and a painted woman did the actual painting, under Ota Qwan's direction. It took an hour, but when they were done Peter was black on one side of his body and red on the other.\n\nBut on his face they had painted something more intricate. He had felt the woman's fingers on his face, around his eyes, her own rapt expression and slightly open mouth oddly transfigured by the fish she wore painted across her eyes.\n\nWhen they were done one of the men brought a small round mirror in a horn case, and he looked at the mask over his face, the divisions of white and red and black like herring bones, and he nodded. It spoke to him, although he wasn't sure what it meant.\n\nHe left them his shirt.\n\nHe walked though the firelit darkness, and the air was cool on his painted skin, and the fires burning at every camp were warm, even from a distance. Ota Qwan led him from fire to fire, and warriors murmured to him. He nodded and bowed his head.\n\n\"What are they saying?\" he asked.\n\n\"Mostly hello. A few comment on how much taller you are now. The old man tells you to keep your paint clean and sharp, and not muddy it as you used to do.\" Ota Qwan laughed. \"Because, of course, you used to be Grundag. Understand?\"\n\n\"Christ,\" Peter said. And yet, the murmured welcomes straightened his spine. He had _triumphed._ He didn't need to wallow in the killing.\n\nHe was alive, and tall, and strong, and he rather liked the paint.\n\nAt his own fire, Senegral had made all of Grundag's belongings into a small display, and she gave him a cup of warm, spiced tea, and he drank it. Ota Qwan stood at the edge of the firelight and watched.\n\n\"She says, look at the good bow you have. Some of your arrows are very poor. You should make better, or trade for them. And she says that she will try not to inflame other men, if you will only keep her the way she wishes to be kept.\"\n\nPeter went through the carefully laid out goods by the bark basket, holding each item up in the firelight. Two excellent knives and a good bow with no arrows to speak of; some furs, a pair of leggings and two pairs of unadorned moccasins. A horn container full of black paint, a glass jar with red paint. Two cups. A copper pot.\n\n\"I thought women made their men shoes?\" Peter asked.\n\nOta Qwan laughed. \"Woman who fancy their men make them magnificent moccasins,\" he said.\n\n\"I see,\" Peter said. He packed everything back into the basket. The woman came and stood next to him, and he put a hand under her skirt and ran it up her leg to her thigh, and then around her thigh, and she made a sound, and soon enough, they were back where they had been when the dead man kicked him in the head.\n\nAt some point she moaned, and later he laughed aloud at the absurdity of it all. He wanted Ota Qwan to translate his thoughts to her, but of course, the man was gone.\n\n_Why is he helping me?_ Peter thought, and then he was asleep.\n\nAnd in the morning, all the painted men rose, took only the equipment they needed for violence, and followed Skadai. Peter took the bow and the best knife, and his paint and a single red wool blanket and strode naked, after Ota Qwan. He found it surprisingly easy to ask no questions, simply follow.\n\nLater, he asked Ota Qwan how to get arrows, and the man silently gave him a dozen.\n\n\"Why?\" Peter asked. \"Is it not every man against every other man?\"\n\nOta Qwan laughed. \"You know nothing,\" he said. \"Do you not follow me? Will you do my bidding when the arrows fly and the steel fills the air?\"\n\nPeter thought about it. \"I suppose I will.\"\n\nOta Qwan laughed. \"Come. Let's go find your name.\"\n\n# South of Albinkirk\u2014de Vrailly\n\nJean de Vrailly clamped down on his impatience and it turned, as it always did, to anger. The blossoming of his rage always made him feel sinful, dirty, and less of a man and a knight, and so, while riding easily through the high ridges and spring flowers of Alba's fertile heartland, he reined up his second charger and dismounted, to the confusion of his brothers in arms, and knelt in the dirt beside the road to pray.\n\nThe mild pain of kneeling for prolonged periods always steadied him.\n\nImages floated to the surface of his thoughts as he imagined the crucifixion of the Christ; as he pictured himself as a knight riding to the rescue of the Holy Sepulchre, or inserted himself in meditation into the adoration of the Magi, a lowly caravan guard sitting on his charger behind the princes who adored the newborn lamb.\n\n_Contempt_ broke through his reverie. He despised the King of Alba, who stopped in each town to play to his peasants, win their sighs and their raucous laughter, still their fears and give them law. It was done with too much drama and it took too much time, and it was obvious\u2014obvious to a child\u2014that something was happening in the north that required the instant application of the kingdom's mailed fist.\n\n_Disgust_. The knights of Alba were slow, slothful, full of vice and barbarity. They drank, they ate too much, belched and farted at table, and never, ever exercised in arms. Jean de Vrailly and his retinue rode from town to town in full armour, cap \u00e0 pied, with heavy quilted cotes under hauberks of ring mail surmounted by shining steel plate\u2014three layers of protection, which every knight in the East wore every day of his life\u2014to town, to church or to ride out with his lady.\n\nThe Wild had not made a major incursion in the East in a century, yet their knighthood stood ready to fight at every moment.\n\nHere, where stands of unkempt trees stood on every ridge, and where an incursion of the Wild threatened a major city just over the horizon\u2014the knights rode abroad in colourful tunics with fashionable, long trailing sleeves, pointed shoes and carefully wrapped hats like the turbans of the far East, with their armour stored in wicker baskets and oak barrels.\n\nRight now, four days from Albinkirk, a party of the king's younger knights and squires were hawking, riding their palfreys along the ridge tops to the west, and he wanted to punish them for their light-hearted foolishness. These effete barbarians needed to be taught what war _was._ They needed to learn to take off their chamois gloves and feel the cold weight of steel on their soft hands.\n\nHe prayed, and praying made him better. He was able to smile to the king, and nod to a young squire who ill-manneredly galloped down the column raising a cloud of dust, mounted on a hot-blooded Eastern mare worth a hundred leopards as a racehorse and worthless in a fight.\n\nBut when the army, which grew every day as contingents of knights, men-at-arms, and archers joined from each town, each county, each manor, settled for the night, de Vrailly ordered his squires to set his pavilion as far from the rest of the army as could be managed\u2014out in the horse lines, surrounded by beasts. He dined simply on soldiers' rations with his cousin, summoned his chaplain, Father Hugh, to hear Mass and confession of his sins of passion, and then, shriven and spiritually clean, he bathed in water from the Albin, the mighty river that rolled by the door of his tent, dismissed his squires and his slaves, and towelled himself dry, listening to the sound of three thousand horses cropping grass on a beautiful spring evening. The smell of the wildflowers overpowered even the smell of the horses.\n\nDry, he dressed in a white shirt and braes and a white jacket, a jupon of the very simplest style. He unrolled a small, precious carpet from far to the east, and opened a portable shrine\u2014two paintings hinged face to face for travel\u2014the Virgin and the Crucifixion. He knelt before their images and prayed, and when he felt empty and clean he opened himself.\n\nAnd his archangel came.\n\n_Child of Light, I salute you._\n\nAs he did every time the angel came, de Vrailly burst into tears. Because he never quite believed a visitation was real, until the next one confirmed all of the past ones. His unbelief\u2014his doubt\u2014was its own punishment.\n\nThrough his tears, he bowed. \"Bless me, Taxiarch, for I have failed you many times.\"\n\nHe tried never to look directly into its shining face which seemed in memory to be made of beaten gold, but in fact looked more like mobile, sparkling pearl. Looking too closely might break the spell\u2014\n\n_It is not your error that the King of Alba did not do as we wished. It is not through you that this kingdom has been untimely assailed by the forces of Evil. But we will overcome_.\n\n\"I succumb to rage, to contempt, to self-righteousness and anger.\"\n\n_None of these will help you to be the best knight in the world. Remember how you are when you fight, and be that man at all times._\n\nNo priest had ever put it to him so well. When he fought, he dismissed all worldly concerns and was only the point of his spear. The archangel's words rang through him like the meeting of blades between two strong men, like the clarion call of a stallion trumpeting.\n\n\"Thank you, lord.\"\n\n_Be of good cheer. A great test is coming. You must be ready._\n\n\"I am always ready.\"\n\nThe archangel placed a shining hand on his forehead, and just for a moment, de Vrailly looked up into the archangel's shining face, his outstretched, perfect hand, his golden hair, so much brighter than de Vrailly's own and yet somehow alike.\n\n_Bless and keep you, my child. When the standard falls, you will know what must be done. Do not hesitate._\n\nDe Vrailly frowned.\n\nBut the angel was gone.\n\nHe could smell the incense, and he felt at peace\u2014his mind comforted, languorous, as he was after he had a woman, but without the sense of shame or dirt.\n\nHe smiled. Took a deep breath, and sang the opening notes of a Te Deum under his breath.\n\n# West of Albinkirk\u2014Harmodious\n\nHarmodius lay on a pack of furs, the pottery mug of warm wine balanced on his chest, and watched Random stir a hot poker into another beaker, adding honey and spices.\n\nBehind the merchant, Gawin Murien sat quietly mending a shoe. He didn't speak, but he was going about the tasks of soldiering, and Harmodius was content to keep a watch on him. His left shoulder was now heavily scaled from the nipple to the neck, and down to the base of his bicep. The scales no longer seemed to be spreading, but they were growing larger and harder. The young man seemed curiously heedless of them\u2014since the first night, he hadn't remarked on them at all.\n\nHarmodius was old in guile, and had known many young men. This one was preparing for death, and so Harmodius watched him carefully. The second ring on his right hand held a phantasm that would drop the boy like a blow from a spiritual axe.\n\n\"I like it sweet. But I have a sweet tooth,\" Random said. He grinned. \"My wife says that all my efforts to win riches and fame are merely to ensure my supply of biscuits and honey.\"\n\nHarmodius drank from his cup again. It was far sweeter than he liked it but on this evening, under the curtain of stars with a dreadful enemy as close as the edge of the firelight, the hippocras was all he could have asked.\n\n_Immanence._\n\nIt was a mild shock, like seeing a former lover walk into a tavern. Somewhere not so very far away, something powerful was manifesting. It might be something supremely powerful, a very long way away, or something merely awesome and terrifying, in the next field.\n\n\"To arms!\" Harmodius said, jumping to his feet. He gathered himself for a moment and extended his enhanced senses.\n\nGawin Morion was already in his leather jupon, and his helmet was going on his head.\n\nRandom had a breast- and backplate on over his travel clothes, and he produced a crossbow from the same wagon bed which had held the makings of their wine.\n\nOther men repeated the alarm, but most were fully dressed, armed and armoured, and Harmodius ignored them all, reaching out\u2014past the orange glow of firelight, past the fields of bracken and fern that surrounded them.\n\nNothing. Not a single boglin.\n\nHarmodius knew the laws of identity in the use of Power. There were two ways to locate another user. He could sit silently, his attuned senses waiting to see if there was another pulse of _immanence._ Or he could send out a pulse of his own power to ring across the night, which would identify him to every creature of the Wild with the slightest sensitivity to such things. Which was most of them.\n\nHe settled for the quieter, more passive option, even though it was not in his nature and although he was all but bursting with power. He hadn't felt so _capable_ in many years. He wanted to play with it, the way a man will swing a new sword about, cutting the heads off ferns and fennel stalks.\n\nHarmodius bore down on his power. And his impatience.\n\nPushed his senses further.\n\nFurther.\n\nWell to the north he found trolls\u2014their large, misshapen forms as horrible in their lack of symmetry as they were in their black crystalline alienness. They were marching.\n\nTo the west, he found a user of much talent and little training. But he had no context for the discovery\u2014a village witch, or a boglin shaman or one of the Wild's living trees. He had no idea, and he dismissed the entity as far too weak to have displayed the power he had sensed.\n\nWhatever it was, it seemed to have left the world\u2014departed by whatever path it had chosen. It manufactured a new _loci_ , or jumped to one that had previously existed.\n\nThe display of power remained like a beacon, and Harmodius was unhappy to find that it was behind them, to the south and the east by many leagues. But he swooped on it like a raptor falling on a rabbit\u2014and fled just as quickly when he sensed the order of magnitude it represented.\n\nWhen he was a small boy in a fishing village Harmodius, who'd had a different name then, had rowed out on the deep in a small boat with two friends to fish for sea trout and salmon with hand lines. Porpoises and small whales shared the sport, and sometimes they caught good fish only to have them snatched away by their aquatic rivals. But late in the day, while pulling in a heavy fish, Harmodius had seen a seal\u2014an enormous seal as long as his boat\u2014flash into a turn and reach for their magnificent fish...\n\n... just as a leviathan, as much larger than the seal as the seal was larger than the salmon, turned under the boat to take the seal.\n\nThe size of the creature beneath the boat\u2014fifty times its length\u2014and its great eye as it rolled, the froth of blood that reached the surface without a sound as it took the seal, the gentle swell it made, and then, perhaps the most terrifying of all, its mighty fluke breaking the surface a hundred yards away and flinging spray all the way over them\u2014\n\nIn all his life, Harmodius had never seen anything that moved him so deeply, or so impressed on him his own insignificance. It was more than fear. It was the discovery that some things are so great that they would not notice you even if they destroyed you.\n\nHe'd brought in the salmon, which died unaware of the role it had played in the death of the mighty seal, and the lesson was not lost on the boy.\n\nAnd all of that came to him as he fled the immensity of whatever creature had briefly been in the Valley of the Albin, fifty leagues to the south.\n\nHe came back into his own skin.\n\nRandom was looking at him with concern. \"You screamed!\" he said. \"Where are they?\"\n\n\"We are safe,\" Harmodius said. But his voice was more of a a sob than it should have been. _No one is safe. What was that?_\n\n# East of Albinkirk\u2014Hector Lachlan\n\nEast of Albinkirk, the sun rose on the western slopes of Parnassus, the westernmost of the mountains of the Morea where the streams rushed down, heavy with the last of the snow and the spring rains to flood the upper waters of the Albin.\n\nHector Lachlan was drinking tea and watching the East Branch. It was high\u2014far too high\u2014and he was trying to figure out how he might get his herds over it.\n\nBehind him, the men in his tail were breaking camp, packing the wagon, donning their hauberks and their weapons, and the youngest, or the least lucky, were already out with the herds.\n\nWhile he watched, his tanist, Donald Redmane, stripped naked at the water's edge and plunged in, using the edge of a ruined beaver dam as a diving platform. He was high spirited and strong and only heartbeats later, he was pulled out by the rope around his waist, his shoulder and collarbone bruised against the rocks.\n\nLachlan winced.\n\nThat night, something killed a stallion in their herd, and Lachlan, who had never fought an irk in his life, had to assume that it was some such creature who was responsible\u2014multiple punctures and slashes from something much smaller than the horse. But the _why_ of it eluded him. He doubled his herd guards, aware of how futile such measures could be. In the Hills, he had stone fences and deep glens with natural fortifications to take herds and guard them, but here, on the road\u2014he was in the country that the drovers held to be _safe._ And something was hunting him. He could feel it.\n\n# Chapter Nine\n\n# Lissen Carak\u2014The Red Knight\n\nThe fog was thin and wispy, but it did its job. It forced whatever was watching them to be more aggressive with the animals it was using. Rabbits came out of the woods in broad daylight. Starlings flew over the new diggings, first in pairs and then in swift flocks.\n\nToward midday, when Ser Jehannes had the double outer ditch dug and when the merchant adventurers of Harndon, Lorica, Theva and Albin were cursing their luck and their temporary taskmaster as the blisters on their hands popped, the Abbess cast again, the fog grew thicker, and the animals grew more numerous still.\n\nBy the time the nearly mutinous merchants were allowed to end their day and go to Mass, the fog was so thick that the watchmen on the fortress towers couldn't see the base of their own wall. They could see the far horizon, though. The captain had no intention of letting his own fog put him at a disadvantage. Despite which precaution, wyverns overflew the fortress every hour or two, and the hearts of the defenders flinched each time the leathery wings passed. Out in the trees beyond the fields there was movement\u2014the kind of movement a hunter sees when his quarry shakes a tree, or when a squirrel leaps to a branch too light to support the weight of the jump.\n\nMichael opened a blank book of bound parchment, and wrote in his best hand:\n\n_The Siege of Lissen Carak. Day One. Or is it Day Eight?_\n\n_Today the captain and the lady Abbess raised a fog with a powerful phantasm. The enemy are all around us, and many have commented that the air seems thick and difficult to breathe. Maddock the Archer was shot dead with a longbow arrow from the cover of a stand of trees when he ventured from the new trenches to retrieve a mallet. He must have left the cover of the lady's fog._\n\n_There is a wyvern in the air over us. I can hear it scream. And I can feel it, even through the roof\u2014a pressure on the top of my head._\n\nMichael put a line through that last, and then very carefully inked it over until not a word was legible.\n\n_The captain has a sortie mounted and ready to ride at all hours. Every armoured man has a turn in the Sortie. He also ordered heavy machines constructed in the towers. The fortress has two heavy towers, and one now holds a heavy ballista and the other, lower tower holds a trebuchet._\n\n_The people of the countryside and the merchants of the caravans have dug a trench from the Lower Town all the way to Bridge Castle. It is deeper than a man is tall, and wide enough to drive a small wagon along the bottom. We are lining it with boards. The captain has ordered bags placed along the bottom and no man knows what is in them._\n\nAt sunset Michael went on to the walls, and joined with every man and woman in the fortress in prayer. They sent their voices up to heaven, and then the lady cast again, a simple sending such as any village witch might make, but aided, Michael hoped, by the wishes and prayers of every man and woman. She worked an aversion\u2014the sort of thing Wise Women did for granaries on farms, which kept the smaller animals from eating the grain. She simply did it on a larger scale, and with a great deal more power.\n\n# West of Albinkirk\u2014Gerald Random\n\nMaster Random's convoy shook out early despite the adventures of the night, or perhaps because of them.\n\nHe was quite proud of them. Men were singing in the dawn\u2014some shaved at mirrors hung from wagon sides, and other men sharpened blades, sharpened arrow heads and crossbow quarrels. Men were rolling their blankets tight against the damp. Others boiled water in copper pots, or heated up a cupful of last night's porridge. At his own fire, the old Magus was heating ale in a copper shoe.\n\n\"You seem content to help yourself,\" Random said.\n\nHarmodius didn't even raise an eyebrow. \"I pay you the compliment of assuming that you are a generous man. And I made some for you.\"\n\nRandom laughed. He was camping with a legend, who was heating him ale on a chilly spring morning.\n\nBirds sang, and men sang, and Random could see young Adrian from the goldsmiths sitting on a wagon box and sketching.\n\nAdrian was a pargeter\u2014an artist in gold leaf. He was a likely lad, just about to leave apprenticeship for journeyman status, which would be a brief stop for him. His father was both talented and rich\u2014one of the goldsmith's coming men. Adrian was medium height, thin and fit, in expensive arming clothes made by professionals. He was wearing his breast- and backplate, his arming hood, and his armoured gloves lay across his lap. More and more of the young men were starting to ape the manners of the sell-swords\u2014wearing their harnesses all day, carefully tending to their weapons.\n\nRandom couldn't see what young Adrian was sketching\u2014it was on the other side of one of the goldsmith's wagons. Warm ale in hand, he went to look.\n\nHe smelled the thing long before he saw it. It had a horrible, sulphurous smell, overlaid with a sickly sweet-shop smell, like sugared liver.\n\nHe smelled the smell, but it didn't warn him.\n\nThe dead thing had been a daemon.\n\nYoung Adrian looked up from his sketch. \"Henry found it in the bush.\" The other goldsmith apprentice stood by the corpse with determined possessiveness, despite the horror of it.\n\nClose up and dead, the daemon was deeply disconcerting. The size of a small horse, it had finely scaled skin, like a river bass or a blue-gill; and the scales varied from white to pale gray with veins of blue and black like fine marble\u2014all surmounted by an opalescent sheen with all the colours of the rainbow. Its eyes were empty pits, the lids collapsed on them as if its death had robbed it of its eyes. It had a heavy, raptor-like head with a snout or a beak, and a crest like the plumage a man might wear on a tournament helm. It lay limp in death, like wilted flowers. It had two arms on its long trunk that were disturbingly like heavy human arms\u2014the muscled arms of a blacksmith, perhaps\u2014and heavy, powerful legs that seemed twice the size of the arms. Upright it must have stood as high as a man on a wagon.\n\nThe legs and torso were balanced by a heavy tail covered in sharp spines.\n\nIt was no animal. The beak and spines were inlaid in lead and gold in fantastic patterns; the bony ridge above the eyes held more inlay, and the dead daemon wore a cote of scarlet leather lined in fur\u2014beautiful work. Random couldn't help himself\u2014he knelt, despite the stink, and fingered the material. Deerskin\u2014dyed brighter and better than any dye he knew of, and tightly sewn in sinew.\n\nThere wasn't a mark on the monster, and the most disconcerting part of it was that its alien face was strangely beautiful, and wore a look of terror.\n\nThe old Magus wandered over, drinking ale. He stopped and looked at the daemon.\n\n\"Ah,\" he said.\n\nRandom didn't know how to broach his thought. \"I'd like the cote,\" he said.\n\nHarmodius looked at him as if he was mad.\n\n\"You killed it. It's yours.\" Random shrugged. \"Or that's how we did things when I was in the king's army.\"\n\nHarmodius shook his head. \"Heh,\" he said. \"Take it then. My gift. For your hospitality.\"\n\nThree more of the goldsmith apprentices helped him roll it over. It took him five minutes to get the cote off. It was the size of a horse blanket, or perhaps slightly smaller and was untouched by whatever wound had killed the monster, and clean. Random rolled it tightly, wrapped it in sacking, and put it in his own wagon.\n\nThe apprentices were eyeing the gold inlay.\n\n\"Leave it,\" Harmodius said. \"Their bodies generally fade rather than rot. I wonder\u2014\" He bent over the corpse. Prodded it with a stick, and despite having just rolled it over, the apprentices stepped back, and Henry hurried to get a quarrel in his crossbow.\n\nThe Magus drew a short stick from his cote. It was like a twig\u2014a crazy twig that looked like a lightning bolt\u2014but it was beautifully maintained with an oil finish that most twigs couldn't hold. The ends had minute silver caps.\n\nHarmodius ran it over the corpse\u2014back and forth. Back and forth.\n\n\"Ah!\" he said. He said a verse of Archaic to the delight of all present, who had never imagined being allowed to watch a famous magus work. It was different in daylight. Men who had hidden away when he cast at night now stared like churls.\n\nRandom could see the power gathering around the older man's hand. He didn't have the talent to cast power, but he'd always been able to see it.\n\nThen the old man cast, flicking his fingers at the daemon.\n\nIt seemed to pulse with colour\u2014every man let go a breath\u2014and then it dissolved to sand.\n\nAnd not very much sand, at that.\n\n\"Fae,\" Harmodius said. \"Something interrupted its decomposition when it died.\"\n\nTheir incomprehension was evident. Harmodius shrugged. \"Don't worry,\" he said. \"I'm only talking to myself, anyway.\" He laughed. \"Master Merchant, a word with you.\"\n\nRandom followed the old Magus away from the wagons. Behind them, Old Bob, the mercenary, rode up fully armed. The pargeter was showing off his sketch and Old Bob was suddenly silent.\n\n\"I've killed two of them in three days,\" Harmodius said. \"This is very bad. I ask your help, in the name of the king. But I warn you that this will be dangerous. Extremely dangerous.\"\n\n\"What sort of help?\" Random asked. \"And for what reward? Pardon me, my lord. I know that all the court think my kind lives only for gold. We don't. But par dieu, messire, I have several men's fortunes in these wagons. My own, first and foremost.\"\n\nHarmodius nodded. \"I know. But there is clearly an incursion\u2014perhaps even an invasion\u2014from the Wild. The daemons are the enemy's most valuable and most powerful asset. I thought it horrifying that I should encounter one. Two means we are watched, and there is a force in behind us. Three... three is unthinkable. Despite which, I ask you to send a messenger to the king. Immediately. One of your best men. And that we continue north.\"\n\nRandom nodded.\n\n\"I have no idea if the king will guarantee the value of your convoy,\" Harmodius said. \"What's it worth?\"\n\n\"Sixty thousand golden nobles,\" Random said.\n\nHarmodius sucked in a breath, and then laughed.\n\n\"Then I can safely say that the king can't replace it for you. Good Christ, man, how can you take so much into the wilderness?\" Harmodius laughed.\n\nRandom shrugged. \"We go to buy a year's produce of grain from a thousand farms,\" he said. \"And beef from the hillmen\u2014maybe fifteen hundred animals, ready to be fattened for market. And beer, small wine, skins from deer, beaver, rabbit, otter, bear and wolf\u2014a year's worth for every haberdasher and every furrier in Harndon. That's the business of the Northern Fair, and that's without their staple of wool.\"\n\nHarmodius shook his head. \"I've never thought of the value of all these things,\" he said. \"Or if I have, I've forgotten.\"\n\nRandom nodded. \"Half a million gold nobles. That's the value of the Northern Fair.\"\n\n\"I didn't know there was that much gold in the world,\" Harmodius laughed.\n\n\"Nor is there. That's why we have helmets and crossbows and fine wines and goldsmith work, and gaudy rings and bolts of every fabric under the sun\u2014and raisins and dates and olive oil and sugar and every other product the north doesn't have. To _trade._ It's why my convoy must get through.\"\n\nHarmodius looked at the mountains, just breaking the distant horizon. \"I've never thought about it,\" he said. \"Now that I do, it seems very\u2014-vulnerable.\" He looked around. \"What happens if there is no fair?\"\n\nRandom had had that very thought several times in the last two days.\n\n\"Then Harndon has no beef; it gets only the grain from the home counties; there are no furs for clothes or hats, no honey for bread, less beer and ale in every house. And the king is less by the tax he collects on the merchandise, and less again by the value of\u2014hard to say, but let's call it half of the wool staple. Small folks would starve. In the East, merchants who buy our wool would break. Most of the money-men of Harndon would break, and hundreds of apprentices would go out of work.\" He shrugged. \"And that's just this winter. It'd be worse in the spring.\"\n\nHarmodius looked at the merchant as if he was being told a fairy tale. Then he shook his head. \"This has been an eventful morning, Master Merchant. We should be on our way. If you'll agree to go.\"\n\nRandom nodded. \"I'll go. Because if I turn this convoy back,\" he shrugged. \"Well, I'll lose a great deal of money.\" _And I'll never be mayor._\n\n# Lissen Carak\u2014Michael\n\n_The Siege of Lissen Carak. Day Two_\n\nMichael licked his pen nib, absently painting the corner of his mouth in tree-gall and iron.\n\n_Today, all of the small folk dug at the trench. I append a small sketch of the work; it runs from the gate of the Lower Town to the out-wall of the Bridge Fort, a distance of four hundred and twenty-four paces. With just under a thousand working men and women, we dug the ditch in two days. The upcast of the ditch has been made into low walls on either side, and the captain has ordered us to plant stakes from our stores\u2014the palisades we use when we encamp\u2014along the edge of the ditch._\n\n_All the day a heavy fog stayed over the length of the ditch\u2014today's phantasm cast by the Abbess and is maintained by the good sisters, who can be heard praying at all hours in their chapel._\n\n_The enemy has sought all day to search out our new work. The air is thick with birds\u2014starlings and crows and doves, but they dare not enter the fog, and the area close in to the castle walls seems abhorrent to them._\n\n_The Enemy has wyverns, and they ride the air currents high above us all day. Even now, there is one above me._\n\n_In the woods to the west, we can hear the sound of axes. Twice today, large bands of men advanced from the wood's edge to within bowshot of the fog, and lofted arrows into it. We did not respond, except that our own archers crept close and retrieved the arrows._\n\n_Nigh on sunset, we released three sorties; one north, one west, and one westerly, but right along the river._\n\nThe captain rode west into the setting sun, his armour gathering what little light penetrated the sun. Grendel had a cote of barding today\u2014two layers of heavy chain falling to the mighty horse's fetlocks.\n\nIt took four valets to lift the cote and get it over the big horse's back. Grendel hated it, but the captain was confident that the Jacks would rise to his raid.\n\nHe had a dozen men-at-arms, fully harnessed, and their archers behind him, and as soon as Grendel's hooves were clear of the Lower Town\u2014empty and sullen but for the two archers on the stone gate towers\u2014he put his spurs gently to the great horse's sides, and Grendel began a heavy canter over the spring fields. The fog hid the light, and the terrain. It was possible to be ambushed in the fog, as he was aware.\n\nBut this was his own fog, and it had some special properties.\n\nHe rode south along the trench, going slowly, looking down to see the work that had been accomplished. It was a broad, deep ditch with a wooden floor. He had hidden a surprise under the wooden floor, but in ground this wet, the floor had its own essential purpose.\n\nThe palisades were too few to stop a determined enemy but, given time, he'd have the workers weave brambles and vines among them, and make a stouter barrier.\n\nHe shook his head. It didn't matter a damn, because the whole thing was a ruse anyway.\n\nThere were five bridges across the trench, each wide enough for two fully armed horsemen to ride abreast without making their horses shy. Again, given more time, he'd arrange mechanisms to raise and lower them.\n\nGiven time, he'd make his opponent look like a complete fool. But he didn't think he was going to get any more time. He could feel\u2014no better explanation that that\u2014his opponent's frustration. He didn't have much experience fighting men. He was arrogant.\n\n_Me too._ The captain grinned and turned Grendel to cross the last bridge before Bridge Castle. Grendel's hooves sounded hollowly, as if he were riding over a coffin.\n\n_Where'd that thought come from?_\n\nHe'd walked down to the apple tree at sunset the night before. She hadn't come. He wondered why. He remembered the touch of her lips on his.\n\n_Best concentrate on the matter at hand_ , he reminded himself.\n\nHe'd left her a note at the tree. She hadn't answered it.\n\nHe was running out of fog. Beyond, the spring fields were green with new grass that would eventually be hay and fodder\u2014or weeds\u2014all tinged red as the sun set.\n\nHe reined Grendel in, and waited for his company to sort themselves out.\n\nTom was at his shoulder, and he raised a gauntleted hand. \"Everyone look around. The fog makes it hard to see, but look at how the ground is clear from here all the way to the wood's edge\u2014not a ditch, not a hedgerow, not a stone wall. Keep that in mind. It we make another sortie it'll be along this path.\"\n\nTom nodded.\n\nSer Jehannes shook his head. \"Let's survive today before we borrow trouble for tomorrow.\"\n\nThe captain looked back at his senior officer. \"On the contrary, messire. Let us plan today for tomorrow's triumph.\"\n\nAnger touched the older knight's face.\n\n\"Peace!\" the captain said. \"We'll discuss this later.\" He kept his voice light, as if the issue were of no moment. \"If we contact the enemy, we ride through them, rally on the trumpet, and retreat into the fog immediately. No more. If we find boats, we destroy them. Is that clear?\"\n\nHe listened carefully. If he was nervous, it didn't show\u2014he seemed merely attentive.\n\nHorses fidgeted. Men spat and tried to appear as unconcerned as their captain.\n\nThe fog seemed too thin to cover so many men. But nothing happened.\n\nAnd then, well to the north, there were the sounds of men cheering, and horses neighing, and the clash of steel on steel.\n\n\" _There_ they are,\" muttered the captain; three words to express fifteen minutes of nervous impatience. Tom grinned. Jehannes reached up and hit the catch on his visor. The sound was repeated all along their line.\n\nBut now the captain seemed in no hurry.\n\nThe cries were redoubled.\n\nAnd then there were coarse bugle calls behind them, and high-pitched horn calls to the north.\n\nIt was all happening as he'd expected, and there, on the edge of battle, he had a moment of panic. _What if this is a trap? How can I expect to predict what they'll do? I'm pretending to know what I'm doing but this can't be so simple._\n\nHis tutor in the art of war had been Hywel Writhe, his father's master of arms. His _supposed_ father's master of arms. A brilliant swordsman, a magnificent jouster. Madly in love with the Lady Prudentia, and to no avail.\n\nA memory crept into place.\n\nRight there, on the edge of battle, the captain realised that he'd been had. His two tutors had been lovers. Of course they had been lovers.\n\n_Why do I think of this sort of thing when I'm about to fight?_ he thought.\n\nLaughed aloud.\n\nHywel Writhe used to say, _War is simple. That's why men prefer it to real life._\n\nAnd his lesson for all six of the boys who would grow to be great lords, masters of armies: _Never make a plan more complicated than your ability to communicate it._\n\nThe captain reviewed his plans one more time.\n\n\"Let's go,\" he said.\n\nThey rode out of the fog at a canter. About half a league to the north, Sauce led the northern sortie out of the shower of arrows sent by the now fully alerted Jacks, boglins and the irks who were gathering like clouds before a storm around her small force.\n\nThe captain led his men west into the setting sun, out of the fog, and right along the river bank.\n\nThere was a unmanned barricade on the road and he rode around it, and up the bank above the road, and round the first bend, and there they were.\n\nBoats.\n\nSixty boats, or more. Farmers' boats, dug-outs, canoes. Rafts of lashed branches. All pulled up out of the water.\n\nEvery archer threw a linen wrapped parcel into a boat. Some got none\u2014some got two\u2014and he heard horns, and trumpets, and some shrill calls to the north.\n\nThey were taking too long.\n\nThe archers down at the far end of the beach received some arrows and charged into the woods on their horses, scattering the enemy archers. Tom set off in pursuit with half the men-at-arms, and the captain suddenly feared he'd been trapped after all. He was over-extended and the size of the bank beneath the ancient trees dwarfed his paltry raid. And now half his men were getting too far\u2014\n\nMore shouts behind him.\n\nHe turned to Michael. \"Sound the recall,\" he said.\n\nMichael's trumpet playing wasn't his strongest suit. He was on his third try when the trumpet rang out clearly, against the sound of screams and heavy crashes from the west of the bank. The captain sat on Grendel's back in a rage of indecision\u2014desperate to get his men back, afraid to commit the rest to the sortie all the way down the bank.\n\nTom emerged from the lowering trees, his sword raised.\n\nThe captain began to breathe again.\n\nMore and more of the men-at-arms and archers emerged from beneath the trees, swords a ruddy green in the failing light.\n\n\"Let's get out of here,\" the captain said. He wheeled Grendel just as two arrows hit the horse's withers, and he reared and grunted and then they were around.\n\nThere were Jacks at the edge of the trees, just to the north, their dirty white cotes shining in the last light of day. The polished heads of their war arrows seemed to flicker as they loosed.\n\nBootlick, one of the foreign horn-bow archers, took a shaft in the neck, right through his aventail. He went down without a croak, and his horse, well trained, kept formation.\n\nBill Hook\u2014Bootlick's man-at-arms\u2014was off his charger in a flash of white armour, lifting the fallen archer onto his crupper. He was struck twice\u2014both blows at long range, falling on his breastplate, and he didn't even stagger.\n\nThe captain pointed Grendel's armoured head at the edge of the wood. If someone didn't stop the Jacks from shooting, his column was going to be dead in heartbeats. Most of the archer's light horses weren't even armoured.\n\nGrendel rose from a heavy canter to a flowing gallop, apparently unemcumbered by a hundred pounds of double mail.\n\nAn arrow struck his visor, and two more struck his helmet. The steel heads screamed against his bascinet and were gone, but each blow rocked him in his high-backed saddle. Another heavy arrow struck the bow of his saddle and another whanged off his right knee cop and then it was like riding through hail, and he put his armoured head down and pressed his long spurs to Grendel's sides.\n\nHe had no way of knowing whether anyone was behind him, and his whole world was narrowed to what he could see from the two eye slits of his helm.\n\nNot much. Mostly, Grendel's armoured neck.\n\n_Clang._\n\n_Clang-clang-clang-whang-ping._\n\nAll hits on his helmet and shoulders.\n\n_Thwak-tick-tock-clang!_\n\nHe sat up in the saddle. Got a hand on the hilt of his war sword and drew, and an arrow caught the blade, shivering it in his hand.\n\nHe got his eyes up, and there they were.\n\nEven as he watched, they broke and ran. There were only six of them\u2014 _All those arrows came from six men?_ and they ran with a practised desperation in six different directions.\n\nHis sword took the nearest neatly, because killing fleeing infantryman was an essential part of knightly training, taken for granted, like courage. He let his arm fall, and the man died, and he used his spurs to guide Grendel after the second man, the smallest of the group. One of his mates stopped, drew, and shot.\n\nCursed when his arrow glanced harmlessly off the captain's right rerebrace, and died.\n\nGrendel was slowing, and the captain turned him. If he exhausted the war horse he'd be stranded and dead. Besides, he loved Grendel. He felt he and the horse had a great deal in common.\n\nA healthy desire to live, for example.\n\nThe four surviving Jacks didn't run much farther than they had to, as they heard the hooves pause.\n\n_Whang,_ came the first arrow off his helmet.\n\nIt was a matter of time before one of those shots found his underarm, his throat, or his eye-slits.\n\nSer Jehannes came out of the woods to the archer's left, at a full gallop. He rode around the great bole of an ancient tree, and the ruddy-haired Jack lost his head in one swing of the knight's great blade.\n\nThe other three ran west, into a thicket.\n\n\"Thanks!\" the captain called.\n\nJehannes nodded.\n\n_He's never going to like me, let alone love me,_ the captain thought.\n\nHe gathered Grendel under him, turned his head, and started moving east.\n\nThe fields to the north of him seemed to ripple and flow\u2014boglins running in their odd hunched posture, low to the ground, irks, their brown bodies like moving mud.\n\nBut they were too late, and the handful of boglins who paused to loft arrows were ineffective.\n\nAt the edge of his effective casting range, the captain reined Grendel in. He stripped the gauntlet off his right hand, and pulled a small patch of charred linen from the palm.\n\nHe _stepped into his palace._\n\n_\"He's waiting for you,\" Prudentia said._\n\n_\"He doesn't know what I can do, yet,\" the boy said. He'd already aligned his symbols. He walked to the door, but instead of opening it, he merely raised the tiny iron plate that covered the keyhole, and a waft of fierce green shot through._\n\n_\"He's waiting for you,\" Prudentia said again._\n\n_\"He's going to have to keep waiting,\" the boy said. He was proud of his work, and his careful preparation. \"Look, it's sympathetic Hermeticism. The wicks on the fire bundles are all made from the same piece of linen and soaked in oil. I have a scrap here, too, already charred.\"_\n\n_The breath of green touched his symbols._\n\n_\"You are the cleverest boy,\" Prudentia said._\n\n_\"Were you and Hywel lovers, Prude?\" the boy asked._\n\n_\"None of your business,\" she shot back._\n\nHe rose in his stirrups, and his charred piece of linen caught and burned red hot.\n\nOn the bank, forty-four firebombs made of oiled tow and old rags and wax and birch bark burst into flame with one, simultaneous roar.\n\n# Harndon City\u2014Edward\n\nEdward cast the first of the master's tubes in the yard. He cast it in sand, and used the same mandril, polished to a mirror shine, as the model for the wax tongue he put in the mould to make it hollow. He cast the walls of the tube a finger's width thick, as the master requested.\n\nWhen it was done it wasn't much to look at. Edward shrugged. \"Master, I can do better. The hole would be better if I bored it, but that would require\u2014\" he shrugged \"\u2014a week to make the drills and other tools. I'd like to add decoration.\" He felt incompetent.\n\nThe master picked it up and held it in his hands for a long time. \"Let's try,\" he said.\n\nHe bored a small hole in the base of the bronze with a hand drill, and Edward was fascinated to watch his careful patience, coaxing the fine steel drill through the heavy bronze. Then he took the tube out of the shop and into the yard, and packed it with his burning mixture\u2014four scoops. He searched for something to put down atop the powder.\n\nSilently, Edward handed him a one-inch hawk bell. It wasn't perfectly round, and it was hollow, but it fitted well enough for purpose.\n\nThe master tied it to the oak tree, put wick in the hole, and lit it. They both hid behind the brick wall of the stable.\n\nWhich was just as well.\n\nThe fizzing, burning mixture made a flash and a bang like\u2014like something hermetical.\n\nIt stripped a handspan of bark from the tree.\n\nThe tube had torn loose from its bindings and had smashed through a horse trough\u2014a solid wood horse trough\u2014flooding the yard with dirty water... and it was a day before the apprentices found the hawk's bell. Even then, they didn't find the bell itself, just the neat round hole it had punched through the tile roof of the forge building.\n\nEdward looked at the hole and whistled.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain had six magnificent bruises from the archery. Other men had worse. Bootlick was dead, despite Bill Hook\u2014known to the gentles as Ser Willem Greville\u2014despite his best efforts at rescue. Francis Atcourt had a Jack's arrow right through the join in his cote of plates\u2014through his gut. Wat Simple and Oak Pew both had arrows in their limbs and were screaming in pain and mortal afraid the heads were poisoned.\n\nIf they hadn't had the nuns, all of them might have died of their wounds.\n\nAs it was, the skill and power of the nuns seemed to mean that any man who wasn't killed outright would be healed. The captain, who was just coming to terms with the idea of a convent of women of power, was staggered by the healing power they poured into his men\u2014Sauce had six badly wounded men, including Long Paw\u2014one of their best men in every respect.\n\nBut the barrage of phantasm was more effective than the barrage of arrows had been.\n\nThe captain walked through the hospital ward in his arming clothes. The wounded were cheerful\u2014as any man or woman might be, waking to find an ugly wound completely banished. Oak Pew, a woman whose dark wood-coloured skin and heavy muscles had given her the name, lay laughing helplessly at one of Wilful Murder's stories. Wat Simple was already gone, the captain had seen him playing at piquet. Long Paw lay watching Oak Pew laugh.\n\n\"Thought I was a goner,\" he admitted, when the captain sat on his counterpane. He showed the captain where a shaft had gone into his chest.\n\n\"I coughed up blood,\" he said. \"I know what that means.\" He raised himself, coughed, and looked at the nun in the corner. \"Pretty nun says if it'd been a finger's width lower, I'd ha' been dead.\" He shrugged. \"I owe her.\"\n\nThe captain squeezed Long Paw's shoulder. \"How do you feel?\" he asked. He knew it was a stupid question, but it was part of the job of being captain.\n\nLong Paw looked at him for a moment. \"Well,\" he said, \"I feel like I was dead, and now I'm not. It's not all bad\u2014not all good.\" He smiled, but it wasn't one of the archer's usual smiles. \"Ever ask yourself what we're here for, Captain?\"\n\n_All the time,_ he thought, but he replied, \"Sometimes.\"\n\n\"Never been that close to being dead before,\" Long Paw said. He lay back. \"I reckon I'll be right as rain in a day,\" he said. He smiled, a little more like himself. \"Or two.\"\n\nThe pretty novice was, of course, Amicia. She was slumped in a chair at the end of the lower ward. When he saw her, the captain realised he'd been hoping to find her in the hospital. He knew she had power\u2014had felt it himself, but he'd finally made the connection to healing when he saw her go in and out of the hospital building that adjoined the dormitory.\n\nHer closed eyes didn't invite conversation, so he walked softly past her, and up the steps, to see Messire Francis Atcourt. Atcourt was not a gently born man; not a knight. Rumour was he'd started life as a tailor. The captain found him propped up with pillows looking very pale. Reading. The parchment cover with its spidery writing didn't offer the captain a title, but closer to, he saw the man was reading psalms.\n\nThe captain shook his head.\n\n\"Nice to see you, m'lord,\" Atcourt said. \"I'm malingering.\"\n\nThe captain smiled. Atcourt was forty\u2014maybe older. He could start a fire, trim meat, make a leather pouch, repair horse harness. On the road, the captain had seen him teach a young girl to make a closed back-stitch. He was not the best man-at-arms in the company, but he was a vital man. The kind of man you trusted to get things done. If you asked him to make sure dinner got cooked, it got cooked.\n\nHe was not the sort of man who malingered.\n\n\"Me, too. You've lost a lot of blood.\" The captain sat on his counterpane.\n\n\"Your nun\u2014the pretty one\u2014\"\n\nThe captain felt himself blushing. \"Not my nun\u2014\" he stammered.\n\nAtcourt smiled like a schoolteacher. \"As you say, of course.\"\n\nIt was odd\u2014the captain had remarked it before. The commonly born men-at-arms\u2014leaving aside Bad Tom, who was more like a force of nature than like a man, anyway\u2014had prettier manners than the gently born. Atcourt had especially good manners.\n\n\"At any rate, the lovely young novice who gives orders so well,\" Atcourt smiled. \"She healed me. I felt her\u2014\" He smiled again. \"That is what goodness feels like, I reckon. And she brought me this to read, so I am reading.\" He made a face. \"Perhaps I'll finish up a monk. 'Ello, Tom.\"\n\nBad Tom towered over them. He nodded to his friend. \"If that arrow had struck you a hand's breadth lower, you could ha' _been_ a nun.\" Then he leered at the captain. \"The tall nun's awake, and stretching like a cat. I stopped to watch.\" He laughed his great laugh. \"What a set o' necks she has, eh?\"\n\nThe captain turned to glare, but it was nearly impossible to glare at Tom. Having sat, the captain could feel every tired muscle, every one of his six bruises.\n\n\"We all saw you charge those archers,\" Tom said, as he turned away.\n\nThe captain paused.\n\n\"You should a' died,\" Tom went on. \"You got hit what\u2014eight times? Ten? By war-bows?\"\n\nThe captain paused.\n\n\"I'm just sayin,' lad. Don't be foolish. You ha' the de'il's own luck. What if it runs out?\" he asked.\n\n\"Then I'll be dead,\" the captain said. He shrugged. \"Someone had to do it.\"\n\n\"Jehannes did it, and he did it _right_ ,\" Tom said. \"Next time, raise your sword and tell someone to ride at the archers. Someone _else._ \"\n\nThe captain shrugged again. For once, he looked every heartbeat of twenty years old\u2014the shrug was a rebellious refusal to accept the reality of what an adult was trying to teach him, and in that moment the captain was a very young man caught out being a fool. And he knew it.\n\n\"Cap'n,\" Tom said, and suddenly he was a big, dangerous man. \"If you die I much misdoubt we will ride through this. So here's my rede: don't die.\"\n\n\"Amen,\" said the captain.\n\n\"The pretty novice'll be far more compliant with a living man than a dead one,\" Tom said.\n\n\"That based on experience, Tom?\" Atcourt said. \"Leave the lad alone. Leave the _captain_ alone. Sorry, m'lord.\"\n\nThe captain shook his head. On balance, it was difficult to be annoyed when you discover that men like you and desire your continued health.\n\nAtcourt laughed aloud. Tom leaned over him, and whispered something, and Atcourt doubled up\u2014first laughing, and then in obvious pain.\n\nThe captain paused to look back, and Tom was taking cards and dice out of his purse, and Atcourt was holding his side and grinning.\n\nThe captain ran down the steps, his leather soles slapping the stone stairs, but she wasn't there, and he cursed Tom's leer and ran out into the new darkness.\n\nHe wanted a cup of wine, but he was sure he'd go to sleep. Which he needed.\n\nHe smiled at his own foolishness and went to the apple tree instead.\n\nAnd there she was, sitting in the new starlight, singing softly to herself.\n\n\"You didn't come last night,\" he said. The very last thing he wanted to say.\n\nShe shrugged. \"I fell asleep,\" she said. \"Which, it seems to me, might be a wise course for you. My lord.\"\n\nHer tone was forbidding. There was nothing about her to suggest that they'd ever kissed, or had intimate conversation. Or even angry conversation.\n\n\"But you wanted to see me,\" he said. _I sound like a fool._\n\n\"I wanted to tell you that you were perfectly correct. I plotted to meet you outside her door. And she used me, the old witch. I love her, but she's throwing me at you. I was blind to it. She's playing courtly love with you and substituting my body for hers. Or something.\" Amicia shrugged, and the motion was just visible in the starlight.\n\nThe silence stretched on. He didn't know what to say. It sounded quite likely to him, and he didn't see a way to make it seem better. And he found he had no desire to speak ill of the Abbess.\n\n\"I'm sorry that I spoke so brusquely, anyway,\" he said.\n\n\"Brusquely?\" she asked, and laughed. \"You mean, you are sorry that you crushed my excuses and made light of my vanity and my piety? That you showed me up as a sorry hypocrite?\"\n\n\"I didn't mean to do any of those things,\" he said. Not for the first time, he felt vastly her inferior. Legions of willing servant girls hadn't prepared him for this.\n\n\"I do love Jesus,\" she went on. \"Although I'm not always sure what loving God should mean. And it hurts me, like a physical pain, that you deny God.\"\n\n\"I don't deny God,\" he said. \"I'm quite positive that the petty bastard exists.\"\n\nHer face, pale in the new moonlight, set hard.\n\n_I'm really too tired to do this_ , he thought. \"I love you,\" he heard himself say. He thought of Michael and winced.\n\nShe put her hand to her mouth. \"You have a funny way of showing it,\" she said.\n\nHe sat down suddenly. Like saying _I love you_ , it wasn't really a decision. His legs were done.\n\nShe reached out a hand to take his, and as their fingers met, she flinched.\n\n\"Oh!\" she said. \"Gentle Jesu, messire, you are in pain.\"\n\nShe leaned over him, and she breathed on him. That's how it felt.\n\nHe opened his defences, running _into the tower. Prudentia shook her head, but her disapproval could be taken for granted for any woman, and he opened the door, secure that the walls of the fortress would protect him from the green storm._\n\n_No sooner did the door open, then she was all around him._\n\n_But the green was right behind her._\n\n_She was very distinct, and she looked the way ignorant men supposed ghosts to look\u2014a pale and colourless picture of herself._\n\n_He reached out and took her hand._\n\n_\"You are letting me in?\" she asked. She looked around, clearly amazed. She curtsied to Prudentia. \"Gracious and Living God, my lord\u2014is she alive?\"_\n\n_\"She is alive in my memory,\" he said, with some dissimulation. He had some secrets too evil to share._\n\n_She twirled. \"It's magnificent! How many sigils have you?\"_\n\n_\"Sigils?\" he asked._\n\n_\"Signs. Workings. Phantasms.\"_\n\n_He shrugged. \"More than twenty,\" he said. It wasn't a lie. It was merely an encouragement to underestimation._\n\n_She chuckled. She was bigger here, her face slightly more elfin and slightly more feral. Her eyes glowed like a cat at night, and were just faintly almond-shaped. \"I knew you when I first saw you,\" she said. \"Wearing power like a cloak. The power of the Wild.\"_\n\n_He smiled. \"We are two of a kind,\" he said._\n\n_She had his hand, and now she took it and put it on her right breast\u2014except that things here were not of the world. His hand didn't find her breast. Rather, he found himself standing on a bridge. Beneath him flowed a mountain stream, burbling a dark, clear brown, full of leaves. The trees on either bank were rich, verdant green, towering into the heavens. Now, instead of the grey raiment of her order, she wore a green kirtel and a green belt._\n\n_\"My bridge risks being swept away by a spring flood,\" she said. \"But your tower is too confining.\"_\n\n_He watched the power flow under the bridge, and he was a little afraid of her. \"You can cast all of this?\"_\n\n_She smiled. \"I'm learning. I tire quickly, and I don't have your twenty workings.\"_\n\n_He smiled. \"You know, unless Prudentia has misled me, now that we have been to each other's places we are bonded.\"_\n\n_\"As long as that armoured door of yours is closed, I can't even find you,\" she said. She gave him a flirtatious frown. \"I've tried.\"_\n\n_He reached out for her._\n\n_As his hands closed on her shoulders, her concentration slipped, or his did, and they were_ sitting on the bench in the apple-scented darkness.\n\nThey kissed.\n\nShe laid her head on his arming cote and he opened his mouth.\n\n\"Please don't talk,\" she said. \"I don't want to talk.\"\n\nSo he sat, perfectly happy, in the darkness. It was some time before he realised she'd magicked his bruises. By then she was asleep.\n\nLater, he had to pee. And the stone bench was icy cold, despite the warm spring air. And the edge of the bench bit into the back of his thigh at a bad angle. Gradually cut off the flow of blood to his leg, which began to go all pins and needles.\n\nHe wondered if it was his duty to wake her up and send her to bed. Or if he was supposed to wake her up and attack her with kisses. It occurred to him that the loss of a night's sleep was not a wise move on his part.\n\nLater still, he realised that her eyes were open.\n\nShe wriggled off his lap. He considered a dozen remarks\u2014all variations on being warmer than her gentle Jesus, but then dismissed them all.\n\nHe was, after all, growing up.\n\nHe kissed her hand.\n\nShe smiled. \"You pretend to be far worse than you are,\" she said.\n\nHe shrugged.\n\nShe reached into her sleeve, and put something in his hand. It was a plain square of linen.\n\n\"My vow of poverty isn't worth much, because I have nothing,\" she said. \"I did a little to ease the tire-woman's joints, and she gave me this. But I've cried in it. Twice.\" She smiled.\n\nHe hoped that he wasn't seeing her in the first light of morning.\n\n\"I think that makes it mine,\" she said.\n\nHe crushed it to his heart, pushed it inside his arming cote, kissed her hand.\n\n\"What do _you_ want?\" she asked.\n\n\"You,\" he said.\n\nShe smiled. \"Silly. What do you want out of life?\"\n\n\"You first,\" he said.\n\nShe smiled. \"I'm easy. I want people to be happy. To live free. And well. With enough to eat. In good health.\" She shrugged. \"I like it when people are happy.\" She smiled at him. \"And brave. And good.\"\n\nHe winced. \"War must be very hard on you.\" Winced again. \"Brave and good?\"\n\n\"Yes,\" she said. She shook her head. \"You don't know me very well, not yet. Now you. What do you want?\"\n\nHe shook his head. He didn't dare tell the truth and didn't want to lie to her. So he tried to find a middle ground. \"To defy God, and my mother.\" He shrugged, sure that her face had just hardened, set in automatic anger. \"To be the best knight in the world.\"\n\nShe looked at him. The moon was up\u2014that's all it was, not daylight\u2014and her face shone. \"You?\"\n\n\"If you can be a nun, I can be the best knight,\" he said. \"If you, the very queen of love, can deny your body to be a nun, then I\u2014cursed by God to sin\u2014can be a great knight.\" He laughed.\n\nShe laughed with him.\n\nThat's how he liked to remember her, ever after\u2014laughing in the moonlight, without the shadow of reserve in her face. She held out her arms, they embraced, and she was gone on soft feet.\n\nHe didn't even stop to shiver. He ran up the steps to the commanderies, drank off a cold cup of hippocras that had once been hot. But before he let himself sleep, he woke Toby and sent him for Ser Adrian, his company secretary. The man came softly, in a heavy wool overrobe.\n\n\"I don't mean to whine,\" said the scribe, \"But do you know what time it is?\"\n\nThe captain drank another cup of wine. \"I want you to ask around,\" he said. \"I don't know what I'm looking for, but I'm hoping you can find it for me. I know I'm not making sense. But there's a traitor in this fortress. I have suspicions, but nothing like a shred of proof. Who here can communicate with the outside world? Who has a secret hatred of the Abbess? Or a secret love of the Wild?\"\n\nHe almost choked on the last words.\n\nThe scribe shook his head. Yawned. \"I'll ask around,\" he said. \"Can I go back to bed?\"\n\nThe captain felt foolish. \"I may be wrong,\" he said.\n\nThe scribe rolled his eyes\u2014but he waited until he was out of the captain's door to do it.\n\nThe captain finished his cup and threw himself, fully dressed, on his bed. When the chapel bell rang he tried not to count the rings, so he could pretend he'd had a full night's sleep.\n\n_The Siege of Lissen Carak, Day Three._\n\nMichael could hear the captain snoring, and envied him. The archers said he'd \"been busy\" half the night with his pretty nun, and Michael was vaguely envious, vaguely jealous, and desperately admiring. And mad as hell, of course. It was _unfair._\n\nThe third day had been so without event that Michael had begun to wonder whether the captain was wrong. He'd told them the enemy would attack.\n\nAll day, the wyverns flew back and forth.\n\nSomething monstrous belled and belled, a high, clear note made somehow huge and terrifying in the woods.\n\n_No action today. We watched the enemy assemble rafts to replace the boats we burned. The captain warned us that they will eventually assemble machines of war\u2014that the men among the Enemy would traitorously teach the monsters to use them. The fog kept up all day, so that, while the sentries on the fortress walls can see many leagues, almost nothing can be seen of the fields immediately around the castle. The men say that the Abbess can see through the fog._\n\n_We heard cutting and chopping all day._\n\n_Towards sunset, a great force moved through the woods to the west. We could see the trees moving and the glitter of the late sun on weapons. And the roar of many monsters. The captain says a force is crossing the river. He ordered a sortie to form when another force, even larger, formed in the woods opposite our trench, but then dismissed us to dinner when there was no attack._\n\nMichael sat back. He wasn't any good at keeping a journal, and he knew that he was leaving out important developments. Wilful Murder had shot a boglin almost three hundred paces away\u2014shooting from a high tower, over the fog, on the dawn breeze. He was now drunk as a lord on the beer ration provided by his mates. But it didn't seem to change the siege. Or be a notable or noble event. Michael had only the histories from his father's library as his examples, and they never mentioned archers.\n\nThe captain came in. He had dark circles under his eyes.\n\n\"Go to bed,\" he said.\n\nMichael needed no second urging. But he paused in the doorway.\n\n\"No attack?\" he said.\n\n\"Your talent for stating the obvious must make you wildly popular,\" the captain said savagely.\n\nMichael shrugged. \"Sorry.\"\n\nThe captain rubbed his head. \"I was _sure_ he'd attack the trench today. Instead, he's sent something\u2014and I worry it's a strong force\u2014south across the river, despite our burning his boats. There's a convoy down there, he's going to destroy it, and I can't stop him\u2014or even try\u2014until I've bloodied his nose in my little trap, and my trap isn't catching anything.\" The captain drank some wine. \"It's all fucking hubris. I can't actually predict what the enemy will do.\"\n\nMichael was stung. \"You've done all right so far.\"\n\nThe captain shrugged. \"It's all luck. Go sleep. The fun part of this siege is over. If he doesn't go for my nice trench\u2014\"\n\n\"Why should he?\" Michael asked.\n\n\"Is that the apprentice captain asking, or the squire?\" the captain asked, pouring himself more wine. He spilled some.\n\n\"Just an interested bystander,\" Michael said, and casually, by mistake done-apurpose, knocked the captain's wine off the table. \"Sorry, m'lord. I'll fetch more.\"\n\nThe captain stiffened, and then yawned. \"Nah. I've had too much. He has to assume I've filled the trench with men and that with one good rush he can overrun it and kill half my force.\"\n\n\"But you have filled it with men,\" Michael said. \"I saw you send them out.\"\n\nThe captain smiled.\n\nMichael shook his head. \"Where are they?\"\n\n\"In the Bridge Castle,\" the captain said. \"It was very clever, but either he saw through the whole thing or he's too much of a coward to try us.\" He looked in his wine cup and made a face. \"Where's Miss Lanthorn?\" he asked. Then he relented. \"Why don't you go see her?\"\n\nMichael bowed. \"Good night,\" he said. And he slipped out into the hallway and pulled his pallet across the captain's door.\n\nHe spent an eternity searching the torchlit darkness.\n\nElissa was sitting on a barrel entertaining half the garrison with a lewd story. But her youngest sister wasn't there.\n\nMary was drinking wine in the Western Tower with Lis the laundress, Sukey Oakshot, the seamstress's daughter, Bad Tom, Ser George Brewes, and Francis Atcourt. There were cards and dice on the table, and the women were laughing hard. All seven looked up when Michael leaned in.\n\n\"She's not here,\" Tom yelled, and guffawed. The other men-at-arms laughed indulgently, and Michael fled.\n\n\"Who's not here?\" Lis asked.\n\n\"His leman. Boy's in love.\" Tom shook his head and his great hand, under the table, chanced against Sukey's ankle. She kicked him. \"Which I'm a'married,\" she said, apparently unafraid of the largest man in the castle.\n\nTom shrugged. \"Can't fault a man for trying,\" he said.\n\n\"Who's his leman, then?\" Lis asked. \"One o' your slatterns? He's too nice for an oyster, ain't he?\"\n\n\"Oyster?\" asked Mary.\n\n\"A lass as opens and shuts with the tide,\" Lis said, and drank more wine.\n\n\"Like you, eh?\" said Mary.\n\nLis laughed. \"Mary, you're a local girl. Boys think you are easy. That's a long chalk from what those girls do.\"\n\nFrancis Atcourt shrugged. \"They're people like everyone else, Lis. An' they play cards and go to church.\" He shrugged. \"Sorry. I got a deep draught of mortality today.\"\n\nTom nodded. \"Drink more.\"\n\nMary looked at Lis, caught beween admiration and anger. \"So what you do\u2014\" she said.\n\n\"What I do is live my life wi'out being ruled by a man,\" Lis said. \"Men is good for play and not so good for anything else.\"\n\nTom laughed.\n\nSer George tossed his cards on the table, disgusted. \"What is this, philosophy hour?\"\n\n\"And it's your fucking sister the young squire's riding,\" Lis said. She wasn't sure just why she was angry.\n\nMary stood up, affronted. \"That's just like Fran\u2014make a rule and then break it herself.\"\n\nLis laughed. \"Not Fran.\"\n\nMary stopped dead. \"Kaitlin? She's not\u2014she wouldn't! She's\u2014\"\n\nLis laughed.\n\nMichael found her in the stable with three other girls, all younger. They were dancing. He went from horse to horse, looking them over. The girls stopped dancing, and one suddenly shouted that she was an evil monster and started shrieking, and the other two were laughing, or crying.\n\nAnd then one of them was screaming, and Kaitlin was soothing her. Michael had been fooled by the screams, but he was over the stall and with them in a moment.\n\nKaitlin's eyes met his. She had the little girl pressed against her.\n\n\"We're going to be _eaten,_ \" bawled the child.\n\nKaitlin rocked her back and forth. \"No, we're not,\" she said firmly. She raised her face to Michael.\n\nMichael knew she was asking something of him, but neither of them were sure exactly what it was. So he knelt with them. \"I swear on my hope of being a knight and going to heaven, I will protect you,\" he said.\n\n\"He's not a knight, he's just a squire,\" said the other girl, with the dreadful truthfulness that afflicts the young. She looked at Michael with enormous eyes.\n\nKaitlin's eyes met his.\n\n\"I will protect you anyway,\" Michael said, keeping his voice light.\n\n\"I don't want to be eaten!\" said the first girl. But the sobs were fading.\n\n\"I'll bet we're gooey and delicious!\" said the second girl. She grinned at Michael. \"And that's why they attack us!\" she said, as if this solved a deep, difficult problem she'd been having.\n\nKaitlin hugged them both. \"I think some people are silly,\" she said.\n\nThe third girl threw a clod of horse manure at Michael and he was caught in an odd dilemma. He wanted Kaitlin alone and yet, watching her with children, he wanted this moment to go on forever. And for the first time, he thought\u2014 _I could marry her._\n\nAmicia reached out. _His door was very slightly open and she slipped through, a wraith in the green light. The Warlock who laid siege to the fortress wasso powerful that he shone like a green sun in her woods, and the green light battered his tower door._\n\n_He was there, standing by the statue of a woman._\n\n_\"I was just coming to look for you,\" he said happily. And yawned._\n\n_She shook her head. \"Go to sleep. You didn't even renew your powers this morn.\"_\n\n_He shook his head. \"One hour with you\u2014\"_\n\n_She backed away. \"Good night,\" she said, and she shut the door. From outside._\n\nHe fell asleep so quickly he dreamed of her.\n\nMichael leaned down and placed his mouth tenderly on hers, and her lips opened under his.\n\n\"I love you,\" he said.\n\nShe laughed. \"Silly.\"\n\nHe grabbed her chin. \"I'll marry you,\" he said.\n\nHer eyes grew huge.\n\nThe door of the next stall flew open. \"Kaitlin Lanthorn!\" shrieked her sister. \"You little bitch!\"\n\nGreen light exploded in the sky outside the stables, and a thunderous concussion shook the walls.\n\n\"To arms!\" shouted twenty voices on the walls.\n\nThe captain leapt from his bed without knowing what had awakened him, and found himself standing by his armour rack with Michael, who had never gone to bed, getting him into his hauberk. He wasn't even awake and Michael was pulling the laces as tight as he could at the back, and then he had his old shoes on over bare legs and was racing along the wall.\n\n\"Bridge Castle,\" Bent shouted from the tower above them. Michael was trying to get into his brigantine while simultaneously watching the starlit sky and the walls.\n\nThe fog was gone\u2014it had been swept away in a mighty gust of wind. The captain felt the wind, and knew it for what it was. He smiled into it.\n\n\"Here we go,\" he said.\n\nTwo beacon fires were alight, and there was a lot of shouting\u2014the distinctive sound of men in danger, or anger.\n\n\"We need a way to communicate with the Bridge Castle.\" The captain leaned on the wall as Michael, now secure in his brigantine and feeling the pain from his ribs, knelt to buckle his knight's metal leg harnesses on\u2014a pair of valets were carrying the armour along behind them as the captain moved. It might have been comical, if the situation hadn't been so terrifying.\n\nMichael gradually got the captain into his harness as the infuriating man moved from position to position throughout the fortress. He made off-colour jokes to nursing sisters and he clasped hands with Bad Tom and he ordered Sauce to mount up in the new covered alley in the courtyard\u2014covered, Michael assumed, to keep the wyverns off the horses. It was the same sortie he'd prepared the night before, and ordered to stand down.\n\nAn hour later, the west tower ballista loosed with a sharp _crack_. As far as Michael could see the bolt had no effect out in the dark.\n\nMichael got the rest of his own armour on, paused to rest, and fell asleep standing up at the corner where the west wall intersected the west tower.\n\nHe awoke to a loud roar. A sea of fire stretched almost to his feet and screams pierced the full-throated bellow of war. The captain's hand closed on his vambrace. \"Here they come!\" he shouted. \"On my mark!\"\n\nMichael looked up, and saw a man leaning far out over the west tower edge, and the sky was not light, but it was grey.\n\n\"Welcome back,\" the captain said cheerfully. \"Have a good nap?\"\n\n\"Sorry,\" Michael mumbled.\n\n\"Don't be. Real soldiers sleep every minute they can, in times like this. Our attackers are making an attempt on the Bridge Castle and the Lower Town, while, I assume, sending men to look at what we built yesterday. Or perhaps to burn it.\" He sounded quite happy about the prospect.\n\nMichael took a deep breath. A valet put a cup of warm wine into his hand and he drank it off.\n\nThe captain leaned well out over the wall. \"Loose!\" he called.\n\nThe trebuchet in the western tower creaked, and the whole tower moved by the width of a finger.\n\n\"Hail shot. Watch this.\"\n\nMichael had sometimes entertained his brothers and sisters by throwing handfuls of stones into water. This was like that, only multiplied many hundreds of times, with larger stones, and instead of striking water most of them hit the ground. The rest fell on chitinous hides and flesh and blood, having fallen several hundred feet.\n\n\"Again!\" the captain called.\n\nDown in the Bridge Castle, both of their heavy onagers loosed together, throwing baskets of stones the size of a man's heart out into the trenches built the day before.\n\nScreams rose out of the churned ground.\n\n\"You seem very pleased with yourself,\" the Abbess said. She was fully dressed and looked exactly the same as she did in the height of a calm day. She had come around the corner by the west tower, attended by stretcher-bearers and a pair of nursing sisters.\n\n\"The enemy has just fallen into our little trap with both feet.\" He turned to Bent. \"We'll get one more round off. Then raise both red flags. At that signal everyone\u2014everyone in the garrison except you and the engine crews\u2014attacks down the road. On me.\"\n\nThe captain then managed to combine a bow to the Abbess with a duck under the lintel of the West Tower door. The valets had Grendel saddled, and the captain took his place at the head of Sauce's column. Michael, still fuzzy headed and with his ribs burning in his chest, tried to keep up with him.\n\nJacques was standing by Michael's horse. \"You looked like you needed your sleep,\" the man said, with a smile. \"Don't get fancy, youngster. Those ribs will kill you.\" He leaned close. \"So will kissing girls, if it costs you sleep.\"\n\nThen Michael was up, Jacques's hand shoving his rump to get him into the saddle, and he was out of the low stable gateway and into the courtyard. Toby was holding the captain's helmet while eating a half-loaf of bread, and the captain was pinning something\u2014a white linen handkerchief\u2014on his cote armour. It was very white against the scarlet velvet.\n\nMichael grinned. \"What is that?\" he began.\n\n\" _Honi soit qui mal y pense_ ,\" said the captain. He winked, took his helmet from Toby, gave the boy a smile, and wheeled Grendel with his knees. \"Listen up!\" he called.\n\nThe sortie quieted.\n\n\"Once we're through that gate kill everything that comes under your sword,\" the captain said. \"The trench edge will be marked in fire so remember your route. If you lose me, follow the route. When you hear Carlus sound the recall, you turn and come back. Understand me?\"\n\nAnd with that as a speech, they rode from the gate as the trebuchet sprayed another rain of death out over their heads.\n\nThe hour was on the knife edge between day and night, and the trebuchet's great baskets of stones had obliterated life over a swathe of ground that was roughly the shape of a great egg\u2014creatures had been turned to a bloody or ichorous pulp, and the ground itself was littered with the stones\u2014softer ground had deep pock-marks. Bushes and grass were pulverized. In the half-dark it was a vision of hell, and the sudden burst of balefire in the new-dug trenches added to the terrible aspect.\n\nEspecially when viewed through the slit of a closed visor.\n\nThere was no fight in the men or the monsters that struggled to win free of the beaten ground, or routed away from the hail of missiles still pelting them from the Bridge Castle. They were streaming for the woods, over a mile distant.\n\nThe captain led his sortie well to the south, right along the river, along the smooth ground, and then formed them up in a single rank and brought up his trumpeter and his great black banner with the lacs d'amour and the golden collars\u2014his personal badge\u2014and drew his sword.\n\n\"All the way to the edge of the wood, and then _form up on me._ \" He had his visor up, and he looked around\u2014Bad Tom was at his back, Sauce to one side, and Ser Jehannes was close.\n\n\"Kill everything that comes under your sword,\" he said again. Michael didn't think they'd lost a single man getting here. The war machines had utterly shattered the enemy attack. He took a deep breath, and the routed enemy flowed past them, running on exhausted feet\u2014or talons or claws or paws\u2014for the woods.\n\n\"Charge!\" he roared, and the banner pointed at the enemy and the trumpets sounded.\n\nMichael had never been in a charge before.\n\nIt was exhilarating, and nothing on the ground seemed to be able to touch them. They swept over the irks and the broken men and a single larger creature, something nightmarish that gleamed a sickening green hue in the first light of the sun, but Bad Tom put his lance tip precisely in the thing's ear-bole as it turned its talons on Grendel, and his lance tip\u2014a spear point as long as a man's forearm and as wide as a big man's palm\u2014ripped its brain pan from its lower jaw.\n\n\"Lachlan for Aa!\" the big man roared.\n\nThe monster died, and the line of knights swept over the pitiful resistance and then into the running men\u2014and things.\n\nBy the time the sun was above the horizon, they had reached the wood's edge, and the creatures and men of the Wild were a bloody mangle on the grass behind them\u2014or rather, any they'd chanced on were a bloody mangle, while hundreds more ran around them to the north or south, or lay flat and prayed as the horses thundered over them.\n\nAnd then the captain led them back to the gate by much the same road, crashing though a line of desperate irks trying vainly to defend themselves with spears which splintered on steel armour. Right through, and on to the edge of the fortress hill, where twenty valets waited with fresh horses.\n\nMichael was mystified. His elation was ebbing quickly to be replaced by fatigue and the thumping pain of his ribs, jarred by the gallop and barely held together by his cote armour.\n\nAll of the men-at-arms and many of the archers were changing horses. The men on the walls were cheering them.\n\nThe captain rode up to him and opened his visor. \"You're moving badly,\" the captain said bluntly. \"In fact, you look like shit. Fall out.\"\n\n\"What? Where\u2014\" Michael spluttered.\n\nJacques took his reins. Michael noted that the valet was in armour\u2014good armour\u2014as Jacques got him out of his saddle and Michael wanted to cry\u2014but at the same time, he couldn't imagine fighting again.\n\nThen Jacques swung up on a heavy horse of his own\u2014an ugly roan with a roman nose. \"I'll keep him alive, lad,\" Jacques said.\n\nSo Michael stood there and watched as they changed horses and formed up, and then to his surprise they turned away from the beaten enemy and rode south, along the edge of the rising sun, moving at a canter. They rode straight for the Bridge Castle's gate, and it opened as if by magic letting them pass through, canter over the bridge, and vanish onto the southern road.\n\nEven as he watched, Gelfred, the master of the hunt, left Bridge Castle with three men and a cart. The men each took a brace of dogs\u2014beautiful dogs\u2014and moved briskly off to the west with a dozen archers covering them.\n\nJust as the first starlings and ravens began to appear, gyrfalcons began to soar into the heavens over Bridge Castle, one after another. Up on the walls, a great eagle leaped into the air with a scream that must have chilled every lesser bird for three leagues.\n\nGelfred had struck, and the Abbess with him.\n\nBraces of hounds emerged from the cover of Bridge Castle, running flat out for the leverets and the coneys and any other animal that lurked at the edge of the woods, and the gyrfalcon, Parcival the eagle and the lesser birds\u2014well-trained birds brought from Theva to sell at the fair\u2014struck the starlings, the ravens, and the oversized doves, ripping through their flocks like a knight through a crowd of peasants, and feathers, wings, blood and whole dead birds fell like an avian rain.\n\nIt took Michael half an hour to climb back to the fortress gate. The valets ignored him, and he stumbled many times, until someone on the walls saw the trail of blood he was leaving and a pair of archers appeared to hold him up.\n\nAmicia cut the sabatons off his feet, and found the flint javelin which had cut deeply into the muscle at the back of his leg. Blood was flowing out like beer from an open tap.\n\nShe was speaking rapidly and cheerfully, and he just had time to think how beautiful she was.\n\n# Lissen Carak\u2014the Abbess\n\nThe Abbess watched the captain's sortie head east along the road, moving so fast that they were gone from sight before she recovered her eagle.\n\n_I have certainly given away my station to every gentleman here,_ she thought. She wondered if the siege would leave her any secrets at all.\n\nParcival, her magnificent Ferlander eagle, was killing his way through the flocks of wild birds like a tiger let loose in a sheepfold. But she could see the big old bird was tiring, and she began to cast her lure. Just to be sure.\n\nShe whirled it carefully over her head, and Parcival saw it, turned at the flash of Tyrian red, and abandoned his pursuit of his defeated enemies. He came to her like a unicorn to a maiden\u2014shyly at first, and finally eager to be caught.\n\nHis weight was far too much for her, but young Theodora helped her, and got a faceful of wings for her trouble as the creature bated and bated again, unused to his mistress having a helper. But she got the jesses slipped over his talons, and Theodora put the hood on him, and he calmed, while the Abbess said, \"There's my brave knight. There's my fine warrior\u2014you poor old thing.\" The eagle was tired, grumpy and very pleased with himself, all at the same time.\n\nTheodora stroked his back and wings and he straightened up.\n\n\"Give him a morsel of chicken, dear,\" the Abbess said. She smiled at the novice. \"It's just like having a man, child. Never give him what he wants\u2014only give him what _you_ want. If he eats too much we'll never get him into the air again.\"\n\nTheodora looked out from the height of the tower. The plain and the river were far below them, and the eagle's sudden stoop from this height had shattered the lesser birds.\n\nAmicia appeared from the hospital with a message from Sister Miram. The Abbess looked at it and nodded. \"Tell Miram to use anything she needs. No sense in hoarding.\"\n\nAmicia's eyes were elsewhere. \"They're gone,\" Amicia said. \"The enemy's spies. Even the wyverns. I can feel it.\"\n\nTheodora was startled that a novice would speak directly to the Abbess.\n\nThe Abbess seemed untroubled. \"You are very perceptive,\" the Abbess said. \"But there's something I don't like about this.\" She walked to the edge of the tower and looked down. Just below her, a pack of nuns stood on the broad platform of the gatehouse and watched the end of the rout below and the disappearing column of dust that marked the captain's sortie.\n\nOne nun left the wall, her skirts held in her hands as she ran. The Abbess wondered idly why Sister Bryanne was in such a hurry until she saw the priest. He was on the wall, alone, and praying loudly for the destruction of the enemy.\n\nThat was well enough, she supposed. Father Henry was a festering boil\u2014his hatred for the captain and his attempts to _discipline_ her nuns were heading them for a confrontation.\n\nBut the siege was pushing the routine away, and she worried that it would never return. And what if the captain went out and died?\n\n\"What do you say, my lady?\" Amicia asked, and the Abbess smiled at her.\n\n\"Oh, my dear, we old people sometimes say aloud what we ought to keep inside.\"\n\nAmicia, too, was looking out to the east where a touch of dust still hung over the road that ran south of the river. And she wondered, like every nun, every novice, every farmer and every child in the fortress, why they were riding away, and if they would return.\n\n# North of Albinkirk\u2014Peter\n\nPeter was learning to move through the woods. Home for him was grass savannah, dry brush and deep-cut rocky riverbeds, dry most of the year and impassable with fast brown water the rest. But here, with the soft ground, the sharp rock, the massive trees that stretched to the heavens, the odd marshes on hilltops and the endless streams and lakes, a different kind of stealth was required, a different speed, different muscles, different tools.\n\nThe Sossag flowed over the ground, following trails that appeared out of nowhere and seemed to vanish again as fast.\n\nAt mid-day, Ota Qwan stopped him and they stood, both of them breathing hard.\n\n\"Do you know where we are?\" the older man asked.\n\nPeter looked around. And laughed. \"Headed for Albinkirk.\"\n\n\"Yes and no,\" Ota Qwan said. \"But for a sailor on the sea of trees, you are fair enough.\" He reached into a bag of bark twine made into a net that he wore at his hip all the time, and drew forth an ear of cooked corn. He took a bite and passed it to Peter. Peter took a bite and passed it to the man behind him\u2014Pal Kut, he thought the man called himself, a cheerful fellow with a red and green face and no hair.\n\nPeter reached into his own bag and took out a small bark container of dried berries he'd found in Grundag's effects.\n\nOta Qwan ate a handful and grunted. \"You give with both hands, Peter.\"\n\nThe man behind him took half a handful and held them to his forehead, a sign Peter had never seen before.\n\n\"He's telling you that he respects the labour of your work and the sacrifice you make in sharing. When we share pillaged food\u2014well, it never really belonged to any of us to begin with, did it?\" Ota Qwan laughed, and it was a cruel sound.\n\n\"What about the dinner I cooked?\" Peter asked, ready to be indignant.\n\n\"You were a slave then!\" Ota Qwan thumped his chest. \"My slave.\"\n\n\"Where are we going?\" Peter asked. He didn't like the way Ota Qwan claimed him.\n\nSkadai appeared out of thin air to take the last handful of the dried berries, he, too, made the gesture of respect. \"Good berries,\" he said. \"We go to look at Albinkirk. Then we hunt on our own.\"\n\nPeter shook his head as the war captain moved on. \"Hunt on our own?\"\n\n\"Yesterday, while you rutted like a stag\u2014wait, do you know even who Thorn is?\" Ota Qwan asked him, as if he were a child.\n\nPeter wanted to rub his face in it, but in truth, he'd heard the name mentioned but didn't know who he was. And he was increasingly eager to know how his new world worked. \"No,\" he said, pouting.\n\nOta Qwan ignored his tone. \"Thorn wishes to be the lord of these woods.\" He made a face. \"He is reputed a great sorcerer who was once a man. Now he seeks revenge on men. But yesterday he was defeated\u2014not beaten, but bloodied. We did not follow him to battle because Skadai didn't like the plan he heard, so now we go east to fight our own battles.\"\n\n\"Defeated? By whom?\" Peter looked around. \"Where was this battle?\"\n\n\"Six leagues from where you rutted with Senegral, two hundred men died and twice that number of creatures of the Wild.\" He shrugged. \"Thorn has ten times that many creatures and men at his beck and call and he summons more. But the Sossag are not slaves, servants, bound men\u2014only allies, and only then when it suits our need.\"\n\n\"Surely this Thorn is angry at us?\" Peter asked.\n\n\"So angry that if he dared he would kill us all, or destroy our villages, or force Skadai to die in torment.\" Ota Qwan cackled. \"But to do so would be to forfeit the alliance of every creature, every boglin, every man in his service. This is the _Wild_ , my friend. If he had won we would look foolish and weak.\" Ota Qwan gave a wicked smile. \"But he lost, so it is he who looks foolish and weak while we go to burn the lands around Albinkirk, which was built on our lands many years ago. We have long memories.\"\n\nPeter looked at him. \"I assume you were not born a Sossag.\"\n\n\"Hah!\" Ota Qwan sighed. \"I was born south of Albinkirk.\" He shrugged. \"It boots nothing, friend. Now I am Sossag. And we will burn the farms of the city, or what Thorn has left of it. He wants the Castle of the Women, which interests us not at all.\" Ota Qwan gave a queer smile. \"The Sossag have never been to war with the Castle of Women. And he has failed.\" Ota Qwan looked into the distance, where the mountains rolled like waves on the sea. \"For now. And Skadai says, let Albinkirk see the colour of our steel.\"\n\nThe words thrilled Peter, who thought he should be too mature to fall for such things. But war had a simplicity that could be a relief. Sometimes, it is good merely to hate.\n\nAnd then Peter thought that Ota Qwan was an injured soul who had fallen into the Sossag to heal himself. But the former slave shook his head and said to himself, \"Be one of them. And you will never be another man's slave.\"\n\nAt nightfall on the second day they were in sight of the town. Peter sat on his haunches, eating a thin rabbit that he'd cooked with herbs, sharing it with his new band. Ota Qwan had complimented his cooking, and had admitted that their new band of war-brothers\u2014Pal Kut, Brant, Skahas Gaho, Mullet and Barbface (the best Peter could do with his name) were gathered as much for Peter's food as for Ota Qwan's leadership.\n\nEither way, it was good to belong. Good to be part of a group. Brant smiled when he took food. Skahas Gaho patted the ground on his blanket when Peter hovered by the fire, looking for a place to sit.\n\nTwo days, and these were his comrades.\n\nSkadai came to their fire towards true dark, and sat on his haunches. He spoke quickly, smiled often, and then surprised Peter by patting him on the arm. He ate a bowl of rabbit soup with his fingers, grinned, and left them for the next fire.\n\nOta Qwan sighed. The other men took sharpening stones from their bags, and began to touch up their arrowheads, and then their knives, and Skahas Gaho, who had a sword, a short, heavy-bladed weapon like a Morean xiphos, made the steel sing as he passed his stone over it.\n\n\"Tomorrow, we fight,\" Ota Qwan said.\n\nPeter nodded.\n\n\"Not Albinkirk,\" Ota Qwan said. \"A richer target. Something to take home with us. Something to make our winter shorter.\" He licked his lips and Brant asked him a question and then guffawed at the answer.\n\nSkahas Gaho kept sharpening his short sword, and men began to laugh. He was stroking it tenderly, with long, lingering strokes of the stone. And then shorter, faster ones.\n\nBrant laughed, and then spat disgustedly and unrolled his furs.\n\nPeter did the same. He had no trouble getting to sleep.\n\n# South and East of Lissen Carak\u2014Gerald Random\n\nRandom had been ready for ambush for five days, and it didn't matter when it happened. His men almost won through.\n\nAlmost.\n\nThey were now riding through deep forest, and the western road was a double cart track with the trees sometimes arched right over the road. However, the old forest was open, the great boles of the trunks sixty feet apart or more with little enough underbrush so his flankers could ride alongside, his advance guard could clear the trail a hundred horse-lengths wide, and his wagons were moving well\u2014it was the fifth straight day without rain, and the road was dry except in the deeper ruts and puddles and some deep holes like muddy ponds.\n\nThe woods were so deep that it was difficult to gauge the passage of time, and he had no idea how far they'd travelled on the narrow track until Old Bob rode back to say that he thought he could hear the river.\n\nAt that news, Random's heart rose. Even though what he was doing was suicidally foolish, to aid an old Magus, and his wife would never approve when she found out.\n\nHe was on the lead wagon, and he stood up to look\u2014a natural thing to do, even when listening would have been more natural. But all he could hear was the wind in the trees overhead.\n\n\"Ambush!\" shouted one of the vanguard. He pointed at a dozen boglins around a young troll, a monster the size of a plough horse with the antlers like a great elk's and a smooth stone face like the visor on a black helmet. It was covered in thick armour plates of obsidian.\n\nThe troll charged them, racing like a mad dog straight at the wagons. The horses panicked but his men did not, and quarrels flew thick as snow, and the troll screamed and slowed, seemed for a moment to be swimming through steel, and then suddenly fell with a crash.\n\nThe boglins vanished.\n\nRandom, standing on the wagon seat, took an arrow in the breastplate. It didn't bite, but it knocked him off the seat and it hurt when he got to his feet\u2014his shoulder hurt and his neck hurt like fire.\n\nJust ahead, the vanguard was abruptly locked up with more boglins.\n\nOld Bob was racing for the melee.\n\nRandom watched as his soldiers crushed the smaller creatures with weight of horse and better weapons, better skill, and the boglins, as was inevitable, broke and ran.\n\nOld Bob shouted something, but his words were lost in the triumph of the moment, and the troopers turned as one to harry the fleeing boglins... and suddenly the trolls were on them\u2014a pair, smashing in from either flank. Blood came off the melee like smoke as they struck, and horses died faster than they could fall to the ground.\n\nRandom had never seen a troll before, but the Wild name for them\u2014dhag\u2014stuck in his mind, as did a picture in a Book of Hours he'd purchased in Harndon for the market\u2014taller than a peasant's house, as black as night or expensive velvet, with plates of black stone like armour and no face, topped with antlers like clubs. A troll could crush an armoured man's breastplate in one blow and behead him in another, could move as fast as a horse and as quiet as a bear.\n\nThe vanguard was dead before Random could close his mouth. Six men gone in a breath.\n\nOld Bob had a light lance, and he lowered the point\u2014one of the monsters turned, almost falling as it skidded to a stop, feeling the vibration of the charging horse. It braced itself, head low, horn-clad feet still churning at the earth, and Random could see the great plate of stone that protected its skull.\n\nAnd then Old Bob's horse was by and his lance, thrown, not couched, went into the beast's side\u2014struck deep between two stone plates, and the meaty sound of the heavy spearhead going home in the flesh carried across the distance.\n\nA dozen bolts hit the creature.\n\nGawin had the rearguard up, forming to the right and left, with companies of guildsmen coming up next to the wagons on either side\u2014not the smoothest, and their faces were as white as snow, and their hands shook, but they were coming.\n\n\"Halt!\" Gawin shouted, and Guilbert came from the other side of the wagons with another five of the wagon guards.\n\nGuilbert took command with one glance. \"Pick a target!\" he called.\n\nThe woods were surprisingly silent.\n\nOld Bob had his horse around, but he never saw the dozen boglins coming. One of them put a spear into his horse effortlessly, like a dancer, the squat thing pirouetting as its spear skewered the beast, and the horse stopped its turn and gave a shrill scream as the wounded troll attacked. Its first blow ripped Old Bob's lower jaw off his face under the brim of his open-faced helmet\u2014then it crushed his breastplate so that a fountain of blood blew out of his open throat.\n\nThe wounded troll slumped. The second one stopped and bent down to feed off both of them, its visor opening and a set of fangs showing sharp against the black of its mouth.\n\nThe wave of boglins charged the line of bowmen and soldiers, and this time his men broke and ran.\n\nRandom watched them with complete understanding, terrified and virtually unable to move his limbs too, and the sight of the old knight being ripped asunder by the troll seemed to have numbed his mind. He tried to speak. He watched as the guildsmen shuffled, cursed, and turned too. The guards had horses, and they put their spurs to their mounts.\n\n\"Stand!\" Guilbert shouted. \"Stand or you are all dead men!\"\n\nThey ignored him.\n\nAnd then Ser Gawin laughed.\n\nThe sound of his laughter didn't stop the horrified men from running. It didn't stop the mounted men from raking their spurs into their panicked mounts... but it did make many men turn their heads.\n\nHis visor fell over his face with a click.\n\nHis destrier took its first steps, already moving quickly, as any horse trained to the joust knows to do.\n\nHis lance, erect in his hand, dipped, the pennon fluttering, and then he was moving like a streak of steel lightning across the ground between the wagons and the boglins. They, in turn, froze like animals hearing a hunter's call.\n\nThe feeding troll raised its head.\n\nThe chief of the boglins raised a horn and it blew a long, sweet note. Other horns echoed, and Random was suddenly freed from the vice of fear that ground against his heart. He got his sword clear of its scabbard.\n\n\"Hear me, Saint Christopher,\" he vowed, \"if I live through this, I will build a church to you.\"\n\nGawin Murien settled the lance in its rest. The boglin chief was standing on the dead troll's chest, the one the crossbowmen had killed, and the knight's heavy lance passed through him so quickly that for a heartbeat Random thought the knight had missed, until the small monster was lifted from its feet, all limbs writhing in a horrible parody of an impaled insect, and a thin scream lifting from its throat, and then it was crushed against the stone wall of the remaining troll's head with a wet sound like a melon breaking, and the stone troll staggered under the impact.\n\nIt roared\u2014a long belling sound that made the woods ring.\n\nGawin thundered away to the right. He kept his lance, passed through a thicket and emerged to the far right of the wagons. His horse was moving at a slow canter.\n\nGuildsmen and soldiers began to gather again, their rout forgotten, and the boglins began to reach them in ones and twos, their desultory chase become a desperate melee in the turn of a card. A dozen guildsmen were cut down, but instead of spurring the others to run, the deaths of their comrades pulled more and more of the townsmen back to their duty.\n\nOr perhaps it was Gawin's repeated war cry that did it, ringing as loud as the monsters roar. \"God and Saint George!\" he shouted, and even the wagons trembled.\n\nThe troll dipped its antlers, and spat something. Great clods of moss flew up, and a smell filled the air\u2014a bitter reek of musk. Then it lifted its armoured head and charged, shoulders bunched from its first massive leap forward.\n\nRandom swung his sword, his right arm seeming to function independently of his mind, and smashed a boglin with the blow. He backed a step, suddenly aware of a dozen of the things around him, and he got his sword up, point in line, left hand gripping it halfway down the blade.\n\nHe charged them. He had the example of the knight before his eyes, and he only had a faint notion that there was more to a charge than bluster. He felt the pain of the first wound and the pressure of the blows on his shoulders and backplate, he also had the time to kill one boglin with the point, break a second with his pommel so that it seemed to burst, and then sweep the legs from a third in as many heartbeats. They had armour\u2014whether it was their own chitinous skin or something made of leather and bone, he had no time to tell\u2014but his heavy sword penetrated it with every thrust, and when it did, they died.\n\nLight flashed, as if lightning had pulsed from the sky.\n\nIn a single heartbeat all of his opponents fell, and as they fell they turned to sand. His sword actually passed through one, and beyond his suddenly evaporated opponents, Ser Gawin rode directly at the troll. A horse length from collision his destrier danced to the right\u2014and Ser Gawin's lance passed under its stone visor to strike it hard in the fanged mouth, plunging his lance the length of a man's arm into the thing's throat and crossing the massive monster's centre-line so that it snapped the lance and tripped the beast, which fell, unbalanced, its armoured head digging a massive furrow in the earth as Ser Gawin and his destrier danced clear.\n\nLightning pulsed again, and two dozen more boglins spattered to the ground.\n\n\"Rally!\" Guilbert demanded.\n\nThe guildsmen were winning.\n\nAnd every boglin they broke, skewered, or sliced reinforced their growing belief that they might win this battle.\n\nMen were still falling.\n\nBut they were going to hold.\n\n... until the horses and the oxen panicked, and shredded their column in ten breaths of a terrified man. A wagon plunged through the largest block of guildsmen, scattering them, and the boglins who had stopped, or slowed their charge, or simply balked at entering weapons range, suddenly surged forward. A dozen more guildsmen died at their hands, and the wall of wagons protecting the right of the column was gone.\n\nRandom got his back to Guilbert's. \"Stand fast!\" he yelled. \"Stand fast!\"\n\nA few feet away, Harmodius pulled a riding whip from his belt.\n\n\"Fiat lux!\" he commanded, and fire raged over the boglins. A guildsman in the process of having his throat ripped out was incinerated in the strike, but the sweet horns were sounding all around them.\n\nRandom estimated that the little knot he was with had perhaps twenty men, and at least one of them was on his knees, begging for mercy.\n\nHarmodius drew his sword. He raised an eyebrow.\n\n\"Damn,\" Random said.\n\nHarmodius nodded.\n\nGuilbert shook his head. \"Wagons punched us a hole,\" he said. \"The mounted men are that way.\" He pointed back along the track. Back the way they'd come.\n\nRandom spat. _I'm going to lose everything,_ he thought.\n\nHarmodius nodded. \"Might as well try,\" he said. \"Ready to run, everyone?\"\n\nRandom felt he ought to contribute something, but it was all happening too damned fast.\n\nHarmodius raised his arms, and a ripple rolled from his hands like a flaw in glass, spreading outwards in a semi-circle like the ripple made by throwing a pebble in a pond, except that trees blackened and grass vanished and boglins fell like wheat under a sharp scythe before it.\n\nGawin, out beyond the edge of the bubble, charged his destrier straight at it. Random saw him put the animal into a jump, and they were up; down again in moments, having leaped the growing edge of the wave of destruction in a bound. And done it apparently unharmed.\n\n\"Oh, well done,\" Harmodius said. \"That's a proper fellow.\"\n\nAnd then they were running back down the trail.\n\nThey ran, and ran.\n\nWhen Harmodius couldn't breathe, Gawin dismounted, put the Magus up on his destrier and ran along with them for a while.\n\nAnd then, as if by common agreement, they all stopped running at a deep stream\u2014the stream they'd crossed that morning at the break of day. There were a dozen wagons there, and all the mounted men on the far bank. One by one, the desperate men scrambled across, soaked to the waist, uncaring. Some stopped in mid-stream to drink from parched throats.\n\nThe mounted men began to weep, and Random ignored them.\n\nBut Gawin, alone of the panicked men crossing the stream, didn't throw himself down in the illusory safety. He sheathed his sword.\n\n\"I have run from terror, too,\" he said to the mounted men. \"And it is three times as hard to regain your honour as it is to preserve it in the first place. But this is where we will all make ourselves whole. Dismount, messires. We will hold the river bank while these good men get to safety, and in so doing, we will find both honour and peace.\"\n\nAnd such was the power of his voice that one by one they dismounted.\n\nRandom watched with disbelief.\n\nThere were nine of them, all well armoured, and they filled the gap of the trail.\n\nGuildsmen took their horses as more men came in\u2014a dozen in one group, wild-eyed, and then in ones and twos, their jackets torn.\n\nAnd then no more.\n\nThere were perhaps fifty survivors from the three hundred men who had awakened that morning.\n\nThey had a dozen wagons\u2014mostly the horse drawn carts whose animals had stuck to the road, or followed the military horses. But as they waited for the next assault of the enemy, whose horns could clearly be heard\u2014a boy appeared on the far bank, no more than fifteen years old.\n\n\"I reckon I need some help!\" he called. \"Can't get these here oxen through the ford on my own!\"\n\nThe boy had saved four wagons. He didn't seem to know that he was supposed to be afraid.\n\n\"They're busy a-killin all the horses and cattle!\" the boy said. He grinned like it was all a great prank. \"So I'm just walking up and taking any wagon ain't got a bunch of 'em aboard!\"\n\nRandom hugged him after they had the oxen across. Then he turned to Gawin. \"I honour your willingness to fight here and get us clear,\" he said. \"But I think we should all go together. It will be a long road back, and as dangerous as these woods\u2014every step of the way.\"\n\nGawin shrugged. \"These men can go\u2014although I believe they owe you a great service.\" A daemon appeared across the river, and a troll belled. \"But I will stand here, for as long as God grants my hands the power to hold this ford,\" he said. And very softly, he said, \"I used to be so beautiful.\"\n\nHarmodius nodded. \"You, messire, are a true knight.\"\n\nGawin shrugged. \"I am whatever I am, now. I hear that daemon across the stream\u2014I think I understand him. He calls for his blood kin. I\u2014\" He shook his head.\n\n\"You saved us,\" said Harmodius. \"Like a knight.\"\n\nGawin gave him a wounded smile. \"It is an estate from which I have fallen,\" he said. \"But to which I aspire.\"\n\nHarmodius grinned. \"All the good ones do.\" He raised his hat. He was still mounted on the destrier, and he seemed a bigger man than he had before.\n\nAcross the river, the trolls belled again, and Random felt bile rise in his mouth.\n\nBut then there was the sound of a horn beyond the sweet horns of the boglins. A bronze trumpet call sounded through the trees.\n\n# South of Lissen Carak\u2014Amy's Hob\n\nAmy's Hob lay still.\n\nHe lay so still that ants crawled over him.\n\nWhen he had to piss, he did so without moving.\n\nThere were boglins at the base of the hill. They were feeding. He tried not to watch, but his eyes were drawn, again and again.\n\nThey went to a corpse, covered it, and when they left it, there was nothing but bone, hair, and some sinew. A few fed alone, but most fed in a pack.\n\nBeyond them, a pair of great horned trolls walked slowly down the ridge. Ten horse lengths from the unmoving scout, the larger of the two raised its head and called.\n\nA dozen boglin horns sounded their sweet, cheerful notes in return.\n\nGelfred appeared at his side, and his face was as white as chalk.\n\n\"How many?\" he breathed.\n\nAmy's Hob shook his head. \"Thousands.\"\n\nGelfred was made of different stuff. He raised himself on his elbows and scanned carefully from right to left. \"Blessed Saint Eustachios stand with us,\" he said.\n\nOne of the trolls' heads whipped round and saw him.\n\n\"Run!\" he shouted.\n\nGelfred aimed his crossbow and the string rang like a bell and the nearest boglin folded. So did the one behind it.\n\n\"We're dead,\" Amy's Hob said bitterly.\n\n\"Don't be such an ass,\" Gelfred said. \"Follow me.\" They ran down the reverse slope of the ridge. The troll was crashing along behind them, much faster in the undergrowth than they were.\n\nAt the base of the ridge they were just a few horse lengths ahead of the thing, but to Amy's Hob's amazement, there were a pair of horses waiting. Both men vaulted into the saddle, and the horses were away, as terrified as their riders.\n\nAs soon as they outdistanced pursuit, Gelfred slowed. \"Go to the captain\u2014he's on the road.\"\n\n\"I'll tell him to get back to the fortress!\" Amy's Hob said, eyes still wild.\n\nGelfred shook his head. He was still pale and his fear was obvious, but he was the kind of man who was afraid and kept on functioning. \"No. Absolutely not. Tell him it can be done. If he's quick.\"\n\nAmy's Hob might have stayed to argue, but staying there was insane. He put his bare heels to the pony's sides, and he was gone, leaving Gelfred alone in the woods with a thousand boglins and a troll.\n\nThe man knelt by his pony, and began to pray, intent on his purpose.\n\nThere was a flare of light, and Gelfred vanished.\n\n# South and East of Lissen Carrack\u2014Bad Tom\n\nVictory can be as much luck as skill, or strength of arms.\n\nBad Tom led the vanguard. They'd heard the boglin horns for a league and had stopped on the trail, a long column of twos, the war horses snorting, the archer's ronceys trying to avoid being nipped by the bigger horses. There was new grass by the road and all the horses wanted it.\n\nAmy's Hob cantered in from the east and he looked as if he'd seen hell come to earth.\n\nTom laughed at the sight of him. \"Guess we've found 'em,\" he said, delighted.\n\nAmy's Hob saluted the captain, who looked remarkably calm, a tall figure in scarlet and steel. \"Gelfred says\u2014\" He shook his head. \"There's a mort of 'em, but Gelfred says it's now or not.\"\n\n\"We're right on top of them,\" Tom said. He nodded to the scout. \"Well done, lad. Must take balls of brass to be out there alone wi' 'em.\"\n\nAmy's Hob shivered. \"Gelfred's still out there.\"\n\nThe captain listened. Sounds can be read as easily as sights, sometimes. He could see the action ahead\u2014the road ran east along the south bank of the river, then south between the hills. Before it turned south and began to climb, it crossed a stream.\n\n\"What's happening?\" Michael asked.\n\n\"The enemy is attacking a convoy,\" the captain said. He and Tom exchanged a look.\n\n_Hywel Writhe used to say, war isn't sword cuts, it's decisions._\n\n\"They're all on this side of the stream?\" he asked.\n\nAmy's Hob nodded. \"Aye.\"\n\n\"Clumped up?\" he asked.\n\n\"Which Gelfred said to tell you _it is now_.\" He shook his head. \"There's a thousand of 'em\u2014\"\n\nThe captain's eyes met Tom's. \"Go,\" the captain said.\n\nBad Tom grinned like a madman. \"On me!\" he roared.\n\nAround him, men checked one more thing. It was different for each one\u2014here, an armour strap, there, the way a helmet sat on your head. Or the check to make sure your dagger was right there, at your hip.\n\nBut men were smiling.\n\nThey said things.\n\nThey were going to do that thing that they did. When they moved like lightning and struck like the hammer on the anvil. Soldiers know, feel, these things. And luck rose about them, as if they were magi casting words of power with the hooves of their horses.\n\nThey rode right for the sound of the horns. Tom only reined in when he saw his first boglin, and he looked back to see Grendel and his rider pounding up the road.\n\nThe captain flipped him a salute. His visor was up.\n\n\"There they are,\" Tom said. He couldn't keep the grin off his face.\n\nThe captain listened and scratched his beard.\n\nTheir eyes met again.\n\n\"Never met a Wild creature that could fight in two directions at once,\" Tom said. \"They don't fight. They hunt. And when they pounce\u2014why, that's all they have.\"\n\n\"You mean, the Wild doesn't keep a reserve,\" the captain said.\n\n\"What you say,\" Tom said. He could tell the captain was of one mind with him.\n\n\"Someday they will,\" the captain said.\n\n\"Not today,\" Tom said.\n\nThe captain hesitated another moment. Breathed deeply, listening. Then he turned back to Tom, and his grin was wide and feral.\n\n\"Let's do it,\" he said. He raised his lance, and pointed with it. Carlus, his trumpeter, raised his long, bronze instrument, and the captain gave him a nod.\n\nTom didn't bother to form up, because surprise was everything. He was sure he knew what was happening ahead, and he led his men forward, armoured in that assurance. And when his destrier leaped a low fallen tree and the track turned and he saw hundreds of the little fuckers plundering wagons, he just raised his sword.\n\n\"Lachlan for Aa!\" he roared, and he began to kill.\n\n# South and East of Lissen Carrack\u2014The Red Knight\n\nThere is a great deal of luck involved in catching an enemy, especially a victorious enemy who outnumbers you twenty to one, flat footed, glutted with spoil, unable to either fight or flee.\n\nThere's even more luck involved when you catch your enemy glutted with spoil and pinned against a roaring torrent of a stream, with only one ford, and that ford held by a desperate madman.\n\nBecause he was in command, and because he feared a trap, the captain was among the last men onto the field, leading half a dozen archers and two men-at-arms and Jacques with all the valets as a reserve. He came forward still full of doubt at his own decision, which seemed rash, and yet full of a sort of certainty\u2014almost like religious faith\u2014that he could feel the enemy's failure.\n\nHe came on the heels of the main battle's charge to cover Bad Tom's headlong rush, and Jacques was less than twenty horse lengths behind the last man of the main battle, and still, by the time he rode under the big oak trees, the fighting was over by the abandoned wagons. He rode by what he assumed had been the convoy's a last stand\u2014a dozen guildsmen face down, some of them looking half eaten or worse.\n\nHe rode past the carcasses of not one, but three, dead dhags. He had only ever seen one, before today.\n\nHe passed down a line of carts, their draught animals dead and partially butchered in their traces. Other wagons had their oxen or their horses untouched, panicked in the traces but alive. There were human bodies among the dead boglins and other things\u2014one corpse looked like a golden bear, cleanly beheaded.\n\nHe shook his head in disbelief.\n\nHe couldn't have planned it this way. Couldn't have coordinated such a victory, not with a pair of magi to handle communications and twice the number of men.\n\nFarther on, they were still fighting. He could hear Tom's warcry.\n\nHe came up to two men holding a dozen fretting war horses and Jacques sent four valets to take their reins. The two men-at-arms grinned, loosened swords in their scabbards, and headed off down the trail toward the sound of belling. The captain took a breath, thinking of the kind of men and women he employed. The kind who smiled and hastened down the trail to battle. He led them. They made him happy.\n\nHe dismounted, handed his horse to Jacques, who gave him his spear. And dismounted himself.\n\n\"Not without me, you loon,\" Jacques said.\n\n\"I have to,\" the captain said. \"You don't.\"\n\nJacques spat. \"Can we get this over with?\" He gestured, and Toby appeared, somehow taller and more dangerous looking in a breast and back and a pot helm.\n\nThey ran forward. There was fighting off to their left\u2014the humdrum sound of blade on blade. And ahead, heavy movement and grunting, like a huge boar in a deep thicket.\n\n\"Don't let it fucking cross the river!\" Tom roared, almost at his elbow.\n\nThe captain came around the great bole of an old elm, and there was the beast\u2014twenty-five hands at the shoulder, with curling tusks.\n\nA behemoth.\n\nIt turned.\n\nLike every creature of the Wild, its eyes met the captain's, and it roared a challenge.\n\n\"Here we go,\" said Tom, with relish. \"Captain's here. Now we can dance!\"\n\nJacques hip-checked the captain. \"Mind?\" he said, and shot the thing, a clean shaft that leapt from his bowstring at full draw and plunged through its hide, vanishing to the fletchings. His war-bow was as long and heavy as Wilful Murder's, and most men couldn't draw it.\n\nSomebody behind it plunged a sword deep into its side, and then a man-at-arms was sawing at its neck, and it was roaring in anger. But the flurry of blows let up, and suddenly it got its feet under it, tossed the man-at-arms free, and put its head down.\n\n\"Oh fuck,\" Jacques said.\n\nA solid lance of fire crossed the stream and struck the behemoth in the head, splintering a tusk and setting fire to the stump. Despite the fear, every man turned to look. Most of them had never seen a phantasm used in combat.\n\nThe captain charged it, because that seemed better than it charging him. His horse had done all the work until now, and his legs were fresh, despite the weight of steel greaves and sabatons.\n\nThe fire was a nice distraction and he slammed his heavy spear into its face, near an eye. It was collapsing back\u2014Jacques, also unaffected by the pyrotechnics, was walking forward, putting arrow after arrow into its unguarded belly.\n\nIt turned away, suddenly less fearsome and sensing the defeat of near death. It tried to burst free across the stream but the rocky bottom betrayed it and it stumbled; a dozen archers, guildsmen and mercenaries alike, poured shafts into it, and its blood swirled in the fast water. It gathered itself up and leaped\u2014awesome in its might\u2014scattered the archers, and killed two guildsmen, massive front feet pounding their bodies to fleshy mush in the spring mud. And still it got its head up when the captain came out of the trees behind it, and it turned at bay. It's great eyes met the captain's.\n\n\"Me again,\" the captain said.\n\nIt raised its head and bellowed, and the woods shook. One of Tom's men-at-arms\u2014Walter La Tour\u2014landed a hard blow with a pole-axe and got swiped by the whole force of its mighty head in reply, crushing his breastplate and breaking all his ribs. He fell without a sound. Francis Atcourt, one day out of the infirmary, struck it with a pole-axe too, and danced aside as it's splintered, burning tusk sought his life. He tripped and fell over a rotten stump, which saved his life as its tusks and fangs passed over his head.\n\nThe captain ran rock to rock across the stream, his sabatons flashing above the swollen water, charging his prey. It turned to finish Atcourt, caught sight of the captain's rush, and hesitated a fraction of a heartbeat.\n\nBad Tom watched his captain rush the monster and laughed. \"I _love_ him,\" he shouted, and leaped after.\n\nThe monster lurched forward, and stumbled, and the captain thrust, catching it in the mouth, cutting up so that ivory sprayed. The splintered tusk caught the back of his rerebrace hard enough to slam him into the stream. He went down, his helmet filled with water, but he got a rock under his backplate and sprang to his feet, stomach muscles screaming as he levered his own weight plus sixty pounds up on his hips, and then he had his feet planted, knee deep in water, and he was cutting\u2014down to the Boar's Tooth guard, his heavy pole-arm cutting from the height of his shoulder all the way down to his hip\u2014then back up the same path, ripping up through its trunk to the Guard of the Woman. He reversed the blade and thrust down into its eye as the creature fell.\n\nBad Tom slammed his fist into the thing before it was done moving. \"I name you\u2014meat!\" he shouted.\n\nThe mercenaries laughed. Some of the men-at-arms were even applauding and the guildsmen began to realise they might live. They began to cheer.\n\nA last arrow flew into the corpse.\n\nThere was nervous laughter and then the cheers swelled.\n\n_\"Red Knight! Red Knight! Red Knight!\"_\n\nThe captain enjoyed it for three heavy breaths. Three deep, lung filling breaths to enjoy being alive, being victorious. Then\u2014\n\n\"We're not out of this yet,\" the captain snapped.\n\nAt the sound of his voice the young knight who'd led the defence of the ford got up from where he'd knelt to pray\u2014or fallen in exhaustion.\n\nThey looked at each other for a moment too long, the way only mortal foes and lovers look at each other.\n\nAnd then the captain turned away. \"Get the horses. Get everyone mounted. Get as many of these wagons as we can save. Move, move, move. Tom, collect wagons. Who's in charge here? You?\" He was gesturing at one of the men of the convoy.\n\nHe turned to Jacques. \"Find out who's in charge of the convoy, get a head count. The knight in front of you\u2014\"\n\n\"I know who he is\u2014\" Jacques said.\n\n\"He looks wounded,\" the captain replied.\n\nThe knight they were talking about rose and hobbled forward. His right leg was shiny and slick with blood.\n\n\"You. _Bastard!_ \" he said, and cocked back his sword to swing at the captain. He collapsed just as Jacques took his sword.\n\nTom laughed. \"Someone who knows you?\" he said. Chortled, and got to work. \"All right, you lot! Archers on me! Listen up!\"\n\nBut the captain, sometimes known as the Red Knight, stood by the young knight's body. For reasons none of them knew, except perhaps Jacques, it was a deeply satisfying moment. A great victory. And a little personal revenge.\n\nRescuing Gawin Murien.\n\nKilling a behemoth. This one, in death, didn't look any smaller. It was still fucking huge.\n\nThe captain threw back his head and laughed and the favour on his shoulder fluttered in the breeze.\n\nTom met his eye.\n\n\"Sometimes, this is the best life I could ever have imagined,\" the captain said.\n\n\"That's why we love you,\" Tom said.\n\n# Harndon\u2014Desiderata\n\nLady Mary stood by the empty bedstead, and watched a pair of southern maids roll the feather mattress.\n\n\"We're taking too much,\" Desiderata said.\n\nDiota laughed. \"My sweet, you won't lie easy without a feather bed. All the knights have them.\"\n\n\"The Archaics slept on the ground, rolled in a cloak.\" Desiderata swirled, admiring the fall of her side-slit surcote and the way the slightest breeze caught the thing. Silk. She'd seen silk before\u2014silk garters, silk floss for embroidery. This was more like something from the aether. It was magic.\n\n\"You cannot wear that without a gown,\" Diota said. \"I can see your tits right through it, sweeting.\"\n\nLady Mary turned away and looked out the window. _I think that's what the Queen had in mind,_ she thought to herself. She exchanged a look with Becca Almspend, who glanced up from her reading to smile her thin-lipped smile.\n\n\"Sleeping on the ground under a cloak doesn't sound any worse than being a maid in the Royal Barracks,\" Becca said. \"In fact,\" she glared at Lady Mary, \"perhaps in a military camp, your friends don't come and steal your blankets?\"\n\nThe Queen smiled at Lady Mary. \"Really, Mary?\"\n\nMary shrugged. \"I have seven sisters,\" she said. \"I don't mean to take other people's blankets. It just happens.\" Her eyes twinkled.\n\nThe Queen stretched, rose on her toes like a dancer, and then settled, arms slightly outstretched, as if she was posing for a portrait. \"I imagine we'll all sleep together,\" she said.\n\nAlmspend shook her head. \"Pin your cloak to your bodice, that's my advice, my lady.\"\n\nDiota snorted. \"She won't sleep under a cloak. She'll have a feather bed in a tent the size of a palace.\"\n\nThe Queen shrugged, and the maids packed.\n\nLady Almspend worked her way down the day's list. The preparations of the king's baggage train\u2014and then of the Queen's\u2014had made Lady Almspend a much more important person.\n\n\"War horses for my lady's squires,\" she said.\n\nThe Queen nodded. \"How goes that task?\"\n\nAlmspend shrugged. \"I asked young Roger Calverly to see to it. He has a head on his shoulders and he seems to be trustworthy with money. But he's come back to report that there are simply no war horses to be had. Not for anything.\"\n\nThe Queen stamped her foot. It didn't make much noise, small as it was and clad in a dance slipper, but the maids stopped moving and stood still. \"This is not acceptable,\" she said.\n\nRebecca raised an eyebrow. \"My lady, this is a matter of military reality. I asked questions this morning at first breakfast in the men's hall.\"\n\nDiota made a spluttering sound of outrage, perhaps she did it too often, but it still effective. \"You was in the men's hall for breakfast, you hussy? Wi'out an escort?\"\n\nAlmspend sighed. \"There aren't any _women_ likely to know much about the price of war horses, now, are there, Diota?\" She rolled her eyes with the effectiveness that only a woman of seventeen can muster. \"Ranald has taken me to the men's hall as a guest. And\u2014\" She paused and cleared her throat a little awkwardly, \"And I had an escort.\"\n\n\"Really?\" Lady Mary asked. \"Sir Ricar, I suppose?\"\n\nLady Rebecca looked at the ground. \"He hadn't left yet\u2014he was eager enough to help me.\"\n\nDiota sighed.\n\nThe Queen looked at her. \"And?\"\n\nAlmspend shrugged. \"Alba doesn't breed enough horses for all its knights,\" she said. \"We import them from Galle, Morea and the Empire.\" She looked at her friend defiantly. \"Sir Ricar explained it to me.\"\n\nThe Queen stared at her secretary. \"Gentle Jesu and Mary his mother. Does the king know?\"\n\nAlmspend shrugged. \"My lady, the past week has revealed that men conduct war without women with all the efficency and careful planning that they do anything else without us.\"\n\nDiota let out a most unladylike snort.\n\nLady Mary laughed aloud. \"Is there beer involved?\" she asked.\n\nThe Queen shook her head. \"You mean to say we don't have enough war horses to mount our own knights, and no one cares?\"\n\nLady Almspend shrugged. \"I won't say no one cares. I could guarantee that no one has taken any thought for it.\"\n\n\"What of remounts?\" the Queen asked. \"Horses die. Like flies. I'm sure I've heard that said.\"\n\nAlmspend shrugged.\n\nLady Mary nodded. \"But Becca\u2014you must have a plan.\" Somewhat cattily, she added, \"You always do.\"\n\nLady Almspend smiled at her, immune to her sarcasm. \"As it happens I do. If we can raise a thousand florins we can purchase a whole train of Morean horses. The owner is camped outside the ditch. I met with him this morning and offered for his whole string. Twenty-one destriers.\"\n\nThe Queen hugged her impulsively.\n\nDiota shook her head. \"We have no money, sweeting.\"\n\nThe Queen shrugged. \"Sell our jewels.\"\n\nDiota stepped up to the smaller woman. \"Don't be an arse, sweet. Those jewels are all you have if he dies. You don't have a baby. If he goes down, no one will want you.\"\n\nThe Queen looked steadily at her nurse. \"Diota\u2014I allow you nearly unlimited liberties.\"\n\nThe older woman flinched.\n\n\"But you talk and talk, and sometimes your mouth runs away with you,\" the Queen continued, and Diota backed away.\n\nThe Queen spread her arms. \"You have it precisely backwards, dear heart. If the king dies, _everyone_ will want me.\"\n\nThe silence was punctuated only by the barking of dogs outside. Diota quailed. Lady Mary pretended to be somewhere else, and Becca opened her book.\n\nBut finally, Diota straightened her spine. \"All I'm saying is let the king look to his own war horses. Tell the squires where they can buy them and let them squeeze their rich parents for the money. When you sell your jewels, sweeting, you will have _nothing._ \"\n\nThe Queen stood very still. Then she smiled her invulnerable smile at her nurse. \"I am what I am,\" she said. \"Sell the jewels.\"\n\n# Chapter Ten\n\n# Otter Creek Valley, East of Albinkirk\u2014Peter\n\nPeter lay on the ground behind a tree as big as a small house, unable to see anything, and waited for battle.\n\nMore than anything, he wanted to piss. From a tiny irritant at the base of his penis, the feeling gradually grew to envelope his every thought. After the first of several eternities, the need to void himself overtook his fear and terror.\n\nFrom time to time, he drifted off on other thoughts\u2014the possibility of moving to a better hiding space; finding a view of the oncoming enemy; finding some actual cover. He had no experience of war in the west, and couldn't imagine what it might be like to face a man in steel armour.\n\nHe had a knife, a bow, and nine arrows.\n\nAnd he had to piss.\n\nIt began to seem possible that he should just let go, and lie in his own urine for however long they lay there.\n\nHe wondered if he was the only one. He wondered if Ota Qwan had meant to tell him to relieve himself before the ambush was set. Or if he had not told him deliberately. The black painted man had some cruelty in him\u2014Peter was already sensing that Ota Qwan had few followers because he enjoyed twisting the knife too much. And he thought that the honeymoon between them was ending\u2014in the beginning, Ota Qwan had been as desperate for his company as Peter had been desperate for an ally amidst the alien Outwallers, but now, with a war-group forming around him, Ota Qwan was undergoing a subtle metamorphosis. And not a pretty one.\n\nAnd he really had to piss.\n\nThere was no way to measure the time. An ant crawled the length of his body, from moccasined left foot to right shoulder. Something larger crossed one of his knees. A pair of hummingbirds came and visited a flower by his head, and he was so still in the agony of his need to relieve himself that the male, bright red in spring plumage, all but landed on his painted face.\n\nThere were three hundred men\u2014more, perhaps five hundred\u2014lying on either side of the road as it led downhill to a ford over a deep running stream. They were somewhere east of Albinkirk. No one made a noise.\n\nHe had to piss.\n\nHe heard the metallic scrape of an iron-shod hoof on stone, and a shriek\u2014a cry, and then a scream that seemed to come from the other side of his tree.\n\nNo one was moving.\n\nThe scream was repeated and suddenly cut off, and its sudden removal left another sound\u2014the sound of shod hooves galloping away.\n\nSuddenly Skadai was on the trail, just a few arm's lengths away, calling softly. \"Dodak-geer-lonh!\" he said. \"Gots onah!\"\n\nAll around Peter, warriors rose from their ambush spots, rubbing themselves, or scraping bark off their skin. Half of them immediately began to relieve themselves. Peter followed suit, previously unaware that urination could be such a great pleasure.\n\nBut Skadai was moving. Ota Qwan swatted Peter on the shoulder. \"Move!\" he said, as if Peter was a child.\n\nPeter gathered his bow and followed.\n\nThey ran east on the trail for twenty horse lengths, and there was a horse, dead, across the trail, and a man pinned beneath it with his face and hair cut away and his throat slit. His blood pooled between the rocks and ran in a sticky rivulet headed downhill to the stream.\n\nAfter running for what seemed a long time, they began to spread out amid big trees. The stream was well behind them, and Peter was terrified\u2014they were running at the enemy, or so it seemed.\n\nSkahas Gaho must have felt the same, because when they stopped running he got in front of Ota Qwan and said something that was clearly remonstration.\n\nOta Qwan struck him\u2014not a hard blow, but a fast one, and the younger warrior bent over with the pain.\n\nOta Qwan spoke quickly, spittle flying from his mouth.\n\nSkadai ran up silently, listened to Ota Qwan, and nodded, running off down the loose line of warriors that extended as far as the eye could see into the great trees on either flank. The trees here\u2014mostly Adnacrag maples and beeches, tall old trees with magnificent tops\u2014were big enough that two men couldn't get their arms around them. But there was little undergrowth because of the high canopy, and despite the sun-dappled forest floor, little grew amidst the carpet of old leaves except the most magnificent irises that Peter had ever seen.\n\nSkahas Gaho got to his feet, glared resentfully at Skadai and spat at Ota Qwan. He said something to the other warriors, and ran off down the line. Brant turned to follow him, and Ota Qwan raised his bow.\n\nPeter acted without thought. He pushed Ota Qwan's bow arm, hard.\n\nThe warrior tried to hit him in the ear with the tip of the bow but Peter caught it and, in a single turn of his arm, he had Ota Qwan's right arm in an elbow lock that threatened to dislocate the man's shoulder.\n\n\"I wasn't _born_ a slave,\" Peter said. \"Don't fuck with me.\"\n\n\"They are deserting me!\" Ota Qwan watched the two men running off.\n\n\"You hit Skahas Gaho when you needed to reason with him.\" Peter wanted to laugh to hear himself explaining basic leadership to Ota Qwan. But he had the arm lock, and he wasn't letting go.\n\nThe other man stiffened, and then went limp. \"He was about to disobey. To disobey Skadai!\"\n\nPeter let the black painted man go. \"I've only been Sossag for three days, but it seems to me that's Skadai's problem, not yours. I think you thought like an Alban, not a Sossag.\" Peter shrugged.\n\nThe other three\u2014Pal Kut, Barbface, and Mullet, watched them warily.\n\n\"You will be loyal to me!\" Ota Qwan hissed at Peter. \"Will you?\"\n\nPeter nodded. \"I will,\" he said, finding that the words made him feel queasy.\n\nPal Kut called something. The line, well-spread, was moving rapidly forward, almost at a run. Most men had an arrow on their bows.\n\nPeter sprinted to make his place in the line, fumbled an arrow and dropped it, and turned back to get it\u2014he had too few to lose one. He bent, and in that moment, the world exploded.\n\nTo the front of the line, off amidst the drover's herd, a bull gave a long, low growl. And suddenly the air was full of arrows flying both ways. And the Sossag gave a great cry, almost like a scream...\n\n... and charged.\n\nPeter had his arrow on his bow. He ran forward, saw Pal Kut take an arrow in the gut, an arrow so big and so powerful that it emerged from his back in a gout of blood, and the head was shaped like a swallow and gleamed with a horrible red-blue malevolence.\n\nPeter ran forward, following Ota Qwan.\n\nHe saw his first enemy\u2014a tall blond boy in ring mail who coolly rose from behind a bush and loosed an arrow into a warrior he didn't know\u2014shot him from so close that the man was knocked off balance by the arrow and stumbled like a beheaded chicken before collapsing in death.\n\nBut Ota Qwan leaped at the man with a dire scream and loosed his own arrow at arm's length, and the barbed head drove through his ring mail at the shoulder. A dozen warriors converged on the wounded boy and he was dead and scalped in a few heartbeats.\n\nOta Qwan took the boy's sword\u2014four feet of shining steel\u2014and brandished it, and all of the warriors who had seen him attack gave a great cry, and then they were pelting forward again.\n\n# Otter Creek Valley, East of Albinkirk\u2014Hector Lachlan\n\nAs soon as the scouts reported, Lachlan knew he was in serious trouble. North of the Inn, in the Hills, the Wyrm of Erch kept the Outwallers at bay. It cost him animals to keep the Wyrm happy, but that was the way of the Hills. For a thousand years or more, the Wyrm had kept the Wild out of the Hills, to the benefit of generations of clansmen and drovers.\n\nHere in the south, the king was supposed to keep the Outwallers away. Otter Creek was taken by some to be the border between the Green Hills and the Kingdom of Alba. But for Lachlan, whoever's territory it was, Otter Creek was safe ground. Not battle ground. Otter Creek ran down into the Albin. Albinkirk itself would be visible from the height of the next ridge but one\u2014even if there was still a long day to drive the beeves to get them to the ford at Southford.\n\nBut the point was that they were _almost there._\n\nBut now\u2014he had scouts, and they knew their business. The clansmen and drovers knew the Outwallers. Outwallers were fierce, savage, and expert in arms. And they'd set an ambush for his drove, which meant they had scouted his herds, knew his strength and felt that they could take him. That meant three to four hundred warriors.\n\nHector didn't hesitate. It was a situation he'd imagined many times, although he'd never had to face it.\n\nHe turned to his tanist, Donald Redmane. \"Go back to the drag guard. Take every animal you can, turn them, and run for the Inn.\"\n\nDonald was a good man\u2014loyal, dogged. Not the smartest, but a wonderful man in a fight, with a beautiful voice and clever hands that made things. \"You go, Lachlan. I can hold them here.\"\n\nLachlan shook his head. \"With your bruised ribs and all? Go. Now.\"\n\nRedmane shook out his hair. \"By the Wyrm, Hector. We're just a day's march from Albinkirk. Let's stampede the cows at the bastards, and put the survivors to the sword.\"\n\nHector looked at the woods under his hand. \"No. My word on it, Donald. They're two to one or more against us, and scattering the herd in these woods\u2014\" He held his peace, lest he lower spirits more than he had to.\n\nHe turned and looked at the scout. \"Ride clear all the way to the Inn. Take two horses so that you can change. Ride like the wind, yunker\u2014they may already be in the High Country. Don't come back unless you can bring a hundred swords with you.\"\n\nThe other men with him in the vanguard loosened their swords in their scabbards. A few checked bows, and one took off his arming cap, replaced it, and went to his mule for his helmet.\n\n\"Bad luck you boys are with me, this day, and not in the rear with the drag,\" Hector said. \"None of us will be dining this evening, I fear.\"\n\nIan Cowpat, a big man with a muddy brown face, gave him a grin. \"Bah. Never met a loon I couldn't kill.\"\n\n\"Outwallers,\" Hector said. \"We'll meet them in the woods where they can't shoot us down. Make a fight of it as long as we can. When I sound the horn, every man to me, and we form a shield wall and make a song of it.\" He looked around. The duty changed every day, because working the back of the herds was so much worse than walking in front, and so he didn't have the oldest or the youngest, or all the best fighters, or even all men he knew. He had a scattering of his own and the Keeper's men; but they were well armed, fifty strong, and not a face betrayed the terror that every one of them must feel. Good men for making a song.\n\nBy which hillmen meant dying well.\n\nHe thought of his new bride, and hoped she had kindled with him because, while he had a few bastards, he didn't have a son to avenge him. He caught the stirrup of his messenger.\n\n\"Listen!\" he said. \"Tell my wife that if she has a son he is to grow tall and strong, and when he is rich and well-loved, he is to take an army north and cut a bloody swath through the Outlanders. I'll take five hundred corpses as my wergild. Tell him when he's old enough. And tell her that her lips were the sweetest thing I ever knew, and I'll die with the taste of them on my own.\"\n\nThe young man was pale. He'd watched a boyhood friend die, and now he was being sent to ride a hundred leagues alone, quite possibly the only survivor of the drove.\n\n\"I could stay with you,\" he said.\n\nHector grinned. \"I'm sure you could, boyo. But you are my last message to my wife and kin. I need you to go.\"\n\nThe messenger changed horses. A bull was lowing, and the cows were turning, the rear of the column was already moving north, away from the line of enemy that was out there somewhere.\n\nThen he turned back to his men, most of whom were helmed and mailed and ready to fight. The lone priest, his half-brother, lifted his cross in the air and all the men knelt, and Paul Mac Lachlan prayed for their souls. When they all said amen, the priest put the cross back into his mail cote and put an arrow on his bow.\n\nHis cousin Ranald had a great axe\u2014a beautiful thing, and he was cutting the air with it. He also had steel gauntlets; having served the king in the south he had fine gear like a knight.\n\n\"Ranald takes command if I fall,\" Hector said. \"We'll go forward into the woods\u2014the youngest ahead as skirmishers. Don't get overrun. Shoot when you can and then retreat. When you hear my horn, retreat. We have to hold until the sun reaches noon, and then Donald will be away and we will have died for something.\"\n\nRanald nodded. \"Thanks, cousin. You do me honour.\"\n\nHector shrugged. \"You're the best man for it.\"\n\nRanald nodded. \"I wish your other brother were here with us.\"\n\nHector looked out into the trees. He could all but feel the oncoming enemy. Perhaps\u2014perhaps they would wait in ambush too long, or balk at a close fight.\n\nBut there was too much movement out at the edge of the meadow. The Outwallers were coming.\n\n\"Me, too,\" Hector said. He looked up and down the line. \"Let's go. Spread well out.\"\n\nThey went forward into the woods, moving quickly. His greatest fear was that the enemy was already at the woods' edge\u2014but they weren't, and he got his fifty into the deep woods where the irises bloomed like crosses in a graveyard.\n\nHe put two men at every tree, and his ten youngest and swiftest a spear's throw in advance of his very open line, and then the bull roared again in the distance and suddenly the arrows began to fly.\n\nHector almost died in the first moments. An arrow hit his bassinet, spinning him, and a second arrow hit the nose guard of his helmet and bent it in\u2014a finger's width from an arrow in the eye and instant death.\n\nHis men did well, although the boys in front were overrun and killed\u2014and it was his mistake. The Outwallers were faster, bolder, and more reckless than he had imagined\u2014but they still took a fearful toll among the savages. When his loose line retreated, running from cover to cover in their heavy mail, the Outwallers hesitated for a moment too long before following them, allowing them a clean break and leaving another thin line of kicking, wounded and gutted corpses.\n\nOne lone Outwaller, painted red from head to toe, stood between two great trees and called, and then sprinted forward. He tackled Ian Cowpat, and Cowpat never rose again\u2014but only a handful of the painted men followed the red one.\n\n_Thanks be to God_ , Hector thought.\n\nHis men had been forced back into the last cover before the meadow, and the sun was not yet halfway into the heavens.\n\n# Otter Creek Valley, East of Albinkirk\u2014Peter\n\nPeter was out of arrows, and he had a great cut across his right shin\u2014he'd been caught by the very end of a wild cut from a fleeing man's sword, but it was enough to send him to the earth for several long minutes.\n\nHe had a dead man's big dagger, almost the size of a short sword, and he had a buckler from the same corpse. He was no longer close to Ota Qwan\u2014the black painted warrior had vanished early\u2014and now Peter was close behind Skadai, who moved with more grace than any warrior Peter had ever seen.\n\nWhomever they were fighting, the men were brave, big, silent, and far too well-armed.\n\nThe Sossag were dying. There were fifty men down already, perhaps more. Peter thought perhaps it was time for the Sossag to admit defeat. But Skadai didn't agree, running right into the enemy line, tackling a huge warrior and slitting his throat with a knife.\n\nPeter couldn't hang back when such daring was shown.\n\nThe next time the enemy turned to run, Peter joined his wild yell to Skadai's, and saw Ota Qwan, who suddenly appeared just an arm's length away, do the same. The three of them rose from their cover, where they had lain to avoid the arrows\u2014and charged. To Ota Qwan's right, Skahas Gaho also rose to his feet, sword in hand, and others joined them\u2014not many, but a dozen all told.\n\nAn arrow flicked out of the sunlight like a hornet and hit Skadai in the groin. He stumbled, tumbled, and lay still.\n\nPeter kept running. The man who had loosed the arrow had lost a step on his companions, and Peter ran for him, his whole self concentrated on that man, a red-haired giant in a fine mail shirt that gleamed in the woodland shade. He had an iron collar, a gorget, and long leather gloves.\n\nPeter opened his mouth and screamed. The man dropped his bow and drew his sword\u2014an arrow stung the inside of Peter's thigh as the head cut his skin, before flitting away between his legs, and Peter reached out with the buckler and the man's sword slammed into it. Peter pushed forward, the buckler pinning the sword, and his own short blade cut hard into the man's face, teeth sprayed and an eye was cut before the man turned away but Peter's sword was past his head, and he grabbed the blade with his buckler hand, locked the blade against the man's throat and sawed back and forth until he crushed his windpipe through the mail and iron collar.\n\nArrows hit his dying opponent\u2014a dozen shot by his friends. But they had loosed unthinking, Peter's rush had spun him around, and every arrow intended for him hit the red-haired man.\n\nHe fell through Peter's hands, dead before he hit the ground, and Peter dropped his long knife and stooped to pick up the great sword from the grass. Ota Qwan screamed in triumph, and the scream was taken up along their line.\n\n# Otter Creek Valley, East of Albinkirk\u2014Hector Lachlan\n\nThe priest, Paul Mac Lachlan, died badly, because he'd never been much of a swordsman, and one of the painted devils was through his guard and into him, slicing his face, choking him, using his body as a shield.\n\nIt demoralized them to watch one of their own carls die so easily, in single combat against an essentially unarmoured man.\n\nOn the other hand, Hector thought, they'd inflicted an incredible number of casualties. All the stories said that Outwallers were averse to taking casualties, and his people had killed fifty, perhaps more.\n\nAnd their red leader was down.\n\nGive the priest that\u2014he'd shot him.\n\nHector grinned at the men around him. \"We all have to do better than that,\" he said.\n\n\"Fucking Paul,\" Ranald said. One of the savages paused to scalp the priest, and Ranald flicked a shaft into the painted bastard. He screamed.\n\nHector held his horn over his head, so all the men left were ready.\n\n\"We're going to charge through their line and make our shield wall over there,\" he said. Retreating any further, into the open ground, was foolish.\n\nThe Outwallers were gaining courage from the success of the last rush, and they were coming forward now. His men were loosing their last shafts. Even as Hector watched, all the Outwallers went to ground again. If he had more woods, he'd retreat again. But he didn't. The wildflowers of the long meadow were at his back.\n\nHe held his horn to his lips and sounded it.\n\nEvery man left to him turned and sprinted towards him. It only took heartbeats for them to join him, and in that time, only a bare handful of enemy shafts flew.\n\nHe didn't wait for the laggards. When he had enough carls to make a song, he started forward.\n\n# Otter Creek Valley, East of Albinkirk\u2014Peter\n\nPeter was running out of courage.\n\nOta Qwan was not. He rose to his feet and dashed forward even as one of their warriors bent over the corpse of the red-haired man, knife in hand, and died for it.\n\n\"Gots onah!\" Ota Qwan roared.\n\nBut the warriors didn't follow him.\n\nPeter could scarcely breathe. The lightning nightmare of the close fight with the red-haired man had taken all his breath, all his energy, all his courage. He wanted to lie down and go to sleep.\n\nThe wound in his leg ached, and worried how deep it went.\n\nOta Qwan went bounding forward as the mail-clad men sounded a horn.\n\nPeter forced himself to follow the black-painted man. As he looked back, he saw Skahas Gaho and Brant rise from the grass as well.\n\nThey were following _him_ , and there were ten more with them. They loped after him, and he ran as hard as he could after Ota Qwan.\n\nTo the right, the enemy shocked all of them by charging\u2014not a handful of them, but a solid wedge, which ran right for the centre of their line.\n\nPeter was so far to the right that the end man of the wedge wasn't even close enough to fight\u2014the wedge ran by him in him moment of indecision and then there were cries deeper in the woods.\n\nOta Qwan continued to run forward. Peter didn't think he'd even seen the enemy charge, but he followed.\n\nSkahas Gaho stooped and scalped the red-haired man.\n\n# Otter Creek Valley, East of Albinkirk\u2014Hector\n\nHector was fresh and unblooded, and the first clump of Outwallers died on his sword point and edge as fast as he could roar his war-cry three times, and then they were down and his wedge was alone in the woodlands.\n\nThe essence of warfare is to force the pace and hope your enemy makes a mistake. That was his father's law of war, and his own. So he didn't stop and form a shieldwall.\n\n\"Follow me!\" he roared, and continued on.\n\nOn, and on.\n\nThe Outwallers were faster but not fitter than the drovers, and tricks of terrain and bad luck\u2014pulled muscles, wounds\u2014left them at the mercy of the hard-faced armoured men, and their mercy had nothing of mercy in it. A dozen Outwallers died in a hundred paces.\n\nHector ran on, his sides heaving and his legs burning. Running any distance in mail was an effort.\n\nRunning five hundred paces was more than an effort. It was like a test.\n\nMost of his men stayed with him. Those few who paused, died.\n\nThe Outwallers fled, but even in panicked flight they ran like a flock of swallows or a school of fish, and those ones unthreatened by the charge recovered first, and arrows started to lick through the trees.\n\n\"Keep going!\" Hector cried, and his men gave him everything they had.\n\nAn Outlander boy tripped over a root and fell, and Ranald beheaded him with a flick of his wrists.\n\nOn and on.\n\nAnd then Hector had to stop. He leaned on the hilt of his great sword, and his sides heaved.\n\nRanald put a hand on his armoured elbow. \"Water,\" he said.\n\nThe length of a barn away they found young Clip, the farmer from the Inn, pinned under his dead horse with his throat slit. A bowshot farther on they came to the ford that they would have crossed. Outwaller arrows had begun to fill the air again, and Hector had perhaps thirty men left when he crossed the ford and won a respite. His men drank water, spread out in the trees, and caught their breaths. Those that had shafts left, or who had pulled them from the ground, began to pick their targets carefully\u2014and it began again.\n\nRanald scratched his beard. He'd taken an arrow in the chest\u2014it hadn't penetrated his fine mail, but it had cracked a rib, and breathing was hard. \"That was worth a song, that run,\" he said.\n\nHector nodded. \"It's noon and we've led them back a mile, anyway. When they come at us across the stream\u2014well, Donald's away.\" Hector shrugged. \"If I'd kept all the boys together, would we have beaten them?\"\n\nRanald spat some blood. \"Nah. They're too canny, and we didn't kill nearly enough of 'em. Hector Lachlan, it's been a pleasure and an honour knowing you, eh?\" Ranald held out his hand, and Hector took it. \"Don't fash yourself, man\u2014I reckon there's five hundred of the loons out in the woods. This way, if you put a boy in that lass\u2014well, he's got a fortune and fifty good men to start him off.\"\n\nHector shook his head. \"Sorry I am I brought you here, cousin.\"\n\nRanald shrugged, despite fatigue and the weight of his chain mail. \"I'm honoured to die with you.\" He smiled at the sunlit sky. \"I'm sorry for a certain lass I love, but this is a good way to die.\"\n\nLachlan looked up at the sun. Arrows were flying thickly, and a few were starting to come from their own side of the stream\u2014the savages had found a crossing too.\n\nDespite it all the sky was blue, the sun was warm and golden, and the flowers of the forest were beautiful. He laughed, and held his sword in the air. \"Let us make a song!\" he roared.\n\n# Otter Creek Valley, East of Albinkirk\u2014Peter\n\nPeter followed Ota Qwan until his lungs were starved for air, and then he slowed. The black-painted man slowed, too, as if they were attached by a string. They had reached an open field, and there was a small herd of cattle, every head facing them\u2014a single horse, and dozens of sheep.\n\nAnd no men.\n\nOta Qwan leaped for joy, dancing on the grass. \"We have beaten them! All their herds are ours!\" He embraced Skahas Gaho.\n\nThe taller warrior didn't address Ota Qwan, but Peter. \"Where?\" he asked. He mimed swinging a two-handed axe or sword.\n\nPeter pointed back the way they'd come. He was bone weary, the wound in his leg was now a cold ache, and all the fury of combat had ebbed away to leave nothing behind. But Peter, having started something, couldn't give it up.\n\nOta Qwan shook his head. \"The cattle! We need to get the cattle, or this is for nothing.\"\n\nPeter looked at the black-painted man wearily. \"Have you not seen the numbers of our dead, Ota Qwan? This is already for nothing. Skadai's death means there is no one to tell the Sossag to stop attacking.\" He shrugged. \"And this is only a tithe of their herds.\"\n\nOta Qwan looked at him. Understanding dawned slowly.\n\n\"We must stop it. We can shoot down anyone who still stands\u2014take our time.\"\n\n_You can be war leader._ Somehow Peter knew that this was Ota Qwan's only thought.\n\nBut together with their two hands of followers, they turned and began to walk back towards the distant screams that marked the current edge of the battle. No one, not even Ota Qwan, had the energy to sprint, so they ran and walked fitfully.\n\nThe sun was just past its height when they scrambled down the last part of the steep glen and crossed the water on rocks slick with blood.\n\nThere were still men fighting.\n\nA dozen of the armoured giants stood in a ring, and some two hundred Sossag stood around them in a ring, and between the inner ring and the outer ring was a wall of corpses, some of which still moved. Even as they crossed the stream, a pair of bold youths leaped at the circle of steel and died, one beheaded by an axe, the other spitted on a four-foot sword.\n\nTheir bodies were cast on the growing barricade of the dead.\n\nAnd then the blood-spattered daemons began to sing. They weren't very good, but their voices rose together, and the Sossag paused a moment in respect. A death song was a great thing\u2014a magic not to be interrupted. Even Ota Qwan was silent.\n\nTheir song went on, many verses, and when it was done their faces, which had been lit with passion, seemed to slump.\n\nOta Qwan leaped up on a stump. \"Shoot them! Back into the trees and shoot! My curse on any man who tries to rush that circle!\"\n\nSome men listened. Arrows began to fly, and when a Sossag arrow hit a mail cote, dust flew, at least, although few shots from their short bows were powerful enough to penetrate.\n\nBut there were many arrows.\n\nPeter saw Sossag die from arrows shot from across the circle. The arrows flew faster and faster, striking hillmen and Sossag alike and the hillmen began to sing again, and they charged and the Sossags ran\u2014again.\n\nBut not far.\n\nPeter had no arrows. He picked up a spear decorated with feathers, and the next time the enemy charged the circle, he chose his moment and hurled the heavy spear into the back of the charging men. The shaft spun out of control, but the weapon hit the back of the man's armoured legs and he stumbled. Peter ran for him, a dozen Sossag with him, and they tore the hillman to bloody rags.\n\nAgain, the hillmen gathered in a circle, and again the Sossag shot them, creeping closer, emboldened by the hillmen's obvious exhaustion and despair. One more time, their leader rallied, whirled his sword and led them at the closest Sossag, bent not on escape but on slaying as many as they could\u2014and again they caught the fringe of the circle, killing a dozen painted men and losing two more of their own. Ota Qwan was roaring for them to fall back and shoot, and Peter joined him.\n\nThe Sossag fell back to the trees and shot their last shafts.\n\nAnother giant fell screaming.\n\nThe Sossag yelled, but it was a tired, thin sound.\n\nOta Qwan looked around. \"When they next charge, we must charge them in turn, and finish them,\" he said. \"We cannot let any of them escape. We must be able to tell the matrons we killed them all.\"\n\nPeter spat. His mouth was dry, and he had never, in all his life, slave or free, been so tired.\n\n# Otter Creek Valley, East of Albinkirk\u2014Hector Lachlan\n\nAlan Big Nose, Ranald Lachlan, Ewen the Sailor, Erik Blackheart and Hector. The last men left.\n\nHector was hit again with a shaft that tickled his ribs. He was ready to die. He had no wind left, no joy in battle, and he was in enough pain that simple cessation seemed like a victory.\n\nEven as he thought it, Ewen took an arrow in the throat and went down.\n\nHe wracked his memory for a song to end with. He was no bard, but he knew some songs. Nothing came to him but drinking songs, but then he smiled\u2014a free smile\u2014to think of his young wife crooning. She'd sung to him, a lullaby.\n\nHe knew it well. Hill folk called it \"The Lament,\" the song of their loss.\n\n_A fine song with which to end._\n\nHector stood straight, took a deep breath, and began to sing. He swung his sword back onto his shoulder and cut an arrow from the air, and Ranald picked up the tune, and Alan Big Nose was there, his voice strong and true on the notes, and Erik Blackheart stepped over Ewen's corpse and roared into the chorus.\n\nAt some point the Sossag stopped loosing arrows.\n\nHector finished the song, raised his sword\u2014a salute to his enemies, who had given him that gift of peace, right at the end.\n\nA warrior, painted black head to toe, raised a sword\u2014just a short bowshot away. And Hector could see that the Outwallers had gathered in tight while his men sang.\n\nGood. It would be a clean end in a straight fight.\n\nRanald sighed. \"Your brother will never forgive himself for missing this,\" he said, and they charged.\n\n# Otter Creek Valley, East of Albinkirk\u2014Peter\n\nWhen it was over, Peter sat on the ground and wept. He didn't know why he was crying\u2014only that his body needed the release.\n\nSkahas Gaho came and put a hand on his shoulder. Brant was meat for the ravens. Ota Qwan had a wound across his chest that would probably kill him, inflicted when the last giant had stumbled forward, dragging three Sossag warriors, shaken them off, and landed one final cut with his great axe before Ota Qwan and Peter had managed to put him down.\n\nThe woods were full of death.\n\nBut even after a day of vicious fighting\u2014and Peter couldn't imagine worse fighting\u2014there were still hundreds of Sossag unwounded, or capable of movement, and Ota Qwan had enough breath to send them to round up any cattle they could find and start them for home.\n\nPeter sat by Ota Qwan and held his hand, watching the blood leak out of the man's chest.\n\nJust at sunset, the faeries came.\n\nPeter had never seen one before but he'd known men who believed in them. He was sitting with the dying Ota Qwan. There were a hundred wounded Sossag groaning or worse, and scavengers had begun to move in on the corpses.\n\nPeter was too tired to care.\n\nThe first one he saw looked like a butterfly, except that it was ten times the size and glowed faintly, as if sun lit. Behind it were four more, in a formation.\n\nPeter had time to wonder whether they were predators, scavengers, or pests, and then the first one alighted on Ota Qwan's chest.\n\n_What is he worth to you, man of iron?_\n\nPeter started, wondering if he had been dreaming.\n\nA faerie is to a man as a hummingbird is to a bumblebee. Or so Peter thought, gazing at the jewel-like being.\n\n_What is he worth to you? A year of your life?_\n\nPeter didn't think. _Yes,_ he thought.\n\nThe pink shape drifted along Ota Qwan's chest, and then reached out, oh, so gracefully, and touched Peter\u2014and that touch was like every slaver's iron ever forged. Something was ripped from his chest, as if red-hot pincers had entered his heart and dragged it out past his ribs, and he vomited over his lap.\n\nAnd the faeries laughed. Their laughter seemed to echo in his empty head like the shouts of revellers in a cave\u2014\n\nAnd Ota Qwan coughed, spat, and sat up.\n\n\"No!\" he said suddenly, his usually too-calm voice alight with wonder. \"No! You didn't!\"\n\nBut Peter was crying, because now he had something to weep for\u2014-whatever it was he'd just lost.\n\nAnd the faeries laughed.\n\n_So sweet, so sweet. So far away! So rare._\n\n_A bargain is a bargain._\n\n_Perhaps we'll give you another, you were so sweet and rare._\n\nTheir laughter sounded more like a curse.\n\n# Otter Creek Valley, East of Albinkirk\u2014Ranald Lachlan\n\nRanald Lachlan rose from the black curse, through pain, and into the soft darkness of an April night. He sat up without a thought in his head, and the arrow that had penetrated his mail fell by his side, and he cut his hand on his own long sword lying in the bloodstained flowers by his side.\n\nAnd then he knew where he was.\n\n_Never say we do not give everything we promise!_\n\n_So sweet, so sweet!_\n\n_Peter saved you. Peter saved you!_\n\nFair folk. And Ranald knew that he had been dead, or close enough as made no matter, and someone named Peter had given them the usual trade. A piece of your soul for the life of a friend.\n\nAnd the Outwallers were all around him in the moonlit dark. Just for a moment, he thought to steal away\u2014but they were looking at him. A hundred of them.\n\nCursing, he dragged himself to his feet.\n\nBlack death was behind him, and in heartbeats would be his again, and he spat.\n\n_Ah, Rebecca, I tried. I love you,_ he thought. He lifted the axe that Master Pyle had made for him\u2014well tested now\u2014and put it on his shoulder.\n\nAt the base of the little knoll where he'd made his last stand, he saw the gleam of moonlight, and one of the dark figures got to its feet, lit by four of the fair folk like some kind of ethereal bodyguard.\n\nThe man was painted black. Ranald remembered him. He came up the knoll, and Ranald awaited him, hands crossed on the haft of his axe.\n\n\"Go,\" said the black man.\n\nRanald had to replay the word again. It was a shock to hear Gothic, and another to be told to go.\n\n\"We are the Sossag people,\" the man said. \"What the faeries return, we do not touch.\" The man's eyes were brilliant in the darkness. \"I am Ota Qwan of the Sossag. I offer you my hand in peace. I was dead. You were dead. Let us both walk away from here and live.\"\n\nRanald was a brave man, veteran of fifty fights, and yet the relief that flooded him was like a mother's kiss and the release of love, and never, ever had he felt he had so much to live for.\n\nHe looked down at the corpse of his cousin. \"May I bargain with the faeries for him?\" he asked.\n\nTheir laughter was derisive.\n\n_Two! We gave two! And we will dine for days!_\n\n_So sweet and rare._\n\nRanald knew what men said of the fair folk. So he bowed. \"My thanks, fair people.\"\n\n_Thank Peter!_\n\n_Hee hee._\n\nAnd they were gone.\n\nRanald reached down and took Lachlan's great sword from his cold, dead hand. He unbuckled the scabbard from the great gold belt, and left the belt for spoil.\n\n\"For his son,\" Ranald said to the black man, who shrugged.\n\n\"I would meet this Peter,\" Ranald said.\n\nThey walked down the knoll together, and the Sossag all moved back.\n\nOne warrior, reeking of vomit, was weeping uncontrollably.\n\nRanald pulled the man to his feet, and put his arms around him. He didn't know why himself. \"Don't know why you saved me,\" he said. \"But thank you.\"\n\n\"He saved me,\" Ota Qwan said, his voice thick with wonder. \"Somehow, the fairies chose to bring you back, too.\" Ota Qwan leaned forward. \"I think you killed me.\"\n\nRanald nodded. \"I think I did.\"\n\nPeter sobbed, and was still.\n\n\"I hurt,\" he said. \"I'm cold.\"\n\nRanald knew the cold to which he referred. He shook the man's hand again, shouldered his dead cousin's sword, and walked away to the east, through a corridor of silent Sossag warriors.\n\n# Lissen Carack\u2014The Red Knight\n\nA league from the convent, the captain began to relax and let the feeling of victory suffuse him.\n\nThey had almost thirty wagons, full of goods\u2014many of them would be of no use, but he'd seen the armour in one, fine helmets, and weapons in another, and wine, oil, canvas cloth\u2014\n\nBut it wasn't rescuing the wagons that lifted his heart. Nor the capture of the wounded knight, a moment that he had yet to allow his mind to savour.\n\nIt was the men. Ten professional soldiers, three dozen guildsmen with bows\u2014almost fifty stout men. If he could make it back to the fortress, he'd have hurt his adversary cruelly and _gained_ in strength.\n\nHalf a league from the fortress, when it was plain that Lissen Carak was not afire, had not fallen to assault of black sorcery, he found himself whistling.\n\nSauce rode by his side. \"A word?\" she asked.\n\n\"Anything you like,\" he said.\n\n\"Do you have to kill every single one of the monsters?\" she asked, and she spat like Bad Tom.\n\nLooking carefully, he could see she was literally spitting mad.\n\n\"I had that tusked thing,\" she said. \"I don't need you stealing my kills. If another man had done it, I'd gut him. Even Tom.\"\n\nThe captain rode in silence for a few paces. \"I can't help it,\" he said.\n\n\"Fuck you,\" she said.\n\n\"I don't mean that the way it sounds, Sauce,\" he said. \"I can't help it. If they see me, they come straight at me. It has been that way for as long as I've faced the Wild.\"\n\nSauce didn't wrinkle her lip\u2014she wrinkled her whole face. \"What?\" she asked, but her tone betrayed that she had noticed something of the sort.\n\nHe shrugged, but he was tired and wearing forty pounds of hauberk and armour, so it wasn't all that evident a movement.\n\n\"Why?\" she asked.\n\n\"I don't know,\" he said, lying.\n\nShe narrowed her eyes.\n\nHe didn't offer any further information.\n\n\"Who's the knight?\" she asked.\n\nThe captain realised he was entering a whole field of cowpats with her questions. \"Ask him when he wakes,\" the captain said.\n\n\"He was going to kill you,\" she said. It was somewhere between a statement and a question.\n\n\"Haven't you ever been tempted yourself?\" Jacques asked from behind them.\n\nSauce's clear, honest laugh rolled across the river and announced them to the Bridge Castle.\n\nAnd the captain rode on, whistling.\n\nIn his head, he saw a beaten, angry adolescent, who said hot words\u2014hot and true\u2014to a man who was _not_ his father, and rode away bent on death. He tried to reach out to that boy, across the years.\n\n_Whatever befalls us,_ he told the broken boy, _today we won a great victory, and men, if any survive, will speak our name for a century._\n\nOf course, the desperate, angry boy simply kept riding. He would ride his horse to death, and then he would walk, and then he would try to kill himself with a dagger, and he would find that he didn't have the stomach for it, and he would fall asleep, weeping. And wake to try again, and fail, hating himself for what he was, and hating himself again for his cowardice.\n\nThe captain knew it. He'd been there. He still had the two sloppy knife scars.\n\n\"Happily ever after,\" he said, with very little bitterness. He touched the white handkerchief at his shoulder and rode to the convent, still whistling.\n\n# Lissen Carak\u2014Mag the Seamstress\n\nMag watched them return from her barrel by the main gate, where she sat with her back against the lead down pipe from the chapel gutters, sewing.\n\nLike many of the farmers and folk in the fortress she had reason to fear the armoured men. But today, they were different. Today, they seemed less like a gang of thugs bent on violence and more like something from a song.\n\nThe young knight who led them was first through the gate, and he paused to call something back to the column\u2014in fact, he shouted to them \"Finish like you started!\" And she saw them all sat up in their saddles, even the ones with blood showing.\n\nThe only difference she could see was that most of them were smiling. But there was something else\u2014a pride to them\u2014that she hadn't felt before.\n\nThe captain swung down from his charger and gave the reins to Toby, and the boy beamed at him, and the captain grinned and said something that made the servant boy grin even harder.\n\nDefeated men wouldn't look like that, the seamstress was sure.\n\nSer Thomas rode in with the female knight by his side, and the two barely fitted through the gate, but neither would give way to the other.\n\nThe courtyard was filling with nuns and farmers and their folk, taking horses, talking\u2014in moments, it was clear that a great victory had been won, and an air of festival filled the fortress.\n\nMag finished her line of stitches quickly, gathering the heady aura of victory against long odds with every stitch and pulling it into the cap.\n\nThe old Abbess came to the steps from the hall, and the young captain, resplendent in his bright red surcote and gilt-edged armour, climbed up, knelt on one knee in salute, and spoke to her.\n\nShe nodded, gave him her hand, and then raised her hands for silence.\n\n\"Good people!\" she called. \"The captain informs me that our little army has won a great victory through the grace of God. But we are to expect an immediate attack, and every one of you is to get under cover now.\"\n\nThe men-at-arms were already pushing people back into the nunnery, dormitory and the great hall. Mag saw the young knight turn, and catch the eye of the novice.\n\n_Oh aye_ , she thought. She smiled, mostly because they both smiled.\n\nWhen the archers on the walls began to look at her pointedly, she gathered her basket and slipped into the dormitory herself.\n\nBut she'd just seen the priest do the oddest thing: he'd taken a dove from a cage and thrown it over the wall.\n\nShe might have said something, or reacted\u2014but even as she watched, the Red Knight appeared and the priest departed. They didn't see each other. The Red Knight spoke to someone who was with him up on the wall\u2014a leg appeared over the dormitory balcony, and suddenly the armoured man held someone in his arms. Someone in the plain garb of a novice.\n\nThe intensity that bound them was blinding. Mag could see it, feel it, the way she could feel the well of power under the dungeons and the Abbess working her spells. It was a magnificent thing.\n\nIt was also private and she turned her head away. Some things, people are not meant to see.\n\n# Albinkirk Citadel\u2014Ser John Crayford\n\nThe Captain of Albinkirk sat at his glazed window, and watched the distant woods.\n\n_My Lord,_\n\n_I must assume that my last messenger has reached you. The citadel of Albinkirk continues to hold. Indeed, it is some days since we have been assaulted, although we are still close-pressed and we can see creatures of the Wild moving about in the town and in the fields._\n\n_Yesterday I felt it was my duty to take a sortie beyond the citadel walls. We scattered the creatures in the main square and rode beyond the city walls, too. As soon as my small force appeared in the fields north of the river, we were joined by dozens of local families who had held one of the outworks and sought admission to the citadel. I had no choice but to let them in\u2014they had no food. Among them were two guildsmen from Harndon, members of the Crossbowmen of the Order of Drapers. They say that a great battle was fought yesterday, south of the Fords, and that theRed Knight prevailed, albeit with a small force, crushing a great ambush of the Wild, for which praise to God. But another pair of refugees from the east informed me that Sossag raiders have burned every town east of the Fords all the way to Otter Creek, and that the hills are crammed with refugees._\n\n_All of this may be rumour. If I can spare the men, I will send a scout west to cooperate with the Abbess and the Red Knight._\n\n_My lord, we face here the very worst of the enemy. I beg you for immediate aid._\n\n_Your servant,_\n\n_John Crayford, Captain of Albinkirk_\n\n# Chapter Eleven\n\n# East of Albinkirk\u2014Thorn\n\nThorn sat cross-legged beneath the tree that bore his name and watched the world.\n\nHe couldn't pretend that he liked what he saw.\n\nHe had suffered a crushing defeat the day before\u2014the little army that the sisterhood had hired, led by the dark sun that could extinguish itself\u2014had combined with the last convoy coming upriver to crush his best mobile force.\n\nEven now, he couldn't reach any of his chieftains among the irks. Boglins were coming back across the river. But the losses had been staggering.\n\nAnd he could feel the waves of sheer power that still rolled across the sea of trees from the fight. Someone almost as great as him had loosed powers that were better left unloosed. That power sang through the Wild like a clarion call. And Thorn knew the taste of that power.\n\n_I should have been there_ , he thought. His stone mouth creased in a near smile. _My great apprentice, free from his tower and loose on the world at last._ He flexed the reins of his spell of ensorcellment, but the reins hung slack, severed at the far end, and he reeled them in. _I wonder how the boy worked it out?_ He thought. But he didn't waste much thought on it. His apprentice had tricked him once and would never, ever best him again.\n\nBut his rebellious apprenctice wasn't the only problem. Someone had killed three of the dhags which men called trolls, the great cave giants armoured in stone of the high mountains. He had only bound a dozen to serve him, and now three were slain.\n\nAnd perhaps the worst blow of all was the Sossag's defection. Their chiefs had deserted him, and gone east to fight their own battle. Had they been present with his force, none of this would ever have happened.\n\nThorn wheeled his starlings and doves in the sky, and looked down from their eyes, and knew that he had been misled by the powers in the old fortress. The assault of the birds of prey had pushed his little helpers away. And he had been blind. _For one scant hour._\n\nBut in his hand was a precious jewel. His friend had, at last, sent him word. Detailed word.\n\nDespite the defeat, he now had the true measure of his enemy, and his enemy was not as strong as Thorn had feared. He didn't like the taste of their power, but he didn't need to fear their soldiers. They were too few.\n\nThorn had not risen to power by ignoring the causes of defeat. He didn't accept false pride. He acknowledged that he had been fooled, and beaten, and immediately altered his plans.\n\nFirst, the Sossag had won a victory that would serve his ends\u2014and they were badly hurt and their leaders looked fools. This was the time to force them back to their allegiance to him. He needed them, and their ruthless human cleverness\u2014so very different, and so much more cunning than the irks and bogglins.\n\nHe needed to consult with his allies among the Qwethnethog daemons, and he needed to convince them, with a show of force, that he was still the master of these woods. Lest they slip away too.\n\nHe savoured the irony. He was attacking the Rock for them, and yet they threatened to defect.\n\nHe sighed, because all these petty inter-plays of emotion and interest resembled the very politics that had driven him away from other men, when he was a man. The Wild had been his escape and now proved the same.\n\nIt was foolish that he needed a victory to convince the unwilling when he could take the lives of most of his allies merely by reaching into the essence of their Wildness and _pulling_ \u2014\n\nHe remembered one of his students admonishing him that you could not convince men by killing them, and he smiled at the memory. The boy had been both right and wrong. Thorn had never been very interested in convincing anyone.\n\nBut reminiscence would solve nothing. He withdrew his attention from the doves and the lynx and the fox, the hares were all dead, taken by dogs, and he moved his thinly distributed consciousness back to the body he had made for it.\n\nA dozen irks stood guard over him, and he acknowledged them. \"Summon my captains,\" he said in the harsh croak he now had as a voice, and they flinched and obeyed.\n\n# West of Albinkirk\u2014Gaston\n\nThe army that now trailed north on the last stretch to Albinkirk, was many times larger than the elite force that had left Harndon a week before. And much, much slower.\n\nGaston sat his horse in the midst of a road blockage bigger than some towns in his home province and shook his head. He was watching four men who sat hunched under a bridge, eating a side of bacon.\n\n\"It's like the rout of a beaten army,\" he said in low Archaic. \"Except that it is still headed towards the enemy.\"\n\nThe king was virtually unapproachable, now, as the entire knight-service of the country had reported in, and all of his great lords surrounded him. No longer could Jean de Vrailly pretend to threaten the king with his three hundred knights\u2014his convoy was no longer the largest. The Count of the Borders, Gareth Montroy, came in with five hundred knights, hard men in lighter armour than the Galles but just as tall, and five hundred archers as well. The Lord of Bain's banner led another two hundred knights, with the popinjay Edward Despansay, Lord Bain, at their head. They were the great lords, with uniformed retinues of professional warriors who trained together, but there also were hundreds of individual knights from the counties under the King's Lieutenant's banner, and almost a hundred of the king's own Royal Knights, his elite bodyguard that also canvassed the countryside as justices and monster hunters under the king's trusted bastard brother, Ser Richard Fitzroy. There were another hundred knights of the military orders, priests and brothers and lay brothers of Saint George and Saint Maurice and Saint Thomas whose discipline was as good or better than any company Gaston had ever seen, riding silently in their black-robed armour under the Prior of Pynwrithe and his marshal.\n\nAll together the king had more than two thousand knights and as many again men-at-arms, plus three thousand infantry who varied in quality from the superb\u2014the green clad Royal Huntsmen rode ahead of the column and covered its flanks, dashing silently through the increasingly dense brush on specially trained horses, although they fought on foot as archers\u2014to the ridiculous: county levies with spears and no armour who served for twenty days or until their side of bacon was eaten.\n\nThe men at his feet were eating as quickly as they could.\n\nHis beautiful cousin was riding at the head of his convoy. He wore his full harness\u2014all the Galles did\u2014and rode a war horse. But the last few days, the Alban knights had begun to do the same\u2014not all at once, but in fits and starts. And in the evenings, they had begun to practise with their lances and with their swords, with their horses formed in great long lines.\n\nAnd de Vrailly went from group to group, praising some and challenging others. He praised the diligent and ignored the lazy, and men began to speak of him.\n\nKnightly men. Not this sort.\n\nGaston watched the men under the bridge, and they watched him, chewing and swallowing as quickly as they could manage, forcing the cooked bacon down their gullets.\n\nHe gave his horse some rein and she picked her way down the grassy bank to the stream. The men under the bridge began to pick up their belongings, but he raised a hand to forestall them.\n\n\"We haven't done nothing,\" a sandy-haired yokel with a short beard said, raising two greasy hands.\n\nGaston shook his head. \"Answer me the one thing,\" he said carefully. Speaking Alban always left him feeling muddled.\n\nThe sandy-haired one shrugged. Gaston noted that he hadn't said one word of polite greeting\u2014neither saluted, nor bowed.\n\nAlbans. A nation of fools and outlaws.\n\n\"Why are you so anxious to eat your cooked ham and scurry home?\" he asked. He walked his mare forward another few steps so that they could hear him better. He looked down at them.\n\nAll four of them looked at him as if he, not they, was the fool.\n\n\"Cause my wife needs me home?\" said one.\n\n\"Cause it's going to be haying in another ten days, if the sun keeps on,\" said the second man. He had a fine linen shirt and a silver ring on his finger. By Galle standards, Alban farmers were rich, fat and very ill-mannered.\n\n\"Cause my duty says I can go home when this here bacon is et,\" said the third, a long-haired old man. His hair was mostly white and Gaston could see the outline of a crusading badge on his tunic, carefully removed.\n\n\"You have fought before, eh?\" he asked.\n\nThe older man nodded, his face still. \"Right enough, boyo,\" he said. Here under the bridge, their voices echoed.\n\n\"Where?\" Gaston asked.\n\n\"In the East,\" the old man said, and took another bite of bacon. \"And before that, under Ser Gilles de Laines, against the Paynim. With Lord Bain, too. And under the old king, at Chevin. Ever heard of it?\"\n\nGaston smiled. \"You are pleased to make game of me,\" he said pleasantly.\n\n\"Nah,\" said the old archer. \"You foreigners don't really know much about war, and you haven't ever seen a big fight like Chevin. If you had, you wouldn't be asking us these tom-fool questions. We're eating our bacon so we can get home and not fight. Because it's going to be horrible, and I, for one, know just fucking how it's going to be. And my son-in-law and his two friends here will all come with me.\"\n\nGaston was shocked by the man's tone, and by the murderous gleam in his eye. \"But you\u2014you have been a _homme arm\u00e9._ You know what honour is\u2014what glory is.\"\n\nThe man looked at him, finished his chunk of bacon, and spat. \"Done. Time to go home.\" He wiped his greasy hands carefully on his leather quiver and the bow case on his six-foot bow.\n\n\"If we lose,\" Gaston said, looking for a way to reason with this arrogant peasant, \"if we lose, your farms will be lost.\"\n\n\"Nah,\" said the younger man with the beard. \"If you'n lose, they'll squash the north flat. We ain't northerners.\" He shrugged.\n\nThe old archer shrugged.\n\nThe other two grinned.\n\nThe old archer came over to the knight's stirrup. \"Listen, ser knight. We stood our ground at Chevin, and a lot of folk died. The old king told us we was done, for our lifetimes. Well, I'm holding him to that promise. Right? Here's some advice from an old soldier. When the boglins scream and charge you, say a good prayer. Cause they won't stop coming, and there's a lot worse behind them. They eat you while you're still alive. There's creatures that're worse, and eat your _soul_ while you're still alive. So it don't even matter if you heard Mass, does it?\"\n\nGaston had considered killing all four of them for their insolence, but the old archer had touched on something, and instead, he found himself nodding.\n\n\"I will prevail. We will prevail,\" Gaston said. \"You will be sorry you were not there, for our day of glory.\"\n\nThe old archer shook his head. \"Nope. That's just what gowps like you never see. I won't be sorry, but I do wish you luck.\" He chuckled. \"We had twenty thousand men when we went into battle at Chevin.\" He nodded again. \"The king has what\u2014four thousand?\" He laughed, and it was a nasty laugh. \"Can I offer you a bite of bacon?\"\n\nTalking to the peasants had caused Gaston to fall behind, and when he rode up the far bank, chewing on bacon, he found himself in the midst of the Borderers. He rode forward until he was among the liveried knights, the professionals, who rode around the Count of the Borders.\n\nA herald spotted him and he was quickly passed from the herald to the captain of the bodyguard, and then on to the knot of men around the count himself. He was riding armed, in a good white harness made in the East, with mail and leather under it. A squire carried his helmet, and he had a green velvet cap on his head with an Eastern ostrich plume sprouting rakishly from a diamond brooch.\n\n\"Gareth Montroy,\" said the great lord, extending his hand even as he reined in his horse. \"You're the Count of Eu?\"\n\n\"I have that honour,\" Gaston said, bowing and clasping the man's hand. He was thirty-five, with dark hair and heavy eyebrows and the absolute air of command that came with great lordship. This was a man who commanded men every day.\n\n\"Your cousin has the big convoy\u2014all Galles?\" Lord Gareth grinned. \"They look like bonny fighters. Big boys every one of 'em, like my lot.\" He jerked a thumb over his shoulder.\n\n\"Your men look like fighters,\" Gaston said.\n\n\"Pour us a cup of wine to cut the dust, eh, Gwillam?\" Lord Gareth said over his shoulder. \"My lads have seen a spot of fighting.\"\n\nEvery man in the count's escort had a facial scar.\n\nGaston felt more at home here than he had in days. \"Where have you been fighting?\" he asked.\n\nLord Gareth shrugged. \"I hold the Westland borders, though there's some awkward bastards at court and elsewhere who don't give me my due,\" he said. A silver cup, beautifully made, with sloped sides and a carefully worked rim, was put in his hand, and another was passed to Gaston, who was delighted to find that it was lined in gold and full of chilled wine.\n\n_Chilled wine._\n\n\"Company magus,\" Lord Gareth said. \"No reason he can't keep some wine chilled until we fight.\" He grinned. \"And sometimes, we fight the Moreans. Bandits, the occasional boglin\u2014we know what boglins look like, don't we, boys?\"\n\nThey laughed.\n\n\"And you, my lord?\" Lord Gareth turned to Gaston. \"You've seen service before, I take it.\"\n\n\"Local wars,\" Gaston said dismissively.\n\n\"How big is a local war, in Galle?\" Lord Gareth asked.\n\nGaston shrugged. \"When my father marches on an enemy he takes a thousand knights,\" he said.\n\n\"Mary, Queen of Heaven!\" Lord Gareth swore. \"Christ on the Cross, my lord. Only the king has a thousand knights, and that only when he sends out Letters of Array.\" He raised an eyebrow. \"I'd heard of such doing, but never from a witness.\"\n\n\"Ah,\" Gaston said.\n\n\"And what do you fight?\" Lord Gareth asked. \"Boglins? Irks? Daemons? Trolls?\" he looked around. \"How many creatures can the Enemy muster, that your father takes a thousand knights?\"\n\nGaston shrugged. \"I have never seen a boglin,\" he said. \"In the East we fight men.\"\n\nLord Gareth winced. \"Men?\" he said. \"That's a nasty business. I admit, I've faced the Moreans on a few fields\u2014but mostly brigands. There's little joy in facing men, when the Enemy is to hand.\" He leaned close. \"Who fights the Enemy in the East, then?\"\n\nGaston shrugged. \"In the north, the military orders. But no one has seen a creature of the Wild for\u2014\" He searched for the words. \"Please do not take this ill\u2014but if you Albans were not so very sure of the Wild, we'd doubt you. None of us has ever seen a creature of the Wild. We thought they were exaggerations.\"\n\nTo a man, the knights around Lord Gareth threw back their heads and laughed.\n\nA tall, swarthy man in a harness of scale armour pushed his horse through the press to Gaston's side. \"Ser Alcaeus Comnena of Mythymna, my lord.\"\n\n\"A Morean,\" Lord Gareth said. \"But a friend.\"\n\n\"Perhaps your convoy needs to be taught about the creatures, yes?\" he volunteered.\n\nGaston shook his head. \"No, no. We'll do well enough. We train very hard.\"\n\nAll the knights around him looked at him as if he'd just sprouted wings, and Gaston had a moment's concern.\n\nAlcaeus shook his head. \"When the boglins get in among the horses, they will give their lives to gut your charger,\" he said. \"A single troll loose in a column can kill ten belted knights as fast as I can tell you this. Yes? And wyverns\u2014in the air\u2014are incredibly dangerous in open ground. Only men with heavy crossbows threaten them, and the very bravest of knights. On foot, horses will not abide a wyvern. And no amount of tiltyard training will prepare you for their wave of fear.\"\n\nGaston shrugged, but now he was annoyed. \"My knights will not succumb to fear,\" he said. The Morean looked at him as if he was a fool, which made him angry. \"I resent your tone,\" he said.\n\nSer Alcaeus shrugged. \"It is of no moment to me, Easterner. Resent me all you like. Do you want your knights to die like cattle, paralysed by fear, or would you like to strike a blow against the enemy?\"\n\nThe Count of the Borders pushed his horse between the two men. His displeasure was evident. \"I think that the good Lord of Eu is saying that we have nothing to teach him about war,\" he said. \"But I do not tolerate private quarrels between my knights, Lord Gaston, so please do not taunt Ser Alcaeus.\"\n\nGaston was flabbergasted. He looked at the man. \"What is it to your knight whether you tolerate his quarrel?\" he asked. \"Surely if a knight's honour is at stake, the least his lord can do is to stand behind him.\"\n\nLord Gareth's face became carefully neutral. \"Are you challenging Ser Alcaeus on his honour, because he tried to tell you that your convoy needs training?\"\n\nHis tone, and the point he made, caused Gaston to squirm in the saddle. \"He suggested that my men would be _afraid.\"_\n\nAlcaeus nodded as though this were a forgone conclusion. All the other men-at-arms around them were silent, and for a long moment the only sound was the jingle of horse harness and the rattle of armour and weapon as the retinue knights walked their horses down the road.\n\n\"You do know that every creature of the Wild projects a wave of fear, and the greater the beast the stronger it is.\" Lord Gareth raised both eyebrows. It made the diamond on his cap twinkle.\n\nGaston shrugged. \"I have heard this,\" he admitted. \"I thought it might be... an excuse...\" He stammered to silence in the face of the massed disapproval of a dozen scarred knights.\n\nSer Alcaeus shook his head. \"You need us,\" he said quietly.\n\nGaston was trying to imagine how he might convince his cousin while he rode up the column.\n\n# North of Lissen Carak\n\nThey came, each with his own tail of followers, because that was the way of the Wild.\n\nThe man known as Jack, the leader of the Jacks, came from the west. His face was masked in ruddy leather, and he wore the same dirty off-white wool jupon and hose of his band. He wore no badge of rank, and carried no obvious symbol of it\u2014no fancy sword, no magnificent bow. He was neither short nor tall, and a greying beard came out from under his mask to proclaim his age. With him were a dozen men with long yew bows, sheaves of arrows, long swords and bucklers.\n\nThurkan came from the south, where he had run the woods with his qwethnethog daemon kin, watching the Royal Army coming up the Albin River. A fifty-mile run through the woods had not winded him. The wave of fear that he projected made the hardened Jacks fold their arms; even Thorn felt his power. With him were just two of his mighty people\u2014his brother Korghan, and his sister Mogan. Each was the size of war horse with jaggedly pointed beaks, inlaid brow ridges, beautiful eyes and long, heavy, muscular legs, long arms tipped with bone scythes, and elegant, scaled tails. With them came the greatest of the living abnethog wyverns in the north woods; Sylch. His people had borne the greatest losses, and his anger was betrayed in bright red spots that moved like flickering fire on the surface of his smooth grey skin.\n\nFrom the east came a party of painted men; Akra Crom of the Abenacki led them. They had harried the suburbs of Albinkirk, taken a hundred prisoners, and were now ready to go home. Such was the way of the Outwallers\u2014to raid and to slip away. Akra Crom was as old as a man could be and still lead Outwaller warriors\u2014his skin betrayed his age. He was hairless, painted a metallic grey that gleamed like silver in the light. He was the rarest of Outwallers\u2014a possessor of power. A shaman, warrior, and a great song-maker among his people, the old man was a living legend.\n\nExrech was the chief paramount of the gwyllch that men called bogglins. His thorax gleamed white, and his arms and legs were a perfectly contrasting ebony black, as was his head. He was as tall as a man and power flickered around his mandibles, far more pronounced than a lower-caste gwyllch; his natural armour was better, and his chain mail, carefully crafted in the far East and taken in war, had been riveted carefully to his carapace to join the living armour. He carried a pair of man-made great swords in his two large hands and wore a horn at his waist.\n\nThorn was pleased they had come, and he offered wine and honey.\n\n\"We have taken heavy losses, and suffered costly victories and humiliating defeats,\" Thorn began. He left it there\u2014the fact of defeat.\n\n\"The Sossag have won a great victory in the east,\" said the painted man. The other warriors with him grunted their approval.\n\n\"They have, at great cost,\" Thorn nodded. Overhead, the stars were rising\u2014a spectacular display of light in the blue-black sky of late evening. But their meeting was not illuminated by fire. Few creatures of the Wild loved fire.\n\nThorn pointed at the heavens. \"The Sossag and the Abenacki are not as numerous as the stars,\" he said. \"And many Sossag fell at the Crossings of the Otter.\"\n\nExrech's jaws opened and closed with a firm click indicating _waste of valuable warrior stock; not easily replaced; no clearly defined target. Strong disapproval._\n\nAkra Crom shrugged. _When you rule the Outwallers, you may choose their wars._\n\nThe black and white gwyllch lord gave an acrid spray of anger. _In deep woods, all soft-skins alike to we._\n\nThorn grunted and both lords settled down.\n\nThurkan spoke, his daemon voice high and badly pitched\u2014a shock from such a large and beautiful creature. \"I blame you, Thorn.\"\n\nThorn had not expected a direct challenge and began to gather power.\n\nThurkan reached out a long forearm and pointed. \"We each act under your order\u2014but we do not mesh. We are not _together._ No gwyllch stand with the Sossag. No gwyllch climb with the Abnethog when we fly against the Rock. Abnethog and qwethnethog and gwyllch fight the same foe in the same woods, but no creature goes to the support of the other. The hastenoch died with gwyllch a few hands away.\"\n\nThorn considered this\u2014full of power, ready for the challenge that criticism usually led to, he was not at his most rational.\n\n\"You have armed yourself against me,\" whined the great daemon. At least, his every utterance sounded like a whine. \"Yet I challenge you not, Once Was Man.\"\n\nThorn let some of the power he had gathered dissipate.\n\nFaeries had been attracted, as they always were by raw power, their slim and elegant shapes flitting suddenly through the air where his release of power glowed a virulent green.\n\nMogan plucked one from the air and ate it, and the faeries' death-curse filled the night as the little thing vanished down her gullet.\n\nExrech nodded. _Strong one. Well taken._\n\nJack of Jacks shuddered. To most men, the killing of a faery was sacrilege. He spat. \"Thorn, we are here for one reason only. You promised us you'd defeat the aristocrats. For that, we have gathered every bow from every farm. Our people suffer under our lords' hammers this summer so that we can defeat them. And yet, the king's army comes closer and closer.\" Jack scowled. \"When will we fight?\"\n\n\"You are a deadly secret, Jack of Jacks.\" Thorn nodded. \"Your long shafts will be the death of many a belted knight, and your men\u2014you said yourself they must stay hidden. They will emerge from decades in the shadows at the right moment, when we play for everything. I will face the king and his army on ground of my choosing. You will be there.\"\n\nHe turned to the qwethnethogs. \"I am guilty of sending each of you to fight your own foes in your own way. This still seems wise to me. Between gwyllch and Outwaller there is no friendship. The Jacks have no love for any creature of the Wild. Every beast in the woods fears the qwethnethog and the abnethog.\" He ate a dollop of honeycomb. \"We should have triumphed by now, and I feel the strong hand of fate on the rim of our shield. I command that you all take more care.\" He'd lowered his voice and imbued it with power from the air around him and the store he held for emergencies, and even so the daemons challenged him.\n\n\"Obey me, now. We will not fight the king at Albinkirk. We let our early victory spread us too far, dissipate our strength. Let Thurkan watch the king and eat his horses. No more. Let Exrech withdraw from Albinkirk. Offer no battle. Let the Sossag and the Abenacki fall back to their camps here. Let the Jacks sharpen their bodkins. Our day approaches, and the king will never reach Lissen Carak.\"\n\nThurkan nodded. \"This is more to my liking,\" he hissed. \"One mighty fight, and a rending of flesh.\"\n\nThorn forced a piece of a smile\u2014it seemed to crack the flesh around his mouth\u2014and all but the daemons quailed. \"We will scarcely need to fight,\" he said. \"But when they have fought among themselves, you may rend their flesh to your heart's content.\"\n\nThurkan nodded. \"Such is always your way, Thorn. But when it comes to teeth and spears I do not like having the Cohocton at my back.\"\n\nThorn hated being questioned, and his anger rose. \"You fear defeat before a single spear is cast?\"\n\nThe great daemon stood his ground. \"Yes,\" he said. \"I have seen many defeats, and many empty victories; my hide bears the scars, and my nest is empty where it should be full. Both of my cousins have died in the last moon\u2014one on the spear of the dark sun, and one with his soul ripped from him by their cruel sorceries.\" He looked around. \"Who will come to my aid? You expect treason\u2014and I agree that humans are born to betray each other. But many will fight, and fight bitterly. This is their way! So I say\u2014who will come to my aid?\"\n\n\"Have you finished whining?\" Thorn bellowed.\n\nJack squared his shoulders. \"If it is your unshared plan that the mighty daemons face the king then my comrades and I will be honoured to share the danger with our scaled allies.\"\n\nThorn wanted to scream in frustration. _My plan is my plan is my plan. I will not share it with the likes of you._ But he narrowed his eyes, banished the bile from his great heart, and nodded.\n\n\"Then gather more boats, and prepare to cross the river. This time, protect them. For unless the king is a great fool, he will advance on the south side of the river, as my brother Thurkan fears. Yes? And if you are hard pressed, I will send gwyllch, at least the lighter kind, who can pass the river.\"\n\nExrech spat a clear fluid. _Waste of resources; conflict of interest._\n\nThorn took a deep breath, and pushed power into his word.\n\n_\"Obey,\"_ he said.\n\nBy the time the fireflies came out, the clearing in the woods was empty.\n\n# Lorica\u2014Desiderata\n\nDesiderata sat on her throne in the Great Hall of the castle of Lorica, still dressed for travelling. She had a dozen minor issues on which to pronounce justice, and all she wanted was dinner and bed. Taking a train from Harndon to Lorica in a day was harder work than she'd expected.\n\nShe worked her way through the cases\u2014the murder of a draper by a woman, the theft of a herd that trailed off into accusations and counter-accusations by the monks of two rival abbeys\u2014and then there was a messenger.\n\nHe wore the royal scarlet and midnight blue livery, and even covered in road dust it commanded instant attention.\n\nHe was young and not particularly handsome, and yet had an air about him. He knelt at her feet and presented a bag.\n\n\"The king sends to you, my lady,\" he said formally.\n\nShe didn't know him, but word of war had made the king increase every part of the household\u2014an action that would affect the royal budget for ten years to come.\n\n\"Royer Le Hardi, my lady,\" the messenger said.\n\n\"The news?\" she asked.\n\n\"All is well with the army,\" Royer replied.\n\nThe Queen took the pouch and opened it, cutting her husband's seal carefully and opening the lead wafers that secured the buckles with the small knife she always wore in her girdle.\n\nThere were four scroll tubes holding about a dozen folded and sealed letters\u2014she saw letters to the Emperor of Morea and the King of Galle\u2014and a thick packet with her name on it in his handwriting, which she snatched up.\n\nShe read a few lines and frowned. \"My lords, ladies, and good men and women,\" she said formally, rising to her feet. \"I will hold court in the morning, and all cases are held over until then. The seneschal and sheriff shall attend me, as will my own lords.\" She smiled, and many in the multitude at her feet smiled back, so personable was her smile.\n\nThe hall's chamberlain smacked the floor with his staff. \"The Queen has dismissed the assembly,\" he said, in case there were those who didn't understand.\n\nBefore the last draper had cleared the portico the Royal Steward and the King's Treasurer\u2014were at her side. \"News?\" asked Bishop Godwin. Lord Lessing\u2014a banker promoted to the aristocracy by the old king\u2014rubbed his beard.\n\nShe tapped the cover note against her teeth. \"We will continue north to join the army,\" she said. \"If we have a tournament at all, at this rate it will be in the face of the enemy, at Albinkirk or even Lissen Carak.\" Her thoughts were clearly elsewhere.\n\nHer king's note sounded desperate, and he had ordered her not to come.\n\n\"Strip this town of carts,\" she said. \"I will leave everything that I don't need\u2014I'll take four maids. No state gowns, no frippery, no clothes. You, my lords, should stay here. You will form the government.\" She paused. \"No. Go back down the river to Harndon.\"\n\nThe bishop breathed a sigh of relief.\n\n\"I might be gone a month,\" she said. \"Or more. I may stay with the king until the emergency is past. Lord Lessing, I would take it as a kindness if you would organise the supply convoys as I have been doing.\"\n\nLessing pulled at his beard. He had gold wire in it, which somehow served only to make it look greyer. \"I will do your will, Lady,\" he said gravely. \"But some of those wagons need to start coming back. We have stripped the southern kingdom bare and I doubt that there is even a wheeled cart to be had in Harndon. If they are lost, the harvest will rot in the fields.\"\n\n\"Best they not be lost, then,\" she said lightly. \"I'll see to it that the wagons I've sent north are turned around\u2014either empty, or full of the northern harvest.\"\n\n\"Boats,\" Lessing said suddenly. \"If he's aiming for Lissen Carak, you should go by boat. The docks here are full of empty hulls\u2014Master Random of Harndon's boats. He's arrayed a mighty fleet of river boats to buy the grain harvest in the north. It's supposed to be a secret, I admit. But I had it from his wife, and you can go faster by oars and sail up the river. And it's safe as houses\u2014never yet heard of a boglin as could swim. Eh?\"\n\nShe loved her lords because they weren't going to try to stop her, and because both of them began immediately to plan for the practical details of her trip to join the army.\n\nAfter they'd made a dozen lists and summoned half the prominent men of Lorica to witness deeds and to become commissioners of this and that, she collapsed at last into the best bed in the royal keep of Lorica.\n\nMary stripped off her silken cote hardie, her kirtle, her shift, and the man's hose she'd worn underneath so she could ride astride. \"You will take me with you?\" Mary asked.\n\n\"You and Emmota, Helena and Apollonasia,\" the Queen said languorously. \"And Becca.\"\n\n\"Bath?\" Mary asked.\n\n\"Perhaps the last for many days,\" the Queen said. \"Oh, par dieu, Mary, we are about to break free of it all and have an _adventure._ \"\n\nLady Mary smiled at her mistress. But her eyes had no smile in them, as if she was looking far beyond their room.\n\n\"Do you still think of him?\" the Queen asked her First Maid.\n\n\"Only when I'm awake,\" Hard Heart admitted. \"And sometimes when I'm asleep.\"\n\n\"He is not with the army.\" Desiderata had received two missives from her husband that included the name of Gawin Murien, but in both his whereabouts was unknown.\n\n\"I will be closer to him,\" Mary said. She sighed. \"I didn't know that I loved him until the king sent him away.\"\n\nDesiderata held her Mary for a few tears, and thought of her husband's letters.\n\n_He was worried._ That came through, either despite his foolish banter or because of it.\n\nHe needed her there. To remind him who he was.\n\nShe fell asleep thinking of Mary and Gawin, and awoke to find that she was the admiral of a fleet of forty river boats, twenty oared boats with sturdy masts and slab sides, capable of a turn of speed and a heavy cargo. By the time the sun was above the river banks, her flotilla was pulling north, and the townsmen were glad to see the backs of the rowers, who had made more trouble than a dozen companies of soldiers. Despite her plans she'd ended up with all of her ladies, a set of pavilions, and a cargo of armour and dried meat for the army. And a company of Lorican guildsmen in horrible purple and gold livery; crossbowmen who had, to the man, never been out of the town before. They were the only soldiers that the bishop could find.\n\n\"Give way, all!\" called the timoneer.\n\nShe lay back under the bright sun, dressed in white, and let the sun turn her hair to gold.\n\n# Chapter Twelve\n\n# Lissen Carak\u2014The Red Knight\n\n_The Siege of Lissen Carak\u2014Day Six_\n\n_The woods around us are silent. Do monsters mourn?_\n\n_Day before yesterday, the captain won a great victory over the Enemy. He took most of the company across the Cohoctorn to the south, where Master Gelfred had located a convoy coming to us. It was hard hit, but the captain's sortie took the enemy in the rear, and destroyed them. The captain thinks we killed upwards of five hundred of the enemy, including four great monsters, to whit, three great Stone Trolls and a Behemoth._\n\n_The men say the captain killed the Behemoth himself, and that it was the greatest feat of arms they had ever seen._\n\n_Yesterday, the company stood to all day, waiting for attacks that never came. Men slept at their posts, fully armed._\n\n_Many of the farmers and as many nuns say this will be the end of the siege\u2014that the enemy will slink away. The Abbess has called a great council of all the officers._\n\nThe Abbess had a table brought in, and the captain thought it might be the longest he'd ever seen\u2014it filled the Great Hall from hearth to dais, space for thirty men to sit at table together.\n\nBut there were not thirty men at the table.\n\nThere were just six. And the Abbess.\n\nThe six were the captain himself, sitting in one chair with his feet on another, and Ser Jehannes, sitting upright in a third; Master Gerald Random, who by virtue of saving almost half his convoy had suddenly become the representative of all the merchants, taking another pair of chairs, and Ser Milus, as the commander of the Bridge Castle, sat with his head propped on his hands. Master Gelfred sat separately from the other men, a self-imposed social distance. And the priest, Father Henry, sat with a stylus and wax tablets, prepared to copy their decisions.\n\nThe Abbess sat to the captain's right, flanked by two sisters, who stood. The captain understood that the two silent figures were her Chancellor and Mistress of Novices, the two most powerful offices in the convent. Sister Miram and Sister Ann.\n\nWhen all the men had settled the Abbess cleared her throat. \"Captain?\" she asked.\n\nHe took his booted feet off a chair and sat up. \"Right,\" he said. \"We are now, at long last, under siege. Our Enemy has finally realised how few we are, and has sealed the roads.\" He shrugged. \"Frankly, this is a harsher defeat than any we have suffered in the field. He should have thought, after yesterday's incredible stroke of luck\u2014\"\n\n\"The work of God!\" Master Random said.\n\n\"The Enemy should have assumed,\" the captain went on, \"we had a big garrison and a lot of potent phantasm to pull off such a coup. Instead, he's used the night to push in all my outposts. I lost three good men last night, gentlemen and ladies.\" He looked around. The cunningly hidden heavy arbalest in the dead ground hadn't been cunning enough, and now Guillaume Longsword, one of his officers, as well as his page and archer were dead, and Young Will, as his squire was known, was weeping his guts out in the infirmary. \"More men than we lost in yesterday's fight,\" he went on.\n\nThe other mercenaries nodded.\n\n\"On a more positive note, Master Random brought us a dozen men-at-arms and sixty archers.\" _Of very variable quality, and every one of them ran yesterday, at one point or another. Every one but one,_ he remembered sourly. Ser Gawin had not yet condescended to open an eye.\n\n\"My guildsmen are not mere archers,\" Master Random said.\n\nThe captain sat back, assessing the man. \"I know they are not,\" he said. \"But for the duration of the siege, Master, we must treat them as soldiers.\"\n\nRandom nodded. \"I, too, can swing a sword.\"\n\nThe captain had noticed that he was wearing one, and reports had it that the merchant had acquitted himself well.\n\n\"So,\" he went on, \"we have forty men-at-arms well enough to wear harness, and our squires; call it sixty knights. We have almost triple that in archers, thanks to the better farmers and the guildsmen.\" He looked around. \"Our Enemy has at least five thousand, boglins, irks, allies and men taken together.\"\n\n\"Good Christ!\" Ser Milus sat up.\n\nSer Jehannes looked as if he'd eaten something foul.\n\nMaster Gelfred nodded when the captain looked at him. \"Can't be less, given what I saw this morning,\" he said. \"The Enemy can cover every road and every path at the same time, and they rotate their forces every few hours.\" He shrugged. \"You can watch the boglins digging trenches out beyond the range of our trebuchets. It's like watching termites. There are\u2014\" he shrugged, \"a great many termites.\"\n\nThe captain looked around. \"In addition, we have another hundred merchants and merchants' folk, and four hundred women and children.\" He smiled. \"In the East, I'd be sending them out right now, to fill the besieger's lines with useless mouths.\" He looked around. \"Here, they'd literally fill the enemy's bellies, instead.\" No one appreciated his humour.\n\n\"You can't be serious!\" said the Abbess.\n\n\"I am not. I won't drive them out to die. But the merchants and their people must be put to work, and I'd like to assign a dozen archers and two men-at-arms to training them. If we cannot be rid of these useless mouths, we must make them useful. We have about forty days' food for a thousand mouths. Double that at half rations.\"\n\n\"And we have all that grain!\" the Abbess said.\n\n\"Grain for two hundred and eighty days,\" he said.\n\n\"The king will be here long before then,\" the Abbess said firmly.\n\n\"Good day to you,\" said a voice from the door, and Harmodius, the Magus, came in. He smiled around, a little unsure of his welcome. \"I received your invitation, but I was in the midst of a dissection. You, my lords, have a plentiful supply of candidates for dissection.\" He smiled. \"I have learned some exciting things.\"\n\nThey all stared at him as if he was a leper newly arrived at a feast. He pulled out a chair and sat.\n\n\"There were rats in the grain, by the way,\" Harmodius said. \"I've disposed of them. Do you know,\" he asked, his eyes on the Abbess, \"who the captain of the Enemy is?\"\n\nShe flinched.\n\n\"You do, I see. Hmm.\" The old Magus didn't look nearly so old, today. He looked closer to forty than seventy. \"I remember you, of course, my lady.\"\n\nThe Abbess trembled\u2014just for a moment\u2014and then forced herself to look at the Magus. The captain saw the effort it took.\n\n\"And I you,\" the Abbess said.\n\n\"Well, three cheers for the air of dangerous mystery,\" the captain said. \"I for one am delighted you both know each other.\"\n\nThe Magus looked at him. \"This from you?\" He leaned forward. \"I know who you are too, lad.\"\n\nEvery head in the room snapped to look\u2014first at the captain, and then at the Magus.\n\n\"Do you really?\" asked the Abbess, and she clutched at the rosary around her neck. \"Really?\"\n\nHarmodius was enjoying his moment of drama, the captain could see it. He wished he knew who the old charlatan was. As it was, he fingered his rondel dagger.\n\n\"If you reveal me, I swear before the altar of your God I will cut you down right here,\" the captain hissed.\n\nHarmodius laughed, and rocked his chair back. \"You, and all the rest of you together couldn't muss my hair,\" he said. He raised his hand.\n\nThe mercenaries were all on their feet, weapons in hand.\n\nBut then he shook his head. \"Gentlemen!\" he said. He raised his hands. \"I beg your pardon, Captain. Truly. I like a little surprise. I thought, perhaps\u2014but please, never mind me, a harmless old man.\"\n\n\"Who the hell are you?\" asked the captain, across his bare blade.\n\nThe Abbess shook her head. \"He is Harmodius di Silva, the King's Magus. He broke the enemy at Chevin. He bound the former King's Magus, when he betrayed us.\"\n\n\"Your lover,\" Harmodius muttered. \"Well\u2014one of your lovers.\"\n\n\"You were a foolish young man then, and you still are in your heart.\" The Abbess settled primly back into her seat.\n\n\"My lady, if I am, it is because he has glamoured me for years,\" Harmodius said. \"I was not as victorious as I had thought. And he is still with us.\" Harmodius looked around the table. \"The captain of the Enemy, my lords, is the former King's Magus. The most powerful of my order to arise in twenty generations.\" He shrugged. \"Or so I suspect, and my guesswork is based on observation.\"\n\n\"You are too modest,\" the Abbess said bitterly.\n\n\"I tricked him, as you well know,\" Harmodius said. \"I could never have even hoped to match him phantasm for phantasm. And less so now, when he has sold himself to the Wild and I have languished in a prison of his making for a decade, at least.\"\n\nThe soldiers and the merchant watched these exchanges\u2014back and forth\u2014like spectators at a joust. Even the captain, whose precious anonymity had teetered at the edge of extinction, was lost.\n\n\"Let me understand this,\" he said. \"Our Enemy is really a man?\"\n\n\"Not any more,\" Harmodius said. \"Now he is an entity called Thorn. His powers are to mine as mine are to the lady Abbess's.\n\nThe priest at the end of the table had stopped writing. Now he looked at them all in horror. The captain almost felt sorry for the man. His aversion to those who possessed the power\u2014Hermetic or natural\u2014was like most men's aversion to coming in contact with disease.\n\nThe captain leaned forward. \"Can we stop the flood of reminiscence and revelation and try to dwell on the siege?\" he asked.\n\n\"He underestimated you, and you hurt him, and that's over now,\" Harmodius said. \"Now he'll hurt us, in turn.\"\n\n\"Thanks for that,\" the captain said.\n\n\"Now that he's closed off our access to the outside world, there will be no more surprise sorties, no more victories.\" The Magus sat back. \"Nor can you imagine that I can face him, because I can't. Although my presence here will make him hungrier to take this place.\"\n\n\"We can still make sorties with every prospect of success,\" the captain insisted. \"With the addition of Messire Random's convoy, we have more men-at-arms and more archers than we had at the start.\"\n\nHarmodius shook his head. \"I don't doubt it. I mean no disrespect\u2014you have done nobly. But the trick with the falcons and the dogs won't work again, and his intellect\u2014pardon me, Captain\u2014is staggering. He'll have traitors inside the walls and he'll be working to get traitors within the ranks of your companies and your merchants. He also has the power to reach out to any person among us who has power. How strong is your will, my lady?\" he asked.\n\n\"Never very strong,\" she answered levelly, \"but where he is concerned, it is like adamantine.\"\n\nHarmodius smiled. \"I imagine that's true, my lady,\" he admitted.\n\n\"Even if he has us locked in a box,\" the captain insisted, \"even if he threw his allies at the walls every day\u2014\" He shrugged. \"We can last.\"\n\n\"He won't,\" Harmodius said. He leaned forward, and it was as if he deflated, the change was so sudden. \"What he will do is seek to undermine us, because that is how he works. He will use craft and misdirection\u2014he prefers to use a traitor to open the gate, because that excuses his own betrayal. And because he likes to imagine his intellect is superior to any other.\"\n\nThe captain managed a smile. \"My old sword master used to say that a good swordsman likes not just to win, but to do it his own way,\" he said.\n\n\"Very true,\" the Magus said. \"Hubristical, but true.\"\n\nThe captain nodded. \"Hubris\u2014a common failing in your profession too, surely?\"\n\nHarmodius smiled bitterly.\n\nThe captain leaned forward. \"I have two questions, and here you are to answer them,\" he said. \"Can he attack the walls directly? With a phantasm?\"\n\n\"Never,\" the Abbess said. \"These walls have half a millennia of prayer and phantasm in them, and no power on earth\u2014\"\n\n\"Yes,\" Harmodius said. He shrugged at the Abbess. \"He is not Richard Plangere, gentleman Magus, my lady, just dressed up in feathers and gone a bit bad. He is Thorn. He is a Power of the Wild. If he puts himself to it, he can assault the very walls of this ancient fortress with his powers, and he will, in time, break them.\" He turned to the captain. \"But in my estimation, and I might be horribly wrong, he won't take that option unless all else fails, because the cost would be staggering.\"\n\nThe captain nodded. \"Not very different from the answer I expected. Second question: you are the King's Magus. Do you have the power to distract him? Or to defeat him?\"\n\nHarmodius nodded. \"I can distract him, I think. Once at little risk to myself, and once at great risk to myself.\" He laughed. \"I can feel him all around us, my lords. He seeks to know our minds and, so far, the power in this convent and in the fortress walls has stopped him. He knows I am here, but as yet I do not think he knows who I am.\" Harmodius shook his head and seemed, once again, to shrink. \"Yet until a few days ago, I didn't really know who I was myself. By God, the extent to which he cozened me.\"\n\nThe captain sat back, already thinking hard. \"Can you imagine any circumstance under which he would abandon the siege?\" he asked. \"If the king comes, will he simply retire?\"\n\nHarmodius looked at all of them for a long time. \"You really have no idea what you are dealing with, here,\" he said. \"Do you seriously think the king will reach us?\" he asked.\n\nThe captain made a face. \"You are the all-knowing Magus, and I'm just the young pup commanding the mercenaries, but it seems to me\u2014\"\n\n\"Spare us your false humility,\" Harmodius snapped.\n\n\"Spare us your overweening arrogance, then! It seems to me this is not a carefully wrought plan, and with due respect, Magus, this Thorn is not as staggeringly intelligent as you seem to think.\" The captain looked around.\n\nSer Milus nodded. \"I agree. He makes beginner mistakes. He knows nothing of war.\" He shrugged. \"At least, not of the war of men.\"\n\nHarmodius started to react and then pulled on his ample beard. There was a heavy silence. The men around the table realised they were prepared for the Magus to react.\n\nBut he shook his head. \"That is\u2014a very interesting point. And quite possibly a valid one.\"\n\nFather Henry came out of the Great Hall with his shoulders slumped, and Mag watched him enter the chapel and sit on a carved chair near the door, his head in his hands.\n\nHe wasn't a bad priest\u2014he had heard her confession and had passed her to God with an endurable penance. She wanted to like him for it, but there was something in his eyes she couldn't like\u2014a quality to his moist hand on her brow that unsettled her.\n\nShe was considering all these things when the archers came by. There were two of them, younger archers she didn't know well. The taller one had bright red hair and a hollow smile. They had their brigantines off and were looking around the courtyard.\n\nThey looked like trouble.\n\nThe tall one with a beard like a Judas goat spotted Lis the laundress, but she didn't truck with men his age, and she turned her back so his attention passed to Amie, the Carters' eldest\u2014a blonde girl with more chest than wit, as her mother herself had said, while her younger sister Kitty had all the wit as well as curly dark hair and slanted eyes.\n\nThe archers headed for the two girls who sat on stools by the convent kitchen, grinding barley for bread in hand mills. It was boring, exacting work that the nuns thought perfect for attractive young women.\n\nThey already had a court of admirers, and the young men\u2014farmers' sons and apprentices\u2014were, naturally enough, doing the work. This was, Mag thought, probably not a common problem among the nuns, but if they didn't wise up to it soon they were going to spoil the Carter girls and the Lanthorns and every other single woman in the fortress who wasn't a nun. And perhaps a few nuns too, Mag thought to herself.\n\nMag had started to get to know some of the senior nuns\u2014\n\nShe never heard what the archer said, but every one of the farm boys and apprentices was on his feet in a heartbeat.\n\nThe archers laughed and sat, and began using tow and ash to polish their helmets and elbow cops to the uniform dark gleam that seemed to mark the men of the company.\n\nMag walked closer. She saw trouble coming, and while the archers didn't seem to be provoking it, they were.\n\n\"Any clod can follow a plough,\" Judas Beard said. He smiled. \"I did, once.\"\n\n\"Who are you, then?\" said an apprentice.\n\n\"I'm a soldier,\" Judas Beard said. Just from his intonation, Mag, who had known some boys, knew that every word he said was aimed at the Carter girls.\n\nAmie looked up from her mill. She'd taken the pestle back from the Smith boy because Mag was there and might tell. \"Did you\u2014fight? Yesterday?\"\n\n\"I killed a dozen boglins,\" Judas Beard said. He laughed. \"It's easy, if you know how.\"\n\n\"If you know how,\" said the other archer, who until now had been silent. He wasn't doing much polishing.\n\n\"Then it ain't any different from any other trade,\" said a shoemaker's apprentice.\n\n\"Except that I'll die rich while you're still be up to your neck in your master's piss,\" Judas Beard said.\n\nKitty put her hands on her hips. \"Mind your language,\" she said.\n\nThe archers exchanged a glance. \"Anything for a pretty lady,\" the quiet one said with a smile. He got up and bowed, a courtly bow, better than any of the farm boys, Mag knew. \"I'm sure you hear too much of that already, eh, lass?\"\n\n\"Don't you lass me!\" Kitty said.\n\nAmie was smiling at the red-bearded archer.\n\nMag didn't know what she felt was wrong here\u2014the tone of it? The anger of the local boys seemed to fuel the archers.\n\n\"If'n you put some tallow on that flax, it'd hold the grit better,\" said another boy\u2014really, a young man. \"Less you're just doing it for show.\" The lad grinned. He was tall, broad in the shoulders, and no more a local than the archers.\n\nSilent gave him a mocking look. \"If I need a yokel to tell me how to polish my armour, I'll ask,\" he said.\n\nThe big lad grinned again. \"Yokel yourself, farm boy. I'm from Harndon, and I can smell the shit on your shoes from here.\"\n\nKitty giggled.\n\nIt was the wrong sound\u2014feminine derision at a critical moment\u2014and Silent turned on her. \"Shut up, slut.\"\n\nAnd suddenly everything changed, like cream turning to butter in the churn.\n\nKitty turned red, but she put a hand on the nearest farm boy. \"No need to do ought,\" she said. \"No need to defend me.\"\n\nMag was proud of the girl.\n\nBut Judas Beard stood up and dusted his lap of tufts of tow. \"That's right,\" he said. \"Be reasonable.\" He smiled. \"Learn to spread your legs like she does, when there's a man about.\"\n\nEvery farm boy was back on his feet, and both archers suddenly had knives\u2014long knives. They took up practised, professional stances. \"Anyone here got balls?\" Judas Beard said. \"Heh. You're just sheep who pay us to guard you. And if I feel like fucking one of your ewes, I will.\"\n\nThe big Harndon boy stepped out of the knot of locals. \"I'll take you both,\" he said. \"And I'll see to it you are taken to law.\" He spat on his hands, apparently in no hurry\u2014but as he spat on his left hand, his left leg shot out. He was in close with Silent, his left knee behind the archer's knee, and suddenly the knife hand was rotating and the knife wielder was face down in the dust, his knife hand behind his back.\n\n\"Christ!\" he screamed.\n\nThe Harndon boy had his knee in the archer's back. He turned to the other. \"Drop your whittle or I'll shatter his shoulder. And I'll still come and break your skull.\"\n\nJudas Beard growled, and a heavy staff hit him in the back of the head\u2014hit him so hard that he dropped like a sack of rocks.\n\nMag was all but nose-to-nose with the mercenaries' commander, who had appeared\u2014apparently out of thin air\u2014and hit the red-haired archer with his staff of office. She squeaked.\n\nHe was standing over the big Harndonner and the smaller archer, who was still locked face down in the bigger boy's grip. \"Let him go,\" said the captain quietly. \"I'll see he's punished, but I need his bow arm working.\"\n\nThe big youth looked up and nodded, and in one fluid motion he rose to his feet and let the archer drop to the cobbled pavement. \"I could have taken your other man,\" he said.\n\n\"I'm know you could,\" the captain said. \"You're a wagoner, aren't you?\"\n\n\"Daniel Favor, of Harndon. My pater is Dick Favor, and he has ten carts on the roads.\" He nodded.\n\n\"How old are you, Daniel?\" the captain asked, as he leaned down and seized Silent's ear.\n\n\"Fifteen,\" the Harndonner said.\n\nThe captain nodded. \"Can you pull a bow, lad?\"\n\nThe big youth grinned. \"And fight with a sword. But a bow\u2014aye. Any kind, any weight.\"\n\n\"Ever thought about the life of a soldier?\" the captain asked.\n\nDaniel nodded solemnly.\n\n\"Why don't you come along and see this miscreant punished,\" the captain said. \"There won't be any carting for some weeks, if I'm any judge, and a boy who can pull a bow can help save his friends. Save some fair maidens, too,\" the captain said, with a pretty bow to the two girls and then to Mag.\n\nWill Carter stepped forward. \"I can pull a bow too, Captain,\" he said. His voice trembled.\n\nThe captain smiled. \"Can you, now?\" he asked. He looked at Mag. \"A word with you, goodwife?\"\n\nShe nodded. The captain took her aside, with the silent archer stumbling after as he kept his grip on the archer's ear.\n\n\"How bad was this?\" he asked.\n\nShe met his eyes. They were very handsome eyes. He was younger than he seemed at a distance. His linens were terrible\u2014the collar of his shirt was ruined and threadbare, and his cuffs were brown-black with grime and a long linen thread dangled from his arming cote. \"Bad,\" she said. She found she was shaken, and her knees were weak. His eyes were not normal eyes.\n\n\"War does not make boys nice,\" he said, giving his man's ear a shake.\n\n\"But you're going to teach it to these young 'uns, anyway,\" she said, while thinking _what's got into you, girl?_ \"My lord,\" she added hastily.\n\nHe considered what she said. The archer tried to move, and the captain twisted his ear viciously. \"I take your point, but the alternative is being eaten alive by the Wild,\" he said. He said it ruefully, as if he understood her point all too well.\n\n\"What will happen to him?\" she said.\n\n\"Sym?\" the captain said, turning the silent archer by means of his ear so that he cried out. \"Sym will have forty lashes on his back\u2014ten a day at two-day intervals, giving him something to look forward to. Unless my marshal thinks it is worth making an example of him.\"\n\nSym cried out.\n\n\"In which case, we'll tie him to a wagon wheel and cut open his back\u2014\" the captain went on, and Sym whimpered.\n\nMag swayed.\n\nThe captain grinned at her. \"It may sound awful, but it is better than rape, and once it starts it will not stop. Sorry\u2014I am too blunt.\" He looked at her, as if seeing her for the first time. \"You are the seamstress\u2014yes?\" he asked.\n\nShe made a curtsy. \"I am, my lord.\"\n\n\"Could you be kind enough to make the time to visit me, Mistress? I need... everything.\" He smiled.\n\nShe nodded. \"So I can see,\" she said. Business straightened her back. \"Shirts? Braes? Caps?\"\n\n\"Three of each?\" he asked. He sounded wistful.\n\n\"I'll wait on you this afternoon, my lord,\" she said with a quick bend of her knee.\n\n\"Well, then,\" he said, towing his archer away by the ear. He walked back to the locals\u2014boys were competing to comfort the Carter girls. Curiously, the Harndon boy was standing uncertainly by, taking no part. Mag flashed him a smile and went about her business.\n\n# Lissen Carak\u2014Bad Tom\n\nTom Lachlan was sitting at his table in the garrison tower. It had become his office\u2014his and Bent's, because Bent was becoming his right hand.\n\nHe looked over his cards, and his ears picked up the unmistakable sound of spurred boots on the stairs.\n\nHe was on his feet, cards in a bag, and looking out an arrow slit at a party of boglins digging in the sun before the captain crested the stairs.\n\nLow Sym was all but thrown across the table. He gave a long squeal as the captain released his hold on the man's ear.\n\nTom sighed. \"What's the useless fuck done now?\" Low Sym was one of the company's leading lights\u2014in crime.\n\nThere were a dozen boys coming up the steps behind the captain.\n\nThe captain indicated them with a shift of his eyes. \"New recruits. Archers.\"\n\nTom nodded. They were likely boys\u2014he'd been eyeing them himself\u2014yeomen's sons, all big, well-fed lads with good shoulders and muscles. At their head was a boy who looked as if he might, in time, be as tall as Tom himself.\n\nTom nodded again, and as he rounded the table to greet the recruits he slammed his fist into Low Sym's head. \"Don't move,\" he said.\n\n\"I'll be in my Commandery,\" the captain said.\n\nTom bowed, and turned to the boys. \"Who here can shoot a bow?\" he asked.\n\n\"There's one other,\" the captain said. \"Red Beve is lying in the courtyard with a busted noggin. Captain's court tomorrow for both. Nice and public, Tom.\"\n\nCaptain's court was official\u2014not a casual ten lashes and no questions asked situation, but for a crime for which the captain might have a man broken, or executed.\n\nThe captain nodded at the boys. \"Tell the truth and do your best. We don't take everyone, and your parents have to agree,\" he said.\n\nTom all but choked on laughter, but the Red Knight was good at this\u2014he was a fine recruiter, while Tom had never been able to recruit anyone for anything unless he had a club in one hand and a whip in the other. _We don't take everyone._ He allowed a laugh to escape his gut.\n\n\"Let's go down to the archery butts and see what you boys are made of,\" he said in what he thought was his kindliest voice. Then he leaned down to Sym. \"Best lie still, laddy. Captain means to have your guts on a stick.\"\n\nThen he followed the boys down the steps to the courtyard.\n\nThe captain leaned on the railing of the hoardings that had been assembled outside his Commandery\u2014in effect, giving him a covered and armoured porch that jutted from the walls four hundred feet above the plain. He was watching a party of men\u2014captives? They had to be captives\u2014under the direction of something horrible. They were digging trenches.\n\nAs far as his eyes could see, men and monsters were digging trenches. It was a maze\u2014a pattern that he suspected was deliberate, and the scope of it was inhuman and both grotesque and awe-inspiring. The trenches were not in concentric rings, like those a professional soldier would have built\u2014they clung to the ground, marking the edges of every contour like a tight fitting kirtle on a curvaceous woman.\n\nSomeone had planned it, and now drove it to execution. In one day.\n\nHe wanted Amicia. He wanted to talk to her, but he was too tired and the fortress was too full to find her. But he knew another way\u2014if she was on her bridge. All it required was that he open his door a little. He reached to\u2014\n\n_Enter the room. He waved at his tutor, Prudentia, and walked to the iron-bound door._\n\n_\"Don't,\" she said._\n\n_She'd been telling him not to do things his entire life and, mostly, he ignored her._\n\n_\"You can't trust her,\" Prudentia said. \"And Thorn is right outside that door. He is waiting for you.\"_\n\n_\"He has to sleep sometime.\"_\n\n_\"Stop!\"_\n\n_He put his whole weight against the door\u2014his whole dream weight\u2014and turned the handle until the tumbler clicked_ \u2014\n\n_And the door slammed back against its hinges and a solid green fog roared into his chamber, enough power to light a city\u2014ten cities_ \u2014\n\n# North of Lissen Carak\u2014Thorn\n\nThorn grinned as he felt the dark sun\u2014felt him surface to the world of power\u2014and he sent all his power along the contact lines to bind him. No more hesitation. Men of power always tried a direct challenge. Thorn was ready.\n\n# Lissen Carak\u2014The Abbess\n\nThe Abbess felt the rising tide of Wild power and stopped\u2014she was feeding bits of chicken to her bird, and the plate of raw chicken fell to the marble floor. There couldn't be _this_ much power in her fortress\u2014she reached out and felt _him_ \u2014\n\n# North of Lissen Carak\u2014Thorn\n\nThorn felt her golden brilliance and he paused, licking at it to taste her, amazed at her potency. Delighted, saddened, angered, guilt-ridden\u2014\n\nUtterly distracted.\n\n# The Memory Palace\u2014The Red Knight\n\n_He lay on the floor, and Prudentia was trying to reach him, her marble hand inches from his own\u2014her hand and the black and white parquetry tiles were the only things he could see in the roiling, choking cloud of green, the green of trees in high summer. He was pinned to the floor\u2014he could see the shape of the cage closing over him, a phantasm so potent that he could only breathe his wonder as it crushed him\u2014it hesitated. He strained, but it was too powerful, even as it seemed to lose its focus, and he pushed against it his mind screaming \"Fool, fool, fool\u2014\"_\n\nThe door slammed shut leaving him lying crumpled in the corner of his armoured balcony.\n\nThe old Magus stood over him, his staff still glowing, and wisps of fae-fire played along its length. \"Well, well,\" the old man said. \"That would be your mother in you, I suspect.\"\n\nThe captain tried to get to his feet and found himself boneless and almost unable to move his arms. \"You have the advantage of me,\" he said softly.\n\nThe old Magus offered him a hand. \"So I do. I am Harmodius, Royal Magus, and you are Lord Gabriel Moderatus Murien\u2014Anna's son.\" He smiled grimly. \"The Viscount Murien. Don't try and deny it, you little imp. Your mother thinks you're dead, but I knew who you were the moment I saw you.\" He got the captain to his feet, and led him across the room to a chair.\n\nJacques came in with a cocked and loaded arbalest. It was smoothly done\u2014Harmodius had no chance to react.\n\n\"Say the word, my lord, and he's dead,\" Jacques said.\n\n\"You heard,\" the captain said. He felt as if he had the worst hangover of his life.\n\n\"I heard,\" Jacques said. The bolt-head on the trough of the crossbow didn't waver.\n\nThe captain took in a shaky breath. \"Why shouldn't I have you killed?\" he asked the Magus.\n\n\"Is your petty secret worth the lives of everyone in the castle?\" the Magus asked. \"None of you will live through this without me. Even with me the odds are long. In the name of the Trinity, boy, you just felt his power.\"\n\nThe captain wished he could think. The Magus's use of his name\u2014Gabriel\u2014had hit him as hard as the green cage had. He didn't even allow _himself_ to think the name Gabriel. \"I have killed, and allowed men to die, to protect my secret,\" he said.\n\n\"Time to stop doing that, then,\" said the Magus.\n\nJacques didn't move, and his voice was calm. \"Why don't you just shut up about it?\" He shrugged, but the shrug never reached the crossbow bolt's tip. \"You being the mighty King's Magus, and all. You stop talking about some dead boy's name, and we can all go on together?\"\n\n\"Three in a secret,\" the captain muttered.\n\nThe Magus pursed his lips. \"I'll give my word not to disclose what I know\u2014if you give me yours to talk to me about it. When and if this is over.\"\n\nThe captain felt as if the floor had dropped from under his feet, and all he wanted to do was jump into the hole and hide. \"Fine,\" he said. He remembered that Gawin Murien was lying in the hospital, almost exactly over his head. _Four in a secret, and one my enemy_ , he thought. _My lovely brother._\n\n\"I so swear, by my power,\" the Magus said.\n\nThe captain forced himself to raise his head. \"At ease, Jacques,\" he said. \"He's just sworn an oath that binds\u2014if he breaks it, his own power will be crippled.\" He turned back to the Magus. \"You saved my life,\" he said.\n\n\"Ah\u2014some shred of courtesy survives in you. Yes, boy, I saved you from a grisly death\u2014he wanted your power for his own.\" The horrible old man grinned. \"He was going to eat your soul.\"\n\nThe captain nodded. \"I feel as if he did. Or perhaps he didn't like the taste?\" he tried to grin and gave it up. \"A cup of water, Jacques.\"\n\nJacques backed up a step, took the bolt from the action and used the goat's foot at his belt to slowly unlever the string. \"Loons,\" he muttered, as he left the room.\n\nWhen he was gone, the Magus leaned forward. \"How powerful are you, boy? Your mother never said a word.\"\n\nThe captain's heart beat faster at the word mother, and flashed on his beautiful mother, drunk and violent and hitting him\u2014\n\n\"Don't mention my mother again.\" He sounded childish, even to himself.\n\nHarmodius hooked a stool over with his staff and sat. \"All right, boy, sod your mother. She was never any friend of mine. How powerful are you?\"\n\nThe captain sat back, trying to recover his\u2014his sense of himself. His poise. His _captainness._\n\n\"I have a good deal of raw power, and I had a good tutor until\u2014\" He paused.\n\n\"Until you ran away and faked your death,\" the Magus concluded. \"Which of course you did with a phantasm. Of course you did.\" He shook his head.\n\n\"I didn't mean to fake it,\" the captain said.\n\nThe Magus smiled. \"I was young and angry and hurt once, too, lad,\" he said. \"Despite appearances. Never mind\u2014cold comfort. I glimpsed your memory palace\u2014magnificent. The entity within it\u2014who is she?\"\n\n\"My tutor,\" the captain said.\n\nThere was a long pause. Harmodius cleared his throat. \"You\u2014?\"\n\nThe captain shrugged. \"No I didn't kill her. She was dying\u2014my mother and my brothers, they... never mind. I saved what I could.\"\n\nThe Magus narrowed his eyes. \"That's a human woman bound to a statue in a memory palace?\" he asked. \" _Inside your head._ \"\n\nThe captain sighed. \"Yes.\"\n\n\"Heresy, thaumaturgy, necromancy, gross impiety, and perhaps kidnapping too,\" Harmodius said. \"I don't know whether to arrest you or ask how you did it.\"\n\n\"She helped me. She still does,\" the captain said.\n\n\"How many of the hundred workings do you know?\" the Magus asked.\n\n\"The hundred workings, of which there are at least a hundred and forty-four, and perhaps as many as four hundred?\" the captain asked.\n\nJacques came in with a tray\u2014apple cider, water, wine.\n\n\"No one comes in,\" the captain said.\n\nJacques made a face that suggested that he was no fool\u2014but perhaps his master was\u2014and left.\n\nThe Magus fingered his beard. \"Hmmm,\" he said noncommittally.\n\n\"I can work more than a hundred and fifty of them,\" the captain said. He shrugged.\n\n\"It was a splendid memory machine,\" the Magus replied. \"Why\u2014if I may ask\u2014aren't you the shining light of Hermeticism?\"\n\nThe captain picked up his cup of water and drained it. \"It is not what I want.\"\n\nThe Magus shocked him by nodding.\n\nThe captain leaned forward. \"That's it? You nod?\"\n\nThe Magus spread his hands. \"I'm keep saying I'm no fool, lad. So your mother trained you all your life to be a magus, I'll guess. Brilliant tutor, special powers. It all but drips off you\u2014you know that?\"\n\nThe captain laughed. It was a laugh full of anger, self-pity, brutal pain. A very young, horrible laugh he'd hoped he'd left behind him. \"She\u2014\" He paused. \"Fuck it, I'm not in a revealing mood, old man.\"\n\nThe old Magus sat still. Then he took the wine flagon, poured a cup, and drank it off. \"The thing is,\" he began carefully, \"the thing is, you are like a vault full of grain, or armour, or naphtha\u2014waiting to be used in the defence of this fortress, and I'm not sure I can let you stay locked.\" He shrugged. \"I've discovered something. Something so very important that I'm afraid I'm not very interested in what men call _morality_ right now. So I'm sorry for the hurt your bitch mother caused you\u2014but your wallowing in self-pity is not going to save lives, especially mine.\"\n\nTheir eyes locked.\n\n\"A vault full of naphtha,\" the captain said, dreamily. \"I _have_ a vault full of naphtha.\"\n\n\"She taught you well, this tutor of yours,\" Harmodius said. \"Now listen, _Captain_. The mind that opposes us is not some boglin chief from the hills\u2014nor even an adversarius, nor even a draconis singularis. This is the shell of a man who was the greatest of our order, who has given himself to the Wild for power and mastery and as a result is, quite frankly, godlike. I don't know why he wants this place\u2014or rather, I can guess at some surface reasons, but I can't guess what he really wants. Do you understand me, boy?\"\n\nThe captain nodded. \"I have a thought or two in my head, thanks. I have to help you, if we're going to make it.\"\n\n\"Even in the moment of his treason, he was too smart for me,\" Harmodius said, \"although, for my sins, I've only had to face my own failure in the last week.\" He shrugged and sat back. He seemed suddenly smaller.\n\nThe captain downed the soft cider in four long gulps.\n\n\"I'd like to survive this, too,\" he said. He sighed. \"I'm not _against_ the use of power. I use it.\"\n\nHarmodius looked up. \"Can you channel?\" he asked.\n\nThe captain frowned. \"I know what you mean,\" he said. \"But I've never done it. And besides, my strength is poor. Prudentia taught that we grow in strength by the ceaseless exertion of muscle, and that the exercise of power is no different.\"\n\nThe Magus nodded. \"True. Mostly true. You have a unique access to the power of the Wild.\" He shrugged.\n\n\"Mother raised me to be the Antichrist,\" said the captain bitterly. \"What do you expect?\"\n\nHarmodius shrugged. \"You can wallow or you can grow. I doubt you can do both.\" He leaned forward. \"So listen. So far, everything he has done is foreplay. He has thousands of fresh-minted boglins; he has all the spectrum of fearsome boogiemen of the northern Wild\u2014trolls, wyverns, daemons; Outwallers; irks. He has the power to cast a cage on you\u2014on you who can tap directly into the Power of the Wild. When he comes against us in full measure he will destroy us utterly.\"\n\nThe captain shrugged and drank some wine. \"Best surrender then,\" he said with a sneer.\n\n\"Wake up, boy! This is serious!\" The old man slapped the table.\n\nThey glowered at each other.\n\n\"I need your powers to be deployed for us,\" Harmodius said. \"Can you take instruction?\"\n\nThe captain looked away. \"Yes,\" he muttered. He sat back and was suddenly serious. He raised his eyes. \"Yes, Harmodius. I will take your instruction and stop rebelling against your obvious authority for no better reason than that you remind me of my not-father.\"\n\nHarmodius shrugged. \"I don't drink enough to remind you of your odious not-father,\" he said.\n\n\"You left out the Jacks,\" the captain put in. \"When you were listing his overwhelming strength. We caught some of them in camp, in our first sortie. Now he's moved them elsewhere and I've lost them.\"\n\n\"Jacks?\" Harmodius asked. \"Rebels?\"\n\n\"Like enough,\" the captain said. \"More than rebels. Men who want change.\"\n\n\"You sound sympathetic,\" Harmodius said.\n\n\"If I'd been born in a crofter's hut, I'd be a Jack.\" The captain looked at his armour on its rack as if contemplating the social divide.\n\nHarmodius shrugged. \"How very Archaic of you.\" He chuckled.\n\n\"Things are worse for the commons than they were in my boyhood,\" the captain asserted.\n\nHarmodius stroked his beard and poured a cup of wine. \"Lad, surely you have recognised that things are worse for everyone? Things are falling apart. The Wild is winning\u2014not by great victories, but by simple entropy. We have fewer farms and fewer men. I saw it riding here. Alba is failing. And this fight\u2014this little fight for an obscure castle that holds a river crossing vital to an agricultural fair\u2014is turning into the fight of your generation. The odds are _always_ long for us. We are never wise\u2014when we are rich, we squander our riches fighting each other and building churches. When we are poor, we fight among ourselves for scraps\u2014and always, the Wild is there to take the unploughed fields.\"\n\n\"I will not fail here,\" the captain said.\n\n\"Because if you are victorious here, you will have finally turned your back on the fate that was appointed to you?\" said the Magus.\n\n\"Everyone has to strive for something,\" the captain replied.\n\n# Albinkirk\u2014Gaston\n\nThere was no battle at Albinkirk.\n\nThe royal army formed up for battle just south of the town, on the west bank of the great river, with the smaller Cohocton guarding their northern flank. Royal Huntsmen had been killing boglins for two days, and the squires and archers of the army were learning to take their guard duty seriously after something took almost a hundred war horses in the dark of the night. Six squires and a belted knight died in the dark, facing something fast and well armoured\u2014bigger than a pony, faster than a cat. They drove it off eventually.\n\nThe army had risen four hours before dawn, formed their battles lines in the dark, and moved carefully forward towards the smoking town. But after all that work, the mouse still escaped the cat.\n\nOr perhaps the lion escaped the mouse. Gaston couldn't be sure which they were.\n\nThe king had almost three thousand knights and men-at-arms, and half again as many infantry, even without the levies who had been left to guard the camp. On the one hand, the force was the largest and best armed that Gaston had ever seen\u2014the Albans had armour for every peasant, and while their mounted knights might seem a trifle antiquated, with too much boiled leather in garish colours over double maille, and not enough plate\u2014the Alban king's force was now larger than any Gallish lord's and well mounted and well-served. His cousin had ceased commenting on them. This close to the enemy, the Royal Host had become slimmer, fitter, and altogether more competent, with well-conducted sentries and pickets. Young men no longer rode abroad without armour.\n\nBut his father King Hawthor had, by all reports, had at least five times as many men when he rode forth against the Wild, perhaps even ten times as many. And the signs were all around them\u2014the lack of plate armour was not just a penchant for the old-fashioned. All along the road, he had seen abandoned farms and shops\u2014once a whole town with the roofs falling in.\n\nIt gave him pause.\n\nBut on this day, as the sun rose behind them and gilded their lance tips and pennons, the enemy melted away before them, abandoning the siege\u2014as if Albinkirk had never truly been under siege after the assault.\n\nThe army halted at the edge of the great river and the Royal Huntsmen finished off any boglins too slow to get down the great earth cliff to the beach below. Heralds counted the dead and debated whether to count the destruction of the small enemy force as a battle or not.\n\nGaston answered his cousin's summons, and saluted, his visor open and his sword loose in the sheath. It seemed possible that there would be an immediate pursuit across the river, even though it seemed odd that the enemy would retreat to the east.\n\nBut Jean de Vrailly handed his great bassinet to his squire and shook his head. \"A royal council,\" he snapped. He was angry. It seemed his mad cousin was _always_ angry these days.\n\nFollowed only by a handful of retinue knights and a herald, they rode across the field, covered in summer flowers, towards the king.\n\n\"We are letting the enemy escape,\" de Vrailly said. \"There was to be a great battle. _Today._ \" He spat. \"My soul is in peril, because I begin to doubt my angel. When will we fight? By the five wounds of Christ, I hate this place. Too hot\u2014too many trees, ugly people, bestial peasants\u2014\" He suddenly reined in his horse, dismounted, and knelt to pray.\n\nGaston, for once, joined him. In truth, he agreed with all of his cousin's pronouncements. He wanted to go home too.\n\nA herald rode up\u2014a king's messenger, Gaston saw. He went back to his prayers. Only when his joints ached and his knees could no longer bear the pain did Gaston raise his eyes to the king's messenger who had been patiently waiting for them.\n\n\"The king requests your company,\" he said.\n\nGaston sighed, and he and his cousin rode the rest of the way to the royal council.\n\nIt was held on horseback, and all the great lords were present\u2014every officer or lord with fifty knights or more. The Earl of Towbray, the Count of the Border, the Prior of Harndon, who commanded the military orders, and a dozen midlands lords whom Gaston didn't know. Edward, Bishop of Lorica, armed cap \u00e0 pied, and the king's captain of the guard, Ser Richard Fitzroy, the old king's bastard, or so men said.\n\nThe king was conferring with a small man with a grizzled beard, who rode a small palfrey and looked like a dwarf when every other man present was mounted on a charger. He was sixty years old and wore a plain harness of munition armour\u2014the kind that armourers made for their poorer customers.\n\nHe had dark circles under his eyes, but his eyes still had fire in them.\n\n\"They were over the outwalls and into the suburbs after three assaults,\" he said. \"They could run _up the walls.\"_ He looked at Ser Alcaeus. \"But you must know the story from this good knight.\"\n\n\"You tell it,\" said the king.\n\n\"The mayor wouldn't send the women to the castle. So I sent out my best men to force them in.\" He shrugged. \"And they did. And by the grace of the good Christ, I took twenty men-at-arms and held the gate to the castle.\" He shook his head. \"We held it for an hour or so.\" He looked at Ser Alcaeus. \"Didn't we?\"\n\nThe Morean knight nodded. \"We did, Ser John.\"\n\n\"How many died?\" the king asked gently.\n\n\"Townspeople? Or my people?\" the old man asked. \"The town itself died, my lord. We saved mostly women and children\u2014a few hundred of them. The men died fighting, or were taken.\" He grimaced as he said it. \"We kept two posterns open the next night\u2014a dozen pole-axes by each\u2014and we got another fifty refugees, but they burned the town to the ground, my lord.\" He bowed his head, slipped from his nag and knelt before his king. \"I beg your pardon, my lord. I held my castle, but I lost your town. Do with me as you will.\"\n\nGaston looked around. The Albans were in shock.\n\nHis cousin pushed forward. \"All the more reason to pursue the enemy now,\" he said strongly.\n\nThe old captain shook his head. \"No, my lord. It's a trap. This morning, we saw a big force\u2014Outwallers, with Sossags or Abenacki, going into the woods to the east. It's an ambush. They want you to pursue them.\"\n\nDe Vrailly coughed. \"Am I to be afeared of a few broken men?\" he asked.\n\nNo one answered him.\n\n\"Where is the main force of the enemy?\" the king asked.\n\nThe old man shrugged. \"We've had messengers from convoys headed west, and from the Abbess,\" he said. \"If I had to guess, I'd say that Lissen Carak is besieged.\" He took the king's stirrup. \"They say it's the Fallen Magus,\" he said suddenly. \"Men claim they saw him while the walls were being stormed, smashing breeches in the wall with lightning.\"\n\nAgain, the Albans muttered, and their mounts started to grow restless.\n\nThe king made a clucking sound, as if thinking aloud.\n\nThe Prior of Harndon pushed his horse forward. He wasn't a big man and he was as old as the Captain of Albinkirk, but something shone from him\u2014power of a sort, based on piety, humility. His black mantle contrasted sharply with the blaze of gold and colour on the other warriors, even the bishop.\n\n\"I would like to take my knights and outriders west, my lord, to see to Lissen Carack,\" he said. \"It is our responsibility.\"\n\nThe Count of the Borders was at Gaston's elbow. Despite the frostiness of their last meeting, he leaned over. \"The Sisters of Saint Thomas are his people\u2014at least at a remove or two,\" he whispered.\n\nThe Captal de Ruth stood in his stirrups. \"I would like to accompany them,\" he declared.\n\nThe Prior regarded him with a smile. It was a weary smile, and it probably wasn't intended to convey insult. \"This is a matter for the knights of my order,\" he said. \"We are trained for it.\"\n\nThe captal touched his sword hilt. \"No man tells me my men are not trained,\" he said.\n\nThe Prior shrugged. \"I will not take you, no matter how bad your manners.\"\n\nGaston put a hand on his cousin's steel clad forearm. In Alba, as in Galle, a man did _not_ threaten or challenge a knight of God. It wasn't done.\n\nOr perhaps his mad cousin thought himself above that law, too.\n\n# Lissen Carak\u2014The Red Knight\n\nA commander is seldom alone.\n\nFor the captain there was paperwork, often done with Ser Adrian. Drills to supervise, general inspections, particular inspections, and an endless host of small social duties\u2014the expectations of a band of people bonded by ties forged in fire. A band of people who, in many cases, are rejects from other communities because they lack even the most basic social skills.\n\nThe captain needed to be alone, and his usual expedient was to ride out over the fields of whatever countryside his little army occupied, find himself a copse of trees, and sit amongst them. But the enemy occupied the countryside, and the fortress itself was full to bursting with people\u2014people everywhere.\n\nHarmodius had left him with a set of complex instructions\u2014in effect, a new set of phantasms to learn, all in aid of defending himself against direct workings from their current enemy. And there was a plan, too\u2014a careful plan\u2014reckless in risk, but cunning in scope.\n\nHe needed time and privacy to practise. And he was never alone.\n\nMichael came, served him chicken, and was dismissed.\n\nBent came to pass a request from some of the farmers that they be allowed to visit their sheep in the pens under the Lower Town walls. The captain rubbed his eyes. \"Yes,\" he said.\n\nSauce came in with an idea for a sortie.\n\n\"No,\" he said.\n\nAnd went somewhere else to find himself some privacy to practise thaumaturgy.\n\nThe hospital seemed like the best bet.\n\nHe climbed the stairs without meeting anyone\u2014evening was falling outside, and he felt as if he'd fought a battle. He had to force his legs to push him up the winding stairs.\n\nHe passed the sister at the head of the stairs with a muttered word\u2014let her assume that he was on his way to visit the wounded.\n\nIn fact, he _did_ visit his wounded first. John Daleman, archer, lay on the bed nearest the far wall with a line of sutures from his collarbone to his waist, but by a miracle, or perhaps by the arts of the sisters, he was not infected and was now expected to live. He was also in a deeply drugged sleep, and the captain merely sat by him for a moment.\n\nSeth Pennyman, Valet, had just come from the surgery, where they had set his broken arm and broken leg. He'd been brushed from the wall by a wyvern's tail. Nothing had set properly, and the sisters had just reset the breaks. He was full of some drug, and muttered curses in his sleep.\n\nWalter La Tour, gentleman man-at-arms, sat reading slowly from a beautifully illustrated psalter. Fifty-seven years old, he wore new glass spectacles on his nose. He'd received a crushing blow from the behemoth in the fight by the brook.\n\nThe captain sat down and clasped his right hand. \"I thought I'd lost you when that thing put you down.\"\n\nWalter grinned. \"Me too,\" he said. \"Don't make me laugh, my lord. Hurts too much.\"\n\nThe captain looked more closely. \"Are those things new?\" he asked, reaching for the glass spectacles.\n\n\"Ground by the apothocary right here,\" Walter said. \"Hurt the nose like anything, but damn me, I haven't been able to read this well in years.\"\n\nThe captain put them on his own nose. They wouldn't really stay, the heavy horn frames merely pinching. There was a fine steel rivet holding the two lenses together so that they pivoted\u2014the captain knew the principle, but had never seen them in action.\n\n\"I... that is, we\u2014\" La Tour looked wistful. \"I might stay here, Captain.\"\n\nThe captain nodded. \"You'd be well suited,\" he said. \"Although I doubt me that you are too old to chase nuns.\"\n\n\"As to that,\" Walter said, and turned crimson. \"I am considering taking orders.\"\n\n_The things you don't know._ The captain smiled and clasped the man's free hand again. \"Glad to see you better,\" he said.\n\n\"I owe God,\" Walter said, by way of explanation. \"They saved me, here. I was dead. That behemoth crushed me like an insect, and these holy women brought me back. For a reason.\"\n\nThe smile was wiped from the captain's face. \"Yes,\" he said. \"I, too, owe something to God.\"\n\nHe moved on down the line of cots. Low Sym lay with his face to the wall, his back carefully bandaged. Justice tended to be instant, in the company. He moaned.\n\n\"You are an idiot,\" the captain said with professional affection.\n\nSym didn't roll over. He moaned.\n\nThe captain was merciless, because next to La Tour and the others, Sym's pain was like the sting of a fly. \"You picked the fight because you wanted the girl. The girl didn't want you, and beating up her brothers and her fellow farm-hands wasn't going to ever make her like you. Eh?\"\n\nMoan.\n\n\"Not that you care, because you are not above a spot of forced love, eh, Sym? This is not Galle. I didn't approve of your way in Galle, my lad, but this is our country and we are all holed up in the fortress together, and if you so much as breathe garlic on a farm girl, with or without her permission, I'll hang you with my own hands. In fact, Sym, let's be straight about this. You are the single most useless fuck in my whole command, and I'd prefer to hang you, because the message that I mean business would cost me nothing. You get me?\" He leaned forward.\n\nSym moaned again. He was crying.\n\nThe captain hadn't been aware that Low Sym was capable of crying. It opened up a whole new vista.\n\n\"You want to be the hero and not the villain, Sym?\" he asked very quietly.\n\nSym turned his head away.\n\n\"Listen up, then. Evil is a choice. _It is a choice._ Doing the wicked thing is the easy way out, and it is habit forming. I've done it. Any criminal can use force. Any wicked person can steal. Some people don't steal because they are afraid of being caught. Others don't steal because it is wrong. Because stealing is the destruction of another person's work. Rape is a violence against another person. Using violence to solve every quarrel\u2014\" The captain paused in his moralizing lecture, because, of course, as a company of mercenaries, they tended to use violence to solve every quarrel\u2014he laughed aloud. \"It's our work, but it doesn't have to define us.\"\n\nSym moaned.\n\nHe captain leaned close. \"Not a bad time to decide to be a hero and not a villain, Sym. Your current line will end on a gallows. Better to end in a story than a noose.\" He thought of Tom. The man was a hillman\u2014easy to forget, but his notions of word fame lingered. \"Finish in a song.\"\n\nThe small man wouldn't look at him. The captain shook his head, tired and not very happy with his job.\n\nHe got up from the nursing stool by the archer and stretched.\n\nAmicia was right behind him. Of course. There he was, the prince of hypocrites.\n\nShe looked down at Sym, and then back at the captain.\n\nHe shrugged at her.\n\nShe furrowed her brow, and shook her head, and waved him on his way.\n\nHe stumbled away, cast down.\n\nHe made an exasperated sound, and stepped out into the corridor that ran from the recovery beds to the serious patients' ward. He walked a few paces and turned the corner only to find himself standing by Gawin Murien's bed. The younger man had one leg bandaged from the crotch to the knee.\n\nHe sat by Ser Gawin's bed. \"No one will look for me here,\" he said in bitter self-mockery.\n\nGawin's eyes opened.\n\n_This is not my day,_ the captain thought.\n\nThere was a pause long enough for vast conversations. For debate, argument, rage. Instead, they stared into each other's eyes like lovers.\n\n\"Well, brother,\" Gawin said. \"So it seems you are alive, after all.\"\n\nThe captain made himself breathe in and out. \"Yes,\" he said, very quietly.\n\nGawin nodded. \"And no one knows who you are,\" he said.\n\n\"You do,\" the captain said. \"And the old wizard, Harmodius.\"\n\nGawin nodded. \"I gave him a wide birth,\" he said. \"Would you help me sit up?\"\n\nThe captain found himself obligingly raising his brother on his pillows\u2014even fluffing one of them. His brother, who had killed Prudentia at his mother's orders.\n\n\"Mother said she was corrupting you,\" Gawin said, suddenly, as if reading his mind. But even as he got those words out, his voice broke. \"She wasn't, was she? We murdered her.\"\n\nThe captain sat back down before his knees could give way. He wanted to flee. To have this conversation another day. Another year.\n\nThe truth was that the truth was too horrible to share. Shameful, horrible, and deeply wounding to everyone it could possibly touch. The captain sat and looked at Gawin, who still believed that they were brothers. That lie, at least, was intact.\n\n\"Prudentia knew something she shouldn't have,\" the captain found himself saying. He sounded remarkably calm. He was quite proud of himself, just for a moment.\n\nGawin made a choked noise. \"So Mater got us to kill her,\" he said, after another mammoth pause.\n\n\"Just as she egged you on every day to torment me,\" the captain said bitterly.\n\nGawin shrugged. \"I realised that, even before you left. Richard never saw it, but I did.\" He looked out the arrow slit by his head. \"I did something terrible, down in Lorica. I got some good men killed and I did something despicable.\"\n\nSuddenly the captain found Gawin's eyes locked on his again. \"When I was kneeling in the mud, acting the craven, I realised that I had to avenge myself or go mad. And\u2014and let me fucking say this, _brother_ \u2014I realised in one flash that I had been the instrument of your destruction, as surely as if I'd killed you myself. You think it didn't touch me? When we found your body, and how did you pull that off?\u2014when we found your body, I rode away into the Wild. I was gone\u2014off my head. I knew who killed Lord Gabriel. I did. Dickon and I did, together. We hated you into death, didn't we?\" He shook his head. \"Except now you are not dead, and I'm not sure where that leaves us. You are a magus?\" he asked.\n\nThe captain sighed. \"Mater had me trained as a magus,\" he said. \"By Prudentia. Even while telling you two how effeminate I was, and what a poor knight I made. I had sworn never to reveal my studies\u2014to her, to God, to all the saints.\" He laughed bitterly.\n\n\"Oh, my God,\" Gawin groaned. \"Prudentia was a magus. So... oh, my God. Mater provided the arrow.\"\n\n\"Of Witch Bane,\" the captain said.\n\nGawin was whiter, if anything, than when the captain had first seen him. \"I'm sorry,\" he said. \"We both knew you loved her.\"\n\nThe captain shrugged.\n\n\"Gabriel\u2014\"\n\n\"Gabriel, Viscount Murien is dead,\" the captain said. \"I am the captain. Some men call me the Red Knight.\"\n\n\"Red Knight? Like some nameless bastard?\" Gawin said. \"You're my brother, Gabriel Moderatus Murien, the heir of the Duke of the North, son of the king's sister.\"\n\n\"Oh, I'm the son of the king's sister all right,\" the captain said, and then clamped down, before any more came out.\n\nGawin choked. He sat up, and cursed. A slow thread of scarlet worked its way across his groin. \"No!\" he muttered.\n\nThe captain nodded. \"Yes. If it makes you feel any better, we're only half brothers,\" he said.\n\n\"Sweet Christ and his five wounds,\" Gawin said.\n\nThe captain came to a decision\u2014the kind of decision he made, where he threw out one set of options and adopted another, like life on the battlefield. He moved his chair closer to his half-brother. \"Tell me this terrible fucking thing you did in Lorica,\" he said. He took Gawin's hand. \"Tell me, and I'll forgive you for killing Prudentia. She already forgives you. I'll explain sometime. Tell me what happened in Lorica, and let's start again, from age nine, when we were friends.\"\n\nGawin lay back, so that their eyes broke contact. \"The price of your forgiveness is steep, brother.\" He was suddenly red as blood. Then he hung his head. \"I am deeply ashamed. I would not confess this to a priest.\"\n\n\"I'm no priest, and I have plenty of which to be ashamed. Some day I, too, will explain. Now tell me.\"\n\n\"Why?\" Gawin asked. \"Why? You'll only hate me more\u2014add contempt to the list of your grievances. I played the caitiff, I was craven and I grovelled under another man's sword.\" Tears came down his face. \"I failed and lost. I was nothing. For my sins, Satan sent this,\" and he pulled down his shirt to show the scales that had grown from his waist to his neck on the right side.\n\nThe captain looked at his brother\u2014still so proud, even after such a thing happened, and all unknowing of his own pride. _So easy to understandothers_ the captain thought with wry amusement. And surprising sorrow. He couldn't keep his emotional distance with Gawin.\n\n\"Losing is not, in and of itself, a sin.\" The captain rubbed his beard. \"It took me years to learn that, but I did. Failure is not sin. Wallowing in failure\u2014\" he hung his own head \"\u2014is something at which I can excel, if I allow it to myself, but that's more like the sin.\"\n\n\"You sound like a man of God,\" Gawin said.\n\n\"Fuck God,\" the captain said.\n\n\"Gabriel!\"\n\n\"Seriously, Gawin, what has God ever done for me?\" the captain laughed. \"If I awaken after a sword thrust with the eternal flames burning my sorry arse, I'll spit in the maker's face, because that's all I was ever offered in a rigged game, and I will have played it _anyway_.\"\n\nThat blasphemy ended all conversation for a long time. The sun was setting.\n\nGawin rolled his hips a little. \"My groin is bleeding again. Can you re-wrap it? I can't stomach the nursing sisters wrapping my groin.\"\n\n\"Crap,\" the captain said. What had been a thread of scarlet was now a rapidly spreading stain\u2014a pool of blood. \"Jesus wept! No, I'm getting expert help.\" He laughed. \"We'll both likely die of the family curse\u2014overweening pride\u2014but I don't have to actively help you die.\" He scraped his chair back. \"Amicia?\" he called. \"Amicia?\"\n\nShe came so quickly that he knew\u2014knew from her face, as well\u2014that she'd heard every word they had said.\n\nAnd she had a length of boiled linen in one hand and a pair of sharp scissors in the other. \"Hold him down and this will go faster,\" she said, all business.\n\nGawin turned his face away.\n\n\"Really,\" the captain said, when the bandage was off, \"you should enjoy having such a beauty work on your groin.\"\n\nAmicia paused. He looked into her eyes for the first time in days and felt like a fool. \"Sorry,\" he muttered weakly.\n\nBut she held his gaze. And then he saw her wink at Gawin. \"A secret for a secret,\" she said, with that not-a-smile in the corner of her mouth. She bent over the long wound on the young knight's leg, and when her lips were a finger's width from his thigh, she breathed out\u2014a long breath\u2014and as she breathed, the wound closed. The captain saw the power flow through her, a great pulse of power, as great as anything he'd ever handled.\n\nIn his sight, it was bright green.\n\nShe looked up from her work and just a flicker of her eyes, and in them was a charge and a promise and in that flicker of a heartbeat he accepted both.\n\n\"What did she do?\" Gawin asked. The captain's broad torso was blocking his ability to see. \"It's all numb.\"\n\n\"A poultice,\" the captain said cheerfully. The room suddenly smelled of summer flowers. She was wrapping fresh linen around the wound, sponging off the fresh blood and the older dried blood.\n\nGawin tried to sit up, and the captain held him down. Under his left hand, something felt very wrong with his half-brother's shoulder, and he rolled the edge of his shirt collar back.\n\nGawin's shoulder was finely scaled, like a fish, or a wyvern. The captain ran his hand over it, and behind him, Amicia's breath came in a sharp gasp.\n\nGawin groaned. \"And you think you are cursed by God?\"\n\nAmicia ran her hand over the young knight's scales, and the captain found himself instantly jealous.\n\n\"I have seen this before,\" she said.\n\nGawin brightened perceptively. \"You have?\" he asked.\n\n\"Yes,\" she said.\n\n\"Can it be cured?\" he asked.\n\nShe bit her lip. \"I really don't know, but it was not uncommon among... among...\" she stammered.\n\nThe captain thought that an astrologer would have said it was a day for secrets, and their revelation.\n\n\"I will look into it,\" she said with the assurance of the medico, and she swept from the room, the pale grey of her over-gown fluttering behind her.\n\nGawin watched her, and the captain watched her. \"She used power,\" Gawin said quietly.\n\n\"Yes,\" the captain said.\n\n\"She is\u2014\" Gawin let his head fall back. \"I was headed north,\" he began. \"The king had dismissed me from court for shooting my big mouth off. I fell in love\u2014oh, I am telling this badly. I was trying to impress the Queen's Maid-of-Honour. She... never mind. I said something I shouldn't have said to the king and he sent me off to the Wild to _gain glory._ \" Gawin shook his head. \"I have a great name as a bane of the Wild. You know why? Because after we killed you\u2014well, we thought we did\u2014I rode away to die in the Wild. Alone.\" He laughed. \"A daemon attacked me, and I killed it.\" His laugh was a little wild. \"Hand to hand. I lost my dagger in the fight, and I battered it to death, and so men call me Hard Hands.\"\n\n\"Pater must have been very proud,\" the captain muttered.\n\n\"Oh, he was,\" Gawin answered. \"So proud he sent me to court so the king could send me away. I rode north to Lorica, and put up in an inn.\" He turned his head away. \"I'm not sure I can tell this while I look at you. I took rooms. A foreign knight came with a retinue\u2014I don't know how many, but it was a hundred knights, at least. Jean de Vrailly, God curse his name. He called me out into the courtyard, challenged me to combat, and attacked me.\" Gawin fell silent.\n\n\"So? You were always a better swordsman than I,\" the captain said.\n\nGawin shook his head. \"No. No, you were the better swordsman. Ser Hywel told me after you died; you'd pretended to be inept.\"\n\nThe captain shrugged. \"Fine. You were, and are, a fine man-at-arms.\"\n\n\"Ser Jean imagines himself to be the very best knight in all of the world,\" Gawin said.\n\n\"Really?\" the captain said. \"How very dangerous.\"\n\nGawin snorted. \"You really haven't changed.\"\n\n\"I have, you know,\" the captain said.\n\n\"I never thought I'd be able to chuckle while I told this. He was in armour\u2014I was not.\"\n\nThe captain nodded. \"He would be, being a Galle. I was just fighting there. They take themselves very seriously.\"\n\n\"I only had a riding sword\u2014by Saint George, I make too many excuses. I held him\u2014took a wound, and he punched my sword into one of my squires. My own sword killed my sworn man.\" Now all the humour was gone, and Gawin was somewhere between toneless and sobbing. \"I lost all sense of the fight, and he mastered me\u2014pushed me down into the dirt. Made me admit myself bested.\"\n\n_How that must have tasted_ , the captain thought. Because he had imagined doing exactly that to this man a thousand times. He sat by the very man's bedside and tried to think what had changed in a few minutes, that now, it seemed impossible that he had imagined his half-brother's humiliation. Desired it. Tasted and savoured it, just two days ago.\n\n\"Then he went into the inn and killed my senior squire,\" Gawin said. He shrugged. \"I have vowed to kill him.\"\n\nThe captain had a restless urge to go follow Amicia. He felt the need to extract a vow of silence. Or was that just an excuse? And the pain\u2014raw, like a visible bruise\u2014in Gawin's voice\u2014he'd only just forced himself to decide in favour of the younger man, and now he was his confessor.\n\nIt was like being the captain.\n\n\"Your enemy is my enemy,\" he said simply, and leaned down, and put his arms around his brother's neck. Amongst the Muriens, a good expression of hate was a way of showing love. Sometimes, the only way.\n\n\"Oh, Gabriel!\" Gawin said, and burst into tears.\n\n\"Gabriel died, Gawin,\" the captain said.\n\nGawin dried his eyes. \"You have problems of your own, no doubt.\" He managed a smile.\n\n\"Where would you like me to begin?\" the captain said. \"I'm engaged in a siege with an enemy who can deploy any kind of creature, who outnumbers me ten or fifteen or twenty to one, and who is led by a ruthless genius.\"\n\nGawin managed another smile. \"My brother is a ruthless genius.\"\n\nThe captain grinned.\n\nGawin nodded. \"You're about to try something insane. I can taste it. Remember the chicken coup? Remember your alchemical _experiment_?\"\n\nThe captain looked around, as if he feared an eavesdropper. \"He's going to hit us hard, tonight. He has to. Up until now, to all intents and purposes, he's been losing the siege. The way the Wild works, eventually, some one of his own will see him as weak and take him down.\"\n\nGawin shrugged. \"They're the enemy. Who knows what they think?\"\n\nThe captain returned a grim smile. \"I do. All too well.\"\n\n\"So?\" Gawin asked, after a difficult moment. \"Why do you know? What they think?\"\n\nThe captain drew a long breath.\n\n_Why do you curse God every morning?_\n\n_Because_ \u2014\n\n\"Maybe sometime I'll tell you,\" the captain said.\n\nGawin absorbed that. \"The man of secrets. Very well. What are you about to do?\"\n\nThe captain shrugged. \"I'm going to try for him. Try to drag him down. The old Magus is in on it.\"\n\nGawin sat up. \"You're going for Tho\u2014\"\n\n\"Don't speak his name,\" the captain said. \"Naming calls.\"\n\nGawin bit his lip. \"I wish I were fit to ride.\"\n\n\"You will be, soon enough.\" The captain leaned forward and embraced his half-brother. \"I'd rather be your friend than your foe. Foe was merely a habit.\"\n\nGawin patted the captain's back gently. \"Gabriel! I'm sorry!\"\n\nThe captain held the young knight until he slept. It didn't take long.\n\n\"I'm not Gabriel,\" he said to his sleeping half-brother. And then he went to find the woman. But he didn't have to go far. She was sitting on a chair in the corridor.\n\nTheir eyes met. Hers said, _Don't come too close\u2014I'm vulnerable just now._\n\nHe wasn't sure what his own said, but he stopped at arm's length. \"You heard,\" he said, far more harshly than he intended.\n\n\"Everything,\" she said. \"Don't offend me by requiring my silence. I hear the confessions of dying men. I care nothing for the secrets of the mighty.\"\n\nIn his head, he knew that her anger was a kind of armour to keep him farther away. But it hurt, anyway. \"Sometimes, secrets are secret for a reason,\" he said.\n\n\"You curse God because your mother was unfaithful to your father and you grew to manhood with the torments of your brothers?\" She spat. \"I thought you were braver than that.\" She shrugged. \"Or do you mean that you intend to sortie out into the night and die?\"\n\nHe took a deep breath. Counted carefully to fifty in High Archaic, and let the breath go. \"You have been in the Wild,\" he said softly.\n\nShe looked away. \"Begone.\"\n\n\"Amicia\u2014\" He almost called her _Love_ , and he stammered. \"I have been in your palace. On your bridge. I'm not making judgment.\"\n\n\"I know, you idiot,\" she spat at him.\n\nHe was stunned by her venom. \"I will protect you!\" he said.\n\n\"I don't want your protection!\" she said, the anger all but forming frost on her lips. \"I am not a suffering princess in a tower! I am a woman of God, and my God is the only protection I require, and I do not know why my power does not come from the sun! I have enough weight of sin on my head without you adding to my burdens!\" She got to her feet and gave him a sharp push. \"I am an Outwaller chit, a slut, a woman lower than a serf. You, it turns out, are some lost prince. You can, I have no doubt, cozen any woman you like, looks, money and _power_ !\" She pushed him again. \"I AM NOT FOR YOU.\"\n\nHe was not a blushing youth of sixteen. He caught her arm as she pushed him, and pulled. He thought she would fall into his arms.\n\nShe almost did. But she caught herself, and his kiss was deflected. His arms pinned her, and she said, with all the ice a woman can muster: \"Shall I tell Sym you forced me? Captain?\"\n\nHe let her go. Just in that moment, he hated her.\n\nJust in that moment, the feeling was probably mutual.\n\nShe walked away to the main hospital room, and he had nowhere to which he could retreat except the dispensary behind him.\n\nOn the other hand, it was empty, and just then what he needed, perhaps more than ever before in his life, was to be alone.\n\nHe collapsed into the heavy wooden chair in the darkened room, and before he knew it, he was crying.\n\n# Lissen Carak\u2014Sauce\n\nSauce had the duty. She was fresh enough to her promotion that she still enjoyed the responsibility\u2014made a special effort to be clean, neat, her armour well-polished, her square-topped cap neat as a pin. She knew that a lot of the older men resented taking orders from a woman, and she knew that a perfect turn-out helped.\n\nShe set the guards on the main gate, and marched the duty detachment to the posterns, relieving each post in turn\u2014challenge, password, posting by the numbers, and accepting the salutes\u2014she loved the ceremony. And she loved to see the effect on the farmers and their families. Farmers clean and oil their tools, tour their livestock, morning and night. Farmers know a patient craftsman when they see one, even when the craft is war.\n\nShe relieved the last post and marched the off-going detachment through the courtyard to the base of the West Tower, where she dismissed them. Two slow-moving archers were detailed to wash the heavy wood piling driven into the ground for sword practice\u2014Low Sym had been tied to it for his punishment, and it had various substances on it that needed cleaning off.\n\nThen she climbed the steps to the tower, listening to the off-going soldiers. She was listening for criticism; she expected it. She wasn't really good enough to be a corporal. She wanted to be\u2014but there was so much to learn.\n\nAnd she knew that this was going to be a tough night. All across the garrison tower men were polishing, sharpening, trimming a belt end, checking the stuffing on a gambeson sleeve. A thousand rituals to conjure safety and luck in battle. And they were all tired.\n\nAt the head of the stairs stood Bad Tom, her nemesis, with his cronies. She straightened her back, noticed that even though he was supposedly off-duty he was still fully armed, wearing full harness but for the gauntlets and the bassinet, which sat together on the plank table. She noted that his armour was as carefully polished as her own.\n\nHe was talking to Bent, and they were smiling.\n\nShe met their looks and glared. \"What?\"\n\n\"Your people look good enough for the Royal Guard,\" Tom said with a rich chuckle.\n\n\"What the fuck does that mean?\" she spat. She looked past him, over the walled balcony that let light and air into the tower from the courtyard. She could see the priest from here, climbing out on the wall. She wondered what he was doing there.\n\nBent slapped his thigh and roared. \"Told you!\" he shouted, and went back to his game, and she forgot Father Henry. \"Can't even take a fucking compliment.\"\n\nShe glared at both of them and went to the roof to watch her posts. \"Where are all the men-at-arms? Captain left a note\u2014\"\n\nTom nodded to her. \"I've got it, Corporal. I'm preparing the sortie.\"\n\nSauce felt a keen disappointment edged with anger. \"A sortie? But\u2014\"\n\n\"You have the duty,\" Tom said. \"It's my turn.\"\n\n\"It's always your turn,\" she shot back.\n\nHe nodded, unrepentant. \"I'm _primus pilus_ , Sauce. I can take the sortie out until Christ returns to earth\u2014maybe after. Wait your turn. Sweeting.\"\n\nShe drew herself up. But Bad Tom shook his head. \"Nay\u2014never mind me, Sauce. That was ill-said. But I want the sorties. The lads need to see me fight.\"\n\n\"And you love it,\" Sauce said. She put her nose very close to his. \"I love it too, you bastard.\"\n\nTom laughed. \"Point taken, _Corporal.\"_\n\nShe backed off. \"I want my turn. Anyway\u2014where is everyone?\"\n\n\"The boys are all off confessing to the priest. Don't worry, Sauce. We probably won't go. But there's going to be a sortie ready all night, every night, in the covered way.\"\n\nSauce shook her head and went up the steps to the roof-top feeling left out.\n\nFull darkness had almost fallen, and the sounds made by the various species of besiegers would have been chilling if she'd let herself think of them that way, but she didn't. Instead she stood with the crew on the great ballista\u2014as of today, re-mounted on a complex set of gimbals designed by the old Magus. She tried it herself. Now it moved like a living thing. No Head, the man responsible for the machine, patted it affectionately. \"The old fuck magicked it, that's what he did. It's alive. Going to get us a wyvern, next time one comes.\"\n\nShe swung it back and forth. It was physically pleasant to move\u2014like playing some sort of game.\n\n\"Sometimes a machine is just a machine,\" said a strong voice, and the old man himself emerged from the darkness. She had never been so close to a real magus, and she started.\n\n\"It's our good luck that we have fifty skilled craftsmen suddenly among us. A pargeter, who can draw precisely. Blade smiths who can make springs. A joiner who can do fine carpentry.\" He shrugged. \"In truth, it is an Archaic mechanism I found in a book. It was the craftsmen who made it.\" Nonetheless, the old man seemed very satisfied with it, and he gave it an affectionate pat. \"Although I confess I gave it a touch of spirit.\"\n\n\"Which he magicked it, and now it's alive!\" said No Head happily. \"Going to bag us a wyvern.\"\n\nHarmodius shrugged as if mocking the ignorance of men\u2014while accepting their plaudits.\n\nHis eyes lingered on her.\n\n_Christ\u2014did the old Magus find her attractive?_ That was a chilling thought. She wriggled involuntarily.\n\nHe caught her movement and laughed. Then stopped laughing. \"Something is moving down between the forts,\" he said.\n\nShe leaned over the tower. \"Wait a little,\" she said. Then, \"How did you know?\"\n\nHis eyes glowed a little in the dark. \"I know,\" he said. \"I can make the sky bright for a moment.\"\n\n\"No need,\" she said.\n\nSure enough, there was a low clash, as if of cymbals, and then another.\n\n\"Captain put lines of tin bangles across the fields,\" she said as the ballista spun, No Head pulled its lever and a bolt crashed out into the darkness.\n\nOn the next tower, the onager released a bucket of gravel, and suddenly the night was full of screams.\n\nA retaliatory bolt of purple-green lightning shot out of the darkness and struck the tower on which the onager rested. Sparks flew as if a smith was pounding red-hot metal.\n\n\"Christ, what _the_ fuck was that?\" Sauce asked the darkness. Her night-sight was ruined by the green bolt; all she could see was a pattern on her retinas.\n\nOld Harmodius leaned over the tower, and a bolt of fire sprayed from his hand\u2014it passed almost exactly down the line of the green lightning, as far as the dancing images on her retinas could discern.\n\n\"Damn, damn, damn,\" he said. Over and over.\n\nHis target caught fire in the distance\u2014a giant of a man, or an oddly misshapen tree. Perhaps two trees.\n\n\"Dear God,\" Harmodius muttered. \"Again!\" he called.\n\nNo Head needed no urging. Sauce watched his crew as they danced through their drill\u2014two men wound the winch, slipped the cocking mechanism into place, removed the winch again, a third carried the twenty-pound bolt as easily as if it was made of straw, dropped it into the charge-trough and pushed it back until the huge nock engaged the heavy string. No Head spun his machine with one hand, gave the burning tree-man a hint of windage, and pulled the release.\n\nAnother line of lightning, this one levin-bright\u2014flashed onto the north tower and rock exploded. Men screamed. Her men.\n\nShe turned and ran for the stairs. And then paused. She couldn't be in both towers at the same time.\n\nBehind her the two valets winding the bow sweated to do it as fast as they could, but No Head didn't look at them or at Simkin, a giant, who dropped the next bolt into the trough with perfect timing, so that just as the string clicked into place on the latch, the nock slid back and engaged the string, and No Head had the weapon aimed.\n\nHarmodius grunted something, and cast fire on the earth. His fire was caught as if by a basket of green light, and cast straight back at them; quicker than thought, his own basket of blue lightning caught it and he threw it back\u2014\n\nNo Head pulled the release.\n\nThe bolt hit the man-tree squarely in the torso-trunk. There was a roar and a burst of ball lightning like a summer night, and the tower trembled. The ball struck the curtain wall over the main gate and there was a cataclysmic explosion\u2014like pouring water on a hot rock, expanded a thousand times. The curtain wall groaned, buckled and collapsed outward, and the new covered way behind the gate started to take hits.\n\nSomeone was alert and still moving on the onager tower, though, because a basket of red-hot gravel\u2014another of the Magus's innovations\u2014flew from the onager, the pebbles flashing through the air like meteorites.\n\nAll the lights went out together, and then there was quiet, punctuated by screams from the plain far below. And moans.\n\n\"Again!\" Harmodius called. \"Same target. Hit him again! Before he can\u2014\"\n\nAnd then there was a wall of green light across the sky, and the onager tower exploded in sparks and a shower of stars. One long scream rang out across the night\u2014and then the top of the tower leaned out, and out, and fell into the night, taking the onager and four men of the company with it. It crashed to the floor of the valley four hundred feet below, a long rumble like an avalanche.\n\nAnd then there was only silence.\n\nSauce had made it to the courtyard when the green fire hit, and she was standing close enough to the gate to be hit by stone chips from the curtain wall. A stone slammed into her shoulder from the broken tower. Up on the main donjon, she could see Harmodius as he leaned out over the wall, with eldritch blue fire coursing over his hands.\n\nThe gate had taken a glancing hit and whole chunks of the crenellations had fallen on the covered way, crushing part of the roof. Inside, men and horses of Bad Tom's sortie were trapped in the pitch black, and there were horse screams of anguish and human shouts.\n\n\"Get torches! Lanterns! On me!\" Sauce shouted.\n\nJust under the back end of the covered way, Ser John Poultney was lying under the ruin of his charger, and his leg was broken. Sauce went with a pair of archers\u2014One Lug and Skinch\u2014to get the horse off him. The archers used spears to raise the carcass and Ser John worked not to scream.\n\nThe roof of the covered way had taken most of the gate's collapse, and it hung askew, and the beams were creaking ominously. It was pitch black under the roof, and men with lanterns appeared at last as the first man-at-arms emerged leading a bucking war horse whose off left foot almost killed the just-rescued Ser John. The horse was wild, and more archers grabbed for his reins to hold his head, and then off-duty valets were pouring out of the main tower.\n\n\"Where's Tom?\" she asked. She plunged deeper into the gloom, and Skinch, usually not a man with any balls whatsoever, followed her. The lantern lit a dozen horsemen fighting their mounts for control in the enclosed space. All of them were dismounted, hauling at their horse's heads, and the horses would calm for a moment and then go off again as another horse continued to panic in the darkness and the noise. Ser John's dead horse was not helping\u2014it smelled of blood and fear...\n\n\"Get them out!\" Tom roared.\n\nHooves were flying. The men were in full armour, but the horses were not calming, and soon enough they'd kill their riders, armour or no.\n\nWith a _whoosh_ the gate behind Tom exploded in flame. It illuminated the narrow space and the plunging horses, the men's armour, like a foretaste of hell.\n\nAlmost as one, the horses turned and ran from the fire. Most of the men-at-arms were knocked from their feet.\n\nSkinch flattened himself against the wooden wall and Sauce, still in her harness, tried to cover him as the great brutes pounded past, leaping the corpse of the dead horse.\n\nOut in the courtyard the valets were ready, and they lunged for reins, threw sacks over the horses' heads and spoke to them calmly and authoritatively, like lords speaking to their serfs. They took control of the horses quickly, kindly, and ruthlessly.\n\nThe men-at-arms began to get to their feet.\n\nSauce realised that the fire at the gate wasn't generating any heat about the same moment that the captain stepped out of the darkness and raised his hands.\n\nThe flames went out like a candle in the wind.\n\n\"Tom? Let's get a head count. Anyone missing?\" he shouted, walking past her. It was dark again, but he seemed to know she was there\u2014he turned unerringly to her. \"We lost a dozen men in the Onager tower. Go and see if anyone can be saved.\"\n\nHis eyes glowed in the dark.\n\n\"M'lord,\" she nodded in the pitch black and went back into the relative light of the courtyard, past a dozen angry war horses and the men trying to calm them. Farmers and their wives and daughters were crowding the door yards and windows.\n\nThe onager tower looked like a broken tooth. About a third of the upper floor was gone, and Sauce thought the only blessing was that it had fallen out\u2014away from the courtyard\u2014and not in.\n\nThe second floor roof had collapsed inward though, showering stones and roof beams on sleeping soldiers. Geslin\u2014the youngest archer in the company\u2014lay dead, crushed under a beam, his broken body horrible in the flickering fire of the fallen floor. Dook\u2014a useless sod at the best of time\u2014was trying to get the beam off him, and was crying.\n\nSauce put on her best command voice, walled off her panic, and shouted, \"I need some help up here!\"\n\nArchers poured up the ladders to her. Men she knew\u2014Flarch, her own archer, and Cuddy, perhaps the best archer in the company, and Rust, perhaps the worst; Long Paw, moving like a dancer, and Duggin, who was as big as a house. They got the beam up off the dead boy, and discovered Kanny pinned under it, unconscious and with a lot of blood under him. And behind him, wedged into a safe space made by a window ledge, was Kessin, the fattest man in the company.\n\nMore and more men came\u2014the Lanthorn men, the Carters from the courtyard, and the other farmers\u2014at unbelievable speed they cleared the heavy timbers and the floor. One of Master Random's men, who had been working with the Magus, rigged a sling mechanism, and before the sun began to rise, the heavy stones that could be saved from the wreckage were being raised over the lip of the ruined tower and laid in the courtyard.\n\nThe captain stood there looking tired, hands on hips above his golden belt, watching the work. He didn't turn his head. \"Well done, Sauce. Go to bed.\"\n\nShe shrugged. \"Lots left to do,\" she said wearily.\n\nHe turned to her with a smile. Very quietly, like a lover, he leaned in to her ear. \"This is the first bad night of a hundred to come,\" he whispered. \"Save your strength. Go to bed.\"\n\nShe sighed and looked at him, struggling to hide her adoration. \"I can do it,\" she said fiercely.\n\n\"I know you can do it,\" he said, rolling his eyes. \"Save it for when we need it. I'm going to bed. You go to bed. Yes?\"\n\nShe shrugged, avoiding his eyes. Walked away...\n\n... and realised that her bed had been in the onager tower. She sighed.\n\n# Lissen Carak\u2014Michael\n\n_The Siege of Lissen Carak. Day Eight_\n\n_Last night the Fallen Magus attacked us in person. The captain said his powers are greater even than those that weave the walls together, and despite our efforts he toppled the south-west tower, where the onager engine was, and killed four men and several boys._\n\n_No Head, an archer, hit the Fallen Magus with a ballista bolt. Many men saw the bolt go home._\n\n_We now have the help of Lord Harmodius, the King's Magus, who duelled with the Fallen Magus with fire. Men hid their heads in terror. The Fallen Magus brought down the curtain wall by the postern gate, but Sauce saved many men and horses with her quick response._\n\nUnder the manuscript page, _No Head_ and _Sauce_ were crossed out. In their place were the names _Thomas Harding_ and _Alison Grave._\n\n# Lissen Carak\u2014The Red Knight\n\nIn the end, they lost six archers and one man-at-arms. It was a hard blow. The captain looked at their names, crossed them off the list, and grunted.\n\nOn the other hand, he had the Carter boys, the Lanthorn boys, and Daniel Favor. And a likely goldsmith's apprentice named Adrian who was a painter and a lanky youngster called Allan.\n\nHe handed the list to Tom. \"Fix the watchbill. Messire Thomas Durrem\u2014\"\n\n\"Dead as a nail,\" Tom said. He shrugged. \"Gone with the tower. Didn't even find his body.\"\n\nThe captain winced. \"We're down another lance, then.\"\n\nTom nodded, and chewed on a lead. \"I'll find you a man-at-arms,\" he said.\n\n# The Bridge Castle\u2014Ser Milus\n\nSer Milus stood with the seven new men-at-arms. They were, in his professional opinion, good men who needed a swift kick in the arse.\n\nHe had a pell in the courtyard; Master Random's apprentices had levered a huge stone out of the flagging, dug a hole as deep as a man's was tall, and put in a post\u2014it was handy to have so many willing hands.\n\nHe walked around the pell, hefting his own favoured weapon. The pole-axe. The hammer head was crenellated like a castle with four miniature spikes projecting from it. On the other side, a long, slightly curved spike protruded, and from the top, a small, wickedly sharp spearhead. A foot of solid steel extended from the butt, pointed like a chisel.\n\nSer Milus spun it between his hands. \"I don't expect we'll fight mounted, from here on out,\" he said conversationally.\n\nGwillam, the sergeant, nodded.\n\n\"Let's see you, then,\" Ser Milus said. He nodded to Gwillam, who stepped forward. By the Company's standard, his armour was poor. He had an old cote of plates, mail chausses, and a shirt of good mail with heavy leather gauntlets covered in iron plates. It was, to Ser Milus's eyes, very old-fashioned.\n\nGwillam had a heavy spear. He stepped up to the pell, chose his distance, and thrust. The spearhead went an inch into the oak. He shrugged, and tugged it clear with a heavy pull.\n\nDirk Throatlash, the next of the convoy's men-at-arms, strode up and took a negligent swipe at it with his heavy double-bladed axe. He embedded his axe head deeply in the post.\n\nArchers were gathering in the towers, and merchants had emerged to watch from their wagons.\n\nJohn Lee, former shipman, also had a double-bitted axe. He swung hard and precisely\u2014matching Dirk's cut and carving a heavy chip out of the post.\n\nSer Milus watched them all.\n\n\"That's what you do at the pell?\" he asked Gwillam.\n\nThe sergeant shrugged. \"I haven't done much at a pell since I was a boy,\" he admitted.\n\nSer Milus nodded. \"Want to kill a monster?\" he said to the men. \"Or a man?\" he asked.\n\n\"Not really,\" Dirk said. His mates laughed.\n\nSer Milus didn't even turn his head. There was no warning. One moment, he was leaning on his war-hammer, and the next, he had tossed Dirk Throatlash into the mud, face first, and still had one arm behind his back.\n\n\"Wrong,\" he said.\n\n\"Jesus _Christ_!\" Dirk wailed.\n\nSer Milus let him up. He smiled, because now he had their attention.\n\n\"We're all going to practise at the pell, every day we don't fight on the wall,\" he said, conversationally. \"Like it was real. I'll teach you how. And if you can cut through it\u2014good!\" He grinned. \"And then you can demonstrate your zeal by helping put in the next pell.\" He pointed to John Lee. \"You have an accurate cut.\"\n\nLee shrugged. \"I cut a lot of wood.\"\n\n\"Try again. But this time, cut as if you were fighting a man.\" Ser Milus waved at the pell.\n\nThe shipman stepped up and lifted his axe, like a man preparing to hit a ball.\n\nSer Milus nodded approvingly. \"Good guard.\"\n\nThe former shipman cut at the pell, and a chip of wood flew. He got the axe back to his shoulder and cut again.\n\nSer Milus let him go on for ten cuts. He was breathing hard, and his tenth cut wasn't nearly as strong as the ninth.\n\nMilus twirled his grey moustache with his left hand. \"Leave off. Breathe.\" He nodded. \"Watch.\"\n\nHe stepped up to the pell, his pole-axe held under hand.\n\nHe cut up with the back-spike, and it just touched the post. He danced to the right on his toes, despite his armour, and his cut finished with the pole-axe head behind his shoulder\u2014a very similar position to that of the shipman's axe. Then he cut down, again stepping lightly, and the hammer-head slammed into the post, leaving four deep gouges. The knight stepped like a cat, back and then forward, powering the spearhead in an underhanded thrust\u2014stepped wide, as if avoiding a blow, and reversed the pole-axe. The spike slammed sideways into the post, bounced, and Ser Milus was close into the pole and shortened his grip for another strike.\n\nLee nodded. \"I could almost see the man you was fighting,\" he admitted.\n\nGwillam prided himself as a good man of arms, and he sprang forward. \"Let me try,\" he said. His own weapon was a heavy spear with a head as long as his arm and as wide as the palm of his hand. He sprang forward on the balls of his feet, cut the pell\u2014twice from one side, once from the other, and backed away.\n\n\"But use your hips,\" Ser Milus said. \"More power in your hips than in your arms. Save your arms; they get tired the fastest.\" He nodded to them. \"It's just work, friends. The smith practises his art every day\u2014the pargeter daubs, the farmer ploughs, the shipman works his ship. Bad soldiers lie on their backs. Good soldiers do this. All day, every day.\"\n\nThroatlash shook his head. \"My arms are tired already,\" he said.\n\nSer Milus nodded. \"The irks ain't tired.\"\n\n# Southford by Albinkirk\u2014Prior Ser Mark Wishart\n\nThe king sent two messengers with the knights when the Prior took his men north-west from Albinkirk's souther suburb, Southford. The Prior moved his men carefully over the ground, their black surcotes somehow blending into the undergrowth. His men rode easily through the densest stands of woods, through thickets of spring briars.\n\nThey halted frequently. Men would dismount and creep forward, usually over the brow of a steep hill, and wave them forward.\n\nDespite the halts, they made good progress. Individual knights would ride away\u2014sometimes at right angles to the line of march\u2014and unerringly find them again.\n\nThe thing the two king's messengers found hardest to understand was the silence. The Knights of St. Thomas never spoke. They rode in silence, and their horses were equally silent. They had no pages, no valets, no servants and no squires. Forty spare horses\u2014a fortune in war horses\u2014followed the main body, packed with forage bags and spares, but otherwise without bridle or lead. Yet the spares followed briskly enough.\n\nIt was, as the older messenger said, uncanny.\n\nStill, it was a bold thing, to be riding through the North Country with the Knights of St. Thomas. Galahad Acon had been named for the saint's church in London, and felt he was almost one of them. His partner, Diccon Alweather, had been a professional messenger in the old king's day, a weathered man with more scars than a badly tanned hide, as he liked to say himself.\n\nThe messengers were used to a hard day riding and no company but their horses, but it was a hard day, even for them\u2014fifteen leagues over broken country that challenged their horsemanship every hour. The knights didn't seem to tire. Many of them were older than Alweather.\n\nTowards evening, one of the youngest of the knights rode back to the main body, and led them off to the right, north, and then up to a steep hill.\n\nWithout a word, every knight dismounted. They drew their long swords from their saddle scabbards, split into four groups of fifteen, and walked off.\n\nThe Prior waited a moment, looking at the two messengers. \"Wait here,\" he said, aloud. The first words Galahad had heard from any of them since they left the Royal Camp.\n\nThe black-clad knights vanished into the woods.\n\nAn hour passed. It was cold\u2014the spring evenings were longer, but not much warmer, and Galahad couldn't decide whether he was cold enough to take his great cloak out of the bundle behind his crupper or not. He didn't want to be caught dismounted at the wrong moment. He cursed the Prior and his silence.\n\nHe kept looking at the older messenger, Alweather, who waited, apparently calm, without fidgeting, for the whole hour.\n\n\"Here they come,\" Galahad said suddenly.\n\nThe Prior walked up to his horse and sheathed his sword on the saddle. \"Come,\" he said. He was smiling.\n\nHe walked off up the steep hill, and all the horses followed him.\n\n\"Uncanny,\" Alweather said. He spat, and made an avert sign.\n\nThey wound around the hill, widdershins, climbing as they went around. It seemed a tedious way of getting to the top, but in the very last light, Galahad could see that the crown of the hill was steep and girt in rock.\n\nThe horse ahead of him shied, and then was quiet. Galahad looked down and saw a corpse. And then another. And another and another.\n\nThey were not human. He wasn't sure what they were\u2014small and brown, with big heads, and cords of muscle, beautifully worked leather clothes and huge wounds made by two-handed swords.\n\n\"Good Christ,\" Alweather said aloud.\n\nThere was the smell of fire, and then they came over a crest.\n\nThe top of the hill was hollow. It was like a giant cup, and the knights had three fires going, and food cooking. Galahad Acon's stomach, outraged by the inhuman corpses and their red-green blood, now seized on the smell of food. Pea soup.\n\n\"Unsaddle your horse, and curry him,\" the Prior said. \"After that, he'll see to himself.\"\n\nAlweather frowned, but Galahad refused to be moved by the older man's caution. Galahad was suffused with joy. He was living one of his secret dreams.\n\nAlweather, clearly wanted to go back to the king.\n\n\"They fought a battle,\" Galahad said, his eyes sparkling in the firelight. \"And we didn't _even hear them._ \"\n\nThe Prior smiled at Galahad. \"Not really a battle,\" he said. \"More of a massacre. The irks didn't see us coming.\" He shrugged. \"Have some soup. Tomorrow will be harder.\"\n\n# Lissen Carak\n\nIt was a quiet night. The besieged collapsed into sleep. Sauce cried out in her dreams, and Tom lay and snored like a hog. Michael muttered into his outstretched arm, sleeping alone. The Abbess wept softly in the dark, and rose to kneel, praying at the triptych that sat on a low podium in the corner of her cell. Sister Miram lay on her stomach to sleep, exhausted from healing so many wounded men. Low Sym woke himself up repeatedly as he shouted, and then lay with his own arms wrapped around him staring at horrors in the dark until the pretty novice came and sat with him.\n\nBut however long and dark the night was, the enemy was quiet, and the besieged slept.\n\nIn the first light of morning, they struck.\n\n_The Siege of Lissen Carak. Day Nine_\n\n_Today, the enemy burned all the country around the fortress, as far as the woods. The men\u2014the traitor Jacks\u2014burned all the farms, all the steadings and barns\u2014even the patches of woods._\n\n_The farmers stood on the walls and watched. Some wept. We were cursed for being poor soldiers, for allowing the fields to be burned._\n\n_The Abbess came out and watched, and then promised that it would all be rebuilt._\n\n_But many hearts turned. And before noon, the creatures of the enemy were in the air over the fortress, and we could feel them again._\n\n# Lissen Carak\u2014Mag the Seamstress\n\nIt was a simple, unstoppable act that changed the nature of the siege, and that cut at the farmers and the simple people of the fortress more effectively than all the military victories that could be scored.\n\nThe first fires were visible to the north-east. Hawkshead, the furthest east of the fortress's communities was put to the torch before morning creased the sky, and the last watch saw the town burn, just two leagues from the walls.\n\nJust as the sun began to cast forth a ruddy light, Kentmere went up to the west. By then, the walls of the fortress were lined in farm folk. Then Abbington.\n\nMag watched her town burn. From this high, she could count roofs and she knew when her own cottage burned. She watched it with a desperate anger until she could no longer see which house was hers. They were all afire\u2014every cottage, every house, every stone barn, every chicken coop. The fields around the fortress ridge were suddenly full of the enemy\u2014all the creatures who hadn't shown themselves in the first days. There were boglins, and irks; daemons and trolls, great things like giants with smooth heads and tusks which the soldiers told her were behemoths. And, of course, men.\n\nHow she hated the men.\n\nThe enemy was now girdling every tree. Orchards of apple trees and pears, of peaches and persimmons, were being destroyed. Vines that had grown for generations were gone in an hour, their roots destroyed or seared by fire, and every structure was burning. As far as the eye could see, in every direction, there was a sea of fire and Lissen Carak a dark island in it.\n\nMag couldn't take her eyes away from the death of her world.\n\n\"Sausage without mustard, eh?\" said a heavy voice at her elbow.\n\nShe started, turned to find the giant black-headed hillman, the company's savage, sitting on the other barrel beside her, watching over the wall.\n\n\"War without fire is like sausage without mustard,\" he said.\n\nShe found herself angry at him. \"That's\u2014my village. My _house_!\"\n\nThe big man nodded. He seemed not to know she was crying. \"Stands to reason. I'd hae' done the same, in his place.\"\n\nShe turned on him. \"War! In his place? This isn't a game! We _live_ here! This is _our land._ We farm here. We bury our dead here. My husband lies out there\u2014my daughter\u2014\" The tears became too much for her, but in that moment, she hated him more than she hated the boglins and their horrible faces and their willingness to burn her life away.\n\nTom looked hard at her. \"Not yours unless you can hold it,\" he said. \"Way I hear it, your people took it from them. Eh? Melike, their dead are buried there too. And right now, I'd say it was theirs. I'm sorry, goodwife, but war is my business. And war involves a lot of fire. He's showing us that we only hold what we stand on\u2014that he can win without taking the fortress. We hurt him last night and now he strikes back. That's war. If you don't want to have your farm burned, you had better be strong\u2014stronger than you were.\"\n\nShe struck him, then\u2014a glancing blow, pure anger without force.\n\nHe let her do it.\n\n\"Not many folk can say they've struck Bad Tom and lived to tell the tale,\" he said. He flashed a crooked smile in the early morning light, and she turned and fled.\n\n# Lissen Carak\u2014Thorn\n\nThorn watched the farms burn with no great satisfaction. It was a cheap victory, but it would help break the will of the farmers to resist him.\n\nHe shrugged inwardly. Or it would harden their resolve to fight to the end. Now they had nothing to save but themselves, and even when he'd been a man, he'd had trouble understanding men. And, increasingly, he felt this contest was too complex for even his intellect. He had made himself the Captain of the Wild, and yet his own interests were scarcely engaged, here. He was far more interested in the puzzle that was the dark sun, and in _her_ , then he was in the prosecution of the siege.\n\nHe wondered, not for the first time, what he was doing here, and how he'd ended up so committed to this action that he was willing to risk himself in combat. Last night he'd taken his invincible new form out onto the field, and the fortress had hurt him. None of the blows he had taken were deadly, but he felt the pain of his exertions and their blows. The pain had angered him, and in anger he had unleashed some of his carefully hoarded power\u2014enough to damage the fabric of the fortress. It had impressed his allies, but the cost\u2014\n\nAgain, he rustled his leaves in what would have been a shrug, in a man.\n\nLast night, he had felt the breath of mortality for the first time in twenty years. He didn't like the smell of it. Or the pain.\n\nBut as the siege continued it was becoming a rallying point for the Wild in the North Country, and despite minor set-backs, more and more creatures were coming in. His prestige was increasing, and that prestige would directly affect a rise in his power.\n\nNone of which would matter if he were dead.\n\nHe thought of _her_.\n\nHe could no longer shake his head\u2014it was now a continuous armoured growth from his neck, and he had to pivot around the waist to look to the left and right. But he made an odd clucking sound as he considered her. She had attempted to hurt him directly, last night.\n\nAnd finally, he considered the third presence in the fortress besides the dark sun. Power\u2014cold, blue power\u2014had struck him. Pure power, untrammelled by doubt or youth. Trained and honed, like fine steel.\n\nIt was his apprentice, of course. Had Thorn been able to smile, he would have.\n\nHarmodius.\n\nThere was a solvable problem.\n\n# Lissen Carak\u2014Amicia\n\nAmicia stood on the wall watching the world burn. She didn't notice him until he was at her shoulder.\n\n\"It was a matter of time,\" he said, as if they had been in conversation all morning.\n\nShe wasn't sure, in truth, if she wanted to say anything. She didn't want to look at him\u2014didn't want him to see how committed she was, or how angry.\n\n\"He has to show his allies that he is making progress.\" The captain leaned on the crenellation and pointed to the western edge of the woods. \"His men are building a pair of trebuchets. Before the end of the day, we'll be feeling their power. Not because it will actually help him win, but because it will make his allies see him as\u2014\"\n\nIf she kept listening to him she would...\n\nShe turned on her heel and walked away.\n\nHe hurried to catch up to her.\n\n\"People are watching,\" she hissed. \"I am a novice in this convent. I am _not_ your lover. Let me go, please.\"\n\n\"Why?\" he asked. He seized her arm in a steel grip. He was hurting her.\n\n\"Let me go,\" she said. \"Or you are no knight.\"\n\n\"Then I am no knight. Why? Why change your mind so suddenly?\" He leaned towards her. \"I have not changed mine.\"\n\nShe hadn't meant to have a conversation. She bit her lip, and looked around for a miracle. Sister Miram. The Abbess. \"Don't you have to do something? Save somebody? Give orders?\" she asked. \"Why not go and save the farms?\"\n\n\"That's unfair!\" he said, and let go of her arm. \"No one is watching us. I would know.\" He shrugged. \"I cannot save the farms. And I'd rather be here, with you.\"\n\n\"You want me to have that on my soul, as well? That in addition to breaking my vows, I am endangering the fortress?\"\n\nHe smiled his wicked smile. \"It's worked on other girls,\" he said.\n\n\"I imagine it works all the time.\" She put her chin as high as she could manage. \"I do not choose to be your whore, Captain. I don't even know your name. Girls like me don't get to know the names of the great lords who try to put their knees between our legs, do we? But I am choosing to say no. You are not afraid of Jesus, and you are not afraid of the Abbess. So I cannot appeal to you along those lines. But by God, messire, I can protect myself. If you lay a hand on me again, I will hit you hard.\"\n\nHe looked at her.\n\nHe had tears in his eyes, and she hesitated. But she'd made her decision, and she carried it through. She walked away, and didn't look back.\n\nIt was difficult for her to decide _why_ she was so angry. It was difficult for her to say\u2014even to herself\u2014 _why_ she was choosing to walk away. But he was not for her, despite the feeling that her very soul was screaming as she walked down the steps.\n\nDespite the look, like agony, on his face.\n\n# Lissen Carak\u2014Harmodius\n\n_Harmodius_\n\nHe couldn't shut it out. Once two entities of power are linked, the link is forever. He couldn't shut Thorn out, but he could wall him off.\n\n_Harmodius_\n\nThat is, he could mostly wall him off.\n\nHarmodius was sitting cross-legged under an ancient apple tree that stood alone on the battlements, in a stone circle. It was a beautiful thing, in full flower, and it was redolent with power. The seat under it was placed to absorb the power that flowed, as if from a well or a spring, around the place. Somewhere just under his feet, was the well spring. It appeared neither green nor golden. It merely _was._\n\nHarmodius drank as deeply as he dared.\n\n_Harmodius_\n\nWould it really hurt to talk to his former master?\n\nIt was dangerous. If he opened the link, Thorn might try to overwhelm him with raw power.\n\nBut sitting here, on the bench by the apple tree, he didn't think Thorn could take him before he could close the link. He wasn't like the boy. The boy\u2014\n\nTo hell with it.\n\n_Hello Richard._\n\n_I knew you would respond._\n\n_It must be satisfying to be right all the time._\n\n_Don't be snide, Harmodius. You hurt me, last night. You have grown very powerful._\n\n_I killed your mortal body at Chevin, old man._\n\n_Yes. But I knew how to deal with that. And I out-subtled myself, of course._ There was a suggestion of smugness. _How was my world of mirrors, boy?_\n\nHarmodius thought for a moment. _Very subtle, you bastard. How did you bind the spirits to the cats?_\n\n_So nice to discourse with someone intelligent. You learned to leave your body then? Ahh! I see you have not. Interesting._\n\nHarmodius didn't think he could damage his cause by honesty. No more than by having any contact with Thorn. _Why are you fighting here?_ he asked. _Must it be war?_\n\n_Harmodius! How unlike you! You wish to negotiate with the power of evil? I thought that you had chosen a different path._\n\n_I have come to realise that there is nothing intrinsically evil about the Wild. Or good about the Sun._\n\n_Ahh._ Thorn gave a suggestion of great pleasure. _You have learned much, then._\n\n_I am still struggling with the concept_ Harmodius admitted.\n\n_The Wild is far more powerful. Men are doomed. They have no role to play in the future. Too fragmented. Too weak._\n\n_That's not how I see it_ Harmodius shot back. _From where I sit, it is the Wild who is losing._\n\n_You delude yourself._\n\n_Not as effectively as you deluded me._\n\n_Let me make it up to you with knowledge. Look. This is how you can possess anybody you choose. And here\u2014this is how to build your own body. See? I give you this knowledge freely. Come. Be a god. You are worthy. And I'm bored_ \u2014\n\nHarmodius laughed aloud. _Bored of monsters and pining for decent company? You betrayed your king and all of humanity, you piece of shit._ As swiftly as he could, with all the borrowed power of the well, he slammed the link closed.\n\nHe sat back against the bole of the tree and examined the conversation.\n\n\"I think that went well,\" he said aloud.\n\nBut Thorn had planted something in him, a seed in damp soil. It was like finding a beautifully wrapped package on your doorstep.\n\nHe put the packages in a room in his memory palace, and he carefully walled that room off from his consciousness. He twinned off a second self to remain in the room.\n\nThe second self opened the first package. A third self stood ready with an axe.\n\nThe phantasm was heartbreakingly beautiful. Thorn had been a great magus, of course.\n\nHarmodius allowed his second self to subsume himself in the complexities of the working.\n\nHe shut down the room, withdrew his second self, and sat in another created room in his memory palace, a comfortable room with a circle of armchairs. His second self sat in another, wrote the phantasms out in longhand, and they discussed them in detail. His third self stood behind the second with an axe.\n\nSuddenly he understood how the cats had been used.\n\nHe understood how his former master was using animals to watch the fortress.\n\nHe understood how he could possess the body of any creature he wanted, unless they had the power to resist him. How he could subsume their essence\u2014in effect, eat that part of a mortal that Harmodius thought of as the soul.\n\n_For power._\n\nAnd take the mortal body for his own, or make one.\n\nHarmodius let the knowledge roll around inside his head for a little.\n\nAnd found himself watching a mongrel dog\u2014one of the mercenaries had brought the animal into Lissen Carak\u2014rooting in the midden heap that was beginning to fill the courtyard. Eventually, the dog would be eaten, if the siege went on.\n\n_I could just try it on the dog._\n\n_The dog is going to die, anyway._\n\nThe dog turned and looked at Harmodius. She tilted her head to one side, watching to see if the man had anything interesting to offer.\n\nPower poured out around him. _No wonder the creatures of the Wild want this place back_ , Harmodius thought. He reached out to the power, took a taste, and ran it through the phantasm\u2014\n\nAnd made a motion of negation with his hands, cancelling the working and draining the power into the walls of the fortress.\n\nHe got to his feet and grinned at the dog. \"You've got to draw the line somewhere,\" he said aloud.\n\n_He did that on purpose, the subtle bastard. He's inviting me to fall._\n\nHarmodius could smell breakfast, and he decided he needed to be with people.\n\n# East of Albinkirk\u2013Ranald\n\nRanald was tired, and he wept a great deal. He wasted an afternoon trying to catch a horse. At every step, he expected to find the drag, the rear guard, or another survivor. But he saw no one.\n\nHe wasted more time at the edge of the battlefield, trying to find his pack.\n\nEventually he gave up and walked, wet when it rained, scorched when the sun shone. He had nothing to cook with, nothing to eat, and no means of acquiring food.\n\nOn the evening of the fourth day after the fight, he walked up the lane to the great Inn. Men shouted when they saw him.\n\nEvery man and woman in the dale came running, when they knew whom he was. And because he was his cousin's tanist, they thought, at first, that his appearance must bode well.\n\nBut when they came closer, they saw the tracks of his tears, and the sword. And they knew.\n\nBy the time he walked the last few paces to the porch of the great Inn, the Keeper alone barred his way, and he was grim-faced. \"Greetings, Ranald Lachlan,\" he said. \"Tell me how many were lost?\"\n\nRanald had no trouble meeting the Keeper's eye. Death made you less careful of such things.\n\n\"They're all dead,\" he said. \"Every man of us. I, too, was dead.\"\n\nThey gasped, the folk of the Dale, and then the tears began, and the wail of loss, the roar of rage.\n\nRanald Lachlan told his story quickly, and without embellishment. And then he turned to the weeping woman who stood by her father. \"Here's his sword,\" Ranald said. \"If you bear him a son, he says the boy is to avenge him.\"\n\n\"That's a heavy load to lay on an unborn bairn's shoulders,\" the Keeper said.\n\nRanald shrugged. \"It's not my choice,\" he said wearily.\n\nLater, he sat in the Keeper's own rooms, and told the story of the last fight. Hector's wife listened through her tears. And when he was done, she looked at him long, and mean.\n\n\"Why'd they send you back, then?\" she spat. \"When they might have sent my love?\"\n\nRanald shrugged.\n\nThe Keeper shook his head. \"Too many men lost, along with the whole herd.\" He put his chin in his hand. \"I'll be hard pressed if they turn on the Dale.\"\n\nRanald didn't even pretend to be interested. And the Keeper let him go.\n\nHe was not interested when the men in the Inn offered him ale.\n\nHe wasn't interested when the woman of the Inn offered themselves, nor when a travelling player offered to make a song of the battle.\n\nHe slept, and the next day he was just as numb as he had been the day before, and the day before that. But he went down from his room to the common room at dawn, and there he faced the Keeper and asked for a horse and gear.\n\n\"You can't mean to go fight the Outwallers all by yourself,\" the Keeper said, gruffly.\n\n\"No,\" Ranald said.\n\n\"You mean to just ride home?\" the Keeper asked, incredulous.\n\n\"I'm a drover,\" Ranald said. \"I have no home.\"\n\nThe Keeper drank some small beer and wiped his moustache. \"Where, then?\" he asked.\n\nRanald sat back. \"I'm going to find the Wyrm of Erch,\" he said. \"I mean to ask why he allowed us to be attacked by the Wild.\" The drover shrugged. \"We pay a tithe to the Wyrm in exchange for protection from the Wild. It's the Law of Erch. Eh? Ancient as the oaks and all.\"\n\nThe Keeper put his beer down slowly. \"You mean to _speak_ to the Wyrm?\"\n\n\"Someone has to,\" Ranald said. \"I might as well; I'm already dead.\"\n\nThe Keeper shook his head. \"I have just a dozen horses left. Your cousin took my herd.\"\n\nRanald nodded. \"I mean to remedy that first, before I go to the Wyrm. Give me twenty men and I'll bring in the herd. There's a lot of it left. A thousand head at least.\"\n\n\"You are like your cousin,\" the Keeper said. \"Always a sting in the tail.\"\n\nRanald shrugged. \"I wouldn't bother, but Sarah's boy will need those beasts, if he means to be a drover.\" He didn't say the other thing that was on his mind. That he was a King's Man, and he owed the king a warning of the Wild.\n\nThat afternoon, with twenty wary men, he rode south.\n\nThey rode quickly, spread in pairs over a mile of ground, scouting every hummock and every stand of trees.\n\nThey made a cold camp and Ranald ate the oatcakes that Sarah had given him, and when the sun was a red disc on the edge of the world they rode on.\n\nBy noon they found the first beasts. The Dalemen were spooked, terrified of the Sossag, and afraid, too, to find corpses grinning at death in the woods, but they were still, by Ranald's reckoning, miles north of the battleground. The herd had turned and headed home, as animals will do.\n\nRanald swept south along the road, and before darkness he found the boy that Hector had sent back as a messenger. He was dead, and he'd either been lost or he'd ridden a long way west to get around something. He lay on his face, a cloud of flies around his bloated corpse, and his horse was still standing nearby. The boy had four arrows in him, and it was clear he'd died trying to fulfil his mission. The Dalemen buried him with love and honour and his cousins, two tall, grey-eyed boys, wept for him.\n\nBut the next day held the greatest shock.\n\nThey were well west of the fight, collecting animals hard against the great Swamp, and Ranald scented a fire and went to scout it himself. It was a foolish risk to take, but he couldn't bear to be the cause of any more Dalemen's deaths.\n\nWhat he found was the drag\u2014twenty of Hector's men, alive, with a third of the herd. Donald Redmane had led them west, and they had fought three times against scattered Outwaller bands, but they had lived and kept a great deal of the herd together.\n\nRanald had to tell the story all over again, and Donald Redmane wept. But the rest of the men in the drag swore a great oath to avenge Hector Lachlan.\n\nDonald took Ranald aside. \"You fought in the south,\" he said. \"You think Tom is still alive?\"\n\n\"Hector's brother Tom?\" Ranald said. \"Aye. Unless the red hand of war has taken him, he'll be alive. On the Continent or in the East, I reckon. Why?\"\n\nDonald Redmane's eyes were red. \"Because he's the Drover, now,\" the older man said.\n\n\"He won't want it,\" Ranald said.\n\n\"He will if it means he can make war,\" Donald pointed out.\n\nThe next morning, scouts killed a strange creature\u2014shaped like a man, short like a tall child, with heavily muscled arms and legs like thick ropes and a misshapen head like a man's but heavier. Ranald had to assume the beast was an irk, a creature somewhere between myth and reality to the men of the hills. Legend said irks, like boglins, came from the deep woods far to the west.\n\nRanald made camp with the whole band\u2014forty-four men. They had more than twelve hundred head of cattle, and all of the goats. Seventy-five head of horses. Sarah Lachlan would not be a pauper, and the clan was not dead.\n\nHector Lachlan was gone.\n\nBut Lachlan was for Aa.\n\n# The Albin River, South of Albinkirk\u2014The Queen\n\nThe Queen watched the banks go by and she smiled at a young guildsman with a crossbow who crouched behind the boat's high sides, watching the banks. In truth, he wasn't really watching them. He was of an age where he was conscious of nothing but Desiderata a few feet away. His eyes flicked to her over and over.\n\nShe watched the banks and smiled inwardly. The rowers chanted, on and on, and the mosquitoes descended on them in swarms unless a breeze rose upriver.\n\nLady Almspend lay next to her in the bow, a wax tablet open across her lap and a stylus ready to hand. \"Another letter?\" she asked somewhat languorously.\n\nThe Queen shook her head. \"It's too hot.\"\n\n\"Pity the poor rowers,\" Lady Almspend answered. She turned her head. Most of them rowed naked to the waist\u2014a few more naked even then that. Their work left them, to a man, with magnificent physiques, and Lady Almspend considered them carefully. \"They are like the Archaics,\" she said. \"I withdraw my former statement. I don't think they are to be pitied, but rather admired.\" She smiled at one in particular, and he smiled back even as he brought his sixteen-foot oar through the end of its sweep and brought it back to the top of its arc.\n\nThe Queen smiled. \"Do have a care, my dear,\" she said.\n\n\"I will only admire from afar,\" Lady Almspend said. \"Do you think the sentries really saw a boglin in the night?\"\n\nThe Queen nodded. \"Indeed, I'm quite sure of it.\" She was not going to enlighten her secretary any further, but the banks were already dangerous, and the boats were now using islands in the river for camps.\n\n\"Could we not arm the rowers?\" Lady Almspend asked.\n\n\"They have weapons; javelins and swords,\" the Queen answered. \"But against a sudden onslaught in the dark, we're safest behind a wall of water.\"\n\nLady Almspend shook her head. \"I cannot imagine what has happened, for the North to be so utterly over run. The king must have his work cut out for him. When will we be at Albinkirk?\"\n\n\"Tomorrow mid-day, at this pace,\" said Almspend. \"If the Queen could wear even less, the rowers might row even harder.\"\n\nDesiderata grinned at her friend. \"I aim to row through the night,\" she said. \"The river remains broad, and we are late.\"\n\nLady Mary looked at her oddly. \"Have you had a message?\" she asked.\n\nThe Queen shook her head. \"I have a feeling,\" she admitted, \"nothing more. If the king has made any pace at all, he'll already be gone west, towards Lissen Carak.\" The Queen lay back, feeling the summer sun on her shoulders. The bugs never troubled her. \"Send a message to the king, Becca. Tell him how close we are,\" She batted her eyelashes at the rowers closest to her. \"Tell him we can be with him in three days.\"\n\nRoyer Le Hardi volunteered, and they put him ashore with his horse and a spare. He received a kiss from the Queen, and he was still red as a beet when he rode west.\n\n# Albinkirk\u2014Gaston\n\nGaston watched the Royal Army break camp and turn west with something very like trepidation. None of the military orders knights had returned, despite Lissen Carak being only two days' march west of Albinkirk. Each night, light rippled in the western sky.\n\nWhatever they were fighting was utterly alien. The boglins had startled him at Albinkirk\u2014even a few of them, they were so ugly and so very _wrong._ He wanted to call them _unnatural_ , except that they were spawned by the Wild.\n\nHis cousin was ecstatic\u2014the flashes of light in the west guaranteed that the castle there still held, and that meant that battle was at last imminent. For Jean de Vrailly, that battle had become the guiding force\u2014the lodestone on which his life turned.\n\nGaston inspected his company and reminded them, for the tenth day in a row, of the lessons they'd learned from the Count of the Borders. To always have scouts\u2014front, flanks, and rear. To ride with the knights inside a strong box of spearmen and bowmen, so that, in case of an ambush, the knights could react instantly, from safety. To put the wagons at the very centre of that box.\n\nAll good sense. But it required a reliance on the low-born men by the knights.\n\nHis scouts rode off into the pre-dawn and he mounted his charger. His squire handed him his weapons and then he sat quietly watching the column form, and waiting for the sound\u2014the shouts, the trumpets\u2014that would signal a fight.\n\nOnce again, he felt homesick. He wanted no part of this strange warfare against fabulous beasts and monsters. At home, he fought men. He understood men.\n\nWhen his company was formed with his cousin's he rode west along the column to the king, who sat mounted amidst a circle of his lords. He had a scroll in his hand, as he did most mornings\u2014the Kings of Alba had a fine express service, and its riders continued to reach him despite the increasingly dangerous roads.\n\n\"She's ignored me,\" the king said happily. He looked up, and greeted the captal with a nod. \"My wife has ignored my advice and is on her way here,\" he said.\n\nThe captal, as usual, mistook his meaning. \"Then I suppose your Majesty must punish her,\" he said.\n\nThe king chose not to take exception, and instead, smiled. \"I think we would be most ungrateful,\" he said, \"to be rude to a lady who brings us a great supply of food.\"\n\nThe Count of the Borders smiled. \"When do you expect her?\"\n\nThe king looked out over the woods that stretched like a sea of green to the west. \"She's three days' march to the south of Albinkirk,\" he said. He shrugged. \"But she's commandeered a flotilla of boats\u2014she's moving much faster than we are.\"\n\n\"But she has to follow the snake of the river,\" said the Count of the Borders.\n\nSir Ricar Fitzroy fingered his beard. \"You Grace, she's got a head on her shoulders. She'll still be faster, and she'll carry a great deal more food and fodder than a wagon convoy.\"\n\nThe constable sat back on his charger and put a fist in the small of his back. \"Am I the only man who thinks he's too old for all this?\" he said. \"Your Grace, I propose that we fall back along the line of the river until we link with the Queen. We only have five days' rations\u2014we're short on meat already, and the woods are scoured of animals. The Royal Huntsmen\u2014begging your Grace's pardon\u2014aren't bringing in enough game to feed the Royal Household.\"\n\nThe Count of the Borders agreed. \"No need to rush to a fight,\" he said. \"Not with the Wild.\"\n\nThe Earl of Towbray shook his head. \"The fortress could fall,\" he said.\n\n\"Lissen Carak will stand or fall,\" the constable said. He looked around, and lowered his voice. \"My lords, we carry the weight of the kingdom on our shoulders. If we lose this army there is no new army to replace it.\"\n\n\"Albinkirk is all but in cinders,\" the king answered. \"I will not lose the Fortress of the North, as well.\"\n\n\"We need food,\" the constable argued. \"We planned to resupply from the magazine at Albinkirk. Or to find the drove coming south from the Hills and buy their beef.\"\n\n\"Can we last five days?\" the king said. \"And how long can the fortress last?\"\n\nJean de Vrailly rose in his stirrups. \"Bah,\" he said. \"The men can last without food. Let us find the enemy,\" he said.\n\nThe Albins looked at him wearily.\n\n\"Let us finally face these creatures!\" the captal insisted.\n\nThe Lord of Bain didn't comment. He merely raised an eyebrow.\n\nThe king's friend, Ser Driant, scowled. \"I'm not the hardiest warrior, and I'm well known to these gentlemen as a lover of my dram.\" He leaned forward towards the captal. \"But we are not going to risk the king's host on a battle where we have unfed horses.\"\n\nJean de Vrailly sneered. \"Of course, you must be _cautious_ ,\" he said.\n\nThe constable narrowed his eyes. \"Yes, my lord. That is exactly what we must be. We must be cautious. We must fight on ground of our choosing, with a well-ordered host in tight array, with secure flanks and a defensible camp to which we can retreat if it all goes awry. We must take every possible advantage over our foes. This is not a game, nor a tournament, my lord. This is war.\"\n\n\"You lecture me?\" Jean de Vrailly allowed his charger to take two heavy-footed steps toward the constable.\n\nThe constable raised an eyebrow. \"I do, my lord. You seem to need it.\"\n\nThe king nodded. \"The captal's willingness to go forward is noted, but I sense my constable would rather dig in here and wait for the Queen. Is that your thought?\"\n\nThe constable nodded. \"It is. I expect to hear from the Prior in the next day. It would be foolish to move forward without word from our most trusted knights.\"\n\nJean de Vrailly's anger was palpable.\n\nGaston put a hand on his arm and his head snapped around like a falcon's.\n\nGaston met his wild gaze.\n\n\"And let us at least travel south of the river. Our best information places the enemy on the north bank.\" The constable was openly begging the king to take these measures, and Gaston felt for him.\n\nThe captal made a grunt of contempt for such precautions. \"If the enemy is on the north bank,\" he said with patronising and deliberate offence, \"surely it is our duty as knights to be on the north bank to contend with them?\"\n\nBut there were quite a few nods of agreement in favour of the south bank, so the king smiled gracefully at the Galle and turned to his knights. \"We cross back to the south bank,\" he said. \"It is my will. We will encamp and dig a fortification on the south bank of the Cohocton, and throw out a heavy screen of prickers and pedites.\"\n\n\"So cautious,\" de Vrailly spat.\n\n\"It is my will,\" the king said. He didn't lose his smile.\n\nGaston had a bad feeling in the pit of his stomach.\n\n# Lissen Carak\u2014Michael\n\nMichael sat and wrote by strong afternoon light.\n\n_The Siege of Lissen Carak. Day Ten_\n\n_Yesterday the enemy destroyed all the villages west of Albinkirk by fire and sword. We were forced to watch. Today, the enemy fills his siege lines withmonsters and overhead his foul creatures fill the air with their cries. When more than two of them are over the fortress, it is as if they darken the sky. And it has disheartened many of the people to see how many our enemies really are. They are literally uncountable. All our efforts to kill them now seem like the efforts of a man with a shovel to move a mountain._\n\n_The captain was tireless today, moving from point to point around the fortress. Our people began to build an artillery platform in the ruins of the Onager Tower. He and Lord Harmodius helped the workmen lay stones in new cement and then worked the cement so that it dried faster\u2014a great miracle, and one that did much to encourage the people._\n\n_Now it is the middle of the afternoon. The enemy had set engines of war to work, but their stones could not even reach the fortress, and we watched them sail uselessly through the air and land well short of our walls\u2014indeed, one killed a creature of the Wild out in the fields. The captain says that the spirit of resistance can be fuelled by things as small as this._\n\n_But an hour ago, using his thousands of slaves, the enemy rebuilt his engines closer to us._\n\n# Lissen Carak\u2014The Red Knight\n\n\"He's going to have a go at the Lower Town,\" Jehannes said.\n\nThe captain was staring out, watching the distant engines as they were cranked back. The enemy had two trebuchets built about four hundred paces from the Lower Town's walls, on a timber and earth mound almost forty feet tall. The speed with which they had built the siege mound had been, for the captain, the most horrifying moment of the siege.\n\nPerhaps not quite the most horrifying. _I am not your lover._\n\nIt was ironic that Harmodius was training him to divide himself, to rule himself, to wall off dangerous elements of spells and counter-spells. He had issued his new apprentice an absolute injunction.\n\n_\"Never use this power on your emotions, boy. Our humanity is all we have._ \" The old man had told him that this morning, as if it was a matter of great moment.\n\nThe captain had used his new talent to wall off his emotions almost the moment Harmodius left. The Mage wasn't attempting to prosecute a siege while feeling as if his leg had been ripped off by daemons.\n\n_Why?_\n\nClearly his control needed work.\n\nHe settled back into the crenellations as a rock struck one of the Lower Town gate-towers squarely. The tower shrugged off the hit.\n\nThe captain breathed.\n\n\"We have men down there,\" Jehannes said. \"We can't hold it.\"\n\n\"We have to,\" the captain said. \"If we lose the Lower Town, he's cut us off from the Bridge Castle. Then he shifts his batteries south. It's like chess, Jehannes. He is playing for the ground just there,\" the captain pointed at a set of sheepfolds to the south and to the west. \"If he can build a siege mound there, and put his engines there, he can destroy the Bridge Castle one tower at a time.\"\n\nJehannes shook his head. He was a veteran of twenty sieges, and he clearly hated it when the captain talked down to him. \"He can build there any time he likes,\" Jehannes snarled.\n\nThe captain sighed. \"No, Jehannes. He cannot. Because he fears our sorties. Despite his immense power and force, we've stung him. If he places engines there without killing the Lower Town, we can sortie out and burn his engines.\"\n\n\"He can build more. In a day.\" Jehannes was dismissive.\n\nThe captain considered this.\n\nJehannes bored in. \"He has limitless muscle power and wood. Probably metal, as well. He can build a hundred engines, in ten different places.\"\n\nThe captain nodded. \"Yes he can, but not if his creatures desert him,\" he said. \"He doesn't want us to win any more victories.\"\n\n\"Why should he care?\" Jehannes said bitterly.\n\nThe captain was watching a party of novices going into the hospital, to take their turn at duty.\n\n\"Why, Jehannes!\" the captain said. His eyes flashed, and his bitterness was evident. \"I thought that you believed that God was on our side.\"\n\nShe hadn't so much as glanced at him as she passed.\n\nJehannes made a fist. \"Your blasphemy is an offence,\" he said quietly.\n\nThe captain whirled at his marshal. \"Make of it what you will,\" he said.\n\nThey were standing, their eyes locked, when a third trebuchet went into action, and they heard the sound of the northern gate tower in the Lower Town collapsing.\n\n\"You need to pull those men out of the Lower Town,\" Jehannes said.\n\n\"No. I will reinforce them. And I'll lead them myself. Who has the Lower Town today? Atcourt?\"\n\n\"Atcourt is still injured. It's Ser George Brewes.\" Jehannes looked out over the walls. \"We're losing too many men,\" he said.\n\n\"We're stronger than when we started the siege.\" The captain was bottling his anger and storing it out of reach.\n\n\"It's time you looked around,\" Jehannes said. \"We have bitten off far more than we can chew. We cannot win this.\"\n\nThe captain turned back to his senior marshal. \"Yes, we can.\"\n\nJehannes shook his head. \"This isn't a time for boyish enthusiasm\u2014\"\n\nThe captain nodded. \"You overstep yourself, Ser Jehannes. Go to your duty.\"\n\nJehannes continued \"\u2014or chivalric daring-do. There are two realistic options\u2014\"\n\n\"And when you are captain, you can act that way,\" the captain went on. \"But let me be as blunt as you seem to be, messire. You can't see the simplest tactical consequence. You play favourites among the archers and the knights. You lack the birth to command men who prize such things. Most of all you don't have power, and I do. So I'm bored with explaining everything to you, messire. Obey. That is all I ask of you. If you cannot, then I will dismiss you.\"\n\nJehannes crossed his arms. \"In the middle of a desperate siege.\"\n\nThe captain's mouth formed a hard line. \"Yes.\"\n\nThey stared at each other.\n\nBy nightfall, the enemy had six engines throwing rocks into the Lower Town.\n\nThe captain collected the relief watch and headed down the slope towards it. There were two routes\u2014the road, which wound in multiple cutbacks down the face of the ridge, and the path, which went straight down the spine of the ridge and had two sets of stairs. Several portions of the path were walled and covered to protect parties going to the Lower Town but, of course, you couldn't take a horse down the path.\n\nThe watch took the path anyway, their feet wrapped in rags to be as noiseless as possible. Given the enemy's dominance of the plain below them, the captain put out scouts to either side of their route\u2014Daud the Red and Amy's Hob were moving carefully down the bare rock.\n\nIt took them an hour to make their way down the ridge. All the while, great rocks fell from the sky on the Lower Town, destroying houses and cracking the cobbles. Sparks flew as each mass of flint struck the town. The heavy _thump-snap_ of the trebuchets sounded every few heartbeats, so clear in the smoky air that the engines seemed near at hand.\n\nThe air was acrid and heavy. Burning barns and roofs on a damp day had saturated the air with smoke.\n\nAn archer coughed.\n\nThey crept on. No stars showed and the darkness had become a palpable thing, an immortal enemy. The choking smoke was far worse down on the plain, and the rocks were raising dust and stone grit with every strike to add to the difficulties.\n\nFar out on the plain, one of the engines loosed its burden. As it rose in a graceful arc, it could be seen dully\u2014it was burning. Its misty appearance showed just how dense the smoke was.\n\nThe burning mass seemed to come right at them.\n\n\"Come on,\" the captain said, ignoring it. \"Follow me.\"\n\nThe fire crashed to earth out in the fields.\n\nAnother engine loosed.\n\nEven the vague light of the burning missiles was enough to help the relief watch move down the path.\n\nThe captain launched into a stumbling run. His sabatons rang on the stone steps as he came to the postern gate.\n\nLink, Blade, Snot, and Hetty caught him up.\n\n\"Relief watch!\" he called softly.\n\nThere was no answer.\n\n\"Fuck,\" the captain said softly. \"RELIEF WATCH!\" he called.\n\n\"Dead,\" said Kanny, softly. \"We should go\u2014\"\n\n\"Shut up,\" Blade said. \"Cap'n, you want me to climb the wall?\"\n\nThe captain was reaching into the postern with his power.\n\nIt was unmanned.\n\n\"Help him up the wall. Kanny, make a bucket. Then onto my shoulders. Stand on my helmet if you have to.\" The captain stood next to Kanny, who grumbled but made a stirrup with his gauntleted hands.\n\nBlade stepped up into Kanny's hands, and then onto the captain's shoulders. The captain felt a shift of weight, and then the man jumped.\n\nAbove him, the archer grunted, swinging from his arms. But on the third swing, he pulled powerfully and got one leg over the lowest part of the wall. And then he was in.\n\n\"Garn, that was too easy,\" Kanny said.\n\nSnot blew his nose quietly. \"You are a useless fuck,\" he said. \"We used to take towns in Galle this way.\"\n\nBlade opened the postern. \"No one here,\" he said.\n\nA rock crashed into the wall, far too close, and all of them had to clamber back to their feet.\n\n\"In,\" the captain said. He rolled in through the low postern gate, and drew his sword. Daud the Red appeared at the wall with Amy's Hob and No Head. \"Get in here. Daud\u2014you and Hob take the postern in case we have to come back through.\"\n\nThe two huntsmen nodded.\n\nMoving across the Lower Town was a new nightmare. Rocks hit the wall\u2014once, an overcast hit a house less than a street away. The streets were already full of rubble, and all of them closed their visors against the rock chips and wood splinters. They fell frequently and cursed too loudly when the did.\n\nThe sky was lightening when the relief watch made it to the northern gate tower. It had taken several direct hits, but the massive fortification was fifteen feet thick at the base and had so far survived.\n\nThe captain hammered at the lower door with the pommel of his sword.\n\nIt took time for a terrified pair of eyes to appear at the grille.\n\n\"Watch!\" the captain hissed. \"We've come to relieve you.\"\n\nThey heard the bar lifted.\n\nA big stone hit, somewhere to their right, and they all cringed. Stone chips rang off the captain's helmet.\n\nBlade began to pant.\n\nThe captain looked back at him\u2014then reached to catch him as he slumped to the ground, a four-inch wood splinter in his neck. Before the captain could lower him to the ground, he was dead.\n\n\"Get the door open,\" the captain roared.\n\nThe door opened outward a handspan and stopped. It was jammed by rubble.\n\nTwo more rocks struck nearby, and then a ball of fire struck fifty paces away, illuminating the smoky air.\n\nNo Head got enough of the rubble off the doorsill to get it open and they piled into the tower, dragging Blade.\n\nScrant, just inside the doorway, flinched at the look in the captain's eye.\n\nThe captain pushed the archer out of the way and stalked along the low corridor. Outside, another rock struck, and the tower gave a low vibration\u2014torches moved in their brackets, and plaster came off the walls.\n\nSer George Brewes was sitting in a chair in the donjon. He had a cup of wine in his hand. He looked blearily at the captain.\n\n\"Are you drunk? Why wasn't the postern manned?\" The captain turned to No Head. \"Round up the off-going watch. Ser George will be staying.\"\n\nKanny lingered in the doorway of the donjon, clearly interested in listening, and No Head grabbed him by the shoulder. \"Move your arse,\" he said.\n\nKanny could be heard grumbling all the way up the stairs.\n\nSer George waited until the archers were gone. \"This can't be held,\" he said. The effect of his statement was largely ruined by a belch. \"It's not tenable,\" he said, as if his careful pronouncement would settle everything.\n\n\"So you thought you'd leave the oncoming watch hanging out to dry?\" the captain said.\n\n\"Fuck you and your righteousness,\" Ser George said. \"I've had a bellyful. It's time someone told you what a posturing arse you are. I pulled my men into the tower to keep them alive. You got here anyway. I was sure someone would. I haven't lost a fucking man, and if I'm drunk, that's no one's business but mine.\" He snorted. \"You were outside. It's hell out there.\"\n\nThe captain leaned over. \"If we abandon the Lower Town, he'll take the Bridge Castle in a day.\"\n\nSer George shook his head. \"You just don't get it, do you? You're playing at being a knight errant\u2014is that because you're doing a nun?\" he guffawed.\n\nThe captain could smell the liquor on the man's breath. The sweet cloying smell of wine and hate. Just for a moment, he thought of his mother.\n\n\"We're mercenaries, not heroes. It's time to find whoever is behind this siege and cut him a deal. Take your girlfriend with us, if that's what it takes. We're done here. And there's no money in the world that would make it worth dying here.\" Ser George hawked and spat. \"Now get out of my way, Captain. I've done my twelve hours in hell and I'm going back up to the fortress.\"\n\nThe captain stood up straight. \"No. You're going to stay right here, with me.\"\n\n\"Like hell I am,\" Ser George said.\n\n\"If you try to leave this room. I'll kill you,\" the captain said.\n\nSer George made a plunge for the door.\n\nHe wasn't in his full harness and he had a good deal of wine in his belly. In a moment, he was kneeling at the captain's feet, with his arm in a lock that threatened to dislocate his shoulder.\n\n\"I don't want to kill you,\" the captain said. \"But to be honest, Ser George, I'd really like to kill someone, and you are the likeliest candidate right now.\"\n\nSer George grunted.\n\nThe captain let go his hold, a little at a time.\n\nSer George backed away. \"You're mad as a hatter.\"\n\nThe captain shrugged. \"I am going to hold this fortress to the bitter end,\" he said. \"I'm going to hold it if I have to do it by myself. When we march away from Lissen Carak\u2014and by my power, Ser George, we _will_ march away\u2014we won't be a nameless company of broken men on the edge of banditry. We will be the most famous company of soldiers in the North Country, and men will bid to have us.\"\n\nSer George rubbed his shoulder. \"We're going to die here, and that's not what we do, boy. We live. Let the other bastard do the dying.\" He looked at the captain. \"You have a very persuasive way with an arm lock.\"\n\nTwo rocks struck close together. _Slam\u2014slam_ , and plaster rained down on their heads.\n\n# The Lower Town, Lissen Carak\u2014The Red Knight\n\nAn hour later, as the light began to grow outside, the off-going watch started up the path with two heavy beams\u2014the rooftrees from collapsed cots\u2014carried high on their shoulders.\n\nThe enemy's machines launched a flurry of stones but the off-going watch was already out of range. They scurried up the ridge, and men came out of the fortress's main gate to help.\n\nAnd then there was silence.\n\nHours passed.\n\nThe captain had been sleeping in his armour, his head down on the table in the donjon. He woke to the silence, and he was up the ladder in a twinkling, his sabatons ringing, his hip armour scraping on the hatch to the first floor of the tower.\n\nNo Head was already on the battlements. He pointed to the enemy machines\u2014just three hundred paces further west. Close enough to touch, or so it seemed.\n\n\"Cuddy could reach 'em with an arrow. Or Wilful Murder.\" No Head grinned. \"I'm tempted to try, myself.\"\n\n\"Even if you caught one or two,\" the captain said, \"there are many, many more of them.\" He was much more exposed here\u2014his Hermetic defences weren't buttressed by the power of the fortress. He could feel Thorn.\n\nHe looked around.\n\nThe Lower Town's curtain wall was breached in four places.\n\n_Harmodius_ he called.\n\nHe felt the old man stir.\n\n_Well sent. I understand you._\n\nThe captain concentrated. _There will be an attack on the Lower Town. I need men. Please tell Ser Thomas._\n\n_You are stronger._\n\n_I am practising_ sent the captain.\n\nHe went back to watching.\n\nSauce watched the beams come through the gate. Skant came over to her\u2014hollow eyed, rubbing his arms\u2014and handed her a note.\n\nShe looked it over and nodded. She had the day watch formed in the courtyard for inspection, and she found Wilful Murder easily. \"Wilful,\" she said. \"On me.\"\n\nHe stepped out of the ranks.\n\n\"Find Bent. And any artificers you can rustle up. Master Random's man is in the dormitory\u2014I think that the pargeter boy is in the Great Hall. These beams are to form the pivot arm of a trebuchet\u2014mounted where the onager was.\"\n\nWilful Murder digested this. Nodded. Chewed on his moustache.\n\nWhile he was looking at the tower and Cuddy was inspecting the duty archers, Bad Tom appeared in his armour. He didn't look like a man who'd been up all night.\n\n\"Captain needs the quarter guard. At the double.\" He nodded.\n\nSer Jehannes came along the wall and down the curtain steps. \"Hold hard, Tom.\"\n\nTom's eyes met Sauce's. \"Now,\" he said.\n\nHe turned to face Ser Jehannes.\n\nThe quarter guard was the watch reserve\u2014half the able men, usually the very best men, but today simply half the available troops. Sauce had more than a dozen men-at-arms in the day watch\u2014most of the rest were kept ready for the sortie\u2014led by Ser John Ansley, a big, cheerful, ruddy-faced young man. \"Ser John, you have the watch,\" she said. \"I'm taking the quarter guard. On me!\" she called, and the quarter guard came; sixteen archers and eight men-at-arms. Most of the archers were guildsmen she didn't know\u2014with all five of the new recruits\u2014the local boys. Ben should have been her master archer, but he was already standing with Wilful Murder.\n\n\"Cuddy\u2014you're the senior,\" she said.\n\n\"Like enough,\" he said.\n\nJehannes raised his voice. \"You are insane!\" he roared at Tom.\n\nTom laughed.\n\nHer senior man-at-arms was Chrys Foliak\u2014one of her own tent-mates. He had the others ready to move.\n\nCuddy made a motion with his hand and Long Paw stepped out of the ranks and joined him.\n\nThey went out the postern. It was obvious to them all that Ser Jehannes disagreed with the order to send them. But then the courtyard was behind them, and they were out in the light.\n\nBelow, on the fields, hundreds\u2014perhaps thousands\u2014of creatures were moving toward the Lower Town. The fields themselves seemed to be moving.\n\n\"Good Christ!\" Chrys Foliack muttered. \"Good Christ.\"\n\nLong Paw spat thoughtfully.\n\nHe paused in the postern, leaned back, and shouted \"Toby! Michael!\"\n\nHe couldn't see the captain's valet or his squire. \"JACQUES!\" he roared.\n\nA nun\u2014tall and pretty despite her hollow eyes\u2014came to the postern. \"May I help?\" she asked.\n\n\"Captain's in trouble. Tell Bad\u2014tell Ser Thomas we'll need relays of arrows and all the men in harness.\"\n\nShe nodded. \"I'll tell him.\"\n\n\"See you do, lass,\" Long Paw spat carefully to one side, flashed her his best smile, turned, and ran down the long path to catch up with the others.\n\n# Lissen Carak\u2014Harmodious\n\nHarmodius watched the bustle in the courtyard as he climbed past the two men-at-arms arguing\u2014reached the wall\u2014\n\nIt was worse than he had thought.\n\nHe ran, barefoot, along the wall to the apple tree.\n\nSummoned power, and raised his staff...\n\n# Lissen Carak\u2014The Abbess\n\nThe Abbess watched the day watch form under her window. There was something particularly well-ordered about the company. Their scarlet jupons, their bright polished armour. They made her feel safe even when she knew that she was anything but.\n\nEven as she watched\u2014looking for the captain, and missing him, and assigning herself a penance for looking, all in one thought\u2014the woman who wore men's armour shouted an order, and all of the men on the right of the formation turned and followed her.\n\nThere was suddenly an air of crisis\u2014men moved in many directions.\n\nShe reached out\u2014\n\n_He was preparing an attack._\n\nShe felt well-slept and immensely strong. She walked across her solar to the windows on the outer wall, three hundred feet above the fields below, and looked out.\n\nHer fields seethed as if covered in maggots.\n\nHer feeling of revulsion was more than physical.\n\nA pair of her novices, alerted by her movements, appeared with a cup of warm wine and a fur-lined robe. She drank the one and shrugged on the other while the older novice brushed her hair.\n\n\"Hurry,\" she said.\n\nShe put light shoes on her feet, pulled the mantle of her profession over her fur robe and was off while the creatures in the fields below were still merely a tide lapping at the foundations, and not a mighty wave.\n\nShe collected the crozier\u2014the crooked staff that the Abbess bore by tradition, with a curious green stone head.\n\nAnd then she ran, like a much younger woman, for her bower\u2014her apple tree.\n\nShe was shocked to find another there. Not just there, but swimming in her power.\n\n\"Master Magus,\" she said, coming to a stop.\n\n\"Lady Abbess,\" he said. \"I'm working.\"\n\nEven as she paused, he raised his staff. His power was visible. His whole form gave off tendrils of power.\n\n# The Lower Town, Lissen Carak\u2014The Red Knight\n\nThe captain watched the enemy's creatures gather. They were well within bowshot, and No Head and his fellows began to pink them. The two youngest archers carried sheaves of fresh shafts from the second floor, and the older men began to loose.\n\nThe captain had seen archers in action before, had watched his men practise at the butts, but he'd never watched a dozen professionals at full stretch.\n\nHe'd fussed at No Head while the older man felt the breeze, and carefully arranged his sheaves in brackets for the purpose set into the wall\u2014little iron buckets.\n\nThe two senior men\u2014No Head and Kanny\u2014raised their bows, loosed, discussed their aiming points, and watched the fall of their shafts.\n\n\"Over,\" said Kanny. It was a different tone of voice from his usual hectoring, barracks-lawyer voice.\n\n\"Over,\" No Head said. \"Ready, lads?\"\n\nHe raised his bow, and every man on the tower raised his in emulation, and they all loosed together. Their arrows rose and rose, and before they had begun to fall the next flight was on its way.\n\nDown on the plain, the distant irks screamed their defiance, showed their hooked teeth, patted their backsides and hefted their spears.\n\nThere were a thousand of them\u2014more, most likely. In their homespun greens and their leathers and brown skin, they looked as if they'd been grown from the earth under their feet.\n\nThe first flight of arrows struck. They all struck together, and tore a small whole in the great patchwork of brown-green irks.\n\nThe phalanx of spears moved a step closer.\n\nThe second flight struck.\n\nAnd the third.\n\nAnd the fourth.\n\nThe regiment of irks started to look like a piece of leather on a shoemaker's bench punched with an awl. And again, and again. The punches only made small holes. But it made a great many of them.\n\nThe irks screamed, their handsome elfin faces contorted into masks of rage, and they charged.\n\n\"Fast as you can, boys,\" No Head called.\n\nHis arms became a blur of motion. He drew and loosed, took a shaft from his bracket, nocked, drew, and loosed so quickly that the captain had difficulty sorting his actions.\n\nBrat, the youngest archer, opened a linen sack and dumped the shafts, points first, into No Head's bracket, and ran to load the next archer.\n\nKanny was grunting with every draw. The sound was so frequent and rhythmic it was obscene.\n\nThe irks had little or no armour, and no shields. As they crossed the three hundred paces to the breaches in the northern wall, they left a trail of wounded and dead creatures behind them. It was as if the whole phalanx was a wounded animal, bleeding little corpses.\n\nThey reached the first breach.\n\nKanny ran dry of arrows, and had to pause to get his own bundle. Brat couldn't keep up. One by one, the bows stopped twanging.\n\n\"They're not going anywhere,\" No Head said calmly. \"Don't rush. Everyone get their quivers full again. Brat, you get one more load up here and join us on the wall.\"\n\nThe captain felt superfluous.\n\n# Lissen Carak\u2014Sauce\n\nCuddy watched the first charge out of the slits of one of the covered ways halfway up the ridge. Then he ran down the steps to Sauce.\n\n\"They're going to need help,\" he said.\n\nShe glared at him.\n\n\"We can hit them from down there,\" he said, pointing to the lower path. \"With arrows,\" he continued. The men-at-arms tended to forget the power of the bows.\n\nSauce paused. \"Yes,\" she said. \"Let's go!\"\n\nThey pounded down the track\u2014over a streambed, down steep steps, around a long curve, and then they were right above the Lower Town. The wall had a fine low parapet, and the Gate Tower was just a hundred paces away and almost at eye level.\n\nCuddy admired No Head's archery for three long breaths. The shooting was continuous, now, and the flow of shafts like a waterfall crashing down on the irks in the field. The creatures died and died.\n\nIt was clear to Cuddy that the irks were defeated. Archery combat had a ruthless logic of its own. Cuddy was an expert in it.\n\n\"Five shafts,\" he said to the men around him. \"Right in the midst of them. Fast as you can.\" Two of his guildsmen had crossbows\u2014not really worth a thing in a fight like this.\n\nOh, well.\n\n\"Ready?\" he called. Every longbowman had five arrows in the ground, ready to hand, and another on the bow. Long Paw had one on his bow, one in his bow hand, and four in the ground.\n\nCuddy raised his bow.\n\n# Lissen Carak\u2014The Red Knight\n\nThe irks broke.\n\nThe new arrows came from behind, plunging down and killing them. In a minute a tenth of their numbers were pinned to the ground, screaming their thin screams.\n\n# Lissen Carak\u2014Sauce\n\n\"Save your shafts,\" Cuddy said. He had only fifteen more. High above, on the ridge, he could see valets starting down with bundles of arrows, but it would be ten minutes before those arrows reached them.\n\nHe pointed to the town. \"Some of them got in,\" he called to Sauce.\n\n\"Are you happy to stay here?\" she asked.\n\nCuddy nodded.\n\n\"Men-at-arms\u2014on me.\" She waved to Cuddy and started for the postern gate.\n\nLong Paw winked at Cuddy as he followed her.\n\n# Lissen Carak, The Lower Town\u2014The Red Knight\n\nThe captain went to open the tower's lower door himself. He and Ser George were the only men without bows.\n\nSauce was outside, with a crowd of armoured men. \"Town's full of irks,\" she said. Her sword was in her hand, and behind her, men were cleaning the dark blood from their blades.\n\nHe nodded. \"We have to keep the street clear for sorties,\" he said.\n\nShe nodded. \"That's going to suck,\" she said in a matter-of-fact voice. And took her party to move stones and fallen roof tiles.\n\nThe captain went with them.\n\nIt was brutal work. As the spring sun rose it burned, distant and orange, through the smoke-filled air. It was growing warm, and inside forty pounds of chain and plate, and a heavy quilted arming cote, it was hot.\n\nJust bending to lift a stone was hard enough in armour.\n\nIt took five of them to lift a fallen roof beam.\n\nWhen they began to complain, he pointed out that it was their horses who would come through here in the dark.\n\nThey went on, picking up rubble, pushing obstructions aside.\n\nAfter an hour, the captain was soaked through. He collapsed on a low stone wall and Toby handed him a flagon of water.\n\n_Thump-snack_.\n\n\"Son of a bitch,\" the captain cursed, and the stone slammed into a church fifty paces distant, blowing a hole through the tile roof and vanishing inside.\n\nHe began to stand up, and the irks attacked.\n\nThere were only a dozen of them; desperate, and brave, and ferocious.\n\nWhen the rush was cleared, the captain found that the armoured man at his back was Ser George Brewes.\n\nThe flagon of water was still unbroken by a miracle. He took a swig, spat, and handed the jug to Ser George.\n\nSer George leaned on his sword. \"Feg,\" he said, shaking his head. \"Irks. I've heard of them.\"\n\nThe captain just panted.\n\n\"Like killing children,\" Ser George said.\n\nThe whole sky was a pink-red. Another rock crashed to earth off to their left.\n\n\"You really think we can hold?\" Ser George asked.\n\n\"Yes,\" the captain wheezed. He'd taken a cut on the back of his shoulder. He could feel the blood mixing with his sweat. _I need to learn to heal myself._ It was trickling down his side\u2014warm, instead of cold.\n\n_Why? Why did she turn her back on me?_\n\nHe made a face.\n\n\"It would be something,\" Ser George admitted.\n\n\"Yes,\" the captain managed.\n\nToby\u2014unarmoured and unarmed\u2014had survived the rush from the irks. He'd simply run away. Now he was back.\n\n\"I've food,\" he said.\n\nHis scrip was packed with beef, bread and good round cheeses and Sauce's men-at-arms fell on him like scavengers on a carcass. His head was patted a dozen times. He had a meat pie for himself. But he always seemed to.\n\nSauce moved among them. \"Drink water,\" she said, as if they were children and turned to the captain. \"Think they'll try again?\" she asked.\n\nThe captain shrugged, and the weight of his armour and the pain in his shoulder defeated the motion completely. So he bobbed his head. \"No idea.\" He took a deep breath. His breastplate seemed to be too small, and he couldn't catch his breath. The smoke in the air was burning the inside of his lungs.\n\nIt was a very small working, an insidious thing. He saw it as soon as he made the effort.\n\nThe air was _full_ of a poison. He couldn't even see how it was done.\n\nSauce started to cough.\n\n_Harmodius!_ He called.\n\n_I see it, lad._\n\n_Do something!_ the captain shouted in his head.\n\n# Lissen Carak\u2014Amicia\n\nHis shout came to her as clearly as his anguish.\n\nShe was working on Sym's back, running her hands along the weels left by the lash, and trying to fix some of the deeper issues, as well. The captain's thoughts were not helping her concentration.\n\nShe reached out instinctively. It was in the air. Poison. She read it from his thoughts.\n\nShe tasted the air through his mouth, and felt it through his lungs.\n\n_She was in him._\n\nThen he slammed his gate shut.\n\nShe was standing over Sym, with her hands clenched into fists. Shaking.\n\n_Captain!_ She sent.\n\nHe responded.\n\n_It's an unhealing. A curse._\n\n_Tell me._\n\n_You cannot banish it. You can only heal it._\n\nAnother voice. The Magus. _I see! Well thought, mistress._\n\nNow it was her turn to raise her defences. _Get out!_ She said it aloud too.\n\nSym looked at her.\n\n\"Not you, silly,\" she muttered.\n\n# Lissen Carak, The Lower Town\u2014The Red Knight\n\nThe captain could feel the poison thickening in the air and he didn't know how to heal. Although now that she showed him, he could see it.\n\nA curse.\n\nThe physical manifestation of a curse.\n\nHe _went into his tower. \"I need help,\" he said to his tutor._\n\n_She smiled. \"Ask me anything,\" she said._\n\n_\"A curse. A physical curse\u2014a poison in the air.\" He went to the door to his tower._\n\n_\"He's waiting for you to open it,\" she said._\n\n_\"I think he's busy, and a lot of people are going to die if I don't act.\" He reached to door._\n\n_\"If it is physical, perhaps we can move it physically,\" Prudentia said. She smiled sadly. \"I don't know healing, either.\"_\n\n_\"That's a fine thought.\" He looked up at his symbols. \"Wind,\" he said._\n\n_\"Yes,\" Prudentia agreed._\n\n_He spoke the names. \"St. George, Zephyr, Capricorn,\" he said, and the great ranges of symbols rotated silently._\n\n_He touched the door._\n\n_He could feel the enemy, and he opened it anyway._\n\n_And slammed it back shut._\n\n# Lissen Carak\u2014Sauce\n\nThe wind came up without warning\u2014first a heavy gust that cooled them, and then a mighty rush of air from the east.\n\nSauce drew a shuddering breath.\n\n\"Get a scarf over your face,\" the captain shouted. \"Anything.\"\n\nThe wind moved the poison\u2014but he could still smell it.\n\nAnd then he felt the sending. It was gentle as snow, and just for a heartbeat the air seemed to sparkle all around them, as if the world was made of magic.\n\n# Lissen Carak\u2014Harmodius\n\nHarmodius watched the Abbess's working and he could only think of Thorn's statement that men were too divided.\n\nIt was beautiful. The sort of mathematical Hermeticism that moved him the most deeply. In it were the rotations of the planets and the paths of the stars across the heavens. And many other things, thought and unthought...\n\n\"You are far more powerful than I had imagined,\" Harmodius said.\n\nShe smiled. Just for a moment, it was the Queen's smile.\n\n\"Who are you?\" he asked.\n\n\"You know who I am,\" she said it playfully. She rose from her seat. \"I think Thorn will find it very hard to use that trick again.\"\n\nHarmodius raised an eyebrow. \"Trick?\" he asked. \"It wasn't Hermeticism. It wasn't a working. Not as I understand them.\"\n\n\"There are more things on heaven and earth than are in your philosophy,\" she said. \"He uses the deaths of the irks to fuel his curse. It is a very, very ancient way to power magic.\"\n\nHarmodius nodded in sudden understanding. \"But you\u2014\"\n\n\"I stand for life,\" the Abbess said. \"Me, and my God, as well.\" She smiled sweetly. \"He will not be back for some time. I need to speak to a novice. Pray excuse me.\"\n\nHarmodius bowed. As she swept past him, he said, \"Lady\u2014\"\n\n\"Yes? Magus?\" She paused. Her attendants paused, and she waved them on.\n\n\"If we linked, lady\u2014\" he said.\n\nShe made a moue. \"Then you would know all my innermost thoughts. And I yours,\" she said.\n\n\"We would be more powerful,\" he insisted.\n\n\"I am already linked to my novices. And to all my sisters,\" she said. \"We are a choir.\"\n\n\"Of course you are,\" Harmodius said. \"Gads, of course you are. I'm a fool.\" It was obvious, when she said it. Forty weak magi would still be very powerful indeed, together. But it would require incredible discipline.\n\nLike monks.\n\nOr nuns.\n\n\"I will think on it,\" she said. She smiled.\n\nHe watched her go, and then sat beneath the apple tree.\n\n# Chapter Thirteen\n\n# Lissen Carak\u2014Michael\n\n_The Siege of Lissen Carak. Day Eleven_\n\n_The captain took the watch to support our garrison in the Lower Town\u2014a small fortified bastion at the base of the ridge. The Enemy has constructed siege engines\u2014catapults and trebuchets\u2014to attack. Because of the rage of our engines atop the fortress, and because we can launch sorties from the fortress through the streets of the Lower Town, the captain says that the Enemy must take the Lower Town first._\n\n_He made two attempts, but both resulted in heavy losses of creatures of the Wild. We lost not a single man or woman yesterday. The Abbess called on the Power of God and defeated the Enemy's poison air. Many men felt lighter at heart after she prayed._\n\n_But the Enemy's engines now throw heavy stones all the time. The air is full of smoke, and many of the farm folk have become angry and downcast._\n\n_During the night boglins assaulted Bridge Castle, but their surprise failed and they were driven off._\n\nMichael put his quill down and shook his head at the ink stain on his forefinger.\n\nKaitlin had not come out to meet him last night, even though he was on his way to the Lower Town. The farmers were angry\u2014he could feel it. Old Seth Lanthorn, an oily bastard in the early days of the siege, was now surly and silent. Farmers muttered when he walked by.\n\nThey resented their boys being taken to be archers. And perhaps resented\u2014\n\n_I will marry her_ , he said to himself. But he couldn't keep his eyes open...\n\n# Lissen Carak\u2014The Red Knight\n\nThe curtain wall around the Lower Town was gradually pounded to rubble.\n\nBefore the sun rose, the stars were obscured, and clouds rolled in. The rain that started wasn't hard, but it was soaking, and cold.\n\n\"Attack coming,\" Toby said, rubbing his cheek. The boy's breath was sweet with apple cider.\n\nThe captain rose blearily, feeling as if he'd been kicked repeatedly. It was an effort of will to run through his Hermetical exercises and it was torture to arm. Toby had to put his harness on him\u2014Michael was down in the Lower Town. Every man and woman had to do their duty, now.\n\nWhen he went out on the wall, the fields were moving again, lines of irks marching to form up opposite the northern flank of the town. Now they had shields\u2014great pavises of heavy bark stripped from downed trees in the deep woods.\n\nThey formed in six deep columns, glistening in the light rain.\n\nBad Tom had twenty men-at-arms and as many squires and valets waiting for them, and twenty archers on the tower. The breaches in the town wall glittered damply with men in harness.\n\nThe enemy's engines were silent.\n\nWilful Murder stepped up on the wall with his captain. \"It's done,\" he said. He pointed to the squat remnants of the former southern tower. Now it was an engine platform, two storeys tall, crowned with a trebuchet whose launching arm was as tall as the spire on the chapel.\n\nThe captain gave him a tired smile.\n\n\"Let's see if we can give Master Thorn another surprise,\" he said. \"Let's go.\"\n\nThe first stone was loaded with some trepidation. The arm of the trebuchet would throw a man in armour five hundred paces. A war horse three hundred paces.\n\nWilful fussed like a mother sending her child to church the first time.\n\nNo Head, who was supposed to be off duty but whose love of engines outweighed his good sense, pushed the loader out of the way and muscled the stone into the great hemp-rope web.\n\n\"Care to do the honours?\" Wilful asked the captain.\n\n\"Everyone off the tower,\" the captain said.\n\nEvery one of the farmers was in the courtyard. They'd worked like draught animals to get the machine built and in place\u2014to level the stump of the tower. Their grumbling was loud and aggressive, and the captain ignored them.\n\nBut he needed them to wind the arm into place. The trebuchet depended on farm women for its motive power.\n\nWhen they were all clear, the captain pulled the lever.\n\nThe trebuchet's arm moved slowly, at first, then rotating faster and faster until the great sling at the end was lifted clear of the deck\u2014the arm and its massive weight passed the centre of rotation and the weight crashed down onto a massive pile of old hordles\u2014 _thump_ \u2014and the sling opened\u2014 _crack_ , and a stone the weight of a man flew free\u2014rising for what seemed an incredibly long time.\n\nAnd of course, the heavy stone started three hundred feet above the fields below.\n\nIt rose and rose, passing over the irks, who had just started to move forward, clearly unsure of the efficacy of their new shields, and then it began to fall. It came down at a steep angle, it passed over the irks, over the deep trench the boglins had dug, over the enemy's artillery platform, the mound on which his engines sat, and vanished into the trees of the woods at the western edge of the cleared ground.\n\nIt did no damage to anyone, or anything.\n\nBut the farmers cheered, and the archers cheered and the captain grinned to see it.\n\nWilful Murder ran back up the ladder and pounded his captain on the back.\n\nThe captain smiled. \"Nice work.\" He turned to No Head. \"Get the engines.\"\n\nNo Head grinned.\n\nThe first assault was retreating by the time the great engine was wound again. Bad Tom's men-at-arms had mangled it, and the great bark shields hadn't done as much to stop the archer's shafts as the irks might have wished.\n\nThe captain gathered a sortie under Ser Jehannes in the courtyard. \"Tom's going to be hard pressed,\" he said to Jehannes. \"A dozen men ahorse will make short work of their next assault.\"\n\nJehannes nodded. \"Yes, ser,\" he said coldly. \"I know my business.\"\n\nThe captain noted that Francis Atcourt was in harness and mounted. He pressed the man's gauntleted hand. \"Good to see you about,\" he said.\n\n\"Good to be here,\" Atcourt said. \"Although, it seems to me another day abed\u2014\" He laughed. \"I'd be strong enough to swim a mountain or climb a river.\"\n\nThe trebuchet released.\n\nThe captain wasn't the only man who ran to the walls to watch the fall of the shot.\n\nNo Head's first round landed out of sight beyond the enemy's engines.\n\nThe captain watched the next assault. It was halfhearted. The irks stayed away from the worst of the archery by bunching up in the front of the central breach, and very few of them went forward all the way to the men-at-arms.\n\nThen one of the enemy's engines released.\n\nThe rock fell like a lightning bolt, into the breach, crushing men-at-arms and goblins alike.\n\n\"Damn,\" the captain said. \"I should have expected that.\"\n\nA creature gave a long, bone-chilling cry\u2014like a trumpet, but louder and more hideous\u2014and irks crept from houses and cellars in the Lower Town. They had crept in during the night, or made it past the archers in the first assaults, and now they struck the rear of Bad Tom's line.\n\nA great armoured troll sprinted from behind the engine platform and pointed its antlered head at the breaches in the curtain wall.\n\nThe irks got out of its way.\n\nAnother rock plunged from the heavens to strike in the central breach. The stone seemed to explode as it hit, spraying attackers and defenders alike with lethal stone chips.\n\nThe men on the walls watched the men in the breach like spectators at a joust.\n\nSer Philip le Beause died when a stone chip caved in the side of his helmet.\n\nRobert Beele fell, stunned, and an irk got its dagger in his eye slit.\n\nSer John Poultney died trying to get his back to the wall, swinging his sword in wide arcs. He stumbled when a stone hit his backplate, and was on his knees; in a heartbeat, a wave of the little monsters were on him. He crushed one with his gauntleted left fist, swung his sword one handed through another pair, and then two were hauling his head back.\n\n\"Release the sortie,\" the captain ordered.\n\nNo Head loosed the trebuchet. The stone flew high, and vanished into the forest of upright machine arms atop the enemy's artillery mound.\n\nWood chips flew, visible even from the fortress.\n\nA half-loaded trebuchet in the enemy's battery was loosed by a panicked boglin and his loader was caught in the casting net and flung a hundred paces to fall wetly to earth.\n\nJehannes galloped down the road from the fortress followed by a dozen knights.\n\nThey flew down the switchbacks, and the troll raced for the breach, and a swarm of irks pushed the defenders of the breach into a knot.\n\n\"Damn,\" the captain said.\n\nHe'd never cast power at this distance, but he had to try.\n\n# The Lower Town, Lissen Carak\u2014Bad Tom\n\nBad Tom was a pebble in a crumbling sand castle.\n\nHe threw back his helmeted head and bellowed.\n\nThe irks quailed.\n\nHe killed them.\n\nHis sword was everywhere, and he was faster than they, taller, longer, stronger.\n\nThey went where he wasn't, but the other men-at-arms knew what Tom was like, and they stuck to him like glue. Francis Atcourt stood at his shoulder, advancing when Tom advanced, retiring when the big man spun away. He had a short spear, and he used it sparingly. He let Tom kill the irks. He only killed those who could threaten Tom.\n\nThey began to retreat off the breach. They couldn't hold it\u2014too many of the men-at-arms were down.\n\nAtcourt saw movement above him on the ridge. \"Sortie,\" he called.\n\nTom was frozen.\n\n\"Troll coming,\" he said. \"Francis, clear what's behind us and open a lane to the tower.\"\n\nAtcourt didn't need to be urged. He tapped the captain's squire and three other men on the helmet as he passes them. \"On me!\" he called.\n\nAn irk appeared in his range of vision\u2014paused, surprised, perhaps to find men in the town, and not on the wall, and died with Atcourt's short spear in its forehead.\n\n\"Michael!\" he called. \"Get to the tower. Tell Cuddy and Long Paw to cover us.\"\n\nThe squire had excellent armour, lighter and better than any of the professionals. Besides, he was the youngest.\n\nThe great troll ran through the irks. At the base of the rubble-strewn slope up into the breach, it paused, glaring around like some eyeless worm seeking daylight or warmth\u2014or human blood. Then it picked its way to the top of the breach, clearly unwilling to move quickly in the bad footing. When it reached the top it paused again, caught sight of the men-at-arms and threw back its head and roared its challenge, its grotesque mouth, back-hooked fangs and black gullet on display as it sounded its challenge.\n\nThe sound rang through the woods, and echoed off the ridge and the walls of the fortress high above. The Abbess heard it at her prayers, and Amicia heard it in the hospital. Thorn heard it and clenched a mighty fist. The captain didn't hear it at all. He was preparing to work.\n\nBad Tom stood his ground, threw back his head, and roared back.\n\nThe sound crashed back and forth\u2014from the fortress walls to the woods, and back.\n\nThey charged each other.\n\nA stride from contact, Tom side-stepped\u2014the monster hesitated, and Tom's sword swept through. The troll's antlers caught him and slammed him to the ground.\n\nThe troll's momentum carried it a dozen steps, and it turned.\n\nTom got a leg under him. He put the point into the ground and used his great sword as a lever to get to his feet.\n\nThe troll completed its turn, and put its armoured head down.\n\nTom laughed.\n\nCuddy leaned out over the tower wall. The troll turned, and he let it turn, reasoning that its arse couldn't be as well armoured as its front. He raised a chisel point above the wall, leaned into his draw, and loosed.\n\nThe arrow struck with a sound like a butcher's blade into a leg of mutton.\n\nThe troll stumbled. The arrow had struck from behind, between its shoulder blades, and sunk in all the way to the fletchings. The troll gave a moan and raised its head.\n\nTom stepped forward.\n\nThe monster flinched, and then punched for Tom's throat with both stone-shod hands.\n\nTom cut.\n\nStruck, and was struck to earth in turn.\n\nSer George Brewes leaped over Tom's body to face the troll in his place. \"Go!\" he roared at the rest of the men-at-arms. \"Run!\"\n\nBut Francis Atcourt came and joined him, and Robert Lyliard too.\n\nThe troll eyed them, pawed at the earth once, twice, and then slumped slowly to it and lay still.\n\n\"Son of a bitch,\" Lyliard said. He stepped forward and slammed his hammer into the thing's head.\n\n\"Get Tom!\" Atcourt called. The irks had the breach, and the troll's death didn't seem to make any difference to them.\n\nThey all got a hand on him. He weighed as much as a war horse, or so they swore later.\n\nAnd then they ran for the tower, the irks hard on their heels.\n\nThe archers shot right into them, Cuddy and Long Paw assuming that their armour would hold.\n\nMostly, it did.\n\nThe irks fell back\u2014flooding the Lower Town, but letting the men have a path to the tower\u2014and the postern opened. Long Paw loosed a shaft right down the line of men-at-arms and then drew his hanger and his buckler, flinging his bow through the door behind him. He stepped out, and the men-at-arms carried Tom past him.\n\nThere was a brief flood of irks. They were all armoured in scale mail and carrying round shields\u2014warriors.\n\nLong Paw's sword and buckler swept up, bound as if they were one weapon\u2014his buckler slammed into the face of one irk's shield, and then, in the same tempo, his sword beheaded another. In the same flow, he swept his sword back into guard, fell back a step, and parried not one but two spear thrusts with a single sweep of his blade. He stepped in, passed his buckler under the spear-wielding irk's arms, wrapped them, slammed his pommel into the irk's unarmoured face, and used his advantage to throw the lighter creature into his mates.\n\nStepped back again, and the postern crashed shut.\n\n# Lissen Carak\u2014The Red Knight\n\nSer Jehannes had halted the sortie two-thirds of the way down the ridge, when it became clear that the breach had fallen. Now the sortie turned and rode silently back up the road.\n\nThe captain was waiting in the gate.\n\n\"Right,\" he said to Jehannes. \"Good call.\"\n\nJehannes dismounted, gave his reins to a farmer\u2014the valets were all in harness\u2014and started to turn away. \"The Lower Town is lost,\" he said.\n\n\"No,\" the captain said. \"Not yet.\"\n\nOver their heads, the trebuchet lashed out again.\n\n\"You are risking everything on the hope that we will be relieved. By the king.\" Jehannes was obviously restraining himself. The words were very carefully enunciated.\n\nThe captain put a hand on his shoulder. \"Yes,\" he said.\n\n\"Christ be with us,\" Jehannes said.\n\n# West of Albinkirk, South Bank of the Cohocton\u2014Gaston\n\nGaston had done his exercises of arms, and had prayed. And now he had little to do. He'd had enough of his cousin, and enough of the army in every way.\n\nHe mounted his riding horse, left his valet at his tent door, and went for a ride.\n\nThe camp was enormous\u2014a sprawling thing as big as a market fair or a small town, with more than two thousand tents, hundreds of wagons drawn up like a wall, and a ditch all the way around it, dug to the height of a man and with the upcast flung back to form a low rampart.\n\nNo man was allowed outside the ditch on pain of punishment. Gaston understood\u2014better than his cousin\u2014that he needed to set an example, so he rode slowly around the perimeter, nodding to the Alban knights he knew, and their lords.\n\nHe saw a pair of younger men with hawks on their wrists, and he was envious.\n\nHe thought of home. Of sun-drenched valleys. Of riding out with his sister's friends, for a day of wit and wine and frolic, chasing birds, climbing trees, watching a well-formed body on a horse, or by a stream...\n\nHe shook his head, but the image of Constance d'Eveaux looking back over her naked shoulder before leaping into the lake haunted him.\n\nThere had been nothing between them. Until that moment, he hadn't even noticed her except as a pretty face among his sister's friends.\n\n_Why am I here?_ Gaston asked himself.\n\n\"See something what you like?\" said a familiar voice.\n\nGaston reined in, his reverie exploded.\n\nIt was the old archer. Gaston was surprised to find that he was happy to see the low-born man.\n\n\"You were going home,\" Gaston said.\n\nThe old man laughed. \"Heh,\" he said. \"Lord Edward asked me to stay. I'm a fool\u2014I stayed. I sent my useless brother-in-law home.\" He shrugged. \"Of the two of us, my daughter probably needs him the more.\"\n\n\"The Lord of Bain?\" Gaston asked.\n\n\"The very same. I was his archer on the crusade, oh, ten years back.\" He shrugged. \"Those were some hairy times.\"\n\nGaston nodded. \"I knew you were an man-at-arms.\"\n\nThe old archer grinned. \"Aye. Well. I meant what I said. It's all foolishness. Why are we at war with the Wild? When I lie out at night hunting I love to have a chat with the faeries. I've traded with the irks more than once. They like a nice piece of cloth, and mirrors\u2014hehe, they'd trade their mothers for a bit o'mirror.\" He nodded. \"Admit I can't stand boglins, but they probably feel the same about me.\"\n\nGaston couldn't imagine such a life. He covered his confusion by dismounting. He was surprised to find the archer holding his horse's head.\n\n\"Habit,\" the old man said.\n\nGaston held out his hand. \"I'm Gaston d'Eu.\"\n\n\"I know,\" the old man said. \"I'm called Killjoy. Make of it what you will. Harold Redmede, it says in the christening book.\"\n\nGaston surprised himself by clasping the man's arm, as if they were both knights.\n\n\"Surely it is a crime against both the King and Church to trade mirrors to the irks.\"\n\nThe old archer grinned. \"It's a crime to shoot Lord Edward's deer. It's a crime to take rabbits in his warrens. It's a crime to leave my steading without his leave.\" The archer shrugged. \"I live a life of crime, m'lord. Most low-born do.\"\n\nGaston found himself smiling. The man was really very likeable. \"But your immortal soul,\" he began softly.\n\nThe old man pursed his lips and blew out a puff of air. \"You're easy to talk to, foreigner. But I don't need to debate my mortal soul with the likes of ye.\"\n\n\"But you are willing to speak with evil.\" Gaston shook his head.\n\nThe archer gave him a wry smile. \"Are all the men you know so very good, m'lord?\"\n\nGaston winced.\n\n\"Stands to reason all the irks ain't bad, don't it?\" he went on. \"What if none of 'em is bad? Eh? What if there's no power on earth as bad as a bad lord?\"\n\nGaston shook his head. \"What bad lord? This is rebel talk.\"\n\n\"Rest easy, m'lord, I'm no Jack.\" The old man sneered. \"Boys playing at causes. And broken men and traitors.\" He nodded. \"Some good archers, though.\"\n\n\"Let's say I'm coming around a little to your way of thinking,\" he said carefully. \"I would like to confess that I want to go home.\"\n\n\"Knew you was a man of sense.\" Redmede laughed. He looked under his hand and shook his head. Pointed at an archer, asleep. \"Swarthy, you useless sack of shit, get off your arse and work.\"\n\nGaston turned and saw the young archer trying to hide in the ditch. He was all huddled up, as if by being very small, he could avoid the old man's wrath.\n\n\"Now I'm the master-archer, and I wear myself out riding these boys.\" He laughed.\n\nGaston didn't think he looked worn out.\n\nRedmede stepped closer to the ditch and bellowed, \"Swarthy!\" at the young man.\n\nHe paused and in a moment Gaston saw what he saw.\n\nThe boy was eviscerated. And very, very dead.\n\n\"Damn,\" the old archer said.\n\n# West of Albinkirk\u2014Galahad Acon\n\nGalahad Acon had never been so cold for so long, and he lay as still as he could lie, watching...\n\nWell, watching nothing at all. Watching the woods. A breath of breeze stirred, moving the new leaves, and the light rain fell and fell. Despite a wool jupon and a wool cote over it, with a heavy wool cloak over all, he was soaked to his linen shirt and colder than he was when riding through heavy snow in December.\n\nThe Prior had left him to watch at the first grey light of dawn. Had said he'd be back.\n\nHe'd taken Diccon with him.\n\nAs time went by, his fancies grew darker and darker. Why would they ride off and leave him?\n\nHe had a fire kit. But the Prior had been very forceful on the subject of fires.\n\n_I'm going to freeze to death._\n\nFor the thousandth time, a twig cracked in front of him.\n\nGalahad wondered how twigs could just crack, in the woods.\n\nA bird fluttered in the wet leaves, and made a low thrumming sound\u2014and then burst out of the leaves and leaped into the air.\n\n_Something had just moved._\n\nGalahad felt his blood still in his veins.\n\nHe scanned his eyes frantically back and forth.\n\n_Oh good sweet Virgin Mary now and in the hour of my death amen._\n\nThey were almost silent\u2014filing along the streambed at the base of the low hill.\n\nBut there were hundreds of them.\n\n_Oh my god dear god ohmygod_\n\nIn the lead was a willowy daemon, all black, which moved like an embodiment of shadow, flitting rather than walking. Behind him, came the hosts of hell, walking, strutting, shambling\u2014\n\nGalahad found he could neither watch nor turn his head away. That when he closed his eyes, he couldn't picture exactly what they looked like.\n\nHe couldn't make his mind work. Run? Stay?\n\nHe was fear.\n\nThey moved along the watercourse, and they scarcely moved the leaves. They travelled quickly, passing from left to right before him.\n\nEventually, he realised they weren't going to turn and rend him limb from limb. But that didn't stop his breath from coming in low pants, nor the deep cold from settling into his bones.\n\nAnd then they were gone, away to the north, towards the river.\n\nIt was a long time before his breathing returned to normal.\n\nWhen the Prior found him, at sunset, still lying there, he burst into tears.\n\nThe Prior embraced him. \"I'm sorry,\" the mailed knight said. \"You did well.\"\n\nGalahad was ashamed of his tears, but he couldn't stop them.\n\n\"They got between us and you,\" the Prior went on. \"I couldn't risk my knights for you. That\u2014that is how it is, out here.\" He patted Galahad. \"You did very well.\"\n\nThey moved camp, in the same silence that the knights did everything. They went north, and Galahad saw that the tracks made by the daemons had the shape of human feet. He looked very closely, and he couldn't see anything but bare feet and soft shoes.\n\nA young Thomasine nodded to him. He cleared his throat quietly and leaned close. \"Sossag,\" he said.\n\nGalahad knew enough to knew that the knight was honouring him by speaking.\n\n\"I thought they were daemons.\" He looked at the knight.\n\nThe young man shook his head. Put a finger to his lips, and rode on.\n\nThat night, Diccon put an arm around him. \"Sorry, lad. It should hae' been me left with the baggage. I don't even know why we're here.\"\n\nThe Prior came and offered each of them a cup of warm mead. He sat on his heels, still armed from head to toe in plate and chain.\n\n\"You are here to take my news to the king\u2014when I have news.\" He looked back and forth. \"Tomorrow.\"\n\nDiccon drank his mead. \"What did you learn today?\"\n\n\"The fortress still holds,\" the Prior said. \"And holds the bridge, as well. The Abbess has done far better than I expected of her, and I owe her an apology.\" He smiled at Galahad. \"The trouble with a vow of silence is that it leaves you vulnerable to talk,\" he said.\n\nDiccon nodded. \"I'll ride at first light.\"\n\nThe Prior shook his head. \"The woods this side of the river are full of the enemy. Sossag, Abenacki, irks, boglins and worse.\" He shook his head. \"Tomorrow night we'll make a demonstration. A loud demonstration. We will draw every creature of darkness like\u2014\" he smiled \"\u2014like moths to a flame.\" He nodded. \"Then you'll ride.\"\n\n# Lissen Carak\u2014The Red Knight\n\nJust a few leagues north of the hillock where the Prior camped, the captain stood in the castle gateway with the Abbess. Behind him were most of the men-at-arms, led by Jehannes, and twenty squires and valets led by Jacques. Every man wore a nun's habit over his harness.\n\nHe gathered them in a circle.\n\n\"What a very scary passel of nuns we make,\" he said. \"The order of Saint Thomas will need to be a little more careful in their selection process.\"\n\nThe Abbess laughed. The men going on the sortie managed a sort of nervous titter.\n\n\"This needs to be fast, so listen up. It's like taking a town in Galle. Sneak to the wall. Ladders up on the whistle. That's all there is. When you are in, head for the towers at the gate. We get the lads there and back we come. Don't leave your wounded behind. You know all this.\" He grinned. Turned to Ser Michael, the sergeant of the original garrison. \"You must keep the gate open until the sortie returns. But don't leave it open for a few men. You hear me? When the sortie is in, close the gate.\" He turned to No Head. \"When you see my blue fire pound the town. Everything you have.\"\n\nNo Head nodded. \"The Bridge Castle has the word, too.\"\n\nBeside him, Harmodius crossed his arms. And winked.\n\nThe captain nodded. \"You all know Tom would come to get you. Let's go get Tom.\"\n\nA murmur.\n\nHe jumped down from his barrel, and led the way\u2014not to the gate, but to the dispensary stairs, and the Abbess walked with him.\n\nShe led them through the lower dispensary, and then down steep steps to a basement, and then down another set to a well\u2014a spring in the deep hillside, a cleft off to the right with lights burning.\n\nThe captain could feel an immense welling of power. Raw power. Neither gold nor green.\n\nHe reached into the well and filled himself.\n\n_You are much stronger, Prudentia said. But not as strong as he is._\n\n_I know._\n\n_You don't. You are arrogant. You are outmatched._\n\n_Fine. Yes, I know._\n\n_Fool! she spat._\n\nHe dropped back into the cleft and came to a long storage room, packed to the rafters with wagon sides and barrels of pork.\n\nIt took long minutes for men to shift the wagon beds.\n\nThere was a door behind them.\n\nThe Abbess drew a key from her girdle. Their eyes met.\n\n\"Now you know all my secrets,\" she whispered.\n\n\"I doubt it,\" he said, and kissed her hand.\n\n\"I am quite sure I should not give you this,\" she said. She smiled bitterly and handed him a small scrap of curled parchment, hard as an old leaf in his hand. Smooth as a woman's skin.\n\n\"I could disapprove, as her spiritual mother,\" the Abbess went on. \"I could just be a jealous woman.\" She shrugged. \"Sister Miram brought this note to me and confessed that she had passed another.\" She met his eyes. \"Amicia is not for you, Captain. She is greater\u2014far greater\u2014than we.\"\n\nHe smiled. \"That is not what I expected you to say.\" He bowed. \"I beg your indulgence.\" He turned aside, and held the scrap up to a torch on the wall in a clamp. He read, and he couldn't control the smile that crossed his face.\n\n_Your gate is closed._\n\n_Meet me._\n\nHe turned back to the Abbess.\n\nShe shook her head. \"You are glowing.\"\n\n\"How is she greater?\" the captain asked.\n\nThe column had begun to move. The door was open, and the lower door, too.\n\nHe kissed her hand again. \"Thank you,\" he said.\n\nShe smiled. \"You have brought me no peace, young man.\" She waved her hand. \"Go\u2014kill our enemies. Triumph.\" She sounded tired.\n\nHe turned and all but leaped down the steps. On his way he stopped to touch the favour he wore on his shoulder.\n\nAmicia felt him, like a touch on her cheek.\n\nShe smiled, and went back to tearing linen into strips.\n\n_I'm a fool_ , she thought.\n\nThe company went down through the Abbess's passage and entered a maze of stone corridors.\n\nTo those who knew what to look for, it was obvious that men had not made these curving corridors.\n\nBut they were empty, although, to the captain, every yard of them reeked of the power that had been used in storming them. More than a hundred years ago. More than two hundred years.\n\nAnd still the power lingered, like the smell of smoke after a fire.\n\nEventually, the Abbess's will-o-wisp led them to a double door of oak, bound with iron, copper, and silver. To the captain's eye, it was covered in sigils\u2014powerful wards drawn Hermetically.\n\nHe'd never seen anything like it.\n\nShe'd given him the key.\n\nHe held it with renewed respect.\n\nSome of the lads were very much on edge. An hour in silent, haunted corridors deep under the earth isn't the best preparation for combat. The sounds behind him were of men on the edge of panic.\n\nHe turned, and cast a soft light.\n\n\"Ready, friends?\" he asked softly.\n\nMore and more men stumbled into the antechamber in front of the great doors.\n\n\"We'll come out into the chapel of the Lower Town,\" he said. \"The roof is collapsed. Don't run. Out here a rolled ankle is a death sentence and we're not coming back this way. So don't linger.\" He couldn't explain why.\n\nHe was about to open the fortress's Hermetic defences, for a moment.\n\nHe imbued his voice with calm. Humour. Normalcy.\n\n\"Let's go get Tom,\" he said. He smiled at Jehannes, who, praise be, smiled back.\n\nAnd he turned the key.\n\n# North of Lissen Carak\u2014Thorn\n\nThorn felt the change. He was busy resighting his battery, wishing again that he had a mathematician or an engineer\u2014some reliable human calculator who could command the tedious business of putting the great rocks on target. Exrech had proven uninterested. And far too slow. Unwilling to _build_ anything.\n\nHe watched the boglins dig, raising a new mound out of range of the new machine on the fortress. He knew this new battery represented a heavy defeat in time and effort.\n\nHe was trying not to acknowledge that he had to go into the debatable ground and destroy the fortress's new machine with his own power. He had no other weapon available with the necessary reach. And he would have to squander power like an angry boy to breach the fortress's millennium-old defences.\n\nThat would leave him weak.\n\nAnd then he felt the shift. He tasted the air\u2014wasted valuable time sending a raven stooping over the walls, and he saw the nimbus of fire on his former apprentice's hands, saw the great engine cranked all the way back, saw\u2014\n\n\u2014nothing.\n\nHis raven was struck by an arrow, and tumbled out of the air.\n\nHe cursed, disoriented by the loss of his connection. Reached for another\u2014\n\n_The fortress's defences were down._\n\nHe stepped out from behind his new siege mound. Raised an arm, and let fly a bolt of pure green lightning.\n\nAnd he laughed.\n\n# Lissen Carak\u2014Harmodius\n\nHarmodius threw a shield in front of the lightning, like a knight making a parry in the tiltyard, and the two castings extinguished each other with a flash of light.\n\nHarmodius stumbled and had to reach for the well of power at his feet. \"Sweet Lord have mercy,\" he mumbled.\n\nOne blow. Thorn could empty him of power in a single blow.\n\n# Lissen Carak, The Lower Town\u2014The Red Knight\n\nThe captain was first out the gate, and Jehannes was on his heels, leading his party of men-at-arms to the right and out of the chapel.\n\nThe nave was full of sleeping boglins.\n\nThe killing began.\n\nHe counted the armoured shapes coursing past him, lost count in the middle, and had to guess.\n\nBut Sauce was true to her promise. She was last.\n\n\"Last out!\" she called, and danced off to the right around the gate.\n\nThe captain slammed the great doors shut, with the key inside.\n\nAs the two doors met, their power meshed, and the gate vanished, leaving a black stone wall behind the altar, only the shape of the two doors burned onto his retinas remaining.\n\nBent and the archers were clearing the nave.\n\nJehannes was already gone over the broken wall.\n\nThe captain began to cut his way to the front of the church.\n\nThorn cast his second levin bolt, and then, without pausing to gather power, he cast a third.\n\n# Lissen Carak\u2014Harmodius\n\nHarmodius's second defence was more refined than his first\u2014a working of his own, weaker than Thorn's but deflective rather than resisting. Thorn's strike bent like a beam of light in a prism and blew a piece of slate the size of a small barn off the side of the ridge.\n\nHis third cover was not quite fast enough\u2014he intended to cast a single line of power like a sword parry\u2014but Thorn's speed left him too late, and he tried to widen his cast, with too little power.\n\nHe still stopped most of it.\n\nThe rest fell on the curtain wall to his left. A section of wooden hoardings twenty paces long burned in a flash, and a section of the wall cracked and fell outward, killing two archers instantly and crushing the two older Lanthorn men to pulp.\n\nHarmodius felt them die.\n\nHis failure made him angry, and anger made him lash out. His riposte was pitiful, small, weak, too late.\n\nIt was also entirely unexpected. Like a slow attack in a sword fight, his flare of anger sailed out into the dark and caught Thorn unprepared.\n\nPain enraged Thorn. It always had.\n\nHe struck back.\n\n# Lissen Carak, The Lower Town\u2014The Red Knight\n\nThe Lower Town square was carpeted in corpses. The captain passed in the chapel doorway looking for his men-at-arms. The archers were spreading out, right and left.\n\n\"On me,\" he said. \"Let's go!\" He ran across the square, and they pounded along behind him.\n\nParties with ladders broke off and headed east, through the rubble.\n\nHe could hear fighting to this left, and more straight ahead. Angelo di Laternum materialized out of the darkness.\n\n\"Ser Jehannes prays your aid,\" he said formally.\n\n\"On me,\" the captain said, and followed the squire. The captain had no time to comment that Jehannes was off course.\n\nA vast burst of light lit the sky, like all the summer lightning ever seen combined in one single burst. The levin flash showed the captain that Squire Angelo was bleeding from the shoulders of his harness; the archers were splashed in red and black and, ahead of him, Jehannes's men-at-arms were caught in the flash, illuminated like a manuscript illustration of knights fighting monsters.\n\n\"Ware!\" the captain shouted. \"Daemons!\"\n\nThe terror struck him like a heavy mall. He set his teeth and pushed himself forward through the terror, and one of the things turned on him with its supernatural speed.\n\nThe captain had supernatural speed, too.\n\nThe daemon's blade met his, so hard that sparks flew from his blade, and he yielded before the creature's awesome strength, rotated his blade around the fulcrum of his armoured wrist, stepped inside its terror and pushed his point into its brain.\n\nIt fell away off his sword, and he was on the next. It turned its head\u2014its beautiful eyes catching his.\n\nThe daemon's taloned hand came up, too fast to block.\n\nHis sword came down.\n\nThe daemon stumbled away, spraying fear the way a skunk sprays scent, and the captain found himself retching. There was blood in his eyes.\n\n_My faceplate is open._\n\n_It got me._\n\nA different fear, colder and heavier, settled on his gut.\n\nBut the daemons were not immortal; their ichor was mixed with the blood of men on the ground and they were retreating. As they began to put distance between them and their foes, the fear abated.\n\nThe captain saw there were fewer than a dozen of the things.\n\nThe archers\u2014frozen in place\u2014suddenly burst into action. The last daemon\u2014the one the captain had wounded\u2014sprouted shafts like a field growing grass.\n\nThe thing turned, its fear welled, and it fell.\n\nJehannes was shouting for his men.\n\n\"Stand!\" called the captain. It sounded like a squeal. But Wilful Murder roared it from behind him. \"Stand!\" he called.\n\nJehannes paused.\n\n\"The tower!\" the captain insisted.\n\n# Lissen Carak\u2014Thorn\n\nThorn's burst of rage fell like a hammer.\n\nHarmodius watched the strike come in, helpless to stop it, a whole heartbeat to see his death wash at him in sickly green radiance.\n\nHe felt the fortress's Hermetic defences go back up, and knew it would never be enough.\n\nThe great works that powered the defence were brilliantly designed\u2014they funnelled what they could, channelled some more, reflected yet more. They were so well artificed that they almost seemed _intelligent._ New practitioners attempted to meet force with force\u2014skilled practitioners knew to meet force with guile, deflecting the opponent's energy like a skilled swordsman. Most static sigils were easily overcome, but this...\n\nIn the moment of his annihilation, Harmodius thought _Who built this?_\n\nThe wards caught, turned, and covered. But there was only so much the ancient sigils could do.\n\nAnd the rest burst through the great wards like a river in flood bursting through a levy.\n\nHe raised a hand.\n\nThe Abbess reached past him, and stopped the overflow of the great spell of wrath just short of their place on the wall. She flung it back down the path of the casting.\n\nShe reached out and put her left hand on his shoulder.\n\n_I know nothing of this sort of war_ she said. _Let me in._\n\n_Through her, he could feel her sisters, singing plainchant in the chapel. Their power did not fuel the Abbess directly. It was far subtler than that._\n\n_Despite the situation, he had to pause to admire the magnificence of the structure. The fortress. The sigils. The sisters, who could maintain the power of the sigils indefinitely, regardless of their individual weakness._\n\n_He wondered, yet again, who made this?_\n\n_Then he gripped her spiritual hand in his own and led her through the great bronze doors of his palace, like a bridegroom leading a bride. \"Welcome,\" he said._\n\n_She was a much younger and less spiritual woman, in the Aethereal. Suddenlyhe had a frisson of memory. Of this same woman dressed for hunting, standing in his master's chamber, tapping her whip on her hand. Trying to get his master to go out riding._\n\n_He dismissed the memory, although here it took on a visible aspect, so that she saw it and smiled. \"He was the worst lover imaginable,\" she said with a sad smile. \"He didn't hunt, didn't ride, wouldn't dance. He was always late, and made many promises he couldn't keep.\" She shrugged. \"I wanted him. And look at the consequences. Some sins do not wash away.\" She spread her arms. \"It is very nice here.\"_\n\n_He flushed with her praise, as if he was a much younger man. Time in the Aethereal had virtually no meaning so he had no sense of urgency. \"Did you ever suspect?\" he asked carefully. \"When he turned?\"_\n\n_The Abbess sat in one of his great leather armchairs. She had riding boots under her voluminous riding skirts, which she crossed over the arm of the chair. \"You know, don't you, that in old age, one doesn't easily adopt positions like this,\" she said happily. \"Ah, to be young.\" She leaned back. \"You must have asked yourself, many times.\"_\n\n_\"I've been largely trapped in his phantasm for many years,\" Harmodius said. \"But yes. I think of it now. All the time.\"_\n\n_\"I only know that in the months before Chevin he discovered something. Something terrible. I badgered him to tell me, and he would smile and tell me that I wasn't ready to understand it.\"_\n\n_Harmodius grimaced. \"He never said as much to me.\"_\n\n_The Abbess nodded. \"But now you know what he knew. I know it too, now.\"_\n\n_There aren't many secrets in the Aethereal._\n\n_\"Yes,\" he said._\n\n_The Abbess shook her head. \"Any servant of the Order of Saint Thomas knows that the green and the gold are the same,\" she said. \"Richard was a fool who saw the world entirely in shades of black and white. He still is. A staggering intellect, a tower of puissance, and no common sense whatsoever.\" She shrugged. \"Enough chatter. My home is being blown to bits. Show me how to use our power to stop him.\"_\n\n_\"Like this,\" he said. \"But it will be more efficient if you pass me power and I cast.\"_\n\n_In a heartbeat\u2014in no time at all, because in the Aethereal, time had so little meaning\u2014they stood on a balcony of his great palace, looking out over the world of solidity._\n\n_In his vision, Thorn stood out like a beacon tagged in green. Harmodius pointed her hand at the thing that had been her lover._\n\n_She flooded Harmodius with power._\n\n_He made fire._\n\n# Lissen Carak\u2014Thorn\n\nFor the first time, Thorn paused to raise a shield. His burst of temper was over, and Harmodius's response had been respectable. No more, but no less.\n\nAnd the fortress's defences were back. He had landed some good blows. But now he was risking himself for nothing. He raised a second shield.\n\nHarmodius's mighty blow rolled away like a child's stick on a knight's armour.\n\nThorn grunted.\n\nIt might have been a laugh.\n\n# Lissen Carak, The Lower Town\u2014The Red Knight\n\nTom's unconscious body took six men to carry and the captain was unwilling to lose the horses that had been left for the Lower Town garrison, so a party of archers cleared the town's upper gate and opened it. The garrison escaped behind the horses, and the sortie went over the walls via ladders.\n\nIt was all going very well, until the daemons struck back.\n\nHis rearguard was slow in forming\u2014understandable, in the conditions\u2014and suddenly three of them were down, dead, and a gleaming monster stood over them with a pair of wickedly curved axes gleaming in the soft spring moonlight. Marcus\u2014Jehannes's valet\u2014and Ser Willem Greville, his armour opened as if he was wearing leather. A third man was face down beside them.\n\nThe fear was like a waft of foul air.\n\nThere were more daemons behind it\u2014fluid and horrible, arresting and beautiful in their movements. And below them, a legion of boglins, irks and men poured into the town they were leaving.\n\nJust like that, the captain was alone.\n\n\"Run, little man,\" the daemon whispered.\n\nThe captain reached _inside and found Prudentia._\n\n_The working was already aligned._\n\n_He opened the door before she could protest\u2014he was so much faster than he had been._\n\n_The green whistled through the crack, a tempest_ \u2014\n\n_\"He can reach you!\"_\n\n_\"He's otherwise engaged,\" the captain told his tutor._\n\n_\"I need to tell you so many things,\" she said._\n\n_He smiled and_ was back in the dark.\n\nHis sword arm was bathed in silver.\n\nThe daemon rotated its two axes, one over each wrist and golden-green light joined the two.\n\n\"You!\" said the daemon. \"Ahh, how I have longed to meet you.\"\n\nThe captain got his blade up into guard, and cast.\n\nThe beam of silver-white light rose into the night like a beacon. And then fell to earth in the centre of the town.\n\n\"Missed,\" hissed the daemon.\n\nThe captain backed away, rapidly.\n\nAbove him on the trail, a crossbow loosed with a snap.\n\nThe daemon grunted as the bolt struck.\n\nLet loose his own spell.\n\nThe captain caught it\u2014marvelling at the ease with which he fielded the blow. In _the Aethereal, his adversary's blow was like the cut of a sword, and he caught it and parried it with a sword of his own power, flicking it away._ And he was back in the solid, because the daemon followed his phantasm immediately with a heavy cut from his right axe.\n\nHe could remember the first time he'd stopped such an attack by Hywel. Had been hit in the next instant because of the sheer pleasure of having accomplished it. Now, as then, he almost died through admiring his own cleverness.\n\nHe passed forward into the attack, his sword at eye level, the Guard of the Window, and the axe fell away harmlessly like rain off a roof.\n\nHe began to cut overhand, his left foot powering forward, and he caught the growth of his opponent's power and he _turned the blow even as it was rising from his adversary's talons._\n\nIn the _solid_ the attack came in, and he drove the power into the stones of the road between them.\n\nThe road exploded, knocking him flat.\n\nWith a high scream the daemon leaped the crater and swung both axes at once.\n\nHe saw Michael step over him and he caught both blows\u2014one on his buckler, one on his long sword. The squire staggered, but the blows fell away.\n\nThe captain was backpedalling from between his squire's knees; using his elbows, steel sabatons scraping the road, he got himself back.\n\nHe rolled to the left, almost falling off the elevated road. The daemon captain was pounding Michael with blow after blow, and the lad was standing his ground, pushing his sword and his buckler up into the blows, deflecting them, using the daemon's strength against it as best he could.\n\nThe other daemons were trying to get around the fight.\n\nThe captain got his feet under him and he cut at the daemon from the side\u2014but the thing parried his blow high with an axe blade\u2014a horrifying display of skill\u2014and flicked his weapon forward. It was all the captain could do to bat the blow aside.\n\nBoth men fell back as the daemon hammered blow after blow, one axe then the other, in an endless rhythm. It might have been predictable, except that it was so _fast._\n\nAnd then, during the moment that the captain had one axe turned on his long sword, and Michael had the other\u2014just for a heartbeat\u2014safely on his buckler\u2014\n\nJehannes punched his pole-axe between them.\n\nThe daemon fell away, folding over the blow. But its armour\u2014or its eldritch skin, or its sigils of power\u2014held.\n\nThe captain stumbled back, and he felt Michael at his shoulder.\n\n\"Let me in,\" Jehannes shouted.\n\nMichael slumped and Jehannes stepped past him.\n\nTwo daemons leaped past their leader, who was just gaining his feet.\n\nFar above them on the fortress, the trebuchet loosed.\n\n_Thump-snack_\n\nThe ballista on the north tower loosed.\n\n_Whack._\n\nThe war engines on the towers of the Bridge Castle loosed.\n\n_Crack!_\n\n_Crack!_\n\nHigh above them, Harmodius leaned out over the wall, hand in hand with the Abbess like lovers, and spread his fingers.\n\n\"Fiat lux,\" he said.\n\nThe Lower Town seemed to explode as a hail of fire fell, a hand of fate that struck buildings flat.\n\nThe daemons were silhouetted in fire. At the back of their company, daemons turned to see what had happened.\n\nThe captain had to fight the vainglorious urge to charge them. He backed another step.\n\nThe two things came at them, and their fear...\n\nWasn't as strong as it had been. Somewhere deep inside, or perhaps above, the fight, the captain had time to smile at the irony. He had lived his entire childhood in fear. He was afraid of so many things.\n\nFamiliarity breeds contempt. He was _used_ to acting while he was afraid.\n\nThe terror projected by the daemons wasn't having any effect on him.\n\nDespite which, it was all he could do to stand his ground, because they remained big, fast and dangerous.\n\nJehannes had a pole-axe. He cut two handed into a blade attack, and his axe-hammer broke the daemon's sword arm. It stumbled back, and he got his haft between the other's legs, and as it stumbled, the captain had all the time he needed to step forward and cut overhand from the garde of the long tail, the sword flashing up, powered by his hips, his arms, his shoulders as he levelled the blow, right to left.\n\nHis blow went under its weapon. Beheaded it.\n\nBeside him Jehannes stepped forward again and rammed pole-axe's spike into the supine daemon, so that it screamed.\n\nThere was a sound very like applause.\n\nThe captain wondered who was watching.\n\nThey were most of the way up the ridge, under the main gate. And still bathed in the silver-white light of his casting. He was breathing hard. His helmet was like a trap over his face, constricting him, the visor was like a hand over his mouth, and he was bathed in sweat.\n\nThe daemons came on again. There were boglins trying to get around them on the left and right, and his archers were shooting with methodical regularity, but he couldn't stop to think about that. They were on him.\n\nThe daemon in front of him swung its axe two handed, and he cut at its hands\u2014its blow turned to a defence, and it's left claw shot out and slammed into his shoulder and he stumbled back in a flash of pain.\n\nHe'd been hit.\n\nAgain.\n\nJehannes threw three fast jabs with his spear point, reversed his pole-arm to bat his opponent's axe out of the way and planted his spike in the daemon\u2014it screamed and fell back, taking the haft with it, planted in its breastbone. Jehannes struggled too long to keep it.\n\nThe captain's adversary swung on Jehannes from the side, catching the knight in the side of the helmet and Jehannes fell.\n\n_He came back for me_ , the captain thought.\n\nHe lunged, his long sword held only by its pommel in his right hand, and raked the point across his opponent's beaked face\u2014an attack of desperation. But the blow landed, and the daemon stumbled off balance. He recovered forward, grabbing the blade near the point, which he rammed into the daemon's scaled thigh, and with that as leverage, he hurled it from the road. It fell away into the darkness.\n\nHe stepped forward again, past Jehannes.\n\nThe one that had spoken jumped forward, shouldering past two of its own kind.\n\n\"I am Thurkan of the Qwethenog,\" it said.\n\nShe hadn't intended to come out onto the wall.\n\nHer place was in the infirmary and wounded men were coming through the gate.\n\nShe told herself that she would only look. Only a moment. People were cheering.\n\nShe ran barefoot through the infirmary's second floor balcony doors, and leaped lightly from the stone balustrade, between a pair of gargoyles that decorated the lower gable ends, and skinned her thigh on the slates as she slid down to the curtain wall. She'd taken this path a thousand times to go out after the nuns blew out the last lights.\n\nShe was a level above the gatehouse. She skidded to a stop when she saw that a section of curtain wall was simply gone, and her left foot hovered over empty space.\n\nBelow her the hillside was bathed in a cruel white light.\n\nWhen she was young, her Outwaller family had called them guardians and worshipped them. North of the wall she had thought they were angels.\n\nNow a mighty one stood on the cobbled road, facing the Red Knight.\n\nHow she hated that substitute for a name. _The Red Knight._\n\nHe looked tired. And heroic.\n\nShe couldn't watch.\n\nShe couldn't look away.\n\nThe guardian struck with two axes, cutting with both at the same time\u2014something a mere man could never hope to do.\n\nHe stepped forward and to the right, and smashed an axe to the ground; the guardian stepped back. She saw it draw power. Guardians were not like men in any way except for their love of beauty. It took in power as if breathing\u2014a natural movement\u2014and then it snapped its working at the knight.\n\nWho turned it. Then he stepped forward, and raised his sword slowly, an elaborate gesture like a salute.\n\nAchieved his guard.\n\nAnd froze.\n\nThe guardian raised its axes.\n\nAnd froze.\n\nTime stopped.\n\nShe couldn't breathe.\n\nWhen one of them moved, it would be over.\n\n# The Ings of the Albin\u2014Ranald Lachlan\n\nDonald came and sat on a rock by Ranald's tiny fire. Half their force was out on picket\u2014the men cooking breakfast spoke in low tones.\n\n\"I've a notion,\" Donald said.\n\nRanald ate a piece of bacon, and raised an eyebrow. He was feeling better. More alive. Ian the Old had made him angry, pissing in the stream where they got drinking water.\n\nYesterday nothing had made him angry, so he savoured that anger as a sign he was alive.\n\nAll those thoughts flitted through his head while he chewed, and then he nodded. \"I thought I smelled smoke,\" he said, and managed a smile\u2014another triumph.\n\nDonald leaned back. \"None of yer sass, now. And you half my age.\" He grinned. \"I think we should push the herd for Albinkirk. It is only twelve leagues, or like enough as makes no difference.\"\n\nRanald was alive enough, and enough of a hillman, to be taken with the boldness of it. \"Right over the same terrain where we fought the Sossag?\" he said. He shrugged.\n\n\"They're gone, Ranald. Nobody's seen dick of them for three days. Not a feather, not a scout, not a bare buttock. It's their way. They don't hold ground.\" Donald leaned forward. \"What's the herd worth at the Inn? A silver penny a head or less? And it's a far longer walk to the Inn than it is to Albinkirk.\"\n\nRanald stared into the flames of his small birch bark fire. He added leaves from a pouch at his belt to his copper cup full of water, stirred honey in, drank it, and gave a quiet thanks to God. His belief in God had suffered\u2014or maybe not. He wasn't entirely sure.\n\n_I was dead._\n\nHard to take. Better not to think of it. Except that, in some horrible way, he could remember the _deadness._ He didn't want to be dead again.\n\nHe sighed. \"Daring,\" he said. But from Albinkirk he could send a messenger to the king. He owed the king that much. More. He sighed.\n\nDonald's eyes sparkled. \"Let's do it.\"\n\nRanald knew that the older man needed to perform a deed of arms if only to justify the fact that he had lived and Hector had died.\n\nBut deep inside, he shared the feeling. And if they could get the herd through\u2014why, then Sarah Lachlan would be rich, and all the little crofters and herders in the Hills would get their shares, and the Death of Hector Lachlan would be a song with a happy ending.\n\nHe drank off the last of his scalding tea, and watched the stream. \"We're loons. And some of the boys may 'decline to accompany us.' \" He gave the last words a distinctive Alban accent.\n\nDonald chortled. \"Good to see you coming back to yourself. Faeries brought my Godmother back from dead\u2014did you know that? Took her months to laugh again, but then she was dead a whole day.\" He shrugged.\n\nRanald gave a little shudder. \"Ouch,\" he muttered.\n\n\"Oh, no. She said that having been dead, life was always sweet.\" He nodded.\n\nRanald was still thinking of that when the herd lumbered into motion, headed west. The boys had muttered about it but none of them turned for home.\n\nFour hours they moved west, down the old drove road through increasingly wooded country. The west slope of the Morean Mountains had been farmed once\u2014grape vines still grew over the new trees, and they passed a dozen farmsteads standing open-roofed and abandoned. None were burned. Men had simply left, one day, and not returned.\n\nRanald had seen it all before. But now he noticed it more.\n\nThat evening, they made camp under the Ings of the Albin. They'd pressed the herd hard and come twenty leagues or better, and the young men were exhausted enough that Donald made up a new duty list, writing slowly and carefully on his wax tablet, making signs for some men and writing the names of others in the old way.\n\nKenneth Holiot was not a bard, but they all knew the boy could play, and that night he sang a few lines to his father's old lyre, and shook his head, and laid down a few more. He was writing the song of the Death of Hector. He knew the death of another Hector, in Archaic, and he had the bit in his teeth\u2014he was going to write the song.\n\nAfter an hour he cursed and went off into the darkness.\n\nRanald cried.\n\nThe other men just let him cry, and when he was cried out, Donald came and put a hand on his shoulder, and then he rolled up in his cloak and went to sleep.\n\n# Lissen Carak\u2014The Red Knight\n\nHe watched his adversary, and waited to die.\n\nHis shoulder was bleeding. His face was bleeding. Jehannes was somewhere half a pace behind him, he didn't dare try to retreat, and for some reason his people seemed to think he wanted this to be a single combat.\n\nWarm blood ran down his side.\n\nThe effort of holding his sword above his head in the Guard of the Window would eventually be too much. He would have to strike, and that would be the end.\n\nIt was faster and stronger than he was. He'd tried attacking, tried thrusting, tried most of his tricks. They all required some advantage\u2014reach, perhaps\u2014that he just didn't have.\n\nThe daemon just stood there, two axes above its head.\n\nAnd then, as sudden as its attack had come, its eyes slipped past him and with a shrug it was _gone._ The air popped as it displaced itself.\n\nHe was damned if he was going to fall over. He stood there looking at an empty road down the hill, and the fires of hell raging in the Lower Town.\n\nHe turned around, and Michael had Jehannes under the armpits and was dragging the knight up the path.\n\nCuddy stood just behind him, with his bow at full draw. Very slowly, the archer let the tension out of the limbs, and the great bow returned to shape. He dropped the arrow back into the quiver at his belt.\n\n\"Sorry, Cap'n,\" he said. \"You wasn't going to win that one.\"\n\nThe captain laughed. He laughed and laughed as they pulled him through the gate and slammed it shut, and Ser Michael lowered the great iron portcullis.\n\nHe slapped Cuddy weakly on the backplate. \"Nor was I,\" he said.\n\nThen Michael had his helmet off his head, and he was sucking in great gouts of fresh, cool air. A dozen men were pulling at his armour.\n\nHe saw the Abbess. Saw Harmodius, who grinned at him.\n\n_Red Knight! Red Knight! Red Knight! Red Knight!_\n\nHe drank it all in for a moment, and then, as his breast and backplate came off, he got to his feet. The men stripping him grinned and backed away, but their grins faltered when they saw how much blood was running down his side.\n\nHe nodded, waved, and ran, unarmed, unaware of his wounds, into the crowd and seemed to vanish. He didn't see Amicia. But he'd felt her there.\n\nHe went to find her.\n\nShe was waiting for him under the apple tree.\n\nShe bit her lip.\n\n\"I won't talk,\" he said cheerfully. \"I\u2014\"\n\nShe pushed him down on the bench with a strong arm, and bent\u2014he hoped, to kiss him. But his hope was cheated. He felt her breath, hot, moist, fraught with magic, on his face and felt his wound heel. She raised her hands like a priest invoking the deity and he saw the power all around her, the well below the tree, the tendrils that connected her to her sisters in the choir and to the Abbess.\n\nShe reached a hand under his arming doublet and her touch was cold as ice. Her hand passed over his chest and his back arched in agony as she touched the edge of his wound\u2014one he hadn't felt.\n\n\"Silly,\" she said. He felt the power go out of her, into his shoulder. For a moment, briefer than a single heartbeat, the pain was infinite. And for that moment, he was her. She was him.\n\nHe lay back. To his shame, a whimper escaped his lips.\n\nShe leaned over him, her hair covering his face. Her lips brushed his. \"Men will die if I stay with you,\" she said.\n\nAnd she was gone.\n\n# Lissen Carak\u2014Michael\n\n_The Siege of Lissen Carak. Day Twelve_\n\n_Last night the watch came and relieved the garrison of the Lower Town. The Red Knight led the watch in person. All the garrison were rescued, but brave knights and men-at-arms were killed and wounded, and the Lower Town, in the end, was lost. The Enemy has limitless creatures._\n\nMichael looked at the parchment and tried to think what he could write. Shook his head, and went to find Kaitlin, who's father had died when the curtain wall fell.\n\nIn the first light, three wyverns came out of the rising sun carrying rocks the size of a man's head in their claws.\n\nThey came in high, and dived almost straight on the trebuchet.\n\nThe watch was just changing and the soldiers were completely unready. The ongoing watch was already tired, the offgoing watch was exhausted, and no one reacted in time.\n\nBefore No Head could even rotate the ballista the first monster's claws opened, and his rock fell\u2014struck the stump of the tower a few paces from the engine, and bounced away with a crack like lightning to fall harmlessly to the hillside below.\n\nThe second wyvern dropped lower, wings folded against his back, but he opened his wings too early, bobbed, and his rock went sailing away to kill one of the hundreds of sheep who were still penned on the ridge.\n\nThe third wyvern was the oldest and the canniest. It swooped off the target Thorn had intended and laid its rock almost gently on the ballista, smashing the engine and throwing No Head off the tower.\n\nThe archer shrieked and grabbed at the gargoyles of the hospital balcony as he fell.\n\nThe wyverns swept away.\n\n# Lissen Carak\u2014The Red Knight\n\nAn hour later the wyverns were back. This time all three imitated the eldest, coming lower along the ridge and rising on the last thermal before the walls of the fortress to unleash their missiles at point blank range.\n\nThis time they were met by a hail of darts, bolts and arrows, loosed from every corner of the courtyard, the towers, and even the hospital balcony.\n\nAll three were hit, and flew away, angry and unsuccessful.\n\nTheir stones knocked a hole in the captain's Commandery, killed two nuns in the hospital, and crushed a war horse and a squire in the stable.\n\nThe captain slept through it.\n\n# Lissen Carak\u2014The Red Knight\n\nHe didn't wake until late afternoon. He awoke in the comfort of his own room, although it felt odd. Air was moving around him.\n\nSomeone had fixed blankets and an old tapestry over a hole the size of a cart. A hole in the wall that went right through to the outside air.\n\nHis little porch was gone, too.\n\nHe got his feet on the floor, and Toby Pardieu had his clothes laid out on the press, and long leather boots over his arm, clean and black.\n\nHis knight's belt was polished, shining like something hermetical.\n\n\"Which the Abbess has invited ye ta' dinner,\" Toby said. \"Master Michael is at his exercises.\"\n\nThe captain groaned as his weight came on his thighs and hips, and just for a moment he had a flash of what old age might be like.\n\n\"Ta semptress ha gi'in me these linens,\" Toby said. He pointed to a basket. \"New, clean, an' pressed. Shirts. Caps. Braes. Two pair black cloth hose.\" Toby pointed at the basket.\n\nThe captain ran his hands over a shirt. The stitches were neat, very small, almost perfectly even but not quite, almost a pattern. The seamstress had used an undyed thread on the glorious new white of the linen\u2014so confident in her skills that the very slight contrast was itself a decoration. A very subtle declaration of skill. Subtle, like the power with which she'd imbued the garments.\n\nHe picked up the shirt. The power was golden\u2014a bright, white gold, the colour of purity. The Sun.\n\nThe shirt didn't burn him, nor did he expect it to. He'd found that out, years ago.\n\nToby interrupted his reverie. \"Wine? Hot cider?\" he asked. He looked at the floor. \"Cider is good,\" he mumbled.\n\n\"Cider. And I'll wear these new things, but with my scarlet cote, Toby. Black is for\u2014\" He sighed. \"Black is for other occasions.\"\n\n\"Sorry, my lord.\" Toby blushed.\n\n\"How could you know? Any word on the wounded? How's Bad Tom?\" He felt the crisp cleanness of the new white shirt. \"I'll have a bath before I dress, if you can arrange it.\"\n\nToby nodded at the challenge. \"Twa shakes of a lamb's tail.\" He vanished. Reappeared. \"Ser Thomas is up and about. An' Ser Jehannes, as well.\"\n\nThe captain heard the boy's footsteps, running. The boy made him smile. Made him feel old.\n\nHe stripped out of his arming clothes. He had had them on for\u2014hmm. Two days now, without rest?\n\nThe shirt was damp and warm and smelled bad. Not like sweat, but like old blood. There was a lot of blood in it. It had a tear, too, all the way down one side.\n\nHe had a mirror, somewhere in his kit. Michael had unpacked his malle and his scrip and the portmanteau he stored in the wagons\u2014he rooted around, vaguely aware that evening was coming, and he wasn't armed.\n\nHe found his bronze mirror in its travelling case, found his razor, and unfolded it from its fancy bronze handle. Looked in the mirror.\n\nHe'd forgotten the wound he'd taken last night. He had a long crease down the left side of his face which was still sweating a little blood. As soon as he looked at it, it started to hurt. It didn't look bad. It merely hurt.\n\nHe shook his head. Felt fuzzy with post-combat shock, and the shock of what he'd just seen in the mirror.\n\nHe tried to look at the wound in his right shoulder. It was a dull ache, and he couldn't locate it, despite the fact that his arming clothes were soaked in blood.\n\nA bit more of a shock, that.\n\nStiff with blood would be more accurate.\n\nHe peeled his braes off. They were stuck to his crotch with blood and sweat, and where his leg met his groin, he had sweat sores. He _stank_.\n\nToby reappeared. \"Which the bath is on its way, m'lord. I told Master Michael and Master Jacques you was awake.\"\n\nJacques came through the door and sniffed.\n\nEven naked, the captain still had authority. \"Toby, take my arming cote out and air it. Give my linens to the laundress and ask her respectfully if they can be saved.\"\n\nJacques was holding one of the new arming caps. \"This is _fine_ work. As good as court.\" He looked at Toby.\n\n\"The tire woman. Mag.\" Toby shrugged. \"She tol' me what the captain had ordered of her. Did I do aught wrong?\"\n\nThe captain shook his head. Jacques smiled. \"I'll go and pay her. And order my own,\" he said. \"You are commanded to dinner with the Abbess,\" Jacques went on. \"As are a number of other worthies. Best dress well and try to behave yourself.\"\n\nThe captain rolled his eyes. After a pause, he said, \"How bad is the wound on my back?\"\n\nJacques looked at the back of his shoulder. \"Healed,\" he said with professional finality.\n\nToby had the arming jacket over his arm.\n\nThe captain snatched at it and held it up.\n\nThe right arm had a slash that ran from just above the underarm voider of chain all the way down to the top of the underarm seam.\n\nJacques gave a sharp noise like a dog's bark.\n\n\"One of the daemons tagged me.\" The captain shrugged. \"I slept... what a sleep!\" Suddenly he picked up the goblet by his bedside.\n\n\"The pretty novice gave me a cordial I was to give you,\" Toby said. He cowered a little.\n\nThe captain found his wallet, a small miracle all by itself, and extracted a silver leopard. He snapped it across the room to young Toby, who scooped it out of the air.\n\n\"I think I owe you a debt of thanks, young Toby,\" he said. \"Now\u2014 _bath.\"_ He scratched himself.\n\nOut in the yard he could see that there were men with swords and bucklers, practising. He walked across the room, and peeled back a corner of the tapestry to gaze out over the fields, the sheepfolds, and the smoking ruin of Lower Town.\n\n\"Wyverns?\" he asked. He was still unbelievably tired.\n\n\"Been pounding us with rocks all day,\" Jacques said cheerfully. \"Gave No Head the fright of his life. Ballista is gone.\"\n\n\"He's moving his engines again,\" the captain said. \"No\u2014he's having boglins dig a new mound, but the engines are still safely out of range.\" The captain found he was scratching things that could not publicly be scratched, not even in front of servants.\n\n\"I need to see Tom, if he's up to it. With the day's reports.\"\n\nThen he squeaked and ripped the coverlet off the bed as two farm girls appeared in the doorway with a tub of steaming water.\n\n\"Coo!\" said the dark-haired one. \"Nothing I ain't seen before.\" She giggled, though, and the other girl blushed, and then they were gone.\n\nBut the water wasn't gone.\n\n\"I'll wash myself, if you don't mind,\" he told Jacques.\n\nJacques nodded. \"You're too old to be bathed.\" He counted the linens in the basket. \"I'll just go pay the lady, eh? And fetch Tom.\"\n\n\"Thanks, Jacques,\" said the captain. The water was hot\u2014nearly boiling hot.\n\nHe got in anyway, hoping to scald some of the dirt and worse away. The captain was sure there was something crawling over him.\n\nHe had just immersed his torso\u2014slowly\u2014when there was a stir behind him.\n\n\"Tom?\" he called.\n\n\"No,\" replied Harmodius.\n\nThe captain wriggled. The water seemed to burn where he had abrasions, and where he had cuts, and where he had sores.\n\nSo pretty much everywhere.\n\nHe realised that his soap\u2014his lovely almond scented soap from Galle\u2014was in his leather portmanteau.\n\nHarmodius came across the room. \"You are stronger,\" he said without preamble. \"I saw you last night. Fast and strong.\"\n\n\"I do your exercises every day,\" the captain admitted. \"And as you said\u2014I try to do everything I can by the arts.\" He shrugged, and the water was delicious. \"When he lets me.\"\n\n\"Our adversary?\" Harmodius nodded.\n\n\"He's camped outside my place of power.\" The captain reached all the way to the well, a long way for him. Thirty paces through rock. But he could feel the power there, now. He reached out, touched it, took a sip, and cast.\n\nThe soap rose, crossed the room, and fell into the bath with a splash.\n\n\"Damn,\" said the captain. Not his soap. The sharpening hones for his razor.\n\nHarmodius grinned. \"Soap? Is it pink?\"\n\n\"Yes,\" said the captain.\n\n\"Still, you are much improved. I know you were well trained, you just have to be less secretive.\" He shrugged. \"An easy thing for me to say.\" He picked up the soap and then held it out of reach.\n\n\"I'd be able to do more if he weren't right outside my door, waiting to come in and rip my soul out,\" said the captain, scratching. \"Soap please?\"\n\nHarmodius looked out from the tapestry. \"Nice new window,\" he said. \"Get your power elsewhere. You know how.\"\n\n\"From the well?\" the captain asked.\n\n\"How about the sun?\" Harmodius asked.\n\n\"I'm a child of the Wild,\" the captain said. \"My mother made me that way.\"\n\nHarmodius wasn't looking at him. He was looking out over the fields. \"Do you trust me, boy?\"\n\nThe captain looked at the tall, proud figure. \"Not really,\" he said. \"Not to give me my soap, anyway.\"\n\nHarmodius barked a laugh. \"Fair enough. Fair enough. Do you trust me as a mentor in Hermeticism?\"\n\nThe captain thought for a long few heartbeats. \"I think so,\" he said.\n\nThe old Magus nodded and ripped the tapestry off its hooks, so that the afternoon sun fell right on the tub. \"Take the soap. With the sun. Do it.\" He held the soap where it could be seen.\n\nThe captain felt the sun against his bare skin like a faint weight. He held up a wet hand, and let the sun lick it.\n\nHe had always liked the sun. Especially in spring.\n\n... _scent of flowers_...\n\nFor a fraction of a heartbeat he'd had it, and then revulsion set in. It was like a gag reflex.\n\nThe soap didn't move.\n\n\"Try harder,\" Harmodius said.\n\n\"You could just give me the soap, and we could do this when I'm dressed.\" The captain felt very much at a disadvantage, naked, wet, hurt and vulnerable.\n\nHarmodius narrowed his eyes. \"Cast.\"\n\nThe captain tried again. He let the sun kiss him. He drank in\u2014\n\nAnd spat up, narrowly avoiding his bath. \"No,\" he said.\n\n\"Better,\" Harmodius said. \"Very good indeed. May I tell you what I admire in you, Captain?\"\n\n\"You're going to try flattery now?\" asked the captain.\n\n\"It's not that you are not afraid of anything, because, as far as I can see, you are afraid of everything.\" Harmodius crossed his arms. \"It's that you overcome that fear every time.\" He nodded. \"Now seize the power of the sun and _cast.\"_\n\nHe let the sun caress him. He felt the power of it, which was rich, like good cheese\u2014thicker than the power of the Wild, and more intense.\n\nAnd then something in his mind slammed shut.\n\n\"Damn it,\" Harmodius said. \"Again.\"\n\nThe captain took a deep breath, and tried again. He could _feel_ the power. And he wanted it. To touch the sun\u2014\n\nTo touch the sun was to be clean.\n\n_I am the child of incest and hate. I was made to be the destroyer. I can never harness the power of the sun._\n\nThe bathwater was warm, and the sun was warm. He pushed his revulsion down, and he reached for it. He thought of riding in the sun. Of horses in the sun. Of Amicia standing in the sun\u2014\n\nJust for a moment, he connected again. The sun falling on his hand was a conductor, and his skin drank in raw power like a sponge.\n\nAnd then he gagged on it again. He coughed, physically, and the soap, halfway across the room, fell to the floor.\n\n\"Ah-HA!\" roared the Magus.\n\n\"I can't do it,\" said the captain.\n\n\"You just did it,\" Harmodius said. He picked up the soap and handed it to the man in the bath. \"There is no limit, boy. There are no rules. You can tap the sun. For a long time, you will resist it\u2014something in you will resist. But by God, boy, you just reached out and tapped the sun in its purist form. I know men who take the sun from water, from the air. Damn few take power straight from the source.\"\n\nHis water was cooling, and the captain began to soap himself.\n\nIt grew cooler, too fast.\n\n\"You bastard,\" the captain said to the Magus.\n\n\"Best do something about it,\" Harmodius said.\n\nThe captain reached out to the well.\n\nHarmodius was there, a tower of blue fire.\n\nHe _went into his palace._\n\n_Don't, said Prudentia. He's waiting._\n\n_\"So he is, said the captain after touching the key hole._\n\n_He could feel_ the bath getting colder. \"You bastard,\" he repeated.\n\nThe sun was all around him, and he reached for it.\n\nAnd nothing much happened.\n\nHe thought of a summer day. But he thought too much and all he saw was sweat and bugs.\n\nAutumn. The colour of pumpkins and standing corn and wheat ready for harvest\u2014so many things golden and orange and ruddy in the setting sun\u2014\n\n_Prudentia laughed aloud. \"Well done, young master!\" she cried._\n\n_\"Pru!\" he said. He was alight with a ruddy gold._\n\n_Without intending it, the windows\u2014the stained glass windows, in the clerestory above the rotating panels\u2014flared to brilliant life. Coloured light fell across the floor._\n\n_\"Son of a bitch,\" he said._\n\n_He pointed to a statue, a panel, a symbol. \"Saint Mary, Herikleitus, Cancer,\" he said._\n\n_The wheels turned. And stopped, with a click._\n\n_Prudentia smiled a solid marble smile. \"Here,\" she said. \"Watch.\"_\n\n_She held up a prism. It took the coloured light, bent it, and sent it as one coherent beam to strike the central panel of cancer._\n\n_Ah!_\n\nThe water was warm. Then warmer. Verging on hot.\n\nHarmodius laughed aloud. \"Well done!\" he said.\n\nThe captain lay back in the bath, tired. Amazed. \"I had help,\" he said, to cover his confusion. \"Magus, that shouldn't have been possible. How is it possible?\"\n\nHarmodius shook his head. \"I have theories. No proof.\" He rubbed his neck. \"I didn't plan to ride out on errantry, two weeks ago. I planned to find some quiet, far from a trap Thorn had set for me. I wanted to perform some experiments.\"\n\n\"Instead, you got the siege.\" The captain was soaping himself shamelessly.\n\n\"I managed a few of my experiments,\" Harmodius said.\n\n\"Like what?\" asked the captain.\n\n\"I got a Wild caster to use sunlight instead,\" Harmodius said smugly. \"I _knew_ you could do it.\"\n\nThe captain shook his head. He ought to be angry. But he felt\u2014\n\nHe felt very powerful, indeed. \"What if you were wrong?\"\n\nHarmodius shrugged. \"It was unlikely. I had reason for my theory in the first place. Besides, I no sooner got here than I found a woman who cast in both colours. Wild and sun. Every time I watch her heal, it is like a miracle.\" He rubbed his hands together in glee. \"Last night I linked with the Abbess,\" he said.\n\n\"You sound like a boy bragging about his first kiss,\" the captain said.\n\nHarmodius laughed. \"You _are_ quick. She used to come around to our rooms\u2014oh, in those days she was the very embodiment of what a woman should be.\" He shook his head. \"It's funny, how you are never too old to be young. But I'm not here to bandy tales of lust and love, lad. The lady has proven what I already suspected. This is going to change the world.\"\n\n\"I like the world fine as it is,\" Tom said from the doorway. \"When you two man-witches are done having your bloody rites, sacrificing babes and eating them or whatever heathen thing you do, I'm ready with the day's muster.\"\n\nThe captain was still lying in the hot water, unmoving. \"Did you come to find me just to experiment on me? Or did you have another motive, Magus?\"\n\n\"Thorn is planning to attack us. Directly.\" The Magus was trying to put the tapestry back over the opening. For a man of such power, he was curiously inept at the task. \"Last night he learned he could overcome our defences. Now he'll come.\"\n\nTom came over, shkk'd him out of the way, and reached the corners out to tug them over heavy iron spikes driven into the end beams of the floor above.\n\n\"Really?\" The captain asked. \"How do you know?\"\n\nHarmodius shrugged and poured himself some wine. \"We are linked to each other, for good or ill. I can feel his fear. And his anger, and his gloating. As can the Abbess.\"\n\n\"Fear?\" Tom asked. \"Fear? Yon mighty godling is afraid o' we?\" He laughed.\n\nBut the captain understood. \"He must be afraid,\" he said. \"I would be.\"\n\n\"He has a great deal to lose,\" Harmodius said. \"But he knows he can destroy our trebuchet with one shot if he gets close enough. Of course, he has to risk himself on the plain to get it, hence his attempt to get it done with the wyverns. But they've failed.\"\n\nTom shook his head. \"You make him sound like he's but an engine himself.\"\n\nHarmodius bobbed his head. \"Not bad, Tom. In a way, the magi aren't much more than siege engines, on a battlefield. Except we move much faster and we are much deadlier. But I agree, the effect is the same.\"\n\nThe captain made a face. \"Why must he get the trebuchet? So he can move his engines against the Bridge Castle?\"\n\nHarmodius nodded. \"I suppose so. That's not my department.\" He put his wine cup down. \"I'll leave you to get ready. The Abbess asked us for sunset.\" He paused in the doorway. \"Don't stop practising, young man. We need you.\"\n\nTom watched him go. \"He's an odd one and no mistake.\"\n\nThe captain smiled. \"This from you?\" He summoned a linen towel from the door. It flew to his hand. He grinned and rose, dripping.\n\nTom rocked back in his seat. \"Don't do that again,\" he said. He had his heavy knife half out of its sheath. \"I'll thank ye to keep that sort o' thing private, where it belongs.\"\n\nThe captain felt himself blush. \"I can cast magic, Tom,\" he said. \"You know I can.\"\n\nTom grunted. \"Knowing and watching is different beasts.\" He shrugged and looked uncomfortable. \"We lost five men-at-arms yesterday and three archers.\" He looked at a wax tablet. \"Nine men-at-arms and nineteen archers since the siege began. \"Twenty-eight, and two valets is thirty.\" He shrugged. \"One man in four.\"\n\nThe captain got his shirt over his head.\n\n\"I'm not saying we should quit,\" Tom said. \"But it may be time to see if we can make a deal.\"\n\n\"You, too, Tom?\" The captain got into his braes. They felt clean and crisp. He felt clean and crisp too. And very tired.\n\n\"We're losing 'em faster every day,\" Tom said. \"Listen. I'm your man. You're a fine captain, and even Jehannes is coming around to that.\" He shrugged. \"But this ain't what we do, lad. One monster; sure. An army of of them?\" He frowned.\n\nThe captain sat on his cot and reached for his new hose. They were rich black wool\u2014a trifle coarse and itchy, but heavy, warm, and stretchy. He took one and pulled it carefully up his right leg.\n\n\"We're not losing,\" he said.\n\n\"As to that...\" Tom said.\n\n\"We're going to hold here until the king comes.\" He grabbed the second leg.\n\n\"What if he's not coming?\" Tom leaned forward. \"What if your messengers didn't get through?\"\n\n\"What if pigs fly?\" the captain said. \"I know the owners of this fortress were notified. I saw it, Tom. The Knights of Saint Thomas will not let this convent\u2014the base of their wealth, the sacred trust of the old king\u2014they will not let it fall. Nor will the king.\"\n\nTom shrugged. \"We could all die here.\"\n\nThe captain started rooting through his clothes for a clean doublet, or at least one without a noticeable smell.\n\nThe one he found was made of fustian and two layers of heavy linen, rumpled but completely clean. He began to lace his hose to it.\n\n\"We may all die here,\" the captain admitted. \"But damn it, Tom, this is worth doing. This isn't some petty border squabble in Galle. This is the North Land of Alba. You're from the Hills. I'm from the Adnacrags.\" He raised his arms. \"These people need us.\"\n\nTom nodded, obviously unmoved by the needs of the peoples of the north. \"You really think the king will come, eh?\"\n\n\"One day's time. Perhaps two,\" the captain said.\n\nTom chewed his moustache. \"Can I tell the lads that? It will help their morale... only once I tell them, that's all the time you get. M'lord.\"\n\n\"Is this an ultimatum, Ser Thomas?\" the captain stood up straight, as if that would make it better. \"Are you telling me that in two days, my troops will demand that I look for another solution?\"\n\nBad Tom sneered. \"Like enough there's some as would. And more every day after that. Yes.\" He stood. Six feet and six inches of muscle. \"Don't you go and mistake me, Captain. I like a fight. I don't really care who brings it. I could fight here forever.\" He shrugged. \"But there's some as can't.\"\n\n\"And they might want to quit,\" the captain said, with a feeling of relief.\n\n\"They might,\" Tom said. He grinned. \"I swear, there's something in the air, like a poison today. Lads are touchy. Every comment has an edge.\"\n\nThe Red Knight took his scarlet cote off the stool and began to lace it. \"I've felt it.\"\n\nTom shook his head. \"I hate your magery. Takes all the sport out of a fight.\" He shrugged his great shoulders. \"I don't so much mind dying, so long as I go down my way. I like a good fight. An' if it's to be my last, well, all I ask is it be good.\" He nodded. \"Good enough for a song.\"\n\nThe captain nodded. \"I'll see what I can do,\" he said.\n\n\"I'll tell the lads,\" Tom said.\n\nAs soon as he passed the door, Michael and Toby came back. His scarlet jupon was brushed, and he saw that the embroidered lacs d'or on the front were repaired.\n\nMichael helped him into it. They each laced a wrist while he stood, thinking.\n\nHe thought more while he pulled on his long boots. Toby did his garters and Michael held his cote.\n\nToby brushed his hair and got the water out of his beard. Michael brought out his riding sword.\n\n\"War sword,\" said the captain. \"Just in case.\"\n\nMichael shortened the belt and buckled it at his waist, and then stood back while the captain drew it three times, testing the hang of the belt. Toby buckled his spurs on. Michael held the heavy gold belt with a questioning air.\n\nThe Red Knight smiled. \"Why not?\" he asked.\n\nMichael buckled it around his waist, handed him his hat, gloves, and baton. \"You'll be early,\" he said, \"but not by much.\"\n\nThe captain walked down the steps to the courtyard. Men and women looked at him\u2014clean, and, although he couldn't see it, glowing.\n\nHe walked across the yard, nodding to all. He stopped to compliment young Daniel on his swordplay; to share a jibe with Ben Carter, and to tell the younger Lanthorn girl that he was sorry for her loss, as both of her parents had died in the night. She rose to give him a curtsy, and he smiled when he saw her eyes slide off him to Michael, who was following him.\n\nHe heard the tale of No Head's near death experience told by a circle of archers who slapped their booted thighs in merriment, and he listened to a complaint that someone was stealing grain from Ser Adrian, who also handed him a piece of parchment rolled very tight.\n\n\"As you asked,\" the clerk said. \"I've spoken to a dozen sisters and some of the farmers.\" He shrugged. \"If you want my opinion, Captain\u2014\" He let the words trail off.\n\nThe captain shook his head. \"I don't,\" he said. He smiled to take the sting out. Tucked the scroll into his cote sleeve and bowed. \"I have an appointment with a lady,\" he said.\n\nSer Adrian returned his bow. \"Count your fingers after you eat,\" he said softly.\n\nThere was a long table, set for thirteen. In the centre was the Abbess's throne, and he sat on her right hand. The table was empty as he was the first to arrive. He went and exchanged glares with Parcival, on his perch and was suffered, with incredible grumpiness, to stroke the bird's head.\n\nA sister came in, saw him, and gave an undignified squeak. He turned, bowed, and smiled. \"Your pardon, sister. A glass of wine, if I might?\"\n\nShe departed.\n\nHe walked over to the _Lives of the Saints._ Now that he knew its secret, he was far more interested and only lack of time had kept him from it. It was so obvious now\u2014a Hermetical Grimmoire. He turned the pages, deciphering them roughly. _Know this one. Know this one. Hmm. Never even heard of this one._\n\nIt was, quite literally, an awe-inspiring tome. Which was sitting in the open, under a window, in a fortress.\n\nHe scratched under his beard.\n\n_Say that every woman here is like Amicia,_ he thought. _And the Order sends them here. To be safe? And to keep them out of common knowledge. Why else_ \u2014\n\nShe was standing at his side. He could smell her\u2014her warmth. And he could feel the golden power on her skin.\n\n\"You,\" she said.\n\nHe turned. He wanted to take her in his arms. It was like hunger.\n\n\"You have come to God!\" she said.\n\nHe felt a flare of anger. \"No,\" he said. \"Nothing like.\"\n\n\"I can feel it!\" she said. \"Why would you deny it? You have felt the power of the sun!\"\n\n\"I tell you again, Amicia,\" he insisted. \"I don't deny God. I merely defy Him.\"\n\n\"Must we argue?\" she asked. She looked at his face. \"Did I heal you?\"\n\n\"You did,\" he said, far more rudely than he meant.\n\n\"You were bleeding out,\" she said, finally moved to anger. \"You scared me. I didn't have time to think about it.\"\n\n_Oh._ He raised a hand. \"I thank you, mistress. Why must we always spar? Of course. Is it the cut on my face you worry about? I scarcely feel it.\"\n\nShe licked her thumb, like a mother removing dirt from her child. \"Don't flinch,\" she said, and wiped her thumb down the wound. There was a flare of intense pain, and then\u2014\n\n\"You should pray when you cast, Amicia,\" said the Abbess from the doorway.\n\nThe captain took a step back from the novice. They had been very close indeed.\n\n\"We are none of us without sin, without need of guidance. A prayer concentrates the mind and spirit. And sometimes His hand is on our shoulders, and His breath stirs our hearts.\" The Abbess advanced on them.\n\n\"Although, in the main, God seems to help those who help themselves,\" said the Red Knight.\n\n\"So easy to mock, Captain. I gather you have tasted the sun. And yet you feel nothing?\" The Abbess tapped the floor with her staff, and two novices helped her onto her throne.\n\n\"It is, after all, just power,\" Harmodius said from the doorway.\n\nThe Abbess nodded at the Magus in greeting. \"There are more things on heaven and earth, Magus.\"\n\n\"So easy to mock,\" Harmodius said. \"And yet\u2014as a seeker after _sophia_ , I confess that when I look inside you, lady, I see something greater than myself. In you and in the Queen.\" He nodded. \"Perhaps, in this novice too.\" He shrugged. \"And in Thorn.\"\n\n\"Name him not!\" said the Abbess, striking the floor.\n\nSer Jehannes came in. With him came Ser Thomas, and the Bailli, Johne, and Mag the seamstress, of all people.\n\nSister Miram sat quietly and with immense dignity, next to Ser Thomas. He grinned at her. Father Henry sat the far right of the table.\n\nSer Milus arrived late, with Master Random and Gelfred from the Bridge Castle.\n\n\"You took a risk,\" the captain said, looking at the Abbess.\n\nShe met his gaze mildly enough. \"They came through your trench, Captain, and through the tunnels. This hill has many rooms and many doors.\"\n\n\"Like your father's house?\" asked the captain.\n\nThe Abbess's look suggested that he wasn't as witty as he wanted to be.\n\n\"And many secrets,\" Harmodius said. \"We are thirteen.\"\n\n\"The number of Hermeticism,\" said the Abbess.\n\n\"Jesus and his disciples,\" Harmodius added.\n\nThe captain gave a lopsided smile. \"Which of us, I wonder, is Judas?\"\n\nThe men at the table gave a nervous laugh. None of the women laughed at all.\n\nThe Abbess looked up and down the table, and they fell silent. \"We are here for a council of war,\" she said. \"Captain?\"\n\nHe rose and stretched a little, still feeling strong. A curious feeling, for him. \"I didn't summon a council of war,\" he said. \"So what do you wish of me?\"\n\n\"A report,\" she snapped. \"How are we doing?\"\n\nHe was being told to mind his manners. Amicia was glaring at him, and Jehannes, too. He thought of Jacques's admonition to be on his best behaviour. Jacques seldom said such things by chance.\n\n\"We're not losing.\" He shrugged. \"In this case, that constitutes winning.\"\n\nJehannes looked away and looked back.\n\n\"Your own men disagree with you, Captain,\" the Abbess said.\n\n\"That's an internal matter,\" the captain said.\n\n\"No, Captain. It is not.\" The Abbess tapped the floor with her staff.\n\nThe captain took a deep breath, looking around to pick up social cues from the audience as he had been taught.\n\nAmicia was very tense. The Abbess gave nothing away, nor did Harmodius, although their blankness contrasted\u2014his a studied indifference, hers an apparently angry attentiveness. Father Henry was nervous and upset. Mag was _willing_ him to do well. To deliver good news. Johne the Bailli was too tired to listen well.\n\nTom was trying to look down Amicia's dress; Jehannes was on the edge of his seat; Master Random was sitting back with his arms crossed, but his whole attention was on the captain.\n\nSer Milus was trying not to go to sleep.\n\nThe captain nodded.\n\n\"Very well, lady. Here it is.\" He took a steadying breath. \"This fortress is ancient, and contains a powerful Hermetic source that is of equal value to magisters of all species. This fortress and the people in it are an affront to the Wild. Events\u2014a slow progression of events that recently reached a crescendo, and include the advent of this company\u2014forced the hand of certain powers of the Wild. And now, the Wild has come to take the fortress.\" He paused.\n\n\"Take it back,\" he said, slowly, for dramatic effect.\n\nEven the Abbess was startled.\n\n\"It was theirs,\" the captain said, in a quiet, reasonable voice. \"They built the well. They carved the tunnels.\" He looked around. \"We took it in a night of fire and sorcery,\" he picked up his wine cup, \"two hundred years ago, I'll guess. And now the Wild is back, because the lines are shifting and things fall apart, and now we're weaker than we were.\"\n\n\"Alba?\" asked Jehannes.\n\n\"Humanity,\" the captain said. \"That's all just background. But it is important, because I have puzzled again and again over why the enemy is taking casualties and engaging us here. It _is_ costing them. Jehannes, how many of the enemy have we killed?\"\n\nJehannes shook his head. \"Many,\" he said.\n\n\"So many that I can only wish I'd signed the Abbess to a per-creature contract,\" the captain said. \"In fact, I was suckered into this contract. My youth was taken advantage of.\" He smiled. \"But never mind that. The enemy has lost several dozen irreplaceable minor powers, as well as hundreds\u2014perhaps even thousands\u2014of the small inhabitants of the High Wilderness. We have lost twenty-seven local people, seven sisters, three novices, and thirty of my soldiers. We have lost all the farms, and all of the animals not penned within the fortress. We have lost the Lower Town.\" He spread his hands and leaned onto the table. \"But we have not lost the fortress. Nor the bridge. Most important of all, we have not _lost.\"_\n\n\"Lost what?\" asked the Abbess.\n\nThe captain shrugged. \"It's spiritual. A matter of faith, if you like. Our enemy depends on success as much as on displays of power to hold his place. It is the way of the Wild. Red in tooth and claw. Wolf eat wolf. Every tiny defeat we hand him, every bee sting, causes his allies to wonder\u2014is he as strong as he seems?\"\n\nThe Abbess nodded. \"Can we win?\" she asked.\n\nHe nodded decisively. \"We can.\"\n\n\"How?\" she asked.\n\nThe captain crossed his arms and leaned against the mantelpiece. \"By hurting him so badly that his allies think he is weak.\"\n\nHarmodius shook his head. \"None of us can take him, lad.\"\n\n\"He's not that bright,\" the captain said. \"I think that all of us, working together, can take him.\"\n\nHarmodius rose. \"You're out of your depth,\" he said. \"He's more powerful than you can imagine. And even if you hurt him\u2014\" He paused, obviously a man on the verge of saying too much.\n\nThe captain sipped wine. \"I've seen him retreat twice now.\"\n\nHarmodius spread his arms. \"I admit he's cautious.\"\n\n\"If his people see him run from us, surely that's enough.\" The captain looked at the Magus. \"Isn't it?\"\n\nThe Abbess slammed her stick on the floor. \"Captain. Magus. Surely you don't believe that we have to raise the siege ourselves?\" She looked at the captain. \"Don't you believe that the Prior is coming? The king?\"\n\nHarmodius didn't turn to face her. \"The king\u2014\" he said. He shrugged.\n\nThe captain smiled at her. \"Lady, I believe the king is a day or two away. But I believe that the essence of a good defence\u2014whether my opponent is a tribe of barbarians, a feudal lord, or a legendary mage, is a good offence planned to keep my opponent off balance. Let me tell you of the next two days.\" He grimaced\u2014for the first time, the others saw the fatigue under his banter. \"Let me guess at the next two days,\" he said.\n\n\"Tonight, the enemy will cross the fields in force, and endeavour to cut us off from Bridge Castle in two ways. He'll try to occupy the trench we built, and he'll seek to destroy our engines.\" He looked at Harmodius. \"He'll try it directly. With powerful workings overloading the Hermetical defences of the walls.\"\n\nHarmodius nodded emphatically.\n\n\"His purpose is so that he can storm Bridge Castle. He is only interested in taking it now because the king is on the south bank of the Cohocton. As long as we hold the Bridge, we have the ability to end the siege in an afternoon.\"\n\n\"You don't know that,\" Jehannes said.\n\n\"Sometimes,\" the captain said, looking at the Magus, \"You know a thing to be true, whatever the evidence. Our enemy is not that good at war. In fact, he's learning to lay a siege _from_ us, as we hurt him. He learned, perhaps three days ago, that the king was coming along the south bank. I'm guessing based on the tempo of his attacks.\" He shrugged.\n\nJehannes shook his head. \"If you are wrong\u2014\"\n\nThe captain slammed his fist on the table. \"When, exactly, have I been wrong? I've done a pretty damn good job here, and we've gone from victory to victory\u2014even when we stumble. We're still standing, at odds of twenty to one.\" He looked around. \"Our magazines are full. Our casualties are acceptable. At this rate, if the worst happens,\" he realised he was growing too angry to sway them but his words were tumbling out, \"then we'll lose the siege engines tonight, but it will be four more days before he storms the Bridge Castle, it will cost him a thousand creatures to take it. And he _still won't have a chance to take this fortress!\"_\n\nSer Milus snorted. \"I think you just condemned my garrison to death.\"\n\nThe captain shrugged. \"I'll go and command the Bridge Castle and you can command here. This is war. We are not losing. Why are any of you considering surrender?\"\n\nJehannes swallowed heavily.\n\n\"Speak!\" the captain insisted. \"Why are you all so silent?\"\n\nAmicia said quietly, \"Your eyes are glowing red.\"\n\nThe Abbess snorted. \"Every young man would have glowing red eyes, if they only could.\" She got up. \"But I agree with you, wholeheartedly, Captain. We will have no more talk of truce, surrender, or accommodation. The Wild will kill us if they penetrate these walls.\" She raised her staff. She appeared to grow. Not taller, nor more beautiful, nor younger, and yet, in that moment, she was greater than any of them.\n\n\"Do not be weak, my friends.\" She smiled, and her smile had the warmth of the sun. \"We are strongest, we mere humans, when we unite. Together we can resist. As individuals\u2014we are no stronger than our weakest.\"\n\nShe diminished, and sat.\n\nHarmodius sat silent.\n\nSer Milus leaned forward. \"Captain,\" he said.\n\n\"Aye, messire?\"\n\n\"I agree. He'll go for us next. Bolster the garrison. Give me fresh troops and more men-at-arms and I'll hold it a week.\" He nodded.\n\nThe captain subsided into his seat. \"Excellent thought. Take them tonight, when you go back\u2014as soon as ever you can.\"\n\nHarmodius shook his head. \"I still think he is too intelligent for all of us, even if we could all cast in concert.\" He rolled his shoulders like a North Country wrestler preparing for a match. \"But I'm game. And I admit that the captain has a point. We don't have to defeat him, only make it look as if he can be beaten.\"\n\nThe Abbess smiled. \"Well said. This is the kind of company I love. Let dinner be served.\"\n\nThe dinner was not rich. There was no roast swan, no peacocks with gilded beaks, no larks' tongues. Duels between torsion engines had killed a dozen sheep on the ridge so every mess in the fortress was eating mutton, and they were no exception.\n\nThe venison sausage was superb, though, and the wines were as ancient as human possession of the fortress.\n\nThe conversation was slow to start but by the second cup of wine, Mag was amused by Tom's ribald story, and Johne the Bailli roared with laughter at the tale of the student and the hornsmith's wife. He told one of his own, about a bad priest who disgraced his vows, and Father Henry glared.\n\nThe Abbess passed wine. She had the captain on her right, and Amicia on her left. When the talk had become general, she turned to the captain. \"You have my permission to engage her in conversation,\" she said.\n\nThe captain tried to smile. \"I'm not sure my eyes aren't still glowing,\" he said.\n\n\"Anger and lust are different sins,\" said the Abbess. \"Amicia is going to take holy orders, Captain. You should congratulate her.\"\n\n\"She has my fullest congratulations. She will make a remarkable nun, and in time, I expect she will make a remarkable Abbess.\" He sipped his wine.\n\n\"She is not for you,\" the Abbess said, but without rancour.\n\n\"So you keep telling me, while dangling her like a tourney prize.\" He took a bite of meat. His tension was only visible in the force he used to cut the mutton.\n\n\"I'm right here,\" Amicia said.\n\nHe smiled at her.\n\n\"Once again, you bite her with your eyes.\" The Abbess shook her head.\n\nAfter dinner, the Abbess held the magi back. Mag was surprised to be invited. \"My working is very slow,\" she said. \"I never even know\u2014\" She shrugged.\n\nAmicia put a hand on the seamstress's shoulder. \"I can feel every stitch you sew,\" she said.\n\nHarmodius snorted. \"You share a mixing of gold and green,\" he said. \"I should have come to this place years ago to have all my notions of Hermetics shattered.\n\nThe Abbess said, \"It is my will that we should stand in a circle, and link.\"\n\nHarmodius winced. \"I'm granting my secrets to every woman in the room!\"\n\n\"You have little time for mere women,\" Amicia snapped at him. \"We're too patient in our castings, are we not?\"\n\n\"Women are all very well for healing,\" Harmodius said.\n\nAmicia raised her head, and a sphere of golden green sat in it. She projected it to a point roughly halfway between herself and Harmodius.\n\n\"Try me,\" she said.\n\nThe captain was surprised by her vehemence.\n\nThe Abbess, on the other hand, merely smiled a cat's smile.\n\nHarmodius shrugged and slapped at the sphere with a fist of phantasm.\n\nIt moved the width of a finger.\n\nThen it shot across the room at Harmodius. He caught it, struggled with it, and it began to move\u2014slowly, but without pause\u2014back.\n\n\"Of course he is stronger than you,\" the Abbess said, and she extinguished the globe with a snap of her fingers. \"But not as much stronger as he would have expected. Eh, Magus?\"\n\nHarmodius took a deep breath. \"You are most powerful, sister.\"\n\nThe captain grinned. \"Let us link. I reserve some memories. But my tutor taught me to hold some walls while opening other doors.\"\n\n\"I give a great deal for very little gain,\" Harmodius said. \"Bah\u2014and yet, the Abbess is right. I am not an island.\" He extended his hand to Amicia.\n\nShe took it graciously. They took hands around the circle, like children in a game.\n\n\"Captain, I intend to pray. Try not to vanish in a puff of smoke,\" said the Abbess.\n\nShe began the Lord's Prayer.\n\n_Prudentia was standing at the door. \"If you were having guests, you might have asked me to sweep up,\" she said._\n\n_The Abbess appeared in his hall. She was young, voluptuous in a tall, thin way, with an earthy power to her face that belied her spirit._\n\n_Amicia was elfin and green._\n\n_Harmodius was young and strong, hale\u2014a knight on errantry, with a halo of gold._\n\n_Miram was shining like a statue of polished bronze._\n\n_Mag looked just like herself._\n\n_He was at once in his place of power, and simultaneously in Amicia's, standing on her beautiful bridge. He sat in a comfortable leather armchair in a great tiled room\u2014that had to be Harmodius\u2014surrounded by chess boards and wheels with wheels. He stood in a chapel surrounded by statues of knights and their ladies\u2014or, as he realised, ladies and their knights, each with a golden chain attaching them. A chapel of courtly love\u2014surely the lady's place of power. He knelt before a plain stone altar with a cup of red blood on it. Miram's place of power._\n\n_He stood in the Abbess's hall, and there was a needle in his hand. Mag's place of power was external\u2014in that moment, he understood how very powerful her making was, because where the rest of them worked the aether, she worked the solid._\n\n_There was a glow or health, of vitality, of goodness, of power. And no time at all._\n\n_He knew many things, and many things of his were learned._\n\n_They made their plan._\n\n_And then, like the end of a kiss,_ he was himself.\n\nHe sagged away from them, tired from the length of the link. Other perspectives were haunting, exhausting\u2014he could see, as quickly as Harmodius had, how a sisterhood of dedicated nuns was the ideal basis for a choir of Hermeticists, because they learned and practised discipline\u2014 _together._\n\nHarmodius was stroking his beard. \"You are taking all the risk, lad,\" he said aloud.\n\nThe captain gave them all a lop-sided grin. \"A single, perfect sacrifice,\" he said.\n\nThe Abbess rolled her eyes. \"Sometimes your blasphemy is just banal,\" she said. \"Try not to die. We're all quite fond of you.\"\n\nAmicia met his eye and smiled at him, and he returned her smile.\n\n\"I have many things to prepare,\" he said. He bowed to the company, and went out into the night.\n\nFirst he walked to the northern tower and climbed the steps to the second floor. He climbed softly, his black leather boots and smooth leather soles giving nothing away. The card players were attuned to the sound of sabatons.\n\nBad Tom was playing piquet.\n\n\"A word,\" he said.\n\nTom raised his head, pursed his lips, and put his cards face down with a start. \"I can leave cards like this any time,\" he said, a little too carefully.\n\nBent was hiding something under his hand.\n\nGiven the circumstances, the captain didn't think he needed to care.\n\nBent shrugged. \"They'll be the same when you come back,\" he said.\n\n\"Better be,\" Tom said. He followed the captain out onto the garrison room's balcony over the courtyard. \"My lord?\" the big man asked, formally.\n\n\"I'm going for a ride tonight, Tom,\" the captain said quietly. \"Out into the enemy. I'd like you to come.\"\n\n\"I'm your man,\" Tom said cheerfully.\n\n\"We're going to try and take him,\" the captain said. He made a sign with his fingers, like antlers or branches growing from his head.\n\nTom eyes widened\u2014just a hair. Then he laughed. \"That's a mad jest,\" he said. \"Oh, the pleasure of it!\"\n\n\"Forget the watch bill. I want the best. Pick me twenty men-at-arms,\" the captain said.\n\n\" 'Bout all we have on their feet,\" Tom said. \"I'll get it done.\"\n\n\"Full dark. You will have to cover me when I\u2014Tom, you know that I will have to use power?\" the captain said.\n\nTom grinned. \"I guess.\" He turned his head away. \"Everyone says you used power against the daemons.\"\n\nThe captain nodded. \"True. If I have to cast, I need you to cover me. I can't fight and cast.\" Then he grinned. \"Well. I can't fight and cast well.\"\n\nTom nodded. \"I'm your man. But\u2014in the dark? After yon horned loon? We need to bring a minstrel.\"\n\nThe captain was lost by the change of subject. \"A minstrel?\"\n\n\"Someone to record it all, Captain.\" Bad Tom looked off into the dark. \"Because we're going to make a song.\"\n\nThe captain didn't quite know what to make of that. So he slapped the big man on the shoulder.\n\nTom caught his arm. \"You can't be thinkin' we can take him with steel.\"\n\nThe captain lowered his voice. \"No, Tom. I don't think so, but I'm going to try, anyway.\"\n\nTom nodded. \"So we're the bait, then?\"\n\nThe captain looked grim. \"You are a little too quick, my friend.\"\n\nTom nodded. \"When there's death in the air, I can see through a brick wall.\"\n\n# Near Lissen Carak\u2014Thorn\n\nThorn had everything he needed to proceed. He'd built his two most powerful phantasms in advance, storing them carefully in living things he'd designed just to store such things\u2014pale limpets that clung like naked slugs to his mossy stone carapace.\n\nHe didn't bother to curse the wyverns who had failed him. It had been, at best, a long shot.\n\nBut now it was down to him, and he didn't want to do it.\n\nHe didn't want to weaken himself by taking on the fortress directly.\n\nHe didn't want to expose himself to direct assaults from his apprentice and the dark sun. However puny, they were not unskilled or incapable.\n\nHe didn't want to fight with _her._ Although his reason told him that when he killed her, he would be much stronger for it. His link to her was a link to his past life. A weakness.\n\nHe didn't want to do this at all. Because win or lose, he'd engaged forces that forced his hand. Made him grow in power. In _visibility._\n\nDamn them all, the useless daemons most of all. It was _their_ fortress, and they were all busy watching him to see if they could bring him down, instead of helping.\n\nAnd Thurkan had failed to take the dark sun.\n\nThorn was not without doubt. In fact, he was full of doubt, and again, for the hundredth time since the siege began, he considered taking his great staff and walking off into the Wild.\n\nBut without him the Wild might fail. And that would be catastrophic. At best it would be fatal for his long term plans.\n\nHe extended his hands, and power flowed smoothly. A cloud of faeries began to gather, so great was the power concentrated in a few yards of air.\n\nHe tried to imagine what it would be like when she was dead. He would miss her. She had once been the standard by which he measured himself. But that self was largely gone, and it was time he did without her.\n\nAnd the apprentice. _It is a weakness, to miss the company of men._\n\nThe Wild had to win. Men were like lice, undermining the health of the Wild.\n\nIt was time to act, and he could imagine all of his actions, a fugue of them extending back to his earliest conscious thoughts, culminating here.\n\nHe surfaced from the tide of his thoughts and looked around, unhampered by the darkness. He looked at Exrech. \"Your people must storm the trench,\" he said. \"And hold it. By holding it, we separate the fortress from the Bridge Castle.\"\n\n\"And then we dig,\" Exrech said.\n\nThorn bowed assent. To Thurkan he said, \"The dark sun will come for me.\"\n\n\"We will lay in ambush for him,\" the daemon promised.\n\nThorn looked at the trolls\u2014mighty creatures which he suspected had been created in the distant past by magi. As bodyguards. He had now acquired two dozen of them, as was the way when one became a power. He was like a beacon, and so they came. He no longer saw them as horrible. Instead, he saw them as beautiful, the way a craftsman views his perfect chisel, the one that fits his hand as if made for it.\n\nThorn tapped his great staff on the ground. \"Go,\" he told his captains.\n\n# Lissen Carak\u2014The Abbess\n\nThe Abbess felt the spells he cast. She had lain down to rest, but it was happening sooner that she expected and she sat up, her mind reaching for the threads of power that bound her to her stone.\n\nShe felt him, in the darkness out there, planning the ruin of her home, and she narrowed her eyes and reached down the link they would always share.\n\n_Traitor!_ she said. She flung the word with a woman's contempt.\n\n_Sophia!_ He cried into the Aether.\n\nShe hurled her defiance at him and she felt her venom strike home, and in the moment of his startlement she read him, and saw that he had a trap prepared\u2014that she had a traitor in her midst, as she had long suspected.\n\nThen she was running, her bare feet slapping the stone floor, her unbound hair trailing behind her like the tail of a comet, running for the courtyard.\n\nShe felt him respond, and she had her defences up. She felt his come up\u2014slowly, but when raised, as strong as a wall of iron. She couldn't even sense him through them, merely that he must be behind that veil. She prayed as she ran\u2014prayed for his ruin.\n\nThe young captain was standing by his destrier in the courtyard, with twenty knights behind him.\n\n\"You cannot go out there!\" she screamed. \"He is waiting for you! It is a trap!\"\n\nThe captain gave her an odd smile, and waved to Michael, who had his bascinet. \"He's coming already, is he?\" he said to her. He turned to his knights. \"Mount!\" he shouted.\n\nShe grabbed his bridle, and his great war horse\u2014quick as lightning\u2014bit at her hand, and only his instant reaction saved her. The Red Knight slapped his hand at Grendel's neck, and the war horse took one step, and tossed his head, as if to say \"could have, if I really wanted.\"\n\n\"He is coming _now_ \u2014\"\n\nHis squire placed his helmet on his head, and pulled the chain of his aventail down over his cote armour. The captain flexed his shoulders and arms\u2014left, right. All through the courtyard, squires held up gauntlets\u2014slid them onto their master's hands, and then reached for the great lances, as tall as small trees and as thick, tipped in long heads of steel.\n\nHis face appeared from under the brow of the helmet. He was smiling. \"Yes,\" the captain said. \"I feel him. Through you.\" He laughed. \"What did you do?\"\n\n\"I told him what I think of him,\" she said. \"A woman scorned\u2014for power?\" She threw back her head and laughed. It sounded mad.\n\n\"I imagine,\" the captain said, even as Michael moved the helmet back and forth, seating it securely on his brow, \"that must have been a shrewd blow.\"\n\nShe shook her head. \"His amour propre will shed it soon enough. But I saw into him. He has a traitor in the fortress.\"\n\n\"I know,\" said the captain. \"I told you,\" he gave a nasty smile, \"and that traitor has been giving our foe a somewhat incorrect version of events for some time now. It is now or never. He can lay all the traps he likes. Sometimes, it all comes down to speed, and audacity. He is cautious. He is sure.\" The captain seemed to glow with the power he'd prepared. \"He wants this fight,\" the captain said. \"So do I. One of us is wrong. We can only try our best, so guard yourself, my lady.\"\n\nThe main gate slid open.\n\n\"Follow me!\" ordered the captain.\n\nShe stood out of the way, and watched him ride out. The hooves rattled with finality, and the knights began to move. Knights reached out to her\u2014Francis Atcourt accepted her blessing and she reached up to pray for Robert Lyliard, who accepted her benison with a salute. Tomas Durrem bowed to her from the saddle and swept by.\n\nThe Red Knight paused in the gateway.\n\nAbove her, on the balcony of the hospital, she saw Amicia. She saw him touch the favour on his shoulder, saw her bow her head.\n\nGrendel reared a little, and plunged through the gate, and he was gone.\n\nShe turned to Bent, who was standing by her. \"Everyone is to go to the basements and lie down,\" she said. \"Everyone!\"\n\nShe ran into the courtyard, shouting orders.\n\nThe alarm bell was ringing, and the archers were pouring out of their barracks, to their battle positions. All of them were in armour. They knew the score.\n\nThe Abbess stopped in the courtyard, and looked around once\u2014the last doors were slamming closed. She nodded in satisfaction, wished she had time to hunt for Father Henry, and ran for the chapel.\n\n# Lissen Carak\u2014Father Henry\n\nFather Henry saw the Abbess talk to her boy\u2014his revulsion showed raw on his face. They were all creatures of Satan\u2014the Abbess, the mercenary, the sisters. He was surrounded with witches and man-witches. It was like hell.\n\nHe was done with inaction. He had the power to destroy them. He had all the tools a normal man had to use against evil.\n\nHe knew he would not survive it\u2014but all his life, he had endured pain and mistreatment for what he knew was right. His only regret was that he could not act directly against the mercenary. That man was like Satan incarnate.\n\nFather Henry went into the chapel, where a dozen sisters were already gathered\u2014not real sisters, he knew it now, but a coven of witches. All gathered to sing their damnable mockery of praise to God.\n\nHe made himself smile at Miram. She was too busy to pay him any heed. Just for a moment, he considered striking with his knife\u2014right here. Taking Miram and a dozen witches\u2014\n\nHe hid his eyes lest they read his mind, and slipped past them to the altar. He reached behind it. Seized the long staff of heavy wood, and his hand unerringly found the one arrow he needed.\n\nBlack as her heart.\n\nIt was a most remarkable arrow. Behind the head and the first three fingers of the shaft, all white bone, the rest of the arrow was of Witch Bane.\n\n# Lissen Carak, The Lower Town\u2014The Red Knight\n\nIn a plan dependent on preparation and planning and Hermetic mastery, it was ironic that the first part required twenty brave men and one middle-aged woman to risk their lives to sweep the road clean. And he didn't even know if they'd succeeded.\n\nBut Thorn couldn't possibly expect him to come on horseback, through the Lower Town. In fact, the captain had seen to it that Thorn would expect him on the covered footpath instead.\n\nOut in the darkness, where the Lower Town had been, a line of lights sprang up. It was a small casting\u2014hardly a ripple on a sea full of heavy waves.\n\nBut when the blue lights sprang up, the captain gave Grendel his head. They marked a sure way through the rubble of the Lower Town.\n\nHe found that the lights heartened him. He wouldn't fail because of a detail. Now, it was a fight.\n\nHe grinned inside the raven-face of his visor, and reached for\n\n_Prudentia. He was in the room, and he didn't want anything to do with the door. He merely touched his tutor, and she smiled._\n\n_\"Find me Harmodius,\" he said. \"Open the link.\"_\n\n_She frowned. \"But I have things I must say to you\u2014\"_\n\n_He grinned. \"Later,\" he said._\n\n_He drew power\u2014just a trickle\u2014stored from the sun and placed it in a ring given him by the Abbess. It had come with power; now he used it in the Aether to ignite his darksight._\n\nBack in reality, and his sense of the night altered. The outline of the trap was clear now, and he smiled like a wolf when the prey begins to tire.\n\nThorn had sent creatures into the ditch beyond the remnants of the Lower Town wall\u2014the ditch his own men had dug to communicate with the Bridge Castle. It was now full of boglins, which suited him just fine.\n\nOff to the south, at the entry to the defended path which the archers had taken and retaken every day of the siege, waited a company of daemons. At least forty of them, enough to exterminate his company of knights.\n\nHe grinned. _I didn't go that way,_ he thought, smugly. The creatures of the Wild were not as clever as men at hiding themselves in the Aether. It occurred to him as he cantered down the steep road that they didn't think of hiding in what\u2014to them\u2014was their natural element. Or something.\n\nAnd out on the plain, moving steadily forward towards the town, was Thorn.\n\nThe great figure towered over his allies. Even at this distance he stood head and shoulders above the trolls who surrounded him, at least twenty feet tall with antlers like a great hart's spreading away on either side of his stone-slab face. He towered, but he was not particularly fearsome from five hundred paces. He was a beacon in darksight, though, and his power wound away in a hundred threads\u2014to the skies, to the creatures around him, to the woods behind him\u2014\n\nTwo-dozen trolls guarded the horned figure, reflecting his power.\n\nEven as the Red Knight watched the horned man he raised his staff.\n\nThorn raised his staff. He could see the dark sun. For a moment he was tempted to lay his great working on the mysterious, twisted creature, but a plan is, after all, a plan. He reached into the slug on his left shoulder, and green fire washed up his right arm, pulsed once on his staff\u2014and it was like joy; like the ultimate release of love.\n\nThe light was like that of the deep woods on a perfect summer day. It was not a pinpoint, a line, a bolt, a ball. It was everywhere.\n\n_The Abbess was in her choir, and she felt the assault on the wards\u2014felt them stumble. She raised her voice with those of her sisters. She could hear them, feel them in the Aethereal, feel Harmodius and Amicia._\n\n_The light was everywhere. It's green radiance was seductive, the siren call of summer to the young, to run away from work and play, instead. The Abbess remembered summer\u2014summer days by the river, her body wet from a swim, her horse cropping grass_...\n\n_Far, far away, the sigils that defended her house were_ \u2014\n\n_Harmodius read the working, and its immense subtlety, and just as he was about to throw his counter, he saw the trap._\n\n_Thorn wanted him to swat the working aside._\n\n_The summer light was an insidious working that struck directly at the sigils from all sides and drained their strength into the Wild itself. The craftsmanship was magnificent._\n\n_The power involved was majestic._\n\n_And any counter\u2014any reinforcement\u2014would drain away with the sigils themselves, into the hungry maw that awaited._\n\nIf I survive this, I'm going to learn that working, _Harmodius thought._\n\n_He took his narrow sword of bright blue power, and severed the Abbess's connection to the fortress sigils._\n\nThe fortress sigils fell. Thorn gave a grunt of satisfaction, tempered by knowing that Harmodius had done the only thing he could have to avoid being sucked down with them.\n\nThe faerie folk danced around Thorn's head, in the sudden accession of power\u2014this ancient power, the very life-blood of wards that had stood for centuries. It was bleeding into the ground at his feet, and they bathed in it, their winged forms like tiny angels flitting in a rainbow of light.\n\nThe final collapse was like the opening of a window. There\u2014and then nothing.\n\nHe didn't pause. His staff swept up, and he released his second working\u2014a simple hammer.\n\nOne Leg and Three Legs and the trebuchet and the top third of the great North Tower vanished in a flash of light. The explosion that followed destroyed every window in the fortress\u2014the stained glass of the saints became a hurricane of coloured shrapnel.\n\nFather Henry, head down behind the altar below the great window, had his back flayed bloody. His robes were all but ripped from his body although his head and arms were covered. He screamed.\n\nThe captain reached into his _palace and drew power through the ring._\n\nHe had the charred cloth in his gauntlet, where he couldn't lose it in the dark, and he funnelled the power through it.\n\nFour feet beneath the duck boards at the base of his trench, beneath the boglin horde, ten fuses sprang alight.\n\nAbove him, in the fortress, a single massive pulse of power ripped through the night air\u2014the concatenation almost cost him his seat on Grendel.\n\nBut the fuses were lit, and now\u2014\n\nNow it was a hundred long heartbeats to Armageddon.\n\nHe had reached the base of the slope and now he followed the path between the first of the blue lights across the rubble to the town's back gate. Grendel couldn't move quickly here, and this was the weakest part of the whole plan. If he could see Thorn then Thorn would see him. Indeed, the whole _point_ was that Thorn should see him. And yet even now, the daemons were starting to shift. They must already know that their trap was in the wrong place. And the huge shapes around the enemy were new.\n\nThorn had already struck the trebuchet, and destroyed it.\n\n_We're too late._\n\nHe was halfway across the town, Grendel was moving at a trot, and one bad step on rubble and he would be down. The risk was insane.\n\nFifty heartbeats.\n\nHe turned in the saddle and looked back. Tom was right behind him, and the sound of the column of knights filled the darkness robbed of other sound by the force of the explosion.\n\nHe rose in his stirrups as Grendel stepped over a downed roof beam\u2014the blue lights seemed to ripple\u2014and then he was over the outer wall and in the field. Bad Tom passed the wall right behind him, and they reined in together.\n\nHe turned Grendel and pointed his muzzle at the horned figure, now at eye level, just two hundred paces away across the plain. Behind him, his sortie shook out into a wedge as they got free of the tumble of rocks and roof tiles that had been a town. In the dark.\n\nThe captain thought, _Damn, we're good._\n\nHe raised his right arm, lance and all. He used a little power to light the tip of his lancehead\u2014not just light it, but make it burn like a star.\n\nHe swept his lance down.\n\nGrendel gave a little start, and went from a stand to a gallop in three strides, as if they were in a tiltyard.\n\nThirty heartbeats.\n\n# Lissen Carak\u2014Thorn\n\nThorn watched the dark sun come at him, and he waited with a curious mixture of elation and loathing for the misshapen thing. It was like a man, but it was not like a man. He was some odd fusion of man and Wild. He might have pitied it, but he hated it, as well\u2014because its fusion was different from his.\n\nIt was coming, just as his secret friend said it would. But not by the path it had said it would take. That meant the secret friend was compromised.\n\nAnd that meant...\n\nThe dark sun held a power that shouted itself to every Wild creature on the battlefield.\n\nThis was his first clear look at the thing, and Thorn felt a tingle\u2014not of fear, precisely. But in that creature was something that bellowed a challenge to him. Like a vast predator roaring defiance across the swamps of the Wild. And every Wild creature felt that call. Some flinched from it. Some were attracted to it.\n\nThat was the Way of the Wild.\n\n... and so the dark sun must be a creature of the Wild, and that meant\u2014\n\nIt was too fast. Thorn's discovery came very, very late. He had allowed himself to ponder the thing's creation for long _thuds_ of his great, slow heart, and in that time the man had crossed the ruins of the Lower Town like a dhag\u2014so fast that even as his hidden ambush of daemons sprang from their concealment and raced to save him they were already too late to strike a blow. The wedge of knights was past them.\n\nSomething was slowing him!\n\n_Bitch_ he roared in his head. _She was working her will on him_ \u2014\n\nHe shook himself free of her enchantment, even as\u2014\n\n# Lissen Carak\u2014The Red Knight\n\nHe put his spurs to Grendel\u2014just a pressure of the pricks to the sides, so that the great horse knew not to stint. This was the great effort.\n\nThorn was standing facing the fortress, and his bodyguard of misshapen horrors were shoulder to shoulder holding massive bill hooks and spiked clubs, wearing armour of wood and leather. They glowed, not with the healthy summer green of Thorn's workings but with a sickly putrescent colour.\n\nThe captain had hoped to save his lance for Thorn with a tiltyard trick, so he gave Grendel the sign to put its head down. He flicked his lance down, and the troll followed the lance tip, cutting up\u2014\n\nGrendel struck the troll as it parried the lance, so that the spike on his great horse's head drove into the monster's stone-armoured chest. It was six inches long, sharp as a needle on its tip and as broad as a man's hand at the base, and the horse weighed more than the troll by several times. The horn broke the stone plate in two and punched through its hide, to shatter the bones of its chest. Grendel crushed the troll flat, and planted a great steel-shod hoof precisely on its hips, the horse's charge virtually unimpeded by the collision.\n\nWith the practise of a hundred jousts, the captain let his lance come down again. Thorn was ten paces beyond his bodyguard, just turning to ward himself.\n\nHe leaned forward, adding the power of his body and hips to the weight of the horse. By luck, or a last second intuition, his lance struck home within a hand's span of where the ballista bolt had struck Thorn hours before and he rocked his enemy back. Thorn tottered, reached out with his staff\u2014\n\nFell backwards and crashed to earth.\n\nThe captain struggled after the impact\u2014it felt much like slamming a lance into a castle, but he kept his seat and swept on, leaving his lance, and the next two men in the wedge\u2014Bad Tom and Ser Tancred\u2014each put their lances into the thing after him; or so he had to hope, because he was riding past, and the rest of the bodyguard were on him. The trolls were as tall as he was, and one blow from one of their weapons would crush his armour and kill him. But he rode as if inspired\u2014he leaned, Grendel danced, and no blow fell fully on him.\n\nGrendel put his spiked head into the next one. The unicorn's horn of twisted steel bit deep again, and again the captain almost lost his seat in the shock\u2014the great horse went from a gallop to a stand, screamed his anger and struck the thing with his hooves\u2014one, two, each landing with greater force than ten belted knights could muster, yet precise as a boxer.\n\nThe Wild monster's sickly green glow was extinguished between the first and second blow to its great stone head, and the horse reared in triumph.\n\nThe captain drew his great sword.\n\nAnother troll screamed from his left, rose to its full height, and was struck in the chest by a lance that knocked it flat.\n\nBad Tom roared, \"Eat me, you son of a bitch!\" at his side and was gone into the green-tinged darkness. Tom was a legend for temper, for ill manners, for lechery and crime. But to see him on a fire-lit battlefield was to see war brought to earth in a single avatar, and as his knights swept past him, the captain watched as Tom's lance, unshivered, swept through the trolls.\n\n\"Lachlan for Aa!\" he roared.\n\nWhen his lance broke in his third victim, he ripped his five-foot blade from its scabbard and the blade rose and fell, catching the fires of the plain on its burnished blade at the top of every cut so that it seemed to be a living line of fire\u2014rose and fell with the smooth and ruthless precision of a farmer scything grain at the turn of autumn.\n\nBy himself, Bad Tom cut a hole through the company of monsters.\n\nThe captain nudged Grendel back into motion. On his sword side, a smooth stone head rose out of the darkness and he swung down with all his might, rising in the stirrups to get the most out of his cut\u2014the sword rebounded from the stone, but the head cracked and dropped away, it's roar changed to the caw of a giant crow as it fell.\n\nAnd then he was through the enemy line. His sword was wet and green with acrid blood, and behind him, the trolls who survived the charge were already gathering to cut him off from the fortress. The crisp spring air was suddenly full of arrows, announced only by their whickering flight\u2014almost unnoticed against the ringing of his ears\u2014but then they began to strike him. And Grendel.\n\n_Whang!_\n\n_Ting-whang WHANG_.\n\nThere were irks behind the trolls, and they were loosing into the melee\u2014unconcerned about their own, or perhaps Thorn was too fully armoured to fear an irk arrow.\n\nMore creatures charged at his knot of knights from either side, and he rode for the long trench he had ordered dug. A trench full of boglins.\n\n_Ready?_ he asked into the Aether, and looked back.\n\nBad Tom had already made his turn. At least a dozen knights were with him.\n\nThey all knew the score, and the plan. He'd lost count of the time. But it had to be close.\n\nHe rode right for the trench, wondering if\u2014hoping that\u2014he had put Thorn down. He had to hope. It had been a mighty blow.\n\nThe trench was only a few strides away. A handful of darts rose to greet him, but the boglins were as stunned as their master by the speed of it, and then Grendel rose, and for a moment, they flew.\n\nHe landed with a thunderclap of strained armour straps and saddlery, a clank and a rattle, his teeth rattled, his jaw hurt, and his helmet slammed into his forehead despite arming cap and padding, and he was blind for a critical moment\u2014\n\n\u2014and Grendel shuddered and stumbled, and all around the two of them, his knights were jumping the trench and the boglins were turning\u2014too slowly.\n\nThe last knight\u2014Tom\u2014cleared the trench. Landed, and passed Grendel, who was slowing under his master's hand.\n\nThe boglins, fooled for a moment by the speed of their passage, came over the lip of the trench in a flood.\n\nThe captain just had time to think _Now would be good._\n\nThe naphtha charge buried under the boards in the trench ignited. It didn't explode. It went with a great _whoosh_ as if God himself had willed it, and then there was only a wall of fire behind them.\n\nThe captain might have laughed in his triumph, but in that moment Grendel died under him. The horse had given his life to get his master over the trench with a dozen well-thrown javelins in him, and he crashed to the earth, and all the lights went out.\n\n# Lissen Carak\u2014Harmodius\n\nA third of the choir was dead.\n\nHarmodius found the Abbess, and got a hand under her elbow, but she levered herself to her feet with dancer's muscles and _reached in the Aether\u2013_\n\n_He was wounded. The boy had hurt him._\n\n_Harmodius had Miram steady on her feet, and the chorus began again\u2014shaky, trembling, but lifting once again. Amicia's voice was clear above them all\u2014for a long minute, she had carried the choir by herself._\n\n_The power was still there\u2014the immense power of the well, wrapped in the working of the choir._\n\n_Harmodius spread his arms, and raised his staff, and began to cast._\n\n# Lissen Carak\u2014Father Henry\n\nFather Henry lay in a pool of his own blood, ears ringing.\n\nThe pain on his back and shoulders was incredible.\n\nHe shrieked.\n\nBut Christ had born pain. Pain was like the Enemy\u2014it could be vanquished.\n\nFather Henry rose to his knees.\n\nBy a miracle, his bowstring had not been cut by the glass that was all around him.\n\nHe nocked his arrow with shaking hands.\n\n# Lissen Carak\u2014Thorn\n\nThorn felt the pain of his wounds, but not as much as he felt the mockery of the attack. The dark sun was taunting him\u2014had ridden through his trap with deliberate mockery.\n\nHatred suffused him.\n\nHe rose to his feet. Tested his strength and grunted.\n\nHe was struck by a crossbow bolt, which didn't even distract him. He spread his fingers, flame crackled and a dome of green power sprang over his head, another flashed into being on his left hand like a verdant buckler, and in his right hand he raised his staff.\n\nHe took a stride toward the trench, and his guards followed him.\n\n_Look, I am an epic hero,_ he thought with bitter irony. _And I have to do everything myself._\n\nHe didn't run. He took long strides to his boglins, surging out of the trench the men had cut like an obscene wound on the earth.\n\nAnd then alchemical fire exploded in front of him. It wasn't a manifestation of power, or he'd have sensed and quenched it. In fact, he tried. It took him wasted seconds to realise that his enemy had filled the ground under the trench with naphtha\u2014they had poured poison into the very veins of earth.\n\n_Men must die._\n\n# Lissen Carak\u2014The Red Knight\n\nHe never quite lost consciousness, although he hit the ground very hard. But he rose before the pain could fill him, and nothing was broken. His sword was lying under Grendel's body, but he got a hand on the pommel and dragged it clear.\n\nHe looked around but the hoofbeats said that it had all worked better than he might have hoped. He hadn't wanted Tom to stay and die. On the other hand, somehow he hadn't ever thought he would lose Grendel.\n\nHe didn't take up his sword because he expected to live, so much as because it seemed appropriate.\n\nFor the first time since the sortie began he had time to breathe. Beyond the confines of his faceplate it was a big, dark, violent night. Many of the boglins in the trench had made it out, and some had started to follow the knights before the naphtha charge went off, and of course he was an infernal beacon to creatures of the Wild. They were coming for him.\n\nSo was Thorn.\n\nThe captain couldn't manage a smile inside his Raven's beak. But he wasn't shaking too badly, and he had control of his head.\n\nHis job now was to hold Thorn's attention as long as ever he could.\n\nBest do a proper job of it.\n\nHe reached out, and summoned the nearest creatures of the Wild to serve him, the way his witch of a mother had taught him to. He'd sworn never to do it. But this was his last stand. Now, for everything, the oaths of an angry boy were thrust aside...\n\n# Lissen Carak\u2014Thorn\n\nThe dark sun's challenge was contemptuous.\n\nHe was forcing the boglins to his will, on the other side of the trench.\n\nThorn shrieked with rage, as if he'd been struck. He threw caution to the wind, and leaped the trench of fire.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain was surrounded by boglins\u2014a crush of them, and their acrid scent filled his helmet.\n\nHe had never been so close to the creatures, and despite his revulsion for them, he found it impossible _not_ to notice things about them\u2014how their soft shells seemed to be formed like armour, their human arms emerging from breastplates.\n\nHe waited for the coup de gr\u00e2ce... But he was holding them, and all their thoughts were his.\n\n_This_ was what he had been made to do. Created. Honed. Polished for it.\n\nAnd he began to work on them.\n\nHe was _in the room of his palace, and Prudentia was off her pedestal, standing by the iron-bound door. She had her stone arms locked against it, and it trembled on its hinges despite her efforts._\n\n_\"He is coming for you,\" Prudentia said._\n\n_\"Open the door,\" he said, trying to master his terror._\n\n_\"He wants you to face him in the Aetherial! He will eat your power, you arrogant child!\"_\n\n_Prudentia said. \"Can't you hear him?\"_\n\n_The captain could hear his bellows of victory, all through the Aether. \"I could use some advice here,\" he said._\n\n_\"Don't stand against the powers of the world until you are much, much more powerful,\" Prudentia said in a matter of fact tone. She shrugged. \"But when brute force will not suffice, consider artifice. Recall, dear boy, that he will not know the limits of your power. He calls you the dark sun.\"_\n\n_Good advice. But he couldn't think of anything he could do with it. He reached for Harmodius and opened the door._\n\nThorn was there.\n\nHe had crossed the trench of fire, and now he stood, smouldering, the acrid smoke of his wounds rising in wisps, and he was backlit by the fire in the trench.\n\nThe captain coughed.\n\nThorn towered over him and even from a horse length away, the captain could see that the sudden shock of lances had hurt him. Something dark and watery oozed form a deep pit in his breast.\n\n_**You thought yourself my peer, you little thing.**_\n\nThe captain was fighting the wave of nausea that came with the fear. Whatever Thorn was, his coming brought terror, revulsion, a deep, sick feeling of oppression and violation. The captain struggled with it. For a long, long moment, all he saw was his mother, promising him\u2014\n\n_**You dared to oppose me. Do you know who I am?**_\n\nDeep in the grip of the horror, the captain writhed. His conscious, rational mind registered that only the most unstable beings asked such questions.\n\nAnd he had a lifetime's experience of pretending courage when all he wanted to do was roll in a ball and weep. It was like arguing with his mother.\n\n_He cast\u2014not an attack, but a subtle reinforcement of his armour._\n\nHe raised his sword. \"Well,\" he said. His attempt at a drawl actually sounded somewhat hysterical. \"Well,\" he said again, and his voice was better. He used to goad his mother this way. \"I understand you used to be the King's Magus.\"\n\nThorn leaned down and one giant, hot hand slapped the captain to the ground. He saw the blow coming, his wrists answered his will, his sword swept up, and the blade shattered as it touched the sorcerer's skeletal hand. The power of Thorn's blow hurt the captain right through the steel harness he wore. Even through the power supporting it.\n\n_**I am infinitely greater than the mere man who was the King's Magus.**_\n\nThe captain couldn't muster a laugh, or even a cackle. But he got back to his feet, as he had when his brothers beat him.\n\nThorn raised his hand.\n\nOne finger fell away.\n\nThe captain felt a wild, foolish joy. He tossed the shards of his sword away and drew his rondel dagger instead. \"You are just one of the many Powers of the Wild, Thorn.\" He took a deep breath against the pain in his ribs. \"Don't get above yourself, or someone will eat you.\"\n\n_Good shot_ , muttered Harmodius, inside his memory palace. _Almost ready._\n\nThere was a pause, as if the earth stood still. The captain tried to see Amicia's face\u2014to think any worthy, noble, or merely human last thought that was not born of fear and would not leave him to die the slave of this creature.\n\nBut he couldn't.\n\n_Hold on_ , said Harmodius.\n\n_**You are challenging me?!**_\n\nThe Red Knight stiffened his spine, stood as tall as he could, and said, \"My mother made me to be the greatest Power of the Wild,\" He managed another breath. And delivered his sentence, like a sword cut. He said, \"You are just some parvenu merchant's son trying to ape the manners of his betters.\"\n\nHe ordered the boglins to _Kill Thorn_ and the crowd of boglins turned their weapons on their former master.\n\nStung\u2014even though none of them could penetrate his glowing green armour\u2014he clenched one gnarled fist.\n\nBoglins died.\n\nThe sorcerer's rage was automatic rage, unthinking rage at being challenged, at insult piled on insult. Thorn bellowed. **_You are nothing!_** Faster than the captain could parry, strike\u2014react at all\u2014Thorn's fist slammed into him and knocked him to the ground again, except this time he felt bones break. Collarbone? Ribs, for sure.\n\nSuddenly he was _in his palace, and Prudentia stood with a handsome young man in black velvet embroidered with stars. So great was his fear and his confusion that he took long heartbeats to see that the stranger was Harmodius._\n\n_But he couldn't hold the palace in his mind. He was too afraid, and even as Harmodius opened his mouth_ he was on his back and the pain was remarkable. His armour had probably saved him from death. But not from pain.\n\nThat was a laugh.\n\nHe used his stomach muscles to roll over, to get to his feet.\n\nThere was Thorn.\n\n_**Why are you not dead?**_ Thorn asked.\n\n\"Good armour,\" the Red Knight said.\n\n_**Aah! I can see your power. I will take it for my own. It is wasted on you. Who are you? You are no different from me.**_\n\n\"I made different choices,\" the captain answered. He had trouble breathing but, just there, he started to be proud. He was _holding his own._\n\nThorn threw a working; bright as a summer day, fast as a levin bolt.\n\nThe Red Knight parried it to the ground with a flash of silver white.\n\n_**I see, now. You were made. You were constructed. Bred. Ahh. Fascinating. You are not an ugly mockery after all, dark sun. You are a clever hybrid.**_\n\n\"Cursed by God. Hated by all right thinking men.\" The captain was gaining strength from sheer despair. With nothing left, he was going to beat his fear, the way he'd beaten it a thousand other times.\n\n_**The time of men is over. Can you not see it? Men have failed. The Wild is going to crush men, and before ten thousand suns set, the young fawn and the bear cub will ask their mothers who wrought the stone roads, and the faerie will weep for their lost playthings. Even now, men are but a pale shadow of what they once were.**_\n\n_**But then, you are scarcely a man. Why do you cleave to them?**_\n\nBreathing was difficult, but he was achieving calm. Calm meant mastery of the Aethereal.\n\n_Hope gave rise only to more fear. But fear was the ocean in which he swam, and he reached through the fear\u2014he used the fear._\n\n_He was in the palace of memory. He reached out to Amicia, who took his hand and Harmodius's, and the Abbess's, and Miram's. And Mag's. And that of every surviving nun singing in the chapel._\n\n_He mastered his thoughts._\n\n_Cast his very favourite phantasm._\n\n_\"Holy Saint Barbara, Despoina Athena, Herakleitus,\" he said, pointing at each statue as he spoke the name, and the great room began to spin._\n\n_Prudentia reached down from the plinth and put a hand on his shoulder. She smiled at him. It was a sad smile. And she reached out and took his hand free hand. \"Goodbye, my lovely boy. I had so many things to say. O Philae pais\u2014\"_\n\n_He was flooded with power\u2014power like pain, when it rises beyond any possible point of pleasure\u2014like victory. Like defeat, like hopelessness and hope. And he stayed there, for an eternity, balanced between all and nothing._\n\n_Like love when love is too much to bear._\n\nWhat did she mean, goodbye?\n\nHe was back in the acrid night air.\n\nHe wondered if the calm that suffused him was artificial.\n\nThorn leaned over him, blocking the stars.\n\n_**You are ours. Not theirs.**_\n\nThe captain laughed, a laugh he treasured. \"There is no _us_ , Thorn. In the Wild, there is only the law of the forest and the rule of the strongest. And if I join you, I will subsume _you_ to my needs.\"\n\nJust to make his point, the captain projected, as his mother had taught him, the imperative. **_Kneel._**\n\nMore than two thirds of the surviving boglins feel immediately to their knees.\n\nHe was deeply gratified to see Thorn twitch so that his singed branches shook as if a strong wind had passed through a forest.\n\nAnd even as he exchanged words with the Enemy, buying precious heartbeats, an agony of power rose inside him\u2014the greatest power he had ever felt, as if love personified drove his phantasm. Between two heartbeats, the captain knew what she had done.\n\nPrudentia had not opened the door, which would have invited Thorn to take him from inside.\n\nShe had ended herself, and as a phantasmal construct, she had poured her own power and the power of her making into his work. It explained the love.\n\nOh, the love.\n\n_I make fire,_ he said in the purest High Archaic.\n\n# Lissen Carak\u2014Thorn\n\nThorn felt the swelling of power\u2014such a sweet power, with a taste he had forgotten. He lost a thousandth of a heartbeat trying to identify it. Only then did he reach for his shield of adamantine will.\n\n_You don't remember that taste, my sweet? That taste is love, and once, you were capable of it._\n\nThe lady was in his head\u2014in his place of power\u2014naked, exposed, and rendering him the same.\n\nConfused\u2014a storm of rage and hate\u2014he struck at her.\n\nIn striking, he did _not_ raise his shield.\n\n# Lissen Carak\u2014The Abbess\n\nThe Abbess took her stand in the ruined chapel, in near darkness, her hair unbound, her feet bare in the shattered glass. Her nuns stood in close array behind her, and their voices rose in sacred music.\n\nHarmodius stood beside her, his staff in his hand, riding the song of power into the darkness, into the labyrinthine mind of the young man on the field below, facing a monster\u2014\n\nShe, too, faced a monster. A variety of monsters, many of them of her own making. That she had loved this thing which now sought the ruin of all she loved\u2014\n\nShe hit him with her frustration and her love, her years of loss. She poured her love of her God into his wounds, and she added her contempt\u2014that he had abandoned her to turn traitor to humanity. That he had taken her gift and made this depravity with it.\n\nShe _hurt_ him.\n\nAnd he struck back. But he was hampered, and still\u2014 _still_ \u2014he hesitated to hurt her.\n\nShe hit him again. She'd had years to expunge her hesitations.\n\n# Lissen Carak\u2014Mag the Seamstress\n\nMag, standing in the former street of the Lower Town, nonetheless felt the old Abbess struggle with the Enemy. It was terrifying, but she felt the Abbess's power and she raised her own hands in sympathy. Unknowing, untrained, the seamstress nonetheless poured her carefully hoarded power into the Abbess.\n\nThe Abbess smiled in triumph.\n\nFather Henry rose from behind the altar, and drew his arrow to his mouth, and loosed.\n\nAnd from the darkness, a cry of rage.\n\nThe Abbess screamed like a soul in torment, and was knocked flat on her face\u2014dead before her head hit the stone floor.\n\nBlood welled from her eyes and she lay still, a vicious black arrow in her back.\n\nFire\u2014a pure fire of crystalline blue\u2014Prudentia's favourite colour\u2014-enveloped Thorn's mortal shell. The heat of it was stupendous.\n\nAnd from the fire, smoke\u2014a rich, bright smoke, luminescent and alive, more than white, more than smoke, and the captain could feel Harmodius sending the smoke through him, through his place of power and down his arm and into the air about him. A subtle working\u2014insidious, clever, a fog of a million mirrors.\n\nShe had hurt him\u2014hurt him so much. And the dark sun had hurt him, and now he was screaming in agony. A moment's remorse\u2014and the cost had been cataclysmic.\n\nBut he was saved\u2014she was dead, her light extinguished, and not by him. Some other power had struck her down and he was innocent of that crime, and he turned\u2014strong enough to finish this pretender.\n\nBut he writhed inwardly in the knowledge that she was dead.\n\nIt had to be done.\n\nIt should not have been done.\n\nAnd then\u2014too late! He felt his apprentice's working, the complex, layered phantasm that was that boy's trademark\u2014a coloured smoke, so quiet, so harmless, so complex\u2014\n\nHe lunged back up the line of Harmodius's casting, as he had attacked along the line of his lover's.\n\nHarmodius felt his former master's power coming.\n\nHis counter-strike was so tiny, so very subtle, that it cost him almost no power. It relied on his enemy's hubris and his sense of his own power.\n\n# Lissen Carak\u2014Thorn\n\nThorn killed the apprentice effortlessly, although he couldn't, for some reason, take the man's not inconsiderable power for his own. Typical of the man\u2014to squander his power rather than let his master have it. His former apprentice fell back amidst a choir of nuns. If he'd had time, Thorn might have exterminated the nest, but the dark sun was still pounding him with his strange blue fire.\n\nIf Thorn had been a man he might have laughed. Or cried.\n\nInstead, his consciousness raced back to the plain below, where his shell faced being consumed by fire.\n\nAnother slow heartbeat while he poured power into the problem and extinguished the blue fire.\n\nHe was surprised\u2014and concerned\u2014to see how badly hurt he was. Again\u2014yet again, he would appear weak.\n\nHe had no time to take stock. Even now he was so badly hurt that any of the lesser powers could take him.\n\nHe raised his staff and was _gone._\n\n# Lissen Carak\u2014The Red Knight\n\n_Run, boy!_ cried Harmodius.\n\nThe captain tried to run.\n\nHe crawled through the prostrate boglins. He forced himself to his feet, and he ran, a broken, stumbling run while waiting for the levin bolt in the back that would end him. _He stood in the palace, and the plinth was empty, and Prudentia's statue lay cold and still on the ground._\n\n_Damn._\n\n_Time to mourn later, if he lived._\n\n_He leapt onto the plinth, and called his names._\n\n_Honorius! Hermes! Demosthenes!_\n\n_Desperation, luck, and a strong will._\n\n_Goodbye, Prudentia! You deserved better than I ever gave you!_\n\n_He ran to the door, and pulled it open._\n\nThe flicker of a casting\u2014Thorn reached out, trying to find its source. The dark sun was still on the battlefield. Still casting?\n\n_I am badly hurt_ , he conceded. He summoned his guards to withdraw.\n\n# Lissen Carak\u2014The Red Knight\n\n_\u2014powered his phantasm and slammed the door shut again._\n\nHis body rose in a leap, sailed through the heated air, and fell to earth again\u2014a hand's breadth clear of the wall of the trench.\n\nThe captain turned away from the fire, and saw a wedge of knights, their mirror-bright harnesses like liquid fire in the smoky darkness. Off to the north, boglins hovered, uncertain.\n\nA daemon raised his axes in challenge.\n\nBut the knights did not pause to fight. Even as the captain ran, strong arms grabbed him, an arm under each shoulder, and he was swept away as cleanly as if he'd been snatched by a great bird.\n\n# Chapter Fourteen\n\n# Valley of the Cohocton\u2014Peter\n\nPeter didn't recover so much as grow used to what was gone, like a man who has lost a hand, or an arm. And it took days, not hours.\n\nOta Qwan scarcely paid him any heed at all\u2014indeed, now that he was paramount war leader, Ota Qwan was loud and definite and far, far too important to waste time on one new warrior. Peter walked all the way back from the Ford Fight, as the Sossag came to call it, to their camp in a haze of fatigue and a darkness that he'd never known, even as a slave.\n\nThree nights in a row, he sat by a dead fire, staring at the cold coals and considering ending his own life.\n\nAnd then he would hear Ota Qwan\u2014instructing, ordering, leading, demanding.\n\nAnd that would give him the strength to go on.\n\nOn the fourth night on the trail back, Skahas Gaho came and sat with him, and offered him some rabbit, and he ate of it, and then together the two of them drank some of the dead men's mead\u2014honey sweet. The Sossag warrior was quickly drunk, and he sang songs, and Peter sang his own people's songs and in the morning his head hurt, and he was alive.\n\nIt was just as well, because they were moving easily along trails as soon as the sun was up, and suddenly, every warrior fell flat on his face, so that\u2014just for a moment\u2014Peter was the only man standing. Then he threw himself flat. He'd been so deep in his pain that he had missed the signal.\n\nScouts wormed their way into the bush and came back to Ota Qwan with reports, and the rumour swept the column that there was a great army on the road. Far too large and well-prepared for the Sossag people to challenge alone.\n\nThey had won the Ford Fight. But they had lost many warriors. Too many warriors. Too much experience, too many skills.\n\nSo they rose as they had fallen to the ground, almost as if a single spirit inhabited many bodies\u2014and they loped off to the north, and they climbed well into the foothills of the Adnacrags, avoiding the enemy by many miles. It was only after three days of gruelling travel over the most difficult terrain that Peter had ever known when they climbed over a low ridge, and saw their camps spread across the woods and green fields of the Lissen where it ran into the Cohocton. From the top of the long ridge, Peter could see thousands of points of light\u2014like the stars in the sky, but every one of them was a fire, and around that fire stood a dozen men, or boglins, or other creatures\u2014such creatures as served Thorn and yet loved fire. And more creatures slept cold in the woods, or slept in streams, or mud.\n\nPeter let Skahas Gaho pass him on the trail and he stood in the deepening twilight at the top of the ridge, and looked down. Almost at his feet rose the great fortress of the lady, which the Sossag called the Rock, and its towers looked like broken teeth, and its arrow slits glowed with fire like a Jack-o'-lantern.\n\nAnd away to the east, at the edge of his vision, he could see another host of fires burning. The army around which the Sossag had slipped. The King of Alba.\n\nThe armies were gathered, and in the last light, Peter watched a tall column of ravens and vultures riding the drafts over the Valley of the Cohocton.\n\nWaiting.\n\nHe sat and watched the play of light\u2014massive pulses of power, flashing back and forth like a summer storm.\n\n# Lissen Carak\u2014Thurkan\n\nThurkan watched the dark sun slip away. He had seen the Enemy captain face down Thorn, pounding him with blue fire until the Wild sorcerer fled. And unlike Thorn, the dark sun's bodyguard came and rescued him, their ranks closing tight around him.\n\nThe daemon had learned much about the skills of the knights.\n\nHe turned to his sister. \"Thorn is beaten.\"\n\nShe spat. \"Thorn is not beaten, any more than you were last night. Thorn said he would kill the great _machine-that-throws-rocks_ and he has done so. Stop your foolish preening.\"\n\nThurkan shivered with suppressed need to fight.\n\n\"I will challenge Thorn,\" Thurkan said.\n\n\"You will _not_ !\" Mogan replied.\n\n# Lissen Carak\u2014Michael\n\n_The Siege of Lissen Carak\u2014Day Thirteen_\n\n_Last night the enemy came with all his might to storm the fortress. The King's Magus and the Abbess and the Red Knight duelled with him and drove him back, but the Abbess died defending her place, shot in the back by a foul traitor._\n\nMichael sat with his head propped on one hand, looking at the hastily scrawled words. He sipped the wine next to him and tried not to go to sleep over the journal.\n\nThe captain was in the hospital. His breastplate had a dent in it the size of a man's fist. They'd lost five men-at-arms.\n\nThe archers were openly saying that it was time to ask for terms.\n\nHe turned on the wooden stool he was using. Kaitlin Lanthorn lay, fully dressed, on his bedroll. She'd come in after the sortie returned, kissed him, and stayed by his side while he saw to little things\u2014like having the armourer get the dent out of the captain's breastplate.\n\n\"You shouldn't be here,\" he said.\n\nShe lay, open eyed. \"I'm pregnant,\" she said. She sat up. \"Oh, I might be wrong, but Amicia says I am. She'd know.\" Kaitlin shrugged. \"I'm pregnant, and the sorcerer is going to kill us all, anyway. So what's it matter if I spend the night with you?\"\n\nMichael tried to think like the captain. To balance it all out. But he couldn't, so he put the quill down, and took her face in his hands. \"I love you,\" he said.\n\nShe smiled. \"That's good,\" she said. \"Cause I love you, too, and we're going to have a baby.\"\n\n\"If we live through the next few days.\" He lay down next to her.\n\nShe turned to him. \"You'll protect me, I think.\"\n\nMichael stared into the dark.\n\nMag stood with her daughter Sukey and a dozen other nuns and local women, laying out the dead.\n\nThis time there was no feeling of triumph. The cost was high\u2014the Abbess was dead, and there was a line of figures wrapped in white linen to show the losses of her community and the losses of the captain's company, intermingled.\n\nAnd the Red Knight was gone, carried into the hospital.\n\nThe Abbess had been killed by an arrow. And no one seemed to be looking into her murder.\n\nMary Lanthorn smoothed a sheet over Ser Tomas Durren. \"He was bonny,\" she said.\n\nFran shook her head. Sukey sobbed, and Mag pulled her daughter's head against her chest. Sukey's husband was dead too. Third winding sheet from the right. She held Sukey for a long time, and then went back to wrapping Third Leg. His body had been crushed\u2014his face almost erased\u2014and yet Mag was gentle in wrapping the fresh white linen tight. Details mattered to Mag.\n\n_God, let these boys come to you swiftly despite the lives they led._\n\n\"I hear the Red Knight's on the verge of death,\" Mary said.\n\nAmy Carter looked up. \"That novice will save him. Amicia.\"\n\nKitty looked at her sister. \"The men are saying she's a witch.\" She looked at Sukey and Mag for a moment, and then back at her sister. \"Ben says she killed the Abbess.\"\n\nAmy's eyes grew wide.\n\nMag put a hand on the girl's shoulder. \"Best not be spreading that kind o' talk, girls.\"\n\n\"It's all around in the stables,\" Kitty said. \"All the boys is saying that some of the sisters is witches.\"\n\nSister Miram was shaking out a winding sheet. Her hearing must have been unnaturally sharp. She turned.\n\n\"Who says we are witches?\" she asked.\n\nKitty blanched.\n\nMiram frowned. \"Child. Who is spreading this poison?\"\n\nKitty looked around, uncertain. \"My brother Ben says the priest said it.\"\n\nSukey looked at her mother. \"Bill Fuller too.\" She spat the words. \"Fuller's been talking crap all night.\"\n\nMiram looked around. She went and touched the first body in the row\u2014smaller then the others. The Abbess.\n\n\"I have been remiss,\" Sister Miram said. \"I let loss cloud my vision of earthly iniquity.\"\n\nKitty Carter looked at her sister. \"I didn't really think Amicia killed the Abbess.\"\n\nAmy rolled her eyes.\n\n# Lissen Carak\u2014The Red Knight\n\nIt wasn't yet morning when he came to. Noise in the corridor had awakened him. He heard armour\u2014and he was in the wrong bed.\n\nThere was no sword by his bed.\n\nThe door opened, and Sister Miram entered his cell, in the full robes of the order; Ser Jehannes in harness, and Michael; Johne, the Bailli of one of the towns, and Master Random.\n\nHe pulled the linen sheet up over his chest.\n\n\"The Abbess died in the enemy attack,\" Sister Miram said. Her face had aged.\n\nThe captain had scarcely heard her speak. What she had said took a moment to register, so that his mind explored the fact of Sister Miram's speaking for heartbeats before he realised the import of what she said.\n\n\"I'm sorry,\" he said. Useless, empty words.\n\n\"There's open talk of negotiation. Of surrendering the fortress for free passage away,\" Ser Jehannes said. The others flinched at his tone.\n\n\"No,\" The captain said. \"There will be neither surrender nor negotiation.\" He was noticing that he'd been bandaged around the ribs, and that all the hair had been shaved away\u2014well. Lots of hair. He winced. The Abbess was dead and he realised that he had, in his way, loved her.\n\n_Always looking for a better mother_ , he thought. \"If you all will leave Michael to dress me,\" he said quietly.\n\n\"Dress quickly,\" Ser Jehannes said. \"It's happening right now.\" He was quiet. \"All the local people. Some of the men.\"\n\nSister Miram withdrew to the door. \"She would never have surrendered,\" she said quietly. \"The men in the courtyard are saying Amicia did it,\" she added.\n\nThe captain winced and met her eye. \"I'll see to it.\"\n\nThe nun closed the door.\n\nThe captain got himself out of bed, despite a touch of vertigo. He had a feeling he knew from childhood\u2014the feeling of having tapped his Hermetical powers utterly. An emptiness, but also a good feeling, like a well-exercised body.\n\n_Prudentia is dead._\n\nIt was not the first time that good people had died to keep him alive.\n\nToby appeared with his old black doublet and his old black hose and his fine gold belt. He looked terrified.\n\nHose took time to get on\u2014he tried to quiet his own pulse. To think about something besides the Abbess and his tutor.\n\n\"She was murdered,\" Ser Jehannes said. \"Someone shot the Abbess in the back.\" He lowered his voice. \"Gelfred says it was Witch Bane.\"\n\nThe thought of it made him physically sick.\n\n\"And no one saw this?\" he asked wearily.\n\n\"Everyone was watching the fight outside the walls,\" Ser Jehannes said.\n\nThe captain sighed. \"Secure the gates and all the passages. There is a passage under the main donjon which leads out of the fortress. Right now, it's blocked by our wagon-bodies, but put a pair of archers\u2014good archers\u2014on the stairs. Give me a nod when this is done.\"\n\n\"When you say I should secure\u2014\" Jehannes paused.\n\n\"As if we were taking the fortress for ourselves,\" the captain said harshly. \"As if we were in Galle. Trust no one who is not one of ours. Use force if you have to. _Secure the exits, Jehannes!\"_\n\nThe old knight saluted. \"Yes, my lord.\"\n\nMichael had his boots. He buckled them around the ankles, laced the tops to the captain's pourpoint.\n\n\"Full armour, gloves, war sword,\" the captain said.\n\nMichael began to arm him. It wasn't a quick process and some parts hurt a great deal. But wearing armour was itself a statement.\n\nThe arming doublet and mail haubergon weighed on him like a shirt of lead and a hairshirt all together. Many knights believed that the very pain of wearing armour was a penance before God.\n\nWell.\n\nLeg harness, starting with the cuisses, and then the greaves and the steel sabatons that buckled so neatly over his boots, right to the shaped and pointed toes. Michael pointed the cuisses into his arming doublet at an amazing speed, while Toby supported him.\n\nHe stood, flexed his legs, and Michael, aided now by Jacques, fitted his breast and back over his head and latched it shut.\n\n\"Had a dent in it like you wouldn't believe,\" Michael said.\n\n\"Oh, I would,\" the captain said.\n\nMichael snorted. \"Carlus says taking the dent out took more strength than he's ever had to use,\" he said. \"Like the steel was magicked.\"\n\nEach of them took an arm harness\u2014vambrace, elbow cop and rerebrace in a single unit on sliding rivets, a miracle of craftsmanship in gilded bronze and hardened steel\u2014and clipped them on, buckling them to his upper arms and then to his shoulders with straps, and then his pauldrons went on, and the circular plates that strapped to the pauldrons and guarded his under arms.\n\nThe golden belt at his waist.\n\nGolden spurs at his heels.\n\nGloves, and a sword, and the baton of his office.\n\n\"There you are, my lord,\" Michael said.\n\nThe captain smiled\u2014it was done as fast and as painlessly as it could have been done by anyone. \"You are a fine squire,\" he said.\n\nHe walked out of the recovery ward, looked down the main corridor, and saw his brother.\n\nGawin had his feet over the edge of the bed.\n\n\"Stay where you are,\" the captain said gently. \"Michael, stay here with this man.\"\n\nMichael nodded. And saluted. He recognised his captain's tone.\n\n\"But\u2014\" Gawin began.\n\nThe captain shook his head. \"Not now, messire.\"\n\nHe walked down the corridor to the other ward. Ser Jehannes had already passed. Low Sym was dressing in his gambeson.\n\n\"Have a sword, Sym?\" the captain said.\n\nSym nodded wordlessly.\n\nThe captain pointed at Amicia's elegant back, standing at the dry sink across the room. \"She is not to leave this ward until I return,\" he said. \"If you harm her you are a dead man. But she is not to leave this room. Understand?\"\n\nAmicia whirled on him. \"What?\"\n\n\"For your own protection, sister,\" he said, his voice quiet. \"Father Henry has killed the Abbess. But he will seek to blame you.\"\n\n\"Father Henry?\" she came towards him, a hand at her chest. \"The priest?\"\n\nHe was at the top of the stairs. \"Obey. On your life.\" He ignored her outcry, and went down the steps, past the commanderies, to the courtyard. At the door, Bad Tom waited, armoured cap \u00e0 pied, a pole-axe in his left hand.\n\n\"It's bad,\" he said.\n\nThe captain nodded. He pulled on his gloves, and took the staff of his command from his belt. \"On me,\" he said, and Tom opened the door.\n\nThe sound hit him. Anger first\u2014then fear.\n\nEvery farmer and tenant was in the courtyard\u2014four hundred men and women packed into four hundred square ells. The noise was like a living thing.\n\nThe dispensary had a wooden step, and two of his men-at-arms were keeping it clear.\n\nOn the other side of the courtyard, a dozen big farmers stood together. With them were some of the merchants.\n\nThe captain turned to Carlus, and he blew his trumpet. It was loud, and shrill.\n\nEvery head turned.\n\nThe captain waved the staff over the assembly. \"Disperse!\" he said into the sudden silence. \"There will be no negotiation, and no surrender,\" he went on.\n\nA dangerous murmur began.\n\n\"Kindly disperse to your stations and your beds, and let's have no more of this,\" the captain kept his voice level and kind.\n\nOne of the merchants raised his head. \"Who are you, messire, to decide for us?\"\n\nThe captain took a deep breath and struggled with the spark of rage that hit him. Why did good men always make him feel like this? \"I will not debate this with you,\" he said. \"If you wish to leave, the gate will be opened for you.\"\n\nAnother farmer shouted \"Fuck you! That's just death! It's our land that's destroyed. Our farms that are burned, you sell-sword. Get out of the way, or we'll put you out.\"\n\nJehannes was waving to him from the portcullis winch. He had a key in his hand.\n\n\"This fortress is under the protection of my company,\" the captain said loudly. \"The lady Abbess charged me with its defence, and I will hold it until I am dead. The power that invests us will not hesitate to lie, deceive, or betray us to our doom\u2014but it will not let anyone here escape alive. The only hope any of you have is to join us in resisting to the last drop of our blood. Or better yet, to the last drop of theirs.\" He looked around. \"The king,\" he almost choked on the title, but he got it out. \"The king is on his way. Do not give way to despair. Now, please disperse.\"\n\n\"You can't fight all of us!\" shouted the farmer.\n\nThe captain sighed. \"In fact, we can kill every one of you.\" He spoke out. \"Look around you. Would the Abbess ever have given in? She isn't even buried yet and look at you. Ready to surrender?\" He pushed his way into the courtyard, ignoring Tom's protests. He pushed his way through the crowd until he was nose to nose with the big farmer.\n\n\"Priest says she was a witch,\" the farmer said.\n\nPeople were shuffling away from him.\n\n\"Priest says all these so-called nuns is witches!\" the farmer insisted. \"Souls black as night.\"\n\nA few men nodded. None of the women did.\n\nThe captain passed his arm through the farmer's arm. \"Come with me,\" he said.\n\n\"I don't have to\u2014argh!\" the farmer stumbled. He was unable to resist the armoured man, and was pulled along through the crowd to the great gate.\n\nThe gate was open, and the sun was shining beyond the walls of the fortress.\n\n\"Look out there,\" the captain said. \"Look out there at what Thorn has done. _He_ betrayed his king. _He_ betrayed his people. _He_ has made himself a construct of the Wild, a sorcerer without compare, unlimited by laws or even friends. _And you think that is better than your Abbess?_ Because a priest told you that black is white, and white is black?\" The captain spat the words.\n\n\"And I should trust you?\" the farmer growled.\n\n\"Since you are so obviously a fool\u2014yes. You'd do better trusting me, the man who fights to defend you, than trusting to the God-damned priest, _who killed your Abbess.\"_\n\nThe crowd was backing away from him, and he had to assume his eyes were burning.\n\nThe farmer stood his ground, but his jaw was trembling. \"You're one of them too. And the priest says the other witch killed the Abbess. For her power.\"\n\nThe crowd muttered again. \"You're one of them!\" shouted a man at the front.\n\n\"I am whatever I choose to be,\" said the captain. \"So are you. What do you choose?\"\n\nTom and Jehannes stepped up behind him. And with them, a dozen other men-at-arms in plate armour, and most of the archers. There were archers on the walls, on the stumps of the towers.\n\n\"Don't make me do this,\" the captain said to the crowd.\n\nSister Miram walked out of the wreckage of the chapel with Mag, the seamstress. Miram raised her arms.\n\nMag spat. \"Look at you, Bill Fuller.\" She put her hands on her hips. \"Playing with fire. Going to stand here and get shot?\" She looked over the crowd. \"Go to your beds. Let go. We've lost the Abbess. Let's not spill any more blood here.\"\n\n\"We can take 'em,\" Fuller said. But his tone suggested he knew he was lying.\n\nMag walked over and slapped his face. \"You always were a weak fool, Bill Fuller,\" she said. \"They'll kill every one of us, if they have to. We wouldn't even hurt them to do it. And for what? The enemy is _out there._ \"\n\nJohne the Bailli came out of the chapel. \"Well said, Maggie.\" He went and stood with Bad Tom. \"I stand for the Abbess. We will not surrender.\"\n\nMaggie's daughter Sukey came and stood with her. She was shaking.\n\nThe Carters started to burrow through the crowd.\n\nDan Favor went and stood with Ser Jehannes.\n\nAmie Carter grabbed her sister's wrist and towed her across the open space. She turned and faced the crowd. \"Don't be a pack of tom-fools,\" she said. \"You been sorcelled. Can't you feel it? Don't be so stupid and pig-ignorant you can't face it.\"\n\nLiz the laundress came and stood by Tom. Kaitlin Lanthorn walked across the open space.\n\n\"Sluts and harlots,\" said a voice.\n\nThe heads of the crowd turned, as one.\n\nFather Henry looked as if he'd been on the cross. His face was streaked in old, dried blood. His robe was flayed and fell around his waist, showing his ascetic body, lacerated with further cuts.\n\nThe people parted for him. He walked between them like a king.\n\n\"Sluts and harlots. Are these your allies, Satan?\" He stopped at the edge of the crowd.\n\n\"Not all of us are sluts, priest,\" said Master Random, and he burrowed into the crowd. \"Adrian! Allan Pargeter! What are you doing with this man? Fomenting mischief?\" Master Random walked into the crowd, looking for other apprentices he knew.\n\n\"You killed the Abbess,\" the captain said.\n\nFather Henry drew himself up, and the captain knew he had his man. He was too proud to deny the crime.\n\n_Fool._\n\n\"She was a witch, a creature of Satan who chose to put her own appetites against\u2014\"\n\nThe stone hit the priest in the head. He snapped around, eyes blazing, and just for a moment, he didn't look like a gentle and crucified Jesus. He looked like a madman. His eyes raged.\n\n\"Take that man,\" the Red Knight said. He pointed his baton.\n\nBad Tom reached out with his pole-axe, caught the priest's foot with the head, and pulled, and the priest fell. Tom kicked him viciously, his armoured foot making a distinctive meaty sound as it connected with the priest's gut.\n\nThe priest retched.\n\nTwo archers grabbed him and hoisted him. He tried to speak, and he got the butt of Tom's pole-axe in the arch of his foot. He screeched.\n\nAnd suddenly, there was no crowd. Just frightened people, looking for salvation.\n\nAnd most of them asked\u2014 _Where is the king_?\n\n# Chapter Fifteen\n\n# Albinkirk (Southford)\u2014Ranald Lachlan\n\nWhen Ranald Lachlan led his scouts down to the edge of the Albin River, he could scarcely believe his eyes.\n\nFifty great boats, like galleys, lay in the river opposite the landing. The river fleet covered the river in four long files of boats, and their oars went back and forth like the legs of water-running insects.\n\nAt his back, the Royal Standard of Alba fluttered in the breeze over the gate-towers of Albinkirk, and the fields by the great bridge were empty of foes. It was like a dream, because the familiar ground was so empty.\n\nRanald sat his horse, watching the big river craft row, and even as he watched, they turned, all together, at a flash of a great bronze shield, and suddenly the whole fleet went from four columns advancing west to four lines heading toward the north shore. His shore.\n\nHe walked his horse out onto the landing stage where the ferry had run, in better times, and waved.\n\nA woman in the bow of the largest galley waved back. An awe-inspiringly beautiful woman in a flowing white overkirtle. It took an effort of will to tear his eyes away from her, and he knew her well, from his years in the south.\n\nQueen Desiderata.\n\nUnbidden, a smile came to his face, and he laughed.\n\n# Albinkirk\u2014Desiderata\n\nWho is that?\" Desiderata said to her maidens teasingly. She was standing in the bow, waving. \"I feel I know him.\"\n\nLady Almspend stood and waved. \"Ranald the barbarous hillman, my lady,\" she said brightly.\n\nDesiderata smiled at her secretary. \"You seem happy enough to see him,\" the Queen said.\n\nLady Almspend sat a little too suddenly. \"He\u2014gave me the most wonderful book,\" she said haltingly.\n\nThe other ladies laughed, but not unkindly.\n\n\"Was it a big book?\" one asked.\n\n\"Very old?\" asked another.\n\n\"Perhaps more like a nice, thick scroll?\" suggested Lady Mary.\n\n\"Ladies,\" the Queen said. The oarsmen were losing the stroke, laughing so hard. But the bank was rushing at them, despite the current.\n\nAs they rowed into the landing, the Queen stepped lightly up on the gunwale and leaped onto the pier.\n\nRanald Lachlan, who she remembered perfectly well, bowed deeply and then knelt.\n\nShe gave him her hand. \"It is a long way, since you were in my bridal guard.\"\n\nHe smiled at her. \"A pleasure, my lady.\"\n\nShe looked past him, up the tall bank, where Donald Redmane had the lads dismounted. \"You have a small army of your people here. Come to aid the king?\"\n\nHe shrugged. \"My cousin lost a small army, my lady. We've already fought the Outwallers. But I have a thousand head of beeves and some sheep, and I'm looking to sell them to the Royal Army.\"\n\nShe nodded. \"I will buy them all. What's your asking price?\"\n\nIf he was surprised by her tone or manner, he hid it well. \"Three silver marks a head,\" he said.\n\nShe laughed. \"You drive a hard bargain,\" she said. \"Is it chivalrous for a knight to bargain with his Queen?\"\n\nRanald shrugged, but he couldn't stop looking into her eyes. \"Lady, I could say I'm no knight, but a drover. And I could say I'm a hillman, and not in any way your subject.\" He grinned, and knelt. \"But he'd be a rude bastard and no kind of a man, who ever failed to acknowledge you as his Queen.\"\n\nShe clapped her hands delightedly. \"You are the very spirit of the north, Ser Ranald. One mark per beeve.\"\n\n\"You, my lady, are the living embodiment of beauty, but for a mark a head, I could have sold them to the Keeper of Dorling. Two silver marks a head.\" His eyes flicked to something behind her, and his smile intensified.\n\n\"You remember my secretary, the very learned Lady Almspend?\" she asked. \"One and a half.\"\n\n\"One and a half, right here, on this side of the river?\" he asked. He made another deep bow, this time to her secretary, who was standing on the gunwale, beaming. \"Two if I have to drive them over the river.\"\n\n\"What's a kiss worth,\" sang Lady Almspend. She blushed, shocked at her own boldness.\n\n\"Everything!\" he shouted back. \"But these aren't my beeves, so I can't trade them for a kiss, my sweet.\" He relented. \"Your Grace, my price is two, but I'll drive them where you like, and pledge my lads to serve your Grace.\"\n\nThe Queen nodded. \"Sold. Fetch me my navarch. I have a thousand head of cattle to ferry over the river.\" She turned back to the hillman. \"So despite your sordid money, you'll do a deed of arms with me?\"\n\nShe put extra effort into her voice. She saw a coldness in him\u2014-something absent, some terror recently passed\u2014and her voice caressed it like liquid gold.\n\nThe hillman looked cautious. \"What kind of deed?\"\n\n\"What knight asks what deed is required of him? Really, Ser Ranald,\" she said, and put her arm through his.\n\n\"I'm no knight,\" he said. \"Except perhaps in my heart,\" he added.\n\nShe smiled at Lady Almspend. \"We must do something to rectify that.\"\n\nOn the bank above them Donald Redmane watched his cousin with the Queen.\n\n\"What's happening?\" asked one of the boys.\n\n\"We just sold the herd to the Queen,\" Donald said. \"What's an Alban mark worth?\" he asked, and then shrugged. \"And now we have to live to spend it.\"\n\n# Lissen Carak\u2014Harmodius\n\nHarmodius listened to the angry crowd and kept his head down. He was almost drained of power\u2014needed more recovery time, and the last thing he needed was a confrontation with ignorant witch-hunters.\n\nLet the boy handle all that.\n\nHe dressed carefully. The old Abbess had never been a friend of his\u2014but now, in death, he had to admire her. She had disclosed power of a level she had never had in youth\u2014and had deployed her power brilliantly. She'd held the Enemy for long moments, while he prepared his masterstroke.\n\nSadly, his masterstroke hadn't quite come off. But she hadn't died in vain. The fortress still stood. And the Enemy's beard had been badly singed.\n\nAgain.\n\nHarmodius imagined himself standing at the Podium at Harnford, staff in hand, lecturing on Hermeticism. _I learned the underpinnings of the nature of reality in the middle of one war,_ he would say, _and I learned to manipulate them myself in the middle of another._ Or perhaps he would say, _I saved the world for mankind, yes, but I only stood on the shoulders of giants._ That was better. Quite good, in fact.\n\nAnd now all of her secrets would go to her grave with her, and her soul would fly to her maker.\n\nHarmodius ran his fingers through his beard.\n\nWhat if\u2014\n\n_What if all the power in the world came from a single source?_\n\nThat's what it was, wasn't it? It was, in a way, a commonplace.\n\nGreen or gold, white or red? Power. It's just _power_\n\n_And that meant_ \u2014\n\nNo good. No evil. No Satan. No\u2014no God?\n\nDid it mean that, in fact? Were there really any fewer angels on the head of a pin, if all power came from a single source?\n\nHis head spun.\n\nWhat if Aristotle was wrong?\n\nHe could hardly breathe. One thing to think it. Another to know it to be true.\n\nHe stumbled down the tight staircase to the common room of the dormitory, and then he forced one foot in front of the other as he walked toward the chapel.\n\nBad Tom appeared at the captain's side. The captain was doing his damnedest to appear to be a member of the congregation. He had just sung a hymn. He had himself well in order.\n\nShe had wanted him to understand.\n\nHe knelt when the other attendees knelt. Sister Miram led the service in the absence of the priest, a matter that seemed to excite no comment.\n\n_I swear on my name and my sword that I will avenge you, my lady._\n\n\"My lord?\" asked Tom, at his elbow.\n\n\"Not now.\"\n\n\"Now, my lord,\" Tom said.\n\nGlaring at his corporal, the captain stood, walked to the aisle and genuflected to the crucified figure that towered over him, and then backed down the aisle to the doors. Every head turned.\n\nToo bad.\n\n\"What?\" he barked, when he was outside. The nuns were singing her to rest\u2014every voice it the woven fabric of music a thread of power. It was incredibly beautiful.\n\nTom looked at the door to the cellars. \"I hae' the priest, God rot his false soul to hell. I put him i' the darkest room wi' a lock.\" Anger made his voice thick.\n\nThe captain nodded. \"You valued her too.\"\n\nTom shrugged. \"She blessed me.\" He looked away. \"That priest, he's going to die hard.\"\n\nThe captain nodded. \"We'll try him for treason, first,\" he said.\n\nTom had his back to the door. \"Why try him? You're the captain of a fortress under siege. Law of War.\"\n\n# Lissen Carak\u2014Gerald Random\n\nGerald Random picked his way fastidiously along the captain's trench, following Ser Milus\u2014clambering over the cooked bodies of a hundred boglins, their charred remnants a testimony to the power of fire. They smelled like cooked meat, and when he lost his balance and stepped on one, it crunched as if he'd stepped on charcoal. He paused.\n\nHis skin prickled.\n\nGelfred the huntsman strode past him, eyes wary, moving faster. The mercenary didn't seem to mind stepping on the cremated boglins.\n\nRandom wondered how long he'd have to do this before he was like Gelfred, or Milus.\n\nBehind him, forty men moved carefully along the trench\u2014company archers, new recruits, farm boys. The reinforcements.\n\nThey came out of the trench under the wall of the Bridge Castle and hollered to the watch to open the postern. Random had answered the call from the fortress before mains, and he wasn't in armour. He grabbed a bite of bread and a sound apple, and one of the young whores who'd come with the convoy handed him some good cheese. He smiled. \"What's a nice girl like you doing in a place like this?\" he asked Dora. She was Dora Candlesomething. Young Nick Draper fancied her, and Allan Pargeter had drawn her naked, which was still a nine-day wonder among the wagons, despite the flying monsters and magic. That made Random laugh.\n\nShe smiled back at him. \"Money,\" she said. \"Same as you.\"\n\nHe shook his head and laughed again. \"If we get back to Harndon, come and ask me for a job,\" he said.\n\nShe looked at him. \"You mean that?\" she asked.\n\nHe made a face. \"Of course.\"\n\nShe rolled her eyes. \"Just when we're all going to die.\"\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain looked out through the hole in his wall and watched the fires burning in a swathe across the enemy's camp. The enemy's men, at least, cooked supper.\n\nThe rest of the camps were dark.\n\nHis back hurt. But then, his side hurt too, he now had cracked ribs on both sides of his ribcage\u2014his shoulders were wrenched from the stress of being plucked off the ground by his knights, and his right hand had odd, numb spots in it and he had no idea why.\n\nHe was supposed to be in bed.\n\nToby stood uncertainly by the door.\n\n\"You want to be in bed, I suppose,\" he said.\n\nToby shrugged. \"I'm hungry.\"\n\nThe Red Knight went to the table in the middle of the room and tossed his valet a biscuit.\n\nThen he looked at the lute on the table. He hadn't played it in\u2014\n\nHe couldn't remember when he played it last.\n\nHe picked it up, suddenly decisive, and walked out the door into the hallway. Toby tried to cut him off.\n\n\"Oh, Toby,\" he said. \"I don't give a fuck.\" He knocked on the door to his Commandery.\n\nIn three heartbeats, Michael was there.\n\n\"Grab your lute,\" he said. \"Good evening, Miss Lanthorn. Michael, these people need some music. Not a grim silence. Let's light a fire.\"\n\nMichael sometimes forgot that his master was only a few years older than he was. He grinned. \"Give me\u2014us\u2014a moment.\"\n\n# Lissen Carak\u2014Mag the Seamstress\n\nMag looked out into the darkness because she'd heard music.\n\nThere it was again, the sound of a southern lute. A wild, joyous sound.\n\nAnd then another, lower lute played back.\n\nThere was a bonfire burning on the cobbles.\n\nAn archer, Cuddy, came and peered out of the North Tower. He shouted something.\n\nAmy Carter peered out of the stable door and saw Kaitlin Lanthorn dancing by firelight, her legs flashing.\n\nShe ran back inside and rubbed her sister's cheek. \"They're dancing!\" she said.\n\nKitty sat up, fully awake.\n\nLow Sym heard music playing below the windows at the end of the hospital room. He threw his feet over the end of the bed and walked softly across the floor and opened one casement, and the sound of the notes raced in like a spell. He leaned out, listening.\n\nThe nun appeared by his side. \"What is it?\" she asked.\n\nSym giggled. \"Capt'n likes to play. Fast.\" He shook his head. \"Leastways, he used to play. On the Continent. Ain't heard him in an age.\"\n\nShe smiled. Leaned out. \"You like him,\" she said.\n\nSym thought about that for so long she thought he wouldn't answer.\n\nFrom their vantage point, they saw the music do its work. Men came out of the stable and down the steps from the towers and the stumps of the towers. Women emerged from the stables and from the nun's dormitory.\n\nSuddenly, there were as many people in the courtyard to dance as had been there for the priest.\n\nThe two instruments were joined by pipes and a drum.\n\nThe dancers began to move in a circle.\n\n\"I don't hate him,\" Sym admitted.\n\nAmicia turned. \"You are not lost, Sym,\" she said. \"You are more hero than villain, even now.\"\n\nHe stepped back as if she'd struck him. But then he grinned.\n\nThen he stiffened. \"Where you going?\"\n\nShe smiled. \"You can come guard me. I'm going to dance. Or at least to watch.\"\n\nIn the courtyard, Sister Miram stretched her arms and smiled wearily at the Red Knight, who stood with his back to the fire playing his lute like a madman. She turned to Sister Anne and ordered a cask of ale opened.\n\nBad Tom put a man-at-arms on the door to the cellars and another on the barracks. He and Jehannes whispered for a moment in the darkness beyond the fire, and Jehannes doubled the watch and forced some unwilling soldiers onto the walls where the farmers could see them.\n\nWhen Jehannes looked down, Tom was dancing with the seamstress's daughter.\n\nMag, Lis, and Sister Mary Rose hauled a great cauldron of beef soup to the door of the dormitory. Cheering archers and farmers hauled it together into the firelight.\n\nLong Paw appeared with a brace of wine jars, and handed them to the first men he saw. They toasted him, and the bottles passed around, soldier to farmer, and farmer to soldier, until they were empty.\n\nA farmer went and burrowed in his belongings in the stable, and returned with a jug that proved to contain apple jack.\n\nAnd the lutes played on.\n\n# Lissen Carak\u2014Michael\n\nAt some point, Michael knew he had never played so well, and he also knew that his fingers were going to hurt all the next day. Kaitlin whirled by, leaped in the air and was caught by Daniel Favor; Bad Tom caught Mag's Sukey around the waist and she, a widow of twenty-four hours, squealed like a girl; Low Sym turned with the eight-year-old daughter of the Wackets, and Sister Miram and Sister Mary turned a somewhat statelier pavane together when Long Paw bowed, very Continental, and took Sister Miram's hand and led her around the yard. Francis Atcourt bowed over Sister Mary's hand and she laughed, and curtsied. Amicia danced with Ser Jehannes, Harmodius whirled Lis like a much younger man, and her feet spun her skirts out around her like a king's cloak. and then Amicia spun by again with Ser George Brewes, and the Red Knight drank off his fourth glass of the Abbess's red wine and played on. Cuddy tilted the apple jack back and back... and rolled off the barrel on which he was perched, and landed flat on his back, and didn't move, and the farmers laughed. Wilful Murder had an arm around Johne the Bailli and a leather flagon in the other hand, and was singing at the top of his voice, his face lit like a daemon's in the firelight.\n\nThe Carter girls began to dance, a fast, flashy dance of their own creation, and the Lanthorn girls, not to be beaten, leaped into the circle, and the music ran away with them. More pipes joined in, and Ben Carter produced bagpipes, and his drunkenness seemed to fall away as he played for his sisters. Fran Lanthorn leaned out of the turning circle and kissed him hard on the cheek as she swept by, and he blushed furiously and his tune tumbled, but he caught it and launched it anew.\n\n# Lissen Carak\u2014Michael\n\nMichael and his master allowed their fingers to fall still. The lutes dropped out of the busy music, which swept on.\n\nMichael felt his captain's arms go around his shoulders. He was afraid he'd cry. The captain had never hugged him before. Or anyone else that he knew of. He'd never seen the man's face so open. So\u2014defenceless.\n\nAnd then he was gone into the swirling darkness and firelight.\n\n# Lissen Carak\u2014Thorn\n\nThorn could hear the music. It drew him the way a candleflame will draw insects and frogs on a still summer night in the deep woods. He walked heavily to the edge of the woods, and listened with his keen senses to the sounds of people laughing and dancing, to the sounds of as many as ten instruments.\n\nHe listened, and listened. And hated.\n\n# Lissen Carak\u2014The Red Knight\n\nThe Red Knight lay with his head in Amicia's lap. She was looking at the firelit scene at their feet, inside the walls of the courtyard, and he was looking at the line of her throat and jaw. She was thinking about how simple happiness could be, and he felt the current of her thoughts through their joined hands.\n\nGradually\u2014glacially slowly\u2014she lowered her mouth over his.\n\nPlayfully, at the last moment, he licked her nose, and they both dissolved into laughter, and he shifted, grabbed her under the arms and began to tickle her and she shrieked and tried to hit him.\n\nHe put her in his lap and bent to kiss her. She arched her back to reach him more quickly, and their tongues touched, their lips touched\u2014\n\nHe drank her, and she drank him. Each of them could feel the contact, real, aethereal, spiritual.\n\nHe had pulled her robes above her hips, and she had not stopped him. The feeling of her naked flank inflamed him, and he pressed on.\n\nShe broke the kiss. \"Stop,\" she said.\n\nHe stopped.\n\nShe smiled. Licked her lips. And then rolled out from under him, as swift as a dancer. Or a warrior.\n\n\"Marry me,\" the Red Knight said.\n\nAmicia stopped. She froze. \"What?\"\n\n\"Marry me. Be my wife. Live with me until we die, old and surrounded by children and grandchildren.\" He grinned.\n\n\"You'd say that to any girl who keeps her legs closed,\" she said.\n\n\"Yes, but this time I mean it,\" he said, and she swatted him.\n\n\"Amicia,\" said Sister Miram. She was standing by the apple tree. She smiled. \"I missed you at the fire.\" She looked at the captain, who felt like a schoolboy. \"She may choose for herself whether to marry a mercenary or be the bride of Christ,\" Miram said. \"But she can choose in daylight, and not on an apple-scented night.\"\n\nAmicia nodded, but her half-hooded eyes concealed a spark that the Red Knight saw, and rejoiced at. He sprang to his feet. And bowed low. \"Then I bid you good night, ladies.\"\n\nMiram stood her ground. \"It was well thought,\" she said. \"They needed to rejoice. And the lady would have wanted a better wake than we were providing.\"\n\nThe captain nodded. \"It _was_ good. I didn't\u2014\" He shrugged. \"I just wanted some music. And maybe to lure this lady into my lair.\" He smiled. \"But it was good.\"\n\n\"There is more heart in us tonight than last night, despite everything.\" Miram looked at Amicia. \"Will you wed her?\"\n\nThe captain leaned very close to the nun. \"Oh, yes,\" he said.\n\nMiram put a hand on the girl's shoulder. \"Then tell us your name,\" she said.\n\n\"I know his name,\" Amicia said. \"He's\u2014\"\n\nThere was a sudden cheer from the courtyard, and then a roar of voices. The captain saw that Ser Jehannes was standing at the edge of the firelight, and behind him were three men in full plate, the fire lighting them like moving mirrors. They had black surcotes with white crosses.\n\nThe Red Knight turned away from the two nuns. He waved to Ser Jehannes and leaned out into the courtyard. \"What's happened?\"\n\n\"Now someone's come _into_ the secret passage,\" Ser Jehannes said. \"From outside. From the king.\" At the word _king_ the courtyard burst into cheers again. Jehannes pointed to the trio of armoured figures standing in the ruin of the covered way. \"Knights of the Order.\"\n\nThe music stopped.\n\nOne of the three knights raised his visor. He was an old man, but his smile was quite young.\n\nThe relief that flooded the captain was palpable, solid. He felt giddy. He felt weak. He said, \"Splendid.\"\n\nThe captain clasped hands with the first man in the long black cloak that marked the Knights of the Order of Saint Thomas of Acon.\n\n\"I'm the captain,\" he said. \"The Red Knight.\"\n\n\"Mark, Prior of Pyrwrithe,\" said the man whose right hand was clasped in his own. \"May we offer you our compliments on a brilliant defence? Although I understand from Ser Jehannes that the lady Abbess is dead.\"\n\n\"She died last night, my lords. In battle.\" Suddenly the captain was hesitant. He had no idea how the fighting orders felt about Hermeticism or any other form of phantasm.\n\nThe Prior nodded. \"She was a great lady,\" he said. \"I will go and pay my respects. But first\u2014the king is across the river, moving carefully. But he should be opposite the Bridge Castle by late tomorrow. The next day at the latest.\"\n\nThe captain grinned with pure joy. \"That is welcome news.\" He looked at the three men, all in full armour. \"You three must be tired.\"\n\nThe prior shrugged. \"The armour of faith is such that we feel little fatigue, my son. But a glass of wine is never amiss.\"\n\n\"Let us go to chapel,\" murmured the central figure. He wore a black tabard with the eight pointed cross of the order.\n\n\"If I may: I'd rather you stood where the people could see you just a little longer,\" said the captain. \"There have been doubts.\"\n\nThe Prior shook his head. \"We're late and no mistake, Captain.\"\n\nThe captain raised his hand for silence. In the courtyard, they cheered and cheered. But after a a few resurgences of spirit, they fell quiet, with Mag shouting \"Shut up, you fools\" and a titter of laughter.\n\n\"Friends!\" the captain said. His voice carried. \"Our prayers have been answered. The king is here, and these three knights of the order are the vanguard.\" There were cheers, but he went on. \"We've had a sip or two and a dance tonight. But when the king comes, we'll have to break this siege. The enemy is still out there. Let's have some sleep while we can. Aye?\"\n\nMen who had cursed him as Satan a few hours before raised wooden jacks now.\n\n\"Red Knight!\" they shouted. And others shouted \"St. Thomas!\"\n\nAnd then, as if by magic, they tottered off to bed. Sym and Long Paw put Cuddy over their shoulders and carried him to bed in the hospital. Ben Carter found himself carried by Wilful Murder and Fran Lanthorn to his pile of straw in the stable.\n\nTogether the four men walked to the chapel.\n\nThe Red Knight didn't say anything. The Abbess lay on her bier, and the three knights knelt around her. After some time, they rose, in unison. The captain led them to his Commandry, which, as he expected, was empty, without a sign of Michael's sleeping gear.\n\n\"This is my office,\" the captain said. \"If you wish to disarm, I can send you a couple of archers.\"\n\nSer John smiled. \"I've been sleeping in harness since I was fifteen,\" he said.\n\n\"Are you three alone?\" the captain asked.\n\nThe prior shook his head. \"I have sixty knights in the woods east of the ford,\" he said. \"Short of direct intervention by the enemy, they won't be found.\"\n\nThe tallest knight nodded and pulled his helmet over his head. He gave a sigh of pure pleasure. \"It's what we do,\" he said. He pulled a cushion off one of the chairs, put it under his head, and went to sleep.\n\n# Lissen Carak\u2014Gerald Random\n\n_The Siege of Lissen Carak. Day Fourteen_\n\n_Yesterday the folk of the towns talked rebellion\u2014but it was all shock at the death of the Abbess, and the captain restored order and none were injured. The priest, Henry, was taken into custody. The Enemy's engines pounded the Bridge Castle, but the enemy was hesitant and careful intheir movements, and we saw a large force crossing the river to the west. We had heavy rain in the afternoon, and at nightfall the captain (crossed out) the people celebrated the Feast of Saint George. After dark a party of Knights of St. Thomas entered and told us we were to be relieved by the king._\n\nIt was a lovely late-spring morning. There was a low fog, and Master Random looked out at it for a moment, enjoying his small beer. He waved to Gelfred, who was fussing with his falcons, and found young Adrian to get armed.\n\nWhile he was still getting his arm harnesses on, the alarm sounded.\n\nBefore the bell had stopped ringing, he was on the curtain wall of the Bridge Castle with the master huntsman. The bridge was still down, and although the bridge gates were closed and heavily barred, it was still the hope of every merchant in the lower fortress that more survivors would stumble in from the Wild\u2014despite all evidence to the contrary.\n\nGelfred had a trio of big hawks with him, and from time to time he flew one away into the morning light. He wasn't much of a conversationalist\u2014mostly he spoke to the hawks, murmuring to them in much the same language that Random's daughters spoke to their dolls.\n\nTwo archers assisted him.\n\nRandom watched the open ground out to the line of trees. Plenty of movement this morning\u2014boglins crawling through the deep grass. They continued to believe that they were invisible in the grass, and Random, for one, hoped they continued to believe it.\n\nHe motioned to one of the small boys who had survived the caravans. \"Tell Ser Milus that there is a boglin attack coming on the curtain wall,\" he said. And was proud that his voice remained steady and professional. He refused to let his mind dwell on how he had seen a line of boglin take his men apart.\n\nThe boy ran along the wall.\n\nThe bell rang again. The new company formed. It was a hodge-podge of men; a dozen goldsmiths with crossbows, with a dozen spearmen, all farmers sons or young merchants in borrowed armour; but the front rank was all men-at-arms, and Ser Milus led them in person.\n\nWhen they were well-formed and he'd inspected their armour, he led them up the ladders onto the curtain wall.\n\n\"Good morning, Master Random,\" he said, as he got to the top of his ladder.\n\n\"Good morning, Ser Milus,\" Random answered. \"Nice of them to announce themselves.\"\n\n\"I've doubled the watch in the towers,\" Ser Milus said. \"Look sharp!\" he said, loud and clear, and the men on the wall stopped their conversations and looked through the crenellations. \"You\u2014Lusty Luke, or whatever your name is. Where's your gorget? Get it _fastened_.\"\n\nOut in the deep grass, irks and boglins began to loose arrows.\n\nOne, lucky or perfectly aimed, struck one of the third rank spearmen and killed him instantly, and he fell bonelessly from the wall into the courtyard behind them.\n\nThe other farmer-spearmen shuffled nervously.\n\n\"And did he have his gorget properly fastened?\" Ser Milus roared. \"And did I _just speak to him about it_ ?\" he bellowed.\n\nGelfred finished lashing his birds to their perches and putting on their jesses and hoods. He went into the north tower followed by his two archers. His calm, unhurried movements contrasted with the spearmen.\n\nTheir shuffling stopped.\n\nThe boglins made their run at the wall. There were enough of them that they covered the ground\u2014it was like a charge by a nest of ants. The grass seemed to come alive, and there they were\u2014hundreds of them, scurrying to the wall, the elfin irks bounding ahead in great leaps.\n\nLike most fortress walls at the edge of the Wild, this one had a slope at the base and then rose sheer for the last few metres. The design had an immediate function beyond stability\u2014as Random had seen in the last four attacks. Boglins misjudged the wall because of the initial slope and attempted to run straight up it\u2014over and over. Apparently, they couldn't help themselves, and they ran at the wall, harder and harder, and very few ever made it to the top.\n\nRandom had come to believe that this, too, was by design, as the success of a few egged the rest on to continue their mostly-fruitless runs.\n\nThe men-at-arms with pole-axes and heavy swords began the slaughter of the soft-bodied things.\n\nThe crossbowmen cleared any that managed to alight on the crenellations, their heavy bolts plucking the creatures off the wall to a body-crushing fall.\n\nThe spearmen were there to catch any who got through the defence.\n\nRandom appointed himself to the third rank. He was much better armoured than the farm kids, and yet\u2014he was more one of them than he was a knight. Or a man-at-arms.\n\nThe fight went very well for two long minutes. The armoured professionals massacred the boglins, and the crossbowmen covered their backs, and one big, fast boglin who knocked Ser Stefan to the ground got a farmer's spear between his limbs and writhed\u2014literally like a bug pinned to paper\u2014until a half-dozen axes finished it. Ser Stefan got back to his feet, unharmed.\n\nRandom was unengaged\u2014almost bored, despite the tide of monsters lapping at the wall. But his boredom saved them, because he was the one who heard the screams of the sentries in the north tower.\n\nRandom whirled and saw boglins on the tower top.\n\nHe turned and went into the tower through the open curtain wall door, drawing his heavy sword as he ran. He had a buckler on his hip and he got that into his left hand.\n\n\"Boglins on the tower!\" he shouted at a huddle of men\u2014Gelfred and his huntsmen.\n\nThen he ran up the ladder to the tower top.\n\n\"Ring the alarm,\" shouted Gelfred\u2014a better response than Random's one-man fire brigade.\n\nRandom threw back the roof-trap and immediately received a blow to his head. It fell on his bassinet and glanced away and he was up another step, buckler over his head\u2014two fast blows to the small shield, and he was atop the ladder and cutting low with his sword, and he felt it cut into the firewood-hard flesh of a boglin's leg and then he pushed with his legs and got clear of the trap door.\n\nA blow to his back plate.\n\nRandom punched with his buckler, the steel rim cracking a boglin's head with the same feeling of a lobster's shell giving under a hard blow, and then he pivoted on his hips\u2014a new move, learned from Ser Milus\u2014and cut with his sword\u2014one, two. The second blow was wasted\u2014his first went home, splitting a head, and the back cut plucked the head off the body and blood spewed from the thing.\n\nBut they were all around him, stabbing with spears. One spear skidded across his back plate and went in under his buckler arm, stopped only by his chain voiders, and another spear-blow hit the side of his head hard enough to make him see stars. He stumbled forward and tangled with yet another of the things, who tried to pin him by wrapping all four limbs around his legs, but he put the pommel of his sword into the centre of the boglin's face and\u2014it's nose seemed to open into a horrible parody of a gullet, lined in spikes\u2014it shrieked in pain, and all four limbs began to scrabble at a tremendous rate.\n\nRandom swept his buckler in a desperate arc, let go his sword, and whipped his dagger from his belt. He rammed it into the leathery parts of the boglin's six-segmented chest, stabbing more times than he cared to count, and the thing almost literally fell to pieces under his hands.\n\nThen he saw a flash of dark green, and Gelfred was there, swinging a short-hafted boar spear with practised efficiency\u2014cut, thrust, cut, thrust, like a weapon's master demonstrating for a class.\n\nAnd then they were done.\n\nRandom was covered in blood\u2014but he felt like a god.\n\nHe leaned over the wall to call down to Ser Milus and saw that the courtyard was full of boglins.\n\nWhite boglins. In armour. _Wights._\n\n\"Gelfred!\" he screamed.\n\n# Lissen Carak\u2014The Red Knight\n\nThe Red Knight woke from a dream of Amicia with a smile on his face and Bad Tom's hand on his shoulder.\n\n\"You look like hell,\" the captain said.\n\n\"Bridge Castle is under assault,\" Tom said. \"It looks bad, and they've stopped signalling.\"\n\n\"Right,\" said the captain. He took a deep breath. Of course the Enemy knew the king was a half-day's march away. Hence their assault. An all-or-nothing assault. And the trebuchet was gone. But\u2014Bent had spent yesterday with the farmers erecting a trebuchet that filled the stump of the old tower. The captain rolled off his bed. He was fully dressed.\n\n\"Bent!\" he called.\n\nThe senior archer came from under the scaffolding. \"My lord?\"\n\n\"Start laying buckets of gravel across the trenchline,\" he said. \"Commence as soon as you can get loaded.\"\n\nBent saluted.\n\nThe captain turned to Tom. \"Tell the archers to start loosing into the field between here and the Bridge Castle. Everything we have. Don't spare shafts now. Someone heat rocks for the trebuchet. Michael! Get me Harmodius.\"\n\nHis squire had, apparently, spent the night in his room.\n\n\"And then armour, helmet and gauntlets,\" he called out.\n\nTom licked his lips.\n\n\"Sortie?\" he asked.\n\n\"Not much choice. Tom, the three gentlemen in my Commandery are knights of the order\u2014see to it they get a cup of wine\u2014\"\n\n\"And horses,\" said the Prior, appearing in the doorway. \"If you will allow me, my lord, I will have my knights meet us in the field below. Which may be a dolorous surprise to our foes, by the grace of God.\"\n\nHe raised a hand and made a sign and spoke a word\u2014a single word that the captain did not know\u2014in Archaic.\n\nSomething definitely happened. But the captain didn't know what it was.\n\nIt did become clear, though, that the military orders used Hermeticism.\n\n\"Wine and war horses, then,\" the captain said. _The king is coming. Let's not get rash._\n\nOverhead, the trebuchet slammed into its supports, and the whole scaffolding creaked.\n\nSeveral hundredweight of gravel flew out into the early morning.\n\nAbove him, on the remnants of the south tower, the heavy arbalests began to _thwack_ away at the creatures in the fields below.\n\n\"You called?\" asked Harmodius.\n\n\"I need to save the Bridge Castle. He's throwing everything at it\u2014and he's waiting for us to respond. I'm hoping that we can pound his attack flat with artillery but I can't count on it. The Prior, here, has offered us another trick up our sleeves, but I need more. What can you do?\"\n\n\"It's the King's Magus!\" the Prior said. \"The king has never ceased to look for you.\"\n\nHarmodius shrugged. \"I was never lost.\" He fingered his beard. \"I think this is a case for misdirection,\" he said. He smiled, and it was particularly nasty smile. \"He thinks I'm dead.\"\n\n# Lissen Carak\u2014Gerald Random\n\nRandom led the valets and the spearmen against the wights. There were fifty of them, and they were bigger and far better armoured than the boglins who had climbed the tower walls.\n\nBy the time Gelfred reached the courtyard many of the merchants who had come in the first convoys were dead. They were no match for the boglins, who were faster and better armoured and whose every limb had a killing scythe or a spike. The merchants did not live in their armour like the mercenaries; they fought unarmoured, and they died.\n\nBut in the light of the sun Gelfred and his archers, high above in the tower, began to slaughter them like rats in a trap.\n\nThe heavy longbow arrows went through their iron armour with a wet slapping sound, and the big boglins shrieked as they died and tried to crawl over each other to reach the tower steps. They were already flowing up the ladders to the curtain wall\u2014up the top and the underside of the ladders. They jammed the open doorway of the tower, and Gerald Random set his feet and fought to hold the door.\n\n\"Fortress is signalling!\" called Nick Draper. \" _On the way.\"_\n\nRandom set his teeth, and slammed his visor shut.\n\nThe arrows flying from the towers were answered by flights of arrows from the ground outside, from the courtyard\u2014the hole there was a yawning maw vomiting monsters.\n\nThere were massive irks, nothing like the slim elfin creatures he'd seen before, but as big as a big man, armoured in ring mail with shields and long swords. There were more boglins as white as the moon, with hooked spears and iron plate. They came at him in one gout.\n\nThe farm-boys slammed spears past him\u2014sometimes they fouled his sword arm, and one pinked him in the buttock, but he was their shield and they were his weapon, their nine-foot spears pinning the armoured things so that Random cut pieces off them\u2014and just past the door, the hail of shafts continued to reap the enemy.\n\nBut there were more and more of the things out in the courtyard.\n\nTo all appearances, the sortie emerged after a concerted volley from all the engines in the fortress\u2014a veritable rain of projectiles from fist-sized rubble to twenty-pound rocks; crossbow shafts two feet long and weighing two pounds.\n\nThe sortie rode down the fortress ridge at top speed, a blur of motion at the edge of the dark, and halted at the foot of the ridge to form its wedge. But they took too long. Men and horses were too far behind\u2014other men had over-ridden the assembly point and had to turn back\u2014and a hundred heartbeats were consumed achieving their formation.\n\nThorn watched the enemy sortie emerge. He watched them ride down the cliff face and he tasted the power of the phantasm that surrounded them. And spat at the taste.\n\nThorn sent the signal to his ambush, and triggered the massive spell he had spent the day preparing. Power leapt across the late morning light, raw and green, and coalesced\u2014\n\nThorn choked.\n\nThat was _not_ the sortie. It was an illusion. The spectre of a sortie.\n\nThe Fallen Magus roared his rage. But it was too late, and the carefully prepared power of his magic fist slammed to empty earth.\n\n# Lissen Carak\u2014Harmodius\n\n\"He didn't used to be this easy,\" Harmodius said, looking up to the captain, who sat on a borrowed destrier. The Magus grinned like a small boy. \"The Wild has sapped his imagination.\"\n\nThe shattering thunderclap of the outpouring of the Enemy's power rang in their ears and the massive flash still burned across the captain's retinas. \"Can he do that again?\" the captain asked.\n\n\"Perhaps, Harmodius admitted. \"I doubt it, though.\"\n\nThe captain exchanged a glance with Sauce, who rode by his side. It was Tom's turn to have the duty, and the big man was fretting about missing the sortie.\n\n\"No heroics,\" the captain called. \"Right across the plain to the castle, then around the walls. Kill anything that comes under our hooves.\"\n\n# The Wild\u2014Peter\n\nPeter had just finished making breakfast when the two boglins came to his fire. They had a pair of rabbits, already skinned, in each arm\u2014eight rabbits in all. They also had a large animal carcass\u2014also field dressed\u2014carried between them on a pole.\n\n\"U kuk fr us?\" said the larger one.\n\nPeter realised with a shock that the larger mammal was a woman\u2014beheaded and skinned. Gutted. _Cleaned._\n\n\"Kuk?\" said the larger boglin.\n\nPeter took a deep breath, pointed at the dead woman, and shook his head. \"I will not cook a person,\" he said.\n\nHe had his fire going, and he had already fed his friends. So he handed the remains of his squash and squirrel stew with oregano to the larger boglin. \"Eat,\" he said.\n\nThe boglin looked at its partner. They touched their heads together for a moment, and a flood of acrid, complex smells filled the air.\n\nThe smaller boglin opened its gullet and swallowed half, and then passed the small copper pot to the larger boglin, who consumed the rest.\n\nPeter didn't watch.\n\nOta Qwan came and stood by him. \"Aren't you two supposed to be in the big attack?\" he asked.\n\nThey remained perfectly still. Animal still. As if they couldn't hear him.\n\n\"Kuk?\" asked the larger.\n\n\"I\u2014will\u2014cook\u2014the\u2014rabbits.\" Peter spoke slowly.\n\n\"Gud.\" The larger boglin bobbed. \"Go kill. Back to eat.\" He made a chittering noise, his partner joined him, and they bent forward and loped off into the gathering night.\n\nOta Qwan looked at Peter. \"Do you have the power, laddy?\" he asked.\n\nPeter shook his head.\n\nOta Qwan shrugged. \"Among the Sossag people it is mostly shamans who can talk to the Wild,\" he said. \"I would like to have boglins to follow me,\" he said. \"If they offer to join us, accept.\"\n\nPeter swallowed. \"You would have them in camp?\"\n\nOta Qwan shook his head in mock anger. \"Boglins are big medicine, you know that?\"\n\n\"Where do they come from?\" Peter asked. \"I had never seen one before\u2014I came here.\"\n\nOta Qwan sat by the corpse of the gutted woman. He didn't seem to notice her, or care. \"I don't know, but I can tell you what men say. The word is that they grow in great colonies like giant termite hills in the deep Wild\u2014way out west. All the creatures of the Wild fear them. The great Powers of the Wild cultivate them, recruit whole colonies, and send them to their deaths.\" Ota Qwan sighed. \"I've heard said they were made\u2014they were created\u2014by a great Power. To fight an ancient war.\"\n\nPeter shook his head. \"That's just a way of saying you don't know.\"\n\n\"Don't I?\" Ota Qwan laughed. \"You have so much to learn about the Wild. Because the Powers pretend that they fear nothing, but they fear the little boglins. A thousand boglins are a fearful sight. A million boglins\u2014\" He shrugged. \"If they could be fed, they could conquer the world.\"\n\nPeter swallowed bile.\n\n\"Maybe you could cook for them, eh?\" Ota Qwan said. \"You know the matrons have given you a name?\"\n\nPeter nodded expectantly.\n\n\"Nita Qwan.\" Ota Qwan nodded expectantly. \"A very potent name. Well done.\"\n\nPeer sounded it out in his head. \"Gives\u2014something.\"\n\n\"He gives life,\" Ota Qwan said.\n\n\"Like your name,\" Peter said.\n\n\"Yes. They see us together. I like that.\" He nodded.\n\n\"What is Ota?\" Peter asked.\n\n\"Take. Like _ota nere_ !\" he paused.\n\n\"Take water. When we are on the march.\" Peter nodded. And then turned. \"You are _Take Life_ and I am _Give Life._ \"\n\nOta Qwan laughed. \"Got it in one. You were Grundag. Now you are Nita Qwan. My brother. And my symbolic opposite.\" He nodded again. \"Now\u2014recruit me those boglins. This siege is almost over; we'll go home as soon as the dead are eaten.\"\n\nPeter shook his head. \"I lack your experience of war,\" he said. \"But the Alban Royal Army is just coming up the Vale of the Cohocton.\"\n\nOta Qwan rubbed his chin. \"That,\" he said, \"is a very good point. But Thorn says we will triumph tonight.\"\n\n\"How?\" Nita Qwan asked.\n\n\"Pick up your bow and spear and come with me,\" Ota Qwan said.\n\nNita Qwan put the rabbits on green stick spits and left his woman to turn them. He took up his bow and his new spear, tipped with the fine blued-steel head that had come to him as a share of his spoils from the Fight at the Ford. He had many new things, and his woman was impressed.\n\nAnd it had only cost him a year of his life. But he spat and followed Ota Qwan, because it was easier to follow than to think. He ran, and caught Ota Qwan by the elbow. The war leader stopped.\n\n\"One thing,\" Nita Qwan said.\n\n\"Be quick, laddy,\" Ota Qwan said.\n\n\"I'm not anyone's lad. Not yours, not anyone's. Got me?\" Nita Qwan's eyes bored straight into the war leader's.\n\nHe didn't flinch. But after several breaths, his nostrils flared, and he smiled. \"I hear you, Nita Qwan.\"\n\nHe turned and ran, and Nita Qwan followed, better satisfied.\n\nAt the edge of the woods, many of the surviving Sossag warriors were waiting\u2014almost five hundred of them. Beyond them, painted fiery red in the sun were Abenacki, and even a few Mohak, in their characteristic skeleton paint.\n\nThe Abenacki war chief, Akra Crom, walked to the centre, between the groups. He raised an axe from his belt and held it over his head.\n\nOta Qwan smiled. \"If he falls today,\" Ota Qwan said, \"I will be war chief of the Sossag, and perhaps the Abenacki, too.\"\n\nNita Qwan felt as if he'd been punched in the gut.\n\n\"Don't be so na\u00efve,\" the older man said. \"This is the Wild.\"\n\nNita Qwan took a deep breath. \"What does he say?\"\n\n\"He says that if we ever want to get home, we must fight well tonight for Thorn, and kill the armoured horsemen as we have so many times. We have a thousand warriors. We have bows, and axes. La di da.\" Ota Qwan looked around. \"In truth, this Thorn doesn't seem to have a serious plan for us\u2014as if he thinks that by ordering us out of the woods and into the fields, we will kill all the knights.\" He shrugged.\n\nNita Qwan shuddered.\n\nOta Qwan put an arm around him. \"We will go and lie in ambush by the enemy back gate,\" he said. He barely waited for the Abenacki man to stop his oration before he rose to his feet, shook his spear, and the Sossag gave a scream of power and followed Ota Qwan into the green of the woods.\n\n# Lissen Carak\u2014The Red Knight\n\nThe horses were all tired, and many of them bore light wounds, muscle strains, scars\u2014and so did their riders.\n\nThere were twenty-five men-at-arms\u2014a pitiful number against a sea of foes.\n\nAnd at the base of the ridge, a perfect circle of cooling glass marked the best efforts of their foe.\n\nThe captain was operating in a haze of fatigue and minor pains that all but subsumed emotion. He knew\u2014at a remove\u2014that the Abbess was gone. That Grendel, almost a friend, was dead and probably eaten down on the plain. That his beloved tutor was cold marble\u2014no longer even a simulacrum of life.\n\nBut at another level, he walled all that away.\n\n_Can you fight every day?_\n\nHe knew he could. Every day, until the sun died.\n\nThe place in his head where his friends were dying was like a bad tooth, and by an effort of will, he didn't run his tongue over it.\n\nNor did he think, _If we win today, we're saved._\n\nHe didn't think that, because he didn't really think much beyond his next stratagem, and he was now pretty much out of tricks.\n\nAll of this went through his head between one leap of his new mount and the next.\n\nHe hurt.\n\nThey all did.\n\nAnd then the sortie was down onto the plain, and forming their wedge.\n\nRandom was more tired than he had ever been, and had he not been wearing first-rate armour, he'd long since have been dead. As it was, blows slammed into him more and more often as the monsters in the courtyard crawled over their own dead to reach him.\n\nTwice, shouts behind him told him that more of the cursed things had made it onto the tower or the wall\u2014apparently using their vestigial wings, or perhaps they were a new and horrible breed\u2014but the spearmen at his back held their ground.\n\nTwice he had a respite from the attacks on the door, but he had no idea why the white things stopped coming. He would pant, someone would hand him water, and then they'd come again. The white boglins were bad. The big irks were worse.\n\nA farmer tried to help him in the doorway\u2014braver or stupider than the rest\u2014and died almost as soon as he took his place, while one of his mates begged him not to go.\n\n\"Ye have no armour!\" a bigger, Harndon accented man called.\n\nHe didn't have armour on his arms and legs, and the wicked scythes on their limbs sliced him to pieces, dragged him down and carved him up. And they ate him\u2014even the dying ones took a bite.\n\nRandom couldn't lift the buckler high any more. He knew it was just a matter of time before he was struck in the visor or the groin\u2014only luck and the efforts of the spearmen kept him at it.\n\nMore irks came. They took their time coming over the low mound of dead, and they all came at him together. A shield caught his outstretched arm\u2014the vambrace held the blow, but he was unbalanced, and the boglins dragged him to his knees\u2014a blow struck the back of his helmet and he was _down._\n\nHe could feel a sharp pain across his instep\u2014something was hacking at his armoured shin\u2014and then, to his horror, he began to be dragged out of the doorway, into the pile of corpses.\n\nHe couldn't help it. He screamed.\n\nAnd then he wasn't being dragged, and a heavy weight crushed him. Only the strength of his breastplate and his backplate and their hinges kept the crushing weight from taking the breath out of him.\n\nThere was a sharper pain in his right foot.\n\nHe tried to call out, and suddenly his helmet was full of liquid\u2014he spat. It was hell\u2014dark\u2014bitter. He choked and spat and realised that he was drowning.\n\nIn boglin blood.\n\nHe tried to scream.\n\nMore pain.\n\n_Christ, I am being eaten alive._\n\n_Christ, save me in my hour of need._\n\n# The Wild\u2014Peter\n\nNita Qwan loped through the woods. The circle of the sun was high overhead. It was a poor time to set a trap, and he wanted to wait for night, but it was late spring, and darkness\u2014true darkness\u2014was still a long way away.\n\nA brilliant emerald flash lit the sky to the south. A titanic concussion rocked the earth.\n\nOta Qwan grinned. \"Our signal. He is mighty, our chief. Let's go! Gots onah!\" The acting war chief ran ahead of the band, and they began to sprint over the grass, angling east, and the summer light threw shadows under them.\n\nThey had almost a mile to go. Nita Qwan was a strong man, and had lived with the Sossag for weeks, but running a mile to fight was the most exhausting work\u2014especially after a morning of food gathering and cooking. He put his head down and tried to seal off his mind from his thighs and his lungs, and he ran.\n\nIt took many long minutes to run all the way to the east of the great ridge, but finally, Ota Qwan raised a hand. \"Down!\" he called, and the People fell to the earth in the tall grass. He turned to Skahas Gaho and another warrior and sent them off farther to the east, and then he lay down by Nita Qwan.\n\n\"Not long now,\" he said. \"We are in the right place. Now we see if Thorn knows what he is doing.\"\n\n# Lissen Carak\u2014Thorn\n\nThorn watched the action develop from the utter safety of the western edge of the woods. He was not strong enough to risk himself today\u2014because he'd thrown too much in a single casting. It rankled. But he had thousands of servants to aid him, and today he was spending them like water, his usual caution forgotten.\n\nMany of his servants would have been disturbed to note that he had already decided to use them all, if he had to. He knew where more creatures of the Wild could be raised. He himself was irreplaceable.\n\nAnd she was dead.\n\nHe had made mistakes, but the end game was going to play out with the inevitability of one of those ancient plays he had once so enjoyed, and now could no longer remember.\n\nThe king would come, and be defeated. That trap was already laid.\n\nAnd then it would all be his.\n\n# Albinkirk\u2014de Vrailly\n\nHe could no longer set his tent away from the army. Tonight, the army was camped hard by a small stream that ran down to the Cohocton; the carcass of a great beast of the Wild lay in sodden and hideous majesty, the bones picked redly clean in mid-stream. A litter of corpses and the screams and quarrels of the animals that fed on the recent dead marked the scene of a recent battle.\n\nThe king ordered the wagons pulled in, trace to axle tree, a fortress of tall, wheeled carts chained at the hubs, and even de Vrailly couldn't fault him for his caution. They were in the very midst of the Wild, and the enemy was palpable, all around them. Many of the footsoldiers and not a few of the knights were afraid\u2014scared, or even terrified. De Vrailly could hear their womanish laughter in the firelit dark, but he himself knew nothing but a fierce joy that at last\u2014 _at last_ \u2014he would be tested, and found worthy. The much-discussed fortress of Lissen Carak was three leagues away to the north, the Queen's flotilla was, by all reports, already lying in mid-stream, ready to support their attack in the morning. Even the cautious old women of the king's council were forced to admit that there would be a battle.\n\nHe was kneeling before his prie-dieu when the angel came. He came with a small thunderclap and a burst of myrrh.\n\nDe Vrailly cried out.\n\nThe angel hovered, and then sank to the earth, his great spear touching the cross-beam of the great tent.\n\n\"My lord de Vrailly,\" the angel said. \"The greatest knight in the world.\"\n\n\"You mock me,\" de Vrailly said.\n\n\"Tomorrow will see you acknowledged as such by every man,\" said the angel.\n\nJean de Vrailly was struggling with his doubt. He felt as a man does who knows he should _not_ mention a certain fact to his wife, but does so, anyway\u2014precipitating an avoidable argument. \"You said we would fight a battle,\" he said, hating the whine of doubt in his voice. \"At Albinkirk.\"\n\nThe angel nodded. \"I am not God,\" he said. \"I am merely a servant. The battle will be here. It _should_ have been at Albinkirk, but forces\u2014circumstances\u2014forced my hand.\"\n\nThe angel's hesitation froze de Vrailly.\n\n\"What forces, my lord?\" asked Jean de Vrailly.\n\n\"Mind your own role, and leave me to mine,\" said the angel. His voice sounded like a whip-crack. Like de Vrailly's own. Beautiful and terrible. Imbued with power.\n\nDe Vrailly sighed. \"I await your orders,\" he said.\n\nThe angel nodded. \"Tomorrow, at dawn, the king will attack. The Enemy has a blocking force on the road between here and the bridge. Let the king lead the attack on that force, and when he falls\u2014\" The angel paused.\n\nDe Vrailly felt his heart stop.\n\n\"When he falls, seize command. Cut your way free, save the king's army, and you will save the day.\" The angel's voice was pure and precise. \"His day is done. He has failed. But he will die well, and you, my lord, will take the woman and be king. She is the kingdom. Her father was the greatest lord of Alba next to the king. With the woman, you will rule. Without her\u2014you will not. Am I making myself clear to you?\"\n\nDe Vrailly's eyes narrowed. \"And what of the north?\" he asked. \"If I am to save the army, am I to let this mighty fortress fall?\"\n\n\"You can retake it,\" the angel said reasonably. \"When you bring an army from Galle.\"\n\nDe Vrailly bent his proud head, shading his eyes from the brightness of the angel. \"Pardon me, my lord,\" he said aloud. \"I have doubted, and been misled by false images.\"\n\nThe angel touched his head. \"God forgives you, my son. Remember\u2014when the king falls, take command, and cut your way clear.\"\n\nDe Vrailly nodded, eyes downcast. \"I understand very well. My lord.\"\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain pointed his wedge south and raised his hand. He could feel the heat coming off the hot glass circle to their right\u2014it went right through his steel gauntlet and his glove.\n\n_Ouch_ , he thought. And thanked Harmodius with a silent nod.\n\n\"Let's ride,\" he called, and they trotted forward, formed tightly. A perfect target for another burst of power.\n\nHis back tingled as he rode _away_ from where he felt his enemy to be, towards the near corner of the Bridge Castle, just two hundred horse-lengths away or less.\n\nThe wedge negotiated the trench\u2014last night it had been an inferno\u2014crossing it carefully and wasting precious time. Some men had to dismount.\n\nIt was still better than riding the other way around the walls.\n\nSome men jumped it, but most men were less flashy and more cautious.\n\nThey reformed on the far side, unopposed.\n\nThe captain rose in his stirrups. He pointed across the darkening grass toward the near corner of the Bridge Castle.\n\n\"It's a trap. If it wasn't, those boglins\u2014\" the captain pointed at a hundred or more boglins who were watching them from a hastily erected earthen assault ramp that rose to the top of the wall of the Bridge Castle \"\u2014those boglins would have tried to hold the trench against us. Instead of watching like spectators.\"\n\n\"Has the Bridge Castle fallen?\" Sauce asked.\n\nThe captain watched it for ten heartbeats. \"No,\" he said.\n\nThe Prior of Harndon came up on his left side. \"If you let me send my signal, my knights will ride to meet us,\" he said. \"They are just there, in the woods closest to the river.\"\n\nThe captain looked a little longer. \"Catching their ambush between two hammers,\" he said. \"Yes.\" He turned to his valet. \"Sound\u2014single rank, full interval.\"\n\n# Lissen Carak\u2014Peter\n\nOta Qwan was on his knees in the high grass. The enemy\u2014a small party of knights in highly polished armour\u2014had hesitated at the edge of the Trench of Fire, as the Sossag called it now, though it was black and cold in the sun.\n\n\"That lordling knows his business,\" Ota Qwan said. \"I don't know him\u2014lacs d'amour? Whose banner is that?\" He spat. \"He's spreading his knights.\"\n\n\"So?\" Nika Qwan asked.\n\n\"So in a tight bunch, his men kill a few unlucky warriors and we massacre them from all sides. In a long line, every one of them kills a warrior\u2014or maybe five. It is a lucky warrior who gets an arrow into one of them.\"\n\nThe knights began to come forward in the strong light, and the blue sky was mirrored in their harness. They looked like monsters from the Aether\u2014like mythical beasts. The overhead sun sparkled from their harness and stung men's eyes.\n\nSkahas Gaho appeared as if by magic from the grass. \"More tin-men behind us,\" he said. \"Forming by the woods closest to the river.\" He shrugged. \"Their horses are wet. They swam the river.\"\n\nOta Qwan made a grunt. Nita Qwan could see he'd made his decision, just in that moment. The war leader stood, put a horn to his lips, and sounded a long call.\n\nThe Sossag stood and ran like songbirds before an eagle. They ran north, even as the two long lines of knights closed on them.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain watched the painted man rise from the grass just a hundred horse lengths in front of him, sound his horn, and begin to sprint north, out of the closing jaws of the counter-trap. He watched with a sense of failure and the vaguest professional admiration. He knew the Outwallers.\n\nHe ordered his valet to sound \"Charge\u2014ahead!\"\n\nHis line caught a handful of stragglers but, obedient to his orders, the line swept east and south, and didn't deviate to pursue the Sossag. Arrows flew as the Sossag rearguard gave their lives for their fellows, and one man-at-arms went down in a tangle of armour plate and dead horse, and then the black-clad knights from the riverside swept over the rearguard, killing every one of them in an instant, no quarter given.\n\nThe Prior moved past him, raised his hand, and summoned the military order knights to him without a word being spoken. It was a magnificent display of power.\n\nThe captain shook his head. \"I thought we were good,\" he said.\n\nSauce had blood on her lance tip, and she reined in. Jacques was sounding the rally, and a wounded knight\u2014Ser Tancred\u2014was being hauled bodily onto Ser Jehannes's horse. She leaned over. \"We are good,\" she said.\n\nTo their left front the whole squadron of black- and red-clad knights went from a galloping charge to a dead stop in a few hoofbeats\u2014then wheeled right around as if performing some gypsy horse trick and halted facing the Bridge Castle in a neat wedge.\n\nSauce shook her head\u2014not a big motion in an aventail and bassinet. \"Sweet Jesu. They are good,\" she admitted reluctantly.\n\nThe Prior cantered to the centre of the new line. \"Well, Captain?\" he asked. \"Shall we relieve the castle?\"\n\nThe captain raised his hand. \"At your command, Prior.\"\n\nSeventy mailed knights made the earth tremble.\n\nThe boglins scattered.\n\n# Lissen Carak\u2014Thorn\n\nThorn watched in weary anger as his useless allies ran rather than face the knights. So many claims\u2014so many boasts that they could fight anything, that they could conquer the maille-clad riders.\n\nHe watched them run, and knew\u2014with the pain of intimate and exact intellect\u2014that his entire plan for the day would come apart.\n\nA burst of power from the field alerted him. The power itself was very low in intensity, but also very tightly controlled. Only someone as imbued with mastery as he himself would detect it.\n\nAnd immediately recognise the wielder.\n\nPrior Mark.\n\nThorn watched as the Prior used his power to pass signals to his knights\u2014to turn them into finely crafted weapons, responsive to his will. Another man who loved power.\n\nFor a moment, he considered using all of his remaining puissance in a single spell to kill the Prior.\n\nBut that was foolish. He needed that power. He reminded himself that there was no hurry. That the king's army would never reach the river.\n\nBut the fall of the Bridge Castle would have made all that unnecessary.\n\nThorn rarely spoke aloud. He had no peers to whom he could speak his mind\u2014voice his indecision, his secret fears.\n\nBut he turned to his startled guards. The shamans who worshipped him. The cloud of midge-like followers who attended his every need. His voice came out as a harsh croak, like the voice of a raven.\n\n\"Thirty days ago, a daemon sought to take this place from an old woman with no soldiers,\" he said. \"Fate and bad luck have left _me_ to contest it with the King of Alba and whole armies of knights, with a dozen able magi and now with the best warriors in the world.\" He laughed, and his wicked croak startled the birds in the trees. \"And yet I will still conquer.\"\n\n# Lissen Carak\u2014The Red Knight\n\nNothing withstood their charge, and the strong band of knights scoured the ground around the Bridge Castle. They rode all the way around it, close against the walls, killing every creature of the Wild that didn't scuttle clear of their path. The lesser boglins rose in brief bursts of flight or lay flat in the tall grass where they were difficult to find, and the greater boglins and irks, those with armour, struggled into their hastily dug tunnels to emerge in one last spurt of violence to the burning hell of the Bridge Castle courtyard.\n\nThe captain raised his hand for his company to halt when they returned to the base of the soft earth ramp that the worker-Boglins had run up to the curtain wall on the north side of the Bridge Castle.\n\n\"Dismount!\" he called. The sun was past noon, but still high. There were streaks of cloud in the west, but hours of daylight remained. Still, experience told him that if he didn't clear the courtyard before full dark he would lose the Bridge Castle.\n\nAnd thus lose his connection to the king.\n\nIf the king was coming at all.\n\nEvery fifth valet took ten horses in his fist.\n\n\"Spears!\" the captain called, and his men formed a tight line at the base of the ramp; men-at-arms in front, valets and squires in the middle, and archers in the rear rank.\n\nThe Prior rode up and saluted. \"We'll cover you!\"\n\nThe captain saluted as Michael handed him his heavy spear. \"If we aren't out before full dark,\" the captain said, \"Assume the bridge is lost.\"\n\nThe Prior crossed himself. \"God go with you, Ser Knight.\"\n\n\"God doesn't give a shit,\" the captain said. \"But it's the thought that counts. On me!\" he called, and started up the slope of new turned earth. It was damp and hard\u2014hardened with something excreted by the boglins, to judge from the smell. Acrid, like naphtha.\n\nThere were fifty boglins on the wall, and they died when the men-at-arms ripped through them.\n\nThe captain looked down into the inferno of the courtyard. All the merchant wagons were afire, and the courtyard crawled with figures like the damned in hell\u2014men stripped of their skin, shrieking their lungs out; armoured boglins in glowing, fire-lit white. Most of them crowded to the door of the nearest tower, but more poured from a gaping wound in the earth where a dozen flagstones had been hurled aside, like maggots in a bloated corpse when it is opened. More boglins on the walls\u2014but on the east wall, a small, disciplined company fought back to back, holding the opposite curtain against assault from both directions.\n\n\"Files from the right!\" the captain called, and led his men down off the curtain wall\u2014down the ramp intended for siege engines to be hauled up to the curtain, and there were a pair of pale boglins gleaming there, each with a pole-axe.\n\nHe had no time for finesse. He raised his spear, point low and butt high and caught the first creature's heavy cut on his haft\u2014wrapped its arm with his own in the high key that men practised when wrestling in armour\u2014and then ripped its arm from its body like a man ripping a crab leg from a new-cooked crab.\n\nThe thing's other arm came at him\u2014he rammed his spear point into its head, let go of the shaft with his armoured left hand and punched into the boglin's throat. It's great maw opened, mandibles flashing at his visor\u2014overhand, he rammed the spearhead down its gullet and acrid ichor blew out of the top of it like lava from a new volcano.\n\n\"Form your front!\" he roared, even as Sauce beheaded the second armoured boglin with her axe.\n\nSer Jehannes came up on his left, and Sauce cleared her weapon and fell in next, tapping her axe-haft against the breastplates of Ser Jehannes and Ser Tancred, and the line was formed.\n\nThe armoured creatures were trying to overrun the defenders of the north tower, and the captain pointed with his spear. \"Charge!\" he called.\n\nTwenty paces into the rear of the things.\n\nHis sabatons rang on the pavement\u2014he stumbled on a corpse.\n\nAnd then\u2014a storm of iron. Skittering screeches and staccato clicks like the beat of an insane drummer as the mass by the North Tower turned and charged him.\n\nIn the first meeting he was head to head with an armoured monster the size of Bad Tom\u2014the complex interlacing of its front armour over the interstices of its six armour plates was like an obscene mouth as the thing reared back, its whole strength bent on a single, crushing blow from its great hammer, its body bent like a bow with the effort.\n\nHe set his feet and took the blow on his haft, rotated the weapon on the pivot of his opponent's blow, and slammed a spike into the middle of its helmeted head. His spike penetrated the thing's face plate, and it spasmed.\n\nBehind his dying opponent towered another, wielding two long swords, and even as he watched, the thing beheaded Ser Jehannes's new squire, the two weapons coming together like a tailor's shears. Jehannes leaped to avenge his squire and took a pommel to the helmet that staggered him, and two lightning fast blows followed it, literally beating him to the ground.\n\nThe captain's command sense shrieked in panic. The boglins had stopped his men-at-arms. It shouldn't have been possible. There was nothing in the Wild that could stop twenty fully armoured men.\n\nNot many things.\n\nThe captain paused and locked eyes with the thing standing over Jehannes, and it _knew_ him. He leaped at the double-sworded thing, but his spear remained lodged in his last kill, and he had to leave it.\n\nDouble Sword turned from his prey\u2014Jehannes\u2014and faced him. It was yet another kind of boglin\u2014sleek, taller than Bad Tom and heavily muscled, with man-made chainmail covering all its joints and feral, organic plate armour that might have been grown, or very finely forged. A wight.\n\nAt the edge of his peripheral vision, Sauce rammed a spike through the carapace of another armoured monster and screamed her war cry.\n\nSer Tancred was locked with another, his arms straining against it as his squire stabbed his long sword into its armpit\u2014rapid, professional stabs that made its limbs thrash.\n\nDouble Sword tapped its blades together and leaped at him with animal rapidity.\n\nThe captain snatched his rondel dagger from his belt and trusted his armour. He entered between the blades, arms high, dagger in both fists, and the longsword blows crashed into his shoulder plates. The hardened steel bent and split\u2014only to cut into the rings of the mail haubergeon underneath, and the blades were held, though the force drove through the thickly padded jupon under the mail, and still managed to bruise his shoulders...\n\nBut he swung the dagger overhand, two handed through the boglin's mail aventail and into its neck.\n\nSix times.\n\nIt's limbs spasmed, but it's forearms tightened like a band of steel around the captain's shoulders. And it lit up with power, eyes glowing cool blue as it prepared\u2014\n\nHe drove his armoured knee in between its legs\u2014nothing there to hurt, but his blow took it off balance, and he pushed his left foot forward and threw the thing over his outstretched right leg. Its wing cases snarled in his knee armour's flanges and ripped free. Its own weight accelerated its fall, but its limbs clasped him fast, and he fell atop it, his rondel dagger a projection from his fists.\n\nHis steel carapace held.\n\nThe monster's didn't. The triangular blade punched cleanly though it, and ichor jetted out.\n\nHe didn't stop, but pulled the foot-long steel dagger clear of the wound and drove it up under the thing's mandibles that were opening and closing with terrific force on the slick metal of his helmet. They ripped his visor off his face, forcing his head around in a painful arc, and he was eye to eye with the thing\u2014its eyes glowing with unfocused power.\n\nHe countered with a lightning blow to its nearer eye-patch. He raked the point through the oblong eye\u2014and again, and again, as a scythed foreleg reached for his face.\n\nIt was not going to die before it cast its phantasm.\n\nHe got his left gauntlet under its head and slammed the dagger into its left eye\u2014through the eye patch, through the skin and bone. He reached for his memory palace to fight its power, even as he stirred its brains with the blade...\n\nAnd a wave of power entered him\u2014a sickly blue wave of chilling intensity, and he writhed\u2014\n\nIts eyes went out.\n\nHe took its force into him, subsuming the alien thing as creatures of the Wild do. He had never done it before, and hadn't known how. He thought it was probably best that Prudentia hadn't been there to watch.\n\nHe bounced to his feet, suddenly awash in concentrated calculations as to the survivability of his host under the conditions of the current combat, and for a fleeting instant, the captain was able to see and calculate as _both sides_ in the courtyard.\n\nBut the balance had shifted.\n\nA third of his men-at-arms were down\u2014dead, wounded, or merely tripped, he had no way of knowing, but the back of the enemy resistance was broken and already the fringes of the melee had become more like a hunt than a fight.\n\nHis archers began to clear the walls, their shafts joined by the dozen archers loosing from the towers, and the pace of victory accelerated. A dozen of the white boglins scuttled down a hole. A man, half the skin ripped from his flesh and trailing down his back, screamed again, and an archer put a shaft into his throat with rough mercy, and stopped his screams\u2014and all through the courtyard, armoured figures opened their visors and heaved air into desperate lungs.\n\nThe captain walked up a ramp of dead bodies to the door of the north tower where a young giant, drenched in acrid boglin-blood, stood leaning on a six-foot bill with a heavy steel head, coated in gore.\n\n\"Well fought, young Daniel,\" the captain said.\n\nThe former carter shrugged. \"Twas Master Random held the door, Cap'n. For most part of an hour, seems to me.\"\n\n\"Dead?\" the captain asked.\n\nDaniel shrugged again. \"They drug him into the pile,\" he said. \"We fought 'em for the corpse but lost him when you charged their rear.\" He stood straighter. \"Deserves finding, I think.\" He seemed to shake off his fatigue, and then he reached out, spiked an armoured boglin on the back-spike of his bill, and flung it from the pile like a farmer moving hay with a pitchfork.\n\nThe captain grabbed another. Dead, the boglins were curiously harmless\u2014disgusting, but less insectoid, and more animal. He tossed one aside, and then another. His hands shook. His knees were weak.\n\nHe was _insanely_ full of power.\n\nSauce joined him. \"What are we doing? Killing the wounded?\" she asked, her voice a little too sharp and bright. This was a fight that men\u2014and women\u2014would relive too many times.\n\n\"Looking for a body,\" the captain said. He was down to waist level, now.\n\n\"I've got his leg!\" Daniel called.\n\nMichael joined them, and suddenly there was Ser Milus, and Ser Jehannes, blood still leaking from the joints of his shoulder, and they hauled, and the corpse of the merchant stiffened, and he screamed.\n\nHis armour was slick with boglin blood, and human, and he popped out of the pile of corpses. The flesh of his left foot was gone at the ankle, and blood was leaking too slowly out of the wound where sharp mandibles had flensed the flesh from his foot.\n\n\"Tourniquet! Cut his greave off!\" the captain shouted.\n\nDaniel already had a small knife in his great paw of a hand, and he slit the straps holding the greave\u2014Sauce opened the catch and the greave came loose with a gout of fresh blood.\n\nThe captain grabbed the stump of his leg. Sauce got her sword belt around the small of the ankle, got it through the buckle, and pulled with all her strength.\n\nThe blood stopped.\n\n\"Tie it off,\" the captain said unnecessarily. Every soldier in his company could be a leach in an emergency.\n\nThen he took a weary breath and ran for the wall.\n\n# Lissen Carak\u2014Thorn\n\nThorn felt the dark sun take Exrech and he cursed. Cursed that he had been fooled again, cursed that every encounter seemed to go against him.\n\nThe accession of power by the dark sun made him far more dangerous than he had been.\n\nThorn reached out to the two Sossag shamans attending him and subsumed them, stripping their essences and their power, feeding on it. Their empty corpses collapsed to the earth. It wasn't much power, but sufficient for him to _see_ and _send._\n\nThe coming darkness was not his friend. He needed light, where he could deploy his superior numbers and his massed archery.\n\nAnd then he sent his powerful senses questing for Clackak. Found him deep in the earth under the stone fort by the water, with a hundred more of his kin.\n\n_Break off_ , he demanded.\n\nThe sun had begun to slide toward evening. There were long hours until night.\n\nThorn shook his massive head and torso. \"Tomorrow,\" he said.\n\n# Lissen Carak\u2014The Red Knight\n\nThe archers opened the gate and the knights rode in, their black hooded surcotes hiding the gleam of armour, their black horses like nightmare creatures in the full dark.\n\nThe Prior rode to the captain, who was sitting on a folding stool, scraping crap out of his sabatons to make the plates work properly. His whole body felt like a badly maintained machine.\n\n\"With God's help, you have conquered,\" the Prior said.\n\n\"If you like,\" the captain said. \"We have conquered, for the moment. But only by the skin of our teeth, as old wives say. And where are the wyverns? Where are the daemons? The Jacks?\" He gazed out into the last light. Killing off the last of the boglins had taken another hour, and now the enemy machines were throwing stones again.\n\nThe valets were stacking corpses outside the gate. The courtyard of the Bridge Castle stank of burned wood, dead boglin and ordure\u2014horses killed in their traces, oxen butchered, men and boglins dead. The rotting meat smell rose like an evil sacrifice in the too-warm evening air, and midges were settling on the working men like an evil plague.\n\nThe Prior dismounted, his own sabatons ringing on the stones of the courtyard. \"Where indeed? I haven't seen so many evil creatures in many years.\"\n\n\"We saw them every day. Now they are gone,\" the Red Knight said. \"Next wave, perhaps?\" he added. \"That's my guess. Wear us out with the boglins. Then break us with the bigger creatures.\" He tested his foot on the ground.\n\n\"Then\u2014\"\n\n\"It's what I'd do. Bleed us with the easily replaced critters and save the others. He needs them to fight the king. This was all just to fix us in place.\"\n\n\"We can hold until the king comes,\" the Prior said. He was pulling his sodden arming cap off his head and paused to slap a mosquito.\n\n\"Despite wyverns and daemons? I hope so,\" said the captain. He got to his feet. \"Michael\u2014tell the valets to serve beer and maple sugar.\" He smiled at the Prior. \"It's going to be a long night.\" He looked around. \"Gelfred?\"\n\n\"My lord?\" Gelfred said.\n\n\"I need you to do something insanely brave,\" he said.\n\nGelfred shrugged.\n\n\"Can you get a message to the king?\" the captain asked.\n\n\"In the dark? Through a host of enemies?\" Gelfred smiled. \"I can with God's help. And by my faith, messire, if you make a crack about God not caring, you can take your cursed message yourself.\"\n\nThe captain got to his feet and gave the huntsman his hand. \"I am rebuked, Gelfred.\"\n\nGelfred shrugged. \"Join me in prayer,\" he said.\n\n\"Let's not get carried away,\" the captain replied.\n\nGelfred laughed. \"Why do I like you so much?\" he asked.\n\nThe captain shrugged. \"The feeling is mutual.\"\n\nHalf an hour later, Gelfred went straight into the river from the docks.\n\nHe swam for fifteen minutes in the dark, and then went with the current for a while to rest. He heard, or felt, a wyvern in the dark air overhead, and he went under the water and stayed down as long as he could. When he surfaced, his heart was beating so fast that he had to head for shore.\n\n\"There goes the bravest man in all my company,\" the Red Knight said to the Prior.\n\n\"Because he faces his fears?\" the Prior asked. \"He has God's aid.\"\n\nThe captain shook his head but said nothing. Only watched the darkness, and wished he was in the castle. He touched the soiled handkerchief pinned to his arming cote. It was no longer white, indeed, it held the blood and ichor of several foes, and it was cut almost in two.\n\n# Lissen Carak\u2014Amicia\n\nAmicia tried _not_ to go to the gate. She tried not to look out the window. When a party of men-at-arms clattered in on exhausted horses, she forced herself to wait until the wounded came in.\n\nSer Tancred told her that the Red Knight was spending the night in the Bridge Castle.\n\nWhen the last wounded were healed, she knelt in the chapel by the Abbess's bier and prayed. She opened herself, as the nuns had taught her, to God. And she made God a hard, heartfelt promise.\n\n# Somewhere\u2014Gelfred\n\nHe was tired and cold and very, very scared when he heard the sound of men's voices on the other bank, and he struck out for them. He swam quietly, as well as he could.\n\nThey had boats.\n\nAfter some time, he swam to the boats, and a sentry saw him.\n\n\"Halt! Alarm! Man in the water!\" A crossbow loosed, and the bolt passed somewhere near him.\n\n\"Friend!\" he spluttered. He was short of breath. \"From the fortress!\"\n\nThey were too alert, but they weren't great marksmen. He swam in, shouting that he was a friend. Eventually, they stopped loosing their bolts at him, and strong arms pulled him into a big barge.\n\n\"Take me to the king!\" he said.\n\nA big man with a hillman's accent pulled him over the side and put him on a bench. \"Drink this, laddy,\" he said. \"You've found the Queen, not the king.\"\n\n# Chapter Sixteen\n\n# Lissen Carak\u2014Michael\n\nMichael watched the captain sleep. It was dawn, or near enough, and he cursed that he was awake. He rose, pissed in a pot, drank half a glass of stale wine and spat it out into the courtyard.\n\nThe place stank like a charnel house, and most of the soldiers had slept in rows in the tower. In their harness.\n\nHe walked to the table, opened his wallet, and took out a pair of wax tablets, withdrew his stylus, and wrote:\n\n_The Siege of Lissen Carak. Day Fifteen_\n\n_Yesterday the enemy tried to storm Bridge Castle and, despite putting monsters inside, was repelled. We lost more than forty men, women, and children of the convoys, and three men-at-arms and two archers, as well as four men of the militia. These are our worst losses so far._\n\n_But the king is coming. Knights of the Holy Order of Saint Thomas came at nightfall on the thirteenth day to tell us we were saved. And yet we fought all day and the king did not come._\n\n_Where is the king?_\n\nMichael looked at the last line. He took the butt of the stylus to rub that line out. Then he shook his head, and went to wake the captain.\n\n# Near Lissen Carak\u2014The King\n\nThe sun was an arc of fire in the east.\n\nThe king's magnificent golden armour and brilliant red and blue heraldry caught the first rays of the sun so that he seemed to catch fire.\n\nBehind him stood three hundred of the most heavily armoured knights Alba had ever seen, their heavy horses left in camp.\n\nThe golden helmet moved to the right and left, examining the dressing of the long line of chivalric warriors that vanished into the woods on either flank, each with his heavily armoured squire at his back.\n\nHis golden gauntlet was raised high, and fell, and the line of the vanguard advanced along the line of the old Bridge Road. The three hundred knights were each a man's height apart, their line was a half-mile long, and the men at either end had hunter's horns\u2014noted horns, which they played back and forth like huntsmen.\n\nThe figure of the king seemed to dance forward joyfully.\n\nHe pressed through the woods, and the woods parted before him. There is nothing in the woods that can impede a man in full harness\u2014no branch, no trailing vine, no bank of thickset canes, no matter how thorny, will stop a man in armour. Or slow him.\n\nThe line ground forward at a walking pace.\n\nHalf a mile.\n\nA mile.\n\nHe raised his hand and his own horn bearer played a long note and the line stopped.\n\nMen-at-arms raised their visors and drank water, but the morning was still early and it was cool in the dark woods.\n\nMen pulled the branches out of their knee-armour, out of their elbow cops, out of the joints in their faulds.\n\nAnd then, with the sounding of two horn calls, the line swept forward again, like a great boar hunt.\n\nA mile behind them, the rest of the army lurched into motion.\n\nThe van pressed forward into the woods. Led by the king, in person.\n\nBill Redmede\u2014Jack of the Jacks\u2014saw the armoured figures coming on foot, armoured cap \u00e0 pied, and the bitterness in his heart was enough to melt steel.\n\nSo much for Thorn and his contempt for men.\n\nJack turned to his lieutenant\u2014Nat Tyler, the Jack of the Albin Plain. \"Bastard aristos have a spy, brother.\"\n\nTyler watched the inexorable approach of the armoured men. \"And we're in deep brush.\"\n\n\"Thorn said they'd be mounted on the road,\" Jack said. \"Fuck!\"\n\n\"Let's loose and get gone,\" Tyler said.\n\n\"This is our day!\" Jack argued. \"Today we kill the king!\"\n\nSeventy yards away, the king stood virtually alone. He stood in a shaft of light in the deep forest, and he raised his arms\u2014he had a four-foot sword in one hand and a sparkling buckler in the other.\n\nRedmede drew his great bow and, suiting thought to deed, loosed.\n\nBeside him, Nat Tyler's bow twanged deep, the harp of death.\n\nAll along the line, Jacks rose from ambush and loosed at the king.\n\nThe king's figure twinkled as he pivoted on his back heel and spun, his buckler sweeping over his head, his sword scything through the first fall of arrows.\n\nAll around him, men-at-arms broke into a dead run, charging the line of archers.\n\nThe king stood his ground\u2014stepped and swung, stepped, cut, and then ran forward.\n\n\"Good Christ,\" Jack muttered. Not a single arrow had gone home. \"Too far\u2014too damned far!\"\n\nBut the Jacks were robbers and partisans, not battlefield men, and they turned and ran.\n\nA hundred paces to the rear, the line of Jacks steadied. Nat Tyler got them into a line at the edge of a meadow of flowers a third of a mile long and two hundred yards deep\u2014an ancient beaver meadow, crisscrossed by a stream. Bill led them over the stream, emerging wet to the waist, and they formed a new line on the far side.\n\n\"Better,\" Nat Tyler said with a grim smile.\n\nThe men-at-arms must have paused to drink water and rest. The sun was much higher when they came\u2014and they came all together. Forward in a line. This time the captain yelled at them to pick their targets and leave the king to the master archers, and the shafts flew thick and heavy over open ground.\n\nHe could no longer cut every arrow in the air, and heavy shafts rang off his buckler, his helmet\u2014he was leaning forward like a man walking into a storm, but his heart was singing, because this was a great deed of arms. He laughed, and ran faster.\n\nThe stream opened under his feet and he fell\u2014straight down the banks and into a thigh deep pool.\n\nTwo peasants stepped to the edge of the pool and loosed arrows at him from a few feet away.\n\nGaston saw the charge falter and blew his horn. Men were falling into something\u2014a line of pits, a hidden trench\u2014\n\nAn arrow rang from his breastplate, denting it deeply, and then he had an armoured fist on the king, and he pulled him straight out of the muddy pool in one long pull. By his side his squire, angered, threw his short spear across the stream and it struck\u2014more by luck than skill\u2014in the torso of a peasant, who folded over it and screamed. And the king got his feet under him and ran straight at the beaver dam\u2014the only clear bridge over the stream.\n\nGaston followed him, and every man-at-arms nearby followed, too. The dam was half in and half out the water\u2014far from solid, just an animal's hasty assemblage of downed branches and rotten wood. But the king seemed to skim across it, even as Gaston's right leg went into water as cold as ice\u2014and he lost his balance, flailed, almost lost his sword and an arrow slammed into his helmet.\n\nThe king ran on, across the uneven top of the dam. The first half ran to a rocky island, and then the dam was even worse, the centre of its span under water, and yet the king ran across it, his feet kicking up spray in a brilliant display of balance\u2014straight across the dam into the archers pouring shafts at him, and one got past his buckler to bury itself in his shoulder by the pauldron, and another rang off his helmet, and then he was among them, and his sword moved faster than a dragonfly on a summer's evening. Gaston was struggling to catch him, breathing like a horse at the end of a long run, soaked, his left leg trapped in mud for a moment and then Gaston was with the king, through the line of archers, and the horns were playing the _avaunt_ and the _mort_.\n\nHe followed the king up the rise to the ridge that dominated the meadow, and more and more men-at-arms crossed behind them\u2014and far off to the left, more men-at-arms had crossed the narrow footbridge on the road, and now the whole line of peasant archers was compromised, and they ran again.\n\nBut even as they turned to run, the wyverns struck.\n\nGaston saw the first one\u2014saw the flicker of its shadow, and looked up in stunned unbelief, even as the wave of its terror struck him and the Alban knights with him. The Albans flowed through the palpable fear\u2014and he refused to let himself pause, although for a moment it was so intense he couldn't breathe\u2014they surged forward even as the carthorse-sized monster killed a dozen of their number in a single flurry of talons and beak.\n\nThere were three of the things.\n\nThat was all Gaston could comprehend\u2014that, and that the king was like a fiend, leaping forward at the first wyvern, and his sword sliced a wing through at the root and his back cut flayed a sword's length of scales from the thing's neck, and it whirled to face him but he was gone, under the flailing neck and his blade went up into its belly\u2014ripped the thing open from anus to breastbone, and was gone again as its intestines fell free.\n\nGaston followed him to the second one, where it crushed the Bishop of Lorica to the ground with one blow and ripped his squire's head from his trunk with its beaked head. Gaston got his spear up, and spiked the head\u2014lost his balance on the uneven ground, broken with the spiked branches the beavers had left\u2014stumbled, and lost his spear, whirled and drew his sword as the head, trailing his spear, went for him.\n\nHe cut into its snout with every muscle in his body.\n\nIts head knocked him flat.\n\nThe head reared above him, with his spear _and_ his sword stuck in it, and the king straddled him. Blood leaked from the arrow in his left shoulder, and the man cut one handed at the monster's neck and severed its head.\n\nThe surviving knights roared their approval and Gaston got slowly to his feet, drenched by the hot blood of the thing, and dug in its jaw for his sword\u2014he had to kick it off his blade.\n\nThe third wyvern was already airborne, leaving a trail of broken knights behind it, but after leaping into the air, he pivoted and collapsed on the king, bearing him to the ground.\n\nEvery knight still alive in the meadow fell on the wyvern, and blows rained on it like a steel hail\u2014pieces of meat flew free like dust rises from the first fall of rain.\n\nThe wyvern hunched and tried to rise again into the air, but Gaston slammed his spearhead into its neck, and a few feet away, Ser Alcaeus hit the thing with a maul and staggered it. The king struggled from beneath it, staggered to his feet, and rammed his sword to the hilt in its guts before falling to his knees.\n\nThe wyvern screamed.\n\nThe king fell to the ground, his golden armour all besmirched with the blood of three mighty foes.\n\nSer Alcaeus swung his maul up over his head, screamed his defiance, and slammed the lead head into the wyvern's skull, and the beast crumpled atop the king.\n\nA dozen gauntleted hands scrambled to pull the dead thing off the king, even as trumpets sounded behind them and the mounted chivalry emerged from the tree line.\n\nGaston ran to the king. He got the king's head on his knee and opened his faceplate.\n\nHis mad cousin's eyes met his.\n\n\"Am I not the greatest knight in the world!\" he roared. \"And no craven, to basely let my liege be slain!\"\n\nHis eyes flickered. \"Get the arrow out of my chest and bandage me tight,\" he said. \"This is my battle!\" And then the light went out of his eyes.\n\nGaston held his cousin tight while a pair of squires tried to staunch the flow of blood, stripping his breastplate and his haubergeon. The remnants of the vanguard pressed on.\n\n\"He demanded it, this morning,\" said a voice behind Gaston, and suddenly the squires were bowing.\n\nThe King of Alba stood there, in Jean de Vrailly's cote armour.\n\n\"He said he knew of a plot to kill me, in an ambush\u2014and he wished the honour of taking my place.\" The king shook his head. \"He is truly a great knight.\"\n\nGaston swallowed his thoughts, and wondered what his mad cousin had done. And why. But the mad eyes were closed forever.\n\n# Near Lissen Carak\u2014Thurkan\n\nThurkan watched the king fall. His eyesight was tremendous and from two ridges away he and his clan watched the abnethog fling themselves on the knights.\n\nOf course, he had told them that he would support their attack.\n\nHe'd told the Jacks much the same.\n\nBut Thorn was doomed, and Thurkan had no intention of letting his people suffer any more.\n\nHe turned to his sister. \"If the men begin fighting among themselves, well and good\u2014we will feast.\"\n\n\"I see nothing of the sort,\" Mogan said.\n\n\"Nor I,\" Korghan said.\n\nBehind them stood forty of their kind\u2014enough Qwethnethog to turn the battle. \"Go tell the Sossag and the Abenacki that the battle is lost,\" Thurkan said to his sister.\n\n\"It isn't lost unless we flee,\" his sister insisted. \"By rock and flowing water\u2014is that your will?\"\n\nThurkan frowned, deep creases appearing in his jaw. \"Thorn must die\u2014now, while he is weak. Otherwise he will hunt us down.\"\n\nMogan poked her snout close to her brother's. \"Do not let me believe that this is all the rivalry of two Powers,\" she spat. \"I have lost kin\u2014you have lost kin. We were promised a feast, and\u2014\"\n\n\"We had a feast at Albinkirk and another on the road.\" Thurkan shook his head. \"I do not do what I do lightly. Thorn must go. We are being\u2014\" he flexed the talons on his forefoot, moving each digit in an intricate arc, \"\u2014manipulated. By something. I can feel it.\"\n\nMogan snorted. \"Very well,\" she said. \"I obey. Under protest.\" And ran off into the trees, as fleet as a deer.\n\n\"West,\" Thurkan told his brother.\n\n\"I can help you,\" his brother insisted.\n\n\"Perhaps. But Mogan cannot lead our clan or fertilize new eggs. And you can.\" The great head turned. \"Obey, brother.\"\n\nKorghan flicked his tongue in anger. \"Very well, _brother._ \"\n\nThe two clan companies started west, even as the Royal Household Knights began to climb the ridge towards them.\n\nBill Redmede ran, loosed an arrow from his dwindling store, and ran again. His bodkin points were all but gone, and he had only hunting arrows.\n\nThe _God-damned_ aristocrats had more plate armour than he'd ever seen. And the monsters\u2014he'd been a fool to ever trust them and no doubt imperiled his soul, as well. He was bitter\u2014tired, angry, and defeated.\n\nBut he'd seen the king fall. It was some consolation, but it didn't seem to slow the rest of the aristos any, and like all his kind, he faced an ugly death if he was caught, so he waited a heartbeat, stepped from behind his tree and put a shaft under the arm of somebody's fucking _lord_ and turned, and ran again.\n\nHe made it up the second ridge, where they had started the morning, where the big daemon lord had issued its orders.\n\nAll the daemons were gone. Sod them, too. Oligarchs. Bad allies for free men.\n\nThe river was close now.\n\nThere were knights in red surcotes at the base of the ridge, and he could see them coming up the hill\u2014most of them had dismounted, and a flurry of arrows told him that his boys were still fighting back. Fighting the Royal Guard.\n\nHe was damned if he was going to lose any more Jacks.\n\nHe turned and ran diagonally across the face of the ridge.\n\nHe came up behind Nat Tyler as the man loosed his last arrow. \"Come on, Nat\u2014the boats!\"\n\nTyler turned like a wild thing\u2014but he got a hold of his wits, paused, and winded his horn and whistles sounded in response.\n\n\"Follow me!\" Bill called, and ran back up the hill\u2014legs labouring, lungs searching for breath.\n\nBehind him, the Jacks loosed a last arrow and ran\u2014the _sauve qui peut_ had been blown.\n\nBill ran, and the Jacks ran behind him. He paused when he saw three of his own trying to face a knight with drawn swords and bucklers, and he put a shaft to his bow\u2014another knight burst from the trees and crossed the crest of the ridge\u2014raised his visor\u2014\n\nToo good a shot to miss.\n\nHawthor Veney made it to the top of the ridge on pride alone. It was his first fight, and he was a King's Guardsman. His red surcote shouted his allegiance, and the Jacks were his enemies, and he pursued them ruthlessly. He caught one and hewed him from behind, a clumsy stroke that buried his point in the man's neck, but the man fell hard, blood burst from the wound, and he ran on, wrenching the sword from the man's corpse.\n\nThe next one he caught fell to his knees and begged for mercy. He was perhaps fourteen years old.\n\nHawthor paused, and an older guardsman beheaded the boy. \"Nits make lice,\" he said, as he swept by, and Hawthor hardened his heart and ran on. Running in armour was hard. Running up a ridge with soft footing and tangled spring undergrowth was worse. His lungs began to labour and, as the Jacks rallied and rallied again, whipping deadly shafts at the guardsmen, Hawthor had to fight the temptation to open his visor.\n\nHe began to pass men when he could see the light through the trees that meant the crest of the ridge was coming. There was shouting to the right\u2014he turned to look, and he heard the sound of steel on steel. He looked back and forth\u2014it was closer, and with his faceplate closed, he couldn't see where. There was a flicker of motion to the front\u2014he looked, ran a few steps, stopped, and looked again.\n\nHeard the scrape of blades. A voice called \"Sauve Qui Peut!\"\n\nHe was breathing like a horse after a race. He was afraid\u2014he was afraid they were behind him. He popped his visor, turned his head\u2014\n\nAnd died.\n\n# Near Lissen Carak\u2014Bill Redmede\n\nBill got another shaft on his bow after putting one through the knight's face\u2014felt better for doing it\u2014but two more of his men were down and he knew better than to join the hand to hand fight. He ran.\n\nThey crossed the ridge, and started down the far side towards their boats. A handful of knights from the vanguard tried to stop them, and the Jacks just ran around them\u2014exhausted men without armour have an advantage over exhausted men in armour.\n\nBill saw the Count of the Borders, close enough to touch, and he cursed his fate, that he should be so close to a mortal foe and be able to do nothing.\n\nBut he ran past the man, down the steep berm, into a broad field\u2014ploughed, until recently. Nat Tyler came out of the trees to his left, and dozens more\u2014a handful, compared to their numbers three weeks ago. But enough to start again.\n\nUp the last dyke, and there were the boats. Fifty light bark boats\u2014it had taken them three careful trips to get everyone over, night before last, and now...\n\nNow they'd all fit in one go.\n\nHe tossed his bow into the bottom of the light boat, pushed it into the water, and stepped in, running lightly down the length of the boat to kneel in the bow. Then he rocked the stern off the muddy beach, and held his position in the current with his paddle until a young blond man in dirty white tossed his own bow into the boat and stepped clumsily into the stern. He almost swamped the light boat, and they were away into the current.\n\nTwenty other boats were putting off behind him\u2014the better boatmen got the boats moving. The less competent men started to die, as the Royal Guard began to close on them.\n\nA few last Jacks dived into the water, abandoning packs and bows, quivers of invaluable arrows, but a few men had had the presence of mind to drag the rest of the boats off the mud and tow them and, safe in the current, they got the swimmers into boats.\n\nMore than a hundred Jacks had been saved from the disaster.\n\nThey began to paddle out into the centre. It was obvious from here that Bridge Castle was still in the hands of the sell-swords\u2014an arbalest bolt skipped across the water to put a hole in one boat.\n\nTyler waved, pointed downstream, waved again, and paddled frantically to turn his boat.\n\nJack looked into the rising sun and it's brilliant reflection on the broad river\u2014and saw flashes. Rhythmic flashes\u2014banks of oars on heavy bateaux, rowing upstream. He counted twenty\u2014counted a second twenty\u2014\n\nDisaster. Disaster after disaster.\n\nHe turned his head. \"Less power and more finesse, comrade. We have to turn this boat and paddle upstream\u2014all your power will serve us well, then.\"\n\nA pair of crossbow bolts, like swallows feeding on insects, skipped by, passing within an arm's length of their boat before sinking out of sight.\n\nThe man in the stern shook his head. \"I'm no boatsman, brother,\" he admitted.\n\n\"Never mind, lad. Drag your paddle on the left\u2014just there. And we're around.\" Bill hadn't risen to leadership for nothing; he was patient, even when everything was at stake.\n\nThen they were around, and his young companion's strong arms were pushing the boat forward like a leaping deer. It was a waste of the man's energy to spend so much but Jack let him tire himself, steering from the bow. Another volley of crossbow bolts from the distant bateaux and he lost a trio of Jacks\u2014they were broadside on to the enemy and all three of them caught bolts.\n\nBill Redmede was an old boatman. And a master archer. He stowed his paddle, took his bow from the bottom of the light boat, wiped the stave and the string\u2014good wax, not too much moisture. He was glad he'd left it strung, and he rose to his feet, the boat tipping\u2014leaped lightly onto the ash gunwales, one foot on each.\n\n\"Good Christ!\" shouted his stern paddle in dismay.\n\nHe drew and loosed in one motion, tipping the boat from side to side, loosing high\u2014a hunting point. Then he knelt as he watched the fall of his arrow.\n\nHe lost it in the sun-dazzle. But he felt better for the shot, and he took up his paddle and gave way with a will.\n\n# Near Albinkirk\u2014Desiderata\n\nDesiderata was in a borrowed chain shirt\u2014worn with a man's hose, a heavy wool kirtle laced as tightly as her maids could manage, and a man's arming cap. The effect should have been ludicrous, but was instead both martial and quite attractive, to judge from the reactions of the guildsmen and the hillmen all around her on the foredeck of the lead row-galley.\n\nLady Almspend stood by her side, also in a shirt of mail, with a sallet on her head and a sword at her waist. She was more ridiculous, but beaming at Ranald Lachlan, whose attention was torn between his lady-love and the approach of combat. The herd was penned in camp, with twenty of his brother's men as guards. He stood in hauberk and leg armour, his open-faced bascinet and leather cote almost barbaric in comparison to the crossbowmen of the guilds of Lorica, most of whom had fancy cotes of plates and visored helmets, the latest fashion from the Continent. His hands rested on the great axe he carried.\n\nThe Queen looked at him. He was quieter than she had known him in year's past. According to her secretary, he had actually been _dead._ The Queen suspected this might be a sobering experience.\n\n\"Boglins on the bank,\" Ranald said, pointing a gauntleted hand.\n\n\"Got them,\" said one of the guild officers. \"Boglins to starboard. Pick your targets. Loose!\"\n\nA dozen bolts flew.\n\n\"The king must have been victorious,\" Lady Almspend said. \"Those aren't our men fleeing across the river in front of us.\"\n\nRanald turned so fast the chain aventail at his neck slapped his helmet. \"Good eyes, my lady.\" He flashed her a smile\u2014pleased to have her company in his favourite pursuit. He looked under his gauntleted hand for a long time. \"They're men\u2014they're in a sort of uniform. Now they turned their boats away\u2014\"\n\nThe guild officer had scrambled up into the bows past the Queen. \"Jacks, by God. Rebels! Traitors! Heretics!\" He raised his arbalest, took careful aim, and loosed a bolt.\n\nThe boglins on the north bank began to flick arrows at them.\n\nThe Queen started. The back of her throat was scratchy. For the first time, she was afraid.\n\n\"We have come too far west,\" Ranald said. \"There are enemies on both banks, and the king won't yet know that we are here.\"\n\nThe Queen had received a message from the king late in the afternoon, and she had ordered the boats to row all night. She'd picked up the messenger at midnight, and his information had been exact. Today was to be the day\u2014she intended to see it.\n\nShe stood on the foredeck and shaded her eyes with her hand\u2014to the front, and to the right, and to the left. To the left, she saw a flash of red, and then another\u2014and then half a dozen Royal Guardsmen appeared on the bank. She waved, and her ladies cheered.\n\n\"Anchor here,\" she ordered.\n\nA half-dozen boglin arrows fell onto the lead galley\u2014most were deflected by the leather curtains that protected the rowers, but one struck home, and the man's oar fell from his hands as he screamed. The arrow was deep in his shoulder.\n\nBoglins poisoned their arrows, and his screams froze her blood. He had laughed and joked with her maids when they lay on the banks of the Alba, eating sausage.\n\nIt was as much of a shock as the sight of a boglin.\n\nAn arrow plummeted from the heavens like a stooping hawk, struck her helm, ripped down her back and knocked her flat.\n\nShe lay on the deck\u2014suddenly the day was darker, and her back was wet.\n\n\"Ware the Queen!\" Ranald called.\n\nShe reached for the golden light of the sun\u2014it was all about her, such a glorious day\u2014\n\n\"She'll bleed out. It is in her back.\" Ranald was doing something.\n\n\"Is it poisoned?\" Lady Almspend asked.\n\n\"I don't think so\u2014give me your pen knife. Wicked bastard\u2014a swallowtail point.\" Lachlan sounded afraid.\n\nShe was floating above them, able to see the hillman digging in the flesh of her back with a knife. He had the mail shirt hiked over her hips having cut the shaft of the arrow. She'd seldom seen herself look less elegant.\n\n\"It's in her kidneys,\" Lachlan said, and sat back on his haunches, suddenly defeated. \"Sweet Jesu!\"\n\nThe captain had slept in his harness like everyone else, his helmeted head in the corner of the curtain wall where the west wall met the north tower. Four assaults had failed to re-take the wall, but he was so tired\u2014\n\n\"Boats on the river, Captain.\" Jack Kaves, senior archer, stood over him. \"I brought you a cup of beer. Young Michael tried to wake you and went off to find more wine.\"\n\nThe captain took the beer, rinsed his mouth and spat onto the mound of dead boglins outside the wall, and then took a long pull. Half of the mound of boglin bodies was still moving, so that the whole pile seemed to writhe\u2014and they made mewling sounds like a pile of kittens, somehow more horrible than the screams of men.\n\nNo more men were screaming. The wounded had been sent up the hill to the fortress during a lull between attacks\u2014the Knights of Saint Thomas, like their sisters, were doctors as well as fighters, and they gave basic care and rigged stretchers between horses. And the enemy killed every wounded man they could.\n\nHe got slowly to his feet. The weight of his armour and his own fatigue combined to make the process of rising painful\u2014his neck hurt like he had been kicked by a horse. \"Michael?\" he asked, confused and looking around.\n\n\"In the store rooms,\" Kaves said.\n\n\"Jack, help me get my helmet off,\" the captain said. He unbuckled his chinstrap, and Jack lifted the helmet clear of his head. The aventail was clotted with gore, which dragged across his face. The visor was gone.\n\nHe unlaced his arming cap. It was one of Mag's, and with the intense interest of total exhaustion, he noted that she had embroidered his lacs d'amour across the crown\u2014lovely work.\n\nThe cap was full of power. He hadn't seen it before\u2014perhaps hadn't been _able_ to see it. He held it closer and saw that every stitch held a tiny rainbow of light\u2014the whole, with the lines of embroidery, was not unlike a set of tiny fish scales.\n\nJack Kaves whistled.\n\nThe captain turned and looked at his helmet, which had a great gouge in it where some weapon had punched right through it. Indeed, with all too little effort, the captain could remember the boglin chief's scythes, slicing at his unvisored face and never quite reaching it.\n\n\"Well, well,\" he said. He leaned forward, and Jack upended a pot of river-water over his head.\n\nThe old archer handed him a rag and he dried his hair, face and beard. While he used the rag, he walked along the wall, feeling the damp spread down inside his breastplate. He could all but hear it rusting. Michael was going to be\u2014\n\nThere were, indeed, boats on the river. Fifty row galleys\u2014obviously crewed by men.\n\nHe stood and watched them for several long minutes.\n\nJack Kaves stood beside him, holding out a sausage. \"What's it mean, Cap'n?\" he asked.\n\nThe captain gave a wry smile. \"It means we win,\" he said. \"Unless we screw it up really badly, we win.\"\n\n# Albinkirk\u2014Desiderata\n\nLady Almspend shook her head. She was tying the points of her sleeves back. \"Don't be a ninny. That's fat. You there\u2014get my kit-bag up from the hold. The barbs\u2014I have a tool for them.\"\n\n\"You do?\" Lachlan asked.\n\nAlmspend took the Queen's hand. \"I know you can hear me, my lady. Stay with us. Take power from the sun\u2014take strength. I can get this out, with a little luck.\"\n\nLachlan grunted.\n\nAn oarsman came up the foredeck ladder with her leather bag.\n\n\"Dump it on the deck,\" she ordered. He did, breaking an ink bottle and putting black ink on every shift she owned.\n\nShe snatched the item she sought\u2014a pair of matching halves, like a mould for an arrow.\n\n\"Hold on, my lady,\" she said. \"This will hurt.\"\n\nShe pushed the mould over the arrow\u2014in and in, along the path of the original wound, and the Queen moaned, and a long line of saliva mixed with blood came out of her mouth.\n\nLachlan spat. \"She'll\u2014\"\n\n\"Shut up,\" said Lady Almspend. She gave her moulds a twist and they snapped over the arrowhead\u2014covering the wicked barbs.\n\n\"Pull it out,\" she said to Lachlan.\n\nHe tugged and looked at her.\n\n\"Pull it out, or she dies,\" Lady Almspend insisted.\n\nLachlan set his shoulders, hesitated, and then pulled. The arrow\u2014moulds and all\u2014popped free with a horrible sucking noise.\n\nBlood spurted after it.\n\n# Lissen Carak\u2014Peter\n\nNita Qwan knew that the great battle had started. But he was cooking. He had built a small oven of river clay, fired it himself, and now he was making a pie.\n\nA third of the Sossag warriors were watching him. Sometimes they clapped. It made him laugh.\n\nThe pair of boglins were back, too. If you didn't look too closely at their bodies they looked like a pair of rough-hewn, slightly misshapen back-country men.\n\nThey lay full length in the grass, beyond the circle of men, so that their wing-cases were atop them like upturned boats. When they approved of his cooking, they rubbed their back legs together.\n\nHis pie was the size of a mill wheel.\n\nHis fire was even larger\u2014a carefully dug pit that he had filled with coals from patient burning of hardwoods.\n\nThere was no reason that the project should work, but it kept him busy, and it entertained the other warriors.\n\nNita Qwan wondered what Ota Qwan intended. The man had touched up his paint, polished his bronze gorget, sharpened his sword and his spear and all his arrows, and now he lay watching Peter cook with the other warriors.\n\nWaiting.\n\nThe problem with a pie was that you never really knew if it was done.\n\nBattle seemed to have some of the same qualities.\n\nNita Qwan went and sat with the pie for a while, and then he went over and squatted on his heels by Ota Qwan.\n\nThe war chief raised his head off his arms. \"Is it done yet?\" he asked.\n\nNita Qwan shrugged. \"No,\" he said. \"Or yes.\"\n\nOta Qwan nodded seriously.\n\nSkahas Gaho laughed.\n\n\"Why are we not on the field?\" Nita Qwan asked.\n\n\"Pie isn't done yet,\" Ota Qwan said, and all the senior warriors laughed. There was a unanimity to their laughter that told Peter that Ota Qwan had passed some important test of leadership. He was the war leader, and they did not contest it. A subtle change but a real one.\n\nOta Qwan rolled over, carefully brushing bits of fern from the grease that carried his paint. \"Thorn is going to fight the knights in the fields,\" he said. \"Fields from which every scrap of cover has been burned.\"\n\nThe older warriors nodded, like a chorus.\n\nOta Qwan shrugged. \"We almost lost a lot of warriors last night,\" he said. \"I won't risk the people on such foolishness again. This time, we will go when it is right for us to go. Or not. And the pie is as good a sign as any.\"\n\nOff by the edge of the clearing, a woman\u2014Ojig\u2014sat up quickly, and her sister, Small Hands, stiffened like a dog at the scent of a wolf, and took up her bow, and suddenly all the people were moving\u2014weaponed, alert\u2014\n\n\"Qwethnethog!\" shouted Small Hands.\n\nNita Qwan never heard an order given but in heartbeats, the clearing was empty, save only his fire, his pie, and the six eldest warriors standing around Ota Qwan.\n\nThe Qwethnethog emerged from the underbrush moving as fast as a racehorse, and she took several long strides to slow. She looked back and forth at the line of men, and at the fire.\n\n\"Skadai,\" she said in her shrill voice.\n\n\"Dead,\" said one of the aged warriors.\n\n\"Ahh,\" she keened. Made an alien gesture with her taloned paws, and turned. \"Who leads the Sossag people?\"\n\nOta Qwan stood forth. \"I lead them in war,\" he said.\n\nThe Qwethnethog looked at him, turning her head from side to side. Nita Qwan noted that her helmet crest was a deep scarlet, and the colour came well down her forehead. But the crest was smaller than on a male. It amused him\u2014even through the terror she broadcast\u2014that he'd become so well-versed in the ways of the Wild as to know male from female, clan from clan. She was of their own clan\u2014the western Qwethnethog, who lived in the steep hills above the Sossag lakes.\n\n\"My brother speaks for all the Qwethnethog of the Mountains,\" she said in her shrill voice. \"We are leaving the field, and will fight no more for Thorn.\"\n\nOta Qwan looked at the men to the right and left. \"We thank you,\" he said. \"Go in peace.\"\n\nThe great monster turned and sniffed. \"Smells delicious,\" she said, to no one in particular.\n\n\"Stay and have a piece,\" Nita Qwan found himself saying.\n\nShe coughed\u2014he assumed that was her simulation of laughter. \"You are bold, little man,\" she said. \"Come and cook for me another time.\" And with a flick of her talons, faster than a deer, she was gone into the woods again.\n\nNo sooner was she gone then a dozen women came out of the woods\u2014matrons, every one. They spoke so rapidly in Sossag that Nita Qwan couldn't understand even single words.\n\nSo instead, he went and opened his temporary oven.\n\nIt was burned all down one side, but the rest had steamed well and the crust was a nice colour\u2014a rich golden brown, shot with darker brown. Perhaps the oven had cracked\u2014he had no idea why part of the outer rim was so singed.\n\nNor did he care, for the people came forward like an avenging army and seized the pie as fast as he could cut slices off it. He had made enough, and it wasn't the way of the people to complain.\n\nOta Qwan took a piece\u2014a burned piece. \"Well done. Now we are fed, and well-fed. We will run all night.\"\n\nHe ate his piece in four bites and drank a cup of water. Nita Qwan emulated him, and noted that his wife had packed his baskets. He took one on his back. She smiled shyly at him.\n\nHe smiled back.\n\nHe shouldered his quiver and his sword, and then\u2014with no further discussion\u2014they were off into the trees.\n\n# Albinkirk\u2014Desiderata\n\nThe row galley landed against the Bridge Fort's dock; the garrison was alert and manned the walls. The captain was waiting on the dock.\n\nThe row galley was full of women, each one more beautiful than the last. It wasn't what he'd expected.\n\nOne woman\u2014short, blonde, and harried\u2014stood on the foredeck. \"I need a healer,\" she said. \"A good one.\"\n\nThe captain turned to Michael. \"Get me a Knight of the Order,\" he said. Then he turned back. \"They are superb healers,\" he said. Unfortunately, they had gone on a sortie to clear the trench at dawn, and they hadn't returned.\n\n\"I know,\" she spat. \"How long?\"\n\n\"A few minutes,\" he said, hopefully.\n\n\"She doesn't have a few minutes,\" the woman said, her face cracking. She seemed to clamp down on a sob. \"She's lost a great deal of blood.\"\n\n\"Who has?\" he asked as he tried to get a leg over the gunwale. A dozen oarsmen reached to pull him into the boat.\n\n\"The Queen,\" she said. \"I'm Lady Almspend. Her secretary. This is Lady Mary, chief among her ladies.\"\n\nThe Queen.\n\nThe Red Knight ignored the people gathered around the figure on the deck. The woman lying on the deck was losing blood at a tremendous rate. He could feel it.\n\nAnd he had very little strength, at least in terms of power. What he had he'd squandered, fighting boglins. And to heal her here, now, would give himself away\u2014at least as a Hermeticist.\n\nSo much blood.\n\nShe was young\u2014imbued with power, herself.\n\nIn that moment, he realised that if she died, he could _take_ her. As he had taken the boglin chief. She was defenceless\u2014wide open, trying to use her power to strengthen herself. She drank in the sun's rays\u2014the pure power of Helios. She was very potent.\n\nHe put a hand on her back.\n\n\"Well?\" Lady Almspend asked, impatient. \"Can you help?\"\n\n_Vade Retro, Satanus_ , the captain thought. He took his arming cap off his head, and pushed it into the wound. Put one finger on the cap as it turned from dirty white to brilliant scarlet.\n\nHe almost grinned. He was linked to a legion of healers. It was easy to forget that.\n\n_The palace seemed empty without Prudentia. He knew the basic phantasms of healing now\u2014he wondered if he could release the power of Mag's bindings to power them. And keep the power\u2014and funnel it through workings he knew mostly from long ago lessons._\n\n_\"Amicia?\" he asked._\n\n_She was there. \"Hello!\" she said. She took his hand, smiled\u2014and let his hand drop._\n\n_\"I need to heal someone.\" He wished_ \u2014\n\n_\"Show me,\" Amicia said briskly._\n\n_He took a moment to kneel by the fallen statue, and brush a hand across Prudentia's marble back. \"I miss you,\" he said. \"Help me, if you can.\"_\n\n_Then he took Amicia's hand and laid it_ on the Queen.\n\n_She pointed to workings he now knew\u2014through her\u2014in a mind-wrenchingmoment, he was on her bridge using her memory palace even as he stood on Prudentia's pedestal and collected what was left of his power._\n\n_It wasn't enough._\n\n_Amicia shook her head. \"I have nothing to give,\" she said. He looked up at her, and even in the aethereal her exhaustion was obvious. \"So many wounded,\" she said._\n\n_Sighing for the loss, he tested the binding of power on Mag's cap. He cast, as Harmodius had taught him, guided by Amicia's sure hand on his\u2014three workings, each contingent on the other, like nested equations on the chalkboard. The loosing, the binding for power, the healing. He used what was left of the life force he had taken from the boglin chief._\n\n_\"Saint Barbara, Taurus, Thales. Demetrios, Pisces, Herakleitus. Ionnes the Baptist, Leo, Socrates!\" he invoked, pointed, pivoted, and the room moved\u2014the gears of his imagined rooms turning at the speed of a man's muscles, so that the room spun like a top._\n\n_It was the most complex conjuring he had ever attempted\u2014and the power that flared from it astounded him, a backlash of released power that rose in the room around him._\n\nThe arming cap immolated itself in a paroxysm of power\u2014a brief flare, and all that power vanished into her.\n\nA red mist crossed her back from her spine to the top of one tanned leg and around to her hip, right across the kidney. A flake of grey-white ash fell away from it.\n\nThe captain fell back away from her.\n\nThe Queen gave a squeak, and then sighed, as if stroked by her lover. And then gave a low moan.\n\nLady Almspend clasped her hands together. \"Oh, by the power of God, ser! That was brilliant!\"\n\nThe captain shook his head. \"That wasn't me,\" he admitted. \"Or not just me.\" His voice was a croak.\n\nThe wound began to bleed again. They bandaged it tightly, being careful of the wound which still seemed to be open.\n\nThe captain shook his head. \"But I _felt_ the power flow,\" he said in frustration.\n\n\"I feel the pain less,\" the Queen said bravely. \"It was well done, Ser Knight.\"\n\nA red-haired giant threw his cloak over the Queen. \"We need to get her ashore.\"\n\nThe captain shook his head. \"I wouldn't. That castle is the lynch pin of the battle, and I've been holding it all night. I wouldn't risk the Queen of Alba in it.\"\n\nBut other boats were pulling up against the pilings of the bridge, anchoring or tying up, their crossbowmen engaging the boglins on the north bank. The bolder boatmen were pulling under the bridge, through the narrows, to further outflank the enemy in the meadows north of the river.\n\n\"I have twenty brave men to add to your garrison,\" Red Beard said.\n\n\"I'd rather have all those nice crossbowmen,\" the captain said. He smiled to take any apparent sting from his remark. \"Very well. Land the Queen. Don't mind the boglin guts\u2014we haven't had time to tidy up.\"\n\nHe rose from the deck, almost unable to walk. He clambered back over the side to the dock, and managed to give the required orders.\n\nHe collapsed onto a bollard. He was aware that Red Beard was standing with him, talking, but he hadn't slept, hadn't recovered any power, and he'd just cast\u2014he was phantasm sick, something about which Prudentia had warned him, over and over.\n\nHe reached out into the wan sunlight. Pulled the gauntlets off his hands and raised them to the sun.\n\n_What would mother think of this?_ He wondered. Because as soon as the sun licked his hands, he felt a trickle of power through his arms. The headache receded. The depression\u2014\n\n_Amicia?_\n\n_Captain? she asked tartly._\n\n_The sun. Reach out and take power from the sun._\n\n_I cannot. It is not given to me._\n\n_Crap, my lady. To paraphrase Harmodius, power is just power. Take it._\n\n_Did I hear my name?_\n\n_Show her what you showed me. Show her the way to the sun._\n\n_With pleasure, as soon as I have a moment in which I am not fighting for my life._ Harmodius's image in the Aethereal was looking tattered.\n\n_Use the well, then,_ countered the captain.\n\n_Without intending, he was on her bridge over her stream. The stream was a trickle, the rocks dry, the foliage wilted._\n\n_He took her hand and she sighed._\n\n_\"We're going to win,\" he said. \"It is close, but we are going to win.\" He wasn't sure just how the well would manifest in her place of power. He conjured a well cover, and a hand pump, just at the end of her wooden bridge. \"Hold out your hands,\" he said._\n\n_She smiled. \"The sun is not for me, but I can use the well.\"_\n\n_He shook his head. \"It's just there. Power is power. Take what you need.\" He pumped the handle and a surge of power shot from the nozzle like water under pressure and soaked her through her green kirtle._\n\n_She laughed. Power sprayed around them\u2014into the pool under the bridge, into the trees._\n\n_The light became richer, the stream began to sing._\n\n_\"Oh!\" she said, and reached out to the well_ \u2014\n\n_The well-cover and the pump-handle vanished, and the stream beneath their feet roared to life._\n\n_\"Oh!\" she said. Her eyes were tightly closed. \"Oh, my God!\"_\n\n_He sighed. It was not the denouement he had hoped for._\n\n_But outside the palaces of the Aethereal, men were calling his name._\n\n_He leaned over and kissed her, all the sweeter for being there._\n\n_\"I must go,\" he said._\n\n\"Those are Royal Guardsmen,\" Red Beard shouted, pointing to the south across the river, and back east of the bridge. \"I know them.\"\n\n\"Horses,\" the captain said to Michael. \"War horse for you, another for me, a mount for the red giant. Ser Milus, you are in command until I return. Send to the fortress for a healer. Tell them that the Queen of Alba is dying.\" He was hard put to leave her. It wasn't his way to turn his back on a project. He had a new reserve of power\u2014but she needed a fine, trained hand. And he needed to have something left for the fight.\n\nThey carried her past him.\n\n\"Fuck it,\" he muttered to himself. He reached out and put a hand on her naked shoulder. He gave her all the power he had\u2014everything that he had taken through Amicia at the well, and all he had taken from the sun.\n\nHe sagged away from her. Spat the taste of bile into the water, and fell to his knees.\n\nShe made a sound and her eyes rolled up.\n\nMichael caught his shoulder, and put a canteen in his fist. He drank. There was wine in the canteen, mixed with the water, and he spat it out, then drank more.\n\n\"Get me up,\" he said.\n\nRed Beard got under his other shoulder. \"You're a warlock?\" he asked brusquely.\n\nThe captain had to laugh. \"I'll forgive you your imprecise terminology.\"\n\nThe wine was good.\n\nMichael handed him a chunk of honey cake. \"Eat.\"\n\nHe ate.\n\nHe let the sun fall on his face and hands, and he ate.\n\nFifteen feet away, Ser Milus was trying to find the bottom of a leather jack of water. He nodded, sputtered. \"Is the fight over?\"\n\nThe captain shrugged. \"It ought to be,\" he muttered. He could hear them fetching horses\u2014could hear the heavy clop-clop of the hooves on the cobblestones of the Bridge Castle's yard, and the rattle-slap of the tack going on.\n\n\"Jacques has him,\" Michael said.\n\n\"I hate that horse,\" the captain said. He finished his honey cake, swallowed more wine and water, and made himself run up the ladders to the top of the Bridge Castle's north tower.\n\nSixty feet above the flood plain many mysteries were explained.\n\nHe couldn't see beyond the ridges south of the river, but the brilliant sparkle of armour told him that the men-at-arms pouring over the last ridge had to be the Royal Army.\n\nTo the west the trees were full of boglins, and north, almost a mile away, a trio of creatures\u2014each larger than war horses\u2014emerged from the woods with a long line of infantry on either hand.\n\nThe new trebuchet mounted in the ruins of the north tower of the fortress loosed\u2014 _thump-crack_ \u2014and the hail of stones fell short of the Wild creatures, but they shied away anyway.\n\nBut as far as he could see, along the woods' edge, the undergrowth boiled with motion.\n\n\"Why are you still here? Even if you win you won't take the fortress. You've lost, you fool,\" the captain muttered. \"Let it go. Live to fight another day.\" He shook his head.\n\nFor a mad moment, he thought of reaching out to Thorn. Because if Thorn stayed to fight, some of his men were going to die, and he'd come to love them. Even Sym.\n\n_I'm tired and maudlin._\n\nHe scrambled down the ladder and found Jacques holding his new charger. Michael was at the postern gate. Jack Kaves waved.\n\nThe captain got a leg over his saddle and groaned. The big stallion shied and tossed his head.\n\n\"I hate this horse.\" He looked down at Jacques. \"Go straight for Jehannes, now.\"\n\n\"Ser Jehannes is wounded,\" he said.\n\n\"Tom, then.\"\n\n\"Tom's the man, aye,\" Jacques said.\n\n\"Get every man-at-arms of the company mounted, and by the foot of the ridge,\" he said. \"All the farmers and all the guildsmen along the trench and to the fort, here.\"\n\nJacques nodded. \"Just for the sake of conversation,\" he said, \"we could keep the fortress.\" His smile was transparently empty of guile, like a boy who has just thrown a rock at a hornets nest and remains unrepentant.\n\nThe captain nodded. \"We could. Hold it for ransom. Sell it to the highest bidder.\" He sounded wistful. \"We could be the baddest. The Knights of Ill-Repute. Rich. Feared.\" He shrugged. \"Sometime in the last month we became paladins, Jacques.\"\n\nJacques nodded. \" 'Bout time, my prince.\"\n\n\"Stow that, Jacques,\" the captain said. He turned his horse's head, backed his charger a few steps, and saluted Smoke, the archer commanding the gate. \"Open it,\" he called. \"And the Bridge Gate.\" He turned back to Jacques. \"Don't forget to bring healers,\" he said.\n\nRed Beard joined them, mounted on an old roncey that had seen better days.\n\n\"Sorry about the horse,\" the captain said. \"I'm the captain.\"\n\n\"That's your name?\" asked the red giant. \"I'm Ranald. Ranald Lachlan.\"\n\n\"You know the Royal Guard?\" the captain asked. Then he paused. \"Lachlan? Tom Lachlan's brother?\"\n\n\"Cousin,\" the other man said. \"You know Bad Tom?\"\n\n\"Doesn't everyone?\" the captain said. \"Let's go find the king.\" His voice was a little shaky.\n\n\"Amen,\" the hillman answered. \"Do you know him? The king?\"\n\n\"What a very interesting question,\" the captain answered. \"No. Not exactly.\"\n\nMichael followed them, and their horses' hooves rang as they crossed the bridge. At the middle the captain reached into the purse on his sword belt and produced a key\u2014intricate, beautiful and apparently solid gold. He leaned out\u2014groaning at the pressure on the muscles of his back and neck\u2014 _how long ago did I fight the God-damned wyvern in the woods?_ He fitted the key into the great gate, turned it, and the gate vanished.\n\n\"Nice trick,\" muttered Ranald.\n\n# Near Lissen Carak\u2014The King\n\nThe king was collecting his guardsmen and the knights of his vanguard\u2014the vanguard had lost fifty men-at-arms and as many squires, the men were exhausted already, and the morning was young. Two of his leading noblemen were dead\u2014both the Bishop of Lorica and the constable had both gone down in the first fighting. The Captal de Ruth had taken a mortal wound defending the king, and was dying.\n\nBut the valets were coming up with the horses and the machines of war were grinding along\u2014surgeons were searching among the wounded for those who could be saved, and his huntsmen, who had swept east to guard the flank of the onslaught of the vanguard, were trickling in. They, too, had lost men fighting monsters in the woods by the river\u2014nor had they been victorious, by all accounts. The Wild creatures had burst through them and run off east. They had lost sixty men. Good men. Trained men.\n\nIt was hardly the great victory he sought. He had been ambushed and his column had survived. That was all.\n\n\"Messengers, Sire. From across the river,\" called a herald.\n\nThe king looked north-west, and saw them\u2014three men crossing the bridge at a fast canter.\n\n\"Sound the rally,\" the king said.\n\nMore and more of his Royal Huntsmen were merging from the west, moving warily.\n\nThe Count of the Borders rode up and saluted. \"The flower of our chivalry is half an hour behind me with the main battle,\" he reported. The man slumped. \"By Saint George, my lord, that was the hardest fighting I ever need to see.\"\n\n\"The guardsmen say there are boglins across the river,\" the king noted.\n\n\"Boglins?\" The count shook his head. \"I struck a blow at a wyvern this morning, sire. This is the Wild, my lord, fighting for its life.\"\n\n\"I thought the Wild was beaten,\" said the king.\n\nThe Count of the Borders shook his head. \"Where is Murien? What has happened to the Wall Castles?\"\n\nThe king's master huntsman, Febus de Lorn, bowed respectfully. \"This isn't from north, my lords. This is from west. I see Gwyllch\u2014boglins\u2014across the river, and Bothere has huntsmen who claim to have faced trolls in the low ground west of the road. Dhag's come from the west, my lords.\"\n\nThe king looked back at the approaching messengers. They weren't messengers\u2014all three in were armour, two cap \u00e0 pied on war horses, and the third\u2014\n\n\"Par Dieu, gentlemen\u2014that's Ranald Lachlan, or I'm a minstrel's son.\" The king turned his horse and rode towards the approaching trio.\n\nLachlan waved. The king had eyes only for him, and they rode together and embraced.\n\n\"By all the saints, Ranald\u2014I never expected to greet you on a stricken field!\" The king laughed. \"How fares your fortune?\"\n\nRanald looked away. \"Aweel,\" he said, and a shadow touched his face. \"I'll tell ye, when we've time, my lord. These gentlemen, now, they seek to parley with you. This is the captain of the company yonder, that holds Lissen Carack for the nuns. And his squire, Michael.\"\n\nThe king extended a hand to the knight\u2014a man of middling height with a black beard and blacker circles under his eyes\u2014absurdly young to be any kind of commander, but wearing superb armour.\n\n\"Messire?\" he said.\n\nThe man was staring at him. Then, as if remembering his manners, the man touched his hand and bowed in the saddle. \"My lord,\" he said.\n\n\"You hold the fortress?\" the king asked eagerly.\n\n\"The fortress and the Bridge Castle,\" the captain replied.\n\nThe king thought there was something familiar about the young man's face, but he couldn't quite place it. Something\u2014\n\n\"My lord, if you would bring your forces across I believe we can relieve the fortress and evacuate the villagers\u2014and leave the Enemy facing a newly victualled and garrisoned fortress they cannot hope to take, without the loss of another man.\" The captain was speaking quickly, and his eyes were on the far wood line. \"The Enemy\u2014your father's magus, or so they say\u2014has made a number of errors. Not the least of which has been his consistent underestimation of our side's intelligence. I believe he intends one more all-out attack, to attempt to restore his fortunes through the heroic exertions of his allies.\" The young man smiled crookedly. \"I built a trench line twenty days ago for just this moment, my lord. If you would place your archers in that trench, and gather your chivalry behind the Bridge Castle, I believe we can hand this arrogant Magus a heavy defeat.\"\n\n\"Might I have your name and style, messire?\" the king asked. The plan was solid\u2014the lad had a head on his shoulders, and his pure Alban speech made him one of the king's subjects, mercenary or no.\n\nThe dark-headed man drew himself up straight in his saddle. \"Men call me the Red Knight,\" he said.\n\n\"I thought you to be a Galle, and a good deal older,\" the king said. He turned to the Count of the Borders. \"My lord\u2014will you take the constable's place? Command the Royal Guard? And where is the Count d'Eu? He must have the command of the vanguard now, eh?\"\n\nThe Count of the Borders turned to the young knight. His banner bore a dozen lacs d'amour. \"How many lances do you have, my lord?\"\n\n\"Twenty-six, my lord Count\u2014and the Knights of Saint Thomas. And several hundred very able militiamen, in the form of a contingent of Harndonner merchants. And I have the pleasure of having the aid of the king's own Magus\u2014Harmodius.\" The young fellow bowed in his saddle again.\n\n\"Harmodius is here?\" the king asked. Suddenly, his day looked considerably brighter.\n\nThe young man looked away. \"He has been a pillar of our defence,\" he said. \"With my lord's leave, I must prepare to receive you.\"\n\nThe king smiled\u2014such an odd young man. \"We're right behind you. Go!\"\n\nThe man bowed, as did his squire, and together they rode back across the bridge.\n\nThe king turned to the Count of the Borders. \"He seems odd but able. Wouldn't you say?\"\n\nThe count shrugged. \"He's held this place for twenty days against Richard Plangere and his legions of Hell. Do you really care if he's odd?\"\n\n\"He reminded me of someone,\" the king said. He glanced at Lachlan, who had stayed with the command group. \"You have something to say about our young sell-sword?\"\n\nLachlan shrugged. \"No, my lord. About the Queen. She was struck\u2014in the back\u2014by an arrow. She is resting and doing well, in part thanks to the young fellow there. He used power. I saw it.\"\n\n\"The Queen? The Queen is hurt!?\" asked the king.\n\n\"She's now resting quietly\u2014in the Bridge Castle. The young captain sent for healers.\"\n\nThe king rose in his stirrups. \"Attend me, guards. Let's go!\"\n\nThe Count of the Borders was left with the Royal Staff, sitting on their horses in the dust stirred by the king's rapid departure.\n\nHe shook his head. \"A great knight,\" he said, watching his king. He sighed. \"Very well\u2014messires, attend me. The Royal Guards will cross the river first, followed by the Huntsmen and the Household. In the second line of battle, the Chivalry\u2014\"\n\n# Near Lissen Carak\u2014Gaston\n\nGaston, Count D'Eu, was as tired as he had ever been, and something was wrong with his left hip\u2014it didn't seem to move as freely as it ought\u2014but he managed to get his leg over his destrier's broad back and he rode forward under his own banner, with his cousin's men arrayed behind them\u2014two hundred knights and men-at-arms. Fully a hundred gentlemen lay dead or wounded in the woods and meadows along the road\u2014an absurdly steep price for his cousin's reckless desire to be the man who broke the ambush his angel had told him awaited the king's army.\n\nHis cousin, who lay in the arms of death. Who only wanted to be the greatest knight in the world.\n\nGaston wanted to go home to Galle, sit in the chair of judgment of his castle, and pontificate on which wine was the best at harvest time. He thought back to the peasants under the bridge, his heart now full of understanding. He vowed\u2014would God accept such a vow?\u2014to go home and beg Constance for her hand in marriage.\n\nAt the top of the last ridge, the king's friend, the Count of the Borders, was sitting with a number of other gentlemen under the flapping folds of the Royal Banner. The Count d'Eu rose in his stirrups\u2014damn it, that left hip hurt\u2014and looked down to the river where the red-surcoted Royal Guard were just marching for the great three span bridge. On the other side, two companies of men-at-arms were formed in neat wedges at the base of the great ridge on which the fortress sat\u2014half a league north of the river. From the Fortress of Lissen Carack to the bridge ran a trench, black, as if it had been burned.\n\nAt the western edge of the meadows and burned-out farms that had marked the demesne of the Abbess, thousands\u2014perhaps tens of thousands\u2014of creatures swarmed like ants from a recently kicked hive.\n\nAs he watched, the long arm of a trebuchet mounted high in the fortress swung. It appeared to swing slowly, but its payload\u2014invisible at this distance\u2014flew at the sudden whip-crack release of the counterweight. The count looked for the fall of the shot, but he couldn't see it.\n\nThe Count of the Borders waved. \"My lord,\" he said. \"You command the vanguard?\"\n\n\"I do. My cousin is wounded,\" Gaston said. \"I have fewer than two hundred lances, and many of my younger knights are spent.\"\n\n\"Despite which, the king begs that you will use every effort to get your men across the river\u2014dismount and occupy the line of works prepared for you.\" The count pointed at the black slash that ran from the fortress's ridge to the bridge.\n\n\"I see it,\" Gaston said. \"But I lack the force to occupy that length.\"\n\n\"You shall be with the Royal Guard and all our archers,\" the Count of the Borders added. \"All dispatch, my lord!\"\n\nGaston could see creatures from the swarm now venturing farther and farther into the fields beyond the wood's edge.\n\n\"A moi!\" he ordered. \"En avant!\"\n\n# Lissen Carak\u2014Thorn\n\nThorn watched the Royal Army begin to deploy across the river. His blow was ready\u2014a single hammer strike to win Alba.\n\nThe Royal Army appeared singularly unharmed by a morning-long ambush. That was unexpected. The Qwethenethogs alone should have done great damage amongst their ranks.\n\nHe felt a ripple of power\u2014identified it, and cursed again. Both the dark sun and his former apprentice had survived. He acknowledged his own hubris in imagining them dealt with. It was the very curse of his existence. Why did he constantly think things would go his way?\n\nBecause they should.\n\nHe felt another use of power\u2014closer to him, and it smelled like Qwethnethog. Like Thurkan.\n\nHe nodded and drew power to himself. The Qwethenethogs' presence on this side of the river was very revealing.\n\nThe great daemon was coming for a trial of power. Thorn rocked his stone head.\n\n_Idiot. Traitor. I undertook this for **you**_.\n\nTurquoise fire began to play along the edges of his stick-like tree limbs and his beard of grey-green moss oozed power, and the faeries flitting through the clearing, excited by the overflow of his vast resources, he now drained of power in a single sip, leaving their fragile bodies to flutter to the ground.\n\nThe magnificent daemon entered the clearing from the south. His hide was still wet from swimming the river, but green and brown lightning played along the sides of his head, down to his long, scythed arms and over his richly inlaid beak and armour.\n\nThorn let him come.\n\nWhen they were a few horse lengths apart, Thorn raised one hoary arm. \"Stop,\" he said. \"If you mean me harm, save it for the defeat of our enemies.\"\n\nThurkan stopped but he shook his mighty head. \"Greater Powers than you or I contend here today,\" he said. \"You are a pawn in the plans of a greater Power.\"\n\nThose were not the words Thorn expected, and they stung\u2014stung with the peculiar power of words that carry their own truth.\n\n\"It cannot be,\" Thorn said.\n\n\"Why else do the humans have every advantage when we have none? That thing you call fortune; we have none. Every turn we make favours the enemy. Let us withdraw from this field.\" Thurkan held up an axe. \"Or we must be rid of you.\"\n\nThorn needed time to test the hypothesis that he had been used. He was the one who used others\u2014the enmity of the Outwallers for the Albans, the needs of the boglins for new ground to live, the hunting instincts of the wyverns and the trolls.\n\nHe was not, in turn, used.\n\n\"We have been used!\" Thurkan insisted. \"Order the retreat, and we will fight another day!\"\n\nThorn considered it.\n\nAnd he considered the great mass of his infantry\u2014the wights in their magnificent armour, the five thousand irk archers, the squadrons of trolls ready to engage the enemy's knights. The Outwallers and the wyverns and the other daemons.\n\n\"Even if what you say is true,\" Thorn said, \"we are about to win a great victory. We will scour the kingdom of Alba from the face of the continent. We will _rule_ here.\"\n\nThurkan shook his great head. \"You delude yourself,\" he said. \"There is no number of boglins who can match this number of armoured men in combat. And Thorn\u2014I call you by name\u2014I call you three times to attend my words. A battle, says my grandsire, is the result of a situation wherein both sides imagine they can win a conclusive fight with one throw of the knucklebones. _And only one side is right._ Today, the King of Alba believes he can defeat us. You believe that you can defeat him, despite everything. I say we will lose on this field. Withdraw and I am your loyal ally. Order this attack and I will fall on you with fire and talon.\"\n\nThorn chewed on Thurkan's words for many heartbeats, and not a breeze stirred the torpid late spring heat in the woods. Insect noises stopped. Not a gwyllch chattered, as if all of nature waited on Thorn's decisions.\n\n\"Not for nothing do men call you The Orator, Thurkan,\" Thorn allowed. \"You speak brilliantly. But I doubt your motives. You want this army for your own. The only good you know is the good of the Qwethnethog.\" He took a breath and let it out slowly, to still his rage. And then he threw a single phantasm, a long prepared blow, like a single punch.\n\nThe daemon reacted instantly, raising all of its not-inconsiderable power in a wall of walls to stop the blow.\n\nQuick as a mountain lion Thorn cast again.\n\nThe single gout of green lightning blasted through his walls like a siege ram through the walls of the wattle and daub house, and the tall daemon crumpled to the ground without a sound. He lay still but for the thumping of his left leg under the command of his hindbrain, still battering the ground in rage and frustration at his own death.\n\n\"Attack,\" Thorn ordered his other captains. To the corpse, he said, \"One of us was wrong, Thurkan.\" He reached out and subsumed the daemon's power. And rose from it more powerful than he had ever been.\n\n_I should have done that a year ago,_ he thought, and smiled. And walked out onto the field at the head of his armies.\n\n# Near Lissen Carak\u2014de Vrailly\n\nJean de Vrailly lay dying, content in knowing that he had performed a marvellous feat of arms\u2014one of which men would speak for hundreds of years. His cousin had left him; a correct action, as the battle continued and the king's standard was advancing, and he lay pillowed on the legs of his squire, Jehan, who had also taken a terrible wound.\n\nThe pain was so great that de Vrailly could barely register thoughts\u2014and yet, he was in an ecstasy of relief to be atoning for sin with every waning beat of his heart. The massive damage to his side\u2014the great puncture wounds that sucked air and spat blood and bile with every breath\u2014were living penance, the very stuff of chivalric legend. He would go pure to his Saviour.\n\nHis only regret was that there was so much more he might have done\u2014and in his darker moments of dying, he reviewed how he might have swayed his hips a little farther, evaded the wyvern's blow, and carried on unhurt. So very close.\n\nThe archangel's manifestation took him by surprise\u2014first, because he had refused the angel's orders, and second, because the archangel had always insisted on coming to him in private.\n\nNow he appeared, glorious in armour, cap \u00e0 pied in dazzling white plate, with the red cross emblazoned on a white surcote so utterly devoid of shadow as to seem to repel death.\n\nAll over the beaver meadow wounded men stopped screaming. Servants fell on their faces. Men rose on an elbow, despite the pain, or rolled themselves over despite trailing intestines or deep gouges\u2014because this was the _heaven_ come to life.\n\n\"You fool,\" the archangel said softly\u2014and with considerable affection. \"Proud, vain, arrogant fool.\"\n\nJean de Vrailly looked into that flawless face in the knowledge that his own had deep grooves of pain carved into it. And that he was going to his death. But he raised his head. \"Yes!\" he said.\n\n\"You were quite, perfectly brilliant.\" The archangel bent and touched his brow. \"You were worthy,\" he said.\n\nJust for a moment, Jean de Vrailly wondered if the archangel were a man. The touch was so tender.\n\nThe words cheered him. \"Too proud to betray the King of Alba,\" he said.\n\n\"There is a subtle philosophical difference between killing and letting die,\" the archangel said softly. \"And thanks to you, all my plan is in ashes, and I must build a new edifice to make certain things come to pass.\" He smiled tenderly at the dying knight. \"You will regret this. My way was better.\"\n\nJean de Vrailly managed a smile. \"Bah!\" he said. \"I was a great knight, and I die in great pain. God will take me to his own.\"\n\nThe archangel shook his head. \"Perhaps,\" he said. \"But I think you should live a while longer, and perhaps learn to listen to me next time.\" He bent low, and stripped the bright steel gauntlet off his hand\u2014a slim, ungendered hand\u2014and ran it along the knight's body. That touch struck de Vrailly like the shock of taking his first wound\u2014and lo, he was healed.\n\nHe took a deep and shuddering breath, and found no pain at the bottom of it.\n\n\"You cannot just heal me,\" de Vrailly snapped. \"It would be unchivalrous of me to walk away healed when my brave people lie at the edge of cruel death.\"\n\nThe archangel turned his head, brushed the long hair back from his forehead, and he stood. \"You are the most demanding mortal I have ever met,\" he said.\n\nDe Vrailly shrugged. \"I will beg and pray, if that's what you require, Taxiarch.\"\n\nThe angel smiled. \"I grant you their healing\u2014those who have not already passed around the curve of life into death. And I grant to you great glory this day\u2014for why would an angel of the Lord visit you except to bring you great power in battle? Go and conquer, arrogant little mortal. But I tell you that if you ever choose to match yourself against the greatest Power that the Wild has ever bred, he will defeat you. This is not my will, but Fate's. Do you hear me?\"\n\n\"Craven fate would never keep me from a fight,\" de Vrailly said.\n\n\"Ah,\" said the angel. \"How I love you!\" The angel waved his spear over the beaver meadow.\n\nA hundred knights and as many squires, men-at-arms, servants and valets were cured, their pain washed away, their bodies made whole. In many cases they were better than they had begun the battle. A peasant-born man-at-arms, a Galle, had the permanent injury to his lower left leg healed and made straight\u2014a valet missing one eye had his sight returned.\n\nAll in the wave of a spear.\n\nSeveral dozen wounded Jacks were cured, as well.\n\n\"Go and save the king,\" the archangel said. \"If that is your will.\"\n\nEvery man in the meadow knelt and prayed until, in a puff of incense-laden displaced air, the armoured angel vanished.\n\n# Lissen Carak\u2014Desiderata\n\nDesiderata lay in a patch of bright sunlight. Her power was dimmed\u2014she herself felt like a candle under a shade. Flickering.\n\n_So unjust!_ That single arrow, plummeting from heaven, and she was done. She had desired to be her husband's support, perhaps to win herself a share of glory. And instead\u2014this.\n\nThe strange young man had put the pain at a distance. That was a blessing. She could feel _his_ worthiness like a bright flame. A knight and a healer\u2014what a superb combination\u2014and she longed to know him better.\n\nAround her, her ladies were silent.\n\n\"Someone sing,\" she said.\n\nLady Mary started, and the others slowly joined her.\n\nDesiderata lay back on the cloaks of a dozen soldiers.\n\nAnd then old Harmodius came. He came unannounced, walked into the castle courtyard and knelt beside her.\n\nShe was pleased to see the look in his eye. Even mortally wounded, he found her pleasing. \"There you are, you old fool,\" she said happily.\n\n\"Fool enough to leave the battle and save you, my dear,\" he said.\n\nCarefully, painfully, with Lady Almspend and Lady Mary, he rolled her over and stripped the linen from her back. \"It's really quite a nice back,\" he said conversationally.\n\nShe breathed in and out, content at last.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain could see the king riding for the bridge at the head of his household, and he could see the king's battles\u2014each with more men-at-arms than he had ever commanded\u2014coming down the ridge.\n\nHe rode along the trench\u2014a trench currently occupied by two hundred archers and valets of his own company, and all the farmers from all the out-villages.\n\nHis sanguine surety that the Enemy had made a tactical error was gone, blown away on the wind, and now he watched an endless line of boglins crossing the open ground toward the trench with something akin to panic. It was hard to breathe.\n\nThe Prior was sitting on his destrier with Bad Tom, in the non-shade of a burned oak tree.\n\nThe captain rode his horse over to them, and then wasted his strength controlling his young war horse as the stallion sought to make trouble with the Prior's stallion. Finally, he curbed the big horse mercilessly.\n\n\"I miss Grendel,\" he said to Tom.\n\n\"Bet Jacques doesn't,\" Tom said. He looked back over the sunlit fields. \"They're coming.\"\n\nThe captain nodded. Overhead, the trebuchet disgorged another load of small stones. Cast from a height, it smashed into the oncoming tide and ripped a hole in the enemy line.\n\nThe hole closed almost at once.\n\n\"It's so _stupid_ , the captain said petulantly. \"When he burned the farms, he did all the damage he needed to do.\" He turned his head to where the king's Royal Guard was pouring into the trench, led by two hundred purple- and yellow-clad crossbowman from Lorica. \"And his attack\u2014whether it carries this trench or not\u2014won't take the fortress.\"\n\nThe endless wave of boglins, and larger, worse things, swept across the burned plain towards the black line of his trench.\n\nThe reinforcements were not going to make the near end of the trench in time.\n\nThe farmers and the guildsmen were spread too thin, and they knew it. And the inexperienced purple and gold Loricans were halting, only a third of the way along the trench, and loosing bolts. Like militia.\n\nOf course, they _were_ militia.\n\n\"The farmers will hold,\" Tom said. He was chewing on the stem of a flower. It was an oddly disconcerting sight. \"The guildsmen will break. They've broken before.\"\n\nThe captain looked at the Prior. \"Messire, you are so much my senior\u2014in years, in experience, and in this place\u2014guide me. Or command me.\"\n\nThe Prior let his horse put his head down to munch grass around the heavy bit. \"Oh, no, you don't. You have led this force to this point\u2014you think I'm going to change commanders now?\"\n\nThe captain shrugged. \"I wish you would,\" he said.\n\nTom was watching the oncoming line. \"You know we have to charge that line,\" he said. \"If we charge the line, we should buy\u2014hmm\u2014ten minutes or so.\" He was wearing a grin that made him look like a small boy. \"A hundred knights\u2014ten thousand boglins\u2014and trolls, and daemons, irks...\" He looked at his captain. \"You _know_ we have to.\"\n\nThe Prior looked at Tom, and then back at the captain. \"Is he always like this?\" he asked.\n\n\"Pretty much,\" the captain said to the older man. \"Will you come? I'm not at all sure any of us will come back.\"\n\nThe Prior shrugged. \"You are lucky,\" he said. \"And luck is better than any amount of skill or genius. I can feel the power in you, young man. And I think your presence here is God's will, and God is telling me to go where you go.\"\n\nThe captain rolled his eyes. \"You're making this up,\" he said.\n\n\"Did you speak so to the Abbess?\" the Prior said.\n\nFor once abashed, the captain looked away.\n\n\"We will follow you,\" the Prior continued. \"If this fortress falls our order will have lost everything.\"\n\nThe captain nodded. \"Have it your way, then. Tom; we'll file across the trench on the two bridges and form line on the far side in open order.\" He looked around\u2014to see Sauce, Michael, Francis Atcourt, Lyliard all looking pale with exhaustion.\n\n\"Kill whatever comes under your sword,\" the captain said with an edge of sarcasm. \"Follow me.\"\n\nThe king entered the Bridge Castle's courtyard to find his Magus, Harmodius, kneeling by the Queen. He was examining a wound in her back, and Lady Almspend put a hand on the king's shoulder and kept him from approaching any closer.\n\n\"Give him a moment, my lord,\" she breathed quietly.\n\n\"Here they come!\" called a voice on the walls.\n\nCrossbows began to release in a series of flat snaps.\n\nThe king didn't know what to do. \"I must see her!\" he said to Lady Almspend.\n\nLady Mary came up. \"Please, my lord. A moment!\"\n\n\"The battle is about to be won or lost,\" the king moaned.\n\n\"Fast as you can, lads! The captain is depending on us!\" called the voice on the walls.\n\n\"My love?\" Desiderata called.\n\nHarmodius stepped back, face pale, and the king came forwards.\n\nDesiderata reached out and took his hand. \"You must go and win this battle,\" she said.\n\n\"I love you. You make me a better king\u2014a better man. A better knight. I can't lose you,\" the king said.\n\nShe smiled. \"I know. Now go and win this battle for me.\"\n\nHe bent and kissed her, mindless of the thread of blood that ran from the corner of her lip.\n\nAs he pulled himself away, Harmodius followed him.\n\n\"I might ask you what you are doing here, but we're in haste,\" the king said.\n\nHarmodius narrowed his eye. \"This battle is a closer run thing than I would ever have imagined, and even now, our enemy has increased his power to a degree I could never match,\" he said. \"If I work to heal her he will know me, and he will assail me here. And I will be destroyed. This is as much a fact as the rising of the sun.\"\n\nThe king paused. \"What can we do?\" he asked.\n\nHarmodius shook his head. \"There are protections in the fortress\u2014-especially in the chapel.\" He shrugged. \"But even if I could get her there, my saving her would deprive the army of my protection, and when he starts to kill, he will devastate us.\"\n\nThe king frowned. \"Save her,\" he commanded. \"Save her. I will form up my knights and guard her to the fortress on a litter, and you can take her to the chapel, though all the enemies in the world stand between us and them.\"\n\nHarmodius considered his king, who was willing to sacrifice the army for the love of his queen.\n\nBut his feelings were very much engaged as well. \"Very well,\" he said.\n\n# Lissen Carak\u2014Father Henry\n\nHe didn't like what he had to do. He didn't like that they all hated him, now, and he wanted to argue with them. To show them what they were going to become.\n\nLike her. Like the witches.\n\nGnawing the ropes was easy. But the archers had hurt him, and his back was flayed raw. It took time, and pain. He paused and rested. Paused and slept.\n\nAwoke when he heard voices coming into the cellars. _From below._\n\n_He gnawd his bonds again, mad with fury like a trapped animal. When he exhausted his muscles, he made himself pray. He overcame the pain._\n\nHe was good at pain.\n\nAfter hours and more hours, he had the ropes off. And then he got through the scuttle\u2014a trap door to the next cellar room. He moved carefully, and he only passed out once and woke again minutes or hours later.\n\nHe made it to the base of the main cellar ramp\u2014where he could hear a pair of archers on duty.\n\nHe prayed... and God showed him the way. Whoever had come up into the cellar had left a door open. He dragged himself to the portal, and looked down.\n\nScrambled and found a lantern with a candle and a tinderbox. It was God's will.\n\nHe dragged himself down the steps into the dark.\n\nThe mercenaries, efficient as always, had left arrows painted on the rock. He began to follow them.\n\n# Lissen Carak\u2014Thorn\n\nThorn watched his great assault sally forth from the edge of the woods, and knew fear.\n\nHe had lost many, many creatures in the weeks of siege and now he feared he lacked the resources to survive.\n\nHis fear hadn't started there, though.\n\nAs his assault began, something whose level of manifested power was to Thorn as Thorn was to a boglin shaman, had appeared on the other side of the river. It had cast a single phantasm of such complexity and power that it beggared the very strongest sending Thorn had ever cast. And then it had vanished.\n\nA Power. A great Power of the Wild.\n\nThorn stood at the edge of the burned fields, watching his massive assault leap towards the hated enemy; seeing the fruition of his revenge on the king and his useless nobles, watching as his boglins finally seized the empty Lower Town and boiled through its streets.\n\nAnd all he could think was\u2014 _Damn the daemon. He was right. I've been had._\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain led his men in single file across the boards laid across the burned, vitrified trench. As he crossed, two farm boys with halberds waved. They gave a cheer.\n\nWhy not? They weren't riding into a horde of boglins.\n\nHe laughed. Turned to find Jacques behind him, Carlus the armourer with his trumpet on his hip, and Michael carrying his banner.\n\n\"Form your front,\" he called.\n\nThe line of boglins was about six hundred paces distant.\n\nHe looked back at Bridge Castle, hoping to see the king.\n\nHe looked across the river, but the main battle was just straggling down the ridge. Two thousand knights.\n\nThe king was just a little late.\n\nHe could see a handful of knights crossing the bridge. The banner was from Galle, and not one he knew.\n\n_Move!_ he thought.\n\nHe looked back.\n\nHis men-at-arms, with the addition of all the military orders knights, formed in two ranks, and took up two hundred yards of front\u2014leaving as much again on either flank.\n\nEmpty air.\n\nHe was the centre man in the line.\n\nThe boglin line was four hundred paces away, give or take.\n\n\"Advance! Walk!\" he called, and Carlus repeated it by trumpet.\n\n\"Remember this, boys!\" Bad Tom called from his place in the ranks.\n\nThe big horses made the earth shake, even at a walk. Their tack rattled and clinked, and the sound of their riders' armour added to it. The sound of a company of knights.\n\nTwo hundred and fifty paces.\n\n\"Trot!\"\n\nEven a hundred and fifty armoured men on destriers make the ground rumble like an earthquake.\n\nOne last time, the enemy had underestimated them. They had more than a dozen of the great trolls, belling and ranting several hundred paces to the rear of the infantry line. They were coming on now\u2014coming quickly. But like the king, they were going to be much too late for the moment of impact.\n\nThe captain had a feeling, though, that the trolls were not at their best in the open, and that they wouldn't be particularly manoeuvrable. Or was that his own hubris?\n\nBut that was all passing away. Strategy and tactics were over, now.\n\nHe turned his head at the cost of some pain, and saw the Gallish knights pushing along the trench. The Lorican crossbowmen were moving too\u2014Ser Milus was visible, roaring orders at them.\n\nThere would be no gap in their line when the Enemy struck.\n\nThe two lines were approaching each other at the combined speed of a galloping horse. The boglins were not going to flinch but they were spread out over the ground, all cohesion lost, like a swarm of insects pouring over the ground.\n\n\"Charge,\" he shouted. Carlus and Jacques might not have heard him over the drumming hooves, but he swept his lance down to point at his first target\u2014locked it into the hook-shaped rest under his arm, and Jacques sounded the charge.\n\nThe captain leaned forward into his lance.\n\nFor a few glorious heartbeats, it was the way he had imagined, when he was a small boy dreaming of glory.\n\nHe was the wind, and the roar of the hooves, and the tip of the spear.\n\nThe slight bodies of the boglins were like straw dolls set in a field, and the lances ripped through them so smoothly that creatures died without dragging the lances down, and the stronger men were able to engage three, four even five of the creatures before their lances broke, or their points touched the ground, dug in and shattered or had to be dropped.\n\nThe horses were spread widely enough to allow horse and rider to thread the enemy line, to take advantage of spaces between boglins, to weave their path.\n\nFor a few deadly heartbeats, the knights destroyed the boglins, and there was nothing the boglins could do to retaliate.\n\nBut like mud clogging a harrow, the very density and sheer numbers of the boglins began to slow the knights' charge and even their heavy horses had to shy\u2014or simply could no longer trust their hooves to ground that was so thickly littered with boglins. The charge slowed, and slowed.\n\nAnd then the boglins began to fight back.\n\n# Lissen Carak\u2014Father Henry\n\nFather Henry paused at the base of the steps to gather his courage. His hate. He was deep underground, and his candle was guttering, and he had no idea how far it was to the outside. And he _hurt._\n\n_He prayed, and then he walked. Walked and prayed._\n\nAnd, of course, it wasn't much farther than walking down the castle road, outside.\n\nHe finally found a pair of double doors, as high as two men, and as wide as a church. He expected them to be locked with all the power of Hell. But the sigils lay cold and empty. He reached for the two great handles. There was a key between them.\n\n# Lissen Carak\u2014The King\n\nThe king had his queen on a litter between four horses, and he and his household knights got out the main Bridge Castle gate even as the garrison shot bolt after bolt over their heads into the oncoming line of creatures.\n\nEven as he watched he saw the Prior and the sell-sword knight lead their men-at-arms over a pair of narrow wooden bridges and onto the plain.\n\nHe looked to the right and left, trying to imagine why they were charging the enemy.\n\nBut it was _glorious_ to see.\n\nThe knights took their time, formed up neatly, and the endless horde of enemies ran at them silently\u2014perhaps the most horrible aspect of the boglin was its silence. He could hear the mercenary captain calling orders, and his trumpeter repeated them.\n\n\"Ready,\" Ser Alan said.\n\nThe king gestured across the front of the trench. \"Since our friends have been kind enough to clear us a path,\" he said, and touched his spurs to his mount.\n\nAs he rode, he watched the charge go home.\n\nIt was superb, and he was annoyed that he wasn't a part of it. He leaned back to Ser Alan. \"As soon as we have the Queen to the fortress, we will join them,\" he said, pointing to charge which was cutting through the enemy like an irresistible scythe.\n\nSer Ricar shook his head. \"My lord,\" he protested. \"We have only sixty knights.\"\n\nThe king watched the charge even as his household trotted across the front of the trench. \"He hasn't much more than that.\"\n\n\"But you are the _king_! _\"_ Ser Alan protested.\n\nThe king began to feel the onset of the indecision that infected him on every battlefield. A lifetime of training in arms as a knight demanded that he lead his knights in that wonderful charge\u2014a charge that even now was beginning to lose its impetus, three hundred paces from the trench at his feet.\n\nHe was also aware\u2014as a man is aware of a distant call\u2014that it was not his duty as king to perform feats of arms.\n\nBut Desiderata had said\u2014\n\nThe fighting was so close.\n\nAnd his queen didn't need him. She had a clear path all the way to the gate of the fortress.\n\n\"Knights!\" roared the king. \"On me!\"\n\n# Lissen Carak\u2014Father Henry\n\nThe priest had the secret doors open, and he stood back and watched the boglins flood through the great opening, squirming in a very inhuman way, to vanish onto the steps which ran up and up into the ridge. He watched for a moment, and then something slammed into his head.\n\nHe started to fall. Out of the corner of his eye he could see some sort of spike.\n\nIn a moment of vertigo, he realised it had to be _through_ his head.\n\nHe tried to move, and couldn't.\n\nSomething hurt more than his back.\n\nSlowly, like a tree falling, he went to the ground. He tried to pray, but he could not, because they pressed all around him and he screamed, trying\u2014\n\nTrying to die before they began to eat him.\n\n# Lissen Carak\u2014Ser Gawin\n\nSer Gawin had risen with the dawn and managed to get himself to the chapel to pray. He remained on his knees for a long time in the morning light, unaware of anything except the pain in his side and the crushing sense of his own failure.\n\nBut, eventually, he roused himself when he heard the soldiers bellowing for every man-at-arms to get mounted. He rose and crossed himself, and then walked as steadily as he could manage out the door of the chapel, and hauled himself in front of Ser Jehannes.\n\n\"I can ride,\" he said.\n\nJehannes shook his head. \"He didn't say the wounded,\" Jehannes said. \"I'm not riding, myself, lad. Stay here.\"\n\nGawin was minded to disobey. The longer he was on his feet, the better he felt. \"I can ride,\" he said again.\n\n\"Ride tomorrow, then,\" Jehannes said. \"Tom's got all the men-at-arms already. If you want to be a help, arm yourself as much as you can and walk around looking confident. It's bad out there.\" Ser Jehannes pointed into the courtyard of the fortress, where the farmwomen and the nuns stood in knots, silent. Most of them were watching the plains below. \"We've perhaps forty men to hold the fortress, and yon ladies feel they've been abandoned.\"\n\n\"Sweet and gentle Jesu,\" Gawin swore. \"Forty men?\"\n\n\"Captain's trying to win the day,\" Jehannes said. \"Stupid bastard. All we had to do was sit tight in the fortress and let the king do as he would. But the little bourc always has to be the fucking hero.\"\n\nGawin gave the older man a lopsided smile. \"Family affliction,\" he said, and went to do his share.\n\nIt took him long minutes to find his armour, left unpolished in a heap and not in the hospital but in a closet off the apothecary.\n\nBut he couldn't seem to get into it.\n\nHe managed, in the end, to get into his arming cote, and to get his breast and back closed by lying full length on the floor and closing it around him like a clamshell. But then the pain in his side kept him from buckling it.\n\n\"I'll do your buckles, if you'll let me,\" said a voice.\n\nIt was the novice. The one whose appearance made his brother squirm. The one who had used power to heal him.\n\n\"You are\u2014\"\n\n\"Amicia,\" she said. She nodded at an archer, who stood quietly across the room. He looked tired and unhappy. \"He was left to guard me, but he's bored, and I haven't turned into a boglin or a dragon yet. Stop moving.\"\n\nHer hands were curiously confident. And strong.\n\n\"You are using power,\" he said.\n\n\"I'm giving you some strength,\" she said. \"Something evil is coming\u2014I can feel it. Something of the Wild. We're going to go and stop it.\" She sounded fey, terrified, and overly bright. Brittle.\n\nGawin took her assertion at face value. He looked at the archer. \"What's your name?\" he asked.\n\nThe boy wouldn't meet his eye. \"Sym, my lord,\" he said sullenly.\n\n\"Sym, can you fight?\" Gawin asked.\n\n\"Anything,\" Sym said assertively. Looked away. \"Only thing I'm any good at, and look at me\u2014left to guard the captain's nun.\"\n\nThe fingers on Gawin's shoulder harness stiffened.\n\nSym looked at the two of them from under his eyebrows. \"Sorry. Know you ain't. But I'd rather be with my mates.\" He shrugged. \"This is the big fight. I never been in one. All the oldsters talk big about this fight and that fight, but this is the biggest the company was ever in, and I want my part of it by fucking God.\" He looked away. \"Want to be a hero.\"\n\nGawin laughed. He surprised himself with the purity, the unforcedness, of his laugh. \"Me, too,\" he said. He slapped his shoulders. He couldn't bear the weight of his arm harness, but he had a breast and back, and she put the gauntlets on his hands, and then, with Sym's help, they put his bascinet on his head, slipping the aventail over his hair.\n\nHe considered saying something flirtatious\u2014 _Best looking squire I've ever had._ But at the thought of _squire_ he choked.\n\nWhile Sym pulled his aventail down over his back plate, she did something\u2014something that started as a word, and rose in pale yellow fire, and ended like the pop of a soap bubble.\n\n\"Mater Mary,\" she said, and crossed herself. \"They are here. Right here. In the fortress. Follow me!\" she called and ran for the door.\n\nSym followed her, leaving Gawin to find his long sword resting in a corner, pick up Sym's buckler, and follow.\n\n# Lissen Carak\u2014The Red Knight\n\nWhatever his other failings, the captain's borrowed young destrier had a great heart, and he loved to fight.\n\nThe horse swung back and forth\u2014pivoted on his forefeet and kicked with his iron-shod back hooves, half-reared and pivoted on his back feet, punching with his front, keeping the captain in the centre of a carefully cleared circle devoid of standing foes. Boglins who tried to get under the horse to hamstring him or worse were trampled to sticky ruin or simply kicked clear.\n\nThe captain had long since lost track of how many of the creatures he'd killed. His arm was tired\u2014but then, he'd _started_ the action almost too tired to lift his weapon.\n\nBut, as they had practised, the companions were drawing together\u2014horse to horse, man to man.\n\nThe captain swung from the shoulder, nipped both arms off an enemy on the foreswing like a farmer pruning vines, leaned well forward using his stirrups for balance, and cut back into another creature's head, clearing his front, and George\u2014somewhere in the combat, the captain had named his horse George\u2014backed a few paces.\n\nAnd tucked in behind Bad Tom, who was like a millwheel of destruction.\n\nHe let Tom do it. Thumbed his visor, and raised his face plate, and drank in great gouts of fresh air.\n\nGeorge wanted to be back at it.\n\nThe captain stood in his stirrups and looked over the battle line. His people had formed up well and althought there were gaps, there were not many.\n\nHis people going to get buried.\n\nHe had no sense of time\u2014no one did, in a hand-to-hand fight. But at his back, the purple and yellow tabards had flowed all the way down the trench to Master Random's guildsmen, and a sturdy line of scarlet was filling in behind them. And beyond them, just crossing the bridge, was solid green. Archers of the Royal Hunt.\n\n\"Jacques!\" he roared.\n\nHis valet was two horse lengths away, fighting for his life.\n\n\"Carlus!\" he roared.\n\nThe trumpeter didn't even look around.\n\n\"Damn,\" the captain said. It was a game of seconds and hard-fought inches, and he was losing time. They needed to ride clear.\n\nHe gave George his head and sent the war horse crashing into one of Jacques's adversaries. A ton of war horse versus a hundred pounds of irk was no contest at all.\n\nHis sword took another, and then Jacques went down as his horse fell\u2014killed by one of the dozen creatures under its hooves. That quickly, Jacques was gone. The captain turned, cut at the irk under George's feet and watched a spear catch Carlus under the jaw, killing him instantly. Down he went, with his trumpet, and with it, their chance to cut their way free. The captain cut down, his sword beheading a boglin even as the hideous thing bit into Jacques's throat\u2014and he roared and looked for help, but there was none.\n\n# Lissen Carak\u2014Desiderata\n\nGuarded by Ser Driant and five knights, the Queen's litter started up the long and twisting road to the great gate of the fortress.\n\nThe king had ordered his knights to form a compact company behind him.\n\n\"Once more, my lord,\" Ser Alan said, \"I'd like to remind the king that if Lord Glendower were alive, he would never allow this.\"\n\nAt the word _allow_ all sense left the king's head. \"I'm the king,\" he said. \"Follow me!\"\n\nMost of the mercenary knights and their retainers had formed in a thick knot, almost dead centre in the field. The king aimed his horse's spiked head at the banner with the lacs d'amour. \"Follow me!\"\n\n# Lissen Carak\u2014Harmodius\n\nHarmodius spat with rage, turned his horse, and followed the king, who was throwing himself into the arms of his enemy when almost any other action would have saved him.\n\nThe Queen would die. And he, Harmodius, loved her in a way the king never could\u2014she was the perfect child of Hermeticism. An angel, come to earth.\n\nBut like an artist with a favourite painting, Harmodius could not bear to see the king die either. Not here\u2014not so close to triumph, or at least to survival.\n\n_We are all making the wrong decisions_ , Harmodius thought. And he realised that if he died here, his new-found knowledge would die with him.\n\nIt was like some ancient tragedy, in which man is granted knowledge only to be destroyed.\n\nBut he didn't have to waste much more time on such thoughts.\n\n# Lissen Carak\u2014Thorn\n\nThorn watched, almost unbelieving, as the target of his campaign threw himself forward, unprotected. He couldn't have manipulated the king into such a foolish move.\n\nThe king.\n\nHe had made a dash for the fortress and Thorn had suddenly seen his defeat\u2014for in the fortress the king would be unassailable.\n\nBut no.\n\nThe fool was now leading his knights forward into the very maw of Thorn's monsters.\n\nAnd his boglins were _in the fortress._\n\nJust for a moment, he was balanced on an exquisite knife-blade of doubt as to whether to kill the king himself, by means of power, or to send his choicest creatures to do his work.\n\nBut in that moment, he decided that, regardless of the campaign, if he killed the king, he had won. No matter which power was using him, killing the King of Alba would place him in the front rank. It would cause civil war. Would weaken the human hold on Alba.\n\nHe gathered power to him.\n\n# Lissen Carak\u2014The Red Knight\n\nThe company was dying around him.\n\nThe anonymity of armour kept him from knowing who\u2014he could never spare more than a glance\u2014but as the boglins surrounded them and hemmed them tighter and tighter, armoured figures went down\u2014either hamstrung horses, spear thrusts, or lucky arrows.\n\nTom continued to be like a hammer at his side, Sauce was like an avenging angel, and the military order knights fought like the legions of Heaven.\n\nEven as he raised and lowered his sword yet again, he would have chuckled at the pointlessness of it all, if he had not been occupied. They had bought the time, and the battle should now be safely won. And the bitterness\u2014had Carlus not gone down with the trumpet, had Jacques lived fifty more heartbeats\u2014\n\nHe slew two more boglins before he saw the troll.\n\nIt reared, its blank stone face smooth and black, and it belled, it's shrill trumpet ringing out above the ring of weapons and the silent intensity of the boglins.\n\nNot just one of them.\n\nSix of them.\n\nAnd the wave front of their fear made the boglins beneath his horse's hooves quail and void their attacks. George rose, kicked out, and then plunged forward.\n\nThe wave of terror passed over them.\n\nThe captain got his sword in a good two-handed grip, and George leaped for the nearest troll as he brought it up high above his head on the left. _You are supposed to use a lance on these things,_ he thought.\n\nThe troll saw him, turned, and put its antlered head down, low, so its antlers covered its neck, and charged, seeking to get its antlers under the Red Knight's sword and unhorse him.\n\nGeorge turned mid-stride.\n\nFaster than human thought, the animals struck.\n\nLike a cat, George pivoted his weight and one hoof licked out and caught the monster a staggering blow in the centre of the forehead, so hard that it cracked its stone face.\n\nThe troll screamed, turned its head, whipping its antlers through a spray of motion and leaped, turning, caught the armoured horse in the right rear haunch. George got his back feet off the ground with a caper and the blow slewed the horse around on his forefeet\u2014\n\nThe line of attack opened like a curtain as the two creatures turned into each other. The captain felt as if he had all the time in the world\u2014as if this moment had been predicted since the dawn of the world. The troll's turn\u2014his destrier's turn\u2014the open line at the back of the monster's neck...\n\nHis sword struck, two handed, like the fall of the shooting star to earth, and cut along the line where two great plates of hardened flesh met; sliced through the troll's spine, and in, down, out and free in a gout of ichor\u2014\n\nGeorge leapt free, stumbled, and the captain was thrown from the saddle.\n\nHe got a shoulder down, landed on something squishy and rolled, the plates of his shoulder harness clanking like a tinker's wagon and the muscles in his neck, injured and re-injured since early spring, wrenched again.\n\nBut he ended his shoulder roll on his knees, and pushed immediately to his feet.\n\nOff to the right, Tom and Sauce were pouring blows into another troll, but behind them the thick knot of companions had begun to dissolve as the remaining trolls ripped into their horses. Armour crumpled; men died.\n\n# Lissen Carak\u2014Ser Gawin\n\nGawin followed Sym as the archer followed the novice\u2014down the stairs, across the courtyard to the entrance to the cellars where the stores were kept.\n\nThere were two archers guarding the heavy oak door to the cellars.\n\n\"The Wild is coming up the escape!\" Amicia yelled, fear and frustration powering her words.\n\nEvery farm wife and nun in the courtyard heard her.\n\nThe two archers looked at each other.\n\nSym came up next to her. \"Captain's orders!\" he yelled, his thin voice shrill and not very heroic.\n\nThe bigger of the two archers fumbled with his keys.\n\nGawin ran across the yard to join them.\n\nThe women were frozen, and he had a moment to consider the looks on their faces\u2014panic, determination, and a sullen kind of anger that it should come to this when they had already lost so much.\n\nYes, he understood those looks of loss. Of failure.\n\n\"Arm yourselves!\" he called to them.\n\nThe bigger archer opened the iron-bound oak door and Sym ran down the steps into the darkness.\n\nGawin pushed past the novice.\n\nThe first cellar was gloomy but well-enough lit. A stack of spears leaned against one of the company's great wagons. Gawin caught one up as he went by.\n\nThere was another door, ahead, which was just opening.\n\nSym was too late to stop it, so he spitted the creature that opened it\u2014ripped his sword out of the boglin's armoured thorax and kicked it so hard that it folded backwards\u2014\n\nGawin caught a glimpse of steps going down and a seething knot of the creatures filling the stairwell.\n\n\"Hold the door!\" Gawin called. He thrust with his spear, and felt the steel head crunch through the soft hide around the boglin's neck and head\u2014just like digging a knife into a lobster. Something popped, it fell off his spear, and he pushed.\n\nSym cut, and cut again, and again, desperation and terror lending wings to his sword arm.\n\nThe stairwell was crawling with them.\n\nHe killed another one.\n\nAnd another one.\n\nAnd the novice turned, raised her hands, and spoke a single word in Archaic, and golden-green light filled the cellar.\n\n# Lissen Carak\u2014Desiderata\n\nDesiderata could scarcely breathe for the immanence of power. And the pain, which was returning. But she could feel the enemy\u2014the centre of the power of the Wild, its emerald intensity shot full of black\u2014gathering force. She could feel it as surely as she could feel the power of the sun on her arms.\n\n\"What's happening here?\" Ser Alan asked. He bent to carefully place her litter on the doorsill of the chapel.\n\nThe woman was older\u2014dressed plainly, like a servant or a farmwife. She had a spear in her hands. \"If it please you, Ser Knight\u2014there's boglins got into the cellars, and all the garrison is trying to hold the doors.\"\n\n\"Good Christ!\" Ser Alan cursed. The other knights of the escort drew their swords.\n\n# Lissen Carak\u2014Thorn\n\nThorn watched as the king and his knights obligingly fought their way into the centre of his range.\n\nSometimes plans did work out.\n\nHis trolls\u2014the magnificent _dhags_ \u2014were cutting the knights to pieces. They were also dying, but he had more. Or he could obtain more. The Wild was fecund beyond human imagining.\n\nHe let the king fight on\u2014on and on\u2014until his reckless charge broke through the ring of bone and hide around the mercenaries. Around the dark sun.\n\nThe king and the dark sun together.\n\nHe took his gathered power, summoning every tendril that he could muster\u2014the might that had been Thurkan, the souls of the fair folk, the convoluted essence of the Sossag shamans\u2014\n\nHe savoured it, for a moment.\n\nThere was nothing to interrupt him, no distractions as he placed his power almost lovingly on a spot just between his two foes.\n\nThe edifice of his memory was no palace but a twisted yarn of ropes and webs, and he braided them in his mind with the mastery of an aeon.\n\nLaid his hand to the completed cord, and cast.\n\nHarmodius felt it, saw it, and cast his counter: a mirror. Even his counter had tails and vestiges\u2014traps within traps. As he had learned.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain felt the moment the great phantasms were loosed as a single instant. It was as if fire or lightning had flashed through every inch of the air between the two casters.\n\nHe was _Harmodius. As, for a moment, he had been Amicia._\n\n_There was no time._\n\n_He had so little left\u2014but he gave it, straight into Harmodius's arms. He reached and took from Amicia, who was herself fighting for her life\u2014from Miram and her choir. And from the very sunlight around him._\n\n_And it wasn't going to be enough._\n\n_The captain reached out to the great iron-bound door, and threw it open, and green light flooded into him._\n\n_He threw it through Harmodius to strengthen the counter work._\n\nThere was a thunderclap\u2014a gout of white-green fire that shot into the heavens. A ripple in the curtain of reality so that, just for a moment, the veil of the world was wrenched aside. The captain saw black night pierced with white stars, and the dawn of chaos, and the rising plume of power that was the coming of the world.\n\n# Lissen Carak\u2014Desiderata\n\nDesiderata felt Harmodius's power rise to meet the emerald giant\u2014and she saw the deep subtlety of his mind in his casting.\n\nBut the emerald's might was twenty times greater than that of the court Magus, and the tide of green rolled over him\u2014dissipated, mirrored, channelled\u2014but overpowering, like a rising river facing a plain full of channels and damns, yet eventually overcoming all of them to spill in one unstoppable flood\u2014\n\nBut vast quantities of the emerald power hung in the air, cast aside by Harmodius's counter spell. Or part of it.\n\nThe ripple of power passed the king, who watched, horrified, as Ser Alan was burned at his side, his armour straps charring, his face a livid red as he screamed\u2014and man and horse collapsed. Beyond him, Harmodius frowned\u2014his hand withered and blew away to ash and then, in a few heartbeats, the Magus was subsumed. He turned to ash, crumpled and was borne away on the wind.\n\nThorn was struck by the mirror in the very moment of completion of his phantasm, and some of his own carefully hoarded power struck right back down the channel of his casting, burning him.\n\nHe screamed. Flinched. But far across the battlefield, Harmodius's essence flickered and went out.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain struck, the sword descending more from the force of gravity than from any power of his shoulders.\n\n_In the Aether he had Harmodius by the hand._\n\n_Take me, boy._\n\nIn one moment, the captain had to understand, and to act. He opened _his way into his palace, seized the spirit of the dead Magus with one Aethereal hand and cast his own phantasm with the other. The air outside was heavy with discarded power, green and ripe for plucking, and he took it, aided by the meticulous ordering of last night's foe, aided by the thaumaturgical knowledge of his tutor\u2014of Amicia's Wild casting_ \u2014\n\n_And there he was. Standing on the plinth, where she had always stood._\n\n_\"Better the slave of a bad master,\" the Magus muttered._\n\n_Suddenly the captain was unsure whether he should have allowed this\u2014entity\u2014refuge to his palace._\n\n_\"Any port in a storm, lad,\" the dead Magus said. \"Go fight monsters, or you'll be as dead as I am.\"_\n\nAnd he lifted his sword again. The air was still redolent with power.\n\nGeorge was behind him, and on his feet.\n\n_Amplify my voice_ , he told the dead Magus.\n\n\"Wedge! On me! Michael\u2014the banner to me!\" His voice rang out like some antique god's.\n\nIn a moment out of time, the captain wondered if this was _exactly_ how the antique gods came about.\n\nNo time like the present.\n\n_Kneel!_ He commanded the creatures of the Wild.\n\n_Hermes Thrice-sainted, boy! You are challenging his control! Stop!_\n\nA third of the creatures around him stopped fighting, fell back or stood, stunned.\n\n# Lissen Carak\u2014de Vrailly\n\nSer Jean de Vrailly led the main battle of the king's host down the last ridge, and their hooves clattered like a fall of hail as they crossed the bridge. He had more than a thousand belted knights, and no one\u2014not even the Count of the Borders\u2014questioned him. An archangel had given him great glory, and every man in the main battle knew it.\n\nJean could see the Royal Standard trapped, far out in a sea of foes, with another standard he didn't know\u2014lacs d'amour in gold on a field of black. A foppish banner.\n\nBut he laughed to see the battle, and led the first files to cross the bridge off to the left, west towards the setting sun.\n\nThe soldiers in the long trench were rising from it, either in loyal determination to save the king, or in eagerness to join his attack.\n\nGood for them. For once, there was to be enough glory for all.\n\nHe continued to ride west, and the long file of knights followed him\u2014gradually enveloping the southern flank of the enemy.\n\nBehind him, the Count d'Eu rose to his feet, and pointed his cut-down lance at the knot around the Royal Standard. \"A moi!\" he roared.\n\nDaniel Favor, former wagoner, climbed over the edge of the trench, to stand on the grass in the wind. Around him, farmers from the villages around Lissen Carack looked at him, and knew they could not let him be a better man.\n\nAdrian Pargeter climbed out of the safe trench, and put his crossbow on the ground to draw his sword. Older guildsmen looked at each other. A draper with a grey beard asked his lifelong business rival\u2014 _we really doing this?_ \u2014and then they were up the vitrified earth too, drawing their swords.\n\nRanald Lachlan leaped up the side of the trench, waved his axe at his comrades, and pointed it at the enemy. \"Come on, then!\" he said.\n\nThe trench emptied in moments, and they came.\n\nLachlan threw his axe in the air, and it spun in a great wheel of light over his head and fell back into his hand.\n\nAnd the thin line of men charged.\n\n# Lissen Carak\u2014Ser Gawin\n\nGawin saw Sym stumble, and a pair of the armoured things took him\u2014dragged him down. Sym's dagger licked out, gutted another boglin which fell atop him... and then the archer was gone, and Gawin was alone in the doorway.\n\nA bright green light flashed, and Gawin was able to see far too much in the illumination. The crawling things beneath him on the stairs turned brown, their eyes burned away and dozens of them sank to the ground, all vitality leaching away as their bodies crumbled.\n\nGawin heaved a breath.\n\nThere were a dozen of the things left\u2014all in a clump, a crawling, rolling mass of legs\u2014he cut and cut at them like a madman, and then forced the door with sheer weight and determination, and he stumbled back...\n\nA swarm of armoured men fell on the knot of boglins, hacking with axes, stabbing with spears\u2014six knights he knew all too well. Ser Driant\u2014the King's Companion\u2014other men of the household.\n\nGawin found himself pulled to the floor. He'd lost a moment's attention and two of the things had him\u2014\n\nBut he was Hard Hands, and he closed his left hand and slammed it into a lobe-shaped eye, keyed his hand around his adversary's arm, and ripped it off the boglin with a tearing like ripping old leather, and then he swung the taloned arm like a club beating the bleeding thing to the ground. Ripped his rondel from its place at his hip, drove his knee into the soft place at the centre of the second boglin's breast, and as its arms closed on him, slammed the dagger home, breaking its back. Spears slammed into the thing from all sides.\n\nHe got to his feet with his dagger clenched like a mantis's claw. But the only figures standing in the green-lit cellar were armoured men.\n\nGawin sagged.\n\nSer Driant reached out an ichor-spattered hand. \"Ser Gawin?\" he said.\n\nGawin was looking for the novice.\n\nShe was slumped against the wall. At her feet were the remnants of Sym the archer\u2014the skin of his face flensed away where they'd swarmed atop him. She was pouring her power into him.\n\n\"You cannot help him,\" Gawin said. \"However great your talent, you cannot help him.\"\n\nShe ignored him.\n\nSer Driant seized his shoulder. \"Is she a healer?\" he asked.\n\n# Lissen Carak\u2014Thorn\n\nThorn felt the challenge as a blow in his gut.\n\n_The dark sun._\n\nThe young Power glowed with fresh vitality. He had taken new prey, and he was stronger for it.\n\nThorn gathered his wits.\n\n_I am hurt. He is not. And I have been duped._\n\n_What if he can best me?_\n\nThe air between them was thick with the misspent green power of his last phantasm, only partially expended. He had only to reach forth and take that power...\n\nBut if he was caught while doing it, it would be the end of him.\n\n_What if this was a Power's plan all along? To lead me to over-extend, so that I might be destroyed?_\n\n_Oh, Thurkan, it may be I owe you an apology._\n\nCarefully, he began to wrap sigils of concealment about himself, even as he roared with false defiance.\n\n_Attack!_ he commanded his creatures.\n\nHigh above him, in the fortress of his enemies, someone seized the power of the Wild\u2014raw\u2014and shaped a mighty phantasm with it.\n\n_So!_\n\nHe wasn't waiting for the trap to close. He fled.\n\n# Lissen Carak\u2014de Vrailly\n\nJean de Vrailly judged his moment well. He had led the chivalry of Alba off to the west almost a league along the river. A handful of boglins had tried to oppose him, his sword was wet with their hellish ichor, and it was as easy as taking the heads off fennel plants in his mother's garden.\n\nAnd now\u2014\n\nOh, the glory.\n\nHe raised his arm, closing his fist\u2014turned his horse. \"Halt!\" he ordered. \"Now turn to face the enemy!\" Not a military command, but he had never led so many knights, and he didn't know their commands in their language. So he turned out of the line, and cantered along the column. \"Face me!\" he called. \"Come! Turn your horses!\"\n\nAs soon as half a dozen knights understood him, they all understood. And the great column, a thousand horses long, turned into a line a thousand horses wide as he cantered down the front, his lance held above his head, the royal arms of Alba sparkling on his chest.\n\n_I will be king_.\n\nHe didn't know where the thought came from, but suddenly it was there\u2014he grinned and turned his horse to face the enemy. He was in the centre of this mighty line. To his right front, his own dismounted knights, led by his cousin, and the men of the King's Guard had just slammed into the enemy fighting line. They were outnumbered badly.\n\nBut it didn't matter.\n\nBecause he lay across the enemy's line, like the crossing of a T, and the enemy had committed all of his reserves. And there was no force on earth, in the Wild or out of it, that could stop a thousand of his kind charging in a line.\n\nHe raised his lance high, feeling the astonishing, angelic vitality that filled him. \"For God and honour!\" he roared.\n\n\"Deus veult!\" cried the knights. Men closed their faceplates.\n\nAnd then the line started forward.\n\nThe battle was over long before the first lance struck home. The enemy's whole right wing had begun to melt back into the forest as soon as the knights emerged over the bridge\u2014and now, as their charge rumbled forward, the wyverns, the trolls, and the handful of daemons edged back too. Some simply turned and ran for the woods. They didn't have the bad judgment of men. Like any animal in the Wild faced with a larger predator, they turned and fled. Wyverns leapt into the air; the remaining trolls ran with stone-footed grace, and the daemons ran at the speed of racehorses\u2014untouchable.\n\nOnly the boglins and irks stood and fought.\n\nAnd in the centre, held by Thorn's will, a dozen mighty creatures and a horde of boglins continued to try to kill the king and the dark sun.\n\n# Lissen Carak\u2014The Red Knight\n\nThe captain could no longer raise his sword to cut. He had the weapon in both hands\u2014his left gauntlet held the blade halfway down, and he used it as a short spear, slamming the point into faces and armoured chests.\n\nMoments of terror blended together\u2014a scythe talon that came inside his visor, luck or skill directing the razor-sharp claw to curve up into his scalp and hair, leaving him alive instead of blind or dead.\n\nA trio of irk warriors dragged him down with their sheer weight, their thin, strong limbs racketing against the steel of his armour in a killing frenzy. As slowly as honey poured on snow, or so it seemed, his right hand burrowed past the hideous strength of their limbs to the rondel dagger at his hip, and then he was on one knee, and they were gone, and his dagger dripped gore.\n\nThe comfort of steel armour rasping against his own\u2014back to back. He didn't know who it was, he was just thankful for steel not chitin.\n\nAnd then, a daemon.\n\nThis lord of the Wild was taller than a war horse. The captain hadn't remarked on their absence from the battlefield, but now that he faced one some part of his brain registered that he hadn't faced one before.\n\nThe crest on its head was a livid blue\u2014utterly different from the one he'd faced in the woods to the west, or in the dark.\n\nIt watched him intently, but it didn't attack.\n\nHe watched it and wished he had his spear\u2014currently leaning against his armour rack inside the fortress\u2014and a horse, and a ballista, and twenty fresh friends.\n\nThe thing had a pole-axe the size of a wagon's axle-tree. The head was flint. It was crusted with blood.\n\nIt turned its head.\n\nHad he been fresh he'd have sprung forward with a mighty attack while it was distracted, but instead he merely breathed again.\n\nIt looked back at him.\n\n\"You are the dark sun,\" it said at last. \"I can take you, but if you hurt me, I will die here. So instead\u2014\" It saluted him with a flourish of the great pole-axe. \"Live long, enemy of my enemy.\"\n\nIt turned and ran.\n\nThe captain watched it go, throwing boglins from its path, with no idea who or what it was. Or why it had left him alive.\n\nBut he was trembling.\n\nHe fought more boglins. He cut some sort of tentacled thing from the Prior, who flicked him a salute and went back to work. Later, he saw the king go down, and he managed to get a foot on either side of the king's head, and then all the monsters in the Wild came for him.\n\nSome time passed, and he was standing between Sauce and Bad Tom, and the King of Alba's body lay between his feet. The last rush of the monsters had been so ferocious as to rob the word of all meaning\u2014an endless rain of blows, which only fine armour could repel, because sheer fatigue had robbed muscles of the ability to parry.\n\nTom was still killing.\n\nSauce was still killing.\n\nMichael was still standing...\n\n... so the captain kept standing too, because that's what he did.\n\nThey came for him, and he survived them.\n\nThere finally came a point when the blows stopped. When there was nothing to push against, no fresh foe to withstand.\n\nBefore he could think about it, the captain slapped his visor open and drank in the air. And then bent down to check the king.\n\nThe man was still alive.\n\nThe captain had had a leather bottle, just an hour ago. He started to search his person for it with the slow incompetence of the utterly exhausted.\n\nNot there.\n\nHe felt an armoured back against his, and turned to find the Captain of the King's Guard\u2014Sir Richard Fitzroy. The man managed a smile.\n\n\"I will build a church,\" Michael chanted. \"I will burn a thousand candles to the Virgin,\" he went on.\n\n\"Get the crap off your blade,\" Tom said. He had a scrap of linen out of his wallet, and he was suiting action to words.\n\nSauce didn't grin. She took a handkerchief from her breastplate and wiped her face. Then she took in what her captain was doing and handed him a wooden canteen of water, pulling it over her shoulder on a strap.\n\nHe knelt and gave water to the King of Alba.\n\nWho smiled.\n\nThe knight who reined in above him provided some shade. His giant war horse had a hard time standing securely on the shifting pile of dead boglins, and his rider curbed him savagely and swore in Gallish. He looked around, as if expecting something.\n\nThe king grunted something, and the captain bent over further, his shoulder screaming at the effort, the helmet and the aventail on his head and neck feeling like the weight of a lifetime of penance.\n\nThe king had a horny talon between the plates of his fauld, buried deep in his thigh, and his blood soaked the ground.\n\n\"I have saved you,\" said the knight who towered over them. \"You may take your ease\u2014you are saved.\" Indeed, as far as the eye could see, a wave of knights were dispatching the last creatures too foolish or too bound by Thorn's will to flee. \"We have won a mighty victory today. Where is the king, please?\"\n\nThe captain was able for the first time in hours\u2014it felt like hours, and later it would prove to be only a few minutes\u2014to look around.\n\nHis company\u2014\n\nHis men-at-arms were gone. They lay in a ring, their white steel armour, even matted with gore, brilliant when surrounded by the green, grey, white and brown of their adversaries.\n\nBut their red tabards were very like those worn by the king's knights.\n\nThe king's household knights were intermixed with them, and the Knights of Saint Thomas in their black. Many of the latter were still standing\u2014more than a dozen.\n\n\"The king is right here,\" Fitzroy said.\n\n\"Dead?\" the foreign knight asked.\n\nThe captain shook his head. He could easily come to dislike this foreigner. Galles were superb knights but very difficult people.\n\nHis mind was wandering.\n\n_Don't give him the king_ , said Harmodius.\n\nThe captain stiffened in shock. _How did you do that? Prudentia never spoke to me outside the memory palace._\n\n_Do I look like Prudentia?_ Harmodius muttered. _Do not give this man the king. Take him to the fortress, yourself. Take him to Amicia, with your own hands._\n\n\"Give him to me,\" said the foreign knight. \"I will see he is well guarded.\"\n\n\"He's well-guarded right here,\" said Sir Richard.\n\nBad Tom leaned forward. \"Sod off, son.\"\n\nThe captain reached out a hand to steady Tom.\n\n\"You need manners,\" said the mounted knight. \"But for my charge, you would all be dead.\"\n\nTom laughed. \"All you did was to lower my body count, pipkin,\" he said.\n\nThey glared at each other.\n\nThe Prior waded over to them. \"Ser Jean? Captal?\"\n\nDe Vrailly backed his horse. \"Messire.\"\n\n\"A litter for the king.\" He waved.\n\nOther knights rode forward\u2014there was the banner of the Earl of Towbray, and there was the Count of the Borders. They came in a rush, now that the king had been discovered. Towbray found the king's squires and the Royal Standard, and raised it, covered in ichor.\n\nThere was a low cheer.\n\nA long line of infantrymen came over the field of the dead. They had to pick their footing, and they weren't quick about it. As they came, the captain and Michael got the king's breast and back off, and got his hauberk up. Bad luck had slit a dozen rings\u2014worse luck to receive a second blow that bent the fauld and penetrated the leg. There really was a lot of blood.\n\n_Do I have anything left?_\n\n_You can stop the blood flow. But I've been squandering your power, keeping you alive, for a long time now. Amicia?_\n\n_I'm right here._\n\nThe captain smiled, knelt, placed his hand on the king's bare thigh when Michael peeled back his braes and his hose, and with no conscious effort he released Amicia's power.\n\nHarmodius did the actual casting.\n\nIt made the captain feel a little sick, as if he was three people.\n\n_You feel sick?_ The dead Magus laughed in his head.\n\nAnd then the footmen of the Royal Guard were there\u2014everywhere around them\u2014and the king was lifted high, placed on a cloak across two spears... and he held onto the captain's hand. So they walked, hand in hand, across the stricken field. It was the longest walk the captain had ever taken\u2014the sun was beating down like a new foe, the insects descended like a plague, and the footing was impossible.\n\nBut eventually, they were free of the corpses and were climbing the long road to the fortress.\n\nSoldiers stopped and bowed, or knelt. Men in the field had begun to sing the Te Deum, and its strains rose like the casting of a mighty phantasm from the fields below. The captain felt the king's hot hand in his own, and tried not to think too much about it.\n\nThe Queen lay in the chapel\u2014on the altar. She raised her head, and smiled.\n\nThe king released a sigh, as if he had been holding his breath.\n\nThe captain saw Amicia. She stood in the light of the window behind the altar. She appeared inhuman, a goddess of light and colour, and she was, to his sight, sparkling with power.\n\n_Christ. Look at her, boy._\n\nThe captain ignored the dead man.\n\nHe couldn't take his eyes off her anyway.\n\nShe was healing each injured person brought to her. The power went into her as easily as breathing\u2014she was drinking the unspent green from Thorn's hammer blow, and from the sun streaming through the broken chapel window, and the well\u2014taking all three streams of power and releasing it in a cloud of rainbow light so that soldier after soldier approached her, knelt, and arose healed. Most stumbled away and went to sleep in the arms of their comrades.\n\nShe passed her hands over the king as if he were any other soldier, any of the women wounded in the desperate defence of the courtyard, any of the children injured in the collapse of the West Tower\u2014and he was healed.\n\nAnd then she turned, and her eyes looked into his.\n\nHe couldn't breathe.\n\nHe had the foolish impulse to kiss her.\n\nShe touched him. \"You must open your powers, or I cannot heal you,\" she said. She gave him a smile. \"You were not this powerful, a few days ago.\"\n\nHe sighed. \"Nor were you,\" he said.\n\n_The room was the same. He was almost afraid to enter it, but it looked better. The moss was gone from the floor, and Prudentia's statue was repaired, and now occupied a niche that hadn't existed before._\n\n_The Magus stood on the plinth in the centre of the room._\n\n_The captain by-passed him, and walked to the door._\n\n_\"Think on what you do, boy,\" said the dead Magus. \"She is a Power, neither more nor less than you.\"_\n\n_The captain ignored him, and opened the iron-bound door._\n\n_And she was there._\n\n_And he was healed._\n\n_She looked at the plinth and her eyes widened in horror. \"My God,\" she said. \"What have you done?\"_\n\n_And she was gone._\n\n# North of Lissen Carak\u2014Peter\n\nThey stopped in a clearing in the woods. The ground had been rising steadily to the north, and they were running almost due north, and that was all Nita Qwan knew except that, as usual, he had never been so tired in all his life.\n\nThey all lay down in a muddle and slept.\n\nIn the morning, Ota Qwan stood up first, and they ran again. The sun was high in the sky before they straggled over a ridge, and young warriors were sent back without their baskets to fetch the matrons and mothers of newborns who had lagged behind.\n\nAnd when the last of the women was over the ridge, fires were lit carefully and the people made food, and ate.\n\nAnd when Nita Qwan felt as if life might be worth living, Ota Qwan came to the centre of the ring of fires with a spear. And Little Hands, the senior woman, came and faced him.\n\nHe handed her the spear. \"Our war is over,\" he said. \"I give you the spear of war.\"\n\nLittle Hands took it. \"The matrons have it, ready for any enemy. Our thanks, Ota Qwan. You have surprised us, and done well.\"\n\nNo one said anything more\u2014there was neither applause nor censure.\n\nAn hour later, they were running north again.\n\n# Chapter Seventeen\n\n# Lissen Carak\u2014The Red Knight\n\nDays passed.\n\nWounded men were healed, and slept.\n\nDead men and women were mourned, and buried.\n\nThe creatures of the Wild were burned, and their ashes spread over the fields.\n\nThe company was not entirely dead. A few men-at-arms were found wounded, and healed. Ser Jehannes and Ser Milus had not been part of the charge; Bad Tom and Sauce were untouched, although they each slept for more than thirty hours after their armour was stripped. But the archers were still alive, many of the valets and a few squires.\n\nThe captain was very difficult to find. Some said he was drunk, and some said he was with his pretty novice, and some said he was taking service of the king, or of the Knights of Saint Thomas.\n\nNone of these things were true.\n\nThe captain spent a great deal of time weeping, and when he buried his company dead, he invited no others. They lay in neat rows, sewn in white linen by Mag and her friends, who now stood silent in a light rain. Dora Candleswain stood with Kaitlin Lanthorn, and the Carter sisters stood watching their surviving brother who, with Daniel Favor, was in the ranks of the company.\n\nAnd the knights of Saint Thomas appeared from the rain. The Prior came out at their head, and said the service for the dead. Bad Tom, Sauce, Ranald and the captain himself lowered the bodies. There was Carlus the Smith, smaller in death but no lighter; there was Lyliard, no longer the company's handsomest man. They went into marked graves with headstones, each of them marked with the eight-pointed cross of the Order. It made a great difference to many of the man and women\u2014better, in fact, than most mercenaries ever imagined.\n\nOne corpse absorbed the captain, and he wept. He wept for all of them, and he wept for his own errors, and the ill-judgement of others, and a thousand other things\u2014but Jacques was his last tie to childhood, and was gone.\n\n_Your mother's still alive, lad. Doesn't she count?_ said the old Magus in his mind.\n\n\" _Could you shut up?\"_ muttered the captain to the interloper in his head.\n\nSauce looked at him, because he muttered to himself a great deal, lately, and because Sauce helped Dora Candleswain to stop screaming every night. She was sensitive to the other men and women in the company who were near the breaking point, or past it. Not all wounds bled.\n\nThey all stood there in the light rain\u2014the survivors. Atcourt, and Brewes and Long Paw. Ser Alcaeus, who wore the red tabbard and stood with the knights; Johne the Bailli. Bent. No Head. Knights and squires and archers and valets, men and women, soldiers and prostitutes and laundresses and farm girls and servants. And to a man and woman, they looked at the captain and waited for him to speak.\n\nLike a fool, he hadn't planned anything. But their need was palpable\u2014like a spell.\n\n\"We won,\" he said, his young voice as harsh as the croaking of a raven. \"We held the fortress against a Power of the Wild. But none of these men or women died to hold the fortress\u2014did they?\"\n\nHe looked at Jehannes. The older man met his glance. And gave him a small nod of agreement.\n\n\"They died for us. We die for each other. Out there in the world, they lie, and cheat each other, and betray, and we, here, don't do that.\" He was all too aware that sometimes, they did. But funerals are the time to speak high words. He knew that, too. \"We do our level best to hold the line, so that the man next to us can live. We\u2014we who are alive\u2014we owe our lives to these, who are dead. It could have been us. It was them.\" He managed a smile. \"No one can do more than to give his life for his friends. Every drink of wine you ever taste, every time you get laid, every time you wake and breathe the spring air, you owe that to these\u2014who lie here in the ground.\" His eye caught the smallest bundle\u2014Low Sym. \"They died heroes\u2014no matter how they lived.\" He shrugged and looked at the Prior. \"I suspect it's bad theology.\" He had more to say, but he was crying too hard, and he found that he was kneeling by the mound of damp earth that was Jacques.\n\nWho had saved his life so many times.\n\n\"Jesus said, I am the way, and the life,\" said the Prior in a calm, low voice.\n\nThe captain shut out the sound of his company praying.\n\nAnd eventually, there was a hand on his shoulder. It was a light hand. But he didn't have to open his eyes to know to whom it belonged.\n\nHe rose, and she stepped back. She smiled at the ground. \"I thought you'd just hurt your back again, with all that kneeling,\" she said.\n\n\"Marry me?\" he asked. His whole face ached from crying\u2014and he knew she didn't care how he looked, or sounded. It was the most remarkable thought.\n\nShe smiled. \"I'll tell you tomorrow,\" she said lightly. \"Open to me?\" she asked, and he thought he heard an immense strain in her voice. He put it down to fatigue, and he opened his _door, and she entered in. She kept her distance from Harmodius, and instead she pulled him out his own door and into the green wonder of her bridge\u2014but it was no longer a simple green. Overhead, the sky was a golden blue, and the sun shone in splendour in heaven and the water that rushed under her bridge was clear as diamonds and the spray was as white as the brightest cloud. The leaves of the trees were green and gold, and every tree was in flower. The smell was of clean water and brilliant air and every flower scent he'd ever imagined or smelled._\n\n_\"God,\" he said, involuntarily._\n\n_She smiled at him with her slightly tilted eyes, and she passed her hands over him, and a dozen small knots were eased, and the lump at the back of his throat passed away._\n\n_\"I'm not so arrogant as to heal your sorrow,\" she said._\n\n_He caught her hands. \"You will heal my sorrow,\" he said._\n\n_She smiled, and put her lips on his, her eyes closed._\n\n_And after a time, she pulled away. \"Goodbye,\" she said._\n\n_\"Until tomorrow,\" he said. \"I\u2014I love you.\"_\n\n_She smiled. \"Of course you do,\" she said with a little of her old tone. Then she softened. \"I love you, too,\" she said._\n\nShe walked away into the rain, and he watched her until the grey of her cloak merged with the sky and the stone and the hillside.\n\nAnd the captain found his services very much in demand. He accepted a contract in the east, serving among the Moreans with Ser Alcaeus. They concluded the contract just a week after the day of the battle, after an hour of loud and apparently angry bargaining that had featured several cups of wine and a warm embrace at the end.\n\nThen he picked up the staff of his command and walked out of his tent\u2014the company were back in their tents on the plain, so that the Royal Household could occupy the fortress\u2014and mounted a pretty Eastern mare that had belonged to Master Random. No amount of miraculous healing could fix his partially-eaten leg, so the merchant would be bed-ridden for some time. He'd been delighted to sell the mare for a profit.\n\nThe captain rode up the familiar road to the main gate. Royal Guardsmen held the post, and he saluted them. They returned the salute.\n\nHe gave his horse to a newly minted Royal Squire\u2014somebody's younger son\u2014and climbed the steps to the Commandery. No longer his office.\n\nThe Prior was at prayer.\n\nThe captain waited patiently.\n\nEventually, the Prior rose and put his rosary back around his waist. He smiled.\n\n\"Your servant, Captain.\"\n\nThe captain smiled back, reached into his wallet and fetched forth a pair of heavy gilt-bronze keys. \"The keys to the fortress and the river bridge,\" he said. \"They were placed in my keeping by the Abbess. I relinquish them to you in peace and triumph,\" he said formally. And then added, with a smile, \"You owe me a sizeable sum of money.\"\n\nThe Prior took the keys and settled into a seat. He waved the captain into another, and the captain had the oddest feeling\u2014one of having lived this moment before, perhaps from the other side of the desk.\n\nThe Prior took a writing set, checked the pen for sharpness, used a little ink and began to write.\n\n\"You would not consider turning to God, my son? Become a knight of my order?\" he asked, raising his eyes briefly.\n\n\"No,\" the captain said.\n\nThe Prior smiled. \"So proud. Amicia tells me that you see God as your enemy.\" He shook his head.\n\n\"Amicia has misinterpreted the information with which she was provided,\" the captain said. Then he shrugged. \"Or maybe not. Your God and I are not friends.\"\n\n\"Ahh,\" said the Prior. He shook sand over the paper, shook it, and blew on it. Then, after struggling with a candle, he managed to drip heavy black wax on the document and he affixed the great seal from the ring on his thumb. \"Your defence here will never be forgotten by my knights.\" He shrugged. \"Even if outside these walls men say the king won the battle, and defeated the Wild.\" He handed over the parchment. \"My God loves you, and every other living thing, Captain. My God loves the sick, the blind, the leper, the unclean\u2014the irk, the boglin, and the witch.\"\n\nThe captain glanced at the sum, drawn on the Church\u2014a draft redeemable at any bank, anywhere\u2014and nodded. He even smiled.\n\n\"This is more than I contracted for,\" he said.\n\n\"I supposed that you contracted for the loss of men and horses, and for the usual victory bonus,\" the Prior said.\n\nThe captain shook his head. \"No,\" he said. \"I had no idea what I was getting into.\"\n\nThe Prior nodded. \"I don't know what your problems are with my God,\" he said. \"But I won't let you add ingratitude to your list of His inadequacies. Without you and the sacrifice of your company, this place would have been lost and all humanity would have suffered for it.\"\n\nThe captain rose and bowed. \"You do me too much honour. For my company\u2014\" He found himself unable to speak. When he was master of himself, he said, \"I will recruit more.\"\n\n\"Easily, I predict,\" the Prior said. \"Listen, young man. You have interests beyond the mundane. You will not turn to God. So be it. But you have a brain, and it's a keen one. Did we win here?\"\n\nThe captain hadn't expected this turn of conversation. He stood in the doorway with his payment in his hand.\n\nThe Prior rose and poured two cups of wine. \"Sit.\"\n\nHe sat and drank. \"No?\"\n\nThe Prior shook his head. \"Of course we did. Had we lost the king would be dead, the Alban border would be south of Albinkirk and the Royal Host would be shattered.\" He crossed himself. \"But of course, we didn't win either, did we?\"\n\n\"Thorn burned every house and barn from here to Albinkirk,\" the captain said. \"And hit the population hard.\"\n\nThe Prior nodded.\n\n\"Most of the survivors will leave. Move south.\" The captain sipped some more wine. \"That's why\u2014I'm guessing\u2014there was no fight at the wall first. Thorn never intended to fight there. He went deep\u2014\"\n\n\"Stop saying his name,\" the prior said. \"He still lives, licking his wounds.\"\n\n\"He still lives, and nothing died out there but this year's crop of boglins,\" the captain said bitterly. \"Sixteen trolls, a dozen wyverns and some daemons.\" He rubbed his beard. \"We're losing the exchange.\"\n\n\"We're losing, period,\" said the Prior. \"In our order we have records that go back six hundred years. We are not winning this war.\" He shrugged. \"If the Wild were not so utterly divided against itself, they'd have swept over us a thousand years ago.\"\n\nIn his head, Harmodius said, _Exactly. Who knew the Prior was a kindred spirit?_\n\n\"What can we do?\" the captain asked.\n\nThe Prior bent forward. \"Well, at least you are interested. Where is your next contract?\"\n\nThe captain leaned back. \"Morea. A rebellion and a magus gone bad.\" He looked out the window. \"What will you do with this place?\"\n\n\"Put a garrison into it, for a while. I don't quit easily\u2014I'll offer a sizeable benefit and a total remission of tithes to any family who will stay here and rebuild. And I, too, will recruit\u2014there must be younger sons south of the river looking for farms. I'll find them.\"\n\n\"That will cost a fortune,\" the captain said.\n\n\"I have a fortune,\" said the Prior. He leaned forward. \"You have power.\"\n\nThe captain shrugged.\n\nThe Prior shook his head. \"Your power comes from the Wild. I've seen it.\"\n\nAgain, the captain shrugged.\n\nThe Prior nodded. \"Very well. But if you ever choose to talk about it there are many knights of the order who channel the Wild. We know more about it than you might think.\"\n\nThe captain finished his wine, rose, accepted the Prior's embrace and even stayed still while the man blessed him.\n\n\"Will you not tell me why you turn your back on God?\" the Prior asked.\n\nThe captain looked at him, smiled and shook his head. \"When you offered to make me a knight of the order, just now\u2014\" he said.\n\n\"The offer remains open,\" said the Prior.\n\n\"\u2014I'll treasure that,\" he finished.\n\n\"Your brother turned me down, as well,\" the Prior said.\n\nThe captain nodded. \"Gawin is riding east with me,\" he said.\n\nHe walked out of the Commandery, and down the stone steps. A valet in de Vrailly's arms stood by the steps up to the Hall, holding a beautiful destrier\u2014tall and grey as steel. The captain didn't feel the slightest need to take leave of the king. Or the Queen. Or, for that matter, their new favourite the Captal de Ruth, already known as the Victor of Lissen.\n\nInstead, he walked to the hospital, up the steps, and to Master Random's bedside. A trio of local farmers stood by his bed, with Master Johne the Bailli.\n\n\"A moment, good sirs!\" cried Master Random. \"This worthy knight must always have first call on my time. Damn my foot,\" he said, trying to twist in the bed. \"How can it hurt so much when it isn't there?\"\n\nThe captain embraced the merchant. \"You look better.\"\n\n\"I am better, my friend. That wonderful young lady poured her spirit into me, and I feel twenty years younger for it.\" His eyes sparkled. \"Though if I was home, I daresay the goodwife might tell you that the deal just struck with these worthies was part to my joy. Eh?\"\n\nThe captain looked around. Master Johne had acquitted himself very well against the enemy, every farmer present had carried a spear or an axe. The captain knew them by name\u2014Raimond, Jaques, Ben Carter and young Bartholemew Lanthorn, a rogue, a scoundrel, and despite that, a very successful farmer.\n\n\"He's bought the whole grain crop,\" Johne the Bailli said. He smiled.\n\nThe captain glanced around. \"Of course\u2014it's all in the cellars.\"\n\n\"A little messed about,\" Random noted. \"But grain's grain, and the need downriver\u2014the price, when they hear of the battle and the burning of farms!\"\n\n\"How will you ship it?\" the captain asked, to be polite.\n\n\"Boats!\" Random said. \"All those boats which brought the Queen? Mine.\"\n\nThe captain shook his head. \"A coup, my friend. You will be rich.\"\n\n\"I'll break even or a little better,\" Master Random said with a smile. \"Drink with me,\" he said.\n\nThe captain nodded. \"May I broach a small item of business, myself?\" he asked.\n\nRandom nodded. \"Always open.\"\n\nThe captain took the Prior's note from the breast of his jupon. \"You are a bank, are you not?\"\n\nRandom sniffed. \"Not of the size of the Etruscan banks, perhaps. But I do my\u2014Gracious God!\" he said. His eyes snapped to the captain's.\n\n\"I'm investing in you,\" the captain said. \"I may have to make some pay outs, and buy some horses, but three-quarters of this sum is at your service for at least a year.\"\n\nThe captain had a cup of wine, embraced all concerned, and met the Bailli's eye. The man nodded.\n\nHe went back through the ward, to the bed where his brother lay reading. He had his feet up, but he was fully dressed and his kit was neatly packed in wicker hampers. \"She's not here,\" he said. \"Don't even pretend you are here to see me.\"\n\n\"I won't, then,\" the captain said. \"Where is she?\"\n\nGawin shrugged. \"I need out of here, Gabriel. I'll kill the foreigner if I stay.\"\n\n\"I'll have another cot put in my pavilion. We ride tomorrow.\" He turned to go. \"Where is she, Gawin?\"\n\nGawin met his brother's eye. \"I'd tell you if I knew,\" he said.\n\nTheir eyes locked, and Gawin motioned with a finger. A woman's form was outlined in the curtain of the courtyard window.\n\nThe captain raised an eyebrow.\n\n\"He's not the enemy, Mary,\" Gawin said, and the Queen's Lady in Waiting emerged. She was blushing.\n\n\"You have other things to take up your time,\" the captain said.\n\nGawin laughed. \"I really don't know where she is,\" he admitted.\n\nThe captain turned with a wave, and headed out. He peeked into the dispensary and the apothecary, and he climbed the steps in the dormitory. No one had seen her. The smiles he left in his wake pained him.\n\nFinally, in the courtyard, he met Sister Miram. She smiled at him, and took him by the hand to her cell in the chapel. \"You are going,\" she said, pouring him wine.\n\nHe tried to refuse the wine but she was a forceful woman, and a pleasant one, and her silence intimidated him. She waited him out. Finally, he drank it. \"Tomorrow.\"\n\n\"We will celebrate the feast of Mary Magdalene tomorrow,\" she said. She smiled. \"We will inter the old Abbess.\" Sister Miram looked at her hands. \"I will be ordained Abbess in her place.\"\n\n\"Congratulations,\" the captain said.\n\n\"There is talk that the whole convent is to be moved south to Harndon,\" Sister Miram said. She looked the captain firmly in the eye. \"I won't have it.\"\n\nThe captain nodded.\n\n\"We will also accept the vows of novices advancing to the sisterhood of Christ tomorrow,\" she said.\n\nIce formed in the captain's stomach.\n\n\"She is performing her vigil at the moment,\" the sister said. \"Drink your wine, Captain. No one is forcing her to.\"\n\nThe captain took a breath.\n\n\"We owe you so much,\" Sister Miram said. \"Do you think we do not know it? But she is not for you, Captain. She is to be the bride of Christ; it's what she wishes.\" She rose, went to her prie-dieu, and opened the triptych. From it she drew a folded piece of parchment. \"She left this for you. If you should come.\"\n\nThe captain took it with a bow. \"Your servant, ma soeur. May I express my congratulations on your elevation, and my\u2014\" He stopped. Swallowed. \"I will make a donation to the convent. Please give Sister Amicia my congratulations and my kindest regards.\"\n\nSomehow, he reached the courtyard.\n\nToby was holding his horse.\n\nThe captain took the reins, and vaulted into the saddle, aware, in that cursed part of him that was always awake, that he was on the stage of chivalry, and that half of the knights of Alba were watching him.\n\nThen he rode down the hill to his camp. He paused at the guard fire.\n\n_Don't be a fool. Read it._\n\nThe Red Knight took the parchment from his breast, and threw it in the fire unread.\n\n_You idiot._\n\nMichael was sitting in his tent. He leaped to his feet, obviously guilty about something. \"Master Ranald is waiting for you,\" he said. \"I was entertaining him!\"\n\nRanald Lachlan sat with a mug of beer, and his cousin Tom sat across the captain's camp table with another. They had dice on the table, and cards.\n\n\"It'd be a pity to stop him playing,\" Tom said. \"Especially as I'm taking all his money,\" he added.\n\n\"I'm so pleased you two feel free to make use of my tent and table,\" the captain spat.\n\nTom raised an eyebrow. \"Brother's got something to say,\" he said.\n\nRanald rose. \"I\u2014need to make a great deal of money,\" he said. \"I wonder if you'll have me as a man-at-arms.\" He looked embarrassed to ask.\n\n\"I'd have thought the king would've knighted you,\" the captain said.\n\nRanald shrugged.\n\n\"All right,\" said the captain, sitting and pouring wine for himself. \"Now deal me a hand.\"\n\n\"But first,\" Ranald said, \"I have to pay a visit to the Wyrm of Erch.\"\n\nThe captain gagged on his wine. \"The Wyrm?\"\n\n\"Our liege lord in the hills, or so we call him,\" Ranald said, and Tom nodded.\n\nThe captain shook his head. \"I don't understand.\" He frowned. \"Possibly because I'm drunk.\"\n\nTom shrugged. \"The ways of the hills are easier on a man with drink in him. Tis like this, my lord: the Wyrm guarantees us peace for a tithe of the flocks. Tis been that way for twenty generations of men or more. These Outwallers that killed Hector\u2014the Sossag\u2014they were serving a Power of the Wild called Thorn. Aye?\"\n\n\"Naming calls. But yes.\" The captain drank.\n\n\"So I call him and he comes and I gut him,\" said Tom. \"So?\"\n\n\"Excellent point,\" the captain said. \"Go on.\"\n\n\"The Wyrm owes us for our loss,\" Ranald said.\n\nThe captain sat back. \"I'm not drunk enough to believe that,\" he said.\n\nTom and Ranald sat with set faces.\n\nThe captain finished his cup. Michael poured him more, and he didn't say no. And then he said, \"She's taking orders as a nun, Tom.\"\n\nTom shrugged as if all women were one and the same. \"Best find another one then,\" he said. And then, as if the collapse of the captain's hopes was not the most important thing in the world, he said, \"So we want leave to go to the Wyrm.\"\n\nThe captain shook his head. \"I have a better idea,\" he said. \"Let's all go.\"\n\nRanald looked at him and raised an eyebrow at his brother.\n\n\"I love him,\" Tom said to his brother. \"He's mad as an adder.\"\n\nRanald smiled. \"So we all go? The company?\"\n\n_Yes. This is important._\n\nThe captain suddenly had a piercing pain between his eyes.\n\n_Be quiet. You're a guest._\n\n_You are getting drunk because you've been spurned by a woman. How romantic of you. Of course, that note might have confessed her undying love for you and her willingness to elope tonight to face the future as a mercenary captain's whore. Hmm? But you burned it, so you'll never know. Youth is wasted on the young._\n\n_Shut up. Fuck off._\n\n_Listen, young man. The Prior is right\u2014humanity is losing. But he is also wrong\u2014as I will endeavour to prove. The world is not as I thought it was, and your going to see the Wyrm is the very best idea I have ever heard. You must go to the Wyrm. The stakes of this game are immense. The consequences of failure are extermination\u2014the death of our race. Your dalliance with some novice\u2014albeit one imbued with power of the very highest order\u2014is not quite in the same league._\n\nThe captain put his head in his hands.\n\nTom grinned at him. \"You're drunk, my lord.\"\n\nThe captain looked around for Jacques, but of course he was dead. The last piece of his old life\u2014the last man to connect him with\u2014\n\n_I'm conveniently dead, too. Prince Gabriel._\n\nThe captain took a deep breath. \"I have a headache,\" he said. \"I find it unfair that I have the hangover before I'm done with the drunk.\"\n\nMichael leaned forward and poured more wine.\n\nSer Jehannes came in with Ser Milus, both of them drunk too. They were singing \"Green Grow the Rushes\" with their arms around Sauce, who seemed to be carrying them.\n\n_Three, three, the lily white boys, clothed all in green, oh, Two, two the rivals._\n\n_And one is one and all alone, and ever more shall be, oh._\n\nTheir attempt at harmony was almost as horrible as a charge of boglins.\n\nTom started to laugh.\n\nJehannes poured a cup of wine, sat on a stool, and raised his cup. \"Absent friends,\" he said.\n\nTom's laughter stopped. He rose to his feet, and so did the rest. \"Victory and defeat are for amateurs,\" Tom said. \"For us, there is only life and death.\"\n\nThey all raised their cups, and drank. \"Absent friends,\" they chanted, one by one.\n\nThe captain put his cup down on the table carefully, because it seemed to be a long way away and it moved slightly, and he leaned on the table to make sure he could stay on his feet. \"They will bury the old Abbess tomorrow,\" he said. \"I'd like every man and woman at that service in their best kit. But with the camp struck first, ready to march.\"\n\nHis corporals nodded.\n\n\"The Prior paid me today,\" he said. \"With a success bonus and a tallage for the horses we lost. A pretty sum. I invested it. But none of you needs to fight for a living. Your shares will be a hundred gold nobles or more. Enough to buy a knight's fee.\"\n\nJehannes shrugged.\n\nTom sneered.\n\nSauce looked away.\n\nMichael laughed.\n\nRanald smiled. \"Wish it was mine,\" he said.\n\n\"It will be,\" the captain said. \"We have a new contract, and I mean to wrap it up quickly.\" He felt a little better. \"Sauce, come here.\"\n\nShe was dressed in old hose and a well-cut man's doublet\u2014something of a brag, since it flattered her figure as much as any kirtle. She leered at him. \"Any time, Captain,\" she said, with a spark of her old sauce.\n\n\"Kneel,\" the captain said. He held out his hand to Michael.\n\nMichael handed him his war sword.\n\nSauce paused and knelt. On the edge of a double entendre, she stopped.\n\nTom nodded. \"Do it.\"\n\nThe captain raised his sword. \"By the virtue of knighthood and my birth, I dub thee knight,\" he said. He didn't slur the words. His sword pressed down hard on each of her shoulders.\n\nShe burst into tears.\n\nTom smacked her, quite hard, on the shoulder. \"Let that be the last blow you ever accept without reprisal,\" he said. He grinned.\n\n\"Michael, kneel,\" the captain said.\n\nMichael knelt.\n\n\"By the virtue of knighthood and my birth, I dub thee knight,\" the captain said.\n\nMichael accepted the slap from Tom, rocked back on his heels, and smiled.\n\nThe captain took his wine cup. \"I meant to do it on the battlefield,\" he said. And shrugged. \"We were busy.\"\n\nMichael stood up. \"I'm a knight?\" he laughed. \"A man-at-arms and not a squire?\" He laughed again.\n\n\"I'll need a new squire,\" the captain said.\n\nSauce was still crying. \"Is it real?\" she asked.\n\nTom put an arm around her shoulder. \"Of course it is, lass. He wouldn't mock you with such.\"\n\nThe captain sat back down. \"We need twenty new men-at-arms. We need as many squires and a dozen valets and some archers.\" He shrugged. \"My brother Gawin is one. Johne the Bailli is another. Both have their own harness, and they'll ride away with us. Ser Alcaeus himself, despite negotiating our contract, will be joining us. Any other prospects?\"\n\nJehannes nodded. \"I have half a dozen younger sons ready to sign articles\u2014all with harness and horses.\"\n\nRanald shrugged. \"All my lads, too,\" he said. \"We have no other means of employment, at least for the balance of the year.\"\n\nTom leaned forward. \"Daniel Favor. Likeliest man-at-arms I've ever seen. He signed with me. And two of the Lanthorn boys\u2014dangerous boys. Murderous.\" He grinned. \"Archers.\"\n\nJehannes nodded. \"I made out a watchbill,\" he said. \"If we go to one man-at-arms, one squire, one valet and two archers to a lance, we have a company.\" He looked at the captain. \"Gelfred should start arming as a man-at-arms too.\"\n\nThe captain nodded. \"We could use twenty more lances,\" he said. \"I wrote a contract for forty, and we only have what\u2014twenty?\" He sat up, decided that was a mistake, and shuffled to his feet. \"Tomorrow night we'll be on the road. Less wine.\" He raised his cup. \"To the company,\" he said.\n\nThey all drank.\n\n\"Now, since it's my tent I'm going to bed,\" he said. And motioned to the door.\n\nOne by one they ducked under the awning and left, until it was Michael and Sauce\u2014each seeming to want the other gone first. Finally Michael spoke.\n\n\"Can I help you, my lord? I'm not above myself yet.\" He laughed.\n\n\"I'm guessing you already have a nice pair of solid gold spurs to go on those heels, and you'll have them on your boots in the morning,\" the captain said, slapping his shoulder. \"Just send me young Toby.\"\n\nMichael smiled. \"Thanks,\" he said. \"I\u2014\"\n\nThe captain waved his thanks away, and Michael bowed low.\n\nThat left Sauce.\n\n\"Good night, Sauce,\" the captain said. He avoided her embrace. \"Good night.\"\n\nShe stood with her hands on her hips. \"You need me.\"\n\nHe shook his head.\n\n\"I won't go all soppy on you, Captain.\" She shrugged and then smiled engagingly.\n\n\"Good night, Sauce.\"\n\nShe grunted.\n\n\"I just made you a knight,\" he said. \"Don't play the woman scorned part.\" Even drunk he could see his refusal hurt her. He raised a heavy hand. \"Wait,\" he said, and stumbled through the curtain to his bed, reached into his trousseau and found his other spurs. The solid gold ones his mother had given him, which he never wore.\n\nHe came back out. \"Take these.\"\n\nShe reached out and took them. Realised they were solid gold. \"Oh, my lord\u2014\"\n\n\"Out!\" he said.\n\nShe sighed, and walked out of the tent, swaying her hips to brush by Toby, who came in, and silently relieved him of his clothes and accoutrements.\n\n\"How old are you, Toby?\" he asked.\n\n\"Rising twelve, my lord. Or perhaps thirteen?\" he said.\n\nThe captain lay his body down on clean linen sheets. \"Would you care to be a squire, Toby?\" he asked.\n\nHe survived the protestations of joy and eternal loyalty, and waved the boy away. When he put his head down, though, the tent spun. So he put a foot on the ground. Gave sleep up as a bad job, sat up, and drank some water.\n\nThe headache was back.\n\nHe stood by his water basin for a full watch. Staring into the dark.\n\nIt was, as such things went, pretty dark.\n\n_You make them love you, and then you tire of the energy they demand_ , the voice said.\n\nHe sighed, lay down, and went to sleep.\n\nThe chapel was magnificent, with all the decoration that could be managed for an occasion that featured the King, the Queen, the Prior, and a thousand noblemen\u2014virtually the whole peerage of Alba.\n\nBut there wasn't room for all of them. The chapel had been built for sixty nuns, as many novices, and perhaps another hundred worshipers.\n\nIn the end the service was held in the chapel, but only a select few were there. The rest waited in the courtyard and were served communion there. It was well-managed, and had a festive air despite the great solemnity of the occasion. The courtyard was full to bursting, and velvet clad gentlemen stood shoulder to shoulder with farmers and farm wives.\n\nThe Prior and the new Abbess had been very mindful of the future in their assignment of places. Only the greatest lords were in the chapel. The King and Queen sat enthroned. By the king's right hand stood the Captal de Ruth; by the Queen stood Lady Almspend and Lady Mary. The Count of the Borders stood with the Count D'Eu; the Earl of Towbray stood with Ser Alcaeus, as the ambassador of the Emperor Basileus. And next to him stood the captain.\n\nThe Prior said the mass, and a thousand beeswax candles burned.\n\nIt was brutally hot.\n\nOut in the courtyard, the company stood in full armour, four ranks deep. With them, by a curious choice of the Prior's, stood the surviving knights of the military orders in their black. Mag stood nearby, with the women of the company. Her home was gone, and Johne the Bailli had made her a proposal.\n\nThe Prior preached about Mary Magdalene. He spoke about sin, and forgiveness. About faith, hope, and charity, and the nuns brought forth the bier on which the Abbess lay. When her corpse entered the chapel, the air temperature dropped, and a smell, like lilacs, wafted in through the doors.\n\nThe captain looked at her and wept.\n\nThe Captal de Ruth looked at him and raised an eyebrow.\n\nThe Queen placed a hand on the captal's arm.\n\nThe captain looked up\u2014he'd surprised himself\u2014and found that he was eye to eye with Amicia. She was standing by the rightmost choir stall, near the altar screen, with six other women in sparkling white-grey. She had, no doubt, been watching him weep.\n\nAnd now, her eyes remained fixed on his.\n\n_She was knocking at the door._\n\n_He left it closed._\n\n_One is one, and all alone, and ever more shall be so._\n\nThe service went on for too long.\n\nWhen the novices had been elevated; when the new Abbess had been formally invested\u2014when the last words had been spoken over the old Abbess\u2014then the whole congregation rose from their knees and walked in procession from the chapel, through the gate, and down onto the plain. The company acted as guards to the bier with the knights. It was a signal honour, subtly granted by the Prior.\n\nShe was lowered slowly into the newly turned earth by six knights.\n\nThe Prior threw a shovel of earth onto her.\n\nThe captain found that he had wandered away into a world of his own, when the king\u2014the king himself\u2014materialized in front of him.\n\n\"I owe you a debt of gratitude,\" the king said. \"You are not an easy man to find.\"\n\nThe captain shrugged. \"Your servant, my lord,\" he said dismissively.\n\nThe king was shocked by the mercenary's rudeness, but he mastered himself. \"The Queen has requested that she meet your company. We know what sacrifices they made for our kingdom.\"\n\n\"Oh, as to that,\" the captain said, \"We were well paid.\" But he turned, and led the king and Queen and a small host of their courtiers through the ranks of the company.\n\nThe first man on the right was Bad Tom, and next to him, his brother. The king smiled. \"Ranald!\" he said. \"I thought that you had returned to my guard?\" He laughed. \"I note the colour of your tabard remains the same.\"\n\nRanald looked straight ahead. \"Business,\" he said, seriously. \"My lord.\"\n\n\"But this is a woman, surely?\" asked the Queen, who had taken a few more steps.\n\n\"Ser Alison,\" the captain said. \"Her friends call her Sauce.\"\n\n\"A woman knight?\" the Queen asked. \"How delightful.\"\n\nBy her elbow, the captal laughed. \"Knighted by whose hand?\" he asked.\n\n\"My own,\" the captain said.\n\nConversation stopped.\n\n\"By what right do you make knights?\" demanded the captal. \"That is reserved for the very highest nobility, members of the greatest orders, and knights of great renown.\"\n\n\"Yes,\" the captain said. \"Yes, I agree.\"\n\nThe king cleared his throat. \"I doubt any knight in this gathering would doubt the captain's renown, Captal.\"\n\nThe captal laughed. \"He is a bastard\u2014a bourc. Everyone says so. He cannot be noble, and he cannot make a knight\u2014most especially not make a knight out of a woman.\"\n\nThe captain felt the tension in his chest\u2014not fear, but something like anticipation.\n\nIn a low voice, he said, \"My lord, you requested to see my company. If you are done, we will take our leave.\"\n\n\"Unsay it,\" the captal insisted. \"Unsay that this woman is a knight. Make her take that golden belt off her hips. It is unseemly.\"\n\n\"Captal!\" said the king. \"Control yourself.\"\n\nThe captal shrugged. \"You are too easy, my liege.\" He looked at the captain and sneered. \"I say you are a bastard, a caitiff, a low-born poseur, and I say before all these gentlemen that you cannot make a knight, that no knighting of yours\u2014\"\n\nThe captain turned to the king. Leaned over, and whispered in his ear.\n\nThe king whirled, looked at the mercenary, and the blood left his face like a tide slipping away from a white sand beach. In three beats of a man's heart, the king aged\u2014he looked as white as parchment. His upper lip trembled. The Queen, who had not been able to hear the words, felt his hand close on her arm like a vice and gave a little grunt of pain.\n\nOn the other side of the grave, Sister Amicia gave a start, and went as pale as the king.\n\nThe silence went on for so long that wasps could be heard droning, and the grunt of the men filling the Abbess\" grave.\n\nThe king looked at the captain, and the captain looked back at the king, and then the king inclined his head\u2014the sort of civil motion that a gentlemen makes to a lady about to proceed him through a door.\n\nIn a hoarse voice, the king said, \"This gentleman has the power to make a knight anywhere within the kingdom of Alba, of anyone, no matter how ignoble their birth or station. Such is my word.\"\n\nThe captain bowed deeply and the captal was silent.\n\nThe king acknowledged the captain's bow, and he took the Queen and led her on, up the hill to the fortress.\n\nThe captain caught the captal's eye. Jean de Vrailly was afraid of nothing\u2014so he stopped.\n\n\"I take it I have managed to offend you?\" he said. \"It is difficult for me to understand how a whore like you can take offence. You fight only for money.\"\n\nThe captain had control of himself. He took his time. Composed his answer while the captal was pinned in place by convention like a butterfly to parchment.\n\n\"Sometimes I fight for free,\" he said. \"But only when it interests me.\" He paused, holding the captal with his eyes. \"But I imagine that in the end, someone will pay me to put you down like the mad dog you are.\"\n\nJean de Vrailly smiled\u2014a beautiful smile that filled his face. \"So,\" he said. And laughed. \"I look forward to see you try.\"\n\n\"I imagine you do,\" the captain muttered. He wasn't sure that he'd had the better of the exchange, but he walked away without falling over his feet.\n\n# Lissen Carak\u2014Michael\n\nThe Earl of Towbray left his tail of men-at-arms and all but ran down the steps behind the Commandery to catch the captain's squire. Former squire.\n\n\"You are a knight!\" he said.\n\nMichael turned. \"Pater. So are you, I find.\"\n\nTowbray couldn't be angry. \"I gather you won your spurs and then some,\" he said. \"Can you come home now?\"\n\nMichael shook his head. \"No, Pater.\" He looked up, and found it easier to meet his father's eye than he had expected. \"I was glad to see our banner. With the king.\" He looked around. \"Surprised. But glad.\"\n\nTowbray shrugged. \"I can't love the king. But\u2014damn it, boy! Who are you to tell me how to play the game of court?\"\n\nMichael shook his head and then bowed. \"A new-minted knight, who makes twenty-eight florins a month in a company of mercenaries. \"He stepped back. \"I must go.\"\n\nTowbray reached out a hand. \"I admire you.\"\n\n\"You won't admire me as much when I tell you that I'm planning to marry a farm girl from Abbington.\" Michael grinned, feeling, for once, that he was master of a conversation with his father.\n\nHis father started, but with grim determination, extended his hand. \"So be it,\" said his father, although his face showed distaste.\n\nMichael took the hand. \"Then may I have my allowance back?\"\n\n# Lissen Carak\u2014The Red Knight\n\nAn hour later, the company was mounted and ready. All week the wagons had been swayed out of the cellars and re-built, rolled down the hill, and loaded. The company's stock had been safe in the fortress, and they were hitched with the company's usual efficiency. The valets mounted the wagons, the archers collected the spare mounts, and the camp followers got their nags and donkeys. At the head of the column, the captain mounted a strange new war horse, just given him by the Prior, and looked back to see Michael\u2014Ser Michael\u2014attending to the banner.\n\nOne by one, the corporals reported in, ready to march. A small crowd formed\u2014mostly Lanthorns and Carters and a dozen guildsmen from Harndon, come to see their boys off as they marched away. And their girls. Amy and Kitty Carter, Lis the laundress, Old Mag\u2014who hadn't looked as young in twenty years. Her daughter Sukey, whose husband had died in the siege. The captain had noted Sukey with Bad Tom. Twice. He made a note to himself to look into that.\n\nThe captain looked repeatedly for a single face in the crowd, but it refused to be there. Many women looked\u2014for an instant\u2014like her. Too many women.\n\nSo when all his people were ready, and the sun was so high in the sky that it made a mockery of his intention to march away, he raised a hand. \"March!\" he said.\n\nWhips cracked, men shouted, and wagons rolled.\n\nGerald Random waved from the walls, and Jean de Vrailly watched silently. The Prior saluted and women cried.\n\nThe king stood alone in the north tower, watching the convoy begin to roll east. His hands shook. And the Queen watched him from the courtyard and wondered what was amiss.\n\nA young nun knelt, her back straight, at the high altar of the chapel.\n\nA mile from the fortress, the captain came upon his huntsman, sitting his horse silently at a bend in the road. It took him a long moment to recognise where they were.\n\n\"We still never caught the man who killed that nun,\" Gelfred said. \"It sticks in my craw. I want justice.\"\n\n\"It was the priest,\" the captain said. \"Sister Amicia and I figured it out\u2014far too late to punish him for it. He's off to the Wild, I suspect.\"\n\nGelfred crossed himself. \"He will go to Hell!\" Gelfred said. \"God will punish him.\"\n\nHe captain shrugged. \"God doesn't give a fuck, Gelfred,\" he said. He touched his heels to his magnificent new charger. \"But I do, Gelfred, and I promise you, the priest will die.\"\n\nAnd with that, he put his horse's head to the east, and rode away.\n\nFar to the west, Thorn paused at the top of a ridge. He could see fifty leagues in the clear air, and he breathed deep. He had twenty wounds, and his powers\u2014greater than they had ever been\u2014were nonetheless spent.\n\nHe looked east.\n\n_That was foolish_ , he thought. The further he got from the rock, the more it was like a bad dream.\n\n_I could have been killed. For ever._\n\n_But I wasn't, and when I return_ \u2014\n\nThe great creature that was Thorn could not smile, but something passed over the heavy bark and stone of his face.\n\nOn the downslope of the ridge, he thought, _Or perhaps I'll do something else. Unify the boglins, perhaps._\n\n# Chapter Eighteen\n\n# The North Road\u2014The Red Knight\n\nThe column rolled east at a good pace and within hours, the captain's precautions were justified by huntsmen flushing creatures of the Wild\u2014a pair of boglins, and a lone irk.\n\nThey made camp early, dug a trench, and stood watches.\n\nThe captain lay awake most of the night.\n\nIn the morning they moved with the dawn, and his heart began to lift. The process of camping, of moving camp, and the sounds of the horses and the wagons\u2014the sounds of people and animals\u2014it all raised his spirits.\n\nIt took them three days to come to the Southford of the Albin. Albinkirk still smouldered, on its hill. The Royal Standard still flew from the castle, and the captain and his officers rode to the town gate, were admitted, and dined with Ser John Crayford.\n\nSer Alcaeus, who was falling into the company as if he had always been there, walked them around the walls. \"This is where we held their first rush,\" he said at the ruined west wall. \"Here's where a dozen of us held the gate.\" And again, with a wry look, \"Here's where we almost lost the wall.\"\n\nCrayford shook his head. \"You're the very King of Sell-Swords, now, I reckon,\" he said. He leered at the captain. \"My squire's older than you, boy! How'd you do it?\"\n\nThe captain raised an eyebrow. \"Clean living.\"\n\nCrayford shook his head. \"Good on you, lad. I'm a jealous old man. If I had another battle in me, I'd follow you.\"\n\nThe captain smiled. \"Even though two of your men are leaving you for my company?\" he asked.\n\nThe old man managed to nod with a good grace. \"Even then, you scapegrace.\"\n\nHe let them go with a fine meal and a hogshead of wine.\n\n\"No one left here to drink it,\" he muttered.\n\nPeople were trickling back into the town. The captain bought bread for the whole company from a young woman with haunted eyes. Haunted, but practical.\n\n\"Burned the house,\" she said, eyes on the west. \"Couldn't burn the ovens, though, could they? Little fucks.\"\n\nThey rode north on the east side of the Albin in the morning, and Ranald told them of having met the Queen at the ford as her boats rowed past.\n\nPast Albinkirk the huntsmen ranged wider, over the hills on either side. Summer was coming and the abandoned farms seemed sinister in their wrappings of verdant life. Grains stood tall and ripe and there wasn't going to be a soul to harvest it.\n\nThe captain watched it go by.\n\nSer Alcaeus rode by his side. \"There were men and women in these farms when I came through in late winter.\"\n\nThe captain shook his head. \"I wonder if men will ever farm here again,\" he said.\n\nTwo days north of Albinkirk, they came to the crossroads and made camp. The East Road ran up over the passes and down into the Vale of Delf, and on into the Morea.\n\nThe North Road ran into the Hills, past the Inn of Dorling and eventually to the Lakes and the Wall.\n\nThat night, over dinner in his tent, the captain put a map on the table. \"Jehannes, you'll take the company east to Morea. Find us a secure camp. I'll join you in a ten-day.\"\n\nJehannes made a face. He looked at Tom Lachlan. \"If this is so important, why don't we all go?\"\n\nTom laughed. \"We're going to see the Wyrm, Jehannes. Not pay a call on a lady, nor smoke out a company of brigands.\"\n\nThe captain leaned over the table. \"The Wyrm is a creature of the Wild. A Power like Thorn. And the company won't impress it. Him.\"\n\n_Not like Thorn_ , Harmodius said in the captain's head.\n\nJehannes shook his head. \"I mislike it.\"\n\n\"Reservation noted,\" the captain said.\n\nTom sat back, his booted feet on one of the captain's stools. \"Ahh. I can smell the hills already.\"\n\nRanald nodded. \"At some point,\" he said, \"we need to talk about the drove.\"\n\nTom nodded.\n\nThe captain looked at Ser Alcaeus. \"We won't be gone long,\" he said. \"And Jehannes can deal with any emergency.\"\n\nThe Morean knight raised an eyebrow. \"I never thought otherwise, messire,\" he said. \"But I will be with you.\"\n\nRanald shook his head. \"No offence. But why?\"\n\nThe Morean shrugged. Twirled his moustaches. \"It is a Deed,\" he said. \"I wish to see a dragon.\"\n\nThe captain smiled.\n\nWhen the company's wagons rolled, the captain sat his elegant riding horse under the shade of a great oak tree and watched them go by. Men saluted him. It made him want to cry.\n\nThere was Bent, riding with Long Paw; behind him rode No Head and Jack Kaves and Cuddy. They were laughing as they passed, but they all gave him a smile and a nod. Behind them were younger men\u2014Tippit arguing with Ben Carter and Kanny about something. They stopped when they saw him, and saluted\u2014Ben Carter drew his sword to salute, and then looked sheepish about it.\n\nDan Favor rode by with Ser Milus and Francis Atcourt, who was explaining a jousting technique using a walking-stick tucked under his arm.\n\nAnd more, and more. Men-at-arms, valets, squires, archers. Wagoners and tailors, prostitutes and seamstresses.\n\nSauce\u2014Ser Alison Graves, now\u2014made her horse rear a little, and flicked him a showy salute. And near the back of the column, Mag the seamstress hugged her man and rode her donkey clear of the column's dust to join the captain. \"If it please m'lord,\" she said.\n\n\"Your downcast eyes are wasted on me,\" he said.\n\n\"I would like to accompany you,\" she said.\n\nHe rolled his eyes. \"Why?\" he asked. \"A few days of sleeping on the ground and bad food?\"\n\nIn his mind Harmodius said, _Excellent_.\n\nSo when the column was gone, headed up the long ridge to the east, Ranald turned his horse's head north. \"I don't know where you are sleeping tonight, Captain,\" he said. \"But I'm for the Inn of Dorling.\" To Mag, he said, \"It's a little more comfortable than the cold, hard ground.\"\n\n# The Inn of Dorling\u2014The Red Knight\n\nThe Keeper came into the yard with eyes as wide as new-minted pennies. His people were on the walls, and the gate was open to receive them.\n\nHis eyes went right past Ranald\u2014wearing armour like a knight, and a red tabard. He nodded to the captain. \"You are welcome here, messire. The best of everything, the most reasonable prices.\"\n\n\"Don't you know your own kin?\" Ranald drawled.\n\nTom kicked free of his stirrups and dismounted in a clash of plate and mail. \"I hear my brother married your Sarah,\" he said.\n\nThe Keeper looked back and forth. \"By God!\" he said.\n\nTom took him in a bear hug.\n\n\"We all thought you were dead,\" said the Keeper.\n\nTom growled. \"Not yet, ye bastard.\"\n\nHe looked past the Keeper at the young woman on the porch. \"Hello, spark. You'll be Sarah. Last I saw you, you was smaller than a pig.\"\n\n\"Now I'm big enough to carry your brother's seed,\" she said.\n\nHe left the Keeper's embrace and gave her a hug.\n\nThe captain hadn't seen Bad Tom as a man who embraced people. It shook him a little.\n\n\"Hillmen,\" Ser Alcaeus said. \"I'm quite fond of them.\"\n\n\"You sound like you are talking about dogs,\" Mag said.\n\nAlcaeus snorted. \"Touch\u00e9, madame. But they are more like us than you Albans. They burn hot.\"\n\nRanald dismounted and kissed Sarah first. Then hugged the Keeper. He went to his malle, slung across the back of his horse, and took out a slim leather envelope, the size of a letter.\n\nHe tossed it to the Keeper.\n\nThe Keeper looked at it, frowning.\n\n\"Six hundred silver leopards,\" Ranald said. \"In a note of hand on a bank in Etrusca. That's yours. And another twelve hundred for Sarah.\" He gave the girl a lop-sided grin. \"I sold the herd.\"\n\nShe clapped her hands together.\n\nMen in the courtyard grinned. There were two dozen hillmen\u2014local herdsmen, small farmers, and the like\u2014and every one of them knew in that instant that his money wasn't lost.\n\nThey grinned. Embraced. Gathered round Ranald and slapped his back, shook his hand.\n\nThe Red Knight laughed, to find himself so far from the centre of attention.\n\nBut the Keeper disentangled himself from the celebrations shaping in his courtyard and came forward. \"I'm the Keeper,\" he said. \"I'm guessing you're the Red Knight.\"\n\nThe captain nodded. \"Men call me the captain,\" he said. \"Friends do, anyway.\"\n\nThe Keeper nodded. \"Ay\u2014Red Knight's a heavy handle to carry and no mistake. Come off your horses, now, and my people will see to you. Leave your cares here, and come and be easy.\"\n\nEasy it was. The captain shucked off his riding armour and left it in a heap for Toby and went down the steps to the common room, where he found his brother and Ser Alcaeus sampling the ale.\n\nMag came and sat by herself, but the captain wasn't having any of it. He walked to her table, and offered his hand. \"Ma dame,\" he said. \"Come and sit with us.\"\n\n\"Mag the seamstress with three belted knights?\" she asked. There was a wicked gleam in her eyes, but the words seemed sincere.\n\n\"Play piquet, mistress?\" asked Gawin.\n\nShe let her eyes drop. \"I know the rules,\" she said, ill-at-ease.\n\n\"We'll play for small stakes,\" Ser Gawin said.\n\n\"Couldn't we play for love?\" she asked.\n\nGawin gave her an odd look. \"I haven't felt cards in my hands for a month,\" he said. \"They need a little fire.\"\n\nMag looked down. \"If he takes all my money\u2014\"\n\n\"Then I'll order a dozen more of your caps,\" the captain said.\n\nLooking at the seamstress, the captain smiled inwardly. _How powerful is she, Magus?_\n\n_Hard to say, young man. Untrained talent. She had to learn everything for herself, from first principles._\n\n_Ah._\n\n_Possibly the greatest of us all, though. She was never trained. She has no chains._\n\nThe captain sat watching Gawin deal the cards. Something about the hawkish expression on Mag's face gave her away.\n\n_But a very limited repertoire_...\n\nHarmodius spluttered in the captain's palace. _Drink some wine, so I can taste it. She may have had a limited grimmoire, but not any more\u2014eh, young man? She has your phantasms, and mine, and all of the Abbess's. And Amicia's, too_\n\n_As do I. As does_ \u2014\n\n_Yes._\n\nMag sorted her cards. A boy brought an armload of sawn oak and started to lay a fire. The smell of lamb filled the common room.\n\nGawin sat back. \"Captain? I need to borrow some money.\"\n\nThe captain looked at him.\n\nMag was grinning.\n\n\"Doubled and rebated,\" Maggie said.\n\n\"I'll never be wed at this rate,\" Gawin said.\n\n\"Wed?\" asked the captain.\n\nSer Alcaeus smiled politely into his ale. \"To the Queen's Lady Mary, if I'm not mistaken,\" he said politely.\n\nThe captain laughed and laughed, remembering her. \"A most beautiful lady,\" he said.\n\n\"Eldest daughter of Lord Bain.\" Gawin looked off into the distance. \"She loves me,\" he said suddenly. He choked on the words. \"I\u2014I'm not worthy of her regard.\"\n\nThe captain reached out to his brother tentatively but Gawin didn't seem to notice.\n\n_Youth. It's wasted on the young._\n\nAlcaeus barked a laugh. \"Listen, messire. I have known a few knights. You cede worthiness to none.\"\n\nGawin said nothing. He drank off the rest of his jack, and raised his cup to the tap-boy. \"Wine, boy. And in truth\u2014\" He rose. \"I need to piss.\"\n\nAlcaeus cleared his throat when Gawin was gone. \"I can't help but note,\" he said with some diffidence, and paused. \"He calls you brother.\"\n\nThe captain laughed. \"He does me that honour.\" _Here we go._\n\n\"I had thought\u2014pardon me, messire\u2014\" Ser Alcaeus sat back.\n\n\"You thought I was some man's bastard. And here's the great Duke of Strathnith's son, calling me brother.\" The captain leaned forward.\n\nAlcaeus met his eye steadily. \"Yes.\"\n\nThe captain nodded. \"I had thought\u2014pardon _me_ , messire\u2014I had thought that you were a free lance, a knight on errantry, joining my company. And yet\u2014\" He smiled. \"Sometimes, I might be tempted to a thought. And that thought...\" He sat back.\n\nMag looked back and forth. \"Men,\" she said quietly.\n\n\"What thought would that be?\" Ser Alcaeus whispered.\n\nThe captain drank some excellent ale. \"Sometimes it seems anything I say to you will go straight to the Emperor.\" He shrugged. \"I mean no insult. You are his liege man.\"\n\n\"Yes,\" Ser Alcaeus admitted.\n\n\"And his cousin,\" the captain went on.\n\n\"Ah? You know this?\" Ser Alcaeus sighed.\n\n\"I guessed. So as to my own parentage\u2014\"\n\nSer Alcaeus leaned forward. \"Yes?\"\n\n\"It is not your business, messire. Am I clear?\" he said leaning forward.\n\nSer Alcaeus didn't flinch. \"Men will speculate,\" he said.\n\n\"Let them,\" the captain said.\n\nMag put a hand on the table and picked up the cards\u2014large squares, beautifully painted. \"People are watching you, my lords. You look like two men about to draw daggers.\"\n\nAlcaeus finished his ale. \"Beer makes men melancholy,\" he said. \"Let's have wine, and I'll think no more about it.\"\n\nThe captain nodded. \"I don't mean to be a touchy bastard. But I am.\"\n\nAlcaeus nodded and extended his hand. \"For what it is worth, so am I. A bastard.\"\n\nThe captain's eyes widened. He reached out and took the hand. \"Thanks for that.\"\n\nAlcaeus laughed. \"No one has ever thanked me for being a by-blow before.\" He turned to Mag. \"Would you like me to shuffle?\" he asked.\n\nShe shook her head. \"You rich boys,\" she said. \"You think bastardy matters? Look at yourselves\u2014gold rings, fine swords, wool cotes worth fifty leopards. Fine horses. By the Gentle Jesu, m'lords. Do you know what a poor man has?\"\n\n\"Parents?\" Ser Alcaeus said.\n\n\"Hunger,\" Mag answered.\n\n\"God's blessing,\" the captain said.\n\nGawin came back. He had a glow on, a brittle humour. His eyes sparkled. \"A fine inn. Maybe the best I've ever seen. Look at that lass\u2014red hair. Red! I've never seen so much red hair in all my life.\" He looked around. \"Their fires burn hotter, or so men say.\"\n\nMaggie smiled, reached under her cap and teased out the end of her braids. Her hair was bright red. \"Really, ser knight?\" she said.\n\nGawin sat back and laughed. The captain laughed harder, and Alcaeus caught it too. It was infectious.\n\nAs if his laughter was a signal, the Inn burst into life. Tom and Ranald came in, and joined their table, and men and women came pouring in. Local farmers and shepherds from the hills arrived as the word spread, and the mercenaries who served the Keeper, and a tinker and his apprentices\u2014the smith, and his apprentices too.\n\nThe common room could hold them all, well enough.\n\nMen called for music, and Tom sang surprisingly well. Gawin turned to the captain amidst the uproar. \"You used to play the harp,\" he said.\n\nThe captain frowned. \"Not in years. And not here.\"\n\nBut the Keeper had heard him. He took a harp down from the wall and put it in the captain's arms. He shushed the room\u2014something he did as easily as a magus might cast a spell.\n\n\"There's a man here as may be a harper,\" said the Keeper.\n\nThe captain cursed Gawin under his breath.\n\n\"Give me some time,\" he said, when it was clear to him they wouldn't let him off. He took the harp and his second cup of wine and walked out into the summer night of the yard.\n\nIt was quiet out there.\n\nSheep baaed, and cattle lowed, and the sounds of men in the Inn were muted, like the babble of a distant brook.\n\nHe started to tune the harp. There was a plectrum in the base-board, just where he would have expected it, and a clever mechanical key for the strings.\n\n_Let me,_ said Harmodius. _It's just mathematica._\n\n_He drew power, and cast\u2014and his power manifested in the strings._\n\n_The rule of eight, rendered in sinew,_ said the dead Magus.\n\n_Thanks,_ said the captain. _I always hated tuning._\n\nHe walked about the yard, plucked out a simple tune\u2014the first he'd learned\u2014and walked back into the Inn.\n\nThey fell quiet when he appeared, and he sat down with Gawin and played some simple stuff. He played _There Was a Squire of Great Renown_ and everyone sang, and he played _Green Sleeves_ and _Lovely On the Water._ He made mistakes, but the audience was forgiving.\n\n\"Play for dancing!\" the young widow called.\n\nThe captain was about to admit he didn't know any dances, but Harmodius forestalled him.\n\n_Allow me._\n\nHis fingers plucked the strings slowly, and a jig peeled out\u2014slowly at first, and then faster and faster, and then it was a reel and then it was a hillman dance tune, sad and wild and high\u2014\n\nThe captain watched his fingers fly over the strings, and wasn't altogether pleased. But the music swept on, higher and higher, and the men fell out of the dance, and the women danced, skirts kirtled up, legs flashing, heads turning and Mag jumped up and leapt into the circle.\n\nThe harp grew warm under his hands.\n\nSarah Lachlan leaped and flashed like a salmon. Mag gave a turn and one of the Inn's servants twirled in billow of skirts. The men applauded wildly as the hands on the harp fell still, and the captain seized control again.\n\n_Ahh,_ said Harmodius. _I had forgotten._\n\n_Please don't do that again, old man._ The captain went to steady his own breathing. People were crowding around him, slapping his back.\n\n\"I swear,\" said the Keeper. \"You play like a man possessed.\"\n\nLater when men and women had paired off, when Mag had gone, bright eyed, to her room, and Ranald had been congratulated by every man and woman there, and when Ser Alcaeus had the Inn's prettiest serving girl in his lap\u2014he went back outside.\n\nHe stood under the stars, and listened to the cattle.\n\nHe played _Green Grow the Rushes_ to them.\n\nHarmodius snorted.\n\nIn the morning, they mounted for the ride north. None of the captain's companions seemed to have a hard head and he was surprised to see the Keeper mount a fine riding horse, as eastern in its blood as the captain's own.\n\nThe Keeper nodded to the captain. \"You're a fair harper and no mistake, m'lord. And a good sport.\"\n\nThe captain bowed. \"Your house is one of the finest I've ever visited,\" he said. \"I could live here.\"\n\n\"You'd need to learn some more tunes first,\" Gawin said.\n\n\"Coming to see the Wyrm?\" Ranald asked the Keeper.\n\nHe nodded. \"This is my business as well as yours an' Tom's.\n\nThey rode.\n\nThere was a good path, the width of two horsemen, and it ran like a snake between the hills, and the bottoms of valleys were damp and the heights were rocky. They didn't go fast.\n\nCrossing the Irkill River took half a day, because the bridge was out. The Keeper begged a favour of the captain and sent Toby back to the Inn with the news.\n\n\"This is my business,\" he said. \"And I don't like it.\" The bridge looked as if a battering ram had struck it\u2014it was beaten to flinders, heavy oak beams now splinters.\n\nThat night they slept in a cot by a quiet burn. The farmer and his family moved out into a stone barn so that the gentles could use the beds.\n\nIn the morning, the captain left a silver penny and they were away with the sun, full to bursting with fresh yogurt and honey and walnuts.\n\nThey rode higher and higher into the hills, and passed a pair of heavy wagons loaded to the tall seat with whole, straight trees\u2014oak, maple, and walnuts, trunks bigger around than a tall man might reach, and straight as giant arrow-shafts. The wagoners allowed as there were lumbermen working in the vales.\n\nGawin sneered. \"It must be all they can do to move these monsters.\"\n\nThe wagoners shrugged. \"Maybe. Maybe not.\"\n\nSer Alcaeus waited until they were past. \"They float the larger logs on the water.\"\n\nThe Keeper nodded grimly. \"That's what happened to my bridge.\" He led them down into the dale and they found the foresters hard at work\u2014not local men, but easterners.\n\nThey had cut a swath through the dale, and a dam on the big stream that fed the Irkill. The leader of the woodsman stood in the new clearing, obvious in his long cloak. He had a heavy axe in his hand, gull winged and long hafted, and his wood-cutters were tall and strong, with long beards.\n\nThe Keeper rode up to him. \"Good day to you,\" he said.\n\nThe man nodded. His eyes were wary. He watched the troop of horsemen\u2014more armoured power than anyone liked to see, especially far from home.\n\n\"What can I do for you?\" he said. His accent was thick.\n\nThe Keeper smiled pleasantly enough. \"Pack and leave. Let the water off your dam slowly.\"\n\nThe woodsman's eyes widened and then narrowed. \"Who are you, then?\"\n\nHis men were gathering, and horns were blowing.\n\nThe Keeper didn't touch his weapons. \"I'm the Keeper of Dorling,\" he said. \"You owe me the cost of a bridge, and more. No one logs these dales without my leave\u2014and the time to cut was early spring, when the last snow lies on the ground.\"\n\nThe captain swatted a black fly.\n\nThe woodsman frowned. \"The woods are any man's, or no man's. This is Wild Land.\"\n\n\"No. These Hills are in the Circle of the Wyrm,\" the Keeper said.\n\nThe woodsmen began to gather. Many had spears, and every man had an axe. They were forming.\n\nGawin dismounted and, as fast as a dancer, remounted on his war horse. He drew his great sword.\n\nThe Keeper raised his hand. \"Peace, ser knight.\" He looked back at them. \"No need for arms.\"\n\n\"You have wisdom, old man,\" called the leader of the woodsmen.\n\n\"You have been warned,\" said the Keeper.\n\nThe woodsman spat. \"I laugh at your warning. What business is it of yours? And if one of your bridges is swept away by my logs\u2014\" He shrugged. \"There is wood everywhere. Build another.\"\n\nThe Keeper looked around at the crowd of woodsmen. \"If you remain here, every one of you will die,\" he said.\n\nThey looked unimpressed.\n\nThe Keeper wheeled his horse. \"Let's ride,\" he said.\n\nThe Keeper led the way, and they rode at a trot until they were out of the dale and up the next green ridge.\n\n\"I feel as if I just ran away,\" Gawin said.\n\nThe captain grimaced. \"Me, too.\"\n\nThe Keeper turned in the saddle. \"If the Wyrm is of a mind, he'll kill them all for this, and us, too, by association.\"\n\nThat night, for the first time, they camped. There was little grass for the horses, and they had to put nosebags on them and use the oats that the pack animals carried. Mag watched Gawin start dinner and then pushed him out of the way.\n\n\"By the good and sweet Christ,\" she said. \"At least use a clean knife.\"\n\nAlcaeus laughed and took the cook knives to the stream and washed them, scouring them with sand.\n\nThe Keeper rode out with the hillmen and came back with two big turkeys.\n\nGawin greeted them with a pair of big trout. \"I take it there's not much in the way of angling in these parts,\" he said. \"Glad I brought a line.\"\n\nMag looked at the birds and the fish. \"What you catch, you clean,\" she said. \"I'm a cook, not a servant.\"\n\nThat made the captain laugh. He'd spent the late afternoon building a shelter and digging her a fire pit and now he helped clean the fish with a good grace. They drank the last of the wine by firelight.\n\n\"Tomorrow,\" said the Keeper.\n\nThey rode with the dawn.\n\nThe next range of hills was bare of trees, as if a horde of sheep had clipped them clean\u2014green grass rippled in the wind like a green sea, and the hills rolled away like a greater sea\u2014from the height of their ridge, they could see twenty more ridges spread out like pleats in green wool.\n\nMag raised a hand. \"Is that an eagle?\" she asked.\n\nFar to the north-east, a great bird rode the air over the hills.\n\nThe Keeper looked under his hand.\n\nThe captain looked too. The great creature was farther away than he had imagined, and he looked and looked until he appreciated what he was seeing, and then his heart beat in pure fear.\n\n\"Good Christ,\" said Mag.\n\n\"My God,\" said Gawin.\n\n\"That's the Wyrm of Erch,\" said the Keeper.\n\nIt was flying. It was larger than a castle, and it was flying over the hills to the north. Even as they watched, the titanic dragon turned\u2014for a moment its immense and spiky tail was clearly silhouetted against the northern sky, and its huge wings swept out on either side.\n\n\"Good Christ,\" Mag said again.\n\nIt was faster too.\n\nThe captain couldn't take his eyes off it.\n\n_So,_ Harmodius said in his mind. _So._ The dead Magus sounded, if anything, more awestruck than the living captain.\n\nThe wind-storm of its wing beats began to echo across the hills. The only sound the captain could imagine like it was the beat of the great mills in Galle\u2014he'd heard them in the low country.\n\n_Whoosh._\n\n_Whoosh._\n\nIt was as big as the hills.\n\nHis riding horse began to panic. Mag's threw her with a sudden twist and bolted, and all the horses went wild. The captain dismounted, hauled his horse's head down, and knelt by the seamstress.\n\n\"Nothing hurt but my pride,\" she snapped. \"And nothing much there to bruise.\"\n\nThe Wyrm was coming right at them.\n\nIts wings swept up, their tips almost touching, and then down, and the power of their passage left a swath of matted grass far below as the Wyrm passed over them. It was enormous. The captain was able to count to ten while the immense thing passed over him. His riding horse stood frozen in terror and the dragon's shadow covered the ground for a hundred paces in all directions\u2014more. It covered the sun.\n\nHe blinked his eyes and looked again.\n\n_Look in the Aether,_ said Harmodius.\n\nThe captain raised his _sight and staggered in renewed awe. If Thorn had been a pillar of green, the Wyrm was\u2014was the sun._\n\nThe captain shook his head.\n\nGawin threw his head back and whooped.\n\nBad Tom laughed aloud.\n\n\"Now that, my friends,\" he said, \"is a Power of the Wild, and no mistake.\"\n\nThey rode down into the next valley as the rain clouds came on, building to the north over the loch. A series of lochs fell away for leagues\u2014larger and larger, until they merged into a sheet of water twenty leagues or more away. It was a superb view. In front of them, just short of the first loch, was a ford over a burn. They got cloaks off their saddles as they came to the stream. No one spoke much.\n\nThe rain came down like a curtain, sweeping from the north end of the valley, cutting off the view of the lochs.\n\nBeyond was only rain, and black cloud.\n\n\"It's like the end of the world,\" Mag said.\n\nThe captain nodded. Ser Alcaeus crossed himself.\n\nThey crossed the stream quickly at a cairn. The captain rode off to the side, and then rejoined them. \"Let's move,\" he said. \"The water here rises very quickly and very high.\"\n\nGawin watched the water. \"Salmon in that loch,\" he said wistfully.\n\nOn the far side was a narrow track that rose on the hillside. It was just wide enough for a horse, and they picked their way in single file, with the Keeper at the head and Bad Tom last.\n\nIt took them an hour to climb the ridge, and the rain caught them in the open again. It was cold, and they were soaked through despite heavy cloaks and hoods.\n\nUp, and up they went.\n\nAt the top of the ridge was a seat of stone facing west.\n\nThe captain looked at it. So did Mag. It held the residue of power.\n\nThe Keeper didn't stop. He rode down the far side.\n\nFrom the very top, just beyond the High Seat, the captain could see the ghostly impression of crags to the north\u2014far away, and gleaming white. Almost everything else was lost in the rain, although they were above it for a few hundred paces, and then they rode back into it.\n\nDown and down, and trusting his horse. His light saddle was soaked, and he worried for his clothes. For summer, this was _cold_ rain.\n\nHis brain was running wild.\n\n\"We're going to _visit_ that?\" he asked, sounding more like Michael than he would have liked.\n\nRanald turned and looked back. \"Aye.\"\n\nIt was afternoon by the time they came out of the bottom of the clouds and could see, through gaps in the rain curtain, another valley of lochs. It was oriented differently\u2014in this one the lochs grew smaller as the valley rose to the east and north, into high crags.\n\nThe Keeper reached the first ford, marked again with a cairn of stones that leaped to the eye in the naked, empty landscape of green grass and rock and water.\n\n\"Water's high,\" he shouted.\n\nThe captain leaned out and watched it for a long minute. They could hear rocks being rolled under the water.\n\nThe stream rushed down a narrow gorge above them, gathered power between two enormous rocks, and shot into the loch on their right\u2014a sheet of water perhaps three hundred paces long and very deep.\n\nBad Tom laughed. He roared, \"Follow me,\" and turned his horse's head south. He seemed to ride straight out into the loch, yet his horse was virtually dry-shod as he rode a half circle a few paces out from the shoreline.\n\nThe captain followed, as did Ranald. Looking down into the water, he could see a bank of rocks and pebbles just under the water.\n\n\"In the spring run-off,\" Ranald said, \"the force of water pushes all the rock out of the mouth of the stream. Makes a bank\u2014like yon.\" He laughed. \"Any hillman knows.\"\n\nTom looked back at the Keeper. \"Aye. Any _true_ hillman.\"\n\nThe Keeper shot him a look, but Tom was immune to looks.\n\nThey started up the valley, wet and feeling surly.\n\nThe trail followed the stream past a magnificent waterfall, and then they climbed the cliff\u2014the trail was just wide enough for an experienced rider to stay mounted, and it cut back and back\u2014nine switchbacks to climb a few hundred feet. Ser Alcaeus's war horse balked, and would not climb until Ser Alcaeus dismounted, walked back, and fetched him.\n\nMag dismounted at a switchback and looked at the captain.\n\nHe understood. She was not going to _ask_ for help. He took her horse by the reins.\n\n\"Thanks,\" she said.\n\nShe began to walk up the track.\n\nHe led her horse.\n\nAt the top of the cliff there was another loch. It was smaller, deeper, trapped in narrow cleft and dammed off by the ridge of rock that made the cliff. Above the loch was a long, grassy ridge that rose and rose. Above it all towered a mighty crag, covered in snow\u2014but the snow line was still as far above them as they had come in two days.\n\nThe trail ran along the banks of the loch, in deep grass.\n\nThere were sheep high on the hillsides.\n\nThe only sound was the muted roar of the waterfall coming off the loch behind them, and the distant babble of the stream off the glaciers running into the top of the loch.\n\nThere was a gravel beach at the top of the loch. The captain caught the Keeper and pointed to it. \"Camp?\" he said.\n\nThe Keeper shook his head. \"He's telling us to go away. This weather's unnatural.\" He shrugged. \"We're in for a bad night.\"\n\nThe captain was looking through the rain at the distant beach. \"I see wood there.\"\n\nMag nodded. \"I saw rowans up in the highest valleys,\" she said.\n\n\"Rowan, alder, and older things,\" agreed the Keeper. \"We can't have a fire, this close to the Wyrm.\"\n\n\"Why not?\" the captain asked.\n\n\"The Wyrm has rules.\" The Keeper shrugged.\n\nThe captain shook his head. \"Taking living wood might incur the wrath of a Power,\" he said. \"Dead wood on a beach, however\u2014\" He managed a smile and shrugged off the rain. \"There's an overhang there. Gather all the horses against it to break the wind.\"\n\nThe Keeper shrugged. \"On your head be it. If we turn back now, we can have better weather before sunset.\"\n\nGawin rubbed water out of his moustache. \"Tell me why we didn't camp by a loch with fish?\" he asked.\n\nThe captain looked out over the rain-swept sheet of water. \"I'd bet a golden leopard to a copper there's salmon in this water,\" he said. \"But I wouldn't be the man to catch one.\"\n\nGawin smiled. \"You don't know much about salmon, brother, if you think they can climb a hundred foot of falls.\"\n\n\"My bet stands,\" the captain said. \"But to catch one would be a deadly insult to our host, and as the Keeper has noted, he's not in love with us at the moment.\"\n\nMag cackled. \"So worried about a bit of wet. I'm twice the age of most of you, and I can roll up in a wet cloak and sleep. My joints will cry in the morning, but what of it? I saw a dragon fly in the dawn.\" She looked at them. \"I'm not turning back, gentles.\"\n\nThey constructed a shelter from spear poles and heavy wool blankets, pinned down with the biggest rocks on the beach. The wind tested it for a while, but didn't seem interested in a real contest.\n\nThe captain rode off with Ser Alcaeus, and together they roamed the long beach and picked up every stick on it\u2014it made a respectable woodpile.\n\n\"And where'd it come from, I'd like to know?\" asked the Keeper.\n\nThe captain shrugged. \"Our host put it out for us to find, I expect.\"\n\nGawin, a practised hunter, took a fire kit from his pack and looked at his brother across the fire pit. \"Like being boys again,\" he said.\n\n\"We never tried to light a fire in a storm like this,\" the captain said.\n\n\"We did, too,\" said Gawin. \"I couldn't get it lit, you used power, and Pater cursed you.\"\n\n\"You're making this up,\" the captain said, shaking his head.\n\nGawin gave him the oddest look. \"No,\" he said. He used his body and his soaking cloak to cover the fire pit, and the captain's quick hands laid a bed of twigs\u2014damp, but dry as drift wood ever is. Gawin put a bed of dry tow from his fire kit inside a nest of birch bark.\n\n\"Bark from home,\" he said.\n\nThe captain shrugged.\n\nGawin laid charred linen deep in the tow, and then struck his fire steel against a small shard of flint until spark flew. The char-cloth lit, he dropped it into the nest in his hand, and blew. Smoke billowed out. He blew a second time, a long, slow breath, and more smoke came.\n\nThe captain leaned over and blew.\n\nBefore his breath was out, Gawin blew, and the whole nest burst into flame. Gawin dropped it onto the waiting twigs, and both men added more, and more\u2014speed and accuracy embodied.\n\nIn two cracks of lightning, they had a fire.\n\nMaggie laughed. \"You could have just magicked it,\" she said. \"Instead of showing off with your woodcraft.\"\n\nGawin frowned.\n\nThe captain smiled. \"I avoided the use of power for many years.\" He shrugged. \"Why waste it?\"\n\nShe nodded, understanding.\n\nThey made tea from the water of the loch, ate cold meat, and curled up to sleep. The stones of the beach were cold and wet, but the wool tent and the warmth of the horses won out in the end.\n\nThey took watches in turns. The captain took the mid watch, and he sat high above the beach on a rock. The wind was gone, and with it the rain, and he watched a thousand thousand stars and the moon.\n\n_May we talk?_\n\n_No._\n\n_You've closed your door and you aren't responding to Mag and she's confused. You are linked to her. The courtesy of mages requires you_ \u2014\n\n_No._ The captain looked out over the loch. _Go away. Not at home._\n\nHis head hurt.\n\nIn the morning, they drank hot tea, ate fresh Johnny cake made in ashes on a flat rock by Mag, and rode on. The horses were tired and cold, but by a miracle none of them were lame or sick despite a cold night on a mountainside. They followed the trail up over the green ridge at the north end of the loch, down into a shallow, high valley of green turf with the stream ripping through, full of rain water. From there down a rocky course at the centre, and then they cut back twice, riding up another ridge. The green of the hills was deceiving\u2014what looked like one endless ridge proved to be a succession of them, one merging to another in the grey light.\n\nThe Keeper shook his head. \"It wasn't like this the last time,\" he said.\n\nRanald laughed. \"Never the same twice, is it, Keeper?\"\n\nThe Keeper shrugged. \"This is only my second trip, Ranald.\"\n\nBad Tom grunted. \"Never been, meself. But Hector said it was different every time.\"\n\nUp and up.\n\nThey climbed the next ridge as the sun struggled through the curtain of cloud, and at the top of the next ridge, in a fold of the earth, sat a shepherd's cot with a curl of peat-smoke coming out of a low chimney.\n\nSheepfolds extended right out from the walls of the stone house, as if the whole place were built for sheep.\n\nThe trail led from their ridge to the door of the shepherd's cot, straight as a lance.\n\n\"Biggest sheep I've ever seen.\" Alcaeus was rubbing the water out of his hair.\n\nThey rode down the track. The stone wall by the cot had a gate with richly worked iron hinges and the captain leaned over and opened it.\n\nOn the far side, hidden by the crest of the hill, was a brick horse barn. It had eleven stalls.\n\nThe captain grinned. \"I'll take this as a sign we're welcome,\" he said.\n\nThe brick horse barn looked very out of place.\n\n\"I know this barn,\" Gawin said. \"This is Diccon Pyle's barn.\" He looked at Ranulf, who nodded.\n\n\"From Harndon,\" Ranulf said. \"I was just thinking of it. Warm, snug\u2014\" He blew out a breath.\n\nThey took the horses into the barn. Their hooves rang on the brick floor, louder than the captain would have thought possible. There were oats in every manger, fresh straw on the floors, clean water in the buckets.\n\nThey unsaddled the horses, and took the gear off the pack animals. The captain curried his new destrier and put a blanket\u2014ready to hand\u2014over him. Gawin and Alcaeus did the same, as did the Keeper and Ranald. Bad Tom stood in the doorway, a sword in his hand.\n\n\"I don't like this. It's fey.\" Tom thumbed the edge of the blade.\n\n\"Not a problem you can solve with a sharp blade,\" said the captain. He got the tack off Tom's big gelding. \"Relax.\"\n\nTom didn't leave the doorway. \"I want to get this over,\" he said.\n\nRanald went and took his arm. \"Not the way to go, Tom. Be _easy_.\"\n\nMag smiled at Ser Alcaeus. \"Would you be so kind as to have the saddle off my horse, ser knight? I'm a poor weak woman.\"\n\nSer Alcaeus grinned.\n\nMag gathered her cloak, pushed past Bad Tom, and walked to the door. She knocked politely.\n\nThe knock sounded as loud as the crack of a trebuchet in the silence.\n\nThe door opened.\n\nMag went in. The Keeper paused at his currying and dropped the brush. \"Damn,\" he said. And ran for the door, but it was already closed. He knocked, and the door opened, and he was gone.\n\n\"I think the rest of us might as well go in together,\" the captain said. He wiped his hands on straw. He walked up to the door. \"You, too, Tom.\"\n\nTom was breathing hard. \"It's _all magick._ \"\n\nThe captain nodded and spoke carefully, as he would to a skittish horse or a scared child. \"It is, that. We're in his hands, Tom. But we knew that.\"\n\nTom stood straight. \"You think I'm afraid.\"\n\nRanald made a motion of negation.\n\nThe captain nodded. \"Yes, Tom. You are afraid. If you weren't, to be honest, you'd be some sort of madman.\"\n\n\"Which you may be, anyway,\" Ranald said.\n\nTom managed a smile. \"I'm ready.\"\n\nThe captain rapped at the door.\n\nAnd it opened.\n\nThe croft was low and close yet surprisingly spacious. The rooftrees were just above the captain's head height, too low for Tom, and the building had a roof-end hearth, not a proper fireplace at all. The fire in it was enormous, filling it like a furnace, so that individual logs couldn't be made out in the inferno\u2014but just enough heat escaped to make the room pleasant on a cool summer evening.\n\nAround the fireplace were heavy wooden chairs, covered in wool cloths. Some cloths were armorial, and one was an ancient tapestry, cut up and sewn to cover the chair.\n\nThe cot beams were black with age, but carving could still be seen on them.\n\nOver the fireplace, a pair of swords were crossed and, on the main beam, a spear was carefully set on a long row of iron nails.\n\nMag sat with the Keeper, her legs crossed. And beyond her sat a small man smoking a long pipe.\n\nHe was so very ordinary that their eyes passed over him, at first. He wore a plain wool cote of coarse wool, and leggings of the same, and his weather-beaten face was neither handsome nor ugly, old or young. His eyes were black.\n\nHe opened them, and they were instantly arresting.\n\n\"Welcome,\" said the Wyrm.\n\nThe captain bowed. He looked around, and none of his companions was moving\u2014except that the men behind him in the doorway were suddenly sitting in chairs, hands on their knees.\n\nHe hung his cloak with theirs, and went to a seat.\n\n\"Why is no one speaking?\" he asked.\n\n\"You are all speaking,\" the Wyrm said. \"It is easier for all of us if I deal with each in turn, in privacy.\"\n\n\"Ah,\" said the captain. \"I'll wait my turn.\"\n\nThe Wyrm smiled. \"I can talk to you all at once,\" he said. \"It is you who needs the feeling that there is structure, not me.\" He took a pull on his pipe.\n\nThe captain nodded.\n\n_Of course time means nothing to them,_ Harmodius said.\n\n\"Are the two of you together?\" the Wyrm asked.\n\n\"There's just one of me,\" the captain said. \"I can't speak for Harmodius.\"\n\nThe Wyrm smiled again. \"Very wise of you to see that. You know that if you do not rid yourself of him, he will, in time, demand control. He cannot help himself. I offer this information free of obligation.\"\n\nThe captain nodded. A cup of mulled wine appeared at his elbow. He picked it up and drank it gratefully.\n\n\"Why have you come?\" asked the Wyrm. \"You, at least, had to know what I was.\"\n\nThe captain nodded. \"I guessed.\" He looked around. \"Are there rules? Do I have three questions? Fifty?\"\n\nThe Wyrm shrugged. \"I don't want visitors. I try never to look into the future. All that is for my busy, busy kin. They plot, and strive. I live. I seek truth.\" He smiled. \"Sometimes I grow lonely, and a lucky traveller is brought in for entertainment.\" His smile became a feral grin.\n\nThe captain drank more wine. \"What of the Lachlans?\"\n\nThe Wyrm pulled on his pipe, and smoke wound to the ceiling and up into the draught of the roaring fire. \"That is your question?\"\n\nThe captain shook his head. \"No, but they are my sworn men and I need to know they are being well served.\"\n\nThe Wyrm smiled. \"The concept of fealty comes so naturally to men and I am having a difficult time being bound by it. But I will deal fairly with Tom and Ranald. Ask your own.\"\n\nThe captain swirled his wine, and clamped down on a question about Amicia. \"Can the conflict between Man and Wild be resolved?\" he asked.\n\n\"Is that your question?\" asked the Wyrm.\n\n\"Yes,\" said the captain.\n\nThe seated figure smoked. \"How delightful.\" He walked to the mantelpiece and opened a stone jar, took out a handful of old leaves and tamped them into the bowl of his pipe. \"Do you believe in free will, prince?\"\n\nThe captain was growing hot, and he stood up and took off his cote and hung it by the mantel to dry with a muttered \"beg your pardon\" to his host. He sat again.\n\n\"Yes,\" he said.\n\n\"Why?\" asked the Wyrm.\n\nThe captain shrugged. \"Either I have free will, or there's no point in playing.\"\n\nThe Wyrm rocked its head back and forth. \"What if I were to tell you that you only had free will in some things, and not in others?\"\n\nThe captain found he was chewing one of his riding gloves. He stopped. \"I'd suggest that my power to affect the universe is about the same whether I have free will in every action or only in one.\"\n\n\"Interesting,\" said the Wyrm. \"Man and the Wild are merely concepts. Philosophical constructs. If they were created to represent\u2014to symbolize\u2014opposition, then could they ever be reconciled? Can alpha and omega switch places in the alphabet?\"\n\n\"Next you will tell me there is no Wild. And there is no Man.\" The captain smiled.\n\nThe Wyrm laughed. \"You've taken this class before, I take it.\"\n\n\"I sat at the feet of some philosophers in the East,\" the captain said. \"I had no idea they were dragons, although, now that I think of it\u2014\"\n\nThe Wyrm laughed again. \"You please me. So I will answer your question. Man and the Wild, while being two sides of a coin, can live together\u2014just as the coin lives perfectly well in the purse.\"\n\n\"Separate?\" the captain asked.\n\nThe Wyrm shrugged. \"Nothing about a coin is separate, is it?\" he asked.\n\nThe captain leaned back in his very comfortable chair.\n\n\"My brother died,\" Tom said. \"He was your liege man, and he died. Tell us who killed him?\"\n\nThe Wyrm shrugged. \"He died outside my circle,\" he said. \"I concede that I wasn't paying very much attention. I further concede that while my mind was taken with other affairs, some of the Wild peoples crossed my lands without my leave. But in truth, Tom, and Ranald, my circle is a creation for my own convenience. I scarcely trouble men, in or out of it, and you two are the first to demand some sort of action of me in a long enough amount of time to be meaningless.\"\n\n\"So you won't avenge him,\" Tom said. \"Just tell me who killed him?\" he asked.\n\n\"Are you telling me what I'm doing, or asking?\" the Wyrm asked politely. \"Is this your question?\"\n\nRanald leaned forward. \"Yes,\" he said. \"It sounds odd but it isn't the Sossag I'm after, though they slayed Hector and me, too. It's Thorn. Thorn sent them\u2014he summoned them. Drove them to war.\"\n\nThe Wyrm threw back his head and laughed. \"Are you simple, Ranald Lachlan? The Wild Peoples do exactly as they please. They are not children. If they raided your brother, they did so apurpose.\"\n\n\"They'd never ha' been at the fords if it hadn't been for Thorn.\" Tom was insistent.\n\nThe Wyrm put his chin in his right hand. \"How much of the truth would you like, hillman? Shall I tell you enough to spark an epic revenge? Or shall I tell you enough to render you incapable of action? Which would you prefer?\"\n\nRanald chewed the end of his moustache. \"What could you tell us that would make us unable to act?\" he asked.\n\nTom glowered.\n\nThe Wyrm sat back and put his pipe down, put his hands behind his head. \"The Sossag who killed Hector is called Ota Qwan. He is a worthy enemy for you, Tom\u2014driven, passionate, highly skilled. Your riddle is that, in time, your captain will want him as an ally.\" The Wyrm smiled.\n\n\"And so you render Tom incapable of action?\" Ranald asked. \"You don't know Tom.\"\n\nThe Wyrm shook his head. \"No. Because behind Ota Qwan was Skadai, who made the decision to risk my wrath and raid the hillmen and the drove. He's already dead, though. Behind Skadai is Thorn, who was pushed into war\u2014\" the Wyrm was smiling, \"\u2014by one of my kind, to whom you and your brother are less than ants, and who wishes to encompass not just the end of your brother, but the death of every man and woman in the entire circle of the world. I should offer you my thanks\u2014I have just realised that I have slept through a cycle of drama. Things are moving out in the world. Damn the lot of you.\"\n\n\"His name?\" Tom said.\n\n\"Tom Lachlan, you are a name of fear among men from East to West. Daemons and wyverns wet themselves in fear at the mere mention of your name.\" The Wyrm gazed at Tom with affection. \"But my kind\u2014nothing in your arsenal can harm us.\"\n\n\"His name?\" asked Tom.\n\nThe Wyrm leaned forward. \"I would like to deal with this myself.\"\n\nTom slapped his thigh. \"Now you're talking, Wyrm. A good lord stands up for his man. But I'll help ye. Tell me his name, and together we'll put him down in the dust.\"\n\nThe Wyrm shook his head. \"Are you to be drover, Tom?\"\n\nTom shook his head. \"I doubt I could. I'd kill every loon as bade me nay.\"\n\nThe Wyrm nodded. \"Ranald?\"\n\n\"I'd be proud to be drover. But I seek to be knighted by the king\u2014to have a little treasure\u2014so I may wed a lady.\" Ranald felt like a small boy confessing to stealing apples.\n\n\"None of these things is my concern,\" said the Wyrm. \"Although the two of you are a pleasure to converse with.\"\n\n\"He's the man of reason,\" Tom said. \"I'm the man of war. Two sides of a coin.\"\n\n\"Nothing about a coin is separate,\" the Wyrm said.\n\nMag sat with her hands folded in her lap.\n\n\"And how may I help you?\" the Wyrm asked her.\n\n\"I'd like to defeat and destroy the sorcerer known as Thorn,\" she said.\n\n\"Revenge?\" asked the Wyrm.\n\nShe shrugged. \"A dog bit one of my children some years ago. He'd bitten other children. My husband went out with his crossbow and put the dog down.\" She met the Wyrm's eyes. \"I'm sure that there was some revenge involved.\"\n\n\"But it was, in the main, it was about the other children?\" asked the Wyrm.\n\nShe nodded.\n\n\"You are a very modest woman,\" said the Wyrm. \"You allow men to speak their minds, and you keep yours to yourself.\"\n\nShe smiled and looked at her hands in her lap.\n\n\"But you, the Goodwife of Abbington, intend to encompass the destruction of Thorn, who has put himself on the path to be a Power.\" His black eyes sought hers.\n\nShe wouldn't let him in. \"That's right,\" she said easily.\n\nThe Wyrm whistled soundlessly. \"This war that you have all just experienced has enhanced your powers to a wonderful degree. Indeed, I was able to see you\u2014really see you\u2014as far away as Albinkirk.\"\n\nMag gave way to a satisfied chuckle. \"I always knew I had the talent,\" she said. \"But thanks to the old magister and the Abbess I know things, now.\" She looked up. \"Terrifying things.\"\n\n\"Do you doubt God?\" asked the Wyrm.\n\nMag turned her head away. \"Who are you to ask that? Satan?\"\n\nThe Wyrm laughed. \"Not hardly, Mistress. Satan's idle young cousin, perhaps.\"\n\n\"Will you answer my question?\" she asked.\n\n\"You haven't asked one,\" he said gently. \"You've implied that you'd like my help in attacking Thorn, and you've implied that you'd like to know if there is a god.\"\n\nShe straightened her back. \"I can find my way to God without you,\" she said.\n\n\"Good,\" said the Wyrm.\n\n\"I'd like your help with Thorn,\" she said.\n\n\"That's the other side of the same coin, surely,\" said the Wyrm. \"If you can decide for yourself about God, you scarcely need me to tackle a mortal sorcerer.\"\n\n\"It would be easy for you,\" said Mag.\n\n\"No argument at all. In the end, that would be me putting down the dog. For _my_ reasons.\" He put his chin in his hands.\n\nShe shook her head. \"I understand, but I'd like you to separate the two sides of the coin.\"\n\n\"Nothing about a coin is separate,\" the Wyrm said.\n\n\"Nothing about a coin is separate,\" said the Wyrm.\n\nThe captain looked around to find all his companions also blinking like people coming out of sleep.\n\n\"It has been a great pleasure meeting you,\" he said. \"The beds are warm, and the fire is real enough, and the food is, if I say so myself, exemplary. Please don't stint with the wine. I'd be affronted if you didn't try the harp on the wall.\" He smiled at them. \"I have little interest in the affairs of the world, but I am choosing to help you, almost entirely to serve my own ends. Which, I will add, are infinitely less threatening to you and yours than any of the rest of my kin's might be. I seek only to be left alone\u2014I have my own ambitions, and they have nothing to do with war, conquest, pain, or hate.\" He smiled, and just for a moment, they saw an enormous head with fangs the length of warships, slitted eyes as tall as church spires. \"You will be my allies. You will go out in the world and serve my ends with your own plans and your free will.\" He smiled. \"I doubt that we will succeed, but if we do we'll have the satisfaction of having been vastly the underdogs.\" He nodded, as if to himself. \"Ah\u2014the party-favours. I've made certain artefacts\u2014or gathered them\u2014for this. To each, her own. And in parting\u2014\" The Wyrm smiled at all of them. \"May I leave you with some genuine wisdom, in place of all the humdrum claptrap? Do well. Act with honour and dignity. Not because there is some promised reward, but because it is the only way to live. And that is as true for my kind as for yours.\"\n\nThe captain was still pondering a smart remark when he realised that the Wyrm was no longer among them.\n\n_That was amazing,_ Harmodius said.\n\nThey lingered over breakfast.\n\n\"The marmalade is like\u2014\" Mag giggled, her mouth full of warm, crusty bread with rich new butter.\n\n\"Like God-made marmalade?\" asked Ser Alcaeus.\n\n\"I feel like a thief,\" Ranald said. He'd taken one of the swords from over the fireplace.\n\nTom took down the other. He grinned. \"God,\" he said, flicking his thumb over the blade. He gave a moan of pleasure as the blade he'd chosen swept through the air.\n\nThe Keeper shook his head. He had a box in his lap. \"I'm afraid to open it.\"\n\nSer Alcaeus rose and took down the sword hanging behind the main roof beam\u2014with a belt and scabbard. It matched his arms\u2014a surprisingly short sword with a heavy wheel pommel. \"These are things left for us. Indeed, unless I miss my guess, the whole cot is made for us.\"\n\n\"I'm not leaving until the marmalade is finished,\" Mag said, and laughed. She picked up her napkin to get the stickiness out of the corners of her mouth, and there was a chatelaine on the table beneath it\u2014gold and silver and enamel, with sharp steel scissors, a needle case full of needles, and a dozen other objects suspended on chains\u2014including a pair of keys.\n\n\"Oh,\" she said, and flushed, her hand to her bosom. \"Oh, par dieu. It is magnificent.\"\n\nGawin tried some of the marmalade. \"I had the most remarkable dream,\" he said. \"I wore a green belt\u2014\" He stumbled to silence. There was a green belt around his hips, worked in green enamel with gold plaques, and from it hung a heavy dagger in green and gold.\n\nThe captain stood under the roof beam, looking up at the spear.\n\n\"Just take it, man!\" said Tom.\n\nThe captain rubbed his chin. \"I'm not sure I want it,\" he said.\n\n_Take it! Take it!_ Harmodius couldn't control himself.\n\nFive feet of ancient blackthorn, knotty and yet straight as an arrow. And at the top, a long, heavy blade gleamed.\n\n\"Someone has taken the magister's staff, and fitted it like a glaive,\" the captain said.\n\n_Take it, you fool._\n\nThe captain rubbed his chin. \"I'm going to see to the horses.\"\n\n_So much of my power. Please? He wouldn't have brought it here unless he trusted us to use it._\n\n\"I can't help but notice that his gifts either bind, are pointed, or are double edged,\" said the captain. \"Belts and blades.\"\n\n_Don't be a fool._\n\n_Am I a fool to be slow to make use of tools I do not understand?_ asked the captain. _The stakes are very high. I will probably take it in the end. But not right now_ \u2014\n\nHe took his time currying the horses. They looked fat and happy. It had been a way of hiding from his father when he was young.\n\nWhen they were all gleaming like the sun on the water of the high loch outside, he went back into the cot\u2014so much bigger on the inside than the outside\u2014and took the spear down from its nails.\n\nIt was a heavy blackthorn shaft, but the butt was spiked in bronze and inlaid in gold, and the head was magnificently worked\u2014folded steel, carefully chiselled.\n\n_Oh. Empty._ Harmodius lost all interest in it. _Not mine at all._\n\nThe captain hefted it for a long time.\n\nThen he frowned and tucked it under his arm.\n\nOne by one they filed out of the cot. Mag left last, and closed the door behind her.\n\nShe looked puzzled. \"I thought it would... vanish,\" she said.\n\n\"He's not showy,\" the captain responded.\n\nThey all mounted, and rode over the ridges. In two ridges, the cot was gone, hidden in the folds of earth.\n\n\"If I ride back, will there be aught there?\" Tom asked.\n\nThe captain shrugged. \"Does it matter?\"\n\n\"You know what?\" Tom said. \"He reminded me of you. Only\u2014more so.\"\n\nHe laughed.\n\nThe captain raised an eyebrow. \"I think I'm flattered, Tom,\" he said.\n\nTom patted the sword at his side. \"I have a magic sword,\" he said happily. \"I want to go try it on something.\"\n\nRanald shook his head. \"Tom, you _hate_ magic.\"\n\nTom grinned. \"Och. You can teach an old dog a new trick, if ye are patient.\"\n\nGawin shook his head. \"Why us?\"\n\nThe captain shook his head.\n\nThey rode on.\n\nThe woodsmen were gone. There was no pile of bodies, no line of graves, no rusting tools. Merely gone.\n\nOver the Irkill a stone bridge stood on heavy pilings, as wide as two horsemen abreast or a single wagon, and on the other side sat a new keep\u2014a square tower\u2014with a small toll house.\n\nIt was solid, and smelled of new masonry. The Keeper sat in the road, looking at it.\n\n\"Open it,\" said the captain.\n\nThe Keeper looked at him.\n\n\"The box\u2014open it.\" The captain crossed his arms.\n\nThere was an anticlimactic moment as the Keeper rooted in his malle and emerged with his box. He opened it.\n\nThe box held a circlet, an arm ring, and a key.\n\nThe key fitted the door of the keep.\n\nThe circlet fit on his brow. He tried it and then snatched it off.\n\n\"Damn,\" he said.\n\n\"He's telling you something,\" said Ranald.\n\n\"The arm ring is for the drover,\" said the Keeper. \"I know it.\"\n\nRanald looked at it. \"Leave it lie, then,\" he said. \"I'll come back in spring, and we'll see.\"\n\nThey rode back to the inn.\n\nToby unpacked his master's portmanteau and appeared at his elbow. \"M'lord?\" he asked.\n\nThe captain was playing piquet with Maggie. He looked up.\n\n\"What do I do with these?\" he asked. He held up two velvet bags. They all but glowed a deep, dark red.\n\n\"Not mine,\" the captain said.\n\n\"Begging your pardon, m'lord, but they was in your bag.\" Toby held them out again.\n\nThe captain looked in one, and laughed. \"Why, Toby, I've just discovered our host was more thoughtful than I had imagined. Come here.\" He gestured to his new squire. \"I assume these are for you.\" He handed the bag over.\n\nIn it was a pair of silver spurs. Rich squires wore such things.\n\nToby gasped.\n\nThe captain shook his head. \"He knew we were coming, but we sent Toby back.\" He looked in the other bag. And frowned.\n\nA small, and very beautiful ring, gleamed in the bottom of the bag. It said \"IHS.\" \"Ah,\" he said. \"This is too much,\" he said quietly, and flung the bag across the room.\n\nIt bounced off the wall.\n\nHe went back to his cards.\n\nIn the morning, when he went to pay the Keeper, he found the ring among his coins.\n\n_Give it up_ said the magister. _He wants her, as well. You two are not done with each other, it seems._\n\nHe embraced the Keeper. \"Got anyone going west to Lissen Carak?\" he asked.\n\nThe Keeper grinned. \"In the autumn, maybe, and then only with twenty swords,\" he said.\n\nThe captain wrote a brief note on parchment. \"Send this, then.\" He wrapped the ring in the parchment. It gave him the oddest feeling.\n\n\"Go well, Captain,\" said the Keeper. \"Stop here when you come west for the tournament.\"\n\nThe captain raised his eyebrows.\n\n\"You are a famous knight,\" the Keeper said with his child-like delight in knowing news the others didn't know. \"The Queen has ordained that there will be a great tournament at Lorica, at Pentecost in the New Year.\"\n\nThe captain rolled his eyes. \"Not my kind of fight, Keeper.\"\n\nThe Keeper shrugged. \"So you say.\"\n\nThey spent five days riding over the mountains to Morea. They came down the pass north of Eva and the captain took them south and then east over the hills to Delf. He didn't seem to be in a hurry. Gawin and Alcaeus were of the same mind, and Tom and Ranald saw the whole trip as an adventure, riding high on the hillsides, searching out caves...\n\n\"Looking for a fight,\" Mag said in disgust. \"Can we get home?\"\n\n\"Home to our company of hired killers?\" said the captain.\n\nMag looked at him and shook her head. \"Yes,\" she said. \"If you must. Aren't you\u2014excited? Hopeful? Interested?\"\n\nHe was watching the two hillmen ranging high above them. Alcaeus had purchased a good goshawk from a peddler and was flying him at doves. Gawin was riding ahead, feet crossed over the pintle of his saddle, reading.\n\nHe shook his head. \"Not really,\" he said. \"I think I've just been enlisted by one mighty Power to fight another in a war not of my making, over things I don't understand.\" He rubbed his chin. \"I swore off being a tool when I was a child.\"\n\n\"The Wyrm is good.\" Mag put a hand on his arm. \"I can feel it.\"\n\nThe captain shook his head. \"Mag, what do my thoughts of good and evil mean to the worms in the road? I can be the most honourable knight who ever lived, and my horse's iron-shod hooves will crush their soft bodies every step, after a rain.\" He smiled at her. \"And I won't even know.\"\n\nDown in the deep valley ahead of them, he could see rows of tents; a palisade; neat circles of heavy wagons, and over all, a banner, black, with lacs d'or worked in gold.\n\n\"Damn you,\" she said. \"Why can't we just act? Why can't we simply win?\"\n\nThe captain sighed. \"Men love war because it is simple,\" he said. \"Winning is never simple. I can win a fight\u2014together, we can win a battle.\" He rubbed his beard. Down in the valley, men were pointing and messengers were mounting horses. \"But turning victory in battle into something that lasts is like building a place to live. So much more complicated than building a fortress.\"\n\nHe pointed at the riders. \"Luckily for me, those men are bringing me word of our contract. A nice little war.\" He forced a smile. \"Something we can _win._ \"\n\n# Harndon City\u2014Edward\n\nEdward finished his first rondel dagger\u2014a fine weapon with a precise triangular blade and an armour-piercing point\u2014and handed it to Master Pyle with trepidation. The older man looked it over, balanced it on the back of his hand, and threw it at the floor, where it stuck with a satisfying _thunk_.\n\n\"Very nice,\" he said. \"Hand it to Danny to be hilted. I'll have a project for you in a few days\u2014until then, cover the shop.\"\n\nWell\u2014shop work was clean and dull, but Edward was courting his Anne in the long summer evenings, and shop work allowed him to dress well\u2014fine hose, a good doublet, not shop-worn linen stained in nameless chemicals and burned with a thousand sparks.\n\nAnne was a seamstress, and her hands were always clean.\n\nMost evenings she would dance in the square by her house, and Edward would swagger his sword and buckler against other journeymen\u2014he was becoming a good blade.\n\nHe was designing himself a fine buckler\u2014sketching in a sure hand with charcoal\u2014when the shop door opened and a small man came in. He was middling. And not very memorable.\n\nHe smiled at Edward. He had odd black eyes, and he tapped a gold coin on the heavy oak table where customers examined the wares. \"Fetch me your master, young man,\" he said.\n\nEdward nodded. He rang a bell for another shop boy and sent him to the yard, and Master Pyle appeared a few minutes later. The dark-eyed man had spent the time looking out the window. Edward couldn't tear his eyes away, because the man was so very _difficult_ to look at.\n\nHe turned just a moment before the master appeared, and met him at the counter.\n\n\"Master Pyle,\" he said. \"I sent you some letters.\"\n\nMaster Pyle looked puzzled. Then he brightened. \"Master Smith?\"\n\n\"The very same,\" said the odd man. \"Did you try my powder?\"\n\n\"I did. Scary stuff, and no mistake. Shot a hole in the roof of my shed.\" Master Pyle raised an eyebrow. \"Not very consistent, though.\"\n\nThe man's dark eyes sparkled. \"Mmm. Well, perhaps I didn't explain entirely. Try wetting it with urine after you've mixed it. Dry it in the sun\u2014far from fire, of course. And then grind it back to coarse powder, very carefully.\"\n\n\"If I was an alchemist, all this might entertain me, Master Smith. But I'm a blade smith, and I have many orders.\"\n\nMaster Smith appeared confused. \"You make weapons, though.\"\n\n\"All kinds.\" Master Pyle nodded.\n\n\"The very best in Alba, I've been told,\" Master Smith said.\n\nMaster Pyle smiled. \"I hope so.\"\n\nMaster Smith rocked his head back and forth. \"Is this a matter of more money?\" he asked.\n\n\"I'm afraid not.\" Master Pyle shook his head. \"It's just not my trade.\"\n\nSmith let out a sigh. \"Why not?\"\n\nEdward looked at Master Pyle very hard, willing him to turn his head.\n\n\"I have more orders than I can manage, and this is very untested.\" Master Pyle shrugged. \"It would take months, perhaps years, to perfect.\"\n\nSmith shrugged. \"So?\"\n\nEdward was all but hopping up and down. Master Pyle turned his head and glared at him. But it wasn't his hard glare.\n\n\"This is my journeyman, Edward. He made both of the test devices. He's very competent, and perhaps he'd be willing to do the work for you.\" Master Pyle looked at Edward. \"Want to try, Edward? Your own commission?\"\n\nEdward beamed.\n\nThe odd, dark-eyed man rocked his head again. \"Excellent, then.\" He put two sheets of vellum down on the shop counter. \"Have a look at these, and see what you think,\" he said. \"Tube, stock, powder, and match. I want you to make them all.\"\n\n\"Just one?\" Edward asked. \"Delivered where?\"\n\n\"Oh, as to that, I'll send you my directions. It is for some friends.\" He laughed. \"Just one, and then you destroy all your notes. Or I will find you. Understand?\"\n\nEdward looked at the man. He didn't seem very dangerous. And yet, he did. And just for a moment, he seemed to have scales on the backs of his hands.\n\n\"How much?\" Edward asked carefully. \"Do I get paid?\"\n\n\"Absolutely,\" the strange man said. \"Fifty gold nobles in advance. Fifty more on completion.\"\n\nEdward had to struggle to breathe.\n\nMaster Pyle shook his head. \"I'll get a notary.\"\n\n# Harndon Palace\u2014The King\n\nJust above them, in the great fortress of Harndon, Master Pyle's friend the king lay with his wife. He had two new scars on his heavily muscled thighs. She had one on her back.\n\nNeither found the other a whit less fascinating.\n\nWhen the king had done his thorough worship of her, he licked her leg and bit her gently and rose. \"Men will mock me,\" he said. \"A king who loves only his wife.\"\n\nShe laughed. Stretched like a cat, fists clenched and turned inward to the best advantage of her breasts and back. \"I,\" she purred, \"beg leave to doubt your Majesty.\"\n\nHe laughed and threw himself back down by her like a much younger man. \"I love you,\" he said.\n\nShe rolled atop him and kissed him. \"And I you, my lord.\"\n\nThey lay for a while in companionable silence, until royal squires in the hall started to make noises that indicated that they had royal work to do.\n\n\"I have set the date for your tournament at Lorica,\" the king said. He knew how much she wanted it. \"It will help\u2014after the battle. After Pentecost next.\"\n\nShe took in a deep breath, also to her advantage, and clapped her hands together.\n\n\"And I ordered Master Pyle to build two of your military carts with the Wagoner's Guild,\" he said. \"To test the concept. I'll show them at the tournament. Ask men with retinues to build to the pattern.\" He shrugged. \"It will be a start.\"\n\n\"And the Red Knight?\" she said.\n\nHe reacted as if he'd been stung.\n\nShe shook her head. \"His company had standard wagons, built to the purpose in Galle.\" She dimpled. \"So I didn't invent the idea, apparently.\"\n\nHe shook his head. \"I hadn't noticed.\"\n\nShe shrugged, again to her advantage.\n\n\"If you don't get dressed, the new ambassador from the Emperor will find me a _most_ tardy host.\" He reached for her.\n\n\"I've taken the liberty of inviting him to the tournament,\" the Queen said. She watched the king like a hawk.\n\nHe didn't flinch.\n\n\"Ah,\" he said.\n\n# Morea\u2014The Red Knight\n\nThe camp was snug on the late summer evening. And the return had been enough like a homecoming to make him cry. He smiled a great deal, and rode through the camp.\n\nGelfred was sitting on a wagon, feeding\u2014\n\n\"Goodness gracious, Gelfred! Do we have Parcival?\" The captain slid down from his riding horse and shocked his hunt master with an embrace.\n\nThe eagle bated and said _squaaack._\n\nGelfred nodded. \"Wonderful bird.\" He looked around. \"Not quite right, of course. Neither you, nor, pardon me, the Abbess is a king. Or queen.\" He grimaced.\n\nThe captain gave him a quick nod. \"We'll ask the Emperor for a special chrysobull, shall we?\" he laughed. \"Although, to be honest, I'm pretty sure the Abbess almost was the Queen.\"\n\nGelfred looked shocked.\n\nSer Alcaeus nodded. \"I suspected the same.\"\n\nSer Gawin looked at the captain. \"I'll be the slow brother. What are we talking about?\"\n\nIn the captain's head, Harmodius laughed. A nasty, gossipy laugh. _So! You did see who she was._\n\n\"The old king's mistress, Gelfred. That's what men called her. Sophia Rae. To whom Hawthor the Great offered marriage after the Battle of Chevin, and was refused.\" The captain smiled. \"Imagine having been Hawthor's lover and Richard Plangere's _at the same time._ \" He shook his head. \"And then an Abbess for thirty years.\" He reached out and smoothed the bird's plumage. \"Hawthor must have given her the bird. He must be quite ancient.\"\n\nThe bird's eyes were fathomless and gold, with a black centre.\n\n\"I've heard of them living fifty years,\" said Gelfred.\n\nThe bird's grumpy eyes locked with the captain's.\n\n\"I see,\" he said.\n\nMag sat with Johne the Bailli in the last of the light. He had camp stools\u2014comfortable enough, but backless, and she wasn't getting any younger. He was watching the stars.\n\n\"I see a lot of unfamiliar faces,\" she said, watching two men-at-arms go by. They paused in the light of Johne's lanterns, gave her an appraising look, and bowed.\n\n\"We did some recruiting,\" he admitted. He ran a hand down her back. Turned his head, and smiled. \"All right, they all but attacked us. As soon as we made camp\u2014every younger son in the North Country. Some Moreans, too. By the Saviour I would expect we have a hundred lances.\"\n\nShe sighed. \"So many,\" she said.\n\nHe sat back. \"Won't he be pleased? The young captain?\"\n\nShe leaned over and kissed him gently. \"I'm a sinful old woman, and I don't need to be seduced, if that's what your hand is supposed to be doing.\"\n\nHe stiffened, but then grinned. \"My lady, I am out of practice.\"\n\nThey didn't talk much, for a moment.\n\n\"Am I clumsy?\" he murmured.\n\n\"No,\" she said. She was thinking of blowing out the lanterns and lying on the carpet shamelessly. \"No,\" she said.\n\n\"What then?\" he asked.\n\nShe made a dismissive gesture and went to blow out the candles.\n\n\"You can tell me,\" he said.\n\n\"I'm just thinking of the captain. Of him being pleased.\" She shrugged. \"You all think he's fine, and he is _not._ He's like a horse that's taken a wound, and keeps going. He looks fine, right up until he falls stone dead.\" She found she was leaning back into him.\n\nHe held onto her. \"When I was young, I wanted nothing so much as to be a knight,\" he said. \"I wanted it, and I fought for it. And I did not get it. And after more time and some bad things, I met your husband, and we survived a bad time. And then I became a decent man in a small town. I had some dark days and some good days.\" He shrugged. \"And now\u2014par dieu, now it seems that I may get to be a knight. And I may have you, my lady.\" He held her tight. \"Which is by way of saying\u2014our little captain will take many hurts. If they break him?\" he shrugged. \"Then they do. That is the way of it.\"\n\nShe nodded. And slipped a little closer to the carpet of their tent.\n\nThe captain sat with Ser Alcaeus and his brother in the last light. The great eagle sat on a perch in the shaded end of the tent, head muffled, squawking softly. The captain went and petted the bird and calmed him, and while he was doing so, Toby poured him wine. Ser Jehannes knocked at the captain's tent poles.\n\n\"Come,\" said the captain.\n\nSer Jehannes had Ser Thomas and Ser Antigone, and Toby poured them all wine. In the distance, Oak Pew slammed a fist into Wilful Murder's head. The archer sat suddenly. The captain shook his head.\n\n\"It's good to be home,\" he said.\n\nJehannes held out a leather wallet. \"I know this is supposed to be a night to revel,\" he said. \"But the messengers who brought these have been like bluebottles on horse manure, m'lord. Dispatches and letters,\" he said. He grimaced. \"Most for our well-born recruit here.\" He motioned at Alcaeus. \"Your uncle seems determined to hear from you.\"\n\n\"Your pardon,\" Alcaeus said, and broke the seal on a scroll tube of dark wood.\n\nWhile he did so, Jehannes handed an ivory tube to the captain. He glanced at the seal and smiled.\n\n\"The Queen, gentlemen.\"\n\nThey all drank. Even Sauce.\n\nHe broke the seal while Alcaeus was still reading.\n\nAlcaeus looked up. \"M'lord,\" he said formally. \"The situation has worsened. I must ask, in the Emperor's name, that we ride with all dispatch.\"\n\nThe captain was still reading his own. \"Relax, gentles,\" he said. \"We aren't riding anywhere tonight.\"\n\nAlcaeus looked at white as a sheet. \"The Emperor has been\u2014taken. Hostage. A week and more ago.\"\n\nThe captain looked up and fingered his beard. \"All right. That _does_ constitute a crisis. Tom?\"\n\n\"Ready to ride at first light it is.\" Tom grinned. \"Never a dull moment.\"\n\n\"We live in interesting times,\" the captain said. \"Everyone get sleep. We will be moving fast. May I assume this is part of the same\u2014er\u2014trouble for which your uncle is hiring us?\"\n\nAlcaeus shook his head. \"I don't know.\" He shuffled. \"I don't even know if he is alive, or still Emperor.\"\n\nThe captain nodded. \"Dawn, then,\" he said. \"We'll pick up information as we go.\"\n\nJehannes looked at the other parchment. \"And the Queen?\"\n\nThe captain sighed. \"An invitation to a Deed of Arms,\" he said. \"In the spring.\" He smiled. He looked out into the darkness. He was smiling. \"Someone has kidnapped the Emperor, and we are going to be called on to save him,\" he said quietly. \"I think we'll have to miss the tournament.\"\n\nHe looked around the table. \"Remember this night, friends. Breathe the air, and savour the wine. Because tonight, it's all in the balance. I can feel it.\"\n\n\"What is?\" Sauce asked. She raised an eyebrow at Tom, as if to say _Is he drunk?_\n\n\"Everything,\" the captain said. He laughed aloud. \"Everything.\"\n\n# Acknowledgements\n\nThis book is the culmination of thirty ears of study, chivalric martial arts, real life, and role-playing. To be fair to all my influences, I'd have to thank everyone I've ever known. There's a Somali man who worked for me in Kenya in this book; a woman I met once in Marseille; a chivalric fighter I sparred with at a tournament a few years back\u2014it's like that.\n\nBut several groups of people deserve my special thanks.\n\nFirst, the friends of my days in university. Joe and Regina Harley, Robert Sulentic, Robert Gallasch, Gail Morse, Celia Friedman, Steven Callahan, Jevon Garrett, and another dozen\u2014who played in the original Alba campaign. I am an unashamed nerd. Without you people, there would be little life on these bones.\n\nSecond, the friends of my reenactment hobby\u2014most especially those who attend our yearly historic trek, where we wander off into the Adirondacks with eighteenth century equipment\u2014or fourteenth century equipment\u2014to learn what it is like to live with the past. We pack it in on our backs and we go places that\u2014in some instances\u2014no person has been in fifty years. These experiences have helped me write this book and I owe you all a debt of thanks for putting up with me. And all the people with whom I spar, in and out of armour\u2014here, and in Ottawa and in Finland and Greece.\n\nThird, the craftsmen who recreate the items that make history and fantasy come alive. Leo Todeschini of www.todsstuff.co.uk deserves a visit online\u2014his stuff is incredible. Magical, even. Ben Perkins at www.barebowarchery.co.uk makes long bows and war bows that look and behave like the originals, as far as we know. Mark Vickers at www.stgeorgearmouryshop.co.uk and Peter Fuller at www.medievalrepro.com reproduce armour that is as near exactly like originals as makes little difference. Comfortable, too. I wear it quite often. www.albion-swords.com make superb, non-nonsense swords. They are not \"like\" the real thing. They are the real thing. Visit my website and you can see a dozen more craftsmen every bit as good.\n\nFourth, the teachers who taught me about history, about life and philosophy, about weapons, and about chivalry; Dick Kaeuper of the University of Rochester; Father William O'Malley, SJ, who may or may not forgive my theology; Guy Windsor, possibly the world's finest swordsman (he runs a school!) and Ridgeley Davis who taught me to be a much better rider. And to use a spear on horseback.\n\nFifth, the many people who have helped me in the publishing world; my Agent, Shelley Power; my gallant publicist, Donna Nopper, and most of all on this book, Gillian Redfearn, who gets credit in every step from creation to actual editing.\n\nAnd last, the other teachers\u2014the hundreds, if not thousands, of writers who inspire me to write. Medieval fishing? Theology? Hermeticism? Memory palaces? Jousting? Singing Neanderthals? Neurology? Ancient Greek philosophy? I owe a debt to the authors of hundreds of books for filling in the gaps in experience, or just teaching me a dying or dead craft.\n\nAnd, of course, there's fantasy itself. I adore\u2014nay, worship\u2014J. R. R. Tolkien. For my taste, it is not just _The Lord of the Rings_ or _The Hobbit,_ but _Sir Gawain and the Green Knight,_ from which I have borrowed shamelessly. And C. S. Lewis and their lesser known contemporary, E. R. Eddison. I will not claim Eddison is the best of the lot, but I will confess that my idea of what fantasy ought to do owes a great deal to Eddison, and to William Morris. Does anyone still read William Morris? Have a go. _The Sundering Flood_ is one of my favorite books, not least because I share Morris's love for the crafts and the material culture. More recently, I love Celia Friedman, Glen Cook, Katherine Kurtz and Steven Erikson. My hat is off to Erikson\u2013I think he did the most magnificent job of plotting in our generation. And C. J. Cherryh and Lois McMaster Bujold. I don't think either has ever written a book that I didn't enjoy.\n\nI could go on. But I have to work on book two\u2014 _The Fell Sword_. If you want to know more, visit my website at www.traitorson.com. And if you want to wear armour in the Wild...\n\nWell, we'll see if we can accommodate you.\n\n_Miles Cameron_\n\n_August, 2012_\n\n# extras\n\n# meet the author\n\nMILES CAMERON is a full-time writer who lives in Canada with his family. He also writes historical fiction under another name. THE RED KNIGHT is his fantasy debut.\n\n# introducing\n\nIf you enjoyed\n\nTHE RED KNIGHT,\n\nlook out for\n\n**PROMISE OF BLOOD**\n\nThe Powder Mage Trilogy: Book One\n\n_by Brian McClellan_\n\n_Field Marshal Tamas's coup against his king sent corrupt aristocrats to the guillotine and brought bread to the starving. But it also provoked war with the Nine Nations, internal attacks by royalist fanatics, and the greedy to scramble for money and power by Tamas's supposed allies: the Church, workers unions, and mercenary forces. Stretched to his limit, Tamas is relying heavily on his few remaining powder mages, including the embittered Taniel, a brilliant marksman who also happens to be his estranged son, and Adamat, a retired police inspector whose loyalty is being tested by blackmail._\n\n_Now, as attacks batter them from within and without, the credulous are whispering about omens of deathand destruction. Just old peasant legends about the gods waking to walk the earth. No modern educated man believes that sort of thing. But they should_...\n\nAdamat wore his coat tight, top buttons fastened against a wet night air that seemed to want to drown him. He tugged at his sleeves, trying to coax more length, and picked at the front of the jacket where it was too close by far around the waist. It'd been half a decade since he'd even seen this jacket, but when summons came from the king at this hour, there was no time to get his good one from the tailor. Yet this summer coat provided no defense against the chill snaking through the carriage window.\n\nThe morning was not far off but dawn would have a hard time scattering the fog. Adamat could feel it. It was humid even for early spring in Adopest, and chillier than Novi's frozen toes. The soothsayers in Noman's Alley said it was a bad omen. Yet who listened to soothsayers these days? Adamat reasoned it would give him a cold and wondered why he had been summoned out on a pit-made night like this.\n\nThe carriage approached the front gate of Skyline and moved on without a stop. Adamat clutched at his pantlegs and peered out the window. The guards were not at their posts. Odder still, as they continued along the wide path amid the fountains, there were no lights. Skyline had so many lanterns, it could be seen all the way from the city even on the cloudiest night. Tonight the gardens were dark.\n\nAdamat was fine with this. Manhouch used enough of their taxes for his personal amusement. Adamat stared out into the gardens at the black maws where the hedge mazes began and imagined shapes flitting back and forth in the lawn. What was... ah, just a sculpture. Adamat sat back, took a deep breath. He could hear his heart beating, thumping, frightened, his stomach tightening. Perhaps they _should_ light the garden lanterns...\n\nA little part of him, the part that had once been a police inspector, prowling nights such as these for the thieves and pickpockets in dark alleys, laughed out from inside. _Still your heart, old man_ , he said to himself. _You were once the eyes staring back from the darkness._\n\nThe carriage jerked to a stop. Adamat waited for the coachman to open the door. He might have waited all night. The driver rapped on the roof. \"You're here,\" a gruff voice said.\n\nRude.\n\nAdamat stepped from the coach, just having time to snatch his hat and cane before the driver flicked the reins and was off, clattering into the night. Adamat uttered a quiet curse after the man and turned around, looking up at Skyline.\n\nThe nobility called Skyline Palace \"the Jewel of Adro.\" It rested on a high hill east of Adopest so that the sun rose above it every morning. One particularly bold newspaper had compared it to a starving pauper wearing a diamond ring. It was an apt comparison in these lean times. A king's pride doesn't fill the people's bellies.\n\nHe was at the main entrance. By day, it was a grand avenue of marbled walks and fountains, all leading to a pair of giant, silver-plated doors, themselves dwarfed by the sheer fa\u00e7ade of the biggest single building in Adro. Adamat listened for the soft footfalls of patrolling Hielmen. It was said the king's personal guard were everywhere in these gardens, watching every secluded corner, muskets always loaded, bayonets fixed, their gray-and-white sashes somber among the green-and-gold splendor. But there were no footfalls, nor were the fountains running. He'd heard once that the fountains only stopped for the death of the king. Surely he'd not have been summoned here if Manhouch were dead. He smoothed the front of his jacket. Here, next to the building, a few of the lanterns were lit.\n\nA figure emerged from the darkness. Adamat tightened his grip on his cane, ready to draw the hidden sword inside at a moment's notice.\n\nIt was a man in uniform, but little could be discerned in such ill light. He held a rifle or a musket, trained loosely on Adamat, and wore a flat-topped forage cap with a stiff visor. Only one thing could be certain... he was not a Hielman. Their tall, plumed hats were easy to recognize, and they never went without them.\n\n\"You're alone?\" a voice asked.\n\n\"Yes,\" Adamat said. He held up both hands and turned around.\n\n\"All right. Come on.\"\n\nThe soldier edged forward and yanked on one of the mighty silver doors. It rolled outward slowly, ponderously, despite the man putting his weight into it. Adamat moved closer and examined the soldier's jacket. It was dark blue with silver braiding. Adran military. In theory, the military reported to the king. In practice, one man held their leash: Field Marshal Tamas.\n\n\"Step back, friend,\" the soldier said. There was a note of impatience in his voice, some unseen stress\u2014but that could have been the weight of the door. Adamat did as he was told, only coming forward again to slip through the entrance when the soldier gestured.\n\n\"Go ahead,\" the soldier directed. \"Take a right at the diadem and head through the Diamond Hall. Keep walking until you find yourself in the Answering Room.\" The door inched shut behind him and closed with a muffled thump.\n\nAdamat was alone in the palace vestibule. Adran military, he mused. Why would a soldier be here, on the grounds, without any sign of the Hielmen? The most frightening answer sprang to mind first. A power struggle. Had the military been called in to deal with a rebellion? There were a number of powerful factions within Adro: the Wings of Adom mercenaries, the royal cabal, the Mountainwatch, and the great noble families. Any one of them could have been giving Manhouch trouble. None of it made sense, though. If there had been a power struggle, the palace grounds would be a battlefield, or destroyed outright by the royal cabal.\n\nAdamat passed the diadem\u2014a giant facsimile of the Adran crown\u2014and noted it was in as bad taste as rumor had it. He entered the Diamond Hall, where the walls and floor were of scarlet, accented in gold leaf, and thousands of tiny gems, which gave the room its name, glittered from the ceiling in the light of a single lit candelabra. The tiny flames of the candelabra flickered as if in the wind, and the room was cold.\n\nAdamat's sense of unease deepened as he neared the far end of the gallery. Not a sign of life, and the only sound came from his own echoing footfalls on the marble floor. A window had been shattered, explaining the chill. The result of one of the king's famous temper tantrums? Or something else? He could hear his heart beating in his ears. There. Behind a curtain, a pair of boots? Adamat passed his hand before his eyes. A trick of the light. He stepped over to reassure himself and pulled back the curtain.\n\nA body lay in the shadows. Adamat bent over it, touched the skin. It was warm, but the man was most certainly dead. He wore gray pants with a white stripe down the side and a matching jacket. A tall hat with a white plume lay on the floor some ways away. A Hielman. The shadows played on a young, clean-shaven face, peaceful except for a single hole in the side of his skull and the dark, wet stain on the floor.\n\nHe'd been right. A struggle of some kind. Had the Hielmen rebelled, and the military been brought in to deal with them? Again, it didn't make any sense. The Hielmen were fanatically loyal to the king, and any matters within Skyline Palace would have been dealt with by the royal cabal.\n\nAdamat cursed silently. Every question compounded itself. He suspected he'd find some answers soon enough. Adamat left the body behind the curtain. He lifted his cane and twisted, bared a few inches of steel, and approached a tall doorway flanked by two hooded, scepter-wielding sculptures. He paused between the ancient statues and took a deep breath, letting his eyes wander over a set of arcane script scrawled into the portal. He entered.\n\nThe Answering Room made the Hall of Diamonds look small. A pair of staircases, one to either side of him and each as wide across as three coaches, led to a high gallery that ran the length of the room on both sides. Few outside the king and his cabal of Privileged sorcerers ever entered this room.\n\nIn the center of the room was a single chair, on a dais a handbreadth off the floor, facing a collection of knee pillows, where the cabal acknowledged their liege. The room was well lit, though from no discernible source of light.\n\nA man sat on the stairs to Adamat's right. He was older than Adamat, just into his sixtieth year with silver hair and a neatly trimmed mustache that still retained a hint of black. He had a strong but not overly large jaw and his cheekbones were well defined. His skin was darkened by the sun, and there were deep lines at the corners of his mouth and eyes. He wore a dark-blue soldier's uniform with a silver representation of a powder keg pinned above the heart and nine gold service stripes sewn on the right breast, one for every five years in the Adran military. His uniform lacked an officer's epaulettes, but the weary experience in the man's brown eyes left no question that he'd led armies on the battlefield. There was a single pistol, hammer cocked, on the stair next to him. He leaned on a sheathed small sword and watched as a stream of blood slowly trickled down each step, a dark line on the yellow-and-white marble.\n\n\"Field Marshal Tamas,\" Adamat said. He sheathed his cane sword and twisted until it clicked shut.\n\nThe man looked up. \"I don't believe we've ever met.\"\n\n\"We have,\" Adamat said. \"Fourteen years ago. A charity ball thrown by Lord Aumen.\"\n\n\"I have a terrible time with faces,\" the field marshal said. \"I apologize.\"\n\nAdamat couldn't take his eyes off the rivulet of blood. \"Sir. I was summoned here. I wasn't told by whom, or for what reason.\"\n\n\"Yes,\" Tamas said. \"I summoned you. On the recommendation of one of my Marked. Cenka. He said you served together on the police force in the twelfth district.\"\n\nAdamat pictured Cenka in his mind. He was a short man with an unruly beard and a penchant for wines and fine food. He'd seen him last seven years ago. \"I didn't know he was a powder mage.\"\n\n\"We try to find anyone with an affinity for it as soon as possible,\" Tamas said, \"but Cenka was a late bloomer. \"In any case\"\u2014he waved a hand\u2014\"we've come upon a problem.\"\n\nAdamat blinked. \"You... want my help?\"\n\nThe field marshal raised an eyebrow. \"Is that such an unusual request? You were once a fine police investigator, a good servant of Adro, and Cenka tells me that you have a perfect memory.\"\n\n\"Still, sir.\"\n\n\"Eh?\"\n\n\"I'm still an investigator. Not with the police, sir, but I still take jobs.\"\n\n\"Excellent. Then it's not so odd for me to seek your services?\"\n\n\"Well, no,\" Adamat said, \"but sir, this is Skyline Palace. There's a dead Hielman in the Diamond Hall and...\" He pointed at the stream of blood on the stairs. \"Where's the king?\"\n\nTamas tilted his head to the side. \"He's locked himself in the chapel.\"\n\n\"You've staged a coup.\" Adamat said. He caught a glimpse of movement with the corner of his eye, saw a soldier appear at the top of the stairs. The man was a Deliv, a dark-skinned northerner. He wore the same uniform as Tamas, with eight golden stripes on the right breast. The left breast of his uniform displayed a silver powder keg, the sign of a Marked. Another powder mage.\n\n\"We have a lot of bodies to move,\" the Deliv said.\n\nTamas gave his subordinate a glance. \"I know, Sabon.\"\n\n\"Who's this?\" Sabon asked.\n\n\"The inspector that Cenka requested.\"\n\n\"I don't like him being here,\" Sabon said. \"It could compromise everything.\"\n\n\"Cenka trusted him.\"\n\n\"You've staged a coup,\" Adamat said again with certainty.\n\n\"I'll help with the bodies in a moment,\" Tamas said. \"I'm old, I need some rest now and then.\" The Deliv gave a sharp nod and disappeared.\n\n\"Sir!\" Adamat said. \"What have you done?\" He tightened his grip on his cane sword.\n\nTamas pursed his lips. \"Some say the Adran royal cabal had the most powerful Privileged sorcerers in all the Nine Nations, second only to Kez,\" he said quietly. \"Yet I've just slaughtered every one of them. Do you think I'd have trouble with an old inspector and his cane sword?\"\n\nAdamat loosened his grip. He felt ill. \"I suppose not.\"\n\n\"Cenka led me to believe that you were pragmatic. If that is the case, I would like to employ your services. If not, I'll kill you now and look for a solution elsewhere.\"\n\n\"You've staged a coup,\" Adamat said again.\n\nTamas sighed. \"Must we keep coming back to that? Is it so shocking? Tell me, can you think of any fewer than a dozen factions within Adro with reason to dethrone the king?\"\n\n\"I didn't think any of them had the skill,\" Adamat said. \"Or the daring.\" His eyes returned to the blood on the stairs, before his mind traveled to his wife and children, asleep in their beds. He looked at the field marshal. His hair was tousled; there were drops of blood on his jacket\u2014a lot, now that he thought to look. Tamas might as well have been sprayed with it. There were dark circles under his eyes and a weariness that spoke of more than just age.\n\n\"I will not agree to a job blindly,\" Adamat said. \"Tell me what you want.\"\n\n\"We killed them in their sleep,\" Tamas said without preamble. \"There's no easy way to kill a Privileged, but that's the best. A mistake was made and we had a fight on our hands.\" Tamas looked pained for a moment, and Adamat suspected that the fight had not gone as well as Tamas would have liked. \"We prevailed. Yet upon the lips of the dying was one phrase.\"\n\nAdamat waited.\n\n\" 'You can't break Kresimir's Promise,' \" Tamas said. \"That's what the dying sorcerers said to me. Does it mean anything to you?\"\n\nAdamat smoothed the front of his coat and sought to recall old memories. \"No. 'Kresimir's Promise'... 'Break'... 'Broken'... Wait\u2014'Kresimir's Broken Promise.' \" He looked up. \"It was the name of a street gang. Twenty... twenty-two years ago. Cenka couldn't remember that?\"\n\nTamas continued. \"Cenka thought it sounded familiar. He was certain you'd remember it.\"\n\n\"I don't forget things,\" Adamat said. \"Kresimir's Broken Promise was a street gang with forty-three members. They were all young, some of them no more than children, the oldest not yet twenty. We were trying to round up some of the leaders to put a stop to a string of thefts. They were an odd lot\u2014they broke into churches and robbed priests.\"\n\n\"What happened to them?\"\n\nAdamat couldn't help but look at the blood on the stairs. \"One day they disappeared, every one of them\u2014including our informants. We found the whole lot a few days later, forty-three bodies jammed into a drain culvert like pickled pigs' feet. They'd been massacred by powerful sorceries, with excessive brutality. The marks of the king's royal cabal. The investigation ended there.\" Adamat suppressed a shiver. He'd not once seen a thing like that, not before or since. He'd witnessed executions and riots and murder scenes that filled him with less dread.\n\nThe Deliv soldier appeared again at the top of the stairs. \"We need you,\" he said to Tamas.\n\n\"Find out why these mages would utter those words with their final breath,\" Tamas said. \"It may be connected to your street gang. Maybe not. Either way, find me an answer. I don't like the riddles of the dead.\" He got to his feet quickly, moving like a man twenty years younger, and jogged up the stairs after the Deliv. His boot splashed in the blood, leaving behind red prints. \"Also,\" he called over his shoulder, \"Keep silent about what you have seen here until the execution. It will begin at noon.\"\n\n\"But...\" Adamat said. \"Where do I start? Can I speak with Cenka?\"\n\nTamas paused near the top of the stairs and turned. \"If you can speak with the dead, you're welcome to.\"\n\nAdamat ground his teeth. \"How did they say the words?\" he said. \"Was it a command, or a statement, or...?\"\n\nTamas frowned. \"An entreaty. As if the blood draining from their bodies was not their primary concern. I must go now.\"\n\n\"One more thing,\" Adamat said.\n\nTamas looked to be near the end of his patience.\n\n\"If I'm to help you, tell me why all of this?\" he gestured to the blood on the stairs.\n\n\"I have things that require my attention,\" Tamas warned.\n\nAdamat felt his jaw tighten. \"Did you do this for power?\"\n\n\"I did this for me,\" Tamas said. \"And I did this for Adro. So that Manhouch wouldn't sign us all into slavery to the Kez with the Accords. I did it because those grumbling students of philosophy at the university only play at rebellion. The age of kings is dead, Adamat, and I have killed it.\"\n\nAdamat examined Tamas's face. The Accords was a treaty to be signed with the king of Kez that would absolve all Adran debt but impose strict tax and regulation on Adro, making it little more than a Kez vassal. The field marshal had been outspoken about the Accords. But then, that was expected. The Kez had executed Tamas's late wife.\n\n\"It is,\" Adamat said.\n\n\"Then get me some bloody answers.\" The field marshal whirled and disappeared into the hallway above.\n\nAdamat remembered the bodies of that street gang as they were being pulled from the drain in the wet and mud, remembered the horror etched upon their dead faces. _The answers may very well be bloody._\n\n# BY MILES CAMERON\n\nThe Traitor Son Cycle\n\n_The Red Knight_\n\n### Thank you for buying this e-book, published by Hachette Digital.\n\nTo receive special offers, bonus content, and news about our latest e-books and apps, sign up for our newsletter.\n\n Sign Up\n\nOr visit us at hachettebookgroup.com\/newsletters\n\n# Contents\n\nWelcome\n\nDedication\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\n\nChapter Sixteen\n\nChapter Seventeen\n\nChapter Eighteen\n\nAcknowledgements\n\nExtras\n\nMeet the Author\n\nA Preview of _Promise of Blood_\n\nBy Miles Cameron\n\nNewsletters\n\nCopyright\n\n# Copyright\n\nThe characters and events in this book are fictitious. Any similarity to real persons, living or dead, is coincidental and not intended by the author.\n\nCopyright \u00a9 2012 by Miles Cameron\n\nExcerpt from _Promise of Blood_ copyright \u00a9 2013 by Brian McClellan\n\nAll rights reserved. In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher constitute unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nOrbit\n\nHachette Book Group\n\n237 Park Avenue, New York, NY 10017\n\nwww.orbitbooks.net\n\norbitshortfiction.com\n\nFirst e-book edition: January 2013\n\nOrbit is an imprint of Hachette Book Group, Inc. The Orbit name and logo are trademarks of Little, Brown Book Group Limited.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to www.hachettespeakersbureau.com or call (866) 376-6591.\n\nISBN 978-0-316-21229-8\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nPublished by American Palate\n\nA Division of The History Press\n\nCharleston, SC 29403\n\nwww.historypress.net\n\nCopyright \u00a9 2014 by Steven Shomler\n\nAll rights reserved\n\nAll photos by author unless otherwise noted.\n\nFirst published 2014\n\ne-book edition 2014\n\nISBN 978.1.62584.866.6\n\nLibrary of Congress Cataloging-in-Publication Data\n\nShomler, Steven, author.\n\nPortland food cart stories : behind the scenes with the city's culinary entrepreneurs \/ Steven Shomler.\n\npages cm -- (American palate)\n\nSummary: \"A survey of Portland's food cart scene\"-- Provided by publisher.\n\nprint edition ISBN 978-1-62619-373-4 (paperback)\n\n1. Food trucks--Oregon--Portland--Guidebooks 2. Cooking--Oregon--Portland--Guidebooks. 3. Portland (Or.)--Guidebooks. I. Title.\n\nTX907.3.O72P677 2014\n\n641.59795'49--dc 3\n\n2014008348\n\n_Notice_ : The information in this book is true and complete to the best of our knowledge. It is offered without guarantee on the part of the author or The History Press. The author and The History Press disclaim all liability in connection with the use of this book.\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any form whatsoever without prior written permission from the publisher except in the case of brief quotations embodied in critical articles and reviews.\n_This book is dedicated to all the chefs out there who are bootstrapping their way to a better life_.\n**CONTENTS**\n\nForeword, by Tiffany Harelik\n\nAcknowledgements\n\nIntroduction\n\nPART I. ICONIC FOOD CARTS\n\nA Life-Changing Plane Ride to LA: KOi Fusion\n\nOpen 'Til 3:00 A.M.: Potato Champion\n\nWe Opened the Cart to Save Our Own Asses: Big-Ass Sandwich\n\nThe Boss You Wish You Had: Grilled Cheese Grill\n\nCaptivated by Portland While Walking Across the Steel Bridge: Whiffies Pies\n\n_Food Writer's Perspective: Jeanna Barrett_\n\nPART II. WE MOVED TO PORTLAND TO OPEN A FOOD CART\n\nWe Moved from Florence, Italy, to Portland, Oregon, to Open a Food Cart: Burrasca\n\nPlay the _Rocky_ Music: The Italian Market\n\nWe Bring the Circus to Your Door: Retrolicious\n\nWe Just Had to Get the Food into Their Mouths: PBJ's Grilled\n\n_Food Writer's Perspective: Amy Burglehaus_\n\nPART III. BREAKFAST!\n\nThis Could Ruin Our Relationship: The Big Egg\n\nStepping into a New Life: The Egg Carton\n\nThe Restaurant that Opened a Food Cart: Yolk\n\nIt Has Never Felt Like Work: Fried Egg I'm in Love\n\nPART IV. AN IMPORTANT PART OF THE PORTLAND FOOD CART SCENE\n\nScott Batchelar: The Mayor of Food Cart Town\n\nRoger Goldingay: The Mississippi Marketplace and Cartlandia Food Cart Pods\n\nTim Hohl and Terry Travis: Friday Is Our Favorite Day of the Week\n\nBrett Burmeister: Telling People About the Cool Things I Find\n\nRick Humphrey: His Card Is in My Wallet\n\nBryan Sebok: Mapping the Mobile Food Movement\n\nOn Many Food Cart Menus: Carlton Farms\n\nBrian Wilke and Eric Stromquist: Being a Nice Person in the Hospitality Business\n\nDoing Away with Throwaway: GO Box\n\n_Food Writer's Perspective: Phil Shen_\n\nPART V. VEGETARIAN FOOD CARTS: DELICIOUS WITHOUT MEAT\n\nFor the Love of Cheese: The Cheese Plate PDX\n\nI Hitchhiked My Way to Portland: Divine Caf\u00e9\n\nPedaling Kale: Moberi\n\nWhole Hawg Vegan: Homegrown Smoker Vegan BBQ\n\n_Food Writer's Perspective: Jennifer Chan_\n\nPART VI. CUISINE FROM AROUND THE WORLD\n\nEat Something Georgian (and We Are Not Talking Peaches): Kargi Gogo\n\nSo Much More than Just Curry: Tiffin Asha\n\nBecause of Meatballs and Grandma Clenora: Viking Soul Food\n\nA Baker's Sense of Cooking: PDX Six Seven One\n\nPortland's Asian Dumpling Champs: Stumptown Dumplings\n\nOut of Egypt: Ramy's Lamb Shack\n\nMore Fulfilled than I Have Ever Been: Bento Box\n\n_Food Writer's Perspective: Aaron Wakamatsu_\n\nPART VII. COMFORT FOOD\n\nCheese Makes Me Happy: Herb's Mac & Cheese\n\nMore than Just a Baker: The Sugar Shop\n\nTenacity and Artistry: Gaufre Gourmet\n\nRugby, Not Soccer: Timber's Doghouse PDX\n\nLaissez Les Bon Temps Rouler: A Cajun Life\n\nStay Caffeinated, My Friends: Ol\u00e9 Latte Coffee\n\nAbout the Author\n**FOREWORD**\n\nDecember in Portland is a little different than December in my hometown of Austin. The wet Pacific Northwest weather didn't stave off my hunger for learning about the food carts, but the daily drizzle and gray chilled my Texas-thin blood to the bone.\n\nWhile there on my initial trip, I slogged on some borrowed galoshes and rain gear and went to as many food carts as a girl could handle in a week. The amount of not just food carts but _really good_ food carts is overwhelming, with several hundred being open each day. I found that Portland, like Austin, is a city that is passionate about supporting the local economy. As an outsider, I networked with Steven Shomler, Brett Burmeister, Scott Batchelar and all the local authorities to connect me to some of their favorite carts. After eating at almost fifty carts, \"You should really come back in the spring,\" is what many who pitied my shivering would say. So, I booked another trip for April of the following year to enjoy Eat Mobile and fairer weather.\n\nBy then, I was working on the second volume of the Portland edition of the _Trailer Food Diaries Cookbook_ series. I had eaten at tons of downtown carts and a few pods in little pockets peppered through the outskirts of main hubs. I conquered Eat Mobile, hugging some of my food cart friends from the previous years and eager to meet more new cart owners. After the festivities ended, it was Steven who really took me under his wing and said, \"All right chick, I see that you want to do this and you want to do it right, so let me help you out.\" He planned my cart itinerary over the week I was there, personally taking me to cart openings and some of his favorite places, as well as some I had high on my wish list. The cart owners were consistently thrilled to see Steven, and he would immediately start photographing and tagging everyone on social media to document and promote the occasion.\n\nAfter interviewing food cart\/truck owners and collecting their recipes in more than ten states, I feel confident in saying that Portland is the true mecca of food carts because of their longevity, diversity and commitment to uniqueness in the market. And after spending time there with Steven, his passion is clear to me: he loves promoting food cart owners in order to share their stories and encourage their successes.\n\nThis book illustrates a portion of people in Portland willing to take their chance at living the American dream. Every walk of life is represented among the food cart owners, and their behind-the-scenes stories are fascinating. In his interviews, Steven found that most cart owners were working for six to seven days a week while they were starting up. They were willing to work eighty-plus hours a week to avoid working forty hours a week, all in pursuit of their own happiness and finding their own version of the American dream. Steven's first book is a tribute to all Portland's cart owners, acknowledging all it takes to be a bootstrap chef.\n\nTIFFANY HARELIK\n\nAuthor of the _Trailer Food Diaries Cookbook_ series\n**ACKNOWLEDGEMENTS**\n\nI am very grateful to the people who helped make this book possible.\n\nAdam Whalen, thank you for insisting that I go with you that beautiful fall day in September 2010 to eat lunch at a food cart for the very first time in my life. My life is very different because of that day, and I am very grateful.\n\nAl Gallo, thank you for all the inspiration and encouragement you have given me for the past twenty-five years or more.\n\nKen Wilson, thank you for being a terrific food cart wingman and for designing the Portland Food Cart Adventures logo, business cards and so on.\n\nKevin Brusett, thank you for providing me with life-changing, brand-legitimizing T-shirts.\n\nTiffany Harelik, thank for being kind enough to sit with me outside a food cart at a picnic table and listen to me tell you about the hopes I had to write a food cart book. Your \"You can write that book, and I am happy to help you any way I can\" comment meant more to me than you will ever know.\n\nMany thanks to Molly Woodstock and Ken Wilson for helping out with taking some of photos found in this book. My deadline was looming, and you both came through.\n\nThank you to Pro Photo Supply and your awesome man behind the counter, Kevin, for helping me learn to take real photos with a real camera.\n\nLori Nold, thank you for all the care and kindness you provided my family and I over the summer, fall and winter of 2013. I don't think that I could have done this book without you.\n\nThank you to the people in my Dream Stoker Nation community and to all of my Facebook friends! The support you gave me as I worked on this book night after night was invaluable.\n\nThank you to my History Press commissioning editors Aubrie Koenig and Will McKay, as well as my production editor Ryan Finn. You were all awesome and incredibly helpful.\n\nThank you to the guys at CityTeam for putting up with my many absences at the end of 2013 as I finished up this book.\n\nThank you to Zayne Shomler for spending many Saturdays with me in the fall of 2013 as I took hundreds and hundreds of food cart photos. You were a big help, and I enjoyed spending those days with you so much.\n\nThank you to Zac, Zayne and Zoe Shomler for your ongoing encouragement and being understanding all those times that dad was monopolizing the computer so he could write.\n\nThank you to the bootstrap chefs featured in this book for trusting me with your precious stories. Much love to each of you.\n\nLastly and most importantly, thank you to my lovely wife, Gayla, for always believing in this book and in my dream to be an author and a speaker.\n**INTRODUCTION**\n\n**W HAT'S IN THIS BOOK?**\n\nWithin the pages of this book, you will find lots of stories\u2014forty of them grouped into eight different parts.\n\nThere are stories of the owners of thirty different food carts. When I visit a food cart, I am always curious to know certain things: How did these people end up here doing this? Where did that name come from? Why this cuisine? And so on. The stories in here will provide answers to some of those questions.\n\nI could have easily filled this book with in-depth stories of just five food cart owners; however, I wanted to give you, the reader, a sense of the incredible diversity that we have in Portland among our food cart owners. I chose to go the route of painting with a broad brush so you could get to know a number of my heroes.\n\nPlease note that I was not able to include every story I wanted to. For example, Nong's Khao Man Gai is a food cart that I really wanted to include but was not able to. Hopefully, I will be able to include the story of Nong and her great food cart in _Portland Food Cart Stories Vol. 2_.\n\n_\"What's Next,\" \"When You Go\" and \"Recipes\"_\n\nAt the end of most of the food cart owners' stories, I have included additional postscripts. Every food cart owner I have met has big dreams, and the \"What's Next\" postscript gives you a window into a few of that particular food cart owner's hopes and dreams.\n\nThe \"When You Go\" postscript tells you what I would suggest that you eat when you visit that food cart.\n\nFinally, I am a big fan of Tiffany Harelik. She is the author of the _Trailer Food Dairies Cookbook_ series. To date, Tiffany has written two volumes featuring recipes from food carts right here in Portland. The \"Recipes\" postscript lets you know if that food cart has recipes in one of those volumes and occasionally other books as well.\n\n_The Eight Parts of This Book_\n\nWe start with an introduction to the Portland food cart scene. This section gives you an insider's view into this eclectic and entrepreneurial community. You are in this section right now.\n\nNext we have the \"Iconic Food Carts\" section. These are some of the most popular and successful food carts here in Portland. These food carts are so well known that tourists visiting Portland seek them out, and their food is good enough to keep us locals coming back time and time again.\n\nThe second part tells you about eight people who heard the siren call and moved to Portland for the express purpose of opening a food cart. It is one thing to already live in Portland and roll the dice on a food cart venture. It is quite another to be daring enough to move across the country or the globe and take that risk.\n\nNext comes breakfast! Portland has a well-deserved reputation as a great breakfast town. The \"Breakfast!\" part of this book will tell you about four carts that specialize in the first meal of the day.\n\nThe fourth part tells the stories of people and companies that do not own foods carts yet still are an important part of the Portland food cart scene.\n\nThe fifth part concerns vegetarian food carts. Of course Portland has vegetarian food carts. One of these four carts is actually vegan, and it is run by someone who wears tie-dyed T-shirts and loves the Grateful Dead.\n\nThe sixth part describes food carts that bring us delectable cuisine from around the world. In Portland, you can enjoy authentic food from places like the countries of Georgia, Norway and India; the island of Guam; the Mediterranean; and Asia.\n\nOur final section is all about comfort food. I don't know about you, but sometimes I eat for comfort, and it does not get better than the food in this section\u2014everything from mac and cheese to hamburgers and gumbo. Closing us out are boozy desserts from the Sugar B's, Li\u00e8ge-style waffles and some of the best coffee that you will ever have.\n\nInterspersed between these eight parts are contributions from other food writers who love food carts as much as I do.\n\n_Go On Your Own Food Cart Adventure_\n\nI hope that this book inspires you to get off the couch and head out on your own food cart adventure! If you happen to visit one the great food carts featured here in the volume, look for the _Portland Food Cart Stories_ sticker and be sure to tell them that Steven sent you! Now, though, it's time for you to learn the facts about food carts.\n\n**W HAT A FOOD CART REALLY IS**\n\nA question that is often asked is, \"What is a food cart?\" I have heard many stories of food cart owners trying to explain to friends and family who live in other parts of the country what it is they have started. Sometimes people envision a small pushcart. Basically, food carts are like food trucks.\n\n_Food Cart vs. Food Truck_\n\nA food cart in Portland is like a food truck in any other part of the country with one _big_ exception. Most of our food carts stay in one place; they don't drive from location to location. In fact, many Portland food carts aren't mobile at all.\n\nSome Portland food carts are actual trucks, with engines and everything; some food carts are modified travel trailers; and other food carts have been built onto flatbed trailers from the floor up. Regardless of what the actual contraption is, in Portland we call them \"food carts\" rather than \"food trucks\" and \"food trailers.\" If you want to eat at a food trailer, you'll have to go to Austin, Texas.\n\nPart of the appeal of the Portland food cart scene is that you know where you need to go to find your favorite food cart. It's almost always there. All you need to worry about is whether or not it will be open when you want it to be. And that leads us to food cart pods.\n\n_Food Cart Pods_\n\nAnother very appealing part of the Portland food cart scene is our food cart pods. A food cart pod is a collection of food carts that are in one place. In Portland, we have two kinds of food cart pods, the first being the downtown pods that provide you with many food cart options in one area. When you buy from a downtown Portland food cart, you generally eat standing up or you take your meal back to your office.\n\nThe second type of food cart pod that we have is a neighborhood pod. The neighborhood pods have anywhere from two to about twenty food carts in one place. Many of these neighborhood pods offer covered seating, and some even have heated seating. A handful of the neighborhood pods have places where you can enjoy a beer with your meal, either from a beer garden built into the pod or from a pub that is adjoining the pod.\n\nIf you are visiting Portland and you want to see the Portland food cart scene, you really need to visit both a downtown pod and a neighborhood pod. Simply seeing one or the other is getting an incomplete picture.\n\nAt the end of 2013, Portland had more than thirty food cart pods and more than three hundred food carts.\n\n_Food Carts Are Startups and Small Businesses_\n\nMost people don't realize that each and every food cart is a small business and, in the beginning, a startup. When you eat at a food cart, you are supporting small business and helping someone with his or her startup. Food cart owners who open a food cart face all the challenges that everyone else that launches a startup faces, and honestly, people with the kind of courage needed to start a business inspire me.\n\n_Bootstrap Chefs: The Best of the American Dream_\n\nAs noted chef and restaurateur Michael Symon said, \"You do not have to go to school to be a chef...the title is earned by running a professional kitchen...although culinary school will help you reach your goals...and may speed up the process...it does not make you a chef.\"\n\nPeople who start food carts do so because they have had a dream to be self-employed smoldering in their hearts and they are finally going for it. Many food cart owners have longed wished to be able to open their restaurant but lacked the $250,000 to $500,000 often required to do so. A food cart gives them a path to begin making that dream come true.\n\nTo me, food cart owners represent the best of the American dream; they are chefs who are bootstrapping their way to a better life. You want to know what a food cart really is? It is someone's hopes and dreams, and that is what this book is about: chefs who had the courage to battle for their dreams.\n\n**M Y FOOD CART STORY**\n\nOn Friday, September 9, 2010, my friend Adam Whalen insisted that I join him at a food cart for lunch. My family and I had moved to Portland in the fall of 2004, and in the six years after that, I had never been to a food cart. In fact, the thought had never even crossed my mind.\n\nThat fateful September day, Adam took me to the Portland Soup Company Food Cart, and I had the mozzarella sandwich. This sandwich came with cart-made mozzarella, fresh basil, marinated Roma tomatoes, balsamic syrup and a roasted garlic aioli. I also bought the chilled tomato gazpacho with sweet corn and cilantro.\n\nI was blown away by the food. It was so good, and I could not believe that something this good came out of kitchen in a little box. The next week, Adam took me to the Lardo Food Cart, and once again I could not believe how good the food was.\n\nI had been toying with the idea of becoming a writer, and I told Adam that he and I should start a website and let everyone know how incredible Portland food carts are. I told him that there had to be a number of idiots like me who have lived in Portland for years and completely overlooked these local treasures.\n\nTime passed, and days turned in weeks. A year later, in the summer of 2011, I decided that I was going to become a speaker and an author and that I was going to start writing to make it happen. I was no longer going to play around with my writing. I was going to take it seriously and make a career of it. By that point in my journey, I had written enough to know the topics that I was going to write about, and food carts were going to be one of those topics.\n\nIn the fall of 2011, I started eating at many food carts. I told my wife that I was doing research for a website that I was going to launch. She smiled and wondered where this would all lead. On January 1, 2012, I launched the Portland Food Cart Adventures website, and I began writing about food carts.\n\n_Food Cart Magic_\n\nThe more I wrote about food carts and the more that I ate at them, the more impressed I became. To me, making such incredible food in a little box was nothing short of amazing, and I began to call this feat \"food cart magic.\"\n\n_How Can I Help These Entrepreneurs_\n\nI also got to know food cart owners, and I began to fall in love with them. I got to know their stories and hopes and dreams, as well as the huge obstacles they faced. I wanted to do what I could to help them make it. I began regularly consulting with food cart owners, helping them with things like having a successful startup, branding, marketing, social media and extending their brand.\n\nI did not charge food cart owners for this advice. I wanted to serve them and help them. Even today, if you are a food cart owner or if you are going to be opening a food cart, if you buy me a cup of coffee (or a beer, or a whiskey), I will give you an hour or more of my time and my best advice.\n\nI began to dream about having a festival in the summer here in Portland where food cart owners could make a killing and keep the proceeds of whatever they sold. In March 2012, I bought the domains PortlandFoodCartFestival.com and SummerFoodCartFestival.com.\n\nThe more I have served, the more good things have come my way. I remember the first time I got a call asking if I was willing to be filmed for show that would appear on the Food Network. My food cart website was less than six months old, and I was sure that I was being pranked.\n\nTo date, I have been privileged to be filmed for various food-related TV shows nine different times. My son Zayne will occasionally get texts from his friends that contain a picture of me on the paused TV, with the question, \"Isn't that your dad?\" One of my TV-related goals is to someday be a judge on _Iron Chef_.\n\n_The Portland Summer Food Cart Festival_\n\nIn the fall of 2012, I was contacted by a wonderful nonprofit that serves adults with disabilities, Adult Learning Systems of Oregon Inc., or ALSO. It had done a food cart event that past summer and wondered if I would help it plan an event featuring food carts for the next summer. I knew exactly what kind of event I wanted to see come together, and I knew exactly where I wanted this event to occur. I had been dreaming about this event since March, and I was very excited to help!\n\nOn June 22, 2013, the first annual Portland Summer Food Cart Festival took place at Mt. Hood Community College.\n\nWe had more than sixteen food carts, and more than 3,200 people showed up. We sold tickets for admission, and the ticket proceeds went to ALSO. The food carts that were part of this event got to sell their food, and they kept every dime they sold that day. Every single food cart owner who took part in the festival reported that June 22, 2013, was his or her best day for food sales ever. I was exhausted that evening when I went to bed; however, my heart was full.\n\n_I Am a Food Cart Fan_\n\nI love being a part of the food cart community, and I am thrilled that the opportunity to write this book came along. I hope that you enjoy reading it as much as I enjoyed writing it.\n\n_What's Next_ : I am going to continue to write about food carts for my website, Portland Food Cart Adventures. I plan to continue to assist with the Portland Summer Food Cart Festival. I will continue to consult with food cart owners and do what I can to help them succeed. I am also going to expand my food writing beyond food carts. In 2014, I will be writing for GreatFoodGreatStories.com. Lastly, I hope that this book will sell well enough to allow me to write a _Portland Food Cart Stories_ , vol. 2!\n**PART I**\n\n**ICONIC FOOD CARTS**\n\n**A L IFE-CHANGING PLANE RIDE TO LA: KOI FUSION**\n\nBo Kwon is the founder of what I call the KOi Fusion food cart empire. KOi Fusion has legions of fans, and Bo is arguably the most successful food cart owner in Portland. I sat down with Bo to interview him for this book, and my first question was, \"When did you first think about starting a food cart?\" This is his answer:\n\n_I was sitting on the couch talking to my dad about what to do because I knew that I was going to lose my job. The TV was on, and a news report came on about food trucks in LA becoming popular. I asked my dad if he would ever open a restaurant because I knew that he could cook. My dad said, \"Why don't you open a food cart?\" I started laughing. \"A Korean food cart? Yeah right!\" I few weeks later, I was sitting on the same couch, and I saw a Korean TV show talking about a new trend\u2014the Korean taco_.\n\nWith that, things started to click with Bo, and he began to think about the possibility of opening a food cart that specialized in Korean tacos.\n\n_Inspired by Roy Choi_\n\nBo then heard about Roy Choi in LA, who was with the Korean taco food truck fleet known as Kogi. Bo began to hound and pester Roy, first with e-mails and then with phone calls. The response Bo consistently got was that Roy was busy.\n\nBo was convinced that somehow this guy in LA whom he had never met could help him with his dream of opening a food cart in Portland. Bo then did what any not-so-sane person with a dream would do: he called in sick one Monday, went to the airport, bought a plane ticket to LA and headed off to check out the food truck scene and hopefully find Roy.\n\nBy this point, Roy had opened a restaurant. Bo went to the restaurant and right into the kitchen, and there was Roy. Bo went over and tapped him on his shoulder. Bo said, \"Hey, I just came from Portland.\" Roy said with astonishment, \"You are that guy who has been bugging me about opening a food truck!\"\n\nLuckily, Roy said that he would give Bo five minutes. Bo's passion and commitment to his dream won Roy over. In those five minutes, Roy gave Bo hope and told him that he already had within himself all he needed to make a Korean food truck work in Portland.\n\nRoy was doing what Bo wanted to do. Even though Bo had not met Roy until that day, Roy Choi had become an inspiration and hero to him. What meant the most to Bo was that Roy told him, \"Go do Korean tacos in Portland. You have my blessing.\"\n\nOn the plane ride home, Bo felt like he had a new lease on life, and he now had the momentum, motivation and strength to start his business because of the blessing his hero had given him. A friendship was started, and along the way, Roy would occasionally give Bo feedback and counsel as KOi Fusion was getting launched.\n\n_No Money and It Is Time to Buy a Cart_\n\nFollowing his meeting with Roy, Bo arrived back in Portland. A week later, all he had was his dream and the encouragement he received from Roy. He had no money and no credit, and now he needed to get a food cart so he could get started.\n\nBo found a taco truck in Hillsboro on the side of the road, and he went up to talk with the owner about buying it. Remember, Bo had no money. He bought a few tacos and then he asked the owner, \"Would you want to sell me your food cart?\" She laughed and said, \"Very funny.\" Bo replied, \"No, I'm serious. I really want to buy your food cart.\"\n\nAs it happened, the \"owner\" who was running this food cart had bought it for her son, who decided after opening the business that he did not really want to own a food cart. She was working the food cart to make the payments.\n\nBo proposed that he would buy the cart and they would hold the note. He would make monthly payments until he had paid the agreed-on amount in full. Bo also proposed that Rosie (the owner) stay on for the next six months, working in the cart and teaching Bo everything she knew. Bo did not know the first thing about running a food cart, and he realized that he needed to be schooled.\n\nTo entice Rosie to say yes to this whole arrangement, Bo said that he would hire Rosie and pay her an hourly wage and that he would make her monthly food truck payments. In addition, any profit the truck made in those six months would also go to her.\n\nBo said that those six months of learning to work in a taco truck selling Mexican food alongside Rosie was priceless. He learned a tremendous amount. He learned everything from dealing with gray water to insurance, hiring people and making handmade tortillas and pico de gallo. That experience working in a taco truck selling Mexican food is why the _fusion_ in KOi Fusion is legitimate.\n\n_The Name_\n\nThe _KOi_ in KOi Fusion is an acronym. It has nothing to do with koi fish or Japanese culture. KOi stands for \"Korean Oregon infusion.\" Bo was born and raised right here in Oregon, and he is of Korean descent. He loves both Oregon and his Korean heritage, and he considers himself to be a fusion of both.\n\n_Extending the Brand_\n\nThe reason Bo made the leap from one food cart to two food carts was because while his first food cart was doing well, he realized that he still had no money to speak of and that if he was going to get where he wanted to go, he was going to have to extend his food cart brand. That second food cart was actually a mobile truck, and it allowed him to start doing events and catering.\n\nThe KOi Fusion Food Cart at Mississippi Marketplace.\n\nBulgolgi BBQ Beef Taco with kimchi from KOi Fusion.\n\n_Kimchi_\n\nOne of my favorite Bo Kwon anecdotes is that when he was attending the University of Oregon, his roommates would not let him keep kimchi in the refrigerator. Now those same people love kimchi and are huge fans of Korean cuisine. The same thing can be said for the thousands of K headz (how KOi Fusion fans refer to themselves; the phrase is a riff on \"sneakerheads,\" people who are rabid fans about sneakers, such as Bo) out there. All because Bo was willing to get on a plane and go to LA to pursue his dream.\n\n_What's Next_ : Bo plans on opening his first restaurant in the first half of 2014 on Division and then, late in 2014, he plans on opening a second restaurant on Burnside, near the Timbers stadium. Yes, those are ambitious goals. However, when I look at his track record, I know that Bo can accomplish whatever he sets his mind to.\n\n_When You Go_ : So many good things! I especially love the Bulgogi beef tacos and the Korean cheesesteak.\n\n**O PEN 'TIL 3:00 A.M.: POTATO CHAMPION**\n\nWhen Mike McKinnon opened the Potato Champion Food Cart in April 2008 at the corner of 12th and Hawthorne, he changed Portland.\n\n_\"I Had No Money and I Wanted to Quit My Job\"_\n\nMike did not set out to change Portland. He simply wanted to be able to become self-employed and make enough money to quit his job in the drum department at Trade Up Music. This is the Potato Champion story.\n\nIn 2005, Mike was driving around Portland with a friend and potential investor looking for a place that might work for them to open a storefront where they could sell pommes frites. Pommes frites are better known in the United States as French fries. In Belgium and France and other places, they enjoy \"pommes frites.\"\n\nMike loves music; to him, playing music is a life-giving endeavor. Following high school, Mike spent some time traveling the United States with a band. While on those adventures, he fell in love with pommes frites. He had them in Holland, Belgium, New York City, San Francisco and Vancouver, British Columbia. Mike soon found himself living in Portland, Oregon, and following concerts with his band, he would wish for late-night dining options beyond the iconic (and delicious) Original Hotcake House. Mike would also lament the lack of friteries in Portland. His friend would tell him, \"You should open one.\"\n\nThat is how Mike found himself in the spring of 2005 in a car with a friend looking for places to open a friterie. Later that summer, Mike's friend and potential investor pulled out. He wished Mike the best and encouraged him to press on with his dream. Food carts began to catch Mike's eye, and he thought maybe, just maybe, a food cart is the way to go.\n\n_\"My Culinary Experience Began and Ended with Delivering Pizza\"_\n\nOne of the reasons why food cart owners inspire me so much is because they pursue their dreams, even when conventional wisdom would say that they have little chance of success. Like Johnny Utah leaving the plane without a parachute in hot pursuit of Bodhi in _Point Break_ , they just go for it!\n\nMike was not going to let a little thing like not having any culinary background stop him from opening a food cart that sold pommes frites. Mike began a three-year research program teaching himself to make pommes frites and sauces. He started with a lot of reading, and then eventually he bought a home fryer and started peeling and frying potatoes. He mentioned to me that once the cart opened and he moved from his home fryer to a forty-five-pound fryer, the learning curve started all over again.\n\nMike said that starting a food cart _was_ his culinary school. For a guy whose culinary journey started with delivering pizzas, he has done very well in the cooking department. Today, the sauces and ketchups, and most everything else they sell at Potato Champion, is made from scratch. They even make the peanut butter that goes into the peanut satay sauce.\n\n_Why \"Potato Champion\"?_\n\nMike struggled with what to call his food cart. \"Naming a food cart is as bad as naming a band,\" he said. Mike began reading about the history of the potato and found his inspiration. He learned that in 1748, the French parliament outlawed the cultivation of potatoes. Potatoes were believed to be dangerous and one of the causes of leprosy. Enter Antoine-Augustin Parmentier. Antoine fought for France in the Seven Years' War, was captured, became a prisoner of war and was put in prison in Prussia. While in prison, Antoine was faced with eating the repugnant hog feed known as potatoes.\n\nThe war ended, and Antoine returned home and began studying nutritional chemistry. He began to promote the potato as a safe source of nourishment. In 1772, the Paris Faculty of Medicine declared potatoes safe to eat. The reason that French cuisine includes potatoes is because of Antoine-Augustin Parmentier, the potato champion.\n\n_\"I Lived for Three Years Like a Robot on Zombie Mode\"_\n\nWhen Potato Champion opened, Mike worked six and seven days a week. That first year, Potato Champion was open from 6:00 p.m. to 3:00 a.m. Back then, the cart was open from Wednesday to Saturday. Mike did the shopping, the books and lots of food prep from Sunday to Tuesday.\n\nMike spent three years working those six and seven days a week. During this period, with rare exceptions, he had to put his music on the shelf. Mike paid the price that many people who succeed in a startup pay. If you aren't ready for that, don't open a food cart.\n\n_\"The Weird Feeling of Sitting Still\"_\n\nAfter three years, Mike was able to shift from working in the cart to running the business. This is also when he finally began having regularly scheduled days off. Mike shared how incredibly hard it was to recalibrate a schedule that allowed him to have some breathing room. Yes, Mike worked very hard and put in some crazy hours building his startup. But he said that it was worth it and that he has found the journey to be very satisfying.\n\n_Changing Portland_\n\nWhen Mike opened Potato Champion in 2008 at 12th and Hawthorne, most food carts were located across the Willamette River in downtown Portland. Those food carts were also, for the most part, open during lunch, Monday through Friday. Very few, if any, were open for dinner or late-night dining.\n\nBo Kwon, founder of the KOi Fusion empire, has inspired many people to believe that they could succeed as a food cart owner. Mike McKinnon has inspired many people to believe that they could have a successful Portland food cart that was not downtown.\n\nI know that before Mike opened Potato Champion, there were a handful of successful food cart that were not downtown\u2014the Flavour Spot Food Cart comes to mind. However, I have heard many food cart owners tell me that it was Mike and Potato Champion that inspired them to believe that they could have a successful food cart in Portland located somewhere other than downtown.\n\nQuite a few food carts owners have also told me that it was Mike's example of staying open until 3:00 a.m. that convinced them that they could do more than just lunch.\n\nMike McKinnon, owner of the Potato Champion Food Cart.\n\n_3:00 A.M_.\n\nBefore Mike got to the Cartopia Food Cart Pod on 12th and Hawthorne, the few food carts that were there were open during lunch. Mike started staying open until 3:00 a.m. In doing so, he was fulfilling a dream that he had years ago. Portland now had a late-night friterie!\n\nFrom the beginning, Mike was very diligent about maintaining his posted hours. Potato Champion would stay open until 3:00 a.m., even if the pod was empty at 1:00 a.m. Back then, other (and now defunct) food carts would close up early on slow nights. Mike persisted with his vision, and a late-night food cart scene was born. Today, all of the Cartopia food carts are open late at night on Fridays and Saturdays.\n\n_The Cartopia Late-Night Dining Scene_\n\nThe Cartopia late-night scene is a blast. If you haven't been, go! If you are visiting Portland, make sure that you visit Cartopia and do so on a Friday or Saturday night just after midnight. Rain or shine, you can get a great meal and enjoy it in covered seating. Cartopia is the perfect place to end a Friday or Saturday evening in Portland. I tip my food cart hat to Mike McKinnon. He is my potato champion!\n\n_What's Next_ : Mike hopes to open up a brick-and-mortar Potato Champion some day. He would be delighted to be able to serve hamburgers to go with his pommes frites. He would also serve Mitraillettes, which Mike first had in Belgium. They are a sandwich commonly available at Belgian friteries and caf\u00e9s containing meat and pommes frites. Mike has also begun to look into packaging some of his ketchups and sauces for retail sale. He gets many requests to bottle his rosemary truffle ketchup!\n\n_When You Go_ : Take a few friends and share at least three orders among yourselves. Get a large order of fries and pick the three sauces that sound most appealing to you. I especially like the horseradish ketchup, the pesto mayo and the rosemary truffle ketchup. Also, get the Pulled Pork Fries\u2014topped with slow-cooked Carlton Farms pork shoulder and Potato Champion's own whiskey barbecue sauce. Lastly, get the PB and J Fries. This is my favorite dish at Potato Champion! Fries topped with Potato Champion's peanut satay sauce and chipotle raspberry sauce. The flavors work so well together.\n\n**W E OPENED THE CART TO SAVE OUR OWN ASSES: BIG-ASS SANDWICH**\n\nThe Big-Ass Sandwich Food Cart opened on December 21, 2009, and it has become one of Portland's most iconic food carts. The press loves it, celebrities stop by and, in 2013, it expanded from one food cart to two. The Travel Channel's Adam Richman declared that it has the best sandwich in the Northwest. I am thrilled that I get to share Brian and Lisa's startup journey with you.\n\n_Laid Off...Again_\n\nBrian and Lisa met in February 2003 at the 5 Point in Seattle, Washington, and they have been together ever since. In June 2009, they were living in Portland, and Lisa was laid off for the second time in two years. They realized that because of the economic downturn, Lisa was not going to find another job in her industry. They were going to have to do something to stay afloat.\n\nFor almost as long as they have been together, Brian and Lisa had talked about opening a restaurant. Both of them had wanted to be self-employed for many years. When Lisa was a teenager, her parents started their first business, and that inspired Lisa to want to start her own as well some day. Brian has wanted to be self-employed since \"the first time the boss unfairly yelled at me for something stupid and ridiculous.\"\n\n_Maybe We Could Open a Food Cart_\n\nRight after Lisa was laid off in June 2009, Brian and Lisa decided that it was time to pull the trigger on being self-employed. Starting a restaurant was not in the cards financially, and they began to consider opening a food cart. Brian's one big hesitation for opening a food cart was that he wanted a cart that had a commercial kitchen that he did not have to fight to get to work. Up to this point, Brian had twenty years' experience working in the restaurant business. He started washing dishes at fourteen and worked his way up the culinary ladder. He spent ten of those twenty years working as a chef in fine dining establishments.\n\nLisa and Brian had been looking at used food carts, but as Brian put it, \"The used carts we looked at were hammered to dog #^%@, and the kitchens in them sucked.\" He added, \"The way we found our food cart was both fortuitous and flukish.\" Brian and Lisa were hanging out with Josh, a chef and a friend of Brian's. An acquaintance of Josh's happened to stop by, and he knew about some guys who were food cart builders. Brian and Lisa met with Rich and Jason, and Brian discovered that he could design his own kitchen.\n\nThey met with Rich and Jason a number of times after that. One of these times, they were meeting in a coffee shop on Belmont that Brian and Lisa liked, and Brian asked Rich, \"Hey, when are we going to get to see one of your food carts?\" Rich said, \"Ummm, actually...Your cart is going to be our first one.\"\n\nBrian and Lisa were lucky enough to be working with the guys who were starting Northwest Mobile Kitchens (NMK). Today, Rich is the owner of NMK, a very good food cart\/food truck builder. Incidentally, NMK built a number of the food carts featured in this book, including Potato Champion, PDX Six Seven One and Yolk. The way they found their food cart was indeed fortuitous.\n\n_The Secret Origin of the Big-Ass Sandwich_\n\nA few weeks later, Brian and Lisa were in that very same coffee shop talking about the menu for their food cart. Brian wanted something that was unique and tasted great. Then he remembered his days cooking on the line. Some days, you would get a whole ten-minute break to smoke a cigarette and eat enough food to keep going for the next nine to ten hours. It was common to throw everything that you were going to eat\u2014even the French fries\u2014between two pieces of bread so that you could consume it quickly before your ten minutes were up and you had to hop back on the line.\n\nAs he was describing this to Lisa, she said, \"Brian, you make incredible roast beef and really good cheese sauce. We could take both of those and some French fries and make a sandwich out of them.\" Brian then held up his hands as if he were holding this sandwich and loudly exclaimed, \"This is gonna be a big-ass sandwich!\"\n\nLisa has a friend who told her before they opened the cart that \"opening a business together will really test your relationship.\" Early on, it was very hard for Brian and Lisa to figure out how to work well together. It was really tough on their relationship, and they fought a lot. Brian remembered more than once thinking, \"What the *&#@ did we do?\" Many nights ended with \"whiskey fights.\"\n\nBrian and Lisa Wood, owners of the Big-Ass Sandwich Food Cart.\n\nThey did gut it out. They pressed on through the adversity and found a rhythm that worked for them. As a couple, they are doing awesome. Spend just fifteen minutes with them, and you can tell that they like each other and that they have done the hard work of building a great relationship. As business partners, they came up with a division of labor that worked for them. In the cart, Brian is in charge; if it is a food question, talk to Brian. Lisa handles the rest. Brian is an excellent chef and prepares wonderful food, while Lisa oversees their social media platforms. She is stellar at it! Lisa uses social media to effectively tell their brand story, and she is very diligent about connecting with the many fans that Big-Ass Sandwich has. \"Social media brought the people in, and the food kept them coming back,\" she noted.\n\n_If You Start a Food Cart, Be Prepared to..._\n\nI asked Brian and Lisa what they would tell someone wanting to open a food cart. Lisa's immediate response was, \"Don't do it; it is harder that you think.\" Here is the rest of what they shared with me in rapid-fire succession:\n\n\u2022 Be prepared to give up three years of your life.\n\n\u2022 Be prepared to work really, really hard.\n\n\u2022 Be prepared to be a pauper for a few years.\n\n\u2022 Be prepared to give up your summers.\n\n\u2022 Be prepared for the struggle.\n\n\u2022 And for the love of God, do some marketing!\n\nBrian and Lisa speak very highly of their crew and of the loyal core group of fans they have. They give much credit to the people who have helped them get this far. \"We wouldn't have made it this far without our crew and our fans,\" said Lisa. I applaud the fact that they are sincerely grateful to the people who have stood by them.\n\n_Grit and Guts_\n\nBrian shared with me that in his younger years, he always marveled at the people who were able to start their own business. It feels good to now be on that very same journey. He wistfully spoke of people he knew in the past who wanted to start their business and never did. Lisa added, \"It takes balls to make that leap.\" Brian and Lisa, I am one of your fans, and speaking on the behalf of the legion of other supporters you have out there, we applaud the fact that you had the guts to make that leap.\n\n_What's Next_ : Brian and Lisa would love to open two brick-and-mortar restaurants. The first one they want to open is a Big-Ass Sandwich restaurant\u2014somewhere people could hear great music, enjoy awesome beer and chow down on a Big-Ass Sandwich. Once that dream has come true, Brian longs to someday open a gastro pub that features rustic comfort food and has a huge fireplace. I wish that you could have seen the look in his eye when he was describing this place to me.\n\n_When You Go_ : Get a Big-Ass Roast Beef with extra b\u00e9chamel, grilled mushrooms and horseradish. I also love the Big-Ass Breakfast Sandwich. I get mine with bacon. The scrambled eggs pair incredibly well with the ciabatta roll and the b\u00e9chamel. My absolute favorite Big-Ass Sandwich only comes around once a year, at Thanksgiving. The Big-Ass Thanksgiving Sandwich has turkey, fries, homemade gravy and homemade stuffing made with sausage, walnuts, sweet potatoes, rosemary and cranberries. Make sure to get extra, extra turkey gravy. Yes, I said \"extra\" twice. You will want _lots_ of gravy. Once your gravy-laden Big-Ass Thanksgiving Sandwich is handed to you, don't unwrap it. Put it down and let it rest for few minutes. Let the fries absorb that glorious gravy. After about four minutes, unwrap your beauty and be swept away into gravy-bathed Thanksgiving heaven.\n\n**T HE BOSS YOU WISH YOU HAD: GRILLED CHEESE GRILL**\n\nOn April 20, 2009, Matt Breslow opened his first food cart at 11th and Alberta. The Grilled Cheese Grill is a very successful food cart business here in Portland. It has four food carts\u2014three Grilled Cheese Grill locations and the Shot Gun Sub Shop Food Cart. Matt Breslow is the entrepreneur who founded it all, and he is the boss you wish you had.\n\n_Moving to Portland_\n\nMatt grew up in New Jersey, and his family visited Portland when he was kid. He had aunts, uncles and cousins here. Matt also spent two consecutive summers here in Portland when he was eleven and twelve years old.\n\nMatt remembered Portland being the place where he felt the most at home. He always assumed that he would move to Portland someday and settle here. That someday came about in the summer of 2007. Matt moved to Portland to open a breakfast place. Matt went to five different Portland restaurants that served breakfast. Screen Door was the fifth. After eating brunch at Screen Door, Matt concluded that \"Portland had breakfast covered\" and began to look for other culinary endeavors.\n\nMatt Breslow, owner of the Grilled Cheese Grill Food Cart.\n\n_Three Questions_\n\nMatt then noticed the food cart scene, and he began to wonder if maybe starting a food cart would make sense. Matt had three questions related to starting a food cart. One, what would his cuisine be? Two, where would he put his first food cart? Three, what about the \"nine months\" of rain?\n\nA little while later, Matt found himself sitting on a couch in his cousin's home. They were both hungry, and in the kitchen was half a loaf of wheat bread, a block of cheese, some turkey, some hummus and an avocado.\n\nMatt and his cousin proceeded to make an eclectic grilled cheese sandwich that was very tasty. Matt began to think, \"You know, maybe grilled cheese sandwiches would be the way to go.\" Matt realized that there are lots of different kinds of bread and cheese and that he could use bread and cheese as a palette for being very creative. At the same time, Matt loves comfort food, and he knew that for many people, a grilled cheese sandwich is quintessential comfort food.\n\nThe Grilled Cheese Grill's first food cart at 11th and Alberta.\n\nMatt was not sure that he wanted to have to start out with a food cart downtown handling the lunch crowd. He was also not sure that a food cart could work somewhere other than downtown. Then Matt began talking to Mike, the founder of the Potato Champion Food Cart. According to Matt, Mike had singlehandedly turned the food cart pod at 12th and Hawthorne (known as Cartopia) into a late-night destination. Matt found Mike to be very helpful and insightful.\n\nMatt decided that he did not have to be downtown to have a successful food cart and that he could serve more than just lunch. He liked the Alberta neighborhood; he found a vacant lot on 11th and decided that a food cart on Alberta might just work.\n\nWhat about the \"nine months\" of rain? In a stroke of pure genius, it occurred to Matt that he could put a school bus on the lot next to his food cart, and people could sit inside it when it was raining.\n\n_The Name_\n\nAll three of Matt's questions had been answered, and now he needed to name his cart. Matt thought through a number of names. He wanted a name that was memorable and clear. He did not want anyone to have to wonder about what it was he was selling at his food cart. He wanted a name that would be good for word of mouth. The name Grilled Cheese Grill fit the bill on all accounts.\n\nI have to say that I love Grilled Cheese Grill's motto: \"So come by for a taste of your childhood. Unless your childhood sucked, and then we'll let ya have a taste of ours.\" That tagline always makes me chuckle, and as a survivor of a very rough boyhood myself, I appreciate the playful and caring acknowledgement that not everyone looks back fondly on their childhood.\n\n_Extend Your Food Cart Brand_\n\nMatt has expanded his Grilled Cheese Grill startup into a thriving enterprise that has four food carts, a school bus and a double-decker bus.\n\nI very much believe that food cart owners need to extend their brand somehow\u2014start selling branded merchandise like A Cajun Life, add more carts like KOi Fusion and Fried Egg I'm in Love or go brick-and-mortar like former food cart Lardo. You can only sell so much from one food cart. If most food cart owners are going to achieve the better life they wanted when they launched their startup, they are going to have to expand some way, somehow.\n\nIf you are going to grow your food cart business from one food cart to two or more food carts, you have to transition from being the person cooking the food and interacting with most of your customers to successfully hiring and then leading a team of employees who will do most of the cooking and interact with most of your customers.\n\n_Hiring and Leading Employees_\n\nSuccessfully hiring and then leading team of employees is a whole different set of skills from those needed to start a food cart, and Matt Breslow has mastered these skills. I believe that the mindset and perspective that Matt has about hiring and leading employees has greatly contributed to the success of Grilled Cheese Grill. Said Matt, \"90 percent of finding a good employee is hiring.\"\n\nGrilled Cheese Grill has a very interesting job application. (You can Google \"Grilled Cheese Grill job application\" to see it.) The most important part to Matt are the six questions. Matt particularly likes question #5 (\"Please describe your last good argument\") and question #1 (\"Please describe either your favorite or least favorite job\").\n\nMatt feels that how someone answers the six questions on his job application will reveal how well someone communicates, and it will also tell whether or not they are a complainer and a whiner. Matts wants employees who are good communicators because they will most likely be talking to his customers.\n\n_Morale Is Key_\n\nMatt does not want employees who are complainers because food carts are small and, as he put it, \"Who wants to spend eight hours in a small space with a whiner and a complainer?\" Morale is a _big_ deal to Matt. \"Morale is the biggest thing a business owner should be concerned with. Bad morale can spread faster that a virus.\" Speaking about morale, Matt went on to say, \"Work is not always fun, and maintaining good morale helps the people who work for you to actually want to come to work.\"\n\nMatt has worked in a number of corporate settings, and he has seen firsthand how much poor morale negatively effects productivity and the bottom line. He went on to make two powerful statements that I have been reflecting on ever since I first heard them. \"My most important job is to treat my staff well and keep them happy and make sure that they look forward to coming for work...I care about my staff more than I care about my customers. I care about my staff so that they will care about my customers.\"\n\nI have had a number of \"bosses'\" in my lifetime, and after interviewing Matt for this book, I realized that very few of them had the kind of mindset toward me that Matt has to his staff. I also realized that I would really enjoy working for someone with Matt's perceptive.\n\n_What's Next_ : Matt is planning on adding a mobile food cart in 2014 so that the Grilled Cheese Grill can do events and add catering to the mix. Matt is also looking at adding a Grilled Cheese Grill brick-and-mortar restaurant sometime in the next year or two. He would love to have a place where people could get French fries with their grilled cheese sandwiches; he would also serve breakfast there. My best guess is that Matt's first brick-and-mortar Grilled Cheese Grill would be like a wonderful old-school diner, breakfast in the morning and grilled cheese creations for lunch and dinner.\n\n_When You Go_ : First, get the Kindergartner. It's a classic grilled cheese sandwich. Don't you dare get wheat bread or cheddar! A proper classic grilled cheese sandwich has white bread and American cheese. Next, get yourself a Cheesus, a delightful cheeseburger. Instead of a bun, you get two distinctive grilled cheese sandwiches bookending a  -pound burger, with lettuce, tomato, ketchup and mustard. One of the grilled cheese sandwiches on the Cheesus is made with pickles and American cheese, and the other is made with grilled onions and Colby jack. Please note: I add avocado to my Cheesus. I grew up near Fallbrook, California, and I love avocado on a burger.\n\n**C APTIVATED BY PORTLAND WHILE WALKING ACROSS THE STEEL BRIDGE: WHIFFIES PIES**\n\nGregg Abbott opened the Whiffies Fried Pies Food Cart in March 2009. Gregg grew up in Upstate New York and gradated from Lyme Central High School. Barry Davis is now the principal at Lyme Central, and back in the day, he was Gregg's social studies teacher. Mr. Davis remembered Gregg and shared that he was a very smart student.\n\nFrom there, Gregg went on to attend Evergreen State College in Olympia, Washington, and when he finished school, he got on a Greyhound bus and came to Portland to see a former girlfriend who had moved here. He left the bus terminal midday and walked across the Broadway Bridge, checking the city out on foot.\n\nGregg remembered walking back over the Steel Bridge just at dusk, stopping on the bridge and looking out across the water. Gregg said that it was right then, standing on that bridge, that he became captivated by Portland. The longer Gregg lived in Portland, the more he liked it. \"Portland was like a city version of Evergreen,\" he noted.\n\nGregg fondly remembered \"that sense of wonder you get when you first come to Portland.\" He found Portland to be like a huge, citywide performance art production, and he was excited to become one of the performers. Gregg believes that Portland is a place where you can do your own thing and make something of yourself.\n\n_I Can Do That: Inspired by Food Carts_\n\nGregg had wanted to open food cart of his own for eight years before he opened Whiffies. For a time, he had a job working downtown, and he used to have lunch at the Honkin' Big Burrito Food Cart and at the Divine Caf\u00e9 Food Cart. Gregg remembered thinking, \"I can do that. I could open a food cart and be in business for myself.\"\n\nWhile Gregg was growing up, his parents ran a catering company out of the family home, and Gregg got an insider's view into what it takes to succeed in the food industry. Gregg's dad was the chef, and his mom handled the books and the marketing.\n\n_This Is a Bad Idea_\n\nFinally, it was time to move ahead with his long-held dream of self-employment. Gregg found a food cart for sale on Craigslist and went to Seattle to pick it up. He met the couple and paid for the cart, and as he was hooking it up to the truck to haul it back to Portland, it started snowing. Gregg told himself, \"This is a bad idea.\" The winters he spent in Upstate New York paid off, and he made it home just fine.\n\n_Fried Pies_\n\nGreg wanted to make something that was easy to prepare and delicious to eat. He wanted to offer gourmet fair food. Gregg noticed that no one was offering hand-held fried pies, and he and his father, Dion, began to perfect recipes and figure out the menu.\n\n_It Was an Inside Joke_\n\nGregg and his dad, who is also a restaurant consultant, had an inside joke between them. Neither of them likes overpriced dishes that are insubstantial\u2014\"Like when a chef puts three spoonfuls on a plate and sells it for fifty dollars.\" They called those kind of meals \"Whiffies.\"\n\nGregg felt that his fried pies were going to offer people good food for a good value\u2014they would be anything but \"Whiffies.\" His dad once asked him, \"What are you going to name your food cart?\" Gregg answered back, mostly teasing, \"Whiffies.\" When it came time to fill out the business paperwork and put down an actual name, Gregg chose Whiffies. He needed a name right then, and that is what came to mind.\n\nGregg Abbott, owner of the Whiffies Fried Pies Food Cart.\n\n_12 th and Hawthorne_\n\nGregg had been to Potato Champion and loved it. He was convinced that his soon-to-be-open food cart would do very well there. Gregg called up the landlord and said, \"I am opening up a food cart, and Mike with Potato Champion says that I should put my cart at 12th and Hawthorne.\" The landlord replied, \"Well, if Mike says that you should be here, you are in.\" Later, when Gregg actually got to know Mike, he shared that story with him, and Mike got quite a chuckle out of it. Food cart owners do what they have to do to make it work.\n\nGregg has done well with his business, and it has been good to him. Gregg even met his future wife, Claire, at Whiffies. She came to the cart in 2009 for her birthday, met Gregg and subsequently sent him a message on Facebook. Greg saw that she had been posting pictures of food on her Facebook page, and he messaged her back on Facebook and suggested that they go out to dinner sometime\u2014they could both take pictures of the meal.\n\nThe next time you are near 12th and Hawthorne, stop by and get yourself a Whiffies fried pie.\n\n_What's Next_ : Gregg is planning to get a second food cart so that he can cater events. He gets a number of requests to have a Whiffies Food Cart come to events.\n\n_When You Go_ : There are so many good options! I love the barbecue brisket and the chicken pot pie. On the sweet side, I am very fond of the peach, the Oregon mixed berry combo and the \"Mounds\" (coconut pudding and chocolate).\n\n_Recipes_ : You can find a Whiffies recipes in the _Food Truck Cookbook_ by John T. Edge.\n\nFood Writer's Perspective:\n\nJEANNA BARRETT\n\nJeanna Barrett is the founder and chief dater at 50 Food Truck Dates (the50dates.com), a site about modern dating amid the national food truck craze. Jeanna goes on blind and \"real\" dates with men to food trucks in cities across the nation and then writes afterward about the date and the food. Part food blog and part dating rag, Jeanna hopes to simultaneously find love and not gain twenty-five pounds from street food. Through her project, she's also launched National Food Truck Date Day (February 14), a movement to change the negative sentiment around Valentine's Day. She resides in San Francisco by way of Seattle, but her food truck travels have taken her to Portland, Austin, New York City, Boston, Chicago, LA and Philadelphia. Where there are food trucks and\/or guys, you can find her.\n\nI first visited Portland's food truck scene in May 2013, ten years after the last time I had stumbled into Portland in my early twenties to stay at a two-story dive motel near the Doug Fir, where I cheers-ed a man in a bathrobe grilling hot dogs in the parking lot and sang Journey's \"Don't Stop Believin'\" at the top of my lungs through the Pearl District. To say my 2013 trip was different is an understatement, yet it was still hilariously similar to grilling hot dogs in a parking lot. This time, nearly a decade later, I came to eat my way through one of the more famous foundations of our nation's street food scene.\n\nWhen I was twenty-one, the food cart scene in Portland was in its infant stages\u2014something I didn't notice or hear about while walking the streets of the City of Roses. Now Portland's street food has grown into something so famous that it was my first chosen stop on a series of \"food truck pilgrimages\" I had planned in half a dozen of our nation's cities. My plan for these pilgrimages was to taste every food truck morsel I could possibly fit into a long weekend, coupled with a taste of the city's dating scene.\n\nI landed in Portland during a gray and drizzly May morning, and my baby sister picked me up so we could immediately start to tackle my list of more than twenty food carts I hoped to squeeze into four days of meals. We drove down SE Hawthorne in search of Fried Egg I'm in Love, my first chosen Portland food cart. I've been a huge virtual fan of the cart since stumbling across its account on Twitter because I love puns, breakfast sandwiches and the Cure. (Yes, social media is super important for food carts, otherwise traveling food writers like little ol' me wouldn't know about them!)\n\nWe saw food carts poking out on nearly every corner we drove by, and as I pointed and shouted, \"There's one! There's one over there!\" my sister exclaimed, \"It's like a mini treasure hunt!\" Yes, driving around to find one of Portland's best food carts really is like a treasure hunt.\n\nOnce we had moved to downtown for a food cart lunch, I was struck by the sheer number of carts that inhabit each corner of Portland. The city has truly been transformed by the pods and blocks of stationary carts painted in bright colors, often paired with picnic tables and community spaces. While very different from San Francisco's food truck scene, where the food is served in full-sized trucks that roam the city, always changing location, I love how Portland has built vibrant, intriguing food cart pod communities found everywhere throughout the city and has turned food carts into a way of life in the Pacific Northwest. My very favorite \"pod\" is located at 23rd and Alberta, which even features a beer cart to wash a drink down with your cart meal (beer and food is the best choice for dates).\n\nDuring my Portland culinary cart exploration, which included eating at nine food trucks in the first day, I found that while the number of food carts in Portland shows years and depth of street food knowledge that far surpasses many of America's cities, there are still a number of mediocre \"roach coach\" carts throughout the city. But take the time to do a little research and make special trips to each of Portland's pods, and you'll find more than a few hidden food cart gems as you explore. Here are some of my favorites.\n\nVada Holes from Tiffin Asha: I was able to snag a visit to Tiffin Asha during the first week it was open thanks to Steven, and I fell in love with food cart owner and chef Elizabeth Golay's south Indian food. Since most of the Indian food you'll eat in Indian restaurants is from northern India, southern Indian food is a unique little treat not to be missed. As a fan of sweet and savory, I was _obsessed_ with Tiffin Asha's Vada Holes\u2014a savory dal donut rolled in coconut-chili fleur de sel and served with sambar. The donut is warm and chewy, as well as perfectly spicy and sweet. They're served in orders of five and eight vadas, but really I wanted twenty-four, so I recommend starting out with four orders of the large vada holes. Don't mistrust an expert.\n\nBlack Pepper Dumplings (Khinkali) from Kargi Gogo: If I tell you that Kargi Gogo serves Georgian food, do you think peach pie, barbecue and sweet tea? Think again. Kargi Gogo is authentic Georgian street food from the Republic of Georgia, and the cart owners Sean and McKinze know how to cook it because they spent two years there while volunteering for the Peace Corps. The story of how the cart came to be is interesting; however, I was most intrigued by the steaming salt-and-pepper dumplings that were being handed out their cart window. There's no shortage of dumplings in different cuisines across the world, but the heavily black-peppered dumplings from Georgia are something different and special. So moist is the meat inside that it's actually swimming in juices and is better eaten upside down, holding the top of the dumpling like an upside-down top. First bite a hole in the dumpling and then suck the juices out and finish off the dumpling in an incredible spinning circle of spiced dough.\n\nChicken and Rice from Nong's Khao Man Gai: I feel like Nong's Khao Man Gai (or, simply, Chicken and Rice) is synonymous with talking about the success and deliciousness of Portland's food cart scene. Often called one of the most popular food carts, the chicken and rice dish is an unsuspecting treat, and if you only had one hour and one food cart to visit in Portland, I'd tell you to go now, run, to Nong's. When I first received the brown paper bag from a tall, loud and somewhat strange man who said to me, \"Today is the 144th day of the year!\" as I approached the cart window, I didn't know what to expect. Could anything sound more boring than \"chicken and rice\"? When you unwrap the white butcher paper, you'll find rice, chicken breast, a few slices of cucumber, pieces of cilantro and a little side cup of sauce. Pour the sauce all over the chicken and rice, stir and prepare to have your mind blown. I have no idea what's in that sauce, but it is by far one of the more exploding mouthfuls of flavor experiences I have ever had. From ginger to lemongrass to chilies, I couldn't place my finger on the dozens of flavors, but I knew that I wanted to eat that chicken and rice dish one hundred times over. You can even buy bottles of the sauce Nong makes now at the cart and specialty stores throughout Portland to bring home a little piece of Nong's Khao Man Gai to your kitchen, which I did. Two bottles in my usually-never-checked suitcase.\n\nNaughty in Nogales Tater Tots from Timber's Doghouse PDX: Often when I think of street food, I think of the greasiest comfort food that makes your mouth water and your belly ache. Who doesn't love a burger, hot dog or fries smothered and covered in cheese and gravy? Even better, who wouldn't love a big serving of tater tots, smothered and covered in jalapenos, grilled onions, pineapple, bacon, pepper jack cheese and spicy sauce, all served up in a dog bowl? The Naughty in Nogales Tater Tot bowl from Timber's is literally the messiest, dirtiest, spiciest, creamiest and naughtiest pile of salty comfort food that you'll ever need to order before you can die happy. You're welcome.\n\nThe FoPo Christo from the Egg Carton: I'm a huge fan of breakfast foods and bacon, so I was over the moon to hear about all the different breakfast food carts located throughout Portland\u2014so much so that I did my own miniature breakfast food cart tour while visiting. Again, being a huge fan of sweet and savory dishes, I fell in love with the FoPo Christo from the Egg Carton, which is an amazing combo of sweet and salty breakfast foods. The FoPo Christo is a twist on a Monte Cristo\u2014an oozy fried egg, bacon, Canadian bacon, cheddar cheese, spicy mustard and strawberry jam between two slices of French toast. A perfect food cart start to a Pacific Northwest morning.\n**PART II**\n\n**WE MOVED TO PORTLAND TO OPEN A FOOD CART**\n\n**W E MOVED FROM FLORENCE, ITALY, TO PORTLAND, OREGON, TO OPEN A FOOD CART: BURRASCA**\n\n\"I have a friend who is moving from Italy to Portland to open a food cart. Would you be willing to meet with him and help him?\" I was asked that question on April 29, 2013. I love a good story, and that sounded like a great one. The thing that amuses me to no end is that I was at the Italian Market Food Cart when I was asked that question. I am always amazed at how the dots connect.\n\nIn the fall of 2012, Erin and Andrew, owners of the Italian Market Food Cart, had a successful Kickstarter campaign. One of the pledge amounts allowed you to shave Andrew's head as a prize. A good friend of theirs, Rob, happily made that pledge so that he could see Andrew sheared like a sheep in the spring.\n\nRob lives in Philly, so I was asked to come over and remove all of Andrew's hair. If you Google \"The Shaving Food Cart TV YouTube,\" you can watch the fun eight-minute short that my friend Ken Wilson made to commemorate this event.\n\nShaving a food cart owner is hard work. To recover, I got an Ellsworth from the Italian Market and headed into the Belmont Station to enjoy a beer with my beef brisket reward. I was enjoying great food and great beer and having a wonderful time. Then I was asked if I would meet with someone name Paolo, who was moving from Italy to Portland to open a food cart. I handed over one of my business cards and enthusiastically said, \"Yes, I would be delighted to. When Paolo gets to America, have him call me.\"\n\nPaolo Calamai, owner of the Burrasca Food Cart.\n\n_Lunch with Paolo_\n\nA short while later, I got a phone call from someone who identified himself as Paolo Calamai. As soon as I heard him say his name, I immediately flashed back to that day at the Italian Market Food Cart. I couldn't wait to meet Paolo and hear his story.\n\nPaolo and I met at the Carte Blanche Food Cart on June 27, 2013. We shared a meal, and I got to hear a little bit of Paolo's hopes and dreams. His passion was palpable, and I knew right then that his still embryonic food cart would be very successful.\n\nAs we visited, I provided Paolo with some pro bono food cart consulting. We chatted about menus, possible names for his food cart, branding, social media and more. As I drove away from that meeting, I was filled with anticipation about what he and his lovely wife, Elizabeth, would develop.\n\n_The Food at Burrasca Is Off the Charts_\n\nPaolo has been cooking since he was a small child. In his youth, he spent many hours in the kitchen with his mama and nonna (grandma), learning how make the dishes he now serves at Burrasca. Each day that Burrasca is open, Paolo makes his own bread, one of the many things he learned long ago while standing on a stepstool in that Florence kitchen.\n\nThe last week of August 2013, Burrasca opened, and I finally got to sample Paolo's cuisine. Most of the dishes left me speechless. Paolo grew up in Florence, Italy, and Burrasca focuses on the food he grew up with. You can get dishes that are common to both the city of Florence and the region of Tuscany.\n\nOne of my favorites dishes at Burrasca is the Arista alla Fiorentina\u2014a roasted pork loin made with fennel pollen. Cooking with fennel pollen is something Paolo picked up from Dario Cecchini, a world-renowned butcher who, like Paolo, is also from Tuscany. What Paolo does with pork is masterful. The pork is tender, rich with flavor and perfectly seasoned.\n\nAnother amazing dish that you can get at Burrasca is Inzimino, a Florentine dish made with squid and spinach that has been simmered in red wine and herbs. Yep, you can get squid at a Portland food cart, and it is exquisite.\n\n_Paolo Fell in Love with the United States_\n\nPaolo first came to America for a vacation in 1984. He fell in love with the United States. He remembered being impressed with the open spaces, the national parks, the freedom and the sense that opportunity abounded. To Paolo, America was the place where people could successfully pursue their dreams.\n\nA few years later found Paolo living in San Francisco, working the front of the house in a restaurant there. It was in San Francisco in 1992 that Paolo met and fell in love with Elizabeth Petrosian. Their first date was to a winery for a picnic.\n\nEventually, Paolo and Elizabeth married and moved to Florence, Italy, where they started a family and had two beautiful children.\n\n_A Long-Held Dream to Open a Restaurant_\n\nSince his early twenties, Paolo has wanted to own his own restaurant. This is a dream that has been smoldering in his heart for many years. One of his biggest obstacles to this dream was that Paolo did not have the financial resources needed to do such a thing.\n\nIn the summer of 2011, Paolo and Elizabeth came to Portland for a vacation because they had dear friends who were living here. While they were here, they checked out the Portland food cart scene, and Paolo began to wonder about the possibility of maybe opening his own cart there. Upon returning home to Florence, Paolo and Elizabeth began researching what it would take to do just that.\n\nBy the end of the fall of 2011, they had decided that they were going to move across the globe and come to Portland to open a food cart so that Paolo could begin pursing his dream of owning a restaurant. They began the process of selling both their home in Florence and many of their possessions. They were going to invest these proceeds.\n\n_Leaving Florence_\n\nTheir last night in Florence, June 16, 2013, Paolo and Elizabeth were robbed by bandits using sleeping gas. They woke up to find that the small amount of jewelry they had\u2014jewelry with deep sentimental value\u2014had been stolen. They discovered that their wedding rings been taken off their fingers while they were knocked out. Elizabeth teared up when she talked about this incident and how violated she felt.\n\n_The Name_\n\nIn Italian, _burrasca_ means sea storm. This definition is very evident when you Google \"burrasca images.\"\n\nBurrasca is also the name of the main character in a much-beloved Tuscan children's book written in 1907 and 1908 by Luigi Bertelli under the pen name \"Vamba.\" Paolo and Elizabeth named their food cart Burrasca in honor of this precocious fictional boy who, like Paolo, loves Pappa al Pomodoro, a traditional thick Tuscan soup made with bread and tomatoes. And yes, you can occasionally get Pappa al Pomodoro at Burrasca.\n\nBurrasca. I love that name. It just works. To me it sounds very Italian and very interesting.\n\nThe Burrasca Food Cart.\n\n_Bone Tired and Very Happy_\n\nPaolo now has the joy that comes from finally being able to pursue a dream that he had held in his heart for a long time. He puts in many long days, working between ten and twelve hours each workday. Paolo comes home from the cart \"bone tired\" but \"very happy.\" Elizabeth says it is wonderful to see Paolo so fulfilled.\n\nI am just one of the many Burrasca fans who are very glad that Paolo and Elizabeth moved from Florence to Portland to open a food cart.\n\n_What's Next_ : For Paolo and Elizabeth, the dream is to see the Burrasca Food Cart become a brick-and-mortar restaurant. They have established a wonderful brand, and I believe that they will make it. I am not alone in this opinion. In October 2013, I was lucky enough to take John and Rita Duyn and Wendy Trunick from Carlton Farms on a food cart tour one beautiful Sunday afternoon. Our food cart adventure took us to four food cart pods, and we ate at seven different carts. John loved the food so much at Burrasca that we \"ran the menu.\" John was blown away by the food. Paolo uses Carlton Farms pork at his cart, and I had asked Paolo to make us Arista alla Fiorentina. Many times, John has eaten Carlton Farms pork that has been prepared at some of Portland's best restaurants by the city's best chefs. John proclaimed that Paolo's pork loin was some the best that he has ever tasted. John visited extensively with Paolo, asking about his story and listening to his hopes and dreams. When our adventure was over and we were in the car, John spoke very enthusiastically about Paolo: \"[He] is going to make it. He has what it takes to succeed as a restaurant owner.\"\n\n_When You Go_ : Burrasca, like the Cheese Plate PDX, has a seasonal menu. Every single item I have had at Burrasca has been stunning. Regardless of what is on the menu, just order the dishes that speak to you. Get at least two dishes. It will be some of the best food you have ever had. Of the dishes that I enjoy, my favorites include the Arista alla Fiorentina, the Pappa al Pomodoro, the Pappardelle al Cinghiale (handmade pasta and wild boar) and Crespelle (crepes filled with spinach and ricotta and topped with a b\u00e9chamel-tomato sauce).\n\n**P LAY THE _R OCKY_ MUSIC: THE ITALIAN MARKET**\n\n_We moved here for this, and this is our life. We aren't just doing this over the summer, and we aren't doing this part time. We are all in with this cart_.\n\n_\u2013Andrew Vidulich_\n\nErin Callahan and Andrew Vidulich opened the Italian Market on March 27, 2013, seven grueling months from when they had first expected to open it. Andrew and Erin met in 2010 at the Plough and Stars, Philadelphia's premier Irish pub. Soon they became a couple, and they discovered that both of them wanted to be self-employed. Erin had visited Portland during the summer in her college years to see friends. Erin took Andrew to Portland in 2010 and again in 2011. He loved Portland\u2014the beer, how clean it was and the slower pace.\n\n_We Can Do This_\n\nThey visited food carts during their first trip to Portland. During their second trip, in 2011, they went to see if maybe they could open a food cart. They visited a number of food carts and even took a very helpful food cart tour with Brett Burmeister. When they got back to Philly, they said, \"Let's do it!\" They began to work to come up with the funds to have a food cart built. Erin took all the extra shifts she could, and Andrew got a second job.\n\n_The Nightmare Begins_\n\nIn March 2012, they sent a food cart builder $9,500. They were to pick up their food cart at the beginning of September and pay the remaining $1,000 then. An empty shell of a small trailer was going to have a working commercial kitchen, with all appliances installed, as well as doors, service windows and so on. Over the summer, they were able to secure a spot in a downtown pod to put their cart.\n\nOn Thursday, August 30, Andrew and Erin's possessions left Philly in a truck headed for Portland. On Friday night, they had a goodbye dinner with family, and by Saturday morning, September 1, they were getting on a plane and moving to Portland to pursue their dream. They even had a house in Portland rented in which to live.\n\n_2,865 Miles Away_\n\nEarly on Friday morning, August 31, everything came crashing down. The phone rang at 5:00 a.m. It was their food cart builder calling to say that she was going out of business and that Andrew and Erin had until 5:00 p.m. at the end of the month to pick up the shell of their unfinished food cart. There was so much bad news to process: the cart was unfinished... _that day_ was the last day of the month...and they were in Philadelphia, more than 2,800 miles away.\n\nThankfully, Erin and Andrew had friends in Portland, Mike and Don, who had a truck with a hitch. The guys went and picked up the shell and put it in the driveway of Andrew and Erin's new Portland home. Andrew remembered being sick to his stomach on Friday night as friends and family were telling him goodbye and wishing them luck with their new business.\n\nOn Saturday, September 1, Andrew, Erin, James Brown and Frank Sinatra (their dog and cat) arrived in Portland. Their unfinished food cart was in the driveway. The money they had earned to finish it was gone, and they had no jobs. Welcome to Portland.\n\n_The world ain't all sunshine and rainbows. It's a very mean and nasty place, and I don't care how tough you are\u2014it will beat you to your knees and keep you there permanently if you let it. You, me or nobody is gonna hit as hard as life. But it ain't about how hard ya hit. It's about how hard you can get hit and keep moving forward\u2014how much you can take and keep moving forward. That's how winning is done_.\n\n_\u2013Rocky Balboa (Sylvester Stallone)_\n\nAndrew and Erin kept battling for their dream. On Tuesday morning, September 4, Andrew hit the streets to find a job. He went into more than one hundred restaurants looking for a job. He kept hearing \"summer is over and we have entered our slow season.\" He kept at it and eventually found a job, as did Erin.\n\n_Kickstarter_\n\nAndrew and Erin decided that they would do a Kickstarter to raise the funds to get their empty shell of trailer turned into a food cart. With no training in film and using the software that came with her computer, Erin made a Kickstarter video and submitted their project. Kickstarter approved their project, and it was scheduled to begin on October 22, 2012, and run until November 12, 2012.\n\nThey had twenty-one days to get strangers in Portland and friends and family on the East Coast to donate $11,000 so they could get their food cart opened. It felt awkward and uncomfortable to them to ask for assistance; however, they did not want to wait a year to once again work double jobs and save up the needed funds a second time.\n\n_Superstorm Sandy_\n\nOn October 29, eight days into the Italian Market Kickstarter, Superstorm Sandy made landfall on the East Coast. The Plough and Stars Pub in Philly had a fundraiser for Erin and Andrew scheduled during its October 31 Halloween party. That party got cancelled. Erin and Andrew had thought that they would get most of their pledges from the East Coast. They had only been in Portland seven weeks, and they did not really know many people here. They kept moving forward.\n\n_Rocky Balboa and Vince Papale_\n\nI asked Erin and Andrew how they kept going through all of these challenges. They told me that that between the day they arrived in Portland and the day seven months later when they opened their food cart, they watched two movies over and over again: _Rocky_ and _Invincible_. Those two inspiring Philly stories kept them going, especially through the first two weeks of their Kickstarter project. They told me that through all of the tough times, they would play the _Rocky_ music, keep going and keep battling.\n\nErin Callahan and Andrew Vidulich, owners of the Italian Market Food Cart.\n\n_Dreams Do Come True_\n\nIn spite of the second-costliest hurricane in U.S. history, friends and family back east did come through. The Plough and Stars rescheduled the fundraiser and helped out. Local media here in Portland found the story of this courageous couple and helped spread the word. Strangers in Portland did contribute. By November 12, their Kickstarter had raised $12,765. They made it. On March 27, 2013, the Italian Market opened for business.\n\n_The Cuisine_\n\nErin and Andrew proclaimed in their Kickstarter that \"cheesesteaks are for tourists.\" They wanted to bring the best of the Italian food they loved in south Philly to Portland. They were inspired by places like Dinic's, John's Roast Pork and Paesano's. They did share that if you are going to go to Philly and you want to get a cheesesteak, go to Steve's Prince of Steaks.\n\n_Paying the Price_\n\nThe first four months that the Italian Market was open, Erin and Andrew worked seven days a week from 8:00 a.m. to midnight. Food cart owners are a special breed, and they do what it takes to make it happen. Erin and Andrew are great people who own one of Portland's best food carts. I highly encourage you to give them a visit.\n\n_What's Next_ : Andrew and Erin hope to some day open up an Italian Market brick-and-mortar Philly-style sandwich shop. The very next step they want to tackle is doing more catering. They get a number of catering requests from people who have loved their take on south Philly food.\n\n_When You Go_ : Everything on the menu is wonderful, even the Passyunk, which is an eggplant sandwich suitable for vegetarians. I particularly love the Ellsworth (an Italian beef brisket sandwich), Uncle Mike's Meatball Sub, the Tasker (a Panzanella salad with crusty bread, tomatoes, cart-made mozzarella, fresh basil, onion, cucumber, olive oil and balsamic) and the pretzels. If you get a pretzel, get it \"wit\" Cheez Whiz.\n\n_Recipes_ : You can find two of the Italian Market recipes (Uncle Mike's Meatball Sub; and the Wet Pretzel, a Philly soft pretzel) in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**W E BRING THE CIRCUS TO YOUR DOOR: RETROLICIOUS**\n\nRoy and Kimmy Swope opened the Retrolicious Food Cart in May 2012. Most of Portland's food carts stay in one place and are not mobile. Roy and Kimmy's journey, though, has led them to have one of Portland's few mobile food carts.\n\n_Curry Chicken Salad_\n\nRoy and Kimmy Swope met in Phoenix, Arizona, where they fell in love and became a couple. Together, the two of them, both chefs who have been to culinary school, operated the Paper Plate Caf\u00e9 located in the Arizona Republic building in downtown Phoenix.\n\nRoy's curry chicken salad became the most popular dish at the Paper Plate Caf\u00e9. People would call and ask if they were going to be serving the curry chicken salad. If they happened to go several days in a row without preparing the much-loved salad, people would complain vociferously. Roy got to the place where he wondered if he would ever make anything else again.\n\n\"I had such good food out of those carts, I thought to myself, 'What a great entrepreneurial opportunity.' It intrigued me and I thought, 'We can do this,'\" said Kimmy. The day after Christmas 2010, Roy and Kimmy came to Portland for a vacation. They stayed at Hotel Lucia for a week. As they were out walking around, they saw some food carts, and Kimmy was drawn to them. She said to Roy, inquisitively pointing at one, \"What is that?\" Every day of their vacation, they ate at food carts.\n\nPhoenix had been good to Roy and Kimmy, and they had enjoyed their time there. Now they were ready for a change. Kimmy wanted rain. She was tired of the heat and the brown. Once they returned back to Phoenix from their vacation, Kimmy told Roy in her ever-so-sweet voice, \"I am moving to Portland, and you are invited to join me.\"\n\nBy the fall of 2011, they had decided to move to the Northwest to open a food cart, and they left Phoenix in January 2012\u2014only one year after that fateful vacation.\n\n_The Vagabond Diner in Corvallis_\n\nRoy and Kimmy had decided that they would renovate a 1957 Century trailer, call it the Vagabond Diner Food Cart and open it in Corvallis, Oregon. They were then offered free housing at a ranch out in Lacomb, Oregon, a small community about thirty miles east of Corvallis. They left Phoenix and headed for Lacomb.\n\nIn short order, they discovered two things: It was going cost too much to renovate the 1957 Century trailer into a food cart, and you can't really operate a food cart in Corvallis. Time to change plans. They found someone to build them a food cart and began looking for a pod in Portland in which to put it.\n\n_Drinking Wine and Playing Gin_\n\nWhile they waited for their new food cart to be built, Roy and Kimmy would sit in the dining room of their Lacomb home, drink wine, play gin and talk about what they would name this new cart. For them, the name Vagabond Diner was attached the Century trailer, and if they weren't going to use that trailer, they weren't going to use the name they had given it.\n\nThey decided that they wanted to cook what they called \"old-school comfort food.\" They wanted it to be retro and delicious. As they talked about those concepts, they hit on the name Retrolicious. That name fit exactly with the brand story they wanted to tell.\n\n_On the Road_\n\nIn May 2012, they opened Retrolicious at a Portland food cart pod. There was one big challenge: they still lived in Lacomb, Oregon. While Lacomb had only been thirty miles from Corvallis, it was eighty miles from Portland.\n\nFor the next three months, they drove an hour and forty-five minutes each way to their food cart. They did this five and sometimes six days a week. Moving from Lacomb to Beaverton in August was a huge relief. As they approached their first winter and things started slowing down at the pod, Roy and Kimmy began looking for opportunities to take their food cart to places where people were because they weren't at that pod.\n\nThey ended up pulling out of the food cart pod and heading to business parks and large corporations to sell lunch to the employees, who were more than ready to try the new food cart that was coming to them. Every new food cart owner needs a viable game plan to survive the first winter, and Roy and Kimmy had found theirs. They did put the Retrolicious Food Cart back into a pod the next spring; however, by the middle of summer, they discovered that they liked being mobile. They ended up leaving that pod as well.\n\n_Your Fingernails Are Never Painted_\n\nOccasionally, I get asked about culinary school. I talked with Roy and Kimmy about this. They both graduated culinary school, they worked in restaurants as employees, they owned a brick-and-mortar restaurant and they now had a successful food cart business.\n\nMy first question was, \"Should someone work in a restaurant before they attend culinary school?\" Both Roy and Kimmy emphatically believe that if you think you might want to go to culinary school in order to pursue a career cooking professionally, work in a restaurant first, even if you have to start out in the dish pit. Kimmy shared, \"It is not glamorous. People without a professional cooking background watch the Food Network and think that cooking is a glamorous profession. It is hard work, at times exhausting; your fingernails are never painted...It is one thing to like to cook at home for family and friends; it is quite another to enjoy cooking professionally and doing it ten hours or more a day, day after day.\"\n\n_Salt Is an Algorithm_\n\nI then asked Roy and Kimmy whether they thought that someone should attend culinary school before opening a food cart. They were quick to say that many times they have enjoyed amazing food at food carts owned by people who did not attend culinary school.\n\nThat being said, they both noted that if at all possible, a person would do very well to attend a \"good\" culinary school, especially if you have plans to grow your food cart business beyond just one cart. Roy explained:\n\n_A good culinary school teaches many, many things that you don't learn watching the cooking channel\u2014mother sauces, how to layer flavors, helpful culinary history_.\n\n_People forget that a food cart is a business. The fewer times you need to touch food before your serve it, the quicker you can get it out to the table and the more profitable you are. Culinary school teaches you systems and fundamentals that allow you to cook more efficiently_.\n\n_It is one thing to cook a dish for four people; it a different thing altogether to scale up a recipe to make it for forty people or four hundred people or two thousand people. You learn how to scale up recipes in culinary school_.\n\n_Take salt, for instance. If a recipe that produces four servings calls for a teaspoon of salt, you do not use ten teaspoons of salt when you scale that recipe up to make forty servings. Salt, like a number of ingredients, does not get used in an exponentially linear fashion as you scale up a recipe. Salt is an algorithm_.\n\n_Being Mobile in Portland_\n\nWe do not have what I call a \"food truck\" culture in Portland. The vast majority of our food carts can be found in the same place day after day, and their customers go to them.\n\nRoy and Kimmy, on the other hand, show up where you are. Four to five days a week, they can be found serving lunch at a business park or at a large corporation. They have settled into a mostly regular schedule. For example, their current schedule has them at Nike every Tuesday and at Xerox every Wednesday.\n\nThey also provide the food at a number of upscale corporate cocktail parties, and they do lots of weddings. It is common for people to decide that they would love to have a food cart come to their wedding and cater it. It is then that they discover that most Portland food carts are not mobile, and the cart you want may not be able to come to your wedding. That is why a number of food cart owners are looking at getting a second cart that they can use for events.\n\nRoy and Kimmy Swope, owners of the Retrolicious Food Cart.\n\nWhen Roy and Kimmy cater a wedding, they love to ask about dishes that would invoke good memories for the family. Given their extensive culinary background, Roy and Kimmy are able customize a menu that fits your wedding and your family.\n\n_The Circus Has Come to Town_\n\nWhen Kimmy was a child, she experienced the joy of the circus coming to her town. It was not always there, but when it rolled into town, you went to see it before it moved on to its next stop. When Retrolicious arrives at a business park that it has not been to since last week, and they open their window to see a line of people ten deep waiting to order, her heart swells with gratitude. Now she has the joy of being the one who brings the circus to town.\n\n_What's Next_ : Roy and Kimmy can imagine someday opening a small, intimate brick-and-mortar restaurant where you could get a glass of wine with one of Roy's amazing entr\u00e9es. I wish that they would bottle their salad dressing that they affectionately have labeled \"salad gravy.\" It is really good, and I would buy it. Roy and Kimmy also cook with a few custom-made spice mixes that are tremendous. I hope that someday they follow in A Cajun Life's footsteps and package a few of them.\n\n_When You Go_ : Given that the Retrolicious is mobile, you can't just head over to the pod and see Roy and Kimmy. However, if you are ever lucky enough to have them cross your path and bring the circus to your town, you will not be disappointed. Roy is one of the best chefs I have encountered. I particularly love their chicken and waffles, the Southern Pimento Mac and Cheese and Damn That Castro\u2014a Cuban-style roasted pork sandwich. If Roy ever has a hamburger on the menu as a special, _get it_! With many apologies to my beloved In-N-Out, I have to admit that Roy makes the best hamburger I have ever had. If Kimmy has whipped up a batch of her red velvet cupcakes, buy them too. Don't just buy one or two; buy a dozen and take them to share with your friends and family. They are that good.\n\n_Recipes_ : You can find three Retrolicious recipes (Creamy Pimento Mac and Cheese with Garlic Kale; Damn That Castro; and Snickerdoodles) in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**W E JUST HAD TO GET THE FOOD INTO THEIR MOUTHS: PBJ'S GRILLED**\n\nWhen Keena Tallman was in grade school in Hood River, Oregon, her grandfather Manuel would often make her carnitas with Serrano chilies. One day, he finished off this dish with local Hood River cherries, and Keena was delighted with each mouthful.\n\nThe PBJ's Grilled Food Cart has an eclectic, delicious menu that can be traced back to that warm summer day when Grandpa revealed that spicy and sweet can be combined to become something unexpected and amazing.\n\nNow when I am at one of the PBJ's Grilled locations and I enjoy a Hot Hood\u2014challah bread, PBJ's peanut butter, roasted jalape\u00f1o, apple wood bacon and black cherry jam\u2014I tip my food cart hat to Keena's grandpa.\n\n_The Oregon Country Fair_\n\nIn 1993, Keena visited the Oregon Country Fair with her brother, James. James had disappeared, and when Keena found him, his hands were full with six peanut butter and jelly sandwiches that had been grilled in butter. He devoured all six and stubbornly would not share.\n\nKeena was so fascinated by this bizarre sandwich concept that she found the vendor and got her own and absolutely loved it. It was creamy and grilled and salty and sweet. That quirky sandwich was life changing. Those two stories are the foundation of the PBJ's Grilled Food Cart.\n\n_Grilled Peanut Butter and Jelly Sandwiches_\n\nYep. We have a magical food cart in Portland that specializes in grilled peanut butter and jelly sandwiches. You cannot believe how good these crazy sandwiches are. My son Zayne loves the Smoking Goat\u2014Kalamata olive bread, PBJ's almond butter, apple wood smoked bacon and apricot jam. My daughter Zoe loves the Oregonian\u2014challah bread, Oregon hazelnut butter, Rogue Creamery blue cheese and cart-made Marion berry jam\u2014and she always includes the duck that is an optional add-on. You read that right. Here in Portland, you can get a grilled peanut butter and jelly sandwich that comes with duck.\n\nThe Smoking Goat from PBJ's Grilled\u2014Kalamata olive bread, PBJ's almond butter, apple wood smoked bacon and apricot jam.\n\n_The Dream to Be Self-Employed_\n\nBusiness partners Keena Tallman and Shane Chapman opened PBJ's Grilled in the summer of 2010, and they have been very successful. Shane and Keena first met in Las Vegas in 2008. Both of them had wanted to be self-employed for as long they each could remember. Together, they began to conspire about opening a business.\n\nShortly after meeting each other, they took a short trip to Portland, Oregon, and upon returning to Vegas, Shane began researching what it would take to open a food cart in Portland. But what cuisine would they have?\n\nKeena remembered back to that fateful day in Veneta, Oregon, at the Oregon Country Fair, and she made Shane a grilled peanut butter and jelly sandwich, telling him, \"Let's do a grilled peanut butter and jelly food cart.\" Shane immediately said, \"Absolutely not! Who would go to a food cart like that?\"\n\nThe next day, Keena began to dream about making gourmet grilled peanut butter and jelly sandwiches that adults would love, and she plied Shane with the kind of sandwiches that she would make. Eventually, Shane cried uncle, and they had a cuisine for their food cart.\n\n_From Vegas to Portland_\n\nIn the fall of 2009, they sold everything, moved to Portland and began the process of finding and opening a food cart. They each got two jobs to raise additional capital. They got their first logo, planned what kind of culture they want the business to have and bought a food cart that they found in Salem. Then they found a location.\n\nThey opened on the corner of Kearney and Lovejoy. That location required them to set up and take down the cart _each_ day. This went on for six months. A customer who was a huge fan one day suggested that they move to a spot that he could lease them at 23rd and Lovejoy. That spot has been home to their first food cart ever since.\n\n_Samples Were Key_\n\nEarly on, people were bewildered and confused by the concept of grilled PBJs. To solve this, Shane and Keena would go out onto the sidewalk and hand out samples. Keena was convinced that \"if people would eat the sample, they would get the flavor. We just had to get the food into their mouths.\"\n\n_What Would Bo Do?_\n\nIn January 2011, they started to hear about this guy named Bo Kwon whose food carts empire was taking off. They decided to do whatever Bo was doing. Bo had a food cart out at Nike for lunch, and so PBJ's Grilled had to sell lunch out at Nike, too. That plan of doing what Bo was doing has paid off handsomely.\n\nToday, PBJ's Grilled has eight to ten employees, a space at the Beaverton Farmer's Market, a busy catering enterprise and two food carts, including one at Cartopia, one of Portland's most iconic food cart pods.\n\n_What's Next_ : Keena and Shane are dreaming about having a brick-and-mortar PBJ's Grilled Caf\u00e9, with both cold milk and cold beer on draft, and I would not bet against them making this happen.\n\n_When You Go_ : Get two PBJ's Grilled sandwiches\u2014a savory one and a sweet one. For savory, I love the Hot Hood or the Smoking Goat, and for sweet, I love the Joy and the Pumpkin Pie.\n\nFood Writer's Perspective:\n\nAMY BURGLEHAUS\n\nHi, I'm Amy, and I blog at EatingMyWayThruPortland.com. I can't promise to resolve any dining debates, but I can help you decide where you might like to go. You see, I like food. I like it a lot! I am always on the lookout for something fun and creative, and if you don't already know, Portland is booming with places that are fun and creative.\n\nYou can also follow me on Facebook as I make an effort to keep an eye on what's going on in the Portland food scene and post upcoming events. There are so many things happening in Portland all the time that it's hard to keep up, but I try. If Twitter is more to your liking, I also send out information there as well.\n\nPortland has so much to offer, but I have to say I really enjoy the Portland food carts. Here's why: these are individuals or teams doing their own thing and doing what they love. They are passionate about food, and all they want to do is share it with you. I love that they choose something specific and really focus on it. You can find food carts that are all about cheese, crepes, pasta, Guam food, waffles, ice cream, eggs, wraps, Norwegian food, sandwiches, pork and fish\u2014the list goes on and on. When you walk up to a cart, you never know what you might be in for. I especially love that!\n\nI enjoy being able to talk with the food cart owners, who are also the chefs. I love hearing their stories, their struggles and their joys. When the space is small and the setting more intimate than a restaurant, you get to have that kind of a conversation with the chef. Struggles, you ask? Remember, it rains in Portland eight months out of the year. This is a tough one for these owners. Many of the pods outside of the downtown carts have really adapted well to the weather. They have covered seating, allowing you to enjoy their fare all year long.\n\nAnother thing I love about Portland food carts is that many are ecofriendly, which is very Portland, and I love it. I have been to a number of carts where my entr\u00e9es have come out on real plates with real silverware. Now I'm eating for a fraction of the price on a patio with amazing food and real plates. I sometimes forget I just walked up and ordered from a window.\n\nWith all the food carts in Portland, it would be nearly impossible to hit every one of them, but it's fun trying. With all the effort I put into finding the best, I would have to say that these are a few of my favorites.\n\nThe cart at the top of my list right now is Carte Blanche. The food that they serve is outstanding. The flavor is bold, beautiful and very memorable. Aside from the amazing food, I love that these girls are in their mid-twenties, and they cook with such incredible depth\u2014it just wows me. They change their menu often, so I can't recommend a particular dish. I just recommend you don't miss this cart.\n\nGaufre Gourmet is a fantastic waffle food cart. Forget about the sweet waffles with strawberries and whip cream (which are wonderful, by the way). Step out of your comfort zone and order a savory waffle. It might sound scary at first, but your taste buds will thank you. I highly suggest ordering the Spicy Goat.\n\nAnother favorite that I send many people to is the Cheese Plate. Order the Cheese Lover's Oregon Cheese Plate. You will get Oregon cheeses and cart-made crackers. The presentation is so wonderful, but I don't want to spoil it for you. Go see for yourself and be prepared to feel like you are on a perfectly planned picnic.\n\nMy family's all-time favorite breakfast food cart (yes, we even have a few of those here in Portland) is the Egg Carton. They had our attention from the moment I read about their Eggs Benedict. Yep, Eggs Benedict from a food cart, and they are perfect every time. Another item you can't miss when you visit this food cart is the FoPo Monte Cristo.\n\nAs you can see, it is easy to become passionate about the Portland food scene, especially the food carts. These vendors make my blogging a fun and easy job.\n**PART III**\n\n**BREAKFAST!**\n\n**T HIS COULD RUIN OUR RELATIONSHIP: THE BIG EGG**\n\nGail and Elizabeth opened the Big Egg Food Cart on September 21, 2009, at the brand-new Mississippi Marketplace Food Cart Pod. When they met with Roger Goldingay to talk about possibly placing their cart at his pod, it was still a field. Roger took more than seventy applications for the handful of carts that first opened at Mississippi.\n\nThe Big Egg was one of the carts that made the cut. One thing is for sure: when it comes to food carts, Roger can pick the winners. Today, the Big Egg is the only food cart left from that original group. It has not only survived, it has thrived as well.\n\nThe Big Egg opened on a Monday, and it was busy from the very beginning. The first two weeks, it was word of mouth that brought lots of people to the cart. Sixteen days from its first day in business, Karen Brooks wrote an article listing the top ten food carts of 2009. The Big Egg made the cut. From the day that article came out, it has had a \"sea of people\" coming to the food cart.\n\nKaren's 2009 top ten list included iconic Portland food carts like KOi Fusion, Nong's Khao Man Gai, Potato Champion and Tabor\u2014heady company indeed. The Big Egg most definitely belongs on a list with carts like these. Not only is Karen Brooks a very talented food writer, she also knows great food cart talent when she sees it.\n\n_Hands Down the Best Breakfast Sandwiches You Will Ever Have_\n\nI know that as Gail and Elizabeth are reading this chapter, they are both proud and slightly uncomfortable. They are very private, behind-the-scenes people. The last thing they want is the spotlight. They are also very humble. Even though it will make them cringe, I have to tell you: they can flat-out cook! There is a reason that they sell out almost every day.\n\n_Why Breakfast?_\n\nWhen I asked them why they feature breakfast at their food cart, Gail and Elizabeth said, \"We are obsessed with breakfast. It is our favorite meal.\" For years, they had wanted to open a small, intimate brick-and-mortar caf\u00e9 together and call it the Big Egg. When they were finally ready to pull the trigger on this dream, it was 2008, and the economy was crashing. They realized that at that point a brick-and-mortar Big Egg was not possible. Then they decided to start with a food cart. (That caf\u00e9 is still in the plans.)\n\n_Sauces, Jams, Curds, Butters and Marmalades_\n\nOne of the things that makes the food at the Big Egg so amazing is the cart-made sauces, jams, curds, butters and marmalades. Every sandwich gets one, and it adds that little extra that makes the meal extraordinary. Here is some of what I am talking about: herbed aioli made with thyme, rosemary and cayenne; balsamic aioli; lemon aioli; lemon tarragon aioli; yogurt lime sauce; roasted poblano sauce; apricot jam; tomatillo-lime marmalade; chili orange marmalade; kiwi serrano marmalade; carrot orange marmalade; quince marmalade; apricot saffron jam; sweet potato jam; chester blackberry jam; strawberry kumquat jam; nectarine jam; rhubarb orange jam; rhubarb kumquat lavender jam; rhubarb apple butter; honeyed tomato habanero butter; pumpkin butter; pear butter; orange curd; ginger lime curd; apple curd; strawberry-lemon curd; and grapefruit curd. See what I mean?\n\n_Seven Feet by Fourteen Feet_\n\nGail and Elizabeth are business partners and life partners. An unsung chapter in the Big Egg story is how Gail and Elizabeth have managed, maintained and nurtured their relationship through the white-water ride that is a startup. Even before the cart opened, they realized that working together ten-, twelve- and, in the beginning, sixteen-hour days in a ninety-eight-square-foot workspace could ruin their relationship.\n\nAs an entrepreneur in a relationship myself, I know how easy it can be to focus on the startup and overlook the relationship. I have seen more than one couple open a food cart and lose the relationship, and I have seen the tears of aguish that come from that path. An excellent resource for people in a relationship with someone pursuing a startup is _Startup Life: Surviving and Thriving in a Relationship with an Entrepreneur_ by Amy Batchelor and Brad Feld. I can't recommend this book highly enough. The philosopher Kahlil Gibran had some wise words on the subject as well:\n\n_Let there be spaces in your togetherness, and let the winds of the heavens dance between you. Love one another but make not a bond of love: Let it rather be a moving sea between the shores of your souls. Fill each other's cup but drink not from one cup. Give one another of your bread but eat not from the same loaf. Sing and dance together and be joyous, but let each one of you be alone...and stand together, yet not too near together: For the pillars of the temple stand apart, and the oak tree and the cypress grow not in each other's shadow_.\n\nWhen they opened their food cart, Gail and Elizabeth did not know what specific challenges their relationship would face, but they knew that rapids were ahead. When those rapids came, they worked through them as they went along. Each relationship will have different needs. Given that they are both introverts, Gail and Elizabeth learned that they each needed alone time and learned to be careful to give each other that space.\n\nSo many food carts owners and people in startups underestimate how hard it will be to keep the relationship healthy and pay the price necessary to lead a startup to success. If you are in a startup and in a relationship, I challenge you to follow the trail that Gail and Elizabeth, as well as many others, have so wisely blazed.\n\n_What's Next_ : Coming soon...the Big Egg Caf\u00e9. Elizabeth and Gail will make the leap to a brick-and-mortar restaurant. If they need to do a Kickstarter to help make this dream come true, I know they can count on me and many of their fans to gladly donate money. Gail and Elizabeth, I hope that you are still reading. Your fans are with you, and we believe in the beauty of your dream to open a Big Egg Caf\u00e9. I am also looking forward to being able to someday buy _The Big Egg Jams, Butters and Marmalades Cookbook_.\n\n_When You Go_ : Get the Arbor Lodge Sandwich. Is it breakfast sandwich perfection. Also, get whatever special they happen have. You will love it.\n\n**S TEPPING INTO A NEW LIFE: THE EGG CARTON**\n\nTim and Sarah opened the Egg Carton Food Cart on Sunday, May 27, 2012. They had wanted to open on Saturday, May 26; however, as is common with food carts, challenges prevented that from happening.\n\nWhen Sarah first started getting serious with Tim in early 2008, she told him that she wanted a new life that included a different career and a good marriage. Those are dreams that many people have. Sarah and Tim have actually pulled it off. Here is their story.\n\nSarah grew up in Wasilla, Alaska. She calls herself \"the other Sarah from Wasilla.\" From a young age, Sarah loved to cook. Growing up in a small town, she had no idea that a woman could go to culinary school and pursue a culinary career. Sarah went to college in Colorado, and following college, she ended up working in retail management and corporate training. Along the way, Sarah was given the opportunity to move to the Northwest and also given a choice between Portland and Seattle. Sarah visited both cities, and Portland was the city she chose. Here is where she would put down roots.\n\n_Are You Being Catfished?_\n\nIn 2007, Sarah met Tim online playing World of Warcraft. Tim had grown up in Michigan, and when he met Sarah online, he was finishing up law school in Lansing, Michigan. In early 2008, they began chatting while playing the game, and that quickly led to e-mailing and connecting via social media.\n\nThey decided to meet face to face in April 2008. Tim flew out to Portland to meet Sarah. Both friends and family of Sarah's and of Tim's were worried that maybe they were being scammed. Tim had friends who were concerned that maybe Sarah would turn out to be a man. Sarah had friends who were worried that Tim would turn out to be a serial killer or a very bad guy.\n\n_The Rules and the Dreams_\n\nWhen Sarah realized that things were getting serious with Tim, she told him her \"rules\" and her dreams. Sarah had had a previous relationship that left her with some pain, and she did not want to go though that again. Even before she met Tim, she told him a number of things, including her desire to someday soon change careers to an as-yet-undetermined occupation.\n\nSarah's ex would not taste her cooking. One of Sarah's rules was that Tim would need to have at least one bite of whatever she made. Tim was fine with all of the rules and dreams that Sarah shared with him. She was really hoping that this face-to-face meeting would go well.\n\nSarah and Tim Arkwright, owners of the Egg Carton Food Cart.\n\n_Do You Want to Go Get Donuts?_\n\nTheir April 2008 meeting went so well that they were married by August that same year. Early in 2008, Sarah told Tim about Voodoo Doughnuts and sent him a link to Voodoo's website. Tim noticed that people actually got married at Voodoo Doughnuts.\n\nAs Sarah and Tim were dating, the phrase, \"Do you want to go get donuts?\" became a euphemism for the possibility of them getting married. When the time came for a wedding, they decided to literally go get donuts. Sarah and Tim got married at Voodoo Doughnuts, and Cat Daddy officiated. Cat Daddy (whose real name is Kenneth Pogson) is one of the founders of Voodoo Doughnuts.\n\n_Occupational Changes_\n\nTim gradated law school and had intended to go into business law. By now, he was living in Portland with his wife, and due to the economic downturn, jobs in his field were scarce. He ended up opening his own firm. In April 2010, Sarah had just finished up her MBA but was laid off due to the recession. She received a small severance package. Tim told her that this the perfect time for her to reorganize and find another career. Sarah was thinking of maybe going into marketing or social media.\n\n_Julia Child and Blogging_\n\nTim and Sarah watched the movie _Julie & Julia_, and Tim very encouragingly said, \"You can cook like that. Let's get you Julia Child's cookbook.\" In the summer of 2010, Sarah got some chickens and started a blog she called the Improbable Farmer. She began to write and cook. She focused on flavor and technique. Her newly acquired chickens began to lay a lot of eggs, and so she found herself making many egg dishes. Tim upheld his agreement and tasted every one of those dishes. He also loved them and cheered Sarah on as her culinary skills grew.\n\n_The Best Hollandaise Sauce I Have Ever Had_\n\nThe first time I ever visited the Egg Carton, I was delighted to see Eggs Benedict on the menu. I love hollandaise sauce, and Sarah's blew my mind. Early on, in December 2010, Sarah accepted a blog challenge to attempt to make hollandaise sauce for the second time in her life. The whole experience was a fiasco that included Sarah cooking in her freezer in an attempt to save her creation. Her sauce was separating, and she needed to cool it down.\n\nSarah ended up switching to Julia Child's recipe, making hollandaise sauce over and over again until it was perfected. When the time came to create the menu for the Egg Carton, it was Tim who strongly encouraged Sarah to put her Eggs Benedict on the menu. I am so glad that Tim did that.\n\n_It Started with One of Tim's Signature Puns_\n\nBy the beginning of 2011, Tim was ever so gently suggesting to Sarah that it would be financially helpful for her to pick a career and start earning some money. In the spring of 2011, they were driving down 82nd Avenue and passed the Cartlandia Food Cart Pod. Tim turned to Sarah and said, \"Hey, you can make awesome egg dishes, and you have a lot of eggs. Why don't you start an egg food cart\u2014you can call it the Egg Carton!\"\n\nThe Egg Carton Food Cart.\n\nOne month later, they decided that they were going to open a breakfast food cart featuring egg dishes. They worked on a name for their food cart for months before finally embracing the \"Egg Carton.\" By August 2011, they had an LLC and had purchased a food cart as well as a domain. Time to open the cart! Or was it?\n\n_Nine Months Battling Fear_\n\nThe Egg Carton did not open until May 2012 because Sarah was nervous. She was afraid of failure, and she worried about how people would receive her food. She wrestled with not being a \"real\" chef. A number of crippling thoughts went through her head. She did not go to culinary school\u2014what business did she have opening a food cart? Will people even like her food?\n\n_The First Customer_\n\nSarah and Tim tried to open on Saturday, May 26, 2012. They had even announced on Facebook and Twitter that Saturday was going to be their first day open. I love it when I get to attend a food cart's first day. I enjoy getting to see them progress from that first day to a grand opening a month or two later and then to their one-year anniversary. I was coaching my son Zayne's baseball team, and we had a tournament game that morning that kept me from visiting the Egg Carton on their first day.\n\nAs it goes with food carts, Tim and Sarah were not able to actually open until Sunday morning. Sarah had not conquered all of her fears, and as she opened her food cart serving window for the first time, she was very anxious. Her fears came crashing down around her a few minutes later when her first customer ever showed up, and she recognized him.\n\nSarah calmly took the order\u2014Eggs Benedict and three sides: a fresh fruit bowl with cantaloupe, strawberries and blueberries; rosemary potatoes; and bacon. Once her customer had gone and sat down in the tent to wait for his order, Sarah began _freaking out_. She was cursing and shaking and trying to cook. She told Tim, \"#@x%. Our first customer is a #$%@!*& food writer!\" She was so nervous that she burned the rosemary potatoes, and what's more, she was so shook up that she ended up serving those very same burned potatoes to her customer by mistake.\n\nSarah put her dishes on a tray and brought them out to the customer, who happened to be me. I fell in love with her hollandaise sauce. The burned potatoes did not even faze me. As far as I knew, it was their second day. Stuff happens when you are brand new. It was not until September 24, 2013, while interviewing Sarah and Tim for this chapter, that I found out the rest of the story and learned that I was their first customer ever.\n\n_In Portland, Dreams Do Come True_\n\nTim and Sarah have a successful food cart business. In July 2012, a few months after the Egg Carton opened, Tim closed down his law firm. Tim has worked side by side with Sarah in the cart since day one. \"In Portland, you can just dream something up and do it!\" said Sarah.\n\nEarly on in their relationship, Sarah told Tim that she wanted a new life and a new career. She was in retail management, and he was a lawyer. Now they are food cart owners and are living the life she always wanted. As she told me, \"Really? People want to buy my egg dishes, and I can pay my bills doing this? How awesome is that!\"\n\nSarah was right. In Portland, dreams do come true.\n\n_What's Next_ : Sarah hopes to write an Egg Carton cookbook. They are also planning on packaging Egg Carton Egg Spice. Someday they would like to open a brick-and-mortar Egg Carton Caf\u00e9 specializing in breakfast and featuring Sarah's amazing egg dishes.\n\n_When You Go_ : Get the Eggs Benedict. It comes with the best hollandaise sauce you will ever have. I also really like the Famous FoPo Cristo\u2014fried egg, chunks of bacon, Canadian bacon, cheddar cheese, spicy mustard and strawberry jam between two slices of French toast dusted with powdered sugar. Occasionally, Sarah will create a French toast special, and they are fabulous too!\n\n_Recipes_ : You can find two of the Egg Carton recipes (the Popeye with Roasted Red Pepper Aioli; and Rosemary Sweet Potato Home Fries) in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**T HE RESTAURANT THAT OPENED A FOOD CART: YOLK**\n\nDonald Kotler opened the Yolk Food Cart on Monday, September 5, 2011, fulfilling a long-held dream to open an establishment on Woodstock Boulevard. Donald grew up on Long Island, New York. He got his first food job at sixteen years old working at a Dunkin' Donuts in Coram, New York. Until Donald moved to Portland in 1993, New York was home base.\n\n_Following the Dead_\n\nStarting in the mid-1980s, Donald began occasionally following the Grateful Dead. He would follow them for a summer tour, a fall tour or a spring tour. This allowed Donald to see the United States and the national parks, camp and travel through most of the states. Donald paid for these trips by being a rogue cook. He would set up outside the concert area and sell food to those heading into or out of the concert. In August 1993, Donald followed the Dead to Eugene, Oregon.\n\nDonald Kotler, owner of the Toast restaurant and the Yolk Food Cart.\n\nWhen Donald told me this story, I smiled. In August 1993, my good friend Jerry Riecken took me from Olympia, Washington, to Eugene, Oregon, for my one and only Grateful Dead concert. I remember seeing those rogue cooks. Kind of funny to think that twenty years ago, Donald and I were both at that same concert. What a long, strange trip it has been.\n\n\"I came to Portland to stay for a month, and I never left,\" said Donald. Following the Grateful Dead concert in Eugene, Donald realized that the only states he had not yet seen were Oregon, Alaska, Washington and Montana. Donald had a friend who lived up in Portland, and he decided to stay with that friend for a month or so. Once Donald arrived in Portland, he realized that that he had finally found the place where he wanted put down roots and call home.\n\n_A Restaurant of His Own_\n\nFourteen years later, in 2007, Donald opened Toast. For many years, Donald had wanted to open his own restaurant. In fact, prior to opening Toast, Donald was working three jobs so that he would have the funds to move ahead with his dream when the right space became available.\n\nDonald kept looking on Woodstock Boulevard for a place to open a restaurant. He had lived in the Woodstock neighborhood since moving to Portland, and for many years, he had dreamed of having his own place on that well-traveled thoroughfare.\n\nIn 2006, a location at 5222 SE 52nd Street, not too far from Woodstock Boulevard, became available. Donald spent nine months swinging hammers and renovating the space that for thirty years had been occupied by the Bad Ass Adult Video store. The neighborhood was very happy with the change that Donald was bringing.\n\n\"Breakfast has always been a part of me. I love breakfast. I can have breakfast for breakfast or dinner or late night. To me, egg yolk is the perfect sauce,\" said Donald. Indeed, from the beginning, Toast specialized in breakfast. I have eaten breakfast at Toast many times, and if you go, be sure to get a side order of griddle coffee cake. It is incredible. Before Toast, I had never had or even heard of fried coffee cake. I asked Donald about this dish, and he sat back, smiled, got a faraway look in his eye and told me a story.\n\nWhen Donald was eight years old and living in New York, his grandfather would take him to a diner in Queens that sat twelve and only had counter seating. You would stand behind someone siting at the counter and wait for them to finish. Sometimes the line for one of those twelve seats was four people deep. Donald's standard order there was three pieces of bacon and a grilled corn muffin. The griddle coffee cake at Toast is an homage to those cherished diner meals Donald had with his grandfather.\n\n_Time to Expand to Woodstock_\n\nIn 2011, Donald had the resources necessary to open a food cart, and to him, that seemed like a great way to expand and grow his business. Donald had always loved food carts and food trucks, and he was excited that he was going to be able to open one himself. When Donald saw a Craigslist ad for a food cart space right on Woodstock, his heart soared. He was finally going to get to be on Woodstock.\n\nPatrick makes the \"food cart magic\" happen at the Yolk Food Cart.\n\n_Patrick Makes the Food Cart Magic Happen at the Yolk Food Cart_\n\nMost days, it is Patrick who is the person doing the cooking at Yolk. He had worked for a number of years at Toast, and it was decided that he would be the face of Yolk. When it came time to fill out the first menu for Yolk, Donald and Patrick worked together picking out the items.\n\n_Bacon on a Caramel Roll_\n\nOne of best occasional specials they have at Yolk is one of Patrick's creations: the Brioche Maple Bacon Hazelnut Caramel Rolls. Oh my, they are good! If you happen see them on the menu, get one and thank me later.\n\n_Restaurant vs. Food Cart_\n\nEven though I love the breakfast at Yolk, there are plenty of times that I have had breakfast at Toast. Sometimes I want a food cart experience, and sometimes I want a restaurant experience. I know that some restaurateurs across the country are opposed to food carts and food trucks. They are worried that they hurt business. I consider that to be a ludicrous position. Like I said, I have cravings for both experiences equally.\n\nDonald's opinion is that good competition can make you stronger and that the strong survive. Donald is a big food cart fan, and he loves it when a food cart is able to make the leap to brick-and-mortar.\n\nI encourage you to visit both Donald's restaurant and his food cart.\n\n_What's Next_ : Donald has plans to open a second Yolk food cart in Portland, as well as open up a brick-and-mortar Yolk Breakfast Sandwich Restaurant. I, for one, hope that they start bottling that hot sauce. I love it, and I would buy it regularly.\n\n_When You Go_ : Two of my favorite items at Yolk involve the scrumptious pretzel bread they get from Little T American Baker. The Brother Bad Ass has maple-glazed pork belly, Beecher's cheddar cheese, an over-easy egg, greens and Dijon mustard between two slices of pretzel bread. Patrick also makes a delectable French toast from the pretzel bread that is served with maple syrup\u2014a sweet and salty delight. Also, get a side order of potatoes just so you can put on them Patrick's signature cart-made hot sauce\u2014roasted red peppers, honey, grape seed oil, smoked paprika and a few secret ingredients. This sauce is wonderful\u2014spicy but not too hot and full of flavor.\n\n**I T HAS NEVER FELT LIKE WORK: FRIED EGG I'M IN LOVE**\n\nJace Krause and Ryan Lynch opened the Fried Egg I'm in Love Food Cart in March 2012, and one year later, they opened a second cart. Even with only two food carts, one could argue that they have the most successful food cart business in Portland. Here is their story.\n\nRyan Lynch grew up in Sonoma, California, and it was there that he met Jace in 2007. Jace, who then called Seattle home, had a band, and Sonoma was a regular tour stop. Jace and Ryan became fast friends who made it a point to stay in touch with each other. Along the way, even though Jace lived in Seattle and Ryan in Sonoma, they spent some time in a band together.\n\n_Relocating to Portland and Looking for a New Career_\n\nRyan had always had a romance with the Northwest, and in 2010, he and his wife moved here to begin a new life. Ryan had a newspaper background that started with the _Sonoma Index-Tribune_. When he arrived in Portland, he got a job with a local Portland paper and began to consider what to do next. Ryan even thought about going back to school.\n\nJace's professional background was working as a corporate copywriter. Jace had family here in Portland, and in 2011, he and his wife moved from Seattle to Portland. He had some money saved, and he began to look at what he could do to start a business here in Portland.\n\nJace began to play around with the idea of opening a food cart that served breakfast sandwiches. He eventually told Ryan about this plan, and they decided to become business partners and make it happen. Ryan left his newspaper job to work with Jace and start Fried Egg I'm in Love.\n\nJace Krause and Ryan Lynch, owners of the Fried Egg I'm in Love Food Cart.\n\n_A Partnership that Started with Music_\n\nJace had no hesitation partnering with Ryan. He knew that they would get along and be able to make it work. It did not hurt that they had played music together before. Jace shared that \"working together in music is intimate, and I knew that if we could do music together, we could run a business together.\"\n\n_The Name and the Weekly Specials_\n\nThe name of their food cart, Fried Egg I'm in Love, is a playful riff on the 1992 Cure song \"Friday I'm in Love.\" Every menu item and every special at the cart has a fun music theme, and Jace and Ryan have a blast thinking up new ones.\n\nMany food carts do specials as a way of expressing creativity and avoiding becoming too bored with making the same thing day in and day out. The specials that Ryan and Jace come up with are not only cleverly named, but they are always delicious too. In my opinion, Fried Egg I'm in Love has some of the best specials of any food cart in Portland.\n\n_Why Breakfast Sandwiches?_\n\nThere are three reasons why Ryan and Jace choose to have breakfast sandwiches be the cuisine for their food cart. One, Jace absolutely loves eggs. Two, both Ryan and Jace love breakfast, and they fell in love with the breakfast sandwiches they were considering as they were kicking around food cart ideas. Three, and this is actually the biggest reason that they choose to feature breakfast, \"from the beginning, we designed the food cart to fit our lives, not the other way around,\" said Jace.\n\nThey wanted to be self-employed, and they did not want their new business to own them. They wanted to be done with their workday long before 5:00 p.m. I have had scores of food cart owners tell me that during their first year in business, they worked twelve to sixteen and even eighteen hours per day for five and six and even seven days a week.\n\nI asked Ryan and Jace a question I have asked every other food cart owner in this book: \"How many hours a week did you put in, on average, your first year in business?\" It took me a while to process their answer, and I even repeated the question before their response finally sunk in. \"One time we worked maybe fifty or sixty hours in a week,\" noted Ryan and Jace.\n\nI could not believe what I was hearing. I kept asking them about this, and they were completely sincere. They each work about forty hours a week total on their food cart business, and that has been the case from the beginning. In fact, they planned it that way; it was in the business plan they wrote before they ever opened the cart. When I told them that many food cart owners worked upward of seventy, eighty or even ninety hours per week during their first year, they were quite taken aback.\n\nGiven that that they have succeeded with their startup, grown their brand and, from the _very_ beginning, worked eminently reasonable hours, one could argue that Jace and Ryan have one of the most successful food cart businesses in Portland. I tip my food cart hat to them and their business acumen.\n\n_What's Next_ : Jace and Ryan plan to get a third food cart that will be mobile and used for events and catering. I hope that someday they bottle the special seasoning they put on every one of their fried eggs. I would buy it to use at home in a heartbeat.\n\n_When You Go_ : I love the Sriracha Mix-a-Lot\u2014fried egg, seared ham, avocado, tomato, Havarti and Sriracha on toasted sourdough. My second favorite is the Yolko Zeppelin\u2014two fried eggs, cart-made pesto, Parmesan and two hand-pressed sausage patties on toasted sourdough bread.\n**PART IV**\n\n**AN IMPORTANT PART OF THE PORTLAND FOOD CART SCENE**\n\n**S COTT BATCHELAR: THE MAYOR OF FOOD CART TOWN**\n\nI can hear Scott right now: \"Hello, Mr. Steven!\" You often hear Scott even before you see him. When I hear him bellow a greeting, my heart is warmed, and I can't help but smile. I love Scott, and it is always a treat to run into him at a food cart.\n\nIn my opinion, you cannot write a book about Portland food carts without including a section on Scott Batchelar. Scott is a fixture in the Portland food cart scene. Almost every food cart owner knows him, and Scott knows about each and every food cart. Scott eats at a food cart at least five to eight times per week.\n\n_Scott Knows Food Carts_\n\nScott doesn't drive. He gets around Portland on one of his two bikes that he rides everywhere. Scott is always wearing his iconic reflective leg bands just above his ankles, and his bike helmet is never too far away.\n\nScott has been riding his bike around Portland, visiting food carts, since at least 2009. His knowledge about the history of Portland food carts encyclopedic. He can tell you with great detail who and when and where and what cuisine. He can tell you about many great food carts that have come and gone. He always knows all of the latest food cart gossip, and if you want to know about the best food carts in Portland, talk to Scott.\n\nScott's ever-present reflective leg bands.\n\n_Scott's Story_\n\nI have Scott's permission to tell his story, and I am very grateful to do so. Scott occupies a special place in my heart. He is incredibly intelligent, and like my eldest son, Hayden, Scott is disabled.\n\nScott wrote about the 2013 Portland Summer Food Cart Festival, and in that article, he said, \"As some of you know I am a disabled individual.\" The thing is, lots of people don't know that about Scott, and truth be told, as Scott himself told me, sometimes he can be a little obnoxious.\n\nI want to tell his story because I want you to get know him and get to enjoy the Scott that I know and love. If you run ever into him, and he is being a little bothersome or loud, you will now have a context with which to understand him.\n\nScott Batchelar, the \"mayor of Food Cart Town.\"\n\nWhen he and I sat down for this interview, he confessed to me that sometimes in the middle of a conversation, he will realize that he is being obnoxious, and that saddens him. During that interview, I said, \"Scott, do you ever realize that you are sometimes being really loud?\" His honest reply was, \"I am loud sometimes?\" I told him, \"Yep, sometimes you are very loud.\" He had no idea.\n\nIf you are ever at food cart and come across a guy with a bike who is telling everyone within earshot that that particular food cart is a great one, you might have just found my friend Scott. First, listen to the man. If Scott says a food cart is worth eating at, it is. Second, go introduce yourself and invest ten or fifteen minutes listening to Scott's wonderful food cart stories.\n\n_Belonging_\n\nIt is my opinion that Scott's love of food carts is all about the community he experiences. During our interview, Scott said three things that deeply touched my heart. I heard these sentences not as a food writer but as the parent of a disabled child: \"I go to the food carts for the people and the community...Food carts let me be okay with myself...Part of the reason I go to food carts is for the community, because I am accepted for who I am.\"\n\nLet me close Scott's chapter with this story. He teared up when he relayed it to me while we were sitting in Teote, a Latin American restaurant specializing in street food that Scott loves. (Incidentally, Teote started out as a food cart.)\n\nScott's birthday is September 21, and on that day in 2011, he was asked by Bo Kwon, founder of the KOi Fusion food cart empire, to stop by the Mississippi Marketplace Food Cart Pod. (Scott loves Bo and considers him to be one of his heroes.) When Scott got to Mississippi Marketplace, Bo presented him with a cake. But not just any cake; this cake was in the shape of a bicycle. \"The cake was shaped like my bike,\" Scott said, his voice wavering a bit.\n\nIf the Portland food cart scene were a town, Scott would be the beloved mayor.\n\n**R OGER GOLDINGAY: THE MISSISSIPPI MARKETPLACE AND CARTLANDIA FOOD CART PODS**\n\nRoger and his wife, Carol, moved back to Portland in 2002. They left Malibu, California, to come to Portland. Some may shake their heads at such a choice. Those of us who know and love Portland completely understand someone choosing Portland over anywhere in California.\n\nRoger did share with me that it was not the weather that drew him back to Portland. It was the culture we have here. \"It was laid-back, less judgmental, more easygoing.\" He added, \"Portland has a creative vibe that was very attractive...every time I read about Paris in the early 1900s, Portland came to my mind.\"\n\nI said \"moved back to Portland\" because Roger first lived in Portland in 1975. There is an amazing little-known fact about Roger.\n\n_Soccer_\n\nRoger was a professional soccer player in the 1970s. In 1974, the first team to ever bear the name \"Seattle Sounders\" took the field for the first time. They were part of the North American Soccer League (NASL). Roger Goldingay played for the Sounders that inaugural season.\n\nIn 1975, the first of four teams to bear the name Portland Timbers took the field in their inaugural North American Soccer League season. Roger Goldingay was on that team as well. He is the only person to have played for both the Sounders and the Timbers during both of their respective inaugural seasons.\n\nPortland is known for a number of things, including the Timbers and food carts. Given Roger's love for Stumptown, it seems fitting that he would have a noted place with two of our most beloved institutions.\n\n_Revitalizing Mississippi Avenue_\n\nWhen Roger came back to Portland in 2002, he moved to Mississippi Avenue, and for a number of years, he worked to buy the lot on the corner of North Mississippi Avenue and North Skidmore. By the time he was finally able to purchase it, the economy had crashed, and the lenders who were going to help him develop that piece of property were nowhere to be found.\n\nRoger was able to get a loan from the Albina Opportunities Corporation to put in a \"parking\" lot with water and electrical hookups for food carts, as well as renovate the dilapidated building that was an eyesore. Roger figured that in the crashing economy, there would be a number of people who would be forced into considering self-employment, and he correctly surmised that some of those people would consider starting a food cart.\n\nRoger's vision for that lot on the corner of North Mississippi Avenue and North Skidmore was to put in a premier food cart pod that featured some the best food carts in Portland, as well as have a great bar in the building that he was renovating.\n\nWhen he was getting ready to open the Mississippi Marketplace Food Cart Pod, Roger took more than seventy applications for ten spots. Roger's goal was to pick the best carts possible. The first two food carts that Roger said yes to were Garden State (a now-defunct but legendary Portland food cart) and the Big Egg. Rogers remembered meeting Elizabeth and Gail with the Big Egg and tasting their food. He knew that they would make it.\n\nThe Mississippi Marketplace Food Cart opened in the summer of 2009, and Prost!, a neighborhood pub that specializes in German beer (or should I say \"bier\"), opened in October that same year. In my opinion, Roger has done a magnificent job implementing his vision. Prost! has great beer, and it provides both outdoor and coveted indoor seating for food cart fans. The lineup of food carts at Mississippi Marketplace is consistently stellar.\n\nRoger Goldingay, founder of Cartlandia and the Mississippi Marketplace.\n\nRoger remembered North Mississippi Avenue before it became a highly desired Portland location. It is very gratifying to Roger to know that he has played a small role in the revitalization of that part of Portland.\n\n_Revitalizing 82 nd Avenue_\n\nWith the positive changes that have occurred on North Mississippi Avenue since 2002 on his mind, Roger bought a lot on 82nd Avenue in October 2010 with the intention of putting in a food cart pod there. Right away, Roger had doubters. 82nd Avenue has\/had a stigma, and people were not sure that a food cart pod could work out there.\n\nRoger saw what others couldn't. He saw parking, bus service, the Springwater Corridor bike trail only three hundred feet away and the more than thirty-five thousand cars that drive past on 82nd every day. The Cartlandia Food Cart Pod opened in May 2011, and it is home to about twenty-five food carts.\n\nIn the spring of 2014, Roger will be opening a pub at this pod. Like Prost!, this pub will have indoor seating and will welcome patrons to bring their food cart food in with them.\n\n_The Landlord_\n\nWhile I like Roger, I would be remiss if I did not address the reality that not everyone involved in the Portland food cart scene likes him as well. Roger is a landlord, and I know firsthand that sometimes the landlord makes decisions that the tenant is not fired up about.\n\nWhen we talked about what it's like to be the landlord of more than thirty food carts at a time, Roger told me that it is like herding cats. I believe him. Being a landlord to food carts is not a job that I would want.\n\nIt is my opinion that Roger feels a heavy and serious responsibility to look out for the food carts to which he rents. He loves it when one of \"his\" food carts succeeds, and his heart aches when one of his food carts goes out of business.\n\n_Talk to Roger_\n\nIf you are with a municipality that is thinking of implementing the Portland food cart model that we have here in the city of Portland and in Multnomah County, Roger is one of three people featured in this book that I would encourage you to talk with. Roger knows firsthand how food carts can play a pivotal role in revitalizing an urban neighborhood, and he has perspectives and wisdom that you would do well to consider.\n\n_Portland Is Better Off Because of Roger_\n\nAs a food cart fan, I am very glad that Roger had the vision to start the Mississippi Marketplace Food Cart Pod and the Cartlandia Food Cart Pod. There are a number of small business owners who would not be where they are now if it were not for Roger and his pods. If your food cart started at one of his two pods, you owe him a debt of gratitude. Portland is a healthier and more appealing city because of Roger Goldingay.\n\nRoger, I tip my food cart hat to you, and I thank you for all that you have done for my beloved city and for the Portland food cart scene.\n\n**T IM HOHL AND TERRY TRAVIS: FRIDAY IS OUR FAVORITE DAY OF THE WEEK**\n\nTim Hohl and Terry Travis have introduced the people of Portland to scores of food carts though the \"Food Cart Friday\" segment on _First Edition_ , the radio show they host on KPAM 860.\n\n_We've Never Had a Bad Meal_\n\n_First Edition_ launched in the spring of 2011, and by the summer of 2011, they had added a weekly food cart\u2013centric segment. Most weeks, a food cart owner comes to the radio station and talks about their cart and brings one of their signature dishes, which Tim and Terry sample live on the air.\n\nTim and Terry have enjoyed more than one hundred early morning food cart meals, and they are always pleasantly pleased by how good the dishes are. Terry is very impressed that each and every dish has been wonderful.\n\n\"I live in the greatest place on Earth,\" said Terry Travis, who grew up in Vancouver, Washington, and graduated from Columbia River High School. Terry's career took her to places like Boston, Brooklyn and Seattle. When the chance came to come back to Portland, she jumped on it. Reflecting on that move, Terry said, \"I was so glad to be coming back to Portland.\"\n\nOne of Terry's cherished hobbies is riding her bike. \"I still remember the first time I got to ride my bike without training wheels\u2014I felt so free!\" She said that she likes to ride a long ways...slowly. One of her favorite bicycle trails is the Springwater Corridor\u2014particularly the section that heads out toward Gresham. Out there, she encounters stunning vistas that have Mount Hood in the distance smiling down at her.\n\nTim Hohl was born in Michigan and lived in Detroit until his family moved to Portland when he was ten years old. He stayed in Portland through high school, graduating from Jesuit. \"Portland is home,\" said Tim. Right out of college, Tim started in radio here in Portland. Eventually, his radio career took him north to Seattle. Thirteen years ago, the doors opened for him to come home. Portland is where his heart is. Tim really appreciates that we have both the beach and the mountains so close. He loves it that he can leave the parking lot at work and be on his skis in less than ninety minutes.\n\nTim's favorite hobby is beer, and he loves the Portland beer scene. \"Beer is an art form,\" he said. His eyes danced when he told me that \"in Portland you can have a different beer, every day, for 365 days, and each one will be from a local brewery and each one will be great!\" Not only does he enjoy having a pint at one of Portland's many local pubs, Tim is a home brewer as well. He would love to someday open up his own local brewery.\n\n_I Am Out of Meatballs but I Will Fix You Up_\n\nTim told me this delightful story about his daughter and food carts. When Tim's daughter does not have school on Fridays, she will come into the station with him to check out \"Food Cart Fridays.\" She was in the station the day that Jason Moreno, owner of J Mo's Sandwich Shack Food Cart, brought in the Dirty Mo. She loved it! I can't blame her, as the Dirty Mo is one of my favorite sandwiches. It has spaghetti and cart-made meatballs on really good garlic bread. Honestly, when I first heard about the Dirty Mo, I was not really excited. Spaghetti and meatballs on a sandwich didn't sound that good to me. However, when I took my first bite, I was enraptured.\n\nTim Hohl and Terry Travis, hosts of KPAM 860's _First Edition_.\n\nFrom that particular \"Food Cart Friday\" on, Tim would have to take his daughter to J Mo's so she could get a Dirty Mo. One day, something awful occurred, followed by something wonderful. Jason was out of meatballs, and consequently, the Dirty Mo could not be made. Tim's daughter was crushed. Jason, being the resourceful chef that he is, said, \"Don't worry. I will fix you up!\" Jason whipped up an eggplant Parmesan sandwich. Tim's daughter took one bite and _bam_ \u2014she loved it! Culinary crisis averted. That sandwich is now on the menu.\n\n_We Get to Celebrate a Food Cart Every Friday_\n\nTim and Terry look forward to the amazing food they enjoy each week during their \"Food Cart Friday\" segments, and they have come to love hearing the stories of the inspiring people who start food carts. As Terry told me, \"Friday is our favorite day of week!\"\n\n**B RETT BURMEISTER: TELLING PEOPLE ABOUT THE COOL THINGS I FIND**\n\nBrett Burmeister is the managing editor of FoodCartsPortland.com. Brett has been writing for this website since just before the beginning of 2009. Brett has written more than six hundred food cart articles, eaten at more than five hundred food carts and dined at food carts more than one thousand times. The man knows food carts!\n\nFor the past five years, Brett has been Portland's on-the-street food cart correspondent reporting to us about all that he finds. The Portland food cart scene has grown and flourished in the city. It is my opinion that Brett is one of the people most responsible for that, and I for one am very grateful for his efforts.\n\nThrough the FoodCartsPortland.com website, Brett has made it possible for the average person to find out about food carts. He has made food carts accessible. Each time he writes about a food cart, he includes what he calls \"the data\"\u2014the location, the hours, the story and a sample menu.\n\n_BeforeFoodCartsPortland.com_\n\nBrett began writing back in 2005. He started a blog called \"Walking in Portland,\" and his first post was January 19, 2005. He wrote about the cool stuff he found as he walked around Portland. In 2007, Brett was recruited to write for Portland Metblogs, which he did until June 2008.\n\nBret wrote a little about food carts for both of those blogs, and in the winter of 2008, he was recruited to come write for FoodCartsPortland.com. Brett had found his niche. He loves to find cool stuff and tell others about it. Going around Portland, finding food carts and then telling us about them was right up his alley.\n\n_Tours, Consulting and Events_\n\nNot only does Brett write for FoodCartsPortland.com, he also has three other food cart\u2013related services that he provides. He provides walking tours of food carts, during which you get small bites from four food carts, and as you traverse from cart to cart, you get the chance to hear Brett tell you about a number of very interesting food cart topics.\n\nBrett Burmeister, managing editor of FoodCartsPortland.com.\n\nA Big-Ass Sandwich.\n\nThe Big Egg Food Cart. _Photo by Ken Wilson_.\n\nThe Arbor Lodge Sandwich from the Big Egg Food Cart. _Photo by Ken Wilson_.\n\nThe Cheese Lovers' Oregon Cheese Plate from the Cheese Plate PDX Food Cart.\n\nThe Divine Caf\u00e9 Food Cart.\n\nThe Killer Quesadilla from the Divine Caf\u00e9 Food Cart.\n\nA fried egg with Magic Egg Dust from the Fried Egg I'm in Love Food Cart.\n\nThe Grilled Cheese Grill Food Cart tagline.\n\nThe Original Cheesus from the Grilled Cheese Grill Food Cart\u2014two grilled cheese sandwiches, one with pickles and American cheese and a second with grilled onions and Colby jack. In between these two sandwiches are lettuce, tomato, ketchup, mustard and a  -pound burger. Oh baby! Who is your big grilled cheeseburger daddy?\n\nThe school bus from the Eleventh and Alberta Grilled Cheese Grill Food Cart.\n\nMac and cheese from Herbs Mac & Cheese Food Cart, topped with broccoli, bacon and tomato. _Photo by Molly Woodstock_.\n\nThe Sampler Platter from the Homegrown Smoker Vegan BBQ Food Cart. _Photo by Ken Wilson_.\n\nThe Federal from the Italian Market Food Cart\u2014Italian pork with sharp provolone and broccoli rabe.\n\nKhinkali from the Kargi Gogo Food Cart\u2014a traditional Georgian stuffed beef and pork dumpling.\n\nA perfect latte from Ol\u00e9 Latte Coffee Food Cart.\n\nThe PDX Six Seven One Food Cart.\n\nChicken Kabob from the Ramy's Lamb Shack Food Cart.\n\nSuper Lamb Gyro from the Ramy's Lamb Shack Food Cart.\n\nLiquid Gold Thai Iced Tea from Stumptown Dumplings Food Cart, held by Nimesh Dayal.\n\nDumplings from Stumptown Dumplings Food Cart\u2014Cheeky Chicken, Pompous Pork and Sassy Spinach.\n\nThe Sugar Shop Food Cart.\n\nStrawberry cake topped with basil buttercream and balsamic drizzle from the Sugar Shop Food Cart.\n\nThe Hot Chick from the Tiffin Asha Food Cart\u2014Pakora fried chicken drizzled with black cardamom-infused honey, pickled greens and creamy yogurt cheese wrapped in a cart-made dosa.\n\nThe Tiffin Asha Food Cart.\n\nThe Yolk Food Cart.\n\nThe Brother Bad Ass from the Yolk Food Cart\u2014maple-glazed pork belly, Beecher's cheddar cheese, an over-easy egg, greens and Dijon mustard between two slices of Little T American Baker pretzel bread.\n\nThe Whiffies Fried Pies Food Cart.\n\nRed velvet cupcake from the Retrolicious Food Cart.\n\nGreek salad from Ramy's Lamb Shack\u2014tomatoes, cucumbers and feta cheese.\n\nThe PB and J Fries from the Potato Champion Food Cart. The fries are topped with Potato Champion's peanut satay sauce and chipotle raspberry sauce.\n\nBrett also does a little consulting. If you are thinking of starting up a Portland food cart, I encourage you to have at least one conversation with Brett. Bring a pad of paper and be prepared to take notes. If you are with one of the many municipalities that come to Portland to find out about the food cart scene, Brett is one of three people with whom you need to sit down and talk. Schedule a consultation with him and ask him to talk with you about O'Bryant Square.\n\nLastly, Brett works with a number of Portland's truly mobile food carts, and for a fee (paid by the food cart owner), he can help you get a genuine Portland food cart to your event!\n\n_Brett's Favorite Food Cart_\n\nOne question that Brett gets asked repeatedly is, \"What is your favorite food cart?\" He smiles when he tells the person that they are asking the wrong question. He goes on to say that the better question is, \"Where would you take a guest from out of town?\" Brett was recently asked to provide a list of the top ten food carts in Portland. He began making that list but put his pen down when he realized that the list already had forty carts and still more to go.\n\n_Food Cart Future_\n\nI asked Brett what some of his next steps will be regarding food carts. He wants to do what he can to help further the cause of street food, and he would love tell people across the country \"the glory of the pod idea.\" Brett has seen with his own eyes how food cart pods can improve a city and rehabilitate a neighborhood. He has a passion for food carts, and Portland is better off because he is willing to tell us about the cool stuff that he finds out there on the streets of Portland.\n\n**R ICK HUMPHREY: HIS CARD IS IN MY WALLET**\n\nRick Humphrey is the assistant parts manager at Curtis Trailer, and if you own a Portland food cart, it is very likely that you have his card in your purse or wallet. Rick has been at Curtis Trailer since 2007, and he has been servicing food carts for many years. He has assisted many food cart owners over the years, and he has an excellent reputation among food cart owners.\n\n_\"What Is a Food Cart?\"_\n\nPrior to 2007, Rick worked at the now-defunct Gresham RV. It was while he was working there that he first encountered a food cart owner. Rick had a customer buy a water pump. This same customer came back a few months later for a second water pump. When this same customer bought a third water pump just thirty days later, Rick's curiosity got the best of him, and he asked this gentleman what he was doing with the water pumps. Rick was told, \"I use them in my food carts.\" Rick then asked, \"What is a food cart?\"\n\nRick Humphrey, \"the Food Cart Guy.\"\n\nRick had been selling water pumps to Saied Samaiel, the founder of Aybla Grill, a food cart business that today has five locations. Saied was excited to tell Rick all about the Portland food cart scene. Rick and Saied still do business together to this very day. The second food cart owner that Rick ever met was Rose Guardino, owner of the Divine Caf\u00e9 Food Cart.\n\n_The Floodgates Opened_\n\nRick did not do very much business with food cart owners while at Gresham Trailer. The owners did not really want to sell to food cart owners, preferring to stick with people who own RVs for recreational use. Once Rick moved over to Curtis Trailer in 2007, he was encouraged to go out to the food cart pods, hand out his card and help as many food cart owners as he could. Rick told me that once word got out that he was a resource food cart owners could depend on, \"It was like the floodgates opened.\" Many food cart owners would show up to see him and get his assistance.\n\n_A Valuable Resource for Food Cart Owners_\n\nRick has become well versed on the health department regulations for Multnomah, Clackamas, Marion and Hood River Counties. Food cart owners regularly turn to Rick for his advice, counsel and, of course, parts! The items for which food cart owners depend on Rick include holding tanks, water pumps, water heaters, concession windows and doors, awnings, electrical cords and hasps (latches used to keep doors locked).\n\n_Rick Humphrey, the Food Cart Guy_\n\nPeople began calling Rick \"the Food Cart Guy,\" and he eventually embraced that moniker. Today, Rick has a \"TheFoodCartGuy\" Twitter account and Facebook page. Rick serves the Portland food cart community at a tremendous level. Many times, he has taken late-night calls to help food cart owners in need. I mentioned Rick in a conversation I was having with Jeff Mendon, owner of the Bento Box Food Cart, and he smiled and said, \"I have his card in my wallet!\" Every successful Portland food cart owner I know has Rick's card. Nuff said.\n\n**B RYAN SEBOK: MAPPING THE MOBILE FOOD MOVEMENT**\n\nBryan Sebok spent most of 2013 filming the feature-length documentary _Cartography: Mapping the Mobile Food Movement_ , and he plans to premiere this film in late 2014.\n\nBryan moved to Portland in 2009 to become a professor of media studies at Lewis and Clark College. Bryan did his undergraduate studies at NC State University. He graduated NC State with a degree in mass communications and film studies. He got his masters in film studies at Emory University and has a doctorate in media studies from the University of Texas\u2013Austin. There is a reason why Bryan is so well spoken and erudite.\n\nBryan also loves film. He has liked movies as long as he can remember. When he was baby, his nursery overlooked a drive-in theater, and his mother told of nursing him in that room while watching movies out the window. During Bryan's childhood, he got to go to the dollar theater and see a movie two to three times per week. Growing up some of his of his favorite movies were _The Goonies, Gremlins_ and _One Flew Over the Cuckoo's Nest_.\n\nBryan Sebok: professor, filmmaker and food cart fan.\n\n_Visually Telling the Portland Food Cart Story_\n\nOnce Bryan moved to Portland, he saw all of the food carts, and given his academic background, he did some research on them with a focus on the impact that food carts have culturally, industrially and economically. Bryan's research revealed lots of interesting data, and he found that the people who owned Portland food carts had some incredible stories. Given his love of film, Bryan began to ask himself, \"Why am I not making a documentary about this amazing phenomenon?\" Bryan's goal was to visually tell the story of Portland's food carts.\n\nBryan began filming on January 1, 2013, and he filmed at hundreds of food carts. As Bryan sat down with food carts owners and interviewed them on camera, he realized how important the story was. He observed firsthand how food carts have had positive and long-term effects on the community as a whole and on the neighborhoods in which they are located.\n\n_Portland: A Model for Mobile Food Success_\n\nBryan's year of research and filmmaking led him to conclude that Portland is a model for mobile food success that can be adopted by municipalities in other parts of the country. If you are with one the many municipalities that regularly visit Portland to investigate our food cart scene, I highly encourage you to sit down with Bryan and consult with him.\n\n_Food Cart Culture vs. Food Truck Culture_\n\nBryan has learned that one of elements that is part of the Portland mobile food success story is something that he defines as the difference between a food cart culture, where the cart generally stays in one place, and a food truck culture, where the truck is mobile and does not have a set location.\n\n_College Students_\n\nBryan is the kind of teacher who cares about students, and twelve of his students got to take part in the making of this documentary. I ran into Bryan a number of times throughout 2013, and I got to know two of those students: Sofia Alicastro (who served as production manager) and Alia Al-Hatlani (who worked on social media and design).\n\n_Cheering One Another On_\n\nOver the course of the year, one of the most pleasant surprises that Bryan found was that there was less competition and more community among food cart owners than he had expected there to be. The community spirit that exists among many food cart owners touched him. Bryan himself is a food cart fan. He has many favorite food carts, including the Egg Carton, Moberi and the Smaaken Waffle Food Cart.\n\n_Go See This Movie_\n\nI enjoy sports radio, and Colin Cowherd occasionally talks about how much he likes a good documentary. _Cartography_ is going to be right up Colin's alley. It has very interesting stories, and it is set in Portland, a town that Colin knows well. When _Cartography_ comes out, go see it! It is going to be a compelling, enjoyable and insightful documentary.\n\n_What's Next_ : Bryan has to get the editing completed and shepherd _Cartography_ through a film festival season. Once all of that has been completed, Bryan would love to make a documentary on the craft brewing culture. Portland definitely has its share of craft breweries, too. I hope that Bryan is able to find college students willing to help make a film about beer\u2014that could be tough.\n\n**O N MANY FOOD CART MENUS: CARLTON FARMS**\n\nI was at the Burger Guild Food Cart in the spring of 2012 when Carlton Farms first came onto my radar.\n\n_The Burger Guild Midwest Pork Tenderloin Sandwich_\n\nMike at the Burger Guild makes this extraordinary Midwest pork tenderloin sandwich. The pork tenderloin is breaded, deep fried and topped with lettuce, onion, pickles and mustard. This sandwich is _huge_ and incredibly good. The pork is so tender and flavorful that you want to eat another as soon as possible.\n\nJohn Duyn and Jake Burns of Carlton Farms, two guys who love and know their sausage.\n\nI noticed that on the menu, Mike was prominently featuring that he used Carlton Farms pork. As the year progressed, I saw a number of other food carts stating on their menu that they used Carlton Farms meats. Food carts were doing this to make it clear to their customers that they were using top-shelf meat from a superb local producer. It was one of the ways that a food cart could set itself apart and convey that it used first-class ingredients.\n\nI could taste the difference, and whenever I saw that Carlton Farms was on the menu, I knew that I was in for a great meal. When it came time to write this book, I wanted to know more about this business that so many food cart owners were attaching to their brand. Given who I am, I wanted to get to know the story of this company and the stories of the people there.\n\n_John and Jake_\n\nI was lucky enough to meet and talk extensively with two of the amazing people who make the magic happen at Carlton Farms: John Duyn, president and CEO, and Jake Burns, VP of operations. Carlton Farms was started in 1956 by Henry \"Hank\" Duyn. In 1958, Carl Duyn, Hank's brother, joined the company. In 1974, John Duyn was recruited to come to Carlton Farms to be the general manager. Under John's leadership, this local family business has flourished to become a very well-known and prominent local brand.\n\n_John's Food Cart Tour_\n\nI first met John in October 2013, when I was privileged to take him on a food cart tour. We visited four food cart pods and ended up eating at seven different food carts. It was quite fun to see how excited food cart owners were to have the president of Carlton Farms at their food cart.\n\nJohn earned a tremendous amount of my respect that day. He was very kind and was genuinely interested in each person we met. The encouraging way he interacted with the small business owners we met with that day was impressive.\n\nIt was apparent that John loved food and also that he was quite knowledgeable about many different types of cuisine. One minute he was conversing with Paolo from the Burrasca Food Cart about Italian food and a Tuscan butcher they both happened to know, and the next he was talking with Alec Harry from the Guero PDX Food Cart about Latin American cooking. It was also clear that John takes great pride in the fact that his company produces high-quality meats. It takes longer and costs more to do that, but for those people who want a top-shelf product, people like food cart owners, the extra investment is more than worth it.\n\nIt was John's grandfather Girard who instilled the value for making a high-quality product into the Duyn family. Girard \"Jake\" Duyn taught both John's father and his uncle to have an old-world charcuterie approach to meat.\n\n\"I love the science of meat and spice,\" said Jake Burns. It was that old-world aesthetic that caused Jake to agree to come work at Carlton Farms eleven years ago. Most large U.S. producers make bacon in about twenty-four hours. The bacon at Carlton Farms takes two to three weeks to make. When Jake learned that Carlton Farms followed the longer process, and when he saw the cure cooler, he knew that he wanted to be a part of this very special company.\n\nJake loves working at Carlton Farms. He graduated college with a degree in meat science, and he has loved that business ever since. If you are ever lucky enough to catch up with Jake and talk with him about meat and spice, his passion will be palpable and you know that he works for the right company.\n\n_Two Guys Who Love Sausage_\n\nWhile I was interviewing both John and Jake, I asked them which Carlton Farms product was their favorite. Both of them enthusiastically proclaimed that it was sausage. Not only does Jake make sausage at work, he also makes it at home in his own kitchen. John loves sausage as well, and he is also a hot dog aficionado.\n\nThe workday starts very early at Carlton Farms. Both John and Jake shared with me that they love to grab a cold hot dog for a snack during early morning breaks. It warmed my heart to discover that John and Jake both love sausage. Why would that matter to me? That fact told me something important about Carlton Farms. Let me explain it this way: Shannon and Autumn own the Sugar Shop PDX Food Cart. This is a dessert cart, and that means that both Autumn and Shannon do a lot of baking. I mean a whole _lot_ of baking, hours and hours of it. The thing is, they both absolutely love to bake, and you can tell that when you enjoy one of their amazing desserts.\n\nThe same is true when you enjoy meat from Carlton Farms. You can taste the care and the joy that goes into what it produces. I want to buy my sausage, ribs, pork, beef and bacon from a company where those in management love sausage the same way that Autumn and Shannon love to bake.\n\n_Truly Handcrafted_\n\nBy the way, I have been to Carlton Farms, and I was privileged to get to see behind the scenes. When they say \"handcrafted,\" they are serious, and you can taste that difference in every bite. In my opinion, the food carts that feature Carlton Farms on their menus are making a very wise business decision.\n\n**B RIAN WILKE AND ERIC STROMQUIST: BEING A NICE PERSON IN THE HOSPITALITY BUSINESS**\n\nI have been working with food cart owners and writing about the Portland food cart scene for two years now, and in that span of time, I would occasionally have a food cart owner gratefully talk about a culinary school chef who trained them named Brian Wilke.\n\nBrian Wilke and Eric Stromquist of the Oregon Culinary Institute.\n\nI also have spoken to many chefs who have gone to culinary school, and while they always think that it is a very good idea for someone who wants to cook professionally to attend culinary school, they always say that someone should go to a \"good\" culinary school. They then either imply (or, more often than not, directly say) that not all culinary schools are ones you would want to attend. I decided to find out for myself who this Brian Wilke guy was and what makes for a \"good\" culinary school.\n\n_Two Nonnegotiable Conditions_\n\nBrian Wilke and Eric Stromquist started Oregon Culinary Institute in the spring of 2006. Prior to starting OCI, they had each spent ten years working at another Portland-area culinary school. They had each started out as instructors, and before they left, Brian had become executive chef and Eric the VP of academics. Their previous culinary school ended up going though a corporate merger, and as all mergers do, this one brought a number of changes. Those changes led both Eric and Brian to part ways with that institution.\n\nPioneer Pacific College approached them and asked, \"How would you like to start a professional culinary program for our college?\" Both Brian and Eric initially said, \"No thanks.\" Pioneer was persistent, certain that it had found the right people for the job.\n\n\"Cooking is like love. It should be entered into with abandon or not at all,\" at the Oregon Culinary Institute.\n\nBrian and Eric started having discussions with each other, and they began to remember what had first led them to fall in love with culinary education. They then said to Pioneer, \"We have two nonnegotiable conditions: one, you can't tell us how to do it, and two, we keep small class sizes.\" Pioneer said yes to both conditions, and they began to build OCI.\n\nBoth Brian and Eric are deeply committed to small class sizes. They believe that in a hands-on environment, you need to have access to your instructor. Brian emphatically told me, \"You can be in a four-hundred-person class for English. That does _not_ work for cooking.\"\n\n_Two Very Different Skill Sets: Cooking Great Food and Teaching Others to Cook Great Food_\n\nEric shared with me that it took him twenty years to learn that what he liked best about the restaurant business was being a customer. His passion, as well as Eric's, is teaching others first how to make great food and then preparing them for a career in the culinary business. Brian and Eric have each worked in the restaurant business their entire adult lives, and at OCI, they were allowed to design a curriculum based on what they know their students will need out there to be successful.\n\n_One-Third of the Incoming OCI Students Struggle with Math_\n\nBrian loves to see the students who struggle with math reach a point where they not only know that they made a really great plate of food, but that they also know how much that plate of food costs and how much they would have to sell that plate of food for to make a profit.\n\n_You Have to Be a Nice Person_\n\nTwo things really impressed me about OCI. The first was why the students do not have nametags. Brian and Eric expect the instructors to know their students' names, and that fits hand in glove with the second thing that really impressed me. If you want to be an instructor at OCI, you have to be a nice person. If during the first interview they get the sense that you are not a nice person, you won't get a second interview.\n\nEric and Brian see no place for the stereotypical rude and unpleasant chef in the hospitality business. As Brian said, \"It's the _hospitality_ business. You _have_ to be a nice person.\" Hearing Brian say that reminded me of what John Duyn, president and CEO of Carlton Farms, said about Paolo, the owner of the Burrasca Food Cart. John said that one of the reasons why Paolo would make it as a restaurant owner (aside from his amazing food) is that Paolo has experience working the front of the house. John went on to say that he has met a number of chefs who were mean, awful people, and they can get away with that because they are hidden back in the kitchen, where the customers don't see them.\n\nPeople who work the front of the house quickly learn that you need to have good people skills to keep the customers coming back. Great food alone won't do that.\n\n_What I Would Tell My Kids_\n\nIf one of my kids wanted to start a food cart, and they asked me what path I would suggest they take, here is what I would say. I would ask them if they wanted to simply start a food cart that would be around for a year or two, or if they wanted to start a culinary brand that would have long-term success.\n\nIf they answered the former, I would encourage them to work in restaurant for at least six months, to work in a food cart for at least six months, and then, if after doing both of those they still wanted to move ahead, I would tell them to go to a \"good\" culinary school like OCI.\n\n\"We teach bakers and cooks to how to make money doing what they love,\" said Eric. Brian added, \"Our joy at OCI is equipping someone to take the passion in their heart for cooking and express it on a plate.\" I have no doubt that OCI is one of those \"good\" culinary schools that my friends who have been to culinary school encourage others to attend.\n\n_When You Go_ : They actually have a restaurant at OCI, where you can enjoy a three-course lunch or a four-course dinner made by the students. I have eaten there, and the food is wonderful.\n\n**D OING AWAY WITH THROWAWAY: GO BOX**\n\nLaura Weiss started GO Box in July 2011 because of a conversation she had in March 2010 with a friend who worked at the City of Portland Bureau of Planning and Sustainability. GO Box is what makes it possible for you to get your downtown Portland food cart food in a reusable to-go container. As of the fall of 2013, the need for more than thirty thousand disposable containers has been eliminated because of the service that GO Box provides.\n\n_Found Her Calling at Thirteen_\n\nLaura grew up in New Jersey, and one day when she was thirteen years old, the _New York Times_ had an article about the environmental tragedy and scandal that occurred in the Love Canal neighborhood in Niagara, New York. Laura remembered feeling upset and outraged that people would have to suffer the way they did in that situation.\n\nLove Canal\u2014a rather large neighborhood with many homes, schools and businesses\u2014was built on top of a site where, years earlier, twenty-one thousand tons of toxic waste had been dumped. Subsequently, many tragic health problems emerged. This environmental disaster, along with a similar situation in Times Beach, Missouri, played a major part in leading to the Comprehensive Environmental Response, Compensation and Liability Act (CERCLA), commonly referred to as \"Superfund.\"\n\nFrom that article on, saving the environment has been Laura's focus. Both her education and career have been guided by her passion to do her part to save the environment. Laura did her undergraduate work at Rutgers, getting a degree in human ecology (environmental science), a master's from UC Berkley in public health and an MBA in sustainable business from the Bainbridge Graduate Institute in Seattle, Washington.\n\nPrior to starting GO Box, Laura worked for twenty-five years in environmental advocacy and policy, working in the corporate world for both state governments and nonprofits.\n\n_The Conversation that Led to a Business_\n\nLaura and her aforementioned friend from the Portland Bureau of Planning and Sustainability were talking about the roughly sixty thousand disposable containers that downtown Portland food carts use each month. Most of those containers end up in a landfill. Laura herself is a big fan of food carts. She had often wished that she could take her own reusable container with her when she ate at food carts; however, health codes prevent that. At the end of that conversation, Laura's friend lamented, \"If only there was a service that provided reusable containers that could be used at food carts.\"\n\nThat thought got Laura's mind working, and she began to think about starting her own business. She was aware of reusable containers made by G.E.T. Enterprises out of Houston, Texas, that would be perfect for food cart owners and customers. With a year, Laura had come up with an innovative business model using those very containers.\n\nGO Box reusable to-go containers.\n\n_The Name_\n\nLaura came up with the name for her new startup while cross-country skiing. Laura said that she does her best thinking while skiing or riding her bike. The \"GO\" in GO Box stands for \"Green Options.\"\n\n_A Community-Facilitated Endeavor_\n\nHere is how GO Box works. A person invests eighteen dollars for a one-year subscription. People can purchase subscriptions online at the GO Box website or from participating food carts. You can find a list of the participating food carts on the GO Box website as well. With your subscription, you receive a GO Box token. When you show up at a participating downtown Portland food cart, you turn in your token with your order.\n\nLaura Weiss, owner of GO Box.\n\nYour order is served to you in one of the GO Box\u2013provided reusable containers. When you are finished with the meal, you take your used GO Box container to one of the drop-off sites scattered throughout downtown Portland. When you turn in your used container, you get another token that you use the next time you visit a downtown Portland food cart participating in GO Box.\n\nIn addition to the public drop-off sites listed on the GO Box website, Laura has agreements with more than twenty other corporate partners that provide drop-off sites for those people who work in that particular office or building.\n\nLaura can be seen riding around downtown Portland on her bike dropping off clean GO Box containers to food carts and picking up the used ones so that they can be washed and made ready to be reused. Laura takes the used to containers to one of the GO Box partners who help with washing and sanitizing the used containers. Laura has two downtown restaurants and one social service organization that help with this key step in the process: the Original, Bijou Caf\u00e9 and the Transitions Project.\n\n_Work that Is Fulfilling and Fun_\n\nLaura loves what she does. She is making a difference for the environment, she gets to ride her bike for work, she gets to work in downtown Portland and she gets to work with food cart owners.\n\n_What's Next_ : Laura hopes to expand GO Box to other parts of the country, and she is looking at providing other entrepreneurs with licenses to use the GO Box name and concept. A city does not need food carts to be able to sustain a GO Box business. GO Box will work in any dense urban environment that has lots of office workers and lots of takeout.\n\nFood Writer's Perspective:\n\nPHIL SHEN\n\nMy fianc\u00e9e, Kim Pham, and I started Behind the Food Carts (behindthefoodcarts.com) because we wanted to show the world the amazing food cart stories we were coming across. As professional photographers and videographers, we wanted to document food carts with the highest quality to match the hard work and love that goes into running a food cart.\n\nWhenever we meet people who have visited Portland, the first thing they always mention is all the food carts they saw and ate at during their trip. Portland's food cart scene is undoubtedly one of a kind in America. The city is friendly toward food carts, and the people are friendly toward eating at food carts.\n\nThe pods conveniently placed throughout the city immediately set Portland apart. These pods allow food cart vendors to have a more permanent home than their more mobile truck cousins and let you frequent your favorite carts over and over again without having to seek them out. Food carts are an ingrained part of Portland culture. This helps push the quality forward; Portlanders are sophisticated food cart consumers. With so many amazing food carts, their expectations are high!\n\nThe food cart scene as a whole in the United States is constantly evolving. I think now that we live in the San Francisco Bay Area, we've been able to see a different side of the food cart industry and get a fresh look at our time in Portland. We definitely appreciate the focus and specialty carts that you see in Portland. Sure, there are fusion carts out there, but the Portland carts we loved most were the ones focused on doing one type of food and doing it perfectly.\n\nA good example is Nong's Khao Man Gai. Nong and her Thai chicken over rice are a staple in the Portland food cart scene. It goes to show you what one dish done perfectly can bring to the table. It's not about having a million options to choose from that are mediocre; it's about that one perfectly comforting dish that keeps people coming back for more.\n\nWe love the opportunity that a cart can present to an aspiring entrepreneur. A new favorite of ours is Caspian Kabob. Victor saw that there wasn't enough authentic Persian cuisine in the area and wanted to fill that gap (and our stomachs) with the real deal. I grew up in the Little Persia area of Los Angeles and ate some of the best Persian food you can find in the United States, so I was extremely excited to try his cart.\n\nVictor wasn't trying to put a radical spin on Persian cuisine or put Persian food into a burrito (which I bet someone is planning right now already). He simply wanted to bring an accurate representation of the food he is passionate about to a new audience. That's the beauty of carts in Portland. They can take a risk and introduce audiences to a whole new cuisine that they didn't even know they were missing.\n\nWe frequently get asked whether food carts are a fad. There are definitely more and more carts every year and, with that, more and more carts that go under as well. There is less room for error and more room for innovation and quality. But this is no fad; this is a lifestyle and a small business dream for a lot of chefs and owners.\n\nIt's been an inspiration to us to talk to these owners about their passion and all the hard work they put in. Sixteen-hour days seven days a week can only be summed up as a labor of love. We've been honored to be able to tell their stories and focus on not just the food but also the down-to-earth, loving people we've met along the way who share a piece of themselves with us every day through their food.\n**PART V**\n\n**VEGETARIAN FOOD CARTS**\n\n_Delicious without Meat_\n\n**F OR THE LOVE OF CHEESE: THE CHEESE PLATE PDX**\n\nFor more than five years every August around the time of Carina's birthday, Nick and Carina would host a gathering at Delores Park in San Francisco. The rule of this gathering was that everyone had to bring his or her favorite cheese, chocolate or sparkling beverage.\n\nOnce a year, they would spend the whole day \"eating the most awesome cheese and choice sparkling wine.\" That event is where Nick and Carina spent many beautiful hours enjoying themselves and cherishing others. Those joy-filled days fed the amazing souls that they both are.\n\nThat annual party featuring quality cheese is the genesis of the Cheese Plate PDX Food Cart.\n\n_From San Francisco to Portland_\n\nIn December 2008, Nick and Carina moved to Portland, Oregon. Carina had taken the position of managing editor and publisher of _Mandala_ , the official publication of the Foundation for the Preservation of the Mahayana Tradition. FPMT is a very impressive charitable organization that was founded by two Tibetan Buddhist masters, Lama Thubten Yeshe and Lama Thubten Zopa Rinpoche. This organization is of the same lineage as the Dalai Lama.\n\nCarina Rumrill and Nick Dickison, owners of the Cheese Plate PDX Food Cart.\n\n_Let's Start by Opening a Food Cart_\n\nIn the winter of 2012, Carina and Nick went away for the weekend to Hood River, and it was there that the idea for the Cheese Plate PDX was born. They decided that they were going start a food cart that focused on cheese. Carina and Nick spent the next few months getting ready to launch their business. For them, the food cart they were working on opening was going to be the first step in a bigger plan.\n\nOn July 6, 2012, the Dalia Lama's birthday, the Cheese Plate PDX opened its doors. The Cheese Plate PDX doesn't specialize in just any cheese. It specializes in serving high-quality local cheese. It features cheese from creameries such as Willamette Valley Cheese, Briar Rose Creamery, Portland Creamery and Rogue Creamery. If you love fine-quality cheese, the cheese plates they have will be right up your alley.\n\n_A Vegetarian Food Cart_\n\nTrue confession time: I love meat. I love bacon. And I have had lots of really bad vegetarian food. In my world, hot dogs that come from a can are just wrong. The Cheese Plate, like two other food carts in this book, is a vegetarian food cart. It is serious about this vegetarian thing.\n\nCheese is made with something called rennet, which is, most of the time, made from the stomach of a cow, a goat or a sheep. Vegetarians are not real big on eating mammal stomachs. The Cheese Plate PDX only sells cheese that is made with rennet that is not animal based. Although the Cheese Plate is seriously vegetarian, let me be clear: I _love_ the food at this cart. When I eat there, I do not miss meat. Not one bit.\n\nYes, the cheeses that they serve at the Cheese Plate are superb. What people tend to overlook is the amazing sides that you can get there. The menu is seasonal, so depending on the time of year, you can get items like olives, spiced nuts, truffle potato salad, roasted beets, mushroom kale p\u00e2t\u00e9, cantaloupe with cardamom salt or watermelon with ginger salt.\n\nThe grilled cheese \"sammiches\" that they have at the Cheese Plate are a revelation. It sometimes has on the menu a lavender caramel grilled cheese sammich made with goat cheese and fontina. It is decadent. It has a seasonal grilled cheese sammich made with smoked mozzarella, roasted Brussels sprouts and caramelized onion aioli. Really? Brussels sprouts on a grilled cheese sandwich? It works. It works really well.\n\n_Cheese for Vegans_\n\nThe Cheese Plate PDX is _very_ popular with vegans. I am talking real vegans\u2014the kind who don't eat cheese. Carina said that some of her friends who are vegetarians sometimes miss meat. Vegans don't miss meat; however, some of them do miss cheese. With that in mind, Nick and Carina set out to come up with a vegan cheese that tastes good. The amazing thing to me is that they did it. They sell a cashew-based vegan fromage fort, and it actually tastes great.\n\n\"Enjoy Yourself...Cherish Others.\" The Cheese Plate PDX Food Cart.\n\n_Dalai Lama Comes to Portland_\n\nFor Nick and Carina, their most significant highlight to date was in May 2013, when the Dalai Lama came to Portland. The Cheese Plate PDX Food Cart and the Gonzo Falafel and Hummus Food Cart were both chosen to have their food served at a reception for the Dalai Lama. I was lucky enough to be there that day, and Carina and Nick were absolutely thrilled to be part of that event. It is funny how the universe works. In July 2012, Nick and Carina had purposely opened their food cart on the Dalai Lama's birthday, and just ten months later, their food was served at a reception for him.\n\nCarina and Nick have a dream, and that dream is coming together. They built that dream on their love for cheese and the joy it brought into their lives back in San Francisco at Delores Park.\n\n_What's Next_ : Before 2014 ends, Carina and Nick plan to remodel their food cart and add a cheese and specialty food shop onto the end. In the long term, they would love to grow their brand to the point that they could have a brick-and-mortar establishment. I personally think that they are on to something with their cashew-based vegan fromage fort, and I will not be surprised if someday the Cheese Plate Cashew Fromage Fort is sold nationwide.\n\n_When You Go_ : First get a Cheese Lovers' Oregon Cheese Plate and whichever seasonal grilled cheese sandwich catches your eye. Then, depending on the time of the year, I have three favorites you would do well to get. If it is late fall or winter when you visit, the cheddar ale soup is on my list of the best things I have ever enjoyed at a food cart. If it is summer, and either the cantaloupe with cardamom salt or the watermelon with ginger salt is available, get it!\n\n_Recipes_ : You can find three of the Cheese Plate PDX recipes (Cilantro and Goat Cheese Spring Rolls with Peach Chutney; Wild Mushroom and Kale P\u00e2t\u00e9; and Fromage Fort) in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 1. Two more of the Cheese Plate PDX recipes (Smoky Caramel Grilled Cheese; and Spicy Kale and Garlic Cheese on Toast) can be found in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**I H ITCHHIKED MY WAY TO PORTLAND: DIVINE CAF\u00c9**\n\nSome food cart owners will say, \"Back when I opened my cart, the Portland food cart scene was just getting started.\" In the fall of 2013, I was privileged to attend the twelfth anniversary of the Divine Caf\u00e9 Food Cart. That's right, I said twelfth. The Divine Caf\u00e9 Food Cart has been around for twelve years. When the Divine Caf\u00e9 Food Cart moved to the 9th and Alder pod, it was the fourth food cart to open there.\n\nOne of the tough parts of being a food cart fan is seeing how many of them go out of business. Often a food cart owner's initial goal is make it through the first winter, and it is quite an achievement for a food cart to make it through two winters. The Divine Caf\u00e9 has been through thirteen winters.\n\nRose Guardino visited Portland in the summer of 1990, and in 1991, she graduated high school, having attending Santa Catalina School for Girls in Monterey, California. Following graduation, Rose knew that she wanted to live in Portland, and she proceeded to hitchhike her way to the \"City of Roses\" itself. She then spent nine years working in the restaurant business. During that time, she worked in both the front and back of the house and attended culinary school. Rose had been professionally cooking for a number of years before attending culinary school, which taught Rose to think in ounces and pennies and helped her realize how important each of those pennies and ounces were to succeeding as a small business owner working in the food industry.\n\n_I Want to Open a Food Cart_\n\nRose opened the Divine Caf\u00e9 Food Cart in 2001. In March 2000, she was going through a tough transition phase in her life, and a friend asked, \"What do you want to do with your life?\" Rose answered, \"You know that Snow White Food Cart downtown? I want to open a food cart, a vegetarian food cart.\"\n\nIn May 2000, while meditating, Rose saw the name that her food cart would have\u2014the concept, the colors and so on. Rose then began putting together a business plan. At this point in her journey, Rose also worked with a SCORE mentor, and she found that experience to be extremely helpful. SCORE is a nonprofit association that provides entrepreneurs access to coaching from active and retired businesspeople who have had success in their field. The company was previously known as the Service Corps of Retired Executives, but it is now recognized as SCORE, \"Counselors to America's Small Business.\"\n\nRose Guardino, owner of the Divine Caf\u00e9 Food Cart.\n\n_Every One of Us_\n\nRose occasionally gets asked, \"What does the name of your food cart mean to you? Are you trying to make a religious statement?\" Rose believes that none of us is perfect and that each of us is divine and special. Rose wants people to know that that they are worth it. She loves providing people with healthy, high-quality food that is delicious, and she hopes that her food helps people realize that good food can also be healthy.\n\n_Leaving 9 th and Alder_\n\nIn 2006, Rose was given the opportunity to open a food cart on the campus of the National College of Natural Medicine located near the Ross Island Bridge just off of Southwest Naito Parkway. She bought a second food cart and, for a time, had two carts opened. Rose ended up selling her first food cart, which was located at 9th and Alder, to Kir Jensen, who opened the Sugar Cube Food Cart in that very cart.\n\nThe Divine Caf\u00e9 Food Cart remained at the National College of Natural Medicine until July 2012, when Rose moved to the Good Food Here Food Cart Pod on 43rd and Belmont. Rose has done very well at her latest location. The Divine Caf\u00e9 Food Cart now has a beer and wine garden, and Rose was among the very first food cart owners to get an Oregon Liquor Control Commission (OLCC) liquor license for a food cart.\n\n_Grimm_\n\nA number of Portland food carts have a connection to the TV show _Grimm_. Rose actually works on the show, serving as a food stylist. She really enjoys seeing food that she has styled end up on TV.\n\n_A Vegetation Paradise and Great Food for the Rest of Us_\n\nAs I have mentioned elsewhere in this book, I love bacon, and there is nothing like the smell of meat cooking over a flame. That being said, when I eat at the Divine Caf\u00e9, I thoroughly enjoy my meal. In fact, only about 25 percent of Rose's clients are vegetarians. The majority of her customers are people who simply go for the wonderful food.\n\n_What's Next_ : Rose plans to expand the catering that she does out of her food cart, and she hopes to continue to do even more food styling for the TV and film industry.\n\n_When You Go_ : Get the quesadilla! It is delightful. The Tempeh Rueben is stellar, as is whatever soup and dessert they happen to be serving that day. I think that I could live on the wonderful soups they have at the Divine Caf\u00e9.\n\n**P EDALING KALE: MOBERI**\n\n_I'm Ryan with Moberi, and we make smoothies with bikes!_\n\n_\u2013Ryan Carpenter from the 2013 Moberi Kickstarter_\n\nRyan opened the Moberi Food Cart in August 2012, launching one of Portland's most innovative food carts. At Moberi, they ride a bike hooked up to a blender to make phenomenal smoothies that taste fantastic and are quite healthy for you.\n\n\"None of the traditional options appealed to me,\" said Ryan. When he was in high school, he would hear about career options that he could pursue following college. It was then that he realized that he would most likely end up being self-employed. Working \"nine to five\" Monday through Friday for the \"man\" for forty years was not a path that Ryan wanted to take.\n\nRyan Carpenter, owner of the Moberi Food Cart.\n\nIn his twenties, Ryan traveled the world. He would say that he was looking for the perfect business idea. Ryan always said this half jokingly and with a smile...but only half jokingly. Someday he would have to start his business; he just did not know what it would be.\n\nRyan traveled the globe, seeing South America, Europe, Vietnam, Australia, New Zealand and many other places. Ryan had a number of adventures along the way, and I hope that someday he writes about them. For example, in Morocco, Ryan met a very special turtle named Mo and even ran out of money and ended up sleeping on a roof for a few weeks.\n\n_Inspiration Struck in the Land Down Under_\n\nRyan is not a trust fund baby, and he funded much of his traveling working odd jobs as he roamed the planet. In Australia, he had part-time job washing dishes, and a coworker showed him a YouTube video featuring a Guatemalan NGO named Maya Pedal. Maya Pedal makes a number of bike-powered machines. Ryan knew then that he had found the business idea that he had been searching the world over for.\n\n_It Started on the Sidewalk_\n\nIn 2010, he returned home to Portland, got a job to pay the bills, bought a journal and began to sketch out his new business. In the spring of 2011, Ryan set up a bike-powered blender on the sidewalk in front of his house. He had a modified Schwinn, a blender, a cooler full of fruit and a dream. People liked his smoothies, and over that summer, he hauled around the bike, the blender and a portable sink and made smoothies at events all over Portland. Moberi was on its way!\n\n_The Name_\n\nMoberi is so named to honor Ryan's turtle Mo, as well as because Ryan likes the awesome berries that we have in Oregon. One day, that name popped into his head, and once he said it out loud, it stuck. He knew that he could build a great brand around that name.\n\n\"Money aside, Kickstarter was the best thing I ever did for my business,\" Ryan noted. He bought his first cart in 2012 and opened up at a food cart pod in August that same year. He soon realized that he would do well to have a second food cart that he could use for events, and on April 12, 2013, he launched a Kickstarter project. Ryan was able to raise the $6,000 he needed for his second cart. In addition to the needed funds, he also got two huge unexpected benefits from his Kickstarter efforts.\n\nThe Delorean from Moberi.\n\nThe first was that in preparing for his Kickstarter, he basically had to sit down and finally write a comprehensive business plan. He found that process to be immensely valuable. It is actually common for food cart owners to open their food cart without a business plan.\n\nThe second unexpected benefit was the Kickstarter video itself. That video allowed people to see what Ryan means when he says, \"Moberi is smoothies powered by bikes.\" I tell food cart owners all the time that you have to tell people your brand story, and the Moberi Kickstarter video does just that for Ryan. Go watch the video for yourself and see what I mean. Ryan is very grateful to Ken Wilson for the work he did filming and editing that video.\n\n_Pedaling Kale_\n\nRyan has done a fantastic job providing amazing smoothies that taste great and have ingredients like kale, beets and spinach in them. I have to admit, the first time I had one of Ryan's healthier-sounding smoothies I had some fear and trepidation. But wow, Ryan's concoctions are great\u2014even the ones with kale!\n\nRyan's two most popular smoothies are the Delorean (strawberries, banana and mango) and the Captain Planet (kale, spinach, mango, banana, hemp protein, ginger and coconut milk). In addition to those smoothies, you can get an Uncle Jessie, a One Night in Bangkok, a Johnny Utah and others.\n\nThe next time you are in Portland and want a smoothie, go to Moberi!\n\n_What's Next_ : Ryan is working on a smoothie cookbook, and then he hopes to launch a website where he will share some of his travel stories and talk about the lessons he has learned on his startup adventure.\n\n_When You Go_ : My favorite Moberi smoothie is the One Night in Bangkok, a savory smoothie made with peanut butter, Sriracha, sesame oil, banana, whey protein and almond milk. For me, it is the cr\u00e8me de la cr\u00e8me of the smoothie world. Whenever I have one, I feel smart enough to play chess.\n\n_Recipes_ : You can find two Moberi recipes (the Delorean; and Turtle Power) in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**W HOLE HAWG VEGAN: HOMEGROWN SMOKER VEGAN BBQ**\n\nJeff Ridabock opened the Homegrown Smoker Vegan BBQ Food Cart in April 2009. He himself became a fully fledged vegan two months later.\n\n_The Economy Was Crashing and the Kids Had Become Vegan_\n\nFor many years, Jeff worked at a corporate job that he loved. He was a chef consultant in the food services industries. In 2008, he lost his job, and as he looked toward finding new employment, he discovered that the only jobs he could get in that industry were going to pay him half or even less than the one he lost.\n\nIn 2006, Jeff's two teenage kids came home one day and announced that they were now vegetarian. Jeff, being a chef in his own right and the one who did all the cooking in the house, started learning to cook vegetarian. Two years later, his kids decided to go vegan, and Jeff was faced with learning to cook without butter, cream or cheese.\n\nThis culinary challenge actually invigorated him. Jeff found that he loved getting to be creative in the kitchen once again. The more he cooked vegan cuisine, the more excited he got. He found that he could make vegan food that tasted great.\n\nJeff had previously been involved with a Westside company that did meat barbecue. The more he cooked vegan, the more the idea of applying barbecue techniques to vegan cuisine appealed to him.\n\n_Perhaps a Food Cart Is the Next Step_\n\nIn late 2008, Jeff began to realize that instead of taking another corporate job, he could choose the self-employment route and open a food cart. Jeff then began to toy with the idea of opening a vegan food cart. His big question was: would people buy his vegan concepts and dishes?\n\nIn early 2009, Jeff did the cooking for his son's birthday party, and all the items were vegan dishes prepared with barbecue techniques. He wanted to see if young people would like his food. The items were a huge hit, and by the time that party was over, Jeff knew that he could make a vegan barbecue food cart work.\n\n_What to Name the Food Cart?_\n\nJeff had the master bathroom mirror peppered with post-its that had potential names on them. Yet he did not like any of the names that he came up with. It eventually came down to three concepts. Jeff felt that this was homegrown business, the food was vegan and much of it was prepared with barbecue techniques. Eventually, he settled on calling his cart Homegrown Smoker Vegan BBQ. To Jeff, this name was memorable, fun and accurate.\n\n_Vegan and BBQ?_\n\nI asked Jeff, \"How can your cart be vegan _and_ barbecue?\" Jeff's answer made sense to me. He uses natural woods and smoke, and he cooks at a low temperature. He uses rubs and marinades and sauces. Jeff employs the same techniques now that he used when he was barbecuing meat.\n\nJeff Ridabock, owner of the Homegrown Smoker Vegan BBQ Food Cart. _Photo by Ken Wilson_.\n\n_The Ups and Downs of Running a Food Cart_\n\nJeff first opened his cart on 23rd and Alberta in 2009. By the time 2012 rolled around, Jeff had two food carts\u2014one downtown and one at the now-defunct Green Castle Food Cart Pod. Jeff was able to make two carts work for nine months before he had to shut down one of his carts and move the other. The year 2012 was a rough one, and Homegrown Smoker Vegan BBQ almost did not make it.\n\nThankfully, Jeff was able to move his remaining cart to the Mississippi Marketplace Food Cart Pod, and the summer of 2013 was his best ever. The future of Homegrown Smoker Vegan BBQ is once again looking bright.\n\n_The Cream in the Coffee_\n\nJeff actually had a tough time himself going vegan. Just before his cart opened in 2009, he had given up the last bit of chicken and cheese that he had occasionally nibbled on. With all the vegan dishes that the had learned to cook, giving those two items up was no big deal. The cart had been open two months, and Jeff was still having two ounces of cream in his coffee each morning. It was then that he told himself, \"Jeff, if you are going to go vegan, you have to go whole hog [or \"hawg,\" as he often puts it].\" With that, he became 100 percent vegan.\n\nJeff is quite a character. He is often wearing a tie-dyed T-shirt, and he calls himself a \"psychedelic relic.\" And yes, I did ask\u2014the Grateful Dead is indeed his favorite band. Jeff and his vegan food cart are very much a part of the always interesting Portland food cart scene, and he has many vegan fans who count on him for their supply of vegan 'cue.\n\n_What's Next_ : Jeff has been working on a cookbook, and he gets many requests to bottle and sell his barbecue sauce.\n\n_When You Go_ : Get the Slosmomofo\u2014smoked soy curls with a maple bourbon barbecue sauce and a chipotle slaw. I also recommend the smoked tempeh burger. If you want to sample a number of Jeff's dishes, the combo platter is the way to go.\n\nFood Writer's Perspective:\n\nJENNIFER CHAN\n\nPortland is a city of many things, but from my visits, I see a city of determined and innovative people. It is a city of intense lovers, so focused in the \"what\" yet cognizant of the need for the social glue to make it also about the community they're raising around it.\n\nPortland's food cart scene is nearly legendary, and yet I still had no idea how robust it was. Nor did I realize how intimidating it would be to even begin. Ultimately, I started with the Frying Scotsman. I couldn't have picked a better cart\u2014James King is such a character and so willing to share his story. Eventually, I would pick out the craziest carts or the ones with dishes I'd not seen. Those are the carts with a story behind them\u2014they're doing something out of the ordinary either with their d\u00e9cor or in their kitchen.\n\nThe maturity in Portland's food cart scene is the thing that I wish for most in other cities. Seeing how pods have fostered a real community space was also really eye-opening. Food trucks in other cities don't have the advantage of a permanent area to stay in, and so they never have the chance to establish something more than transient with the people they serve. I can only hope that other cites embrace the unique Portland pod concept\u2014it's a great way to create real ties beyond service.\n\nIn the end, great food carts are not just about a good product; they are about the people behind it. There are plenty of carts out there that are there only to feed your hunger, but the ones that stand out are the ones that share their enthusiasm and connect with you. Whether by verbal means or beyond by using social media, it's the connection that will keep people coming back. Social media plays heavily into that these days, but it's more because that is where a large portion of their audience lives. Sometimes just talking face to face is the best way.\n\nFor those visiting and wanting to get right into food carts, I'd tell them to do some research first. Having an idea of what's worth eating will cut down on all the aimless wandering. Here are my top food cart dishes:\n\nKhachapuri from Kargi Gogo. This hot bread oozing with cheese and with a lightly crisped exterior is the perfect carb-bomb. With salty melted butter running down my chin, this just hit me in all the right ways right from the first bite.\n\nLi\u00e8ge Waffle from Gaufre Gourmet. It's the smell that gets you first\u2014that distinctive smell of caramelizing sugar and yeast. But what spurs you on is the waffle's crunch from sugar and an extra toasty exterior. Then there's the \"just right\" give of a perfectly chewy interior. I love these over Belgians as the li\u00e8ge-style waffle gives your teeth something to sink into.\n\nFish and Chips from the Frying Scotsman. This classic dish is just done really well here. The batter is airy and not greasy, the fish is perfectly done and the fries are amazing. Love that hint of turmeric in the batter.\n\nCarnitas al G\u00fcero from Guero. I've had little experience with Mexican sandwiches, but these were just flat-out delicious. Great punchy flavors, but they are still really well balanced, encased in these fluffy breads with just the right give. So good.\n\nEggs Benedict from the Egg Carton. Love their take on the classic eggs benny. Tangy lemon hollandaise sauce over beautifully poached eggs and a cart-made sausage patty. Then topped with a hit of saltiness from the kale chip.\n\nKelaguen Mannok from PDX Six Seven One. Cold and smoky chicken cubes liberally mixed with coconut, peppers, onions, lemon and a good kick of chili. Ridiculously good. My mouth is watering as I type this.\n\nArbor Lodge Sandwich from the Big Egg. A perfect breakfast sandwich\u2014meaty, but not in a way that would weigh you down. The garlicky goodness and peppery bite of arugula are perfect touches. And of course, it's finished with a fried egg. Terrific.\n**PART VI**\n\n**CUISINE FROM AROUND THE WORLD**\n\n**E AT SOMETHING GEORGIAN (AND WE ARE NOT TALKING PEACHES): KARGI GOGO**\n\nMcKinze Cook and Sean Fredericks opened the Kargi Gogo Food Cart on March 20, 2013. Their path to 9th and Alder took them from Iowa to Akhaltsikhe, Georgia, to Portland.\n\n_We Wanted to See the World_\n\nBoth Sean and McKinze attended the University of Iowa in Iowa City, Iowa\u2014Sean for his graduate degree (an MBA) and McKinze for her undergraduate degree (in anthropology). When they met in 2006 in Iowa City, McKinze was working for the chamber of commerce, and Sean was the manager of the Englert Theater, a nonprofit performing arts venue.\n\nMcKinze and Sean became a couple in 2007, and they were married in 2009. Both of them wanted to see the world, and they applied to the Peace Corps. They were accepted and were notified that they were going to spend two years serving in the country of Georgia. They left the United States in April 2010 for an amazing adventure.\n\nWhen they landed in Tbilisi, they knew how to count to ten in Georgian, and they could say, \"I am not hungry\" and ask, \"Where is the bathroom?\" Sean and McKinze spent three months in language training in Kvibisi, a small village outside Borjomi, home of Borjomi Mineral Water. It was there in Kvibisi that they fell in love with khachapuri, one of the best things that you will ever eat.\n\n_Making a Difference and Enjoying Great Food_\n\nOnce their training was complete, Sean and McKinze moved to Akhaltsikhe, where they lived with a host family until July 2012. Both of them worked with nongovernmental organizations\/nonprofits in Georgia, helping them (the NGOs) to become more effective and efficient. McKinze worked with organizations that dealt with women's health issues, and Sean worked with a few NGOs, including a youth center that taught teenagers leadership and computer skills.\n\nOne of the ways that McKinze and Sean connected with their host family was learning to cook Georgian dishes. Inga, their host \"mom,\" taught them to make the dough for khinkali. Sean and McKinze also became friends with Nino and Dato, who owned the Saghighino restaurant. It was in the kitchen of Saghighino that Sean and McKinze learned to fold khinkali. Making this amazing food in Georgian kitchens is the reason why Sean and McKinze can prepare authentic Georgian cuisine today. Little did they know how these skills would serve them later.\n\nFeasting is a huge part of Georgian lifestyle, and this makes perfect sense when you discover that Georgia is the birthplace of wine and home of the oldest wine-producing regions in the world. Sean told me that wine is an integral part of Georgian culture. Almost every weekend, Sean and McKinze got to enjoy a traditional Georgian supra (feast).\n\n_Vacation in Portland, May 2011_\n\nIn May 2011, Sean and McKinze took a vacation back to the United States, and they came to Portland. They were trying to decide where they might move to once their service in Georgia was over. It was down to Chicago and Portland. While here in Portland, they even went to the 9th and Alder food cart pod and had a meal. By the summer of 2011, they had decided to move to Portland. Now they just needed to figure out what to do for income once they got here.\n\nKhachapuri from Kargi Gogo, the best cheese bread that you will ever have.\n\n_The Culinary Beauty that Is Khachapuri_\n\nKhachapuri is Georgian cheese bread. When you get khachapuri, the cheese is between two pieces of fried bread, and it is melted and warm and ready for you to devour. Khachapuri is a salty, cheesy delight. Some people say that it is the best cheese bread in the world. It is certainly one of the best things that I have ever eaten. There is only one way that khachapuri could get even better.\n\n_A Stack of Khachapuri Gave Them Their First Answer_\n\nOn a fall Saturday morning in 2011, McKinze and Sean were in their bedroom talking about what they could do for income once they moved to Portland in 2012. Sean had always wanted to open a restaurant or do something with food, and this long-held dream was a part of the conversation.\n\nInga, their host mom, called them down for breakfast. As was usual for a Saturday morning, there was stack of khachapuri. As soon as McKinze saw the steam rising from the stack and the melted cheese oozing from the sides, she turned to Sean and said, \"We need to do Georgian cuisine!\" Within two months, they had decided that when they got to Portland, they would open a food cart serving their favorite Georgian dishes. Now they just needed a name for their food cart.\n\n_Tony Gave Them Their Second Answer_\n\nIn May 2012, some of their friends from Iowa came to Georgia and stopped and visited with Sean and McKinze. A very common expression in Georgia is \"kargi gogo,\" which means \"good girl.\" Upon hearing the explanation for this phrase, Tony said, \"That would be a good name for your food cart.\" Tony is also the person who later designed the Kargi Gogo logo.\n\n_Six Months to Open a Food Cart_\n\nSean and McKinze arrived in Portland on September 25, 2012, and they began making preparations to open their food cart. They both got jobs in the restaurant industry, Sean as a line cook and McKinze as a hostess. They did this to learn just a little more before opening their food cart. In February 2013, they found a food cart that was available right at the same 9th and Alder food cart pod where they had eaten just twenty-two months earlier.\n\n_Ambassadors of Georgian Cuisine_\n\nEven before they opened Kargi Gogo on March 20, 2013, Sean and McKinze received support from Georgia. Georgians are fiercely proud of their culture and cuisine, and they were very honored that Sean and McKinze were promoting both here in America. I am delighted that Sean and McKinze chose Portland over Chicago, and I look forward to seeing how they grow their fledgling startup.\n\n_What's Next_ : Sean and McKinze have big plans for the Kargi Gogo brand. I myself am hoping to one day be able to go into a brick-and-mortar Kargi Gogo and have a cold beer with a plate of warm khachapuri. In my opinion, the only way to improve on the melty, cheesy goodness that is khachapuri would be to enjoy it with a cold beer and a playoff football game.\n\nMcKinze Cook and Sean Fredericks, owners of the Kargi Gogo Food Cart.\n\n_When You Go_ : Need I say it? Get some khachapuri! I also love the khinkali (a stuffed beef and pork dumpling) and the badrijani (a vegan dish that has a puree of walnuts, garlic and Georgian spices wrapped in strips of saut\u00e9ed eggplant).\n\n_Recipes_ : You can find the Kargi Gogo recipe for Khinkali in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**S O MUCH MORE THAN JUST CURRY: TIFFIN ASHA**\n\nElizabeth Golay opened the Tiffin Asha Food Cart on May 17, 2013. Since then, it has received quite a bit of well-deserved attention, including being named one of the top-ten new food carts of 2013 by Michel Russell, the restaurant critic for the _Oregonian_ , as well as being featured in a very complimentary August 2013 _Portland Monthly_ story written by Karen Brooks.\n\n_Inspired by a Cookbook and a Restaurant_\n\nElizabeth met her future wife, Sheila, in 2002 in Seattle, Washington. In 2004, they began dating, and in 2006, they moved to Boston. While there, they were married in Cambridge. Sheila's family is from South India, and back when they were dating, Sheila loaned Elizabeth her copy of _Dakshin_ , a cookbook written by Chandra Padmanabhan.\n\nElizabeth Golay, owner of the Tiffin Asha Food Cart, and her wife, Sheila.\n\n_Dakshin_ features cuisine from South India, and Elizabeth quickly fell in love with the food from this part of the globe. Sheila and Elizabeth lived in Boston from 2006 to 2009. During this time, Elizabeth, a culinary school\u2013trained chef, worked at Oleana for the James Beard Award\u2013winning chef Ana Sortun. Elizabeth was able to get a front-row view to what a chef could do with a cuisine he or she loved. It was then that Elizabeth began to dream about what she could do with the cuisine of South India.\n\nElizabeth noted, \"I was sick of working hard for someone else and not getting anything for me. I wanted to work hard for myself, building my own business.\" In 2009, Elizabeth and Sheila moved back to Seattle, and Elizabeth went to work as a pastry chef at Poppy, a much-acclaimed restaurant on Capitol Hill started by Jerry Traunfeld, the James Beard Award\u2013winning chef who worked as the executive chef at Herbfarm from 1990 to 2007.\n\nBy late 2010, Elizabeth had decided that she was going to open her own restaurant and that she was going to begin this endeavor by starting a Seattle food truck. Before she could pull the trigger on this decision, Sheila came home one day and said, \"How would you like to move to Portland?\" Sheila had been offered a job in Bridgetown.\n\n_Falling in Love with Portland_\n\nSheila and Elizabeth moved to Portland in January 2012, and they loved what they found here. \"The people are wonderful, it is small business oriented and it is a place where you can be yourself,\" said Elizabeth. Even before they arrived in Portland, it was decided that Elizabeth would open a food cart here. Seattle's loss became our gain.\n\nFor the next year, Elizabeth's \"job\" was perfecting the items that she would put on her menu. Sometimes a food cart can be opened a few weeks or even a few months before the owner has nailed down the cooking side of their business. I first ate at Tiffin Asha on May 17, 2013, and from the first day that they were open, the food has been incredible. Elizabeth is a _very_ talented chef.\n\n_Dosa Is the Canvas_\n\nWhen you go to Tiffin Asha, you will find a number dishes with which you may not be familiar: dosa, sambar, vada and so on. Dosa is a paper-thin fermented rice and dal (lentil) crepe. Start out with the Hot Chick\u2014most everybody does. The Hot Chick is a pakora fried chicken drizzled with black cardamom-infused honey, pickled greens and creamy yogurt cheese wrapped in dosa.\n\nSambar is a delightful soup that reminds me of a tomato soup for grown-ups. Get some idli and dip it in your sambar. Vada is a savory donut, and the vada at Tiffin Asha are made with coconut-chili fleur de sel. They are scrumptious.\n\n_You Can Get Very Creative: Chutney and Gunpowder_\n\nTwo of the fun things about eating at Tiffin Asha are the chutneys and the gunpowder. You can get one of four cart-made chutneys that allow you to change the flavor profile of each bite simply by switching from one chutney to another. I love to spread the coconut chutney on my vada.\n\nYou can also get three different flavors of \"gunpowder,\" which is Elizabeth's take on podi. Podi is a mix of dry spices that you combine with sesame oil just before you enjoy it. At Tiffin Asha, you are given a packet of gunpowder and a little cup that has sesame oil in it. You empty your spice packet into the little cup and voil\u00e0! You have yet another flavor with which you can experiment.\n\nChef Elizabeth Golay making dosa at the Tiffin Asha Food Cart.\n\nBetween the four chutneys and the three gunpowders, you have seven extra ways to personalize and season your food. Make sure that you get extra dosa and vada to enjoy with your combinations. Have fun with it. Remember when you were a young child, and you got to finger-paint and would fearlessly swirl different colors together to see what would happen? You can do the same with your chutney and gunpowder.\n\n_The Art of Making Dosa_\n\nElizabeth had come up with her own recipe for dosa that she spent a year perfecting. She combines two kinds of rice (basmati and sona masoori) and three different kinds of lentils (chana dal, urad dal and fenugreek seed) and soaks that mixture overnight in a big container. Over night, it rises and ferments.\n\nIn the morning, Elizabeth whisks up this mixture and puts it in the fridge to chill for a few hours. When it is time to make dosa, the chilled liquid is poured very thinly onto the hot grill. Watching Elizabeth make dosa is quite mesmerizing, and you can tell that you are seeing an artist at work.\n\n_Bringing the Cuisine of South India to Portland_\n\nElizabeth loves it when people who are not familiar with food that she specializes in come to her cart, learn about it and enjoy it. She told me that when many people in the United States think of Indian food, they think of curry, and she relishes helping people discover that there is much more than just curry to embrace when it comes to this cuisine.\n\nI have to say that I think it is time for a Portland food cart chef to win a James Beard Award, and I think that Elizabeth Golay is a worthy candidate.\n\nBy the way, \"Tiffin\" means \"snack,\" and \"Asha\" means \"hope\" or \"wish.\" It is my hope that Elizabeth will be able to see her food cart dreams come true.\n\n_What's Next_ : From the very beginning, the plan was to build the Tiffin Asha brand to the point that a small, intimate brick-and-mortar Tiffin Asha was feasible. I hope along the way that Elizabeth packages her gunpowder and chutneys. I would buy them to use at home in an instant.\n\n_When You Go_ : Take a friend with you, order three or four dishes, get some gunpowder and some chutney and have a blast enjoying amazing food the likes of which you may have never had before.\n\n_Recipes_ : You can find the Tiffin Asha recipe for Coconut Chutney in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**B ECAUSE OF MEATBALLS AND GRANDMA CLENORA: VIKING SOUL FOOD**\n\nMegan Walhood and Jeremy Daniels opened Viking Soul Food on August 19, 2010. When a food cart first opens, the owner is hoping to be able to make it through the first winter. Viking Soul Food has made it through four winters and shows no signs of stopping.\n\nViking Soul Food features Norwegian cuisine, and the menu is primarily built around the Norwegian flat bread know as lefse. This food cart has received quite a bit of press. It was in a _Food and Wine_ magazine feature on Nordic food, and in 2012, the _Smithsonian_ magazine named it one of the twenty best food carts in the United States.\n\nMegan Walhood and Jeremy Daniels, owners of the Viking Soul Food Cart.\n\n_Where All Good Love Stories Start_\n\nThe Viking Soul Food story started where many good love stories have started: the kitchen. In 2006, Jeremy and Megan met in the kitchen at Nostrana, one of Portland's best restaurants. By 2007, they were a couple. \"I am a chef, and I figured out that I was not really good at working for someone else and that I was not good at middle management. They left me needing to start my own business,\" said Megan, who grew up in Portland.\n\n_To the Antarctic and Back_\n\nFollowing high school, she began to see the world, cooking along the way. The first stop was Olympia, Washington, to attend Evergreen State College. \"Greeners\" (as the locals refer to the students at Evergreen) are a special bunch.\n\nOne common Evergreen joke suggests that \"patchouli oil is considered an aphrodisiac at Evergreen.\" Archibald Sisters is a local Olympia business that sells many different fragrances, including peach, sandalwood, wisteria and blackberry. There is a reason why two of the fragrances you can get at Archibald Sisters are patchouli and patchouli vanilla.\n\nFollowing her time at Evergreen, Megan began cooking. Her first stop was a restaurant cherished by locals and now gone: Olympia Restaurant, also known as Ben Moore's. In 1999, Megan went to Antarctica to cook and hang out during the approaching Y2K crisis. The world did not end, and Megan headed off to New Zealand for three years. Finally, it was time to come home to Portland. Megan missed her family and the vibrant Portland food scene.\n\n_Jeremy's Culinary Ascent_\n\n\"My first job after culinary school was working as a dish washer. Those were the longest three months of my life,\" said Jeremy, whose family moved to the Portland area from Colorado when he was ten years old. Following high school, Jeremy attended Western Culinary School. Once Jeremy graduated from culinary school, he was ready to take on the world and become the head chef at a restaurant in Portland. No one ever accused Jeremy of lacking confidence. Getting hired to work as a dish washer was quite a shock. Humbled but not bowed, Jeremy began working his way up the Portland culinary ladder.\n\nOf Megan, Jeremy said, \"I never worked so well with anyone before [her].\" Megan, in turn, said of Jeremy, \"I love the way that [he] and I make food together.\" Within six months of beginning their relationship in 2007, Megan and Jeremy began talking about starting a culinary business together. They wanted to open a restaurant; however, they just did not have the financial resources for that. They eventually began talking about opening a food cart. Megan noted that \"a food cart had the fewest barriers to entry.\"\n\n_Christmas Meatballs!_\n\nWhen Megan was child, her Norwegian family would enjoy meatballs made from a family recipe at Christmas. The meatballs were topped with Gjetost (\"yay-toast\"). Gjetost is a decadent goat cheese sauce made from caramelized goat milk. The Gjetost that Megan enjoyed as a child, and that you can get today at Viking Soul Food, is Ekte Gjetost, or \"true\" Gjetost, and this means that the sauce is made _only_ with goat milk and not cow's milk.\n\nYears later, when Megan and Jeremy were invited to a Christmas party, Megan harkened back to those Christmas meatballs from her childhood. She made them and took them to the party. The meatballs covered in Gjetost were a huge hit. On the way home, Jeremy told Megan, \"Let's sell those meatballs at a food cart!\" Megan loved the idea and continued the concept, saying, \"We can also have lefse and use them as a wrap for all kinds of things.\" With that, they had a cuisine. Now all they needed was a cart.\n\n_The Fine Art of Making Lefse_\n\nLesfe is a Norwegian flatbread that is made with potatoes that have been boiled and riced, butter, cream, salt and flour. This mixture is stirred, kneaded like bread dough, formed into patties and hand-rolled out to be very thin rounds about twelve inches or so in diameter. These rounds are then moved to a grill to be quickly cooked. After about thirty seconds, your lefse is ready to be flipped, and you need to use a lefse stick to effectively execute that delicate maneuver. Needless to say, lefse is very difficult to make. A fan named Susan Cersain wrote on the VSF Facebook wall, \"I grew up eating lefse every Christmas, but tonight I think I had the best lefse I've ever had.\"\n\nGudrun, \"She Who Knows the Secrets of Battle,\" at Viking Soul Food.\n\nMegan was four years old when she first got to make lefse with her grandma Clenora. It was Grandma Clenora who taught Megan to make lefse years ago, and Megan, in turn, was the one to teach Jeremy. I have had Jeremy's lefse, and while he may not be Norwegian, he sure can cook like one.\n\n_Gudrun, \"She Who Knows the Secrets of Battle\"_\n\nIn March 2010, Megan began looking on Craigslist for a food cart they could buy. Megan had started with Portland Craigslist, and she was just not finding what she wanted. Megan was on the hunt for a food cart that had character and charm. She expanded her search to include western Washington as well.\n\nFinally, in Snoqualmie, Washington, she found a cart that she thought might work. On May 17, Norwegian Independence Day, Megan and Jeremy went to see it. Everything was perfect! The person selling the cart actually knew what lefse was, and she was delighted that her 1958 Streamline Duchess was going to be used for Norwegian cuisine.\n\nWhen Megan and Jeremy went into their new cart for the first time, they found a picture of Julia Child on the wall. They took this to be a good sign. That picture is still up in the cart to this day. Megan and Jeremy brought their newly acquired food cart home to Portland and named it \"Gudrun.\" They chose this name for two reasons. First, they loved what this Nordic name means (\"she who knows the secrets of battle\"). The second reason had to do with a successful Portland businesswoman who was friends with Megan's parents and who had inspired Megan as a teenager to believe that someday she could be a successful businesswoman herself.\n\nGudrun \"Goody\" Cable is the dynamic woman who is the founder and owner of the Rimsky-Korsakoffee House in Portland and the literary-themed Sylvia Beach Hotel in Newport, Oregon. Gudrun the Viking Soul Food Cart is named after her.\n\n_While Having Martinis_\n\nMegan had been taking business classes at MercyCorps Northwest, and those classes drove home the point that their new business needed a name that was effective and memorable and reflected who they both were. In other words, the name they gave to their food cart business needed to fit with their brand story.\n\nMegan was at her parents' having martinis when inspiration struck. She called Jeremy and excitedly told him, \"I got it!\" All she had to say were three words\u2014\"Viking Soul Food\"\u2014and Jeremy knew that their food cart business now had a name.\n\nThe first of those three words, \"Viking,\" referred of course to Megan's Norwegian heritage, and the last two words, \"Soul Food,\" were a tip of the hat to Jeremy's African American heritage. Megan and Jeremy felt that \"soul food\" not only referenced a type of cuisine that originated in African American culture but also evoked the idea that you could take the ingredients you had on hand and make food that was comforting and incredibly pleasing to the palate.\n\n_They Just Walk Away_\n\nAn interesting and sad phenomenon I have come across while talking with Portland food carts owners is something that Nimesh with Stumptown Dumplings described this way: \"People seem to want you to cook in your lane.\" Let me explain.\n\nTimber Adamson is the founder of Timber's Doghouse PDX Food Cart. However, that is Timber's second food cart. She intentionally closed down her first food cart, which featured Korean food, because of the grief she received for being a white woman who was cooking authentic Korean food.\n\nJustin and Nimesh sometimes get questions about their being Indian and having a food cart that sells Chinese dumplings. Jeremy has had similar experiences. More than once, Jeremy has seen an entire family\u2014mom, dad, kids and even an aged grandma\u2014come to the pod, walk right over to the prominent Viking Soul Food menu and began excitedly talking about the items on it.\n\nIn hushed, almost reverent voices, they say things like, \"They _do_ have lefse!...You can get lingonberries on your lefse\u2014just like Grandma Jorun used to make...They have Gjetost!...Look\u2014pickled herring!\" People start naming off what they want to get, and someone starts keeping track of the dishes they are going to order. At that point, they look over to the cart window and see Jeremy's smiling face\u2014his smiling _black_ face. He has actually been asked, \"Where is the viking?\" Jeremy kindly answers, \"She is not here right now, but I can help you.\" What they don't know is that Jeremy made that lefse and pickled that herring. In fact, his lefse is most likely way better than Grandma Jorun's ever was.\n\nOnce it sinks in that the only person in the cart is not Norwegian enough for them, they quietly confer with one another and then just walk away. My heart ached when Jeremy told me about these occurrences. It is incredibly painful to him when people who come to his food cart to enjoy his food end up walking away because of his skin color. Portland has the reputation as a liberal and progressive city. If Jeremy experiences that kind of racism here in my beloved city, I have to wonder what it is like in other parts of the country.\n\n_Culinary Alchemists_\n\nOne of the things I love the most about Viking Soul Food are the specials that Jeremy and Megan put on the menu. They are incredibly flavorful. Check out some of the lefse wraps they have served: the Winter Wrap, filled with roasted pears, Portland Creamery ch\u00e8vre and sherry gastrique; the Ambrosia Wrap, with roasted peaches, Portland Creamery vanilla ch\u00e8vre, walnuts and sherry syrup; the Polse, filled with chicken sausage, Jarlsberg cheese, mustard and lemon fennel slaw; and the Bird Brain, slow-cooked chicken, roasted mushrooms, asparagus, Jarlsberg cheese, walnuts and greens, all rolled into, you guessed it, a lefse. I could go on, but you get the picture.\n\nThe dish that convinced me that that these two chefs are culinary alchemists was when I had a simply delightful dish that included beets and vanilla. Who would ever think to combine beets and vanilla? As odd as it sounds, it was a really good dish. I am so glad that Grandma Clenora taught Megan to make lefse all these years ago.\n\n_What's Next_ : Megan and Jeremy would love to open up a brick-and-mortar Viking Soul Food restaurant some day. They would have an expanded menu, with plated entr\u00e9es, and Megan has her heart set on having a fireplace with a large hearth. They also would like to have a small shop where people could buy lefse-making equipment and Scandinavian pastries and cookies that would be made right there in the kitchen.\n\n_When You Go_ : Come prepared to order a number of dishes. Of course, get an order of the pork and beef meatballs. They come with the Gjetost, that scrumptious sauce made from caramelized goat cheese. Get a sweet lefse with lingonberries and order a side of pickled herring. No really, order the pickled herring. Lastly, order whichever special most tickles your fancy.\n\n_Recipes_ : You can find three Viking Soul Food recipes (Lefse; Surkal, or sweet and sour cabbage; and Lemon Curd with Spiced Pecan Lefse) in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 1.\n\n**A B AKER'S SENSE OF COOKING: PDX SIX SEVEN ONE**\n\nEd Sablan opened the PDX Six Seven One Food Cart on May 25, 2010. Ed was born in Turkey and raised in Taguac, a community in the Guamanian village of Dededo, and now he is thriving in Portland. Here is his story.\n\nEd's parents are both Chamorro, and Ed's dad joined the U.S. Air Force. Ed was born in Turkey while his dad was stationed there. Chamorro is the name of the indigenous people of the Mariana Islands, which are located south of Japan and north of New Guinea, about 5,600 miles from Portland and 3,900 miles from Hawaii.\n\nEd graduated from Father Due\u00f1as Memorial High School in 1993, and in August of that same year, he moved to Portland to pursue both an undergraduate degree and a very special woman named Marie. In 2000, Ed and Marie married.\n\n_A Pizza Dough\u2013Fueled Epiphany_\n\nOnce Ed got to Portland, he proceeded to spend about seven years attending three different local colleges, each time taking classes related to art and\/or music. Ed eventually found himself working at the McMenamins Kennedy School making pizza dough, and it was then that he remembered that he loved to bake. Ed had received his first baking cookbook in third grade, and he had done quite a bit of baking prior to high school.\n\nEd Sablan, owner of the PDX Six Seven One Food Cart.\n\nThe precision and science of baking deeply resonated with Ed, and he decided to attend culinary school. He graduated from Western Culinary Institute with a degree in baking and patisserie. Even before he graduated, Ed was hired by St. Honor\u00e9 Boulangerie, a prominent French bakery here in Portland.\n\nEd started at the bottom and worked his way up to production manager, and when St. Honor\u00e9 Boulangerie opened its Lake Oswego location, it was Ed who oversaw the opening. At St. Honor\u00e9 Boulangerie, Ed learned how to manage a culinary business.\n\n_A Restaurant of His Own_\n\nEd has wanted his own restaurant since he was a teenager living in Guam. Here in Portland, Ed would often find himself making the food he grew up enjoying in Guam for family and friends. He began to wonder, too, if maybe his dream to open his own restaurant could actually happen. In 2007, Ed decided that he would open a food cart featuring that cuisine, and he even bought the domain pdx671.com. PDX is the airport code for Portland, and 671 is the area code for Guam, though the business name uses \"Six Seven One\" officially.\n\n_A Baker's Sense of Cooking_\n\nReally good bakers are meticulous, and they pay close attention to details, knowing that things like slight changes in humidity or room temperature can drastically effect what you are cooking. Ed began to apply that same approach to his dream to open PDX Six Seven One. In 2009, he quit his job and invested one year working full time on the business plan, menu and recipes for PDX Six Seven One.\n\n_Chamorro Cuisine or Guamanian Cuisine?_\n\nThe answer to this question is probably just \"yes.\" On the PDX Six Seven One website, it mentions \"Guamanian cuisine,\" and I have heard Ed talk about how the history of the Mariana Islands has shaped the cuisine the Chamorro people have grown up with. There are even differences between the cuisine of the Chamorro people from Guam and the cuisine of the Chamorro people from the northern Mariana Islands. Even though those two U.S. territories are only 128 miles apart, they each have had a very different history, and history affects flavor and food.\n\n_Triple D_\n\nPDX Six Seven One has the honor of being the very first Portland food cart to ever be featured on Guy Fieri's Food Network show _Diners, Drive-Ins and Dives_. Ed is a very humble man, and he does not like to call attention to himself. Ed was both thrilled and a little uncomfortable to receive this honor.\n\n_A Wonderful Ambassador_\n\nAt the risk of embarrassing Ed, I have to say that he has done a stellar job of sharing the cuisine of his childhood with the world. Because Ed had the courage to pursue his dream, thousands of people now know about and enjoy items like Kelaguen Mannok, hineksa' agaga' and ensaladan kiukumba.\n\n_What's Next_ : Ed has a detailed plan to open up a PDX Six Seven One brick-and-mortar restaurant. Look for it sometime in 2015.\n\n_When You Go_ : Ed's most popular dish is the Kelaguen Mannok, a chopped, spicy grilled chicken that is served cold and features onions, grated coconut, chili peppers and lemon. My favorite dish at PDX Six Seven One is the Nengkanno' Gupot, or Fiesta Plate. This dish comes with a grilled chicken thigh that has been marinated overnight, red rice, shrimp fritters and Fina'dene', a Chamorro sauce that is made with soy sauce, lemon, chili peppers and onions. I also love the grilled chicken that comes with the Nengkanno' Gupot. It has a fantastic flavor that I have never had before, and it is some of the best barbecue chicken I have ever had. Whatever else you get at PDX Six Seven One, make sure that you get an order of Titiyas\u2014hand-made, grilled, coconut milk flatbread. It is terrific and goes perfectly with every dish that Ed has on the menu.\n\n_Recipes_ : You can find two of PDX Six Seven One recipes (Kelaguen Mannok, chopped grilled chicken; and Bo\u00f1elos Uhang, shrimp fritters) in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**P ORTLAND'S ASIAN DUMPLING CHAMPS: STUMPTOWN DUMPLINGS**\n\nJustin Prasad and Nimseh Dayal launched the Stumptown Dumplings Food Cart on May 31, 2012. I was lucky enough to be at their grand opening that night, and I have been very impressed with the way they have grown their brand ever since.\n\nThey now have two food carts, one located at a food cart pod and a second mobile cart that is used for events and catering. They also have a service window location that is just down the wall from Voodoo Doughnuts and is open daily from 9:00 p.m. to 2:00 a.m., serving the late-night crowd. They will be opening a brick-and-mortar Stumptown Dumplings Restaurant in the first part of 2014. Not bad, not bad at all.\n\nNimesh Dayal and Justin Prasad, owners of the Stumptown Dumplings Food Cart.\n\n_Fate Brought These Partners Together_\n\nNimesh first came into Justin's life in 2000, when Nimesh began to date Jaishree, a friend of Justin's. When Jaishree became Nimesh's bride a year later, Justin was at that wedding. Within a few years, Justin and Nimesh drifted apart and lost touch.\n\nFlash-forward to 2011. Justin is now the general manager of a restaurant, and he wants to open a food cart. He wants to have a business of his own to oversee and grow, and he is convinced that he can make a food cart business work. Later that same year, Justin is in the Pearl District at a bar for a birthday party. Lo and behold, Nimesh is there. He said to Justin, \"We need to talk...I have been thinking about opening a food cart. Would you have any interest in partnering with me on a project like that?\"\n\nJustin remembered thinking, \"Wow! What are the odds that I would run into Nimesh and that he would also want to open a food cart?\" Justin had enough respect for Nimesh's business skill to seriously consider entering into a partnership with him. In Hindi, they call that \"bhaagya.\" In English, we call that \"fate\" or \"destiny.\" The funny thing is that in Gujarati, \"bhaagya\" means partnership.\n\nBoth Justin and Nimesh grew up here in the Portland area, and both are of Indian descent. Justin's family is from India via a two- to three-generational stopover in Fiji. Justin's second language is Fijian Hindi, which is similar to the traditional Hindi they speak in India. Nimesh's family is Gujarati, and his second language is Gujarati.\n\nBhaagya (\"fate\" in Hindi) brought them together into a bhaagya (\"partnership\" in Gujarati).\n\n_The Last Hotel Had Been Sold_\n\nNimesh had been working in his family's business. It is quite common for Gujarati to be involved in the hotel business. Some reports suggest that more than 40 percent of all motels in the United States are owned by Gujarati. The book _Life Behind the Lobby: Indian American Motel Owners and the American Dream_ by Tufts University professor Pawan Dhingra chronicles this fascinating phenomenon.\n\nNimesh had overseen the business side of the hotels that his parents owned. They sold their last hotel and retired, and Nimseh needed to find something else to do. Succeeding as a food cart owner involves both culinary skills and business skills. Justin knew culinary, and Nimesh knew business. Sounds like bhaagya to me.\n\n_Making the Partnership Work_\n\nMany partnerships don't end up working out. I have observed and been around Justin and Nimesh for more than a year and half now, and I have seen qualities that helped their partnership flourish. They have separate duties but the same goals. While their duties occasionally overlap, they tend to focus on their strengths. Nimesh is more of the business manager, and Justin is the head chef or chef de cuisine. This division of duties helps them avoid the problem that they playfully refer to as \"too many chiefs and not enough Indians.\"\n\n_Liquid Gold_\n\nI would be remiss if I did not mention the fantastic Liquid Gold Thai Iced Tea they have at Stumptown Dumplings. I love it, and my two younger sons, Zac and Zayne, consider it their hands-down favorite beverage of all time. I am very lucky that you can't yet get take-home growlers of that Thai iced tea. If they could, Zac and Zayne would drink it by the gallon.\n\nThe Stumptown Dumplings Food Cart.\n\nJustin spent quite a bit of time perfecting his Thai iced tea recipe. He said that he can't stand it when you order a Thai iced tea, and you taste the milk and the sugar rather than the tea itself. The Thai iced tea at Stumptown has that distinctive tea flavor that comes through very nicely.\n\n_Chinese Dumplings and Boa_\n\nThai iced tea aside, the dumplings and boa really are the stars of show at Stumptown Dumplings. Every once in a while, Justin and Nimesh get asked, \"What are you guys doing cooking Chinese dumplings?\" Once someone tastes one of their dumplings, that question is completely answered.\n\nRegularly, Justin and Nimesh are told things like, \"I used to live in Hong Kong, and these are the best dumplings I've had since living there!\" As one fan wrote on the Stumptown Dumplings Facebook wall, \"Best I've had outside of Beijing. Just phenomenally great.\" I encourage you to go check out Stumptown Dumplings and see for yourself why it is Portland's Asian Dumpling champ.\n\n_What's Next_ : Justin and Nimesh are going to open the Stumptown Dumplings brick-and-mortar restaurant in the first half of 2014. I am hoping that their new restaurant will have a bar so that I can enjoy some Thai iced tea cocktails. I can think of a number of ways to make a great cocktail using Stumptown Dumplings Thai iced tea. One of my ideas is to add coconut rum to the tea. Once the restaurant is open, Justin and Nimesh hope to bottle three of their sauces for retail sale\u2014the Asian chipotle, the Thai mustard and the chili hoisin.\n\n_When You Go_ : I love the chicken dumplings with the creamy peanut sauce. Get a coconut custard bao for dessert. Lastly, make sure that you get some of that Thai iced tea.\n\n_Recipes_ : You can find a recipe for Stumptown Dumplings' Thai iced tea and Asian Chipotle Sauce in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**O UT OF EGYPT: RAMY'S LAMB SHACK**\n\n_Far, We've been traveling far. Without a home, but not without a star. Free. Only want to be free. We huddle close. Hang on to a dream. On the boats and on the planes, they're coming to America. Never looking back again, They're coming to America. Home...To a new and a shiny place. Make our bed and we'll say our grace. Freedom's light burning warm, Freedom's light burning warm_.\n\n_\u2013Neil Diamond, 1981_\n\nRamy Armans opened the Ramy's Lamb Shack Food Cart in March 2012. With this step, he fulfilled his childhood dream of being self-employed. I am inspired by the fact that eleven years earlier, Ramy arrived in the United States as a seventeen-year-old kid with no assets to his name other than an unwavering belief in the American dream.\n\nRamy Armans, owner of the Ramy's Lamb Shack Food Cart.\n\nRamy was born and grew up in Cairo, Egypt. He was born into a Christian family, and as early as he can remember, the members of his family were persecuted for following the \"wrong\" religion. Ramy remembered being a small child and watching while an adult family member was kidnapped off the street in front of him, shoved into a minivan and taken away. It was weeks before that family member was returned home, having been severely beaten and abused\u2014all because the family were not Muslim and worshiped Jesus rather than Allah. Some people in America read about religious persecution and feel for those in such situations, while others aren't sure if it's really that bad \"over there.\" But it is that bad. Just ask Ramy what it was like to grow up Christian in a Muslim country.\n\nIn America, the Constitution provides for religious freedom. If you were born a Minnesota Lutheran, you can move to San Francisco and become a Buddhist. Not only can you do that, it is perfectly legal to do so. In many Muslim countries, the law of the land declares it a crime punishable by death to convert from Muslim to Christianity.\n\nThe stanza from Emma Lazarus's poem adorning the bronze plaque on the Statue of Liberty is a fitting reminder of those who leave one home to find another:\n\n_Give me your tired, your poor_ ,\n\n_Your huddled masses yearning to breathe free_ ,\n\n_The wretched refuse of your teeming shore_.\n\n_Send these, the homeless, tempest-tost to me_ ,\n\n_I lift my lamp beside the golden door!_\n\nThere were two bright spots in Ramy's childhood. The first was cooking with his mom. He began cooking at an early age and loved working with spices and trying new combinations. The second was the hope that someday his family could move to the land of the free and the home of the brave.\n\nRamy yearned to move to America, a place where he could have the freedom to publicly practice his faith and be able to start his own business. Ramy knew that in Egypt, it would very difficult, if not impossible, for him to be a native-born Egyptian man _and_ a Christian, as well as have the self-employment success that he dreamed about.\n\nIt was a plane that brought Ramy to America, and after a brief layover at Dulles International Airport, he landed at PDX to begin a new life. His first job was at a gas station at 181st and Burnside. Eventually, Ramy went to work as a meat cutter at the Winco grocery chain, where he learned skills that would later serve him well in his food cart. From there, Ramy went into retail management, working at both Best Buy and OfficeMax.\n\nIn 2007, Ramy began going to the SW 9th and Alder food cart pod. There he began to think that maybe a food cart would be the way he could start his own business. Ramy even began running spreadsheets in 2007 to try to figure out what it would take to succeed as a food cart owner.\n\n_What Do You Want to Do?_\n\nIn 2009, Ramy met Leila Ramezani, a beautiful woman who grew up in Tehran, Iran. Leila had her own story of her journey to the United States, immigrating here with her family in 2001. Leila quickly became his girlfriend, and in 2010, she asked Ramy, \"What do you want to do?\" Ramy had a good corporate job, but his desire to have his own business left him feeling restless. Ramy told Leila that he had wanted to be self-employed since he was child, as well as that he thought that he could do it by opening a food cart.\n\nLeila said, \"Well, you should do it. I believe in you.\" From that moment on, Ramy began making plans to open his food cart. The first cart that Ramy bought was not especially attractive, to put it mildly. He paid $3,500 for it and spent months renovating it. It was ugly, but it was his. As Ramy says, \"The American dream is real. You have to bust your ass, believe in yourself and work for it. Do those things and you can make it!\"\n\nRamy eventually replaced his first food cart with a newer, much better-looking one, and within less than a year, he had opened his second food cart. In December 2012, when he was tearing down that first cart, Ramy became so overcome with emotion that he sat down and cried. He realized that he had come a long way from the dusty streets of Cairo, where he wished that he could be free and have his own business. That first cart, ugly as it was, had helped his dream come true.\n\nIt drives Ramy crazy to hear people born in America complain about the lack of opportunity. He does not understand people who were lucky enough to be born here and yet complain about their life instead of working to improve it. If you ask Ramy about this, he will passionately tell you that in America, if you want to better yourself, you can, but only with hard work and determination.\n\nBy the way, Ramy and Leila were married on October 14, 2013, on the beach in Maui, Hawaii.\n\n_What's Next_ : Ramy plans to continue to build his brand, and he has a number of plans for how to expand it, including opening up a brick-and-mortar restaurant someday.\n\n_When You Go_ : Get a lamb kabob and a chicken gyro\u2014both are wonderful. For dessert, get some of Ramy's signature baklava. My grandmother first introduced me to baklava when I was small child, and I have loved it ever since. I have enjoyed it all over the world, including places like Thessaloniki, Greece, and Los Angeles, California, and Ramy's baklava is the best I have ever had.\n\n_Recipes_ : You can find the Ramy's Lamb Shack recipe for Beef and Lamb Kofta in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 1.\n\n**M ORE FULFILLED THAN I HAVE EVER BEEN: BENTO BOX**\n\nJeff and Madeline Mendon opened the Bento Box Food Cart in the dead of winter in January 2013. In the course of one short year, Bento Box has developed numerous fans and attracted a plethora of people who consider themselves part of the Bento Box community. In other words, they have succeeded as a startup, they have established a great brand and they are the proud owners of a successful, thriving food cart in downtown Portland. How have they succeeded where so many others have failed?\n\nOne of the most interesting aspects of their food cart success story is that unlike many food cart owners, they had _no_ previous professional culinary experience. Jeff worked in the medical device field, and Madeline is an elementary school teacher.\n\n_Choosing Portland_\n\nJeff and Madeline both grew up in California. They met each other at Chico State University, which is located ninety miles north of Sacramento. They finished college, married and moved to Redwood City, California. Right out of college, Jeff talked about opening a restaurant. He really wanted to do this. And then, as he put it, \"life happened,\" and that dream got put on the shelf.\n\nJeff and Madeline Mendon, owners of the Bento Box Food Cart.\n\nThey lived in Redwood City for four years, and it never felt like home. They did lots of research on where they might want to live and raise their family. They picked the Pacific Northwest, and it came down to Seattle and Portland. They visited Seattle and really liked it. Then they visited Portland and liked this city even more. In 1990, they both quit their jobs and moved to Portland. They had come \"home.\"\n\n_The Portland Food Scene_\n\nBoth Madeline and Jeff are foodies. They watched the Portland food scene develop and eventually flourish over the next twenty years. They ate at and loved most of the restaurants that people around the country read about. They were true fans of the local scene.\n\nAlong the way, as each of those twenty years rolled by, Jeff would wistfully remember his long-ago dream to open a restaurant of his own. Jeff and Madeline would sit in those amazing Portland restaurants and talk with each other and dream about how much fun it would be to not just be fans of the Portland food scene but to actually be a part of it.\n\nIn 2003, Jeff began to think about opening a food cart of his own. He did not have the financial resources necessary to open a restaurant. Maybe, just maybe, a food cart was a viable way to finally stretch his culinary wings. Things started to come together for Jeff and Madeline's culinary dreams in 2010. By then, Jeff had decided on the name, concept and menu for his food cart.\n\n_Bento_\n\n_Bento_ is the Japanese name for a takeout or home-packed meal for one. A traditional bento has rice, meat (or fish) and vegetables. Jeff became convinced that a food cart that served simple, healthy food with a limited menu would work here in Portland. Jeff loved bento, and he loved grilled chicken, so he decided that his food cart would serve what he loved.\n\nKeeping with the \"simple\" theme, Jeff and Madeline decided to call their food cart Bento Box. Their cart now had a name, a concept and a menu. Just a few more pieces of the puzzle were needed.\n\n_More than Six Years in the Making_\n\nThe cart-made teriyaki sauce served at Bento Box is not a glaze; it is a light sauce that Jeff invested many years perfecting. This teriyaki sauce was a hit right from the get-go. As soon as Bento Box opened, Madeline encouraged Jeff to hand out samples of his teriyaki sauce. Jeff revels in the weekly expletives he receives. Someone will sample his teriyaki sauce for the first time and exclaim, \"F@#%!...I need to get me some of that.\"\n\nWith the teriyaki sauce recipe finalized, all they needed was a cart and a location for that cart.\n\n_Finding a Food Cart_\n\nIn December 2012, Jeff saw a food cart for lease on Craigslist on Alder, just past 10th. They had been looking for a food cart near 10th and Alder. Jeff spent a few hours considering this location before he picked up the phone to try to lease this cart. By the time Jeff called, the cart had already been taken.\n\nThe very next week, Jeff saw another ad on Craigslist for a food cart for lease. This time, the cart was right on 10th between Alder and Washington. Jeff immediately picked up the phone. He was not going to let this cart slip through his fingers.\n\nFricken Good Chicken at the Bento Box Food Cart.\n\nMadeline remembered driving with Jeff to their prospective cart to meet the person they would be leasing from. Would this be it? After ten years of planning, would they finally have a food cart business to call their own? After more than twenty years of dreaming, would they actually become part of the Portland food scene?\n\nYes, yes and yes. They signed a lease for that food cart, and they were ready to open their window and offer their version of bento to the world.\n\n_Paying the Price_\n\nIn the process of getting Bento Box up and running, Jeff had to quit his corporate job. Coworkers told him that he was \"completely out of his mind.\" In his first summer, Jeff worked sixteen to eighteen hours per day, and he was open seven days a week. In its first spring and fall, Bento Box was open six days a week, and Jeff put in twelve-hour days.\n\nI asked Jeff what it feels like to work that hard. He said, \"At the end of the day, my whole body hurts, especially my wrists, my knees and my back. I come home bone tired. And...I wake up excited to get out of bed and do it again. I am more fulfilled than I have ever been.\"\n\n_What's Next_ : Jeff and Madeline would love to see their food cart brand grow to the place where a brick-and-mortar Bento Box is feasible. I have strongly encouraged them to look into bottling Jeff's teriyaki sauce. I know that I would regularly buy it if I could.\n\n_When You Go_ : Get a Chicken Bento Box. They really do have \"Fricken Good Chicken.\" The grilled chicken is incredible. Make sure that you get yours with extra teriyaki sauce. Trust me on this this: Jeff's teriyaki sauce will be some of the best teriyaki sauce you have ever tasted.\n\nFood Writer's Perspective:\n\nAARON WAKAMATSU\n\nAloha! I'm Aaron Wakamatsu, the man behind Aaron's Food Adventures, a blog that I started back in May 2011. Originally, I intended my food blog to be a simple private journal to document the places I went to for food. Over time, my blog gained a following that includes many Portland food cart owners.\n\nI launched a YouTube channel in August 2012 to expand the scope of my audience. Today, I have followers worldwide. I usually do two to three blog posts and two to three videos per week. I am very active on social media, so be sure to follow my adventures. I just might show up in your town!\n\nI grew up in Hawaii on the island of Oahu. Not many people might know that Hawaii also has a long history in the food cart scene. Many of us growing up on the islands have known them as \"lunch wagons,\" as these owners usually served the general public during lunch hour.\n\nAlthough I have lived in the Pacific Northwest since 2007, I did not realize that Portland had a bustling food cart scene until 2010. I was going to school in Salem, Oregon, at the time, and one of my classmates said that I had to check out the Portland food carts one day.\n\nPeople have told me that it is their dream to go to Hawaii for the food and the culture. Interestingly, those are the reasons I love staying in and around Portland. Like Hawaii, Portland has become a melting pot of delicious dishes from around the world. Where else in the United States can you find food carts offering dishes from Guam, Poland, Ethiopia, Mauritius, Russia and the country of Georgia within the city limits?\n\nWith more than five hundred food carts open for business in Portland at any one time, I've been asked what sets one food cart apart from the others. First, I look at the personality of the food cart owner or the employee running the cart. A simple \"Hi,\" \"Thank you\" or a smile goes a long way. I believe that the person should be full of energy and doing anything possible to make the customer feel welcome. To me, Todd Edwards at Ol\u00e9 Latte Coffee is a prime example of a charismatic individual. I have visited his cart multiple times just to get a coffee and chat with the man for a few minutes. Frankly, we need more food cart owners like him.\n\nSecond, a great Portland cart has delicious, addicting food. Just about everyone I know has a favorite dish or finds something on the menu that sounds incredible. Think of something you ate one day that you still dream about many months or years later.\n\nFinally, I believe that a great food cart has that \"it factor\" or some other intangible that innately makes you keep coming back. It is a purely subjective criterion unique to each individual.\n\nHere are some of my top Portland food cart dishes, those menu items that have stood out and left an indelible mark on my palate:\n\nOriginal Breakfast Tacos from the Pepper Box: Jim Wilson brought the flavors of New Mexico to Portland, including adding New Mexico green chile and red chile to various menu items. Breakfast lovers will enjoy the abundance of cage-free eggs, potatoes and cheddar cheese. The best part: Jim makes his own flour tortillas from scratch.\n\nThe Dirty Mo's Super Q Pulled Pork Sandwich from J-Mo's Sandwich Shack: While \"J-Mo\" and others flock first to the \"Dirty Mo\" sandwich (think spaghetti and meatballs inside garlic cheese bread), I got hooked on J-Mo's massive pulled pork sandwich that should satisfy the hungriest of appetites. Heaps of tender, juicy, slow-roasted pulled pork inside a roll, topped with sweet-tangy barbecue sauce. I highly recommend adding the homemade coleslaw for added crunch.\n\nChicharron Sandwich from La Sangucheria: Maribel, owner of this Peruvian food cart, recently won the Judges' Choice Award at Eat Mobile 2013. The honor is even more significant when competing against at least fifty other carts for the coveted crown. Here, \"chicharron\" is not simply fried pork skin but rather skin-on slices of seasoned pork loin hitting the deep fryer. The exterior crisps up slightly to leave me wanting more. The refreshing sarza criolla, a South American salsa featuring sliced red onions, cilantro and habaneros, tickled my taste buds and blew my mind.\n\nKelaguen Mannok from PDX Six Seven One: Ed and Marie Sablan provided my first introduction to Guamanian food. Their version of Kelaguen Mannok is more than a cold chicken salad; it simply makes me feel like I'm enjoying a fabulous meal in Guam. While the minced bird's eye chili tickles my taste buds, the refreshing lemon in the dish provides the extra bite and makes this dish pop. Enjoy Kelaguen Mannok with their signature hineksa' agaga' (red rice) or titiyas (flatbread), plus some fina'dene' (a Chamorro sauce) on the side. Not surprisingly, this food cart was featured on Guy Fieri's _Diners, Drive-Ins and Dives_ in September 2013. I couldn't be happier for Ed and Marie. _Si yu'os ma'ase_ (thank you) for bringing Guamanian cuisine to Portland!\n\nPeople should experience the world of Portland food carts at least once in their lifetimes. Even after three years, my food adventures are still just beginning. Perhaps I will see you at a Portland food cart in the near future.\n**PART VII**\n\n**COMFORT FOOD**\n\n**C HEESE MAKES ME HAPPY: HERB'S MAC & CHEESE**\n\nHerb Acken opened Herb's Mac and Cheese Food Cart in January 2011 after having moved from Florida to Portland in the fall of 2010. Herb had originally intended to sell his mac and cheese at the Portland Saturday Market the first part of the 2011 and then open his food cart in the spring.\n\nWhen Herb got to Portland, he discovered that the Portland Saturday Market was closed the first two months of the year for its winter break, and he realized that he couldn't go two months without income, so he decided to get his cart open right away.\n\nHerb was able to get his cart configured for making his intended menu, and he found an open spot at the now defunct D-Street Noshery Food Cart Pod at 32nd and Division, and he was off and running.\n\nHerb noted, \"I like the rain. I was a lifeguard on the beach for fifteen years. I am good with less UV.\" Prior to his big move, Herb had lived in Fort Myers, Florida. While there, he owned a few businesses, including the EB Caf\u00e9, an outside-seating-only restaurant called Burrito a Go Go and a shaved ice cart. He spent his last two years in Florida selling shaved ice right on the beach.\n\nHerb's Mac & Cheese Food Cart. _Photo by Molly Woodstock_.\n\nWhen Herb left Florida, he brought his shaved ice cart with him. He changed the graphics and the equipment and added some diner blocks across the top. What was once a cart on a beach was now a Portland food cart.\n\nFor Herb, it was time for a change, and Portland had everything that he wanted, including \"rainy weather and beautiful scenery.\" For Herb, Portland was a place that \"encourages those with an entrepreneurial spirit.\"\n\n_Why Such a Creative Name?_\n\nHerb chose to call his food cart Herb's Mac & Cheese. He had learned the hard way that the name of your business should clearly communicate to people what it is, as well as what you sell. Back in Florida, he had many people think that Burrito a Go Go only sold burritos to go. He also got very tired of people jokingly asking, \"Do your burritos make you go?\"\n\n_The First Year_\n\nI asked Herb how many hours per week he worked his first year. He answered, \"All of them.\" That first year, Herb worked mostly seven days a week and put in seventy to eighty hours per week\u2014some weeks even more than that.\n\n_Vegan Mac and Cheese with Bacon_\n\nHerb sells a tofu-based vegan \"cheese\" sauce, and he sells quite a bit of it. He still remembers the first time someone ordered bacon with the vegan mac and cheese. Herb was very confused and could not help but give his customer a double take. She told him, \"I am not a vegan. I am lactose-intolerant.\" It is common for people to order his vegan mac and cheese with one of the meat toppings that Herb offers.\n\n_Build Your Own!_\n\nAt Herb's Mac & Cheese, you can add a number of awesome toppings to your dish: baked breadcrumbs, chicken, hot dogs, garlic, broccoli, sun-dried tomatoes, feta crumbles and blue cheese crumbles. I have two favorite combinations. The first is chicken, Frank's Red Hot Cayenne Pepper Sauce and blue cheese crumbles. The second is mushroom, feta cheese and bacon. The beauty of Herb's menu is that you can build whatever you want.\n\n_Why Mac and Cheese?_\n\nHerb had made his signature mac and cheese for family for years, and it was always greeted with rave reviews. When it came time to choose a cuisine for his food cart, mac and cheese won for a number of reasons. He believed that mac and cheese fit Portland and was convinced that Portland would support such a food cart. To Herb, mac and cheese is quintessential comfort food, and he loved the idea of selling comfort food. Most importantly, Herb said, \"Cheese just makes me happy.\" The idea of having a business where he could pass on that happiness sounded perfect to him.\n\nHerb Acken, owner of Herb's Mac & Cheese Food Cart. _Photo by Molly Woodstock_.\n\n_What's Next_ : Herb gets many requests to package and sell his vegan cheese sauce, and he is looking into doing that. He would also love to someday open up a brick-and-mortar Herb's Mac & Cheese. He would serve his mac and cheese in small cast-iron skillets and add breadsticks, cheesy breadsticks, beer and wine to his list of offerings.\n\n_When You Go_ : Pick the toppings that resonate with you and enjoy some of the best mac and cheese you will ever have.\n\n**M ORE THAN JUST A BAKER: THE SUGAR SHOP**\n\nAutumn Burns and Shannon Scully opened the Sugar Shop Food Cart in August 2012, and with this step, they were finally professional bakers. They were finally \"making cookies for a living.\" So many people want to do something different with their life and don't because of fear, doubt or peer pressure\/family pressure to take the \"traditional\" route. I am very fond of these two courageous women, and I am honored to share their story.\n\n_What Does Your Heart Say_\n\nAutumn attended Louisiana State University and graduated with a degree in kinesiology. The goal was medical school. Autumn took some time off from that path once graduation was over. One thing led to another, and she ended up back in her home state of New Mexico dating a chef who had been to culinary school.\n\nAutumn came to realize that if she could, she would become a baker. The question was not whether she could become a baker. The question was whether she would allow herself to. Up to this point, she had been pursuing one of those careers that parents brag about\u2014doctor, lawyer and so on.\n\nWho wants to say, \"I was going to be doctor, but now I roll dough out for a living?\" I will tell you who\u2014people who love to roll dough. Sometimes, when faced with this tough kind of crossroad, people say no to the dream that smolders in their heart and pursue the career that they think they should.\n\nThe sad news is that you can't extinguish those kinds of dreams. They just keep burning and are always there. Unpursued dreams keep burning in your heart, and eventually they become a dull, ongoing ache.\n\n_Not Doctor or a Lawyer\u2014Just a Baker_\n\nAutumn wrestled with the fear of being \"just a baker\"\u2014baker is not one of those cool careers. Thankfully, when she talked to her mom, her mom got it and affirmed the idea that Autumn should pursue her dream, even if it wasn't as cool as being a doctor. Autumn and that chef she was in love with headed off to Portland so Autumn could attend the Oregon Culinary Institute and become a baker. This chef who believed in Autumn's dream is named Jason. He's a great chef and an even better guy. I hope that my daughters end up with someone as awesome as he is.\n\n\"I have wanted to be a baker as long as I can remember,\" said Shannon Scully. Before moving to Portland in 2011, Shannon had a tough journey. One constant was her love of baking. In her late teens, Shannon got a job at a bakery working in the front as a barista. She was finally able to convince the owner to let her go to the back to bake. Shannon loved it, even though the person who was the head baker spoke no English\u2014Shannon's first lessons in professional baking came via hand signals.\n\nLater, Shannon entered culinary school in San Francisco and made it three-fourths of the way through. Things did not work out, and one day Shannon left and never went back. At this point, most people would have given up on their dreams. Shannon kept battling for her dream, and a few years later, she moved to Portland to attend her second culinary school: the Oregon Culinary Institute.\n\n_The Very First Week_\n\nAutumn and Shannon met at OCI their first week. Both of them had been in Oregon less than a week, and they both were there to become bakers. They became fast friends and discovered that they had much in common. They both love the rain. They both love Portland \"for the food, the vegetables, the neighborhoods and the wine.\" Both of them remembered baking for the first time as a small child, making chocolate chip cookies with mom. They both have very quirky senses of humor. Most importantly, both of them love to bake. As Autumn said, \"You know that you are a baker if, when you are really stressed out, you bake.\" Shannon added, \"There is something very therapeutic about rolling dough.\"\n\n_Cinco de Mayo 2012_\n\nJason was working, and Autumn and Shannon found themselves together on Thursday night, May 5, 2012, drinking margaritas. It was during that evening that they decided to open a food cart together. They knew that they could work together because they \"are both really not nice people.\"\n\nActually, they _are_ both very nice people. They just prefer to be blunt and direct. To me, that is part of their charm. I also like it that when I am around them, I can be more open and not filter myself so much. They both revel in the fact that they have a business partner with whom they can have the freedom to be very honest. They have a great working relationship.\n\nThe Sugar B's: Autumn Burns and Shannon Scully, owners of the Sugar Shop Food Cart.\n\n_The Boozy Bakers_\n\nBoth Shannon and Autumn love to bake with booze. They love the challenge, the creativity and the science. While all of their desserts are truly amazing, the ones made with booze are on a whole other level. To get an idea of what these incredible bakers can do, here are some the desserts that I have enjoyed from their food cart: chocolate chip bacon cookies, cheddar biscuits filled with cart-made bacon jam, strawberry cake topped with basil buttercream and balsamic drizzle, pumpkin bourbon mousse on a nutmeg shortbread, lemon cake with cranberry Riesling compote and chocolate cake with a goat cheese frosting and port drizzle.\n\n_The Sugar B's Can Flat-Out Bake_\n\nEarly on in their food cart journey, Autumn and Shannon were given the nickname \"the Sugar B's.\" You will have to ask them about this nickname. While you are at it, ask them about the eye patch of shame, fruit box shoes, yogurt yeti and dormant mayonnaise. Remember, they spend many hours a day in a little metal box baking. Sometimes I think that they go a little mad. But who wants a completely sane baker anyway?\n\nI am so glad that each of them had the guts to battle for her dream to become a baker. To me, Autumn and Shannon are not \"just bakers.\" They are the kind of inspiration that the world needs to know about.\n\n_What's Next_ : Autumn and Shannon are working on a cookbook, and someday they would like to have a brick-and-mortar business that has the Sugar Shop bakery on one side and the Sugar B's Bar on the other. The bakery would be open during the day and have cakes, pies, scones, cookies and so on. The Sugar B's Bar would be open nights and have cocktails and the boozy bakers' signature desserts for grown-ups.\n\n_When You Go_ : The menu is seasonal. Everything is mind-blowingly good. Just order four or five items and worry about your diet tomorrow.\n\n_Recipes_ : You can find three of the Sugar Shop recipes (Ginger Carrot Cake with Orange Cream Cheese Frosting; Lemon Cake; and Molasses Spice Cookies) in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**T ENACITY AND ARTISTRY: GAUFRE GOURMET**\n\nCharlene and Michael opened the Gaufre Gourmet Food Cart on November 22, 2010, a day they consider to be the worst possible day to open your food cart. In 2010, November 22 was the Monday before Thanksgiving. Their opening-day sales totaled three dollars, but they have come a long way since then.\n\n_While Watching the Travel Channel_\n\nMichael and Charlene met in 2008. They were both working for a catering company. They became a couple shortly after that first meeting. By 2009, the catering company they worked for had become severely affected by the economic crash that affected so many of us. They knew that eventually both of them would be laid off.\n\nMichael and Charlene decided that maybe opening a food cart together would be a viable economic path for them to take. The next decision they needed to make was deciding what cuisine they would sell at this food cart.\n\nMichael Susak and Charlene Wesler, owners of the Gaufre Gourmet Food Cart.\n\nOn Wednesday, May 13, 2009, Charlene and Michael were watching the Travel Channel, and they learned about Li\u00e8ge-style waffles. Nothing was ever the same after that, but more on Li\u00e8ge-style waffles in a bit. A few days later, they decided that they were going to open a food cart selling these specialty waffles. Michael and Charlene began researching how to make Li\u00e8ge waffles. They got a waffle iron and spent the next year perfecting their recipe. In May 2010, they purchased and began to renovate their food cart.\n\n_That Tree Is a Deal-Breaker_\n\nMichael and Charlene began looking for a place to put their food cart. They really wanted to be downtown at the 9th and Alder food cart pod. A spot there opened, and they went to look at it. Unfortunately, there was a tree that would be right in front of their cart, and Charlene vetoed this otherwise perfect spot.\n\n_The Tenacity to Press On_\n\nThey ended up at 4th and Burnside. That pod got closed down in June 2011, and they had to move, this time to the Green Castle Food Cart Pod just off 19th and Burnside. That pod was not a good match for their cuisine. Sales were awful, and they almost went out of business. Then that pod closed down, and they had to move again. \"There were so many times that first year that we could have quit and many more times that we wondered if maybe we should,\" said Charlene.\n\nThey were now a year into their food cart business. They hadn't really made any money, and they needed to move their cart for the third time. I asked Michael why they did not simply throw in the towel and find something else to do. Michael said that they believed in their product and knew that if they persisted, they would reach a tipping point where they had enough fans to sustain their business.\n\n_The Tree Is Back_\n\nThey heard that a spot for a food cart had opened up on 9th and Alder\u2014the pod they had originally wanted to be at. They were very hopeful as they quickly hurried to see if they could secure it. As they approached the open spot, Charlene realized that this was the exact same spot she had vetoed more than a year ago. The tree was still there. You can see that tree when you visit Gaufre Gourmet because it is right in front of their cart. Charlene was not going to let a tree stand in the way of their dream.\n\n_Culinary Artistry_\n\nMichael loved the idea of serving Li\u00e8ge-style waffles because it takes real culinary skill to pull them off. The artistry required appealed to him. Li\u00e8ge waffles are not made from batter. They are made from a yeasted dough that you have to first let rise for two hours or so. Next, you punch the dough down and let it rise for a second time. Finally, you have to portion it off, and then you are ready for the waffle iron\u2014assuming, of course, that you have the right one.\n\nTo make authentic Li\u00e8ge-style waffles, you need a special cast-iron waffle iron that weighs one hundred pounds and takes from thirty to forty-five minutes to heat up. Lastly, you need to use a decadent substance known as Pearl Sugar. Pearl Sugar is a coarse, hard sugar that caramelizes when it encounters the heat of the waffle iron. Unless your grandma is Belgian, these waffles are not your grandma's waffles. These are way better.\n\n_The Smell Will Lure You In_\n\nThere is nothing quite like the smell of these waffles as they are being cooked. The appealing aroma will grab ahold of you and not let go! Don't resist. Just follow your nose and be prepared to become enraptured in culinary ecstasy. These waffles are chewy and tender at the same time. And they are finished with a sprinkling of vanilla powered sugar.\n\n_Gow-Free?_\n\n_Gaufre_ is a French word that means \"waffles.\" I have heard many Portlanders pronounce this word \"gow-free.\" I have watched a director with the Food Network who had a French background painfully cringe when he heard this pronunciation. I have been told that you can pronounce this word with a few acceptable variations: \"go-ff,\" \"go-f-fray\" or \"go-f-fra.\" The nice thing is that Charlene and Mike don't care how you pronounce it. They just want you to enjoy their amazing waffles!\n\n_No One Will Want Goat Cheese on a Waffle_\n\nGaufre Gourmet has a very innovative menu. You can get a classic waffle (my favorite), and you can get a waffle drizzled with Belgian chocolate sauce and topped with whipped cream. You can also get the Milk and Honey Waffle\u2014topped with goat cheese mousse, honey-roasted pistachios and balsamic caramel sauce.\n\nWhen Carlene first suggested adding this waffle to the menu, Michael was sure that it was a bad idea. He was convinced that no one would want goat cheese on a waffle. He also opposed the Monte Cristo and the Spicy Goat. All three are top sellers, and Charlene came up with all three of those winners.\n\nThe Monte Cristo has turkey, ham, Swiss cheese and strawberry preserves, all on a waffle. I knew that I would love the Monte Cristo even before I tasted it. Disneyland was one of the few bright spots in my childhood, and I loved getting to have lunch at the Blue Bayou Restaurant that is inside the Pirates of the Caribbean attraction. I had a Monte Cristo sandwich for lunch at the Blue Bayou many times. I was delighted to discover that Charlene had been inspired by her own visits to the Blue Bayou! Her version does not disappoint.\n\nThe Spicy Goat is a waffle topped with arugula, ch\u00e8vre cheese, pistachios, salami and Kelly's habanero jelly. This waffle is spicy and sweet and perfect. It is so good that it helped Gaufre Gourmet beat out more than fifteen other food carts for the grand prize at the 2013 Summer Portland Food Cart Festival. When you visit Gaufre Gourmet, make sure to ask to see the trophy.\n\n_Courage and Creativity_\n\nMichael and Charlene had the courage to keep pursuing their food cart dream even when the outlook sometimes looked very bleak that first year. They also had the creativity to combine things like goat cheese and balsamic caramel sauce and put them on a waffle. I applaud their tenacity and artistry.\n\n_What's Next_ : Michael and Charlene have a number of big dreams, including a second food cart for events and catering, opening a brick-and-mortar location and packaging some of their items.\n\n_When You Go_ : No matter what else you get from Gaufre Gourmet, buy yourself the Original. It is simple elegance. The Original is one of my all-time favorite dishes. I also love the Milk and Honey Waffle, the Monte Cristo and the Maple Bacon Waffle.\n\n_Recipes_ : You can find the Gaufre Gourmet recipe for ABC Waffle\u2014Arugula, Bacon and Camembert with Fig Sauce\u2014in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 1. Three more of Gaufre Gourmet's recipes (Ch\u00e8vre Mousse; Balsamic Caramel Sauce; and Buttermilk Biscuit Waffle Dough) can be found in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n\n**R UGBY, NOT SOCCER: TIMBER'S DOGHOUSE PDX**\n\nTimber Adamson has been a food cart owner since April 1, 2010. She opened Timber's Doghouse PDX in its current configuration on April 1, 2012. Her journey has taken her from Springdale, Arkansas, to Portland, Oregon, with important stops in Fayetteville, Chicago and Korea. Each of those stops along the way has led her to where she is now.\n\nTimber graduated from Western Culinary School here in Portland in 2006. She went to culinary school so she could become self-employed and work in molecular gastronomy. In case you were wondering, her food cart does not sell hot dogs. The Doghouse PDX (aka Timber's Doghouse PDX) sells some of the best hamburgers you will ever have. Trust me, it all makes sense.\n\n_Ice Does Cut: Becoming Timber_\n\nTimber grew up in Springdale, Arkansas, and while growing up, she enjoyed a life-changing hamburger, but more on that important hamburger in a minute. Timber had a different first name until college. In college, she played rugby for the University of Arkansas. In that sport, when you make your first try (a score), you are expected to \"Zulu\"\u2014to strip down naked and run around the field. When Timber scored her first try, she was too shy to strip down right on the pitch (field) in front of the fans and the other team.\n\nTimber's team told her that she could Zulu at a later date. The date was picked, and the time was set for the middle of the night. As fate would have it, there was an ice storm preceding the selected date, and the pitch was a sheet of ice. Timber dutifully stripped down to just her cleats and proceeded to run across the pitch\u2014she immediately fell and slid across the ice. Timber shivered a little when she told me this story, and she ended it by saying, \"Ice does cut.\" I left that comment alone.\n\nAt the very next practice, while doing a common rugby drill, Timber tripped herself up and fell down once again. At the end of practice, her whole team lined up and called out to her, and all of them proceeded to fall over, each one yelling, \"Timberrr!\" From that day on, Ms. Adamson had a brand-new first name.\n\nTo this day, Timber loves rugby, and she played here in Portland in 2007 and 2008, on an Oregon Sports Union (ORSU) rugby team.\n\n_The Rain, the Produce and Safety_\n\nTimber eventually left Arkansas for Portland to attend culinary school. When she was considering culinary school, she had three options: Scottsdale, Arizona; Atlanta, Georgia; and Portland. Portland won for two reasons: Timber was done with being in the hot sun, and Portland had the best produce. When she got here, she discovered a wonderful third benefit to Portland. Here in Portland, Timber felt safe for the first time in her adult life.\n\nTimber found that Portland was a place where people accepted you for who you were and where her sexual orientation was not an issue. In Arkansas, Timber had not ever felt safe. As we sat at a picnic table next to her food cart, Timber told me some of the experiences she had had in Arkansas, and I cried. No one should have to go through the things that she did.\n\nTimber Adamson, owner of the Timber's Dog House PDX Food Cart.\n\n_Chicago and Korea_\n\nFollowing culinary school, Timber had an externship at Alinea, a Chicago restaurant started by Grant Achatz, the leading American chef in molecular gastronomy. Upon returning to Portland, she followed a girlfriend to Korea, where she spent almost two years. In Korea, Timber fell in love with Korean food, and she spent day and night applying both her previous ten years cooking professionally and her culinary school education to learn how to make this amazing food.\n\n_Food Carts, Plural_\n\nTimber returned to Portland on December 31, 2009, and on April 1, 2010, she opened the Yogio Food Cart specializing in authentic Korean cuisine. Yogio received much press and acclaim when it opened; it was even featured on TV. In the fall, Timber got the opportunity to buy another food cart that sold hot dogs at the pod where she was located. She bought that cart, changed the name to the Doghouse and kept going. Timber lovingly refers to that cart as the Red Dragon, unless the transmission goes out\u2014then it is the Red Beast. For a while, Timber ran both food carts.\n\nThe Red Dragon, home of Timber's Dog House PDX Food Cart.\n\nTimber closed down Yogio on Halloween 2011 because over the time that it was open, she received continuous and ongoing grief for being a \"white girl\" who was cooking authentic Korean food. Eventually, Timber decided that it was not worth it. Timber actually closed down both carts that day, and she planned to sell one and reopen the other in the spring of 2012. The Yogio cart was sold, and it later became the first Fried Egg I'm in Love food cart.\n\n_A Ten-Year Burger Quest_\n\nWhile growing up in Springdale, Arkansas, Timber loved the burgers they had at the local hamburger joint. That place went out of business while she was in high school, and Timber discovered that no other place in town had patties as good as she was used to. Following college, Timber started cooking professionally, and she began trying to come up with recipes for a seasoning and marinade that matched the burger patties of her childhood.\n\nOver the winter of 2011, Timber finally nailed those recipes, and her ten-year quest was now complete. She was ready to make burgers. When she opened her cart back up on April 1, 2012, it was renamed Timber's Doghouse PDX, and while it still sold incredibly creative tot bowls, the hot dogs were replaced with equally creative hamburgers. Timber's hamburgers are everything that you want in a burger\u2014juicy, tender and perfectly seasoned. I could eat the patty all by itself.\n\nYou can get the classics at Timber's, including a cheeseburger and a bacon burger, but you can also get burgers with items like peanut butter, pineapple or roasted garlic truffle aioli. Timber's Doghouse PDX has nothing to do with soccer or hot dogs, but it does have great hamburgers.\n\n_What's Next_ : Timber always has something creative and delicious up her sleeve. I am a big fan of hers, and I can't wait to see what is next in her journey.\n\n_When You Go_ : Get whichever burger strikes your fancy and make sure to get a tot bowl. Timber's is one of my daughter Zoe's favorite food carts, and Zoe loves the Unexpected Perfection Burger\u2014candied bacon, onions, blue cheese and house three-way mustard. Zoe's favorite tot bowl is the Truffle Treasure Tot Bowl\u2014classic tots topped with chanterelle salt, Asiago cheese and roasted garlic truffle aioli.\n\n**L AISSEZ LES BON TEMPS ROULER: A CAJUN LIFE**\n\n\"Have I made a mistake?\" That was the question that Chris Fontenot asked himself at the end of the day on July 6, 2012, the first day that his new food cart was open at its Damascus, Oregon location. He had made a _lot_ of food, and he had no customers. Not one. Not a singe customer.\n\nChris's incredible wife, Hillary, told him, \"That's ridiculous! They will come.\" The next day, they did show up, but only three of them. So far, the A Cajun Life Food Cart had been open at its Damascus location for two days, and it was averaging 1.5 customers per day. Let the good times roll!\n\nChris and Hillary Fontenot, owners of the A Cajun Life Food Cart.\n\nFrom those inauspicious beginnings, a great food cart has come into its own. It turned out that Hillary was right. By the fall of 2012, the customers had found out about this food cart on the outskirts of the Portland metro, and they now come in droves from far and wide all the way out to Damascus to get food that is more than worth the drive.\n\nI have worked with scores of small business owners\u2014chefs with big dreams who have started food carts with the hope that they will be able bootstrap their way to a better life. Every week, I tell at least one of these determined entrepreneurs that they need to follow Chris's enterprising example. In my opinion, every food cart owner needs to do what Chris has done, and I am excited to tell you his story.\n\n_Eunice, Louisiana: The Heart of Cajun Country_\n\nThe A Cajun Life Food Cart specializes in Cajun cuisine. It only has four menu items: Cajun Gumbo, Crawfish \u00c9touff\u00e9e, Bayou Boogie and Crawfish Pistolettes. Chris is serving the food he grew up eating using family recipes. Chris is from Eunice, Louisiana, the heart of Cajun country.\n\nHis gumbo is one of my favorite dishes of all time. I love to bring the white round container that Chris serves his gumbo in up to my lips and take a sip of that deep, rich broth. His gumbo is ethereal. The broth is perfectly seasoned.\n\nEvery day that it is open, the food cart has people who used to live in Louisiana and now live in the Northwest make the trek out to Damascus to get themselves some truly authentic Cajun cuisine. Chris has regulars who come from as far away as Salem, Washougal, Beaverton and even Eugene\u2014a two-hour drive.\n\n_Authentic Cajun Cuisine_\n\nSo what makes the food at A Cajun Life Food Cart truly authentic? A number of things contribute to the authenticity\u2014the recipes, the seasonings and definitely the ingredients. Chris has his sausage and tasso flown in from Eunice, Louisiana. Chris has a Delta cargo account, and he regularly drives out to Portland International Airport (PDX) to pick up his shipment of Cajun delicacies.\n\nA Cajun Life All-Purpose Cajun Seasoning.\n\n_Extending the Brand_\n\nOne of things that sets A Cajun Life apart from so many other food carts is that in his mind, Chris did not start a food cart\u2014he started a brand, and the food cart was the first step in that brand. In January 2013, Chris asked himself a question that, in my opinion, every food cart owner should ask: \"How do we take the next step in developing our brand?\"\n\nFor Chris and Hillary, the answer was to package and sell A Cajun Life All Purpose Cajun Seasoning. They did a Kickstarter campaign in April to help fund this step, and by July 2013, Chris had 1,100 units of seasoning in his garage ready to sell. There are a number of stores now carrying this awesome seasoning, and that number keeps growing. I encourage every food cart owner to follow in Chris's footsteps. Develop not just a food cart but a brand, and sooner, rather than later, ask yourself how to best extend that brand.\n\nChris did _not_ make a mistake when he started A Cajun Life; in fact, I predict that it will turn out to be one of his best decisions. My food cart hat is off to you, Chris, and as they say in Louisiana, \"Laissez les bon temps rouler!\"\n\n_What's Next_ : Chris has quite a bit planned for A Cajun Life. He is going to add two more flavors to his seasoning line: a Cajun blackening rub and a Cajun chicken rub. He also hopes to add a branded sausage and a branded tasso to his packaged food line. The space that A Cajun Life occupies in Damascus is going to become a four-cart pod, and he is planning to open a second A Cajun Life Food Cart in Portland.\n\n_When You Go_ : Get some gumbo and buy a bottle or two of A Cajun Life seasoning.\n\n**S TAY CAFFEINATED, MY FRIENDS: OL\u00c9 LATTE COFFEE**\n\nTodd Edwards opened the Ol\u00e9 Latte Coffee Food Cart in June 2012. The crazy thing is that it was March 2012 when he decided that he was going to open his food cart. He owned a cart by April and began retrofitting, and he was open by June.\n\n_From Wine Shop to Coffee Cart_\n\nTodd is passionate about wine, and he knows his way around a wine list. Todd is the person you want to have with you at the restaurant when it comes time to pick out a wine to go with your meal. He can look at a wine list and pick a wine that will pair perfectly with your dish. I have seen him use this skill with my own eyes.\n\nFor many years, Todd had wanted to open a wine shop. By March 2012, Todd realized that just wasn't going to be in the cards for him to do this. Todd Loves wine, and he also loves coffee. At that time, he did not know coffee like he knew wine. Todd began to educate himself, and he has become quite the coffee aficionado. Today, his food cart is regularly praised in the press as having some of the best coffee in Portland.\n\n_Time to Name the Cart_\n\nTodd began looking for a name for his coffee cart, and every time he came up with a name he liked, he would see if the domain was available and learn that it was already taken. This happened to Todd repeatedly. Meanwhile, Todd had a running catch phrase that he and his girlfriend, Monica, used. Every time something was great or amazing, as in \"that was an amazing breakfast,\" they would say, \"Ole!\" One day, it just clicked. He wanted to serve amazing coffee, and the name Ol\u00e9 Latte Coffee was spot on.\n\n_The Coolest-Looking Espresso Machine You Will Ever See_\n\nAs Todd began to educate himself on coffee, he learned that some of the best espresso machines come from La Marzocco, a company that has been making espresso machines since 1927. Todd was able to get his hands on a beautiful red La Marzocco. Ask Todd about his machine when you go to his food cart, and he will passionately tell you why it makes such excellent lattes and espressos.\n\n_The First Year_\n\nTodd opened his food cart at a neighborhood food cart pod and hoped for the best. The first year, Todd worked seven days week twelve to sixteen hours a day, and halfway through that first year, they almost closed. Toward the end of 2012, Todd was offered the chance to move to the 10th and Alder food cart pod, and he thought that maybe this location would be the place where they could flourish.\n\nTodd then remembered that his cart serves from the side, and that meant he would need to rent two spaces if he moved to that pod. As fate would have it, two spaces right next to each other were open at 10th and Alder. Todd took both, and the rest is history. His new spot has been wonderful for his business. Downtown turned out to be the perfect place for Todd's cart.\n\n_Burnt Toast_\n\nTodd is very proud of the beans they use at Ol\u00e9 Latte, some of which he roasts himself and some he gets from a local Portland roaster, Ristretto Roasters. Todd is very passionate about how long coffee beans should be roasted, and he is not a fan of dark roast coffee. He said that if you toast bread, the flavor improves, but if you burn the toast, you lose the benefit of tasting it. He considers dark roast coffee to be burned.\n\n_Suspended Coffee Comes to Portland_\n\nTodd was very grateful for the way that downtown Portland embraced his fledgling business, and he began to look for ways to give back. When he learned about the suspended coffee movement, he immediately put up a suspended coffee board and implemented this wonderful practice, which originally started in Italy, at his cart. The way it works is that if you pay for a suspended coffee, a mark is made on the suspended coffee board indicating that there is a paid-for coffee available for whoever needs it. It is a simple and easy way to give back.\n\nThe people who order coffee from the suspended board are people who are homeless or people who are simply down on their luck and not able to currently afford a treat like a latte. For the past year and a half, I have worked extensively with the homeless here in Portland, and doing something like this is much better than giving the homeless cash. Todd has seen people go from being homeless and taking coffee off the suspended board to being gainfully employed and paying for coffee to go onto that board for others in need.\n\nTodd Edwards, owner of the Ol\u00e9 Latte Coffee Food Cart.\n\n_The Coffee Truly Is Ol\u00e9!_\n\nI love the coffee that Todd serves, and his lattes are some of the smoothest and most flavorful I have ever had. Todd has a phrase that he often uses when he ends a social media post: \"Stay caffeinated, my friends.\" I always smile when I see that phrase and think about his amazing coffee.\n\n_What's Next_ : Todd's current plans are to open a second cart, also to be located downtown, and then get a third cart that will be mobile and be used for events. Some day, he hopes to open a brick-and-mortar Ol\u00e9 Latte Coffee and own his own roaster.\n\n_When You Go_ : Get a latte, and if you like espresso, get a doppio from Todd. I love his doppio. He proudly carries pastries from the Bake Shop Bakery, and you need to buy two of their sweet and salty cookies. They are scrumptious! Get one for now and one to eat when you get home.\n\n_Recipes_ : You can find the Ol\u00e9 Latte Coffee recipe for Cold Brew Coffee in _Trailer Food Diaries Cookbook_ , Portland edition, vol. 2.\n**ABOUT THE AUTHOR**\n\n_Photo by Zayne Shomler_.\n\nSteven Shomler is a Portland food writer. He writes about food carts at Portland Food Cart Adventures, a website he founded in 2012. As Steven began writing about food carts, he fell in love with the people inside the carts, as well as their stories. Steven regularly consults with food cart owners on how to have a successful startup, coaching them on topics such as branding, marketing and social media. Steven is also one of the founders and organizers of the Portland Summer Food Cart Festival, which takes place each summer on the campus of Mt. Hood Community College.\n_Visit us at_\n\nwww.historypress.net\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n_Godless: 150 Years of Disbelief_\n\nEdited by Chaz Bufe\n\nThis edition \u00a9 2019 PM Press.\n\nAll rights reserved. No part of this book may be transmitted by any means without permission in writing from the publisher.\n\nISBN: 978-1-62963-641-2\n\nLibrary of Congress Control Number: 2018949082\n\nCover by John Yates \/ www.stealworks.com\n\nInterior design by briandesign\n\n10 9 8 7 6 5 4 3 2 1\n\nPM Press\n\nPO Box 23912\n\nOakland, CA 94623\n\nwww.pmpress.org\n\nPrinted in the USA.\n\n# Contents | Introduction\n\n---|---  \n|\n\n_Dan Arel_\n\nCHAPTER I | The God Pestilence\n\n|\n\n_Johann Most_\n\nCHAPTER II | Woman, Church, and State\n\n|\n\n_Matilda Gage_\n\nCHAPTER III | The Devil's Dictionary _(excerpts)_\n\n|\n\n_Ambrose Bierce_\n\n|\n\nAmerican Heretic's Dictionary _(excerpts)_\n\n|\n\n_Chaz Bufe_\n\nCHAPTER IV | The Failure of Christianity\n\n|\n\n_Emma Goldman_\n\nCHAPTER V | Twelve Proofs of the Nonexistence of God\n\n|\n\n_S\u00e9bastien Faure_\n\nCHAPTER VI | The Meaning of Atheism\n\n|\n\n_E. Haldeman-Julius_\n\nCHAPTER VII | How Christianity Grew Out of Paganism\n\n|\n\n_Joseph McCabe_\n\n|\n\nChristianity and Slavery\n\n|\n\n_Joseph McCabe_\n\n|\n\nJudeo-Christian Degradation of Woman\n\n|\n\n_Joseph McCabe_\n\nCHAPTER VIII | Why Science Leaves Religion in the Dust\n\n|\n\n_Chaz Bufe_\n\n|\n\nTwenty Reasons to Abandon Christianity\n\n|\n\n_Chaz Bufe_\n\nCHAPTER IX | May the Farce Be with You: A Lighthearted Look at Why God Does Not Exist\n\n|\n\n_Pamela Sutter_\n\nCHAPTER X | Disbelief 101: A Young Person's Guide to Atheism _(excerpt)_\n\n|\n\n_S.C. Hitchcock_\n\nCHAPTER XI | Dogspell\n\n|\n\n_Earl Lee_\n\n|\n\nAfterword\n\n|\n\n_Chaz Bufe_\n\n|\n\nABOUT THE AUTHORS\n\n# Introduction\n\n# by Dan Arel\n\n\"No Gods, No Masters,\" a mantra that almost seems long forgotten in today's radical political landscape. We've lost some of our fervor to call religion what it is and to hold it accountable for the terrible things it has caused humanity to do. Some of this disconnect is understandable when some of the loudest voices claiming to speak out against religion are showing themselves to be in bed with white supremacists or are flat-out white supremacists themselves.\n\nToday's new wave of atheist figureheads\u2014better known as New Atheists\u2014are writers and public figures such as Sam Harris, known these days for calling the Nazis marching in Charlottesville \"peaceful\" and advocating for the racial profiling of Muslims, or retired biologist Richard Dawkins, who has defended \"light\" pedophilia and referred to the phrase \"Allah Akbar\" as \"violent sounding.\" This is not intellectually challenging religion, this is racism and bigotry.\n\nYet, while these people may mask their racism and bigotry behind the criticism of religion, we often allow religion to act unchecked. Christian extremists continue to terrorize and murder abortion doctors and staff. Christian politicians continue to use their religious beliefs to fight for laws that limit a women's abortion rights or fight to weaken protections for the LGBTQIA community.\n\nOnce upon a time, these critiques of religion were an everyday part of radical left politics and theory. In our modern life, we tend to ignore the impact religion has on climate denial. Ken Ham, the evangelical Christian owner of the Creation Museum, claims that while the climate is changing, there's nothing humans can do about it, God is in control. Others believe there is no climate change at all and attempt to root this ignorance in biblical claims, much like that of Ham, that God controls the heavens and earth and gave humans fossil fuels to burn and animals to slaughter.\n\nSchools throughout the U.S. continue to debate whether or not evolution or sex education should be a part of a child's education. Tax dollars from the working class have been sunk into Noah's Ark theme parks, and so-called leaders in this country continue to push religion into our daily lives.\n\nMaybe we have forgotten this, maybe we are too afraid to critique religion because of the New Atheist bigotry, or maybe we have simply forgotten how important this discussion can be. This is what makes _Godless_ such an important anthology. It takes us back to a time before atheism was overrun with rampant bigotry and racism. This was a time when the discussion around the existence of God or the importance of Jesus was discussed and\/or debated in a philosophical way that focused not on the believer but on the belief. This was when religion was properly challenged by those wanting to create a better world.\n\nThis does not mean the left doesn't believe in a person's right to religion or to believe in god. What it does mean is working to remove the shackles of control religious leaders have over people and removing a politician's ability to use their religious beliefs to dictate how we live our lives.\n\nThe authors within these pages spoke out before atheism was a profitable business that got you invited on cable news networks or took you on campus tours around the world. They spoke out not for fame or fortune, but because it was the right thing to do. They spoke out because someone had to.\n\nThe new atheist movement, though important to challenge, is also static. It's disrupting our message, and it has put us off course. This book rights that ship; let it be our compass.\n\nWe've forgotten the importance of what \"No God, No Masters\" means. For me, this collection of works serves as a reminder. It reinvigorated my atheism, for a lack of a better term. It's a return to the naturalist argument for the existence of everything, but it also serves as a reminder that just like our desire to abolish the state and return to a world of autonomy, as long as humans continue to believe that they must answer to a higher power and allow the self-proclaimed masters of that power to dictate so many aspects of their lives, we are still dealing with a force with the same kind of controlling power as that state.\n\nReligion, at its most powerful, is used to control and subjugate women, turning them into property. It's used to excuse the abuse of children, by turning away from lifesaving medicine or from \"sparing the rod,\" giving parents permission to physically abuse their children. It was and in many ways still is used to condone rape and slavery.\n\nWe know that the world can exist without the good that religion claims to provide, and oftentimes exists in spite of it. It is we, those fighting for self-determination, free association, and, above all, mutual aid, who understand best that we as humans must care for one another. We don't seek a higher power to grant us this authority. We don't want authority. We simply want to build a better world, and it's the gods and the masters who are standing in our way.\n\nToday, we begin tearing them down.\n\n# CHAPTER I\n\n# The God Pestilence\n\n# by Johann Most\n\nAmong all mental diseases which man has systematically inoculated into his cranium, the religious pest is the most abominable.\n\nLike all things else, this disease has a history; it only regrettable that in this case nothing will be found of the development from nonsense to reason, which is generally assumed to be the course of history.\n\nOld Zeus and his double Jupiter were still quite decent, jolly, we might even say, somewhat enlightened fellows, if compared with the last triplet on the pedigree of gods who, on examination, can safely rival with Vitzliputzli as to brutality and cruelty.\n\nWe won't argue at all with the pensioned or dethroned gods, for they no longer do any harm. But the more modern, still officiating cloud lollers and terrorists of hell we shall criticize, expose, and vanquish the more disrespectfully.\n\nThe Christians have a threefold God; their ancestors, the Jews, were content with a single simpleton. Otherwise both species are quite a humorous crowd. \"Old and New Testament\" are to them the sources of all knowledge; therefore, willing or not, one must read the \"holy writ\" if one wants to fathom their shallowness and learn to deride them.\n\nIf we only take the \"history\" of these deities, we find an ample sufficiency for the characterization of the whole. In a short sketch the case stands thus: in the beginning God \"created\" heaven and earth\u2014consequently he found himself next of all in a complete void, where it surely may have been dreary enough to bore even a deity. And it being but a trifle for a god to conjure worlds out of nothing by magic, like a juggler shaking eggs or silver dollars out of his coat sleeves, he (God) \"created\" heaven and earth. Somewhat later he molded the sun, moon, and stars to suit himself.\n\nIt is true, certain heretics, called astronomers, have established long since that the earth neither is nor could have been the center of the universe, nor could its existence have antedated that of the sun, around which it revolves. These people have proved it to be sheer lunacy to speak of sun, moon, and stars and with the same breath of the earth as being, compared with the former, something special and of great preponderance. It has been taught to every schoolboy that the sun is only a star, the earth one of its satellites, and the moon an under-satellite of the earth; and, furthermore, that the earth, compared with the universe, far from acting a conspicuous part, is only an atom, looking like a rain of dust.\n\nBut why should a God concern himself about astronomy? He does what he pleases and poohoos science and logic. For this reason, he made, after manufacturing the earth, first the light and afterwards the sun. Today even a Hottentot can understand there can be no light on earth without the stuff, but God\u2014well, he is no Hottentot.\n\nBut let us continue to investigate. Thus far the \"creation\" was quite a success, but there was still something lacking\u2014things were not lively enough. The creator wanted some pastime, therefore he finally made man. Curiously enough, he now deviated entirely from the method previously applied. Instead of accomplishing this creation by a simple and imperative \"Let it be!\" he made it exceedingly troublesome. He took a prosaic lump of common clay in his hand, modelled it into the figure of a man \"after his own image\" and \"breathed into his nostrils the breath of life.\" God being of infinite wisdom, benign, just, in short\u2014amability itself\u2014it occurred to him that this Adam, as he had named his last article, being alone, would find life exceedingly tedious (perhaps he remembered his own former lonely existence to \"Nothing\"), and so he made him quite a nice, enticing little Eve. But, in the meantime, experience had evidently taught him that the handling of a lump of clay was a rather unclean business, especially for a god, therefore he applied another new method of manufacture. He tore (dexterity is no witchcraft, least of all for a god) a rib out of Adam's body and changed it into a charming female. Whether this rib, extracted from Adam was restored at a later period or whether, after the performed operation, Adam had to run about in the world as a \"one-sided\" individual is a matter upon which the polite historian says nothing.\n\nModern natural science has established that animals and plants have, through the most manifold ramifications, developed during the course of millions of years from simple molluscous matter to their present forms. Man is nothing but the most perfect form of this development, and that he not only had, some thousands of years ago, a very brutish appearance, without language, but also that he\u2014every other supposition excludes itself\u2014must have developed from inferior animal species.\n\nConsequently, natural science stamps God with his self-proclaimed creation of man as a preposterous braggart. But of what avail is all this?\u2014God won't have any tomfoolery. Whether his tales have a scientific ring, or whether they sound like foolish babble, he commands belief in them, otherwise he will let it come to pass that his competitor, the \"devil,\" will get you into his clutches, which is supposed to be quite uncomfortable. For in hell there is not only moaning and the gnashing of teeth but an eternal fire burns, an indefatigable worm is gnawing your soul, and a dreadful stench of burning pitch and sulfur fills the air. To all these discomforts the bodiless man is supposed to be exposed. His flesh, of which he is void, is stewed; his decayed and fallen-out teeth clatter; he howls without a throat or lungs; he smells without a nose\u2014and all this eternally. A devil of a god! Taking all in all, God is, as he candidly informs us in his autobiographical chronicle\u2014the Bible\u2014extremely whimsical and revengeful; actually, an ideal model of a despot.\n\nHardly were Adam and Eve in existence, before God took it as a matter of course that this rabble must be governed. He decreed a penal code, which said categorically, \"Thou shalt not eat of the fruit of the tree of knowledge!\" Since that time no tyrant has existed anywhere who did not lay down the same decree for the people.\n\nAdam and Eve did not respect this prohibition, therefore, they were exiled and sentenced to hard labor for life\u2014they and their descendants for all time to come. Beyond this, \"civil rights\" were taken away from Eve, she being declared to be a bond servant to Adam, whom she was to obey. Besides, both of them were to be under eternal divine police surveillance. Verily, not even William (the German Emperor) has got ahead of him as a haberdasher of human affairs.\n\nBut in spite of God's useless severity to mankind, they angered him more and more in proportion as they multiplied. How rapidly this multiplication took place is demonstrated by the history of Cain and Abel. After the former had slain his brother, he went into \"a strange land, and took unto himself a wife.\" Where the location of this strange land and whence the women that were to be found there, God has forgotten to mention, a matter of no surprise, considering the burden of overwork he had to perform at that time.\n\nAt last the cup was overflowing. God resolved to destroy all mankind by means of water. Only a few specimens of the race were excepted, with which to make another trial. Unfortunately, notwithstanding all his wisdom, he had made a misgrab, for Noah, the chief of the saved, was soon unmasked as a drunken sot who disported with his own daughters. What good could ever come of so degraded a family?\n\nAgain mankind spread; again they developed to such simpletons and rapscallions in sin\u2014about whom the renowned Mecklenburger psalmbook states so much viciousness\u2014that God felt like bursting with heavenly wrath, the more so, as all his exemplary local punishments, such as the destruction of whole cities with fire and brimstone, were entirely disregarded and \"thrown to the dogs.\" He resolved to destroy the whole mob, root and branch, when a really remarkable event occurred which toned him down considerably. Otherwise mankind would have been done for long ago.\n\nOne fine day, a certain \"Holy Ghost\" suddenly appeared upon the stage. He came hither like a wart overnight, nobody knew whence. The Bible scribe (God) merely says he himself was the Holy Ghost. All at once we have to deal with a dual deity. Said Holy Ghost took a notion to descend in the shape of a dove, or rather of a cock pigeon, and to enter into an intimacy with an obscure woman named Mary. In a sweet hour he \"overshadowed\" the selected of his heart, and lo! she gave birth to a baby boy, which occurrence, as God positively avers in the Bible, never encroached the least bit upon her virginity. Now, this boy was not only human, he was also God, being the Son of God (of the Holy Ghost). The first mentioned God now called himself God the Father, at the same time assuring us of his identity not only with the Holy Ghost but also with God the Son. The father to be his own son, the son his own father, and either or both the Holy Ghost. Thus the \"Holy Trinity\" was shaped.\u2014Neat!\n\nAnd now, poor human brain, stand firm, for what now follows is enough to stagger a horse. We know that God the Father had resolved to friccasee the whole human rabble. This intention filled the Son with unbearable sorrow. He (being his own father), shouldered all the guilt of man, and allowed himself, to appease the fury of the Father (being his own son), to be cruelly put to death by the \"to be redeemed\" rabble\u2014of course, not without subsequently ascending hale and hearty to heaven. This sacrifice of the Son (who is one with the father) tickled the Father (who is one with the Son) to such an extent that he immediately proclaimed a general amnesty\u2014under conditions\u2014which partly remain in force today.\n\nThat is the \"historical\" part in the \"Holy Scriptures.\" Here we see that absurdity and nonsense are put on so thick that those who are already idiotic enough to digest such stuff are susceptible to the most crazy hallucinations. Among these must be classed first and foremost the doctrine of reward and punishment of mankind in the \"great hereafter.\" It has long ago been scientifically proved that there is no existence of a soul independent of the body. That which the religious humbuggers call soul is nothing more or less than the seat of thought, the brain, which receives impressions by means of the living senses, and by such impressions becomes active; and, consequently, at the moment of physical dissolution this action necessarily must cease. But what care the deadly enemies to human reason for the results of scientific research? Just as much as is necessary to prevent their promulgation among the people.\n\nAnd so they preach the immortality of the soul. Woe to it in the \"hereafter\" if the body which here held it has not punctiliously respected God's penal code during life. As these folks assure us, their \"all bountiful,\" \"all righteous,\" \"all benevolent,\" \"all merciful\" God is a super highly developed pokenose, sniffling into the minutest affairs and trifles of each and every individual and making entries of all their shortcomings in his blacklist. He is quite a queer coon anyway. Under danger of giving newborn babes a bad cold, he desires that, to his glory, they be drenched with water (otherwise, baptized).\n\nHe takes a heathenish delight in hearing an innumerable herd of his faithful sheep bleat their litanies to hire from their church stalls consecrated to such practices, or when the most devoted of his adherents send aloft without cessation their pious \"caterwaulings\" and pray, or rather beg, for all things possible and impossible, while he participates in bloody wars and receives the thanksgivings and frankincense of the victors as the \"God of battles.\"\n\nHe gets rip-roaring mad if anybody doubts his existence, or when a Catholic eats meat on Friday or does not by repeated application to the confessional chair scrub off his sins; or when a Protestant does not hold in contempt the bones of saints and the paraphernalia and images prescribed for Catholics, or when he does not, as a rule, toddle through the world with a face long enough to stop a clock, turned up eyes, bent back, and folded hands.\n\nIf such a person dies impenitent, then his \"all merciful\" God decrees a punishment to him, in comparison to which all the scourges of the knout or cat-o'-nine-tails, all pangs and sufferings of prison life, all privations of deportation and exile, all emotions of those sentenced to the scaffold, all pains of the rack and other instruments of torture that human tyrants ever invented are only pleasant, agreeable, tickling sensations. This God exceeds in bestial cruelty everything malignant that we know of on earth. His prison is hell, his handman the devil, and his punishments last forever. He employs worms that never cease gnawing, fires which are never quenched, and other deviltries a thousandfold as chastisements and only shows mercy for minor transgressions after long periods of time, provided the transgressor died a Catholic; for these he has under certain circumstances provided a \"purgatory,\" which differs from hell about as much as a jail from prison. It is fitted up only for comparatively transient inmates, with somewhat more lenient regulations\u2014but, at any rate, even in purgatory you will be singed mercilessly.\n\nThe so-called \"cardinal sins\" are never punished with purgatory, always with hell. These include, among others, \"blasphemy\" perpetrated by word, by writing, or by thought. Consequently, in this direction God permits neither freedom of the press nor speech, not even of the unspoken thought. This in itself is enough to stamp him from the outset as a successful competitor in churlishness with the basest despots and tyrants of any country or time, but the means and the duration of his punishments augment the baseness of his nature to the utmost. Consequently, this God is the most atrocious monster conceivable.\n\nHis attitude is the more infamous in that he allows it to be said of him that the entire world, and especially mankind in their behavior, are regulated by his omnipotent divine providence. He maltreats man for actions of which he himself is the originator or prime cause. How amiable, compared with this monster, are the tyrants of earth of the past and present time. Should it please God, however, to permit a person to live and die happy in accordance with his (God's) conceptions, he maltreats him still worse, because the promised \"heaven,\" examined by gaslight, is a good deal worse institution than hell itself; for there you have no desires, you are always satisfied, without ever having a longing for anything. (But as without desire and attainment thereof no gratification is possible, so existence in heaven is without enjoyment.) Eternally employed in \"beholding the Lord,\" eternally listening to the same strains from the same harps, eternally singing that same new and entrancing song in the same melody, if not the one of \"Gabriel blowing the trumpet in the morning,\" surely not anything more exhilarating.\n\nThis is the highest degree of \"getting tired after a whilishness.\" The occupancy of an isolated prison cell would be decidedly preferable. It is not surprising, then, that those who are rich and mighty enough to enjoy paradise on earth, should laughingly proclaim with Heine:\n\nThe angels and the birds may own\n\nThe heavens for themselves alone!\n\nAnd yet the rich and mighty foster and nourish divine idiocy and religious stupidity. It is, in fact, part of their business; it is really a question of life or death to the domineering and exploiting classes whether the people at large are dumbfounded religiously or not. With religious lunacy stands and falls their power. The more man clings to religion, the more he believes\u2014the more he believes, the less he knows\u2014the less he knows, the more stupid he is\u2014the more stupid, the easier he is governed\u2014the easier to govern, the better he may be exploited\u2014the more exploited, the poorer he gets\u2014the poorer he, the richer and mightier the domineering classes get\u2014the more riches and power they amass, the heavier their yoke upon the neck of the people.\n\nThe tyrants of all time and of all countries were always well acquainted with this train of thought, and for that reason always were on good terms with the priesthood of all creeds. Casual quarrels between these two kinds of enemies to mankind were at all times only of a domestic character, merely a struggle for supremacy. The priests or preachers know that they are done for, unless they have the support of the circle of cronies at the governmental top. It is no secret to the rich and powerful that mankind can only be enslaved and exploited when the necromancers of the churches ingraft sufficient servility into the hearts of the masses of the people to make them look upon the earth as a vale of tears, to imbue their minds with the justness of the godly decree; \"Serve ye your master\" (those in authority), and to buy them off with an alleged \"sparerib,\" of which the people will get the soup in that home beyond the skies, the \"Nobodyknows.\"\n\nMr. Windthorst, member of the German Parliament, arch-Jesuit, and champion of the clerical faction, once in the heat of debate gave us plainly to understand what the frauds and charlatans of society think in regard to this matter. \"When the people lose their faith\"\u2014said he\u2014\"they will no longer bear their intolerable misery, but rebel!\"\u2014That was to the point, and should have aroused the minds of all working people to earnest reflection, if\u2014yes\u2014if the great majority of them had not become religious imbeciles to such an extent that it were an utter impossibility to comprehend the simplest ideas, though heard with normal ears.\n\nIt is not in vain that the priesthood of all sorts\u2014i.e., the \"black gendarmerie\" despotism\u2014have always so strenuously exerted themselves to prevent the retrogression of religious sentiment, although, as is well-known, when among themselves they could burst with laughter over the nonsense they preach for ready cash.\n\nDuring thousands of years these brain defilers have instituted a reign of terror, without which the religious craze would long since have been abolished. Scaffold and sword, dungeon and chains, poison and poignards, assassination and judicial murder, these were the means by which the religious insanity was upheld, forever a blot of shame in the history of the human race. Hundreds of thousands have been slowly roasted to death at the stake \"in the name of God,\" because they dared to find a stink in the biblical mire. Millions have in tedious wars been compelled to break each other's heads, to burn and sack entire countries, and, after wholesale murder and incendiarism, to spread disease and pestilence, all to maintain religion. The most excruciating tortures were invented by the priests and their accomplices to scare those who had ceased to fear God into religion by the application of mundane devilishness.\n\nWhen a man maims the hands or feet of another, we say he is a criminal. What shall we call those who maim the intellect of man and, failing in that attempt, with refined cruelty destroy the body inch by inch?\n\nIt is certainly true that today they cannot carry on their nefarious \"godly\" brigandage in the manner formerly in vogue, but in lieu thereof have taken to worming into domestic affairs of families, influencing women, kidnapping children, and misusing the schools for the furtherance of their ends. Their hypocrisy has rather increased than diminished. After their attempts to abolish the newly discovered art of printing had signally failed, they, with their usual cunning and craftiness, utilized it and have gradually to a great extent made the press of today servile to their cause.\n\nAn old adage says, \"Where a priest has trod, no grass will grow!\" That means, in other words, if a person is once in the clutches of the priests, his intellect becomes barren\u2014his intellectual functions cease to operate in a normal way, and instead religious maggots and divine worms wriggle through his brain. He resembles a sheep that has the staggers.\n\nThese misguided, unhappy wretches have been defrauded of the real object of life; and, what is worse, they form the great crowd in the train of the opponents to science and the march of reason, to evolution and liberty. Whenever new chains are to be forged for mankind, they are willing to work at the anvil as if possessed by demons. Whenever the road of progressive development is to be blockaded by obstacles, these Kaffirs oppose in their broadest front to the spirit of the times. The attempt to cure such imbeciles is not only a piece of good work to them, it is really an attempt to cauterize a cancer which brings suffering to the whole people and which must ultimately be unconditionally extirpated if this earth is to become a fit habitation for mankind instead of being a playground for gods and devils to torment us.\n\nOut then with religion from the heads of the people, and down with priesthood! The latter are in the habit of saying, \"The aim sanctifies the means.\" Very well, let us apply this precept against them. Our aim is to make mankind independent of every condition of slavery, of the yoke of social servitude, as well as of the shackles of political tyranny, and, not least because last, of all bane of religious darkness. All means to attain this object and made use of at all opportunities offering will be acknowledged just and right by every true philanthropist.\n\nEvery person possessing common sense in place of religious insanity, neglecting to do the utmost in his power daily, hourly, to overthrow religion, shirks a duty. Every person, released from deistic superstition, forbearing to oppose priesthood where, when, and however an opportunity presents itself, is a traitor of his cause. Therefore, war to the black hounds! Implacable war to the knife! Incite against the seducers of man, enlighten the seduced! Let us make every means of strife subservient: the scourge of derision and scorn, the torch of science and knowledge, and where these are insufficient\u2014weightier arguments\u2014those that will be felt.\n\nFor the ignorant, or rather those craftily made and kept so, if they appear to have a little sense left, the following questions will be proper: \"If God desires that we know, love, and fear him, why does he not show himself?\u2014If he is as good as the clergy tell us, what reason is there to fear him?\u2014If he is omniscient, why bother him with private affairs and prayers?\u2014If he is omnipresent, why build him churches?\u2014If he is just, why the supposition that man, whom he created full of faults, shall be punished?\u2014If man does good only by the grace of God, why should he be rewarded?\u2014If he is omnipotent, how can he permit that we blaspheme?\u2014If he is inconceivable, why shall we occupy ourselves with him?\u2014If the knowledge of God is necessary, why does he remain obscure?\" Such questions are puzzles to them.\n\nEvery thinking person must admit that not one single proof of the existence of a God has ever been found; and, besides this, there is not the least necessity for the existence of God. As we know the inherent properties and laws of nature, the presence of God, either within or beyond this nature, is really to no purpose, quite superfluous, and evidently untenable. Morally the necessity for his existence is still more insignificant.\n\nThere is a large empire ruled by a potentate whose demeanor creates differences of opinion in the minds of his subjects. He wants to be known, loved, and honored, and that all shall obey him, but he never shows himself. Everyone endeavors to confound the conception of him by individual nations. The people, subjected to his power, have only such ideas about the character and laws of their invisible sovereign as his ministers see fit to make known, although the latter admit, at the same time, that they are unable themselves to form a conception of their master, that his will is inscrutable, his views of things and his properties unfathomable, and that even his servants disagree about the decrees sent forth by him; for in every province of his empire, the laws differ, and they accuse each other of having altered and forged them. These edicts and commandments, which they claim to have authority to promulgate, are obscure; they are conundrums which the subjects for whose special benefit and enlightenment they are issued can neither understand nor solve. The laws of this hidden monarch require explanation, but those who explain are ever at variance themselves.\n\nEverything that they relate about their concealed sovereign is a chaotic mass of contradictions. They do not say one word that could not at once be proved as a lie. They speak of him as exceedingly good, but still there is no individual existing who does not complain of his mandates. They speak of him as infinitely wise, but yet in his administration everything opposes common sense and reason. They praise his justice, and still the best of his subjects are, as a rule, the least favored. They assure us that he sees everything, still his omnipresence alleviates no distress. He is, they say, a friend of order, yet in his domain everything is confusion and disorder. All his actions are self-determined, yet occurrences seldom, if ever, bear out his plans. He can penetrate the future but does not know the things that will come to pass. He permits no insult to himself to go unpunished but still submits to them from everyone. They are amazed at his knowledge and the perfection of his works, yet his works are imperfect and of short duration, for he creates, destroys, and constantly improves upon that which he has made, without ever being satisfied with his productions. All his enterprises are for the sake of glory, yet his purpose, to be universally glorified, is never attained. He labors incessantly for the welfare of his subjects, but the most of them are in dire distress for the necessities of life. Those whom he apparently seems to favor most are the least satisfied with their lot. We see them nearly all refractory to a master whose grandeur they admire, whose wisdom they praise, whose benevolence they honor, whose justice they fear, and whose commandments they revere but never keep. The empire is the earth; this sovereign is God; his vassals are the priests; his subjects are mankind\u2014a fine conglomeration.\n\nThe God of the Christians, as we have seen, is the God who makes promises only to break them; who sends them pestilence and disease in order to heal them; a God who demoralizes mankind in order to improve them. A God who created man \"after his own image,\" and still the origin of evil in man is not accredited to him. A God who saw that all his works were good and soon after discovered that they were bad; who knew that man would eat of the forbidden fruit and still eternally damned him therefore. A God who is so dull as to allow himself to be outwitted by the devil; so cruel that no tyrant on earth can be compared with him\u2014that is the God of the Jewish-Christian theology. He is an all wise bungler who created mankind perfectly, yet could not keep them in that state; who created the devil, yet could not keep him under control; a God who is omnipresent, yet descended from heaven to see what mankind was doing; who is merciful, and yet has, at times, permitted the slaughter of millions. An Almighty who damned millions of innocent for the faults of a few, who caused the deluge to destroy mankind excepting a very few with whom to start a new generation no better than the preceding\u2014who created a heaven for the fools who believe in the \"Gospel\" and a hell for the enlightened who repudiate it. A divine charlatan who created himself through the Holy Ghost, and then sent himself as mediator between himself and others, and who, held in contempt and derided by his enemies, was nailed to a cross like a bat on a barndoor; who was buried\u2014arose from the dead\u2014descended to hell\u2014ascended to heaven, and since then for eighteen hundred years has been sitting at his own right hand to judge the living\u2014and the dead when the living cease to exist. A terrible despot! whose history should be written in letters of blood, because it is a religion of terror.\n\nAway then with the Christian theology! Away with a God invented by preachers of the bloody faith, who, without their important nothing, by means of which they explain everything, could no longer revel in superfluity; no longer glorify poverty and live in luxury themselves; no longer preach submission and practice arrogance; but who would, through the march of reason, be hurled into the deepest depths of oblivion.\n\nAway then with the malignant Trinity\u2014the murderous Father\u2014the unnatural Son\u2014the lascivious Ghost! Away with all the debasing phantasies in whose name man is degraded to miserable slavery, and through the almighty power of falsehood has been deluded into hoping for the joys of heaven as an indemnification for the miseries of earth. Away then with those who with their sanctified hallucinations are the curses of liberty and happiness\u2014the priesthood of all sorts!\n\nGod is merely a specter fabricated by designing scoundrels through which mankind is tyrannized and kept in constant dread. But the phantom instantly dissolves when examined under the glass of sober reflection. The defrauded masses become impatient... and no longer fear the bugbear, but will rather hold out to the priesthood the word of the poet:\n\nA curse to the idols to whom we pray'd,\n\nThat in winter our hunger and cold be stay'd,\n\nIn vain did we hope, in vain did we wait,\n\nTo be humbugg'd and fool'd was ever our fate.\n\nIt is to be hoped that they will not stand humbugging and fooling much longer but will, one of these fine days, throw their crucifixes and saints into the fire, \"transform their crucifixes and chalices into useful utensils, and make of their church\" theatres, concert and assembly halls or, should they not be serviceable for that purpose, use them as corn bins or stables\u2014find useful work for the priests and nuns, and then be surprised at themselves for not having done it long before.\n\nThis short and terse method will, of course, only be consummated in the storm of the coming social revolution; in fact, at that moment, when the conspirators of priesthood, the princes, the nobility, the bureaucrats, the capitalists, and the exploiters of all kinds are swept away as by a whirlwind, thereby cleaning state and church with an iron broom of the accumulated mire of centuries.\n\n# CHAPTER II\n\n# Woman, Church, and State\n\n# by Matilda Gage\n\nThe most important struggle in the history of the church is that of woman for liberty of thought and the right to give that thought to the world. As a spiritual force the church appealed to barbaric conception when it declared woman to have been made for man, first in sin, and commanded to be under obedience. Holding as its chief tenet a belief in the inherent wickedness of woman, the originator of sin, as its sequence the sacrifice of a God becoming necessary, the church has treated her as alone under a \"curse\" for whose enforcement it declared itself the divine instrument. Woman's degradation under it dating back to its earliest history, while the nineteenth century still shows religious despotism to have its stronghold in the theory of woman's inferiority to man. The church has ever invoked the \"old covenant\" as authority, while it also asserts this covenant was done away with at the advent of the new dispensation. Paul, whose character as persecutor was not changed when he veered from Judaism to Christianity, gave to the church a lever long enough to reach down through eighteen centuries in opposition to woman's equality with man. Through this lengthy period, his teaching has united the Christian world in opposition to her right of private judgment and personal freedom.\n\nEach great division of Christianity alike proclaims the supreme sinfulness of woman in working for the elevation of her sex. In this work she has been left outside of religious sympathy, outside of political protection, yet in the interest of justice she claims the right to tear down the barriers of advancing civilization and to rend asunder all beliefs that men hold most sacred. Freedom for woman underlies all the great questions of the age. She must no longer be the scapegoat of humanity upon whose devoted head the sins of all people are made to rest. Woman's increasing freedom within the last hundred years is not due to the church but to the printing press, to education, to free thought and other forms of advancing civilization. The fashions of the Christian world have changed but not its innermost belief. The power of the pulpit, built up by a claim of divine authority, with the priest as an immediate representative of God, has been reacting upon the priesthood itself, and now while vainly struggling for light this order finds itself bound by chains of its own creating. Today the priesthood is hampered by creeds and dogmas centuries old, yet so fully outside of practical life that the church has become the great materialistic force of the century; its ideas of a God, its teachings of a future life, all falling within the realm of the physical senses; the incorporeal and spiritual are lost in the grossest forms of matter. Although a body professing to inculcate pure spiritual truths, the church teaches the grossest form of materialism. It asserts principles contradictory to natural laws; it presents chaos as the normal condition of the infinite; it bids people live under faith outside of evidence, and in thus doing is guilty of immeasurable evils to mankind. A bark without compass, it steers upon a sea of night, no star illumining the darkness; the control and guidance by humanity of the psychic part of being, generally spoken of as \"supernatural,\" although the truest to nature, has become nearly lost through the materialization of spiritual truth by the church, the worst form of idolatry. Christianity was a stern reality to the men of the Early and Middle Ages, who believing themselves to have been created nearer to God than woman also believed themselves to have lost earthly immortality through her. Permeated with this idea, it is not strange that men through many hundred years taught that woman was especially under control of the Evil One. The devil was an objective form to the clergy and people alike. Nor, under such belief, is it strange that priests should warn their flocks from the pulpit against the wiles of woman, thus degrading her self-respect and teaching men to hold her in that contempt whose influence is felt today. The result of this teaching has been deplorable to humanity; men equally with women having sunk under this degradation of one half of the race.\n\nThe most stupendous system of organized robbery known has been that of the church towards woman, a robbery that has not only taken her self-respect but all rights of person; the fruits of her own industry; her opportunities of education; the exercise of her own judgment, her own conscience, her own will. The unfortunate peculiarity of the history of man, according to Buckle, is that although its separate parts have been examined with considerable ability, hardly anyone has attempted to outline them into a whole and ascertain the way they are connected with each other. While this statement is virtually true as regards the general history of mankind, it is most particularly so in reference to the position of woman in its bearings upon race development. A thorough investigation of her connection with our present form of civilization, or even with that of the past, as compared with each other, or as influencing the whole, has never yet been authoritatively undertaken. This failure has not been so largely due to willful neglect as to incapacity upon the part of man to judge truly of this relation. Woman herself must judge of woman. The most remote feminine personality is not less incomprehensible to man than the woman of today; he now as little understands the finer qualities of her soul or her high intuitive reasoning faculties as in the past. Reason is divided into two parts, theoretical and practical; the former appertains to man; the latter, composed of those intuitive faculties which do not need a long process of ratiocination for their work, inhere in woman. Although the course of history has given many glimpses of her superiority, and the past few decades have shown in every land a new awakening of woman to a recognition of her own powers, man as man is still as obtuse as of yore. He is yet under the darkness of the patriarchate, failing to recognize woman as a component part of humanity, whose power of development and influence upon civilization are at least the equal of his own. He yet fails to see in her a factor of life whose influence for good or for evil has ever been in direct ratio with her freedom. He does not yet discern her equal right with himself to impress her own opinions upon the world. He still interprets governments and religions as requiring from her an unquestioning obedience to laws she has no share in making and that place her as an inferior in every relation of life. Ralph Waldo Emerson, with keen insight into the fallibility of lawmakers, declared that \"good men must not obey the laws too well.\" Woman is showing her innate wisdom in daring to question the infallibility of man, his laws, and his interpretation of her place in creation. She is not obeying \"too well,\" and yet man fails to analyze her motives in this defection. The church and the state have long done man's thinking for him, the ideas of the few, whose aim is power, have been impressed upon the many; individualism is still characterized as the essence of evil; self-thought, self-control as heretical. The state condemns both as a crime against itself, the church as a sin against heaven. Both church and state claiming to be of divine origin have assumed divine right of man over woman; while church and state have thought for man, man has assumed the right to think for woman.\n\nAs man under fear of eternal damnation surrendered to the irresponsible power of church and state, so woman yielded to that power which closed every external avenue of knowledge to her under pretext of her sinfulness. One-tenth of the human race, within the period covered by modern civilization, has compelled the other nine-tenths to think their thoughts and live lives according to their commands. This has been the chief effort of governments and religion. The most formidable general evil under which woman has suffered during the Christian ages has been that of protection; a non-recognition of her ability to care for herself, rendering watchful guardianship over her a recognized part of man's law; not alone to prevent her sinking into depths of vice but to also prevent her entire subversion of government and religion. Buckle and other writers have recognized the protective spirit as the greatest enemy to civilization, its influence causing the few to establish themselves as guardians of the many in all affairs of life. The American Revolution in proclaiming the rights of humanity struck a blow at the protective system. This system has ever based itself upon a declaration of the supreme rights of a God and certain rights as pertaining to certain classes of men by virtue of authority from that God. The defense of such authority has ever been the chief business of church and state, and thus religions and governments have neither found time nor inclination to uphold the rights of humanity. Under the Christian system, woman as the most rebellious against God, in having eaten a forbidden fruit, has found herself condemned through the centuries to untold oppression in order that the rights of God might be maintained. Yet while constantly teaching that woman brought sin into the world, the church ever forgets its own corollary; that if she brought sin she also brought God into the world, thus throwing ineffable splendor over mankind. The whole theory regarding woman, under Christianity, has been based upon the conception that she had no right to live for herself alone. Her duty to others has continuously been placed before her, and her training has ever been that of self-sacrifice. Taught from the pulpit and legislative halls that she was created for another that her position must always be secondary even to her children, her right to life has been admitted only in so far as its reacting effect upon another could be predicated. That she was first created for herself, as an independent being to whom all the opportunities of the world should be open because of herself, has not entered the thought of the church; has not yet become one of the conceptions of law; is not yet the foundation of the family.\n\nBut woman is learning for herself that not self-sacrifice but self-development is her first duty in life; and this, not primarily for the sake of others but that she may become fully herself; a perfectly rounded being from every point of view; her duty to others being a secondary consideration arising from those relations in life where she finds herself placed at birth or those which later she voluntarily assumes. But these duties are not different in point of obligation, no more imperative upon her than are similar duties upon man. The political doctrine of the sovereignty of the individual, although but partially recognized even in the United States, has been most efficacious in destroying that protective spirit which has so greatly interfered with the progress of humanity. This spirit yet retains its greatest influence in the family, where it places a boundary between husband and wife. Of all circumstances biasing the judgment and restricting the sympathies, none have shown themselves more powerful than physical differences, whether of race, color, or sex. When those differences are not alone believed to be a mark of inferiority but to have been especially created for the pleasure and peculiar service of another, the elements of irresponsible tyranny upon one side and irremediable slavery upon the other are already organized. If, in addition, that inferior is regarded as under an especial curse for extraordinary sin, as the church has ever inculcated in reference to women; and when, as in the case of woman and man, an entire separation of interests, hopes, feelings, and passions is impossible, we have reached the extreme of injustice and misery under the protective system. Consequently, no other form of \"protection\" has possessed so many elements of absolute injustice as that of man over woman. Swedenborg taught, and experience declares, that morality cannot exist except under conditions of freedom. Hence, we find much that has been called morality is the effect of dependence and lessened self-respect and has really been immorality and degradation. While in every age the virtues of self-sacrifice have been pointed to as evidence of the highest morality, we find those women in whom it has been most apparent have been those doing least justice where justice first belongs\u2014to themselves. Justice as the foundation of the highest law is a primal requirement of the individual to the self. It is nonetheless a serious impeachment of the religious-moral idea that the doctrine of protection and the duty of woman's self-sacrifice were taught under the theory of divine authority. No faith was more profound, none could be more logical if resting on a true foundation, than the church theory regarding woman. Life assumed a sterner reality to men who believed themselves in point of purity and priority nearer their Creator than woman. Thereafter, she was to be protected from herself, the church and man cheerfully assuming this duty. Under the protective spirit it is not so very long since men sold themselves and their families to some other man in power, either lay or religious, under promise of protection, binding themselves to obey the mandates of such lord evermore. The church protected and directed the thought of the world. To think for one's self is not even now the tendency of mankind; the few who dare, do so at great peril. It will require another hundred years of personal and political freedom for men to appreciate what liberty really is\u2014for them to possess confidence in their own judgment upon religious questions\u2014for the man of humble station to fully believe in himself and in his own opinions when opposed to the authority of church or state.\n\nWomen of the present century whose struggle for equal opportunity of education with men for a chance to enter the liberal professions, for a fair share of the world of work, for equal pay in that work, for all demands of equality which make the present a noted age in the world's history, have met their greatest opposition from this protective spirit. No less than during the darkest period of its history does the church still maintain the theory that education and public life are not fitting for woman\u2014indelicate for herself and injurious to the community. During the Christian ages, the church has not alone shown cruelty and contempt for woman but has exhibited an impious and insolent disregard of her most common rights of humanity. It has robbed her of responsibility, putting man in place of God. It has forbidden her the offices of the church, and at times an entrance within its doors. It has denied her independent thought, declaring her a secondary creation for man's use to whom alone it has made her responsible. It has anathematized her sex, teaching her to feel shame for the very fact of her being. It has not been content with proclaiming a curse upon her creative attributes but has thrust the sorrows and expiations of man's \"curse\" upon her, and in doing these things the church has wrought her own ruin. A religious revolution of the most radical kind has even now assumed such proportions as to nearly destroy the basic creeds of various sects and undermine the whole fabric of Christendom. It everywhere exists, although neither the world nor the church seem to realize the magnitude of its proportions. As a legitimate result of two opposing forces, a crisis in the life of the church is at hand; nay, even upon it. While we see it making organized effort for extension of power and entire control of the state, we also find great increase of radical thought and development of individual conscience and individual judgment. With thought no longer bound by fear of everlasting punishment, mankind will cease to believe unproved assertions, simply because made by a class of men under assumed authority from god. Reason will be used, mankind will seek for truth come whence it may, lead where it will, and, with our own Lucretia Mott, will accept \"truth for authority and not authority for truth.\"\n\nIn knocking at the door of political rights, woman is severing the last link between church and state; the church must lose that power it has wielded with changing force since the days of Constantine, ever to the injury of freedom and the world. The immeasurable injustice of woman and her sufferings under Christianity, her intellectual, moral, and spiritual servitude, will never be understood until life with its sorrows shall be opened to our vision in a sphere more defined than the present one. The superstitions of the church, the miseries of woman, her woes, tortures, burnings, rackings, and all the brutalities she has endured in the church, the state, the family, under the sanction of Christianity, would be incredible had we not the most undeniable evidence of their existence, not alone in the past but as shown by the teachings, laws, and customs of the present time. \"She has suffered under a theology which extended its rule not only to her civil and political relations but to her most significant domestic and personal concerns, regulating the commerce of husband and wife, of parent and child, of master and servant, even prescribing her diet and dress, her education and her industries.\" Edmund Noble speaks in like manner of the ancient Russians under the tyrannical provisions of the Greek church, saying, \"clearly, such a system of theocratic supervision and direction as this is compatible only with the lowest possible spiritual condition of the subject, or the lowest possible conception of God.\" Possessing no proof of its existence, the church has ever fostered unintelligent belief. To doubt her \"unverified\" assertion has even been declared an unpardonable sin. The supreme effort of the church being maintenance of power, it is but recently that woman has been allowed to read history for herself or, having read it, dared to draw her own conclusion from its premises. Ignorance and falsehood created a sentiment in accord with themselves, crushing all her aspirations. In the family, man still decides the rights and duties of the wife, as of old. As legislator and judge, he still makes and executes class laws. In the church, he yet arrogates to himself the interpretation of the Bible; still claims to be an exponent of the Divine will, that grandest lesson of the reformation, the right of private interpretation of the scriptures not yet having been conceded to woman. The premises upon which the church is based being radically false, and this, most especially in everything related to woman. Trained from infancy by the church to a belief in woman's inferiority and incapacity for self-government, men of the highest station have not hesitated to organize societies in opposition to her just demands. As early as 1875, an anti-woman's franchise association was formed in London under the name of \"Association for Protecting the Franchise from the Encroachment of Women\"; Hon. Mr. Bouverie, a leading opponent of woman suffrage in the House of Commons, being its chairman. Among the promoters of the movement were Sir Henry James, formerly attorney general (for the Crown), Hon. Mr. Claflin, and Mr. Leathers, correspondent of the _New York Tribune_.\n\nSince this period, a number of women distinguished as \"the wives of\" have petitioned legislative bodies for protection against freedom for themselves and all others of their sex, in asking that legislatures shall not recognize woman's self-governing right. The deepest depth of degradation is reached when the slave not only declares against his own freedom but strives to tighten the bonds of fellow slaves; and the most cruel wrong resulting from such slavery is the destruction of self-respect in the enslaved, as shown by the course of these women petitioners. The protective theory reached its lowest depth for woman by an attack upon her already vested rights of the ballot in the former territory, now State, of Washington, on the Pacific coast, in the case of Nevada M. Bloomer (a woman) against John Wood and others, to have the women of that territory deprived of their already existing right of suffrage. In line with the general opposition to the enfranchisement of woman, men of even the most liberal tendencies declare that her political freedom will be used to sustain the church, apparently forgetting that man alone has placed the church in power, and that man alone holds it in power. And proof of man's complicity is even greater than this. Despite what is said of the larger church membership of women, the most noted modern evangelist, Moody, recently declared that he \"found men tenfold, aye, an hundredfold\" more receptive of his preaching than women. While speaking in Farwell Hall, Chicago, 1886, he said, \"For fifteen years I have preached to women in the afternoon and very often as near as I could, have preached the same sermon to men at night, and in ninety-nine cases out of a hundred have had five times more result in preaching to men than to women.\" This pseudo-argument, as to woman's susceptibility to church teaching, brought up by the enemies of her freedom, possesses no more real value than the pseudo-political argument sometimes presented in opposition to woman's admission into active politics; that is, her emotional temperament. To one who has been present at four great presidential nominating conventions and several large state conventions, knowledge upon this point is practical. When one has seen a cordon of police, enforced by the mayor upon the platform, protecting the officers of such a convention, while its members, standing upon seats, stamped, shouted, gesticulated, threatened with revolvers, acting more like uncaged wild beasts than like men, when one has witnessed the wildest enthusiasm at the mention of a name, the waving of flags, of hats, of handkerchiefs, the shaking of umbrellas, chairs, canes, with violent stamping, amid a hubbub of indistinguishable voices, all shouting, screaming so loud that people for blocks away are roused from slumber in affright of a fire or the approach of an ungovernable mob, such objections to woman's freedom as her \"emotions\" fall to their lowest value.\n\nIn church and in state, man has exhibited the wildest passions, the most ungovernable frenzy\u2014has shown himself less controlled by reason than possible for woman under the most adverse circumstances. Judaism, and its offspring, Christianity, show the results of the patriarchate in some of its most degenerate forms; industrial servitude, educational restrictions, legal thralldom, political slavery, false religious teachings are but a portion of the evils existing under its most enlightened forms, and equally with the more pronounced polygamy and infanticide they show a total perversion of moral ideas. Woman dearly pays for the rights she has secured. Labor opposes, in less pay for the same work; literature, at first welcoming her only through the cookbook, next compelled her to conceal her sex under a male pseudonym, in order that her writings might be received with the same respect as those of man; art has given her similar experiences, and while today admitting her to the same advantage of study with man, yet compels her to pay twice the price for the same instructions.\n\nThe careful student of history will discover that Christianity has been of very little value in advancing civilization but has done a great deal toward retarding it. \"Civilization, a recognition of the rights of others at every point of contact,\" has been carried forward by means of rebellion against church teaching and church authority. The experience of science is familiar to all, even school children quoting Galileo and Dr. Faust. What are called reformations in religion, the work of Huss, of Luther, of the Waldenses, the Huguenots, are equally familiar instances to the youngest student of rebellion against the church. These and a myriad of others known to the historian have all been brought about by refusal to accept the authority of the church as final. The Peasant War in France, the struggles of Wat Tyler and of Hampden in England, the French and the American Revolutions looking toward equality of rights, and a thousand minor forms of political progress have all been opposed by the church as rebellions against its teachings, yet all have been marked steps in civilization. The church and civilization are antipodal; one means authority, the other freedom; one means conservatism, the other progress; one means the rights of God as interpreted by the priesthood, the other the rights of humanity as interpreted by humanity. Civilization advances by free thought, free speech, free men. The uprising of the women of all peoples in assertion of their common humanity with man is exemplification of that fact recognized in the Declaration of Independence, that while patient endurance of wrongs to which persons are accustomed, always long borne, rather than by change, perhaps to meet evils they know not of, shows its absolutely certain ultimate effect, no matter how long delayed, in rebellion. A time comes in the history of souls, as of nations, when forbearance ceases to be a virtue, and self-respecting life is only to be retained through defiance of and rebellion against existing customs. The soul must assert its own supremacy or die. It is not one woman or the women of one nation that have thus suddenly shown desire to rule themselves\u2014to act for themselves alone. A strange identity of thought pervades all parts of the world\u2014India, China, Japan, Russia and all of Europe, North and South America, the vast continents of the southern seas and the isles thereof, and even barbaric Africa, all evince proof of the wide psychic undercurrent which, seething through women's souls, is overthrowing the civilizations built upon the force principles of the patriarchate and will soon reinstate the reign of truth and justice. During those long ages of priestly intolerance, of domestic and governmental tyranny, in which woman seemed to accept the authority of the priest as that of God, there still existed a consciousness hardly perceptible to herself that she was an independent being to whom by virtue of her humanity all opportunities in life belonged. From century to century mothers transmitted this scarcely developed perception to daughters, until suddenly within the past fifty years these dominant ideas woke to thought, and the women of all nations began to proclaim their same right to self-control as that claimed by man.\n\nIt is impossible to write of the church without noticing its connection with the great systems of the world during its course of life. The history of Christendom is the history of the myriad institutions which have arisen through its teachings or that have been sustained by its approval. The world has not grown wise under it, except with a wisdom that is leading the purest humanitarian thought in a direction contrary to its footsteps. Slavery and prostitution, persecutions for heresy, the inquisition with its six hundred modes of torture, the destruction of learning, the oppression of science, the systematized betrayal of confiding innocence, the recognized and unrecognized polygamy of man, the denial to woman of a right to herself, her thought, her wages, her children, to a share in the government which rules her, to an equal part in religious institutions, all these and a myriad more are parts of what is known as Christian civilization. Nor has the church ever been the leader in great reforms. During the anti-slavery conflict, the American church was known as \"the bulwark of American slavery.\" Its course continues the same in every great contest with wrong. A memorial history of the American Episcopal Church, an extensive work in two volumes of seven hundred pages each published within the past few years devotes but seven pages to the \"Attitude of the Church during the Civil War,\" and the general refusal of the church to take part in the great struggle for national life is referred to with complacent satisfaction. Penitentiaries and prisons, asylums and reformatories, all institutions of a repressive character which the church prides herself as having built up are no less evil than the convents, monasteries, and religious orders belonging to it. They have all risen through perversion of nature. Crimes and criminals are built up and born because of the great wrong first done to mothers; they are the offspring of church and state. Science now declares crime to be a disease, but it has not yet discovered the primal cause of this disease. It is an inheritance from centuries of legalized crime against woman, of which the church in its teachings is prime factor.\n\nWoman will gain nothing by a compromising attitude toward the church, by attempt to excuse its great wrong toward her sex, or by palliation of its motives. On the contrary, a stern reference to facts, keeping the face of the world turned toward its past teachings, its present attitude, is her duty. Wrongs of omission equal in magnitude those of commission.\n\nAdvance for woman is too well-established, woman has had too much experience, has borne too much ridicule, misrepresentation, and abuse to now hesitate in an attack upon the stronghold of her oppression\u2014the church. She possesses too full knowledge of its subtle touch upon civil law to dare leave it alone; it has become one of woman's first duties, one of her greatest responsibilities, to call public attention to its false doctrines and false teachings in regard to the origin, condition, and subjection of woman. She has engaged in too many battles, weathered too many storms, to longer hesitate in exposure of its stupendous crimes toward one half of humanity. Let those who fear hide themselves, if they will, until the storm is past. Let those who dare, defiantly rejoice that they are called upon to bear still more, in order that woman may be free. A brighter day is to come for the world, a day when the intuitions of woman's soul shall be accepted as part of humanity's spiritual wealth; when force shall step backward and love, in reality, rule the teachings of religion; and may woman be strong in the ability and courage necessary to bring about this millennial time. The world is full of signs of the near approach of this period; as never before is there an arousing sense of something deeper, holier in religion than the Christian church has given. The world has seemingly awaited the advent of heroic souls who once again should dare all things for the truth. The woman who possesses love for her sex, for the world, for truth, justice, and right, will not hesitate to place herself upon record as opposed to falsehood, no matter under what guise of age or holiness it appears. A generation has passed since the great struggle began, but not until within ten years has woman dared attack upon the veriest stronghold of her oppression, the church. The state, agent and slave of the church, has so long united with it in suppression of woman's intelligence, has so long preached of power to man alone, that it has created an inherited tendency, an inborn line of thought toward repression. Bent in this line before his birth, man still unwittingly thinks of woman as not quite his equal, and it requires a new creation of mind to change his thought. A second generation has arisen, in whom some slight inherited tendencies toward recognition of a woman's right to herself are seen. In the next generation this line of inherited thought will have become stronger, both church and state more fully recognizing woman's inherent right to share in all the opportunities of life; but at what cost to all who have taken part in the great struggle.\n\nHas woman no wrongs to avenge upon the church? As I look backward through history I see the church everywhere stepping upon advancing civilization, hurling woman from the plane of \"natural rights\" where the fact of her humanity had placed her, and through itself and its control over the state, in the doctrine of \"revealed rights,\" everywhere teaching an inferiority of sex; a created subordination of woman to man; making her very existence a sin; holding her accountable to a diverse code of morals from man; declaring her possessed of fewer rights in church and in state; her very entrance into heaven made dependent upon some man to come as mediator between her and the Savior it has preached, thus crushing her personal, intellectual, and spiritual freedom. Looking forward, I see evidence of a conflict more severe than any yet fought by reformation or science; a conflict that will shake the foundations of religious belief, tear into fragments and scatter to the winds the old dogmas upon which all forms of Christianity are based. It will not be the conflict of man with man upon rites and systems; it will not be the conflict of science upon church theories regarding creation and eternity; it will not be the light of biology illuminating the hypothesis of the resurrection of the body; but it will be the rebellion of one half of the church against those theological dogmas upon which the very existence of the church is based. In no other country has the conflict between natural and revealed rights been as pronounced as in the United States; and in this country where the conflict first began, we shall see its full and final development. During the ages, no rebellion has been of like importance with that of woman against the tyranny of church and state; none has had its far-reaching effects. We note its beginning; its progress will overthrow every existing form of these institutions; its end will be a regenerated world.\n\n# CHAPTER III\n\n# The Devil's Dictionary (excerpts)\n\n# by Ambrose Bierce\n\n_and_\n\n# American Heretic's Dictionary (excerpts)\n\n# by Chaz Bufe\n\n# The Devil's Dictionary\n\n**Adore** , v.t. To venerate expectantly.\n\n**Altar** , n. The place whereon the priest formerly raveled out the small intestine of the sacrificial victim for purposes of divination and cooked its flesh for the gods. The word is now seldom used, except with reference to the sacrifice of their liberty and peace by a male and a female fool.\n\n**Christian** , n. One who believes that the New Testament is a divinely inspired book admirably suited to the spiritual needs of his neighbor. One who follows the teachings of Christ in so far as they are not inconsistent with a life of sin.\n\n**Divination** , n. The art of nosing out the occult. Divination is of as many kinds as there are fruit-bearing varieties of the flowering dunce and the early fool.\n\n**Evangelist** , n. A bearer of good tidings, particularly (in a religious sense) such as assure us of our own salvation and the damnation of our neighbors.\n\n**Faith** , n. Belief without evidence in what is told by one who speaks without knowledge of things without parallel.\n\n**Heathen** , n. A benighted creature who has the folly to worship something that he can see and feel.\n\n**Immoral** , adj. Inexpedient.\n\n**Indigestion** , n. A disease which the patient and his friends frequently mistake for deep religious conviction and concern for the salvation of mankind. As the simple Red Man of the western wild put it, with, it must be confessed, a certain force: \"Plenty well, no pray; big bellyache, heap God.\"\n\n**Infidel** , n. In New York, one who does not believe in the Christian religion; in Constantinople, one who does.\n\n**Koran** , n. A book which the Mohammedans foolishly believe to have been written by divine inspiration, but which Christians know to be a wicked imposture, contradictory to Holy Scriptures.\n\n**Magic** , n. The art of converting superstition into coin. There are other arts serving the same high purpose, but the discreet lexicographer does not name them.\n\n**Orthodox** , n. An ox wearing the popular religious yoke.\n\n**Pantheism** , n. The doctrine that everything is God, in contradistinction to the doctrine that God is everything.\n\n**Pray** , v. To ask that the laws of the universe be annulled in behalf of a single petitioner confessedly unworthy.\n\n**Rack** , n. An argumentative implement formerly much used in persuading devotees of a false faith to embrace the living truth.\n\n**Religion** , n. A daughter of Hope and Fear, explaining to Ignorance the nature of the Unknowable.\n\n**Reprobation** , n. In theology, the state of a luckless mortal prenatally damned. The doctrine of reprobation was taught by Calvin, whose joy in it was somewhat marred by the sad sincerity of his conviction that although some are foredoomed to perdition, others are predestined to salvation.\n\n**Revelation** , n. A famous book in which St. John the Divine concealed all that he knew.\n\n**Reverence** , n. The spiritual attitude of a man to a god and a dog to a man.\n\n**Satan** , n. One of the Creator's lamentable mistakes, repented in sash-cloth and axes. Being instated as an archangel, Satan made himself multifariously objectionable and was finally expelled from heaven. Halfway in his descent he paused, bent his head in thought a moment, and at last went back. \"There is one favor that I should like to ask,\" said he.\n\n\"Name it.\"\n\n\"Man, I understand, is about to be created. He will need laws.\"\n\n\"What, wretch! You his appointed adversary, charged from the dawn of eternity with hatred of his soul\u2014you ask for the right to make his laws?\"\n\n\"Pardon; what I have to ask is that he be permitted to make them himself.\"\n\nIt was so ordered.\n\n**Scriptures** , n. The sacred books of our holy religion, as distinguished from the false and profane writings on which all other faiths are based.\n\n**Universalist** , n. One who foregoes the advantage of hell for persons of another faith.\n\n**Virtues** , n. pl. Certain abstentions.\n\n**Worship** , n. Homo Creator's testimony to the sound construction and finish of Deus Creatus. A popular form of abjection having an element of pride.\n\n**Zeal** , n. A certain nervous disorder afflicting the young and inexperienced. A passion that goeth before a sprawl.\n\n# American Heretic's Dictionary\n\n**Atheist** , n. A person to be pitied in that he is unable to believe in things for which there is no evidence, and who has thus deprived himself of a convenient means of feeling superior to others.\n\n**Believer** , n. One who wisely chooses to avoid unnecessary work, especially that of thinking for himself.\n\n**Catholicism** , n. A popular form of self-degradation involving ritual cannibalism.\n\n**Christian** , n. One who generously seeks to transfer his expertise in morality into provisions in the penal code.\n\n**Compassion** , n. Mercy. A virtue so rare in Christian lands that when discovered by chance it is always deemed newsworthy.\n\n**Cruelty** , n. The most important and highly valued element in entertainment in Christian and Muslim countries, especially in entertainment provided by the legal and penal systems.\n\n**Death** , n. For Christians, a blessing\u2014the gateway to heaven, the portal to paradise. It speaks volumes of the generosity of Christians that they so freely bestow this blessing upon their enemies, yet routinely do all in their power, even in extreme old age, to deny this same blessing to themselves.\n\n**Faith** , n. An attribute of desperation. An attempt to make the intolerable tolerable. When achieved, it gives its holders a satisfying feeling of superiority over those so unfortunate as to see things as they are.\n\n**Fundamentalist** , n. One in whom something is fundamentally wrong.\n\n**God** , n. 1) An unsavory character found in many popular works of fiction; 2) A three-letter justification for murder.\n\n**Hell** , n. A place of everlasting torment, much like the United States during an election year.\n\n**Islamophobia** , n. 1) The unspeakable act of pointing out that acts committed by Islamists in the name of Islam just might have something to do with Islam; 2) The ever popular American mass participation sport of scapegoating and further victimizing an already victimized minority. While both definitions are accurate, those who accept one will never accept the other. This provides an example of yet another popular American mass participation sport: rejection of reality.\n\n**Jesus Christ** , int. A common exclamation indicating surprise, anger, disgust, or bewilderment.\n\n**Life** , n. 1) A series of minor annoyances and missed opportunities punctuated by blinding pain; 2) According to pro-lifers, the most beautiful, precious thing in the world, from the moment of conception to the moment of execution.\n\n**Moral** , adj. 1) In the conventional religious sense, anything tending to increase human misery, as in \"moral conduct\" and \"moral standards\"; 2) A term of self-approbation used by those who enjoy locking other human beings in cages.\n\n**Mystic** , n. A man or woman who wishes to understand the mysteries of the universe but is too lazy to study physics.\n\n**Pascal's Wager** , n. A cunning stratagem based on the premise that God's last name is \"Ed,\" his first name is \"Special,\" and that he can't tell the difference between lip service and belief.\n\n**Piety** , n. An uncommonly strong fear of death.\n\n**Prayer** , n. 1) A form of begging, unusual in that it's often practiced as a solitary activity. When practiced in groups, it is normally referred to as \"worship\"; 2) A humble supplication, the premise of which is that the omnipotent, omniscient creator of the universe will benefit from the guidance of the humble supplicator.\n\n**Priest** , n. A holy individual who follows\u2014often to excess\u2014the divine injunction to \"love the little children.\"\n\n**Rapture** , n. An eagerly awaited religious event in which the world will be rid of much unwanted rubbish.\n\n**Religious Extremist** , n. A term used by religious liberals to condemn anyone who shows the poor judgment to follow to the letter the inerrant word of God.\n\n**Spirituality** , n. 1) A meaningless but uplifting term of self-congratulation often found in dating site profiles.\n\n**Televangelist** , n. Ideally suited to his profession, the televangelist makes a career of denouncing greed, gluttony, dishonesty, drunkenness, drug abuse, prostitution, adultery, fornication, and homosexuality\u2014and thus speaks from a wealth of personal experience.\n\n**Ten Commandments** , n. pl. The fundamental moral precepts of the Christian and Jewish faiths and a fine set of guiding lights for us all. Some low-minded skeptics have suggested that God could have improved upon these inspired moral principles if it had dropped the commandments concerning swearing, idol worship, sexual exclusivity, and resting on the sabbath, and had instead instituted bans on slavery, torture, and cruelty, but these suggestions are obviously sacrilegious. If slavery, torture, and cruelty were true moral evils, organized Christianity would not have condoned, instigated, and practiced all three for the better part of two millennia. Fortunately, most Christians realize this and do not bother themselves about such trifles. Instead, they rightly concentrate their moral outrage on the true evils of \"obscene\" books and pictures, \"filthy\" language, and the sexual practices of their neighbors.\n\n**Witch** , n. To fundamentalist Christians, a burning curiosity\u2014and a curiosity which many of them, even at this late date, hope to reignite.\n\n# CHAPTER IV\n\n# The Failure of Christianity\n\n# by Emma Goldman\n\nThe counterfeiters and poisoners of ideas, in their attempt to obscure the line between truth and falsehood, find a valuable ally in the conservatism of language.\n\nConceptions and words that have long ago lost their original meaning continue through centuries to dominate mankind. Especially is this true if these conceptions have become a commonplace, if they have been instilled in our beings from our infancy as great and irrefutable verities. The average mind is easily content with inherited and acquired things or with the dicta of parents and teachers, because it is much easier to imitate than to create.\n\nOur age has given birth to two intellectual giants who have undertaken to transvalue the dead social and moral values of the past, especially those contained in Christianity. Friedrich Nietzsche and Max Stirner have hurled blow upon blow against the portals of Christianity, because they saw in it a pernicious slave morality, the denial of life, the destroyer of all the elements that make for strength and character. True, Nietzsche has opposed the slave morality idea inherent in Christianity in behalf of a master morality for the privileged few. But I venture to suggest that his master idea had nothing to do with the vulgarity of station, caste, or wealth. Rather did it mean the masterful in human possibilities, the masterful in man that would help him to overcome old traditions and worn-out values, so that he may learn to become the creator of new and beautiful things.\n\nBoth Nietzsche and Stirner saw in Christianity the leveler of the human race, the breaker of man's will to dare and to do. They saw in every movement built on Christian morality and ethics attempts not at the emancipation from slavery but for the perpetuation thereof. Hence, they opposed these movements with might and main.\n\nWhether I do or do not entirely agree with these iconoclasts, I believe, with them, that Christianity is most admirably adapted to the training of slaves, to the perpetuation of a slave society; in short, to the very conditions confronting us today. Indeed, never could society have degenerated to its present appalling stage, if not for the assistance of Christianity. The rulers of the earth have realized long ago what potent poison inheres in the Christian religion. That is the reason they foster it; that is why they leave nothing undone to instill it into the blood of the people. They know only too well that the subtleness of the Christian teachings is a more powerful protection against rebellion and discontent than the club or the gun.\n\nNo doubt I will be told that, though religion is a poison and institutionalized Christianity the greatest enemy of progress and freedom, there is some good in Christianity \"itself.\" What about the teachings of Christ and early Christianity, I may be asked, do they not stand for the spirit of humanity, for right and justice?\n\nIt is precisely this oft-repeated contention that induced me to choose this subject, to enable me to demonstrate that the abuses of Christianity, like the abuses of government, are conditioned in the thing itself and are not to be charged to the representatives of the creed. Christ and his teachings are the embodiment of submission, of inertia, of the denial of life; hence, responsible for the things done in their name.\n\nI am not interested in the theological Christ. Brilliant minds like Bauer, Strauss, Renan, Thomas Paine, and others refuted that myth long ago. I am even ready to admit that the theological Christ is not half so dangerous as the ethical and social Christ. In proportion as science takes the place of blind faith, theology loses its hold. But the ethical and poetical Christ myth has so thoroughly saturated our lives that even some of the most advanced minds find it difficult to emancipate themselves from its yoke. They have rid themselves of the letter but have retained the spirit; yet it is the spirit which is back of all the crimes and horrors committed by orthodox Christianity. The Fathers of the Church can well afford to preach the gospel of Christ. It contains nothing dangerous to the regime of authority and wealth; it stands for self-denial and self-abnegation, for penance and regret, and is absolutely inert in the face of every [in]dignity, every outrage imposed upon mankind.\n\nHere I must revert to the counterfeiters of ideas and words. So many otherwise earnest haters of slavery and injustice confuse, in a most distressing manner, the teachings of Christ with the great struggles for social and economic emancipation. The two are irrevocably and forever opposed to each other. The one necessitates courage, daring, defiance, and strength. The other preaches the gospel of non-resistance, of slavish acquiescence in the will of others; it is the complete disregard of character and self-reliance and, therefore, destructive of liberty and well-being.\n\nWhoever sincerely aims at a radical change in society, whoever strives to free humanity from the scourge of dependence and misery, must turn his back on Christianity, on the old as well as the present form of the same.\n\nEverywhere and always, since its very inception, Christianity has turned the earth into a vale of tears; always it has made of life a weak, diseased thing, always it has instilled fear in man, turning him into a dual being, whose life energies are spent in the struggle between body and soul. In decrying the body as something evil, the flesh as the tempter to everything that is sinful, man has mutilated his being in the vain attempt to keep his soul pure, while his body rotted away from the injuries and tortures inflicted upon it.\n\nThe Christian religion and morality extols the glory of the hereafter, and therefore remains indifferent to the horrors of the earth. Indeed, the idea of self-denial and of all that makes for pain and sorrow is its test of human worth, its passport to the entry into heaven.\n\nThe poor are to own heaven, and the rich will go to hell. That may account for the desperate efforts of the rich to make hay while the sun shines, to get as much out of the earth as they can: to wallow in wealth and superfluity, to tighten their iron hold on the blessed slaves, to rob them of their birthright, to degrade and outrage them every minute of the day. Who can blame the rich if they revenge themselves on the poor, for now is their time, and the merciful Christian God alone knows how ably and completely the rich are doing it.\n\nAnd the poor? They cling to the promise of the Christian heaven as the home for old age, the sanitarium for crippled bodies and weak minds. They endure and submit, they suffer and wait, until every bit of self-respect has been knocked out of them, until their bodies become emaciated and withered, and their spirit broken from the wait, the weary endless wait for the Christian heaven.\n\nChrist made his appearance as the leader of the people, the redeemer of the Jews from Roman dominion; but the moment he began his work, he proved that he had no interest in the earth, in the pressing immediate needs of the poor and the disinherited of his time. what he preached was a sentimental mysticism, obscure and confused ideas lacking originality and vigor.\n\nWhen the Jews, according to the gospels, withdrew from Jesus, when they turned him over to the cross, they may have been bitterly disappointed in him who promised them so much and gave them so little. He promised joy and bliss in another world, while the people were starving, suffering, and enduring before his very eyes.\n\nIt may also be that the sympathy of the Romans, especially of Pilate, was given Christ because they regarded him as perfectly harmless to their power and sway. The philosopher Pilate may have considered Christ's \"eternal truths\" as pretty anemic and lifeless, compared with the array of strength and force they attempted to combat. The Romans, strong and unflinching as they were, must have laughed in their sleeves over the man who talked repentance and patience, instead of calling to arms against the despoilers and oppressors of his people.\n\nThe public career of Christ begins with the edict, \"Repent, for the Kingdom of Heaven is at hand.\"\n\nWhy repent, why regret, in the face of something that was supposed to bring deliverance? Had not the people suffered and endured enough; had they not earned their right to deliverance by their suffering? Take the Sermon on the Mount, for instance. What is it but a eulogy on submission to fate, to the inevitability of things?\n\n\"Blessed are the poor in spirit, for theirs is the Kingdom of Heaven.\"\n\nHeaven must be an awfully dull place if the poor in spirit live there. How can anything creative, anything vital, useful, and beautiful come from the poor in spirit? The idea conveyed in the Sermon on the Mount is the greatest indictment against the teachings of Christ, because it sees in the poverty of mind and body a virtue, and because it seeks to maintain this virtue by reward and punishment. Every intelligent being realizes that our worst curse is the poverty of the spirit; that it is productive of all evil and misery, of all the injustice and crimes in the world. Everyone knows that nothing good ever came or can come of the poor in spirit; surely never liberty, justice, or equality.\n\n\"Blessed are the meek, for they shall inherit the earth.\"\n\nWhat a preposterous notion! What incentive to slavery, inactivity, and parasitism! Besides, it is not true that the meek can inherit anything. Just because humanity has been meek, the earth has been stolen from it.\n\nMeekness has been the whip which capitalism and governments have used to force man into dependency, into his slave position. The most faithful servants of the state, of wealth, of special privilege, could not preach a more convenient gospel than did Christ, the \"redeemer\" of the people.\n\n\"Blessed are they that hunger and thirst for righteousness, for they shall be filled.\"\n\nBut did not Christ exclude the possibility of righteousness when he said, \"The poor ye have always with you\"? But, then, Christ was great on dicta, no matter if they were utterly opposed to each other. This is nowhere demonstrated so strikingly as in his command, \"Render to Caesar the things that are Caesar's, and to God the things that are God's.\"\n\nThe interpreters claim that Christ had to make these concessions to the powers of his time. If that be true, this single compromise was sufficient to prove, down to this very day, a most ruthless weapon in the hands of the oppressor, a fearful lash and relentless tax gatherer, to the impoverishment, the enslavement, and degradation of the very people for whom Christ is supposed to have died. And when we are assured that \"Blessed are they that hunger and thirst for righteousness, for they shall be filled,\" are we told the how? How? Christ never takes the trouble to explain that. Righteousness does not come from the stars nor because Christ willed it so. Righteousness grows out of liberty, of social and economic opportunity and equality. But how can the meek, the poor in spirit, ever establish such a state of affairs?\n\n\"Blessed are ye when men shall revile you and persecute you, and say all manner of evil against you falsely, for my sake. Rejoice, and be exceeding glad: for great is your reward in heaven.\"\n\nThe reward in heaven is the perpetual bait, a bait that has caught man in an iron net, a straitjacket which does not let him expand or grow. All pioneers of truth have been and still are reviled; they have been and still are persecuted. But did they ask humanity to pay the price? Did they seek to bribe mankind to accept their ideas? They knew too well that he who accepts a truth because of the bribe will soon barter it away to a higher bidder.\n\nGood and bad, punishment and reward, sin and penance, heaven and hell, as the moving spirit of the Christ gospel, have been the stumbling block in the world's work. It contains everything in the way of orders and commands but entirely lacks the very things we need most.\n\nThe worker who knows the cause of his misery, who understands the makeup of our iniquitous social and industrial system, can do more for himself and his kind than Christ and the followers of Christ have ever done for humanity; certainly more than meek patience, ignorance, and submission have done.\n\nHow much more ennobling, how much more beneficial is the extreme individualism of Stirner and Nietzsche than the sickroom atmosphere of the Christian faith. If they repudiate altruism as an evil, it is because of the example contained in Christianity, which set a premium on parasitism and inertia, gave birth to all manner of social disorders that are to be cured with the preachment of love and sympathy.\n\nProud and self-reliant characters prefer hatred to such sickening artificial love. Not because of any reward does a free spirit take his stand for a great truth nor has such a one ever been deterred because of fear of punishment.\n\n\"Think not that I come to destroy the law or the prophets. I am not come to destroy, but to fulfill.\"\n\nPrecisely. Christ was a reformer, ever ready to patch up, to fulfill, to carry on the old order of things; never to destroy and rebuild. That may account for the fellow feeling all reformers have for him.\n\nIndeed, the whole history of the state, capitalism, and the Church proves that they have perpetuated themselves because of the idea \"I come not to destroy the law.\" This is the key to authority and oppression. Naturally so, for did not Christ praise poverty as a virtue; did he not propagate non-resistance to evil? Why should not poverty and evil continue to rule the world?\n\nMuch as I am opposed to every religion, much as I think them an imposition upon and crime against reason and progress, I yet feel that no other religion has done so much harm or has helped so much in the enslavement of man as the religion of Christ.\n\nWitness Christ before his accusers. What lack of dignity, what lack of faith in himself and in his own ideas! So weak and helpless was this \"Savior of Men\" that he must needs the whole human family to pay for him unto all eternity, because he \"hath died for them.\" Redemption through the Cross is worse than damnation, because of the terrible burden it imposes upon humanity, because of the effect it has on the human soul, fettering and paralyzing it with the weight of the burden exacted through the death of Christ.\n\nThousands of martyrs have perished, yet few, if any, of them have proved so helpless as the great Christian God. Thousands have gone to their death with greater fortitude, with more courage, with deeper faith in their ideas than the Nazarene. Nor did they expect eternal gratitude from their fellow men because of what they endured for them.\n\nCompared with Socrates and Bruno, with the great martyrs of Russia, with the Chicago anarchists, Francisco Ferrer, and unnumbered others, Christ cuts a poor figure indeed. Compared with the delicate, frail Spiridonova, who underwent the most terrible tortures, the most horrible indignities, without losing faith in herself or her cause, Jesus is a veritable nonentity. They stood their ground and faced their executioners with unflinching determination, and though they too died for the people, they asked nothing in return for their great sacrifice.\n\nVerily, we need redemption from the slavery, the deadening weakness, and humiliating dependency of Christian morality.\n\nThe teachings of Christ and of his followers have failed because they lacked the vitality to lift the burdens from the shoulders of the race; they have failed because the very essence of that doctrine is contrary to the spirit of life, exposed to the manifestations of nature, to the strength and beauty of passion.\n\nNever can Christianity, under whatever mask it may appear\u2014be it New Liberalism, Spiritualism, Christian Science, New Thought, or a thousand and one other forms of hysteria and neurasthenia\u2014bring us relief from the terrible pressure of conditions, the weight of poverty, the horrors of our iniquitous system. Christianity is the conspiracy of ignorance against reason, of darkness against light, of submission and slavery against independence and freedom; of the denial of strength and beauty against the affirmation of the joy and glory of life.\n\n# CHAPTER V\n\n# Twelve Proofs of the Nonexistence of God\n\n# by S\u00e9bastien Faure\n\nComrades,\n\nThere are two ways to examine and resolve the problem of the non-existence of God.\n\nThe first involves eliminating the hypothesis of God from the field of plausible or necessary conjectures, through a clear and precise analysis, through the presentation of a positive system of the universe, of its origins, successive developments, and ends.\n\nThis account would make the idea of God useless and would destroy in advance all the metaphysical scaffolding with which the spiritualist philosophers and the theological scholars have propped it up.\n\nNow, _in the present state of human knowledge_ \u2014limiting ourselves, as is proper, to what is demonstrated or demonstrable, verified or verifiable\u2014that analysis is missing and the positive system of the universe is lacking. Of course, there are some ingenious hypotheses, which in no way shock the reason. There exist some more or less plausible systems which rest on a mass of observations and draw from those abundant observations an impressive degree of probability. We can boldly assert that these systems and suppositions compare favorably with the assertions of the deists, but, in truth, they are only theories, still not possessing scientific certainty. Each individual remaining free, in the end, to give preference to any given system\u2014or to any other that is opposed to it\u2014the solution of the problem would appear, _at least_ from this point of view _and at present_ , necessarily elusive.\n\nThe followers of all the religions so clearly grasp the advantage conferred upon them by the study of the problem when posed in this way that they all constantly attempt to bring it back to that position; and if, even on that terrain\u2014the only terrain on which they can still maintain their composure\u2014they do not leave the encounter with the honors of battle\u2014not by a long shot\u2014it is still possible for them to perpetuate doubt in the minds of their coreligionists. And, for them, that is the essential point.\n\nIn this combat, where the two opposing theses grapple and strive to lay each other low, the deists receive some hard blows, but they also give them. For better or worse, they defend themselves and, the outcome of this duel remaining uncertain in the eyes of the crowd, the believers, even when they seem to have been defeated, can cry victory.\n\nThey do not hesitate to do so with that impudence that is the public sign of their piety; and that comedy manages to keep the vast majority of the flock under the crook of the pastor.\n\nThat is all these \"bad shepherds\" desire.\n\n# **The Problem Posed in Precise Terms**\n\nHowever, comrades, there is a second way of examining and attempting to resolve the problem of the non-existence of God.\n\nThat involves examining the existence of the God whom religions offer for our adoration.\n\nCan any sensible and thoughtful man be found who could accept the existence of this God\u2014of whom we speak as if he was not shrouded in any mystery, as if we were ignorant of nothing about him, as if we had penetrated all his thought, and as if we had received all his confidences: \"He has done this and done that, and then this, and then that. He has said this and that, and then again that. He has acted and spoken with this aim and for that reason. He desires this thing, but he forbids this other thing. He will reward these actions and punish those others. And he has done this and wants that because he is infinitely wise, infinitely just, infinitely powerful, infinitely good\"?\n\nJust think of it! Here is a God who makes himself known! He abandons the realm of the inaccessible, clears away the clouds that surround him, descends from the heights, converses with mortals, confides his thought to them, reveals his will to them, and charges some privileged few to spread his doctrine, to communicate his law, and\u2014let's be clear\u2014to represent him here below, with full powers to bind and release, in heaven and on the earth!\n\nThis God is not a God of force, intelligence, will, and energy, which, like everything that is composed of energy, will, intelligence, and force, can be in turn, according to the circumstance and consequently indifferently, good or bad, useful or harmful, just or unjust, merciful or cruel. This god is the God in whom everything is perfection and whose existence is and can only be compatible\u2014since he is perfectly just, wise, powerful, good, and merciful\u2014with a state of things of which he would be the author and through which he would affirm his infinite justice, his infinite wisdom, his infinite power, his infinite goodness, and his infinite mercy.\n\nYou will recognize this God. He is the one who is taught through the catechism to the children. He is the living and personal God, the one to whom we raise temples, towards whom prayer mounts, in honor of whom we make sacrifices, and the one whom all the clergy, all the priestly castes, claim to represent on the earth.\n\nHe is not an \"unknown,\" an enigmatic force, impenetrable power, incomprehensible intelligence, unknowable energy, or mysterious principle. He is not a hypothesis that the mind of man, in its continuing powerlessness to explain the how and why of things, is happy to employ. He is not the speculative God of the metaphysicians but a God abundantly described to us in luminous detail by his representatives.\n\nHe is, I repeat, the God of religion, and, since we are in France, the God of the religion that has dominated our history for fifteen centuries: the Christian religion.\n\nThis is that God that I deny, and it is only this God that I wish to discuss and this God that you must study, if you wish to draw from this lecture a positive profit and a practical result.\n\nWhat is this God?\n\nSince those who manage his business here below have been good enough to describe him to us in lavish detail, let us profit from the graciousness of his duly authorized representatives; let us examine him up close; let us go over him with a magnifying glass. In order to discuss him well, we must know him well.\n\nThis God is the one who, with a powerful and life giving gesture, has made all things from nothing, the one who has called the void to being, who has, by his will alone, substituted movement for inertia, universal life for universal death: _he is the Creator!_\n\nThis God is the one who, having accomplished this act of creation, far from returning to his age-old inaction and remaining indifferent to the thing created, concerns himself with his work, takes an interest in it, intervenes when he judges it appropriate, manages, administers, and governs it: _he is the Governor, or Providence_.\n\nThis God is the one who, as Supreme Tribunal, summons each of us after our deaths, judges us according to the acts of our lives, determines the balance of our good and bad actions, and pronounces, in the last resort and without appeal, the judgment that will make us, for all the centuries to come, the most fortunate or unfortunate of beings: _he is the Justice Bringer, or Magistrate_.\n\nIt goes without saying that this God possesses all attributes and that he does not just possess them to an exceptional degree; he possesses them all to an infinite degree.\n\nThus, he is not only just: he is infinite justice; he is not only good: he is infinite goodness; he is not only merciful: he is infinite mercy; he is not only powerful: he is infinite power; he is not only learned: he is infinite science.\n\nThis, I repeat, is the God whom I deny and whose impossibility I will demonstrate by twelve different proofs (although, strictly speaking, just one would suffice).\n\n# **Division of the Subject**\n\nHere is the order in which I will present my arguments. They form three groups: the first of these groups will most specifically apply to the Creator God and will include six arguments; the second will most specifically concern the Governor God, or Providence, and will encompass four arguments; finally, the third and last of these groups will focus on the Justice Bringer God, or Magistrate, and will be composed of two arguments. Thus: six arguments against the Creator God; four arguments against the Governor God; two arguments against the Justice Bringer God. That will make twelve proofs of the nonexistence of God. The plan of my demonstration being known to you, you can better and more easily follow its development.\n\n# **First Series of Arguments**\n\n_Against the Creator God_\n\n# First Argument\n\n_The Creative Act Is Inadmissible_\n\nWhat does it mean to create?\n\nWhat is creation?\n\nIs it to take scattered, separate, but existing materials, and then, making use of certain tested principles, applying certain known rules, to bring together, group, classify, combine, and modify these materials, in order to make something of them?\n\nNo! That is not creation. Examples: Can we say of a house that it has been created?\u2014No! It has been constructed. Can we say that a piece of furniture has been created?\u2014No! It has been manufactured. Can we say that a book has been created?\u2014No! It has been composed, printed.\n\nThus, to take existing materials and make something of them is not to create.\n\nWhat, then, is creation?\n\nTo create... My word! I am embarrassed attempting to explain the inexplicable, to define the indefinable. I will, nonetheless, attempt to make myself understood.\n\nTo create is to draw something out of nothing; it is to make something with nothing at all; it is to call the void to be something.\n\nNow, I imagine that not a single reasonable person will be found who could imagine or accept that something could be drawn from nothing, that it would be possible to make something with nothing.\n\nImagine a mathematician; choose the most experienced calculator and place before them a gigantic blackboard; ask them to draw on that blackboard zeros and more zeros; they will add and multiply in vain; they may engage in all the operations of mathematics, but they will never succeed in extracting a single unity from the accumulation of these zeros.\n\nWith nothing, we make nothing; with nothing, we can make nothing, and the famous aphorism of Lucretius, ex nihilo nihil, remains the expression of a self-evident certainty.\n\nThe act of creation is an act that is impossible to accept, an absurdity.\n\nTo create is thus a mystical, religious expression, which may possess some value in the eyes of persons happy to believe what they do not understand and for whom faith is that much more necessary the less they understand; but to create is an expression void of sense for every astute, attentive man, in the eyes of those for whom words only have value to the extent that they represent a reality or possibility.\n\nConsequently, the hypothesis of a truly creative Being is a hypothesis that reason rejects.\n\n_The Creator Being does not exist and cannot exist_.\n\n# Second Argument\n\n_The \"Pure Spirit\" Could Not Have Determined the Universe_\n\nTo the believers who, in defiance of all reason, persist in accepting the possibility of creation, I would say that it is, in any case, impossible to attribute that creation to their God.\n\nTheir God is _pure spirit_. And I say that the pure spirit, the immaterial, cannot have determined the universe, the material. Here is why: pure spirit is not separated from the universe by a difference in degree, in quantity, but by a difference in nature, in quality. So that the pure spirit is not and cannot be an amplification of the universe, any more than the universe is or can be a reduction of pure spirit. The difference here is not only a distinction but an opposition\u2014and an opposition of an essential, fundamental, irreducible, and absolute nature.\n\nBetween pure spirit and the universe, there is not just a gap, more or less wide and deep, which it would be, in a pinch, possible to fill or cross; there is a veritable abyss of such depth and expanse that whatever they might try, no one could fill or bridge it.\n\nAnd I challenge the most astute philosopher or the most consummate mathematician to build a bridge, to establish a relation of any kind (and particularly a relation as direct and close as that which links the cause to the effect) between pure spirit and the universe.\n\nPure spirit accepts no material alloy; it assumes no form, body, line, material, proportion, extent, duration, depth, surface, volume, color, tone, or density.\n\nNow, in the universe, everything is, on the contrary, form, body, line, material, proportion, extent, duration, depth, surface, volume, color, tone, and density.\n\nHow could we accept that the latter was determined by the former?\n\nIt is impossible.\n\nHaving reached this point in my demonstration, I firmly plant the following conclusion atop the two arguments that have preceded it: we have seen, first, that the hypothesis of a truly creative power is inadmissible; we have seen, secondly, that even if one persists in believing in that power, one could not accept that the essentially material universe had been produced by an essentially immaterial pure spirit; if, despite it all, you persist, believers, in maintaining that it is your God that has created the universe, the moment has come to ask you where, given your theory of God, we are to find matter at the origin, at the commencement.\n\nWell! One of two things must be true: either the matter was outside of God or else it was within God. (You could not assign it a third place.) In the first case, if it was outside of God, God had no need to create it, since it already existed; it coexisted with God, was concomitant with him, and, therefore, your God is not a creator; in the second case, if it was not outside of God, it was _within_ God; and, in this case, I conclude:\n\n1) That God is not pure spirit, since he bears within him a bit of matter\u2014and what a bit: the totality of the material worlds!\n\n2) That God, bearing matter within him, has not had to create it, since it existed; he had only to bring it forth; and, thus, the creation ceases to be an act of true creation and is reduced to an act of externalization.\n\n_In both cases, there is no creation_.\n\n# Third argument\n\n_The Perfect Cannot Produce the Imperfect_\n\nI am certain that if I asked a believer this question: \" _Can the imperfect produce the perfect?_ \" that believer would respond to me without the least hesitation and without fear of being mistaken: the imperfect cannot produce the perfect.\n\nNow I say: the perfect cannot produce the imperfect, and I maintain that my proposition possesses the same force and the same exactitude as the previous one, and for the same reasons.\n\nHere again, between the perfect and the imperfect there is not only a difference of degree, of quantity, but a difference of quality, of nature, an essential, fundamental, irreducible, absolute opposition.\n\nHere again, between the perfect and the imperfect, there is not only a ditch, more or less deep and wide, but an abyss so immense and so deep that nothing could cross over or fill it.\n\nThe perfect is the absolute; the imperfect is the relative. Compared to the perfect, which is everything, the relative, the contingent, is nothing. Compared with the perfect, the relative is without value; it does not exist, and it is not within the power of any mathematician or philosopher to establish a relation\u2014of any sort whatever\u2014between the relative and the absolute; a fortiori, this relation is impossible when it is a question of a relation as rigorous and precise as the one that must necessarily unite cause and effect.\n\nSo it is impossible the perfect could have produced the imperfect.\n\nThere exists, however, a direct, inevitable, and, in some sense, mathematical relation between the work and the one who is its author. The work is only as good as the worker, and the worker is only as good as the work. It is by the work that we recognize the worker, as it is by the fruit that we recognize the tree.\n\nIf I examine an essay that has been written badly\u2014where errors in French abound, where the sentences are constructed badly, where the style is poor and careless, where the ideas are scarce and banal, where the knowledge is inexact\u2014I could not think of attributing this page of bad French to a sculptor of phrases, to one of the masters of literature.\n\nIf I look at a poorly executed picture, where the lines are badly drawn, the rules of perspective and proportion violated, I would never think to attribute that rudimentary sketch to a teacher, to a master, to an artist. Without the least hesitation, I would say: it is the work of a student, of an apprentice, of a child. And I am certain of not having committed an error, so true is it that the work bears the mark of the worker, and that through the work we can appraise its author.\n\nNow, nature is beautiful; the universe is magnificent, and I passionately admire, as much as anyone, the incessant spectacle of splendor and magnificence that it offers us. However, as enthusiastic as I am about the beauties of nature and whatever homage I may render them, I cannot say that the universe is a work without defects, irreproachable and perfect. And no one would dare to maintain such an opinion.\n\nSo the universe is an imperfect work.\n\nConsequently, I say: between the work and its author there is always a strict, close, mathematical relation; now, the universe is an imperfect work; thus, the author of that work can only be imperfect.\n\nAs a result of this syllogism, the God of the believers is stamped with imperfection and, consequently, denied.\n\nI can then reason as follows: either it is not God who is the author of the universe (I express thus my conviction).\n\nOr else, if you persist in claiming that he is its author, the universe being an imperfect work, your God is himself imperfect.\n\nSyllogism or dilemma, the conclusion of the reasoning remains the same: _the perfect cannot produce the imperfect_.\n\n# Fourth Argument\n\n_The Eternal, Active, Necessary Being Cannot, at Any Moment, Have Been Inactive or Useless_\n\nIf God exists, he is eternal, active, and necessary.\n\n_Eternal?_ He is eternal by definition. It is his reason for being. We cannot conceive of him confined within the limits of time; we cannot imagine him beginning or ending; he can neither appear nor disappear. He exists at all times.\n\n_Active?_ He is and cannot not be active, since it is his activity that has produced everything, since his activity is affirmed, say the believers, by the most colossal, the most majestic, of acts: the creation of worlds.\n\n_Necessary?_ He is and cannot not be necessary, since without him nothing would be; since he is the author of all things; since he is the original source from which everything has flowed; since, alone, sufficient unto himself, it is his will alone that has determined whether there is something or nothing. He is thus eternal, active, and necessary.\n\nI maintain and I will demonstrate that if he is eternal, active, and necessary, he must be eternally active and eternally necessary; that, as a consequence, he cannot, at any moment, be inactive or useless; that, as a further consequence, he has never created.\n\nTo say that God is not eternally active is to admit that he has not always been active, that he has become active, that he has commenced to be active, that before being active, he was not; and since it is through creation that his activity is manifested, it is to admit at the same time that during the billions and billions of centuries that perhaps preceded the creative action, God was inactive.\n\nTo say that God is not eternally necessary is to admit that he has not always been necessary, that he has become necessary, that he has commenced to be necessary, that before being necessary, he was not, and since it is the creation that proclaims and attests to the necessity of God, it is to admit at the same time that during the billions and billions of centuries that perhaps preceded the creative action, God was useless.\n\nGod idle and lazy!\n\nGod useless and superfluous!\n\nWhat a position for the essentially active and essentially necessary Being!\n\nSo it is necessary to confess that God is at all times active and necessary.\n\nBut then he cannot be responsible for the creation; for the idea of creation implies, absolutely, the idea of commencement, of origin. A thing that commences cannot have existed at all times. There was necessarily a time when, before being, it did not yet exist. As short or as long as that time may have been preceding the thing created, nothing can eliminate it; in any case, it exists.\n\nIt follows that God is not eternally active and eternally necessary; and, in this case, he has become active and necessary through creation. If this is the case, then before the creation, God lacked the two attributes of activity and necessity. This God was incomplete; he was a portion of God and no more; and he needed to create in order to become active and necessary, in order to complete himself.\n\nOr else God is eternally active and necessary; and, in this case, he has created eternally; the creation is eternal; the universe has never commenced; it has existed at all time; it is eternal, like God; it is God himself and becomes mixed up with him.\n\nIn this case, the universe has had no commencement; it has not been created.\n\nThus: in the first case, God, before the creation, was neither active nor necessary, was incomplete, which is to say imperfect; and, thus, he did not exist; in the second case, God being eternally active and eternally necessary, he cannot have become active and necessary; and, thus, he has not created.\n\nThere is no escape from that.\n\n# Fifth Argument\n\n_The Immutable Being Cannot Have Created_\n\nIf God exists, he is immutable. He does not change; he cannot change. While, in nature, everything changes, is metamorphosed and transformed; while nothing is once and for all, and everything becomes, God, a fixed point, immobile in time and space, is subject to no modification, does not know and could not know any change.\n\nHe is today as he was yesterday; he will be tomorrow as he was today. If we envisage God in the distance of the centuries passed or in that of the centuries to come, he is constantly identical to himself.\n\nGod is immutable.\n\nI maintain that if he has created he is not immutable, because, in this case, he has changed twice.\n\nTo make up one's mind to want is to change. Quite obviously there is a change between the being who does not yet desire and the being who wants.\n\nIf I desire today what I did not want, what I had not even dreamed of forty-eight hours ago, it is because there have been produced in me or around me one or more circumstances that have determined my desire. This new desire constitutes a modification: there is no room to doubt it: it is indisputable.\n\nSimilarly: to decide to act or to act is to be change.\n\nIt is, furthermore, certain that this double modification\u2014to want and to act\u2014is that much more considerable and marked, as it is a question of a more serious resolution and a more important action.\n\nGod has created, you say?\u2014So be it. Then he has changed twice: the first time, when he resolved to create; the second time, when, executing that determination, he accomplished the act of creation.\n\nIf he has changed twice, he is not immutable.\n\nAnd if he is not immutable, he is not God; he does not exist.\n\n_The immutable Being cannot have created_.\n\n# Sixth Argument\n\n_God Cannot Have Created without Motive; Now, It Is Impossible to Discern a Single One_\n\nHowever we consider it, the creation remains inexplicable, enigmatic, void of sense.\n\nIt is blindingly obvious that if God has created, it is impossible to accept that he has accomplished this grandiose act, the consequences of which must inevitably be proportionate to the act itself and, consequently, incalculable, without being set on that course by a motive of the highest importance.\n\nWell! What can that motive be? For what reason has God been able to resolve to create? What cause has impelled him? What desire has gripped him? What plan has he formed? What aim has he pursued? What end has he proposed?\n\nMultiply, in that order of ideas, questions upon questions: turn the problem over and over; consider it from all sides; examine it in every way\u2014and I challenge you to resolve it, other than by quips or quibbles.\n\nLook: here is a child raised in the Christian religion. His catechism tells him, his teachers teach him, that it is God who has created him and put him in the world. Suppose that he asks himself this question: Why has God created me and put me in the world? And suppose that he wishes to find a serious, reasonable response. He will not succeed. Now suppose that, confident in the experience and knowledge of his educators, persuaded that, through the sacred character with which they, priests or pastors, are clothed, they possess special insights and particular graces, convinced that, through their sanctity, they are closer to God than him and more fully initiated into revealed truths, suppose that this child was curious enough to ask his masters why God had created him and put him in the world\u2014I maintain that they could make no plausible, sensible response to that simple question.\n\nIn truth, there is none.\n\nLet us examine the question up close, look deeply into the problem.\n\nThrough thought, let us examine God before the creation. Let us take him in his absolute sense. He is all alone; he is self-sufficient. He is perfectly wise, perfectly happy, perfectly powerful. Nothing can increase his wisdom; nothing can augment his happiness; nothing can bolster his power.\n\nThis God can feel no desire, since his happiness is infinite; he can pursue no goal, since his perfection lacks nothing; he can form no plan, since nothing can extend his power; he cannot resolve to want anything, since he feels no needs.\n\nCome now! Profound philosophers, subtle thinkers, prestigious theologians respond to this child who questions you, asking why God has created him and placed him in the world.\n\nI am confident that you cannot respond, unless you say, \"The plans of God are impenetrable.\" And you cannot consider that response sufficient.\n\nAnd you would be wise to abstain from answering, for every response\u2014I warn you charitably\u2014would be the ruin of your system, the collapse of your God.\n\nThe conclusion, logical and pitiless, imposes itself: God, if he has created, has created without a motive, without knowing why, without an aim.\n\nDo you know, comrades, where we would inevitably be led by the consequences of such a conclusion?\n\nYou will see.\n\nWhat differentiates the acts of a man endowed with reason from the acts of a man afflicted with madness, what makes the one responsible and the other not, is that a reasonable man always knows, or in any event can know, the motives that have motivated him when he has acted, the reasons that have caused him to act. When it is a question of an important action, the consequences of which can seriously engage his sense of responsibility, it is enough for the man in possession of his reason to withdraw within himself, to give himself up to a serious, persistent, and impartial examination of conscience; it suffices for him to reconstruct, in memory, the framework within which events have enclosed him; in short, he need only relive the hour that has passed in order for him to discern the mechanism of the movements that have led him to act.\n\nHe is not always very proud of the reasons that have motivated him; he often blushes at the motive that have caused him to act; but whether these motives are noble or vile, generous or cheap, he always manages to uncover them.\n\nA lunatic, on the contrary, acts without knowing why. Question him when his act accomplished; interrogate him; press him with questions; insist; badger him. Even when the act is one most laden with consequences, the poor madman will stammer out some extravagance, and you will not draw him out of his delirium.\n\nThus, what differentiates the acts of a sensible man from one who is insane is that the acts of the first are explicable. They exist for a reason, and we can distinguish their cause and their aim, their origin and their end. The acts of a man deprived of reason are inexplicable, and he is himself incapable of discerning their cause and aim; they have no reason to exist.\n\nWell! If God has created without an aim, without motive, he has acted in the manner of a lunatic and _the creation would appear to be an act of insanity_.\n\n# **Two Crucial Objections**\n\nIn order to finish with the God of creation, it appears indispensable to examine two objections.\n\nYou can be certain that, here, objections abound, so when I speak of two objections to study, I talking about two crucial, conventional objections.\n\nThese two objections are that much more important since we can, with the tools of debate, reduce all the others to these.\n\n# First Objection\n\n_God Eludes You_\n\nSomeone says to me:\n\n\"You have no right to speak of God as you have. You present to us a caricature of God, systematically shrunken to the proportions that your understanding deigns to grant him. That is not our God. Our God could not be conceived by you, for he surpasses you; he slips from your grasp. Know that what would be fabulous for the most powerful of men, the men most endowed with strength, wisdom, and knowledge, is for God only child's play. Do not forget that humanity cannot move on the same plane as divinity. Do not lose sight of the fact that it is as impossible for man to understand the ways in which God operates as it is for minerals to imagine the modes of action of animals and for animals to understand the actions of men.\n\n\"God soars in heights that you can never attain; he occupies summits that remain inaccessible to you.\n\n\"Know that however magnificent a human intelligence may be, no matter the effort achieved by that intelligence, whatever the persistence of that effort, human intelligence can never raise itself up to God. Consider, finally, that, as vast as it might be, the brain of man is still finite and, consequently, cannot conceive of the infinite.\n\n\"So be faithful and modest enough to confess that it is not possible for you to understand or explain God. And what you can neither understand nor explain, you have no right to deny.\"\n\nAnd I respond to the deists:\n\nGentlemen, with your recommendations of faithfulness I am entirely disposed to comply. You remind me of the legitimate modesty that befits the humble mortal that I am. I have no desire to stray from it.\n\nYou say that God surpasses me, that he eludes me. So be it. I agree to recognize this fact and affirm that the finite can neither understand nor explain the infinite. This is a truth so certain, and even obvious, that I haven't the slightest urge to oppose it. We are, thus far, very much in agreement, and I hope you are quite happy.\n\nBut, gentlemen, allow me, in my turn, to urge the same faithfulness; permit me, in my turn, to recall you to the same modesty. Are not you men, as I am? Does not God surpass you, as he surpasses me? Does he not elude you, just as he eludes me? Would you pretend to move on the same plane as the divinity? Would you have the impudence to think and the foolishness to declare that with a stroke of your wings you have climbed to the summits that God occupies? Would you be presumptuous enough to say that your finite brain has embraced the infinite?\n\nI will not insult you, gentlemen, and believe you stricken with such extravagant vanity.\n\nSo, like me, be faithful and modest enough to confess that if it is impossible for me to understand and explain God, you meet with the same impossibility. Have the integrity to recognize that if I am not allowed to deny God, because I can neither understand nor explain him, you, who likewise are unable to understand or explain him, are not allowed to affirm his existence.\n\nAnd do not imagine, gentlemen, that we are, from now on, in the same boat. It is you who first affirmed the existence of God, so it is you who must first put an end to your affirmations. Would I have ever thought to deny God, if, when I was very small, belief in him had not been imposed on me? If, as an adult, I had not heard his existence declared all around me? If, having become a man, I had not constantly observed churches and temples raised to God?\n\nIt is your affirmations that provoke and justify my negations.\n\nStop affirming, and I will stop denying.\n\n# Second Objection\n\n_There Is No Effect without a Cause_\n\nThe second objection would appear as formidable. Many still consider it as without reply. It comes to us from the spiritualist philosophers.\n\nThese gentlemen say to us, sententiously: \"There is no effect without a cause; now, the universe is an effect; thus, that effect has a cause, which we call God.\"\n\nThe argument is well presented; it appears well constructed and seems solidly framed.\n\nThe whole question is to know if this is truly the case.\n\nThis form of reasoning is what is called, in logic, a syllogism. A syllogism is an argument composed of three propositions: the major, the minor, and the consequence; and including two parts: the premises, made up of the first two propositions, and the conclusion represented by the third.\n\nIn order for a syllogism to be unassailable, it is necessary: 1) that the major and minor propositions be exact; 2) that the third flow logically from the first two.\n\nIf the syllogism of the spiritualist philosophers meets these two conditions, it is irrefutable and I can only bow; but if one of these two conditions is lacking, it is null and void, without value, and the argument falls apart completely.\n\nIn order to know its value, let us examine the three propositions of which it is composed.\n\nFirst, the major proposition: \" _there is no effect without a cause_.\"\n\nPhilosophers, you are correct. There is no effect without a cause; nothing is more precise. There is not and cannot be an effect without a cause. The effect is only the consequence, the extension, the outcome, of the cause. Whoever says effect says cause; the idea of effect _necessarily_ and _immediately_ calls for the idea of cause. If it was otherwise, the without cause would be an effect of nothing, which would be absurd.\n\nThus, on this first proposition, we are in agreement.\n\nSecond, the minor proposition: \" _now, the universe is an effect_.\"\n\nAh! Here I demand reflection and call for explanations. What is the basis of such a flat, peremptory affirmation? What is the phenomenon or collection of phenomena, what is the observation or collection of observations, that allows you to express yourself in such an unequivocal tone?\n\nFirst of all, do we have sufficient knowledge of the universe? Have we studied, examined, searched, and understood it well enough that we can be so positive? Have we penetrated its inner workings? Have we explored its immeasurable spaces? Have we descended into the depths of the oceans? Have we scaled all the heights? Do we know everything to be found in the domain of the universe? Has it delivered up all its secrets? Have we stripped away all the veils, penetrated all the mysteries, and solved all the riddles? Have we seen everything, heard everything, touched everything, smelled everything, observed everything, and noted everything? Do we no longer have anything to learn? Does there remain nothing for us to discover? In short, are we in a position to make a positive appraisal, a definitive judgment, or an incontrovertible decision regarding the universe?\n\nNo one could answer in the affirmative to all these questions and we would feel deeply sorry for the reckless fool, we could say the madman, who would pretend that they know the universe.\n\nThe universe!\u2014we mean by that not only this tiny planet that we inhabit and over which we drag around our miserable carcasses, not only the millions of stars and planets that we know, which are part of our solar system or which we will discover in the fullness of time, but also those worlds upon worlds of whose existence we know or suspect but whose number, expanse, and distance remain incalculable!\n\nIf I say that the universe is a cause, I am certain that I would spontaneously unleash the hoots and protests of the believers; and yet my assertion would be no more ridiculous than their own.\n\nMy temerity would be equal to their own; that is all.\n\nIf I study the universe, if I observe the known facts to the extent allowed to the man of today, I note an incredibly complex and involved ensemble, an inextricable and colossal tangle of causes and effects, which determine, enchain, succeed, repeat, and penetrate one another. I perceive that the whole is formed like an endless chain, the links of which are unbreakably connected, and I note that each of these links is at once cause and effect: the effect of the cause that has determined it, the cause of the effect that follows.\n\nWho can say: \"Here is the first link, the link called cause\"? Who can say: \"Here is the last link, the link called effect\"? And who can say: \"There is necessarily a cause we can number the first, there is necessarily an effect we can number the last\"?\n\nThus, the second proposition\u2014\"now, the universe is an effect\"\u2014lacks the indispensable condition of precision.\n\nConsequently, the famous syllogism is worth nothing.\n\nI add that even in a case where the second proposition was exact, in order for the conclusion to be accepted, it would still have to be established that the universe is the effect of a unique cause, of a first cause, of the cause of causes, of a causeless cause, of the eternal cause.\n\nI await that demonstration without distress, without worry. It is one of those that has been attempted many times but has never been made. It is one of those of which we can say, without being too rash, that it will never be seriously, positively, scientifically established.\n\nI add, finally, that even in a case where the entire syllogism was flawless, it would be easy to turn against it, and in favor of my demonstration, the thesis of the Creator God.\n\nLet us try. There are no effects without a cause?\u2014So be it. Now, the universe is an effect?\u2014Okay. So that effect is a cause and it is the cause that we call God?\u2014Again, okay.\n\nDo not be in a hurry to triumph, deists, and hear me well.\n\nIf it is obvious that there is no effect without a cause, it is as completely obvious that there is no cause without an effect. There is not, and there cannot be, a cause without an effect. Whoever says cause says effect; the idea of cause _necessarily_ implies and _immediately_ calls to mind the idea of effect; if it were otherwise, the cause without an effect would be a cause of nothing, which would be as absurd as an effect of nothing.\n\nSo it is well understood that there is no cause without an effect.\n\nNow, you say that the universe has God as a cause. So it would be appropriate to say that the God Cause has the universe as an effect.\n\nIt is impossible to separate the effect from the cause; but it is equally impossible to separate the cause from the effect.\n\nYou maintain, finally, that the God Cause is eternal. I conclude from this that the universe effect is equally eternal, since to an eternal cause there must inescapably correspond an eternal effect.\n\nIf it were otherwise, if the universe had commenced during the billions and billions of centuries that have perhaps preceded the creation of the universe, God would have been a cause without effect, which is impossible, a cause of nothing, which would be absurd.\n\nConsequently, God being eternal, the universe is also eternal, and if the universe is eternal, it is because it has never commenced, because it has not been created.\n\nIs this clear?\n\n# **Second Series of Arguments**\n\n_Against the Governor God, or Providence_\n\n# First Argument\n\n_The Governor Denies the Creator_\n\nThere are those\u2014and they are legion\u2014who, despite everything, persist in believing. I understand that, strictly speaking, one can believe in the existence of a perfect creator; I understand that, strictly speaking, one can believe in the existence of a necessary governor; but it seems impossible to me that one can reasonably believe in both at the same time. These two perfect beings categorically exclude one another: to affirm one is to deny the other; to proclaim the perfection of the first is to admit the uselessness of the second; to proclaim the necessity of the second is to deny the perfection of the first.\n\nIn other words, we can believe in the perfection of one and the necessity of the other; but it is unreasonable to believe in the perfection of both: we must choose.\n\nIf the universe created by God had been a perfect work; if in the ensemble and in the least details this work had been without defect; if the mechanism of this gigantic creation had been flawless; if, in addition, its arrangement had been so perfect that there was no fear that it would produce a single derangement or a single bit of damage; if, in short, the work had been worthy of that brilliant worker, that incomparable artist and wonderful builder whom we call God, the need of a governor would never have been felt.\n\nThe initial helping hand once given, the formidable machine once set in motion, there would be nothing to do but to leave it to itself, without fear of any possible accident.\n\nWhat need for this engineer, this mechanic, whose role is to monitor the machine, direct it, intervene when it is necessary, and provide the necessary modifications and successive repairs to the machine in motion? This engineer would be useless and this mechanic without purpose.\n\nIn this case, there is no place for a governor.\n\nIf the governor exists, it is because his presence, supervision, and intervention are indispensable. The necessity of the governor is like an insult, a challenge hurled at the creator; his intervention attests to the clumsiness, incompetence, and powerlessness of the creator. The governor denies the perfection of the creator.\n\n# Second Argument\n\n_The Multiplicity of the Gods Demonstrates That No Gods Exist_\n\nThe Governor God is and must be powerful and just, infinitely powerful and infinitely just.\n\nI claim that the multiplicity of religions testifies to his lack of power and justice.\n\nLet us ignore the dead gods, the abolished cults and extinct religions. Those will number in the thousands. We speak only of the current religions.\n\nAccording to the best estimates, there are presently eight hundred religions that fight for control of the sixteen hundred million consciences that occupy our planet. There is no doubt that each imagines and proclaims that it alone is in possession of the true, authentic, indisputable, and unique God, and that all the other Gods are laughable false Gods, cheap contraband Gods, whom it is a pious work to battle and crush.\n\nI add that if there were only a hundred religions instead of eight hundred, if there were only ten, if there were only two, my argument would retain the same force.\n\nWell! I claim that the multiplicity of these Gods demonstrates that none exist, because it certifies that God lacks power or justice.\n\nPowerful, he could have spoken to all as easily as to some. Powerful, he could have shown himself, revealed himself to all, with no more effort than would have been required to reveal himself to a few.\n\nA man\u2014whoever he may be\u2014can only show himself, can only speak to a limited number of men; his vocal cords have a power that cannot exceed certain limits; but God... !\n\nGod can speak to all\u2014however great the multitude\u2014as easily as to a small number. When it is raised, the voice of God can and must resound to the four cardinal points. The divine word knows neither distance nor obstacles. It crosses the oceans, scales the summits, and traverses spaces without the least difficulty.\n\nSince he has seen fit\u2014as religion affirms\u2014to speak to men, to reveal himself to them, to confide his plans to them, to indicate his will and make known his law, he could have spoken to all without any more effort than to a handful of the privileged.\n\nHe has not done this, since some deny him, since others are unaware of him, and since others, finally, oppose this God to one of his competitors.\n\nIn these conditions, isn't it wise to think that he has not spoken to any and that the multiple revelations are only multiple impostures; or else that, if he has spoken to some, it is because he could not speak to all?\n\nIf this is the case, I accuse him of powerlessness.\n\nAnd, if I do not accuse him of powerlessness, I accuse him of injustice.\n\nWhat, indeed, are we to think of this God who shows himself to some and hides himself from others? What are we to think of this God who speaks to some and to others maintains his silence?\n\nDo not forget that the representatives of this God maintain that he is the Father, and that we are all, by the same title and to the same degree, the well-loved children of the Father who reigns in the heavens.\n\nWell! What do you think of this Father who, full of sympathy for a privileged few, snatches them, by revealing himself to them, from the anguish of doubt and the tortures of hesitation, while he intentionally condemns the immense majority of his children to the torments of uncertainty? What do you think of this Father who shows himself to some of his children in the dazzling radiance of his majesty, while for the others he remains shrouded in darkness? What do you think of this Father who, demanding a cult, respect, and worship from his children, calls a few elect to hear the word of truth, while he deliberately refuses that singular favor to the others?\n\nIf you reckon that this father is just and good, you will not be surprised that my judgment is different.\n\nSo the multiplicity of religions proclaims that God lacks power or justice. Now, God must be infinitely powerful and infinitely just; the believers affirm it; if he lacks one of these two attributes, power or justice, he is not perfect; if he is not perfect, he does not exist.\n\n_The multiplicity of gods demonstrates that no gods exist_.\n\n# Third Argument\n\n_God Is Not Infinitely Good: Hell Demonstrates It_\n\nThe Governor God, Providence, is and must be infinitely good, infinitely merciful. The existence of hell proves that he is not.\n\nFollow my reasoning closely: God could\u2014since he is free\u2014refrain from creating us; he has created us.\n\nGod could\u2014since he is all-powerful\u2014make us entirely good; he has created the good and the wicked.\n\nGod could\u2014since he is good\u2014allow us all into his paradise after our deaths, contenting himself with the times of trials and tribulations through which we have passed on the earth.\n\nGod could, finally\u2014since he is just\u2014allow into his paradise only the good and refuse access to the wicked, but destroy the latter at their time of death, rather than condemn them to hell.\n\nFor he who can create can also destroy; he who has the power to give life also has the power to destroy it.\n\nLet us see: you are not gods. You are not infinitely good nor infinitely merciful. I am, however, certain, without attributing qualities to you that you perhaps do not possess, that if it was in your power, without costing you a painful effort, without the chance of it causing you material harm or moral injury, if, I say, it was in your power, under the conditions that I have just indicated, to prevent one tear, one pain, one hardship for your brothers in humanity, I am certain that you would do it. And yet you are neither infinitely good nor infinitely merciful!\n\nWould you be better and more merciful than the God of the Christians?\n\nFor, in the end, hell exists. The Church teaches it; it is the horrific vision with the aid of which they terrify the children, old folks, and timid spirits; it is the specter installed at the bedsides of the dying, in the hour when death's approach robs them of all energy and lucidity.\n\nWell! The God of the Christians, a God known as a being of pity, pardon, indulgence, goodness, and mercy, casts some of his children\u2014forever\u2014into this place occupied by the cruelest tortures, the most unspeakable ordeals.\n\nHow good he is! How merciful!\n\nYou know that line from the scriptures: \"Many are called but few are chosen.\" That phrase means, if I am not mistaken, that the number of elect will be tiny and the number of damned considerable. This affirmation is so monstrously cruel that some have attempted to give it another meaning.\n\nIt matters little: hell exists and it is obvious that the damned\u2014in large or small number\u2014will endure the most agonizing torments there.\n\nLet us ask ourselves who can profit from the torments of the damned.\n\nWould it be the elect?\u2014Obviously not! By definition, the elect will be the most just, the virtuous, the fraternal, and the compassionate, and we could not imagine that their bliss, already inexpressible, would be increased by the spectacle of their tortured brethren.\n\nWould it be the damned themselves?\u2014Again, no, since the Church maintains that the torture of these wretches will never end, and that in billions and billions of centuries their torments will be as intolerable as on the first day.\n\nSo... ?\n\nSo, apart from the elect and the damned, there is only God; there can only be God.\n\nIs it then God who would profit from the sufferings of the damned? Is it this infinitely good, infinitely merciful Father who would revel sadistically in the sorrows to which he has deliberately condemned his children?\n\nAh! If that is the case, then God appears to me as the most savage executioner, the most implacable torturer that one could imagine.\n\nHell proves that God is neither good nor merciful. The existence of a good God is incompatible with that of hell.\n\n_Either there is no hell, or God is not infinitely good_.\n\n# Fourth Argument\n\n_The Problem of Evil_\n\nIt is the problem of evil that presents me with my fourth and final argument against the Governor God, as well as my first argument against the Magistrate God. I do not claim that the existence of evil, physical or moral evil, is incompatible with the existence of God; but I do claim that it is incompatible with the existence of a God who is _infinitely powerful and infinitely good_.\n\nThe reasoning is known, if only by the multiple refutations\u2014always powerless, however\u2014that have been opposed to it.\n\nIt can be traced back to Epicurus. So it has already been in existence for more than twenty centuries; but no matter its age, it still maintains all of its vigor.\n\nHere it is: evil exists. All beings capable of feeling know suffering. God, who knows all, cannot be unaware of this. Well! One of two things must be true: either God would like to eliminate evil, but he cannot; or else God could eliminate evil, but he does not wish to.\n\nIn the first case, God would like to eliminate evil; he is good and sympathizes with the suffering that oppresses us, with the evil we endure. Ah! If it only depended on him! Evil would be wiped out, and good would flourish on the earth. Once more, he is good; but he cannot eliminate evil, and, thus, he is not all-powerful.\n\nIn the second case, God could eliminate evil. He need only wish it for evil to be abolished, as he is all-powerful, but he does not wish to eliminate it; and, therefore, he is not infinitely good.\n\nHere, God is powerful, but he is not good; there, God is good, but he is not powerful.\n\nNow, in order for God to exist, it is not enough that he possesses one of these perfections: power or goodness. It is indispensable that he possesses both.\n\nThis reasoning has never been refuted.\n\nNow, listen: I do not claim that no one has ever attempted to refute it; I say that no one has every succeeded.\n\nThe best-known attempt at refutation is this: \"You pose the problem of evil in an entirely erroneous manner. You are quite wrong to hold God responsible. Yes, certainly, evil exists and it is undeniable; but it is man who should be held responsible for it. God did not want man to be an automaton, a machine, whose actions are inevitable. In creating man, he gave him his liberty; he made him an entirely free being. In every circumstance, God has left him the ability to make whatever use he wishes of the liberty that he has generously granted him; and if it should please man, instead of making a judicious and noble use of that inestimable good, to make an odious and criminal use, it is not God that we must accuse. That would be unjust; it would be equitable to accuse man.\"\n\nThat is the objection; it is classic.\n\nWhat is it worth? Nothing.\n\nI will explain: let us first distinguish physical from moral evil.\n\nPhysical evil is sickness, suffering, accident, and old age, with its process of flaws and infirmities; it is death, the cruel loss of those we love, of children born who die some few days afterward, without having known anything but suffering; it is a host of human beings for whom existence is only a long series of sorrows and afflictions, so that it would be better if they had never been born; it is, in the domain of nature, the plagues, cataclysms, fires, droughts, famines, floods, tempests, and the whole sum of tragic inevitabilities that are measured in sadness and death.\n\nWho would dare say, regarding this physical evil, that man must be held responsible for it?\n\nWho does not understand that if God has created the universe, if it is he who has endowed it with the formidable laws that rule it, and if physical evil is the ensemble of these inevitabilities that result from the normal play of the forces of nature, who does not understand that the author responsible for these calamities is, in all certainty, the one who has created this universe, the one who governs it?\n\nI suppose that on this point no dispute is possible.\n\nThe God who governs the universe is thus responsible for physical evil.\n\nThat alone would be sufficient, and my response could stop there. But I maintain that moral evil is imputable to God by the same title as physical evil, since, if he exists, he has presided over the organization of the moral world, just like that of the physical world, and, consequently, man, victim of moral and physical evil alike, is no more responsible for one than for the other.\n\nBut I must connect what I have to say regarding moral evil to the third and last series of my arguments.\n\n# **Third Series of Arguments**\n\n_Against the God of Justice_\n\n# First Argument\n\n_Being Irresponsible, Man Can Be Neither Punished nor Rewarded_\n\nWhat are we?\n\nHave we presided over the conditions of our birth? Have we been consulted on the simple question of whether we would like to be born? Have we been called to settle our own destinies? Have we, on a single point, had a say in the matter?\n\nIf we had had a say, each of us would have been, from the cradle, furnished with all the advantages: health, strength, beauty, intelligence, courage, generosity, etc., etc. Each of us would have been a living summary of all the perfections, a sort of God in miniature.\n\nWhat are we?\n\nAre we what we wish to be?\n\nUnquestionably not!\n\nSupposing there is a God, we are, since he has created us, what he wanted us to be.\n\nGod, since he is free, could have chosen to not create us.\n\nHe could have made us less wicked, since he is good.\n\nHe could have made us virtuous, healthy, and splendid. He could have lavished upon us all sorts of physical, intellectual, and moral gifts, since he is all-powerful.\n\nFor the third time, what are we?\n\nWe are what God has wanted us to be. He has created us as he pleased, according to his whim.\n\nThere is no other response to that question\u2014what are we?\u2014if we accept that God exists and that we are his creatures.\n\nIt is God who has given us our senses, our capacity for understanding, our sensibility, our means of perceiving, feeling, reasoning, and acting. He has anticipated, willed, and determined our conditions of life: he has conditioned our needs, our desires, our passions, our fears, our hopes, our hatreds, our tender feelings, and our aspirations. The whole human machine corresponds to his wishes. He has designed and arranged all aspects of the environment in which we live; he has prepared all the circumstances that, in each moment, will mount an attack on our will and determine our actions.\n\nBefore this formidably armed God, man is irresponsible.\n\nThe individual who is dependent on no one is entirely free; the one who is _slightly_ dependent on another is _slightly_ enslaved and is free in other respects; the one who is _largely_ dependent on another is _largely_ enslaved and is only free in a few respects; finally, the one who is _entirely_ dependent on another is _entirely_ enslaved and enjoys no liberty.\n\nIf God exists it is in this last position, that of slavery, that the individual finds himself in relation to God, and his slavery is that much more complete as there is a greater distance between him and the Master.\n\nIf God exists, he alone knows and wills; he alone is capable and free. Man knows nothing, wants nothing, has no power; his dependence is complete.\n\nIf God exists, then he is everything, and man is nothing.\n\nThe man thus held in slavery, placed in full and complete dependence on God, can bear no responsibility.\n\nAnd if he is irresponsible, he cannot be judged.\n\nEvery judgment entails a punishment or a reward; and the acts of an irresponsible being, having no moral value, are subject to no judgment.\n\nThe acts of the irresponsible can be useful or harmful; morally, they are neither good nor bad, neither meritorious not reprehensible; they could not be equitably rewarded or punished.\n\nIn setting himself up as a judge, by punishing or rewarding the irresponsible man, God is only a usurper; he has assumed an arbitrary right and has used it in ways contrary to all justice.\n\nFrom what I have just said, I conclude:\n\na) that the responsibility for moral evil is imputable to God, like the responsibility for physical evil;\n\nb) that _God is an unworthy magistrate, because, being irresponsible, man can neither be rewarded nor punished_.\n\n# Second Argument\n\n_God Violates the Fundamental Rules of Equity_\n\nLet us concede, for a moment, that man is responsible, and we will see that, even in this case, divine justice violates the most elementary rules of equity.\n\nIf we admit that the practice of justice could not be exercised without involving a sanction and that it is the mandate of the magistrate to establish that sanction, there is a rule regarding which our sentiments are and must be unanimous: just as there is a scale of merit and culpability, there must be a scale of rewards and punishments.\n\nAccording to this principle, the magistrate who best practices justice will be the one who most exactly makes reward proportionate to merit and punishment proportionate to guilt; and the ideal, impeccable, perfect magistrate will be the one who establishes with a mathematical rigor the relation between the act and the sanction.\n\nI think that this elementary rule of justice is accepted by all.\n\nWell! God, through heaven and through hell, misjudges that rule and violates it.\n\nWhatever the merit of man, it is limited (like man himself ) and yet the sanction of reward\u2014heaven is without bounds\u2014is such only through its character of perpetuity.\n\nWhatever the culpability of man, it is limited (like man himself ) and yet the sanction of punishment\u2014hell is without limits\u2014is such only through its character of perpetuity.\n\nThere is thus a disproportion between the virtue and the recompense, a disproportion between the fault and the punishment. There is disproportion everywhere. Thus, _God violates the fundamental rules of equity_.\n\nMy thesis is completed; nothing remains for me but to sum up and conclude.\n\n# **Summary**\n\nComrades, I had promised you a tightly argued, substantial, decisive demonstration of the nonexistence of God. I believe I can say that I have kept that promise.\n\nKeep in mind that I have not proposed to provide you a system of the universe that renders useless any recourse to the hypothesis of a supernatural force, of an otherworldly energy or power, of a principle superior or prior to the universe. I have had the fidelity, as I should, to tell you that, considered in this way, the problem does not entail, _in the present state of human knowledge_ , any definitive solution, and that the only attitude appropriate for cautious and reasonable minds is uncertainty.\n\nThe God whose impossibility I wished to establish, whose impossibility I can now say that I have established, is the God of the religions, God as Creator, Governor, and Magistrate, God infinitely wise, powerful, just, and good, the God whom the clergy claim to represent on earth, and whom they attempt to set up for our veneration.\n\nThere is not, there can be no ambiguity. It is this God whom I deny; and if one wants to debate usefully, it is this God who must be defended against my attacks.\n\nEvery debate on another terrain will be\u2014I warn you of it, for you must be on guard against the ruses of the enemy\u2014every debate on another terrain will be a diversion and will be, moreover, a proof that the God of the religions cannot be defended or justified.\n\nI have proven that, as Creator, he would be inadmissible, imperfect, inexplicable; I have established that, as Governor, he would be useless, powerless, cruel, horrible, despotic; I have shown that, as a Justice Bringer, he would be an unworthy Magistrate, violating the essential rules of the most elementary equity.\n\n# **Conclusion**\n\nSuch is, however, the God who, since time immemorial, has been taught and in our days is still taught to a multitude of children in so many families and schools. What crimes have been committed in his name!\n\nWhat hatreds, wars, and calamites have been furiously unleashed by his representatives! Of what sufferings has this God been the source! What evils he still engenders!\n\nFor centuries, religion has held humanity bowed in fear, wallowing in superstition, prostrate in resignation.\n\nWill the day never come when, ceasing to believe in eternal justice, in its imaginary judgments and questionable reparations, humans will work with tireless ardor for the advent on earth of an immediate, positive, and fraternal justice?\n\nWill the hour never strike when, disillusioned with the consolations and fallacious hopes suggested by the belief in a compensatory paradise, humans will make our planet an Eden of abundance, peace, and liberty, whose doors will be fraternally open to all?\n\nFor too long, the social contract has been inspired by a God without justice; it is time that it is inspired by a justice without God. For too long, the relations between nations and individuals have stemmed from a God without philosophy; it is time that they proceeded from a philosophy without God. For centuries, monarchs, rulers, castes and clergy, conductors of people and directors of consciences have treated humanity like the vile flock, good only to be shorn, devoured, cast into abattoirs.\n\nFor centuries, the disinherited passively endured poverty and servitude, thanks to the deceptive mirage of heaven and the horrific vision of hell. We must put an end to this horrible enchantment, to this abominable trickery.\n\nOh, you who hear me, open your eyes and look, observe, understand. The heaven of which you speak endlessly, the heaven with the aid of which they try to numb your misery, anesthetize your suffering, and stifle the groan that, despite it all, rises from your chest, that heaven is unreal and deserted. Only your hell is peopled and positive.\n\nEnough lamentations: the lamentations are in vain.\n\nEnough prostrations: the prostrations are pointless.\n\nEnough prayers: the prayers are powerless.\n\nStand up, oh man! And, upright, trembling, rebellious, declare an implacable war on the God whose mind-numbing veneration has for so long been imposed on you and your brethren.\n\nRid yourself of this imaginary tyrant and shake off the yoke of those who claim to be his deputies here below.\n\nBut remember that, this first act of liberation accomplished, you have completed only a part of the task that falls to you.\n\nDo not forget that it will do you no good to break the chains that the imaginary, celestial, and eternal gods have forged for you, if you do not also break those forged for you by the passing, positive gods of the earth.\n\nThese gods prowl around you, seeking to starve and enslave you. These gods are only men like you.\n\nRich men and rulers, these earthly gods have peopled the earth with countless victims, with unspeakable torments.\n\nMay the damned of the earth finally rebel against these villains and found a city where these monsters will be forever rendered impossible!\n\nWhen you have chased the gods from heaven and earth, when you have rid yourselves of the masters from above and the masters from below, when you have accomplished this double act of deliverance, then, but only then, oh, my brother, you will escape from your hell and you will achieve your heaven!\n\n# CHAPTER VI\n\n# The Meaning of Atheism\n\n# by E. Haldeman-Julius\n\nAtheism is accurately defined as the denial of the assumptions of theism. The theist affirms that there is a God running the universe; he declares that the idea of such a God is necessary to an understanding of life; he offers various arguments or, as he rather presumptuously calls them, evidences for his God idea.\n\nWhat is the position, logically, of the atheist? He will not say in a mild, uncertain fashion that he doesn't know whether the idea is true or that it is an open question. He has studied carefully the case for and against theism. He finds that case utterly insupportable, lacking any real or positive evidence, defended by arguments which are easily discovered to be casuistic and fallacious, and linking itself with other supplementary ideas which are incredible.\n\nThe atheist perceives that history, in every branch of science, in the plainly observable realities of life, and in the processes of common sense there is no place for the picture of a God; the idea doesn't fit in with a calmly reasoned and realistic view of life. The atheist, therefore, denies the assumptions of theism, because they are mere assumptions and are not proved; whereas the contrary evidences against the idea of theism are overwhelming. He takes a clear-cut position. To proclaim himself an agnostic, while to some it might appear more respectable and cautious, would be to say in effect that he hadn't decided what to believe.\n\nWe can understand, of course, why many prefer to call themselves agnostics. They don't wish to appear bigoted. Or they are honestly in doubt and feel that the idea of God may or may not be true; yet with scarcely an exception the attitude of the agnostic is the same as that of the atheist\u2014he denies the assumptions of theism\u2014his disbelief in God, as an agnostic, is quite as strong really as the atheist's disbelief.\n\nBut atheism is not in the least bigoted. It is a conclusion reached by the most reasonable methods and one which is not asserted dogmatically but is explained in its every feature by the light of reason. The atheist does not boast of knowing in a vainglorious, empty sense. He understands by knowledge the most reasonable and clear and sound position one can take on the basis of all the evidence at hand. This evidence convinces him that theism is not true, and his logical position, then, is that of atheism.\n\nWe repeat that the atheist is one who denies the assumptions of theism. He asserts, in other words, that he doesn't believe in a God because he has no good reason for believing in a God. That's atheism\u2014and that's good sense.\n\n# **Atheism Is the Realistic Answer to the God Idea**\n\nWe are not fanatics on the subject of religion. If it were merely a matter of abstract argument, we should not be so interested. Ideas, if they could be quite separated from actual influence in living issues, might be regarded with an air of detachment. They might in such case be discussed mildly and dismissively. One might be indifferent to such ideas or only amused by them.\n\nBut religion has always asserted and it does yet assert a very direct and commanding interest in the conduct of men. It is true that, fortunately, there are old terrors and powers that religion no longer can exercise so effectively as it did only a few score years ago. But the atmosphere and the attitude of bigotry remain. If religion cannot ordinarily invoke the armed force of law to punish heretics, it still plays upon the psychology of fear, and predominantly its influence is to frighten men and distort their views and poison every process of their reasoning.\n\nThe remnant of religion that is cherished by a few educated and urbane men\u2014the philosophical or poetic religion that one observes here and there\u2014does not concern us so acutely. Such a provisional or partial belief in religion is baseless logically, and it is confusing; but we may grant that it is relatively harmless; we can point out its fallacy and continue cheerfully on our way about other things. But this philosophical or poet religion is not, after all, the religion of the masses.\n\nThere are many cultured people who do not realize that among the masses\u2014among millions of honest but deluded people\u2014the most extravagant, fanatical, and obviously dangerous notions about religion are prevalent. One of the malign emotional and prejudicial influences that helped to lend menacing strength to the late Ku Klux Klan, for example, was the spirit of religious prejudice. We all know how that vicious organization was strengthened by a Protestant tone of creedal fanaticism. On the other hand, the Catholics have their own extreme tone of fanaticism; and they still assert, moreover, that the Catholic religion should be and rightfully is supreme in belief and power\u2014Catholicism, that is to say, is definitely opposed to the modern principles of political liberty and intellectual freedom.\n\nProtestantism is not, in its definite official statements, so brazenly intolerant. Probably this is because Protestantism includes so many creeds\u2014and these religious people feel that they must be protected against one another. They are not so kindly toward atheists.\n\nIn a number of American states atheists cannot testify in a court of law. Blasphemy laws are still on the statute books; and occasionally they are enforced. Our laws regarding marriage and sex are sadly distorted by religious prejudice; and a few of these distortions and absurdities are ably summarized by Anthony M. Turano. Bible reading (which means Bible teaching) in the public schools is compulsory in Pennsylvania, Arkansas, and other states. In Tennessee and Mississippi a medieval law bans the teaching of evolution\u2014the teaching, in a word, of the most serious principle of truth in modern science\u2014in the public schools. The circulation of a responsible, scholarly, important sex questionnaire at the University of Missouri was followed by a ridiculous campaign of prejudice in which the chief element, plainly enough, was a religious attitude of obscurantism on the sex question.\n\nOur laws and customs are still deplorably handicapped and corrupted by the ideas of religion. These ideas are no longer of valid currency in the intellectual world. They are centuries behind the times. They are not insisted upon with such vicious and perilous persistency as was the case a few centuries ago. But they remain\u2014these terribly wrong and menacing ideas\u2014and it is the part of a civilized program of enlightenment to combat these ideas with all the force possible.\n\nWe, of course, believe in the force of reason and argument and persuasion; yes, and the force of ridicule and denunciation, all legitimate and free weapons which we can employ against religion; in short, we believe in the clarifying conflict of ideas, and as region cannot be defended intelligently we know that in the long run it must he conquered. It remains yet, however, as a serious and major issue in the thoughts and actions of men. Granting, we naturally do, the fullest right of every man to believe in any theory of religion or politics or social conduct which is preferred by him, we do not forget that we have an equal right to promote our own ideas and to attack, relentlessly and clearly, ideas which we recognize as vicious in theory and inevitably vicious also in practice.\n\nWe are well aware that religion is not as bad an influence as it was a short time ago, as history is counted. But it is a sufficiently bad influence even in modern times; and its reduced viciousness (in practice) is due plainly enough to its reduced power. We want to reduce that power to an absolute nullity. We want religion to be entirely outgrown by the advancing intelligence of mankind. Universal education is our ideal; and this means, in our convinced opinion, that the philosophy of atheism (which is also the philosophy of realism) will displace with complete sanity and wholesomeness the dark and morbid and unintelligently fanciful ideas of religion.\n\nWe advocate the atheistic philosophy, because it is the only clear, consistent position which seems possible to us. As atheists, we simply deny the assumptions of theism; we declare that the God idea, in all its features, is unreasonable and unprovable; we add, more vitally, that the God idea is an interference with the interests of human happiness and progress. We oppose religion not merely as a set of theological ideas; but we must also oppose religion as a political, social, and moral influence detrimental to the welfare of humanity.\n\nWe attack religion, because religion is not true\u2014because religion is an obstacle (or a set of obstacles) in the way of progress\u2014because religion foments strife and prejudice\u2014because religion is the breeding ground of intolerance\u2014because, in short, religion is essentially hostile to mankind.\n\nReligion glorifies the dogma of a despotic, mythical God. Atheism ennobles the interests of free and progressive Man. Religion is superstition. Atheism is sanity. Religion is medieval. Atheism is modern.\n\n# **Preacher Urges the Establishment of Religious Despotism**\n\nThat religious fanaticism is a modern menace and not merely a medieval memory, that steady propaganda on behalf of freedom of thought is a most serious necessity, we have proved again for our warning in the sermon of Rev. W.D. Lewis, pastor of the Second Presbyterian Church of Wheeling, West Virginia. This preacher, who occupies the pulpit of an important city church, declares that religious liberty must be ended in America and that a system of compulsory religion must be established. \"I shall never be in full sympathy with our system of compulsory education,\" he said, \"until there is set up side by side with it a system of compulsory religion.\"\n\nIn suggesting a course of despotic religious procedure for modern times, Rev. Lewis goes way back to the days of ancient Israel. He turns to the Bible and its Old Testament code of theocratic laws. Modern Americans, he says, must be compelled to acknowledge the sovereignty of a personal, autocratic, all-ruling God, even as did the ancient Israelites\u2014and, according to the scheme of this preacher, this God of Bunk must be worshipped by all and no argument permitted.\n\n\"The whole scheme of things in Israel,\" says Rev. Lewis, \"revolved around the idea of a personal God. The first leaders of the Jews saw that it would never do to attempt to create a national solidarity without the establishment of a fixed authority.... So those first leaders of Israel did the wisest thing ever done by any group of men aspiring to bring forth a nation: they invested all authority in God. They took neither responsibility nor credit for themselves.... They were simply his mouthpieces and his agents.\"\n\nThat the priests and rulers of Israel \"took neither responsibility nor credit for themselves\" is of course a ridiculous bit of sophistry. They had a very imposing prestige and very profitable revenues in their role as the \"mouthpieces and agents\" of their mythical God. Clearly it was a great stroke of clever exploitation (clever enough to deceive primitive tribes and clever enough to fool many moderns who nevertheless do not live intellectually in the modern age) for the priests to put over the faction that a big, strong, mysterious, and fearsome God was behind their words and actions; that piece of fiction made the priests seem far greater than mere men, greater than merely human rulers, and they have fought and schemed jealously through the centuries to retain that advantage.\n\nIt is the prestige and power of clericalism that Rev. Lewis is eager to have restored fully in America. This is clear in what he says about the specific command to worship (i.e., to patronize the clerical shops of superstition). \"One day in seven, the Sabbath,\" he says, \"was made holy unto God and set aside solely for his worship (in ancient Israel). There was no choice about it. In those first days there was no such thing as religious liberty in Israel. A man had to go to worship whether he liked it or not. The fact that he didn't like the priests didn't matter.... The excuse that he was intellectually superior to the congregation of Israel didn't work.... Religious liberty was given no thought in Israel. I sometimes wonder if it isn't given too much thought in our own America.\"\n\nWe might indeed remind Rev. Lewis that in modern America we have many features of life which were unknown in ancient Israel. We have not only religious liberty but also political liberty, and the two are inseparable. The Old Testament Jews, that primitive and superstitious tribe, had no conception of modern democracy. They had no glimmering of the materials of modern education. For instance, those old Jews whom Rev. Lewis would have us follow in their system of religious despotism had the most ridiculous notions of life\u2014they believed in creation by a God and in all the farrago of legends which are sprawlingly conspicuous in the Old Testament. They believed that the earth was the center of a very small universe (they had really no conception of a universe) and that the sun, moon, and stars were merely conveniences to illuminate the earth. They had the most absurd, strangely twisted, cruelly barbaric, and superstitious ideas of morality\u2014the conception of moral law as social law, while it was necessarily followed by them to some extent, was not fully understood by them. Crude indeed were the ideas prevalent in ancient Israel about religion and about government and about morality and about the earth and man. If we were really compelled to follow the ways of ancient Israel, as this West Virginia preacher insists we should, we have should have to scrap our system of education and embrace the system of despotic religion in its stead.\n\nIt may be doubted if Rev. Lewis has much concern for education, save as it can be used spuriously as a support for religion. His fixed idea seems to be the importance of compulsory religion. \"I shall never be in full sympathy with our system of irreligious education. Why should we be compelled to attend and support our schools if there is nothing that can be done to compel us to attend and support our churches?... If education is absolutely necessary for our community life so is religion. Or yet why should we be compelled to support the idea of government if we are at liberty to treat the idea of God with contempt?... You will never make a full success of a compulsory government or a compulsory education until you give the same dignity to religion and make it compulsory; at any rate compulsory enough to make it respected throughout the land. The nation that plays fast and loose with its idea of God will soon or late play fast and loose with its idea of education and its idea of government.... If God doesn't matter, then nothing else matters, and all the compulsions of life might just as well be set aside.\"\n\nWhat Rev. Lewis does not understand (and presumably does not care about) is the truth, well illustrated in history, that no system of education can survive as educationally free and genuine if it is loaded with the chains of a compulsory religion. A religious despotism is utterly incompatible with the freedom and dignity and progressive achievements of social life. As a matter of fact, religion is an eccentric revival from ignorant earlier periods in the life of mankind. It is not in sympathy with modernism (of course not), and it cannot be reconciled with modernism. The right to believe in religion and practice its forms of worship as an individual affair is one that, on modern principles, we must grant. Religion, however, must be kept in its place as a private matter. It is too dangerous when it goes beyond that and presumes to command or threaten the state. Rev. Lewis is an exponent, bold yet typical, of a sentiment of religious bigotry which we cannot afford lightly to dismiss nor to ignore. We must expose these bigots and fight them with a sternness that is uncompromising and a sweep of propaganda that is irresistible.\n\n# **The Problem of Evil**\n\nThe problem of evil has always been a mischievous, difficult trap of tormented logic for theologians. They have affirmed dogmatically the existence of an all-powerful and omniscient and benevolent God\u2014but in explaining the evil things in the world they have been not at all deft but rather desperate.\n\nWe have been told that God created only the good and not the evil\u2014but that doesn't jibe with the theory of a God who has complete power. If he can't prevent evil, then he is a limited God with a grave element of weakness.\n\nOthers have argued that God permitted the evil for purposes of his own, which were really good purposes but beyond man's finite comprehension. But that is a harassed recourse of a man who is in a corner and can think of nothing better to say. It is an argument that admits of no demonstration. It assumes something that can't be proved. It isn't satisfactory.\n\nAgain, we are told that there is no evil in the world\u2014that when we regard certain phenomena as evil it is only because we have a distorted view\u2014that all things are good if we could only understand them truly. And that again is wild assertion without even the appearance of logic.\n\nYes, the problem of evil is too much for theologians. It can't be reconciled with the God idea. It is understandable only in a naturalistic, atheistic view of things.\n\nAfter all, the principle objection which a thinking man has to religion is that religion is not true\u2014and is not even sane.\n\nThe fear of gods and devils is never anything but a pitiable degradation of the human mind.\n\n# **Can God Lie?**\n\nThis question is put to Christians who believe that the Bible unerringly describes God and reports the commands and the characteristics of God. If there is a God, it is natural that we should wish to be quite correct in our understanding of that God's nature. So we ask: Can and does God lie?\n\nLooking this point up in the mazes of Holy Writ, we discover confusion. In Numbers 23:19, we are told: \"God is not a man, that he should lie.\" This is put even mere strongly in Hebrews 6:18, where we read: \"It was impossible for God to lie.\"\n\nBut do these citations settle the matter? Ah, no, we are upset in our calculations the moment we turn to 2 Thessalonians 2:11, where we read: \"For this cause God shall send them strong delusions, that they should believe a lie.\" And in I Kings 22:23, God is thus reported: \"Now, therefore, behold, the Lord hath put a lying spirit in the mouth of all these thy prophets, and the Lord hath spoken evil concerning thee.\"\n\nCan God lie? Can the Bible lie? Anyway, there is a mistake somewhere. The big mistake is in entertaining the idea of a God.\n\nWhen we read that some minor scientist (usually a skilled technical worker but not a thinker in science) has \"found God\" somewhere, we are not excited. We know this is only a form of words, meaning only that the scientific worker, turning away from science, has rediscovered the stale old assumption of theology, \"There is a God.\" We find invariably (as we should expect) that there is no satisfactory definition or description or identification or location or proof of a God. \"God\" is merely a word, whether it is used by a preacher or a mystic in a laboratory.\n\nThe fact that millions of people still believe in a hell of eternal punishment for sinners and unbelievers is a drastic reminder of the need for persistent, progressive education of the masses. We have as yet only begun to realize the possibilities of progress. But science, rationalism, and humanism have pointed the way; they have taken the first great steps, and we must keep right ahead on the highway of modernism.\n\nDon't take our word for it. Read the Bible itself. Read the statements of preachers. And you will understand that God is the most desperate character, the worst villain in all fiction.\n\nCommonly, those who have professed the strongest motives of love of a God have demonstrated the deepest hatred toward human joy and liberty.\n\nTheism tells men that they are the slaves of a God. Atheism assures men that they are the investigators and users of nature.\n\nBelief in gods and belief in ghosts is identical. God is taken as a more respectable word than ghost, but it means no more.\n\nReligion, throughout the greater part of its history, has been a form of \"holy\" terrorism. It still aims its terrors at men, but modern realism and the spread of popular enlightenment has progressively robbed those terrors of their old-fashioned effectiveness. Wherever men take religion very seriously\u2014wherever there is devout belief\u2014there is also the inseparable feeling of fear.\n\nChristian theology has taught men that they should submit with unintelligent resignation to the worst real evils of life and waste their time in consideration of imaginary evils in \"the life to come.\"\n\nPriests and preachers have tricked, terrified, and exploited mankind. They have lied for glory of God.\" They have collected immense financial tribute for \"the glory of God.\" Whatever may be said about the character of individuals among the clergy, the character of the profession as a whole has been distinctly and drastically anti-human. And, of course, the most sincere among the clergy have been the most dangerous, for they have been willing to go to the most extreme lengths of intolerance for \"the glory of God.\"\n\nPerhaps religion might be dismissed as unimportant if it were merely theoretical. It is difficult, however, if not impossible to separate theory and practice. Religion, to be sure, is full of inconsistencies between theory and practice; but there is and has always been sternly and largely a disposition of religion to enforce its theory in the conduct of life; religion has meant not simply dogmatism in abstract thinking but intolerance in legal and social action. Religion interferes with life and, being false, it necessarily interferes very much to the detriment of the sound human interests of life.\n\nFor centuries men have fought in the most unusual and devious ways to prove the existence of a God. But evidently a God, if there were a God, has been hiding out. He has never been discovered or proved. One would think a God, if any, should have revealed himself unmistakably. Isn't this non-appearance of a God (the non-appearance of a God in the shape of a single bit of evidence for his existence) a pretty strong, sufficient proof of non-existence?\n\nA God of love, a God of wrath, a God of jealousy, a God of bigotry, a God of vulgar tirades, a God of cheating and lying\u2014yes, the Christian God is given all of these characteristics, and isn't it a wretched mess to be offered to men in this twentieth century? The beginning of wisdom, the beginning of humanism, the beginning of progress is the rejection of this absurd, extravagantly impossible myth of a God.\n\n# **Hidden Gods**\n\nLook at the God idea from any angle, and it is foolish, it doesn't make sense but extravagantly proposes more mysteries than it assumes to explain. For instance, is it sensible that a real God would leave mankind in such confusion and debate about his character and his laws?\n\nThere have been many alleged revelations of God. There have, indeed, been many Gods as there have been many Bibles. And in different ages and different lands an endless game of guessing and disputing has gone on. Men have argued blindly about God. They still argue\u2014just as blindly.\n\nAnd if there is a God, we must conclude that he has willfully left men in the dark. He has not wanted men to know about him. Assuming his existence, then it would follow that he would have perfect ability to give a complete and universal explanation of himself, so that all men could see and know without further uncertainty. A real God could exhibit himself clearly to all men and have all men following his will to the last letter without a doubt or a slip.\n\nBut when we examine even cursorily the many contradictory revelations of God, the many theories and arguments, the many and diverse principles of piety, we perceive that all this talk about God has been merely the natural floundering of human ignorance.\n\nThere has been no reality in the God idea which men could discover and agree upon. The spectacle has been exactly what we should expect when men deal with theories of something which does not exist.\n\nHidden Gods\u2014no Gods\u2014all we see is man's poor guesswork.\n\n# **Take Your Choice**\n\nIf the Bible, which Christians believe is the word of God, is inspired and infallible, why does it have two distinctly opposite versions of many things? God's nature and God's opinions and God's wishes are contradictorily reported in Holy Writ.\n\nIt is stated, for example, in Genesis 1:31, as follows: \"And God saw everything that he had made, and behold it was very good.\" But in Genesis 6:6, it is stated: \"And it repented the Lord that he had made man on the earth, and it grieved him at his heart.\" Does the good Christian believe both statements?\n\nIn Chronicles 7:12, 16, we read: \"And the Lord appeared to Solomon by night, and said unto him: I have heard thy prayer, and have chosen this place to myself for a house of sacrifice.... For now have I chosen and sanctified this house that my name may be there forever; and mine eyes and my heart shall be there perpetually.\" Then in Acts 7:48, we read: \"Howbeit the Most High dwelleth not in temples made with hands.\"\n\nWhether God preferred the darkness or the light seemed to be uncertain to the Hebrew prophets of the Most High; but if the Bible were thoroughly inspired there should have been perfect agreement. But in I Timothy 6:16, God is referred to in this manner: \"Dwelling in the light which no man can approach.\" On the other hand, in I Kings 8:12, this reference is contradictorily made: \"The Lord said that he would dwell in the thick darkness.\" And in Psalm 18, we are told about God: \"He made darkness his secret place.\" And in Psalm 47:2, we are told: \"Clouds and darkness are round about him.\"\n\nSuch contradictions are common in the Bible. Naturally this happened, as the Bible was a collection of books written at different times by different men\u2014a strange mixture of diverse human documents\u2014and a tissue of irreconcilable notions. Inspired? The Bible is not even intelligent. It is not even good craftsmanship but is full of absurdities and contradictions.\n\n# **\"God's Will\"**\n\nThoughtful men have always observed that \"God's will,\" as that amusing expression has been employed by theologians and by lay commentators, has been nothing more nor less than a reflection of human impulses and desires and fears and whimsicalities. Whoever interprets this so-called will of God always presents a picture of his own, the interpreter's, way of looking at things.\n\nA sober, devout man will interpret \"God's will\" soberly and devoutly. A fanatic with bloodshot mind will interpret \"God's will\" fanatically. Men of extreme, illogical views will interpret \"God's will\" in eccentric fashion. Kindly, charitable, generous men will interpret \"God's will\" according to their character.\n\nAnd, of course, this means that whatever happens in life and in the world of nature, entirely independent of the will of any supposed God, such happenings (of the most immensely variant and complex kind) are ascribed to the will of God\u2014a blanket phrase and a bombastic one too, which explains absolutely nothing. Back of the phrase \"God's will\"\u2014and back of the idea, such as it is, which is reflected by this phrase\u2014there is the old, sound, and really (to the thinking man) obvious truth that gods and all that appertains to them are fashioned by man in his own image or, that is to say, by men in the images cast by their fancies and fears. What we have under observation, always, are human impulses and schemes of action: to say that \"God's will\" is behind them is to say exactly nothing.\n\n# **Incredible Instances**\n\nAs the Bible is regarded as a holy and inspired book by practically all Christians, a book absolutely without errors by many Christians, and the most important proof (through alleged revelation) of the existence of a God by many Christians, it is very important to point out incredible instances recorded in the Bible which no man can sensibly believe.\n\nColonel Robert G. Ingersoll did a very useful work in exposing the folly of believing that the Bible was inspired. \"One can scarcely be blamed,\" he said, \"for hesitating to believe that God met Moses at a hotel and tried to kill him [Exodus 4:24]; that afterward he made this Moses a god to Pharaoh and gave him his brother Aaron for a prophet [Exodus 7:1]; that he turned all the ponds and pools and streams and all the rivers into blood [Exodus 7:19] and all the water in vessels of wood and stone; that the rivers thereupon brought forth frogs [Exodus 8:3]; that the frogs covered the whole land of Egypt; that he changed dust into lice, so that all the men, women, children and animals were covered with them [Exodus 8:16\u201317]; that he sent swarms of flies upon the Egyptians [Exodus 8:21]; that he destroyed the innocent cattle with painful diseases; that he covered man and beast with blains and boils [Exodus 9:9]; that he so covered the magicians of Egypt with boils that they could not stand before Moses for the purpose of performing the same feat [Exodus 9:11]; that he destroyed every beast and every man that was in the fields, and every herb, and broke every tree with storm of hail and fire [Exodus 9:25]; that he sent locusts that devoured every herb that escaped the hall, and devoured every tree that grew [Exodus 10:15]; that he caused thick darkness over the land and put lights in the houses of the Jews [Exodus 10:22\u201323]; that he destroyed all of the firstborn of Egypt, from the firstborn of Pharaoh upon the throne to the firstborn of the maidservant that sat behind the mill [Exodus 11:5], together with the firstborn of all beasts, so that there was not a house in which the dead were not [Exodus 12:29\u201330].\"\n\nDo these marvels read like inspiration? Or do they read like superstition? Remember that millions of Christians still base their belief in a God upon the words of the Bible, which is a collection of the most flabbergasting fictions ever imagined\u2014by men, too, who had lawless but very poor and crude imagination. Ingersoll and numerous other critics have shot the Christian holy book full of holes. It is worthless and proves nothing concerning the existence of a God. The idea of a God is worthless and unprovable.\n\n# **Blind Alleys**\n\nMyself when young did eagerly frequent\n\nDoctor and Saint and heard great argument\n\nAbout it and about evermore\n\ncame out by the door as in I went.\n\nThis well-known stanza by Omar, the agnostic Persian poet, expresses the simple truth that he learned nothing from all the arguments about God\u2014nothing, that is to say, except that the arguments were aimless and meaningless. The doctors and the saints were floundering amid unrealistic abstractions. God was merely a name. It had scarcely the solid dignity and comprehensibility of an idea\u2014even a false idea.\n\nThis argumentation which taught nothing to Omar\u2014which left him with as little evidence for a God as before he heard a word of the argumentation\u2014was a vain, wordy repetition of fears, fancies, assumptions, dogmas, and whimsically elaborated nonsense. And so it has always been. The efforts of theism, intellectually speaking, have been a chasing up blind alleys. They have arrived nowhere\u2014but, on the contrary, the more argument there has been about the idea of God, the more steadily have men grown in the conviction that the idea is obviously untrue and unrealistic.\n\nTalk of God leads by a direct road to the conclusion of atheism. The only sensible attitude is to dismiss the idea of God\u2014to get it out of the way of more important ideas. The wide dissemination of this intelligent atheistic attitude is one of the leading features of any program of popular education which is completely worthy of the name.\n\nWith its fears and superstitions and prejudices, religion poisons the mind of anyone who believes in it\u2014and even the best man, under the influence of religion, cannot reason wholesomely. Atheism, on the contrary, opens the mind to the clean winds of truth and establishes a fresh air sanity.\n\nNobody has ever taken notable pains to locate the legendary heaven; but probably that is because nobody ever thought seriously of going to a heaven.\n\n# **Is God a Joker?**\n\nA few weeks ago, a hurricane struck the little religious community of Bethany, Oklahoma. A number of pious citizens of the little town were killed. Houses were destroyed\u2014homes in which prayer and devotion reigned. A church was demolished.\n\nOnly a few miles away is the large, wicked city of Oklahoma City\u2014at least we can certainly assume that, from the religious viewpoint, many sinners live in Oklahoma City. Assuming also (which is a great deal riskier assumption) that there is a God, why should he perpetrate this grim and sardonic joke? The sinners in the big city were left untouched. The godly folk in the little nearby village were punished by the evidences of God's wrath. How do the religious people interpret this calamity? Often they explain such calamities as flood, fire, and storm by saying that God is angry at the sinful people and is warning them or destroying them for their sins. Was the hurricane in Bethany a sign of the love of God for his faithful worshipers?\n\nAnd God missed an even better chance, if there were a God who wished to punish rebels against his majesty and inscrutability. Just a few hundred miles north and east of Bethany, Oklahoma, is Girard\u2014the home of _The American Freeman_ and _The Debunker_ and _The Joseph McCabe Magazine_ and the _Little Blue Books_ \u2014the center of American free thought where an enormous stream of atheistic literature and godless modern knowledge pours forth to enlighten the masses. If there were a God directing hurricanes and he wanted to really \"get\" an uncompromising foe, whom he has no chance of persuading in the ordinary way, it would have been a devastating stroke for him to send his howling punitive blasts through the town of Girard. It would be a more remarkable suggestion of the avenging act of a God if only the Haldeman-Julius plant were destroyed and the rest of the town left unhurt\u2014and, as good neighbors, we shouldn't wish the Christian and respectable people of Girard or those who are respectable and not so Christian or those who are Christian and not exactly respectable to suffer from our proximity and our propaganda of atheism.\n\nIs God a joker? No\u2014let us whisper it\u2014the joke is that there is no God. Hurricanes come upon the just and the unjust, the pious and the impious.\n\nTo be true to the mythical conception of a God is to be false to the interests of mankind.\n\n# **God as a Gamble**\n\nOne of the most amusing arguments frequently offered in defense of belief in the idea of a God is that such a belief is a way of playing safe. It is said that even though a man is not sure of the existence of a God and a future life beyond the grave, it is the part of caution for him to believe; then, as the argument goes, the man believing is safe whether there is or is not a God and a future existence; if there is no God, the believer will be no more dead than the unbeliever; while if there is a God, the believer will have preferential treatment in the judgments of the celestial tribunal.\n\nThis queer argument makes the matter of belief in a God an intellectual gamble. It is, of course, an utter denial of intellectual integrity. Proceeding on this basis, the appeal to belief is not made on the score of truth. One is urged to consider the God idea not from the standpoint of its reasonableness but, rather, from the standpoint of blind faith and a chance bet on an idea.\n\nDoesn't the religious person who uses this appeal to a particularly low form of intellectual cowardice? What men need is courage in their thinking. They need to be trained in facing facts frankly. They need to learn that all ideas should be judged with strict regard for the evidence. Instead religion harps on the emotion of fear and tells men that they should treat ideas merely as gambling chances, and that it is safer (not intellectually the better but the more craven part) to believe in a God.\n\nThis argument has other fallacious aspects. It assumes, for instance, that the evidence for and against the idea of a God is equal; whereas the vast preponderance of evidence is against the idea, there being, in fact, no genuine evidence for the idea. It is overlooked, too, that belief is genuine or it is not; and that a belief which is frankly grounded on a gamble\u2014a belief affirmed for safety's sake\u2014cannot be a real belief. One believes or one does not; real belief can only assert the truth of an idea. In short, the man who bases his belief on such a principle is bordering close to hypocrisy and is certainly revealing a striking lack of mental integrity.\n\nSuch weak arguments exemplify the decline of religion and show its utter intellectual bankruptcy. It has all the air of a desperate and last plea for a set of ideas which, ordinarily and reasonably, cannot be defended. It is, after all, a virtual admission of the charge of the atheist that the idea of a God is merely an assumption and has no ground of truth upon which firmly to plant itself.\n\n# **Credulity\u2014A Crime**\n\nCredulity is not a crime for the individual\u2014but it is clearly a crime as regards the race. Just look at the actual consequences of credulity. For years men believed in the foul superstition of witchcraft, and many poor people suffered for this foolish belief. There was a general belief in angels and demons, flying familiarly, yet skittishly, through the air, and that belief caused untold distress and pain and tragedy. The most holy Catholic Church (and, after it, the various Protestant sects) enforced the dogma that heresy was terribly sinful and punishable by death. Imagine\u2014but all you need do is to recount\u2014the suffering entailed by that belief.\n\nWhen one surveys the causes and consequences of credulity, it is apparent that this easy belief in the impossible, this readiness toward false and fanatical notions, has been indeed a most serious and major crime against humanity. The social life in any age, it may be said, is about what its extent of credulity guarantees. In an extremely credulous age, social life will be cruel and dark and treacherous. In a skeptical age, social life will be more humane. We assert that the philosophy of humanity\u2014that the best interests of the human race\u2014demands a strong statement and a repeated, enlightening statement of atheism.\n\n# **\"Spiritual Realities\"**\n\nWhen preachers talk about \"spiritual realities,\" what do they mean? They do not mean the emotions of men. At least they do not mean these emotions as realistically observed and interpreted human emotions. Love, hate, fear, greed, malice, envy, ambition, dreams, and desires\u2014these are human emotions which the rational, scientific mind takes as themes for analysis. They are understood not in any \"spiritual\" sense but in terms of heredity and environment and constitutional (physical and mental) makeup. Their causes and their expressions are, so far as science has been able to trace them, essentially material.\n\nAll of mankind's art, mankind's morality, mankind's experiments with and yearning for beauty can be and are explained in terms of human cause and effect and are placed in the evolutionary pattern worked out by science. They are not mysterious in the theistic sense; they are not, that is to say, mystic. An emotion in human nature is as realistic a fact as an object in nature: and science deals with both emotions and objects materialisticly, experimentally, analytically.\n\n\"Spiritual realities\" mean nothing to science. This is the special and unrealistic lingo of the clerical bunk shooters, who depend upon sweeping (but empty) phrases and pious dogmas and a large spooky and spoofy atmosphere of aimless mystery for the maintenance of their prestige. That their belief is often sincere does not affect the case.\n\nBy \"spiritual realities,\" if you probe the phrase, you will discover that the preachers mean some mystic working of the mind of a God in the minds and motives of men. They intend us to believe that human emotions are something more than human\u2014that back of them is the shadowed and obscure and awesomely immense loom on which is woven a divine pattern.\n\n\"Spiritual realities,\" according to the preachers, are the reflections of the most unreal of all myths, namely, the myth of a God. These so-called \"realities,\" said to be the highest conceivable, are seen to be the most unreal and the most inconceivable.\n\n# **Is God Fair?**\n\nThat's a funny question. But still we ask it: Is God fair? The Christians say that God damns forever anyone who is skeptical about truth of bunkistic religion as revealed unto the holy haranguers. What this means is that a God, if any, punishes a man for using his reason.\n\nIf there is a God in existence, reasons should be available for his existence. Assuming that such a precious thing as a man's eternal future depends on his belief in a God, then the materials for that belief should be overwhelming and not at all doubtful.\n\nYet here is a man whose reason makes it impossible for him to believe in a God. He sees no evidence of such an entity. He finds all the arguments weak and worthless. He doubts and he denies.\n\nThen is a God fair in visiting upon such a skeptic the penalty for his inevitable intellectual attitude? The intelligent man refuses to believe fairy tales. Can a God blame him? If so, then a God is not as fair as an ordinarily decent man. And fairness, we think, is more important than piety.\n\n\"Faith,\" said St. Paul, \"is the evidence of things not seen.\" We should elaborate this definition by adding that faith is the assertion of things for which there is not a particle of evidence and of things which are incredible.\n\n# CHAPTER VII\n\n# How Christianity Grew Out of Paganism\n\n# Christianity and Slavery\n\n_and_\n\n# Judeo-Christian Degradation of Woman\n\n# by Joseph McCabe\n\n# How Christianity Grew Out of Paganism\n\n# **How the Creed Was Synthetically Formed**\n\nIt is one of the ironies of the history of religion that what we call the great, historical, or organized religions took their rise from prophets whose mission in life was to denounce religion in the sense in which these organized bodies use the word. The very corrupt and hypocritical Buddhism of Japan and its less offensive but equally elaborate counterpart in China and other Asiatic countries takes the name of a moralist, Buddha, who was, according to all the leading authorities, an atheist and urgently warned folk to keep clear of priests, temples, and idle speculations about a future life. The vast army of Buddhist priests and monks, the most sordid priesthood in the modern world, now make a gorgeous living chiefly by inspiring simple folk for the next world.\n\nI do not include Confucianism, because it never was a religion, and it has therefore consistently preserved the atheism and high moral code of its founder. Taoism, the second religion of China, pretends to have been founded by Lao Tzu, a contemporary of Confucius, who, though inclined to mysticism, equally warned his followers against organized religion with its ceremonies and devil dodging. What is broadly called Zoroastrianism in Persia takes the name of another prophet, Zoroaster or Zarathustra, who set out to purify religion by stripping it of all that the modern Zorastrians call religion. Even Mohammed was a rebel against all organized and priestly religion, yet Islam, while rejecting the title of priests and remaining comparatively simple in its services, really has a powerful and tyrannical body of priests.\n\nBut the most notable instance of this, shall we say, ingenuous development in the religious world is Christianity. As I said, if there is one thing that we may on sound historical principles consider probable about Jesus, it is that he was an independent moral preacher\u2014as hundreds of the Essenes were in Judea at the time\u2014who so angered the Jewish priests, presumably by his criticisms of them and their temple religion, that they had him put to death as what we now call a Bolshevik. When that fine leader of the French Revolution Camille Desmoulins was asked his religion, he said: \"The same as that of the sansculotte Jesus.\" If you want me to give the proper American equivalent of the French expression, I shall have to paraphrase it: \"ragged seat of the pants.\" Fortunately, we have no quarrel here with those who think that the Gospels are real biography. Except for a ridiculous, late interpolation about his \"church,\" they uniformly represented Jesus as a violent opponent of priests, temples, services, set prayers, and every element of sacerdotal and ritual religion. Paul has the same idea.\n\nWe have nothing to do here with modern explanations of how or why this iconoclastic Gospel, which is still read in church (in the midst of the most gorgeous and elaborate ceremonies) every Sunday, became the basis of the Episcopal or the Roman Church. One of the funniest explanations, which you often hear today, is that Jesus took his fishermen lieutenants up one of those high mountains of which he was so fond and explained to them that he \"spoke in parables to the multitude so that they would not understand\"\u2014in the next page you read that he spoke to the multitude in parables, because it was the simplest and most beautiful way to get his ideas over to them\u2014but he would tell them something. What he said to the people about an approaching end of the world was\u2014well, anything you like to call it. The world was going to last for thousands of years, and for security of Christian morality they must found a great Church with buildings and choirs (including soprano eunuchs), bishops, archbishops, cardinals, priests, etc., vestments, incense, and dog collars, the power to kill heretics and dictate to princes and presidents.... Tens of millions of Christians believe that. But there is no need for any guessing about what happened. Early Christian literature makes it clear. In one of the most important, yet one of the least frequently quoted, documents of the first century is a \"Letter from the Christians of Rome to those of Corinth,\" which we may confidently date about the year 96. That \"Letter to the Corinthians\" says that the Romans have just simple meetings\u2014in each other's houses, for we know that they had no sort of chapel until after 200 AD\u2014at which they said prayers and sang a hymn (probably Jewish) or two. To keep order they had an overseer (which in Greek is the word we have corrupted into bishop) and a few assistants (deacons). The \"Letter\" says nothing about priests, who were pagan officials, much less a pope. Pliny's letter to the emperor about Christians in Bithynia suggests the same simplicity.\n\nBefore the end of the second century, the simple supper they had periodically held had grown into \"the Mass,\" and the priests who performed this very august ceremony were, of course, a special and consecrated caste. The word Mass is taken from a Latin phrase which the priest still addresses to the people at the end of the ceremony. It means, \"off you go: dismissal.\" Now we know that in ancient Rome this formula was used (in Greek) at the end of the service in the temples of Mithra and of Isis, and the temple of Mithra particularly interests us. At Rome it was almost next door to the Christian quarter, and it was a fierce rival. To be brief, if you will look up the English translation ( _The Mysteries of Mithra,_ 1903) of the smaller work on Mithraism of Franz Cumont, the highest authority on the subject, you will find that in the underground temples of Mithra, blazing with candles and stuffy with incense, priests in vestments consecrating bread and wine at an altar and communicating them to the people. The evolution of the Mass out of the Lord's Supper is clear enough, and it gave a mighty lift to the priestly caste over the laity.\n\nThe Christians still, and until the fourth century, refused to celebrate Christmas... or pay any honor to Mary. The reason again is clear. All the pagan religions celebrated either one or the other or both. If you want the details and the references in the writings of the Fathers, I must refer you to my _Little Blue Books_ 1102 and 1104, since I must confine myself here to an outline.\n\nTake Mithraism. The simpleminded Christian Father Firmicus Maternus says candidly\u2014he explains that it is all a trick of the devil to seduce Christians\u2014that Cumont has verified from the inscriptions that Mithra was a Savior God who brought \"eternal life\" to his votaries (especially through baptism and communion). They celebrated his birth in a cave, with great pomp and blaze of lights, at midnight (Midnight Mass) December 25\u2014the date when the Romans themselves celebrated \"The Birthday of the Unconquered Sun\"; and Mithra was a solar god. In the spring they celebrated the death and resurrection of Mithra with equal solemnity, as Firmicus describes in detail. The image of the dead god was exhibited on a bier in the temple and the worshipers mourned. Then\u2014Firmicus does not say how long the mourning lasted\u2014the priests bade them rejoice for the god had risen again.\n\nIn another part of Rome was the temple of Isis, and the early Christian Paschal chronicle tells us that in midwinter the temple of Isis used to exhibit a sort of tableau (as Catholic churches still do at Christmas) of the mother Isis and her newborn divine child, Horus, lying in a manger. The Greeks at the same time made great parade of their god Dionysos (or Bacchus), represented by the statue of a newborn infant lying in a basket (some say manger). At Alexandria, another early Christian writer tells us, there was a cult of a virgin mother Kore, who was particularly honored as giving birth to a divine child in midwinter. In other words, since these Greek, Persian, and Egyptian religions had temples in all cities, and December 25 was a gold-letter day in the Roman calendar as the birthday (virtually the resurrection) of the sun, there is not much doubt how the simple early Christian religion came (when it suppressed all its rivals by force) to have its most popular and characteristic festival and its cult of a virgin mother and divine babe.\n\nBut the celebration of the death and resurrection of a god was even more familiar to the Greeks and Romans. I have told how the Mithraists, who were for three centuries more numerous than the Christians and spread from Persia to Britain, celebrated annually the death and resurrection of their god. Even more familiar to the Romans was the annual celebration of the violent death and resurrection of Athis, son of the Great Mother (Cybele), because the devotees were permitted to hold their picturesque processions in the streets. The ceremonies lasted a week, which is very clearly the model of the later Holy Week. St. Augustine saw them in Rome and described them. It is enough here to say that on the first day the priests carried reeds (compare palms) in procession; the fifth day\u2014remember that Good Friday is the fifth day from Palm Sunday\u2014was \"the Day of Blood,\" when the eunuch priests gashed themselves with knives (in the East they publicly emasculated themselves) and howled as they bore a statue of the dead young god bound on a pine tree; and the last day was the Day of Rejoicing, because he had come to life again. All over the East that was one of the most famous religious ceremonies, the god being variously called Athis, Adonis, and Tammuz. The prophet Ezekiel (8:15) refers to it: \"And behold there sat women weeping over Tammuz,\" and as late as the fourth century, Jerome, Augustine, Cyril, and other Church Fathers speak of it.\n\nThis covers most of the ancient world, from Rome to Mesopotamia, but the Egyptians had another version of what we may almost call the universal myth. Firmicus says that \"they buried a statue of Osiris in the temple and honored it with an annual lamentation,\" and a few days later \"they lay aside their grief and rejoiced.\" Frazer's masterly _Adonis, Attis, Osiris_ will give you all details. And\u2014to complete the circle of the ancient world\u2014the Greeks, besides having a legend (Demetese Persophone) of their own that comes near to the common myth, borrowed the mourning festivals over both Osiris and Tammuz (Attis). Plutarch describes them holding the ceremony as early as 415 BC ( _The Life of Alcibiades_ , XV:111), and the Roman writer Ammianus Marcellinus _(Res Gestae,_ XXII) describes it in Antioch, in 362 AD.\n\nThe birth, death, and resurrection of a god, which simpleminded believers imagine to be so characteristic of Christianity, were almost as familiar in the Greek-Roman world for two or three hundred years before Christ as the Fourth of July is in America. There is not the least obscurity about the matter. In midwinter, the sun begins to wax strong again and, as the Roman writer Macrobius says, these temple representations of the birth (obviously rebirth or resurrection) of a divine infant refer to the solar deity (Mittra, Horus, etc.). At the beginning of winter, the spirit of vegetation, the divine son of Mother Earth, is stricken and dies. It is interesting that often in the myth and ceremony he dies by losing his testicles or penis, figuring the earth's loss of fertility. But he rises to full vitality again in the spring. On these great phenomena of the year, pre-civilized man built his winter and spring nature festivals and, for convenience and greater effect, later priests compressed the death and resurrection festivals into three days.\n\nThese are by no means all the cults of the ancient world that influenced Christianity. The city of Tarsus had its own god, Sandan, who died and rose again annually. Late Babylonia and Phoenicia had a savior god, Esmun, whose cult could give spiritual features to the new religion. But what I have said will suffice to illustrate how the austere creed of Paul, which was developed out of late Jewish ideas of what the Messiah was to do for them, and the still simpler religion ascribed to Jesus in the Gospels, once it won power to stay all its rivals by imperial decrees, borrowed their myths and services, their art and temple paraphernalia (vestments, incense, holy water, etc.), and attracted the old pagan worshipers.\n\n# **The Facts About Early Christianity**\n\nFor several centuries the early Christians, especially those who lived in the first century after the death of Jesus, would have none of these pagan robes and trimmings. Paul was the real founder of this early Church, and his grim spirit haunted it for a century. But a change occurred before the second century was out, and the myth that for several centuries these early Christians remained shining examples of virtue in a wicked world, ready at all times to sit on red-hot gridirons or meekly face the lions for their faith, is another element that must be set aside if you want to get a correct attitude to Christianity. It is the first point to which those who talk about our Christian civilization would appeal, and it has to be flatly and comprehensively denied.\n\nAlthough, as Paul's letters to the Corinthians plainly tell, some of the new communities were by no means solid in virtue, we are pleased to learn from his other letters and from the letter of the Romans to the Corinthians that the latter community was exceptional; and it must have been particularly difficult to sustain the Christian sex code in gay Corinth. Anyhow, we find the Roman Christians still virtuous in the year 96 AD. You may, of course, suggest that if they expected the end of the world at any moment and the alternatives being an eternity in boiling oil or amongst cool clouds, it ought not to have been difficult for a man to be satisfied with one bed; but we have nothing to do with those things. Temptation was, as a matter of fact, not as alluring in the early Church at Rome as the preachers seem to think. Apparently, they do not know that the Christians did not live in Rome at all but outside the walls in a poor foreign settlement, and that their language was Greek. On the other hand, the common picture of them as a communistic body is a libel. I say libel, because this would bring them into line with those dreadful Bolsheviks. Some of them were rich and some poor. Some had slaves, and some were slaves. Paul sends his kind regards to two of them who were high officials in the imperial palace, and later Christian historians claim that the community included cousins of the Emperor Vespasian.\n\nBut the luster of their virtue was very much dimmed before the end of the second century. A learned man arose in the Christian Roman community\u2014the only one in six or seven centuries\u2014and, as he is a \"saint\" in the Roman calendar, we must read him with respect. In his _Refutation of All Heresies,_ this Hippolytus has a few chatty pages about the life of the Roman Christians. Pope Victor, the first pope who tried to be papal and got very nastily rapped on the knuckles by the other bishops, was, it seems, a friend of a lady named Marcia, who lived in the Emperor Commodus's palace. Not to put too fine a point on it, Marcia was the lewdest concubine in the spacious harem (which also included three hundred handsome boys) of Commodus, who could have given lessons to Nero in sex matters. However, Marcia and her friend and tutor in vice, Hyacinth, who is claimed to have been a Christian, got many favors for the Christians when Pope Victor went for tea or something to the palace.\n\nHippolytus broadens the picture. One of the rich Christians directs his Christian slave Callistus to open a bank in the city, and the faithful, although the Christian code declared all interest on money to be usury and a mortal sin, all rushed to put their money in it and make a bit. Callistus embezzled the lot and went to jail, and a few years later he became pope and applied his talents to the humanization (and enrichment) of the Church. It was, he said, time they abolished this musty old idea that if a Christian sinned after baptism he or she must be expelled and considered damned forever. He and his priests could forgive sins, he said. So by this and other humane measures he opened the door of the Church to rich Roman women, and they brought in a good many things besides money. In fact, anybody who represents the Roman community after this time as an oasis of virtue in a desert of sin ought to be on the staff of a Ministry of Information.\n\nDocuments of the third century show that the dry rot spread quickly and very thoroughly to the whole Church. About the middle of the century, St. Cyprian, a very stern man, was leader of the African Church, and his letters to the pope describe how a large part of his clergy and bishops were unmitigated scoundrels: fornication, murder, embezzlement, and all the rest of it. For the Spanish Church, we have, about the end of the century, the canons, and, my word, the women were as gay as the cigar girls of modern Seville. For the East, we have a letter (in Bishop Eusebius's _History_ ) of the bishop of the place describing the behavior by the Christians of Alexandria when the general persecution opened. With mordant irony, the bishop describes how, when his Christians were summoned to the tribunal, they provoked the jeers of the crowd by nervously disowning the faith. Of course, the bishop himself, being a very necessary person, had had to avoid martyrdom.\n\nThis question of martyrdoms gives you a general indication of the character of the Christians of the thirteenth century, when the only two real general persecutions of Christians occurred, and at the same time gives you the measure of the fairy tales that are still told about the early Church\u2014told, that is to say, to the people and cherished by politicians and editorial writers, though Catholic scholars who have specialized in martyr literature (the Jesuit Father Delehaye, the Austrian Professor Ehrhard, Bishop Gregg, etc.) have to admit that it is the finest collection of forgeries that we have. For instance, Catholic historians claim that by the middle of the third century, when the Decian persecution occurred, the Roman Christians numbered about 30,000 or more. But in his special study of this persecution, Bishop Gregg finds that only half a dozen of them won the golden crown of martyrdom. Of about 150 priests and clerics of the Church, only 6 were arrested. And in the next and greatest general persecution, under Diocletian, the Catholic historian Duchesne can find only a score of genuine martyrdoms in the whole Church, and only 2 of these were at Rome.\n\nThe African very orthodox Bishop Optatus has a pleasant little story about what happened in his _History of the Donatist Schism_ (of the fourth century). A bunch of the African bishops met to discuss the appalling general apostasy of their Christians when the persecution was over. They fell to violent quarreling, and it transpired that they had all dodged the golden crown. One had, presumably with a wink at the presiding pagan official who was bribed, handed in a medical work pretending that it was the Bible. Another of the bishops was accused by his brothers in Christ of murders. \"Yes,\" he said\u2014and Optatus is copying a stenographic report of the proceedings, \"I did, and I'll knock off anybody who gets in my way.\" In short, modern experts on this literature find that only a few dozen of the supposed \"acts\" (accounts of trial and execution) of the martyrs are not blatant forgeries, and beyond these we just have the vague reports of local bishops that a number of the more zealous of their people\u2014generally enthusiasts who regarded martyrdom as a sure ticket to paradise\u2014were executed. Delehaye has shown in a special study that all stories of Christians being exposed to lions in the Roman amphitheater (which so moved Mr. George Bernard Shaw that he based a play on them) are bogus. Of three generations of Christians in the third century\u2014certainly at least six or seven million\u2014only a hundred or two did not deny the faith or take to flight; and the learned Origen himself says that for the first two centuries you could count the martyrs on your fingers.\n\nThat will give you some idea of the colossal fabrication of the early Middle Ages, on which the first part of the myth of our Christian civilization\u2014the character of the early Church\u2014is based. Naturally the corruption deepened when, after the conversion of the Emperor Constantine, the Church became rich. The legend is that the Christians were now able to build churches in Rome and attract the pagans by their virtuous lives. Let me state, very briefly, three notorious facts.\n\nIn the year 366, there was an election for the papacy, which was now very rich. The successful candidate was \"St.\" Damasus, and his methods were such that in one day his men left the corpses of 160 of his rival's supporters on the floor of a small church. The war lasted a week and was so furious that the Roman \"police\" were swept aside and the prefect driven out of the city. The second fact is that St. Jerome, who then lived in Rome as a sort of secretary to Pope Damasus, has left us a large number of letters in which he describes the character of the Christians of Rome. In almost incredible language, he insists that clergy, monks, consecrated virgins, widows, etc. are monstrously and, with very few exceptions, comprehensively corrupt. \"St.\" Damasus himself was denounced by his priests to the civil power for adultery and was only saved by the emperor. And the third undisputed fact is that there was no \"attraction\" of the pagans at all. In the extant Theodosian Code we have ten decrees which the bishops got from the emperors suppressing all rival religions and sects under pain of fine, imprisonment, or death.\n\nThe corruption of character was general in the Church. St. Augustine in his sermons and letters describes it in North Africa, which was then more flourishing and populous than it now is. St. John Chrysostom in his sermons paints an equally dark picture of the people of Constantinople and Antioch. At Antioch, he says, there are one hundred thousand Christians but he doubts if one hundred of them will ever see heaven. They laugh, he says, when he preaches on chastity. St. Gregory of Nyssa in two extant letters forbids Christian women of his diocese to make the pilgrimage to Jerusalem, because it is, he says, a hotbed of crime and vice. And so on. The Christian world was sinking into the Dark Age in which not only all attempts at restoring the shattered Greek-Roman civilization were suspended for seven centuries, but Europe fell to a level which most historians describe as barbarism.\n\nThat is [but] part of the reply to those who glibly talk about our Christian civilization. In its primitive purity Christianity was, like Paul and (possibly) Jesus, quite indifferent to what we call civilization.\n\n# Christianity and Slavery\n\n# **Paganism and Slavery**\n\nAbout the year 100 AD, a remarkable lecture on slavery was delivered in Rome. The central part of Rome was a very broad open space, the Forum, crowded with statues and works of art, lined with beautiful marble temples and public halls. In these halls lectures were delivered, just as they are in New York and Chicago today; and, as the Romans knew and practiced shorthand as well as we do, many of the lectures have been preserved for us.\n\nThe orator to whom I now refer was the eloquent Greek Stoic, Dion Chrysostom, or \"Dion of the Golden Mouth.\" He was no demagogue. At times you would see him driving about Rome with the great emperor, Trajan, of whom he was an intimate friend. He was the idol of the thoughtful section of the Roman nobility. And for the two days\u2014the subject was too large for one day\u2014Dion had announced as his subject \"Slavery\": a delicate topic, one would imagine, if pagan Rome were quite the slave driving city it is commonly supposed to have been, unless the aristocratic orator intended to justify the institution for his aristocratic audience, every member of which owned many slaves.\n\nBut Dion, as we read in the extant lectures, denounced slavery as unjust. But no religious writer in the world seems ever to have heard of Dion Chyrsostom and his denunciation of slavery.\n\nIt is quite formal, explicit and lengthy. It fills two lectures. Here is an express and honorable condemnation of slavery by a well-known friend of the emperor in the most public and effective circumstances, at a time when the Christians were a mere handful of obscure folk, mumbling a Greek liturgy and debating whether the end of the world was not at hand.\n\nIt is the reverse of the truth to say that Christianity abolished slavery and gave the world education; and I say this knowing well that H.G. Wells has endorsed the Christian claim. No one admires Wells's ability and service to this generation more than I do, but here he made, or borrowed, a statement which he had never examined. The undisputed historical facts are that:\n\n1) The Greek and Roman moralists perceived the injustice of slavery, often denounced it, and rendered great services to the slave.\n\n2) No Christian leader denounced slavery until the ninth century, when the age of slavery was over.\n\n3) In the Christian Middle Ages the workers were far worse off, because nearly everyone was a serf, and serfdom was slavery under another name.\n\n4) The betterment of the condition of the workers has been won quite independently of religion and to an enormous extent in spite of the churches.\n\nLet me underline a truth which is a simple historical fact. There have in history been two great periods of benevolence and social services: one was under the pagan Stoics and the other is under modern paganism. The Christian Era lies between these two paganisms, and it has as poor a record of social service as one can imagine. By the first century, the Stoics openly condemned slavery. Other Greek moralists besides the Stoics condemned it. Plutarch condemned it. Epicurus had come near to condemning it three centuries earlier when he had defined the slave as \"a friend in an inferior condition\"; and the Epicurean Hegesias had maintained that slaves were the equals of free men. Florentinus and Ulpian, the two famous Stoic jurists, declared that the enslavement of a man was against the law of nature, the supreme standard of the Stoic. Seneca insisted that the slaves were our \"lowly friends,\" and he pleaded repeatedly and nobly for them. Pliny shows us in his letters that by the second century the slaves were very humanely treated, even on provincial estates. Juvenal fiercely attacked inhumanity to slaves.\n\nYet I presume that all that any religious reader is likely to know about Roman slavery is that the rich patricians had large armies of slaves on their estates and treated them like cattle. He is never told that this refers to the early period of Roman expansion, and that before the end of the first century the slaves were protected by law.\n\nHe has probably heard how Cato made some callous remark about his slaves; and he is not told that the pagan writer who has preserved it for us gives it expressly as an instance of \"a mean and ungenerous spirit.\"\n\n# **The Gospel and the Slave**\n\nThere can be no doubt that if the Roman Empire had continued and developed normally slavery would have been abolished. Abolition would, as every American knows, have been a colossal task. It would have been far more terrible in Rome than in the southern [United] States, because the entire empire rested to a great extent upon slave labor. The immense privileges even of the Roman working men were based upon the labor of slaves in the provinces.\n\nYet public feeling was profoundly affected by the Stoic principle, and the \"manumission\" of slaves\u2014the grant or sale of freedom to them\u2014was a daily occurrence. Even before Christ this liberation proceeded on so large a scale that the Emperor Augustus checked it for a time, on political grounds. The Stoics urged it and facilitated it, and the final term of the movement was certain.\n\nRome, however, fell upon evil days just at the time when the humanitarian gospel was accepted. The manhood of Italy, then of the provinces, was almost exhausted in war. The empire was so vast, its frontiers so far-flung, that the military burden was terrible; and frontier wars naturally increased as the military forces weakened. The third century was one of great poverty and confusion. In the fourth century there was a recovery, but the empire was bleeding to death, and new formidable forces were advancing upon it.\n\nEarly in the fifth century it fell. The great slaveowners, the imperial estates, and the wealthy Romans were ruined. The whole economic system was shattered. The old slaves were not \"freed\": they found themselves free. No one \"broke their fetters.\" They had no fetters. But the barbarians slew or sent into exile the owners, destroyed the connection of the provinces with Rome, and wrecked the administration of the estates. The slaves dispersed and there were now no Roman troops to prevent them.\n\nThus, we can write the history of ancient slavery without any reference to Christianity. If it were not for this religious controversy which perverts the facts of history, the Christian religion would hardly be noticed in any complete and impartial study of Roman slavery. All that would be noted would be that some of the Christian emperors of the fourth century issued edicts about the condition of slaves; though they are much less important than the great measures of the pagan emperors. It would then be recorded that the new Christian masters of Europe\u2014petty princes, bishops, abbots, and landowners\u2014continued to use slave labor. But it was comparatively easy to deal with this new kind of slavery, and Christendom, tardily recognizing a little of the Stoic ethic, turned it into serfdom: which would have horrified the Stoics.\n\nHow, then, has this persistent belief that Christianity broke the fetters of the slave originated and been maintained? Naturally, in the same way as the belief that the Church emancipated woman. It is a quite modern belief. Until recent times nobody cared two pins about the social services of religion. Its business was to save souls. When men could no longer be prevented from attaching importance to social interests, however, the cry arose that religion was just the thing to serve us. The history of the past was caricatured. Already everybody believed that the era before Christ was dark and impotent, and the Christian Era brought a wonderful transformation. Part of this transformation, it was now said, was the uplifting of woman, the emancipation of the slave, the opening of schools, the purification of morals, the beginning of charity, and so on. Neither preachers nor their hearers read the facts of ancient history.\n\nWhat is there in the Bible that even tends to discourage or condemn slavery? Not a word from cover to cover. Apologists manage to find a word or two which they can twist into a desperate defense of woman, but there is not a single phrase of Jehovah or Jesus or Paul that they can, with all their ingenuity, represent as a condemnation of slavery or war, the two most colossal evils of the ancient world.\n\nAs I have said, one of the ablest of the apologists actually turns this silence of the Bible into a piece of high diplomacy. Jesus did not want to cause the economic ruin of the empire, so he did not condemn slavery! How religious readers permit such stuff to be presented to them one cannot imagine.\n\nThroughout the Bible slavery is as cheerfully and leniently assumed as are war, poverty, and royalty. In the English Bible there is frequent mention, especially in the parables, of \"servants.\" The Greek word is generally \"slaves.\" Jesus talks about them as coolly as we talk about our housemaids or nurses. Naturally, he would say that we must love them: we must love all men (unless they reject our ideas). But there is not a syllable of condemnation of the institution of slavery. Fornication is a shuddering thing; but the slavery of fifty or sixty million human beings is not a matter for strong language. Paul approves the institution of slavery in just the same way\u2014he is, in fact, worse than Jesus. He saw slaves all over the Greco-Roman world and he never said a word of protest.\n\nAs to the customary quibble, that these reforms were \"implied\" in the teaching of Jesus, it reminds me of Disraeli's famous joke. Asked his religion, he (being a rationalist, yet a politician) said that he held \"the religion of every sensible man.\" And to the question what that was, he replied that no sensible man ever tells. It reminds me also of the great achievement of Pope Leo XIII, who at last (in the eighteenth century of papal power) found the courage to declare that the worker was entitled to \"a living wage.\" But when the clergy found that working men of the nineteenth century were not so easily duped by phrases and wanted to know what was a living wage, the pope refused to answer the questions privately submitted to him.\n\nHere is another historical truth to underline: for eight hundred years no Christian leader condemned slavery. And here is one for the Roman Catholic: no pope ever condemned slavery. In Rome, the pope saw more slavery than in any other city in the world. The life of Rome was based upon the labor of millions of slaves in the provinces. All the dreadful things quoted about pagan slavery are from Roman writers. And no pope ever uttered a syllable of condemnation of slavery.\n\nNegative statements are a little dangerous. I borrowed this statement\u2014that no Christian writer condemned slavery until the ninth century\u2014from Ingram's _A History of Slavery and Serfdom_ , which is the best authority on the subject. Then I waited for the reply. It came in a shabby booklet or pamphlet from the Christian Evidence Society; and it reminded me of the Irishman's complaint about his sandwich, that there was \"so much mustard for so little meat.\" In quite a fury of righteous indignation the clerical writer exposed my \"lies\" to the contempt of the Christian world. He had found\u2014or he confessed that some industrious theologian had found for him\u2014one Christian condemnation of slavery in those eight hundred years!\n\nNow, I did not profess to have read every page of every Christian work for eight centuries. I know the Migne collection of this literature as well as anybody and have spent, in all, many weary months over it. But it was fair to assume that theologians would long ago have quoted Christian condemnations of slavery if there were any; and none had appeared. The great search now yielded a sort of condemnation of slavery in a work ascribed to Gregory of Nyssa, one of the least influential of the Fathers. How I would have treasured that solitary gem; but, alas, it was spurious. The authorship of the work is disputed, and the author, whoever he is, does not so much condemn slavery as an unjust institution but attacks all holding of property, including slaves.\n\nThe true and typical attitude of the churchman is seen in Pope \"St. Gregory the Great.\" Possibly some Catholic may be surprised at my effrontery in quoting Gregory. Did he not say in one of his letters that all men are \"born free,\" that slaves are only such by \"the law of nations,\" and that it is proper to free slaves? Oh, yes. I know the letter well: much better than the Catholic writers (and even Ingram, who, being a positivist, favors the Church when he can) who quote it. The pope is writing to two of his slaves. He is giving them their freedom. But this is the little suppressed fact\u2014they have inherited money, and Gregory secures the money for the Church!\n\nPope Gregory, my Catholic friend, was the greatest slaveowner in the world in the sixth century. Announcing that the end of the world was to come in 600 AD, he kindly allowed landowners and slaveowners to hand over their property to the Church\u2014God would not damn the Church for its wealth\u2014and enter monasteries. The papacy soon had an income from land of about two million dollars a year; a stupendous sum in those impoverished days. Enormous numbers of slaves tilled the 1,800 square miles of the Church's property. Gregory freed them occasionally: when they got money. He never condemned slavery. He would not allow any slave to become a cleric, and he expressly reaffirmed (Epp. 7:1) that no slave could marry a free Christian.\n\n# Judeo-Christian Degradation of Woman\n\nWoman could not hold any property, either earned or inherited. If unmarried, she was obliged to place it in the hands of a trustee, to whose will she was subject. If she contemplated marriage, and desired to call her property her own, she was forced by law to make a contract with her intended husband by which she gave up all title or claim to it. A woman, either married or unmarried, could hold no office of trust or power. She was not a person. She was not recognized as a citizen. She was not a factor in the human family. She was not a unit but a zero, in the sum of civilization.... The status of a married woman was little better than that of a domestic servant. By the English Common Law [in force in Boston] her husband was her lord and master. He had the sole custody of her person and of her minor children. He could punish her \"with a stick no bigger than his thumb,\" and she could not complain against him.... The common law of the State held man and wife to be one person, but that person was the husband. He could by will deprive her of every part of his property, and also of what had been her own before marriage. He was the owner of all her real estate and her earnings. The wife could make no contract and no will, nor, without her husband's consent, dispose of the legal interest of her real estate.... She did not own a rag of her clothing. She had no personal rights and could hardly call her soul her own. Her husband could steal her children, rob her of her clothing, neglect to support the family; she had no legal redress. If a wife earned money by her labor, the husband could claim the pay as his share of the proceeds.\n\n\u2014Elizabeth Cady Stanton on the situation of women in Boston, 1850\n\n# **Woman Before Christ**\n\nIn the year 586 BC, King Nebuchadnezzar of Babylon destroyed Jerusalem and carried away most of the Jews of the better class to his great city on the Euphrates. Let us imagine the dark-eyed maid Rebecca or the portly matron Susannah blinking in the light of the brilliant metropolis and then inquiring what the position of woman was.\n\nWe know well what the position of woman was in Judea. It is pithily put in Leviticus 7:1\u20135. This book had, in the year 586, not yet been forged, it is true, but it clearly gives an old law:\n\nAnd the Lord spake unto Moses, saying, Speak unto the children of Israel, saying, if a woman have conceived seed, and borne a man child; then she shall be _unclean seven days;_ according to the days of the separation for her infirmity shall she be unclean. And on the eighth day the flesh of his foreskin shall be circumcised. And she shall then continue in the blood of her purifying _three and thirty days_ ; she shall touch no hallowed thing, nor come into the sanctuary, until the days of her purifying be fulfilled. But if she bear a maid child, then she shall be unclean _two weeks_ , as in her separation; and she shall continue in the blood of her purifying _threescore and six days._ (emphasis added)\n\nIt's ironic to describe this bit of primitive tribal barbarism and superstition as a special revelation from the Most High! It just expresses woman's position under this \"foregleam of Christianity.\"\n\nThe female was an inferior creature. She never had a lover or chose a husband. Her parents handed her over to a youth who became her very despotic lord and master. She was \"unclean\" about ten times in twenty years, as a rule, to say nothing of shorter periods. She had no property, no personality. Her husband could divorce her when he willed; she could not divorce him when she willed. Her husband could take a second wife or a concubine or dally with painted ladies. Rebecca had to disguise herself as a prostitute if she wanted a change (Genesis 38:14) _._ And when she had fulfilled the whole Law, she was peppered with spiteful aphorisms (Proverbs, Ecclesiasticus, etc.) about her malice and odiousness.\n\nRebecca would not find this inspired code in Babylon. She would see the women living under the code of King Hammurabi, which the old monarch refused to ascribe to any god. And, as laws are very solid documents, let us quote at once a few clauses about woman's position, to contrast with the Jewish law:\n\n127. If a man has pointed the finger against a priestess or the wife of another man unjustifiably, that man shall be thrown before the judge, and his brow shall be branded.\n\n134. If a man has been taken prisoner, and there is no food in his house, and his wife enters the house of another; then that woman bears no blame.\n\n137. If a man has set his face to divorce a concubine who has borne him children, or a wife who has presented him with children, then he shall give back to that woman her dowry, and he shall give her the usufruct of field, garden, and property, and she shall bring up her children.\n\n138. If a man divorce his spouse who has not borne him children, he shall give to her all the silver of the bride-price, and restore to her the dowry which she brought from the house of her father. 139. If there was no bride-price, he shall give her one Mina of silver for the divorce.\n\n142. If a woman hate her husband, and says, \"Thou shalt not possess me,\" the reason for her dislike shall be inquired into. If she is careful and has no fault, but her husband takes himself away and neglects her; then that woman is not to blame. She shall take her dowry and go back to her father's house.\n\n171b. The spouse shall take her dowry, and the settlement which her husband made her and wrote in a tablet for her and she shall dwell in the domicile of her husband. While she lives she shall enjoy it; she may not sell it for silver, but after her it is her children's.\n\nNot bad for the year 2100 BC is it? In fact, immeasurably superior to the Christian law under which Christian women were living in England and New England four thousand years afterward and far better than the Jewish law of 500 BC.\n\nYet this is the oldest law of Babylon. Rebecca would, in short, find women as free and respected in Babylon in 586 BC as they are in Boston today. I show in another book how the deciphering of the literature of ancient Babylon has completely discredited those picturesque ideas of the vice of the great city which are still used to give purple patches to sermons. So far were the Babylonians from enjoying a remarkable looseness in sexual relations that they incurred sentence of death by adultery. We will hope that their practice was not as savage as their law. And there was not one law for the man and one for the woman (as in Christendom). The man and woman were bound together and thrown into the Euphrates. A man was burned alive for rape. A mother and son were burned alive for incest. A man was drowned for intercourse with his daughter-in-law. No woman was forced to prostitute herself at the temple, and there was probably no temple of that kind in Babylon.\n\nIn other words, if we were to return tomorrow to the \"morals of ancient Babylon,\" as preachers somberly announce that we may if their income is cut off, a woman would find herself protected from man's \"lust\" by a series of drastic laws which no section of Christendom ever knew!\n\nWoman was, at any rate, quite free in ancient Babylonia. Her rights were splendidly protected by law. She held property as legally as man did. She was not an inferior creature; she had no periods of uncleanness. She did not miss the Christian Mary, because Ishtar was one of the greatest deities of Babylonia and Assyria, and, at least in later Babylon, was an ethical deity. Woman had, of course, no vote, as there were no votes for either sex. In a word, she was in every respect the equal of man, free to own and control her own property and therefore be independent, able to divorce an unworthy or even merely neglectful husband at any time, protected against any encroachments upon her rights by other women. That is what Rebecca, coming from the female slavery of Judea, would find in benighted Babylon.\n\nAnd woman was just as free, equal, independent, and respected in ancient Egypt. Every single authority on ancient Egypt will tell you that. Flinders Petrie even observes that in the home the husband was merely \"a sort of boarder, a visitor, who had to keep up the establishment.\" Maspero, another of the chief authorities, puts it that the husband was \"a privileged guest\": that the many tomb inscriptions we have show that the wife was \"the beloved of her husband and the mistress of the house.\" There was no \"stick as thick as a man's thumb\" in the corner, no exploitation or bullying, no appropriation of her earnings or property. In the _Maxims of Ptah-Hotep,_ a middle-class treatise [written] four thousand years ago or more, it is said: \"She will be doubly attached if the chain is sweet to her.\" Of course there was a chain, a contract of marriage; but it was mutual. \"Make glad her heart during the time that thou hast,\" the counsel to the husband ran; and even in case of misconduct he was urged: \"Be kind to her for a season, send her not away, let her have food to eat.\"\n\nPolygamy was permitted in Egypt, but a wife could, and did, stipulate in her marriage contract that there should be none. It is one of the \"higher religions,\" Mohammedanism (in this respect borrowing from Judaism), which has brought degradation upon the woman of Egypt. For the four thousand years of life of the old pagan civilization, she was the equal of man in all respects. She could not only bring an action on her own account in court, but she could plead. She could practice medicine. She had a great career as priestess; and the female deity Isis was one of the most famous and revered, and in later times one of the most ethical, in the pantheon. The Christian Madonna and child is based most particularly upon the popular Egyptian representation of Isis and Horus. Mary is an understudy of Isis and Ishtar.\n\nAs in Babylonia, there was no political opportunity for either sex, since the form of government was a despotic monarchy, so that woman had no special injustice here. In the palace she was treated as an equal, and there were queens who made great names in Egyptian history. It was most probably Queen Tu, the mother of Amen-Hotep IV, who brought about the religious revolution which substituted the worship of one God for the polytheism of the Egyptian temples.\n\nThe beginning of civilization is dated by different authorities at various periods between 3000 and 4000 BC. This means that the stretch of time during which Egypt and Babylon were the chief representatives of civilization is far greater than the whole of subsequent history; and during all that time woman was free, independent, and the equal of man. She was \"treated with justice and respect.\"\n\nThe only other civilizations were those of the Cretans, the Hittites, and the Chinese. The Cretan is the most interesting, since it began as early as the great kingdoms of Egypt and Mesopotamia. We have not yet discovered the key to the Cretan language, and our knowledge is therefore imperfect; but if there is one point on which the authorities are agreed and upon which they feel justified in expressing an opinion, it is the position of woman. The fact that Crete had no male deity but only the Mother Earth goddess would dispose us to expect to find woman in an excellent position. That sex cannot be held inferior which is the sex of the deity.\n\nBut we have ample indications of the fact. The frescoes on the walls of the Knossos palace plainly show that there was no oriental confinement of women. They are depicted enjoying themselves on the terrace of the great palace just as ladies do on the terrace of the modern mansion. The impression of every student of Cretan remains is that women probably had in the Cretan civilization an even better position than they had in Egypt and Babylonia.\n\n[Here] we are concerned with that particular evolution of the human race into which Christianity entered as an influence. Looking back, in the light of what I have said, upon that evolution, and taking the position of woman as a test of civilization, we should have to divide the whole into two eras, the era of light and justice and the era of darkness and injustice; and it is an elementary historical truth to say that the era of light is the period before Christ and the era of darkness the time which we proudly call the _Christian Era._\n\nIt is also an elementary and uncontroverted historical truth that the recovery of woman, the removal of her wrongs, did not begin until the Christian domination of the world was profoundly shaken and reduced; it made progress in proportion as the churches grew weaker; it received no assistance whatever from Christianity; and it was brought to a triumphant issue only when the majority of men in the cities of the world had thrown off their allegiance to Christianity. These are statements of fact, not rhetoric. I have shown this for the period before Christ, and later chapters will establish the rest. And it would hardly be possible to frame a more deadly reply than these facts give to the claim that Christianity has been \"woman's best friend.\"\n\nThere is only one qualification to be made in this general statement. Egypt and Babylon did not bring the story of woman's equality down to the beginning of the Christian Era; and the influence of Christianity, on the other hand, did not begin until about the end of the fourth century. This leaves a period of eight or nine centuries between what I have called the era of light and the era of darkness. That period is filled with the stories of Greece and Rome, and we have next to see whether it may not have been these which inaugurated the degradation of woman, and whether the Church did not simply inherit an unjust social order for which it was not responsible. The further question, what the effect of the destruction of the Roman civilization by the northern barbarians was likely to be, will be considered in a later chapter.\n\n# **The Greek and Roman Woman**\n\nLet us say at once that in the Greek and Roman civilizations woman had not the position of equality and freedom which she had had in Egypt, Mesopotamia, and, apparently, Crete.\n\nThe earlier attitude of the Greeks toward woman does not concern us much. She was in a position of dependence and inequality but certainly not of degradation. Polygamy was abandoned before the historical period opened, and another change of which we have some trace improved the lot of the wife. Originally it had been the custom, as in many savage and barbaric tribes, for the wooer to make a present to the girl's father. Very early this was changed into a gift from her father, so that she was no longer a chattel, an article to be bargained for and bought.\n\nIn the earliest Greek literature, in fact, woman already has a position of respect, if not honor. Some noble types of wives are described in the Homeric poems, and the great tragedians of the earlier part of the brilliant career of Athens have given the world some of the finest types of womanhood in all literature: not scheming assassins like Judith, not bloodless and loveless automata like the Christian saints and ascetics, but normal women bearing their part in life bravely and honorably and sweetly.\n\nWhen the full light of history falls upon the Greek community we find woman in a position that certainly would not accord with modern standards. A special and secluded part of the home was set apart for the women, and while their excursions from the home were restricted, the men had full liberty. Athens and most of the Greek city-states were democracies, yet woman had no part whatever in the political life. Her place was the home.\n\nGirls, it is true, had a life of comparative freedom and, one feels that they would say, happiness. They had excellent athletic training, music, games, and graceful dancing. The old idea that a woman was a man's property to be carefully guarded from a defilement that lowered her value persisted; but there was no note of contempt, no insinuation, as in Judea, that she was unclean and useful only as a breeder of men. She was the companion of man; but it was understood that politics and war were not her concern. She was excluded from public life.\n\nQuite early in Greek life, however, a movement began for the removal of whatever wrongs and disabilities she had. The _Medea_ of the great tragedian Euripides is one of the most poignant presentments of the case for woman that was ever given to the world. Its exaggeration is so great, yet so sincerely and profoundly felt, that no woman genius could have penned a more formidable complaint. Already, also, the Greeks had the poetry of Sappho, a brilliant practical demonstration of woman's independence. The historian Herodotus also shows, Mahaffy says, a \"fair and gentle attitude toward women.\"\n\nIn fact, in the best days of Athenian prosperity there was a \"women's movement\" akin to that of modern times.\n\nIt seems, as we should expect, to have been connected with the famous Aspasta, one of the most brilliant and most respected ladies of antiquity, a friend of the leading writers and statesmen of Athens. The comedian Aristophanes ridiculed the movement, but the philosopher Plato defended it, as, apparently, Socrates had done before him. We may, in fact, infer that this \"women's rights\" movement of 2,200 years ago had very solid support amongst the educated men of Athens, and it was clearly of sufficient importance for the greatest of the comedians to devote two plays to it, the _Lysistrate_ and the _Ecclesiazusae_ (a word which might almost be translated \"suffragettes\"). In the latter play Aristophanes ironically describes the women of Athens donning male dress, taking over the work of making laws, and reforming the \"man-made\" state.\n\nThus, the fight which agitated England and America in the last part of the nineteenth and early part of the twentieth century really began and reached important proportions in Athens in the fourth century before Christ! It was the mass of the people who ridiculed the movement, as modern crowds ridiculed the suffragettes. Educated men realized that, although it was unquestionably part of the ideal of a Greek gentleman to see that his wife was happy, the position of woman was unjust. Plato eloquently pleaded that woman had, though in a less developed form, the same faculties as man and ought to have the same education. Her supposed unfitness for political life he caustically ridiculed. Epicurus, who closes the long line of Greek thinkers, had no interest in politics. He urged both men and women to avoid the corrupt and turbulent life of politics and cultivate a sober tranquility. Within the limits of his own ideal, however, he welcomed men and women on equal terms.\n\nThus, Greece, to which religious writers appeal as an example of the antagonism of the most brilliant of the pagan civilizations to woman, proves to be exactly the contrary. For three or four thousand years, in Crete, Egypt, and Mesopotamia, woman had been free and respected. Then for a few centuries we find her in Greece, not degraded or vilely used, for nearly every great Greek writer treats her with respect, but certainly in a position of dependence and inferiority. But at the very dawn of the Golden Age of Athens, a movement for her emancipation begins, and it has the support of all the best elements of Greek life.\n\nUnfortunately, Athens was ruined before the movement could reach a successful issue. Yet its ideals continued. The chief Greek writer about the time of Christ, Plutarch, maintained that woman was mentally and morally equal to man and ought to have, as Plato had said, the same education. He denied that the moral law should be interpreted more liberally in the case of man than of woman. And the last glimpse that we have in history of Greek culture before it is entirely lost in a Christianized and barbarized world is a picture of the philosopher Hypatia taking a leading part in the life of the great city of Alexandria, and by her culture and personality rising high above all her contemporaries.\n\nThe murder of Hypatia by a Christian mob is a fitting allegory of the murder of the new hope of women by the new religion. That may seem a harsh sentence, but even the broad historical facts must give the modern Christian woman ground for reflection. A movement for the emancipation of woman from grievances far lighter than those of a century ago began in Greece nearly 2,300 years ago. It gathered force and was endorsed by the most influential Greek writers. But it completely disappeared when Christianity became the religion of Europe, and it did not reappear until skepticism about Christianity spread through the civilized world.\n\nWe shall see that the prejudice which this broad fact creates against the new religion is fully confirmed by a study of Christian teaching. First, however, we must carefully consider the position of woman in the Roman civilization with which Christianity came into actual contact.\n\nIt is usual in religious literature to divide Roman history into two parts: a first part, until a century or two before the birth of Christ, in which woman was very virtuous but a slave, and a later part, in which she was free but very wicked.\n\nThis generalization is as false as most of the other \"historical\" statements upon which the supposed service to the race of the new religion is based. The women of the Roman Republic (in its earlier centuries) were assuredly very chaste and virtuous. The names of some of them rank with the names of Christian saints. But just as the chastity of the saint is a kind of commercial venture, the price of a colossal reward in heaven, so the virtue of the early Roman maid or matron may be attributed to fear of the lash or the knife. The women were the property of the men. They ranked with the children. The law did not enter a Roman's house. He had power of life and death over his wife, his children, and his slaves. Small wonder that the wife and daughters were very \"virtuous.\"\n\nYet even here woman was far better treated than she was in Judea. One of the Roman historians, Valerius Maximus, makes the almost incredible claim that there was no divorce in the Roman Republic for 520 years after its foundation! The Jewish civilization\u2014the real, not the legendary, civilization of the Hebrews\u2014was practically a contemporary of the Roman, and a record of woman's experience in the two would be an instructive document. Roman women were not confined in special quarters of the house, were not forbidden to go out to dine or to the theater, and had not separate places in the temple. They were treated with the greatest respect at home and abroad.\n\nMoreover, the tyranny of the older Roman custom broke down long before the time of Christ. Greece had been civilized only a few centuries\u2014not 1,500 years, like Christian Europe\u2014when it started a movement for the emancipation of woman. Rome, similarly, was civilized only some three or four centuries when its women began a formidable movement for emancipation and admission to political life.\n\nIn the second century before Christ scenes curiously like those of the suffrage struggle of modern times were witnessed in Rome. Crowds of women obstructed the way of the reactionary senator and loudly demanded their rights. And I may add that their greatest opponent, Cato the Elder, the personification of the old Roman discipline, is nevertheless reported to have said: \"A man who beats his wife or his children lays impious hands on that which is most holy and most sacred in the world.\" As to property, Roman women had already, in 215 BC, become so wealthy that a law\u2014a special war measure\u2014was passed restricting their property.\n\nThe suffrage, or a share in political life, was not won in Rome, because long before the birth of Christ the men themselves lost or surrendered their political power. The Republic became an Empire. There were still municipal elections, and, as \"election addresses\" which were found in the ruins of the buried city of Pompeii informed us, women took an active part in these. In the provinces, the inscriptions show, women sometimes held high municipal offices. At Rome there were \"women's clubs\" ( _conventus matronarum_ ) where the affairs of the city and the state were keenly discussed. A Roman woman could hold property just as well as a man and was often very rich and independent. Men, in fact, grumbled (as in ancient Egypt) that their wives charged an excessive interest on the money they lent their husbands!\n\nNot only the later and more liberal ideas of the Greek philosophers but the humanitarian precepts of the Stoic ethic had entered into the blood of Rome and softened the harshness of its old discipline. \"The old law,\" says the famous jurist Sir Henry Maine, \"had fallen into complete discredit and was verging on extinction.\" The end of the first century and greater part of the second century of the Christian Era were a well-recognized \"Stoic Period\" at Rome, and the emperors, a splendid series of monarchs, allowed the Stoic lawyers to infuse their humanitarian ideals into the law concerning woman and the slave. Education was, as we shall see, organized on a most generous basis, and philanthropy covered Italy with orphanages and homes for the aged and ailing. \"Anyone who knows the inscriptions,\" says Sir Samuel Dill, \"may be inclined to doubt whether private benefactions under the Antonines were less frequent and generous than in our own day.\" (Samuel Dill, _Roman Society from Nero to Aurelius_ [London: Macmillan and Co., 1925], 234).\n\nWe will return to this later. But this \"great Stoic doctrine of the brotherhood and equality of men,\" as Sir Samuel Dill calls it, completed the work of the emancipation of woman from the tyranny of the old institutions. Sir Henry Maine, perhaps the highest authority on Roman law, says of Gaius, one of the famous Stoic lawyers of the time: Woman had, in short, just been freed from the grievances which older Greek and Roman tradition had laid on her at the time when the Christian Church was spreading through the Roman world. By the end of the second century she was, socially and legally, as free as the modern woman; and she remained free to the end of the Empire. When the influence of Christianity began in the fourth century, paganism had recovered from its temporary injustice to woman. Throughout the range of Roman law, which means throughout civilization at the time, there was no degradation or subjection of woman. Yet fifteen centuries of degradation and subjection followed.\n\nThe great jurisconsult himself scouts the popular apology offered for it [the old law] in the moral inferiority of the female sex, and a considerable part of his volume is taken up with descriptions of the numerous expedients, some of them displaying extraordinary ingenuity which the Roman lawyers had devised for enabling women to defeat the ancient rules. Led by their theory of Natural Law the jurisconsults had evidently at this time assumed the equality of the sexes as a principle of their code of equity.\n\n\u2014Henry Maine, _Ancient Law_ (New York: Henry Holt and Company, 1906), 154\n\nCould there be a more ironic comment on the preacher's eloquent and constant plea that Christianity brought into the world the doctrine of the brotherhood of men? It was precisely the great Stoic doctrine of the brotherhood and equality of men (as the Protestant historian calls it) which had emancipated woman. It was the new Christian doctrine of the _inequality_ of the sexes which degraded her. The Christian scholars who claim that at least the new religion taught men a \"reverence\" for woman are almost completely ignorant of the facts. They rely only on the usual rhetoric about the vices of the pagans and the virtues of the Christians.\n\nAll this rhetoric is based upon the most scandalously loose quotation of particular instances. Even the best Christian writers ask us to blush at the crimes of Nero or Elogabalus, and never mention that during three-fourths of the Empire its rulers were good men. They say dark and vague things about the vices of Messalina and Faustine (which are grossly exaggerated), and they never tell that there were ten good pagan empresses for every bad one. They quote St. Jerome about the virtue of his score of Christian pupils, and they entirely ignore his assurance, in the same letters, that the Christian world generally was vicious and corrupt.\n\nThere was no such general contrast of pagan vice and Christian virtue; and the notion that at the adoption of Christianity the world passed from an era of vice to one of virtue, from a period in which woman was the toy of \"brutal lusts\" to a period in which she was respected because of her Christian virtues, is, as we shall see, one of the most fantastic and unjustifiable beliefs that zeal ever engendered.\n\n# **The Effect of Christianity**\n\nLet me at once oppose to this frothy rhetoric, this fanciful construction of history by men who will not take the trouble to study the facts, the sober and deliberate testimony of a great lawyer, based upon the solid foundation of the actual tenor of the law.\n\nI have quoted Sir Henry Maine's recognition of the position of justice and freedom won for women by the Stoic lawyers. \"Christianity,\" he says (p. 156), \"tended somewhat from the very first to narrow this remarkable liberty\"; and \"the latest Roman law, so far as it is touched by the Constitutions of the Christian Emperors, bears some marks of a reaction against the liberal doctrines of the great Antonine jurisconsults.\"\n\nI try always to let my reader know what the best-informed religious writer or scholar would at each step say to my arguments and facts. For the mass of popular Christian literature, I have too much disdain to answer it. It is appallingly untruthful; and, unfortunately, it is just from this literature that preachers and the great bulk of the faithful derive their ideas. Yet even the more scholarly Christian writer is apt to betray an inaccuracy of knowledge and a straining of the evidence which deprive his conclusions of value and entitle them to little or no respect. Here he would remind me that the new liberty of the Roman women was won by weakening the institution of marriage, and that Christianity strengthened marriage and so fortified and uplifted the position of woman.\n\nIt is true that the freedom and the right to own property of the Roman women were won by an attack upon the old form of marriage ( _confarreatio_ ) _._ This ancient ceremony conferred upon the husband his despotic power over the wife, and no enlightened generation would preserve it. But Roman law and religion were, as they always are, conservative, and the ceremony was evaded rather than replaced. Looser forms of marriage were introduced. They were, however, recognized in law, and were as valid as a modern marriage. All that the Christian writer can mean, if he knows the facts, is that divorce was comparatively easy and frequent, and that the Church now came forward with the doctrine that the union of man and wife was a holy thing, a sacrament, an indissoluble association. Sir Henry Maine has this in mind chiefly when he speaks of the \"reaction\" caused by the Church.\n\nAnd the distinguished Christian lawyer calls the change, in so far as there was any change\u2014there was very little in the fourth century\u2014a \"reaction\" not an \"advance,\" because he knows well that it tended to restrict the rights of women without in the least securing for them greater respect.\n\nSir Henry Maine was no theologian. In point of fact, marriage was not a sacrament to the Christians of the fourth century. An ascetic priest here and there might publish the speculations of his dreamy hours about \"sacraments,\" the Christian leaders generally might pour their disdain upon the flesh and its joys, but the great mass of the Roman people, even after their conversion to Christianity, regarded marriage and divorce as civic and human affairs and did not listen to the clergy. The Church did not obtain control of marriage until the eleventh century. Until that time the clergy themselves were commonly married, and the laity had almost as much liberty of divorce as the pagans had had. I have written a special work on this subject, _The Influence of the Church on Marriage and Divorce_ (London: Watts & Co., 1916) and carefully studied every phase of the transition.\n\nThe only meaning, therefore, that one can read into this common claim that the Church won respect for woman is that the religious writer supposes that the Church induced women to be more chaste and thus earn the respect of men.\n\nThis, as I amply show in other books, is a double untruth. It is a repetition of the threadbare legend of pagan vice and the discredited legend of Christian virtue. I have, I trust, amply shown that the charge against the Romans rests on vague and unsatisfactory grounds. I will merely give here one or two authoritative opinions. Dr. Ludwig Friedlander is one of the highest authorities on the morals of the Romans. He was a Christian and a Puritan, yet in his classical _Darstellungen aus der Sittengeschichte Romo_ , he says:\n\nThere is nothing to show that in imperial Rome shamelessness ever went so far as it did in parts about the middle of the last [eighteenth] century.\n\nWhatever the proportion of virtue and vice in this pagan world was, there is not one single bit of evidence of improvement when Christianity triumphed. There is, on the contrary, ample evidence of actual deterioration. On the one hand, the pagan ideas so far remained that we find St. Augustine himself permitting a man to take a concubine into his home if his wife proves barren _(De Bono Conjugali,_ XV\u2014Augustine, of course, refuses a corresponding right to the wife of an impotent man); we find Paulinus of Pella boasting that he guarded \"the treasure of chastity\" and was \"content with the enjoyment of the female slave of the house\" ( _Eucharisticos,_ line 166); we find the First Synod of Toledo under the archbishop laying it down in its seventeenth canon that a man who has a mistress but not a wife also may be admitted to communion. On the other hand, every contemporary Christian authority assures us that from the fifth century onward the morals of Europe suffered a swift, appalling deterioration.\n\nThus, the main plea of the writers who try feverishly to show that the new religion improved the position of woman is demonstrably false. Neither men nor women became more virtuous. We should, in fact, find any such service to woman reflected in the legislation of the Christian Emperors, and, as Sir Henry Maine says, every innovation in their laws tends to restrict woman's rights and liberties. I have in my work on marriage and divorce collected and analyzed such innovations as there were, and they constitute, as Professor J. Muirhead says ( _Historical Introduction to the Private Law of Rome_ , [London, A & C Black, Ltd., 1916], 356), \"a miserable chapter in the history of law.\"\n\nAfter the severe verdicts of two such experts it is needless to dilate on the various changes of the law by Christian emperors. Constantine restricted the grounds of divorce to three, but his successors allowed his law to lapse. Theodosius, the brutal and superstitious soldier who, as I describe in _The Triumph of Christianity_ ( _Little Blue Book_ , no. 1110, [Girard, KS: E. Haldeman-Julius Company, 1926]), bludgeoned the rival religions out of existence, applied the ideas of the clergy; and certainly no modern American will see anything but the most cruel harshness to woman in his enactments. He forbade the marriage of cousins; he imposed severe disabilities on any woman who married again after the death of her husband; he forbade a woman to remarry for five years, and then restricted her right to inherit if she had divorced her husband for a very grave cause; and, if she divorced her husband for one of the lighter causes recognized in the law, she lost her dowry, she was strictly forbidden to marry again, and she was liable to a sentence of banishment. The husband, I may add, was, in the same circumstances, permitted to marry again!\n\nThis is the new \"fairness\" to woman, as seen in the legislation of the most priest-ridden of all the Roman emperors. His laws were so brutal that his successors were compelled to repeal them, yet each in turn attempted some new interference with the right of divorce. In practice, during the whole of this period the people followed the old Roman custom of divorce by mutual consent or for any of a dozen serious reasons. In the west the ruin of the Empire reduced everything to a chaotic condition. Synods of bishops repeatedly attempted to impose indissoluble marriage, but they failed. The prayer books of the Anglo-Saxon Church as late as the ninth century recognize several grounds of divorce, and it was the same in France.\n\nIn the east, where Roman law continued in force, it was remodeled by Justinian, and six or seven specific grounds of divorce were admitted. It was not until the eleventh and twelfth centuries that the clergy got control of marriage: with what awful results you may read elsewhere.\n\nIn point of historical fact, therefore, we find little more than the failure of the Church during seven or eight centuries to achieve what it proposed to do: to set up indissoluble marriage. Its real influence was apart from legislation, and we must ask how far its teaching was calculated to improve or did in fact improve the position of woman.\n\nThe crop of feminist literature which the rapid progress of the women's movement brought into existence a few years ago contains more than one work of a Christian character. As I will tell in the last chapter, this task had been for decades left almost entirely to non-Christian writers and agitators. In their books\u2014the works of Mrs. Cady Stanton, Mrs. Gage, A. Bebel, etc.\u2014the grievance against the Christian religion was very frankly, if not very vehemently, expressed. Neither Jesus nor any follower of Jesus before the nineteenth century had helped woman; and even in three-quarters of the nineteenth century, in spite of the flagrant degradation of woman's position, very few Christians\u2014and the world was still overwhelmingly Christian\u2014and no clergymen would open their lips. In spite of them the movement throve, and then Christian accounts of the evolution of woman's position and her debt to Christianity began to appear.\n\nI have one of the most orthodox of these before me, and I seek with some interest the page in which the lady describes woman's indebtedness to Christ. It is amusing. Jesus, it seems, insisted that man and woman were equal, because he described them as \"twain in one flesh.\" I should have thought that Christian delicacy would prevent any lady from trying to read a mystic meaning into that very broad remark.\n\nNext, we are reminded that after his resurrection Jesus appeared first to a woman: which is one of the least historical statements in the latest parts of the Gospels. Thirdly, and this completes the case, we are reminded that Jesus saved an adulterous woman from death and bade her \"go, and sin no more,\" which is one of the most flagrantly apocryphal stories in the New Testament; for any such interference with the course of Jewish law would have been drastically resented. The writer discreetly omits to notice that in text after text Jesus speaks harshly of his mother, and in no single text has he a tender or affectionate word for her.\n\nJesus, the dreamer who saw the end of the world close at hand, the ascetic who saw in the sweet charm of woman only a snare of the devil, never troubled for a moment about woman's position. Not one word of his was calculated to affect it. The Gospel writers, it is true, attribute to him certain words about divorce (not mentioned by the Protestant lady to whom I have just referred) which have had a profound influence on woman; but these words are given in two contradictory versions, and this plainly shows that they are merely the interpolated opinions of two rival schools in the early Church. The Jews themselves in the time of Jesus were divided as to whether divorce should be permitted for adultery or entirely forbidden. The quarrel passed on into Christianity. What the devout Christian ought to ask himself is why Jesus, who (he believes) foresaw all things, permitted this ambiguity on a fundamental issue to be given to the world in his name?\n\nOn this, however, we will not linger. Protestant America in practice rejects both versions of the words of Jesus. The indissolubility of marriage, or even the restriction of divorce to adultery, is so patent a cause of immorality that whether Jesus said it or not, the good Protestant demands a more proper and humane and less disastrous rule of life.\n\nI have said that Jesus never gave a thought to the question of woman's position, and it is usual to blame Paul for the profoundly mischievous spirit which got into the early Church. But Paul, here at least, does not differ from Jesus. It is the asceticism or puritanism of both that led to the degradation of woman. She is the chief implement used by the devil in his scheme of temptation. The gentle touch of her hair upon your cheek, the ripe lips that draw you, the tender pressure of her breasts, the white rounded limbs and finely molded neck\u2014the devil smiles cynically in every one of her allurements, and the ascetic sees only that imaginary smile.\n\nWhat Paul adds to this general asceticism is a reminder of the childish, yet fateful, references to woman in Genesis. She was made out of man and for man; and she at once dragged him down from a state of semi-divine bliss to all the horrors and labors of earthly life. Man, says Paul, not only may, but must, remove his hat in church, \"for as much as he is the image and glory of God\"; but woman must veil her head, for she was made from man and for man, not in the image and likeness of God (I Cor _._ 11:5\u201310). There is no need to quote Paul's other depreciations of woman; though, if there is anything in Christian tradition, he found the devoted attentions of maids like Thecla very comforting amidst his labors.\n\nWe have only to glance over the works of the Fathers to see how these two ideas\u2014the ascetic scorn of woman's sweetest ministry in life, her love, and the belief in her inferiority on account of her origin and her primary share in the fall\u2014brought into the life of Europe the element which caused and explains the degradation of woman. Scores of these quotations from the Fathers are given in feminist histories, while even the Christian historians can find no pleasant or tender sentiment to put in the scale against these profoundly mischievous aphorisms.\n\nAlready in the second century, Justin pronounced that \"a man who marries a divorced woman is guilty of adultery,\" and Athenagoras sneered at a second marriage as \"a decorous sort of adultery.\" St. Basil said: \"The wife will take back her husband when he returns to her from his whoring, but the husband will cast forth a sinful wife from his house.\" Origen said that \"bigamists are saved in the name of Christ, but are no means crowned by him.\"\n\nContempt of marriage is associated with contempt of woman. St. Gregory of Nazianzum, supposed to be the most tender and emotionally the least flinty, I should say, of the Fathers, gives counsel in poetic form to the maid Olympias. She is not to study philosophy, for \"Woman's philosophy is to obey the laws of marriage.\" She must take no interest in public affairs or attend even innocent entertainments. She may marry, but \"blessed is the man who soils not the divine image within him with the filth of concupiscence.\" And at last it comes: \"Fierce is the dragon, cunning the asp: but woman has the malice of both.\"\n\nTertullian is almost as eloquent. For Christian women he writes a book, _De Cultu Feminarum_ [On the Adornment of Women] _._ \"What do they want with finery?\" he asks.\n\nIf, my beloved sisters, your faith were as firm as its eternal reward, no one of you, after learning of the living God and her own condition as a woman, would dare to seek gay apparel, but would dress in rags and remain in dirt as a sorrowful and penitent Eve.\n\nAnd a few lines later he hisses this fine Christian sentiment: \"Thou art the devil's gate, the betrayer of the tree, the first deserter of the divine law.\" Marriage, even, he says, is \"a sort of fornication.\"\n\nThe learned Clement of Alexandria said: \"It brings shame to a woman even to reflect of what nature she is.\" Gregory the Wonder-Worker said: \"You may find one chaste man among a thousand, but not a woman.\" St. Ambrose said: \"It becomes not the modesty of a maid to choose a husband.\" Paulinus of Nola invented the phrase \"weaker vessel\" on the ground that the devil found woman easier to tempt.\n\nSt. Augustine said, in his _Commentary on Genesis_ (XIX), that \"man was made to rule, woman to obey\"; and in a second _Commentary_ he asks with portentous seriousness why woman was created at all and can find no decent answer. Possibly, he reflects, it was just part of the divine scheme to create a being of inferior personality so as to insure that the fall would take place!\n\nHis contemporary, St. Jerome, who was the spiritual director of the prettiest and wealthiest young ladies of the Roman Church, habitually scorned woman and all tender sentiment. Woman is \"the root of all evil,\" and marriage is \"good for those who are afraid to sleep alone at night.\" In one beautiful letter, which his lady pupils learned by heart, he says: \"Though thy father cling to thee, and thy mother rend her garments and show thee the breasts thou hast sucked, thrust them aside with dry eyes to embrace the cross.\"\n\nThese were the sentiments which the great interpreters of the Christian message gave to the world. At first, it is obvious, the world at large took little or no notice of them. But to say, in the face of them, that the new religion uplifted woman and taught man a respect for her which the pagans had not known is worse than empty rhetoric. It is stupid.\n\nIt is very little use to remind us how the Church exalted Mary. She was not placed even so high as Isis and Cybele. But the main point is that the special treatment of Mary did not affect women generally. There was to be no second Mary. Woman was the devil's gate, the cause of the curse and of Christ's death, the weaker vessel, the worst occasion of sin and damnation, and so on, except in one special case which no other woman could imitate.\n\nNowhere else is Christian rhetoric quite so insincere. I have shown that from Paul to Augustine all the Christian leaders used contemptuous language about woman, and no language, either of Jesus or any distinguished follower of his can be quoted in the opposite sense. On the other hand, as we shall now see, the historical facts are as plain as facts can be. Woman was not uplifted, but she sank, when Europe became entirely and thoroughly Christian, to a position which she had not known throughout the whole history of civilization.\n\n# **The Middle Ages and Woman**\n\nThere is no need to attempt here to trace the gradual descent of woman. Her position at the beginning of the nineteenth century is enough. It was a legacy from the Middle Ages. Pagan Rome had made of her a personality. Christian Rome degraded her to a chattel, a dependent, a weaker vessel, a wholly inferior creature. From the fifth century onward she sank steadily, though differently in different countries.\n\nIn England women had a good deal of political influence\u2014I mean women of the noble families\u2014almost down to the Reformation, though the position of the mass of the women throughout the Middle Ages was vile. In most countries, the days of chivalry, which were plainly due to the old pagan spirit of the Teutons, maintained for a time the position of the aristocratic women. Incautious Catholic writers sometimes quote this age of chivalry as one of the proofs that their religion won respect for woman. All that they know about it, as a rule, is from Tennyson's puritanical version of the legendary King Arthur and his legendary knights. Chivalry was a glorification of lust, the lust of fighting and the lust of love. If a woman was pretty and aristocratic, she met very courtly deference. If she was aristocratic and not pretty, she had a worse time than the Greek matron had had. If she was pretty and not aristocratic\u2014well, the adventures of Rebecca in Scott's _Ivanhoe,_ which comes nearer to the historical truth than Tennyson's pretty _Idyls,_ fairly illustrate her chances in life.\n\nIn short, this argument from chivalry is one more good illustration of the feebleness and carelessness of all religious arguments. As I said, a beautiful and noble woman had quite a good time. Knights fought for her \"favors.\" You may or may not trace the influence of Jesus of Nazareth either in the fight\u2014which was a real deadly combat\u2014or in the tender reward of the victor, but it is more important to keep this clearly in mind: not one woman in tens of thousands during the age of chivalry got the least advantage from its amorousness. You pick out the baron's pretty wife or daughter and remind us of the knights bowing to the dust before her; and you say nothing whatever about the wives and daughters of the ten thousand serfs or yeomen, or even craftsmen and merchants. _Their_ life could be hell. _They_ were protected neither by law nor religion against the \"lusts\" of these very hot-blooded and masterful knights. And then you boast about the religion of the brotherhood and equality of men!\n\nThere has been a good deal of talk about the medieval _fits primae noctis:_ the right of the lord to have the peasant bride in his own bed for the first night, or the first few nights, after the wedding. Someone discovered that there was such a right in medieval Europe, and for a time it figured as an important item in the indictment of the Christian Middle Ages. Now, religious writers say it has all been exploded. \"Modern historians\" find that there was no such right. The Church has been libeled, as usual.\n\nThere was no libel whatever. The _jus primae noctis_ was for centuries, and over a large part of Europe, a recognized institution. Sometimes it was the bishop or abbot who had the right to the bride for a night\u2014if his retainers reported her a virgin and attractive to the episcopal palate. As late as the sixteenth century we find the French peasantry rebelling against the odious custom. It was well-known in medieval England. It was not generally a formal law for the simple reason that barons and abbots did not dream, as a rule, of waiting until the peasant's daughter married. She was rarely a virgin when she did marry. Bede, the English clerical chronicler, tells us that in his time in England it was the custom for the lord to take to his harem any attractive peasant girl on his lands, then sell her when she became pregnant. There was hardly such a thing as chastity in Europe during the Middle Ages. A wench was a wench.\n\nDoes anybody imagine that a peasant or a smith would \"go to court\" in the Middle Ages when a soldier or a noble manhandled his daughter?\n\nMrs. Cady Stanton hardly exaggerates when she says that \"mankind touched the lowest depths of degradation\" in the medieval treatment of women. The brutality and grossness which the Church suffered to spread over Europe were bound to degrade woman when the strong arm of Roman law no longer protected her. Europe had undergone one of the most amazing transformations in history. Streams of barbaric Goths and Vandals had poured all over it. Their fertility seems to have been stupendous. They crossed into Africa and swamped the Roman province of North Africa as well as Spain. Law and justice and education disappeared. The barbaric chief became a feudal noble and was a law to himself. His women became as loose and vicious as any wealthy Roman woman who had ever lived. \"Conjugal morals returned to brutality,\" says Legouve, the historian of female morality ( _Histoire morale des femmes_ [Paris: Librairie Acad\u00e9mique Didier et Cie, 1869], 183).\n\nNo one questions that several centuries of moral chaos, of the densest ignorance and most unbridled license and violence, followed the Christianization of Europe. But how, I am asked, could one expect the Church to tame at once these myriads of hot-blooded barbarians from the north and the mixed population they bred everywhere in Europe? The answer is quite easy, as far as the treatment of woman is concerned. The pagan myths of these barbarians had persuaded them to treat woman with respect. If you hold that the Christian myth was even more favorable to women, how is it that it entirely failed where paganism had succeeded? The plain truth is that the old Teutonic religion had taught men that woman is a superior being, and Christianity taught exactly the opposite. One modern historian has gone so far as to say that the Goths and Vandals embraced Christianity because it taught the inferiority of woman! They now had a free run.\n\nIt is at all events clear that the teaching of the Fathers continued to be the belief of the bishops; that is to say, when they happened to be really religious, which was not often. It was precisely the best Christians, the men most eager to apply Christian doctrine to life, who damned woman. All through the period of chaotic transition from Roman civilization to the later civilization of Europe, we find gatherings of bishops depreciating women, like the Council of Auxerre, in 578, which said that women must not, like men, take the sacramental cup in their own hands, because they are \"impure.\"\n\nThe decrees of these councils of bishops and of the popes make up what is called canon law or Church law. When Europe settled down again to a rudimentary sort of civilization, its law consisted of fragments of the old Roman law and the native laws of the barbarians with a good deal of canon law. The Church was all powerful. Abbots and bishops were dispensers of justice.\n\nNow both the Roman and the barbaric laws bad been favorable to woman, yet the new law of Europe was grossly unjust to her. It was then that she was legally and socially degraded; that she lost control of her person and property, lost her legal personality, and lost all respect except that which a man pays to the woman whose caresses he desires. Sir Henry Maine and every other authority trace the degradation to the Church law. Maine says:\n\nNo society which preserves any tincture of Christian institutions is likely to restore to married women the personal liberty conferred on them by the middle Roman law; but the proprietary disabilities of married females stand on quite a different basis from their personal incapacities, and it is by the tendency of their doctrines to keep alive and consolidate the former that the expositors of the Canon law have deeply injured civilization.\n\nSir Henry shows that the British law, which, through colonial days, was responsible for the state of things in Boston as late as 1850, owes all its enormities to Church law, and therefore to Christianity. And no one with the least authority on the subject questions this. One of the recent feminist historians who certainly does not write from an anticlerical point of view, Mrs. Georgiana Hill, says:\n\nAlthough women appear to have had a wider field of activity than they afterwards enjoyed when social life became more complex, there was a counteracting influence which told against the development and free exercise of their energies. This was the influence of the Church. It was the policy of the Church to keep women in a subordinate position.\n\n\u2014 _Women in English Life,_ [London: Richard Bentley & Sons, 1896], vii)\n\nThe \"wider field of activity\" in the Middle Ages means that all sorts of trades were open to women. Some, like brewing, were almost monopolized by them. As the population was kept almost stationary by war, famine, and disease, there was no sex rivalry in work. Women have always had liberty to work. Let us grant man that.\n\nWhen Mrs. Hill and others remind us that women might become abbesses\u2014a new trade opened by Christianity\u2014one must again deplore the lack of sense of proportion. For every abbess who had a little power, there were tens of thousands of women whom the Church had degraded to the level of children or cattle. It is as if the world had discovered what Genesis really taught. \"Male and female created he them,\" says the English Bible. But the Hebrew word for \"female\" is, literally, \"A thing to be perforated.\" That was all woman had become. You may say that the Church provided an escape from all the brutality: the nunnery. Alas, most nunneries during the Middle Ages were totally corrupt. In any case, it is a rather curious service to the race to make a country desolate and then provide one or two quiet retreats amidst the desolation.\n\nBut it is needless to brood any longer over the period of woman's degradation. It was in every respect except art a foul period, a state of semi-barbarism; and, while the Church can be indicted only for permitting the hourly violence, the fearful cruelty, the sordid ignorance, the universal exploitation, the incredible filth and disease, and the almost equally incredible growth of prostitution and monastic hypocrisy, it has a direct and clearer responsibility for the injustice to woman.\n\n# **The Clergy and the Modern Struggle**\n\nAfter about the year 500, \"human life was suspended for a thousand years,\" says a brilliant French writer. Something like that certainly will be the unanimous verdict of historians when our scholars have shed the last trace of subservience to the clergy. At present, some of them have an affectation of showing that the Middle Ages were not quite so bad as the older historians had said. It is wrong, it appears, to call the early Middle Ages the Dark Ages, because, by diligent search, we can find a lamp in it here and there!\n\nBut we may see enough about the Middle Ages in other books. As far as our present subject, the degradation of woman, is concerned, no one is quite so foolish as to try to defend the Church. By the year 300 AD, woman was in a position of freedom and respect. She had enjoyed that position throughout nearly the whole four thousand years of civilization. After the year 500 AD\u2014allowing two centuries for the application of the principles of the new religion\u2014woman fell to a state of degradation which has no parallel in the history of any pagan nation. For more than a thousand years, during which Christianity absolutely dominated the life of man, she remained in that condition of degradation. That requires a good deal of explaining\u2014if you are reluctant to admit the obvious fact that Christianity degraded woman.\n\nAnd there is no room here for the familiar quibble that it was not Christianity but the men who professed it that injured woman. It was quite plainly the doctrine. It was the morbid puritanism about love and the legends of Genesis _._ The men who most drastically relegated woman to an inferior position were the men whom the Churches regarded as their religious heroes and oracles.\n\nThe true story of woman's recovery of the position she had held under paganism can be told in a few lines, and it is actually more significant and instructive when it is so told. From the fifth to the fifteenth century, from the death of Hypatia to the time of Petrarch at least, no one had a good word to say for woman. Not a scholar in Christendom, not a priest or writer, was inspired to make a syllable of protest against the disgraceful injustice of the system. It was the literary men of the Renaissance who began to raise woman\u2014the woman of their class\u2014to a position of equality; and the Renaissance was, notoriously, the rebirth of paganism and skepticism.\n\nThen came the Reformation and what Catholics humorously call \"the Christian Renaissance,\" or a half-hearted attempt to reform the morals of Rome under the lash of Protestantism. Europe became again intensely interested in religion. Many millions of people cut each other's throats in the name of religion. The civilization of Europe was put back a hundred years by the zeal for religion. And the attempt to emancipate woman was at once crushed.\n\nLuther gave a shrewd and healthy blow at the Catholic glorification of virginity and all the hypocrisy caused by it\u2014but he also said such things as: \"No gown worse becomes a woman than the desire to be wise.\" To say that he robbed women (how many?) of opportunities by suppressing nunneries is fatuous; but he certainly provided no other opportunities for them. The \"three Ks\" (Kinder, K\u00fcche, Kirche\u2014children, kitchen, church) were stereotyped as the ideal of the German woman.\n\nThe Reformation did nothing for women on the continent of Europe. In England, in the Elizabethan age, educated women (a tiny minority) had more freedom socially, though they lost their last hold on public life. But their new freedom was plainly due to the fact that in England the Reformation and the Renaissance occurred together. The Reformers, through a statute of Henry VIII, forbade \"women and others of low condition\" to read the Bible. The humanist's invited them to read.\n\nBut the historical facts are clear enough. Protestantism of a pure or puritanical type was as deadly to woman as Catholicism. What did she get from the Puritans of England or New England? From the Calvinists of Switzerland? From the Lutherans of Germany and Scandinavia? Nothing whatever. Protestant divines were as blind to the injustice of the system as Catholic divines were. The service of Protestantism was indirect; and I would stress that in this sense it was mighty. It smashed the tyranny of Rome and could not set up a lasting tyranny of its own. Yet to use a phrase of Emerson's in a different connection, \"Luther would, if he had foreseen the revolt of the women, have cut off his right hand rather than nail his theses to the door of the cathedral.\"\n\nThis is the stark truth about the redemption of woman from all the injustices which Christianity had brought upon her. _Not one single Christian clergyman the world over raised a finger in the work until it had so far succeeded that the clergy had to save their faces by joining it._ No amount of pulpit rhetoric, no amount of strained apology from Christian feminist writers, can lessen the significance of that fact. And to it you must add another of equal significance. The men and women who started the revolt against the injustice and carried it to the stage of invincibility were non-Christian in the proportion of at least five to one.\n\nTake the movement in America. Three of its greatest leaders, Mrs. Cady Stanton, Mrs. Gage, and Miss Susan B. Anthony, have described it minutely and conscientiously in their monumental _History of Woman Suffrage._ It began in 1820, when Frances Wright, a deist, a pupil of the British agnostic Robert Owen, invaded the [United] States. She was joined by the brilliant Ernestine L. Rose, a Polish Jewess who had cast off all theology; by Lucretia Mott, a Quaker whose views were regarded as \"heresy\" even in the Society of Friends; by Abby Kelly, another rationalistic Quaker; and by the sisters Grimke, also Quakers. I have shown in my _Biographical Dictionary of Distinguished Rationalists_ that Mrs. Cady Stanton, Mrs. Gage, and Miss Anthony, who led the fight in the next generation, were all agnostics. And for fifty years, as this detailed history shows, the clergy of America were the most deadly enemies of the movement, basing their opposition expressly upon the Bible.\n\nI smiled when, in 1917, 1 was invited to speak for the movement in New York. It was then respectable. Parsons were available by the score. Few in the movement had ever heard of Fanny Wright or Abby Kelley or Ernestine Rose and the other splendid pioneers.\n\nNone knew of the time when pastoral letters had circulated amongst the American clergy calling their attention to \"the dangers which at present seem to threaten the female character with widespread and permanent injury.\" That was all over. Preachers were now assuring them that Christianity was the best friend, the only friend, that woman had ever had!\n\nIt was the same everywhere. In Britain, the pioneers were Mary Wollstonecraft, Fanny Wright, George Eliot, Harriet Martineau\u2014all rationalists\u2014supported by Godwin, Robert Owen, Jeremy Bentham, G.J. Holyoake, and J.S. Mill\u2014all agnostics or atheists. In Germany, the work was done by Max Stirner, Karl Marx, B\u00fcchner, Engels, Bebel, and Liebknecht\u2014all atheists. In France, it was Siey\u00e8s and Condorcet\u2014atheists\u2014who first pleaded for the emancipation of woman, and George Sand, Michelet, Saint-Simon, and Fourier\u2014all deep-dyed heretics\u2014who raised the plea again in the nineteenth century. In Scandinavia, Ibsen and Bj\u00f8rnsen and Ellen Key\u2014all rationalists\u2014led the protest.\n\nLet the women of the world read their remarkable story once more, with open eyes.\n\n# CHAPTER VIII\n\n# Why Science Leaves Religion in the Dust\n\n_and_\n\n# Twenty Reasons to Abandon Christianity\n\n# by Chaz Bufe\n\n# Why Science Leaves Religion in the Dust\n\nKnow-nothingism has become fashionable on the religious right. Many right-wing fundamentalists insist that assertions contained in an ancient mishmash of a book are every bit as valid as carefully arrived at, repeatedly tested scientific theories and conclusions.\n\nIn a striking bit of irony, some go even further and (unconsciously) mimic academic postmodernists, insisting that all \"opinions\" (including scientific conclusions) are equal. Thus, willful ignorance among the least educated mirrors willful ignorance among the most educated.\n\nGiven all this, it's good to remind ourselves of why facts matter and why science is superior to religious faith.\n\nFailure to take facts into account has real-world consequences. To cite a trivial example, if you believe you're invulnerable because you believe you are, test your hypothesis by stepping in front of a truck. To cite a sadder, all too real example, science has established that the similarities between human beings vastly outweigh the differences, and that there's no basis for assertions that any race is superior to any other. So, are the opinions of racists just as valid as the scientific conclusion that the differences between racial groups are trivial?\n\nTo cite still another example of why facts matter, in the Middle Ages in Europe, with science at a standstill, many believed that disease and bad weather were caused by witchcraft. End result? Tens, perhaps hundreds, of thousands of \"witches\" were brutally murdered for \"causing\" storms and disease.\n\nThere are innumerable other examples demonstrating why facts matter. And, yes, you can't absolutely prove anything, but probabilities are so high in so many cases that it's reasonable to act as if the probability is 100 percent.\n\nSo, facts do matter. But why does science trump religion?\n\n**1. The scientific method is the only reliable way to arrive at the most probably correct explanation of almost _anything_.** Scientists reach conclusions by formulating hypotheses, checking those hypotheses against observed phenomena, devising experiments to test the hypotheses, checking them for internal consistency, and checking to see if the hypotheses can generate accurate predictions. Then doing all this over and over again, with different scientists repeatedly testing the hypotheses (\"theories\" if they consistently pass all these tests over a prolonged period of time) through experiment, observation, and analysis. This is a bit different than pointing to a hoary book written by iron-age slaveholders and asserting, \"This is a fact! It says so here!\"\n\n**2. Science is self-correcting. Religion isn't.** Science continually tests and refines hypotheses and theories to arrive at more accurate explanations. Religion doesn't.\n\nA good example of this is provided by scientific exploration of racial differences between humans. In the nineteenth century, some scientists asserted that whites were superior to other races. By the middle of the twentieth century, other scientists had definitively debunked those assertions through observation, experiment, and analysis. (Yes, there are still a few racist scientists, but their assertions are knocked down almost as soon as they make them, and the vast majority of scientists now accept, in line with scientific research, that assertions of racial superiority or inferiority are baseless.)\n\nThe overt racism of the Book of Mormon slightly predates the racist assertions of some nineteenth-century scientists, with the Book of Mormon referring to Caucasians as \"white and exceedingly fair and delightsome\" (2 Nephi 5:20\u201321); and as late as 1935, Mormon prophet Joseph Fielding Smith asserted that \"because of [Cain's] wickedness he became the father of an inferior race\" ( _The Way to Perfection_ [Salt Lake City: Genealogical Society of Utah, 1931], 101).\n\nFinally, in 1978, in response to widespread social condemnation (and undoubtedly a desire to increase the number of potential converts), then prophet Spencer W. Kimball announced a new \"revelation\" that the church should abandon its racial restrictions on the priesthood (but not the \"revealed\" racist passage in 2 Nephi, nor the racist statements of previous \"prophets\"). That's a bit different than the way science handled the matter, eh?\n\n**3. Science improves daily life. Religion doesn't.** One clear example of this is in the field of medicine. Scientists discovered the microbial nature of disease. That discovery led to use of antiseptics and the later development of antibiotics, which have saved the lives of untold millions. In contrast, religion has led to no developments that improve daily life. (And please don't start talking about the power of prayer and the peace it supposedly brings\u2014we're speaking here of demonstrable physical improvement.)\n\n**4. Science leads. Religion lags.** A good example of this is our understanding of the universe beyond the earth. Early scientists (Copernicus, Galileo et al.) led the way to accurate description of the physical universe. At the same time, the Church was insisting that the sun revolves around the earth, and hauling scientists who dared to state the opposite before the Inquisition.\n\nAnother example is the scientific versus religious attitude toward women. Science has established that while there are obvious and not so obvious differences between men and women, their intellectual abilities are almost identical (with a few \"end of the bell curve\" differences in a few specific areas). In contrast, religion has insisted on the inferiority and consequent subordination of women from antiquity. To cite but two of a great many Bible verses denigrating women, \"How then can man be justified with God? or how can he be clean that is born of a woman?\" (Job 25:4) and \"These [redeemed] are they which were not defiled with women\" (Revelation 14:4).\n\nToday, some religions have acknowledged reality and accept the equality of men and women. Others have dug in their heels and still insist upon female subordination, though most are now wary of openly stating that women are inferior. And it's safe to say that the more conservative the religion\u2014that is, the more literally its members take their scriptures\u2014the more likely they are to insist upon the inferiority and subjugation of women.\n\n**5. Finally, science opens doors and religion closes them** , as Neil deGrasse Tyson famously remarked. Science not only leads to improvement in daily life but to broader intellectual horizons; it encourages people to think for themselves, to question everything; it leads to one question after another. Religion insists that all the answers are contained in ancient holy books, and that it's wrong, even dangerous, to question those answers\u2014that you have an intellect, but you shouldn't use it. It's hard to conceive of anything more stultifying.\n\n# Twenty Reasons to Abandon Christianity\n\nThis essay looks at many of the reasons that Christianity is undesirable from both a personal and a social point of view. All of the matters discussed here have been dealt with elsewhere at greater length, but that's beside the point: the purpose of \"Twenty Reasons to Abandon Christianity\" is to list the most outstanding misery producing and socially destructive qualities of Christianity in one place. When considered in toto, they lead to an irresistible conclusion: that Christianity must be abandoned, for the sake of both personal happiness and social progress.\n\nAs regards the title, \"abandon\"\u2014rather than \"suppress\" or \"eliminate\"\u2014was chosen deliberately. Attempts to coercively suppress beliefs are not only ethically wrong, but in the long run they are often ineffective\u2014as the resurgence of religion in the former Soviet Union demonstrates. If Christianity is ever to disappear, it will be because individual human beings wake up, abandon their destructive, repressive beliefs, and choose to be here now in the only life we have.\n\n**1. Christianity is based on fear.** While today there are liberal clergy who preach a Gospel of love, they ignore the bulk of Christian teachings, not to mention the bulk of Christian history. Throughout almost its entire time on earth, the motor driving Christianity has been\u2014in addition to the fear of death\u2014fear of the devil and fear of hell. One can only imagine how potent these threats seemed prior to the rise of science and rational thinking, which have largely robbed these bogeys of their power to inspire terror. But even today, the existence of the devil and hell are cardinal doctrinal tenets of almost all Christian creeds, and many fundamentalist preachers still openly resort to terrorizing their followers with lurid, sadistic portraits of the suffering of nonbelievers after death. This is not an attempt to convince through logic and reason; it is not an attempt to appeal to the better nature of individuals; rather, it is an attempt to whip the flock into line through threats, through appeals to a base part of human nature\u2014fear and cowardice.\n\n**2. Christianity preys on the innocent.** If Christian fearmongering were directed solely at adults, it would be bad enough, but Christians routinely terrorize helpless children through grisly depictions of the endless horrors and suffering they'll be subjected to if they don't live obedient Christian lives. Christianity has darkened the early years of generation after generation of children, who have lived in terror of dying while in mortal sin and going to endless torment as a result. All of these children were trusting of adults, and they did not have the ability to analyze what they were being told; they were simply helpless victims, who, ironically, victimized following generations in the same manner that they themselves had been victimized. The nearly two thousand years of Christian terrorizing of children ranks as one of its greatest crimes. And it's one that continues to this day.\n\nAs an example of Christianity's cruel brainwashing of the innocent, consider this quotation from an officially approved, nineteenth-century Catholic \"children's\" book, _Tracts for Spiritual Reading_ , by Rev. John Furniss, C.Ss.R.:\n\nLook into this little prison. In the middle of it there is a boy, a young man. He is silent; despair is on him.... His eyes are burning like two burning coals. Two long flames come out of his ears. His breathing is difficult. Sometimes he opens his mouth and breath of blazing fire rolls out of it. But listen! There is a sound just like that of a kettle boiling. Is it really a kettle which is boiling? No; then what is it? Hear what it is. The blood is boiling in the scalding veins of that boy. The brain is boiling and bubbling in his head. The marrow is boiling in his bones. Ask him why he is thus tormented. His answer is that when he was alive, his blood boiled to do very wicked things.\n\nThere are many similar passages in this book. Commenting on it, William Meagher, vicar-general of Dublin, states in his approbation:\n\nI have carefully read over this Little Volume for Children and have found nothing whatever in it contrary to the doctrines of the Holy Faith; but on the contrary, a great deal to charm, instruct and edify the youthful classes for whose benefit it has been written.\n\n**3. Christianity is based on dishonesty.** The Christian appeal to fear, to cowardice, is an admission that the evidence supporting Christian beliefs is far from compelling. If the evidence were such that Christianity's truth was immediately apparent to anyone who considered it, Christians\u2014including those who wrote the Gospels\u2014would feel no need to resort to the cheap tactic of using threats to inspire \"belief\" (more accurately, \"lip service\"). That the Christian clergy have been more than willing to accept such lip service (plus the money and obedience that go with it) in place of genuine belief is an additional indictment of the basic dishonesty of Christianity.\n\nHow deep dishonesty runs in Christianity can be gauged by one of the most popular Christian arguments for belief in God: Pascal's wager. This \"wager\" holds that it's safer to \"believe\" in God (as if belief were _volitional!_ ) than not to believe, because God might exist, and if it does, it will save \"believers\" and condemn nonbelievers to hell after death. This is an appeal to pure cowardice. It has absolutely nothing to do with the search for truth. Instead, it's an appeal to abandon honesty and intellectual integrity and to pretend that lip service is the same thing as belief. If the patriarchal God of Christianity really exists, one wonders how it would judge the cowards and hypocrites who advance and bow to this particularly craven \"wager.\"\n\n**4. Christianity is extremely egocentric.** The deep egocentrism of Christianity is intimately tied to its reliance on fear. In addition to the fears of the devil and hell, Christianity plays on another of humankind's most basic fears: death, the dissolution of the individual ego. Perhaps Christianity's strongest appeal is its promise of eternal life. While there is absolutely no evidence to support this claim, most people are so terrified of death that they cling to this treacly promise insisting, like frightened children, that it must be true. Nietzsche put the matter well: \" _salvation of the soul_ \u2014in plain words, the world revolves around _me_.\" It's difficult to see anything spiritual in this desperate grasping at straws, this desperate grasping at the illusion of personal immortality.\n\nAnother manifestation of the extreme egotism of Christianity is the belief that God is intimately concerned with picayune aspects of and directly intervenes in the lives of individuals. If God, the creator and controller of the universe, is vitally concerned with your sex life, you must be pretty damned important. Many Christians take this particular form of egotism much further and actually imagine that God has a plan for them, or that God directly talks to, directs, or even does favors for them.1 If one ignored the frequent and glaring contradictions in this supposed divine guidance and the dead bodies sometimes left in its wake, one could almost believe that the individuals making such claims are guided by God. But one can't ignore the contradictions in and the oftentimes horrible results of following such \"divine guidance.\" As \"Agent Mulder\" put it (perhaps paraphrasing Thomas Szasz) in a 1998 _X-Files_ episode, \"When you talk to God it's prayer, but when God talks to you it's schizophrenia.... God may have his reasons, but he sure seems to employ a lot of psychotics to carry out his job orders.\"\n\nIn less extreme cases, the insistence that one is receiving divine guidance or special treatment from God is usually the attempt of those who feel worthless\u2014or helpless, adrift in an uncaring world\u2014to feel important or cared for. This less sinister form of egotism is commonly found in the expressions of disaster survivors that \"God must have had a reason for saving me\" (in contrast to their less worthy of life fellow disaster victims, whom God\u2014who controls all things\u2014killed). Again, it's very difficult to see anything spiritual in such egocentricity.\n\n**5. Christianity breeds arrogance, a chosen-people mentality.** It's only natural that those who believe (or playact at believing) that they have a direct line to the Almighty would feel superior to others. This is so obvious that it needs little elaboration. A brief look at religious terminology confirms it. Christians have often called themselves \"God's people,\" \"the chosen people,\" \"the elect,\" \"the righteous,\" \"the saved,\" etc., while nonbelievers have been labeled \"heathens,\" \"infidels,\" and \"atheistic Communists\" (as if atheism and Communism are intimately connected). This sets up a two-tiered division of humanity, in which \"God's people\" feel superior to those who are not \"God's people.\" That many competing religions with contradictory beliefs make the same claim seems not to matter at all to the members of the various sects that claim to be the only carriers of \"the true faith.\" The carnage that results when two competing sects of \"God's people\" collide\u2014as in Ireland, the Indian subcontinent, and Palestine\u2014would be quite amusing but for the suffering it causes to the innocent.\n\n**6. Christianity breeds authoritarianism.** Given that Christians claim to have the one true faith, to have a book that is the Word of God, and (in many cases) to receive guidance directly from God, they feel little or no compunction about using force and coercion to enforce \"God's Will\" (which they, of course, interpret and understand). Given that they believe or pretend that they're receiving orders from the Almighty, who would cast them into hell should they disobey, it's little wonder that they feel no reluctance, and, in fact, are eager, to intrude into the most personal aspects of the lives of nonbelievers.\n\nThis is most obvious today in the area of sex, with Christians attempting to deny women the right to abortion and\u2014ignoring overwhelming scientific evidence\u2014to mandate ineffective abstinence-only sex \"education\" in the public schools. It's also obvious in other areas of education, with Christians attempting to force biology teachers to teach their creation myth (but not those of Hindus, Native Americans et al.) in place of (or as being equally valid as) the very well-established theory of evolution. But the authoritarian tendencies of Christianity reach much further than this.\n\nUp until well into the twentieth century, in the United States and other Christian countries (notably Ireland), Christian churches pressured governments into passing laws forbidding the sale and distribution of birth control devices, and they also managed to enact laws forbidding even the description of birth control devices. This assault on free speech was part and parcel of Christianity's shameful history of attempting to suppress \"indecent\" and \"subversive\" materials (and to throw their producers in jail or burn them alive). This anti\u2013free speech stance of Christianity dates back centuries, with the cases of Galileo Galilei and Giordano Bruno (who was burnt alive) being good illustrations of it. Perhaps the most colorful example of this intrusive Christian tendency toward censorship is the Catholic Church's Index of Prohibited Books, which dates from the sixteenth century and which was abandoned only in the latter part of the twentieth century\u2014not because the Church recognized it as a crime against human freedom, but because it could no longer be enforced (not that it was ever systematically enforced\u2014that was too big a job even for the Inquisition).\n\nChristian authoritarianism extends, however, far beyond attempts to suppress free speech; it extends even to attempts to suppress freedom of belief. In the fifteenth century, under Ferdinand and Isabella, at about the time of Columbus's discovery of the New World, Spain's Jews were ordered either to convert to Christianity or to flee the country; about half chose exile, while those who remained, the \" _conversos_ ,\" were favorite targets of the Inquisition. A few years later, Spain's Muslims were forced to make a similar choice.\n\nThis Christian hatred of freedom of belief\u2014and of individual freedom in general\u2014extends to this day. Up until the late nineteenth century in England, atheists who had the temerity to openly advocate their beliefs were jailed. Even today in many parts of the United States laws still exist that forbid atheists from serving on juries or from holding public office. And it's no mystery what the driving force is behind laws against victimless \"crimes\" such as nudity, sodomy, fornication, cohabitation, and prostitution.\n\nIf your nonintrusive beliefs or actions are not in accord with Christian \"morality,\" you can bet that Christians will feel completely justified\u2014not to mention righteous\u2014in poking their noses, often in the form of state police agencies, into your private life.\n\n**7. Christianity is cruel.** Throughout its history, cruelty\u2014both to self and others\u2014has been one of the most prominent features of Christianity. From its very start, Christianity, with its bleak view of life, its emphasis upon sexual sin, and its almost impossible to meet demands for sexual \"purity,\" encouraged guilt, penance, and self-torture. Today, this self-torture is primarily psychological, in the form of guilt arising from following (or denying and, thus, obsessing over) one's natural sexual desires. In earlier centuries, it was often physical. W.E.H. Lecky, in _The History of European Morals_ , relates:\n\nFor about two centuries, the hideous maceration of the body was regarded as the highest proof of excellence.... The cleanliness of the body was regarded as a pollution of the soul, and the saints who were most admired had become one hideous mass of clotted filth.... But of all the evidences of the loathsome excesses to which this spirit was carried, the life of St. Simeon Stylites is probably the most remarkable.... He had bound a rope around him so that it became embedded in his flesh, which putrefied around it. A horrible stench, intolerable to the bystanders, exhaled from his body, and worms dropped from him whenever he moved, and they filled his bed.... For a whole year, we are told, St. Simeon stood upon one leg, the other being covered with hideous ulcers, while his biographer [St. Anthony] was commissioned to stand by his side, to pick up the worms that fell from his body, and to replace them in the sores, the saint saying to the worms, \"Eat what God has given you.\" From every quarter pilgrims of every degree thronged to do him homage. A crowd of prelates followed him to the grave. A brilliant star is said to have shone miraculously over his pillar; the general voice of mankind pronounced him to be the highest model of a Christian saint; and several other anchorites [Christian hermits] imitated or emulated his penances.\n\nGiven that the Bible nowhere condemns torture and sometimes prescribes shockingly cruel penalties (such as stoning and burning alive) and that Christians so wholeheartedly approved of self-torture, it's not surprising that they thought little of inflicting appallingly cruel treatment upon others. At the height of Christianity's power and influence, hundreds of thousands of \"witches\" were brutally tortured and burned alive under the auspices of ecclesiastical witch finders, and the Inquisition visited similarly cruel treatment upon those accused of heresy. Henry Charles Lea, in _A History of the Inquisition of the Middle Ages_ , reports:\n\nTwo hundred wretches crowded the filthy gaol and it was requisite to forbid the rest of the Conversos [Jews intimidated into converting to Christianity] from leaving the city [Jaen, Spain] without a license. With Diego's assistance [Diego de Algeciras, a petty criminal and kept perjurer] and the free use of torture, on both accused and witnesses, it was not difficult to obtain whatever evidence was desired. The notary of the tribunal, Antonio de Barcena, was especially successful in this. On one occasion, he locked a young girl of fifteen in a room, stripped her naked and scourged her until she consented to bear testimony against her mother. A prisoner was carried in a chair to the auto da f\u00e9 with his feet burnt to the bone; he and his wife were burnt alive.... The cells in which the unfortunates were confined in heavy chains were narrow, dark, humid, filthy and overrun with vermin, while their sequestrated property was squandered by the officials, so that they nearly starved in prison while their helpless children starved outside.\n\nWhile the torture and murder of heretics and \"witches\" is now largely a thing of the past, Christians can still be remarkably cruel. One current example is provided by the Westboro Baptist Church of Topeka, Kansas. Its members picket the funerals of victims of AIDS and gay bashings, brandishing signs reading, \"God Hates Fags,\" \"AIDS Cures Fags,\" and \"Thank God for AIDS.\" The founder and long-time pastor of this church reportedly once sent a \"condolence\" card to the bereaved mother of an AIDS victim, reading \"Another Dead Fag.\"2 Christians are also at the forefront of those advocating vicious, life destroying penalties for those who commit victimless \"crimes,\" as well as being at the forefront of those who support the death penalty and those who want to make prison conditions even more barbaric than they are now.\n\nBut this should not be surprising coming from Christians, members of a religion that teaches that eternal torture is not only justified, but that the \"saved\" will enjoy seeing the torture of others. As St. Thomas Aquinas put it:\n\nIn order that the happiness of the saints may be more delightful and that they may give to God more copious thanks for it, they are permitted perfectly to behold the sufferings of the damned.... The saints will rejoice in the punishment of the damned.\n\nThus, the vision of heaven of Christianity's greatest theologian is a vision of the sadistic enjoyment of endless torture.\n\n**8. Christianity is anti-intellectual, anti-scientific.** For over a millennium Christianity arrested the development of science and scientific thinking. In Christendom, from the time of Augustine until the Renaissance, systematic investigation of the natural world was restricted to theological investigation\u2014the interpretation of biblical passages, the gleaning of clues from the lives of the saints, etc.; there was no direct observation and interpretation of natural processes, because that was considered a useless pursuit, as all knowledge resided in scripture. The results of this are well-known: scientific knowledge advanced hardly an inch in the over one thousand years from the rise of Orthodox Christianity in the fourth century to the 1500s, and the populace was mired in the deepest squalor and ignorance, living in dire fear of the supernatural\u2014believing in paranormal explanations for the most ordinary natural events. This ignorance had tragic results: it made the populace more than ready to accept witchcraft as an explanation for everything from illness to thunderstorms, and tens, maybe hundreds, of thousands of women paid for that ignorance with their lives. One of the commonest charges against witches was that they had raised hailstorms or other weather disturbances to cause misfortune to their neighbors. In an era when supernatural explanations were readily accepted, such charges held weight\u2014and countless innocent people died horrible deaths as a result. Another result was that the fearful populace remained very dependent upon Christianity and its clerical wise men for protection against the supernatural evils which they believed surrounded and constantly menaced them. For men and women of the Middle Ages, the walls veritably crawled with demons and witches, and their only protection from those evils was the Church.\n\nWhen scientific investigation into the natural world resumed in the Renaissance\u2014after a thousand-year-plus hiatus\u2014organized Christianity did everything it could to stamp it out. The cases of Copernicus and Galileo are particularly relevant here, because when the Catholic Church banned the Copernican theory (that the earth revolves around the sun) and banned Galileo from teaching it, it did not consider the evidence for that theory: it was enough that it contradicted scripture. Given that the Copernican theory directly contradicted the Word of God, the Catholic hierarchy reasoned that it must be false. Protestants shared this view. John Calvin rhetorically asked, \"Who will venture to place the authority of Copernicus above that of the Holy Spirit?\"\n\nMore recently, the Catholic Church and the more liberal Protestant congregations have realized that fighting against science is a losing battle, and they've taken to claiming that there is no contradiction between science and religion. This is disingenuous at best. As long as Christian sects continue to claim as fact\u2014without offering a shred of evidence beyond the anecdotal\u2014that physically impossible events (\"miracles\") occurred or are still occurring, the conflict between science and religion will remain. That many churchmen and many scientists seem content to let this conflict lie doesn't mean that it doesn't exist.\n\nToday, however, the conflict between religion and science is largely being played out in the area of public school biology education, with Christian fundamentalists demanding that their creation myth be taught in place of (or along with) the theory of evolution in the public schools. Their tactics rely heavily on public misunderstanding of science. They nitpick the fossil record for its gaps (hardly surprising given that we inhabit a geologically and meteorologically very active planet), while offering absurd interpretations of their own which we're supposed to accept at face value\u2014such as that dinosaur fossils were placed in the earth by Satan to confuse humankind, or that Noah took baby dinosaurs on the ark.\n\nThey also attempt to take advantage of public ignorance of the nature of scientific theories. In popular use, \"theory\" is employed as a synonym for \"hypothesis,\" \"conjecture,\" or even \"wild guess,\" that is, it signifies an idea with no special merit or backing. The use of the term in science is quite different. There, \"theory\" refers to a testable, well-developed, logically consistent explanation of a phenomenon and an explanation that is consistent with observed facts and that has withstood repeated testing. This is very different from a wild guess. But fundamentalists deliberately confuse the two uses of the word in an attempt to make their religious myth appear as valid as a well-supported scientific theory.\n\nThey also attempt to confuse the issue by claiming that those nonspecialists who accept the theory of evolution have no more reason to do so than they have in accepting their religious creation myth, or even that those who accept evolution do so on \"faith.\" Again, this is more than a bit dishonest.\n\nThanks to scientific investigation, human knowledge has advanced to the point where no one can know more than a tiny fraction of the whole. Even the most knowledgeable scientists often know little beyond their specialty areas. But because of the structure of science, they (and everyone else) can feel reasonably secure in accepting the theories developed by scientists in other disciplines as the best possible current explanations of the areas of nature those disciplines cover.\n\nThey (and we) can feel secure doing this because of the structure of science and, more particularly, because of the scientific method. That method basically consists of gathering as much information about a phenomenon (both in nature and in the laboratory) as possible, then developing explanations for it (hypotheses), and then testing the hypotheses to see how well they explain the observed facts and whether or not any of those observed facts are inconsistent with the hypotheses. Those hypotheses that are inconsistent with observed facts are discarded or modified, while those that are consistent are retained, and those that survive repeated testing are often labeled \"theories,\" as in \"the theory of relativity,\" \"the theory of gravity,\" and \"the theory of evolution.\"\n\nThis is the reason that nonspecialists are justified in accepting scientific theories outside their disciplines as the best current explanations of observed phenomena: those who developed the theories were following standard scientific practice and reasoning\u2014and if they deviate from that, other scientists will quickly call them to task.\n\nNo matter how much fundamentalists might protest to the contrary, there is a world of difference between \"faith\" in scientific theories produced using the scientific method and subject to near continual testing and scrutiny and faith in the entirely unsupported myths recorded three thousand years ago by slaveholding goat herders.\n\nNearly five hundred years ago Martin Luther, in his _Table Talk_ , stated: \"Reason is the greatest enemy that faith has.\" The opposite is also true.\n\n**9. Christianity has a morbid, unhealthy preoccupation with sex.** For centuries, Christianity has had an exceptionally unhealthy fixation on sex, to the exclusion of almost everything else (except power, money, and the infliction of cruelty). This stems from the numerous \"thou shalt nots\" relating to sex in the Bible. That the Ten Commandments contain a commandment forbidding the coveting of one's neighbor's wife but do not even mention slavery, torture, or cruelty\u2014which were abundantly common in the time the Commandments were written\u2014speaks volumes about their writer's preoccupation with sex (and women as property).\n\nToday, judging from the pronouncements of many Christian leaders, one would think that \"morality\" consists solely of what one does in one's bedroom. The Catholic Church is the prime example here, with its moral pronouncements rarely going beyond the matters of birth control and abortion, and with its moral emphasis seemingly entirely on those matters.\n\nAlso note that the official Catholic view of sex\u2014that it's for the purpose of procreation only\u2014reduces human sexual relations to those of brood animals. For more than a century the Catholic Church has also been the driving force behind efforts to prohibit access to birth control devices and information\u2014to everyone, not just Catholics.\n\nThe Catholic Church, however, is far from alone in its sick obsession with sex. The current Christian hate campaign against homosexuals is another prominent manifestation of this perverse preoccupation. Even at this writing, condemnation of \"sodomites\" from church pulpits is still very, very common\u2014with Christian clergymen wringing their hands as they piously proclaim that their words of hate have nothing to do with gay bashings and the murder of gays.\n\n**10. Christianity produces sexual misery.** In addition to the misery produced by authoritarian Christian intrusions into the sex lives of non-Christians, Christianity produces great misery among its own adherents through its insistence that sex, except the very narrow variety it sanctions, is evil, against God's law. Christianity proscribes sex between unmarried people, sex outside of marriage, homosexual relations, bestiality,3 and even \"impure\" sexual thoughts. Indulging in such things can and will, in the conventional Christian view, lead straight to hell.\n\nGiven that human beings are by nature highly sexual beings, and that their urges very often do not fit into the only officially sanctioned Christian form of sexuality (monogamous, heterosexual marriage), it's inevitable that those who attempt to follow Christian \"morality\" in this area are often miserable, as their strongest urges run smack-dab into the wall of religious belief. This is inevitable in Christian adolescents and unmarried young people in that the only \"pure\" way for them to behave is celibately\u2014in the strict Christian view, even masturbation is prohibited. Philip Roth has well described the dilemma of the religiously\/sexually repressed young in _Portnoy's Complaint_ as \"being torn between desires that are repugnant to my conscience and a conscience repugnant to my desires.\" Thus, the years of adolescence and young adulthood for many Christians are poisoned by \"sinful\" urges, unfulfilled longings, and intense guilt (after the urges become too much to bear and are acted upon).\n\nEven after Christian young people receive a license from Church and state to have sex, they often discover that the sexual release promised by marriage is not all that it's cracked up to be. One gathers that in marriages between those who have followed Christian rules up until marriage\u2014that is, no sex at all\u2014sexual ineptitude and lack of fulfillment are all too common. Even when Christian married people do have good sexual relations, the problems do not end. Sexual attractions ebb and flow, and new attractions inevitably arise. In conventional Christian relationships, one is not allowed to act on these new attractions. One is often not even permitted to admit that such attractions exist. As Sten Linnander puts it, \"with traditional [Christian] morality, you have to choose between being unfaithful to yourself or to another.\"\n\nThe dilemma is even worse for gay teens and young people, in that Christianity _never_ offers them release from their unrequited urges. They are simply condemned to lifelong celibacy. If they indulge their natural desires, they become \"sodomites\" subject not only to earthly persecution under religion-inspired laws but to being roasted alive forever in the pit. Given the internalized homophobia Christian teachings inspire, not to mention the very real discrimination gay people face, it's not surprising that a great many homosexually oriented Christians choose to live a lie. In most cases, this leads to lifelong personal torture and gross unfairness to their spouses, who deserve someone who desires them sexually, but it can have even more tragic results.\n\nA prime example is Marshall Applewhite, \"John Do,\" the guru of the Heaven's Gate religious cult. Applewhite grew up in the South in a repressive Christian fundamentalist family. Horrified by his homosexual urges, he began to think of sexuality itself as evil, and eventually underwent castration to curb his sexual urges.4 Several of his followers took his anti-sexual teachings to heart and likewise underwent castration before, at Applewhite's direction, killing themselves.\n\n**11. Christianity has an exceedingly narrow, legalistic view of morality.** Christianity not only reduces, for all practical purposes, the question of morality to that of sexual behavior, but by listing its prohibitions, it encourages an \"everything not prohibited is permitted\" mentality. So, for instance, medieval inquisitors tortured their victims, while at the same time they went to lengths to avoid spilling the blood of those they tortured, though they thought nothing of burning them alive. Another very relevant example is that until the latter part of the nineteenth-century Christians engaged in the slave trade, and Christian preachers defended it, citing biblical passages from the pulpit.\n\nToday, with the exception of a relatively few liberal churchgoers, Christians ignore the very real evils plaguing our society\u2014poverty; homelessness; hunger; militarism; a grossly unfair distribution of wealth and income; ecological despoliation; corporate greed; over-population; sexism; racism; homophobia; misogyny; freedom denying, invasive laws against victimless \"crimes\"; an inadequate educational system; etc., etc.\u2014unless they're actively working to _worsen_ those evils in the name of Christian \"morality\" or \"family values.\"\n\n**12. Christianity encourages acceptance of real evils while focusing on imaginary evils.** Organized Christianity is a skillful apologist for the status quo and all the evils that go along with it. It diverts attention from real problems by focusing attention on sexual issues, and when confronted with social evils like poverty glibly dismisses them with platitudes such as: \"The poor ye have always with you.\" When confronted with the problems of militarism and war, most Christians shrug and say, \"That's human nature. It's always been that way, and it always will be.\" One suspects that two hundred years ago their forebears would have said exactly the same thing about slavery. This regressive, conservative tendency of Christianity has been present from its very start. The Bible is quite explicit in its instructions to accept the status quo: \"Let every soul be subject unto the higher powers. For there is no power but of God: the powers that be are ordained of God. Whosoever therefore resisteth the power, resisteth the ordinance of God; and they that resist shall receive to themselves damnation\" (Romans 13:1\u20132).\n\n**13. Christianity devalues the natural world.** In addition to its morbid preoccupation with sex, Christianity creates social myopia through its emphasis on the supposed afterlife, encouraging Christians not to be concerned with \"the things of this world\" (except, of course, their neighbors' sexual practices). In the conventional Christian view, life in this \"vale of tears\" is not important\u2014what matters is preparing for the next life. (Of course, it follows from this that the \"vale of tears\" itself is quite unimportant\u2014it's merely the backdrop to the testing of the faithful.)\n\nThe Christian belief in the unimportance of happiness and well-being in this world is well illustrated by a statement by St. Alphonsus:\n\nIt would be a great advantage to suffer during all our lives all the torments of the martyrs in exchange for one moment of heaven. Sufferings in this world are a sign that God loves us and intends to save us.\n\nThis focus on the afterlife often leads to a distinct lack of concern for the natural world, and sometimes to outright anti-ecological attitudes. Ronald Reagan's fundamentalist secretary of the interior, James Watt, went so far as to actively encourage the strip mining and clear-cutting of the American West, reasoning that ecological damage didn't matter because the \"rapture\" was at hand.\n\n**14. Christianity models hierarchical, authoritarian organization.** Christianity is perhaps the ultimate top-down enterprise. In its simplest form, it consists of God on top, its \"servants,\" the clergy next down, and the great unwashed masses at the bottom, with those above issuing, in turn, thou shalts and thou shalt nots backed by the threat of eternal damnation. But a great many Christian sects go far beyond this, having several layers of management and bureaucracy. Catholicism is perhaps the most extreme example of this, with its laity, monks, nuns, priests, monsignors, bishops, archbishops, vicars general, cardinals, and popes, all giving and taking orders in an almost military manner.\n\nThis type of organization cannot but accustom those in its sway\u2014especially those who have been indoctrinated and attending its ceremonies since birth\u2014into accepting hierarchical, authoritarian organization as the natural if not the only form of organization. Those who find such organization natural will see nothing wrong with hierarchical, authoritarian organization in other forms, be they corporations, with their multiple layers of brownnosing management, or governments, with their judges, legislators, presidents, and politburos. The indoctrination by example that Christianity provides in the area of organization is almost surely a powerful influence against social change, a powerful influence against freer, more egalitarian forms of organization.\n\n**15. Christianity sanctions slavery.** In the eighteenth and nineteenth centuries, the African slave trade was almost entirely conducted by Christians. (Today, it's almost entirely conducted by Muslims.) Christian slavers transported their victims to the New World in slave ships with names such as \"Mercy\" and \"Jesus,\" where they were bought by other Christians, both Catholic and Protestant. Organized Christianity was not silent on this horror: it actively encouraged it and engaged in it. From the friars who enslaved Native Americans in the Southwest and Mexico to the Protestant preachers who defended slavery from the pulpit in Virginia, the Carolinas, and Georgia, the record of Christianity as regards slavery is quite shameful.\n\n(Those who point out that many abolitionists were Christians should remember that the abolitionists were a very small group and were well hated by most of their fellow Christians.)\n\nThe Christians who defended slavery and engaged in it were amply supported by the Bible, in which slavery is accepted as a given, as simply a part of the social landscape. There are numerous biblical passages that implicitly or explicitly endorse slavery, such as Exodus 21:20\u201321: \"And if a man smite his servant, or his maid with a rod, and he die under his hand; he shall be surely punished. Notwithstanding, if he continue a day or two, he shall not be punished: for he is his money.\" Other passages that support slavery include Ephesians 6:5, Colossians 3:22, Titus 2:9\u201310, Exodus 21:2\u20136, Leviticus 25:44\u201346, 1 Peter 2:18, and 1 Timothy 6:1. Christian slave owners in colonial America and the preachers who provided them with \"moral\" guidance were well acquainted with these passages.\n\n**16. Christianity is misogynistic.** Misogyny is fundamental to the basic writings of Christianity. In passage after passage, women are commanded to accept an inferior role and to be ashamed of themselves for the simple fact that they are women. Misogynistic biblical passages are so common that it's difficult to know which to cite. From the New Testament we find \"Wives, submit yourselves unto your own husbands, as unto the Lord. For the husband is the head of the wife, even as Christ is the head of the church\" (Ephesians 5:22\u201323) and \"These [redeemed] are they which were not defiled with women\" (Revelation 14:4); and from the Old Testament we find \"How then can man be justified with God? Or how can he be clean that is born of a woman?\" (Job 25:4). Other relevant New Testament passages include Colossians 3:18; 1 Peter 3:7; 1 Corinthians 11:3, 11:9, and 14:34; and 1 Timothy 2:11\u201312 and 5:5\u20136. Other Old Testament passages include Numbers 5:20\u201322 and Leviticus 12:2\u20135 and 15:17\u201333.\n\nLater Christian writers extended the misogynistic themes in the Bible with a vengeance. Tertullian, one of the Early Church Fathers, wrote:\n\nIn pain shall you bring forth children, woman, and you shall turn to your husband and he shall rule over you. And do you not know that you are Eve? God's sentence hangs still over all your sex and His punishment weighs down upon you. You are the devil's gateway; you are she who first violated the forbidden tree and broke the law of God. It was you who coaxed your way around him whom the devil had not the force to attack. With what ease you shattered that image of God: Man! Because of the death you merited, even the Son of God had to die.... Woman, you are the gate to hell.\n\nOne can find similarly misogynistic\u2014though sometimes less venomous\u2014statements in the writings of many other Church Fathers and theologians, including St. Ambrose, St. Anthony, Thomas Aquinas, St. Augustine, St. John Chrysostom, St. Gregory of Nazianzum, and St. Jerome.\n\nThis misogynistic bias in Christianity's basic texts has long been translated into misogyny in practice. Throughout almost the entire time that Christianity had Europe and America in its lock grip, women were treated as chattel\u2014they had essentially no political rights, and their right to own property was severely restricted. Perhaps the clearest illustration of the status of women in the ages when Christianity was at its most powerful is the prevalence of wife beating. This degrading, disgusting practice was very common throughout Christendom well into the nineteenth century, and under English Common Law husbands who beat their wives were specifically exempted from prosecution. (While wife beating is still common in Christian lands, at least in some countries abusers are sometimes prosecuted.)\n\nAt about the same time that English Common Law (with its wife beating exemption) was being formulated and codified, Christians all across Europe were engaging in a half millennium\u2013long orgy of torture and murder of \"witches\"\u2014at the direct behest and under the direction of the highest Church authorities. The watchword of the time was Exodus 22:18: \"Thou shalt not suffer a witch to live,\" and at the very minimum tens of thousands of women were brutally murdered as a result of this divine injunction and the papal bulls amplifying it (e.g., _Spondit Pariter,_ by John XXII, and _Summis Desiderantes_ , by Innocent VIII). Andrew Dickson White notes:\n\nOn the 7th of December, 1484, Pope Innocent VIII sent forth the bull _Summis Desiderantes_. Of all documents ever issued from Rome, imperial or papal, this has doubtless, first and last, cost the greatest shedding of innocent blood. Yet no document was ever more clearly dictated by conscience. Inspired by the scriptural command, \"Thou shalt not suffer a witch to live,\" Pope Innocent exhorted the clergy of Germany to leave no means untried to detect sorcerers.... [W]itchfinding inquisitors were authorized by the Pope to scour Europe, especially Germany, and a manual was prepared for their use [by the Dominicans Heinrich Kr\u00e4mer and Jacob Sprenger]\u2014\"The Witch Hammer,\" _Malleus Maleficarum_.... With the application of torture to thousands of women, in accordance with the precepts laid down in the _Malleus_ , it was not difficult to extract masses of proof.... The poor creatures writhing on the rack, held in horror by those who had been nearest and dearest to them, anxious only for death to relieve their sufferings, confessed to anything and everything that would satisfy the inquisitors and judges.... Under the doctrine of \"excepted cases,\" there was no limit to torture for persons accused of heresy or witchcraft.\n\nGiven this bloody, hateful history, it's not surprising that women have always held very subservient positions in Christian Churches. In fact, there appear to have been no female clergy in any Christian Church prior to the twentieth century (with the exception of those who posed as men), and even today a great many Christian sects, most notably the Catholic Church, continue to resist ordaining female clergy. While a few liberal Protestant Churches have ordained women in recent years, it's difficult to see this as a great step forward for women; it's easier to see it as analogous to the Ku Klux Klan's appointing a few token blacks as Klaxons.\n\nAs for the improvements in the status of women over the last two centuries, the Christian Churches either did nothing to support them or actively opposed them. This is most obvious as regards women's control over their own bodies. Organized Christianity has opposed this from the start, and as late as the 1960s the Catholic Church was still putting its energies into the imposition of laws prohibiting access to contraceptives. Having lost that battle, Christianity has more recently put its energies into attempts to outlaw the right of women to abortion.\n\nMany of those leading the fight for women's rights have had no illusions about the misogynistic nature of Christianity. These women included Mary Wollstonecraft, Victoria Woodhull, Elizabeth Cady Stanton, and Margaret Sanger, whose slogan \"No God, No Masters\" remains relevant to this day.\n\n**17. Christianity is homophobic.** Christianity from its beginnings has been markedly homophobic. The biblical basis for this homophobia lies in Genesis, in the story of Sodom, and in Leviticus. Leviticus 18:22 reads: \"Thou shalt not lie with mankind, as with womankind: it is abomination,\" and Leviticus 20:13 reads: \"If a man lie with mankind as he lieth with a woman, both of them have committed an abomination: they shall surely be put to death; their blood shall be upon them.\"\n\nThis sounds remarkably harsh, yet Leviticus proscribes a great many other things, declares many of them \"abominations,\" and prescribes the death penalty for several other acts, some of which are shockingly picayune. Leviticus 17:10\u201313 prohibits the eating of blood sausage; Leviticus 11:6\u20137 prohibits the eating of \"unclean\" hares and swine; Leviticus 11:10 declares shellfish \"abominations\"; Leviticus 20:9 prescribes the death penalty for cursing one's father or mother; Leviticus 20:10 prescribes the death penalty for adultery; Leviticus 20:14 prescribes the penalty of being burnt alive for having a three-way with one's wife and mother-in-law; and Leviticus 20:15 declares, \"And if a man lie with a beast, he shall surely be put to death: and ye shall slay the beast\" (which seems rather unfair to the poor beast). (One suspects that American Christians have never attempted to pass laws enforcing Leviticus 20:15, because if passed and enforced such laws would decimate both the Bible Belt and the cattle industry.)\n\nCuriously, given the multitude of prohibitions in Leviticus, the vast majority of present-day Christians have chosen to focus only upon Leviticus 20:13, the verse calling for the death penalty for homosexual acts. And at least some of them haven't been averse to acting on it. (To be fair, some Christian \"reconstructionists\" are currently calling for institution of the death penalty for adultery and atheism, as well as for \"sodomy.\")\n\nThroughout history, homosexuality has been illegal in Christian lands, and the penalties have been severe. In the Middle Ages, strangled gay men were sometimes placed on the wood piles at the burning of witches. One member of the British royalty caught having homosexual relations suffered an even more grisly fate: Edward II's penalty was being held down while a red-hot poker was jammed through his rectum and intestines. In more modern times, countless gay people have been jailed for years for the victimless \"crime\" of having consensual sex. It was only in 2003 that the Supreme Court struck down the felony laws on the books in many American states prescribing lengthy prison terms for consensual \"sodomy.\" And many Christians would love to reinstate those laws.\n\nThus, the current wave of gay bashings and murders of gay people should come as no surprise. Christians can find justification for such violence in the Bible and also in the hate filled sermons issuing from all too many pulpits in this country. If history is any indication, the homophobic messages in those sermons will continue to be issued for many years to come.\n\n**18. The Bible is not a reliable guide to Christ's teachings.** Mark, the oldest of the Gospels, was written twenty to forty years after Christ's death, at a time when historical records were exceedingly rare and the vast bulk of the population was illiterate. Worse, the authors of the Gospels had no direct knowledge of Christ\u2014they never met him; they almost certainly relied exclusively upon oral tradition. That's analogous to a contemporary author writing a history of the Reagan Administration while having no access whatsoever to research materials and having never even seen Reagan on television. In fact, it's worse than that, given that the other Gospels were written decades after Mark. As well, the Gospel texts have been amended, translated, and retranslated so often that it's extremely difficult to gauge the accuracy of current editions, even aside from the matter of the accuracy of texts written decades after the death of their subject. This is such a problem that the Jesus Seminar, a colloquium of over two hundred Protestant Gospel scholars mostly employed at religious colleges and seminaries, undertook in 1985 a multiyear investigation into the historicity of the statements and deeds attributed to Jesus in the New Testament. They concluded that only 18 percent of the statements and 16 percent of the deeds attributed to Jesus had a high likelihood of being historically accurate. So, in a very real sense fundamentalists\u2014who claim to believe in the literal truth of the Bible\u2014are not followers of Jesus Christ; rather, they are followers of those who, decades or centuries later, put words in his mouth.\n\n**19. The Bible, Christianity's basic text, is riddled with contradictions.** There are a number of glaring contradictions in the Bible, in both the Old and New Testaments, including some within the same books. A few examples:\n\n...God cannot be tempted with evil, neither tempteth he any man (James 1:13).\n\nAnd it came to pass after these things, that God did tempt Abraham\" (Genesis 22:1).\n\n...for I am merciful, saith the Lord, and I will not keep anger for ever (Jeremiah 3:12).\n\n\"Ye have kindled a fire in mine anger, which shall burn forever. Thus saith the Lord\" (Jeremiah 17:4).\n\nIf I bear witness of myself, my witness is not true (John 5:31; Jesus speaking).\n\n\"I am one that bear witness of myself... (John 8:18; Jesus speaking).\n\nAnd last but not least:\n\nI have seen God face to face, and my life is preserved (Genesis 32:30).\n\nNo man hath seen God at any time (John 1:18).\n\nAnd I [God] will take away mine hand, and thou shalt see my back parts... (Exodus 33:23).\n\nChristian apologists typically attempt to explain away such contradictions by claiming that the fault lies in the translation, and that there were no contradictions in the original text. It's difficult to see how this could be so, given how direct many biblical contradictions are; but even if these Christian apologetics held water, it would follow that every part of the Bible should be as suspect as the contradictory sections, thus reinforcing the previous point: that the Bible is not a reliable guide to Christ's words.\n\n**20. Christianity borrowed its central myths and ceremonies from other ancient religions.** The ancient world was rife with tales of virgin births, miracle working saviors, tripartite gods, gods taking human form, gods arising from the dead, heavens and hells, and days of judgment. In addition to the myths, many of the ceremonies of ancient religions also match those of that syncretic latecomer, Christianity.\n\nTo cite but one example (there are many others), consider Mithraism, a Persian religion predating Christianity by centuries. Mithra, the savior of the Mithraic religion and a god who took human form, was born of a virgin; he belonged to the holy trinity, he was a link between heaven and earth, and he ascended into heaven after his death. His followers believed in heaven and hell, looked forward to a day of judgment, and referred to Mithra as \"the Light of the World.\" They also practiced baptism and ritual cannibalism\u2014the eating of bread and the drinking of wine to symbolize the eating and drinking of the god's body and blood. Given all this, Mithra's birthday should come as no surprise: December 25; this event was, of course, celebrated by Mithra's followers at midnight.\n\nMithraism is but the most striking example of the appearance of these myths and ceremonies prior to the advent of Christianity. They appear\u2014in more scattered form\u2014in many other pre-Christian religions.\n\n# **A Final Word**\n\nThese are but some of the major problems attending Christianity, and they provide overwhelming reasons it should be abandoned. Even if you discount half, two-thirds, or even three-quarters of these arguments, the conclusion is still irresistible.\n\n1 A friend who read the first draft of this essay notes: \"My moronic sister-in-law once told me that God found her parking spots near the front door at Walmart! Years later, when she developed a brain tumor, I concluded that God must have gotten tired of finding parking places for her and gave her the tumor so that she could get handicapped plates.\" As Nietzsche put it in _The Anti-Christ_ : \"That little hypocrites and half-crazed people dare to imagine that on their account the laws of nature are constantly broken\u2014such an enhancement of every kind of selfishness to infinity, to impudence, cannot be branded with sufficient contempt. And yet Christianity owes its triumph to this pitiable flattery of personal vanity.\"\n\n2 The Westboro Baptist Church directly addresses the question of its hatefulness and cruelty on its website (www.godhatesfags.com): \"Why do you preach hate? Because the Bible preaches hate. For every one verse about God's mercy, love, compassion, etc., there are two verses about His vengeance, hatred, wrath, etc.\" Unfortunately, they're right about this.\n\n3 The repeated mention of this sin in medieval ecclesiastical writings leads one to wonder how widespread this practice was among the Christian faithful, including the Christian clergy. One eighth-century penitential (list of sins and punishments) quoted in Arthur West Hadden, _Councils and Ecclesiastical Documents_ , 3 vols. (Oxford: Clarendon Press, 1869\u20131878) states: \"If a cleric has fornicated with a quadruped let him do penance for, if he is a simple cleric, two years, if a deacon, three years, if a priest, seven years, if a bishop, ten years.\" Similar lists of sins and penalties can be found in many other types of penance.\n\n4 Given his religious background, and that his cult mixed Christianity with UFO beliefs, Applewhite was quite probably aware of the divine approbation of self-castration in Matthew 19:12: \"For there are some eunuchs, which were so born from their mother's womb: and there are some eunuchs, which were made eunuchs of men: and there be eunuchs, which have made themselves eunuchs for the kingdom of heaven's sake. He that is able to receive it, let him receive it.\"\n\n# CHAPTER IX\n\n# May the Farce Be with You: A Lighthearted Look at Why God Does Not Exist\n\n# by Pamela Sutter\n\n# **Introduction**\n\nAtheist. The word has shock value, but it simply means \"without belief in God.\" Unlike agnostics, who suggest that God is unknowable, and that they cannot or do not know whether God exists, atheists discount any and all supernatural entities. Atheists are not waiting for evidence of God any more than they are waiting for evidence of purple unicorns.\n\nIt is no coincidence that humans are the only species that practices religion, for we are also the only species that is aware of death. Soon after evolving the ability to think (and to think about death), we began to believe in gods: sun gods, fertility gods, volcano gods, animal gods, and the god of special interest to atheists, the monotheistic Judeo-Christian, omnipotent, universe creating God of the Bible. God belief seems to be an inevitable cultural byproduct of self-awareness.\n\nWe may be smart enough to believe in (more accurately, hope for) a higher power, but our unique human brain is also stubborn enough to continue believing\/hoping despite the discoveries of science that pretty much put God out of a job. Recent polls indicate that roughly 10 percent of Americans consider themselves to be atheists, making nonbelievers a larger minority than Jews or Mormons. That still leaves 90 percent of citizens claiming to believe in God.\n\nGod's existence, it is said, cannot be disproven. In fact, _anything_ that the mind can imagine can never be disproven. Think up something fanciful, something you know can't possibly exist, such as an invisible elf that resides in your refrigerator or a table with ten thousand legs. \" _Poof!_ \" it automatically has an above zero chance of existing, because it cannot be proven absolutely that this invention of your mind doesn't, hasn't, or won't exist somewhere at some time in the universe. All of space and time would have to be searched to prove it 100 percent false.\n\nThe logic seems absurd, but it's a necessary disclaimer. A believer who says \"prove God doesn't exist\" could just as easily demand, \"Prove there isn't a colony of Flatlanders living in Planet X's core.\" God's existence is more problematic than that of Flatlanders, for God is supernatural, not physical. The whole cosmos could be searched atom by atom and He'd never be found. Because of this, atheists consider belief in the supernatural, and hence in God, to be irrational\u2014as irrational as backwards time travel or claiming to know the final digit of pi.\n\n# **Paradoxical Prayer**\n\nA comatose young boy hovers near death. Family and friends pray he'll survive. The boy dies. The family muses that God needed him; he's been called \"home\" and is being embraced by all his loved ones that preceded him. Perhaps his parents ask, \"Why God, why?!\" and then concede that the answer would be beyond their understanding. Suppose the child survives. It's a miracle; their prayers were answered! Now let's say the boy survives, but with a severe handicap. Were the prayers only partially answered? No, the boy becomes an inspiration to others as part of God's grand plan. What about a patient down the hall who dies. Not enough prayers? God needed her more? (And for that matter, how could God \"need\" anyone?)\n\nNotice that no matter what the outcome, believers twist events to justify their sense of hope. No matter what happens, God comes out smelling like a rose.\n\nA glance at the newspapers reveals God's odd prayer answering priorities. Apparently, He favors granting record-breaking home runs over saving a doomed commuter train. He allows a planeload of praying passengers to smash into the ocean while He puts someone's multiple sclerosis into remission. He looks the other way when a toddler is kidnapped and murdered but gladly answers a farmer's prayers for rain.\n\nWhen planes don't crash or when a person doesn't get a disease, prayer again gets the credit, and God gets points for allowing life to proceed normally.\n\nPraying before a sporting event is among the most ridiculous forms of \"worship\" imaginable. To think that the Creator of the universe would care about the outcome of a high school football game is ludicrous. Did God favor the winning team? Was He angry with the player who made the losing fumble? Many believe so: one in four Americans believe that God intervenes in sporting events.\n\nHe must have really had fun with Superbowl 49 (Seahawks vs. Patriots), especially in the final minute. That God\u2014what a prankster!\n\nThe motive behind prayer is especially revealing. Prayer begs God to intervene. Isn't that second-guessing God's intent? Isn't prayer a suggestion to God that this present course might not be the best way to handle things? \"Do you really need to take little Timmy now?\" Praying implies that God's divine plan is in error, and that we have useful advice for him.\n\nWhy interfere with the Lord's plan for us? Don't pray for a loved one's cancer to be cured, for it's God's will that this disease will claim her, after which she'll be on her way to heaven. Might unanswered prayers be a test of faith? \"God won't give me more than I can handle.\" My, what a nice God, always testing us, always toying with our emotions.\n\nThe September 11, 2001 terrorist attacks on the World Trade Center and the Pentagon spawned a surge of prayers, though all too late to have changed God's mind about allowing the tragedy. Inevitably, the \"Where was God?\" question surfaced. Clergy answered, \"It's not a question of where was God or why did He let it happen, but rather, why wasn't it worse?\" So, if we can imagine something worse, we then praise God for not letting it happen? Religious believers used every excuse in the book. The alternative was notably absent: \"Maybe there is no God.\"\n\nThe fact is no prayers are answered. No prayers go unanswered. People are simply seeking meaning in otherwise inexplicable events, in much the same way that they see shapes in clouds. Prayer can be used as a form of calming meditation, which in turn can aid the body to heal, but there's nothing divine behind this placebo effect. Scientific studies show similar results from petting cats and watching fish swim in an aquarium.\n\nPrayer is simply a human reaction to apparently meaningless physical and emotional pain.\n\n# **Murky Miracles**\n\nA horrible accident took place in December 2000 in which an eight months pregnant woman was hurled through the windshield of a truck onto the pavement; the jackknifing vehicle then sliced her body in half. This caused the expulsion of the baby, still attached by the umbilical cord and very much alive. The child's survival was hailed as a miracle by the woman's husband, the driver of the truck. \"Something else must be at work here besides luck,\" he commented.\n\nA miracle? It was a miracle that the mother died an ugly death by being cut in half by a truck? A miracle that the baby will grow up without his mother and, when old enough, will be saddened by the horrible way she died? A miracle that the baby was jettisoned onto cold asphalt as his father became a widower?\n\nSorry, there was no miracle here. The human trait of seeking comfort is a strong one. Notice that the father focused only on the good: the baby survived the freak accident. People look for reasons to maintain their faith in God, and they will convince themselves they've found it, no matter what.\n\nTime and again folks overlook the negative and latch on to whatever is left. An infertile couple praises the Lord for the miracle He bestowed when the fertility drugs kicked in to give them septuplets. Why was the couple plagued by infertility to begin with? Why weren't they content with the destiny He chose for them? Maybe they were meant to adopt a special needs child instead? No. Truth be told, all \"miracles\" are miracles only in hindsight.\n\nA tornado skips along a residential neighborhood, wiping out some houses, sparing others. Those with intact houses will praise the Lord, while the neighbor next door standing amidst rubble demands, \"Why me?!\" Random happenstance is assigned meaning by pattern seeking humans. The result is a \"miracle.\" No god need apply, for the human mind has filled the position.\n\nA man has a near fatal encounter with (insert your own mayhem). In an interview he humbly explains, \"It wasn't my time.\" Has he forgotten about all the people who were snuffed out in the prime of life and infants who never got to know life at all?\n\nWatch what happens when trouble befalls a church. An earthquake in El Salvador destroyed the church in a small town and also killed hundreds under a landslide of dirt and rubble. Survivors quickly declared, \"God loves us, He left some of us alive to rebuild the church!\" Huh?! God let people die horrible, claustrophobic deaths in order for the others to see themselves as \"survivors\"? This is a good thing? Why wasn't the church spared? Come to think of it, why did the quake occur at all? Was God busy curing someone's canker sore?\n\n\"God only knows why this happened\" is another common utterance, often spoken in dismay. Indeed, He must be one sinister God to allow or to create a natural disaster. Or maybe He's a nice God, taking whole families up to heaven at once so they can discuss his \"acts of God.\"\n\n\"It's a miracle that more people weren't killed.\" Uh, it was a miracle that the earthquake killed a lot of people, but not everyone?\n\nMiracles are in the eye of the beholder.\n\n# **Illogical Heaven**\n\n\"Everybody wants to go to heaven, but nobody wants to die.\" What's the matter, do we have doubts? We should, for the concept of heaven is a nightmare.\n\nTake free will, the decision-making capability that lets us make personal choices and gives us our sense of self. As nice as free will is, it has the potential to lead to crime, lying, cheating, and other unheavenly behavior. Will we retain our free will in the afterlife?\n\nIt's a Catch-22 situation. Without free will, we become sheep. Keep our free will, and heaven would have all the problems found on earth. Either way, what's the point?\n\nPerhaps spirits won't want to rebel? Ever hear of the fallen angel Satan? And look at Adam and Eve, the first humans given a shot at paradise; they blew it. Apparently, God didn't even see that one coming.\n\nThen there's the slew of sticky social situations. The murder victim who sees her \"born-again\" attacker stroll through the Pearly Gates. Families waiting for their earthbound loved ones to die so they can all be reunited. Spouses sitting on a cloud _Family Circus_ \u2013style who spy their mates remarrying or pawning heirlooms.\n\nAnd boredom. With death out of the picture and without goals or purpose, we are going to be a restless bunch. Humans like to overcome obstacles. We thrive on new challenges and on the unexpected events of a physical world. An eternity is a very long time, and the prospect of forced immortality will have winged residents wishing they could hang themselves with their halos.\n\nSome believers claim boredom will be unheard of, that our souls will be free to embark on any endeavor in the universe. Hmmm, as long as we're making it up, might as well make it sound good, right? But, given infinite time, _anything_ would wear thin eventually.\n\nWhat about the Bible's own easy way of achieving eternal life, by believing in Christ (John 3:16)? Under this premise, a repentant rapist\/serial killer who has \"found religion\" at the last moment will get his own harp, while any of his victims who rejected Christ will be doomed to eternal torture in the fiery pit. It seems that God cares more about the flattery of His own ego than the unfairness of a killer getting into heaven while his victims do not.\n\nHeaven is nothing more than invention, false hope to make the living feel better. \"Frank Sinatra is now singing for a heavenly audience\" sounds a lot better than \"He's six feet under and never coming back,\" doesn't it?\n\nObserve the contradiction when dealing with the death of a loved one. In between sobs, people state the deceased has gone to a better place, and that there was a reason he died in the manner in which he did. Then shouldn't we be happy? We'll see them again someday? Right?\n\nHeaven has a way of making a mockery out of life. So what if Grandma died? She's with Jesus now. One little girl even stepped in front of a train to \"join\" her recently deceased mom up in heaven. If heaven is so great, you'd think \"believers\" would enthusiastically welcome death for themselves and everyone they know who is \"saved.\"\n\nTo underline what a selfish, fabricated notion heaven is, consider those who believe their pets are \"crossing over\" too. Not former chicken nuggets or last year's Christmas ham, but Fluffy, Fido, and Polly. Just because they own it, the creature must be deserving of heaven. How convenient to overlook dinosaurs, gazelles, and every trout that ever lived. But, by George, Poopsie will be there waiting for me. Hopefully not that mean pit bull next door, but Poopsie for sure. No doubt the owner of the pit bull is wishing the same thing for Fang.\n\n# **Evolution and God: Incompatible? Yes!**\n\nDarwin's \"dangerous idea\" should have made atheists out of everyone. Yet the vast majority of students who learn about evolution in public schools still believe in God. Either they doubt what they are taught in biology class or are content keeping evolution and religion in separate parts of their brain, and never the two shall meet.\n\nMany scientists say that science has nothing to do with belief in God, but let's be realistic. If God played no role in formation of the human species, then what else didn't He do? Attempts to reconcile facts with faith have led to fanciful conjectures filled with contradictions that paint believers into corners.\n\nThese attempts to reconcile religion with science include:\n\n  * _God put evolution into motion, knowing that its random path would produce humans someday._ Why use a wasteful, indirect process when He could just as easily have zapped everything into place at once? This scenario means apes are our kin, a point many people find distasteful; gone would be the special creation of Man, starting with Adam and Eve and their \"original sin.\"\n  * _God made each species separately but in stages to prepare the earth for future epochs._ This is a poor attempt at explaining the indisputable evidence of the fossil record that shows simple life preceded more complex life. There is no getting around the fact that trilobites came before fish, fish before birds, birds before mammoths, and dinosaurs before humans. In this scenario, God's first shot at creating life would have to have been something like blue-green algae. But why would God have needed to make such primitive life first? Did He need eons of practice to reach the level of mammals? Is He not powerful enough to make snails, flowers, reptiles, and birds all at once? God's supposed work parallels what fossil dating shows, which is well explained by the theory of evolution. So \"God\" as an explanation is entirely unnecessary.\n  * _An Intelligent Designer is responsible._ This argument claims that an \"Intelligence\" of some sort made life, and that this \"Intelligence\" doesn't necessarily have to be God or even be supernatural. It opens up the farcical notion that there could be multiple Intelligent Designers, yet no one attempts to explain what made them. Presumably God stands by and merely watches the Designer(s) work animal creating magic.\n  * _God created the illusion of evolution to throw scientists off._ Then He is a manipulative, deceitful god who is not to be trusted.\n  * _Evolution explains the origin of all species except for man, who was specially created in God's image, complete with eternal spirit._ Then why did He use primates as a template? We look like chimps, our DNA is closer to chimps than chimp DNA is to orangutans, a chimp fetus is a spitting image of a human fetus, and we share many behaviors. In fact, chimps are so close to us that biologists have considered adding \"homo\" to their Latin scientific name.\n  * _Evolution is fact, but God exists too._ For starters, the Bible says God created the world and all its creatures in six days (and, strangely for an omnipotent being, rested on the seventh). It doesn't say a mindless process is involved. To believe in God and to accept the fact of evolution is to invalidate an important part of the Bible. To acknowledge that life got a foothold on earth all by itself is also to acknowledge that God is uninvolved, if not entirely absent. If God doesn't care enough to choose what life to seed a planet with, then He's surely not going to care about a hairless primate's cancer or hope for an afterlife. Such a God would be distant, not the personal savior so many people want to believe in. In all honesty, evolution voids the entire supernatural content of the Bible and its petty, intrusive God.\n\nAn alarming percentage of the general public remain ignorant of the theory of evolution. They find it hard to accept that such complex creatures as humans evolved from a chemical soup. Perhaps our ignoble beginnings foster a fear that we aren't so special after all.\n\nWhy do Christian fundamentalists focus on this field of science? They feel it is a threat to their faith. And they are right.\n\n# **Got Proof?**\n\nIronically, lack of evidence is a reason that some people believe in God, for it automatically designates God as being supernatural, and hence no physical evidence is necessary. The trouble is this logic leaves the door open for magical gnomes, invisible twelve-dimensional sentient trees, and other figments of the imagination.\n\nMuch more convincing than lack of evidence is the fact that humans believed in gods long before the Bible was written. It is our nature to want to believe in something higher than ourselves. But why should the biblical God be any more credible than Zeus, Ra, or a volcano god?\n\nNot surprisingly, theists have an answer. They claim their god is purely spiritual and not limited by superstitious beliefs based on ignorance of physical phenomena like erupting volcanoes or the changing seasons. Show how a volcano works, and, poof, there go the volcano gods. Show how species arose on their own and every god's existence is in danger; so it's no surprise that Muslims and Christians oppose evolution, though the supernatural status of God allows even people who accept evolution to maintain their faith.\n\nOnce a culture has concluded that some ultimate \"higher power\" must be \"out there,\" a fear and reverence of this power, this God, naturally follows. Such a God is a product of believers' faith alone, which puts it conveniently out of reach of the physical world. This mental construct effectively makes various cultures' Supreme Being a single deity. Muslims, Jews, and Christians believe in the same single God, despite variations in doctrine, and Muslims openly acknowledge this. Such a God becomes by default, the creator of the universe. Clearly, this points to God as being simply an end product of thinking about abstract \"ultimates\"; but is this a rational basis for believing that God actually exists.\n\nAnother flaw involves circular reasoning. God exists through blind faith, but why is blind faith needed to believe in God? The Bible is the inspired Word of God. How do we know? The Bible says so.\n\nAn episode of _The Simpsons_ involves Homer becoming a genius. One of the feats he accomplishes with his enhanced brain is writing a proof showing that God does not exist. Alas, there can never be such a proof, because the supernatural is supposed to be beyond proof. God \"exists\" because He must be supernatural in order to be God. Such logic in effect says God must exist, because He cannot be disproven. This demonstrates that God is a thought experiment and nothing more.\n\nStill, this doesn't stop us from believing. Lack of evidence has never been a hindrance to belief. When prayers aren't answered or when nothing happens, we think there must be a purpose to it. But what's the difference between \"not intervening\" and \"there is no God to intervene\"? God is a product of faith, faith is a product of the mind, the mind is a product of evolution, which in turn is a product of other non-divine physical processes.\n\n# **Watering the Seeds of Doubt: Where Did God Come From?**\n\nIf some people have a hard time accepting that simple life formed from non-life in a natural chemical process and diversified into species from there, imagine the giant leap one must make to suppose that a complex intelligence such as God arose ex nihilo, or that such an entity had no beginning at all! Some have shoved God back to the beginning of time and left Him there as a remote, non-personal Cosmic Cause, in an attempt to explain the origins of the universe. But this is not an explanation. Rather, it's the use of a convenient label (\"God\") as a means of _evading_ explanation. While the circumstances \"before\" the universe existed are still unknown to science, ignorance is no excuse to invoke the supernatural.\n\n# **Why Is Life Unfair? Why Is There Evil in the World?**\n\nThis seems a tricky spot for God to wriggle out of. If He made everything, then He made evil too, so how can he be \"all good\"? If He allows evil to continue, He's negligent. If He can't fully control evil, He's not omnipotent.\n\nThere really is no such thing as \"evil\" per se, just the undesirable choices of a species with free will. But it's hard to explain why millions were murdered during the Holocaust, while God is credited with tiny \"miracles\" like putting a tumor into remission. Can the same God really allow children to starve to death in Third World countries and yet help someone in California find a missing wedding ring or win a golf tournament? Disturbingly, the \"answer\" is all too easy to fabricate: \"We wouldn't understand.\" \"God works in mysterious ways.\" \"The meaning will be clear later.\" Yada, yada, yada.\n\n# **Free Will or Sock Puppet?**\n\nMonotheistic religions posit that an all-knowing, all-seeing, omnipotent God is everywhere and everywhen. If so, is our free will an illusion? A sham? Are we living a predestined lie? It would seem so if God knows our every action in advance. He must laugh at the deep thought we put into our puny daily decisions and presumptuous prayers. He's always known the outcome; He knew it trillions of years ago. An eternity ago.\n\n# **Anthropocentric Humans**\n\nWe humans have always thought of ourselves as special. Since ancient times, we've assumed that the earth was the center of the universe. When the ancients discovered other planets, they assumed that the earth was at the center of the solar system, with the sun revolving around it. As recently as the early twentieth century, we believed ours was the only galaxy. Now we know there are billions of galaxies, and that our solar system is in the outskirts of an arm of an average spiral galaxy. The evidence has continually dashed our hopes of being center stage.\n\nToday, many people wrongly assume that evolution had us in mind as its ultimate goal, and some charts depict humans occupying the top branch of life's tree. We continue to be pompous by anthropomorphizing animals. We project ourselves onto other creatures in order to make them less like animals and more like us, as if that somehow improves them.\n\nAnd, of course, we think ourselves the special creation of a single perfect God, who thought so much of us that he granted us an eternal soul which lives on in a glorious afterlife. This last bit of self-flattery cannot be disproven by science but neither can believers prove it, and the burden of proof lies on those making assertions not those denying them.\n\n# **Nails in the Coffin**\n\nGod's coffin has more than enough nails in it. Notice how we have given it, God, a gender, a male gender: He, Him, the Heavenly Father, etc. This is still more evidence that God is a fabrication of self-centered humans.\n\n# **To Err Is Divine**\n\nFor a deity who is supposedly inerrant, Jehovah has a lot of explaining to do. Mistakes were made.\n\nExtinction is a prime example. A large majority of earth's species have gone extinct. Estimates put it at 99 percent, with today's species representing a mere 1 percent of all life-forms in the planet's history. It seems that most of God's \"perfect\" creatures weren't quite perfect enough. They couldn't adapt quickly enough, compete well enough, or weather natural disasters successfully enough (disasters that God could have averted).\n\nThis is not to say that today's species are perfect. They are simply adapted to current conditions.\n\nBut God is said to have created each life-form \"as is\" using His infinite powers of wisdom and perfection. Why then do giraffes have a hard time bending down to drink? Female kiwi birds struggle to deliver an enormously large egg. Mammalian eyes have the optic nerve passing through the retina, and, consequently, a blind spot. Humans suffer from obsolete wisdom teeth and from lower back problems caused by relatively recent bipedalism.\n\nVestigial organs also point to less than perfect workmanship. They betray a creature formed by evolution not a perfectionistic Designer. The whale's shrunken pelvis. A python's residual back leg stumps. A human's pesky appendix. Vestigials can still be utilized by an animal's body for other purposes than they originally served, but they are the telltale signs of evolution not of a perfectionist Creator.\n\nGod's planning ability is as dubious as His designing. How do the dinosaurs fit in? Why create such a diversity of \"terrible lizards\" to dominate the planet for so long when He planned to make humans later on? Did He tire of them and allow an asteroid to wipe them out? Dinosaurs are not given the play in the Bible one would expect of such incredible creatures, undoubtedly because the authors were unaware of them. God must not have thought much of them if He left them out of his \"dictations.\"\n\nGod's trouble with planning becomes obvious in the story of Adam and Eve. When God allegedly created humans, He initially made only Adam. It was after Adam expressed loneliness that the Lord created Eve. This shows an incredible lack of foresight. God gave each animal a mate but felt Adam could go it alone? Could He not foresee that Adam would need a partner too? Poor Adam had to bring this to God's attention, after which he awoke, minus a rib.\n\nAre such oversights and imperfections the intent of God's \"mysterious ways\"? No. The answer is much more straightforward. There is no need to second-guess God, for the world behaves exactly as it would if there was no God at all.\n\nWhich there isn't.\n\n# **Paradoxes**\n\nCan God create another God equal to Himself? This feat would be impossible, since the newly minted \"God II\" would be \"younger\" than the original and hence would not be equal in duration to the God that had no beginning. The very notion that this second God had a creator makes such a deity lesser. So, no, God could not make an equal God. If God did decide to produce multiple copies of Himself, how would we know? If we were to be oblivious to two, twenty, thirty, or sixty thousand Gods, why believe a single God exists?\n\nCan God kill Himself? Let's suppose an eternity of being all-knowing and all-seeing wore thin on Yahweh. Could He commit suicide? Any method that worked would have to be more powerful than God Himself, an impossibility for the Almighty. As long as we're speculating, say He did manage to pull it off. Again, how would we know? Why then believe he exists right this moment? Perhaps the universe would vanish as soon as God ceased to exist. We'd be unaware of this as well. An impossible premise yields impossible questions. God is impossible.\n\n# **But... the Universe Is Fine-Tuned for Life!**\n\nThe universe seems like Goldilock's porridge: not too hot, not too cold, but just right for life. Some take this to mean the cosmos was designed. Earth is just the right distance from the sun to support life, and the sun is just the right type of star. Conclusion: a Creator was needed. This \"anthropic universe\" scenario is a case of using an unnecessarily complex answer when a simple one is better. Hindsight in this case is blinding, not 20\/20. If the universe didn't turn out the way it did, we wouldn't be here to discuss it. We focus on how astonishing it is that we exist, the same way a lottery winner is astonished that he won, despite a 100 percent chance that somebody would win. If we weren't here or if no sentient life existed at all in the universe, something would be around, even if just a hodgepodge of atoms or empty space, just as \"lucky\" to exist as we are. We are a self-centered species that gasps, \"Golly gee!\" The universe doesn't know it's \"suited\" for life and that humans would come along and ask such questions.\n\n# **Life Is Too Diverse and Complex to Have Formed on Its Own!**\n\nBut it only had to happen once. Again, we're like the starstruck lottery winner mentioned above. A God isn't needed to explain any natural phenomenon. The only place He is needed is to give false hope to a death fearing primate species.\n\n# **Then Is Human Life Purposeless?**\n\nLeave it to us prideful humans to think that we are so wonderful that there must be a reason we are here. We are inventors and designers who do things with intent, so we figure since we are here, something made us for a purpose. Some take this \"purpose\" as a personal destiny or \"calling,\" while others think their purpose awaits them in heaven. There is no reason why we are here or why some people die sooner than others or were spared from harm. The \"purpose\" of living organisms is to pass along their DNA, but even then, there is no reason why it must happen. The universe and everything in it is an ongoing natural process, and we evolved brains to ponder what we consider \"big questions.\" This need not be a bleak view. Our purpose is what we make it. Who needs a God for that?\n\n# **Why Believe? Fear of Death\/Hope for an Afterlife**\n\n\"There are no atheists in foxholes,\" goes an old quip. We don't want to die. We don't want this life to be the only one. We yearn to be reunited with loved ones.\n\n# **Justice**\n\nIf a wrongdoer escapes punishment here on earth, it feels good to think the person will have a personal \"judgment day\" with God.\n\n# **Experience**\n\nMany people swear that they feel God's\/Jesus's presence, have witnessed miracles, experienced divine healing, dreams, or visions, and that their personal relationship with God is as real as any tangible event or object. The human mind has enormous aptitude for believing in the non-verifiable. Psychics, horoscopes, and alien abductions all have their true believers. The mind can convince itself of anything. It can invent significance out of thin air. Personal experience is subjective and in no way can be used as evidence that something is real.\n\n# **The Bible**\n\nThe Bible is the Word of God, so there must be a God. Sure. Just as the Koran is the Word of God, so there must be a God. Just as the Book of Mormon is the Word of God, so there must be a God...\n\n# **Upbringing\/Social Pressure**\n\nMany folks indoctrinated into believing in God from an early age never even conceive of questioning their faith. This is why conservative Catholics, fundamentalist Muslims and evangelicals, and other religionists are so insistent upon giving their children a religious \"education.\" The desire to please one's parents or to fit in with the traditions of society also plays a large role.\n\n# **Belief Gives Comfort**\n\nThe security of hoping one's life has meaning in some bigger picture is a huge inspiration to maintaining belief. It helps many to cope during a crisis to imagine God's \"hands\" holding them and the world. Who wants to believe that one's life is a purposeless, one-shot event? Apparently, not many! Insisting that a higher power is \"out there\" is a comforting crutch.\n\n# **Loose Ends**\n\nAsk people if they believe in gods, and they'll likely answer, \"No, that's for superstitious villagers or the ancient Greeks.\" Ask people if they believe in God, and they'll likely answer, \"Of course, what kind of question is that?\"\n\nAnnouncing that you don't believe in gods is a moot point, but stating you don't believe in God will raise eyebrows. Why? Atheism is viewed as bleak, cold, cynical. To be \"Godless\" is to be on par with being immoral and uncaring. \"What went wrong to make you an atheist?\" they'll ask. Evolution, which is often equated with atheism, has been blamed for fostering an attitude of \"we're just animals,\" which in turn leads to social decay. But social problems have always been around, long before the theory of evolution was formulated. In fact, some of the very worst problems, such as slavery, rampant infectious diseases, and the routine use of torture (all abundantly common in the Age of Faith), have markedly declined since the theory of evolution was propounded.\n\nOften people will remark, \"I'm not very religious,\" and then hasten to add, \"but I believe in God,\" as if it's horrible not to believe. Yet a recent survey by Public Agenda reported that 58 percent of respondents said, \"It's not necessary to believe in God in order to be moral and have good values.\"\n\nThe pious find it hard to swallow that atheists do not have some form of belief. \"Surely you believe in something?\" they'll inquire. Well, atheists believe that all religions have it wrong. Atheists believe there is no God, or, to be more precise, that it's silly to believe in something for which there is no evidence.\n\nAgnostics like to declare that atheists and preachers share something in common: both are positive their respective beliefs are correct. Agnostics claim it's illogical to be strongly certain either way. How illogical of them! It's a black-and-white issue. There either is a God or there isn't. There are only two options, and one must be the correct one; and given the total absence of the desperately longed for evidence supporting the existence of God, not to mention all of the above listed paradoxes and absurdities, it's simply not reasonable to beg the question by saying, \"No one has disproved the existence of God, so it's an open question.\"\n\nBy saying the subject is open, agnostics are actually giving believers the benefit of the doubt that God might exist. But note the absence of magical elf agnostics or supernatural gnome agnostics. No one has disproven their existence, either.\n\nSo, one strongly suspects that agnostics maintain their stance for social reasons\u2014to be acceptable to their religious neighbors and, perhaps, to flatter themselves that they're more \"open-minded\" than atheists.\n\nBelievers ask, \"What would it take for you to believe in God?\" A giant hand parting the clouds and resurrecting the dead would help. As would spying a purple unicorn in one's neighbor's yard bolster the case for the existence of unicorns. But that evidence has never appeared and never will. Only in your mind can you imagine a day when trumpets blare and a legion of angels descends from the heavens, followed by a bearded man on a white horse.\n\nAtheists can still be in awe of the cosmos and of a delicate flower and other wonders of nature. Atheists have more reverence for life and nature than most religious believers, because we know that we're here only by chance.\n\nWe don't consider the world a \"vale of tears,\" the testing ground of a capricious God. Rather, we treasure our brief and only lives, and we treasure the beauties and wonders of the world in which we live.\n\n# CHAPTER X\n\n# Disbelief 101: A Young Person's Guide to Atheism (excerpt)\n\n# by S.C. Hitchcock\n\nOne of the more annoying aspects of talking to believers in god is that they always consider the god hypothesis to be the default position in any argument about the natural world. For example, a believer might ask you if you know how life could have \"sprung up\" in the first place. If you say science isn't sure yet but has some pretty good and testable hypotheses about how it might have happened, the religious person will seize on this. \"But you're not sure, are you? Nobody was there!\"\n\nAll of a sudden, people who will believe anything the Bible says on faith become the most careful skeptics when it comes to science. The implication of the above comment is that if there isn't any conclusive evidence on a topic involving the natural world, then \"god must have done it.\" The religious seem to be saying, \"If you can't prove it in front of my eyes, then my belief must be true.\" Or, if you can't absolutely show me how science explains this beyond a shadow of a doubt, then my religious ideas and your science must be on equal footing.\n\nNothing could be further from the truth. In fact, the claim that science is killing god is not true. God committed suicide a long time ago. The sharpest arguments put together to \"prove\" god's existence end up cutting god to pieces.\n\nI'll go into the three most common arguments for the existence of a god, but before I get to them let me explain why they are so deadly to the very god idea they are designed to protect. Normally, when people argue endlessly about a topic it is because they are arguing from different beginnings or prepositions. (As you've probably heard, there has to be some original point of agreement before two people can argue about anything.) For example, the argument over whether abortion should be legal or not will likely never go away because people cannot agree on the proposition. For someone who thinks that human life begins at conception, abortion at any stage is \"murder.\" However, for people who think that human life begins at birth, then abortion is not murder. (The abortion debate is more complex than this, but the purpose here is to explain how arguments work.)\n\nWe are fortunate, then, that we have no such problem when dealing with the three most common arguments for god's existence. I disagree with all of their premises, but that won't be an issue, because even when I pretend that I agree with them they obliterate the concept of a god. Here, then, are god's \"suicide arguments.\"\n\n# **God's Suicidal Arguments**\n\n# 1. The Prime Mover\n\nIt is a religious trick to dress absurdities up with solemn ceremonies in the hope that no one will notice their silliness. The taking of communion, for example, where Catholics eat a very sacred wafer that is supposed to change, at some point, into the flesh of Jesus is about as bizarre as you can get. Yet if everyone goes through the procedure with a solemn face, as if this is all very serious and important, then it appears to have some weight.\n\nSometimes atheists fall into the same trap and treat religious arguments with the same type of seriousness. So any discussion about the prime mover argument generally begins with a long preamble about the deep philosophical thought of Aristotle, St. Thomas Aquinas, and blah, blah, blah. It's like putting a mule in a tuxedo.\n\nEssentially, the prime mover argument makes the case that something had to cause the universe. That something, according to the religious believer, must be god. No effects exist without a cause, so there must have been some \"first cause.\"\n\nThe reason this argument is so suicidal to god is that it brings up an even bigger question. Where did god come from? How does adding an all-powerful invisible entity at the beginning help us at all? (In fact, this argument is flawed scientifically. The universe did not need a first cause, and this will be explained shortly.)\n\nLet's all just accept the Big Bang theory for a moment. The Catholic Church does, saying that it was the finger of god that sparked the creation of the universe. Well, we might then ask the scientific question of what existed before the Big Bang? We can have a lot of fun just throwing out theories without evidence. I could say that before the Big Bang all that existed were two wart-covered aliens playing video games. And one of the aliens got angry and threw his controller through the television set, thus providing the spark for the Big Bang. So, what we see now as \"reality\" is simply the video game running on, broken and partially haywire. You would laugh at this. Where did the aliens come from? Where did the television set and video game console come from? Sure, it's possible that this happened. After all, no one can prove conclusively that it didn't, but the absurdity of it makes this hypothesis seem unlikely.\n\nAnd yet, people who would laugh at the alien\/video game theory will simply believe in the existence of a being who is infinitely more complex than they are, and who has created reasons for our existence which are equally bizarre. In fact, every \"effect\" we see in the universe has an easily explained cause. When I see a baseball flying through the air, I don't have to look very far to see what caused it\u2014someone threw it or hit it. We could then ask a seemingly endless set of questions, such as, \"Where did the thrower come from,\" etc., and we would have unremarkable answers all the way back until the Big Bang. Now, if we accepted god as the prime mover, he would be the only cause in this massive chain for which we have no explanation or hope of an explanation. This, somehow, makes him more likely to exist?\n\nScience, as always, will simply say, \"Well, we won't know until we can gather enough evidence to create a decent theory.\" Science does not insist upon anything that it can't prove. The idea of god being the first mover in the universe is silly and utterly without evidence. It should be of no more intellectual merit than the notion that a talking tractor, video game playing aliens, invisible flying clowns, or angry raccoons created the universe. None of the arguments for a god are any better than the arguments for any of these things.\n\nWhen talking about the origins of life, that is, when life actually began, religious people will often bring up a concept related to the prime mover argument. This is the principle of Occam's Razor, which is maybe the most misunderstood concept in philosophical history. (Even many professional philosophers miss what Occam's Razor implies.) According to the medieval philosopher William of Occam, if you are presented with two explanations for a phenomenon, then the explanation with the fewest assumptions is generally the right one. This is usually paraphrased as, \"All else being equal, the simplest explanation is best.\"\n\nTo put this another way, Occam's Razor simply means that we have to consider the evidence without any \"extras.\" We'll see what this means in a moment and also why Occam's Razor is so dangerous to religion. (The medieval church, by the way, at least seemed to sense what Occam was up to and was not pleased\u2014he died in exile. So, I'm not sure why religious people think his razor should be used on their behalf.)\n\nWhen talking about the creation of the universe, religious folks will often invoke Occam's Razor and say: \"What could be simpler than 'god did it'?\"\n\nWell, this is not such a simple explanation. First of all, we must make the mother of all assumptions\u2014the existence of an all-powerful, invisible deity. Then we have to tack that notion on to any explanation we give for anything. So, god becomes a tumor that grows on the back of any explanation.\n\nLet me show how Occam's Razor can be used to cut away the tumor from a more sophisticated argument made on god's behalf. The passage below is from a liberal and respected theologian who spoke up on behalf of evolutionary theory (and hence against the \"intelligent design\" proponents) at a trial in Dover, Pennsylvania. Catholic theologian John Haught was arguing that there was no conflict between science and religion, because, he believes, the two do not intersect:\n\nSuppose a teapot is boiling on your stove, and someone comes into the room and says, \"Explain to me why that's boiling.\" Well, one explanation would be it's boiling because the water molecules are moving around excitedly and the liquid state is being transformed into gas.\n\nBut at the same time you could just as easily have answered that question by saying, \"It's boiling because my wife turned the gas on.\"\n\nOr you could also answer the question by saying, \"It's boiling because I want tea.\"\n\nAll three answers are right, but they don't conflict with each other because they're working on different levels. Science works at one level of investigation, religion at another.... The problems occur when one assumes that there's only one level.\n\n\u2014quoted in Matthew Chapman [Charles Darwin's great grandson], _40 Days and 40 Nights_ (New York: HarperCollins, 2007), 106\n\nThis line of reasoning made me think for a while, which is something that the god arguments have been inspiring people to do for years. However, it seems that Dr. Haught's argument cannot survive Occam's Razor. The idea that explanations can work on different levels is interesting. In fact, most actions do have several layers of explanation. The fallacy in this is in thinking that this includes a supernatural, rather than natural, explanation.\n\nIf I stated that the water was boiling because I wanted tea, then my want of tea could be easily explained through biological means. My body needs moisture and sends me signals to make sure that I get it. Ancestors that didn't have such signals would have died of thirst. Perhaps I was tired and my intellect, derived through evolution, would remind me that tea has caffeine.\n\nIf my wife turned on the stove for me, it may be because we have found that doing small favors for one another makes our marriage work better. Where, exactly, would I need god in any of these explanations? You see, if I was trying to observe and explain why water on a stove boils, the simplest explanation would be to say that heat causes the particles inside the water to move, that heat was caused by the electricity flowing through the heating element, and that the electricity came from a power plant, etc. If I believed in god, I would still have to explain the water particles, heat, etc., but I would have to tack on an extra layer of explanation at each point. Instead of saying, \"The particles in the water are moving quickly because of the heat source,\" I would have to say, \"The particles in the water are moving quickly because of the heat source and because an eternal, invisible deity of unlimited complexity designed this.\" Or, to use Occam's Razor to cut through Dr. Haught's several layers of explanation, I would have to say, \"I want tea because I'm thirsty, and because an eternal invisible deity of unlimited complexity wanted me to be.\" Adding a god or an invisible flying clown or any other supernatural cause to an explanation makes it more complicated, not less. Occam's Razor cuts no tumor more deeply than the one called god.\n\nSo, we see that the \"prime mover\" argument holds the seeds of its own destruction, as does its companion, the misused Occam's Razor. If everything that exists must have a cause, and god exists, then where did he come from? (By the way, if he's all powerful, could he make a rock so big that he couldn't lift it?)\n\n# 2. The Watchmaker, er, Cell Phone Maker\n\nThe second of god's suicidal arguments is roughly two hundred years old and was first put forth by the Anglican philosopher William Paley. The argument is rather simple. If something looks designed, then it must have a designer. A watch must have a watchmaker. In other words, if something is complex then it requires something more complex to create it. The universe, obviously, is very complex, therefore its creator must be, well, you get the picture.\n\nNow, let's refer back to the believer's assertion that god as a prime mover, in keeping with Occam's principle, is a simple solution. As the great science writer\/teacher\/atheist Richard Dawkins has pointed out, any \"creator\" must be at least as complex as his creations. It makes no sense to explain how something became complex by invoking an invisible, undetectable something that is infinitely more complex. In other words, if the universe is too complex to not have had a creator, then what does that say about the creator? If a watch needs a designer, then how could an incredibly complex creator just have sprung into existence?\n\nIn fact, the watchmaker argument, often used in conjunction with the human body, is not just suicidal but deeply flawed. For one thing, it ignores that many people are born with harmful birth defects that almost immediately cause suffering and death. Was the watchmaker drunk? Secondly, the most complex things actually have teams of inventors, so it would seem that this argument actually is a better proof for the existence of many gods.\n\nThirdly, this argument is historically preposterous. Imagine a watch just popping into existence, fully formed. This is absurd. Everything that is complex in the universe has less complex origins. Modern cell phones include cameras, video games, telephones, radio transmitters and receivers, and computers. The cell phone was not created before any of these other inventions but was made up of them. And each of the cell phone's components had ancestors which were less complex. Digital cameras did not come out, indeed could not have come out, before Polaroids. Flat screen televisions with DVR capabilities did not come out before black and white television. The Grand Theft Auto video games did not come out before Pong. Nothing that is complex just pops into existence. The very nature of complexity is that it is made up of things that are less complex.\n\nFurther, if you grasp this, you would understand that the phones from the 1950s, the kind that Andy Griffith spoke into when he was calling Aunt Bee and Opie from the sheriff's office, could be considered a different species from the modern cell phone. Andy's phone did not want to become another type of phone. It did not consider itself to be a \"transitional species\" of phone but merely survived for its time, and then eventually found itself outcompeted, and then extinct. Only its fossil record survives in thrift stores, attics, and museums.\n\nYou might say, yes, the phone has evolved, but it took humans (something more complex) to guide that evolution. Humans are a good metaphor in this case, but not for god, since no human being who grew up on an isolated island away from modern technology could ever hope to create a cell phone, or even conceive of what it is. (For that matter, imagine telling people fifty years ago that one day they would carry a phone in their pocket; I would bet that, almost universally, their first comment would be about how long the extension cord would have to be.) Human \"inventors\" aren't a good metaphor for god; but they are a good metaphor for natural selection. All that humans can do is look at all the phone designs that don't work (experiment) and then pick the ones that do. This is just a faster version of what nature does through environmental and sexual selection.\n\nIf I were ever brought into court to refute the watchmaker argument, I would find it an easy task. Every part of a watch had, at one time, uses that were entirely unrelated to its use in a watch in the same way that many of the features of a modern cell phone were once completely unrelated to being part of a cell phone. Numbers obviously had uses other than just being on a watch face. Gears were used for mills. The glass on the face of a watch was used for windows, and the strap is just a shrunken belt. When they all came together they turned out to have another use, at least partially unrelated to their original uses. Better yet, I could prove this. It would not be difficult to show the origins of numbers, of glass, of gears, of straps and to show that they had other uses prior to being part of a watch. I could even show \"fossil\" evidence of the antecedents of the modern watch. Egyptian sundials, Chinese water clocks, and the great designs which came from John Harrison's workshop (google his name; he's pretty cool) could all be used to prove the point that the watch evolved from smaller, less complex pieces. The evidence table would be full, and the court reporter would have to stop and massage the tendons of her wrist when I was through.\n\nAnd I could be confident that god would never show up to testify on his own behalf.\n\n# 3. Why Is There Something Rather Than Nothing?\n\nThe third of god's suicidal arguments is not really an argument. It actually involves an interesting question that theologians hope has an assumed answer. That is not to say the answer that believers give is interesting; it isn't. Religious people simply have to hope that believers will assume that the uninteresting answer is true.\n\nWe will now examine the biggest question of them all: Why is there something rather than nothing? Personally, I don't know why people assume there ever was nothing. What if the natural state of the universe is to be here? I don't know why people who live in a world of something should assume that there was at one point nothing.\n\nHere's the problem with this question: Religious people don't believe there was ever really nothing, do they? According to them, god was hanging around, just waiting to create a universe so that he could make humans and play his little \"faith or hell\" game with us. This, then, is the problem. You cannot ask the question of why is there something rather than nothing if you don't assert that there actually was nothing. Instead, religious believers assert that rather than nothing, there was an Ultimate Something. This argument is the most persuasive when it is being used against the notion of a god. Why is there a god rather nothing? Again, this argument is suicidal for the god idea.\n\nIt is here that we can also address the point I raised earlier about the universe not needing a \"first cause\" or \"prime mover.\" Stephen Hawking (the really smart guy in a wheelchair who is often depicted in _The Simpsons_ ) addressed the issue of the universe's creation from nothing to something and stated this in his best-selling _Brief History of Time_ :\n\n[T]he quantum theory of gravity has opened up a new possibility, in which there would be no boundary to space-time and so there would be no need to specify the behavior of the boundary.... The universe would be completely self-contained and not affected by anything outside itself. It would be neither created nor destroyed. It would just BE.\n\n\u2014Stephen Hawking, _A Brief History of Time_ (New York: Bantam Books, 1988), 141\n\nDr. Hawking is careful to point out that this is just a proposal based on the mechanics of quantum gravity and will remain a proposal until all the evidence is in. That being said, doesn't it make much more sense to theorize about the universe's \"beginnings\" (if the word even applies) from the standpoint of science than it does to theorize about it from the standpoint of religion? What, exactly, makes anyone think that a religious proposal is likely to be helpful here?\n\nI prefer to address the question in this way. The question \"Why is there something rather than nothing?\" answers itself, since it is not possible to ask its opposite.\n\n# **And**\n\nIn our culture, perhaps no single word conjures up such positive overtones as the word _faith_. Religious people often proudly proclaim themselves to be \"people of faith.\" Children are taught that faith is an important aspect of their upbringing. \"We're raising our child in faith\" is a common parental boast.\n\nFaith is a central aspect of Christianity and Islam. But even religions that claim not to put a high premium on faith (that is, those that don't claim it's the golden ticket to heaven) still need faith. Religious Jews need to have faith that there is worth in carrying out the same rituals as people who lived and died thousands of years ago in a desert. They need to have faith that there really is a savior on the way. Hindus must have faith in myriad (it doesn't hurt to look up a word now and then) gods and to have faith that good works will move them up the reincarnation chain to enlightenment. Buddhists have to have faith that the serene feeling that they get from meditation is somehow more spiritual than the feeling I get after going for a jog. In one way or another, faith, the ability to \"believe,\" is an important aspect of all religions.\n\nBut why is \"faith\" held in such high regard. What does it mean anyway? The Oxford Dictionary defines faith like this: \"1. complete trust or confidence. 2. firm belief, especially without logical proof. 3. a system of religious belief; b. belief in religious doctrines.\"\n\nIsn't this a strange set of definitions? First of all, faith implies a firm belief. The person of faith is usually a person who strongly believes in something. People of faith have little or no doubt about what they believe in.\n\nOkay, you would think that people would have this kind of faith in things that have been proven to work, that is, things that have been proven true. I have unswerving faith that dawn will come tomorrow. It's come every day for billions of years! And I have faith that the gravity which has kept the earth swirling around the sun will continue to work tomorrow and the next day and the next. Why? It's always worked. It's easy to have firm belief in something that is proven by evidence to be true.\n\nAnd yet, the next part of the Oxford definition makes the first part seem bizarre. Faith is not just a firm belief; it's \"a firm belief, especially _without_ logical proof.\" (emphasis added)\n\nNow wait a minute. How can anyone have a firm belief in something without proof for it? If I were to say that I had a firm belief in the existence of sound-carrying angels or an invisible flying clown, people would think I was weird. If I said I had firm belief that the invisible flying clown would rescue me if I jumped off a building, I'd be considered insane. After all, gravity is known to exist. One can have real faith in gravity. If something goes up, it comes down (unless it leaves the earth's gravitational pull), but to put so much faith in something for which I have no evidence, to the point where I'd wager my life on it, would be nuts.\n\nLet's revisit a slightly different version of an example I used earlier but go into greater detail. Let's say there's a very religious farm family living out in the middle of nowhere, and the youngest boy has his leg mangled in a farming accident. The boy's mother and father, having been raised to have faith in the power of god, simply cover the boy's leg with a sheet and pray for him.\n\nAfter one day of this, the boy is in extreme shock and delirium from pain and loss of blood. The mother calls her pastor who contacts the other members of the church, and they gather to pray for the boy's healing. The boy gets worse and worse despite the prayers. One of the members of the congregation finally suggests that the mother and father take the boy to the hospital. They don't, thinking that if they show a lack of faith in god he will take the boy's life. The boy is only appearing to get sicker and sicker to test his and their faith.\n\nThe boy dies.\n\nAny rational person, even a deeply religious one, would consider the actions of everyone involved, except for the injured boy, to be criminal. Why? Isn't faith a good thing, and the more faith the better? Why should having so much faith in god be a criminal and negligent act? Why should it be considered child endangerment?\n\nMany religious people would answer that, while they believe in a god, they also believe that he wants them to visit a doctor. Most people see no contradiction in taking a loved one to the hospital and then praying to god for his or her recovery while that loved one gets all the help of modern medical science.\n\nBut if you really have faith, why go to the hospital?\n\nIn his wonderful book, _The God Delusion_ , Richard Dawkins, wrote about a study called \"The Great Prayer Experiment.\" It involved 1802 patients in six hospitals. All were recovering from the same type of heart surgery. The patients were divided into thirds. One group was prayed for and knew it; the second group was prayed for and didn't know it; and the last group was not prayed for and didn't know it.\n\nIt turns out that those in the two groups that were unaware of their prayer status had no difference in their health or recovery. The only ones to show a difference were the people who were prayed for and knew it. They \"suffered significantly more complications than those who did not [know they were being prayed for]\" (Richard Dawkins, _The God Delusion_ [Boston: Houghton Mifflin Company, 2006], 63). Dawkins attributes this to the added stress the knowledge brought.\n\nThis should have been an unnecessary study. A simple look at history shows that prayer does nothing. The infant mortality rate used to be much higher than it is now. Plagues used to ravage medieval Christian Europe and the Islamic nations of the Middle East. I presume that many people in these deeply religious societies prayed often and fervently and died horrible deaths regardless. Nobody in America dies of plague or smallpox anymore. Is it because we pray harder than our medieval predecessors? Or is it because science has given us better medicines?\n\nDo we no longer have epidemics in this country because god likes us better than people in the past, or is it because we discovered that sanitation and hand washing are effective in preventing outbreaks?\n\nAsk yourself why it is that you're supposed to have just the right amount of faith. You're not supposed to have so much faith that you actually expect god to do anything useful, like heal the sick or rescue the helpless, but just enough so that you believe in very old texts and in wild stories. Believe just enough, the church seems to be saying, to get your behind in the seats every Saturday or Sunday.\n\nWhy do all religions put such a high premium on faith? Why do they ask their followers to \"grow in faith,\" when clearly growing too much in faith can become a problem and can even land you in jail? What do the religions want? They want your faith to grow but only in certain untestable areas. Religious Jews want you to have faith that a messiah is coming to save the world and that the books of the Old Testament are literally or figuratively true. They want you to have faith that Jesus was not a messiah or savior. The Christians want you to have faith that Jesus was the son of god and that he was born of a virgin, walked on water, healed the sick, talked to Satan, was crucified for the sins of humanity, and that he died and came back three days later. The Muslims want you to have faith that Jesus was a prophet, but not the son of god, that god is named Allah, and that his true prophet was Mohammed, who was visited by an angel in a cave and spent his life conquering territory to spread Islam, taking many underage \"wives\" along the way.\n\nHow can these linked religions all exist? Simple: none of them have any evidence for their claims, so there's no way to test their validity. Each claim, without any real evidence to back it, is just as valid as the next. It is the absence of evidence, of logic, of reason, that forces all religions to put a high premium on faith. Because they have no evidence for their claims, they have to make it a virtue to believe in things that are illogical\u2014even though in any other area to have faith in something without evidence is crazy.\n\nEvery single religion in the world teaches that you exist for only one reason: to find and believe in that religion, whatever it may be. They all teach that god went through all the trouble of creation just so you can have free will and discover his one true religion.\n\nAnd most people believe that the one true religion is whichever one they grew up with.\n\nHow do the many churches of the world sell something as blatantly stupid, as sadistic, as faith? Well, they promise a lot, don't they? These religions claim that faith is the one thing that god requires. Many believers of various sects of Christianity and Islam think that god requires that we believe in him despite all evidence against his existence and the truth of the scriptures. It is a virtue, for example, to believe in creationism and not evolution because of all the evidence stacked against the idea of creationism; and the fact that there is not a single shred of real evidence in favor of creationism is only a test of faith. God wants to see if you will trust your own mind over the ancient holy texts he had written on his behalf. If you trust your own mind, if you lose faith that these comically flawed documents are the actual truth, then you will be punished. If you keep your faith, then you will be rewarded.\n\nWe might ask, \"What are the rewards of having such faith?\"\n\nHeaven. A place of eternal bliss.\n\nWhen do you go to heaven?\n\nAfter you die!!\n\nOf course, our next question is obvious: If you don't go to heaven until after you die, how does anyone know it exists? I think you know the answer: you have to have _faith_!!!\n\nAnd if you don't believe in all of this nonsense, what do the many religions of the world say awaits you?\n\nHell. A place of eternal torment.\n\nTake a wild guess when you go. That's right: after you're dead. (You're getting good at this.) And how do we know that hell exists if people only go there after they're dead.\n\nDrum roll, please...\n\nYou have to have _faith_!!!\n\nOddly enough, these extremes of reward and punishment, heaven, on one hand, and hell, on the other, are enough to scare many people into being religious. Many people go to church and give money to it like they put money into a retirement account, hoping to do just enough to get into the nice gated community that is heaven and, perhaps more importantly, avoid that nasty slum called hell.\n\nIsn't this insane? Isn't this a crazy wager? What if you picked the wrong religion? What if you're Catholic and god is a Southern Baptist? What if you're a Hindu and god is an African Animist?\n\nWhat if god wants you to conclude he's not there, and the only people who get into heaven are those wicked atheists? Why not? If you believe in a god who enjoys playing little games, how hard is it to believe in a god who tells everybody he wants them to believe in his holy books, but who really wants them to buck the system and not believe?\n\nOf course, I don't believe any of this for a second. Many Christian and Muslim religious people are put off by the notion that people of other faiths, billions of people, are going to go to hell. And yet, if religious people want to believe that god lets people of all faiths into heaven, then what's the point of believing in any particular religion?\n\nYou see the problem? If you can get into heaven being a Buddhist, Muslim, Jain, or whatever, then why should you come to\u2014and give money to\u2014some Christian church? From the point of view of members of any particular religion, it makes no sense to say that everyone gets into heaven; and yet it seems cruel to condemn most of humanity to hellfire for believing, with total faith, in whatever holy book and religion happened to be fashionable in the area in which they were born.\n\nThis is a real problem for those seeking to sell religion, so they mostly ignore it. In America, it is a social convention not to argue about religion. We seem to have a policy of \"if you don't mess with my nonsense, I won't mess with yours.\" It's downright impolite to bring up the topic of logic to a religious person.\n\nIf you ever ask people who attend a \"megachurch\" why they give money to it, when it's plain for anyone to see that the tax-free cash is being used to build media empires and to line the pockets of already wealthy preachers, they'll probably look at you funny. The truth is they don't care where the money goes. They give the money because they have faith that god is pleased with them for giving it and is building them a nice retirement condo in the clouds. Because religion is a business built on faith, it has to make faith into a virtue. Religions have to get to you when you are young and plow into your impressionable mind the idea that faith is a good thing, that it's the only thing that matters, that it's important. Not total faith, no, just enough faith to believe what god's spokesmen (and it is almost always men) are telling you, even when what they're telling you is complete rubbish.\n\nReligions have to do this. After all, faith is their only product. Faith may consume your whole life and a good deal of your money and your intellect, but it costs religions next to nothing to produce it. And the best part: Every indoctrinated child grows up to sell the product to the next generation.\n\n# CHAPTER XI\n\n# Dogspell\n\n# by Earl Lee\n\nA new book recently came across my desk, called _Dogspell: The Gospel According to Dog_. At first I thought maybe _Dogspell_ was a new version of the musical _Godspell_ , but using barking dogs\u2014kind of like the Christmas music CDs put out by \"Jingle Dogs,\" or the even more annoying \"Jingle Cats\" music CDs, where a group of cats meow out Christmas classics like \"Here Comes Santa Claws\" or \"Meowy Christmas.\"\n\nAs it turns out _Dogspell_ is a book that wants to guide Christians to a better understanding of their relationship with God. According to the publisher, the author \"uses [the] metaphor of [a] dog's unconditional devotion to its human and the joy it finds in [this] relationship.\"\n\nThis idea is disturbing in so many ways.\n\nCan this be the spiritual goal of most Christians\u2014to view the universe on all fours while sniffing the crotch of God? And think of all the theological questions it raises. Is it appropriate to hump God's leg only on Sundays? Or Saturdays? Or should this be a daily ritual?\n\nAnd then there is the question of evil. How can we address the fact that I have fleas? Why doesn't God do something about this? Get me a flea collar! Buy some flea powder! Please, God, do something to clean up all these horrible problems in the world.\n\nI love my neighbor. So can I ask God to send the city's Mobile Spaying Unit to my neighbor's house and \"fix\" them all? (Just my idea for cleaning up the local gene pool.)\n\nWhat if it turns out that my God is violent and brutal, and he beats the hell out of me with a two-by-four and sells me out to dogfights, like Michael Vick? Am I still expected to lick his hand?\n\nWhy is it that people look up to the sky, searching for some invisible master and abase themselves like dumb animals?\n\nWhat is it about this idea that makes me want to lie down and lick my own ass, just to get the taste of this out of my mouth? Oops, I can't reach. A little help here!\n\nThe religious never cease to amaze. Sometimes their weird ideas are pretty funny. Other times their violence and senseless bigotry are downright shocking. From female circumcision to abortion clinic bombings, these people are seriously disturbed. Maybe they're trying to work out overwhelming feelings of worthlessness? At least that would explain the self-identification with dogs.\n\nAnd their ideas and practices are truly crazy. Ritual cannibalism on Sundays. Confessing one's \"sins\" to a pedophile in a dress. Not using birth control. Praying in front of candles and statues. Wishing for miracles. Denying dying children medical care. Then they accept whatever crazy shit the preacher tells them and reject the evidence of their own eyes.\n\nI know of a Baptist preacher in our town (pop. eighteen thousand) who has a congregation of only twenty-five people. Every Sunday (and Wednesday and Friday) he subjects them to two hours of yelling and personal abuse (I am _not_ exaggerating). Why do they put up with this jerk? And worse yet, a few months ago he emptied the church bank account and left town. And some church members still want him to come back! How sick is that?\n\nI guess it does all comes back to the self-identification with dogs: _Here boy! Good boy! Kill that unbeliever! Roll over for your heavenly daddy!_\n\n# Afterword\n\n# by Chaz Bufe\n\n# **Pamphlet**\n\nA booklet that consists of up to several dozen pages printed on both sides, folded in half, and bound with a staple in the center. (Pamphlets are distinguished from fliers and handbills in that pamphlets are bound and contain at least a few pages printed on both sides, whereas handbills and fliers consist of single pages and often are printed only on one side.)\n\n# **Historical Background**\n\nPamphlets have been around nearly as long as the printing press. They've been, and still are to some extent, a handy and cheap means of spreading information and oftentimes controversial opinions. In sixteenth-and seventeenth-century Europe, early pamphlets often dealt with political and religious matters, a famous example being by John Knox, the Calvinist father of Presbyterianism: _The First Blast of the Trumpet Against the Monstrous Regiment of Women_ , which appeared in 1558. (It's not _quite_ as bad as it sounds: Knox was railing against rule [\"regiment\"] by female monarchs.)\n\nAt about the same time, what could well be considered the first anarchist pamphlet appeared: _Discourse on Voluntary Servitude_ ( _Discours de la servitude volontaire_ ), by \u00c9tienne de la Bo\u00e9tie. While he apparently wrote the essay during his university years, it was only published in 1578, fifteen years after his tragically early death in 1563. De Bo\u00e9tie's _Discourse_ was very much an outlier, and nothing remotely similar in political tone appeared until well into the eighteenth century.\n\nIn Britain's American colonies, a similar situation prevailed from the seventeenth century on, with pamphlets published on religious and political matters, often mixing the two topics, as with Knox's _First Blast_. The most famous and by far the most influential of these pamphlets was Tom Paine's _Common Sense_ , an impassioned plea for American independence, which can rightly be considered the spark that ignited the Revolutionary War. In all likelihood, it remains the biggest selling pamphlet of all time. During its first year, at least one hundred thousand copies were sold in the U.S. (with a [white] population of three million), and several hundred thousand more were sold in Great Britain and (in translation) France.\n\nPaine followed this up with another famous pamphlet, _Rights of Man_ , in 1790, which argued that human beings have inherent (\"natural\") rights, and that the sole purpose of government should be to safeguard those rights. After that, in 1794, his influential book _The Age of Reason_ appeared. In it, Paine attacked organized religion, especially Christianity, and advocated deism. Because of his attack on Christianity, he was forced to flee to the United States, where he was again ostracized for his views and died in poverty in 1809.\n\nIn both the U.S. and Europe, atheism had long been proscribed, and its few open advocates often cruelly tortured and murdered for \"blasphemy.\" Jean Meslier, a French priest, was the author of the first openly atheist book, in part titled _Clear and Evident Demonstrations of the Vanity and Falsity of All the Religions of the World_ ; like Copernicus, he had the good sense to delay publication of the book until after his death, in 1729.\n\nIn the late eighteenth century, especially in the wake of the French Revolution, openly atheist and deist works began to appear, including, in 1782, what was likely the first atheist pamphlet in England, _Answer to Dr. Priestley's Letters to a Philosophical Unbeliever_ , of which the authorship remains a matter of dispute.\n\nStill, it remained dangerous to openly express atheist opinions while blasphemy laws remained on the books in Britain, the United States, and much of continental Europe. Atheist publishers continued to be persecuted for blasphemy throughout the nineteenth century, with the final British atheist jailed for the \"crime\" of publishing atheist materials being George Jacob Holyoake, in 1842. Nonetheless, suppression of \"blasphemous\" materials and their authors continued, though not systematically, in the British Isles, with blasphemy laws wiped off the books only in 2008 in the UK and 2018 in the Republic of Ireland.\n\nIn the U.S., blasphemy and \"obscenity\" laws were used throughout the nineteenth century, in contravention of the First Amendment, to suppress atheist materials and their authors and publishers. The last person convicted for blasphemy in the United States was Charles Lee Smith in Little Rock, Arkansas, in 1928, for giving away atheist pamphlets he had written. He was sentenced to twenty-six days in jail after the blasphemy charge was thrown out, and he was convicted instead of publishing \"scurrilous\" or \"seditious\" materials. It wasn't until 1952, in Joseph Burstyn, Inc. v. Wilson, that the U.S. Supreme Court declared blasphemy laws unconstitutional. Despite the Supreme Court ruling, blasphemy laws remain on the books, unenforced, in a number of American states.\n\nDespite the danger of blasphemy laws, atheist and political\u2014especially socialist and anarchist\u2014pamphleteering greatly increased in the latter part of the nineteenth century in both the U.S. and Britain. In Britain, atheist publishing picked up steam with the founding of the Rationalist Press Association (now the Rationalist Association; in the early days, it underwent several name changes) in the 1890s, which published both books and pamphlets, including, in 1898, the pamphlet _God Eating: A Study in Christianity and Cannibalism_ , by John T. Lloyd.\n\nIn the United States, where pamphleteering\u2014especially anarchist and socialist pamphleteering\u2014was quite common by the late nineteenth century, there was no corresponding organization; and there still isn't. Instead, small independent publishers took up the slack. One notable early atheist pamphlet was Johann Most's _The God Pestilence_ (included in this volume), which appeared in print in Most's newspaper in 1883 and was a blistering attack on both religious belief and capitalism, as can be seen in the following passage:\n\nThe more man clings to religion, the more he believes\u2014the more he believes, the less he knows\u2014the less he knows, the more stupid he is\u2014the more stupid, the easier he is governed\u2014the easier to govern, the better he may be exploited\u2014the more exploited, the poorer he gets.\n\n# **Twentieth-Century American Pamphlet Publishing: The Haldeman-Julius Company**\n\nIn 1895, the pamphlet publishing situation in the United States began to change. In that year, Julius Wayland founded the socialist _Appeal to Reason_ , a weekly newspaper, in the small town of Girard, Kansas. (The reason for this apparently odd choice of location was that Girard had a station on the Atchison, Topeka, and Santa Fe Railway, had a socialist local government, and was near the geographical center of the United States.) By 1910, the _Appeal_ had a circulation of over five hundred thousand and occasionally printed special editions with print runs in the millions. It was the largest-circulation political publication in the world and, according to at least one estimate, was the largest-circulation periodical in the world, period.\n\nThe _Appeal_ also published pamphlets, and within a decade of its founding had published hundreds of them, a good majority bearing on various aspects of socialism; authors included such luminaries as Marx, Engels, Eugene V. Debs, and Clarence Darrow.\n\nBy 1915, the _Appeal_ was in decline, though it and its publishing operation would shortly undergo a metamorphosis. In 1916, one of the _Appeal_ 's editors, Emanuel Julius, married Marcet Haldeman, a banker's daughter and reputedly the richest woman in Kansas. In 1919, they purchased a majority interest in the _Appeal_ and changed the emphasis of the publishing operation from the severely weakened newspaper (now titled the _New Appeal_ ) to pamphlet publishing.\n\nAlong with this change in direction, they also drastically increased the number of pamphlets published and drastically decreased the price of most to only five cents. These very cheaply produced 3.25\" \u00d7 4.75\" pamphlets set in 8-point type (this is 8-pt. type) became the famous _Little Blue Books_ issued by the Haldeman-Julius Company. The company also issued a smaller number of 5.5\" \u00d7 8.5\" pamphlets, so-called _Big Blue Books_ (though most didn't have blue covers), priced at between twenty-five cents and a dollar. In all, during its active pamphlet publishing period of 1919 to 1951, the company issued roughly 2,500 titles with a press run, according to one historian, of 500,000,000 copies. That seems optimistic, but it's almost certain that the Haldeman-Julius Company sold at least 100,000,000 copies. (They sold the pamphlets by mail and took out large ads in major newspapers across the country listing hundreds of pamphlets in a single ad and selling them twenty for a dollar.)\n\nOne notable aspect of this pamphlet publishing empire was that once they assumed control of it in 1919, Emanuel Julius (name changed to E. Haldeman-Julius [EHJ] at the time of marriage) drastically diversified the topics of the pamphlets, with the issues addressed now including philosophy, self-improvement, classic works of fiction, poetry, and any other subject EHJ thought would sell.\n\nImportantly, the published topics now included atheism. For the first time in U.S. history, a major (though highly unorthodox) publisher was issuing atheist pamphlets as both _Little_ and _Big Blue Books_. EHJ published a lot of such pamphlets\u2014almost certainly well over one hundred\u2014and even wrote one _Little Blue Book_ on the issue himself, _The Meaning of Atheism_. In it, he outlines the primary reason he considered religion a menace:\n\nPerhaps religion might be dismissed as unimportant if it were merely theoretical... but there is and has always been sternly and largely a disposition of religion to enforce its theory in the conduct of life; religion has meant not simply dogmatism in abstract thinking but intolerance in legal and social action. Religion interferes with life.\n\nAtheist titles EHJ published included _The Anti-Christ_ , by Friedrich Nietzsche; _Religion: A Dialogue_ and _A Few Words on Pantheism,_ by materialist philosopher Arthur Schopenhauer; and _Christianity, a Continuing Calamity_ , by Clay Fulks, in which Fulks states, in an unfortunately still relevant passage:\n\nHaving fundamentalists in a nation is like having congenital imbeciles in the family\u2014it's a calamity. Allow their mountebank, swindling leaders enough control over society and though religious faith would flourish fantastically, society would revert to the sheep-and-goat stage of culture.\n\nBut by far EHJ's most prolific atheist author was former Catholic priest and theologian Joseph McCabe. McCabe wrote dozens upon dozens of both _Big_ and _Little Blue Books_ attacking religion on many, many fronts. His topics included Christian misogyny ( _Degradation of_ _Woman_ ), sexual prudery and authoritarianism ( _The Catholic Church and the Sex Problem_ ), Christianity's syncretic nature ( _Pagan Christs_ and _How Christianity Grew Out of Paganism_ ), and Christianity's woeful record as regards slavery ( _Christianity and Slavery_ ).\n\nThe Haldeman-Julius Company's glory days were in the 1920s and 1930s, when it undoubtedly sold a majority of its hundreds of millions of copies. But sales began to go downhill with the approach of World War II and the appearance of the still familiar 4.25\" \u00d7 7\" mass-market paperback book\u2014importantly, a cheap book with a _spine._\n\nSales were already in decline by the beginning of World War II, and following the war they plummeted. With the postwar economic boom and the arrival en masse of cheap paperbacks often priced at twenty-five cents, the day of the cheap-looking _Little Blue Book_ had passed. Following the war, EHJ all but ceased publishing new titles, and with his death in 1951 (while being hounded by the IRS during the post\u2013World War II anti-communist hysteria), his company entirely stopped publishing new works. It continued to slowly sell off its remaining stock until 1978, when its printing plant\/warehouse was destroyed by fire. Today, the funky-looking _Little Blue Books_ have become cheap collectibles, oddball reminders of early twentieth-century America.\n\n# **The Post\u2013World War II Period**\n\nFollowing the downfall of the Haldeman-Julius Company, pamphlet publishing fell to a low ebb in the United States. Undoubtedly fewer were being published than even in the latter part of the nineteenth century prior to the rise of _Appeal to Reason_ 's publishing program. One of the primary reasons for this is that in the post\u2013World War II period leftist political activity in the U.S. all but ceased, but for the heavily persecuted and deeply deluded (Stalin worshiping) Communist Party USA (CP). Politically, the CP had sucked all the air out of the room\u2014aided and abetted by reactionaries who ceaselessly promoted the false dichotomy of Soviet-style \"communism\" vs. American-style \"free enterprise\"; and in seemingly endless economic boom times, with a rising standard of living, the appeal of both socialism and anarchism had waned.\n\nThis began to change in the 1960s with the Vietnam debacle and the rise of the largely inchoate leftist opposition to it. With that opposition, weekly newspapers sprang up in every major American city and in many medium-sized ones, and books and pamphlets by \"alternative\" publishers began to appear. One amusing example from the 1970s was produced by the Fifth Estate group in Detroit. Its title was a play on the classic _Twilight Zone_ episode, \"To Serve Man.\" The title of the pamphlet was _To Serve the Rich_ , and it featured recipes for such delicacies as \"Rocky Mountain Oysters Rockefeller.\"\n\nThe pamphlet's authors comment:\n\nA note on brains: \"You are what you eat\" should not be taken too literally; unlike some primitive tribesmen, you needn't fear that the brains of the wealthy will endow you with their personality traits and transform you overnight into an avaricious, exploitative parasite.\n\nAs for atheist pamphlet publishing, very little was going on in postwar America prior to the founding of American Atheists (AA) in 1963 by the Elmer Gantry\u2013like figure, Madalyn Murray O'Hair (MMO'H). She was a confirmed and charismatic atheist, but she used the organization and its membership to enrich herself, squirreling away millions in offshore bank accounts before her murder in 1995 by a disgruntled ex-employee during a kidnapping gone wrong.\n\nBy the 1970s, American Atheists had begun publishing books, pamphlets, and a monthly magazine, _American Atheist_. Under MMO'H, American Atheists published reprints by historical atheist figures such as Joseph McCabe and Robert Ingersoll, as well as original books and pamphlets, some by MMO'H herself.\n\nAmerican Atheist's pamphlets tended to be pricey and poorly produced; one reason for this was that MMO'H ran the organization with an iron hand, and, among other things, decreed that all American Atheist publications be typeset in the Souvenir typeface, a display face very common in the 1970s that veritably reeks of hot tubs and granola, and is difficult to read in extended passages. (This is Souvenir.)\n\nSales for American Atheists books and pamphlets have never extended much beyond its membership, largely because they've apparently made little effort to expand sales beyond that base, but also because very few bookstores (and no chains) would accept pamphlets, and because their books (especially the covers) are usually so poorly designed that they'd be unsalable in bookstores. As for the pamphlets, since sales have always been almost exclusively to members, at most we're looking at total sales in the low six figures and quite possibly below that.\n\nFollowing MMO'H's death in 1995, American Atheists has continued under much more ethical leadership, and the group continues to occasionally publish books and pamphlets. Their prices are now much more moderate than in the time prior to 1995, but the design problems continue.\n\nOther smaller atheist groups and publishers also began to issue pamphlets in that period, including See Sharp Press. (Disclosure: I'm the publisher.) From the mid-1980s through the first decade of the twenty-first century, See Sharp issued fifty pamphlets with a total press run of well over one hundred thousand. The predominant publishing areas were anarchism and atheism, with atheist titles comprising probably 60 percent of the total.\n\nDuring the 1980s and 1990s, See Sharp's pamphlet sales boomed, largely because of the presence of well-organized anarchist distributors (who also carried the atheist pamphlets), such as A Distribution, Left Bank Distribution, and AK Press in the United States, and Freedom Distribution, Active Distribution, and AK Press in the UK.\n\nSales began to dip in the late 1990s, and by the end of this century's first decade, See Sharp's pamphlet sales had all but ceased. There were three reasons for this: first and most importantly, the internet spelled doom for short-form publications, because it offers easy and often free access to such short-form materials; the abovementioned distributors went under or drastically curtailed their activities; and the Republicans in Congress kneecapped the U.S. Postal Service in 2006, with the fallout from that including astronomically higher overseas shipping rates, which made sales to the UK all but impossible.\n\nToday, See Sharp is selling off its remaining pamphlet inventory but continues to publish atheist books.\n\nAs for other post-millennium atheist pamphlet publishers, as mentioned above, American Atheists continues to put out the occasional pamphlet, and PM Press publishes a couple of pamphlets per year, including an atheist pamphlet every now and then (next up, reportedly, S\u00e9bastien Faure's _Twelve Reasons for the Nonexistence of God_ , which is included in this volume). Beyond that, atheist pamphlet publishing is all but dead.\n\n# **The Materials in This Collection**\n\nThe original intention with this collection was to feature only materials that originally appeared as pamphlets. However, upon reflection I decided to also include several pieces that appeared in other forms, in order to provide a more well-rounded compilation. (Restricting this to a pamphlets-only anthology would, among other things, have precluded including Emma Goldman's \"The Failure of Christianity,\" and it would have precluded including almost anything from the current century.)\n\nSo, while many of the following essays first appeared as pamphlets, there are also excerpts from books by small publishers, essays that originally appeared in newspapers, and two short pieces that originally appeared as blog posts.\n\nAs for the materials and their authors:\n\n\"The God Pestilence,\" by Johann Most, first appeared in 1883 in German in Most's anarchist paper, _Die Freiheit_ (Freedom). A fiery anarcho-communist and proponent of armed insurrection, Most was viciously persecuted in both his native Germany and in the U.S. following his exile from Deutschland.\n\n\"Woman, Church, and State,\" by prominent early feminist Matilda Gage, was published in 1893 by The Truth Seeker Company, which also published the very long-running (over a century) freethought magazine _Truth Seeker_. The excerpt included here was the final chapter of the book.\n\n\"The Devil's Dictionary\", by muckraking journalist and author Ambrose Bierce, appeared originally in newspapers and magazines in San Francisco over a period of more than thirty years in the late nineteenth and early twentieth century. \"Bitter Bierce's\" collection first appeared in book form in 1906 and in more complete form in 1911. The accompanying \"American Heretic's Dictionary\" was written by the editor of this collection, Chaz Bufe, and published by See Sharp Press in 1992.\n\n\"The Failure of Christianity,\" by America's most famous and most admired anarchist, Emma Goldman, was first published in her magazine _Mother Earth_ , in April 1913. Although Goldman's anarchist and feminist views are well-known, many are still unaware that she was also a militant atheist.\n\n\"Twelve Proofs of the Nonexistence of God,\" by S\u00e9bastien Faure, was published in Paris as a pamphlet in 1914. Faure was a leading member of the French anarchist movement for over five decades, from the late nineteenth century to his death in 1942. The first English version that we're aware of appeared in the 1930s, and was translated by two Italian anarchist refugees from fascism, Aurora Alleva and D.S. Menico. The version in this compilation is a new translation specifically for this volume.\n\n\"The Meaning of Atheism,\" by publisher and socialist E. Haldeman-Julius, appeared as a pamphlet ( _Little Blue Book_ ) issued in 1942 by the Haldeman-Julius Company, the most important American pamphlet publisher of the twentieth century. (See above for more information on Haldeman-Julius.)\n\n\"How Christianity Grew Out of Paganism\" (1926), \"Christianity and Slavery\" (1926), and \"Judeo-Christian Degradation of Woman\" (1943) all appeared as Haldeman-Julius _Little Blue Books_ and are all by ex-Catholic priest and theologian Joseph McCabe, Haldeman-Julius's most prolific author. For this compilation, I've edited down all three pamphlets to some extent in order to eliminate tangential passages entirely unrelated to the topic (of which there were many). The most extreme elision occurred in \"How Christianity Grew Out of Paganism,\" where I eliminated the entire first half of the pamphlet, because, as McCabe himself states in his introductory remarks, the first half is completely unrelated to the title topic.\n\n\"May the Farce Be with You: A Lighthearted Look at Why God Does Not Exist,\" by contemporary graphic artist and feminist Pamela Sutter, was published as a pamphlet by See Sharp Press in 2001. While the pamphlet is no longer available, \"Farce\" is now \"in print\" as an e-book.\n\n\"Twenty Reasons to Abandon Christianity,\" by Chaz Bufe, was published as a pamphlet in 2001 by See Sharp Press, and is currently available in both pamphlet and e-book form. Chaz is currently expanding it to book length, with the tentative title, \"Twenty-Four Reasons to Abandon Christianity.\" Chaz's other piece, \"Why Science Leaves Religion in the Dust,\" appeared in 2014 on the See Sharp Press blog: https:\/\/seesharppress.wordpresss.com.\n\n\"Disbelief 101: A Young Person's Guide to Atheism (excerpt),\" by contemporary American educator and labor activist Christ Edwards (as S.C. Hitchcock), was published as a short book in 2009 by See Sharp Press. (I insisted on his using a pseudonym, as at the time he had a young family, lived in the Bible Belt, and had received anonymous death threats for writing letters to the editor opposing the Iraq War.)\n\n\"Dogspell,\" by recently deceased American author and humorist Earl Lee, appeared as a post on the See Sharp Press blog in 2013. It's an excellent example of Lee's acidic, earthy humor.\n\n# About the Authors\n\n**Chaz Bufe** is the author, editor, or translator of over a dozen books, including: _Alcoholics Anonymous: Cult or Cure?_ ; _An Understandable Guide to Music Theory_ ; _The Heretic's Handbook of Quotations_ ; _The American Heretic's Dictionary_ ; and the science fiction novel _Free Radicals: A Novel of Utopia and Dystopia_ (under the pseudonym Zeke Teflon). Chaz has also been the publisher and primary editor at See Sharp Press since 1984.\n\n**Dan Arel** is a journalist, activist, and author of the critically acclaimed book _Parenting without God_.\n\n# **ABOUT PM PRESS**\n\nPM Press was founded at the end of 2007 by a small collection of folks with decades of publishing, media, and organizing experience. PM Press co-conspirators have published and distributed hundreds of books, pamphlets, CDs, and DVDs. Members of PM have founded enduring book fairs, spearheaded victorious tenant organizing campaigns, and worked closely with bookstores, academic conferences, and even rock bands to deliver political and challenging ideas to all walks of life. We're old enough to know what we're doing and young enough to know what's at stake.\n\nWe seek to create radical and stimulating fiction and nonfiction books, pamphlets, T-shirts, visual and audio materials to entertain, educate, and inspire you. We aim to distribute these through every available channel with every available technology\u2014whether that means you are seeing anarchist classics at our bookfair stalls, reading our latest vegan cookbook at the caf\u00e9, downloading geeky fiction e-books, or digging new music and timely videos from our website.\n\nPM Press is always on the lookout for talented and skilled volunteers, artists, activists, and writers to work with. If you have a great idea for a project or can contribute in some way, please get in touch.\n\n**PM Press**\n\n**PO Box 23912**\n\n**Oakland, CA 94623**\n\n**www.pmpress.org**\n\n**PM Press in Europe**\n\n**europe@pmpress.org**\n\n**www.pmpress.org.uk**\n\n# **FRIENDS OF PM PRESS**\n\nThese are indisputably momentous times\u2014the financial system is melting down globally and the Empire is stumbling. Now more than ever there is a vital need for radical ideas.\n\nIn the years since its founding\u2014and on a mere shoestring\u2014PM Press has risen to the formidable challenge of publishing and distributing knowledge and entertainment for the struggles ahead. With over 300 releases to date, we have published an impressive and stimulating array of literature, art, music, politics, and culture. Using every available medium, we've succeeded in connecting those hungry for ideas and information to those putting them into practice.\n\n_Friends of PM_ allows you to directly help impact, amplify, and revitalize the discourse and actions of radical writers, filmmakers, and artists. It provides us with a stable foundation from which we can build upon our early successes and provides a much-needed subsidy for the materials that can't necessarily pay their own way. You can help make that happen\u2014and receive every new title automatically delivered to your door once a month\u2014by joining as a Friend of PM Press. And, we'll throw in a free T-shirt when you sign up.\n\nHere are your options:\n\n  * **$30 a month** Get all books and pamphlets plus 50% discount on all webstore purchases\n  * **$40 a month** Get all PM Press releases (including CDs and DVDs) plus 50% discount on all webstore purchases\n  * **$100 a month** Superstar\u2014Everything plus PM merchandise, free downloads, and 50% discount on all webstore purchases\n\nFor those who can't afford $30 or more a month, we have **Sustainer Rates** at $15, $10 and $5. Sustainers get a free PM Press T-shirt and a 50% discount on all purchases from our website.\n\nYour Visa or Mastercard will be billed once a month, until you tell us to stop. Or until our efforts succeed in bringing the revolution around. Or the financial meltdown of Capital makes plastic redundant. Whichever comes first.\n\n# **_Parenting without God: How to Raise Moral, Ethical, and Intelligent Children, Free from Religious Dogma, Second Edition_**\n\nDan Arel with a Foreword by Jessica Mills\n\n**ISBN:988-1-62963-708-2**\n\n**176 pages**\n\nChildren inevitably turn to their parents for more than just food and security; equally important are assurance, recognition, and interpretation of life. A child develops best in an environment where creativity and discovery are unimpeded by the artificial restrictions of blind faith and dogmatic belief. _Parenting without God_ is for parents, and future parents, who lack belief in a god and who are seeking guidance on raising freethinkers and social-justice-aware children in a nation where public dialogue has been controlled by the Christian Right.\n\nDan Arel, activist and critically acclaimed author, has penned a magnificently practical guide to help parents provide their children with the intellectual tools for standing up to attempts at religious proselytization, whether by teachers, coaches, friends, or even other family members. _Parenting without God_ is also for the parent activist who is trying to make the world a better place for all children by first educating their own children about racism, sexism, and all forms of discrimination that continue to serve as a barrier to the fundamentals of human dignity and democracy. It's for parents who wish for their children to question everything and to learn how to reach their own conclusions based on verifiable evidence and reason. Above all, Arel makes the penetrating argument that parents should lead by example\u2014both by speaking candidly about the importance of secularism and by living an openly and unabashedly secular life.\n\n_Parenting without God_ is written with humility, compassion, and understanding. Dan Arel's writing style is refreshingly lucid and conveys the unmistakable impression of a loving father dedicated to redefining the role of parenthood so that it also includes the vitally important task of nurturing every child's latent human impulse for freedom and autonomy. This second edition has been expanded with new material from the author.\n\n_\"_ Parenting without God _is not just about the absence of religion\u2014it's about the glorious space that opens up for secular parents and their lucky kids once the clutter and smoke of religion is gone. Dan Arel's voice is clear, smart, and a welcome addition to the growing chorus of parents taking the hands of their children and running at full speed into the real world.\"_\n\n\u2014Dale McGowan, author\/editor of _Parenting Beyond Belief_ and _Raising Freethinkers_\n\n# **_Anarchy and the Sex Question: Essays on Women and Emancipation, 1896\u20131926_**\n\nEmma Goldman\n\nEdited by Shawn P. Wilbur\n\n**ISBN: 978-1-62963-144-8**\n\n**160 pages**\n\nFor Emma Goldman, the \"High Priestess of Anarchy,\" anarchism was \"a living force in the affairs of our life, constantly creating new conditions,\" but \"the most elemental force in human life\" was something still more basic and vital: sex.\n\n\"The Sex Question\" emerged for Goldman in multiple contexts, and we find her addressing it in writing on subjects as varied as women's suffrage, \"free love,\" birth control, the \"New Woman,\" homosexuality, marriage, love, and literature. It was at once a political question, an economic question, a question of morality, and a question of social relations.\n\nBut her analysis of that most elemental force remained fragmentary, scattered across numerous published (and unpublished) works and conditioned by numerous contexts. _Anarchy and the Sex Question_ draws together the most important of those scattered sources, uniting both familiar essays and archival material, in an attempt to recreate the great work on sex that Emma Goldman might have given us. In the process, it sheds light on Goldman's place in the history of feminism.\n\n_\"Emma Goldman left a profound legacy of wisdom, insight, and passionate commitment to life. Shawn Wilbur has carefully selected her best writings on that most profound, pleasurable, and challenging of topics: sex. This collection is a great service to anarchist, feminist, and queer communities around the world.\"_\n\n\u2014Jamie Heckert, coeditor of _Anarchism & Sexuality: Ethics, Relationships and Power_\n\n_\"Shawn Wilbur has done a great job assembling and introducing Emma Goldman's writings on women, feminism, and sexuality. As he notes, Goldman's essays continue to provoke and inspire. The collection artfully documents the evolution of Goldman's views on freedom, sex, and human liberation.\"_\n\n\u2014Robert Graham, editor of _Anarchism: A Documentary History of Libertarian Ideas_\n\n# **_No Gods, No Masters, No Peripheries: Global Anarchisms_**\n\nEdited by Raymond Craib and Barry Maxwell\n\n**ISBN: 978-1-62963-098-4**\n\n**408 pages**\n\nWas anarchism in areas outside of Europe an import and a script to be mimicked? Was it perpetually at odds with other currents of the Left? The authors in this collection take up these questions of geographical and political peripheries. Building on recent research that has emphasized the plural origins of anarchist thought and practice, they reflect on the histories and cultures of the antistatist mutual aid movements of the last century beyond the boundaries of an artificially coherent Europe. At the same time, they reexamine the historical relationships between anarchism and communism without starting from the position of sectarian difference (Marxism versus anarchism). Rather, they look at how anarchism and communism intersected; how the insurgent Left could appear\u2014and in fact was\u2014much more ecumenical, capacious, and eclectic than frequently portrayed; and reveal that such capaciousness is a hallmark of anarchist practice, which is prefigurative in its politics and antihierarchical and antidogmatic in its ethics.\n\nCopublished the with Institute for Comparative Modernities, this collection includes contributions by Gavin Arnall, Mohammed Bamyeh, Bruno Bosteels, Raymond Craib, Silvia Rivera Cusicanqui, Geoffroy de Laforcade, Silvia Federici, Steven J. Hirsch, Adrienne Carey Hurley, Hilary Klein, Peter Linebaugh, Barry Maxwell, David Porter, Maia Ramnath, Penelope Rosemont, and Bahia Shehab.\n\n_\"Broad in scope, generously ecumenical in outlook, bold in its attempt to tease apart the many threads and tensions of anarchism, this collection defies borders and category. These illuminating explorations in pan-anarchism provide a much-needed antidote to the myopic characterizations that bedevil the red and black.\"_\n\n\u2014Sasha Lilley, author of _Capital and Its Discontents_\n\n_\"This wonderful collection challenges the privileging of Europe as the original and natural laboratory in which anti-statist ideas developed as well as the belief that anarchism and Communism could not intersect in fruitful ways. Drawing on non-Western locations (from Latin America, the Middle East, North Africa, and South Asia) its authors demonstrate how antiauthoritarian movements engaged with both local and global currents to construct a new emancipatory politics\u2014proving that anarchy and anarchism have always been global.\"_\n\n\u2014Barry Carr, La Trobe University\n\n# **_Shout Your Abortion_**\n\nEdited by Amelia Bonow and Emily Nokes with a Foreword by Lindy West\n\n**ISBN:978-1-62963-573-6**\n\n**256 pages**\n\nFollowing the U.S. Congress's attempts to defund Planned Parenthood, the hashtag #ShoutYourAbortion became a viral conduit for abortion storytelling, receiving extensive media coverage and positioning real human experiences at the center of America's abortion debate for the very first time. The online momentum sparked a grassroots movement that has subsequently inspired countless individuals to share their abortion stories in art, media, and community events all over the country, and to begin building platforms for others to do the same.\n\n_Shout Your Abortion_ is a collection of photos, essays, and creative work inspired by the movement of the same name, a template for building new communities of healing, and a call to action. Since SYA's inception, people all over the country have shared stories and begun organizing in a range of ways: making art, hosting comedy shows, creating abortion-positive clothing, altering billboards, starting conversations that had never happened before. This book documents some of these projects and illuminates the individuals who have breathed life into this movement, illustrating the profound liberatory and political power of defying shame and claiming sole authorship of our experiences. With _Roe vs. Wade_ on the brink of reversal, the act of shouting one's abortion has become explicitly radical, and _Shout Your Abortion_ is needed more urgently than ever before.\n\n_\"_ Shout Your Abortion _reflects what makes me most hopeful for this next century, that there's a whole new generation of young people who refuse to be judged and shamed about who they are, whether it's their gender identity, their sexual orientation, or what they do with their body.\"_\n\n\u2014Cecile Richards, president of Planned Parenthood Federation of America\n\n_\"By presenting a collection of nuanced narratives,_ Shout Your Abortion _aims to advance a message of broader acceptance: If your abortion experience was hard and sad, that's okay. If your abortion experience wasn't hard or sad, that's also okay. This marks a significant tonal shift in the cultural conversation about abortion.\"_\n\n\u2014Caitlin Gibson, _Washington Post_\n\n_\"_ Shout Your Abortion _has altered the landscape. We all knew that abortion stigma was a central tool in keeping women silent and isolated. Amelia Bonow busted down that door and women poured through it like a dam had burst.\"_\n\n\u2014Martha Plimpton, actor and cofounder of A is For\n\n# **_Witches, Witch-Hunting, and Women_**\n\nSilvia Federici\n\n**ISBN: 978-1-62963-568-2**\n\n**120 pages**\n\nWe are witnessing a new surge of interpersonal and institutional violence against women, including new witch hunts. This surge of violence has occurred alongside an expansion of capitalist social relations. In this new work that revisits some of the main themes of C _aliban and the Witch_ , Silvia Federici examines the root causes of these developments and outlines the consequences for the women affected and their communities. She argues that, no less than the witch hunts in sixteenth\u2013and seventeenth-century Europe and the \"New World,\" this new war on women is a structural element of the new forms of capitalist accumulation. These processes are founded on the destruction of people's most basic means of reproduction. Like at the dawn of capitalism, what we discover behind today's violence against women are processes of enclosure, land dispossession, and the remolding of women's reproductive activities and subjectivity.\n\nAs well as an investigation into the causes of this new violence, the book is also a feminist call to arms. Federici's work provides new ways of understanding the methods in which women are resisting victimization and offers a powerful reminder that reconstructing the memory of the past is crucial for the struggles of the present.\n\n_\"It is good to think with Silvia Federici, whose clarity of analysis and passionate vision come through in essays that chronicle enclosure and dispossession, witch-hunting and other assaults against women, in the present, no less than the past. It is even better to act armed with her insights.\"_\n\n\u2014Eileen Boris, Hull Professor of Feminist Studies, University of California, Santa Barbara\n\n_\"Silvia Federici's new book offers a brilliant analysis and forceful denunciation of the violence directed towards women and their communities. Her focus moves between women criminalized as witches both at the dawn of capitalism and in contemporary globalization. Federici has updated the material from her well-known book_ Caliban and the Witch _and brings a spotlight to the current resistance and alternatives being pursued by women and their communities through struggle.\"_\n\n\u2014Massimo De Angelis, professor of political economy, University of East London\n\n# **_Re-enchanting the World: Feminism and the Politics of the Commons_**\n\nSilvia Federici with a Foreword by Peter Linebaugh\n\n**ISBN: 978-1-62963-569-9**\n\n**240 pages**\n\nSilvia Federici is one of the most important contemporary theorists of capitalism and feminist movements. In this collection of her work spanning over twenty years, she provides a detailed history and critique of the politics of the commons from a feminist perspective. In her clear and combative voice, Federici provides readers with an analysis of some of the key issues and debates in contemporary thinking on this subject.\n\nDrawing on rich historical research, she maps the connections between the previous forms of enclosure that occurred with the birth of capitalism and the destruction of the commons and the \"new enclosures\" at the heart of the present phase of global capitalist accumulation. Considering the commons from a feminist perspective, this collection centers on women and reproductive work as crucial to both our economic survival and the construction of a world free from the hierarchies and divisions capital has planted in the body of the world proletariat. Federici is clear that the commons should not be understood as happy islands in a sea of exploitative relations but rather autonomous spaces from which to challenge the existing capitalist organization of life and labor.\n\n_\"Silvia Federici's theoretical capacity to articulate the plurality that fuels the contemporary movement of women in struggle provides a true toolbox for building bridges between different features and different people.\"_\n\n\u2014Massimo De Angelis, professor of political economy, University of East London\n\n_\"Silvia Federici's work embodies an energy that urges us to rejuvenate struggles against all types of exploitation and, precisely for that reason, her work produces a common: a common sense of the dissidence that creates a community in struggle.\"_\n\n\u2014Maria Mies, coauthor of _Ecofeminism_\n\n# **_Romantic Rationalist: A William Godwin Reader_**\n\nEdited by Peter Marshall with a Foreword by John P. Clark\n\n**ISBN: 978-1-62963-228-5**\n\n**192 pages**\n\nWilliam Godwin (1756\u20131836) was one of the first exponents of utilitarianism and the first modern proponent of anarchism. He was not only a radical philosopher but a pioneer in libertarian education, a founder of communist economics, and an acute and powerful novelist whose literary family included his partner, pioneering feminist writer Mary Wollstonecraft, and his daughter Mary Godwin (later Mary Shelley), who would go on to write _Frankenstein_ and marry the poet Percy Bysshe Shelley.\n\nHis long life straddled two centuries. Not only did he live at the center of radical and intellectual London during the French Revolution, he also commented on some of the most significant changes in modern history. Shaped by the Enlightenment, he became a key figure in English Romanticism.\n\nThis work offers for the first time a handy collection of Godwin's key writings in a clear and concise form, together with an assessment of his influence, a biographical sketch, and an analysis of his contribution to anarchist theory and practice. The selections are taken from all of Godwin's writings including his groundbreaking work during the French Revolution, _An Enquiry Concerning Political Justice_ , and arranged by editor Peter Marshall to give a coherent account of his thought for the general reader.\n\nGodwin's work will be of interest to all those who believe that rationality, truth, happiness, individuality, equality, and freedom are central concerns of human enquiry and endeavor.\n\n_\"Peter Marshall has produced the most useful modern account of Godwin's life and now the most useful modern anthology of his writings. Marshall's selection is sensible and valuable, bringing out the important points.... His introduction is a good summary of Godwin's life and work.... Marshall is right to see him as 'the most profound exponent of philosophical anarchism.'\"_\n\n\u2014Nicolas Walter, _New Statesman_\n\n_\"A handsome and handy little book, excavating nuggets of Godwinian wisdom from the whole range of his writings.\"_\n\n\u2014Colin Ward, _Times Educational Supplement_\n\n# **_Kropotkin: The Politics of Community_**\n\nBrian Morris\n\n**ISBN:978-1-62963-505-7**\n\n**320 pages**\n\nThe nineteenth century witnessed the growth of anarchist literature, which advocated a society based on voluntary cooperation without government authority. Although his classical writings on mutual aid and the philosophy of anarchism are still published today, Peter Kropotkin remains a neglected figure. A talented geographer and a revolutionary socialist, Kropotkin was one of the most important theoreticians of the anarchist movement.\n\nIn _Kropotkin: The Politics of Community_ , Brian Morris reaffirms with an attitude of critical sympathy the contemporary relevance of Kropotkin as a political and moral philosopher and as a pioneering social ecologist. Well-researched and wide-ranging, this volume not only presents an important contribution to the history of anarchism, both as a political tradition and as a social movement, but also offers insightful reflections on contemporary debates in political theory and ecological thought. After a short biographical note, the book analyzes in four parts Kropotkin's writings on anarchist communism, agrarian socialism, and integral education; modern science and evolutionary theory; the French Revolution and the modern state; and possessive individualism, terror, and war.\n\nStanding as a comprehensive and engaging introduction to anarchism, social ecology, and the philosophy of evolutionary holism, _Kropotkin_ is written in a straightforward manner that will appeal to those interested in social anarchism and in alternatives to neoliberal doctrines.\n\n_\"Peter Kropotkin has been largely ignored as a utopian crackpot, but Brian Morris demonstrates in this wide-ranging and detailed analysis that Kropotkin addressed significantly and perceptively the major issues of the present day.\"_\n\n\u2014Harold B. Barclay, author of _People without Government: An Anthropology of Anarchy_\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Contents\n  5. Introduction--Dan Arel\n  6. Chapter I The God Pestilence--Johann Most\n  7. Chapter II Woman, Church, and State--Matilda Gage\n  8. Chapter III The Devil's Dictionary (excerpts)--Ambrose Bierce\n    1. American Heretic's Dictionary (excerpts)--Chaz Bufe\n  9. Chapter IV The Failure of Christianity--Emma Goldman\n  10. Chapter V Twelve Proofs of the Nonexistence of God--Sebastien Faure\n  11. Chapter VI The Meaning of Atheism--E. Haldeman-Julius\n  12. Chapter VII How Christianity Grew Out of Paganism--Joseph McCabe\n    1. Christianity and Slavery--Joseph McCabe\n    2. Judeo-Christian Degradation of Woman--Joseph McCabe\n  13. Chapter VIII Why Science Leaves Religion in the Dust--Chaz Bufe\n    1. Twenty Reasons to Abandon Christianity--Chaz Bufe\n  14. Chapter IX May the Farce Be with You: A Lighthearted Look at Why God Does Not Exist--Pamela Sutter\n  15. Chapter X Disbelief 101: A Young Person's Guide to Atheism (excerpt)--S.C. Hitchcock\n  16. Chapter XI Dogspell--Earl Lee\n  17. Afterword--Chaz Bufe\n  18. About the Authors\n\n## Landmarks\n\n  1. Cover\n  2. Title Page\n  3. Table of Contents\n  4. Start of Content\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"# Cover\n# About the Author\n\n* * *\n\nTed Barris is an accomplished author, journalist, and broadcaster. As well as hosting stints on CBC Radio and regular contributions to the Globe and Mail, the National Post, and various national magazines, he is a full-time professor of journalism at Centennial College in Toronto. Barris has authored seventeen non-fiction books. In 2011 he received the Canadian Minister of Veterans' Affairs Commendation and in 2012 the Queen Elizabeth II Diamond Jubilee Medal.\n\n# Also by Ted Barris\n\n* * *\n\nmilitary history\n\nBehind the Glory: Canada's Role in the Allied Air War\n\nDays of Victory: Canadians Remember, 1939\u20131945\n\n(with Alex Barris, 1st edition, 1995)\n\nDeadlock in Korea: Canadians at War, 1950\u20131953\n\nCanada and Korea: Perspectives 2000 (contributor)\n\nJuno: Canadians at D-Day, June 6, 1944\n\nDays of Victory: Canadians Remember, 1939\u20131945\n\n(Sixtieth Anniversary edition, 2005)\n\nVictory at Vimy: Canada Comes of Age, April 9\u201312, 1917\n\nBreaking the Silence: Veterans' Untold Stories from the Great War to Afghanistan\n\nother non-fiction\n\nFire Canoe: Prairie Steamboat Days Revisited\n\nRodeo Cowboys: The Last Heroes\n\nPositive Power: The Story of the Edmonton Oilers Hockey Club\n\nSpirit of the West: The Beginnings, the Land, the Life\n\nPlaying Overtime: A Celebration of Oldtimers' Hockey\n\nCarved in Granite: 125 Years of Granite Club History\n\nMaking Music: Profiles from a Century of Canadian Music\n\n(with Alex Barris)\n\n101 Things Canadians Should Know About Canada (contributor)\n\n# Dedication\n\n> To air force veteran Charley Fox and military history\n> \n> buff Dave Zink\u2014both gone now\u2014who challenged me\n> \n> to properly retell this great, great story.\n\n# Contents\n\n* * *\n\n  * Acknowledgements\n  * Introduction: \"Heroes Resurface\"\n  * 1 The King's Regulations\n  * 2 Bond of Wire\n  * 3 \"Spine-Tingling Sport\"\n  * 4 Escape Season\n  * 5 Servant to a Hole in the Ground\n  * 6 \"Shysters and Crooks and Con Men\"\n  * 7 The Play's the Thing\n  * 8 \"Through Adversity to the Stars\"\n  * 9 The Hate Campaign\n  * 10 Long Road Home\n  * 11 \"A Proud, Spectacular Distraction\"\n  * Notes\n  * Photograph Credits\n\nThe author and publisher gratefully acknowledge the permission granted to reproduce the copyright material in this book. Every effort has been made to trace copyright holders and to obtain their permission for the use of copyright material. The publisher apologizes for any errors or omissions and would be grateful if notified of any corrections that should be incorporated in future reprints or editions of this book.\n\nThe author would like to acknowledge the following for the use of published and unpublished works as follows:\n\n_A Gallant Company_ , Jonathan F. W. Vance, permission from copyright holder.\n\nAir Force Association video interviews, 1970s and 1989, permission from rights holders National Air Force Museum of Canada, Trenton.\n\n_Bonds of Wire_ , Kingsley Brown, permission from copyright holder Ethel Alle.\n\n_Forced March to Freedom_ , Robert Buckham, permission from copyright holder Nancy Buckham.\n\nFrank Sorensen collection, permission from Glenn, Stephen, and Vicki Sorensen.\n\n_Goon in the Block_ , Don Edy, permission from copyright holder.\n\n_In Enemy Hands_ , Daniel G. Dancocks, permission sought through Hurtig, Random House.\n\n_It's All Pensionable Time_ , George Sweanor, permission from copyright holder.\n\nJohn Colwell diary, permission from copyright holder Harold Johnstone.\n\nJohn Weir letters, permission from Mrs. Frances Weir.\n\n_Lonesome Road_ , George Harsh, permission from publisher W. W. Norton, New York.\n\n_One Man's War: Sub Lieutenant R. E. Bartlett, RN Fleet Air Arm Pilot_ , Stuart E. Soward, permission from copyright holder Sheila Soward.\n\n_Serving and Surviving: An Airman's Memoirs_ , John R. Harris, permission from copyright holder.\n\n_The Great Escape_ , Paul Brickhill, permission from rights holders David Higham Associates, UK.\n\n_The Great Escape, Stalag Luft III (from the original drawings made by Ley Kenyon 1943)_ , permission from copyright holders RAF Museum, Hendon, UK.\n\n_The Tunnel King: The True Story of Wally Floody and the Great Escape_ , Barbara Hehner, permission from copyright holder.\n\n_They Were So Young_ , Patricia Burns, permission from copyright holder.\n\n\"Tom, Dick and Harry of Stalag Luft III,\" Bob Nelson, unpublished manuscript, permission from Sally Hutchison.\n\n# Acknowledgements\n\n* * *\n\nTHE CONTEMPORARY ROAD to Zagan is nearly as inhospitable and neglected as it must have been in 1942, when the town became neighbour to a prisoner-of-war camp. Even when I travelled on it in 2010, most of the one hundred miles of road southeast of Berlin, primarily in western Poland, didn't seem to have ever enjoyed priority status. The post-Soviet-era asphalt was still as patchy, the lanes still as poorly marked, and the rough countryside terrain still encroaching the roadside shoulders right to the road surface as it likely had when the Nazis occupied Poland during the Second World War.\n\nLikewise, the Zagan (the Polish spelling of Sagan) train station at the edge of town looked as if it hadn't enjoyed any remodelling since it was built in the early 1900s. When I walked inside, I could almost see the first wave of Great Escape fugitives hurrying through ticket queues and platform document checks by guards in the morning gloom on March 25, 1944, to get aboard the Breslau-to-Berlin express train without raising suspicion. And the trees\u2014the omnipotent pine forest to the south\u2014between the railway platforms and the prison camp looked as dense and claustrophobic as they must have been to the air force officers trying to escape a generation ago.\n\nWhen I approached the actual North Compound site at Stalag Luft III, now overgrown with mature trees, dense underbrush, and weeds, I could see scattered bricks and concrete pads where the barracks huts had stood on blocks. None of the buildings remained. I could see the foundations of the infirmary, the cooler, and the coal store. Beyond it, farther south, I could see the fire pool, the cement floors of the kitchens (with scorched circles where huge soup cauldrons had boiled every day). Beyond them lay the brick foundation of the North Compound theatre (where a fourth tunnel, \"George,\" has just recently been unearthed). And finally, on the surface of the still very sandy soil, I walked along a walkway of crushed stone with wooden borders, just twenty inches wide (the same width of the tunnel), showing above ground where tunnel \"Harry\" had stretched underground\u2014some 336 feet\u2014from the concrete pad beneath Hut 104 to beyond the Stalag Luft III fence, but just short of the woods.\n\nI imagined about eighty POWs who had all served in one capacity or another in the manufacture of the tunnel excavation and its ancillary requirements (sand dispersal, security, intelligence, forgery, document production, tailoring, language study, compass manufacture, ration supplies, et cetera) making their way from inside the North Compound. I visualized one POW disappearing into \"Harry\" every three minutes through the night of March 24\u201325, travelling along the trolley way northbound (beneath where I stood in 2010), and then popping up from the other vertical shaft beyond the prison camp wire. This was the home of the Great Escape, or at least an attempt to cause enough havoc behind German enemy lines to suck away valuable manpower in the search and recapture of the escapers.\n\nA few thousand feet away, I came to the cemetery of those who'd died in captivity during their existence at Stalag Luft III. Next to the individual tombstones, I found the stone memorial to the fifty Commonwealth air force officers murdered by the Gestapo after the breakout. Here, too, I sensed the blood of six Canadians, or at least the ashes in urns buried there during a ceremony on December 4, 1944 (the ashes later exhumed and reburied in the military cemetery at Poznan, Poland). This was the home of the Great Escape. This was the place where myth and reality had inspired books, documentaries, and Hollywood movies. Arriving there, walking there, remembering there in 2010, moved me emotionally, and moved me professionally to fulfill a longtime promise to my veteran friend, RCAF fighter pilot Charley Fox, to tell the story of the Great Escape the way it could and should be told\u2014as a Canadian story.\n\nNot only have I chosen to write this story as an homage to Charley Fox, who died in 2008, but here also I want to offer my gratitude to others who have assisted this labour of love both recently and over many years of preparation and research:\n\nAmong other veterans and their families, I want to thank George and Joan Sweanor, and their daughter Barbara; Albert Wallace and his daughter Barbara Trendos; Don Edy and his daughters Barb Edy and Jane Hughes; John R. Harris; Vicki, Stephen, and Glenn Sorensen, who gave the gift of their father Frank Sorensen's correspondence; David (and Cathy) Pengelly for remembering brother Tony Pengelly; Chris Pengelly, Tony's son, for the treasure trove of his father's personal records of Stalag Luft III; and friends Mary and David Ross (and their trusty Facebook account), who helped track down the Pengellys; Fran Weir for letters written home by her husband John Weir; Margaret Bartlett, and her daughter Anne Dumonceaux and grandson Nick Dumonceaux, for remembering Dick Bartlett; Don McKim, with the assistance of son Al McKim, daughter Wendy Johnson, and friend Bernice Marsland; Catherine Heron, sister of Wally Floody, for the scrapbook and photo files of her brother Wally Floody's career, and son Brian (and Lorraine) Floody for their interview collections of Wally; Ethel Alle and Kingsley Brown Jr. for access to their father Kingsley Brown Sr.'s memoirs; Barry Davidson Jr. for access to his father Barry Davidson Sr.'s logs; Fred and Susan Bendell, as well as daughter Katie Bendell, for material from Gordon Venables' experience in the prison camp; Nancy Buckham for access to her husband Robert Buckham's diaries and sketches; Marjorie Acheson (with help from Kitchener Public librarian Karen Ball-Pyatt) for access to John Acheson's memoirs; Kim and Kelly Crozier for access to their father John Crozier's diaries; Sally and John Hutchinson for the gift of her father Bob Nelson's writings; Harold Johnstone for granting access to John Colwell's precise images and diary; Keith Ogilvie Jr. and Jean Ogilvie for photos of their father Keith \"Skeets\" Ogilvie; Ted Nurse for permission to refer to his account of his father Edward Nurse's experience at Stalag Luft III; Marilyn Walton for assistance in obtainiing US Air Force Academy photos; and Don and Linda Jarrell for stories about Don's father, John MacKinnon \"Mac\" Jarrell as a kriegie.\n\nThanks to support and data (files and photographs) provided by Marek Lazarz and the volunteers at the Museum of Allied Forces Prisoners of War Martyrdom at Zagan, Poland.\n\nAs well, a debt of gratitude to James Taylor, Parveen Kaur Sodhi, and Sally Richards at the Imperial War Museums in the UK for assistance in gaining access and rights to their Stalag Luft III photo archives. Also in the UK, my thanks to Andrew Dennis and Vinit Mehta at the Royal Air Force Museum at Hendon for their help securing the rights to the Ley Kenyon sketches of The Great Escape.\n\nNo one writes who doesn't also read and recognize the previous work of fellow authors (specific references and permissions are contained in the endnotes). Among those key to my interpretation of this story are Paul Brickhill (The Great Escape), Kingsley Brown (Bonds of Wire), Robert Buckham (Forced March to Freedom), Patricia Burns (They Were So Young), Andrew Carswell (Over the Wire), H.P. Clark (Wire Bound World), Art Crighton (Memories of a Prisoner of War), Daniel G. Dancocks (In Enemy Hands), Ian Darling (Amazing Airmen), Don Edy (Goon in the Block), Hugh Godefroy (Lucky Thirteen), Philip Gray (Ghosts of Targets Past), John R. Harris (Serving and Surviving), George Harsh (Lonesome Road), John Hartnell-Beavis (Final Flight), Barbara Hehner (The Tunnel King), Stuart G. Hunt (Twice Surreal), Harold Johnstone (John Colwell), Phil Marchildon (Ace), Wayne Ralph (Aces, Warriors and Wingmen), Stuart E. Soward (One Man's War), George Sweanor (It's All Penshionable Time), Tyler Trafford (Almost a Great Escape), and Jonathan Vance (A Gallant Company).\n\nIn various ways, but drawing on their unique expertise, I wish to thank Susan Hall for her wisdom about music; Dave Zink (who died in 2012) for his encyclopedic knowledge of war history and militaria; Barb and Stuart Blower for their assistance in restoring old photographs; Don Young for his sense of Canadian storytelling; and Marian Hebb for her legal acumen and commonsense approach to getting books from writers to readers.\n\nIn terms of the special assistance of colleagues and friends in journalism, creative non-fiction writing, and publishing, I owe much to authors Malcolm Kelly and Byron Christopher, journalism professors Lindy Oughtred, Stephen Cogan, and Ellin Bessner, broadcaster Rick Cluff, publisher Marc Cot\u00e9, and editor Don Loney. Special thanks to my neighbour, Navy veteran Ronnie Egan, for her homemade sandwiches that helped me maintain my daily writing quota. I thank my team of transcribers, Octavian Lacatusu and Michael Laing-Fraser. At Thomas Allen Publishers, I'm grateful to editors Janice Zawerbny and Linda Pruessen, to proofreader Ruth Chernia, to marketing team Krista Lynch and Catherine Whiteside, and to my favourite TAP triumvirate of David Glover, Bonita Mok, and Heather Goldberg. Special thanks to Beth Crane for making our photos sparkle. And as he has done for all of my TAP books, Gordon Robertson has contributed his exquisite visual sense to the design of this project.\n\nAnd the first shall be last: I thank my wife, Jayne MacAulay, who edits by profession, and thinks clearly by nature.\n\n\u2014Ted Barris 2013\n\n# Introduction\n\n\"HEROES RESURFACE\"\n\n* * *\n\nAHEAD OF HIM in the distance, lie the Alps. Disguised in a German dispatch rider's battledress and stahlhelm helmet,  \na fugitive hero brings his motorcycle to a sudden halt in the middle of this back road in wartime Germany. Nervously jerking the bike's throttle, he wheels his body around in the saddle, searching for his pursuers. A close-up catches his face as his eyes settle on the mountains ahead.\n\n\"Switzerland,\" he says under his breath.\n\nHe throttles up, pops the bike back into gear, and roars off. Minutes later, his disguise revealed, the motorcyclist bashes through a checkpoint, peels off the now useless German uniform, and blasts cross-country toward the distant mountains and, he hopes, freedom. Behind him, military trucks, motorcycles with sidecars, and what seems to be half the German Army are in hot pursuit. All the while, Elmer Bernstein's stirring film score accelerates in tempo and rises with crescendo. Then, in his character's final moments on the run, actor Steve McQueen launches himself and his 650 Triumph over the first of two barriers of sharp pickets and barbed wire. Unable to make it over the second barrier, US Army Air Force Captain Virgil Hilts crashes into the fence, entangles himself in the wire, tries to break free, then realizes the futility of his struggle and surrenders to the hordes of German troops who've now overtaken him.\n\nThese few minutes of flight and pursuit, caught on 35-millimetre colour Panavision film, remain among the most thrilling in the canon of Hollywood war moviemaking and viewing. Indeed, this climactic sequence, among many crowd-pleasing, escalating jolts of action and drama in The Great Escape, earned Hollywood film editor Ferris Webster an Academy Award nomination.* Based on Paul Brickhill's non-fiction book The Great Escape, published in 1950, the movie became an immediate hit when it was released in North American theatres in the summer of 1963. On the whole, it also garnered upbeat critical reviews.\n\n\"With accurate casting, a swift screenplay, and authentic German settings,\" Time magazine raved, \"producer-director John Sturges has created a classic . . .\"[1]\n\nWas it a classic? Yes. In 2006, more than forty years after the film's release, a poll asked TV watchers and moviegoers in the United Kingdom which flick they would most want to view over Christmas. The Great Escape came in third among families, first among male viewers. Was it authentic? In part it was. A group of veterans returning to the site of the escape on the occasion of the sixty-fifth anniversary, in 2009, told journalists reporting the event that Hollywood had depicted much of the life in the infamous Stalag Luft III prisoner-of-war camp accurately.\n\nOn the other hand, there was no Captain Virgil Hilts, \"The Cooler King,\" in the actual Great Escape. Nor were there a lot of Americans in the prison's North Compound when the breakout occurred. Nor was there a motorcycle chase. When writers James Clavell, W. R. Burnett, and Walter Newman originally scripted the fictitious USAAF Captain Hilts into the screenplay, they had him attempting to escape aboard a train among other fleeing Allied POWs. However, Steve McQueen, contracted by Sturges and the Mirisch Company to co-star in the film, had such a passion for motorcycle racing that he insisted on having the chase scene built into the film.\n\nDid the movie present authentic settings? Apparently, it didn't matter to the producers that the actual Great Escape occurred near the town of Sagan in southern Poland, not in southern Germany, near the Swiss Alps. Nor was it relevant to McQueen or the movie's creators that the escape happened on March 24, 1944, when a metre of snow still lay on the ground, where no motorcycle could easily travel, let alone leap barbed-wire fences.\n\nHere are the facts of the escape. On the night of March 24, 1944, eighty Commonwealth air officers crawled through a 360-foot-long tunnel and slipped into the darkness of a pine forest beyond the wire of the North Compound at Stalag Luft III near Sagan. The intricate breakout, more than a year in the making, involved as many as two thousand POWs, extraordinary coordination, and a battle of wits inconceivable for the time. As dawn broke on March 25, however, German guards outside the compound spotted prisoners emerging from the exit hole, set off an alarm, and over the next few days managed to recapture all but three of the escapers. In a rage over the incident, Adolf Hitler called for the execution of all the escapers; instead, the death list was adjusted downward and fifty Commonwealth air officers were executed, with perpetrators claiming the prisoners were shot while attempting to escape Gestapo custody. Their bodies were cremated and buried in a remote corner of the Stalag Luft III grounds to hide the truth.\n\nA hint about the accuracy of Hollywood's version of the story is evident in one of the movie's final scenes. It shows the fifty officers gunned down en masse by a German machine-gun crew in an open field somewhere in Germany. In truth, after their recapture, imprisonment, and interrogation, the officers were taken out and shot in twos and threes by Gestapo death squads hand-picked and given licence to execute the officers in cold blood by a German High Command edict known as the Sagan Order. The little-known origins of the order add much intrigue to this story.\n\nBut was the casting as accurate as the rave reviews said? True, the hiring of Steve McQueen, Richard Attenborough, James Garner, Charles Bronson, Donald Pleasence, and James Coburn among a cast of hundreds made a lot of sense. The on-screen ensemble of principal actors represented the cream of Hollywood idols in the early 1960s. They ensured The Great Escape would become among the best box-office draws of the year. But did the movie escape committee accurately represent the real escape committee?\n\nIn an effort to make the movie plot and its characters more inviting and palatable to an American audience, the writers invented Flight Lieutenant (F\/L) Bob Hendley, an American in the RAF, as the scrounger inside Stalag Luft III. Sturges cast US screen and TV star James Garner to play the part. In fact, the scrounger was a twenty-eight-year-old Blenheim bomber pilot from Calgary, Alberta, named Barry Davidson.\n\nFor the key roles of the tunnel designers and diggers, Sturges's creative team invented RAF F\/L Danny Velinski and RAF F\/L Willie Dickes and cast American actor Charles Bronson and British actor\/singer John Leyton in the roles. The actual tunnel king was a downed Spitfire pilot, twenty-five-year-old Wally Floody, originally from Chatham, Ontario. His tunnel digging partners were fellow RCAF fighter pilots: twenty-four-year-old John Weir from Toronto, Ontario, and twenty-six-year-old Hank Birkland, from Spearhill, Manitoba.\n\nWhen it came to portraying the chief forger\u2014the POW who designed many of the fake documents used by the air officers during the escape\u2014the screenplay writers manufactured another British flyer named Colin Blythe and cast seasoned British film and TV actor Donald Pleasence (who had actually been a POW during the war) in the role. The actual forger behind much of the document fabrication, however, was twenty-four-year-old Whitley bomber pilot Tony Pengelly, from Truro, Nova Scotia.\n\nNext, the Hollywood production team imagined one of the intelligence chiefs in the camp and parachuted into the script a British air officer named Andy MacDonald, casting Scottish-born actor Gordon Jackson to play him. In fact, among the officers conducting much of the intelligence activities was thirty-two-year-old Kingsley Brown, a journalist and father of four from the Toronto area.\n\nTo portray an officer in charge of the security of the three tunnels\u2014\"Tom,\" \"Dick,\" and \"Harry\"\u2014the movie producers conceived an RAF F\/L Sorren and cast British actor William Russell in the role. In fact, the security team inside the wire at the North Compound included thirty-three-year-old RCAF air gunner George Harsh, originally from Milwaukee, Wisconsin, and twenty-four-year-old RCAF bomb-aimer George Sweanor, from Port Hope, Ontario. And that doesn't include the air officer in charge of security at the entrance of the main tunnels, the so-called trapf\u00fchrer, twenty-six-year-old Patrick Langford from Edmonton, Alberta.\n\nFor organizing what were called \"diversions\" inside the compound, the movie producers invented RAF F\/L Dai Nimmo and then hired English horror and mystery film actor Tom Adams to play him. Among the actual diversionary geniuses, however, was twenty-five-year-old George McGill, an RCAF navigator from Toronto; McGill helped orchestrate boxing matches and other sporting attractions to distract German guards during some of the earliest escape attempts at Stalag Luft III. As well, the escape committee inside the North Compound had organized diversionary \"culture appreciation sessions\" in the prison library, where twenty-nine-year-old RCAF navigator Gordon Kidder from St. Catharines, Ontario, actually taught conversational German to the soon-to-be escapers. Adding to the real diversionary linguistics staff was another Canadian, twenty-one-year-old Spitfire pilot Frank Sorensen, who taught the head of the escape committee, Roger Bushell, basic conversational Danish as the two men exercised on the walking path inside the warning wire at the compound.\n\nIn a delicious irony of casting and accuracy, The Great Escape screenplay writers chose to change Roger Bushell's name to Roger Bartlett. Coincidentally, the man responsible for servicing and hiding an inside-the-camp short-wave radio that delivered nightly newscasts from the BBC was yet another unheralded Canadian, twenty-four-year-old dive-bomber pilot Richard Bartlett, from Fort Qu'Appelle, Saskatchwan.\n\nIn other words, contrary to Hollywood's Anglo-American version of the Great Escape story, the actual mass breakout owed much of its design, execution, and tactical success to an extraordinarily talented crew of air force officers from Canada. Again, the producers' rationale for stacking the script with US and UK character names and stars was clear: to help the audience connect to the story and to give the marketing and promotional team plenty of recognizable talent to build box office. And it worked. A film that cost roughly four million dollars to produce grossed more than eleven million dollars in ticket revenue during the summer of 1963.\n\nMotorcycles and the Hollywood dream factory aside, recognizing that the actual Great Escape brain trust was largely Canadian is not the only reason for revisiting this unique war story. The wider picture, occupied by other Canadian air officers shot down during the war and sent to Stalag Luft III, illustrates just how remarkable this escape feat really was. As Kriegsgefangenen (prisoners of war)\u2014or, as they called themselves, \"kriegies\"\u2014the air officers were not required to work inside the wire. They had to report for appell (roll call) several times a day and abide by all the other rules administered by their Luftwaffe (German Air Force) jailers. But beyond that, they were spared the forced labour, wrath, and deprivation of concentration camps or non-commissioned officer (NCO) compounds.\n\nConsequently, the Commonwealth air officers were left to their own devices day and night\u2014feeding themselves, clothing themselves, educating themselves, entertaining themselves, keeping themselves fit, and conspiring among themselves. For the months and years that they were imprisoned at Stalag Luft III, the POWs\u2014many of them Canadian\u2014built, seemingly from thin air, an extraordinarily orderly world inside the wire. There was \"the Tin Man,\" John Colwell, who built a pot, or chair, or escape tool nearly every day of his imprisonment. There were the extroverted officers, such as James Wernham, Don Edy, Arthur Crighton, and Tony Pengelly, whose penchant for performing led them to become regulars in the musicals, concerts, revues, and full classical productions staged weekly at the North Compound theatre. And since plenty of the officers had played international or professional sports\u2014including big league pitchers Phil Marchildon and Bill Paton\u2014the sports grounds at the North Compound buzzed with tournaments on the baseball diamond, football pitch, or ice-hockey rink, no matter the season. There was even an inside-the-wire press corps to publish results and colour commentary. But beneath that veneer remained a secret society of officers\u2014about a third of whom were Canadians\u2014intent on breaking out of the camp, or at least disrupting the German war machine sufficiently before being recaptured, sent to solitary confinement for a time, and finally returned to the barracks huts inside Stalag Luft III.\n\nIn addition to a review of the existing accounts, telling the Canadian story of the Great Escape has yielded new and disturbing information about some little-known aspects of the breakout and its aftermath. For one, the struggle for power between the Luftwaffe and the Gestapo may have doomed the escape plot and escapers long before the breakout on March 24\u201325, 1944. For another, it appears that no intervention by Red Cross, Luftwaffe prison administrators, or articles of the Geneva Conventions, could have prevented the murders of the fifty Commonwealth officers, including six Canadians. Conditions of the Sagan Order, and the inbred hostility of those hired to carry it out, meant certain death. Ultimately, as resilient and innovative as the kriegies proved to be throughout their incarceration, the forced march during the last months of the winter in 1945 and on the eve of their liberation proved a final test of mind and body some could not endure.\n\nIf there remains any doubt that the Great Escape myth and reality continue to resonate and demand Canadians' attention, consider the appetite for the story on the eve of the seventieth anniversary of the historic breakout. Across the country today, the legacy of the kriegie experience lives on in local, regional, and national reunions (albeit in ever decreasing size and frequency). Fascination for the story persists because the surviving wives, sisters, and brothers kept letters and news clippings. It lingers because daughters and sons and grandchildren never stop seeking the answers to unresolved questions about their kriegie parent's experience. Meanwhile, the Great Escape tale grows exponentially; every year, fresh stories emerge in newly discovered postcards and letters, never before published photos, unearthed tunnel archaeology at the site near Zagan, Poland, and remembrances that are finally shared by kriegie offspring and dedicated collectors with the rest of us. Documentarians and authors continue to expand the library of discoveries and views. Canadians and others never stop asking why this kriegie lived or that one died. And then, on Christmas break, near the anniversary, or online, there always seems to be a constituency\u2014young and old\u2014eager to watch Hollywood's version, right down to Steve McQueen's mythical motorcycle attempt to beat the odds.\n\nAbout the longevity and indelibility of The Great Escape story at Stalag Luft III, US Air Force veteran and historian Arthur A. Durand may have assessed it best.\n\n\"In a day when we don't have many heroes, it's kind of nice to see some heroes resurface,\" he wrote. \"And while there's a part of it that says instinctively, 'Yeah, but that's Hollywood,' there's the other part that says if even a fraction of that was true . . . then here's something to make us] take notice and find inspiration.\"[[2]\n\n* * *\n\n* During the Academy Awards ceremony at the Santa Monica Civic Auditorium on April 13, 1964, actor Sidney Poitier awarded the best editing Oscar to Harold F. Cress for his work on How the West Was Won. While it was eligible for best picture in 1963, The Great Escape earned no other Oscar nominations.\n\n# 1\n\nTHE KING'S REGULATIONS\n\n* * *\n\nTHE ESCAPE SEQUENCE from a Whitley bomber about to crash seemed pretty straightforward in the procedure manual. Pilot Tony Pengelly had practised it often enough with the other four members of his crew, although they had only done it when the Whitley was stationary and sitting on the ground. When the bail-out order comes and the Whitley is flying straight and level, the Royal Air Force (RAF) manual said, Pengelly's co-pilot, seated behind him in the cockpit, had to move quickly down a couple of steps to the cabin escape hatch below and to the right of the pilot. By the time he got there, the observer-bombardier, positioned below the pilot in the nose of the Whitley, would have opened the escape hatch door on the floor next to him, and the door would have dropped open with gravity. That would allow Pengelly's co-pilot to be the first to crouch, fall backwards through the escape hatch and free of the aircraft, open his parachute, and descend safely to the ground. He would be followed by the observer-bombardier, next by the wireless radio operator-gunner, and finally by the pilot himself. The manual stipulated that the tail gunner had to extricate himself from the rear turret, fit on his parachute, and climb to the escape hatch in the roof of the rear of the fuselage. The rear-gunner was always pretty much on his own.\n\nThe problem remained, however, that a Whitley wouldn't necessarily be flying straight and level in such an emergency. It could be side-slipping, diving, spinning out of control, or upside down in its unscheduled descent. It could be stricken by icing on its wings, hit by lightning, or buffeted by upward or downward turbulence. Nor did the procedure manual take into account such variables as a power failure, fire in the fuel tanks, explosions in a bomb bay, blocked passageways, or any other unexpected impediments to an orderly escape. Finally, the official RAF instructions for bailing out of a Whitley bomber in its death throes did little to account for the final variable in such an event\u2014the nighttime skies over enemy-occupied territory.\n\nWhen Armstrong Whitworth Aircraft introduced its state-of-the-art prototype Whitley bomber aircraft in 1934, the Royal Air Force adopted it as its first heavy night bomber. Among its unique characteristics, to offset the absence of flaps, the Whitley's main wings were set permanently at a high angle to potentially improve its takeoff and landing capabilities. Aircrews recognized right away, however, that the Whitley seemed to fly with a pronounced nose-down attitude, and its pilots sensed this added to the aircraft's considerable drag in flight. By the time he began piloting them in 1938, Flight Lieutenant (F\/L) Pengelly was flying a version of the Whitley bomber that included higher performance Rolls-Royce Merlin engines, modified fins, de-icing on the wings' leading edge, manually operated turrets armed with .303 Browning machine guns, and an extended tail section for a better field of fire for the tail gunner. But all that made his Whitley considerably heavier and less manoeuverable. Pengelly and his comrades considered the Whitley\u2014with a crew of five, capacity for a seven-thousand-pound payload of bombs, and a maximum speed of 230 miles per hour at sixteen thousand feet\u2014vulnerably slow, notoriously cold, and, if forced to shut down an engine, unable to maintain altitude. Whitley crews came to refer to their bomber as \"the flying coffin.\"\n\nTo offset the Whitley's alleged shortcomings and intimidating nickname F\/L Pengelly\u2014being a stickler for detail\u2014ensured that his aircrew was fully trained to cope with them. But as meticulously as he prepared his crew, Pengelly prepared himself even more so. Whenever new navigational aids appeared at his home aerodrome, the Canadian bomber pilot learned to use them as well as his navigator could. He paid close attention to the way meteorological officers read cloud formations and wind velocities so that he could read them equally well. And because the airworthiness and efficiency of his Whitley bomber depended so directly on the skills of his Erks, the ground crew, Pengelly developed tight working relationships with the mechanics, artificers, armourers, and riggers at RAF Topcliffe, where 102 Squadron was based in Yorkshire, England.\n\nWhen his Whitley Mk V arrived at Topcliffe, Pengelly studied all its attributes and idiosyncrasies\u2014engine revolutions, gun armament, bomb loads, petrol capacity, and the location of everything from the wireless radio to the evacuation dinghy. On days the squadron wasn't briefed and dispatched to bomb targets in German-occupied Holland or France, he even took to blindfolding himself and his aircrew, simulating nighttime conditions inside the Whitley in an emergency. Pengelly insisted that if their aircraft were hit by flak or night fighters, all members of his crew had to be able to find any piece of equipment or reach an escape hatch by touch alone. As much as he could, he wanted to inspire an esprit de corps among the other four members of his crew\u2014J. F. M. Moyle, C. P. Followes, H. Radley, and T. Michie. During downtime at the station, Pengelly even challenged his crewmates to motorcycle races on the aerodrome tarmac to sharpen the crew's competitive edge. The skipper of the Whitley bomber, nicknamed \"M for Mother,\" wanted everyone on his crew at the top of his game. No doubt, some of the British prime minister's oratory rang in Pengelly's ears.\n\n\"Only one thing . . . will bring Hitler down,\" Winston Churchill wrote, \"and that is an absolutely devastating exterminating attack by very heavy bombers from this country upon the Nazi homeland . . . without which I do not see a way through.\"[1]\n\nForty-eight hours before he was shot down, Tony Pengelly felt fully prepared and sharply motivated to take on the enemy. At age twenty, he'd already served as an officer in the Royal Air Force for two years, since the fall of 1938. He'd seen action as a bomber pilot as soon as the war broke out. His RAF 102 Squadron had flown operational sorties (ops)\u2014dropping propaganda leaflets on the Ruhr River in Germany\u2014on September 4, 1939, the second day of the war. Then, for the first year of fighting, his bomber squadron had played mostly a supporting role. His station's Whitley aircraft had flown operations to Norway in a losing cause. They had bombed German supply lines inland from Dunkirk as the British Expeditionary Force retreated from France in late May 1940. Through the rest of that year, including the crucial Battle of Britain period, which tested principally the Fighter Command aerodromes around London, 102 Squadron was on loan to Coastal Command, escorting naval convoys to sea from Prestwick on the west coast of Scotland. Like most of his air force comrades, F\/L Pengelly\u2014a Canadian in the RAF\u2014felt as if Bomber Command was flying in circles. He was sick of Britain's taking it on the chin. He was itching to go on the offensive.\n\n\"For the first six months of the war,\" Pengelly lamented, \"I flew at night mostly over Germany to gain operational and navigational experience dropping leaflets.\"[2]\n\nIf the fate of aircrews seemed up in the air, so too was the leadership and bomber strength of RAF Bomber Command. While it didn't concern Pengelly and his crew directly, in the fall of 1940, Sir Charles Portal moved up to commander-in-chief of the RAF, Air Marshal Sir Richard Peirse took over Bomber Command, and Air Vice-Marshal Arthur Harris became Deputy Chief of the Air Staff, second in command to Portal; Harris would presently assume total charge of Bomber Command's strategic bombing campaign.[3] Ironically, as its role suddenly seemed suspended, so did Bomber Command's aircraft strength suddenly diminish. Late in 1940, the British Air Ministry reported a total inventory of 532 bomber aircraft\u2014217 Blenheims, 100 Wellingtons, 71 Hampdens, 59 Whitleys, and 85 Fairey Battles.[4] RAF leadership deemed the Blenheims and Battles obsolete and began moving them to Training Command or the scrap heap. Wellingtons still proved reliable so the RAF called for another hundred of them. Meanwhile, the Hampdens and Whitleys would soon be ready for replacement by more modern aircraft; but for the moment their numbers remained  \nthe same. The bottom line was that Richard Peirse's Bomber Command, in the fall of 1940, was reduced by more than half, to 230 aircraft. Operations would be fewer, but more specific and with less impact.\n\nThat autumn, nighttimes brought heavy frosts and penetrating cold. Like many crews waiting for news, Pengelly's men dressed in their flying suits for warmth and huddled inside crew huts at the station. They had hung thick curtains as much to hold in heat as to black out the windows and doors. Spartan wall decor included propaganda posters, diagrams reminding aircrew about emergency procedures, and a few favourite pin-ups. And the men sat at linoleum-topped tables marked with indelible glass and cup stains and cigarette burns from hundreds of nights like these. Their talk\u2014of pubs, of home, of women who couldn't resist the attraction of an RAF uniform\u2014reflected their bravado, their boredom, and their apprehension.[5] They all just wanted the CO to come in, announce offensive ops, and get on with it. Inevitably, he did arrive with those orders.\n\nMid-November brought what aircrew called the moonlight period, a time most Royal Air Force bomber crews welcomed during that phase of the war. If skies were clear, ops to a target would  \nbe smoother without clouds buffeting them en route, targets would be more discernable, and the damage they inflicted on the ground would be more photographable. Aircrew also said a moonlit sky seemed to release them personally from the oppressiveness of flying in total darkness or dense, endless cloud. The British Air Ministry had earmarked German marshalling yards and iron smelters as high priority targets, but at the top of the list were synthetic oil plants in western Germany. A nighttime sortie against the refineries at Wesseling, near Cologne, revealed many of the shortcomings of 102 Squadron's weapon of war\u2014the Whitley bomber. But the emergency response aboard one of Pengelly's sister aircraft illustrated the capability of 102 Squadron bomber crews to offset that deficiency.\n\nOn November 13, 1940, over Cologne, German anti-aircraft batteries on the ground hit the Whitley bomber piloted by Pilot Officer Leonard Cheshire. The enemy flak ignited a fire near the fuel tanks and among the target flares still inside the fuselage. With fire and smoke filling the cockpit and his bomber descending rapidly, Cheshire wrestled with the controls to keep the Whitley aloft while his crew ejected the remaining flares and battled the fires. Eventually, his crew extinguished the flames, and after nearly nine hours in the air, Cheshire brought the shredded Whitley back to base safely. For Cheshire, leaving the aircraft\u2014even in a dire emergency\u2014was not an option. The action earned Cheshire a Distinguished Service Order[*] and inspired the entire 102 Squadron, including F\/L Pengelly. Explaining the successful return of his Whitley, Cheshire credited the variable that had overcome any of the Whitley's inefficiencies.\n\n\"These eighteen-year-olds,\" said Cheshire, himself just twenty-three, \"are a remarkable breed of men.\"[6]\n\nIf Cheshire's courageous example and the exhilaration of taking the war deep into enemy territory hadn't boosted the sense of purpose around 102 Squadron's crew huts at Topcliffe, the news from northwest of London the next evening certainly would. On November 14 came intelligence reports that the Luftwaffe had delivered its most lethal bombing raid of the war. The same clear, moonlit skies RAF bombers relished had led four hundred German bombers to the heart of Coventry in Britain's West Midlands. The enemy bombers had dropped five hundred tons of high explosives and thirty thousand incendiaries. The attack had destroyed three-quarters of the city's munitions, aircraft, and armament plants, the centuries-old St. Michael's Cathedral, and four thousand houses, more than half the city's residential area. Britons were stunned and appalled. Pengelly had experienced the blitz himself. Earlier in the war, his private apartment had been bombed in a Luftwaffe raid that destroyed most of his photographic equipment. But cameras, lenses, and photo development gear could be replaced; nearly a thousand people had died in Coventry that night\u2014the night immediately preceding Pengelly's last combat flight of the war.\n\nAt the briefing for his operation, F\/L Pengelly discovered that of the three ops targets slated\u2014Berlin, Hamburg, and the airfields at Schipold and Soesterberg\u2014the largest, Berlin, would be theirs. He'd been to Berlin first on the night of September 23\u201324 and three times since. This would be his fifth sortie to the German capital. In all, eighty-two Hampdens, Wellingtons, and Whitleys would set out across the North Sea that night and of them fifty bombers would attack sites in and around Berlin. The idea of security in numbers was not something Whitley crews experienced or even preferred. For most of his previous thirty trips\u2014and commonly on Whitley sorties\u2014Pengelly's \"M for Mother\" Whitley had flown alone. As the Luftwaffe had over Coventry, the RAF operation to Berlin on November 14\u201315, 1940, enjoyed the assistance of moonlight, but for various reasons only half the attacking force of Whitleys actually reached the city.\n\nAt approximately ten o'clock, Pengelly and Moyle, his co-pilot, homed in on their Berlin-area objective and released their bomb load. They then decided to make a second pass over the target so that the Whitley's newly mounted cameras could record the accuracy and damage of their attack. That's when German anti-aircraft batteries around Berlin found their mark. Flak penetrated one of the bomber's two engines and ignited a fire, just as it had two nights earlier aboard Cheshire's Whitley. Pengelly reacted quickly, shutting down the engine in an attempt to prevent the fire from spreading to the fuel tanks inside the wing. Desperately, he turned the aircraft westward for home, but he knew the odds were prohibitive. The loss of one of his Merlin engines effectively cut his power in half. And the results were as predictable as a mathematical equation. Cheshire's heroic dash for home on November 13 had succeeded because he still had two serviceable engines. Pengelly knew the technical specifications of his now one-engine Whitley\u2014even empty of bombs\u2014would prevent him and his crew from making it to friendly soil in England.\n\nTo his credit, the experienced bomber pilot did manage to keep the crippled bomber airborne long enough for his crew to prepare for the end. The extra minutes in the air gave each crewman enough time to tighten harnesses, secure parachutes and survival kit, and open escape hatches for evacuation. An hour west of Berlin, the radio operator tapped off an SOS on the wireless to alert RAF Coastal Command of their final descent. Five hundred miles from home, flying on one engine over German-occupied territory in a heavy bomber destined to crash or be shot down, and with just enough altitude to bail out successfully, Pengelly ordered his crew to abandon the aircraft. Then, just as the procedure manual stated, and as they had practised regularly on the ground at Topcliffe, first Moyle the co-pilot crouched and fell backward through the forward escape hatch, next Followes the observer-bombardier, then Radley the wireless radio operator (as well as Michie the tail gunner on his own out the rear hatch), and finally, Pengelly, the skipper, bailed out of the nose of their crashing Whitley. All got out safely.\n\nWhen Tony Pengelly ran away from his Canadian home in Weston, Ontario, in 1938, two-and-a-half years earlier, to join the Royal Air Force, he'd resigned himself to the RAF life. It meant dedication to a life of service of indeterminate duration. Typically in the Commonwealth air forces, thirty combat operations or about two hundred hours were considered a complete tour of duty. But reaching that plateau merely entitled an airman to a six-month rest from operations (often instructing at a training station during the break) and then a return to combat. Early in the war, when 102 Squadron accumulated the third-worst casualty rate in the RAF, fewer than half all bomber crews were surviving a single tour.\n\nMiraculously, by the night of November 14, 1940, Pengelly had completed thirty-one ops. Had he made it home that night, he might have been entitled to some leave time or at least service behind the lines as a respite. But at that stage of the war, Pengelly's oath of service meant he was in for the duration. The expatriate Canadian had chosen life and loyalty to a centuries-old system of King's Regulations that governed British society, commerce, and military service. And in the Royal Air Force, Pengelly had sworn to uphold its list of directives when serving in a theatre of war. If nothing else, Pengelly was a master of detail, and even as he floated to earth in Germany, he considered his actions methodically and exactly.\n\n\"I took my chute off and buried it,\"[7] Pengelly said. \"Being very optimistic\u2014which you were at twenty\u2014I was going to head west and walk to England, probably try walking on the water.\"\n\nWith equal na\u00efvet\u00e9, Pengelly started to travel on the wrong side of a public road and nearly collided with a volunteer police officer on his bicycle. Before long he was in the custody of two policemen, in a car, and en route to a local Luftwaffe station. He learned from his captors that they had detained at least two other \"M for Mother\" crew members, but because Pengelly was an officer, he was destined for a train ride to the district Durchgangslager der Luftwaffe (Dulag Luft)\u2014interrogation centre\u2014in Frankfurt-am-Main. Inside a cell with no window on the door, but a view through bars to a nearby wooded area and wire fence, Pengelly again reflected on the appropriate actions as he prepared for his first interrogation. They were lines he'd memorized among the King's Regulations and Air Crew Instructions at initial training school, reminding him \"to protect the Security of the Royal Air Force by every means.\"[8]\n\nHis captors brought him coffee and bread, probably to invite his co-operation. He felt unkempt. He hadn't shaved. He was still in his flying suit. Then a German interrogator entered the room. Pengelly thought he looked as if he'd just stepped from a tailor's shop and was terribly polite.\n\n\"Good day, Flight Lieutenant Pengelly,\" the man said in plain English. \"How's life in 102 Squadron?\"\n\nPengelly hadn't recalled giving anybody his squadron number. The reference in the regulations to the Geneva Conventions of 1929 came back: \"a prisoner of war is only required to give his name, rank, and number.\"[9]\n\nThe interrogation officer offered descriptions of other members of Pengelly's squadron; it proved he knew their names and some of their activities. It also suggested the captor actually knew everything and needed nothing from the prisoner.\n\nPengelly would have to, as he recalled from the interrogation manual, \"stand correctly to attention, maintain rigid silence, avoid attempts to bluff or tell lies, address any officer senior to himself as 'Sir' and . . . don't fraternize; the enemy is not in the habit of wasting his time, whisky and cigars on those who have nothing to give him in return.\"[10]\n\nThe interrogation officer explained to Pengelly that he had been a professor in Austria, had majored in British history, and had even travelled widely in England. And, oh, by the way, he happened to have English cigarettes apparently captured from the British Expeditionary Force during its hasty withdrawal from Dunkirk.\n\nAs attentive to the interrogator's methods as Pengelly felt he was, every time his captor edged toward sensitive information, Pengelly reminded himself of the regulations that required him not to try \"to fool your interrogators; they will be experts at their job, and in any battle of wits you are bound to lose in the end.\" Pengelly repeated that he did not have to tell the man anything.\n\n\"That's all right,\" Pengelly's interrogator insisted. \"I'll find out some way or other. We're shooting lots of you chaps down.\"[11]\n\nHe couldn't let on, but Pengelly was amazed by the volume of information his interrogator spouted, including turnover figures on the squadron, casualty numbers, and that on one of his combat operations, Pengelly had flown to Italy and back.\n\nEven though the interrogation had ended, and Pengelly felt some relief at having emerged without divulging any secrets of the Service, there were still other parts of that RAF manual at play. Yes, he had offered only his name, rank, and number. Yes, he had seen through the German officer's offer of food and idle chatter and not shown his surprise at how much the German knew about 102 Squadron and its crews. Yes, he had rebuffed the interrogator's attempts to fraternize with him, no matter how good those cigarettes might have smelled. But no, he had not yielded any data about his squadron, where it was stationed, what its strength was, or about the performance of any new designs in aircraft or armament. He hadn't revealed a speck of information about air force training, tactics, or defence systems. F\/L Pengelly had indeed lived up to all the King's Regulations stipulated and memorized . . . except one.\n\nAll the regulations, while never specifically saying so, implied that included in an airman's duty to King and country, particularly when captured and imprisoned on enemy territory, was an obligation to escape.\n\n\"Don't betray those who help you to escape,\" the RAF document stipulated. It added, \"Do keep your eyes and ears open after capture. You may learn much which may be of value both to your country and yourself if you succeed in escaping.\"[12]\n\nEscaping was a military obligation for which Pengelly admitted there had been no training. None of his instructors at initial training, elementary, or service training in the RAF had explained how to cut barbed wire at night, steal timely documents that would allow him to travel undetected, disguise himself as a German civilian, or dig a tunnel under a fence. All this and more\u2014if he cared to live up to all the obligations of his contract with the King\u2014he would have to learn by trial and error. But Pengelly was a quick study. After just four months in England, in the summer and fall of 1938, he had earned his \"certificate of competency for private flying machines\"[13] which meant he could fly \"all types of land planes.\" In addition, within his first year in the UK Pengelly had earned his commission as a flying officer in the RAF, making him equally capable in the cockpits of all military aircraft of the day.\n\nLater that first full day in German captivity\u2014November 16, 1940\u2014F\/L Pengelly was moved from the interrogation centre to the Dulag Luft prison camp. Curiously, as far as Pengelly was concerned, everybody inside the wire gathered at the gate to see the latest arrival. Among the greeters was a great surprise for Pengelly: the Senior British Officer (SBO) in the camp, Group Captain Harry \"Wings\" Day. Pengelly hadn't seen his wing commander for over a year; a decorated veteran of the Great War and already age forty when the Second World War began, Day had been shot down on his first op on October 13, 1939. The welcome from Wing Commander Day provided Pengelly's first official lesson as a prisoner of war: each incoming POW had to be scrutinized to verify he was who he said he was. Because his wing commander knew Pengelly and could prove it, any suspicion was immediately lifted about his identity.\n\nBut Pengelly's reacquaintance with Day initiated the next phase of his POW education. At Dulag Luft, the German prison camp officials had designated Day as Permanent Staff. They assigned him the responsibility of acclimatizing incoming captured aircrew to their new POW lives. It turned out to be a perfect ploy. Together with captured RAF fighter pilot Roger Bushell and Fleet Air Arm pilot Jimmy Buckley, as well as Middlesex Regiment infantryman Johnny Dodge, Pengelly would help cover Day as he dug a tunnel out of the camp from beneath his prison bed. A precedent had been established.\n\nIn June 1941, eighteen POWs broke out of Dulag Luft; all were recaptured, but the effort was the first mass escape of the war.[14] A few months later, when the Germans had transferred all the RAF POWs to Stammlager Luft I\u2014or, more commonly, Stalag Luft I\u2014in Barth, the ad hoc escape committee, dubbed \"X Organization,\" masterminded another tunnel breakout of twelve officers. Again the prisoners were recaptured. But by then the escape committee had started pushing back. The Canadian bomber pilot who'd prepared his crewmen carefully should they have to evacuate their crashing Whitley now began preparing to overcome the next adversity.\n\n\"Active participation in escape] work and planning for escape became the most important thing in my prison camp life,\"[[15] Pengelly said. \"The two and a half years I spent behind barbed wire before we began to plan the big escape was all training for that opportunity.\"\n\nFor Pengelly and the others, intent on living up to that final obligation of their Royal Air Force service, the trial-and-error period would seem horribly long, tedious, and frustrating. Pengelly was an officer, a skilled bomber pilot, capable photographer, and born leader. But for him the air war was finished. He would have to focus his talents on the new tasks at hand. Over the next year and a half, the escape committee at Barth successfully dug forty-eight tunnels. Their German captors found every one of them and thwarted every escape.\n\n\"That was because at Barth escaping was strictly private enterprise,\" Pengelly said. \"But] a man can't forge his own identity papers, dig his own tunnel, make his own wire clippers, escape clothes, maps [and] compasses. . . . From our futility, we knew we would have to organize to be successful.\"[[16]\n\n* * *\n\n1. The DSO citation read: \"Showing great coolness, Pilot Officer Cheshire regained control of his aircraft, which had lost considerable height and was being subjected to intense anti-aircraft fire, and although the explosion had blown out a large part of the fuselage and caused other damage he managed to regain height. . . . Although the aircraft was only partially answering the controls Pilot Officer Cheshire succeeded in returning to his aerodrome.\"\n\n# 2\n\nBOND OF WIRE\n\n* * *\n\nFRANCES MCCORMACK got her first letter from her imprisoned fianc\u00e9, a downed Spitfire pilot, just before Christmas 1941. She had met John Weir on a blind date a couple of summers earlier, when friends in Toronto arbitrarily matched them for a night of dancing at the Palais Royale on the Toronto waterfront. Weir cursed his friends for tricking him into the double date, but fell in love with Frances as the pair embraced on the outdoor terrace dance floor overlooking Lake Ontario. The war broke out in the middle of their courtship and because the Royal Canadian Air Force (RCAF) sent John to Winnipeg for elementary flight training, the couple began a long-distance relationship by correspondence. Even when John successfully soloed and graduated to service flight training, the air force posted him to Borden, Ontario, more than an hour's drive north of Toronto, where Frances lived. Still, the two managed to communicate in creative ways.\n\nOn a day when John planned to dodge the curfew at the training station and drive to Toronto for an evening date with Frances, he decided to alert her to his plans for a rendezvous. During the day's instructional flight near Toronto, the young pilot trainee simply detoured over the Forest Hill area of the city and \"bombed\" his girlfriend's residence with a message wrapped in a handkerchief. Neighbours picked up the note attached to a small parachute and rushed it to the addressee: Miss Frances McCormack, 61 Heathdale Road.\n\n\"Be down about 8 o'clock or 8:30. If not, I'll phone,\" Weir said in the note. \"No news about a 48 hour leave of absence]. P.S. Don't say anything about this to anyone.\"[[1]\n\n\"You should be a bomber pilot, not a fighter pilot,\"[2] Frances later told him.\n\nAs much as Weir's after-hours dash to Toronto illustrated his youthful exuberance and ingenuity, it nearly ended his air force career. His Borden flight instructor found out about his student's illegal antics. He lambasted the sprog (novice) pilot about his disregard for the King's Regulations and lack of respect for his fellow pilots. Then he told Weir he'd be reassigned\u2014i.e., washed out of pilot training. When Weir pleaded his case, the instructor decided to redirect his student away from the multi-engine training that would have streamed him toward becoming a bomber pilot. Instead, Weir would train to eventually fly the single-engine, solo cockpit Hurricane fighter aircraft. John Weir couldn't have asked for a more appropriate punishment.\n\nCollisions with authority and protocol seemed routine for Weir, but they never clouded his drive or independence; among many things, he was an avid outdoorsman capable of fending for himself if and when he had to. Later that year, during an instructional flight at Trenton, the young trainee had struggled to bring a Fairey Battle bomber with a burning engine down to earth safely. The episode left his uniform singed, covered in grime, and stinking of glycol from the Battle's burst coolant system. No one bothered to acknowledge that young Weir had chosen to save the King's property and put his own personal safety at risk; nevertheless, when a visiting RAF officer spotted him in a parade lineup with a glycol-stained uniform, he pointed him out as \"a rather scruffy looking individual.\"[3] The nickname stuck.\n\nFrances McCormack felt so moved by Scruffy's passion to join up and serve his country that she decided to resign her paying job as a personal shopper at Simpson's department store to look for war work herself.[4] She found it at Research Enterprises, the company manufacturing ASDIC, the navy device that used sound waves to detect other ships\u2014principally enemy submarines\u2014at sea. Frances knew how to drive so she landed work as the company chauffeur and took great pride in contributing to the war effort this way. Meanwhile, the couple received their parents' blessing to marry and were engaged October 2, 1940, a few weeks ahead of Weir's overseas posting. The two travelled with friends to Ottawa and shared final words at the train station. Frances knew her fianc\u00e9 felt an allure for the excitement of the war.\n\n\"John, I just hate you going away to this war,\" she recalled saying to him before he embarked.\n\n\"Fran, I've had everything my own way all my life,\" he had told her. \"I'm not concerned in the least.\"[5]\n\nPartly to prove himself and partly because it was serious now, Weir moved deftly into the cockpits of Hurricanes in operational training at Sutton Bridge, northeast of London. Then he moved to active duty with 1 RCAF Squadron, the first Canadian squadron in Fighter Command that daily joined RAF squadrons scrambling to beat back German Dornier bombers during that historic Battle of Britain summer. By the time Weir went operational with 1 Squadron, it had logged an impressive record\u20141,694 sorties, 29 downed enemy aircraft, another 43 damaged or destroyed\u2014and its Canadian fighter pilots had earned three Distinguished Flying Crosses (DFCs). However, losses had also been severe\u2014three killed in combat, two in accidents, and ten wounded\u2014and it was generally acknowledged that life expectancy of green pilots was six hours of combat or less. Nevertheless, a primed and combat ready Flying Officer (F\/O) John Weir reported for active service in October 1940, just as 1 Squadron relocated to Thurso, Scotland, to protect the Royal Navy base at Scapa Flow.\n\nWith 1941 came changes. The squadron had replenished its core of experienced fighter pilots. RAF Fighter Command had relocated its aircrew to Digby, in east-central England, and then Biggin Hill south of London. The Canadian squadron had been renumbered 401 and taken into service its first Supermarine Spitfire fighter aircraft. John Weir had also reconnected with one of his oldest pals from Toronto, Hugh Godefroy. Their families had been close. They had grown up, played shinny hockey, and attended school together. They had both enlisted the same day, September 3, 1939, and for the same reason\u2014the German torpedoing and sinking of the passenger liner SS Athenia. They had been posted overseas within months of each other, and they arrived at Digby together. They paired up in combat formations to look out for each other and at Biggin Hill flew their maiden Spitfire sorties together. And they got their baptism of fire over the Channel together. But as uplifting as banding together this squadron of eager young Canadians felt, Fighter Command victories were few and far between that fall. On October 27, 1941, a Biggin Wing sweep over Dunkirk and the resulting dogfight with scores of Messerschmitt Bf 109s produced an all too frequent result.\n\n\"The ground crew from 'A' Flight came into the dispersal hut and sat around in dead silence. None of their aircraft had returned,\" Godefroy wrote. \"We had lost Wally Floody, Johnny Small, Blake Wallace, Stan Thompson and Brian Hodgkinson.\" Three of them had been shot down \"on their first ever sweep.\"[6]\n\nIt got worse in November when, during the first week of operations into France, the Biggin Wing lost ten of twelve fighter pilots. Then, on November 8, a flight of Spitfires from 401 conducted a sweep over the River Somme Estuary, where German fighter aircraft swooped down and broke up the Spitfire formation. Godefroy was chased by four Bf 109s. He twisted, turned, and threw his aircraft around the sky to the point of near exhaustion, all the while firing at whatever he saw until his ammunition ran out. He barely made it home in one piece. At the dispersal hut he turned in his escape kit and gave his combat report to the duty intelligence officer (IO).\n\n\"Weir and Gardiner are missing,\"[7] the IO said.\n\n\"Weir missing?\" Godefroy shot back, admitting he'd never worried about his friend not returning.\n\n\"Blakeslee[*] reported seeing two Spitfires astern, both on fire,\" the IO added.\n\nIt was Wally Gardiner's first ever sweep, but John Weir had accumulated a thousand hours in operational flying. To Godefroy, not seeing the \"tough as nails and perpetual clown\"[8] John Weir home safe for a celebratory drink and slap on the back \"seemed impossible.\" Nevertheless, the job of informing Weir's parents and Frances McCormack, his fianc\u00e9e, fell to Godefroy. Of course, the air force would send the obligatory \"we regret to inform you\" form letter, but Godefroy worried that whatever words of sympathy and explanation he might cobble together would fall short. Nothing seemed appropriate or adequate. And the silence that followed proved equally painful. Later, Frances would learn that when her fianc\u00e9's Spitfire was attacked at an altitude of twenty-six thousand feet over Caen, France, Messerschmitt shells had ignited one of his fuel tanks. The resulting fire had burned his hands and face, nearly fusing his eyelids shut. A combination of adrenaline and shock must have masked any pain; he'd managed to bail out and, once on the ground, he'd immediately set about burying his parachute and attempting to disappear into the countryside. To no avail. Soon after, F\/O Weir was captured, processed at the Dulag Luft, and became a prisoner of war\u2014Kriegsgefangenen or \"kriegie\"\u2014en route to the airmen's prison at Barth.\n\nLuftwaffe-run prison camps, such as Stalag Luft I, housed captured air force personnel. As opposed to the Offizier Lager or Oflag prison for non-commissioned officers, the Luft camps generally imprisoned Commonwealth officers shot down in bombers or fighters. As a rule, those officers, even as POWs, were treated with respect. The morning after he was shot down over Boulogne, France, in January 1942, RAF Spitfire pilot Robert Stanford Tuck was wakened by a German lieutenant, who first saluted him smartly. \"We are taking you by train to Germany,\" the young German officer said. \"You are regarded as an important prisoner . . .\"[9] Throughout his trip to Dulag Luft, Stanford Tuck was fed hot drinks, soup, plenty of bread, and potatoes. He was, after all, a squadron leader.\n\nThe architects of Germany's military prison camps could not have conceived of a better containment area than the facility on the outskirts of the medieval town of Barth. Situated on a peninsula jutting out into the Baltic Sea on the north coast of Germany, the grounds and location of Stalag Luft I alone posed a deterrent to escape. Its sandy soils lay flat and virtually without contour; sightlines were perfect for prison guards. A lagoon to the southeast and the open sea to the northeast meant the water table would be relatively high, only six to eight feet underground; so any thoughts of deep tunnel digging appeared to be out of the question. Some pine tree growth beyond the camp perimeter broke the skyline, but not the flow of frigid north winds off the Baltic.\n\nThe man-made containment, built in the summer of 1940, consisted of barbed-wire fencing with watch towers manned around the clock by armed German guards. Initially, the camp was a dusty cage, roughly one hundred yards long and seventy yards wide.[10] The perimeter was further accentuated with a ditch excavated just inside the outer wire. Then, just inside that ditch, a second line of wire was strung\u2014at about boot-top level. This wire marked the boundary between the inner camp and a no man's land; prisoners were warned if they ventured beyond the inner wire they would be shot. The compound housing, initially built to hold about two hundred prisoners, was spartan at best. The Germans had hastily constructed low, wooden barracks huts, some with partitioned areas (rooms) containing double-decker bunks for nine to a dozen men (changed to triple-deckers as the number of POWs swelled). The huts lacked insulation, which meant the inside temperature matched the outside in summer and winter, unless the room offered a stove with a modest amount of fuel. And if the chilling winds and primitive accommodation weren't demoralizing enough, the knowledge that just forty miles northwest, out in the Baltic Sea, lay the international border between Germany and Denmark, made Stalag Luft I the epitome of frustration.\n\n\"My darling, here I am safe and sound in Germany,\"[11] John Weir eventually wrote in the first letter to Frances, in mid-December 1941.\n\nThe letter was written on a single, prison-issue, skin-thin piece of paper, which was the allowable limit for POWs. The page measured ten inches by five-and-a-half inches\u2014enough for about three hundred words\u2014and folded in thirds for mailing. Weir wrote that his letters wouldn't be frequent or very explicit, what with the Germans censoring everything. He appeared to be cut off from the rest of the camp, likely in some kind of solitary. He was not the usual upbeat \"Scruffy,\" kidding and romanticizing about things, and he encouraged Frances and the Weir family back in Canada to contact Red Cross authorities to send him parcels because, he wrote, he really needed socks and a shaving brush. Finally, perhaps to ease her mind, he wrote that he had her photograph close by.\n\nBy the time he'd composed and mailed his Christmas 1941 letter, ten days later, Weir's demeanour seemed to have improved. Instead of sounding lost, forlorn, and forgotten, he explained that the Germans had been treating them civilly, that his captors had laid out a skating rink area for hockey (all they needed was ice and skates) and that the entire camp was envious of his colour photo of Frances. In addition to wishing her a Merry Christmas, however, Weir added \"as education and entertainment\" that he was taking language instruction and that he had been put into a barracks with other downed airmen.\n\n\"There are quite a few Canadians (twenty-eight or twenty-nine) here,\" he wrote, sounding more upbeat. \"In fact, my darling, it's not bad at all. . . . I hope I'll be on my way back to you soon.\"[12]\n\nPerhaps the last thing John Weir might have found inviting about imprisonment in an enemy POW compound, Frances would soon learn, was the camp's education and entertainment facilities. She knew that as a younger man her fianc\u00e9 had actually travelled with his family in pre-war Germany. Even so, discovering that he was enrolling in language study in a POW camp must have seemed a bit odd. Yet there it was, his sudden attraction to studying Spanish, French, and German, and even trigonometry and calculus, if he wanted. Perhaps less odd to his fianc\u00e9e was Weir's delight that some of his air force mates\u2014Hank Birkland and Wally Floody\u2014were barracked in the same prison hut. Frances would learn later that attending German language instruction inside the camp was preparing her fianc\u00e9 for life on the run outside the wire, if he could get there. Meanwhile, reacquainting himself with his air force comrades was offering him the means to fulfill that off-handed promise he'd made in his Christmas letter to \"be on my way back to you soon.\"\n\nUnlike John Weir's relatively comfortable upbringing in Toronto, Hank Birkland's background told a tougher, more school-of-hard-knocks tale. Hank was born in 1917, one of a carpenter's seven children, in Spearhill, Manitoba. Like many of his generation\u2014born around the time of the Great War and raised during the Great Depression\u2014Birkland rode the rails, worked for his keep, and chased any and all opportunities as a farm hand, meat packer, door-to-door salesman,[13] and labourer in the ore mines of Ontario and the gold mines of British Columbia. In Sheep Creek, BC, he even returned to one of his favourite childhood pastimes\u2014playing lacrosse. With his size and strength came the nickname \"Big Train.\" When the Second World War broke out, Birkland's enlistment might well have offered him as much a way out of the Dirty Thirties\u2014with three square meals a day, a new suit of clothes, and a paycheque\u2014as it presented a way to defend King and Empire. At any rate, in the fall of 1941, Flying Officer Birkland was in the RAF and piloting Spitfires with 72 Squadron; during a sweep over the coast of France, German anti-aircraft gunners brought his Spitfire down to a crash landing on the beach and ultimately brought him to the same Stalag Luft I barrack hut as John Weir.\n\nAlso sharing that hut was Weir's former 401 Squadron wing mate, Wally Floody, shot down just days before him. The six-foot-tall Ontarian was an athlete in every sense. Born in Chatham in 1918, Floody had grown up mostly in Toronto, but during his boyhood summers he enjoyed the family's access to farm holidays, camping out on Toronto Island, and competitive team sports such as basketball, football, and baseball. As a teenager in search of work in the mid-1930s, Floody travelled to north-central Ontario for shift work shovelling mud, rock, and precious ores excavated from mines near Timmins and Kirkland Lake. But so too did Floody find time to  \nplay for the mining companies' sports teams[14] \u2014the Preston East Dome Mine baseball team and the Lake Shore Mines Blue Devils basketball team. Floody did not have the matriculation diploma from high school to gain entry to the RCAF, but a letter from his high school principal convinced the enlistment officer, and Floody joined soon after Canada declared war on Germany. In the short span of eighteen months in 1940 and 1941, Floody got married, graduated from the British Commonwealth Air Training Plan, got his commission in the RCAF, and was shipped overseas, only to be shot down on his first op, October 27, 1941.\n\n\"I'm having my picture taken . . . in an all-Canadian group,\" John Weir signed off his Christmas 1941 letter to Frances.\n\nThe photo depicted a short but broad-shouldered jack-of-all trades, Hank Birkland, aged twenty-four, imprisoned at Stalag Luft I six weeks; a hard-rock miner and natural team leader, Wally Floody, aged twenty-three, in the Barth camp eight weeks; and the youngest, at twenty-two, John Weir, a natural outdoorsman with a scruffy attitude to match, and just a month inside the first prisoner-of-war camp exclusively built for captured British Commonwealth airmen. Since the Luft was run by Luftwaffe, it's likely the photograph of the three Canadians was snapped by a German airman, who on any other day would have been the Canadians' mortal enemy. But with his prisoners safely contained inside the wire, the Luft guard likely took out a personal camera and snapped the picture as a gesture of ambivalence and respect.\n\nBehind that ambivalence and respect lay one of the strangest contradictions of Germany's prisoner-of-war system. In spite of the racial intolerance and obsessive ideology of Germany's ruling Nazis, who systematically brutalized prisoners from the East, the regime appeared to deal with prisoners from the West with a degree of deference. With some exceptions,[*] in dealings with POWs from the Commonwealth countries and the United States, the Germans recognized certain clauses of the Geneva Conventions. Early in  \nthe war there was a very practical reason for such regard; some of those same Commonwealth nations, Britain and Canada for example, held German POWs on their side of the battle lines. As well, though the overall responsibility for wartime prisoners lay in the hands of the German supreme commander of the Wehrmacht, control of the camps themselves fell to army, navy, and air force chiefs where the prisons were located. In the case of captured Allied aviators, all camp administration, food and clothing allotment, accommodation, and day-to-day concerns were governed by the command of the air district, or Luftgau.[15] Nor did it hurt downed Allied aviators that Reichsmarschall Hermann G\u00f6ring held great sway with Hitler; it was no secret that G\u00f6ring had high regard for fellow fliers no matter which side they fought for. And up until 1944 that attitude at the top filtered down and was reflected in all treatment of captured Allied air officers. It meant, for example, that captured enemy airmen from the Commonwealth or the United States generally did not have to work inside prisons. Allied senior officers would be saluted by equals and lower ranks on the German side. Allied chains of command would be acknowledged even inside the prison wire. Allied officers could receive mail (including regular Red Cross parcels) and send it. They would be entitled to recreation and entertainment of their own making. That G\u00f6ring chivalry, a holdover from the Great War, at least early in Germany's war with the West, influenced daily life in Luftwaffe-run POW camps.\n\nWhen that first photograph of John Weir arrived with a letter at Frances McCormack's Toronto home, it shocked her. For the first time she saw how severely Weir's face had been injured when he bailed out of the burning Spitfire near the River Somme in France. She could see that his eyelids were virtually gone. No doubt she also recognized the starkness of conditions at the prison camp\u2014the barren setting, the primitive huts, the wire-bound world that was his home for the foreseeable future. She marvelled at her fianc\u00e9's strength, his sense of humour, and his very clear vision of the future.\n\n\"I've been playing hockey a little and I'm really in fine fettle,\" he wrote, and then added a cryptic note about his living arrangements that may have had more to do with escape activity at the camp than building their future dream home. \"I'm still working on a design for our cabin, dear. There are a few architects here who'll help once I get a decent idea to work on. . . . I still have trouble with the children's rooms\u2014how many, four?\"[16]\n\nNext to the Britons imprisoned at Stalag Luft I, the Canadians were in the minority and\u2014even inside the prison wire\u2014obliged to respect the authority of the Senior British Officer and his adjutant. At Barth, Group Captain Harry \"Wings\" Day was the SBO. Formerly a member of the Royal Marine Light Infantry in the Great War, Day had been awarded the Albert Medal for saving the lives of crewmen aboard the torpedoed HMS Britannia in November 1918. In the RAF, as the Second World War began, Day served with 57 Squadron, where he earned his nickname, but was shot down five weeks into the war on a reconnaissance operation near Essen, Germany. His adjutant on the Permanent Staff at Stalag Luft I, Lieutenant Commander Jimmy Buckley, had trained for naval aviation and served aboard the Royal Navy's aircraft carrier HMS Glorious before the war; however, as British aircrew attempted to provide cover for the evacuation of the British Expeditionary Force from France in May 1940, Buckley was shot down over Calais. Another member of the Permanent Staff at Stalag Luft I was RAF Squadron Leader Tom Kirby-Green, who had piloted thirty-seven combat operations when his Wellington bomber was shot down in October 1941. Inside the wire at Barth, he assisted the SBO as a one-man welcoming committee by entertaining newcomers on his bongo drums, offering bits of exotic food from his Red Cross parcels, and delivering lectures in the library room.\n\nInside Stalag Luft I, Group Captain Day and Lieutenant Commander Buckley appeared quite compliant to German demands\u2014helping inbound prisoners acclimatize to their new surroundings. Following interrogation by the Germans, a newly arrived POW met with the SBO and his RAF Permanent Staff, received some necessary toiletries from a Red Cross parcel, and was assigned to a barracks hut. The Luftwaffe administration anticipated that the civilized nature of relations between prison camp officials and the kriegies would defuse any hostility inmates might feel and might even invite a POW to allow vital information to slip out. Conversely, the RAF Permanent Staff expected the Germans to think that British Commonwealth prisoners were resigned to their fate to sit out the war without resistance. In fact, Day as SBO and Buckley as \"Big X\" (chief of X Organization) were hard at work scouring the camp for ideas, arranging the escape expertise, and executing each new plan.[*]\n\n\"I had worked in the gold mines in northern Ontario,\" Wally Floody said. \"But if you had an air force] commission and had worked in a mine, the Englishmen figured you had to be an engineer, and if you had worked in a mine you knew a lot about tunnelling in sand. But there was absolutely no similarity between the two.\"[[17]\n\nThousands of feet below ground level in the Lake Shore Gold Mines near Kirkland Lake, Ontario, Floody could pass other miners in tunnels that were seven feet high and chiselled through solid Canadian Shield. Sub-surface at Stalag Luft I, escapers dug through unstable sand to create passageways barely two feet wide by two feet deep; they worked on their backs or their stomachs and faced the threat of being buried alive with every cut into the sand. Nevertheless, the trio of Weir, Birkland, and Floody became the principal architects and work crew for the first tunnelling expeditions at the Barth prison compound. Since the huts included boards that skirted the exterior walls to the ground, the digging crews simply crawled unnoticed into the two-foot space under the huts. First they built a trapdoor. Then they excavated a shaft straight down and cribbed it with wood scavenged from the huts. When excavation ended each day, the diggers would simply cover the entrance with boards, pile sandbags around it, and brush earth on top of the entrance.\n\nOnce underground, the dig crews determined where the water table lay; at Barth it was situated just a few feet beneath the surface of the compound sand. But since it proved extremely difficult to keep a tunnel perfectly level, the digging crew often returned to its work on a new day to find depressions in the tunnel filled with water. That forced the diggers to work virtually naked so as not to reveal a pair of pants or a shirt or a jacket covered in wet sand. Additionally, the farther the tunnel proceeded from the hut, the staler the air became. At first, Floody dug narrow shafts upward to allow fresher air into the tunnel. Then the tunnel crew designed its first air pump, consisting of a German jam can with a flop valve and a bicycle pump.\n\n\"Geez, if you went down below about four feet you were swimming in fucking water,\" Floody said. \"And putting up air holes wasn't a total success either.\"[18]\n\nThe Germans discovered the first tunnel when it caved in from flooding and multiple air holes. That prompted a sea change among the Luftwaffe guards running the camp. German work crews began systematically stripping the Luft huts of their skirting boards to reveal the space between the floors of the huts and the ground. They also began driving heavy wagons around the compound to collapse any shallow tunnels.[19] They organized a counter-intelligence team called Abwehr, or \"defence,\" dressed the men in blue-grey-coloured overalls and equipped them with steel probing rods. The kriegies dubbed them \"ferrets\" and fought back with a primitive form of security. A prisoner was assigned the job of taking his laundry\u2014perhaps a pair of underwear and socks\u2014close to the front gate of the compound. He would erect a clothesline with a string stretched between two nails to apparently dry his laundry. The moment vehicles, guards, or ferrets approached the front gate, the laundry man would simply remove his shorts and socks from the line. That alerted a system of prisoners, bucket brigade style, that there was a potential threat to the tunnel. Digging could be shut down and a tunnel trapdoor concealed in minutes. What emerged from the desolation of the camp and the persistence of the POWs to work together was an esprit de corps that went beyond the protocol of squadron barracks in Fighter and Bomber commands. The kriegies had fashioned a prison camp collaboration that knit them together into a single cause\u2014theirs was a bond of wire.\n\n\"I tried digging in the tunnels,\" Pilot Officer (P\/O) Barry Davidson said, \"but I got claustrophobic.\"[20]\n\nCompared to the other Canadians imprisoned at Stalag Luft I, Davidson, at age twenty-eight, was the old man of the group. He had learned to fly in the middle of the Great Depression and earned his private flying licence on a Tiger Moth at the Calgary Municipal Airport in 1937. The moment he got the certificate, he wrote a letter to Chinese General Chiang Kai-shek, who was attempting to repulse attacks from the Imperial Japanese armed forces; Davidson offered his services to the Chinese \"Flying Tigers,\" but the general politely declined. No matter. Next, he travelled to England, joined the RAF, and became a Blenheim bomber pilot with 18 Squadron. In July 1940, as the German blitzkrieg gobbled up the Low Countries and northern France, Davidson's squadron was dispatched to slow the enemy offensive.\n\n\"We found one of the forward new aerodromes they were building . . . around fifteen miles from Paris,\" Davidson said. \"So we lined up on the equipment at the end of the aerodrome with two of our two-hundred-and-fifty-pound bombs.\"[21]\n\nThe German anti-aircraft batteries finally responded and shrapnel ripped through the centre of Davidson's Blenheim, destroying its wing and tail controls. Disoriented and with a dead stick (virtually no control over the aircraft's direction or attitude in the air), he managed to crash-land the bomber on a beach. He hoped it might be a friendly Allied coast. But moments later, his plane was surrounded by German troops informing him and his crew they were prisoners of war. His first letter home to family in Calgary was a simple one.\n\n\"Looks like I am in cold storage for the duration,\"[22] he wrote.\n\nBut if his talents had not properly been put to the test in the air over France, they more than adequately served him on the ground inside Stalag Luft I. Davidson quickly blended into the Canadian kriegie contingent and came to understand that the war had deprived the prison guards of much the same amenities it had the POWs. So, as soon as Red Cross parcels became available to the prisoners of war, Davidson used some of their contents to advance the priorities of the escape committee. Davidson managed to put parcel staples, such as chocolate, coffee, and cigarettes, into the hands of the guards in return for items that would come in handy for the kriegies. Back came tools for tunnelling, the loan of a camera, and the raw materials for forged documents. An avid sportsman, Davidson helped secure sports gear\u2014tennis rackets, baseballs and bats, and hockey skates and sticks\u2014from the YMCA in Morgan's department store back in Canada. The equipment may well have provided kriegies some welcome recreation, but the hardwood from tennis rackets burned long and hot in a barracks stove and a used skate blade often enjoyed a second life as a cutting or digging tool.[23] From the moment he got settled in the POW compound, P\/O Davidson became known as \"the scrounger.\"\n\nHe wasn't the only one skilled in the art of scrounging. MacKinnon \"Mac\" Jarrell, from Armow, Ontario, joined the RAF in 1940 and trained as a navigator and bomb-aimer serving aboard Blenheim aircraft. In July 1941, on his twenty-second operation, Jarrell was shot down and badly burned in the episode. The hospital where Mac was admitted happened to have a German-born doctor raised in the United States. Seconded to military medical work, the doctor administered to Jarrell's burns with plastic surgery[24] and nursed him back to health. When Jarrell was finally discharged and sent to a POW camp in central Germany, he joined an escape committee that built a tunnel and, months later, nineteen escaped through it. After his recapture, six days later, Mac was considered high-risk to escape, so the Germans kept moving him from prison camp to prison camp\u2014eight different camps over the next forty-five months.\n\nOnce he was cleared by the escape committee at each prison compound, Jarrell joined a crew of scroungers getting to know the German guards and taming them\u2014i.e., bartering with them for the equipment and supplies X Organization needed for its long-term escape agenda. Among other bribing tools, Mac used the flow of tobacco sent from home, including cigarette packs from the Ontario Chinese Patriotic Federation. Also sent from Canada, and innocently cleared by the Luftwaffe administration at the compound, were small wind-up phonograph machines and a regular supply of recordings.\n\n\"One record called 'Corn Silk,'\" Mac noted in his log, \"came in almost every package. It was terrible and the prisoners hated it.\"[25]\n\nBut that was the intention. Eventually, one POW in the compound became so agitated over the horrible sound of \"Corn Silk\" that he smashed the record only to discover a map of Germany and area had been hidden inside. Suddenly, Jarrell's scrounging crew had additional contraband to assist in the long-range escape plans.\n\nAnother western Canadian joined the Stalag Luft I escape committee about the same time that Barry Davidson did. Of course, it had never been Dick Bartlett's intention to become a POW. Born in 1919 and raised on a dairy farm near Fort Qu'Appelle, Saskatchewan, as a boy Dick raised silver foxes, the assets of which underwrote his passage to England and provisional entry into the Royal Navy Fleet Air Arm (RNFA) in 1938. By October of 1939, a month into the Second World War, Bartlett had received his RNFA graduation wings and joined the torpedo training unit perfecting airborne sorties against German shipping across the Channel. By the spring of 1940, as the Nazi occupation of northwestern Europe and Scandinavia began, Sub Lieutenant Bartlett was posted to 803 Squadron flying Skua dive-bombers attacking targets in occupied Norway. The Skuas were manned by a pilot facing forward in the cockpit and, right behind, a gunner facing aft. As the sun rose ahead of them on June 13, 1940, during an attempted attack on the German battle cruisers Scharnhorst and Gneisenau in Trondheim harbour, Bartlett and his gunner Lloyd Richards took return fire from German Bf 109s.\n\n\"A heavy blow felt like] a kick from a mule,\"[[26] Bartlett explained later.\n\nMachine-gun bullets and cannon-shell shrapnel had penetrated the Skua's fuselage, its fuel tank, and Bartlett's flight suit up his left side. Unable to evade the fighters and rapidly losing consciousness, Bartlett barely made it over the rooftops of Trondheim, and as the Skua disintegrated\u2014ultimately losing its engine\u2014he crashed-landed into a field. Only three of the fifteen Skuas had completed the attack. Bartlett and his gunner survived their crash, but even as he received treatment at a Trondheim hospital, it didn't take Bartlett long to consider his escape options. Two attempts later, he was put under guard and sent to a series of POW camps in northern Germany and Poland, including Stalag XX-A, where he learned an important covert skill.\n\n\"Through friendly relations with Polish labourers at the camp,\" Bartlett related, \"we] bribed one of the Poles to sneak the components for a small radio receiver into the camp.\"[[27]\n\nThe resulting wireless set allowed the POWs to hear BBC broadcasts. But to ensure that the radio was never discovered by German guards or ferrets, Bartlett regularly disassembled it and placed its parts inside a medicine ball; the ball, a bit larger than a basketball and usually weighted with sand, was used by prisoners of war for calisthenics and other sporting pursuits. Without the guards or ferrets ever realizing it, the wireless set was always out in the open, being carefully tossed among the \"sports enthusiasts,\" and consequently under close surveillance by members of the escape committee. Only when the kriegies could safely open the medicine ball was the radio pulled from its hiding place and assembled to catch the BBC news from England.\n\n\"That way,] the radio-equipped medicine ball subsequently travelled from camp to camp [becoming] a continuous source of war news and intelligence,\"[[28] Bartlett pointed out, and always with S\/L Bartlett as its custodian. Despite his standing assignment at Stalag Luft I\u2014protecting the radio\u2014Bartlett was never far from the escape planning among his fellow Canadian POWs.\n\n\"I haven't missed any German lectures yet,\" John Weir reported to Frances in his first letter of 1942 from the Barth prison camp. \"I keep myself fairly busy, so the time doesn't drag too much . . . what with learning to cook and stuff.\" Then, in what seemed a Freudian slip apparently missed by both the German and British censors, Weir concluded that Hank Birkland, Wally Floody, and he \"will dig in for the winter;\" then he signed off with his now routinely optimistic, \"it won't be long before this war is over and we will be home again.\"[29]\n\nWally Floody noted from his experience inside several German POW camps that there were only three ways of escaping\u2014over the wire, through the wire, or under the wire. He, Birkland, and Weir had chosen tunnelling as the safest and most efficient way out. At Barth, however, the kriegies tried every possible scenario and the RAF Permanent Staff wasn't about to blunt their initiative. In January 1942, a prisoner tried to escape by hiding in a cart that had just emptied the compound's latrines. Several others attempted to disguise themselves in mock-ups of German uniforms. Another jumped into a snow bank, but was spotted when steam rising from the snow where he was hiding gave him away. Later that winter, three airmen used the cover of a blizzard to cut their way through the wire at  \nthe NCO's compound, but a guard spotted the trio and shot Sergeant Johnny Shaw, making him the first fatality of the escape campaign.[30]\n\nArrivals at Barth continued to mount. During the Battle of Britain, RAF fighter pilots who were shot down most often parachuted to safety on British territory; after Dunkirk and the fall of France, however, surviving fighter pilots who bailed out over occupied France and Belgium most often became prisoners of war. In 1941, for example, RAF Fighter Command claimed 711 Luftwaffe fighters shot down, while sustaining the loss of four hundred Hurricanes and Spitfires, mostly over enemy territory. Similarly, between July of 1940 and January of 1941, RAF Bomber Command lost 330 aircraft, resulting in fourteen hundred aircrew killed, missing, or captured. Over the course of the entire war, more than six hundred Fighter Command and nearly ten thousand Bomber Command aircrew became prisoners of war.[*] Most of them were captured in Europe and most came through the Luft containment system.\n\nThe flow of downed and captured Commonwealth aircrew\u2014perhaps a dozen or two a month at the beginning of the war\u2014became a torrent of hundreds a month in 1941 and 1942; the tide of downed airmen had grown too with the entry of the United States into the war after the Japanese air attack on Pearl Harbor in December 1941. The impact of this ever-growing prisoner-of-war population on Luftwaffe resources\u2014the availability of trained interrogators, transit guards, and bureaucrats, not to mention food, accommodation, and living space\u2014became more than one prison facility could handle. But camp administrators recognized they were facing perhaps an even greater problem\u2014the growing number of dedicated escape artists the camp appeared to be fostering. And with the growing number of escape attempts, Kommandants at POW camps across Germany feared reprisals from their superiors should any of the large-scale escape plots succeed.\n\nMeanwhile, Wings Day took full advantage of that fear, even if in truth as few as 5 per cent of the POWs might be described as dedicated escapers. Day's intelligence told him that maybe a quarter of the camp's inmates considered escape an option. To further motivate kriegie escape attempts, he made it known that escaping was an \"operational function\"[31] in the camp, reinforcing the perception that attempting an escape was an RAF airman's duty. The result was nearly a fever pitch of escape activity at Stalag Luft I; or, as Day described it, he'd been incarcerated \"in a kennel with a good pack of foxhounds\" all sniffing to get out.\n\nJohn Weir's letters to his fianc\u00e9e trailed off slightly in the winter and spring of 1942. In a few notes during January and February, he wondered if Hugh Godefroy (still flying with 401 Squadron) had found the gold pin John had purchased for Frances as a Christmas present just two days before he was shot down on November 8, 1941. Then, in just a couple of letters in late winter, he explained that hockey was over because the rink had become a mud hole and that the reduced number of Red Cross parcels had left him in need of basic \"clothing\u2014pajamas, sox sic], shoes, slippers, gloves and cap.\"[[32] Yes, Frances had received her Christmas present, but she sensed her fianc\u00e9's calls for additional clothes were more for a stockpile than his own personal use. He was up to something.\n\nAs it turned out, so were the German prison authorities. Scuttlebutt around the Barth compound suggested most of the officers and air force NCOs were about to be packed off to somewhere in Silesia. In mid-March, Stalag Luft I guards instructed the prisoners to pack their belongings, assembled them for parade, and marched them to the trains outside Barth. Along with the instructions came warnings against any attempts to escape. Roommates Birkland, Floody, and Weir became separated when the Germans loaded POWs into the train cars alphabetically, but that didn't dissuade Scruffy. He teamed up with another kriegie named Mike Wood and the two worked out the details of Weir's escape plan.\n\nBased on maps the escape committee had stowed away in the camp, Scruffy expected the eastbound train would pass through Stettin, which had access to the sea; if the two men could loosen the train windows and jump from the moving train as it slowed through Stettin, maybe they could make it to the harbour and stow away on a ship. Everything went according to plan\u2014preparation of the windows, the jump from the train, safe passage to the waterfront, and refuge overnight until they could find a ship leaving the port. It nearly worked until they ran into an SS officer and were arrested, jailed, interrogated, and shipped off to their intended destination\u2014the Silesian town of Sagan and the new home of imprisoned aircrew, Stalag Luft III.\n\n\"Hello, Darling,\" John Weir began his twelfth letter home to Frances McCormack in the spring of 1942. \"We are now well installed in the new camp . . . situated in the middle of a reforestation area with lots of fresh air.\"[33]\n\nScruffy complained that it had been almost two months since he'd heard from anybody in Canada. His fianc\u00e9e and the Weir family had no idea he had just been shipped from a prison camp where he'd participated in forty-eight tunnel escape attempts, daily goon-baiting (terrorizing the German Officers or Non-coms, or \"goons\"), and then a nearly successful leap to freedom from the prison train.\n\n\"We've been clearing stumps for a kitchen garden (I only hope the Germans give us some seed) and for basketball, volleyball, and tennis courts. . . . Besides that I'm slinging the discus. I hope to be fairly healthy by the time I get back to you. Then all you'll have to do is fatten me up.\"[34]\n\nFrances McCormack wouldn't necessarily have understood what her husband-to-be was up to, but clearing stumps, gardening, and pursuing sport while inside the wire at this new state-of-the-art prison camp in Silesia were not at all about filling a cold cellar with fresh produce for POWs' diets or even about keeping fit. These were the grassroots pursuits of a transplanted escape committee already cultivating its next plan.\n\n* * *\n\n* American-born Don Blakeslee joined the RCAF in August 1940, arrived in England in January 1941 to serve with 401 Squadron (earning a DFC), and was transferred to 133 Eagle Squadron in June 1942 as a flight leader. In August 1943 he flew four sorties over Dieppe and was promoted to squadron leader. By the time the three Eagle Squadrons were turned over to the USAAF as the heart of 4th Fighter Group, Blakeslee had flown 120 Spitfire sorties and accumulated 240 combat hours.\n\n* In the summer of 1944, German authorities delivered as many as 168 Allied airmen shot down over occupied Europe to the Buchenwald concentration camp. Since many of these Allied aircrew, including twenty-six Canadians, had been assisted by members of European underground groups (and were therefore disguised as civilians carrying non-military identity papers), their German captors considered them spies not protected by the Geneva Conventions. At Buchenwald they were subjected to the same torture, deprivation, confinement, work details, and even human medical experimentation that others faced in the death camp. By October, most air force prisoners were transferred to German prison camps under Luftwaffe control.\n\n* The first official escape meeting took place at Dulag Luft near Frankfurt-am-Main, in August 1940, when Royal Navy airman, Lieutenant Peter \"Hornblower\" Fanshawe, directed the construction of a tunnel through the fall of the year, but the Germans discovered it just as it reached the prison camp fence.\n\n* Between 1939 and 1945, from a total of 125,000 aircrew, Bomber Command lost 55,573 airmen killed, 8,403 wounded, and 9,838 became prisoners of war; 8,325 aircraft were lost. Fighter Command lost 3,690 airmen killed, 1,215 wounded, and 601 became prisoners of war; 4,790 aircraft were lost. Total RCAF casualties included 3,150 killed, 9,890 presumed dead, and 49 who died as POWs.\n\n# 3\n\n\"SPINE-TINGLING SPORT\"\n\n* * *\n\nTREES used for reforestation, especially pines planted closely together, are cultivated that way to allow fast growth upward, not much outward. Stuck into the ground as seedlings perhaps a few feet apart, and with little or no human attention\u2014water or fertilizer\u2014reforestation pines shoot skyward. Their purpose is not to grow freely, but to keep loose, sandy soils from eroding and to create, overnight, what foresters refer to as a \"green wall.\" For some years, farmers along the Bobr River near the town of Sagan in Silesia had protected their sandy soils by planting rows and rows of these fast-growing pines. By the 1940s, thousands of tall, palisade-like pines appeared to cut off the town of twenty-five thousand inhabitants from the outside world. Whether by accident or design, when Sagan's new arrivals\u2014several hundred aircrew prisoners of war\u2014began debarking at the train station and marching down the road to their new quarters, the pine forest seemed to gobble them up for good.\n\n\"Situated in the middle of Sagan forest,\" one aircrew POW said, \"one could never see more than a few hundred yards in any direction, except upwards. . . . The lack of view rather enhanced the fenced-in feeling.\"[1]\n\nIn fact, the new prison compound\u2014Stalag Luft III\u2014was carved out of those woods, hundreds of miles from any ocean coastline and even farther from existing battlefields; there was little chance that an escaper could get far enough west to contact the French or Dutch Underground, or the other direction to find refuge with Soviet allies beyond the Eastern Front. For a year before the prison camp opened in March of 1942, Russian forced-labour workers and eventually some of the first non-commissioned Allied aircrew (whom the Germans could force to work) who were brought there chopped down many of the reforested pines to form a clearing\u2014about three hundred acres of grey-coloured sandy topsoil. The fine consistency of the soil would pose an obvious problem for tunnellers, while assisting any camp officials thwarting them. Working in the tunnellers' favour, however, was the virgin ground, without any previous excavation and (unlike Barth) with a water table hundreds of feet under the surface. In addition, Sagan, less than a mile from the camp, was a busy train terminus with six rail lines intersecting in its marshalling yard. And while that meant smoother delivery of prisoners from trains to platforms and then to the compound, for any observant POW the presence of so many railway lines seemed to provide countless ways to exit town, should he get that far.\n\nThe Germans had built two separate and parallel wire fences around the roughly half-mile-square compound. The double-jeopardy fences each stood about ten feet tall and six to eight feet apart. Atop the fences, like a thorny crown, they had installed a barbed-wire overhang that pointed inward toward the prisoners. And between the two fences they had piled and strung additional barbed wire to impede any passage. Illustrating the immense size of the prison, about thirty feet inside the outer fence was an inner warning wire, strung about eighteen inches off the ground; beyond the knee-high wire was no man's land, and any POW caught in it, they were told, would be shot without warning. Guard towers were positioned on each corner and every hundred or so yards along the perimeter of the compound; all were armed with searchlights and machine guns. Eight barrack buildings were built a further forty yards inside the warning wire (each hut sat several feet off the ground on concrete pillars so that ferrets could inspect beneath them at a glance), as well as a wash house, a cookhouse, a couple of latrines, and a pool for rainwater in case of fire. Meanwhile, outside the entire enclosure, the pine forest had been clear-cut back as much as a hundred feet beyond the double fences. The message was evident everywhere: this camp was built to prevent escaping.\n\nWhen the majority of their occupants began arriving on the first day of spring 1942, there were two compounds ready and waiting. The East Compound contained eight barracks huts; these would house approximately nine hundred air force officer prisoners. The Centre Compound had a dozen barracks to house approximately sixteen hundred non-commissioned air force prisoners. Every POW barrack was a wooden, single-storey hut containing twelve good-sized rooms, and each of those held six double-decker bunk beds and a stove for heat in winter (whenever fuel was provided); before they became overcrowded, two of the dozen rooms in each hut offered a library and recreation room. West of the Centre Compound was the Kommandantur,[2] the German administrative compound for the whole camp. To the northeast stood the Vorlager, the utility section of the camp housing a bathhouse, an infirmary, buildings to store fuel and supplies, housing for Russian prisoners (the camp's forced labour), and the notorious \"cooler\"\u2014solitary confinement cells.\n\nFor several days freight trains pulled into Sagan Junction disgorging their human cargo\u2014Allied air force prisoners that none of the German Kommandants at POW camps across Germany ever wanted to see again. Emerging from the \"forty-and-eights\"\u2014boxcars built for either forty men or eight horses but often fit for neither\u2014were hundreds of the most experienced escape artists the European air war had produced. German authorities figured it was safer to dump all the bad apples into one basket, a basket that could contain them, than to have them causing havoc at thirty different prison camps all over Germany.\n\nAmong the prisoners marched into the new facility near Sagan was the old guard of Royal Air Force men, those who had been locked up in German POW camps for two or three years already. Of course, there was Harry \"Wings\" Day, who had led 57 Squadron's Blenheim bombers right from the beginning of the war, but was brought down in October 1939\u2014on a Friday the thirteenth. Perhaps the Germans had been the unlucky ones for shooting him down that day; as Senior British Officer, Day entertained most of the hundreds of escape plans while appearing to toe the line with German camp brass. There was Peter Fanshawe, a Royal Navy pilot shot down June 13, 1940, while attacking the German battle cruiser Scharnhorst. One of the original kriegies, \"Hornblower\" Fanshawe had organized the first escape meetings inside Dulag Luft at Frankfurt-am-Main in August 1940.\n\nThere were RAF aircrew men such as Blenheim pilot Mike Casey, shot down over Emden, Germany, on October 16, 1939; and Fleet Air Arm flight commander Jimmy Buckley, shot down near Calais, France, six days later on May 29, 1940. Buckley and Casey had organized the first breakout attempts even before RAF airmen were transferred to Stalag Luft I at Barth. From Warburg and Oflag VI-B prisoner-of-war camp came fighter pilot Douglas Bader (shot down in August 1941), whose Lazarus-like recoveries from combat and crashes had earned him the rank of wing commander, but whose lust for escaping had his German captors threatening to take away both his artificial legs.[3] And from Spangenberg prison came Battle of Britain ace and RAF squadron leader Robert Stanford Tuck, shot down January 28, 1942; he had also made numerous individual escape attempts.\n\nSimilarly experienced and equally undeterred by the imposing look of the prison camp were the Canadian POWs. In fact, some of the Canuck prisoners being transported from Spangenberg prison to Sagan had become quite notorious, even among the German hierarchy. When Ottawa-born RAF fighter ace Keith \"Skeets\" Ogilvie was shot down over Lille, France, on July 4, 1941, William Joyce (a.k.a. Lord Haw Haw, the British-born fascist who became the Nazis' chief English-language broadcaster) announced the victory immediately; normally, processing and publishing such information took days or even months to accomplish.[4] Even earlier, there was the flurry of attention around the capture of Alfred \"Tommy\" Thompson, from Penetanguishene, Ontario. Like Ogilvie and Tony Pengelly, the twenty-one-year-old Thompson had packed up for England in the 1930s to take advantage of the RAF's Commonwealth-wide enlistment option. Posted to 102 Squadron in 1937 and assigned the job of bombing a German target with propaganda leaflets from his Whitley bomber on September 8, 1939, he and his crew were shot down just five days into the war. Tommy Thompson had the dubious distinction of being the first Canadian airman captured in the war and as such was taken to a special meeting with Reichsmarschall Hermann G\u00f6ring in Berlin.\n\n\"He said they discussed hockey,\" Tommy's son Andrew Thompson said. \"G\u00f6ring] liked Canadian hockey. That was one topic they talked about. And [G\u00f6ring] mentioned that my father had gotten him out of bed the night before, when his bomber had flown over Berlin.\"[[5]\n\nNot only was Thompson exposed to G\u00f6ring, but also to German propaganda cinematographers. Shortly after his capture, they shot newsreel footage and sound of Thompson in his RAF uniform standing at a microphone and speaking about living conditions as a POW.\n\n\"Since being taken prisoner here in Germany,\" Thompson said haltingly to the camera, \"my brother officers and myself have been treated with every courtesy and consideration.\"[6]\n\nAlso en route from Spangenberg prison to Sagan was RCAF navigator George McGill, just twenty-three and from Toronto Island. During a combat operation over Wilhelmshaven in January of 1942, McGill was ordered by his pilots to bail out of their burning Wellington bomber. Immediately after the premature order, the two pilots managed to put out the fire and keep the bomber aloft all the way back to England[7] , leaving McGill to face years as a POW. And from Barth came F\/L Henry Sprague, from Nelson, BC, shot down in November of 1941, when his 96 Squadron began using Defiant aircraft as night fighters.[8] Another Barth alumnus was 102 Squadron bomber pilot Tony Pengelly, as well as Spitfire pilots Hank Birkland from 122 Squadron and Wally Floody from 401 Squadron. Floody's wing (and tunnelling) mate, Scruffy Weir, would arrive a few days later.\n\nAll these men had one thing in common. They had experienced years inside the German stalag system. Sometimes alone, other times in teams, they all had plotted and dug tunnels to escape it, and while nearly every one of their attempts had failed, none of the perpetrators had ever given up. Their jailers felt equally assured they could keep their prisoners inside the wire and, with a few exceptions, they had been successful.\n\nWhen he arrived at the end of his march from Sagan Junction train station, on March 21, and surveyed what his German captors had prepared for the officers and NCOs, Pilot Officer Wally Floody didn't see all the sophistication of the new Sagan camp so much as a preventative system. He considered all the German-designed features of the new facility a compliment to the determination and ingenuity of the kriegies' abilities to find new ways of breaking out.\n\n\"It wasn't a punishment camp,\" he said. \"It was the one camp they'd built with all the knowledge they'd acquired during two or three years of holding us prisoners of war. It was supposedly escape-proof.\"[9]\n\nThe Germans called the new camp Stalag Luft III. The prisoners called it G\u00f6ring's luxury camp,[10] but did so sardonically. Even in the earliest days of captivity there, some of the Commonwealth aircrew POWs weren't prepared to sit still; they set about testing the weakest points of the compound right away. On March 21 , the very first day inside the wire, Wings Day joined two other officers in transforming their RAF uniforms to look like German Luftwaffe uniforms (the pants and tunics were not dissimilar in colour), and\u2014being fluent German speakers\u2014tried to talk their way to freedom through the main gate; the bluff didn't work and the three got the obligatory fourteen days of detention in the cooler.\n\nMeanwhile, Jimmy Buckley re-established X Organization[*] inside the East Compound and began receiving a steady flow of escape proposals. One scheme\u2014worked out by Bill Nichols, an American in the RAF, and Irish airman Ken Toft\u2014claimed that the mass of coiled wire between the double fences and the distance between German sentry positions created blind spots along the outer barrier. They theorized if the two of them could get across the warning wire to a blind spot at the main fence, they could cut their way through the wire before being spotted. All they needed was a diversion. So, Wings Day and the escape committee organized a series of them. There was a staged fight. A prisoner approached a key sentry box and asked that a phone call go through to the Kommandant for an interview. A POW at another sentry box asked permission to retrieve a ball in no man's land. And elsewhere, one prisoner dumped a bucket of water over the head of another. All this distracted the guards long enough to allow Nichols and Toft to snip their way through both fences and the obstruction wire in between. Once the two men emerged outside the double-jeopardy fences, other prisoners orchestrated more diversions to help the escapers get into the pine forest beyond. It all worked. But they were soon recaptured on the run and sent to the cooler too.\n\nIt didn't take the seasoned tunnellers long to begin sampling the soil inside the compound and testing the Germans' detection systems. Just as the Canadians had organized themselves at Barth to design, construct, and employ specific tunnelling approaches, a New Zealander and his group had pioneered their own unique tunnelling technique. Air officer Henry Lamond's team would first construct a tunnel about four feet underground and twenty feet long, capable of accommodating three men head-to-toe in a line and closed at each end. Inside the tunnel, the lead digger would pull soil from the front and pass it to the rear. So would the second and the third, advancing the face of the tunnel while enclosing the space behind. The men would work in the nude, pulling their clothes in bundles behind them. They would dig in complete darkness and survive by poking sticks to the surface for air. Lamond's \"mole\" concept got the green light from Buckley and X Organization that first spring inside the new camp.\n\nIn order to make the plan work, the group needed a legitimate excuse to be digging near the double-wire fence. They found one when they realized water from the wash huts drained close to the fences. Day convinced a German duty officer to let the POWs use shovels to trench the water deeper and toward the wire. Over several days Lamond and his two mole mates\u2014Jack Best and Bill Goldfinch\u2014dug the drainage ditch deeper and deeper until they estimated they could begin their horizontal shaft toward the fences without being spotted. Following the evening appell (roll call) on June 30, the three buried themselves in their horizontal shaft and began pulling and pushing the soil to advance their tunnel under the wire. After moling 150 feet, the following night they emerged on the other side of the fence and dashed into the woods. They got all the way to the river near Sagan, stole a rowboat, and managed to make it some distance toward the Baltic when the boat's owner caught up with them and had them arrested.\n\nOn most days, since the Geneva Conventions prevented German captors from putting the officer captives to work, any gatherings in the East Compound attracted a lot of kriegies either by accident or design. One September afternoon in 1942 proved no different. Seemingly uninvolved, Jimmy Buckley and Wings Day watched as members of the escape committee assembled a primitive boxing ring in the extreme southeastern corner of the compound. Moments later, as the kriegies began cheering and jeering their arrival, Canadian airmen George McGill and Eddie Asselin entered the ring, donned boxing gloves, and began to spar. Of course, not only was the match a spectacle for the kriegies, but it also captivated the attention of the German sentries, most notably in the guard towers. As it did, and with a nod from Buckley and Day, two POWs dressed like compound workers slipped over the warning wire, quickly hit the dirt in one of those blind spots next to the fences, and began cutting their way through the wire. In just a few minutes, with all eyes riveted on the McGill\u2013Asselin boxing exhibition, the two so-called workers were through the wire and into the woods for at least a temporary escape.\n\nTo deter any future above-ground escape attempts, the Germans expanded their sentry staff. The Kommandant instituted foot patrols outside the double fences. At night the patrols were augmented with hundf\u00fchrer, armed guards with dogs (twenty-four dog handlers and thirty-two dogs). While a guard could survey an area of perhaps thirty-five feet beyond the wire, a watchdog could easily pick up movement or scents across a thousand-foot area.[11]\n\nMeanwhile, to counter the subterranean escape attempts, the German guards dug an eight-foot-deep ditch inside the double fencing in an attempt to deter further moling excavations. That just drew more challenges to the trench; it wasn't long before Canadian digger Joe \"Red\" Noble and a friend attempted a mole dig from the security ditch, but failed. Then another three-man team tried and was caught. That prompted the Germans to undo the ditch. They filled it back in and resorted instead to a subterranean preventative device they had pioneered at Barth. They buried microphones deep in the ground along the perimeter inside the double fences and posted ferrets on headphones around the clock, listening for the sounds of tools cutting through the soil.[12] It was turning into a tit-for-tat summer. With sentries mounting in number and listening devices going underground, X Organization met to discuss the means of circumventing them. Several familiar tunnelling experts inside the East Compound brought new ideas to the table for consideration.\n\n\"The basics are these,\" Floody told Harry Day and Jimmy Buckley. \"First you've got to find a place to sink a shaft. Next, you've got to build a tunnel very deep so the Germans can't hear any digging. You've got to dispose of the sand. And most important, you've got to be able to do all this under the very noses of the Germans.\"[13]\n\nAt the escape committee meeting, Floody suggested they dig two distinct tunnels at the same time, each one thirty feet underground to avoid detection by the buried microphones. If the Germans discovered one of the tunnels, Floody said, the ferrets might let their guard down while the diggers continued with the second tunnel. The tunnelling committee worked out schemes for camouflaging trapdoors in the floors of the barracks huts where the tunnels began. It also hatched a scheme for hiding the real escape tunnels with dummy tunnels. Tunnel diggers would excavate shallow camouflage tunnels that went down a few feet and out about thirty feet before they were terminated. In the floors of the camouflage tunnels the kriegies rigged a trap system through which they would dig an additional shaft fifteen feet deeper underground. That shaft would lead to the entrance of the real escape tunnel. If the Germans discovered the camouflage tunnels, they likely wouldn't spot the deeper shafts. The tunnel crews could then reconnect with the hidden shaft from another location and continue work on the real escape tunnel undetected. Wally Floody and Robert \"Crump\" Ker-Ramsey began a tunnel from Hut 67, some four hundred feet from the fence line; they figured the Germans would least suspect a tunnel that far from the wire.\n\n\"I had started tunnelling in 1941,\" Floody said. \"But by the time we got to Sagan, we were getting pretty expert at it.\"[14]\n\nDigging deeper into the Silesian sand triggered new problems, however. Each new shift underground meant more diggers emerging with more bags and washbasins of dirt. The volume of soil excavated from camouflage tunnels, vertical shafts, and the two actual escape tunnels began to mount. And because the sub-surface sand was damp and bright yellow in colour, which differed from the fine, dusty grey-coloured sand on the surface of the compound, the escape committee recruited kriegies whose sole job became finding new places and ways of dispersing the excavated sand. Initially, they raked the subsoil into the surface sand beneath the barracks huts, but when ferrets discovered one of the camouflage tunnels, the dispersal crews needed to find new, less obvious hiding places.\n\nThe farther down Floody and his diggers burrowed, the tougher it was to breathe. They improvised by fitting sticks together end-to-end and pushing them up to break the surface of the soil above, allowing some fresh air to pass twenty feet down into the escape tunnel. Tunnel sentries, called \"stooges,\" watched for the sticks breaking the surface of the compound soil; their job was to hide the holes\u2014first with their bodies and then by placing a stone temporarily over the air hole. But the air holes proved a half measure. Tunnel diggers, such as Wally Floody, John Weir, and Hank Birkland, could work only a few hours without light or fresh air. When they emerged from their shifts underground, they had to cope with headaches or vomiting. Improvising a solution, Jimmy Buckley found an accordion among the prisoners and had it configured to force fresher air into the tunnel system.\n\nTo assist in the dispersal of sand, the diggers changed the direction of one of the main escape tunnels toward the cookhouse so they could hide sand under it. Ferrets discovered the camouflage tunnel and destroyed it. In the process, Floody was caught in a cave-in and nearly suffocated. When Ker-Ramsey and Floody dug their way back toward the main escape tunnel and replaced the camouflage tunnel, they inadvertently weakened the hut foundations above them and the entire dummy tunnel collapsed with Floody still inside. The only thing that saved the veteran tunneller was that his face happened to be over the trapdoor to the vertical shaft; while his teammates dug furiously to reach him, Floody was able to breathe the air from the entrance to the shaft.\n\nWhen the escape tunnel reached three hundred feet in length, just a hundred feet short of the wire, ferrets raided Hut 67. They tore the building apart and dug far enough to unearth both the dummy tunnels and the main escape tunnel. They blew them all up.\n\nFloody told the escape committee that the elaborate excavation scheme was missing an important ingredient: an equally elaborate scheme to protect the tunnels, the trapdoors, the sand dispersal, and the diggers themselves. In his view, the escape committee had to pay closer attention to security. It needed someone who could be as ruthless as the German prison architects and ferrets. It required the know-how of an individual who could instinctively predict the actions of a prison's inmates and its keepers. It had to find somebody with first-hand knowledge of prison life, well before he fell from the sky and into the Germans' stalag system. It found those qualifications in an RCAF air gunner who had arrived with a civilian record as infamous as his peers' military ones were famous.\n\n\"Of the thousand prisoners . . . of Stalag Luft III,\" George Harsh wrote, \"I was the only ex-convict, and so, for me, the psychological adjustment to being imprisoned] was basically a minor one.\"[[15]\n\nHarsh had lived an entire lifetime, it seemed, before he ever considered risking his second chance at life in the air over German-occupied Europe. The son of a well-to-do family in Milwaukee, Wisconsin, eighteen-year-old George Harsh\u2014fulfilling a \"thrill\" pact with some buddies at Oglethorpe University in Atlanta\u2014held up a grocery store in 1928. During the robbery, Harsh shot and killed a clerk, was himself shot, then later arrested, charged, tried, found guilty, and sentenced to hang. Plea bargaining saved his life, but relegated Harsh to existence on a Georgia chain gang. For the next twelve years he shovelled dirt for fourteen hours a day, slept in an iron cage, fought off sexual attacks, and eventually knifed another prisoner in a fight over a cake of soap.\n\nIn prison Harsh learned how to endure punishment, how to bargain, whom to trust, when and why to fight back, and eventually how to campaign to reform the penal system. Transferred to less severe internment at a Fulton County prison, near Atlanta, Harsh became a trusty (an inmate seeking freedom via legal means), serving as an attendant at the prison hospital and eventually conducting an emergency appendectomy to save a prisoner's life. In 1940 Harsh won a pardon, was freed, and considered his options. He felt the need to begin again with a clean slate. He travelled to Montreal and presently found himself at an air force recruiting centre pleading his case to become an air gunner in the RCAF.\n\n\"In one flamboyant beau geste,\"[16] Harsh wrote, \"I was trying to counterbalance my entire past. . . . I was a man trying to prove something.\"\n\nOverseas Flying Officer George Harsh was posted to 4 Group as a gunnery officer aboard Halifax bombers with 102 Squadron. In Bomber Command's history, early 1942 proved a key period when its commander, Arthur Harris, ramped up the strategic bombing offensive with spectacular one-thousand-bomber raids on Cologne, L\u00fcbeck, and Rostock, Germany. This period of Harsh's service, however, also witnessed the improvement of German defences. Harsh and his squadron mates knew well the 4.3 per cent loss rate[17] of Bomber Command raids. That summer, Canadian army, armoured corps, and aircrews had also joined Operation Jubilee, the attack on the French port of Dieppe. On August 19, 1942, 4,963 Canadians spearheaded the morning raid on the seaport in the nine bloodiest hours of Canadian military history; the Second Infantry Division suffered 907 dead and 1,946 captured; the RCAF lost thirteen aircraft, its highest single-day total in the war so far. The setback resonated for everyone in a Canadian service uniform.\n\nSome weeks later, on October 4, RCAF F\/O Harsh was not scheduled to fly an operation from 102 Squadron's station at Pocklington, England; however, as gunnery officer he joined the operations officers that night directing the stream of Halifax bombers into the air with signals from an Aldis lamp. The third to last bomber in the flight had a problem. Its rear gunner\u2014on his first op\u2014had a broken gun sight and had no idea what to do. Harsh ordered him out of the turret and went in his place to ensure an experienced gunner was aboard to protect this rookie crew. Everything went wrong that night. Instead of the lightly defended Dusseldorf target, Harsh and his crew found themselves over Cologne and all its night fighter and anti-aircraft defences. His bomber was coned by several searchlights and flak converged on the aircraft, igniting fires and forcing its crew to bail out over Germany. Harsh spent two weeks in hospital for treatment of shrapnel wounds. Then they moved him on to Dulag Luft for interrogation and eventually to Stalag Luft III, where he was photographed, given a kriegie dog tag, and handed over to the SBOs inside the wire.\n\nOver a barracks table, Wing Commander Harry Day conducted the question-and-answer session to verify Harsh was who he claimed to be. Once he'd passed the SBO cross-examination, the serious questioning began. A New Zealand squadron leader wanted to know everything Harsh had witnessed on his way into Sagan: What were the trains like currently? What were present procedures at railway stations? What passes did the guard escort show en route? As Harsh offered his recollections of what he'd seen and heard, the squadron leader made notes on a small \"chewable size\" piece of paper. Finally, at the end of a very long day that effectively returned him to the life he had led for twelve years in the US penal system, Harsh commiserated with himself about a prospect as dim as it had been his first day shackled to a Georgia chain gang in 1928.\n\n\"They tell me you're an American,\"[18] a friendly voice suddenly interrupted.\n\n\"Yup,\" Harsh had said, using his best Gary Cooper impression.\n\n\"I'm Wally Floody. I got my monthly letter [from home] from the Germans today,\" he said. \"I'm writing my wife, Betty, in Toronto. If there's anyone in the States you want to know you're a prisoner, I'll get her to contact them.\" Floody explained that by passing on the information in this way, Harsh's family would know he was alive faster than if they waited for official word from the Red Cross or Air Ministry. He was right; Betty Floody was able to notify Harsh's sister of his whereabouts and status six weeks before the official air force form letter arrived with its \"we are pleased to inform you\" language.\n\nA kriegie friendship and collaboration began in that moment. Floody's generosity left an indelible impression on the latest arrival to Stalag Luft III. In the days that followed, Floody and Harsh often walked the perimeter of the camp, up against the warning wire. It was the only safe place to talk. They shared their histories, their hopes for survival, and the current state of escape committee activity under X Organization. Floody brought Harsh up to date on the growing list of problems facing the tunnel designers and diggers\u2014too much sand to disperse in secret, too many ferrets to track, too many tunnel operations to protect, too limited an inside security force. At the end of one of their exchanges, Harsh walked right into Floody's verbal trap.\n\n\"Just how the hell do you guys plan to secure all] this?\"[[19] Harsh asked.\n\n\"That's going to be your problem,\" Floody said with a grin. \"You, my friend, are going to be in charge of security.\"\n\nHarsh was not impressed by being dragooned into the job, or by what he realized would become a mammoth responsibility. Floody pointed out that Harsh would be assisted by Canadian officer George McGill, who had earlier delivered such a convincing escape diversion with his impromptu boxing exhibition that the escape committee immediately recruited him for security. Tom Kirby-Green, who spoke myriad languages (including dialects from his homeland of Nyasaland\u2014now Malawi\u2014 in Africa), would become a third member of the security team. And Harsh's final security co-chief would be a fellow American. Albert \"Junior\" Clark had risen quickly in the military; at age twenty-seven, he was already a lieutenant colonel, and by the summer of 1942 was serving in the RAF. In July he'd been flying a sweep over Abbeville, France, where Luftwaffe fighters shot his Spitfire down; he'd arrived at the East Compound of Stalag Luft III in August, a few months before Harsh, and was also befriended by tunnel designer Floody.\n\n\"You know something?\" Floody said finally to Harsh. \"You're probably the only man in the world who got a job because he's an ex-convict.\"[20]\n\nAnother item on the escape committee's wish list was its own version of MI5, an intelligence service designed to recognize any threat to the secret activity of the escape committee, conceive and organize defence against that threat, and carry out counter-intelligence to mislead or derail all efforts to discover the day-to-day operations of X Organization. Some of the expertise the escape committee needed arrived at the Sagan prison in the form of a Canadian airman shot down over Germany that summer. Born in Toronto and educated in England, Kingsley Brown had two passions as a young man\u2014writing and flying. Beginning in 1928, he wrote for the Toronto Star, the New York Herald-Tribune and the Halifax Herald, and while employed at the latter he learned to fly at the Halifax Aero Club. When war broke out he enlisted in the RCAF and was serving as a Hampden bomber pilot when he was shot down over Bremen, Germany, on July 2, 1942.\n\nBrown recalled his first night at Stalag Luft III in the East Compound barracks hut adjusting to his new surroundings. As new boy he'd been assigned a top bunk, so when the German guards locked the hut doors for the night and the prisoners drew the blackout curtains shut, Brown just sat on his bed, legs dangling. He recalled how, with his head near the ceiling, he was nearly overcome by the stench of body odours and the cloud of smoke from cigarettes and burning candles made of margarine and shoe polish. Despite the haze, however, he began to notice activity at the opposite end of his room, where the stove had been moved aside to reveal a hole in the floor.\n\n\"In this eerie light, I watched naked and half-naked men, their bodies shiny with sweat, slipping in and out of the black hole,\"[21] Brown wrote.\n\nThis was his first exposure to a tunnelling crew, assembling, disappearing down the excavation hole, and then emerging with cloth bags, pails, and tin cans full of sand. Brown was allowed to witness this parade of diggers entering and exiting the trapdoor for the night's excavating only because several kriegies had vouched for him; they had known him in Toronto or at his Bomber Command aerodrome in England and knew he could be trusted. After a few days of orientation, Brown was summoned to a meeting with Group Captain H. M. Massey. Then the SBO, Massey was in his sixties and (before being captured in 1942) had been appointed to liaise with US strategic bombing personnel in Washington, DC; but Massey had felt he ought to witness a bombing operation over Germany first-hand and was shot down on his inaugural op. Massey knew Brown had worked for newspapers and concluded his journalism skills might help the escape committee's propaganda activity.\n\nThrilled at the prospect of potentially working in connection with those underground operations he'd witnessed, or perhaps joining clandestine radio transmitting, or assisting in production at a secret printing press, Brown reported immediately to his superior, Wing Commander Taffy Williams. Brown found Williams alone in his barracks room. He was seated at a table in front of an open window, working with what looked like empty jars, a pair of scissors, and tiny pieces of paper.\n\n\"You're just in time to lend me a hand,\" Williams told Brown. \"We've got a propaganda job.\"[22]\n\nWilliams handed Brown one of the German jam jars he'd procured from a guard. Held inside the jar by a piece of cheesecloth were several bumblebees\u2014very much alive, but not very happy about the arrangement. Williams had a crew of kriegies capturing the bumblebees near the cookhouse and passing them along for propaganda duty. Williams handed Brown a pair of gloves and instructed him to open the top of the jar and gently grab each bumblebee by the wings. Williams then slipped a noose of thread over the abdomen of the bee to where the abdomen joined the thorax and gently tightened the noose; attached to the thread was a pennant-shaped piece of tissue paper on which was carefully written, in tiny script, the propaganda Germans later discovering the bees would read. The kriegies hoped the messages would surprise and offend.\n\n\"Deutschland kaput,\" one side of the pennant said, and on the other, \"Hitler kaput.\"[23]\n\nGraduating from his initiation into the kriegies' hierarchy to the escape committee enterprise (with relatively few stings to show for it), Brown moved to a new level of intelligence service. Upon his arrival at the compound, he'd been given a German language textbook that apparently was standard issue to the thousands of non-German workers transported inside the country to work as forced labour. Language was among Brown's specialties and he soon learned that his conversational German was an asset inside the wire. Brown could initiate small talk with the guards to discover valuable bits of information or to determine which guard might be a potential bribery prospect, a so-called \"tame\" guard. And that work led to joining a sub-committee of the X Organization\u2014known as \"Dean and Dawson,\" after the British travel agency\u2014which forged such documents as identity cards, travel passes, and labour permits. Dean and Dawson, led by Briton Tim Walenn and Canadian Tony Pengelly, put Brown's journalist skills to work, assembling a complete card index of police stations, labour recruiting centres, and industrial addresses for every major town and city in Germany.\n\nAs a consequence, Kingsley Brown became something of a fixture in the library or reading rooms in the East Compound. When the latest German newspapers\u2014V\u00f6lkischer Beobachter and the Frankfurter Zeitung\u2014arrived in the camp, the former Canadian journalist who was apparently hungry for news (if any guard asked) pored over each edition of the newspapers' classified ads, obituaries, and public service announcements. From his daily readings, Brown might learn the names of real people whose identities could be stolen, the names of police officials who might sanction travel papers, and which industries in what locations might be recruiting migrant workers in Germany or any part of the occupied territories of the Third Reich. While building the card index in the service of Dean and Dawson, Brown met Gordon Brettell, a British pilot officer from Fighter Command. They both spoke conversational German. And that helped them hatch an escape plan for the coming winter.\n\n\"The prospect of escape had all the exhilarating fascination of a spine-tingling sport,\" Kingsley Brown wrote. \"And it was this aspect more than any other that beckoned to captives from the free side of the wire.\"[24]\n\nIf digging tunnels, hiding excavated sand, dodging guards and dogs, or tying propaganda notes to the abdomens of bees weren't enough of a sporting challenge, some kriegies tried out-and-out theft. In addition to the data that Brown and the rest of the Dean and Dawson volunteers were compiling from newspapers and public service announcements in the library, Keith Ogilvie contributed on his own. Since his official job inside the wire involved handling the Red Cross parcels[25] arriving for the POWs, Ogilvie had regular contact with the Luft guards and vice versa. One day, he spotted a wallet sticking out of the back pocket of an older guard's uniform. Ogilvie silently lifted it, rushed it along to the forgers to copy, and then informed the distraught guard he'd discovered the wallet on the floor.[26] Worried about the consequences of prison authorities discovering he had lost his papers inside the compound, the guard thanked Ogilvie profusely. In one short, deft act of thievery, Ogilvie had gained temporary possession of valuable identity papers and tamed a prison guard; the man was forever in the Canadian officer's debt.\n\nIn spite of the spirited nature of the kriegies engaged in escape attempts, there was no escaping some of the harsh conditions of prison camp life. When his Halifax bomber was shot down by a German night-fighter aircraft over Frankfurt-am-Main, Germany, in December 1942, navigator Don \"Tiger\" McKim (from Lynedoch, Ontario) jumped into the frigid night sky and a 185-mile-per-hour wind. In addition to the cold and fear he felt, he smashed his head going through the escape hatch and was temporarily knocked out. He came to soon enough to activate his chute, but on the ground was quickly arrested and interrogated at Dulag Luft, arriving at Stalag Luft III on Christmas Eve. His present, on arrival, was a Red Cross parcel and the bottom tier of a bunk bed, closest to the floor and consequently closest to the outside December air.\n\n\"I was never so cold in my life,\" McKim said. \"The mattress was made of bags of wood chips, so the cold would work its way through. . . . I didn't take off my clothes. I put all the clothes I had on. My greatcoat. My mittens. Everything.\"[27]\n\nFollowing appell outside each day, McKim could anticipate the kriegies' breakfast ration\u2014a slice of German black bread, which often had fine wood chips or sawdust inside to make the loaf go further; the bread ration was a loaf per person per week. There might be a tiny piece of butter or cheese with it and their Red Cross tea or coffee. For later meals, meat and fresh vegetables were rare. The Germans doled out either potatoes or kohlrabi (a coarse vegetable that resembled turnips) to make sauerkraut or thin vegetable soup. Generally, the Stalag Luft III officers remained hungry, but had enough to sustain life and health; in fact, with the aid of Red Cross parcel staples, they generally digested more nutrition and calories than the Germans guarding them.[28]\n\nBut Flying Officer McKim never escaped the cold. Because he was only five foot two[29] , when he volunteered to help on the escape committee he was assigned to stooge work. That winter and the one that followed, McKim would walk outside discretely noting the arrival, location, and direction of a guard or ferret entering the compound. He would alert the next stooge in the network as to the nature of the goon in the block and which direction he was headed. All the while, he would try to look as if he were just strolling for exercise and pass along his signals while (thanks to his diminutive stature) staying out the sightlines of the guard towers.\n\n\"It was certain jobs for certain people,\" McKim said. \"My job required me to be out in the cold. I wasn't capable of doing anything else.\"[30]\n\nFlying Officer Bob McBride wrote his wife back in Montreal about coping with the cold. Shot down during a torpedoing operation off the coast of France in September 1942, McBride's first postcard to his wife, Jean, simply said, \"Wait for me a little longer.\"[31] As a consequence, she joined the Prisoner of War Relatives' Association, packing parcels for the Red Cross. At one point she managed to pack up and send her husband a bowling set she hoped would provide exercise and diversion.\n\n\"The bowling set was marvelous,\" McBride wrote back. \"It burned forever.\"[32]\n\nAnother new arrival at the East Compound that fall was a no-nonsense fighter pilot from Port Alberni, British Columbia. Just twenty years old, Arthur \"Jack\" Moul had learned to fly on the West Coast at age eighteen. Flying sorties with 416 Squadron, he quickly earned a reputation for his quick hands on the Spitfire's control column and its machine guns. However, on October 23, 1942, during a trip over occupied France strafing German freight trains, a locomotive blew up beneath him, damaging his Spitfire. Moul ditched in the Channel, but was picked up by a German patrol before he could be rescued by RAF seaplanes. Moul arrived at Stalag Luft III in time to become a valued scrounger, working with fellow Canadians Barry Davidson, Red Noble, and Keith Ogilvie searching for guards who could be tamed for the good of the escape committee.\n\nThe end of 1942 brought top-to-bottom transformation to X Organization at Stalag Luft III. What had been for Wally Floody and Robert Ker-Ramsey trial-and-error excavation would shortly be elevated to a scale of construction Floody remembered from the mines of Northern Ontario. What George Harsh and George McGill had launched as a ragtag group of inmates scrambling to pinpoint goons and ferrets approaching barracks huts would eventually become so timely a network of spying, tracking, and locating that it would even outstrip the Germans' control of their own guarding system. And the intelligence that had created Dean and Dawson and a library of information for future forgery would soon start generating such sophisticated documentation as to pass for the real thing for hundreds of potential escapers. The catalyst for revamping, expanding, and improving every aspect of X Organization was the arrival of Roger Bushell, the South African-born RAF squadron leader who'd been captured twice\u2014once after being shot down in France in May 1940, and again following his escape from Dulag Luft all the way to the Swiss border the following summer. It was his third escape, during transport to the prison at Warburg, however, that nearly got him free.\n\nEn route in a cattle truck, Bushell and several others managed to pry up the floorboards of the truck and drop to the roadway below. One of the three escapers fell under the wheels and was killed, while Bushell and a Czech officer in the RAF, Jack Zafouk, made it all the way to Prague. Meantime, Operation Anthropoid had landed two Czech patriots (trained in Britain) back in the Czechoslovak capital. On May 27, 1942, they ambushed Reinhard Heydrich, the Gestapo chief in charge of the Holocaust of the Jews in Europe; he died of his wounds a week later. Zafouk and Bushell went into hiding but were eventually recaptured, the former being sent to Kolditz Castle prison, the latter rushed to Gestapo headquarters in Berlin for torture and interrogation. His interrogators tried to implicate Bushell in the Heydrich assassination. The Czech family that had hidden the two escapees was executed.\n\nWeighing forty pounds less than when he arrived at Dulag Luft and wearing the scars of his Gestapo grilling in Berlin, Squadron Leader Bushell miraculously emerged from the interrogation ordeal. At the time, rumours circulated that the chief censor officer at the Sagan prison camp had a brother in the Gestapo who intervened and returned Bushell to the Luftwaffe prison system. On the day he arrived at Stalag Luft III, toward the end of 1942, Bushell was wearing his now-tattered air force battledress. That same German censor officer warned the squadron leader if tried to escape and was caught again, the Gestapo would probably shoot him. If he escaped again, Bushell shot back, the Gestapo would never catch him. Unnoticed in a bag of belongings the testy air force officer had tucked under his arm was a civilian suit he'd received from the family that had hidden him in Prague.[33] No doubt Bushell, now the new chief of escape operations, or Big X, was already calculating what he would wear during his fourth escape attempt.\n\n* * *\n\n* In March 1942, when Commonwealth air force officers arrived at Sagan, Harry Day remained the Senior British Officer and Jimmy Buckley as Big X, chief of the escape committee. In June 1942, Group Captain H. M. Massey arrived at Stalag Luft III and assumed SBO duties. In October 1942, Wings Day, Jimmy Buckley, and others were purged to Oflag XXI-B at Schubin, and Massey to Obermassfeld hospital to treat his injured foot, so the SBO became newly arrived Group Captain D. E. L. Wilson of the Royal Australian Air Force, until Massey returned from treatment in November 1943 and was restored as SBO until May 1944. In March 1943, Day and Buckley participated in an escape from the Schubin prison. Day was recaptured and returned to Sagan just as the North Compound opened, while Buckley disappeared in his attempt to navigate a small boat to Sweden. By that time, Roger Bushell had arrived and had become Big X.\n\n# 4\n\nESCAPE SEASON\n\n* * *\n\nBY MARCH of 1943, Kingsley Brown had become somebody else. He was quite sane, or as sane as any RCAF airman shot down over Germany, interrogated, and sent to a prison in the middle of Silesia could be. It's just that besides his kriegie dog tags and German prison ID file, Brown also now owned a set of false identity papers. His alter ego was Goleb Plasov[1] , a steelworker from Liegnitz, in the former Poland. His father's name was Jakov. His mother's name Natasha. And he\u2014according to a rehearsed story\u2014was in Germany en route to new steelworks in Strasbourg, France. Among his false documents Brown also held a piece of stationery on which the police chief of Liegnitz had inscribed his signature and affixed his official stamp. All the details were accurate. Brown knew they were, since he had personally tracked down the police chief's autograph and seal while doing research for Dean and Dawson, the forgery team. At the same time, Brown and fellow kriegie Gordon Brettell had also received tailored civilian clothing that made the two of them appear to be Bulgarian immigrants. A combination of the papers, German currency, and the clothes\u2014all prepared by X Organization\u2014as well as the two airmen's conversational German to help them deal with checkpoint officials, gave them nearly all they would need to pull off an escape from the East Compound of Stalag Luft III.\n\nConstruction that had begun over the winter to the west and a little north of what had been the main prison grounds for Commonwealth air officers appeared to give Brown and Brettell an ideal opportunity. Kriegies in the East Compound spotted crews of Russian workers hacking down more of the reforested area beyond the Kommandantur. In a casual conversation with one of the guards, Squadron Leader Bushell had learned that Luftwaffe authorities planned to move inmates in the East Compound to the new North Compound[2] sometime in the spring. As work accelerated, some of the non-commissioned ranks of Allied airmen joined the work crews building the new facility. And since the construction zone had no prisoners housed in it, German guards were few and far between. The escape committee arranged to have Brown and Brettell switch places with two NCOs on the construction gang. On March 27, 1943, four days before the North Compound was due to open, German guards watching over the work crew, including Brown and Brettell in disguise, were distracted with some cigarette trading. The two kriegies quickly hid in wood shavings, waited for the commotion of their apparent escape to blow over, and in the dark climbed the North Compound fence and slipped into the Sagan forest.\n\nBrown remembered the first night as a dash through a Grimm's fairy tale, as the two men used the stars and a compass to navigate their way through forested areas. In the town of Sorau they bought tickets, boarded a train, and hunkered down en route to Leipzig. By coincidence, they shared a cubicle with six soldiers of the Wehrmacht. On the cubicle wall, a poster featured two troops in conversation being overheard by a suspicious-looking civilian. The text on the poster read: \"Beware the Third Person! The Enemy has Ears!\"[3]\n\n\"I've never known the pure joy of living, tempered so deliciously by a sense of danger,\"[4] Brown wrote.\n\nIf an escape bid seated in the same railway coach compartment as their enemies and the propaganda poster weren't ironic enough, their interrogation four days later sure was. A routine documents check at a railway station in Chemnitz, southwest of Berlin, revealed their masquerade and the two fugitive air force officers were escorted to the local Gestapo headquarters. They expected to be shot. In an office decorated with carpets and potted palms, the district police boss smoked a cigar and fired questions at them.\n\n\"Why did you escape?\" he demanded.\n\n\"It is our duty,\" they said.\n\n\"Correct,\" he acknowledged. Apparently more interested in talking than listening, the police chief regaled them with his own biography. \"I was a prisoner in France during the First World War. I also escaped, but unlike you, I got away.\" He chatted some more about prison life and then signalled guards to take the kriegies away and to have them sent back to their prison camp. He added finally, \"Better luck next time.\"[5]\n\nHaving experienced the thrill of getting beyond the wire and a few days of freedom in his first escape attempt, Brown had another crack at it in December of the same year, with Czech prisoner Joe Ricks along for the ride. That escape was thwarted just as quickly. But it left Brown with an indelible memory. As he and Ricks travelled with armed guards back to Sagan, this time they sat in a train compartment with a member of the Hitler Youth, a Wehrmacht officer who'd lost a leg on the Russian front, and three young schoolteachers\u2014all heading home for Christmas. Someone began to sing. Brown offered a version of \"Alouette.\" Ricks chipped in with a Czech folksong. And the three women began \"Silent Night.\" Everyone joined in, and for those moments, Brown said, the war was forgotten.[6]\n\nThe same winter that Brown and Brettell were brushing up their German and getting fitted with their escape disguises for the March attempt, the long-range mass escape plans of X Organization took shape, very much in the image of the new escape chief, Roger Bushell. Whether inside barracks huts far from the probing ears of German ferrets, or pounding the exercise circuit along the inside of the warning wire, during the non-tunnelling months of the winter, Big X had assembled and consulted with his section heads\u2014security bosses Harsh, Kirby-Green, and recently arrived US Lieutenant Albert \"Junior\" Clark, forgery chiefs Walenn and Pengelly, sand dispersal leader Fanshawe, and, most urgently, tunnel kings Floody and Ker-Ramsey.\n\nKnowing the Germans would soon move most of the Commonwealth and American prisoners to the new North Compound, Bushell calculated the committee's next move. It was decided the current SBO, Group Captain Massey, should approach the camp Kommandant and, in a spirit of co-operation, suggest some East Compound officers provide work groups to assist in the construction of the new camp. Feigning goodwill and enthusiasm for the construction work, Floody, Ker-Ramsey, Fanshawe, and Bushell himself all joined the work parties. Between shovelfuls and hammer swings, however, the expert earth-movers calculated the length and breadth and depth of their new home. Floody and Ker-Ramsey paced out and recorded the distances and angles of the place to determine where future tunnels might be built and how creative they'd have to be to put men beyond the wire. One of them even managed to smuggle back a stolen diagram of the projected underground sewage lines.\n\nNot unlike the Centre and East compounds, the North Compound was three hundred yards square, completely surrounded with those reforested pines and with two fences about nine feet high and five feet apart. More bundles of barbed wire filled the space between the double-jeopardy fences, and the warning wire was strung thirty feet inside that fencing. Guard towers\u2014or, as the kriegies described them, \"goon boxes\"\u2014were situated every one hundred or one hundred and fifty yards. The living quarters consisted of fifteen wooden huts, a kitchen hut (with caldrons for boiling water), a fire pool, and a large sports field, which doubled as the appell area. Each hut contained eighteen rooms; each of the rooms had a dining area, living area, table, stools, lockers, a stove on a tile base in the corner, and double-decker bunk beds (with palliasse, a mattress of woven paper or wood shavings) for eight prisoners. Every hut also had a washroom, lavatory, and small kitchen with a coal stove. The fifteen barracks huts could house up to fifteen hundred air force officer prisoners. Again, north of the barracks, the Vorlager contained a coal shed, the hospital, and a building with both the cooler and a room where Red Cross parcels were warehoused and inspected by German guards, and then passed along to the Commonwealth officer in charge of distribution.\n\nMoving was scheduled to take place on April Fools' Day 1943, and the kriegies lived up to the date's moniker. Seven hundred officers (by this time including some three hundred Americans) gathered their worldly possessions\u2014plates, mugs, cooking pans, makeshift gadgets, thread-bare clothes, and Red Cross parcel boxes filled with personal effects, such as photographs, pieces of string, and nails\u2014and assembled for appell. The roll call among the officers that day had the look and feel of juvenile boys lining up for their first day back at school. The POWs broke ranks and clustered around the guards who were searching them. Articles were passed in jest from one man to the next as the inspection descended into chaos. That allowed some vital escape tools\u2014pens and ink, paper, tin shovels, and civilian clothing (including Bushell's civilian suit)\u2014to pass through the screening unnoticed. Miraculously, the antics had also kept one of the escape committee's most valuable possessions undetected. Since the earliest escape committee days at Barth in 1940, pilot Dick Bartlett had carried his exercise medicine ball with him everywhere. Seemingly the kriegies' physical fitness director, the Canadian Fleet Air Arm sub lieutenant was still guarding the hidden wireless radio in that ball.[*]\n\nMany kriegies knew X Organization had the wireless set, nicknamed \"the canary,\" and that it could receive the evening BBC broadcasts offering the latest world news. Once the canary had been successfully smuggled into the North Compound via that medicine ball, Bartlett and two other officers assumed the responsibilities of its round-the-clock protection. RAF officer Nellie Ellan operated the radio itself. A second officer recorded the BBC broadcast content in shorthand; the contents were later read aloud in each compound barracks. What most kriegies did not know, however, was that relaying the BBC news was a secondary function of the radio. Once the three custodians of the canary had transcribed the BBC news, Bartlett changed the radio coils to receive signals from the British Air Ministry. These encrypted messages contained intelligence for X Organization. When the canary went silent after each broadcast and intelligence message, it was up to Bartlett to hide it in a most unlikely location\u2014under a latrine toilet in Hut 101[7] . Bartlett and his two companions practised the emergency response if the Germans suddenly descended on the wireless hiding place. If the canary's capture were imminent, Bartlett could destroy its coils, eat any written messages, conceal the wireless under the toilet, and be innocently sitting on the toilet in half a minute or less.[8]\n\nOnce inside the new North Compound and dismissed, that April 1, kriegies dashed into the comparatively spacious huts to claim their rooms, where, one man said, there was \"almost enough room to swing a stunted cat.\"[9] Big X ensured that the escape committee was represented in every hut; in each barracks he appointed one officer, Little X, to receive and process all officers' schemes for breaking out of the compound, and Little S to deal with security.\n\nThe excitement of the move and the confusion on both sides made this time ripe for all manner of escape attempts. Pilot Officer Gordon King, from Winnipeg, had arrived just in time for the move to new quarters. At age nineteen, in 1940, he knew Morse code, so the air force streamed him into the wireless air-gunner trade, but he was upgraded to pilot training. The RCAF rushed him overseas and sent him, as Second Dickie (observing pilot), on several large bombing operations, including the first thousand-bomber raid on Cologne, Germany, on May 31, 1942. A few nights later, without the security in numbers and piloting a slower Wellington bomber, he and his crew were shot down and captured.\n\nKing arrived at Sagan barefoot, having lost his boots when he bailed out. At the train station he faced the mile-long walk to the compound with nothing on his feet; somebody loaned him footwear he'd never seen before\u2014wooden Dutch clogs. They served him well as he joined the officers' work crews preparing the North Compound and then on the walk to his new home on April 1, 1943. King spotted plenty of escape hijinx that first day, including an attempt by Joe \"Red\" Noble, one of his Canadian barracks mates. Noble spotted a truck loaded with pine tree boughs leaving the camp, jumped into the back, and buried himself among the logs and branches. But a guard in a goon box had spotted him and passed the word to a guard closer to the truck.\n\n\"Joe, we know you're in there,\" the guard shouted. \"Come on out.\"\n\nNoble stayed put, figuring the commotion of moving day would distract the guard.\n\n\"Come out, or we'll start shooting,\"[10] King remembered the guard shouting more emphatically. And out came Noble.\n\nNot long after he arrived in the North Compound, Gordon King joined a group building its own tunnel to the cookhouse. He remembered it as a scheme to establish a secret passage to the food stores of the building so that they could fatten their meagre rations whenever they wished. The digger at the face of the tunnel was a former fighter pilot, while King worked as \"the dish,\" taking the excavated sand to the entry point and handing it off to a dispersal man.\n\n\"We got about halfway to the cookhouse,\" King recalled, \"when a team of horses pulling a wagon walked on the sand above the tunnel and it caved in. The Germans thought it was a big joke. They were elated they'd found it. But at that point tunnelling was just something to do.\"[11]\n\nWhile such tunnelling efforts seemed short-sighted, they succeeded in creating a diversion for other escape committee enterprises. Bushell used a similar diversion to send four-foot-three-inch Ken \"Shag\" Rees through the manhole into the North Compound sewage system to see if human passage was possible. The Welshman returned moments later with bad news; the sewage pipe was only six inches around. At this point in his wartime service, Rees was into his fourth military aviation career. He'd completed a tour with 40 Squadron in 1941, added a second tour based in Malta, then contributed as an RAF instructor back in Britain, and was flying Wellingtons with 150 Squadron in October 1942 when he was shot down and eventually sent to Stalag Luft III. Inside the North Compound, he was effectively commencing his fifth tour of duty, as a member of the principal tunnel-digging team for Roger Bushell's escape committee.\n\n\"There were escape attempts going on all the time,\" Wally Floody said. \"The escape committee was vetting people who wanted to try anything . . . even one chap who wanted to go out the main gate disguised as a German shepherd dog.\"[12]\n\nTwo creative kriegies\u2014Czech Ivo Tonder and Australian Geoff Cornish\u2014let their beards grow for a few days. Then they traded Red Cross parcel cigarettes and sweets for some Polish greatcoats, smeared their faces with dirt, and joined a Russian work crew leaving the compound through the main gate. The guard there noticed that the numbers entering and exiting didn't add up, but couldn't distinguish the real workers from the imposters. The seasoned Abwehr duty officer, Hauptmann Hans Pieber, arrived on the scene. Formerly a member of the Austrian Nazi Party, Pieber had worked at Stalag Luft I outside Barth and then was transferred to the complex near Sagan with the Commonwealth officers the previous year. Pieber recognized the kriegies in disguise and despite the Russians' protestations that the two men were truly members of their work crew, carted Cornish and Tonder off to the cooler. Another escape attempt about that time, a kriegie clinging to the chassis of a work truck, was stopped by the chief of the German anti-escape troops. Staff Sergeant Hermann Glemnitz (dubbed \"Dimwits\" by the kriegies) had also served the German military prison system for several years. He had worked and lived in England before the war. Nevertheless, he took great pleasure in unravelling escape schemes, whether it was a man hiding in the undercarriage of a truck or tunnel crews burrowing toward the double fence.\n\nMeanwhile, Squadron Leader Roger Bushell seemed equally intent on winning the war of wits. Even if all escape activity at Stalag Luft III did not deliver a \"home run\"\u2014actually getting Allied air force officers back to Britain\u2014Bushell planned to disrupt as much of the Third Reich's momentum as he could from inside its occupied territory. In the first eleven days of April 1943, Big X and the section chiefs of the escape committee met to determine the number of tunnels to be built, their locations, their trapdoor entrances and projected exits, their depths and direction, their designers, their engineers, their diggers, their soil dispersal units, their security overseers, their routines, and the communications support. Settling into their new quarters in the North Compound were more than seven hundred officers, with just as many skills and aptitudes among them. Each one could now be called on to fulfill every job X Organization required. It all began with Roger Bushell's original meeting among his section chiefs.\n\n\"My idea is to dig three major tunnels,\"[13] he told the committee. He added that teams would dig simultaneously, that it would take five hundred officers in the camp to do the job, and that if the goons and ferrets found one or two of the tunnels, his view was that one would eventually deliver two hundred or more POW officers outside the wire.\n\n\"Now you're talking,\" Floody said. Already a seasoned tunneller with experience at three German prison compounds, he recognized the brilliance of the plan. Later he commented, \"We didn't dream that getting one or two or three people back to Britain was going to change the outcome of the war. But we realized that larger escapes would make the Germans tie down a lot more troops.\"[14]\n\nThe committee had decided a crew would dig one tunnel originating in Hut 123 and running westbound from a trapdoor in concrete and brick that formed the foundation for the chimney in that barracks. It was a location the committee recognized the Germans would suspect, because it was closest to the double fence, but farthest from the main gate of the compound. It offered the most seclusion and therefore aroused the most suspicion.\n\n\"We're going to call this one 'Tom,'\" Bushell said.\n\n\"Dick,\" the second tunnel, would originate in Hut 122 and also run westbound. Its trap would be concealed beyond a concrete wall and below a pool of water that was run off through an eighteen-inch-square grating in the floor of the hut's washroom shower. Less obvious because it was located inside an inner barracks hut, the tunnel was perfectly concealed.\n\nMeanwhile, the third tunnel, \"Harry,\" would head northbound from Hut 104, under the warning wire, the entire Vorlager, and, in fact, beneath the cooler itself en route to the pine forest beyond the northern perimeter of the compound. Its trapdoor had to be excavated beneath a stove in Room 23, through a square of tiles, then through the solid brick and concrete foundation that supported the weight of the stove and chimney, ultimately into the topsoil and sand below. Building the traps to each tunnel entrance took days of planning, manpower, tool assembly, security, and precise timing. Canadian Flight Lieutenant Henry Sprague witnessed the birth of tunnel \"Harry.\"\n\n\"That night we staged a diversion, which means we had a party in the block,\" said Sprague, a veteran of Dulag Luft and Stalag Luft I imprisonment. \"The stove rested on tiles in a six-foot-square configuration . . . so during the party some Polish chaps officers Minskewitz, Wlod Kolanowski, and Zbigniew Gotowski], who were expert cement men, they chipped away the tiles carefully, constructed a platform underneath, re-cemented the tiles\u2014that's all they did that night\u2014and put the stove back in place. That gave us an access point to what later became the vertical shaft to 'Harry.'\"[[15]\n\nThe next step, cutting into the concrete beneath the now movable tile flooring, demanded an equally tricky manoeuvre. First, a pick-head borrowed from the Russian work crews was strapped to a baseball bat handle that Ker-Ramsey used to pound through the concrete. Under Floody's supervision, the digging crew laid blankets to catch the debris and muffle the noise. To mask the smacking sound of each blow, Junior Clark and others organized a team of diversionary kriegies outside the window of Hut 104. There they began pounding pieces of tin and wood as if manufacturing cookware, not an out-of-the-ordinary sight for POWs just settling into their new barracks. Meanwhile, George Harsh's kriegie guards, or \"stooges,\" watched every move the Germans made, whether they were armed guards in the towers, goons on foot patrol, or ferrets wandering the compound looking for evidence of tunnelling.\n\n\"We had close to two hundred people in our security force,\" Floody said. \"The moment a German guard or ferret came through the gate, we had one of our security people spot him and signal his whereabouts. . . . I could say to Harsh at any given time, 'How many Germans are in the camp?' And he knew exactly how many, where they were, and when they were due to go off shift. . . . It was complete surveillance.\"[16]\n\nBy contrast, as many kriegies as there were in the ranks of the security system, only a handful knew where the trapdoors to the three tunnels were located; in fact, few knew which huts had been chosen as starting points for the digging. Soon after the move to North Compound, most officers in camp spotted notices going up, inviting kriegies to participate in this baseball game or that cricket match, when really the escape committee was recruiting volunteers for duty in the day-to-day escape operations. Patrick Langford joined right away. As far as he was concerned, too much of his time in the air force had been spent preparing, and not enough spent doing. Born in Edmonton in 1919, Langford had taught himself to ski, ride horses, and play the piano. When the war broke out, he left a good chauffeuring job to join the air force and get into the fighting; instead, because of his high grades in the British Commonwealth Air Training Plan in Canada, the RCAF made him an instructor. After sixteen months of teaching others to fly, he finally got an overseas posting to heavy bombers himself, but in July 1942 his Wellington was shot down over L\u00fcbeck, Germany. He bailed out safely, but fire in the disintegrating aircraft burned him across his upper body.\n\nAfter lengthy hospitalization, Langford arrived at Stalag Luft III just as his fellow officers prepared to move to the North Compound. He seemed eager to join the parade of escape attempts\u2014one plan he presented to Big X involved using a trench the Russians had built as an avenue of escape, and another involved hiding in an exiting garbage wagon. However, the escape committee found a job more demanding and valuable for Pat Langford and his workmate Henry Sprague. They shared the responsibility for protecting the trapdoor to \"Harry.\" As Ker-Ramsey picked his way deeper into the concrete foundation beneath the stove, early in April 1943, the floor in that corner of Hut 104 was exposed for almost ten days. Langford and Sprague used spare palliasses to cover the area whenever guards came near. And with the ferrets staging more surprise inspections all the time, the two kriegies trained themselves to open or close the trapdoor system with lightning speed.\n\n\"Sprague and Langford were the fastest trap men in the world,\" Floody said. \"They could open or close the trap in twenty to thirty seconds.\"[17]\n\nGenerating some good-natured chaos during the move to the North Compound seemed innocent on the surface. But it too was serving a greater purpose. As the general population of officers moved into new barracks, claimed bunks, and established routines for sharing kitchens, washrooms, and all other facilities under a barracks roof, the inner circle of the escape committee never rested. Bushell and the section heads began looking for any and all weak points in the Germans' day-to-day maintenance of the stalag. Even the smallest of details\u2014such as haphazard document checks at the main gate in the weeks immediately following the move\u2014might offer the best escape opportunity.\n\nSo it was when the first insect infestation of the barracks buildings occurred in the spring of 1943. All it took was the senior Commonwealth medical officer to alert the German medical officer to an outbreak of lice, fleas, or bedbugs and, with not unexpected precision, the prison system responded. Prisoners living in the infested hut would be paraded in parties of twenty-five to the shower house in the woods\u2014four hundred yards outside the main gate\u2014for hot showers and bug inspection. Meanwhile, a delousing team would move into the hut and fumigate it. On June 11, just after 2 p.m. (immediately following a routine relief of every guard in the compound), a delousing party of twenty-five kriegies (each carrying a towel covering a second set of clothes) and two German Unteroffizier (NCO) guards approached the main gate. Meanwhile, inside the compound in full view of the guards at the gate, two kriegies\u2014Bill Geiger and Henry \"Johnny\" Marshall\u2014began a fencing demonstration. Tony Pengelly assembled an apparently impromptu audience that began cheering on the fencers.\n\n\"The Germans, who love fencing, had only one eye on business,\" Pengelly said. \"The delousing party reached the gate, its guards shouted something to the guards on the gate, and off it marched into the woods toward the showers.\"[18]\n\nBefore the Germans at the main gate realized anything was wrong, part two of the caper had already kicked into gear. A second group under guard approached the main gate. This time the party consisted of seven Commonwealth wing commanders and group captains and an American lieutenant colonel\u2014allegedly on their way to an emergency meeting with the camp Kommandant\u2014with a German corporal escorting them through. One guard gave little notice to the pass the corporal handed him. A second gatekeeper paused and examined the corporal and his pass more closely. A phone call to the Kommandant confirmed that the emergency officers' meeting had been fabricated, and a closer examination of the corporal escorting the officers revealed that it was really kriegie Bob van der Stok,[19] a Dutch-born fighter pilot in RAF 41 Squadron shot down a year earlier, in July 1942; as well as the fake German uniform, van der Stok had a dummy gun in a cardboard holster.[20] The camp Kommandant, Colonel Friedrich Wilhelm von Lindeiner-Wildau, was furious. The sixty-one-year-old ex-cavalry officer from the Great War and former member of Hermann G\u00f6ring's personal staff was notorious for his quick temper, but also his sense of humour.\n\n\"How do you think of these things? You British!\" von Lindeiner said. Then, he began to laugh. He did not attempt to suppress his admiration for the kriegies' fabricated meeting, nor his sense of satisfaction that his men had discovered it. \"But we are too smart for you. . . . Of course, we'll have to put you in cells for ten days. Rules, you know.\"\n\nMeanwhile, it slowly dawned on the guards at the gate that the two guards escorting the previous delousing party might also have been imposters. Indeed, they were Belgian airmen fluent in German and each equally well outfitted in a mocked-up guard uniform, gate pass, and sidearm.\n\n\"It was all we could do to keep from looking down the road,\" Pengelly said. \"If there was no commotion there we would know the twenty-seven men were clear\u2014twenty-five who had carried various civilian outfits maps and concentrated food cakes] wrapped in their towels and two guards, our own men, dressed as German soldiers.\"[[21]\n\nEventually, a guard ran up to the Kommandant and explained he had found towels in the woods and that that shower party had vanished. Von Lindeiner went pale. He immediately dispatched search parties into the woods, had the entire camp population paraded to the sports grounds for roll call, and sent machine-gun armed troops through every hut in the North Compound. Even though it began to rain heavily, he kept the POWs standing on the sports field for five hours.\n\nWhat Pengelly referred to as \"Operation Bedbug\" still did not deliver Bushell's home runs\u2014escapers who made it all the way back to Britain\u2014but it proved to be the first significant victory in the psychological warfare between X Organization and the Luftwaffe administration of the North Compound at Stalag Luft III. The temporary escape of twenty-seven POWs required the mobilization of additional German troops to track them down; as many of the kriegies put it, the short-term goal of these escape plots was to tie up German manpower. The events of June 11 had also tarnished the otherwise pristine record Colonel von Lindeiner had enjoyed thus far in the war. Worse yet, even if the Kommandant refused to admit it, Operation Bedbug had exposed an important chink in the defensive armour at Stalag Luft III. X Organization's seemingly endless array of ruses, diversions, and masquerades continued to fool the compound staff, at least temporarily. This time, credit for the successful breakout, beyond the wire at least, went to a simple but very powerful section in the escape committee hierarchy\u2014its tailoring crew.\n\nRAF airman Tommy Guest and his master seamsters had, in secrecy, manufactured both civilian clothes and convincing German uniforms from little or nothing. Under normal circumstances, the Germans had allowed sewing equipment inside the barracks so that prisoners could repair their own clothing. However, by procuring blankets, sports jackets, and other garments arriving in Red Cross parcels, Guest and his sewing-machine operators had turned a repair room into a veritable men's garment factory providing the escape committee with all manner of look-alike clothes, from plain business suits to the top hat and overalls of a town chimney sweep. All it took was time, Pengelly added, and \"time was our cheapest commodity.\"[22]\n\nIf Colonel von Lindeiner thought the bedbug caper represented the oddest diversion his Commonwealth POWs could create, he was mistaken. A combination of available time and the kriegies' now-expert capability to create something from nothing led to another extraordinary diversion inside the wire. In April 1942, Canadian Flight Lieutenant Arthur Crighton had bailed out of his burning Wellington bomber over Hamburg; all but his tail gunner, Dick Howard, had survived, and eventually the Luftwaffe interrogation and transporting system delivered him to Stalag Luft III. When he discovered, the following spring, that a kriegie had received a set of golf clubs from the YMCA in Norway, Crighton and a number of golf-starved officers in the compound decided to transform a portion of their prison real estate into a nine-hole golf course. One vital element remained missing: golf balls.\n\n\"The] thin elastic strips used to provide insulation for jam, fish, and meat tins in Red Cross parcels [were] rolled around a pebble,\"[[23] Crighton wrote. \"It was finished by sewing on a protective leather covering from an old shoe [around the elastics].\"\n\nThe resulting golf rounds, in which kriegies could smack the makeshift ball up to fifty yards, were played between the huts, amid kriegies walking along the circuit and across the shifting expanses of sand inside the wire. The only hazards the golfers faced were shots that rolled under the barracks huts and those that ended up beyond the warning wire into territory that truly was \"out of bounds.\" For a while, Crighton and his fellow golfers devised a system that allowed a kriegie with a red scarf on his shoulders to step over the warning wire to retrieve errant fairway shots. Crighton admitted he was never courageous enough to don the red scarf and test machine-gun toting tower guards, but he did face a logistics problem as challenging as manufacturing golf balls from pebbles and elastic bands.\n\n\"I was left-handed and left-handed clubs were not available,\" he said. \"We] melted down a leaden Keintrinkwasser jug and fashioned it in a sand mould to become a club head. And a shaft? As factotum of everything musical in the compound, I appropriated the upright staff that supported the lid of the grand piano [in the prison camp]. Thus was constructed the most elegant left-hand five iron in the whole of Kriegiedom.\"[[24]\n\nWhile Art Crighton appeared to fill his time with repeated rounds of golf on the makeshift compound links, during the remainder of his POW time he occupied himself with his second great passion: music. Before the war, he had studied to play the organ and lead choirs; growing up in Calgary he had earned the prestigious Licentiate diploma of the Royal Schools of Music. In peacetime Crighton had become a respected music teacher and proficient on the organ and the trumpet. Inside the wire, he asked to play trumpet in a concert band and later took over as bandleader.\n\nSince German prison officials appeared to encourage constructive prisoner pursuits, the Germans welcomed Crighton's requests to bring additional live music to the prisoners when the North Compound opened in April 1943. Then, when he sought permission to purchase such instruments as violins and cellos, Stalag Luft III authorities agreed. Crighton also requested wind instruments for a forty-piece symphony orchestra. They opened channels for the YMCA and Red Cross to send reed instruments, trumpets, trombones, and a tuba. Eventually, Crighton the kriegie music director not only led symphony concerts of Mozart, Bach, and Beethoven (which were of course acceptable to the German Kommandant), but also organized band performances, marching bands for outdoor events, and a pit band for productions at the compound theatre when it opened in September 1943. Crighton went so far as to assemble an orchestra wardrobe; if prisoners received new uniforms from home (which was permitted) but didn't want to wear them, Crighton commandeered them for his musicians. He insisted that orchestra members wear them.\n\n\"The Germans encouraged us,\" he said. \"They knew if we had an orchestra, we weren't digging tunnels.\"[25]\n\nEscape fever heated up that summer, and not just at the North Compound. For as long as Commonwealth and, more recently, American officers had been imprisoned in the East and North compounds at Stalag Luft III, a growing number of NCO aircrew (by the summer of 1943 they numbered nearly nineteen hundred men) had been penned inside the Centre Compound. By comparison to the officers' huts, the condition of the NCOs' barracks reflected German administration's neglect and the POWs frustration with conditions. Huts had broken windows. Wiring had been torn from the walls. Latrines had been stripped of wallboards for firewood and for shoring up tunnel projects.[26] That June, the Germans transferred all but fifty of the NCOs to Stalag Luft VI in Lithuania. The fifty who remained at Luft III appeared to be the most respectful of the premises; the German prison-keepers decided these fifty men would become orderlies to the incoming officers. They turned out to be the nucleus of the tunnelling effort well underway that summer. The Germans discovered and destroyed it that September.\n\nThe same summer, a British air force officer, a Royal Artillery officer, and a Canadian air force officer concocted an escape scenario that would take place virtually in plain view of their compound captors. Serving as crew navigator on a Stirling bomber and shot down over Germany in December 1942, RAF Flight Lieutenant Eric Williams had originally been imprisoned in Oflag XXI-B at Schubin. There he met an artillery officer, Lieutenant Michael Codner. They quickly built a tunnel and just as quickly were recaptured and sent to the East Compound at Stalag Luft III. Just weeks inside their new prison camp, Codner had an epiphany.\n\n\"A vaulting horse, a box horse like we had at school,\" he told Williams. \"One of us inside digging while the others vaulted over it. It's foolproof.\"[27]\n\nEventually, the two of them worked out the details of their Trojan horse scheme and sold the idea to the British escape committee. With the help of a carpenter who was legitimately constructing frames for stage flats at the theatre inside the compound, they built a strong wooden vaulting horse with stolen wood for the frame and light plywood taken from Red Cross parcel boxes to sheath the horse. And for the comfort of the vaulters, they used cigarette packaging to fashion a cushion on top.\n\nThe scheme required a dedicated group of accomplices. Flight Lieutenant Edward Nurse, originally from Newfoundland, then an RCAF instructor, and eventually a Halifax bomber pilot shot down in April 1943, joined the plot. Initially, each day early in the summer, Nurse and the others carried the exercising horse to the spot chosen to begin the escape tunnel. Once the Germans got weary of inspecting the horse,[28] the team began to carry it\u2014containing one tunnel digger, his tools, and about a dozen empty sock-like containers\u2014to a corner of the compound next to the warning wire. Coincidentally, it was the same spot where George McGill and Eddie Asselin had staged their boxing match as a diversion for an escape attempt the year before. As Ted Nurse and the rest of the dedicated POWs conducted vaulting exercises over the horse, the tunneller cut into the sand under the horse, built a trapdoor, eventually a vertical shaft, and then a horizontal tunnel. At the end of each vaulting\/digging session, Nurse and the others carried the horse containing the tunneller and the day's excavated sand to a nearby hut to let the tunneller out, dispose of the sand, and plan the next vaulting session.\n\nIn the early spring, Oliver Philpot, a thirty-year-old Canadian torpedo bomber pilot who had been shot down over Norway in December 1941, joined the Trojan horse scheme. Originally from Vancouver, British Columbia, and a veteran of escape attempts from Dulag Luft, Oflag IX at Spangenberg, and Oflag XXI-B at Schubin, Flight Lieutenant Philpot was left behind in the East Compound when the big move of Commonwealth officers to the North Compound occurred in April 1943. Considered a loner, he approached the concept of escaping from a different direction\u2014build an outside identity first, then build a means of escape. And so, as Codner and Williams worked on their two-man tunnel beneath the vaulting horse, Philpot began manufacturing his outside-the-wire persona\u2014a Norwegian margarine salesman named Jon J\u00f6rgensen. By mid-summer, Philpot took on the additional task of helping to disperse the sand that Williams and Codner were packing into bags inside the horse. Before long they invited him to join the tunnelling as well. From then on it became a three-man escape plan. After 114 days, the tunnel (just thirty inches beneath the surface) extended about one hundred feet eastward under the fence and was ready to deliver its diggers outside the wire.\n\nEscape day was October 29. That morning, the vaulters carried Codner and Philpot inside the horse to the top of the concealed trap to the tunnel. At the end of the vaulting exercise, Codner stayed underground to dig the last leg of the tunnel while Philpot retrieved the last of the full sandbags. At that evening's appell, the escape committee ensured that it could cover Codner's absence. Then the vaulters carried three men inside the horse to the top of the tunnel\u2014Philpot, Williams, and a man to close the trap and seal the three men inside. At six o'clock, after dark, Michael Codner broke through the soil outside the wire with his fist.\n\n\"It was our] first glimpse of the stars . . . in the free heavens beyond the wire,\"[[29] Williams wrote.\n\nMoments later the three men, shrouded in dark clothes and carrying bundles that contained their civilian suits, travel documents, and food, made off into the woods, undetected. Philpot (as J\u00f6rgensen, the salesman) caught a train from Sagan, arriving twenty-four hours later at the Baltic Sea port of Danzig. A day later he managed to stow himself aboard the Swedish vessel Aralizz. The captain discovered him and almost had him thrown off the ship, but the ship's sympathetic chief engineer hid him until the Aralizz was at sea bound for Sweden. Meanwhile, Codner and Williams, disguised as French workmen, travelled to Stettin where they managed to make contact with the Danish Resistance. Through its members they gained passage on a ship to Copenhagen, and on to Gothenburg in neutral Sweden. The three Trojan horsemen completed their home runs via the British Legation in Stockholm, arriving in Britain by Christmas. They were the only prisoners of war to successfully escape the East Compound.\n\nAll three received the Military Cross. For Eric Williams, however, it was small compensation. His wife had died in Liverpool during a German air raid, and two of his three brothers serving in the Royal Air Force had been killed in action.[30] A year of his life lost, caged as a POW at Stalag Luft III, somehow seemed a smaller price to pay.\n\n* * *\n\n* NCOs brought the first radio to the Centre Compound at Stalag Luft III when they arrived from Barth. They had assembled it from parts gathered by bribing guards and hid it in a functioning accordion until it was confiscated during a snap inspection in January 1943. Officers in the East Compound also brought a radio from Warburg or Barth; it was confiscated in July 1942. In April 1943, parts for another radio were smuggled (inside luggage, a medicine ball, and a biscuit tin) into the officers' barracks. Later, in January 1945, following the evacuation of the POW camp and during the forced march westward, the German Abwehr officer, Hans Pieber, actually carried the radio in his briefcase for the Commonwealth airmen.\n\n# 5\n\nSERVANT TO A HOLE IN THE GROUND\n\n* * *\n\nFrank sorensen served his adopted country in deed, with oratory, and to carry on a military tradition. Born in Denmark in June of 1922, he moved with his family from the Danish town of Roskilde to Canada just a week before the Second World War broke out. Frank's father, Marinus Bonde Sorensen, had served in the Canadian Expeditionary Force during the Great War, married Frank's mother, a wartime nurse, and in August of 1939 escaped Denmark and settled the family\u2014two adults and six children\u2014on a farm in the Eastern Townships of Quebec.[1] However, the war drew several Sorensen family members back overseas: Marinus representing Canadian and Danish interests in Britain; Eric, Frank's older brother, in the Canadian Army preparing for the invasion of Italy; and Frank as a Spitfire pilot in the Royal Canadian Air Force with 403 Squadron in Suffolk, England. Just days before he was transferred to RAF 232 Squadron en route to combat operations in North Africa, Frank Sorensen became a voice of freedom on the public airwaves. Because of Marinus Sorensen's Danish ancestry, his status as a Canadian Pacific Railway agent, and his connections with the BBC, his son, air force pilot Frank Sorensen, was invited to speak to Danes in occupied Denmark via the BBC's Radio Free Europe broadcasts.\n\n\"Hello, all you Roskilde boys,\"[2] Sorensen began his October 15, 1942, broadcast. \"Let me first of all speak to those who were boys with me, friends from school and Boy Scout camps . . . I know that there's not one who would not consider it his duty and his part of the burden and fight to make Denmark a free country again.\"\n\nJust twenty, but already worldly enough to assess what a fascist dictator and his armies had inflicted on the country of his birth, Sorensen blamed German \"parasites and thugs\" for annexing his homeland, eliminating free speech, and imposing martial law on a passively resistant Danish population. He didn't profess to understand why an older generation had chosen to resolve Europe's differences this way. But in his broadcast, he cited his Danish-born brother, now an engineer in the Canadian Army, and himself, a Danish-born fighter pilot in the RCAF, as models for the way young Danes should respond to German occupation.\n\n\"I know how fortunate I and my comrades out here are that we are actively engaged in the battle,\" he concluded. \"For us the problem is simple and straightforward, but . . . it doesn't pay to wait. Let us young people fight together now . . . until] the Germans once again are driven out of Denmark.\"[[3]\n\nA month later, on November 24, 1942, his squadron boarded SS Antenor sailing from Liverpool to Algeria to assist Lieutenant General Bernard Montgomery and the British Eighth Army mount a counter-offensive in the North African desert against Erwin Rommel's Afrika Korps. By mid-January of 1943, with the Allies advancing quickly, 232 Squadron had established a landing strip at Tingley in Algeria, from which its Spitfires conducted sweeps and reconnaissance operations. By March the squadron was flying patrols and escorts in and out of Victoria airstrip in Tunisia. In three weeks, Sorensen flew just thirteen operational sorties[4] and was growing weary of harbour patrols, tactical reconnaissance, and escort operations. However, a letter to his brother Eric described a sudden change in the intensity of the squadron's activity.\n\n\"With no more excitement than ground strafing enemy transport once in a while, I welcomed the order by my flight commander one Sunday morning April 11, 1943] over Tunis to break away and polish off a few Junker 52s flying low over the Bay of Tunis,\"[[5] Frank Sorensen wrote.\n\nDuring his descent from an original altitude of twenty-three thousand feet, his flight of four Spitfires encountered enemy fire; one Spit was hit, caught fire, and crashed into the sea. The three remaining fighters in his formation opened fire on the German transport aircraft. Sorensen saw his flight commander fire cannon shells into one of them and looked back to see it crash in a ball of fire; he recalled seeing the troops previously inside the Junker's fuselage flailing about in water mixed with burning fuel and wreckage. Sorensen then focused on catching up to another of the transports heading north toward Italy.\n\n\"I sent two of them crashing in flames into the drink,\" Sorensen continued. \"Bags of fun, I thought, until I was reminded by half a dozen Me. 109s that crime does not pay. I wouldn't have met these 109s, if my commander upstairs had not told us to climb] for I had in mind to go home on the deck [just above the surface of the water].\"[[6]\n\nAs he climbed, the first Messerschmitt attacked him from behind. Sorensen tried out-turning his opponent three times, expending the remainder of his ammunition as he manoeuvred around the sky. Now there were six more Me. 109s. In ones and twos the enemy fighters engaged him and he dodged fire from a dozen of those attacks. The thirteenth attack found the target as a burst of machine-gun fire hit his engine. Oil sprayed across his windscreen. He was too low to bail out, so he quickly decided to take his chances with a controlled crash landing. He'd forgotten, however, that he regularly left his cockpit safety belts loose (only tightening them for landings), so unable to see ahead clearly and not properly secured into the cockpit, Sorensen prepared for a rough landing on hilly desert terrain.\n\n\"I closed my eyes,\" he wrote. \"I hit. I felt I hit again. I could still think and feel. Then I turned a somersault and with a final crash my flying carrier had come to an abrupt stop\u2014upside down. I thought I was dead . . . I took off my flying helmet and goggles, oxygen mask . . . and had a look through a hole in the ground. I had ploughed right into the ground so my cockpit and myself were completely under the ground. I started digging my way out when] I heard footsteps and then, 'Hands up.'\"[[7]\n\nSorensen had crashed right next to a German infantry post. At gunpoint he was pulled from beneath the half-buried Spitfire, quickly searched, thrown into a vehicle, and hauled off to Tunis for interrogation. The Americans bombed the city heavily overnight, but the next day he was put aboard a transport, not unlike the four German aircraft his squadron had shot down the previous day. He and a British officer shot down the same day were escorted to Italy by twenty German paratroopers inside the aircraft and a flight of Me. 109s outside the aircraft. They travelled through Rome, then on to Munich, and to Dulag Luft at Frankfurt, all the while refusing to offer any information and paying for their resistance by receiving very little food and plenty of solitary confinement. P\/O Sorensen arrived at Sagan on April 20, \"on Hitler's birthday, when every house had a flag out,\" he wrote.\n\nAs it was for so many families on the home front, word of Frank Sorensen's survival of the crash outside Tunis, his transport through Italy, his interrogation by German authorities at Dulag Luft, and his eventual deposit at Stalag Luft III took weeks to arrive. In the interim came the Department of National Defence letter to Frank's mother saying he \"is reported missing,\"[8] then a long gap with little or no information from either the air force or the government about his fate, and finally\u2014a month after he was shot down\u2014word to Frank's father from the RCAF Overseas office in London that \"your son . . . is a prisoner of war.\"[9]\n\nMeanwhile, Frank Sorensen's letters began wending their way to his grandparents in Hj\u00f8rring, Denmark, to his father in the UK, and to his mother and siblings now living in Kingston, Ontario. His earliest written thoughts from inside German prison camps were predictably melancholy. His first letter from Dulag Luft, just eight days after his capture, said he was happy, enjoying the good company of other \"unfortunate chaps,\" that he'd asked for his $175-a-month allowance to be sent home to his mother, and that he wanted his father \"to write to my squadron and tell them how happy and proud I was to have been one of them.\"[10] By his fifth letter, notably, the dispirited tone appeared to be gone. He seemed no more or less resigned to his POW plight, but there was something more substantial contained in his letter home than just the words of a kriegie waiting for his imprisonment to end.\n\n\"I find time goes by very fast,\" he wrote in May 1943. \"I find plenty to do besides reading and studying chemistry and math. . . . There is plenty of exercise we get when we feel like it\u2014basketball, football, discus, shot put, horizontal bar, horseshoes, and walking 'round the perimeter.\"[11]\n\nJust as Roger Bushell had done with escape committee section heads Floody, Harsh, and Fanshawe\u2014discussing high-priority issues affecting the progress of tunnelling\u2014by walking the perimeter with newly arrived officer Frank \"Sorry\" Sorensen, Bushell acquired valuable new intelligence about Erwin Rommel's defeats in North Africa during the winter of 1942\u201343 and plans for the invasion of Sicily that summer. But having access to Sorensen on a regular basis actually gave Bushell something he hadn't expected; Big X was fluent in German, French, and a few Russian phrases, but he was deficient in Danish. And he recognized if he had any hope of talking his way aboard a vessel sailing the Baltic, simple Danish phrases would be essential. So, pounding the exercise circuit inside the warning wire with P\/O Sorensen allowed Bushell to pick up everyday Danish.[*] But Big X wasn't the only fellow officer Sorensen tutored in Danish. Some time later, Sorensen befriended Flight Lieutenant Eric Foster. As a means of expediting his way out of the camp, Foster feigned insanity. But in the meantime, partly because Sorensen knew that Foster had a Danish wife, he tried to help Eric learn some important phrases. It proved to be a frustrating exercise.\n\n\"I tried teaching him Danish,\" Sorensen wrote, \"but for all he learnt, I might as well have taught a horse.\"[12]\n\nUltimately, Foster's performance around the North Compound convinced the Germans he was mentally unstable and he was repatriated in 1944. But Sorensen's impromptu language instruction, if Big X hadn't already recognized it, illustrated the growing need for X Organization to make conversational training a higher priority among monolingual kriegies. This, in turn, explained one of Frank Sorensen's repeated requests of his family in his regular correspondence.\n\nGiven that each kriegie was rationed to roughly four cards and three letters per month, and that the space on official stalag-issued writing paper\u2014ten inches by five-and-a-half inches\u2014was limited to between two hundred and three hundred words, Sorensen's priorities must have appeared odd in his letters home. No doubt, the censors would not have found his references the least bit troublesome. But after the weather, his health, and general POW activity, in both his May 18 and July 12 letters Sorensen reserved space to request certain items from home or via a Red Cross parcel. He asked his mother to send summer season underwear, dehydrated meat, fruit, milk, sugar raisins, oatmeal, maple sugar, chocolate, powdered egg, and onion. Finally, in a quick sentence, he asked a favour of his father.\n\n\"Dad, would you send me the thesaurus dictionary, please?\"[13] he wrote.\n\nRequesting clothing and food items was commonplace, although emphasizing that the foods be high in energy and dehydrated suggested they might be packed more easily for the trip into the compound, or, conceivably, in the pocket of an escaper, out of it. The special request to his father for the thesaurus, however, had little to do with enhancing Sorensen's literacy or even filling his leisure time with stimulating reading. Contained within the back pages of every thesaurus since its creator, English physician Peter Mark Roget, had first published the book in 1852 was a section called \"Foreign Phrases,\" translated with reference to the applicable English categories into French, Latin, and German. A thesaurus or two or more might well make their way to the North Compound library and reading room, but their circulation would be less about offering readers a lexicon of valuable synonyms and antonyms than about offering them a guide to phrases used in everyday conversation outside the wire.\n\nAssembling a library of language textbooks, including thesaurus donations from the Sorensen family library, likely helped escape preparations a great deal. But finding officers who could properly pronounce the phrases and then get English-speaking kriegies to parrot those phrases back with some credibility was something else again. In that regard, the escape committee had received a remarkable gift, appropriately the previous Christmas, when a twenty-eight-year-old RCAF officer arrived at the North Compound. Gordon Kidder, born in St. Catharines, Ontario, grew up a scholar. He learned French and could speak fluent German as a result of his studies at the University of Toronto in the 1930s; in fact, in 1937, Johns Hopkins University in the US invited him to finish his master's degree in German. Instead, he worked in Ontario's education department and also translated language for an insurance company.[14] When war broke out, he trained as a navigator, and was on his ninth combat operation aboard a 156 Squadron Wellington when it was hit by flak and crashed into the North Sea. Kidder and the wireless air gunner survived in a dinghy until a German minesweeper captured them the next day. Kidder arrived at Stalag Luft III in December 1942.\n\nFlying Officer Kidder was no doubt surprised to find his POW compound not only had kriegies fluent in as many languages as he, but also library facilities to match. Those book rooms became a hub for language classes that not only weren't disguised, but were also encouraged. Had he wanted to, Kidder could have continued his linguistic studies inside Stalag Luft III. By the time kriegies had moved into the East Compound (in 1942) and North Compound (in 1943), some of the most ambitious among them had established courses in business, social science, and the humanities. The courses proved so sophisticated that examining boards in the UK and Canada allowed prisoners of war to earn full credits. While he was imprisoned at Stalag Luft III, RCAF officer Ian Tweddell, from Lashburn, Saskatchewan, received engineering textbooks from the University of Saskatchewan so that he could advance his studies.[15] Textbooks from the same university helped another kriegie, who'd only received his senior matriculation before the war, complete a credit in political science.[16] College certificates were even available for proficiency in German, Russian, and French. To support the kriegies' educational pursuits, the libraries began to assemble sizable inventories. Thanks to Red Cross and personal family parcels, the libraries boasted hundreds of texts and reference books, as well as detective novels, westerns, and biographies.[17] The Centre Compound library had as many as nineteen hundred books, which explains why Frank Sorensen's emphatic pleas for more specific reading materials passed the censors without arousing any suspicion.\n\n\"Yes, the thesaurus dictionary,\" he wrote on August 11, 1943. \"I sure long to get my fingers on that book. No better opportunity learning German, except time and German newspapers. Am studying it though.\"[18]\n\nBy the time Sorry Sorensen had started teaching Roger Bushell and other kriegies some elementary Danish expressions, Big X was receiving the first reports of progress underground. Within days of completing the trap entry to the chimney in Hut 123, Wally Floody, Robert Ker-Ramsey, and Johnny Marshall had penetrated the topsoil into the yellow-coloured sand beneath it, and had begun to excavate the vertical shaft of tunnel \"Tom\" about three feet square and down thirty feet. To reinforce the walls of the shaft, when he had dug down about the height of a man, Floody had \"borrowed\" the bedposts from unused double-decker bunk beds around the compound, bolted them together, and then inserted \"borrowed\" bed-boards as walls behind the posts to shore up the shaft against the loose sand around it.\n\n\"We didn't have springs in our beds.] Each bunk . . . had ten or fifteen boards varying in width from three inches to six inches,\" Floody said. \"We had fifteen hundred officers. That meant fifteen hundred beds. So if you took one board out of each bunk . . . that's a lot of wood for shoring.\"[[19]\n\nWith each new five- to six-foot section of the vertical shaft, Floody would need more wood to shore up the walls before continuing his descent. To supply the raw materials, the escape committee had assigned Australian officer John E. \"Willy\" Williams to lead a crew that constantly cruised the barracks on the lookout for available bedposts, bed-boards, and even nails and screws that could be liberated without being noticed by German guards. When Williams' scavengers liberated the lumber and fasteners, they passed along their found supplies to expert metalworkers and carpenters who engineered the shaft and tunnel construction.\n\nOne such expert had come from the same theatre of war as Frank Sorensen. Initially an apprenticed mechanical engineer from Leeds, England, Bob Nelson had branched into aeronautical engineering before joining the RAF in 1937. A talented pilot, Nelson had instructed in Rhodesia until 1941, when he was posted to the Middle East, piloting Wellington bombers against Rommel's Afrika Korps. On his twenty-second operational flight inland from Tobruk his aircraft was hit. He ordered his crew to bail out and crash-landed the Wellington 150 miles behind enemy lines. He endured three weeks in the Libyan Desert (surviving by licking dew from rusty gas cans[20] ) and was within three hundred yards of the British lines when he was captured and sent to Stalag Luft III in November 1942. A man of his resourcefulness proved invaluable helping to shore up tunnel \"Tom\" in the spring and summer of 1943.\n\nBy late April, \"Tom's\" shaft was down thirty feet, \"Dick\" was nearly that deep, and \"Harry\" was down about twenty feet. Once Floody had completed the vertical dig in each of the tunnel shafts, Nelson and John Travis, formerly a Rhodesian mining engineer, fashioned a kind of crossroads where the vertical shaft ended and the horizontal tunnel began. At the foot of the shaft, they constructed chambers into three of the walls\u2014one chamber for equipment storage, a second for temporary storage of sand from the tunnel before dispersal, and a third (about six feet in length) to accommodate an air pump. Through the fourth wall of the shaft was the entry to the horizontal tunnel out of the North Compound. For \"Tom\" and \"Harry,\" the engineers had organized a multi-purpose basement workshop, with fresh air drawn by an air pump from the natural flue of the chimney above, down through the shaft, and along to the tunnel face where the diggers were extending the tunnel.\n\n\"The air pump consisted of two canvas kit-bags attached to a central wooden valve box,\" Nelson wrote. \"A reciprocating movement of a wooden frame caused one bag to be compressed in a delivery stroke while the other expanded in a suction stroke. A suction pipeline made from Klim cans the Red Cross tins about four inches in diameter that had contained powdered milk] ran up the shaft to a fresh air supply and the delivery pipeline was then laid along the floor of the tunnel as it was dug.\"[[21]\n\nNelson explained that a second Klim can pipeline was laid underground from below each trapdoor to the nearest chimney in the barracks hut. That provided a hidden outlet to expel used air, which otherwise on cold days would have revealed a condensation trail. This way, the regular chimney of the barracks hut gave the tunnel operation a natural draught while a bypass valve in the pump allowed the tunnel to ventilate itself naturally even when no one was underground. The ventilation system also made it possible for a team of diggers, dispersal men, and pump operator to work below with the trapdoor completely sealed above. It meant the work could progress continuously between roll calls.\n\nDigging the horizontal tunnels presented its own set of problems, including excavating on a level plane and in a straight line. The engineers fabricated a level and stole prismatic compasses from the Germans.[22] To calculate the distance travelled underground, they came up with primitive tools, including a premeasured ball of cord. They melted margarine and other combustible fat into candles with pajama string as wicks, ensuring that the sooty by-product went safely up the Klim can chimney; the resulting fat lamps lit the way up each tunnel. Once \"Tom\" began on the horizontal, Nelson and the other engineers worked with Floody to shore the walls of the tunnel with box frames in a trapezium shape for strength. Each of the bottom and two side boards of the frame, he said, was 21.5 inches long, while the top board was 20 inches long. Because most of the bed-boards were conveniently that length, and featured tongue-and-slot joints, they fit together like a prefab floor, ceiling, and walls. The weight of the earth above the ceiling and against the outside of the walls kept the entire frame rigid. That meant no nails were needed, nor was any hammering required.\n\n\"It wasn't very difficult to dig,\" Nelson said. \"Floody] digging the tunnel would lie inside the existing frame. He would scoop out the earth and the whole roof would collapse on him, but he'd be protected by the wood frame. The tunnel sand would collapse into an arch-like shape. All the sand that had fallen out would be packed behind the wooden frame and the shaft would move forward with the wooden support. . . . The big problem was the risk of collapse because you could get trapped thirty feet underground.\"[[23]\n\nIt was about the time the digging team had nearly completed the three workshop chambers at the base of \"Dick's\" vertical shaft that near-disaster struck. On a day in late April, just after the 9:30 a.m. appell, as the digging shift began, Ker-Ramsey joined Floody underground to finish shoring up the storage chamber and the sand dispersal chamber. A third digger, Norman \"Conk\" Canton, was shoring the pumping chamber with bunk-bed timber. Suddenly, about twenty feet above the men, a bed-board in the shaft wall cracked and broke, then a frame broke and sand began cascading through the gap and down the shaft. Fortunately for the diggers, the ladder up the shaft hadn't broken, so Ker-Ramsey and Canton clambered upward through the spilling sand, with Floody bringing up the rear. In seconds Floody was up to his waist in sand and unable to move. The first two grabbed his arms and heaved him out before the sand smothered him.\n\n\"Much sputtering and shrieking on my part,\" Floody said, and he cursed for minutes afterward, but then realized his good fortune. He'd become \"a little greyer, a little wiser, and a bit more cautious.\"[24]\n\nIt took Floody, Ker-Ramsey, and the others four days to regain the ground they'd lost in the cave-in down the shaft of \"Dick.\" The same kind of setback occurred in the shaft of \"Harry\" within a few days. Moreover, digging and re-digging down the throats of \"Tom\" and \"Harry\" not only put tunnellers' lives in peril, it multiplied exponentially the problem facing the sand dispersal teams. German Staff Sergeant Glemnitz, the chief of the anti-escape guards, had warned all his troublemaking kriegies there was no way to hide the damp, bright yellow-coloured sand from beneath the surface of the compound unless they could find a way to \"destroy\" it.\n\nAt the outset, that apparently impossible piece of magic had fallen to Peter Fanshawe, head of the sand dispersal section. Engineers, such as Nelson, had calculated that each stretch of tunnel, roughly three-and-a-half-feet long by two-and-a-half-feet wide (or twenty-two-cubic-feet), yielded about a ton of sand. Initially, Fanshawe's dispersal men used pots and pans, jugs, washbasins, socks, and makeshift sandbags to dispense with sand in manageable quantities. They could hide it under huts, in latrines, behind wallboards, and even inside Red Cross parcel boxes. But with three tunnelling teams excavating in three sites from morning appell to evening appell, in total likely to generate maybe a hundred tons of sand, his dispersers were running out of places to hide it all. And each time Glemnitz and his team of ferrets found traces of the telltale yellow sand, they increased their spot searches, added unexpected roll calls, and ratcheted up the tension everywhere. The answer?\n\n\"Trouser bags,\"[25] Fanshawe said at a section meeting of X Organization.\n\nFrom beneath his tunic and pants, he pulled out a set of suspenders from which hung two cut-off legs of a pair of long, woolen underpants, looking like two sausage bags. (Suspenders and underpants had been sent by the score to POWs in Red Cross parcels in anticipation of the coming cold winter.) A pin was stuck into the lower end of each sausage. By placing the suspenders around his neck and sticking the sausage bags (full of sand) inside his trousers, he could pull a string concealed in his pants pocket. It would release the pin and allow the sand contents of the two sausage bags to flow down the inside of his trousers and onto the ground, where it could be scuffed into the ground relatively unnoticed.\n\nThe dispersal team members\u2014now numbering in the hundreds\u2014could fan out across the compound. They could march across the parade square and disperse sand as they conducted a drill. They could cultivate their gardens and hoe the evidence in among the plants. They could even stage fights and mix surface and sub-surface dirt in the resulting dust, and make both nearly disappear. Noting the way his dispersers tended to waddle with eight or ten pounds of sand down their legs, X Organization dubbed Fanshawe's men \"penguins.\" The trick was to find equilibrium, to ensure that, each day, the penguins could disperse roughly the same volume of sand the diggers excavated. Wally Floody noted the diggers were clawing out ten feet of sand per day, while Peter Fanshawe said the penguins could get rid of about six feet per day. One POW noted astutely that kriegies had become servants to an ugly hole in the ground.[26]\n\nJohn Colwell joined that servitude in June of 1943. Not that he had an aptitude for hauling and hiding sand. But Flying Officer Colwell had experience at just about everything else. The son of a medical missionary, John grew up in India through the 1930s, when his father, a veteran of the Great War in the British Army, decided to move the family to Canada's west coast to own and operate a chicken farm. Riding to and from Nanaimo, on Vancouver Island, to take school and sell the farm's eggs didn't appeal, so when the war began Colwell joined the RCAF, where his skills in math and geometry moved him quickly through the air training plan. By 1942 he was posted as a navigator to 405 Squadron and flying operations with Coastal Command, chasing U-boats and guiding rescuers to downed airmen. Grounded due to bad September weather and a broken engine radiator, Colwell was told replacement parts were a week away.\n\n\"No matter,\" a fellow airman wrote. \"John and his crew . . . scrounged a radiator and did the repairs themselves.\"[27]\n\nBy the spring of 1943, his squadron had joined the newly formed 6 Group of Canadian squadrons, with his 405 Squadron flying seven-man Halifax bombers against submarine pens on the coast of France and industrial targets along Germany's Ruhr River valley. On April 3, F\/O Colwell was navigating one of the lead bombers in the stream to attack Essen, Germany. They delivered their bombs, but anti-aircraft flak had damaged Colwell's astro compass; instead of a path northward away from heavily populated areas and Luftwaffe night-fighter bases, they were headed west and were shot down near the Maas River, southwest of Rotterdam.\n\nColwell's first entry in his POW diary was a simple one.\n\n\"Made my first parachute jump at 11 p.m.,\"[28] it read. Nine days later, Colwell noted that he and a couple of the officers in his Halifax crew, pilot Jim Lago and wireless-operator Bill Hoddinott, \"arrived in Sagan about 8 a.m. It was about noon before we were finally allowed into the compound . . . got our Red Cross parcels to start housekeeping . . . in Room 14, Block 120.\"[29] A few days after he and his roommates figured out who cooked, who cleaned, and who took care of retrieving Red Cross parcels, Colwell began to focus on the rhythm of the North Compound, its periodic escapes, its scheduled and unscheduled appells, and contributing to the escape committee's greatest needs.\n\n\"Two Americans got out under the south fence. . . . They searched the camp just after 1 a.m., and identified everyone by card and picture,\" Colwell wrote on June 17. \"We were then moved out into the hall while they searched the room carefully. They lost . . .\"[30] Three nights later, \"another night search. Jamie Jim Jamieson], Art [Hawtin], Ach [John Acheson], Mull [W. D. Mullins], and I had our hair all cut off.\"[[31] A day later, he simply noted, \"Night search. Started work as a penguin.\"[32]\n\nColwell learned about the penguin work shortly after arriving at the compound that spring of 1943. Since none of the officers was compelled to work, he enjoyed the notion of joining a friendly horseshoe match after appell one day.\n\n\"Suddenly, these two Dutch POWs came along and sort of scuffed around in the middle of our game,\" Colwell said. \"I remember thinking it wasn't very considerate of them. And then I saw this sand trickling out of their pant legs and I realized what was going on.\"[33]\n\nColwell's diary reflected the experiences of an officer newly introduced to Stalag Luft III and feeling his way into the routine of POW life. He sensed his first priority, self-preservation, was a matter of being accepted by his roommates, assuming a role in Block (or Hut) 120 life, and attending to his health and nutritional needs as best he could. The Allied doctors in the compound recommended that a grown, fairly active man needed three thousand calories in his diet each day; the German rations at best delivered fifteen hundred to nineteen hundred calories in the form of bread, some portions of vegetables, and even fewer of meat. Colwell noted early on that he weighed 148 pounds. His lower calorie intake and his penguin activity consumed those limited calories very quickly. For penguins, each circuit began at the tunnelling hut. One penguin loaded another's two inner pant legs with sand; at capacity that could mean columns of sand two feet long and three inches in diameter down each leg. Then, controllers sent the penguin on a casual walk to a specific dumping site. Controllers also directed penguins to various loading points and dumping sites so that the ferrets wouldn't spot a kriegie repeating a circuit. It was hot, continuous work that summer as Colwell and the other penguins tried to keep pace with the diggers.\n\nMoving sand underground had also become more sophisticated. As each of the three tunnels crept farther and farther from the base of its entry shaft, diggers abandoned their washbasins and jugs in favour of higher-volume transport. The engineers had scrounged the makings of an underground railway system. Willy Williams had retrieved battens that lined the ceilings and walls of the barracks huts; split lengthwise, the battens became one-inch-wide rails nailed to the floor of the tunnel. Bob Nelson and the other carpenters built sturdy trolleys\u2014consisting of chassis and detachable boxes built from beechwood bed-boards, axels liberated from barracks stoves, and wheels of wood covered in tin from discarded food containers. When a shift started, the first digger would wheel himself on the trolley chassis to the face of the tunnel. A second man reeled the empty trolley back with its draw rope and wheeled himself forward. The first digger, facing forward, cut into the face, drew the sand down his body to the second digger, who faced backwards and scooped the sand up his body and into the trolley box. When the box was full he tugged the rope and a third man back at the base of the shaft reeled in the trolley to remove the sand. All the while, a fourth man pushed the bellows in the air-pump chamber, delivering a steady flow of fresher air through the Klim can ventilation system under the railway tracks and floor boards to the face of the tunnel. Gordon King, who had worked as the dish man on the impromptu tunnel to the cookhouse a couple of months earlier, gladly took on the role of air-pump operator.\n\n\"I was small and in good shape,\" King said. \"I could stay on the bellows . . . for a shift of eight hours if I had to. We stopped from time to time to rest. Using the cart with the long rope attached, we could bring the sand out much faster. They even put carpet on top of the rails to deaden the sound.\"[34]\n\nMatching the digging pairs required a bit of psychology and physics. The section bosses matched John Weir with Hank Birkland, for example. First of all, they were both Canadian. More important, they had dug tunnels together since the first escape efforts at the Stalag Luft I camp back in 1941. But they also compensated each other's work in a unique way. Weir had a tendency to dig harder to the left when he was at the face of \"Tom,\" while Birkland often veered to the right; by the end of a shift one would balance the other. On other shifts, Floody worked at the face of the tunnel and Weir was his second, pulling out the bags of excavated sand. The bond these fellow diggers shared, going all the way back to Luft I at Barth, was paying off.\n\n\"One time, Wally was digging and I was passing the sand] back,\" Weir said. \"Suddenly the candle blew out and there was a helluva wind in my face. I knew damn well what it was. I just went forward as fast as I could and got [Wally] out in a matter of minutes.\"[[35]\n\nBirkland preferred to dig fully naked and that aggravated Floody; he was afraid the ferrets would notice the scrapes on Birkland's elbows and knees and start paying closer attention to his activities. To hide any telltale scars, tunnel boss Wally Floody insisted his diggers wear long underwear and vests. John Weir's fianc\u00e9e Frances McCormack might well have wondered why the flow of letters from Weir had slowed that spring; his digging shifts underground increased and lengthened. Further, she was probably puzzled as to why he seemed to be going through his underclothes so quickly.\n\n\"This is my forty-seventh letter and I've had a hundred and twenty-three from you so far,\" Weir began his letter of April 30, 1943. \"You wondered what clothes I need? I could do with some light pajamas. . . . Five pair underwear and shirts, sox sic] could be used. . . . I don't think another May will come with us separated.\"[[36]\n\nFear of cave-ins, vomiting from pockets of foul underground air, and elbows and knees scraped raw by the sand were just some of the occupational hazards the diggers tolerated. Fat lamps proved extremely helpful in guiding the tunnellers up to and back from the tunnel face, but they too were a half measure. To address the problem, some of the escape committee's engineers came up with a partial solution. They calculated that the wiring in the barracks huts probably had some slack, so clandestinely they stripped away sections of the wall and ceiling boards to track down any excess.[37] When they finished their wire roundup they had spliced together enough wire to bring electricity and lighting to the mouths of all three tunnels. The Germans regularly turned power to the huts off during the day, so for some of the evening tunnel shifts, the electric light was a psychological lift.\n\nMeantime, and suddenly, on June 10, 1943, the Russian work crews\u2014with their axes, picks, and debris wagons\u2014were back at work inside the wire, this time beyond the south fences of the North Compound. The then-SBO, H. M. Massey, learned from Colonel von Lindeiner that the upper command of the prison administration had decided the ever-increasing numbers of American officers flowing through the gates of Stalag Luft III would require a separate compound\u2014a new South Compound\u2014to house as many as four or five hundred USAAF POWs. The inner circle of X Organization met in an emergency session. Bushell and Day were feeling guilty that so many USAAF officers had contributed to the design, creation, and protection of the three tunnels, and it now seemed likely von Lindeiner would complete the South Compound before the kriegies finished any of the three North Compound tunnels. Big X recognized that a lot of American officers might feel cheated out of their chance at freedom. He felt obliged to find a solution.\n\n\"We close up 'Dick' and 'Harry' for now,\" Bushell said. \"We go full out on 'Tom.' . . . We might make it before South Compound is ready.\"[38]\n\nAt that point \"Tom\" had advanced some sixty feet from Hut 123. There were still at least a hundred feet to cover to get the face of the tunnel beyond the double fence and into the protection of the reforestation pines. Bushell asked Wally Floody how much they might advance the tunnel by September if all three digging shifts concentrated on \"Tom.\" Floody told him that digging advanced each tunnel about five to ten feet a day with one team, but simple arithmetic meant three digging teams might move \"Tom\" westward up to thirty feet a day. Fanshawe agreed that the pace would put pressure on the penguins, but it was manageable.\n\n\"It was decided that 'Tom' should be finished before the Americans went,\" Bob Nelson said. \"With the necessity for speed, however, greater risks were taken and the Germans' suspicions were aroused.\"[39]\n\nWith the priority shifting to \"Tom,\" digging and dispersing sand from the two other tunnels came to a stop. The Polish officers who had so expertly created the invisible trapdoors for \"Dick\" in the showers of Hut 122 and the rebuilt tile flooring at the entrance to \"Harry\" in Hut 104 returned to seal them up as if they'd never been there. Bushell and Floody then chose the fifteen best diggers, included all of the American diggers available, and divided them into three teams of five for the new push at the face of \"Tom.\" Wally Floody's predictions of ten feet per day proved accurate, and so did Fanshawe's promise to \"disappear\" the sand as quickly as it came to the surface.\n\nOut of urgency, or perhaps by accident, that summer a number of penguins stumbled on an ideal dumping ground for the now relentless flow of yellow sand gushing from \"Tom\" like Niagara Falls. And it was incredibly obvious. With few exceptions, neither the Russian workers nor the escape committee \"volunteer\" officers had disturbed much of the sub-surface yellow sand during the felling of the pine forest or the construction of the barracks huts and sports grounds. The one exception was the structure known as the fire pool, which had required a deep cut a dozen feet into the sand and then installation of a solid brick lining to catch and supply rainwater in the event of a fire. While much of the yellow sand excavated from the pool hole had been hauled away, around the edge of the structure some still lay on the surface. But the pool was right in the centre of the compound, under direct and constant observation by goons in the towers and by some with binoculars hiding in the woods spying on every move the kriegies made inside the wire. The pool perimeter suddenly became a focal point for Fanshawe's dispersers.\n\n\"Each penguin nonchalantly sauntered to the edge of the pool with his hands in his pockets,\" Bob Nelson said. \"On reaching the edge of the pool, he pulled the cords which opened the bottom of the containers inside his trousers and allowed the sand to trickle slowly out as he walked. . . . During the summer, many tons of sand were disposed of. Although the sand around the pool got gradually deeper, it was very imperceptible.\"[40]\n\nBeyond the edges of the pool, Fanshawe and his legion of penguins created numerous other scenarios for making the sand disappear. Whenever a pine stump was removed, exposing yellow subsoil on the surface of the compound, penguins congregated. If a new latrine were dug, penguins would wander by and contribute. One of the American air force officers in the North Compound, Jerry Sage, who had served as a paratrooper in North Africa, used the sports grounds to help the penguins keep up with the flow of excavated tunnel sand. One day, he would organize scores of his USAAF officers to kick up a curtain of dust by demonstrating techniques in unarmed combat. Any and all penguins were invited to participate or spectate; in the midst of the melee, none of the German guards would notice tunnel sand being emptied and mixed into the surface sand of the sports grounds. Other times, Sage staged volleyball matches with penguins two and three deep excreting sand as they jumped up and down cheering on their favourite team.\n\nThe day after the summit, \"Tom\" advanced ten feet and Fanshawe's penguins kept pace. The next day, Floody accidentally triggered another cave-in and had to be pulled out by his ankles before leading a repair of the tunnel frame; they only advanced eight feet that day. The following day it was the same thing, only this time hot margarine from a candle burned his leg. Floody seemed cursed. Nevertheless, by mid-June \"Tom\" was more than a hundred feet in length. At that point the diggers widened the tunnel to create a halfway house about ten feet long and half a foot wider than the tunnel itself. Floody believed the chamber, roughly beneath the compound fencing, would allow a changeover of crews and trolleys so there was less danger of a long tow rope getting tangled or rubbing against the tunnel shoring and loosening a wooden frame. The engineers calculated that another forty feet would put \"Tom\" beyond the double wire and under the woods. Then, at a section meeting to discuss timelines, digger Johnny Marshall suggested gradually sloping the tunnel to the surface. Floody considered that too risky; a rope could break or a trolley might run away on them and inflict a real setback. He was sure they could complete the tunnel before the Germans completed the South Compound and moved the US POWs into new huts there. Big X sided with Floody. It was July 3.\n\nShortly after dawn the next morning, when the Germans had unlocked the barracks doors, and just about the time POWs were beginning to assemble for appell, two men appeared on the sports grounds in a costume\u2014the front and hind quarters of a horse. On the horse's back was USAAF officer Jerry Sage portraying Paul Revere with a tricorner hat and knee breeches made of woolen underpants.\n\n\"The British are coming!\"[41] he shouted as he brought the two-man horse to a halt.\n\nAt first startled, Hauptmann Hans Pieber went along with the prank and commenced the roll call, whereupon the American inside the hind quarters poured out the contents of a can of water; the horse was urinating during Pieber's appell.\n\n\"Zwei und achtzig . . . und ein pferd,\" he shouted to the recorder, noting eighty-two prisoners and one horse.\n\nBut they weren't done, as John Colwell noted. \"Yankee Doodle and his horse . . . then paraded through the huts with a band.\"[42]\n\nDrummers and buglers came marching through the barracks with about forty other US POWs masquerading as native tribesmen on the warpath. While the horse and tribesmen wreaked havoc on the roll call, Paul Revere, a.k.a. Jerry Sage, and George Harsh threw Roger Bushell out of bed, sat on him, and offered him a sample of their latest creation. The Americans had scrimped and accumulated their rations of sugar and dried fruit from Red Cross parcels for weeks and distilled them into a raisin-flavoured wine. The party continued all day, ending with many of the officers, including Wings Day, being tossed into the fire pool during a water fight. It was Independence Day and the American kriegies weren't going to let it pass without notice.\n\n\"The Goons are afraid of something,\" Colwell wrote in his diary a few days later. \"We are having four appells per day and they've been around the last four nights between 1 and 4 a.m.\"[43] Two days later, he recorded, \"Last night, the Germans] were in the hut several hours. Each room was turned out in turn and searched.\"[[44] Every other day for the rest of the month, Colwell reported searches where \"there were dozens of guards and barbed wire used.\"[45]\n\nDuring one of those staged volleyball matches with penguins gathered around to empty their hidden sand pouches, Glemnitz came prowling. Bushell and the rest of the escape committee realized he was on to something. Security boss Harsh added he'd seen German guards in the towers all equipped with binoculars. Digger Hank Birkland reported seeing a ferret walking in the pine forest beyond the fences; the German didn't come out. So Harsh and Junior Clark took a closer look and realized the Germans had strategically placed blinds in the woods\u2014piles of brush through which they could use field-glasses to spy on the kriegies without the prisoners realizing they were under surveillance. This sparked a series of after-dark visits from Hans Pieber. The ferret barged into Hut 101, throwing everybody out of bed with shouts of, \"Aus! Aus!\"\n\nIt prompted digger Ker-Ramsey to tell Pieber he was wasting his time.\n\n\"You think I know fuck nothing,\" Pieber defended himself in a broken English malapropism, \"but I actually know fuck all!\"[46]\n\nThe war of wits continued. Glemnitz resumed the snap searches. He had narrowed his focus to the western perimeter of the compound, with particular interest in Hut 123. Each time the Germans raided, the escape committee resealed the trapdoor at the entrance to \"Tom,\" waited for the searchers to retreat, and then re-opened the trap and resumed the digging and sand dispersal. Next, Glemnitz arrived with Hauptmann Broili, a member of the Abwehr counter-intelligence, and about forty soldiers. Following another hut search, he ordered the troops to dig a trench between Hut 123 and the outer double fencing. They went down four feet, where the ferrets used five-foot steel rods to probe even deeper. By that time, \"Tom,\" which was twenty feet below the probing ferrets, had come within about fifty feet of what the escape committee figured was the safety of the pine forest. Floody constructed another halfway house and prepared to dig the last horizontal section and finally the vertical shaft to the exit in the woods. With Fanshawe's penguins under close surveillance from the forest and towers, he suggested redirecting the sand dispersal teams to Hut 122, where Mike Casey and Ker-Ramsey began filling \"Dick\" with the sand from \"Tom.\" The pace picked up; \"Tom\" was now two hundred feet long and, Floody figured, within striking distance of the woods.\n\nThe ferret chief's next move was uncanny. It was almost as if he knew \"Tom\" was beneath his feet. Suddenly there were workers felling pine trees all along the pine forest directly in front of where \"Tom\" was heading. The tunnel would have to go another hundred feet to find safety under the trees. But the penguins' secret weapon, tunnel \"Dick,\" was running out of room; sand had now filled \"Dick's\" horizontal space, and Bushell refused to let Fanshawe's crew fill the vertical shaft. The next sand dispersal area was dangerously obvious\u2014all the empty Red Cross parcel boxes lying about in the huts\u2014but they would try it anyway. Five days later, August 21, penguin Colwell noted the inevitable.\n\n\"The Goons found over a hundred and fifty Red Cross boxes of sand in Hut 101 ,\"[47] he wrote. And at the end of August, \"Long morning appell. They searched all the huts and took hundreds of Red Cross boxes. Also my soldering outfit.\"[48]\n\nFor safety's sake, the Poles sealed the trap at the entrance to \"Tom\" once again. The tunnel was 260 feet long, but forty feet short of the recently extended no man's land to the woods. Still, the escape committee decided that was far enough. Floody would build the vertical shaft to the surface; while potentially out in the open, \"Tom's\" exit might well be far enough away from searchlights to allow men to escape at night. Since April, the hundreds of kriegies had excavated, hauled, hidden, and disguised the dispersal of more than seventy tons of sand from \"Tom\" alone. They had built an underground railway and ventilation system, not to mention a security and intelligence system. And while they hadn't delivered the mass escape they'd planned, they had at least escaped detection for more than twenty-one weeks.\n\nRight after appell on September 8, as Wally Floody prepared to send his latest digging crew into \"Tom\" again to finish excavating the vertical shaft to the surface, Glemnitz suddenly threw up a wall of guards around Hut 123 and led a team of ferrets into the barracks. A couple of hours later, the German staff sergeant jabbed a probe into the concrete floor around the hut's chimney and discovered a chip in the concrete that revealed the trapdoor to the tunnel. Glemnitz beamed with pride at his victory. He had thwarted the largest escape attempt the German prison camp system had ever faced. Other German officials saw the discovery of a 260-foot-long escape tunnel at the heart of the inescapable Luft III quite differently.\n\n\"They were amazed, appalled, and at the same time very cocky about their discovery,\" Bob Nelson wrote. \"Nothing like it had been seen. And high officials of the Luftwaffe and Wehrmacht came into the camp to inspect it.\"[49]\n\nDuring their inspection, the Germans evacuated the hut and placed a guard at its entrance around the clock. Having unearthed \"Tom,\" it appeared that neither the Luftwaffe Kommandant nor the Gestapo chief knew what to do next. So on September 16, they turned \"Tom's\" fate over to a demolition team.\n\n\"The German sappers entrusted with the job . . . did it] very effectively,\" Nelson continued, \"but at the same time blew a hole through the roof of the hut above the late 'Tom's' trap.\"[[50]\n\n\"Tom\" (if not Big X) had had the last laugh.\n\nThat same week in September 1943, the German police and guards completed the move of American POWs to the South Compound. Without missing a beat, X Organization held its next meeting, with Bushell already planning its next move. Anticipating further Gestapo searches, the escape committee ordered the collection of several thousand more bed-boards and hid them in the still-empty vertical shaft of tunnel \"Dick.\" For the time being, work was suspended on \"Dick\" and \"Harry.\" In Hut 120, John Colwell, despite losing some of his tools and soldering gear in the purges, began work on a cuckoo clock that would amuse his roommates as well as hide his diary.[*] And while he had worked diligently as Big X's personal Danish language tutor, Sorry Sorensen was stricken with appendicitis that fall and wound up on a dining table used for emergency surgeries; there, without benefit of anesthesia or sterilized surgical tools, a Russian doctor (also a POW at Stalag Luft III) performed the appendectomy.\n\n\"A dog circled the table and] was rewarded with the appendix,\"[[51] Sorensen related later. Still, this brush with death didn't seem to blunt the twenty-one-year-old Canadian's drive to escape. Writing home to his parents that fall, he said he would refrain from betting on when the war would end, the way his roomies were, but concluded, \"I don't intend being here the winter after this.\"[52]\n\n* * *\n\n* In his 2011 article, \"From Roskilde Cathedral School to Stalag Luft III\u2014And the Great Escape,\" Danish writer Mikkel Plannthin points out that Roger Bushell learned Danish from Frank Sorensen and Arne B\u00f8ge, who'd both been born and schooled in Denmark.\n\n* John Colwell not only hid his diary and his tin-bashing tools from any German ferrets rummaging through barracks huts at Stalag Luft III, he also hid messages in his writing. He regularly wrote letters home to his mother, Fern, in Hindi\u2014a language in which they were both fluent, but which German censors couldn't decipher.\n\n# 6\n\n\"SHYSTERS AND CROOKS AND CON MEN\"\n\n* * *\n\nEACH OF THEM had plenty of reasons why they shouldn't fall in love, much less get married. Joan Saunders already had a boyfriend, although he seemed noncommittal. Her family had suffered loss in this war; her cousin and his wife had died in the Luftwaffe attack on Coventry in 1940. She had a good wartime job as a bookkeeper at Lougheed, Berg, and Beck's, in her hometown of Leamington Spa, about ten miles from Coventry. The British company manufactured parts for the Royal Navy, including the hydraulics for motor torpedo boats. Like so many things in the UK then, however, the job was temporary. The plant could be hit by German bombing attacks and put her out of work. The contracts from the Royal Navy wouldn't necessarily last. She might be called up to enlist in the armed forces. Or, she might only be able to work there until a man needed the job.\n\n\"I was an only child,\"[1] she said, and that meant whatever income she could provide her family was minimal, but important.\n\nGeorge Sweanor was an RCAF airman at an operational training unit (OTU) in Britain when he met Joan. Born in 1919 in Sudbury, Ontario, George had grown up through the Great Depression\u2014the eldest of three children\u2014in Port Hope. He'd idolized his uncles, both veterans of the Great War, but read anti-war literature and even wrote an essay entitled, \"Who Wants War?\"[2] In spite of that, George was fascinated by aviation, periodically coaxing his father to take him down the road to Trenton, Ontario, to see training aircraft airborne. After high school, he took a job with the bank and volunteered for the militia. By 1941 he had enlisted in the RCAF. By the winter of 1942, he'd graduated as a sergeant-observer. And by that summer, he had crossed the Atlantic and been posted to train as a bomb-aimer at the Wellesbourne OTU. On the training course, he got to know a fellow Canadian, Pilot Officer Pat Porter from British Columbia. They would later crew up for combat operations, but that summer they flew training ops and spent time on leave together.\n\nNeither Porter nor Sweanor smoked or drank. They generally avoided the pubs, preferring the dances put on by the towns adjacent to their OTU station. They felt guilty sometimes, since so many local young British men were away fighting in North Africa or at sea with the Royal Navy and merchant navy. The dances usually featured a four-piece band playing soft music, which made the atmosphere conducive to having a conversation. Most military men and local women went stag. And the drill was to select a partner (often a wallflower), dance three numbers with her, escort her back to her seat, then select a different partner.\n\n\"If there was one partner you'd like to know better,\" Sweanor said, \"you would try to get her in the home waltz, which permitted you to ask if you could walk her home.\"[3]\n\nThis particular night, Sweanor and Porter arrived by bus at Leamington Spa in search of the main dance hall. As it turned out, a sign on the town hall announced a special dance to raise funds for Red Cross food parcels to be sent to prisoners of war. Admission was more expensive than the regular dance, but the two Canadians wanted to help POWs. Coincidentally, local resident Joan Saunders, age twenty-three, and her girlfriend decided to support the dance fundraiser too. Joan was already seated when the two airmen arrived. On his way into the hall, Sweanor nudged Porter. \"That's for me,\" he said, pointing to a potential dance partner who had \"looks, poise, and figure.\"[4] And for the rest of the night, he plotted to make sure he got Joan Saunders for the home waltz.\n\n\"Everybody knew it was absolutely stupid to get married during the war,\" Sweanor said, \"because if you survived, you'd probably be minus a leg or an arm or an eye, and be a burden to your wife the rest of your life. On the other hand, life is so temporary.\"[5]\n\nEventually, however, they each realized they had little control over their feelings, and that the war would affect both of them whether they courted or not. In fact, shortly after the couple met, Joan's office was strafed by a Luftwaffe fighter pilot shooting up whatever came into his gun sights.\n\n\"His bullets came right through the window, right through my office and my boss's office,\" Joan said. \"It would have gone right through my head, if I'd been sitting there. But it was six in the morning and I always got to the office at eight.\"[6]\n\nMeanwhile, George realized how close he was coming to the death that the war dispensed daily. Early in October of 1942, Bomber Command was directing Wellington bombers from his 419 Squadron against German targets for the first time since the squadron had arrived at its Yorkshire operational stations. On this night Sweanor's crew was not on ops. Dressed in his best blues to go to a dance, he stopped to speak to F\/O Arthur Morlidge, who was going on the bombing raid.\n\n\"Can I be of any help?\" Sweanor asked.\n\nMorlidge pointed to a letter on his dresser and said, \"If I decide to stay over there, would you post that letter for me?\"[7]\n\nThe letter was addressed to his parents in Lloydminster, Saskatchewan. As readily as he'd agreed to Morlidge's request, Sweanor admitted to an uneasy feeling as he watched his squadron mates leave the Nissen hut for the flight line and the bombing attack to Krefeld that night. The next morning Sweanor saw the padre packing Morlidge's belongings. Bomber Command had lost nine aircraft that night, the one from 419 Squadron had twenty-year-old observer Arthur Morlidge aboard. Sweanor took the letter for posting.\n\nJoan Saunders and George Sweanor were married just after New Year's, on January 6, 1943. As with most events in the UK at that time, the actual wedding happened when the war dictated. At the time, 419 Squadron (now flying brand new Halifax bombers) was standing down, waiting for what Sweanor called \"delightfully foul weather\"[8] to clear before it resumed flying. His squadron leader, D. W. S. Clark, loaned Sweanor a bicycle to get to and from Middleton St. George, where the couple had rented a single room. His wing commander volunteered his own crew to fly in place of Sweanor's to give the couple two free nights away from the war as a wedding present. His aircrew mates bought the couple a chest of silverware. Air force friend and artist Ley Kenyon designed a wedding card. His pilot, Pat Porter, stood up as best man. And as the couple was married, George Sweanor got a promotion to Flying Officer. Three nights later his Halifax \"K for Kitty\" was back in the battle order, dropping mines in German shipping lanes off the East Frisian Islands. But the inclement weather persisted, limiting his combat flights to three in January. They made up for the operations scrubbed in January with seven ops in February. Then he was away on a four-week bombing leaders' course, then back on bombing operations against submarine pens at St. Nazaire and German industrial targets at Duisburg and, on March 27, Berlin.\n\nSweanor counted the op to Berlin that Saturday night among numerous \"wrong decisions\" in his life. In the first place, although the operation didn't need their services, pilot Pat Porter volunteered his crew to replace one with lesser experience. Although they had flown most operations in Halifax \"K for Kitty,\" she was in for repairs, so they settled for an unfamiliar bomber \"E for Edward.\" Sweanor was sick with stomach flu and could have opted out. As bomb-aimer, however, he was the most mobile member of the crew\u2014serving as second pilot on takeoff and landing, manning the front turret, taking astro shots for the navigator, changing fuel tanks for the engineer, and ultimately lining up and bombing the target. Sweanor didn't want to let his mates down. A decision not to go would have changed his life. Instead, he joined the op.\n\nEarly in their outbound flight southeast of Bremen, their bomber was hit by flak, knocking out one of the Halifax bomber's four engines. Unperturbed, Porter felt that they should press on to the target, despite the danger of losing altitude on three engines and perhaps becoming a target themselves. Sweanor released the bombs at a target near Berlin, muttering \"my usual, useless prayer that my bombs hit only military targets,\"[9] and Porter banked for home. But they'd been sighted by an airborne night fighter that quickly caught up with them and raked their aircraft from behind. The attack set fire to the wireless radio area, and shrapnel wounded several of the crew. With the Halifax losing altitude, hydraulic oil fires breaking out, and wounded aircrew, Porter ordered his six crew members to abandon the aircraft. Sweanor went to the cockpit.\n\n\"I don't know if I've got any engines left,\" Porter said, working the throttles.\n\n\"Look at the altimeter,\" Sweanor said. It was unwinding at a terrific pace. \"We're in a plunge. We've got to get out of here.\"[10] He made his way to the forward escape hatch so the crew could bail out, but the hatch had been fused shut by the enemy cannon fire. The two gunners aboard reported on the intercom that the rear hatch was fused too, and frozen solid. Sweanor made one more stop in the cockpit to give the pilot his parachute and to open the hatch over his head so Porter could bailout. \"Good luck, Pat,\" he shouted, and then, with an axe in his hands, Sweanor made his way to the rear hatch to help the rest of the crew get out.\n\nRear gunner Scottie Taylor, who happened to have been a lumberjack in Quebec before the war, grabbed the axe from Sweanor's hands and hacked at the hatch until it fell open. One by one the crew evacuated the Halifax\u2014Murray Bishop, the flight engineer; Gerry Lanteigne, the wireless air gunner (he hesitated and Taylor pushed him out); Alan Budinger, the navigator; Danny London, the mid-upper gunner; and finally Sweanor, who paused for a second.\n\n\"I knew I was leaving my last ticket home,\"[11] he said. One further thought haunted him this night: Joan was pregnant with their first child, and Sweanor revisited the reasons why they shouldn't have gone ahead with a wartime marriage. Would he be maimed? Would he a burden to his wife, if he got back? And if he didn't return, how would his widow and fatherless child manage?\n\nAs he fell from the hatch into the night, Sweanor felt a sheet of flame pass over his head. He hadn't realized how much of the Halifax was ablaze. He pulled his rip cord and immediately felt the jerk of the chute; it was barely open when there was a second jerk as the chute caught a tree. Had he delayed just a few seconds longer, he quickly realized, he would have ploughed into the ground with the burning bomber; the chute catching the tree as rapidly as it did probably saved his life. Back in Middleton St. George, in the early hours of March 28, Joan was awake.\n\n\"He used to come in after a bombing operation] about four o'clock in the morning,\" she said. \"I used to reach over and there he was. But this morning I woke up and it was daylight and there was nobody there. And I thought, 'Oh God, he's not coming home.'\"[[12]\n\nThe 419 Squadron padre notified Joan later that day that her husband had, in fact, been shot down. A few days later, the wing commander wrote a letter encouraging her to hope he was a POW. The International Red Cross confirmed that to be true on April 19, 1943. By that time, however, Sweanor had gone from downed airman on the run, to police prisoner, to the subject of repeated Luftwaffe interrogations, and, finally, to prisoner of war and inaugurated kriegie in the newly opened North Compound of Stalag Luft III. During that transition he learned that crew mates\u2014Bish, Bud, Gerry, Danny, and Scottie\u2014had all survived the crash with contusions and shrapnel wounds, but that his pilot, Pat Porter, who had apparently kept the Halifax airborne long enough for his crewmates to chop their way through the rear hatch and bail out, had died in the burning wreckage.\n\nLater, Sweanor learned that forty-three bombers had been shot down in two nights of Bomber Command attacks against Berlin, and that of 301 aircrew aboard those bombers, only fifteen men had survived to be interrogated at Dulag Luft\u2014six of them from his Halifax \"E for Edward.\"[13] Sweanor considered himself fortunate to be among the 5 per cent who survived those end-of-March raids.[*] He felt well treated by the civilians who'd captured him, sufficiently respected by the police and Luftwaffe officials who'd questioned him, relieved upon his arrival in Sagan that German civilians hadn't lynched him, and grateful that his captors promised there would be Red Cross food once he arrived inside the compound.\n\n\"Freed from operations, we could cease contemplating how few hours we had yet to live,\" Sweanor said. \"Our taut nerves could relax.\"[14]\n\nIn spite of the relative ease he felt as he was about to enter the prison camp, Sweanor learned that relief was ephemeral. There was commotion at the gate as a prisoner was caught trying to escape disguised as a Soviet worker. A dozen German guards marched the new arrivals into a large shack for a routine naked search. Sweanor remembered he still had some escape aids\u2014a cloth map of Europe, currency, hunting knife, compass, German-English dictionary[15] \u2014in his pockets; he handed away his jacket to distract the guard and palmed the escape aids as he removed the rest of his clothes. When the guard returned his jacket, Sweanor dumped it on the floor with the escape aids underneath. The final indignities during his introduction to Stalag Luft III were a search of body orifices, a head-and-shoulder mug shot, and fingerprinting. Passing through the main gate into the compound for the first time, he realized the might of the place\u2014its double-fence outer barrier, guard towers, guard dogs, searchlights, and rifle-toting guards. The often-repeated words of Welsh kriegie Shag Rees felt more haunting than humorous.\n\n\"So this is where we are going to lose the best years of our wives!\"[16] Rees reminded his fellow POWs.\n\nIn his first permitted letter home to his wife, George Sweanor wrote a tribute to his Halifax pilot, Pat Porter, who \"had not left the aircraft] via the top hatch as I had thought, but had deliberately sacrificed his life to save ours. I felt deep guilt being alive. . . . Would I ever be worthy of his sacrifice?\"[[17] Sweanor and others thought Porter's heroism warranted his receiving the Victoria Cross posthumously, but since so many pilots had died similarly in an air offensive experiencing such high casualty rates, Porter received recognition only by Mentioned in Dispatches. Tangible evidence that his first letter got to Joan arrived at the prison camp about a month later. One day, Sweanor spotted a familiar face coming through the main gate of the North Compound. He waited patiently while the outside guards processed the new arrival and the inside kriegie interrogators passed him. Then he pounced on his fellow 419 Squadron crewmate, Jack Fry. The two men exchanged the latest 419 news. Fry explained that he had been shot down over Stettin on April 20; despite being fluent with his high school German, he'd been unable to get through German checkpoints. Fry assured Sweanor that Joan had received his first letter home as proof he was alive.\n\nSeveral weeks passed before Sweanor learned about the escape activities that abounded in the compound. His initial contributions were the few bits of escape kit he'd managed to smuggle past the search guards and into the compound. He joined the outdoor gardening units, mixing regular surface soil with the yellow tunnel sand that passing penguins deposited in their vegetable patches. But Sweanor saved some of his best work for the camp-wide security network, serving X Organization as a stooge. He worked in shifts spying on the whereabouts of ferrets, the Abwehr men in their blue overalls. Stationed outside at the northeast corner of Hut 101, with a full view of the main gate into the compound, Sweanor would look, for all the world, as if he were just reading a book or playing a game of chess with another kriegie. In fact, he was the \"duty pilot,\" logging every German who entered or exited the compound via the main gate, especially the chief ferret Hermann Glemnitz and his Unteroffizier (Corporal) Karl Griese. Because of the nature of his appearance\u2014he had a conspicuously long neck\u2014the kriegies nicknamed Griese \"Rubberneck.\"\n\nOne day, the Germans appeared to have vacated the compound, but Sweanor's log showed Rubberneck still somewhere inside the prison camp. The Commonwealth officers working as security stooges narrowed their search to the cookhouse, which they discovered was locked from the inside. They responded by closing the shutters, so anyone inside the cookhouse could no longer see out. Within an hour, Rubberneck emerged embarrassed and fuming. It wasn't long before the duty pilot position at the main gate was openly accepted by both sides. Even by Glemnitz. One afternoon, the Oberfeldwebel himself came through the gate, walked up to Sweanor, saluted, and shouted, \"I am in. Mark me down.\"[18]\n\nSweanor marked Glemnitz in.\n\n\"Who else is in?\" Glemnitz asked and took Sweanor's logbook. It showed no other ferrets in the compound. Further, it showed Rubberneck and another guard had just left the compound. Glemnitz was not happy. \"Book me out!\" he bellowed.\n\nSweanor learned later that the two ferrets had left their posts too early. Glemnitz punished them\u2014one got four days in the cooler, the other no leave for two weeks. The stooges were beating the ferrets at their own game\u2014surveillance. In fact, in early September 1943, when tunnel \"Tom\" was discovered, X Organization bounced back even when the Gestapo briefly took over the compound. The secret police, who descended on Hut 123 and its occupants, were accustomed to pushing around civilians, not experienced air force officers. At the North Compound, when the Gestapo men set about searching an area, they armed themselves to the teeth, surrounded the offending area, and then ransacked it, placing anything they fancied\u2014pens, pencils, rope, and anything resembling a tool\u2014in a \"swag bag\" off to one side. As soon as they put the swag bag down, the kriegies created diversions and began pilfering the contents of the bag back. The net gain invariably favoured the prisoners, not the Gestapo.\n\n\"To any but the Gestapo,\" Bob Nelson wrote, \"it was a well-known fact that if a workman was sent into a prisoners' compound, two guards also had to be sent to watch the workman's tools.\"[19]\n\nA perfect illustration of Nelson's assessment occurred about the same time as the Italian army in Sicily capitulated to the Allies in the summer of 1943. Colonel von Lindeiner, sensing a need to compensate for the Mediterranean setback and ramp up German morale, sent in a work crew to begin stringing electrical cable for a loudspeaker to broadcast German radio propaganda into the prison camp. The escape committee noted the workers in their midst and organized a diversion to attempt to steal some of the electrical cable. They were too late. Coincidentally, Canadian officer Red Noble had just completed a term in the cooler, so German guards were releasing him back into the North Compound. In strolled the redheaded Noble, carrying his blanket over one arm. He spotted a spool of wire lying unattended on the ground. Going just a few steps out of his way, but not missing a beat, Noble snatched the wire, tucked it under his blanket, and disappeared into a hut, leaving the workmen flummoxed as to where they'd mislaid the wire. Noble had pilfered more than eight hundred feet of wire, which immediately went into the vertical shaft of tunnel \"Dick\" for storage and later provided power for the lighting system inside tunnel \"Harry.\"[20]\n\nThe impromptu Gestapo inspection of the North Compound that autumn may well have appeared to be a bumbling affair, with the kriegies quickly reversing every measure the secret police instituted or recapturing any contraband their jailers seized. As Nelson observed, the Gestapo did not appear to understand the nature of a Luftwaffe prison nor the means of containing its inmates; he ridiculed the Gestapo \"searches as] very ineffective compared to those of the experienced Luftwaffe ferrets.\"[[21] On the other hand, allowing the kriegies to believe they had the upper hand inside the wire may have been part of a greater Gestapo game plan.\n\nIn November 1943, Max Wielen, the head of the Kriminalpolizei (Kripo) at Breslau visited von Lindeiner at the North Compound and toured the site. A system of buried microphones, which the Germans had installed a year earlier, conclusively revealed that throughout the previous six months \"large-scale digging was being carried out by the prisoners.\"[22] However, Wielen's search of huts and other buildings, according to von Lindeiner, \"failed to discover where the tunnels were situated, or at least left that impression in the camp.\"[23] The Kripo chief's visit to Stalag Luft III was followed by renovations to several of the compound facilities, including construction to enlarge the camp area. Not surprisingly, during such disruptive renovation (and perhaps not coincidentally as escape activity began to intensify), Luftwaffe administrators were forced to remove the microphone listening equipment. Still more remarkable was that the microphones remained disconnected through the fall and into the new year.\n\nThen, early in 1944, von Lindeiner convened a meeting of district security personnel to discuss ways to prevent camp breakouts. He brought in local security chief, SS Major Erich Br\u00fcnner, who listened to von Lindeiner's concerns that a large-scale escape was imminent. However, Br\u00fcnner stayed at the meeting barely an hour. He merely chatted with von Lindeiner about the problem, carried out no inspections of anti-escape measures, and refused to order the redeployment of the underground microphone listening system. Dissatisfied with the security chief's response, von Lindeiner went so far as to arrange a meeting among Group Captain Massey, the senior Commonwealth officers, chaplains, and doctors and delivered a special warning to be passed to the prisoners not to undertake a major escape plan.\n\n\"Escapers will] in future suffer very severe penalties,\"[[24] Kommandant von Lindeiner told one of the RAF officers.\n\nThis pre-emptive approach was reinforced when the camp adjutant, Hauptmann Hans Pieber, spoke to F\/L Henry \"Johnny\" Marshall early in February. Pieber liked Marshall and hinted that the RAF reconnaissance pilot and his fellow POWs might face horrible consequences from higher up the chain of command if they attempted a mass breakout. If they did, he suggested, the Gestapo might respond with lethal force. Pieber's warning to Marshall, as well as Br\u00fcnner's failure to re-activate the anti-escape defences, and von Lindeiner's veiled warning against mass escapes all suggest that German High Command had already decided it would turn a blind eye to or even encourage a larger escape. And if such a breakout occurred, that would prompt the Gestapo to take matters into its own hands. The Gestapo, it appeared, was setting a trap for both the Commonwealth prisoners and the camp's Luftwaffe administrators. What made the secret police involvement a greater threat to those involved in an attempted escape was that the German High Command was drafting a restraining order called \"Stufe R\u00f6misch III,\" or Grade III, which decreed that any recaptured escaped POW\u2014whether he escaped in transit, via a mass escape, or on his own\u2014would not be returned to his military jailers, but instead be handed over to the secret police.[*]\n\nInside the wire, with the Gestapo seemingly out of the picture and von Lindeiner and the Luftwaffe guards apparently back in control of Stalag Luft III during the autumn of 1943, X Organization's tunnel activity went on hiatus. While the diggers and penguins enjoyed a well-deserved rest, the pause did not signal a decline in escape committee activity. To be sure, Big X entertained one-shot \"wire jobs.\" Roger Bushell welcomed kriegies presenting reasonably sound escape plans and travel arrangements that might deliver them out of Germany. To assist, the escape committee would supply potential escapers with some currency, compasses, civilian outfits, and wire cutters. Among the attempts, a New Zealander and a Canadian thought they had found another blind spot at the far south end of the sports grounds. On the designated night, they crawled for seven hours, undetected by searchlights or tower guards, and had all but two wires of the outer fence snipped when a patrolling guard spotted and recaptured them. Meanwhile, the escape committee ramped up production in the mapmaking, tailoring, and forgery sections.\n\nIf Bushell's grand plan to have hundreds of Allied POWs escape and fan out across German-occupied Europe was going to succeed, he needed assurance that his escapers would have the appropriate documentation.[25] Through his own experiences and listening to tales from fellow officers who'd also tried to get back to Britain, Big X knew his escapers would need at least a light grey identity card (Kennkarte) or, better, a visa (Sichtvermerk), plus a pass (Ausweise), and likely a brown card (Dienstausweise) legally allowing the holder to be on Wehrmacht property. In addition, if a man were disguised as a foreign worker, he would require Polizeitliche Bescheinigung, a police permit authorizing the worker to be in a specific area; Urlaubsscheine, a yellow paper entitling the worker to be on leave to get there; or R\u00fcckkehrscheine, a pink-coloured form that signified the worker was legally en route to his home country.\n\nFor Dean and Dawson, the escape committee's forgery section, obtaining accurate samples of these documents amounted to only half the challenge. Equally daunting, the forgers had to procure tools and materials\u2014pens, inks, brushes, and paper stock\u2014to generate the fakes. Initially, Tim Walenn's forgers worked in an empty room in Hut 120, in the row of huts farthest from the main gate. His team of artists, mapmakers, printers, and even carvers spent the better part of a year assembling master documents, finding the ink and paper with which to duplicate them, and then painstakingly replicating. They even built a mimeograph and used razor blades to carve designs into a boot heel for recreating official Nazi stamps with the swastika and eagle emblems. In Hut 120 there were relatively secluded windows, guarded by stooges, where forgers could practise replicating the documents in bright daylight.\n\nIn one respect, one member of the Dean and Dawson document factory owed the quality of his forgery to Canada's Group of Seven artists. Born and raised in Toronto, Robert Buckham had been attracted to sketching and painting as a young man. In the 1930s, he signed up for art classes at the Art Gallery of Toronto and attended Saturday lectures from established artists such as Arthur Lismer of the Group of Seven.[26] Before the war he worked in advertising as an art director. When the war interrupted his career, the six-foot-four Buckham was twice refused by the RCAF for his height; but with the war not going well for Britain in 1941, he was finally accepted. He'd flown ten combat operations with 428 Squadron when his Wellington bomber was shot down by a night fighter in April 1943. At Stalag Luft III, X Organization forgery section leaders Tony Pengelly and Tim Walenn were delighted to learn Buckham was an artist.\n\n\"I said I was going to major in art there,\" Buckham said, \"and I did.\"[27]\n\nHis steady hand with pen and ink as well as his penchant for colour painting made Buckham an ideal candidate for the reproduction of the Kennkarte and Ausweise passes. Using the best quality paper they could find in the compound\u2014often the fly-leaves of the library Bibles[28] \u2014Buckham and the others cut the paper to the exact dimensions of the pass. They replicated the background swirl of the master document's watermark (much like currency notes) with pen and ink, and brush and watercolour, then penned in the category headings (first name, surname, date, and place of birth, et cetera). If the document needed a hard back, the forgers glued tracing linen over cardboard and coloured it to match the original. Meanwhile, Jens Muller, a Norwegian air force officer, replicated any official stamps\u2014including date or swastika insignia\u2014by carving a mould in a bar of soap. Once he'd begun sketching and painting for the escape committee, Buckham never stopped. He employed his artistic skills \"from morning to night\"[29] to paint posters advertising theatre and lecture events. And in his bound, Red Cross\u2013issue diary he drew images of kriegies, their tools, and their surroundings to distract himself from \"an empty belly.\"[30]\n\nWhen the forgery room drew guards' suspicions, Walenn moved the crew to a kitchen next to where the compound orchestra rehearsed. When the orchestra finished practising, the forgers wrapped up their work too and stowed it in a violin case. The musicians left the hut with their instruments in hand; one carried his violin case,[31] with the forgeries-in-progress enclosed, to Hut 104, where Tony Pengelly hid them in a secret compartment behind a removable wall board.\n\nAs well as serving the escape committee as custodian of the forged documents, Canadian bomber pilot Pengelly took on the responsibility of coaxing original documents and other things of value from the German guards who regularly patrolled the compound. Some of the guards weren't necessarily highly educated men, but Pengelly had learned that he could occasionally befriend one by offering him a cigarette and even a short visit to his barracks room for a cup of freshly steeped tea. Pengelly did regular shifts dispensing Red Cross parcels to the kriegies and had been given the authority to \"borrow\" such luxury items from the parcels to hasten the process. During such visits, Pengelly might share snapshots of his family. A month or two later, Pengelly would invite in the same man and repeat the pleasantries. When the Canadian pilot knew his prey might soon be going on leave, he would offer the bait-covered hook.\n\n\"How would you like to take some coffee home?\"[32] he would ask the guard.\n\nThe German usually jumped at the chance.\n\nPengelly knew the average German hadn't tasted real coffee since 1936; nor had he enjoyed other delicacies. \"And some chocolate for your little boy?\"\n\nOften the gift might draw the offer of a favour from the guard, such as, \"Can I bring you anything from outside?\"\n\n\"Yes, if you wouldn't mind,\" Pengelly would say. \"I'd like a hundred toothpicks.\" Something that inconsequential would be sufficient the first time, but it was oil for the machine. With each trip the guard made, Pengelly repeated the exchange until it got to be habitual\u2014taking a little booty home each leave. And having broken the rules once or twice, the \"tame\" guard wouldn't likely refuse any of Pengelly's requests, fearing the POWs might expose him. Coffee and chocolate yielded a camera, developing, and printing equipment, and even the short-term loan of passes and visas.\n\n\"It was the psychology of binding a man with a thread,\" Pengelly said, \"and gradually strengthening the thread until it was far easier to submit to our bondage than to rebel. They never foresaw where it led . . . and we paid them in wartime Europe's best currency\u2014food that Big X had commandeered from our Red Cross parcels in any quantity he believed necessary.\"[33]\n\nAs well as his police permit, his leave papers, or his permission to be going home, any potential escaper disguised as an immigrant worker, needed paperwork that validated his transit for the purpose of seeking work. Working from published advertisements in German newspapers, Pengelly directed the forgers to draft letters addressed to a real German, in a real company, from a real German in another real company, all on forged letterhead and with forged signatures and corporate stamps. If a guard on the outside needed proof that the immigrant worker was indeed travelling to legitimate employment in Germany, occupied France, or occupied Holland, the fake letter and a kriegie-manufactured map proved it.\n\n\"A bigger bunch of shysters and crooks and con men you'd never find anywhere in the world than in a prison camp,\"[34] said Don MacDonald, a pilot officer kriegie from Winnipeg.\n\nFor some of the POWs at Stalag Luft III, the lengths to which X Organization went to dig tunnels, forge documents, and hide the evidence was all part of a cat-and-mouse game conducted inside the wire. For others, the activities comprised what Wings Day had once called an \"operational function\" of an officer's duty to try to escape while in enemy hands. For those such as Roger Bushell, it was a combination of one-upmanship, spite, never allowing oneself to be idle, and an ideology of never accepting defeat. George Sweanor hadn't experienced the number of years of imprisonment that Bushell had, but he resented Big X's control over the cultivation of German guards and ferrets. If a kriegie were not directly involved in using the Germans for the objectives of the Organization, Bushell made it clear the POW was to back off, remain polite, and stay aloof.\n\n\"Underhanded tactics were all part of this war,\" Sweanor wrote, \"but I argued there was room for those of us who wanted to lessen the enemy's will to war by showing him that we were just plain folks, willing to make friends, and to share a few luxuries that the Germans had not seen for years. I never was very good at obeying orders I did not like.\"[35]\n\nAmong the luxuries Sweanor and others seemed eager to share were those found in the new building going up just beyond Hut 119 and in front of the sporting fields. In fact, from his window, Sweanor watched the construction of a theatre in the North Compound.[*] Under a parole system, the Germans had provided the materials and lent kriegies tools, provided at the end of each day's work every tool was returned intact. Some of the bricks that formed the foundation of the theatre may have come from buildings knocked down during bombing attacks on nearby Sorau, but some of the theatre's other components had obvious origins. Kriegie carpenters took the plywood from all the crates containing Red Cross parcels from Canada and fashioned it into 350 theatre seats, complete with armrests, sloping backs, and tip-up seats. Just like a professional facility, the house floor was raked from a projection room at the rear of the theatre down to an orchestra pit in front of the stage. And beyond the proscenium, the stage itself featured trapdoors, wings, two dressing rooms, and space for lighting, flats, costumes, and props. An extension of the building backstage included a reference library, a lecture room, the chapel, and an ops room.\n\nThe magic ingredient at the theatre, however, was the ingenuity of the POWs themselves. Two RAF non-commissioned officers with electrician skills installed indirect lighting in the auditorium and a switchboard panel for stage lighting.[36] Air force officers who had fine arts degrees or who had joined university theatre productions before the war were suddenly in demand as producers, directors, playwrights, set designers and builders, props and wardrobe creators, and actors.\n\n\"There was even an electric sign in the foyer of the theatre,\" remembered Canadian pilot Don Edy, who arrived at Stalag Luft III just after the theatre opened. \"They took a wheel and put a strip of tin on the wheel. As the wheel turned powered by the water], the electric current would pass along the tin. When it came to the end of the tin, the current would break and the light would go dark. The light illuminated cut-out letters. . . . So the light would flash on and off announcing the play.\"[[37]\n\nThe only memorable entertainment Edy had experienced since his overseas posting in 1941 came while on operations with the RAF in North Africa. As a Hurricane pilot with 33 Squadron, on leave in Cairo, for instance, he'd enjoyed high tea once at the Mena House resort hotel[38] across from the pyramids. In Alexandria, at the Grand Trianon Bar, he'd marvelled at the \"Gilli Gilli\" boys who passed the hat while one of them placed a baby chick inside Edy's shirt one moment, and used sleight of hand to pull out a garter snake in its place.[39] In February 1942, while strafing a truck convoy with his wing mate Lance Wade near Msus, Libya, Edy's Hurricane took return fire in the engine and radiator; the engine stopped dead and he was forced to crash-land on the desert, where he was quickly captured by German ground troops. Treated for head wounds in Tripoli, he was put aboard a tramp steamer bound for a prison camp in Italy. But the steamer was torpedoed and sunk at sea; Edy clung to debris, was picked up by an Italian cruiser, and eventually delivered to a POW camp in Sicily, then another at Certosa di Padula in southern Italy. He was imprisoned in the monastery (Camp 35) there long enough that he joined some of the British POWs staging musicals, comedies, and revues. One revue, written by POW Neville Lloyd, included a song about life in the camp:\n\n> We're the Padula boys of Camp 35\n> \n> going rapidly 'round the bend.\n> \n> One of our habits is digging like rabbits\n> \n> on tunnels that never end.\n> \n> We lie in bed quite quietly, while the guards\n> \n> are on their rounds.\n> \n> But the moment they're gone, we're at it again\n> \n> with a joy that knows no bounds.\n> \n> We're the Padula boys, Hey, Hey.[40]\n\nWhen the Italians capitulated to invading British, Polish, Canadian, and American forces in September 1943, Don Edy's group was on the move again. The Germans assumed control of the POWs and during the next two months transported thousands of them aboard trains from northern Italy to camps in Germany or German-occupied territory; Edy's imprisonments included Stalag VII-A at Mossburg, Fort Bismarck, Oflag V-A at Weinsberg, and, finally, on November 1, 1943, Stalag Luft III.\n\n\"I doubt if there is a lonelier feeling in the world than when . . . first taken prisoner of war,\" Edy wrote. \"Everything seems completely hopeless and the thought of being behind barbed wire for God knows how long, maybe years, brings on an immediate depression.\"[41]\n\nBy that time, Edy found himself in a Silesian POW compound that housed about two thousand Allied prisoners of war. He recognized the monotony he would have to fight off, the fear he might never see home or loved ones again, and the anxiety over the civilian hostility that surrounded the camp. He was haunted by the story of a recently arrived Irish officer who'd bailed out over Berlin; he'd survived only because German soldiers, cutting down Allied airmen whom Berlin civilians had strung up on a telephone pole following an air raid, got to him first.[42]\n\nEdy fought off his demons by taking on the role of permanent cook in Room 11 of Hut 123 with roommates Bill Stephenson, Johnny Taylor, Cliff Thorpe, and John Crozier. Edy spent hours preparing the meals, in anticipation of the short time he would have cooking on the top of the one hut stove or in the oven below. He worked with the rations officer to have jam tins cut and bashed into additional cooking and heating surfaces for the hut. With the Klim cans that arrived in the Canadian Red Cross parcels, Edy tried his hand at tin-bashing and soldering; he joined a couple of tins end to end, wound some tin around a pencil for a perking tube, perforated a tobacco tin with nail holes, put a specimen bottle upside down in the lid, and manufactured a coffee percolator.[43] It lasted more than a year. All these important distractions helped Edy forget the state of the war and his state of mind. He enjoyed some immediate relief from his travels and travails as a POW on his very first night at Stalag Luft III. Macbeth was playing at the theatre and special arrangements were made for the new arrivals to see the show before it closed.\n\n\"Tickets were made available, neatly typed with the date, seat number, and row,\" he wrote. \"I thought this was a little far-fetched in a prisoner-of-war camp, but] the theatre only held three hundred people; there were nearly two thousand in the camp and everyone wanted to see the shows.\"[[44]\n\nEven German officers who wished to enjoy the inside-the-wire productions had a difficult time securing tickets. George McKiel, a navigator on Lancasters shot down in November 1943, joined the theatre production crew as soon as he arrived at Stalag Luft III; he said the Commonwealth officers quickly decided to reserve the front two rows for the German officers. Further, the kriegies suggested that the plays be recorded for posterity,[45] so in return the Germans gave the actors film to snap photos of the casts. Of course, most of the film made its way to Dean and Dawson for the manufacture of forged passports. The shows ran one or perhaps two weeks at most before the next production went into rehearsal and opened at the theatre. The same night Don Edy attended Macbeth, George Sweanor managed to scrounge an extra ticket for another recent arrival to the North Compound\u2014Ley Kenyon, the same RAF gunnery officer who had graciously donated his artistic talent to create a card for the George and Joan Sweanor wedding back in Middleton St. George at New Year's.\n\n\"I was amazed at how well the theatre] transported us out of kriegieland,\"[[46] Sweanor wrote.\n\nDon Edy nearly had to pinch himself to his senses at the sophistication of the kriegie productions. Most shows had thirty to forty musicians (in air force uniforms) in the orchestra, conducted by Canadian Flight Lieutenant Arthur Crighton, and officers ushering audiences to their seats. There was a full curtain across the proscenium which rose on cue, full lighting on the performances, staging, props, and actors who seemed as if they'd stepped out of Toronto's Royal Alexandra Theatre or Drury Lane in London's West End onto a stage in far-off Silesia in the middle of a world war. Under appropriate lighting, cardboard that the kriegies cut out and painted black became a wrought-iron fence for The Importance of Being Ernest. For Messalina, the theatre carpentry shop took scrap wood, cardboard, and tin, plus paper swans, and created a fountain with flowing water lit by coloured lights. And the artistic crew took a Red Cross crate, put homemade wheels on it and turned it into the wheelchair for character Sheridan Whiteside in The Man Who Came to Dinner. Backstage, actors used oleomargarine as grease-paint makeup.[47] If anyone received colourful wrapping paper in his Red Cross parcel, it immediately found its way onto flats for set decoration. Meanwhile, the same tailors who secretly worked on civilian escape clothes moonlighted by transforming uniforms into suits and coloured sports shirts or pajamas into women's dresses for the dramas. They created wigs from re-braided and re-stitched rope. Perhaps the most astonishing transformations Edy found were in the portrayal of women on stage.\n\n\"Our girls were excellent,\" he said. \"At first the boys took quite a kidding, but after the first few productions the kriegies all realized there was nothing funny about it. The plays and dances needed girls and these fellows worked really hard to create feminine characters.\"[48]\n\n\"I was slight . . . and I was young, twenty-one,\" said Gordon King, one of the air-pump operators inside tunnel \"Tom\"; King jumped at the chance to be on stage. \"They made me up] into a girl very easily, so I was in The Man Who Came to Dinner as the daughter. I had falsies, a wig . . . real professional people doing everything.\"[[49]\n\nGeorge McKiel did his fair share of penguin and stooge service for the escape committee, but among the toughest roles he faced was learning the Katherine Hepburn part in the kriegie production of The Philadelphia Story, because \"it took me three months to learn to walk and talk like a woman.\"[50]\n\nWhen the house lights dimmed and the curtain went up, Don Edy said Belgian airman Bobby Laumans made a beautiful Elaine D'Argent in Arsenic and Old Lace, and New Zealand officer Michael Ormond won storms of praise for his \"Connie Boswell\" act in a revue. Meanwhile, John Dowler made a fitting Frankie in George and Margaret, and Kenneth Mackintosh was stellar as Lady Macbeth. Edy remembered playing a lowly executioner in Saint Joan, but Malcolm Freegard was memorable in the starring role; and, he remembered, Tony Pengelly was a perfect Clara Popkiss in Rookery Nook.\n\nFlight Lieutenant Tony Pengelly claimed that time was a kriegie's cheapest commodity. He expended a lot of it at the theatre as stage manager, producer, and cast member. He most enjoyed memorizing his lines, walking through his roles in rehearsal, and, when it came show time, using makeup, prostheses, wigs, costumes, and a higher register voice to transform himself into countless female characters on the theatre stage. He played ing\u00e9nues, young wives and old, dancers, singers, goddesses, and witches. Like so many of the men taking on those alter egos on the theatre stage, Pengelly was creating a gift for his fellow POWs and a fanciful distraction from the realities of life in a wartime prison.\n\n\"I spent most of the Second World War in drag,\"[51] Tony Pengelly later joked.\n\nStill, Pengelly performed his most vital roles elsewhere at Stalag Luft III. From the moment he arrived in the compound outside Sagan\u2014and under the escape committee's direction\u2014he began to pool kriegie artists, calligraphers, cartographers, photographers, and printers to generate a whole array of forged papers. Beginning with documents, letters, and fake identification cards, he worked with four officers. He led ten others drafting the maps that escapers would need once they got beyond the Sagan area. They started their forgery\u2014an hour or two every day at first\u2014making sure that every detail was noted and replicated with utmost accuracy. If a forger happened to use the wrong colour for a forged rubber stamp, he could ruin a year's worth of work. By the time the Dean and Dawson forgery group was operating at capacity, Pengelly had 137 on his staff. Naturally, Pengelly controlled any and all incoming items that might offer information or technology required for the forgery shop. At one point he oversaw the smuggling of an entire typewriter\u2014one piece at a time\u2014into the North Compound.\n\nThis led to another of Pengelly's high-priority responsibilities: acting as a chief parcel officer. Almost from the very beginning of the war\u2014when Wings Day was first SBO at Dulag Luft in late 1939\u2014British and Canadian kriegies had decided to pool and redistribute the contents of the Red Cross parcels.[52] In this way the older prisoners of war gave up their seniority for food and toiletries in favour of new arrivals, who had just been shot down, interrogated, and shipped to the prison compound with nothing but what they had on their backs. When parcels arrived at Stalag Luft III, they were not immediately distributed for consumption; they were stored in the building that housed the sick quarters in the Vorlager adjacent to the compound.[*] On average, each week a POW received the equivalent of a full parcel\u2014from the US, the UK, or Canada. Generally, the parcel contents consisted of basic ingredients (what the British called \"housewife treats\") such as soup, cheese, corned beef, salmon, sardines, raisins, pudding, coffee, tea, butter, jams, biscuits, powdered milk, and occasionally sweets such as chocolate or candy. Since Pengelly was also on the front lines of kriegies taming bribery prospects among the guards, his work, of necessity, required bait to entrap the prospects, or, as he put it, to \"oil the machine.\" Pengelly had the authority from Roger Bushell himself to confiscate any and all parcel items.\n\n\"Big X had full control over everything that came into the camp,\" Pengelly said. \"Each Red Cross] parcel received bore with it a list of contents, and from those lists, Big X commandeered anything he thought the organization could use.\"[[53]\n\nConsequently, civilian shirts, sports jackets, plain blankets, and sheets became fair game for X Organization's tailoring group. Picture books, pen and ink, toilet paper, coloured wrapping paper, and even newspaper clippings helped supply the forgery group. Meanwhile, chocolates and any item considered rare in wartime Germany went right into the hands of those blackmailing the guards. In one instance, Pengelly said, they had one guard so well tamed they asked for his passport for an afternoon; the kriegies promised in turn that the guard would not be caught without it and that it would be back in his hands by nighttime. It was, and the guard got a gift for his trouble.\n\n\"The camp Kommandant's secretary had a boyfriend, a German guard we had tied up pretty well,\" Pengelly remembered. \"The secretary would get correspondence files from the Kommandant's desk, give them to her boyfriend, who would give them to us. Big X would read them and then send them back along the same route.\"[54]\n\nThat autumn, fighter pilot Don Edy signed up for a couple of productions at the North Compound theatre\u2014a small part in the operetta Messalina and some parallel bar gymnastics in a show called Six to the Bar. He had been a POW for just under two years. That November, bomb-aimer George Sweanor acknowledged his twenty-fourth birthday. Back in June, he'd been buoyed by a letter from Joan telling him she'd given birth to a healthy baby girl she named Barbara. More than once Sweanor found himself seated on a stump in the compound, glaring at the barbed wire and vowing he would survive and one day return home to his wife and daughter. He had been a prisoner of war for eight months. Meanwhile, Roger Bushell also got involved in the kriegie theatre productions. Partly to draw the Germans' attention away from his escape committee activities, and no doubt to seek psychological escape from the summer setbacks of 1943, that fall Big X appeared in a production of George and Margaret. At age thirty-three and more than three years a captive of the German stalag system, Bushell agreed to prepare the part of Professor Higgins for an upcoming production of Pygmalion. He would rehearse periodically through the coming winter in anticipation of an opening night scheduled for March 24, 1944, a night during which he would ultimately be otherwise engaged.\n\nLike so much else at Stalag Luft III, the theatre, while appearing to offer harmless and distracting entertainment to young POWs awaiting the war's outcome, became a disguise for greater pursuits.\n\n* * *\n\n* George Sweanor cited statistics from his own research indicating that the life expectancy for a Bomber Command aircrew was five operations, and that of those shot down only 17 per cent survived.\n\n* In interviews and analysis compiled in 1946 by an RAF War Crimes Interrogation Unit, and under the auspices of the Judge Advocate General at the War Office in London, England, authors of the report ask: \"Was it a matter of high policy that, in the view of the reprisal measures already planned and in operation in the form of the 'Stufe III' order and the '[Aktion] Kugel' order, at least one major camp break-out was not to be discouraged?\"\n\n* According to George Sweanor, the Canadian government took thirty-five dollars a month (deducted from his Canadian pay accumulating in Canada) and transferred the funds to the Germans so that Canadian POWs could rent theatrical costumes and musical instruments; similarly, in Canada, German POWs used their allotment to order by mail from the Eaton's catalogue.\n\n* While kriegies generally endorsed Wings Day's edict of pooling the Red Cross parcel foods, Canadian F\/L Ted Kipp and RAF F\/L Ken Toft (fellow-POWs at Warburg) founded Foodacco, an inside-the-wire emporium, which used cigarettes as the primary currency to purchase additional food or delicacies they accumulated. Controversy emerged over whether Foodacco should be free enterprise or non-profit; there was a referendum, but it was overruled by SBO Day, who nationalized the Kipp-Toft free enterprise operation to run as a co-op.\n\n# 7\n\nTHE PLAY'S THE THING\n\n* * *\n\nIT SEEMED a world away and it was, but that fall of 1943 the Harlem Globetrotters maintained their winning ways\u2014 2,163 victories versus 162 defeats in fifteen seasons. The war was, however, having an impact on the calibre of all professional sports back in North America. At the end of the 1942\u201343 NHL season, Boston's entire Kraut line of Milt Schmidt, Bobby Bauer, and Woody Dumart enlisted in the forces, which prompted Bruins coach Art Ross to say, \"the best forward line today couldn't make the Bruins' third line a couple of years ago.\"[1] And that fall the New York Yankees won the fortieth World Series, four-games-to-one over St. Louis. The Cardinals committed ten errors in the series. Ironically, pro baseball's commissioner promised the game would go on as long as there were at least eighteen men left to play.[2]\n\nFor perhaps that one season, members of the Canadian All-Star Baseball Team, playing games inside the wire at Stalag Luft III, could have thumped the '43 champion Yankees. They had that much talent in their lineup. With the Canadians and Americans housed in separate compounds (since September 1943), contact between the two nationalities of air force officers was limited. That changed, however, when the Americans asked the camp Kommandant if they could meet their Canadian rivals in a challenge for softball supremacy.\n\n\"The Americans always figured they were better ballplayers than the Canadians,\" Flying Officer Art Hawtin said. \"They were just next to us in the South] Compound and they got permission to bring their team over.\"[[3]\n\nWhile baseball was a sport Art Hawtin had enjoyed in the 1930s (he had a Babe Ruth baseball card in his collection), it would not have been his first choice. The fifth of six children born on a farm near Kinmount, Ontario, Art did his fair share of chores, but his favourite physical activity was pole-vaulting; in fact, he stayed in Beaverton High School an extra year as a junior athlete to accumulate more victories and won ten track events in 1938. Along with his blue ribbons, Art gained a valuable skill landing repeatedly in all those high school pole-vaulting pits. On the night of May 13, 1943, as a navigator on a 405 Squadron bombing mission to Germany, Hawtin's Halifax was shot down; he bailed out of the crippled aircraft and landed in the Dutch countryside.\n\n\"Landing with a parachute or pole-vaulting over a bar ten or twelve feet high is just about the same,\" Hawtin said. \"I came down in a farmer's field near Dedemsvaart, Holland. . . . I was completely relaxed and I landed safely.\"[4]\n\nWith the likelihood that any Dutch family hiding an evader would be shot, Hawtin agreed to turn himself in. A week later he came walking through the main gate at Stalag Luft III and was assigned to the same room as John Colwell, the dedicated diarist and tin-basher originally from Vancouver Island. Hawtin and Colwell had both served as navigators with 405 Squadron stationed in Yorkshire, so Colwell was able to vouch for his friend in front of the compound SBO. Hawtin's kriegie card identified him as a student. And when he was asked about participating in escape committee activities, he readily volunteered to work as a penguin. But on the day the Americans decided to show the Canadians the finer points of their softball prowess, Art Hawtin turned out to be a pretty agile, hard-hitting left fielder on the Canadian All-Star Baseball Team roster.\n\nBefore F\/O Hawtin arrived on the scene, baseball had already become an all-consuming pastime at most of the prison camps where North American POWs were incarcerated. Originally, the prisoners of war played sandlot baseball. The game didn't require a lot of equipment, so the kriegies improvised by using bedposts as bats and baseballs fabricated from old shoes or scraps of leather wrapped together with the rubber seals from empty coffee cans. Eventually, the YMCA back home shipped bats, balls, and some gloves into the camps so that the games at least looked like legitimate softball competitions. Initially, barracks hut played against barracks hut, senior officers versus junior officers, married men against bachelors.[5] The games took on greater significance when the editors of the Gefangenen Gazette camp newspaper began publishing box scores and commentary. Eventually, across the six compounds of Stalag Luft III, there were as many as two hundred teams playing on the improvised baseball diamonds of the various sports grounds.\n\n\"The Americans made arrangements to play us twice,\" Hawtin said. \"But we had Bill Paton pitching for us. He'd pitched senior ball for the Beaches League in Toronto. They had a pretty good pitcher too . . . but Paton struck] out sixteen batters.\"[[6]\n\nAbout the fifth or sixth inning of that first game, umpire Larry Wray went out to the mound to have a confab with Paton. Wray was Senior British Officer and also a Canadian, but he wanted the game to be more competitive. The Americans hadn't hit the ball out of the infield to that point.\n\n\"Please let them hit, Bill,\"[7] Wray pleaded.\n\n\"One ball was hit into the outfield in that game,\" Hawtin said, adding that the most strenuous part of the game for him was walking to and from the outfield. \"The first game we didn't get many runs, but we got enough to win. The second game wasn't even close. I think we won fourteen to one.\"[8]\n\nNot ones to miss out on such sporting spectacles, the German guards gathered along the baselines to witness the games too. Art Hawtin wasn't sure for whom the Germans cheered\u2014the Americans or the Canadians\u2014but Colonel von Lindeiner was frequently seen beating his cane on his leather boots enthusiastically.[9] While he was quite content to leave the rules on the diamond up to Allied air force officers, the colonel nevertheless kept tight control on other aspects of kriegie baseball, no matter who was winning or losing. Art Hawtin noted in his diary some of the rules the Kommandant wanted obeyed rigidly in and around the barracks, but especially near the warning wire.\n\n\"Care must be taken that no window panes are broken,\"[10] Hawtin quoted from one rule. \"Panes broken through carelessness will not be replaced.\"\n\n\"Opportunity will be given twice weekly under the supervision of an interpreter to collect small balls from between the warning wire and the fence,\" stated another rule. \"POWs must furnish a man to collect balls. . . . He must wear some mark which will be easily seen and readily distinguishable at a distance by the guards.\"\n\nAs awkward and silly as the wording of the ball-reclamation rule seemed, its application occasionally proved deadly. Phil Marchildon, from Penetanguishene, Ontario, pitched seventeen winning games for the Philadelphia Athletics in the 1942 major league baseball season before enlisting in the RCAF; a pilot officer and tail gunner aboard a Halifax bomber, he was shot down in 1944. During a kriegie baseball game he witnessed, a ball bounced over the warning wire. A Canadian asked permission from a guard to retrieve the ball; Marchildon wrote that the guard waved the POW to get the ball, but \"when the Canadian was] no more than a foot over the fence [the guard] shot him dead.\"[[11]\n\nAnd there were flying objects other than fly balls that could get kriegies into trouble near the warning wire. Canadian Frank Sorensen had been a regular \"circuit basher\" when he first got to Stalag Luft III, walking along the safe side of the warning wire with Roger Bushell, teaching him Danish. But later, when his Danish relatives sent him a glider kit, he test flew the model with its seven-foot wingspan and worried more about its landing than its flight.\n\n\"I'm afraid to tow it too high, as it might hit a window or fly over the wire,\"[12] he reported in one of his letters home. In fact, on its maiden flight from the appell grounds, the glider caught a gust of wind and flew over the fence. Fortunately, the guard outside the wire wasn't as volatile as some, and tossed it back into the compound.\n\nMeanwhile, letters posted the other way\u2014from his Danish family connections\u2014helped keep Frank Sorensen and fellow kriegies up-to-date on wartime events closer to the front lines. He learned, for example, about the so-called \"clearing murders,\" in which civilians such as Danish author Kaj Munk[13] were executed in retaliation for Danish resistance murders of German soldiers. BBC broadcasts received on the kriegies' hidden wireless were also a lifeline to the Allies' progress in 1943. The balance of the war was tipping in their favour. They learned about Montgomery's pursuit of Rommel at El Alamein in January, the eventual victory of Soviet forces over the German Sixth Army in Stalingrad in February, the successful Bomber Command attacks on Ruhr River dams in May, and the capitulation of the Italian army to joint Canadian-British-American forces in Sicily that summer.\n\nNot surprisingly, there was more at stake on the kriegie baseball diamonds than just the Canadian All-Stars beating the Americans at their favourite pastime. The escape committee took full advantage of the movement of people and sports gear around the compound to camouflage its activities. Some camera and radio receiver parts were small enough to be smuggled inside baseballs, and if a bat were hollowed out, it too could help X Organization deliver contraband between compounds. In addition, prior to the discovery of tunnel \"Tom\" in Hut 123, in September 1943, American paratrooper Jerry Sage's scheme of mingling penguins and fans helped disperse tons of sand along the sidelines of kriegie football scrimmages, soccer matches, track meets, and baseball games. But with the first snows and nighttime temperatures below freezing in November and December, only the Canadians' maintained outdoor activities, either circuit bashing or playing shinny hockey on the sports grounds. However, neither sand dispersal in the snow, nor above-ground escape seemed practical or inviting during that time of year.\n\nAcross from baseball player Art Hawtin in Room 14 of Hut 120   \nlived the man Hawtin called \"the Tin Man.\"[14] In his diary, F\/O John Colwell continued to keep a record of POW activities. More important, he kept fabricating tools and, with those tools, a steady flow of practical utensils\u2014some overt and others covert\u2014made of tin, wood, wire, cloth, glass, and any other raw materials he could scavenge. Colwell had arrived April 12, 1943, but within a few months he had tailored ten pairs of shorts, an airman's tunic, socks, and a sleeping bag. With the woodworking and tinsmith tools he'd fabricated from scratch, he then became a virtual assembly line, manufacturing useful items for any kriegies in need. Using pieces of tin and steel, and mislaid screws and nails, he built appliances such as coffee percolators, kettles, cooking pots, pie plates, baking pans, toast racks, a fireless cooker, an ice cream freezer, picture frames, a Klim clock, and a metal brace for Group Captain Massey's injured leg. Then, from any Klim tins left over, he secretly began building a suitcase. It took eighty-one tins, but by November the bag was finished. Somehow, he sensed he'd be going somewhere.\n\n\"He could make anything out of anything,\" Hawtin said. \"There were twelve of us who moved into a bigger room. There wasn't a thing in the room when we got there. Seven o'clock the next morning he rounded up all the tin cans he could get and we had every utensil we needed within a week. He was a master tinker.\"[15]\n\nWhen he moved into a room that doubled their numbers from six to twelve POWs, Colwell felt he needed to build a new oven. At some point in the manufacturing process, he drew a diagram in his diary showing that his Klim tin stove required \"120 Klim tins, 25 pounds of clay, a grate and two fire bricks.\"[16] Colwell also helped organize the menu for Christmas that year. It wasn't lavish, but following appell on December 25, 1943, Colwell's roomies enjoyed a breakfast of sausages with a cheese souffl\u00e9, toast and jam, and coffee. At 12:30, they had a lunch of toast and jam and tea. And for Christmas dinner, the cooks for the day\u2014Bill Hoddinott and Art Hawtin\u2014served up macaroni with cheese, roast beef, baked potatoes, Christmas pudding, and coffee. As well, they enjoyed chocolate sent to them by the Canadian Prisoners of War Relatives Association and a package forwarded from the Canadian government with a greeting from the prime minister.\n\n\"All Canada joins in warmest Christmas greetings and good wishes to you,\" the attached card read. \"Arrangements have been made to forward to all Canadian prisoners of war a Christmas gift . . . comprising articles such as gramaphone sic] records and cooking utensils.\"[[17] It was signed W. L. Mackenzie King.\n\nIt's difficult to know whether the prime minister realized it at the time, but the gramophone records in the Christmas packages could not have come at a better time. Two huts away from John Colwell's one-man utensil production line, in Hut 103, Al Hake, an Australian airman with a penchant for three-dimensional designing and construction, was hard at work building a more covert utensil\u2014compasses for the escape committee. Chief among the basic components of the compasses, Mackenzie King might have chuckled to discover, were melted-down Bakelite (plastic) gramophone records. Hake and his co-workers cut the plastic into pieces, heated it until it was dough-like, and pressed it into a mould for the compass base about an inch in diameter. They created the directional device with either a sewing needle or a razor blade strip stroked repeatedly with a magnet; the magnetized needle or razor was carefully mounted on a gramophone needle, which stuck up vertically from centre of the Bakelite base, creating the pivot point for the needle or razor. A compass card inside the casing indicated magnetic north (later models actually had luminous chips stolen from the Kommandant's alarm clock). From broken windows, Hake then fashioned a glass face and gently heated it and the gramophone plastic to create a waterproof case around the compass mechanism. It was further sealed with reused window-frame putty. And if there was any question about the authenticity of the compass, Hake and his team pressed a manufacturer's inscription into the bottom of the compass while the Bakelite base was still warm.\n\n\"Made in Stalag Luft III,\"[18] it read.\n\nIt may well be that Mackenzie King had no hint his government's thoughtful gift of gramophone recordings was playing right into the hands of the escapers. But some of the kriegies' relatives received direct, if a bit perplexing, appeals to help supply them with the tools of escape. Since December 1941, Flying Officer John Weir had been writing regularly to Frances McCormack at the family's residence in the Forest Hill area of Toronto. Naturally, Weir's fianc\u00e9e had become accustomed to his words of longing to be with her. But she also began to understand he wasn't sitting idly by waiting for the war to end. He had told her of his German language lessons, that he had managed to have pictures taken of his fellow POWs, and that he needed several pairs of long johns. Harmless on the surface, the correspondence clearly indicated Weir was up to something. And while Frances might not have deduced her fianc\u00e9e was one of X Organization's most reliable and productive tunnellers, she probably sensed from his Halloween letter of 1943 that there was a reason behind all of his requests.\n\n\"The pajamas you sent in the July parcel just came in time,\" wrote Weir, referring to what likely became his tunnelling clothes. And then he added a new, emphatic request. \"Darling, what we really need are gramophone needles or are they practically non-existent at home too? There are two hundred or more of us here in the compound now. We are all hale 'n hearty, but itching to be home.\"[19]\n\nIronically, just weeks later, Scruffy Weir's lingering war wounds would remove him from both the task of finishing \"Harry\" and the entire escape plan. Dating back to November 8, 1941\u2014the day Weir's Spitfire was hit by Messerschmitt shells and the resulting fuel and cockpit fire nearly blinded him\u2014Weir had coped with no eyelids over his eyes. In the fall of 1943, after \"Tom\" was discovered and excavating underground was halted, tunnel boss Wally Floody recommended that Weir take an opportunity to have surgery to prevent potential blindness. Weir was transported to a German hospital near Frankfurt-am-Main, where he underwent skin-graft surgery performed by a British surgeon captured by the Germans in the Mediterranean in 1942. Weir would not return to Stalag Luft III until June 1944.\n\nMid-upper gunner Albert Wallace arrived at the North Compound in late May of 1943. By coincidence, he was shot down the same night as Art Hawtin. Aged twenty-two, Wallace had already worked at two or three civilian jobs, enlisted in the RCAF, graduated as a gunner, become a pilot officer, and completed fifteen and a half bombing operations with 419 Squadron in Bomber Command when he was shot down over Duisburg, Germany. But if he thought spending the night in the mid-upper gun turret of a Halifax bomber was a lonely experience, Wallace discovered, when he first arrived at Stalag Luft III, that life as a POW could be even more isolating. Once he'd been deloused, photographed, and issued a blanket and cutlery by the Germans, he was assigned to Hut 101 and \"a friendless room,\"[20] where the occupants spoke only South African dialects. Cut out of any conversation, he asked for a transfer and was eventually moved to Hut 104 and Room 23, the very place where tunnel \"Harry\" had begun.\n\n\"I had no idea it was the tunnel room,\" he said. \"I didn't know for weeks that goddamn tunnel was seven feet from my bunk bed. Periodically, the sand would come out, but at first I didn't realize.\"[21]\n\nWallace never actually saw the tunnel opened or closed, although he did notice that one of his roommates, Pat Langford, was \"extremely security conscious,\"[22] almost always sitting on his top bunk, legs dangling over the side and surveying things all the time. Wallace eventually learned that Langford was \"Harry's\" trapf\u00fchrer, responsible for opening and closing the entrance to the tunnel at a moment's notice. In time Wallace would participate in escape activities, but as winter approached he and his fellow kriegies faced a more immediate problem: keeping warm behind the thin, poorly insulated, and under-heated barracks walls. Taking a page from the tunnelling efforts, Wallace and several roommates began a nightly ritual\u2014sneaking out of their hut, breaking into the German kitchen facility, filling their kitbags with coal briquettes, and stockpiling them for use during the colder nights.\n\n\"We had so much coal in our room, but there was nowhere to hide it,\" Wallace said. \"We put it under our bunks. We put it in our Red Cross boxes to hide it. We'd be in our shirt sleeves, hotter than hell, and guys would come in muffed up to their necks with tuques on and we'd say, 'Oh, it's our new chimney.'\"[23]\n\nThe supply of extra briquettes dried up when the Germans noticed the sudden depletion at the kitchen and padlocked the coal bin. But Wallace, after a further move to Hut 107, would soon find himself busy with another wintertime activity, in the ranks of the penguin sand-dispersal team. Right after New Year's, Roger Bushell assembled his section heads for the first escape committee meeting of 1944. He sensed that a combination of the ferrets' success in discovering \"Tom\" in September and the kriegies' relative inactivity through the fall had lulled the Germans into a false sense of security. It was January 7, and he wanted work to resume on \"Harry\" within days, while the goons' defences were down. The greatest barrier to any progress underground, Wally Floody pointed out, was trying to get rid of the sand with snow all over the compound. The meeting wrapped up without a solution. Then, Fanshawe and Ker-Ramsey hit on an idea. They told Bushell they would explore the space between the raked floor and the earthen foundation of the North Compound theatre. Fanshawe took a fat lamp into the crawl space. He reported that there was enough space under the floor to handle as much sand as \"Harry\" could ever deliver.\n\nAs usual, the toughest part of the transaction\u2014the journey between \"Harry\" and the theatre\u2014was getting the penguins from one to the other without bumping into ferrets. On January 10, when appell was done, three non-permanent residents of Hut 104 assembled in the large room at the north end of the building. Wally Floody, Robert Ker-Ramsey, and Pat Langford spent a couple of hours removing the square of tiles under the stove that concealed the trapdoor, swinging it open, and descending into \"Harry's\" vertical shaft. The air was cold; although the trapdoor had been sealed shut for three months, Ker-Ramsey had left a bypass open to allow air to circulate below. Floody discovered that the kitbags on the air pump had rotted through and needed replacing. Still, the greater problem was lack of air circulation farther up the tunnel. A couple of the Klim tin ducts had collapsed a short distance up the tunnel; they were soon repaired to restore the flow of fresh air. Three days after the section bosses reopened \"Harry,\" Floody took his first shift digging and the sand began to move again. About that time, he took a few minutes to dash off a postcard to his sister, Catherine, at home in Toronto.\n\n\"My mail has taken its usual winter lapse . . . a four-month gap,\" Floody wrote late in January, suggesting he might have been slightly distracted. Then he added an ironic sign-off. \"Life goes on in its predestined rut . . . Your brother, Wally.\"[24]\n\nMeanwhile, Group Captain Massey got the Kommandant to ease the outdoor winter nighttime curfew; kriegies were allowed to walk between the barracks huts until 10 p.m. Massey was reminded, however, that one of the ball-retrieval rules, enacted during the previous season's baseball games, was now in effect. \"Snowballing must cease one-half-hour before roll call,\"[25] the order said. The kriegies would abide by the snowball rule, but under cover of darkness, a steady line of penguins moved between \"Harry\" and the theatre carrying something else. Unlike the previous summer's covert activity, the winter penguins didn't have to hide their sand pouches. When a penguin arrived at the mouth of the trapdoor in Hut 104, Langford laid a kitbag or trouser pouches full of sand over his shoulder. When security boss George Harsh told him the way was clear, the penguin carried his load out the door of Hut 104 straight to Hut 109. And if the route was safe from there, a stooge directed him around Hut 120 through the snow and into the theatre.\n\nGiven that Flight Lieutenant Tony Pengelly was stage manager for all the kriegie productions, the goons had little reason to be suspicious of his late evening presence at the theatre. But when Big X announced the push to renew digging in \"Harry,\" Pengelly worked with the stage carpenters to install false nails in the base of a seat in the back row of the theatre so that it looked secured to the floor but was actually movable. The chair back was also not secured in place. When a trap operator slid the chair back up in the air, then flipped the seat itself forward toward the row in front, a small trapdoor in the floor under the seat was revealed.\n\n\"That was a lifesaver, because about a hundred tons of sand came out of \"Harry\"],\" Albert Wallace said. \"I remember going into the theatre one night with my bags full of sand. I was told where to sit because that's where the trapdoor was. I sat in seat number thirteen, pulled my little tickies and out went the sand.\"[[26]\n\nPengelly explained that when the theatre was dark (had no productions running), dirt was dumped through the trap all evening. Beneath the floor boards of the theatre, meanwhile, the Tin Man (John Colwell) and five other sand-dispersal men worked feverishly to keep the trap under seat thirteen clear and move the sand to the far reaches of the theatre's crawl space.[27] When each penguin unloaded his kitbag or pouch through the trap, the sand fell into a basin that was pulled by rope to a packing point. There, Colwell's crew tamped the sand up tight to the floor between the joists and against the walls. No space was left empty. At the end of an evening's work, as they left the crawl space, each of Colwell's crew would stand over the open trapdoor, slap the dirt from his clothes, and sweep the floor clean of the evidence. Pengelly and his crew would close the trap and reposition the seat and chair back, completing another successful night of making sand disappear.\n\n\"One day, we estimated we got rid of a total of twelve tons of sand,\"[28] Pengelly said.\n\nIf the raked floor and Red Cross crate seats in the theatre weren't all they seemed, neither was the chapel, a room of prayer and reflection built into the backstage area toward the eastern end of the theatre building. Most afternoons between 1 and 4 p.m., when the autumn and wintertime light was at its best, RAF pilot Des Plunkett led a staff of about a dozen draftsmen (with about a dozen stooges guarding in the hallways and outside the windows of the chapel) working on the manufacture of escape maps. The men took the \"flimsies\" (thin paper linings) from gramophone record sleeves or cigarette packs and drew escape routes on them. Using inks they acquired or manufactured in the camp, the mapmakers then created negatives in jelly, which in turn generated map prints. The final editions of the maps had five legible colours: green for wooded areas, black for railway lines, red for roads, blue for rivers, and yellow for autobahns. Among the map copyists was former architect turned RAF pilot John Hartnell-Beavis, who knew the value of accurate maps. Shot down in occupied-Holland, he used a silk map from inside his flight jacket to evade German troops for several days before being captured in July 1943. By November he had volunteered his drawing skills to create the jelly stencils that printed up to thirty-five copies of a specific map.\n\n\"All our work inks, pens, rulers, dividers, paper, etc., had to be used so that] it was possible for everything to be concealed in a matter of a few seconds especially when [ferrets were] in camp,\" Hartnell-Beavis wrote. \"If the padre had seen all the secret hiding places and panels in the floor and walls of the chapel, he would have had a severe shock.\"[[29]\n\nThe signalling system used by X Organization stooges could be as simple as a red book set in a window as a sign that a ferret was in the area, or a blue towel on a clothesline indicating that all was clear. During the fall and winter of 1943\u201344 , however, the system depended on signals communicated from inside a window in one barracks hut to inside a window in another, and on down the line. Australian air force officer Paul Brickhill[*] served the escape committee in the security section. He worked out a system of signals passed from one stooge to the next to protect the forgery work going on in the library in Hut 110 and the secret mapmaking going on in the chapel of the theatre. A window or a blind suddenly opened by one stooge would signal danger to a second stooge; that stooge would hold a folded piece of paper in his window, signalling a full shutdown of activity; and those signals would prompt a third stooge near the covert production to tap on the wall. All the forgery in the library or mapmaking in the chapel would disappear in seconds.[30] None of X Organization's stooges considered his work glamorous, but most learned how vital it was, including Flying Officer John R. Harris.\n\n\"Many a weary hour I spent peering out from behind windows and doors and in all kinds of weather,\"[31] Harris wrote.\n\nBorn and raised in Toronto and eager to land any job he could after high school, during the Great Depression, John Harris found work as an office clerk with Canadian General Electric. But when the Nazis invaded western Europe and threatened to keep going across the English Channel, he sensed his new job was \"to shoot down those Germans blitzing England.\"[32] As a new recruit with the RCAF in 1941, however, he was streamed to observer training. While still in Canada, he faced greater danger on the ground than in the air. Right after his graduation in August 1942, during a trip home, a freight train crashed into his passenger train on a siding; nevertheless, all passengers survived. Overseas he was posted to 419 Squadron of Bomber Command and his penchant for survival stayed with him. On September 5, 1943, during their eleventh bombing op over Mannheim, Germany, and immediately following the release of their bomb load, Harris and his crew came under attack. The Halifax began to shudder and descend uncontrollably.\n\n\"My poor mother isn't going to like hearing about this,\" was his first thought; then, just as suddenly, all he could see around him were the stars of the night sky. \"My God, I'm out of the kite.\"[33]\n\nMiraculously, either by being sucked through an open machine-gun turret beneath him or blown free by the explosion of the aircraft, Harris found himself floating to earth; he pulled the rip cord for his guide chute, which released the main parachute, and he landed in a forest, the only survivor of his seven-man crew. Following his capture, he was interrogated and eventually packed aboard a Luftwaffe truck for transfer to Stalag Luft III. With him and also shot down that night was F\/O John Crozier, flying as a Second Dickie (observing pilot). Crozier was so recently assigned to the station (flying his first op with 620 Squadron) that he was still in his RCAF Blues tunic[34] , not his battledress, and considered by the others to be a German spy. The two men\u2014Harris and Crozier\u2014both served X Organization as stooges. The watches could be as long as a couple of hours, and the stooges were moved around so the ferrets wouldn't notice a pattern.\n\n\"The system wasn't foolproof,\" Harris said. \"One time I was peering out through a door and up a hallway at what was going on. A German guard came right up beside me. I don't know where he'd come from.\"[35]\n\nPerhaps a stooge's ability to recover when caught off guard proved as valuable an asset as passing along an alarm. Harris managed to collect his wits quickly and pretended that he had been worrying about the weather before sauntering out of the building. He hoped his performance was sufficiently convincing that the ferret wouldn't bother to trigger a complete search of the building where Harris had been standing guard. It was, and Harris proved he could survive yet again.\n\n\"I remember Pat Langford, my immediate superior, he came and chewed me out for not having done my job,\" Harris said. \"Then they discovered that somebody else had slipped up and he apologized.\"[36]\n\nFrom the moment \"Harry\" was reopened, security boss George Harsh spent his days seated within view of the room containing the trapdoor to the tunnel. Stooges on the job\u2014from the duty pilot watching the main gate to those shadowing ferrets\u2014remained in touch with Harsh at all times, so that the second any German entered the compound with Hut 104 in his sights, Harsh could let Pat Langford know. In turn, the trapf\u00fchrer warned diggers below to stop any work that might be heard. Then he installed the grill at the top of the tunnel shaft, tucked blankets over it to muffle the sound, closed the trap with the tiles on top, and moved the stove back on top of the trap, simultaneously replacing the extension flue (above the stove) with its usual short flue. All of this happened within twenty or thirty seconds of Harsh's call for a shutdown.\n\nBy the third week of January 1944, about ten days after the resumption of digging, \"Harry\" had advanced fifty feet. Right there\u2014about halfway between Hut 104 and the warning wire\u2014Wally Floody built a halfway house they called \"Piccadilly.\" He planned a second one when \"Harry\" was two hundred feet in length. They reached that point\u2014roughly beneath the cooler in the Vorlager area of the compound\u2014the first week of February, and finished the \"Leicester Square\" halfway house by February 10. During that time, they had even been forced to shut down for a week when the moon was full in a cloudless sky. Had they continued to dig and disperse the sand as usual under the theatre, the penguins would have stood out against the moonlit snow. But the escape committee put the excavation downtime to good use.\n\nBy this time, Al Hake's assembly line in Hut 103 had manufactured as many as 250 compasses and hidden them down \"Dick's\" vertical shaft. Des Plunkett's mapmakers had mimeographed approximately four thousand escape maps. Meanwhile, the men in Tommy Guest's tailoring section were making headway on the task of outfitting scores of potential escapees in clothing that would help them blend into the civilian populations of Europe. Using shirts, pants, and jackets from Red Cross parcels, linings from greatcoats, and old uniforms, they shaved the rough surfaces of the cloth and re-coloured them with beetroot, shoe polish, or the dye from book covers. They used the broadsheets of German newspapers to cut out the patterns and sized each piece of clothing to accommodate the escaper.\n\nBy the time tunnellers had dug \"Harry\" as far north as the tunnel would go, Guest's tailors had manufactured as many as fifty civilians suits. They would be worn by POWs made up to look like businessmen, professionals, and travellers going about their daily lives, boarding trains that stopped at the Sagan Junction station. They would be worn by kriegies with sufficient skill in several languages to talk their way through their documents and German checkpoints fluently. For most of the rest\u2014the so-called \"hard-arsers\"\u2014there would be some documentation, some clothing, and the basic tools of travel: maps, compasses, and kitbags. Without multilingual skills, however, the hard-arsers would have to rely principally on wits and good luck. John Harris, whose forged documents would identify him as Antoine Zabadose, and whose set of stencilled maps would guide him to the Czechoslovak border, was outfitted to look like a Hungarian ironworker.\n\n\"I made some effort, though not very successful, to alter the appearance of my greatcoat,\" he said. \"It was almost the same khaki colour as the Canadian army uniform, except that it had flares for sitting on horseback. It was worsted or tan-coloured heavy wool that covered me from the neck to the calf. Underneath, I wore my battledress, which I'd had since I was shot down.\"[37]\n\nElsewhere in the North Compound, kriegie life\u2014as far as the guards and ferrets could tell\u2014looked normal. The baseball, soccer, volleyball, and other field sports fields beyond the appell area had given way to the Canadians' wintertime pursuit. With the surface of the ground nearly frozen or covered in snow, conditions were ideal for flooding the field into a regulation-sized skating rink for hockey. Players would grade the soil for the rink with homemade shovels and a homemade level\u2014water in a pan\u2014then haul the water from the fire pool several hundred feet away. The first skates were entirely homemade; kriegies took angle irons from benches and screwed the steel to the bottoms of their boots.[38] But true to his kriegieland reputation, Alberta-born-and-raised pilot Barry Davidson managed to scrounge the real equipment needed to outfit teams for shinny.\n\n\"I wrote Don Mackay, the mayor of Calgary,\" Davidson wrote. \"They got skates and hockey equipment and sent them to the camp. We flooded our rinks with buckets and they were regular sized rinks, so it was a lot of work.\"[39]\n\nHockey sticks were hard to come by and maintain. In addition to the city of Calgary's contribution, the YMCA came through with some as well, but depending on the calibre of players and intensity of the play, keeping the sticks in one piece was a challenge. To protect the players from injury, some groups came up with special rules, such as only allowing body checks or shot blocks within a certain distance of the net. However, there were cases of hockey games, indeed an entire season at Stalag Luft III, coming to an end when the supply of sticks simply ran out.[40]\n\nIn February, the North Compound theatre staged a homegrown revue. Between Ourselves, produced by Peter Butterworth, came complete with comedy skits, dance routines, and short dramatic works. Among the highlights, Bobby Laumans was back on the boards in the role of a torch singer and Tony Pengelly joined a Latin dance number with six couples, featuring the female performers made up in Carmen Miranda-like head gear, capes, and miniskirts. Taking one of the male dancing roles was another Canadian airman, James Wern\u00adham. Born in Scotland in 1917, Jimmy had emigrated and settled with his family in Winnipeg, Manitoba. He had worked in a general store and for an accounting firm during the Great Depression years, but enlisted in the RCAF in 1940 and was trained as an observer. Overseas on ops, he became something of a celebrity in May 1942, when newspaper reporters photographed him and his crew[41] (from 405 Squadron) as veterans of \"Operation Millennium,\" the first thousand-bomber raid on Cologne, Germany. Three ops later he was shot down over Holland and captured. At Stalag Luft III, when not tending to other escape committee duties, he devoted his time to the theatre as a means of boosting POW morale.\n\nAnother kriegie helping out onstage at the theatre that winter was twenty-two-year-old George Wiley, from Windsor, Ontario. More of a free spirit than most, Wiley wasn't the best student in school and suffered from rheumatic fever as a youth. And he appeared much younger than his years, as if he were underage. Nevertheless, he had the credentials when he enlisted in the RCAF in 1940, about the same time James Wernham did. Wiley flew Kittyhawks for 112 Squadron in North Africa, surviving a crash landing in October 1942 and repeated close calls in dogfights with German fighters through the winter. He was finally shot down in March 1943 in support of the British Eighth Army over Tunisia. At Stalag Luft III, he joined escape committee preparations by assisting John Colwell with sand dispersal under the theatre. That winter, Flying Officer George Wiley took time to write home about his activities on the theatre floor (and below it).\n\n\"I've got an important part to play in one of our kriegie plays,\" he wrote, \"and am a bit nervous about doing my part well. May see you sooner than expected.\"[42]\n\nBy the middle of February, Unteroffizier Karl Griese, the ferret the kriegies had nicknamed \"Rubberneck,\" was snooping more suspiciously than usual around the North Compound barracks. He periodically ordered impromptu appells in the middle of the morning or the middle of the afternoon. Big X had warned the section chiefs to be prepared for these unscheduled roll calls. To help attract suspicion to himself\u2014and away from the others\u2014Bushell made certain he was spotted in innocuous pursuits, such as attending language classes or rehearsing his role in that upcoming production of Pygmalion. When the spot searches came, the kriegies made sure they dawdled en route to the assembly area, a tactic that allowed the tunnel crews enough time to be pulled from \"Harry\" and cleaned up before appell. That month Rubberneck sprang a sudden search in Hut 104, then one in Hut 110. Then he assembled Wally Floody, George Harsh, Wings Day, and Roger Bushell and strip-searched them. Next, he brought in a diviner who passed the divining rod over the ground around several of the huts. There wasn't the slightest twitch. In the last days of February, when \"Harry\" was perhaps a hundred feet short of its run beyond the wire, the escape committee learned that its chief nemesis, Rubberneck, would be on leave for two weeks.\n\n\"I told Bushell] we could finish 'Harry' before he got back,\"[[43] Floody said.\n\nBut Rubberneck had a parting shot and delivered a nearly fatal blow to X Organization before taking his leave. On February 29, during the morning appell, the pesky ferret appeared with Hauptmann Broili and thirty additional guards. They called out the names of nineteen kriegies, including Wally Floody, George Harsh, Peter Fanshawe, Kingsley Brown, MacKinnon \"Mac\" Jarrell, Gordon \"Nic\" Nicoll, Robert Stanford Tuck, Jim Tyrie, and Gwyn Martin. The entire appell ground of kriegies held its collective breath as Broili led his select group to Hut 104, on the very doorstep of \"Harry.\" The nineteen were searched for two hours. Then, without any opportunity to go to their rooms to gather belongings, the column of kriegies was marched under guard through the main gate and down the road to a satellite POW camp at Belaria[44] , about five miles away. In one short, sharp dragnet, Rubberneck had hauled away a half-dozen key members of the escape committee, some of whom had been in the service of X Organization since 1940, investing those four years in one real chance to gain their freedom.\n\n\"They just wanted to get rid of us,\" Wally Floody said. \"But they had a pretty good shot at it, because they got the man in charge of sand dispersal, the man in charge of security, an intelligence specialist], and myself, a tunnel digger.\"[[45]\n\nDarker more deadly counter-escape measures were occurring in Berlin, even as the dust settled on Rubberneck's surprise purge. Because Heinrich Himmler, Reichsf\u00fchrer of the Schutzstaffel (SS) and architect of the Holocaust, wished to put greater control of prison camps into the hands of the SS, he gave his blessing to Aktion Kugel, or Bullet Operation. Clearly in violation of the Geneva Conventions, it stated that any recaptured escapee officers who were not American or British were to be chained and handed over to the Mauthausen Concentration Camp and either gassed or shot.[46] Himmler's directive also used the code phrase \"Stufe III\" or \"Grade III\" to cover up all reference to activities surrounding any recaptured POWs. The directive was not handed down as a document, but passed by word of mouth to Colonel Friedrich von Lindeiner-Wildau, the Kommandant at Stalag Luft III. The order also gave Lindeiner the authority, should he choose to use it, to execute recaptured POWs in his camp. Von Lindeiner apparently had no intention of resorting to summary executions. In fact, when he received the Bullet Order, he assembled senior officers, medical officers, and padres from all the Stalag Luft III compounds. As explicitly as he could, von Lindeiner relayed the content of the order, almost entreating the officers to halt all escape activity. For the North Compound, however, the die was already cast.\n\nOn March 1, 1944, with Rubberneck on leave and several Canadian veteran section heads now removed from the scene, Robert Ker-Ramsey took the lead in the tunnel. Deflated by the loss of key men, but bolstered by the opportunity to complete the final push while their ferret nemesis was away, the tunnellers went back to work. The crew underground doubled, with two diggers at the face of the tunnel, two in each of the halfway houses, a carpenter preparing the shoring, and a man on the air pump continuously. The parade of penguins to the back row of seats in the theatre and John Colwell's packers under the floorboards made the most of the shortened daylight hours and the 10 p.m. curfew. In just nine days, by March 9, the tunnel had extended the one hundred feet that\u2014based on the underground measurements\u2014they figured would put \"Harry\" beyond the wire, beyond the road, and well into the pine forest. On the tenth day the diggers carved out what would be the base of the vertical shaft to the surface. Over the next five days they gingerly dug upward and\u2014just as Wally Floody had done in the first hours of the tunnelling downward a year before\u2014at the upper end of their vertical dig they built a final solid box frame around four bedposts and a wooden ceiling. It was positioned right below some pine-tree roots, to remain in place until the night chosen for the breakout.\n\nThey had tunnelled for eleven months\u2014from April 11, 1943, to March 14, 1944. They had removed and dispersed several hundred tons of sand from three major tunnels. Scrounging from every corner of the compound, kriegies had incorporated 4,000 bed-boards, 90 double bunk beds, 1,212 bed bolsters, 1,370 battens, 1,699 blankets, 161 pillow cases, 635 mattresses, 192 bed covers, 3,424 towels, 76 benches, 52 twenty-man tables, 10 single tables, 34chairs, 30 shovels, 246 water cans, 1,219 knives, 582 forks, 478 spoons, 1,000 feet of electric wire, 600 feet of rope, and 69 lamps[47] into \"Tom,\" \"Dick,\" and, mostly, \"Harry.\" According to the measured ball of string the diggers unravelled in the tunnel, \"Harry\" covered 336 feet (nearly 400 feet including the two vertical shafts). They were just six inches away from the sod and roots of the forest floor well outside the wire\u2014six inches to freedom.\n\nThe theatre troupe made a couple of offbeat choices to complete its winter playbill. In March, they presented the farcical black comedy Arsenic and Old Lace.[*] Next came Escape, a 1926 play by celebrated British novelist and playwright John Galsworthy. The storyline followed the life of a law-abiding man who met a prostitute, accidentally killed a police officer defending her, and then escaped from prison. The POW production featured longtime kriegies Peter Butterworth as the shopkeeper, John Casson as the parson, and, of course, taking on the female leads were John Dowler, Malcolm Freegard, and Tony Pengelly.[48] No doubt the irony of the plot occurred to the kriegie performers as well as the German officers seated in the first two rows. But nobody made mention of it. Not even Pengelly had much of the play's subject matter on his mind during the production.\n\n\"Up until this last great escape plan was well underway, none of us knew how many were to go out in it, or who,\"[49] he said.\n\nDuring a two-hour meeting in the library room of Hut 110 on March 14, the same day Rubberneck returned from leave, Big X led discussion about the timing of the breakout. The escape committee considered three possible dates\u2014March 23, 24, and 25\u2014the next three nights without potential exposure by bright moonlight, the New Moon period. March 25 was a Saturday, which likely meant additional train traffic and potential congestion along some rail routes through Sagan. They would wait to see what the weather brought on March 23 and 24. The section heads debated whether a mass escape in bad weather, freezing nighttime temperatures, and with snow on the ground might jeopardize any hard-arsers' attempts to get away. Did it make sense to delay a month? The tunnel experts couldn't guarantee the construction integrity of \"Harry's\" shafts, ceilings, and walls, not to mention the continuing danger of the tunnel's discovery. The decision was to go either March 23 or 24, depending on the weather. The committee hoped between nine o'clock on the night of the escape and 5:30 the next morning it could spring more than two hundred kriegies\u2014one every three or four minutes.\n\nThe final agenda item involved Big X and the section heads drawing the names of those who would comprise the list of escapers. The first thirty names selected came from a list of the best German speakers. The next twenty names came from the most prominent escape committee workers. Then, thirty more were drawn from a list of stooges, penguins, tailors, compass and mapmakers, and forgers. Finally, all remaining names of escape workers were pulled from a hat to bring the total number to about two hundred.[50] Escapers would go through \"Harry\" in the same order the names were drawn.\n\n\"When the time came close,\" Pengelly continued, \"we drew lots, intensely, in small groups. Mere slips of paper they were, holding the 'yes' or 'no' of freedom\u2014and for the lucky ones, how long he would be after the first to leave . . . I drew number ninety-three.\"[51]\n\nThe mass escape scenario that the committee chiefs and about six hundred other Commonwealth prisoners of war had built from scratch was just days away from its final act.\n\n* * *\n\n* Paul Brickhill was born and raised in Melbourne, Australia; during the war, he was shipped to Canada to train in the British Commonwealth Air Training Plan. Serving as a Spitfire pilot in RAF 92 Squadron, he was shot down over Tunisia in 1943 and sent to Stalag Luft III, where he joined the security section of X Organization. Post-war, his book, The Great Escape, provided the first comprehensive telling of the POW and escape experience at Luft III.\n\n* An air force officer arrived at Stalag Luft III in the winter of 1944 with unused tickets in his pocket to a production of Arsenic and Old Lace being staged at the Hudson Theatre in London. His tickets were honoured at the North Compound theatre.\n\n# 8\n\n\"THROUGH ADVERSITY TO THE STARS\"\n\n* * *\n\nHolding the slip of paper with his escape number on it, and sensing all his work falsifying documents for dozens of his comrades at Stalag Luft III was nearly done, F\/L Tony Pengelly still agonized over his shot at freedom. He hadn't seen his fianc\u00e9e in England for more than three years. He hadn't seen Canada since 1938. All the amenities he'd gone without\u2014the food, the people, the shows, the lights, the freedom to open a door and walk down any street whenever he wished\u2014could be waiting at home in Toronto.\n\n\"It was the greatest decision of my life as a prisoner of war,\"[1] he said.\n\nWeighing on his mind, however, were the details of duty. He had directed the production of many of the escape documents. He knew their design, detail, and delivery better than almost anyone inside the forgery section of X Organization. He wondered whether\u2014on the night of the escape\u2014somebody in his branch of Dean and Dawson should stay behind to check that every identification card was in the right hands of the right escaper as he entered \"Harry\" on his way out. Nothing could be left to chance. No man could leave with papers that didn't match his disguise. And if Pengelly took his position\u2014ninety-third in the tunnel\u2014and left the job to someone else, might one vital detail be omitted? Could there be a mistake he might have caught? Should someone with his seniority stay behind to help the next escape?\n\n\"In the nights when the barracks] were quiet,\" Pengelly said, remembering that crucial moment in Hut 104, \"I ground it over in my mind. . . . I realized in those nights I lay awake and those days I pounded the circuit inside the wire, that more than a high wire fence had me prisoner now. There was this responsibility, and on my acceptance or rejection of it, depended my chance at freedom.\"[[2]\n\nIn the end, Pengelly decided to forfeit his spot. Just twenty-four, engaged to be married to Pauline Robson, and a POW in Germany since the fall of 1940, the wily pilot who'd cajoled countless German guards to loan him their legal papers, and then stepped on the North Compound theatre stage to flawlessly portray female roles, had decided to bow out of the finale. Instead of joining the coming mass escape, he would stay behind. In doing so, he would help give the escape committee a nucleus of old hands to ensure the success of the current breakout and to build up X Organization again for another.\n\nThe same kind of decision weighed heavily on navigator George Sweanor's mind. A year older than Pengelly, he had even more to hurry home to, including a new bride and a newborn daughter he'd never seen. But the young RCAF flying officer considered other factors. Despite the sophistication of the security operations in which he'd participated (as the duty pilot inside the main gate), Sweanor still concluded that escaping to Britain was a pipe dream. He realized Stalag Luft III was too deep in occupied Europe and that only those who spoke German had a legitimate chance of getting away. He was also realistic enough to fear the Gestapo's brutality should he be recaptured. He also sensed he had reason to fear German propaganda that had painted Allied aircrew as Luftgangsters (air gangsters) and their bombers as Terrorflugzeuge (terror aircraft); survival in cities decimated by Bomber Command was not a sure thing either.[3]\n\n\"I argued that a mass escape would cause a desired disruption to the German war effort because it would take a lot of people to track us down, but there would be little hope of anybody getting home,\" Sweanor said. \"So] I felt relieved when my name was not drawn.\"[[4]\n\nA couple of Sweanor's barracks mates did have their names drawn, including George McGill, who had conjured up diversions during earlier escape attempts and then worked as tunnel security assistant to George Harsh; he would be seventy-fifth into the tunnel.[5] Other Canadians' names drawn or chosen by Big X included Gordon Kidder, the Johns Hopkins University master's prospect who spoke German and French fluently; he was thirty-first on the escape list. James Wernham and George Wiley, who had both joined the theatre production crew during the past year, would be thirty-second and thirty-third. Hank Birkland, the Big Train, had dug tunnels in every German compound that had imprisoned him since the fall of 1941; he would go through \"Harry\" for the last time in the fifty-first position. Pat Langford, the trapf\u00fchrer responsible for opening and closing \"Harry\" almost every day for the previous eleven months was number fifty-eight. Tommy Thompson, the Canadian pilot who'd personally earned the wrath of Hermann G\u00f6ring for waking the Reichsmarschall the September night he was shot down in 1939, would be sixty-eighth; once outside the wire he planned to team up with Flying Officer Bill Cameron, who was in the sixty-second spot. Fighter pilot Keith \"Skeets\" Ogilvie had won a Distinguished Flying Cross (DFC) for intercepting a Dornier bomber en route to bomb Buckingham Palace during the Battle of Britain; he was seventy-sixth on the escape list. Bob McBride, who'd burned the bowling set his wife sent him the winter before to stay warm, would face the March cold in the eightieth position. A few more Canadians held escape numbers in the eighties and nineties, including Jack Moul, Red Noble and Mac Reilley. Further down the list, Gordon King, who had regularly operated the air pump, held escape number 141. Meanwhile, John Colwell, the Tin Man, who'd hand-built many of the tools of escape while making so much of the sand disappear under the theatre, would be 147. All set to travel outside the wire as Hungarian ironworkers, John Crozier and his roommate John R. Harris had drawn escape numbers 179 and 180.\n\nAs Pengelly's forgers put the finishing touches on the fake identity Kennkarte and Ausweise passes that Crozier, Harris, and many others would need, X Organization moved into fine-tuning mode on other fronts. John Travis's engineers weren't building tools and equipment for the tunnelling anymore; they were transforming sheets of food tins into portable water bottles for the escapers. Des Plunkett's assembly line had completed its last multicoloured, mimeographed maps. For Tommy Guest's tailors there was still civilian wear to stitch and colour, but Al Hake had punched his Stalag Luft III manufacturer's imprint on the last escape compass. Robert Ker-Ramsey and Johnny Marshall began assembling the escapers in small groups to explain how to get through the tunnel. Roger Bushell called for one last levy on the Red Cross parcels, with every ounce of sugar, cocoa, raisins, milk, and biscuits being confiscated and poured into a stewing pot and then a baking oven; the resulting fudge, prepared in Hut 112, could provide a man with enough caloric intake to last him two days.\n\nOtherwise, Bushell remained away from the escape production centres as much as he could\u2014conducting lectures in the library and rehearsing his lines as Professor Higgins in the upcoming theatre production of Pygmalion\u2014so that the guards could see him in a normal, innocuous routine. But Big X had one final piece of the puzzle to orchestrate: the logistics for getting more than two hundred men into Hut 104 on the appointed night and through the tunnel in an orderly escape. The first thirty to make their way through \"Harry\" would attempt to get to the Sagan station in time to catch the earliest trains. Then, using their natural linguistic skills, civilian disguises, and forged papers, they would board passenger coaches and scatter in a half-dozen different directions. The next seventy to make their way through, the so-called \"hard-arsers,\" would be dressed in work clothes, looking like migrant workers in transit. If they couldn't get tickets for third-class passenger seats, they would resort to using compasses, maps, intuition, and darkness to make their getaway individually and in small groups.\n\nAir gunner Albert Wallace had no seniority with the escape committee, but he had served as a penguin hiding sand under the theatre, and could have put his name in the draw. He chose not to. He recognized in himself and others the pent-up frustration of extended imprisonment in the German Straflager system. When life boiled down to twice-a-day roll calls, scrounging for food, and shivering inside poorly insulated barracks, he saw comrades become \"barbed-wire happy,\" obsessed with getting out. A Canadian kriegie who had notoriously attempted an escape by cutting through the wire was put in solitary for a month. The resulting claustrophobia turned a trip to the toilet into a run for it; he was shot, though not fatally.[6] Two others had committed suicide, one at Stalag Luft I and the other at Luft III.[7] As depressing and debilitating as compound life became, however, Wallace felt he'd kept his head.\n\n\"I had no interest in escaping whatsoever,\" he said. \"We were eight hundred miles from England. Escape through Germany with eighty million people all speaking German and I spoke English? It would take a miracle to get back.\"[8]\n\nHaving spent a little less than a year behind wire at the North Compound, Canadian Spitfire pilot Frank Sorensen was nonetheless feeling the tyranny. Through the winter of 1944, his letters home more often reflected the symptoms that Albert Wallace had seen in some of his roommates. Sorry Sorensen wrote home that the winter had forced the POWs indoors and that \"indoor life in a kriegie camp does not make time go any faster.\"[9] Fortunately, Sorensen had the benefit of fluency in Danish and therefore had received the nod to exit through \"Harry\" in the first or second wave of escapers on the night of the escape. However, several factors influenced Sorensen's eventual decision to forfeit his higher position on the list. X Organization intelligence suggested the first escapees\u2014fluent in German and dressed like businessmen or travellers\u2014stood the best chance of getting away safely if they caught the fast morning trains leaving Sagan. Sorensen deduced those same express trains would also undergo the greatest scrutiny and surveillance by German police and railway guards. He therefore considered going through the tunnel lower on the list to catch a later, slower train, where his presence would attract less attention.\n\nBut Sorensen weighed yet another consideration. Among his closest friends inside the wire was James Catanach, an Australian officer who'd grown up in a tightly knit family, and Arnold Christensen, a New Zealand officer who shared Sorensen's Danish heritage. Both Catanach and Christensen had been imprisoned longer than Sorensen had, and in the grand scheme of the mass breakout, their successful escape seemed to hold greater emotional significance, at least in the way Sorensen looked at it. So, for strategic purpose or matters of the heart or both, he chose to trade his earlier spot on the escape list for a higher number and later exit through the tunnel.[10]\n\nCanadian Fleet Air Arm pilot Dick Bartlett faced a slightly different dilemma before the breakout. For nearly four years he had successfully moved the kriegies' wireless radio, hidden inside a medicine ball, from one prisoner-of-war camp to another, right under the Germans' noses. At the North Compound, he had safely concealed \"the canary\" in a non-functioning toilet. Based on his service, Bartlett was assigned the sixteenth position on the escape list, and paired with Norwegian pilot officer Halldor Espelid in the fifteenth spot. Then, in the weeks leading up to the breakout, another Norwegian flying officer, Nils Fugelsang, was shot down and imprisoned at the North Compound. Bartlett figured that Espelid and Fugelsang, with their linguistic advantage, together stood a better chance of evading capture and getting home. Bartlett chose to give up his spot to Fugelsang and stay behind.[11]\n\nCircumstances leading up to the breakout also put an end to Barry Davidson's hopes of escaping through \"Harry.\" The former Blenheim pilot shot down in July 1940 had served on the original escape committee and in time had become X Organization's leading scrounger. A born baseball and hockey player, but also a poet, diarist, and artist, Davidson had shone brightest in the art of friendly persuasion, taming Germans guards\u2014offering them Red Cross chocolate, coffee, and cigarettes\u2014to lend him the makings of tools or identity documents. In the pre-breakout lottery, he had won the seventy-eighth position on the escape list. But then, almost on the eve of the breakout, his strength became his weakness.\n\n\"I had been seen talking to Fischer, one of the guards, shortly before the escape,\" Davidson said. \"He hated the Nazis and had sympathy for the POWs. We had such a good security system that X Organization] knew the Germans had seen me talking to him. . . . My relationship with this guard would have risked his life had I gone. . . . So Roger Bushell asked me if I'd step back and not go out.\"[[12] Reluctantly, Davidson agreed.\n\nA couple of days past the official arrival of spring, the Sagan area of Silesia still had six inches of snow on the ground. But the air above the ground was mild. The escape committee met in Hut 104 on March 23 and decided to delay the breakout one more day. There was more snow that night, but when the committee met in Hut 101 on the morning of March 23, the members knew a decision had to be made right away to give Ker-Ramsey the day to prepare \"Harry\" for the wear and tear of the escape and to allow Pengelly the time to have all required documents signed and date-stamped \"March 24.\" More discussion focused on the plight of the hard-arsers in the snow and cold of the night. Wings Day and Roger Bushell agreed the hard-arsers' chances of escaping were slim anyway, but even if they were only on the loose for a few days, the resulting chaos across Germany rounding them up would have as desirable an effect as if they all got back to Britain. Bushell gave the decision his blessing and then walked to the North Compound theatre where his understudy, Kenneth Mackintosh, got the word Big X would not be onstage as Professor Higgins that night. Pygmalion would have to open without him.\n\nAll day long the atmosphere was electric at the North Compound. Behind closed barracks doors\u2014all with stooges at the watch\u2014kriegies collected their forged maps, compasses, and food rations and stitched them into clothing pockets. Meanwhile, Ker-Ramsey, staying behind like Pengelly as an escape committee veteran, made last-minute adjustments underground\u2014covering the trolley tracks with fresh blankets to muffle the sound, installing new tow ropes (passed through the main gate by the Vorlager on the premise they would be used for a North Compound boxing ring) on the trolleys, and installing light bulbs (taken from the huts) in every socket available along the full length of \"Harry.\"\n\nAfter sunset, about six o'clock, those on the escape list had last meals prepared by their roommates. John Travis, the tunnelling engineer, cooked up a concoction of bully beef fritters and a gruel of boiled barley, Klim powder, sugar, and raisins for two of his barracks mates\u2014Roger Bushell and Bob van der Stok. Van der Stok, the Dutch flyer, was going out in the first twenty of the escape order. When they'd finished eating as much of Travis's fritters and gruel as they could, Bushell got into the suit he'd smuggled into the compound from Prague a year before and van der Stok emerged in his escape apparel; unlike the fake German corporal's uniform he'd used during Operation Bedbug, van der Stok wore a civilian business suit, handmade by Tommy Guest's tailors.\n\n\"How do I look?\"[13] he asked his roommates.\n\n\"Immaculate,\" Gordon King told him as they examined the tailoring and the quality of his passes and identity documents forged by Dean and Dawson. King knew van der Stok had been a medical student in Holland prior to the war and spoke several different languages, including German. He planned to connect with the French Underground to get across the Pyrenees to Spain en route to Britain. He was going through \"Harry\" in the eighteenth spot.\n\nIn Hut 112, George Wiley had gathered up his things and joined fellow Canadian James Wernham in 104 to await their call. Wiley appeared anxious about his first escape attempt and approached his roommate Alan Righetti. He handed Righetti his watch and a few other personal things, asking him to pass them along to his mother back in Windsor, Ontario, if things didn't work out. Righetti, a veteran of earlier escape attempts, joked that Wiley would likely be home before Righetti, but he accepted the watch and Wiley's final wish.[14]\n\nJust before 7 p.m., the clockwork movement of men to Hut 104 began. Under the direction of block commanders at each hut in the compound, and with stooges positioned at windows, men trickled into Hut 104 at thirty-second intervals. They were assigned rooms in which to wait for their escape number to be called. Once that happened, they were guided into Room 23 and to the trap under the stove. The committee had appointed marshals among the escapers; each marshal ensured his allotted ten men were all set. Within an hour the priority escapers were in position, ready to go. And even though their spots were way down the list, so too the hard-arsers began to make their way to Hut 104.\n\n\"In the room where we were, we tried to play bridge,\" John Harris said. \"I was with three other would-be escapers. I was with Johnny Crozier because he was the one with the set of maps.\"[15]\n\nAt 8:45 p.m., the first man in the escape sequence, Les \"Johnny\" Bull, hustled down the ladder in the shaft beneath Hut 104, stretched himself face down on the trolley, and dog-paddled his way to Piccadilly, the first halfway house, a hundred feet up the tunnel. Once there, he jerked the rope for Johnny Marshall, who would act as an underground conductor for the first hour and\u2014once his ten men were through\u2014exit the tunnel in the number eleven position. Marshall retrieved the trolley by reeling in the rope, climbed aboard, jerked the rope ahead, and Bull reeled him up to Piccadilly. They repeated the exercise to get to Leicester Square, and finally to the base of the exit shaft at the north end of \"Harry.\" Their job was to remove the ceiling boards at the top of the exit shaft and make the final cut through the sod in the pine forest. Behind them, Czech flyer Arnost \"Wally\" Valenta (second), Roger Bushell (third), and Bernard Scheidhauer, a Free French officer and fourth on the list, prepared to do the same. At the bottom of the entry shaft two Canadians got into position in the pump room. Trapf\u00fchrer, Pat Langford, would help get men up the tunnel and reel in the empty trolley; then, with his group of ten on their way, he would take his turn as the fifty-eighth on the list. Meanwhile, Gordon King, the diminutive Wellington pilot from Winnipeg, volunteered to pump the bellows through the night until his turn came, way down the list.\n\n\"I was a hard-arser,\" King said. \"I had a map of the area, a little package of food, and my compass, just waiting my turn.\"[16]\n\nAlso in position along the escape route through \"Harry\" were experienced tunnellers Red Noble, Shag Rees, and Hank Birkland. They had agreed to position themselves at the halfway houses and haul ten men to their location before joining the escape. The eleventh, twelfth, and thirteenth men in the order\u2014also with underground experience\u2014would haul the next ten through and hand off the responsibility to the next three.[17] The plan called for a controller to remain outside the exit hole in the woods to get twenty men on their way and then hand off the controller job to the twenty-first man. Everybody was in place. The only piece missing was the completion of the exit hole up through the sod in the woods.\n\nAbout ten o'clock, the German lock-up guard from the Vorlager went through the North Compound closing shutters and barring the hut doors from the outside for the night. Shortly afterward, King, Ker-Ramsey, and Langford began to wonder what was wrong. Seventeen escapers had been in position since 9:30, but nobody was moving through \"Harry\" yet. The first two men up the exit shaft\u2014Les Bull and Johnny Marshall\u2014had been battling for thirty minutes to open the exit to the tunnel, but the wooden ceiling at the top of the shaft was wet, swollen, and wouldn't budge. Bull and Marshall, both dressed as civilians bound for the early morning trains, had to strip down to their underpants so as not to soil their business suits and try to free the wood and break through the soil above. Bull finally succeeded in prying the boards loose, and then, using a small shovel, broke a hole through the soil and snow above. He felt the cool air of the winter night on his face as the hole widened and he could see the night sky above, the stars unobscured by wire. Up the last rungs of the ladder, he poked his head through the hole. The sight was both awesome and terrifying.\n\nJust as the tunnel engineers had planned, the exit to \"Harry\" was well outside the wire. However, the reason Bull could see so many stars as he peered up the exit hole was that there was no tree canopy blocking his view. There was nothing overhead. \"Harry\" was short of the pine forest by ten feet. Worse, the goon tower was just forty feet away. Worse still, the path of two sentries posted outside the wire\u2014one pacing between the tower and the gate to the east, the other walking to the west fence and back\u2014passed within thirty feet of the exit hole. Even without moonlight, at ground level a sentry would spot anyone moving in the open against the blanket of snow in an instant. All this Bull reported to Marshall and Bushell, who'd by now reached the base of the exit shaft. Amid the frustration and panic rising among the lead group, Bushell weighed the options. They could put the whole thing off and dig farther, but the digging and dispersal units had virtually shut down and were in escape mode. And with the compound-wide final preparations, there was a good chance ferrets would stumble across something to draw them to Hut 104 and \"Harry's\" trapdoor. Then it dawned on Bushell: they couldn't postpone the escape. All the documents were dated March 24, 1944.\n\nJust as quickly, the solution dawned on Les Bull. At the edge of the woods, closest to the exit hole, lay a blind, a tangle of timber and brush. Birkland, Clark, and Harsh had spotted the ferrets hiding behind it spying on kriegies during the push to complete tunnel \"Tom\" the previous summer. Bull concluded if they attached a signal rope from the top of the escape shaft ladder, up through the exit hole, and out to the blind, a controller behind the blind could signal with a tug on the rope when the way was clear of German sentries. It was settled in an instant. They tied a new rope to the top rung of the ladder and\u2014when the sentries were out of view\u2014Bull unravelled the rope to the blind and prepared to signal the next man, Johnny Marshall, through. Bushell quickly scribbled the sequence on the wall at the base of the shaft, so each man would read the note and know what to do.\n\n\"Pause at top of shaft. Hold signal rope tied to rung. Receiving two tugs, crawl out. Follow rope to shelter,\"[18] it said.\n\nBull got into position behind the blind in the woods and waited for the sentries to come back into view, stop at the tower, turn, and retrace their steps out of view. He tugged twice on the rope and Wally Valenta popped up, wriggled through the snow across the distance to the blind, and entered the woods. A few minutes later Roger Bushell, number three, did the same. Big X had planned to travel with Robert Stanford Tuck, but with the British ace purged to Belaria a month earlier, his new partner was Bernard Scheidhauer, the fourth man through the exit and into the woods. The improvised \"controller in the blind\" system meant the interval between men was now longer than the planned two or three minutes, but at least the system was working.\n\nDeeper into the pine forest, the kriegies stood up without fear of being seen. For two years those pine trees\u2014a green wall\u2014had been a physical and psychological barrier keeping them from the outside world. Now it concealed them from the compound they'd just fled. Among the first groups assembling in the woods were Wally Valenta, Johnny Marshall, Des Plunkett, and Freddie Dvorak, all bound for Valenta's and Dvorak's homeland, Czechoslovakia. Next, Roger Bushell and Bernard Scheidhauer, who planned to hook up with the French Underground in Alsace and Paris. Shortly afterward came the two Norwegians, Halldor Espelid and Nils Fugelsang, who'd taken Canadian Dick Bartlett's sixteenth spot, along with New Zealander Arnold Christensen and James Catanach, one of whom had switched with Canadian Frank Sorensen; they were headed through Berlin to the Danish frontier. The eighteenth man out of the tunnel, in his \"immaculate\" suit, was Bob van der Stok; he too was making his way toward the station with intentions of getting across France and through Spain back to England. They exchanged \"see you in London\" send-offs, and then dashed into the night. They had all been instructed to vary their routes into the railway station. There were three options: stairs to a walkway west of the station, directly across the tracks to the eastern entrance of the station, or through a subway that ran under the tracks into the station.\n\nBushell and Scheidhauer as well as Plunkett and Dvorak emerged from the woods and approached the station just as an express train en route from Breslau to Berlin arrived at Sagan. The first two entered the station in plenty of time to purchase their tickets and board. Meanwhile, Plunkett and Dvorak were blocked at the eastern entrance to the station by a passing Russian work party; when they diverted to the subway under the tracks, they were stopped by a railway worker and a guard. However, at exactly that moment, there was confusion all over the station and marshalling yard because an air-raid siren sounded and the station lights went out. The two kriegies took advantage of the moment and dashed aboard the express without tickets. Dutch airman Bob van der Stok got caught in the same confusion; he was just outside the southern entrance to the subway under the station tracks. A soldier questioned him, and van der Stok explained he was a foreign worker trying to board the train. Conveniently for van der Stok, the soldier insisted he follow instructions and enter the air-raid shelter, which in turn led him conveniently to the subway under the tracks to the station and boarding platforms.[19] But by now, the entire Sagan area was in blackout against a looming Allied bomber attack.\n\nIt was just before midnight when the sirens sounded at Stalag Luft III. Wings Day, twentieth in the escape sequence, had just begun his climb down the entrance shaft. By the time he reached the bottom of the ladder, the Germans at the Kommandantur had cut electrical power to the entire prison facility. Everything above and below ground fell into darkness. The single benefit from the blackout was that it snuffed boundary lights and tower searchlights too, which expedited the escapers' dash from the exit hole into the woods.\n\nBack inside the North Compound, however, \"lights out\" required hut shutters to be opened and fat lamps in the hut rooms doused; it also sent the hundf\u00fchrer and their dogs patrolling through the compound. Inside Hut 104, dozens of hard-arsers lay still in hallways and in bunks throughout the perceived danger of the air raid. With electricity cut off to \"Harry\" as well, tunnel travellers couldn't judge the distances, the whereabouts of the halfway houses, or even how close they were to the tunnel walls. Ker-Ramsey was quick to realize the additional danger of throwing already jittery escapers into total darkness in such a confined space, and he immediately began lighting fat lamps along the tunnel. The escape schedule was falling further behind. Instead of taking two or three minutes, it was now taking a man at least six minutes to travel the full length of \"Harry\" and up the exit shaft.\n\nAs he moved northbound through the tunnel, Wings Day helped Ker-Ramsey light the lamps, then completed a hauler shift at the base of the escape shaft before exiting the hole himself. But the anxiousness around the blackout amplified the tension in the tunnel. Most of the first thirty-five men, including Canadians Gordon Kidder, James Wernham, and George Wiley, carried suitcases; while they were only made of cardboard or plywood, some were bulky and difficult to pass through \"Harry\" without making separate trolley trips. Some of the men and bags caused derailments of the trolley. Others bumped into shoring and caused small cave-ins. Tom Kirby-Green, thirtieth in the tunnel, derailed between Piccadilly and Leicester Square and his shoulders bumped a weakened shoring. It broke and sand caved in, burying him from his waist to his shoulders. Canadian digger Hank Birkland, on duty at Leicester Square, realized the problem, crawled back to Kirby-Green, and freed him from the sand. Birkland then worked feverishly in the darkness to clear the sand and replace the shoring boards. Bob Nelson faced a similar problem a short time later near Piccadilly.\n\n\"Due to the cold weather, people were wearing warmer, thicker clothes, and being hauled through . . . they pulled down some of the roof support and collapsed the sand,\" Nelson said. \"When I was hauling James Long] through, the roof fell on top of him. I had to pull him out and then when he got past me, I then had to go up the tunnel on my elbows and toes to repair the roof and clear the sand that fell in.\"[[20]\n\nAs Birkland completed his repair and dispersal of the sand that had collapsed on Kirby-Green, above ground the air-raid sirens were sounding the all-clear across the POW complex and the Germans restored the electricity to their tower searchlights and consequently to \"Harry's\" underground lights. That's when Birkland, the Canadian who'd spent much of his waking life the past year digging inside Stalag Luft III tunnels, travelled through \"Harry\" for the last time and got his first taste of life beyond the wire. He moved out of the tunnel and into the woods, marshalling the next ten men on their way. Then he joined British air officers Les Brodrick, Denys Street, Edgar Humphreys, and Paul Royle as they made their way westward, away from Stalag Luft III and Sagan.\n\nWith the lights back on inside \"Harry,\" at least those in transit could now see where they were going. However, now the escapers weren't lugging suitcases through the tunnel, but blanket rolls, additional layers of clothing, and pockets bulging with extra food and survival supplies. And if, in addition, the hard-arsers happened to be broad-shouldered, passage along the tunnel and through the halfway houses proved just as difficult and sluggish. Word went back up top to Ker-Ramsey, the above-ground controller, who began inspecting men as they were about to climb down the first shaft. As difficult a decision as it became, Ker-Ramsey had to relieve some men of their bulkier bedrolls, extra clothing, and survival supplies. That meant that hard-arsers such as Canadians Pat Langford, Bill Cameron, Tommy Thompson, George McGill, Keith Ogilvie, and Bob McBride had to dump some extra clothing or blankets they were carrying for warmth, but they did so sensing a stronger urge to escape than to carry protection against the March cold.\n\n\"They hoped for the sake of fellas like myself, going hard-ass, that . . . our chances of hiding out in the woods or getting something to eat would be a little better,\" Ogilvie said, \"but it was really cold and frosty.\"[21]\n\nBy four o'clock in the morning, the experienced members of the escape team had been clearing documents, hauling trolleys, rescuing escapers, re-shoring walls and ceilings, re-dispersing sand, and coaxing the system for more than seven hours. About sixty men had been relayed through \"Harry\" in that time. Dawn was less than two hours away, so Pengelly and Ker-Ramsey made the decision to call off anyone holding an escape number greater than one hundred. They were all told to stow their forged documents away, eat their escape rations, and hide whatever escape clothing they could. The escape committee hoped it could squeeze twenty or twenty-five more men through \"Harry\" by 5 a.m. and then close the tunnel down before the Germans learned anyone had escaped.\n\nMeanwhile, there was another holdup at the exit hole. Roy Langlois had just relieved Canadian George McGill at the ferret blind in the woods when he signalled another halt in the flow of escapers; he'd heard a shout coming from the guard tower. One of the sentries pacing along the fence was summoned by the tower guard\u2014the two had apparently agreed to switch places for some reason. Then the tower guard climbed down, crossed the snow-covered road, and began walking directly toward the tunnel hole. Langlois, still hiding behind the blind in the woods, thought for sure the goon had spotted the steam rising from the tunnel hole. He was coming straight for it. Then, suddenly, the guard stopped, took down his trousers, and squatted to relieve himself. Five minutes later, the guard departed, Langlois tugged on the rope, and escapers began moving again.\n\nIt was close to 4:50 a.m. when the controllers in Hut 104 decided it was time to shut things down. They needed to determine who the final escapers would be and give the trolley haulers\u2014Shag Rees, Red Noble, and Tim Newman\u2014enough time to haul them to the exit hole and then get themselves up the escape ladder and out as well. Then, the trap over \"Harry\" back in Room 23 of Hut 104 could be resealed and rooms around the stove returned to normal. It appeared the last group of escapers would include Lawrence Reavell-Carter, Keith Ogilvie, Michael Shand, Len Trent, Bob McBride, Roger Maw, Michael Ormond, Ian Muir, Clive Saxelby, Jack Moul, and Frank Sorensen. Mac Reilley had taken Tony Pengelly's number, ninety-three, but just missed the cut-off.\n\n\"There I sat in Hut 104, waiting the night away,\" Reilley wrote. \"Frank Sorensen . . . was sitting with his legs reaching for the ladder to descend into the tunnel when it all ended.\"[22]\n\nLanglois was still positioned at the blind just inside the pine forest, marshalling his group of ten men to safety. He directed Spitfire pilot Ogilvie and burly air-gunner Reavell-Carter to his position just hidden by the pines. Next, Michael Shand, another Spitfire pilot, emerged from the tunnel exit and began crawling through the snow toward Langlois at the end of the signal rope. Right behind him was Len Trent, the Victoria Cross winner from New Zealand. Suddenly, Shand and Trent felt the rope guiding them to the blind jerk again; it was Langlois, trying to get them to stop. A guard, formerly pacing close to the fence, had deviated from his sentry's path and was stepping through the snow right toward the escape hole, as if something had attracted his attention. Shand and Trent stopped dead in the path. The guard kept coming, nearly stepping into the hole; he still hadn't seen the two air officers prostrate on the snow. Then, recognizing Shand's human shape in the slush path to the woods, the guard lifted and aimed his rifle.\n\n\"Nicht schiessen! [Don't shoot!]\" Reavell-Carter shouted as he jumped from behind the brush pile in the woods.\n\nStartled by the kriegie shouting and leaping into the open, the guard pulled the trigger on his rifle, but the bullet didn't hit anybody. Reavell-Carter had no choice but to surrender. Trent did the same. When the guard finally shone his flashlight down at the exit hole, next to where he was standing, there was Bob McBride, perched on the top rung of the ladder and waiting his turn to crawl out; with no other option available, he also surrendered. To add to the alarm, the guard began blowing his warning whistle.\n\nThat night, Don Edy had slept fitfully in Hut 123, where tunnel \"Tom\" had originated. He remembered the eerie stillness of the dawn broken with that single rifle shot at the north end of the compound. He and his roommates didn't dare go to the windows to see the Germans' reaction. They just stayed in bed and listened.[23] George Sweanor had gone to bed fully clothed, wearing socks, boots; he'd even stuck food in his pockets[24] because he figured when the Germans got wind of the escape there would likely be an all-day appell and probably ration cuts. When he and his roommates heard the gunfire and whistle blowing, they pried open the shutters covering a window at Hut 119 and saw some of the escapers in Hut 104 attempting to race back to their home huts. More gunfire and the hundf\u00fchrer racing into the compound soon stopped that.[25] Inside Hut 104 the secretive quiet was suddenly broken.\n\n\"All hell broke loose,\" said Gordon King, who'd been at the base of the entry shaft pumping air through the bellows much of the night. \"Some of the kriegies] stupidly ran out of the hut. They could have been shot doing it.\"[[26]\n\nTunnel controller Ker-Ramsey and forgery chief Pengelly reacted to the rifle shot in an instant. They passed along the order throughout Hut 104 for all forged papers and escape kits to be destroyed. Within minutes there were small fires burning up and down the halls and in every room of the hut. The next order was a full retreat from the tunnel. In moments, Tim Newman was thundering back from Piccadilly halfway house to the entry shaft, Ian Muir right behind him. Michael Ormond wasn't long after. Finally, the trolley haulers\u2014Red Noble, Roger Maw, and Shag Rees\u2014began to withdraw. As he turned to retreat from Piccadilly, Rees was squeezed out of the way by the two escapers who'd been at the base of the exit shaft\u2014Jack Moul and Clive Saxelby. With the tunnel mouth discovered, Rees felt for sure there'd be a ferret on his tail in seconds. And yet each time he looked ahead all he saw was \"Sax's bum blocking the way.\"[27]\n\nWith the last man up the shaft and out of \"Harry,\" the trap was sealed as quickly as it ever had been over the previous year. The stove was replaced. And blankets that had been used to muffle the sound and hide any excess sand quickly disappeared into nearby bunks. By the time a hundf\u00fchrer entered the block, a few minutes later, much of the obviously incriminating evidence had vanished. The German guard gathered a few greatcoats lying around in the halls and waited. His dog curled up on the coats and went to sleep.\n\n\"Our mad haste was really unnecessary,\" John R. Harris said. \"A German guard coming on duty had accidentally sighted escapee Michael Shand] leaving the tunnel exit and fired a shot, more to attract attention . . . than in hopes of hitting the fleeing prisoner. By the time he succeeded in summoning support and the Germans realized what was going on, we could have burned half the camp down.\"[[28]\n\nFor the moment, the real action was happening at the exit end of the tunnel. The sentries were marching Bob McBride, Len Trent, Roy Langlois, and Lawrence Reavell-Carter to the guardhouse. In spite of the rifle shot, most of the compound guards inside were still asleep. As the guards were roused by the sentries, the escapers took advantage of a nearby stove and jettisoned their forged papers into the fire. When he arrived, Kommandant von Lindeiner was livid. He demanded to know the number that had escaped, which kriegies had led the escape, and where the tunnel had originated. Finally, he told the four foiled escapers the Gestapo would soon be involved.\n\n\"They will shoot you,\" he predicted, \"get rid of the lot of you.\"[29]\n\nOn the involvement of forces outside the camp he was right. Within an hour of the discovery of the tunnel, seventy German troops equipped with helmets, machine guns, and Schmeisser automatic pistols arrived at the compound and surrounded Hut 104. Even von Lindeiner, Hans Pieber, and Broili had their revolvers drawn. Over a hundred kriegies began to emerge from the barracks building to face an armed force that didn't seem the least bit interested in upholding the Geneva Conventions. Outside Hut 104, snow began to fall. The ferrets forced many of the men to strip to their skin. Every man was searched. All the civilian clothes accumulated or altered by Tommy Guest's tailoring crew were thrown into a pile, leaving most of the kriegies in little more than their underwear against the cold. John R. Harris was wearing clothes that looked more like his uniform, not civilian clothing, so he didn't have to strip. He tossed a blanket that hadn't been confiscated to a fellow kriegie without any clothes. Harris had forgotten, however, that he and his partner still possessed an obvious connection to the escape.\n\n\"Did you burn the maps?\"[30] Johnny Crozier, his escape buddy, suddenly asked.\n\n\"What maps?\"\n\n\"The ones I gave you inside,\" Crozier reminded him.\n\nDes Plunkett's mapmakers had given most of the hard-arser kriegies maps that would guide them away from Stalag Luft III. Since Crozier and Harris would be travelling together outside the wire, the pair had been given a set of nine stencilled maps showing them the country from Sagan south to the Czech border. But in the confusion following the rifle shot and the closing of \"Harry\" an hour earlier, both men had forgotten to destroy the evidence.\n\n\"You never gave me any maps, Johnny,\" Harris insisted.\n\nThat's when Crozier realized they were still in his pocket. The Germans could quickly determine the routes the other escapers might be taking if they suddenly came into possession of the stencilled maps in Crozier's pocket. He and Harris had to get rid of them. And fast.\n\n\"Fortunately, I had read many thrillers in my boyhood,\" Harris wrote. \"We could eat the maps. And that's just what we did, along with the help of any other prisoners to whom we could surreptitiously pass a map. Some of those fellows never spoke a friendly word to Johnny or me again.\"[31]\n\nNine stencilled maps were the least of the Kommandant von Lindeiner's worries. Inside Room 23 of Hut 104, a ferret who had entered the tunnel from the exit hole outside the wire had made his way back to the entrance shaft. With \"Harry's\" entire length now exposed and nothing left to hide, Red Noble opened the trapdoor to let the man out. Noble and Shag Rees were then ordered outside, where Rubberneck relished the moment of humiliating his nemeses, tearing off their civilian clothes. Noble and Rees pushed back, ready to fight. Rubberneck reached for his revolver. Von Lindeiner intervened and had the two kriegies marched, in the nude, to the cooler. But the German colonel had yet to face the toughest truth of the entire episode.\n\n\"At last an appell was called,\" Don Edy remembered, \"and everyone was turned out onto the sport field under heavy guard. The Germans were beyond themselves with rage. So we did nothing to provoke them further.\"[32]\n\nOut came boxes with the name and photograph of every prisoner in the compound neatly catalogued and filed. It took two hours, but every man was identified and accounted for as he stood in the snow. More stressful to von Lindeiner was the process of elimination: Who was missing? As the list of escapers rose eventually to the seventy-six who had managed to get away from the North Compound, George Sweanor watched the Kommandant's face grow pale. Eventually, he left the compound. Von Lindeiner knew that not only had he failed to keep prisoners of war from escaping, but he was also now in danger of violating the Aktion Kugel and Stufe R\u00f6misch III orders by not immediately handing over the four escapers his men had recaptured at the exit hole to the Gestapo.\n\n\"Gentlemen,\" Hans Pieber said to the kriegies as von Lindeiner left the compound, \"you should not have done that to him.\"[33]\n\nAfter the camp offices had completed the inventory of those present or absent, all remaining kriegies were dismissed back to their huts, except those from Hut 104. They stood virtually naked under guard for another hour before the Schmeisser-wielding guards allowed them to dress and re-enter their block. That afternoon, von Lindeiner learned that sixteen escaped officers had been recaptured and were being held at the Sagan civil police station. In spite of the two German High Command orders to the contrary, he demanded his prisoners be returned to the camp. He placed dozens more phone calls, notifying railways stations, airfields, border crossings, and even port authorities along the Baltic about the breakout.\n\nMeantime, Max Wielen, the head of Kripo at Breslau, issued Kriegsfahndung, a nationwide manhunt order, and then Grossfahndung, the highest alert to police stations across the region, advising them to hold recaptured prisoners under the Stufe R\u00f6misch III order. Just as Big X had hoped and expected, the manhunt diverted the energies of police, SS, armed forces, Hitler Youth, the Home Guard Air Raid Precaution personnel, and civilian searchers\u2014some seventy thousand Germans[34] \u2014away from the war effort to the recapture of the officers who'd escaped from Stalag Luft III.[*] Later the same day, Saturday, March 25, the German counter-intelligence chief, SS Major Br\u00fcnner arrived at the Sagan-area prison compound with orders to arrest von Lindeiner in preparation for his court-martial over the escape. The sixty-four-year-old colonel suffered a non-fatal heart attack over the matter.\n\nThe North Compound experienced an eerie calm after the original storm on Saturday. All the stories of the events of the mass escape the previous night circulated from hut to hut around the camp. However, scuttlebutt about the seventy-six escapers remained scarce. Von Lindeiner was replaced with a temporary Kommandant. Initially, that didn't change things, but the sequence of events that had sealed von Lindeiner's fate was about to overtake the kriegies and the escapers as well.\n\nA Canadian flying officer from Winnipeg was shot down the same weekend as the breakout. At twenty-six, Gordon Venables was older than most RCAF aircrew, and when he tumbled into a farm field that night he broke his leg. F\/O Venables came through the prisoner-of-war delivery system in the middle of all the post-escape upheaval. Luftwaffe medical staff set his leg, but he was soon on the move to Stalag Luft III. He was fortunate not to have been swept up and dispensed with in the Stufe R\u00f6misch order, but still endured a rough passage to the POW camp. German civilians were riled up by increased Allied bombing attacks on their cities and by the highly publicized mass escape.\n\n\"I had a broomstick for a cane,\"[35] Venables said. \"We arrived at the train station. People were everywhere. To my surprise, a compartment was cleared of passengers for [the guard and me. We] had to change trains at Frankfurt. The city had been heavily bombed . . . with severe damage and loss of lives. He told me to stay close to him as the people were very angry and liable to turn on me. He put me in the cab of the engine, away from the civilians [until] we arrived at the POW camp near Sagan.\"\n\nOn April 1, 1944, exactly a week after the breakout, the Gestapo arrived at the North Compound and the mass searches of men, belongings, and barracks huts resumed. As usual, the Gestapo men appeared ham-fisted as prison guards, until they started handing out summary judgments. When they inspected \"Harry,\" the first thing they spotted was the string of lights up the tunnel. They grilled all the German electricians involved in maintaining the camp, forcing two to confess that the theft of the wire had gone unreported. Gestapo men promptly shot the two men. Next they shot the supervisor for not acknowledging the loss and punished two more prison guards for possessing morsels of food that had come from the prisoners' Red Cross parcels. The newly installed Kommandant, Oberst Braune, announced there would be four appells a day. He closed the theatre, stopped all mail in and out, and dumped wagon-loads of raw human sewage down the entry shaft of tunnel \"Harry.\"[36]\n\nWhat kriegies had known as a \"spine-tingling sport\" at best, and an uncomfortable existence at worst, was taking a deadly turn.\n\n* * *\n\n* Paul Brickhill suggests information gathered from the tamed German guards at Stalag Luft III put the total number of Germans\u2014military and civilian\u2014involved in the search for the escaped POWs closer to five million.\n\n# 9\n\nTHE HATE CAMPAIGN\n\n* * *\n\nTHE FATE of the hard-arsers\u2014as Big X had accurately predicted\u2014was pretty much in their own hands. Or, in the case of New Zealand officer Michael Shand and Canadian officer Keith Ogilvie, their feet. When the guard outside the Stalag Luft III fence fired his rifle, Shand, who was already in the woods, bolted in one direction while Ogilvie crawled for about a hundred and fifty feet, stood up, and started to run in another direction.[1] Whether from fear of being caught by prison guards and interrogated by the Gestapo or just from the shock of the late winter conditions on his system, Skeets kept on running for most of the next forty-eight hours. Perhaps the last kriegie to get away that morning, he had planned to make his way toward Czechoslovakia, but all that mattered initially was to get clear of the compound and the search parties that would inevitably be on his trail. He simply ran west, but since snow covered much of the countryside, he tried to find less travelled roads to put as much distance between himself and the compound as quickly as possible.\n\n\"I ran out of food and it was still wet in the trees and the snow. And you couldn't sleep,\" Ogilvie said. \"Bloody miserable.\"[2]\n\nBy Sunday night, he'd managed to cover almost forty miles and had reached a major road heading south toward the Czech border. But the moment he left the relative cover of remote roads and forested areas, Ogilvie\u2014dressed in army battledress and a greatcoat\u2014was spotted by members of the German Home Guard near Halbau, Germany, and taken to a local inn. Knowing his escape bid was over, en route to interrogation, he tore up the maps and forged identification in his pockets[3] and discreetly dropped the pieces when the guards weren't looking. The inn proved to be a holding cell for several recaptured kriegies. Before long, Ogilvie was joined by RAF officers Charles Hall and Brian Evans as well as Canadian countryman Tommy Thompson. The three had linked up with another Canadian, Bill Cameron, outside the wire in the pine woods shortly after the Germans sounded the alarm at 5 o'clock on the morning of the escape. Within a few hours, Cameron began suffering from exposure to the cold. Hall, Evans, and Thompson wrapped him in all the warm clothing and blankets they could spare, left him some rations, and pressed on.[4] Cameron was soon recaptured near Sagan, the others by the Home Guard in the same region as Ogilvie.\n\nUntil early 1944, German police had held very little jurisdiction over recaptured prisoners of war and generally, after brief questioning, returned them to German Armed Forces; in short, the Gestapo could not punish them. In February 1944, however, Heinrich Himmler's Aktion Kugel, or Operation Bullet, had wrenched control of recaptured POWs from the military, in this case the Luftwaffe at Stalag Luft III, and given it to the Gestapo. Under Bullet, recaptured prisoners other than British or American were to be taken to Mauthausen concentration camp and exterminated. It didn't matter whether the escape occurred in transit, in a mass breakout, or singly\u2014all recaptured POWs would be turned over to the secret police, not the military authorities, as quietly as possible.\n\nBy coincidence, the same weekend of the mass breakout from Stalag Luft III, the leaders of the German Prisoner of War Directorate were en route to Berchtesgaden. They arrived at the F\u00fchrer's Bavarian headquarters in the midst of a tempest. Hitler, Heinrich Himmler, Hermann G\u00f6ring, and Wilhelm Keitel had been conferring since word of the mass escape from the North Compound had arrived. Amid the accusations and laying of blame, Hitler decreed that all recaptured escapers would be shot. G\u00f6ring protested. Ultimately, the weekend conference at Berchtesgaden decided on the number of Commonwealth air officers to be executed. That night, Himmler issued the \"Sagan Order\" to the POW Directorate and quickly altered the operations at police and military prisons across Hitler's occupied Europe. Those moves had life and death consequences for the prisoners of war detained in them.\n\nThe Sagan Order began by describing the increased number of POW escapes as \"a menace to internal security.\" Adolf Hitler had initially demanded all eighty Commonwealth air officers who made it out of the tunnel be shot. Ultimately, however, the Sagan Order decreed that \"more than half of the escaped officers . . . after interrogation . . . are to be returned to their original camp and to be shot en route.\" Had Hitler's order been carried out to the letter, the Sagan Order might have meant forty killings. But the arithmetic appeared to be lost in the Gestapo's eagerness to retaliate. Fifty would be killed. Finally, the order spelled out how the killings would be covered up, by declaring that \"the recaptured officers were shot whilst trying to escape.\"[5]\n\nThe man given the job of choosing which fifty men would die was SS Gruppenf\u00fchrer Arthur Nebe. At age fifty, the former First World War explosives soldier turned police detective had served the Nazi Party, the state police, Kripo, and the SS Reichssicherheitshauptamt, or RSHA. Three days after the escape from Stalag Luft III, Nebe began receiving daily telegrams on those kriegies the manhunt had recaptured. He then sat in his Berlin office with each POW's identity card and his record sheet and played god. The factors influencing Nebe's decision were the prisoner's age and life status: middle-aged with a family, he lived; not too young and unmarried, he died. In addition, if the POW had led groups of escapers because of his fluency in various languages\u2014a so-called \"linguist\"[6] \u2014the Gestapo considered him dangerous and therefore expendable. The air officer's place of birth was a deciding factor too: almost all men of non-British origin and \"an unduly high percentage of men from the Dominions\"[7] received Nebe's death sentence. His decisions were passed to subordinate Gestapo officials, first Max Wielen, the Kripo chief, and then to the chief of the Breslau Gestapo, Oberregierungsrat Wilhelm Scharpwinkel. It was up to Scharpwinkel to organize the execution squads and fulfill the Sagan Order.\n\nLate in the evening on Sunday, March 26, security guards escorted Keith Ogilvie and thirty-four other escapers to a Gestapo prison at G\u00f6rlitz, ironically near the Czech border where so many of them had been headed. Three days later, Gestapo interrogators there questioned Ogilvie about the escape and its organization.\n\n\"I'm a British officer and it's my duty to escape,\"[8] he parroted back, and then remained silent to further questioning.\n\nDuring his interrogation, an interpreter translated his statements as a female typist recorded them. When the questioning ended, the interpreter turned to Ogilvie and said, \"The young lady typist] said you are lucky. You have escaped in a soldier's uniform. Therefore you will be tried before a military court. The others will not be so lucky.\"[[9]\n\nTommy Thompson's interrogation at G\u00f6rlitz, also on March 29, proved more confrontational and frightening over the issue of whether he was who he said he was and the protection of his rights as a military officer. When the Gestapo interrogator demanded information about the breakout, Thompson refused to answer.\n\n\"I must warn you,\" the interrogator said. \"You are not the in hands of military authorities, but . . . the secret service. Anything might happen to you without protection and you may never go back to your camp.\"[10]\n\n\"Despite whose hands I am in, I [am] protected by the Geneva Convention,\" Thompson protested.\n\nThe interrogator half laughed, and even when Thompson presented his identity tags the man waved them aside. Ultimately, Thompson managed to convince his questioner that his clothes were indeed military dress. \"You are lucky,\" the Gestapo man said finally. \"You are recognized as military. The rest are wanted for civil investigation.\"\n\nThe contradiction of the next hours for Ogilvie, Thompson, and others, proved to be watching Commonwealth officers clearly dressed in the air force battledress (he saw George Wiley in his Canadian blue battledress and John E. \"Willy\" Williams in his Australian airman's tunic with flight-lieutenant stripes visible) being led away from the prison. Following his interrogation, Keith Ogilvie was returned to a cell with fellow escapers Charles Hall, Neville McGarr, and Paul Royle; the cell was so small that all four men had to either stand or lie down at the same time. A few days later, Ogilvie spotted a German corporal he knew from Stalag Luft III outside his G\u00f6rlitz cell.\n\n\"Say, Horst, when are you gonna get me out of this place?\" he asked.\n\n\"Oh, Mr. Ogilvie, tomorrow morning you'll go,\"[11] the guard said. Indeed, the next day Luftwaffe guards arrived and escorted Ogilvie and seven others back to Sagan. Only one of his cellmates\u2014Australian Paul Royle\u2014was with him; Briton Charles Hall was taken from G\u00f6rlitz prison March 30, and South African Neville McGarr disappeared April 6. Some time later, the eight survivors of the G\u00f6rlitz imprisonment and interrogation learned the fate of the other twenty-eight kriegies held there.\n\nOn Thursday, March 30, Oberregierungsrat Scharpwinkel and his staff arrived at G\u00f6rlitz to carry out the Sagan Order. First the Commonwealth officers\u2014Squadron Leader Ian Cross, Flight Lieutenants Mike Casey, Tom Leigh, and George Wiley, and Flying Officers John Pohe and Al Hake\u2014were collected by the dozen Gestapo men in civilian clothes and pushed into three waiting cars. Other recaptured officers looking through prison windows knew Hake was suffering from frostbite and assumed the men in civilian clothes were escorting the air force officers in the cars to a hospital. They were actually going to the local Kripo headquarters for interrogation.\n\n\"I went to G\u00f6rlitz,\" Scharpwinkel[*] said, \"for the purpose of getting a picture of the prisoners. As I speak English, I put one or two questions to the prisoners . . . Were they married? Had they children? I did not ask any questions about how the escape was organized, because I was not interested in that.\"[12]\n\nThe interrogations of the half-dozen Commonwealth officers went on for three and a half hours. Kriminalinspektor Richard Max H\u00e4nsel[*], who was in charge of the local Gestapo office at G\u00f6rlitz, attended the interrogations. He recalled that interrogators questioned the prisoners one at a time. He said there was no torture, but that the questioning covered name, rank, place and date of birth, previous occupation, unit, targets bombed, where shot down, duration of captivity, organization of the escape, source of civilian clothing, origin of false papers, names of other escapees, and how each had been captured. H\u00e4nsel said the interrogation included the confiscation of the prisoners' watches and then a blanket threat from Scharpwinkel directed at everybody\u2014prisoner and guard alike.\n\n\"Take care they don't get away,\" he said. \"Otherwise something unpleasant will happen to you. Or something unpleasant will happen to them.\"[13]\n\nThe interrogations wrapped up at 12:30 in the afternoon, when the POWs were again pushed into the waiting cars. As there was insufficient room in the three cars for all the Commonwealth officers and the Gestapo men, Scharpwinkel told H\u00e4nsel he would have to drive a service vehicle with two of the POWs and the Gestapo chief to their next destination. The three cars and service truck travelled along the autobahn to a wooded area five miles past Halbau. Scharpwinkel ordered the column of vehicles to stop and the prisoners got out to stretch their legs. H\u00e4nsel said he led the two officers in his service truck to the head of the column where the others had gathered and returned to the truck to retrieve his lunch from a briefcase. Two of Scharpwinkel's staff carried sub-machine guns as the Gestapo chief ordered the POWs deeper into the woods.\n\n\"The prisoners were placed in position,\" Scharpwinkel said, \"and it was revealed to them that the sentence was about to be carried out. The prisoners showed considerable calm, which surprised me very much.\"\n\nThe six Commonwealth officers\u2014several of them wearing their air force blue battledress, clearly military POWs\u2014stood stationary, side by side in the woods. With a hand movement, Kriminal Obersekretaer Lux[*] gave the order to fire and the execution squad fired a burst into the unarmed prisoners of war. Lux shot the prisoners as well, and by the end of the second salvo, the officers were dead.\n\n\"While I was eating my slice of bread and butter,\" H\u00e4nsel said, \"several shots were fired. I ran at once to the place in the wood, which was about 150 metres 500 feet] away and learnt . . . that the prisoners had attempted to escape and had been shot in the attempt. They lay sprawling in the wood about four or five metres [15 feet] from each other. . . . I myself do not suppose that the prisoners attempted to escape.\"[[14]\n\nAfter the shootings, Scharpwinkel ordered H\u00e4nsel to drive the service truck into Halbau to arrange for an undertaker to take away the corpses for cremation. When reports of the shootings leaked to the British Government via Switzerland, H\u00e4nsel was ordered to another conference, at which Scharpwinkel told him that they were to say that on March 30 their vehicles had broken down along the autobahn and that \"the prisoners used this opportunity to attempt an escape.\"[15]\n\nThe great escape had ended in murder for these six Commonwealth air force officers. Mike Casey, the brilliant RAF Blenheim pilot who in the camp had managed to hide the priceless forgery tools from the earliest days of the North Compound, died at that roadside. Also killed there was Australian-born Spitfire pilot Al Hake, who mastered technical drawing and metalwork so well that at Stalag Luft III he created the most sophisticated assembly line for the manufacture of compasses ever generated from thin air. Fellow Australian Tom Leigh was also murdered that afternoon. And New Zealand Flying Officer John Pohe, who had maintained a steadfast sense of humour throughout his imprisonment at Sagan by signing all official papers as a Maori tribesman named Porokoru Patapu.[16] British-born Squadron Leader Ian Cross had previously made an unsuccessful escape attempt, but this time paid with his life. As did Canadian George Wiley, who had no valuables for the Gestapo to confiscate that March 30 afternoon since he'd left his watch and a goodbye letter with his roommate back in North Compound the night of the escape. Somehow Wiley sensed it might end this way.\n\nThat night, Scharpwinkel's execution squad made its way back to the G\u00f6rlitz prison. Through the same evening hours, the undertaker's vehicles arrived at the roadside where Casey, Hake, Leigh, Pohe, Cross, and Wiley had been shot, and the bodies were carried away to G\u00f6rlitz to be cremated. Two days later, on April 1, the same group of henchmen, led by Kriminal Obersekretaer Lux, handcuffed ten more Commonwealth air force officers, drove them away from G\u00f6rlitz, and killed them en route to Sagan in a similar fashion. This group of murdered officers included Flight Lieutenants Edgar Humphreys, Cyril Swain, Charles Hall, Brian Evans, and Flying Officers Wally Valenta, Wlod Kolanowski, and Bob Stewart.\n\nAlso killed in this execution were Canadian X Organization committee members George McGill, Pat Langford, and Hank Birkland. McGill was the RCAF navigator told to bail out of his burning Wellington in January 1942 only to learn later that the pilot had managed to get the bomber safely home to Britain; he'd spent twenty-six months behind wire, providing diversions for other escapers, and had co-led the team keeping all three tunnels secure night and day; he was dead at twenty-five. Also twenty-five, Alberta-born Pat Langford had survived severe burns bailing out of his Wellington in July 1942; a self-taught pianist and multi-faceted athlete, he'd served Canada as an air training instructor, a combat pilot, and at Stalag Luft III was the man responsible for keeping tunnel \"Harry's\" trap entrance secret and safe for a year. But perhaps none had given as much of himself to the \"operational function\" of escaping than Manitoba's Hank Birkland; shot down in his Spitfire in 1941, Birkland had tunnelled at Dulag Luft, Stalag Luft I, and Stalag Luft III, and coped with claustrophobia in a hole, vomiting from lack of oxygen, and the challenge of tearing tons of earth from beneath his captives' feet to help his air force mates escape. At the end of three years underground, Big Train had tasted only a few hours of freedom for himself.\n\nIn total, Lux's shooters were responsible for murdering twenty-seven of the officers recaptured following the mass breakout on the morning of March 25. Simultaneously, another execution squad assembled at a police prison in Zlin, Moravia, where two more of the Commonwealth officers had been brought for interrogation. Despite being temporarily buried in sand during the cave-in the night of the escape, Tom Kirby-Green had emerged from \"Harry\" thirtieth on the escape list. Canadian Gordon Kidder came through the exit hole right behind him. The two were masquerading as migrant Spanish workers; both were fluent in German, French, and Spanish, so their linguistic abilities put them high on Nebe's hit list. They had managed to get through the air-raid blackout at the Sagan station and over the next three days travelled by train unnoticed as far south as Hodonin, in Moravia, where they were recaptured. Severe interrogation went on for twenty-four hours. Then, about 2 a.m. on March 29, under the direction of Polizeiassistant Erich Zacharias, Kidder and Kirby-Green were loaded into two cars, apparently for the trip back to Breslau. One of the two Gestapo drivers, Friedrich Kiowsky, recalled the prisoners being manacled with their hands in front of them.\n\n\"As I was driving, I asked Erich] Zacharias what was going to happen to them,\" Kiowsky reported. \"Zacharias sat beside me and said nothing, but turned his thumb downwards. . . . At the same time he told me to drive slower and looked around the countryside.\"[[17]\n\nAbout six miles from Moravska Ostrava, Zacharias ordered Kiowsky to stop his vehicle at the side of the road. Without realizing it, Gordon Kidder played into Zacharias's hands by asking if he could relieve himself. Zacharias directed Kidder and Kirby-Green (from the second vehicle) with their guard Adolf Knueppelberg to the curb. It was all going according to the Gestapo plan.\n\n\"Knueppelberg raised his right hand holding the pistol, with the barrel pointing in the direction of the back of Kirby-Green's] head,\"[[18] Zacharias said. \"I drew my service pistol, which was all ready for firing . . . and fired obliquely in the left side of my prisoner Kidder] in order to hit his heart.\"[[*]\n\nTo complete his action, Zacharias fired a second shot at Kidder's head, then crouched to check for a pulse and called to Knueppelberg[**] to make sure that Kirby-Green was also dead. When the first shots were fired, Kiowsky, then lighting the cigarette of the fellow driver, turned to see what had happened. First he saw blood all over the snow, then the two air force officers lying dead in the ditch, and finally the two senior Gestapo officials removing the handcuffs from Kidder and Kirby-Green. Half an hour later, a van from the Czech police force arrived to pick up the bodies. Gestapo higher ups told all those present to report that the two air force officers had been shot while attempting to escape.\n\n\"I saw nothing that gave me the impression that the officers had wished to escape or had made the attempt,\"[***] Kiowsky said. [19]\n\nAmong the other Commonwealth officers murdered outside Hirschberg by Lux, was James Wernham, the Winnipegger who had survived the first thousand-bomber raid on Cologne in May 1942 and then the rigours of numerous productions in front of kriegies at the North Compound theatre. Meanwhile, Nils Fugelsang and fellow Norwegian officer Halldor Espelid, who'd been given Canadian pilot Dick Bartlett's escape number just before the breakout, together had attempted to make their way to Denmark; they were recaptured near Kiel on March 26 and then handed over to a hired Gestapo gunman, SS Sturmbannf\u00fchrer Johannes Post.[*] In a most macabre sequence of events, Post interrogated Espelid, Fugelsang, New Zealander Arnold Christensen, and James Catanach, the Australian bomber pilot decorated with the DFC and promoted to squadron leader in 1942 when he was only twenty.\n\nFollowing his orders with particular zeal, Post had manacled the four air force officers with handcuffs behind their backs and loaded them into two cars\u2014Espelid, Fugelsang, and Christensen into one vehicle, and Catanach into the other, Post's own staff car. En route through the city of Kiel, Post had directed his driver stop at a residence so that he could pass on a set of theatre tickets that (because of Post's orders to execute the prisoners that afternoon) he would not be able to use. Post even chatted to Catanach about the landmarks the vehicle passed, Catanach noting that he recognized them from previous combat operations over Kiel. Suddenly Post remarked, \"I am going to shoot you.\"[20]\n\nCatanach smiled at what he thought a tasteless joke and countered, \"I have an appointment in the cooler at Stalag Luft III.\"\n\n\"Those are my orders,\" Post confirmed. At a field outside of Kiel, the destination Post had chosen to carry out the order, he ordered his prisoner out of the car and repeated his statement, \"I have orders to shoot you.\" In the awkwardness of moving into the field on the premise of allowing the POW to relieve himself, Post's nervous assistant gunman accidentally fired a shot from his pistol. Afraid the man might botch the execution, Post pulled out his own pistol and shot Catanach through the heart. Moments later, when the second car arrived, Christensen, Espelid, and Fugelsang were quickly escorted to the same spot and shot at close range. Post later claimed proudly it was he who had killed \"these terror-fliers. . . . For the glory of the F\u00fchrer I have killed any number of sub-humans.\"[21]\n\nWhile the Gestapo squads translated Nebe's paperwork to physical execution for some, those spared his death sentence were returned to do time in the cooler at Stalag Luft III. First, Bernard \"Pop\" Green and Doug Poynter arrived from Hirschberg. Alex Neely arrived from Berlin. Next, the first of the Canadians, Keith Ogilvie and Tommy Thompson, came back from G\u00f6rlitz with Alistair McDonald and Paul Royle. Also transported from G\u00f6rlitz back to Sagan were Shorty Armstrong, Tony Bethell, Les Brodrick, Dick Churchill, Johnny Marshall, Michael Shand, Bob Nelson[*] and Canadian Bill Cameron. The cooler at the North Compound was so busy following the mass escape, each cell contained four or five men. But the arithmetic quickly became obvious. Of the eighty kriegies who'd emerged from the tunnel the week before, only fifteen were back in the compound. And when the officers emerged from the cooler to share their individual stories, they tried to calculate what had happened to the scores of others they had seen recaptured.\n\nIt took nearly two weeks, but the ripple effect of the Sagan Order finally arrived at the North Compound on April 6, 1944. As George Sweanor remembered it, Hans Pieber came into the compound late that morning and summoned Group Captain Massey to a meeting with Oberst Braune at the Kommandantur. Leaning on his cane and accompanied by Squadron Leader Philip Murray, his personal interpreter, Massey accompanied Pieber out of the compound. The two Senior British Officers were inside the Kommandantur less than thirty minutes. Just after noon, word spread that a senior man from every room was expected in the theatre on the double. In minutes, three hundred kriegies occupied the Red Cross crate seats.\n\n\"As senior man in my room I left on the run,\" Sweanor wrote. \"Massey limped onto the stage and came right out with, 'The new Kommandant has just informed me of the shocking, unbelievable news that forty-one of the escaping officers have been shot.'\n\n\"I was half expecting such an announcement. I raced back to my room to report. 'The bastards have shot forty-one!' I repeated the SBO's words.\"[22]\n\nOne of Sweanor's hut mates mocked him for being so gullible. He said the announcement was just a bluff to stop the remaining kriegies from attempting another escape. Sweanor hoped his fellow officer was right, but knew the news of the executions had to be true. Amplifying his certainty, Sweanor heard Hans Pieber imploring the kriegies not to blame the Luftwaffe. The shootings were committed by the Gestapo in response to the escape, he emphasized, not by his air force.\n\nWhen Keith Ogilvie finished his term in the cooler, he immediately ran into Red Noble and got the latest news.\n\n\"They shot all those guys,\" Noble said, \"trying to escape.\"\n\n\"That's not so, Red,\" Ogilvie retorted. \"These guys were fine. I saw them. There's no way they could re-escape.\"[23] Until then, Ogilvie considered his brush with the Gestapo as little more than a fling away from the camp. All he and his mates wanted was to get back to the air force camp and wait for the end of the war.\n\nA few days later, the Germans posted a comprehensive list of the officers shot. Someone counted the names. There were forty-seven names on it, not forty-one! Among the revised list of dead was Big X. Roger Bushell and Free-French officer Bernard Scheidhauer had been captured en route to France at Saarbr\u00fccken station and executed nearby on March 29. Also shot that day by the Danzig Gestapo were Gordon Brettell, who had escaped previously with Canadian Kingsley Brown in 1943, and Tim Walenn, the man who had directed many of the camp's printers, journalists, artists, cartoonists, photographers, and calligraphers in the production of official documents\u2014all perfect forgeries\u2014to ease the passage of his fellow escapers.\n\nNews of the shootings reached Britain in mid-May. Anthony Eden, the British secretary for foreign affairs, informed the House of Commons that word of the deaths had come from Switzerland. Then, in late June, he rose in Parliament to announce that official word had come from the German government about the deaths of fifty officers. The official German note claimed \"the escapes were systematically prepared, partly by the General Staffs of the Allies, with] both political and military objectives [that endangered] public security in Germany.\" Eden scoffed at Germany's contention the officers had met their deaths while escaping Stalag Luft III or resisting recapture. He accused Germany of \"cold-blooded acts of butchery\" and vowed His Majesty's Government would track down \"these foul criminals . . . to the last man. When the war is over,\" Eden said finally, \"they will be brought to exemplary justice.\"[[24]\n\nThere were still Stalag Luft III prisoners of war who believed the trickle of information, the posted lists of dead officers, and even the Gestapo purge at the camp had all been a bluff to cow the remaining POWs into total compliance. The Germans couldn't have stooped to murdering all those air force officers, some kriegies insisted. Then, pieces of military kit, suitcases, and some personal effects the escapers had carried through the tunnel and into the Silesian countryside were delivered to huts in the North Compound. Belongings such as shoes, handkerchiefs, and even bloodstained personal photographs were brought to the SBO. Ultimately, all doubt about the finality of events following the breakout of March 24\u201325 were put to rest the day Kommandant Braune delivered urns of cremated ashes to Group Captain Wilson. And it didn't take the SBO to deduce why the fifty had been cremated. Ashes would offer no evidence of the cause of death.\n\n\"It proved] that this had been a deliberate massacre,\"[[25] George Sweanor wrote. \"I could not forget my old school chum, George McGill, nor the boyish face of Tom Leigh, nor my boss, a man with an envious war record, Tom Kirby-Green. . . . What a terrible loss to humanity.\"\n\nMcGill's ashes came from Leignitz, Sweanor remembered, Tom Leigh's from G\u00f6rlitz, and Tim Walenn's from Danzig. He concluded from the locations of the crematoria engraved on the urns that many of the escapers had covered great distances before being recaptured and killed. Sweanor found some solace in knowing that the Germans had expended millions of man-hours away from the frontlines tracking down George McGill, his classmate from St. Clair Public School in Toronto. Eventually, as if reuniting the dead men with their wartime comrades, the remaining Commonwealth officers gathered and housed the fifty urns in a building inside the wire of the North Compound. One of the kriegies who had created so many of the theatre production sets, Wylton Todd, volunteered to design a permanent memorial and the new Kommandant provided stone for the masons among the POWs to build a crypt that would contain the urns.\n\n\"A committee was formed to collect the belongings of the fifty and to put all up for auction with the proceeds going to their families after the war,\"[26] Sweanor wrote. \"We gave promissory notes for payments and bid outrageous prices to show our sympathy: $200 for a pair of running shoes, $50 for a razor, and $15 for a handkerchief.\"\n\nEven Staff Sergeant Hermann Glemnitz, still on duty at the North Compound, attended the highly publicized auction. He got swept up in the spirit of fundraising and could be heard urging the kriegies to bid higher for the benefit of the dead airmen's families. Among the auction items that drew particular attention, Tom Kirby-Green's wooden suitcase went for twenty-five pounds and some of the personal clothing of Canadian airman Pat Langford earned 104 pounds. To commemorate the fifty murdered air force officers, kriegies wore whatever black insignia they could find\u2014black ties, black hats, black diamond cutouts sewn to their sleeves\u2014to indicate they were in mourning. Though it was forbidden to sing it, at every opportunity in front of their captors, the kriegies sang \"God Save the King,\" if only because, as Tony Pengelly put it, \"we sang it and felt better.\"[27] The POWs considered every possible act of stubbornness, inaction, and passive resistance they could conceive as a protest to the killings.\n\nGeorge Sweanor didn't agree with the tactics, fearing their campaign of hatred might bring even more reprisals. He was right. The tit-for-tat psychological warfare inside the wire continued seemingly without end. Kommandant Braune responded to kriegie insolence by withholding incoming mail for six weeks. When Pieber conducted his roll calls, the prisoners made life miserable for him, ridiculing his appell, resisting his demands, and interfering wherever they could. One day, when the innocuous guard looked totally demoralized, George Sweanor took pity on him. They happened to be walking side by side across the compound.\n\n\"Cheer up, Pieber,\" Sweanor said. \"We can't keep this up forever.\"\n\nPieber flashed a quick smile, but that was all he could muster.\n\nThe spring and summer of 1944 brought a welcome rush of good news to the inmates of Stalag Luft III, news many of them wished had come six months earlier; it might have persuaded even diehard escapers such as Big X to wait out the war. Still, the BBC broadcasts received by the canary, Dick Bartlett's wireless radio receiver, gave officers in the North Compound a needed lift. Kriegies learned of the liberation of Rome on June 5, the Normandy landings on June 6, the main Soviet offensive in the Baltic region on June 22, the Chindits' victories in Burma on June 26, the attempt on Hitler's life on July 20, the liberation of Paris on August 25, the Canadian capture of Dieppe on September 1, and, on September 17, the bold airborne Operation Market Garden attempt to catapult the Allies to the enemy's side of the River Rhine. That same summer, letters addressed to Stalag Luft III prisoners from fictitious relatives informed kriegies around the North Compound that three Commonwealth air officers had actually completed Roger Bushell's so-called \"home run.\" Per Bergsland and Jens Muller had made it back to Britain via Sweden within a week, while Bob van der Stok had reached his occupied homeland, the Netherlands, within thirty-six hours. He soon pushed on across the Pyrenees to Madrid and arrived back in Britain four months after emerging from tunnel \"Harry.\"\n\nA lot had changed at Stalag Luft III since the March 1944 escape. Von Lindeiner was gone; he would be court-martialled in October for allowing as many as 262 escape attempts during his command of POW camps and for defying the Sagan Order to hand over escapers directly to the Gestapo.[*] Von Lindeiner's nemesis Roger Bushell was also gone, executed under that same order. What remained, however, was X Organization, which both men had grappled with since April 1943. Bushell had determined and manipulated its objectives, its tactics, and its timing like a battle force pursuing an enemy. Von Lindeiner had fought back, trying to destroy its gains and blunt its resolve with unexpected appells, relentless surveillance by ferrets, and isolation punishment in the cooler. In some senses, both von Lindeiner and Bushell failed to control the fate of X Organization. True, the organization had delivered Bushell his greatest objective\u2014a mass escape and a resulting nationwide manhunt\u2014and ultimately von Lindeiner's Gestapo successors appeared to have buried its material gains by filling \"Harry\" with human waste and by exterminating X's ringleaders with the Sagan Order. And yet the Germans had never found tunnel \"Dick,\" which still housed tunnelling tools, bed boards, and wire, and they had not annihilated all of X's brain trust\u2014men such as Robert \"Crump\" Ker-Ramsey, Norman \"Conk\" Canton, and Tony Pengelly, who had given up his number on the escape list expressly so that the escape committee might carry on.\n\n\"Immediately after news of the executions reached England,\" Pengelly said, \"Air Ministry told us over the BBC, via our smuggled radios, to stop escaping. . . . [But] with the incentive of escape work gone, many were losing the comparative peace of mind with which we had lived and endured the long years. Our morale was dangerously low.\"\n\nThe Normandy invasion in June and the attempt on Hitler's  \nlife in July re-inspired the kriegies and breathed new life into a dormant X Organization. Pengelly and what was left of his mapping and document forgery team decided to reintroduce the war of wits to the education room of the North Compound library. They manufactured and posted a huge, detailed map of Europe on the library wall and began sticking in pins (thread strung between the pins graphically illustrated the front lines of the war). Based on the BBC reports, they updated the location of the pins and thread daily. That spring and summer of 1944, the thread lines showed Allied gains in mainland Italy, in Scandinavia, a toehold in France, and significant Soviet advances along the Eastern Front.\n\n\"On the Russian front there were two sets of thread lines, one red and one black,\"[28] Don Edy recalled. \"The one represented German news from the front . . . and the other based on the BBC news, which always showed the Allies closer than the German news. We often saw the German ferrets coming in to take a look at the map, then walking out shaking their heads.\"\n\n\"It gave us infinite satisfaction to show the Germans] the ring drawing ever tighter,\"[[29] Pengelly said.\n\nThe war of the pins gave momentary satisfaction, perhaps, but the rapid westward advance of the red thread on the library wall map\u2014representing victories for the Soviet armies over German armies along the Eastern Front\u2014posed a number of new and potentially ominous threats to Stalag Luft III's imprisoned airmen, officers and NCOs alike. The first impact of the declining fortunes of the German Army in Lithuania, East Prussia, and eastern Poland arrived in the form of other prisoners of war that summer of 1944. Kriegies under guard began arriving from POW camps east of Sagan; they described their captors force-marching them ahead of the Soviet onslaught. With no new huts available for the newcomers, the Germans forced them into existing barracks. They supplied lumber and the loan of tools for the kriegies to convert double bunk beds to triple bunks in existing huts. Rations were halved. Sick parades ballooned. And Red Cross parcels for the original North Compound POWs became few and far between.\n\nIf the flight of POWs ahead of the Soviet advance didn't put the kriegies nerves on edge enough, the sudden arrival of a purge of Commonwealth and American airmen from a place called Buchenwald certainly did. George Sweanor learned that in previous months German SS had had rounded up the 168 downed airmen, including twenty-six RCAF aircrew, stripped them of their identities, and shipped them off to the Buchenwald extermination camp. Starved, tortured, and bearing the scars of Nazi medical experiments, these latest additions to the North Compound shocked Sweanor and his roommates with their first question:\n\n\"How many do they shoot each day?\"[30] they asked.\n\nGeorge Harsh discovered that such fears were based more in reality than in fiction. Shot down, interrogated, and sent to Stalag Luft III in October 1942, the American-born RCAF gunner had served X Organization inside the wire as a tunnel security boss. A sudden purge at the end of February 1944, just weeks before the planned mass escape, had then whisked Harsh and eighteen others away from the North Compound. He had learned about the March 24 breakout and the reprisals of the Sagan Order from their prisoner-of-war huts at Belaria, a satellite compound five miles from Stalag Luft III. Harsh had survived POW imprisonment for nearly two years, but he recognized that his experience on a chain gang in the 1930s had prepared him more than most to live one day at a time. Even so, he admitted that he still lay awake nights worried that Gestapo guards would suddenly descend, line them up, and shoot every tenth man unless the kriegies surrendered the radio the Germans knew was hidden.\n\n\"Toward the end of the war,\" Harsh wrote, \"Count Folke Bernadotte of Sweden, as a representative of the International Red Cross, was allowed into the camp. He] slipped word to us that Hitler had ordered Himmler to have every one of us shot rather than let us be liberated by the approaching Russians.\"[[31]\n\nSuch rumours and fears sparked a reconstituted X Organization to a predictable response. In July, those comprising the core of the escape committee gathered at the now reopened North Compound theatre. Yes, there would be new productions that summer and fall, if the war went on that long. Airman John Casson would produce a version of J. B. Priestley's I Have Been Here Before, while airman David Porter would rehearse a musical comedy called Palina Panic. Since the Germans had never discovered the repository of all the tunnel sand excavated from \"Harry\"\u2014in the space beneath the raked floor of theatre seats\u2014the escape committee reopened the theatre's secret subterranean enterprise. This time, X Organization diggers began excavating tunnel \"George\" from beneath the theatre proscenium eastbound toward the North Compound exterior wire. It was a short distance from the new tunnel to the sand dispersal site, all under the same roof, and there was still plenty of room to store \"George's\" excavated sand under the same theatre floor.\n\nKommandant Oberst Braune may well have suspected that news passed along by new kriegies arriving in the North Compound, or indeed from BBC radio reports of Allied successes across Europe, would embolden his prisoners. In September, the new Stalag Luft III administration adorned the walls inside the camp huts with threatening posters. In broken English, the lengthy proclamations first accused Britain of instituting \"illegal warfare in non-combat zones in the form of gangster commandos, terror bandits, and sabotage troops even up to the frontiers of Germany.\" The poster claimed that a captured British booklet, entitled \"The Handbook of Modern Irregular Warfare,\"[32] encouraged the English soldier to be \"a potential gangster (with) the sphere of operations (to) include the enemy's own country . . . and neutral countries as a source of supply.\" The poster described the areas of Europe in which English soldiers might consider operating as a \"death zone.\" Finally, if the message wasn't clear enough, the narrative concluded with a series of warnings to prisoners \"against making future escapes. . . . All police and military guards have been given the most strict orders to shoot on sight all suspected persons. . . . Escaping from prison camps has ceased to be a sport!\"\n\nBut the Kommandant hadn't finished his propaganda offensive. Soon after the death zone posters went up came another blizzard of proclamations with a provocative assessment of the war and a most peculiar invitation to \"soldiers of the British Commonwealth and the United States of America.\"[33] The poster began with some apparent revisionist history, turning the retreat of German armies from occupied areas of the Soviet Union and Eastern Europe into \"the great Bolshevik offensive [crossing] the frontiers of Germany. The men in the Moscow Kremlin believe the way is open for the conquest of the Western world. This will certainly be a decisive battle for us. But it will also be the decisive battle for England, for the United States, and for the maintenance of western civilization. Or whatever today remains of it.\"\n\nThe posters' rhetoric painted an even bleaker picture of Europe under Bolshevik domination than under Nazi occupation. Suddenly, German captors appealed to all POWs, \"regardless of your rank or of your nationality,\" to recognize \"the danger of Bolshevik-Communism\" and (in bold type) to see \"the consequences of the destruction of Europe\u2014not just of Germany, but of Europe\u2014[and what] it will mean to your own country.\" Unable to disguise its inherent racism, the document positioned its authors and readers not as captors and captives, but \"as white men to other white men.\" Then it laid out a specific offer to the kriegies: \"Whether you are willing to fight in the front-line or in the service corps, we make you this solemn promise: Whoever as a soldier of his own nation is willing to join the common front for the common cause, will be freed immediately after the victory of the present offensive and can return to his own country via Switzerland.\"\n\nAnd finally, in bold-type exclamation, the poster completed its call-to-arms with, \"Are you for the culture of the West or the barbaric Asiatic East? Make your decision now!\"\n\nGeorge Sweanor remembered that the German press joined the propaganda initiative as well. Every time a V\u00f6lkischer Beobachter or Frankfurter Zeitung arrived at the North Compound library, Sweanor devoured every detail, including the one-sided reports of \"Soviet atrocities.\" Then, during one appell that autumn of 1944, representatives of Kommandant Braune called for volunteers to help the German war effort against the perceived Communist threat.\n\n\"We were surprised when one man volunteered,\"[34] Sweanor said, \"saying he would be glad to help. But we could not conceal our mirth when they asked him his civilian trade. He answered, 'Funeral director.'\"\n\nBomb-aimer Sweanor had always considered the mass escape plan to be futile. He had served in the protection and completion of the tunnels from the moment he arrived at Stalag Luft III in 1943, but his highest priority was always survival. Consequently, he rejoined the X Organization service, assisting in the completion of tunnel \"George\" and preparing his fellow kriegies for the kind of chaos the propaganda posters had predicted . . . or worse.\n\n\"We knew that as the Soviets advanced there would be bedlam outside our compound. We would have no Red Cross food. The Germans would be evacuating and leaving weapons behind,\"[35] Sweanor said. \"I joined a group we were training as commandos. I had had some army training, artillery training, and so I was put in charge of a small platoon of people to find German weapons . . . so we could provide an armed united group. . . . We decided to use 'George' to store the equipment. The tunnel was considered to be our after-Soviet occupation outlet, our last survival exit.\"\n\nAs the snows of late November began to accumulate around Sagan, tunnellers under the North Compound at Stalag Luft III brought the face of \"George\" to a position just beyond the east perimeter wire and within feet of the surface. X Organization planners agreed the tunnel would not be used for another escape attempt, merely, as Sweanor had considered it, an emergency exit from the North Compound, a bolt-hole should either the retreating Germans or advancing Soviets decide to take out their frustrations on the kriegies. Meantime, under SBO Wilson's direction, the entire compound population was reorganized into sections, platoons, and companies of the commando self-defence force. The Klim Klub, as it was code-named, prepared for an expected final confrontation with either the camp guards or a hostile invading army.\n\nAs it turned out, however, there was a more lethal enemy than either the Germans or the Soviets in the final battle to survive Silesia: the natural elements, among the very reasons the creators of Stalag Luft III had located the prison there in the first place.\n\n* * *\n\n* Oberregierungsrat Wilhelm Scharpwinkel, chief of the Gestapo at Breslau, was arrested by the Soviet Army on May 10, 1945. In August 1946, the Soviets allowed Captain M. F. Cornish of the British Intelligence Corps to travel to Moscow to interview Scharpwinkel (under Soviet supervision). His voluntary statement (quoted here) was taken September 19, 1946. He died in a prison in Moscow, October 17, 1947.\n\n* Kriminalinspektor Richard Max H\u00e4nsel, in charge of the Gestapo sub-office at G\u00f6rlitz, was in British custody by June 20, 1946, and four days later, the Royal Air Force War Crimes Interrogation Unit took a voluntary statement (quoted here) from him. A war crimes trial began in Hamburg on July 1, 1947, and lasted fifty days. The Judge Advocate General determined there was insufficient evidence to convict H\u00e4nsel; he was formally acquitted on November 6, 1948.\n\n* Kriminal Obersekretaer Lux was killed during fighting at Breslau in 1944.\n\n* In 1945, Gestapo officer Erich Zacharias managed to acquire papers from American occupation authorities in Germany identifying him as a Customs official with no Nazi record. He was eventually traced to a refrigeration plant in Weserm\u00fcnde and arrested; he escaped and was recaptured April 1, 1946. His voluntary statement (quoted here) was taken April 12, 1946. He was found guilty of murder and hanged at Hameln jail on February 27, 1948.\n\n** At the end of the war in Europe, Gestapo NCO Adolf Knueppelberg was in the (Soviet) Red Army Camp 33 , near Brno, and prematurely released. He was never arrested.\n\n*** In the fall of 1945, Gestapo driver Friedrich Kiowsky was arrested by the Czech police. His testimony and February 22 , 1946, voluntary statement (quoted here) implicated Zacharais and Knueppelberg in the murders of Kidder and Kirby-Green, but he was found guilty and executed in Czechoslovakia in 1947.\n\n*Sturmbannf\u00fchrer Johannes Post was executed at Hameln jail on February 27, 1948.\n\n* When Bob Nelson and Dick Churchill considered their good fortune at being sent back to Stalag Luft III and not shot, the two men speculated that perhaps SS Gruppenf\u00fchrer Arthur Nebe recognized their surnames had historical significance to Britons and didn't want to tempt fate.\n\n* Colonel von Lindeiner was court-martialled and sentenced to eighteen months in prison. Reassigned to a mental hospital during the German capitulation, he was shot and wounded by Soviet troops, then handed over to the British. On July 1, 1947, the first of two trials in the Stalag Luft III murders began at the War Crimes Court in Hamburg. Testifying for the defence, Colonel von Lindeiner was asked if under the Aktion Kugel and Stufe R\u00f6misch III orders he would have shot the prisoners himself. \"I should have put a bullet through my head,\" he said. He was exonerated, but remained in prison until November 1947.\n\n# 10\n\nLONG ROAD HOME\n\n* * *\n\nART CRIGHTON escaped Stalag Luft III for about an hour in late 1944. The twenty-five-year-old peacetime musician and wartime Wellington pilot was about to enter his third winter of captivity at the POW camp near Sagan. To occupy himself, during his first summer inside the wire, he had fashioned a left-handed five iron and golfed a nine-hole course inside the wire with his fellow kriegies. Evenings he had played trumpet in the Commonwealth officers' band and eventually led the North Compound orchestra through every musical genre from Beethoven to big band music revues. He'd been dead against escape activity, conducting the orchestra on stage at the theatre even as sand dispersal teams packed tons of earth from tunnels \"Harry\" and \"George\" beneath his feet. But suddenly, on December 4, eight months after the mass breakout, word arrived that he was to retrieve his trumpet and report for a detail leaving the compound.\n\nCrighton joined a group of about two dozen officers assigned to participate in an official ceremony commemorating the fifty dead escapers whose remains had been housed in an outdoor crypt just to the north of the North Compound wire. As both the leader of the orchestra and an accomplished trumpet player, F\/L Crighton had been chosen to play \"The Last Post\" at the memorial. He gladly accepted the assignment, particularly since RAF pilot Les \"Johnny\" Bull\u2014the first down \"Harry\" the night of the escape and among the fifty murdered airmen\u2014had been Crighton's roommate at the North Compound. And yet, as honoured as he felt to be asked to play at the memorial, Crighton recalled something even more indelible about his march to and from the service.\n\n\"It was] my first walk out into the woods,\"[[1] he said. \"I was excited as hell. A real tree. I could actually touch it.\"\n\nStill surrounded by the pine forest, but beyond the wire, the members of the ceremonial party included the Senior British Officer, Group Captain D. E. L. Wilson, and fifteen other officers representing the home countries of the executed airmen. No Americans were permitted to be there. But seven officers from both the East Compound and the Belaria POW camp (including Wally Floody) were allowed to attend, as well as two representatives from the Swiss legation and an adjutant from Kommandant Braune's camp staff.\n\n\"The memorial is in the form of a large altar table with three scroll-like stones sweeping up at the back with the censored] names on it,\" Floody wrote to his wife, Betty, that Christmas. \"We all lined up around it while the R.C. and C. of E. padres read a burial service, then 'The Last Post,' after which the three group captains put wreaths on. . . . It was well done. Tell Betty McGill and others it was quite an impressive service.\"[[2]\n\nAfter he had performed his role on the trumpet and stood silent for the wreath-laying, Crighton and the other Commonwealth officers from North Compound were escorted down the dirt road and back through the main entrance.\n\nA guard began shouting in German to his partner at the gate: \"How many went by?\"\n\n\"Twenty men,\" the guard replied.\n\n\"We're twenty officers,\" insisted one of the kriegies, who resented being downgraded by the compound guards. \"Don't call us men. We're officers!\"[3]\n\nThat same week, Flying Officer John Weir started his final letter of 1944 from Stalag Luft III to his fianc\u00e9e, Frances McCormack, back home in Toronto. He began in very much the same way as he'd begun his first letter to her, soon after his Spitfire was shot down over Caen, France, in November 1941.\n\n\"Hi darling. I thought I'd end this day the best way possible, by writing you. I've had two-hundred-and-forty-six letters from you,\" he wrote. \"This is my ninety-fifth epistle . . . so near and yet so far\u2014the end of the war and you. I get more impatient every day . . . but it will be soon now.\"[4]\n\nHe wished Frances a Merry Christmas and Happy New Year and got her caught up on harmless news from around the camp, news that he expected the censors wouldn't blacken out on his letter paper. Several times in his letters sent during his hospitalization over the previous fall, he had reassured her that the surgery to graft new skin to his eyelids, burned away in the fire when his Spitfire went down, had gone reasonably well; however, he hinted any future portrait photographs wouldn't be particularly flattering. He kibitzed about his new roommates' names\u2014Lorne, Pappy, Sam, Hank, and Pop, as well as his own, Scruffy\u2014sounding \"like something out of Snow White.\" But to his now savvy fianc\u00e9e, he was also passing on valuable information about the state of affairs in the camp. First, he was alerting her to the whereabouts of the downed air officers now living in his hut, in case any of their families back in Canada didn't know. But he was also signalling something else, and Fran would certainly have spotted it.\n\n\"The members of our room have changed,\" he pointed out, meaning people were on the move. \"Lorne Chamber (came in from the west) and Pappy Plant (also west), Sam Sangster, Hank Sprague, Pop Collette and Scruffy (that's me). Wally Floody] is at Belaria now. All ye boys are hail and hearty. . . . If the optimism of this camp is right, I'll beat this [letter] home to you.\"[[5]\n\nSuddenly his barracks in the North Compound\u2014previously offering its veteran kriegies fairly spacious sleeping, eating, and living quarters\u2014were becoming overcrowded with newcomers, he was telling her indirectly. She would certainly have recognized, with five additional men in her fianc\u00e9's hut room, that living space at Stalag Luft III had become cramped, restful sleep less likely, and meagre rations stretched even more thinly than in previous months. His suggestion of beating the letter home offered more a hint of things being in flux at the camp than his honest belief that liberation was at hand. Then, in the same letter, which turned out to be his second last from Stalag Luft III, John Weir inadvertently acknowledged perhaps a greater threat to the well being of the kriegies than even he realized.\n\n\"Darling, it's so hard to write anything 'cause so little happens here of interest,\" he explained. He commented on an American movie called The Spoilers that had been screened in the theatre; it featured Marlene Dietrich and a lively bar room brawl. \"I've been very lazy the last six or eight weeks, due mainly to bad weather, just reading and doing the occasional circuit walk].\"[[6]\n\nIn other words, Scruffy Weir and the rest of the Stalag Luft III kriegies were leading a much more sedentary life without the exercise they had all experienced while digging, maintaining, and protecting the tunnels. Their idleness was reducing, if not eliminating, their higher physical fitness level, dulling their alertness, and certainly taking the edge off their preparedness for the unexpected. And if the officers weren't able to notice their fitness slipping, they certainly recognized that the quality of their nutrition was almost non-existent. George Sweanor, who had taken on the job of training the Klim Klub for potential commando action to the death, woke up one morning during the fall of 1944 and felt too weak to stand. Half rations, a leg infection, and several physical blows he'd sustained playing football were taking their toll on Sweanor's heart; his pulse was spiking inexplicably to 150 beats per minute. A South African Army doctor in the compound examined him and deduced that Sweanor was suffering from a severe case of malnutrition. His condition wasn't uncommon among the POW population.\n\nRobert Buckham, the Toronto-born artist shot down about the same time as Sweanor in 1943, had served the Dean and Dawson forgery factory for a year during the lead up to the mass escape. However, because he had not drawn an escape number on March 24, he'd been forced to leave his bunk in Room 23 of Hut 104 as his fellow kriegies made their way down the shaft under the stove to freedom. After the breakout, with the need for forged documents gone, Buckham didn't stop drawing, but toward the end of 1944 he did begin to write and sketch in a diary. Some of his first entries noted they'd heard the population of Sagan had swollen from twenty thousand to one hundred thousand as civilians, fleeing westward, sought refuge from the fighting and the cold in the town. He noted in early 1945 that the Soviets were steadily advancing. But his more immediate diary subject matter focused on the basics of survival.\n\n\"A man's eyes betray his hunger,\"[7] he wrote as he surveyed the faces of his eight roommates. \"Watch the eyes recede and narrow as they probe deeply for the taste of remembered meals. Watch them . . . comparing the size of portion, measuring the width of bread slice.\"\n\nHe described the typical daily menu among the Commonwealth officers that January of 1945. It consisted of one weak cup of Nescaf\u00e9 for breakfast, one cup of turnip soup containing well-boiled white maggots for lunch, perhaps a few slices of bread, and for dinner the boiled pulp of potatoes retrieved from a waste pile outside the huts. The tastiest parts of the ration, if they could salvage them, were the potato peelings salted and fried in margarine. If he and his hut mates could acquire a Red Cross parcel, the contents filled in the remaining ration gap. And though there were still a few evenings of deliciously distracting music or drama at the North Compound theatre that winter, Buckham noted that his mind ricocheted back to reality when his stomach rumbled with hunger during the performances.\n\n\"An empty belly is a very basic thing,\" he wrote.\n\nThen, suddenly, on January 22, 1945, the bread ration stopped. The same day, Buckham described seeing a Soviet pilot\u2014still clad in his helmet, boots, and other flying gear\u2014wandering outside the Vorlager of the camp. This was followed by a large explosion just outside the wire. More and more, the approach of the Soviet armies became evident. Reports came to the camp that Breslau was brimming with evacuees from the east, and that both shelling and minus-eight-degree temperatures were taking a toll on civilians there. Some of the kriegies openly predicted the evacuation of Stalag Luft III and began preparing. Buckham described large crowds gathered at the cookhouse to read German news bulletins, but \"equally large crowds were on the circuit, toughening feet and legs in case the threatened forced march becomes a reality.\"[8]\n\nRCAF navigator John Colwell was as much a barometer of the compound's focus and tempo as anybody. The Tin Man, who had miraculously supplied the escape committee with so many of its working utensils the previous year, was busily supplying kriegies with homemade survival gear\u2014items they would need during the next phase of their wartime experience. His diary entries became shorter and shorter, with simple references to the daily-life tools he was manufacturing or repairing from scraps of metal and wood\u2014potato mashers, cooking pots, water jugs, cigarette containers, and slop pails. He also retrieved his Klim-tin suitcase from a hiding place in the wall of Hut 120.\n\n\"Half-soled my boots,\" Colwell wrote on January 25, 1945. \"Everyone is sewing.\"[9]\n\nThat same day, the spearhead of the westbound Soviet armies reached the River Odra at Steinau, just forty miles east of Stalag Luft III. Though everyone inside the prison camp knew it, both the Germans and the POWs appeared to go about their daily routines ignoring the obvious. The Soviets were getting closer. In the Centre Compound, for instance, the guards carried out a barracks inspection. On January 26, teams of kriegies in both the North and West Compounds staged the first hockey games of 1945. Performances in the compound theatres went on as normal; in fact, that evening, the North Compound theatre troupe was in dress rehearsal for the first performances of The Wind and the Rain.[10] As late as Saturday morning, January 27, Kommandant Braune received orders from Berlin that the prisoners were not to be moved.\n\nThat night in Hut 119, Don Edy and his roommates\u2014Joe Noble, Larry Somers, Ken Rees\u2014had tidied up their room and were seated on stools at the group's wooden dining table. Fellow officers Barney Barnes and George Smith reclined in their bunks, and Jack Probert fussed with what little food the group had on hand. It was another bitter January night outside. Edy and company anticipated a visit from the Senior Canadian Officer Group Captain, Larry Wray. When he arrived, he joined them at their table for tea to ward off the nighttime chill. Wray initiated conversation by posing a few questions about the coming days: What did they think the Germans were up to? How soon before they reacted to the Soviet advance? For about an hour Wray listened to Edy and the others offer observations and opinions. Everybody had his say. Then Wray offered his view.\n\n\"Personally, I think the Germans have left it too late to move us,\" he said. \"Their armies will be around and we could only clutter up the roads. I think they'll leave us here to be overrun by the Ruskies.\"\n\nAt about 8:30 that Saturday evening, Don Edy noted that Wray took a final swallow of tea and stood up to leave. Just then the door flew open and Bill Jennings, the group's liaison officer, rushed in.\n\n\"The camp has to be ready to move out in one hour's time!\" Jennings roared.\n\nWray absorbed the shock of Jennings' announcement in momentary silence. The senior Canadian officer had come to the wrong conclusion. The Germans had in fact received orders to evacuate the entire prison camp almost immediately. And all the camp's officers would be force-marched away, it seemed, sixty minutes later. Group Captain Wray put down his empty tea cup, made his way to the door, and said, finally, \"Good night, gentlemen, and good luck.\"\n\n\"I guess this is the beginning of the end,\" one of Edy's roommates said. \"I wonder what's in store for us now?\"[11]\n\nThen, like every other hut and room inside Stalag Luft III, Edy's quarters were thrown into pandemonium. Edy admitted his first gut reaction wasn't to the forced march per se, but to the reality of being thrown out of his barracks. The last place he wanted to be on a late January night was outside in the elements. The thought sickened him, but fear propelled him. Edy had a tin suitcase (likely made by Colwell), so he began packing his blanket roll, extra socks, shirts, and a sweater inside. The room that had been cleaned up for Wray's visit became a shambles. Whatever rations of biscuits, cigarettes, and chocolate the men had saved were divided equally and thrown about as bunk mates in Hut 119 assembled their survival kits. A former RCMP officer in the barracks had shown some of the officers how to manufacture backpacks; another man gave out instructions for nailing bed-boards into a wooden box with skis and a tow rope to create a makeshift sleigh. And because their incarceration at Stalag Luft III had taught all kriegies the meaning of resourcefulness, an officer from each hut was dispatched to the library to tear pages of the thinnest paper he could find in the books to serve as toilet paper during the march.\n\nWhen word of the exodus reached Colwell's hut, he and his roommates\u2014Bill Hoddinott, Jim Jamieson, Art Hawtin, and John Acheson\u2014assembled their rations and all their worldly possessions and methodically went about preparing to leave. Hoddinott divided the so-called \"iron rations,\" necessities, equally among the officers. Each man had pack boards for their backpacks. Colwell seemed to go into a trance, as if he had rehearsed his departure from the North Compound over and over in his mind. First, he changed into the clothes he was going to wear on the march. Next, he laid out two blankets on a couple of stools in the middle of the room and began piling his supplies on them. He laid out two shirts, six pairs of socks, six handkerchiefs, razor blades, a shaving set, toothbrushes, pyjamas, two pencils, shoelaces, his logbook, photographs, matches, and a towel. Then, the Tin Man gathered his portion of the food rations\u2014raisins, biscuits, sugar, prunes, cheese, meat, and iron rations\u2014that went into the centre of his pack. Finally, he retrieved a couple of pounds of chocolate, his sketches of camp life, and his daily diary, all hidden in his Klim clock. By 10 p.m., he was ready to go.[12]\n\nBut departure was delayed by a series of postponements to 10:30, then 11:30, then midnight. Rumours added to the confusion. Someone said the Allies had broken through on a one-hundred-mile front and the Germans were negotiating for an armistice. Another said POWs could hide in the bush and take their chances with the Soviets when they arrived. The suppressed panic and the prolonged delay were sapping the officers' strength. Colwell's group conferred and it was decided their colleague Bill Hoddinott was not well enough to travel. He left for the compound hospital in the Vorlager to be liberated, he hoped, by the approaching Soviets. Also during the delay, many of the kriegies tried to consume as much nutritional food as they could find, but couldn't pack. Remaining members of X Organization had last-minute loose ends to tie up, such as collecting the maps from the library walls in case they came in handy on the march. They also set bonfires to burn old clothes, furniture, and any leftover escape committee documents. In the rush to destroy anything the Germans might consider useful, Hut 104 caught fire[13] and the last of tunnel \"Harry's\" entrance went up in smoke.\n\nAt 1 a.m. on Sunday, January 28, the kriegies began moving through the main gate of the North Compound. Robert Buckham took a last look back and considered the significance of the moment. In addition to memories of imprisonment and lost comrades, the kriegies left behind thousands of books in the library; a theatre full of props, tools, and musical instruments; cupboards loaded with sports equipment; and 2.5 million cigarettes.[14] As they passed the stores buildings, each man received one last Red Cross parcel; the Germans had stockpiled as many as fifty thousand of them. But since the POWs had packed their kits to overflowing, most of the departing officers just tore open the boxes right there at the gate and hurriedly selected only items they didn't have, leaving thousands of partially opened and tossed parcels \"bleeding their contents into the snow,\" Buckham wrote.\n\n\"I was outside the wire for the first time in twenty months,\" he went on. \"Ahead the road disappeared into the darker mass of the forest, the trees soft in silhouette and taller than imagined from inside the camp.\"[15]\n\nFor the moment, Buckham had forgotten about his empty stomach, as Colwell had forgotten about the scores of utensils he'd bashed together\u2014a veritable museum of inmate ingenuity in two years of imprisonment\u2014ultimately left inside the wire. Don Edy had fond memories of his bit parts on stage at the North Compound theatre doing gymnastics in the Six to the Bar review and dressed up in a toga in the operetta Messalina. There were even photographs as evidence he had been featured on stage, but they never crossed his mind the night they left the prison camp. Art Crighton, the North Compound orchestra leader, said he had played countless solos during North Compound concerts on \"a hell of a great instrument,\"[16] a trumpet the YMCA had sent from Canada, but he somehow felt the trumpet ought to stay behind. He learned later someone had wrapped it around a tree in anger. But every memento, every memory faded at least temporarily as the kriegies marched away from Sagan. Even Hans Pieber was selective about what he carried out of the camp. The German duty officer, who'd been perhaps closest to the kriegies going back to their days at Barth in 1941, agreed to carry the canary\u2014the illegal radio\u2014since he too wanted to know the BBC's latest reports on the war.\n\n\"Although the sky was covered with clouds, it was quite light since the moon was up,\" John Colwell wrote in his diary. \"Although my pack was heavy, I enjoyed the march, especially when we passed through small towns and villages where there were things to see besides barbed wire fences and Goon boxes.\"[17]\n\nKriegies completed their evacuation of the West Compound by 12:30 Sunday morning. The last of the North Compound prisoners passed through the main gate by 3:45 a.m. The Centre Compound was empty shortly afterward. By 6 a.m. Sunday, the final group of POWs from the East Compound was on the road. Stalag Luft III\u2014once home to nearly ten thousand prisoners of war\u2014was like a ghost town. Only those kriegies left behind in the compound infirmaries remained in the Vorlagers. They were under guard, attended by a doctor, and would wait for the Soviets to arrive to determine their fate. Because of his irregular heartbeat, for example, George Sweanor had been placed back on sick parade. He sat in the North Compound hospital as his fellow kriegies trekked out of the camp.\n\nLater on Sunday, however, Sweanor, the one-time duty pilot inside the North Compound gate, got permission from the Germans guarding the hospitalized POWs to retrieve his personal belongings. As he dashed into the compound back to Hut 119, all he could think of were his letters from his wife Joan, his personal photographs, his books, and the food he'd stashed inside his palliasse. The food and some of his scribblers were gone, but the rest was intact. In fact, he noticed that some optimistic kriegies had even bundled their belongings and inscribed them with some Russian words requesting their personal effects be forwarded to home addresses later. The guards then let Sweanor enter the parcel shed to gather rations for his imminent departure. From rations such as Berger's Food, condensed milk, egg powder, dried fruit, and sugar, he cooked up a nutritious concoction and poured it into small tins, sealed the tops with tape, and sewed the tins into the lining of his greatcoat. Much the way Colwell had assembled his backpack, Sweanor prepared his. Then, on orders from the infirmary doctor to vacate the compound and attend to two other patients in the process, Sweanor too made his final exit from Stalag Luft III.\n\n\"I glanced back for a last look at the compound that had been my home and where I had met so many fine human beings,\"[18] he wrote.\n\nBut Sweanor didn't have much time to reflect on either his nearly two years of imprisonment at the compound near Sagan, or the hundreds of men with whom he'd shared a prison camp bond. He suddenly faced different responsibilities on the road to survival. He was a patient tending other patients. He had to keep a USAAF officer with pneumonia comfortable as they marched. And he regularly had to apply a calamine-like lotion to the oozing sores of a Rhodesian airman in his care. And when the elderly German assigned to guard the sickly patients en route to a train siding a full night's march away could no longer carry his heavy rifle, Sweanor agreed to carry it part of the way. Dick Bartlett carried more than his fair share too. The Canadian Skua dive-bomber pilot shot down over Norway in 1940, and the custodian of the secret radio, had befriended fellow Fleet Air Arm pilot John Nicholson at Stalag Luft III. Sub Lieutenant Nicholson had been shot down at Dunkirk the same year, but a bullet had remained lodged in his chest near his heart throughout his time in German prison camps. On the march from Sagan, Nicholson kept asking Bartlett if he could just stop and lie down in the snow.\n\n\"Hang on and keep moving,\"[19] Bartlett encouraged his friend as he half dragged and half carried the still wounded Dunkirk veteran.\n\nWhen they did stop, Bartlett would loosen his bootlaces to relieve pain and swelling in his feet. After the short rest, when Bartlett's hands were too numb to retie his own boots, Nicholson was able to reciprocate his friend's care and concern during the forced march that winter.\n\nSimilarly, Edward Nurse and Mac Reilley\u2014two officers who'd crewed up together in 1943, flown ops together with 405 Squadron, bailed out together from the same doomed Halifax bomber,[20] and vouched for each other upon arrival at Stalag Luft III\u2014turned to each other for support in the exodus from prison camp a year and a half later. In moments when they felt too frozen to continue, too hungry to find that extra burst of energy, and too exhausted to keep moving, both Nurse, the pilot, and Reilley, the navigator, leaned on each other to keep going.\n\nIt was Sunday evening by the time the Germans emptied the satellite prison camp at Belaria, a few miles from Sagan. Since February 1944, just a month before the mass escape, the Belaria camp had been home to three members of the original Stalag Luft III X Organization\u2014tunnel architect Wally Floody, intelligence specialist Kingsley Brown, and security boss George Harsh. It hadn't taken the three RCAF men long to realize that Colonel von Lindeiner's purge of officers to Belaria had likely spared them the wrath of the Sagan Order and murder at the hands of the Gestapo. Now it was up to them to fend for themselves as their German guards rounded up kriegies by the thousands to \"save us from the Bolshevist terror,\" Kingsley Brown remembered a camp officer saying. Brown added that \"we found it difficult to appreciate their solicitude,\"[21] and concluded that their German captors actually hoped to buy their way out of the war using the kriegies as human bargaining chips.\n\nGeorge Harsh imagined the future outside the Belaria prison and immediately drew several conclusions. Though he had endured twelve years on a chain gang in the United States prison system back in the 1930s, and nearly three more as a POW during the war, when the evacuation order came Harsh recognized his best shot at survival depended on his two resourceful roommates. Floody told Harsh to line the insides of his jacket with newspapers[22] for warmth, and to fill his pockets with chocolate for quick energy. Brown told him to stash cigarettes as barter for food along the way. Each man tied a Klim tin to his belt with a loop of string so that anything liquid, hot, and edible could be contained in it for a meal. Together, the three men tore apart a table and stool, and with bed-boards for skis and belts and blankets for harnesses, fashioned a crude sleigh for transporting their clothing, bedding, and survival food supplies.\n\n\"As Wally, Brownie, and I marched out the gate our pockets were bulging, but our hands were free,\" Harsh wrote, \"except for the tins of bully beef we were wolfing.\"[23]\n\nDuring his pre-war civilian days, Kingsley Brown had enjoyed his life as a journalist gathering and publishing stories for newspapers in Toronto and Halifax. When the X Organization at Stalag Luft III learned of his researching and writing talents, it immediately dispatched Brown to gather intelligence information from library newspapers and magazines for the Dean and Dawson forgery section. By his own admission, Brown did most of his gathering for the escape committee, not for himself. In the mad dash at Belaria to assemble his own survival gear before he marched out the gate, however, he allowed himself the luxury of salvaging one keepsake: a German beer stein with a delicate silver top. But the first hours of marching through drifting snow and penetrating cold imposed a sudden reality among the prisoners. Out in the elements and in the POWs' weakened state, the supply sleighs became heavier by the hour. The kriegies began to unburden themselves of extra tins of food, packs of cigarettes, and blocks of chocolate\u2014all but the absolute minimum luggage needed to survive.\n\n\"The beer stein . . . was my one souvenir of my prison years,\"[24] Brown wrote. \"But I tossed it into the ditch. The ditch was strewn with violins and guitars, books, trumpets, framed family pictures\u2014precious items in a prisoner's 'life savings.'\"\n\nThough his colleagues often called him a loner, Wally Floody left Belaria in close company with his two tunnel co-conspirators, Harsh and Brown. Aside from survival supplies, however, the only mementoes he took with him were a small journal and a photograph of his wife, Betty, whom he hadn't seen in almost four years. Leaving Belaria prompted mixed emotions for the Tunnel King. Still bitter about the way German authorities had hauled him away from the North Compound and potential escape through the tunnel he had designed, excavated, and protected, Floody recognized that the twist of fate had likely extended his life. And as costly as the Gestapo reprisals had been, he continued to insist the escape had achieved its ultimate objective. When news of the mass escape reached the German population, Floody continued to point out, it was the first time since the war began in 1939 that every German in uniform had been called back from leave. Every fifteen minutes on German radio authorities broadcast the latest on the escape of the Terrorflieger (terror flyers).\n\n\"The slowdown of the German economy caused by the tunnel and the breakout had been the equivalent of dropping a couple of divisions of paratroopers into German-occupied Europe,\" Floody reminded his comrades. \"I think the cost was worth it.\"[25]\n\nOnly outside the wire did Commonwealth air officers come face to face with the other realities of the war. Most of the east-west highways had become the exclusive domain of Germany military traffic, moving troops and weapons to and from the rapidly approaching Eastern Front. So the kriegies and their guards made their way along secondary dirt roads, where they were soon caught in the backwash of the war. Roads were clogged with slave labourers from occupied countries, starving civilians pushing carts of their moveable possessions, homeless women and children huddling from the cold, nuns and priests uprooted from their parishes, and domesticated animals wandering beside the traffic. The kriegies told the civilians where they could find thousands of Red Cross boxes of provisions abandoned in the Sagan pine forest. For at least the moment, the former inmates of Stalag Luft III had the advantage of provisions they had packed in their pockets and packs, but as one kriegie noted, on the road \"we were just people now, all members of the human race . . . sharing common levelers of cold, lice, misery, and despair. . . . This was truly G\u00f6tterd\u00e4mmerung.\"[26]\n\nThe first night outside the wire, the aircrew officers and their guards coped with temperatures well below freezing. Few prisoners had the kind of winter gear the conditions demanded. Greatcoats proved too thin. Summer boots cracked and leaked in the cold and snow. Mitts and scarves were in short supply. In addition, though the moon had originally cast some light en route, before long the weather closed in and the men were marching through falling snow that deteriorated into nearly white-out conditions. Marching into the teeth of a blizzard slowed everybody down, and within a few hours the columns of POWs stretched over twenty miles of road.[27] Stragglers feared they would be shot. Even as the kriegies marched from the Stalag Luft III camps, the BBC was broadcasting an order each afternoon that the officers and men should not risk escape attempts, and further, that they should try to stay together for safety in numbers and better identification.[28] But the German guards fared no better. Most were older men who grumbled about having to escort Luftgangsters across the frozen countryside. It appeared the guards were constantly in search of shelter for themselves and their prisoners.\n\nSometimes, survival on what kriegies soon dubbed \"the Death March\" came down to the individual strength of fellow air officers rising to the occasion. Musician Art Crighton said the cold wasn't nearly as penetrating when he was walking; when his body was moving, his circulation seemed to fend off the freezing temperatures, the winds, and the driving snow. But when the columns had to stop and he had to stand for hours out in the open waiting for congestion to ease or an order to be issued, he could feel himself going numb at the extremities. That's when Crighton remembered Scruffy Weir coming to the rescue.\n\n\"Wave your arms!\"[29] Weir shouted as he ran up and down the columns of men. Then he'd cuss and add to his call to action: \"Wave your legs!\"\n\nLater in the march, Crighton said his German guards occasionally found shelter for their prisoners in empty or nearly empty barns. Hungry, exhausted, and sick, the kriegies were jammed like sardines into stalls, troughs, and lofts of straw for the night. At one end of the barn, the guards placed a rain barrel full of water. They were fearful of the possibility of fire ignited by careless prisoners sneaking a smoke during the night. Next to the rain barrel was also the spot where prisoners could relieve themselves in an emergency. One night in just such a setting, Crighton got an attack of \"squitters,\" prompting him to dash from his straw bed through the dark to the relieving spot next to the rain barrel.\n\n\"In pitch black, I stumbled . . . down the hall and crashed into the rain barrel. I fell head first into two feet of icy water. Then I collapsed on the floor. Losing control, disaster followed,\" he said. \"I remember nothing more except] my comrades wiped me clean and dry and laid me on my straw bed.\"[[30]\n\nOver the next few days, all of the prisoners from the various Stalag Luft III compounds\u2014as many as ten thousand airmen of the Allied air forces\u2014made their way via back roads south from Sagan and west about fifty miles to the German rail centre at Spremberg. Men from the South Compound, finding ample shelter at a brick factory along the way, arrived first on January 29. Two days later, the kriegies from the West Compound made their way into the town. A group of five hundred Americans from the Centre Compound had joined up with POWs from the East Compound by the time they reached Spremberg on February 4. Of the larger groups, the last to arrive at the railway yards were the North Compound air officers. At Spremberg, German authorities divided the prisoners into new groups, loaded them into boxcars, and sent them in different directions\u2014the Americans to Stalag XIII-D outside N\u00fcrnberg; those from Centre and South Compounds to Stalag VII-A near Moosburg; the POWs from the Belaria Compound were transported to Luckenwalde, a prison camp southwest of Berlin; and prisoners from the North Compound (including most of the six hundred Canadian kriegies) travelled to Marlag-Milag, a naval facility in northwestern Germany.\n\nIn that first week of February 1945, the ancient locomotives and boxcars\u2014the infamous \"forty-and-eights\"\u2014threaded their way from Spremberg westward across Germany, away from Soviet armies advancing from the east and toward Allied armies advancing from the west. Apparently dodging higher priority military trains and both daylight and nighttime air raids, the POW trains chugged from one marshalling yard to the next. The lack of food and water were bad enough, but darkness and confinement compounded everybody's anxiety and ailments. Occasionally, when the trains pulled into sidings, guards unlocked the doors and allowed the prisoners to exit, stretch, and relieve themselves. John R. Harris was positioned at a boxcar door when it was suddenly thrust open.\n\n\"I had no sooner alighted from the car than I promptly fainted,\"[31] he said. \"The others picked me up, but . . . I keeled over once again. This time, I was carried back to the [box]car. Someone brought me a cup of water. I don't know where it came from. . . . It was full of rust particles, so it could have come from the train's engine.\"\n\nIn his boxcar, Robert Buckham's group also had to do without water, artificial light, or straw for bedding, but they found a margarine lamp and lit it. The kriegies tried to make the best use of the lack of space. They hung bags and sundry gear on the walls and from the ceiling of the car. Any available blankets were spread on the floor so they could take turns resting in a prone position. Others slept sitting. The rest stood, attempting the same. When his train got to Hanover, Buckham and his group could see water being rationed to the two cars ahead of theirs. The Canadians began shouting and banging on their boxcar door to get a water ration too. Through a crack in the door, they could see a guard approaching with a bucketful of water. He unbolted the door and slid it back a few inches. But before the kriegies could reach the water, the train lurched forward. The door slid shut. The guard re-bolted the door and tossed the water away. To add to his discomfort, Buckham ended up beside an Australian airman who screamed and groaned through the night and banged his fists on the boxcar door during the day.\n\n\"We could do little for him. Dysentery,\" Buckham said. \"A Red Cross box served as his toilet, barely ten inches from my head. Endurance was our only resource.\"[32]\n\nThe trio of Canadian kriegies from the Belaria compound had stuck together all the way to Spremberg, and even as the Germans divided the prisoners into groups for train transit, Kingsley Brown, George Harsh, and Wally Floody managed to get aboard the same boxcar. That was about the only redeeming aspect of the trip. Fifty men were crammed into their railway car and struggled to get comfortable in the shared space. There was a bit of straw on the floor and a single wooden box in the middle of the car to serve as its latrine. There was no food and no light except what entered through cracks between the wallboards. A determined group of card players enlarged a crack in the wall with a penknife so that they could carry on their game. And when night came, so did the endless struggle to organize arms, legs, and heads into any degree of comfort to sleep. The odour of fifty unwashed bodies mixed with the stench of the \"thunderbox.\" And when the train stopped on a siding to wait out a Bomber Command attack some distance away, the assault on the senses heightened the claustrophobia, fatigue, and fear. Several nights into their trip west, men awoke to cries in the darkness of the boxcar.\n\n\"I want my mother,\"[33] a man called out. When he heard the plea again, George Harsh knew it came from one of the youngest airmen among the prisoners. Just twenty, with several kills in the Fighter Command books, the fighter pilot had been shot down over France in 1942. Otherwise a dynamic and bright young warrior, this night he called out like small child for his mother.\n\n\"Okay, boy, okay,\" a voice answered from across the boxcar. The consoling response came from a man making his way through the tangle of arms and legs and bodies. \"It's all right now.\"\n\n\"I want my mother,\" the first man repeated, nearly weeping.\n\n\"We'll get you to your mother,\" came the assurance. It was Wally Floody. The big former Spitfire pilot and X Organization leader had reached the younger man in the dark and was rocking him gently in his arms. \"There now. Get some rest.\"\n\nIn the three years George Harsh had known the hard-driving tunneller with nine lives, he had never seen this caring side of Wally Floody. Like John Weir shouting encouragement up and down the columns of men in the cold, George Sweanor nursing two fellow patients' medical needs while trying to stay alive himself on the forced march, and those who had cared for Art Crighton during his dehumanizing bout of dysentery, Floody playing mother to a distressed young pilot on a prison train in the middle of Germany illustrated the brotherhood that bound these longtime prisoners of war together. If one were going to make it safely to the end of this long road, each man would somehow try to ensure that everybody else did as well. Harsh witnessed further proof of such loyalty even as the numbers of the POWs trekking west from Belaria dwindled to less than one hundred. One morning, weeks into their overland marching and gruelling boxcar transit to an unknown destination, Harsh and Floody found themselves \"sleepwalking, silently lost in our own personal miseries\"[34] as a twosome. It slowly dawned on them that their number three, Kingsley Brown, was missing. For as long as they could muster the strength and not drawing unnecessary attention from their guards, the two kriegies searched high and low for their absent comrade. They had all but held a requiem for their missing friend when he suddenly reappeared.\n\n\"Where in the hell . . . ?\" Floody started in on him.\n\n\"Never mind,\" Brown laughed. \"There's a group of Frenchmen back there . . . and they've got a small cart with 'em loaded with bread.\"[35] And with that he opened a blanket he'd been wearing like a shawl to reveal three loaves of German black bread. He informed his buddies he'd swapped his wristwatch for food.\n\n\"A Rolex watch for three loaves of bread,\" Harsh protested.\n\n\"They promised for the next watch to throw in a bottle of Schnaps too,\" Brown quipped.\n\n\"That's different,\" Harsh grinned.\n\n\"Jesus,\" Floody sighed. \"What a set of values!\"\n\nAfter several days cooped up in their boxcars bound for Marlag-Milag, northeast of Bremen, the kriegies from Spremberg arrived in the marshalling yard at Tarmstedt. Peering through cracks in the boxcar walls, they could see a changing of the guard out in the pouring rain. Since Marlag-Milag\u2014a naval prison\u2014was to become their new home, their Luftwaffe guards now handed the Commonwealth air officers over to German marines. It was a two-hour walk from the railway yard to the gates of Marlag-Milag. There, outside the barbed-wire fencing, along a cinder roadway, the kriegies chain smoked cigarettes and stood \"like cattle with] our backs to the wind and rain,\"[[36] waiting to experience a naval guard search of their belongings, one man at a time. The marine Kommandant seemed determined to conduct as thorough a search as any the prisoners had received from the German air force. But John R. Harris discovered that the marine guards detailed to do the inspection were just as frustrated as the kriegies; a judicious bribe to the guards resulted in a perfunctory search, and the Canadians were admitted to the compound. It still took six hours.\n\nAs best they could, Harris and some of his former roommates from the North Compound searched out a barracks block where they could sleep comfortably. In the following days, they discovered how well off they had been at Stalag Luft III. They now had to live\u2014fifteen or twenty to a room\u2014in huts lit by a couple of naked light bulbs, with no furniture and no bunks, but plenty of rats. All they had to sleep on were bags of damp wood shavings. There were very few stoves and even less firewood to burn in them. So the kriegies began stripping the floors and walls\u2014not to procure shoring for an escape tunnel, like the old days, but to fuel fires to heat their rooms and cook their meagre rations. Despite the Spartan surroundings, John Colwell seemed perfectly at home. The morning after he'd arrived and settled into a room in the Marlag-Milag barracks, the Tin Man had managed to visit all the dumps in the camp to collect any discarded tin. On that first day, he manufactured a soldering lamp with a blowpipe and bashed together a two-gallon water pail and stew pot.[37] By the second day, his fellow kriegies from Hut 120 at Stalag Luft III had cooked a small meal for themselves and a dozen roommates. Meanwhile, Don Edy's first edible intake at Marlag came from the dregs of his kit bag and some boiled water.\n\n\"My powdered milk was gone,\"[38] he wrote. \"And there was only a tablespoon of Nescaf\u00e9 left. So I put it all together with all the sugar I could find and poured in the hot water. It was a good brew all right, but it was too strong for my poor old stomach. Seconds after it went down, I was up and outside, sicker than a dog.\"\n\nThe infirmary at Marlag-Milag rarely had fewer than a hundred patients. The sick bay seemed continuously crammed to the four walls with patients on cots or lying on the floor under any available greatcoats and blankets. There was one shower in the entire camp. It was located in the ablution shed and consisted of one wall, a tin roof, a cement floor, and a cold-water shower with a pretty much unobscured view of the great outdoors. And the north Germany dampness seemed to seep into everything. Robert Buckham said the rain in Tarmstedt would \"take first prize in density;[39] \" it could penetrate broken windows, roofs, floors, and walls, as well as socks and shoes, no matter how dried out they seemed. The continuous rain even forced cancellation of numerous appells\u2014remarkable in a naval prison. The Commonwealth aircrew kriegies, mostly Canadians, would spend the next ten weeks at Marlag-Milag. At least half their stay occurred under oppressive grey skies that pelted them with rain and snow from the nearby North Sea. The kriegies at Marlag-Milag didn't see sunshine until March 8.\n\nClear skies brought a clear view of things other than the farthest reaches of their prison compound, however. Most of the kriegies who'd been imprisoned at Stalag Luft III hadn't been close to an Allied bomber since the day or night they were shot down\u2014often months or even years ago. But suddenly, in those first days of March, they found themselves in front-row seats for some of the final airborne operations of the war. On March 8, the kriegies at Marlag-Milag were witness to a massive nighttime bombing attack on Hamburg. Robert Buckham watched the stream of hundreds of bombers almost circle the camp overhead en route to the target (U-boat construction pens in the harbour). When the visual show seemed over, Buckham went to bed to get warm, but was thrown back out of bed when the ground began to shake with the bomb explosions at Hamburg that continued for ninety minutes. The air battle that followed left quite an impression on fighter pilot Don Edy.\n\n\"The bomber stream headed home directly over the camp,\"[40] he wrote. \"We were thrilled at the sight and felt very close to those friends of ours just overhead. Suddenly, we heard the [German] night fighters on the attack. We followed the course of the battle . . .\n\n\"We would see a pattern of twinkling lights . . . cannon shells from the fighters. . . . Then all of a sudden an orange ball of flame as a bomber caught fire . . . then dropped towards the earth in a heart-breaking arc. I saw ten planes shot down that night in as many minutes. I don't think any other incident of the war shook me quite so deeply.\"\n\nIn contrast, John Colwell was again busily grinding out pots and pans and stools and bunk beds. And because he sensed both the winter and the war were nearing an end, he fashioned a pair of shorts for sitting outside in the sunshine and tore apart his flying boots to make a pair of bedroom slippers.\n\n\"The spring is sprung. The grass is riz. I wonder where the armistice is,\"[41] he wrote in his diary on the first day of spring.\n\nThen, just a few days later, the Kommandant called an unscheduled afternoon appell, ordering all the prisoners to pack for a 6:30 p.m. evacuation of the compound. That night, April 9, Colwell reported in his diary that Group Captain Larry Wray instructed the kriegies to break into the compound kitchen, steal as much white soap powder as possible, and write \"R.A.F.\" and \"P.O.W.\" in block letters on the sand of the parade ground,[42] indicating with a huge arrow their likely march route out of the camp, so that Allied fighters wouldn't fire on the kriegies in transit. Wray also attempted to delay departure. He'd learned that British ground forces were just seven miles from Bremen. One day's delay might allow the Desert Rats (British 7th Armoured Division) to overtake the evacuation march and hasten the kriegies' liberation. But it didn't happen. The next day, all of the prisoners from Stalag Luft III were on the march again, heading northeast from Tarmstedt and bound for L\u00fcbeck.\n\nThe first deadly strafing of kriegies happened on the road near the town of Zeven the next day, April 10. Several marines were killed when a flight of RAF Tempest fighter aircraft attacked one of the marching columns. They didn't realize they were shooting at POWs. Other strafings occurred near Harsefeld over the next few days; two kriegies were killed and seven were wounded. A few days earlier, the group of kriegies that included RCAF officer George Sweanor, housed at Stalag XIII-D near N\u00fcrnberg, were ordered to evacuate. Depending upon which rumour he believed, Sweanor's group was either en route to Hitler's mountain retreat at Berchtesgaden to become his personal hostages or toward Dachau, the Nazi death camp just outside Munich. In southeastern Germany too, however, Allied aircraft supporting General George Patton and the US Third Army had virtual air supremacy. In fact, Allied bombing of the bridge over the Danube had stopped Sweanor's train at Ingolstadt for several hours.\n\n\"Kriegies, who still had some cigarettes, bartered them for red and white paint,\" Sweanor said, \"and we painted large red crosses on white backgrounds on the roofs and sides of our boxcars.\"[43]\n\nBut the POW train had barely left Ingolstadt when the kriegies aboard heard sirens warning of an imminent air attack. Their guards flung open the doors of the boxcars and the prisoners in the railway cars watched the last waves of Flying Fortresses and Liberators bombing Ingolstadt. A flight of P47 Thunderbolts flew low over the prisoner train, but then climbed steeply, banked, and lined up the train for a strafing run. Sweanor and his kriegie comrades instinctively knew the boxcar was the least safe place to be under these circumstances and dashed toward ditches about three hundred feet away. The Thunderbolts were gaining too quickly, so Sweanor went to ground and felt their strafing bullets cut a deep furrow a few feet from his prone body. When he tried again to make it to the ditch, more Thunderbolts began strafing from the other direction. Kriegies by the hundred were scattering in every direction, Americans among them.\n\n\"You God-damn trigger-happy idiots,\" one US pilot yelled up at them. \"You can see our Red Crosses.\"[44]\n\nSeveral other kriegies on the periphery of the attack stood and waved their arms wildly as the fighters swooped in for their final attacks at treetop level. One of the Thunderbolt pilots apparently spotted the Allied uniforms on the people scattering before him and ceased firing. He climbed to rejoin the rest of his flight, likely shared his discovery, and the flight raced away before any of the kriegies lying on the ground could record their squadron markings. An American pilot taking fire on the ground had flown with the same squadron over Italy.\n\n\"I know those bastards,\" he swore out loud. \"I'll have them all crucified!\"\n\nBut the damage was done. A South African air officer near Sweanor had most of his wrist shot away. A doctor in the group supplied a tourniquet to stop the bleeding and a cigarette to calm the man, while the surviving German train guards raced around herding the non-wounded kriegies back into the boxcars. It turned out the locomotive hauling their boxcars was mortally wounded by the strafing and barely managed another dozen miles before dying right there on the tracks. Meanwhile, inside the boxcars the kriegies experienced a new ambiance as a result of the air attack.\n\n\"At least we could thank the Yanks for the ventilation,\"[45] Sweanor wrote.\n\nThe American Thunderbolt pilots weren't the only \"trigger-happy\" airmen in the final days of the European war. On April 19, 1945, RCAF pilots flying Typhoons had inadvertently strafed and killed twenty-nine kriegies[46] as they marched under German guard near Gresse, on the road to L\u00fcbeck. Among the five Canadians killed in the attack was Sergeant Robert Douglas, one of Sweanor's 419 Squadron comrades. During a bombing operation against St. Nazaire in March 1943, Douglas's aircraft was thrown on its back and into an inverted spin. Bomb-aimer Douglas managed to bail out, but the pilot got the Halifax out of the spin and flew the aircraft home; a month later, the same crew (minus Douglas, who was then in a German POW camp) was shot down over the Skagerrak; all crew were killed. Douglas had survived the near crash, and made it through the rest of the war imprisoned, but he died at the hands of Allied fighter pilots just twenty days short of German capitulation in Europe, May 8, 1945.[47]\n\nThere were too many close calls for Commonwealth air officers who had come so far, but had not yet been liberated. On April 20, John Colwell's group heading north from Marlag-Milag managed to get makeshift tents erected just before the heavens opened on their campsite near Elmenhorst. Then, during the night, Mosquito night fighters from Bomber Command shot up the town. Colwell and the others watched cannon tracer bullets coming out of the sky during the attack. With the rain continuing into the next day, Colwell's group\u2014about thirty-five kriegies\u2014sought shelter in a barn near Neritz, until two SS army officers showed up to move the POWs along.\n\n\"Aus! Aussteigen!\" came the order from the Germans. They wanted everybody out of the barn in ten seconds or they would start shooting.\n\nThe kriegies realized how serious the SS officers were and scrambled out through windows and doors as fast as they could. One man missed the top rung of a ladder coming from the loft and slid to the bottom in a heap. Then, the SS men lined up Colwell and the others.\n\n\"Terrorflieger,\" they shouted at the air officers, criticizing them for bombing women and children. And the more they shouted, the more they flashed their guns.\n\n\"I really thought it was the end,\"[48] Colwell told himself. The POWs turned to a pastor who had been trekking with them, but the religious man was frightened to the point of being speechless. Finally, somebody made it clear the air officers were not fugitives, but that they were billeted in barns and sheds in the area under instructions from their guards. They were supposed to be there.\n\n\"We didn't stir from the barn for the rest of the day. We ate cold meals,\" Colwell wrote in his diary. Nor did it make any sense to enter L\u00fcbeck. A Red Cross medical officer, the SBO, and a German Kommandant had inspected what were to be the kriegies' final quarters, but pronounced them medically unfit. In his final war diary entry, on May 2, Colwell wrote, \"Goons deserting. Tanks arrived at noon. FREE!\"[49]\n\nDon Edy, John R. Harris and the kriegies in their group were liberated nearby, on the Trenthorst Estate, about the same time. They had occupied a two-storey barn sturdy, stately, and well stocked with dry straw. Edy had set up a kitchen for cooking in the barnyard. Bags of flour from Red Cross parcels arrived and some of Edy's buddies found a bakery in a village and began baking. The resulting white bread was the first he'd tasted since he'd been shot down in North Africa in February 1942, three years before. On the night of May 1, 1945, Edy heard the British guns booming closer, saw Spitfires overhead, and awaited release. By daybreak the German guards had disappeared.\n\n\"We lined the road west of the estate like children waiting for the Santa Claus parade,\"[50] he said. \"Sure enough, about four o'clock in the afternoon, an armoured car came careening down the road. . . . We cheered like mad and swamped the car trying to shake the hands of the men in it. The demonstration was one of exuberance and relief.\"\n\nFrank Sorensen rushed those same tanks that afternoon. Earlier in the day, May 2, a German guard had handed him a Luger as a sign of surrender. But that paled in comparison to seeing the convoy of Scottish armoured vehicles arrive at Wulmenau farm near Trenthorst.\n\n\"If I hadn't been so keen to get back to England in a hurry, I would have jumped on the tanks . . . and gone with them through Denmark,\" he later wrote his family. \"They were some of the happiest moments in my life when I climbed up on the first tank and had my picture taken by the tank commander together with a whole tank-load of yelling and crying kriegies.\"[51]\n\nRobert Buckham, Ley Kenyon, both artists and forgers inside the wire, and Les Brodrick, the former trapf\u00fchrer for tunnel \"Dick,\" experienced liberation somewhat differently. With no accommodation available for them in any of the barns outside L\u00fcbeck, the three air force officers were taken at gunpoint to another prison, this one holding several thousand French officers. In the final days of April they burned straw in a chip heater to cook food, slept in cellars with double bunks, and scrounged for food around the barracks. On May 3, 1945, tanks from Bernard Montgomery's Second Army clattered out of cloud of smoke and dust. Then a khaki-clad commander emerged from the lead tank and waved at the POWs gathered on the parapet of the prison. Buckham made his last diary entries.\n\n\"A roar of cheers; crudely made flags waving; laughter and tears mingling; the guards running off, weaponless; men climbing the wire to run to the tanks; men embracing each other, shouting incoherently; men kneeling to pray; men staring vacantly, bewildered; thousands of men in a state of hysterical, blessed release. It continued for minutes,\"[52] he wrote.\n\nThe first Allied officer dispatched to enter the prison near L\u00fcbeck to liberate the thousands of Allied POWs, including hundreds of Commonwealth air officers, arrived after 6 p.m. that day. The kriegies watched with some sense of irony as finally the tables turned. All German troops captured there were paraded in the square, without gear, and marched off. The liberators found a German general hiding in the basement and took him away too. But the kriegies were told they would have to stay in the prison ten more days, until arrangements for flights to Britain could be arranged.\n\n\"We have been liberated,\" Buckham wrote, \"but we are not free.\"\n\nLiberation proved equally bittersweet for George Harsh, Kingsley Brown, and Wally Floody, who had trekked and trained across Germany from the satellite prison camp at Belaria to one of the worst Wehrmacht prison camps in Germany, at Luckenwalde, south of Berlin. When the three RCAF officers were incarcerated there at Stalag III-A, in February, they were already hungry, filthy, and in a generally weakened state.[53] And conditions at the prison didn't help. For most of the winter they had survived on a ration of a cup of soup and a few slices of bread a day. They had slept in bug-infested bunks and had heated their quarters by burning twigs and bed shavings in a tin can. Then, on April 22, 1945, two days after Adolf Hitler's birthday, Soviet Army troops pushed the Germans out of Luckenwalde and the prison camp. However, the Canadians (along with other Allied air force officers) were separated from civilian prisoners at Stalag III-A, and kept for another six weeks while the Soviets negotiated with the Americans on the west side of the River Elbe to be traded for an equal number of Soviet prisoners the US Army had liberated.[54]\n\nFinally, at the end of May 1945, the three Canadians arrived at the pontoon bridge over the Elbe where they would be exchanged for three Soviets. Harsh led the trio, followed by Brown and Floody. When they reached free soil at the west end of the bridge, an American infantry colonel greeted them.\n\n\"Here you are, chaps,\" the American said in an exaggerated British accent. \"Right this way to Tokyo!\" And he waited a moment for their reactions.\n\n\"Fuck you, Jack,\" retorted the American-born George Harsh. \"And the same . . . for any friends you may have in California.\"\n\nBrown just leapt down from the bridge and shook his head in disbelief. Then Floody, with greater reverence for the significance of the moment, joined his two ex-kriegies on the ground and said, \"By God, we made it.\"[55]\n\nMeanwhile, Tony Pengelly's liberators nearly killed him with their kindness and efficiency. Once in the hands of the British Army, north of Marlag-Milag, Pengelly and the rest of his group of Commonwealth officers became just another lot of freight that had to be dealt with. When army drivers arrived at the frontlines, they simply unloaded ammunition and other battle supplies; then with equal dispatch they loaded POWs into the empty space and made the return trip. Similarly, once inside an Allied military base, British officials were convinced the Commonwealth air officers had typhus and had them deloused.\n\n\"They assumed we were totally contaminated,\"[56] Pengelly said. \"So they took all our clothes, burned them . . . and we took hot showers using special lice soap.\"\n\nThe next day, the kriegies were moved to a new location. They were stripped again and their new battledress from the day before was also burned, while their bodies were deloused with hot showers and lice soap yet again. On May 7, the day before V-E Day, the Commonwealth officers, who hadn't been airborne in several years, boarded Lancaster bombers and were flown back to England.\n\n\"We landed at a big reception field,\" he said. \"And they assumed we were all lousy and they did the same routine. Took our army battledress, burned it, and gave us air force battledress.\"\n\nThen, the British reception centre greeters led the former kriegies into aircraft hangars filled with card tables and chairs for an afternoon social. Local community ladies, Red Cross women, and nurses soon descended to entertain the officers over tea and gin rummy.\n\n\"From seeing no women for four years to being swamped\u2014hundreds of them\u2014and here we were with teacups and scones. It was the most absurd scene.\"[57]\n\nNow ex-kriegies, the air force officers of the Commonwealth who had somehow succeeded in surviving to this first week of May 1945 marked the achievement in personal ways. George Sweanor, penned up in a barracks in Moosburg, Germany, remembered his best friend Pat Porter[58] , who had kept their Halifax aloft long enough for the crew to bail out two years before. When Don Edy arrived at the rehabilitation centre in Bournemouth, England, he took the small tin suitcase in which he had carried his belongings for more than three years\u2014from the first POW camps in Italy, to Sagan at Stalag Luft III, and throughout the forced march across Germany\u2014and chucked it under his bed for good; he was shipping out to Canada and the tin case \"looked silly and rather childish.\"[59] Art Hawtin, who credited his dedication to staying physically fit for his survival, salvaged one souvenir of his POW experience\u2014a photograph of the Canadian All-Star Baseball Team that had routed the Americans in that summer tournament of 1943.[60] Darrell Larlee saved a photo too, one depicting the POW roommates for whom he had cooked; the picture hung on a wall at the family home in Campbellton, New Brunswick, as a constant reminder to be thankful and frugal\u2014to the point that his son, Peter, said, \"nothing would get thrown out. Everything got put back in the fridge. . . . We'd eat Sunday dinner until Thursday.\"[61] Alan Righetti, an Australian air force officer, carried a piece of Canada with him almost all the way home. On the night of the breakout, his roommate, RCAF fighter pilot George Wiley, had entrusted Righetti with his watch in the event he didn't make it. En route home to Australia in 1946, Righetti stopped in Windsor, Ontario, to fulfill the promise and delivered the watch to Wiley's parents and sister.\n\nThe memento snapshots, the jettisoned belongings, the lost comrades, the legacy of the murdered fifty, and the recollection of a thousand other experiences in airborne combat or inside the compound wire stayed with the ex-kriegies\u2014sometimes by design, often not. Most former POWs returning to Canada chose to leave any tangible evidence of being shot down or being imprisoned tucked inside flight logs, shoeboxes, or basement cupboards. Best forgotten, they thought. Phil Marchildon returned to Penetanguishene, Ontario, walked the familiar streets of home, and noted that everything looked exactly the way he remembered, \"as if the war had never happened.\"[62] As with other returning vets, the former air force POWs tried to get their lives back on track\u2014retooling in school, returning to jobs that had been held for them, and restoring relationships with families and friends. For many ex-kriegies, there were understanding spouses who learned to tolerate their nighttime outbursts, idiosyncratic eating habits, or their need to sleep in a single bed. Often there was little or no other post-trauma help. When enough time had passed, there were air force reunions they might join, Remembrance Day observances they might brave, and special POW gatherings they might attend. During a memorial service held in London, England, several months after the spring breakout and Gestapo murders in 1944, a chaplain offered an assessment of the kriegies' plight: \"Their sacrifice was touched by the finger of God,\" he said. \"Their freedom in a measure lost, they fought on, doing their duty twice over.\"[63]\n\nVolunteers, they had trained for war, served in aerial combat, been blown out of the sky, withstood interrogation for secret intelligence, and been imprisoned for the duration. Prisoners of war, some had joined The Great Escape to spite their captors or stave off boredom, others out of a sense of duty. Still others viewed the plot to dig to freedom as simply futile. But all faced its brutal aftermath. Ultimately, as hostages to a dying regime, the air force officers faced one last test of their fortitude and sense of duty, trekking toward a liberation that always seemed just one more day away. Theirs was service many times over.\n\n# 11\n\n\"A PROUD, SPECTACULAR DISTRACTION\"\n\n* * *\n\nTHE IRONY of his May 8th experience took a while to sink in. RAF pilot Philip Gray had spent most of the previous four years preparing for, enduring, and feeling grateful for surviving the nightly combat operations of Bomber Command. He flew his sixteenth and final bombing raid against a marshalling yard between Bremen and Hamburg, Germany. It was not a piece of cake. En route home, Gray's Lancaster had to evade heavy anti-aircraft fire. The crew discovered one five-hundred-pound bomb still hung up in the bomb bay, but safely released it over the North Sea. Then, on final approach to his home airfield, Gray grappled with falling pressure in the landing-gear brakes. But when he brought the Lanc to a dead stop halfway down the runway, it dawned on him. His war was over. The tension of living life on the edge was gone. Any anxiety about surviving evaporated. One day, Bomber Command's strategists led an all-out bombing campaign, dispatching hundreds of aircrews and their four-engine bombers from aerodromes all over the country. The next\u2014May 8, 1945\u2014with all those airmen and their aircraft on hand, there was nothing for them to do.\n\nSuddenly, Gray received a briefing on Operation Exodus, an overnight solution to a problem he and his RAF bomber crew hadn't even known existed. Air officers and men by the thousands[*] had been cut loose from prison camps all over the Reich. They were holed up in former prisons, fenced compounds, hospital wards, warehouses, and makeshift tent cities next to recaptured aerodromes waiting for a ride home. Learning of their plight, Prime Minister Winston Churchill gave their repatriation the highest priority. The luxury of commanding such a well-organized machine as Bomber Command, its ready aircrews, and now all its available aircraft, seemed too good to ignore. Nearly five hundred round trips of Lancasters, Halifaxes, and other heavy bombers would do the job.\n\n\"We were] ideal magic carpets to whisk the ex-POWs back to freedom in a hurry,\"[[1] Gray wrote.\n\nNot unlike each of his bombing runs over Europe, Gray's first flight bringing kriegies home to Britain proved instructive, if a bit unsettling. Serving in the war as \"Lords of the Air\" and never having experienced the daily task of survival in a POW camp, Gray admitted that he was ignorant of the trauma the former prisoners of war had known. Only when he loaded twenty-four passengers that first day of Operation Exodus did he learn that some of them had been four, five, and six years in enemy stalags, mostly cut off from the Allied war effort, and all of them entirely reliant on their own resourcefulness to make it through the war. To learn that freedom and home were less than an hour's flight away proved overwhelming for the physically weakened and emotionally drained ex-kriegies.\n\n\"The sun was shining during that first Exodus touchdown,\" Gray wrote as he watched the POWs step from his Lancaster onto the tarmac at Westcott Airfield in Buckinghamshire. \"Some got down on their hands and knees and kissed the concrete. Others simply burst into tears where they stood, while others lay on the grass and sobbed. I was thunderstruck . . .\"[2]\n\nWith an awakened recognition that these former prisoners of war needed time and space\u2014even aboard their flights back to Britain\u2014Gray had a greater sense of empathy at the controls during his second Exodus sortie. He dropped his jovial \"we'll-get-you-home\" act. He simply nodded and smiled as they climbed aboard. He let them be by themselves. He made sure that he and his crew delivered as smooth and uneventful a ride home as possible. He merely wondered about \"the stories that lay behind the eyes, the docile manner, the resigned, rather sad, expressions that came] back at me from these men, stories which would possibly never be told in full.\"[[3] Never, indeed.\n\nThe pilot of the Lancaster bomber with ex-kriegie Kingsley Brown aboard that first week of May embellished passage to England slightly more. And the former Nova Scotia journalist soaked up the trip like a kid on a carnival ride. For his flight back to the UK, Brown got a front-row view, seated in the bomb-aimer's bay. Leaving the Belgian coast, the Lanc pilot brought the bomber down to a few hundred feet off the North Sea as he followed the coastline and flew right over the French port of Dunkirk. Brown recognized the burned out vehicles and the sunken hulls as the sight of the British Expeditionary Force retreat in 1940. Then, on the opposite side of the Channel, the Lanc made an equally dramatic flypast over the white cliffs of Dover.\n\n\"To me they seemed a little blurred,\"[4] Brown wrote, \"but actually they hadn't changed a bit.\"\n\nLater stationed at Bournemouth, on the south coast of England, Brown and his forced-march comrade George Harsh waited for the RCAF repatriation system to process them home to Canada. Not surprisingly, the bureaucracy didn't move quickly enough for either of them. So Brown, the former intelligence expert for X Organization, and Harsh, the former security boss for tunnels \"Tom,\" \"Dick,\" and \"Harry,\" hatched a plan to solve their immediate problem. They decided to liberate a local rowboat and paddle out to a four-engine RAF Sunderland flying boat[5] moored in the harbour at Poole. The fact that Brown had only piloted twin-engine Hampdens and Harsh had served as a tail-gunner didn't seem to faze them. They crept aboard the Sunderland and had gotten one engine started when the military police arrived. It took the pull of two decorated wing commanders to spring Harsh and Brown from jail, and a week or so later the two veterans had been booked aboard \u00cele de France for their trans-Atlantic passage.\n\nBack home and back in civilian life, George Harsh chose big city publishing while Kingsley Brown appeared to avoid contact with a lot of people in a confined space. He took over his great uncle's general store in the community of West Jeddore (population two hundred) on the south shore of Nova Scotia. By 1950, he had returned to life as a professional communicator\u2014in public relations, then reporting and editing, and eventually the civil service as a special assistant to Ellen Fairclough, Canada's first woman cabinet minister in the John Diefenbaker government.\n\nPerhaps because he'd been a journalist first, Brown eventually put his memories of Stalag Luft III to paper\u2014from harnessing bumblebees with propaganda on onion skin paper, through several escape attempts, to his role in stockpiling intelligence for the forged documents, and eventually the forced march across Germany with Wally Floody and George Harsh. When he finished his account of the experience he titled Officers Travel First Class, he sent the manuscript to publishers, but kept receiving rejection letters. In the late 1980s, Brown had given up. \"I never expected to see a book of mine make the light of day. And this proves it,\"[6] Brown told his wife, Marion. \"Nobody is going to buy this manuscript. Nobody.\"\n\n\"That's not true,\" Marion interrupted. \"Be patient. Someone will buy it.\"\n\n\"Who?\"\n\n\"I will,\" Marion said. \"I'll buy it.\"\n\n\"And just how much are you prepared to pay for the manuscript, Angel?\"\n\n\"Five hundred dollars,\" she said without hesitation.\n\n\"Then consider it sold to Marion, the highest bidder.\"\n\nKingsley signed over the work to Marion, and in February 1989, his story of life at Stalag Luft III, renamed Bonds of Wire, was published by Collins in Toronto. His daughter Ethel recalled watching Kingsley flip through books at home as if he were speed reading. When she asked him what he'd read, he illustrated chapter and verse how well and how much he had digested, perhaps a skill developed in the North Compound library, skimming German newspapers for information that could be used in forged documents for the escapers.\n\n\"He had the intellect of an encyclopedia,\"[7] Ethel (Brown) Alle said.\n\nIn one of his last letters home in January 1945, just before the Germans marched their prisoners out of Stalag Luft III, John \"Scruffy\" Weir remembered he'd better wish his fianc\u00e9e, Frances, a happy birthday for the upcoming April, just in case the vagaries of the mail system delayed his best wishes until after the fact.\n\n\"Perhaps I'll be there in time to give you a birthday kiss,\"[8] Weir wrote. \"I find it impossible to visualize what our reunion is going to be like. But I still dream of it.\"\n\nAlmost five months later to the day\u2014June 1945\u2014Hugh Godefroy, Weir's former squadron mate, got word that Scruffy had been shipped home to Canada and would arrive in Lachine, Quebec, for a formal repatriation ceremony. By the time the RCAF officers arrived at Lachine to be marched into an assembly hall and officially released to their families, Weir's father and fianc\u00e9e, Frances McCormack, had travelled up from Toronto. First the repatriation was set for 9 p.m. But thanks to air force protocol, it was postponed and then postponed again.\n\n\"Finally, at two in the morning,\"[9] Frances remembered, \"John and his crew marched in. John was in the front line, about dead centre and he spotted his father. And I was beside his father. Well, John just walked out of the lineup and came to us. Broke up the whole ceremony.\"\n\nBut John Weir's triumphant return to civilian life in Canada still had a few bumps to endure. After the ceremony, they all hopped into Colonel Weir's automobile and began the drive to the Godefroy home. Minutes into their trip, they got a flat tire; so while Frances sat in the backseat of the car, John and his father jacked up the car, removed the flat, and replaced it with the spare. It was the middle of the night when the trio reached its destination. And while the rest of the household went to bed, John and Frances sat in the household sunroom and talked the night away. \"We never drew a breath all the way home,\" Frances said of the train ride back to Toronto. \"It was like we were [in Toronto] in a minute.\"\n\nWeir had warned his fianc\u00e9e he didn't look the same with his eyelids gone, burned in the descent when his Spitfire was shot down four years before.\n\n\"But it didn't change him as far as I was concerned,\" she said. \"He never came to terms with it. I didn't see what he was talking about. We got married October 2, 1945. Love is blind.\"[10]\n\nThere were only a few important women in Don McKim's life. The youngest of three children, Don had grown up on a farm near Lynedoch, in southern Ontario. At age sixteen, Don was told by his father he would be leaving the farm for a job at the Canadian Bank of Commerce. Just before he joined the RCAF late in 1940, he was working at a branch of the bank in Sault Ste. Marie, Ontario. There he had enjoyed an acquaintance with Doreen Olson, a telephone company employee, but bank employees were not allowed to marry or else they would have to forfeit their jobs. Still, McKim found creative ways to enjoy Olson's company. On each day she came to the bank to deposit the cash receipts of her telephone company, he always found a way to be the teller who counted her cash.\n\n\"Did you bounce?\"[11] McKim always asked her, meaning was the cash amount different from the total on the deposit slip.\n\n\"Yes,\" she would generally reply.\n\n\"I'll count it again after we close,\" he'd say. While he confirmed that the cash and deposit slip balanced, they could have a conversation about anything but money.\n\nAfter McKim was shot down in December 1942 and introduced to German air force prison camps, he depended on any supplies his mother would send from home. Agnes McKim was a lifeline for her son, sending clothing, dry goods, and letters. Following his liberation and repatriation to England, McKim finally got passage back to Canada aboard a troopship, where he ran into a Canadian Army corporal named Sid Olson. It was Doreen's brother, who had also been captured during the war and liberated about the same time Don was. And though McKim's air force officer comrades frowned on it for a while, throughout the trans-Atlantic voyage home when meals were served, Don would always retrieve Sid from his Spartan third-class accommodations onboard the ship and bring him up to the first-class officers' dining area to eat.\n\nHome in Ontario again, Don returned to civilian life and to his teller's wicket at the Canadian Bank of Commerce. He left Doreen Olson behind, as the bank reassigned him to a branch in Binbrook, Ontario (near Hamilton). There, he connected with a previous acquaintance, Grace Crozier. Soon after they began to see each other, she told him the grim coincidence that had occurred during the war.\n\n\"Grace's husband was in the air force like I was,\"[12] McKim said. \"When she heard that I was missing in action [in December 1942], she sat down to write a letter to tell her husband Dave [Crozier] who was also overseas. The very moment she was writing that letter, a knock came at her door. A man had arrived to inform her that her husband was also missing. [She learned later] he'd been killed . . .\n\n\"When I got back, I went to see Grace . . . and I married her,\" McKim said.\n\nFormer RAF Spitfire pilot and scrounger at Stalag Luft III Keith Ogilvie finished his war near Bremen, where his forced march had ended in March 1945. Like so many air officers during the trek, he'd survived thanks to employing a buddy system with fellow officer Samuel Pepys. Perhaps what had contributed equally to his survival was that a year earlier, on the morning of March 25, 1944, F\/L Ogilvie had been the last officer out of tunnel \"Harry\" to get away from the North Compound. He was probably the hard-arser who'd covered the greatest distance on foot\u2014about forty miles\u2014before being recaptured by German Home Guards near Halbau, Germany.\n\nBack in England after VE Day, Ogilvie was hospitalized in Gloucester, as much to ensure that fractures in his arm, sustained in July 1941 when his Spitfire was shot down, had healed, as to aid his recovery from the forced march. Nevertheless, during his convalescence, Ogilvie made up for lost time on a number of fronts. He reconnected with a Canadian friend who had worked with the British Ministry of Information censoring the letters of Canadian servicemen, Irene Lockwood.\n\nAs well, Ogilvie filled out the forms to transfer from the RAF to the RCAF. And finally, he met with British Intelligence officers of MI9 to recount his experiences at G\u00f6rlitz prison following his recapture from the breakout on March 24\u201325 , 1944. In question was the German assertion that the Commonwealth air officers had been shot while attempting to escape custody and that, at the time, they had been disguised as civilians.\n\nDuring the debrief, a British Intelligence official asked Ogilvie to describe what he witnessed on March 29\u201330, 1944, from his G\u00f6rlitz cell.\n\n\"I saw F\/L Mike] Casey, [S\/L Ian] Cross, [F\/L George] Wiley, [F\/L Cyril] Swain, and maybe two others [F\/O John Pohe and F\/O Al Hake] leave handcuffed under the control of civilians,\" Ogilvie said. \"After this, other parties also left at night usually in fours or sixes.\"[[13]\n\n\"What clothes were the officers wearing?\" the MI9 man asked.\n\n\"They were almost entirely dressed in Air Force uniform. [F\/O Denys] Street for instance, had an RAF officer's tunic with wings, rank, and buttons.\"\n\n\"As far as you [are concerned] had they committed any criminal offences?\"\n\n\"Absolutely none!\"\n\nF\/L Ogilvie returned home to Canada in July 1945 and married Irene Lockwood in the summer of 1946. He dedicated much of his early return to active service in the RCAF, officially welcoming those aircrew members who returned to Canada through the ports of New York, Montreal, and Halifax. He served in the RCAF another eighteen years. He died in Ottawa in May 1998.\n\nOgilvie is remembered largely for his DFC, his eight damaged or destroyed victories as a fighter pilot, and his nearly quarter century of service in the air force. Equally important in the story of the Great Escape, however, was that account given to MI9 . Based, in part, on Ogilvie's specific recounting of events on March 30 at G\u00f6rlitz prison, Oberregierungsrat Wilhelm Scharpwinkel of the Gestapo and Kriminal Obersekretaer Lux were found to be complicit in the murders of the Commonwealth officers at Halbau and found guilty (in absentia) during the war crimes trials in 1947\u201348. Scharpwinkel was traced to a prison in the Soviet Union in 1946, and interviewed by Capt. M. F. Cornish of British Intelligence. The Gestapo chief at Breslau never faced trial at Hamburg, but died in a Moscow prison in 1947. Lux died during the advance of the Soviet Army at Breslau in 1944. Ogilvie had at least helped to deliver the promise that the British Foreign Minister had made in June 1944.\n\n\"These foul criminals,\" Anthony Eden had said, \"will be brought to exemplary justice.\"[14]\n\nThe fifty murdered air force officers weighed heavily on Wally Floody's mind for a long time afterward. Perhaps the impact of those bad memories took away Floody's impulse to speak about the escape and its aftermath for many years. Unlike other kriegies, Floody the Tunnel King wasn't liberated until the end of May 1945, when the Soviets exchanged him for Russian POWs the Americans had liberated. Floody had spent his twenty-seventh birthday, April 28, awaiting his freedom. He missed VE Day in England. He was just as glad to get home and leave the war and its experiences behind. Indeed, a year later, when his first son was born, Wally learned he was to receive the Order of the British Empire.\n\n\"Flight Lieutenant Floody . . . became one of the leading organizers and most indefatigable workers in the tunnels themselves,\" the OBE citation read. \"Time and time again, projects were started and discovered by the Germans, but despite all dangers and difficulties, Floody persisted, showing a marked degree of courage and devotion to duty.\"[15]\n\nThe announcement put Floody in the spotlight. Reporters wanted his story again and again. Buckingham Palace wanted him to come to London for the OBE investiture. As far as he was concerned, however, he didn't deserve the fuss. He never considered his tunnel designing and digging heroic. He turned the invitation down and attended to his chartered air service based on the islands along the Toronto waterfront instead. Eventually, Floody abandoned the flying business altogether.\n\n\"I can still see their faces,\"[16] Floody told reporters years later, \"especially the six Canadians. The Gestapo and SS took the fifty out, two by two and . . . dispatched them with shots in the back of the head.\"\n\nDuring the time of the trials of those complicit in the murders of the fifty officers, Floody happened to be employed as a marketing manager and living in Britain. There he reconnected with Wings Day, who, after the Great Escape, had been recaptured and thrown into the Sachenhausen concentration camp. The two ex-POW comrades watched with fascination as the work of S\/L F. P. McKenna and the RAF Special Investigation Branch yielded convictions of the Gestapo gunmen during trials at Hamburg in 1947 and 1948. Floody steadfastly refused to talk about his memories, even when Paul Brickhill's book, The Great Escape, was published in 1951. It took another decade before Floody felt comfortable enough to openly reflect on events at Stalag Luft III.\n\nA phone call from moviemaker John Sturges came when Floody was in the right frame of mind. The Hollywood director requested Floody's expertise as a technical advisor during the shooting of his $4-million feature film, The Great Escape, in the spring and summer of 1962. Then in his mid-forties, Floody visited the set at Geiselgasteig, in Bavaria near Munich. Initially for two weeks, he offered his impressions and suggestions on many aspects of the production\u2014the way the Commonwealth officers' uniforms should look, how the underground air pump worked, and in particular the way set designers had reconstructed the tunnel \"Harry\" for the digging and escaping scenes. At one point, before the filming began, Floody was asked to crawl into the tunnel replica. He noted it had a little too much room, so the production designer lowered the tunnel ceiling to make it believably claustrophobic. Just before Wally and Betty Floody left the film location, they enjoyed a dinner with some members of the production crew, who wondered about the production's authenticity.\n\n\"I know you're getting everything right,\" Floody said, \"because I had terrible nightmares last night.\"[17]\n\nThe Great Escape opened in the summer of 1963. Its Canadian premiere in Toronto on July 3 featured a march past by an RCAF band, and attendance by the Ontario lieutenant governor and as many former kriegies as Wally Floody could contact. As well as initiating a successful summer of box office receipts in Canada, the opening netted $10,000 for the RCAF Ex-Prisoner of War Association. Wally Floody regularly participated in POW reunions and memorials in Canada and abroad. One in Toronto in 1970 reunited not only ex-kriegies, but also Hermann Glemnitz, the former staff sergeant at Stalag Luft III. As Floody posed for a Globe and Mail photographer with Glemnitz, the two men offered an exchange for reporter Arthur Moses.\n\n\"I didn't know anything about any tunnels,\"[18] Floody grinned.\n\n\"Come on, Floody,\" the former German guard said, \"I won't put you in the cooler now.\"\n\nGeorge Sweanor never fully endorsed the tunnel escape plan. Nor did he feel comfortable with Roger Bushell's confrontational strategy when dealing with the Luftwaffe guards at Stalag Luft III. Nevertheless, he co-operated fully with every demand the escape committee made of him. Almost from the day the North Compound became home to two thousand Commonwealth flyers, in April 1943, Sweanor had committed to serving X Organization as a duty pilot, making note of everyone who entered or exited the main gate. In addition to those security duties, he had also served as a penguin, dispersing his share of excavated sand, and as a stooge, spying on the ferrets who were spying on the prisoners. As much as anybody else inside the wire, Sweanor felt motivated to get home, where his wife, Joan, and the daughter he'd never seen awaited his return. However, when it came time to draw numbers for the order of escape down tunnel \"Harry,\" George Sweanor had refused to enter his name.\n\n\"I was all for it initially,\" he said. \"You were a member of the military. You were expected to carry on degrading the enemy's ability to make war, and escaping would be . . . degrading their manpower.\"[19]\n\nBut somewhere between his drive for survival and his conception of trekking through hundreds of miles of enemy-occupied territory back to England, Sweanor discovered a reality that shaped his days of captivity and the rest of his life. Aside from the obvious confinement, surveillance, and deprivation he experienced at the Luftwaffe prison camp near Sagan, Poland, he came to believe that his existence inside the wire had had a lasting intellectual impact on him too.\n\n\"I consider Stalag Luft III my alma mater,\"[20] Sweanor said. \"With years to discuss life with intelligent aircrew from well over a score of countries, and with interaction with the enemy, it was evident that people are people\u2014good, bad, and indifferent\u2014in every culture.\"[21]\n\nWith the help of textbooks that were specially imported from Canadian universities, Sweanor upgraded his education with a political science course. In the library he read newspapers the Luftwaffe regularly supplied because their content endorsed the German position in the war. Together with the BBC broadcast content passed along by the kriegies operating the wireless radio hidden inside the compound, the library books and magazines helped fill in gaps that his wartime imprisonment had created.\n\n\"We knew much more about the war than the people who were still fighting it,\" he wrote.\n\nAnd while it was a chore, he dutifully attended German language class, in part to be conversationally capable should the need arise, but mostly for the discipline of attending and learning. He participated as actively in sports in the prison compound as he would have at a Canadian university; he realized, if nothing else, that the physical exertion maintained fitness and health when the lack of adequate nutrition sometimes did not. And while he only assisted in the functioning of the North Compound theatre peripherally, helping out with set construction once in a while, Sweanor felt the regular weekly drama and musical productions constantly boosted the kriegies' morale.\n\nGeorge Sweanor also took up the pencil and paper while imprisoned at Stalag Luft III. As with so many other rituals developed inside the wire, writing became a daily habit he has continued throughout his life. When the first edition of his memoirs, It's All Pensionable Time: Twenty-five Years in the Royal Canadian Air Force, was published in 1967, fully two-thirds of the book's content was drawn from notes on his experience at Stalag Luft III. And in the years following the war, when he continued to serve in the RCAF\u2014in Interim Force, the Arctic, the Korean Airlift, the Distant Early Warning (DEW) line, and NORAD at Colorado Springs\u2014he never stopped writing his observations and thoughts. Preparing his journal for publication, assembling articles for periodicals, and composing notes for speeches, he downplayed as \"verbal diarrhea,\" but as of 2013, George Sweanor, age ninety-three, continues to prepare the monthly newsletter of 971 Air Marshal Slemon Wing of the RCAF Association, in Colorado Springs, Colorado. His sign-off is a regular feature of the publication: \"Ye Olde Scribe.\"\n\nRichard Bartlett's landing back in England proved nearly as bumpy as his crash landing in Trondheim, five years before, during the Allied defeat in Norway. After hiding \"the canary,\" the radio that brought BBC broadcasts to the POWs, Bartlett then survived the forced march to L\u00fcbeck and was liberated that first week of May 1945. But as had become the routine on arrival in the UK, ex-kriegie Bartlett had his uniform virtually stripped from him and his body subjected to repeated delousing showers. When Sub Lieutenant Bartlett emerged from the final medical and debugging sessions, he did not receive the appropriate Fleet Air Arm uniform he required, but a British Army uniform instead. Nevertheless, the RAF put him on a plane with three other Royal Navy personnel and sent ahead a message to Portsmouth to the effect that four naval POWs were en route and transport had to be arranged. The Portsmouth officer in charge misinterpreted the note and sent a paddy wagon to meet \"the prisoners.\" To add to the insult, Bartlett soon learned British authorities had shipped all his uniforms and kit to his family's Canadian home in Fort Qu'Appelle, Saskatchewan.\n\nBartlett arrived at the Royal Navy air station at Lea-On-Solent in southern England on the eve of VE Day with no uniform, no back pay, and no access to the victory celebrations about to begin all over the UK. Not unaccustomed to scrounging and making do, however, Bartlett liberated a white shirt, collar, and tie that substituted for his navy whites and he joined the party suitably dressed anyway. Next, Bartlett wondered if he might be allowed to get to Canada. He learned he was entitled to three months leave, but the rules blocked his way home.\n\n\"Royal Navy personnel are not allowed to take foreign leave in wartime,\" the regulations stipulated.\n\nAs it turned out, there was a way around that barrier too. Bartlett's commanding officer simply arranged to have him posted to Northern Ireland. With Royal Canadian Navy ships regularly passing through the port of Londonderry, his CO surmised that with his \"on the job training in escape and evasion,\" Bartlett would have no trouble getting aboard a westbound ship. Indeed, he hitched a ride with a North Atlantic convoy corvette and several days later landed in Halifax. He was nearly stopped with insufficient domestic currency in his pocket, did a quick exchange with a stranger, and bought train fare to Regina en route to being reunited with his family in Fort Qu'Appelle.\n\nBut that ended Richard Bartlett's career path of dipping and dodging, since by summer's end he was back on station in the UK, ready for his next assignment with Fleet Air Arm. The Allies had secured victory in Europe and it would soon be theirs in the Pacific as well. Bartlett received his full lieutenant commission in addition to a transfer to the Royal Canadian Navy, and in early 1946 he joined the squadron serving aboard carrier HMCS Warrior. By 1947, he had married Margaret Falconer and assumed peacetime command of the Firefly squadron. It didn't seem to matter. In spite of his exemplary career in the RCN\u2014from 1946 to 1964\u2014he couldn't seem to shake his kriegie past nor the interest from family and friends about his imprisonment at Stalag Luft III. In 1992, Dick and Margaret Bartlett attended the Remembrance Day assembly where their grandson, Nick Dumonceaux, attended elementary school. During the observance, the students were asked if their grandparents had fought in the Second World War. Seven-year-old Nick gladly went to the microphone.\n\n\"My grandfather spent the war in jail,\"[22] he announced proudly.\n\nBoth the assembly and his grandparents got quite a chuckle out of Nick's perception of Dick Bartlett's POW time from 1940 to 1945. But the youngster grew up to understand his grandfather's war and, in a world that offered more than a microphone to express himself, in 2011, after his grandfather had died, a twenty-six-year-old Nick Dumonceaux posted a tribute on his Facebook page.\n\n\"You are the reason I am here today,\" the post said. \"You gave so much and never asked anything in return. You were a part of what made this country great and we will never forget.\"\n\nIt took just as long for Frank Sorensen's three children to understand his Stalag Luft III experience. They were all at least young adults before the real story of their father's Second World War air force record\u2014his training in the BCATP, his propaganda speech from England to the Danish Resistance on Radio Free Europe in 1942 , his posting with RAF 232 Squadron to North Africa, his victories over the Tunisian desert, then being shot down in 1943, and his eventual participation as a language specialist in the Great Escape\u2014all came to light. The three Sorensen children were born within a dozen years of the end of the war: Glenn in 1948, Stephen in 1950, and Vicki in 1957. All they knew was that their dad was a successful dentist married to Betty Bodley, a former schoolteacher. But that was the Frank Sorensen whom most knew outside his Kingston, Ontario, home.\n\nInside their home, the three Sorensen kids never went without, but they came to know a father who had a short fuse (sometimes sparked by his weekend drinking) and who seemed haunted by something or someone. At night Glenn, Stephen, and Vicki all recalled their parents suddenly shouting\u2014their father screaming at demons of some sort, their mother trying to wake him with the assurance he was safely in bed in his own home. The three children were specifically told never to stand in a doorway where their father slept because he might suddenly attack the silhouette by kicking the door closed without realizing a family member was standing there.\n\n\"Things got worse as we all grew up,\"[23] Stephen said. \"Glenn and I [had bedrooms] in the basement. We could hear all the fighting and bickering through the basement wall. Vicki was literally across the narrow hallway next to my parents' bedroom.\"\n\n\"I asked my mom, 'Why did you stay with him]?'\"[[24] Vicki said. \"And she told me with the veterans returning from the war . . . all the women knew they would be traumatized, but they had a sort of unspoken pact, that they were going to stand by them no matter what.\"\n\n\"When I was a teen,\"[25] Glenn said, \"to suggest that [my dad] get psychiatric help would be admitting he had problems. . . . It was never really addressed throughout his lifetime.\"\n\nIn school in the late 1950s, Stephen read Eric Williams' The Wooden Horse, the fictionalized version of his own escape, with Michael Codner and Oliver Philpot, from Stalag Luft III in 1943. Then, when The Great Escape Hollywood movie was released, Frank Sorensen took Stephen to see it at the Grand Theatre in Kingston, just the two of them. Little by little, Frank began to share some of his own Great Escape remembrances with his two sons\u2014the food he ate, his work as a penguin dispersing sand, teaching Roger Bushell conversational Danish on the circuit, even the horrors of the forced march in the winter of 1945. The boys eventually learned about their father's extraordinary athleticism at Stalag Luft III, such that he could throw a golf ball farther than almost every other kriegie in the North Compound.\n\n\"[Dad] was very thin at the end of the war,\" Stephen said. \"When he got out [of the service,] he started lifting weights, building his body back. There are pictures in which he looked nicely chiselled and cut. But my mother never had any interest . . . in his physique. He would say, 'I married the wrong woman. She didn't appreciate it when I started putting muscles back on.'\"\n\nIn some respects it took a discovery after Vicki's father died to put him, his wartime past, and his postwar life of torment into focus for her. Like her brothers, Vicki had absorbed the passed-down stories about her father's career as a fighter pilot and a prisoner of war, but she said it was all a bit like watching a sub-title without any detail. Then, late in 2010, she gained access to some photographs of her father in the air force and the letters exchanged between her father and family members while he was a POW at Stalag Luft III.\n\n\"[From his letters] I learned . . . he was keen. He wanted to serve his country. . . . One of his greatest fears was not being posted overseas. He never complained about doing what was asked of him. He was fearless.\"\n\nSince reading the correspondence and viewing the photos, Vicki Sorensen has willingly dedicated countless hours on the phone, on the internet, and on her own time to discovering as much about her father's role in the Great Escape as she can. She methodically contacts other former kriegies or their families all over the world in pursuit of any details about her father's time in the prison camp. Perhaps what motivates her most is a drive to understand Frank Sorensen's decision to trade his position on the escape list; since she surmises her father gave his spot to either James Catanach or Arnold Christensen (both shot by the Gestapo), she believes that the demons he experienced came from knowing he survived and they did not.\n\n\"My father never let go of the guilt, saying, 'That should have been me,'\" she said. \"My father never got over the execution of the fifty. . . . A good friend of his told me when he was visiting, my father showed him the Tunnel Martyrs memorial picture and said, 'These were my buddies. They're all gone.'\"\n\nFamily correspondence continues to shed light on the silent battle that Frank Sorensen waged as his wife and children struggled to understand him. Among the passages Vicki Sorensen turns to for explanation and comfort is her grandfather's assessment of Frank at an early reunion as the war was ending.\n\n\"There he was tall and smiling,\" her grandfather noted in his diary. \"He was in his battledress with a kitbag under his arm, thin and tanned. We had lunch and got out for the table some of the things we have been saving up just for this occasion. His appetite is not great, but it is growing. He needs building up. His spirits are rising and he has improved much . . .\"\n\nA final observation from Vicki's Uncle Ben about his brother concluded that \"after the war Frank was . . . without his former zest for life. He suffered [from] post-traumatic stress syndrome, which in my opinion affected the change in Frank from his youthful exuberance to essentially an unhappy life and old age.\"\n\nAs she grew up in London, Ontario, in the 1960s, Barbara Edy recognized her father Don Edy had two careers. She knew he was involved in business, earning a living, paying the bills, and raising a family. But her dad had also devoted plenty of time to an equally important avocation\u2014getting his wartime memories off his chest and down on paper. The resulting book, called Goon in the Block, was published in 1961 and it recounted Don's experiences inside Stalag Luft III from his arrival there in November 1943 until the kriegies were marched out of the compound and across Germany in the winter of 1945. When The Great Escape movie hit the theatres in 1963 , for the first time in her life, Barbara could see (even if distorted by Hollywood fabrication) \"a visual perspective\"[26] of her father's war inside German prison wire.\n\nUnlike the children of other kriegies, who wouldn't (or couldn't) talk about their POW experiences, Barbara and her four older siblings grew up hearing about Stalag Luft III as told and written by their father. Not only did he record the events with precision, Barbara noted that he told the stories with the flair of a raconteur. What's more, word of Goon in the Block had spread and more than just Don Edy's family members wanted to read it; but there weren't any more copies of the book around and Don had no interest in a reprint. That's when Barbara sensed he was passing the torch. She felt compelled to help her father stay in touch with his wartime comrades at the same time she wanted and keep his stories in circulation among historians and journalists. She became Don's \"information gal.\" More than that, her work became a personal crusade.\n\n\"The families and offspring of the kriegies all express the same sentiment,\" Barbara said, \"to carry on the memory of not only the fifty escapers] whom Hitler ordered murdered, but, as well, the two hundred, huddled in Hut 104 waiting their turn to escape, the hundreds [of others] who assisted, and all the men of Stalag Luft III and its famous, proud, spectacular distraction to create havoc in the midst of Germany's war effort.\"[[27]\n\nVia correspondence and personal contact, Barbara has built a rapport with ex-kriegies and their children in Australia, New Zealand, the UK, the US, and Canada. In 2012 she and her sister Jane Hughes helped to recover and restore a collection of published photographs originally compiled in a book of remembrances and reflections, called Wire Bound World, to redistribute to other survivors of Stalag Luft III. Then, through a series of emails, she initiated an internet exchange of information called The Beginning of List 200, designed to pool stories, images, biographies, and communications about the two hundred men on the original escape list.\n\nAlong the way, like other kriegie offspring, Barbara believes she has learned how that \"proud, spectacular distraction\" succeeded in binding her dad and the rest of the Commonwealth flyers together, and why that common bond made the events leading up to March 24\u201325 , 1944, so significant. Just as bomber pilot and Stalag Luft III forger Tony Pengelly had deduced when he was first imprisoned at Barth in 1940, that \"we would have to organize to be successful,\"[28] Barbara Edy concluded that none of the kriegies' accomplishments at Stalag Luft III would have been possible without that large group of captives working together as one.\n\n\"The Great Escape would not have been possible,\" she said, \"save for the absolute co-operation from over a thousand POWs' non-stop secrecy, vigilance, persuasion, distraction, cunning, bravery, spirit, and talent.\"[29]\n\nDavid Pengelly was three years old in 1938 when his older brother left the family home in Weston, Ontario. At eighteen, Tony Pengelly couldn't vacate the household fast enough. He had not enjoyed a close relationship with his father, besides which, he had his heart set on a career in the Royal Air Force whether his father endorsed his decision or not. A few weeks later, the older Pengelly son stepped off a cattle boat in the UK and was quickly accepted into RAF training, which had him flying combat operations in Bomber Command from the first week of the war in September 1939. Periodically, Tony sent mementos to his little brother, the first being a picture postcard depicting a Fairey Battle bomber. Tony had addressed the card \"To Liney,\" since Lionel was David's middle name. He cherished the card. On David's fifth birthday, just a few months before F\/L Tony Pengelly and his Whitley bomber crew were shot down over Germany, David received a Dinky toy model of a Whitley in the mail.\n\n\"From the time I was five,\" David Pengelly said, \"I adored the thought of my brother. Spitfire and bomber pilots were like rock stars or astronauts flying these wonderful machines.\"[30]\n\nIn July 1945, Tony Pengelly, former flight lieutenant and forgery chief at Stalag Luft III, came home to Canada. The family made plans to help Tony decompress on Pengelly Island, a one-acre rock outcropping on Sawyer Lake in the Haliburton Highlands of Ontario. There was no electricity on the island, nor indoor plumbing, just a family cottage accessible only by boat. With limited accommodation inside the cottage, David and Tony were encouraged to camp outside in a tent. The first night alone together, older brother regaled younger brother with some of his wartime yarns. Then, the boys' mother prepared a meal with all of Tony's favourites\u2014roast beef, corn-on-the-cob, and fruit pies\u2014and watched the former kriegie dig into his first home-cooked meal since he'd left Weston in 1938.\n\n\"Do you have a little more?\"[31] Tony asked when he'd finished his first portion.\n\n\"Sure! Sure!\" his family all said. And he dove into a second portion and a third until he couldn't eat another mouthful.\n\n\"There's two or three cobs of corn left,\" Tony noticed. \"What's going to happen to those?\"\n\n\"Oh well, they'll be cold,\" his mother said. \"We'll just throw them out.\"\n\n\"Don't do that. I'll have them for breakfast,\" Tony insisted. And he promised he'd do the same with the beef and the pies.\n\nMementos that prisoners of war brought home were few and far between. Any personal items Tony Pengelly may have carried with him when he was shot down on November 14, 1940, were destroyed in the crash of his Whitley bomber or confiscated during questioning by Luftwaffe interrogators as they processed him for imprisonment in Germany and later Poland. And since he became one of X Organization's principal forgers, Pengelly would have shown or shared very few of his possessions publicly inside the compound either. In other words, despite helping to win the war, Pengelly had little to show for it outside of service medals and replenished air force insignia. Consequently, when he married and had his own children, Tony came to his younger brother David for a favour.\n\n\"Do you still have that Whitley [Dinky toy] I sent you?\" he asked.\n\n\"Of course,\" David told him. It was the favourite memento from his big brother's service overseas as a bomber pilot during the war.\n\n\"Can I have it for a while? I'd like my kids to see it,\" Tony said.\n\nDavid agreed to loan it, but then never saw it again. Some of the other keepsakes that Tony did manage to salvage from his time at Stalag Luft III ended up in a small briefcase passed down to his son, Chris Pengelly. A few RAF certificates, diagrams, letters, newspaper clippings, and photographs of Tony posing in a group in front of barracks huts or on stage at the North Compound theatre survived in the leather case. Of his father's role in the Great Escape, Chris Pengelly knew just a little. His father had shared more with his Uncle David than with him. Chris was a teenager in 1963 when The Great Escape was released; his mother Pauline reported the movie disrupted Tony's sleep with recurring nightmares. In contrast, when the sitcom Hogan's Heroes\u2014depicting life in a mythical German POW camp\u2014appeared on television between 1965 and 1971, Chris recalled that his parents loved the series.\n\n\"They watched it all the time,\"[32] Chris Pengelly emphasized. \"He laughed so hard each time [the farcical German Sergeant Schultz] said, 'I see nothing. I hear nothing.' He considered it very funny, but quite realistic. You don't get shot down and say, 'Oh, I'll just take my digital camera with me.' He knew they had to make things from scratch and bribe the guards for things like the camera . . .\n\n\"The Great Escape movie gave him nightmares,\" Chris said finally. \"The TV show let him laugh about it.\"\n\nVery little of the North Compound that the six hundred Canadians knew from 1943 to 1945 exists intact today. On the actual site, just outside the town of Zagan (the Polish spelling of Sagan), the double fencing is gone. So are the watchtowers, the Vorlager, and any above-ground evidence of the cook house, the theatre, or the fifteen barracks huts. All gone. Only concrete pads and some masonry walls remain the way they were when the Commonwealth kriegies were transferred there from the East Compound in the middle of the war. The rest of the former prison camp, the forest and weeds have pretty much reclaimed. Periodically, Polish groundskeepers chop back the brush that pokes through the bricks of the fire pool or the theatre foundation so that visitors passing through each summer can get an idea of what they once looked like.\n\nAt the northern edge of the property, a walkway of crushed stone with wooden borders, twenty inches wide, runs 336 feet north-south the full length that tunnel \"Harry\" did\u2014from the concrete pad where Hut 104 stood to the approximate exit hole just shy of the woods. At the edge of that same pine forest (that stands very much as it did during the war) are sun-faded commemorative plaques. Near the end of the walkway nearest Hut 104 rests a series of flat stone markers with the names of the fifty executed air force officers engraved on them. The markers include the names of Hank Birkland, Gordon Kidder, Pat Langford, George McGill, James Wernham, and George Wiley\u2014the Canadians murdered after the breakout.\n\nTo their credit, the volunteers at the Museum of Allied Forces Prisoners of War Martyrdom periodically welcome groups of tourists, school children, and some of the kriegie offspring who occasionally stop to explore and imagine on their own. West of the former prison compound, at the museum site, a replica of Hut 104 gives visitors an approximation of the Commonwealth air officers' barracks experience. There's a stove (like the original that covered the trap to tunnel \"Harry\") sitting in the appropriate corner of Room 23, as well as bunk beds, a dining area, and the \"To All Prisoners of War! The escape from prison camps is no longer a sport!\" propaganda poster tacked on the hut wall after the Great Escape. A nearby pavilion contains a small library and an exhibit room. Out in front of the pavilion, a reproduced watchtower lords over a stretch of tunnel containing replica bed boards, trolley, and tracks. This is a facsimile of about fifty feet of \"Harry\" constructed a few feet down and covered in a see-through plastic ceiling. The replica gives a false sense of accessibility and ease of passage.\n\nNowhere on the old compound property nor among the museum exhibits can visitors experience the claustrophobia that tunnel designer Wally Floody and diggers John Weir and Hank Birkland knew underground . . . or realize the audacity and skill of scroungers Barry Davidson, Joe Noble, and Keith Ogilvie . . . or recognize the volume of intelligence Kingsley Brown amassed for Dean and Dawson . . . or appreciate the precision of Tony Pengelly's work forging documents and performing female roles on the theatre stage . . . or witness the speed with which trapf\u00fchrer Pat Langford opened and closed the entrance to \"Harry\" each day it moved the kriegies closer to a shot at freedom . . . or hear the conversation basics that language trainers Gordon Kidder and Frank Sorensen gave potential escapers . . . or comprehend the nerve that security men George McGill, George Sweanor, and Dick Bartlett exhibited to protect the escape committee's greatest secrets . . . or understand the efficiency of the penguins, stooges, and duty pilots all running interference at the camp their German captors described as \"escape proof\" . . . or feel the helplessness the six Canadians murdered by the Gestapo must have known in their last moments.\n\nProof of their contribution to the Great Escape is recorded in these pages and in the stories yet to be gathered and verified by historians, the families of ex-kriegies, and an apparently ever-growing community that refuses to let this story die.\n\n* * *\n\n* The official Bomber Command records show that Operation Exodus aircrews from 1, 5, 6, and 8 Groups carried out 469 flights, principally from Brussels to the UK, and repatriated approximately seventy-five thousand officers and airmen. Aircrews completed the operation without a single mishap.\n\n# Photo Section 1\n\nSpace inside an RAF Whitley bomber (above left) was extremely limited. In 1940, after he bailed out of his crashing Whitley, RCAF pilot Tony Pengelly found himself in an equally confined space (above right, seated on bed, left) in a POW hut at Stalag Luft I (Barth, Germany) with downed Canadian Spitfire pilot Wally Floody (seated next to him).\n\n(above left) Long before The Great Escape, a tunnel designing\/digging team\u2014(clockwise from lower left) Wally Floody, Sam Sangster, John \"Scruffy\" Weir, and Hank \"Big Train\" Birkland\u2014experimented in ad hoc escape attempts at Stalag Luft I. Eventually, under Big X, Roger Bushell (above right), at Luft III, X Organization developed plans for a sophisticated mass breakout.\n\nThis photo, taken from the American South Compound (Sagan c. 1943) and looking northeast toward the German administration area (Kommandantur), shows the barren landscape of the appell area of the North Compound broken physically and psycologically by the theatre.\n\nRCAF pilots Barry Davidson (left) and Dick Bartlett, once shot down, became chief scrounger and custodian of the secret wireless radio for the escape committee respectively. They developed such skills early in 1940 at Stalag Luft I and were seasoned pros by the time they were sent to Luft III.\n\nMeanwhile, fellow RCAF pilots Keith Ogilvie (left) and Kingsley Brown at Stalag Luft III worked in intelligence\u2014the former fleeced German guards for their wallets, the latter searched the prison library for German officials' identities to steal and for police protocol at train stations and borders\u2014all to assist camp forgers manufacture fake documents for the escapers.\n\nBefore the mass breakout on March 24\u201325, 1944, RAF F\/L Ley Kenyon was asked to record images of escape tunnel \"Harry.\" Underground, sometimes on his back, by the light of fat-fuelled lamps, he sketched (above left) a digger cutting into sand at the face of the tunnel; (above right) an underground workshop (tin over worker's head, activated from tunnel trap above, contained pebbles as signal for quiet). In Hut 104 (below) he drew the entrance to tunnel \"Harry\" through the trap in the concrete foundation under the stove (the entire operation required lookouts\u2014\"stooges\"\u2014to warn tunnel crews of any approaching German guards).\n\nAerial intelligence photo (c. 1944) shows the evolution of Stalag Luft III prison camp from original East and Centre compounds, opened in 1942, to additional North, South, and West compounds; the prison camp eventually held 10,000 air force POWs. Despite hundreds of attempts, only six POWs completed the \"home run\"\u2014getting from Luft III back to the UK.\n\nRCAF airman George Sweanor (left) vowed not to get romantically involved while training as a bomb-aimer in the UK, while Joan Saunders felt her work in the British war industry was a top priority. Nevertheless, they fell in love, married, and were separated a month later when George was shot down and sent to Stalag Luft III for the duration.\n\nFor RCAF Spitfire pilot Don Edy (left), barracks in the North African desert were rustic at best; it got worse in February 1942, when he was shot down, imprisoned first in Italy, and then sent to Stalag Luft III. On the other hand, severe conditions for RCAF gunner George Harsh were nothing out of the ordinary; before being shot down in October 1942 he'd been an inmate in US prisons for a dozen years.\n\n# Photo Section 2\n\nStalag Luft III POWs\u2014particularly those from North America\u2014organized dozens of baseball teams; this one was typical, featuring (back l to r) \"Gee\" Rainville, Slim Smith, Joe Loree, Stephens, Jimmy Egner, Jimmy Lang, Randy Ransom; (front l to r) Art Hawtin, Earl Clare (team captain), Tommy Jackson, Glen Gardner, Ernie Soalier.\n\n(middle) Ice hockey games didn't end with regulation time or even sudden death goals, but when the last of the hockey sticks (donated by the Red Cross or stores in Canada) broke beyond repair. (bottom) Boxing matches inside Stalag Luft III, such as one with George McGill and Eddie Asselin in 1942, offered exercise and a diversion while other POWs attempted to escape.\n\n(above) RAF officer Ley Kenyon captured the climax of The Great Escape in his sketches, including February 29, 1944, when German guards fortuitously transferred key players of the escape committee to a neighbouring prison, and (below) the night of March 24, 1944, when escapers realized their tunnel \"Harry\" came up short of the pine forest, requiring a rope system to signal when German sentries had passed and another escaper could crawl out and make a run for it.\n\nAfter German guards discovered the mass breakout and recaptured 77 of the 80 escapers, they displayed the contraptions the POWs had built inside tunnel \"Harry.\" (above) Prison guard Karl Griese (left) and a second guard show off a tunnel trolley inside shoring box frames (they've got them upside down\u2014frames were narrower at top). (below right) Griese (a.k.a. \"Rubberneck\") operates a ventilating pump that kept fresh air flowing underground through nearly 400 feet of Klim tins soldered together. (below left) Guard demonstrates sand-dispersal sacks hidden under POWs' coats and inside trousers. In over a year of tunnel excavation at Stalag Luft III, \"penguins\" dispersed several hundred tons of sand from tunnel \"Harry\" alone.\n\n(above) Senior British Officer G\/C Herbert Massey (right) speaks to German officer Hans Pieber, who guarded Commonwealth air officer POWs at both Stalag Luft I and III; also in charge during the forced march in the winter of 1945 (he then knowingly carried the POWs' secret radio to receive BBC broadcasts about the war). (below) Controversial crime investigator and SS officer Arthur Nebe is said to have soured on Nazi philosophy (extermination of European Jewry and executions of Russian civilians) and joined the plot to kill Hitler in July 1944; however, he was guilty of choosing 50 of the recaptured Great Escapers to be executed.\n\n(above right) This secretly photographed image of Allied air force officers being force-marched from Sagan, Poland, was taken in January 1945. Indicative of their skill to adapt, in just hours, the POWs transformed prison camp tables, chairs, and bed boards into sleds and backpacks to carry survival gear. (above left) Just before their liberation in western Germany, the POWs posted signs to prevent being shelled or strafed by friendly forces.\n\nThroughout the winter forced march, RCAF pilot Frank Sorensen carried in his pocket this Ley Kenyon sketch of the memorial (built in 1944) to the 50 murdered officers\u2014the stone monument remains intact northwest of the Stalag Luft III site today.\n\n# Notes\n\n* * *\n\nintroduction: \"heroes resurface\"\n\n1. \"has created a classic\": Time magazine, July 1963.\n\n2. \"take notice and find inspiration\": Arthur A. Durand, quoted in The Making of the Great Escape documentary, produced by Prometheus Entertainment in association with Van Ness Films, Foxstar Productions, Fox TV Studios, and A&E TV Network (Metro Goldwyn Mayer, Home Entertainment Inc.).\n\nchapter one: the king's regulations\n\n1. \"a way through\": Winston Churchill quoted in Charles Messenger, \"Bomber\" Harris and the Strategic Bombing Offensive, 1939\u20131945 (London Arms and Armour Press, 1984), p. 39.\n\n2. \"dropping leaflets\": Tony Pengelly, diary notes, courtesy of Chris Pengelly collection.\n\n3. strategic bombing campaign: Martin Middlebrook and Chris Everitt, The Bomber Command War Diaries, An Operational Reference Book: 1939\u20131945 (Penguin, London, 1985) p. 92.\n\n4. Fairey Battles: Ibid., p. 93.\n\n5. apprehension: Russell Braddon, Cheshire V.C., A Study of War and Peace (Evans Brothers Limited, London, 1954), p. 60.\n\n6. \"remarkable breed of men\": Leonard Cheshire quoted in ibid., p. 69.\n\n7. \"chute off and buried it\": Daniel G. Dancocks, In Enemy Hands, Canadian Prisoners of War 1939\u20131945 (Hurtig Publishers, Edmonton, 1983), p. 8.\n\n8. \"protect the Security\": \"The Responsibilities of a Prisoner of War\" (RAF Air Publication 1548, RAF Command of the Air Council, March 1936).\n\n9. \"name, rank and number\": Ibid.\n\n10. \"give him in return\": Ibid.\n\n11. \"you chaps down\": Dancocks, p. 10.\n\n12. \"succeed in escaping\": \"The Responsibilities of a Prisoner of War\"\n\n13. \"private flying machines\": Certificate of Competency and Licence to fly Private Machines, C.A. Form 64, issued by Great Britain and Northern Ireland.\n\n14. first mass escape: Sydney Smith, Wings Day: The Man Who Led the RAF's Battle in German Captivity (William Collins, London, 1968), p. 77.\n\n15. \"prison camp life\": Tony Pengelly, \"X For Escape,\" by Flt.-Lieut. Tony Pengelly, as told to Scott Young, with permission from Maclean's magazine, November 1 and 15, 1945.\n\n16. \"organize to be successful\": Ibid.\n\nchapter two: bond of wire\n\n1. \"anything about this\": John Weir, note to Frances McCormack, 1940, with permission.\n\n2. \"bomber pilot\": Frances Weir (n\u00e9e McCormack) interview, June 19, 2012, Toronto.\n\n3. \"scruffy looking individual\": Sandra Martin, \"He played a role in The (real) Great Escape,\" Globe and Mail, November 11, 2009.\n\n4. war work herself: Frances Weir interview, June 19, 2012.\n\n5. \"in the least\": Ibid.\n\n6. \"first ever sweep\": Hugh Godefroy, Lucky Thirteen (Canada's Wings Inc., Stittsville, Ontario, 1983), p. 100.\n\n7. \"Weir and Gardiner are missing\": Ibid., p. 107.\n\n8. \"a perpetual clown\": Ibid., p. 107\n\n9. \"an important prisoner\": Robert Stanford Tuck, quoted in Larry Forrester, Fly For Your Life: The Story of R. R. Stanford Tuck (Nelson Doubleday, New York, 1956), p. 275.\n\n10. seventy yards wide: Arthur A. Durand, Stalag Luft III: The Secret Story (Patrick Stephens Ltd., UK, 1989), p. 79.\n\n11. \"safe and sound in Germany\": John Weir, letter to Frances McCormack, December 13, 1941, with permission.\n\n12. \"back to you soon\": John Weir, letter to Frances McCormack, December 22, 1941, with permission.\n\n13. door-to-door salesman: Jonathan Vance, A Gallant Company: The Men of the Great Escape (Pacifica Military History, Pacifica, California, 2000), p. 120.\n\n14. mining companies' sports teams: Barbara Hehner, The Tunnel King, The True Story of Wally Floody and The Great Escape (HarperCollins, Toronto, 2004), pp. 3, 7.\n\n15. Luftgau: Durand, p. 134.\n\n16. \"how many, four\": John Weir, letter to Frances McCormack, February 26, 1942, with permission.\n\n17. \"no similarity between the two\": Wally Floody, interviewed by the National Air Force Museum, CFB Trenton, c. 1970.\n\n18. \"a total success either\": Wally Floody, interviewed by the National Air Force Museum, CFB Trenton, 1989.\n\n19. collapse tunnels: Durand, p. 81.\n\n20. \"I got claustrophobic\": Barry Davidson, \"Barry Davidson\u2014Prisoner of War,\" Bomber Command Museum of Canada, courtesy of Barry Davidson Jr., 2000.\n\n21. \"pound bombs\": Ibid.\n\n22. \"for the duration\": Ibid.\n\n23. cutting or digging tool: Jean Morrison McBride, quoted in Patricia Burns They Were So Young: Montrealers Remember World War II (V\u00e9hicule Press, 2002), p. 239.\n\n24. plastic surgery: Eric Howald, \"Mac Jarrell Remembers: Spent 45 months as prisoner of war,\" The Kincardine Independent, November 11, 1979.\n\n25. \"the prisoners hated it\": Ibid.\n\n26. \"kick from a mule\": Dick Bartlett, quoted Stuart E. Soward's One Man's War: Sub Lieutenant R. E. Bartlett, RN Fleet Air Arm Pilot (Neptune, 2005), p. 53.\n\n27. \"into the camp\": Ibid., p. 90.\n\n28. \"news and intelligence\": Ibid., p. 90.\n\n29. \"be home again\": John Weir, letter to Frances McCormack, January 21, 1942, with permission.\n\n30. \"escape campaign\": Vance, p. 32.\n\n31. \"operational function\": Smith, pp. 58\u201360.\n\n32. \"slippers, gloves, and cap\": John Weir, letter to Frances McCormack, April 29, 1942, with permission.\n\n33. \"lots of fresh air\": Ibid.\n\n34. \"is fatten me up\": Ibid.\n\nchapter three: \"spine-tingling sport\"\n\n1. \"fenced-in feeling\": John Hartnell-Beavis, Final Flight (Merlin Books, Braunton, Devon, UK, 1985), p. 31.\n\n2. Kommandantur: Durand, p. 103.\n\n3. both his artificial legs: Paul Brickhill, Reach for the Sky, The Story of Douglas Bader (Collins, London, UK, 1954), p. 335.\n\n4. months to accomplish: Les Allison, Canadians in the Royal Air Force (self-published, Roland, Manitoba, 1978), p. 177.\n\n5. \"flown over Berlin\": Andrew Thompson, quoted in documentary \"The Great Escape: The Canadian Story,\" 2004, courtesy of producer Don Young.\n\n6. \"courtesy and consideration\": Tommy Thompson, quoted in Ibid.\n\n7. back to England: Vance, p. 39.\n\n8. night fighters: Ibid. p. 192.\n\n9. \"escape proof\": Floody interview, 1970.\n\n10. G\u00f6ring's luxury camp: Paul Brickhill, The Great Escape (Faber and Faber Ltd., London, UK, 1951), p. 24.\n\n11. thousand-foot area: Durand, p. 258.\n\n12. cutting through the soil: Brickhill, The Great Escape, p. 28.\n\n13. \"noses of the Germans\": Floody interview, 1970.\n\n14. \"expert at it\": Ibid.\n\n15. \"a minor one\": George Harsh, Lonesome Road (W. W. Norton & Co., New York, 1971), p. 176.\n\n16. \"flamboyant beau geste\": Ibid., p. 132.\n\n17. 4.3 per cent loss rate: Middlebrook and Everitt, p. 297.\n\n18. \"you're an American\": Harsh, p. 187.\n\n19. \"secure all this\": Ibid., p. 191.\n\n20. \"he's an ex-convict\": Ibid., p. 192.\n\n21. \"the black hole\": Kingsley Brown, Bonds of Wire, A Memoir (Collins, Toronto, 1989), p. 50.\n\n22. \"propaganda job\": Ibid., p. 57.\n\n23. \"Hitler kaput\": Ibid., p. 58.\n\n24. \"free side of the wire\": Ibid., p. 65.\n\n25. Red Cross parcels: A. K. Ogilvie, \"Tigers in the Tunnel,\" Air Intelligence Training Bulletin, Vol. XIV, No. 1, January 1962.\n\n26. wallet on the floor: Ian Darling, Amazing Airmen: Canadian Flyers in the Second World War (Dundurn, Toronto, 2009), p. 27.\n\n27. \"my mittens. Everything\": Don McKim, interview Simcoe, Ontario, January 2, 2011.\n\n28. Germans guarding them: Durand, p. 159.\n\n29. five-foot-two: Lyn Tremblay, \"Veteran prisoner of war took part in the Great Escape,\" Simcoe Reformer, December 3, 2009.\n\n30. \"doing anything else\": McKim interview, 2011.\n\n31. \"a little longer\": Bob McBride, quoted by Jean Morrison McBride in Burns,  \np. 239.\n\n32. \"burned forever\": Ibid., p. 240.\n\n33. hidden him in Prague: Brickhill, The Great Escape, p. 32.\n\nchapter four: escape season\n\n1. Goleb Plasov: Brown, p. 63.\n\n2. North Compound: Brickhill, The Great Escape, p. 34.\n\n3. \"the enemy has ears\": Kingsley Brown, \"The Good Grey Days of Stalag III,\" The Globe Magazine, May 18, 1957, p. 10.\n\n4. \"a sense of danger\": Ibid., p. 10.\n\n5. \"better luck next time\": Kingsley Brown quoting arresting police officer, Bonds of Wire, p. 75.\n\n6. war was forgotten: Op cit, p. 11.\n\n7. Hut 101: John S. Acheson, A World War II Tale of Stalag Luft III and of The Great Escape (unpublished manuscript housed at the Kitchener Public Library, 2002, with permission). p. 9.\n\n8. half a minute or less: Soward, p. 114.\n\n9. \"swing a stunted cat\": Brickhill, The Great Escape, p. 42.\n\n10. \"or we'll start shooting\": Gordon King, interviewed by Byron Christopher, Edmonton, October. 20, 2011.\n\n11. \"just something to do\": Ibid.\n\n12. \"German shepherd dog\": Floody interview, 1970.\n\n13. \"three major tunnels\": Brickhill, The Great Escape, p. 34.\n\n14. \"a lot more troops\": Floody interview, 1970.\n\n15. \"vertical shaft to Harry\": Henry Sprague, interviewed by the National Air Force Museum, CFB Trenton, 1989.\n\n16. \"complete surveillance\": Floody interview, 1970.\n\n17. \"twenty to thirty seconds\": Floody interview, 1989.\n\n18. \"toward the showers\": Pengelly, \"X For Escape.\"\n\n19. Bob van der Stok: Vance, p. 138.\n\n20. cardboard holster: Larry Forrester, Fly For Your Life: The Story of R. R. Stanford Tuck (Nelson Doubleday, New York, 1956), p. 282.\n\n21. \"dressed as German soldiers\": Pengelly, \"X For Escape.\"\n\n22. \"our cheapest commodity\": Ibid.\n\n23. \"rolled around the pebble\": Arthur Crighton, Memories of a Prisoner of War (privately published memoirs, 3rd edition, February 2012), p. 21.\n\n24. \"whole of Kriegiedom\": Ibid., p. 22.\n\n25. \"weren't digging tunnels\": Arthur Crighton, interviewed by Byron Christopher, Edmonton, Alberta, March 18, 2012.\n\n26. tunnel projects: Durand, p.116.\n\n27. \"it's foolproof\": Michael Codner, quoted in Eric Williams, The Wooden Horse (Collins, London, UK, 1955), p. 38.\n\n28. weary of inspecting the horse: Edward Patrick Nurse, The Last Glorious Flight of Halifax Bomber Lima Quebec Papa 741: A chronicle of the wartime experiences of Edward Patrick Nurse (unpublished manuscript, 2012, with permission), p. 48.\n\n29. \"heavens beyond the wire\": Williams, p. 122.\n\n30. had been killed in action: Ian Crofton, Great Escapes (Quercus, London, UK, 2009), p.145.\n\nchapter five: servant to a hole in the ground\n\n1. Eastern Townships of Quebec: Betty Sorensen, family wartime letters transcription, 1939, courtesy of Vicki Sorensen.\n\n2. \"Roskilde boys\": Frank Sorensen address to Denmark via BBC Radio Free Europe, October 15, 1942, courtesy of Vicki Sorensen.\n\n3. \"driven out of Denmark\": Ibid.\n\n4. thirteen operational sorties: Mikkel Plannthin, \"Fra Roskilde Katedral Til Stalag Luft III\u2014Og Den Store Flugt\"\/\"From Roskilde Cathedral School to Stalag Luft III\u2014and the Great Escape,\" published in Flyvehistorisk Tidsskrift\/Journal of Aviation History, Dansk Flyvehistorisk Forening\/Danish Aviation Historical Society, Forening, December 2011.\n\n5. \"Bay of Tunis\": Frank Sorensen letter to brother Eric Sorensen, July 23, 1945, courtesy of Vicki Sorensen.\n\n6. \"home on the deck\": Ibid.\n\n7. \"Hands up\": Ibid.\n\n8. \"is reported missing\": letter to Mrs. M. B. Sorensen, from F\/L W. R. Gunn, RCAF Casualties Officer, Department of National Defence, Ottawa, April 17, 1943, courtesy of Vicki Sorensen.\n\n9. \"a prisoner of war\": letter to Mr. M. B. Sorensen, from S\/L A. B. Matthews, Air Officer Commanding-in-Chief, RCAF Overseas, London, May 11, 1943, courtesy of Vicki Sorensen.\n\n10. \"one of them\": Frank Sorensen letter to parents, April 19, 1943, courtesy of Vicki Sorensen.\n\n11. \"'round the perimeter\": Frank Sorensen letter to parents, May 8, 1943, courtesy of Vicki Sorensen.\n\n12. \"taught a horse\": Frank Sorensen letter to parents, July 21, 1944, courtesy of Vicki Sorensen.\n\n13. \"thesaurus dictionary\": Frank Sorensen letter to parents, May 18, 1943, courtesy of Vicki Sorensen.\n\n14. insurance company: Vance, p. 172.\n\n15. advance his studies: Linda Tweddell, correspondence with Vicki Sorensen, February 13, 2012.\n\n16. in political science: George Sweanor, interview Colorado Springs, Colorado, July 6, 2011.\n\n17. westerns and biographies: Durand, p. 227.\n\n18. \"studying it though\": Frank Sorensen letter to parents, August 11, 1943, courtesy of Vicki Sorensen.\n\n19. \"wood for shoring\": Floody interview, 1970.\n\n20. rusty gas cans: Brickhill, The Great Escape, p. 57.\n\n21. \"tunnel as it was dug\": Bob Nelson, \"Tom, Dick and Harry of Stalag Luft III\" (unpublished manuscript, 1948, with permission).\n\n22. from the Germans: Vance, p. 125.\n\n23. \"thirty feet underground\": Bob Nelson monologue on Great Escape, recorded by family, 1987.\n\n24. \"a bit more cautious\": Floody interview, 1970.\n\n25. \"trouser bags\": Peter Fanshawe quoted in Brickhill, The Great Escape, p. 52.\n\n26. hole in the ground: Brickhill, The Great Escape, p. 56.\n\n27. \"repairs themselves\": Harold Johnstone, preface to unpublished diary of John Colwell, 2001, with permission, p. 2.\n\n28. \"jump at 11 p.m.\": John Colwell, unpublished diary, April 3, 1943, with permission.\n\n29. \"Room 14, Block 120\": Ibid., April 17, 1943.\n\n30. \"they lost\": Ibid., June 17, 1943.\n\n31. \"hair all cut off\": Ibid., June 20, 1943.\n\n32. \"as a penguin\": Ibid., June 22, 1943.\n\n33. \"what was going on\": John Colwell, quoted in Lynn Welburn, \"A witness to the Great Escape,\" Harbour City Star, May 17, 2001.\n\n34. \"deaden the sound\": King interview, 2011.\n\n35. \"a matter of minutes\": John Weir, quoted in \"The Making of the Great Escape,\" Prometheus Entertainment, 2007.\n\n36. \"with us separated\": John Weir letter to Frances McCormack, April 30, 1943, with permission.\n\n37. any excess wiring: Brickhill, The Great Escape, p. 94.\n\n38. \"South Compound is ready\": Alan Burgess, The Longest Tunnel: The True Story of World War II's Great Escape (Grove Weidenfeld, New York, 1990), p. 45.\n\n39. \"suspicions were aroused\": Nelson, unpublished manuscript, 1948.\n\n40. \"very imperceptible\": Nelson monologue, 1987.\n\n41. \"British are coming\": George Sweanor, notes on \"The Great Escape\u201424 March 1944,\" with permission.\n\n42. \"huts with a band\": Colwell diary, July 1, 1943, with permission.\n\n43. \"1 and 4 a.m.\": Ibid., July 13, 1943.\n\n44. \"in turn and searched\": Ibid., July 15, 1943.\n\n45. \"barbed wire used\": Ibid., July 24, 1943.\n\n46. \"know fuck all\": Brickhill, The Great Escape, records the exchange with the expletive \"damn,\" p. 106; several kriegies, including Pengelly, Wallace, and Sweanor, report Pieber used the word \"fuck,\" making the exchange all the more ridiculous.\n\n47. \"in Hut 101\": Colwell diary, August 21, 1943.\n\n48. \"my soldering outfit\": Ibid., August 31. 1943.\n\n49. \"to inspect it\": Nelson, 1948.\n\n50. \"late Tom's trap\": Ibid.\n\n51. \"with the appendix\": Frank Sorensen, quoted in Plannthin.\n\n52. \"the winter after this\": Frank Sorensen, letter to family, September 17, 1943, courtesy of Vicki Sorensen.\n\nchapter six: \"shysters and crooks and con men\"\n\n1. \"an only child\": Joan Sweanor interview, Colorado Springs, Colorado, July 5, 2011.\n\n2. \"Who Wants War\": George Sweanor interview, July 5\u20137, 2011.\n\n3. \"could walk her home\": George Sweanor, correspondence, February 6, 2013.\n\n4. \"looks, poise, and figure\": Sweanor, It's All Pensionable Time: 25 Years in the Royal Canadian Air Force (self-published, 1967, with permission), p. 36.\n\n5. \"life is so temporary\": Ibid.\n\n6. \"the office at eight\": Joan Sweanor interview, 2011.\n\n7. \"letter for me\": Arthur Morlidge, quoted in George Sweanor, p. 58.\n\n8. \"delightfully foul weather\": Ibid., p. 75.\n\n9. \"only military targets\": Ibid., p. 103.\n\n10. \"get out of here\": George Sweanor interview, 2011.\n\n11. \"last ticket home\": Ibid.\n\n12. \"not coming home\": Joan Sweanor interview, 2011.\n\n13. \"E for Edward\": Sweanor, It's All Pensionable Time, p. 120.\n\n14. \"nerves could relax\": George Sweanor, notes on \"The Great Escape\u201424 March 1944,\" with permission.\n\n15. German-English dictionary: Ibid., p. 107.\n\n16. \"years of our wives\": Shag Rees, quoted by George Sweanor in It's All Pensionable Time, p. 121.\n\n17. \"worthy of his sacrifice\": Sweanor, p. 112.\n\n18. \"mark me down\": Hermann Glemnitz, quoted by George Sweanor in It's All Pensionable Time, p. 133.\n\n19. \"the workman's tools\": Nelson.\n\n20. inside tunnel Harry: Brickhill, The Great Escape, p. 135.\n\n21. \"Luftwaffe ferrets\": Nelson.\n\n22. \"by the prisoners\": von Lindeiner, reference in \"Sagan,\" Royal Air Force Special Investigation Branch report, Report No: WCIU\/LDC\/1460, JAG Ref: MD\/JAG\/FS\/22\/2(2a) War Crimes Interrogation Unit, London, December 1946, p. 17.\n\n23. \"impression in the camp\": Ibid., p. 17.\n\n24. \"very severe penalties\": von Lindeiner, quoted by John E. (Willy) Williams in Ibid., p. 21.\n\n25. appropriate documentation: Durand, p. 291.\n\n26. Group of Seven: Ken MacQueen, \"A Brutal March: Wartime diaries record a trek of 10,000 POWs,\" Maclean's, January 13, 2003, p. 48.\n\n27. \"and I did\": Robert Buckham, quoted in Ibid.\n\n28. the library Bibles: Durand, p. 291.\n\n29. \"morning to night\": Frank Sorensen, letter home, June 26, 1943, courtesy of Vicki Sorensen.\n\n30. \"an empty belly\": Robert Buckham, diary excerpt quoted in Op cit.\n\n31. the violin case: Brickhill, The Great Escape, p. 122.\n\n32. \"some coffee home\": Pengelly, \"X for Escape.\"\n\n33. \"he believed necessary\": Ibid.\n\n34. \"than in a prison camp\": Don MacDonald, quoted in Dancocks, p. 112.\n\n35. \"orders I don't like\": Sweanor, p. 142.\n\n36. panel for stage lighting: H. P. Clark, Wire Bound World (self-published, 1946), p. 13.\n\n37. \"announcing the play\": Don Edy interview, London, Ontario, March 6, 2012.\n\n38. Mena House resort hotel: Don Edy, Goon in the Block (self-published, 1948, with permission), p. 52.\n\n39. garter snake in its place: Ibid., p. 14\u201315.\n\n40. \"Padula boys, Hey, Hey.\": Ibid., p. 95.\n\n41. \"an immediate depression\": Ibid., p. 57.\n\n42. \"got to him first\": Ibid., p. 130.\n\n43. a coffee percolator: Ibid., p. 140.\n\n44. \"to see the shows\": Ibid., p. 136\n\n45. recorded for posterity: George McKiel, quoted in Doug Pricer, \"Survivor of the Great Escape,\" Military History, May 2005, p. 30.\n\n46. \"out of kriegieland\": Sweanor, It's All Pensionable Time, p. 141.\n\n47. grease-paint makeup: Durand, p. 243.\n\n48. \"create feminine characters\": Edy, Op cit., p. 161.\n\n49. \"people doing everything\": King interview, 2011.\n\n50. \"walk and talk like a woman\": McKiel, quoted in Pricer, p. 30.\n\n51. \"Second World War in drag\": Rick Cluff interview, Vancouver, B.C., May 25, 2012.\n\n52. Red Cross parcels: Durand, p. 77.\n\n53. \"organization could use\": Pengelly, \"X for Escape.\"\n\n54. \"along the same route\": Ibid.\n\nchapter seven: the play's the thing\n\n1. \"couple of years ago\": Art Ross, quoted in Brian McFarlane, Stanley Cup Fever: 100 Years of Hockey Greatness (Toronto, Stoddart, 1992), p. 97.\n\n2. men left to play: Kenesaw Mountain Landis, quoted in Time Capsule (Time-Life, 1943), p. 138.\n\n3. \"their team over\": Art Hawtin interview, Beaverton, Ontario, January 8, 2011.\n\n4. \"landed safely\": Ibid.\n\n5. married men against bachelors: John E. Dreifort, \"Anything but Ordinary: POW Sports in a Barbed Wire World,\" Journal of Sport History, Vol. 34, No. 3, Fall 2007, p. 420.\n\n6. \"struck out sixteen batters\": Hawtin interview, 2011.\n\n7. \"fourteen to one\": Ibid.\n\n8. \"let them hit, Bill\": Harold Garland, quoted in Dancock, p. 101.\n\n9. leather boot enthusiastically: Durand, p. 248.\n\n10. \"panes are broken\": Art Hawtin, \"Grandpa's Wartime Album: Memories Surrounding the Great Escape\" (unpublished diary prepared by Janet Hawtin, with permission).\n\n11. \"shot him dead\": Phil Marchildon with Brian Kendall, Ace Phil Marchildon, Canada's Pitching Sensation and Wartime Hero (New York, Viking, 1993), p. 138.\n\n12. \"over the wire\": Frank Sorensen, letter home, June 15, 1944, courtesy of Vicki Sorensen.\n\n13. author Kaj Munk: family letter to Frank Sorensen, March 10, 1944, courtesy of Vicki Sorensen.\n\n14. \"the Tin Man\": Hawtin interview, 2011.\n\n15. \"a master tinker\": Ibid.\n\n16. \"two fire bricks\": John Colwell diary diagrams, September 26, 1944, with permission.\n\n17. \"and cooking utensils\": card contained in John Colwell's diary, with permission.\n\n18. \"in Stalag Luft III\": Brickhill, The Great Escape, p. 129.\n\n19. \"itching to be home\": John Weir, letter to Frances McCormack, October 31, 1943, with permission.\n\n20. \"a friendless room\": Albert Wallace, correspondence, March 15, 2013.\n\n21. \"didn't realize\": Albert Wallace, interview, Toronto, January 5, 2011.\n\n22. \"extremely security conscious\": Ibid.\n\n23. \"our new chimney\": Ibid.\n\n24. \"Your brother, Wally\": Wally Floody, postcard to Catherine Floody, January 26, 1944, with permission.\n\n25. \"before roll call\": Hawtin diary.\n\n26. \"and out went the sand\": Wallace interview, 2011.\n\n27. theatre's crawl space: Harold Johnstone, \"More Memories of Stalag III Days,\" additional notes to John Colwell's diary, 2001, with permission.\n\n28. \"twelve tons of sand\": Pengelly, \"X for Escape.\"\n\n29. \"a severe shock\": Hartnell-Beavis, p. 36.\n\n30. disappear in seconds: Brickhill, The Great Escape, p. 140.\n\n31. \"all kinds of weather\": John R. Harris, Serving and Surviving: An Airman's Memoirs, self-published, 2004, with permission, p. 117.\n\n32. \"Germans blitzing England\": John R. Harris, interview, London, Ontario, March 6, 2012.\n\n33. \"out of the kite\": Harris, Serving and Surviving, p. 71.\n\n34. RCAF Blues tunic: John Crozier, memoir in Aircrew Memories: The Collected World War II and later memories of members of the Aircrew Association, Vancouver Island Branch, Victoria, B.C. (Victoria Publishing Co., 1999), p. 88.\n\n35. \"where he'd come from\": Harris interview, March 6, 2012.\n\n36. \"and he apologized\": Ibid.\n\n37. \"I was shot down\": Ibid.\n\n38. bottoms of their boots: Dreifort, p. 425.\n\n39. \"a lot of work\": Davidson.\n\n40. sticks simply ran out: Dreifort, p. 426.\n\n41. him and his crew: Vance, p. 47.\n\n42. \"sooner than expected\": George Wiley, quoted in Vance, p. 210.\n\n43. \"before he got back\": Floody interview, 1989.\n\n44. camp at Belaria: Sweanor, p. 156.\n\n45. \"a tunnel digger\": Floody interview, 1989.\n\n46. gassed or shot: \"Sagan,\" Royal Air Force Special Investigation Branch report, Report No: WCIU\/LDC\/1460, JAG Ref: MD\/JAG\/FS\/22\/2(2a) War Crimes Interrogation Unit, London, December 1946, p. 67.\n\n47. and 69 lamps: Johnstone.\n\n48. and Tony Pengelly: H. P. Clark, Wire Bound World, self-published, 1946, p. 32.\n\n49. \"in it, or who\": Pengelly, \"X for Escape.\"\n\n50. about two hundred: Alan Burgess, The Longest Tunnel: The True Story of World War II's Great Escape (Grove Weidenfeld, New York, 1990), p. 124.\n\n51. \"number ninety-three\": Pengelly, \"X for Escape.\"\n\nchapter eight: \"through adversity to the stars\"\n\n1. \"prisoner of war\": Pengelly, \"X for Escape.\"\n\n2. \"chance at freedom\": Ibid.\n\n3. a sure thing either: Sweanor, It's All Pensionable Time, pp.156\u201357.\n\n4. \"name was not drawn\": Sweanor interview, July 5\u20137, 2011.\n\n5. into the tunnel: Jacek Jakubiak, names and escape numbers are published in Guide to the Site of the Former POW Camps in Zagan (Museum of Allied Forces Prisoners of War Martyrdom, Zagan, Poland, 2008) pp. 42\u201343.\n\n6. though not fatally: Brickhill, The Great Escape, p. 133 (the Canadian's name was Probert).\n\n7. the other at Luft III: Barry Davidson, A Wartime Log: A Remembrance from Home Through the Canadian YMCA, unpublished, with permission of Barry Davidson Jr.\n\n8. \"miracle to get back\": Wallace interview, 2011.\n\n9. \"time go any faster\": Frank Sorensen, letter to parents, March 20, 1944, courtesy of Vicki Sorensen.\n\n10. through the tunnel: Vicki Sorensen, Frank's daughter, has searched for his exact number on the escape list and has anecdotal information to support that he drew a number in the high teens, but forfeited the spot to either James Catanach or Arnold Christensen.\n\n11. Fugelsang and stay behind: Margaret Davidson, interview Victoria, BC, August 13, 2012.\n\n12. \"and not go out\": Davidson.\n\n13. \"how do I look\": Bob van der Stok, quoted in King interview, 2011.\n\n14. Wiley's final wish: Vance, p. 214.\n\n15. \"the set of maps\": Harris interview, 2012.\n\n16. \"just waiting my turn\": King interview, 2011.\n\n17. to the next three: Brickhill, The Great Escape, p. 185.\n\n18. \"rope to shelter\": Ibid., p.179.\n\n19. station and boarding platforms: Mirek Walczak, guide, Museum of Allied Forces Prisoners of War Martyrdom, Zagan, Poland, June 22, 2011.\n\n20. \"sand that fell in\": Nelson monologue, 1987.\n\n21. \"cold and frosty\": Keith Ogilvie, quoted in Dancocks, p. 130.\n\n22. \"when it all ended\": Mac Reilley, \"Another Perspective of the Great Escape from Luft III,\" The Kriegie Eagle newsletter, January 1995, the National Air Force Museum, CFB Trenton.\n\n23. bed and listened: Edy, p. 158\u201359.\n\n24. food in his pockets: Sweanor interview, July 5\u20137, 2011.\n\n25. soon stopped: Sweanor, p. 161.\n\n26. \"shot doing it\": King interview, 2011.\n\n27. \"Sax's bum blocking the way\": Rees, Ken, \"Stalag Luft 3\u2014The Great Escape: The Wartime Experiences of Wg. Cdr. H.K. Rees,\" at Rob Davis website, www.elsham.pwp.blueyonder.co.uk\n\n28. \"half the camp down\": Harris, Serving and Surviving, p. 119.\n\n29. \"get rid of the lot of you\": Friedrich von Lindeiner, quoted in Sweanor, p. 161.\n\n30. \"burn the maps\": Harris, p. 120.\n\n31. \"Johnny or me again\": Ibid., p. 120.\n\n32. \"provoke them further\": Edy, p. 159.\n\n33. \"done that to him\": Hans Pieber, quoted in Sweanor, p. 162.\n\n34. some 70,000 Germans: Thomas Fleming, \"The Great Escape,\" Boy's Life, March 1997, p. 46.\n\n35. \"broomstick for a cane\": Gordon Venables, from interview by (granddaughter) Katie Bendell, c. 1990, with permission.\n\n36. entry shaft to Harry: Sweanor, p. 162.\n\nchapter nine: the hate campaign\n\n1. run in another direction: Darling, Ian, Amazing Airmen: Canadian Flyers in the Second World War (Dundurn, Toronto, 2009), p. 30.\n\n2. \"bloody miserable\": Keith Ogilvie, quoted in Dancocks, p. 133.\n\n3. in his pockets: Darling, p. 31.\n\n4. and pressed on: Vance, p. 240.\n\n5. \"whilst trying to escape\": Kriminal Kommissar Peter Mohr, quoted in \"Sagan,\" Royal Air Force Special Investigation Branch report, Report No: WCIU\/LDC\/1460, JAG Ref: MD\/JAG\/FS\/22\/2(2a), War Crimes Interrogation Unit, London, December 1946, Appendix 'C' p. 57.\n\n6. \"linguist\": Hugh A. Halliday, \"Flyboys in the Great Escape,\" Legion magazine, July 1, 2007.\n\n7. \"from the Dominions\": Ibid., p.19.\n\n8. \"my duty to escape\": Ogilvie, quoted in Dancocks, p. 134.\n\n9. \"will not be so lucky\": transcripts entitled \"Testimonies of Prisoners of War who escaped on March 25th from Stalag Luft III,\" courtesy Don Young.\n\n10. \"back to your camp\": Ibid.\n\n11. \"morning you'll go\": Ogilvie, quoted in Dancocks, p. 134.\n\n12. \"interested in that\": Wilhelm Scharpwinkel, quoted in \"Sagan,\" Royal Air Force Special Investigation Branch report, Report No: WCIU\/LDC\/1327(a), JAG Ref: MD\/JAG\/FS\/22\/2(2a) War Crimes Interrogation Unit, London, December 1946, p. 1.\n\n13. \"will happen to them\": Richard Max H\u00e4nsel, quoting Scharpwinkel, in \"Sagan,\" Royal Air Force Special Investigation Branch report, Report No: WCIU\/LDC\/1221, JAG Ref: MD\/JAG\/FS\/22\/2(2a) War Crimes Interrogation Unit, London, December 1946, p. 2.\n\n14. \"attempted to escape\": Ibid., p. 2.\n\n15. \"attempt an escape\": Ibid., p.3.\n\n16. Porokoru Patapu: Allen Andrews, Exemplary Justice (Harrap & Co., London, 1976), p.49.\n\n17. \"around the countryside\": Friedrich Kiowsky, quoted in Ibid., Report No: WCIU\/LDC\/1450 MFC\/GEP, JAG Ref: MD\/JAG\/FS\/22(2a), p. 2.\n\n18. \"back of [Kirby-Green's] head\": Erich Zacharias, quoted in Ibid., Report No: WCIU\/LDC\/1133(a) MFC\/LH, JAG Ref: MD\/JAG\/FS\/22\/2(2a), p. 2.\n\n19. \"made the attempt\": Kiowsky quoted in Op cit.\n\n20. \"to shoot you\": quoted in \"Sagan,\" Royal Air Force Special Investigation Branch report, Report No: WCIU\/LDC\/1460, JAG Ref: MD\/JAG\/FS\/22\/2(2a), War Crimes Interrogation Unit, London, December 1946.\n\n21. \"number of sub-humans\": Johannes Post, quoted in Allen Andrews, Exemplary Justice, p. 10.\n\n22. \"the SBO's words\": Sweanor, It's All Pensionable Time, p. 163.\n\n23. \"they could re-escape\": Keith Ogilvie, quoted in Dancocks, p. 135.\n\n24. \"to exemplary justice\": Anthony Eden, in House of Commons Hansard, May 19 and June 23, 1944, quoted in Andrews, p. 211\u201313.\n\n25. \"a deliberate massacre\": Sweanor, p. 166.\n\n26. \"families after the war\": Sweanor, p. 167.\n\n27. \"we sang it and felt better\": Pengelly, \"X for Escape.\"\n\n28. \"one red and one black\": Edy interview, 2012.\n\n29. \"ring drawing ever tighter\": Pengelly, \"X for Escape.\"\n\n30. \"they shoot each day\": Sweanor, p. 172.\n\n31. \"the approaching Russians\": Harsh, p. 177.\n\n32. \"The Handbook of Modern Irregular Warfare\": poster quoted in Sweanor,  \np. 174.\n\n33. \"the United States of America\": poster quoted in Sweanor, p. 175.\n\n34. \"one man volunteered\": Sweanor, p. 173.\n\n35. \"leaving weapons behind\": George Sweanor interview, 2011.\n\nchapter ten: long road home\n\n1. \"out into the woods\": Crighton interview, 2012.\n\n2. \"quite an impressive service\": Wally Floody, letter to his wife, Betty Floody, December 6, 1944, with permission.\n\n3. \"we're officers\": Crighton interview, 2012.\n\n4. \"it will be soon now\": John Weir, letter to Frances McCormack, December 5, 1944, with permission.\n\n5. \"this letter home to you\": Ibid., June 28, 1944.\n\n6. \"the occasional circuit\": Op cit., December 5, 1944.\n\n7. \"eyes betray his hunger\": Robert Buckham, Forced March to Freedom (Canada's Wings Inc., 1984), p. 10.\n\n8. \"becomes a reality\": Ibid., p. 12.\n\n9. \"everyone is sewing\": John Colwell, unpublished diary, January 25, 1945, with permission.\n\n10. The Wind and the Rain: Harris, p. 127.\n\n11. \"for us now,\": conversation recorded in Edy, p. 166.\n\n12. he was ready to go: Colwell.\n\n13. Hut 104 caught fire: Durand, p. 328.\n\n14. 2.5 million cigarettes: Ibid., p. 329.\n\n15. \"from inside the camp\": Buckham, p. 18.\n\n16. \"a great instrument\": Crighton interview, 2012.\n\n17. \"wire fences and Goon boxes\": Colwell.\n\n18. \"many fine human beings\": Sweanor, p. 180.\n\n19. \"and keep moving\": Bartlett, quoted in Soward, p. 136.\n\n20. same doomed Halifax bomber: Nurse, p. 31.\n\n21. \"appreciate their solicitude\": Brown, p. 195.\n\n22. his jacket with newspapers: Hehner, p. 99.\n\n23. \"beef we were wolfing\": Harsh, p. 214.\n\n24. \"souvenir of my prison years\": Brown, p. 196.\n\n25. \"I think the cost was worth it\": Floody interview, 1970.\n\n26. \"truly G\u00f6tterd\u00e4mmerung\": Harsh, p. 216.\n\n27. over twenty miles of road: Stuart Gardner Hunt, Stuart Gardner, Twice Surreal: A Memoir of World War II and Korea (self published, 2005), p. 123 (author reports the column as 20 miles long).\n\n28. safety in numbers and better identification: Durand, p. 330.\n\n29. \"wave your arms\": John Weir, quoted in Crighton interview, 2012.\n\n30. \"on my straw bed\": Crighton, p. 19.\n\n31. \"when I promptly fainted\": Harris, p. 133.\n\n32. \"endurance was our only resource\": Buckham, p. 37.\n\n33. \"I want my mother\": officer quoted in Brown, p. 208.\n\n34. \"own personal miseries\": Harsh, p. 219.\n\n35. \"loaded with bread\": Brown, quoted in Harsh, p. 220.\n\n36. \"to the wind and rain\": Buckham, p. 40.\n\n37. water pail and stew pot: Colwell, diary, February 5\u20136, 1945, with permission.\n\n38. \"powdered milk was gone\": Edy, p. 173.\n\n39. \"take first prize in density\": Buckham, p. 43.\n\n40. \"directly over the camp\": Edy, p. 177.\n\n41. \"where the armistice is\": Colwell, diary, March 21, 1945, with permission.\n\n42. \"R.A.F. and P.O.W.\": Ibid., April 9, 1945.\n\n43. \"sides of our boxcars\": Sweanor, p. 195.\n\n44. \"see our Red Crosses\": USAAF pilot, quoted in Sweanor, p. 196.\n\n45. \"Yanks for the ventilation\": Sweanor, p. 196.\n\n46. twenty-nine kriegies: Hugh Halliday, correspondence, November 2007, with permission.\n\n47. May 8, 1945: Sweanor, footnote p. 134.\n\n48. \"it was the end\": Colwell, diary, April 21, 1945, with permission.\n\n49. \"arrived at noon. FREE!\": Colwell, diary, May 2, 1945, with permission.\n\n50. \"Santa Claus parade\": Edy, p. 184.\n\n51. \"yelling and crying kriegies\": Frank Sorensen letter to family, May 15, 1945, courtesy of Vicki Sorensen.\n\n52. \"It continued for minutes\": Buckham, p. 92.\n\n53. generally weakened state: Floody, journal February 8, 1945, quoted in Hehner, p. 104.\n\n54. US Army had liberated: Harsh, p. 221.\n\n55. \"By God, we made it\": Ibid., p. 222.\n\n56. \"we were totally contaminated\": Pengelly, quoted in Dancocks, p. 207.\n\n57. \"most absurd scene\": Ibid., p. 209.\n\n58. best friend Pat Porter: Sweanor, p. 200.\n\n59. \"and rather childish\": Edy, p. 186.\n\n60. summer of 1943: Hawtin, unpublished diary prepared by Janet Hawtin, with permission.\n\n61. \"Sunday dinner until Thursday\": Peter Larlee, quoted in \"Son recalls his father's fight for freedom,\" Vancouver Sun, c. 1995.\n\n62. \"war had never happened\": Marchildon, p. 152.\n\n63. \"their duty twice over\": quoted in documentary The Great Escape: The Canadian Story, 2004, courtesy of producer Don Young.\n\nchapter eleven: \"a proud, spectacular distraction\"\n\n1. \"freedom in a hurry\": Philip Gray, Ghosts of Targets Past: The Lives and Losses of a Lancaster Crew in 1944\u201345 (MPG Books, England, 2005), p. 166.\n\n2. \"I was thunderstruck\": Ibid., p. 167.\n\n3. \"never be told in full\": Ibid., p. 168.\n\n4. \"a little blurred\": Brown, p. 262.\n\n5. RAF Sunderland flying boat: Vern White, \"The Sunderland Caper,\" RCAF 427 Squadron Association website, 2007.\n\n6. \"and this proves it\": Kingsley Brown, quoted in correspondence from Ethel (Brown) Alle, Barrie, Ontario, March 19, 2013.\n\n7. \"an encyclopedia\": Ibid.\n\n8. \"a birthday kiss\": John Weir, letter to Frances McCormack, January 21, 1945, with permission.\n\n9. \"two in the morning\": Frances Weir (n\u00e9e McCormack) interview, June 19, 2012, Toronto.\n\n10. \"love is blind\": Ibid.\n\n11. \"did you bounce\": McKim interview, 2011.\n\n12. \"air force like I was\": Ibid.\n\n13. \"in fours and sixes\": transcripts entitled \"Testimonies of Prisoners of War who escaped on March 25th from Stalag Luft III,\" courtesy of Don Young.\n\n14. \"exemplary justice\": Anthony Eden, House of Commons, Hansard, May 19 and June 23, 1944, quoted in Allen Andrews, Exemplary Justice (George H. Harrap, London, 1976), p. 211\u201313.\n\n15. \"courage and devotion to duty\": Member of the British Empire citation, October 1946.\n\n16. \"see their faces\": Wally Floody, quoted by Ron Lowman, \"Airmen still see the faces of comrades slain by Nazis,\" Toronto Star, March 25, 1984.\n\n17. \"terrible nightmares last night\": Hehner, p. 122.\n\n18. \"about the tunnels\": Wally Floody, quoted in Arthur Moses, \"Prisoners and guard from Stalag Luft III reminisce about war and the Great Escape,\" Globe and Mail, August 1970.\n\n19. \"degrading their manpower\": George Sweanor interview, July 5\u20137, 2011.\n\n20. \"my alma mater\": Ibid.\n\n21. \"in every culture\": George Sweanor, \"Bomber Command, 1939\u20131945,\" 971 Air Marshal Slemon Wing, RCAF Association Newsletter, November 2011.\n\n22. \"the war in jail\": Nick Dumonceaux, quoted in email correspondence with his mother, Anne Dumonceaux, Port McNeill, BC, September 12, 2012.\n\n23. \"as we all grew up\": Stephen Sorensen, interview, Howe Island, Ontario, August 13, 2011.\n\n24. \"why did you stay\": Vicki Sorensen, interview, Howe Island, Ontario, August 13, 2011.\n\n25. \"when I was a teen\": Glenn Sorensen, interview, Howe Island, Ontario, August 13, 2011.\n\n26. \"a visual perspective\": Barbara Edy, correspondence from Calgary, Alberta, April 6, 2013.\n\n27. \"Germany's war effort\": Ibid.\n\n28. \"organize to be successful\": Pengelly, \"X for Escape.\"\n\n29. \"bravery, spirit, and talent\": Op cit.\n\n30. \"these wonderful machines\": David Pengelly, interview, Dundas, Ontario, June 19, 2012.\n\n31. \"do you have a little more\": Ibid.\n\n32. \"watched it all the time\": Chris Pengelly, interview, Milford, Ontario, May 8, 2011.\n\n# Photograph Credits\n\n* * *\n\nfirst section\n\npage\n\n1. Whitley interior and barracks room, Tony Pengelly collection; four POWs, Imperial War Museums hu-1604; Roger Bushell, US Air Force Academy, McDermott Library, Stalag Luft III Collections.\n\n2. East Compound, Imperial War Museums hu-21013; Sentry tower interior, Barry Davidson collection; North Compound, US Air Force Academy, McDermott Library, Stalag Luft III Collections.\n\n3. Barry Davidson, Barry Davidson collection; Dick Bartlett, courtesy Anne Dumonceaux; Keith Ogilvie, courtesy Keith Ogilvie Jr. and Jean Ogilvie; Kingsley Brown, 1945 sketch by Lieut. John Lundquist, USAAF, courtesy Ethel Alle.\n\n4. Sketches of digger, workshop, and trap entrance to tunnel, sketches by Ley Kenyon, Royal Air Force Museum, Hendon, UK.\n\n5. German guard and prisoner assembly at Stalag Luft III, Imperial War Museums hu-21043; Biewer and Lindeiner, Imperial War Museums hu-21052; German \"ferret\" guards, Imperial War Museums hu-21190.\n\n6. John Colwell POW card, courtesy Harold Johnstone; Frank Sorensen, courtesy Vicki Sorensen; Canadians at North Compound, Tony Pengelly collection.\n\n7. Aerial intelligence photo of Stalag Luft III 1944, Don McKim collection.\n\n8. George Sweanor and Joan Saunders, courtesy Sweanor family; Don Edy in desert, courtesy Barb Edy and Jane Hughes; George Harsh, courtesy W. W. Norton & Co., New York.\n\nsecond section\n\npage\n\n1. Theatre sketch by Ley Kenyon, courtesy Barry Davidson Jr.; Tony Pengelly in female role and theatre interior seating area, Tony Pengelly collection; Arthur Crighton conductor from Wire Bound World by H. P. Clark.\n\n2. Baseball team \"Clare's Cards,\" courtesy Art Hawtin; hockey game, Barry Davidson collection; boxing match, Imperial War Museums hu-21164.\n\n3. Don McKim, courtesy Al McKim and Wendy Johnson; Albert Wallace, Albert Wallace collection; Red Cross parcel, Wire Bound World by  \nH. P. Clark; wireless radio, photographer Harold Kious, US Air Force Academy, McDermott Library, Stalag Luft III Collections.\n\n4. Sketches of February purge to Belaria and March escape from tunnel \"Harry,\" sketches by Ley Kenyon, Royal Air Force Museum, Hendon, UK.\n\n5. Guards show off trolley, Imperial War Museums hu-21219; guards and ventilating pump, Imperial War Museums hu-21215; guard with sand-dispersal sacks, Imperial War Museums.\n\n6. Massey and Pieber, Imperial War Museums hu-1603; Arthur Nebe, courtesy documentarian\/filmmaker Don Young.\n\n7. The Great Escape 1962 movie publicity stills, Catherine (Floody) Heron collection.\n\n8. POWs marching from Sagan, Poland, courtesy Marilyn Walton, US Air Force Academy, McDermott Library, Stalag Luft III Collections; British POWs sign, Wire Bound World by H. P. Clark; Ley Kenyon sketch, Frank Sorensen collection, courtesy Vicki Sorensen.\n\n# Copyright\n\nCopyright \u00a9 Ted Barris, 2013\n\nAll rights reserved. No part of this work may be reproduced or transmitted in any form or by any means\u2014graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems\u2014without the prior written permission of the publisher, or in the case of photocopying or other reprographic copying, a licence from the Canadian Copyright Licensing Agency.\n\nLibrary and Archives Canada Cataloguing in Publication\n\nBarris, Ted, author\n\nThe great escape : a Canadian story \/ Ted Barris.\n\nIncludes bibliographical references and index.\n\nIssued in print and electronic formats.\n\nisbn 978-1-77102-272-9 (bound).\u2014isbn 978-1-77102-474-7 (epub).\u2014\n\n1. Stalag Luft III. 2. World War, 1939-1945\u2014Prisoners and prisons, German.\n\n3. Prisoners of war\u2014Canada\u2014Biography. 4. Prisoners of war\u2014Germany\u2014\n\nBiography. 5. Prisoner-of-war escapes\u2014Poland\u2014Zagan. 6. World War,\n\n1939-1945\u2014Personal narratives, Canadian. I. Title.\n\nd805.5.s73b37 2013 940.54'72430922 c2013-902860-9\n\nc2013-902861-7\n\nJacket and text design: Gordon Robertson\n\nJacket image: Tony Pengelly Collection\n\nEndpaper illustrations: Lightfoot Art & Design Inc.\n\nPublished by Thomas Allen Publishers,  \na division of Thomas Allen & Son Limited,  \n390 Steelcase Road East,  \nMarkham, Ontario l3r 1g2 Canada\n\nwww.thomasallen.ca\n\nWe would like to acknowledge funding support from the Ontario Arts Council, an agency of the Government of Ontario.\n\nWe acknowledge the support of the Canada Council for the Arts, which last year invested $157 million to bring the arts to Canadians throughout the country.\n\nWe acknowledge the Government of Ontario through the Ontario Media Development Corporation's Ontario Book Initiative.\n\nWe acknowledge the financial support of the Government of Canada through the Canada Book Fund for our publishing activities.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n**Iwo Jima**\n\n**World War II Veterans  \nRemember the Greatest Battle  \nof the Pacific**\n\nLARRY SMITH\n\nSome images in this e-book are not displayed owing to permissions issues.\n**TO**\n\n**_Barney Barnum, John Ripley,  \nIron Mike Mervosh, and the veterans of Iwo Jima  \nand to all marines,  \neverywhere_**\n\nCONTENTS\n\nMaps\n\nIntroduction\n\nPART ONE: THE INVASION\n\n1. Corporal Richard Nummer\n\n2. Seaman James Bush\n\n3. Corporal James \"Salty\" Hathaway\n\n4. Captain Fred Haynes\n\n5. Corporal Hershel Woodrow \"Woody\" Williams\n\nPART TWO: IRON MIKE'S POSSE\n\n6. Corporal Glenn Buzzard\n\n7. Private First Class Pete Santoro\n\n8. Private First Class Domenick Tutalo\n\nPART THREE: RUSSELL, TSO, ABBATIELLO, AND WATERHOUSE\n\n9. Captain Gerald Russell\n\n10. Private First Class Samuel Tso\n\n11. Corporal Al Abbatiello\n\n12. Private First Class Charles Waterhouse\n\nPART FOUR: THE FLAG RAISINGS\n\n13. Warrant Officer Norman Hatch\n\n14. Corporal Charles \"Chuck\" Lindberg\n\nPART FIVE: THE LST, WORKHORSE OF THE INVASION\n\n15. Machinist's Mate Third Class Perle \"Dusty\" Ward\n\n16. Pharmacist's Mate Third Class George Wahlen\n\n17. Sergeant Cyril O'Brien\n\n18. Sergeant Thomas Haywood McPhatter\n\nPART SIX: THE PLANES THAT CAME\n\n19. Lieutenant Robert Merklein\n\n20. Staff Sergeant Valentine Chepeleff\n\n21. Lieutenant Phil True\n\nPART SEVEN: AFTERMATH\n\n22. The Japanese Commanders\u2014Lieutenant General Tadamichi Kuribayashi and Captain Tsunezo Wachi\u2014and the Years Afterward\n\n23. Patrick Mooney and the Graves\n\n24. Colonel John W. Ripley, USMC (Ret.)\n\nAppendix: Military Terminology\n\nAcknowledgments\nOLD MEN FORGET, yet all shall be forgot but he'll remember, with advantages, what feats he did that day.\n\n\u2014WILLIAM SHAKESPEARE, _Henry V_\n\nCARLISLE: My Lord, wise men ne'er sit and wail their woes, but presently prevent the ways to wail. To fear the foe, since fear oppresseth strength, gives in your weakness strength unto your foe, and so your follies fight against yourself. Fear, and be slain. No worse can come to fight. And fight and die is death destroying death, where fearing dying pays death servile breath.\n\n\u2014WILLIAM SHAKESPEARE, _Richard II_\n\nINTRODUCTION\n\nThey are all in their eighties now. They have had occupations, sweethearts, wives, children, even great-grandchildren. But once they stood trembling, hearts pounding, laden with weapons, ammunition, and assault packs as the amtracs and Higgins boats careened over sickening waves toward the black sands of Iwo Jima. They are survivors of one of the great battles of history, the campaign to conquer a small island in the far Pacific. Most were eighteen or nineteen years old when the boats dropped them off at the surf's edge. They were teenagers charging up those horrible ashy, slippery beaches into a torrent of artillery, mortar, and machine-gun fire, with death and destruction all around them. Hundreds surrendered their lives on the spot while those who were to survive struggled frantically to get off the beach. They were kids. They are old now. They are dying by the hundreds, age taking them instead of bullets or shrapnel. Yet thousands hold Purple Hearts, the decoration of the wounded. Twenty-two of them tell their stories in this book.\n\nFebruary 19, 1945, dawned bleak but manageable. That morning nearly eight hundred vessels, ranging from battleships, cruisers, and destroyers to transports and LSTs (landing ship, tanks), lay offshore. Aboard the transports were seventy thousand marines from three divisions charged with conquering eight square miles defended by twenty-two thousand Japanese soldiers fighting out of caves, bunkers, and tunnels. The Japanese knew the Americans were coming and the emperor personally had sent the brilliant general Tadamichi Kuribayashi to defend Iwo Jima. Implicit in the order was a fight to the death. The general arrived in June 1944, eight months prior to the invasion. _Iwo Jima_ means \"Sulfur Island,\" and the Japanese had carved an extensive network of mining tunnels that ran in all directions. Kuribayashi had his own soldiers work alongside Korean laborers to widen, deepen, and extend the sandstone tunnels so that there were more than sixteen miles of passageways. Some of the tunnels ran three stories underground. The island is volcanic, and the heat must have been dreadful.\n\nEarly that morning the amphibious craft, amtracs and Higgins boats, embarked from a fleet of forty-three transport ships. The fleet was a floating city, serving fifty thousand meals of steak and eggs to marines about to go ashore. The Fifth Marine Division had trained more than a year for this battle alone. Fleet Admiral Chester Nimitz was in overall charge in the Pacific. Admiral Ray Spruance was in command of the Fifth Fleet. Under him was Admiral Kelly Turner, the commander of the Fifth Amphibious Corps. General Harry Schmidt was in command of the landing force. Superfluous in the command structure but superior to Schmidt was General Holland \"Howling Mad\" Smith. Planning for the battle had been under way for more than a year. The Marines were _on_ the ground; the Japanese were _in_ the ground, and they were ready for the siege. Each man was told to fight to the death, but not before taking at least ten marines with him. They survived on half a cup of water daily and a handful of rice, yet they held out for thirty-six days. The last five days they had neither food nor water.\n\nOn the first day alone the Marines suffered 2,420 casualties, including more than 500 killed. Before the campaign was over, 13 of 24 battalion commanders fell, while 15 doctors were killed, along with 195 Navy corpsmen, who were medics on the battlefields. In those thirty-six days, 28,000 marines and soldiers\u2014American and Japanese\u2014were killed, and 16,000 were wounded. In the first ten days the Fourth Division lost 4,000 marines getting from the landing beaches to an assembly point you could walk to in twenty minutes\u2014if you were not under fire. There have been few more disastrous military encounters in American history. Iwo Jima remains the Marine Corps's deadliest campaign. Some have compared Iwo Jima with the battle of Gettysburg, where 40,000 died, and with Belleau Wood in World War I, where the casualties, killed and wounded, reached 9,770. At the end of the fighting, the American dead totaled 6,821. It was the only campaign the Marines ever fought in which they took more casualties than the enemy. Every foot of the island was contested, and the fighting was so intense that several battle sites became legendary: Nishi Ridge; the Quarry; Cushman's Pocket; Hills 362 Able, Baker, and Charlie; the Meatgrinder, consisting of the Amphitheater, Hill 382, and Turkey Knob. Then there were Bloody Gorge and Kitano Point.\n\nMost famous of all but no more difficult than these places was Mount Suribachi, where on the fourth day a team of six men went up in the morning and raised the American flag. Then, a few hours later, a second team of six raised a second flag because the first was not large enough. Joe Rosenthal, an Associated Press photographer, captured an indelible image of this moment. Indeed, Rosenthal's shot became one of the most famous images of all of World War II, if not the entire twentieth century. The photo shows six men straining to raise a heavy pole on a bleak hilltop against a gray sky. None of their faces can be seen, but you can see the flag flapping over them.\n\nThe photo also generated considerable controversy. The story has been told more than once, notably by James Bradley in _Flags of Our Fathers_ and by Hal Buell in _Uncommon Valor, Common Virtue: Iwo Jima and the Photograph that Captured America._ More recently of course, _Flags of Our Fathers_ has been filmed by Clint Eastwood. While the old-timers were glad to see a movie about Iwo finally come to the screen more than sixty years after, they were dismayed by its lack of focus on the battle itself and in its portrayal of A. A. Vandegrift, the commandant, who in obscene terms sends the alcoholic Ira Hayes, the Pima Indian who was one of the flag raisers, back to his unit. They were also disappointed by other liberties taken with the facts. Chuck Lindberg, the last living flag raiser, tells the story from his perspective in Chapter 14. Just before, Norman Hatch, a combat cameraman, explains in Chapter 13 how he clarified for the commandant of the Marine Corps how the flag raisings took place.\n\nBetter received was a companion film, _Letters from Iwo Jima,_ based on letters written by Kuribayashi and others, depicting the battle from the Japanese point of view. It was shot almost exclusively inside caves and tunnels like those occupied by the Japanese soldiery. Another film foray that received mixed reviews from veterans was _Windtalkers,_ about the Navajo Indians who served the Marines in radio communication, using their indecipherable tongue in a code the Japanese could not break. One code talker, Samuel Tso, recounts his story in this book. Rather than tell the dramatic story of this elite group, the movie focused on a character who was instructed to shoot the code talker in the event of capture.\n\nSuch unwelcome publicity, or perhaps the unwelcome memories, inspired by the film may have encouraged the Japanese in June 2007 to restore to the island its old name of Iwo To, which has the same written characters and meaning. The Associated Press reported that it was known as Iwo To to the thousand or more civilians who lived there prior to the outbreak of the war and that it mistakenly came to be called Iwo Jima by Japanese Navy officers who moved in to fortify it after the civilians were evacuated in 1944.\n\nThere were three thousand yards of beach, designated green, red, yellow, and blue, and essentially three axes of attack, right, center, and left, each with its own division, numbering twenty thousand men. Elements of the Fifth Division were to cross the skinny south end of the island, just eight hundred yards. Their mission was to turn left, isolate, capture, and cut off Mount Suribachi, then do an about-face and move up the left, or west, side. The Third Division was assigned to the center of the island, and the Fourth was to seize Motoyama No. 1, the airfield, then pivot and go all the way to the north. One regiment of the Third Division, roughly thirty-three hundred men, was held in reserve and never released for the battle. That remains a sore point to this day because their presence and firepower were so badly needed. Units in combat could not be given any rest, and so many were being killed and wounded that raw replacements were funneled directly into the battle. These green soldiers were slaughtered because there was no time to assimilate them into experienced units.\n\nColonel John Ripley, USMC (Ret.) (Chapter 24), a former director of the History and Museums Division of the Marine Corps and a leading authority on the campaign to conquer Iwo Jima, is careful to avoid saying that one division or another bore the brunt of the action \"because it's simply not true.\" He adds, \"All three divisions shared equally in the struggle, along each axis of attack.\" The Fifth Division captured Suribachi and raised the flags, he notes, but \"nobody talks about the hell it went through after Suribachi. It lost six flag raisers, including Mike Strank, Harold Hansen, and Harlon Block (three from each event), plus the battalion commander, Colonel Chandler Johnson, the commander of 2-28, in the seizure of Hill 362 Able, one of the toughest nuts of all, right off Nishi Ridge.\"\n\nRipley adds, \"From this point on these guys are nose down, fighting like hell to get up here to Bloody Gorge at the northern tip of the island and the last pocket of resistance. Then it takes nine full days for an entire division, the Fifth Marine Division, to seize a piece of ground so narrow you could throw a baseball across it. I have been down in there. There are reinforced bunkers opposing each other; the ground is just unbelievable. So the Fifth Division was bled white up here. They never get any credit for that.\n\n\"We used damn near four tons of TNT and also plastic and ten thousand gallons of thickened fuel for flamethrowing, predominantly Zippo tanks, every day for nine straight days. Day after bloody day, a whole division. The Fifth Division had four thousand seventy-two casualties trying to take this position. And it's never talked about. It's one of the toughest battles ever, for any service, anywhere: the seizure of Bloody Gorge.\n\n\"Seldom in Marine Corps history has there been a comparable battle for such a tiny piece of ground, requiring so much effort, so much in the way of casualties, and yet the history books don't identify it as such. It's just part of the battle of Iwo Jima, and perhaps that's as it should be.\n\n\"Bloody Gorge alone would convince you of Kuribayashi's genius. By this time no Japanese had eaten or had any water for five straight days. They never had any sort of sustenance compared to what our marines had, but at the same time they fought and fought and fought, and what a hell of a job they did.\"\n\nOn March 14, after more than three weeks of horrific fighting, the island was declared secure. The invasion of Okinawa had been planned for April 1, and it was important to get the amphibious shipping back to prepare. So Iwo was declared secure on Day twenty-four. \"That's when they had the ceremonial flag raising down there at Fifth Division cemetery,\" Ripley says, \"yet we would fight like banshees for another two weeks, including Bloody Gorge.\"\n\nFinally, on D plus thirty-four, or thirty-five days after the invasion began on February 19, the island was declared secure for a second time, and various units were directed to turn in their weapons and ordnance, link and ball ammunition, ranging from belted machine-gun rounds to eight-shot rifle clips, all live rounds that could not be taken aboard ship, plus exploding ordnance such as grenades, mortars, rockets, TNT, and plastic explosives. Just when everyone thought the battle was finally over, a force of three hundred enemy made its way down near Motoyama or Airfield No. 1 and slaughtered forty-four pilots sleeping in tents before they themselves were annihilated. Lieutenant Robert Merklein, a P-51 pilot, narrowly escaped death that night. In Chapter 19, he describes what happened.\n\nAfter the Marines departed, according to Patrick Mooney, another authority on the campaign, Army garrison forces of the 147th Infantry took over. They continued the work of hunting stragglers, recovering weapons and equipment, and disposing of Japanese bodies. More than 1,600 additional Japanese surrendered while at least 800 more were killed. The Marines captured only 216 of the enemy during the entire battle. Perhaps not widely known is the fact that, after March 26, 1945, Army occupation troops captured 867 and killed 1,602 over the next few years. The Japanese would come out at night to steal food, clothing, and water. Many of the later casualties were Korean laborers who had been impressed by the Japanese to carve out the tunnels. On January 8, 1949, the last known enemy walked out and surrendered to a Navy lieutenant. This Japanese soldier had apparently found a copy of the military newspaper _Stars and Stripes_ that showed on its front page Douglas MacArthur and Emperor Hirohito standing together in Tokyo. He figured that if Hirohito was standing, and MacArthur wasn't bowing, the Japanese had lost the war.\n\nI visited Iwo Jima with Colonel Warren Wiedhahn USMC (Ret.) and his Military Historical Tours on March 8, 2006. We met in Los Angeles before flying on to Hawaii and Guam as a unit. \"You will never be the same once you set foot on the black sands of Iwo Jima,\" Colonel Wiedhahn said. Many Iwo veterans were on the trip, and he had them stand and tell a little about themselves. After several had spoken, he noted how among the veterans there almost invariably recurred the phrase \"And then I was wounded.\" As I grew more familiar with what had taken place, my old newspaperman's curiosity got the best of me. A number of questions began to emerge, and I have sought to answer them through the stories of the men in this book. When talking to each of them, I sought to focus on personal experience as well as what I saw as the four major elements to the story: the landing; the raising of the flags on Suribachi; the at times seemingly hopeless attempt to conquer the rest of the island; and the arrival of the B-29s, which had been one of two major objectives of the invasion.\n\nFirst, how did the invading American force keep from being annihilated and driven off the beaches? General Kuribayashi placed his artillery, machine guns, and riflemen from Mount Suribachi on the south end all the way to the Quarry on the north, so that the entire four thousand yards of shoreline were utterly bracketed, enfiladed by guns and artillery from every direction except the east, where the American ships lay offshore. Even after the assault was under way, he waited an hour or more, until several waves of marines had landed, before unleashing his artillery. The carnage was appalling. Joe Rosenthal said later the bullets and artillery were so thick it was like trying to run through rain without getting wet.\n\nSecond, what really happened with the flag raisings, and how did it get sorted out? There's the lingering false assertion that the Rosenthal photograph, the model for the Marine Corps Memorial in Washington, D.C., was faked. And who actually raised the flags? At last count, the number of men claiming to have helped raise them numbered 1,632. The last living flag raiser, Chuck Lindberg (no relation to the pilot), died in Edina, Minnesota, on June 24, 2007, only a few weeks after recounting his story for this book (Chapter 14).\n\nThird, what happened to the twenty-one thousand defenders? And what became of Lieutenant General Tadamichi Kuribayashi, the Japanese commander? He had accepted the assignment of defending Iwo Jima, which was considered part of the Japanese homeland, to the death. He knew he could not win; he could only delay the American advance. During his nine months there he inspired his troops, devised a brilliant strategy of defense, and found time to write forty-one selfless and tender letters to his wife and children in Tokyo. No one has solved the mystery of his death. Recent accounts that have him commanding that last attack on Motoyama No. 1 conflict with prevailing knowledge over the previous fifty years. You'll read about the discovery of Kuribayashi's chief of staff's remains in Chapter 22; the general's remains have never been found.\n\nFourth, was it worth it? At the time the Allied commanders offered two reasons for the invasion. The Army Air Corps was sending B-29 Superfortresses by the hundreds to bomb Tokyo and other targets in Japan. These planes flew out of Saipan and Tinian, more than a thousand miles one way to Japan, and then back. Iwo, a territory under Japanese mandate, was at the halfway point, 750 miles from Tokyo. Forces there could alert the mainland that the B-29s were coming. That had to be stopped. The Air Corps strategists also knew that Iwo had two fully equipped airfields. These essential strips would provide a place where B-29s damaged or with engine trouble could stop on the way back.\n\nFinally, what has happened to the place since? More than twenty years after the end of hostilities the Johnson administration returned ownership of Iwo Jima to Japan. The American flag was lowered there at 12:15 p.m. on June 26, 1968, and was replaced minutes later by the Rising Sun. Thousands of old marines, including many of those in this book, are still angry about it. Though the Japanese were reluctant to allow it, the terms of what was called the Status of Forces Agreement stipulated that the Marine Corps would have access to the island in perpetuity. I wanted to see how many men had gone back and how they felt about that tiny island they had nearly given their lives for.\n\nThe poet William Butler Yeats wrote, \"An aged man is but a paltry thing, a tattered coat upon a stick,\" and so it is with many of these old vets, some widowed, some getting about with walkers, toting hearing aids, memories uncertain, while many others remain vigorous and feisty, full of energy, their recollections sharp. Many, like Domenick Tutalo and Norman Hatch and Cyril O'Brien, are still working. Mike Mervosh, eighty-two, can still do nine one-handed pull-ups. More than sixty years have passed since they labored onto the beaches of Iwo Jima. It is poignant to see pictures of them as dashing, uniformed, handsome young men and to encounter them now. The war that shaped their lives is far in the past, yet still vivid in their memories. Where relevant, I have supplied specific dates, times, and facts without intruding otherwise. This book does not presume to offer a definitive account of what has been called the thirty-six days in hell of Iwo Jima, although it does strive for accuracy. Rather, it is a series of snapshots offering a glimpse into the lives of twenty-two men who took part in various aspects of the conflict and how they have fared since. Their stories speak for themselves.\n\nIwo Jima\nPART ONE\n\nThe Invasion\n\n**There were four** assault regiments, representing the Fourth and Fifth Divisions. The run from ship to shore took about half an hour. The first landing came at 9:02 a.m., and in a short while there were six thousand marines on the beach. The first waves came in on LVTs or tracked amphibious vehicles called amtracs. Subsequent units came in on Higgins boats, known as LCVPs (landing craft, vehicle, personnel). The beach segments were designated green, red, yellow, and blue.\n\nThe Japanese commander Kuribayashi had spurned conventional wisdom of protecting the shoreline from invaders and instead held his fire till the three thousand yards of beach were crammed with marines, and then shortly after 10:00 a.m., he opened up from caves, tunnel mouths, and fifteen hundred pillboxes with heavy artillery, giant mortars, rockets, and antitank weapons. All the weapons had been preregistered for accuracy.\n\nAmerican intelligence had reported the sand would cause no problems, but this was dead wrong. The steeply terraced soft black volcanic sand immobilized every type of vehicle, and soon there was a terrible jam-up along the beach and the marines themselves had trouble moving in it. The effect of the barrage was devastating, but still the Marines managed to land 30,000 men the first day. Casualties on the first day totaled 2,420, including 501 killed.\n\nThe weather turned bad, the beaches were closed, and there was chaos everywhere, but the marines kept coming. Colonel Joseph Alexander, in his definitive account of the campaign, quotes one observer: \"The wreckage was indescribable. For two miles the debris was so thick that there were only a few places where landing craft could still get in. The wrecked hulls of scores of landing boats testified to one price we had to pay to put our troops ashore. Tanks and half-tracks lay crippled where they had bogged down in the coarse sand. Amphibian tractors, victims of mines and well-aimed shells, lay flopped on their backs. Cranes, brought ashore to unload cargo, tilted at insane angles, and bulldozers were smashed in their own roadways.\"\n\nOn the third night, fifty kamikaze planes from Katari Air Base near Yokosuka attacked offshore, crashing into and sinking the escort carrier USS _Bismarck Sea,_ damaging several others, and putting the USS _Saratoga_ out of the war. All the kamikazes were lost. The number of casualties from these attacks and shore fire during the battle totaled 2,798, including 1,917 wounded.\n\nIn the first five chapters, an infantryman, a sailor, an amtrac driver, an operations officer, and a flamethrower talk about the opening days of the fight.\n\nCORPORAL RICHARD NUMMER\n\n**Weapons Company, Twenty-eighth Marines, Fifth Marine Division**\n\n_For forty years I never even talked about it or nothing, but soon as I seen that flag I knew I was the one put the hole in it. Not too many people know there's a hole in the flag._\n\nStanding in the backyard of his home north of Detroit in the summer of 2006, eighty-year-old Richard Nummer holds the bayonet he took from the body of Siguo Kubo, a Japanese soldier he shot dead during the campaign for Iwo Jima in February\u2013March 1945.\n\nCorporal Richard Nummer, right, and his foxhole buddy Al Esposito, from New York, have their picture taken atop Mount Suribachi on February 24,1945, with a camera recovered from the body of a slain marine.\n\n_I met with Richard Nummer at his modest home in East Detroit in the summer of 2006 before the Marine Corps opened its new museum in Quantico, Virginia. He was eighty years old, lively and talkative. We hit it off immediately. It turned out he had quit school to join the Corps in 1942. He added that sixty-three years later, in 2005, he received an honorary diploma from his alma mater, East Detroit High School. \"They finally got the government to give you a thing saying they appreciate what you did. So I got my honorary diploma. All those years later I finally became a high school graduate.\" Richard laughed when he said this. \"I still got the cap and gown they gave me.\"_\n\n\"I went in with the fifth wave. When we got off the ship, we went around and around in circles, rendezvoused, so they could start letting different waves go in, you know, one right after another. So when we went in with the other waves, it seemed like there was no action at all. But just about the time I got there, around ten o'clock or ten-thirty, we start running in, that's when they really opened up.\n\n\"I was in a Higgins boat. The ramp went down in front. Our gun, a thirty-seven millimeter on wheels you pulled with a truck, was on the other boat, an LST [landing ship, tank], I think. At that time I was with Headquarters Company of the Twenty-eighth Marines.\n\n\"When I hit the shore, like I say, we ran up. There were not too many bodies when we first landed. I saw about ten. The first wounded guy I seen his jaw was gone. He was running back, and a couple corpsmen were trying to tackle him so they could get morphine into him. I don't know if he made it or not.\n\n\"But as you ran up, you'd tap a guy on the shoulder, and he'd run up a little farther and you'd get in his hole. He was supposed to get up and go to next hole when you tapped him, like leapfrogging. So I get in this hole and tap the guy on the shoulder and, nothing. He was gone. There was shell holes all over from the aircraft. I jumped in this one hole. It was pretty good, but right above me I could see all this sand getting hit with bullets from up above, coming right down. Anyhow, I laid there. You couldn't move. Every time you'd just move an arm or something you'd draw fire.\n\n\"I was pinned down there for maybe five minutes. I had a New Testament with some of the Psalms included, here in my pocket, so I opened that up, and I read the Twenty-third Psalm. And I just finished that, put it back in my pocket, when a shell hit, and I don't know nothing till the next day. [Richard had been knocked out cold.] I guess it was the concussion from that shell hitting close by. There were bodies all around there. I lay there twenty-four hours. Anyway, next morning I woke up, and when I got up to where my group was, the rest of the company, the sergeant just shook his head. He said, 'We had you down KIA.' Killed in action. 'Cause I was laying there with the dead.\n\n\"Our gun maybe didn't get hit, but it could have got stuck in sand. You couldn't use them until they got some bigger vehicles to pull them out of the sand and up onto the beach. Each ground section of artillery was supposed to be with one of these companies.\n\n\"Anyways, Combat Team Twenty-eight crossed the neck at the base of the island and turned toward Suribachi. They got the Thirty-sevens up, and they were firing at anything they could see on Suribachi. They kept putting the shells in so fast and shooting them that the lines and grooves were just round after a while, and the barrels were no good at all [heat melted the rifled grooves inside the barrels, so the shells could not rotate. The grooves give them accuracy and velocity]. They were almost disabled. So we got stuck with infantry in the attack on Suribachi, and then, four days later, on February 23, the first flag went up. Louis Lowery got that first photograph. I knew him real well. He was on the ship with me. He slept almost right next to me. So, oh boy, that didn't mean the war was over for us, but at least we got something going. The whole thing was supposed to take three days. I was there thirty-seven days, the whole time, and one day more.\n\n\"I didn't go up with Lowery on the first flag raising. Some of us decided we was going to climb up Suribachi for a closer look, so we started going to the top. It probably took us forty-five minutes. Suribachi was pretty much taken out when we went up. We got there about the same time as this group arrived for the second flag raising. And they was just putting the flag up. We saw it go up, and we were just walking around up on top, looking down. A lot of pictures were being taken, but nobody knew this was going to be anything famous. We went up there because that's where our group was. We were up there maybe an hour or so, and they decided our outfit, the Twenty-eighth, had got hit so bad we had to wait for replacements. And they said, 'You're going to stay right here.'\n\n\"So we were the first marines to sleep on top of Suribachi, six feet from the flag. I have a picture of my buddy and me sitting there. My buddy Al Esposito from New Jersey found a camera in the pack of a dead marine, and we had our picture taken with that. I was six feet tall and weighed a hundred sixty. We were taking turns on guard duty. Every hour we'd wake the other one up, kept changing like that. We was the closest foxhole to the flag, which was like right here, off my right shoulder. So he just got to sleep, and I kinda dozed off. At that time this was probably the safest place on the island because the rest of it was all downhill. There were about forty of us up there. Anyway, I heard this noise and I turned around and I thought it was a Jap, so I shot. But it wasn't a Jap; it was the flag, snapping in the wind. I told Esposito next morning, I said, 'Jeez, look what I did.' He said, 'What the heck did you do that for?' I says, 'Well, I thought it was a Jap.'*\n\n\"If you ever go to Washington, D.C., that flag is in the Navy Yard. Both flags are there [they have since been moved to the new Marine Corps Museum in Quantico, Virginia]. The first time I went for an Iwo reunion was twenty years ago. Reagan was president then, and we went to White House, met him and his wife and the first George Bush, and then we went over to Arlington Cemetery, where they got the flag raising statue, and soon as I went in that museum and seen the flag, I knew.\n\n\"For forty years I never even talked about it or nothing, but soon as I seen that flag I knew I was the one put the hole in it. Not too many people know there's a hole in the flag. It's right there in the second stripe. Here it's probably the most famous picture ever taken, and I stuck my little hole in it. I should have been court-martialed. At that time, of course, nobody knew it was going to be famous, not until we got back to Hawaii a month later and seen the pictures and all.\n\n\"I knew right away, but all those years I never told anybody. In fact, the Marine Corps don't even know about it. This guy was here last December [2005] it was the first time I ever told anybody about it. He come here and like interviewed me. They're coming out with a picture, and this is the guy that does Clint Eastwood's sound and all that. He was here, and he came here to try get this worked into the picture. But after he left, I got a phone call from him saying the picture was too far in the making, so it probably wouldn't be mentioned.\n\n\"We were up on Suribachi five days while we waited for reinforcements. We lost five hundred ten men in four days of fighting. The reinforcements came on February 27, which was my birthday. I turned nineteen on Iwo Jima. We'd go down every day to work on the graves detail. They'd give you big black gloves that reached up to your elbow, for picking up the dead. That was our job. We'd pick up these bodies and put them on the truck, and off they'd go. They'd bury them in trenches. They were all brought back to the States later on.\n\n\"You saw some real bad sights there. No masks, and the smell and the flies were terrible. This one guy we picked up, somebody said, 'There's John Basilone' [Basilone was a notable marine who'd received the Medal of Honor for action on Guadalcanal]. He wasn't in the Twenty-eighth. I think he was in the Twenty-sixth or Twenty-seventh. He got it the first day I guess, but he laid there for all that time. A lot of the bodies were there for five days already. We didn't fool with dog tags at all. You got your name stenciled across your back. For every dead Jap, I seen twenty-five dead marines easy. The Japanese would pick up their dead at night and drag them inside the caves, and we never knew exactly how many we got.\n\n\"A lot of times we'd lose two or three marines and cover them up with a blanket or a poncho. During the night the Japs would come and crawl in there, and some would take their uniforms off so you didn't always know if it was a marine or not. Then the burial detail would come around. When we was on burial detail, we always had to watch out there wasn't a Jap among 'em waiting for us, with a rifle or a grenade, whatever they could find. They'd try to get as many of us as they could.\n\n\"We just piled the bodies on the truck, and then they took them to where the cemetery was going to be. We did that for five days. We were walking around almost like zombies, you know? We had all this death on our mind, and we knew it was going to go on farther, and is this the end of us or what? We did this, and went back and forth to the beach to get supplies and stuff.\n\n\"We finally got into combat when we got down to the other end. We were on the front lines there. I had an M1 all the time. At night we'd put our thirty-seven millimeters in a row and put canister in, so if the enemy came, you could wipe out a whole bunch of them. But they never made a banzai attack except on that last day. Some days you stayed right where you was. All day long you didn't move.\n\n\"We were down to the other end about a week, and one night off in the distance I could see somebody running around. We had different password codes. You'd yell 'tree,' and the person was supposed to answer with the name of a tree. And then you'd yell, 'president,' and the answer would come back: 'Roosevelt!' So quite a ways off here this guy come running. It was just getting dark. Two buddies in the hole with me, and I had the first watch that night. It was going to be an hour off, an hour on.\n\n\"This figure kept coming closer and it was getting darker, and I said to the guys, 'There's somebody out there, but I think it's a marine.' And they said you don't want to shoot another marine, but he shouldn't be out there. So he come closer. A hundred yards away was a lot of rocks and stuff, and he kept coming, coming; he had a rifle, got closer and closer, so pretty soon he got close enough to where he could hear me, and I yelled out, 'Tree!' And he's supposed to yell back, 'Oak,' or whatever. 'President!' Nothing. 'Car!' He's supposed to yell back, 'Ford,' or whatever. So then I started over. About twice I went through the thing. Nothing. So then I said, 'Bukyosterol.' That means 'Drop your weapon.' But he kept coming, and he's almost right here. And the guys got up and said, 'Shoot!' My finger just froze on the trigger, and down he went. So that was the first one I got.\n\n\"Now he laid in the rocks, and I must have got him in the belly somewhere. I thought I killed him, and now I'm really shaken up because I thought it was a marine. So he started talking in Japanese. We didn't know what he was saying, but farther on you could hear the other ones answering him. So he was trying to give away our position. We started throwing flares out there to see where he was. Meantime the whole group was all lined up with our guys. We knew they were coming. One of them hit the trip flare, and that lit up the field right in front, so quite a few Japs got killed. Next morning my lieutenant came up, his name was Manning, Robert Manning. He come up, and he said, 'Who had the machine gun last night that started all this?'\n\n\"They said, 'That was no machine gun. That was Nummer over there, with his M1.' So he come over to me and shook my hand and said, 'Good going, kid.' He was from Bougainville and different campaigns before that. He says, 'Go on out there now, and any souvenirs on him, they're yours.' So I went out there. 'Wait a minute!' he says. One of the guys brought a rope, put the rope around his leg and pulled it, and sure enough, he had two or three grenades underneath. If he hadn't told me that, I would have gotten blowed up. That was one of the tricks they had. He probably shoved them under himself as he was dying. Those grenades were little black things as big as your fist that just had a piece of rope, and you pulled that out and put it underneath you and it wouldn't blow. Sometimes they'd hit them on their helmet to get them started.\n\n\"The grenades blew him up. I got his wallet and his bayonet. I still got the bayonet in the other room there. Then the lieutenant says, 'Well, you probably saved some of these guys' lives. I'll talk to you about it later on.' I didn't know if I was going to get some kind of medal or what.\n\n\"He left, and this was a terrible thing: He left, and he wasn't gone long. He said, 'You try to get some sleep now.' So I got in the hole and I just about went to sleep and they woke me up. They said the lieutenant just got killed. And he wasn't that far away from us. The reason he got killed he was crawling out to another guy laying there wounded and he was pulling him back. A sniper got him. It took us another two, maybe three days just to get to his body.\n\n\"One of our sergeants was named Duffy, from New York. There was this other kid named Hood. He drove the lieutenant's jeep. Here the war's going on, and the two of them get into a fistfight. They couldn't get the lieutenant's ring off because his hands were all swelled up, so the sergeant said, 'We're going to cut his finger off to send his wife back his ring.' And the other guy, Hood, said, 'No, just cut the ring off.' So they got in a big fistfight over this. We were aboard ship when we heard his wife just had a baby. He never seen the baby.\n\n\"Before we landed on Iwo, I had boils all over, so they put me in sick bay aboard ship, and the doctor come in and said, 'We got a new drug we're going to try on you; it's called penicillin.' So I was one of the first to get penicillin. They shot you in the rear end; I think I got three of them, and all the infection concentrated in one big boil on my knee. Even when I was on Iwo, I had boils pretty bad. A corpsman said, 'You're in more pain than some of these guys that got hit,' and he told me, 'We can get you off the island.' I said, 'No, you're not getting me off the island, not for boils.' So I stayed right there. Why leave if I was still in good enough shape to fight? Just because you had boils? Did I think I was going to make it or not? Every night you'd say, 'Well, I wonder if this is going to be it.' You never knew.\n\n\"I got a little story about Ira Hayes [Hayes was the Pima Indian who helped raise the second flag and whose life was romanticized after he died drunk in an irrigation ditch on the reservation in New Mexico in the 1950s. A movie about his life was made starring Tony Curtis, and Johnny Cash had a hit song called the \"Ballad of Ira Hayes\"]. There were six guys in that flag-raising picture, and three of them got killed before we left the island. Bradley was left, Ira Hayes and Rene Gagnon. Bradley, his son came to one of the reunions. He's the one that wrote the story about _Flags of Our Fathers._ But Ira Hayes was the only one I knew real well. After we got back to Hawaii, they went on the seventh war loan drive in the States, and he got drinking quite a bit, had his problems with drinking. They were making a big hero out of these three guys, and Ira Hayes just couldn't see it. He said, 'Anybody could do the same thing we did. It's the dead guys that should be . . . ,' so anyway, they sent him back to us.\n\n\"He was probably in the States three, four months, and he come back to our outfit. In our tent we had one of the code talkers, his name was Thompson. And Ira come into the tent, and everybody shook his hand, one thing and another. He started drinking right away, beer, you know, we had it in the tent. And he wouldn't leave until all the beer was gone. If there was any left, he'd shove it in all around his belt. He'd be gone for days, nobody knew where he was at. It's a shame.\"\n\n[Richard shows me a Japanese flag inscribed with various cities and explains.]\n\n\"After the war ended, I had occupation duty in Japan, nine months in Sasebo. Iwo Jima was the only battle I was in, but right after that we was getting ready to hit Japan and the bomb went off, which saved a lot of our lives. From Sasebo, not long after the war was over, we went to Nagasaki, where the second atom bomb was dropped. We were drinking their water and everything else. I was home about five years when I woke up with a gray streak. What the heck? The doctor said, 'I think you had a slight stroke.' My one eyebrow on the side was gray. They didn't know about this radiation stuff at that time. I got out a corporal in July of '46. When I got back, I couldn't even get a beer: I wasn't old enough.\"\n\n[When the book _The Spearhead,_ relating the history of the Fifth Marine Division, came out, Richard's name was not in it. He wrote to Washington and received a letter from Lieutenant Colonel H. W. Edwards at the Historical Branch, written on March 15, 1954. The key paragraph reads: \"As a member of Weapons Company, 28th Marines, Fifth Marine Division, you sailed on board the USS _Talledega_ for Iwo Jima, at sea 1\u20134 February 1945, at Enwietok _(sic)_ in the Marshall Islands 5\u20136 February, at sea 7\u201310 February, at Saipan 11\u201316 February, at sea 17\u201318, arrived and disembarked for action against the enemy at Iwo Jima 18 February. You were engaged in action on Iwo Jima from 18 February _(sic)_ until 26 March on which date you embarked on the USS _Winged Arrow_ to return to base. You sailed from Iwo Jima on 27 March.\"]\n\n[Richard has one last story.]\n\n\"I had this wallet from this guy that I shot that night when the lieutenant was also killed, different things, so forty years later I saw all this stuff that I had. It was no good to me, so I thought I'll send it back. I was all over different places, Japanese restaurants, and I was trying to get this stuff sent back. I was in Denver at a car show, and there were these three Japanese. I didn't know if they were Japanese, Koreans, or Chinese so I asked, 'Any of you guys Japanese?' One says, 'Yeah, we're Japanese.' So I says I got some stuff I'd like to send back to Japan. One guy says, 'I'm going back in a month, back to Japan,' and he says, 'I'll take the stuff with me.' So I packed it all up and sent it to him in Denver, where he was at, and he left for Japan. About a year later I still hadn't heard anything, so I was really disgusted. I had his address and everything, so I wrote him.\n\n\"And lo and behold, the same day I wrote him I got the first letter, from the daughter of this guy that I'd shot. She was forty years old, born after he left. Never knew her father at all. I got the letter here from her. She was so happy that she knew finally what happened to her dad.\n\n\"Besides the wallet, I sent diaries and different stuff, flags that had names on them.\"\n\n[Here is the handwritten letter from the daughter.]\n\n_Dear Mr. Nummer:_\n\n_Allow me to write to you. My name is Mrs. Kimie Sato, a daughter of Siguo Kubo, who was a soldier died on Iwo Jima. I received my father's notebook and three articles from you via Mr. Yamamoto six years ago [_ sic _]. I heard you will go to Iwo Jima on March 14. How I wish I could go there and see you and say my thanks to you. However I cannot go there because of many restrictions. I was very excited and my heart was choked with the memories of my grandparents and my mother who died in 1981 when I handed the articles left by my father six [_ sic _] years ago. They are my treasures for me now._\n\n_As you know I was born ten days after my father had departed for the front._\n\n_Thank you very much for your kindness and also I want to express my heartfelt thanks for Mr. Yamamoto and the Welfare Ministry. I received many congratulations from my friends and my relatives after the newspaper reported the ceremony of the return. I hear you are ill in bed. How do you feel these days?_\n\n_Although I guess you had a pain of both physical and mental injuries I am sure that your warm and heartfelt kindness will impress the deceased and their families. I am praying for you that you may get better quickly for your friends and your family and you will have good days. If you go to Iwo Jima on March 14, please contact Mr. Takada if possible who will join the ceremony there._\n\n_Thank you again and I hope you will feel better soon._\n\n_Sincerely,  \nKimie Sato_\n\n\"That was the year I went to Iwo Jima\u20141995, the fiftieth anniversary. When I got there, the plane landed, and we was the last ones to get to where they had the ceremony. They announced my name on the loudspeaker to come to the podium. So I went to the podium, and there was a package for me. Now here I get a package and I don't know\u2014a hand grenade?\u2014what's in there. My son and I got on the side and we opened it up and she had a nice tie for me, coasters, a tablecloth, different stuff she was so proud for me to have. Some thought I was wrong by sending the stuff back, but I don't think so. It was no good to me, that stuff, but to her it meant the whole world.\"\n\nDid she know Richard Nummer had shot her father?\n\n\"No, I never did tell her that. Couldn't do that.\"\n\n* * *\n\n*The use of the word _Jap_ to describe the enemy was commonplace during the war, and it is repeated throughout this text by the marines who fought at Iwo Jima. The correct word, of course, is _Japanese._\n\nSEAMAN JAMES BUSH\n\n**Water Tender Third Class  \nUSS _Terror,_ Cruiser Mine Layer**\n\n_We had the flag over the body in the canvas, and we'd go through the whole service, and they'd say, \"Now we commend this body to the deep,\" and we'd pick up the board, and the bag would slide straight out and down into the water. It would disappear in a hurry._\n\nCivilian James Bush, eightyone, at home in Ajo, Arizona, in October 2006.\n\nWater Tender Third Class, James Bush, in a photo taken in San Diego in 1944.\n\n_Jim Bush and I were roommates on Guam in March 2006 on his first return to Iwo Jima since 1945. He was affable, talkative, and friendly, and we got along just fine. I interviewed him in our hotel room on Tuesday before we made the day trip to Iwo. I asked what had brought him back to the site of the battle. \"I saw something about it on the History Channel one time,\" he said, \"and I wondered how you got there. I couldn't afford it then anyway. Then a few months back I saw a little mention in the paper about this Military Historical Tour, and I called, and they sent me this literature, and I'm out here\u2014sixty-one years later.\"_\n\n\"I'll be eighty May 16, 2006. I was born in New Mexico, and we moved to Arizona when I was nine. I joined the Navy in May of 1943, stayed in a little over two years. I joined at seventeen so I would not have to sign up for the draft.\n\n\"We brought all our wounded from Iwo Jima to Saipan at the end of February in 1945, resupplied, and went and anchored in a big lagoon at Ulithi, where we went ashore to swim and dive and eat and drink beer. There was nothing there but beach. The beer was Iron City.\n\n\"The story of how we got the beer started in Pearl Harbor in January, when we were all fueled up and loading the last of our supplies. Some new young officer pulled up alongside the ship in a weapons carrier and parked it near the end of our gangplank. We told him, 'Don't park there,' because we were unloading trucks and putting supplies on the ship. He said he would park where he wanted to. He was a real starchy-looking guy with a uniform that was too large for him. He didn't look any older than me, and I was going on nineteen.\n\n\"We'd already put all the supplies we could down below. Back on the fantail we had a big old space with some tie-downs. We'd put a hundred tons of potatoes back there. A weapons carrier has lifters on it so it can be picked up and set aboard ship. Well, guess what? I stood up there and watched them guys look around all over the place, no other officers watching them, and they reached over and picked that weapons carrier up and set it onto the ship next to the potatoes, covered it with a big tarp. An hour after that we were backing out of dry dock so we could get out of Pearl Harbor before they put the gate up. They had cables down there to keep enemy submarines out of the harbor.\n\n\"When we got to Saipan, they set that weapons carrier off onto the dock, and everybody was riding around. I even went out in it for an hour or two. Some of the guys who pulled that stunt struck a deal with some of the guys on Saipan, military people. They liked that weapons carrier. They were moving to the war zone, and they didn't have anything like that.\n\n\"Our guys said, 'Well, what have you got to trade?' They said, 'We know where there are about four pallets of Iron City beer. Dozens of cases.' Done. 'Let's haul it down to the ship.' So they went on down to the ship, waited till the officer of the deck left his post, and they picked those pallets up, brought them aboard, and moved them to a walk-in cooler. Iron City beer was nasty-tasting stuff, but when we got to Ulithi after Iwo, it was really good, I'll tell you that. It was worth that weapons carrier.\n\n\"I pitied that pore little officer, though, having to walk all the way up through that shipyard, back to his commander saying, 'Guess what? I lost the weapons carrier.' It took us four days to drink up all the beer.\n\n\"I served on the USS _Terror,_ a cruiser minelayer (CM-5). It was big, four hundred seventy-five feet long, with close to five hundred on board, including the admiral and all his staff. We were the flagship of the mine fleet. When I came on in Hawaii, I asked when they laid mines, and someone in the crew told me they had used it to lay mines at Casablanca early in the war but after that we were taking the battle to the enemy and we weren't laying mines anymore. They changed its function and moved the _Terror_ to the Pacific to service minesweepers because we were island jumping by then.\n\n\"We went with a large number of these little boats, minesweepers, to Iwo Jima six days before the invasion. They were only forty or fifty feet long, each served by a crew of ten. They were like PT boats [fast-moving patrol boats] except their job was to cut loose the mines that had been anchored off the invasion beaches.\n\n\"It was my first combat action. I was a water tender third class. I was usually down in the boiler room. Everybody had a station to go to in the battle. I was in the mess hall, high up, so I could see out. My job was to regulate, cut off, or open valves, down below hydraulically if there was bomb damage.\n\n\"A storm went by, and we got there early in the morning, woke up to a lush green island called Iwo Jima, except it wasn't lush green. It had been bombed to smithereens.\n\n\"Some of the minesweepers had wooden hulls so they could get close to the magnetic mines without setting them off. We had demagnetizers too. The _Terror's_ job was to take care of those boats, pump fuel into them, service them, help with repairs. They'd come back alongside of us with mattresses hanging out of holes that had been shot in the sides not too far above the waterline. Some of the holes were big, made by forty-millimeter artillery.\n\n\"We were like the old mother hen with a bunch of chicks out there. We had a big old wooden float, like a barge, on one side, and they'd pull over and tie up to that. It had ladders going up, and we had a crane that could reach over and go down. As soon as they came in, our guys in damage repair would go over the side and see what they needed and measure it all off. They'd string acetylene torches over the side and use them to cut pieces out of the ship so they could weld plate over the holes.\n\n\"There were mines in the invasion beaches besides those in the water. I heard a lot of explosions. We had some boats with flat bottoms that could get in close, and they had rows and rows of rockets on deck, pointed out, and they'd fire a row of them, walk them up and along the beach so they covered everywhere. When one boat would run out of rockets, another would come right in behind and take over, and they marched right up the beach, taking out mines as they worked their way north, starting from Suribachi.\n\n\"This went on for three days, and we got so much damage to our sweepers from artillery fire out of the bunkers that they put the invasion off three more days. It was supposed to start February 16. I didn't see the UDT [underwater demolition teams] guys, but I knew they were out there.\n\n\"They never managed to sink any of our sweepers, but by the end of the second day we had twenty-one killed on the small ships, and numerous wounded. We had to replenish some of the crews with people off our ship.\n\n\"We buried our dead at sea. We steamed to another volcano that had stinking smoke coming out of it, and we backed into a little cove between two reefs. I think we put an anchor down. The place we were working on deck was all covered with blood, so we washed it over the side with hoses, and when we did, that whole lagoon come up live with sharks. We shot them up real good, and they all disappeared.\n\n\"We heard later it had to do with shooting them through the liver, which exuded some kind of chemical that told the rest they were in danger, so they all left. I think that liver extract was used later to create a shark repellent to be given to pilots who could spread it out in the water when they were shot down. I don't know if it worked or not.\n\n\"We put the dead in body bags made out of real heavy canvas. We'd cut out a section, put a body on one side, then throw it over and sew the sides together. We used big old hook needles, like carpet hooks. You'd flop the canvas over the head and stitch the needle and string with pliers. We sewed a thirty-five-pound practice shell between their feet.\n\n\"We had a board rigged up, like a heavy piece of plywood. We put hinges on it and attached a lifeline. We had the flag over the body in the canvas, and we'd go through the whole service, and they'd say, 'Now we commend this body to the deep,' and we'd pick up the board, and the bag would slide straight out and down into the water. It would disappear in a hurry.\n\n\"We did it that night and left. The next morning we had to come back and do three more who had died during the night. That's why I think that volcano was something like thirty-eight miles south of Iwo Jima.\n\n\"The first day the Japanese didn't fire at our boats at all. The first day the Japanese thought they were decoys, didn't know what they were doing. They didn't want to give the location of their guns away by firing at us. But when the sweepers got in close enough by the second day, the Japanese would take a chance and shoot at them. They didn't want to fire because it would expose their gun positions to ships offshore, which could then blast them.\n\n\"How did the sweepers work? There's a cable under the bow that goes out at an angle, like a pyramid, and out at the end is a torpedo-looking device that's got a big old cutter in it. It's called a paravane. Your sweeper cable catches the cable holding the mine in place and slides on down to the paravane, which cuts the mine loose so it floats away or up to the surface, where you can fire at it and blow it up.\n\n\"They didn't shoot every one. Some drifted to beaches, where they were recovered and parts of them were taken to the translator, who could read when they were made and stuff like that. I saw some that were made in 1931. Some of them were three or four feet high, and when they'd drift ashore, some of them would bust one of their horns and blow themselves up.\n\n\"When the invasion flotilla came in the dark on the morning of the nineteenth we didn't leave; we just moved over a little ways, northeast off the Quarry, always on the move, always zigzagging. When they started in, I could look down the whole beach and see the waves of marines coming in. Some other pretty good-sized cruisers showed up too although when we arrived, it was just our ship and the minesweepers.\n\n\"On the fourth day there come an announcement that the flag was going up, and quite a few people come out on deck to see it. We headed for Saipan soon after that as fast as we could because we had so many wounded, and our doctors were working day and night on the wounded and the burned. We had people taking mirrors down so guys couldn't see how badly they were burned.\n\n\"The _Terror_ made twenty-five or thirty knots, always zigzagging. We made it down there that night or two days later. There were so many wounded that we had them in our bunks. The one in my bunk had a leg wound from shrapnel. They operated on him the night before we got to Saipan. That piece of shrapnel, it was about the size of a cigarette, went in right behind his knee, and they took it out way down in his calf muscle.\n\n\"There was a big sick bay. I think we had three doctors, one a nerve specialist. They were operating day and night. Corpsmen were doing all they could to help plus evaluating the injured to see who needed surgery the most. If somebody wasn't wounded too bad, they'd come down looking for a place to put him. We probably took several hundred back to Saipan. Most of them were sailors in blue denims. I didn't see anybody in camouflage like the invasion marines wore. I figured they were all off the minesweepers.\n\n\"When we got to the dock in Saipan, it looked like a sea of white down there, all the medical people waiting for us, ambulances lined up.\n\n\"After our R and R with all the beer in the lagoon at Ulithi we went back to Saipan, picked up more supplies, and headed toward Okinawa, then moved south down along the Retto chain, south of Japan, to Kerama Retto, to wait for the seizure of Okinawa, getting ready for the invasion of mainland Japan or the China coast.\n\n\"We were hit by a kamikaze carrying one or two five-hundred-pound bombs early the morning of May 14, 1945. They liked to come in real low, in the sunrise, but he was a little early. We had smoke screens laid down, but the wind blows that smoke around, and he saw an opening, made a loop up in the smoke, and came back and hit us right in the after stack. It made a big gaping hole in the stack and went on down through the middle of the ship, exploding down inside.\n\n\"It killed all our doctors and corpsmen, took out the sick bay, part of the mess hall and the kitchen. The officers' quarters were right there, and all the ship's records, everything that would burn, was up there. Losing the records caused us a lot of trouble because every shot we ever had for whatever ailment, we had to take over. I'd have to look it up, but I think we had about fifty killed and over a hundred injured. One of the MIAs [missing in action] was not found until we got all the way back to San Francisco. We had to put cables on the stack to keep it from falling over and camouflaged the damage areas with canvas painted to look like the rest of the ship.\n\n\"I was about three decks down when the plane hit. There was a terrible compression like you were inside a big bin. A vertical pipe sheltered me from the worst of it, but the explosion vaulted me down the passageway thirty-five to forty feet like a bullet, taking off skin wherever it was close to the bone. It knocked me out. I had a helmet on, but it did not protect part of my neck. The pipe protected me except for the back of my head, and I still get junk out of there, must be paint or dirt or wood from something, I don't know what. They've looked at it a few times and don't see anything, but every once in a while it'll swell up and I'll get some junk out of it.\n\n\"It hit us around four a.m. and we fought the fire till almost nine. We put the dead in body bags, and they were buried ashore. A friend of mine, John J. Epping, was killed, and I unzipped his bag to make sure he had his dog tags with him.\n\n\"When somebody got killed, they took all the pictures, all the address books, and stuff and threw it in the garbage, because you might have a wife at home, and you got a girlfriend somewhere else, and you don't want them to get together. You don't want them to see the address book and all that stuff. But I salvaged a book out of the garbage, a little diary, showing when he left, starting in Hawaii, all the way to Iwo Jima and wherever we stopped.\n\n\"I thought I'd keep that because someday something might happen and someone might want it, and sure enough, long afterward\u2014just two or three years ago\u2014his brother wrote the people that put on the ship's reunion, asking to hear from anybody who knew the status of his brother John. So I sent him that diary, and he called me about two weeks later, and he was so happy I sent him that thing. Over fifty years later.\n\n\"We thought the kamikaze was a twin-engine Betty. There were two guys in it, and unbelievably their bodies were in better shape than some of the guys I saw. One was an old guy, real raggedy-dressed, and the other was younger, in a nice uniform, probably went along for the ride. They didn't get the treatment ours did, though. We just swept them over the side with the rest of the junk.\" Jim Bush got out of the service in the summer of 1946. He and his wife, Patty, were married on June 9, 1950. They had two children, David, born in 1952, and Tammi Jo, in 1958. They lived in Ajo, Arizona, where Jim worked as an electrician for the Phelps Dodge Mining Company and later operated a sand and gravel company. He has served as a Santa's Helper, driving the sledful of toys, since 1946. He and Patty treasure their granddaughter, Tatiana, who was fifteen in 2006.\n\n_James V. Bush Jr. died in Phoenix, Arizona, November 3, 2007. He was eighty-one._\n\nCORPORAL JAMES \"SALTY\" HATHAWAY\n\n**Amtrac Crew Chief  \nTenth Amtrac Battalion, Fourth Marine Division**\n\n_I saw the first flag. I didn't see the second. Someone said, \"Those crazy damn marines are putting a flag up on Suribachi,\" and I turned and looked and seen it go up. The second flag went up several hours later, I understand. All hell broke loose when that flag went up too. The Japs just shelled the shit out of us._\n\nJames Hathaway, in uniform, at age seventy-eight.\n\nJames Hathaway as a young marine. The Bronze Star is pinned to his chest.\n\n_In the name of the President of the United States, the Commanding General, Fourth Marine Division, Fleet Marine Force, takes pleasure in awarding the Bronze Star Medal to Private First Class James S. Hathaway . . ._\n\n_For heroic achievement in connection with operations against the enemy while serving as a crewman with an amphibian tractor company on Iwo Jima, Volcano Islands, on 20 February 1945. Although he and his crew had worked during the preceding twenty-four hours under enemy fire and without cessation, Private First Class Hathaway volunteered to transport an urgently needed load of ammunition to a front line infantry unit. He fully realized the danger of the mission, it [_ sic _] being necessary to pass through mined areas and return to the beach in the dark, but he knew also that no other amphibian tractors were available for the task. He assisted in unloading the ammunition despite enemy fire in the area, then helped in loading casualties on board the tractor and started the hazardous return trip to the beach, where he left the casualties at an evacuation station. His initiative and courageous conduct were in keeping with the highest traditions of the United States Naval Service._\n\n\"I'm seventy-nine years old, I'll be eighty in a couple months. I enlisted in the Marine Corps at eighteen years of age in 1943 off a farm north of Rocky Mount, North Carolina. My dad had five hundred seventy-one acres. We plowed with mules and raised corn, cotton, and tobacco, but I never was a smoker or a drinker. I come off a tobacco farm and never took a cigarette. I went through the whole Marine Corps and never took a drink. It didn't do nothing for me.\n\n\"We had twenty-six mules, two horses, and a pony. We went to town in a horse and buggy. It was June or July and hot as the devil. I took a bus from Charleston, South Carolina, to Yemassee, where we were greeted by a corporal with a swagger stick, and we were told right then who our mothers and fathers were and who God was and everything else. I went on from there to Mainside, Parris Island, and was dropped off solo at the receiving barracks before daylight. Then we got the bucket and the haircuts, the whole nine yards, went in one end of the building, took a shower, and come out the other end buck naked. They issued us uniforms.\n\n\"Boot camp was twelve weeks. Our DI's [drill instructor's] name was McCarty, I believe. They used to beat us on the head with the swagger stick and boot us in the butt and all that good stuff. I often wondered then: What in the world has that boy James got himself into now? But I got through OK. We slept in Quonset huts. One time they had us down eating grass, and the commanding general of the base drove by and stopped and backed up and asked the drill instructor what was he doing. He said, 'Sir, they march like cattle, and I'm going to make them eat like cattle.' The commanding general said, 'Carry on,' and drove away.\n\n\"During our two weeks on the rifle range we slept in tents with the mosquitoes and sand fleas eating us all up. There was only one paved parade deck, and the rest of it was dirt. We drilled in sand. As we got better, we were allowed to get up on the parade deck.\n\n\"I understand we were the second or third platoon to go through with M1s. I shot Marksman the first time out. I couldn't find the target. The front sight of my rifle was off to one side or something. After I got it fixed, I shot Expert.\n\n\"We got out of boot camp mid-August. There was no fancy graduation. They gave us the eagle, globe, and anchor and shook hands with us when we left on the bus. I made PFC out of boot camp. When I come home on leave, my dad said some civilian people in cheap suits had been there checking on me at the farm and in town, with my schoolteachers and all. When I went back I got sent to Courthouse Bay Camp in Lejeune. This was an amphibious tank school at that time, amtracs, top secret.\n\n\"We were in a restricted area down there. We weren't allowed to talk about the amtracs or anything. Nobody even knew what it was, a tractor that floated even when it was full of men or equipment. We went though two or three months of training with Alligator amtracs, which had aluminum tracks and truck engines in them. I think we were the first or second group to go through because we knocked down all the trees and vegetation and even tore up a bunch of houses. The amtracs were painted orange so we wouldn't run into each other in the woods.\n\n\"The amtrac weighed sixteen tons loaded. It was a beautiful piece of equipment, cost thirty-two thousand dollars. It was eight or nine feet wide, nine or ten feet tall and about twenty-seven feet long. It was called the LVT, for landing vehicle, tracked. They kept improving them as the war went on. We had an LVT3 on Iwo. It was powered by a seven-cylinder aircraft engine and had a bilge pump that wouldn't quit. Later the LVT4 had twin Cadillac engines, but the LVT3 was faster.\n\n\"We finished training on one called a Water Buffalo, which could carry eighteen or twenty guys. It had two thirty-caliber machine guns and one fifty, and a tracking mount for one of the thirty calibers on the back. We got all kinds of training in ammo handling and demolition and then took a train across the country. The train had to break up and go across one of the rivers out there in a boat. Our company of one hundred twenty was formed in Camp Oceanside, California, where we took more advanced training.\n\n\"We had a corporal there, an Italian kid, who took a dislike to me and gave me every shit detail there was, partly because he knew I'd do a good job. He gave me a hard time continuously. My dad had warned me some Yankees were tough, so I took all I could from the guy and then got mad and knocked him down three or four times. After that the guys started to call me Salty. That stuck with me all through the war.\n\n\"There were twelve or fifteen amtracs in a platoon, with three men assigned to each one. They were planned as troop carriers at that time. The first seven or eight waves of troops in a landing were brought in by amtracs, before the Higgins boats. The amtracs would arrive on LSTs [landing ship, tanks], drive off and swim to shore, and go right up the beach.\n\n\"We went up to San Francisco to get ready to board our ship on Thanksgiving of 1943. I remember because we had three turkey dinners in a row, one in the train station, one on the train, and another when we got to San Francisco. We were there probably a week before we went aboard the heavy cruiser _Boston,_ one hundred twenty of us. It was a beautiful ship, brand new, and had just come through the Panama Canal and up the coast. We had two destroyer escorts on the trip to Pearl Harbor, where we went aboard LCIs [landing craft, infantry] that took us over to Maui. They let us off in water waist deep. Our amtracs were already there, waiting for us on the beach. We fixed them up like we wanted them and trained there for several weeks. We set up in pup tents. We had a water tank, dirt and dust.\n\n\"We joined the Fourth Marine Division when they came over, went on maneuvers and did several practice landings with them on Maui. We'd go out and pick up troops off the cargo nets and bring them ashore in waves, like a real landing. We'd bring seventeen to twenty troops ashore in one amtrac, and there'd be maybe twenty amtracs in a wave. When we got to the big islands like Iwo and those places, there'd be a whole string of companies joined in one wave. The first waves would come two minutes apart, and then they would extend it to five-minute intervals, then seven, then fifteen. I happened to be in A Company, First Platoon, First Squad the whole dern war, which meant I was in the first wave in every dern landing we made.\"\n\n[Roi and Namur are twin islands in the Kwajalein atoll of the Marshall Islands. Roi is 1,200 by 1,250 yards and Namur is 800 by 900 yards, neither a square mile in size. They were defended by about 3,000 Japanese. Following heavy bombing, the two islands were invaded by the Twenty-third and Twenty-fourth Regimental Combat Teams of the Fourth Marine Division on February 1, 1944. Marine casualties consisted of 190 dead and 547 wounded, while 264 Japanese were taken prisoner and 2,472 were killed. The Fourth Marine Division was the first to go directly into combat in the Pacific Theater from the United States and the first to capture Japanese-mandated territory\u2014Iwo Jima\u2014in the Pacific.]\n\n\"Our first landing was on Roi and Namur, and it was like hitting a sandbar, just a maneuver for us. Very few amtracs blowed up, and we didn't lose but one or two drivers. The Japs didn't know what we were. We rode up on the beach and let the troops out. We got shot at a few times, but it was not too bad on those islands. The amtrac people had it real easy for some reason. We took Roi in six hours and Namur in just about a day. We came back from there to Maui, an eighteen-day trip around February 21.\n\n\"Back in Maui, we went through another phase of training, more maneuvers, and then on June 15 we hit Saipan, which is sixteen miles long. That was a big operation for me; we got shot at a lot more and everything. The amtracs landed eight thousand marines in twenty minutes. I was the crew chief, Ed Kaminsky was the radio operator, and Goule ran the ramp. You did get scared, though. Anybody who said they weren't scared is a damned liar. My foot would start jumping up and down on the accelerator as we went in, and I had to hold it down with my hand on my knee. Guys would go berserk before we'd hit the beach, and Goule had to knock them out or tie them down. I remember going in at Saipan. They were just shelling the daylights out of us and I told Kaminsky to call the Navy and tell 'em to lift their fire, they're killing our own troops. And he come back and said 'That ain't Navy. That's Japs.'\n\n\"The first assault wave hit the beach and continued inland right up to edge of the airfield, and the second assault wave had shotguns and regular rifles. The infantry carried twelve-gauge pump shotguns with double-ought buckshot along with their M1s, so they wouldn't hit us up front. The shotgun fire wouldn't carry like an M1 slug, you see? Because we had just overrun the Jap defenses on the beach. At that time we had a little bit of armament on the amtracs, a quarter inch of steel plate up front. We just overran the defenses. I remember one time hitting a foxhole and just putting it on one track and spinning around and just screwing this Jap right out of the ground. He got hung in the tracks. We went on into the airport. When the second wave caught up with us, they throwed the shotguns away.\n\n\"We stayed with them at the airfield for two or three hours. Then we went back out for more troops and continued to ferry them in. We brought ammo up and wounded back. Saipan was a heavy-duty operation; it was more wounded and everything. I remember the third night an Army outfit, the Twenty-seventh Division, came ashore. I remember they relieved us, and we dropped back to the beach. We were getting hit all the time. We woke up the morning of July 2 on the front line. The Japs had a big banzai attack, and the Army had quit and run. Almost two battalions of the division came running through. They just weren't trained; they were just nothing. They throwed down their weapons and ran.\n\n\"Someone passed the word on the radio that the damn Japs had killed the damn doggies. A whole lot of enemy were killed that night. It was bodies in every direction; we were just running the machine guns all night, killing Japs. I don't know. It looked to me like it was thousands of 'em. I may be exaggerating, but it was at least hundreds, lying around like cordwood. We burned up one machine-gun barrel and had a second one halfway burn up. The fifty caliber about burned out. We were on the edge of the water almost, and the amtracs were so full of holes they wouldn't float. We had no place to go. They almost broke through. We just kept killing them, throwing grenades and ever' dern thing. The Japs were all hopped up on sake. One had a rubber hose, and he was beating the amtrac with it. They had sticks with bayonets on them.\n\n\"We had flares in the air all the time. It was just like daylight almost. Amtracs in line blasting as the Japs came charging. The Marine amtrac people were in kind of a staggered line, a good fifteen or twenty amtracs. The Army guys ran past us right into the water, and the Japs came right after them, and the marines stood up there and met them with thirties and fifties and burned their barrels out. I was too damn busy to be scared, for some reason. You get a grim feeling over your body.\n\n\"The Twenty-seventh Army Division was green troops, a New York Guard outfit, is what I was told. We heard that General Smith, Lieutenant General Holland [Howlin' Mad] Smith, was chasing the Army General Ralph Smith with his forty-five, trying to kill him. That's a true story too. I think he shot at him two or three times but he never did get him. That Army general was relieved, and the Army guys were sent back to Maui and told they'd pull guard the rest of the war. [Two battalions of the 105th Regiment were evacuated.] I heard they court-martialed a hundred and some officers from the top man down in that outfit. General Cates was commanding general of the Fourth Marine Division. Next morning we pushed forward and recaptured the airport and went on from there.\n\n\"About the fourth day there was a big naval battle out there somewhere, and every ship in harbor left except for one LST. What happened was we got into a warehouse of sake and beer and Japanese whiskey, I guess, and loaded up the amtracs. We had been aboard an LST for thirty days or so and got to know these guys real well, so we took an amtrac load of that beer out to the ship and give it to the guys. They all got drunk. Every ship was going next morning for that big battle that got known as the Great Marianas Turkey Shoot except for one LST, and it was sitting out there all by itself. The whole crew was drunk.\n\n\"We went to visit them later on, went down to see the crew, and do you know the skipper of that ship came out with a forty-five pulled, and he stood on the ramp and said, 'I'll shoot the first son of a bitch that steps aboard this ship!'? That was after we got 'em all drunk. Little things like that happened during the war.\"\n\n[Saipan was secured by July 9, nineteen days after the great turkey shoot. More than 3,400 Americans were killed while 13,100 were wounded. Enemy dead totaled 16,525.]\n\n\"After taking Tinian, we went back to Maui to refit, retrain, and take on new troops, and what made us so hot was that all the rank, corporals and sergeants, was being made back in the States, and we couldn't even get promoted. We'd get back to Hawaii, and I'd see a whole bunch of new faces, corporals and sergeants, green troops, shiny faces, you could just see them. You can tell how many times a man had been in combat by his actions, the way he sits, his looks, his appearance; it's just different. As a combat veteran you'd go to the mess hall on Maui and forget to wash your mess gear. You'd just hang it up in the tent, and all of sudden you'd look up and say, 'Damn, I got to go wash this mess gear,' and stuff like that. And everywhere you went you looked for a hole to dive in or someplace to get behind. You'd just get crazier and crazier.\n\n\"When we got back to Maui after Saipan, we struck out to find those Army guys. We were really bloodthirsty. We were going to get as many as we could, but they took 'em to another island so we couldn't get at them. The Seabees were down the beach from us, and they were out looking for them too. There was a few scraps, but they got rid of most of them pretty soon. The court-martials of those guys in the Twenty-seventh Division followed.\n\n\"We went through another phase of training on Maui, going to obstacle courses, machine gun school, throwing grenades. We even went to the rifle range. We had rifles on the amtracs. I'd picked up a thirty-caliber carbine on Saipan, and I unloaded a fifteen-round clip into an enemy soldier. and he kept coming. Then some guy raised up with an M1 and knocked him back about three feet. From that day on nobody would take a damn carbine rifle. It'd kill a man [if you hit a vital organ], but it wouldn't stop him. That one I shot was just like hamburg. I got his Jap rifle. My son has it in North Carolina.\n\n\"Going to Iwo, we were aboard ship before we found out where we were headed, just like Roi-Namur, Saipan, and Tinian. Nobody knew what was coming. The convoy, hundreds of ships, zigzagged continuously, changing direction every fifteen minutes. We stopped in the Bay of Guam; some of the convoy dropped off there. From there to Iwo took about ten days, so altogether we were thirty-some days aboard ship, didn't do a darn thing but sit on our butts.\n\n\"The three days of shelling went on as we were approaching. We had these TCS radio sets, and we'd take 'em up topside on the LST and listen to the Navy talking to its planes, so we knew pretty well what was going on. We just steamed straight in on D-day. We glimpsed the island out at sea; it was just a shadow.\n\n\"When they served steak and eggs, we knew that would be our last meal aboard ship. Every operation we went on they'd give us steak and eggs, and then you had all them dead marines with steak in them. General Clifton Cates gave us the godspeed farewell message over the ship intercoms. We had heard two Navy pilots had been captured and tied to poles on Iwo and the Japs ran by, cutting them with swords. General Cates said in his farewell speech, 'You know what went on ashore. Take no goddamn prisoners.' Those were his exact words. All the time I was on Iwo Jima I saw one prisoner, and a chaplain had him.\n\n\"We ran the amtracs single file off the LSTs at roughly eight o'clock, formed our waves, and went straight into the beach, I believe it was Blue One [various sections of the landing beach were described by color]. We had it all timed, precisely on schedule. A battleship sitting off the coast with anchors out bow and stern was lobbing sixteen-inch shells onto the island. You could actually see a sixteen-inch shell come out of the muzzles of those ships. I looked back, and I could see this big old brass thing coming out of the muzzle. The power of that shell would pick you right up out of the water, and the track would speed up on the amtrac. I understood the ships would back up twelve feet in the water when they let off a broadside.\n\n\"My wave split; my two guys and I went around the bow of my ship, and the amtrac on my left went around the stern, and when we rejoined, I was in the first assault wave. I was in A Company, First Platoon First Squad, the Tenth Amtrac Battalion. Every landing. [Here Hathaway laughed.] They just never did change it. A Company goes first every time. We had quite a few killed. We lost more on Iwo Jima than we did Saipan. We had the new modified amtracs, LVT3s. They had aircraft engines, seven cylinders, air cooled engines. They made eight knots.\n\n\"The water was always rough from the bombardments going on, and the waves were bouncing you around. The Japanese tried to confuse the first wave by ordering us back but we had two Navajos, Isaac and Sam, in the headquarters platoon as radiomen. They didn't call them wind talkers then. We'd had orders that once we hit the line of departure, to go straight in, no one stop for anything, wounded, hurt, killed, anything, and here all of a sudden came this voice telling us to go back to the line of departure. But then Isaac came on, telling them in Navajo to repeat the last transmission, and the Jap didn't know what to say. He started speaking Japanese then, and we just kept going. We wouldn't have turned around anyway. I did let up on the gas, I said, 'Darn, this sounded kind of official,' but I didn't turn. I think one or two amtracs did swing around, but they caught right back up. Those Navajos were pretty good. They'd grunt two or three times, and it'd mean a whole sentence.\n\n\"It was real dangerous close to the beach. You had to kinda kick in the clutch and time yourself, wait for the wave to settle down. Every third wave was a big one. You popped the clutch and headed in with that one, went right onto the beach. The sandbank on Iwo Jima was so high that we couldn't get up very far, but we had protection.\n\n\"The amtracs we had on Saipan you had to go over the side, but the LVT3s had a ramp in back, and guys would stream off, cut around both sides, and charge up the beach. We'd have the machine gun going, and we also had daisy cutters, twenty-four rockets mounted on the back, all fired by electrical charge. They just flowed out. They'd go zoop. We had blisters, periscopes, sticking above the armor plate in front. The water would hit, and the periscope would fog up, and you'd have to go in with your head stuck up again.\n\n\"One of the boys had a camera. You weren't supposed to have them. It was against all kind of regulations, but he took some pictures. I got one or two of them here at the house.\n\n\"One time a Navy Corsair was strafing right above us. He evidently got hit as he went in. He circled around with two of his machine guns going, strafing our own troops. You could see him dead at the stick, all bent over, with his guns still firing. He hit the water between me and the amtrac next to me, threw water all over us. Darn thing almost drowned us.\n\n\"We were not too far from the base of Suribachi, maybe six hundred yards. We hit the beach and were supposed to go inland, but we couldn't get up the bank. The sand was too soft. So we just dropped our troops off and continued down the beach single file to our left, then went back out to sea while the second wave hit two or three minutes later. They had beachmasters who would direct us, and they had little boats off the beach, LCIs I guess they call 'em. They'd direct traffic. We'd come out, and they'd tell us which direction to go and what ship.\n\n\"Getting back aboard ship was a job in itself. The ramp would go down, and you had to back on, catch it with your tracks, and the bow of the amtrac would go under. You had to sit soaked in water and pop the clutch when the tracks caught the ramp.\n\n\"We made five or six trips that day, taking troops and whatever they needed in, hauling wounded back to the hospital ships. We lost quite a few of our own outfit, amtracs getting shelled or hitting land mines. My boat was all shot up. We could carry eight to ten wounded. The crewmen would help load and then hold up plasma bottles or do whatever was necessary. It was pretty messy, you know. Somewhere around the third day an eighty-eight shell hit near my amtrac when we were a couple thousand yards out in the ocean, spun it around, and it sank. We bailed out, of course. We had life preserver belts. You'd mash a button, and they would inflate. We figured the war was over for us; we were going aboard a ship. We were just all smiles; we were out of it, you know, we were going home. A small boat picked us up and took us out to another LST. And shit, we got out there and there was a damn brand-new amtrac sittin' there on the dern ship, waiting for us.\n\n\"On the third or fourth day I was on top of my amtrac, getting ready to get off and get a little rest; we had been running day and night. A kid digging a foxhole next to the amtrac hit a land mine with his shovel, or else a shell went off, but it blowed me through the air off the amtrac. When I come to, I was stooped down in a hole, I was numb, my teeth were loose, and my head was splitting. I threw up my hand and counted my fingers. Then I pulled the other hand up and run it over my other arm. I felt of my legs, and I started smiling then, you know, because both my legs were there. I just laid still until I finally come to all the way. The kid who'd been digging the hole wasn't around. It was either a mine or a shell, an eighty-eight. I think it was a mine, but I'm not sure. A piece of shrapnel had hit me in the back and cut my blouse. A corpsman put some Merthiolate and a bandage on it. I never even turned into sick bay. I didn't even know it left a scar until I got home after the war. My mother, I had my shirt off, she says, 'Why, son, where'd you get that scar on your back?' The other two guys from my amtrac were sleeping or resting out on the ship. Lots of times we'd go out by ourselves, just one guy, and bring in a load of ammo. I was on top fixing to get off. I was going to get under the amtrac or something because they were shelling us pretty heavy.\n\n\"There was no Purple Heart for that, but the three of us did get the Bronze Star award for action on Iwo Jima. The citation is wrong. It says we were the only amtrac around, but that wasn't true. There were others. It says I was a PFC, and that isn't right either. I was a corporal.\n\n\"But anyway, I was the first amtrac to get up the sand dunes and make a road. We called it Roosevelt Boulevard. We had a company of marines cut off at the one end of the island. We loaded up a bunch of ammunition and water and broke through the line to bring them supplies. We just drove right over them damn bastards, firing machine guns. It must have been about the fifth day. [The citation says February 20, the second day.] What was left of our company was surrounded, completely out of ammo. We had a dump established on the beach by then\u2014ammo, food, water, and supplies\u2014kept bringing it in continuously. We took on a load, then went up, through, and over the Jap defenses. We could see their land mines because all the sand had blown off, and we'd scoot around them with the amtrac. That volcanic ash was something else.\n\n\"Our top speed on land was twenty-eight miles an hour. We were running in second gear and hitting them with machine guns. We ran right over the top of them. They were firing back and could have got us with bazooka or mortar. We stayed with the marines a little while. They kept hollering at us, 'Get that damn amtrac out of here! You're drawing fire!' They filled us up with wounded. We had fifteen, I guess, some of them real seriously hurt. One or two of them died before we got back to the ship. We came back the same trail we went up. When we got in the ocean on the way to the hospital ship, I was holding one guy, and he died. His mouth come open, and the blood just shot right out and soaked me. I jumped out in the ocean water and washed off. It was pretty bad, but we got the most seriously wounded out.\n\n\"I saw the first flag. I didn't see the second. Someone said, 'Those crazy damn marines are putting a flag up on Suribachi,' and I turned and looked and seen it go up. The second flag went up several hours later, I understand. All hell broke loose when that flag went up too. The Japs just shelled the shit out of us.\n\n\"I was there all thirty-six days and kept doing mainly the same thing the whole time. All three of us came back together, which was kind of remarkable because so many of them didn't, you know. I lost two or three of my real good buddies.\n\n\"We didn't get too excited toward the end because you knew it was phasing down. A couple B-29s came in with landing gear out before the fighting was even quit. Sometimes mortar would hit right in front of a plane as it was landing. The Army Air Corps, the Mustang fighters, came in. A bunch of them landed there. Those guys were a bunch of alcoholics. They'd come in so drunk they couldn't even get out of their planes. They carried the booze with 'em, I guess. They'd bounce three or four times while landing, and we'd debate if he was drunk and would he be able to walk when he got out of the plane.\n\n\"The A-bomb was quite a thrill for us. After Iwo we went back to Maui to train for the invasion of Japan. We were told we were going to hit the southern island of Kyushu and to be prepared to lose ninety-eight percent on the initial landing. It was quite a thrill when they told us that much. We had loaded up the ships at Maui when we heard they had surrendered. We had our amtracs aboard and ever'thing. There was celebration going on all over the place. Some guy opened up with a thirty-caliber machine gun right down the company street.\n\n\"I left the Marines when I returned to the States, but after ten months on the farm I told my dad, 'I know something better than this.' And I reenlisted in the Marine Corps, as a PFC. I had been a corporal when I got out. I came back in motor transport because of my experience with the amtracs. I checked in at Cherry Point, did tours overseas and different things, but I never became a drill instructor. I did go to Korea. We were more or less in combat, stationed right behind the infantry with the trucks and jeeps, doing maintenance and all.\n\n\"We were in the division move up near the Chosin, and I remember it snowed nearly a foot and we were sleeping out in it. Korea was so darn cold that guys were losing their feet and fingers. I put some sandbags on my feet, two or three of 'em, and tied them up around my ankles. I looked around, and everybody was doing it. It saved many a foot.\n\n\"I got discharged at Camp Lejeune in '63, opened a filling station. I later built a shop in Jacksonville. I had ten or twelve mechanics working for me. I retired again twenty years later. My second wife, Inez, and I got married down here in Florida seventeen, eighteen years ago. I have a boy and a girl from my first marriage. The boy, James Alfred, is in his fifties. He's in real estate, up near Pinehurst, North Carolina. My daughter, Judy Yvonne, is a pilot, a captain with Hawaii Airlines.\n\n\"I stayed in touch over the years, seeing the guys at different times, anniversaries and the like. When you go to reunions, it's sort of amazing to see all them old guys from Iwo or Saipan messed up in their feet and legs, but when the 'Marine Corps Hymn' is played, they all get up.\"\n\nCAPTAIN FRED HAYNES\n\n**Operations Officer  \nTwenty-eighth Marines, Fifth Marine Division**\n\n_Some days, if we made one hundred yards, we were really doing it. On our side we had to cross corridors, and when you do that, you get enfiladed. At Iwo it was from the back, the front, and both sides. That's why we lost so many people in the north end, particularly in fighting for Hill 362. We attacked every goddamn morning._\n\nFred Haynes holds a photo of the flag raising, signed by members of his outfit. His right hand points to his fellow Texan Harlon Block. Haynes is the author of _The Lions of Iwo Jima_ , the story of Combat Team 28 on Iwo Jima.\n\nYoung Captain Fred Haynes takes a cigarette break on Iwo Jima, March 3, 1945. The photo was taken during the attack on Hill 362A and the taking of Nishi Ridge. His Bronze Star citation speaks of his \"outstanding service\" in helping plan the invasion and for \"coordinating the regiment's ship-to-shore movement,\" adding that he frequently \"braved heavy enemy fire to obtain valuable tactical information\" on terrain over which his regiment was to attack.\n\n_Fred Haynes, who was eighty-five in 2006, was a captain and operations officer during the Iwo Jima campaign. He later became a general. When I interviewed him in January 2006, he was one of three from the battle who had become a general and was still alive. As an operations officer during the battle, he was involved in day-to-day planning and indeed devised during the battle a strategy that may have saved a great many lives. Haynes joined the Twenty-eighth Regiment out of Quantico and was never wounded, though he served in three different wars. He retired a major general in 1977, after having commanded the Second Marine Division at Camp Lejeune, North Carolina, and the Third on Okinawa. In 1958\u201359 he founded the Combat Veterans of Iwo Jima, with a membership of about seven thousand veterans and their families. A little-known blood disorder, hemolytic uremia, hospitalized him from December 2003 to May 2004. He was unable to walk for a year and half afterward. With his balance uncertain when he did get back on his feet, he fell and suffered compression fractures of three thoracic vertebrae, leaving him unable to stand up straight. He was strong and active, nonetheless, and got around as well as most on the return to Iwo Jima on March 8, 2006. I was on that trip as well._\n\nCaptain Fred Haynes received a Bronze Star for his service on Iwo Jima even though in all thirty-six days of the battle, he never fired a shot. \"I carried an M1 rifle because I wanted to look as much like a private as I could; the Japanese were very careful about targeting officers. They would bang away at 'em very quickly. I was on Hill 362 with Lieutenant Colonel R. H. Williams at the peak of that battle early in March. I was right on top of the hill and a Japanese ran out of a cave intent on doing bodily harm to us. Williams's runner, Yates, carried a thirty-eight-caliber revolver in a shoulder holster. Where he got it, I have no idea.\n\n\"Anyway, I went to shoot this Japanese, pulled the trigger, and nothing happened. So Yates pulled out his thirty-eight and killed him. The body was out in front of the hill. We had another runner with us, and he wanted to shoot this corpse. I gave him my rifle, and he was starting to aim it when Williams told him\u2014and this was an indicator of how we indoctrinated people\u2014Williams said, 'God damn it, you don't shoot corpses anytime! So put it down.' So he did. Then I checked the weapon. There was no ammunition in it. I had gone all the way to that point in the battle without ammunition.\n\n\"I came on active duty with the Marines not long after Pearl Harbor. I had been doing graduate work at Southern Methodist University for a master's in the new science of limnology, the study of inland waterways, now a well-known environmental science. I had actually graduated, in June of 1941, and later that year, probably December, I was doing some research in the library stacks and I saw a book called _Fix Bayonets!_ by John Thomason. He was a captain in the Fifth Marines at Belleau Wood and Ch\u00e2teau Thierry. The book is a classic, supergood. I took it off the shelf and read it. I made up my mind that if I had to go to war, I'd be a marine.\" Colonel Charles Waterhouse ([Chapter 12) was similarly inspired by _Fix Bayonets!_ ]\n\n\"Two Marine officers came through our campus to sign up potential graduates. I got on their list and was called to active duty the first of February 1942. I went to Quantico with three guys from Texas Christian University. All of us had played football. I played freshman ball at SMU before the coach urged me to give it up, and I became a track athlete. I played center and linebacker, and I only weighed one sixty-three. Our SMU team played Stanford in the Rose Bowl January 2, 1941. A lot of guys with significant football backgrounds fought at Iwo Jima. Our regimental commander, Colonel Harry Liversedge, had been an All-American tackle at the University of California at Berkeley. Liversedge was one of these guys like Cheney [Lieutenant Colonel Chandler] Johnson. Liversedge always wore his helmet, and he never carried a knapsack. He rarely used leggings. After D-day he took his leggings off. He wore a tank officer's shoulder holster and carried a forty-five. He was one of those guys you need on the battlefield. He was a big man. He weighed about two fifty-five, two sixty. He was six-three or six-four. Even for those days he was pretty big. He had a kind of a loping stride. We called him Harry the Horse. He was a relatively quiet man.\n\n\"Combat Team Twenty-eight consisted of three infantry battalions, a weapons company, a headquarters company, and the Charlie companies of each of the combat support units, the pioneers, the tanks, the medics and the engineers, and usually we had the Third Battalion of the Thirteenth Marine Regiment, which was an artillery support unit. So we had about four thousand men. We walked off the island with about six hundred when the battle was over.\n\n\"Combat Team Twenty-eight landed on Green Beach at H hour. Our initial mission was to cross the narrow neck, about seven football fields wide [seven hundred yards], as rapidly as we could, then turn south to assault and seize Suribachi. I was responsible for making sure our boat waves were properly organized at the line of departure and dispatched on time to make the assault at nine o'clock. The first waves, the amtrac waves, were followed by, as I remember, two waves of Higgins boats, known as LCVPs [landing craft, vehicle, personnel]. Those were the organized waves. The Navy had patrol craft at both ends of the seaward line of departure, four thousand yards off Green Beach, and we had a ship-to-shore communications net operating from the command ship. It would indicate when each wave was to be at the line of departure and then the time, like at nine-five, nine-ten, and so forth, that the waves were to leave for the various beaches.\n\n\"There were probably seven waves ahead of me when I landed. I went ashore about nine-fifty with Colonel Bob Williams, who was the regimental executive officer. The last scheduled wave landed about nine-thirty.\n\n\"The Fourth Division was on our right, and on my right was the Twenty-seventh Marines. They had two battalions abreast. We landed in a column of battalions. The first battalion had the immediate mission of getting across the narrow neck. We had an excellent naval gunfire plan, really superior, and we pushed our troops up against the naval gunfire. We even had a few casualties from the gunfire, but not many. The combination of our previous training and the high caliber of the people we had in the First Battalion enabled us to cross this narrow seven-hundred-yard neck in about an hour and thirty minutes. They really smashed it.\n\n\"Back in California and Hawaii, when we were training, only about ten of us knew we were going to attack Iwo and Suribachi. We did amphibious landings at Pendleton and San Clemente, and when we got to Hawaii, we went through the whole cycle again before we left for Iwo. We picked a bubble on the slopes of Mauna Kea about the same height as Suribachi. We took engineer tape, used it to delineate lines where mines had been cleared, and formed the end of Iwo around that bubble. We would line up our troop units by boat team and walk them at the same speed, roughly about three miles an hour, and when they came to this tape, each boat team would deploy and follow the maneuver to get across the island. And then the Second Battalion would move in on the left. We had excellent, totally realistic training, and while the youngsters didn't know where we were headed, they knew damn well they were going to do something like that. We were just going to go hell-for-leather straight across. We thought the First Battalion was practically ruined by the time they crossed the island. And of course the confusion on the beach as the troops moved inland was incredible. This was far worse than anything I ever experienced in three different wars. It took us about three or four days to figure out how many we had lost. I think we lost like three hundred men out of thirty-two hundred in the regiment on D-day. I'm one of the few people who lived through this battle without a wound and who started from scratch with the formation of the regiment and the combat team and continued with it through the occupation of Japan.\n\n\"Kuribayashi, the Japanese commander, was a really smart general. He waited until our units got on the beach, including Williams and Liversedge, the commander of Combat Team Twenty-eight. Within minutes, Kuribayashi put all his artillery and his heavy mortars on these beaches. It was chaos.\n\n\"But we got in and formed a command group on the beach with Liversedge and Williams. In the very late afternoon\u2014and I'll never understand this\u2014Liversedge said, 'You know, we'll have to move to a better place where we can command this unit.' So about one hundred of us, communicators and all, stood up and walked across the neck, almost halfway, to where we could see both beaches, and not a soul was hit.\n\n\"It was not quite dusk. And the next morning the Japanese pinpointed us and dropped a mortar barrage on our command group, killed our regimental surgeon, and blew the legs off one of our uniformed correspondents.\n\n\"I'd never been in battle before, and I didn't know whether I'd panic or what would happen, but I found out quickly that I didn't flap very easily. About ten-thirty we felt that it was absolutely essential to get our Third Battalion, which was being held back as a division reserve, back in under our control because a gap in the line had developed. The First had crossed the island, the Second had turned left, and casualties were very heavy. There was a lot of confusion, and this gap had opened between the right flank of the Second and the left flank of the First.\n\n\"We told General [Major General Keller] Rockey, the commander of the Fifth, it was absolutely essential to get the Third in, so we started landing them, about ten-forty, and they came and filled in the gap. By the close of D-day we had all three infantry battalions, all of our thirty-seven-millimeter guns and all our seventy-five-millimeter half-tracks, with the exception of one thirty-seven that had been blown up and sunk coming in.\n\n\"The weasels, small tracked vehicles, were the prime movers for the thirty-seven-millimeter guns. The Navy would not allow us to put a weasel and a gun in the same boat because they said it would overload it. Colonel Liversedge wanted to land eight thirty-sevens in the scheduled waves. We persuaded the Navy to let us overload the landing craft, so we devised a means of putting the gun on top of the weasel so we were able to double the number of thirty-seven millimeters that we got ashore in the initial waves.\n\n\"The seventy-five-millimeter half-tracks were good-sized weapons, and we had trouble getting them off the beach. We got a couple bulldozers and were able to move the half-tracks and a couple tanks that had got stuck. I've never seen anything like that black sand since. It was the consistency of percolator-grain coffee, and you'd sink up above your ankles in it. There were three terraces, or three levels of sand, on the beach. We got steel mats put down by the close of D-day, but the beach continued to be just utter confusion.\n\n\"The beachmaster for the division was a Navy commander who later became the first violinist of the San Diego Symphony. He was really an interesting guy. He's dead now, but he was terrific. He was a no-bullshit type, and he got it done, but how he did it I'll never know. Our supply system was a little bit different than it had been earlier in the war. We pushed everything up. We didn't wait for somebody to say, 'I need more ammunition.' We pushed ammunition based on rough calculations, straight on up to the battalions as quickly as we could.\n\n\"Each amphibian tractor carried a predetermined packet of essentials, like a crate of grenades, two boxes of sixty-millimeter ammo and two boxes of eighty-one and about four boxes of machine-gun belted ammunition. The minute that amtrac got above the waterline, their instructions were to throw this stuff out on the beach, which they did. And that saved our butt on that first night and the next day because the marines all knew they could go back down there and get whatever they needed and get their butts back up there.\n\n\"We were able to dig foxholes about two feet deep. This was the middle of the winter, in February, and it was in the northern Pacific and it could have been very, very uncomfortable, but the whole island was a volcano, and once you got down six or eight inches, the temperature was very pleasant. Many of us would dig a little hole and put a can of C rations in it, and next morning you'd open it up and you'd have warm beans.\n\n\"There was no natural water on the island, and we were quite lucky because our Navy made water aboard ship. The Japanese only had what little they could catch, and it didn't rain very much. By the time we were about a third of the way up the island, they were getting a hard biscuit a day and about a cup of water, terrible water. In my judgment, the Japanese officers were criminal in continuing this battle; they tortured their own men. These poor bastards got a biscuit and a cup of water a day, and they had to work underground in temperatures of one hundred ten to one hundred twenty degrees. As letters from Kuribayashi pointed out, they'd work maybe ten or fifteen minutes, and then they'd have to get out. Otherwise they'd collapse from heat exhaustion.\n\n\"They were going to die anyhow, and he told them, 'Look, guys, I want you to kill ten Americans before you die.' That was the order. And it looked like for a little while like they might be able to do that. It was the only battle in which we lost more than they did. They lost about twenty-one thousand and we had more than six thousand dead, nineteen thousand wounded.\n\n\"Prior to the landings, we had asked for ten days of bombardment, but the Navy said they couldn't do that. We said, 'Well, give us five.' They couldn't do that. So they gave us three, but the naval bombardment was what got us ashore and across the narrow neck. It saved our butt. The Navy guys were very courageous, and they came in close in gunboats, snuggled right up against the beach, firing directly into the caves on Suribachi. Each of our battalions had a destroyer in support, and the regimental combat team had a cruiser in support. The division had a battleship. There were actually two battleships, five cruisers, and nine or ten destroyers in overall support. And it was effective in the southern third of the island; it was less effective up by the Quarry to the north and on the ridgelines. Naval gunfire is a flat trajectory weapon and difficult to use in many places. About all you could use on the ridges and hollows was artillery and mortars. Artillery tubes began to wear out when Combat Team Twenty-eight started to fight in the north end of the island. We ordered new tubes, and one of the unfortunate incidents of the war was losing those tubes. They were being delivered by air from Guam, and the wind shifted with the first batch, and we sat there and watched the tubes parachute right in the water. So we had to deal with inaccurate artillery tubes.\n\n\"Many of us thought Cheney Johnson, who commanded the Second Battalion, was probably killed by our own artillery. He was blown to smithereens. I think all they found was a piece of his torso and his shoes. Eventually we got new tubes, and the artillery became more accurate. Cheney wore a combat fatigue hat. He never wore his helmet. My personal view is that Johnson tripped the detonator on a U.S. aerial bomb dud the Japanese had rigged just north of Hill 362. Captain Naylor, the F Company CO, had warned Johnson not to go into the ravine, but he did and met his demise.\n\n\"But you need a few guys like Johnson, who sort of wander around the battlefield without their helmets and get things done. They become an inspiration to the troops. That was true of Liversedge, who was our combat team commander, and of Williams, who was his exec, and Johnson.\n\n\"Suribachi was captured on the fifth day. We landed on the nineteenth of February, and the flag went up on the twenty-third. The first flag went up at ten-twenty a.m. The second and most famous one went up about two that afternoon. We didn't pay much attention to the second one because you know, we were all busy, fighting the war, and the big impact was the first flag. You could barely see it, just a little tiny thing. Major Oscar Peatross\u2014I was his deputy\u2014and Williams and I went up the mountain shortly after that first flag was raised, when the second patrol was getting ready to come up.\"\n\nFrom the top of Suribachi, Haynes and his group got their first view across the battlefields.\n\n\"We saw the mess on the beach and what we had ahead of us. You could see the real challenge was going to come once we got past the airfields, where we had one hell of a fight. That photo of me was shot while we were taking Hill 362. We had more casualties in three days taking 362 and Nishi Ridge than we had taking Suribachi. We lost two hundred fifty men a day up here. That's more than a rifle company a day, _boom boom boom._ \"\n\n[Haynes described the landscape around Hill 362 Able in an article for the _Marine Corps Gazette_ in 1953: \"Northern Iwo was a treeless wilderness of cave-studded, jagged, frequently ill-defined compartments, radiating from a central plateau toward the coastal cliffs. The terrain around Hill 362 constituted the right (west) one third of the main cross-island position. It dominated the western side of the island south to Suribachi.\"]\n\n\"That was the meanest fighting because whereas we could get naval gunfire into vertical sections where all the caves were, here you couldn't do it. We had to depend on infantry guts, just getting up to these places and then cross them.\n\n\"The battle became a slogging thing, almost a depressing proposition, because there had been all the excitement and progress: We took the mountain and put a flag up, and when we went into the line, we fought on the left flank. Progress from midway on was just tiny. Some days, if we made one hundred yards, we were really doing it. On our side we had to cross corridors, and when you do that, you get enfiladed. At Iwo it was from the back, the front, and both sides. That's why we lost so many people in the north end, particularly in fighting for Hill 362. We attacked every goddamn morning. We'd jump off around seven-thirty or eight o'clock. Wherever Williams went he always took me. In the evenings at the end of each day it was our responsibility to make sure our lines were tied in, particularly with the Fifth Division units on our right.\n\n\"I went up to our OPs [observations posts] frequently, and I could see what we were faced with. The proposition I made to Colonel Williams and Liversedge was we should try to form a horseshoe, so we could go down these corridors instead of across and avoid the enfilades. So Colonel Liversedge said, 'OK, get yourself an airplane and go look.'\n\n\"So I went to Airfield No. 1 and asked a pilot of a grasshopper, one of these little Piper Cubs, to fly me down the corridors and then back up the corridors from the sea. I told both colonels what I had seen made it seem right. The idea was to hold the line steady and swing the right-flank units around to positions from which we could attack down the corridors.\n\n\"Then you're going the length of the corridor instead of across it. And when you did that, instead of going where you got fire from the flanks, like in the case of 362, where we lost a huge number of casualties by fire from the rear because these bastards had caves up here on the south side, and then they had these interlocking caves, underground tunnels, and they came out on this side and they could shoot you in the back. So if you came over the top, you not only got hit from the back and the front, but you got the sides also. There were probably fifteen or twenty of these corridors, each six hundred to seven hundred meters long. So the aim was to go up them instead of across. Then you could put machine guns and observation over on both sides, and you could shoot this way and that way, and it was a lot easier: set up interlocking fields of fire and close off the caves. And that was what we did ultimately. There's no way you could tell for sure how many casualties we reduced, but whereas we had huge casualties initially, once we started that maneuver, the numbers dropped off.\n\n\"I had no frame of reference from past experience. Nobody did for this kind of thing. This was really the first time that any of our officers and men had been in anything like this. In fact, nobody did after that either, even though Okinawa was a tough place. Very few Japanese surrendered on Iwo. My Combat Team Twenty-eight captured a total of sixteen prisoners in the thirty-six-day campaign, and all but two or three had been wounded or were physically incapacitated from lack of food and water.\"\n\nWhy were there so few? Were captured Japanese simply shot?\n\n\"We conducted an extensive indoctrination program prior to Iwo in which we made a strong point of the need to take captives. This had to be done with great care because the Japanese were very canny: They'd put hand grenades under bodies of their own people so that when you moved them, the grenades would go off; they also booby-trapped artillery shells, kind of like these guys the jihadists are doing in Iraq. So we urged the men to be very careful in how they took people. Nearly everyone we captured begged to be shot. They begged to be killed.\n\n\"During the interrogation of one prisoner, he said, 'You know, as far as the Japanese are concerned, I'm dead.' Which was another way of saying, 'I can't come out of the woodwork. I've got to be very careful.' It was hard to find any POW who would talk. But I well remember an instance in which a Japanese was buried up to his neck in debris, from an artillery round, probably, and he was alive. One of our Japanese-language officers gave this guy a cigarette and then very gingerly removed the dirt from around him. He confirmed some of the order of battle information we already had.\n\n\"The most useful intelligence, after we landed, came mainly from our own OPs, our aerial observers, and from documents that we found in a couple of caves in Suribachi. So we had a pretty good idea of what the order of battle was. We also had two excellent interrogators, who had been to the Japanese-language school, and we had a third, John Lloyd, who had been raised in a missionary family on Honshu. He knew Japanese street language.\n\n\"By the time we finished 362, we were not Combat Team Twenty-eight anymore. Our leadership was gone, we had only a couple company commanders left, most of our lieutenants had been either killed or wounded, many of our NCOs had been killed or wounded, and I liked to say we walked by faith. Casualties in the companies of the Twenty-eighth were running well above fifty percent by the evening of March 1. We just believed in the guy on our right and the guy on our left, and we had been told to do it, and we were gonna do it, do it to the end.\"\n\n[Haynes has written his own account of Combat Team Twenty-eight's ordeal, _The Lions of Iwo Jima,_ published by Henry Holt.]\n\n\"Yates, the runner who killed the Japanese when my rifle didn't fire, later had his leg broken by sniper fire and had to be evacuated. He had had this damn little black and white terrier, a mascot he called Tige, short for Tiger. He smuggled him aboard ship to go to Hawaii and then smuggled him aboard ship to Iwo. Yates was primarily a runner for Williams, and the night before D-day I was in the wardroom. I couldn't sleep, and neither could Williams. He said to me, 'What the hell are we going to do about Yates's dog? We can't take him ashore.'\n\n\"And I said, 'Well, you know the Navy kids will take care of him, so we can tell Yates to leave him aboard.' But I remember saying to Williams, 'You know he's not about to do that. He'll smuggle him somehow.' And he did. He put the dog in his knapsack and took him ashore. Later, after Yates was wounded and evacuated, we got one of the communicators to take care of Tige, and he lived through the battle. We decorated him with an Asiatic-Pacific Campaign Medal with one star, put it on his collar when we got back to Hawaii.\n\n\"What was the Japanese leadership like? The commander at Iwo, Tadamichi Kuribayashi, had been an attach\u00e9 in the United States. He was also a cavalryman, and he wrote a song to the horse cavalry. Both he and Lieutenant Colonel Takeichi Baron Nishi, who commanded a Japanese tank unit at Iwo, at various times had indicated that the Japanese were dumb as hell to fight America. Both knew they couldn't beat us. But nobody paid any attention to them. So Kuribayashi ultimately committed hara-kiri toward the end of March. We know this mainly from intercepted communications.\n\n\"He sent a message on the fourteenth telling the emperor that he was sorry but he didn't think he could hold Iwo. We're not sure of this, but we have fairly good evidence of Kuribayashi's death from a Major Horie, who was on nearby Chichi Jima during the battle for Iwo. He thought Kuribayashi stuck himself in the gut with a knife and had his aide chop his head off.\n\n\"Nishi dug all of his tanks in and used them as pillboxes on Iwo. Between the fifteenth and the twentieth of March, he got hit in the leg and then committed seppuku. We never found his body. These are things that we found out from the very few Japanese who lived holed up in caves.\n\n\"Kuribayashi's eldest son is still alive. His name is Taro, and he is a member of the Upper House in the Japanese Diet. Kuribayashi had two daughters, one of whom died of typhoid in Tokyo during a B-29 raid there, and the other became a movie star, believe it or not. His grandson is in the foreign service of Japan.\"\n\nWhat happened to the island after the battle? It provided an emergency landing strip for crippled B-29s returning from attacks on the mainland, for one thing. Haynes described another purpose.\n\n\"Iwo Jima became a critical navigation point for the atom bomb attacks on Nagasaki and Hiroshima. Colonel Paul Tibbets, who was in charge of both bombings, had Charles W. Sweeney, the pilot for the Nagasaki bombing, go to Iwo and dig a bomb transfer pit near Suribachi, down at the end of the runway. This was so that if Tibbets's plane, the _Enola Gay,_ had trouble, she'd be able to get back to Iwo, come over the transfer pit, and lower the bomb into the pit, so he could get another B-29, pick it up, and go finish the mission.\n\n\"Second, three B-29s followed Tibbets. One, McKnight's _Top Secret,_ landed at Iwo as a standby aircraft in case the _Enola Gay_ had trouble en route. Chuck Sweeney in the _Great Artiste_ and George Marquardt in _Number 91_ flew the entire mission with Tibbets from the Marianas, and they used Iwo as a navigation point. After Tibbets made the drop, they were the ones who did the photography and the radiation checks. And of course Sweeney then turned around and three days later flew the second A-bomb to Nagasaki.\n\n\"What gives the battle for Iwo Jima such lasting significance? First of all, it was the bloodiest and most difficult of any battle that the Marines fought, ever. Even though Okinawa was greater in terms of casualties, it was a battle where three Marine divisions and several Army units participated. And the Iwo battle had enormous significance nationwide and worldwide because of the photo of the flag raising taken by Joe Rosenthal. It was a huge boost to morale back in the U.S. People were getting weary of the war. The European war they were pretty sure was going to come out OK, but here was Iwo, a little tiny goddamn island, eight and half square miles. Tarawa had been a tiny island, and it had been terribly costly. Peleliu had been costly. The Marianas had been costly. Guam had been costly. The war tolerance in the U.S. was pretty low, so this was a huge boost to the morale in the U.S. The photo epitomized courage, cooperation, patriotism, working together. It had a Navy corpsman in it. You remember, there were five marines and one Navy guy.\n\n\"A second feature was this: You can't identify the flag raisers. You can only see a little part of James Bradley's face, hardly enough to recognize him. The others are hidden, and they could all have been unknown if Roosevelt hadn't said you got to bring them back. We didn't know who they were. This was not a posed picture, and it was so symbolic because of where these marines came from.\n\n\"The guy putting the flag in the ground is Harlon Block, another high school football player from Texas, who joined the Marines with all the senior members of his ball team. Bradley was a rural kid from Wisconsin. Ira Hayes was a Pima Indian from one of the poorest tribes. Strank was probably the best of all the group. He was a sergeant, an immigrant who came across as a baby with his parents from Czechoslovakia. His dad was a coal miner, and Strank became a coal miner until he joined the Marines. He was killed, of course. And Gagnon was a Canuck from New Hampshire or Vermont. He's the one you can't see. You can see a hand, I guess. It was a representation of America. The only thing not there is a black. We didn't have black Marines in combat.\n\n\"We had a black DUKW company [see page 255] that was Army, and we had ammunition and depot companies, most of them black, on the shore party, and they did a lot of the digging of the graves. They were in that last fight, and Harry Martin was given a Medal of Honor. He was killed there.\"\n\n_Iwo Jima is the signature battle of the Marine Corps because it epitomizes keeping going in the face of fatigue and an enemy who fought to the death. It epitomizes the Marine Corps' esprit, which is a lot like love. It's impalpable. You cannot touch it. You cannot exactly get your mind round it, but you certainly know it when it's there. Marines trust one another. You have this animating spirit, this belief in the guy on your right and the guy on your left. They've all been through a similar kind of training, and it's tough training done by people who have been in some pretty tight spots. You understand that these guys know their job, and they will cover you and you them. It's a belief, too, that you can't beat us because we're in it until the end, and we're in it together as Marines. You may knock a few of us off, but if you do, watch out. If you kill Marines, one thing is for sure, there will be other Marines coming along soon, and they will keep coming until they find you.*_\n\nAfter World War II Haynes rose to the rank of major, serving as executive officer of an infantry battalion in Korea. As a colonel he became military secretary to the commandant in 1968. After being promoted to major general, he was given command of the Second Marine Division on the East Coast and, in 1973, took command of the Third Division in the western Pacific.\n\n\"Headquarters had moved from Vietnam to Okinawa, and we did the evacuation of Phnom Penh. Later I was given command at Camp Lejeune, North Carolina, and went from there to Washington for my last tour of active duty. I retired January 1, 1977.\"\n\nFollowing retirement, Fred Haynes joined an aerospace firm, and for several years he was a military commentator for CBS News in New York. He is chairman emeritus of the American Friends of Turkey, which later became the American Turkish Council. He also is a member of the Council of Foreign Relations. He and his wife, Bonnie, a TV producer, have four children between them from prior marriages.\n\n* * *\n\n*Fred Haynes speaking in _American Spartans_ by James A. Warren.\n\nCORPORAL HERSHEL WOODROW \"WOODY\" WILLIAMS\n\n**Medal of Honor  \nTwenty-first Marines, Third Marine Division**\n\n_Our objective was the first airfield. We didn't hardly move for a day or so. There were bodies everywhere, absolutely, stacked up like cordwood, just unbelievable._\n\nHershel Woodrow Williams displays his Medal of Honor, awarded by President Harry Truman on December 5, 1945.\n\n_Woody Williams spoke to me at length from his home in Ona, West Virginia, in early January of 2006. Eighty-two at the time, he said he would never go back to Iwo Jima. \"I guess it's an idiosyncrasy. We paid a terrible price for that thing, and if we'd have kept it, I'd have gone back on the first invitation. It was a mistake to give it back to the Japanese. We paid a terrible price, eighteen thousand wounded plus sixty-eight hundred killed, and we should have kept it, as a memorial, if nothing else. Now, I just don't want to make that association. I want to remember it as it was. I know it doesn't look anything like it did then. I have no desire to go back.\"_\n\n\"I weighed three pounds when I was born October 2, 1923. I don't know how I made it. I was the last and smallest of eleven children. I got all the hand-me-downs. Only four boys survived. The flu got them. My folks were in the dairy business, and when I reached five years of age, I became subjected to it. This was long before the machines came in, and we milked by hand. My job was to get out to the barn around five a.m., with a bucket of soapy water so I could wash the cows' teats before the others got there. We milked thirty to thirty-five cows on a sixty-eight-acre farm. We were about seven miles out of a town called Fairmont, West Virginia, and all our business was there, so each morning my father or my brothers would take the pickup truck, a Model A Ford, and we would head to town and make house-to-house deliveries.\n\n\"We bottled the milk on the farm, in pints and quarts, and of course we had a separator, so if you wanted a quart of pure cream, we could give that to you too. Our milk sold for seven cents a quart. My job, along with a couple of my brothers, was to ride on the running board and my father would yell out, 'One quart of milk!' or a dozen eggs or a pound of butter; we'd grab it and run it to the customer's door, gather up the empty milk bottles, and run back to the truck.\n\n\"I went to grade school in a little country community called Quiet Dell in Marion County. It started out as one room, but they added a second one eventually, so you'd go from first to fourth grade in one room and then from fifth to eighth in the other.\n\n\"Then you'd have to go to high school in Fairmont, and of course there was no bus service or anything, so you got to school however you could. The morning milk deliveries made it easy for those of us in high school. We also had evening deliveries, and they'd pick us up and bring us home. Any sports or extracurricular activity was completely out of the picture for us. There was too much to do at home.\n\n\"My father's name was Lloyd Dennis. He died when I was nine years old. My mother was named Lurenna, and everybody called her Rennie. I don't know her background. Maybe we just had some unusual names. One of my brothers was named Nora, and I don't think I've ever met another man with the name Nora.\n\n\"Of course there was very little money, but we would usually get a dime on Saturday, and we would go to town if we weren't putting up hay or didn't have something else to do. They wouldn't take us, so we'd either walk or hitchhike. A nickel would buy an ice-cream cone, and a nickel would get you in the movies.\n\n\"I did not graduate from high school. My brother next to me dropped out and went into the Civilian Conservation Corps in the late thirties. He was making about twenty-one dollars a month. That was a lot of money then, so I decided I wanted to do the same thing. I dropped out of high school in the eleventh grade and went to the CCC. I had the hope they would send me where my brother was, but they sent me to another city, Morgantown. I stayed there ninety days and then they shipped that whole group of us all the way to Whitehall, Montana. I thought I was clear out of the world.\n\n\"We were building jack fences\u2014barbed-wire fences with a post like a tent pole notched at the top. We put barbed wire on each side of them. The purpose was to keep animals off the government reservations. My job was hauling men to the forest who did the cutting and then eventually hauling those pine trees to the mill, where they would be creosoted and notched and turned into fence posts.\n\n\"I was there when Pearl Harbor happened December 7, 1941. A lot of the men enlisted right away. I was still coming up to my eighteenth birthday. Of course the Army was in charge of the CCC. They furnished the uniforms, did all the medical treatment, did all the chow. We worked under civilians off the base, contract people who supervised the road building or fences or whatever. So a lot of these CCC fellows went straight into the Army right after Pearl Harbor was hit. I wasn't eighteen yet, so I requested my discharge from the CCs and came home. I wanted to go in the Marine Corps. I did not want to go in the Army.\n\n\"Why? [Williams laughs at the memory.] During the time I was twelve or thirteen, we had two fellas that had gone into the Marine Corps from Quiet Dell. Of course ever'body knew ever'body, and most people were related. I had one aunt and five uncles in the area and first cousins by the dozens. But these two guys had gone on active duty in the Marine Corps\u2014at that time you had to enlist for six years\u2014and they would get a thirty-day furlough once a year. They would come home in their dress blues. The Marine Corps was trying to make a name for itself so it could attract people. These two guys were tall, very polite, just really squared away, with spit-shined shoes and all that stuff. We kids would hang around and go with them wherever they would let us. They'd tell us these great big wild stories, probably most of them not true. It was good entertainment anyway.\n\n\"But I decided, apparently in my subconscious, if I ever became a military person, that was what I was going to be. That's all I knew. We didn't see any other military people.\n\n\"I went home to enlist in Fairmont right after my eighteenth birthday in October 1942. My mother did not want to sign for me. My father was already deceased, and I was the last of the Mohicans. I was still the baby. You know how that is.\n\n\"But I went to enlist anyway, to the Marine Corps office in Fairmont, sometime in November. I filled out a piece of paper and took it up to the marine sitting at the desk, and I handed that paper to him, and he kind of looked me up and down and said, 'We can't take you.'\n\n\"He hadn't even looked at my papers. I said, 'How come?' He said, 'You're not tall enough. You've got to be five feet eight or better. We can't take ya.' So I went home. I decided if I couldn't go in the Marine Corps, I wasn't going anywhere.\n\n\"So I went home and back to farming. But then, after the first of the year in 1943, the Marine Corps did away with that height requirement and began taking little runts like me. I was five feet six, and weighed one hundred thirty-five pounds. But I was muscular. I'd worked on a farm all my life.\n\n\"That recruiting sergeant had kept my paper, and he looked me up and then, in May of '43, I went in. I had really enlisted before that, but they put me on a waiting list because so many people were going in that Parris Island couldn't handle them all. So when my number came up, they sent me to Charleston to catch a train. I thought we were going to Parris Island, but they couldn't handle any more recruits. They had big tent cities, and guys who were there told me there was mud clear to your hip pockets. Anyway, they started a troop train down in Georgia or Florida and came up through the southern states, picking up people. They picked up six of us in Charleston, and we went all the way to California. So I'm a Hollywood marine.\n\n\"Training was tough. There were times I was laying out there in the mud or dust, dirty, thirsty, and bone weary, and I thought I might not get through it, but I never thought about quitting. They could kill me, but I wasn't going to quit.\n\n\"After basic, I went to a place called Jacques Farm, a tank farm at Camp Pendleton where they taught advanced infantry training with tanks. I didn't get into demolition until I went overseas.\"\n\nA ten-day furlough did not give him enough time to go home following training, so Williams wrote his sweetheart, Ruby Meredith, that their wedding would have to wait until after the war. \"I wouldn't marry her,\" he said, in an earlier interview with West Virginia writer Wayne Sheets. \"I did not know if I'd be coming back, and I didn't want to saddle her with that kind of burden.\" He also told Sheets how he devised a way to evade the censors and let Ruby know where he was, by employing the first letter of each paragraph to spell out his location. His first overseas base was New Caledonia. \"By the time I'd written enough to disguise my code, I'd written a letter the size of the Sears, Roebuck catalog. And my next assignment was Guadalcanal.\" He learned later from a censor that many other servicemen had used the same system.\n\n\"They shipped us to New Caledonia as a replacement group, where hundreds of marines were sorted out and sent to the various divisions as replacements. I got sent to Guadalcanal to join the Twenty-first Marines, Third Marine Division, which at that moment was on Bougainville.\n\n\"The First Division had captured Guadalcanal. Bougainville was the first campaign of the Third. They'd been to New Zealand, shipped to Guadalcanal, and now they were up in Bougainville. We were going to go up there, but those guys did us a favor and secured the island before we could leave. We joined them when they came back to the Canal, and that's when I got into flamethrower demolition. We trained there till June of 1944.\n\n\"I was a Browning automatic rifleman previously, before they sent me to this demolition flamethrower school. The guy who took my place on the BAR got killed when we got to Guam. So if I had stayed with the BAR, you wouldn't be talking to me, unless you got a line direct to heaven. We had a banzai one night, and they overran his outpost and got him.\n\n\"We went to Guam July of '44. It was our first action, jungle-type fighting after we got off the beach. The jungles were very thick. You just had to beat or cut your way through. You didn't see the enemy until after you killed him. But one of the memorable things, there were thousands of frogs on Guam. It rained a lot on Guam, and the shell craters were full of water, and there would just be hundreds of frogs in those shell craters. At night those things would move, and of course, to a scared marine, he thought it was a Japanese. I expect we wasted more ammunition on frogs than we did on the Japanese. Whatever noise happened, we shot at it.\n\n\"When we hit the beach, we picked up our flamethrowers, but we didn't need them after the initial invasion, so they stayed in Headquarters Company. You were a rifleman again. We had M1s. They issued me a forty-five, which I carried on my hip, and a carbine, but as soon as I could find an M1 that somebody had dropped or lost or belonged to a guy who had been wounded, I threw the carbine away and got me that M1. I couldn't hit the side of house with that carbine. I didn't like it. I'd always pick up an M1 and take that with me.\"\n\nAlong with maneuvering the flamethrower, demolitions consisted of learning how to set fuses and fling satchel charges.\n\n\"We also used a lot of pole charges, about a ten-foot two-by-two piece of lumber or anything else you could find. Marines always improvise some way. If you haven't got it, you make it or steal it. Instead of carrying the satchel, you'd tape or fashion or tie the satchel charge to the end of a pole so that once you burned out a pillbox, then you could run up and stick the pole charge through the aperture of the pillbox and blow it up rather than have to go up there and try to stuff a satchel charge in there by hand. Or you'd take a pole charge and stick it down a cave. You were more accurate that way than you were just throwing it.\n\n\"Most of the explosive was composition C2. We did use a lot of quarter-pound blocks of TNT when we didn't have C2, which was very secure. You could shoot through a composition C2 and it wouldn't go off. A blasting cap fastened to the C2 would make it go off. The fuse set off the blasting cap.\n\n\"They issued the flamethrower operator an assistant because somebody has to carry all his gear. He can't carry a pack and ammunition and all that other stuff. The fellow who became my assistant was from Floyd, Montana. With my having been in Montana, we had a little rapport with each other. He was six feet six. Now I'm five-six and he's six-six, and he carried everything I couldn't: my canteen, my roll pack, and everything else. His name was Vernon Waters, and he was my assistant when we got to Guam. He was still my assistant on Iwo, and we lost him there.\n\n\"We stayed at Guam after capturing it and continued to train for jungle-type warfare. Then we got notice we were going someplace new. The whole division was boarding ship. Once we got aboard, they brought out this mock-up on a piece of board that showed the island of Iwo Jima. Looked like a pork chop. That's when they told us where we were going.\n\n\"The Fourth and Fifth Division would be the primary force. We were in reserve. We'd probably never even get off ship, we'd probably be gone for four or five days, and then we would come back to Guam. Well, you know intelligence, ha. It's not always good intelligence.\"\n\n\"The Fourth and Fifth went ashore February 19 [Williams told Wayne Sheets]. On the twentieth somebody decided more marines were needed. At three in the morning of February 20, we off-loaded onto Higgins boats. My scariest moment was getting off that ship by rope ladder. There was no light and we were carrying a backpack that weighed sixty or seventy pounds, and we had to go down that rope ladder forty or fifty feet. I knew if I fell into the water it would be the end of me. I couldn't swim.\n\n\"We stayed in the Higgins boats and amtracs all day long. Everybody was sick, either from seasickness or because the guy next to you vomited all over you. Conditions were beyond description. There was no room for us on the beach. Units there were pinned down, suffering unbelievable losses, and couldn't move inland. About ten that night they took us back aboard our ship. They gave us some food and everyone collapsed and slept wherever they were. At three the next morning we did it all over again, aiming for Red Beach.\"\n\n\"Fortunately most of the group I was with were in amtracs [Williams told me]. They hit the beach and stopped because those things couldn't go in that volcanic sand. They just bogged down. You got off when the front ramp dropped. Trying to walk or run in that volcanic ash was like trying to move on a floor covered with BBs. You couldn't get any traction, and there was no cover. The enemy was firing straight across the ground, and there was nothing to stop the bullets. If you tried to dig a hole, the sides caved in. The only protection was shell holes that had been blasted in the ash. There'd be six or ten guys in each crater.\"\n\nWilliams said he did not know how he conquered his fear. \"I had a job to do, and I suppose I got so wrapped up in it I didn't have time to think. I knew deep down there was a power greater than I was. I didn't know what it was, but I prayed to it lots of times. Many years after I left the Marine Corps I came to realize that power was God Almighty and that He alone had the power to save a life\u2014mine. When I was going after the pillboxes, I absolutely thought that I would die.\"\n\nWilliams's C Company, First Battalion, Twenty-first Marines, consisted of 279 men when it hit the beach on February 21. By March 5, thirteen days later, only 17 were left.\n\n\"Because of the tremendous losses in men and equipment suffered by the Fourth and Fifth Divisions, our division became the spearhead. Our job was to punch a hole in the Japs' line of defense, mostly pillboxes, so the tanks and personnel carriers could come ashore and start inland.\n\n\"Our objective was the first airfield. We didn't hardly move for a day or so. There were bodies everywhere, absolutely, stacked up like cordwood, just unbelievable. And the airfield of course had been bombed and strafed and bombed until there was nothing left except shell craters and the pillboxes I got involved in. Somebody reported later there were about eight hundred of them in the area where we were headed. Those pillboxes had a deadly field of fire over that airfield. You'd just run from one shell crater to another, and if you made it, you felt pretty lucky. That's all the protection you had.\n\n\"Then, on the twenty-third, of course, the flag went up. We were still on the beach at that point. I didn't know what was going on. I guess I had my head under the sand about as far as it'd go. People around me started yelling and screaming and jumping up and down and firing their weapons, and for a few seconds I couldn't tell what in the world was going on. And then I looked up and saw Old Glory flying above Mount Suribachi, so I just joined and fired mine too. I jumped up and down and screamed.\n\n\"It was about that time the CO asked me if I thought I could do anything with the pillboxes that had us stymied. I was in Headquarters Company, C Company, and when I hit the beach, I had six flamethrower and demolition people with me. Each had an assistant like Waters, so there were twelve of us. But by the twenty-third they'd either all been killed or wounded, and we hadn't used a flamethrower up until that point. My job primarily was to keep them supplied with demolition and flamethrowers as they needed them.\n\n\"I had not had any occasion up until then to use any demolition or flamethrowers. Since I was a corporal and all the other guys were PFCs or privates, I'm the man in charge, and I was acting sergeant. We were all called to a meeting in a big shell crater. We had tried and tried to get across that airfield and get through those pillboxes, but we had been kicked back every time. Pillboxes surrounded the airfield, and they were self-protecting with what I call interlocking fields of fire; you couldn't get to one without exposing yourself to the others.\n\n\"So the CO, Captain Beck, asked if I thought I could do anything about some of those pillboxes. Somebody else said later I told him, 'I'll try.' I have no idea what I said. So he assigned me two BAR men and two riflemen, and I picked up the flamethrowers and started knocking some of them out. The BAR and riflemen would cover me as I went from crater to crater, until I reached the point where I had to crawl. You could only run from one place to another for so long, and when you get so close, you got to get down and crawl on your belly. If you raise up, somebody's going to get you.\n\n\"We were using a mixture of high-octane gasoline and diesel fuel, so if you fired the thing into the air and you had any air blowing toward you, it'd burn all the hair off your arms and eyebrows. It would kick back at you, so we fired it in two- or three-second bursts and rolled the flame on the ground. It would roll across the ground, twenty to twenty-five yards depending on terrain, and right into a pillbox.\n\n\"We had quit using what we now call napalm; back in those days it was called phosphorus gel because it had phosphorus in it and it stuck to whatever it hit. But it was a single stream. You only had about seven seconds of fuel, and you'd fire it, and by the time you got on target you've already shot up everything you had. We didn't like it.\n\n\"The guy in charge and the rest of us came up with this mixture of diesel fuel and high-octane gasoline. It burned about thirty-four hundred degrees Fahrenheit, and it wouldn't burn a body up, but it would catch your clothes all on fire and take all the oxygen out of the air. That's what he [the enemy] died of: suffocation.\n\n\"You had to get within about twenty yards of a pillbox, with machine-gun bullets kicking up around and everything else. I don't know how I lived; I have no idea. I was just reading this morning somewhere in an interview that I had where I mentioned these thoughts come and go, and some of what happened that day I never have remembered. But at one point I do remember I was in a ditch. They had ditches that connected so they could go from one pillbox to another and still stay below. I was crawling up one of those ditches, and they were shooting at me with a Nambu [Japanese machine gun], and the bullets were ricocheting off my flamethrower.\n\n\"It was just like a jackhammer, you know, _drrrrrrrrrrr._ I guess I had enough sense to know if I backed up, I'm going to get in trouble. So I went forward, and when I got forward, I was at a point where he couldn't lower his guns far enough down to get me, so I got him. He was in a pillbox. You had to hit the aperture. The fireball is probably twenty foot in diameter, so it's a great big ball of flame. It would roll clear over the pillbox, but a lot would penetrate the aperture, giving you time to get up there and throw a satchel charge or a pole charge and blow the pillbox up.\n\n\"I've jokingly said over the years those marines back at headquarters apparently didn't like me because not a one of them ever volunteered to bring me a full flamethrower. I had to go back and get them myself, each time. When it's empty, you just roll out of it and git, go back and get you another flamethrower, and go back again.\n\n\"I got seven pillboxes with six flamethrowers. How many in each one? I have no idea. I guarantee you I wasn't counting, but somebody reported in some piece of literature that there were twenty-one Japanese in those seven pillboxes. I never got a scratch that day. One time the men in one pillbox came out and came running straight at me. I don't know whether they had run out of ammunition or whether they had decided that as a group they could get me. As they came running toward me with their rifles and bayonets poised, they ran straight into the fire from my flamethrower. As if in slow motion they just fell down.\n\n\"I was out there alone: Waters was back taking care of the flamethrowers, and the riflemen were laying down fire. One of the four was my pole charge man, so once I got the thing burned out, he would grab a pole charge and run forward with that and blow the thing up. He survived. His name was Schlager; I don't remember his first name.\n\n\"That was February 23, exactly the same day the flag went up. I lost two marines there that day who were giving me protection, and when I wear the Medal of Honor, which I do on many special occasions, and I'm giving a talk, I make a point that the medal really does not belong to me. It really belongs to those guys because they gave their lives. I do not recall their names.\n\n\"I got wounded March 6, a leg wound from shrapnel. I've always said it was a piece of our own; I still have it. I was really behaving myself at the point. I was in a foxhole that somebody else had dug. It was very shallow, and I couldn't get my body in it. I got my upper body in, but my knees were doubled up in the air and a piece of shrapnel come along and caught me just above the left knee. I called for a corpsman because I didn't know how really severe it was. He came and pulled the shrapnel out and gave it to me. He bandaged me up, gave me some morphine and put sulfa on it and said, 'I'm going to write a ticket and get you safely back.' And I said, 'I'm not going back.' At that point, Vernon was still living. I said, 'I'm not going back.' And he said, 'Well, I put a ticket on you, you have to go back.' And I said, 'You better take it off me, or I'll just tear it off. I'm not going.'\n\n\"So he said, 'OK, if you're that stupid.' He just let me go, and a few hours after that Vernon was out in front of me\u2014he was also just a rifleman at that point\u2014and the Japanese fired what they called a knee mortar. It came over a ridgeline and hit him smack-dab in the center of the helmet, never knew what hit him.\n\n\"After we got through the pillbox area, I was more of a rifleman than a flamethrower for the rest of the campaign. The next big flamethrower operation was right at the end, probably the thirty-fourth day. We had reached the other shore, and we were down in beach area securing the place, and they captured a Japanese who supposedly said there was a cave containing three hundred Japanese. It was on the side of a hill that went down to the ocean.\n\n\"My CO asked me to seal that cave. As a rifleman I'm still carrying two satchel charges on each shoulder. They were like satchels that you would carry to school, canvas bags with a strap on it and you'd just hang 'em over your shoulder and take off. They weighed eight pounds each. There were eight blocks of composition C2 in each one. They gave me some rifle protection so I could advance, throw the satchel charges in, and blow it up. It was a huge hole. Big.\n\n\"They got to shooting, and I'm getting my satchel charge, and I put only a ten-second fuse on it because I wanted it to explode pretty quickly because I don't want them to throw it back at me. So I ran up to the cave and threw the thing like you were throwing a softball, a big underhand throw. I sailed that thing into the entrance of the cave and ran back on the beach, and just as I hit the deck, I heard a pop. And it didn't explode. The cap went off, but it had drawn dampness and didn't have enough power to set the explosive off. So I prepared the second one, and we went through it a second time. I threw it in there and ran back as hard as I could run, and I didn't even get to the deck that time because I had shortened the fuse a little bit and it popped. So I knew I had to have some new caps.\n\n\"The CO sent me back to headquarters for a new box of caps, and I came back and prepared a third charge. I went up there and throwed it in, and boy, when it went off that time, it set the other charges off, just lifted the side of that hill off the mountain and sealed the cave. We never knew whether there was anybody in there or not.\"\n\nThe island was declared secure on March 27.\n\n\"We came off Iwo April 1 and went back to Guam. When we got back, the interesting thing, at least to me, was they had changed the whole method of training. We had been jungle fighting before; now we're going to go street fighting.\n\n\"Scuttlebutt was running around we're going to Tokyo. They'd built up some false-front buildings with windows and doors in them and a street between two houses. They were training us how to get through a window, how to go through a door, how to approach a house, that kind of stuff. We knew we were going to a city, we didn't know where, so the dropping of the atom bomb ending the war was a big relief.\n\n\"You know, your averages do run out and with two hundred fifty thousand civilians the Japanese had trained, women, children twelve and up, with every kind of a weapon you could think of, an ax or sharpened bamboo or whatever; they were going to fight us to the last person, so I'm not sure my number wouldn't have come up.\n\n\"I was on Guam when I was ordered back. The first sergeant said General Erskine had requested my presence in his office, and of course, as a lowly corporal, you go quickly. He congratulated me. I was going to be sent back and recommended for a medal. He never did tell me it was the Medal of Honor, and even if he had, I don't think it would have meant a thing. I didn't know it existed.\"\n\nWilliams was flown to Hawaii, where he was given a seat on a planeload of prisoners of war returning from camps in Japan. A seat came open when one of the prisoners died.\n\n\"Those people did not even look human. I'll never forget that as long as I live. It was just unbelievable, the condition of those marines, right out of prison camps in Japan. Some of them had weighed one hundred sixty or one hundred eighty pounds, and now they were down to eighty or ninety pounds. They just looked like a skeleton with skin drawn over it.\"\n\nHe went home from San Francisco by train to Fairmont, where he picked up his mother, Rennie, and his girl, Ruby, and went on to the White House, where he and twelve other veterans\u2014three Navy and the rest of them marines\u2014received the Medal of Honor out on the lawn from President Harry Truman on October 5, 1945.\n\n\"There were at least four from Iwo, myself and Jack Lucas and George Wahlen, a corpsman [Chapter 16]. They're still living. Of course my mother wasn't scared, but I was. I think I was more scared there than I was on Iwo.\"\n\nWoody and Ruby got married twelve days after he received the medal. They were to have two daughters and five grandsons. He left the corps that November and in January went to work for the Veterans Administration, where he stayed thirty-three years, looking after veterans from West Virginia.\n\n\"In 19 and 62 my life changed because God took hold of me and turned me and headed me in some other direction, and I became a lay speaker for the Methodist Church. If a preacher is on vacation or gets ill, we replace him.\"\n\nHe became chaplain of the Congressional Medal of Honor Society in 1986, replacing Desmond Doss, and served into 2005, when Woody stepped down.\n\nHe was active throughout in the Marine Corps League, serving as commandant and in other capacities. He also served the Veterans of Foreign Wars and the American Legion and helped raise money for scholarships for the children of slain marines or league members who had died.\n\nWoody also raised and trained horses for thirty years, first Morgans and then American saddlebred. A grandson took over when Williams grew too old to continue.\n\n\"West Virginia has the record of having more people per capita serve in the armed forces than any other state. It's always been a very patriotic group of people. For whatever reason, few West Virginians were involved in the Civil War. The line goes right through us, North and South. But patriotism has always been one of the high-water marks of West Virginia. Many cities put on tremendous parades for Veterans Day and Armed Forces Day. Every year I'm invited to speak.\n\n\"In fact, dating from the Civil War, West Virginia has had sixty-eight of her sons receive the Medal of Honor. I'm the last one left. If I could write my own epitaph, it would probably say, 'Here lies Woody Williams, who loved people and the United States of America.' \"\n\n**_Medal of Honor citation_**\n\n_For conspicuous gallantry and intrepidity at the risk of his life above and beyond the call of duty as demolition sergeant serving with the First Battalion, Twenty-first Marines, Third Marine Division, in action against enemy Japanese forces on Iwo Jima, Volcano Islands, on 23 February 1945. Quick to volunteer his services when our tanks were maneuvering vainly to open a lane for the infantry through the network of reinforced concrete pillboxes, buried mines, and black volcanic sands, Corporal Williams daringly went forward alone to attempt the reduction of devastating machinegun fire from the unyielding positions. Covered only by four riflemen, he fought desperately for four hours under terrific enemy small-arms fire and repeatedly returned to his own lines to prepare demolition charges and obtain serviced flamethrowers, struggling back, frequently to the rear of hostile emplacements, to wipe out one position after another. On one occasion he daringly mounted a pillbox to insert the nozzle of his flamethrower through the air vent, killing the occupants, and silencing the guns; on another he grimly charged enemy riflemen who attempted to stop him with bayonets and destroyed them with a burst of flame from his weapon. His unyielding determination and extraordinary heroism in the face of ruthless enemy resistance were directly instrumental in neutralizing one of the most fanatically defended Japanese strong points encountered by his regiment, and aided vitally in enabling his company to reach its objective. Corporal Williams' aggressive fighting spirit and valiant devotion to duty throughout this fiercely contested action sustain and enhance the highest traditions of the United States Naval Service._\n\n_Harry S. Truman  \nPresident of the United States_\nPART TWO\n\nIron Mike's Posse\n\n**I met** Glenn Buzzard, Domenick Tutalo, and Pete Santoro at a drill instructor's reunion in 2004, when I went down to Parris Island, South Carolina, to do research for a book about the place. That was where I met Iron Mike Mervosh, a veteran of Iwo Jima who had also served two tours as a drill instructor at Parris Island. Neither Glenn nor Domenick nor Pete had served on the drill field, so why were they there? They all had been in the Fourth Marine Division, had served at Iwo Jima, and had known Mervosh back then, so the drill instructors' reunion offered them a chance to get together as well. Mike lived in California, Glenn in Ohio, Domenick in New Jersey, and Pete in Massachusetts. They doted on Mike, who was profiled in _The Few and the Proud,_ a book about drill instructors that grew out of my visits to Parris Island and the Marine Corps Recruit Depot in San Diego, the two places where marine recruits are trained.\n\nMervosh saw combat in three wars, holding every enlisted rank from private to sergeant major over a thirty-five-year career. He was the Fourth Division middleweight champion until wounds suffered on Saipan and Iwo Jima ended his boxing career. He holds Navy commendation medals for action on Iwo Jima, in Korea, and in Vietnam, a Bronze Star for action in Korea, and three Purple Hearts. He served more than nineteen years as a sergeant major in battalion, regiment, brigade, station, base, and division. His last assignment before retiring on September 1, 1977, was Fleet Marine Force Pacific Sergeant Major\u2014the largest field command in the Marine Corps\u2014over eighty thousand marines.\n\nI found Buzzard, Tutalo, and Santoro to be sensitive, self-effacing, thoughtful men whose stories deserved to be told. They're not really a posse, of course. They're just three old pals, each of whom knew Mike on Iwo Jima and who got acquainted and became friends as a result of attending reunions in the years long after the war.\n\nCORPORAL GLENN BUZZARD\n\n**Machine Gunner  \nTwenty-fourth Marines, Fourth Marine Division  \nTwo Purple Hearts**\n\n_I didn't have to shave. I wasn't old enough._\n\nStanding in the yard of his Ohio home in August 2005, Glenn Buzzard holds his dress uniform in one hand and, in the other, the thirty-eight-caliber pistol his brother Erett sent him. He purchased his dress uniform when he graduated from boot camp in 1942.\n\nThe decorations on young Glenn Buzzard's blouse include a Purple Heart with one gold star, signifying two recognitions. In the center is a presidential unit citation with a star, again indicating two citations, and an Asiatic-Pacific Campaign ribbon with three stars, signifying four battles. On the left shoulder is the Fourth Marine Division patch, the ruptured duck, an eagle in a circle, signifying combat experience.\n\n_I spoke with Glenn Buzzard several times over a period of months. We met at a few different reunions, and I visited him and his wife, Needra, at their home in Ohio. One time he told me how Pretty Boy Floyd, the notorious bank robber and killer, had been shot to death on October 22, 1934, by the authorities on the farm across the road from his father's place in East Liverpool, Ohio, just two months before Glenn himself came there to live with his dad. \"His [Floyd's] car had run out of gas, and he walked a short distance and came up and asked this older couple for a meal. Back in them days they would feed you.\" Floyd wanted a ride to Columbiana and was talking to the couple about it when the authorities \"got wind of where he was at and surrounded the place,\" Glenn recalled. \"He went out the window and down past the corncrib and the barn, and that's why them two buildings were all shot up. And then he got out into the field, and they finally hit him and got him down, and then, according to what I hear, they didn't give him any sort of a chance, just kept shooting at him while he was laying there on the ground. They wasn't going to take any chances. My dad was home, working, and he heard gunfire and a hell of a commotion. By the time he got over there they wouldn't let nobody around, naturally, but then they brought his body right up to the edge of the road to be picked up. They drug it up there, and he was laying there on the ground, my dad said, riddled with tommy guns. He was a modern-day Robin Hood is the way it's been put. He had killed people, no question.\"_\n\nGlenn Buzzard was born in Chester, West Virginia, on December 15, 1925. His mother died in childbirth just before he turned five. There were two other boys, and his father was a bit of a hell-raiser, as Glenn put it, so \"an old maid\" named Nancy Lee Conkel took him home to live with her in Canonsburg, Pennsylvania, in the spring of 1931. His older brothers, Arnett and Erett, were taken in by others. \"People didn't keep track,\" Glenn recalls. \"Today you couldn't get away with that, but back in them days people farmed kids out and everybody was happy. Nobody stuck their nose in it. I know it's farfetched. The Depression was on, and Dad had to move out of his farm. Dad was a wild fella. He'd get out and get in fights. He was pretty rowdy. He had moonshine. But he never mistreated us kids.\n\n\"Nancy had a place across from my dad's, but she just came down in the summertime. She had a big old Buick, big wooden steering wheel on it. She was just a little person; she'd sit and look out through the steering wheel. She and my dad knew each other; maybe he farmed some of her property. They was just friends. She was a businessperson, as they called her, and never had been married.\n\n\"My mother died in November when I was four, and I turned five December 15. Because he knew the authorities would be coming out in connection with her death, Dad got rid of the moonshine still he kept upstairs. He kicked out the back of the gable end of the house and dumped everything, sour mash and all, out there to the chickens and the pigs and the ducks. He had to dismantle the whole still. I don't remember that personally, but my brother Arnett is eighty-seven this February [2007]. He's still got a real good mind, and I went out of my way to ask him questions. He said the pigs were wallowing around drunk.\n\n\"I was with Nancy till I was thirteen, about eight years. I'd come down in summertime and stay for a week with my dad or with her relatives in Chester. I went to school in Canonsburg. She was associated with the WD Wade Company and sold women's clothes out of the back of her car. She would go to Pittsburgh and buy the stuff wholesale. I remember she had big suitcases full of jewelry, men's jewelry: watch fobs\u2014those were a big seller\u2014stickpins, sleeve garters. That's how she made a living. She'd buy everything by the dozen and go door to door, sell one at a time. She had magazines, books, and cloth samples. Corsets was a big one back then.\n\n\"I delivered stuff for her after school, and on weekends I'd run clear across town. She kept her table completely set for eight people at all times, knives and forks and everything laying just so, and what she done she lay back one corner of it and we eat there every day. On Sunday I had to go to church. I wasn't allowed out to play because it was the Sabbath. She was strict about that.\n\n\"Long dresses was in, naturally, and she went up a stepladder to get up into the cupboard\u2014I think it was my birthday in December\u2014and she got her heel in her dress and pulled herself off that ladder. She went to the hospital, probably there for a couple weeks. I was home alone going to school.\n\n\"Canonsburg wasn't a big town. Everybody knew everybody. You didn't lock your doors or nothing. There was only two police. Attis was the chief, and Haney was his assistant. He was a little short fella and wore those knicker pants with big black leather boots, and he rode that motorcycle all over town. Down below Canonsburg was a children's home, Morganza.\n\n\"Haney come out on his motorcycle and said, 'You know they're coming to get you in the morning, to take you to Morganza. Don't you have a father somewhere?' I said, 'Yeah. Down in Chester, West Virginia.' So he took me and my dog Spanky back down to the station and put me to bed up on the fire truck on them hoses, and then he called Chester and got Doc Lyons, the chief of police in Chester. It was the dead of winter, and Doc Lyons walked up Middle Run Road to Cunningham's house. He knew Cunningham and my dad knew each other. My dad was at work in Weirton, West Virginia, in the mill, but Leonard Cunningham was a good friend of my brother Junior, and he says, 'I'll go get him. I know where it is.' So he did, in the middle of the night. He brought me down there, and I went right with my dad. It sounds complicated, but that was how things played out.\n\n\"I was in eighth grade, and my dad had remarried and had three more children, and me and my stepbrother would ride this red horse, a gelding named Charley, to Clarkson school and then tie its head up so it couldn't bend down to eat and let it loose. It would go back to the farm, and Dad would work it all day. Then we'd walk home, four or five miles, after school.\n\n\"I quit school at fifteen and went to work in a pottery, carrying molds. I was running around, and I got my first car, a Model A Ford truck. It would pull Logan's hill in high gear. That's how you could tell if it was a good vehicle. I didn't have to double clutch to get over neither. I gave one hundred fifty dollars for it.\n\n\"Everybody older than me was going to the service, and I was partying and getting in at two o'clock and having to go to work at six, and I got tired of that. So one day I just went to Pittsburgh and enlisted. There was a local football player who joined the Marines and got shot up on Guadalcanal. I suppose that had some influence on my decision to join the Marines. Anyway, I took my old Ford truck and went to Pittsburgh, found the post office building, enlisted on Monday, and they said come back up on Tuesday with my birth certificate and paper signed by my dad and the chief of police, and you'll go right straight to Parris Island. It was August of '42. My age, which was sixteen, never come up. Monday they gave me some physical and stuff and everything seemed be OK. Only thing was, I was humiliated to death because I never wore undershorts.\n\n\"The only time my age came into play, I was in boot camp on Parris Island, and the drill instructor, his room was right at the end of our big long wooden barracks, he called me into his room. I'm sure he wasn't too polite about it, but anyway, he just said, 'Do you want to get out of the Marine Corps?' I said, 'No.' He said, 'OK, go back to your bunk.' I suppose they had to ask that question to keep it legal, because somebody had found it out.\n\n\"I started boot camp in August, took the train to Yemassee. We got off there, got in them trucks. It was dark, like they always say, dead of night. You had to stand up on the trucks. Once they got you in, it was so crowded you didn't need worry about falling over. Boot camp didn't bother me much. I didn't have to shave. I wasn't old enough. I was underage, but also, having the farm background, I was probably in better shape than most of them there. I was six foot, and I weighed one hundred forty-four pounds. I probably knew more for my age than most kids that had a mother and father did. We had a lot of boys from New York and West Virginia. They didn't get along at all. There was always a lot of trouble in our platoon. Consequently, when two guys got into a fuss, we all had to pay for it.\n\n\"When we got out of boot camp, we went straight to New River. We had rifles at Parris Island, but we did not get to fire them until we got to New River at Hadnot Point, all part of Camp Lejeune now. We trained with Springfields at Parris Island, but when we went to the range they gave us M1s. We lived in tents.\n\n\"We trained and trained, and they built us up, and when they had enough people, they called us the First Separate Battalion. Mike Mervosh was in it too. We got six days' furlough, and when we came back, they put us on a troop train to Camp Pendleton. Then they brought other outfits in, and in the spring of '43 we became the Fourth Marine Division.\n\n\"When they had enough battalions, they'd make a regiment and then it takes three regiments to make a division. The Twenty-third, Twenty-fourth, and Twenty-fifth regiments became the Fourth Marine Division. [A platoon contained about 45 men, a company had 250, a battalion contained 1,100, and a regiment accounted for 3,300, or three battalions. There were 20,000 men in a division.] Plus there were headquarters companies, artillery, mortars. We were in machine guns. We didn't even train with the riflemen until we went on maneuvers; then we fit in because that was the way you were going to fight.\n\n\"That's how I got to know people in A Company, B Company, and C Company. We had three platoons in D Company, which was a weapons company. They would take the First Platoon in D Company, put it with A Company, machine guns. Second Platoon, B Company. I was in the Third Platoon, and we'd go to C Company. Next time we went out they'd maybe put us in a different company. That's how I got to know those guys. But after Roi-Namur they got so low on officers they couldn't shift us around. So they changed that. Put First Platoon in A Company and so on. I was in C Company from then on.\n\n\"You had four squads in a platoon, two sections, a gun in each squad, so that's four heavy water-cooled machine guns and four light machine guns. You used the light air-cooled machine gun in the daytime; then at night you set up your air- and your water-cooled machine gun and you'd cross your fire, set up interlocking fields of fire in a solid wall right down the line. The heavies had a big water case. We trained with them a lot. We had carts to pull them on. The gun weighed ninety-one pounds, and we used it strictly at night. It was always right behind you on a jeep or something. We carried the air-cooled guns and had Browning automatic rifles, the next best firepower. You needed firepower. That was the name of the game.\n\n\"The BAR was lighter than our light machine gun, but it was a damn good weapon. It was part of the rifle platoon. They let us fire the machine gun from the hip. It would work, and I did it. You had to have an asbestos glove which they gave you. We'd tape that asbestos glove right on that barrel, and then we wouldn't lose it. Then you'd put an ammunition belt around your neck and tie it on to that gun. The worst thing was you couldn't put a whole belt through it because you couldn't control the belt feeding into it, so you cut the belts, which were canvas. The fifty calibers had metal belts. We cut it to maybe thirty shots. You only fired when you really needed it.\n\n\"I was a machine gunner all the way through. I never carried anything else. We had side weapons. They tried shotguns, and they tried carbines. The shotgun was a good weapon because it scattered, especially in jungle, but was hard to keep clean, and the shells were paper, so when they'd get wet, they'd swell up and wouldn't go in the chamber. When I became a machine gunner, I needed a sidearm. They came out with forty-fives, but they didn't have enough, so they give them to the squad leaders. Junior, my brother Erett, sent me a thirty-eight through the mail along with a Bible. The thirty-eight had no grips. It was up in the attic, just layin' up there, a six-shot Smith and Wesson. I knew it was there, and I asked him for it. I was on liberty somewhere, and I bought an old black holster for it. I had to get special permission to carry it. You peart near had to be in combat because I couldn't wear it with my dress uniform any way, shape, or form. I must have bought the ammo for it in Hawaii. They were lead bullets, which would splatter, which was good. It didn't have no handle grips, and I made a pair of wooden ones and taped them on. I carried it through all my campaigns.\n\n\"The first time I used it was in Saipan, on a banzai attack when a Japanese tripped over my machine gun in the dark. Needless to say, somebody killed him. I don't know whether I did or not. I fired at him with that thirty-eight. I know I had it going at him because it was close quarters. I couldn't get the machine gun on him, but whether I or somebody else did it, I kinda thought maybe I did it, but then it's nothing to brag about, taking a man's life. They was there just like we were, probably didn't know any more than we did. Anyway, that's life, that's the way wars are. People that know everything ain't in the war. They're sittin' back somewhere.\n\n\"I was wounded later that night in a mortar attack. Somebody took the thirty-eight off me because I had to go back to the beach next day on an ammunition truck to get patched up. It was in my back and shoulder blade area, but it was pretty well cauterized. Adrian DeWitt from New York somewhere pulled that shrapnel out. It was long, about an eight-inch sliver, and it was spent to a good degree. Otherwise it would have took my shoulder off. Another thing saved me on that one was my poncho. It was all tore up, so it took a lot of the shrapnel down in the kidney area. But I did get my thirty-eight back.\n\n\"For the trip to Iwo we boarded ship at Maui, had maneuvers at one of the other islands, then went to sea again. Then we went to Guam but didn't get off. Finally they told us where we were going.\n\n\"It was pretty devastating when we landed that morning, February 19, 1945. My first thought was: Why the hell didn't they know the island was this way, because of the sand and stuff like that? We wasn't just fighting the Japanese; we was fighting the elements, the island itself, the make of the island, worse than anything else. The devastation was terrible, bodies everywhere. They put nine thousand marines ashore the first day. You can figure it out yourself, nine thousand guys, and they weren't more than three hundred feet off the beach. They couldn't get far. It was devastating, just carnage everywhere.\n\n\"Not knowing about that sand was a big failure by our intelligence people. The other was they didn't realize the Japanese people could go underground that far. The effect of the bombing was almost nil. We thought it was doing a hell of a job, wiping them out every time we dropped a bomb, but we weren't.\n\n\"We landed up at the boat basin, as far away from Suribachi as you could get, in about the third wave. Everything didn't go according to Hoyle. They were trying to get nine thousand people on there in eight or ten hours. I'm not trying to take any glory from any wave. I was not in the first, but when I got there, the first wave was laying right there, waiting for us. Bodies everywhere; equipment couldn't move. The main thing was to get out of that boat and get the hell off that beach, three hundred feet in if you could. You're still right out in the open. It wasn't any safer, with every firing mechanism pointed at that beach. Nine thousand people they said put ashore the first day. That's a hell of a lot of people; it was elbow to elbow.\"\n\nWhy wasn't the landing force annihilated?\n\n\"Just sheer determination, I guess, or maybe they didn't have the ammunition to expend on a full-blown assault. Maybe they were conserving ammo, but I don't see any reason why they couldn't have blowed us clear off that beach. Only thing is we just overwhelmed them, I suppose, bodily. It boiled right down to the individual person. The company commander wasn't telling him what to do; the first sergeant wasn't telling him what to do. If Kuribayashi'd had more weapons than we had, we probably wouldn't have made it. But say eight or ten of you came to a hole in the ground and it was a pillbox, and if you was getting trouble with it, you congregated on it and took it out. You couldn't call for tanks, couldn't call for rockets. It was just sheer determination, small-arms initiative.\n\n\"The Quarry had guns sticking out of it, and the friendly fire damage was about the worst I ever seen. They were trying to get at them guns and they'd call in an air strike, and our kids would just roll dead off them hills because the planes hit short. We had frontline panels, but things got mixed up, you know what I mean? It was right out in the open, just like a theater, and we was off to the left and a little bit inland, and we could look right around. When we first went in, the Japanese guns were right above us, but they were six-inch guns, not for use on us but mostly to shoot the boats. They had plenty of other stuff. They didn't even have to point a gun. All they had to do was pull the trigger because it was all zeroed in. So the Japanese'd send up some guys, and they'd go pull triggers, and then the guys'd disappear. Then they'd send up some more because they had the people there, and it was all under cover. Most of the guns were in caves, and they slide right up to the opening, fire, then slide back. They'd put camouflage over them, brush, netting, whatever they had.\n\n\"The strafing planes hit the marines. A frontline panel is an orange plastic sheet you can roll up. One guy in every squad had to have one, and when they'd call to put out the frontline panel, an air strike was coming in, it was up to that one guy to run out there as far as he could beyond the front line out in no-man's-land. It was dangerous as hell to try to roll that thing out. It worked pretty good from up in the air. You could see that thing, but short firing was bound to happen especially from the face of that cliff because they'd come in off the water.\n\n\"When an air strike wasn't happening, the company commanders would be sending guys like Domenick Tutalo with flamethrowers up there to try to get into that cave and seal it off. Machine guns would be firing right into the opening, trying to keep them down till our guys worked up there, and then, all at once, an air strike would come in. They were bound to get hurt. I think that was the worst I ever saw of something like that. First, second day. Third day I'd say we'd probably taken those guns out of there, and we were past that. Starting to make the swing up the island. We didn't have to go inland too far before we started to swing north.\n\n\"The Fourth and Fifth were abreast, and the object was to move right across that short pork chop thing and secure that part and then swing to the right and to the left, that'd be Fifth Division on Suribachi, and then we went to the right, and that's where we spread out and needed some people, and that's when we brought in a few of the Third Marine Division.\n\n\"You didn't see too many Japs. Once in a while they'd run from one cave to another. You more or less seen their fire. You could see dust coming. As soon as we'd see that, we'd zone right in, and when we got up there, they'd be layin' there.\n\n\"The terrain got rougher and rougher because of the catacombs and stuff where the water had washed in amongst it over the years. Some places you could step over a crack and you'd see a big gap deep down in there. Or you'd go around the corner and they'd be standing right there face-to-face. Whoever shot first was the winner. I saw one marine shoot another marine bone dead right in my squad because he went around this way and the other went around that way and, it was just like I said, you don't have a split second. You just pull the trigger. Shoot first. Whoever does, they're one's going to win. We had to take the guy that shot the other marine, take him clear out because he just went berserk.\n\n\"The first day of March I was wounded and evacuated out to a hospital ship. It was shrapnel like I'd got in Saipan. I was taken aboard ship and cleaned up. They picked that stuff out of me. It was mostly like buckshot. One thing about shrapnel: If it hits you the right way and it's a little bit on the spent side, not with full force, it'll burn you, and when it burns you, it cauterizes the wound. That's what it did to me on Saipan because Adrian DeWitt pulled it out of my back and they put some sulfa powder on it and I never had a minute's trouble with it. But the corpsman would bandage it every day for me, check it, and I stayed in action.\n\n\"My big problem on Iwo came from a concussion, not that shrapnel. I was in a big hole with ten or twelve other guys, and a shell came right in there. Near as I can remember, I was the only one came out of there alive. We were in reserve. They kept two platoons on the line, and C Company just happened to be in there getting a rest. Give you a little chance to do what you wanted, sit down and cry, or smoke, whatever.\n\n\"We were in there grouped up. You didn't group up because it was a bad thing to do, but we did. Otherwise there wouldn't have been as many killed by one shell. One hand grenade'll get you all, if you don't stay spread out. Otis Boxx was my gunner at the time. As people get bumped off, people move up in command. I had moved up to squad leader, and I had to give up the gun for a few weeks. So Otis Boxx from Florida was the gunner. The only thing I really remember is that all that was left of his head was his lower jaw. It was just settin' there. Never moved. Just settin' there.\n\n\"I was shocked, no question about it. I come up out of the hole, and somebody grabbed me, and I was just blood everywhere. They thought I was hit worse than I was, but they started getting my dungarees off me and seen that I wasn't really hurt, but my mind was gone.\n\n\"I couldn't hear. I remember giving my thirty-eight away to Elmer Neff. They took me back to the beach and took me out to the big boat, and first thing I remember is a corpsman waking me up. Or I came to, apparently. I was talking with him, and I said, 'Where's my clothes?' He said, 'We'll give you clean stuff.' I said, 'Well, get 'em for me. I'd like to get some clothes on.' So I did. Then daylight came, and they were bringing casualties in. They had a stairway up the side of the ship to bring casualties up. They couldn't bring them up the cargo net, but there was a cargo net down there, and I clumb down that cargo net and got in the front of that boat and went ashore. Nobody asked any questions.\"\n\nWhy did he go back?\n\n\"I have no idea. I have no idea. They'd dug out shrapnel, give me sulfa, and I had a hell of a headache. My vision and my hearing were bad. When that shell hit, I was out on my feet. A lot of people were killed that way, by concussion. It would just roll them along the ground like a ball. They was dead. Everything confined in that little shell goes out when it explodes, so therefore you go with it.\n\n\"I got my thirty-eight back. Elmer Neff had got killed, and somebody picked it up. A day or so later just out of the clear blue sky somebody walked up to me and said, 'Hey, I got that thirty-eight off Elmer when he got hit, so here it is.' I walked right back up there and found out where C Company was, I kept asking, 'Where's C Company?' and I finally found where they were, and I went right back to work.\n\n\"I'm not sure who was company commander at that time. It might have been Mike Mervosh. I just don't know who had got killed and evacuated in the line of officers. Mike could very well have been my company commander at that time. We knew each other, hell, yes. We met in New River. He was in First Separate Battalion. We weren't in the same squad, but we were all in C Company. You'd get to know maybe fifty guys. Mike was my section leader at first.\n\n\"He became company commander on Iwo Jima. A company commander could have been a major. Mike didn't become a major. You just became the company commander even if you were a buck-ass sergeant because there was nobody else to do it. Then, when you went back to Maui, they'd call you out on the parade deck and have a pretty nice ceremony out of it; they'd call everybody out in formation and either give them their promotion or their Purple Heart or whatever. You didn't know you was getting a Purple Heart because the only thing that gave you a Purple Heart them days was your medical records. The first sergeant wrote in the book you were treated, and you were in line even though your wounds were not serious. They had what they called the walking wounded. And that's the way you were put: WW. And there was WE, wounded and evacuated. So first I was WE or walking-evacuated, and then I was WW or walking wounded. When I come back off the ship, he put me in as WW.\n\n\"Day to day you'd have riflemen go out in front. The number one guy would go so far and keep track of the guy on his left and the guy on his right. There wasn't no talking going on. It seemed like we had to put bait out there to draw their fire. Then we'd find out where it was coming from and focus on that. If their fire was severe enough and determined enough, we had to call for more support. Rockets were a great thing. The Dodge power wagon had these things mounted on them. They'd bring up seven or eight in a bank, and there was probably fifty rockets on each of 'em in a canister and they'd go _chu chu chu,_ a hell of a noise, and boy, they just blanketed the target area. They didn't have to be too accurate. And grenades were good. You could damn near throw a grenade ten feet, and it would go down a crevice and you'd hear somebody scream or moan or groan or something. You knew somebody was in there. Grenades were a good thing, and we used a lot of them.\n\n\"If your skirmishers found a cave, it would take a considerable amount of time to work your way up to its mouth. The machine gunner would cover the front of it so you could get your flamethrower close enough to shoot flame in. As long as you're firing into that cave, you're keeping their heads down, allowing the flamethrower time to get close, and then the demolition man had to be right there within arm's reach to throw the damn satchel charge. Most of them are still in those caves. We killed more accidentally than we did on purpose, put it that way. Sure we was trying to kill 'em all, but we didn't realize when we sealed a cave how many we killed. We killed thousands.\n\n\"Some days we didn't move. One side couldn't move past the other. We had to keep order, especially when the island started getting smaller as we moved up. They'd spell us off, give us a little rest. I can't remember sleeping, but I'm sure I did, and I don't remember ever going to the bathroom. I'm sure I did. There's things you don't remember. I think you're programmed, and you're just like a zombie, but you have to pay attention to your surroundings at the same time. Then you live longer. I think sometimes guys would just forget where they were at, and bingo, they were gone.\n\n\"Believe it or not, as a person on the front line you had it better than them in the rear. You were relaxed, you might say, because you knew where the enemy was. They were right in front of us. But if you went into reserve and were trying to crap out like we were in that hole, there's enemies all around you all the time. They can lob a shell at you or get behind you through tunnels going everywhere. You didn't know where the hell they were. That was one small advantage of being on the front line: We knew where they were.\n\n\"I was nineteen on Iwo. I came back from wounded and evacuated to walking wounded and finished out the campaign with no more trouble than an infected fingernail.\"\n\nGlenn married his wife, Needra, in 1949, and they live today in Hubbard, Ohio. They have five children. Glenn farmed most of his life. Did his survival of the battle of Iwo Jima affect the way he lived afterward?\n\n\"I don't know. I always tried to do the right thing. I didn't get in too much trouble when I was a kid. I never drank to any extent. Never ran around much. That's why I went to farming. That was a solitary life. I come home from the service, and I broke three vertebrae in my neck falling off a load of hay. That could have killed me right there. I got hurt worse falling off a load of hay than I did in four campaigns in the Pacific. I had to wear a brace eighteen months. I was seventy-nine years old, last year, I'm eighty now, and I fell down eight steps backwards there in the cellar. Could have broke my neck or hit my head on that cement.\"\n\nWhy did he survive the battle?\n\n\"When I stop to think about it, I have no idea. I'm a Christian. I believe in God. I believe in the hereafter. But why me? I was no better or no worse than Cooksey or Elmer Neff or Bowman or any of those guys. But yet they took it, you know what I mean? They're gone. Why, I don't know. I can't answer it. I didn't do anything any better than they did. I wasn't a better scholar, I wasn't a better . . . anything. They say every hair of your head is numbered. Well, the fellow who's got that kind of account is in charge, and if he's in charge, then maybe he can answer that question. I don't think God created man to do this to each other, although men have been doing it ever since Christ walked the face of the earth. Otherwise I have no idea. I cannot answer it.\"\n\nPRIVATE FIRST CLASS PETE SANTORO\n\n**Boxing Champion; Two Purple Hearts  \nTwenty-fourth Marines, Fourth Marine Division**\n\n_In 1938, when I was twenty-one, I won the New England heavyweight amateur championship. I won the Golden Gloves and the Diamond Belt Tournament. I kept that belt in a drawer until I met my wife, Maria Lydia LesCault, and then I took the diamond out and had the jeweler make a ring for her._\n\nSantoro holds a sword he took from a Japanese soldier he killed during the battle of Roi-Namur.\n\nPete Santoro and his wife, Maria, also known as Bunny, on their wedding day in Norwood, Massachusetts, July 29, 1946. They had five children. Bunny died in 1997.\n\n_I was in Pete Santoro's home in Wrentham, Massachusetts, three days before he turned ninety on January 22, 2007. We talked about his life and his military service, and as I got ready to leave, I pulled out my camera and asked if I could take a photo. I saw a sword on the wall. Pete took it down and showed it to me. He said he had seized it from a Japanese soldier he killed one night during the battle of Roi-Namur. Ships offshore were sending up parachute flares. The enemy soldier came up to a foxhole occupied by Pete and two other marines and, speaking in Japanese, brandished the sword over them. \"I could see his feet,\" Santoro recalled. \"I just turned and shot him three times. He went down. There was blood on the handle of the sword. I wiped it off. He had no rifle. I was told later I couldn't keep the sword, so when I went aboard ship, I gave it to the chaplain, and he took it home for me, wrapped up with my name on a tag. It was at my sister's when I got home a couple years later. She said a priest had brought it over. It was the chaplain.\"_\n\nPete Santoro was twenty-seven, old for an enlisted man, when he fought at Iwo Jima in February and March 1945. \"I was born January 22, 1917, in Norwood, Massachusetts. By the time I got to tenth grade in 1933 I was sixteen, and my father said, 'OK, get a driver's license. You don't need no more school. You know how to read and write. Drive the truck.' My two brothers Benny and Charlie had the trucks. They worked construction. My brother Benny taught me to drive. My father didn't know how.\n\n\"My brother said, 'No, he's going to high school and play football. He can play football in college.' But my father said, 'You can't eat the football. Drive the truck.'\n\n\"My father was a bootlegger. He made wine, five or six fifty-gallon barrels a year, both red and white. He also mixed muscatel with California grapes. We'd go to the freight yard with one of the trucks and load up with grapes. My father taught me how to taste them. This one guy had all overripe grapes, and that was what he wanted. They were like raisins, sweet. They made the best wine. Most of the grapes were black.\n\n\"We'd drive the truck home, back it into the driveway by the cellar window, and they'd pass the grapes down. My job was to grind them up. The grinder was on a stand. I stood on top. We'd grind seventeen boxes of grapes for each barrel. My father would put a canvas over the cover so it would generate heat, let it all settle, and when the bubbles stopped, he'd drain off the liquid and put it in a separate container and let it ferment for a few months. People would come and buy it by the gallon, three or four dollars a jug. The priest would come down, and we'd give him a gallon, for the altar. Maybe that's how my brother Henry got mixed up in driving a moonshine truck for the Capone gang later on.\n\n\"Both my father and mother spoke broken English. His name was Pasquale. He came from the Abruzzi. My mother too. Her name was Carmen; they called her Carmella. Her father and two brothers were fishermen in Pescara, on the Adriatic across from Rome. They had boats. She and my father married and had some kids in Italy, and he came to the States, through Ellis Island.\"\n\nPete thought this happened in 1912, but Pasquale actually came in 1909, according to Pete's nephew Patsy Santoro of Branford, Connecticut, the son of Pete's brother Henry and the keeper of the family records, who was born in 1924. Patsy said Pasquale went to Elmira, New York, and worked with the railroad long enough to be able to send for two daughters, who came over and worked with him there, saving money until 1912, when they could afford to send for Carmella and the remaining six children, including Guarino, whom they called Henry once he got to the States.\n\nThrough a family connection, Pasquale got a job with Bird and Son, a floor covering and paper products company in Norwood and moved the family there. The Santoros stayed in Norwood, except for a six-month stint in Chicago in the 1920s, according to Patsy. It was here Pasquale had some contact with the Capone gang. \"The Italians were everywhere, like horseshit,\" Patsy observed.\n\nPasquale found Chicago sooty and unpleasant and so returned to Norwood, where he and Henry got to know associates of Capone's in the Reedville area outside Boston. This was how, both Pete and Patsy maintained, Henry wound up driving a bootlegging truck for gangsters.\n\n\"But then one day the truck got shot up by a rival outfit. They didn't kill Henry. They figured they'd be in big trouble if they shot the driver. They just wanted to ruin all the booze. That ended Henry's driving career.\n\n\"In 1938 my father and brothers decided to open a bowling alley, with roller skating upstairs, in Norwood. They called it Roll-Land. I worked construction during the day and put my skates on at night. I became a professional skater. I used be able to pick up a partner and skate around, do a deep knee bend with him in my arms. He'd hold a match in his teeth and light it on the floor while I spun him around.\n\n\"I started boxing around the age of fifteen. I had a friend I went to school with named Dewey Tronti. He was a welterweight, and he changed his name to Trunny Hogan because you'd get along better if you were Irish. He was pretty good. We trained at night.\n\n\"In 1938, when I was twenty-one, I won the New England heavyweight amateur championship. I won the Golden Gloves and the Diamond Belt Tournament. I kept that belt in a drawer until I met my wife, Maria Lydia LesCault, and then I took the diamond out and had the jeweler make a ring for her. We got married July 29, 1946, after the war.\n\n\"I had four pro fights after I won the belt, and then I had to quit. I was too small for a heavyweight. I was like Marciano. I wasn't as good as him, but I was pretty good. I fought at one eighty-nine. Marciano was a half a head taller than me. Same build, but he hit harder. He saw me fight when I won the New England. I beat this guy six foot six, two hundred thirty-five pounds. I saw his belt; it was right at my eyes.\n\n\"I won three pro fights, lost one. You know who beat me? Al McCoy. It was a six-round warm-up fight for him. He was going to fight Joe Louis, who was heavyweight champion of the world. McCoy took it a little easy on me because he knew I was just coming in out of the amateurs. I hit him a hard shot, and he come back at me to show who's the boss.\n\n\"I joined the Marines in November 1942. What happened was, after I had served three years in the National Guard, I got these papers telling me to report to the Army. I went to the recruiting office in Boston, and I found this Marine major and said, 'Sir, can I speak to you?'\n\n\"I told him I didn't want to go in the Army because my mother and father came from Italy, and Italy was fighting against us, and I had relatives in Mussolini's army. I'd said I'd be fighting my own relatives and I'd feel bad shooting at them.\n\n\" 'Oh,' he says, 'now I understand. Follow me, son.' He puts his hand on my shoulder, leads me into an office, passes me over to another marine, and says, 'I got a ripe one for you.' I went to Parris Island in November 1942. Boot camp was eight weeks. We trained with the '03 Springfield, but they gave us the M1 before we left. Eight shots, you just kept pulling the trigger. You had to crank each shot with the Springfield. We got on trucks and went to New River for a month and a half of infantry training.\n\n\"I was part of a unit they called the Second Separate Battalion, which was to become part of the Fourth Marine Division. We went by train to Camp Pendleton, California, in about March or April of '43. Then it was all ship-to-shore training.\n\n\"I was a rifleman. I would have been a squad leader, but I had a little trouble. A guy picked a fight with me, and I had to put him away. I was training boxers, but I couldn't fight because of my pro background. I wouldn't fight those kids; they were too green. I was in G Company, Twenty-fourth Regiment, Second Battalion. Iron Mike Mervosh, who I met back at New River, was in the First Battalion. The First, Second, and Third Battalions made up the Twenty-fourth Regiment. Mike was a good boxer. He used to spar with some of my guys. He was a middleweight. He won the Fourth Division middleweight championship, but then he got wounded and couldn't fight anymore.\"\n\nPete saw action at Roi-Namur, where he was slightly injured when a bullet pierced his canteen and grazed his hip.\n\n\"They just put a Band-Aid on it, but that kept me from going to Saipan. I was in the rear echelon as backup until we went to Iwo. At that time I was a squad leader, had twelve men. Four of the guys from the boxing team were in my squad.\n\n\"We found out aboard ship that we were going to Iwo. It was going to be a piece a cake, they said. It was a piece of cake all right. We got the cake. My outfit, the Twenty-fourth Regiment, Second Battalion, landed side by side with Mike Mervosh's unit. He was in the First Battalion.\"\n\nPete had no idea that his nephew Patsy had been at Iwo Jima a week before the invasion began aboard the light cruiser USS _Birmingham._ Patsy, who was a cook, said the ship stayed there and shelled the island throughout the battle before going on to Okinawa, where she was hit by a five-hundred-pound armor-piercing bomb dropped by a kamikaze. \"It killed all the doctors and the dentist,\" Patsy said. \"The corpsmen had to take over. They all deserved medals.\"\n\n\"The Japanese let the first wave come in without much firing. I was coming ashore in the second wave around nine-thirty when they opened up. I saw a boatload of marines take a direct hit. Bodies were flying all over. My heart went _boom boom boom._ You're ready to go in, but you see somebody right beside you get blown up and . . .\n\n\"Our company captain, Joseph J. McCarthy, who got the Medal of Honor for what he did on Iwo, says, 'Santoro, get your squad. We got to go around this thing here.' It was D plus three, and the Japs were zeroed in on F Company, under Captain Walter Ridlon, and they were tearing hell out of it.\n\n\"There were six or seven of us with my captain, McCarthy. I'm behind, trailing a guy, and I said, 'Wait a minute.' There was this big hill they were going around. I thought: I'm going to go over the top; I'm going to see what the fuck's on the other side. So I get up over this bank and look down on the entrance to one of those sulfur mines. Two Japs were lying there, crawling, so I put a shot in each one from up above. Then McCarthy shows up. He doesn't see me up there, and then, bing, bing, he says, 'I got them, I got them.' I was waiting for some more to come out. He spoiled it all. He didn't know they weren't alive. He just saw them and shot.\n\n\"We secured that whole area. McCarthy got the medal for killing two dead Japs. [As his Medal of Honor citation makes clear, McCarthy did a whole lot more than shoot two Japanese soldiers dead that day.] I got a Purple Heart.\n\n\"I was on Iwo till February 23, the day the flag went up. I was at the second airfield when I saw a hole in the ground, and I was trying to drop a rifle grenade in there when I got a bullet in the back. It hit me like a sledgehammer, tore up the poncho, and shattered an eight-round brass clip on my back. The powder didn't explode, but the brass of the shells shattered, peppering my back. It felt like bee stings. I couldn't move my legs. I thought: What the fuck? Jesus Christ. But if that bullet hadn't hit my clothes and that ammo, I wouldn't be here today. I managed to roll over as some guys came up, and I said, 'Hey, my back, my back.' And they said, 'What the fuck did they shoot you with, a shotgun?' I was all torn up. The X rays at the Quonset hospital in Guam showed a bunch of pieces, and they just left them in there. My wife was pulling that stuff out with tweezers for years afterward.\"\n\nIn 1975 Santoro's commanding officer, Joseph J. McCarthy, who retired a lieutenant colonel, wrote a To Whom It May Concern letter recommending Santoro for disability benefits. Naming Santoro as a private first class with G Company, Second Battalion, Twenty-fourth Marines, Fourth Marine Division, McCarthy described landing on Iwo with 257 men and receiving 90 replacements for a total of 347, only 35 of whom were able to walk off the island when the fighting ended.\n\nWriting about February 21, the day Santoro was shot, McCarthy said his company \"was suffering heavy casualties from fortified positions and bunkers.\" His right assault platoon was pinned down, and he was at his \"wit's end, trying to effect an advance of a few yards.\" He added, \"My entire area was also subjected to intermittent large caliber mortar and artillery fire and we were in direct line of fire from one or two high velocity weapons.\"\n\nAs the afternoon wore on, casualties increased until finally McCarthy assigned an assault squad to clean out some pillboxes that were holding back the advance. \"Private First Class Santoro was a member of this group,\" he wrote. The reason he remembered was that the action of the assault team \"led to the demolishing of the enemy positions and the advance of my Company to the nearby newly captured ground.\" Further, he said, the platoon leader was killed in the assault, and several enlisted men were wounded. \"I distinctly remember Peter Santoro was one of these men . . . he appeared to be paralyzed from the waist down, from gunshot wounds he received near his spine.\"\n\nSantoro was evacuated to the USS _Solace,_ but despite his injury, he volunteered to return to duty five days later. \"I was later wounded myself,\" McCarthy added. \"However, at the time of my evacuation on 6 March 1945, Private Santoro was still serving with the company in spite of his painful wounds.\"\n\nSantoro remembers: \"That slug scraped my vertebrae right at the waist. It was three pieces of lead. They took it out on Guam in a Quonset hut hospital. 'Lay down over here, Santoro. We got to take that piece of shrapnel out of your back.' The surgeon was there. They froze it with a needle. Clunk, clunk, they stitched it up. The surgeon goes, 'You want these?' What I want those for? 'Can I have 'em?' Yeah, you can have the fuckin' things. He had a canful of shrapnel. He had the whole bottom of a hand grenade; he took it out of a guy's jaw.\n\n\"They took me off in a stretcher to an LST. They picked up three stretchers at a time with a crane and a cable. I got on board, and they said, 'Move your toes.' I moved my toes. 'OK, you be good.' If I couldn't move my toes, I'd be paralyzed. Later I transferred from the LST to a hospital ship, the _Solace._ Two days later I started walking, crooked like, you know? They called thirty or forty of us back to the fantail. The colonel wants to talk to you. Then he's telling us how bad guys are needed for backup, with everybody getting slaughtered. He asked for volunteers, walking wounded, to go back to the aid station and wait for the next ship. They had too many critically hurt that had to go to Guam. When I stepped out, three or four more stepped out. That was it. So five days after I was hit I went back to rejoin the company.\n\n\"When I came ashore, the ground crew from the P-51 Mustangs were putting up sandbags near the beach. What was that for? Turned out there was a sniper up on the hill, hiding in the rocks lining the edge of the first airfield, Motoyama No. 1, and every night around dusk, about five o'clock, he'd pick off one of the guys between the airfield and the beach. We had people looking for him, but they couldn't find him. They looked all over hell. Had to be up there because if he ran across the field, someone would have seen him. He had to be up there in those rocks.\n\n\"I said, 'Gimme a rifle.' I picked one up and walked up there and sat and test fired it. The ocean was here to my right and the runway to the north, and Suribachi was behind me. He wouldn't be there. I sat between two rocks for about two hours. My back just felt numb from the bullet in it. It was just getting dusk when I thought I saw something move. I'm sitting down this way here, I'm looking. The airfield's beside me, I'm watching both sides. Sure enough, I see something move between two rocks. He had half his head out. I shot and blew his face off, killed him. He was camouflaged in a barrel, a big hole, with twigs and stuff. Guys come running up: 'Hey, the marine got him.' I picked up one of his own hand grenades and threw it in there. I hit with my thumb and dropped it in. I didn't want no souvenirs. He was buried right there.\n\n\"I got one more Jap after that. We had a couple young marines on ship, replacements just off the ship, and when I caught up with the company, Captain McCarthy told me to take these two guys, keep them separated and make sure they didn't get hurt.\n\n\"The three of us were walking down this hill, and I said, 'Stop talking, for Christ's sake, and stay far apart.' Downhill from us, there was a ridge with some rocks sticking out. To me it was a hiding place for a Jap. I told those two kids to stay five feet apart and wait until I circled around. I said, 'I'm going to walk ahead and see what's between those rocks. You two guys stand by.' So I backed up and circled around, and I spotted this Jap watching those two marines. I shot him through the head and then called them up for a look.\n\n\"Before he went off with his wound on March 6, McCarthy put me in charge of an ammo and food dump. I had a young fella, baseball player, there with me on March 9, when a big mortar came over, killed him, blew his head off, and I got a concussion out of it. They took me back out to the same ship, the _Solace._ It had come back from Guam for us walking wounded. It took a bunch of us to Guam, and I was there maybe two weeks, before they sent me to a hospital in Honolulu. Then I went on a ship to the States along with a hundred Japanese prisoners under Army guard. We went to Treasure Island at San Francisco, then to San Diego. From there we took a hospital train to Massachusetts. I was discharged at the Boston Navy Yard in November 1945. I had been overseas two years. The doctors listed me with a ten percent disability for the back wound.\n\n\"I went back to work at the skating rink, where I met Marie LesCault. Her parents had died, and she was an orphan. She had been brought up by the state. We got married July 29, 1946. We had five kids, four boys and a girl, Diane, Peter, Robert, Bernard, and William Timothy. I stayed with Roll-Land, the roller rink we had bought in 1938 and sold my share a few years ago. My brothers still have it. Now it's called Workout World. It's a gym for physical fitness. Marie died of bone cancer in 1997, two months before our fiftieth anniversary. It was too bad she had to die like that. She would have been with me now.\"\n\nPRIVATE FIRST CLASS DOMENICK TUTALO\n\n**Replacement, Flamethrowers and Demolition  \nTwenty-fourth Marines, Fourth Marine Division**\n\n_Some guys get it, and some guys don't. That I could never figure out. I'm just thankful it worked out the way it did. There was no way to explain the bullets coming by and other people getting shot and yet they're not hitting you. Why, I don't know. I prayed. I prayed one time, I says, \"I'm too young.\" I says, \"I'll never be bad if you let me live.\"_\n\nDomenick and his wife, Maryellen, stand in front of the West Orange, New Jersey, home where they have lived for more than fifty years.\n\nDomenick Tutalo holding a carbine at Camp Linda Vista, California, December 1943. On his left cheek is a bandage covering a boil.\n\n_Domenick Tutalo turned eighty-two on February 6, 2007, one day before the fifty-ninth anniversary of his marriage to Maryellen. The three of us were sitting at the dining room table, talking over a glass of wine. He was still working, repairing machines for a screen printing business. Domenick gets to set his own hours, though the owner of the printing business keeps calling him back. He is paid twenty-five dollars an hour. \"The money's good; he treats me good,\" Tutalo said. Besides, he considers the owner's two sons good friends. They accompanied him to the drill instructors' reunion in April 2007. \"Yes,\" said Maryellen, who turned eighty on November 2, 2006, \"he's giving it to the grandchildren.\" Domenick smiled. \"I don't mind,\" he said. \"It helps them out.\"_\n\n\"I was born February 6, 1925, and I've lived in the same house in West Orange, New Jersey, for fifty-one years. I grew up in Jersey. I quit school in ninth grade and went to work for a butcher, for six dollars a week. It was pretty good money, but I had to put in ninety hours, seven days a week. I was thirteen years old. I used to go to the market with him at four a.m., to buy stuff. That's when they butchered the whole animal. He bought it from Swift, the meatpacking company, hung it, then cut it up.\n\n\"My dad worked for the WPA, the Works Progress Administration. He was born in the United States, but my mother came here from Foggia, Italy, at the age of nine.\n\n\"When Pearl Harbor came, I was in the Embassy Theater with my uncle. He was in the Army. They stopped the movie and told all the men to report back to base. I don't remember the movie.\n\n\"I joined in March of 1943 as soon as I turned eighteen, a year and a half after Pearl Harbor. I was a fairly good meat cutter by then. The whole family was in the Marines: my cousin Phil Petrillo and Jimmy Zarrilla, my father's first cousin, who was killed at Roi-Namur. He was in the same platoon with Lee Marvin, who got shot in the butt on Saipan. Jimmy was supposed to get the Congressional Medal of Honor, but he got the Navy Cross. They said it was better if the colonel got the Medal of Honor instead.\n\n\"I was five feet five, and I weighed one hundred twenty-six. To me Parris Island, boot camp, wasn't hard. I was in good shape from being a butcher. We did a lot of running. We trained with sticks, then got the M1. I got out the end of April, and after leave I was put on gate duty at the Philadelphia Navy Yard. I was bored and kept trying to get out. Finally they sent me to New River, which is now Camp Lejeune.\n\n\"I joined a replacement outfit, and we trained there, then we took a train to San Diego and the boat to Maui. The Fourth Division went to Roi-Namur, but I went right to Maui. After Saipan they needed machine gunners, so I signed up for that. I was in that about a month; then I volunteered for a demolition outfit being formed by Sergeant Harry Kaff. He was tough. I liked him later, but at that time I thought he was a son of a gun. He was by the book, a Jewish man, but he was religious. You couldn't curse or nothing. When he was there, you watched what you said. We trained for six or eight months, learning how to blow things up with satchel charges and burn with flamethrowers. They used a naphthalike jelly. It shot about twenty feet. You had to be careful, fire in real short bursts. You could run it all out in maybe ten seconds. It weighed about seventy pounds.\n\n\"One guy I grew up with was with the Twenty-third Regiment. He thought I was nuts. He said, 'You're cracking up. Don't do it.' He said I had a good thing going, but I wanted to see combat. Why, I don't know. If I had it to do over, would I still do it? Yeah, I think so. I don't know why. . . . Maybe I wouldn't have went into flamethrowers or something like that, but why I did it, I couldn't tell you. I did believe in serving my country. We were all very patriotic.\n\n\"We found out on the transport from Maui that we were going to Iwo, but first we went to Guam, where they put out some smaller boats and let us swim off them. Whoever wanted to swim could just jump in. I did it. I never forgot that because it happened on my birthday, February 6. I was twenty years old.\n\n\"We left for Iwo the next day with a company of demolition guys and Harry Kaff. During the landing we were getting shelled as we went in on Higgins boats. There were bodies all over the place.\n\n\"What did I feel? Scared. You had the fear, but you didn't have it. I think it's a stupid thing to say it, but whatever they wanted you just did. There was no place to hide, and you have to function if you can control yourself. We got off the boats from the front and charged up the beach. There were cannons, artillery, machine guns, mortars. I'm not that good at relatin' what went on from there. A lot of stuff I forget. I don't know. What I know is we kept doing things on the right flank, we kept moving up by the Quarry. I do remember we spent six days in the same foxhole on Hill 362.\n\n\"We were a company of demolition guys, and if they needed somebody, they'd call. If a platoon was pinned down and needed somebody, you'd go in and help. In our first engagement up on the right flank at Iwo everybody got killed but the sergeant and another flamethrower.\n\n\"I knew Mike Mervosh, but I didn't know him, you know? You never got that close. We got friendly after the war. I was in C Company, Twenty-fourth Regiment. Up near the Quarry. Our squad leader took us up toward the Quarry, and that's where this sniper just shot nine guys right out in the open dead before a rifleman named Scowie got him. The only reason Jeffries and I survived was we made a run to a cliff where he couldn't shoot us, but we couldn't do nothing either. We were stuck there the whole night. We were in the Quarry most of the battle. We kept moving up the island. I heard about the flag going up, but I didn't see it.\n\n\"There was no real strategy on the caves or bunkers. The riflemen gave you cover fire, and your job was to charge as close as you could, then blast it. At a bunker you would try for the slit; if not, the flame would stick wherever it went. You didn't carry a flamethrower all the time. Sometimes you had to carry a rifle. You only got to be a flamethrower when they called for you. They bring you the flamethrower. Then, when you run out of fuel, you just drop it, and they reload it for you. You don't have to run back because somebody else followed you. Once you blasted the opening with flame, somebody would run up and throw a satchel charge. They were mostly caves. I only saw an enemy once. It was hard to stomach. He came running out on fire. I saw a lot of dead bodies. Did I see our guys taking teeth and ears? I'd rather not comment. I picked up a couple souvenirs, like a notepad and money, and I got some wooden dog tags. Tell the truth, I haven't gone through it.\n\n\"I was on the island the whole time and was never wounded. I used flamethrowers and threw satchel charges, saw a lot of action, charging tunnels and bunkers. I never went inside.\n\n\"They said Iwo was really bad for replacements, who got killed easily because they were green and never had time to get integrated into the units, but I never got hurt. Some guys get it, and some guys don't. That I could never figure out. I'm just thankful it worked out the way it did. There was no way to explain the bullets coming by and other people getting shot and yet they're not hitting you. Why, I don't know. I prayed. I prayed one time, I says, 'I'm too young.' I says, 'I'll never be bad if you let me live.' The worst thing I didn't like was the mopping up. You'd walk the whole island looking for stragglers. That's where a lot of fellows got killed, I think.\n\n\"When it was over, we went back to Maui to train for the invasion of Japan. Meantime whoever had enough points were the first ones to go home. It was figured by combat and total time overseas. After Japan surrendered, it didn't matter so much. I was overseas twenty to twenty-two months. I got there January of 1944 and did not get back to the States till November of '45. I was discharged December 10, 1945, in Quantico, Virginia. I still weighed one twenty-five. All my cousins made it through except Jimmy, the one who was killed on Roi-Namur.\n\n\"I had three hundred dollars when I got home, and I took a chance and opened a restaurant, a luncheonette, short-order cooking, hamburgers, coffee. I called it Tut's. My mother and sister helped, but I couldn't pay them, couldn't make ends meet. I bought everything on the black market. It closed after about a year.\n\n\"Maryellen McDonough lived upstairs with her parents in the same building where we had the restaurant. Her father didn't want her dating Italians, but he and I got to know each other before I started going out with her, and I guess he changed his mind. He was a good man, her father. We got married in February 7, 1948, the day after my birthday. We had two boys and a girl. We bought the West Orange house in 1955. We were married fifty-nine years as of February 7, 2007.\n\n\"I started excavating after that, burying four-hundred-fifty-gallon fuel oil tanks for fifty bucks each. I had one helper. We dug the hole and the trench by hand. They supplied all materials; I supplied the truck. I bought a brand-new Ford dump truck for twenty-seven hundred dollars. I don't know why I didn't go into the butcher business. I guess I thought there was more in trucking. From there we went into hauling dirt for contractors, and then we got into landscaping. Tutalo's Contracting, we called it. I had my brother John with me. We got into grading and planting trees for towns. We made a good living.\n\n\"I went to my first reunion in 1950, but raising a family, I couldn't afford to go too often. In 1978 I hooked up with Mike [Mervosh] and Glenn Buzzard at the reunion in Philly. We had a little more money and a little more time by then. I didn't meet Pete Santoro yet. I just met all of C Company. Most of them were there, guys I hadn't seen in thirty years.\n\n\"I enjoy the reunions. All the years we've been coming we never even mention the service or combat, even with all my friends. Nothing ever came out. Never talked about it, never really realized what it was. We go to reunions, even Parris Island, and there's no talking about the war. Whether the mind went blank or what, I don't know.\"\n\n[Iron Mike remembers:]\n\n\"Tutalo is very modest, and he forgets a lot too. He was a flamethrower and demo man, and that takes a lot of doing. I was in a machine-gun platoon, divided into three sections to support a rifle company. When that became decimated, I became commander of the rifle platoon. The demo men or flamethrowers were available to whichever platoon needed them. They had their own section within the company.\n\n\"Tutalo was a replacement. Replacements usually didn't last long. I had experienced a lot of battle before I went there, and I knew what to expect more than the replacements did. Even if you took care, it was still deadly. Every damn part of the island was covered with something, interlocking fire, artillery, mortars, and rocket fire. You just had to be lucky, damn lucky.\n\n\"I seen him in action, applying demolitions and throwing satchel charges in those caves and bunkers. He'd go up to those pillboxes, and we'd give supporting fire, and someone would throw a satchel charge to blow the thing, and then Tutalo comes up and flames it, and anyone left alive comes running out burning from head to toe, you know what I mean? Guys were getting killed left and right, but we considered that routine because hell, that's the job. We didn't think there was anything heroic about it. But we said to Tutalo, 'For what you did, we're going to give you a reward when we get back to rest camp.' But he said, 'I don't need none of that shit.' After that island was secured, he said, 'Hell, I got five medals now: One head, two arms, and two legs. That's the award I got.'\n\n\"We got back to Maui to rest camp and he's called front and center with several other marines, and we promoted him to private first class, and that was his reward. He was happier than shit. He says, 'Boy, four dollars more a month.'\n\n\"Tutalo was an ordinary marine; that's the way I looked at it. We were all ordinary. I ended up as the company commander, and what was my reward? I got a handshake from the company commander: 'Mervosh, well done.' And that was good enough for me. Hey, we were all ordinary. It was a huge reward just to be alive.\n\n\"We built strength among us. We strengthened each other. That's how we survived. The guys would strengthen me, I would strengthen them, and it worked pretty damn good. That's the whole key. You're damn right.\"\nPART THREE\n\nRussell, Tso, Abbatiello, and Waterhouse\n\n**The four men** in this section had revealing, widely divergent experiences. Gerald Russell commanded a battalion, Samuel Tso sent out in Navajo word that Suribachi had fallen, Al Abbatiello was instrumental in the capture of Suribachi, and Charlie Waterhouse never got to fire a shot. Starting out as an artilleryman, Russell survived Guadalcanal, came back to the States, and transferred to the infantry so he could have more time with his wife. But then, as a brand-new infantry officer, he had to prove himself to the troops right up to the minute he landed on Red Beach One in the third wave at Iwo.\n\nSamuel Tso, nicknamed Chief, had to contend with discrimination everywhere he went. He was denied contact with his family for years because none of them could read or write and there was no telephone on the reservation in northern Arizona.\n\nAl Abbatiello didn't want to become a raider because he thought it would be too dangerous, but then he confronted almost ceaseless combat in his thirty-six days on Iwo.\n\nCharlie Waterhouse, an expert marksman who was to go on to a splendid career as a Marine Corps artist and illustrator, had barely got into the fight when a bullet in the arm stopped him cold, and he was out of it by the third day. After many operations he went home to art school, and eventually made hundreds of paintings of Marine Corps subjects.\n\nCAPTAIN GERALD RUSSELL\n\n**Battalion Commander  \nTwenty-seventh Marines, Fifth Marine Division**\n\n_The most vivid memory I would mention out of all that time was the flies and the smell of the dead. That's something they don't mention in documentaries, but boy, when you have that many bodies all over the place, it's sometimes days and days before they can bring them out._\n\nGerry Russell, eighty at the time, standing in front of a situation map of the island at the annual Combat Veterans of Iwo Jima Symposium in Alexandria, Virginia, on February 17, 2007.\n\nMajor Gerald F. Russell accepts the surrender of the Tsushima Islands in Japan from Lieutenant General Bukei Nagasi in November 1945.\n\n_I first learned it was possible for a civilian to visit Iwo Jima from Colonel Harvey \"Barney\" Barnum, USMC (Ret.), who directed me to Military Historical Tours, founded and operated by another former Marine colonel, Warren Wiedhahn, in Alexandria, Virginia. I signed up for the Iwo trip and went down to Virginia to check the place out. Colonel Wiedhahn managed to have on hand Gerry Russell, one of three battalion commanders from the Iwo campaign who were still alive. Russell had worked for Penn State in State College, Pennsylvania, for many years, and it turned out he knew Joe Paterno, the Penn State football coach, very well. Russell and I had a long, satisfying visit in Colonel Wiedhahn's offices._\n\n\"I grew up outside of Providence, Rhode Island, and went to La Salle Academy, a Christian Brothers school, and ran track, the thousand yards, indoors. I won the national championship in a race in Madison Square Garden, and by virtue of that I got a scholarship to Boston College. They had me do an extra year, redshirting, although they didn't call it that. [Redshirting means holding an athlete back from intercollegiate competition his or her first year, so it takes five years to graduate instead of four. This way the participant has a year to get bigger and stronger before entering competition.]\n\n\"I majored in history, prelegal, and graduated in 1940. Law was my intention, and I applied for Harvard Law School and Boston College Law School. I was accepted into both, and my goal was to get enough funding somehow to go to Harvard. If not, I knew I could make it at BC.\n\n\"Then an interesting thing happened: I had an old Jesuit priest for an adviser. His name was Father Finnegan. He said, 'You know, you're a college graduate, and you're single. This guy Roosevelt wants to get us into a war, and you're going to be the first to go. They're going to pass a draft law.' And he said, 'What you should do is try to get into one of these officer reserve programs in the Army or the Navy and then go through and get the commission, and then, if they don't need you, you can just sit and go through school. They'll know they have you.'\n\n\"Well, I had some interest in the Marine Corps, so I wrote to the commandant, Major General Thomas Holcomb\u2014this was early summer 1940\u2014and told him what a splendid prospect I was for an officer. I got a response back quite quickly saying, 'We don't want you.' I said, 'Well, I guess that's that.' But then I get another letter from the same outfit saying we're going to start a program. If you're still interested, fill out this form. So I filled it out and was immediately sworn in. I was ordered to Quantico in October of 1940 as part of the very first officer candidate class the Marine Corps ever had.\n\n\"Up to that time the Marine Corps would get officers from the Naval Academy or from excess of the various Army Reserve Officers' Training Corps classes around the country. There hadn't been much opportunity for military advancement during the Depression years. The Marines usually got their number one choice because things were so limited.\n\n\"The Quantico program was envisioned as nine months, three months as a private first class, three months as a reserve second lieutenant in a classroom environment, then three months of duty with the troops. The fine print said you could go inactive subject to call if the exigencies of the time no longer required your services. I was actually in the first platoon of the first company in the candidates' class. They brought some drill instructors up from Parris Island, and those guys' attitude was: These men are going to be pushing us around one of these days; let's get 'em now. Actually their goal was to make us good Marine officers, and they wanted us to be tough. That three months we were really on the go all the time. It was not unusual for them to break us out late in the evening and have us run five miles in skivvy drawers holding the rifle at high port, you know, all kinds of things. Sometimes we would either march or be trucked over to the Manassas battlegrounds. I think a lot of our tactical problems were based on some of the action that had transpired there during the Civil War, so that was pretty interesting.\n\n\"I think we ended up with about two hundred commissioned on 20 February 1941. We were supposed to be commissioned on the first, but the Department of the Navy decided to short-tour the Naval Academy class, so they would be ahead of us on the lineal list, which was very important. They were commissioned on the seventh, and we were delayed until the twentieth.\n\n\"We stayed right there for three months after that, until the month of May, and learned tactics in the classroom. Some of my classmates went to troops immediately because the Marine Corps had been so very short of officers. They needed artillery officers. We had base defense units that needed officers. They even had a barrage balloon outfit that needed officers.\n\n\"I was selected to go to artillery school. At that point there was no First Marine Division or Second Marine Division. There was no Camp Lejeune, no New River. They hadn't even acquired it yet. I'm sure they were in process. The Fifth Marines was a battalion in Quantico. The Seventh Marines were at the Charleston Navy Yard, and I think the First Marines were at Norfolk. They were not the huge outfits that they later became.\n\n\"The Eleventh Marines, the artillery, was situated at Parris Island, so my first duty with troops was at Parris Island, probably August or September. After New River was acquired, we went up there, where the First Division was beginning to form as various units were brought in. The artillery was placed at Courthouse Bay. Of course all the roads were dirt, and the vehicles that were the prime movers for the one fifty-five howitzers had hard rubber tires. In short order those roads were just a muddy mire, and they had to cut trees and corduroy the roads, which was rather primitive but a good way of getting the units together. Not too long after, this battalion of the Eleventh Marines was divided into three. My first assignment was as a supply officer with the Third Battalion.\n\n\"Lieutenant Colonel James Keating from Philadelphia was designated battalion commander, and he called me over and said, 'Russell, you're going to be the supply guy. Go down to the quartermaster supply outfit, see where it is. They must have some kind of a list telling you what we rate [qualify for in the way of supplies].' He said, 'Start as soon as you can because we're going to need stuff.'\n\n\"Fortuitously we got a supply sergeant who knew the business, so I depended heavily on him. God bless the noncoms, all of them. That's one of the first things I learned, and it helped me through a lot. He was so good we decided to make him a warrant officer, so he became the supply officer. And I wanted to get into a firing battery, so I became the exec of the H Battery of the Third Battalion, Eleventh Marines.\n\n\"Things were accelerated once December 7 occurred. I was at Quantico visiting my fianc\u00e9e, Eileen Herlihy, when Pearl Harbor was attacked. I'll never forget her father. He was old Navy, with the Department of Yards and Docks. He supervised all the barracks building for the Marines. She had spent almost her whole life at Quantico. She told me, 'Well, I'd like it very much if you made a career of the Marine Corps.' As it turned out, we spent the next five or six years involved in it [the Marine Corps] with no flexibility [because of the war]. I liked the Marines. She didn't have to make the decision for me.\n\n\"We got married on the twenty-seventh of December, 1941. We were going to get married in April, but we decided it would be good to do it now. Even though she was never officially in the corps, she was a great marine, the greatest asset I could have had. If I have had any success, she was a very significant part of it. We were married fifty-seven years. She died, unfortunately, six years ago from ovarian cancer.\n\n\"I was sitting in a hammock on her porch on the base at Quantico when her father came rushing out, saying, 'Those goddamn Japs.' He was a good old Irishman. He kept us posted, listening to the radio. Finally he came out and said all military personnel should report immediately back to their camps. I had ridden up with one of my friends stationed at New River. He had gone on to New York, and he came back shortly after and we went on back to New River. There were four officers in the H Battery. I was of course the junior one. One officer had to stay there for the Christmas holidays. That was me.\n\n\"I had the next section beginning on the twenty-seventh, so she came down with her mother and we were married in a church about forty miles from Jacksonville. Her father was sick and couldn't make it. For a honeymoon we got on a bus for a thirty-hour ride to Rhode Island, where I introduced her to my parents. Then we zipped back to North Carolina.\n\n\"Our artillery unit had started with French seventy-fives, the seventy-five-millimeter field gun, and then we got seventy-five-millimeter pack howitzers. We got the word to move out in March. It took us a long time to get loaded. I was in charge of my part of the battery. We had the guns and the troops in about eight freight cars and one Pullman. Officers were in the Pullman, and the troops were in the rail cars, where they just nailed bunks together out of two-by-fours. It was miserable. We had our regular field stoves, and they just screwed those into these big old boxcars. We zigzagged back and forth. It took eleven days to get across the country. We had four pieces of artillery, the pack howitzers, all on flatcars. We'd stop at a siding so the cooks could prepare meals. There were no heads or anything like that. They had to use the bushes.\n\n\"We finally got to San Francisco, then left there in early May of '42. The First Marine Division's job was to defend New Zealand, so we landed in Wellington. The complexion of things changed drastically after the battle of the Coral Sea and the battle of Midway, and then they discovered the Japanese were building an airstrip on this unknown place, which was a key position to cut off the supply line from the States to Australia and New Zealand. The airstrip came to be known as Henderson Field, and the island was called Guadalcanal. It was in the Solomons. And everybody said, 'Where the hell is that?'\n\n\"It had to be taken, so instead of going ashore and into camp in New Zealand, we unloaded. We didn't have all these sophisticated amphibious shiploading techniques. We had one basic approach, and that was: What you want to get off first, put in last. Everything was in these paper cardboard cartons. It was rainy season, so we had to unload on the wharves. They got wet, and we lost an awful lot of stuff, but we got loaded up and took off, probably in July. If you can believe it, when we were leaving the States, we were issued five-gallon aluminum cans of hardtack from World War One.\n\n\"Nobody had any idea about Guadalcanal, no charts or anything. And our information was based on a sketch from the manager of a plantation that had been owned by the people who made Palmolive soap. There was no resistance. Carrier aircraft would come in and bomb, and we just sat there and waited, and then we got in the boats and had a successful landing, but it was strictly without opposition. The Japanese were away.\n\n\"I went in on the _George Elliott,_ which was a transport. The second day the Japanese sent over a big flotilla of twin-engine bombers, and they were after the transports. My battery was attached to the Second Battalion, First Marines, H Battery. The transports had started unloading. Their holds were open, and apparently one of the planes got hit and crashed into the superstructure of the _Elliott,_ killed all the people on the bridge. The flaming gasoline spilled into the open hatches, and the ship caught on fire, so I lost everything from the start.\n\n\"Fortunately we had the guns off and some of the rations and a good amount of the ammunition. They continued unloading all the other ships. The _Elliott_ was the only one that was hit, and the second night was the battle of Savo Island, August 8\u20139, 1942. I don't think it's possible to describe the thunderous noise and the brightness of the light. It was about midnight, and of course we thought our Navy was really clobbering them. In the morning we had switched positions. We could fire out to the ocean or fire inland, depending on the need. We were watching along the ocean, and we could see sailors swimming or bodies floating, and a couple ships drifted down by there, superstructures all knocked over. This was my first combat.\n\n\"We learned fairly soon that we had lost four cruisers, The _Quincy,_ the _Historia,_ the _Vincennes,_ and the _Canberra,_ and fortuitously the Japanese didn't follow through. They clobbered them and took off. They could have just wiped everything out. Next day the fleet commander, Admiral Ghormley, decided to husband what units he had, so he pulled everything out. Well, we weren't unloaded completely, so we ended up very short of supplies.\n\n\"It was just about this time that I got to know Joe Foss. [Foss received the Medal of Honor for shooting down twenty-six enemy aircraft during the Guadalcanal campaign.] I got to know him because my battery was along this fighter strip, and those guys, aviators, you know, they don't have much sense, and they didn't have any place to protect themselves. So whenever the Japs came in to bomb, they'd come running over to my battery position, and I took care of them. So I knew Joe Foss, I knew him very well. I'm so sorry he passed away; he was a great guy.\n\n\"They tried to run supplies in, but they couldn't until finally they did begin to get a few in at night. They had some of these old World War One destroyers they had taken the stacks off. They'd get in about dusk and unload and get out of there before dawn. And even they got hit. We were subject to air bombardment every day, oftentimes several times a day. It was not unusual for a Japanese naval vessel to stand offshore and just lob shells in. One shell landed very close to my position, and a hunk of shrapnel hit me. I saved one piece. It got me in the arm and the side. By that time we all had malaria too, you know. Malaria was endemic. The shrapnel didn't stop me. It wasn't that serious, and you couldn't evacuate casualties anyway. I got a Purple Heart out of it. I got one on Iwo Jima too. I was promoted to captain a year and half after being commissioned and served as battery commander for quite a while on Guadalcanal.\n\n\"Finally the First Marine Division units were being relieved by Second Marine Division units, and then some Army elements came in. I think the Army took over responsibility probably in March or April of '43. We went to Brisbane, then Sydney and finally Melbourne. We didn't suffer the high rate of casualties we were to have at Iwo. I'd say about eighty percent of the battery was intact. From Melbourne we moved out to an old Australian army camp, and I got sent back to the States because of my malaria.\n\n\"They were just starting the First Marine Corps Command and Staff School at Quantico, and of course my wife was there. I was assigned, and about that time I got promoted to major. I was there six or eight weeks and then got ordered to an advanced officers' course at Fort Sill, Oklahoma, for ten weeks. My wife was able to accompany me; this was still '43. When I got back to Camp Lejeune, they were just beginning to form the Fifth Marine Division. The artillery for the division was the Thirteenth Marines, and I was ordered to the Third Battalion. The rest of the Fifth Division was being formed at Camp Pendleton in California, so we got the Third Battalion fairly well formed and then moved to Pendleton in the late summer of '44.\"\n\nRussell was the battalion executive officer, but there was a great deal of rotation going on as more senior officers came back from the field. He was pushed out of the exec job to operations officer, somebody else came in, and he was pushed out of that.\n\n\"They were talking about sending me back out. I hated like heck to have to leave my wife when we were finally getting acquainted after three or four years of marriage\" [so he transferred to an infantry battalion and became the exec there]. So I became an oh-eight\/oh-three, artillery-infantry.\n\n\"It wasn't easy because the Marine Corps had these raider and paratroop outfits, and I think, wisely, they decided to disband them. A marine is a marine. They didn't need all this fancy raider stuff. But all those people, these prima donnas, were thrown into this regiment, and they were horrified to have an artilleryman in the outfit. I had to really work and work and work to gain their respect. Any time there was a hike I was always leading it. They had a lot of swimming pools, for amphibious training. They had a platform twenty or thirty feet high. You'd have to climb up that with your full pack on and either be pushed off or jump off into this pool. And of course the pool was crystal clear, and it looked like you were up about four hundred feet.\n\n\"When it came time, I was the one designated to take the battalion up. And you know, all the troops are looking and wondering what the heck's going on, so who has to go up the damn ladder? First one was me. Oh, I tell you when I looked down there, I thought there were a lot of places I'd rather have been. But I said, 'I gotta go, and I'll show these so-and-sos.' So I went. Full pack, combat boots, everything. It gradually got to the point where I had the confidence of the troops and the company commanders, so everyone was comfortable with me by the time we got the word for Iwo Jima.\n\n\"We made the landing on Red Beach One. I landed in the third wave, which was the first of the LCVPs [landing craft, vehicle, personnel] fourteen to twenty minutes after the first waves landed at eight-twenty. The first two waves were amtracs. Our mission was to cut across and go up the other side.\n\n\"When I got out, I had my jeep in this boat, and when we got out, I kept yelling to the troops, 'Get off the beach! Get off the beach! Get up on the high ground!' And of course they were trying to climb up that soft volcanic sand. It was extremely difficult. You'd go up, and you'd go sliding back, and already there were a lot of casualties. There was firing on the beach, not the heavy volume that came later, but there was firing.\n\n\"I got the troops out, and the driver of the jeep got it across the ramp, and as soon as the front wheels hit the soil, it started to spin. He couldn't get it any farther than that. We were trying to get up the hill, and when I looked back he had got out of the jeep. One of the kids had been hit, and I was seeing how bad he was hurt. When I looked around, the driver had gotten out, and he must have got hit; there was this enfilade fire coming up the beach, and it looked as though he was just broken in half. He must have got it in the midriff. One of the sailors on the ramp just reached over and dragged him in. The coxswain then took the LCVP and jiggled it back and forth back and forth to get rid of the jeep and I just remember seeing my jeep just gradually slipping down into the sand. It became part of that pileup of debris all along the beach.\n\n\"We were under fire from all directions, heavy fire from just all over. I don't remember being appalled. I was kind of dismayed, I guess you could say, but I was more interested in getting the troops up and on the higher ground where they wouldn't be so vulnerable. I did not know Fred Haynes then see [Chapter 4]. I may have known him as officer in the Twenty-eighth Regiment. My regiment was the Twenty-seventh. My job as executive officer was to get the troops in and coordinate the command post.\n\n\"Iwo Jima was an action of constant, constant fire all the way through, twenty-four hours a day even to the thirty-sixth day. It was just constant. Guadalcanal was spasmodic; there were times you could just take it easy. Korea you were in trenches. But at Iwo there was artillery, mortars, machine guns, rifles, everything, and they would pop up from places you least expected. Of course they don't mention it much, but we thought this was gonna be a piece of cake. We were going in there for ninety-six hours just to kind of, you know, wipe up what was left, and go on and support the Okinawa operation.\n\n\"The fact that they had all these troops in tunnels came as a big surprise. When I went back the first time, I was almost confused. There was something different. All of a sudden I realized that when we landed, there wasn't a blade of grass on that thing. They had pulverized it. I mean everything. It was just bare soil, and it was hard to imagine that anybody could survive in that, but they were so far down. Last time I went out I was amazed that the caves held up so well even after all these years, because of the consistency of the rock. They just chipped away and chipped away. They didn't need anything to support it.\n\n\"I recall having two code talkers with my unit. My experience with them went back to our training base in the Hawaiian Islands. We worked for quite a while with them when they first joined us. I had one, and the battalion CO had one. He might send a message to me saying, 'I got a call from the Baker Company or Easy. They need some ammunition. Get it for them.' He'd write it out in longhand, and the code talker would go off and transpose it. Then he'd get on the radio and send it and nod to the battalion CO that it was on the way. Then I'd see that Navajo on my end pick up his phone and start writing. Then he'd go off to the side, transpose it into English, and hand me the thing. They were very well organized. They went through the routine and knew exactly what to do.\n\n\"But the first five days on Iwo were so heavy I really don't recall using them too much. That action was so concentrated. It was just day and night. You hear people say, 'Well, when did you sleep?' You didn't sleep; you just got by. It rained the first couple days. You'd stick the poncho over your head, and after a while you'd get soaking wet, but you just took it. That's the way it was.\n\n\"For meals, you grabbed something when you could. We did have some canned C rations. It was interesting, you could dig down six or eight inches, put the can in, and in a half hour it would be cooked. If you dug a foxhole, you had to wait for it to cool off before you could get in it. It was an amazing phenomenon.\n\n\"I was taking care of all the details that came up, making sure there was ammunition or rations or that casualties were taken care off, radios, runners, telephones. Hell, the ground looked like spaghetti, with telephone wire going everywhere. My battalion CO was a major named John Antonelli. He got through OK. About the sixth or seventh day we had been pulled off the line. Of our original twelve hundred, we were down to around four hundred men. By this time Suribachi had been taken, so they could bring in replacements. So we were pulled off the line and brought down, and we got our replacements, and we were trying to filter them in where we needed them. We got a good meal for the first time, had a chance to bathe a little bit.\n\n\"Next morning the regimental CO called the battalion commanding officer so he could show him where the battalion would be moved back up to the front. The battalion CO went up in his jeep, and I led the troops up. I was out in front of them, and I had them spread out with a good interval between each one, two rows going up this full road. I came up on the top of the hill, and there was a cut in the road, not quite as wide as this room, and when I got to the top, I could see all the way to the end of the island, and just as I got there, a shell hit the side of the hill.\n\n\"It threw shrapnel all over, and I got hit in the face. It barely missed my eye. I landed in the ditch on the side of the road. I was groggy, but I told the headquarters company commander to take the troops down around the back of the hill and then notify the battalion CO. Next thing you know I was being driven down to the beach by the regimental CO. I was sitting in the front seat with blood all over me, and the battalion CO was holding me so I wouldn't fall over.\n\n\"We used to get these two-ounce bottles of Lejeune medicinal brandy, and they were giving that to me, plus I think the regimental CO had a bottle, so by the time I got to the beach I didn't know much of anything. Anyway, they claim I insisted that I not be evacuated. I remember vaguely going through the next few days with a big patch on my face. Three or four days after that, halfway through the operation, well into March, Antonelli, the battalion CO, got hit, and he was evacuated. That's when they talk about me being one of the last remaining battalion commanders because Antonelli died four or five years ago. At any rate, I took over the battalion and brought it all the way through for the rest of the campaign, right to the end.\n\n\"We always tried to dig in for the command post. We tried to be as close to the front as we could, and usually we were one hill behind the front lines. Take the case of First Lieutenant Jack Lummus, who was to receive the Medal of Honor for all the stuff he did on March 8. His action was up front of us, on sort of a mound, you might say, and the CP was behind it. I remember when they brought him back. He had stepped on a mine, and both of his legs were blown off. They weren't bleeding.\n\n\"He was on a stretcher, and I lit a cigarette and held it for him. Jack had been an All-American at Baylor and was an end for the New York Giants. He talked reasonably cogent, and of course I was shocked that he wasn't bleeding. I learned later that shock just clamps things off. Anyway, he said, 'Well, I guess the Giants have lost a good end.' And I said, 'Jack, the Marine Corps has lost one hell of a good officer.' They took him in the jeep down to the clearance station. I talked to a doctor down there afterward, and he said he didn't have a chance. He said what you didn't see was the fact that his stomach was all ripped open. He said he could reach in and pull a handful of dirt out of it.\n\n\"We went through the whole thing on the front lines all the time. On the morning of the fifth day we had swung up to the north, and we were having a hell of a time. Our two code talkers got clobbered that fifth day, and one of them was in a shell hole for two days before we could get to him, the action was so intense.\n\n\"We were facing away in a kind of crevice, and one of the kids yelled, 'Look!' He pointed up, and there on the top of Mount Suribachi we could see this small group of men and Old Glory. It was very emotional. You can't imagine how I felt. There was an old gunnery sergeant standing near me. He was about six feet two and had been in the Marines for I don't know how many years, the Old Corps, you know?\n\n\"This guy had the most colorfully profane vocabulary I've ever heard. How he could conjure up some of these things was just amazing. He never showed any emotion or anything else, and on the fifth day we were coated with that black grime. We barely had enough water to drink, let alone to wash. When the flag went up, I couldn't say anything. I had a lump in my throat, and I don't know if I had any tears, but I looked at this guy who I never thought had an ounce of emotion in his body, and he looked at me and you could see tears coming down through this grime on his face, and he said\u2014and I'll never forget it\u2014he said, 'God, that's the most beautiful sight I have ever seen.'\n\n\"I've said this in Flag Day speeches and stuff that up to that point, we weren't sure whether we were going to succeed or not. But from that moment on, when the flag went up, we knew we were. It didn't get any easier, but we knew we were going to win. We were reminded of what we were there for. I don't know which flag it was, but it was in the morning. We just kept going. We were just relieved that they had taken it.\n\n\"Iwo was a place where you had to go in and dig the enemy out, where you had to use flamethrowers and satchel charges. It wasn't just huge barrages of fire. For example, we were in an area where the rocky formations were such that every time our troops would try to maneuver, the Japs could come out of these caves and fire on them. We needed some kind of heavy fire, and there was no way we could get a tank in there.\n\n\"Being an old artilleryman, I suggested we get a pack howitzer, because it breaks down into seven or eight pieces. It was originally designed to be carried by donkeys or mules. It was easily portable. This weapon was designed primarily as mountain artillery that could be broken down and taken ashore in six loads from a boat. It became an important support weapon of the Marine Corps at every major landing in the Pacific. It was crewed by five men and could throw a sixteen-pound shell almost ten thousand yards. So I recommended we get one up there at night, set it up, and start using it during the daytime.\n\n\"But the regimental executive officer of the artillery said, 'That's damn foolishness; it'd never work.' But apparently General Rockey, the division commander, said in no uncertain words, 'If they want it, get it and give it to them.' So they brought it up, just this little group with one gun. They put up the gun at night, camouflaged and put rocks around to protect it. We used it, and we moved right through after that. But again it was that individual action, individual initiative.\n\n\"We kept losing men after I took over the battalion, and we got a few more replacements. But by the end of the operation there were so few left in the whole regiment that they consolidated the First, Second, and Third into a sort of temporary battalion. There was just one lieutenant colonel left, a guy named Don Robertson. He took over.\n\n\"The most vivid memory I would mention out of all that time was the flies and the smell of the dead. That's something they don't mention in documentaries, but boy, when you have that many bodies all over the place, it's sometimes days and days before they can bring them out. The flies and the stench were pretty bad.\n\n\"I carried a forty-five, but I never fired it. Three or four days before the end of the operation I had the remnants of my companies together, and we were dug in. I was in a foxhole myself, and I had a phone from each one of the companies. They were having trouble, and we needed some heavy fire. I had requested a tank. When it came up, I went around to the side away from where the Japanese were, so I had some shelter. I got the phone from the side of the tank, and I told them inside exactly what fire we needed and where. I vividly remember that when the sixty-millimeter small-caliber mortar shells would explode, they gave off that yellow picric acid kind of smell. When they acknowledged my message, I ran like heck back and jumped into my hole. Well, while I was gone, a shell had landed in there and chewed up all three phones and everything else. We always have an expression: If your name is on it, no matter what happens, you're going to get it. I guess mine wasn't on it. To this day I kinda sweat a little bit when I think about that one, right up almost to the end. They didn't give up. They fought all the way. Talk about a hunk of hell, it really was.\n\n\"I think the battle of Iwo Jima is one of the truly great landmarks in the overall history of the Marine Corps. It was an achievement tremendous in its significance because of the way we were able to handle it, win it.\n\n\"The performance of the troops became a defining statement for what marines can accomplish because at times it seemed hopeless. We were still able to keep going, day after day, with an unbelievable number of casualties. It was unsurpassed. We never had anything like it before. The troops, those marines proved that no matter what, we would succeed.\n\n\"After Iwo, and as soon as we got back to the base camp at Hilo in the Hawaiian Islands, we immediately started getting ready for Operation Olympic, to land on Kyushu. We trained, got replacements, then the two atomic bombs were dropped, and the Japanese sued for the armistice.\n\n\"They took the First Battalion, Twenty-seventh Marines, and the First Battalion, Twenty-sixth Marines, to make a preliminary landing during the occupation. We didn't know what to expect from the Japanese. They had been treacherous, mistreated our prisoners of war, and so we went in ready to fight. But they were the most docile, cooperative people you could ask for. Honor is so important to them, and the emperor said, 'Do what you're told, and cooperate.' I never had one bit of trouble with them.\n\n\"So I spent almost a year in the occupation, and I got a couple key assignments. One was to accept the surrender of the Tsushima Islands under Lieutenant General Bukei Nagasi. The Tsushimas were the pair of islands in the Sea of Japan with Korea on one side and Kyushu on the other. I got some pictures of the guns we had to blow up, fourteen-inch naval guns.\n\n\"I went in with my battalion on LSTs. I had a group of twelve interpreters from the Home Ministry. General Nagasi was from the old warrior class, very distinguished, a very proud, very pleasant guy. He had eight thousand troops, and I had a little over a thousand, and his were crack troops, and I wanted to get them the hell out of there. I commandeered every damn fishing boat, everything I could find, and put them aboard and sent them back to the mainland.\n\n\"I'm still just a major in my mid-twenties, and this lieutenant general kept saying, 'Well, when can we make arrangements to have a surrender?' He kept saying, 'Well, when is your general coming?' And I had to tell this poor old\u2014you know honor would dictate somebody of the same rank would accept the surrender, and here he has to surrender to this punky major. By virtue of that, I got to know the Japanese people and everything, so I could see their side. There were some things I had come to understand about them where they felt justified in a way.\n\n\"Japan was a small country, very short of minerals, and all the metals of the world were right there in their backyard, all controlled by countries half a world away. They got it through colonial efforts; why can't we do the same thing? I think they felt we'd back down if they showed any force. They thought the United States was a bunch of pussycats, wouldn't be willing to fight. Of course Kuribayashi and I think Yamamoto knew better, but nobody listened to them.\"\n\nGerry Russell retired from the Marines in 1968 as a full colonel following service in World War II and Korea, then as a regimental commander after that. He commanded the ground transport in Guant\u00e1namo during the Cuban missile crisis in 1962. He and his wife had two daughters, Eileen and Maureen. He spent his postmilitary years at Penn State University in State College, Pennsylvania. He has been the chief finish judge of the Millrose Games for many years. He never made it to Harvard.\n\nPRIVATE FIRST CLASS SAMUEL TSO\n\n**Code Talker  \nReconnaissance Company, Fifth Marine Division**\n\n_They called me Chief, and I resented that for a long time. I don't know what they mean by that at the time, but later on I decided, oh, they give me a promotion from PFC, and from then on I don't bother with it._\n\nSamuel Tso in Albuquerque, New Mexico, in October 2006. The Navajo code talkers were honored by President George W. Bush in 2001 with the Congressional Gold Medal, the highest civilian honor Congress can bestow.\n\nPrivate First Class Samuel Tso, displaying one battle star for service on Iwo Jima, is photographed in Camp Pendleton, California, in 1946.\n\n_There were twenty-nine original code talkers, four of whom were still living in the spring of 2007. Sam Tso was recruited for a later group. He and I were at Iwo Jima in March 2006, but I didn't really get acquainted with him until we met again in Albuquerque in October of that year. He was there with several other code talkers to sign copies of three different books to help raise funds for their association. I bought all the books. They signed them. You couldn't help liking these guys._\n\n\"I'm from Arizona, above Window Rock. It's way up by a place called Black Mountain, about eighty miles from Gallup, New Mexico. We didn't live on top but down below, on the east side of the mountain. The nearest town or store was twenty-five miles away. My father and mother raised sheep. They'd sell wool and sometimes sell the lambs. I had brothers and sisters; there were ten of us.\n\n\"I don't know what year I was born. The only thing my mother remembered was it was sometime close to the midsummer, probably around 1926. When I finish eighth grade, I'm ready for high school, so they send me to Fort Wingate, New Mexico. In that four years I never came back home. I learned to read and write. It was a boarding school paid for by the federal government. We lived in dormitories. We were mostly Navajo, but other Indians were sent there too. During the summer everybody goes home except me. I had to go look for a job. I found out the government was running an experimental sheep station close by there. So I went and herded sheep for them. They were experimenting with wool. Old Navajo sheep wool had a hole through every strand, and to get rid of that, they brought in Rambouillet rams to crossbreed and change the wool so it was solid.\n\n\"After high school I didn't get a chance to go home at all. The federal government just told me, 'There's a road down that way. Somewhere you'll find a job.' So I picked up my suitcase and started walking.\n\n\"You know what? I sure hate to tell anybody, but I really resented it. I really resented it. I just kept it inside me all those years. In fact, when I started school, they told me not to speak my language, Navajo, at Indian school. We used to get punished for it.\n\n\"It took me approximately nine years before I got back home. I did not get home until after the war. When I got home, I found out my father, my mother, and one of my sisters, that I used to herd sheep with all the time, they were all dead. There were no phones, and no one ever wrote; they couldn't write. They were uneducated.\n\n\"When I come out of high school, the only job available was with the railroad company in Gallup, so I went there and asked for a job. They asked how old I was. I said nineteen. Then I found out they didn't employ nineteen-year-olds, so in the morning I was nineteen, but by the afternoon, when I got hired, I became twenty-one years old and signed a contract saying I was born June 22, 1922. This was 1943. It was an easy birthday to remember. But that's how I came to be drafted.\n\n\"When I got the job from Gallup, they transported me all the way to Barstow, California. We were replacing old ties underneath the tracks. While we were out there, the draft board came around and told me to report to the San Bernardino board. I went over there and signed in. About a month later they called me. I was drafted and sent back to Arizona. Instead of going home, I hitchhiked from Gallup over to the draft board at St. Johns, Arizona, to report. I didn't know a bus had been set aside for the draftees, so I hitchhiked all day, a long ways. It was in 1943, March.\n\n\"The guys I worked with on the railroad had bought me a charcoal-colored civilian suit, and they told me to wear that coming back. I was hitchhiking with that suit on when I got to St. John. I found the guys there resented it. Right after lunch they ganged up on me and tore some of my suit, took my hat away from me and stomped on it. They were all Navajos. Jealous. The guy in charge, I guess he was the sergeant, came out and broke it up. Next morning they put us on the bus to Phoenix, for our physical exams. Only three of us were selected out of those two busloads of guys. We found out the rest of them don't know how to speak English very good. I found out that to go in the service, you had to speak good English, read and write. Those others did not know how to read or write.\n\n\"After we were drafted, they found out we were Navajos, so they put us into the Marines because the Marines had the [code talker] program. Two of us went through boot camp in San Diego together. The third one came later, in the next platoon. It was 1943, March 13, when we started. We finished in April, then went to Camp Elliott, east of San Diego. We trained as infantrymen. We were ready to be shipped out when all of a sudden they pulled my seabag to the side. The rest of the guys I trained with, all white guys, were hauled off and sent to the Pacific.\n\n\"They took me to Camp Pendleton and put me in the code talker school. I said, 'Hey, you didn't want me to speak my language. How come? Now you want it.' At least this time they said please. Before that they never said please.\n\n\"The way I heard it there were thirty original code talkers. Something happened to one, so there were only twenty-nine. A guy by the name of Johnston ran the school. Later on I found out he was the son of a missionary that was on our Navajo reservation. The first day I reported to Pendleton I met Mr. Johnston, and all of a sudden he talked to me in Navajo. I was really surprised that he could speak Navajo. His terms and tenses were all mixed up, but I could understand him.\n\n\"I believe I was there about a month, maybe more. We were at least one class, close to twenty. As soon as they got trained there, they got sent out. We were all from Arizona, New Mexico, and Utah. I did not know any one of them. I learned all the code signals, and they gave me a partner and took us out in the field to set up the radio and practice sending messages back and forth. We noticed that some of the Anglo kids that taught us how to set up the radios were listening to us and memorizing all the coded words. Right there me and my partner said, 'Hey, if these kids can memorize that in such a short time, the Japanese will do the same thing.'\n\n[The original code, invented in 1942, contained 236 terms. By 1945 it had grown to more than 400, all memorized. The expanded code uses alternate words for common letters of the alphabet. The letter _A_ could be \"ant,\" \"apple,\" or \"ax.\" The word for _America_ was _Ne-he-mah,_ meaning \"our mother.\" The word for _battleship_ was _lo-tso,_ meaning \"whale.\"]\n\n\"We started making another set of twenty-six symbols for the letters in the alphabet, and then we started on a third set, and somewhere along the third set we were ready to be sent out for further training. So that's where we started the other sets. And when we started it, we started jumping around between those sets, and the kids that memorized the first set got confused. We mixed it all up, so the Japanese could not memorize it.\n\n\"We were sent out as soon as we were trained. Six of us went to a recon company. We trained up in mountains, down on the beach, out on an island, even in submarines. The sub would come in offshore; we would paddle over there and go into the sub. It dives, takes us out somewhere, comes in, and surfaces, and then we get out and use rubber boats to go along the shore. They told us to look for military installations, ammo dumps, or fuel dumps. When we got back to the submarine, we would dive again, then get together and report what we saw. After about half a year they sent us to Hilo, Hawaii, in early 1944.\n\n\"We left Hilo on Christmas night of '44 and went to Honolulu. They told us to go to town and do what we liked. I didn't know anybody, at the time, to go into bars and drink beers and stuff. As I remember, we Navajos were told not to do that because the federal government put restrictions on us. So I didn't bother to go to any bars. I went to one show and then another one. It was real close to sundown, so I went by myself to the ship, I got aboard, and in about two hours there was all this commotion. All those guys were coming in, some so drunk they couldn't walk up the gangplank. So they let the cargo nets down, put 'em in there and hoisted them up and dumped them on the deck. This was the Fifth Marine Division. There were six of us code talkers in that company. All the rest got drunk, and I was the only one sober. Some got so sick they just puked all over the place.\n\n\"The boat left for Guam that same night, sneaked out, zigzagging all the way. Some guys said later the reason was to avoid Japanese submarines. From Guam we went to Saipan and practiced landing there. A recon company was practicing landing there, and I found out that some of these guys who could really swim good belonged to an underwater demolition team, UDT. I got to be real good friends with a guy named Al Mertz, one of the UDT experts. We sort of stayed together. He was from the Ozarks in Arkansas, Pine Bluff. I remember because he used to get letters. Me, I never got no letters. My father, my mother, my brothers and sisters were uneducated. They couldn't write anything. I never got a letter from anyone all that time. I didn't even have a girlfriend. Mostly I kept by myself.\n\n\"We didn't know we were going to Iwo until we were out there at Saipan. I can't remember what wave I went in with, but when we landed, there was no fire from the Japanese. But after we went on top and started spreading out, they opened fire. Some of the guys jumped in an artillery crater. We jumped in on the south side of it, and the guys who jumped in on the north side got shot because they were exposed. My personal sergeant was a guy named Barnes; when we started moving forward, he got blown up. He told me to go around to the other side and stay behind. He went straight ahead and stepped on a mine. If I'd followed him, I'd have been killed.\n\n\"Let me tell you, I was scared stiff. The only thing that helped me go on was the fact that I was committed to the fellows that I trained with. We were told that you go in as a team, that you must watch out for each other. That's what kept me going, even though I was scared.\n\n\"When we went ashore, our mission was to cut the island in half, but they held some of us behind. They put us by the airfield and said, 'You hold this for a certain day and then follow.' My job was to receive and send messages from the ships or the command post or whatever it is. You receive it and send it on. All in Navajo. All the radio guys were Navajos doing code. I don't know how many there were altogether. I know my recon company had six. Sam Billison was supposed to be with us, but they selected him to be with headquarters command. All messages went in code. Major Howard Connor said he had six Navajo networks going twenty-four hours, and they sent and received eight hundred messages without an error.\n\n\"On February 23, 1945, just somewhere close to noontime, all of sudden the radio signaled, 'Message for Arizona.' So I just grabbed my papers and my pencil and just sent it. They sent this message: _DIBE BINAR NAAZII:_ 'Sheep's eyes is cured: Mount Suribachi is secure.' _Sheep Uncle Ram Ice Bear Ant Cat Horse Itch_ spelled _Suribachi._ And it was encoded too. It was sent out, and I caught it there by the airfield. And the marines that were there saw me writing it down, and they all said, \"What's up, Chief?\" All I did was just point up to the flag, and they saw it. Oh, gosh, those guys just jumped up and started celebrating there. They forgot the Japanese were still shooting. As I remember Sergeant Thomas screamed at us, said, 'Damn you knotheads! Get back in your foxholes there!' And then the guys stopped celebrating, and they jumped back into their foxholes.\n\n\"They called me Chief, and I resented that for a long time. I don't know what they mean by that at the time, but later on I decided, oh, they give me a promotion from PFC, and from then on I don't bother with it.\n\n\"One night, sometime early in the morning, my buddy Paul Blatchfield kicked me, and he said, 'Hey, Sam! Are you having a nightmare?'\n\n\"And I woke up, stood up, then sat down. I was dreaming about a young Indian maiden who came to me, and gave me something. She says, 'Here, you wear this, and you'll come back to us.' It was so real that I got up and just set there thinking about it. And all of sudden they called the rest of the marines to go eat. Everybody went to go eat, but I just set there and thought about it.\n\n\"They came back and said, 'Hey, Chief, are you still here?' I said, 'Yeah, I'm just thinking.' Then, mail call. They all went over there. I know I don't get no mail, so I don't go. All of sudden, here this Mertz came running back, and he said, 'Hey, Chief, you got a letter! You got a letter!' He brought it over, and he said, 'There's something in it.' We opened it and here is the necklace that Indian maiden tried to give me in my dream, it's in that letter. That letter, I was so stunned I didn't know what to do about it . . . oh, yeah . . . I was supposed to wear it. I put it on; as soon as I put it on, all that fear disappeared.\n\n\"The necklace was made out of what we call cedar tree berries. The berry has seeds in it, and it's formed into something like one of those Catholic Church rosaries, and at the end it has a cross made out of cedarwood. I wore it and wore it and wore it. I don't know when that thing broke. It just fell apart.\n\n\"Who sent it? I never knew. My name is on the envelope, but there's no return address or nothing. No note. Just plain the necklace. And the dream. After that I wasn't afraid anymore. That's the reason why. We were sent to run across a place called Death Valley. This Death Valley is on the north side; there's a valley there the marines tried to cross and on the other side a Japanese machine gun. When the marines tried to cross, they'd just mow them down. We were sent to cross that valley and locate the Japanese machine gun . . . the recon guys. I was selected to take the farthest point. Me and another guy were both assigned over there.\n\n\"They told us, 'At noontime we will give you the signal to cross the valley and get to the machine-gun nest.' So we strung out all the way. I believe a platoon, don't know which one it was. Sergeant Thomas told us, 'If you get hit, we're going to come and get you; we won't leave you out there.' Gave us that pep talk. At noontime they gave us the signal, and we took off. Just beyond the boulders, oh, man, there were bodies laying on top of each other, some still alive, some reaching out for help. We tried to stop and help, but the sergeant screamed at us to keep going. You complete your mission first. I took off again. We ran smack into a Japanese submachine gun. They told us those Japanese soldiers would get their food ration at noontime. When they go out there, that's when they sent us out, and we ran into that submachine gun. I think my buddy just threw a hand grenade over there, and we beat it back to report.\n\n\"The sergeant called me over and just chewed me up and down: 'Don't you know how to take orders?' and all that stuff. He was mad because we had tried to help the wounded. That was not our mission; our mission was to locate the machine gun. The other two code talkers sent the coordinates and ordered mortar fire, rocket fire, and artillery fire, all three at the same time, at one machine-gun nest. Within five minutes you could hear all the explosions coming in, you could see rockets coming over, mortars coughing and going over. That place was just completely . . . Rocks were flying all over the place. Within maybe twenty or thirty minutes they stopped. Before the dust and all that settled they ordered the marines to cross. The marines they just walked right across without any Japanese fire. We found out stretcher bearers were all kneeling down waiting to go get our wounded out.\n\n\"Anyway, that's where that dream helped. It kept all my fear away, and I said, 'I'm going home, I'm going home.' And sure enough, I came back.\n\n\"I was on Iwo for the whole battle. We carried M1s. Guys don't believe me, but one time we were out on a ledge at the front and they told us to hold that point at all costs. Sometime late in the afternoon the sun was almost down, and the guys become real spooky, and one said, 'You hear that?' 'Hear what?' 'That popping sound?' As soon as we heard that popping sound, we could see that Japanese hand grenade coming. And we just grabbed the M1, used it like a bat and whammed the grenade back instead of trying to pick it up. Then we found out the M1 was a good bat too.\n\n\"I was never wounded. [Gestures left.] A guy got shot here [gestures right], another guy got shot there. Why I wasn't hit is beyond me. The maiden in the dream? I believe that's what it is. I believe I know for sure of three code talkers that got killed on Iwo Jima.\n\n\"They found out that carrying the radio on Iwo Jima was all right for a wide-open area but too risky in real close fighting. They used the telephone, a regular telephone receiver, mostly. Most of the code talkers had to string the wire. If it broke, you had go back and repair it or string another one. Some guys set up radios for relays in certain areas; that's where I intercepted that message coming in about Suribachi.\n\n\"The radiomen, they always said, 'Message for Arizona!' And that meant a code talker needed to receive it, even on the radio. Message for Arizona. Can you hear it? A lot of people don't understand that this Navajo code language was not written. It's only a spoken language. You can't read it, and you can't write it. You have to decode it in your head and change it to English. It's only in the head here. A rich language.\n\n\"Now they're trying to pass a law that all Indian languages should be taught in public schools. It is being taught on the reservation and on the university level, at New Mexico State, University of Arizona, Northern Arizona, and at Arizona State. But back then they wouldn't let us speak it; they wanted everybody to be Anglo. I went to speak in Washington, D.C., last April 12 [2006]. I talked to some congressmen, Senators Daniel Inouye and Daniel Akaka and some others. We talked about the Native American language. I told them it was very valuable. This was about legislation saying that Native American languages will be taught in states where they have reservations. The other was to recognize code talkers from other tribes, like the Sioux, the Choctaw, Comanches, Sac and Fox, and some others. They were used as translators in their own languages, but ours was coded.\n\n\"During the war there were guards assigned to protect us. Some code talkers had bodyguards. I wasn't told I had one, but there was a guy who stayed with me all the time: 'Hey, Chief, where you going?' 'Oh, I got go to the restroom.' And he comes along. He followed me all over that place. The guard was supposed to protect you.\n\n\"That _Windtalker_ movie? One of the guys who saw it, one of the original twenty-nine, said that's eighty percent correct and twenty percent Hollywood. But another guy says it's fifty percent OK and fifty percent bull hit the snake. That's Navajo code for _bullshit._ When Tokyo Rose used to come on the radio, this guy by the name of Chester Soul always used to say, 'Oh, that's bull hit the snake.' That means 'bullshit.'\n\n\"Chester trained with us. Groups came in later. Some joined and learned the code, but they never saw action. Altogether there's about four hundred twenty code talkers, all marines, in the First, Second, Third, Fourth, Fifth, and Sixth Divisions. They were at Okinawa too. Every theater. Last March, when we were back over there, we stopped at Okinawa, and man, oh, man, there were people out there who really appreciated the Navajo code talkers.\n\n\"I didn't go to Okinawa during the war. Iwo was my last action. When we got back to Hawaii, they were getting ready to invade the mainland. Japan. I came back to the States sometime in February of 1946. If my family had said, 'Write to me,' I would have seen my mother again. But nobody could write, no telephone. That's the one that really hurts.\n\n\"I got out March 29, 1946. They tried to keep me in the Marines. They said, 'We'll give you a corporal rating if you stay in.' I said, 'No, thank you. You don't give me no promotion or nothing all this time, and I'm supposed to be a specialist. I'll take my chances on the outside.'\n\n\"I was released in San Diego, and from there they took us to Camp Pendleton, and we waited a whole month to get discharged. The bus depot, the train station, everything was all jammed. I had to stand in line almost two nights just to get on a bus. Some would get so disgusted they'd take their seabag and try to hitchhike. I got on a bus as far as Winslow, Arizona. It turned south there, so I took my seabag and hitchhiked to Gallup. A guy from New York picked me up. He wanted to stop at Petrified National Forest before we got to Gallup, so I went along with him. We saw it, came back, and he decided to eat. We went and sat at a restaurant, and to my surprise he ordered beer for us. Then the guy that owned the place said, 'No beer for this Indian!'\n\n\"And the guy from New York got into an argument with the owner. He was mad at that guy. He said this man just risked his hind end for you guys, and you won't even give him a beer. So they argued. I was beginning to think they might get into a fight, and I said, 'Forget it, forget it, let it go. I'm used to it now.' But he finally talked the owner into it. He says, 'OK, then you be responsible for him.' 'Responsible for what? It's just one beer.' So we had a beer, and then we went on to Gallup.\n\n\"From Gallup there was no transportation to the reservation, and it was more than ninety miles to Black Mountain. I had to wait for the mail truck to Chinle early next morning. There was no transportation from Chinle to Black Mountain. Nobody knew anything about where I was.\n\n\"So I had to walk again, had to walk all day. Just before sundown I got to a place where they have rocks piled up on the south side. There was a mirror hidden underneath, so I took that mirror out and started flashing out toward the Black Mountain, where my home was. All of sudden I saw a horse coming. I put the mirror back in there and walked a little ways. The horse came bearing my brother. I hadn't seen him in nine years. I left my seabag there till they could come for it with a wagon. We rode back home, and I found my father had died.\n\n\"Then just before dawn I got up and walked out of there because there was nothing for me to stay for. A big cedar tree is still standing just a little ways from where we lived. Underneath that tree is where I was born. When you stay with the sheep, you have to move here and there every month so they don't overgraze. You don't move the hogan, which is your home. At that time we didn't have any tent. All we had was sheepskins spread over a lean-to. I was born in a lean-to under that tree.\n\n\"I looked over there at that tree and saw there's a family living there. I asked, 'Who's that?' They said, 'They own that place now.' And here that was supposed to be mine. And I talked with the guy who lived there, and he told me, 'This is mine because you been away all these years. To own this land, you're supposed to live on it, and tend to it but you've been gone all these years.'\n\n\"I said, 'Hey. I been fighting for that land. The government took me away.' But that's tribal law. He still lives on it.\n\n\"I just walked out of there. My brother found out I just left. He brought a horse over to me and asked where I'm going. I said, 'I don't know where I'm going. I'm just going. The first thing I need to do, I need to get to Chinle and that mail truck.' So he gave me the horse and gave me instruction. 'You ride this over there and leave the saddle and the bridle at the church and put my name on it and turn the horse loose. He'll come back,' he said.\n\n\"I rode the horse to Chinle, got on the mail truck, and went back to Gallup. From Gallup I bought a bus ticket all the way to San Francisco. I had found out one of my sisters got married and she and her husband lived there. She didn't know our mother, our father, and my other sister had died. She couldn't communicate with them either.\"\n\nSam worked in San Francisco and then back on the railroad, and then he went to Lawrence, Kansas, where he spent two years at the Haskell Institute vocational school, a trade school for Indians. He learned electrical work and refrigeration. Here he met his wife, Anne, also a student. He went on to Oregon while Anne found a job at a new school for Navajo children in Brigham City, Utah. Eventually Sam joined her there and obtained a degree in elementary education. They were married in June 1952 and had a son and a daughter. For thirty years he taught Navajo children at the Intermountain Schools and up in Lukachukai, \"a little bitty settlement,\" about twenty miles from where he was born.\n\nSo how does Sam Tso feel today about the treatment he received at the hands of the government and about his service on the Island of Iwo Jima?\n\n\"Let me tell you. We are still part of the United States, whether it's called a reservation or not. We are still part of the United States. When I thought about it, I felt that, if were not for the United States, some other country would have come in and taken over, and what would we be? I doubt we would be a democracy. I found out in high school that a democratic form of government is best for people. And that's what made me decide to go and help in the war.\n\n\"People aren't stupid. Sooner or later they are going to learn to respect everybody. Not only one group, but everybody. It's best that we be patient. Coming back from the war, I found out that the people of America were really thankful. The reason I say this is because one time President Clinton invited us to go with him to Hawaii to celebrate V-J Day. We followed him over there, and we were at the Punch Bowl, where all those dead soldiers were buried. It was a lot of people. He made a speech, and then he introduced a lot of the top brasses from different parts of the services.\n\n\"At the end he introduced us Navajo code talkers. Man, oh, man, I just couldn't believe it. All the people in that bowl just stood up, yelling and clapping. Some even started running up here to shake our hands, and then one right after another had to have us sign autographs. People wanted to take our pictures. I just couldn't know how to take that.\n\n\"That same year, 1995, the _National Geographic_ paid my transportation back to Iwo Jima. They made a documentary film over there, _War Code Navajo._ They sent me a copy. I later found out it won an Emmy award.\"\n\nCORPORAL AL ABBATIELLO\n\n**Combat Engineer  \nTwenty-eighth Marines, Fifth Marine Division**\n\n_If they took Truman out of his grave, I'd kiss his ass right now for dropping that bomb because he saved our hide._\n\nAl Abbatiello, eighty-one, displays the Purple Heart he never wanted and he never wore. He came home from Iwo Jima with two samurai swords, one of which he took off a dead officer near the end of the battle and the second from a supply cave for noncommissioned officers; three Japanese rifles; and a couple of Nambu pistols. He hid them in a trailer that held demolition supplies and had them shipped to Hawaii, then sent home from there.\n\nCorporal Al Abbatiello sports a goatee along with a very large grin shortly after reaching Hawaii on the return from Iwo Jima. \"The big reason I'm smiling,\" he said, \"is because I'm still alive.\" The whiskers came off shortly afterward.\n\n_I met with Al Abbatiello early in 2007 in his Lighthouse Point, Florida, condominium, not far from Fort Lauderdale. He was eighty-one at the time. I was inspired to contact him by the former Marine Corps historian John Ripley, who said, \"Al is one of my favorite Iwo veterans from all the trips I have taken there. He is the real deal, authentic and accurate in all his recounted stories. He was also the stereotypical marine in his day, big, tough, hard, and a real professional. I also like his entire family, all of whom admire him, which says a lot about any family.\"_\n\n\"I was born in New York June 26, 1925. For some reason my mother didn't want to put me in kindergarten when the time came, so she lied about my age and put me in first grade, and as a result, I graduated from Christopher Columbus [High School] in the Bronx in January, before I turned seventeen. I wanted to go into the Marine Corps after my birthday in June, but my dad would not sign the papers. He was the straightest guy. If a nickel ever came out of a phone booth, he'd put it back in again. 'Don't screw around,' he said. 'That's the way you gotta be.' So later the following year, when the draft was coming up, he finally agreed to sign for me. I was one month short of eighteen. It was 1943.\n\n\"I went down the Marine Corps recruiting station and they said enlistments were closed because you couldn't be more than seventeen and ten months, but then they arranged it finally, and I left August 1. I took a train to Yemassee (we used to sing 'Beautiful Beaufort, by the sea, just twenty-eight miles from Yemassee'). We took a truck to Beaufort. Then I actually went by barge out to Parris Island, even though they had the causeway from the mainland. I think they wanted us to feel isolated.\n\n\"Boot camp was tough. I was a six-footer from the Bronx, one hundred eighty-five pounds, and I was in good shape. I did an awful lot of swimming. I was supposed to be a tough guy, so I got picked on all the time by the drill instructor. He'd come over and say, 'You think you could handle this or that guy?' He was always on my butt. I had done a little boxing with one of the guys in a neighborhood youth organization, and somebody mentioned it, so all of sudden the DI said, 'You are the boxer.' You'd compete between platoons. It worked that way.\n\n\"After boot camp we went to New River, part of Camp Lejeune, and that was where they'd give you what they now call your MOS, military occupational specialty, and with a name like Abbatiello, I'm the first one in line. ABBA, you know? The only guy who ever beat me was Aaron, AA. They called my name and said I was going for raider training. That's when they had Carlson's Raiders. I said, 'Oh, my God, I joined the Marine Corps, but I don't want to be a raider,' you know? I had a ten-day furlough, and when we reported back, some guy was reading the roster off, and he looked at me and said, 'What's your background?' I said I'd worked with my dad as a carpenter's helper. He said, 'OK, we're going to put you in engineers. You got a good strong back.' I said to myself, 'Thank God, I don't have to be a raider.'\n\n\"At first we did pontoon bridges, building stuff like that, and then we got into demolition. The majority of demo guys were engineers, and as an engineer, what are you going to do in combat? Build something? No. You blow things up. I took a course and did a lot of demolition work along with everything else. We worked with blocks of TNT. Later on we trained with C2, which was a plastic explosive we used on Iwo Jima most of the time.\n\n\"I was in the Fifth Division, Fifth Engineer Battalion, attached to G Company of the Twenty-eighth Marines as assault demolition. We did flamethrowing, we blasted pillboxes, blockhouses, caves and mines and took care of minefields, anything to do with demolition. Mainly we worked with the frontline troops. We did lay out mats to keep vehicles from getting stuck in the sand. We had armored dozers. One platoon had all the heavy equipment.\n\n\"We used C2 when we could get it. When you couldn't, you used blocks of TNT, which was not as stable. The satchel charges contained six sticks of C2, two inches square. You'd put a fifteen-second delayed detonator in there and then throw it. One stick would blow the others, so the whole sack would go. It was heavy, I'd say ten or fifteen pounds.\n\n\"I landed with two of them when we hit the beach. I had the demolition kit, with all the stuff in it, different fuses, and it was on my chest, and I had two sacks of C2 and the regular half a pack they gave you and a gas mask. I was so covered with stuff when I hit the beach the first thing that went was the gas mask. Everybody threw that away.\n\n\"We were supposed to be reserve, but they called us in early. We must have landed an hour or two after H hour. We got into Higgins boats, and the coxswain was going around in circles, waiting to hit the line of departure, and he said, 'They're walking up the beach in formation; the place is secured.' The first few waves didn't get no resistance. Then, when they called us in, it got hot. I was on the last boat on the left on Green Beach. There was nothing between me and Suribachi, and we were getting all the fire from Suribachi. They were really giving us hell.\n\n\"I had a hole a shell had made, a big crater, so I dove into that. I hit about four guys on the side. Then another guy comes flying in. When he turns around, you can see he's got his helmet on backwards. This was my first combat, but he's a veteran. He says, 'Man, this is a hot fucking place. This is going to be great!' Holy Christ. I said, 'You got your helmet on backwards.' He said, 'Yeah, it gets my face deeper in the sand.'\n\n\"We were right on the base of Suribachi. We went up to the right a little bit. We had a rendezvous area where we were supposed to meet some people. You couldn't just walk there. You were doing flopovers behind dead guys and in and out of holes till you got to where you were supposed to be. We spent the first night right above the beach, then moved across the island the second day. We found the G Company commander by word of mouth. They set up the battalion CP, we dug in, and that was it. I was the assistant squad leader. We moved out to work with G Company on the right side of Suribachi the following morning, blowing up tunnels and cave openings.\n\n\"I got wounded on the twenty-third, the same day the flag went up. Actually I was in battalion aid at that point. We had been working on a cave with a big coastal gun emplacement. One of the guys set a couple of charges up on top because it was surrounded by concrete, and our stuff wouldn't do anything but make a big loud noise. We figured if we could get something up high, we could drop half the mountain on it.\n\n\"The guy with the charge climbed up the side and got it set. We were covering him, and the infantry was covering us. They even brought up a couple tanks to give us cover. Anyway, he got up there and came back down, but the charge didn't go off. Something was wrong with the detonator. So I took a charge myself and climbed up and put it on top of the other charge. I waited a decent amount of time and put it on the charge, and I wanted to get away from there in a hurry. Coming down, I tripped. I slipped, fell, and rolled all the way down. There were huge explosions going all over the place. When I hit the hole, somebody said, 'Oh, my God, your face is gone.' I said, 'What are you talking about?' Turned out I was full of blood. Since I fell, I figured all the pain was from the fall, but actually it was a piece of shrapnel, probably from a Japanese grenade that was rolled down there.\n\n\"They dug something out of my nose and out of the side of my cheek . . . over here. Something ripped out the side of my nose and gum, and my cheek was cut wide open. I thought it came from the fall. The lieutenant checking in on us said, 'Get yourself to the aid station,' so I went over to battalion aid midway across the neck. You know what a million-dollar wound is, where you get hurt, but not bad, but bad enough so you have to pull out? This young kid corpsman was treating me. He had been on the ship with us. He patched me up, a couple Band-Aids, this and that and the other thing. 'You didn't sew it, though,' I said to him. 'A million-dollar wound, huh?' He looked at me and said, 'Get the hell out of here.' I was there the whole thirty-six days and was not hit again.\n\n\"As he was patching me up, somebody said, 'There goes the flag,' so I broke away, and I looked outside. I was still bloody. There were these big field binoculars you could look into, and I saw where first flag had gone up. Nobody paid any attention to the second flag. Later somebody said they put a bigger flag up. Nobody cared.\n\n\"Near the end of the operation, we had secured the island pretty close, and we were mopping up. I had the squad just going around, blasting anything that could be bad. We went out on patrol, and they put a corpsman to go with six of us. It was that same kid who treated my face injury. He was running around all over the place with a forty-five, and my BAR guy said, 'Get him off my back; he's going to shoot me.' Anyway, a charge goes off, and I hear this screaming. There's a big rock right over the corpsman. He's lucky, having just enough space under it so it broke his leg but didn't crush him. We drug him out and sent him to battalion aid, and when we got back that evening, somebody said, 'Hey, a guy wants to see you over in sick bay.' So I go to the battalion aid station, and he's laying on the floor. He's got a cast on him, and he looks at me and goes like this [waves his hand]. I figure he can't talk loud, so I lean over, and he kisses me. He says, 'Million-dollar wound!' I say, 'You son of a bitch!'\n\n\"We worked with C2 till we ran out, and then they dropped us in some dynamite from Guam. Sixty dynamite, it was called, and it's bad stuff because I think it's sixty percent nitro, which is not very stable. Forty is what everybody uses for construction. But they dropped us some sixty, and we were using that to blow up caves, blockhouses, pillboxes until we got some more C2.\n\n\"Here's how we operated: The engineer battalion broke down into companies, then platoons, then squads. My unit, C Company, was attached to the Twenty-eighth Marine Regiment. They had four battalions. I was in the third platoon, so we were attached to the Third Battalion. The first platoon was attached to the First Battalion, the next to the Second, and the fourth to the Fourth. Each platoon works out of a battalion command post. The fourth platoon goes to the Headquarters Battalion. Within the Third Battalion, the first squad goes with G Company, the second squad with H Company, the third with I Company, and the fourth squad goes with the Headquarters Company. That's one platoon.\n\n\"Nobody wore bars or any kind of rank insignia. It was all just numbers. Number 414 was our outfit; it meant engineers. So Number 1414 meant a private in engineers. Mine was 2414 because I was a corporal. It was stenciled, with your name, on the chest of your jacket. Everybody wore the same cover [hat, or helmet].\n\n\"When we got to other end, when there were not so many people around anymore\u2014I'm talking a lot of casualties\u2014we used to bleed over. If another company needed a little more demo, then we'd work with them, although at night we always went back to the battalion CP and stay there as protection for them, plus we'd blast caves around them at night. It wouldn't be dark yet when we'd pull back and dig around the CP, blast caves and stuff, find out what could be the worst thing on the perimeter and lay charges in there. But not after dark. That was the given rule: Nobody got out of that hole after dark. If there was anybody walking around, and it was dark, you shot him. That's the way it went. Then, when morning came, you'd go to your designated area and meet with the head guy of whatever company you were assigned to and he'd tell you where to go.\n\n\"After a while we hit an area where they had sunk these antiboat mines, which were like ones you float, with ears sticking out. Only they were half of a complete mine, and they would bury them in the beach. There was a minefield inland with all these things on it, so they called us in to clear it. When you go for mines, you dig all around with a bayonet to make sure there's no connections that are going to explode when you pull the mine up.\n\n\"The artillery was giving us cover as we were doing that, but then we had a short round, which happened to be a dud, hit behind us. It went flying through this minefield, and luckily it didn't hit any of the mines, but from that moment on we didn't fool around with the bayonet. We just reached out and picked them up. They had handles. There's acid inside, and when you break it, a battery is activated, and that sets off the charge. But you could grab them by the handle and set them to the side, so everybody could go around them. They were semicircular, heavy, and had handles on both sides. Did I ever wish I had gone into the raiders instead? At times, yes.\n\n\"The thing I remember most vividly was the steam and stuff coming out of the ground on the upper part of the island. You couldn't sleep in a foxhole\u2014you got burned. It was that hot. We used to take a canteen and bury it, and it would come up boiling. And, like I said, you never got out of your foxhole at night, but you're sitting there and you're seeing all these things, people moving, but actually it's steam coming out of the ground but you're picturing people. I shot more steam than I did anything else, you know? You very seldom saw the enemy. You never saw them. You could hear them; you could hear them digging. You'd close up the cave, and you could hear them digging at night, trying to get out.\n\n\"I didn't recognize the place when I went back in 2005. The whole island was completely green, and I didn't see one little bit of green in that place when we were there the first time. I did find the cave where we had buried two of my guys. It was just something that bugged us. We had just a few men left in the squad; they had put a sergeant from headquarters in charge, a guy named Joquist. I had the other half of the squad.\n\n\"This was near the end, almost mop-up, you know? They said on March 15 the island was secure, but it wasn't. We had a lot more fighting after that. Anyway, it was near Kitano Point on the north end, and we were coming down around a rock wall, and up there a little bit was a cave, and above it was another cave, a small hole.\n\n\"Joquist turned around, motioned to me, and pointed up. A Jap was trying to get into the other cave. I signaled my BAR guy, and he took a shot at him with the BAR, but the gun jammed on him. The Jap blew himself up with a grenade. We threw smoke in there, and smoke came out around the bend. So a couple of the guys went over there to take care of it. I guess they assumed there was enough rock and debris on top so that if they blasted, it would drop down on the cave. For some reason, two of them went in, and they both got shot. One was killed, and the other crawled out. Why did they go in? I don't know. Maybe to set a charge. To this day we don't know. That would have been a no-no and a half.\n\n\"The guy who got killed was my foxhole buddy John Babich. He looked like a kid. They called him Young Babich. The other guy was Sam Queen, the Mississippi gambler who had come in as a replacement. He was in a different squad, but they had shoved him from headquarters as a replacement into my squad because we had lost three or four guys. This Joquist put a call out to the lieutenant, and he came down with a runner, Madsen, who was originally from my squad. They tried to get Babich out. They pulled Sam Queen out. He had a bullet through his belly, and he said, 'Don't go in; he's dead. Babich is dead. Don't go in there.' But you could see him laying there. The lieutenant came, and they got a piece of pipe, put a rope on it, and tried to get to him.\n\n\"Anyway, Madsen was trying to get a rope on the feet to drag Babich out, and the Japs reached around and shot him in the head. He fell in there, and so now the lieutenant orders the cave shut with a bulldozer. He didn't want to blast it. So we got Kingsbury to come down, and he scraped dirt off the top of it and buried it, closed the cave.\n\n\"When you leave somebody, you got to give the location to Graves and Registration so they can reclaim it. So when I was planning on going back, I always had the\u2014I wanted to go to that cave. I wanted to go see if the thing had been opened and if they got the bodies out. I talked to the lieutenant, he's over in Oregon, and I got his letter here somewhere showing where he had marked the coordinates. He had a letter from Graves and Registration saying they could not locate him. I always wondered did they ever get him out. You hate to leave anybody; you know how the marines are. There was somebody who said they went and saw Babich's grave up in upstate New York, but I never could verify it.\n\n\"When we got to Iwo, my son, Don, and daughter Lisa and I went looking for it. We were part of a Discovery Channel film. I'm ninety-eight percent sure I located the cave, but it didn't look the same because the whole roof had been caved in. The lieutenant had given me the coordinates. It looked like somebody had dug it out, and then the roof had collapsed, probably because they used a backhoe. There were tracks all over the place. There was no sign of Babich or Madsen. I mailed that Discovery tape to my lieutenant. I wanted to verify the site. I haven't heard from him yet.\n\n\"Why didn't we throw smoke to rescue Babich? You don't know why you don't do nothing. First of all, the lieutenant's in charge. This sergeant, Joquist, was the one who summoned the lieutenant, and he made the decision on what we were going to do. After it's all over, you think, why the heck didn't we reach down and flip in a couple smoke grenades or get a flamethrower and turn it around the corner? There were ways of getting them out; it just didn't happen.\n\n\"We blew some of Kuribayashi's cave. There was a small entrance, but inside were a series of rooms that were all connected. My son went in one, and he said it was enormous. He went in that small entrance, passed cutouts for munitions, for a radio room, and it all opened up into an enormous cave. We blasted all those caves hard. There was a blockhouse too we assumed was his headquarters. We had a hell of a time trying to blow that up.\n\n\"My Purple Heart? That's another story. We all refused the Purple Heart. Most of the guys in the outfit had some sort of damage, but they never put in for it. But when I got out, my discharge papers were one year shy in showing the time I had spent overseas. In New York they had a bonus so that the more time you had overseas, the more money you got. So I sent to Washington to have my discharge corrected, and when they fixed the time and wrote me back, they also sent me the Purple Heart. It reads the twenty-second of February but it was actually the twenty-third.\"\n\n_The citation reads: \"Dear Mr. Abbatiello, I am directed by the Commandant of the Marine Corps to inform you that you are entitled to a Purple Heart, awarded in the name of the President and by order of the Secretary of the Navy.\"_\n\n\"I never wore it. I don't even mention it when I go to the convention because every one of those guys rates one. They interviewed my lieutenant, and he said if it was the Army, the whole platoon would have Purple Hearts. The company I was attached to came out with about one hundred ninety men. They supposedly had one hundred eight percent casualties, meaning the entire company was replaced, with a few more thrown in. And my name was first on the wounded list.\n\n\"When we first went in on D-day, I was scared shitless, but then after that I don't know if I was scared or not. I don't remember being frightened. You just hoped it worked out the right way. I did what I was told to do, and I did it when I was told to do it. I'm not saying I was gung ho and not scared, but I don't remember it being so bad that it affected my ability to do what I was supposed to.\n\n\"One instance that happened came when my friend Red Brown got killed. I had half the squad, and we were backup, waiting to rejoin the action, and here came a couple guys dragging Red Brown. So I jumped in and grabbed a hold of him. He was still alive, but he had a bullet in the head. Part of his head was . . . anyway, I got in and helped carry him down to the battalion aid station. The lieutenant, Herb Hammond, was there.\n\n\"And what bugged me so badly was I'm up at the battalion aid station and I had left four guys down there, and here I was the assistant squad leader, when I all at once thought: Oh, my God, the lieutenant's going to think that I'm scared, that I'm yellow. It hit me hard that I had left my post because Red Brown was my best buddy and he's lying there with half his head blown away. He's still alive, still breathing, but he died when we got him up there.\n\n\"Anyway, I took off down the road and went back to them, but that bugged me for years. It always bugged me that maybe they thought I ran out of that area because it was hot, that I had come back to the battalion aid station and left my men down below. I mentioned it to the lieutenant not too long ago, and he said, 'What? Are you kidding me, Abbatiello?' But that always bugged me.\n\n\"They said it was going to cost one million lives to conquer Japan. If they took Truman out of his grave, I'd kiss his ass right now for dropping that bomb, because he saved our hide. I got sent to China, sort of shanghaied, to fix roads for a spell after the war and then got discharged on Mother's Day, May 12, 1946. When I got sent home, they said I would automatically be raised to sergeant if I came back within ninety days.\n\n\"But then I met this gal, Mary Ganguzza, a Sicilian, and that was it. I was a Neapolitan. I loved swimming. We lived right off Pelham Bay. I was down swimming, and she happened to be there with a couple friends. She was a good-looking gal, tough. We got friendly, and it escalated from there. We got married in 1947.\n\n\"We had two daughters, Lisa and Angela, and a son, Don. All live locally. Moved to Florida in 1974 from Puerto Rico. I worked forty years for a contractor as a superintendent in charge of concrete construction. I worked in New York, New Jersey, and then in Puerto Rico for ten or eleven years. We did all the bank buildings. I had the whole family with me. We moved to Florida in 1974. But they kept calling me back to Puerto Rico as a consultant. I was there four years ago; we did an addition to the Hilton.\n\n\"I'm eighty-one now. I retired more or less when I was sixty-six or sixty-seven, but it didn't last long. I'd stay home a couple months, and they would call me for a job, and I would go back. I worked up until just before my wife passed away three and half years ago. She died very peacefully. I went to call her, and she didn't wake up. We were married close to fifty-nine years. I sure miss her.\"\n\n[A newspaper article written in February 1945 by a combat correspondent credited Al's Company C, Fifth Engineer Battalion with a \"a major assist in the seizure of Mount Suribachi.\" He wrote that the company killed six hundred Japanese, \"three times their own number,\" and that they moved in front of the infantry, \"smashing pillboxes, blockhouses, minefields, tank traps and emplacements.\" He added: 'With explosives lashed to their backs and dangling from their belts, the engineers knocked out more than 165 concrete pillboxes and blockhouses; they blasted 15 strong bunkers, naval gun positions and dug up or exploded 1000 mines and booby traps, filled in 200 caves with bulldozers, trapping more than 100 Japs in one cave alone. Some of the caves were three stories high and blocked by heavy reinforced steel doors. In addition, they evacuated hundreds of wounded Marines, buried more than 200 Japs, built more than 1500 yards of roads and tank paths around the crater, saved six trapped tanks in one day, built retreat roads for the tanks with broken ammunition cases, strung barbed wire at night and kept illuminating flares soaring over the enemy.\"\n\n[Asked if this account was accurate, Abbatiello replied, \"I don't know. Like I said, I was only a little part of the company. We did our part, which would be a portion of that. I was just a run-of-the-mill guy doing a job. No big heroics. You did what you had to do.\"]\n\nPRIVATE FIRST CLASS CHARLES WATERHOUSE\n\n**Wounded at Iwo Jima  \nTwenty-eighth Marines, Fifth Marine Division**\n\n_Al Abbatiello was in my squad, the engineers. He was very tough, a tough, tough guy. There used to be an ad in the magazines in the forties, something about shaving cream, how it was \"tough but ever so gentle.\" That was Al. He was big, and he could break you in half, but he was a tender guy. Very nice guy. We keep in touch._\n\nColonel Charles Waterhouse, USMC (Ret.), eighty-two, stands beside his painting of a young marine facing a cemetery and Mount Suribachi, one of several works Waterhouse has done on the Iwo Jima campaign.\n\n_I spent an afternoon with Colonel Waterhouse in his museum in Toms River, New Jersey. He was extremely gracious and cordial, granting me all the time I wished even though his wife was home and not feeling well. We talked at length, and he gave me a tour. I saw action illustrations of the Medal of Honor recipients Hector Cafferata at Fox Hill in Korea and Mitchell Paige at Guadalcanal, among others. He also had a compelling work of Colonel John Ripley hanging off the bridge at Dong Ha. I said I couldn't understand why Ripley had not received the Medal of Honor for his feat that day. He said he couldn't either. I asked how he liked being described as the Norman Rockwell of the Marine Corps. \"That's the ultimate compliment,\" he replied._\n\n\"The first wave shoved off at Iwo at nine o'clock. The waves came in about five minutes apart, and we were in the seventh or eighth. I came in at 0937. We didn't expect to go in that fast, but the first couple waves got in so quickly, zip, they sent us in. Kuribayashi opened up while we were in the water: machine guns, mortars, artillery, everything. We were in the left-flank landing craft, Abbatiello too, and the coxswains got mixed up. They saw this marker they thought was in the center, but it was the left end of the beach.\n\n\"We were fortunate to be on the left flank because the machine-gun fire and stuff were going over us at boats to our right. We landed much closer to it than we were supposed to. When the ramp went down, there were no footprints, but there were lots of people shooting at us from Suribachi. It seemed like we were getting fire from three directions, from Suribachi, from the neck of Iwo, and from up north, by the Quarry. We were immediately in a firefight, all this shooting, and you could not see where any of it came from.\n\n\"We faced a series of at least three terraces. Our orders were to go two hundred fifty yards inland, turn left and dig a hole, and sit there until we had orders what to do. As we went in, we kept edging to the right. We found a nice-sized ridge thirty feet high or so, and that's where we dug in.\n\n\"We saw a few people getting hurt, but luckily nobody from my platoon did. We had a guy named Danaluk from Brooklyn, New York, whose draft number had come up. He wanted to get in the Coast Guard because he lived in Brooklyn and figured he could get a job on a ship patrolling New York Harbor, see? So he said to them, 'I want the Coast Guard.' They said, 'You're in the Marines.' 'No, no, no, I want the Coast Guard.' They finally convinced him he had no say in the matter and that he was going to be a marine. So every morning, as he threw the blankets off, his first words, the first thing he'd say was, 'Oh, that effing [fucking] draft board!' Every day. So, in his honor, when the ramp let down on Green Beach, the whole boatload of us hollered, \"Oh, that effing draft board!\" That was for Danny. The Japs must have thought: Here comes a bunch of nuts. Guys I haven't heard from for thirty years, I'll suddenly get a Christmas card, and it'll say, 'I'll never forget, Oh, that effing draft board!'\n\n\"The first day we made our two hundred fifty yards up the beach; the second day we didn't make fifty yards. The third day we got to the base of Suribachi. Each one of our engineer squads was assigned to a different company in the Twenty-eighth. Harold Pierce was a buck sergeant in my company. The lieutenant said to him, 'There's Easy Company. You're going to support them. Crap out here with your people until I get back and tell you what to do.'\n\n\"So the lieutenant's gone, and somebody walks up behind Pierce and gives him a kick and says, 'Hey, why don't you go down there and blow up some pillboxes or something?'\n\n\"And Pierce says, 'Well, sir, I can't order my squad to do that. My lieutenant told me to stay here with my men until he came back and told us what to do.'\n\n\"Well, the guy who gave Pierce the kick was none other than Colonel Chandler Johnson, they called him Jellybelly, and he sent Pierce and some of the squad up to where Easy Company was going to jump off. Pierce caught up to the rear elements of the assault and said, 'What do you want me to do? I'm a demo man.' And they said, 'Well, there's a big cave over there.' He'd picked up a BAR from a dead marine, and he started for the cave with that.\n\n\"Pierce spent the whole day taking out caves. He wasn't supporting an attack. He was making attacks. Three or four times he came back to get more explosives, and he finally got hit by fragments from a knee mortar. Then a machine gun stitched him up one arm. He wasn't going to have somebody take him back to the aid station. Instead he helped another wounded guy get back. Chandler Johnson has been watching this all day, and at the end he said, 'I sent that guy down there to do something, and he did it. Put him down for a Navy Cross.' A week later Johnson was dead. I think Pierce was from Braintree, Massachusetts.\n\n\"Why did I want to join the Marine Corps? When I was twelve years old, they took us to the library for the first time, a real library. Naturally you want an exciting book, and I'm looking and looking, and finally I see something. It says, ' _Fix Bayonets!_ ' I figured that would be exciting. It was by John Thomason. His writings influenced many, many young men between World War One and World War Two to join the Marine Corps. He treated his characters like they could walk on water. I took it out so often they wouldn't let me take it out anymore. He drew pictures too, and I'd try and copy his pictures. I wanted to be one of those good guys.\n\n\"I was born September 22, 1924, and I grew up in Perth Amboy, New Jersey. I was seventeen when I graduated from high school in 1942. When I got to be eighteen, a group of kids I hung out with, five of us, were going to join the Marines. So we went up to North Jersey to sign up, and we all got separated, and when we came outside, they said, 'Are you in?' 'Yeah, I'm in.' But they're all different. One's in the Air Force, one's in the Army, one's in the Navy, one's in the Seabees. It's fate.\n\n\"It took thirty-three hours to get to Parris Island. They put us on a Higgins boat even though the causeway was built. They wanted to make you unhappy. And while you were going, you'd hear the chant 'You'll be sorreee!' I couldn't imagine what they meant. But then I thought I was in jail; I thought I signed the wrong papers. Boot camp was twelve weeks. I shot expert with the M1, got an extra five bucks a month.\n\n\"After boot camp we got a ten-day leave, then showed up at Camp Lejeune. My application form asked what did I do when I graduated high school. Well, I wrote I'd worked four or five months in the engineering room of a copperworks factory, copying these big old drawings on transparencies to be put in a mountain in Pennsylvania for protection in case the Germans bombed us. So they put me down as a draftsman and said, 'OK, you're an engineer.'\n\n\"I met Abbatiello right away. We moved to a tent camp out in the woods and did all kinds of things. I remember it snowed like crazy. They had us riding around in trucks shoveling out the officers' quarters. They issued us long winter underwear, and everybody's running around singing 'The Man on the Flying Trapeze.' A week after New Year's 1943 we got on a train and went to Camp Pendleton in California. That was where our training really started.\n\n\"The Fourth Marine Division was loading its ships to go as we were unloading our trains. They went right into combat while we trained at Pendleton for a year and a half or more and didn't ship out until September of 1944. We finished training cycles a couple of times, and they still didn't ship us out. We were going to be reserve for the battle of Saipan, but they never called for us.\n\n\"We went to the Second Division's Camp Tarawa on the big island of Hawaii that September, and here's something you never read about: We ended our training cycle that September, and we were supposed to hit Iwo in November, but MacArthur had tied up all the shipping in the Philippines and they couldn't get adequate space aboard ship. This was going be a big deal, eight hundred ships, equipment for three divisions, and we didn't get it. Now you read about how Kuribayashi in that time doubled the number of tunnels, doubled the number of pillboxes. If we had invaded in November, it might have been a different story. It might not have been the greatest battle in the world.\n\n\"It was always the same drill: The Twenty-seventh and Twenty-sixth would be on the right, and we would land in a column of battalions, in a line. We'd train, go so far inland, turn left, and here's this big hill. We would assault it over and over again. It was Suribachi. It got to be like walking in your sleep. You knew exactly what you were supposed to do: Go two hundred fifty yards and turn left.\n\n\"They closed the Green Beaches the afternoon of D-day. A destroyer came damn near into shore, and he kept pumping five-inch shells. There was one huge vertical cave slit in the mountain, and you could see the shells go right inside. The beaches were closed the second day also.\n\n\"I was wounded at one o'clock in the afternoon of the third day. They finally decided to go inland another two hundred fifty yards and make a left turn to support the attack on Suribachi. The two forward holes facing Suribachi were occupied by Barney Barnstein, with the radio, and me. I was thanking God just the other day because if Kuribayashi had said, 'Hey, banzai banzai banzai,' mine was the first hole they would have hit.\n\n\"Anyway, Barney had a radio aerial sticking up, and the machine guns were really working over us two and the foxholes in back. All this lead was flying right above your nose, and when it stopped, the platoon sergeant come running over. He stood right where the beating zone was, where the bullets were flying. He says, 'Waterhouse, you follow the lieutenant. Don't let anything happen to him. We're moving inland.' So I got my stuff together, and by the time I got up there I could see the sergeant and the lieutenant. The sergeant got hit in the legs. They ran with blood pumping out of him, and then he fell down. The lieutenant was on top of him, trying to put first aid on his legs, and the sergeant reaches up and yells to the lieutenant to get down. They were very close. The sergeant had been on Guadalcanal and Cape Gloucester. He was real tough.\n\n\"Meanwhile I get up almost to them, and the lieutenant hollers to everyone to leave their packs in the holes and to come with weapons and shovels. I took my pack off and ran back with it. I'm waving the shovel and the rifle, and I go to jump in the hole, and something hits me in the left shoulder. The corpsman offers to come over, and I say no, but then I figured maybe he'd better because they say if you don't feel it, it's bad. And I didn't feel any pain. I was completely numb in the hand, from the shoulder down. So he ran over and put a dressing on it, and then they had to go. The last guy that went by said, 'Charlie, give me your carbine,' and he hands me his rifle, and they were gone into the smoke.\n\n\"I got out of there in a quiet time and edged sideways and went down beside a pillbox, and I took this rifle and I'm trying to work it. It's all coated with ashes, and it won't work. I wanted to just shoot a clip, just to whom it may concern, you know? At Suribachi. I swore at that guy who took my carbine. If I ever saw him again, I'd kill him.\n\n\"So I never fired a shot. At what? I was the best shot in the platoon, expert, and you couldn't see a thing. It was starting to rain again, so I went down past a thirty-seven-millimeter gun armed with canister, persuaded them not to shoot me, and then made my way down to the beach. It got messier and messier; it looked like a junkyard.\n\n\"There was a hole not quite as big as this table, and it had two shelter halves over it. I figured that must be the aid station. I tried to get in, but these two doctors said it wasn't big enough for any more people. Told me to find myself a spot to crap out. I figured office hours are over, and the doctor's not in, so I looked around for a spot that wasn't littered and lay there. I laid there in the rain thinking if they had a barrage or started sending mortars, it wouldn't look too good. Finally a Higgins boat came in, and they hollered, 'Casualties! Casualties!' People started getting up from different piles of stuff with litters and bringing them, so I shuffled down. I think I was the only ambulatory guy. The rest were all on litters. They closed the ramp and started backing off. I'm in the boat water, and it's red as can be: All the guys are bleeding. I found out what it was like to get seasick immediately.\n\n\"Gradually the sound of the motors were louder than what was going on ashore, but then they couldn't find the ship. They started getting rejection slips from ships that were loaded with casualties, and it was raining. The LCVP [landing craft, vehicle, personnel] crewmen were getting worried. It's getting dark; it's raining. They could float out to the sea, you know? We went the whole range of the beachhead, from Green all the way near the end to Blue. The Fourth Division command ship was there with generals on it, and they took us aboard.\n\n\"And I'm thinking: How the hell am I going to climb the nets? I'm dead from the shoulder down. They sent down a hoist with five or six wooden things on it and started loading stretchers. I climbed on and held on to a greasy cable. When we got to the top, a corpsman lifted me over the rail, and that was it. I was the first man down because I could walk, and they checked me over, then sent me down a hatch. I went down, found a bottom bunk, and I figured this is it, I'm off that [out of the battle].\n\n\"I was in the dark, like, and I felt like an animal licking his wounds in the cave. I went to sleep, and after a while I came to. There's only one little red light on, and here from the sack above me is an arm with a knife. He's swinging this thing, and I'm huddled against the bulkhead and the corpsmen are scattered all over. He's having nightmares: 'You effing Japs!'\n\n\"That scared the heck out of me, but they finally quieted him down and then, about two hours later, I wake up, and here's a guy six feet away, directly across from me, and he's got a submachine gun. The corpsmen are trying to take it from him, and he says, 'The goddamn Japs can't do it. You can't do it.' And he's cocking it, and I'm thinking, holy mackerel. He finally went to sleep with it.\n\n\"Two days later two officers came down and said, 'Guys, we need people on the beach. This is the worst one yet, the very worst, and we need people who think they can go back.' And everybody started, 'Oh, my head's bleeding, oh, my guts, oh, my legs, oh.' But an hour later this bunk's empty, that one's empty, hey, I'm going back. I didn't want to. It's only supposed to take five days, you know. They said it would take three days of fighting, like Tarawa, two days to mop up, and off we go to Guam.\n\n\"I had to see a doctor for real first. It turned out I had a bullet through the shoulder, under the arm, at an angle that cut these nerves. The first thing the doctor said was, 'Make up a cast.' I said, 'Sir, my arm's not broken.' He said, 'Make up a cast.' They put me in a cast with my hand up in the air, fingers curled over. I'm still complaining. What the hell do I need that for?\n\n\"The bullet had cut the nerves, and he knew it. He punched me with needles. I couldn't feel a thing. He knew. Anyway, I get two cartons of cigarettes, a fountain pen, and whatnot in a little bag, and I'm going back to the beach. A half dozen guys had got up on deck early, waiting for the boats to come alongside, and I'm there, and then it hit me: How do I get down the nets? They're not going to lower something just for me.\n\n\"And I cried.\n\n\"It was the only time I ever felt like crying. I prayed: Help me! I never cried again after that. It was a big tragedy for me. I didn't want to go, but I had it all figured out: I know they're going to land on the Fourth Division beaches and I'm going to have to make my way down to Green Beach because those people are still filling in the holes around Suribachi. I know you can pick up helmets and guns and all kind of things [so he can get all the gear he needs].\n\n\"It didn't happen. [Charlie did not go ashore.]\n\n\"I was aboard ship when the flag went up on the fifth day. They announced it on the loudspeaker. The ship had been hit that morning, and I went up to get some fresh air and saw splinters on the bulkhead. Looking ashore, I could see demo teams working in the north. We were on the extreme north, right off the Quarry. I could see teams blowing caves, and I'm thinking: Gee, if I can see them, they can see me, but they were busy. I walked up to the front of ship and looked to the south, and I could see a speck of a flag, and I'm thinking: Hey, that's my people.\n\n\"Later, on March 1, the day the Twenty-eighth came north to attack Hill 362, they sent all the wounded topside, and there was a Higgins boat . . . on deck . . . and they just walked us onto it, lifted it up, set it down in the water, and took us out to a Coast Guard boat loaded with wounded, going to Guam. They transferred us same way, just picked up the boat and set it on deck. That ship didn't have room for anything. It was packed with wounded.\n\n\"We left that night, and next day, during a rainstorm, they buried a number of marines at sea. I went out on the deck, getting soaking wet to watch.\n\n\"On Guam the first thing you know they got me in a cast with my arm raised up with a brace wrapped around the chest to keep it up in the air. I looked like a secret weapon. You know Bob Dole? He's got a hand like that. Once they leave it that way, it stays. With the cast I couldn't get in the sack by myself. I was dangerous. Walking like this, I could knock your brains out. Sleep wasn't easy.\n\n\"I had two operations, one on an artery immediately. This doctor says, 'I haven't checked you.' So he looks me over and says, 'How long have you been walking around?' Since it happened. He said, 'Lie down.' Twenty minutes later I was in the operating room. Turned out that bullet had nicked an artery, and it could have gone, just like that, any minute. He picked it up with the stethoscope.\n\n\"I ended up in a series of hospitals. I flew back to the States, took a medical train to the East Coast, and ended up in St. Albans Hospital on Long Island that July, where they operated on the nerves. I was released from the Marine Corps in May of 1946.\n\n\"I had met my wife-to-be, Barbara Andersen, on a blind date in 1945. When I got out of the car to let her out that night, there was no running board, I still had that cast on, and I fell into the gutter. My future mother-in-law said to her daughter, 'You'll never go out with that drunken marine again.' And I wasn't drunk. We got married June 6, 1948. We had two daughters, Amy and Jane.\n\n\"How did I get into drawing? They had the GI Bill, and you couldn't beat it. If you served three years' time, you qualified for three whole years of study, three hundred and sixty-five days. You could go to college, and they gave ninety dollars a month if you were married, eighty if not. That was a big deal. I had been drawing a cartoon strip of the battle of Iwo Jima, and I went to King Features with it, and they said, 'Christopher Columbus! We just got rid of four years of war, and you're drawing a war thing? Why don't you go to art school?' So I did. I started school the following January 1947. I found out there was an art school in Newark, New Jersey. I figured, that's close to home. I studied for three years, didn't get a degree. I was hired immediately by the Prudential Insurance Company. They had eighteen artists doing everything from cartoons to paintings. It was a marvelous chance for experience. I worked there over five years. Then I figured: Someday I'm going to be an old man, I'll be in my sixties, I'll be smoking a cigar, and I'll have the eighteen people running around, but I won't be making any pictures, and I'll say I always thought I wanted to make pictures for magazines. So I talked it over with my wife, and she said, 'Do it.'\n\n\"It was the day my second daughter, Amy, was born. I went to work, and I said, 'Mr. Gasser'\u2014he was a famous watercolor painter, head of the art department\u2014I said, 'Sir, I quit.' His cigar fell right out of his mouth, just like in the movies. Because they had it all figured out: Hey, he's got another mouth to feed. He's gotta come in for a raise, and we'll say, 'In a little while we'll see that you get a raise.' Instead I quit.\n\n\"I had an arrangement with the director of the school I had gone to. I taught one day a week, a class in illustration, something to base an income on.\n\n\"I had gone out and got some freelance jobs by showing pictures I had done at Prudential. I was working on illustration jobs for three different magazines while Amy was in the hospital with her mother\u2014a story from a detective magazine, a story from a children's grade school publication, and one for an adventure magazine\u2014and they all were due on the same day. One paid on delivery, one on publication, which was four months later, and one had a different deadline, so you were never sure when you were going to get paid. I did that for twenty-odd years. I didn't work the slick women's things, but _Saga, Argosy, Man's True Whatever, True Man's Whatever._ They were always starting magazines. If they caught on in four months, that was OK, but if they didn't, they dropped them. You always hoped you got paid before they dropped them.\n\n\"I belonged to the Society of Illustrators and the Salmagundi Club, and both had connections with the military, and you could get jobs to go paint weather balloons or ships, so when you did something for a magazine about ships, you had some idea of what to do. It was cashless, but you got experience.\n\n\"Then one day this friend of mine said, 'Hey, Charlie, why don't you go to Vietnam?' He worked for the Air Force. The war was on then, and he had just come back. He was in the Society of Illustrators. I said, 'I don't paint airplanes; I paint marines.' He said, 'Well, go down to the Salmagundi Club.' I sent down a portfolio and got a call from the man who ran the program, and he asked if would go to Vietnam.\n\n\"I said, 'I will if my wife will let me.' This was in January of 1967, and I went for a month and a half, February and March. I did a book of sketches and things. I figured I could use some of the ideas. Vietnam was just becoming a dirty word, and I had people saying, 'I shouldn't look at this,' or what was I doing? Was I trying to sell the war? I said, 'No, I'm trying to show that I can draw pictures on the spot.'\n\n\"I went a second time for the Air Force, flew on a B-52 mission over Laos, hitting some Ho Chi Minh trails, a thirteen-hour roundtrip from Guam. We refueled over the Philippines. I went a third time for Navy medicine, and the first person I met was a Navy corpsman. He said, 'Come on down to my village. I've got five marine advisers.' So off I went. There was a patrol boat a hundred yards across a very muddy beach. He said, 'OK, follow me.' I'm going in mud sucking up to my knees, slothering in it with my paint stuff.\n\n\"Three years later he calls, says he wants to come visit me with his Vietnamese wife. He says, 'You know, Mr. Waterhouse, I hated you. I thought you were one of those damn media people.' He says, 'There was a dock out there we could have walked out on, but I made you walk through the mud.' I said, 'You had to walk through the damn stuff too.'\n\n\"Eventually I went all over. I ended up on missile cruisers, carriers, destroyers. They lowered me in a monsoon from a helicopter to the fantail of a destroyer, and when my feet hit the deck, a wave hit me this high. I thought: I'm not going to like this. I was seasick from that moment until the next morning, when a helicopter came, dropped a life preserver, and took me off.\n\n\"The pay was terrible, fifteen dollars per diem in the States, and when I got to Vietnam it was nine dollars and fifty cents, and then, when I got back, the Navy Department docked me four dollars fifty cents for chow and a buck a night for quarters, even though I slept in foxholes at times, five hundred yards from the demilitarized zone. I didn't go with the idea of making money, but I always thought at least Barbara would get a check while I was gone, to buy some groceries. It didn't happen.\n\n\"Finally I get a call from a Major Dyer at the Marine Corps Museum in Washington, asking me to paint something for the bicentennial. I said sure. Then he said, 'Write us a letter saying you can't. We want to convince the powers that be that this has to be funded.' So I wrote a letter to General Simmons, said I was thoroughly professional, had worked at all the magazines and I had a wife, two daughters, two cats, and a dog, and we all ate, and while I would love to do this, I could not accept the job unless it was funded.\n\n\"I had a double-page book jacket for the Rutgers Press showing Washington and his men slogging through the snow, going to fight the battle of Princeton. This thing arrived while I was writing the letter, so when I finished, I folded up one of the proofs and put it in there. The general was impressed, thank God. If it had been Chesty Puller, he would have said, 'What the hell do we need pictures for?'\n\n\"A week later Major Dyer calls and asked me to make seven paintings. I said, 'Last week it was paint one, then don't do any, and now it's seven. I'm confused.' He said, 'So are we.'\n\n\"Then he says, 'Could you come on active duty?' I said, 'Well, if my wife doesn't mind.' So now I'm forty-seven years old, and I'm going to the draft board in Newark to get drafted and all these eighteen-year-old bushy-headed kids are standing in line. It was just like a TV show. 'What the hell do you want?' I says I come for my physical. You know how Jack Benny used to do double takes? Everybody's head turned. They thought it was _Candid Camera._ It took four months, paperwork, FBI checks, all kinds of junk.\n\n\"And then it was, 'Oh, by the way, sir, we can only make you a major.' At the end of the war, I had been a corporal, just like Hitler and Napoleon had started, and I thought if they can do it, I can do it. They weren't marines, though.\n\n\"I didn't have to do any training. I'd have been dead. First day I came down, they drove me to lunch and there's thirty marines running in their red skivvies, and they say, 'Here's what we do at lunchtime, Major.' I said, 'You do that to me, and I'll be in Arlington, not running through it.'\n\n\"I became the only artist in residence the Marines ever had. I always loved historical things, and I did over five hundred paintings for the Corps. When I came on duty, they asked me what I liked to paint, and how big. I said I paint historical pictures, and I like 'em big. The first picture had fifty-three people in it. All my work went to the museum from that point on. It is not on exhibit, however. It's in Washington, D.C., in the archives.\n\n\"The new museum at Quantico? They're exhibiting one picture, probably. I had a wonderful relationship with the Marine Corps. I did dozens of covers for the _Marine Corps Gazette,_ for officers, for _Leatherneck,_ which was for enlisted, and for the _Marine Corps League_ magazine.\n\n\"Everything is acrylic. Sometimes it's thin, like watercolor; sometimes it's black and white, it's just black acrylic instead of ink. It's the magic medium for me. All art students and artists look for the one that works; well, this works. Acrylic won't turn black; it won't crack and eat the canvas.\n\n\"I retired for the second time 19 February 1991. I had seven guys from my platoon at retirement at the museum in Washington, D.C. The commandant, Al Gray, made a surprise visit, gave a speech. He knew more about me than I did. He was very nice to us.\n\n\"My museum in Toms River opened six years ago. I needed a place to hang stuff. It's all supported by contributions. I'm working now on a series featuring Marine Corps recipients of the Medal of Honor.\n\n\"I've been back to Iwo Jima three or four times. I spent at least a week there on two occasions. I painted a number of little pictures on the spot, including the admiral's cave. I got friendly with some Japanese who were there, including one who had been wounded twenty-seven times. He had an artificial nose. He said it was 'made in the USA.' \"\nPART FOUR\n\nThe Flag Raisings\n\n**Inseparable from the campaign** to conquer Iwo Jima is the photograph by Joe Rosenthal of the flag being raised on Mount Suribachi. Five marines and a Navy corpsman put it up on February 23, 1945, five days after D-day. Back in the states, the photograph was an instant sensation. It appeared on the front page of dozens of newspapers across the country on Sunday, February 25. The success of Rosenthal's image obscured the fact that it merely recorded the second flag raising on the mountain. Six other marines had erected the first flag a few hours earlier. The planting of the second flag was a routine incident, not even mentioned in the daily logs.\n\nThat first flag, fifty-four inches long and twenty-eight inches deep, was put up by Lieutenant Harold Schrier, Platoon Sergeant Ernest T. Thomas, Sergeant Henry O. Hansen, Corporal Charles W. Lindberg, and Privates First Class Louis C. Charlo and James Michels. Three of the marines who raised the first flag were to die in combat, just as three from the second flag raising were also killed. Staff Sergeant Lou Lowery, a staff photographer for the Marine Corps magazine _Leatherneck,_ took a picture of three marines tying the first flag to a pipe and another after it was in place, with some marines standing around. The second photo is static and uninspired.\n\nLindberg, the last living member of the twelve flag raisers, told me shortly before he died in June 2007 that it went up at 10:30 a.m. Much was made in the press at the time about the ferocious fighting the patrol endured in order to get the flag to the summit, but it was all grossly exaggerated. Both the first and the second group of flag raisers scaled the 556-foot elevation with minimal interference from the enemy.\n\nNo one is quite sure why they had such an easy time. There were still plenty of Japanese in various caves and tunnels all the way to the summit. For whatever reason, however, none emerged until after the first flag had gone up. Lindberg thought the Japanese troops stayed inside because Suribachi had taken such a terrible pounding from the ships before the patrols ascended its slopes. In any case, most of the fight for Suribachi occurred around its base in the four days preceding the flag raisings.\n\nThe second flag, which was eight feet long and four feet eight inches deep, almost twice the size of the first, was raised early in the afternoon. Present besides Rosenthal were Sergeant Bill Genaust, a Marine motion-picture photographer, and Private Bob Campbell, a still photographer. Rosenthal took several pictures, including what he called a posed \"gung ho\" group shot. The story of this posed photograph later helped foster confusion over the assertion that the famous flag raising shot had been staged. Genaust shot a sequence of the flag going up that later confirmed the validity of Rosenthal's photo. Rosenthal took his picture with a 4 x 5 Speed Graphic at 1\/400th of a second, lens aperture between f8 and f11 on sheet No. 10 of a twelve-sheet film pack, according to Hal Buell. Campbell got a nice shot showing the new flag in place while the first one was being gathered in.\n\nHal Buell, who spent forty years as a photo editor with the Associated Press, including twenty-five as the head man, reports in his authoritative book _Uncommon Valor, Common Virtue: Iwo Jima and the Photograph that Captured America_ that three other photographers were present before the second flag went up.\n\nOf course it was Rosenthal's photo that got the attention. Three of the flag raisers\u2014the Navy corpsman Pharmacist's Mate Second Class John H. Bradley, Private First Class Ira H. Hayes, and Private First Class Rene A. Gagnon\u2014were later brought home and feted as heroes as they led a nationwide bond drive to help raise money for the war effort. The other three\u2014Private First Class Franklin R. Sousley, Sergeant Mike Strank, and Corporal Harlon H. Block\u2014were all killed in the weeks following the flag raising.\n\nLowery initially resented the attention Rosenthal received and thought his shot had to be a fake because he wasn't even there when the first flag went up. Chuck Lindberg too maintained that Rosenthal was a self-promoter who helped set it all up, but this was not the case. Rosenthal was just a news photographer, trying to get a good shot, and things went his way. His material got back to Guam quickly, got sorted out and sent to the States right away, where it became an immediate hit.\n\nI interviewed Joe Rosenthal in 1995, when _Parade_ magazine ran his photo on its cover to accompany a fiftieth anniversary piece on the battle of Iwo Jima, written by James Brady. You always hear about famous photos being staged, so we ran a sidebar with Joe explaining how he had come to get the shot, emphasizing that it had not been posed. \"I backed off thirty-five feet and stood on a couple of old Japanese sandbags and some rocks,\" he told me. It was around noon, he said. \"The light from above tends to sculpt your subject. The tension in the picture comes from the weight of the pole, which was about one hundred fifty pounds. Other guys jumped in to help when they saw it was heavy. I never would have used that many in the shot.\"\n\nSince he worked for the AP when he took the photo, the agency owned the rights. He got a raise, though, and won a Pulitzer Prize. Ten years later, on February 18, 1955, he told the full story with W. C. Heinz in an article for _Collier's_ magazine. Rosenthal downplayed his part in the story, and his last line was the best. Yes, he said, he took the picture, but the marines took the island. Rosenthal went on to a full career with the _San Francisco Chronicle._\n\nThough the Associated Press owns the copyright to the photograph, the Marine Corps has the right to distribute it. According to Hal Buell, proceeds from commercial use go to the Navy Relief Fund. Though it is one of the most memorable images of the twentieth century, the AP doesn't make any money off it. \"Everybody knew the picture was going to be around forever,\" says Buell.\n\nRosenthal was \"a delightful human being, witty, insightful, easy to talk to, very friendly, and unassuming to a point of distraction,\" Buell recalls. He was a small man, five feet five, 130 pounds. He wore horn-rimmed glasses. He had volunteered for all the services, and they all had turned him down because his eyesight was so bad. Buell adds, \"As I say in the book, he was painfully modest. There was a story I heard after I did the book: He'd been out on assignment with some young reporter at the _Chronicle_ , and she introduced Joe this way: 'You know he's the man who made that famous picture.' And Joe took her aside and said, 'Don't ever do that again. Don't introduce me that way. I'm Joe Rosenthal, a photographer for the _San Francisco Chronicle_.' He was very modest about it; I used to give him a hard time because he always said the picture was a lucky shot.\n\n\"Serendipity is part of every picture ever made, but there's a lot more to it than that. He wanted at first to make a picture of one flag going up, the other one coming down, and then he thought: No, that wouldn't, might not work. It was too much like a football play. Go this way or that way, and you could miss it. So he went around to the front, then had to step back so he could get the whole swing of the flag. If you notice in the [Genaust] film, the flag goes up and out of the frame, and he didn't want that to happen, so he stepped down.\n\n\"Then he built a little platform that cut down the foreground, and then Genaust walks in front of him and they get involved in a 'Am I in your way?' 'No, you're not\u2014whoops, there it goes,' and he raised the camera and\u2014bang\u2014right at the perfect peak of the action. So there's a lot of skillful photography involved. It wasn't lucky. There was some luck, but it was mostly Joe making a good picture, taking the time to make it right.\n\n\"Some photographers go to war because it's a way to make a career; a great war picture can make a career. Some photographers go because it's a way to make a buck; there's a story to be covered, and you can sell your pictures and make some money. And some photographers go because they sincerely feel that the story should be told because they can help change the world. A lot of photographers are motivated by that. Photographers are basically emotional journalists. The nature of the work is emotional, instinctive, and intuitive, and so the mind works that way all the time. There's a swashbuckling element involved as well, something that takes you to the place where the action occurs.\n\n\"Joe's flag, the second flag, flew till March 15. There's a lot of irony in that too. They said the island was now secure. So they declared a headquarters company and put a flag up, and that became the flag on Iwo Jima, and the flag on the mountain was taken down. But there were a lot of people died after March 15.\" Both flags now hang in the Marine Corps Museum in Quantico.\n\n\"You know Rosenthal's favorite picture from the invasion days is not that good, but Joe was crazy about it. [The photo shows two slain marines in the foreground, with another marine advancing over the sand from right to left.] The reason he liked it was, he said, he was so impressed by the fact that no matter how many marines fell, they just kept coming. They just kept coming. They just kept coming. And I think that's why they weren't annihilated, because the Japanese couldn't kill them all. There were too many of them. Once they got up off the beach, it was a little tougher to get them.\"\n\nJoe Rosenthal died in California on Sunday, August 20, 2006. He was ninety-four.\n\nWARRANT OFFICER NORMAN HATCH\n\n**Photo Officer, Fifth Marine Division**\n\n_[T]he brave ones were shooting the enemy; the crazy ones were shooting film._\n\nNorman Hatch, eighty-seven, in the study of his Alexandria, Virginia, home in March 2007, in front of a poster for the movie _Sands of Iwo Jima_ with John Wayne. Behind and to his right is a colored chalk portrait of the twenty-three-year-old Hatch, then a tech sergeant, in 1942, after he returned to Boston from Tarawa. It was commissioned by an uncle.\n\n_A cassette tape of \"Pennsylvania 6-5000,\" a tune made popular during the war by Glenn Miller, began playing in Norman Hatch's car as he started it up. He was driving me to the Metro station in Alexandria, Virginia, so I could catch the subway to my train at Union Station in Washington, D.C., in mid-March 2007. We had just spent nearly three hours in his basement office, a long railroad car of a room filled with memorabilia and files. His backyard contained several very large boxwood shrubs, which he said had been very small when he planted them after he and his wife, Lois, moved into the house in 1951. On the way to the Metro, he said something about how you just have to take chances in life and make things come to you, even if you do fail now and then._\n\nPerhaps not many people know the Marine Corps won an Oscar, but in 1944 it was awarded an Academy Award for the best short documentary of 1944, _With the Marines at Tarawa._ One of ten men assigned to film the bloody invasion, which took the lives of one thousand marines and forty-seven hundred Japanese, was Norman Hatch, who is credited as the cinematographer. Hatch also landed on Iwo Jima on D-day, February 19, two years later and stayed there eighteen days until he left for Washington to play a crucial role in sorting out for the commandant of the Marine Corps charges that Joe Rosenthal's photograph on Suribachi was faked.\n\nThe invasion of Tarawa took place November 20, 1943.\n\n\"I went in practically with the first wave. I was the only cameraman to get on the beach for the first day and half. It was not the fault of the other cameramen. After the first three waves of amphibious tractors hit the beach, all hell broke loose. It wasn't just Kuribayashi on Iwo who did this. The Japanese always waited for you to get on the beach. That was their philosophy: Let 'em get on the beach, and then annihilate them.\n\n\"I was sitting on the engine hatch of a Higgins boat with Major Jim Crowe, the commander of the Second Battalion, Eighth Marines, and he saw he was losing his beachhead there at Tarawa. He was not due to come in until later. His exec was on the beach with the first waves, but they were losing the beachfront because a Japanese guy in a tank turret buried in the sand was firing a machine gun at the amtracs. They were like tin; they didn't have any armor, and those thirty-caliber and fifty-caliber rounds would go right through them. The drivers were getting scared and pinching in away from the gun, and Major Crowe said, 'Jesus Christ, I'm losing my beachhead. Coxswain, put this goddamn boat in right now!'\n\n\"So we were the only boat to hit the beach at that particular time. We got in and walked ashore with no problem. A couple guys got hit and dropped in the water. My assistant, Obie Newcomb, and I are lugging black-and-white camera gear on our shoulders, and we get onto the beach, and then the fire opens up. We could lie there dug in and watch a landing craft come in, drop the ramp, and then a shell would land right on the ramp.\n\n\"The Japanese knew where the reef was, and they knew where the boats would have to stop, and they had it all zeroed in. After they blasted a half a dozen boats, the guys on the beach radioed out to the ship and said, 'Jesus, don't send any more until we figure this thing out.' I doubt we had six hundred or seven hundred men on the beach at that stage of the game.\n\n\"Anyway, all the rest of the Higgins boats, which had been loaded up at nine o'clock in the morning, were held off overnight, so everybody had to spend the night in those things getting seasick and dashed with spray. And there I am on the beach. I don't know that these guys are all afloat, and I'm shooting the war as it's happening pretty much in front of me.\n\n\"It winds up that I've got the only real combat footage of the fight because, when they come in about ten o'clock the next morning, there's some fighting to be done but it's mostly wrap-up stuff. The first nine hundred guys who got ashore did most of the action, and I filmed it in black and white. Why is the movie in sixteen-millimeter color? The kids that we trained shot color, a lot of the boat stuff, island shots, and planes dropping bombs, but when you get to the combat, it's my thirty-five-millimeter black and white that you see. We tinted it so there wouldn't be such a sharp contrast.\n\n\"I was able to catch the only shot, not just in the Pacific but the European war, that had both the enemy and our guys in the same frame of footage. I'd attended a briefing on Tarawa with Major Crowe and with the battalion exec, Major Bill Chamberlain, who said to me, 'We're going to take that big blockhouse up there. Do you want to come and cover it?' Well, you know, a staff sergeant doesn't argue with a superior, so I said sure. We had been sitting there in his CP, his command post, which was a big shell hole, and he's got his senior NCOs and a couple of officers, and they plan the attack. What they're going to do is go over the top of the blockhouse and do some damage to the air vents, drop grenades down the smoke pipes, and it's going to happen at nine o'clock. It was just like a World War One movie, with everybody synchronizing their watches.\n\n\"All the people went back to their command units, and Chamberlain looks at me and says, 'Are you ready?' I says, 'Yup.' He gets up and yells, 'Follow me!' He waves his hand, and the two of us run up, climb onto the top of this thing and get all the way across and look down the other side, and there's a dozen Japanese looking up at us, wondering what the hell we're doing on top of their blockhouse.\n\n\"We turn around, and there's nobody in sight. We're the only two people on top of the world entirely. And I looked at the major. He didn't have a weapon. I said, 'Where the hell is your rifle?' He said, 'I gave it to somebody that lost his coming in.' Where was his pistol? He lost it. I had one, but it was in back of me behind all the camera gear. And I said, 'I think we better get the hell out of here.' He agreed. We ran back and down off the side. He held another meeting, with a little ass chewing, and then we started to take the place. I was standing at the foot of it at that point in time, filming some stuff, and somebody yells. 'Here come the Japs!' So I just stood in position, moved over, and that's when I got the shot. I was within thirty or forty feet of the enemy.\n\n\"I had a marine in the foreground, pretty much in the middle, with a machine gun, while the Japanese can be seen running across from right to left in the background. When I got back to Pearl Harbor, I sent a telegram to my wife saying I think I got a shot of both of our forces fighting and, if I did, it's a lucky shot. It turned out that way. Andy Warhol said everybody should have his fifteen minutes of fame. Well, I got fifteen seconds of fame. That's about the length of time it took, sixteen seconds, to go through the camera. Later people asked me how I could walk through a battle like that, taking pictures as I went, and that was when I said the brave ones were shooting the enemy; the crazy ones were shooting film. You have to remember the island of Tarawa is only about one-third the size of Central Park in midtown Manhattan. Within seventy-six hours, over forty-seven hundred Japanese and one thousand marines were killed. We only took seventeen Japanese prisoners.\"\n\nThe color film went to Washington, where it was put together by Warner Brothers' Office of War Information. Hatch's black-and-white went to San Francisco and eventually wound up in newsreels that were shown in movie theaters, which is how Hatch's name ended up on marquees. \"Of course, being a dumb photographer, I never took a picture of it,\" he recalled. Hatch eventually worked with the director Frank Capra to combine Hatch's three thousand feet of black-and-white with the color footage to make a training film, _Army-Navy Screen Magazine No. 21,_ to be shown to troops all over the world.\n\n\"Tarawa had hit the press in a heavy way with all those casualties occurring in such a short space of time. It was also the first time in the history of the world that anybody had assaulted a heavily fortified beachhead from the sea and succeeded. There's a rumor\u2014I can't validate it, but it's a hell of a good story\u2014that Churchill was having trouble with Eisenhower and his own staff about the landings they were planning on the French coast. He couldn't get them to agree that it should be done. He was watching what happened at Tarawa, and after he saw we had done it, he came to both commands, Eisenhower and the Brits, and said if they can do it, we can do it. Did that happen? I don't know, but it's a hell of a good story.\n\n\"I was born March 2, 1921, in Gloucester, Massachusetts. I enlisted July 7, 1939. My father had told me I should join the Navy because it could teach me a vocation, and I signed up, but they kept putting me off. They were only accepting twelve or fifteen a month from all of New England. In June of '39 I walked in again and said, 'When are you going to take me?' I'd been doing odd jobs to earn a little money. I bought a Ford dump truck for seventy-five bucks with another guy, and we were hauling loam all over the city of Boston. The Navy rep told me it would be another three to five months, so on the way out I went by the Marine Corps office, walked in on a whim, and said to the sergeant, 'If I wanted to join the Marine Corps, how long would it be before you took me?' He said, 'Do you want to leave Friday or two weeks from Friday?'\n\n\"I left on the seventh of July 1939. I was eighteen. When we got off the train at Yemassee, we were met by a sergeant who picked us up and took us by gig, by boat, from Beaufort to Parris Island. My senior drill instructor was a sergeant named Johnny Watkins. He wasn't cruel, but he had his own ways. He carried the steel shaft of a golf club, which he would use to whap you on the back if you didn't do things just right. In the early stages of training, if you didn't do a facing maneuver properly, while marching, he would likely run at you and jump and hit you in the middle of the back with his elbows and his knees. You'd have a rifle, you'd go flying, and you might take three guys with you. And he'd say, 'Now will you do it correctly?' He was nimble. He was a little guy too. There was a kid, a little Jewish boy, who couldn't get anything right. Drilling to him was just anathema, and so Watkins had him pushing a peanut with his nose on the sidewalk all the way around the barracks.\n\n\"One time Watkins didn't like the way we were drilling, so he had us go behind the barracks, stack our rifles, then march out into the bay. It was all mud, just icky, gooey mud, and you'd get it up to about waist high, and you'd begin to founder. You couldn't stand up straight. Some of the kids were getting scared because they really couldn't swim that well. Finally we turned around and walked in and took showers with all our clothes on.\n\n\"Another day I was corporal of the barracks guard for the evening, and Watkins came in and said, 'Will you mail this letter for me tomorrow?' I saw it was addressed to Mrs. Johnny Watkins, such and such, Shanghai, China. But Johnny also had a wife over in Beaufort. That's Old Corps for you. They could get away with things like that. There would be girls coming on the base to clean various offices, and when they passed you, they would say very quietly, so the DI couldn't hear them, 'Ah'm available for twenty-five cents; Ah'm available for twenty-five cents.' They knew all these guys would be horny anyway, and they were just trying to get some guys to slip out, which was damn near impossible. Parris Island was something else in those days.\n\n\"Boot camp ended in September, and I was scheduled to go to sea school in Norfolk, which meant I'd be trained in how to react aboard a ship and wear blues and look pretty all day long. But coming from Gloucester, I'd spent a lot of time aboard ship, and I knew what shipboard life was like, and I thought: Hell, I don't want to get sent into that. I didn't think I could plan my future very well that way.\n\n\"While I was in boot camp, they told us to religiously read the bulletin board every day because it would tell you what your commander was doing and what the Marine Corps was doing. I followed this advice and saw an opening for an English instructor at the Marine Corps Institute at Eighth and I in Washington, D.C. I applied and forgot about it, and two days before I was supposed to ship out for Norfolk, Johnny Watkins comes up to me with a piece of paper and says, 'God damn it, Hatch, what have you been doing? This says you're going to teach English. What the hell kind of job is that for a marine?' \"\n\nNorman Hatch then found himself in a succession of jobs in the Marine Corps, from English teacher to an assistant editor at _Leatherneck_ to public affairs, at which time he got to know the radio personality Arthur Godfrey, and finally, on October 1, 1941, to a class in photojournalism in New York City, with an emphasis on motion pictures and still photography. His film career began with another trip to the bulletin board, but it took three tries. Nothing came of these three trips. \"Everything sort of happened because I either slept next to somebody or knew somebody,\" he said. He was lucky enough to meet an important person. In this case it was the director Louis de Rochemont, who got him into the class. The idea was to learn how to tell a story with a camera and not just settle for the \"grip and grin.\"\n\n\"Even back in high school I'd liked photography. To get our so-called degree, in New York, we had to take one roll of film out into the city and come back with a complete story done in one roll, which ran through the camera at ninety feet a minute and there was only one hundred feet in there. I went out to one of these concrete chess boards in Manhattan where a lot of old men sat and played. That was my story. You had to learn to think out ahead what you were going to do.\n\n\"From there I went to Quantico, where we were doing training films, and all of a sudden two of us were ordered to the Second Marine Division. I was promoted to corporal, and we were supposed to be on the train to California Saturday night. I came up to Washington and told my girlfriend, Lois, I was getting ready to go, so we decided to get married that Saturday morning, September 19, 1942. Some cousins drove us down so she could say good-bye to me at the train. I called Johnny Ercole and said, 'Pack everything. I'm not coming back to the barracks.'\n\n\"We were at Camp Elliott while the Second Division completed its forming, and then we sailed for New Zealand in late September. We were there eleven months and then landed at Tarawa November 20, 1943. The director John Ford had won an Academy Award in 1942 and again in 1943, for his filming of the battle of Midway, and we used to say that when we got to the front, we were going to make a film and get an Academy Award, just like Ford.\n\n\"After the filming at Tarawa and my return to the States and the war loan drive, I wound up as a photo officer with the Fifth Division on the big island of Hawaii the summer of 1944. I had insisted that we forget the thirty-five-millimeter black-and-white camera and go to the sixteen-millimeter color. It was a lot easier to carry, and the success of the Tarawa film proved its worth. So we were the first ones in the services to go completely all color. We were on the Parker ranch, one of the largest in the world, training for Iwo even though we didn't know it. I made frequent trips back and forth to Oahu for staff meetings and got to know Herb Schlosberg, the photo officer for the Fourth Division, and he and I became very good friends. I had been promoted to warrant officer by then, the highest rank you can reach as an enlisted man.\"\n\nBill Genaust, who shot the motion-picture film of the second flag raising, joined Hatch's group in Hawaii.\n\n\"Genaust was one of the supernumeraries I got in the pool from Cincpac. We knew action was coming, and I had sent a request to the Marine Pacific headquarters at Pearl for more photographers and motion-picture cameramen for the next action. We ended up with probably ninety cameramen on the beach at Iwo, thirty for each division. Genaust had been wounded on Saipan and had come back on active duty. He didn't want to return to the States, so he was assigned to me when I requested more people. I wrote up the order of photographers prior to taking off from Hilo. We sent him up Suribachi even though nobody knew for sure if there was even going to be a second flag raising.\n\n\"I went ashore on Iwo Jima right after the first wave. Once I knew the philosophy of the Japanese, I promised myself I'd always go in the first wave, I wanted to get on the beach and dig in before things hit the fan, and sure enough, that's what happened. They let 'em get on the beach and gave them hell after that.\n\n\"I was near Suribachi when we hit the beach. I had been briefed by the colonel as to where they were going to set up once they got ashore, and it was right near the tail end of Motoyama No. 1. I had Obadiah Newcomb, the still man who had been with me on Tarawa, as my assistant photo officer now. I said, 'Obie, we got to get up there by Motoyama No. 1. I don't know how we're going to make it.' We had the broad expanse of Iwo in front of us. We'd gotten up over the hump, and I said to Obie, 'I want you to look out behind us, and I'll look in front of us. We don't want some goddamn Nip coming out of a hole in the ground and shooting us in the back. We'll take pictures as we move.' We didn't have any trouble. We walked right up to Motoyama No. 1, probably twenty minutes after we landed. I found a fifty-caliber gun mount circled by fifty-gallon drums loaded with sand. The mount was sticking up, but the gun wasn't there. It made a perfect tent pole, and I covered the whole thing with a big tarp I found nearby. That was my command post.\n\n\"I was all dug in and Obie was out shooting stills when finally the guys began to infiltrate in. They'd also been told we'd be somewhere near the end of Motoyama No. 1. We had ten guys shooting motion-picture film. The rest were still men except for a camera repair guy.\n\n\"The Navy had noticed in looking at all the film from previous actions that everybody's footage was the same in every battle: Guys running here, guys running there. So we determined to select and assign various topics to various men. I had a tank battalion, a medical battalion, and a signal battalion I would cover. I had one still man and one movie man assigned to them and nothing else. Out of that came several good training films. One film has a very distinctive name, _Glamour Gal,_ which was painted on the barrel of a howitzer. We also had free roamers, guys attached to a unit, a battalion or a regiment, to document whatever was going on. This way we avoided duplication.\n\n\"The whole thing was planned out to the extent that three times a day an LCVP came in from the command ship. It had great big letters across both sides: PRESS. Anybody, meaning civilian press or our public affairs writers or combat correspondents, would go down to that boat and turn their stuff in. It would go to the command ship and then be packaged individually and flown to Guam, where the Navy had a big photo lab set up. They processed all the film. There was total censorship all through the Pacific war, and all military and civilian stuff had to be looked at before it was released.\n\n\"That's how Joe Rosenthal's shot of the second flag raising and his other stuff was sent in. He went back to the ship in the press boat that night. He went back and forth for four days. There wasn't any sense in him sleeping on the beach at nighttime, so he'd go back, write his captions, turn his stuff in, and it'd be flown to Guam. So he never saw it until he arrived there around D plus eight. Herb Schlosberg, the photo officer for the Fourth, was on that boat with him. Schlosberg had orders to take all the Marine motion-picture film from D plus one to D plus eight to Washington and to Hollywood, where Warner Brothers could edit the film. We figured it would be smart to have the film edited and ready by the time the battle was over. I took the film between D plus eight and eighteen to Washington.\n\n\"For the second flag raising, my boss, Colonel George Roll, the division intelligence officer, came to me and said General Rockey wanted a bigger flag up there because nobody could see the first one, it was so small. I said, 'Where the hell are they gonna get a bigger flag? Nobody's carrying any holiday or Sunday flags.' He said there were LSTs on the beach, and maybe they would find one there. The official Marine Corps position is quite simple: Lieutenant Tuttle, who was an aide to the battalion commander, was sent out to get some supplies from some of those ships and to look at the same time for a larger flag. He was lucky because LST 799, I think it is, had one, and the lieutenant JG gave it to him. It was eight feet long, and if he'd tried to fly it on the stern of the LST, it probably would have dragged in the water.\"\n\nHatch, aware that the marines were going to try to raise a second flag, knew he ought to have someone there to film it.\n\n\"I didn't have anybody to send out, but then in walked Bill Genaust and Bob Campbell to turn in film and get new stuff. I had assigned them to the Twenty-eighth Marines.\n\n\"I said, 'You guys have to turn around and go right back. There's a possibility that a second flag is going to be put up. Just go up there and sit on top of the mountain and shoot whatever you see, and if a flag goes up, shoot that too.'\n\n\"They headed back down. It was kind of dangerous, getting from my CP at the foot of Motoyama No. 1 airstrip over to the mountain. It was a good quarter mile, and there was nothing but firefights going on. But they got over there and ran into Rosenthal going up. He and Bob Campbell had worked on the same newspaper in San Francisco, and they were friends. They joked about going up, and Rosenthal apparently said, 'Oh, you guys are carrying guns. You can protect me.' But it was pretty secure on the mountain at that point. They get up there, and it was only seconds later that the second flag went up. They were already putting it on a new pole, and Bob Campbell shot the perfect picture of the one flag coming down, the other going up, crossing each other. This disproved a lot of arguments people have brought up that there weren't two flags.\n\n\"For his part on the summit, Joe had built a little rock cairn to stand on, and Genaust was over to the left of him. But Genaust figured that was not a good shot, so he started to move across, and that's when Joe said, 'Hurry up, it's going up.' Genaust got on Joe's right, and that's the way the shot worked out. Campbell didn't want to stand next to Rosenthal and get the same picture, so he moved over to the side and down at a lower angle, and that's how he got that beautiful shot of the two flags.\n\n\"When Rosenthal's stuff came through processing at the Cincpac lab in Guam, somebody, and I don't know who\u2014nobody's been able to put a finger on it\u2014whether it was the lab man or the Associated Press photo editor on the island, but somebody made a decision. I always say in a talk, 'There's another man involved in the Rosenthal picture.' He is the person who changed that picture from a horizontal to a vertical, and that made the photo. Because if you look at the horizontal, it shows the top of the mountain, it shows the debris around the place, it shows part of the beach over here, you can even see very faint lines of ships out to sea. It's got too much crap in it. I've said often that if that picture as taken were sent out and printed that way, I don't know if very many people would have done anything with it. But in turning it into a vertical, he did something that artists, true artists, really like: It's got a triangle in it; it's got action in a still photo. You get the feeling of the movement up there, and it tells the story without the caption.\n\n\"Of course the picture was a sensation. It was put out the Sunday after Rosenthal took it, and it was on the front page of damn near every major newspaper across the country. Then word got out that it was phony, that it had been staged.\n\n\"What happened was Lou Lowery, who shot the first flag raising, got together with a Time-Life correspondent, Bob Sherrod, on the island after we'd heard about this wonderful picture via radio. We hadn't seen it. This was part of that thing we'd determined at Pearl beforehand: We'd get lots of info back on what the press was saying. Everybody was extolling the wonderful picture. Now Lowery had been up there and was part of that little attack that went on up there and jumped off the side of the mountain and busted both his cameras. He couldn't understand why this guy who had gone up and just shot a replacement flag was getting all this credit. No combat, no nothing, you know?\n\n\"He was pissed off enough that he sat down with Sherrod and he told him the story. He said, 'There's something phony about this. It's just not right.' Plus a lot of professional photographers thought it was a staged picture from the start.\n\n\"What Bob Sherrod told me later was that when he sent it in, he put a note on the piece he had written as a result of talking with Lou. He said, 'Don't run this until you check with AP.' Now that's good reporting. And they didn't publish it per se. But there's always a glitch somewheres. Time-Life in New York at Thirty Rock [Rockefeller Center in midtown Manhattan] was as big as the Pentagon. It was on a great number of floors, and there was also a radio station in there that received copies of all the correspondences going through Time-Life from around the world. So Sherrod's story comes through all the various stations, goes through this radio desk, and there's no hold on it; the hold had dropped off somewhere along the line. So they ran with it, U.S. wide, a phony story, phony flag. And then, when Joe hit Guam, he was met by a press conference of his pals saying, 'What a wonderful picture.' Well, he hadn't seen it yet. Someone in the course of asking questions said, 'Did you pose any pictures?' He said, 'Oh, yes, sure.' Well, he didn't pose the flag raising, but he did get one of everybody standing around the flagpole. We called that the banzai (or gung ho) shot. It was taken all over the goddamn Pacific. That's how the misunderstanding arose.\n\n\"And that's what I was up against. The reason I went to Washington was I was taking the film in, from D plus eight to D eighteen, following Herb Schlosberg. I was met at National Airport on March 17 by a lieutenant colonel who was deputy director of public affairs for the Marine Corps. I knew him. His name was Ed Hagenah. I said, 'Jesus, this is really nice, for a colonel to come over in a car to pick up a warrant officer. I'm really proud.' He laughed and said, 'No, I've got to take you into headquarters.' I said, 'I can't go like this.' I hadn't shaved for a day and a half, I was partly in uniform, still had combat gear on. I had been five days getting from Iwo to Washington.\n\n\"He said, 'Get in the car, and I'll explain.' I did so and he told me we were going to the commandant's office. I said, 'Oh, for chrissake, Ed. Let me go home and shower and shave, put on a uniform. I only live five minutes away.' He said no, because in the office waiting for us was the bureau chief of Time-Life in Washington, the executive vice-president of the Associated Press, a bunch of horse holders, and the commandant of the Marine Corps, General A. A. Vandegrift. He said they all wanted to know about the two flag raisings. There was contention about the fact that Time-Life had said Rosenthal's shot was a phony picture. Now AP wants to sue Time-Life, and it wants to sue Sherrod, and I'm supposed to settle the goddamn argument.\n\n\" 'That's all I need,' I said, 'to be in the middle of two warring major news organizations.' So I get there, and the commandant looks at me and kind of grins. He said, 'Gunner [in those days warrant officers wore the bursting bomb on their covers, or hats, and they were called gunners even though they had nothing to do with guns], he said, 'Gunner, we don't know the real story about the two flags. You're the first one to come back here who might be able fill us in on it. Tell us what the story is.'\n\n\"I told them the first flag was put up by the patrol that went up there, and it was considered too small. There was a desire to put up a larger flag, but nobody knew where the hell they'd find one, but they finally did find one on the LST. They took it up and put it up. I said that was all there was to it. It was a replacement flag. I said I thought one of the problems was simply that the AP was calling it _the_ flag raising, and it's not. _The_ flag raising referred to the one that went up at ten-twenty, the first flag. This one, Genaust and Rosenthal's, was a replacement flag. Lowery shot the first one. In fact, there is nothing reported in the records about that second flag going up. If you didn't have a picture of it, you wouldn't know it was there. The battalion log has nothing; the D-2 log, intelligence, has nothing, the D-3 log, operations, has nothing. It was just a replacement flag, a routine action.\n\n\"There was a definite second flag raising, I explained, that had been photographed by Joe Rosenthal, and he hadn't planned it, he hadn't promoted it, he hadn't staged it. So that cleared things up. I hadn't seen Genaust's film at all. He was killed March 4, and his film was on the West Coast. The crux came when the commandant said to Alan J. Gould, the senior vice-president who pretty much ran AP, 'This is a wonderful photo, and I think we could use it a lot in the future. Would you give us permission to use it?'\n\n\"The guy said something typical, normal and natural for anybody in that business, with a stock photo library, to say. He told the commandant, 'Well, yes, I'll give you a couple duplicate negatives, but it will cost you a dollar apiece for every eight-by-ten that you make.'\n\n\"Well, there was dead silence in the room. It felt like icicles. You could feel the horse holders sitting around thinking: If it hadn't been for the Marine Corps, Rosenthal wouldn't have been up there taking the picture in the first place, and now they want to charge us for it? So the commandant turned to me, and said, 'Gunner, what do you think about that?'\n\n\"And I thought: Well, here goes nothin'. I said, 'That is a normal position, for them to come back with a normal response to a request for a stock photo, but if we don't decide to go with that, we have the sixteen-millimeter color film of the flag raising taken by Bill Genaust and,' I said, 'you can always find one frame that's fairly sharp. We can blow that up to eight-by-ten, make color or black-and-white prints of the exact same shot.'\n\n\"Now I didn't know whether Genaust's film was underexposed, overexposed, or whether there was dirt or scratches in the film or what the hell might have happened. His motor might have been running slow. I could see the wheels turning over in Gould's eyes as I was saying this. I could see him thinking there was going to be two pictures out there, one a little fuzzy and the other good and sharp, but the public wouldn't know the difference. He very quickly came back and said to the commandant, 'Well, as an afterthought, I'll give you the two dupe negatives and you can use them in perpetuity at no cost.' That ended the whole conversation.\n\n\"The first thing I did when I got home was call Schlosberg. I got him in the cutting room, and I said, 'Have you seen Bill Genaust's film yet?' He said yes. I said, 'How is it?' He said, 'It's beautiful.' I said, 'Thank God,' because all I could picture was that if by any chance, the decision had gone the way I suggested and the film was bad, I'd have been shipped to Iceland or someplace.\n\n\"At first Lou Lowery was embittered that Joe got all the attention, but once he saw the photo, he didn't have any doubt that it was a good one, so the animosity sort of drained out of him. But he was pretty bitter there on the beach. He came to me and said, 'God damn it, what the hell is this?' I said, 'Lou, don't get excited until we find out what it is. We haven't seen it.' Later on he and I were very friendly. We worked together on several Marine Corps photo contests as judges. He and Rosenthal became good friends. In fact, Rosenthal came all the way east for Lou's funeral.\n\n\"I never got to know Rosenthal real well, but we talked on the phone over the years. It'd be like, have you heard about this? One of the funniest stories we traded was about some Air Force captain or lieutenant who was having trouble with his P-51 fighter. He put down on the Iwo airstrip, and while he was waiting for the mechanics to fix it, he said, 'Well, as long as I'm here, I might as well go get a good look at the island.' So, he said, he walked over and climbed Suribachi just in time to help put the flag up.\n\n\"Besides Genaust, we lost one other film guy, Fox, on Iwo after I left. I never did a lot of filming there myself although I went out a couple times. I stayed pretty close to the CP, the command post, in case the colonel needed me.\n\n\"I had left the island by the time Genaust was killed March 4. A funny thing happened about ten years ago, 1997 or thereabouts, here at my home in Arlington. I heard a knock on the door one Saturday morning, and a tall gentleman introduced himself as former Lieutenant John K. McLean. He lived right up the street. He said he was an interpreter who was on the patrol when Genaust was killed. A report had come back to division intelligence that a Japanese was sitting at a table in a cave with lots of papers, working on something, but nobody could talk to him. It was thought the papers might be valuable; it was thought it might be a trap, so McLean was sent out with a patrol to find out what this guy was doing. Bill tagged along, looking for a story. It was a rainy, misty day. They got there, and here was this guy sitting at a table down inside this cave, which was quite large. The lieutenant tries to talk to him: 'Why are you there? What are you doing?' No answer.\n\n\"So the patrol leader says, 'We'll go down there and find out what's happening.' but he didn't have a flashlight, and nobody else had one, except Bill. He said, 'Oh, hell, I got the flashlight. I'll go down there.' He was that type. I mean, he'd been in combat. He knew his way around. He knew what to do. But why he did this one, I'll never know.\n\n\"To get into the cave, you had to climb over a large berm formed by dirt taken out of there. There was just enough room to squeeze through. Bill went down into the cave while McLean and two or three other members of the patrol squeezed through behind him. But they didn't descend. Genaust got two-thirds of the way down there, and a Nambu machine gun opened up from a tunnel off to the right, which they could not see from the entrance. It killed him.\n\n\"The Japanese guy starts to pick up his papers, and McLean immediately ordered his men to kill him. But the Nambu scared the living shit out of them, and they turned around and had a hell of a time climbing back out through all that volcanic ash, plus it was so narrow. They were scared to death, and they just piled ass out of there. They got out, but nobody fired at them. Genaust was too far down for anybody to reach without getting shot. So they called for a dozer to close up the cave. McLean, who has since died himself, said he went back a couple days later, but he couldn't find the cave, and I guess it has been lost ever since.\n\n\"Just last week [March 2007] I was at a meeting with a group of Japanese who had come from an organization that looks for bodies, and they were going to make their fifty-fourth or fifty-fifth trip to Iwo and wanted to talk to some marines who had been there during the fighting. The question of Genaust came up, and I gave them a copy of McLean's story. Of course our own guys have looked for Genaust for years. [At the time this book was going to press, there had been no word from the Japanese contingent.]\n\n\"Was the Marine Corps thrilled when the Tarawa film won the Academy Award in '44? I guess they were. I haven't the slightest idea. I wasn't there. General Julian C. Smith, who was the division commander on Tarawa as a major general, was a lieutenant general by then and director of the Department of the Pacific, in San Francisco. He and his wife were invited down to the awards to sit at the table of Twentieth Century Fox, even though they didn't have anything to do with the film, which was produced by Warner Brothers.\n\n\"He accepted the award. I don't know if you know it or not, but this was not _the_ award. It was a little wooden plaque. Because of the metal shortage, they weren't making any Oscars during the whole war period. After the war they went back and made Oscars for those who didn't get any. The Marine Corps Museum's got it in Quantico. I often used to ask Happy Smith, the general's wife, whose real name was Harriet, 'Would you let me take that so I can keep it on my mantel? Whenever the general passes on, I'll turn it in, to the Marine Corps historical people.' But she never would agree to that. It was kind of a joke between us. Still, I thought it would be kind of nice to have that plaque sitting there on my mantelpiece.\"\n\nAfter a hitch in Japan during the occupation, Norman Hatch wound up back in Washington as an operations officer for photographic services for the Marine Corps, first as a military man, then in 1946 as a civilian. He and his wife, Lois, had a son, Skip, and later a daughter, Colby. Following that, he worked in procurement, then joined Bell & Howell in Chicago for a stretch, did some freelancing back east, and finally, in July 1956, became the chief of the Defense Department's newsreel setup, where he spent the next twenty-three years. As newsreels phased out, he created a radio-TV-newsfilm branch and then became operations officer for the audiovisual division. There he came to work with folks like John Wayne, who wanted Marine Corps authenticity in his film _The Green Berets._\n\n\"John Wayne was fine to work with. He was emotionally involved with his project, and he thought it was wonderful and great, and it was pretty good in the sense that it didn't hurt the service any, but there were little things that aggravated us, relationships he portrayed that were farfetched. But then he built a whole Vietnamese village at Fort Benning and gave it to the Army to train guys.\n\n\"I retired as a major in January 1979, after forty-one years' service. There were several retirements in there: the Marine Corps, the Defense Department. It's like I've been on the crest of a wave ever since I got out of the Marine Corps. I went from one good job to another. Now I am doing something like this interview for this book, I just did a similar thing for somebody writing an article, and they're asking me now to write for the Second Marine Division newsletter. Everybody's asking me to write a book, a fellow in New Zealand is writing a book, and I got to dictate something for him, so I'm busier now than I was before.\"\n\nCORPORAL CHARLES \"CHUCK\" LINDBERG\n\n**Last of the Flag Raisers; Silver Star, Purple Heart  \nTwenty-eighth Marines, Fifth Marine Division**\n\n_It still bugs me that we never got recognition for being the first to raise the flag on Iwo Jima._\n\nChuck Lindberg in his Richfield, Minnesota, home on May 8, 2007, when he granted an interview to the author, who took this photograph. Lindberg's Silver Star and his Purple Heart rest beside him on the arm of his easy chair.\n\nCorporal Charles W. Lindberg, grenades hanging off his jacket, flamethrower in hand, looks for cave entrances along the summit of Mount Suribachi shortly after the first flag was raised on the morning of February 23, 1945.\n\n_I was sitting talking with Chuck Lindberg and his wife, Vi (for Violette), in the living room of the modest Minneapolis home they had shared for more than fifty years when I noticed after some time that half the forefinger on his left hand was missing. Thinking it got shot off in the war, I asked about it. But no, he had lost it as a twelve-year-old reaching into some rocks at a dam, trying to catch a bullhead, when the rocks loosened, came down, and crushed the front part of the finger. \"Them days,\" Chuck said, \"they didn't have what they have today or they could have saved it. But when I got to the Marine Corps recruiter in Seattle in 1942, he took a look at it and said, 'Is that your trigger finger?' I said, 'Nope.' And he said, 'You're accepted.' I couldn't get in now, if I wanted to, with a finger like that.\"_\n\n\"I'm from Grand Forks, North Dakota. I was born June 26, 1920. I quit high school in tenth grade, in 1937, and joined the Civilian Conservation Corps, the CCCs, planting trees, down around Fargo, mostly, for about six months or a year. When that ended, I went to work on a farm outside Grand Forks, getting a whole ten dollars a month in the winter and twenty-five dollars in the summer. One winter day this big Lincoln Zephyr pulled up in the yard. As soon as the driver got, out I recognized him as Archie Simonson, a man I used to take care of horses for back in my school days. He owned three or four gas stations in Fargo, Grand Forks, and St. Cloud. He kept some big workhorses up there and sent them down to the woods for the winter. I took care of them for him. He remembered me and where I was. He said he was starting a gas station in Fargo and he'd give me ninety dollars a month and a room. Oh, boy! I jumped this high.\n\n\"I worked for him about a year and then got acquainted with the road boss of some truckers hauling cars. I asked him one day, 'Hey, I'd like to get another job and get out of here.' He says, 'I'll give you a job. You can make yourself two hundred forty bucks a month.' Pretty soon I was driving a big truck hauling cars from Detroit to Spokane.\n\n\"That was my big move. I drove truck till war was declared, and then, when I found out two months later there was no more cars to be delivered, I went up to the Marine Corps. It was January of '42, and I was stranded in Spokane. I always had great respect for the Marine Corps, from books, posters, and movies. I joined in January 8,1942. I knew the draft would get me eventually anyway. I was single; I was twenty-one. I had six brothers and three sisters. My folks didn't know I was in the Marines until I wrote home. By the time they got it I was on my way overseas.\n\n\"I had to go to Seattle to get sworn in. They sent me to boot camp in San Diego. When you're driving trucks all the time, you get weak, and boot camp was tough even though it only lasted five weeks. Them marines don't monkey around. The day we finished boot camp they marched us down to the parade grounds, and here was a big platform and Carlson (Lieutenant Colonel E. F. Carlson) and Jimmy Roosevelt was there. But Carlson was talking, recruiting men for his raider outfit. He says, 'I want men who ain't afraid to die, I want men who ain't afraid to kill. I want men who'll walk fifty miles in a day.' I raised my hand. I put my like in him right away. I liked his way of talking. They sent us up to Jacques Farm for training; they put us in pup tents on the crick. There was a house sitting there, and I thought, first thing, I thought there's where the officers are going to sleep. But about ten o'clock that night here come the officers, all of 'em, and they get in the pup tents. That's the way he run it. He'd eat, sleep, and go with the men all the time. The officers didn't get fed any better. We liked that.\n\n\"I stayed with them until they were disbanded, after Guadalcanal and Bougainville. We come into Guadalcanal November 2 at Aola Bay and went on what they called the Long Patrol, thirty-two days behind enemy lines, from November 4 to December 4, 1942. It was said to be the longest patrol of World War Two. I was a rifleman. [Lieutenant Colonel Carlson led the guerrilla raid, for which he received a Navy Cross. His men were credited with killing 488 Japanese while 16 raiders were killed and 18 wounded.]\n\n\"After that we went to Hebrides, and then they called us back north again. I was a private first class. Later we made the invasion of Bougainville in 1943. We were there three months, and then they brought us back to the States, disbanded the raiders, like they did the paratroopers, and put us all in the Fifth Marine Division. After I returned from leave to Camp Pendleton, I saw the flamethrower unit and volunteered for it. Everybody said I was nuts, but I made corporal, and they gave me an assault squad. I was five feet ten and weighed a hundred seventy pounds. The flamethrower weighed seventy-two pounds. It carried five gallons of jellied gasoline, and you could empty it in six seconds.\n\n\"After several months at Pendleton, we went to the Big Island at Hawaii and trained more there, at Camp Tarawa. From there it was on to Iwo. We didn't have too good info on that island. From Hawaii we stopped at Guam and switched from big ships to LSTs. Was it bumpy? Oh, Jesus, you can say that again. It went plop plop plop through the sea.\n\n\"I was in the Third Platoon of E Company, Second Battalion, Twenty-eighth Marines, under John Keith Wells until he got injured. Then Lieutenant Harold Schrier took over. We came in next to Suribachi in the eighth or ninth wave. Kuribayashi opened up when we come ashore. It was really bloody, with guys lying all over the place. They mortared us up and down the beach. They had it all bracketed. I found out later their plan was to let us get on the beach and annihilate us. But we got off the beach and up behind the sand ridges. There were tunnels and mines and booby traps all around Suribachi. I was used to combat, so that never worried me too much. We got to the base of the mountain February 21. We got across the island and cut Suribachi off that first day. We thought we were really doing something.\n\n\"We spent the next few days attacking the mountain, taking out caves, pillboxes. I never seen so many caves and bunkers. Ever bunker was attached to another one, and they'd go by one connector on down fifteen feet, and then someone would come up behind and shoot you in the back. Our casualties was high on that Third Platoon.\n\n\"With the flamethrower you had to sneak up on them, and once you got within thirty or forty feet you could use it. It would reach sixty feet under good conditions. Once you sent that load of gas in there, you didn't have to worry too much, as long as they didn't have another way out. I can't say how many I hit, maybe ten or fifteen. The biggest one I got was on top of the mountain when we went up at eight a.m. on February 23, forty of us. I was on that first combat patrol with Schrier. They came the night before and said we were going to start climbing in the morning. We had a kind of jittery night. The next morning we reported to Colonel Chandler Johnson, who handed Schrier a flag and said, 'If you get to the top, raise it.' We thought it was going to be a slaughterhouse, but it turned out we went up that mountain with no resistance.\n\n\"I was back a bit. The scouts and riflemen go first. We didn't hit the cave right away. The first thing they told us was to get the flag up. When we got there, I took the flamethrower off and laid it beside me. We found a water pipe to tie the flag to laying on the ground. It was about four inches thick and twenty feet long. It was heavy. I helped tie the flag on with Thomas and Schrier and Hansen. Then we went to the highest spot we could find and stood it up. It hit my helmet going up and knocked it off. There were horns honking, guys cheering; it was just wild. All the ships out there tooting their horns. We felt very good about it.\n\nIn his excellent account of all aspects of both flag raisings, _Uncommon Valor, Common Virtue: Iwo Jima and the Photograph that Captured America,_ Hal Buell quotes Chuck Lindberg from an oral history collection, _Never in Doubt:_ \"It was a great patriotic feeling, this chill that runs through you. . . . My proudest moment of my time in the Marines was raising the American flag on Iwo Jima. My feeling of being a Marine is I served with the finest, and I feel proud every day that I can tell somebody that.\"\n\n\"Once we put up the flag, the Japs started coming out of the caves. I know the first one we got wasn't more than one hundred or two hundred feet from where we put up the flag. And we were right near the edge where you go over to go down where that cave went in. I went around the back of it and found another entrance. Somebody had dropped something big and knocked a hole in the cave. I sent a good blast in there. The other guys were standing on top, trying to get something to shoot at, but nobody came out. We thought that was kind of funny, so we blew both ends of that cave shut. Then two days later we dug it open, went in there, and found seventy-two dead Japs. My theory has always been that before we started up that mountain, they ordered barrages and Colonel Johnson just pulverized it, with planes, battlewagons, the whole thing. I think they were afraid to come out. Then, after we got up there, they started shooting at us from this one cave, and that's the one we went after first. It sat right on the rim of the volcano, a big rocky opening.\n\n\"You know about Donald Ruhl, who got the Medal of Honor? Before we went up the mountain, he and Hansen were down on edge of a big hole, looking for targets for me, and all of a sudden a grenade came over, right between them. I was right directly back. Ruhl he just gave Hansen a shove, and he fell on it. He was a good man; there was no doubt. The trouble is, you know, then Hansen was killed March 1, the same day I was shot. [In addition to falling on the grenade on February 21, Ruhl was cited for his bravery over the previous three days in attacking the enemy and rescuing under heavy fire a wounded fellow marine.]\n\n\"Sergeant Lou Lowery, the _Leatherneck_ magazine photographer who took the pictures of our flag raising, was trying to get away from a grenade when he fell down the side of the mountain and busted his camera. I knew him very well. Lowery was with us off and on. He was unhappy, he told me after, when Rosenthal's photo got all the attention. He was unhappy over that. I was very good friends with him after the war too. I even went to his funeral at Quantico, after he died in 1973. The thing that got me was here come Joe Rosenthal too. He took the second flag raising picture. 'Oh,' he says, 'you were on the real one, wasn't ya?' He wasn't told to do that, in my opinion. He wanted a picture, and he was going to get a picture. Oh, you bet. It's a good picture. I even told Rosenthal it was a good picture. But me and him got into a few arguments.\n\n\"How did I feel about it? It still bugs me that we never got recognition for being the first to raise the flag on Iwo Jima. Why, here you were doing the dirty work, and they get the credit for it. They were called heroes. The mountain was secured. All they did was come up, put the flag up, and go back down again. And when I got home and started talking about this, I was called a liar and everything else. It was terrible.\n\n\"We spent from the twenty-third to the twenty-eighth on the summit. I reloaded and went back up several times because there were a lot of places we wanted to burn up there. You didn't know if they were in there or not and you couldn't blast them out, so you used the flamethrower. I and Robert Goode were flamethrowers for the company. I had a whole squad, and where we were needed, we went.\n\n\"We moved on the twenty-eighth, went up to Hill 362 Able. We stayed there the first night, and then the next night Hansen was shot. We were all settin', kind of got to talking, when all of a sudden a sniper nailed him right there, at noon. And that's where I was shot, March 1, that same afternoon. I was trying to get to a mortar position, kind of running across the side of a ridge there, and all of sudden\u2014 _boom_!\u2014a bullet went right straight through my right forearm and on through my jacket. I was bent over like this, and if I'd been standing straight, another inch or two, it would have got me. But because I was bent over, it missed my stomach. It shattered my forearm bone. You bet I went down. I hollered, 'Corpsman,' and here come John Bradley, from the second flag raising. I knew him very good. He was our platoon corpsman, a wonderful corpsman, I'll tell you that. He was from Wisconsin [Bradley's son James, with Ron Powers, wrote _Flags of Our Fathers,_ the best-selling story of those who raised the second flag and the basis for the Clint Eastwood film]. We saw each other after the war, and I saw them in Chicago when they were on that bond drive. I was back in Great Lakes Hospital by then.\n\n\"Anyways, I'm on the ground lying there, not feeling too bad. He got some splints on it and gave me a shot of some kind. He was going to take me back, but I said, 'No, I'll go by myself,' which I did. I left the flamethrower behind. I walked from Hill 362 to the beach; I didn't know if I could make it or not. It was a long way too. When I got to the Higgins boat, they stripped me of all my equipment, my rifle, pistol, the works, and away I go. They set my arm on the hospital ship, but I got the best deal when I got back to Pearl Harbor. They really worked on it.\n\n\"I went from the hospital ship at Iwo to Saipan. After about a week a guy says to me, 'You want to see a picture of the flag raising?' I said sure. He hands me a picture, I look at it, and that's not the one we put up. I didn't know a thing about it. Nobody said anything about it. I was astounded. After they sent me on to Pearl Harbor, _Yank_ magazine came out with pictures of the first flag and the second flag, and then I saw what happened.\n\n\"But we hadn't known the flag was being replaced. Nobody said anything about it. We were busy hitting caves when the second patrol came up with big flag. We had the mountain secure by one o'clock, and I went down to the bottom because my tanks were empty. I was being assisted by Robert Goode. We left the mountain at one-thirty, went straight down, and Rosenthal came up over to the other side. I didn't see him go up. I don't think anybody knew the flag was being replaced. Nobody cheered. Everybody knew the flag was up. They weren't paying attention to that anymore. If it were not for the photo, nobody would have noticed. The mountain was secure when he came up there. He didn't go under fire. We didn't even go up under fire. Nobody fired at us.\n\n\"Rosenthal pulled a rotten trick there, I thought. He was behind that second flag; he was going to get a picture. They weren't going to use ours. Give us our share of credit for what we did. People today don't know the difference. They called him a hero, called the six men who raised the second flag heroes when they sat at the bottom while we went up. To their credit, they didn't consider themselves heroes.\n\n\"I talked to them guys on the bond drive when I was at the Great Lakes Naval Hospital in Chicago, where I spent several months after being treated in Guam, Hawaii, and Camp Pendleton; that's where they cured me. We found out they were going to be at Wrigley Field, so I and another guy went down there. Huge crowd. They were really eating that up. That Gagnon, he got me. He introduced me around as a guy that killed a lot of men. I didn't like him as well. He wanted to be in the movies. Bradley introduced me to other people too. We didn't talk too long. My arm was still in a cast. I never talked to Gagnon.\n\n\"I kind of felt sorry for Ira Hayes, the Indian. He did it wrong [lost control of his life], but he was a good marine. He was a paratrooper. I talked to him quite a bit. We came back on the same ship together, seventeen days from Guadalcanal to San Diego. He'd drink, but it wasn't that way [not to excess]. I imagine everybody had some. I took a drink myself.\n\n\"After that they sent me down to Charleston, South Carolina, and made me a guard in the naval brig for two or three months. That was an easy racket. I was discharged down there January 16, 1946. When they pay you, they pay you from where you enlisted at. I had enlisted in Seattle and was discharged in Charleston, so that was a nice check, including travel pay for all that distance.\n\n\"I met my wife, Vi, in November of 1946 at the Legion Hall in East Grand Forks, Minnesota. I belonged to it. I was a past commander of the post. I was the bouncer, and I let them in free, four good-looking girls. She's from Minnesota. We got married in October 1947. We had two sons and two daughters.\n\n\"I was an electrician by then, wiring farms up there. I had run into an old friend in Grand Forks that I used to work for long ago, and he says, 'Can you wire a house?' I said, 'I don't know.' I tried it, and I guess I got it wired all right, with a few mistakes. That was my start in the electrical trade, and I did that for thirty-nine years, IBEW Local 292 Minneapolis. I retired in 1985. Two years ago they put up a new building in St. Michael's for training apprentices, and they told me they were going to name it after me, the Charles Chuck Lindberg Training Center.\n\n\"We had some attention for raising the flag but just not what you'd think, not like those other guys got. I was invited to Washington quite a few times. I was there for the dedication of the flag raising monument in 1964. Harold Schrier was there, and Lou Lowery. Nobody recognized us at all. Eisenhower was there. Nixon was vice president. I shook hands with him. He was shaking hands with me and looking over at somebody else. I met President Clinton and went to the dedication of the new Marine Corps Museum in Quantico November of 2006. I got sent a lot of stuff, and you ought to see the statue they got of me in Bemidji. It's beautiful, six feet tall, over in the center of the state. Vi and I went to Washington on a private plane, just the two of us for the Veterans Day dedication of the new museum last November. Seven of us were there from the Third Platoon. That's all that were left.\n\n\"It's taken half a century, but I think I've finally helped set the record straight.\"\n\nOn February 24, 2001, then eighty-year-old Charles Lindberg was presented with a letter from the commandant of the Marine Corps honoring him for his role in the raising of the U.S. flag at Iwo Jima. Lindberg also holds a Purple Heart, of course, and the Silver Star, awarded for his performance at Suribachi and on Hill 362. Part of the citation reads:\n\n_Repeatedly exposing himself to hostile grenades and machine-gun fire in order that he might reach and neutralize enemy pillboxes at the base of Mount Suribachi, Corporal Lindberg courageously approached within ten or fifteen yards of the emplacements before discharging his weapon, thereby assuring the annihilation of the enemy and the successful completion of his platoon's mission. While engaged in an attack on hostile cave positions on March 1, he fearlessly exposed himself to accurate enemy fire and was subsequently wounded and evacuated._\n\nChuck Lindberg died on Sunday, June 26, only weeks after I visited with him and his wife at their Richfield, Minnesota, home on May 8. He and Vi had been married fifty-nine years.\nPART FIVE\n\nThe LST, Workhorse of the Invasion\n\n**The LST, or landing ship, tank,** was 327 feet long and 50 feet wide and was manned by a crew of seventy plus seven officers.\n\nAbout a thousand were built during the entire war, at Evansville, Indiana, and floated down the Ohio to the Mississippi and New Orleans.\n\nSeventy-five to one hundred were used for the invasion of Iwo Jima, according to William E. Jayne of Hobe Sound, Florida. \"We were the dungaree Navy,\" he said. \"I was seventeen years old, in charge of all foodstuffs. I served on LSTs for two and a half years as a gunner and storekeeper. We spent ten days at Iwo Jima.\" LSTs could carry up to 350 troops plus tracked landing vehicles, such as the amtracs, the Higgins boats, and the DUKWs (see page 255). They also carried pontoons for causeways, bulldozers, and Marston matting, a pierced steel planking that could be laid across the sand for traction.\n\nWith their relatively flat bottoms, they could pull right up to the beach, set a stern anchor, pump out the ballast, and open their bow doors so a vehicle theoretically could drive right out onto the sand. It should have been easy. With the steep, terrible sands and rugged surf on February 19, 1945, at Iwo, it wasn't that simple.\n\nMACHINIST'S MATE THIRD CLASS PERLE \"DUSTY\" WARD\n\n**133rd Seabee Battalion**\n\n_Because I was born and raised on a farm and could run a tractor and drive horses, different things like that, they thought that I ought to be able to run a bulldozer._\n\nThe young Seabee Seaman First Class Dusty Ward leans against a palm tree in Honolulu in 1944. He received an official commendation for helping evacuate casualties on D-day on Iwo Jima, working all night as a stretcher bearer to bring in wounded marines. \"His activities contributed materially to the saving of many lives,\" the citation says.\n\n_I did not know, when I reached Dusty Ward by phone at his home near San Diego, that Mabel, his wife of fifty-four years, had died only days before, on April 7, 2007. \"We just finished her memorial last Saturday at the Elks Lodge here in El Cajon,\" he said. \"It turned out to be a beautiful one. I'm just sitting here looking at a bunch of flowers, and I started out this morning to write thank-you cards to all the people that sent flowers. I made up my mind I was going to get to it this morning and not stop till I'm done. She wasn't that old. She was only seventy-two. The poor gal had a lot of things went wrong with her.\"_\n\n\"I came into Iwo Jima February 19, 1945, on an LST. Going into shore, I remember an LST coming in next to me, and as soon as they dropped the damn ramp, a mortar shell hit right on the ramp as guys were coming off. I don't know how many were killed, but it was right alongside me. So talk about being lucky, I feel that way. And that was the first day. No heavy machinery came in. They probably started moving it late in the second day. Our job was to help get other equipment inland.\n\n\"I got involved in a deal late in the afternoon of the first day where some officer from the Fourth Marine Division came and asked for volunteers to carry some wounded back. I was one of the four that volunteered. We carried them on litters, maybe as many as thirty guys, one after the other, after they had been treated by the corpsmen. Of course there were shells they were bombarding us with, but none came too close. The Japanese could shoot artillery or come out of their caves and rapidly fire off a couple mortars and duck back inside. They felt pretty safe. They had the range to hit any place on the island. Everything was bracketed.\n\n\"Well, this went on longer than we anticipated, way into the wee hours of the next morning, till two-thirty or three o'clock. We had left our outfit on the beach because we hadn't been landed too long, and when we came back, our outfit had moved. So there we were four of us, alone in enemy territory, and it was dark. We felt the enemy had a big advantage over us, knowing the terrain and one thing and another, so we were smart enough to dig a foxhole for the night. And that's where we stayed until the next morning.\n\n\"When daylight came, we went looking for our outfit, and after asking different people, we finally found them a few hundred yards up the beach, going away from Suribachi. When we got there, my CO [commanding officer] said one of us had to report. They designated me, and then he pointed up a hill and said, 'See those two officers standing up there? Report to them, and tell them briefly what happened last night and why you weren't with your outfit.'\n\n\"So I walked those fifty yards up the hill and talked to a Marine lieutenant colonel and a Marine major, probably from the Fourth Division. I briefly told them what had happened, how we came ashore and they asked for volunteers to carry wounded marines from inland to the beach to be sent out to the hospital ship. Turned out to be a longer job than we anticipated and we had to dig a foxhole and stay the night. When morning came, we started looking, and here we are, I said. One of them thanked me.\n\n\"This whole thing took probably a minute and a half. So I turned around and walked down the hill toward where my guys had already dug a foxhole for me near the beach. I got forty or fifty foot from them when a mortar landed just where the three of us had been standing. Of course it killed the major and the lieutenant colonel of Marines, and it blew me from that forty or fifty foot away clear on down, head first, into my foxhole. When I come up, I had a little blood on my left hand. Guys were kidding me about going down to the beach to the hospital tent and get it fixed. Anyway, I still can see a couple scars, so I don't know whether a piece of shrapnel hit me from that far away or whether the sand cut it when the concussion blew me into the foxhole. Turned out to be minor. But anyway, that was our first day and the first morning of our second day on Iwo. Why were they killed and me spared? The good Lord was looking out for me. It was also a good thing I didn't take too long to make my report.\n\n\"Seabees was kind of a moniker. It came from the NCB, which stood for Naval Construction Battalion. Our job was to unload ships and build or repair roads and airstrips. We were part of the Navy, but on Iwo Jima we were attached to the Fourth Marine Division. We wore Marine uniforms and were subject to Marine regulations.\n\n\"We also had the carryall, as we called it. Road graders they call them nowadays. I was born in 1926. I'm one of these dummies. I had some illnesses when I was young, and I quit school in eighth grade, when I was sixteen, and never got back to it. My dad had a sixty-acre farm, cows, horses, pigs, chickens, and a John Deere. Because I was born and raised on a farm and could run a tractor and drive horses, different things like that, they thought that I ought be able to run a bulldozer.\n\n\"I grew up near Armada in the Thumb district of Michigan. My given name was Perle, but somewhere along the line they took to calling me Dusty, because I used to get pretty dirty when I was a kid. I'm sure glad they didn't call me Dirty. But I lived with Dusty, and that's what most people know me as ever since.\"\n\nWouldn't Perle just confuse people anyway?\n\n\"Yeah. I was always mad at my grandmother on my father's side. That was her responsibility for giving me that name. I give my four kids all simple names like Pam, Deb, Bob, and Mike.\n\n\"Anyway, my older brother was in, and I knew it was just a matter of time before they'd draft me. It looked like the war was going to last for a while, so I just went down to the recruiting office in Detroit one day. I was going to join the Navy, but they talked about the Seabees and convinced me that was the way to go. I enlisted August 14 of 1943. I didn't know any better. I'd do it all over again, but I don't know if I'd have that much luck.\n\n\"We had basic training at Camp Perry in Virginia, and they give us the same training as they give the Marines. You had these drill instructors putting you through the ropes. Mainly it was just indoctrination, getting used to what the service was going be like, fundamentals. We didn't get into guns until we got to Camp Endicott at Providence, Rhode Island. We were there until the first part of 1944, and then we went to Biloxi, Mississippi, for more training, as I recall.\n\n\"We were trained with guns just like the Marines because we might get in a situation where we needed to defend ourselves, and then we were learning heavy equipment for road building and basic construction. The equipment was not armored. We crawled under live fire, through barricades, and learned how to knock down wire barriers. It was pretty advanced training as we moved along.\n\n\"We went from Biloxi to a camp in California between Oxnard and Ventura. We had bulldozers; we called them Cats for Caterpillar and road graders and road wheelers that tamped dirt on the airstrips. From there we went to Hickam Field in Hawaii. We were organized in platoons and companies. I was in the One Hundred Thirty-third Seabees, which turned out to have the most casualties, as I understand it, of any Seabee unit in history [reports vary, from 245 to 370, with at least 3 officers and 39 enlisted men killed].\n\n\"We had several more weeks' training at Hickam and then got on ships, headed out, and rendezvoused with the Fourth Marine Division, going to Iwo Jima. It was my first combat. I was just a wet-nosed kid, eighteen years old, a machinist's mate third class. The job was to defend yourself and build roads and airstrips. The equipment bogged down real fast in that coral ash when they tried to get it off the LSTs. We had these landing mats you put together to cross a bad area. They were folded and fit together by hooks. They were wide enough to drive over with jeeps or heavy equipment. We did the same thing on the airstrip, Motoyama No. 1, later on as temporary measure. We had the dozers or Cats; road graders and road tampers\u2014heavy rollers\u2014come in the belly of the LSTs later on.\n\n\"Before we could get our nose into things, you had to move the enemy out of there. We tried to stay alive and help organize things on the beach as we moved inland. Our job was to set up whatever they wanted, to assist the Fourth Marine Division, whatever their commanders wanted us to do, relayed to us through our officers.\"\n\nAccording to a Seabee staff correspondent, Robert V. Evans, there were two battalions from the 41st Seabee Regiment at Iwo Jima, the 133rd, Dusty Ward's, attached to the Fourth Division, and the 31st Naval Construction Battalion, which was attached to the Fifth Marine Division. He reported that, following Seabee repairs, the southern airfield, Motoyama No. 1, was in use by February 26, a week after the invasion.\n\n\"We were there from February until the end of November. We stayed there long after the battle was over, built the airfields back up for B-29s to land, all that sort of stuff. We got shot at while driving the equipment, but that was about it. Those bulldozers were heavily made, so generally bullets would just glance off the blades and the metal parts. You were totally exposed, no armor other than the blade. Some guys got shot. Like I say, I was one of the lucky ones. We worked on Motoyama No.1 and No. 2, either rebuilding or refurbishing when there were bomb and shell craters, and then we maintained them as long as we were on the island. I saw _Dinah Might_ land March 4, when it took out the telephone pole with one wing. I also saw B-29s come later with three motors gone, and I saw one hit the end of the runway where it dropped off onto a pretty steep bluff. It went right into the end of that thing and blew itself up.\n\n\"As far as what was on the island was concerned, there was no material value. Nobody would want that piece of land. But from the standpoint of strategic maneuvering, it was a good decision on America's part to take that island. They saved a lot of planes and a lot of lives coming back from bombing runs to Japan. They feel it saved something like twenty to twenty-five thousand lives, plus all those planes.\n\n\"When we left that November, we stopped by Guam on our way back to the States. We were originally supposed to go to San Francisco, but for some reason, after we passed Hawaii, they decided to go to Seattle. We got there 19 December 1945. They gave us the choice to stay there and get discharged, which would delay our getting home for Christmas, or we could leave immediately, maybe make it home by Christmas, and get discharged later on. I chose to leave immediately. We took a train, and I'll always remember, after coming from the sunny South Pacific all that time, riding through these big snowbanks in Montana and places like that.\n\n\"Anyway, I walked into my family farmhouse at seven o'clock Christmas Eve 1945. My mom said I made quite a Christmas present. She was there and my father and younger brother would have been there. My older brother was Air Force, stationed in the Pacific. My discharge was delayed till the fourteenth of March 1946.\n\n\"I left Michigan before the winter of 1947 rolled around. I was spoiled by the Pacific, and I couldn't stand that cold weather anymore. My brother's got the farm, and he's got that same John Deere my dad owned. I ran it for a while after I got out of the service. It's still going strong. But then I went to California, and I've been here ever since.\"\n\nDusty fetched up in the San Fernando Valley and owned a gas station for a while, then went into real estate. He sold vacuum cleaners door-to-door, connected with Sears in the late forties and spent thirty-five years with the company, most of it in management, working all over California and Arizona. His last assignment took him to San Diego, and that was how he ended up in El Cajon, where he lives to this day.\n\nHe met Mabel in Riverside, California, during his first department manager job. He was told he was being promoted to a position in San Jose. Shortly after, he and Mabel were at dinner with one of her girlfriends one night.\n\n\"They knew I was going to San Jose, and I said something like, 'How are we going to work this?' And the girlfriend said, 'Well, why don't you guys up and go to Vegas and get married, and then you can go to San Jose together?'\n\n\"So that's what we did. It happened to be a Saturday night, and it happened to be my birthday, March 19, 1954. We went to Las Vegas and got married at the old Hitching Post, and that was it. We didn't think about it at the time, but had it worked out differently I wouldn't have wanted it on my birthday.\n\n\"I think the Seabees were a good outfit. Back on Iwo, we didn't have time to socialize much with other units. We stuck to our own area. It was pretty fast-paced, and they kept you pretty busy. I liked the bulldozer. All those years the Seabees never got credit for what they did. We should have received a Presidential Unit Citation, though. But I enjoyed my part in them. I'm glad it's behind me now; I feel lucky to be alive.\n\n\"My most vivid memory of Iwo was probably those six or seven seconds when I walked away from the major and the lieutenant colonel on the beach and how lucky I was. If I had been a long-winded guy and stood there another six seconds, I wouldn't be here today. I have probably lived that moment a thousand times since then and realize that no matter how good or bad in life you might be, luck has to play a very important part. The full realization of what happened to me that day has made me a better person, and I thank the good Lord for guiding me and putting me down the hill away from that artillery explosion. Of course the good part would be that I would have never known about it, because that Marine lieutenant colonel and major never knew what hit 'em. Their lives went out. There was no suffering there. There wouldn't have been with me either. I got to have a life instead.\"\n\nPHARMACIST'S MATE THIRD CLASS GEORGE WAHLEN\n\n**Navy Corpsman; Medal of Honor  \nTwenty-sixth Marines, Fifth Marine Division**\n\n_[T]he last week aboard ship I think is the first time I ever prayed in my life. And in my prayer I always asked the Lord, I said, \"If there's anything at all you can do, don't let me let one of my buddies down.\"_\n\nGeorge Wahlen and Melba, his gracious and charming wife, in their Roy, Utah, home in May 2007. By the following August they had been married sixtyone years. A photo of George as a young corpsman hangs on the wall to his left.\n\nKnees shaking, the last of fourteen men honored that afternoon, a petrified young pharmacist's mate, the first living corpsman to receive the Medal of Honor in World War II, is greeted by President Truman, who snatches his hand and says, \"It's mighty good to see a pill pusher here in the middle of all these marines.\" Then he put the medal around George Wahlen's neck.\n\n_Two days before I was scheduled to fly to Roy, Utah, to visit George Wahlen in May 2007, he called to say he had just been hospitalized and maybe we should reschedule. When he found out my plane tickets were fixed, however, he offered to let me conduct the interview in his hospital room. Happily, he had been released by the time I got there, and I had a most enjoyable visit with him and his wife, Melba. Though George was not initially religious, he eventually joined the Church of Jesus Christ of the Latter-day Saints, and he and Melba went on to have five children, twenty-six grandchildren, and thirty-seven great grandchildren. I asked Melba how she handled all those birthdays. \"It's simple,\" she said. \"I just send each one a card and a two-dollar bill.\"_\n\n\"I was born August 8, 1924. I was one of three boys. My father farmed, and later we moved into Ogden, Utah. My father had never been in the military, and he was kind of scared of what would happen to me if I went in, but he did let me quit school my senior year to take an aircraft engine mechanic's course at a college in Logan. Pearl Harbor was attacked the day after I enrolled. It was a six-month course, but I went for three and then had an opportunity to go to Hill Air Force Base near Ogden. I worked there several months. My dad still wouldn't let me join the service, but I went down and volunteered in June of 1943. I tried to join the Army Air Corps, but they said they were all filled up, so then I tried the Navy because I was told they had planes.\n\n\"I went by train to the Naval Training Center in San Diego for basic training June 18. I had a hard time learning to march. When I finished, they sent me to hospital corps school instead of aircraft mechanic's school. I tried to talk them out of it, but no one would listen. They told me if I did well, they might reconsider after I graduated, but that was baloney.\n\n\"I learned basic first aid; then they put me in a medical ward at the Naval Hospital in Balboa Park, San Diego, changing bedpans, cleaning floors, and being ordered around by female nurses. I was promoted to pharmacist's mate third class. I remember some nurse I didn't like got mad at me for not emptying some bedpans, and she said, 'If you don't shape up, I'm going to send you to the Marine Corps!'\n\n\"I thought about that and said to myself, 'I'm not going to be sent; I'll go volunteer.' I went and done it that day, and next morning I had my seabag all packed and went to the Marine Corps for seven weeks of further training. They sent me to Camp Elliott, gave me a Marine Corps uniform, and we had seven weeks of training, learning how to dress shrapnel wounds, splint bones, and treat an open belly wound. Instead of taking care of people in a ward and following a doctor or nurse's instruction, you had to learn how to treat wounds on the battlefield. Lots of times you get involved with sick ones too.\n\n\"I was in the Navy, but I was attached to the Marines and wore a Marine uniform. The Marine Corps medics were all Navy corpsmen. We had three months of training, including weapons. I trained with a carbine, the Browning automatic rifle, machine guns, hand grenades, forty-fives, everything. We got the same training as an infantryman. You were expected to know how to act in combat because you were in combat, same as the Marines. You were being shot at and exposed just the same as they was. You had no insignia showing you as a medic at all. You didn't want to; they claimed that was more dangerous. My pharmacist's rating translated me to a three-stripe sergeant. I was assigned to the Fifth Marine Division, which was activated January 1944. I was in Fox Company, Second Battalion, Twenty-sixth Marine Regiment.\n\n\"We sailed to Hawaii in late July 1944 and spent six months there at Camp Tarawa on the Big Island. We reached Saipan February 11, 1945, and left for Iwo on February 16.\n\n\"By now we knew where we were going. I remember the last week aboard ship I think is the first time I ever prayed in my life. And in my prayer I always asked the Lord, I said, 'If there's anything at all you can do, don't let me let one of my buddies down.' For a nonreligious person, the comradeship you had with those marines was something that doesn't often happen in your lifetime. The thought of letting one of them down just really bothered me, and I think that's the feeling inside that I had. I would probably rather have been dead than let one of those guys down and know that I did. I worried about it before we went into combat.\n\n\"We went ashore late afternoon at Red Beach Two. There were bodies everywhere. We were in reserve, but we were also supposed to be protecting the left flank of the Twenty-seventh Marines. Three days went by without me having to treat a bad wound in my unit. The first one I treated, Lieutenant James Cassidy, died of a bullet wound near the heart.\n\n\"Then on D plus three, February 22, a shell blew the right leg off my platoon sergeant, Joe Malone. It also took the fingers of his right hand and the right side of his face. There was sand and thread from his dungarees in the stump of his leg, and I got a tourniquet on it and a battle dressing, wrapped his hand, and gave him a shot of morphine. Then I wrapped his face. The skin was peeled away, and his teeth, gums, and cheekbone were exposed. But he survived. A litter crew came and took him out of there.\n\n\"I was trying to catch up with my platoon after that, but then I heard another wounded marine moaning, and I crawled over to him. A machine gun had cut across his stomach, exposing his intestines. The important thing with this kind of wound is to keep it moist, so I soaked a battle dressing from my medical bag with canteen water and put that on and then gave him a shot. I went on from there and treated a man from my platoon. It went like that.\" [There were 120 killed from George's battalion that day alone.]\n\n\"I quit carrying my carbine, gave it away to a BAR man, because it just got in the way. About ten a.m. on the seventh day Fox Company was sent forward to relieve Easy Company. It started to rain. We'd move forward, but then the Japanese would follow in tunnels underneath and come out and attack us from the rear in areas we thought were secure.\n\n\"The Second Platoon was crossing a flat, open area about an acre in size when the Japanese waited for the right minute, then hit us with mortar and machine guns. I was trying to stay with my unit, but then I turned and saw about fifteen marines down, still under attack from mortars and artillery fire. My friend Eddie Monjaras was their corpsman, and I didn't see him out there.\n\n\"I crawled out and dragged a marine with a leg wound into a shell hole, wrapped him, then went out and crawled from man to man. I found Eddie, finally. He was hit in the chest and stomach. I treated him, but he died later. They tried to have two corpsmen assigned to a platoon, but they was lucky to have one, I think. Mortar kept falling, but I kept going until I got to them all, through about twenty minutes of it.\n\n\"I assessed the injuries, usually by the state of their clothes and by how much they were bleeding. As I remember, we did have scissors, but I don't remember using them. Under combat you'd do whatever you could to take care of the problem. Getting shot at and everything, you just didn't want to do any more than you had to to get the job done. Those I treated were all laying out there pretty much not moving, all exposed. I just crawled to each one of them, I think there were fourteen, and done that, took care of them. I remember I finally got back to my own team. I remember laying there thinking about what had happened. I was still there. I was amazed I survived. Not even scratched. How do you explain it? I've often wondered.\n\n\"I have no idea how or why I survived, without even a scratch. I crawled into a shell hole and got the shakes as I thought about it. I was worn out. But the platoon was moving again, so I followed. We were right near Hill 362A. Later that day a grenade went off as I was crawling, and several pieces of shrapnel cut my face. I couldn't see out of my right eye and I was bleeding, and I got a dressing out and wrapped the side of my face.\n\n\"I heard a call for a corpsman and I started crawling toward a wounded marine, but grenades kept landing near him. I finally saw a Japanese soldier come out of a cave on the hill and throw a fourth grenade. All I had was my forty-five, so I yelled down to some guys to throw me some grenades. They did so. Then I took some shrapnel in the rear and legs from another grenade as I crawled up toward the cave. I wanted to throw mine when he was coming out, but I couldn't pull the pin. The marine had bent it over, for safety. I straightened it with my Ka-Bar. I crawled up close enough, and after he threw out the next one, as I remember, I pulled it out and waited for a couple seconds before I threw it in, so it would go off right away. I knew those things could come back at you, and I didn't want to give him time to grab it and throw it.\n\n\"He jumped out, and my grenade got him. Then I went to treat the wounded marine, and I dragged him down the hill. His leg was torn up, the calf all exposed. Somebody came with a litter after a while, and we got him down and out. After that George Long, another corpsman, and I took care of several casualties. Then I went back for supplies because Long refused my order to go.\n\n\"My captain, Frank Caldwell, was at the command post, and he saw me all bloody with a bandage over my face and told me to go down to the aid station. But instead I just grabbed a bunch of bandages and battle dressings and things and headed back to my unit. He saw me two or three days later. I was still with my platoon, and he said, 'I thought I told you to go back.' I said, 'Sir, I was still needed.'\n\n\"I'll always remember one particular time when we had this marine brought back to me screaming and out of it. I was in a shell hole, and we held him down, and I gave him a shot of morphine. He got pretty quiet after laying there awhile; then I got involved taking care of more casualties up ahead. I finally come back to see if they got him evacuated, which I'd hoped to do. But he wasn't there, and I never seen him again until the reunions years later back here in the States. Our company commander asked everybody to speak a little bit, and so I did. A guy come up from the back of the group and started to cry. He said, 'Doc, you saved my life.' I didn't know who it was because he had a beard. I talked to him afterwards, and it turned out he was the guy who had cracked up. He says, 'If you hadn't of left me there, I would never have made it.' But he was one of the five out of two hundred fifty in my company that survived the whole thing.\"\n\nBy that night Wahlen had treated more than twenty of the forty-nine casualties suffered by Fox Company, according to his story, told in the book _The Quiet Hero,_ by Gary Toyn. Eighteen were killed in action. The company was relieved the next day.\n\n\"We stayed in reserve until February 28, D plus nine, then it was back to 362A. We wound up alone on the front after the 327 pulled back, and Fox came under attack that night. Hill 362A was declared secure March 1. Fox Company had almost one hundred casualties with twenty-seven killed.\"\n\nOn D plus eleven Wahlen's company was in support of the 128 when several marines were shot and the corpsmen were busy. Near Hill 326B on Nishi Ridge, Wahlen crawled out to treat a marine whose legs were badly injured by a mortar shell. He was dragging the man to a safer spot when a shell went off behind and knocked George flying. When he came to, he could barely move, but he finally managed to crawl down the hill and into a shell hole, where he asked another marine to look at his back, which was badly gouged. He had the marine clean the wound and sprinkle sulfa powder on it, then wrap it with a battle dressing.\n\nHe continued to treat injured marines the next day as Fox Company was assigned to circle around Hill 362B as part of the assault. By noon it had gained three hundred yards, and by midafternoon it was moving up the hill. Then, around 4:00 p.m., Wahlen was rocked by a large artillery shell that landed on a number of marines in a shell hole.\n\n\"My strongest memory of Iwo was what turned out to be my last day in combat. As we were going up north, a group got hit with heavy fire, and as I was crawling up there, I got hit in the leg. There was casualties right in front of me, so I started to get up, but I couldn't. I looked down at my foot, and part of my boot had been torn away, and my right leg was all bloody and broken just above the ankle. I pulled my boot off and put a battle dressing on it and give myself a shot of morphine. Then I crawled up to where the marines were. As I remember, there were about five of them, and they were all pretty well shot up. I think one guy lost a leg, and others were all beat up. I worked with them and bandaged them and give them morphine as long as I could. Finally they were evacuated. Then somebody out to our left flank got hit and started hollering for a corpsman, so I crawled out on hands and knees and took care of him too. He could have been forty or fifty yards out there, so I crawled out and bandaged him up, and we crawled to a shell hole. Finally we was both evacuated.\n\n\"The stretcher bearers came for us but then dropped me when rifle fire came. I got out my forty-five and started crawling toward the enemy. It was the morphine. They finally came and got me and took me to the aid station. Four of us went from there on a truck to the field hospital. My war was over. I think it was March 3. I was scared myself plenty of times. I always remember that feeling of being scared, but the thought of letting somebody down scared me even more.\" [Only 82 remained of the 250 in Fox Company that landed on D-day.]\n\n\"They finally got me on a ship next morning and took me to Guam. After several days there they flew me to Pearl Harbor. The leg was not healing well, and they sent me from there to a naval hospital in Oakland and then down to the U.S. Naval Hospital at Camp Pendleton. I had several surgeries because they couldn't get the blood to flow downward, and I was afraid I was going to lose my lower leg. It was months before I could walk on it.\n\n\"When I got home, I met my wife, and she was Mormon, so later on I joined the church. I had to give up smoking first, and that was a problem. I learned to smoke when I was with the Marines. I think when I got started, I was in the hospital there at Camp Pendleton. I remember spending all that time with those marines out on the porch, and they're all smoking. Every time you got up in the morning there'd be a pack of cigarettes on your bunk. I smoked whatever they gave me; I wasn't particular.\n\n\"I got the Navy Cross with a gold star signifying two] when I was at Pendleton, and then in September they told me about the Medal of Honor. I flew to Washington, D.C., alone, and met my mother and father there and two aunts and uncles. There was fourteen Medals of Honor awarded that day, October 5, 1945, including Woody Williams [[Chapter 5] and Franklin Sigler, who was in my company. Neither of us knew the other would be there. Here were all these generals and admirals and brass, and I was nervous. I believe I was the first living Navy corpsman in World War Two to get the Medal of Honor. I couldn't imagine something like that happening to me.\n\n\"As I remember, I was the last of the fourteen to receive my medal, and all that time here were all these high-ranking people sitting out there watching us. For a Navy corpsman to receive the Medal of Honor is almost unbelievable, so I was pretty nervous. Here's me, a lowly corpsman, shaking hands with the president. It was almost unimaginable. So when President Truman finally got to me, I started shaking. He saw it and grabbed my hand and said, 'It's mighty good to see a pill pusher here in the middle of all these marines.' Then he put the medal around my neck. I was not tall, five feet eight, and I weighed probably one hundred twenty.\"\n\nWahlen had violent nightmares for months after he came home. He went to junior college. Then an Army recruiter came after him, and once he was assured he could stay around as a recruiter himself, Wahlen went into the Army as a master sergeant.\n\n\"I kinda hesitated, thinking about going back to school. Then he promised me a tour on the recruiting side, and I wanted to get married, so I decided to go in. The medal was the reason they wanted me. Public speaking was a little tough on me, and I didn't like publicity, but I had to get over that.\"\n\nHe and Melba Holley married in August 1946, before he went back into the service. They had met on a blind date. Her father was opposed to the relationship at first because she was only seventeen, and George was a sailor, he took a drink now and then, and he smoked.\n\n\"I got sent to Japan from 1952 to '54, went to Korea in 1963, and I was in Vietnam during the Tet offensive in 1968. Somewhere along the line I ended up getting commissioned as a lieutenant. I retired in 1969, worked for the state, then spent ten years with the Veterans Administration in Salt Lake City.\"\n\nYears went by before Melba even knew about George's Medal of Honor.\n\n\"She found out about it when we got our first invitation to go to a medal ceremony in the Rose Garden with President Kennedy. I didn't pass the word around.\"\n\nNobody ever asked him to return the Navy Cross and Gold Star, which were superseded by the Medal of Honor. And of course he holds two Purple Hearts.\n\nGeorge finally managed to quit smoking and became a full-fledged Mormon.\n\n**WAHLEN, GEORGE EDWARD**\n\n**Rank and organization _: Pharmacist's Mate Second [_ sic _] Class, U.S. Navy, serving with 2d Battalion, 26th Marines, 5th Marine Division._ Place and date _: Iwo Jima, Volcano Islands group, 3 March 1945._ Entered service at _: Utah._ Born _: 8 August 1924, Ogden, Utah._**\n\n**_Medal of Honor Citation:_**\n\n_For conspicuous gallantry and intrepidity at the risk of his life above and beyond the call of duty while serving with the 2d Battalion, 26th Marines, 5th Marine Division, during action against enemy Japanese forces on Iwo Jima in the Volcano group on 3 March 1945. Painfully wounded in the bitter action on 26 February, Wahlen remained on the battlefield, advancing well forward of the frontlines to aid a wounded marine and carrying him back to safety despite a terrific concentration of fire. Tireless in his ministrations, he consistently disregarded all danger to attend his fighting comrades as they fell under the devastating rain of shrapnel and bullets, and rendered prompt assistance to various elements of his combat group as required. When an adjacent platoon suffered heavy casualties, he defied the continuous pounding of heavy mortars and deadly fire of enemy rifles to care for the wounded, working rapidly in an area swept by constant fire and treating 14 casualties before returning to his own platoon. Wounded again on 2 March, he gallantly refused evacuation, moving out with his company the following day in a furious assault across 600 yards of open terrain and repeatedly rendering medical aid while exposed to the blasting fury of powerful Japanese guns. Stout-hearted and indomitable, he persevered in his determined efforts as his unit waged fierce battle and, unable to walk after sustaining a third agonizing wound, resolutely crawled 50 yards to administer first aid to still another fallen fighter. By his dauntless fortitude and valor, Wahlen served as a constant inspiration and contributed vitally to the high morale of his company during critical phases of this strategically important engagement. His heroic spirit of self-sacrifice in the face of overwhelming enemy fire upheld the highest traditions of the U.S. Naval Service._\n\nSERGEANT CYRIL O'BRIEN\n\n**Combat Correspondent  \nNinth Marines, Third Marine Division**\n\n_Going ashore at Guam on D-day all I had was a forty-five-caliber pistol and a little Hermes; it was in my backpack. . . . Sheets of paper and everything. My whole office was in the backpack._\n\nCyril O'Brien at the Marine Corps Museum in Quantico in February 2007. He was eighty-seven at the time and writing regularly for such military magazines as _Leatherneck_ and the _Marine Corps Gazette_.\n\nCombat correspondent Cyril O'Brien interviews a marine holding a carbine \"probably on our first day on Guam,\" O'Brien recalls, in June 1944. In O'Brien's left pocket is a hand grenade, still in its carton. \"I guess I figured I could use it if I got in real trouble.\"\n\n_Cy O'Brien, known to friends and acquaintances as Obie, is a vigorous sprite of a man, not quite five feet five inches high, energetic and vivacious at eighty-seven, still writing, still raising his voice. We met at the Iwo Jima Veterans Symposium in Arlington, Virginia, in mid-February 2007, hit it off, and started hanging out. He ended up telling me his story in my hotel room. I offered him a scotch. He accepted. I joined him. As our talk went on, we had a couple more, and then, as we went down to dinner, I said something about martinis. \"Martinis!\" he exclaimed. \"I just love martinis!\" So we each had a martini. Dinner followed, accompanied by wine. We both got crocked._\n\n\"My mother ran away from home in Newfoundland in the 1890s, at the age of sixteen, and went to Camden, New Jersey, where she worked as a chambermaid for several years. She said she saw the great star Lillian Russell and her entourage sweep into some fancy hotel in Atlantic City. Eventually she went back home and met and married my father, who was a salty old son of a bitch. He was a fisherman, and a good one. He had about a fourth-grade education, but intelligence and learning are not the same. He was very intelligent. At the age of sixteen he went to sea, to Australia, all over, and then came home to fish for cod. He was good at it. I was born in Newfoundland in 1919, and we moved down to Massachusetts and then down to Camden, New Jersey, when I was nine. I grew up in a blue collar part of town where nobody got any higher education. My dad insisted I go to college. I don't think there was anybody in four or five city blocks around me that went to college.\n\n\"There was a Newfoundland guy in Camden who built docks. Dad was used to the sea, and Dad was a carpenter, so it was easy for him to get a job as a dock builder. I graduated from high school in Camden 1938 and went to St. Joseph's, a Jesuit college in Philadelphia, across the Delaware River. I majored in English and philosophy. The Jesuits are big on philosophy. You had philosophy three days a week. You had Descartes! You had Spinoza! The Jesuits! Boy, they run you through it.\n\n\"At the same time I worked at the _Courier-Post_ in Camden as a copyboy and then a reporter. I finished usually at one-thirty in the morning. I'd leave the college and go there at four in the afternoon. After they put the paper to bed, I studied there, or I went home and studied, or I didn't study at all. I made up for it on the weekends, cloistered myself, never went out, no fishing, no girls, no nothing. It was supreme discipline. You had to be motivated. But I knew it was the only way I could get through college. I think I made twenty-five dollars a week. My parents didn't have any money. I worked there the whole time I was in college. I did box while I was there. I always loved boxing. I took it for two years and never did learn to use the right hand.\n\n\"I managed to get hired because I always wanted to be a writer. Unlike other people who want to run things, I wanted to be a writer. At that time I wanted to do fiction, and right now I don't give a shit about fiction. I don't even read fiction. I thought I wanted fiction, but now I don't like it. I prefer reality. I was somewhere between quasi-reporter and copyboy. I took a course in typing in high school. I also learned shorthand. I wrote an article that appeared in _Family Circle_ magazine; it was a handout in grocery stores.\n\n\"I wasn't heroic. I wasn't interested in the military. I didn't have a military mind. Had there been no war, I wouldn't have been a marine any more than I would have been a baker. But I knew I was going to be drafted. The only reason, really, why I went into the Marine Corps was . . . it's a shame sometimes how you do things just because you're challenged: The recruiters came out to the college seeking people for OCS, officer candidate school, and when they got to me, this recruiter said they wouldn't take me because I was too short, by less than half an inch! 'You're a half inch too short,' he told me. Those were his exact words. How chickenshit could you be? That was so chickenshit it was ridiculous! I mean that's childish, isn't it? Less than a half inch too short? That's crazy.\n\n\"So I got pissed off and I went down to the Marine Corps and complained. Rank didn't bother me. I didn't give a shit who they were. So I went into the colonel and shared my opinion with him, and he said, 'Well, if you're so interested in going into the Marine Corps, why don't you just enlist?' and I said, 'OK, I'll enlist.' So I went to Philadelphia, enlisted, had a month free, and I went in on July 21, 1942, and I landed on Guam July 21, in 1944, and I also got married on July 21\u2014in 1950. Just coincidence.\n\n\"Parris Island was tough, it was challenging, but it was a wonderful experience. One night, in comes a runner: 'O'Brien!' He didn't say Mr. O'Brien. You weren't anything; you were a boot. He said, 'The record has it that you're a college graduate; why don't you go to officer candidates' school?' I was chagrined. I said, 'You turned me down!' He said, 'Well, look, you're in the Marine Corps now, so it doesn't make any difference. You wanna go to OCS?' I said sure, but I had a tendency to high blood pressure and I kept getting tested and everything got delayed, and I was training with the same guys, boondocking, and we completed boot camp in October.\n\n\"They had me doing sentry duty in Cape May, New Jersey, watching for subs, for a few months and then they sent us to New River and out to San Diego and Camp Pendleton, for more training, running up and down hills. Then we were all ready to go overseas. A day or two before we're going to leave, this guy comes into my tent and says, 'Obie, you're not going overseas. You're going back to OCS.' And I said, 'Horseshit! I'm going overseas!' Wouldn't you do the same thing? After all this training, I'm going for training again? Horseshit. So I went overseas. I don't know what kind of authority I had that way. I guess you can reject officers' school, can't you? Well, I did. I just rejected it.\n\n\"We arrived on Bougainville November 1943. It was fascinating going over the South Pacific, oh, my God! Why, I saw flying fish coming right up out of the water. I was a boot, and we came in as reserves, and they gave us all the lousy work. I was the number three mule in a mortar squad, humping ammo, out on patrol every day. You couldn't see, hear, or smell a man five feet away. It was the most dense jungle in the world. But I got off easy on two campaigns, Bougainville and Iwo Jima. But on Guam I made the assault, as a combat correspondent. I wanted to be a correspondent. It was in my mind, but I never knew where to go.\n\n\"Here's how that happened: After Bougainville we were sent to Guadalcanal for more training. Bill Burnette was a military correspondent, who had worked for the _Baltimore Sun._ He didn't know me at all, but I was walking by, and I saw he had a tent that said combat correspondent, and I went in and told him I was a newspaperman, and were there any chances, any opportunity for me? He says, 'You hit at just the right time. We're looking for a guy.' It was completely fortuitous, but that was how I became a combat writer. This was 1944. Pat Osheel was a newspaperman too, but he was a captain. He was the public relations director, and he took me on because I had been a reporter on the _Courier-Post_ back in Camden. It was just completely fortunate that they needed somebody, a total accident. Correspondent! Just like that. The first job they gave me was a whole lot of medal citations. I had to find the guys who were being decorated and interview them and write them up and send them back to the hometown newspapers. It was doing rewrite in the city room. Most of them were for Bronze and Silver Stars. I'd write maybe five or six a day.\n\n\"I want to tell you what happened on Guam, but first I'll explain about Iwo Jima. I came in about March 12, after the second airfield was taken, and stayed about three weeks, writing up medal citations and going out with patrols. I had a forty-five caliber pistol and a little Hermes typewriter, and when I wanted to write something, I put it on my lap or on a box, anything I could find to hold it. They flew me in with a combat pack. We got fired on when I landed, and I had to get in under this jeep that was picking me up.\n\n\"Were they firing at me? I don't know, but whenever you hear firing, you get down. The fight was slowing down, but there were patrols that went out looking for Nips, and I went with them. This was mostly cleaning up, although there was some bitter fighting still to come. I used to love patrols. I went on patrols all the time. We never had night patrols. The reason I went on patrols is that you knew damn well something was going to happen, and the safest place you could be was behind a platoon of marines. I was behind the greatest bunch of marines in the world.\n\n\"And where was I going to use that forty-five? I did get into a fight. We were on a patrol, and all of a sudden I'll be damned the Japs started shooting at us from the bushes. The marines immediately put skirmishers out, and I had nothing but this goddamn forty-five. Soon they're heading toward the Japs. What the hell have I gotta move up for, with a forty-five? So I'm waitin' there, and all of a sudden, from behind this pile of logs, a Jap jumped up, and he's firing at me, and I said to myself, 'This son of a bitch, doesn't he know I'm a correspondent?'\n\n\"I know I was on Iwo March 26, the day the island was declared secured and those three hundred Japanese came down and killed all those pilots. I had been up around Kitano Point, but I went down the next day, when the bulldozers and three big Army trucks came loaded up with bodies. They buried them right down by Motoyama No. 1 airfield. They scooped out a big trench, laid them in neatly, then came along and dozed them over. The only reason I know I was there was because I saw them being buried.\n\n\"I had much more action at Bougainville, where I was infantry, and on Guam, July 21, 1944, by which time I had become a correspondent. I landed with the third assault wave on Guam. I wanted to cover the landing. I was thinking of the great epic of the war. But it was just a lot of fire, all confusion. I didn't know where to go or anything, so to play safe I jumped into a shell hole. All I had was a forty-five pistol and my Hermes; it was in my backpack. The top pack was my gear; the bottom pack was my Hermes. Sheets of paper and everything. My whole office was in the backpack.\n\n\"It was very hot, by which I mean artillery fire and mortars and machine guns were blazing from shore. The only thing is, it was unknowable. You landed, but you didn't know what was going on. You didn't know where to go. You only knew you had to get the hell off the beach because mortars are falling, and there's machine-gun fire all over hell, sparkling in the water. I therefore knew it was dangerous if I stood up, so I jumped into that shell hole. I stayed there awhile, and in true bravado I saw marines going by. Remember, I wasn't like them, I could go where I wanted to. But when I saw them going by, I said, 'Well, it must be safe,' so I got up and went with 'em.\n\n\"As I found my way around, a very interesting thing happened. Up on Chonito Ridge a marine says, 'Look over there, Obie.' And I saw a body tumbling down the ridge. He said, 'You know what, Obie? There's a Jap machine gun up there. A guy gets out there and shoots, and when he gets killed, they push him out and another guy takes over.' It was a kind of cliff face, and I just saw the body come tumbling down.\n\n\"I got ashore that first day with a guy who was the intelligence officer of the regiment, a very bright guy. I followed him. He says, 'Obie, we got a good spot.' He went up to the side of a cliff, side of a hill, and I dug the foxhole where the hill was going up. Well, he was so smart. You can't get hit by mortars there. You aren't getting attacked right there. It was like sleeping at home in bed. So all that night they battered the hell out of the beach, but they didn't bother me at all.\n\n\"I told you one time how, after you get used to danger, danger is danger? Later in that same day I remember a conversation. A marine was talking about Peoria, Illinois, and he says, 'Well, there's a church up there\u2014' and all at once machine guns open up, and mortars are screaming in, and we all get down. Then it goes away, and he gets up and starts going on about Peoria again, calm as could be. What I mean to say is how even something as deadly as that can be kissed off. If that happened to you once, if you were a civilian, you'd remember it forever.\n\n\"The mentality amazes even me. Here I wanted to write this epic story of the landing, so I started to type, and I'll be damned, mortars started to fall in. And I said, 'Oh, shit!' I didn't say, 'Oh, shit,' because I was going to get killed. I said, 'Oh, shit,' because they were interrupting my writing. I hadn't even dug in: I was just typing. So when the mortars started falling, I jumped into a foxhole with a guy, and he said, 'Fine, Obie. It's fine.'\n\n\"It stops, and I go back to typing, and\u2014God damn!\u2014the mortars come again. So I jump in again. The third time he says, 'Obie, God damn, you got to dig your own foxhole!' The point I want to make is you got so interested in doing what you wanted to that you forgot about the danger. Isn't that something? They called me Obie, short for O'Brien. In the military all the emphasis is on the last name.\n\n\"I was wounded only once, on Guam during the landing. I felt wet and looked, and blood was running down my hand. I didn't know what had happened, never knew I was hit, never felt I got hit. All I did was see blood. As a correspondent you knew everybody, so I went up to the corpsman, and I didn't say 'wound,' I said, 'I got a cut here. Better put something on it.' I wouldn't have gone to a lot of trouble, but I was right with him, and he fixed it. Then he says, 'Obie, you're gonna get a Purple Heart.' I liked that, but then I looked around, and here was a guy with his leg split open, and there was a guy with his belly open. I says, 'Horseshit!' For a cut? I couldn't take it. He even wrote my name down, but I said no. Wouldn't you do the same thing? Had I been out on patrol and had been hit like that and nobody was around, I might have accepted it. But being right there with guys battered on the beach, I couldn't do it.\n\n\"I'll tell you when I felt bad. On Guam the biggest banzai charge that ever happened in the war was on July 26, 1944. I think they found three thousand bodies. It was like a flood. They came down at us, screaming and yelling. Somebody who wrote about it later said some of them knew they were going to die, so they tore up the pictures of their loved ones before they attacked. Maybe the banzai worked against the Chinese, but it would never work against serious marines. It was like shooting ducks in a barrel.\n\n\"You know I think the most wonderful, most highest journalistic praise I ever received, was on Guam. My company, Easy Company, Second Battalion, Third Marines, went up to outflank the Japanese who were holding the top of Chonito Ridge. Able Company had spent a day and a half in a frontal attack and couldn't do it. The Japs were so close, they couldn't throw grenades. They had to roll them down.\n\n\"My company went up the side. Three times our guys tried to assault that hill to get up there and was thrown back. Finally Easy Company flanked them, and we took the hill. I wrote about it, and it got published back in the States. It came back, and Captain Moore brought the company out, and with me standing there, the company commander had someone read the story out loud. Isn't that great? You couldn't get a better tribute than that.\n\n\"I went back to Guam from Iwo and stayed there till September of 1945. I got out in October of 1945. Back in the States, I returned to my old job at the _Courier-Post_ in Camden. After a strike in 1947, I wound up at the _Long Island Press,_ the most colorful newspaper I ever worked for. From there I went to the paper in Stroudsburg, Pennsylvania, where a combat correspondent name of Hamilton offered me a job covering the Capitol for the Irwin News Service.\n\n\"Journalism is fascinating, isn't it? I show up at the office, and he says, 'Obie, get up to the Capitol and to the Senate Cloakroom. Senator Taft's having a press conference on the price of natural gas at the wellhead.'\n\n\"What the hell did I know about natural gas at the wellhead? But as a reporter, when you're told to do a story, you do it, I don't care what it's about.\"\n\nCy O'Brien eventually moved on to do media work for the Applied Physics Lab at Johns Hopkins University in Baltimore. He and his wife adopted four children. She died in 1975. He never remarried. Obie retired from Johns Hopkins in 1985 but still maintained a weekly column for two newspapers and went on to write extensively for the _VFW Magazine._ He continues to write for such Marine Corps publications as _Leatherneck._ He is the author, among other things, of a definitive thirty-eight-page account of the battle of Guam for the World War II Commemorative Series entitled, _Liberation: Marines in the Recapture of Guam._\n\nHere is a sample of his reporting on Chonito Ridge, taken from a carbon copy he saved and donated to the Marine Corps. It is the piece Captain Moore had read out loud to the company:\n\n_Guam July 25 (Delayed) Nearly half my old company lies dead on the barren slopes of Chonito Cliff. Four times they tried to reach the top. Four times they were thrown back. They had to break out of a twenty yard beachhead to make way for later landing waves. They attacked up a 60-degree slope, protected only by sword grass, and were met by a storm of grenades and heavy rifle, machinegun and mortar fire._\n\n_The physical act of forward motion required the use of both hands. As a consequence they were unable to return the enemy fire effectively. Most of the casualties were at the bottom of the slope. They had been hit as they left cover._\n\n_There was Pappy, his name stenciled on his canteen cover. A bullet had ripped away the first \"P\" in his name._\n\n_My former assistant squad leader was beside him. He often had me on his working parties. I had seen those arms of his, which in death still clutched a splintered rifle, throw full ammunition cases about as if they were empty._\n\n_There was the first scout of my squad. We had shared the same tent for months. He was always promising himself \"a white Christmas in forty-five.\" He was facing the sky, his hands at his sides. You'd think he was dreaming_\n\n_Willie, who volunteered for mess duty so he \"wouldn't have to stand inspections,\" was lying halfway up the slope. His feet were dug into the dirt. His arms were bent as if he were ready to charge again, but his Garand rifle was empty and thrown aside. The pistol in his hand was empty too. Perhaps he intended to club the enemy with the empty weapon._\n\n_The \"Beast\"\u2014we called him that because he was so big\u2014had charged his big frame to within five yards of an enemy machine gun nest. He caught a blast in the chest. The fancy lettering he always placed on the back of his dungaree blouse was torn by bullets._\n\n_There was Frankie, who had received a shiny, chrome-plated pistol from home. He had boasted he would get many Japs with it. Now the sun's rays from over the ridge glinted on the handle. The pistol was still in its holster._\n\n_Peter had a strong voice in camp. He had it in the face of death. He was halfway up the ridge, yelling something about the \"bastards\" on the top, when their fire cut him down._\n\n_The lieutenant we called \"Chicken\" because he seemed so young was the only one to reach the crest. A grenade smashed in the side of his head. Those skinny legs which had led me so often to exhaustion were white in the dried grass. Two Japs, five feet in front of him, had holes in their heads. An American grenade which the Chicken evidently had thrown was lying between the Japs. How often the lieutenant had drawled in his slow hesitant tone: \"Now, a grenade\u2014it explodes five seconds after you heave it.\" His hadn't._\n\n_Eddie was lying in a bed of mountain flowers. He was fond of flowers. He used to put them in his helmet for camouflage when others used grass. The hand that was not on a rifle was crushing a flower._\n\n_Behind the lieutenant, his face anxious as if awaiting an order, was Angelo. He loved to sing\u2014but couldn't. He and I were penalized once for singing \"Put on Your Old Gray Bonnet\" after taps._\n\n_The company was still under fire when, on the ridge, I talked to the men who had made it. Private First Class Leon Slicner of Perth Amboy, New Jersey, tried to tell me how \"Smokey\" could have been saved had they been able to pull him out of the fire lane in time. His words came slow. Finally, he stopped in the middle of sentence, leaving the story in midair. He really didn't want to talk. Besides, he was pressing low into a foxhole, and he couldn't breathe well, for a machine gun was spitting fire over our heads._\n\nFollowing are some samples of Sergeant O'Brien's combat correspondence on Iwo Jima, culled from carbon copies he had saved and donated to the Marine Corps:\n\n_Somewhere in the Marianas \u2014 How a Marine machine gun section repulsed a Jap night attack on Third Marine Division lines on Iwo Jima was told by Corporal Herbert J. Bovia, of New Orleans, La., the squad leader._\n\n_The 20-year-old Marine, who lives at 518 General Pershing Street, led a six man team which with two machine guns filled a gap between two platoon defenses. Due to the nature of the terrain, Corporal Bovia's unit dug in 20 yards beyond the rest of the front line._\n\n_Enemy small arms fire ceased at darkness, and, except for the pounding of artillery on the right flank, the front was quiet. It was nearly two in the morning when Bovia's gunner saw a figure crawling toward their position._\n\n_Bovia scanned the area to his front. Other figures hopped over a low ridge not twenty five yards from the muzzle of the corporal's machine gun. Crouching, the Japs moved toward the Marine lines. They did not see the machine guns that lay in their path until the raking fire cut into them. Three figures slumped dead on the volcanic rock._\n\n_Yelling and throwing grenades the others charged the exposed machine gun positions. Grenades exploded near Bovia's parapet and he was sprayed with dirt and stones. Two grenades exploded in the No. 2 gunpit, wounding the gunner and killing his two assistants._\n\n_The machine gun fire disorganized the night attackers, but they still rushed his position, and Bovia was in danger of being cut off from the rest of the Marine line._\n\n_\"The Nips were trying to get behind us,\" Corporal Bovia said. \"We killed three who almost made it. Our only choice was to get back to the lines, so I disabled the machine gun, and abandoned it._\n\n_We dashed back to the company and the Nips were hot on our heels._\n\n_\"From the lines we blasted the Nips again, but it was nearly fifteen minutes before the attack stopped.\"_\n\n_When morning arrived fifteen Japs were found dead in front of the Marine guns. Blood trails on the jagged ledges showed that at least six Japs had staggered away seriously wounded._\n\nOverseas for twenty-seven months, Bovia saw action on Bougainville and Guam. Before entering the Marine Corps in December 1941, the New Orleans marine was a doffer in a cotton mill near his home.\n\nThe next example describes an ambush:\n\n_Iwo Jima \u2014 We planned to trap the Japs tonight in a small ravine that sheltered several waterholes._\n\n_Under cover of dusk and the rising evening mists, we crept to our ambush position behind the boulders that ringed the waterholes, and waited._\n\n_Enemy stragglers, driven by thirst from their cave hideouts in the surrounding volcanic rock cliffs, ventured here nightly to drain the muddy puddles into their canteens. Their sandal tracks dotted the the moist sandy margins of the holes._\n\n_I hid with a 9th Marines machine gun squad, whose weapons pointed down a foot trail that led to the cliffs. The gunner, Private William E. Winkler, whispered a curse to the cold and pulled his blanket over his shoulders._\n\n_He had no time to enjoy the blanket's warmth, for he threw the cover off and stared behind him. Footsteps sounded to our rear. Japs were coming for water._\n\n_Privates Leo M. Chabod and Jack Woenne covered the rear approach from an adjacent foxhole._\n\n_Five figures, hugging the side of the trail, walked past the muzzles of the ambushers' rifles. The leader carried a mess tin, which tinkled on its handle as the Jap moved. Behind him the others carried canteens and bringing up the rear a stubby Jap, who continually glanced behind him, carried a water bucket._\n\n_Chabod let them enter the center of our ring. They cluttered around the waterhole and the leader was looking into it when Chabod's automatic rifle stuttered. The Jap carrying the bucket grabbed his back with both hands. The bucket rolled between his legs and the Jap pitched on his face. Another ran for the back trail but Woenne's fire followed him there, and he slumped in a heap on the trail's edge._\n\n_The Jap who carried the mess gear was already running down another trail, his mess gear clanging like a cheap bell. He swerved toward the cover of an overhanging ledge but was affronted by the foxhole of Pvts. Raymond Tenavitz and Thomas Cashin._\n\n_Cornered, the Jap pulled a grenade and was extracting the safety pin when Cashin's rifle fire sprung him into a somersault and the Jap rolled down an incline dead, the grenade still in his hand._\n\n_Two Japs still hid somewhere in the center of our ambush but Private Victor Schmidt soon routed them with a hand grenade._\n\n_The Japs leaped from the base of a dead tree. Resigned to sell their lives dearly, they rushed the foxhole occupied by Private First Class Robert Wolf and Private Fred Herrick Jr. The leader flung a grenade and was within ten yards of the foxhole with another in his left hand when Herrick's M-1 barked. The attacker spun and fell on his back. Herrick sprang from the hole and blasted the Jap again, to make sure._\n\n_Trapped and confused, the last Jap ran to the tree, changed his mind and charged back over the body of his dead companion and out of the ambush._\n\n_Silence fell again except for the occasional rasping scratch of a land crab or the moan of a tortured tree. An animal ran across the trail that was our fire lane, but that was all that came during my watch._\n\n_I had awakened Private Duane Wills to relieve me, when two carbine shots cracked in quick succession to our right. We turned in time to see Private First Class Dale Beckett dive into a rock pit as a hissing grenade passed over his head and exploded behind him._\n\n_In a draw below, a Jap slumped over an abandoned enemy satchel, two bullets through his neck. Another Jap hugged the shadowed sides of a draw from where he had thrown his grenade._\n\n_The Jap could not be seen in the shadows, but he made a frantic dash into the moonlight to escape from the draw. Two .30 caliber slugs passed through his head. Private First Class Harper R. Rudge guarded the ravine from the opposite wall. Rudge crawled to the edge of the draw, tossed a grenade, then disappeared behind the rock barricade._\n\n_Star shells were falling continually now over the beach area to our front, and in the distance a machine gun clattered._\n\n_\"Doggies,\" [Army soldiers] Wells said. \"The Nips are giving them trouble again.\" He stared back on the trail and hunched his body over the machine gun. I curled up at his feet in an attempt to sleep, but he soon tapped my helmet. Japs were again back on the trail._\n\n_Four walked boldly into our ring, gibbering among themselves. From behind a stone wall a burst of fire cut into the Japs. Two doubled over and fell. Private Patrick J. Cleary Jr. stood upright in his foxhole and cradled his Browning Automatic Rifle._\n\n_Shot through the legs, a Jap dragged his body with his elbows toward a grenade bag but before he had moved three feet another burst from Cleary's weapon caught him in the chest._\n\n_Another Jap, his right leg shattered, moved with surprising swiftness toward Chabod's position. The Marine dropped on the ground beside his companion as a grenade bounced off the parapet and exploded. The Jap was still rushing with a second grenade when a shot from Woenne's rifle caught him in the middle. He dropped in a sitting position, dead._\n\n_On the road, the first Jap caught in Cleary's surprise fire raised his body on his left arm. A grenade sputtered in his hand but this Jap was through fighting. He exploded the missile under his chest._\n\n_Dawn, and the ambushers stirred from cover in the crypts and behind the rocks. Through habit they still spoke in low tones._\n\n_Private First Class Ferdinand Leon found a bloody trail. Someone had dragged a wounded Jap away. He followed the trail for 20 yards but lost it on the jagged slope._\n\n_We filed back past the waterholes, and for the first time I looked in them. Eleven Japs had come carrying canteens, buckets and mess tins. Nine had died here._\n\n_There was not enough water in the well to fill a single canteen.\u2014Sergeant C. O'Brien._\n\n[In a note accompanying this story, O'Brien wrote that after the ambush in the dawn hours, \"I went to the Japanese bodies and found one had been taken away. I found and followed the tracks of a dragged body up into a kind of field where I lost it. Whoever had rescued that body or person had come in under our guns to do it, and got away with it. There's heroism here. That was a waterhole and you know what water meant there and then.\"]\n\nFinally, this one:\n\n_Somewhere in the Pacific \u2014 A Memphis, Tenn., Marine, Corporal Kenneth R. Sigman, 22, is alive today because a Jap on Iwo Jima forgot to wear a helmet._\n\n_The Jap was a member of a band of night attackers which crept from the well stocked caves in the north section of the island to harass Third Marine Division lines. All the Japs wore helmets except the soldier who attacked Sigman's position._\n\n_The Japs carried grenades, but before throwing them, they had to strike the missiles on their helmets to ignite the fuses. The bareheaded Jap hit the grenades on his bare head, then flung the explosives at Sigman._\n\n_But the Jap's head was not hard enough. The grenades landed harmlessly on the foxhole parapet, three feet from Sigman's head._\n\nSERGEANT THOMAS HAYWOOD McPHATTER\n\n**Eighth Marine Ammo Company**\n\n_Nobody in combat can live without food or ammunition._\n\nCommander Thomas H. McPhatter, fifty-three, of the Navy Chaplain Corps, photographed in San Diego in 1977. He retired as a captain in 1983.\n\n_I asked Thomas McPhatter over the phone at his home in San Diego if he had been a Montford Point Marine and whether he had known Hashmark Johnson and Edgar Huff. Yes, and yes, he replied. Montford Point was the segregated base where blacks were sent to be trained by the Marine Corps following the issuance of an order by President Franklin Roosevelt in June 1941 stipulating \"full participation in the defense program by all persons regardless of color, race, creed or national origin.\" Johnson and Huff became well-known leaders at Montford Point. McPhatter and the roughly twenty thousand who trained there during the war were not considered suitable for combat, although they were trained to fire the rifle. Between fifteen hundred and two thousand blacks served on Iwo. Each Marine division had a regimental service battalion or field depot staffed by black marines, not to be confused with the Army DUKW drivers. There were stevedore units that were part of the shore party battalions. The depot companies' job was to move supplies from beachfront to supply dumps and from dumps to tractors and weasels (small tracked vehicles) and moved to the front by motor transport battalions. Blacks also worked in Graves Registration._\n\n\"I come from Lumberton, North Carolina, but I've called San Diego my home for over fifty years. We trained at Montford Point and practiced landings at New River. Second Lieutenant John D'Angelo had the first section of our platoon, and I was a sergeant and leader of the second section, First Platoon, Eighth Marine Ammo Company. We came ashore at Iwo Jima D plus one in an LST.\n\n\"I did not go up on the summit of Suribachi, but we were established at the foot of it. We had crawled from the LST up near Suribachi to establish an ammo store to supply the Third, Fourth, and Fifth Divisions. They were running ammo on a conveyor out of the mouth of the LST onto the sand. My lieutenant, D'Angelo, and I crawled up there to find a secure place for the dump. We chose a spot, and he went back to get the rest of the troops.\n\n\"After growing up in Lumberton, I became a pretheological student at Johnson C. Smith University in Charlotte. I was the eleventh child in our family and the only male. My mother had ten girls. I didn't have to go into the military. I also worked during the summer for a dry dock company in a Newport News shipyard, so I had an exemption there too. But I was back on campus, and all my friends had got into the ERC, the Enlisted Reserve Corps, which was supposed to allow them to stay in college till they had finished their education and then go into the military. Well, the buses came up from Fort Jackson, South Carolina, one day and pulled them right out of class and took them away.\n\n\"This left me one of the very few males on campus, and I didn't feel right because I didn't know whether I was ministerial material. My mother wanted me to be a minister because she felt I was a godsend. My pastor wanted me to be a minister, but I didn't know if that was where I should be.\n\n\"I'm thinking: I don't want to be here because the war will be over and I'll be considered a draft dodger. I went to the Merchant Marines, but I was too young, I wasn't twenty-one. So then I went to the Marine Corps, and they said you have a 4D deferment. We can't take you unless you're reclassified to 1A. So I got myself reclassified and joined the Marines. So here I was on the train to New Bern with these black marines strutting through the train, walking on me like I was grit. At Jacksonville a big truck came, and the driver made me get in the back. It was cold, January of '43, and that was how I landed in Montford Point. They told me to say yes, sir, and no, sir, to every uniform and to salute them all.\n\n\"Hashmark Johnson and Edgar Huff were there. Hashmark was the top man, Huff was the battalion sergeant major. Boot camp lasted twelve weeks, and it was not difficult for me. I had just pledged a fraternity, and they did so much hazing on me that the way they treated me at Montford Point didn't bother me. I was used to it. I came out a PFC because I fired expert at the rifle range.\n\n\"I almost became a chaplain out of boot camp in March, and I was just getting snapped into the job when a man came in whose wife was about to have a baby, and he wanted the job. I said he could have my place. I said I'd rather be with the fellas I had come through boot camp with. So I gave that up to go with Eighth Ammo.\n\n\"We did part of our training at nearby New River, making landings and getting combat swimming instruction. They saw I had some college and they sent me to ammo tech school to learn about fusing rockets and mortars and propellants, how to store ammo, TNT, bangalore torpedoes, small arms, firing caps, that kind of stuff. We learned how to store ammo and how to move it. I had taught water safety back in Winston-Salem, so I volunteered and began teaching troops combat swimming, how to make life preservers out of mattress covers, pillowcases, or trousers. I was at Montford Point perhaps a total of six or seven months.\n\n\"We went to the West Coast late in 1943 and from there to Hawaii, at Camp Catlin, where we didn't feel we were being trained properly. The white boys were getting overnight liberty and weekends, and we had liberty for five, six, or seven hours. I wrote a letter to my mother and told her the very things I had got into the Marine Corps to get away from I'd run right into the middle of, which was racism. Well, all the letters were censored, and when they read that, they had me go see the captain and executive officer. The executive officer said, 'Did you write this, McPhatter?' 'Sir, yes, sir.' 'Why did you write it?' 'That's the way I felt, sir.'\n\n\"He said, 'Well, you're from Lumberton, and I'm from Asheville, and you've had more college than I.' I said, 'Sir, that's exactly what I'm talking about. You're the executive officer and a lieutenant, and I'm a sergeant.' He said, 'Well, the captain wants to see you.'\n\n\"The captain said, 'Did you write this letter?' 'Sir, yes, sir.' 'Well, what do you mean by you came into the corps and had more racism? Didn't you know this letter was going to be read? Why did you do that?' I said, 'It was the only way to let you know how I felt without being charged with insubordination.'\n\n\"I had more trouble when they wanted me and my men to move a big stack of corroded ammunition from an underground magazine, to be taken out to sea and dumped. I refused because it was too dangerous. Lieutenant D'Angelo stuck up for me, and that's what made me love the man. He was a man. He knew what I did was right, but he never said anything to me about it, just led us on a forced march to get us off the hook. My career was loaded with that.\n\n\"I was run up to the captain so many times, even on the ship to Iwo Jima. They had me up there, trying to get me busted. I had to do it. I had to insist on my rights.\n\n\"We landed on D plus one on Blue Beach right at the foot of Suribachi. We were standing by to go in the first day, but the Japanese just turned all hell loose after the first three waves came ashore. They came out of caves and holes, gun emplacements, everything else. You could hear men from ship to shore; you could see planes flying overhead trying to drop stuff and the big guns firing off the carriers. They killed a lot of our own men on that island.\n\n\"The beach was still very hot when we went in on the second day. Men were there dead and dying, still trying to get up off the beach, bodies bobbing in the surf. We worried about broaching and had to get pontoons off on the side so we could anchor the LST and get the tractors and trucks and guns out of the belly of the ship. They dropped some mats down as they landed. I had to go set up the ammo dump. You had to be very careful where you walked because of the mines.\n\n\"We were waiting for the DUKWs, big rubber-tired vehicles that could go through water, to move ammo up forward where we were putting it in store up by the first airfield. We had men who didn't want to go. They were afraid. They were saying, 'Shoot me, I ain't going up there.' Lieutenant D'Angelo was my hero. He never asked me to do anything he wasn't willing to do first. So we went up there and got this dump and a perimeter set up. Nobody in combat can live without food or ammunition.\n\n\"We kept moving that ammo on up until one night here's a little gift, _pip pap pup,_ mortar dropping into our ammo dump. Pretty soon a charge went off that shook the whole island. A captain came running up and told me to get a detail and go in there and put the fire out. I said, 'Sir, it is burning. I don't know how I can stop it. We need an armored bulldozer.' Lieutenant D'Angelo said, 'Let's go in,' so I started to run with him, but then that whole dump started going off, shells flying everywhere. A smoke shell fell near me and burned my face. We all started running for the beach, hollering the password for that night, which was American Trees. You didn't know who would shoot you. If you didn't give the password, they'd just blow you away.\n\n\"The next day the lieutenant told me to see how many men we had left, and I was going around to the ships in the harbor to see how many I could find. I told them we were going up to the dump to see what was left. Four or five of our men never got out of their holes. They were dead.\n\n\"Then a priority order came through for hand grenades, howitzer shells, and we didn't have any left. So they flew it in from Saipan, but the planes couldn't land, so they dropped it in parachutes. Lieutenant D'Angelo told me to get my sections up on the side of the airport and chase the parachutes. Now we're running after these parachutes of ammo scared to death. Then the Japanese spotted us and came in with a barrage of mortar fire, and now everybody's running for cover, and I jump into a bunker. A young white marine was there, holding on his chest pictures of his girlfriend, mother, and all. He had blood coming out of his mouth, his ears and his nose. He was dead. I had to try and scrape out a hole with my helmet out of that ash big enough to put my head in because that was the only thing I could protect.\n\n\"I lay there next to him and said the Lord's Prayer over and over, along with the Twenty-third Psalm, until the all clear was given. I made a promise to God if he let me live, I'd give my life to him in service. I never looked back. It may sound heroic, but it really wasn't. I wasn't in for the flash. I was a driver of men, and I knew if I got it, they would do the same thing.\"\n\nAfter the war McPhatter went back to Johnson C. Smith University, became a pastor, and later entered the U.S. Navy Chaplain Corps and served in Vietnam as a captain. He retired from the military in 1983. He is a retired Presbyterian minister with a Doctor of Divinity from the Interfaith Theological Center in Atlanta.\n\nHave things improved for blacks in the military since the days of Iwo Jima?\n\n\"They have improved, but you know I'm a marine and, if it's wrong, I think it ought to be right. There are no shades of right and wrong. Justice has no shades. Was it King who said justice prolonged [delayed] is justice denied? I'm eighty-three years old. If it's not here now, when am I going to see it? I just cannot be at ease with anybody being treated unjustly. I'm a champion of the underdog, right from the beginning of my life. I was known as a black Moses in the Chaplain Corps. They even kicked me out for a spell. I don't care what color they are. If it's unjust, let's make it just. My Lord says, 'Be in the world but not of the world.' I want the world to be different than it was before I came in here. I can't help myself.\"\n\n**The Army DUKW Drivers**\n\nPat Mooney, who is associated with both Military Historical Tours and the Iwo Jima Veterans' Association, as well as the Marine Corps Museum in Quantico, has this to say about the Army's black soldiers, who were associated with the Marines on Iwo Jima:\n\n\"Each Marine division had an Army DUKW company assigned to it. These consisted of black enlisted personnel and black NCOs commanded by white officers. Sergeant Fred Gray was one of the unsung heroes of Iwo Jima. Fred was part of the 471st Amphibious, attached to the Fourth Marine Division.\n\n\"The contribution of these three Army amphibious companies really has gone unnoticed. They labored from D-day all the way through the battle with these vehicles, called DUKWs [pronounced _ducks_ ], which were absolutely phenomenal but very tippy. [The _D_ meant it was designed in 1942, the _U_ meant \"utility,\" or also amphibious, and the _K_ indicated four-wheel drive while the _W_ meant there were two powered rear axles.] They were fifteen feet long, eight feet wide, usually mounted with a machine gun, and crewed by a driver and codriver.\n\n\"Their preliminary job was to bring guns in for the Marine artillery regiments, and they brought these ashore on D-day as well.\n\n\"They were plagued by the heavy plunging surf on the east beaches. The high freeboards were not meant for heavy surf, and the DUKWs themselves were not really meant for oceangoing traffic. Add to that the heavy loads of artillery and ammo and then float it ashore, bring it on land, and then unload a heavy one-oh-five-millimeter howitzer from the back of this DUKW via an A-frame that was either manhandled by a bulldozer or other vehicle to hoist this one-oh-five millimeter off the back. It was built on the chassis of the famous deuce and a half, the six-by truck, and classified as a two-and-a-half-ton truck. Propulsion through water was provided by an additional transmission with two screws, or propellers, in the back. Because of plunging surf and tidal conditions, the DUKWs had a tendency to broach to port. They would be pushed against the beach, and once a DUKW or any ship broached in that surf, it was only a matter of time before wave action would cause the vehicle to flip over.\n\n\"There were also many documented instances of these DUKWs coming heavily loaded off the bow of an LST and sinking. You had a combination of horror and excitement of coming off the bow of a ship in an amphibious vehicle where the vehicle actually submerges and you fervently pray it will rise back to the surface. Amtracs as well came off these ramps, and for whatever reason, the nose went under, and they plunged to the bottom of the ocean with their loads of equipment and any people who could not get out.\n\n\"This was about two miles offshore. I can only imagine the confusion. You watch the films and listen to the accounts and see these things plunge off the big boats and circle around and move in under fire. And they'd be stuck at sea, sometimes ramming into each other, missing the control boat, sailing around for hours in choppy water because crafts ahead had broached.\n\n\"Particular actions that strike me came after they had brought in supplies or artillery, and then their job was to reload under fire on that beach from some of the finest artillery ever massed in the Pacific. They had to load wounded on board and take them back to the ships, making that round-trip all day long. In one case, the stern anchor on this DUKW broke loose and the ship was broaching with wounded aboard, and the brave actions of the African American crew to go out and resecure anchors to bring that DUKW back from broaching saved the lives of every wounded marine and sailor aboard. Had they not done so, the DUKW surely would have been flipped in the surf and the men would have been crushed or drowned.\"\n\nBruce Jacobs of Alexandria, Virginia, was a combat historian from the Army's Central Pacific Command in Hawaii, assigned to record Army participation in the Iwo Jima campaign. But the only Army involvement initially consisted of the three DUKW companies. \"They were the Four Hundred Fifty-second Amphibian Transportation Company and the Four Hundred Fifty-third and the Four Hundred Fifty-fourth. The Four Hundred Fifty-second, the one I was attached to, was with General Rockey's Fifth Marine Division. My recollection is that there were eighteen to twenty vehicles in each company, so sixty would have been involved altogether. I only remember hearing about two sinking on the way in from the ships, but I wasn't that peripatetic. I didn't get around that much.\n\n\"When you hit the beach, it was pretty much necessary to make sure the tires were deflated so you didn't skid on the sand, and then you have to know when to inflate them so you could keep moving. The requirement on D-day was to get some of General Rockey's seventy-five-millimeter-howitzers ashore. But when they got to our part of the beach, it wasn't soft and sandy, like the photos had led everybody to believe, it was an eighteen- to thirty-inch escarpment, and the tires couldn't make it up and over.\n\n\"Almost out of nowhere a Seabee in a little tractor of some kind came down the beach, and he was able to hook on to the DUKWs and get them up over the escarpment, where they could inflate their tires and take off. They worked with that Seabee all afternoon the first day until he got killed.\n\n\"The DUKW drivers were a terrific bunch of people. They had a lot to prove, and they knew it. A lot of the Army guys, between you and me, expected them to fail, and I think that when the word started to get back to our bosses in Hawaii as to how well they were doing, there was a considerable amount of surprise. You got to think a little bit about what the Army was like in those days. It was not necessarily the grandest moment in our national history.\n\n\"The Four Hundred Fifty-second got a Naval Unit Citation based on the recommendation of an amtrac battalion commander. The stories I heard about the other two companies indicated that they did just as well as the Four Hundred Fifty-second.\n\n\"The thing I have to say is that to me, Iwo Jima is the Marine Corps story, and I have always resisted anything that looked like the Army was trying to edge in. The Army contribution was tiny, and when you look at the casualties, it's just staggering when you consider the number of young marines who got killed up there. So I don't think the participation of the Army in the battle is worth more than a footnote to history, including the participation of the black DUKW drivers.\"\nPART SIX\n\nThe Planes That Came\n\n**The principal reason** for invading Iwo Jima was to provide emergency landing strips, as Bob Merklein and Victor Chepeleff and Phil True point out, for the thousands of planes flying the three-thousand-mile round trip from Tinian and Saipan. Both True's and Chepeleff's B-29s stopped for repairs. Altogether 2,251 B-29s with 24,761 crew members landed on Iwo Jima. The first B-29, _Dinah Might,_ landed on March 4, in the thick of the battle, made repairs, and got out of there. A number of other B-29s landed on Iwo before March 15. At one time there were as many as 300 B-29s on Iwo.\n\nThe _Midpacifican_ , published in Honolulu during the war, ran an article on March 24, 1945, about a \"blind\" B-29 crash landing on Motoyama No. 1 on March 10. Its lead paragraph said: \"Last night more than 200 Superforts raided Tokyo and today five of them landed on Iwo's recently won Airfield No. 1. One of them cracked up after an amazing seven-hour trip with only two engines running, no compass and no other navigational aids.\" It added that clouds were so thick the plane was unable to fly above them to get a fix from the stars and the pilot, Lieutenant G. S. Savage of Collinsville, Illinois, \"couldn't give enough praise\" to its navigator, Lieutenant. W. D. Born, \"who just got us here somehow.\"\n\nDeparting from the Marianas at 2:00 a.m., the plane had bombed Tokyo's \"new business district\" from a height of only fifty-eight hundred feet, the article said. It was picked up by searchlights from the ground, \"and the Japs opened up with everything they had. It looked like the Fourth of July and the end of the world combined,\" Lieutenant Savage said.\n\n\"Sniping at the wounded giant,\" fighter planes pursued the crippled B-29 as it headed back toward Iwo Jima. Somehow the navigator got them there, and with the radio out, the pilot circled Iwo Jima and, working only with hand signals from the ground, saw to it that his plane \"became the second Flying Fortress landing on Motoyama No. 1.\"\n\nAirfield No. 2 was extended for B-29s, veteran William Glaser recalls, but it also housed P-51 fighters, as did Airfield No. 3.\n\nThe other reason for the invasion was to stop the radio transmissions warning Tokyo that the bombers were coming.\n\nWas it justified? John Ripley points out that prior to the seizure of Iwo, seventy-seven B-29 Superfortress were lost and that this number dropped off drastically once Iwo Jima was captured.\n\nLIEUTENANT ROBERT MERKLEIN\n\n**P-51 Pilot, Iwo Jima  \nForty-sixth Fighter Squadron, Twenty-first Fighter Group, Seventh Air Force**\n\n_You hope you have the skill, that you can shoot straight and fly right and keep out of trouble, and that the bad one doesn't get you._\n\nLieutenant Robert Merklein gets ready to take off in his P-51, _Available Jayne_ , on the very first land-based fighter mission departing from Iwo Jima to attack mainland Japan on April 7, 1945.\n\n_I really wanted to find someone who had flown a plane at Iwo Jima during the war, and I first learned about Bob Merklein from his nephew, Jim Stanek, of Newport, Rhode Island. Merklein and I later spoke at length on the telephone in November 2006. He and his wife had recently moved from Houston to a retirement home in Sugar Land, Texas. In talking about his experiences during the war, he used the word_ luck _a lot, as it applied to his survival. But as his story indicates, it was clear that he had made a lot of his own luck. He was a handsome pilot, and he named his plane_ Available Jayne. _This chapter tells why._\n\n\"The battle started February 19, as everybody knows, and I arrived March 22, while it was still going on, four days before that last banzai attack, March 26. We'd been sitting down at Tinian, and that attack scared the hell out of us because we weren't prepared for it, you know, at five o'clock in the morning. The Japs would take over Motoyama No. 2 at night and the Americans had it during the daytime. There was a lot of fighting after they said the island was secure, before we even got there. From our tent area on up there was fighting on that north end, and then in the evening the marines would come right by our area with their dead and wounded. There was a bunch of 'em, I'll tell you. But it's war, you know. They still fought a hard battle for more than a month after I arrived.\n\n\"I was right in the middle of that banzai attack. For some reason or another when we landed there on the twenty-second, they had put up tents to the northwest. Why they put us up there, toward the area where the majority of fighting was still going on, I don't know. But anyway, on the twenty-sixth at five o'clock in the morning these three hundred Japs came out of the caves and hit our tent area.\"\n\nWhile they were not supposed to take part in combat, black marines serving on Iwo played a significant part in defeating and killing the attackers. The Japanese, armed with automatic weapons, bayonets, swords, and hand grenades, had made their way underground from the northern end of the island to an assembly point just north of Motoyama No. 1, near the western beaches. Attacking in a gap between two garrisoned units preparing to debark the island, they struck the bivouac area, a tent city for the Seventh Fighter Command. Patrick Mooney, the historian and an expert on the battle, notes: \"The Army Air Force units in the bivouac did not have observation or listening posts out. They were not prepared for combat. They were living aboveground, in tents. There were also Marine shore parties, supply troops, antiaircraft gunners, and Seabees. The Japanese came into that area and killed forty-four pilots in their sleep with swords and bayonets. Eighty-eight were wounded. The quick reaction of the Fifth Pioneer Battalion, a white unit, and the Eighth Ammo and Thirty-sixth Depot, both black, pinched off the Japanese and then swept through, methodically killing, wounding, or capturing the Japanese. They saved what could have been a real catastrophe for the troops in the area. Nine marines were killed, and thirty-one wounded. Eighteen Japanese were taken prisoner.\" Private James M. Whitlock and Private First Class James Davis of the Thirty-sixth Depot, both blacks, were awarded Bronze Stars. Private First Class Harold Smith was killed while several other black marines were wounded. First Lieutenant Harry L. Martin of Bucyrus, Ohio, led the fight against the infiltrators and was himself killed. The last marine to die in action on Iwo Jima, he posthumously received the twenty-sixth and last Medal of Honor awarded for the battle.\n\n\"It took the marines three hours to get us out of our tent because we had Japanese right outside in a shell hole,\" Merklein recalled. \"We could hear them talking just like I'm talking to you. Of course bullets and shrapnel were flying around, and all we could do was hit the deck in our tent. Finally, when the sun came up, we heard a loudspeaker coming from the south, the direction our tent was facing, and they said, 'Anybody left in the pilots' area, yell out!'\n\n\"Well, the Japanese didn't know we were in the tent, and of course it was just a piece of canvas. They were so close, right on the other side of the canvas at the back wall of my tent. All they had to do was lift the tent and roll a hand grenade in there and we were done for. Later in that shell hole they found over twenty-five Japanese dead. Around our tent there were six more, just spread around, and everything from the ground up was just shrapnel holes and blood and guts on the tents and the ground pretty well blown apart. Most had already been wounded during the battle for Iwo because they came out of the caves. They had some American equipment on them. They came out of tunnels from the north.\n\n\"Anyway, finally, we saw some movement to the south, and we waved a towel or something, and they spotted us. They yelled over the speaker to get ready to get up and run. The order finally came about nine o'clock and we had one wounded guy in the tent. We drug him, and I remember diving into a garbage pit they had made. I landed on my flight surgeon, and he had a marine who was split wide open on his lap.\n\n\"A couple guys in my squadron actually did get to fire at the Japanese because they were on the farther side of the tent. My tent area got hit first, and that's why we were basically trapped there and couldn't do anything. We were lucky to get out. I was in the second row of tents. When we got there, I picked a tent out in the first row and then changed my mind and went back to the second row. Everybody in that first tent I picked out got killed. So you know you just luck out sometimes.\"\n\n\"I was born August 7, 1922, and graduated from high school in Madison, Wisconsin, in June of 1941. I wanted to join the service well before Pearl Harbor was attacked because my two brothers had already been in for a year. One was drafted, and the other belonged to the National Guard.\n\n\"The general sense was that we were going to be in the war; it was just a matter of time. Then, when the war broke out, I joined. At that particular time I was not interested in politics, so I wasn't paying much attention to the isolationist debate. I signed up with the Army Air Corps right after Pearl Harbor. They called me on January 20, 1942. I always wanted to fly, but I never expected it because I didn't have any college.\n\n\"After the preliminaries at Fort Sheridan, Illinois, we went down to Keesler Field in Biloxi, Mississippi, which they had just opened up. We took all kinds of exams to help decide what guys were going to be doing. It was decided that I would go to the Casey Jones School of Aeronautics in downtown Newark, New Jersey. It was a civilian school with a government contract. I was there about six months learning aircraft mechanics, and I enjoyed every minute of it. Then they sent me to Columbus, Mississippi, where I started working on advanced trainers, the AT-17 and AT-10. It was a flying school. The planes weren't armed.\n\n\"Things were building up fast in Army Air Corps. But I always had this thing about flying, and as far as I knew at that time you had to have the two years of college. I got my ratings pretty fast, and by the time I made buck sergeant I decided to sign up for gunnery school and at least be a crew member on a bomber, just a gunner or radio operator on a B-24 in combat, something like that. They had ten or eleven members in a crew, four officers and the rest enlisted. That's what I was going after. Turned out all you needed was a physical.\n\n\"Well, I got to know the girl at post headquarters, and I'd go over there every noon at lunchtime to check and see if they'd given me my orders, and she got to know me. I have no idea what her name was. This was November of 1942, and one day she said, 'Bob, why don't you try for aviation cadets?' And I said, 'I don't have the college.' She said, 'Well, they got this new program where you take eight hours of written exams, and if you pass, you can be a pilot. Why don't you do that?'\n\n\"I had just taken all this mechanical stuff, and I wasn't that long out of high school, so I was used to taking tests. Anyway, I passed and got accepted into the program, and January 1 of '43 I was at Nashville, Tennessee, for a month. That was where they decided if you were going to be a pilot or a bombardier or whatever. Of course most of the guys wanted to be pilots. When it was time for the list to come out, we all ran to the bulletin board, and I was lucky enough to be chosen as a pilot. So nine months later, on October 1, 1943, I got my wings and became a second lieutenant.\n\n\"Of course it was one of the great experiences of my life, but the training was so intense and we'd all worked so hard as cadets we almost felt like we deserved the wings. But it was quite a thrill. Becoming a second lieutenant was fine, but the main thing was getting the silver wings. It was a dream come true because I had never thought it could happen.\n\n\"They sent us to Tallahassee [Florida], and from there they sent me to P-39 school, in Thomasville, Georgia. I had about ninety hours by the time we graduated in January. By the time I graduated I had the P-17 Stearman, an open-cockpit biplane, and the BT-13, the basic trainer, the AT-6, and ten hours of P-40 time. So I was on the road to fly anything. I hit it at just the right time. They moved you around, to Maxwell Field, to Douglas, Georgia, back to Alabama, then Marietta, Georgia, where I graduated from. The washout rate was 50 percent.\n\n\"It was up in Tallahassee where I met Sarah Jayne Pitts. Tallahassee was a pilot replacement pool, and they had the state teachers' college for women there, and all the guys would date the gals from the college. She said she was available for a date, so when I got to Iwo Jima, I decided to call my P-51 _Available Jayne._ That's the way she spelled her name. I dated her for several weeks.\n\n\"Then they shipped me to California to go overseas. We were there two or three weeks on Angel Island in San Francisco Bay, right next to Alcatraz. In fact, the ferry that took us out there stopped by Alcatraz to drop off prisoners.\n\n\"We got on a big Navy boat and thought we were going to combat. Everybody wanted to go to combat. I got off at the Hawaii Islands and said, 'Gee, I wouldn't mind spending a week or two here,' and by God I got stuck there for a whole year. I mean it was fun, but it wasn't combat. We went from P-39s to P-38s, and that's when we thought we were going to combat. Then, about November of '44, they said, no, you're going to get 51s. They put us aboard an aircraft carrier the first of February, I think. We went to Guam, where they off-loaded the 51s. We went up to Tinian, then to Saipan, and sat there waiting for number two airfield to open up on Iwo Jima.\n\n\"After the banzai attack, we lived in foxholes for three or four weeks. Hell, everybody was gun happy after dark, and everybody was shooting. So we dug down deep and lined the hole with sandbags so our heads would still be below the surface when we'd lie on an army cot. We were living on C and K rations.\n\n\"I was in the Forty-sixth Fighter Squadron of the Twenty-first Fighter Group of the Seventh Air Force. There were three different squadrons to a group. There were about thirty in a squadron, so there were eighty to ninety planes in a group. They were all P-51s until the P-47s came along later.\n\n\"The P-51 was the most advanced fighter that I flew, and it was actually designed after the British got into the war with Germany. It carried six fifty-caliber guns, three on each wing. We had what we called practice missions to Chichi Jima and Haha Jima, one hundred to one hundred fifty miles away. We could take five hundred pounders with us and strafe because we didn't need drop tanks. But on the long missions we started off with heavy one-hundred-ten-gallon wing tanks, two of 'em. Then we graduated to two one-hundred-sixty-five-gallon tanks. We'd carry them as long as we could and then drop them before we went in to strafe.\n\n\"We had two airfields, all 51s, and the P-47s started coming in just about the time I left there. It was a big deal. You don't read about it, but we had a lot of planes there. Of course the Navy fighters were going in off aircraft carriers and strafing three or four weeks before we hit because they could get up pretty close.\n\n\"I was lucky enough to be on the very first land-based fighter mission to Japan. That was April 7, and we were one hundred fifty planes flying out of number two airfield on Iwo Jima, escorting B-29s, which came from Guam and Tinian, over the target. It was seven hundred fifty miles to Japan from Iwo, fifteen hundred round trip. We were to rendezvous over Japan.\n\n\"They had a stripped-down B-29 we would meet over Iwo, and that plane would do the navigation for us on the way up. He'd get us up there, and we'd break off for our mission. We'd have a rendezvous point afterward, but a lot of times we never caught up to the B-29 on the way back. We also had what we called our Uncle Dog, a radio in the plane that we could home in on the B-29. All we had beyond that was a compass. It was all dead reckoning, and I was good at it.\n\n\"The missions, to different parts of Japan, lasted almost eight hours. Sometimes we escorted B-29s, sometimes we just strafed, depending on the mission. Then we had sea patrol off Iwo. Weather was our biggest problem. We always hit at least one front going up there or coming back. On June 5, 1945, we lost twenty-seven airplanes and twenty-five pilots just due to the weather. I happened not to be on that mission. There was always at least one weather front between Iwo and Japan, and we'd try to fly under it or we'd try to fly over it, and they just got discombooberated that day. The weather was so bad they shouldn't have even gone.\n\n\"But you do just luck out sometimes. My plane got hit three different times, but I didn't even know it till I got back. It was tracer fire from the ground: It was pretty; it had a curved arc coming up at you. I remember on one mission we strafed this airfield and pulled out over some water, and I happened to look down, and I said, 'My God, it looks like somebody's throwing gravel on the water,' and hell, those were shells hitting the water that they were firing at me.\n\n\"I finally did get to shoot down a plane. It was my last mission. It was on the east side of Tokyo Bay on May 29, 1945. I didn't know it was my last mission, but I was lucky enough to get one Japanese aircraft. It was a Tojo. I can't remember the number. We had a flight of four, and we had already escorted the B-29s, and we were heading for the rendezvous point where everybody shows up to fly back to Iwo, and as we were approaching the rendezvous point, we saw these two planes above us. We realized they were enemy planes, so we took off after them. Our flight leader took the one to the right, and I took the one to the left and just chased him down.\n\n\"He didn't fire back at me. He was trying to escape. He dove toward the ground, and I caught up with him and got him with the fifties. My wingman said he bailed out. We saw flames on the airplane where I hit him. I never did actually see him bailing out.\n\n\"The Tojo looked like a P-47, but it was one of the newer fighters. But by that time most of their pilots, the good ones, had been killed.\n\n\"For the return to Iwo, we didn't have cross runways. There was only one, and the fighters had to come in first because they had less gas than anybody: On that last mission all I had was five or ten gallons left when I landed after shooting down the Tojo. You had to be an optimist back then. You felt confident; you figured it was going to happen to somebody else, not yourself. That's why they wanted young guys, I guess.\n\n\"It was all very interesting. If we weren't on a mission and the B-29s were coming in low on fuel or damaged, we'd go up and watch them land with the engines shot up. We all gathered around for that. I don't know how many were saved, but it was a bunch. They were flying a lot of missions, and there were a lot of wrecked B-29s on Iwo Jima. They were just junked, too bad to try to fix up after they made their emergency landings.\"\n\nWhile the seizure of Iwo Jima was intended largely to provide an emergency halfway stop for B-29s returning from attacks on Japan, space was extremely limited, and pilots were strongly encouraged to continue on to Saipan or Tinian if at all possible.\n\n\"They discouraged the B-29s from landing. They had a colonel in the tower, and if he thought they could make it back to Guam or Saipan, he wouldn't let them land. But I've seen B-29s bellying off the shore of Iwo. There were too many landings going on at one time, and they had to do something, so they ditched them. The B-29 was a good airplane to ditch in shallow water. I saw one float for at least five days before it sank. The cabins were pressurized.\n\n\"When the war started, they used to give you a Distinguished Flying Cross for shooting down a plane. But then it got so common they'd give you the Air Medal instead. All I got was three Air Medals and some campaign ribbons, nothing real exciting. But anybody who's been in combat has been through that kind of crap. There's no hero stuff. I just lucked out. You hope you have the skill, that you can shoot straight and fly right and keep out of trouble, and that the bad one doesn't get you. I figured I at least paid for my cadet training because I shot one down. It was hard to find them. The higher ranks were always out in front. They were trying to get their glory, and you can't blame them.\n\n\"The reason that eighth trip was my last mission was we got back that night and the squadron commander came up to four of us who had been in the squadron over a year and he said, 'I've got to send somebody to Victoria, Texas, to learn about the K14 gun-sight, a brand-new self-computing gunsight, and I need a volunteer to go learn it and then come back out here,' Well, the other guys were married, and I wasn't. He said 'I'll give you till tomorrow morning, and if nobody volunteers I'll have to choose somebody.' We'd all been out there at least eighteen months, and everybody was looking for a rotation back home. If you went to this gunnery school, you had to come back to Iwo. You had to learn this new gunsight and then come back, and that would extend your tour. So I talked it over with my buddy, and I said, 'Hell, I'll come back and probably get my captaincy.' I wanted to stay with the squadron.\n\n\"By golly, they sent me back and the war ended three days before school was out. I went to sign off the base to go back to Iwo, and I cleared the post, but the squadron commander wouldn't sign my papers. And all my records were on Iwo Jima. So I sat there and just junked airplanes up to Arkansas, stuff like that. I never got any of my stuff back.\n\n\"I had to wait from August 15 until November 29, and I finally got my records from Iwo, and by that time I was pretty well PO'ed from the delays and being pushed around, and then they closed the base. I couldn't even get my four hours of flying time in.\n\n\"I'd met my wife-to-be by then, so I said, 'Hell, I'll get out.' Which I did. We got married March 2, 1946, and then I joined the National Guard for about three and half years. That was the extent of my flying career.\"\n\nIt turned out his wife-to-be, Laura, had herself dated a pilot while she was a student at the University of Texas in Austin during the war. And just as Bob had christened his plane _Available Jayne,_ that fellow Laura was dating had named his P-51 for her. There must have been a lot of that going around.\n\n\"I stayed in Texas. She was an only child, and I'd been out of snow country four years and just decided to stay down here in Houston. We raised our family here and made a living. I was in the lumber business about forty years. It was my father-in-law's. He didn't have any sons, and one day he called me up and said, 'Would you like to join me?' So my wife and I talked it over, and he was a man of his word and well liked in Houston. It just worked out.\"\n\nBob and Laura had a boy and two girls. By the winter of 2006 they had one grandchild, about the time they were downsizing into a retirement home.\n\n\"It's not home like we used to know, but it's working out. There's a bedroom and a study\u2014nine hundred and twenty-five square feet. When you're downsizing from twenty-six hundred square feet and four bedrooms and a den, there's a lot to get rid of.\n\n\"I had wanted to rejoin my squadron because I figured they'd be going to Japan. I wanted to stay in the Air Corps. I loved flying, you know. But things just didn't work out, so the story's not too interesting.\"\n\nSTAFF SERGEANT VALENTINE CHEPELEFF\n\n**Radar Bombardier, B-29  \nFortieth Squadron, Sixth Bomb Group**\n\n_Had Iwo Jima not been there, we would either have had to ditch or bail out. We'd have never made it back to Tinian._\n\nValentine Chepeleff, eighty-five, in his Russian Orthodox priest's clothing in February 2007. Pinned to his jacket are his Distinguished Flying Cross and other medals relating to his service in the Pacific. \"How did it feel, an individual bound for service in the church, to be dropping bombs on the Japanese people? I felt that since they had attacked us, we were defending ourselves. They were trying to take control of our country, which had not attacked them. What they were trying to do was get control of people. Germany was doing the same thing. Just like we saved all those lives that would have been lost invading Japan by dropping the atomic bomb.\"\n\n_I spoke with Valentine Chepeleff a couple of times on the phone and then visited him at his home in Wrentham, on the Massachusetts\u2013Rhode Island border in February 2007, not long before he turned eighty-five. He walked with two canes, but he had dressed up in his black priest's clothes and had pinned smaller versions of five medals on his breast. One represented a two-time award of the Distinguished Flying Cross. He lived with his daughter Christine and her husband, Charlie, who made me a cup of tea. Christine brought a pizza home, and they shared it with me._\n\nValentine Chepeleff, who went on to a full career as a Russian Orthodox priest, was the radar bombardier on a B-29 that led dozens of bombers on thirty-five missions from Tinian to Japan between January and June of 1945. His plane was named _Earthquake McGoon,_ after the Li'l Abner character. It made one stop at Iwo Jima.\n\n\"After several missions we had an emergency on the plane, and Iwo Jima is halfway between Tinian and Japan. We were coming back, and we had to land someplace. There was just water all over the place, and here they're fighting on Iwo Jima, but we called in and asked if they would have an airstrip maybe available.\n\n\"The answer was, 'Well, we got half an airstrip, but if you could wait a little bit, maybe we can get the rest of it.' We said, 'Thank you very much; we'll take the half.' And we landed the plane. We had a terrific crew. They repaired it, and we left right away. I don't remember what was wrong with the plane, but had Iwo not been there, we would either have had to ditch or bail out. We'd have never made it back to Tinian.\n\n\"My father and mother were Russian. I was born here, in Cohoes, New York, 1922, February 28. My father, Theodore, came over as a missionary priest in 1913, planning to stay for ten years, but then of course the Communists took over, so he never got back to Russia.\n\n\"When I was getting into the Army, I spoke Russian. I heard they were ferrying planes from Alaska to the Russians, and so I thought I might as well try that. As I was signing up for it, I talked to a Navy pilot, and he was trying to talk me into getting into the Navy program for pilots. And I said to myself: 'Let me think this over.' I had never thought of flying, but it sounded so interesting, flying a plane. But the thing that cautioned me was that if you're on a carrier and you go on a mission and you come back all battered and what if the ship is not there? Where do you land, in the water? So I thought: No, I'm going to go into the Army Air Force. You know, you're over land. [Laughs.] Well, there's a big surprise. I got into the Army program and got sent to the Pacific, where there was nothing but water.\n\n\"I signed up for a fifteen-week air cadet pilot training program September 8,1943, at Dickinson College near Hershey, Pennsylvania. But then they decided to take us out of there to learn radar bombing, this secret program for the B-29, which was brand-new at the time, and they needed a lot of things. You never saw a more dejected group of fellas. We were intent on flying and becoming pilots, and here we were being transferred to something else. We had to take special courses in radar, learning to apply it in combination with bombing. We were the first group when the whole thing started and we were supposed to get commissions, but they never came through. They sent us to Tyndall Field in Florida, May 8, 1944.\n\n\"In August they sent us to Lincoln, Nebraska, and from there we made trips to Puerto Rico to get the bugs out of the Superfortress and to get the crew accustomed to long-distance flights and to give the radar operator practice in navigating and bombing over land and water. We flew through some violent electrical storms, and it felt like the wings might be torn off. I was the only one armed to protect the equipment. If we had to land at some other airfield, I'd have to wait there till the MPs got there to protect the equipment because it was all so secret.\n\n\"I was in the middle of the plane in a little room with my radar equipment. In fact, they had a little control there where I could move the plane left or right after we were at altitude. I had the screens in front of me, and I had to make my calculations for altitude, airspeed, drift, from the IP [initial point] to the target; then you directed the plane either left or right. The bombardier was up front with his regular radar bombing equipment. I would just go along then, and if he could see something in a break in the clouds, he could make corrections.\n\n\"My job was to get to the target. I would get the target on my radar, zero in on it, make my calculations left or right, and release it. I don't know how it worked. As far as I know, I was releasing the bombs. I felt I was running the whole mission. [Actually, the radar operator or the navigator would provide the bombardier with the course to the IP. From there on, it was up to the bombardier to bring the plane the next fifteen or twenty miles to the target.]\n\n\"We went on to the West Coast in December and left for Hawaii January l7, arriving at Guam and then Tinian January 21. We were supposed to get commissions when we got through with this, but everything went so fast after we got out to the Pacific that they never came through. But after many missions, the replacements were coming in with commissions. I got to staff sergeant, but I should have been commissioned. At first we lived out of our helmets and canteens. Eventually we got water and power. We were stationed with the Twenty-first Bomber Command in the Marianas with the Three Hundred Thirteenth Bomb Wing. I don't remember when the raids started.\n\n\"There were eleven in the crew. The pilot, copilot, bombardier, navigator, flight engineer, and radar operator; the rest were gunners. When we first got there, we escorted some of the Marine fighter planes up to Iwo Jima so they could strafe it. So they must have had some navigational problems. We would bring them up, then hang out while they went in and did their job, strafing the targets. Then we would lead them home. Navigational problems were great for them. To navigate, you had to use your stars, your compass, anything you could pick up, any of the islands, because it was all water. It was not like in Europe, where you could find landmarks. In the Pacific you'd have to find an island or something. We had radar that would reach an island we could recognize fifty or one hundred miles out there so we knew where we were. I said that was a twist. In Europe the bombers were escorted and protected by fighter planes, and here we are escorting the fighter plane. We were just one plane; they didn't need any more than that to take them up there.\n\n\"We made thirty-five missions to Japan. Our first mission was a big operation, and I was deputy lead, meaning we were the second plane. But something happened to the lead plane on our bombing run, so we had to take over. The first plane is the one that picks out the target, and the rest drop their bombs when the first plane releases. Ours was the lead plane for the rest of my thirty-five missions. We were so good that after twenty-five missions we were supposed to get a rest leave in Australia, but instead they asked us to keep flying. We called our plane _Earthquake McGoon._\n\n\"Originally we were going to fly at very high altitudes. The B-29 was built to fly at 33,000 feet where it would be out of reach of the ack-ack, the antiaircraft fire. But we found that the winds up there blew from one hundred fifty miles an hour so that, if you got a tailwind, you'd get over the target too fast, and if you got a headwind, you were practically standing still. So we started lowering, and we wound up flying much much lower.\n\n\"Our targets were mainly airfields, military installations, and industrial plants. You couldn't help but hit some civilian population in industrial areas, but they were not our target. We had strict instructions to avoid ancient temples or the palace in Tokyo.\"\n\nIn a raid on March 10, 1945, nearly 280 B-29s destroyed 267,171 buildings, one-fourth of urban Tokyo, killing 83,793 and wounding 40,918, exceeding the number of civilians killed in the firebombing of Dresden in Germany as well as those who died in the atom bomb explosion over Hiroshima. Fourteen planes were lost. All came from the 21st Bomber Command's 73rd, 313th, and 314th Bombardment Wings, based in Guam, Tinian, and Saipan. The planes flew at altitudes ranging from forty-nine hundred to ninety-two hundred feet.\n\n\"Whenever we went up to hit a target, I would take radar photos of that city so I could get more familiar with it in case we came back. I don't know what the other radar operators did, but whenever we left the target, I'd be scanning the area for any planes because we were all on our own to get back. We didn't go back in formation. We'd get together before the target, but we all flew back on our own.\n\n\"After I made my bombing run, I switched over to scanning the area three hundred sixty degrees. Many times I'd see a target [an enemy plane] coming in and we were on a collision course. I'd warn the crew and the pilot because they couldn't see through the cloud cover. Now I'm thinking: My God, I wonder how many planes we lost in collision with one another in a cloud cover? Later on we had Japanese suicide planes come after us, and I could pick them out as they were coming in. They would just try to crash into us. We used to laugh about it, say, 'Oh, they must have been pilots who washed out.'\n\n\"Many times, when we got back to base, the fellows would come to me and say, 'How are you?' I'd say, 'Why?' 'Well, we saw the flak burst right under your seat.' I said, 'I know. I felt the concussion.' We got a lot of flak, but it never penetrated. We took a look, but they were just scratches. None of it had penetrated. I said, 'Thank you, St. Nicholas.' My father had given me an icon of the saint, who is supposed to be the protector of seamen and maidens, and I had it with me on every mission. So many times the fellows came to ask and the plane was not damaged. The saint had to be out there praying for us. I had to miss one mission when my arm swelled up from an insect bite, and I thought: Oh, my God, the crew is going to think it is their time to go and I'm being saved by being left behind. So I took off my cross and gave it to my navigator. I said, 'You bring this back to me.' He did.\n\n\"So we never got our R and R in Australia, and on top of it all, after our thirty-five missions, I flew a War Weary back while the rest of the crew went by ship. A War Weary was a plane that needed repairs, had a lot of engine time, and was pulled out of combat to be sent back to the States. But when we were leaving Hawaii, this B-29 War Weary developed engine trouble. One engine was having trouble, and another engine was beginning to go, and I thought: Oh, my God, I got through thirty-five missions, and here I am on my way home, and we're going to have to ditch?\n\n\"On top of that we got news on the radio that the war had ended. So the war is over, and I'm in trouble on a B-29. Ha! They're celebrating in Hawaii, and I thought: Oh, my God . . . We made it eventually, but these little incidents kept your attention.\"\n\nBack in the States, Valentine went to the seminary and became a priest and worked for years with his father, the Right Reverend Theodore Chepeleff, a mitered archpriest, in the Boston area. Valentine met and married Zinaida, or Zena, Martinoff, and they had five children. Zena died a few years ago.\n\n\"We were very active with the Roman Catholic Church. Father got to be well known among various denominations. He knew Cardinal Cushing. I knew Cardinal Cushing, Cardinal Law, and was close to a lot of the chaplains at the Veterans Administration. They know me and want me to speak here and there, but right now I got to get my medical health program set up. My mind is still active, but my body has slowed down quite a bit. I never officially retired. I'm the chaplain of the Disabled American Veterans here in Wrentham.\n\n\"When I got out of the service, I put it all behind me, never talked to anyone about what I did or anything for fifty years until I met Pete Santoro [Chapter 7] in a rehab pool and found out he had been at Iwo Jima. Then I started to think about it.\"\n\n_Earthquake McGoon_ 's first citation for a Distinguished Flying Cross, dated June 1, 1945, cites her officers and one enlisted man, Valentine Chepeleff, \"of the 40th Bombardment Squadron, 6th Bombardment Group\" as \"members of a lead crew bombing team which led three highly successful formation bombing missions against the Japanese Empire on the 17th and 22d of April and 10th of May 1945.\" It says they performed under rapidly changing and oftentimes adverse weather and under extremely difficult navigational problems over great expanses of ocean and unknown enemy territory. It also cites \"varying degrees of enemy antiaircraft fire and fighter opposition.\"\n\nThe second, this one an oak-leaf cluster awarded in lieu of the DFC, is dated August 2, 1945. It describes a daylight incendiary raid on June 5 against important industrial and shipping facilities in Kobe, Japan, in the course of a seven-hour flight. It says Valentine's plane was attacked \"fourteen times by aggressive enemy fighters, and encountered heavy, intense and accurate anti-aircraft fire.\" It adds that this crew \"led an exceptional formation, releasing their bombs over the briefed target with great accuracy\" and says further that \"over four square miles of the built-up portion of Kobe was destroyed.\" The captain, or airplane commander, is identified as Irvin M. Parsons. Lieutenant Hubert B. Connell was the navigator, and Lieutenant Carl J. Manone the bombardier. Also cited is Staff Sergeant Valentine Chepeleff of Brookline, Massachusetts.\n\nLIEUTENANT PHIL TRUE\n\n**B-29 Navigator  \nNinety-ninth Squadron, Ninth Bomb Group**\n\n_I gave a talk about the significance of Iwo for those of us in the Twentieth Air Force and the many lives saved, including mine, because of the Marines' heroism in taking that ugly piece of rock and sand. Yes, I did land there three times, and I am eternally grateful for the island and the Marines._\n\nPhil True at eighty-one near his home in Fairfax, Virginia, in the fall of 2006\n\nSecond Lieutenant Phil True, nineteen, in a photo he says was probably taken at home in Jackson, Michigan, when he was given two weeks' leave after receiving his navigator wings, with a globe in the center, on December 15, 1944. On his lapel is the Army Air Forces insignia, a wing with a propeller through it.\n\n_I spoke with Phil True at length over the phone, and his feeling for the Marines was evident. In 1997, when he was talked into being chairman of a reunion for his Ninth Bomb Group, he chose to honor the Marines for capturing Iwo Jima. \"I organized a tribute featuring the Marine Drum and Bugle Corps,\" he told me, \"and located members of the Third, Fourth, and Fifth Marine Divisions, who had invaded Iwo. They spoke, as did our commanding officer during the war and the colonel in charge of the Marine Barracks. We held it at the Marine Corps Memorial on August 28. We had close to three hundred people. Afterward I was asked to be an honorary Marine with Chapter Twenty-five of the Fourth Marine Division.\" He has continued his association with the Corps._\n\n\"I live in Fairfax, Virginia, now, but I grew up in southern Michigan, near Jackson. I was born April 14, 1925. I was the first person in my immediate family to graduate from high school. My father and mother, like most farmers, quit school in ninth or tenth grade. There was a junior college in town, and when I got out of high school, I decided to go there. When Pearl Harbor came, the draft age was twenty, and I really didn't think about going into the service. But then it was lowered in the fall of 1942, while I was in junior college, and I realized I'd be facing the draft the following spring, after I turned eighteen.\n\n\"Very few people went to college in those days, and the Navy and the Army assumed anybody in college was 'smart' and therefore officer training material. They had these specialized training programs, and I think in February of '43 the Army Air Forces, as it was called then, set up their aviation cadet program. Although almost every kid my age was interested in flying, for me it wasn't that I just _had_ to fly. It wasn't that kind of thing. They had a little bonus. They said if you signed up, you could be called up sometime after you were eighteen and there would be a five-month college program.\n\n\"I took the physical and a written test, and I was sworn into the Army Air Forces three days before I turned eighteen. In fact, I was the first seventeen-year-old to do that in Jackson. I was called up August of 1943 and went through the entire training program. The aviation program was roughly a year. You had to go through basic, and then it depended on how you were classified: If you were going to be a pilot, you went to primary training, then advanced training. You'd be commissioned. Then, if you were going to be a bomber pilot, you'd go through multiengine school.\n\n\"Of course taking it didn't mean you were in it. The washout rates were maybe fifty percent. A lot of people didn't have the ability to do this.\n\n\"We did our basic training, from early September through early November, in Miami Beach. The reason was that when the war started, they simply did not have enough Army barracks built throughout the country to house everybody. So they used the War Powers Act to requisition hotels after the tourist season was over. I think in April of 1942 about ninety percent of the hotels in Miami Beach were being used. They had close to a hundred thousand people undergoing basic training in South Florida.\n\n\"While I was down there, they decided to give us classification tests, two days of written tests and what they called psychomotor tests, devised to try to determine if you were best suited to be a pilot, navigator, or a bombardier. Some people didn't qualify for any, so they were sent to radio school or gunnery school.\n\n\"Anyhow, I qualified for all three, but my highest score was for navigation. We had ten hours of dual flight instruction in little two-seater Piper Cubs before we got sent up to Allegheny College in Pennsylvania in mid-November of '43. I learned after the war the college training program, the Army Air Forces cadet program, was set up because the small liberal arts colleges had been drained of students except for women. They sort of struck a deal that they would put these training detachments in from the Air Forces, and provide them with money. There were about four hundred of us and about a thousand coeds, so you could see the ratio was actually pretty good.\n\n\"As for the flying, I sort of enjoyed it, I guess, but I was rational enough to realize I wasn't a natural. I could put the plane in a stall and recover, but getting it to spin was not my forte. When I got down to preflight school in San Antonio in March of '44 I already knew my scores because when you were a student at Allegheny, you had to pull junior officer of the day, and so late at night, when no officers were around, you looked up your file. I discovered what my scores were and decided I would be a navigator.\n\n\"I went through preflight school, which ended in June of '44. Normally most people went on to gunnery school, but about four hundred of us in our class stayed on and went to the San Antonio Aviation Cadet Center, which is now Lackland Air Force Base. We waited around for shipping orders, and it got to be mid-August, and there was hardly anybody left when they received a direct call for people up at San Marcos near Austin. I was in this remaining group, so I went directly to navigation school instead of gunnery school. We were still not commissioned. We were essentially privates in the Army. There were fifty in my class, and I think thirty-one of us graduated. The attrition rate was high.\n\n\"We had actually four basic ways to navigate. One was through dead reckoning; that was simply through your instruments, using the compass and airspeed. You had to compute your drift, of course. You had to use maps, to figure out your declination in terms of your compass heading. We had a log, we called it a navigator's log. I had eight or ten columns of data that had to be entered in for any particular time to get your true airspeed. Then you got your ground position from that. Then you had radio, depending on where you were. Then you had celestial navigation, based on figuring your position from the stars. And finally you had what you called pilotage, which was just looking out the window. 'What do you see out there?' There were sectional air maps that provided good detail for what you were seeing on the ground.\n\n\"We actually had twenty practice missions. Each mission had three student navigators, and we'd have a three-legged triangular mission. One student would be lead navigator on each leg while the other two would follow what he was doing. The next student would plot the next leg and so on.\n\n\"We flew in a twin-engine Beechcraft, and it wasn't until the final mission or two that we were allowed to use all four navigational tools. If you had a zero zero mission, that meant you were perfect in figuring your ETA, your estimated time of arrival, and reaching your destination. You might be two minutes and five miles off, but that was considered reasonably good. It was a sixteen-week course. I graduated in December of 1944 and became a second lieutenant.\n\n\"We got some home leave, and I went back to San Marcos about New Year's. Somebody got sick, and an opening came up so I went to Lincoln, Nebraska, which was a classification center. They were making up crews to send to various bases for training. I didn't have any gunnery training, so they put me in a B-29 program because the navigator in a B-29 did not handle a gun. Our crew assembled about the first of February in Tucson, Arizona, and trained there until about the first of May.\n\n\"From there we went up to Wichita, Kansas, to pick up a B-29 to fly overseas. It took a couple of weeks to get it ready. We had to do a series of things, make sure the instruments were calibrated and so on. The citizens of Manhattan, Kansas, had had a bond drive to pay for a new B-29. It took about six hundred thousand bucks, if you can imagine that. They wanted to name the plane, and they had a contest and came up with _Nip Finale,_ and that was painted on its nose. It's not politically correct now, of course, but that was our plane, and we enjoyed it.\n\n\"In late May we flew to California and took off from a field near Sacramento, then to Oahu, on to Johnson Island, and eventually we reached Kwajalein, which the Marines had taken I guess, in late 1944. It was just a flat atoll. The control tower there wasn't much. The airstrip was probably eight thousand feet long. I remember they had a sign up there in the operations office. It said: KWAJALEIN ATOLL, ELEVATION EIGHT FEET, NO WHISKEY ATALL, NO WOMEN ATALL, NO NOTHING ATALL.\n\n\"Next day we flew out to Tinian, where the Three Hundred Thirteenth Wing was located. We landed, and the pilot went to report, and one of the ground crew came up, dressed only in a pair of shorts, and he says, 'Where's the whiskey?' One of the guys, I guess the engineer, said, 'We didn't bring any whiskey.' What? No, we didn't bring anything. I heard the guy say, 'Didn't bring anything?' Whiskey was a great barter item. Actually one of our gunners had wanted to, but our pilot\u2014he was a major\u2014was very strict on rules and regulations.\n\n\"Anyway, we did not keep the plane. You make the mistake of thinking just because you fly a plane, it's yours. That plane was taken to one of the groups that needed it more than the Ninth Bomb Group, where we were assigned. A number of years later, when I got reunited with my crew, I did some checking and discovered that _Nip Finale_ had been taken over by the Five Hundred Fourth Bomb Group, which was one of four in the Three Hundred Thirteenth Wing. There was the Ninth, the Five Hundred Fourth, the Five Hundred Fifth, and the Sixth Bomb Group. Each group had about forty planes and crews. I learned the _Nip Finale_ got in twenty missions. It was the third plane for that ground crew. Their previous two had been shot down.\"\n\nPhil True was in the 99th Squadron of the Ninth Bomb Group in the 313th Wing. Each bomb group consisted of three squadrons, each with fifteen planes. There were five bomb groups in the Pacific at the end of the war: the 73rd, in Saipan; the 313th, on Tinian; the 58th, on Tinian; the 314th Wing, on Guam; and the 315th, on Guam. Each wing had four bomb groups, for a total of twenty.\n\n\"We flew whatever plane was available. Crews didn't necessarily fly the same plane every time. Engines needed replacing frequently, and when your ship was in for repair, you flew whatever was available. Some crews flew as many as thirty-five missions, but they would fly maybe twenty-five or so on 'their' plane. So you never knew what plane you were going to fly. There would be an announcement that such and such crews would be briefing at eleven o'clock that night, twenty-three hundred, whatever; it was going to be a daylight mission. Then you would have a little breakfast, get in the truck, and go down the line. Then you'd find out which plane you were going to fly.\n\n\"We'd get down there, and one of the problems for the navigator was you had to take the astro compass and make sure everything was aligned correctly. You used the North Star normally to do that. On some planes the compass reading was maybe a couple of degrees off from what you were actually doing. You always had to make little corrections. Occasionally planes got caught in smoke thermals, and some of them got twisted two or three degrees, so you and the compass in the plane didn't quite agree.\n\n\"We flew some practice missions, bombing the airfield of an island called Pagan, in the Northern Marianas, about halfway between Iwo and Saipan and Tinian. I think there were a few Japanese left but no defenses. The B-29 could carry eight tons of bombs, but most missions you carried seven, depending on whether you had incendiaries. There were five hundred, thousand, and two thousand pounders.\n\n\"Our first mission over Japan was June 4 to 5. We took off on the fourth and bombed Kobe. We didn't fly in formation to Japan. That would have taken too much gasoline. We flew up separately, then used some sort of reference point; often it was just a compass point in the ocean. It was a very, very tricky job to get so many planes in formation once you got there. Each group had a slightly different certain altitude and a slightly different location. Then, at a certain time, they all had to fly to the IP, the initial point, from where you went on your bomb run. It was an intricate job.\n\n\"About four hundred fifty planes went over the target, in formation. There was a high overcast, so we flew a daylight mission at fourteen or fifteen thousand feet, which was very low. This put you in easy range of all the antiaircraft they had. The city was well defended, and the Japanese put up a couple of hundred fighter planes. About a dozen B-29s were shot down, and seventy-five were damaged.\n\n\"I went along as observer on that particular raid. I sat in a nose hatch right behind the pilot, copilot, and bombardier. Apparently what happened was the fighters came in after we dropped the bombs.\n\n\"They didn't attack during the bombing run, when they could get in the flak themselves. Normally the fighters would attack before you got in, or just after. Fighters came at us, usually in pairs. Mainly they'd fly head-on through the formations. The Japanese had three or four aces on B-29s. Some had shot down as many as fifteen planes.\n\n\"When the main force of Japanese fighters were coming from, say, twelve o'clock, straight overhead, there would be two or three of these aces, or better pilots, who would fly a couple thousand feet below a formation. They'd pick out a plane, and because the gunners were all looking ahead and because there was a slight dead spot on the B-29 right directly below, they would come right up under it to within a couple hundred feet and then fire their twenty-millimeter cannon shells into the wing tanks.\n\n\"That happened to our plane, although we didn't realize it at first. Then one of the gunners called in and said gasoline was coming out behind number three. Gasoline was from a wing tank, of course, and number three was the inboard engine on the right-hand side of plane. The pilot immediately shut down the engine and then feathered the propellers. Feathering meant you altered the pitch of the blades so they offered less air resistance. We feathered that engine, and then, ten or fifteen minutes later, the supply of gasoline to number four, which was the outboard engine on the right-hand side of the plane, ran out because the fuel tank had been hit. There was no way to transfer any additional fuel out there on this particular model B-29.\n\n\"So we feathered number four, and tried to restart number three, but I think the starter motor had burnt out. It may have been hit. Anyway, number three and number four engines were not working. By that time we were just leaving Japan, and a buddy ship flew with us for three or four hours. The pilot put the plane on what you'd call a climb setting, put the nose like he was going to climb, but because he didn't have enough power we just kind of mushed our way.\n\n\"We lost altitude till we got down around four or five thousand feet. Here the air was heavy enough so we could level off. But we still did not have much gasoline, and the pilot thought about having us bail out when we got to a picket ship, a Navy destroyer escort. But then he decided the water was too cold for him to jump. So he said we'd try to make Iwo.\n\n\"It was a stormy day. We got to Iwo, and the pilot was being very careful. He was too high, so the tower told him to do a go around, which was a little tricky. In fact, it was not recommended at all on a B-29. We had to go around the island with two dead engines. With that right-side wing down, it was very hard to control the plane.\n\n\"After the first go-round, we came in with flaps down, wheels down, and then, just a couple hundred feet above the runway, another B-29 cut right in front of us. In fact, I read a report many years later that said the air control was not good that day; pilots were not obeying the tower. Apparently this guy had wounded on board.\n\nSo we had to pull up. Any kind of little flicker on the remaining two engines would have been the end of us. The pilot said, 'If we can't get in this time, I'm going to dump you out over the ocean. I'll try to climb to fifteen hundred feet and let you out.' So we came around a third time, and this time, as we came in to land, visibility was essentially zero. But the tower had elementary radar, and they picked us up and guided us in. We screeched to a halt at the end of the fighter strip. We had maybe ten minutes of gas left. That was my first, and almost my last, mission. That's my story of getting saved on Iwo Jima.\n\n\"We landed twice again in June while on two other missions, but it was primarily because we were on the margins with our gas supply. If you were based on Tinian and you were returning with less than two thousand gallons, you were supposed to land on Iwo to take on more.\n\n\"Something wrong with the engines was discovered when we landed, and they said we had to hitchhike aboard another 29 that was going back to Tinian. And just as we took off, _boom,_ one of its engines conked out. The pilot said, 'Hell, we can get back on three.' But then, an hour later, _ping_! another engine went out on the other side, and he said, 'Well, we gotta go back to Iwo now.' A friend of mine in college after the war had been in B-24s in Italy, and he was talking about planes. I said, 'Well you know, a B-29 can fly with two engines out on the same side,' and he said, 'You're nuts, you can't do that.' But I know, because we did.\n\n\"We flew twelve missions before our crew was selected in late July to come back and be trained as a lead crew to lead a group or squadron in bombing. There was so much bad weather over Japan you had to rely on radar bombing. About a dozen crews were being sent back each month. Then in August the war ended.\n\n\"Of course the first B-29 to land on Iwo was _Dinah Might,_ on D plus thirteen, March 4. Lieutenant Raymond Malo, the pilot, was from my Ninth Bomb Group, Ninety-ninth Squadron. They had flown a mission over Tokyo, and their bomb bay doors had frozen open, and they had a fuel transfer problem. The Seabees said, 'You give us a couple hours, and we'll have another thousand feet of runway.' Malo said, 'No, thanks, I'm going to take off. I want to get out of here.' He didn't stay long.\n\n\"His plane was later shot down, on April 14, over Kawasaki. Malo and the entire crew were killed. Our group lost four planes that night. Another navigator took the cot of Malo's navigator in the Quonset hut where we were going to be living. Three or four days before we arrived in _Nip Finale,_ the second navigator's plane was taking off on a night mining mission, and something happened on the runway: The plane skidded off, and the mines exploded. The tailgunner survived, but everybody else was killed. So when I got to this Quonset hut, I took this empty cot. After a few days the guy next to me told me the story. He said, 'You know, Malo's navigator had this cot, and then Caldwell's navigator had this cot, so welcome to the cot.'\n\n\"Do I think the seizure of Iwo Jima was worth the cost? Yes, I do, because around twenty-two hundred B-29s landed there after it was taken, although it's difficult to say how many crewmen would have been lost otherwise. Some planes were low on gas but would have got back anyway. And if you had to bail out, you had maybe a forty percent chance of survival. And the P-51 wing was located there as well.\n\n\"While we were sitting there at that reunion at the Marine Corps Memorial in 1997, two buses with Japanese calligraphy on the side came along, and out jumped a bunch of Japanese tourists. We had a dozen marines there, and one rushed over and told them to get back in the bus. I thought it was kind of ironic.\n\n\"I got discharged in mid-December 1945. I was only in for not quite two and a half years, but it seemed long enough to me. Besides, I was only eighteen when I started.\n\n\"I went back to Jackson, took a semester of junior college, then went to Western Michigan University in Kalamazoo. I graduated from there in 1948, ending up with a major in geography. I went on to the University of Chicago, where I got a master's in geography and taught for a year before a friend said they were hiring for the CIA, and I said, 'What's the CIA. Is that the CIO?' Anyhow, that's how I came to Washington.\n\n\"I worked forty-seven years in the CIA. I was on the analytic side, which meant I didn't go into phone booths and leave pencil marks. I worked primarily on China, and then I retired and worked on contract and then worked primarily in training and ran some courses. Most of the people I found at the CIA were very dedicated people. It's not a perfect organization, it's obvious. They certainly made a serious mistake in the 2002 October NIE [National Intelligence Estimate]. That's another story.\n\n\"I got married to Fern Brooks September 1949. I told this story at our fiftieth wedding anniversary: I was taking a speech class at the junior college in Jackson. It met early in the morning. It was wintertime; it was February 1946; it was cold. There were not many people in the class.\n\n\"I sat in the back row, and there was a young lady named Lenore Larsen. She was blond, pretty good-looking, liked to wear sweaters. She'd come in late most of the time, and she'd sit in the back of the room as well. She'd take off her coat, and then she'd stretch. So this friend, Warren Tisch, and I would kind of eyeball that. And after two or three weeks the professor said, 'Why don't we just move people around?' He said, 'Mr. True, there's an empty chair right up here.' And that's where my wife-to-be was sitting, though I didn't know her at that time. So it was proximity and Lenore Larsen that led me to my bride. She came from a farm family too. We had two boys and a girl.\n\n\"I was invited along with some other people four years ago when they opened up the Air and Space Museum out at Dulles. The _Enola Gay,_ the B-29 that dropped the atom bomb on Hiroshima, was there. Eight or nine of us from the Ninth Bomb Group were going to get a picture taken next to the _Enola Gay,_ and a Japanese TV crew came up and asked if they could interview me. I simply said, 'I think it [dropping the bomb] was cruel in many ways, but I think in the long run it saved more lives than were lost.' Despite the fact that their navy had been largely defeated, the Japanese were not in the mood to surrender.\n\n\"I was asked to say a few words at a memorial service on behalf of B-29s at a Fourth Marine Division reunion in Washington, D.C., in 2002. And at an Iwo symposium in 2005 I gave a talk about the significance of Iwo for those of us in the Twentieth Air Force and the many lives saved, including mine, because of the Marines' heroism in taking that ugly piece of rock and sand. Yes, I did land there three times, and I am eternally grateful for the island and the Marines.\"\n\n**The characters on this monument,** erected on the summit of Mount Suribachi after Iwo  \nJima and the Ogasawara (Bonin) Islands were returned to Japan on June 26, 1968, say:  \n\"The Monument to Pay Tribute to the Soldiers Who Fought and Died on Iwo Jima.\"  \nThe map on the surface of the monument consists of stones from each prefecture  \nof Japan, and an epitaph on an accompanying memorial reads:\n\nA small force pitted against a mighty army  \nOn a lonely island, where even the deities are in awe.  \nFor thousands of years, love and hate disappear  \nAnd the fragrance of sincerity spreads over eternity.\n\n(translated by Masahide Mizoguchi)\n\nThe small bottles and objects at the base of the  \nmemorial represent offerings to the dead.\nPART SEVEN\n\nAftermath\n\n**There was no glamour** on Iwo. There were millions of flies. The bodies burned black. The smell was overpowering. The burials began as soon as possible. Two cemeteries were established for the thousands of marines slain. The Japanese dead lay where they fell or else were covered in anonymous graves plowed out by bulldozers. To this day thousands remain entombed in the caves and tunnels. All were memorably eulogized by Rabbi Roland B. Gitelsohn. The text of his remarkable speech is included in this part.\n\nThe Army took over as the Marines departed in March and April. There were up to thirty-five thousand Army and medical personnel stationed on Iwo until September, when the numbers began to decrease, according to William A. Glaser of New York City, who served as an emergency surgical nurse in an Army hospital and later as a clerk on Iwo from February 1945 to February 1946. Substantial numbers of Japanese soldiers remained in small groups in caves in the northeast. Many came out at night to steal food and clothing. Glaser tells of a group of soldiers coming out to surrender from behind a movie screen set up near Suribachi. He said they had been living in caves inside Suribachi for months. Many who surrendered were Korean slave laborers rather than uniformed soldiers. A huge hospital, the 232nd General Hospital, was planned to treat wounded expected from the invasion of Japan, scheduled for late 1945.\n\nThe island was returned to Japan by the United States in 1968. Various individuals from both nations, such as Tsunezo Wachi and John Ripley, acted to preserve the heritage of Iwo Jima, to venerate the dead and preserve the island as a shrine to the sacrifice of the fallen soldiers. Colonel Ripley was determined to find a way to sustain access to the island by Americans in general and the Marines in particular. Captain Wachi founded the Association of Iwo-Jima (Iwo-to Kyokai) and helped make possible the Reunions of Honor, which now occur annually on Iwo.\n\n[THE JAPANESE COMMANDERS\u2014  \nLieutenant General Tadamichi Kuribayashi and Captain Tsunezo Wachi\u2014and the Years Afterward](05_Con.xhtml#ch22)\n\n_My girl is waiting for me  \nAnd does not know  \nThat my body will stay here  \nOn the rocks of Mount Kamo._\n\n\u2014Hitomaro (From _100 Poems from the Japanese,_ translated by Kenneth Rexroth)\n\nLieutenant General Tadamichi Kuribayashi, Imperial Japanese Army, commanding general, 109th Division and commander, Ogasawara Army Group. He was fifty-three years old at the time of the invasion. He fired eighteen senior army officers, including his chief of staff, for opposing his strategy for the defense of Iwo Jima. He is believed to have died by ritual suicide in his cave near Kitano Point on March 23, 1945.\n\nIwo Jima today is a shrine, actually a vast graveyard where between 14,000 and 17,000 Japanese soldiers lay entombed in its honeycomb of caves and tunnels. Only the immediate family members of those soldiers are permitted to visit from Japan. There is no restriction on American visitors. Only about 20 of the original 1,023 Japanese survivors are alive today. Thousands of marine veterans of the campaign still live, but they are dying off as age overtakes them.\n\nKey figures from the Japanese side of the story, alongside the more than twenty-one thousand soldiers who died there (some bodies have been recovered), are Lieutenant General Tadamichi Kuribayashi, their commander, and his predecessor, Captain Tsunezo Wachi, who became a Buddhist priest after the war and dedicated his life to the Japanese soldiers and sailors killed on Iwo Jima.\n\nKuribayashi, a brilliant strategist and an exemplary individual, earned the respect of his troops and very high regard of his enemy, even after his death in March 1945. Kuribayashi wrote a series of letters, forty-one in all, to his family during the nine months he was there. Much of his legacy is based on these letters, and they offer the best way to understand the Japanese commander. Clint Eastwood's second film about the battle, _Letters from Iwo Jima,_ leans heavily on Kuribayashi's correspondence, as well as letters from the soldiers. For a man facing certain death, Kuribayashi's letters are a marvel of restraint amid expressions of concern for his wife and fatherly advice to the children. From the way they read, he might be returning home in a few days.\n\nKuribayshi actually took over from Captain Wachi, who commanded the naval garrison on Iwo during the early part of 1944 and left in October. Wachi left a lengthy memoir, which has been translated by his daughter, Rosa Ogawa. In May 2007, I wrote to Ms. Ogawa, who then sent me poems, photos, and other accounts of her father's life. He is also warmly remembered by American marines who worked with him to establish the Reunion of Honor, now held once a year when American veterans come to visit.\n\nKuribayashi's story has been eloquently captured by Kumiko Kakehashi in _So Sad to Fall in Battle_ (Presidio Press, 2007), based on his letters. She notes that as a captain in his late thirties Kuribayashi spent more than two years studying and traveling in the United States, from March 1928 to April 1930. He audited classes in English, American history, and current affairs at the University of Michigan and Harvard. He bought a Chevy and drove it from Kansas to Washington, D.C., where he met several American military men. Among his papers are forty-two letters, many illustrated by his drawings, that he wrote home to his family from the States.\n\nKuribayashi was samurai (a powerful military caste in feudal Japan), and some Japanese authorities resented his imperial connections. Kuribayashi bred greater animosity among his peers for his conviction that the United States was an unbeatable enemy and that Japan should steer clear of direct conflict with the Americans. It is thought that at least part of the reason he was sent to defend Iwo Jima was to get rid of him. This clearly was a fight that the Japanese were not going to win.\n\nIn 1944, as Guam and Saipan fell to the Americans, the Japanese knew very well that Iwo would be next. Its airfields, situated halfway between Saipan and Tokyo, would be crucial for the American bombing campaign. Wachi noted that many of the draftees on the island were civilians with no military training. He set about trying to remedy that before Kuribayashi even got there.\n\nKuribayashi arrived in June and began drastically reordering defensive priorities, assigning troops to help dig tunnels, a task that was purgatory itself since the troops had to endure sulfur gas and heat so intense they could work only in five- and ten-minute shifts. As he began to implement his defensive strategy, Kuribayashi established fields of fire that would blanket the invasion beaches.\n\nBecause the Americans had control of air and sea, he also abandoned the standard tactic of heavily fortifying the water's edge because there was no way his men could sit there and fire at the incoming landing craft without being wiped out by planes and shelling. He also dismissed any notion of a banzai charge because it wasted too many men and had proved only marginally successful against American marines. Sensing the coming onslaught, Kuribayashi focused on his underground defensive positions, which could withstand bombing from planes and shelling from the ships. Indeed, the island was bombed for seventy-four straight days prior to the invasion. It caused a lot of headaches and obliterated every shred of greenery on the island but seems to have caused few, if any, casualties among Kuribayashi's men.\n\nKuribayashi also wrote and distributed six statements known as the \"Courageous Battle Vows\":\n\n_1. We shall defend this place with all our strength to the end._\n\n_2. We shall fling ourselves against the enemy tanks, clutching explosives to destroy them._\n\n_3. We shall slaughter the enemy, dashing in among them to kill them._\n\n_4. Every one of our shots shall be on target and kill the enemy._\n\n_5. We shall not die until we have killed ten of the enemy._\n\n_6. We shall continue to harass the enemy with guerrilla tactics even if only one of us remains alive._\n\nOn March 16, Kuribayashi sent a farewell telegram to the Imperial General Headquarters:\n\n_The battle is entering its final chapter. Since the enemy's landing, the gallant fighting of the men under my command has been such that even the gods would weep. In particular, I humbly rejoice in the fact that they have continued to fight bravely though utterly empty-handed and ill-equipped against a land, sea and air attack of material superiority such as surpasses the imagination._\n\n_One after another they are falling to the ceaseless and ferocious attacks of the enemy. For this reason the situation has arisen whereby I must disappoint your expectations and yield this important place to the hands of the enemy. With humility and sincerity, I offer my repeated apologies._\n\n_Our ammunition is gone and our water dried up. Now is the time for us all to make our final counterattack and fight gallantly, conscious of the emperor's favor, not begrudging our efforts though they turn our bones to powder and pulverize our bodies._\n\n_I believe that until this island is recaptured, the emperor's domain will be eternally insecure. I therefore swear that even when I have become a ghost, I shall look forward to turning the defeat of the Imperial Army into victory._\n\n_I stand now at the beginning of the end. At the same time as revealing my inmost feelings, I pray earnestly for the unfailing victory and the security of the Empire. Farewell for all eternity._\n\n_As regards Chichi Jima and Haha Jima, I am sure the men under my command there can completely crush any attack the enemy might make. I entrust that matter to you._\n\n_Finally, I append an inept work for your perusal below. Please forgive its clumsiness._\n\nKuribayashi's death poem follows:\n\n_Unable to complete this heavy task for our country_\n\n_Arrows and bullets all spent, so sad we fall._\n\n_But unless I smite the enemy_\n\n_My body cannot rot in the field._\n\n_Yea, I shall be born again seven times_\n\n_And grasp the sword in my hand._\n\n_When ugly weeds cover this island,_\n\n_My only thought will be the Imperial Land._\n\nKakehashi notes that the Imperial General Staff edited the telegram before releasing it to the newspapers, eliminating the phrase \"utterly empty-handed and ill-equipped,\" and changing \"so sad we fall\" from his poem to \"mortified, we fall.\" It's easy to see why. She goes on to suggest that Kuribayashi died leading the last charge, the attack on the pilots by Motoyama No. 1. This has been disputed, as Colonel John Ripley makes clear in Chapter 24 of this book.\n\nCaptain Wachi ended the war in southern Kyushu, in charge of a suicide torpedo boat training base, in which young volunteer trainees were to attack enemy vessels with only enough fuel for a one-way mission. In October, with demobilization under way, Wachi got off the home-bound train in Kyoto to visit a Buddhist monk. Wachi's daughter says that after he spoke of his determination to dedicate the rest of his life to mourn the war dead, including those lost on Iwo Jima, the monk ordained him on the spot. After a week's practice he became Jushoan Koami. He went on to write several poems under that name. During the occupation American troops imprisoned him for five months, presumably because of his wartime involvement.\n\nAs Jushoan Koami he founded and became president of the Association of Iwo-Jima and finally gained permission to return to Iwo to perform religious rituals. He wrote:\n\n_It was still under occupation when we first landed on Iwo Jima Feb. 1952 by L.S.T. While there I put up two Kannon (a popular Buddhist deity) statues at the north and south of the island for all the departed souls without distinction of nationality. My idea was to carry out religious services and pray and mourn for them as I had stated in my application based on Buddhism principle: to pray wholeheartedly for all the souls evenly for their eternal rest beyond human resentment._\n\n_In the meantime we went down into the raid-beaten and flamethrower ruined caves to search and collect bodies and relics of the departed. We discovered to our sorrow that quite a number of skulls had been removed from the bodies. We immediately reported to the U.S. Marines there and asked them for measures to prevent further mishaps._\n\n_Later in December 1955 I was informed by a staff officer of the U.S. Pacific Fleet Headquarters of the fact that the caves had been sealed with the bodies lying as they were. That might have helped prevent further removal of the skulls but it . . . made it almost impossible for us to collect other relics which had been sealed in with the bodies unless we used heavy machinery._\n\n_As the result of the defeat in the war which had never taken place in the long history of our nation, so-called \"Democracy\" came along. The idea had developed in the West but it was totally new to our nation. Many people took it as their golden rule to replace the august virtue of his imperial majesty which had always existed ever since the dawn of our national history._\n\nHis daughter adds that he offered prayers for the departed, crept into caves to pour water or sake, or smoked cigarettes as requested by the bereaved survivors aware of how the soldiers had died in \"extreme shortage of their needs.\"\n\nMs. Ogawa notes that while the marines did not remove heads during or after the battle, soldiers recuperating on Iwo during the Korean conflict took skulls as souvenirs. Two American writers, Richard Newcomb, a wartime correspondent and author of six books, including _Iwo Jima,_ and Bill Ross, who wrote _Iwo Jima: Legacy of Valor,_ published with Wachi's address an appeal for the return of skulls and other relics, such as diaries, photos, and letters.\n\nWachi was instrumental in helping arrange the first Japanese-American memorial ceremony conducted on Iwo in February 1985, the fortieth anniversary of the invasion. The retired Marine general Jarvis Lynch was the liaison with Wachi for that event, and I heard him speak about Wachi at an Iwo Jima symposium in Arlington, Virginia, in February 2007. Lynch has also written about Wachi for _Naval History_ magazine. He said it was doubtful that without Wachi's prestige in Japan, the memorial service or those that followed would have occurred. He also said that in the forty years before his death, Wachi had accounted for the remains of more than eight thousand Japanese soldiers and sailors.\n\nWachi died at the age of eighty-nine on February 2, 1990. Here is one of his poems, in both the original script and in translation, provided by Masahide Mizoguchi of the Bronx in September 2007, who says the poem was written in Kanji, one of the Japanese writing systems that use Chinese characters. According to Mizoguchi, this poetry format was very popular among the samurai class and, consequently, in the military before and during the war. The use of Kanji would enable one to convey deep meanings and nuances in a limited amount of space. Mizoguchi adds that notes written underneath the poem, part of a package sent to me by Wachi's daughter, indicate that Wachi wrote it while imprisoned by the Americans after the war. Wachi indicates in his notes that he wrote the poem after seeing a photo in the military newspaper _Stars and Stripes_ showing the American flag flying from the summit of Suribachi. The Kanji script following the English version was reproduced by Mizoguchi on a computer.\n\n**To the Suribachi Monument**\n\n_A new monument stands_\n\n_At the top of the hill,_\n\n_Where the Stars and Stripes still wave,_\n\n_As it dominates the sea._\n\n_The foul smell of blood arises_\n\n_From the stones it stained;_\n\n_While mounds of bones_\n\n_Cover the mountaintop._\n\n_The battle has been lost,_\n\n_A year has now come around;_\n\n_All ill will and vengeance_\n\n_Have already ceased to exist._\n\n_'Tis hard to console the souls_\n\n_Of the defeated,_\n\n_While only the souls of the victor_\n\n_Ascend to heaven._\n\nA veteran Japanese journalist, Toshio Aritake, reports that it is not easy to get people to talk about the battle today. As a whole, they are concerned that if they speak out, they might end up bragging about their experiences or telling wrong or misleading stories. \"History does not convey facts,\" says Kiyoshi Endo, eighty-four, a lieutenant of the Imperial Navy at the time of the battle and the chairman of the Association of Iwo-Jima.\n\nA resident of Yokosuka, Endo expresses mixed emotions about the sudden resurgence of interest in Iwo Jima after Clint Eastwood's movie. Endo says the movie did help collect more information from Japan and the United States and contributed to the increasing return of belongings of the dead on the island. At the same time, he and the association began receiving commercially oriented inquiries. People suddenly wanted to travel to Iwo Jima. They wanted interviews for books and the chance to make films.\n\n\"I feel at a loss to see what happened in the island, what we did there, how we felt about our families and how our families and relatives felt about us during the battle are written and shown like ads. The battle had different meanings for everyone, and it is impossible to generalize those meanings and we should avoid such generalization,\" Endo says.\n\nFor young Japanese, the Hiroshima and Nagasaki atomic bomb attacks, which caused a huge number of casualties, are on the threshold of becoming ancient history. Likewise, for most Japanese, the Iwo Jima battle evokes blurred monotone images of soldiers from both countries sacrificing their lives for the causes of their respective countries.\n\nFor the Iwo Jima survivors, their families, and dependents, however, the battle is fresh and real, as the A-bombs are for those concerned, Endo says. \"American troops were rescued [when they were injured], but for us, there was no rescue and we were just left to perish.\" He did not offer details but probably was referring to wounded Japanese soldiers who could not be treated or saved for the lack of medical supplies.\n\nEndo feels awkward talking about his experiences of the battle because . . . \"one person may say something while others may see it differently and say different things. Personal experiences cannot be accurately conveyed [in writing and images]. And when they speak, they may boast about their experiences, even though they did not do it.\" He says, \"The only fact regarding that battle is that it was over and the war was over. Which side won mattered when it was being fought, but now, sixty-two years after the battle, won or lost\u2014there is no such thing.\"\n\nThe journalist Aritake adds that like Endo, his late father, Asazo Aritake, who was assigned as a Navy Marine to Guadalcanal and other South Pacific islands \"never talked about his experiences to us children. When he was running a sushi restaurant in downtown Tokyo, his wartime colleagues would come visit him begging him to tell war stories. He never reciprocated. He was among a very few who returned from there.\"\n\nThe bond between the veterans and other people concerned from both countries remains strong; they all actively participate in the annual memorial service held on the island. It is because \"our mission is a common one\u2014that is, to search and collect the belongings and jointly with the Japanese government, look for the remains of the dead,\" Endo says.\n\nThe Iwo Jima annual memorial is, as I understand it, the only ceremony for the war dead in the world that peoples of two former enemy countries hold. That in itself underscores how close the two countries are, says Yoshitaka Shindo, a member of the House of Representatives and a grandson of Lieutenant General Kuribayashi.\n\nVeterans and others from the two countries hold an annual meeting at Yasukuni Shrine, the controversial Shinto pavilion where the war dead, including world war criminals, are buried. Endo visits the shrine every year to pay respects to the Iwo Jima battle dead buried in the shrine. \"They dedicated their lives to the country.\"\n\nEndo says there still are more than thirteen thousand remains uncollected, and the task is made difficult by the lack of regular air or sea travel. \"I do not know how long it will take to collect all the remains, but we will continue. And if it is over, we want to collect the remains of Japanese soldiers and others who died in the South Pacific islands too.\"\n\nYasunori Nishi, eighty, is the son of the late Takeichi Baron Nishi, an Imperial Army officer and a 1932 Los Angeles Olympic equestrian athlete who died in the battle. He says the Iwo Jima Association, which at its peak had about five thousand members, has shrunk to about twelve hundred. But this does not mean the Japanese interest in the battle has waned. \"It is going the opposite direction and becoming active again because of participation of grandsons and granddaughters,\" he says. The survivors and their families are becoming fewer, but \"there are grandsons and daughters asking to take them to the island to pay respects. Some of them, since they cannot go there for various restrictions, would go to Ogasawara [Bonin] Island, and there they would pray in the direction of Iwo, which is about two hundred fifty kilometers away.\"\n\nCaptain Wachi's daughter, Rosa, declares, \"We appreciate that more people are developing an interest in Iwo Jima, but we do not want it to be used for business. . . . My father was sometimes bad-mouthed that he was collecting the remains of the Japanese soldiers for money, but he just kept on doing it.\" She also questions the famous photo of the U.S. Marines' flag raising on the island, saying that it might be \"a show probably asked by a photographer to model. . . . It is too good to be true!\"\n\nPATRICK MOONEY AND THE GRAVES\n\n**Deputy Director, Combat Veterans of Iwo Jima; Director, National Museum of the Marine Corps, Docent and Visitors' Services**\n\n_In the loading order for the LSTs, you see the list for Graves Registration, you see the items that were needed to process the dead. The crosses and Stars of David were already manufactured and stacked and brought in aboard the LSTs._\n\n_I grew acquainted with Patrick Mooney, himself a former marine, during my visit to Guam and Iwo Jima with Military Historical Tours in March of 2006, and spoke with him at length over the course of an Iwo Jima symposium in Alexandria, Virginia, in February 2007. He was invariably gracious and accommodating, and extremely knowledgeable in all aspects of the campaign._\n\n\"Every marine wore two dog tags. One was on the main cord, and the secondary tag was attached by a short suspension loop below the other dog tag, and that was the one that went on the cemetery cross.\n\n\"We [historians] count five thousand nine hundred thirty-one combat ground casualties on Iwo Jima, consisting of marines, Navy guys, and one coast guardsman. They all were buried on Iwo or lost on the island. The remains of one hundred seventeen are still missing, meaning they have been declared dead but their remains have not been recovered. A lot of them were lost in the sea when their boats were hit or shot as they tried to wade ashore. Others were incinerated by artillery explosions or blown to pieces. A few, like Bill Genaust and others, we know what happened to them, but we never recovered their remains. The actual number of graves in the two cemeteries came to forty-seven hundred.\n\n\"The burials began under fire. The battlefield was still proximate, and on Iwo Jima there was no rear area. The Third and Fourth Divisions adjoined each other in one large cemetery that was split in half. The Fifth Division cemetery was separate. The Third and Fourth Division cemetery was just off Blue Beach One and Two. The Fifth Division cemetery was on other side of the island south of Airfield No. 2, just north of Airfield No. 1.\n\n\"The first burials I recorded were dated about four days after the battle started. Until that point they were gathering the dead in the beachhead area. The bodies were sprayed with insecticide to keep the bugs off and the odor down, then wrapped in ponchos. There are photographs of them doing that. Then, about D plus three or four, they began to bury the first dead in temporary cemeteries around the regimental hospitals and divisional hospital sites. They established the sites of the permanent cemeteries even before the landing went in.\n\n\"In the operational orders, the sites of the hospitals and cemeteries were already established. In the loading order for the LSTs, you see the list for Graves Registration, you see the items that were needed to process the dead. The crosses and Stars of David were already manufactured and stacked and brought in aboard the LSTs.\n\n\"After about the fifth day they began to to inter the dead in the permanent cemeteries. The process was very well organized and regimented. You have to realize that by February of '45 we had been in ground combat for eighteen months and the Marine Corps had suffered sixty thousand killed and wounded. So they were well prepared by this point. About a quarter of that number would have been dead and buried in various places as they moved from island to island. And each island had its own cemetery or several cemeteries. The dead were always arranged by the unit, so you had the different divisional cemeteries, and within the cemeteries at each plot you can look at the burial dates _vice_ the date of death. Date of death can vary by two weeks. There were cases of bodies not recovered for as long as two weeks before they were buried. That's how the plats go.\n\n\"The divisional battalions, mostly black units, would dig the trenches. They'd bring in bulldozers, D8 Caterpillar bulldozers, and dig long trenches. The width of the bulldozer blade was about eight feet. They dug from a sloping height down for a certain length, and then they would slope upward on the other end. The bottom height would be anywhere from four to six feet below surface, depending on the stability of the sand or earth they were on. They would dig down and scoop out that long section.\n\n\"The bodies would be arranged, again very regimented, even in death. The bodies were carried in ponchos or in mattress covers. There were no body bags; that was an invention for later wars. The bodies would be laid in these trenches with a uniform width between each body, I think it's twenty-four to twenty-eight inches depending on the different unit and how they buried. That also dictated later on when the graves were filled how the cemeteries were laid out and how the graves were marked and where the crosses went. Every marine had two dog tags, and a dog tag was left with the body and a dog tag was taken by the Graves Registration personnel. All personal effects on the body were gathered and put in a small green canvas bag with a white linen tag on the front. The white linen tag was filled out as completely as possible with the name, rank, serial number, date of death, any information that could be garnered from the body. All the personal effects on the body would be placed in this bag. All the combat equipment, helmet, weapons, web equipment would be placed in the salvage depots, where they were processed for destruction or reuse. Personal effects were sent home.\n\n\"The body was buried in whatever uniform it was in at the time so they would have their boondockers, leggings if they wore them, dungaree trousers, dungaree shirt, and then they were placed in the poncho or the mattress covers. They left the boondockers on. They did not remove footwear or any clothing items. Even if there was a field jacket on, they left that but took all the personal effects. The Marine Corps poncho was big enough to encircle the body. You can see in the photos that if the poncho was spread out from top to bottom, it would cover from head to toe everybody except the tallest individuals. Priority was always given to covering the head and the face, with the boondockers sticking out the other end.\n\n\"Depending on the layout of the different parts of the cemetery, the trenches could contain as many as forty or fifty bodies or as few as twenty bodies. Each segment would be dictated by the quality of the soil they were digging in. The top layers of sand, that volcanic pumice, is very loose, but as you dig down, it gets into sandstone; that was what gave the Japanese the ability to tunnel so extensively. The bulldozer dug right down into the sandstone and gouged it out. It would take numerous passes to get to the requisite depth.\n\n\"The Pioneer battalions were all white: the Fifth Pioneer Battalion for the Fifth Division, the Fourth Pioneers for the Fourth Division, and the Third Pioneers for Third. The Pioneer battalions did the road engineering, the building, the construction of what we would normally think of as an engineer unit. In the Marine Corps combat divisions in World War Two, combat engineers performed combat engineering tasks, which were road clearing and demolition of obstacles. On Iwo that would include the demolition of caves and also obstacle clearance like the antitank ditch at Hill 362 Able. That was done by the Fifth Combat Engineer Battalion, the one Al Abbatiello was in. Tom Cox; Colonel Charles Waterhouse, the famous Marine Corps artist: these guys were all Fifth Combat Engineers.\n\n\"The Fifth Pioneers were the same guys involved in that shootout the last day when the three hundred Japanese came down and killed all the pilots. One of their leaders, Lieutenant Harry Martin, received the last MOH on Iwo Jima. These guys are often mistaken for being black. In World War One, large Pioneer battalions, mainly black African American units, were used as road clearance. It's often assumed they operated the same way in the Marine Corps in World War Two. Gentlemen from the Fifth Pioneer Battalion will be quick to tell you it was an all-white outfit. African American units were very limited in the Marine Corps. It was still a very segregated service.\n\n\"Getting back to the graves. . . . As the bodies were brought in, they were carried down this long slope, not lowered, but carried in, and then the trench would be back-filled from one end to the other. From the photos and from what I've been able to uncover, after every other body was laid, the chaplain would be about two bodies behind the registrar, who was moving along as the bodies were laid to make sure they were put in the right order. So the men would call out, \"Smith, John, Sergeant, two-one-four-six-nine-seven-two, and the registrar would make the note that he was in Plat one, Row one, Grave twenty-four.\n\n\"Two bodies behind, the chaplain would be giving the appropriate benediction for the Catholic, Protestant, or Jewish faith. So they would move along that line, and once the bodies were all laid in and the appropriate benedictions were made and services and honors rendered, the grave would be filled in from head or foot depending on where the dirt had been pushed and plowed out. That was done mostly by the bulldozers; it was a tremendous task to fill that. The crosses in the photos were placed at the head of the trench so they knew approximately where the body was laid. They knew when they put Private Jones in that his cross was laid here.\n\n\"After the trench was filled, they would raise the cross and sink it into the ground. It was wooden construction, a cross or a Star of David. The second dog tag for that individual was affixed to the cross. The first dog tag was on the body. A number was stenciled on the back of the cross. Sometimes you'll see the dog tag affixed to the front and sometimes to the back of the cross. The name of the individual was stenciled, his service, his service number, and the emblem appropriate to his branch: the eagle, globe, and anchor, a Navy shield, or the Coast Guard emblem.\n\n\"The notched dog tag was an Army invention, and the notion that it was designed to be placed in the teeth upon death and then kicked shut was a myth. The notch was meant to hold the tag steady in the machine when it was being made. Throughout World War Two, the Marine Corps by and large did not have the notched dog tags. They used solid aluminum ovals that would either be etched or stamped. By Iwo Jima they were stamped, a flat oval cylinder about the size of a silver dollar, with a single hole. They were affixed around the neck by a linen-coated wire which had a screw fixture. The Marine Corps did not come out with chains to hold the dog tags until very late.\n\n\"Starting in 1946, all the graves were exhumed and repatriated by the end of '47. There was a continuing effort in the latter half of the 1940s to go back and resolve as many missing in action cases, the unknowns, as possible, and a very elaborate process was followed. Burial mistakes were made under combat that might not have occurred in a calm peacetime situation.\"\n\nAccording to Army Graves Registration, the coordinating entity for all services, what it dug up varied from mummified corpses to just bones and other human remains. This depended on the location of the grave, the cemetery, soil conditions. Because it was so dry and there was so much heat on Iwo, there was a wicking effect of moisture into that volcanic pumice that in effect pulled the fluids out and mummified the body.\n\n\"A number of graves were lost in the Fifth Marine Division cemetery because of the ground subsiding and collapsing in different areas because of the vast Japanese tunnel complex. Mortuary specialists and embalmers and forensic people from all the services were recovering bodies, and when they dug down the requisite depth to recover the bodies, they weren't there. They continued to dig, and they finally found that the tunnels below had collapsed in a particular section and a half dozen bodies had fallen four and half feet below the area where they were originally buried. They were recovered.\n\n\"In another case, they went to grave one and dug down where there was supposed to be Private Smith, but he wasn't there. Then they dug grave two, and there was Private Smith. So a row that was supposed to be fourteen individuals wide was actually thirteen wide, and all the bodies were off by one set. All were recovered. It was just that somehow, in the rush and mix of everything, they had shifted the bodies one down and incorrectly recorded the grave as being marked on another so that number two was actually number one. ID's were found on the bodies as they were exhumed, and forensic exams were done, compared with extensive personal data from the casualty reports, dental records, everything. They were very thorough.\n\n\"So we had fifty-nine hundred guys buried in two cemeteries in a five-week period. They stayed there until 1946, when the War Department began to repatriate the remains of battlefield casualties. At that point we had battlefield cemeteries all across the Pacific. Every island battlefield and base area had a cemetery. So New Caledonia had a cemetery. Ulithi. In New Zealand and Australia. Not only combat areas. An effort was begun to gather these remains, come up with a consolidated cemetery plan, and to repatriate some remains whose survivors wished to have the bodies brought back.\n\n\"Family members were given one chance: They could elect to have the bodies brought back to their hometown cemetery or they could have the bodies reinterred at a national cemetery. And the two sites chosen in the Pacific were Honolulu and Manila. Or back to their hometown. You had one chance to do it. It was decided that the remains of those who fought in the central Pacific and northern Pacific would be brought to Honolulu. Those who served in the southwest Pacific or in the Philippines, to include Peleliu, those remains would go to Manila. Those who died in, around, and over Iwo Jima were brought back to Honolulu or back to the States. The vast majority of the Iwo Jima casualties were brought back to Honolulu, where they rest to this day.\n\n\"I have not found an instance in the record of a combat casualty who was brought back to Arlington. The family could choose where they buried their marine, and if the family chose Arlington, well, I would assume there are some. Never say never. The Basilones and folks like that and of course one of the flag raisers, Ira Hayes, are at Arlington. He died well after the war. A lot of postwar guys ended up there. You had to be a combat vet. But space is so limited now you have to have a medal of valor, such as a Bronze Star, or a Purple Heart to qualify for interment now.\"\n\nEd White of New York City, a theater critic for the _Wall Street Journal_ for many years, remembers this:\n\n\"A small group of us landed on Iwo two years to the day of the first landing, February 19, 1947. I had gone into the Army on an eighteen-month enlistment in the fall of '46. They kept putting the draft on, then calling it off, and I wanted to get it over with.\n\n\"I was put to work in the port director's office, a Quonset hut, overseeing things that were wrecked just offshore. The Japanese were reclaiming vehicles and weaponry and taking them back to Japan for scrap metal, which was kind of ironic. There was very little to do. I taught myself to type on a manual typewriter there in the office.\n\n\"People had advised me get into Special Services, and there was a small radio station in two or three Quonset huts under the shadow of Suribachi. They had generators and played large platter-type records with fifteen minutes of shows that we were getting from the Armed Forces Radio Service, popular music, big bands like Tommy Dorsey's. It was strictly for entertainment. I volunteered and learned how to run the station. Then the man who had been running it was reassigned to the Philippines, and I was the only man on the island who knew how to run the place.\n\n\"It was while I was doing that that the gravedigging unit came in March or April of '47. They were an all-black unit, come to dig up the graves for movement back to the States or Hawaii. I vividly remember this graveyard of white crosses with a few Stars of David standing out against that black volcanic ash. It was quite an incredible sight, a field of crosses, very stark.\n\n\"One of the men in charge, a master sergeant or somebody, came to see me one day at the radio station and said, 'We got a chorus. We sing spirituals.' I said, 'That's terrific.' Because we had very few live shows on the station. So I put them on the air every Sunday afternoon. Eighteen or twenty of these guys would crowd into our little studio and do thirty minutes of spirituals and harmony, a cappella, of course. I don't think we even had a piano.\n\n\"They were still working on the graves when I left in July for Guam.\"\n\nAbout fifteen hundred marines of Jewish faith took part in the battle of Iwo Jima. Rabbi Roland B. Gittelsohn, the first Jewish chaplain appointed by the Marine Corps, was assigned to the Fifth Marine Division, according to an article appearing November 6, 1998, in the newspaper _Forward._ He was asked by the division chaplain, Warren Cuthriell, a Protestant, to deliver the memorial sermon at a combined religious service dedicating the Fifth Division cemetery in March 1945. But, the article goes on, a majority of Christian chaplains did not want a rabbi preaching over predominantly Christian graves and blocked his participation. Ultimately three separate services took place. The rabbi delivered the eulogy he had written for the combined service. About seventy attended, including three Protestant chaplains incensed by the prejudice displayed by their colleagues.\n\nAs a consequence of their anger, copies of Gittelsohn's talk were widely circulated, including back in the States; it was read into the _Congressional Record._ The talk remains famous in Marine Corps circles and elsewhere to this day. Here is what the rabbi said:\n\n_This is perhaps the grimmest, and surely the holiest task we have faced since D-Day. Here before us lie the bodies of comrades and friends. Men who until yesterday or last week laughed with us, joked with us, trained with us. Men who were on the same ships with us, and went over the sides with us as we prepared to hit the beaches of this island. Men who fought with us and feared with us. Somewhere in this plot of ground there may lie the man who could have discovered the cure for cancer. Under one of these Christian crosses, or beneath a Jewish Star of David, there may now rest a man who was destined to be a great prophet\u2014to find the way, perhaps, for all to live in plenty, with poverty and hardship for none. Now they lie here silently in this sacred soil, and we gather to consecrate this earth in their memory._\n\n_It is not easy to do so. Some of us have buried our closest friends here. We saw these men killed before our very eyes. Any one of us might have died in their places. Indeed, some of us are alive and breathing at this very moment only because men who lie here beneath us had the courage and strength to give their lives for ours. To speak in memory of such men as these is not easy. Of them too can it be said with utter truth: \"The world will little note, nor long remember what we say here. It can never forget what they did here.\"No, our poor power of speech can add nothing to what these men and the other dead of our Division who are not here have already done. All that we even hope to do is follow their example. To show the same selfless courage in peace that they did in war. To swear that by the grace of God and the stubborn strength and power of human will, their sons and ours shall never suffer these pains again. These men have done their jobs well. They have paid the ghastly price of freedom. If that freedom be once again lost, as it was after the last war, the unforgivable blame will be ours, not theirs. So it is we the living who are here to be dedicated and consecrated._\n\n_We dedicate ourselves, first, to live together in peace the way they fought and are buried in this war. Here lie men who loved America because their ancestors generations ago helped in her founding, and other men who loved her with equal passion because they themselves or their own fathers escaped from oppression to her blessed shores. Here lie officers and men, negroes and whites, rich men and poor\u2014together. Here no man prefers another because of his faith or despises him because of his color. Here there are no quotas of how many from each group are admitted or allowed. Among these men there is no discrimination. No prejudices. No hatred. Theirs is the highest and purest democracy. Any man among us the living who fails to understand that will thereby betray those who lie here dead. Whoever of us lifts up his hand in hate against a brother, or thinks himself superior to those who happen to be in the minority, makes of this ceremony, and of the bloody sacrifice it commemorates, an empty, hollow mockery. To this, then, as our solemn, sacred duty, do we the living now dedicate ourselves: to the rights of Protestants, Catholics and Jews, of white men and negroes alike, to enjoy the democracy for which all of them here have paid the price._\n\n_To one thing more do we consecrate ourselves in memory of those who sleep beneath these crosses and stars. We shall not foolishly suppose, as did the last generation of America's fighting men, that victory on the battlefield will automatically guarantee the triumph of democracy at home. This war, with all its frightful heartache and suffering, is but the beginning of our generation's struggle for democracy. When the last battle has been won, there will be those at home, as there were the last time, who will want us to turn our backs in selfish isolation on the rest of organized humanity, and thus to sabotage the very peace for which we fight. We promise you who lie here: we will not do that! We will join hands with Britain, China, Russia in peace, even as we have in war, to build the kind of world for which you died._\n\n_When the last shot has been fired, there will still be those whose eyes are turned backward, not forward, who will be satisfied with those wide extremes of poverty and wealth in which the seeds of another war can breed. We promise you, our departed comrades: this too we will not permit. This war has been fought by the common man; its fruits of peace must be enjoyed by the common man! We promise, by all that is sacred and holy, that your sons, the sons of miners and millers, the sons of farmers and workers, the right to a living that is decent and secure._\n\n_When the final cross has been placed in the last cemetery, once again there will be those to whom profit is more important than peace, who will insist with the voice of sweet reasonableness and appeasement that it is better to trade with the enemies of mankind, than by crushing them, to lose their profit. To you who sleep here silently, we give our promise: we will not listen! We will not forget that some of you were burnt with oil that came from American wells, that many of you were killed with shells fashioned from American steel. We promise that when once again men profit at your expense, we shall remember how you looked when we placed you reverently, lovingly, in the ground._\n\n_Thus do we memorialize those who, having ceased living with us, now live within us. Thus do we consecrate ourselves the living to carry on the struggle they began. Too much blood has gone into this soil for us to let it lie barren. Too much pain and heartache have fertilized the earth on which we stand. We here solemnly swear: this shall not be in vain! Out of this, and from the suffering and sorrow of those who mourn this, will come\u2014we promise\u2014the birth of a new freedom for the sons of men everywhere. Amen._\n\nCOLONEL JOHN W. RIPLEY, USMC (RET.)\n\n**Director of History and Museums, Marine Corps  \nIwo Jima, Then and Now**\n\n_Our flag flies [over Iwo Jima] one day a year. The Marines bring it. We're not permitted to leave it there when we leave the mountain. We have to take it down._\n\nColonel John Ripley's most recent photo in uniform was taken around 1999, while he was president of Hargrove Military Academy. The large medals on the top row, from left to right, are the Navy Cross (blue with a white stripe), the Silver Star (red stripe), the Legion of Merit (two), the Bronze Star with Combat V (two), and the Purple Heart.\n\nColonel John Ripley stands before the Fifth Marine Division Memorial on the summit of Mount Suribachi in 1987. In the colored version of this photo, the flag, on top, and the bas relief of the flag raising, just below, were shining brightly because, Ripley said, \"My marines spent the day Brassoing it.\" The original flag stood just behind the monument on which now hang dozens of dog tags left by visiting service personnel. Ripley recalls: \"This was the end of a number of 'survey trips' to Iwo Jima when my small survey party made very detailed explorations of the Japanese defenses; went into bunkers, caves, tunnels all over the island; and helped them with their recovery efforts of the remains of an estimated twenty-two thousand defenders, still in progress. They were most grateful.\"\n\n_Colonel John Ripley, USMC (Ret.), was five years old when the Marines invaded Iwo Jima. After graduating from the Naval Academy in 1962, he served two years in Vietnam, participating in twenty-six major operations, and eventually was awarded the Navy Cross for almost single-handedly blowing up the bridge at Dong Ha during the 1972 North Vietnamese Easter invasion, a feat for which he should have received the Medal of Honor. He became involved with Iwo Jima as a colonel stationed in Japan and has visited the place more than fifteen times, always sleeping on the summit of Mount Suribachi. He later served as the director of history and museums for the Marine Corps. I was with him on March 5, 2006, as he knelt to pass a coin across the sawed-off circle at the base of the flagpole where the American flag first flew that morning of February 23, 1945. \"To the Marines,\" he said, \"this is Lourdes.\" Sacred ground._\n\n\"When we returned Iwo Jima to the Japanese in 1968, the Status of Forces Agreement and the treaty made it very clear that we, the Marine Corps, would always have the right to train there and we will always have the right to visit there.\n\n\"It all started during my watch, in the fall of 1987 in Okinawa, when I was a colonel, the G3 in charge of operations and planning for the Third Marine Amphibious Force. I said I wanted the Japanese to understand that although we hadn't been there in quite a while, I wanted them to understand that we were going to do this, establish more of a presence on the island. Prior to this, we would train in Iwo Jima very infrequently. An outbound Marine MEU [Marine Expeditionary Unit], a battalion aboard ship, would do a landing, wander around a bit, and then backload and proceed to Okinawa. It was training, but it was very limited.\n\n\"The G3 is the mover and shaker of everything that's happening in the force. So all the operations were set up and planned by my organization. I was responsible for everything the MEF [Marine Expeditionary Force] did. It was the Third MEF, which consists of all the elements\u2014the division, the wing, all of the supply and ordnance. The purpose was to make sure the Japanese knew we were going to reassert our presence there. They were more than a little reluctant.\n\n\"The Navy wasn't all that keen to do it either, and finally the Marine Corps said, 'These are our MEUs, they need to have amphibious landings, to get as much training as they can, so we're going to schedule regular landings, training exercises, at Iwo, on the way out to Okinawa or on the way back.' This was 1987\u201388.\n\n\"So when I went back, the whole purpose of my going back was to walk around there, be seen, not be abrasive or offensive, but to let them know we were going to be around. We did send three Third Marine Division vehicles there. I've since made about a dozen trips to Iwo Jima, and I've stayed on the island for a couple weeks at a time.\n\n\"The Japanese used it for a similar purpose themselves, which is to say their war college and other units would come down and do battle studies at Iwo Jima. Every time I went over there I'd run into these guys. I was very cordial, very nice to them, never interfered with what they were doing. I would always take a party down there, say, three or four of our people, sometimes as many as ten, and we would do our own exploration.\n\n\"Soon the Japanese could see we were not a threat, and then two things happened that endeared us to them. One had to do with our using a wider area of the ocean there to practice Navy flights and simulated carrier landings, so we could keep away from Tokyo and Yokosuka. Our wing worked that out for them. The second thing was, their Home Office had just begun to disinter Japanese remains, all twenty-two thousand. That's a disputed figure, but I point out that at least a couple thousand were Korean laborers, and they're counted among the total losses on the Japanese side. In some cases they were killed, but mainly they were sealed off the same as the Japanese Army.\n\n\"My gang was going down into these caves in the fall of 1987 before the Japanese Home Office even got into it. We knew where they were, and we would go down there and find these incredible scenes. You would not believe the amount of equipment we found down in those caves.\n\n\"We found a hospital cave jammed with medical stuff, tons of medical supplies, cooking gear, rifles, Browning automatic rifles, a Lewis gun, records. We could only stay down there for twenty minutes. We had to rope up because it was so bloody hot you would collapse and couldn't get out of there. An artillery cave was packed with stuff, surveyors' kits, helmets.\n\n\"Then two things happened: We began to show these guys where this stuff was. They had nothing there but a backhoe or a couple diggers, and we never saw more than, say, two of these things with one or two other guys. I don't know what their progress was, but it was going to take forever to disinter them all. We'd go down there and find some remains, Japanese dead, and we'd go over and tell the guy, bring him over, and show him, and they appreciated that. And of course we were very solemn about it. We would uncover [remove hats] and render a salute, and I think that impressed them.\n\n\"The other thing was Hartzell. Staff Sergeant Hartzell was the one who found the body of what we called the chief of staff of General Kuribayashi. It was not so much the remains of the chief of staff, which actually had mummified, as something else he found.\n\n\"Everyone knew where Kuribayashi's cave was. It's still there, still marked. It was directly across the street from the Coast Guard station, and there was a beautiful mowed grass field right in front of the cave. It's all jungled up now, but then we stayed at the Coast Guard station. I kept all my vehicles there. Of course we all went over to the cave. Hartzell went in a couple times. He had done a considerable amount of exploring in Okinawa. He had an Okinawan wife; he could read and speak Japanese. He had a real sixth sense. This guy was quite a talent. When he got there, it was a treasure trove for him, and he walked into the cave. To enter, you had to lie on your back and turn around and come up over like a cornice into the main chamber of the cave.\n\n\"What wasn't seen initially was it had planking, kind of like duckboards, all covered with paper. Somebody had gone in there and trashed the hell out of it over the years, just souvenir hunting, I'm sure. Hartzell gets in there. It had a much higher ceiling than most of the caves we'd been in. You didn't have to stoop down. It was a command bunker. And almost as if he's getting some message, some divine message, Hartzell gets down and scoops all the paper away, and he's looking at the boards. Finally he gets all that out of the way, and there's the mummified remains of Kuribayashi's chief of staff.\n\n\"This was the man who had the honor of helping his chief with seppuku. Of course, knowing the importance of this, we back out. And Hartzell has found not Kuribayashi's diary but his combat journal, and that was a real treasure. We went right over to the airfield and told the air station commander of the maritime defense force there. He was a very nice fella. He couldn't speak English, but we showed him, opened it up, and he was dumbstruck.\n\n\"He called the Home Office, and they said they were going to send an aircraft down to pick it up and they wanted Hartzell to come with it. They wanted to honor Hartzell, which wasn't possible. I've seen him on a number of Iwo programs since he left the Marine Corps. I think he has some kind of official position in Okinawa at present.\n\n\"The last entry in the log was made the day Kuribayashi died. He had diagrams in there, exceedingly well done, with a fountain pen, for heaven's sake. The only thing we saw were diagrams of the defenses of Iwo Jima. It was in the possession of the chief of staff.\n\n\"How do we know the chief participated in seppuku? That brings up another very interesting question. They're now saying Kuribayashi died leading that last attack of the three hundred on the pilots, and I think all of this is a very obvious reconstruct or revisionist history. They want to somehow glorify Kuribayashi beyond the amazing man that he was. Yet virtually everything, sixty years of history, gives the account of him being killed by proper seppuku, and this chief of staff is mentioned in all this history, most of it Japanese.\n\n\"I have read in these various histories, which are translations, myself. I'm sure there's a degree of interpretation involved. I have never seen an actual Japanese document, and I'm not sure our Marine Corps archives would have it either. But over and over and over you see this account. The chief of staff killed himself. His body appeared to be undamaged, but there might have been a puncture somewhere.\n\n\"The thing which is the most irrefutable evidence of all is that Kuribayashi's body, which was a hot item, was never found. Every single body involved in that last attack was not necessarily identified, but they knew it wasn't Kuribayashi. Every single one. Because they were looking for him. And he wasn't there. Period. So that is the premier fact that refutes his involvement in that battle.\n\n\"We know the last attack originated in the naval forces area over by the Quarry. They came all the way down, heaven knows how, it had to be through tunnels. Because there was Hill 362 Charlie, Cushman's Pocket, the amphitheater, amazing fighting still going on over in those places. So how in the world did they manage to get past all these marines? The only way it can be understood is by tunnels. Because the actual fighting took place way over by the airfield on the other side of the island.\n\n\"Was Kuribayashi brilliant? No question. It's an overused term, but nevertheless it applies. He was a brilliant and a very determined strategist. He was chastised very heavily for his nondefense along the beach. This is all written in Japanese memoirs as well. His intent was to permit the landing force to get ashore, to let it get concentrated on the beach and then open up, which he did.\n\n\"Another point you never see written anywhere is this: If you stand at Suribachi to the south or at the Quarry to the north and look down that beach, you have linear plunging fire from any kind of weapon. He had both sides enfiladed from one end to the other, multiple lines of fire, and if he could hold that landing force right there, he could do a hell of a lot of damage, and he did.\n\n\"He got into a big brouhaha over that eight months beforehand. When he began to fortify the island we had not yet captured the Marianas. His superior was Lieutenant General Hideyoshi Obata, the Thirty-first Army commander, who was trying to get through the Marianas and back to Saipan in order to relieve Kuribayashi. Along comes the battle June the fifteenth, 1944, and he gets stuck in Guam. But his intent was to relieve Kuribayashi because he wasn't following bushido, the warrior's code. He refused to defend from the water's edge.\n\n\"That tells you Kuribayashi dug his heels in. He said, 'I know a hell of a lot about fighting and we damn sure ain't going to lose what paucity of people I have here by trying to defend at the water's edge, in open air.'\n\n\"Why weren't our Marine units annihilated on the beach? Several factors were involved. First of all the presumption of Kuribayashi and essentially the whole Japanese force was that we would, under fire, go to ground, dig in. Natural. That's what people do, go to ground. But that's not what we did. We kept moving. Our mentality at this point, from Guadalcanal all the way forward, was to get the hell off the beach.\n\n\"Well, this was one hell of a big wide beach. You couldn't get off just by running out of the boat, down the ramp and across the sand. It was too wide. He started that attack at ten-oh-three, fifty-nine minutes, one minute less than an hour after the landing force first came ashore.\n\n\"Another factor was we had almost two-thirds of the landing force ashore. That excludes Third Marine Division, which was still division reserve, but in that one hour's time we had gotten a hell of a lot of people on the beach. And Combat Team Twenty-eight, Fred Haynes's outfit, was assigned to get off Green Beach and cut the island. His battalion went straight across, in an hour and a half. They were by the airfield, not all the way across, but they were essentially unhampered by the bombardment.\n\n\"Those who had business to perform on the beach, beachmasters and mortarmen and artillerymen, were pretty much positioned, and their first reaction was to dig in. Unfortunately a number of them were lost because there was no way to dig in those damn terraces. They said that sand was like walking through wet coffee grounds.\n\n\"The Marine Corps never differentiated between the bombardment losses and the total losses, but I would say over half the losses were from the bombardment, anything twenty-five millimeter or greater. We could deal pretty much with infantry small arms, and we did. What Kuribayashi did was restrict the first hour to infantry small arms, and although we lost marines\u2014no question about it\u2014it was only when he started his bombardment an hour later that heavy casualties began to appear, and we're talking about not just dismounted infantry out of the boats, trying to get up the terraces. We're talking about the boats themselves; we're talking about ships, LSTs.\n\n\"He had hull-down tanks [buried up to the turret], we call them, which are not terribly effective. A tank weapon is designed for direct fire, so you can't really fire it like an artillery piece. It doesn't plunge. All of his eight-centimeter weapons, his big guns, he made the terrible mistake on D minus one of firing on the eight Navy gunboats, thinking that was the invasion. He fired on those gunboats, and we lost every single one, forty-three KIA, one hundred fifty-three WIA. Almost one hundred percent, all the gunboats were lost, and almost one hundred percent of the personnel were killed or wounded. These were wave guide boats, and they were the ones that accompanied the UDT, the underwater demolition teams, into the beach. They were spread all the way across the beach, two miles of beach, eight gunboats, and they accompanied the UDT in, and they were meant to protect.\n\n\"Kuribayashi thought this was the landing, so he unmasked his antiboat guns, and as a result, we knocked them out. The one we didn't knock out was one still standing at the bottom of Suribachi. I think he had eight or ten of those guns. The sacrifice of our boats saved a hell of a lot of lives. What if all those damn shore batteries had opened up on the landing force on D-day?\n\n\"But the biggest single reason our force was not annihilated goes to, we like to say, leadership, meaning gunnery sergeants: 'Get your ass off the beach! Get moving!' Small unit action at its finest. A very major factor. And if you bounce that off Normandy, they didn't. Those poor guys, especially the National Guard divisions, Twenty-ninth Division, they just went to ground, and they just got slaughtered because the enemy owned the heights. They just directed all their fire right smack down. These guys went to ground, and they were stuck. The only ones who had any degree of protection were the ones that had come far enough across the beach so they were masked from the cliff there.\n\n\"Another big irony was the sand, which intelligence had said would not pose a problem to men or machinery. It turned out to be a partial blessing in disguise because it absorbed shelling. You didn't get what we call point detonating rounds. That's a factor a lot of people don't understand. A point detonating round has all of its splash horizontal right when it hits. The round doesn't penetrate at all. The softer the soil, mud, or something like that, it's going to absorb part of that blast. These rounds, especially artillery rounds, even if they were point detonating rounds, still would have gone in deep enough so that almost the whole front end of the round would have been absorbed in that sand. That was a major factor.\n\n\"The biggest killer on the battlefield is the mortar. Not small arms. Overwhelmingly. Mortars got a bow end on 'em, and they don't penetrate. They generally hit pretty much vertically, and they stop right at the deck, and all that splash is horizontal. A sixty mortar's killing radius is fifteen meters, meaning anything within fifteen meters or forty-five feet is almost assuredly a casualty.\n\n\"Another thing to be aware of is this, and it's very important: There were no such things as snipers on Iwo, although you hear again and again that any man shot individually was hit by a sniper. A sniper is a trained marksman with a scope with a special weapon. Most of the so-called sniper losses on Iwo Jima were nothing more than a rifleman who happened to hit a guy. It's a good point because you hear it constantly, but these were nothing more than infantrymen doing exactly what they normally do, and they take out people.\n\n\"Kuribayashi opposed the attack on the United States. Another aspect of Kuribayashi that is not talked about and I'm not sure how it figures overall, but Kuribayashi was nobility. He was very highly regarded. He had access to the emperor, and that always causes problems with your superiors, and I daresay his were a bit jealous of him. He was sent to Manchuria. He spent most of the war in Manchuria. He was up there and went from there to China. The important thing about that is, these guys he brought to Iwo never had any really serious combat against marines in the Pacific. It was all against an inferior Chinese enemy.\n\n\"It [the slaughter in China] was one of the world's greatest atrocities when you stop to think about it, just the numbers alone and what they did to the Chinese. The Japanese forces up there were very arrogant, very self-assured, tremendously loyal to Kuribayashi. A bunch of them went to Iwo with him. The whole purpose of pointing this out is if you consider his presence in China in the light of the atrocities that took place there, which the Japanese have been very careful not to do, he would not be the hero that he is.\n\n\"They have managed to totally wipe over the fact that he was part of the Kwantum Army (Japanese armed forces in Manchuria), which participated in all that genocide. His reputation is almost exclusively Iwo Jima. If we were to make an effort to tie him to the Kwantum Army for the two or three years he was up there, he would probably have been hanged had he survived the war. That also has impact here. None of his troops had any experience in tough warfare such as the ones who had been in the Pacific from the very beginning. None of them. The Chinese were so poorly armed that it was murder. When they got to Iwo Jima, things were radically different.\n\n\"It was a great honor to be sent to Iwo Jima by the emperor, who personally wanted Kuribayashi to go there. That's when he wrote his letter to his wife and said, 'I shall not return. I know this is my last posting. Don't expect to see me anymore. Say hello to the kids.'\n\n\"Was it an honor to die? Sure. Still, having been washed in victory from all the time they were in China, it had to be a pretty stark change when they were sent to Iwo Jima knowing that this was the last stand. We do have some unit names and identification. They had the equivalent of eight infantry battalions, a tank regiment, two artillery and three heavy mortar battalions, naval infantry, a mixed brigade.\n\n\"Kuribayashi arrived at the East Boat Basin in June of 1944. They had been working on Iwo since 1938, fortifying it, building the airfields, getting it prepared. It was totally off-limits to anyone other than military. Kuribayashi got there, and he was unsatisfied with the defenses, and I think he was predominantly unsatisfied with surface defenses, how the artillery was unrevetted, weapons not dug in properly. So he declared Japanese soldiers had to do this in addition to what the Koreans were doing and had already done. He said everybody's got to work. He made a profound impact on the island's defenses.\n\n\"Rations up until the invasion were satisfactory, and that is a stretch, but it means everyone had a meal. Most of their supplies came down from Chichi Jima. They had an airfield there also. Of course everything was shut off once the battle started. Food was limited, and the soldiers received a half a cup of water per day. There's no natural water, but they had cisterns and runoff from rain, but even that was inadequate. It's amazing what you can survive on, and for the Japanese, it literally was survival. All or most of the reports you read about involving night attacks or night activity were Japanese going out trying to take water off American casualties. We put deep wells there when we left.\n\n\"Why is the battle significant? It is now considered the most important bloodletting of any battle of that size, of that scale, in the war. We consider seventy-five thousand marines in what was called the Fifth Amphibious Corps, consisting of three divisions, to be veterans of Iwo Jima. Of that number, twenty-five thousand became casualties, nearly seven thousand of whom were killed. Thirty-three percent of all the Marine Corps losses in World War Two took place at Iwo Jima.\n\n\"Of eighty-two Medals of Honor earned in World War Two, almost a third of them were earned on Iwo Jima. Of the total losses at Iwo Jima, eighty-two percent were solely marines. If you add the corpsmen and the doctors, we lost over three hundred corpsmen. We had fifteen doctors killed in action. These guys were brutally set upon the whole time they were ashore. There were three hundred sixty-five Seabees lost, and ships lost. The _Saratoga,_ the _Bismarck Sea,_ the _Lunga Point,_ all these were attacked by Kamikazes on D plus one. We lost the _Bismarck Sea_ damn near with all hands. Eight hundred went down. These are stunning figures when you stop to think about it.\n\n\"The writer James Bradley, whose father was one of the flag raisers, points out that by noon at Normandy, the day of the invasion, you could have had a picnic on Omaha Beach, whereas on Iwo Jima people were fighting and dying from the very beginning to the end, thirty-six days later. And an additional several thousand Japanese would be killed after we declared victory on the fifteenth of March, when they ran the colors up.\n\n\"That of course was subterfuge because Nimitz needed to get all that shipping back to put the force aboard to go invade Okinawa April 1. We'd lose another seven thousand people killed and wounded between there and the end of Iwo Jima, after Nimitz had determined the island was secure enough to pull the shipping out. It was a profound decision. Not only did he secure the shipping and take back . . . Well, the hospital ships left, a lot of the mat\u00e9riel that might have been used, might have been necessary, left.\n\n\"Of course we had the airfields in operation by then, and that was important because we could evacuate casualties by air and bring in critical material by air, but under no circumstances would that ever take place today, nor did it prior to Iwo Jima. You just would never do that. And with them went the Third Marines, the reserve. The Third Marines, that's probably the biggest remaining controversy. The Third Marine Regiment never went ashore. And they were asked for five times. Two division commanders and the force commander, General Schmidt, asked Holland Smith to please bring the Third Marines ashore.\n\n\"The kindest thing that has been said about that is Holland Smith expected we would be invading Japan and we needed to have some experienced troops to build a corps around. We had lost the equivalent of an entire Marine division in casualties. There were three divisions, but the Third Marine Division put only two regiments ashore, Ninth Marines and Twenty-first Marines. An entire regiment, the Third, was withheld.\n\n\"The biggest problem with that was the replacements. They went in there, honest to God, just like cannon fodder, because there were no real veterans there to show them how to do things. We lost replacements, I would say, on a scale of two to one compared to our veterans. Holland Smith's not permitting the force reserves, the Third Marines, to come ashore and enter the fight did several things: It relied too heavily on replacements who were untrained, grossly inexperienced, and not attached to the leadership. They had not bonded in their units the way they should have, and therefore their losses were horrendous. In addition, Smith's decision forced the units that stayed there to become even less combat-efficient, because they never got a break.\n\n\"If he had brought that additional regiment ashore, then we could have pulled units off the line and let them rotate and rest and get back in shape, move them in and out, which we always did in every other battle. That was simply not possible without a division reserve ashore.\n\n\"I view Iwo Jima as the signature battle in the history of the Marine Corps. There are several things that make Iwo Jima different. One, it was a tiny little three-by-five, eight-square mile island with two airfields on it, every inch of which had to be captured, every inch of which was contested. There was no such thing as Japanese falling back to another line, falling back and giving ground in order to stretch out the forces. There wasn't any consideration for surrender. They were a ferocious, violent, very, very formidable enemy that had one purpose: Kill Americans. There was no place to get out of their way to cover yourself from their fire. And they had an enormous amount of time to prepare the defenses for the island, which was not the case in several other battles we'd fought.\n\n\"Fighting and dying took place from D minus two (February 17) all the way through March and beyond. The last Japanese gave up in 1949.\n\n\"There were anywhere from eleven to seventeen miles of tunnels, also individual chambers, eight hundred of them, used as headquarters command positions, hospitals, barracks, living quarters, and so forth. In some of the tunnels we found mummified casualties lying on litters with intravenous drips still in their arms.\n\n\"The chambers were not interconnected. There would be a tunnel leading to this chamber, but it had a totally separate purpose. The tunnels were a combination of what we would call barracks or a position for troops to stay. The chambers, on the other hand, had a very separate purpose. When I went down in the artillery position on Nishi Ridge, the tunnel there was surrounded by maybe fifteen chambers, primarily for ammunition storage. Then there were spokes off that for the troops who supported that position.\n\n\"There were three levels of tunnels. I have a diagram of Hill 362 Able, and there are three separate levels thirty and forty feet down. Everyone asks about heat, and I did as well when I first went there. It is ferocious, worse than a damn sauna. I finally figured it out: They found a way to ventilate them, but that was closed off when we got there. We got the Japanese diggers to open one up, and all of a sudden we could feel this breeze coursing through. And that's the only way you could possibly survive in there.\n\n\"They used ladders to reach the different levels. There's a funny story about one of them. I was coming down a slope with no steps, just a carved slope. This was volcanic rock and sandstone, easy to shape or sculpt. This ramp went down, and at the bottom there was a vertical descent of maybe twenty feet, a big hole, and I could see the tunnel below. Their ladder was gone by this time, but I had us all roped up, and when we got to that point, I just did a seat rappel down to the tunnel.\n\n\"When I came back a month or so later, I had the commanding general with me, Lieutenant General Norm Smith. And he didn't want to rappel; he wanted to jump. I said, 'General, we don't know what's in these damn things. There's still ordnance in here.' Well, he still wanted to jump. The surface of the lower tunnel was sort of pebbled. I was down there and shined the light up so he could see when he jumped. He landed, and his foot, the arch of his foot, was right on a Japanese grenade. His boot was right on that damn grenade. And of course the fuse sticks out of the upper end, and it's a pressure detonator. They'd bang it against their helmet, and it would start the timing chain.\n\n\"And I said, 'Don't move.' I got the grenade out from under his foot. So the damn things were full of ordnance. Those caves actually went in one direction to a huge cliff face that looked over Hill 362 Able and Nishi Ridge itself. I have identified that as the location where most of Combat Team 2-28, all the heavy losses, all the flag raisers, were killed in that area.\"\n\nI raised the question of atrocities committed by marines during the battle and told Colonel Ripley of one veteran who related to me how he had come home with a jarful of gold-studded teeth and dried ears he had cut off Japanese soldiers. He had said it was not uncommon for those back at Camp Lejeune to tell marines departing for the Pacific, \"Bring us back some teeth and ears.\" Richard Nummer knew of a marine who collected teeth, and Sergeant McPhatter told of marines driving prisoners into the sea and shooting them.\n\n\"Ears didn't become popular until Vietnam, believe it or not, but teeth definitely. As for actual torture, I would say without question these things probably happened once or twice, but marines have a different focus. We're here to kill you and get on with the next guy and the next guy. We're not going to sit here and play cat and mouse with a casualty. It doesn't make any sense. I'm not saying it doesn't happen, but I am saying it is rare because there are too many other things we are dealing with. Now, at Iwo Jima, the Japanese atrocities that are probably more prominent to me involve James Bradley's example of his father's good friend Ignatz, Iggy, who was hideously tortured and mutilated, and then we think Bill Genaust was probably tortured. But he was shot. They saw him get shot. There wasn't any question that he was down. See Norman Hatch, [Chapter 13].\n\n\"I would say there are eighteen thousand soldiers still entombed on Iwo Jima, which the Japanese naturally consider a shrine. An easy way to determine for sure would be to find out how many they've shipped back, but I'd be surprised if they've even reached five thousand yet. [Some reports say eight thousand have been recovered.] There's the ossuary, where they store bones, on that little peak just over from 362 Charlie and right in front of Cushman's Pocket. Over the last five years they've built a massive memorial right next to the ossuary. The whole hill is now a very sacred spot. Even Japanese citizens are not permitted to go to the island unless they are the father, mother, or brother of someone who served. This is the Japanese definition of 'immediate family.'\n\n\"Of course Japanese are on the island full-time, probably not much more than two hundred. And American carrier pilots fly over from Japan and stay there while they practice touch-and-go's and maneuvers that wouldn't be possible over Japan because of airspace and noise restrictions. We have an American barracks there that will accommodate two hundred. That took place after I left, but I'm pretty proud of that because I'm sure all the work we did made it amenable to the Japanese.\n\n\"Our flag flies one day a year. The marines bring it. We are not permitted to leave it there when we leave the mountain. We have to take it down.\n\n\"The Japanese permit Military Historical Tours in Arlington, Virginia, to bring visitors one day a year, usually in March. A support element comes ashore with vehicles and tents and things solely to support this day, which is referred to as a Reunion of Honor. There was a lot of talk in the Marine Corps that after the sixtieth anniversary we'd have to stop doing that because our business now is war [in Iraq] and we can ill afford to be bringing that element over to Iwo Jima. The Navy complains about the shipping. 'We got a lot of things to do, and we don't want to have to bring all you guys to Iwo Jima.' Three ships generally. I would say there is constant complaining about it. One thing indisputable is the Japanese will not permit any other organization, so it's not as if they were beating out the competition.\n\n\"The reason we do it is everyone knows that Iwo Jima is _sanctum sanctorum_ to the Marines, and the value that we get out of bringing those marines there, mainly from Okinawa, is so enormous that the commandant every single year says, 'Hey, we're going to continue this as long as we can.' For current and old marines. There's no frills associated with this, and it's the only way these veterans can get back to Iwo Jima. Even so, it would be hard to justify if we were just doing it for these old warriors. We love it, but it'd be damn hard to justify that kind of commitment by itself. But we can clearly justify it for the morale, the reenlistments, the tremendous lift these marines get by getting to go to Iwo Jima and support this reunion.\n\n\"Misconceptions? The main one, after you clear up the death of Kuribayashi, would be the battle for Suribachi. You constantly see references to the bloody battle for Suribachi, and it's always the flag raisers fighting their way up the mountain, fighting their way down, with terrible losses and so on. Well, the battle for Suribachi was at the bottom, and it was a hell of a fight: Nine hundred marines out of one regiment were lost in those first five days. In fact, the most decorated platoon in the Marine Corps was Third Platoon Easy 2-28 that took the colors up the mountain. A Medal of Honor and two Navy Crosses, just getting their way clear to get up the mountain. The greatest Marine of all, the only surviving flag raiser in 2007, was Chuck Lindberg, a flamethrower [Chapter 14]. He won a Silver Star that day.\n\n\"So although there was some action at the top of Suribachi, it was the three days prior to the actual patrol that went up on the fourth. The predominance of Suribachi's threat was really at its base. The USS _Pensacola_ came in there and was whacked right smack in the bridge with an artillery shell fired from Suribachi, killing fifteen men, including the skipper, so Suribachi was lethal, a very tough nut. But it was not a tough trip when the very first patrol, Sergeant Watson's patrol, and Lieutenant Schrier's final flag raising patrol, which was a whole platoon, went up there. There are a billion myths and inaccuracies associated with the flag raising. Over a thousand people have laid claim to taking part in it.\n\n\"Talk to any veteran of Iwo Jima, and he's going tell you about his fight and the importance of his fight and the brutality and the difficulty, and, in other words, we had it far worse than anyone else. And you know what? Every single one of these guys is right. Their fight was tough. There wasn't any such thing as an easy run on Iwo Jima. The way that plays out now is the Fifth Division gets so much attention because of the flag raising, rightly or wrongly.\"\n\nLatter-day revisionists have contended that the battle for Iwo Jima was wasteful and unnecessary and that only a small proportion of the 2,251 landings by B-29s afterward were for genuine emergencies.\n\n\"One of the things that such contentions overlook is that we lost seventy-seven Superforts before we even took Iwo, flying out of Tinian or Saipan straight to Tokyo and back. We never anticipated that kind of volume. Although American industry was magnificent in war production, they certainly couldn't sustain those kind of losses. Frankly the biggest issue with Iwo Jima was early warning, so Tokyo knew they were en route, and more important, there was the fighter cover, the CAP, the Combat Air Patrol, that they could put up from Japan to intercept the bombers. This originated when they saw them going over Iwo Jima.\n\n\"As to talk about how only a small portion of the two thousand two hundred fifty-one landings were genuine emergencies? Bullcrap! A landing for whatever reason is an emergency landing. Why the hell touch down on Iwo Jima unless you need to go to Iwo Jima? That wasn't their home base; they would have achieved nothing by being there. There was no room on the ramp for all these huge damn B-29s. They had to get back to the Marianas, had to get back to Tinian.\n\n\"So to hear some latter-day bobby-sox historian come along and say we didn't need Iwo Jima is just an ultimate insult. It makes no sense. I could go on and on with other examples, but the truth is this was nothing more than an academic exercise by some researcher poring over old unit archives and trying to make a case where none exists. Academics are the only ones that support this, meaning some professor says, 'Aha! This is recently revealed information.' They have no combat experience or understanding of our war aims, our strategy at the time. And then sixty years later they take the entire battle out of context and try to find fault with its underlying importance to victory, and even those who fought the battle! There's nothing new in here.\"\n\nA question to Colonel Ripley on whether we were correct in returning the island to Japan led me to the following letter, which he wrote to a friend, Ross Mackenzie, on the night of November 17, 1987, from the summit of Suribachi.\n\n_Dear Ross:_\n\n_From this most unlikely spot I am inspired to write you for reasons I can't fully explain. Certainly you have received no other letters from here I would wager, and you may find this interesting. It's the middle of the night\u2014cold, windy, uncomfortable & profoundly moving. I'm looking down on a tiny island three miles wide and five miles long. Down there, and here where I'm writing by flashlight, over 7,000 Marines died. The mountain is Suribachi, the island, Iwo Jima. Of the hundreds of thousands of words written about this place, nothing close to describing its starkness, its inestimable cost and now, sadly, the poverty of its abandonment._\n\n_The entire island is a shrine, mostly Japanese, but a few Americans\u2014only a few. Americans don't seem to care about such things when, as is the case here, it's inconvenient. And yet this island, its name and most especially this very spot where I sit\u2014where the flag was raised\u2014is immortalized in our national consciousness for as long as there is an America._\n\n_The debris and detritus of war remain even after nearly 43 years. Rusty vehicle hulks, wrecked boats, sunken ships, canteens, mess kits, thousands of rounds of corroded ammunition, blockhouses, pillboxes, trenches, abandoned airfields, large naval shore guns, artillery, etc. And beneath my feet remains of 22,000 Japanese defenders, brave men who died at their posts; hated then, respected now._\n\n_Rupert Brooke said it perfectly: \"Here in some small corner of forgotten field will be forever England.\"_* _And this brutally stinking sulfuric rock, depressing to see, demoralizing as it has lost its once vital importance and our nation's once great concern, will be forever America. It will be forever in the memory of those 75,000 Marines who fought here, the 25,000 who suffered wounds here and the 7,000 who gave their blood and lives to its black soil. Again Rupert Brooke. . . . \"In that rich earth, a richer dust concealed.\" Their hopes, their happiness, their dreams ended here. And if we fail to honor them in our memory and our prayers, we should be damned to hell for such failure._\n\n_I brought a small team here to survey the island for future exercise use. The Japanese would prefer we did not exercise here, but that will be over my dead body. I find it hard to believe (and impossible to accept) that our government gave the island back to them. It's as if we gave them Gettysburg or Arlington National Cemetery. Americans died in such numbers here that in nine and a half months, had the battle lasted that long, it would have equalled our losses of 10 years in Vietnam._\n\n_The Marine Corps must never lose its right to exercise here, and I'm damned proud of having something to do with assuring that will be so._\n\n_Yours aye,_\n\n_John_\n\nIn June 2007 Japan announced that it was going to restore to the island its original name, Iwo To, which means essentially the same thing as Iwo Jima, or Sulfur Island. I asked Colonel Ripley what he thought about that.\n\n\"My feeling is that the name change, perhaps important to the Japanese, is disingenuous and ultimately will make no difference. The Burma Road is still the Burma Road, and all of Napoleon's battle sites bear their original names despite the renaming of each locale. This is replicated worldwide.\n\n\"Many years ago the island center in Micronesia called the Gilberts decided to change the name of our famous Tarawa to Kiribati. Well, it has made no difference whatsoever as we, and everyone else, still call it Tarawa. By the same token, Iwo Jima is a hell of a lot more than just a Japanese piece of property. Changing the name sure as hell won't change the history\u2014especially recent history\u2014of the island. It will forever be Iwo Jima in the hearts of all marines and the American public as well, for all time.\"\n\n* * *\n\n*Brooke's original lines, from \"The Soldier,\" read:\n\n_If I should die, think only this of me:_\n\n_That there's some corner of a foreign field_\n\n_That is for ever England._\nAPPENDIX\n\nMilitary Terminology\n\n**amtrac:** Tracked amphibious vehicle used to transport the first waves of the invasion on Iwo Jima. Called an LVT for \"landing vehicle, tracked,\" it was eight feet wide, nine feet tall, and twenty-seven feet long.\n\n**banzai:** Usually suicidal infantry assault, which the commander of the Japanese forces refused to employ on Iwo Jima. Ironically, had Kuribayashi ordered a banzai attack on the first night of the invasion, it may very well have succeeded, because the landing forces were so disorganized.\n\n**BAR:** Browning automatic rifle. This was the principal rapid-fire weapon for a regular infantry squad. It was a gas-operated .30-06 caliber and could be fired via tripod or standing up. Basically a \"walking machine gun,\" its twenty-round magazine could be emptied in three seconds. Each BAR man had an assistant who carried ammunition for it. The .30-caliber round was interchangeable for the M1 rifle, the BAR, and the light and heavy machine guns, according to Mike Mervosh, who was on Iwo Jima for the entire campaign.\n\n**bazooka:** Common name for the tubular 2.36 rocket launcher, fired over the shoulder, used against tanks or fortified positions. The 2.36 refers to the diameter of the shell. Each platoon had at least two. The assistant inserted the shell from the back, then tapped the gunner on the shoulder. Effective range was about one hundred meters.\n\n**Betty:** Principal heavy bomber of the Japanese Navy, the G4M was referred to as the Betty by the Allies. It was remarkable for its long range, achieved by depriving it of armor and equipping it with huge fuel tanks in the wings. Because it tended to catch fire readily when hit, it acquired the nicknames One-Shot Lighter and the Flying Cigar.\n\n**boondockers:** Marine field shoes, usually worn with canvas leggings.\n\n**boot:** Marine recruit.\n\n**bulkhead:** A wall.\n\n**C2:** A puttylike plastic explosive that could be molded by hand; favored for use in satchel charges. A fuse set off a blasting cap that caused the C2 to explode.\n\n**cannon:** A large, heavy piece of artillery of four basic calibers: the .57 millimeter, which could be dragged up to the line of combat by hand; the .75-millimeter pack howitzer, which could be broken down and carried by several marines; the 105; and the 155. The .57 millimeter could fire ten rounds a minute; it employed mostly canister, small bullets in one shell that spread out like shotgun pellets but with much more lethal effect.\n\n**carbine:** This was a .30 caliber rifle with a fifteen-round clip. It lacked the stopping power of the M1.\n\n**Cincpac:** The sort of acronym loved by the military, this one stood for Commander in Chief, Pacific Command.\n\n**Corsair:** Single-engine Marine fighter plane.\n\n**D-day:** Debarkation (invasion) day.\n\n**deck:** The ground or the floor.\n\n**dog face or doggie:** Soldier in the U.S. Army. Negative term used by marines to label Army counterparts.\n\n**DUKW:** Manned by black soldiers and also white marines in separate DUKW companies\u2014this amphibious vehicle, pronounced _duck,_ was fifteen by eighteen feet, with four-wheel drive, rubber tires, and a propeller that drove it through the water. One company of Army DUKWs was assigned to each Marine division on Iwo.\n\n**ear-banger:** Someone who seeks to curry favor with a superior.\n\n**flamethrower:** A portable weapon carried by hand or fired from a (Zippo) tank, projecting an incendiary fuel such as jellied gasoline or a mixture of high-octane gas and diesel fuel. The hand-carried version weighed seventy-two pounds. It held five gallons of fuel, which burned up in about seven seconds.\n\n**gung ho:** Based on a Chinese phrase that means \"to work together,\" this phrase was adopted by the Marines to describe an overly zealous member of the Corps.\n\n**hand grenades:** There were three types: fragmentation, incendiary, and smoke. After you pulled the pin\u2014by hand and never with your teeth\u2014and released the lever, you had three to five seconds prior to explosion. It was customary to let two seconds go by before throwing the device, so it could not be flung back by the enemy. Mike Mervosh said he threw more grenades than he fired bullets. The effective radius was about ten yards. The Japanese version was activated by rapping it on one's helmet or another hard object.\n\n**hatch:** A doorway.\n\n**H hour:** The hour at which an operation begins.\n\n**Higgins boat:** The flat-bottomed landing craft devised by Andrew Higgins of New Orleans. Various types included the LCI (landing craft, infantry), LCT (landing craft, tank), the LCVP (landing craft, vehicles, personnel), and the LCM (landing craft, medium). The LVT was the amphibious tracked ship-to-shore vehicle known as the amtrac.\n\n**house apes:** Children.\n\n**Ka-Bar:** a solid, six-inch fixed blade knife carried by virtually every marine, primarily for combat but also useful for opening K rations.\n\n**KIA:** Killed in action.\n\n**line of departure:** A suitably marked offshore line intended to coordinate landing craft so they could land on designated beaches according to predetermined schedules.\n\n**LP:** Listening post.\n\n**M1 Garand:** The infantryman's favorite weapon, it was a gas-operated .30-06 caliber with an eight-shot clip. Called by General George Patton \"the greatest battle implement ever devised,\" it could be fired repeatedly by just pulling the trigger. With its sixteen-inch bayonet attached, the rifle weighed almost ten pounds. \"You snapped the bayonet on the end of the muzzle when you hit the beach, and you put it on when you were snooping around caves,\" Mike Mervosh said.\n\n**machine gun:** There were two basic types: a .30 caliber, air-cooled version that could be fired standing up or on a ten-pound tripod (each gunner had an assistant to help carry the gun and the webbed canvas belt that held 250 rounds) and the .30 heavy version, which was water-cooled and rested on a fifty-one-pound tripod. It was brought up at night for perimeter defense. Each gun was manned by seven marines, including four ammo humpers. Two machine guns were assigned to each platoon. The water-cooled gun weighed ninety-one pounds.\n\n**Maggie's drawers:** Red flag or disk raised at a firing range to indicate the shooter has completely missed the target. It was said to come from a 1940s song entitled \"Those Old Red Flannel Drawers That Maggie Wore.\"\n\n**Marine units:** A division included 20,000 men, consisting of four regiments, engineer, pioneer, tank, service, motor transport medical, amphibian tractor battalions, signal and laundry company, a war dog platoon, an observation squadron, and two replacement battalions. A regiment consisted of 3,300 men, broken into three battalions. A battalion numbered 1,100, consisting of three assault companies of about 250 each, plus a headquarters company and various support units. A rifle platoon held 45 men, 43 marines and 2 Navy corpsmen. Mortar platoons were smaller, 18 to 20, and machine-gun platoons held up to 56 men.\n\n**mortar:** a short smooth-bore gun for firing shells at high angles. The sixty-millimeter mortar was manned by six: the gunner, his assistant, and four carriers. It was muzzle-loaded and could be elevated to forty-five degrees. It was very accurate from three hundred to seven hundred yards, and it would reach as far as eighteen hundred. There was also the eighty-one-millimeter mortar.\n\n**mustang:** Enlisted man who leaves those ranks to become an officer.\n\n**Nambu:** Japanese light machine gun.\n\n**noncom:** Noncommissioned officers are the enlisted men, especially gunnery sergeants and the like, who really enable the Marine Corps to function.\n\n**Quonset hut:** Building made of corrugated metal with semicircular cross section, first manufactured at Quonset Point, Rhode Island.\n\n**rifle grenade:** An adapter on a rifle muzzle could launch this grenade up to 150 yards, whereas 30 yards was about maximum for one thrown by hand.\n\n**salt:** A marine who has been in the Corps a long time; an old hand.\n\n**satchel charge:** A hand-carried charge of dynamite or C2 or C4 plastic explosive that could be thrown once a flamethrower had been used to blast the occupants in a cave into retreat.\n\n**scuttlebutt:** Drinking fountain gossip or rumor.\n\n**slop chute:** Enlisted men's bar.\n\n**Springfield '03:** The principal combat rifle used before the M1 Garand. It was a .30-caliber bolt action, which meant each shell had to be ejected manually. The M1 was semiautomatic, meaning it could be fired repeatedly by simply pulling the trigger. The Springfield held a five-round clip.\n\n**squadron:** Consisted of two or more groupings of aircraft or divisions of ships.\n\n**swab jockey:** Any sailor, also referred to as a swabby, squid, or anchor clanker. The term was not complimentary.\n\n**Thompson submachine gun:** This was a hand-held .45-caliber submachine gun equipped with a stick magazine that held twenty rounds. It was useful in close combat at point-blank range; otherwise it was not much good. It was favored by tankers and artillery men.\n\n**Tojo:** Single-seat, single engine aircraft used to intercept B-29 bombers. With its poor visibility on the ground, weak armament, and high landing speed, it was generally disliked by its pilots. This was the kind of plane shot down by Lieutenant Robert Merklein. Tojo was also the name of Japan's wartime prime minister.\n\n**tracer:** Bullet with a phosphorous coating, designed to burn in flight to provide visual indication of trajectory.\n\n**weapons carrier:** A one-ton vehicle, manufactured by Dodge, that looked like an oversized pickup truck. It was designed to carry its crew and mortars or machine guns.\n\n**Weasel:** Small tracked vehicle used for towing .57-millimeter cannon and hauling supplies to advanced areas, among other things.\n\n**WIA:** Wounded in action.\n\n**WW:** Walking wounded.\nACKNOWLEDGMENTS\n\nThis book could not have been written without the splendid help and advice of the Medal of Honor recipient Colonel Harvey H. \"Barney\" Barnum, USMC (Ret.); the kindness and courtesy of Colonel Warren Wiedhahn, USMC (Ret.), of Military Historical Tours; and Colonel John W. Ripley, USMC (Ret.). I was led to its subject by Sergeant Major Mike Mervosh, USMC (Ret.), and some of his Iwo Jima comrades, all of whom were gracious, accommodating, and helpful. In addition to donating their time and their invaluable memories to this book, many of the men gave me permission to reproduce pictures from their personal collections and allowed me to photograph them. I am grateful for their generosity. I also wish to thank Gerry Byrne of _Parade,_ John Butler, Keiko Hirano, Masahide Mizoguchi, Rosa Ogawa, Lee Kravitz, and Lou Leventhal, for helping me with facts, and Jonathan Au, for rescuing vital copy that seemed irretrievably lost in the computer.\nALSO BY LARRY SMITH\n\n**B EYOND GLORY:**  \n _Medal of Honor Heroes  \nin Their Own Words_\n\n**T HE FEW AND THE PROUD:**  \n _Marine Corps Drill Instructors  \nin Their Own Words_\n\n**T HE ORIGINAL:**  \n _A Novel_\nCopyright \u00a9 2008 by Larry Smith\n\nThe poem on page 295 by Hitomaro, translated by Kenneth Rexroth,  \nfrom _One Hundred Poems from the Japanese,_ copyright \u00a9 1955 by New  \nDirections Publishing Corp. Reprinted by permission of New  \nDirections Publishing Corp.\n\nThe extracts on pages 298\u2013299 from _So Sad to Fall in Battle_  \nby Kumiko Kakehashi, translation copyright \u00a9 2007 by Shinchosha  \nCo., Ltd. Copyright \u00a9 2005 by Kumiko Kakehashi. Used by permission  \nof Presidio Press, an imprint of The Ballantine Publishing Group, a  \ndivision of Random House, Inc.\n\nAll rights reserved  \nFirst Edition\n\nFor information about permission  \nto reproduce selections from this book,  \nwrite to Permissions, W. W. Norton & Company, Inc.,  \n500 Fifth Avenue, New York, NY 10110\n\nFor information about special discounts for bulk purchases,  \nplease contact W. W. Norton Special Sales at  \nspecialsales@wwnorton.com or 800-233-4830\n\nBook design by Charlotte Staub  \nProduction manager: Julia Druskin\n\nThe Library of Congress has cataloged the printed edition as follows:\n\nIwo Jima : World War II veterans remember the greatest battle  \nof the Pacific \/ [edited by] Larry Smith. \u2014 1st ed.  \np. cm.  \nISBN 978-0-393-06234-2 (hbk.)  \n1. Iwo Jima, Battle of, Japan, 1945\u2014Personal narratives,  \nAmerican. I. Smith, Larry (Larry Earl)  \nD767.99.I9I95 2008  \n940.54'2528\u2014dc22\n\n2008001301\n\nISBN 978-0-393-33491-3\n\nISBN 978-0-393-28563-5 (e-book)\n\nW. W. Norton & Company, Inc.  \n500 Fifth Avenue, New York, N.Y. 10110  \nwww.wwnorton.com\n\nW. W. Norton & Company Ltd.  \nCastle House, 75\/76 Wells Street, London W1T 3QT\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n#  \n\n#  \nCopyright \u00a9 2018 by Erica Sage\n\nAll rights reserved. No part of this book may be reproduced in any manner without the express written consent of the publisher, except in the case of brief excerpts in critical reviews or articles. All inquiries should be addressed to Sky Pony Press, 307 West 36th Street, 11th Floor, New York, NY 10018.\n\nThis is a work of fiction. Names, characters, places, and incidents are either the products of the author's imagination or used fictitiously.\n\nSky Pony Press books may be purchased in bulk at special discounts for sales promotion, corporate gifts, fund-raising, or educational purposes. Special editions can also be created to specifications. For details, contact the Special Sales Department, Sky Pony Press, 307 West 36th Street, 11th Floor, New York, NY 10018 or info@skyhorsepublishing.com.\n\nSky Pony\u00ae is a registered trademark of Skyhorse Publishing, Inc.\u00ae, a Delaware corporation.\n\nVisit our website at www.skyponypress.com.\n\nwww.EricaSageBooks.com\n\n10 9 8 7 6 5 4 3 2 1\n\nThe Library of Congress has cataloged this book as follows:\n\nNames: Sage, Erica.\n\nTitle: Jacked up \/ Erica Sage.\n\nDescription: New York : Skyhorse Publishing, [2018] | Summary: Sent to Jesus Camp by his nonreligious parents, Nick gains a new perspective on his sister's suicide and the secret he has been keeping about it, with the help of new friends and the ghost of Jack Kerouac.\n\nIdentifiers: LCCN 2018001603 (print) | LCCN 2018009408 (ebook) | ISBN 9781510730069 (eb) | ISBN 9781510730052 (hc) | ISBN 9781510730069 (ebook)\n\nSubjects: | CYAC: Camps--Fiction. | Grief--Fiction. | Christian life--Fiction. | Brothers and sisters--Fiction. | Kerouac, Jack, 1922-1969--Fiction. | Humorous stories.\n\nClassification: LCC PZ7.1.S242 (ebook) | LCC PZ7.1.S242 Jac 2018 (print) | DDC [Fic]--dc23\n\nCover design by Kate Gartner\n\nCover illustration by Pete Ryan\n\nPrint ISBN: 978-1-5107-3005-2\n\nEbook ISBN: 978-1-5107-3006-9\n\nPrinted in the United States of America\n\nInterior design by Joshua Barnaby\nThis is for you, Mom.\n\nI miss you like mad.\n\nSUNDAY\n\nJack Kerouac was sitting cross-legged on the end of my bed. Again.\n\n\"BOO!\" he said.\n\n\"I hate it when you do that.\"\n\n\"I didn't scare you, Nicolas,\" he said, skipping across my name and dropping the s. I rarely heard his accent, except at the end of some sentences. Or when he was cursing me up and down in French.\n\nHe peered at me with his glacier-blue eyes and took a drag on a cigarette.\n\n\"I asked you not to smoke in here,\" I said, reaching for the cigarette. But there was nothing to grab. Not a cigarette, not a hand.\n\n\"I opened the window,\" Jack said.\n\n\"My parents are going to think I'm sneaking cigarettes and giving myself lung cancer.\"\n\nHe looked at me, holding the cigarette loosely between two fingers. \"Tell them the truth, then.\"\n\n\"Yeah right.\" I picked up the cheap blue pen from my nightstand and the mini composition book underneath it, and wrote down the date. Jack Kerouac, the great American author, had been dead for fifty years, but this was the twenty-seventh time he'd visited me in three months. I put the notebook back.\n\nJack looked over my shoulder at the radio on my dresser. It popped on and rolled across stations, settling on jazz. Always jazz. \"I like this cat.\" Jack bounced his leg across his knee. \"Charlie Parker.\"\n\n\"Just don't turn it up so high this time.\"\n\nThe music got louder, the saxophone crying like a lone bird.\n\n\"Knock it off.\" I turned around and jabbed at the off button. I sighed. \"I thought you were going off to work on a new book or something.\" At least, that's what he'd said the last time, accusing me of taking up too much of his time, saying I didn't understand anything about beauty or art, certainly not the truth! All I cared about was rules and correcting everybody. You'll grow up to be a gawddamn editor one day! And then he stomped himself out in a fury, taking his near-empty bottle of whiskey with him.\n\nNow, he snuffed his cigarette out in the ashtray on my windowsill and rolled back onto my bed. \"When are you going to take that down?\" He nodded at the poster hanging above my pillows and began reading it aloud mockingly. \"'The only people for me are the mad ones, the ones who are mad to live, mad to talk, mad to be saved, desirous of everything at the same time, the ones who never yawn or say a commonplace thing, but burn, burn, burn like fabulous yellow roman candles exploding like spiders across the stars.'\"\n\n\"I'm not going to take it down, I told you,\" I said.\n\nHe propped his boot on the poster. \"Gawddamn book,\" he murmured, and ground the sole of his boot across his signature at the bottom like it was the butt of a cigarette. We'd been arguing about the poster ever since he showed up the night of my sister Diana's funeral. I'd been scared shitless that first time\u2014confronted by a brooding drunk man. But now I was just irritated because he wouldn't go away.\n\n\"Well, everyone else seems to love the book,\" I said.\n\n\"Poseurs,\" he said.\n\nI'd never read On the Road, so I didn't care so much about the book or the fact that Jack Kerouac, the guy who wrote the thing, hated the poster. All I knew was that it was \"an American classic,\" but not the kind teachers would ever assign. The kind teenagers read to piss their parents off and prove to the world how rebellious they were.\n\nMy sister had given me both the poster and the book. I didn't plan on reading the book, not even her paperback copy, the cover bent and rubbed pale by her hands after years of rereading. I didn't care about the 1950s or the \"Beat Generation.\" I didn't care about Jack Kerouac, some icon of nonconformity and road trips and wild nights. Some author who drank himself to death under the weight of his success, like every other literary great who put her head in the oven or a shotgun to his head. It was clich\u00e9d. Jack Kerouac and every teenager's life-changing experience with On the Road was clich\u00e9d.\n\nBut I wasn't taking the poster down.\n\nBecause I did care about my sister.\n\nI missed her like hell.\n\nI tried to shove Jack's leg over to make room for myself on the bed, but lost my balance instead. There was nothing to actually shove.\n\nJack smirked. I resigned myself to the edge of the mattress and tried to ignore him.\n\nHe folded his hands behind his head, and I knew what was coming. \"Is today the day?\"\n\n\"No. I keep telling you.\" Why couldn't it be someone funny like Mark Twain or Voltaire showing up in my bedroom? Why did it a have to be a moody, nagging alcoholic?\n\n\"Confession is good for the soul,\" Jack said.\n\n\"Is that what your nuns at Catholic school said?\" I mocked.\n\nThe jazz clicked on again behind me. Maybe the same song, maybe different, but now the saxophone battled with a piano.\n\nI glared at him. \"Or was it the Jewish kids at Horace Mann?\"\n\n\"Are you making fun of me?\" He asked as the volume went up. \"Or the Jews?\" And up.\n\n\"Don't,\" I hissed.\n\nAnd up.\n\n\"Seriously, you're pissing me off.\"\n\n\"Am I making you mad?\" He winked at me, but without a smile. \"The burning-madness of it.\"\n\nThe jazz blasted. I plugged my ears.\n\nJust then, my door swung open, my dad on the other side.\n\nThe saxophone cried.\n\nMy dad jabbed at the radio till he succeeded in turning it off. \"We need to talk.\"\n\n\"I just like the song, it's fine.\" I insisted, \"I'm fine.\"\n\n\"It's jazz, Nick, not gangster rap.\" He sighed. \"Your mom and I just need to talk to you.\" He set his hand on my shoulder and led me out of my room.\n\nI turned around to f lip Jack Kerouac the bird, but he'd slipped out the window, the way he sometimes did.\n\nA few minutes later, my parents had me trapped in the living room, dreading whatever was coming next. They sat across from me on the couch, my mom's legs tucked neatly against my dad's, her hand in his, her eyes brimming with anticipation. It looked happy, this scene. It looked all Norman Rockwell. But I'd heard their arguments. I'd heard the blame. That's what tragedy does to families. It holds a f lame to what binds you, and sometimes you learn what you thought were rods of steel are actually mere threads.\n\nI was the only thread they had left, and I knew how thin it was.\n\nI stared. I'd figured they'd pulled me out of my room to discuss the too-loud music, or the liquor bottle my mom had found the week before (I'd snuck it in for Jack Kerouac, of course), or just the fact that I'd holed up in my room for too many days.\n\nBut no. We were on to Phase Three of the Fix-Nick-Quick Plan. If we could fix Nick, maybe that would mean this family wasn't broken.\n\nBut they didn't know. They couldn't know.\n\n\"It's already paid for,\" my dad announced, like a sports commentator, all loud and certain. But he was rubbing his thighs nervously with his palms. This had clearly been my mom's idea.\n\nCounseling had been my dad's idea, back in April. And I'd done that. (We'd all done that.)\n\nBut Jesus camp?\n\nHow was a week with zealots going to cure my \"depression,\" as the psychologist had diagnosed it? (Isn't everyone depressed when their sister dies? Does that need a diagnosis?)\n\nThe prescription had been the doctor's idea, back in May. And I'd done that too. (I think my mom was still doing that.)\n\nBut I was not doing a wrathful God and his hobo-hippie son.\n\n\"I don't want to go,\" I said, staring down at the brochure my dad had handed me.\n\nEden Springs, the camp was called. WHERE HAPPINESS HAPPENS!\n\n\"I'm already happy,\" I said. \"I'm fine.\" And I cared about God as much as He cared about me, which didn't seem like a whole lot.\n\n\"Oh, honey.\" My mom slid from the couch and squished into my chair next to me. \"We know you're happy, but we also know you're sad.\"\n\n\"I'm fine, Mom. And besides, I can't do a week with these people.\" I f lipped open the brochure. \"They're not even real. Look at them. They're like mannequins. It's going to be me and a bunch of plastic dolls, like, praising the Lord or whatever. Like...\" I straightened my arms and then bent them in perfect right angles at the elbows. \"Like Barbie, you know? 'Hallelujah,'\" I said in a girly voice, lifting my arms straight into the sky. \"'Hallowed be thy name.'\"\n\nMy parents stared at me gloomily.\n\n\"You know how you have to move Barbie's arms and stuff?\" I tried again. I lifted my arms straight this time.\n\n\"We get it, Nick,\" my dad said.\n\nMy mom slid back to the couch with my dad. Side by side, they looked like an advertisement for a leading detergent. All pastel and khaki.\n\n\"Look, we don't even go to church,\" I reasoned. Sure, there was a Bible in the living room. But it was on the top shelf, way out of reach, next to Beowulf, The Tao of Art, and a book of poetry by some Spanish guy. Like part of the foreign language section.\n\nThe only reason our Bible was even on my radar was that I'd heard on the bus in elementary school that the Bible had swear words. I'd wanted to see a swear word printed out for the masses, so I'd looked for the part where Jesus rode in on an ass. And there it was.\n\nIt turned out there were more swear words in the dictionary, though, so I'd ultimately opted to read that for more scandalous text.\n\n\"Honey,\" my mom started again. \"We know you miss her. And we know this summer is going to be hard. If you can't do your annual road trip with her, then\u2014\"\n\n\"Then I can be with her in spirit?\"\n\nShe looked at me optimistically. \"Well, yes, I suppose so.\"\n\nI rolled my eyes. \"Because you know she's the last person in hell\u2014\"\n\n\"Nick\u2014\" my dad chided.\n\n\"\u2014who'd go to church camp. They wouldn't even let her, you know. Those people would not have even let her set foot on their campus. Or their revival tent. Or whatever.\"\n\n\"Well, yes, because it's for high schoolers,\" my mom pointed out. \"She's too old.\"\n\n\"She's actually dead,\" I snapped, and my mom winced. What was wrong with me? It was her daughter, for God's sake. I softened my tone. \"Look, even if she were in high school\u2014and alive\u2014she would never have gone.\"\n\n\"You'll make new friends,\" my dad recited.\n\n\"Where'd you even hear about this camp?\" I glared at them as realization dawned. \"Charlotte.\"\n\nCharlotte was my oldest sister. Throughout her teen years, she'd been that girl (i.e., a lot of boys got to \"know\" her, Biblically speaking). Then she'd gotten involved in some youth group, graduated high school, and moved to Nebraska, or some other f lat, dry state with rolling waves of corn and a high percentage of Republican voters. I'd lost track. She was cultish and hated anything with a rainbow on it (not post-Noah's ark kind of rainbow, the gay-pride kind). I'm sure she didn't even let her kids eat Lucky Charms, and it would have nothing to do with gluten.\n\nBoth Charlotte and Diana had been in middle school when I was born, and they acted like extra moms when I was growing up. They'd treated me like their doll, painting my nails and putting me in tutus. I'm sure she'd since repented for encouraging cross-dressing.\n\nAnd, yes, once she'd gotten all religious Charlotte had tried to convert me. She'd dragged me to her youth group like a pet. At first it had been okay\u2014what boy doesn't like free peanut-butter Ritz crackers and endless cups of lemonade, and being the center of all his sister's older friends' attention? But pretty soon, I'd been forced to memorize verses and perform in skits. The day a bunch of youth group kids lowered me from a ceiling to be healed by Jesus\u2014and dropped me\u2014I was done.\n\nShe'd never tried to convert Diana. No, Diana was clearly a lost cause. Not fit for the fold.\n\nThe sad thing was, Charlotte had five kids. I had five nieces and nephews, and I'd never met them. She'd send pictures sometimes. Now and then she'd call our parents, but only brief ly. Nothing real.\n\nSo, it had been me and Diana.\n\nAnd then she left me too.\n\nI took a deep breath and looked at my parents, who at least had the decency to look a tad bit contrite about the Charlotte factor. They knew my feelings about her\u2014regret and creepiness.\n\nBut she was still their daughter. \"Honey, this camp was very good for Charlotte when she was hurting.\" I hated it when my parents did this, when they justified Charlotte. When they defended her.\n\n\"She wasn't hurting, she was\u2014\" humping the whole town, I wanted to say. But I didn't. We all knew she was \"Charlotte the Harlot\" at her high school.\n\n\"Aren't you afraid I'll end up like Charlotte? I mean, what if I come back and think you're devil worshippers too?\"\n\n\"That's not how it works...\" my mom sighed.\n\n\"That is how it worked.\"\n\n\"Charlotte needed something. She found that something at Eden Springs.\"\n\nShe had found God. She had found justification to hate our sister. The rest of the family hadn't been unholy enough in her eyes to hate, but we hadn't been righteous enough to love either.\n\n\"Wait,\" I said, holding up the brochure. \"Did you talk to Charlotte?\" She hadn't even come to Diana's funeral. She'd called us maybe once in the months since Diana died. \"Did you call her? Or did she call you?\"\n\nMy dad's face folded up all neat. \"It'll be fun. End of discussion.\"\n\nHe wiped his palms on his khakis and stood up. My mom stepped toward me and touched my cheek. They both walked out of the room.\n\n\"Why would you listen to her?\" I asked no one.\n\nBut I thought I knew. Just to hear her voice. Charlotte and Diana were nothing alike, but their hearts beat the same blood. When you lose someone, you find the closest thing to that person and hold on.\n\nI left the house. I wasn't going back to my room, where Jack would be waiting, broody and antagonistic. And I wasn't going to sit with my parents. They'd always loved their daughters, but Charlotte had loved God and Diana had loved women. And Charlotte's God did not love women who loved women. It was a love pentagon, which seemed appropriate given Charlotte was certain Diana was in hell. And my parents had thought they could love both of them anyway.\n\nMy parents didn't hate Diana. She was their daughter, pure and simple. But every time Charlotte had come up and they'd told Diana \"just let it go,\" or \"she'll grow out of it\" (like she was a bed-wetting schoolgirl), it had been like arsenic.\n\nI walked around the garage, through the gap between the building and the neighbor's fence. At the back of the yard stood a handful of tall fir trees that shaded a green Volvo.\n\nAfter my sister died, I had avoided the car, avoided the shadows where it sat altogether. After a while, I was able to walk by it, but I kept my eyes down, always staring at my feet. I didn't want to look inside the windows. I was afraid of suddenly seeing her there\u2014maybe what she looked like after she died in it, or what her rotting corpse would look like now. But I was also afraid of not seeing her behind the wheel. Where she'd always been.\n\nI snapped open the door and slid inside. Just me, the hollow silent interior of the car, and not even the smell of her anymore. Not the patchouli oil she wore or the coffee she drank too much of. It used to smell like her. In the way that clothes and bedrooms and anything smells like someone. The smell you especially notice when someone leaves.\n\nDiana bought the car when she'd turned sixteen, and she'd planned to hand the keys over to me on my sixteenth. I still had a year to go, but I didn't want it anymore. It just sat in the yard under the dripping fir trees, the interior smelling of mildew, the rain slipping through the poorly sealed windows and the two little holes in the roof.\n\nI leaned back on the ugly, plaid driver's seat and stared at the roof, the small places where the rain always seeped in, so tiny that I couldn't see the sky through them.\n\nOn the day Diana died, I hadn't been paying attention.\n\nShe'd walked into the house\u2014she'd still had a key even though she'd moved out years before. After a few minutes without one of my parents hollering for me to do this or that, I'd figured it was my sister. I'd pulled myself away from the old-school Atari game, Asteroids I think it was, to walk upstairs and find out what she wanted. She'd been so erratic by then, and I remember feeling dully irritated with her. Just under the skin.\n\nShe'd been planning to move to northern California to work as the gardener at a spiritual retreat. And she hadn't really been hanging out with me much, which\u2014yeah\u2014had started to piss me off. We'd go listen to some music at the coffee shop, but she'd be on her cell phone the whole time. One of the last times we hung out, she just left me to go hang out with her girlfriend, Leah, trying to make up for some feud between the two of them. Diana hadn't hung a picture of her and Leah up at work, and Leah had been pissed. Diana had argued that Leah didn't understand how conservative it was there. She wouldn't get fired, but she'd be second-guessed and scrutinized.\n\n\"Leah's right, just hang the picture,\" I'd said. \"Nobody cares.\" Then I'd joked, \"You're being gay.\" We both hated that phrase. That's gay. This is gay. You're gay.\n\nDiana hadn't laughed or rolled her eyes with me that time, though. She'd put her head in her hands. \"Nick, you have no idea how small this world's box is. You've never had to try and fit in it.\"\n\nShe was right. That's what I would tell her now, if I could. And I'd tell her that I didn't know how small the world was because she made it big for me.\n\nBut she was gone, and the world had shrunk and shaded.\n\nJack was waiting for me back in my room.\n\n\"I have to pack,\" I said. \"Why are you still here?\"\n\n\"Are you going to tell your parents about what happened? Before you go to this religious camp of yours?\" He eyed me from under his heavy brows. His black hair was glossy, perfectly in place.\n\nI ignored him and got my suitcase out of my closet.\n\nHe kept on behind me. \"Yes, the Great Editor must pack his suitcase full but leave the truth out!\"\n\n\"I can't just tell them and take off.\"\n\nThe truth was, I was never going to tell my parents. I was never going to tell anyone.\n\nWhen I turned around, the window was open wide, and Jack was gone. Thankfully, the empty bottle of liquor and pack of cigarettes had gone with him.\nMONDAY\n\nMy alarm didn't go off\u2014probably Jack messing with the radio\u2014but my parents cajoled me out of bed, lest I arrive late to my own salvation.\n\nI threw on the T-shirt I'd picked out to protest this whole camp thing. It featured a doe-eyed cat and read EVERY SINGLE TIME \"YOU'RE\" IS MISSPELLED \"YOUR,\" GOD KILLS A KITTEN.\n\nI didn't make it halfway down the stairs before my dad said, \"Change your shirt.\"\n\n\"Why?\"\n\nHe raised his eyebrows.\n\nMy mom came around the corner. \"Honey...\" She looked a bit misty. Maybe she always did.\n\n\"Freedom of speech,\" I said. \"Freedom to be hilarious.\"\n\n\"You don't have the freedom to be a jerk,\" my dad said with a chuck on my shoulder.\n\n\"Honey, you don't want this to be your first impression,\" my mom added.\n\nActually, I did. It was best everyone at Jesus camp knew that God's role in my life had been relegated to punch lines on grammar T-shirts. But, whatever. My parents weren't going to change their minds. I'd made my point to them, at least.\n\nI changed into my GOT MILK? shirt. The \"got\" was crossed out, and \"Do you have any\" was scribbled over it. I had a plethora of nerd shirts thanks to Diana, who'd thought my obsession with correctness in English usage was fun and quirky. Unlike Jack, who found it infuriating and smug.\n\nI didn't talk on the drive to the church. I just watched the houses get closer and closer together as we made our way toward the center of our small city, which is situated just south of cool (Seattle) and just north of mentionable (Tacoma).\n\nWhen we arrived, I carried my backpack to the check-in line. My dad rolled my suitcase over to a cluster of others, as directed by a guy in a lime-green T-shirt that cheered HAPPINESS HAPPENS HERE! in rainbow-colored letters. (Rainbow! And to think, my homophobic sister chose this camp for me.) There was a heart over the i in \"Happiness.\"\n\nAt the front of the line, a girl and boy with the horsey smiles that dentists dream of handed me my own T-shirt, and I vowed to save it for Charlotte. A thank-you gift.\n\nMy parents lingered behind me. I hugged my mom. I was quite a bit taller than her, but still too skinny to offer much comfort. Her shoulders shook lightly beneath my arms, and I realized she was crying. Christ. I felt bad for her pretty much all the time. My dad put his hand on my shoulder, and then pulled us all in for a big hug.\n\nLike most parents, they were na\u00efve and misinformed. Having spent the evening before discussing the relative merits of confession with a dead author, it was clear I needed antipsychotics, not Jesus. But of course my parents didn't know what was going on with Jack Kerouac. I didn't even know what was going on.\n\nMy mom cried a bit more and hugged me again before my dad half carried her to the car, and I waved to them as they drove away. And then I was alone.\n\nExcept for the fifty other kids teeming around the church.\n\nI was struck by the sheer volume. The squeals, the guffaws. They'd all pulled their Happiness camp shirts on over whatever they'd worn, and their parents fumbled with cell phones to take pictures. The girls nearly snapped their moms' wrists to get the phones back and hyperanalyze the photos.\n\n\"Not good.\"\n\n\"Oh, my gosh, look at my face.\"\n\n\"Weird raccoon face.\"\n\n\"I need a new face!\"\n\n\"Cute!\"\n\n\"Oh, my gosh, tag me! Tag me!\"\n\nAnd then the Instagram bender. The Snapchat binge and purge.\n\nI felt a tap on my shoulder and turned.\n\n\"Good morning!\" The suitcase-director in the Happiness camp shirt held out his hand.\n\nI shook it. \"Hi.\"\n\n\"I'm Jason!\" He continued to pump my hand.\n\n\"I'm Nick.\"\n\nThe overenthusiastic stranger pulled me into a side hug. \"It's a blessing to have you here. T-G-I-J, am I right?\"\n\n\"Perhaps... ?\"\n\n\"Thank God it's Jesus!\" He squeezed my shoulders. \"Right?\"\n\n\"Right... I guess. But redundant.\" I thought Jesus was God.\n\n\"Ha! You from Valley Christian?\"\n\nI was trapped in his embrace. \"No, Mountain View High School.\"\n\n\"So glad to have you! You ready for an awesome week?\"\n\nI shrug-nodded under the weight of his side hug.\n\n\"It's gonna be awesome!\" He threw his hands into the air, then shouted toward the group, \"Because HAPPINESS... ?\"\n\n\"HAPPENS HERE!\" everyone shouted back, even the parents, cheeks pink with abandon.\n\n\"Awesome, awesome!\" Jason laughed ever so joyously. He slapped my shoulder, unaware of the brute strength of his enthusiasm.\n\nJason stood next to me, hands together, smile beaming as he looked over the throng of teenagers. I waited for more conversation. When nothing happened, I, too, looked over said throng of teenagers and their hundred-watt smiles.\n\nSuddenly, I realized: I was that kid. The awkward kid with no one to talk to. The Shadow Kid. The parent's shadow, the teacher's shadow, the camp leader's shadow, the one whose face beseeches his peers to please, please, please be his partner for the science lab. The one who writes poems in Elvish for his English teacher.\n\nI was not a super popular guy by any means, but I'd never been that kid before.\n\nAlthough, my name in Elvish is Turwaithion. Toor-wye-thee-on. Of course I looked it up.\n\n\"I guess I better get on the bus,\" I said. I had to escape from my role in this drama. And from the PCP-strength of Jason's cheerfulness. Luckily, he moved off to herald more campers.\n\nI headed over to the line of four buses: three regular school buses and the ubiquitous short bus.\n\nThe chanting continued. Like a commune. Like a cult. No wonder the camp was held in Oregon.\n\n\"HAPPINESS!\"\n\n\"HAPPENS HERE!\"\n\n\"HAPPINESS!\"\n\n\"HAPPENS HERE!\"\n\n\"HAPPINESS!\"\n\n\"HAPPENS HERE!\"\n\nI weaved through the \"members of the movement\" and approached a high schooler who looked particularly power hungry with his clipboard. His f loppy black hair and eyeliner contradicted his Happiness camp T-shirt. \"Which bus do I get on?\"\n\nGoth kid looked at me, read the front of my shirt. \"Where's your camp T-shirt?\"\n\nI held it up.\n\nHe nodded. I waited. We locked eyes. And then, with the help of his intimidating eyeliner, he won the T-shirt standoff. I slipped the camp shirt over my own. In exchange, he pointed to my bus.\n\nI was on the short bus. Of course I was. I did, in fact, have a religious disability. There was no federal No Soul Left Behind Act for me, though. Just the short bus on earth and hell in the afterlife.\n\nI mounted the steps, and the bus driver held out a piece of paper toward me. I tried to take it, but he pinched it tight, eyeing me dubiously before he let go. \"You chewin' gum?\"\n\n\"No.\"\n\nHis lips tightened. Then, \"You sure?\"\n\nI opened my mouth, lifted my tongue. \"Any other cavities you want to see?\" Pun intended. \"Cavities, get it?\"\n\nHe didn't get it, or he wasn't impressed. \"Go sit down,\" he said.\n\nI spotted a seat at the back of the bus and walked down the aisle, avoiding eye contact with the clan of short-bus unfavorables:\n\n1. a girl in a wheelchair\n\n2. a boy with a broken arm\n\n3. a girl with too-red lips (but superhot)\n\n4. a girl with a red bandana on her head (but prettycute)\n\n5. a twin brother and sister\n\n6. a girl who I later watched pick her nose for hours. At this age, nose-picking struck me as a disability, so assigning her to the short bus made sense.\n\nI fell into my seat and peeled the Happiness T-shirt off. Then, I pulled my cell phone from my back pocket, opened my text messages, and scrolled to the feed between me and Leah.\n\nAll the messages from her were old, from before my sister's funeral. I hadn't talked to her since. I tapped out a new message: You know what week this is. Bro-sis road trip week. Parents are sending me to church camp. Can we talk when I get back?\n\nMaybe if she answered, I could explain what happened at the funeral. I could tell her how she was the closest thing to my sister I had left. My parents had Charlotte. I had Leah.\n\nMaybe I'd even tell her the truth.\n\n\"Anything in that writing-box there?\" Jack was peering over the seat in front me. I darted my eyes to see if anyone heard him. No one on the bus had even turned in my direction.\n\nHe added, \"Just makin' sure you're feeling okay amidst this joy-collecting.\"\n\nI shook my head, unimpressed with his word magic. \"Do not follow me to camp,\" I whispered. If my tally marks were correct, Jack had appeared twenty-nine times so far. His visits were getting more frequent.\n\nBandana Girl turned around. \"Pardon?\"\n\nI glared at Jack, but he was gone. \"Um, do we take this to camp?\" I f lashed the paper the bus driver had given me.\n\n\"You have to confess,\" the girl said. She turned back around.\n\nI looked at the paper. I looked at the back of the girl's head.\n\nWTF.\n\nJack popped his head back over the seat and winked. \"I told you so.\"\n\nPRAYERS AND CONFESSIONS\n\nBill: Jesus? He's no son of God. He's an alien. Yeah, he came down to Earth. And, yeah, he rose again. Back to a spaceship where his little green friends live. Listen, I just drive the bus. I don't give a shit. As long as the little angels don't put gum all over the seats.\n\nJason: Once again, Father, I just thank you. Thank you for these kids, for sending each one of them here this week. I know You have a purpose. You see their hearts, Lord. You know who needs the Happiness. Happiness happens here because of You. And also because I did cheer in college. Ha! Praise Jesus!\n\nHolly: If Payton wants to break up with me because he thinks I'm the Whore of Babylon, then fine. Whatever. I'll work it like the Great Harlot.\n\nSomewhere around hour three, after Jack had convinced me that Bandana Girl was a real person and not some paid pro-confession accomplice of his, after we took a break at a rest stop where I nearly hyperventilated in a tampon bag in the unisex bathroom, a mechanical shout jolted me from my anxieties. Goth stood at the front of the bus with a walkie-talkie in his hand.\n\nHe was pushing buttons and turning knobs, clearly frustrated by the crackle, then silence. Crackle, then silence.\n\nThe other bus riders weren't paying attention to his technological woes. The kid with the broken arm had his headphones in. Nose-Picker was lecturing Bandana Girl about mermaids while she passed a bottle of sunscreen to Red Lips. (Sunscreen on a bus?)\n\nOutside, the land was f lat and brown. In the distance I thought I saw a patch of green and a barn, but it could've been a mirage. Or the ghosts of the infamous Rajneeshee commune.\n\n\"Goddamn it,\" Goth mumbled, and everyone's heads shot up in shock and horror. Shame splashed his face, but he quickly recovered. \"No, I mean that literally. Like, this thing is of Satan and deserves damnation.\"\n\nEveryone around me nodded.\n\n\"What are you trying to do?\" I asked Goth, who swayed with the motion of the bus.\n\n\"It's Lottery time, man.\"\n\n\"Which you need to explain, Stewart,\" Red Lips called up to him. (Stewart. Could Goth have a less scary name than Stewart?) \"He's a Pamper.\" She turned all the way around in her seat, resting her chin on her hands and grinning at me. Wait, was she f lirting with me?\n\n\"A Pamper?\" I asked.\n\nBandana Girl chimed in: \"Pamper. Diaper. Baby\u2014\"\n\n\"You're a new camper,\" Red Lips said, side-eyeing Bandana Girl. \"Pamper, camper, get it?\" Her eyelashes got all batty.\n\nRhyme was the lowest level of humor in my book. But this girl was the highest level of hot, so I didn't roll my eyes.\n\n\"So, like, are you from Mountain View?\" she asked.\n\nThe kid with the broken arm was suddenly watching me. His eyes f licked to Red Lips, then back to me. Red Lips smirked at him. Smiled at me. Smirked at him.\n\nOh, is that what was going on. Broken Arm and Red Lips\u2014a drama couple. Or a drama ex-couple.\n\n\"It's the Donkey Lottery,\" Stewart interrupted, then slammed the walkie-talkie on the seat twice, like he was un-jamming a stapler.\n\nI reached for the walkie-talkie and Stewart handed over the cheap, plastic, China-crafted toy. \"See this button here? It's to change channels,\" I explained.\"This button is to talk. You just kept changing channels every time you tried to talk.\"\n\nI knew Stewart had found the right channel when I heard crackly cheering through the speakers. He turned the volume way up until the tinny sound threatened to rupture my eardrums. What was I hearing? The last screeching death throes of a donkey being sacrificed to God\u2014the donkey that \"won\" the lottery? Like that Shirley Jackson short story we read in English, where the winner was stoned to death?\n\nI tried to plug my ears, and then I saw the mouths of my fellow passengers begin moving. They were chanting.\n\nHee-haw Hee-haw Ha Ha Ha,\n\nOn a donkey, no need to walk.\n\nHee-haw Hee-haw Ha Ha Ha,\n\nJust like Jesus, no need to walk.\n\nHaw-hee Haw-hee He He He,\n\nWho's gonna win the lottery?!\n\nAnd then they started shouting, \"Meeeeeee!\"\n\n\"Meeee!\"\n\n\"Meeee!\"\n\nAnd I recognized the screeching from the walkie-talkie. What I'd thought were the death throes of the donkey had just been the campers' desperately gleeful responses to an apparently not-rhetorical question. Bandana Girl set down her book and improvised a little dance to go with the chanting.\n\nI did not join their sacrificial rites.\n\nFinally, they ended their chant in laughter and applause\u2014my fellow passengers just as delighted as those I could hear from the other buses through the walkie-talkie. Then everyone on the short bus got quiet, looking expectantly at Stewart. He held out the walkie-talkie.\n\n\"Genevieve!\" shouted a voice\u2014I recognized Jason's\u2014through the speaker.\n\nSomeone squealed on the other end before being drowned out by my bus and the rest of the campers. \"Hee-haw... Hee-haw...\" they brayed. And this time it was not a chant; they actually sounded like donkeys.\n\n\"Chris Cooper!\"\n\nThe boy twin across the aisle startled me with a shout and a fist pump, and I heard the deep tones of, \"Cooooop! Coooooooooooooooop!\" from the speaker before they were again drowned out by the brays of my short-bus compatriots.\n\n\"Hee-haw... Hee-haw...\"\n\nSo, now I knew that guy's name, and that my bus was not completely ostracized.\n\n\"Kaitlyn J.!\"\n\nSqueal!\n\nBray.\n\nI still had no idea what was going on, but my busmates were on the edge of their seats. Literally. Rubbing their sweating, anxious palms into their thighs.\n\n\"Nick S.!\"\n\nMy bus friends whooped.\n\n\"Isn't that your name?\" Bandana Girl asked.\n\n\"Is that you?\" the twin sister chimed.\n\n\"Nick Sampson, or the new kid?\" Goth called through the walkie-talkie.\n\nI didn't know whether to be excited or terrified.\n\nThe walkie-talkie crackled. \"Sampson,\" Jason said f latly.\n\nMy fellow passengers groaned and looked at me with pity. I was not the Chosen One.\n\n\"Hee-haw... Hee-haw...\" The tinny chant echoed through the walkie-talkie.\n\nThen the bus stopped.\n\nThe driver cut the engine, the bus parked right in the middle of a dirt road. Dust billowed up from below the tires. Through the yellow hue, I saw only more yellow and brown. We were in a canyon, just as barren and lifeless as the land we'd driven through to get here.\n\nI was going to die with my fellow travelers, \u00e0 la the Donner Party. Except, instead of freezing to death, we were going to bake. Which camper would I eat first?\n\nThe door to the bus opened and Jason bounced up the steps. \"T-G-I-J! Am I right?\"\n\nSqueals and cheers from the other eight people on my bus. I was sure this was more convincing on a normal bus of thirty-five.\n\n\"Coooooooop,\" he growled. \"Chris Cooper, my man! Praise Jesus!\"\n\nFist pumps from Chris. Glares from his sister. I could almost see the devil on her left shoulder shank the angel on her right.\n\nJason looked at her, and he must have caught on to her disdain. \"Jesus has a plan! Next year for you, Christina!\"\n\nChristina? And Chris? Had their parents totally lacked creativity, or just thought that was a cute idea?\n\nI caught movement from outside my window.\n\nFour donkeys stood docilely next to the camp counselors holding their reins. The donkey-toting teens donned long white robes and sandals. Several other counselors stood next to wagons loaded with our luggage and yoked to pairs of Clydesdales.\n\nChris Cooper retrieved his backpack, slipped it over a shoulder, and followed Jason off the bus.\n\nThe rest of us grabbed our belongings and stepped off the bus. Well, the girl in the wheelchair didn't step. We waited for her while she whirred slowly to the ground.\n\nIt was when we finally walked (or rolled) around the bus that I understood what the Donkey Lottery was. The campers I could only assume were Genevieve, Nick Sampson, and Kaitlyn J., as well as our own Chris Cooper, each scrambled up onto the back of their own donkey.\n\nThey were not in robes or sandals like the teenage counselors. They did not have a beard or halo like the real Jesus riding on an ass into Jerusalem. But that was definitely what was being simulated there.\n\nI stif led a laugh and looked around for someone with whom I could commiserate, but I saw only earnest and enraptured faces. Even the prettycute Bandana Girl appeared at peace with it all. There was nothing ironic going on here.\n\nAnd then we started down a dusty road into Eden Springs, in the blazing afternoon sun.\n\nThe road went on forever. I was without water. I was without food. I was without the sunscreen that had been passed among my busmates.\n\nIt didn't take me long to consider prayer.\n\nI knew we'd arrived in Eden Springs when I spotted a long line of counselors, a tunnel of them, a gauntlet of them, all in robes and sandals. They each bent over and picked something up off the ground, and then they started waving the things. And I saw the whole donkey metaphor had not been played out.\n\nAs the donkeys walked through the gauntlet, the robed counselors waved palm fronds. The rest of the campers followed those on donkeys through. Triumphant smiles abounded.\n\nSurely they knew the end of the story, the part where the notorious donkey-jockey got taunted, maimed with a cat-o'-nine-tails, mocked, and stabbed in the skull with thorns. And that was before he finally asphyxiated on the cross and was pierced in the side. You know, just to make sure he was not playing dead.\n\nI was hot and dehydrated. My skin was pink with a burn. I walked around the gauntlet in order not to endanger these well-intentioned Christians with my homicidal disposition.\n\nAnd then I was at camp. Which was, indeed, a paradise when compared to the walk.\n\nGreen grass everywhere. Somehow a breeze. In the distance, a pool and waterslides. Perfectly leaved trees. Paved sidewalks. Freshly painted cabins with porch swings.\n\nSomeone in a robe handed me a Dixie cup of water, and for a minute I didn't care about his costume. I shot the water like I'd seen Jack do with cheap tequila.\n\nThe same someone handed me another cupful, and another, all the while just smiling, glowing really, as he watched me drink and drink. And then he said something and pointed to a banquet table stocked with vegetables and sandwiches and more water.\n\nAnother counselor, this one a girl with her long blond hair pulled back using twine, handed me a plate. I didn't even have to dish up. She served me.\n\nMusic shouted through loudspeakers. Modern music. Finally, something normal. I sat down in the grass and ate my sandwich. The bread f laked like it was fresh from a bakery. The vegetables must've been plucked from a vine moments ago, so juicy and perfect.\n\nEveryone else was milling about the grounds, newly energized, hopeful. We had water. We had food. We would not have to resort to cannibalism.\n\nI was thrilled and relieved and... Yes, happy!\n\nHappiness did happen here! At least for the near-death desperate.\n\nI lay back in the grass, the cold smooth green grass, and looked at the big blue sky.\n\nWhen I looked around the camp again, people had their cell phones out and were taking selfies.\n\nChris was taking a selfie with his donkey and disciple.\n\nAnother camper leaned in with a bunch of palm-toting counselors.\n\nI took another bite of my sandwich.\n\nA camper stood next to a woman\u2014another one of the counselors? She definitely didn't seem like one, the way she was dressed. Or not dressed. Black lace and red silk. Her dark hair wrapped up with perfect ringlets hanging down over her shoulders and those breasts. I stared at those breasts. I didn't think I'd ever seen anything like them. Were we at church camp or the Playboy Mansion? The camper took a selfie with her.\n\nI finished my sandwich as the boy moved away and another boy moved in for the next selfie.\n\nThen, as my blood sugar normalized, my senses returned.\n\nI saw everything.\n\nOh, dear gawd.\n\nNext to the breasty woman lay a man on a mat. Dirty and haggard, he didn't get up. The campers took selfies by lying on the ground with him.\n\nThis could not seriously be what I thought it was. Perhaps it was only my post-traumatic stress from being forced into Bible skits by Charlotte's youth group.\n\nAnother man, with a cane, wandered around the camp with his eyes closed, bumping into campers. They didn't mind. They took selfies with him and the \"blind\" man smiled for the camera.\n\nOh, it was indeed what I thought. This was not PTSD.\n\nAnother woman, face horror-stricken, was only wrapped in a sheet, a Jewish rabbi nearby glaring at her, a pile of stones by his feet. They paused to take selfies with the just-arrived campers. Sometimes the characters posed together, sometimes apart.\n\nI spotted a disciple with a coin purse and a noose dangling from his neck. Judas.\n\nThere was a man robed in purple, velvet finery. He stood by a sign that read, PICTURES WITH PONTIUS! Wow, the camp even repped the Romans.\n\nNext to him, a character from The Walking Dead. Huh?\n\nAnd then I saw Jesus.\n\nI knew it was Him because He was surrounded by the masses. The Harry Styles of this band of religious figures. Like, hair and everything. But Jesus's hair was f lattened by a crown of thorns. And, oh yeah, there was a tree-sized wooden cross behind him, stuck in the ground. He wore only a loincloth. I worried about the security of his junk.\n\nWhen I was eleven, my parents took me to Disneyland. Before going on the rides, which is all I'd cared about, we'd walked through the village with all the shops. And there had been characters everywhere. Alice in Wonderland and Mickey Mouse and the Beast and Cinderella.\n\nThat was what I was seeing now. Bible Disneyland, every character alive and greeting the campers. But this was more Rob Zombie than Disney.\n\nSuddenly, I was certain I was hallucinating. My Dixie cup must have been laced with shrooms. Perhaps this was some neo-Jonestown-cult-mass-homicide plot wherein hundreds or thousands of kids perished in the name of Jesus camp. Did cyanide make you hallucinate? Did I drink anything at the church before we left?\n\nBut no. Those were real donkeys. Real palm fronds. Real breasts on the fake prostitute.\n\nA robed and sandaled counselor stood over me. \"Somethin' for the PC Box, my man?\" He held out a box, padlocked shut, with a slit on top.\n\n\"Uh...\" Tithing? We had already overpaid to be here. But whatever. The disciple-counselor looked homeless, so I'd just tell myself he was buying himself some whiskey to keep the DTs away. I stood up, reached into my pocket for a dollar, and slipped it into the box.\n\nThe counselor squinted at me, but then shook his head and walked on. I heard someone laugh behind me and turned.\n\nBandana Girl. Her book still in hand.\n\n\"That's not a tithing box. That's the PC Box,\" she said.\n\n\"Politically correct?\"\n\n\"Prayers and Confessions.\" Bandana Girl grinned. \"The paper you got when you got on the bus. I told you, you have to confess. You have to write down a prayer or confession and then put it in the box.\"\n\n\"What happened to the good old days of talking to a priest?\" I asked.\n\n\"You're Catholic?\"\n\n\"Ha. No. I'm not into confessing anything.\" I looked around for Jack Kerouac, hoping he'd heard me. Actually, I was hoping he wasn't at camp at all, but that was likely too much to hope for. I didn't want Jack interrupting while I talked to this prettycute girl. \"I'm just saying, at least with a priest, it's only you and some guy who's so old he won't remember anything you say anyway.\"\n\n\"So you're worried about it being written down?\"\n\n\"I'm not worried about anything. I just don't want to do it.\" The pitchiness of my voice gave me away.\n\nShe laughed. \"You look petrified.\"\n\n\"Well, did you see the zombie back over there?\"\n\n\"That was a leper. Maybe you do need some help in the politically correct department.\"\n\n\"I'll write that down as my confession.\"\n\n\"That's cheating. You've already confessed it to me. Anyway, you'll get used to camp. It's actually fun.\" She smiled. I eyed the book in her hand.\n\n\"What're you reading?\"\n\n\"I'm not.\" She opened the book and (gasp!) there was writing all over it.\n\n\"You broke the cardinal rule DON'T WRITE IN BOOKS.\" Dear gawd, she probably ran with scissors.\n\n\"I was marginalizing it,\" she explained.\n\n\"Literature is not to be marginalized.\" I glared. \"Unlike zombies. They need to be put in their place pre-apocalypse.\"\n\nShe laughed. \"I'm just writing in the margins.\"\n\n\"That's called annotation.\"\n\n\"If you're a nerd.\" She raised her eyebrows and nodded toward me. \"But, then, nerds would wear grammar T-shirts.\"\n\nI shrugged. \"Or read a book on the way to summer camp.\"\n\n\"You cannot deny your attire. I have already denied reading. I was marginalizing.\" She opened her book to a random page and pointed to some scribbling. \"It's a play on words. The fact that I am writing in the margins\u2014and more so, what I am writing\u2014marginalizes the text.\" She f lipped the book closed. \"Thus, marginalizing.\"\n\n\"And that's not nerdy at all.\" I smiled; she smiled back. Then we both watched the prostitute walk by, followed by a girl wearing a bikini of leaves. It was hard to tell Eve from a tree-hugging working girl. \"So, what book was that?\" I continued.\n\n\"I'm working on Steinbeck right now.\" She showed me the cover\u2014The Grapes of Wrath. \"I already did Of Mice and Men and The Red Pony, but this takes way longer. And needs way more.\"\n\n\"But you're not reading them?\"\n\n\"I've already read them. But I go back through and write in the margins. Like, things to think about and questions and aha! moments. That way, the next poor soul to suffer through torture-by-classics may find some reprieve in the snarky or insightful remarks of a classics-survivor.\" She laughed. \"I mean, have you read the turtle chapter?\"\n\n\"It's symbolic.\"\n\nShe snorted. \"And boring. I made up a bunch of Why'd the turtle cross the road? jokes.\"\n\n\"That's literary blasphemy.\"\n\n\"Yes, I'll be going to hell. Me and everyone on Wattpad.\"\n\n\"What do you do with the books after you've defiled them?\"\n\n\"Pay the fine to the school for destruction of property and sneak the book back into the pile for next year.\"\n\n\"Poor kids never get to read the untainted story,\" I joked.\n\n\"What? No way. I feel bad for the twenty-five kids who don't get the marginalized copy. I think I'm going to publish a line of them, like those books For Dummies. Except it'll be called something like Classics for Bored Kids.\"\n\n\"I guess it's a step up from SparkNotes.\"\n\n\"It's a stepladder up.\" She laughed. \"I'm Natalie, by the way.\"\n\n\"I'm Nick.\"\n\nIt got quiet between us. Clearly, we had met our quota of clever for the day.\n\nThen Natalie held up her paper between her finger and thumb. \"Okay, well, I'm going to walk over there now. You know, drop this in the box. Because that's what we do here at camp.\" She sauntered away, smiling back at me. \"Do it, Nick... You have to do it...\"\n\nI pulled my own paper from my pocket and waved it at her like I might actually do something with it. I wished I'd had something else interesting to say to keep her around. I wished I'd asked for a copy of The Bored Kids Guide to The Red Pony, or something.\n\nI watched her drop her paper into the PC Box. And I wondered what sort of prayer or confession a girl like her would write.\n\nNo one can blame me for checking the mattress for hay when we got to our cabins. Jesus was born in a manger, and I thought there'd be something of that experience in our sleeping quarters. Straw, the donkeys, three creepy old men gifting herbs, Mary's placenta.\n\nI peeled back my sheets, checked the bathroom, looked under the bed. I wasn't sure what I was looking for, but I didn't want any surprises. Visits from Jack Kerouac were unpredictable enough; I didn't need unexpected Bible metaphors.\n\nFinding nothing unusual, I sat on my bed and rested my head in my palms and wracked my brain for Bible stories. What skits had my sister forced on me as a child? Would an ill man be dropped from the ceiling? The pig with the demons in it, did it go into a house before it ran into the lake to drown? I had to know what was going to happen, what could happen.\n\nMy cabinmates rolled their suitcases in, took out Twizzler packages and cans of Monster, and shelved everything in cubbies next to their beds.\n\nA short, skinny kid unpacked kale chips and Vitaminwater. He gave me a critical once-over. \"Are you homesick?\"\n\n\"No,\" I said. \"Just a headache.\"\n\n\"Sometimes I get homesick, but then I just take a rest and do some problems in my math book to calm my nerves.\"\n\n\"That's messed up, Charles,\" a boy said as he walked out of the bathroom in swim trunks. Then he looked at me. \"Are you the kid from Mountain View?\"\n\nI nodded. \"Are you all from Valley Christian?\"\n\n\"I'm not,\" Charles said. \"I'm homeschooled.\"\n\n\"Thus the math book to find your Zen,\" I said.\n\nNobody laughed.\n\n\"I'm Matthew,\" the kid from the bathroom announced. \"You want to go down to the waterslides?\"\n\n\"Is that what we're supposed to do?\"\n\n\"Did you already drop a prayer or confession? Then it's free time,\" he said. \"God gave us free will, and we get free time to practice it.\"\n\nI couldn't tell if he was saying all that in earnest or not, but I was certainly ready for the pool after the near-death experience of walking to camp.\n\n\"Free time isn't until after the Silent Three,\" Charles corrected.\n\n\"Yes, Charles, yes. But, let's get our stuff on so we're ready.\"\n\n\"The Silent Three?\" I asked as I unzipped my suitcase and dug around for my swim shorts.\n\nMy cabinmates answered:\n\n\"It's hell.\"\n\n\"Hell comes before Eden in this case.\"\n\n\"Three silent hours of hell.\"\n\nThey still hadn't answered my question.\n\nI pulled my beach towel out from under my clothes. Someone gasped. All moaning and groaning stopped.\n\nCharles stood at my bed, mouth agape, Matthew at his shoulder, face grave. The other boys slowly moved into a semicircle around my bunk.\n\n\"Harry Potter...\" Charles whispered.\n\nYou could barely see Hogwarts on the towel anymore, it was so faded. \"Yeah, it's kinda old\u2014\" I started.\n\n\"And Satanic,\" Charles said.\n\nI laughed. Nobody laughed with me.\n\n\"It's a book,\" I said.\n\n\"About magic,\" Charles insisted.\n\nI assessed the other six boys, all of whom stared at my towel dubiously.\n\n\"I'm sure it's fine,\" Matthew said. \"The rules don't actually say you can't bring magic paraphernalia to camp.\"\n\n\"Did you bring a voodoo doll?\" Charles asked me.\n\n\"Why would I bring a voodoo doll?\"\n\n\"Why would you bring a Harry Potter towel?\"\n\n\"Because it's a hundred years old, so my mom won't kill me if I leave it here accidentally.\"\n\n\"But you've read the books?\"\n\n\"Yeah, when I was, like, ten.\"\n\nCharles pursed his lips. \"This is bad. This is a bad sign.\"\n\nA couple of the other boys went back to their bunks, apparently not too concerned.\n\n\"I'm sure it's fine,\" Matthew said again. \"It's just a book to you, right?\"\n\n\"Technically, it's just a freakin' towel to me.\"\n\n\"The language,\" Charles groaned. \"I might need to call my mom. I might need a cabin change.\"\n\n\"Shut up, Charles,\" one of the boys said from behind his closet door.\n\nJust then, Jason popped his head into the cabin. \"Gentlemen! It's gonna be ACES today, praise God. The sky shines blue, the slides await!\"\n\nNo amens chimed back.\n\n\"Happiness does not seem to be happening here. Lay it on me, my friends with a capital F. What's goin' down other than those smiles?\"\n\nCharles pointed at my towel.\n\nJason pursed his lips. \"Neal is a Pamper, Charles.\"\n\n\"Nick,\" I corrected.\n\n\"Right, right.\" He waved me away. \"Do you remember your infancy in Christ? In camp? Conviction is like a tulip slowly pushing through the earth. Dormant, then slow, and then boom! Pop of color!\"\n\nThis is what I was afraid of. If, in their eyes, a fictional character on a towel was a ticket to hell, then I can only imagine what they'd say about Diana. \"It's a towel. About a book. I'm sorry. I don't care. Can I trade with someone?\"\n\n\"I'm not trading!\" Charles near-shrieked.\n\n\"No, I meant with the camp. Is there an extra?\"\n\n\"It's already springtime for this tulip! Am I right? Awesome, awesome! I'll find something in lost and found. Just use your towel wrong side out for now. No worries, Nea\u2014Nick. Got it that time!\" he said with a display of jazz hands. \"We don't judge.\"\n\n\"Amen,\" Charles said.\n\n\"Really?\" I said to Charles.\n\nA face appeared behind Jason. And I was actually relieved to see someone I knew. Or kind of knew. It was the kid from the bus with the broken arm.\n\n\"Payton! P-Dog!\" Jason shouted.\n\n\"What's up?\" Payton said and rolled his suitcase between Jason and the door. He eyed me as he walked by, then took the last lower bunk. There was still an upper bunk left.\n\n\"What's up? Fist pump for Christ, is what's up!\" Jason fist-pumped. All by himself. \"Not to mention, the smiles in this room now that we've solved the Beach Towel Dilemma! Amen. Let's get to the pool. It's awesome out there, folks! HAPPINESS!\"\n\nSilence.\n\n\"Gentlemen, it's day one. Let's try that again. HAPPINESS!\"\n\n\"Happens here,\" a few mumbled.\n\n\"It's just hard, Jason, with the towel still in the room.\"\n\nMatthew put his hand on Charles's shoulder. \"It's fine.\"\n\nJason held out the PC Box. \"Any last-minute Ps and Cs for this bad boy before I put it in the sanctuary?\"\n\nI thought of Natalie. How easy it was for her to just drop her paper in the box. Then I walked into the bathroom to put on my swim trunks. When I returned, Jason had left.\n\nThere were seven of us left in the cabin:\n\nMe: the devil-worshipping Pamper.\n\nMatthew: who quickly took me under his wing (of an angel, obviously).\n\nCharles: math savant and HP-hater (likely a Slytherin).\n\nPayton: who ignored me altogether except for the occasional dagger-tossing glare. (He's on scholarship, Matthew whispered. Everybody knows the scholarship kids. Matthew was not on scholarship. He was a state tennis champion, for God's sake. Tennis players are never on scholarship.)\n\nStewart the Goth.\n\nChris Cooper: a.k.a Coooooooooop: The Chosen One. The Donkey Jockey. The Prodigal Son.\n\nDan: our counselor, who was at a meeting.\n\nFrom the clock by the counselor's bed, I surmised I had eight hours left till lights out. Anyone could survive eight hours, right?\n\nBut it was the first three hours I had to worry about. The Silent Three. Death by boredom. Hell did come before Eden. And there were demons. Or at least one.\n\nPRAYERS AND CONFESSIONS\n\nCharles: Dear Lord, thank you for this week. May Your holy plan for me reveal itself, and may I pass my AP Euro test, if it be your will. And my AP Chem. Though I probably don't need prayer for that one.\n\nDan: I worry about these kids. They're such puss\u2014sorry. They need Your strength, Jesus. Life doesn't get any easier than in their entitled little worlds. Real life is war. Or at least a battlefield. Thanks, Pat Benatar. Yeah, one of my dad's favorites.\n\nMatthew: Forgive me for renaming all my songs on iTunes. I mean Drake to Lecrae doesn't seem that bad, but Jason Derulo to Chris Tomlin? It's just lies. I feel bad for using talented Christians as a cover for my sinful listening habits. But, dude, my parents blame secular music for my downfall.\n\nJason: These kids can be tough, Papa. But I know you're with me. It's because of you that I can keep this smile pasted to my face. And because of all those summers at drama camp! Ha! Praise Jesus!\n\n\"You will not believe who's here,\" I heard from behind me. A fence rattled and, lo and behold, Jack Kerouac was scaling the chain link. He f lopped over the fence and squatted beside me.\n\n\"What are you doing here?\" I whisper-hissed. We were all supposed to be silent for three hours to ruminate on Jesus's sacrifice, which the Stage-Jesus had described in slasher-film detail before sending us on our way. The beating, the crown of thorns, the death, the three days in the tomb while His disciples mourned. \"I told you not to come.\"\n\n\"And miss all this?\" Jack looked out at the masses of teens, who had so far not exhibited any signs of ADD.\n\nSome had their Bibles open in their laps. Others had journals and pens and lay in the grass, scribbling their ref lections. Some had their eyes closed in prayer.\n\n\"Where's that little white paper you got? The one the girl on pretty road asked you about?\"\n\n\"The girl on pretty road? You mean, the one with the bandana?\"\n\n\"The one that got you all wondering and dreamy.\"\n\n\"Her name's Natalie.\"\n\n\"The very one.\" He nodded. \"Where's the paper?\"\n\n\"I put it in the box.\"\n\n\"Untrue. It's in your pocket.\"\n\n\"Why do you ask if you know everything?\"\n\n\"I'm evaluating your truth-saying.\"\n\nHe pulled a pack of cigarettes from his jacket pocket.\n\n\"Are you seriously going to smoke at Jesus camp?\"\n\n\"They were a gift.\"\n\n\"Who would give you a gift here?\"\n\n\"I got connections.\" He lit his cigarette. \"Like I said, you wouldn't believe who's here. Muhammad and Buddha. Some old Indian fella\u2014\"\n\n\"Native American.\"\n\n\"Sure, yeah. He talks like he's got a barrel in his belly, all round and echo-like. I got these here from a fine young woman called Ereshkigal.\"\n\n\"Oh yeah?\" I mocked. \"And where's Jesus?\"\n\n\"Well, He's with them, obviously. All friends, doing their thing, one man on earth at a time.\"\n\n\"Right.\" I rolled my eyes and snarked, \"Then, please, say hi to my sister.\"\n\n\"I can't do that. I've already told you.\"\n\n\"How convenient.\"\n\n\"Believe what you want.\" He took a long drag and exhaled. \"You can be such an arrogant little shit.\"\n\n\"Can you at least smoke downwind?\"\n\nHe surprised me by standing up and sitting down on the other side of me.\n\n\"Look at all those beautiful people, Nicolas. Imagine the depths in their hearts, the secrets.\"\n\n\"I don't want to know their secrets.\"\n\n\"All that truth f lying around their heads like birds.\" He was staring off, unfocused, slipping into his poet-self. I wondered for a moment if it was only tobacco he was smoking.\n\n\"And we go on shooting it down,\" I said in mock poet-stoner voice.\n\nHe snapped his head in my direction. \"Ah hell, what's that supposed to mean?\"\n\nJack was always digging around my psyche, and usually I just walked out of my room to get away from his prying. But I was stuck in pretend prayer, so I told him what I thought. \"I mean those truth-birds are being shot down by our word-bullets.\"\n\n\"Ah yes. Metaphor on, Nicolas.\"\n\n\"Seriously. All our words shooting out of our mouths. Always trying to tell everyone this or that. Hunting down the things we want to say. The things we want people to hear. Just leaving dead bird truths all over everywhere. Nobody knows how to say anything right.\"\n\nHe clicked his tongue at me, obviously disappointed by my pessimism. He fingered his cigarette while he stared out at all the campers sitting around, all meditative. \"Man, you are rolling fast down sad hill.\"\n\n\"I should've realized what was happening with Diana. I should've known. She should've told me. Maybe she did. Maybe words just fail us.\"\n\n\"We misuse them, misunderstand, misrepresent, miss the target,\" Jack singsonged. \"And now we're back to it. This is about your sister.\"\n\n\"Of course it's about my sister.\" It was always about my sister, and he knew it. He was prodding me. This is why I hated talking to him. Because his words were sharp, and all my packed-up-and-put-away sadness was going to spill out everywhere.\n\n\"Well, I can't always be sure. You've been all curled-over obsessing about Leah.\"\n\n\"Because she's alive! My sister's dead. I can't talk to her.\" I sighed. \"Leah's what's left of my sister. And to Leah, I'm what's left of Diana.\"\n\n\"I know\u2014yes\u2014I do know.\"\n\nI felt def lated. \"At the funeral\u2014\"\n\n\"Oh yes, I saw what happened at the funeral.\" Of course he did. Stalker.\n\n\"That's what I'm saying. That's all it was. Just grabbing on to what's left.\"\n\nJack smoked, pondered.\n\nA cricket started off in the dry grass. Someone cleared his throat. The wind picked up the leaves, hanging there like paper coins, the light shifting the color of them.\n\n\"Nicolas, that's what we've got. Words. That's all we've got. And if that chasm from soul to soul is as wide as the Grand Canyon, and perhaps it is\u2014ah hell, maybe even wider\u2014then we gawddamn better have a bridge. That's what our words are. They're the bridge from me to you, from you to your parents, from you to those lovely girls you met today\u2014\"\n\nI rolled my eyes.\n\n\"\u2014and that bridge might be rickety-rackety, broken, dangerous to cross. But, ah man, that's all we have.\" Jack stood up, brushed off his pants. \"Diana is walking a different road now\u2014long gone and down\u2014and there are no more words or nonwords to puzzle on and figure out.\"\n\nI looked up at him. \"But I was wrong. I didn't understand what Diana was saying.\"\n\n\"No, that's not it, not it at all,\" he said, shaking his head. \"There is no right or wrong. There's just what you do, and what happens after.\"\n\nThe tears needled. \"I don't like this after.\"\n\n\"You have something to say, Nicolas, and those words are gonna tear you up or rot you from the inside out if you don't tell it out. That's a worse after.\"\n\nMy throat tightened. I swallowed to keep all the words and all the sadness down. I looked back out at the people with the books and the pens, all the words.\n\nJack walked past me, following the fence up the hill. \"I don't think I care to sit around and witness your tearfall. There are such times when a young man needs some privacy. But I think you might've heard me this time.\"\n\nI was not going to cry at church camp. The clich\u00e9 was too much.\n\nCharlotte would've been too pleased.\n\nI swallowed the thick sadness in my throat. I took out my pen and composition book, scratching two lines on the Jack Kerouac tally page. I'd seen him thirty times now.\n\nF lipping to a blank page, I held the pen over the paper.\n\nIt hovered.\n\nThen I wrote it down. Just to see how confession felt. The words screamed back at me, rattling around on the page like prisoners. I ripped the paper out of the notebook and stuffed it in my pocket.\n\nA microphone screeched from the center of camp. I looked down the hill into the field, where a counselor stood in the center of the grassy area and tapped on the mic. Thump, thump, thump. \"Test.\" Screech. \"Can you hear me?\" Screech. Thump, thump.\n\nThe Silent Three was definitely over.\n\nI stood up and headed down the hill toward the counselor calling us to join him. The characters were coming back out of hiding. The leper. The blind man. The lame man on his mat. Jesus milled about. Music blasted from the speakers.\n\nI felt a tap on my shoulder and turned to find two counselors, both with long, dark hair, smiling down at me. Both wore robe-like garb, and one had fake wrinkles drawn on her forehead and around her mouth. I hadn't seen these two when we'd first arrived.\n\n\"Have you seen Him?\" the young one asked.\n\n\"Who?\" I asked.\n\n\"My son, the Son of God!\" the old one said.\n\nMary Magdalene and Mother Mary?\n\n\"He's over there.\" I pointed.\n\n\"Come closer,\" they urged, all smiles.\n\nI swiped at their hands, unwilling to be part of this skit. Unwilling to drink the Kool-Aid.\n\nThey tugged at my T-shirt.\n\n\"I got it. I'm going.\" I batted Mary Magdalene's hand away. Both girls fell out of character, struck by my unwillingness to play make-believe.\n\nThey lifted their robes to keep them from dragging on the ground and sauntered over to another camper on the outskirts, a girl who willingly followed them into the swarm. I weaved through the campers and characters.\n\nThe phones were back out. Mine registered no service, and I wasn't about to join the throngs for the photo shoot. But I watched.\n\nA group of about ten campers posed with some famous sinners, who stood next to signs the camp had clearly provided to help the Pampers get it. I didn't need the signs for all the characters. You didn't need to go to church to know the guy with the noose around his neck was Judas. But there were more difficult characters to peg, like the two thieves from the crosses, plus Barabbas. And the adulteress and her sheet again.\n\nAnd then I saw Jack, photo bombing the campers, his fingers making bunny ears behind Barabbas's head. I walked quickly toward him. He saw me coming and darted away like an unruly toddler.\n\nHe sauntered across the lawn, and I followed him. I took the paper out of my pocket. I'd show him I'd done it\u2014I'd written it down\u2014and then he could leave. He'd done his job, got me to confess.\n\nBut then the music clicked off and everyone turned toward a stage. I couldn't see what they were looking at.\n\nThe chant from the church that morning started up again.\n\n\"HAPPINESS!\"\n\n\"HAPPENS HERE!\"\n\nJack had vanished. I shook my head and walked back toward the crowd of campers.\n\nA man ascended some steps with a microphone in hand. \"Goooooooood afternoooooon, Christiaaaaaaans!\" A sad attempt at an impression of Robin Williams in Good Morning, Vietnam. Way out of character, because the man with the mic looked less like Robin Williams and more like Robert Redford in his prime. He'd been peeled right off an Abercrombie & Fitch poster. He was the guy they hired for Viagra or Crest commercials.\n\nEveryone cheered and shouted and whooped.\n\n\"Where are my Returning Champions?! My reigning Super Bowl Christians!\"\n\nCheers and whoops.\n\n\"Where are my Pampers?!\"\n\nDelayed whoops from a few, including the Marys fangirl, prompted by a nudge from Mother Mary.\n\n\"Are you ready for a Christastic weekend?!\"\n\nEarsplitting cheers.\n\nChristastic?\n\nI was finding it hard to know if my parents had signed me up for church camp, drama camp, or cheer camp.\n\n\"Then let's pump it up! Fist pumps for Christ!\"\n\nThey all threw their arms in the air and pumped their fists, chanting.\n\nFist pumps for Christ!\n\nFist pumps for Christ!\n\nThese people had an infinity worth of cheers.\n\nThe Marys stood at the back corner of the stage, fist-pumping along with them. Their enthusiasm would've been more convincing if their beloved Jesus hadn't died at the end of this story.\n\nThe pastor guy gave them each a high five and the three of them exited the stage just as the prostitute walked up, carrying an enormous Bible\u2014like if the Bible took steroids.\n\nJoining her on the stage were two counselors, a girl and a guy, both wearing Hammer pants and gold chains. It was not MC Hammer music that echoed through the canyon, though, but Sir Mix-A-Lot. I recognized the notes of \"Baby Got Back,\" and I waited for God to send lightning bolts and strike us all dead. Then I saw the girl put the mic to her face and, glaring at the prostitute character, begin her rap:\n\nOh, my gosh, Becky, look at her Bible.\n\nIt's, like, so big.\n\nShe looks like one of those Jesus followers.\n\nThe girl stepped aside, and the boy started rapping. As he did so, counselors walked among the crowd, distributing Bibles. Some campers held out their own big Bibles.\n\n\"Fun, right?\"\n\nI turned around.\n\n\"Did I scare you?\" Natalie laughed, and her eyes f lashed. \"Boo!\" She shimmied to the beat and sang along.\n\nI like big Bibles and I cannot lie,\n\nYou other followers can't deny\n\nThat when a friend walks in with a shiny leather case\n\nAnd those pages in your face\u2014\n\n\"And, look, you've got your confession ready, even,\" she said, nodding at my hand.\n\nI looked down. There was the paper, in my hand. I crumpled it, freaked that it might fall on the ground, or she might grab it.\n\n\"Here!\" Natalie called to Jesus, who was walking through the crowd, carrying the PC Box.\n\nI tried to turn around. \"That's okay,\" I mumbled. But then Jesus was right there. His dark eyes waiting, the slit in the box wide open. Natalie smiled and looked at me expectantly. And there was Charles, walking behind Matthew, headed right toward me.\n\nDeep in the Book we're reading,\n\nI'm hooked and I can't stop needing\u2014\n\nMatthew put his hand on my shoulder. \"Hey, dude, hurry up. Let's hit the pool.\"\n\n\"He's got to drop his confession in first.\" Charles pointed at my clenched fist. Was this payback for Harry Potter?\n\nThe paper wilted in the heat of my hand. Natalie's smile never wavered. Jesus shifted his weight. Patience was barely his virtue.\n\nAnd Jack appeared, leaning up against the fence across the field, smoking, watching, waiting.\n\nI took the paper out of my hand\u2014\"Oh yeah, this. Sure.\"\u2014and uncrumpled it. Folded it once, twice. Slipped it in the box.\n\nMy churchboys tried to warn me\n\nThat Book you got makes me so holy\u2014\n\nThey rapped on.\n\nLater, I would think about the sheer number of secrets dropped into that locked box.\n\nBut at that moment, a counselor took my cell phone, and everyone else's, exchanging them for Bibles.\n\nMy soul died a little.\n\nPlease forgive them, Sir Mix-A-Lot. They know not what they've done.\n\n\"Behold Eden Springs, Nick,\" Matthew said as we walked through the gate to the pool. \"I know Eve was totally nude in Eden, but this is a close second.\"\n\nHe tossed his beach towel on a deck chair. He surveyed the sea of half-naked campers like a captain taking stock of his ship. Meanwhile, I kept my eye out for Jesus and the PC Box.\n\n\"Asstastic,\" Matthew said, more to himself than me. \"Everywhere, fine, fine ass.\" He yanked his eyes away from the fine, fine ass. \"I know that's what you're thinking. I can see you making your assessments.\"\n\nI'd actually been pondering how much money I could pay Jesus to let me pull my paper out of the box. And wondering where I could stash my Harry Potter towel to avoid further scrutiny. But I said, \"I was thinking titillating. Literally titillating.\"\n\n\"Yes! I love me some SAT vocab!\" Matthew shouted. \"This dude's going Ivy League!\"\n\nI laughed and folded my towel into an inside-out bundle so that no one could see the boy wizard.\n\nMatthew shoved me. \"Quit doing your laundry and see the sights!\"\n\nThey didn't tell you on the \"What to Pack\" list not to bring a Harry Potter towel. They also didn't tell you not to bring your tarot cards, your crystal balls, your voodoo dolls, or your statues of Aphrodite. They didn't tell you not to bring alcohol or cigarettes or cyanide tablets or LSD or knives or guns either.\n\nAnd they didn't tell you not to pack bikinis.\n\nApparently not all temptation is equal.\n\n\"Titillating!\" Matthew repeated, beaming at me. \"Nick's a boob guy.\"\n\nI wasn't a boob guy. I wasn't an anything guy, really. I mean a hot girl is a hot girl. But even Diana used to give me a hard time about my lack of romantic experience. You should be able to diagram the female anatomy at least as well as a sentence, buddy, she used to say. And not because of health class worksheets.\n\nAnd of course Jack was chagrined by my lack of sins of the f lesh. I imagined him sitting in a tree with binoculars, pedophile-like, staring down at all those leggy ladies.\n\nNext to me, Matthew's voice took on the Martin Luther King Jr. boom. \"I have a dream, Nick. I have a dream that one day, in the sweltering heat of Eden Springs... well\"\u2014he went back to his own voice\u2014\"I don't know how the speech goes. But pretty much there're two girls to every guy, so that's my dream.\"\n\nAs he stood there in his swim trunks with his tennis player\u2013lean physique, I was sure that there were more than two ladies for every Matthew. More like thirty-two girls for every Matthew, and negative eleven girls for every Nick. After adjusting for those numbers, I needed Homeschool Charles to figure out how the odds played out for the rest of the guys.\n\nI continued Matthew's rendition of \"I Have a Dream\" scandalously, at least as far as my memory would serve. \"Let freedom ring... from the mountainsides.\"\n\n\"And prodigious hillsides.\"\n\n\"And curvaceous slopes.\"\n\nClearly, MLK had been a boob guy.\n\nAnd Eden Springs was worried about my Harry Potter towel.\n\nMatthew hollered out to the swimmers and sliders, \"I love you, Eden Springs!\" He threw his arms into the air and whooped. The counselors, the campers, everyone looked over at him. \"Fist pumps for Christ!\" He yelled once more. A couple kids pumped their fists. \"Hells to the yes.\"\n\nA girl with a book\u2014she was too far away for me to make out her face\u2014looked over and gave a slight wave. There was only one girl with a book. Natalie. I waved. Then I remembered that I was standing next to Matthew.\n\n\"Is that how they do it at Valley Christian? MLK-turned-porn?\" I asked, pretending I hadn't just thought prettycute girl was waving at me.\n\n\"Hells to the no, Nick. My parents pay for me to go there so that I can be 'positively inf luenced' by all these blessed lovers of Jesus. And then the private-school girls take their clothes off for one week at Eden Springs, and it's all worth it.\"\n\nNatalie was in a bikini, just like most of the other girls, but it wasn't the bikini. It was the way the light shimmered off the pool and winked from her sunglasses. It was the way her painted toenails bounced to the pop song blasting from the speakers. It was the way the navy blue of her bikini bottom lay against her pale thigh. Frankly, it was the way she read\u2014or wrote in!\u2014a book by the pool, while everyone else played volleyball and chicken and Marco Polo. \"I get it,\" I said to Matthew, just to say something.\n\n\"Oh no, young Nicolas, I don't think you do. My life lacks the titillating pleasures of your public school experience.\"\n\n\"There's nothing titillating about my experience, personally.\" My only experiences with girls were awkward things you needed to apologize for.\n\n\"Ah, but Nick, there's at least the possibility...\" His voice fell away in reverence. \"My experience is pent-up and buttoned underneath starchy blouses.\" He grabbed my arm, and I stopped pretending not to stare at Natalie. \"Speaking of asstastic and titillating both...\"\n\nIt was hard not to see who he was talking about. Long golden legs, long dark hair. White bikini. Red lips. \"Ho-Lo.\"\n\nIt was the superhot girl from the bus. \"Ho-Lo?\" I asked.\n\n\"Her name's Holly. Ho... well, that's obvious. Lo, as in like J-Lo.\"\n\n\"Jennifer Lopez? She's our mom's age.\"\n\n\"Don't overthink it. We couldn't figure out how to combine her name and Kylie Jenner's,\" Matthew said, watching Holly saunter to a deck chair.\n\nWatching Holly was like watching one of those slow-motion shampoo commercials. Slo-Mo-Ho-Lo.\n\nMatthew interrupted his own reverie: \"Speaking of fast girls.\" He led me to the waterslides. \"The Jezebel,\" he said, pointing to the slide's name.\n\nI laughed. At the top, kids lay on their bellies on mats and readied themselves for a race straight down the yellow slide to the water below.\n\n\"Head first, that's how she likes it,\" Matthew quipped. \"Right, Holly?\" he called over to her.\n\nHolly looked up. Matthew waved innocently. Holly just looked away.\n\n\"The line is long,\" he said to me, \"but it's worth it.\"\n\nI laughed because he was funny\u2014supersmart funny.\n\n\"I mean, I'm sure Payton thought the wait was worth it, but he clearly got sick of the long line of male suitors she entertained,\" he continued.\n\nI didn't really know this girl Holly\u2014though to hear Matthew chatter on and on, others seemed to know her quite well. Just like Charlotte.\n\nI didn't want to think about my oldest sister. I loathed how she had treated Diana after coming back from Eden Springs. I despised that this camp, with its list of rights and wrongs, had snipped the threads that tethered my sisters to each other. I detested that nothing on this earth could tie them back together\u2014not even regret after Diana died, if Charlotte felt that at all.\n\n\"Yeah, this line's too long,\" I said. \"Let's catch this one later.\"\n\nWe headed over to the Belly of the Whale slide. Along with a couple other kids, we f lipped in loops and circles through the dark tunnels on the extra-large tube. Over and over, I raced down that slide, and I refused to think about Charlotte, and I refused to think about my confession.\n\nMatthew and I spent the rest of the afternoon at the pool, mostly going down the seven waterslides. We jumped off the ten-foot platform-cliff into the deep end of the pool. And when we'd had our fill of snacks and ogling and water sports, we walked back to the chairs to grab our towels and head to the showers.\n\nExcept, my Harry Potter towel was gone.\n\nMatthew rubbed his hair dry with his towel while I searched the pool deck. It had cleared out a bit as people left to get ready for dinner. Goth was lying out in a pool chair next to Nick Sampson\u2014my Donkey Lottery nemesis\u2014and the wheelchair girl. Charles was talking to a group of bikini-clad girls. I headed over to him, sure he'd swiped my towel. Maybe burned it. Like the Joan of Arc of beach towels.\n\nThen a hand on fell on my shoulder and spun me around. Jesus. I immediately panicked. Had he read what I'd dropped in the PC Box? They'd all assured us that nobody read them. And I'd seen the size of that padlock but... what if?\n\n\"You the kid with the towel?\" he asked, his voice lacking the God-boom one might expect.\n\nI nodded. There could be only one towel of import.\n\nHe handed me a brand-new\u2014really, there was a Polo-brand price tag on it\u2014yellow towel.\n\n\"I already wrote about this and put it in the PC Box,\" he said.\n\n\"About the Harry Potter towel?\"\n\nHe nodded. \"A prayer,\" he said and waved me away. \"No need to thank me, man.\"\n\nThe evening service in the sanctuary, where I'd expected venomous snakes and raising of hands and healing of the crippled, was oddly normal. The speaker from the morning, the guy who had welcomed us into camp, turned out to be the pastor, Pastor Kyle.\n\nHe looked nothing like any pastor I'd ever seen. Though, in truth, the only ones I'd ever seen were featured on the nightly news. The ones accused of child rape. All pornstache and thick glasses.\n\nBut Pastor Kyle looked like Adonis. Like Paul Walker (may he rest in peace). After seeing the plastic-perfect kids modeling on the cover of the camp brochure, I was prepared for his tanned body and white teeth, and yet I found the pastor's beauty\u2014yes, that's what I said, beauty!\u2014unearthly. I'm sure he led a lot of campers to God; girls would follow this guy anywhere. And boys? Well, they'd just follow the girls. I mean, everyone knows the Adam and Eve story.\n\n\"I know the Silent Three was an amazing time for you. I know the Lord was with you in your time of ref lection and prayer. All those sins and secrets, friends.\" He talked about sin, as they all do, but without the hellfire and brimstone I was anticipating. We would have another chance with the PC Box tonight, he told us, gesturing to an older woman on stage, who shuff led off in her too-short shorts\u2014presumably to retrieve it. Then Pastor Perfect told us a story from his own childhood where he did something wrong. \"But that's a mute point,\" he said, \"because Jesus saves!\"\n\nFor God's sake, Jesus needed to save him from his grammar.\n\n\"Moot,\" I said to Matthew, who sat next to me, rapt. I nudged him. \"He meant it was a moot point.\"\n\nHe shrugged me off, apparently not interested in the irony of a point in a speech, which you say so that people can hear you, being mute.\n\nI felt like the pastor's grammar failings had to be a sign that the universe was balancing itself, though. Like, Pastor Kyle couldn't have it all. To have those bright teeth, he needed to sacrifice his understanding of the English language.\n\nI had the latter, thus my need for braces and a tan.\n\nWhile everyone jotted down some scripture that f lashed on the screen, I took out my little composition book, turned past the Jack Kerouac tally sheet, and started a list: MUTE\u2014MOOT, I wrote on the next page.\n\nThe older woman scuttled back onto the stage, interrupting Pastor Kyle's massacre of vital grammatical rules. She stood on her tiptoes, gesticulating and whispering something into the pastor's ear. The pastor leaned down, pushing the mic as far away from her as possible. Then she hurried away, nearly tripping down the stairs.\n\n\"Okay, my Champions,\" he said, but it lacked his Lord-lovin' fervor of moments before. \"Now, there is no need to panic.\" Which is a sure way to incite panic. Immediately there were whispers, murmurs, oh god, oh god, oh god. \"There's been an incident.\"\n\nEyes darted. Faces paled.\n\nWas it a terrorist attack?\n\nAnother 9\/11, on West Coast soil?\n\nA president assassinated? (Actually, that one might warrant a celebration\u2014at least, for the eleven Democrats in the crowd.)\n\n\"We're hoping someone will come forward immediately.\" Pause. \"We'll get to the bottom of this atrocity.\" Pause. \"Happiness will happen here, despite this grave situation.\" He was getting his evangelical groove back.\n\nHe cleared his throat.\n\nThere was a collective intake of breath.\n\n\"Someone has stolen the PC Box.\"\n\nYou know those movies where the plane f lies over unsuspecting civilians (often in rags and farming rice paddies or shepherding goats, because people in rags are worth more emotionally in film and worth less in wartime economics... at least to the people pulling the trigger)? The propellers beating against the sky, loud and violent. Then the whistle of the falling bomb and the explosion as it hits the earth.\n\nAnd then the vacuum of all sound. The silence, complete and irrevocable and soul sucking.\n\nPastor Kyle's words dropped down on me. And then the heat. My nerves f lashed fire through my arms and face and left me tingling, then numb.\n\nSomeone was going to read my confession. My brain scrambled for the words I'd scrawled on that paper. What did I write? What did I say? I mean, exactly. Could I explain it away? Could I make it a joke?\n\nThe campers were silent. Blank eyes. Clenching jaws. They were in their heads. What did we write? Exactly? Oh no. But did we write the name? Did we say our names? Her name? His name? Their names?\n\nShit. Shit. Shit.\n\nAnd then the f lush of shame as we realized what we wrote. Exactly.\n\nThe sound filtered back into our brains, our faces blanching, our eyes searching those around us for sympathy.\n\nI looked for Jack. Not because he had stolen the box. But because this was his fault regardless. I shouldn't even have to care. I wouldn't care, if it hadn't been for his constant blathering.\n\nThe one time I really wanted him, and he'd vanished.\n\nWho the hell stole the box? Suddenly, I wasn't seeing potential sympathizers in the crowd around me. I was seeing culprits. We were all seeing culprits. Our shame had faded to fury, our eyes narrowing. Who the holy hell stole that box? Our confessions, our secrets, the things we refused to say aloud to people, not even in the dark.\n\nI looked at Matthew, but he stared at the carpet, unblinking. Passively terrified. I don't know what he'd written on his paper, but no way he'd stolen the box. That, or he was a fantastic actor. Was he in drama at his private school? Did he say that? No, tennis.\n\nPayton's face had turned rage red.\n\nHolly's mascara was running.\n\nCharles watched me, nose f laring.\n\nHe'd picked me as the culprit, clearly.\n\nEyes f lashed from face to face. We searched for the offender now, not the sympathizers.\n\n\"It's okay, I didn't write my name on either of mine,\" a girl in pink shorts said to her friend behind me.\n\nI turned.\n\n\"Neither did I,\" her friend said back, and then her mouth fell open. \"But I wrote his. I wrote his. Oh, my God.\" She slid back down into her seat and hid her face in her hands.\n\nMurmurs, condolences, rumors washed through the crowd.\n\nI saw Natalie then, just behind pink-shorts girl and her friend. She scanned the back of the room, mouth slightly open\u2014searching for something, maybe. Or someone. She was agitated, but not crying like most of the girls. She brushed her hair back and pushed through the aisle. I watched her turn around once more. She took us all in, scanned the whole room, then she walked out the door.\n\nI told Matthew I needed to go to the bathroom and then excused myself down the row of campers. Most of them were out of their seats, standing with their arms around someone else who thought they had it worse than the next person. Groups had formed up the aisles and in front of the stage. The hugging, the tears.\n\nI stepped out into the night, the door closing slowly behind me, trapping the chatter and prayer and sobs in the sanctuary. I stood in relative silence. The wind whispered its own secrets through the leaves of small trees. I listened for footsteps. I peered into the shadows cast by light posts surrounding the field. Natalie was nowhere.\n\nNo. Natalie was somewhere, and I wanted to find her. Maybe it was worry. Maybe it was curiosity. Or maybe it was suspicion. Maybe it was all three.\n\nI walked the path through the middle of camp, scanning the field, the empty porches rolling out from each building. There was no one. Every camper, counselor, pastor, worker was in the sanctuary dealing with the fallout of the PC Box theft.\n\nI turned to face the cabins. Maybe Natalie had snuck out to be alone, to figure out her own mess without the crying and condolences, without the it's so much worse for me though sentiments that (accidentally?) slipped out or emanated from superficial hugs.\n\nI didn't know which cabin was hers. Light poured through the curtains of some windows; others were dark. I knocked on the first cabin door, waited. No answer. I skipped steps up to the second f loor and knocked. No answer.\n\nWhen I turned around to head back down the stairs, a silhouette of someone walking toward the hillside caught my eye. I couldn't tell if it was her. I couldn't even tell if it was a girl or boy. But I hurried down the stairs and jogged across the field anyway.\n\nThe hill turned out to be more like a cliff. It was dry, rocky, sagebrush\u2013covered terrain, clearly not maintained by the crew who kept Eden Springs green and fresh. Unauthentically perfect, just like humans prefer. Nothing's perfect\u2014but it can at least look that way, right? Facebook, Snapchat, friendships, marriages, dinners, family vacations, gardens, houses, kids. I thought about my own family. Fa\u00e7ade after fa\u00e7ade.\n\nThere was no sign of a trail, or of the person who'd been walking up it. No sign of anyone else outside, either. They were all still locked away with their sorrows, it seemed.\n\nI walked along the hillside and toward the barns, and realized there was an old, barbed wire fence surrounding the camp. The terrain, the cliff, the fence. Clearly I wasn't supposed to leave, even if I found the trail. And neither was the person on the cliff. But there must have been way to get up there.\n\nSuddenly, I heard the scrape of rocks. A shadow moved behind the barns. I changed course again. Slinking along the wall of the building, I looked for Natalie. I looked for any shape.\n\nAnd then I found it.\n\nIt was a donkey.\n\nIt shifted its weight, sending more rocks down the embankment. The donkey barely registered my presence. It was probably exhausted from carrying the lucky lotto winners to camp.\n\nI leaned up against the barn. I had no idea what I was doing, besides looking for a curious, pretty girl.\n\nI wondered if Natalie had left the building because she knew something about the PC Box. There was no other reason to slip out so quietly. And, unlike everyone else, she hadn't looked traumatized. She hadn't been crying or hugging.\n\nWhat were the odds that Natalie had done it?\n\nBut then I thought, maybe Pastor Kyle was behind this. Maybe it was all a trick. I'd heard one time about a church where the pastor had arranged for gunmen to burst into the building. They'd lined up members of the congregation and asked them if they believed in God. The idea was to see who would deny Christ at gunpoint. I don't know what happened. I don't know how many people stood their religious ground. I do know that no one got killed, but it was still some sick shit.\n\nMaybe that's what Pastor Kyle was doing. I mean, this camp went all-out. Maybe this was all leading up to some kind of (divine!) intervention. Maybe our parents were all going to show up, sit at tables with us, and talk us out of our drug addictions, drinking problems, and gambling issues.\n\nBut I had no idea how I'd find out if he was doing that. And for the moment, Natalie was the one who was acting suspiciously.\n\nThe donkey shifted, closed its eyes.\n\n\"Have you seen a girl? She's about this tall...\"\n\nThe donkey continued his silent treatment. Useless.\n\n\"Jack?\" I asked the night. He was constantly telling me what to do. For once, I could actually use his intrusive line of reasoning. \"Jack.\"\n\nHe didn't appear.\n\n\"Are you Jack?\" I asked the donkey. \"Because he is a total jackass, so you kind of fit the bill. No offense.\"\n\nThe donkey said nothing. Clearly no offense had been taken.\n\n\"Hey.\" A hand landed on my shoulder, and I spun around.\n\nIt wasn't Jack or Natalie. It was the Jesus character.\n\n\"What're ya doin' out here?\" He didn't sound at all holy. He sounded like a regular man, like at the pool earlier.\n\n\"I'm looking for someone.\"\n\n\"There ain't nobody out here.\" He sounded like a regular hillbilly man.\n\n\"I saw somebody come out here, so\u2014\"\n\n\"Like I said, there ain't nobody out here.\" He was like Jesus appearing on an episode of Duck Dynasty.\n\n\"Okay...\" I wanted to ask him what he was doing out here. Jesus was born in a manger, and I was in fact standing next to a barn, but this dude was a wee bit bigger than an infant and had no signs of frankincense or myrrh or Bible-times diapers.\n\nI eyed the man. As Jesus, he carried the PC Box around all day. He could very well have stolen it and hidden it out in the barn.\n\n\"Lemme walk ya back,\" he said, suddenly all friendly. And, quite frankly, acting a little suspicious.\n\nJesus followed behind me as we walked around the barn. I craned my neck to peek into the shadows, hoping to catch a glimpse of the box. His hand landed on my shoulders and he ushered me forward a bit more quickly.\n\nHe was an older guy, probably in his thirties. Hair pulled back in a ponytail and beard a little gray. His skin was pale, too pale for the real Jesus. And he didn't even bother with bronzer.\n\n\"You know Jesus was from the Middle East, right?\" I snarked.\n\nHe looked down at me, but kept his hand on my shoulder till we'd both stepped out onto the grassy field. \"How ya know I ain't?\"\n\nI fixed my eyes on the hillside, searching for signs of movement, and pondered how to answer. I wanted to point out the obvious, namely his hillbilly accent. But\u2014who knows\u2014maybe hillbilly accents were a thing in the Middle East. Maybe the goat herders of the Middle East sounded a lot like the crawfish hunters of rural America. Maybe there was a global hillbilly community. An Iraqi Duck Dynasty. Mexican Duck Dynasty. Tibetan Duck Dynasty. A dynasty of Dynasties!\n\nHillbilly Jesus probably wasn't going to follow my train of thought, though, so I answered, \"Because of Donald Trump and the Muslim ban.\"\n\n\"Touch\u00e9.\" He laughed. Whoa. Hillbilly Jesus used a French word. Was he trying to impress me? Shepherd me into his f lock with his broad sense of language and culture? What's French for gag? \"How you likin' camp?\" he asked.\n\n\"It's fine,\" I said.\n\nThere was no sign of Natalie at all anymore. Not up on the hill, not down here in the field.\n\n\"Listen,\" I said, \"I can find my way back from here.\"\n\nHe ignored me. \"You on scholarship?\" he asked.\n\n\"Me? No. Why?\" I mean, I didn't go to some rich private school like most of the kids at the camp, but my family was doing just fine. \"Do I look like a poor kid?\"\n\n\"No, but yer attitude is poor.\"\n\nOh, jokes! Snarky bastard. \"Well, clearly they don't pay you enough here, because your Jesus accent is poor.\"\n\nJesus laughed and laughed. \"I asked about scholarship because I was wonderin' if yer on work crew.\"\n\n\"Uh, no.\" I tried to walk faster, to shake this guy, but he kept at it.\n\n\"Hey, it ain't bad. Some of yer friends over there are on my crew.\"\n\n\"Doing what?\"\n\n\"I'm Jesus by day, custodian by night. I was feedin' the donkeys jus' now. Toilets next.\" He nodded toward the sanctuary. \"Yer friends help out.\"\n\n\"Well, I don't need the money.\"\n\nHe raised his eyebrows at me. \"Maybe you jus' needda work.\"\n\nI got the feeling he was calling me an entitled ass, but I decided to go the way of a docile donkey and take no offense. \"Okay, cool, thanks.\"\n\nThen he added, \"An' forget that business with the towel. Have a good time. You got yerself a new towel, donchya?\"\n\n\"I never cared about the towel,\" I said. I could feel nerves tugging at my insides. I was running out of time. \"I care about that PC Box.\"\n\n\"That's some messed up sh\u2014uh, business.\" Jesus raised his eyebrows and swiped a hand across his forehead.\n\nI wondered if he'd been in that barn turning too much water into wine. Maybe Jack had been in there with him. \"Are they going to do something about it?\" I asked, hoping I could shake him before we got to the sanctuary. I wanted to get on my detective way.\n\n\"What're they supposed to do?\"\n\n\"Search the camp? Appeal to the goodwill of kidkind? Interrogate campers, cabin by cabin, with bamboo up the fingernails? Crucify someone?\" My helpless fury stirred. \"I don't know. Something.\"\n\nThe camp had managed to provide me with a Polo towel in order to conquer such evil as magic, but they had no plan for the evil of stealing. And, as always, Jesus was some white dude. The hypocrisy confounded me.\n\n\"Look, it's going to be okay.\"\n\nI shook my head, defeated. He didn't understand, and it was clear I wasn't going to be able to go anywhere tonight without this character.\n\n\"Everything happens for a reason.\"\n\nOh, gawd. Not that platitude. Hypocrisy and na\u00efvet\u00e9. The reason bad things happened was that people were bad.\n\n\"Besides. It ain't so bad. John 1:9, Nick.\"\n\nHillbilly Jesus quoting the Bible. Whoa. Actual whoa. Too bad I had no idea what he was talking about.\n\nWe were halfway across camp when the doors to the sanctuary burst open and all the campers poured out of it. Some bass pumped through the speakers and out the door, but I couldn't tell what song was being bastardized this time. The campers were in pairs and groups, and I could hear them laughing and singing.\n\nMaybe they'd found the box. I quickened my step, leaving Jesus to clean up after the humans and the asses.\n\nMatthew was nowhere to be found. Natalie was still AWOL. Everyone was forming groups and sitting in circles on the well-lit patio, Bibles and notepads open on their laps. Holly was the last person to come out of the sanctuary. She somehow didn't look like the girl I'd seen at the pool that afternoon. She was smaller somehow, chin lower, eyes downcast. It didn't take away from how sexy she was though.\n\nBold as it was to approach Red Lips, something about Matthew's rumor report at the pool that afternoon made me feel like I could. If she'd Netf lix-and-chilled with every other guy, she could talk to me, right? \"What's going on?\" I asked her.\n\n\"Nothing. How're you?\"\n\n\"No, like, what are we supposed to be doing now?\"\n\n\"Oh.\" She looked around. \"I don't know. I was in the bathroom.\" She sounded different than she had on the bus. Different than at the pool.\n\nThat's what it was: she sounded real. No Pampers\/campers rhymes. No lipstick. She was shell-shocked, like the rest of us. The disappearance of the box had knocked us all down a peg, I guess.\n\n\"Are you okay?\" I asked.\n\n\"I just. You know. What people could say.\" She looked back at the sanctuary door. \"What people do say.\"\n\nI thought of how Matthew had described her. I wondered if it annoyed her. But she had to know what it looked like, the way she dressed, those red lips.\n\n\"Maybe\u2014I mean, probably\u2014I need to just pray about it,\" she said, and she sounded sincere.\n\n\"Where've you been?\" someone asked from behind me.\n\nHolly looked up, and I turned to find Payton. I remembered the way he had glared at me on the bus. Oh no.\n\nWe answered simultaneously:\n\n\"Barn.\"\n\n\"Bathroom.\"\n\n\"Together?\" The WTF on his face made the reason for the question clear. But it was a stupid one anyway. Obviously barn and bathroom meant we were not together.\n\n\"I was looking for you,\" Payton said to Holly. \"We're supposed to be in groups.\"\n\nHolly's demeanor changed back to Red Lips from the bus. She crossed her arms and stepped a little closer to me. Double down on the Oh no.\n\nStep away from the nerdy kid, Ho-Lo, I wanted to say. I like chess\u2014not that I'm going to tell anyone that\u2014but please don't make me your pawn in this game with Payton.\n\nSaved by the Goth. Stewart walked over. \"Anyone see Natalie?\"\n\nI said nothing. I was standing in the middle of a romantic showdown of some kind between Payton and Holly, and Natalie may or may not have been scaling a dusty cliff, and I didn't want any part of it.\n\nGoth led the three of us to the group of kids from our bus: brother and sister (Chris and Christina, Christastic!), Wheelchair Girl, and Nose-Picker.\n\n\"We were just debriefing about what happened tonight,\" Goth informed us.\n\n\"What are they going to do about it?\" I jumped in.\n\n\"What do you mean, what are they going to do about it?\" Goth asked, crossing his arms over his chest.\n\nClearly he'd misunderstood my tone. \"No, I mean, I want them to do something about it. That's what I'm saying. I'm keenly interested in their plans.\"\n\nThe group eyed me.\n\nI said, \"Look, are we going to look for the box, or are we going to interview people, or what?\"\n\n\"That's what the pastor is working on. We can only control what we can control.\" Goth sounded like the counselor my parents sent me to. \"Let's talk about how we're feeling. How are we doing right now?\" So, so counselor-y.\n\nAnd so that's what we did. For about thirty minutes. Chris. Christina. Nose-Picker. Goth. Holly. Payton. Everyone's feelings, not surprisingly, could be summarized this way: super-duper-Christastically pissed and scared.\n\nI wanted to punch a wall, sitting there. It was like counseling all over. But it was so stupid. I'd done this all before, and here's what I knew: Talking about feelings doesn't change anything. It doesn't change your sisters, and it doesn't change your parents, and it doesn't change anything that's happened in a family. It doesn't change the weight of your parents' marriage on your shoulders.\n\nBut my confession, so recklessly stuffed into a box that was now missing, could change what remained of my family.\n\nI had to find that box.\n\nJust as I moved to get up, Goth knelt in front of Wheelchair Girl. I'd forgotten about her. I assumed she hadn't written a confession. I hadn't seen her write one, and I didn't know if she could. She hadn't spoken. From what I could tell, she communicated with her eyes. \"Monica,\" he said, and touched her arm. Goth asked her yes\/no questions, and her eyes moved right for yes, left for no. Yes, she was having fun. No, she was not sad. No, she was not scared.\n\nI envied this girl one thing. And it was that. She was not sad, and she was not scared.\n\nI was both of those things. All the gawddamn time.\n\nAnd then Natalie appeared, joining the group with her light step. And, of course, a book in her hand. \"Sorry. I'm really sorry. I had to get out of there after that debacle. I did some praying. And I really think it's going to be okay.\" She sat next to Nose-Picker and gave her a side hug.\n\nThey all did the hug thing. I did not. I watched Natalie. I looked for signs of dust on her knees, on her shoes. I didn't buy her story about needing to pray.\n\n\"I really do think it's going to be okay,\" she repeated. \"Not just for me. For everybody here.\"\n\n\"Well, praise Jesus, then, right?\" And just like that, Holly turned on the super chipper voice I'd heard her use on the bus.\n\nIt seemed our debriefing was officially over. Natalie spotted Pastor Kyle and headed his way. Goth sat quietly by the girl in the wheelchair. The kid he was hanging out with by the pool earlier came over and sat by them. Payton wandered over to another group, but he kept his eye on ours. Well, on Holly.\n\nHolly glanced at him.\n\n\"Is that your boyfriend?\" I asked her.\n\n\"No.\" Then, quite loudly, \"He's my ex-boyfriend.\"\n\n\"Are we praising Jesus for that?\" I asked. \"T-G-I-J and all that?\"\n\nShe kind of laughed. The smile on her face seemed honest again. She was pretty in that moment. Pretty in the way being real makes someone pretty.\n\nShe f lipped her notebook closed, and I noticed the guitar design on the front.\n\n\"Do you play the guitar?\" I asked.\n\nShe considered me. Then, \"Um, yes.\"\n\n\"You do?\"\n\n\"What, is that hard to believe?\" she asked, still smiling.\n\n\"Are you, like, in a band and stuff?\"\n\nShe blushed and looked away. \"I wouldn't call it a band. I play with a couple of people, yeah. If we had a drummer, we might be a band.\"\n\n\"That's so cool.\" It was cool, and it made her even hotter. She had the perfect body, and she played an instrument.\n\nShe rolled her eyes, \"Yeah, well,\" she said, dismissively, but she was smiling. \"I play mostly by myself. I actually like to write songs too,\" she continued, \"but I've never really tried to sing any of them. I don't sing. Like, really don't sing. But, I think\u2014\" And then she stopped.\n\nI turned to see what she'd seen. Payton. Her eyes darkened for just a split second before she asked, \"Do you want to pray with me?\"\n\nShe and I hadn't moved. We still sat cross-legged on the ground while the others milled about. I started to decline, but she grabbed my hand. \"Lord, Your power is so great, so grand, and so big. So, so big.\" She talked too loudly. She definitely wanted everyone (especially those with broken appendages) to hear. Her other handed rested on my thigh. \"We are just so blessed. Thank you for bringing Nick to camp, Jesus. Funny Nick.\" She held my hand quite tightly. The other one brushed my shorts against my leg. \"Thank You for breakups and the rough roads and the hardships. All the hard, hard things.\" Her hands squeezed the parts of me she touched. \"In Your name, Amen.\"\n\nShe opened her eyes, and they f lashed golden mischief. Shit. I was waxing poetic.\n\n\"A\u2014\" I croaked. Jesus. My voice hadn't cracked since eighth grade. \"Amen.\"\n\nHolly looked back at Payton, and smiled even more triumphantly. Then she stood up, brushed off her shorts, and walked off toward some other group.\n\nI stayed where I was, abandoned to think about big and stiff and firm and hard, and her hand making it so. I could not move.\n\nNatalie was staring at me, and her wry smile was growing. Now we'd both seen something we didn't want the other to see. I'd seen her leave the building, and she'd seen the effect a hot girl's hand has on a virgin grammar nerd, known as Turwaithion to his Elvish buddies.\n\n\"Do you need a book?\" she said, holding one up. \"You know, for your lap?\"\n\n\"Where'd you go tonight?\"\n\nShe narrowed her eyes, a touch of her smile remaining. \"Why are you asking?\"\n\n\"Because I saw you leave.\"\n\nShe laughed.\n\nI continued, \"Why did you leave right after the box was stolen?\"\n\nShe bent over in laughter. People were watching. \"Wait, are you interrogating me?\"\n\n\"No, I'm just asking.\"\n\n\"Is it weird that I'm f lattered by your suspicion?\"\n\n\"Uh, yeah.\"\n\n\"What's my motive?\"\n\n\"I have no idea.\"\n\n\"You're a terrible detective, Nick.\" She explained, \"The motive is the path to reasonable suspects.\"\n\n\"So what's your motive?\"\n\n\"You're so bad at this!\" She laughed some more. \"I need to marginalize some Sherlock Holmes for you.\"\n\nShe was right; I did need that book. Thankfully I no longer needed it to cover my lap, but I did need some crime-solving mentorship. \"Look,\" I said, \"it's just that I didn't even want to put my confession in that box, and now it's gone.\"\n\n\"Then why did you do it?\"\n\nBecause of you, I wanted to say. Her bright eyes. The eager way she'd called over Jesus and the box. The way she danced to the songs. The way she believed in it all. \"Because everyone else was,\" I said, immediately regretting it. \"And yes, I jump off bridges when my friends do.\"\n\nShe laughed. \"I'm going to tell you a secret, even though you are the worst interrogator on this planet.\"\n\n\"Okay.\"\n\n\"I just got accepted to City School. Have you heard of it?\"\n\n\"No.\"\n\n\"Okay, well, it's a public school, but you have to apply, and you have to sign a contract and stuff.\" It surprised me that she would switch from a private school to a public school. \"It's a school of the arts.\"\n\n\"Oh, you're into the arts?\"\n\n\"Well, yeah, like design. I want to go to Rhode Island School of Design. Have you heard of that?\"\n\n\"No.\"\n\n\"That's right, you're about grammar.\" She smiled. \"I want to do some kind of design with books. Like covers or something.\"\n\n\"Or marginalizing.\"\n\n\"Exactly.\" She laughed, and then it died away. \"Anyway, I have a confession in that box that violates that contract.\"\n\n\"Why are you telling me this?\"\n\nShe sighed. \"Because you're not the only one with something to lose.\"\n\nMy turn to sigh.\n\nI appreciated the fact she was trying to make me feel better, but she didn't know.\n\nLater that night, as I was walking back to the cabins, I thought about what I could lose. What she could lose. She was worried about school, but there were lots of schools. This confession was about my family. You only get one.\n\nAnd then I realized, Natalie may have told me a \"secret,\" but she still hadn't told where she'd gone that night.\n\nAnd there was no doubt in my mind she'd done that on purpose.\n\nGawd, I really was a terrible detective.\n\nBack at the cabin, Matthew and I found everyone except Charles sitting around in a circle. Jeez, this camp had a lot of sitting around in circles.\n\nBut this circle was a little different. Donkey Lottery Winner Chris and Goth sat in the center, both with their fingertips resting on the plastic planchette of a Ouija board.\n\nI had been hoping Payton wouldn't be in the cabin yet, but he was there, all f lared nostrils, pursed lips, and dropped brow line. I wanted to say, Mr. Neanderthal, talk to your girlfriend\u2014ex-girlfriend. But he could probably kick my ass even with a broken arm, so instead I inquired about the \"demon worship\" going on center-cabin.\n\n\"It's the Holy Ghost we're after here,\" Chris explained.\n\n\"It is the Holy Ghost. It already said it was,\" Payton barked.\n\n\"How old are you?\" Chris asked the board.\n\nI watched, but nothing happened. So I went over to my cubby and grabbed my toothpaste.\n\n\"Holy shit,\" someone said.\n\n\"Holy Ghost, you asshole,\" someone else said back, laughing.\n\nI looked over and saw Chris's and Goth's hands moving with the planchette.\n\n\"It's just going in circles,\" Payton said. \"You guys are pushing it.\"\n\n\"No, it keeps going back to the A and then the Z,\" Matthew said.\n\nI watched as my bunk mates theorized.\n\n\"A-Z.\"\n\n\"Azzz...\"\n\n\"What starts with A-Z?\"\n\n\"From A to Z?\"\n\n\"Infinity,\" I interrupted.\n\n\"Yeah, A to Z. The whole alphabet. Infinity,\" Matthew confirmed.\n\n\"No,\" I said. \"The thing is making the infinity sign. Watch it move.\" I pointed.\n\nSilent, we watched the ivory-colored game piece move around and swoop down and to the right, around and down and to the left. A sideways eight over and over.\n\nGoth pulled his hand back, and the planchette stopped moving immediately. \"No way,\" he said.\n\n\"Ask it something else,\" Chris said.\n\nMatthew sat next to Chris and put his fingers on the planchette. \"Is there a God?\" Matthew asked.\n\n\"Whoa, right off like that?\" Payton said.\n\n\"What's it gonna say?\" I jested. \"It's the Holy Ghost. It has to say yes, or it has no credibility.\"\n\nAnd, indeed, it said yes.\n\nThen it spelled out \"A-L-L H-O-L-Y.\"\n\n\"All holy?\" Chris asked.\n\nThe planchette moved on. We watched as the letters spelled out \"E-V-E-R-Y-T-H-I-N-G.\"\n\n\"What an existential game,\" Charles mumbled from his bunk. He looked like he'd been crying. What had he written in his confession?\n\n\"Fine,\" Chris said. \"When am I gonna die?\"\n\n\"Shit, you guys are not messing around,\" Payton said.\n\nThe planchette slid across the board to land on the eight. Then it crept in small circles for a moment before sliding back to the two.\n\n\"Eighty-two. Not bad,\" Chris said.\n\nPayton shouldered Matthew out of the way and took a seat next to Chris.\n\nHe placed his two fingers on the planchette with Chris's. \"What's your name?\"\n\n\"Holy Ghost doesn't need a name,\" Chris chastised.\n\nBut the planchette moved. \"J-\"\n\n\"E-\"\n\n\"A-\"\n\n\"N-\"\n\n\"Jean?\"\n\n\"The Holy Ghost is a girl?\"\n\n\"It's obviously not the Holy Ghost, dumbass. I mean, Jean? That's not even biblical.\"\n\n\"It is now.\"\n\nThe planchette continued to move.\n\nL-O-U-I-S. My face got warmer with every letter.\n\n\"Louis? Do we have two ghosts? Jean and Louis?\"\n\nWhat we had was a French-Canadian dead man.\n\nThe planchette was moving to NO.\n\n\"This is dumb,\" I said. \"Ask a new question.\" I could not let the board reach the last name, a name most of the guys probably would recognize, even if they hadn't read his books.\n\nK-\n\nE-\n\nR-\n\n\"Ask a new question,\" I said, a little too loudly.\n\nO-\n\n\"Like what?\" Chris asked.\n\nU-\n\nIt was still moving, but they couldn't keep track of the letters if I kept talking.\n\n\"Like anything that's not boring, like a name. Like who cares about the ghosts' names? Obviously we have a man and a woman.\"\n\nThe planchette stopped.\n\n\"Okay,\" Chris said, his tone serious. \"Tell us a secret.\"\n\n\"Not that,\" I said in a panic. I had to stop Jack from answering that. \"I mean, you can just get those in the PC Box.\"\n\nDamn it. Why did I always keep talking way past the time for shutting up?\n\nHeads whipped in my direction faster than if I'd admitted to tossing puppies in a blender.\n\n\"Do you know where it is?\" Charles asked, upright in his bunk now.\n\n\"I'm joking, for God's sake.\" But they kept staring. \"Keep going.\"\n\nChris cleared his throat. \"Tell us a secret,\" he said again. Matthew looked around at the group. The planchette didn't move. And then it did.\n\nIt started out doing the infinity sign and then it stopped on S.\n\n\"This better not be about me,\" Goth said. I was confused till I remembered his real name: Stewart.\n\nThe planchette moved to the I.\n\nS-\n\nIt landed on the T.\n\nI took a step back.\n\nE-\n\nR.\n\nIt stopped.\n\n\"Seriously. This isn't helping,\" I said, making sure not to look anyone in the eye as I grabbed my toothbrush.\n\nIn the bathroom, I brushed and brushed and brushed. I stared at my complexion in the mirror: the purple half moons under my eyes, the way the f luorescent lights pierced the skin and revealed everything underneath\u2014the blood, the bone, the grief, the guilt.\n\nIs this what confession felt like? I didn't want people to know what I stored in the dark parts of my psyche. I did not want lanterns and headlamps searching in there for the hidden things.\n\nThe door swung open, and I spit my toothpaste into the sink. \"What's wrong?\" Matthew said.\n\n\"Nothing. Just tired.\"\n\nMatthew eyed me.\n\n\"You guys done out there?\" I asked, nonchalant.\n\n\"Must be you,\" he said.\n\n\"What?\"\n\n\"Nobody else out there knows anything about a sister.\"\n\n\"Well, Chris might. His sister was pretty pissed when he won the Donkey Lottery on the bus.\" Matthew leaned against the wall while I rinsed my toothbrush. \"Do you seriously think the Holy Ghost hangs out on the Ouija board, whispering secrets like some middle school girl at a slumber party?\" I said.\n\n\"No. But you're the one who snuck away at the mention of a sister.\"\n\nI rinsed my mouth out, my teeth more than clean, my gums brushed raw.\n\n\"And, dude, you're awfully defensive,\" he added.\n\nI sighed. \"It's not really a secret.\"\n\nMatthew waited.\n\n\"Just nobody here knows.\"\n\n\"Knows what?\"\n\n\"My sister died. That's the thing. That's why I'm here.\"\n\n\"Because your sister died?\"\n\n\"Pretty much. My parents thought this would be better than mourning at home because that's just awkward. All that silence and tears.\"\n\nMatthew squinted at me for a beat, then grinned. \"Yeah, I hate it when people act all sad when someone dies.\"\n\nWe both laughed.\n\nMatthew brushed his teeth while I took a pee. When I came out to wash my hands, he was done and sitting on the counter waiting for me. \"Do you really think that was the Holy Ghost?\"\n\nObviously, I didn't. \"I don't know,\" I said.\n\n\"It said it was.\"\n\n\"Yeah, but then it said Jean and Louis,\" I said, careful to make it two separate entities, careful to mispronounce the names into ugly hard English consonants and sharp vowels. Even though no one could Google it on their phones while we were here anyway.\n\n\"That plastic thing really moved under my hand.\"\n\n\"I believe you.\"\n\n\"It spelled out sister when I asked for a secret.\" Matthew looked legitimately shaken.\n\n\"Pretty much any noun could've led to some secret,\" I explained. But obviously Jack had picked just the right word for me. I put my toothbrush up in my cubby. \"But, listen, I really don't want to talk about my sister, okay? So...\"\n\n\"Yeah, I get it,\" he said. \"It's private.\"\n\n\"Well, it was,\" I said. I imagined for a moment the scene\u2014me, my mom, my dad, Pastor Kyle, the truth. I felt the pang as though it were already happening.\n\nWhen Matthew and I emerged from the bathroom, the Ouija board was gone, but the group was still in a circle, and our counselor Dan had joined them. Dan was a college kid who also, as the brochure promised, was a specimen of orangish-tanned, bleach-white, four-hours-per-day-in-the-gym perfection. The unfairness of it all. Perfect godly lives. Perfect football champions. Perfect complexions. And then there was me\u2014nerdy blaspheme.\n\nI hated Dan.\n\nEspecially when he told us to sit down.\n\nIn something called a \"Trust Circle.\"\n\n\"What happens in the cabin stays in the cabin,\" Dan said, as if strippers, shots of tequila, and lucrative poker games were imminent.\n\nHe summed up the story of Jesus and then demanded that we share our own anecdotes of mischief, like the pastor had done during his sermon.\n\nWere we really doing this right after the PC Box got stolen?\n\nApparently so.\n\nCharles: didn't do the dishes once when his mom asked.\n\nMatthew: lied to his girlfriend. (Lied next to her, if you know what I mean, he said to me later. Lay, I corrected. I know, he said, but bad grammar is part of the punch line.)\n\nStewart, a.k.a. Goth: had fallen in love and gone out on a secret date.\n\nI got the distinct impression no one was totally trusting the Trust Circle, because these stories were lame. But who would? The worst of our truths were out there somewhere with someone of questionable intent.\n\n\"What about you, Nick?\" Dan asked.\n\nI looked at Matthew. Did he tell Dan about my sister being dead? It wasn't a secret, but I'd said I didn't want to talk about it. And I certainly wasn't going to share that in the Trust Circle.\n\nI thought of one of the nights Jack and I stayed up too late, one of the nights he actually made me laugh instead of making me mad. The night he told me he'd lost his virginity to a prostitute.\n\n\"I drank beer a couple weeks ago,\" I admitted. \"I snuck it into my room.\"\n\nCharles nodded, his suspicions confirmed.\n\nPayton perked. \"I got shit-faced a couple weeks ago too.\"\n\nCharles gasped. Dan tsk-tsked.\n\n\"I didn't get shit-faced,\" I clarified.\n\n\"Language,\" Dan snapped at me.\n\nReally?\n\n\"Oh, because you're better than me?\" Payton glared at me.\n\n\"No, I just wanted to clarify.\"\n\n\"Just admit it. You think you're better than me. I saw you out there with Holly.\"\n\n\"Wha\u2014\" I started.\n\n\"Okay, let's pray,\" Dan said. He did. We did. Or, at least we all bowed our heads and closed our eyes. I closed one eye, lest Payton attack. \"Now, speaking of doing bad things...\"\n\nI eyed Dan\u2014with both eyes now\u2014finding his segue odd for Jesus camp.\n\n\"We've got some pranks to pull, gentlemen. As most of you know.\"\n\nMatthew nodded vigorously and leaned in.\n\nCharles left the Trust Circle and climbed into his bed. \"This is when I go back to my workbook.\"\n\n\"We know,\" Dan said, rolling his eyes. \"Nerd,\" he added under his breath.\n\n\"Okay, Nick,\" Dan said. I was the only kid new to Dan's cabin. \"This is a Christian camp. But it's still a summer camp. Thus, pranks must ensue. There are some rules to these camp pranks, however.\" Dan looked around. \"Gentlemen?\"\n\nThe other eight campers recited the rules:\n\n\"We will not physically or emotionally hurt anyone.\n\n\"We will not spend too much money on any one prank.\n\n\"We will not damage the camp property or the property of others.\n\n\"We will not harm children or puppies, but rodents might suffer.\n\n\"We will not overshadow the purpose of our time here, which is to worship our Lord Jesus.\"\n\n\"Good. So, tonight I've got a new one for us. The Garbage Can Cleanup.\"\n\nWe all changed into black clothing, which apparently had been on the camp packing list for glow-in-the-dark mazes and Frisbee golf, or something, but Dan had also employed them annually for his cabin's pranks. He'd been doing this for three years.\n\nAs he explained the tradition, I suddenly realized I'd actually been blessed to have this asshole as my counselor. He was giving me direct permission to be out and about, all over the camp, under the cover of darkness.\n\nI didn't care about the pranks. I cared about finding that box.\n\nIn the dining hall, we snagged three garbage cans. Then Dan led us to the amphitheater, where we found two more. Two more in the sanctuary. I made a mental note of nooks and crannies to search later.\n\nBy the time we slid through the dark space behind the cabins, all nine of us had garbage cans.\n\n\"We need buckets,\" Dan whispered. \"Nick, follow me.\" We snuck along the side of the sanctuary, through a door, down a hall. We opened one door, but Dan closed it fast. \"Backstage.\" He went to the next. It opened into a small office. Shelves, a desk, boxes.\n\n\"Is this the pastor's office?\" I asked, my heart beating wildly. Breaking into the pastor's office seemed like a capital offense.\n\n\"Custodian's, you idiot.\" He sighed. \"Grab the buckets.\"\n\nSo, this was Jesus's domain. Well, when he was less the scrub-your-feet guy and more the scrub-your-toilets guy.\n\nI grabbed the bucket by the door. I didn't see buckets, plural.\n\n\"That one,\" Dan growled, pointing to one half-hidden behind the custodian's apron, or jacket, or whatever. I fumbled with the bucket handle and we slipped back out the door, down the hall, through another door, and along the side of the sanctuary.\n\nI noted more rooms, more corners to investigate. And then I saw it, in the corner, like a moth. A small, white, f luttery piece of paper. I dropped to my knee and pretended to tie my shoe. Dan spun. \"Hurry up, newb.\" When he turned back around, I snatched the paper off the ground and slipped it into my pocket. I couldn't read it now. Maybe it was nothing but a phone number or a Bible verse, but it looked a whole lot like the papers I had seen people slip into the PC Box over and over that afternoon.\n\nAfter rejoining the group, Dan led us to the far end of the campgrounds. There was a small building with a spigot. We filled the buckets with water as fast as possible, then dumped them in the garbage cans until each was three-quarters full. I kept checking my pocket to make sure the paper hadn't wiggled out. Dan chose nine cabins at random, and we rolled the garbage cans to the doors, then leaned them there to wait for morning.\n\nOur cabin was on an upper f loor, and so were half the other campers'. They would be very happy in the morning that they had to trudge up and down stairs each and every day, because it spared them from our prank. Those garbage cans were just too heavy to carry up the stairs.\n\nFinally, we snuck back to our cabin and closed the door. We didn't turn on any lights. We followed the beams of small f lashlights throughout the cabin and kept the curtains closed. If anyone spotted the tiny lights moving, they would assume people were just going to the bathroom.\n\nI slipped out of my shorts and tucked the paper under my pillow.\n\nMatthew leaned over the side of the bunk. \"Dan's been doing this for years, and he's never been caught.\"\n\nI feigned interest and crawled into my bunk. Under my blanket, aided by my own small f lashlight, I unfolded the paper and read.\n\nI realized quickly that what was written there had never been intended for someone else's eyes. It was not a number, or a Bible quote. It was simply this: My uncle molested me. I think I'm going to tell.\n\nHoly crap. This wasn't some girl drama that had campers crying and hugging and hoping they hadn't included names. This was real. And it had been f loating around in the hallway.\n\nI doubted some camper had just missed the hole on the top of the box when they dropped this in. And slips of paper didn't just fall out of slots like the one the confessions had been dropped into. That box had been opened, that's for sure, or that paper wouldn't have been on the ground. Maybe this secret had blown out when the box was opened. Or maybe the papers had been moved from the box to a different receptacle.\n\nBut that box had been opened. And whoever opened it was reading those secrets.\n\nSomeone was reading our secrets.\n\nI clicked the f lashlight off and slipped the confession under my pillow. That little paper had been in the hall outside the sanctuary. Lots of people traveled that hall because it linked the sanctuary to the camp offices, but the closest office was the custodian's.\n\nI thought of Jesus, who had been out by the barn when everyone else had been praying and crying and hugging one another. He'd said he'd been feeding the donkeys, but I didn't remember seeing any hay on him. Of course, I didn't remember not seeing any hay on him.\n\nI tried to remember the conversation, if he'd said anything odd. He'd told me it'd all be okay; he'd quoted the Bible. John 1:9.\n\nBut Natalie had been out that night too, and she hadn't told me where she'd gone. She'd said she had a secret in that box, but she'd also urged me to put my secret in that box three times earlier the same day. Three times. Like a holy trinity.\n\nAnd then there was Dan. Matthew had just said he was a master at sneaking around, never been caught. And the dude was an unrelenting prick. But nasty enough to steal people's prayers and confessions?\n\nI needed motives. All I could see was opportunity. A real detective would know motive was more important.\n\nI needed to get into the sanctuary. I needed to read the Bible. And I needed to get into the girls' cabins, but not for reasons Jack Kerouac would like.\n\nIn the middle of the night, I stumbled across our already-messy cabin f loor to the bathroom. The light automatically clicked on when the door swung open, the f luorescents blinding. I squinted and pushed open a stall door.\n\n\"What the hell?\"\n\nJack was sitting on the toilet, painting the wall, his antiquated suitcase tucked between his legs.\n\n\"Hello, Nicolas,\" he said, not looking at me. There was a half bottle of some white liquor at his feet. The cap lay next to it.\n\n\"You need to leave.\" I reached for the bottle, to get the cap on it and throw it away before someone saw it, but my hand slipped right through it.\n\n\"Why did you do that on the Ouija board?\"\n\nHe ignored me.\n\n\"For God's sake, I put the paper in the box. I did what you said. And now it's gone. And you're still stalking me?\"\n\nHe looked at me then, shook his head, and went back to painting.\n\n\"Ugh,\" I sighed as I let the stall close behind me. I stood at the urinal. \"Jean-Louis,\" I mumbled, murdering the French accent all over his name.\n\nWhen I turned back around, I couldn't hear anything or see his feet under the stall door. But when I pushed it open, he was still there.\n\nI leaned in to get a look at what he was creating. It was a red and black, very sharp, very gory depiction of the crucifixion of Jesus.\n\n\"What is wrong with you?\"\n\n\"I'm Catholic,\" he said, and then he let out a short, bitter laugh.\nTUESDAY\n\nOur first alarm the next morning was Dan. Six a.m. \"Going for a run! Who's going with me?\"\n\nMumbles, grumbles, heads under pillows.\n\n\"Athleticness is next to godliness!\" he shouted.\n\n\"Athleticism,\" I mumbled.\n\n\"You're the one who wanted the wake-up call,\" he said right in my ear. I swore I heard him call me a douche before walking out the door.\n\nMy head throbbed thanks to Jack, who had left me wondering late into the night what it would take to get him to leave me alone for good.\n\nDan shouted one more time outside my window, but I rolled over and went back to sleep.\n\nThe next alarm was the first scream. I opened my eyes. Matthew was scrambling to the door. Chris behind him.\n\nCrap. I'd meant to get up with Dan so that I could check out the sanctuary before the first sermon. Now I'd have to sneak around after breakfast.\n\nI heard another scream. I sat up while my cabinmates laughed. Guffawed, really, till they lost their breath with it.\n\nI rolled out of bed and walked out onto our deck. Matthew leaned over the railing. Charles hung back, lips pursed.\n\n\"Yessss!\" Matthew said as a garbage can of water fell inside the Zion Cabin to our right. No screams, just expletives from the boys.\n\n\"This is your handiwork?\" Charles asked me.\n\n\"Well, Dan gets the credit,\" I said.\n\n\"This does not glorify the Lord.\"\n\n\"Charles,\" Matthew said, \"if you have an issue with it, why do you keep signing up for Cabin D? Cabin Dan? Cabin Of-the-Devil?\"\n\nCharles shook his head and walked back inside, the slamming of the cabin door punctuated by a third scream.\n\nMatthew and I stood at the deck and enjoyed six more screams and ensuing expletives. For a minute, I considered telling him about the confession I'd found the night before. The guy was smart; maybe he could help. But something held me back.\n\nI thought about Holly, how he'd made jokes and told me things and I'd laughed and played along. Was it the same thing? Maybe I didn't want Matthew to know I had a secret.\n\nThere was that feeling about Charlotte again. And Diana was right there too. She wasn't like Holly and Charlotte. She'd dated some, but she only loved Leah. And that was enough for people to pay too much attention.\n\nDiana had gone out on a date one time in high school. She'd tried to pass it off like she and the other girl were friends, but someone had seen them kissing. It wasn't even like Charlotte and Holly. It was just a kiss, and she hadn't meant for that secret to get out there, either. But everyone had talked about it, and it had caused giggles and scrutiny and worse.\n\nI knew about the gossip because Diana had gotten beaten up so badly for it. I had still been a little kid, and her face had been so swollen she'd looked like a monster. I was afraid of her. She was bruised and broken because her whole being couldn't fit in the world's small box.\n\nNo, I couldn't tell Matthew about the confession I'd found. I knew what would happen to my family if my parents found out about what I'd written. I could only imagine what would happen to this kid's family if they found out about this uncle. It was up to that kid if they would eventually confess. That was their truth to set free or to tame.\n\nWe headed to the showers, and on my way I checked the bathroom stall. Jack had responsibly cleaned up his ode to the Renaissance.\n\nBy the time we were ready to head to the cafeteria, Charles was fully absorbed in his math workbook. We waited for him to find a derivative, and then we left, Charles a few steps behind.\n\nAs we walked into breakfast, the counselors were singing Vanilla Ice's \"Ice Ice Baby.\" Or rather, \"Christ Christ Baby.\"\n\nDan served us eggs and bacon. Counselors all around the room served their campers\u2014if I remember correctly from the brochure, it was something about demonstrating servitude. Dan's face was all scrunched up in disdain. I hope Jesus hadn't made that face at the Last Supper.\n\nAs he poured our orange juice, Dan asked, \"Where's Stewart?\"\n\nPayton pointed. Goth was sitting next to that same guy and Wheelchair Girl.\n\n\"Stewart, your love triangle can wait!\" Dan shouted across the dining hall. \"Dude, I thought he and Nick broke up.\"\n\nI about choked on my orange juice. \"He's gay?\"\n\n\"Gay as the day is long,\" Dan said. \"Why, you got a crush?\"\n\n\"No, I'm just\u2014\"\n\n\"Tongue-tied with desire?\"\n\nGoth meandered over, and I stared till he was in his chair.\n\nI looked around the table, at all the guys just chewing their food, not one of them bearing stones or torches or pitchforks or other implements of the riot before the exile. I had no problem with him being gay. I was just shocked nobody else seemed to have a problem with it. I was at Jesus camp, after all. The very one that converted Charlotte, the Queen of Homophobelandia. So I had pretty much just assumed that Eden Springs would be filled with more Royalty of Judgment.\n\nI was genuinely shocked, but in a good way. Like the universe just got a little bigger and warmer. I didn't know how to explain that, so I finally said, \"I was just curious.\"\n\n\"Bi-curious?\" Matthew laughed.\n\nDan didn't seem to get Matthew's joke. He had his own. \"You heard about the cat that died of that?\" Dan said. \"Curiosity killed the cat?\"\n\nI didn't look at him. I just chewed my food. \"Curiosity's not a disease,\" I mumbled, my mouth full of egg casserole. \"It's not malaria.\"\n\n\"So be it. It's a fatal car crash. A school shooting.\"\n\nThe f lame of hope I'd just felt f lickered. He really was an asshat. A racist, misogynist asshat with decent pranks and my key to searching the campus. He turned around and said something to one of the girls in too-short cutoffs. Then, back to us, \"All right, you pansies! We got our schedule for the week. We've got crafts in fifteen. Ropes tomorrow.\" And the next part he whispered, \"And dirty work every night.\"\n\nMy cabinmates stood up. I stayed seated. Matthew sat back down next to me. \"He's a douche. Dan the douche.\"\n\n\"I was just asking,\" I said.\n\n\"Which is as deadly as a zealous antiabortionist. Stop doing dangerous things, Nick. Questions bad. Curiosity bad. Only happiness happens here!\"\n\nI laughed.\n\n\"Okay, let's go craft!\" he added.\n\nI started after him, then remembered my morning mission. \"I have to go do something really quick,\" I said.\n\n\"Do something?\" He looked around us in a wide, dramatic circle. \"Oh, I didn't see Holly here this morning.\"\n\nI had to think about that for a second. \"Wow.\"\n\n\"I know, dude. That's what she's gonna say when she sees your tiny manhood.\"\n\nHe turned around and skipped away, laughing.\n\nHe would make Kerouac proud on so many levels.\n\nI pushed the sanctuary door open. The dark and the AC cooled me off. The door closed behind me, and I couldn't see anything. I waited for my eyes to adjust for a moment.\n\nIt didn't take long for the stage to take shape down front, a sliver of light from behind a curtain illuminating it. This is where Pastor Kyle had expected to see the PC Box last night. It had obviously traveled down the hall after being stolen and opened. I would start in the sanctuary and move backward.\n\nI walked down the aisle to get a closer look around the stage. The box wasn't off to the side or on the ground in front. Not that I really expected it to be. But wouldn't that be a prank of sorts? Just steal the box for the night, freak everybody out, then put it back. But no. If that confession had fallen out of it, that meant the box was definitely opened.\n\nShuff ling from backstage startled me.\n\n\"Hello?\" I waited.\n\nNothing.\n\n\"Jack?\" I asked, a little quieter.\n\nI looked up and behind me into the sound booth to make sure no one was there before making my way around the stage and up the stairs. I wasn't really doing anything wrong, but this is what empty rooms do to me. Even if a teacher hands me her keys to get something from her classroom, I'll unlock the door and step in, and there will be this tingle of nerves, of guilt. The silence, the darkness, the lack of witnesses. It's the temptation to be tempted. Like standing on a cliff and wanting to jump.\n\nMy footsteps echoed as I made my way to the closed curtains. I peered between them to glance backstage.\n\n\"Hello?\"\n\nStill nothing.\n\nI pushed the heavy fabric aside and slipped through.\n\nI walked behind a screen. Looked on shelves. The door that led to the hallway where Dan and I had been the night before was propped open with a bucket.\n\nDisappointed I hadn't found the box, I walked through the curtain and back onto the stage.\n\n\"Hey, you're not supposed to be in here,\" an older kid called out from the back of the auditorium. As he walked closer, I saw his counselor badge.\n\n\"I was just doing some prayer stuff.\"\n\n\"In the sanctuary?\" The kid said as if this was baff ling. \"It's for skits.\"\n\nThe irony was not lost on me. \"I know. I just\u2014\"\n\n\"Where are you supposed to be?\"\n\n\"I'm on my way to crafts.\"\n\n\"Crafts are outside.\"\n\n\"I know. I just\u2014\"\n\n\"Who's your counselor?\"\n\nPlease don't get him involved. \"Dan.\"\n\n\"What's your name?\"\n\n\"Nick.\"\n\n\"So, listen, buddy. Nick\u2014\" He stopped and his mouth dropped open. \"Oh... Nick.\"\n\n\"Yeah... ?\"\n\n\"You're here for prayer?\"\n\n\"Um, well\u2014\"\n\n\"For the devil-worshipping stuff.\"\n\nHarry Potter again. \"Is this about\u2014\"\n\n\"Listen, I can lay hands. I took a class.\"\n\nOh, hell no. I wasn't going to let him pray for me, let alone touch me while he did it. \"I'm a verbal person, not kinesthetic,\" I said. \"I learned that in class.\"\n\n\"I get it, I get it.\" But he stepped closer as he said this.\n\n\"No, seriously.\"\n\n\"Yes,\" he said and reached for me.\n\n\"I mean, no hands. I have a rash.\"\n\nHe stopped, hands in the air like a traffic cop. Breathing deeply, he reached for me again. \"Jesus healed the lepers.\"\n\nI stepped back. \"You're not Jesus.\" Then, for good measure: \"And I'm not a leper.\"\n\nIf the Harry Potter gossip had traveled that fast and turned into devil-worship, God only knows what would become of the leper comment.\n\n\"So, you don't want me to touch you?\"\n\n\"Just pray for me from over there.\" I pointed to a corner. \"I'll be back after crafts to let you know how it all turned out.\"\n\nI spotted Matthew easily enough and headed over for crafts. At the Creation Station. No, I'm not joking. That's what it was called. The Creation Station.\n\nCrafts. Nobody does crafts, except kindergarten kids in colorful classrooms and kindergarten kids in church basements. But Eden Springs was bringing it back old-school\u2014gangster rap and childhood experiences alike.\n\nI could see as I walked over that our cabin had been paired with a cabin of girls, which made the whole crafting thing that much worse. Boys don't craft, and our incompetence is made ever more obvious when juxtaposed with girls\u2014who do craft\u2014crafting.\n\nMatthew patted my stool. As soon as I settled in it, someone slid The Grapes of Wrath across the table. Natalie smirked at me. I hadn't even noticed her and another girl sitting there.\n\n\"Care to read?\" she asked.\n\n\"Do I look like someone who would read during a riveting session of crafts?\"\n\nShe read my BAD GRAMMAR MAKES ME [SIC] T-shirt.\n\n\"Yes. Definitely yes.\"\n\nMatthew looked at me. \"Do you two know each other?\"\n\n\"We're old dance partners,\" she said.\n\nI f lipped absentmindedly through the pages of the book. \"And nightwalkers,\" I said, and made eye contact.\n\nNatalie's smile disappeared and then reappeared. \"Nick always needs a book when he prays. It makes him feel grounded.\"\n\nReally? Matthew looked at me.\n\nI shook my head.\n\n\"Sorry it's not Sir Arthur Conan Doyle or Agatha Christie,\" she added.\n\nA counselor dropped large piles of clay in front of us, and everyone at my table got to work. I eyed Natalie for a bit. She was beautiful and happy this morning, not at all the aloof girl I had seen walk out of the sanctuary. She was the breezy girl who told everyone it would be okay even though our secrets were out, and sincerely meant it. The one who hugged the girl who picked her nose\u2014All. The. Time. And she understood that we were sparring to see who'd tell the truth\u2014the whole truth\u2014first. But we'd have to continue later, sans audience.\n\nI looked down in front of me, at the beige mass of what looked like Play-Doh.\n\nClay at the Creation Station.\n\n\"What am I supposed to do with this?\" I asked. Recreate Adam and Eve? Remove ribs? Breathe life into little dolls?\n\n\"Kim, slide that over here,\" Matthew said to the other girl. It was a menu of clay-crafting options.\n\nI read the list: plate, handprint (\u00e0 la kindergarten), coffee mug, candleholder, and wait a minute\u2014\"An ashtray?\"\n\n\"Just 'cause you smoke doesn't mean you're going to hell,\" Payton said, glaring at me.\n\n\"I thought the body was a temple?\" I asked.\n\n\"Play nice, Nick,\" Natalie said.\n\n\"My grandfather smokes,\" Charles said.\n\n\"And you're worried about my beach towel?\" I mumbled.\n\n\"Satan and smoking are very different things. My grandfather is a godly man.\" Charles grabbed his pile of clay and moved to a different table.\n\nNatalie smiled at me. \"I heard about you and the devil towel.\"\n\n\"Who hasn't?\" I looked at the kids around me. \"Here's what I don't get.\"\n\nMatthew leaned in. Kim looked up from her mass of clay.\n\n\"First off, the girls and their bikinis are no problem, but my towel\u2014which actually covers the body\u2014is satanic. Second, just about every counselor serving food in the lunchroom is female. Has anyone noticed it's like 1950 here? And now it's cool again to smoke and James Dean is all that and all the hot housewives look like you?\" I pointed to Natalie with that bandana in her hair.\n\n\"Wait, did you just point at me?\" She smiled. \"You think I'm hot?\"\n\nMy face f lushed, and I attempted to dig myself out of a pit of embarrassment. \"All the girls serve us in the dining hall and poolside, like we're JFK and they're Marilyn Monroe about to sing us 'Happy Birthday' with their bathing costumes blowing in the wind, full ass exposed. And no one thinks this is odd?\"\n\n\"You didn't answer the question about thinking I'm hot,\" Natalie said.\n\n\"And the gold chains and gangsta rap? Seems kind of racist,\" I continued.\n\n\"Last year, it was cowboy boots and country,\" Payton said.\n\nMatthew smirked. \"You sure had a lot to add to our MLK speech.\"\n\nIt was about boobs, not race, I almost said. Oh, gawd, was I just as bad? I looked down at my clay, formless and impotent.\n\n\"Make a handprint for your mom, Pampers,\" Matthew said. \"It'll soothe the hypocrisy blues.\"\n\n\"You're just placated by bikinis.\" I decided on a candleholder, and stewed in my idiocy as I rolled my clay out f lat, following the directions.\n\n\"So many bikinis, I'm easy to please,\" Matthew singsonged.\n\n\"Save your rhyming for the poetry session,\" I quipped.\n\nMy table laughed.\n\n\"Proverbs Slam is tonight,\" Matthew said.\n\n\"No effing way,\" I said.\n\n\"You're right,\" Matthew laughed. \"No effing way.\"\n\nNatalie beamed. \"Oh my gosh, please can we start that?\" I couldn't tell if she was even joking. She concentrated on her clay and massaged something to life.\n\nThat moment felt like an in with the table. Like maybe we could broach the subject and solution of our secrets. Surely they wanted to get to the bottom of this instead of twerking to rap-turned-gospel and kindergarten throwback time. \"So, last night\u2014\" I started.\n\n\"Like Demi Moore, right?\" Kim interrupted. She had wrapped her arms around Payton to help him sculpt despite the cast on his arm. \"In Ghost. Have you seen that old movie?\" Her lips were right by his neck.\n\nAt this point, nobody cared what I had to say. Even I barely cared. All eyes turned to watch the amateur porn going on across the table.\n\n\"I never saw that movie, but you make it sound good,\" Payton said, their hands working the clay. He caught me watching them and smirked. Then he looked around the crafting area, no doubt searching for Holly. He spotted her at the same time I did. She stood at the edge of the covered area, hands on her hips, watching it all.\n\nPayton got up from his stool, but Holly marched over to our table.\n\n\"Oh, hey, Holly,\" Kim said sweetly. \"Did you come to play?\"\n\nHolly rolled her eyes. \"I don't even have my guitar.\"\n\nKim laughed. \"That's not the kind of\"\u2014air quotes\u2014\"playing I was talking about.\"\n\nHolly's mouth dropped open. Before she could say anything, Payton took her by the arm and led her away. As they got farther away, I could see her hands whipping through the air while she talked. I didn't know if she was crying or yelling.\n\n\"I can't believe she still even comes here,\" Kim said, sitting back in her stool. \"Whatever. Jesus loved the whores too.\"\n\nMatthew laughed. \"Are you talking about you or Ho-Lo?\"\n\nKim glared. \"Duh, who do you think?\"\n\nI squinted at her. She was the one who'd just made a (sex) scene at the crafting station, and she was making a crack about Holly?\n\nNatalie sat watching the scene, then looked down at her clay project.\n\nI wasn't about to step in. I didn't know these people, and I definitely didn't want to be a part of this. So I changed the subject. \"Did your cabin get hit last night?\" I asked, careful that I didn't look guilty. \"You know, with the pranks?\"\n\nNatalie shook her head extra hard, like she was clearing it of the Kim ugliness at the same time she was answering my question. \"No. We're on the second f loor. Only the first f loor got it.\"\n\n\"Oh,\" I said, and my voice cracked. I am no better a liar than I am a detective.\n\n\"This looks like a pile of shit,\" Matthew said, staring at his project, and thankfully changing the subject yet again.\n\nCharles gasped from the neighboring table.\n\n\"That's the point, isn't it?\" Natalie said.\n\n\"'Pile of Shit' is not on the list of options,\" Matthew said, holding out the menu of craft choices.\n\n\"Everything we touch turns to shit,\" Natalie said.\n\nShe had no idea.\n\n\"I think that's the point,\" she added. \"Of this whole Creation Station business. Like, God can create the real stuff. You and me, if we try to do it alone, it's going to turn out like... well, like your pile of shit.\"\n\n\"You people and your overthinking things,\" Matthew said. \" I'm just bad at crafts.\"\n\nWe worked our clay a little longer.\n\n\"I'm out,\" Matthew dropped his clay on the table. \"I'm going to weave some socks out of grass.\" He walked over to a station where, in fact, grass-weaving was taking place.\n\n\"I don't think you can really make socks there,\" Natalie said to me. \"Place mats, yes, but socks seems a bit far-fetched.\"\n\nAfter a while, Payton returned to our table, shoulders slumped. Kim watched him, but he didn't look up. He mumbled something and took his clay over to Charles's table.\n\n\"I do like your outfit,\" I said to Natalie, trying to cover up the earlier awkwardness, the public announcement about her hotness. \"How you look like you stepped right out of that World War II 'We Want You' poster.\"\n\n\"Wait. I look like the old guy with the pointy beard and accusing finger?\" Natalie asked.\n\n\"No, you know. The one with the lady f lexing her biceps.\" I f lexed my biceps. \"'Yes, We Can!' That poster.\"\n\n\"'Yes, We Can'? That's Obama, Nick.\"\n\nKim laughed.\n\n\"I thought he said I was hot earlier,\" Natalie said to Kim. Then, to me, \"I look like a black man with a pointy white beard? An Obama\/Uncle Sam mash-up.\"\n\n\"Obama's kind of hot,\" Kim said.\n\n\"Not with f luffy white hair, he wouldn't be,\" Natalie chastised.\n\nI shook my head. \"No, you know who I'm talking about. The one with the full red lips and dark hair all pulled up.\" I gestured toward Natalie's face. \"And she has those cheekbones and bright eyes. And she's strong, but not masculine. Even though she's f lexing, she's feminine. And she looks out at everyone all strong and dares you to do the thing...\" I trailed off, noticing Kim and Payton watching me. \"Whatever the thing is,\" I mumbled. \"You want to do the thing.\"\n\nEven drop your worst secret in a box, lest you disappoint the courageous girl hailing Jesus.\n\nSilence as Natalie and Kim stared at the f lush eating up my face.\n\nNatalie smiled. \"Her name's Rosie the Riveter.\"\n\nI nodded. \"Yeah. That one.\"\n\nNatalie held up a clay piece. \"I'm done.\" I recognized a frowning face molded there. \"I made the drama faces.\" She held up another molded face, this time smiling.\n\n\"Are you in drama?\" I asked.\n\nShe smiled. \"Aren't we all?\"\n\nShe kept looking at me, so I pretended to fix something in my candleholder-to-be. \"I meant at school.\"\n\n\"No,\" she said, standing up. \"No, I'm not.\" She picked up the two drama faces, one in each hand.\n\nShe and Kim stood up and walked away. Payton walked behind them. Kim called something over to him, but he just waved it away. Maybe she apologized. Or maybe she said another mean thing. Matthew sat down next to me again.\n\n\"We can put your candleholder on this.\" He held out a small square of unevenly woven straw.\n\n\"What happened to the sock idea?\"\n\n\"This is it. It's a two-dimensional, sock-like mini place mat.\"\n\n\"It's not even shaped like a sock.\"\n\n\"Don't be so critical. Your candleholder looks more like the model of a vagina one might find in an anatomy classroom.\"\n\n\"They have models of vaginas at Valley Christian?\"\n\n\"Or, look\"\u2014Matthew f lipped the vagina-shaped candleholder upside down\u2014\"a penis.\"\n\n\"It's like a mood ring,\" Matthew said, f lipping the clay back and forth, \"but it's only got two moods. Penis...\" Matthew f lipped it over. \"Vagina.\"\n\nWe headed over to the basketball courts.\n\n\"Kim and Natalie. Titillating and asstastic, respectively,\" he said, grabbing a ball.\n\nI took the ball out of his hands and got lucky on a shot. Matthew watched the girls round the corner till they were out of sight. He wasn't Pastor-Kyle perfect, but he was tall, broad shouldered, and funny. I was sure he had no trouble with the \"fine ladies.\"\n\nI, on the other hand, couldn't say anything normal, as evidenced by the Crafting Station debacle. And I couldn't even get Leah to talk to me again\u2014even though I wasn't interested in her the way I was interested in Natalie. I just wanted to apologize. Or something. I wanted to know she was still there.\n\n\"Kim's obviously into Payton, though.\" Matthew continued. \"Fool's errand, that one. He'll never give up what Holly's puttin' out.\"\n\n\"I thought they broke up.\"\n\nHe scoffed.\n\n\"What about Natalie?\"\n\n\"Natalie is not a slut.\"\n\n\"I didn't mean that.\"\n\n\"Oh... I see. Yeah. She's hot. Like, in a cool way.\" Matthew grabbed a basketball. \"And smart.\" He laughed. \"Yeah, I could see how you'd be into that. She's no doubt Ivy League-bound. Parents have tons of money, is what I heard. I think her brother's on a row team at the University of Washington or something.\"\n\nI didn't correct him about the Ivy League thing, or tell him that she planned to attend RISD, the number two art school in the nation. \"She doesn't act like a snob.\"\n\n\"Oh, no, I'm not saying that. I'm just saying. She's smart and rich and perhaps\"\u2014Matthew took a shot and it swished into the net\u2014\"out of your league.\"\n\nPretty much every girl was out of my league.\n\n\"I think I saw her walking up on the hill yesterday. It was weird.\" I didn't tell him it was right after the box was stolen.\n\n\"Yeah, dude. She's kinda weird. Does her own thing. Just kinda a closed book.\"\n\nThe ten-minute whistle sounded.\n\n\"A closed book with a hot cover,\" Matthew added and tossed the basketball over to the bin.\n\nPRAYERS AND CONFESSIONS\n\nDan: My dad's not going to AA anymore. Yeah, so. I'm thinking of hiring some singing strippers to come bang on his door dressed up as those princess fairy girls from Frozen. They could sing some \"Let it Go\" carols. It's like the theme song for AA. Like that slogan, \"Let Go and Let God.\" Maybe that would get his attention and he'd go back to AA. And then maybe my mom will move back home (if I promised there'd be no stripper encore).\n\nPayton: I didn't break up with Holly for the stuff she's done. I broke up with her for what everyone says she's done. I guess that makes me a jerk, but I don't want to deal with all the rumors.\n\nPastor Kyle: Thank you for the Sampson & Delilah salon idea. What camper doesn't love a good pedicure, where your feet are oiled and dried with hair, just like Mary of Bethany did for Jesus. And Sampson and Delilah? Come on, that's clever! Even if Delilah was a bad guy.\n\nMatthew: Please forgive me for how I've blamed my parents in all that stuff. Please forgive me for how I treated Sarah. And maybe also for the MLK porn speech. That probably wasn't cool. But it made Nick laugh, and I like that guy.\n\n\"The Disciplettes,\" Matthew explained, nodding toward a large group of girls that had gathered at the outdoor amphitheater. They shouted something, got into a formation, ran out of it. Shouted. They were all dressed in matching skirts. It was a cheer squad. Each sweater had a capital J for Jesus on the front.\n\n\"BE SO RIGHTEOUS!\n\n\"B-E SO RIGHTOUS!\n\n\"B-E-S-O-R-I-G-H-T-E-O-U-S\n\n\"BE RIGHTEOUS!\"\n\nThe Disciplettes leapt and danced and cartwheeled for Christ. Charles f litted among the girls now, talking very intently to the bounciest, likely condemning them to an eternity in hell for their tight sweaters.\n\nThe bass pumped from the speakers in the amphitheater, calling us over to our cabin groups. The rappers came to the stage and made a disaster of Young MC's \"Bust A Move\":\n\nThis is a camp for all you followers\n\nTryin' to do what the Lord does tell us.\n\nAvoid lukewarm and get super zealous,\n\nLeave nonbelievers feeling jealous.\n\nThe rappers had done their part, and the whole camp started singing:\n\nSo open your Bibles because the Word does prove,\n\nJesus is here to bust a move.\n\nI did not sing with them. But I spotted Natalie, singing and hip wiggling with Kim and a couple other girls. After a few verses, she suddenly bent over with laughter.\n\nThat's how Diana and I used to laugh. Till it hurt. Till our sides cramped, and the hilarity rendered us silent and shaking. Till our parents would yell at us to keep it down in there. Cease and desist. Enough of that racket.\n\nNatalie caught me watching her, and I looked away just as Pastor Poseidon came up to the stage. He joined the rap, his bright teeth a lantern of truth. Counselor-servants wove among us with cups of what looked like wine. But it was just grape juice. Because laws.\n\n\"Fellas!\" Douchey Dan called us over to him.\n\n\"With an F,\" I said, more into my Dixie cup than to anyone.\n\nThe rap ended, and Pastor Liam Hemsworth instructed us to sit in the grass. As we did so, Payton kept his eyes on Holly, who seemed to be looking around for someone. She caught my eye and waved.\n\nI felt a stir of disgust and pity. I hadn't for one second thought Holly's extra prayers and attentions the night before had meant she was actually into me. I knew she'd f lirted with me to get to Payton. This was her game, and Holly was that girl. The punch line of promiscuity jokes, Matthew's and Kim's and probably the whole camp's.\n\nBut she also played the guitar and wrote music, this girl that everyone gossiped about. She sat alone in her bedroom strumming songs she made up in her head. I couldn't figure out how both the sweet artist and the slutty girl could be a part of this one person. Could she keep both? Could she be both? At the same time? Maybe some parts of us have to move out to make room for the new parts.\n\nLook what Charlotte had pushed out to make room for God.\n\n\"Ladies and gentlemen,\" the pastor said. \"Jesus does INDEED bust a move! Jesus was so fun (such a party animal!) that he turned a hundred twenty gallons of water into wine at his mom's request.\" He was careful to add an emphatic, \"Not that we condone underage drinking,\" yada yada. \"But in this doggy dog world, we need someone on our side!\"\n\nI was pretty sure he didn't mean Snoop Doggy Dogg, though the camp's dedication to nineties rap did make me wonder. \"'Dog eat dog world,'\" I said aloud to no one in particular, reminding myself to add that to my list under MUTE\u2014MOOT.\n\nHe talked about Jesus, and Hillbilly Jesus came to the stage and mimed a Bible story. The pastor concluded with an update: they had no leads on the location of the PC Box, and they were still debating whether or not to create a new one.\n\nA new one? WTF?!\n\nI looked around. Head nods. A couple amens.\n\nHell no.\n\nI started to head toward Natalie, but just then a hand landed on my shoulder. When I spun, Jason stood there with Pastor Abercrombie & Fitch. Dan walked over.\n\n\"Where's your stuff?\" Jason asked in a tone that said no happiness was happening here.\n\n\"What stuff?\"\n\n\"You don't have anything with you?\"\n\n\"My candleholder,\" I said, pulling it out of my pocket.\n\nDan cough-laughed. \"Is that a candle in your pocket, or are you just happy to see me?\" he said.\n\nJason glared him into silence.\n\nSuddenly, I heard a scraping behind me and turned around to see an older man dragging a large piece of wood across the concrete. \"First one of the camp,\" the man said.\n\nJason nudged me over to the man. \"Turn around,\" he said.\n\nI did what I was told and saw that the piece of wood was actually a large cross. Before I could even process that, they were situating the crossbeam against my back and across my shoulders.\n\n\"You've got to be kidding me,\" I said.\n\nNobody said anything.\n\nThe men just nodded at each other and let go, sure I could hold the thing. It wasn't too heavy, not like the one I supposed the actual Jesus had carried. But then, I hadn't been whipped to near-death like the actual Jesus.\n\nPlus, they probably wanted to avoid a lawsuit for spine injuries.\n\n\"What's this for?\" I argued. Was this really happening?\n\n\"Transgression,\" Jason said.\n\n\"What'd I do?\"\n\n\"Walk it on over there,\" Jason said, pointing to the sanctuary.\n\nThe campers close to me watched, nudged their neighbors. Heads turned like a wave of falling dominoes to get a look at the spectacle. I didn't want to make out the faces of anyone I knew, but I saw Natalie, her smile fading. Goth was right there too. Holly was so close, she started to reach out to touch my shoulder, but then seemed to think better of it. Culpability was contagious.\n\nI walked, my cheeks burning with humiliation, which I supposed was the point. But I thought the crown of thorns had been for that. The cross had a more practical purpose: execution.\n\nI didn't dare look again at the campers, who were no doubt fist-pumping for Christ at my exit.\n\nWhen I reached the grassy lawn, soft with fresh water and hidden divots, the cross got harder to drag. But I didn't fall down. Not once. Not thrice, like the holy man himself.\n\nAfter I crossed the lawn, the four men led me down the sidewalk to the side of the sanctuary. They lifted the cross off of me. Jason and the older man carried it through the door to the backstage area.\n\nPastor Prince Charming led me a bit farther. Dan followed us into a room with two desks and a bank of computers and phones. There was a sign above it that read BABYL ON.\n\nI stared at all the phones. I hadn't seen mine since the day before. None of the campers would know if anyone had texted or called or emailed or tagged them in a post for nearly a week. I wondered for a beat if Leah had answered the text I had sent from the bus.\n\nDan cleared his throat. He and the pastor waited expectantly.\n\n\"What'd I do?\" I asked.\n\n\"I think you know,\" Dan said. He'd been with me all morning, basically, so I doubted he actually knew. But he was enjoying his authority. As he delivered that line with his arms crossed over his chest, it looked like he'd been practicing in the mirror.\n\nThe pastor put his hand on Dan's shoulder and asked if he'd give us a moment of privacy.\n\nI waited for a lecture.\n\n\"Nick, I like you, and I know it is God's will that you are here. I don't get the sense, from what happened here today, and what happened in the cabin yesterday, that you do.\"\n\n\"It's a beach towel.\"\n\n\"Irregardless,\" the pastor said.\n\nI opened my mouth to correct him, to tell him regardless, but I thought better of it. There'd probably be some kind of follow-up punishment for my contempt. What happened to campers who had already carried their cross? Were they nailed to it for a second offense?\n\n\"But today it is not a beach towel.\"\n\nI looked at him dubiously.\n\n\"Yesterday was a beach towel. An oversight, you argue. But today...\" His words faded out. He shook his head.\n\n\"The candleholder? I didn't mean for it to look like a vagina.\" The pastor looked puzzled. \"Or a penis. Or whatever part you're seeing.\"\n\n\"This isn't about a penis.\" At the mention of the word, the pastor's face curdled into the antithesis of an Adonis face. \"Or a vagina.\"\n\n\"The sexism comment at the Creation Station?\"\n\nThe pastor shook his head. \"We can't move forward until we confess our sins and hold ourselves accountable.\"\n\n\"I did confess my sin!\" I pulled my hair back from my forehead. \"I put it in the box. The box you said was stolen.\"\n\n\"Now you've hit the nail on the hand.\"\n\nIt's nail on the head. Nail on the head. Nail in the hands was Jesus's fate. I looked around for someone, something that made sense.\n\n\"Where's the box?\"\n\n\"What?\" I rubbed madly at my hair. \"How would I know?\"\n\n\"Young man, calm down.\"\n\n\"I'm not going to calm down. I didn't steal that box! I didn't even want my confession in that damn thing, and now it's there and I'm shitting my pants and you're accusing me of stealing it.\"\n\n\"Nobody's sh\u2014defecating.\" Pastor Hot Stuff put his hands on his hips and took a deep breath. \"I was told you threatened to steal the PC Box last night during the Trust Circle.\"\n\n\"Are you talking about the Ouija board thing?\"\n\n\"There was no mention of a demonic relic.\"\n\n\"Other than my towel,\" I sighed. \"Look, I was joking about stealing the box.\" I was sure Charles had brought this bit of information forward.\n\n\"And this morning, a very reliable source said that you were in the sanctuary alone, sneaking around backstage before crafts.\"\n\n\"Yeah, looking for the PC Box,\" I said.\n\nPastor Ryan Gosling considered me. \"I didn't want to have to do this.\" He handed over a piece of paper, and my heart beat against my ribs. Somehow they must have found my secret. They'd probably already called my parents.\n\nI unfolded the paper. And it wasn't mine.\n\nIt wasn't mine.\n\nBut it was the one from under my pillow. It must've fallen on the f loor when I made my bed or rummaged through my clothes this morning.\n\n\"If you didn't steal the box, how'd you get this?\"\n\n\"I found it.\"\n\n\"Where?\" Pastor Kyle would make Natalie proud with his grilling.\n\n\"In the hall, outside the sanctuary.\"\n\n\"Why were you there?\"\n\nOh, crap. \"I was in the sanctuary this morning, and I went out in the hall,\" I lied. Despite Dan's douchiness, I would not snitch. I would not tell him about the pranks.\n\nThe pastor eyed me.\n\nAnd then it occurred to me. \"That could be my confession. How do you know it's not something I wrote?\"\n\n\"Because the person who wrote that confession came to us as soon as the box was stolen.\"\n\n\"What for?\"\n\n\"To pray, Nick. And to ask for advice.\"\n\nI thought of Natalie searching for Pastor Kyle the night the confessions were stolen, how she'd left our circle to seek him. I'd watched him pray with the kids who were crying. I saw him hug them, pat their shoulder, sit with them.\n\n\"Well, I didn't steal that box,\" I insisted. \"I found that confession. I don't know how to prove it, but you have to believe me.\" I handed the paper back to him.\n\nHe sighed. \"The Lord calls me to trust you.\" He pursed his lips. \"For now, that is.\"\n\nI exhaled. I hadn't realized I'd been holding my breath. \"Can you please go tell the masses that I've been exonerated?\"\n\nHe thought for a moment. \"I will tell them you bore the cross for a different transgression.\"\n\nI could see campers walking down the path outside the window. These kids thought I'd stolen the box. I didn't really care what Pastor Kyle told them. I didn't care about the towel or the clay vagina, or whatever other minor thing I'd done. But I did not want people to think I stole their secrets. It was enough that I was already an outsider. I didn't want to be despised.\n\nI thought of Diana for a second. Of what it must feel like to doubt what people were thinking of you, for the connections between you and others to be so tenuous. To feel so apart from the crowd, to not know who condemned you and who accepted you. I didn't want to feel that.\n\nThe pastor looked a bit contrite for a second. I thought maybe he'd apologize then, but he didn't.\n\nWe were all choking on apologies. Even the reverent.\n\n\"Can we use those phones?\" I asked.\n\nHe cocked his head. \"Well, now, phones are for emergencies at camp.\"\n\n\"The PC Box is kinda an emergency.\"\n\nHe started to say something.\n\n\"I just want to talk to my family,\" I said, hoping that sounded appropriate and true.\n\n\"That's the thing, Nick. We're trying to disconnect you from your family here at camp. Only for a bit, so that you can get in touch with your real Father. Your Heavenly Father.\"\n\nI could only stare. Disconnect us from our family? \"I'm upset,\" I said. \"I'm embarrassed.\" These two things were true, but they weren't really why I wanted to use the phone.\n\nEither that clued him into my imperviousness to his holy intentions, or he succumbed to whatever guilt he felt about his wrongful accusation. \"Okay. This is obviously an emergency,\" he said, lifting the receiver of an old phone that actually had a cord attached. He looked at me and waited.\n\nI told him the phone number. Outside the window, Payton and Holly walked by. He had a hose over his shoulder. She had a computer bag. They didn't even glance inside. I wondered if anyone knew where I was.\n\nPastor Kit Harington (the hair was wrong, but the jawbone was right) put the receiver to his ear, and I thought for a second he was going to do the talking, in order to make sure I wasn't checking in with my dealer or pimp or something. But after the first ring, he handed me the phone.\n\nIt rang and rang and rang, and I thought about that confession I'd found. Whoever that person was, they had sought out someone to confide in. I wanted to talk to Leah.\n\nBut the phone just kept ringing.\n\nI hung up. I had hoped she would answer if she didn't recognize the phone number. I didn't want to talk to Pastor Kyle, even if he was probably an okay guy. I wanted to talk to someone who actually knew Diana.\n\nMy confession settled back in my throat.\n\nBut I didn't hand the phone back to Pastor Kyle. I didn't want to go back outside yet. The haze of accusation still lingered. I wanted to talk to someone.\n\nAnd if I couldn't talk to Leah, then there was someone else I wanted to hear from. Someone I hadn't talked to in months. I didn't need to apologize to her like I did Leah. In fact, the reverse was true. I wanted to hear an apology. If I had to be at this camp, she should at least have to explain herself. After breakfast that morning, when I had realized that there were Christians on this planet\u2014and even at this camp\u2014who weren't pink-swastika-carrying homo-haters, I kept thinking about her. Eden Springs welcomed Goth, who was openly gay. It welcomed Dan who, while a bully, didn't hate Goth. It also welcomed Charlotte years ago, and I didn't understand how she became the way she was, and why she couldn't accept people for who they were.\n\nOr maybe I just hadn't understood her. Maybe I just needed to hear her out.\n\n\"Um, can I try a different number?\" I didn't tell Pastor Kyle the new number was for someone else entirely. But this was actually for family, so I didn't feel too bad.\n\nThe phone rang three times before she answered.\n\n\"Hello?\"\n\n\"Hey.\"\n\n\"Who's this?\"\n\n\"Your brother.\"\n\nSilence.\n\n\"Really?\"\n\n\"Yes, Charlotte. It's Nick.\"\n\n\"Wow. Nick. Hi. How are you?\"\n\nAwkward is how I was. I hadn't talked to her since before the funeral.\n\nCharlotte had called the house after the funeral. She'd asked for me after talking to my parents. But I'd refused. But as I stood there at Eden Springs, with the old phone to my ear, I thought about how maybe refusing her calls was something like Leah refusing mine. I thought about Diana trying to talk to me in those last few weeks. How I'd been too irritated with her to really listen.\n\nMaybe when people reach out to you, you should reach back. Like Jack said, our words are our bridges.\n\n\"I'm good, I guess,\" I said.\n\nMaybe her apology was lodged in her throat.\n\n\"I was just calling to say hi,\" I continued. \"I'm at that camp you sent me to. Well, the one you told Mom and Dad to send me to.\"\n\n\"That's great. I thought you'd like that.\"\n\nReally? \"Yeah, it's great.\" Lie. The apology, the explanation, nothing was forthcoming. Maybe she needed a hint. \"I was wondering if I could ask you something.\"\n\n\"Sure.\"\n\nIt was silent as I tried to think of the exact words, to build up the courage for any words at all. How do you ask for an apology? It wasn't even for me. She owed it to Diana.\n\nIn the background, I could hear her kids\u2014my nieces and nephews\u2014arguing or playing. It was hard to tell the difference.\n\n\"Nick?\"\n\n\"I'm here.\" And then I just asked it: \"Why didn't you come to Diana's funeral?\"\n\nSilence. I watched Pastor Kyle shuff le papers, put pens in boxes. Then she said, \"You called me from Eden Springs to ask me that, Nick?\" Her voice was high-wire tight, and part of me wanted to take the question back. I didn't want to hear the wrong answer. I didn't want it to be what I'd thought. I wanted Charlotte to be better than I'd thought.\n\nI didn't want that hope-f lame to die out completely.\n\n\"Well, I've been wondering for a while, but\u2014why didn't you go to the funeral?\"\n\nA deep sigh through the phone lines. \"Nick, you know why. I didn't approve of her lifestyle.\"\n\nAnd there it was. \"Her lifestyle?\"\n\n\"Or how she died, frankly.\"\n\n\"How someone dies is a matter of approval?\" The pastor looked up. I turned around so that he couldn't see my face. I knew my eyes would give away my desperation.\n\n\"Suicide is a sin, Nick. And so is homosexuality. And the last thing I'm going to do is go to some funeral where people think I condone that.\"\n\n\"It's Diana. It's your sister.\" This was not the reaching out and reaching back that I'd expected. That I'd hoped for.\n\n\"And I sure as heck didn't want to have to see that Leah person. Her...\" Charlotte couldn't even say it.\n\n\"Her girlfriend, Charlotte. And you didn't even know Leah.\"\n\n\"How cool her girlfriend is doesn't make the sin less of a sin. Suicide and homosexuality, Nick.\"\n\n\"It's how you die and how you're born,\" I nearly shouted. I could feel the pastor watching, but I refused to turn around.\n\n\"We choose whether or not we go to hell. She chose.\"\n\nI couldn't believe I was related to this person. \"Did you choose to be such a bitch?\"\n\nAnd here it was again.\n\nIn that millisecond of silence, I heard the decrepit bridge between me and Charlotte collapse, falling into the rushing river below, debris breaking on the rocks on its way down.\n\n\"Okay, Nick,\" I heard my sister say, and the pastor was reaching around me to grab the phone, an echo of Charlotte: \"Okay, okay, that's enough.\"\n\n\"It's not okay,\" I said to both of them. \"None of this is okay. It's not fine. None of this is fine. Not you, and not you. I'm not fine.\"\n\nAnd I wasn't. I'd had two sisters my whole life, and now I had none. And I was stuck at some camp for God, and He was the one who'd stolen them both. He stole Charlotte, feeding her some kind of bullshit where she could trade in being judged for judging others. And He stole Diana entirely. He swiped her off the earth.\n\nCharlotte had already hung up, but I gripped the handset and kept saying the truth, \"I'm not fine,\" aware that there were tears on my cheeks. By the time the pastor wrested the phone from me and said good-bye into the receiver, I was panting.\n\nHe stood there, red-faced. \"Let me appraise you of the situation.\"\n\n\"It's apprise,\" I said, pressing the heels of my hands to my eyes. \"The word you are looking for is apprise, not appraise. Appraise is to assess value. Apprise is to inform.\"\n\n\"I don't know what that phone call was about, Nick, but I do know that cursing out a family member\u2014anyone at all\u2014is disrespectful and certainly not the will of God.\"\n\nThe f lood of anger was sinking back into my brain matter, leaving me too rational. Too aware of what I'd just done.\n\n\"I can tell you're struggling, Nick. Is there something you want to talk about?\"\n\nI looked at this man, who likely had never had a problem in his life. Perfect complexion, perfect teeth, perfect physique. God had given him everything and taken nothing. He'd been blessed by God since slipping out of the womb with a perfect smile through a painless birth.\n\nWas there something I was struggling with, besides the fact that he'd forced me to walk across campus carrying a cross for a crime I didn't commit? At least I'd just given him something to tell the campers. That I'd carried the cross for cursing my sister out. Definitely not following the Golden Rule.\n\nI shook my head. \"I'm sorry,\" I said to the pastor. \"I'm fine.\"\n\nBefore sending me back to my cabin for a big-kid time-out, he put his hand on my shoulder and closed his eyes. His lips started moving, and I could tell he was praying.\n\nI didn't have the energy to shrug him off.\n\nApparently the story about my rash hadn't reached him yet.\n\nWhen I opened the cabin door, Jack Kerouac stood in the middle of the room applauding.\n\nI stared at him. Finally, the clapping slowed to nothing.\n\n\"Is this ovation for swearing at my sister?\" I asked.\n\n\"I liked that, indeed.\" Jack appeared on the top bunk above me. \"But, more so, you have them convinced you'd actually steal something.\"\n\nI sighed and walked over to my bed. \"I wouldn't do that.\"\n\n\"Oh, that I know. That's your problem.\"\n\nI grabbed my notebook from my backpack and sat on my bed. I added a tally mark for Jack, as well as DOGGY DOG WORLD and IRREGARDLESS and APPRISE\/APPRAISE to my list of verbiage faux pas.\n\nHe looked over my shoulder to see what I wrote. \"One of your problems.\"\n\nJack's legs dangled over the bed above mine. He held a licorice stick between his fingers like a cigarette. He dropped an open bag of Twizzlers down to me. I took one, chewed on it as I lay back on the bed.\n\n\"You got any of that piss-beer like the other day?\" he asked.\n\n\"Seriously?\"\n\n\"Sadly, no. Unlike the handsome pastor, I do not think you'd risk a rule for a fun time.\"\n\n\"It's not a fun time when someone is reading all of our shit. You know that, right?\" I kicked at the top bunk. \"The stuff about my sister is in that box. I'm in full panic mode down here and you're bummed I didn't steal the gawddamn box myself.\" I sighed, chewed my licorice. \"For God's sake, what is so wrong with being a rule follower?\"\n\n\"Oh, Nicolas... these words. This idea. This is your soul-leech.\" He sat before me now, a sofa chair appearing suddenly underneath him. He was going for the full philosopher effect now. \"You're not meant to be a rule follower at all. You're not supposed to follow rules. They're not a trail up a mountain. They're not even a sidewalk through a dirty city. Rules don't lead anywhere. They're tools, is what they are. Yes, yes. You learn them, know them, pick them up, and put them in your pocket. Use them when the time's right. But good lord, do not follow them. You follow what you love.\"\n\n\"Are you done?\"\n\n\"Well, one more thing. Don't pick up too many rules either. They're heavy as stones. And you gotta travel light. Yes, you do.\"\n\nI lay back on my bed and closed my eyes, imagining soft leeches crawling around in my chest, searching for a little life left to suck out of me. \"You stole the box, didn't you?\" I said.\n\n\"Now, I would do that. But I did not.\"\n\n\"But you talked about how you wanted to know what was going on in everyone's heads. You just said that yesterday.\"\n\n\"I cannot steal that box any more than I can walk into a store and purchase a fifth of whiskey.\"\n\n\"Then who did? You said you see everything.\"\n\n\"I saw you call Leah.\"\n\n\"Yes, because I needed to talk to her.\"\n\n\"I know what you want,\" he said quietly. Kindly, in fact.\n\nI sighed and admitted why I'd called Charlotte. \"I wanted my sister to apologize.\"\n\n\"You can't ask for people to react any way, this or that. Not apologies, not forgiveness, not grace. You can only extend these things.\"\n\nYou can reach out, but you can't expect them to reach back.\n\n\"There a radio here?\" Jack asked, suddenly across the room, peering into cubbies.\n\nI shook my head.\n\n\"No cigarettes, no beer, no bebop, no women, nothing to do but roam and ramble and stare out into this vast expanse of half-naked girls\u2014\"\n\nAnd he was back at it. \"You can leave.\"\n\n\"\u2014most of whom are available for sweet breath and lips and words\u2014\"\n\n\"You can leave,\" I said louder.\n\n\"\u2014none of whom are Leah.\"\n\n\"You can leave!\" I sat up straight in my bed and threw my pillow.\n\nBut Jack was already gone.\n\nI lay back on the bed, but I was too angry to sleep. I had to carry a damn cross for something I didn't do. It was humiliating, and everyone saw. And even if Pastor Pettyfer (sans British accent) told everyone I didn't do it, they would only half believe him. You can't take that back. The accusation lingers, the condemnation a veil between you and everyone else.\n\nAnd Charlotte. Here I was at Jesus camp with gay people, and my own sister couldn't wrap her mind around homosexuality. Not even to understand her own dead-and-gone sister. Not even to go mourn her at her funeral.\n\nAnd then Leah. If she would just let me apologize. If I could just talk to her, to hear the way she talked. Just like Diana. If I could just say good-bye at least. But her absence just festered. To my parents, Charlotte was what they had left of Diana. I understood why they had spoken to her after the funeral\u2014she was their only daughter now. She was all they had left. And to me, Leah was what I had left.\n\nWhen my sister died, I spoke at her funeral, which was held in an old cathedral near where she lived. She didn't go to St. Mark's\u2014she wasn't a practicing Catholic\u2014but she liked to go there on Sunday nights for Compline, to hear the sonorous chants of the choir. To listen to the rites, to ponder her mortality. She'd brought me a couple times. I had lain on the f loor of the church, which you could do, and allowed the organ notes and the choir's voices to slip into me and resonate in my chest. I loved it there, and when she died I knew it was the only place for her funeral, but I had worried that the priest would have nothing to say because she didn't fellowship, or whatever, with the congregation.\n\nI stood up there, looking out into the mourners one last time in search of Charlotte, indignant that she hadn't come.\n\nI told the story of how, for my tenth birthday, Diana had kidnapped me and drove me in the green Volvo to the coast. She'd planned it with my parents, but I hadn't known that till later, when it was getting dark, and I had been afraid that my parents would be worried.\n\n\"It's okay,\" she'd said. She always said that to me. \"You're fine. I'm going to take care of you.\"\n\nShe'd parked her car on the beach, where the loamy waves beat against the hard gray sand. We had woven through the clumps of seaweed, crab shells, and barnacle-covered driftwood, picking up every sand dollar we could find, turning each over and over and inspecting the pale disks. It had taken a long time to find ten sand dollars that hadn't been broken apart by the seagulls. We'd put them in a yellow plastic bucket that Diana had pulled from her trunk, and then in a falling-apart cottage she'd rented for the night. We'd painted each sand dollar with a different scene, a scene that represented one wish for every year I'd lived.\n\n\"If only you had ten lives,\" Diana had said. \"You are ten-lives cool, Nick. Maybe one-hundred-lives cool. You're bigger and better than the rest of us.\"\n\nShe was crazy, and she said crazy things I barely understood. I told her so.\n\n\"I hope you live ten lives in this one. Live till you lose your breath.\"\n\nI didn't quote that conversation in my eulogy. Instead, I'd just said, \"Diana was a mad one.\" And then I'd meant to recite the Jack Kerouac quote that was printed on the poster on the wall, but I couldn't. My throat squeezed shut till I couldn't breathe, and I tried to talk and a sob came out, and my mom, whose mascara had run all over her face, came up to the podium with my dad and held me up as I walked back to the pew.\n\nAt the wake, I sat by the fireplace and looked out at all the beautiful women my sister had known. Her girlfriend\u2014or ex-girlfriend\u2014Leah sat down in a chair next to me and put her arm around me, telling me what a beautiful job I'd done.\n\n\"She was a Roman candle,\" Leah said, choking on her own tears at the end. I loved Leah for many reasons, least of which was her finishing the Kerouac quote with me.\n\nHer hair f lowed down her shoulders and she wore this tight black dress like she was going to a cocktail party. Leah was beautiful, and I had no childhood memories without her in them. She'd been my sister's girlfriend for a decade. I was fifteen, and she was twenty-three, and we were lost. I couldn't feel what held us together. I didn't know if it was a thread or a branch or a steel rod. I just knew I needed her to not leave me too.\n\nWhen Leah had broken up with my sister, I hadn't cared that Diana had been sad. I had been sad.\n\nAnd I had been mad at Diana for whatever she'd done to cause the breakup. It had to have been her fault. Diana had been too much a butterf ly. Leah had been on more solid ground. Leah had read books. She'd talked about them. She'd cooked dinner. Her interests hadn't waxed and waned. She'd had a job. She'd had a career.\n\nThe day my sister died, Diana drove to Leah's house. I don't know if she drove out there to talk, to win Leah back. Or maybe she'd always intended to punish Leah for ending it. Maybe there had never been a conversation one way or the other. Maybe there had never been any hope for either of them. Maybe the plan all along had been punishment, never atonement.\n\nIt didn't matter. Leah was punished.\n\nBecause my sister shot herself, and Leah had to see it. And see it. And see it. And see it. For all the long years that she would live. She would be punished.\n\nDiana was a mad one.\n\nAnd I was mad at her.\n\nAt the wake, Leah started talking, started retelling that day.\n\n\"It's my fault,\" she said. \"I should've just talked to her, or pretended I still wanted her... It's my fault.\" And then she just kept on saying it, repeating it like a mantra, like the prayers any of the left-behind say. Like my mom said, like my dad said.\n\nIt's my fault.\n\nIt's my fault.\n\nIt's my fault.\n\nBut they were wrong. They didn't know.\n\nLeah cried, her mascara snaking in sad rivers down her cheeks that somehow made her more beautiful. Desperate and beautiful, and I still loved her because I just always had, because she was always there. And I knew then that I wouldn't see her again. She would leave me too.\n\nI walked around the house, sipping off people's abandoned cocktails. Everyone seemed to drink strong at wakes. Soon, I was stumbling to the bathroom to take a piss. I was holding walls for balance, and then Leah was there, in the hall by the bathroom door. She'd cleaned off the mascara, but her eyes were still glassy, from drink and sadness.\n\n\"It's my fault,\" I said with a heavy tongue. I almost told her then. About the gun and the bullets.\n\nBut, instead, I leaned forward and kissed her. I didn't wait, and she didn't stop me. She opened her mouth, and I tasted her, the warm darkness of what it meant to love her. And the love mixed with my tears and her tears. And I loved her like my sister did, kissed her like my sister did. Sadly and wholly.\n\nThe only time I cried.\n\nThe only time I was being true.\n\n\"It was all sin and holiness,\" Jack said from the corner of the room.\n\nI was awoken by the door bouncing open against the wall and my cabinmates storming in.\n\n\"It's because of his towel,\" Charles said. \"Even if he didn't steal the box.\"\n\nThere was only one towel worthy of this level of attention.\n\n\"I don't even have my towel anymore,\" I said from my bed, startling them all.\n\nThe guys\u2014Charles, Chris, and Payton\u2014turned toward me.\n\n\"Oh, hey.\"\n\n\"What's up?\"\n\n\"You okay?\"\n\nCharles continued, unfazed by the fact that I'd heard his accusations. \"But you did. You did have the towel. And you brought it here.\"\n\nI looked meaningfully at Chris, but didn't say anything about his Ouija board.\n\nI sat up on the edge of my bed. \"Next it'll have brought a plague.\"\n\n\"Like smallpox,\" Chris said.\n\n\"Don't be ridiculous,\" Charles said to Chris. \"That's been eradicated.\"\n\n\"Probably swine f lu,\" Payton offered, straight-faced except for the twitch at the corner of his mouth that Charles clearly didn't see. Whoa\u2014he was on my side. I felt a sting of guilt for the Holly stuff. Did he know people said stuff about her?\n\n\"Ebola,\" Chris said.\n\n\"I'm serious,\" Charles said. \"All of this\u2014the PC Box\u2014this is what happens when you bring the relics of demonic worship to camp.\"\n\n\"Fine. What do you want me to do? I got rid of the towel. I didn't steal the box. I mean, I'm super pissed too. I have a confession in there.\"\n\n\"Maybe. If your confession is still in there,\" Chris said.\n\n\"What do you mean?\" I looked from face to face, every one of them drawn. \"What are you talking about?\" I felt my throat tighten in panic (or cardiac arrest\u2014as opposed to the symptoms of an imminent plague).\n\n\"Look outside.\"\n\nI stepped out onto the porch. About fifty political signs were stuck into the hillside. Campers walked back and forth on the lawn in front of the fence. Reading, waving, pointing, while the counselors frantically plucked the signs from the ground.\n\n\"Mine wasn't there,\" Payton said from behind me.\n\n\"What are those?\"\n\n\"Confessions,\" Chris said.\n\nMy face f lushed. \"Printed onto signs?\"\n\n\"No, printed on paper and taped to signs.\"\n\nI squinted to read, but I couldn't see from where I stood. At least they were taking them out.\n\n\"Mine was on one of those,\" Charles said.\n\nI heard him. I understood him. It explained his indignation.\n\nBut mine could still be out there.\n\n\"It was pretty easy to tell it was yours,\" Payton said to Charles. \"We all know you love the Disciplettes.\"\n\nCharles's eyes darted from Payton to Chris. \"But you didn't know I wanted to be one.\"\n\n\"Yeah we did,\" Chris said.\n\nHe swallowed, looked at me. He wanted me to make him feel better, but I couldn't.\n\nI kept my eyes on the scene. The barbed wire fence clearly indicated the hillside was off-limits. Campers were supposed to stay on the green grass. Stay inside the Garden of Eden, lest a serpent lead you to the tree of the fruit of good and evil, and all that jazz. Surprisingly, most campers still abided the rule.\n\n\"When did those signs even get there?\" I asked. The counselors kept yanking them from the ground, one by one. Gawd, I hoped mine wasn't out there.\n\n\"They were there when Pastor Kyle finished the sermon. You know, after he led you away. After they called us back.\"\n\nI turned away from the few signs that remained on the hill, my cheeks still hot from panic.\n\n\"Did he tell everyone I didn't steal the box?\"\n\nThe guys nodded.\n\n\"Did he tell everyone I found that confession, that it wasn't something I stole?\"\n\nWhen no absolutions were forthcoming, I asked, \"Did everyone believe him?\"\n\nShrugged shoulders, eyes avoiding mine. Matthew was the only one who ever stood up for me. But he wasn't here. I was the leper after all.\n\n\"You guys, it was in the hall when we were getting buckets for the prank.\"\n\n\"Sure,\" Payton said. \"You'd better not post my secret on one of those signs.\"\n\nThe signs were gone now. If mine had been out there, I'd have to wait till it blew through camp like their gossip before I'd know about it. I'd have to wait till someone put it all together.\n\nThe guys started getting ready for the pool, slipping into their swim trunks and grabbing towels. I lay back on my bed and stared at the bunk above me. The door shut without a single one of them inviting me.\n\nThey didn't believe the pastor. They didn't believe me. That truth curdled in my stomach like a slow poison.\n\nI took the Bible the camp had given me off the windowsill.\n\nI hadn't stolen that box. I hadn't dropped that confession in the hall. Hillbilly Jesus had said everything was going to be okay, and he'd quoted the Bible. He'd wanted me to know something. I turned to John, f lipped to chapter one, and read the ninth verse:\n\nThere was the true light, even the light which lighteth every man, coming into the world.\n\nI read it again and again.\n\nWhy would he give me this verse? Maybe it was a clue. Maybe Jesus knew who stole the box, knew that I was onto something, but couldn't tell me. Maybe he was speaking in code. Maybe the person who did the lights in the sanctuary was the culprit. Was there a lighting person? This wasn't Broadway.\n\n\"That doesn't make sense,\" I said aloud.\n\n\"What doesn't make sense?\" I sat up. Charles was still in the room. I'd thought he'd gone out with the other guys.\n\n\"I was just reading John 1:9, and it doesn't make sense to me.\" I wasn't going to tell him why I was reading the Bible. Maybe I'd get some credit for reading \"the Word.\" You know it's bad when you're hoping for homeschool credit.\n\n\"Which one?\"\n\n\"Which one, what? I asked.\n\n\"There's John, 1 John, and 2 John. Which John 1?\"\n\nFor eff's sake. I'd read plain John 1.\n\nI f lipped to 1 John 1:9 and read: If we confess our sins, he is faithful and righteous to forgive us our sins, and to cleanse us from all unrighteousness.\n\nI read it again. And I read it again.\n\nI'd found motivation.\n\nI set the Bible down and walked out the cabin door. From the porch, I could see most of camp. It was eerily quiet. Some of the Super Bowl Christians were back to normal, jumping off the high dive at the pool. Some played volleyball. A couple girls bought ice cream bars from the freezer bike rolling around blaring Christian rock. I saw many people walking in and out of the sanctuary. Small groups were sitting together on the patio where we'd prayed in small groups the night before. There was a line on the sidewalk to Pastor Kyle's door.\n\nI scanned from the barns to the sanctuary, looking for that familiar, long robe. I had to find Hillbilly Jesus. He'd had the opportunity. He'd been outside of the sanctuary the night the box was stolen. His office was by the sanctuary. He carried that box around all day. And now motivation\u2014he was a guy who played Jesus. A guy who had a bad accent, who scrubbed toilets by night. A guy with no actual power, who had found a way to really play Jesus. A written confession was one thing, but putting it out into the world, making us accountable to everyone. To Christ. That was a thing. I'd read about it on the Internet. And that's what he was doing. Charlotte would've been all about it.\n\nThe counselors were on high alert, peering into campers' faces, scanning the grounds, all a bit twitchy. Or perhaps the one riding the bike just didn't ride bikes that well.\n\nThe girls with their ice cream wandered past my cabin, and I followed them with my eyes. A guy with long hair was setting up a small table on the lawn outside the first set of cabins.\n\nJesus.\n\nI went downstairs and headed for the table just as some volleyball players left their game to walk over.\n\nJesus was performing a card trick\u2014an elementary one at that. He dealt four rows of cards, a girl pointed to a card, and then he picked them all up, shuff led, and stacked them in fours. He pulled out her card, apparently, because the round of golf clapping ensued.\n\nJesus shuff led the cards, looked around at his audience, then spotted me. \"Come in a bit closer,\" he said. His hillbilly accent was gone. I know he recognized me\u2014I was the Harry Potter kid, the barn kid, the carry-his-cross kid. But he was in character now, so he was all aloof and ominous, \u00e0 la Old Testament God. He looked around us, chin high, hands out like he was preparing to feed us loaves of bread and fish.\n\nI did as he said, and the girls around his table grudgingly stepped aside.\n\n\"Would you like a turn?\" he asked, his voice resonating. He must've taken vocal lessons to do that. He included all his syllables, no \"ain't\" to be heard.\n\n\"A card trick?\"\n\nDiana had bought a book of card tricks when she was in high school. She actually made money doing kid birthday parties and setting up makeshift stands down by the Pike Place Market in Seattle.\n\n\"I can guess your card.\"\n\n\"I'm sure you can,\" I said. Anyone could learn tricks.\n\n\"Don't you wanna witness it?\" he asked.\n\n\"I can watch other people while you\u2014\"\n\n\"Jack,\" he interrupted.\n\nI looked at him sharply. \"What?\" I'd come down here to talk straight, but he'd knocked me off balance. I needed him to explain that Bible quote, but not with all these other campers around.\n\n\"Jack,\" he said again. \"Always.\"\n\nI looked around for Kerouac. Were other people finally seeing the things I saw?\n\n\"Jack of Hearts for you.\"\n\n\"What are you talking about?\"\n\nJesus nodded toward me. \"In your pocket.\"\n\nI reached into both pockets.\n\nThere was the Jack of Hearts, in my right hand.\n\nThe audience gasped.\n\n\"Now do you believe?\" he asked.\n\nI shrugged. \"I already did.\" It was sleight of hand. Then again, he'd transformed overnight\u2014that wasn't just some cheap trick. What did he really have up his sleeve? I tried to hand him the card back.\n\n\"You keep it,\" he said, pulling a Jack of Hearts off the top of the deck in his hand. \"I already have one.\" He shuff led the cards. \"You like deception?\"\n\n\"No.\"\n\n\"You lie to yourself,\" he said. Then he winked at me. \"We all like deception. We like where the hunt takes us.\" I was sure he was talking about finding me in the barn, but it was still a weird-as-hell thing to say.\n\nHe folded up his table, slipped his cards into his robe pocket, and walked away. The girls with the ice cream cones followed him, chattering in awe. The volleyball girls waved over their friends who'd remained at the net. They all followed the man across the camp, to the end of the cabins where he set up his table once again.\n\nI started after them. I would hunt down this deception.\n\n\"You rose from the dead already?\" I turned around and saw Matthew, his hair still wet from the pool.\n\nIt took me a second to realize he was talking about dragging the cross for my punishment. \"It turns out they didn't crucify me.\"\n\nMatthew laughed. \"Nope. You just have to do the carrying the cross bit. You won't be the last. I had to do it my first year here too.\"\n\n\"Whatever, it's pretty effed up.\"\n\n\"Kind of, but it's like an initiation.\"\n\nI didn't think it was an initiation. An initiation was something everyone had to do, and I hadn't seen hundreds of kids dragging crosses that morning. Just me. And I felt that fine burning point of exclusion keenly.\n\n\"I didn't steal that confession.\" I regaled Matthew with the story.\n\n\"Why didn't you tell someone? Or turn it in?\"\n\n\"I wish I had,\" I said, but that was only true in retrospect. Only because I ended up having to carry a cross. Finding the paper had not expedited my search for the culprit. And now the confessions were all over the campus. Well, except for the one I'd found. Lucky kid.\n\nI looked over at Jesus setting up his table again, a new audience forming around him. \"Why's that guy doing card tricks?\" I asked.\n\n\"Dude, that's Jesus.\"\n\n\"I know,\" I said. \"Doing card tricks.\"\n\n\"He's performing miracles,\" Matthew corrected. \"He'll go hang from a cross on the second-to-last day. He's ripped under that robe. It's sick.\"\n\n\"He's not performing a miracle. He's performing a card trick that eight-year-olds learn.\"\n\n\"The tricks get better,\" Matthew said earnestly. \"And he does this cool magic where he lifts a table\u2014\"\n\n\"Wait, magic?\"\n\n\"Well, yeah, he's not literally Jesus doing miracles.\" Matthew chuckled.\n\n\"Magic. As in Harry Potter?\"\n\n\"No, as in Jesus. It's different.\"\n\nI looked at Matthew, who was a cool guy, the kind of guy who made this otherwise shitshow of a summer experience actually fun. He didn't see the hypocrisy. Or he'd settled in. Maybe that's what people did for a sense of belonging.\n\n\"Card trick miracles? Seriously, the garbage-can prank had more pizzazz.\"\n\n\"Sh, Nick, keep your head, man,\" he said, looking around. \"I'm gonna go shower and get a pedicure or ice cream or something. Wanna come?\"\n\n\"A pedicure?\"\n\n\"Sampson & Delilah's Salon. It's new this year.\"\n\n\"Maybe later,\" I said. Right now, I wanted to talk to Jesus again. He'd had motive and opportunity and now he'd made that weird Jack reference and said that thing about deception. He was playing me.\n\nI watched Magic Jesus (gone were the days of Hillbilly Jesus) pack up his card table and walk away a second time, his entourage following right behind. He raised his hands to them in what appeared to be a blessing, and they scurried away.\n\nHe walked around the sanctuary at the far end of camp and disappeared down the path by the medical building (aptly named Healing Hands!). It was the same place we'd filled the garbage cans with water the night before. Not far from the custodian's office, and the hallway where I'd found that confession. I walked quickly down the sidewalk after him. But when I rounded the corner of Healing Hands!, he wasn't there. I saw the back door of the sanctuary propped open. I didn't know if Magic Jesus was in there or the in the medical building.\n\nI started with the sanctuary. It was lit up, and the stage curtains were open. I could hear chatter and laughter and tears. Standing at the side of the stage, I saw groups of campers in prayer or chattering lightly. Jesus was not among them.\n\nI headed backstage next.\n\n\"Hey, what're you doing back here?\" It was Dan. \"Now's not the time for pranks, dumbass.\"\n\n\"Have you seen Jesus?\"\n\n\"No, but I feel him in my heart.\" He thumped his chest and held up a gang sign. Or a God sign.\n\nI rolled my eyes. \"The magician.\"\n\n\"Look,\" Dan whispered. \"I'm getting some shiz ready for some pranks, and you're making it look all suspicious. Go pray, or go home. He's not back here.\" He stood there till I walked out the door to the sidewalk.\n\nI crossed over to Healing Hands!. The door was closed. I imagined the PC Box inside the building, broken open, with confessions strewn like straw across the f loor. I tried the handle. The door was locked.\n\nI banged on the door. Magic Jesus had to be hiding in there\u2014unless he'd slipped away while I was searching the sanctuary. If he was in there, then he'd have to come out sometime. Jesus was essential to the pastor's sermons.\n\nI could have sat and waited for him to come out, but I didn't. Because, just then, I spotted Natalie walking up on the hillside again.\n\nThis time, I knew it was her, not just some nameless silhouette. I could see her red bandana.\n\nAnd a backpack. I imagined Jesus stuffing that backpack with confessions, Natalie making the signs with her clever designs. Marginalizing our confessions.\n\nCo-conspirators.\n\nI set off after her.\n\nI started up the hillside, following that familiar backpack. Yes, I could get in trouble for being on the hill, but I already had gotten in trouble, and for something I hadn't even done. At least the small trees dotting the hillside partially hid us from view.\n\nWe climbed, Natalie still unaware of me behind her. After scrambling up and over the ridge, Natalie kept walking, but I stopped. Not because I was out of breath. I was stunned into place by the view. Three giant crosses stood beyond the hill in a small valley in the vast, desertlike landscape. They hadn't been visible from camp, and they were huge\u2014way bigger than the one I had to carry.\n\nNatalie slowed as she passed between two crosses, resting her hand on the middle cross. She was not in a hurry. Not to bury the backpack or remove its contents.\n\nI called out to her, hoping my voice wouldn't sneak out of the valley and down the hill toward camp.\n\nNatalie turned and spotted me, clearly surprised. She didn't smile like usual, and she didn't wave.\n\nInstead, she looked back over her shoulder, in the opposite direction from camp, as if someone else was waiting for her. I looked past her, but all I saw was more gray-brown hills of brush and short trees. The sky was turning a soft pink. Natalie turned back around to face me.\n\nFinally, she turned on a smile, the smile of the bubbly, breezy Natalie. But I could tell that's what she'd done. Folded up whatever expression had been on her face and put on her smile like a mask. She walked a few steps toward me and called, \"You do know you're breaking a rule right now?\"\n\nI caught up to her. \"Birds of a feather and all that,\" I said, eyeing her pack.\n\n\"Yeah right.\" She smiled.\n\nI shrugged and smiled back at her. \"What is this place?\" I asked, tilting my head back to look up at the crosses. They were at least twenty feet tall.\n\n\"Golgotha, obviously.\"\n\nIt wasn't so obvious to me. \"What?\"\n\n\"Not what. Where,\" she said. \"It means 'hill of skulls,' or something like that. It was the place outside Jerusalem where the Romans crucified their criminals.\"\n\n\"Like Jesus.\" I hadn't known the place had a name.\n\n\"Exactly. But it wasn't just Him. There were thousands of others. It wasn't a special way to die or anything. It was normal. Like lethal injection.\"\n\n\"Your definition of 'normal' concerns me.\"\n\nNatalie laughed. \"You know what I mean.\" She shifted the pack off one shoulder.\n\n\"Well, I can say from personal experience that I'm glad we've moved away from crucifixion.\"\n\nShe raised her eyebrows. \"Oh, it still happens. Just like people are still stoned and burned and beheaded.\"\n\nI imagined giant nails hammered through my wrists and feet. Hanging naked and bloody for hours, until the weight of my body crushed my lungs. The time it would take. How long you'd have to be alone in your pain.\n\nIt was the last moments of death that obsessed me, especially after Diana died. What did the brain think? What did it feel? How much did pain hurt in the last minute of life? In the last millisecond? Was it the worst kind of hurt, or did it not matter at all?\n\nI put my hand on the thick, wooden pole that reached way up to the crossbeam. \"But there are only three crosses that really matter,\" I said.\n\n\"Well, technically one,\" Natalie said, nudging me with her shoulder. Then she stepped away from me, but continued talking. \"We go on a hike up here the day before we leave. You know, to help us really see the sacrifice.\" She set the pack on the ground.\n\n\"So you came up here early to really feel it without the masses of bored teens?\"\n\nShe narrowed her eyes, looked out and away from camp, decided something. I could see it cross her face. Some concrete plan. And it was just like I'd told Jack during the Silent Three. She had some secret I couldn't know. The nonwords.\n\nShe unzipped the backpack, and I knew what was going to happen next. Those papers.\n\n\"Picnic?\" she asked.\n\nThe pack, which she now held out to me, was full of food. So she was not some kind of confession-runner for Jesus.\n\nI was relieved because I liked her, disappointed because I wanted finding my confession to be that easy. \"Food is always good.\" I shrugged.\n\nNatalie sat down on the ground with her back to the camp and faced the pink sky in the West. She unzipped the small front pocket and took out two protein bars.\n\n\"You like to hike?\" she asked, handing me one.\n\n\"Not really. I was just following you.\"\n\nShe smiled.\n\n\"I like to fish, though, and hunt.\" I said so that she knew I liked the outdoors, in case the answer mattered. \"My dad and grandpa go a lot. I'm the only one who packs a book.\"\n\n\"That you don't write in.\"\n\n\"Am I belaboring that point?\"\n\nShe smiled, bit into her protein bar. \"I like to hike. My favorite part of camp is the hike up here actually.\"\n\n\"Seems kind of like a nonevent. You just walk up a steep hill.\" All hiking seemed like a nonevent to me. Fishing could be boring at times, but it had a purpose. Hunting too. Though I wondered if anyone in my family would go hunting this fall. I didn't know if my dad or grandpa would be keen on it anymore. After Diana shot herself.\n\n\"No, they take us this really long way around at sunrise, telling stories, preaching, and all that. It's cool. Then we get up here and someone tells his testimony, and the sun comes up and we all walk back in the light.\"\n\n\"The way, the truth, and the light.\"\n\n\"Amen.\"\n\n\"So when's this hike?\"\n\nShe swallowed another bit of the protein bar. \"Friday.\"\n\n\"What's today?\"\n\nShe laughed. \"Tuesday.\"\n\n\"It's summer; they took my cell phone. Time is lost on me.\"\n\n\"As it probably should be all the time.\"\n\nI took a couple bites of my protein bar. It tasted like berry-infused paste.\n\n\"Should I not admit that I followed you because I thought you were a nefarious confessions trafficker for my number one suspect?\"\n\n\"I'm f lattered yet again,\" she laughed. \"But, wait, you have a suspect?\"\n\nI told her about Magic Jesus and his opportunity and his motivation. She listened and complimented me on my cleverness, but she didn't seem entirely convinced or deeply concerned.\n\n\"Don't get me wrong,\" she said. \"I want them to find this person and put a stop to our secrets being out there. I really, really do. But we have to face the fact that we might not figure it out. We need to plan in that direction. We need to pray in that direction. We can only control what we can control, and I don't know if you're going to catch the guy.\"\n\nI didn't like that answer. I was going to hunt down Magic Jesus. By the end of the day, if I could manage it.\n\n\"So why'd you come up here if we're hiking here soon?\" I asked.\n\n\"This is the prettiest place.\"\n\nWe were sitting in an off-limits area, where the grass didn't grow green, where there were no slides or rap songs or ice cream bicycles or lush cabins or family-style meals. We were sitting in the dirt, looking out at dry sage and too-short trees on a dusty landscape with the horizon painted pink. We were outside of Eden Springs, the part of the desert that men built to show us the beauty of God.\n\nNatalie was right, though. This was, in fact, the prettiest part. \"My brother's about twenty miles that way,\" she said, pointing in the direction we were facing. The opposite direction of camp.\n\n\"You have family that lives down here?\"\n\n\"He's at camp too.\"\n\n\"That's cool.\" What does a University of Washington rower do in the desert? \"Same kind of camp, or... ?\"\n\n\"He's at zombie camp,\" she said and took a bite of her protein bar.\n\nI laughed. \"Seriously?\"\n\n\"Yeah,\" she said.\n\n\"Well, I guess it is the same kind of camp then. Jesus being the first zombie and all.\"\n\nShe smiled, shaking her head. \"You're going to hell.\"\n\n\"Raised from the dead, walking around scaring his enemies, and his friends too, frankly.\"\n\n\"Lightning bolts are coming for you right now.\"\n\nShe pulled four bottles of water from her pack, handed me one, stuffed two back.\n\n\"You were prepared,\" I said, indicating the water bottles.\n\n\"I'm always prepared.\"\n\n\"For an apocalypse?\" I asked as she took out two oranges and some trail mix. I saw a couple of cans of soup or chili or something still in her pack.\n\nWe peeled our oranges in silence. I popped a full slice in my mouth. We sat for awhile, and Natalie's smile disappeared. The one she'd put on for me. \"You know what it's like to miss someone so much that you feel like you can't breathe?\" Natalie asked. \"Like their very absence stole all the air out of your lungs?\"\n\nThe sudden question, the fact I knew exactly that feeling, stole my air.\n\nWhen my dad first brought me to counseling after my sister died, the psychologist did this weird exercise where he had me tap on my sternum. He instructed me to say certain things I was thinking as I tapped. He told me, You'll know you've hit on the truth when you cry.\n\nI have no idea what the chest-tapping thing was. I don't know if we hold our tear bucket under our collarbone, and bruising the shit out of our rib cage is the equivalent of punching a hole in it. Or if it's some physical distraction that lets the mind wander to what we really believe. I never cried with the counselor, not when tapping my chest or just talking.\n\nI'd cried only once since my sister died. At the wake. Only two people as my witnesses\u2014Leah and Jack. If Jack counted as a person.\n\nNatalie had hit on the truth just then. Afraid to talk now, afraid months of tears would spill everywhere, I simply nodded yes. I knew what it felt like to miss someone so much that the breath was snatched from your lungs. To feel claustrophobic with loss, like the universe's walls were falling in on you. To feel trapped in a world where your favorite person no longer existed.\n\n\"That's how I feel about my brother,\" she said.\n\nI ate another orange slice, chewed it a long time, focused on the movement of my jaw until I was sure I wouldn't cry. Finally I said, \"You don't see him?\"\n\n\"He's pretty busy with school,\" she said before looking away, and again I sensed words still in her mouth, tucked in her cheeks, and under her tongue. \"I go to this camp because he goes to that camp. That way I can be near him.\"\n\nI was obviously missing something. It wasn't just a college thing. Maybe that family black hole. \"Divorce?\" I asked, and the word felt sticky in my throat.\n\nShe nodded vaguely. Like Matthew said, a closed book.\n\n\"Sometimes, I just talk to him, and it's like a prayer and I think maybe I can hear him talk back. And then I think God probably calls that a sin.\"\n\nI didn't really know what to say. \"I don't think He judges you.\" I didn't know how to talk about God, so I changed the subject. \"I sometimes do that with my sister. I miss her so much that I sit in her car and talk to her.\"\n\nShe looked away from me, her eyes settling somewhere across the valley. \"You know that feeling when you watch someone walk away? They just get smaller and smaller on the horizon till they're just a f licker. And even when they aren't there anymore, you can still swear you see them. Maybe that's how you know you love someone.\"\n\n\"There's something my sister used to say.\" Gawd, I hoped Jack Kerouac wasn't listening when I quoted, \"'A pain stabbed my heart, as it did every time I saw a girl I loved who was going the opposite direction'\u2014\"\n\n\"\u2014'in this too-big world,'\" Natalie finished.\n\nI thought of Leah then. Walking away from my sister and eventually walking away from me.\n\nNatalie's face spread out with a smile so big. Real this time. \"You like Jack Kerouac?\"\n\n\"My sister kind of worshipped him,\" I said.\n\n\"I think I like your sister.\" Natalie used the present tense, and I reveled in it for a moment, could almost pretend it was true. \"I kind of worship him too,\" she said, then closed her eyes and raised her hand to the sky. \"I'm just hyperbolizing. Sorry, Jesus.\" Then she laughed and looked at me with this kind of amazed glowiness all over face. \"I can't believe you know Jack Kerouac. Like, who knows Kerouac these days? I mean, people our age anyway.\"\n\n\"I don't actually like him. Not his writing, and not him really. I mean, from what I know, he was kind of an asshole to people. But I can quote him at dinner parties,\" I joked. \"I don't really even know what it means half the time.\"\n\n\"Dinner parties\"\u2014Natalie laughed at this\u2014\"and church camp.\" Then she took a deep breath and settled into this peaceful smile. \"You're a funny one,\" she said.\n\nShe was too. The way she looked around, all bright-lights-in-her-eyes, all that delight when the pages of herself were f lipping from this to that. With her hair and cheekbones all sexy and strong like Rosie the Riveter.\n\n\"I suppose you've marginalized On the Road?\" I asked.\n\n\"No, I haven't, actually. I've read it, but haven't taken pen to page.\"\n\n\"No doubt it needs marginalizing for anyone to like it,\" I said, hoping Jack would hear that part.\n\n\"You're quite the harsh critic.\"\n\n\"He just talks and talks and talks, and never really says anything.\"\n\n\"So you've read it?\"\n\nCrap. \"Just parts of it. And then, you know, my sister and her obsession.\"\n\n\"Why does she like him so much?\"\n\n\"She wanted to join the circus and hit the road. And he did. Well, hit the road, not the circus part.\"\n\n\"She doesn't want to anymore?\"\n\nAnd I said the next part before I could stop myself. \"She committed suicide.\"\n\nNatalie's mouth fell open. \"Oh, my gosh, Nick, I'm so sorry.\"\n\n\"No, it's okay. I mean, it's not okay. I just\u2014\" I stopped rambling before the rest of the story spilled out. With the way people talked about everyone at camp, maybe she'd already heard about the Ouija board. But nobody knew the rest of it.\n\nNatalie had stopped eating and was looking at me. Really looking, in a way that made me suspicious that she might see the nonwords, and she'd read me anyway.\n\n\"My sister was this crazy beautiful thing,\" I said. \"She kind of butterf lied through life.\"\n\n\"Butterf lied through life... I like that.\"\n\n\"New verb.\"\n\n\"From Urban Dictionary?\"\n\n\"It is pretty gangster.\"\n\nShe nudged my shoulder. \"Your sister was a mad one then?\"\n\n\"Pretty much,\" I said. \"And she dragged me around with her like a toy and I had some vicarious fun.\"\n\nNatalie was smiling. \"She danced, and you stood against the wall?\"\n\n\"Well, she was always trying to get me to dance.\"\n\n\"At least you showed up?\"\n\n\"Can't take credit for that either. She forced me.\"\n\n\"If it's any consolation, I didn't used to dance, or even stand against the wall. I was, like, on the other side of the wall, peeping through a small hole. I was scared to do anything.\"\n\n\"I can't imagine Rosie the Riveter not dancing.\"\n\n\"I was shy, scared.\"\n\nI couldn't imagine that of the girl from pretty road. \"What changed?\"\n\n\"I don't know really. I guess there was something I wanted bad enough. Like, I lived behind that wall. And then one day, I walked around it. Whatever I was afraid of stopped scaring me.\" She was looking off again. \"But honestly, I don't remember what it was.\"\n\n\"No epiphany?\"\n\n\"Nope. I just started trying things and doing things, and finding I loved the things. And the world just kept getting bigger and bigger.\"\n\n\"Like what?\"\n\n\"Like, the f lute. I tried that. I was horrible. And fishing. Worms, gross. But I like swimming, and I like painting, and I like\u2014\"\n\n\"Marginalizing.\"\n\n\"I'm an expert marginalizer.\"\n\nWe both laughed.\n\n\"You know how they say eyes are the window to the soul and all that? For me, art is like the door out of my soul. It's safe to come out that way.\" She looked at me, searching. \"You know?\"\n\n\"Yeah, I think so.\"\n\n\"It's why I have to go to City School, why I have to go to RISD. It's just another kind of church, for another part of my spirit.\" She paused. \"What about you? What are you good at?\"\n\n\"Not much.\"\n\n\"What?! You can't not be good at something.\"\n\n\"Okay, I'm good at grammar.\"\n\nShe shook her head and smirked. \"This situation is grave.\"\n\n\"I never met a dangling participle I couldn't nail back into place.\"\n\n\"You just keep talkin' sexy.\"\n\n\"Apostrophes, pronoun usage, split infinitives.\"\n\n\"So hot.\"\n\n\"I have very strong opinions about the Oxford comma.\"\n\n\"Doesn't everybody?\"\n\nWe laughed.\n\n\"I never should've admitted to writing in books,\" she said.\n\n\"My stomach curdled to witness it.\"\n\n\"There are the people who write in books and then there are the people who don't.\"\n\nI nodded. \"I'm Team Not Writing in Books.\"\n\n\"But you're Team Secretly Correcting Grammar.\"\n\n\"I'm not so secret about it.\"\n\n\"Charming.\"\n\nWe sat there for a while, but I didn't want to go just yet. I liked being around her. The easy banter.\n\nI cleared my throat. \"So what do they do at zombie camp? Like, costumes and stuff? Crawl around on the ground like their limbs are falling off, perhaps?\"\n\nShe looked off to where the sun was dipping below the horizon. \"I guess.\" She started packing up our picnic garbage. \"I'm gonna head back, okay?\"\n\nHer mood had changed. Just like that. \"I'm sorry.\"\n\n\"Oh. No, it's not you.\" She smiled weakly. \"You didn't do anything.\" But I had done something. Or hadn't done something. I didn't even know.\n\nShe stood up and brushed off her shorts.\n\n\"I'm gonna go the long way back, okay?\" She pointed to a trail that I supposed was the one we would return by when the whole camp walked up here for testimonies. I could tell it wasn't an invitation to follow her.\n\n\"Yeah, I'll just slip down my little hill over there like a ninja.\"\n\nShe didn't laugh. Just swung her backpack over her shoulder and slipped her other arm through. She took a couple of steps toward the trail and then turned around. \"I'm glad you broke the rules to stalk me,\" she said, which made me feel a little better.\n\nI smiled, nodded, put my hands in my pocket. \"Thanks for the snacks.\"\n\nShe waved and resumed her walk toward the trail. I looked up at the three crosses, no longer casting much of a shadow as dusk descended. I leaned against the middle one and watched Natalie wind her way into the valley until I couldn't see her anymore.\n\nI was well aware that I was still looking for her when she was long past the point of being seen.\n\nBefore Dan returned from his nightly counselor meeting, the Ouija board was out again. But this time, I had an idea. No search for the Holy Ghost this time.\n\n\"Let's ask about the PC Box,\" I suggested. I had returned to Healing Hands! after hanging out at Golgotha with Natalie, but found the door still locked. I'd peeked into the sanctuary, visited the barn, walked circles around the camp. Jesus was nowhere to be found, and he hadn't shown up for the evening's sermon.\n\nSurely Jack, know-it-all and stalker, had seen who took the box. And he insisted he was roaming around camp with the likes of Jesus and Ra and the Buddha. Surely, one of them would be able to shed some light on the truth here.\n\nMaybe Jack would talk to me through the board. Maybe he'd stop with his word puzzles.\n\nWe were all in the cabin\u2014Chris, Payton, Goth, Charles, Matthew, and me. Chris and I put our fingers on the planchette. After doing the infinity thing like the night before, it started settling on letters.\n\nF-\n\nR-\n\nA-\n\nN-\n\nK-\n\n\"Frank?\" Chris said. \"Someone named Frank stole the box?\"\n\n\"There's no Frank at camp,\" Goth said.\n\nMatthew added, \"Though it does sound very criminal. Very 1920s mafia.\"\n\nBut the planchette had still been moving while we talked. There'd been something else, and we'd missed it.\n\nM-\n\n\"Matthew?\" Charles called from his bunk.\n\nOur heads spun toward our very own Matthew.\n\n\"Dude, are you serious?\"\n\n\"I'm just making inferences based on the text,\" Charles defended. He had that look on his face again, the darting eyes and fidgety hands. Was he concerned for us? His cheerleader confession was already public. But worrying on behalf of other people didn't seem like Charles.\n\nY-\n\nR-\n\nR-\n\nH\n\n\"Common Core failed you,\" I called up to Charles.\n\nSomeone made a buzzer sound.\n\n\"That's not even a word. That's the sound you make while taking a shit,\" Payton said.\n\n\"It's myrrh,\" Goth pointed out.\n\n\"Is that even a name?\" We all looked at Payton.\n\nThe board spelled out B-E-T-H.\n\n\"Beth and Frank?\" Matthew's brow furrowed in disgust. \"Nobody names their kids that anymore. Those are grandma and grandpa names.\"\n\n\"Bethlehem,\" Charles said. He'd been watching when we hadn't. Like before, the planchette had kept moving.\n\n\"Wait, then,\" I said. Bethlehem, myrrh... \"Frankincense. I bet that's what the other word was. Not Frank.\"\n\nThey all nodded.\n\nThe Ouija board spelled MANGER. WISE MEN. MORMON TABERNACLE CHOIR.\n\nThis was absolutely Jack Kerouac having fun with us.\n\nThe planchette circled. Then, S- I- L- E- N- T- N- I-.\n\n\"Silent Night!\" Payton shouted.\n\nJ- I- N- G\n\n\"Jingle Bells!\" Again, Payton.\n\n\"F- R- O- S-\"\n\n\"Frosty the Snowman!\"\n\n\"Seriously, let us have a chance,\" Chris whined.\n\n\"It's like Name That Tune, without the tune,\" Payton said.\n\nChris added, \"And with the Holy Ghost.\"\n\nT- A- N- N- E- N- B- A- U- M.\n\n\"The Royal Tenenbaums? That's not a Christmas song.\"\n\n\"It's a movie. And not a Christmas one either.\"\n\n\"That's the German name for 'Oh, Christmas Tree,'\" Charles said. \"'O Tannenbaum.'\"\n\nClearly we'd gotten too cocky. Jack had upped the ante.\n\nThe Ouija spelled FELIZ NAVIDAD.\n\n\"The Holy Spirit is going multicultural,\" Matthew said.\n\nHow PC of the PC Box.\n\nThe planchette spelled song after song, like a YouTube Yule log playlist.\n\nYULETIDE.\n\nCAROL OF THE BELLS.\n\nTHE HOLLY AND THE IVY.\n\nThe planchette stopped moving.\n\nPayton begged for more. \"Come on...\"\n\n\"Are you serious right now?\" Chris actually shouted.\n\nI shook the game piece across the board. \"I guess the Holy Ghost got bored,\" I said. But what I really wanted to say was, Jack is an asshole.\n\nPRAYERS AND CONFESSIONS\n\nPastor Kyle: Lord, thank you for bringing these kids here to camp. Please help me guide them through this dark time, Heavenly Father. We've all experienced dark hours. And You've been there for me in the worst of times.\n\nDan: My dad's nightmares are getting worse. Last week I woke up and found his shotgun leaning up against the wall by my bed. He has no recollection of putting it there. Alcohol + PTSD = a negative equation. I don't need Charles's help to know that.\n\nHolly: I'm not sorry for going to that stupid party. I just should've known better. They're always looking for a story.\n\nDan was back and ready for action. He talked about Jesus as our BAE. We said a prayer. \"Okay, let's trust it out, pray it out, and prank it out, fellas!\"\n\n\"What about the PC Box?\" Charles asked.\n\n\"What about it?\"\n\n\"Well, doesn't it seem inappropriate to continue your pranks?\"\n\n\"Yeah, seems kinda weird,\" Goth said.\n\nWhile I didn't say it aloud, I was ready to get back out there. But this wasn't pranks for me. I was going to search for clues.\n\nIf only Charles knew. He could be the Robin to my Batman on this camp-wide hunt for justice. The Pythagoras to my Euclid.\n\n\"Inappropriate,\" Dan sang back in a baby voice. \"Kinda weird.\" He put his hands on his hips and returned to drill sergeant mode. \"Don't be a bunch of pussies.\"\n\nSo the cabin chanted the prank rules while Charles crawled up into his bunk. I mouthed watermelon, watermelon, watermelon so that it would look like I knew the rules.\n\n\"Lower cabins got it last night. Upper cabins tonight.\"\n\nDan led us through the darkness to the sanctuary. \"Nick,\" he whisper-growled. \"Get in there and get duct tape and superglue.\"\n\n\"Me?\"\n\n\"You know where the custodian's office is. Now go.\"\n\nDan seemed to want me to get in trouble. As if carrying that cross hadn't been enough. \"Payton's on work crew; he knows that office better than me.\"\n\n\"I do grounds. I don't go in there,\" Payton said.\n\nBut then I realized: this was something like divine intervention. I had just been given permission to snoop for evidence. Magic Jesus had had the opportunity to steal those confessions. And Dan had just directed me to dig around in his office.\n\nI walked through the door and down the hall. I knew this time which door was for backstage and which was the custodian's. The light was on in Magic Jesus's office, but he wasn't in there. So I tiptoed in and slowly closed the door behind me, to keep any of my shadows or noise from the hallway.\n\nI looked on the shelves. Leaf blower, several large toolboxes, toilet paper. I opened the cupboard, looked through boxes, shoved cans of paint aside. Nothing.\n\nNo doubt Jack was loving this. Two-time rule breaker.\n\nUnlike when I'd snuck up the hill after Natalie, though, this time I had clear purpose. If Jesus was the one who stole the PC Box, it would be in this office. That little paper confession from the hall the night before would not be the only one f loating around. While sign after sign of spilled confessions had already been posted, there had to be hundreds more. And this would be the perfect place to store them; no one else would be in here.\n\nI checked the desk. Pens, Post-its, a f lash drive, and a deck of cards in the center drawer. Magic Jesus's magic cards.\n\nI opened the side drawers one by one. The top drawer had the tape. Second one had the glue. I didn't need to open the big bottom drawer. I had the supplies. But just in case.\n\nI slid the drawer open to reveal a set of hanging file folders with campers' names on them. Maybe ten. Payton's name was there. So this was his work crew. Not surprisingly, Holly's name was in there too. Maybe she was on crew for a scholarship, or maybe she just wanted to follow Payton around. I didn't recognize any of the other names.\n\nThe door swung open.\n\n\"What're you doing?\"\n\nI whipped around, snagged my foot on a cord.\n\nDan caught the laptop as it slid across the desk.\n\n\"What is taking so long?\" he snarled.\n\nMy heart raced. \"I couldn't find it at first.\"\n\nDan situated the laptop back the way it had been. \"Wanna bet this is full of child porn?\"\n\nHe sat down in the chair. He was an asshole, and he was really going to check the laptop.\n\n\"Let's just go,\" I said.\n\nAnd he did, thank God.\n\nI hadn't been caught searching. But I also hadn't found anything. Yet.\n\nDan explained our mission and then assigned each of us a partner and a cabin. I was with Goth, pranking Jericho.\n\n\"That's Holly's cabin.\" Payton balked.\n\n\"Which is why you're not going in, man-whore,\" Dan said.\n\n\"We'll get our hands on the box while we're in there, Payton,\" Goth said.\n\n\"Shut the hell up,\" Payton said.\n\n\"Get in, get out,\" Dan laughed, dragging Payton to their cabin.\n\n\"That's what he said,\" Chris joked.\n\n\"Or she.\"\n\n\"Depends on if it's good or bad.\"\n\n\"Depends on if it's hetero or homo,\" Goth said.\n\n\"Holly definitely says that to Payton.\"\n\n\"Hurry up!\" Matthew imitated a girl's voice. \"Get in, get out.\"\n\nSomething heavy settled inside me. I didn't care about Payton. But I kept thinking about Kim's nasty remark to Holly. \"Come on, you guys,\" I said.\n\nThey stif led their laughter as we reached the cabins.\n\n\"Okay, but seriously. Look for the box,\" I said.\n\n\"We're always looking for the box,\" Matthew said.\n\n\"I'm not,\" Goth said.\n\nMore laughs.\n\n\"Okay, but seriously,\" I said.\n\n\"Dude, okay, we're going to.\"\n\nWe split up.\n\nWhen Goth and I reached the back of Jericho, one of the girls' cabins, we walked carefully around the side and up the stairs to the second story. This was one of the bigger cabins. Two entrances, and way more campers with stuff to mess with. More potential thieves.\n\nI held the duct tape, aware my hands were sweaty. Goth had the superglue.\n\n\"Let's start with the glue in the bathroom,\" Goth whispered.\n\nThe bathrooms were farthest from the front door, our escape route. It made sense to get the hardest part of the job done first. We climbed the cabin stairs like cartoon robbers and stopped at the first door, but I signaled right, and we crept over to that door. I put my hand on the knob and nodded. He nodded back. I opened the door a crack, and Goth sneaked in. I shut the door and waited outside for a brief moment. When he didn't come out screaming, I opened the door and slipped inside.\n\nWith the curtains closed, it was too dark to see Goth's whereabouts. I waited for my eyes to adjust, listening to the sleep-breathing of the girls. One girl snored, but it was a soft sound, not like guys' snores, which always sound like we're chomping at the air.\n\nTheir light snoring was one reason why I couldn't imagine the PC Box being in a girls' cabin. Plus, I didn't know who was staying in this cabin, and I didn't know about motives and opportunities for these girls. But then I thought about talking to Natalie up at Golgotha. She hadn't seemed that interested in finding the PC Box at all. So poised, for someone whose secrets could be leaked at any moment.\n\nI definitely wasn't going to pass up the chance to poke around.\n\nMy eyes weren't really adjusting, so I walked straight through the dark with my hands out in front of me.\n\nThen the bathroom light f lickered on, automatically triggered by the door, and Goth slipped through. I followed. It smelled like apple-vanilla-lavender sweetness.\n\nI really couldn't imagine the PC Box in a girls' room.\n\nWe started at the cubbies, each with a name taped on the top. From every cubby, we grabbed one bottle or compact or comb. We put three dabs of glue on each item and set the objects back into the cubbies.\n\nI slid Natalie's bottles of this and that around in her cubby, looked in the little zippered bag there. No evidence.\n\n\"You do Holly,\" Goth whispered, interrupting my search. \"That's not my style.\"\n\nAt Holly's cubby, there was a manila folder. I didn't want to ruin her love poetry or song lyrics or whatever she was storing, so I left that alone and glued the rest of her stuff to the sides of the cubby.\n\n\"I hope this holds with the moisture,\" Goth whispered. \"You know, the heat and the shower.\"\n\nThe door suddenly swung open. Goth crouched in a shower. I backed into a stall and locked it. My blood stormed in my ears.\n\n\"Sarah, is that you?\" a girl asked.\n\n\"Mm-hmm,\" I mumbled in my best high-pitched, tired-girl voice.\n\nShe peed for a long time. More like tinkled.\n\nWhich was the third reason I couldn't imagine the PC Box here. Girls tinkle. They don't rob and steal.\n\nShe f lushed the toilet, washed her hands. We remained still. After a few minutes, the lights went out. We left our hiding places and exited the bathroom, the light popping on behind us.\n\nGoth and I jungle-crawled across the f loor to the other side of the cabin to the second bathroom, where we glued more girl paraphernalia to the cubbies.\n\nTime for the duct tape.\n\nOutside the bathroom, we found shoes and snacks and notebooks, and duct taped them to the ceiling and to the sides of clothing cubbies.\n\nI was disappointed to not find the PC Box. Not in the cubbies, not under the beds, including Natalie's. But I was relieved to get out of the dorm. I didn't want to get caught, and the girly perfumes were giving me a headache.\n\nWe walked back toward our cabin. \"It's way too fruity in there,\" I said.\n\n\"No such thing.\"\n\nIt took me a second to get his joke. When I laughed, Goth did too.\n\n\"Can I ask you something?\"\n\n\"Go for it.\"\n\nI prefaced the question by telling him about Charlotte. How she was a self-proclaimed \"righteous follower of Jesus\" who simultaneously hated certain groups of people. \"But everyone here knows you're gay?\"\n\nHe shrugged. \"I assume so.\"\n\n\"Even the pastor?\"\n\n\"Yep.\"\n\n\"And they're just... okay with that?\"\n\n\"It's not really their business to be okay or not okay.\"\n\n\"Yes, totally,\" I said. That was it. That was thing with the whole world judging one another. \"That's what I tell Charlotte. But I swear she's on some mission. She goes on about how God hates homosexuals and all that.\" I air-quoted the last part, lest he think me earnest.\n\n\"Well, they can hate me and not have me at camp. Or they can love me and have me at camp. If you claim to be a Jesus-lover, there's only one choice.\"\n\n\"So, they accept you.\"\n\n\"Girl, I said love. They love me here.\" He said the last part with a lisp and a click of his tongue.\n\nI laughed again.\n\nI so wanted Diana to be there to laugh too. I wanted her see how it could've been. How it should've been. Not necessarily being at Jesus camp. I didn't really want to be here, and I doubted Diana would either. But just being anywhere. Being anyone. She could've lived as big as she wanted, as openly and joyously as she wanted.\n\nI had known Diana wasn't happy in the months before she'd died. It wasn't just the breakup with Leah. She'd never finished stretching out her own skin and realizing it fit.\n\n\"My other sister was gay.\" I cleared my throat. \"She died, though.\"\n\n\"That sucks.\"\n\n\"Yep.\" It was an understatement, but it was true. \"Not everybody loved her,\" I said, thinking of Charlotte. Thinking of the first time I ever told a kid at school that my sister was a lesbian, assuming it was as interesting and as significant as the fact that my house was two stories. But as I'd watched that girl's face curdle and pinch, I'd realized it wasn't just being gay that was a problem. It was also a problem to love gay people, to find no fault. To pass no judgment was to be judged too. At least where we lived. And it wasn't just Christians. The kids who beat up my sister in high school weren't doing it in the name of God. Maybe they went to church, maybe they didn't. The only certain thing was that they wanted my sister to know she was not free to love who she wanted. Or even kiss who she wanted.\n\nThey beat her up in the name of hate.\n\n\"Not everybody loves me either,\" Goth said, shrugging. Their loss.\n\n\"And Nick's your boyfriend.\"\n\n\"I met him here.\"\n\n\"That's funny.\"\n\n\"So, you know, they just ask us not to be in the same cabin or perform poolside gay sex tricks or start riotous gay orgies.\"\n\n\"But straight sex tricks and orgies are okay?\"\n\n\"Absolutely.\"\n\nWe got to the cabin, but no one was back yet.\n\n\"So how do you know...\"\u2014I realized I didn't know her name\u2014\"the girl in the wheelchair?\"\n\n\"Monica? That's Nick's sister.\"\n\n\"What happened to her?\"\n\nHe looked at me like I was being a dick. \"She was born.\"\n\n\"No, I just mean, like what kind of disabilities does she have?\"\n\n\"All kinds. It's creatively called multiple disabilities.\"\n\nI wanted to ask more questions, but no matter what I said, I sounded like an idiot.\n\n\"It's cool she comes to camp with Nick, though.\"\n\n\"Yeah. Monica is easy to love.\"\n\nThe guys returned to the cabin pair by pair. There was good and bad news. We'd successfully decorated the cabins, but no one had found the box.\n\nThere were other cabins. But there was also the elusive Jesus.\n\n\"That was a pale shadow of debauchery.\"\n\nJack was in the bathroom again that night, sitting on the shower f loor with an empty bottle next to him.\n\n\"Breaking and entering.\" His mouth moped across the words. \"What about pretty road girl?\"\n\n\"Are you drunk?\"\n\n\"She's a fine, fine girl, ample breasts, brown from the sun.\"\n\n\"You're a pervert.\"\n\n\"No, no, I'm a dead man who hasn't loved a woman for half a century,\" he slurred.\n\n\"It's like the opposite of necrophilia.\"\n\n\"When will you decide to live?\" he asked, ignoring my insults.\n\n\"I am living.\"\n\n\"You exist. You breathe and walk around and wait for whatever to happen.\"\n\n\"I'm doing something. I'm looking for the PC Box.\"\n\n\"It's the problem your sister had with you.\"\n\nThe anger burned in my chest. \"My sister had a lot of problems, the least of which was me. So don't even go there.\"\n\n\"It's what your sister wanted for you. To live so hard you'd lose your breath from the thrill of it all.\"\n\n\"I'm fifteen! It's hard to live so big.\"\n\n\"No. You spend too much time looking for wings before you f ly.\"\n\n\"I think being careful is part of living through your twenties. Why don't you go ask my sister about that?\"\n\nI locked myself in a stall to pee.\n\n\"Why don't you kiss that girl?\" Jack said from the shower.\n\n\"Well, I'm not gonna walk up to some girl and grab her ass and stick my tongue in her mouth, like you would likely advise,\" I said as I washed my hands.\n\n\"Maybe that'd cure what ails you.\"\n\n\"Nothing ails me.\"\n\n\"Your spirit's a pale thing. You're formless.\"\n\n\"I don't even understand you half the time.\" I looked down at him. \"Look at you. You're drunk on the f loor of a church camp bathroom.\"\n\nI kicked at his shoe and nearly lost my balance when my foot hit nothing.\n\n\"Why won't you help me find the box?\" I asked, indignant.\n\n\"I did, I did. You don't hear the words when they drift in.\"\n\n\"On the Ouija? That was just songs. A bunch of Christmas carols.\"\n\n\"You don't see the things hiding in plain sight.\"\n\n\"Are you seriously going to sit there and throw my words back in my face? About my sister? Fuck you, Jack.\" I slammed my hand into the wall.\n\n\"You're so angry, Nicolas.\"\n\n\"What are you doing here if you're not helping me?\"\n\nJack opened his mouth as if to speak. Then he puked all over the f loor.\nWEDNESDAY\n\nThe next morning, Natalie was waiting at the bottom of my cabin stairs.\n\n\"Totally ignore my hair,\" she said. She still had the bandana, but her hair was pulled up in a bun. \"I couldn't wash it. We got hit by the Superglue Fairy. Shampoo deemed inaccessible.\"\n\n\"Crazy,\" I said. I had no wit or charm. I was exhausted and angry from the night before.\n\nShe eyed me. \"You certainly look well-groomed.\"\n\n\"Yeah? Thanks.\"\n\n\"But your cabin's upstairs.\"\n\n\"Yeah...\" Her eyes sliced through me. She suspected me. \"But my shoes were hanging on the ceiling. Laces duct taped and everything,\" I covered.\n\n\"Mm-hmm.\" Her scalpel-eyes continued their exploratory surgery.\n\nI quickly changed the subject. \"Have you seen Magic Jesus?\"\n\n\"Magic\"\u2014an air quote there\u2014\"Jesus?\"\n\n\"Well, you know, the robed guy who does magic,\" I explained. \"But, have you seen him lately?\"\n\n\"I've heard he's been praying with the campers and stuff. I haven't seen him recently.\"\n\n\"Where? In the sanctuary?\"\n\n\"I don't know\u2014Wait, what are you wearing?\"\n\nI looked down, afraid I had committed some heinous fashion crime. Or worse, left some evidence of last night's prank. But it was just a striped T-shirt.\n\n\"Where's your uniform?\"\n\nAh yes, my nerd shirts. \"I only have so many T-shirts.\"\n\n\"Excuses.\" She got oddly serious for a beat. \"I wanted to give you something before\u2014\" She looked away. \"Breakfast.\"\n\n\"Okay...\"\n\n\"But now I doubt you can handle the necessary nerdery.\" And just like that, she was back. Matthew had called her a closed book. But she was more like a book that kept f lipping from page to page.\n\n\"I can handle it.\"\n\n\"Okay, then, speaking of duct tape and hanging things from the ceiling.\" She handed me a book\u2014Sixty Scenes of Sexy.\n\n\"I take that back,\" I said. \"I can't handle it.\"\n\n\"It's perfect for you.\"\n\n\"What?\" I looked around, afraid of witnesses, afraid of heavy crosses. \"Why?\"\n\n\"I think you'll have a lot to say about it.\"\n\n\"I don't need to read it.\" I was a boy. My brain wrote new chapters of erotica every minute. I just didn't know what to do with all of it.\n\n\"You said you were good at grammar and all that.\" She handed me a red pen. \"Marginalize it.\"\n\n\"It's already marginal.\"\n\n\"Right. So you have a duty to do something with what you love.\" She waved her hands wide. \"This world must be saved from the shlit.\"\n\n\"Is that Yiddish?\"\n\n\"Shit literature. Shlit.\"\n\n\"I thought that was your job.\"\n\n\"What? No. I make the classics more accessible.\"\n\nI laughed.\n\n\"I'm serious. No more of this shlit.\"\n\n\"I cannot walk around here with this.\" I held the book up. \"This is a crock of shlit.\"\n\n\"Tough shlit.\"\n\n\"The shlit's gonna hit the fan.\"\n\n\"Shlit happens.\"\n\n\"I'll be up shlit creek without a paddle.\"\n\n\"Don't lose your shlit. Hold on a sec.\"\n\nShe took off, jogging to her cabin. She ran up the stairs of Jericho, the very cabin I'd been in the night before. Had she seen me? Is that why this morning's suspicion?\n\nWhen she returned, her hair falling out of its bandana, her face red, she took Sixty Scenes of Sexy out of my hand and wrapped it in a new book cover.\n\n\"Twilight? This is a piece of shlit too.\" I pushed the book back at her. \"They confiscated my Harry Potter towel, for God's sake. This is a supernatural love triangle. Written by a Mormon.\"\n\n\"Mormons love Jesus too.\"\n\n\"She has sex in book four. They break the windows with their passionate screeching.\"\n\nShe smiled. \"So you've read them.\"\n\n\"I've seen the movies,\" I admitted. Diana had said it was part of my romantic education. To know every woman just wanted a man-eating monster who could quote Shakespeare.\n\n\"Just take the book,\" she pled and started backing away from me so that I couldn't give it back. \"It'll feel so good to write all over it. And honestly, it's a fun, fast read. Don't be such a snob.\"\n\n\"Okay, I'll take it.\" She was walking away, heading toward the dining hall. I called after her. \"But can you please find me a different cover? Maybe Suzanne Collins or James Dashner or something?\"\n\n\"I'll try!\" she called back.\n\n\"Find something that hasn't been banned in public schools!\"\n\nShe laughed maniacally. \"Impossible!\"\n\nPRAYERS AND CONFESSIONS\n\nDan: I miss my mom. How does a mom just up and leave? I get that my dad's crazy. But then, you're my mom. Take me with you.\n\nPayton: Holly thinks I broke my arm skateboarding. Right. I broke my arm in a fight with some kid who called her a slut. Look, I know what they say. Everybody does. Because even though our school is super Christian, we all hear the same rumors. But Holly's not a slut.\n\nMatthew: I'm not stupid. I know why I say all that stuff. And I probably shouldn't. But if I just keep talking about tits and ass, then no one will think I really care about what happened between me and Sarah.\n\nI went back to my cabin to stash the book. But before I slipped it under my pillow, I f lipped through it.\n\nNatalie had inscribed a note on the inside cover:\n\nDearest Nick,\n\nYou are likely the Messiah of Grammar. Messiahs are called to save mortal man. Unsheath your pen (mightier than the sword, they say!) and do this thing.\n\n\u2014Natalie\n\nP.S. I started it off so that you could see how it's done.\n\nAnd she had, mostly crossing out repetitive words (burgeoning; sardonic) and phrases (oh dear; she twirled her hair) and replacing them with something new. For a visual dynamic, she'd doodled a picture of the sorry kid in the stationery store who had a crush on Miss Bronze (at least, that's what I figured the doodle was later, when I read it).\n\nI didn't want to read the book, but I wanted more of Natalie. And I liked being talked to as I read. I liked being inside someone's head at the same time I was in mine. I liked being with Natalie even when she wasn't there.\n\nAfter stashing the book in my pillowcase, I headed over toward the dining hall. Magic Jesus was nowhere to be found, but I spotted the Disciplettes, who had gathered near the pool. They ran around in different formations.\n\n\"GIVE ME A J!\"\n\n\"J!\"\n\n\"GIVE ME AN E!\"\n\n\"E!\"\n\nOne cheerleader was wearing pants.\n\nCharles.\n\nIn polyester pants that matched the cheer skirts.\n\nAnd a cheer shirt with a capital J.\n\nCharles was a Disciplette.\n\nOr a plain old Disciple. I wasn't sure how it all panned out with the gender differences.\n\nI wandered over. \"It's early for cheer practice,\" I said.\n\nHe narrowed his eyes.\n\n\"I'm not teasing you, Charles. I think it's cool,\" I said. Even if it was cheerleading, it was cool to him.\n\nCharles stepped away from the girls. \"Is everything okay?\" he asked.\n\n\"Yeah, why?\" My friendliness seemed to have caught him off guard, or something.\n\n\"You're just wandering around?\"\n\n\"Well, I guess.\"\n\nHe scanned the camp, biting his lip. \"There're no more secrets out?\"\n\n\"No.\"\n\nCharles nodded. \"Okay, that's good.\"\n\nHe didn't actually seem like he'd settled into his role as a Disciple(tte) yet, but I felt jealous of him nonetheless. Charles was lucky. His secret getting out actually provided him the liberty to live his dream, but my secret getting out would be a family nightmare.\n\n\"Charles, get over here,\" one girl called. \"We need our bases.\"\n\n\"I designed a stunt,\" he explained to me, walking away.\n\n\"Wait, you're going to throw that girl in the air?\"\n\nHe turned back around, the side of his mouth turning up in an almost-smile. \"I calculated the initial velocity and angle of the trajectory prior to designing the stunt.\"\n\nAnd then he and two sturdy girls, standing in a threesome, a holy cheer trinity, positioned themselves to throw another girl into the air.\n\n\"This is not safe,\" I called out.\n\nBut the girl f lew, and the Disciplettes caught her.\n\n\"See,\" Charles called back. \"It's math, so it is safe.\"\n\n\"It's not math.\" I joked. \"It's cheer, for God's sake.\"\n\n\"No,\" he said. \"It's cheer for God's sake.\"\n\nIn the sanctuary, Pastor Tom Brady kicked things off with Christian rap. I wasn't paying too much attention. Mostly I was staring at the pastor, thinking Russell Wilson would've been a better sports reference, because everyone at camp was pretty loyal to the Seahawks. After all, most of us were Pacific Northwesterners. And it's probably more responsible of me to make some multicultural comparisons when describing the unnatural perfection of the pastor. Plus, according to Twitter, Russell Wilson is a born-again virgin. But then, the pastor, though tan, tanned more orange. And didn't quite have the puppy smile of the famed quarterback.\n\nThe song changed to something slower and the screen blipped. I expected new lyrics to appear on the screen.\n\nBut the words didn't match the song at all.\n\nIt took me a second. It took us all a second before we realized what we were seeing.\n\nScreen after screen after screen of confessions.\n\nI don't know if I should tell my dad about my mom and the grocery clerk. I mean\u2014\n\nI pray for my dad, but he keeps hitting her. And I want to move out, but I can't just lea\u2014\n\nI'm called to lead my brother, but he doesn't know how much I drink. Nobody knows I\u2014\n\nI smoked it last time, but I know it's a matter of time before I\u2014\n\nI love him, so is it really so wrong\u2014\n\nThere were screams. Oh, my Gods. The pastor scrambled up the aisle. A woman raced him into the sound booth. The screen went black.\n\nThen it was just holy music and holy tears.\n\nThe pastor walked slowly back to the stage. A couple counselors moved to sit by some of the kids who were crying, hiding their faces in the shoulders of their friends. It made it worse that they cried so publicly. That the counselors rushed over. They were basically admitting that one of those was theirs\u2014that their dad was a cheater or a wife beater, or they'd had sex or done drugs.\n\nI was relieved my confession hadn't been there. And I was in the room, too, so they couldn't accuse me again.\n\nOr so I thought.\n\nPastor Kyle apologized, prayed. As he did so, a couple girls kept their eyes open, watching me.\n\nI looked away.\n\nSomeone behind me kicked my back. When I turned around, a kid sitting a couple feet behind me glared.\n\n\"I gotta be real with you,\" the pastor said. \"We gotta talk about sin anyway, so I'm going to lay it out for you. We're seeing each other's sins in a way that was not meant to be. You're seeing it, and you're tempted to judge one another.\"\n\nI spotted Holly. She and Payton were actually sitting next to each other.\n\nPayton must've felt me staring. He turned, then pursed his lips and shook his head.\n\nHe thought I'd done this. But how could I? I was in the sanctuary with them.\n\nThe pastor continued, \"Not today, Champions. And, Pampers, you're in the Super Bowl now. Listen up.\"\n\nHe pulled a rolling whiteboard to the stage and drew a diagram up on a screen. A cityscape. It even had the Space Needle, so I knew he was drawing Seattle, a city most of us had at least visited, since the majority of the audience had come from Washington, Oregon, and California.\n\n\"And this is how we all see sin,\" Pastor Kyle continued. \"We see some sins as worse than others.\" He pointed to a squatty building. \"Maybe this is telling a little white lie to some of you, not that big. Peeking at someone's paper on a test, maybe.\"\n\nI was trying to pay attention, but I kept thinking about Payton's accusing eyes. The two girls.\n\n\"And maybe this is adultery, cheating on your girlfriend or husband,\" he said, pointing to the Space Needle, which isn't that tall, though it does stand out.\n\nSomething bounced off my head. A crumpled piece of paper. I didn't open it to see if it was a hate note.\n\n\"And this is murder.\" The pastor pointed to the tallest building, which would've had to be the Columbia Tower, since that's the tallest in Seattle.\n\nThen the pastor paused to draw an aerial view of Seattle\u2014a design like a plate for the top of the Space Needle. The dot in the center for the needle turned it into a breast. With a nipple.\n\nI couldn't laugh, though.\n\nSomeone cleared his throat. I happened to look up, just in time to see a kid f lip me off.\n\n\"This is how God sees sin,\" Pastor Kyle said when he was done with the bird's-eye view. \"You can't recognize the height of any of the buildings. None of them are taller, bigger, greater than the others. Your lie is as great as the adultery is as great as murder.\"\n\nAnd I swear he was looking right at me.\n\n\"Remember that when you look at one another. Before you judge. No matter what you see on this screen.\"\n\nAmens from the audience.\n\n\"T-G-I-J!\" someone shouted.\n\nSeveral campers had their arms around each other. They were feeling His call.\n\nThey weren't going to judge each other for what appeared on the screen.\n\nBut they were going to judge me because they thought I'd put it up there.\n\n\"And now, you know what time it is?!\" Pastor Kyle called.\n\nTime to get ill? I wondered, anticipating the Beastie Boys.\n\n\"Time to get dirty!\" he shouted. Which I did not expect. \"Fist pumps for Christ!\"\n\nAnd the kids exploded from their seats, sprinting for the door, nearly trampling me. Matthew held onto my elbow and dragged me out. \"Here comes the best part.\"\n\nRight. He hadn't seen the glares or the spit wads. Even now, the other kids jostled past me too hard.\n\nMatthew was out the door when Dan stepped in front of me. \"After we get dirty,\" he said, jabbing his finger at my chest, \"you better come clean.\"\n\nPRAYERS AND CONFESSIONS\n\nPastor Kyle: I think about that afternoon often. I think about what I preach to kids about wisdom. I think about what I preach at the church about obedience. It seems small. Just pick up the phone really quick and take a peek. The highway is straight. It's no big deal. But that second could have destroyed a life.\n\nDan: I found my dad passed out at the table. The steak was burned. Smoke filled the kitchen. I went to school the next day smelling like a house fire. \"Your mom needs to wash your clothes,\" a nerdy kid said. I smashed his face. That doesn't mean I don't love Jesus.\n\nHolly: I don't hear them talking about me. I feel them talking about me. Like insects on my skin.\n\nMatthew: I sent Sarah a card. I asked if I could see her. I don't want to get back together or anything. I just want to tell her I didn't abandon her. That it was my parents who sent the money and said I couldn't see her anymore.\n\nThe doom and gloom of the sin sermon gave way to a glee stampede as campers raced to their cabins and changed into grubby clothes. Apparently, we were literally going to get dirty.\n\nI didn't run back to my cabin. I just walked.\n\nI thought about finding the PC Box; if I did find the box, would it matter? Would they assume I was just giving back what I'd stolen?\n\nOur cabin was empty when I got there; everyone had already changed clothes and headed out. I considered just hiding in my bunk, but I felt like that would be admitting to something. I had to keep participating. I had to keep putting myself out there, even if it meant judgment and cruelty, because I hadn't done anything wrong.\n\nI dug through my clothes, set aside my HYPERBOLE: BEST. THING. EVER. shirt for the next day. I didn't want to ruin it, and I wanted to show Natalie that I'd returned to the nerd uniform.\n\nIn too-short basketball shorts and a faded middle school honor society shirt (yep, I could still fit into a shirt from eighth grade. Barely, but yes), I followed a small mob of campers from another cabin. We headed past the donkey barn, where I'd found Magic Jesus that first night, to a field that I had assumed was off limits because it hadn't been watered. Dry grass and tumbleweeds meant out-of-bounds.\n\nWe arrived at a giant mud pit, an Olympic-sized swimming pool of a mud pit.\n\n\"Let the sins begin!\" someone shouted.\n\nCampers whooped and fist-pumped; Pastor Kyle beamed. I stood there baff led until Matthew called me over to our group.\n\nWe were divided into teams, each cabin split into two, for a total of thirty-two teams. We lined up at the edge of the mud pit to compete in a basic relay race.\n\nThe three kids before me surged through the mud, their faces getting splattered, their white teeth the only clean thing as they laughed, maniacal and delirious. Then it was my turn. I sank past my ankles in the mud, tiny pebbles and sticks scratching at the soles of my feet. The earth gripped me, and I pulled. I could barely move. And I was hot, and I didn't want to be doing this.\n\nIn the first relay, my team came in second\u2014second to the other half of our cabin. Dan called directives from the side of the pit to assist us through the quagmire. He only stopped shouting at us to coach the other team representing our cabin. Either way, he was ensuring his cabin's victory.\n\nRelay after relay, we churned through the mud, new obstacles being set up each time to challenge us.\n\nThe campers played the games, rallying to the pastor's anti-judging-sin war cry. It was like nothing had ever happened. It was like we hadn't just seen secrets f lashed across the big screen. Everything they'd worried about, everything they'd done, just out for everyone to see.\n\nThey all judged each other. I'd witnessed it. They'd judged one another before anyone stole the PC Box.\n\nThey judged me.\n\nBut just like that, they could put it all aside thanks to the pastor's message. Thanks to a game. It made me sick, the hypocrisy. Right now I was on a team; in ten minutes, they'd be glaring at me again.\n\nIn ten minutes, we'd all be waiting for more confessions and prayers, more secrets. No matter the sermon, it was not okay. We couldn't ignore the consequences. The afters. Because there were always afters, and mine would damn me, far beyond their judgments.\n\nWhen a counselor announced the next tournament, I walked away from the group.\n\n\"Get back over here.\" Dan grabbed my arm.\n\n\"I think I'm done.\"\n\n\"Don't be a douche. We're almost done.\"\n\nHis words ate at me. I didn't want to be a douche. That was the point, wasn't it? I wasn't a douche, even if they thought I was. I shook my head and walked back to my team.\n\nI looked for Natalie, but I didn't see her. She wasn't running the latest relay with Kim's team. I couldn't find the other team that would've come from their cabin.\n\nThe next game was an individual race. There were twelve lanes marked with caution tape from one end of the mud pit to the other, like in a swimming pool. The object was to get to the other end of the pit, where twelve girls waited for us, all dressed in shining red devil leotards.\n\n\"What are they holding?\" I asked Matthew. Each girl had something in her hands, but they were too far away for me to tell what.\n\n\"Prizes, for the winners. Like, there's cash and minutes to use the Internet in Babyl On and candy bars and stuff.\"\n\nA whistle blew, and it was my turn to race. The first few steps were easy. Or easy-ish. I yanked my feet from the mud and got ahead of the other eleven campers. But the farther I went, the harder the mud pulled me down. The mud pit didn't get deeper, but I was definitely sinking more, the closer I got to the finish line.\n\nIt was like all the stories I'd heard about quicksand. If you struggled in the muck, you basically ensured your death.\n\nI lifted my legs and worked my body through the mud, realizing two campers had passed me. I looked at the devils at the end of the pit, their faces all made-up with caked-on foundation and fake eyelashes. Their mouths shone with sexy-red lipstick, and their hips and breasts threatened to burst through the silky material of their costumes.\n\nTwo more campers passed me in the mud.\n\nAnd ahead of me, there were twelve girls barely older than me, dressed like prostitutes. Waving around a calling card, or a piece of candy, or a dollar bill.\n\nI stopped racing. All eleven other campers reached the end before me, breathless but proud. Slowly, I worked my legs and hips through the muck to the end of pit and pulled myself out.\n\nI heard a whistle and knew Matthew would be plowing ahead in the lane behind me.\n\nThe devilish vixen at the end of my lane looked at me with pity and held the cash behind her back. \"Here's some candy,\" she said in condolence, and I swore she licked her lips at me. I walked away, leaving the Hershey's bar in her pale, long fingers, painted blood-red.\n\nThen it occurred to me: these counselors in scantily clad attire were the embodiment of sin. Not a counselor in a Freddy Krueger mask. Not somebody in a mobster suit. Not a thief, not a drug dealer, not an altar boy\u2013molesting priest. A f lirty, sexy girl.\n\nJeez.\n\nHolly never stood a chance here, did she?\n\nIn the heat of the morning, the mud dried and cracked on my skin, leaving it tight. I had scratches on my arms and a cut on my shin from something in the pit. My stomach felt thick, and breakfast felt too close to my throat, from the heat or the exertion. Or the sin-and-devil metaphor.\n\nI was done, and I didn't care what Dan said. I walked away from the pit toward the cabins.\n\nBut my devil girl stopped me, her costume glistening in the sunlight, the conciliatory Hershey's bar no longer in her hands.\n\n\"You have to clean off,\" she said. Her makeup was so heavy, she looked like a Shakespearean player.\n\n\"Where do you think I'm going?\" I said.\n\n\"You can't go into the cabins like that. You have to go clean off first. Over there.\" She pointed back toward the mud pit, where the male counselors had joined the show. Dressed in long robes with crowns on their heads, each one held a green plastic hose in his hands. More Jesuses. Counselor Jesuses. \"They clean you off.\"\n\nOne of the Jesuses approached me. \"Looks like you're ready for grace.\"\n\nThe devil girl touched my arm. \"Kneel.\"\n\nI was tired of their games. Tired of the fact that they didn't seem to be doing anything about the PC Box except blaming me. The fakeness of all of it. And the hose in his hand looked distinctly like a snaky penis.\n\nSo I said, \"No.\" I did not want to kneel before him and be washed clean by his pee-jizz.\n\nThe devil and Jesus exchanged a look.\n\n\"I'm not going to get on my knees.\" I reached for the hose.\n\nJesus stepped back.\n\n\"You have to ask him to clean you off. Just kneel,\" the devil-girl continued.\n\n\"I'll just spray you down real quick,\" Jesus offered in a softer voice, like he knew I was on the brink. \"You don't have to kneel. Just hold still.\"\n\nI grabbed the hose and yanked, but he held on. \"I'll do it myself.\"\n\nDevil-girl grabbed onto the hose and pulled with Jesus, grunting, \"That's. The. Point.\" It was a holy tug-of-war, with both good and evil against me.\n\nSomeone, somewhere, must have turned the faucet on, because the water sprayed me in the face all of a sudden. The hose twisted, and the water fountained into the air.\n\n\"You're not supposed to do it yourself!\" the devil-girl yelled.\n\nI let go of my end of the hose, and she fell back onto her butt in the now-soaked grass. Jesus stumbled back too, still gripping his end of the hose.\n\nI launched myself at him to pry it from his hand, but his sandals slipped out from under him, and we both toppled together, the hose falling impotent to the ground. Jesus snatched it with his free hand, and I yanked it and twisted it in my direction, but only a light spray reached me\u2014not nearly enough to get me clean.\n\nDevil-girl stood up, her leotard dark where the water had saturated it. She stormed off, her face wet from the hose, or perhaps frustrated tears.\n\nI let go of the hose. Jesus rolled away with it, his robe dragging across the ground.\n\nI sat there, alone, no cleaner. I felt a hand grab my shoulder, and I knew it would be Pastor Kyle or Leader Jason or Counselor Dan come to drag me away.\n\nBut it was Matthew.\n\n\"Dude, what's your problem?\"\n\nThe sun was too bright and hot, the grass too green and wet, the sky too blue. The kids skipped and frolicked away from the mud pit with their relay rewards. They danced and laughed at the feet of the Jesuses and their plastic green hoses.\n\n\"This is my problem.\" I waved my arms around at the circus scene. \"I don't want some kid holding some hose over my head waiting for me to give him a blow job or whatever.\"\n\nMatthew shook his head. \"It's just a game.\"\n\n\"All these girls dressed up like she-devils with their boobs all over the place. Like, do you see the irony?\"\n\n\"No, I see the metaphor.\"\n\nOf course, I saw the metaphor too. The mud was our sin. We got dirty, we got stuck, yet we continued on to win the prize. In our case, food and Internet and money.\n\n\"Fuck this place!\" I shouted.\n\nAnd that's when People magazine's Sexiest Pastor Alive arrived. \"Let's go,\" he said, grabbing my elbow, not even trying to be nice anymore.\n\nHe didn't call for the cross for me to carry. He didn't try to lay hands and pray. Even he could see I was too far gone for salvation and games.\n\nI looked back to see Matthew heading over to kneel at the foot of a Jesus. Charles positioned himself at the top of a pyramid of still-muddy Disciplettes having their picture taken. The twins, Chris and Christina, were hugging out their forgiveness over the donkey inequality, I assumed.\n\nI followed the pastor to the medical building, where he handed me a hose.\n\n\"They could've just let me have it,\" I said. \"I get the Jesus washing sins away part.\"\n\nHe ran his hands through his hair. \"Those counselors are just kids like you.\"\n\n\"I don't want to play these crazy games anymore. They're not real. It's not real,\" I said. \"What happens in real life is real. What happened on those signs and on that screen, that's real.\"\n\n\"I think you take for granite\u2014\"\n\n\"Take for granted,\" I corrected.\n\nThe pastor just stared at me.\n\n\"It's granted, not granite. Do you know how many words you mess up? I have a list in my journal. You know that? I don't write down your scripture. I write down your misuse of the English language.\" I shook my head and turned off the hose.\n\nThe pastor didn't get mad, like I thought he would. Like I hoped he would, frankly. I wanted him to punch me. I needed him to punch me. I needed everyone to take their hardest shot at my face.\n\n\"It's your job to know words, but you mess them all up,\" I said, daring him.\n\nHe continued to look at me\u2014not with fury, but with what I thought might be pity. Then he nodded and said, \"I may mess them all up. But you're an asshole.\"\n\nMy mouth dropped open.\n\nHe raised his eyebrows at me. \"Did I get that word right?\"\n\nI tried to muster indignation, but he was right. He had that one right.\n\nJack was no doubt somewhere nearby, offering another standing ovation.\n\nPRAYERS AND CONFESSIONS\n\nMatthew: Sarah wasn't tits and ass. She was\u2014is\u2014beautiful. Her voice when she sang. Her smile when she looked at me. I do miss her.\n\nDan: The next time I punched someone, it was my dad.\n\nPastor Kyle: It's the impact, that metal on metal, that jars me from sleep. It's thinking about Gabe in his car seat in the back that keeps me from falling asleep at all. A simple text and I could've killed my favorite little guy.\n\nPastor Kyle led me back from the medical building, down the sidewalk to the sanctuary. He fiddled with the lock on a door, and I stole a glance into the custodian's office. Holly was inside, stacking toilet paper on a cart covered in canvas. She started when she heard the pastor's keys jingle. When she saw me, her face softened, and she dropped the sheet over the cart as she stepped toward the door.\n\n\"You missed the mud pit?\" I asked, kind of hoping she hadn't witnessed the scene.\n\n\"Not the whole thing. Just had to get this for\u2014\"\n\n\"For Jesus.\"\n\n\"Let's go,\" the pastor said, tugging at my arm.\n\nHolly smiled slightly. \"I'm sorry about all this.\"\n\nThe pastor pulled me into a room I hadn't been in before. A dim light barely glowed from the ceiling; the walls were gray and rough.\n\n\"Is this a cave or something?\"\n\n\"A tomb, yes.\" He waved toward the risers. \"Have a seat.\"\n\nI picked the ground-level bench in the center and wondered about the room's purpose.\n\nAnimal sacrifice, perhaps.\n\nPastor Kyle sat across from me and leaned forward, his elbows on his knees, his hands together. He looked into my eyes for a moment. \"I'm sorry for calling you an asshole.\"\n\nI shrugged. \"I deserve it.\"\n\n\"Something is going on with you.\"\n\nWell, it didn't take a behavioral expert. I had wrestled with Jesus in a puddle in the grass raging against a hose.\n\n\"Nothing is going to get better if you don't confide in someone.\" He sat up, held up his hands. \"I know it doesn't seem that way, with what's going on at camp right now. And I'm not saying I'm that guy for you. Not at all. I believe grace comes through Jesus. But I also believe that we can be that for other people too. I know you reached out to your sister the other night on the phone, and obviously you didn't get what you were hoping for.\"\n\nI shook my head. But I hadn't really been hoping for grace from her. I had wanted her to ask for mine, honestly. But like Jack said, maybe you couldn't ask that of people. You just had to say the thing and see what happened.\n\n\"Maybe when you get home, you can talk to your parents. I can help with that if you want.\"\n\nThere was no way I was talking to my parents. I'd gone months withholding what would only cause more hurt to them. But I said, \"Yeah, sure,\" just to make the pastor happy.\n\nHe was dubious. He didn't know all his grammar, but I think he knew people.\n\n\"You know where we're sitting?\" he asked.\n\n\"A tomb,\" I repeated. And then I got why he brought me to this creepy place.\n\n\"That's right. The tomb where Jesus lay for three days while the world waited for Him.\" He nodded, leaned back, and looked at me again. \"We all need the dark times to recognize His light.\"\n\nI didn't believe in what he believed. I didn't believe in Jesus as the son of God. I didn't believe in a God who thought we had to hurt just so we could heal. Who set us up to hurt for the benefit of healing us. That just sounded like codependency.\n\n\"We have to wrestle with our demons, we have to spend forty days in the desert, we have to carry our own cross. This is religious jargon to you, isn't it?\"\n\n\"Yeah, I guess. No offense or anything.\"\n\nPastor Kyle smiled. \"I believe it, though.\"\n\n\"I would hope so, given your job and everything.\"\n\n\"Yes. But that's not what I mean. Not as a man of God. I believe it as a human. I don't think you can see if you haven't been blind. I don't believe you can recognize grace if you haven't been stuck in a tomb.\"\n\n\"Wait.\" Panic cracked my voice. \"You can't lock me in here.\"\n\n\"Is that what you think?\" He laughed. \"Now that's crazy, locking you in some jail cell.\"\n\nMy jaw dropped again. I wanted to point out the Donkey Lottery, the selfies with prostitutes, and Judas with his noose and coin purse. Rap turned into gospel. I had just been told to kneel before a teenager dressed like Jesus while some other hot teenager strutted around in a shiny red leotard.\n\nLocking me in a tomb seemed par for the course.\n\nHe slapped his knees. \"I am going to leave you in here now, but I'm not going to lock the door. You stay as long as you like. Or follow me out.\" He scratched his jaw. \"Anyway, I don't think I need to lock you in a dark place. I suspect you're in one now. Quiet time is a gift, so take it if you want.\"\n\nQuiet was torture. Boring at best, it invited memories and ghosts and guilt at worst.\n\nBut I guess that was his point. The being alone with yourself, the silence poking at you, beating you, torturing the truth out.\n\nPastor Kyle stood up and opened the door. He turned around one more time.\n\n\"My job is not about words, by the way. Preaching is not about all those words.\" He pulled out a small book from his back pocket. \"It's about the Word. And I pray before I use it so that I get it right.\" He slipped the little Bible back into his jeans. \"Grammar gods be damned!\" At this, he laughed and pulled the door closed behind him. It clicked in place, and then everything was silent.\n\nReally silent.\n\nI couldn't hear the campers or counselors, birds or planes.\n\nI wanted to stand up and leave, but I honestly had nowhere to go. I'd embarrassed myself out by the mud pit, and I was afraid to face Matthew. I owed the two counselors an apology, but I wasn't ready for that either.\n\nI lay down on the bench, and I felt the silence and memories tear into me like bamboo under my fingernails.\n\nAfter my sister's wake, after everyone had left what remained of my family in our silent house, we sat at the kitchen table while my mom poured some wine for herself. My dad drank something dark with whiskey.\n\nIt's my fault hung like cigarette smoke over the whole house, heavy and thick.\n\nMy dad pulled something from his front pocket. He slid it across the table at me. At first I thought it was a coin, from the metallic drag it made across the wood, but then I recognized it as a key.\n\n\"What's this for?\" I asked.\n\n\"Hide it,\" he said.\n\nI looked at my mom, and then we both looked at my dad.\n\n\"Hide it anywhere in the bedroom.\"\n\n\"Honey, what're you doing?\" my mom asked.\n\n\"I want to try something.\"\n\n\"How much whiskey have you had?\" My mom asked. My parents didn't drink often, and when they did, they didn't drink a lot.\n\n\"What does it matter?\" My dad was not the belligerent type, but he'd just been to the funeral for his daughter, who shot herself. \"Nick, I said hide the key.\"\n\nI picked up the key. \"I don't really want to do this,\" I said.\n\n\"It's like hide-and-seek, Nick. You hide it. I seek it. How hard is it to find this key?\"\n\nMy mom teared up, sank into a chair, but she didn't tell him to stop.\n\nThere was something else thick in the room, beyond the guilty haze. I was scared of it. It was a tension I'd never felt. My parents had stood united their whole marriage\u2014through the chaotic kid-taxiing, my dad's layoff, selling the house, the Charlotte drama.\n\nBut the death of their child.\n\nDiana's suicide.\n\nLike a sharp blade ready to cut the fragile string that kept everything peacefully tethered.\n\nI scooted my chair back, the legs scraping against the linoleum, and walked out of the room.\n\nMy dad watched me, dark-eyed, but eager for some sick game where he tried to find a small key hidden in a small room. My mom walked over to the kitchen sink and poured her wine out. They both followed me to the living room. She stood against the wall, arms crossed, and watched my dad sink into the couch. I continued down the hall to my parents' bedroom, where my dad had always hidden the key.\n\nThe shades were drawn, the lights still out, the air stale. I sat on their bed and looked around the room. I wasn't pondering a hiding spot. I was waiting for the requisite amount of time to make it seem like I was pondering a hiding spot, to make my dad think I was playing his game.\n\nIt was ordinary, this room. The closet door, open a bit. Dark dressers, a jewelry box, a photo album. It was an old album, filled with Diana's school pictures, the one my mom pored over regularly.\n\nI counted thirty more seconds and then walked over to closet, pulled out the old Monopoly board game. I opened the box\u2014the edges of the lid were broken\u2014and set the key next to the metal dog and car game pieces, the little plastic hotels. I put the box back up on the shelf and walked out of the room.\n\nThat's where he always kept the key to the gun cabinet, and I knew he'd never look there.\n\n\"I'm not one for Bible-thumping, but I dig what that cat's got to say.\"\n\nI opened my eyes.\n\nJack.\n\n\"Nice demonstration out there, by the way,\" he said to me. \"You really showed that hose.\"\n\n\"I'm really exhausted.\"\n\n\"You just napped.\"\n\n\"By you.\" I sat up. \"I'm exhausted by you.\"\n\n\"Yes, yes, yes. Where's your catalyst friend, Natalie? She has great taste in literature.\"\n\nI rolled my eyes. He'd clearly heard what she'd said about him up on the hill.\n\n\"What was that book she gave you?\"\n\n\"It wasn't one of your books. You wouldn't call what she gave me 'literature.'\"\n\n\"She's a fire under your ass, that's what she is.\"\n\n\"Don't say something perverted here.\"\n\n\"She burns bright, doesn't she? All firef ly and firelight and sunglare.\"\n\n\"I guess, yeah.\"\n\n\"Like your sister. Matthew too, a little bit, though I'm not sure you like him the way you like Natalie.\" Jack winked at me. \"Or maybe so. I do not judge.\"\n\nAnd it was true he didn't. But on this point, he couldn't. Rumors f lew all over about his romances, with women and men alike.\n\nHe stood up and walked around the room, running his hand against the wall.\n\n\"You know that poster in your room. The people who read that poster, who read that book\u2014\" he spit the last part out, his mouth curled in disdain.\n\n\"On the Road,\" I said for him.\n\nHe nodded. \"They're thinking that was me, that I was mad and crazy and capricious and destructive and distrustful and irrevocably wild and wistful. Because I wrote the damn book, they think I was a Roman candle,\" he said.\n\nI hadn't read the book. But yes, I believed that too. He was the author, after all. Of a book about freedom and living and lust and debauchery. And finding yourself within and without all of it. About liberty.\n\n\"I was and was not that. I was and was not a Roman candle.\" He continued, \"I was quiet, I observed, I appreciated everything around me. I loved God, I missed my brother, I looked for angels, I was scared of God and ghosts and my mother.\"\n\nHe paced for a moment. \"If it weren't for the mad ones, if it weren't for the Roman candles, I might never have taken to that road. You hear me?\" He turned from the wall to make sure I had.\n\nI nodded.\n\n\"I was as much the guy on that road as I was the man stuck in his bedroom, mute fingertips against the typewriter, digging around in my brain for the Great American Novel. I needed Neal, I needed Alan\u2014you don't know who they are, but trust me on this point\u2014or I don't hop in a car and go back and forth across the continent and down into Mexico.\n\n\"And I see that in you. That dichotomy. You've got the madness in you. I see it. Your sister saw it. But you need other people.\"\n\n\"Like my sister,\" I said.\n\n\"And like Natalie. They light you up, make you brighter. We're not all Roman candles, but we're all meant to burn bright.\"\n\n\"My sister burned herself out.\"\n\nJack looked at me warmly, offered a slight nod. \"That she did. But you won't.\"\n\nI wanted to believe him. I wanted to believe what my sister had told me when I was a boy, that I had ten lives and I just had to choose to live big. \"But I don't do anything. You say it yourself. So how do you know I'm mad too?\"\n\n\"You must be, or I wouldn't be here.\" He sauntered down the risers. \"We're all mad here. You're mad. I'm mad.\" He was clearly reciting something.\n\n\"Did you write that?\"\n\n\"The Cheshire Cat.\"\n\n\"Lewis Carroll?\"\n\n\"The very one.\"\n\nI hadn't read that classic either, but I'd seen the movie. A rabbit's hole, pills and libations making Alice bigger, then smaller, talking f lowers, a caterpillar smoking an opium pipe. \"He was pretty messed up,\" I said.\n\n\"The great ones always are. And they are vilified and adored for it.\" Jack was suddenly standing at the door. \"So, how long are you going to hide in this dungeon?\"\n\nI shrugged. \"I don't know what else to do.\"\n\n\"What do you want to do?\"\n\n\"Find the box.\"\n\n\"Have you been listening and seeing?\" he asked, dark brows raised.\n\n\"Uh, yeah, but it doesn't help.\"\n\n\"Then you're listening without ears.\" He opened the door. \"And seeing without eyes.\" When he walked out, he left the door open.\n\nListening without ears. Seeing without eyes.\n\nI'd been able to investigate the custodian's office. Maybe I needed to check the barn. I'd been in one cabin; I could sneak into the others. I needed to keep at this. I needed to find Magic Jesus. After all, I was in too much trouble to get into more, and everyone already hated me.\n\nThere was liberty in being the outsider\u2014like standing at the edge of a cliff. It kind of made nothing matter. I wouldn't let Dan tell me to stay and play games. I'd skip the pastor's evening message. I wouldn't show up for the Trust Circle or the pranks. Instead, I'd stay up all night, if that's what it took to find that robed savior-custodian.\n\nBut first, I had to change my clothes. Bits of mud still stuck to my legs. And I needed my shoes.\n\nI walked down the sidewalk and was halfway through the grass when I stopped dead.\n\nThere was Magic Jesus. Just ahead of me, setting up his table. I walked toward him as he settled in the middle of the field next to the volleyball courts. His entourage hadn't spotted him yet. Perfect timing.\n\n\"Hey,\" I said.\n\n\"How's it goin'?\" It was his Hillbilly Jesus voice.\n\nThe campers were done with the mud pit. Some were being rinsed off in an industrial setup like a car wash. Some had already done that and were heading into their rooms to change. Doors were slamming. Open, closed. No one was approaching us.\n\n\"I looked up that Bible verse,\" I said.\n\nNo one was at the pool. Everyone was going into the cabins. Coming out, running down stairs.\n\nJesus looked around him. Now he'd noticed the running. \"There's a few of those.\"\n\n\"The one you told me about,\" I said.\n\nNo one was playing volleyball or buying ice cream. They were just running in and out, to their own and then other people's cabins. In and out.\n\nWhat the hell?\n\nShouting, tears.\n\n\"Nick!\" Matthew shouted. \"Get up here.\"\n\n\"We'll talk later,\" Jesus said. He folded up his table and headed off across the green.\n\nI ran up the stairs to our room. Charles was sitting cross-legged on the ground, crying.\n\n\"What's going on?\"\n\n\"The confessions,\" he sobbed. There was red paint on his palms.\n\nI heard voices in the bathroom and walked in, Matthew behind me. Payton and Chris stood looking into a stall.\n\nOh, gawd. Had Jack left another graphic illustration of the crucifixion?\n\nI peeked in, over their shoulders.\n\nNo. It was words. Blood-red words painted on every stall wall.\n\nCharles had paint all over his hands. \"And he blamed me,\" I said. It was obvious from the get-go that he'd wanted me to get in trouble. But he'd been caught. Literally red-handed.\n\nChris said, \"This is bad.\"\n\nThey were all staring at one painted confession:\n\nPlease forgive me for threatening the school. I wasn't really going to shoot the teachers. I want to go back to school.\n\nWait.\n\nThere was only one kid who wasn't in school, public or private.\n\nOh.\n\nThat's why Charles had been crying. He hadn't been caught painting. And that's why he'd been all fidgety lately. He'd had another confession in the box.\n\n\"Charles was the first one in here,\" Goth said.\n\n\"Trying to wipe it off.\"\n\nAnd they kept talking. But I wasn't listening anymore.\n\nThe paint.\n\nThe custodian's office.\n\nThe paint was all on that shelf in his office. That's why Jesus had been just standing there. He'd been watching the chaos he'd created unfold. And then he'd f led the scene.\n\nHe had opportunity. He had motivation. And now I had the evidence. But before going to the pastor, I was going to go to that office and get my confession.\n\nI headed toward the door, but Payton grabbed my shoulder. \"Where were you when this was painted?\" Payton asked.\n\n\"What?\"\n\n\"You left the mud pit.\"\n\n\"I know. I was with the pastor.\"\n\n\"We saw the pastor come back.\" Matthew watched me.\n\n\"I know. He left me in that tomb place.\"\n\n\"By yourself?\"\n\n\"Yeah.\"\n\n\"So you were alone?\" Matthew continued.\n\n\"I didn't paint these,\" I said. \"Why would I paint these?\"\n\n\"Why would anyone paint them?\"\n\n\"I told you. My confession's in that box, and I want it back just as bad as you.\"\n\n\"We haven't seen it,\" Matthew argued. \"We haven't seen anything about your sister.\" Matthew was looking at me, just as I'd looked at him when the box disappeared. \"That's gotta be what your confession is about. It's the thing you don't want to talk about.\"\n\nAnd I glared at him as if my eyes could shoot bullets. He wasn't on my side anymore. \"Everybody already knows my sister shot herself. It's not a secret.\"\n\n\"Yeah, and it's also not a secret you hate Charles. And it wouldn't be hard to figure out this was his confession,\" Payton said.\n\n\"Are you serious right now? Listen, I think I know who did it.\"\n\n\"Dude, are you serious right now? Are we supposed to believe you?\"\n\nYes, Matthew was supposed to believe me. He and Natalie had been the only ones left who weren't throwing spitballs at me or kicking me or glaring. \"Someone did see me in that tomb place,\" I continued.\n\n\"You just said the pastor left. You're lying. Or you're insane.\"\n\nI was. I was insane. Jack had been the only one who saw me in that tomb.\n\nBut Holly had seen me go in.\n\n\"Okay, if you really think I did it, go tell on me. But I have to do something.\"\n\nI ran out of the bathroom, past Charles sitting on sad road. I was now one of hundreds of kids running out of my cabin. But I wasn't off to read another gossipy confession. I wasn't seeking solace with my bestie. I wasn't praying to some god.\n\nI ran past the other campers. Past the empty volleyball court, past the empty pool, past Magic Jesus or Hillbilly Jesus, or whatever role he wanted to play. I had him now. But I wanted my confession\u2014and the evidence\u2014in my hand before I turned him in.\n\nHe stood watching the melee, arms folded, no one seeking his miracles.\n\nThe door was closed when I got to the custodian's office, but I let myself in. It was my third time entering without being asked.\n\nThe cart with the paint buckets wasn't there. But the laptop was.\n\nI opened it up. And there it was.\n\nNot child porn. The slideshow. It started out as rap, then contemporary lyrics. And then confession after confession. It had a hundred twenty slides.\n\nSo it was Jesus. Jesus by day, custodian by night, confession bandit behind the scenes. I scanned the shelves again for the PC Box. Maybe I had just missed it the night before. I spun around, looked up and down. No, it wasn't there. I opened the cupboard. The paint cans were gone, and there was no box.\n\nI spotted the two buckets from the night of the garbage can pranks. They were stacked where I'd found the one behind the custodian's work apron. The apron thing wasn't there now, so it was easy to see them. And easy to see they weren't stacked properly.\n\nWhen I lifted the one out of the other, I saw why. The bottom bucket was a foot deep with little scraps of paper.\n\nI set the bucket down and knelt next to it. I dipped my hands in, and the papers shifted away like sand. Every confession.\n\nI would turn them in, turn in Custodian Jesus. But I wanted mine out first. Surely, the pastor wouldn't let me dig through them later.\n\nI dumped them all out on the f loor. Then I read and read and read, discarding each into the bucket if it wasn't mine. Some of the prayers were ridiculous to me\u2014bigger boobs, bigger penis, hotter girlfriend, any boyfriend. But some were devastating\u2014cancer and Alzheimer's and abuse and poverty and meanness and guilt and loneliness.\n\nSo much loneliness. Like nobody understood one another. Like nobody loved each other.\n\nAnd then I saw one that I recognized. Not the confession, but the handwriting.\n\nI knew the handwriting.\n\nI'd seen it in the book she'd given me, the handwritten inscription and sample margin notes.\n\nI was holding Natalie's confession in my hands.\n\nPlease pre-forgive me for running away to find my meth-y brother. And then there was a sketch of Frankenstein's monster.\n\nOr a zombie.\n\nI put my head in my hands.\n\nThat was, in fact, the nickname of meth heads. Zombies. Heroin addicts were sleepwalkers. Meth heads were zombies. Zombie camp wasn't some drama thing. I hadn't really heard her.\n\nThere was another sentence on her confession. I read it. My stomach clenched. I read it again.\n\nForgive me for shooting up too.\n\nMy face f lushed. I should not know this. I should not have seen this. I felt some kind of sick shame for her.\n\nShe hadn't lied about her confession. She would not be going to City School if this got out. It violated every contract any high schooler ever signed. It violated the law.\n\nAnd she was running away.\n\nI'd seen her on the hillside twice. I'd even found her on the hillside with her backpack on. Full of food and water. Had she been running away then? Or trying to?\n\nI'd seen her just that morning. She'd given me that book, and she'd seemed weird for a second. Then she'd gone back to normal.\n\nBut I hadn't seen her the rest of the day. She wasn't at the mud pit. I'd seen Kim, but I hadn't seen Natalie.\n\nI had to tell Pastor Kyle.\n\nRight after I found my confession.\n\nI set Natalie's on the desk, then sifted through another handful of papers and found mine\u2014all caps and a lot shorter than what other people had written.\n\n\"What are you doing here?\"\n\nI turned around, hands up in surrender.\n\nNot Dan this time, but not Jesus either.\n\nIt was Holly.\n\n\"Hey, I know this looks bad. But I found all the confessions,\" I said, sweeping my hands over the pile on the f loor.\n\nShe stepped in the room. \"I see that.\"\n\nI stood up, my confession in hand.\n\n\"The custodian took them,\" I said. \"I know everyone thinks it's me because I said weird things and I found one of the confessions, but it wasn't. It was that Jesus guy, which makes sense because he has access to them. You know, the guy with the master key always has the most power.\" I was rambling.\n\n\"Everyone had access to them,\" she said. \"They were on the stage.\"\n\n\"Right, yeah, but I meant at night.\"\n\n\"The doors were all unlocked.\" She looked at me. \"You would know, right?\"\n\n\"Know what?\"\n\n\"That, with the exception of the tomb room, all the doors are unlocked.\"\n\nI stared at her. \"Are you talking about the pranks?\" She was not acting alarmed. \"You know about the pranks?\"\n\n\"You mean the night you snuck in here and accused Jimmy of being a pedophile because my laptop was open on his desk?\"\n\n\"Who's Jimmy?\"\n\n\"The custodian?\" She sighed. \"Jesus?\"\n\nHillbilly Jesus's name was Jimmy. Dear gawd, the universe was so perfect sometimes.\n\n\"Dan said that, not me.\"\n\n\"Whatever, Nick. I saw you in here.\"\n\n\"Yeah, okay. So I'm going to go talk to Pastor Kyle.\"\n\n\"Nick, please don't do that,\" she moved to the center of the doorway, her hands out, stopping me from walking out.\n\n\"Wait... your laptop,\" I said. And that's when I saw the paint\u2014a little on her palms, a little on her fingers. Red. And smeared. \"You had opportunity...\" I said, mostly to myself. She had access to this office. And she'd come out of the sanctuary last, later than anyone that night. She'd been so distracted. She'd been more Holly and less Red Lips Ho-Lo. She had the slideshow with all the confessions.\n\n\"Opportunity?\" she asked.\n\nI'd seen her with that computer bag. I'd seen her loading the cart with paints. She'd had a folder of papers in her cubby. Maybe more confessions, to tape to more signs and post all over camp again. \"Yes,\" I said. \"But motivation?\" I looked from her hands to her face. \"Why would you steal these confessions? Why would you put them all over?\"\n\n\"Oh, my God.\" She pulled her hands through her hair. \"You have no idea, do you?\"\n\n\"What don't I know?\"\n\n\"Do you know what it's like to be condemned, over and over and over again?\"\n\nThat word. Condemned. What had Diana said once? They condemn me for the love I have to give.\n\n\"All. Year. Long. They just keep telling stories about me. Do you even know how many true stories there are to tell about me? Two, Nick. Two true stories. I've slept with two guys.\" She bit her lip. \"And maybe that's a lot. Maybe that's a lot for a sixteen-year-old Christian girl. Or maybe it doesn't matter. Or maybe everyone is fucking everyone else too, but the goings-on in my vagina are of utmost interest.\"\n\nHer face folded, like she was going to cry. I reached out for her and she shrugged me away.\n\n\"Do you want to know the worst part of all of it? Those aren't even the stories I really hear about myself. I don't hear the truth. Nobody talks about me and Payton. Nobody talks about the other guy, who also goes to our school.\" She laughed. \"No, it's all just made-up stories. If I sigh, I'm breathing all hot and heavy. If I put my hand on someone's shoulder, I'm f lirting. If I walk just so, I'm shaking my ass.\"\n\nShe put on a California-girl accent, \"Oh, my God, did you see how much makeup she wore? Oh, my God, she's not wearing any makeup...\" Back to her normal voice. \"Yeah, 'cause even ugly me is scandalous.\" And then California Girl returned. \"Oh, my God, she curled her hair today. Oh, my God, her hair's straight today. Who is Holly after next? Who's on Holly's 'play' list? Who's on her greatest 'hit' list?\" She looked at me, \"Oh, you haven't heard those? Get it?\"\n\nShe squinted her eyes and put on a look of fake interest. \"Have you heard about why Payton broke up with me? Let me tell you the boring truth. I didn't have sex with anyone. I got too drunk at a party one night. And, guess what? So did everyone else at my school. But that was too boring, so let's make up a new story\u2014according to my classmates, I slept with like five guys that night. All of us! At one time! In the parents' bedroom or some shit!\"\n\nShe raised her eyebrows at me. \"Yeah, that's why Payton and I broke up. Because I\"\u2014air quotes\u2014\"cheated on him. I didn't fucking cheat on him! I went to a party and got drunk and probably batted my eyes at cute boys. But even Payton, who's supposed to love me, believed the stories instead of me.\"\n\nMy stomach twisted. I didn't want to hear this. I felt guilt, and shame. I had listened to the stories about Holly. I had laughed at the jokes. Even when I started to feel uncomfortable, I had never stood up for her. I never really asked her if she was okay.\n\nMy shame and her shame churned inside me.\n\nI felt Charlotte's shame too. As much as I didn't want to, I felt it. She had been judged over and over. Everyone around her had insisted she could only be one thing, just like they tried to fit Holly into a tiny box. And she must've felt the same anger and desperation. The solution, for her, had been to choose something new. To grab on to God\u2014no, to grab on to a list of rules that she got from her understanding of God.\n\nThat's not how everyone did it. But that's what Charlotte did. I didn't agree, but all of a sudden I saw, through Holly, why my sister grabbed on so tightly. Why she still hadn't let go.\n\n\"Holly...\"\n\n\"You mean Ho-Lo, right? Yeah, I know what they call me.\"\n\nShe shook her head.\n\n\"Do you know what Matthew's confession is? Or do you know what it should be? Have you ever asked him? Ask him why his parents sent him to Valley Christian. Ask him about his pregnant girlfriend.\"\n\n\"I didn't know about that,\" I said.\n\n\"I know you didn't. People from school know, but we don't talk about it. It's impolite.\" Holly air-quoted. \"But my business is open to everyone twenty-four seven. Do you know all the shit in that box? Well, you do now, because I put it out there.\"\n\n\"Holly\u2014\"\n\n\"You probably think I'm so, so, so evil. And maybe I am. But more than evil, I am tired. I am so fucking tired of being the story that everybody tells. Look around at what's going on in people's private lives. I am not the only one. And that's what I want them to see. I want them stop talking about me. Because everybody has a fucking story!\"\n\nHolly's face was splotchy, from tears and fury both.\n\nShe'd stolen every hope and every secret. She'd publicized them.\n\nAnd they would crucify her for it. She would lose everything\u2014certainly her work scholarship to camp. Her boyfriend, every friend, her private school.\n\n\"I won't tell.\"\n\nShe wiped snot from her nose. It was not hot. It was real. \"What?\"\n\n\"I promise. I won't tell anyone.\"\n\nShe laughed then. Like a crazy-witch laugh. \"Yeah right. You have every reason to tell. You're the one serving the sentence for everything I've done. They made you drag a cross.\" She shook her head. \"Like you'd steal anything.\"\n\nShe sounded like Jack.\n\n\"I won't tell. But you need to,\" I said.\n\n\"Oh, hell no.\"\n\n\"Not everyone. Just Pastor Kyle.\" Then I added, \"John 1:9.\"\n\n\"Did you just quote the Bible to me? Like, actually?\"\n\n\"I mean 1 John 1:9.\" I wouldn't make her think there was some light she couldn't see. Though, I guess that was true of all of us. We were constantly searching.\n\n\"Are you serious right now?\"\n\n\"Listen, everybody knows this place isn't for me. But, Pastor Kyle, he's... I don't know... he's looking out for everyone.\" And I believed what I was saying. I had seen him working with the campers. I had seen them come to him in tears and in laughter. I had seen him stand up on that stage and speak. He trusted in the Word, and he believed his own words, and he wanted to share the beauty and truth he knew with everyone else. He was what this camp was really about. Not Charlotte, not people like her. The pastor\u2014grammar abuse and all. \"If you tell him, I won't tell anyone.\"\n\nShe considered. \"Give me one more day, Nick.\"\n\nI stared at her. My confession. Natalie's confession. I would need Charles to math out the chances that any one confession would be pulled out in a twenty-four-hour period. It was the Hunger Games all up in here. The odds may have been in our favor, but we had all seen what happened to Primrose Everdeen anyway. \"Why would I give you one more day?\"\n\n\"I'll make you a deal. Take your confession. I'll give you that. You give me another day.\"\n\nAnd I could hardly believe myself, but... I wanted to give her something. Because we\u2014all the campers\u2014had taken and taken and taken. We told stories that stole her truth. We told jokes that stole her dignity. We talked about her like she was our story to tell. And maybe we owed her something: the chance to tell her truth. The choice to tell her story. What Holly was doing was wrong, but she still deserved what she'd wanted all along. The power to say what really happened.\n\n\"If I give you a day, will you tell Pastor Kyle?\"\n\nShe sighed and looked down at the f loor. \"Yes.\"\n\nI clenched my confession in my hand and thought of my parents. What these words would do to their barely-there connection. The way they argued, the way they turned sullen, the contempt that festered between them. The silence that had grown so large that it had pushed my dad out for a while. They'd told me he was traveling, but I wasn't stupid. And now he was back. I didn't know the chances of them getting a divorce, but if they knew what I'd done, they had to be exponential.\n\nI hesitated. \"I take one confession, you get a day, and you go talk to Pastor Kyle tomorrow?\"\n\nHolly nodded.\n\nI picked up the slip of paper on the desk. I read it, and I read mine. Finally, I dropped my own confession back into the pile.\n\nI stirred up the papers before I changed my mind. I was battling math. Fighting the probability that Holly would pull that confession.\n\n\"This is the one,\" I said.\n\nMy confession shouted at me as I ran out the door. It beseeched me to come back for it. But I didn't.\n\nI held on to Natalie's.\n\nNatalie's probability of having her confession pulled was the same as mine. But if it was pulled, the chances of her getting kicked out of her school were 100 percent. My parents divorcing... well, there was no way to know. And I couldn't tell people how to live their lives. I couldn't control their afters.\n\nPastor Kyle was in Babyl On counseling a sobbing girl. I could see them through the window. The door was locked; I knocked on the glass.\n\nHe opened the door and leaned his head out. \"Nick, give me a few minutes.\"\n\n\"I just have to talk to you about Natalie really fast.\" I figured he'd respond quicker to her name than to some heathen like me.\n\n\"Okay, buddy. I've got a situation. Make it fast.\"\n\n\"I'm worried about her.\"\n\n\"I'm worried about a lot of these kids, Nick. Look around. A lot of hurting people.\"\n\nI wanted to tell him about Holly. But I couldn't.\n\nBut if I told the pastor about Natalie, he could help me save her. And it's not like he was going to post her secret for the campers to see.\n\n\"I think she ran away.\"\n\n\"What makes you say that?\"\n\n\"Well, she wanted to see her brother.\"\n\n\"You know about her brother?\"\n\n\"Kind of. He's a\u2014\"\n\n\"A methamphetamine addict, yes. At a clinic nearby.\" And then he patted my shoulder.\n\nHe already knew the truth about her. Imagine everything else he knew. I bet half those confessions weren't even surprises to him.\n\n\"But don't worry, Nick,\" the pastor said. \"He checked out.\"\n\n\"Wait, what? He checked out?\"\n\n\"Of rehab.\"\n\nOn the hillside the day before, she'd said he was there, at zombie camp.\n\nAnd she was running away to go see him. Because she thought he was still there.\n\n\"Did you tell her he checked out?\"\n\n\"Hold on a second, Nick. This is a longer conversation than I can do right now. Give me a few.\"\n\nHe slipped back inside and sat down across from the still-sobbing girl.\n\nI stood there and waited.\n\nNatalie was going to walk twenty miles to see her brother, only to find out he wasn't there. And she was going to be alone when she heard he was gone.\n\nI waited.\n\nNatalie was going to learn her brother was out on the street, probably doing drugs again. What would she do? Would she keep running, or walk back to camp alone?\n\nWould she shoot up again? Did she have something on her?\n\nI didn't know what she would do, but she shouldn't have to be alone when she heard this.\n\nPastor Kyle kept looking over at me through the window, f lashing his hand, holding up fingers. Giving me a five-minute sign. A ten-minute sign. The minutes were definitely going the wrong way. \"How long will you walk down waiting road?\" It was Jack.\n\n\"Someone's going to see you!\" I grabbed at him to pull him toward the wall. But my hand slipped through his jacket.\n\n\"Nobody sees me but the madness in you.\"\n\n\"Other people might see ghosts!\"\n\n\"Ghosts.\" He laughed.\n\n\"This isn't funny.\"\n\n\"You found the doer of wrongs. I told you so.\"\n\n\"You didn't tell me. I figured it out.\"\n\n\"I whispered through the dark swirl of the Ouija.\"\n\n\"Seriously? You gave us a list of Christmas songs.\"\n\n\"I saved it, best\u2014and truth\u2014for last.\"\n\nI went through the list in my head. \"The Holly and the Ivy.\" That's what he'd said last.\n\nHe smiled. \"The end, Nicolas. Everything is settled in the end. The end is the it of it all.\"\n\nHe swayed. \"Did I tell you who was at camp?\"\n\n\"Yes, Muhamm\u2014\"\n\n\"Muhammad, Buddha, Jesu\u2014\"\n\n\"Are you drunk?\"\n\n\"Whoop, look at that! Jesus can turn it all into wine.\"\n\n\"For God's sake, you are now.\"\n\n\"That ol' Indian\u2014\"\n\n\"It's Native Ameri\u2014 Can you just leave so the pastor doesn't catch you?\"\n\n\"Catch you, you mean. Talking to the nothing-wind.\"\n\nI glared at Jack, then looked through the office window at the pastor and the sobbing girl. \"Just say a prayer and get on with it,\" I mumbled at the glass.\n\n\"Waiting road goes down a winding thing toward everywhere, which is nothing.\"\n\nI was tired of his word-puzzling. \"What am I supposed to do?\" I snapped. \"I just want him to help Natalie and he won't open the door!\"\n\nI expected Jack to snap back, to explode into his torrent of impatience. To call me the Great Editor and spin into nothing like he sometimes did. But his voice stayed calm. \"You found the box, and the doing's not done.\" He sounded almost kind again. \"I can't do the doing for you.\"\n\nOh.\n\nHe wouldn't say it outright\u2014he was still waiting for me to hear it and see it without him ever saying it. And most of the time, I couldn't see or hear what I was supposed to. Not just with Jack. With people. With my family. With Diana. Most of the time, I felt surrounded by broken bridges. But I understood him then. I actually understood his word-bridge.\n\nHe wanted me to live a little.\n\nHe wanted me to go a little mad.\n\nHe wanted me to break the rules.\n\nAnd so I did.\n\nI walked quickly back to my cabin to check the camp schedule. I needed to know how long I'd have before someone suspected me gone. There were several more hours till free time, then a service, then dinner. Nobody would be looking for me in particular at the service. We didn't have assigned seats or anything. They might wonder at dinner, but then they might assume I was still imprisoned for stealing the PC Box, since my cabinmates still thought I had. Only the pastor and Holly knew for sure I wasn't still being punished, but they were both preoccupied.\n\nI looked through my cabinmates' cubbies and gathered any unopened snacks. Sour Patch Kids, two bags of chips, two cans of Monster. Now I was actually stealing something from them, and I didn't care. I filled my water bottle from the bathroom sink and then went over to Dan's bunk, knowing he'd have something slightly more sustaining. And he did\u2014a half-filled, Costco-sized bag of trail mix.\n\nI'd never walked twenty miles.\n\nAnd that was just to get to zombie camp. There was also the twenty miles back. Forty miles.\n\nForty miles through the desert.\n\nI stole three more water bottles from my fellow campers and filled them up too.\n\nI tossed a sweatshirt and sweatpants into my backpack. I figured I'd be spending the night, and it was bound to be cold in the desert. Well, colder than the ninety degrees it was during the day. I couldn't take my blanket off my bed because that would be too obvious. I considered taking the pillow, but I doubted it would fit in my backpack.\n\nAnd then, to cheer Natalie when I found her\u2014because I assumed she'd need the cheering\u2014I shoved Sixty Scenes of Sexy into the bag too.\n\nI slipped my arms into the shoulder straps and opened the door.\n\nAnd that is when I heard the voice of the Lord booming from the heavens.\n\nOr rather, I heard the donkeys braying for their afternoon snack.\n\nI'd never walked twenty, or forty, miles before. And I wasn't about to now, either.\n\nThe donkey braying got louder as I approached the barn. I pulled an armful of hay off a bale and threw it into the shade of the barn's overhang. They ate, and I assessed. They all looked of the same build, the same motivation, the same intelligence\u2014which is to say, not much of any of it. But then a brown one made eye contact.\n\nHe accidentally volunteered himself.\n\n\"You,\" I said to the brown donkey. \"You're the one from the other night. You're coming with me.\"\n\nThe donkey shook his head and ducked down for more hay. Now, he might've been trying to rid his mane of f lies, but that seemed like overt defiance of my command. Stubborn ass.\n\n\"It'll be an adventure.\"\n\nThe donkey chewed.\n\n\"As opposed to getting my ass chewed, my ass is doing the chewing. Get it?\" The donkey did not laugh. \"Fine, you eat. I'm looking for the bridle.\"\n\nIt was too much to hope for a bridle and a saddle. Did they even make that kind of stuff for donkeys, or were they assed out of the equestrian tack business?\n\nI put a halter over Donkey's ears. \"You look like a horse mated with a rabbit.\" And he did, all ears and teeth.\n\nThe hay was almost gone, and I was polite enough to wait for him to finish chewing his last bite. I attached the lead rope the counselors had used the first day for the Donkey Lottery, and then I walked him through the gate. It was unlocked. Apparently donkey theft wasn't an issue. Yet.\n\nI looked at the hill behind the barn. It was steeper, but we had to go that route or we'd be spotted. We might still be spotted. A donkey is not exactly inconspicuous.\n\nI prepped my travel companion. \"We gotta haul ass.\"\n\nMy donkey was humorless. Maybe he'd been the butt of too many jokes in his lifetime.\n\n\"Actually,\" I said, and closed the gate behind me. \"Ass has to haul me.\"\n\nDonkey and I made it up the hillside unnoticed. Maybe because he is a part-time ninja. But mostly because a lot of the trail runs through a crevasse where no one could see us.\n\nI stumbled up over the ridge, my sure-footed companion by my side. I led him down the dip into the valley and walked over to the crosses, still out of breath from climbing so quickly. I took my first sip of water and stared at my donkey.\n\n\"Wait here, Ninja.\" He didn't argue about waiting, or about his new name. I figured there had to be some trademark issue with calling him Donkey, thanks to the creators of Shrek.\n\nI wanted to take a peek at camp again. I lay on my stomach and crawled across the dusty ground till I could see it sprawled out before me.\n\nEden Springs was an oasis in this desert. A man-made paradise of lush grass and trees that revealed the glistening underside of the leaves when the breeze blew. The pool water was crystalline, the cabin and paths and outbuildings all perfectly situated and planned. Anywhere you stood in camp, you could see the whole property. You'd always know where everyone was and what they were doing. It took effort to disappear.\n\nI scooted back down the decline and then stood up, looked in the direction Natalie had been headed when I'd caught her with her backpack. That's what I had to go on\u2014straight toward the horizon in the direction she'd pointed the day before. It was barely noon, and it was already hot. And I didn't see a single tree. I knocked on the wood of the middle cross for good luck, then walked through the small strip of shade it offered.\n\nI had about eighty ounces of water. That wasn't even enough for one full day of healthy hydration, but I knew it would be enough to survive twenty miles, which I'd cover by dark, if my calculations served me. If Charles didn't hate me, perhaps I could've had him double-check my formula. It was hot, and that's what worried me most. There was bound to be minimal shade. The land was all shrubs and sharp rock and dry earth, everything dusty yellow and sage. I knew there'd be rattlesnakes, and I anticipated maybe some coyotes, nothing too vicious.\n\nWithout a compass, I planned to find landmarks to keep my direction. The three crosses would be in my sight for a long time, and they always had to be to my back. If I got to a point where I couldn't see the crosses, and there was nothing else to track my location by, then I would abandon this mission and head back toward the crosses.\n\nI led Ninja over to a rock.\n\n\"Don't move.\"\n\nHe didn't. Or if he did, I couldn't see it because, well, he's a ninja.\n\nI swung my leg over his broad back. And slipped right off. I swung again, and slipped off again. The third time, I got enough leverage to hold on. After some grunting and shifting and pulling\u2014Ninja just standing there all the while\u2014I got myself onto his back.\n\nI reconsidered naming him Job because, like the dude in the Bible who came down with the plague and his kids all died, the donkey had a shit-ton of patience.\n\nUnfortunately, he just kept standing there. I didn't have a counselor to lead us.\n\n\"Go,\" I commanded.\n\nHe didn't.\n\nI rocked back and forth. \"You have to walk.\"\n\nHe did not walk. Perhaps because he's a silent assassin and just walking requires no stealth. Or perhaps because there were no shadows in this godforsaken desert. A ninja needs shadows.\n\n\"Please?\"\n\nA ninja need not have manners.\n\nSo I kicked my heels into his sides and hoped he'd seen enough cowboy movies to know this is how it was done. Clearly he was a fan of John Wayne, because off we went.\n\nI considered renaming him The Duke.\n\nThe sun beat down. After a while, I took my shirt off, but the heavy rays only got worse when they landed directly on my skin, so I put it back on.\n\nI made sure to look back often to check my direction. Only once did I think we would have to go back, when I turned around and noticed I couldn't see the crosses, but then we came out of a dip and my landmark reappeared.\n\nAfter about an hour, I clambered off Ninja to give him some water and a break. I saw a building, maybe an old barn, way off in the distance and felt good that I had another landmark to keep our path true.\n\nI took hold of the lead rope, and we walked on.\n\nThe scraping of my feet, the clank of Ninja's hooves on shale-like rocks, and my breathing were all that I could hear. I stopped to sip water, the unzipping of my backpack a rip in the silence, my gulps intrusive.\n\nEverything was beige and brittle for as far as I could see. Everything silent without the drag of feet and hooves across the dust. The crosses were specks on the horizon behind me, the barn to my left just as small. The air coughed up a breeze and a light rattle of leaves on the bushes. I'd never been so alone.\n\n\"No offense,\" I said to Ninja.\n\nNo one knew I was there. I decided I had better not think about that too long, lest I go insane.\n\nAnd then I realized something.\n\nWhat I was doing was insane. Stealing a donkey and then hiking out into the desert wilderness with little food and water, looking for a girl in a place I was not even certain was out here.\n\nZombie camp. Rehab.\n\nI slipped my water bottle into my backpack again, and the pack back over my shoulders. I led Ninja to a rock, and it only took me two tries to get astride his back.\n\nI was a mad one. At least on this day.\n\nDiana would've been proud.\n\nThe miles drifted away in the dust of Ninja's hooves. When my butt hurt, I walked. When my feet hurt, I got back on.\n\nThe sun was sharp in my eyes, hot on my face. My breath was louder in my head, and I was so hot. I couldn't see the crosses behind me anymore, but I caught sight of another outbuilding a ways off. I didn't know how far we'd walked, but it had been over three hours. I had missed lunch, which my stomach reminded me of just as much as the position of the sun did. But I wanted to keep walking.\n\nI started seeing f lashes of figures, small little people, bright and f lickery, in the bushes. Desert trolls, I decided to call them. I'd been walking into the sun too long. I made an effort to look off to the left for a while, to give my eyes a break from the brightness.\n\nOut of the corner of my eye, I saw two figures\u2014two men\u2014walking toward me.\n\nI turned to get a good look, but then I couldn't see them anymore.\n\nI wasn't dying. They weren't a death mirage, but maybe, like the desert trolls, they were shadows of my vision from the sun.\n\nBut then I saw them again. Two men. Maybe a man and woman, hard to tell. And what looked like a child. I stared straight ahead and realized there must be some hilly areas on the plateau up ahead. I kicked at Ninja's side, but he wouldn't go any faster, so I slipped off his back and jogged toward the figures. Maybe they knew how far I still had to go. Maybe they were from zombie camp.\n\n\"Hello?\" I called.\n\nNinja's ears perked up, startled at the break in the silence.\n\nI waited, but no one called back.\n\nMy eyes were playing tricks.\n\nThe sun was blinding me and fooling me.\n\nThe shimmering bush creatures reappeared again. I kept walking, Ninja grumpy behind me, until I knew I'd reached where I'd seen the two people and the child, but they were nowhere.\n\nWhat was I doing here? Here in this hot desert, but more so, here at this camp, this camp for people who loved a God I could not believe in.\n\nI couldn't believe in some old guy sitting in the clouds.\n\nI could believe in some guy in a white robe living two thousand years ago and being killed for being politically insane. But I couldn't believe God sent him down, that he was God in a man's body.\n\nThere had been men before Jesus who had claimed to be the Messiah. They were similarly killed on a cross.\n\nHow did Jesus get to be so famous?\n\n\"You're just the son of man,\" I said out loud. \"Just like me. I'm the son of man too, but I'm stuck in the desert, not hanging out in the air-conditioned sky eating grapes fed to me by virgins and angels.\"\n\nI heard the sharp cry of a bird. Kree, it whistled, fast and hard and lonely across the sky. Like a bullet.\n\nAnd Muhammad. I'd read once that Jesus told people of the coming of a great prophet. They had been in cahoots.\n\nI walked a bit faster, breathing hard now. It wasn't dark, but the loneliness of it all was eerie.\n\nAnd what about Joseph Smith? Now there was a weird dude. Finding magic plates and reading prophecy with magic rocks. In Upstate New York!\n\nAnd the Jews. Still waiting for something. Someone. Something holy.\n\nThe bird called again. Sharp, like the wail of a saxophone.\n\nI stopped, looked around. Where was that stupid thing? I couldn't see it.\n\nA rattling in the bushes behind me.\n\n\"If the bush lights on fire, I'll believe in You,\" I said to all the holy men in the sky.\n\nNothing. Just the heat of the sun.\n\nI was losing my mind in broad daylight.\n\nNinja, on the other hand, being a donkey and a trained killer, was unfazed by questions of morality and mortality.\n\nI led him to a tree up ahead that promised a bit of shade. The tree was huge, its trunk thick, splitting at one point and breaking off into large branches and then smaller ones, pewter-green leaves f lickering in the sunlight.\n\nI sat against the trunk and opened my pack for some water. I poured some water in the bottle cap and held it up to Ninja's muzzle. I offered him some Sour Patch Kids, but he demurred, probably watching his figure. I popped a few in my mouth while I fished out the bag of Tim's Cascade potato chips. Salt. I grabbed a Monster too. Perhaps it would clear my head of the voices and visions.\n\nI leaned back and kept my eye out for the people I'd seen earlier, and the desert trolls. I remembered the bird and looked up into the branches. It wasn't there.\n\nChewing the plasticky candy, I remembered this news story about two old people who'd gone out in Utah to hike the Wave, a protected area where they only allowed twenty people to hike each day. Ten permits were reserved in advance, and the other ten were issued in a daily lottery. On a day in July, this sweet old married couple had been two of the ten people to win permits to hike the Wave. It'd been over a hundred degrees and they were in their seventies, a point in life when the body didn't regulate itself properly. But it was a once-in-a-lifetime thing, and they'd won the lottery. The authorities\u2014under heavy scrutiny\u2014had let them go. Nobody had known what happened exactly, but they'd pieced together a sad story of a dying woman and her chivalrous husband. They'd found the man a few hundred feet away from the woman. It appeared as if he'd started off to find help after she succumbed to the heat. He'd left her in the shade of an anomalous tree, and that's where a hiking party had found her. Still leaning up against the trunk, the shadow of the tree protecting her body, still and silent and just as dead as her husband lying on the desert ground not far away.\n\nHow does shit like that happen?\n\n\"Like this,\" I said. \"Being a stupid ass.\"\n\nNinja took no offense. He was clearly of the mind that ignorance was bliss.\n\nThe bird's cry woke me. I looked at my watch; not even an hour had passed, but I grabbed my backpack, threw it over my shoulders.\n\nNinja stood in the exact same spot. I led him beneath a tree branch, where he stood as I climbed up the branch and onto his back. On the first try.\n\nWe walked on, and I was thankful for the tree as a new landmark.\n\nNow that I'd rested, now that I'd cleared my head, I noticed that it had cooled. The sun was closing in on the horizon, and I was getting worried I wouldn't make it anywhere before dark. But I had to. It couldn't be much farther. It wasn't like I was literally in the middle of a desert, in the middle of nowhere. We'd passed buildings on our way into camp. We'd driven through dusty towns with houses decked with peeling paint and trailer homes.\n\nBut I didn't see any then.\n\nKree, the bird called, and I wondered if he was following me, waiting for me to die like the old woman under the tree. Or now like the old man.\n\nThe thing about the Wave, the thing that killed the man and his wife and others who had gone before and after them, was not just the heat or the perilous cliffs you can fall over if you're not paying attention, or if it gets dark. It's that people get lost. It all looks the same. There's no clear way to go. There are no landmarks. There are no guides pointing you in the right direction. There is no clear path.\n\nYou do not know the Way.\n\nIt's the price you pay for wanting to go where man has not gone.\n\nIt's a little dangerous, it's a little scary. You might not make it.\n\nYour senses must be keen. You have to pack well and pack smart and listen to your instincts.\n\nYou have to be willing to take the risk.\n\nI saw a f lash of something on the horizon, but I couldn't discern it because the sun was right there about to touch down. It had to be someone or something. It wasn't one of the imps behind the bushes; it was still a bit too far off. But I knew I was going the right way. I knew that was zombie camp up ahead, and I was going to make it in time for dinner.\n\nSurely a rehab facility would take pity on me.\n\nNinja walked on, his hooves kicking up more dust, scraping hard on the sharp stones. I watched for that f lash, and it didn't take long before it came from different points. The sun was catching on a whole line of metal. It was a chain-link fence with three lines of barbed wire at the top.\n\nI hopped off Ninja. \"Follow me.\"\n\nI walked the perimeter, knowing it would lead to the entrance. If I didn't find Natalie in one of those buildings, I still needed water, or to call my parents, or to get a ride back to Eden Springs. I needed a horse trailer.\n\nNow that I'd found this place, I knew I wasn't going to have to spend the night alone in the desert, likely being eaten alive by yet-to-be-discovered desert carnivores. Not that I didn't trust Ninja to kill everything that approached, but...\n\nWe didn't get very far along the fence before I saw a swatch of color on the ground up ahead. A big piece of plastic, like a tarp, or a coat.\n\nOr a backpack. It was Natalie's backpack. I walked quicker, worried that she'd dropped it, worried what that meant.\n\n\"Natalie?\" I called out.\n\nAnd then I saw legs, and I saw her shorts, and I saw her lying there. With earbuds in. Her eyes were closed, her head rocking back and forth. I'd thought she was dead or seriously hurt, and here she lay on the desert f loor, jamming to a song that I could actually hear through the buds.\n\nI shook her shoulders, called her name again, and her eyes popped open.\n\n\"What\u2014\" she practically yelled, sitting up quickly and yanking out the earbuds. \"Holy crap, Nick. You scared me.\" Her eyes were swollen, her cheeks red. She'd clearly been crying.\n\n\"Are you okay?\"\n\n\"I'm fine,\" she snapped. Which meant she wasn't. Those were the only words I really understood, because that was my standard answer to everything.\n\n\"What the hell is that?\"\n\n\"A donkey.\"\n\n\"I see that.\"\n\n\"His name's Ninja.\"\n\n\"What are you talking about?\" Natalie tossed her iPod down on top of her backpack and stood up. \"What are you doing here?\"\n\n\"You're mad?\"\n\n\"What the hell are you doing here?\" Her eyes blazed anger.\n\n\"I found the box. I know who did it.\"\n\n\"That has nothing to do with me.\"\n\n\"It has to do with all of us,\" I said.\n\n\"How'd you even know I'd be here?\"\n\nI handed her her confession.\n\n\"You read our confessions?\"\n\n\"Well, no, not really. I was looking for mine and\u2014\"\n\n\"But you read them.\" Her eyes were daggers. \"You read them.\"\n\n\"I didn't want to know that stuff. I just wanted to get mine. And then I saw yours and\u2014\"\n\n\"I don't care. You read it!\"\n\n\"And then I took it out so that no one else could!\" I shouted back. \"I could've taken mine. I can explain the whole thing, but listen. I got yours out to save you.\"\n\nShe scoffed. \"To save me?\"\n\n\"Well, yeah. I mean\u2014\"\n\n\"Have you met yourself?\"\n\n\"Not save you, but save you from\u2014\" Shame. Everyone's shame\u2014mine, Diana's, Charlotte's, Matthew's, my parents', Holly's. Natalie's.\n\n\"If I were you, I'd worry about saving yourself.\"\n\n\"What's that supposed to mean?\"\n\n\"How about calling your sister a bitch? Let's start there.\"\n\n\"How do you even know about that?\"\n\n\"Everybody knows everything about everybody!\" She threw her hands up in the air.\n\n\"People didn't know that stuff in that box. I didn't know about your brother. I didn't know you were planning to run away. I didn't know\u2014\"\n\n\"You weren't supposed to!\"\n\n\"I know I wasn't supposed to!\" I shouted. \"I just didn't want you to lose the chance to go to your school. My confession is still in that box. I left mine in to save yours!\"\n\n\"Look, Nick.\" Her voice was steady and cold. \"This may surprise you, but not all of us feel so dismal and hopeless and melancholy or whatever it is. We have hope. We have something you don't want to believe in. You might think we're fools, but we're doing okay. We're not all wallowing in self-pity.\"\n\n\"Self-pity?\" Rage f lared in the marrow of my bones. \"You think what I feel is self-pity? You think I'm wallowing? My sister killed herself, Natalie. She shot herself in the head.\" I ran my hands through my hair. \"You wanna know where she got the gun? Me. I gave her the gun.\" I was breathing hard. \"Yep, that's what happened. I handed her the gun and the bullets, and she shot herself. That's my confession: I KILLED MY SISTER.\"\n\nNatalie's eyes widened, and her mouth moved as if to speak.\n\n\"And you think I feel 'self-pity'? No. I fucking wish that's what I felt. I feel self-loathing. I hate myself. And when my parents find out\u2014they're not going to survive. So I can chalk up their impending divorce to me too. And your God and this camp can screw off with your crosses and your mud pits and your Magic Hillbilly Jesus and your she-devils and rap songs and family dinners served by second-class citizens. I'm done!\"\n\nI threw my hands up and whipped around, walking quickly away, back into the darkening desert from which I'd come.\n\nFrom which we had come. Me and Ninja. Who I left behind because I was mad, and let's face it, he doesn't walk very fast.\n\nThe rage roared through my skull, loud and thick and pulsing, and I hated that I'd walked twenty miles to\u2014what? Save her? And I'd just told her. Gawddamn it, I'd just told her the thing that I swore I'd never tell. And I'd shouted it at her. I'd thrown it at her, every word bludgeoning and true and sharp. And now it was out there, and she knew. And the trees knew, and the earth knew, and the whole gawddamn world knew!\n\n\"Ahhhhhhhhhhhhh!\" I shouted. And the rage kept coming. From the pit of me, from my stomach, from my bowels, from my chest, from my throat. It was everywhere, and it was pouring out.\n\nAnd then the bird came again. Kree, it screamed with me.\n\n\"Go away!\" I shouted at the bird. I picked up rocks, chucked them into the darkening space. I threw them at the bird I couldn't see, that wasn't there.\n\nSomething stirred behind me\u2014a footfall, the kick of a rock. Like it had that afternoon. I looked over my shoulder, but there was nothing there. At least nothing I could see in the dusk.\n\n\"Go away!\" I shouted. \"Leave me alone!\"\n\nWhy was I here? How did I get here?\n\nI tripped on a large piece of shale, caught my balance.\n\n\"Ugh,\" I growled.\n\nI was hearing things, but I couldn't see a gawddamn thing. Not even the stones in front of me.\n\nEchoes of rocks kicking up behind me. I spun around. It was too dark. I could see, but not enough. I walked on, the drag and scrape of the rocks, my breath, the bird, my words, something in the bushes\u2014all the sounds tumbling around like a crazy chant of craziness.\n\nYou're going mad. I heard these words, but not my words. Not my voice, but in my head.\n\n\"I am mad!\" I was pissed. I was incensed. I was furious.\n\nMadman road, I heard.\n\nI stopped moving. Listened. Listened inside my head. Nothing. Outside my head. No sound. No bird. No voice.\n\n\"What do you want?\" I shouted.\n\nWe're all mad here, the voice said. And this time, I knew who it was. I missed that voice, but it couldn't be real. Diana was dead. I don't even know if her dead body had a mouth and a tongue left after the bullets ripped through her face.\n\n\"Leave me alone,\" I said. \"Leave me alone, leave me alone, leave me alone,\" I chanted.\n\nThe bird cried, Kree.\n\n\"Just go,\" I said, to the bird, to the voice, to the footsteps behind me. \"You left me.\" The words tumbled out, a mumble, half in my head, half out of my mouth. \"You left me here, and now you want to call me mad, sending some bush-rattling creature or some bird. Whispering in my head.\" I rattled on.\n\nYou're talking to yourself, her voice said.\n\n\"I'm talking to you!\"\n\nBut I'm not there. Nobody's there. Just you.\n\nAnd you made it that way! How could you leave me?\n\nI left myself.\n\nYourself? You're so selfish.\n\nYou're mad at me?\n\nOf course I'm mad at you! \"Yes, I'm mad at you,\" I said out loud at the nobody in my head, at the trees, at the earth, at the sky. \"I'm fucking mad at you!\" Then I shouted into the sky. \"I'm mad at you both!\"\n\n\"I'm mad at you,\" I pointed in front of me, like she was there, like the Diana in my head had a space on this earth by my side.\n\n\"And I'm mad at You,\" I pointed up at the sky. \"You want people to believe in You, stop being such an unfair asshole.\"\n\nYou're seeing it wrong, Diana said.\n\nThe bird cried, and I hoped I was near the tree.\n\nI'm going to die out here, and I'm not going to see you in heaven.\n\nIs that what you're afraid of? Because I was gay? Because I died by suicide? Like Charlotte said. You think God makes a heaven only just big enough?\n\nI'm not going to see you in heaven because I can't look you in the face.\n\nBecause of what happened with Leah? So, you kissed her. So, you loved her. How could you not?\n\nNo, I loved you!\n\nSo, it's not shame?\n\nNo, it's your face! I can't look you in the face. Your literal face!\n\nBecause of the gun.\n\nYes, the gun I gave you! Why did you let me give it to you?\n\nHow many bullet holes are in your head anyway? How many shots did you fire into your skull? Did you miss? Where's your face? Where's the top of your head? Where's your jaw?\n\n\"Why were there two bullet holes in the car?\" I shouted.\n\nSilence.\n\n\"Did you do a practice round?\"\n\nSilence.\n\n\"Did you mess up the first time?\"\n\nSilence.\n\n\"Did you put the gun in your mouth?\"\n\nSilence.\n\n\"Did you put it to your temple?\"\n\n\"To your forehead?\"\n\n\"Under your chin?\"\n\n\"Did you hear the sound of it when it went off? Did it hurt your ears? Did you feel the bullet go in before your brain exploded?\"\n\nI was crying. I could feel the tears pouring down my face.\n\nDid you think for one second about buying your own gun? Did you think about what you were doing when you walked into our house? Did you even think of me for one second?\n\nI stopped. I looked up at the sky. \"Why are You punishing me?\" And I knew I was yelling at Diana. But I was also yelling at God.\n\nI didn't do anything to You! Is it because I don't believe in You? Is this Your jealous revenge? The dark night of the soul?\n\nKree. The bird's bullet-cry was close. The tree was visible.\n\nI was sweating. The anger in my head beat inside my skull. The yelling hurt.\n\nI didn't want to talk in my head anymore.\n\nIt's called prayer.\n\nAnd that's the last thing my sister said to me inside my head.\n\nI reached the tree, slid down the trunk. The bark lifting my shirt and scratching the skin off my back. I was out of breath, exhausted, bitter, bitterly exhausted. I leaned my head back.\n\n\"Man, oh, man, she's a mad one.\"\n\nJack.\n\nHe sat up in a branch, his back to the trunk.\n\n\"Get away from me.\"\n\nJack rolled onto his stomach, slipped down from the branch, and hung there for a moment. Then he dropped to the ground and walked behind the tree.\n\n\"Did you hear that bird?\" He came around the trunk with a bottle in his hand. \"That, my friend, is an instrument of heaven. That sound, so pure, so sharp.\"\n\n\"Seriously, I can't do any more madness.\"\n\n\"You seem undone.\"\n\n\"Really? I seem undone to you? I'm in the freaking desert in the dark with a man who's been dead for half a century while my dead sister talks in my head!\"\n\n\"But you did it.\"\n\n\"Went mad like all the great ones before me? Yeah, I did.\"\n\n\"Yes, yes. And no. You said it all out. You told the truth. You told it out.\"\n\n\"So? It doesn't make it go away.\"\n\nJack set down a bottle between us. \"Have some wine.\"\n\n\"I don't want wine.\" I unzipped my backpack and reached in for my sweatshirt and sweats. \"I'm not getting drunk with you.\" I pulled on the extra clothes.\n\n\"The wine is for forgiveness.\" Jack reached into his pocket for his pack of smokes. \"And it'll warm you up.\"\n\nI sighed, reached for the bottle, put it to my lips. It was lukewarm, like the beer had been, but it seemed okay. It tasted like my grandpa's barn, all wood shavings and damp earth. It wasn't bad. It was rich and warm.\n\n\"I thought you might want a sip,\" he said.\n\nI returned the bottle to an even spot on the ground.\n\nJack crossed his legs. \"It's time to put your cross down now, Nicolas.\"\n\nI shook my head.\n\n\"This isn't about Jesus. Or it is, if you want it to be. It doesn't rightly matter. But, you're a boy, you're not meant to carry anything. You got some guilt, some shame, some thing you did in this world that's got you believing you're some heavy sinking stone, not worth the freedom of lightness, weightlessness. That's some bull. Put your damned cross down.\"\n\n\"I can't,\" I muttered.\n\nJack softened his voice. \"Your role in whatever goes down on this earth's already dead. You carrying on like you own a bit of any of that is indulgent and, frankly, selfish. You can't do anything more than what's already been done for you, so drop your tiny bitch-ass cross.\"\n\n\"Bitch-ass cross? Seriously? From beatnik to gangsta?\"\n\nHe blew his cigarette smoke in my face. \"Don't change the subject.\"\n\nI put the heels of my hands to my face. \"Did you see Natalie's face when I told her?\"\n\n\"Yes. Shocked.\"\n\n\"Disgusted,\" I said through my hands.\n\n\"That was hurt. Hurt for you. Horrified for the story.\"\n\n\"It'll kill them.\"\n\n\"Kill who?\"\n\n\"Whom.\"\n\n\"The Great Editor avoids the question.\"\n\n\"My parents. Leah. Anyone who knows me.\"\n\n\"No, the truth never kills anything. The truth only grows things. It connects things. It is your bridge.\"\n\nHe took a sip of the wine, wiped his mouth, and handed the bottle to me. \"Now take one more sip of that liquid forgiveness.\"\n\nI could use forgiveness of any kind. So I did as Jack said. I had a few more sips than I should've.\n\nI lamented leaving Ninja behind. Jack made jokes about being a smart-ass.\n\nAnd then Jack and I talked a good long time about my sisters, his brother, my parents, his parents. His wives, his daughter, his friends. We talked about religion and regret, forgiveness and friendships.\n\nWhen we got tired of the serious, I took out Sixty Scenes of Sexy.\n\nWhat happened to the Great American Novel, he wanted to know. Did anyone aspire to that anymore?\n\nWe squinted in the moonlight so we could read parts aloud to each other. We mostly laughed, though he did get morose a couple times, indignant that this was what the American reader craved. But then we'd find a sentence, a paragraph, read it aloud in accents of varying sophistication, and we'd fall into a fit of giggles. The laugh-till-you-cry kind. The kind that stole our breath. The kind that threatened to kill us with the absurdity of it all. The kind that left us spent and determined to sleep.\nTHURSDAY\n\nSomething velvety, cool in my face. Then a prickle and a snuff of hot air. I opened my eyes, the sun too bright, my back stiff from the ground, from not shifting during the night. Hooves scuffed on rocks near my head, and dust blew into my eyes.\n\n\"Ninja.\"\n\nI sat up and spotted an alien-long shadow coming around from behind the tree.\n\nNatalie walked past me, nudging something with her toe. A bottle clanked across the ground, and I panicked that Jack had left the empty wine bottle behind. But it was just a metal water bottle. She stood over me, studied me.\n\nShe handed me a protein bar.\n\nI unwrapped it, took a bite. \"Thanks.\"\n\n\"Welcome,\" she said, and we both ate our bars in silence.\n\nMy eyes felt swollen. I had a headache. From the sadness as much as from the wine I drank.\n\n\"I can't believe you stole a donkey,\" she said.\n\n\"His name's Ninja.\"\n\n\"I can totally see why.\"\n\nI stuffed the wrapper into my backpack and pulled my sweats from over my shorts. It wouldn't be too long before I got hot. I kept my sweatshirt on.\n\n\"Did you bring an extra toothbrush?\" I tried, hoping to break the tension.\n\n\"Look,\" she said. \"I'm sorry about yesterday.\"\n\n\"No. I'm sorry. I didn't even listen to you. I came to be a friend, and then I made it all about me.\"\n\n\"I'm sorry about your sister.\"\n\n\"I'm sorry about your brother.\"\n\n\"I'm sorry you walked twenty miles for a tongue-lashing.\"\n\n\"I didn't walk the whole twenty miles. Give Ninja some credit.\"\n\n\"We stopped for grass, by the way.\"\n\n\"There's grass?\"\n\nNatalie nodded.\n\n\"Where? I didn't see any\u2014\"\n\n\"Never mind about the donkey,\" Natalie said. \"I need to explain my confession.\"\n\nI held up both hands. \"No, it's private. You don't have to explain.\"\n\n\"Nick, please.\"\n\nI had been selfish the night before, so consumed with my own burning fury. I'd been so caught up in the idea of helping Natalie that I hadn't even done the bare minimum. I hadn't even listened to her.\n\nShe was looking straight into my eyes. \"I did heroin one time. But I did do it.\"\n\nI nodded. \"So, then, can I ask why?\"\n\n\"Gah, it's just so gross.\" She sighed. \"I swore I would never tell a soul.\"\n\n\"I won't tell anyone.\"\n\n\"I know. But, you know, it's even hearing my own words.\" Her face pinched. \"This isn't the first time my brother's disappeared from rehab. He's been in and out.\" She looked off into the distance, seeing something in her own mind. \"So, one time, I went looking for him. I knew where he stayed sometimes, when he was using. Murder Block.\"\n\n\"What?\"\n\n\"I'm not even joking.\"\n\n\"And you went there?\"\n\n\"Sh. It's my story.\" She smiled brief ly. \"It was Fourth of July. And I knew where my brother stayed sometimes, in this camper a couple blocks up from the highway. So, on this particular day, I went to go find him. But he wasn't there. One of the guys I'd met before was around, though. Big Rev.\" She raised her eyebrows. \"As in 'reverend,' yeah. He said I could wait for a while. I was really nervous, like sweaty and jumpy. Kids up the street kept lighting off fireworks, but they sounded like gunshots. All these people kept coming to the camper. Everyone was really nice. Asking me questions about school. Telling me stories. Except these two older woman shouted stuff about the 'little white girl' hiding out. But then Big Rev dismissed them and told me not to worry, they were just drunk. There was an older guy everyone was taking care of. He was in a wheelchair. This lady with marks all over her face kept getting in and out of cars. It was fascinating and horrifying. I was there for hours.\" She looked away. \"I didn't go there to do drugs. I went there to find my brother.\"\n\nShe stopped talking.\n\n\"It's okay.\"\n\n\"None of it is okay. I couldn't save my brother. I can't save him. But, that day, I thought I could maybe understand him better.\" She bit at her lip. \"I didn't shoot up myself. Big Rev did it for me.\"\n\nI wanted to throw up, imagining this teenage girl putting out her arm.\n\n\"I kind of slept or passed out or whatever. I never found my brother. I got sick and threw up and then went home. It was the stupidest thing I've ever done.\"\n\n\"Do you understand him better?\"\n\nShe shook her head. I had no idea what to say, so I didn't say anything. I put my hand over hers and held on.\n\nWe both analyzed the dirt for awhile.\n\n\"Do you believe in signs?\" she finally asked, her voice a little lighter.\n\nI shrugged.\n\n\"I do,\" she said.\n\n\"This does not surprise me about you.\"\n\nShe smiled, taking it as a compliment, which is how I had intended it. \"I'd written the confession, but I didn't know if I should really run away. I mean, here I was going to look for my brother again, after what happened last time, right? But when the box was stolen, I knew it meant I had to go. Remember when you found me on the hillside?\"\n\n\"You were running away then.\"\n\n\"I was. But then you appeared over the hill and I thought, this is a sign too. I wasn't supposed to go. There was something I was supposed to wait for. I even thought you might have some wisdom from God to tell me. Because that's how I hear Him, mostly. In the words of others. And you and I'd been talking and really connecting, or so I thought.\"\n\nShe'd thought right. I thought we had too. The words between Natalie and me f lowed, made sense, were easy. They actually felt like a bridge.\n\nShe fake-glared at me. \"Don't laugh. I'm not even done yet. And I'm not crazy.\"\n\n\"I'm not laughing, I swear.\" I was way crazier than reading signs. I was hearing the voices of people who weren't even on earth anymore. I was hanging out with the dead. I was talking to stolen barn animals.\n\n\"So, I stayed. But the confessions kept coming. And I knew I had to go. That was a sign. Or, that was a counter-sign. The universe redirecting me back up that hill to my brother.\"\n\n\"A bad sign.\"\n\n\"There aren't bad signs, just signs.\"\n\nShe kind of sounded like Jack on that point. No right or wrong. No good or bad. Things just were what they were.\n\n\"Anyway, you showed up last night after everything, and I realized I'd misread everything. I didn't understand any of what I was supposed to do and not do.\"\n\n\"After everything what?\"\n\nShe sighed. \"My brother wasn't there.\"\n\n\"I know. That's why I came.\" I told her about first going to Pastor Kyle, and how he knew her brother had checked out of rehab.\n\nShe nodded, sat thinking for a moment. Then she said, \"I don't see him often, ever since he was sent to this place. You know, it's far away and my great-aunt won't drive me down here. She's old, you know?\"\n\n\"I was under the impression you lived with your parents, who owned yachts and mansions, so no, I don't know.\"\n\n\"Right. Pastor Kyle is the only one who knows the deal. Both my parents are actually meth heads.\"\n\n\"Are your parents in rehab?\"\n\n\"Doubt it. Sometimes they show up at my aunt's and ask for money. Sometimes they're in jail. I don't know.\" She took a deep breath. \"My brother really tried. He's been in and out of rehab for like three years, and we write and stuff. But his letters got shorter and just off. So I suspected he'd left again. And then, yeah, he wasn't there. But don't worry, I won't be going to search for him on Murder Block.\"\n\n\"How can you just leave rehab?\"\n\nShe teared up. \"You just can. Because it's a choice.\"\n\n\"But how did Pastor Kyle know he left if you didn't?\"\n\n\"I have no idea, but I intend to find out.\"\n\n\"Maybe he's friends with the Big Rev.\" I squeezed her hand. \"Men of God and all that.\"\n\n\"Ha, yeah, right,\" Natalie said. \"So, do you think less of me now that you know about my sorry-ass life?\"\n\n\"She didn't mean that, Ninja!\" I called over my shoulder. To Natalie, I said, \"You are as cool as an Obama\/Uncle Sam mash-up with a Rosie the Riveter attitude. There's really nothing sorry about that.\"\n\nShe laughed and nudged my shoulder. \"Do you remember when I told you that when I was a little girl, I was scared and self-conscious all the time? I was the girl who didn't even stand against the wall at dances?\"\n\n\"You peeked through the wall from the other side.\"\n\n\"I meant that literally. One time, I spent Christmas in an attic in some crack house, because everyone was freaking out that the police were coming, so they hid us up there. My brother and I just lay there on the f loor, peeking through this little hole in the ceiling, waiting for Santa to come because we knew he'd sneak us out in his sack of toys.\"\n\nWe both laughed softly, a sad-funny laugh.\n\n\"I am Obama\/Uncle Sam\/Rosie the Riveter strong and interesting and cool\u2014if I do say so myself.\" She winked at me. \"But, I am also scared and nervous.\"\n\n\"At the same time.\"\n\n\"Simultaneously, two opposite humans.\"\n\nLike Jack, who was and was not a Roman candle. \"I get that.\"\n\n\"Which brings me to what I want to say. When you showed up last night, and I freaked out on you like a crazy person and you freaked out on me like a crazy person? That was a counter-counter-sign. The sign I was meant to see.\"\n\n\"I'm afraid to be implicated in this.\"\n\n\"I recognized your crazy. When you were standing there, yelling at me, but really yelling at the universe, I knew that feeling. I knew maybe you'd understand why I would ever put a needle to my vein. What I did, it was pure stupid crazy, based on my own guilt. I survived, my brother didn't, right? You survived, your sister didn't. And all that grossed-outness and self-loathing. I get it.\"\n\nShe was right. It drove me mad. It did.\n\nDiana hadn't had it that bad. She'd had food and clothing. She'd had two parents, and they loved her. Sure, she'd struggled with her identity\u2014or rather, the world struggled with her identity. And yeah, she'd had some emotional things going on. But she'd been surrounded by people who loved her.\n\nNatalie's brother was born to drug addicts, grew up in a meth house, bounced around from crack house to crack house. He never had a Santa, or anyone else who could save him. Obviously worse off than my sister.\n\nBut then, Diana was ostracized and judged by half the planet. She could never be her full self, full-time. She was told the love she had for others was dark and deviant, that she was not entitled to inalienable rights. She'd been hiding in an attic too.\n\nIt wasn't even fair to compare two people's hurts. To say this was worse than that. It just was.\n\n\"You know how you said you talk to your brother sometimes, like a prayer?\" I asked.\n\n\"Yeah.\"\n\n\"I heard my sister last night, like a prayer.\" I swallowed. \"Like a reverse prayer.\"\n\nNatalie didn't say anything, just reached out and took my hand. I held her hand in my lap, looked at her thin fingers, felt the warmth of her palm against mine.\n\nShe'd reached out. And I reached back.\n\nI told her the story that no one had heard. Not my parents or my grandpa or Leah. I told her to be even. To find grace. To share my crazy.\n\nThe day before Diana came over, the day before she died, Leah stopped by the house. She'd come to talk to me, not my parents, she explained. She stood in the doorway in her skinny jeans, high boots, long sweater. Her hair was back, wisps in her face. But her face was wrong\u2014it was tight and dark, not like normal. Still beautiful, but not eager to laugh, to joke, to say something self-deprecating that put people at ease, made you feel like you fit in and mattered.\n\nI was so relieved to see her. I hoped it meant she and my sister were getting back together. And not even for Diana. For myself. She was\u2014and maybe is\u2014the person I hold everyone up to.\n\nI loved Diana, but she was way far out there on the crazy spectrum. Leah was a willing participant with a whimsical streak and an ability to balance that with a college degree and a job.\n\nI wanted her around, and she hadn't been around. I ached for her to come back into our lives. And here she was, and she needed my help.\n\n\"Your sister's being really weird,\" she said.\n\n\"My sister is always weird.\"\n\n\"Ha, right, yeah, but... weirder. She wants to drive across the country. 'Go East, Young Man,' she keeps saying. I don't even know what she's talking about.\"\n\nShe'd been asking Leah to get back together, and every day she called or texted with some new, far-fetched plan. Cool ideas, but erratic.\n\nLeah continued, \"She wants to take me fishing or golfing or scuba diving in Hawaii. Hunting, go-carts, horseback. Like adventure stuff.\"\n\n\"That's normal,\" I said.\n\n\"It is, yeah, but quitting our jobs and crossing the country? That's weird.\"\n\n\"She used to want to join the circus. Weird is relative.\"\n\n\"Nick, she put half her life on Craigslist. She's really going. She really thinks I'm going too. And I'm not.\"\n\nShe asked me if Diana was drinking\u2014which she was; she'd always drunk heavily\u2014or on drugs. I wouldn't know that part.\n\n\"Just talk to her, will you? I know you're her baby brother\"\u2014and that part stung\u2014\"but she respects the hell out of you.\"\n\nI would have done anything for Leah. I loved her, despite the near-decade between us. I promised her I'd talk to Diana, I'd text her when I did.\n\nThe day my sister shot herself, I had woken up late. I was playing video games. Diana interrupted me, and I was tired.\n\nI'd been irritated with her for months. She'd been distracted every time we hung out. She'd stopped taking me cool places in favor of darting off with people I didn't know. Leah had broken up with her, and it was her fault.\n\nAnd the last time I'd seen her she'd been drunk. Breathing heavily in my face, telling me she loved me. Don't worry, you're fine. Like she always said.\n\nAnd I hadn't seen her since.\n\nThe day my sister died, we stood in the kitchen, and she was talkative, but not being funny. Just rushed, not wanting to hang out.\n\n\"What's up with you driving across the country?\" I'd asked.\n\n\"Oh, you know.\"\n\n\"Not really. Leah just told me about it.\"\n\nShe shrugged. \"I'm not going to follow it anymore.\"\n\n\"Follow what?\"\n\n\"Kerouac's path.\" She held up her phone. \"Have you seen the app?\"\n\nI had no idea who Kerouac was, what path or what app.\n\nShe pulled On the Road out from her purse and handed it to me. \"There's an app. You can do his whole trip.\"\n\n\"Huh. But you don't want to do that anymore?\"\n\n\"No.\"\n\n\"Because Leah said you were trying to get her to go. And selling your stuff.\"\n\n\"No, I'm not going.\" She leaned back in her chair and crossed her arms.\n\n\"Okay. So everything's okay? I tried to hand the book back.\"\n\n\"You keep it.\" She waved her hand at me. \"I don't want it. You can do it someday. Hit the road.\"\n\n\"Okay.\"\n\nAnd it was. Because we did do a summer road trip thing every year. So what she'd said, that made sense.\n\nI'd talked to her, like Leah had asked. And it was okay.\n\nAnd because it was okay, I wanted go back to my game.\n\n\"I'm going to the gun range,\" she'd said. \"Leah and I are going.\"\n\nAnd I believed her.\n\nI believed her because it matched Leah's story, about the cool but erratic ideas she was having.\n\nI believed her because I wanted it to be true, to have Leah back.\n\nI believed her because I was tired and impatient. I was relieved she wasn't asking me to go. Because I was also tired and impatient with her.\n\n\"I couldn't get ahold of Dad.\"\n\n\"Dad won't care.\"\n\n\"I know, but I need to borrow his gun,\" she'd said.\n\n\"They're in the case.\"\n\n\"Yeah, but he moves the key.\"\n\nI scoffed. \"No, he doesn't.\"\n\n\"You know where it is?\"\n\nOf course I knew where it was. My dad trusted me. He'd taught me to use the guns. I'd grown up hunting. He'd taken me to the range. Taken me to the woods. Diana had gone with us. If I'd been listening\u2014if I hadn't been tired and impatient and hopeful and lazy\u2014I would've noticed this question was odd. I would've noticed.\n\nI was being emotionally lazy. I wasn't hearing the full truth. The truth means you have to do something, and I didn't want to do something for her that day. I wanted to do the easy thing.\n\nI walked back into my parents' bedroom, and she didn't follow me. Why didn't she follow me? I went to the closet, slid open the door, and pulled out the Monopoly game. I lifted the lid, found the key amid the game pieces, and unlocked the case.\n\n\"Which one do you want?\" I called out.\n\nThere were several guns. A shotgun, a .22, an antique handgun, a .38.\n\nShe didn't answer.\n\nMaybe she didn't hear me. That's what I want to believe. I want to believe she didn't hear me ask that question. Because if she did, and she paused for those seconds, what filled that time?\n\nMaybe she was reassessing. Maybe she was reconsidering. Maybe she wouldn't have done it if I hadn't unlocked the cabinet and pulled out the .38. If I hadn't grabbed the box of bullets.\n\nI walked out of the bedroom. I had a gun in one hand and bullets in another, and I held them out to her.\n\nAnd what was that look that passed over her face? Because there was a look.\n\nMaybe it was disappointment that I didn't see what she was going to do.\n\nMaybe it was realization that now she had to do it. She had to take the gun, because it matched the story she'd just told me. And since she had the gun, she had to follow through.\n\nOr, maybe it was resolve. I like to tell myself it was resolve. That nothing I did mattered in the end.\n\nBut it all mattered.\n\nYou can do that some day. Hit the road. She should've said we. And I should've noticed.\n\nBut I didn't.\n\nI went into the room. I unlocked the cabinet, chose the gun, and put in her hands.\n\nIt was my fault.\n\nIt was my fault.\n\nIt was my fault.\n\n\"When my mom told me\u2014because the police called them, or the hospital\u2014I don't even know, she told me in the kitchen. She said, 'Diana shot herself.' My dad was at the hospital with my grandpa. And, I thought\u2014I reasoned\u2014it occurred to me\u2014she shot her toes. My dad and grandpa were there to visit because she shot her toes. Because the gun accidentally went off.\n\n\"That's what I remember. Standing in the kitchen, looking at my own toes.\"\n\nThe tears fell as I told Natalie this.\n\n\"She was in the hospital because they were keeping her alive. To save the organs. She was a donor. There was no Diana to visit.\n\n\"I don't remember when I understood that she shot herself in the head.\" I swallowed. \"I imagine it all the time. Where she put the gun. If her hands were shaking. The only thing that makes me feel better is that the doctors could use her eyes. She still had her eyes. Something of her was still there.\"\n\nI wiped the tears off my cheeks.\n\n\"Someone has her eyes,\" I said. \"Someone sees the world how she saw it. She saw it so big.\"\n\nI leaned my head against the tree trunk.\n\n\"She wanted to see it all, do it all, have it all. And she couldn't. She couldn't even have half of it. That thing she said to Leah\u2014'Go East, Young Man.' I looked it up. I thought it would reveal something, something to make it better. It was just a play on a famous quote 'Go West, Young Man.' Horace Greeley. He was all about Manifest Destiny. My sister was too, in a way.\"\n\nAfter a while, Natalie said, \"Your parents will figure this out for themselves. They're adults.\"\n\nAnd she was right. But at the same time, I'd wanted them to be adults and protect Diana. To hold Charlotte accountable, to stop feeding the arsenic idea that we should consider the perspective of intolerance. They let their other daughter's hatred exist, let it eat away at Diana. They didn't protect her, and I was afraid they couldn't protect me.\n\nMaybe I should hate them. But I didn't. I needed them. And I needed them to stay with me.\n\n\"They're going to hate me,\" I said, which wasn't exactly what I meant, but I just couldn't explain it all.\n\n\"They are not going to hate you.\"\n\n\"Look what I did to their daughter. Don't you hate your parents? What they did to your brother?\"\n\nI noticed then that she was crying too. She'd been crying for me, and now some of those tears were almost certainly about herself. \"I don't hate them. They didn't do that to him. He had a choice. We all have a choice. Look at me. I'm not a meth addict. Or a heroin addict.\"\n\n\"You got lucky.\"\n\n\"I didn't get lucky. I made a choice. To survive. To survive whatever happened.\"\n\nThis conversation sounded so much like the one I'd had with Charlotte, about all the little things in our lives being choices. But Charlotte's conviction about our sister's choices was judgment. Natalie's was not.\n\nBut what about my sister?\n\n\"My sister goes to hell,\" I argued, \"because the Bible says. That's what people say.\"\n\n\"Why don't you let God do His thing? Quit worrying about what it all says, what it all means? It is what it is. She was a homosexual. She died by suicide. That doesn't mean she's going to hell. You don't know her heart, and God is too big for our comprehension. You'll drive yourself crazy trying to do His job.\"\n\nI sighed. \"You want to know the worst part?\"\n\nThe look on her face suggested nothing could be worse than my complicity in suicide.\n\n\"When Diana died, I kept wishing it was Charlotte. I prayed I would wake up and it would be swapped. I told God I'd do anything to bring her back. I'd trade anything.\" I put my hands to my eyes. \"And I still wish it would've been Charlotte. If I have to hand someone a gun\u2014\" I broke off and pressed my hands into my forehead.\n\nNatalie put her hand on my back until I had no more tears.\n\n\"He doesn't bargain,\" she said quietly.\n\n\"He doesn't understand,\" I said.\n\nShe didn't correct me.\n\nWe sat for a long time against that tree, and the bird shot song bullets through the sky.\n\n\"Forty miles, not forty days, but... crazy things happen in the wilderness,\" she said after a while. \"Satan tempts you, Gabriel tells you to read scrolls, you reach Nirvana.\"\n\nShe was talking about Jesus, Muhammad, and Buddha. I wanted to tell her about Jack. How he said they were all here. Plus some Native American god and Ereshkigal and who knows who else.\n\n\"We could try it out, under this tree,\" Natalie said. \"I mean, I don't think this is a banyan tree, but maybe enlightenment doesn't require particular f lora.\"\n\nThis girl. This mad, crazy girl on pretty road.\n\n\"Why are you looking at me like that?\" she asked.\n\n\"Because you never say a commonplace thing.\" A line from the poster on my bedroom wall.\n\nShe nudged me. \"That might be the nicest thing anyone has ever said to me, Nick.\" Then she stood up, dusted off her shorts, and tugged my arm. It was time to leave.\n\nNinja, as always, didn't argue.\n\nAs we walked side by side, Ninja lumbering behind, a lead rope completely unnecessary at this point. We seemed to be walking slower on the way back than I did on the way there, like neither of us really wanted to return. But, then, what awaited but a frustrated pastor with rockin' abs (I'd seen him in the pool) and an unknown punishment?\n\n\"What are you going to do about your brother?\"\n\n\"What's to be done?\"\n\n\"Go find him?\"\n\nShe laughed drily. \"He's in a city somewhere with stringy hair.\"\n\nI didn't know what to say to that. \"What about Eden Springs? Will you come back? Even though he's gone?\"\n\n\"Yeah. I mean, I have a scholarship. I kind of have to come back.\"\n\n\"Well, you don't have to.\" I thought of Holly. \"So, you're on work crew too?\"\n\n\"No. I'm indebted to some mysterious philanthropist.\"\n\n\"How did you get out of sweeping stray pubic hairs off the bathroom f loor?\"\n\n\"I just got a call from the camp office one time. I was in sixth grade or something.\"\n\nThat seemed really weird. \"That's not really how it works for normal people,\" I started to explain.\n\n\"I think we've established that things don't work normally for me.\"\n\n\"Maybe by next year he'll come back to zombie camp.\"\n\nOur shoes scraped across the rocks, and that was the only sound. I noticed we were walking toward the sun again and worried I'd see the shimmering faces in the bushes. The desert trolls. But I felt safer with her there, like I wasn't going to lose my mind.\n\n\"So, will you be coming back to Eden Springs next year?\" she asked.\n\n\"Ha. I don't think so.\"\n\nNatalie smiled. \"You seem kind of judgmental about it all.\"\n\n\"I think they're crazy, that's what I think. I feel like I'm stuck in some Salvador Dal\u00ed painting half the time.\"\n\n\"They are crazy. That's the best part.\"\n\n\"If you like that kind of crazy, I guess.\"\n\n\"You of all people should understand. You're the one who likes Jack Kerouac.\"\n\nI shook my head. \"No, I do not like Jack Kerouac. I've already said that. My sister likes him.\"\n\n\"Okay, but you like the quote about the mad ones.\"\n\nI nodded. \"I'll concede that.\"\n\n\"Then concede this. Those people down there. Those people who operate Eden Springs and the people who go\u2014Charles, Kim, Matthew, Dan, Jason. Me. They're just Mad Ones. They're mad to live, mad to be saved, desirous of everything.\"\n\n\"But it's all wrapped up in religion,\" I argued.\n\n\"Yes, it's about Jesus for them. They're mad for Him. Mad to be saved by him. So, you're not. That doesn't make their madness less cool just because you don't share in it.\"\n\n\"What about those guys who blow up abortion clinics in the name of religion?\"\n\n\"Not okay. Right-wing terrorism,\" she said and kind of laughed. \"But, Nick, who cares what other people are mad about if they're not infringing upon the rights of others? Why do you let them bother you so much?\"\n\nI knew why it bothered me. Or, at least, I knew part of the reason, but it was so much, a garbage bag of things to explain.\n\n\"I don't understand how they can all be happy,\" I said.\n\n\"Are you jealous?\"\n\n\"Ignorance is bliss and all that.\"\n\n\"But they aren't ignorant. Jeez, Nick, listen to yourself. What is the difference between you hating the conservative Bible-thumpers and them hating you because you don't believe? And those people, down at Eden Springs? They don't hate anybody. They love everybody. They just want people to come to their little oasis and party for a week in the name of Jesus. Don't you get that?\"\n\n\"It's stupid, Natalie. They're stupid. The guy was a guy. Jesus Christ, the real one, not the hillbilly dude at camp, was a guy. That's it.\"\n\n\"Yeah, the maddest guy of them all.\" She looked at me like she felt a little sorry for me. \"Whether you believe in him as a religious figure or as a historical one, he's mad as hell.\"\n\n\"Well, that makes two mad Jesuses.\"\n\nFinally, she said, \"Nick, don't let man's rules and religion get in the way of the divine mysteries of the universe.\"\n\nPRAYERS AND CONFESSIONS\n\nHolly: Please give me the strength to be who I am. All of me. Even the parts other people can't accept.\n\nDan: I didn't say sorry. My dad did, over and over. I pray he really is, Lord. Please help him get sober.\n\nMonica: Thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you, thank you...\n\nThe three crosses stood tall and expectant on the horizon. I was relieved to have the forty-mile journey in the wilderness over. But I was also disappointed to part with Natalie.\n\nNot long after first seeing the crosses, I saw the silhouettes on the horizon. Two men waiting for us.\n\n\"Are they going to send us home?\" I mean, we'd both run away.\n\n\"Maybe you. Not me.\" She wasn't joking. I was just the new kid. And I'd actually stolen camp property. Ninja didn't like to be objectified like that, but he didn't understand the human legal system.\n\nWe kept walking. The men headed toward us.\n\n\"Two more things before they nab us,\" Natalie said.\n\nWe hadn't been talking much the last hour. I raised an eyebrow at her.\n\n\"You need to believe in something.\"\n\n\"Me? Or everyone?\" I asked.\n\n\"You and everyone.\" Natalie tugged at my sleeve to stop walking. \"In a way, we're all at zombie camp.\"\n\n\"We're meth heads?\"\n\n\"Don't be stupid. I think we're all a little like zombies. Walking around a little dead inside, looking for something to fill us up.\"\n\n\"Drugs or... Jesus... or...\" I held my hands up in a shrug.\n\n\"Horses...\"\n\n\"Horse people are really weird.\" We both laughed.\n\n\"Cats...\" Natalie suggested.\n\n\"Old lady cat lovers!\"\n\n\"Video games.\"\n\nI narrowed my eyes. \"Child porn.\"\n\n\"Ew.\" Her face curled in disgust. \"Anyway, believe in something, Nick.\"\n\n\"Anything?\"\n\n\"Okay, not child porn. Too far.\"\n\nWe laughed and kept walking.\n\n\"Okay, so?\" I asked.\n\n\"So... ?\"\n\n\"You said there were two things.\"\n\n\"Oh, yeah.\" Natalie blushed. \"Do you like Holly?\"\n\n\"I mean, I like her, but...\" I couldn't finish that statement.\n\n\"You're not into her?\" She looked at her feet. \"I mean, a lot of guys like her and she seems kinda into you.\"\n\nIt was f lattering that Natalie thought for even a second that a beautiful girl like Holly would sincerely be interested in me.\n\n\"She's actually really smart,\" Natalie added. \"Like STEM-nerd smart. Like MIT smart. Her IQ happens to be triple her bra size.\"\n\nI knew she was smart, and I wished I could tell Natalie the whole truth about the confessions, but I'd made a promise. So I told her a different truth. \"She might be smart, and she might also be the girl that everyone thinks is hot. But I like your theory.\"\n\n\"My theory?\"\n\n\"About the horizon,\" I said, blushing. \"About watching people walk away.\" That maybe the way you can tell if you love someone is when you keep looking for them, keep seeing them, even when they're not there anymore. I looked at her. \"I don't just like your theory. I believe it.\"\n\nShe moved closer. Her arm bumped mine.\n\n\"So, there you go. I believe in something.\"\n\nWe walked in silence the rest of the way.\n\nWe soon heard Pastor Kyle and Jason, the quiet deep rumbles of their conversation rolling toward us.\n\nThen it was the four of us, standing there awkwardly. They breathed hard; we barely breathed. Their lips were tight with disappointment; our lips hung open in anticipation of the smackdown.\n\n\"Why don't you walk with me, Natalie,\" Pastor Kyle finally said, and they stepped out of the square and headed off toward camp, Ninja on a rope behind them.\n\nJason watched me watch them go. I didn't want to look at him.\n\n\"I'm sorry,\" I said, probably for the trillionth time in four days. My list of sins against Jason and his zealous love for all things Jesus and Eden Springs exceeded my lifetime list of all other sins combined.\n\n\"Let's go,\" was all he said, and we started walking toward camp. He didn't say anything for a long time. I just watched Natalie, Pastor Kyle, and Ninja disappear and reappear over small hills and valleys, their silhouettes becoming smaller. They seemed to be walking a lot faster than the two of us.\n\nWhen we reached the ridge above the three crosses, Jason stopped. He looked over the camp. \"We contacted your parents yesterday.\"\n\nMy face burned. They would be freaking out, thinking I was dead, imagining me being eaten eyeball by eyeball by something awful. \"Were they freaking out?\"\n\n\"We assured them you were safe.\"\n\n\"That's good.\"\n\n\"And they're here,\" he added.\n\n\"They're here?\"\n\n\"We asked them to come.\"\n\n\"To pick me up?\"\n\n\"To talk. And then, likely, to take you home, yes.\"\n\nIt made sense. The sending me home. Running away from camp was a major no-no.\n\n\"Does Natalie have to go home?\"\n\nJason sucked in his lips. Finally, \"No.\"\n\n\"Because she's on scholarship?\" That kind of kid always got second chances.\n\n\"It's complicated.\"\n\nJason didn't take us down the steep decline, the way I'd come up to the crosses both times I'd followed Natalie. Rather, he led me down the wide path that snaked slowly into the camp.\n\nNobody noticed us, at first, as we crossed the green manicured lawn. But as we got closer to the pool, more faces turned in our direction. I swore I heard whispers, though that couldn't have been true, considering the bass that quaked from the speakers.\n\nJust the day before, I'd felt a part of the camp experience, though still separate because I didn't believe what they believed. But just then, coming out of the desert, after the visions and the voices, after confessing that I'd handed my sister the gun that she used to blow her brains out... now I felt kept behind a glass window. Like camp was a fishbowl, and I could only look in.\n\nThe campers weren't smaller, by any means. I didn't feel superior in a fish-owner kind of way. I just felt far away, behind a cloud.\n\nI caught sight of Natalie walking through the door to Babyl On, Pastor Kyle ushering her in, probably for a round of counseling and prayer. She didn't have a cross over her shoulders.\n\nAnd Jason didn't drag one over to me. He led me in the direction of the cabins. I didn't know if my parents were in there, or if I was supposed to pack first.\n\nThen Matthew was running toward me from the sanctuary. I could see him coming across the field. He was pointing at the fresh confessions painted on the sides of cabin doors.\n\n\"So you know I didn't do it now, right?\" I said to Matthew when he reached me. \"I wasn't even on campus.\"\n\n\"Not literally, right?\"\n\n\"What? Literally, what?\"\n\nMatthew pointed to the door of our cabin. And there it was. On the door, my confession was painted in scarlet letters.\n\nI killed my sister.\n\nI looked at Jason. Swallowed hard. \"Where are my mom and dad?\"\n\nIt was time to find the right words. To cross the bridge between me and them.\n\nI told my parents. Told them everything they didn't know already. They already knew about Leah coming over. They already knew about my conversation with Diana. They knew about the mistaken pronoun. They knew that she'd planned on going to the gun range. All of these things we'd talked about in the days and weeks after she shot herself. But they didn't know the part about the key. That I went in for the key, that I unlocked the cabinet, that I chose the gun, that I selected the bullets. That I put them in her hand.\n\nMy parents didn't cry at first. Their mouths hung open, their eyes wide, their heads shook slightly. I knew they wanted me to stop, that they didn't want to hear the end of the story. But the real end, they already knew. Diana dies in the end. She always dies in the end. She was always going to die in the end. That was her after.\n\nMy after is this.\n\nI confess. The sour words, the sour truth.\n\nMy after is also this.\n\nMy parents sob and reach for me and draw me in and surround me. Because there are both of us to mourn for. There always was. Me and Diana. Me without Diana. Them without Diana. And somewhere, there is Charlotte. There and not there, in all of it.\n\nI don't have to tell them about the arsenic, about the slow poison. I don't have to explain to them how their support of Charlotte seemed like taking sides, that it ate away at Diana's spirit. They already know. We're all very good at taking blame after a tragedy. If only, if only.\n\nThey knock from my hands the bags of sadness and guilt and regret. Later, after they tell me it's okay, that they don't blame me, that there is no one to blame, that it's not my fault, that she would've found a way, they will pick up those bags and add them to their own burlap sacks of sadness and guilt and regret.\n\nBecause they're my parents.\n\nBecause that's what humans do. We step out onto broken dilapidated untrustworthy scary-as-shit this-bitch-is-gonna-break bridges, and we reach out. We say, give me that thing you're carrying. Let me carry that thing for you. Let me do this kindness for you.\n\nThat's humanity. That's compassion. That's grace.\n\nAnd that's my after.\n\nThere are two things I don't tell my parents, though.\n\nI don't tell them about Jack. About the wise madness.\n\nAnd I don't tell them about kissing Leah. About how holding on to her was just my way of holding on to Diana.\n\nThere are some things that weigh heavily, but simply are not burdens.\n\nWhen it was all over, Pastor Kyle handed me a slip of paper.\n\nHolly had done as she promised. She'd confessed. She'd put the truth of what she'd done into the otherwise empty PC Box and handed it over to the pastor. She'd also told the pastor that I'd encouraged her to do it. Pastor Kyle hugged me. Hugging is not my cup of tea, but I let him do it.\n\nPastor Kyle and Jason, and my mom and dad, decided it would be good for me to stay the last night of camp. It was late; we'd cried till our eyes were mere slits. My parents would stay in the counselor housing, and I would stay that last night with my cabinmates.\n\nOn the quiet walk through the dark to the cabin, I wondered about Holly. About her after. I hadn't seen Payton all day, and I wondered if she'd told him too.\n\n\"Psst.\"\n\nI turned around. The campus was empty, everyone in their cabins. Lights out was in a few minutes.\n\nSomething came f lying at me from behind a bush. It landed at my feet. Twilight. Or rather, Sixty Scenes of Sexy.\n\n\"Where are you?\" I whispered.\n\n\"Get me the second one.\"\n\n\"The second what?\"\n\n\"You said it was a trilogy.\"\n\n\"Seriously?\"\n\n\"I'm as serious as handcuffs.\"\n\nI picked up the book, f lipped through the pages. Jack had not only read it, but he'd finished marginalizing the whole thing. He'd done a remarkable job of making his handwriting look like mine.\n\nI tucked the book into the back of my jeans and headed up the stairs to my cabin. When I opened the door to my cabin, all the guys burst with welcome and praise and the words teenage boys offer as apologies for their false accusations.\n\n\"Giant balls, dude,\" Matthew said, an expression of awe as he sat on my bunk. \"I cannot believe you ran after Natalie.\"\n\nI didn't know what I could say and what I couldn't, what I should say and what I shouldn't. I wanted to protect Natalie's privacy. And I didn't want them to know I hadn't really gotten in trouble. I felt like keeping that to myself was protecting Jason.\n\nSo, I told them about my walk, how I caught up to her. I didn't talk about the fence or the zombies. I didn't talk about the voices I heard in the desert. I didn't tell them Natalie's brother was gone, or why my sister was gone.\n\n\"You took a donkey? What, to like, go faster?\" Everyone turned to look at Dan, standing outside the circle, leaning his muscle-swollen shoulders and arms against the opposite bunk. \"You're an idiot.\"\n\nI held my breath for the next part, the insulting part, the part that would make me shrivel, like Dan had been aiming to do this whole week. But Dan said nothing more, and everyone just kind of chuckled and moved away from me to get ready for bed.\n\n\"It's cool you went after her, Nick,\" Dan said. \"I still think you're an idiot, and I don't want you in my cabin next year, so don't bother requesting it. But it's cool you didn't let her go out there alone. Seriously, life's pretty much crapped all over her.\"\n\n\"So you know about Natalie's family?\" I whispered to him.\n\n\"We're friends.\" He shrugged. \"I'm not a dick to everyone.\" Then Dan looked at me for a beat more, like he wanted to say something else, but he shoved off and headed toward the bathroom.\n\nWhen he came back carrying a briefcase, I realized that the guys hadn't been getting ready for bed. They were dressed head-to-toe in black, ready for the final prank.\n\nDan f lipped latches on the briefcase and revealed ten gallon\u2013sized Ziploc bags with large bumper magnets inside. He handed one to each of us.\n\nI opened mine and pulled one out one of the magnets. I BRAKE FOR CUTE SHEEP, it said.\n\nMatthew held one up, \"'MY OTHER RIDE'S A DILDO'! I'm gonna piss myself. Where did you get these?\"\n\nDan didn't answer. As usual, he was acting like he had nothing to do with any of us.\n\n\"'FIFTY SHADES OF GAY'! That's awesome!\"\n\nCharles shook his head and crawled up into his bunk.\n\n\"You still out on this?\" I asked Charlie as I read the next bumper magnet: SPANK A VETERAN.\n\n\"Nick, God is out on this.\"\n\n\"What if we find some that aren't so offensive?\" I dug around in my Ziploc. \"'TRUMP SUPPORTER'? Wait, someone might actually get shot for this one,\" I said to no one in particular as I shoved it back in the Ziploc and pulled out another. \"What about 'RIBBED FOR HER PLEASURE'?\"\n\nCharles rolled his eyes and lay down.\n\n\"It's about condoms, Charlie,\" I reasoned.\n\n\"Don't call me that.\"\n\n\"Because it refers to the Viet Cong?\" I laughed.\n\n\"No, because it's not my name.\"\n\n\"Okay, Charles. But the magnet is just about condoms.\"\n\n\"I know what it's about,\" he said.\n\n\"Sex is natural. And condoms are an environmentally responsible way of limiting offspring.\"\n\n\"Shut up, Nick.\"\n\nMatthew laughed and pulled me toward the door. \"Come on, let's go.\"\n\n\"I just feel bad that he's left out,\" I said.\n\n\"He's fine. He participates by never telling the authorities.\"\n\n\"Wait,\" I held up my last bumper magnet. \"I don't get this one.\"\n\nEveryone stared at the magnet, then at me, like I was an idiot.\n\n\"You're an idiot,\" Dan murmured.\n\n\"All it says is 'PRO-CHOICE,'\" I insisted.\n\nMy cabinmates shook their heads and a chorus of mumbling circled the room:\n\n\"That might be crossing the line.\"\n\n\"I can't seriously put that on someone's car.\"\n\n\"Have you seen those abortion posters on the hill by the hospital?\"\n\nI shrugged. The hypocrisy among my fellow campers was widespread, but it was what it was. I was not going to be angry about it anymore.\n\nThen I had an idea. I grabbed a pen and altered my magnet to my liking. I marginalized it.\n\nAfter waiting the requisite hour to ensure all the campers\u2014or, more importantly, all the counselors\u2014dreamed of sugarplums, we finally slipped into the dark and walked silently to the employee housing. By the time everyone had emptied their Ziploc bags, every car had at least two magnets on their bumpers.\n\nDan even told us to put magnets on his car so it wouldn't be obvious.\n\nI gladly offered to do that job. For Dan's car, I had saved my marginalized bumper magnet, which now read IF ONLY YOUR MOM WAS PRO-CHOICE. Then, for good measure, I slapped on the one that had a picture of Mount Rainier. I had revised it to say I SUMMITED YOUR MOM.\n\nOn our way back to the cabin, we weren't all that silent, despite Dan's admonition. We giggled like girls, then fell into the cabin, wondering how far down the highway the camp directors and pastors and counselors would get before somebody shouted at or propositioned them through a car window.\n\nPRAYERS AND CONFESSIONS\n\nPastor Kyle: Thank You for Your mercy, Jesus. Thank You for keeping me safe, for keeping Gabe safe, for keeping the family in the other car safe.\n\nPayton: Yeah, I really do want to be with Holly.\n\nHolly: I am sorry for what I did with the confessions. Please forgive me, Lord. Thank you for Payton and Pastor Kyle, and even for Nick.\n\nMatthew: I will never see the baby that Sarah and I had. I know I'm not half of who I need to be to be a dad. But thank you for the family who took him.\nFRIDAY\n\nWe were going to walk on water, Pastor Kyle announced at the morning sermon.\n\nThe pastor had invited my parents, but they had declined, opting for the camper-less pool area. I was relieved. I wanted to have more time with Natalie. I'd have the whole car ride home with my mom and dad. And I did want to talk to them. I wanted to talk to them about Charlotte, about how maybe I understood something about her now. Listening to Holly, bearing witness to how we'd all hurt her, it all had made me think about my older sister. How she'd treated Diana was inexcusable, and I couldn't forgive her for that, but maybe there was something left of the fabric of us. I didn't know what. But, maybe.\n\nIn the meantime, I was determined to make the last day of camp \"Christastic!\"\n\nThe counselors serenaded us out the door in the style of Rick James:\n\nHe's all right, he's all right.\n\nThe Son's all right with me...\n\nSuper Jeez, Super Jeez, He's Super Jesus!\n\nKids were supposed to demonstrate their trust in the Lord. We would go out in the boat and \"walk on water.\" We didn't have to if we didn't want to, and I was dubious. I could see them out there, getting out of the boat, from my vantage point definitely appearing to walk on water, but then something would happen and they would fall in.\n\nSometimes they made it, though. They didn't fall in. And many of those kids got into the boat with their leaders and said a special prayer and accepted Jesus as their personal Savior.\n\nI already knew I wouldn't do that, but fist pumps for Christ anyway.\n\nMatthew and Charles and I swam and pushed and shoved our way through the lake. There was a water trampoline to jump on, a slide off the dock, and a diving board. I liked it more than the water park at camp, honestly. I liked the smell of real water when it dried on my skin. The way it collected sunshine. The way it smelled a little of dirt and a little of grass.\n\nWhen Charles ran over to jump around with the Disciplettes and I lost Matthew to Kim's f lirtations, I walked out of the lake to sit on my towel on the beach. The sun was perfect on my shoulders, and I closed my eyes for a moment.\n\n\"'What's your road, man?'\" I heard from behind me, and my eyes opened as Natalie sank down onto my towel next to me. \"'\u2014holyboy road, madman road, rainbow road, guppy road, any road.'\"\n\n\"Hey,\" I said to the girl on pretty road.\n\n\"Hold on, I'm not done.\" Natalie smiled. She was reciting On the Road. \"'It's an anywhere road for anybody anyhow.'\"\n\nHer eyes were bright, as bright as brown could be. Her eyes were like gold f lecks some forty-niner scooped up out of a creek.\n\n\"I'm done,\" she said, nudging me with her shoulder.\n\n\"Hey,\" I repeated, nudging her back.\n\n\"Hey.\" She smiled, blushed a bit, and looked out toward the lake.\n\n\"Probably madman road,\" I answered. \"That's the road.\" And it was true, though I'd never tell her about the Jack Kerouac visions.\n\n\"Not guppy road?\"\n\n\"Possibly. I like guppies.\"\n\n\"I think kitten road for me. They're so cute.\"\n\nI laughed.\n\nWe both sat looking out at the lake, at all the kids screaming and splashing and dunking.\n\n\"I thought maybe they sent you home.\"\n\n\"Nope,\" she said.\n\n\"You didn't get in trouble?\"\n\n\"Did you?\" Her eyes widened.\n\nI shook my head.\n\n\"I heard about your confession.\"\n\n\"Yeah. It sucked, but my parents are here.\"\n\n\"And so are you, so that's good.\"\n\n\"Happy endings and all that,\" I said. The end of it is the it thing? Is that what Jack had said? \"But where've you been? What happened?\"\n\nShe pursed her lips. \"I found out about my scholarship. Where it comes from.\"\n\n\"Really?\"\n\n\"They weren't supposed to tell me till I graduated or something. I don't really know, there was some contract. But my brother left zombie camp, so they don't owe him anything.\"\n\n\"Wait... what?\"\n\n\"The King of Zombie Camp is my benefactor.\"\n\nI thought I got it.\n\n\"My brother.\"\n\n\"Seriously?\"\n\n\"Yep, the whole time.\" She shrugged. \"Until he left, and then the money wasn't there.\"\n\n\"Which is how Pastor Kyle knew?\"\n\n\"Yeah. He called the rehab place, and he was gone.\"\n\n\"So then, how'd you get to come if there wasn't any money?\"\n\n\"Pastor Kyle paid for it.\"\n\nBecause despite our differences, he was a good man.\n\n\"Why didn't your brother just tell you?\" I asked.\n\n\"I don't know. I kind of like that mystery.\"\n\n\"Does your aunt know that he's behind the scholarship?\"\n\n\"No. Apparently she knows he's sending her money\u2014he sends her what's left of the money he earns there, after the scholarship\u2014but she never told me that part either.\"\n\n\"Wow. Are you glad you know?\"\n\n\"He picked this camp because it was close to him.\" She smiled. \"So, yeah, I'm glad to know. And I'm glad you were here.\" She leaned her head on my shoulder, and I looked around for Dan, Jason, Kyle, someone who might yank me away and beat me with cat-o'-nine-tails like a Roman soldier.\n\nWhen no one came for me, I leaned my head down. Her hair smelled of the sun and something f loral.\n\n\"There's a billboard on madman road,\" I said, my stomach a heavy nervous thing.\n\nShe kept her eyes closed when she replied, \"Read it to me.\"\n\n\"It says, 'I really want to kiss you.'\" It wasn't outright kissing her, like Jack wanted me to. But I was living a little bigger, like my sister had encouraged. I was doing something. Something that I wasn't going to have to carry a cross for.\n\nShe smiled and nodded slightly against my shoulder, under my chin. \"You will.\"\n\nI wanted to kiss her right then, my stomach all lit up with the idea. \"So you think kitten road intersects madman road?\"\n\n\"If it doesn't, I'll just have to take the exit and meet you under that billboard.\"\n\nI noticed Matthew and Kim heading toward us, and Natalie must've too, because she lifted her head off my shoulder. I immediately missed her. Then she stood up and walked toward the water, meeting Kim halfway. They both started laughing, and Matthew threw his arms up, fed up with something.\n\nThen he shouted over to me, \"Get in the boat with us.\"\n\nI shook my head.\n\n\"Come walk on water,\" he insisted. \"It's fun.\"\n\nSo I did.\n\nPRAYERS AND CONFESSIONS\n\nJason: Holy Father, it's because of You that I have the patience and strength to be a rock for these kids. And because of my minor in geology. Double ha! Praise Jesus!\n\nAuthor of Sixty Scenes of Sexy: I am aware of the disdain readers feel for me and my success. It's not like I won the Nobel Prize for Literature, though, so could we all calm down? The entertainment industry is burgeoning with art and fun alike.\n\nCharles: I want to be a cheerleader in college like Jason. I have a lot of respect for him as a man of God. And, I admit, I like the feeling of polyester on my legs.\n\nPastor Kyle: I believe love and service are modes of worship. But lettering in volunteerism? As in a letterman's jacket? Varsity Volunteer?... Huh... That actually has a nice ring to it. T-shirt idea? Never too soon to start thinking about next year.\n\nNinja: Is it me, or are the kids just getting fatter and fatter? I cannot do this lottery shit one more year. And I sure as hell am not carrying some asshole through the desert ever again.\nSATURDAY\n\nThe day after I returned from camp, I walked out to the car, the green Volvo. I stood some feet away from it, analyzing the rusting bumper, the dull paint.\n\nI walked to the other side of the car, slipped my hands into the handle of the passenger side. When I opened the door, the must of the thing leaped out of the car.\n\nJack sat in the back, leaned against the side, his feet propped on the window. He had a bottle of something between his legs, but he'd passed out. He looked older than usual. Unshaven, the wrinkles above his eyes deeper. He blinked at me, then rolled to a sitting position and crawled out of the car, leaving the bottle behind.\n\nHe had a notepad and a pencil in the pocket of his button-down.\n\n\"The passenger side, eh?\" He looked me up and down. \"Lettin' go. Just remembering now. Not searching.\" He nodded. \"That's good. That's really good.\"\n\n\"Yeah.\" I looked over at his profile, strong jaw, clear eyes. \"I read your book.\"\n\n\"I'm sorry,\" he said.\n\nI chuckled. \"It's actually good. I mean, it is a bit rambly, true. But... I liked it.\"\n\nHe sighed.\n\n\"I don't understand why you have such a problem with it.\"\n\n\"The book.\" Jack contemplated. \"It's about awe. It's about looking out at the ever-unpredictable mountains and vast plains and mad people and unbelievable stories.\"\n\n\"Yeah, exactly, I liked that.\"\n\n\"You know how I keep telling you, there is no right or wrong, there's just the after?\"\n\n\"Uh-huh.\"\n\n\"It's not the book. It's the after of that book. I couldn't handle the after.\"\n\n\"But you'd made it. You were famous. After so many years of rejection.\"\n\n\"That's right. And that's good, isn't it?\"\n\n\"Well, yes, that's good. You finally wrote the Great American Novel. I mean, you've got cult fans.\"\n\nJack stared out the windshield. \"And after that book, I drank and I rambled and I fell down in alleys and shit myself and challenged my friends to suicide plots.\" He exhaled, his lips a thin line. \"That's not an after one aspires to.\"\n\nHe reached over and rested his hand on my shoulder. It was the first time he had touched me, and it was as real as anything.\n\nHe looked at me a long time.\n\nI reached up and touched his hand, expecting liquid-chill. Or nothing. But there was something. Something warm and real.\n\nHe nodded. \"You're good to go, kiddo.\"\n\nI was surprised by the reality of his skin, by the tears that sprang to my eyes. \"You can't just stop haunting me.\"\n\n\"I can.\" He opened the car door and stepped out.\n\nI got out too, and walked around to head him off. \"But\u2014I'm troubled\u2014or whatever you said. I'm smug and judgmental.\"\n\n\"You're fine,\" he said.\n\n\"No, I'm not. I'm doomed to a publishing house someday, you said it yourself.\"\n\nHe chuckled. \"The Great Editor lets the truth bleed through that masquerade of words.\"\n\n\"I don't want you to go.\"\n\n\"I'm just a speck, Nick. I'm not the real thing.\"\n\nI wiped the tears off my cheek.\n\n\"I'm going to walk that way.\" Jack pointed past the house, toward the trees. \"And you can watch me till I disappear. But I am going to ask you to stop watching the horizon when you can't see me anymore.\"\n\nThen he stepped forward and hugged me, which was not at all like the Jack I'd known. \"Don't worry,\" he said. His body was real and warm, the fabric of his shirt rough on my chin. I just breathed in the smell of it, which wasn't vanilla or rose. It was just her. It was Diana. \"You're fine.\"\n\nAnd then he let go, patted my shoulder, and walked away.\n\n\"'Lean forward to the next crazy venture beneath the skies,'\" was the last thing he said to me. And it was the only time he'd ever quoted himself.\n\nHe never turned around. He never waved. And his khakis disappeared into his button-down and were soon a shadow and a f leck that bounced and then steadied and then vanished. And I knew I couldn't see him. He wasn't there. It was just me, here, by the side of the Volvo.\n\nI went around to the driver's side of the car, slipped into the seat, and looked up at the ceiling, at the holes where the rain seeped in. I put my fingers in those holes, those two holes where the bullets had pierced the roof of the car. From the gun I put in her hand. From the bullets I put in her hand.\n\nI pulled my fingers away and looked up through the holes into the sky, dark and black. And small. Something so massive, made so small by two tiny holes in the roof of a car.\n\nThe stars were there. The moon was there. But I'd been trying to see the whole sky, the universe\u2014the grand, inexplicable, magic universe\u2014through my damage. Through tiny, man-made holes.\n\nI stepped out of the car, looked up at the billions of bright stars that lit and dimmed in the summer night. That moon, giant and whole and golden, a sentinel in the twilight.\n\nAll the magic in the universe. Planets and beauty, comets and humanity, black holes and love.\n\nThe finite and the infinite.\n\nIt was all a divine mystery.\nAuthor's Note\n\nThis book was inspired by my Aunt Dea, who died by suicide when I was fourteen. She never knew how \"big\" she made me. Even in her death, she continues to inf luence me.\n\nTragically, while I was editing this book, a young man very dear to my heart, Brooks Rolfness, also died by suicide. The loss shattered our very close-knit community. He is deeply loved and desperately missed.\n\nAny one of us who knew these two precious people would have done everything in our power to help them and get them the support they needed. If you are struggling emotionally, reach out to the people around you, be it family, friends, or strangers. Please reach out so that we can reach back.\n\nIf you or a friend are in crisis, please call the National Suicide Prevention Lifeline at 1-800-273-8255. The National Suicide Prevention Lifeline is a national network of crisis centers around the United States that provides free and confidential emotional support to people in suicidal crisis or emotional distress twenty-four hours a day, seven days a week. For more information, visit SuicidePreventionLifeline.org.\nAcknowledgments\n\nI am just so grateful. I will never be able to convey all my thanks, but here are a few thank-yous...\n\nThe best writing advice I ever got was from my friend Mark Teppo. He said, \"Sit your ass down in a chair and write.\" He's an author and a publisher, so I believed him. And, lookie here! A book.\n\nThe best publishing advice I ever got was from my mom. She said, \"Don't give up after a hundred rejections. Not even two hundred.\" She is my mother, so I believed her. Her love of reading (she always had a book in one hand, and a cigarette in the other), and my dad's love of crossword puzzles, meant I grew up surrounded by beautiful words. It is my deepest heartbreak that my mom will not read this book. She is why I read, and she is why I write.\n\nI am the luckiest person in the world to be surrounded by people who have loved me and supported me, whether it's with great advice or the space to write or the willing ears to listen to my rants. My family, my friends, and my community make a tribe of creativity cheerleaders. Andrew Sage built me an office and shooed the kids out of it so that I could write. He provided a special place and priceless time. Christy Johnson, Michele Goode, Melissa Murschall, Sarah Olson, and Aaron Richards listened, read, and answered my questions. More importantly, they asked me the questions. My aunts, uncles, cousins, grandparents, in-laws, nephew, niece, mom, dad, and brother believe in me, no matter what crazy idea I have, including writing a book. And all my students\u2014past and present\u2014offer me the joy and inspiration to tell stories.\n\nA huge thanks to Karen Johnson, Jessica Lewis, Leif Peterson, and Lindsey Keaton for reading the first draft and saying yes to my ideas, then feeding me their better ones. Thanks to Amber Walker for fostering the love of literature among my students. Her book club members became my beta readers, and nothing motivates me more than a group of teenage girls (and food!). Thank you to the Book Worm girls: Analise Walker, Abi Baker, Liz Racine, Carleigh DeLapp, Jessica Reitan, Salena Scoccolo, and Alison Barry. And thank you to the Yak, Snack & Read girls: Sophie Walker, Haley Yandt, Makena Miles, Elly Mark, Mya Wagner, Hailey Wagner, Samantha Lucier, Olivia Levchak, Petria Russell, Tara Hale, and Gaeby Wilson. Thank you, Knowledge Bowl, especially Emily Paris, for asking and asking about covers and titles and how it was going. All of you are the best kind of readers.\n\nKatie Reed and Elizabeth Staple held my hand when I took my first steps in this publishing world. Melanie Jacobsen, Nikki Urang, Kimberly Derting, and Patrick Swenson gave great advice at every turn.\n\nSky Pony Press is full of rock stars! Rachel Stark is a lifesaver and a world changer, and I am #blessed to have her as my editor. My life had been in turmoil throughout 2016, but on Election Day of that year, she gave me hope. She offered to take my weird little book and put it out into the world. She has worked tirelessly to make this story the best book it can be. And then she gave me the greatest team of creatives. Emma Dubin believed in this book, and she has been its cheerleader and my wise counselor. Kate Gartner, Pete Ryan, and Joshua Barnaby designed a book that\u2014inside and out\u2014is more perfect and clever than I ever could have imagined. Seriously. Thank you, Sky Pony Press.\n\nNatalie Lakosil took a chance on this absurd tale, and she took a chance on me. She is a supreme navigator, and I would be lost and stepping all over my own feet without her. Thank you, Bradford Literary Agency.\n\nAnd best for last... McKay and Cole, you are the reason I do any of it. My favorite stories have all been for you, starting with the nightly installments of \"Dream Boy\"! All I ever want is to bring you as much joy and delight as you bring me. And maybe not embarrass you too much along the way. I hope you always have a lot of wisdom and a little madness.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nBegin Reading\n\nTable of Contents\n\nA Preview of _Titans_\n\nNewsletters\n\nCopyright Page\n\nHachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.\n\nThe scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.\nTo Arthur Richard the Third\n\nin whom I have two kings\n\n# A Letter to My Friends, Fans, Readers of My Later-in-Life Novels, and Newcomers to the Books of Leila Meacham\n\nDear Ones,\n\nI'm writing to share with you the history behind _Ryan's Hand_ as my first, long-ago attempt to put on paper a story of fiction and to say (caution?) that you will find this book\u2014an out-and-out romance\u2014a departure from the more recent historical sagas of _Roses_ , _Tumbleweeds_ , _Somerset_ , and _Titans_. That _Ryan's Hand_ and the other two romances subsequently published in the mideighties have been resurrected to a second life has both thrilled and somewhat concerned me. I am thrilled at the interest of my readership in the books that spurred their republication and concerned that the books may disappoint my readers' expectations. Therefore, without a whiff of apology, I believe a little explanation is in order.\n\nIn 1982 I became aware of the Harlequin and Silhouette tidal wave sweeping the country, to name a couple of romance publishers in the forefront of the genre. The eye-opener happened in my classroom at a local junior high school where I was a teacher of ninth-grade English. I began to notice that many of the girls hurried to their seats to open up small, white-jacketed books to read before the bell rang. Curiosity\u2014and the thrill of seeing my students reading something other than surreptitiously passed notes\u2014drew me to their desks to see what had claimed their riveted attention. The handover was usually accompanied by a blush. \"Ah, Mrs. Meacham, I don't know that this is the type of thing you read,\" my students would say, or something on that order.\n\nTo which I'd reply, \"As long as you're reading, I don't care what the subject matter is.\" Well, of course I did, but the books\u2014romances, they were called\u2014seemed harmless enough.\n\nSo I decided to read several for myself, an experience that led me to express to a colleague that while I understood the allure of the books to teenage girls and the women who flocked to the well-stocked shelves in bookstores to buy them, I found the usual dissension between the main male and female characters implausible. I distinctly remember saying, \"Why, their silly conflict could be settled over a cup of coffee at Denny's.\" To which she seriously replied, \"Well, then, why don't you write one yourself and show 'em how it should be done?\"\n\n\"Oh, I couldn't,\" I said. \"I don't know the first thing about how to write a book, romance or otherwise.\"\n\n\"I bet you can,\" she said, \"and I'm willing to put money on it. If you try and can't, I'll buy you a steak dinner at the San Francisco Steak House, and if you try and can, you'll buy me one.\"\n\nWell, to my utter shock, considering I didn't even know how to chapter a novel, I lost the bet. That summer during school vacation, confident that my friend would be picking up the tab at the San Francisco Steak House, I sat down at my old Smith-Corona electric typewriter ('twas the predawn of the PC, and I wasn't awake yet) to give the romance genre a whirl, if for no other reason than to appreciate the writers of the category who tried and succeeded. So _Ryan's Hand_ was conceived. I was determined that the enmity between hero and heroine (who, of course, as the formula dictated, were really secretly hot for each other) was well deserved, and the story evolved from there. Novice that I was, I broke the cardinal rule of fiction: I wrote about what I _didn't_ know. I never knew anybody the likes of Jeth and Cara, had never been to Boston, sat on a horse, twirled a rope, or been on a cattle roundup. I did, however, know about West Texas, land of sandstorms, blistering heat, pumping jacks, and fabulous people. A perfect setting where I could allow my imagination to run wild, I thought, as long as it did not stray too far from what I could see were the established guidelines of the romance.\n\nWhen the book was completed, the same colleague suggested I send it to a local literary agent, and before I knew it, six weeks later, _Ryan's Hand_ was acquired for publication. As a result, I was put under contract for two more romances, _Crowning Design_ and _Aly's House_ , which were published in the following two years. But I'd had enough of writing and publishing. I did not care for the solitary life and isolation of a writer, and I found the experience of meeting deadlines unnerving. I returned to the classroom, and my books went the dusty way of many an unknown author's first literary efforts, without my ever learning whether I had showed anyone \"how it should be done\" or not.\n\nBut that was then, and this is now, thirty-two years later. In the sweeter light of now, I elected to let the books stand as they were then, warts and all. I ask only that you read with the understanding that at the time of their creation, I did not know what in the world I was doing. Please treat kindly and do be well.\n\nLeila Meacham\nChapter One\n\nIn a lounge at Boston's Logan International Airport, Cara Martin waited for the newly arrived 727 to release its passengers. The sun streamed through the wide expanse of glass facing the runway, but it was a cold February day and Cara pulled her old wool coat closer as she waited eagerly for the ramp door to open. Ryan would be among the first to disembark, she knew, for he always traveled first class.\n\nShe hoped he'd been able to get some rest on this trip to Texas to visit his brother. Each evening the weather news had said that temperatures had been mild there, and maybe Ryan had had an opportunity to bask in the sun on that remote ranch of theirs. He'd looked so tired when he left Boston ten days before. He lived too hard, Cara thought. She was convinced the only times he ever relaxed from the fast pace he set for himself were these quiet Sundays they shared together.\n\nAn attendant arrived to open the door to the ramp. Cara, a thrill of excitement lighting her too-serious, violet-flecked eyes, moved to the mouth of the roped-off area that funneled passengers into the concourse, afraid that in the milling crowd Ryan might not see her. She was easy to overlook. Standing less than medium height, she was a slightly built, rather severe young woman whose faded jeans and outdated brown coat gave her a faint air of poverty. She wore no makeup to enliven the classic features of her winter-pale face, and the rich bounty of her golden hair had been sternly drawn into a ponytail at the base of her slender neck.\n\nSuddenly the reserved features exploded with a smile as a tall, dark-blond young man, his gait self-assured and buoyant, strode through the ramp door. \"Ryan!\" Cara called out happily and waved to catch his eye. The smile had brought an astonishing beauty to her grave face, lighting up the violet-blue eyes, drawing the primly set lips back from her small, perfectly aligned teeth. Ryan saw her and responded with a one-sided grin, which gave his urbane good looks a boyish appeal, and lifted a slim briefcase in greeting.\n\nCara moved back away from the crowd so she and Ryan might have a private reunion, her joy still lighting her face. He had gotten some sun, she saw, but a pallor remained beneath the light tan. There was still that hollow look about his eyes.\n\nShe looked up at him with sudden shyness when he reached her. It was still a marvel to her that this handsome, popular, immensely wealthy, transplanted Texan had become the closest friend she had ever had. \"Hi,\" she said. \"Good trip?\"\n\nRyan gazed down at her affectionately, one hand holding a costly leather suit bag over a well-tailored shoulder. \"One to last forever. Miss me?\"\n\n\"Bunches and bunches. How did you find your brother?\"\n\n\"Impressive as usual. He looked great.\"\n\nCara's expression sobered slightly. \"Did...he think the same about you?\"\n\n\"He thought I looked a little green around the gills. Too much salt air, he said. I assured him it was nothing of the kind. Just overwork. Did you bring my car?\"\n\n\"I did, but very reluctantly. The snooty beast doesn't like me, Ryan. I think it knows I drive a Volkswagen.\"\n\nRyan chuckled and let her take his briefcase. Holding hands, they strolled off down the concourse, too engrossed in conversation to be aware of the curious contrast they made.\n\nOnce outside in the unseasonably bright sunlight, Cara looked up at Ryan's drawn face with concern. \"I had to park rather far away. If you want to wait here, I'll bring the car around.\"\n\n\"Don't be silly. My legs need a good stretch. Tell me what you have planned for us on this beautiful Sunday.\" He began the fast clip to which Cara had grown accustomed in the year she had known him.\n\n\"I thought we might go down to Devereux Beach and see if we can find some oak,\" Cara replied. \"We had a nor'easter night before last, and I'm sure if any lobster traps were left in the water, they're driftwood now. We should be able to replenish our store if no one else has beaten us to them.\"\n\n\"You're quite a scavenger, you know that?\" Ryan looked down at her indulgently.\n\n\"Part of my seafaring heritage,\" Cara answered, unperturbed. \"Something you landlubbers wouldn't know anything about. I'll bet you don't have our kind of oak in Texas to burn in your fireplaces.\" She was referring to the oak sections of destroyed lobster traps that Massachusetts residents were fond of collecting from the beaches after a storm to add to their firewood. The salt lodged in the wood produced flames of brilliant hues.\n\n\"You'd get no bet from me on that, Puritan. Scrub oak is what you'd burn in Texas,\" Ryan commented.\n\n\"Not me.\" Cara shook her golden head. \"I can't imagine myself ever being in Texas.\"\n\nRyan did not reply to that, and Cara began a rundown of what had transpired in Boston during his absence. Her summary included amusing gossip about the society lions with whom Ryan hobnobbed as well as some stories from the subsidiary of the Boston City Library where she was a reference librarian. She kept up a steady stream of chatter to spare Ryan the effort of talking, certain that he had come down with a severe virus, and fight it though he might, he was losing the battle. It was only as they arrived at Ryan's car, a sleek, red Ferrari, that she realized he had not been listening to a word she'd said.\n\n\"Ryan?\" Cara spoke worriedly and touched his arm. \"I don't want to harp, but I can tell that you don't feel well. Why don't we forget about going to the beach today? I can leave you the picnic lunch, or make you some soup and then pop on off to my apartment so you can get some rest.\"\n\nRyan looked down at the upturned face, caught once again by the sheer innocent beauty of it, which had always caused a throb deep in his heart. He longed to hold her, to pull her to him and crush her against him until the pain inside his body subsided once and for all. Now that, he thought wryly, would be the way to go.\n\n\"Not a chance, Puritan,\" he said easily and took the keys Cara handed him. \"I'll get all the rest I need soon. I just want to go to the place and change; then we'll hit that beach.\"\n\nRyan's \"place\" was an elegantly masculine town house overlooking Marblehead Harbor. While he changed, Cara stood in the sparkling sunshine on his balcony and looked beyond the harbor toward the Atlantic, which her forebears had sailed. The ocean was calm today. Seagulls cried and soared against a bright blue sky, and waves played gently among the marble-like rocks that had given the harbor its name. It was one of those unexpectedly beautiful days rare to Boston in winter, and Cara was annoyed at the feeling of uneasiness that prevented her enjoyment of it.\n\nShe turned restlessly from the railing and walked through French doors into Ryan's well-appointed living room. Her eyes fell on a silver-framed, enlarged photograph of Ryan and his only brother, Jeth, whom she had never met. The young Bostonian picked up the photograph and studied the faces of the two brothers. Both were wearing Western cords and shirts, and had their backs against a corral fence, Jeth with an arm propped in casual affection on the shoulder of his brother. Their kinship was hard to discern. Ryan, elegant and slim and blue eyed, was an urbane contrast to his taller, more rugged brother. Ryan was handsomer than Jeth, but the older brother with his dark hair and eyes, his strong, masculine features only slightly relieved by a smile for the camera, dominated the photograph. Cara would never tell Ryan that she didn't think she would like Jeth. The older brother looked the epitome of what he was: a feudal rancher living in the remote reaches of West Texas, where he was a law unto himself. He would be a man whose heart would be hard to penetrate, the kind of man who would have held her family in contempt.\n\nCara replaced the photograph in its position of prominence on the mantel, aware that Ryan was standing in the doorway of the living room watching her. She turned with a smile and surveyed him. \"You look better, but are you sure you want to tackle the beach? Even on a day as fine as this one, it's bound to be blustery.\"\n\nBarefoot, Ryan came into the room carrying thick socks and fleece-lined boots. He had changed into a heavy sweater and down-filled pants. With a start, Cara noticed that he had lost quite a bit of weight in the last few weeks. \"Did you fellows eat anything this last week,\" she asked, \"or did you simply drink your meals?\"\n\nRyan chuckled as he sat down in a leather chair near her and began pulling on the socks and boots. \"What foolish notions you have about us bachelors. Why do you ask?\"\n\n\"Because you've lost so much weight, Ryan.\" Cara looked down at the bent head, at the slim fingers busy with laces, and a wave of affection rushed through her. She knelt down in front of him and covered his hands with hers. \"Ryan, what's wrong? Is your brother all right? Is everything okay at the ranch? You didn't quarrel, did you?\"\n\n\"Whoa, Puritan!\" Ryan laughed into the bright eyes that at times seemed to have stars behind them. \"Your concern is sweet, but it's unfounded\u2014\"\n\n\"No, it isn't, Ryan!\" Cara argued. Earnestly, her hands went to his shoulders and found them startlingly thinner than she remembered. \"You don't look well, my friend. You obviously haven't been eating or getting enough rest. You've been such a compassionate ear for me this year, so please let me do the same for you. Tell me what's wrong.\"\n\nRyan lay back in his chair with a sigh and studied her from under his thick, sandy lashes. \"Okay,\" he said cryptically, \"I'll come clean if you'll do one thing for me.\"\n\n\"What's that?\"\n\n\"Take off that monk's cassock.\"\n\nCara looked down in surprise at her coat, then back at Ryan with a grin. \"Am I to assume that you do not appreciate old faithful here? I'd like to remind you that this is the only coat still around from my more affluent days.\"\n\n\"Fidelity has not improved its appearance, love.\"\n\n\"Oh, all right!\" She shrugged out of the coat, letting it fall behind her, and settled on the floor at Ryan's feet. Her over-large thick gray sweater hung from the shoulders but roundly molded the high fullness of her breasts. \"Now shoot,\" she ordered, giving Ryan her attention, unaware of the flicker that had appeared briefly in his blue eyes.\n\nRyan rested his head on the high back of the chair and pondered the ceiling for a few seconds before speaking. Then he admitted, \"I have been ill lately. I've been battling a stomach virus for nearly a month. That's why I've lost weight and look a bit drawn. It's nothing serious, but it is debilitating.\"\n\n\"Have you been to a doctor?\"\n\n\"Yes,\" Ryan said shortly, his tone implying that the subject of his health was closed. \"Also\"\u2014he turned his head to the photograph on the mantel\u2014\"I can't help but worry about that rawhide-tough brother of mine. Jeth seems so alone to me. The big house is as empty and quiet as a tomb. With all that gray tile Jeth chose for the floors, it looks like one, too. Our mother never had a chance to decorate the house. Only the construction was finished when she and Dad were killed in that plane crash. I'd like to see Jeth marry a wonderful woman who will give him children to make that house a home.\"\n\n\"Then why doesn't he?\" Cara asked. \"He certainly can't be without choices.\"\n\n\"He has those, for sure. But Jeth thinks the kind of woman he'd want to share his life with doesn't exist.\"\n\n\"Perfection rarely does,\" Cara commented dryly, then felt herself flush as Ryan lifted his head to look down at her quizzically. \"I'm sorry. I shouldn't have said that. I've no business passing judgment on your brother.\" Cara slipped her hand into his. \"Please go on, Ryan.\"\n\n\"The kind of woman Jeth needs would have to be special, not perfect,\" Ryan explained patiently. \"She'd have to be the kind of woman who could love the land like he does, accept its demands and flaws\u2014the heat and wind and sandstorms and isolation. She'd have to respect its people too, all the different breeds and cultures of them. And she'd have to love Jeth, really love him, not his fortune or his power or his empire, but the man who lives behind that rock-hard exterior of his that's gotten thicker through the years.\"\n\n\"That's quite an order,\" Cara said softly. \"Ryan, have you never thought of marrying and having those children to make that house a home? I'd hate to lose the best friend I'll ever have\"\u2014she smiled wistfully\u2014\"but still that's a possibility. The ranch belongs equally to both of you, doesn't it?\"\n\nRyan shook his head. \"On paper, yes, but the land, the house, the oil, the cattle\u2014they really belong to Jeth. He has a philosophy that land, like horses\u2014anything wild held captive by man\u2014must be cared for or he has to let it go. I remember the day he demonstrated that to me. I was eighteen that summer. I'd been so busy studying for the entrance exams for Harvard I'd completely withdrawn from the ranch's operation. One day Jeth asked me to go out to the stables with him. Texas Star, my horse, was snorting around the corral, and I realized then what was on Jeth's mind. He'd found out that one of the hands had been taking care of him. At La Tierra, a man takes care of his own horse. He asked me how long it had been since I'd ridden Texas, and I had to confess I couldn't remember. I could see that he was reverting to the wild stallion he'd been when I captured him for the roundup...\"\n\n\"So what happened?\" Cara urged as Ryan paused.\n\n\"Jeth walked over to the corral gate and opened it. Then he slapped Texas's flank, and my horse took off for the mountains where I'd found him.\"\n\n\"But that was cruel!\" she gasped.\n\n\"It was _kind_ , Cara! Don't you see? Texas could never belong to anyone else, and he didn't belong to me anymore, so he had to be set free. I can still see that horse. He was a three-year-old palomino and at the base of his mane was a perfectly formed white star. He raced off for a distance, then he stopped and looked back. I started to go after him, but Jeth stopped me by saying something I've never forgotten...\"\n\nWide-eyed, Cara prompted on held breath, \"What was it?\"\n\n\"He said that I should never tame anything I wasn't prepared to love.\"\n\nRyan's eyes closed tiredly, and there was silence in the room except for the faraway cries of the seagulls and the opulent sound of a fine antique clock ticking on the mantel. The sunlight filtering through the French doors did little to dispel the sudden chill that had crept into the room.\n\n\"I understood what he was telling me.\" Ryan spoke again, laying his head back. \"The name of our ranch, you know, is La Tierra Conquistada. Translated, that means The Conquered Land. I suppose that after four generations, the land can be considered conquered, but it wouldn't stay that way long without care and dedication. Jeth was saying I had to let it go. I couldn't divide my loyalties.\"\n\n\"And so you left to study law,\" Cara stated quietly.\n\n\"Yes. I went away to Harvard that fall, and I never went back to stay. We both knew that I was never cut out to be a rancher. Jeth was. I had the blood of my music-loving, aesthetic mother in me; Jeth had our father's. My brother has always insisted that the ranch was as much mine as his, but also that it couldn't be run from a distance. He's made all the decisions concerning La Tierra, scrupulously dividing the profits. He's made me a very rich man.\"\n\n\"You could have done that for yourself even without your brother,\" Cara said warmly. \"You're a brilliant lawyer!\"\n\nRyan opened his eyes to stare down at her in amusement. \"What is there about Jeth that nettles you so, Puritan?\"\n\nCara reddened in embarrassment and withdrew her hand. \"Forgive me again, Ryan. It's just that your brother sounds so...so high-handed. I've always been a little prickly about arrogance of that kind.\"\n\n\"Arrogance is often the unavoidable twin of power, Cara, and Jeth never had a chance to be anything but what he became: a very powerful man. When our parents were killed, I was eleven and Jeth eighteen. He had a very different kind of dream then; he wanted to be an Olympic swimmer. That ended when he had to take over the reins of La Tierra. There was no one to help him. The very men he thought he could trust\u2014lawyers, bankers, other ranchers\u2014proved to be the most unscrupulous. They saw a chance to get their hands on La Tierra and they tried every conceivable chicanery to do it. But they didn't figure on Jeth's brains and guts. He proved too smart for them and too tough. He wasn't a compassionate winner, either. Every one of those men lived to regret the day he ever crossed Jeth Langston.\" Ryan paused to give Cara his crooked grin. \"I guess I do make the guy sound high-handed and hard-nosed, don't I?\"\n\n\"I think you're worrying about him unnecessarily,\" she said. \"He sounds like a completely self-sufficient man who will marry when he feels ready to. Besides, he's young yet, only thirty-four. I'm surprised he doesn't have to fight the women off. He has a...certain virility that's very attractive to most women.\"\n\n\"But not to you?\" Ryan queried, and laughed when her eyes dropped in discomfort. \"Don't be embarrassed, Puritan. You're right in thinking that the two of you would lock horns\u2014at least at first.\" He sat up suddenly and shook his head as if tired of his thoughts. \"Come on,\" he said, hoisting Cara to her feet. \"Let's go down to the beach before the light begins to fade.\"\n\nDevereux Beach, a thin neck of land separating Marblehead Harbor from the Atlantic Ocean, was a favorite Sunday spot for Cara and Ryan. On the first Sunday after they met, Cara had taken the handsome Texan beachcombing there. It had been a blustery day in February almost a year before. They had arrived to find the beach deserted except for the seabirds that scurried about on the wet sand and cawed their plaintive cries overhead. Cara had explained to a curious Ryan that the salt-logged oak burned in spectacular colors. \"I'll show you this evening,\" she promised.\n\nThey had made quite a haul and finished the day at Cara's modest one-room flat, which was perched atop a three-story house. The room featured a widow's walk, a narrow balcony facing the Atlantic where seamen's wives of old would go to watch for their husbands' ships returning from the sea. While a casserole bubbled deliciously in the oven and she opened a bottle of wine, Ryan had stretched his legs out on the floor before the old stone fireplace, fascinated by the brilliant colors in the leaping flames. \"You've made a believer out of me,\" he remarked as he accepted a goblet of chilled wine.\n\n\"You must take home half of what we collected today,\" Cara said, sipping her wine comfortably beside him. \"These colors will be dramatic in your white marble fireplace.\"\n\n\"Only if you promise to come to my town house next Sunday and share the fire with me. I won't promise to cook, but I know an excellent caterer.\"\n\nCara had been surprised that Ryan would extend an invitation to her so soon. He was in great demand by Boston society hostesses, and she was also aware of his reputation as a ladies' man. She was definitely not in his social circle, nor was she like the glamorous, leisured women he was accustomed to seeing. They had met when Ryan came into the library to research a legal matter.\n\nShe had recognized him immediately as the popular young attorney the society columns linked with the names of some of her former school friends. She supposed it was a form of self-torture, but she could not resist reading the social news that not so long ago had occasionally featured the names of her own family members.\n\nWhen Ryan Langston asked at the reference desk for help in finding a certain volume, Cara had been impressed by his manners, his soft Texas drawl, and his clean-cut, boyish good looks. She went off duty at five o'clock, and, as she came out of the library, the sleek red Ferrari parked next to her secondhand Volkswagen told her that Ryan Langston was still inside working on his research.\n\nCara did not notice that her right front tire was flat until she attempted to drive out of the parking space. There was no mistaking the significance of the peculiar list on the right side, so she reparked to assess the damage.\n\nBy the time she had cut the motor, Ryan was standing beside his Ferrari. \"You have a little problem there, I see,\" he said, indicating the tire. \"Do you have a jack?\"\n\nCara not only did not have a jack, she did not have a spare tire.\n\nIt had been one of those days when everything had conspired to remind her of the losses she had suffered in her short twenty-four years. She longed to put an end to the day, to get to her apartment and build a fire, have a light supper, and maybe play the piano until she was too sleepy to lie awake with her memories.\n\nNow, sitting behind the wheel of her shabby car, hearing the voice of a man who had easy access to the world that had turned its back on her, she felt the sudden horrifying urge to burst into tears. She controlled her emotions by rigidly gripping the wheel and staring straight ahead, but the handsome dark-blond head of the man had bent down to peer at her through a closed window. \"Are you all right?\" he asked, and she could hear the sincere concern in his voice.\n\nShe had swallowed hard and given him a polite smile while praying that she wouldn't cry in front of a stranger, especially not _this_ stranger.\n\n\"I'm fine,\" she assured him, opening her door. The night had folded about the neighborhood very thick and cold, and she drew her brown coat closer. \"I thank you for your concern,\" she said to Ryan, \"but you needn't bother. I'll go back inside and call a garage.\" Not for the world would she have him know that she did not have a spare tire or money to buy one. She would figure out what to do when she got rid of him.\n\n\"That won't be necessary,\" he insisted, looking very affluent in his tailored overcoat. \"I can have the tire changed in a jiffy. If you'll just open the trunk\u2014\"\n\n\"No, please\u2014\" she protested, raising delicate hands in a gesture of panic.\n\n\"Look, young lady,\" Ryan said, brushing aside her protests. \"I'm not about to let you wait alone in this parking lot when I can change that tire for you in a few minutes!\"\n\nThere was nothing to do but yield as gracefully as possible. \"Well, but you see, I...don't have a spare tire\u2014\" She could feel the heat flooding her face.\n\n\"I see...\" He spoke softly, and she could tell from the almost imperceptible flick of the blue eyes over her worn coat that he understood the nature of her embarrassment. \"Well, in that case, you must let me take you home.\"\n\n\"Oh, no, I couldn't!\"\n\n\"My name is Ryan Langston,\" the tall young man said calmly. \"I am an attorney practicing here in Boston, and I assure you you're far safer with me than waiting for a bus or a taxi on a street corner.\" He reached inside a breast pocket for a narrow, tan wallet. Cara saw some kind of gold insignia discreetly embossed on one corner. He extracted a card and handed it to her. \"I think there's enough light for you to read that I am who I say I am. You must let me take you home.\"\n\nCara glanced at the card. It wasn't necessary since she knew who he was. \"Yes, I see that. You are kind to trouble with me. I'll get my bag.\"\n\nThe next day Cara had shared a ride to the library with a colleague. As she arrived at the parking lot, her mouth had dropped open when she saw that the flat tire on her Volkswagen had been replaced with a new one. Opening the trunk, she found that the old tire had been mended and beside it lay a new jack.\n\n\"Mr. Langston, you were kind to bother with my car,\" Cara said when she got a line through to him at the law firm. \"I will mail you a check first thing in the morning for the items you purchased.\" She spoke confidently, thinking of several pieces of sterling still to sell.\n\nRyan did not reply immediately. Presently he asked, \"Miss Martin, you're a native Bostonian, aren't you?\"\n\n\"Yes.\"\n\n\"Would you be willing to trade your time as a tour guide for the money you feel you owe me?\"\n\n\"I beg your pardon, Mr. Langston?\"\n\n\"I would like to see Boston through the eyes of a proper Bostonian, Miss Martin.\" Cara wondered if he were laughing at her. Proper Bostonian, indeed! But he continued, \"I've been here a number of years now, but I've yet to see the city the way I want to see it. From the short time I spoke with you last night, I could tell that you have a thorough knowledge of the nooks and crannies of the area, the kind of thing you don't read about in tour books.\"\n\nCara hesitated. He wanted to see her. The tour-guide business was a line, fed to her with subtle good humor, if she had been any judge of the young Texan. And suddenly she wanted to see him. It had been so long since she had enjoyed the company of someone like Ryan Langston.\n\n\"It seems to me, Mr. Langston,\" Cara replied, \"that you are making a poor trade. However, I would enjoy showing Boston to you, and I dislike debts. When shall we begin?\"\n\nHe had surprised her by saying, \"This evening, Miss Martin, if you have no objection. I'll pick you up at your apartment at six o'clock.\"\n\nRyan had allowed her a brief moment to refuse, then wished her good morning and hung up. That evening at six, the Ferrari swung into the drive before her private entrance. Still in business suit and overcoat, Ryan looked at her casual slacks and sweater and remarked that he'd like to go home first so that he might change. \"Tell me where we're going,\" he said, \"so that I can dress for it.\"\n\nThey were going for an evening of chowder and cards with an old sea captain friend of her late grandfather's. \"He's a widower,\" Cara enlightened Ryan in the living room of his town house. \"You'll like him. He can remember when the people of Boston depended upon the sea for a living.\" While Ryan dressed, she wandered admiringly around the beautiful room, resisting the urge to try the baby grand piano that filled a corner of it.\n\nAt the end of the evening when Ryan brought her to her door, they shook hands and agreed to meet again on Sunday. This time, Cara told him, he must allow her to prepare a meal for him as a small token of gratitude for the car.\n\nAfter their next meeting they fell into the habit of seeing each other regularly on Sundays, and a surprising relationship evolved that satisfied them both. Ryan referred to her as his \"Sunday girl\" and called her \"Puritan\" after her New England ancestors, or so he said. But Cara suspected that in his experienced, man-of-the-world way, Ryan knew that she was still a virgin. She was relieved that he preferred and needed her only as a friend. She could not become involved with anyone until her family's debts were paid. There were other women in his life, she knew\u2014beautiful women he squired around to the expensive, public haunts he never suggested taking her. Cara feared he might be ashamed of her. Her clothes were abysmally old, her hair unfashionably long. But in time she realized that Ryan would never be ashamed of a friend. For her sake their Sundays were confined to picnics or walks on the beach, to country drives and out-of-the-way, nose-poking places. With typical compassion, Ryan had known that she did not want to be seen by her former crowd, to once again become the subject discussed over teacups or martini glasses.\n\nCara found it a release to be able at last to confide in someone of Ryan's sensitivity the series of events that led to her living a solitary existence in near poverty. As she came to trust Ryan, she revealed that she was a direct descendant of one of the first ship-building magnates to settle in Boston. He had built the fine mansion where she had spent carefree years growing up as an only child. Through succeeding generations, the fortune had dwindled, but there had been enough money to sustain the family in the highest echelons of Boston society, and for Cara's father to pursue his literary career without the need to work for a living. His written histories of the Boston area achieved for him a modest fame but little remuneration. There had been enough money for Cara to attend the Juilliard School of Music in New York with hopes of becoming a concert pianist. And then, in the second year of her studies, her mother's heart stopped one day in the garden as she bent to welcome the first crocus peeping through the snow.\n\nCara flew home often that year, and each visit she found her father looking more feeble, more lost and bewildered. Because he seemed to be rapidly losing his grip on reality, she took it upon herself to sort through the family's financial records, and made a shocking discovery. There was no money and had not been for some time. The mansion had been mortgaged to subsidize her father's latest literary effort and to provide money for her Juilliard education. Insurance premiums had been allowed to lapse. Even her mother's funeral expenses had not been paid.\n\n\"What are we going to do?\" she asked her family's lifelong attorney after he had read the long list of outstanding debts.\n\n\"Declare bankruptcy, of course,\" he advised smoothly. \"You've no other choice.\"\n\nCara had left her father sleeping in a deck chair on the porch that caught the sea breezes from the Atlantic. \"Dad.\" She shook him gently, determined that he would not be evicted from his home. His thin, blue-veined hand fell lifelessly from his lap, and with a cry Cara knew that he had been spared.\n\nAfter seeing her father buried beside her mother, Cara had gone away to analyze the numbing realities of her situation. Her family was gone. The world she had once known was closed to her\u2014she refused to live in it on credit\u2014and a mountain of debts remained to be paid.\n\nShe would pay them. That decision made, Cara withdrew from Juilliard, then went to each creditor and pleaded for time. She could not declare bankruptcy, she explained. She considered it immoral not to pay back what had been loaned her father in good faith. Could they extend her time to earn a degree in library science\u2014at a cheaper, state-supported university\u2014which would then give her greater financial opportunity to settle the debts?\n\nThe creditors, themselves of the same Puritan stock as she, recognized her plea for what it was: the cry of her Yankee pride to clear her family's name and salvage her self-respect. They agreed. Cara sold the house and auctioned its furniture, including the treasured Steinway that had been the joy of her life. Except for a strand of pearls that had been her mother's and a gold chain her parents gave her the day she had been accepted at Juilliard, all of Cara's personal possessions went on the block. By the time Ryan came into her life, she had been out of college three years, subsisting on a shoestring budget that had allowed for only the two luxuries of a small piano for her one-room apartment and a good pair of fleece-lined boots for walking the beaches. She had totally withdrawn from her former world, knowing that she was regarded, if remembered, as an object of pity and condescension.\nChapter Two\n\nOh, look, Ryan!\" Cara exclaimed as they drove into the sandy parking area that led to the beach. \"We have the place to ourselves.\"\n\n\"Let's get cracking, then,\" Ryan said. \"The light will be gone in no time.\"\n\nHis skin still had an unhealthy pallor, and Cara had some misgivings about her suggestion to come here today. She reached for his gloved hand, and his fingers curled tightly around hers. Ryan looked down at her and smiled, and she smiled back. Then they began to walk toward the beach, Cara swinging the canvas bag she always brought along for what Ryan called her loot.\n\nThey unearthed quite a number of broken sections of oak, one piece a foot long. In addition Cara found several pieces of \"bottle glass,\" broken portions of bottles that the salt had imbued with delicate color and whose edges the seas had worn smooth.\n\n\"You must have a bag of that by now,\" Ryan commented when she held up the broken neck of a bottle for him to examine. \"What are you going to do with it all?\"\n\n\"I don't know yet. But I'll think of something special\u2014you'll see.\"\n\n\"Maybe,\" said Ryan unexpectedly, and turned away from her to walk farther up the beach, having spied something that drew his attention.\n\nCara looked after him. A funny chill played round her heart. What did he mean by that? she wondered. Surely he wasn't leaving Boston, going back to Texas after all? He had become such an integral part of her life. She had only realized how much this last week when she had been almost bereaved by his absence. The prospect of Sundays without Ryan would be unbearable. He meant everything to her. Surely he wasn't planning on leaving?\n\n\"What are you staring at?\" Ryan asked as he returned to her. She gazed up at him, her violet eyes catching the glow of the sunset. He was so tall, and even on a day as blustery and cold as this one, he refused to wear a head covering of any kind. The wind lifted his sandy hair and played with it. The dying sun shadowed the gaunt concaves of his cheeks. Her heart contracted with a kind of fear, and she faltered, \"Ryan, I\u2014\"\n\n\"What is it, love?\" His voice was gentle and very kind.\n\n\"Ryan, I am very fond of you.\"\n\nHe drew her into his arms. \"I know that. I am of you, too, Cara.\" He tilted her chin and searched her disturbed eyes with tender amusement. \"You look sad. Don't be.\" But the lips he pressed to her forehead for reassurance were as cold as death.\n\nLater, in the town house, they picnicked on the floor before the fire, and afterward, with their legs stretched out side by side and their backs against the davenport, Ryan spoke of his brother again.\n\n\"When Jeth graduated from high school, my father presented him with a gold wristwatch. On the back of the watch he'd had inscribed, 'To my beloved son in whom I am well pleased.' That watch and inscription meant the world to Jeth. It was Dad's way of telling him that he approved his dream of becoming an Olympic swimmer.\n\n\"The night before I left for Harvard, Jeth and I had a last drink together in the study. I was feeling pretty low. I felt as if I were deserting my heritage, disavowing my Langston blood, not to mention leaving Jeth alone to run the ranch at a time when he really needed me. The only people left that he could count on were Fiona and Leon. They're our housekeeper and ranch cook, who have been at La Tierra since before Jeth was born.\n\n\"Jeth handed me a small wrapped box. 'If Dad were here,' he said, 'he would have given this to you.' I unwrapped it and removed the lid. Inside the box was Jeth's gold watch.\"\n\nCara remained silent, too moved by the simple story of brotherly love to comment. She watched him stroke the face of the gold watch encircling his arm, his expression slack in the firelight.\n\n\"It was a fine moment between us,\" he continued. \"Jeth couldn't have given his blessing more eloquently\u2014or magnanimously. He stood awfully tall to me in that moment.\"\n\nShe said softly, \"You love him very much, don't you?\"\n\n\"Yep,\" Ryan said, rising to his lanky height. \"He's still the tallest man I'll ever know. I'll get us some more wine.\"\n\nHe had been gone for a while before her thoughtful state was penetrated by the sound of retching from the bathroom. She sat up, alarmed. \"Ryan!\" she exclaimed, getting up. \"Ryan!\" She found him doubled over the basin, holding a wet cloth to his mouth. His face was ghostly gray. \"My God, Ryan, what is it?\"\n\n\"It's nothing to worry about,\" Ryan gasped between heaves. \"Just this stomach virus.\"\n\n\"Let me call your doctor!\"\n\n\"No, there's nothing he can do. I'll be all right in a few minutes. Just get me a glass of cold water, will you?\"\n\nShe did as he asked and watched as he took a couple of pills. \"Get ready for bed,\" she ordered. \"Right now.\"\n\nRyan didn't argue. While he was in the bathroom, Cara turned down the covers and switched on the electric blanket. She filled a glass with fresh water and placed it on his nightstand. Ryan came out of the bathroom a few minutes later wearing pajamas. \"These pj's are for your benefit, I want you to know.\"\n\n\"Come on,\" she said, holding the covers for him to slip under. His sick pallor frightened her.\n\n\"You won't get any argument from me.\" He attempted a grin, and Cara tried to smile back.\n\n\"Do I sound bossy?\" she asked.\n\n\"Yes, but I like it,\" Ryan murmured as she pulled the covers up to his chin.\n\n\"Is there anything I can do for you before I leave?\"\n\nRyan's sandy lashes fluttered sleepily. The pills had been sedatives. \"If you'll just open the blinds to let the moonlight in. Thanks for a great day, Puritan.\"\n\n\"Thank _you_ , Ryan.\" She stooped to kiss his forehead. \"Sleep well.\"\n\n\"No fear of that,\" he mumbled drowsily as his face closed in sleep.\n\nCara looked down at the ashen face and wished suddenly, inexplicably, for this man's brother. The strange desire persisted when she went downstairs into the cold February night to her Volkswagen, parked beside the red Ferrari. \"If you two aren't a classic example of Lady and the Tramp!\" she observed in wry amusement.\n\nShe drove out of the parking lot into the evening flow of traffic toward Boston, hardly aware of the other motorists. Images of the day, scraps of remembered conversation floated like random leaves in her mind. The chill around her heart would not go away. The familiar taste of loss was in her mouth. Ryan's response to her query about his trip came back to her. \"One to last forever,\" he had said\u2014almost as if he had no plans to go again. Other phrases floated disjointedly, lazily, teasing at her memory like a haunting concerto whose name eluded her. \"I'll get all the rest I need soon...\" \"Let's go before the light begins to fade...\"\n\nSuddenly her foot slammed down on the brake pedal. The small tires squealed in protest at the suddenness of the illegal U-turn in the middle of the street. Cara barely heard them. The sound of her own cry had filled her ears.\n\nUnder the covered parking ramp the Ferrari seemed to be waiting for her, as if it had expected her return. Cara fumbled in her purse for the key to the town house that Ryan had insisted she have. She let herself in, listening. Then she walked to the door of Ryan's bedroom. A band of moonlight, like a mask, lay across his eyes. They were open and observed her standing in the doorway without surprise or alarm. The rest of his face was in shadow. \"Hello again,\" he said softly, and Cara thought he smiled in the darkness. \"Why did you come back?\"\n\n\"You know why, Ryan.\"\n\n\"And why is that, love?\"\n\n\"I know how ill you are, Ryan. It's terminal, isn't it?\" Her entire being pleaded with him to deny it, but the answering silence confirmed what she dreaded. \"How much more time?\" she asked, but her knees had turned to water.\n\n\"Not much. I'm living on the borrowed end of it.\"\n\nIncredulous, Cara walked to the bed and gaped down at the handsome, gaunt face. \"It can't be,\" she whispered, but the unblinking blue eyes stared the truth back at her, and the bottom fell from her heart. \"Ryan...\" She knelt beside the bed, next to his pillowed head. \"Tell me it isn't true.\"\n\nWhen he did not reply, tears slowly welled in her disbelieving eyes and began to slip unchecked down her cheeks. \"Ah, love...\" Ryan consoled her, drawing the golden head down to the thin hollow of his shoulder. In silence, his cheek against her hair, he held her until the first bitter wave of sobbing had ceased. Then she pulled back to look at him, her breathing erratic in the aftermath of grief. \"I'm\u2014not\u2014leaving you, Ryan!\"\n\n\"That's comforting to know. There's a set of pj's in my armoire, and I think you'll find an unused toothbrush in the medicine chest. I always keep a few on hand for...er...\"\n\n\"I know, Ryan. They must think it awfully thoughtful of you.\" She rose on unsteady legs. \"I won't be a minute.\"\n\nWhen Cara was ready for bed, she came once again to his doorway. \"I don't want to go to the guest room, Ryan.\" She spoke obstinately, like the child she appeared in the baggy blue pajamas, her cheeks red from the salt of sea wind and tears, her long golden hair loose about her shoulders.\n\nRyan studied her without expression for a long moment; then his mouth softened in a slight smile. \"Come here, then, Puritan,\" he said, throwing back the covers on the other side of his bed. Immediately she went and crawled in beside him, snuggling close and wrapping her arms in desperate protection around him, as if to imbue his body with the health that flowed in her own. Cradled together, Ryan drifted into a peaceful sleep but Cara lay awake and vigilant throughout the long night, listening to the distant sound of the Atlantic and the precious beat of his heart.\n\nThe next morning she rose and dressed before Ryan was awake. When he awoke, she had hot tea ready, which she served to him in bed. \"I've called the library to say I won't be in today,\" she informed him. \"Is that all right with you?\"\n\n\"Need you ask?\" He sipped his tea. \"Thanks for helping me make it through the night.\"\n\n\"Like the song says,\" she said simply.\n\n\"Well, not quite, Puritan.\" He laughed when he saw her blush. \"This relationship of ours is really something. Who would ever believe that I slept in the arms of the most beautiful woman in Boston and nothing happened?\"\n\n\"Oh, Ryan!\" Cara fussed. \"Your fondness for me has affected your objectivity. I'm not in the least beautiful.\"\n\n\"You've just forgotten you are. You probably haven't really looked at yourself in years.\"\n\n\"There's no reason to. I can't afford cosmetics and clothes, and my job doesn't require them. Someday I'll be in a position to let my appearance matter again.\"\n\n\"But I want your appearance to matter now, Cara. Do something for me?\"\n\nCara looked at him curiously, realizing he was serious. \"Why, of course, anything.\"\n\n\"I would like for you to buy yourself a new wardrobe. Also a new hairstyle, cosmetics, anything that will show off that beauty of yours to its best advantage.\"\n\nCara tried not to show her shock. She said gently but firmly, \"Ryan, you know I can't do that.\"\n\n\"You said anything for me, remember? Surely you can shelve that Yankee pride of yours to indulge a dying man's request. Besides, it's all been arranged anyway. I knew you would balk at charging anything to me, so I've had money transferred to your account. Please don't refuse what I ask, Cara.\" Ryan's blue eyes were imploring.\n\n\"But, Ryan\u2014\"\n\n\"I took the liberty of making the first purchase for you. Will you look on the top shelf of that closet?\"\n\nReluctantly, apprehensively, Cara went to the closet door and opened it. On the top shelf was a large silver box. Her breath stopped when she hauled it down and recognized on the cover the embossed name of a furrier her mother had once patronized. She threw Ryan an alarmed look. \"What have you done?\"\n\n\"Open it, love.\"\n\nCara slowly removed the ribbon, lifted the lid, and pulled aside the silver tissue paper. Her mouth parted in awe as she drew out a superbly cut raincoat the color of pearls, fully lined in sable. \"Oh, Ryan...it's the most beautiful thing I've ever seen!\"\n\n\"Try it on, love. Let's see if I chose the right size.\"\n\n\"Ryan, I couldn't possibly accept this.\"\n\n\"Of course you can, and you will. I'm a dying man. You must humor me. Now try it on.\"\n\nCara obeyed him, and in spite of the leaden pain within her, the sensuous feel of the coat stirred long-denied memories of the pleasure of lovely clothes. \"It's so light, yet so warm,\" she said in fascination.\n\n\"And far more suitable for you than that monk's robe.\" Ryan held out a hand to her, and she took it and sat beside him on the bed. \"Now about the rest of the things you'll need\u2014\"\n\n\"That I'll need, Ryan?\"\n\n\"When I'm no longer here to see after you,\" he said reasonably. \"I want you to buy clothes for every season. The cruise clothes are out now, so there should be a good choice of summer clothes. Buy lots of those.\"\n\n\"Ryan\u2014\" Cara stopped him determinedly. \"We should concentrate on getting you ready to go home rather than having me gallivanting around buying clothes.\"\n\n\"I'm not going back to Texas,\" Ryan said quietly.\n\nCara stared at him. \"What? You're not planning to stay in Boston, surely. What about Jeth?\" Then a shocking thought struck her. \"He doesn't know, does he, Ryan?\"\n\n\"No. He believed I had a stomach virus and wanted to come back to Boston to see my doctor.\"\n\n\"You can't mean that you would keep this from Jeth! He'll be irreparably hurt, Ryan!\" Cara got up from the bed abruptly and flung off the coat.\n\n\"All this concern for a man you don't even like!\"\n\n\"He's your brother, Ryan! Think how you would feel!\"\n\nRyan's features tightened stubbornly. \"I know what I'm doing, Cara. You have to believe that and trust me. Someday Jeth will understand why I didn't tell him, why I didn't stay at La Tierra. Let's not discuss it anymore, if you don't mind.\" It was the closest they had ever come to a quarrel, and Ryan softened the atmosphere with his engaging smile. \"Now get out of here and go shopping. I want to see a style show this evening.\"\n\nNumbly, Cara spent the day doing as she had been ordered. She went to a dozen dress shops in order to make a dent in the staggering amount of money she had been told to spend, knowing that if she didn't, Ryan would send her out tomorrow on another expedition when all she wanted was to be with him.\n\nBy the end of the afternoon she had completed her purchases and the little Volkswagen\u2014she had refused to drive Ryan's Ferrari\u2014was filled to its bug top with boxes and bags. On her way back to Marblehead, she stopped by the library to speak to the woman who had been her supervisor for three years. The iron-gray head of the librarian nodded in understanding as Cara explained that she had to take emergency leave and didn't know how much time she would need. Should she resign now, Cara wanted to know, or could she take an indefinite leave of absence and return to her job when she was free to do so?\n\n\"We don't have to decide that now,\" the librarian told her. \"Call me at the end of the week when you have a better idea of how much time you'll need. Then we'll discuss your options.\"\n\nDriving to the town house, Cara thought that she had only one option: even if she lost her job and the few remaining debts remained unpaid for a while, she was not going to leave Ryan to die alone.\n\nThat evening Cara turned and pivoted before an admiring Ryan as she modeled the dozens of dresses, separates, and suits she had purchased that day. \"Tomorrow,\" Ryan told her tranquilly, \"you're to have your hair styled. Also, afterward, you have an appointment with Boston's best makeup artist.\"\n\nCara sighed. There was no point in arguing. Ryan was clearly enjoying his benefactor's role, and if it kept his mind occupied, then she would submit to anything.\n\nLater in the evening, Cara prepared a meal from a diet prescribed for Ryan's condition, which she had found tucked away in a kitchen drawer. After dinner, with the brilliant flames throwing their reflections on the white marble fireplace, Cara played Ryan's favorite classical selections while he listened from the leather chair that now seemed to swallow him.\n\nEventually she saw sleep begin to take hold of the handsome features, and, trailing her fingers off the keys, called softly, \"Ryan?\"\n\nHe opened heavy lids, somewhat startled that she had spoken. \"Yes, love?\"\n\n\"Shall I stay again tonight?\"\n\n\"Need you ask? Actually, I was hoping you would move in with me until I have to go to the hospital.\"\n\nWithout hesitation, Cara replied, \"Tomorrow I'll go get a few things from my apartment.\"\n\nIn the week that followed, Ryan grew weaker each day, but still he was quick with a laugh or a joke. The weather still held, and he sent Cara out on another shopping spree. When she returned, she dumped the armload of parcels on his bed where he sat propped up reading and declared, \"Now, Ryan, I've gone through that money you put in my account, and I'm not spending another cent for clothes. I have enough for years!\"\n\n\"Good,\" he said, eyeing her with approval from head to foot. Her hair had been cut shoulder-length and styled to emphasize the oval shape of her face. Artfully applied makeup enhanced her remarkable eyes, the exquisite beauty of her classic features. In the sable-lined coat she was a captivating mixture of sophistication and innocence, and Ryan said with satisfaction, \"Now your appearance is worthy of you.\"\n\nOne afternoon while they were sitting on the balcony and Ryan was comparing the endless expanse of the Atlantic to the plains of West Texas, the phone rang. Cara answered it, and after a brief pause, a male voice, deep and unequivocal, asked to speak to Ryan Langston.\n\n\"Who's calling, please?\" she asked, intrigued by the voice but not wishing to disturb Ryan for a casual caller.\n\n\"His brother\u2014Jeth Langston.\"\n\nFor some reason, a chill swept her spine. \"Oh!\" she exclaimed involuntarily. \"Just\u2014just a moment and I'll get him.\"\n\nCara watched Ryan assume a smile before he spoke into the phone. In a jaunty voice that belied the fatigue and pain that racked his body, he chatted genially with his brother while Cara returned to the balcony. When he joined her again, she turned on him accusingly, her voice breaking. \"Ryan, you still didn't tell him, did you? For God's sake, why not?\"\n\nBut Ryan was unable to answer her. Clutching his stomach, he gave a cry of intense pain and slumped to the floor of the balcony.\n\nCara ran for blankets and pillows and made him as comfortable as possible before going to the phone to call the number she had written beside it for just this moment. Then she went back to Ryan to await the ambulance.\n\nThe next few days were a nightmare of despair for Cara as she sat beside Ryan's bed, knowing that his life was ebbing away and that there was a brother in Texas who did not know it. Her one source of comfort was the soft-spoken law partner from Ryan's firm who had arrived at the hospital shortly after his younger colleague had been admitted.\n\nThe man, who appeared to be somewhere in his midthirties, had approached Cara with deeply distressed eyes and handed her his business card. \"I am Harold St. Clair,\" he told her, \"a friend and colleague of Ryan's. His doctor had instructions to call me.\"\n\nOut of the maze of grief through which she wandered during the remaining three days of Ryan's life, one fact emerged clearly: Ryan had his business affairs in order. The firm, Harold told her, had been named to handle Ryan's estate. His personal effects would be sent to his brother in Texas. The firm would take care of the disposition of Ryan's town house and furnishings. It would see to the sale of the red Ferrari, unless, of course, she wanted it. It had been a stipulation of Ryan's that she was to have anything in the town house she desired.\n\nCara was aghast. None of Ryan's things were hers, she made it clear to the lawyer. Then she remembered the photograph on the mantel. \"There is one thing,\" she hurriedly amended. \"A picture of Ryan and his brother. I\u2014I'd like to have that.\"\n\n\"Of course,\" the lawyer agreed, making a note in his small leather book. He cast a contemplative glance at the averted profile of the girl. She was exquisite, no doubt about that. No wonder Ryan had completely lost his head over her.\n\nIn the three days, Ryan became lucid only once. Cara was sitting beside the bed, dozing. Harold had gone to the cafeteria for a cup of coffee. Ryan opened his eyes and looked at her. \"Hi,\" he said, and Cara, thinking she was dreaming, lifted her blond head.\n\n\"Ryan...\" She smiled and drew her chair closer to the bed. \"I'm so glad you're awake.\"\n\n\"Thank you for not asking me how I'm feeling.\" He gave her his ironic grin. \"I wouldn't want to lie to you.\"\n\n\"It's bad, is it?\"\n\n\"Yes, very bad. I almost waited too late to ask you something.\"\n\nCara's throat closed painfully. She reached for Ryan's cold, inert fingers, careful not to disturb the tubes taped to the back of his hand. \"Ask me what, Ryan?\"\n\n\"Do you trust me, Cara?\"\n\n\"With all my heart.\"\n\n\"Then would you promise to do something for me after I'm gone, even though I can't tell you now what it is? Think before you answer, love. I know that Yankee determination of yours well enough to know that once you give your word, the devil himself couldn't make you break it.\"\n\n\"Is it important to you that I do whatever it is you're asking?\"\n\n\"Yes. It means that I can rest in peace.\"\n\n\"Well, then, I promise, Ryan.\"\n\n\"Thank you, Puritan. You won't be sorry. You will be at first, and your courage will try to desert you, but don't you let it. See your promise through to the end. You'll be glad you did. I am confident of that.\"\n\n\"How\u2014how will I know what it is you want me to do?\"\n\n\"Harold will give you an envelope after my death. I have left instructions in it. Remember always that I had only at heart the interests of those I...loved.\"\n\nThe words trailed off. His lids closed in quiet finality. \"Ryan, dearest\u2014\" But Cara knew that Ryan had slipped forever beyond the sound of her voice. Already the beloved features had assumed an eternal stillness. Tenderly, as the tears began to come, she lifted Ryan's hand to her cheek and cradled it there for a few private seconds before the door burst open and blurred images in white surrounded the bed. Someone in a business suit spoke gently in her ear and eased Ryan's hand away, then led her from the room.\n\n\"Jeth has to be told,\" Cara said dazedly to the man whose arm was around her. \"Someone has to tell Jeth that his brother is gone.\"\n\n\"Shh,\" Harold St. Clair spoke soothingly. \"Don't concern yourself with that, Cara. The firm will inform him. It would be more appropriate for us to do so.\"\n\nA week later on the first day of March, Cara sat in Harold St. Clair's office. Sleet struck the windows, making the shapes of things beyond them gray and indistinct. In her lavender wool coat, the neck designed to reveal a matching dress beneath, she was like a splash of spring in the somber office, and Harold thought that he had never seen a more beautiful woman. \"How have you been this past week, Cara?\" he asked, observing her with his astute eyes.\n\n\"Empty,\" she answered briefly. \"Quite empty.\"\n\n\"Yes, I can understand that,\" the lawyer responded sympathetically, but in fact he did not understand at all. What had been the relationship of this lovely woman to Ryan? Had she been his mistress? Harold was now inclined to think not. This girl possessed an indefinable quality of sexual innocence, which made him believe that she had never warmed any man's bed. Yet Ryan had loved her above all the women in his life, of that he was certain. Why else would he have arranged his will against Harold's legal counsel and in direct defiance of his brother, whom Harold knew to be one of the most powerful men in Texas?\n\nThe lawyer's hands fidgeted with the legal document on the desk before him. Thank the saints that the two people it concerned would never meet. This fragile young thing in a clash with Jeth Langston, a man notorious for his ruthlessness, was almost obscene to contemplate. At least she would have the firm behind her as well as the courts. Together they would protect her from the vindictive rage that Jeth Langston was bound to be feeling at this moment.\n\n\"Cara,\" the lawyer began, clearing his throat, \"did Ryan ever discuss with you the provisions of his will?\"\n\nHer large eyes regarded him in surprise. \"Of course not. Why should he?\"\n\nThe lawyer returned her gaze with equanimity. \"Because you have been remembered very handsomely in it.\"\n\n\"What do you mean?\" Cara was puzzled. Ryan would have known that she wanted nothing material from him.\n\n\"You have inherited Ryan's share of La Tierra Conquistada.\"\n\nCara sat like a stone figure in the chair, her eyes riveted on the man before her, hoping to see something in his face that would betray his words as a horrible joke. \"You can't mean that,\" she said slowly in disbelief. \"Ryan would never have done that to his brother.\"\n\n\"I'm afraid that he has,\" Harold answered her quietly, in that moment utterly convinced of the girl's sincerity. He would have taken bets of any amount that she had not known about the will.\n\n\"I'll give it back. I can do that, can't I?\" she demanded earnestly, her voice rising. \"I don't want any part of the ranch. It belongs to his brother. I can't imagine Ryan doing such a thing!\"\n\n\"Before you make any decisions about giving up your inheritance, Cara,\" Harold advised her, \"I think you'd better read this. I was instructed to give this letter to you after I informed you of the will's contents.\"\n\nWordlessly, her heart accelerating, Cara took the envelope. \"I'll leave you alone for a few minutes,\" the lawyer said, and pressed her shoulder as he left the room.\n\nHer mouth dry, Cara opened the envelope and drew out a brief letter in Ryan's handwriting. She began to read:\n\nDear Cara,\n\nWhat must you be thinking now that you have learned that you've inherited one half of La Tierra Conquistada? No doubt, knowing you, your first scandalized reaction was to tell Harold that you want the land returned to Jeth.\n\nYou cannot release the land to anyone, Puritan, not until you have lived for one full year, beginning the first day of spring, in the big house on La Tierra. At the end of that time, you may do as you wish with your inheritance.\n\nYou are not to divulge to anyone, especially not to Jeth, that I asked this of you until after your year's tenure in the house is fulfilled. This is the promise you made me, love, and the one that I trust you to keep.\n\n_Vaya con Dios,_\n\nRyan\n\nCara looked up from the letter at the sleet-encrusted windows, unaware that Harold St. Clair had returned to the room. \"Cara?\" He spoke her name close to her chair, and she jumped nervously. Conscious that the letter was exposed to his view, she folded it quickly and slipped it back in the envelope.\n\n\"Mr. St. Clair, does Mr. Langston, Ryan's brother, know the terms of Ryan's will?\"\n\n\"I'm afraid so. I've been on the phone with his attorneys all morning. It was a great shock to him to discover that the original will had been altered in favor of someone other than himself.\"\n\n\"And he probably thinks I used\u2014undue influence is the correct term, isn't it?\u2014to get Ryan to change the will?\"\n\n\"I am afraid that is his opinion.\"\n\n\"Can the will be contested?\"\n\n\"Ryan was an attorney. He would never have drawn up a will for himself that could be contested.\" Seated once again at his desk, Harold assumed an expression designed to ease her misgivings. \"I imagine that you wish to either sell your share of the ranch to Mr. Langston or restore it to him once the estate clears probate. The firm, of course, will take care of all the necessary transference of ownership. No need for you to concern yourself with the\u2014er\u2014unpleasant possibility of confronting Mr. Langston or his attorneys.\"\n\nCara Martin sat straighter in the chair and tried to sound braver than she felt. \"I'm afraid I will not be able to avoid that confrontation, Mr. St. Clair. I'm going to Texas. I plan to live at La Tierra Conquistada.\"\nChapter Three\n\nThe silence that hung in the senior partner's office of the Dallas law firm representing the vast interests of La Tierra Conquistada was thick with tension. John Baines, the senior partner, along with another of the firm's attorneys, regarded Harold St. Clair and his mystifying client with frowning, tight-lipped censure.\n\nFinally, the senior partner broke the silence with one last appeal. \"Miss Martin, _please._ I beg of you to reconsider your decisions. Sign these papers. Mr. Langston wants back only what rightfully belongs to him. He is willing to pay you a more than fair price for the guarantee that once the estate is settled the land becomes his. Since the estate will take at least a year to go through probate, and since he is willing to pay you _now_ , you must surely see how generous he is being.\"\n\nCara's reply, her face pale and set, was a negative shake of her honey-blond head.\n\n\"Miss Martin\u2014\" The frustrated attorney decided to try a different tack and left his chair to sit on the corner of his desk in cozy proximity to Cara. After all, this young woman was the same age as his youngest daughter, and from time to time, he had been able to reason with her. \"Miss Martin\u2014\" He chose a soft, imploring timbre. \"Surely you can imagine what Jeth Langston must think of you?\"\n\nHearing it stated like that, Cara could not prevent the convulsive swallow from moving down her throat. \"Yes,\" she nodded, lowering her eyes from the penetrating gaze. She had been trying to avoid thinking about that question ever since she had read Ryan's letter. If she had thought about it, she could never have resigned her job, sublet her apartment, shipped to La Tierra the belongings she would need for a year, and flown with Harold St. Clair to Texas, all in less than three weeks' time.\n\n\"Well, then.\" The smooth tone had an edge of exasperation to it. \"Why can you not see that it is sheer madness even to consider taking up residence on La Tierra Conquistada? You will be like the biblical stranger in a strange land. You will have no protectors, no one to see after either your person or your interests\u2014\"\n\nCara raised her gaze to his. Her eyes darkened with some strongly felt emotion that intensified their beauty. The lawyer, nonplussed, drew back from their stunning assault as Cara said, \"You paint Mr. Langston as quite a savage, Mr. Baines.\"\n\n\"Mr. Langston is not a savage, Miss Martin.\" The lawyer's patience was strained to the breaking point. \"He is a fair man noted for his ruthlessness toward those who would pose any threat to La Tierra Conquistada. You must admit that you are doing that. By insisting on living under his roof, you are rubbing salt in the wounds of a man who has recently and quite unexpectedly lost his dearly loved brother, the only family he had. He believes you used your, ah, relationship with Ryan to persuade him to leave to you half an empire that has belonged to the Langstons for generations.\" The lawyer peered at her over his glasses, sensing the nodding agreement of his colleague. \"Quite frankly, I would not wish to be in your shoes at the moment. For your own sake, I plead with you to reconsider your decisions.\"\n\nCara heard her even reply as if she were disembodied from it. \"I understand what you are saying to me, and I appreciate both your advice and concern. However, my mind is made up. I will not consider any negotiations for the sale of my half of the ranch to Mr. Langston or to anyone else until the estate is settled. During that time, I choose to live at La Tierra. If Mr. Langston wants my cooperation, he will have to abide by that wish.\"\n\nIn the long silence that followed her little speech, Cara thought shakily that the trio of lawyers, even Harold St. Clair, was staring at her as if she were Joan of Arc just renouncing her last chance to escape the stake.\n\nThe hush was broken when the senior partner gave a defeated sigh and straightened his tall, brittle frame from the desk. He stared down at Cara coldly. \"I do not know what unseemly charade you are playing here, Miss Martin, but I must make sure you understand one thing: Jeth Langston is not a man to be trifled with. You are too young and inexperienced to engage in a contest of any sort with a man of his enormous power and prerogative. He does not merely defeat his enemies\u2014he destroys them. And you, my dear, as lovely as you are, have given him no reason whatever to make an exception of you.\"\n\nThe hard conviction of his words held them all enthralled. While he spoke, the color completely drained from Cara's face and her stomach began to churn. For the thousandth time she wondered what in the world had possessed Ryan to force her into such a dangerous and untenable situation.\n\nThe sudden sound of a buzzer on the desk startled them all. Leaning around, John Baines jabbed an intercom button and barked, \"Yes!\"\n\nHis secretary's crisp voice announced, \"Mr. Langston has arrived, sir.\"\n\nThe senior partner, with the resignation of Pontius Pilate having washed his hands of the whole matter, spoke into the intercom. \"Kindly show Mr. Langston in, Louise.\"\n\nCara was grateful that her back was to the door. Her position gave her time to try to calm her racing heart, which was threatening to burst through its walls. She heard the door open, then close with a quiet, emphatic click. A force flowed into the room, drawing the men at once to their feet. She sensed from their fixed, apprehensive gazes that the man had paused just inside the door, no doubt for dramatic effect.\n\nJohn Baines did his best to smile. \"Come in, Mr. Langston, come in!\" he said in the hearty voice of a businessman welcoming a preferred customer. \"Allow me to introduce Harold St. Clair of Boston, who was a legal partner of your late brother's, and, uh, Miss Cara Martin, also of Boston.\"\n\nCara found that her legs would not permit her to rise. She could feel the man's presence, strong and hostile. Suddenly angry and resolute, Cara stood and turned to meet the steady gaze of Jeth Langston.\n\nThe impact took her breath away. She had expected, of course, an imposing man\u2014similar perhaps to the legendary breed who sailed her great-grandfather's ships and answered to none but the sea. But no knowledge of those long-ago sea lords, and certainly no male of her acquaintance, not even Ryan, could have prepared her for this man. He was easily the most awesome human being she had ever seen. Tall and powerfully built, he stood like a man accustomed to power\u2014strong legs in razor-creased slacks spread imperiously apart\u2014and took her measure from beneath the low brim of a superb fawn Stetson. His presence seemed to flow across the room, almost suffocating her with apprehension.\n\nLooking at him, noting the narrow black band of mourning around the soft crown of the hat, Cara was tempted to speak to him of his sorrow\u2014of their sorrow\u2014but the icy contempt in his eyes froze the words on her tongue. Her hammering heart pounded in her eardrums. Mute and paralyzed, she felt as helpless as a dreamer caught in an inescapable disaster. For the man had begun a slow, deliberate approach toward her, his gray eyes glacial and still. She could not find in his hard, handsome face a single similarity to the brother they both had loved.\n\nJohn Baines waded into the silence by clearing his throat and saying in a tone of accusation, \"Miss Martin refuses to change her mind, Mr. Langston, in spite of our reasoning.\"\n\nJeth, pausing a few feet from Cara, spoke softly. \"Perhaps I can change Miss Martin's mind. Gentlemen, would you mind leaving us?\"\n\n\"Not at all, Mr. Langston,\" acquiesced the attorney, who nervously shuffled a few papers on his desk before relinquishing his turf. He and his colleague filed past, but the Boston lawyer went to Cara's side. \"If you like, I'll remain, Cara.\"\n\n\"That won't be necessary, Mr. St. Clair.\" Cara spoke for the first time and attempted a weak smile. \"I'll be all right.\"\n\nHarold's heart moved queerly at the sight of the brave little smile. He touched her shoulder comfortingly. \"You've only to call. I'll be just outside the door.\" Sidestepping Jeth, he nodded to the rancher and left the two antagonists staring at each other.\n\nAs the door closed, Jeth's eyes left hers and went to her hair, his stony expression betraying nothing of his thoughts. It was the color of pure honey and framed a face that could stop the heart of any man. He had experienced many griefs, but he felt a new kind of sorrow as his gaze lowered in a merciless descent down her body. He had thought that he had known them all, had experienced every variety of alluring fortune hunter known to man, but this one was of a singular cast. He could see how Ryan had been taken in; certainly he would have been, too.\n\nCara drew a sharp breath and resisted the urge to cover herself from Jeth's disturbing eyes. He said slowly, \"Yes...now I understand. Who would ever take a girl like you to be what you are?\"\n\nCara could endure no more. \"Jeth, I\u2014\"\n\n\" _Mister_ Langston to you, lady!\" The words were rapped out like gunshots. \"We're going to keep this conversation on a strictly formal footing, do you understand?\"\n\n\"Very clearly,\" Cara said with rigid dignity, determined not to give any ground during this initial, crucial interview.\n\nJeth regarded her in silence for a few seconds, and Cara thought she saw a flicker of surprise beneath the chilly stillness of the gray eyes. \"Well, now that that's settled, let's talk, you and me.\" He tossed his hat to a couch and chose for himself a deep armchair to accommodate his tall frame. He had thick, dark hair, Cara noted, the kind with a tendency to curl.\n\nCara, following his lead, sat down in her original chair and remained waiting for him to speak, outwardly calm. \"So you want to live at La Tierra, do you?\" he asked conversationally, lifting brows as dark as his hair.\n\n\"Yes,\" she answered with as much force as her taut throat would allow. It was very difficult to meet his eyes. In all fairness, she could not blame him for thinking of her as he did. What in the world _had_ Ryan been thinking to extract such a promise from her?\n\n\"Why?\" Jeth asked bluntly, watching her face carefully as if he did not trust her words to reveal the truth.\n\n\"My reasons are personal.\"\n\n\"Ah\" was Jeth's only reaction before reaching inside the inner pocket of his Western-cut leather jacket for a slim cigar case. Cara could see the same discreet insignia in gold on it that Ryan's wallet had borne, but now she knew it was the brand of La Tierra Conquistada, a _T_ crossed with a _C._ Jeth selected a long, slender cigar and returned the case to his pocket.\n\n\"Ordinarily,\" he said, biting off the tip of the cigar with strong white teeth, \"I ask a lady's permission before smoking in her presence. However...\" The implication hung in the air along with the tendrils of smoke that fanned from his narrow nostrils. Cara felt a surge of heat on her cheeks. Let him insinuate anything he wished! she thought angrily. Knowing she had no cause to feel ashamed gave her inestimable strength. He could blow as much smoke as he liked!\n\n\"You've made your point, Mr. Langston,\" Cara stated with a trace of hauteur. \"And as a matter of fact, you're not the first...\"\n\n\"You'd better get used to it, Miss Martin. The kindest name I've heard in reference to you lately is Ryan's whore.\"\n\nHe had hurt her there, Jeth thought without pleasure, watching the blood drain quickly from her delicate face. She had to look away from him, her eyes apparently seeking refuge in a painting on a far wall. It was a seascape of sand and seagulls and ocean. _Home_ , he surmised, wondering if she missed Boston, if she regretted this course upon which she was embarked. Her expression when she turned to him again was completely composed, revealing nothing. Like him, she too had learned the value of concealing her vulnerabilities.\n\n\"Now,\" he continued in the tone of a father who has just satisfactorily reprimanded a child, \"back to my original question. Why do you want to come to La Tierra?\"\n\nSuddenly, quite thoroughly, Cara hated him. She fought to keep her body from quivering in cold anger at his overbearing manner. \"Back to my original answer, Mr. Langston,\" she replied icily. \"My reasons are my own.\"\n\n\"Shall I take a stab at what those reasons may be, Miss Martin?\" Jeth suggested amicably, his mouth quirking in a slight smile that held no humor whatever.\n\nYou may take a straddling leap at a high fence, Cara silently suggested, but refrained from voicing her thought. \"Why ask, Mr. Langston, since you intend to tell me anyway?\" She squared her shoulders and raised her chin, exposing the smooth, vulnerable line of her throat. Waiting for him to continue, she willed herself not to be affected by his words, however harsh they were. You're no stranger to pain, she reminded herself. You can bear whatever _he_ says!\n\nJeth drew on his cigar, regarding her narrowly through the smoke. For the first time she noticed the handsome, masculine ring he wore on the ring finger of his left hand. It had been designed with a black face on which was engraved the brand of La Tierra Conquistada, set in pav\u00e9 diamonds. There seemed to be nothing about the man, she grudgingly admitted, that did not declare his wealth and position. The boots, which matched the tan leather of his jacket, were obviously hand sewn. The sharply creased slacks were of fine wool, the complementing beige tie of the finest silk.\n\n\"I can think of three possible reasons for wanting to ensconce yourself at the ranch,\" Jeth began, settling comfortably in the chair. \"One, you feel that by living there for a year you will better be able to determine the true value of what you've inherited to set your price once the estate is settled. Second, I understand that you've quit your job. Without an income, you need free room and board for a year, so what better place to nip into than La Tierra? A year without having to work for a living will prepare you for the kind of life to which you anticipate becoming accustomed. How am I doing so far?\"\n\nCara could only stare at him, too appalled to answer. She was forced at last to appreciate fully how she must appear in his eyes, in the eyes of all of those who had loved and respected Ryan. They thought her lazy and opportunistic, a fortune hunter who now had the gall to demand living accommodations under the very roof of the man it appeared she had swindled.\n\nWhy would Ryan have demanded something from her that would place her in such a light? He knew she loathed freeloading. He had often become exasperated with her because she would accept nothing from him that she could not return in kind. And she was nothing if not a hard worker. She had not wanted to give up her job, leaving unpaid for yet another year the final debts that clouded her family's name. Ryan, whom she had trusted, whom she had loved\u2014why had he extracted a promise that would compromise her very soul in the eyes of others?\n\nInwardly she sighed. Now there would be another name to add to her list of debtors. She would pay Jeth Langston back for the cost of her room and board if it was the last thing she ever did!\n\n\"You mentioned a third reason,\" she reminded him.\n\n\"Yes,\" Jeth said slowly. Without hurry, he pulled toward him a bronze ashtray on the desk. The sensuous leather of his coat sleeve defined the hard, virile line of his arm. Cara sensed a sudden and dangerous change in him that made her look at him warily. When Jeth gave her his attention once again, her skin tingled with an ominous chill.\n\n\"I think that somewhere in that scheming little head of yours, you actually entertained the idea that I may be induced to pick up where Ryan left off\u2014two halves are better than one, so to speak\u2014\"\n\nCara was horrified. \"No!\" she gasped. \"What an insane idea!\"\n\n\"Is it, Miss Martin?\" Jeth returned with icy calm. \"Unlike my brother, who preferred tall, statuesque women, I have always had an inclination toward the Dresden type, the kind who are all cool fragility without but fire and passion within\u2014like the kind of woman I suspect you are, Miss Martin. But then you were aware of that. You probably pumped Ryan plenty before he died.\"\n\nAt the mention of Ryan's death, a sudden shadow flitted across Jeth's sun-browned face. For a brief moment Cara saw naked pain etched there and remembered what she had forgotten in their bitter interview\u2014that Jeth was suffering, too. Nonetheless, she jumped to her feet, small fists clenched, instinctively knowing that she must make clear her position on this vital point or lose a foothold that she could never regain. \"I knew nothing of the sort about you! I couldn't care less about your preferences in women! You are reading far too much into why I want to come to La Tierra. I can understand how you must feel about my living in\u2014in your home, and I don't blame you, but I promise you that I will sell back to you my share\u2014\"\n\n\" _Ryan's_ share,\" the man across from her corrected softly, the gray eyes very still.\n\n\"Ryan's share,\" Cara allowed. \"And for a fair price.\"\n\n\"And what do you consider fair?\"\n\n\"That will have to be discussed when the estate is settled. You have my word, though\u2014however little it means to you\u2014that the sum will be reasonable. In exchange\u2014\" She faltered and bit at the soft flesh of her lip, feeling herself blush.\n\n\"Yes?\" Jeth pressed, with unnerving patience.\n\nCara drew a deep breath. \"In exchange for the guarantee that I will sell to no one but you, I must have your guarantee that no harm will come to me while I am living at La Tierra.\"\n\nThere was a short silence, broken when Jeth instructed, \"Do sit down, Miss Martin. Your height is inadequate to provide you much advantage. Besides, you look tired enough to drop.\" She did, too. He had just noticed the delicate blue tinges of fatigue beneath the startling eyes. \"Now tell me, why do you think you'll need my protection?\"\n\n\"Mr. Langston!\" Cara regarded him coldly as she sat down. \"I may look a fool, but I assure you I'm not! Neither do I think I am addressing one. You know perfectly well why I would want such a guarantee. I could be\u2014I could be\u2014\" Desperately she searched for a word that was less graphic than the one that sprang to mind.\n\n\"Molested in some way?\" he suggested politely, a small smile playing about his strong mouth.\n\n\"Yes!\" she said in angry embarrassment. \"That, or\u2014or beaten and starved\u2014\"\n\n\"My dear Miss Martin!\" Jeth could not suppress his laughter. It had a nice, hearty ring to it, and had he not been laughing at her, she might have enjoyed it. She seethed while, still amused, he blew a final stream of smoke and tamped out the cigar in the ashtray. \"You've been seeing too many Italian Westerns,\" he chuckled.\n\n\"I see no Italian Westerns, Mr. Langston. I do not care for them. I am merely stating the obvious vindictive approach you and the people who work for you might take toward me for what you suppose I did to Ryan\u2014\"\n\n\"Suppose? Did you say _suppose_ , Miss Martin?\" He was out of his chair before she could blink, all humor vanished, the arctic coldness back in his eyes. \"Let's get a few facts straight,\" he said very clearly, bending down to imprison her in the chair by clasping each of its arms. \"I don't like dealing in suppositions.\"\n\nCara shrank back from him, the closeness of the granite features and the unaccustomed male scents of cologne and leather and tobacco sending her senses spinning. \"Now these are the facts as I see them. I am sure you will correct me if I'm wrong.\"\n\n\"Given the opportunity,\" Cara managed, pressing back against the chair.\n\n\"You prevented my brother from coming home to die. Oh, he came back for a last token visit, but he never mentioned he was dying. If I had known his illness was terminal, I would have kept him there, and that would have meant curtains for you. I would have found out about the altered will.\"\n\n\"That's not true!\"\n\n\"Isn't it, Miss Martin? Then why didn't he tell me about you, the woman he loved? Why didn't he tell me about the change in the will? Ryan would have known that I would have accepted any decision he made concerning his half of La Tierra. It was his to do with as he chose.\"\n\n\"Mr. Langston, I honestly don't know the answers to those questions\u2014\" He was so close. If she moved, they would touch.\n\n\"Then try this one. Why didn't _you_ tell me he was dying? You had to have known that I didn't know. You were the woman who answered the phone a few weeks ago when I called, weren't you? Why didn't you tell me?\"\n\nCara could not answer. Helplessly, she stared into the suddenly bleak eyes. No wonder Jeth Langston despised her. It was not the loss of the land that sharpened the edge of his hate against her, but the belief that she had denied him the last days of his brother's life.\n\n\"What power you had over him, Cara!\" Jeth said in soft anguish. \"A man doesn't need much imagination to know how you made sure he returned to Boston. I'm sure you had your ways of convincing him that your arms were better for holding him in his final days than mine would have been.\"\n\nA stab of pity for him brought the shine of tears to her eyes. She would not, could not, add to this man's grief by telling him that Ryan himself had refused to return home to die. Without meaning to, she looked longingly at the broad set of shoulders encased in the buttery soft leather. She was so desperately tired. How pleasant it would be to slip her arms around that strong neck and rest her cheek against the leather's yielding softness. Instead, she closed her eyes and lowered her head wearily, feeling a strand of hair brush Jeth's chin.\n\n\"I\u2014I can well understand how all this must look to you, but\u2014but\u2014\"\n\n\"But I'm wrong, is that it?\" Jeth finished for her, his tone almost gentle.\n\nShe shook her head.\n\n\"Oh, Miss Martin\u2014\" He straightened up, an impotent rage filling his soul. Long ago he had dispensed with dreams, especially those about women. But occasionally, when he felt especially lonely and the long evening hours in the study stretched out before him, he wondered what it would be like to know, like his father, the love of a devoted woman. Sometimes his thoughts wandered further, and he envisioned what she would look like, this woman of his dreams. A small, shapely figure, eyes that could melt the needles from a cactus, honey-gold hair, and a mouth so sweet and passionate that it was like drinking ambrosia to kiss her\u2014that was the description of the woman he yearned to give his heart and soul. A woman who looked like Cara Martin.\n\n\"Let's see if I'm wrong about this, too, Miss Martin,\" Jeth said, his voice dangerously soft. He reached down and slipped an arm around her waist. Cara was in the leather enclosure of his embrace before she could close her astonished mouth.\n\n\"Let me go!\" she demanded, aware of the sudden intimate pressure of his chest against hers. His move had been so sudden, he was pressing her so close that her arms dangled uselessly. They had nowhere to go but to his shoulders, and she must not put them there.\n\n\"This is your chance to prove me wrong about you, Miss Martin, that you are not what I think you are, that you were never Ryan's\u2014\"\n\n\"Don't say it!\" Cara said desperately. \"I can't bear to hear you say it.\"\n\n\"Then prove to me how wrong I am.\"\n\n\"Don't\u2014\" The word was just forming when Jeth's lips closed over her mouth.\n\nCara stiffened against him, tightened her lips in rigid protest against such a violation of her privacy. Small fists pummeled his shoulders with powerless blows that drained her remaining strength. Jeth, his hand a gentle vise under the silken fall of her hair, felt the tension suddenly leave and released her mouth. Cara's lids fluttered open, the depths of her eyes starry and deeply violet. Jeth stared down into them, and she was conscious of a strange, frightening desire asserting itself deep within her. \"Please let me go,\" she pleaded, her mouth so close to him that her lips stroked his when she spoke.\n\n\"No,\" he murmured and kissed her eyes. She whimpered\u2014to Jeth's ears like a kitten lost in a storm\u2014but he could not afford to be merciful. He pressed her closer and she gasped and tensed as his lips closed over hers again. He might have let her go then, but she did not pull away. Against her mouth Jeth groaned in gratitude, for he could not have borne the sudden release of her from his arms, the denial of her lips, the feel of her body. The fragrance of her filled his nostrils and drifted down into the hollow of his heart where he had conceived the image of her likeness. Exultantly, hungrily, tasting and devouring her, he led her deeper into a world of sexuality where he could not have known that she had never been before.\n\nAnd Cara, the sudden, unexpected need of him destroying her defenses, could not prevent the ardor with which her flesh responded.\n\nLong after her body had helped Jeth to prove his point, she stayed within his embrace. Finally, he pushed her from him. Shame would not let her meet his eyes. To finish her humiliation, tears began to run down her cheeks.\n\n\"Believe it or not,\" he said quietly, \"I wish I'd been wrong. It would be comforting to know that Ryan had loved a woman who could have remained faithful until his body was cold.\"\n\nJeth brought out a folded white lawn handkerchief and tossed it to her. \"Now let's do a little reconsidering, shall we? I'm sure that you realize that it's out of the question for you to live on the ranch.\"\n\nCara dabbed at her eyes. \"I have to come, Jeth,\" she said. \"I have to. I don't expect you to understand, but be assured I won't ask a thing from you. I won't be in your way. What happened just now will not happen again\u2014\"\n\nJeth asked in astonishment, \"You mean you _still_ intend to go through with this? What the hell for? What can you possibly hope to gain? I'll pay you now for Ryan's share of the ranch!\"\n\n\"It\u2014it's not for sale until the estate is settled, which will take approximately a year, or so I'm told. I'd like to arrive March twentieth, two days from now. Probably by this time next year, the paperwork will have already been drawn up to restore Ryan's portion to you. You have my word that I will ask no more than a fair price for it. And you have to promise\u2014\"\n\n\"Yes, I know,\" Jeth grated. \"My protection from physical abuse. Okay, lady, you have a deal, but I hope your psychological health is in good shape. You'll need it where you're going.\" He turned to pick up the Stetson. \"By the way,\" he asked, \"just how do you expect to get to the ranch?\"\n\n\"I intend flying to Midland Air Terminal. I'd like for you to have someone pick me up when I arrive. I'd rent a car, but I would have no way to return it.\"\n\n\"Suppose I say no.\"\n\nCara had to moisten dry lips, but she stood her ground. \"Jeth, you have to cooperate with my inconsequential requests if you want that land back.\"\n\nHe came back to all but gape at her, his strong brown fingers curved around the brim of the Stetson. Cara found herself gazing at them in fascination. \"I won't bore you with the results of the last attempt to coerce me, Miss Martin, but let it suffice to say that the individual regretted his impulse. You will hand over that land no later than next March twentieth with or without my cooperation to your _inconsequential_ requests, do you understand? And another thing: you have lapsed twice and called me Jeth. Don't do so again.\"\n\nWithout another look at her, he strode from the room and closed the door behind him with the finality of an exclamation mark. Cara stood staring at it with a strange sense of loss, raising to her lips the monogrammed handkerchief he had forgotten.\nChapter Four\n\nHe wouldn't see me, you know,\" said Harold St. Clair after he and Cara had been seated in the dining room of the Dallas Hilton where he had reserved rooms for them.\n\n\"Mr. Langston considers you a traitor,\" Cara said regretfully, practicing the form of his name that she'd been ordered to use. \"But even so, I'd think he would want to hear about his brother from one who had been his colleague and friend.\"\n\n\"Jeth Langston is a hard man, Cara. Apparently he feels that I am partly responsible for this unpleasantness since I knew about the will and didn't tell him in time for him to exert his influence. I plead innocent of knowing that Ryan's illness was terminal.\" He looked across at Cara with a despondent smile. She was ravishing in a red dinner dress that heightened the translucent glow of her skin. Under the flattering lights of the chandeliers, her eyes and hair were dazzling. Harold was aware that glances from other diners kept returning to their table. He only wished Cara's admirers could have the pleasure of seeing her smile. The lovely heirloom pearls encircling her throat were nothing to the pearl perfection of her smile.\n\nGently, Harold covered the small hand toying with the stem of a wineglass. \"If he had agreed to see me, I could have told him that he is mistaken about you.\"\n\n\"How can you be sure of that, Harold?\" The lawyer's first name sounded unfamiliar to her still. This afternoon he had asked her to use it. \"You know no more about me than he does.\"\n\n\"I know that when I first read to you the contents of the will, your face lost all color, and you immediately wanted to give\u2014not sell\u2014the land back to Jeth. Your specific words were 'Ryan would never have done that to his brother.' Remember? You only changed your mind after I left you alone with Ryan's envelope. What was in it, Cara? What was in it that made you change your mind about the inheritance and decide to wait out the settlement of the estate at La Tierra?\"\n\nCara drew her hand away and clutched the napkin in her lap. She was becoming adept at hiding her thoughts, and now a curtain seemed to descend over her features. If Harold probed any further, he might guess at the promise she had made to Ryan. Warmth remained in her eyes, however, and Harold was rewarded with a faint smile. \"I'll have to tell you what I told Mr. Langston, Harold. My reasons are personal. A year will go fast and then I'll have to impose upon you again by asking that you arrange the sale agreement.\"\n\n\"Do you...know what you'll be asking?\" Harold inquired blandly, studying her covertly over the rim of his martini glass.\n\n\"A reasonable price,\" was her evasive answer. \"Now tell me, when are you going back to Boston?\" The question made her heart move strangely. Harold was her last link with home, with what was familiar and safe. He had been a steady and comforting presence at her side throughout the strain of the last few weeks. To be suddenly without his counsel and support, his easy companionship, would be especially hard after the tragedy of Ryan's death and the fear of the ordeal ahead.\n\n\"Tomorrow morning,\" Harold told her reluctantly, reading her thoughts. \"Cara. I want you to consider me your friend. You have my card. If for any reason you need me, you've only to call. I could not bear to think of your needing help and having no one to turn to. I'm only a return flight away.\" He looked at the girl as gravely as he dared. Heaven knew, she was frightened enough for all the composure she was trying to show. \"And my dear\u2014\" He took her hand into his smooth, comforting one. \"If I may offer some advice?\"\n\n\"Please do,\" Cara invited quietly, but her heart fluttered sickeningly at his tone. Into her mind leaped the image of Jeth and the hatred and repugnance that had been in his eyes in the last few moments of their interview. She did not fear anything he could do to her if only he did not prey on the weakness he had discovered, the weakness she had not even known she possessed.\n\n\"Then it is this, Cara. Do not love anything while you are out there. No man, woman, or child. No horse or dog\u2014not even an armadillo\u2014\" His attempt at humor failed. The beautiful eyes darkened with anxiety, but he pressed on. \"Care for nothing or no one through which he can hurt you\u2014\" Harold broke off as a waiter appeared to present them with menus.\n\n\"You don't have to go, you know.\"\n\n\"Yes, I do.\"\n\nHarold sighed. \"I have transferred a sum of money to your account on orders from Ryan. No, don't protest, Cara, and don't be foolish, either. You're broke. I know it, and Ryan knew it. You will have no income while you're in Texas, so don't let that New England pride of yours prevent you from spending it for the things you need.\"\n\nCara opened the menu. Ryan had thought of everything. Everything but an explanation for why she was here. \"I don't think I'll have anything but a salad. I seem to have lost my appetite,\" she said.\n\nThe next morning Cara saw Harold St. Clair off on his return trip to Boston from the huge, modern Metroplex airport that sprawled between Dallas and Fort Worth. As the aircraft lifted off, a bleak depression settled over her, and she clutched even tighter the small box she held. It contained a gold charm in the design of a seagull that Harold had given her just before boarding.\n\n\"For luck,\" he explained, looking down at the golden head bent in surprised pleasure over the trinket. \"You can attach it to that thin gold chain you wear around your neck. If you have any bad moments, just reach up and touch it. I hope it reminds you that you have a friend.\"\n\n\"Harold\u2014\" Words failed her, and so she reached up and kissed his smooth-shaven cheek. Blinking at the tears that threatened, she stammered, \"You've been so kind. I can never repay\u2014\"\n\n\"Shh.\" He stopped her gently by touching a finger to her lips. \"Take care of yourself now, and let me hear from you.\"\n\nOnce he had gone, Cara found a secluded seat in a vacant passenger lounge and cried out the grief and despair that had needed release for weeks. She felt better afterward and resolutely drew a breath. That will have to do me, she thought. No way can I afford to do that once I'm on The Conquered Land!\n\nThe next morning she was wide awake long before the desk called to awaken her. She lay in bed staring at the ceiling and tried to calm herself by mentally lining up the defenses she could call on to protect herself from Jeth Langston's expected vengeance. There was the matter of the land, which would be hers by law within the year. She must not be squeamish when it came to holding that over his arrogant head in case he decided to get rough with her. Also, she would watch her decorum carefully and in no way give anyone reason to call her\u2014she could hardly bear to think of it\u2014Ryan's whore! She would stay out of everyone's way, but if allowed, she would certainly pitch in and help with whatever needed doing.\n\nBut for all the practical advice she gave herself, the knot remained tied in her stomach. Not even the elegant suit she chose for her flight helped to soothe her anxieties. She had come to take pleasure and comfort in the large assortment of beautiful clothes that soon would be hanging in her closet on La Tierra. They reminded her of Ryan and brought him close to her in memory. She wore them proudly, knowing that he would have wanted her to.\n\nCara preferred time to drag, but it did not. By the time she had finished packing and forced herself to eat several bites of melon for breakfast, she had to leave for the airport. She dressed warmly in the sable-lined raincoat, for spring was late arriving in Texas, and in no time at all she was deposited before the flight desk and her bags were being checked.\n\nA flurry of worrisome questions besieged her as the airliner winged its way over the vast reaches of Texas. Were the two big boxes containing her clothes and the belongings she had sent by air freight waiting for her in the small airport where she'd be landing? Would there be someone there to meet her? Cara cringed at the thought that it might be Jeth Langston. She shrugged off that worry immediately, thinking it unlikely that the owner of La Tierra Conquistada performed such menial chores. How would she get to the ranch if no one was there? She could rent a car, but how could she return it? Finally, already exhausted from the burden of her anxieties and a sleepless night, she laid her head back, closed heavy lids over troubled eyes, and slept.\n\nThe steward woke her, it seemed to Cara, just a few minutes later, and yet she felt a surge of fresh strength and well-being. The young man smiled down at her, enjoying her beauty. \"I thought you'd like to be awake before we land,\" he said, \"especially if this is your first trip to West Texas.\"\n\nCara thanked him, and the steward remained at her seat to get her reaction when she looked out of her window. The sight below made her gasp. The handsome young steward smiled. \"That's something, isn't it? Everybody has that reaction the first time they see West Texas from the air. Someone once said this part of the state can best be described as 'miles and miles of miles and miles.' \"\n\nAn accurate statement, Cara agreed, as she gaped down at the vast, seemingly endless desert that surrounded two oasis-looking patches of green. Cara assumed they were the only towns of any size in the area. In between them was the airport, but beyond and around them was nothing\u2014no trees, lakes, or highways\u2014to break the sweeping brown panorama of the West Texas plains. A tough, rugged land, she decided\u2014like the man who had conquered it.\n\nThe thought of Jeth Langston brought shadows to her eyes, and the steward, who had already summed her up as some rich man's toy, ventured curiously, \"Somehow you don't look like you belong out here. Are you just visiting?\"\n\n\"Yes,\" Cara replied, giving him a brief smile before turning back to her window. The steward took the hint and moved off down the aisle, wondering about the man rich enough to afford something like that.\n\nA dry, stiff wind lifted Cara's blond hair when she stepped off the departure ramp at Midland Air Terminal. There was no amenity of a covered ramp from plane to terminal, and she pulled the warm fur at her neck closer. Quickly she scanned the assembled group of relatives and friends for anyone who might be from La Tierra Conquistada. Subconsciously, Cara realized, she was looking for Jeth. No one nearly that tall or dominating was among the group who waited inside the terminal building. She looked searchingly around, but her gaze was met only by those arrested by her striking appearance.\n\n_He hasn't sent anyone!_ she thought in dismay. So this was to be her first taste of what she could expect as Ryan's whore.\n\nShe went into the small restaurant for a cup of coffee and to plan her next move. There was a car rental service here. Perhaps she would have to rent a car and simply hope that she could prevail upon someone from the ranch to return it. Oh, for her trusty old Volkswagen, she was thinking, just as someone tapped her on the shoulder.\n\nCara looked up in surprise. A tall, rangy young man about her age, wearing low-slung jeans, scuffed boots, a sleeveless fleece-lined jacket, and a frowning expression, was regarding her uncertainly. He had a dusty cowboy hat pushed back on his curly blond head. \"Yes?\" she inquired.\n\n\"You Miss Martin?\"\n\n\"Yes, I am. Are you from La Tierra?\"\n\n\"Yeah. The boss sent me to pick ya up.\"\n\n\"Well, that's wonderful!\" Cara exclaimed, a brilliant smile of relief lighting her eyes.\n\nThe young man looked away, momentarily disconcerted. Cara suspected that the tough-guy pose did not come naturally and was being worn for her benefit. Orders from headquarters, she surmised with a flash of temper.\n\n\"Let's go, then,\" he said gruffly.\n\n\"There are a couple of things that I must do first\u2014\"\n\nThe young man dug his heels into the carpet, thrust fingertips into tight jean pockets, and surveyed her with disapproval. \"Like whut?\"\n\n\"Paying my check for one thing,\" she said pleasantly. \"And then I have to pick up my luggage. After that, I have to go to the air freight office to collect the boxes that I sent from Boston.\"\n\n\"That take long?\"\n\n\"No-o-o.\" Cara's eyes rounded innocently. \"Not nearly as long as it would take to make a return trip here and back.\"\n\n\"Well...\" The young cowboy thought this over. \"I guess it's all right. But I have strict orders from the boss to pick ya up and head right on back to the ranch. This is roundup time, ya know.\"\n\n\"No, I didn't know,\" Cara said congenially, fishing in her bag for money. She indicated her bags. \"Would you mind getting those while I pay the check?\"\n\nThe young man picked up Cara's weekender and cosmetic case. \"My name is Cara,\" she said, taking the cosmetic case from him as they were heading for the luggage pickup.\n\n\"Mine's Bill, but I don't think we oughta get too friendly, miss. Let's just get what ya got to get and quit the jawin'.\"\n\nStung, Cara remained silent while the luggage was collected. Bill's only words were, \"The jeep is out here.\"\n\n\"Jeep?\" she cried, following him out to an immaculately painted light-gray jeep with the name of the ranch in small yellow letters on its side. The wind was beginning to pick up. Cara's ears already felt cold, and she did not relish a ride in an open vehicle. She was glad that she'd remembered to tuck a light wool scarf into her handbag.\n\nAll his attention on the road, Bill drove the jeep to the freight office where to her relief the two big boxes from Boston were awaiting her. Without a word, Bill loaded them into the back of the jeep with her other luggage, then looked impatiently at her standing beside the vehicle tying on her head scarf. \"Let's go, miss. We're late enough already.\"\n\nWith Cara clutching the side of the jeep with one hand and her scarf with the other, they tore off down the road leading out of the airport. They headed west on a wide modern interstate for a few miles until Bill turned left onto a two-lane highway. The wind tore at Cara's scarf, stung her eyes and cheeks, and carried away all attempts at conversation. Finally, receiving no response, she fell silent, trying to make as much as possible of the terrain they were passing through. Still in its wintry pall, it was indeed a bleak-looking landscape. Little vegetation grew from the hard, sandy ground, and what there was appeared stunted and sparse. She recognized the gnarled mesquite trees that Ryan had described to her. \"They won't bud until the last freeze is over,\" she remembered his telling her. \"Everything else out there can be fooled by Mother Nature, but not the mesquite.\" Cara had no idea what mesquite looked like when in bloom, but since there was not a single speck of color on the barren landscape, she deduced that winter was not yet over.\n\nShe was managing to hang onto her seat and the scarf until Bill turned off the highway across the open plains. \"Shortcut!\" he yelled, driving the jeep at full speed. Cara glanced back in alarm at the boxes jostling around on the backseat. If Bill hit a bump, they could easily be bounced over the side of the jeep onto the hard ground, and already they seemed to have had all the abuse they could stand. One look at the grim satisfaction on the young cowboy's face, the malicious delight he was taking in her discomfort, and the whole picture became clear.\n\n\"Stop this jeep this instant!\" she shouted, and when he ignored her, she simply reached for the keys and jerked them out of the ignition. The jeep ground to a halt, and Bill turned to her in stupefaction. \"Now you listen to me, you ill-mannered smart aleck!\" Cara exploded. \"You need reminding of a fact you seem to have forgotten. I own half of La Tierra Conquistada, and you will drive this jeep at a sane speed and get us to wherever we're going in one piece, or I may have to exercise a prerogative of my position that I'd just as soon not. Do I make myself clear?\"\n\nBill looked across at her uncertainly, trying to decide if she was bluffing. The furious brilliance of the violet-blue eyes convinced him she was not. \"Yes, ma'am,\" he conceded gruffly, and held out his hands for the keys.\n\nUltimately, out of the vast ocean of nothingness, there appeared in the far distance a white sprawling structure that momentarily gave the young Bostonian an impression of the Taj Mahal planted in the middle of the Sahara. The suddenness of its appearance was relieved by the beginning of a fence, made not of wood but of white steel pipe, which suggested that civilization was not far off. The white fence contrasted peaceably with the green winter pastures it bordered. In them here and there, groups of healthy-looking russet-colored cattle grazed placidly.\n\nAs all of this came into Cara's awed view, the jeep reached a well-paved road that ran beside the fence, and Bill turned left, heading, Cara supposed, to a drive that had access to the shining edifice sitting in the middle of the plains.\n\n\"Is this where the ranch begins?\" she asked.\n\nThe young cowboy shot her a disgusted glance. \"You been on La Tierra since we left the airport,\" he stated scornfully, but he could not conceal the note of pride Cara heard in his voice. She recalled that Ryan had spoken of the loyalty and devotion of the cowhands to the ranch. Many of them, she remembered, represented the fourth generation to work at La Tierra. Cara wondered if Bill was one. If so, she could understand the contempt that he was trying hard to show her.\n\nThey drove for several more miles before they reached the massive wrought-iron gate flanked by equally imposing limestone posts. The gate was heavily scrolled, but the intricate metalwork did not interfere with the brand of La Tierra, which had been worked into the two joining centers of the gate. When the gate opened, the brand was divided. Bill pointed this out to her, adding meaningfully, \"The boss had it designed that way on purpose. He wanted everybody to understand that half was his and half was his brother's.\"\n\nHis words brought home to her what for a short while she had forgotten in the curiosity of her new surroundings: Jeth Langston. Sometime this day she would see him, and the thought chilled her blood. She wished that she could appreciate the beauty of the wide paved drive that was leading her closer to a man who detested her, who already had begun his vengeance upon her. Nonetheless, she noted that the drive had been lined with tree after tree of fuschia oleanders just beginning to bloom. Apparently they did not take their cue from the mesquite. She imagined, once they were in bloom, the profusion of blossoms that would greet the visitor through that exalted gate, bowing and swaying in the wind like a receiving line of plumed courtiers welcoming guests to the throne of a king.\n\nThe drive led uncompromisingly to the broad, wide-porticoed entrance of the house. The impeccable white stucco finish and sloping red tile roof did not surprise her. She was somewhat knowledgeable about Spanish architecture and recognized the style as that of the Spanish grandees who had settled in this area. This one, however, rather than having the usual low, long lines with thick walls to preserve the maximum temperature comfort, was two-storied. A scrolled, black wrought-iron terrace ran the circumference of the top floor with French doors opening to it. Interesting, thought Cara, and very impressive, but somehow unseasoned. She recalled that Ryan's parents had died before they had occupied the house.\n\n\"Where is everyone?\" she asked Bill, for there was no sign or sound of any human activity. The young man was struggling with her big boxes impatiently. He set them down on the limestone porch and rang the doorbell. \"At the roundup of the remuda in the high country,\" he told her, his tone implying that that was where he should be.\n\nCara shaded her eyes to better see the low range of mountains toward the north, far beyond the oasis of the ranch.\n\n\"By high country, I suppose you mean over there?\" She pointed. Bill followed her finger with derisive amusement.\n\n\"Yes, ma'am. You're sure a greenhorn, ain't ya?\"\n\n\"I'm afraid so.\" She smiled, determined not to be nettled.\n\nBefore she could inquire about the whereabouts of Bill's employer, the wide double doors opened. A weathered, stern-faced Mexican woman of indeterminate age, smaller in stature even than Cara, stood surveying her with cold dispassion.\n\n\"She's here, Fiona!\" Bill announced grimly, as if she were some dreaded tornado they had been watching for on the horizon. Cara stared at the little woman. So this was Fiona, the housekeeper that Ryan had spoken of with such affection.\n\n\"So I see,\" the woman said abruptly, turning her glance from Cara. \"Bring her things in, Se\u00f1or Bill, and take them up to the first bedroom on the left. Then you better go on up to Diablo Canyon where the trap is. He caught Devil's Own, but that son of Satan slipped his noose and got away again. He is not in a good mood.\"\n\nWhich did not bode well for her, Cara thought, whatever it was that they were talking about. \"He\" must be Jeth Langston. She did not know whether to be happy or dismayed that she had been reprieved from an immediate meeting with the ruler of this isolated empire. Maybe the initial confrontation was better over as soon as possible so they could go their separate ways. The house looked big enough to allow that arrangement.\n\nThe woman called Fiona returned her inhospitable gaze to Cara as Bill brushed by carrying her things. \"You can come in,\" she said.\n\n\"Thank you,\" Cara responded pleasantly, and walked into the manor house of La Tierra Conquistada.\n\nShe was shocked immediately by its monastic severity. A tomb, Ryan had called it, and Cara felt obliged to agree with him. Immense and silent, the house had an almost menacing sterility about it, like a sanitarium. Furniture was sparse and utilitarian. No paintings or portraits enlivened the stark white walls. And everywhere, in all the rooms open to her view from the spacious entrance hall, she could see the gleaming gray terrazzo, creating an impression of cold, obsessive cleanliness.\n\n\"Come,\" said Fiona, waiting for her at the foot of the wide staircase.\n\nThe bedroom the housekeeper led her to, however, surprised her in another way, for it was obvious that feminine considerations had gone into its decor. Gray and yellow had been used, which Cara now assumed must be the colors of La Tierra. The inevitable gray tile was on the floor, but it had been covered with a large yellow area rug. Draperies and a bedspread in a sprigged print of the ranch's colors matched the window seats of the two small deep-set windows that flanked a slender French door. The big four-poster bed and other furniture\u2014a dressing table, writing desk, chairs for both, armoire, and chest of drawers\u2014were of mellow oak. Two upholstered yellow wingback chairs sat on either side of the fireplace, which had already been laid with a supply of wood\u2014scrub oak, Cara supposed, and remembered her comment on the day she had met Ryan at the airport.\n\nHe had known then that she would be in Texas now.\n\nThe memory sobered her surprised pleasure in the room and made her eyes reflective. Fiona had nearly slipped away before Cara realized she was leaving. \"Oh\u2014I\u2014thank you,\" she said quickly to the unsmiling woman. \"The room is quite lovely, very cozy and feminine. Is it someone's special room?\"\n\nFiona's hand was on the doorknob. \"You've missed lunch,\" she said with undisguised hostility, and was gone before Cara could reply.\n\nThe young woman was left facing the closed door, and in the silence, like bugs scampering out in a house when the occupants leave, all of her fears crawled out from the woodwork to assail her. She looked about her at the luggage and boxes that needed unpacking but was reluctant to begin the task. Reaching inside the neck of her sweater, she fingered the small seagull. A whole year in the remote silence of this monastery? Would there be no one who would talk with her, nowhere to go for relief from loneliness, from the animosity of the man in whose cold eyes she was condemned beyond any reclamation of her innocence?\n\nDeterminedly putting those questions from her mind, she shed her coat and took a penknife from her purse. Cutting the tape of one of the boxes, she began to search for the three items her mood dictated she unpack first.\n\nWhen Cara found the bag of sea glass, she held it up to the gentle March sunlight, which had come into her room to play. The pieces of glass glowed softly like a cache of dull gems recovered from the sea. She found another bag, this one filled with the broken pieces of the lobster traps that she and Ryan had collected on their last visit to Devereux Beach. Glancing at the fireplace, she thought how comforting the fire would be with these reminders from home added to the flames. Lastly, she found the enlarged snapshot that had belonged to Ryan. She gently touched the glass that covered his face, swallowing at the ache that filled her throat. How hard to believe that he was gone, that she would never see that boyish smile again or hear his laughter or feel his friendly arm around her shoulder. \"Do you trust me, Cara?\" he had asked. \"Remember always that I had only at heart the interests of those I loved.\"\n\nThat's you and me, Jeth, she said silently to the other man in the picture. I intend to carry out your brother's wish no matter how hard you make it for me to do so.\n\nA strange rumble that gave her the sensation the earth was shaking brought Cara's head up, and she held still a moment, listening. Placing the picture on the mantel, she went out on the terrace that faced the mountains, and an awesome and unforgettable sight met her eyes.\n\nA great herd of horses, their manes and tails flowing behind them, had come from the mountains to begin a thundering, dusty trek across the plains toward the ranch. Dozens of whooping, hollering cowboys on their own galloping mounts rode at their sides, keeping them maneuvered into a V-shaped formation by waving hats and coiled ropes. Cara looked for the objective toward which horses and men seemed to be headed, but her vantage point told her nothing. Compelled by the rough, masculine drama she was witnessing, she followed the terrace past other French doors until she had a view of the maze of corrals that had been erected beyond the grounds of the house.\n\nSo that's their destination, she thought, feeling suddenly sorry for the animals, whose life of freedom in the mountains would soon be at an end. As the horses drew abreast of the series of lanes that would feed them into the corrals, sounds that she had never imagined filled the air along with the dust. Leather saddles and chaps popped, ropes slapped, men cursed and yelled orders, horses whinnied and screamed.\n\nCara was so fascinated with the unusual scene that she was unaware of having been spotted on the terrace. First one man and then another jerked a head in her direction, but several minutes went by before she realized she was the cause of the gradual decline in activity. None of the men looking at her nodded or tipped his hat. They merely stared, and even from that distance Cara could read the stony unfriendliness on their weathered faces.\n\nA rider on a huge bay, whose back had been to her, turned his mount swiftly to see what had distracted the men, and Cara saw with a sharp intake of breath that the man was Jeth Langston. He sat immobile for a few seconds, staring straight at her from beneath the brim of his black hat, and Cara cursed herself for not having recognized that imperious back. Suddenly one of the horses, sensing an opportunity for freedom, reared and bolted from the orderly line. Other horses quickly followed suit, sending the men scrambling after them. Shouting an order, Jeth wheeled his horse sharply, at the same time uncoiling a rope from the saddle horn. With held breath she watched him streak after the escaping ringleader, his rope twirling above his head until he was close enough to the animal to throw a noose cleanly over its head. Jeth led the horse back to the line without looking in her direction again, and Cara saw that the men had resumed their work with even more fervor than before.\n\nShe turned quickly and sought the sanctuary of her room, wondering if there was to be no end to the trouble she caused Jeth Langston. A strange sensation had begun to play in the pit of her stomach, one that had nothing to do with the fear that she would be blamed for the mishap. The sunlight was chilly, but she was suffused with warmth, and her cheeks felt hot. She began to unpack with furious energy, trying to keep from her mind the sight of that dominant figure on horseback whose hatred she had felt even across the distance that lay between them.\nChapter Five\n\nThe sun was gone, the rough voices of the cowboys were silent, and her clothes hung neatly in her closet when a sharp rap came at her door. \"Yes?\" Cara called.\n\nThe door opened and Fiona stood in the doorway. She had changed from her faded jeans and flannel shirt into a neat cotton dress worn under a starched yellow apron. The unsmiling face, however, was the same. She announced with cold disdain, \"El Patr\u00f3n will see you now.\"\n\n\"I'll only be a few minutes,\" Cara said evenly, hoping the woman could not hear the rapid beating of her heart. El Patr\u00f3n? _El Patr\u00f3n?_\n\nThe woman said warningly, \"It would not be wise to keep him waiting, se\u00f1orita,\" and closed the door with a sharp click.\n\nHe can wait long enough for me to freshen up, declared Cara to herself. She was not going to face him from the disadvantage of a pale face. In the adjoining bathroom, she brushed her teeth and applied fresh makeup. She was still wearing the suit skirt and cowl-neck sweater from her flight and saw no need to change. She brushed her hair, letting it fall into the easy, natural style that she had come to enjoy. \"There,\" she said aloud in satisfaction to her reflection in the mirror when she was ready, then added tremulously, \"Go with me, Ryan.\"\n\nFiona was waiting for her at the foot of the gray-tiled stairs. The housekeeper's face registered nothing as she watched Cara descend. \"He's in there,\" she said, indicating with a stern jerk of her silver-threaded head two double doors off the living room.\n\nCara pondered Fiona uncertainly. Was Fiona to announce her? When the housekeeper moved off to other regions, with only a last disapproving glance over her shoulder, Cara decided she was on her own. Approaching the heavy, forbidding doors, she knocked decisively on one of them.\n\n\"Come in,\" came the deep voice that Cara remembered, and the young woman took one last steadying breath before entering Jeth Langston's inner sanctum.\n\nShe did not see him at first in the darkly paneled room. Besides the light cast from the fireplace, the only other illumination in the large room came from a lamp on a massive desk to the right of the door. The leather chair behind it was empty.\n\nJeth was standing at the fireplace, in the process of lighting a cigar with a glowing piece of kindling. He drew on it, and the smoke wafted across to her, its aroma recalling to Cara the memory of his arms around her and the demands of his lips. \"Good evening,\" she said.\n\nJeth turned to her, the gray eyes beneath the dark brows steady and assessing as she stood in the center of the room with her arms calmly at her sides. He looked as formidable as she had feared. The light from the fireplace cast its shadows over the granite face, throwing in relief the high cheekbones and the cold, metallic brilliance of his eyes. He had changed from the black range wear of the afternoon to casual slacks and shirt in a deep blue. Black boots of superb leather caught the flames in their sheen and added inches to his already intimidating height.\n\nThe rancher removed the cigar from his mouth and reached on the mantel for a glass of heavy cut crystal. \"Welcome to La Tierra Conquistada,\" he said, without meaning it, and lifted his glass in a mock toast. The amber liquor glowed like liquid fire.\n\n\"Thank you,\" Cara returned in a neutral tone.\n\nJeth's lip curled mirthlessly. \"I see you survived Bill's jeep ride.\"\n\n\"Yes. It was...typical of the welcome I expected.\"\n\n\"Then I'm glad we didn't disappoint you, Miss Martin.\"\n\n\"I doubt that you could do that, Mr. Langston.\"\n\nTheir gazes struck and sparked, like the opening parry of swords in battle. Finally Jeth walked to his desk and said amiably, \"Sit down, Miss Martin, sit down. This is likely to be a long conversation.\"\n\n\"I hope not,\" Cara rejoined briskly. \"I'm very tired.\"\n\n\"In that case a glass of wine would be in order. I'm aware that you do not drink hard liquor.\" Ignoring her faint look of surprise, he pressed a button on his desk and indicated with a motion of his cigar that she was to take a leather chair opposite his desk. Cara sat down on the edge of it. Her nerve endings were beginning to quiver. The subtly patronizing tone of his voice sent unpleasant tingles down her spine. Perhaps the wine would ease the tension gripping her neck and shoulders.\n\nJeth sat down and leaned indolently back in his sumptuous chair, the cigar in one hand, the fingers of his other toying with the glass on his desk. The diamonds in the black-faced ring winked derisively at her.\n\n\"Well, Miss Martin, how do you like your room? Adequate in size, I hope?\"\n\n\"Yes, indeed. It's a lovely room, very feminine. I\u2014it seems to have been decorated for someone special. Was it?\"\n\n\"No one in particular, Miss Martin. It's for female guests.\"\n\nIn any other household, the statement would have been innocuous, but Jeth's meaning did not escape her, and he had not meant it to. He took a draw on the cigar, hooding his eyes against the smoke and observing with cold amusement the two bright spots Cara felt flare to her cheeks. _Devil!_ she thought, acknowledging in spite of herself that many women would find his type of rugged virility and wolfish lean looks irresistible. No doubt the guest room she occupied was seldom vacant. \"How kind of you to let me have it, Mr. Langston,\" she returned with equanimity. \"I hope I won't be inconveniencing any of your women guests.\"\n\n\"You won't be, Miss Martin. I'll have no trouble finding a room to their liking when they visit.\"\n\nHer cheeks glowed brighter at this rejoinder, and she was relieved that Fiona entered just then with a tray bearing her glass of wine. \" _Gracias_ , Fiona,\" said Jeth as the housekeeper bent down to let Cara take it from the tray.\n\n\" _De nada_ , Patr\u00f3n,\" murmured Fiona and she left the room on silent feet.\n\nJeth sipped his drink while he waited for Cara to try the wine. She knew vintages, and this one was excellent. The bouquet tingled her nostrils pleasurably, and she said, \"How very nice,\" in honest appreciation after she had taken a generous sip.\n\n\"I'm glad you like it, Miss Martin. I've ordered a case for your enjoyment while you are here, knowing you to be quite a connoisseur of wines.\"\n\nCara showed her surprise. That was thoughtful of him, she granted. \"But how could you possibly know that if, as you say, Ryan never mentioned me to you?\"\n\n\"Well, now, Miss Martin,\" drawled Jeth, reaching for a brown folder that had been in evidence all the while on his desk, \"one of the advantages of wealth is that it provides the means to find out about one's enemies.\"\n\nA detective! He had hired a detective!\n\nCara placed the wineglass carefully on the slate portion of the desk's gleaming surface before standing up. The fury mounting within her did not affect the crystal clarity of her next words. \"How dare you, Mr. Langston! How dare you pry into my private life!\"\n\n\"I will dare anything I choose when it comes to you, Miss Martin. When I say anything, you'd better believe it, so sit down like a good girl before I prove that, too, is an open book for my enjoyment and...perusal.\"\n\nCara sat down, violet eyes flooded with anger and dismay. They shot daggers at him while, unperturbed, Jeth opened the folder and glanced at several pages before enlightening her of their contents. \"I understand that your parents died within the same year when you were still in college, Miss Martin. Is that right?\"\n\nCara did not reply. She picked up the wineglass and defiantly pushed herself back into the supple comfort of the leather chair. Why should she care what he knew about her background? There was nothing in it to incriminate her further.\n\n\"I don't blame you for not responding, Miss Martin,\" Jeth said understandingly. \"I can appreciate their loss must be painful to you. Let's push on to other areas. Your family incurred a large number of debts, which were left to you upon their deaths. Your father, it would appear, thought working for a living far too common a responsibility for the blue-blooded aesthete that he was. He preferred to live off the fortune made by his forebears, and when that ran out, to live on credit.\" Here Jeth paused but did not lift his eyes from the file. Cara interpreted the lull as an opportunity to defend her family and refute what he was reading. She had been right guessing that he would hold her parents in contempt. She chose to remain silent.\n\n\"Since your father did not take the precaution of providing life insurance, you were left virtually penniless. You took it upon yourself,\" Jeth went on, \"to clear your family's financial name rather than declare bankruptcy\u2014very noble of you\u2014and you worked very hard for several years as a librarian, which I understand is your second choice of vocations. Gradually the mountain of debts began to be whittled down. In that time you lived frugally, allowing yourself few luxuries\u2014\" Here Jeth glanced up at Cara and let his gaze linger pointedly on her expensive attire. \"And then,\" he continued smoothly, \"you met my brother...\" He contemplated her again, the affable manner gone, the pupils contracting into two deadly omens of danger.\n\nCara stiffened in her chair. \"What are you implying?\"\n\n\"I'm not implying anything, Miss Martin. I'm _stating_ that you saw my brother as a way to pay off your debts. You were suddenly tired of living in a one-room apartment. You were tired of your old clothes and your old car and of trying to stretch each paycheck to make ends meet. You knew Ryan was dying when you met him. You learned how much he respected integrity. God knows he'd seen little of it in his lifetime, especially from women. How cunning of you to make him think that you were too highly principled to let him pay your debts while he lived, but you sure as hell made sure he would pay them after his death, didn't you?\"\n\n\"No!\" Cara denied, jumping up from the chair. \"Those are terrible, unjust accusations! I didn't know Ryan was dying! I had no\u2014\" She had begun to say that she had no knowledge about the will, but she could not defend herself too strongly. She had to be careful in the heat of these confrontations not to blurt out why she was here. Cara moistened her lips. \"Mr. Langston, I know it looks like that, but\u2014\"\n\n\"Where did you get that outfit you're wearing, Miss Martin?\"\n\nCara looked askance at him. What did that have to do with anything? \"I beg your pardon?\"\n\n\"Ryan bought it for you, didn't he, as well as a\"\u2014he consulted a sheet in the dossier\u2014\"sable-lined raincoat that I believe you wore upon your arrival here. Isn't that correct?\"\n\nCara stared at him, stricken speechless. How cleanly the noose slipped over her head, just like the one thrown over the head of the hapless horse this afternoon.\n\n\"I have nothing to say to you,\" she declared at last. \"You may believe what you like.\" How could she convince him that Ryan had insisted on buying her the clothes? That she alone would continue to pay her family's debts with money she earned with her own hands? That she had no more intention of using La Tierra Conquistada to pay off what was her obligation than she could fly to the moon without a rocket. She would not waste her breath trying to tell this galling, overbearing, full-of-himself land baron _that_!\n\nThe wine on her empty stomach had made her tipsy, she realized, as she set the wineglass down. \"Good night, Mr. Langston. This interview is over.\"\n\n\"No, it isn't, Miss Martin, and if you don't sit down, I will come around this desk and make you sit down.\"\n\n\"You do, and I will scream bloody murder, you rude, arrogant...jerk! How somebody like you could be related to Ryan Langston should be documented as another wonder of the world. Not that anyone would be in the least interested outside Texas, which, in case you do not know, is not the end-all, be-all universe!\"\n\nWas she reeling? She rather thought so, because the desk and the dreadful man behind it had begun to weave before her blurred vision. The big leather chair, even as she looked at it, was all at once empty, and she wondered where the awful man could have gone when suddenly there he was beside her, taking her arm rather gently and lowering her into the chair. \"When did you eat last?\" he demanded gruffly.\n\n\"Last night, I think.\" She pursed her soft lips in complex thought. \"No, I had a bite of melon this morning. Why do you ask?\" She looked up in sudden suspicion at the tall form.\n\nWas it her imagination, or were the gray eyes actually glinting with something related to humor? \"Because you are drunk on a partial glass of wine. Finish it, Miss Martin, while we talk some more. Then you may go have your dinner. Fiona will bring a tray to your room.\"\n\nAn urgent question occurred to her. \"I won't be expected to stay in my room, will I? I will be allowed to come and go about the house?\" Not to do so was a prospect even bleaker than any she had imagined about living at La Tierra.\n\nJeth sat down on the edge of his desk near her. His eyes roamed over her at will, taking in the clean-lined beauty of her features, the glowing hair, the round fullness of her breasts softly outlined by the cashmere sweater. Cara was concentrating on her wine. He had said to drink it. What was there about him that commanded and others did?\n\n\"You may come and go as you like, Miss Martin, but stay away from the working compound. That little exercise this afternoon should be proof to you that you invite disaster wherever you are.\"\n\n\"I\u2014I'm sorry about this afternoon.\" Cara bit her lip, keeping her eyes on her glass. \"I had no idea that I would be the cause of those horses bolting. I was so far away.\"\n\n\"Not so far that you couldn't take the men's minds off their work, Miss Martin, something that I don't intend to allow while you're here. Cowboys suffer accidents when they're distracted; so do animals. I don't want you anywhere near the breaking corrals in the next few days, not even watching from the terrace.\"\n\n\"Is that what you're going to do to those horses you brought in this afternoon\u2014break them for riding?\"\n\n\"Yes. We need them for the roundup that will be taking place in the next few days.\"\n\nCara couldn't imagine why she said aloud the thought that next popped into her head. She did not think it was the wine that provoked her impudence but rather the way Jeth Langston sat on the desk, handing out orders to her like some feudal lord. \"How unfortunate for you,\" she said innocently, \"that you lost that horse you so had your heart set on capturing. His name says a lot about him. Devil's Own, isn't it?\"\n\nThe sudden stillness in him communicated itself to her, a coiling tension that had the potential to unleash like a whip. \"Watch it, Miss Martin,\" Jeth cautioned, his voice soft as a feather along her spine, raising goose bumps.\n\n\"Oh, I intend to,\" she assured him, deliberately ignoring his meaning. \"Next time I'll be prepared for the punch behind this marvelous wine.\"\n\nJeth observed the bewitchingly beautiful face she turned up to him. A muscle along his jaw twitched. With great control, he reached down and took the glass from her hand. \"You'd better leave now,\" he advised. \"Go on up to your room. I'll have Fiona bring your dinner up immediately.\"\n\n\"How kind of you,\" said Cara demurely, giving him a smile prompted more by the alcohol than by any sincerity. He followed her to the door, but at it she thought of something and turned unexpectedly. \"Mr. Langston\u2014oh!\"\n\nShe found herself caught in his arms. He held her steady against an immense, well-remembered chest and looked down at her almost indulgently. \"Yes, Miss Martin, what is it?\"\n\n\"Mr. Langston, where is Ryan buried?\"\n\nThe arms fell from her immediately, and she nearly fell against the heavy doors. \"At La Tierra,\" he informed her coldly, \"where he should have died.\"\n\nThat night Cara slept deeply but fitfully. Her dreams seesawed between two fuzzy realms in which she heard the whinnying lament of horses mixed with the cry of seagulls. Ryan appeared often. Each time he did, she cried his name in delight and ran excitedly after him down a long sandy shore only to have him disappear in the waves that washed his footprints from the sand. \"Ryan! Ryan!\" she called time and time again, flailing her arms in disappointment and bereavement. Once when she cried, someone else came to her, someone whose shadowy form hovered over her and spoke her name softly. The form bent and released her from the tentacles pulling her down into deep warm water where she wept for a nameless fulfillment eluding her heart.\n\nThe next morning Cara woke to a room bathed in sunlight. She had forgotten to pull the draperies the night before, which wasn't surprising when she remembered how exhaustedly she had climbed into bed. She lay in the warm nest of covers trying to remember where she was, and the events that had brought her here. The last vestiges of her dreams faded away and left her with the feeling that she had wrestled with them more than she had actually slept. Her neck and face felt sticky, as if she had cried sometime in the night.\n\nAlmost immediately after swinging her feet to the yellow rug she heard the sounds of horses and men. \"They're breaking the horses today,\" she remembered, and recalled that Jeth Langston had forbidden her to go anywhere near the breaking pens.\n\n\"Fine,\" she said aloud to Ryan's photograph on the bedside table. \"That leaves me free to explore the rest of the ranch without running into your brother!\"\n\nAfter she had dressed in slacks and tailored shirt, she wondered what to do about breakfast. Her dinner the night before had been excellent, but she had been too tired and woozy, she remembered ruefully, to eat much, so she was hungry this morning. Will I be allowed to eat outside my room? she wondered. Then a chilling discovery presented itself. She remembered with certainty leaving the photograph on the mantel just before going out to investigate the origin of that strange rumble. She had not moved it since, of that she was positive. Fiona had not even glanced at it when she came for her last night.\n\nThen what was it doing on her bedside table?\n\nPuzzled, Cara pulled on a sweater matching her blue slacks, then went out into the wide hall. Now that she did not have the disapproving Fiona at her shoulder, she could inspect her surroundings leisurely. Light streamed in through the series of arched windows in the white stucco wall facing her room, and she went to one and peered out. The layout below was as she should have expected. Indeed, La Tierra's big house had been built in the tradition of a Spanish grandee's hacienda, for its inner wall surrounded a tiled, verdant courtyard, enormous in size.\n\nAn Olympic-sized swimming pool, its clear blue water and deck of colorful tiles twinkling in the fresh morning sunlight, commanded the largest area. Set back from it was a cabana with a red Spanish-tiled roof like that of the house, and nearby was an entertainment area with a stage. Across on the far side was a commodious brick pit for barbecuing, its gleaming enamel hood the same bright yellow as that of the table umbrellas dotting the deck. Other matching patio furniture and an abundance of tropical plants providing greenery and shade completed what in Cara's mind was an opulent picture of Southwestern relaxation.\n\nThe pool reminded Cara that Jeth had once aspired to become an Olympic champion, a dream that had forever been deferred when he'd had to assume responsibility for La Tierra and a little brother named Ryan. Ryan had told her that Jeth still swam religiously every day, no matter what the weather, which accounted for the corded, well-toned body of the man and the lack of a pale forehead due to the constant wearing of a hat in the sun.\n\nShe followed the horseshoe corridor to the other wing, then came back to stand before heavy double doors leading to a room the width of the top floor. This has to be _his_ room, she thought, wishing that her room was in the other wing so that she would not have to hear him pass by her door each night. She had not heard him last night, apparently having fallen asleep before he came to bed.\n\nGoing down the stairs, Cara encountered a fresh-faced young Mexican woman who actually smiled at her. She was carrying an armload of fresh sheets and towels. \" _Buenos dias_ , se\u00f1orita,\" the young woman greeted her, and Cara's face showed her pleasure at the first friendliness she had been shown.\n\n\"Good morning to you,\" she responded cheerfully, but when she would have introduced herself, the young woman hurried away up the stairs as if she had been warned about speaking to the yellow-haired intruder.\n\nIgnoring the sharp little pain from the rebuff, Cara went on down the stairs in the direction she guessed the kitchen to be.\n\nShe found Fiona at the gleaming kitchen counter busily chopping peppers and onions. The tantalizing smell of coffee came to her. \"Good morning, Fiona,\" she offered politely. \"May I help myself to coffee?\"\n\nFor answer, Fiona pointed with her knife to a large stainless pot on the stove. \"Thank you,\" Cara said, and then, \"Fiona, could you tell me what's expected of me concerning my meals? Do I have to eat in my room or may I eat here in the kitchen? I'm quite sure\u2014er\u2014El Patr\u00f3n would not care for my company at mealtimes.\"\n\nWhen Fiona did not answer, Cara persisted. \"It would be silly for you to have to climb those stairs bringing my meals to me. As a matter of fact, I can even prepare my own\u2014\"\n\nThat statement got the attention of the impassive-faced Fiona. The eyes she turned on Cara had fire in them. \" _My_ kitchen!\" she declared, pointing the sharp knife at herself for emphasis. \"Nobody cooks here but me. You may eat here, but you\u2014you don't cook here!\"\n\nCara smiled at the feisty little woman and tried what little Spanish she knew. \" _Gracias. Yo comprendo._ \"\n\nWithout asking her, the housekeeper prepared for Cara a fluffy omelet containing a sharp Mexican cheese and topped with a spicy sauce made from onions and fresh tomatoes and peppers. With it were a small breakfast steak, so tender that Cara could cut it with a fork, and steaming, freshly made flour tortillas.\n\n\"Oh my,\" sighed Cara appreciatively when she had finished, \"you cook as superbly as you keep this house, Fiona.\"\n\nFiona did not respond to Cara's compliment. She had kept her back to the Bostonian all the while she had been eating, but Cara sensed the woman was aware of every bite she took.\n\nThe housekeeper was obviously in charge of all matters pertaining to household personnel and maintenance, with the kitchen the office from which she dispensed her orders. Several Mexican workmen came into the kitchen by the back door while Cara was eating. They raised brows when they saw her, then nodded curtly and addressed Fiona in Spanish. The same was true of the maids who wandered in and out. Pointedly ignoring Cara, they discussed in their native tongue their duties for the day as well as, she was certain, their thoughts and opinions about the intruder who sat at the kitchen table.\n\nWith a sigh, Cara took her coffee through a swinging door into a large formal dining room. Again the icy atmosphere of the white walls and gray-tiled floors oppressed her. What a remote, cold, unfriendly house, she thought. No wonder Ryan had called his home a tomb.\n\nThe dining room let out through carved double doors into an alcove that probably had been meant for a sitting room. The furniture, though costly, appeared never used. Even the spring sun shining through a wide arched window could not dispel the pervasively Spartan atmosphere.\n\nCara strolled across the entrance hall to the living room she had passed through last night to Jeth's study. Her eyes were taking in the ample proportions of the austere room when an object in a far corner made her gasp with disbelief. A Steinway! A real, honest-to-goodness Steinway! She set her coffee down on a marble-topped table and flew to the majestic grand piano that sat augustly in a pool of sunlight.\n\n\"Oh...\" she breathed, hardly daring to believe her eyes. Reverently, she pushed the cover from the keyboard. Her fingers gently touched the ivory keys without striking them, savoring the moment when she would summon forth the quality of tone for which this aristocrat of all pianos was renowned. Cara pulled out the bench and sat down. She flexed her hands\u2014it had been a long time since she had played\u2014then ran her fingers up and down the keyboard in a series of chromatic scales to limber both her fingers and the tone of the piano.\n\nBorne away on the strains of a Chopin concerto, she had only been playing a short while when suddenly there appeared beside the bench an incensed and ferocious Fiona. Cara looked up quizzically, barely removing her hands before the housekeeper vehemently pulled the lid down over the keyboard. \"What's the matter with you?\" Cara cried, beginning to get angry, too.\n\n\"Se\u00f1ora Langston's!\" the housekeeper explained explosively. \"No touch! Se\u00f1ora Langston's! Not Ryan's whore!\"\n\nCara was up from the bench instantly, her body shaking in rage at the presumption of this woman to call her such a name, but more importantly to deny her the exquisite release the piano would have provided from the horrors of her confinement.\n\nCara faced the housekeeper levelly. \"Don't you ever do that to me again, Fiona,\" she said in a quiet voice that carried conviction. \"And don't you ever refer to me by that name again\u2014ever!\" Leaving the housekeeper standing stonily at the piano, Cara marched from the room and up the stairs. Once in her room, trembling with fury, she searched in the closet for a warm jacket. She saw that her room had been tidied, the bed made. She could still hear sounds of loud activity coming from the corrals, but she had to get out of this house. Surely there was somewhere on this vast ranch where she could go without causing trouble.\n\nSkirting away from the house, Cara walked in the direction of a tree-shaded rise of land some distance away. Hands in pockets, face up to welcome the sun and the dry, brisk wind that blew across the plains, she tried to deal with the aching disappointment that welled inside her. How could she live in a house with a Steinway and not play it? It was a crime to regard an instrument like that more as a monument to the dead than as a source of joy to the living.\n\nUntil now she had not realized how much she missed the piano that once graced her childhood home. On it she had learned to play the music that would later bring such solace to her life. The day the Steinway had been sold, she walked the beach for hours, mourning the loss of an old friend.\n\nCara was sickened, too, by the encounter with Fiona. She had hoped she could come to care for the irascible little woman whose industry and devotion to La Tierra impressed her. She doubted now whether the housekeeper could ever be induced to like the outsider from Boston, the woman she thought of contemptuously as Ryan's whore.\n\nCara reached the foot of the small hill and was intrigued by its number of trees and lush carpet of young grass when the surrounding land stretched bare and treeless. Staring up at its top, she glimpsed between the green, feathery branches of the mesquite trees something that looked like a wrought-iron fence, and her breath caught. Jeth's answer came back to her from the night before when she had asked where Ryan was buried: \"At La Tierra\u2014where he should have died.\"\n\nCara, certain of what she would find, climbed the hill to the black iron enclosure of a small, private cemetery. New spring grass grew tenderly between the stones to the dead, and Cara gave a sudden, startled cry when she saw the fresh earth that indicated a new grave. A monument, yet unbleached by wind and storms and time, rested at its head. Cara stumbled forward calling, \"Ryan! I've come, Ryan. I've come.\"\nChapter Six\n\nShe did not know how long she had sat on the ground with knees drawn to her chest, forehead resting on folded arms, before she became aware of a pair of black boots and silver spurs planted apart on the other side of Ryan's grave.\n\n\"Oh!\" Cara exclaimed, caught by surprise, and blinked up at the dark countenance of Jeth Langston. He frowned at her from beneath the firm set of his black Stetson, and for the first few seconds she did not know who or what he was. With the sun behind him, he looked menacing in black leather chaps and vest, and she thought at first that he was some angry god come to wreak his vengeance.\n\nCara got to her feet without his offering to help and braced herself for what was to come. When neither spoke after several seconds, she offered lightly, \"You go first.\"\n\n\"Miss Martin, you cannot seem to stay out of trouble.\"\n\n\"Well, so it seems. I'm sure you're referring to my run-in with Fiona over the Steinway a while ago. She must have gone immediately to tell on me, although I will say that surprises me. I would have bet that she was one to fight her own battles.\"\n\n\"You would have won that bet, Miss Martin. Fiona did not _tattle_ on you. I overheard the two of you when I came to see who was playing my mother's piano.\"\n\nCara brushed at the sand adhering to the seat of her slacks. \"So which am I to be strung up for\u2014playing your mother's piano or insulting Fiona's sense of propriety?\"\n\n\"Neither, Miss Martin,\" Jeth answered in cold rebuke. \"But for an intelligent girl, your willful ignorance of the shaky position you're in at La Tierra is astounding.\"\n\n\"I understand clearly the shaky position I'm in\u2014I saw only that it was a Steinway,\" Cara defended soberly. \"It didn't occur to me that it might have been your mother's.\"\n\n\"In that case, don't take liberties with the possessions of my house, Miss Martin, not unless I give you permission. Is that clear?\"\n\n\"Quite,\" said Cara, finding it hard to look at him against the light. Her eyes stung miserably. She wanted to believe the sun or the dry, cold air responsible, but her honesty would not permit it. There was something about Jeth that transcended her growing fear of him, that forced her to admit that he was a man who stirred strange and bewildering emotions within her that she did not understand. \"Is that all?\" she asked him.\n\n\"No, Miss Martin, it is not.\" The words were exactly delivered. \"Since this game is being played with a deck stacked in favor of the house\u2014and since you can't seem to figure out the obvious for yourself\u2014I am going to give you a little advice. You need Fiona. Don't antagonize her. You can avoid it if you understand that she reveres the Langstons, especially the memory of my mother. She is enraged that an outsider like you should try to usurp what belongs to the family. When you sat down at that piano this morning, when you began to play her patrona's most loved possession, you committed what to her amounted to a sacrilege\u2014\"\n\n\"I get the picture!\" Cara broke in, unable to bear any more. She turned her head away so that he could not see the dejection sweeping through her. Hands in pockets, jacket billowing open, slight form buffeted by the wind, Cara presented a vulnerable picture to the tall, powerful man looking down at her. He saw how the sun played in the waves of her tossed hair, exposed the clear purity of her skin and the tender curve of her throat. He saw her blink at the sting of tears she was too proud to shed.\n\nJeth said in a less steely tone, \"Miss Martin, sell Ryan's share to me and leave La Tierra. We'll call it a draw, and you will have heard the last of me. You would have no need to ever fear me again.\"\n\nCara shook her head obstinately. \"No.\"\n\n\"Then you're asking to be broken, you know,\" he warned her gently, \"just like all the other enemies of La Tierra have been.\"\n\n\"Rather like those poor creatures you brought down from the mountains will be, I suppose,\" Cara remarked with distaste, thinking of the proud, spirited horses who right now were feeling the grip of saddles, the dig of spurs.\n\n\"Not at all. Here at La Tierra we're rough with horses, but never punishing. With you, I would be both. Once our horses have earned their yearly keep during the roundup, they're set free to roam until the next one. But you, Miss Martin, I would never free to enjoy the spoils of your relationship to Ryan. I would make sure you carried the brand of La Tierra all of your life.\"\n\nCara, who had kept her head averted, faced him defiantly, her blood running cold. \"You would never get the land back.\"\n\n\"Oh, yes, I would, Miss Martin. Have no delusions about that.\" He turned to go.\n\n\"There is one horse that got away from you, that doesn't wear your brand\u2014one that you won't be breaking for the roundup!\" The words were out before she could stop them. Shocked at her outburst, she watched in dismay as the rancher paused, then turned slowly. His eyes gleamed with surprise and the thrill of challenge.\n\n\"You, Cara Martin, will not be so fortunate. Am I to take that as your final answer to my request that you leave La Tierra?\"\n\n\"Yes,\" she said quickly. She would give anything to take back her taunt. What a fool she had been to ruffle his king-of-the-walk feathers. What could she gain from it?\n\n\"Then I'm looking forward to the pleasure of your company under my roof, Miss Martin. This evening you will have dinner with me. Wear something red, a most appropriate color for you in more ways than one. I'm sure that among all those dresses Ryan bought you there is something suitable.\"\n\n\"No,\" Cara said firmly.\n\n\"You will if you want to eat. Afterward you will play for me. Come to my study at seven and we'll have a drink. Now if you'll excuse me\u2014\" He touched his hat brim in mock respect. \"I've wasted enough time for one day.\" Once again he turned to go.\n\n\"Mr. Langston?\"\n\nWith a sigh of impatience the rancher paused, keeping the broad back to her. \"Yes, Miss Martin?\"\n\n\"Is...this place off-limits too?\"\n\nWithout turning he answered, \"No, Miss Martin, not unless I'm here.\"\n\nCara watched him descend the hill with supple ease to the untethered bay waiting patiently below. The man was so sure of himself, so sure of her. She was sure of neither.\n\nCara spent the rest of the day in her room. She wrote a letter to Harold St. Clair assuring him that she was still in one piece, infusing her comments with a humor she did not feel. Afterward she thumbed through a book she had brought with her on conversational Spanish, thinking that if nothing else was gained from the year, she could at least learn a new language. But her attention persisted in wandering, and after trying to read a few pages she put the book down and went outside to sit on the terrace in the sun. Her thoughts drifted to Ryan. \"Do you trust me?\" he had asked as he lay dying. Even now, with all of her heart, she did. But why had he left her his share of La Tierra? Why had he made her promise to come here, where he knew she would be at the mercy of his brother's vengeance? Had Ryan hoped to play matchmaker? But that was preposterous under the circumstances. He had known how she felt about men like his brother, and he would certainly have foreseen how his brother would regard and react toward her. The situation was impossible.\n\nThe lunch hour came and passed, and Cara's hunger pains reminded her of the evening ahead. Her pride rebelled that in order to eat she had to join Jeth Langston for dinner, but she knew that the rancher was perfectly capable of letting her go hungry. Cara flipped through the dresses hanging in her closet and found the red dinner dress she had worn in Dallas. Angry at her cowardice, she admitted she did not have the courage not to wear it.\n\nLate in the afternoon when she was tired of her room, she took a stroll down the horseshoe hall to the other wing. A door was open to one of the bedrooms, and since she knew that no one occupied this floor but herself and Jeth, she peered in.\n\n\"Ryan's room!\" she exclaimed to the lofty silence, for even though it was a cavernous room, it wasn't totally devoid of the warm, vital presence of the man who had once lived here. At one end was a small library she knew Jeth had ordered built for his brother when Ryan became interested in law. The shelves still contained some of his books.\n\nThe room echoed a loneliness that struck an unhealed wound, and she left quickly, closing the door behind her. She felt closer to Ryan at the cemetery where the wind blew freely across the wide Texas plains.\n\nA splash in the swimming pool brought her to one of the arched windows overlooking the courtyard. Looking down on the pool, she saw a long, tanned figure swimming underwater. A dark head surfaced, wet and sleek as a seal's. She watched him begin a routine of laps, cutting the water effortlessly with long, powerful arms, until suddenly she had to back away from the window, unable to bear watching him any longer. Her heart had begun a fierce beat. A strange longing throbbed in her stomach, forcing her to lean against the cool surface of the stucco wall to steady her breath. A sense of helpless anger flooded her. Was not even her own body to be an ally in this alien house against the enemy below? Would it, too, seek to destroy her?\n\nAt precisely one minute until seven, Cara descended the stairs. She had not heard Jeth come up to his room from the pool, but she did hear him go down. He had passed her door as she was finishing dressing, and her heartbeat stilled when she heard the firm tread of his boots striking the tiled corridor.\n\nNot even the knowledge that she looked her best in the red dress that Harold had admired could inspire Cara with confidence. She drew in deeply as she knocked on one of the double doors, and pressed to her breast the framed photograph that she had brought with her from Boston.\n\nJeth himself opened the door. He was dressed in black tonight, the Western cut of his attire emphasizing the broad shoulders and trim waist and hips. With the lamplight behind him, he filled the doorway with a sinister presence. \"Well, good evening, Miss Martin,\" he drawled mockingly, the cool gray eyes marking the red dress, then sliding down her from head to foot. \"How nice of you to come.\"\n\n\"Did I have a choice?\"\n\n\"No,\" he said dryly, \"but let us observe the amenities as if you did.\" He moved aside just enough to allow her room to pass. \"I suggest you sit by the fire. A norther is coming out of the Panhandle and will be here before we sit down to dinner.\"\n\nCara was happy to do so. She was chilled through and through, and her knees felt trembly. Taking a seat in one of the two tall-backed chairs flanking the fireplace, she asked, \"When does spring actually arrive in Texas?\"\n\nJeth had gone to the bar where a silver wine cooler waited with the exposed neck of what Cara assumed was a bottle of the wine she had been served the night before. She watched as he withdrew a crystal goblet from a bed of ice, then uncorked the napkin-wrapped bottle. \"That depends on what part of Texas you're asking about,\" he informed her, pouring a clear, sparkling stream of wine into the glass. He refilled his own with bourbon and brought both to the fire.\n\n\"Thank you,\" she said, taking care not to touch his fingers when he handed her the glass. \"You were saying?\"\n\n\"Texas is a big state, Miss Martin. In our coastal areas, spring has already arrived. In the Panhandle it won't come until the last of May. Here we'll be lucky to see our last frost by Easter.\"\n\n\"Texas can be quite overwhelming.\" She smiled politely, hoping she didn't sound critical. The state had begun to fascinate her, and she wanted to know more about it.\n\n\"Like its people?\" Jeth asked with a trace of mockery, settling in the chair opposite her.\n\n\"I never found Ryan overwhelming,\" she said. \"That reminds me. I brought this for you. It meant a great deal to Ryan. He kept it on the mantel of his town house.\" She handed Jeth the photograph.\n\nJeth reached for it, the firelight flashing on the diamond brand in his ring. \"Yes,\" he mused, studying it. \"I saw this last night.\"\n\nCara straightened in her chair. \"You were the one who moved it! Then\u2014then you were the shadow in my dream...You were the one who...rescued me.\"\n\n\"Did I?\" Jeth raised a cynical eyebrow. \"We fight such awesome demons in our dreams, don't we? You were crying out in your sleep. I was on my way to my room and heard you. You sounded desperate, so I went in. You were wound in the covers, so I loosened them. That's when I saw this picture.\"\n\nCara couldn't resist saying in surprise, \"I'm amazed that you bothered.\"\n\nImmersed in the study of the picture, Jeth said, \"I might not have except that you were crying Ryan's name over and over. I've had a few of those nights lately myself.\"\n\n\"Of course you have,\" Cara said quietly, feeling sympathy for him. She knew how lonely it was to be facing a future with no family whatever. Ryan had been right about Jeth. This man needed a loving wife who would give him children to make this austere house a home.\n\nJeth placed the picture on the wide stone hearth, then stunned her by saying, \"You have beautiful breasts. You are well-endowed for someone of your small frame, aren't you?\"\n\nRemembering the flimsy nightgown she had pulled on last night in her exhausted haste to get to bed, Cara choked on the wine, almost spilling some of it on the red dress. \"How dare you!\" she sputtered, holding the dripping glass over the hearth. Jeth produced another of the white lawn handkerchiefs like the one he had given her in Dallas, which she had not yet returned. Taking it, she said indignantly, \"You had no right to\u2014to look me over while I slept!\"\n\n\"Why not?\" he asked calmly. \"And Miss Martin, don't use the phrase 'how dare you' again. You are in my state, on my land, in my house, and I will dare anything I damn well please. Thank you for the picture. Tell me about Ryan. Were you with him when he died?\"\n\nCara's head swam, and not from the wine. This man had the power to provoke the most quixotic feelings within her. In the space of a few moments, he had roused her fear, hatred, sympathy, anger, and now she found herself wanting desperately to comfort him, to tell him how much he had been loved by Ryan\u2014that, like him, she did not understand why his brother had not told him about his illness, why he had left to her what rightfully belonged to him. But she could not share those thoughts with this skeptical man, and so she carefully took a sip of her wine and answered simply, \"Yes, I was with him, Mr. Langston. He was in some pain, and had to be sedated the last few days of his life. But at the end he was very lucid. He spoke of you often, and I\u2014I know he loved you very much.\"\n\nJeth tossed off his drink in one swallow. A smile, cold as an Atlantic swell, curved the well-shaped mouth. \"He loved you more, Cara Martin, or he would never have stayed in Boston to die\u2014or left half of La Tierra to you.\"\n\nCara had no reply to this. She sat in uneasy silence, fingering the small seagull charm on the gold chain she wore with her single strand of pearls. When she offered no response, Jeth asked abruptly, \"Is that the red dress you wore with Harold St. Clair the other night?\"\n\n\"Why\u2014why, yes. How did you know?\"\n\n\"There's little about you that I don't know, Miss Martin\u2014or can't accurately guess. I'm sure he thought you looked ravishing in it.\"\n\n\"He liked it, yes,\" Cara replied evenly, becoming apprehensive about the direction of this conversation.\n\n\"And that seagull you keep reaching for\u2014he gave that to you, too, didn't he?\"\n\n\"It was a token of friendship, Mr. Langston\u2014something to remind me of home.\"\n\n\"I'm sure. And the pearls? Did he give those to you or were they from Ryan?\"\n\n\"Neither. These pearls came from the Orient on one of the first clipper ships over a century ago. They've been in my family ever since. They were given to me on my sixteenth birthday.\"\n\n\"Ah.\" Jeth sipped his drink. \"You've come a long way since sweet sixteen, not in years perhaps, but for sure in kisses and...other skills. Did you sleep with Ryan?\"\n\nNow that was a hell of a question! he reproached himself. Why had he asked her that? Of course she had slept with Ryan. There was no way his brother wouldn't have had her in his bed. Yet part of him wanted to hear her deny it, while the other\u2014the part of him that tolerated no threat to his empire\u2014knew that such a denial would continue to seal her doom.\n\nCara began hesitantly, wondering how honestly to answer him without compromising her further in his eyes. \"I\u2014yes, we did sleep together, Mr. Langston, but not in the way you imagine. Ryan was very ill. We shared the same bed, but not to\u2014to make love. It was just a matter of comforting each other, of being near each other through the long nights. We used to joke about how no one\u2014certainly not you\u2014would believe that we could do that and not\u2014and not\u2014\"\n\nCara fell silent as she sensed the dangerous stillness that had possessed Jeth. In the richly black clothes, his eyes upon her steady and penetrating, the powerful body tensed in deep attention, he reminded her of a jungle cat watching his prey. \"I'm telling you the truth!\" she declared, the prolonged silence snapping at her nerve endings.\n\n\"You're trying to sell me a bunch of bull!\" Jeth thundered, getting up to return to the bar. His back to her, Jeth in frustration sent his empty glass skimming along the polished surface until it came to rest in a padded leather corner. God. How could lips like those, eyes that guileless, concoct such a story and expect him to believe it? Since she was not a naive or stupid woman, she must be very sure of herself to take him on.\n\n\"Why can't you believe me?\" she pleaded to his back, her voice very soft and small, projecting still the role of the innocent. He sighed.\n\n\"Because I knew my brother, Miss Martin. He would never have kept his hands off you. Why would he want to?\" From the way she rose to meet him, clutching the handkerchief, he must present a terrible sight.\n\n\"He never touched me!\" she cried as Jeth approached her. \"You know, Mr. Langston, I was very hurt when you told me that Ryan had not mentioned me to you. I wondered why not. We were the best of friends, in the deepest, finest way. But now I can understand why he didn't discuss our relationship with you. A man like you isn't capable of understanding the way it was between us.\"\n\n\"You're frightened, Miss Martin. I wonder why. Is it because you think I might harm you? But I promised I wouldn't hurt you, remember? Do you think I'm not a man of my word?\"\n\n\"I\u2014oh, I believe you're a man of your word, but you are...misguided. You have the wrong impression about Ryan and me.\"\n\n\"Why should it matter to you that I think Ryan was your lover?\"\n\n\"Well, because he wasn't!\" Her eyes, he saw, had taken on the color of smoky lavender and widened in the alarm of an animal being circled by a predator. \"I don't know why it's important to me that you believe that, but it is!\"\n\n\"Don't you know why, Miss Martin?\" Jeth's voice was as smooth as the sliding of a snake across her skin. Before she could register his next intention, he had reached out and pulled her into the inescapable confines of his arms.\n\nCara struggled like a wild thing caught in a trap, but it was too late. Jeth's embrace pinned her arms to her sides, and the boots protected his shins from the kick of her evening shoes.\n\n\"It would seem that it has become necessary one more time to make clear to you that I know what your game is, Miss Martin\u2014why you are so hell-bent on staying here.\"\n\n\"You promised!\" Cara choked, her heart beating like a maddened bird.\n\n\"I promised not to hurt you. Am I hurting you?\"\n\n\"This is physical abuse.\"\n\n\"Nonsense. The denial of what you want is physical abuse. Come here.\"\n\nHis dark head was descending as he spoke, and Cara, held steady by the grip of his hand entwined in her hair, could not escape a kiss she knew was meant to punish and humiliate. She squeezed her eyes tightly and compressed her lips before he could reach them. Her hands gripped his back, intending to dig nails of protest into his flesh. She waited. The kiss did not come. A harsh chuckle reached her ears. \"Miss Martin, you are the world's greatest actress.\"\n\nThe violet eyes flew open to find the kiss still hovering, the sensuous lips quirked in a slight grin.\n\n\"Is all of this necessary?\" she whispered. \"Can't we just have a nice dinner?\"\n\n\"Are you hungry?\" he asked softly, kissing each side of her mouth.\n\n\"Very,\" she answered, trying to ignore the traitorous flutter beginning in the pit of her stomach. Her heavy lashes lowered, and Jeth drew her deeper into the cradle of his shoulder. The contact of his breath on her earlobe sent a shock of awareness through her. Warm and sensuous, the fingers that he thrust through her hair moved to cushion her neck for the gentle play of his lips over her features. Slowly Cara began to yield to the soothing caress of his other hand moving down to her hips.\n\n\"Then let me feed you,\" Jeth suggested huskily, as his hand moved down her back. Expertly he clasped her to the waiting convexity of his body.\n\nCara yelped in dismay, but not before he had felt her response. \"Cara\u2014\" Jeth said her name like a prayer, finding her mouth and drinking of it hungrily, cupping her tightly to him.\n\nEnveloped in the sensual male warmth of him, conscious only of her need for him, Cara could not resist the desire that surged up from the most hidden depths of her being to meet the tide of Jeth's passion. Her fingers dug into his hard-muscled back, not to hurt but to press him closer, closer, with a terrible and urgent craving.\n\nThen, unbelievably\u2014just as she was poised on a crest of incredible longing\u2014the hands that had caressed her with such finesse clamped a grip on her forearms and pushed her violently away.\n\nSurprise exploded within her. Jeth's eyes bore into her with glittering anger. \"Now will you deny that you and Ryan were lovers?\" he raged. \"You're too hot to stay out of many beds, Miss Martin\u2014certainly not Ryan's, so don't ply me with any more Florence Nightingale stories. What really galls me is that you don't even have enough feeling for Ryan's memory to admit that you had an affair with him. You still hope to buy some kind of chance with me if I can be convinced that Ryan never touched you\u2014though what kind of chance only that devious little mind of yours knows. Well you can forget it, Miss Martin. I don't take any man's leavings, not even Ryan's.\"\n\nIn a daze, Cara heard him. How could one human being have this kind of devastating power over another? What is happening to me? she wondered, recognizing the searing pain of grief.\n\n\"Cara?\" Jeth spoke her name warningly and shook her for her attention. \"Whatever role you're playing at now, cut it out, do you hear me?\"\n\n\"Yes,\" she answered dully.\n\n\"Look at me!\" he ordered. Cara lifted dazed eyes. \"Tomorrow morning dress warmly in the oldest clothes you have. Come out to the breaking corrals. You are not getting a free ride this year, lady, no matter what you may have hoped. This is a working ranch and everybody works here, including you.\"\n\nWhen she said nothing, he left her to open a closet door in the paneled wall, taking out a split-cowhide jacket lined with fleece. He buttoned into it while she watched, then returned to her carrying the fawn Stetson with the black band.\n\n\"I won't be joining you for dinner. I seem to have lost my appetite. I hope you haven't. You'll need your stamina tomorrow. Enjoy your meal and the piano, and be at the corrals by eight o'clock. Pleasant dreams.\"\n\nCara watched him stride from the room without looking back, much as he had left her the first day they met. At her feet was another of the white handkerchiefs, which, again, she picked up and held to her lips. This time, however, he had left her with the deepest pain and confusion she had ever known.\n\nThe next morning Cara threaded through the compound of ranch buildings to the maze of fences she assumed were the breaking corrals. Cowhands were assembled around one huge corral, unmindful of the rising dust and fresh manure that choked the brisk air. As she drew closer, she guessed that over fifty horses were in the enclosure, and standing in their midst, his back to her, was a tall, slim man with a clipboard. He happened to turn as she approached and surprised her with a nod.\n\nShe scanned the group of men apprehensively. Nearly all of them were eyeing her, some with boldly inquisitive eyes, others in embarrassment. Where in the world was Jeth? Even his contempt was better than standing awkwardly before the curious gazes of thirty or more strange men.\n\n\"Miss Martin?\"\n\nCara turned in relief at the sound of the familiar voice. Jeth had come up behind her, apparently from a long, low building that bore the name \"Feedtrough\" over its entrance. \"Mr. Langston, what am I doing here?\" she demanded in a low voice.\n\n\"Right now, nothing. In a minute, you'll be working.\" The cold, clear eyes raked her up and down. \"Those are certainly not the kind of clothes you'll be needing for the roundup, Miss Martin. We'll have to do something about that this afternoon, if there's time.\"\n\nSome of the men were within hearing distance, and she felt their sharp surprise along with the jolt that hit her. \"What do you mean?\" she asked tensely, determined to keep her composure, but a tremor of foreboding rippled through her. Jeth's face wore the cynically hard expression that she had come to recognize as trouble for her.\n\n\"You're going on a roundup with us, Miss Martin\u2014as Leon Sawyer's assistant. He's our cook. The man who usually helps him is in the hospital recovering from an emergency appendectomy. I can't spare a man to replace him, so I've decided that you will go in his place. When Toby gets well enough to ride out, he'll relieve you. But until he does, consider the next month to six weeks of your life reserved.\"\n\nBy now, Cara and Jeth had everyone's attention but the horses. They were cantering about the corral, stirring up dust around the tall man in the center. The men appeared to be busy with ropes and saddles, but Cara knew they were listening. \"You can't be serious,\" she said between her teeth.\n\n\"Do I not look serious, Miss Martin?\" Jeth took a step closer to her.\n\nCara restrained an impulse to draw back. \"I thought you didn't want me to distract the men.\"\n\n\"You won't. That I can promise you. I've just been to tell Leon the good news. He wasn't any more excited about it than you are, which is too bad because I like to keep the cook happy.\"\n\n\"Mr. Langston, you surely don't mean what you're saying.\"\n\n\"I always mean what I say, Miss Martin, as you will discover to your grief if you decide to sit this one out. Your only alternative is to leave La Tierra\u2014 _after_ you sign the papers releasing Ryan's half of the ranch.\"\n\nCara stared up at him. The sensuous mouth was an adamant line, the gray eyes the color of slate. There was no doubt in her mind that he would force her to go on the roundup. \"So that's what this is all about,\" she said quietly.\n\n\"You got it, lady. I'm giving you one more minute of my very valuable time to make up your mind which it's to be. I'll take as your answer either your heading back to the house or going into the Feedtrough to find Leon.\"\n\nIt took Cara less than the minute Jeth gave her to make her decision. With a last defiant look at him, she did an about-face and marched into the Feedtrough.\nChapter Seven\n\nThe Feedtrough was the ranch kitchen where food was prepared and served family style to the men who worked for La Tierra. Leon Sawyer, the cook and chuckwagon master, ruled over his spotless domain with an absolute authority that even Jeth Langston took care not to breach. The appendectomy that had claimed Leon's helpmate and dishwasher the afternoon before had left the cook facing the coming roundup in a foul temper.\n\nCara walked into the Feedtrough to interrupt a profane berating the irascible cook was heaping onto the thin shoulders of a young Mexican cowboy picked as a temporary replacement.\n\n\"Now get outta here!\" the small, wiry man finished, taking a booted swipe at the fast-retreating backside of the hapless young cowboy. \"No-good worthless young pup!\" he added, scowling at the swinging back doors through which the cowboy escaped.\n\nCara struggled not to grin. Leon was exactly as Ryan had described. He was indeed a caricature of the legendary chuck-wagon boss of the late-night movies. Behind the ample white apron, the jutting, whiskery chin, the fighting-rooster stance, there had to beat the soft heart of a Gabby Hayes.\n\n\"Hello,\" she said, and Leon spun around on his boot heels to discover Cara. With eyes as blue and round as robin eggs, he peered at her over the tops of his rimless glasses, dropping a jaw to reveal tobacco-stained teeth. \"Who in the Sam Hill holy Moses are you?\" he exclaimed.\n\n\"Mr. Langston's idea of an assistant for you, I'm afraid,\" Cara replied with a smile, and extended her hand. \"Cara Martin, Mr. Sawyer. He said you were expecting me.\"\n\nThe bottom jaw snapped shut. \"Oh, I see...\" He gave her hand a swift shake. \"You weren't what I was expectin' a'tall. However, young lady, ideas are for usin'. Especially the boss's.\"\n\nBy midmorning Cara found that for all his gruff manners, Leon was her kind of boss. He was an orderly man, accustomed to giving explicit, no-nonsense instructions that Cara's quick intelligence appreciated. She learned at once that the two of them had the double duties of getting the men fed for the next two days as well as preparing La Tierra's modern version of a chuckwagon for the roundup that was to take place the day after tomorrow. By lunchtime the shambles that had been breakfast had been cleared away, and an edge of Leon's temper soothed.\n\nCara was too busy to be nervous about serving the men or to be conscious of the stares she received when they trooped into the dining room. Each man seemed to have his designated place at one of the tables that surrounded a longer table in the center of the room. Every table was occupied when Jeth and the tall man she had seen in the corral entered with Bill and several others. They took their places at the central table. Jeth and Bill ignored her when she carried in their food, but the tall man again gave her a friendly nod. They were the last to be served, and afterward Cara had to pause for a tired sigh when she went back to the kitchen. Leon, standing at the huge stainless-steel counter, heard her. \"I must say, for a li'l un, you got a lot of work in ya,\" he said, and Cara flushed with unexpected pleasure. That was probably as close to a compliment as she'd ever get from the wizened old fellow.\n\n\"What do you want me to do now?\" she asked him.\n\n\"Rest yore feet awhile. Eat somethin'. Then we'll start packin' the vans.\"\n\nThat afternoon, as she was packing the last pan into a box for loading, Jeth Langston's big shadow fell across her. She had pulled her hair back with a string, rolled up the sleeves of her silk blouse, and was wearing an oversized pair of rubber gloves. Cara looked up quizzically, aware that her heart had begun to thump.\n\n\"I'm going up to the big house now,\" he said, \"and I want you to come with me. Leon can carry on from here. One of the boys will help him serve supper.\"\n\nCara knew better than to argue. She pulled off the rubber gloves and walked without comment into an adjacent pantry where Leon was checking off supplies from a list he had clamped to a clipboard.\n\n\"Mr. Langston tells me I'm to go now, Mr. Sawyer,\" said Cara, watching the cook of La Tierra pause from his counting to moisten the tip of his pencil with his tongue.\n\n\"That's reason enough to go, child,\" he informed her tranquilly. \"He anywhere around?\"\n\n\"I'm right here, Leon,\" Jeth said, coming up behind Cara to lean against the doorway. She had stepped just inside the pantry, and when she turned to leave, she found that Jeth had blocked her exit. Their eyes met in an impasse\u2014his challenging, hers cold and still. Jeth did not budge. Over Cara's head, he said to the cook, \"Did you need me?\"\n\n\"Looks like I could use one more case of coffee, if it's no trouble.\"\n\n\"No trouble,\" Jeth said. \"I'm going into town this evening. Is there anything else?\"\n\n\"It can wait,\" said Leon, still counting his supplies, and Cara had the distinct impression that she was the matter that could wait.\n\n\"Good night, Mr. Sawyer,\" she said over her shoulder. \"I'll see you in the morning.\" Pointedly she turned back to Jeth. The rancher eyed her with hard mockery for a few seconds before lowering his arm to let her pass. Cara walked outside into the cold dusk. The norther that had hit the night before still had its bite. She did not wait for Jeth but started toward the house.\n\nCara was halfway there before he caught up with her. \"Wait up,\" Jeth ordered, clamping a hand on her upper arm.\n\nCara tried to pull away. \"Let go of me!\" she snapped.\n\nJeth looked down at her in feigned surprise. \"Miss Martin, what do you think I'm going to do?\"\n\n\"Mr. Langston, I could easily learn to loathe you.\"\n\n\"I'm sure you could, Miss Martin. You'll have plenty of reasons to.\" He steered her toward a large six-car garage that held an immaculate fleet of vehicles, all the same light gray. One was the jeep in which she had ridden from the airport and another was a gleaming Lincoln Continental, its Texas license plate bearing discreetly in one corner the brand of La Tierra Conquistada.\n\nJeth released her arm and strode around to the driver's side of the Continental. \"Get in,\" he told her.\n\nCara looked puzzled. \"Why?\" she asked.\n\n\"Miss Martin\u2014\" Jeth drew a weary sigh. \"Haven't you learned by now not to try my patience?\"\n\nCara with sullen grace got into the passenger side of the luxurious car and felt immediately enveloped by a velour cloud. Her tired, aching limbs all but sighed in appreciation of the sumptuous comfort of the contoured seat. \"Where are you taking me?\" she asked warily.\n\n\"Into town to get you some work clothes,\" Jeth informed her. \"I doubt seriously that you have anything suitable for working around men on a roundup. What you'll need are jeans and flannel shirts\u2014loosely fitting ones,\" he added grimly. \"You need boots and socks, also a jacket that you can move around in. You didn't just happen to bring anything like that with you, did you?\"\n\n\"Actually no. Roundups were not quite the in thing in Boston.\" She turned to him contrarily. \"And this is a wasted trip. I don't have my checkbook with me.\"\n\n\"I will take care of the bill, Miss Martin.\"\n\n\"Oh no you won't! I don't want anything from you!\"\n\n\"Really?\" The dark brows rose satirically over a long, level look.\n\nEnraged, Cara turned away from him to stare out across the flat plains. Dear God, let me hate him! she prayed. \"I'll write you a check when we get home,\" she said, unaware of how she had referred to La Tierra.\n\nCara, her cheeks flaming and awash with loneliness, walked out of the old-fashioned dry-goods store of the small prairie town to wait for Jeth while he settled the bill with the gray-haired storekeeper he called \"Miss Emma.\" Miss Emma had clearly not liked Cara. The woman's eyes had sparkled with pleasure at seeing Jeth, but they had turned hostile when lighting upon Cara. \"So you're taking yourself off on a roundup, are you?\" she had asked with a disapproving purse of her lips. \"I must say, from what I've heard I'm surprised.\"\n\n\"We're in a hurry, Miss Emma. Let's get some things together for this tenderfoot here.\"\n\nWithout consulting Cara, Miss Emma and Jeth selected for her a wardrobe of Western work wear. The boots presented a problem because of Cara's small, narrow shoe size, but eventually a pair was found.\n\n\"Why is it so important for me to have boots?\" Cara asked.\n\n\"Because you're going to be doing some riding, Miss Martin.\"\n\nCara closed her mouth without further comment. She would take this up with him later, away from Miss Emma's well-tuned ears.\n\nThe disapproval of the woman had hurt, and Cara felt cheap and soiled. When Jeth joined her, they walked in silence to the car where he chucked the packages to the backseat. Then they drove to a local grocery store to fill Leon's request for a case of coffee. Unwilling to face another Miss Emma, Cara remained in the car while Jeth made the purchase.\n\nDuring the drive back to the ranch, Jeth broke the silence by reminding her, \"You knew what kind of reception you would receive in these parts when you elected to come to La Tierra, Miss Martin. Surely you didn't expect to be greeted with open arms.\"\n\n\"Indeed I didn't, Mr. Langston,\" Cara said with crisp dignity, staring straight ahead with her chin raised an extra inch.\n\n\"I'll have Fiona wash the stiffness out of those jeans for you. Also I bought you a dozen pairs of rubber gloves that will protect your hands while you're washing dishes.\"\n\nCara's head turned in surprise. \"That was very kind of you.\"\n\n\"Not at all. I don't want you crumping out on Leon because your hands can't take the soap and scalding water he uses.\"\n\nRebuffed, Cara gazed out the window at the star-filled night. Only an occasional pumping jack, outlined by the afterglow of the sunset, disturbed the endless prairie. To her, the strange-looking monsters that pumped oil from La Tierra's lucrative acres were a symbol of the land itself: proud, remote, relentless\u2014like the man who sat beside her. She gave a silent sigh and withdrew into the folds of her coat.\n\nEventually the wrought-iron gates came into view and then presently the Continental was pulling to a stop before the porticoed entrance of the house. Cara's breath of relief was cut short when Jeth said, \"You go on in, Miss Martin. I'll take care of the packages. You'll eat with me tonight. No need to change. We'll make an early night of it.\"\n\nCara could not face another evening with the rancher. Her eyes clouded with dismay as she spoke. \"Mr. Langston, haven't we seen enough of each other for one day?\"\n\nJeth's cool gray eyes held a hard gleam. \"You'll be seeing a lot more of me, Miss Martin, so you'd better get used to the idea. As for me, I don't think I would ever tire of looking at you. It's your black heart I can do without.\"\n\nDispiritedly, Cara went upstairs to wash, thinking more about the ruin of a good silk blouse than those last words. When she came back down to the kitchen, she heard Jeth giving Fiona instructions about the jeans. They had been taken from their wrappings, and the bill had floated to the floor. Cara picked it up and tucked it into her pocket. \"Good evening, Fiona,\" Cara said and was rewarded with a nod of the stern gray head. \"I appreciate your washing the jeans for me.\"\n\nJeth turned from the counter to hand her a glass of wine. He had poured himself a bourbon. \"Thank you,\" she murmured without meeting his eyes. She was acutely uncomfortable under his gaze and knew that he was aware of her discomfiture.\n\nFiona rescued her by saying, \"Some mail came for you today, se\u00f1orita. It is with El Patr\u00f3n's in the hallway.\"\n\nCara followed Jeth to the entrance hall where a lone letter lay beside a bundle of correspondence neatly stacked on the refectory table. \"Why, it's from Harold St. Clair!\" Cara cried delightedly when she saw the return address on the envelope. Eagerly she opened it and drew out a letter, absently fingering the gold seagull at her throat while she read.\n\nJeth thumbed through his collection of mail. \"Anything to do with the estate?\" he asked casually.\n\n\"Oh, no,\" Cara assured him with a happy smile. \"It's just a friendly letter, that's all\u2014a breath of sea air from home.\"\n\n\"Is it now?\" Jeth's voice had hardened. \"Finish that later, if you don't mind. I'm hungry, and Fiona is waiting to serve us.\"\n\nWhat a rude, bad-tempered man! Cara thought angrily, folding the letter and slipping it into her pocket next to the bill. She followed Jeth to the table, thinking how changeable his moods were and that the woman who married him was to be pitied.\n\nThey were served in a small dining alcove off the main one, and Cara hoped the food and wine would induce sleep. They ate in silence for the most part, Cara apprehensive of the even blacker mood that had come over Jeth since she told him about Harold St. Clair's letter. He probably thinks we're conspiring against him, she surmised to herself. Well, let him stew!\n\nWaiting for coffee, Jeth pushed his chair back and remarked, \"You know, of course, that there are no bathing or bathroom facilities at a roundup camp. When you pack, make sure you take that into account.\"\n\nCara gave him a dumbfounded stare, which Jeth met with an unruffled air of supreme indifference. \"You've got to be kidding!\" she exclaimed.\n\n\"It's not likely that I would ever kid you.\" He struck a match to his after-dinner cigar and drew on it. \"Enjoy your bath tonight. It will be one of the last you'll have for a while.\" He smiled, quite pleased with himself.\n\nShe pushed back her chair and got up. \"If you'll excuse me, it's been a long day. I'll have to deprive you of any further dubious pleasure you might get from my company this evening.\"\n\n\"Pity,\" said Jeth idly, tapping the ash from his cigar. \"I had hoped you would play for me.\"\n\nCara paused in her escape from the room. Her expression when she turned back to him held its own irony. \"You wouldn't enjoy my playing, Mr. Langston. The piano is one place where you cannot make a fool of me.\" She left him gazing after her, his eyes expressionless behind the smoke.\n\nThe next day passed too rapidly for Cara. Between serving meals, packing the vans, and watching what was going on outside the ranch kitchen with horses and men in preparation for departure to the campsite, Cara could hardly believe it when Leon said, \"That's it for today, li'l lady. You go on up to the big house 'fore it gets too dark to see. I imagine you still have yore own packin' to do. Get plenty of rest tonight, now. Yore gonna need it.\"\n\nTiredly Cara removed the big white apron that Leon had let her use. \"You won't have to tell me that twice,\" she said. \"Good night, Leon. I'll see you in the morning.\" But as she stepped out of the swinging back doors, she collided with the tall man who had nodded pleasantly to her yesterday.\n\n\"Whoa there,\" he said in a friendly voice, steadying her. \"You okay?\"\n\n\"Of course.\" Cara smiled up at him. \"How about you?\"\n\n\"No harm done.\" He grinned. \"This gives me a chance to introduce myself. I'm Jim Foster.\" With obvious reluctance he removed his arms from around her to hold out a hand.\n\n\"Cara Martin,\" she said, feeling her hand swallowed as he took it. \"You're the foreman, aren't you?\"\n\n\"That's right. I run things when Jeth's not around. You must be sure and let me know if there's anything I can do for you when we're out there.\"\n\nLeon was at the sink still tidying up, his back to them, but Cara sensed he was taking great interest in the conversation. \"I'll remember that, Mr. Foster. Thank you very much.\"\n\n\"Jim,\" he corrected with a smile, and Leon turned from the sink.\n\n\"Time you were goin', Miss Martin. Daylight be gone soon.\"\n\nWith a polite nod to the men, Cara left, buttoning her new jean jacket against the stiff night wind as she walked across the ranch yard. There had been some sort of unfriendly undercurrent back there between the foreman and Leon. She was sure of it. She must be careful not to become inadvertently drawn into ranch politics.\n\nCara had glimpsed Jeth only once during the busy day. He had not eaten either breakfast or lunch in the Feedtrough. Now she looked back at the saddling pens that skirted the big central corral. In the pens were all the horses, the remuda, that Jim had assigned to each man for the roundup. Leon had said that each ranch hand would need a change of five horses a day for the work he must do. All day the riders had been shoeing them as well as preparing their own range gear for a month's stay on the open plains. Tomorrow there would be a giant exodus of men and horses, trailers, and vans to the first roundup site fifty miles away. In spite of herself, Cara felt a thrill of excitement about the coming adventure.\n\nWhen she entered her room, Cara found neatly folded on the bed the new jeans and shirts laundered to an old-clothes softness and fragrance. She picked up the flannel shirts and buried her nose in them, inhaling the freshness with appreciation after a day of smelling horses, sweat, and manure. There was no way of knowing how many times these clothes had been washed to acquire the comfortable texture they had now. She must find a way to express her thanks to Fiona.\n\nA knock came on the door. \"Come in,\" she called, but it was Jeth Langston, not Fiona, who entered her bedroom. The hard light in his eyes warned her that he was in an irritable mood, possibly because he was as bone-tired as she was. A deep brim crease around his head suggested that he had not taken his hat off until a few minutes ago, and dust caked his clothes. Without preamble he said abruptly, \"Here is a list of things you'll need. Have everything packed and ready outside your door no later than six o'clock in the morning, earlier if you can manage it. Do you have any questions?\"\n\n\"Why\u2014I haven't had time to think of any\u2014\"\n\n\"Too late now,\" he said curtly, turning to leave.\n\n\"That's all right,\" Cara said to the broad-shouldered back. \"Jim Foster can answer any questions I might have.\"\n\nSlowly Jeth turned back around to face her, and Cara could have kicked herself for the remark. Why had she said such a thing? she scolded herself. The rancher's eyes glinted like sun off metal as he walked back to her. \"What do you mean by that, Cara?\" he asked softly.\n\n\"Why, nothing!\" Cara said, wide-eyed. She pressed the clothes protectively against her. \"What else could I have meant?\"\n\n\"You tell me,\" Jeth said, so near to her now that she could see the stubble on his face, smell the rough male scents of him. \"You wouldn't be thinking of playing your little games out there with any of my men, would you?\"\n\n\"I don't know what you mean\u2014\" Jeth stopped her protest by grasping her jaw in a firm hold.\n\n\"Because if you are,\" he went on as if she had not spoken, \"just remember that I would take a dim view of such a fool-hardy idea. That should dampen your enthusiasm considerably.\" He gave her jaw a stern little shake. \"Those men will be without women for over a month. They don't need you to remind them of what they're missing.\"\n\n\"Then why am I going?\" Cara demanded angrily, clutching his wrist.\n\n\"I told you why.\" He released her and she retreated against the writing desk. Something fluttered to the floor, and he reached down and picked it up. \"What's this?\" he asked, frowning.\n\n\"It's my check to you for these clothes,\" Cara said, rubbing where his fingers had been. What a beast he was!\n\nJeth looked at it with contempt. \"Written on money that Ryan transferred to your account?\" His scorn was as cutting as a scalpel. So he knew about that, too, did he? thought Cara. As he pocketed the check, she said in a futile, childish attempt at some revenge, \"You are such a dreadful man.\"\n\n\"That is an opinion shared by a number of my enemies. Fiona will bring your meal. I suggest you turn in early. Now no doubt you will excuse me. I'm going for a swim.\"\n\nThe next morning was a virtual beehive of activity in the ranch yard as men gathered with their equipment to be stowed in the caravan of vehicles leaving for the campsite. The remuda had been assembled, and Cara overheard Jeth giving instructions to Jim about which men were to ride in the trucks and which were to drive the remuda to a canyon close to where the cattle would be gathered.\n\nThe atmosphere crackled with excitement. Cara could feel the eagerness in horses and men to get started. \"I should be frightened, I suppose,\" Cara told Leon, \"but actually, this is all very thrilling.\"\n\n\"The novelty will wear off for you after a day or two,\" Leon told her, \"but for most of those men out there, this 'n' the fall roundups are the best times on a ranch.\"\n\nWhat, Cara wanted to know, was the purpose of a roundup?\n\n\"To gather up for brandin' and inoculation all the new calves born this spring,\" Leon answered. \"On a ranch the size of this one, roundin' up the cattle is about the only way to count 'em. At the same time we do that, we drive 'em up to the high country for the summer where the grass is more plentiful. Jeth believes in modernization, but there ain't nothin' like men on horseback to gather cattle. Some ranches have gone to usin' helicopters for roundin' up their herds. It wouldn't work for us. We got too many cattle. Them helicopters 'ud just start a stampede.\"\n\nBy eight o'clock the kitchen had once again been cleaned after breakfast, and Leon told Cara to climb into the pickup truck that would lead the two customized, refrigerated vans that made up the chuckwagon. Leon tooted the horn and yelled out of his window, \"We'll have the chow waitin'!\" as the three-vehicle cavalcade pulled out of the ranch yard. The cowhands cheered and waved their hats and lariats. Cara laughed, caught up in the excitement of the new adventure, and searched among the group for Jeth. She caught instead the eye of Bill, who couldn't suppress a grin when she waved at him, and then the rather stern, speculative gaze of Jim Foster. The foreman nodded to her without smiling and touched the brim of his hat. Puzzled, Cara gave him a brief smile before settling back to experience her second ride across the open range of La Tierra Conquistada.\n\nFifty miles later, in a high clearing fringed by scrub oak and mesquite trees, Leon drew up beside a great blackened pit dug in the earth. Beside it was stacked an enormous supply of firewood, cut and piled, Leon explained, before the roundup began. \"This is where the first campsite was last year,\" he told Cara as they climbed out of the truck. \"We'll have to get the fire goin' so we can get the coffee on and the steak fried 'fore the men get here.\"\n\nCara took a minute to stretch and take stock of her surroundings. Her eyes swept acres of rolling, semiarid dun hills and mountain slopes, still under the last dull wash of winter. With a trick of the mind's eye, Cara thought, you could almost imagine you were looking at the Atlantic; the land had the same unbroken endlessness. She took a deep breath of the snappy morning air, letting some of the tense excitement ease out of her shoulders. If she could manage to keep from incurring Jeth Langston's wrath, maybe this wouldn't be such an unpleasant month after all.\n\nBy noon the chuckwagon was in operation. Tiered shelves had been unfolded from the back end of the covered pickup truck, and the earthen pit was crackling with red coals. A ten-by-ten-foot tarpaulin, in the gray and yellow colors of La Tierra, had been stretched over four metal posts anchored in the earth. Kettles of beans, chili, and stew hung from an iron bar over the campfire, simmering for the evening meal. Their spicy smells blended richly in the pure mountain air with those of coffee and fried steak. Lunch was a catch-as-catch-can kind of meal. As their work permitted, the men came in twos and threes to eat quickly the huge slabs of fried steak served between thick slices of bread. They washed the food down with scalding cups of coffee before mounting up to ride back to the draws and mountain passes to flush the cattle and lead them to a holding pen.\n\nIn midafternoon, when no kettle needed stirring or seasoning, Cara strolled over to an enclosure where three young calves were penned. They had healthy, russet-colored bodies and white faces, and the sun shone pinkly through their short, perky ears. One of the calves ambled up to Cara and let out a plaintive bawl. \"What are you doing here, little fella?\" she soothed. \"Sounds like you need your mother.\" The calf seemed mollified by Cara's attention and let her continue to stroke it, batting tender brown eyes that she found endearing.\n\nAfter a while she went in search of a place where she could wash and dress privately away from the hub of the campsite, and found an outcropping of brushy rocks that screened a shallow hollow. There were several flat boulders in the depression, perfect for holding a mirror and a pan of water. Cara returned to the pen and patted her new friend, then carried water and her clothes satchel to the depression to freshen up before she had to help with the final preparations for supper. She was able to manage a thorough wash, she'd like Jeth Langston to know, and after a change of clothes she felt as clean and refreshed as if she'd had a soaking bath. She applied fresh makeup and brushed her hair until it shone, securing it away from her face with a blue ribbon that matched the blue in her eyes.\n\nLeon surveyed her over the top of his glasses when she rejoined him under the tarpaulin, but she could not tell from his permanent scowl if he approved her appearance or not. Busy ladling out flour into a huge bowl for sourdough biscuits, he remarked, \"Put on that big white apron there and wrap it around ya two, three times, Miss Martin\u2014that's a good girl.\" Cara did as he instructed, smiling to herself. In his own gruff way he was trying to protect her from the too-curious eyes of the men.\n\nShe was rolling out biscuits when the men began returning to camp. Her heart skipped a beat when she saw for the first time that day Jeth's tall figure astride the big bay. He dismounted without glancing toward the chuckwagon and strode quickly to a gray pickup that Cara knew contained a telephone for communicating with his office at the ranch.\n\nBusy with her chores, Cara barely noticed Leon leave her to join a group of two Mexican cowboys, _vaqueros_ , and a plump, merry-faced man who earlier in the day had arrived bumping over the plains in a white van. \"Harry's Meat Market\" was emblazoned in red on the door of the van, and Cara had thought the man had come to dicuss an order for beef. Leon had greeted him jovially, and the two had enjoyed a gossip session over steaming cups of coffee.\n\nNow it was obvious they were discussing the calves in the pen, and Cara began to get uneasy. What could be of such interest about them? She watched one of the _vaqueros_ walk cautiously toward the pen, twirling his rope. He threw the noose over the head of one of the calves\u2014her calf\u2014which immediately set up a bawling protest and tugged at the rope.\n\n\"What's he doing?\" Cara demanded of Leon, but he didn't answer her. Intent on the calf, Leon pursed his lips to whistle. Cara saw the other _vaquero_ raise a rifle to his shoulder. \"No!\" she screamed, just as Leon's whistle split the air. The calf turned its head inquiringly in their direction, and in that second a bullet slammed into its white forehead between the dark brown eyes.\n\nIn shock Cara whirled to avoid seeing what happened next and staggered into a pair of arms that held her comfortingly against a rough-vested chest. Jeth, she thought, but the voice she heard bent low to her was that of Jim Foster.\n\n\"Easy now, Miss Martin, no need to carry on so over a little old dogie like that. He's only good for eating. Come on, now. Let's walk a bit. Leon can do without you for a few minutes.\"\n\nTrying to shut out of her mind the picture of the young calf crumpling into the dust, surprise still in its eyes as blood spread over its white face, Cara let herself be led away from the campsite. \"This is no place for you,\" Jim said as they paused behind a small bluff that shielded them from the camp. \"Jeth ought to have his head examined for making you come out here.\"\n\n\"I should have known why those calves were penned,\" Cara said numbly. \"It was stupid of me not to realize\u2014\"\n\n\"You couldn't be expected to know they were for butchering,\" Jim cut her off. \"You just content yourself here for a little while 'cause they're quartering that little fella right now. No use going back to camp until it's all over and done with.\" He took a package of cigarettes from a shirt pocket. \"Want one?\"\n\nCara shook her head. \"No, thank you. I don't smoke.\" She was composed now and was worried about leaving Leon alone with the meal preparations. Besides, Jeth Langston might be wondering where she was. \"I really must be getting back,\" she said.\n\n\"What you'll see won't be pretty, Miss Martin. Give it a few more minutes.\"\n\nCara shuddered, thinking of the merry-faced man in the white van. Now she understood his purpose in the camp.\n\n\"Here now, you're cold,\" Jim said, coming closer to her to put an arm about her shoulders before she could move away.\n\n\"Jim! Miss Martin!\"\n\nThey both whirled guiltily at the sound of Jeth Langston's voice. Jim's arm dropped immediately and Cara felt the blood drain from her face. In the growing twilight, Jeth loomed down at them from the rise of land overlooking the ravine where they stood. His eyes, glinting like rapier points, impaled her as he addressed his words to the foreman.\n\n\"Jim, go down to the truck and phone in your cattle count to headquarters. I'd like a word in private with Miss Martin.\"\n\n\"Right, boss!\" Jim said with alacrity and hurried past Cara without another word, not even to offer an explanation on her behalf to the rancher.\nChapter Eight\n\nCara tried to fight down the immobilizing terror that rooted her to the spot. Jeth was down the incline before she could move, his chaps making harsh leathery sounds as he spanned the distance between them. \"Miss Martin, I warned you! I told you that you were to leave my men alone, that if you didn't\u2014\"\n\nThe rest of his reproof never had a chance for delivery. In fear of the fury that deepened the tan of his handsome face, Cara spun away from him, managing only a few steps before an ankle twisted. She heard the startled cry of her name before she went splaying, stomach side down, on the hard, stony ground. A sharp stone cut into the underside of her chin, but she lay oblivious to everything except the spinning carpet that offered to take her away from the demon towering above her. As he lifted her to her feet, she had a blurred glimpse of her blue hair ribbon lying on the ground.\n\nWeakly, she flailed at him. \"Leave me alone!\" she sobbed. \"Take your hands off me!\"\n\n\"I will when you're calm,\" he said, pulling her into his arms and holding her steady against his chest. Spent and dizzy, Cara clung to him, wrapping her arms around his waist.\n\n\"What have you done to yourself?\" Jeth asked sorrowfully above her head.\n\n\"No more than you would have done to me,\" she said into his chest.\n\n\"Oh, lady\u2014\" He sighed. \"I was mad as hell, yes, but I wouldn't have hit you. A good shake was what I had in mind\u2014to make you understand that while you're on La Tierra soil, you'll remain faithful to Ryan's memory. I will not tolerate your making a fool of him\u2014\"\n\nShe raised her head to look at him. \"I wasn't making a play for Jim!\"\n\nHis mouth hardened. \"So you say.\"\n\n\"It's the truth!\"\n\nSuddenly the emotional and physical events of the past half hour overwhelmed her. Her chin and palms throbbed. She wanted desperately to sag against Jeth's chest again and rest there, but she could not afford such a balm. Her arms dropped from around him. \"You are wrong about what you saw, Mr. Langston. I don't expect you to believe me. But surely you can believe that I would never do anything to hurt Ryan's memory.\"\n\nJeth's embrace loosened. Her small face was very pale, the smooth cheeks smudged with dust and the streak of tears. A thin line of blood had appeared beneath her chin. \"Just your being here does that. Now go down and ask Leon to take a look at that chin.\"\n\nLeon had already begun serving the evening meal. \"Sorry,\" she muttered at his elbow. The cook turned around to find her gazing helplessly at her grazed palms. \"I had an accident.\"\n\nLeon took in the abrasions and the dusty apron, the disheveled hair that had been as smooth as polished gold a short while ago. \"So I see,\" he commented without inflection. \"Here's some clean water.\"\n\n\"I've not been much help, I'm afraid, Leon. I'm sorry.\"\n\n\"No need to be. You've been the best help I've ever had on a roundup, and tomorrow is another day. Here's some salve for your hands. Now let me dab a little of this on that cut.\"\n\nLeon was dabbing when Jeth came up under the canopy behind Cara. She felt the rancher's presence without turning around, and Leon looked from one to the other with a speculative tightening of his eyes. \"How about some food, Jeth?\" he asked, capping the medicine bottle.\n\n\"Pour me a glass of bourbon first, Leon. And open a bottle of Miss Martin's wine for her. I'm sure she can use it.\"\n\n\"Sure thing,\" Leon agreed, going to the van where his employer's private stock of bourbon was kept. Embarrassed, Cara kept her back to him. It was considerate of him to include her wine. What a fool she had been, running from him like that!\n\nShe finished washing her hands, discovering that the stinging cuts were only surface deep, and dried them on the clean towel that Leon had left her. She would wait until Jeth left before applying the ointment. She did not want him to know about her hands. Tomorrow she would wear makeup to hide the graze under her chin.\n\nCara felt Jeth's eyes follow her as she moved out from under the tarpaulin to clean up a spill on a portion of the long folding table where food was served. She took her time at it and presently Leon returned with the bourbon. With relief she saw Jeth stroll to the campfire around which the men were seated.\n\nWhen she returned to her station, she found a cold glass of wine poured and beside it the blue ribbon that had fallen from her hair.\n\nCara ate her supper standing up and did not know what to do with herself when all the chores were completed. The men were sitting around the campfire exchanging jokes and yarns, and their rough, raucous laughter drifted to her in the night air. The stars had come out. Behind them were lingering traces of the sunset, which filled her heart with a strange melancholia that made her want to cry. She strolled a little way from the camp, afraid to go too much farther because she had overheard the men talking about rattlesnakes coming out of hibernation now. She remembered the pocket light in her gear, and thought that in the coming nights she would find a place to read to fill the time between the end of her chores and bedtime.\n\nWhen Cara returned to the chuckwagon, Leon was waiting for her. \"The men are beginnin' to bed down, Miss Martin,\" he said. \"The boss give you any idea about where yore to sleep?\"\n\n\"Why, no, he hasn't,\" Cara replied. With the busy activities of the day, that question had not occurred to her. \"I don't seem to have a bedroll. Do you have any suggestions about what I should do?\"\n\nThe cook studied the young woman's drawn face in the flickering light of the kerosene lamp. He had insisted she wear, at least for the night, ointment-soaked gauze pads taped to her palms. Now his jaw tightened. \"Nobody said anythin' to you about a bedroll, Miss Martin? That don't seem quite right to me.\"\n\n\"Here's her sleeping bag, Leon,\" said Jeth Langston behind them. He had come up in the darkness, and now stepped into the glow of the light. \"Don't worry so about Miss Martin. Believe me, she is very capable of looking after herself. Follow me, Miss Martin.\"\n\n\"Good night, Leon,\" Cara said gently to ease his worried frown, and followed Jeth's tall, striding form to a spot of ground just beyond where several men were already stretched out in their blankets. \"You'll sleep here,\" he told her brusquely. \"You should be warm enough this close to the fire.\"\n\n\"Thank you,\" she said stiffly, watching him unroll the long length of gray quilted wool trimmed in yellow. Jeth unzipped the bag and extracted a small pillow in a crisp white case. She had never been so tired in her life; everything inside and out of her ached.\n\nWithout another word to her, Jeth strode away to the truck that served as his office. He never seemed to rest from the duties of his ranch. Cara wondered where he was to sleep.\n\nThe sleeping bag was as warm as an embrace and imbued her with a sense of peace. Just before drifting off to sleep, she discovered a name sewn in yellow just inside the neck opening: Ryan Langston.\n\nSometime in the night she was dimly disturbed by something brushing her hands. Immediately afterward a welcome warmth spread through the chilled regions of her upper body, and she sighed gratefully in her sleep, the sound mingling with the cacophony of men's snores and the nocturnal noises of horses and prairie creatures.\n\nThe next morning before daybreak Cara was awakened by the aroma of coffee trailing beneath her nose. \"Wake up, child; coffee's on,\" said Leon, setting a mug beside her head. \"Mind you, don't knock that over.\" He was already dressed and in the long white apron he had worn yesterday, only this morning it was reversed. \"There's time to wash 'fore you have to help me with breakfast.\"\n\nCara struggled out of her sleeping bag. She did not remember having zipped it all the way up under her chin the night before. Her chin! Gingerly she touched it, and winced. Something that sore had to show a bruise, and now all the men would think that their boss had worked her over. Despairing at the thought, she carefully picked up the hot mug in her padded hands and hurried away to her own nature-created dressing area. Bless Leon! He had left her a pan of hot water on one of the flat boulders. Better hope that Jeth Langston did not find out about this preferential treatment. She could not bear for Leon to get in trouble because of her.\n\nSurreptitiously, Cara searched the campsite for Jeth as she ladled batter out on the hot grill for the pancakes the men would have for breakfast. The aroma was mouth-watering in the cold, bracing air, and she felt hungry for the first time in days. Jeth was nowhere to be seen, and she thought he had already left camp when suddenly the familiar voice ordered behind her, \"Turn around, Miss Martin.\"\n\nThe tone was low, controlled. She picked up a drying towel to cover her hands before turning to find him very near her, conscious that Leon was deliberately leaving her alone with him on the pretense of going for more water at the windmill.\n\nThe rancher's gaze probed her chin, but when he made to touch it, Cara drew a sharp breath and stepped back from him. Jeth dropped his hand and eyed her grimly. \"You must think the very worst of me.\"\n\nIt had been too dark to use a mirror for dressing. In the black hour before dawn, Cara had combed her hair and washed as well as she could, deciding not to worry about the scrape. Now she felt a flush of embarrassment. \"Is it very noticeable?\" she asked in a whisper.\n\n\"I'm afraid so. Not that it impairs your looks any, if that's what's bothering you.\"\n\n\"How like you to assume that's why I'm concerned,\" Cara spoke coldly. \"Please excuse me. I'm busy.\" She turned her back on him, and after an interval of feeling his penetrating stare, she heard him leave.\n\nIn midmorning Jim Foster appeared unexpectedly at her side as she was returning from the windmill carrying a pail of water. No one but she and Leon were in camp, and she greeted the foreman in surprise.\n\n\"That cut under your chin\u2014that come from Jeth?\" he asked, taking the pail of water from her.\n\n\"Of course not!\" Cara sounded horrified. \"I turned my ankle yesterday after you left me and fell right on a sharp rock. Whatever gave you the idea that Mr. Langston hit me?\"\n\n\"Because he was so hot at you yesterday when he found us together. I got the impression he suspected us of some hanky-panky and didn't like it. If I've ever seen a man in a jealous rage, he was one\u2014although why, I wouldn't know. He makes no secret of the way he feels about you.\"\n\n\"Well, yes, that's true,\" Cara agreed, as a quick little pain darted between her ribs. \"But Mr. Langston would never strike a woman, for whatever reason. Did you explain to him why you were with me?\"\n\nJim averted his eyes. \"It wouldn't have done any good, Miss Martin\u2014believe me. Jeth believes what he wants to believe, and anything I said would have fallen on deaf ears.\"\n\nYou could have tried anyway, thought Cara, glancing at the foreman in a new, critical light. They had reached the long table, where Jim set the pail. \"Thank you, Mr. Foster,\" she said, her tone cool. She faced him directly. \"I believe, however, that we should avoid any kind of contact while we're out here. I wouldn't want to jeopardize your job, and I feel certain you wouldn't want Mr. Langston to suspect me of something that wasn't true.\"\n\nThe tanned, regular-cut features of the foreman slackened in disappointment. \"But, Miss Martin\u2014\"\n\n\"What are you doin' back at camp?\" Leon demanded, suddenly appearing from behind one of the vans parked close by. The wiry cook regarded the foreman with undisguised dislike. \"I'll bet the boss don't know yore back here.\"\n\n\"So what?\" Jim challenged. \"Not that it's any of your business, but I brought a lame horse back to the corral.\" He touched his hat brim to Cara. \"I'll say so long for now, Miss Martin. We'll talk again soon.\" He gave Leon a stony glance before stalking away to his horse tied to a corral post.\n\nAs they watched the lanky figure mount, Cara could feel the older man bristling at her side like a porcupine. \"You don't like him, do you?\" she stated quietly.\n\nThe cook's eyes narrowed on the diminishing horseman cantering across the plains. \"Don't trust him,\" came the clipped reply. \"That lame horse was an excuse to come back here and see you.\"\n\nThe cook and she were close in height, and, moved by affection for her bewhiskered new friend, Cara impulsively put an arm around his shoulders. \"Leon, you mustn't get yourself involved in my battles. Like Mr. Langston says, I can take care of myself.\"\n\nLeon spit a short burst of tobacco juice into the dust away from her, a gesture that Cara had come to recognize as a preamble to one of his terse to-the-point statements. \"Yore about as capable of takin' care of yoreself as a lamb in a den of wolves, young lady. Not that you don't have plenty of grit, mind you. But you ain't got a smidgin of hardness in you, nothin' to protect you against either the likes of Jeth or Jim. Somethin' else I'm thinkin', too, child\u2014\" Another burst of tobacco juice and then Leon's words were tumbling over each other in embarrassment. \"You ain't no tramp, neither, and yore not here to harm La Tierra. I ain't got it all figured out yet, but somehow I see young Ryan's hand in all of this. If that's so, knowin' him like I did, and knowin' Jeth like I do\u2014and Miss Martin, there ain't no finer man in the whole world, even though he can be more ornery than a cooped-up bull in a barbed-wire pen\u2014why, I intend to trust the hand that dealt this confusin' hand of cards.\"\n\nHe peered at Cara over his glasses, his eyes on the bluish tinge, which had begun to spread along her jawline. \" 'Course it would rile me if I knew he'd mistreated you, child. Not to excuse him, but he'd be actin' out of ignorance, you understand, and 'cause he's hurtin' so inside.\"\n\n\"I know.\" Cara smiled in quiet appreciation of his loyalty to Jeth. \"But Mr. Langston never laid a hand on me, Leon. It was my own doing.\" Cara related her shock about the calf and Jim's attempt to console her. \"I confess I thought he was going to hit me. Mr. Langston was very angry, but he was concerned about Ryan's memory and how it would look for\u2014for\u2014\"\n\n\"For you and Jim Foster to be seen keepin' company together,\" the cook said, concluding the narrative. \"I can understand Jeth's thinkin'.\"\n\n\"Me, too,\" Cara said. Gently, Cara put a hand to the cook's whiskery cheek. \"Thanks for your vote of confidence, Leon, and you are right about my not hurting La Tierra\u2014or Mr. Langston. That I can promise you.\"\n\nAfter that conversation with Leon, Cara saw the owner of La Tierra Conquistada only at meals, and often not then. At the end of the day when he rode into camp with the rest of the men, Jeth would frequently make for the truck that kept him in communication with the rest of his empire. On such evenings Leon would take a glass and a bottle of his employer's bourbon to the truck, then, after an interval, a plate of hot food.\n\nThe days began to grow longer and warmer and, for Cara, flowed into each other as tranquilly as sea swells bringing in the tide. She learned to ride again. True to either his promise or his threat, Jeth provided Cara with a gentle Appaloosa mare, which she liked immediately. \"What's her name?\" she asked Bill, who had apparently been tapped to take her out on her first rides to reacclimate her to the saddle.\n\n\"Lady,\" Bill answered, giving her a leg up to the saddle. He had softened considerably toward her in the weeks since the roundup began and had even haltingly apologized for the jeep ride across the plains. \"That was my idea, and not the boss's,\" he admitted sheepishly, and Cara's heart had felt ridiculously lighter upon hearing the truth.\n\nThey began to ride every night after supper when her chores were done and while the twilight provided light enough to see. \"The boss doesn't want us out after dark,\" Bill admonished her, giving yet another indication to Cara that Bill would rather do just about anything than disobey his boss. Even though the rancher was away from camp during many of their twilight rides, she knew that he must have approved them, or the young cowboy would never have accompanied her.\n\nTwice the chuckwagon was moved higher into the mountains to be nearer the men who were driving a huge herd of cattle to its summer pastures. Now when Jeth left the camp, he did so by plane, a shining gray Beechcraft Bonanza with the brand of La Tierra painted in yellow on its fuselage.\n\nOne morning when she was out riding alone, she came across Jim Foster searching the brush-choked draws for strays. Thinking he had not seen her, she reined Lady in the opposite direction. \"Hello there, Miss Martin!\" he called to her, and with a sigh of reluctance, Cara waited for him to ride to her.\n\n\"Don't rush off,\" Jim said when he drew up beside her, his eyes roving in frank appreciation over the golden hair that flowed across her shoulders. It had grown since her last cut, and the sun had begun to streak it with platinum.\n\n\"I really must, I'm afraid,\" she said lightly. \"It's nearly time to begin lunch.\"\n\n\"You own half of all this\u2014\" Impatiently, Jim's long arm swept the limitless, rolling rangeland. \"Why don't you act like it instead of jumping every time Jeth or Leon pulls your string? You can do anything you damn well please.\"\n\n\"Why should you care if my string is pulled?\" They had shared only a few words since their last conversation. Was it for her protection or his that the foreman exchanged only brief, impersonal pleasantries with her when she served his plate at mealtimes?\n\nShifting in the saddle, Jim answered candidly, \"I care because I happen to think you are a gracious, beautiful lady who's getting pushed around. All I'm doing is reminding you that you don't have to take it. Use your power to keep Jeth Langston in his place.\"\n\n\"Mr. Langston's place has always been as owner of La Tierra, Jim. I am the usurper here. The problem is not so much what his place is, but mine. As for Leon, he has been the soul of propriety and courtesy toward me. I like him. And I'm enjoying the roundup. No one is abusing me or, as you put it, pulling my string.\" She dug her heels into Lady's sides and gave the foreman an impersonal smile. \"Now I really must be off. Leon needs me.\"\n\nAs she cantered away, Cara felt a twinge of remorse. Maybe she was allowing Leon's judgment of the man to cloud hers. After all, Jim Foster had been the first to try to make her feel welcome. He had tried to comfort her the day the calf was shot. He didn't owe it to her to jeopardize his job by defending her to his boss. Maybe this backdoor friendship was all he could offer her in the light of the circumstances, all he had the courage for.\n\nAnother morning Cara had reined Lady high above where the men were working cattle and was able to watch without being observed how Jim and several other men maneuvered calves to be branded from among the herd in the holding pen. Fascinated, she watched as Jim rode unobtrusively into the milling cattle, then quietly pointed to the animal he wanted. The ears of his cutting horse perked up expectantly, for this was the work he had been trained for. In a few minutes' fast work, they had the calf edged to the outer rim of the herd, near the corral gate. A man lifted the corral bars, and another cowboy, ready on a roping horse, streaked after the bewildered calf to throw a noose around its neck. Instantly the horse reared against the rope, backing surefootedly until he was practically sitting on his own tail, holding the rope taut until his rider could dismount and finish tying up the animal.\n\nA soft neigh from the brushy thicket to her left drew Cara's attention, and she felt Lady tense under the saddle. \"Easy girl,\" she soothed, and patted the animal's neck. The nicker came again, this time accompanied by a considerable rustling of the thicket, and Lady backed away nervously as a great black stallion emerged to stand calmly eyeing them across a distance of a few yards. \"Take it easy, Lady,\" Cara spoke gently. \"It's all right. He just wants to say hello to us. Easy, girl.\"\n\nThe stallion was an awe-inspiring sight. Coal black with a full mane and tail, head held with the proud, graceful carriage of a Thoroughbred, he was the kind of horse that raised goose bumps just looking at him. Cara could easily understand how this equine king of the range had been able to outrun the fleetest of La Tierra's horses, and outwit the most cunning of her men, Jeth Langston. \"So you are Devil's Own,\" she breathed softly. Beneath her, Lady's muscles twitched coquettishly. The mare's ears perked and her tail swished in outright flirtation.\n\n\"I can certainly see why Jeth Langston would like to get a rope around you,\" Cara said to the great horse. \"But don't you ever let him. The likes of you were born to be free. Don't you ever let him put his brand on your flank. You'd never be the same.\"\n\nDevil's Own gave a soft responding neigh and moved with a graceful rippling of muscles farther out from the thicket. \"You'd better go now,\" warned Cara, realizing that the stallion had probably used this as a hiding place to observe the remuda corralled below on the canyon floor. She wondered if the horses were aware of their leader's presence, if in some kind of equine way he was able to communicate to them that he had not deserted them. \"Go on, boy,\" Cara urged. \"Go on, before the men find you here.\"\n\nDevil's Own whinnied softly, then turned his beautiful body swiftly, catching the sun full on the sheen of his magnificent, unmarked flanks before he raced toward a mountain slope behind which he was soon out of sight.\n\nCara gave herself up to the routine of camp life and found that she loved it. She and Leon came to be a well-oiled machine working together in harmony and respect. Her bathroom anxieties were alleviated by the simple solution of using the time between the completion of her lunch chores and the beginning of the evening meal to bathe. It was then she washed her clothes and hung them to dry on a mesquite tree that had now budded out. No one but she and Leon were ever in camp at that time, and she could take her time washing and drying her hair. By the time the men returned to camp for the evening, she was decked out in a fresh set of clothes, hair brushed and shining, makeup\u2014what little she used, for now her skin was lightly tanned\u2014freshly donned. Sometimes a cowboy, his wit and tongue emboldened by an extra shot of bourbon before dinner, would sniff the air around her and announce, \"It shore do smell better 'round here than when Toby was here, Leon!\"\n\nGradually the roundup crew came to accept her presence in the camp without suspicion or hostility. They began to call her \"Miss Cara\" and made room for her at the campfire when Jeth was not in camp. They asked Cara to tell them about Boston and the sea, a topic that captivated them, and Cara with amazement learned that most of her audience had never seen a body of water larger than the famous Rio Grande.\n\nWhen the roundup was five weeks along, she lay in her sleeping bag one night wide awake and gazed at the brilliant, low-hung stars that now seemed as familiar as old friends. She thought of Jeth, whom she had not seen for a week, and of Devil's Own, who must miss his favorite mare, now penned up in the remuda. Bill had told her that there had been evidence of the great stallion following the roundup.\n\n\"Really?\" Cara had asked, round-eyed.\n\n\"Yep! And he'd better watch out, too! The boss'll get a rope around that jasper's neck yet. No horse gets free space and chow at La Tierra. They all have to earn their keep!\"\n\n\"But what about Texas Star?\" Cara asked. Bill had told her when she'd first inquired about Ryan's now thirteen-year-old stallion that the men had orders not to capture the palomino for the remuda.\n\n\"Oh, well...that's a different story. That was Ryan's horse, ya know. I figure the boss thinks that as long as Texas roams La Tierra, a part of Ryan does, too.\"\n\nNow as she lay sleepless, watching the stars, she prayed silently, \"Please, Lord. Do not let me come to love it here. Do not let me come to care too much for Leon and Fiona and Bill and...Jeth\u2014for La Tierra\u2014so that always, when I'm no longer here, my heart will be...\"\n\nThe next evening after supper, Cara told Leon she was going for a ride by herself. \"Bill hurt his leg and doesn't need any extra riding,\" she explained. All day she had felt strangely depressed and at loose ends with herself. She needed to be alone.\n\nThe leathery old cook gave her a worried frown. \"I don't much like the idea of ya doin' that, child. My rheumatism is actin' up. A storm's brewin' and ya don't wanta be caught out on the high plains on horseback in lightnin'.\"\n\n\"I won't go far. If I see that it's going to rain, I'll come in.\"\n\n\"You do that, child. I wouldn't want anythin' to happen to ya out there. Yore comin' to mean a lot to me.\"\n\nShe smiled at him. \"You too,\" she said, and went to the corral to saddle Lady.\n\nCara had been out less than an hour when dark clouds began to boil up over the mountains. Rain was such a rarity in this country that she couldn't take Leon's admonition seriously. But the cook's rheumatic warning had been correct. In another thirty minutes, lightning began to flash in zigzagged streaks buried deep in the gray clouds that obscured the remaining sunlight. Cara was too far from camp to return to it in the storm, so she looked around for a place where she might shelter until the clouds decided to formulate themselves into a full-fledged storm or simply dissipate into another disappointing promise of rain.\n\nThe elements made up their mind while she was still deciding what to do. The rain, bringing darkness with it, came down in buckets, drenching her and Lady, who protested mildly, having come to trust Cara as having the better sense of the two. This time, however, Cara was at a loss where to find shelter, and the mare was fast losing confidence in her mistress. She was nervous and high-strung, straining at the bit in her mouth, when a voice cut through the darkness, biting it in two. \"Miss Martin, is that you?\"\n\nOh, God, thought Cara, as Jeth Langston, glimmering in a yellow rain slicker, emerged through the pouring rain into her vision. \"Yes, I'm here,\" she called.\n\n\"Follow me\" was the terse order, and Cara, aware that Lady knew a friend when she saw one, allowed the horse her head to follow after the owner of La Tierra Conquistada.\n\nThey found shelter in a cave whose mouth, covered with brush, she and Lady had passed dozens of times in their twilight sorties.\n\n\"Get down,\" Jeth ordered when they were in the safety of the cave. His own horse stood patiently, eyeing the duo with the faint suggestion that they were in trouble, while Cara, hair streaming with rain, dismounted to stand in the narrow space between Lady and Jeth Langston. Jeth did not move an inch to accommodate her, and Cara had to look nearly directly up at him from the disadvantage of her height, blinking rain-matted lashes.\n\n\"Of all the damnfool, irresponsible\u2014\" The rancher seemed at a loss for adjectives.\n\nTaking advantage of the momentary lapse, Cara remarked, \"I thought you were at the ranch.\"\n\n\"Which you interpreted as, while the cat's away, the mouse can play.\"\n\n\"I'm not a mouse.\"\n\n\"No. At the moment you look more like a drowned rat. Get out of those clothes.\"\n\n\"I will not!\"\n\n\"Miss Martin, you have a choice of getting out of those wet clothes yourself, or _I_ will relieve you of them. I'm not going to look. You can use my rain slicker to cover you.\" Jeth ignored the look she gave him and pushed her down on one of the large, weather-smoothed rocks that ringed a pit laid with fresh firewood that Cara supposed had been used countless times in just such situations as these. She snapped the slicker around her while Jeth went to work on the fire. Soon bright flames were crackling in the pit, and smoke was spiraling toward an overhead opening in the cave. The horses stood quietly, discerning perhaps, thought Cara huffily, that here was a man who knew how to take charge of things. Beneath the slicker she slipped out of her clothes, then spread them on another rock to dry while Jeth unsaddled the horses. She still had on her bra and panties, which felt cold and cloying beneath the rainwear. Jeth came back to the fire and sat down, shooting a glance at her spread-out clothes. \"You don't have underwear?\" he asked in surprise.\n\n\"Yes,\" she said through clenched teeth. \"I happen to have some on at the moment. Do you mind?\"\n\n\" _I_ certainly don't, but you might. The important thing is for you not to get a chill.\"\n\n\"Why?\" she asked. \"That would put an end to your problem, wouldn't it\u2014if I caught pneumonia and died?\"\n\n\"That would certainly not be in my best interests,\" the rancher replied, kneeling down to stoke the fire. \"You're worth more to me alive than dead. I need you alive to sign over Ryan's share of La Tierra.\"\n\nCara fell back into the folds of the slicker, abashed. Ryan's share of La Tierra was all he cared about. _She_ had not been the reason he had braved the storm. _She_ was not the concern of the moment. How could she be so in love with a man whose only interest in her was her signature?\n\nWith a muted cry, Cara stood up.\n\n\"What is it, Miss Martin?\" Jeth glanced up at her in alarm. \"You look as if you've been struck by lightning.\"\nChapter Nine\n\nTragically, Cara stared down at the dark head, the high cheek-boned face, the puzzled eyes caught in the flickering glow of the flames\u2014and slowly sank to her seat again.\n\n\"What's wrong?\" Jeth asked.\n\n\"Nothing,\" she whispered. \"Nothing at all.\"\n\n\"Women always say that. They can be drowning in tears, or wringing their hands off, or staring into tomorrow\u2014like you're doing right now\u2014and still say 'nothing' when they're asked what's wrong. So what's wrong?\"\n\nSlowly she answered, \"Ryan was on my mind\u2014no, my heart\u2014all day, or so I thought...\"\n\nJeth turned back to the fire, his expression grave. He finished stoking it, then threw the stick he had used into the pit. Straightening up, he said, \"You know how to ruin a good evening, don't you?\" and went to the mouth of the cave to observe the storm.\n\nCara watched the tall figure gazing out into the lightning-illumined night, an ache within her so intense that she thought she would die from it. \"I love you,\" she whispered. \"I love you,\" the revelation so soft that it was lost in the sound of wind and brush lashing at the mouth of their shelter.\n\nA bright crack of lightning struck near the cave. \"Jeth!\" She was on her feet, shaking. \"Come away from there! It's dangerous to stand so close to the opening!\"\n\nStartled, Jeth turned to her, his stature so great that it blocked the light from the storm. His gaze held hers intently for a brief moment before the horses nickered uneasily, and he went to them, speaking low. Cara watched him run a hand along their quivering flanks, heard his deep murmur, and sat down again, consumed with envy.\n\n\"How did you know where to find me?\" she asked, almost sullenly, when he had joined her.\n\n\"I saw you from the plane when we were coming in to land. If I hadn't, the entire roundup crew would have been out looking for you\u2014led by Leon,\" Jeth added wryly. \"You showed bad judgment in going out on horseback with a storm coming.\"\n\n\"You cut it pretty close yourself. A plane is as susceptible to lightning as someone on horseback. Doesn't that pilot of yours know when it's safe to fly?\"\n\nJeth gave her a long, measuring look. \"No, Miss Martin. That isn't going to work.\"\n\nPerplexed, Cara asked, \"What isn't going to work?\"\n\n\"This sudden interest in my safety.\"\n\nCara sighed. \"Can't you take anything I say at face value?\"\n\n\"I'd be a fool to, wouldn't I? You're proving the most formidable enemy I've ever had to fight.\"\n\nTaken aback, Cara exclaimed, \"Me? What have I done now to make you think such a thing?\"\n\n\"You're trying to beat me at my own game, as if you didn't know, and you've very nearly succeeded. I bring you up here, expecting you to last maybe a week before you begged to sign on the dotted line. I expected you to turn tail the first time a scorpion crawled out of your boot, the first time you heard the squeal of a rabbit being eaten alive by a coyote. But you turned the tables on me. You made yourself an asset to the roundup rather than the liability I anticipated. You made yourself indispensable to Leon. You endured without complaint what has sent some cowboys packing their bags. You've been cheerful and agreeable when you could have been sullen and bitchy. Oh, Miss Martin\"\u2014Jeth shook his head in wonder\u2014\"the more I'm around you, the easier it is for me to see how you got to Ryan. The devil himself would have a hard time holding out against you.\"\n\nSpeechless, Cara thought sickly, He's twisted everything! \"But why?\" she demanded. \"What would be the motive for my behavior except to survive the roundup?\"\n\n\"To confuse the men's thinking about you, and in that way to drive a wedge into their loyalty to me\u2014to La Tierra. You knew what they were expecting you to be, so you cleverly set out to present yourself as just the opposite\u2014a dignified lady whose manners and conduct would be beyond reproach. Now the men don't know quite what to believe about the brave, lovely _Miss Cara._ They've become quite protective of her, as proved a while ago when they all wanted to come looking for the lost lady in the storm. They're beginning to think of her as the next patrona of La Tierra\u2014of a La Tierra _divided_ , Miss Martin, which I will never allow.\"\n\nChills had begun to sweep Cara from head to foot. She had to clench her teeth to keep them from chattering. Beneath the rain slicker, she hugged her body tightly to stanch the hurt spreading within her.\n\n\"But the cleverest move of all,\" Jeth continued, \"is how I've been made to look like the heavy in this little drama.\"\n\nCara spoke through her clenched teeth. \"What do you mean?\"\n\n\"That bruise you wore for a while, your grazed hands\u2014the men thought I was responsible for them.\"\n\n\"But I explained to Jim and Leon that I _fell_!\"\n\n\"Leon believed you. Apparently Jim didn't. He must have intimated to the men otherwise.\"\n\n\"Oh, Mr. Langston, I am _sorry_! Truly I am. Jim thought\u2014would you believe\u2014that...you were _jealous_ of us, and apparently that you had struck me out of\u2014well, jealousy.\" Warmth flooded her face. She huddled miserably in the raincoat.\n\n\"I see. Well now\u2014\" He paused as if deciding whether to divulge his next thoughts. Then he resumed casually, \"He was right, you know. I was jealous. I owe Ryan's memory an apology for using it as the reason for my reaction when I saw you and my foreman together. And while I'm on the subject of apologies, Leon told me why you were with Jim. If it makes you feel any better, I was doubly sorry that I had misjudged you when I saw your bandaged hands that first night when I zipped you in your sleeping bag.\"\n\nCara was stunned. Jeth Langston jealous? And it had been he who had zipped up her bag that first night? \"Uh, Mr. Langston\u2014\" She wet her lips. \"There's something here I don't understand\u2014\"\n\nJeth scoffed harshly. \"Oh, come off it, _Miss Cara._ You know damn well Jim was right. I was jealous, and you knew it even before I did. I wouldn't put it past you to have set the whole thing up, just to get a show of feeling out of me. You're such an expert on men, you knew exactly how I would react.\"\n\n\"F-for your information\"\u2014her chattering teeth made it impossible not to stutter\u2014\"I w-would not be fool enough to risk y-your wrath by consorting with any man in y-your employ. F-furthermore, I don't know the foggiest thing about m-men. The only man I ever really knew w-was your brother, but not in the w-way you are determined to think!\"\n\nShe was beginning to shake visibly from a gripping cold that had penetrated to her bone marrow. Giving her a stern glance, Jeth went to a dark recess in the wall of the cave where Cara could see an ancient wooden box. The lid creaked open as Jeth lifted out a blanket and something that resembled a towel. He brought them to her and explained, \"That box is kept here with emergency supplies for La Tierra riders caught in a storm. Unsnap that slicker and wrap yourself in the blanket.\" He shook out the towel and inspected it. \"This seems clean enough. Dry your hair with it. You've gotten a chill. And you can stop looking at me in such wide-eyed astonishment. I'm not deceived.\"\n\n\"Maybe you're not, but I certainly am!\" Cara snapped, snatching the towel to her. \"How could I possibly have known that you would be jealous of Jim and me? Why would you be?\"\n\nOnly a small distance separated them, and Cara felt the volatile tension growing between them, heightened by the crackling, hissing flames. She countered his direct gaze as bravely as she dared. Then the tension seemed to drain from the broad shoulders.\n\n\"All right\u2014\" He turned his back to her with a sigh. \"Suppose you wrap yourself in that blanket and dry your hair, then tell me about you and Ryan\u2014and how a desirable twenty-four-year-old woman like yourself doesn't know anything about men.\"\n\nCara, warm at last, her hair and body securely wrapped in the towel and blanket, wondered where to begin. Jeth looked so disturbingly male in the way he sat with his elbows on his knees, long fingers locked. The fabric of his Western shirt gripped the breadth of his shoulders and arms, and the leather chaps emphasized the power of his long legs. \"Well?\" Jeth's dark brows rose. \"Begin,\" he ordered.\n\nHaltingly at first, Cara began to tell Jeth of her childhood, of how her first passion had been music. Her parents, she explained, had encouraged her to become a concert pianist. She had been educated, until Juilliard, in private girls' schools where, she realized now, her family's aspirations for her were not likely to encounter competition from the opposite sex. At Juilliard, she had just become aware that she was interesting to men when her world suddenly fell apart, went dark. The obligations she had assumed afterward precluded men. After several long years, there had been a light in the darkness. Ryan. He had offered her friendship, nothing else. His death had left her devastated and more alone than she had ever been. Jeth should know there had been no men in her life. They would have been named in that detective's report.\n\nA silence, broken only by the crackling flames and an occasional whinny of the horses, stretched between them when Cara finished her narrative.\n\n\"So,\" reviewed Jeth, \"you are telling me that you've never been with any man, not even Ryan.\"\n\nHeat surged to her cheeks independent of the fever alternating with chills attacking her body. \"Yes,\" she whispered. \"You can make what you wish of that information.\"\n\n\"What I wish is to find out if you are telling me the truth.\"\n\nCara was snapped out of the musing introspection into which she had wandered. \"What do you mean?\"\n\n\"You know what I mean. No, maybe you don't, not if you're as innocent as you claim. I'm prepared to believe that you are\u2014in that way. That doesn't change the fact that you schemed to get La Tierra. You didn't need experience with men to figure out that you'd be quite a prize to a man like Ryan. You held out on him until he was too sick, or too noble, to take what you promised. However, Miss Martin, I am neither.\" With lithe grace, Jeth rose to his full, awesome height.\n\nCara's heart began to race as she realized his meaning. She stood up also, clutching the blanket tightly around her. She was wearing nothing beneath it. \"No, Mr. Langston, you wouldn't.\"\n\n\"Not here, I wouldn't. This is neither the time nor the place. But I intend to find out just how innocent you are, Miss Martin, and then we'll go from there. I'll have at least one straight answer to this puzzle.\"\n\n\"If you didn't insist on twisting everything I say and do, you'd have all the answers!\"\n\n\"I twist everything, do I? Do I twist the need I feel in you every time I've held you in my arms? Have I misread the message in those beautiful eyes, misunderstood those soft little moans\u2014\"\n\nHer pride made her say it. \"Yes, damn you!\" Cara gritted, chilled from head to foot.\n\nJeth laughed down into her indignant eyes as he reached her. \"You're such a liar, Miss Cara. I'll just take a moment to prove it to you.\"\n\nHis arms were wonderfully warm and strong. She could have basked, easily died, in them, but she had to resist. \"You're taking advantage of me!\" she wailed, gripping the blanket.\n\n\"Taking advantage of you? Never!\" He trailed a series of warm kisses along her neck. \"You'll come to me willingly and gladly. You know it and I know it.\"\n\n\"I'm inexperienced. You'll be disappointed\u2014\"\n\n\"You could not possibly disappoint me, that I can promise you.\" His lips had begun the return journey to the hollow of her throat.\n\n\"Mr. Langston?\"\n\n\"Yes, Miss Martin?\"\n\n\"I am going to sneeze.\"\n\nJust in time he handed her another of the white folded handkerchiefs. While she sneezed into it, he took the slicker and snapped it around her. \"That cold coming on is not going to get you off the hook. It just buys you some time. Sit down by the fire until I saddle the horses. The storm is over. You can wear that blanket beneath the slicker back to camp. Tomorrow you're going back to the ranch.\"\n\n\"But Leon can't possibly manage the chuckwagon by himself!\"\n\n\"He won't have to. Toby came in the plane with me. He can take over now. I'd be taking you back with me in any event. I can't risk your splitting any more loyalties, now can I? No matter how innocently. And, Miss Cara, be convinced that I intend to find out just how innocent you are. If that prospect frightens you, you can always sign over Ryan's share to me and leave. The choice is up to you.\"\n\nThe next morning as they flew over the vast, pumping jack\u2013studded acres that made up Jeth Langston's empire, Cara saw that in her absence spring had arrived at La Tierra Conquistada. The cactus, all varieties and shapes, were flowering, and the rangeland grass shone tender and green under the spring sun. She had forgotten how huge and sprawling the house and ranch compound were. From the air, the swimming pool sparkled blue and clear, and she wondered if Jeth had been able to get in his daily swims on his visits back to the ranch.\n\n\"Lucky for me your cold didn't materialize,\" Jeth said when he handed Cara down from the plane. The cool gray eyes held a mocking glitter. \"You'll have dinner with me tonight. You still haven't played for me. Wear something pretty and join me in the study at seven.\"\n\nBefore she could reply, he was striding off toward the ranch headquarters. The pilot, a wizened, middle-aged man who served as a cowhand when he wasn't flying his employer's plane, taxied the Bonanza toward its hanger.\n\nLeft alone, Cara began the long walk to the house. It was true she did not have the usual symptoms of a cold, but her joints ached and she had a headache.\n\nWhen Cara greeted her in the kitchen, the housekeeper instantly snapped, \"What's the matter with you? Your eyes look bleary.\"\n\n\"I\u2014I think I'm coming down with something, Fiona. I got caught in a rainstorm yesterday.\"\n\nFiona went to a cupboard and took down an aspirin bottle from which she shook two tablets into Cara's palm. \"Take those with a big glass of orange juice and then go up and have a hot bath. Maybe you're just needing the comforts of civilization.\" A thin smile curved her lips. \"I hear you managed fine.\"\n\n\"Who told you?\"\n\n\"El Patr\u00f3n. Off with you now.\"\n\nCara soaked in a hot tub, but the aches in her muscles did not loosen their grip. \"I'll just crawl into bed for a little while,\" she said to herself. Her last thought was to wonder what she would wear that evening.\n\nCara sensed a dark presence looming over her and opened her eyes. At first she thought she was dreaming, for Jeth Langston often occupied the thoughts of her sleep, but then the dream materialized into reality and placed a tray from which steam rose on her bedside table. \"You'll do anything to delay the inevitable, won't you?\" Jeth said dryly. \"Try to sit up. I've brought you some soup.\"\n\n\"What time is it?\" Cara wanted to know. Her throat was sore and scratchy. The room spun dizzily when she tried to rise up.\n\n\"Eight o'clock. You've slept nearly twelve hours.\"\n\n\"Twelve hours!\" As she spoke, Jeth thrust a thermometer into her mouth and indicated that she should move over so he could sit beside her on the bed. The mattress depressed under his weight, and Cara's hip rolled against his thigh. With a large hand that covered one side of her face, he felt her for fever, then slipped it inside her night shift to the supple curve of her neck and shoulder. When she tensed, he said, \"Relax, I'm not going to take advantage of a girl in her sickbed.\"\n\nPresently, he removed the thermometer and studied it with a frown. \"You do have a fever, a respectable one. I want you to stay in bed for the next few days. A good rest and a diet of Fiona's soups should do the trick. They're worth getting sick for.\" After he had capped the thermometer, Jeth's eyes went back to her, moving over the clean, sun-streaked hair and flushed cheeks, the luminous eyes in the softly tanned oval of her face. \"Did I say I wouldn't take advantage of a girl in her sickbed?\" he mused, positioning both hands on either side of her hips and gazing deliberately into her eyes. \"I would very much like to. Right now. You look deliciously enticing, cuddly as a kitten.\"\n\n\"And sick, too,\" Cara reminded him. \"Probably with something highly contagious.\"\n\nJeth's lips twitched in amusement. \"A good point. I'll just have to keep a tight rein on my ardor, won't I? Get well quick, little girl.\"\n\nBut though she rested and dutifully ate the delicious soups Fiona brought her, Cara was a full week in bed. After the second day, Jeth had gone back to the roundup, and Cara had felt a sharp disappointment. Lying in bed, she thought of him every waking moment and knew that she wanted him more than she'd ever wanted anything in her life. There was an aching void in her that only he could fill. She knew she would be incapable of preventing his making love to her. Indeed, she didn't want to. And perhaps when Jeth had positive proof that she had never been...Ryan's whore, he would then have to look at her in a different light. He would probably even intuitively perceive why she had come to La Tierra. She could not lead him to the truth, of course. Her promise to Ryan must be kept. But Jeth had known his brother better than anyone, and once he came to know her as well...then who knew where their mutual need of each other might lead once Jeth guessed the truth?\n\nFinally Cara woke one morning and knew her illness was over. She threw the covers back and got out of bed. The early sun was streaming through the bay windows. She padded out to the terrace and followed it around to Jeth's bedroom, vacant now for nearly a week. She looked out toward the mountains, and her vision fell upon a caravan of horse trailers and pickups followed by a group of men on horseback. \"The roundup is over!\" she said aloud to the spring sky, eager to dress so that she could meet Jeth out of bed and on her feet.\n\nIn the kitchen, Fiona turned from her work to survey Cara with pursed lips. \"You look better, but how do you feel?\"\n\n\"Healthy,\" Cara answered, \"and hungry.\"\n\n\"Good sign. El Patr\u00f3n left word that you are to begin eating solid food.\"\n\n\"Left word?\"\n\n\"He's gone to Dallas on business. Won't be back for a week or more. The roundup is over; so is the cold weather. The planting has already begun.\"\n\nCara barely heard her. She was suddenly not hungry anymore.\n\nLeon greeted her with warmth and relief, and the members of the roundup crew with comradely good humor when she joined them for lunch in the Feedtrough. She ate with Bill and afterward he led her to the stable where the horses of the headquarters staff were stalled, including Jeth's. \"The boss didn't want us to turn her loose like we did the rest of the remuda,\" Bill explained when Cara, spotting Lady, ran to the mare's stall with a joyous cry. \"I figure he meant her to be yours to ride as long as you're here.\"\n\n\"That was kind of him,\" she said, her back to Bill. He didn't see the shadow cloud her eyes.\n\n\"Why is there no flower garden?\" Cara asked Fiona that evening as they were eating their supper in the kitchen. Cara had gone exploring over the grounds of the house in the afternoon and found that, except for the oleanders bordering the formal approach to the entrance, no flowers of any kind had been included in the landscaping.\n\nThin shoulders shrugged. \"Nothing at La Tierra is here for beauty's sake, se\u00f1orita. Everything must have a function and be productive, be it man or horse, woman or child. The care of flowers takes up valuable time and soil and water. El Patr\u00f3n has never ordered a flower garden be planted, only the vegetable fields and orchard.\"\n\nThere should be flowers at La Tierra, Cara decided, thinking of the barren graves at the cemetery. The house needed flowers to enliven its rooms with beauty and color.\n\nThe next day she found an ideal location for a flower garden. It was a bare, unused portion of land outside the ten-foot walls, facing the desert. \"Do you think you could buy this list of flower seeds for me when you go into town tomorrow?\" Cara asked Fiona.\n\nThe small brown eyes peered at the list. \"You intend planting these? Without El Patr\u00f3n's permission?\"\n\n\"Yep!\" Cara said emphatically, using the vernacular she had picked up from the roundup. The list contained the names of regional flowers she had read about in a book from Ryan's room.\n\nThe garden plot would be hard to clear. There were weeds to pull, rocks to be moved, and rocky, sandy soil to be improved with manure and topsoil she'd have to persuade Bill to bring her from the vegetable fields. She had never seen them, but she knew they were the source of the vegetables she'd helped to prepare for the Feedtrough's tables. \"Keep a cowboy's stomach happy,\" Leon was fond of saying, \"and you keep him happy.\" Apparently that was one of the strategies that Jeth Langston employed to keep his men loyal and contented. Flowers were not a big seller.\n\nThat afternoon, Cara, wearing shorts and a halter top, began to clear the land for the planting of the flower seeds that Fiona promised to bring her. For several days she hauled out the larger rocks, which could serve, her mind ran ahead, for a natural limestone fence to protect the garden from the encroachment of grass. As she worked, the sun evened the light tan that she had already acquired on her forearms and at the V-neck openings of her shirts.\n\nBill, seeing her go into the barn to shovel manure into plastic bags, grabbed a shovel and helped her. \"Boss know you're doin' this?\"\n\n\"Nope! But what kind of guy would object to a flower garden?\"\n\nAt the end of a warm day, she would look longingly at the pool. It would be just like him, she thought, to return unexpectedly and find me in it. \"Miss Martin,\" she mimicked the rancher's deep voice, \"didn't I tell you not to use the possessions of my house unless I give you permission to do so?\"\n\nMay was nearly gone. The seeds of zinnias and portulaca, achillea and bachelor buttons had been planted and waited for the miracle of germination. Cara lay in bed in a thin, short nightgown, her limbs still warm and silky from her evening bath, her scalp still tingling from a vigorous brushing. But though she was bone-tired, sleep would not come. Pushing back the covers, she decided that Ryan's room might offer something to read until she grew sleepy.\n\nPattering in slippered feet back along the hall with an armful of books, Cara came to an abrupt halt. Jeth Langston, looking every inch the wealthy Texan in an impeccable light gray Western suit and Stetson, stood at his door, one hand on the doorknob, the other holding a leather briefcase. He registered her presence without expression for an interminable length of time, it seemed to Cara, long enough for her to wonder if he were having difficulty remembering who she was. \"Oh, I\u2014\" she stammered, like a car starting up without the least idea of its destination. Her knees were weak from the sudden sight of him. \"You've been gone for over two weeks\" was all she could think of to say.\n\n\"You've been keeping count?\" he asked dryly.\n\n\"Yes, I...have a calendar\u2014\" She had thrown it away only yesterday when she could no longer bear to keep track of the swiftly passing days of her tenure on The Conquered Land. Jeth's eyes had left hers and were roaming in cool calculation over her figure. Cara realized suddenly how scantily she was clad.\n\n\"Excuse me,\" she said, hurriedly moving past him. \"I have forgotten my robe.\"\n\nJeth blocked her passage by simply stepping in front of her. \"Not quite yet, Miss Martin. How are you? Over your bout with the flu, I see.\"\n\n\"Yes. I hardly remember it now.\"\n\n\"So it would seem from that glowing tan. Have you been riding Lady? If you have, it's been with nothing on.\"\n\n\"I have not been riding Lady with nothing on, Mr. Langston!\" Cara was shocked. \"I\u2014I've been planting a garden.\"\n\nDark eyebrows rose. \"A garden? What kind of garden?\"\n\n\"A flower garden. I\u2014I found a small area that wasn't being used for anything, and I planted some flower seeds.\"\n\n\"Why did you do that, Miss Martin?\"\n\nCara hesitated. Why _had_ she done that? \"Why, I...thought your house should have cut flowers in the rooms. They're so...austere. And there are no flowers for the cemetery\u2014\"\n\nBiting her lip, Cara bent her head in sudden embarrassment. Who was she to decide that his home and the graves of his family should be adorned with flowers? He had every right to think her presumptuous.\n\n\"You think my house austere, Miss Martin?\"\n\n\"Well, I\u2014it's a very imposing house, Mr. Langston, and...immaculately maintained\u2014\"\n\n\"But austere.\"\n\n\"Well, uh, yes, actually.\" Cara felt the light touch of two cool fingertips beneath her chin. They lifted her head to meet an inspection that showed a surprising trace of humor.\n\n\"Tomorrow morning you will show me this garden of yours.\"\n\n\"Yes, of course. Is it all right if I look these over?\" In a fluster, she indicated the books. Anything to be rid of the disconcerting fingertips. \"I wouldn't have taken them if they were yours, but since they were Ryan's\u2014\"\n\nThe humor vanished. \"These belong to me now, Miss Martin. Everything that was once Ryan's belongs to me now\u2014with one exception. However, take them along. Good night.\"\n\nThe rancher let her pass, and she hurried along to her room, conscious of his gaze following her. His last words lingered in her ears. She wondered which of Ryan's possessions was the exception to which he was referring: the land or her?\n\n\"He wants you to join him for dinner tonight,\" Fiona announced to her the next morning. \"Seven o'clock in the study.\"\n\n\"I was to show him the garden plot this morning,\" Cara said.\n\n\"He's been gone since before daybreak,\" Fiona answered. \"I don't know where.\"\n\nAimlessly, disappointment like a sharp knife inside her, Cara roamed around the kitchen. She was wearing a wraparound cover-up over a matching pair of shorts and halter top. The mornings were still cool, but even if they hadn't been, Cara would have worn the cover-up to show Jeth the garden.\n\n\"Where does Mr. Langston stay when he's in Dallas?\" she asked Fiona. The housekeeper was sitting at the table drinking coffee and reading the Dallas _Morning News._\n\n\"He has a town house there. Most often, though, he stays at the ranch of the Jeffers. They are longtime friends of the Langstons. El Patr\u00f3n will be marrying Se\u00f1orita Jeffers, the daughter, this year.\" Fiona folded the paper to a section she had been reading and handed it to Cara. \"This is a picture of her. Very beautiful, no?\"\n\nSilently, her heart halted in midbeat, Cara took the newspaper. It was folded to the society page and showed a picture of Jeth with a stunning brunette who was looking up at him and smiling. They were in evening clothes, and the caption explained that they were at a charity ball. The accompanying article said wedding bells would be ringing for the handsome pair as soon as the estate of the famous La Tierra Conquistada was settled.\n\n\"Yes, she's very beautiful,\" said Cara tonelessly, returning the paper to Fiona. \"I'll go on with my work since I don't think Mr. Langston will be coming.\"\nChapter Ten\n\nThe sun was shining in all its spring benevolence, but it could not penetrate the cloud of despair that descended upon Cara as she walked with bent head to the garden plot. So, she reasoned with sick bitterness, he wanted to make love to her for the sole purpose of divesting her of Ryan's share of the land. The sooner she signed, the earlier he could marry. No wonder he had looked less than happy to see her last night. Having just come from the warm arms of his fianc\u00e9e, he could not relish having hers around him so soon. And to think that she had actually hoped that their lovemaking would resolve their conflicts and lead (Cara could hardly stomach the idea now) to Jeth loving her as\u2014as she did him!\n\nJeth did not appear in the garden, and Cara worked strenuously in the sun, having long discarded the restricting wraparound. She was bursting with a bitter anguish that released itself in energy, and she pulled weeds and removed rocks on yet another section of land she now proposed planting. In her present state, she felt capable of clearing the entire desert. The garden and Lady, she had already concluded, would be her means of surviving the rest of the year.\n\nHe came in the late afternoon, just as Cara had decided to call it a day. Her skin tingled from the sun and shone with a thin film of perspiration. Knees, shorts, and halter top were smeared with dirt. Earlier she had wrapped the long swaths of her hair on top of her head, tucking the ends under in a way that secured them without pins. Brushing at the sand that clung to her golden legs, she did not see Jeth until he straightened up from the fence by which, she realized, he had been watching her for some time. The unexpected pleasure of seeing him momentarily arrested her, and Jeth's eyes glided over the golden swell of her full breasts to the long, shapely legs that gleamed richly in the sun.\n\nCara, clamping down hard on the absurd eruption of joy within her, stomped past him without speaking. \"Whoa, little hoss\u2014\" Jeth gave an uncharacteristic chuckle and caught her upper arm, stopping her in her tracks. \"Am I responsible for that long face? I apologize if I am. I couldn't come this morning. There was a problem in the Santa Cruz division.\"\n\nVaguely, the facts registered that El Patr\u00f3n of La Tierra Conquistada had not only apologized to her but was also bestowing upon her what amounted to a smile. It affected his entire countenance, making it seem more youthful, less severe. \"I wasn't really expecting you,\" she lied. \"I know you're a busy man. However, I'm finished for the day. I'll show you some other time.\"\n\n\"Show me now,\" Jeth said. He looked at her in puzzlement. \"Why so cranky? Maybe a good swim would cool you off. I came out here to ask you to join me for one.\"\n\nCara stared up at him, unsuccessfully willing herself to hate him. He was wearing summer range clothes: cords under chaps, of course, but in addition, a light cotton shirt, cut in the Western style so suited to his broad shoulders and tapering waist. Winter's black Stetson had been replaced with a soft gray one in lighter-weight felt. As her eyes traveled in longing over the beloved face, she realized she was memorizing its every detail to hold in her heart against the day when she was gone.\n\n\"What's wrong?\" he asked softly, concern furrowing his brow. \"Why are you staring at me like that?\"\n\n\"May I take a rain check on the pool, Mr. Langston? As for the garden, here it is. Unless you know something about flowers, the names of what I've planted won't mean anything to you. In August, I'll plant cape daisies and calendulas in the area I cleared today. They're fall flowers and should bloom even past frost. There will be flowers blooming until Christmas if the winter isn't too severe.\"\n\n\"You plan to be here then?\"\n\nThe query came mildly and could have meant anything. In Cara's frame of mind, she thought she heard a note of chagrin beneath the bland tone. \"Yes!\" she avowed belligerently. \"No matter how hard you try to drive me away!\"\n\nHe read her intention before she moved, so that when Cara made to march past him, Jeth's long arm shot out simultaneously to snare her waist and bring her back to him. She pushed at his chest and wriggled pugnaciously, toppling the topknot of hair about her shoulders. Cara heard Jeth's quick intake of breath, saw a fire ignite in the depths of his gray eyes. \"You let me go, you monster!\" she demanded indignantly, but Jeth's fingers interlaced in the platinum-streaked fall of her hair to hold her head still.\n\n\"Miss Martin, stop struggling or I will have to kiss you. I'm going to anyway, but first tell me about this burr under your saddle. What's got your dander up? You're generally pretty even-tempered.\"\n\nCara's heart fluttered like a covey of caged birds. In horror she felt her breasts hardening against Jeth's warm, male chest. She could tell by the amused twist of his lips that he felt them, too. \"Mr. Langston, let me go. I'm very tired. I'm also hot and sticky...\"\n\n\"You feel cool and refreshing, Miss Martin, better than a swim on a hot afternoon.\"\n\nOffended, Cara squirmed like a puppy held too tightly. \"Take your hands off me! I'm not your afternoon diversion, Mr. Langston. I'm afraid you've been entertaining some wild illusions about me.\"\n\n\"Shh, be quiet, Cara.\" Jeth lowered his head and the shadow of the Stetson spilled over her face. His hand under her hair propelled her toward him. \"You most certainly are a diversion. Morning, noon, and night, I find myself thinking about the uncommonly beautiful lady in the room only a few doors from me.\"\n\n\"Even when you're in Dallas?\" she asked contemptuously. Immediately she could have bitten her tongue. She must not let him know that she was aware of his marriage plans. He would then see through her resistance and merely increase his attention. And she would rather die than let him know she cared.\n\n\"Especially when I'm in Dallas,\" he answered, his voice deep and husky. It stole around Cara's heart like a warm, fondling hand. \"When I'm there, I find that I can't wait to get back to the ranch and you.\"\n\n\"To check up on me?\" The derisive note she was reaching for failed. Her breathing grew shallow. The sound of her pounding heart filled her ears.\n\n\"No,\" Jeth said, \"to do this...\"\n\nThe kiss was like nothing Cara had ever thought to experience. Though she strained briefly against the iron embrace, her resistance capitulated to her need of him, and she let him take her lips any way he chose, first gently, then exploringly, then with a mounting urgency that sent her blood throbbing through her veins with an unleashed passion that cried for him to take her, take her. She pulled him down to her, her arms wrapped around his neck, the hat brim a shelter for the long, fiery intimacy of their kiss. She was standing on tiptoe to better reach him, yielding to the hands that now molded her tight against the hard muscular frame, exulting in the feel of his chest against her, the warmth of his chaps against her bare legs. When Jeth finally lifted his mouth, it was only for a fraction of space, of time, so that he might quiz her with his eyes. Cara's own fluttered open, very near the intent gaze, and she saw something in it that lust could not corrupt, something like a...shock of rapture. \"God, Cara,\" Jeth groaned. \"You are unbelievable. I must have you. I will have you. You're like a drug I need to live.\"\n\nHe brought his mouth down again, this time with a savage hunger that sought to consume and overpower her. But Jeth's fevered declaration had penetrated the sensual oblivion in which she was lost. Cara's pride, the legacy of her New England forebears, surged to the fore. She went down off her tiptoes and pushed at the arms engulfing her. What had she been thinking of, melting in his arms like that? She could not let Jeth use her like a common tramp to reunite his beloved La Tierra. He thought her nothing but a fortune hunter, his brother's whore. Once he got her into his bed, he would have the double satisfaction of kicking her out of it\u2014as well as her signature on the papers in his desk drawer. Once he made love to her, she could not trust herself to deny him the land. And once she signed the release papers, she could not stay at La Tierra Conquistada. Her promise to Ryan would have failed.\n\nThere was only one thing to do\u2014she must make him not want her. The idea came to her with the resurgence of her pride. As Jeth's lips withdrew questioningly from hers, she was already marshalling her tactics and praying for the courage and expertise to use them.\n\nWith a slow, triumphant smile, Cara forced herself to meet the stunned query in Jeth's eyes. Instantly the embrace tightened into a prison. \"What the hell are you doing, Cara?\" he asked darkly. \"You didn't open that door just to slam it in my face, did you?\"\n\nFor answer, Cara leaned languidly back in his arms. \"That's one way of putting it,\" she purred, her eyes brilliant and gloating from beneath seductively lowered lashes. \"I just wanted to get an idea of how much you wanted me.\" She sent the pink tip of her tongue on a teasing exploration of her lips, tasting Jeth's kiss. \"Very much, I'd say. But I've decided that you'll just have to wait, cowboy. I never mix business with pleasure. With Ryan I had to, of course, and you are very tempting, and it _has_ been so long...but I think I'll just stick to my old tried-and-true rule.\"\n\nJeth, his expression registering total shock, released her as if he'd been burned. \"You mean that you and Ryan\u2014? You're saying that story you gave me in the cave was all a lie?\"\n\nCara gave a light, mocking laugh and slid her hands slowly up Jeth's shirt front, feeling the hard muscles tense, recoil. \"Well, now, that's for me to know and you to find out, cowboy. But not until the estate is settled. Then if you're still interested, why, you'll find me more than willing\u2014\"\n\nThe name he called her resounded in the still afternoon. She stepped back from the explosion of his rage, even her ears burning from the insult of the expletive. \"I'd rather snuggle up to a female coyote!\" he thundered, wrath cording the muscles in his strong neck. \"I wondered when the whore in you would finally surface. Dear God, to think Ryan loved the likes of you!\" He took a step toward her, clenched fists held rigidly at his sides, repugnance so distorting the features of his handsome face that Cara had to shut her eyes from the sight. \"You just blew it,\" Miss Martin,\" Jeth said inches from her bowed head, his deadly soft voice flowing over her like a malediction. \"I almost fell into the same trap that snared Ryan. Lucky me that your curiosity tripped you up. Unlucky you, lady, that it didn't.\"\n\nJeth stalked away from her back to the house, and Cara, dejection coursing through her, watched him go. A cool, consoling little breeze played in her hair and along her legs, but Cara was beyond solace. She felt cheapened and debased, but her plan had worked. She was repugnant to Jeth now. Not even the return of the land\u2014his real mistress\u2014was worth the price of seducing her.\n\nBut there had been that one, inexplicable moment\u2014so brief that it had flashed like a vein of gold buried deep in a mountain, lost with the blink of an eye\u2014that Jeth's soul had shone in his eyes. Bewildered, desolate, she began the walk to the house, steeling herself for what was bound to come.\n\nThe knock came on her bedroom door at nine o'clock, just as she had toweled herself dry from her bath and slipped on a floor-length robe. Hurriedly, Cara pulled on a pair of briefs as the door began to open. \"Se\u00f1orita!\" came Fiona's harsh whisper, and Cara could have fainted from relief when she saw that it was the housekeeper's head that poked around the door.\n\n\"Oh, Fiona, you scared the liver out of me! I thought that you were\u2014\"\n\n\"He wants to see you immediately. He's in the study.\" The housekeeper drew into the room, her usually impassive countenance frightened and worried. \"Please do not keep him waiting. I have never seen him like this. He is very angry, very dangerous.\"\n\n\"But I'm not dressed!\"\n\n\"Se\u00f1orita\u2014\" the brown eyes beseeched her. \"I beg you to go to him at once. You would not wish him to come here.\"\n\nCara stared at the grim face of the housekeeper. She would never have expected to hear such words from Fiona. A cold terror began to grip her. \"Very well,\" Cara said, following Fiona out. \"Is he drinking?\"\n\n\"The devil's blood from the looks of him, se\u00f1orita.\" At the bottom of the stairs, she regarded Cara levelly. \"I will be in the kitchen.\"\n\n\" _Gracias_ , Fiona,\" Cara whispered in understanding.\n\nThe owner of La Tierra Conquistada was standing at the mantel of the cold fireplace when she entered his study. He held a glass of bourbon, and she could smell cigar smoke. His narrowed gaze traveled the length of the long terry cloth robe before he spoke. \"Did I get you out of your bath?\"\n\n\"Just nearly,\" she answered, her voice cool. \"I was through, though. What did you wish to see me about?\"\n\n\"I wish to see you about you, Miss Martin. No, don't sit down. I prefer that you stand. However, I will sit down. It's been a tiresome day.\"\n\nCara's scalp tingled. Fiona had been right: danger was here. The atmosphere was fraught with it. Jeth finished his drink in a long, deliberate swallow, then reached for his cigar burning nearby. When he turned to her, his eyes were like ice. \"I have been lenient with you for my brother's sake, Miss Martin, because he cared so deeply for you. However, even he must by now be aware of what you are, so I see no further reason to show you consideration on his behalf.\"\n\n\"You have shown me consideration?\" Cara queried, her brows raised faintly, but in the pockets of the robe her hands clenched.\n\nJeth's lips twisted in a cold, distorted smile. \"I believe you will think so, Miss Martin, when you hear how you're to pay for your room and board the remainder of your stay here.\" He drew on the cigar, watching her, reading her immediate thought. He laughed without mirth. \"Relax, Miss Martin, you are safe from me. I've never been one for whores, not even Ryan's. No, I have better uses for that capable little body of yours. Tomorrow morning at seven, you will report to the tack room. The stable manager is Homer Pritchard. He will give you the equipment you will need to clean the stalls of the quarter horse stables daily. There will be other tasks involved, of course. Homer will explain. You're to work there until noon, and then you may have your lunch. Where, is up to you. At one o'clock, you will present yourself to Pepe Martinez, who is in charge of La Tierra's vegetable fields and orchard. He has an office of sorts about a mile from the stables. Homer will drive you out there tomorrow to show you where it is, but after that you'll have to get out there the best way you can. You will follow Pepe's orders concerning your chores. This will be your daily routine until something more...suitable turns up that I feel requires your time.\" The rancher studied her long and hard. \"Miss Martin, you did hear what I just said?\"\n\n\"Very clearly.\"\n\n\"Excellent. Of course\"\u2014he tapped a red coil of ash into the fireplace\u2014\"you can always exercise your option to leave, although I'm hoping you won't. I rather look forward to making your stay with us as memorable as possible.\"\n\n\"I'm sure you will, Mr. Langston, and be assured I've no intention of leaving. Is there anything else?\"\n\n\"Yes. In regard to the piano. You have my permission to play it. It's an instrument that should be played. However\"\u2014his look was grazing\u2014\"as much as I am sure I would enjoy your artistry, I don't want you at that piano while I am in this house. My mother was a lady. I don't think I could stomach hearing her piano played by a woman who so obviously is not.\" He took a long draw on the cigar while Cara remained silent.\n\nAfter exhaling a spiraling stream of smoke, Jeth went on. \"And one other thing, Miss Martin. You have committed a piece of my land to a flower garden. Make sure it produces. I do not tolerate waste on La Tierra, certainly not the waste of water or time on dabbling efforts at an unproductive diversion. Is all of that very clear?\"\n\n\"As crystal,\" Cara replied. \"Will that be all? As you say, it's been a tiresome day.\"\n\nHer composure proved her undoing. \"No, by God, that will _not_ be all!\" Jeth threw the cigar into the yawning fireplace and reached Cara before she could take two steps toward escape, at the same time dexterously yanking at the belt that cinched her robe. \"Now,\" he said grimly as the belt fell away, \"I think I'll satisfy _my_ curiosity and see what I'll be turning down when our business is finished\u2014\"\n\nTo her horror, Jeth wrenched the robe back from her shoulders, pinioning it in such a way that made her arms helpless to ward off his next intent. She tried to scream, but only a strangled whimper made it past the terror in her throat. Ruthlessly, his face a mask of scorn, Jeth commenced his slow, degrading inspection, unhurriedly traveling to explore, inch by inch, the lovely privacies of her body. Cold and numb, knowing better than to struggle, Cara closed her eyes in an agony of shame to wait for the long, painful seconds to crawl by.\n\nAt last she felt the robe jerked back over her shoulders. Jeth's voice, incisive, final, ordered, \"Fix your robe, Miss Martin, and get out of here. But before you go, here's another collector's item for your vanity. You are every inch as desirable as I knew you would be. For that reason, I can forgive my brother for being besotted enough with you to divide our land. But you, Miss Martin, I will never forgive. You are going to find that regrettable while you're on La Tierra.\"\n\nAfter she had gone to bed, Cara lay a long time in the darkness waiting to hear the rancher go past her door. Long after midnight, she heard the firm tread of his boots on the tiled corridor, and her breath held in fear. She thought he paused at her door, and she strained to see if the door handle was turning. He had not. Her imagination and her sense of hearing were both playing tricks on her.\n\nThe next morning Cara went to the huge stable complex that housed the quarter horses used by the ranch hands between roundups. Jeth's big stallion and Lady were stalled in the smaller stable closer to the big house, and Cara was relieved that she would not have to see Jeth each day when he came to saddle Dancer, his bay. With a quick glance around as she entered the stable yard, Cara estimated there must be nearly one hundred stalls built around the well-kept compound. She wondered if she was to be responsible for cleaning them all.\n\nHomer Pritchard was an unsmiling, tobacco-chewing string-bean of a man who let her know immediately that he disapproved of the presence of women in his domain. \"But the boss's orders is the boss's orders,\" he grumbled, handing Cara a pitchfork and indicating that she follow him. He led her to a stall in which a quarter horse eyed her curiously. \"Scared of horses?\" Homer asked belligerently. Cara shook her head. \"Well, that's a plus anyway. Ever clean a stall?\" When Cara replied yes, Homer spit tobacco juice emphatically into one of the many brass receptacles for that purpose attached to the bridling posts. Cara shuddered inwardly. Surely her job would not entail cleaning _those._ \"That's another plus,\" Homer said, his voice holding doubt. \"These thirty stalls are yours. This is your wheelbarrow. The dumpsters are behind the stable. We try to be through with the stall cleaning by noon. That's when the truck comes by to unload the dumpsters and take the manure out to the fields. You'll probably need a few days to get the hang of it around here, miss, but after that, the boss wants you to pull your own weight.\"\n\nCara's lip curled. \"You may tell Mr. Langston that he need have no fear of that!\" she assured Homer curtly.\n\nAt noon Cara rode out to the vegetable fields in the cab of the dumpster truck with an untalkative driver who kept his eyes on the road. She had not had time to eat the sack lunch Fiona had thoughtfully prepared for her that morning, and now she discovered she had left it at the stable. Well, she thought with a sigh, I'm too tired to chew anyway and the day's only half over.\n\nPepe Martinez was a man of short stature, as plump and friendly as Homer was thin and hostile. The Mexican overseer of La Tierra's vegetable acreage looked her over sympathetically and gave an eloquent shrug when she introduced herself. \"I am sorry, se\u00f1orita, but I have my orders.\" He handed her a long instrument with two sharp prongs at one end. \"For weeds,\" he explained, apologetically gesturing toward the countless rows of young beans among which she recognized blades of Johnson grass waving in the sun. His meaning was at once clear, and Cara swallowed.\n\n\" _All_ of them?\"\n\n\" _Si_ , se\u00f1orita.\"\n\nAs the days passed, it became apparent to Cara that in her new duties she was not to know the camaraderie that she had enjoyed on the roundup. Jeth Langston's orders concerning her were clearly expressed in the way both ranch hands and fieldworkers shunned and ignored her, leaving her to struggle with her chores on her own. Ranch vehicles, driven by men who had laughed with her on the roundup, passed her on the long trudges to and from her labors without stopping to offer a ride. She was not invited to join the coffee klatch of ranch hands who met each morning in the stable office, nor at lunch to eat her sandwich with the other workers gathered around the picnic tables beneath the yellow-trimmed gray canopy near Pepe Martinez's office trailer.\n\nCara learned that Bill, whom she missed, had been sent as foreman to run a subsidiary ranch in another county. Happy for the young cowboy, she could not help but wonder if the sudden promotion had not been designed to sever their friendly ties. Cara was confident that Bill would have remained friendly toward her in spite of his loyalty to Jeth. She rarely saw Leon, busy in the Feedtrough these days with the extra duties of butchering calves and preparing the daily bounty of fresh vegetables for La Tierra's freezers. Jim Foster alone remained accessible, but his commiserative manner made Cara uncomfortable. It suggested they shared a mutual alliance against Jeth Langston, an attitude that forced her to avoid the foreman whenever possible.\n\nJune passed into July and there were days when Cara did not hear the sound of her own voice. August came, and La Tierra baked under the hottest, driest sun that she had ever known. She worked steadily and hard, determined not to give Homer or Pepe reason to criticize her to their employer. She grew accustomed to her solitude and the loneliness of her days. The sun deepened her tan and lightened her hair to purest platinum. In her garden, the flowers broke through the caliche-stressed soil and bloomed, and in delight she cupped their colorful heads in her work-roughened hands, thrilling at their beauty and abundance. Great bouquets began to appear on gleaming tabletops in the house and before the headstones of the Langston graves.\n\nCara discovered that Jeth had not forgotten her garden. One evening when she went to tend it, she found a man-sized pair of bootprints embedded in the moist sand where someone had stood to survey her handiwork. _Jeth!_ she thought, and her heart had held in her throat.\n\nSince the evening in the study, Cara had been able to avoid a face-to-face meeting with the owner of La Tierra. She knew his routine by now and was able to circumvent his comings and goings in the big house. At her request, Lady had been moved to one of the thirty stalls she had been assigned to maintain. When Cara's day was over, just as Jeth was finishing his end-of-the-day swim to change for dinner, she was saddling Lady for a ride in the long summer twilight. Afterward, while she was on the dusty trek to the house, Jeth, she knew, would have finished dinner and gone to his study for the evening. It was then, after a visit to her garden, that she would climb the stairs to her room and eat in solitude the dinner that Fiona had left her.\n\nOn the rare occasions when Jeth was away from the ranch, Cara spent her evenings before the Steinway, expressing her pain in selections written for the kind of deep despair she felt. Sometimes Fiona, who had come to have a grudging affection and sympathy for her, would come to lean in the doorway of the living room to hear her, her ever-busy hands motionless around the dish she meant to dry while she listened. One evening when Jeth was gone Cara sat down before the keyboard. Her fingers drifted into the haunting bars of \"Full Moon and Empty Arms,\" from Rachmaninoff's Second Piano Concerto. The piece suited her mood somehow. That afternoon she had ridden Lady into the foothills and had come across Devil's Own again. In majestic splendor, the black horse had gazed down at them from the crest of a mountain, and Cara's flesh had prickled with a sudden portentous chill as she returned the stallion's stare. The message in the dark, equine eyes seemed quite plain: _You wear the brand of La Tierra Conquistada. You will never be the same again. You will never be free._\n\nSo now she released into the music the sudden grief that had made her turn Lady sharply and knee the horse into a fast gallop back to the ranch. It was only as she was stroking off the last chords that Cara became aware of a familiar scent in the room\u2014the aroma of Jeth Langston's cigar. Startled, she wheeled around on the piano bench to find the room empty. Afraid that her imagination was assuming dangerous proportions, Cara rose and walked slowly over to the large formal chair near the study door. Several coils of cigar ash smoldered in the ashtray. Jeth was home. He had been listening to her play the Second Piano Concerto.\n\nIn late August the knees of her jeans gave out. Cara trimmed the legs off above the knee, and while she was at it, decided to cut off the long sleeves of all her shirts. They had been fine when the weather was cool, but now they were confining and hot. She hemmed the edges as best she could, but her skill with sewing was limited, as the shirt hems testified.\n\n\"Fiona,\" she asked shortly thereafter, \"will you cut my hair for me?\"\n\nFiona's impassive face gave way to one of its rare moments of expression. \"Cut your hair, se\u00f1orita?\" The housekeeper was dumbfounded. \"But your hair is beautiful. It is like white gold!\"\n\n\"It is unbearably hot, and I can't keep it out of my way. I can't wash it as often as I would like because it takes too long to dry.\"\n\n\"Very well, se\u00f1orita,\" Fiona agreed reluctantly, \"but it is a pity.\"\n\nAnd so one hot Saturday afternoon after her chores in the stables were completed (like La Tierra's other employees, she was free until Monday morning), Cara sat in the kitchen on a stool, a towel draped around her, and submitted herself to Fiona's scissors. Snip! snip! went the scissors. Down, down fell the hair.\n\n\"What the hell are you doing!\" demanded a voice from the doorway, and the razor-sharp, pointed scissors arrested dangerously close to Cara's eyes as Fiona stammered, \"I\u2014I am cutting Se\u00f1orita Martin's hair, Patr\u00f3n. She asked me to.\"\n\n\"Stop it!\" he ordered, but it was too late. A heap of hair lay on the floor, soft as silk, as shining as the most precious of metals.\n\nCara sat in total silence, staring straight ahead, as Jeth came to stand in front of her, his expression one of horrified surprise. \"My God...\" He let out a deep breath, and Cara wondered what in the world she must look like. Like a waif, she decided, feeling the blunt ends of her hair. Fiona had simply begun at one ear and cut around to the other. The towel did not cover the cutoff jeans, the flannel shirt with its amateurishly hemmed sleeves. \"Your hair, your hair\u2014\" Jeth spoke almost in anguish.\n\nIt will grow again, she thought. It will darken by wintertime. A Texas sun will never again bleach it platinum. The thought made her heart close like a shamrock at dusk. Defiantly, her voice cold, Cara spoke for the first time. \"My hair interfered with my work. It was hot and annoying.\"\n\n\"Yes,\" Jeth conceded. \"I suppose so.\" With a swift movement he reached for one of her hands and inspected it critically. Cara flushed and snatched it away in embarrassment, hiding it under the towel. Her hands were rough and red, the nails broken and unkempt. She had once taken such care of her hands. \"Don't you wear gloves anymore?\" the rancher demanded. \"What happened to the rubber gloves I bought you?\"\n\n\"They were used up long ago. I don't need them now.\"\n\nThe look he gave her made Cara want to curl up and die. It held a mixture of pity and disgust. She was sure that Sonya Jeffers's hands were as soft as kitten fur and that she would never have worn cutoff jeans and a tattered shirt.\n\nJeth left the kitchen, and the housekeeper and Cara stared at each other.\nChapter Eleven\n\nThere's a party this Saturday that I'd like to take you to,\" Jim Foster told Cara. He had followed her into a stall where she was filling a trough with hay. \"Will you go with me?\"\n\nIt was the end of September and in the mornings a crisp touch of fall was in the air. \"Why\u2014why, I don't know, Jim...\" She was startled that things like parties still existed.\n\n\"Why not?\" he demanded impatiently. \"Are you forbidden to do anything but work?\"\n\nGood question, Cara thought, and looked up at him with a stirring of compassion. She had to admit that he had made every effort to be kind to her. He had even defied Jeth's orders by openly befriending her, a surprise move that had made her ashamed of her earlier suspicions. She wondered if Jeth was aware of their limited association. There was no reason for him to be jealous of Jim now. Impulsively, she said, \"I'd love to come. What should I wear?\"\n\nShe should wear, Jim told her without hesitation, a dress! She chose a dusky blue one with a scooped neckline and short sleeves and a full skirt that swirled just below her knees. To complement the dress she selected a pair of high-heeled suede sandals in the same shade of blue.\n\nCara spent all of Saturday afternoon readying herself for her date. She gave herself a beauty treatment from head to foot, rolling the short bob of platinum hair for the first time since Fiona had cut it and pedicuring her feet. The appearance of her hands, she was relieved to see, had greatly improved since the box of work gloves had mysteriously appeared in her room the day after her last conversation with Jeth. As the time neared to meet Jim at their prearranged spot outside the house, Cara found herself getting more excited about the evening ahead. She hummed to herself as she put the finishing touches to her makeup and slipped on her shoes. It had been so long since she'd dressed up, fussed with her hair\u2014had fun! When she had finished dressing and stepped back from the mirror for her first full view of herself, Cara had to blink twice, she was so shocked at the woman staring back at her. Could that platinum-haired, golden-skinned, violet-eyed stranger possibly be her!\n\nA knock came at the door, breaking into her bemusement. She drew away from the mirror, still enrapt, and opened the door, expecting to find Fiona on the threshold. Instead Jeth Langston stood there, taller and even more commanding than she remembered, dressed for dinner in a shirt and slacks of gray twill whose color was reflected in the hard clarity of his eyes. Cara stood stock-still. A faint fear that he had come to prevent her from going made her heartbeat quicken.\n\nJeth spoke first. \"Jim Foster just called from the bunkhouse. Something has come up, and he won't be able to take you to the dance tonight. I told him I'd tell you. He sends his apologies.\"\n\nCara did not reply immediately. Disappointment cut sharply, and when at last she spoke, her voice was strained. \"Is Jim still on the phone?\"\n\n\"No.\"\n\n\"I see. Thank you.\" She moved back to shut the door, glad of the excuse to avert her face.\n\n\"Miss Martin...\" Jeth put out a broad hand to prevent the door from closing.\n\nCara forced herself to meet his eyes with dignity, knowing she would find them alight with mockery, or worse\u2014softened with pity. He was not the least deceived by her cool manner. The man knew exactly how she must be feeling. After all, she had been stood up, and now here she was, all dressed up with nowhere to go. But to her surprise, the gray eyes were sweeping over her in undisguised admiration.\n\n\"It would be a shame for all of that to go to waste,\" he said politely. \"Fiona is visiting relatives tonight and has left me on my own to cook. I know it's not my company you'd hoped for tonight, but maybe you'd consider joining me for one of my steaks and a bottle of that wine you like?\"\n\nCara's heart began to race at the temptation of the offer. The thought of spending the evening alone in her room, where she spent all of her nights and weekends, was abhorrent to her, especially since she had so looked forward to the evening.\n\n\"I'm surprised you don't have plans for the evening,\" she hedged, unsure of Jeth's motives. Was this invitation offered to give him another opportunity to hurt and humiliate her?\n\n\"Mine fell through, too.\"\n\n\"I hope you were not especially looking forward to them.\"\n\n\"I think I can rightly say that my disappointment is less than Jim's. How about it?\"\n\n\"You are suggesting a truce for the evening?\"\n\n\"Why not? It beats spending it alone in separate trenches.\"\n\nCara gave him a small consenting smile, her teeth as white and luminescent as pearls in contrast to the dark honey of her skin and the soft pink lipstick. Jeth took an audible breath.\n\n\"That's the first time I've seen that.\"\n\n\"What?\"\n\n\"Your smile.\"\n\n\"Then as usual you're one up on me, Mr. Langston, for I've never seen one of yours.\"\n\nJeth lit the grill by the pool and prepared their drinks while Cara tossed a salad and put two potatoes into the oven to bake. There had been a tense moment in the kitchen when Jeth had returned to inquire about lighter fluid for igniting the mesquite. \"Fiona keeps extra supplies of that sort up here,\" Cara told him, and made to get it, automatically pulling up the kitchen stool the two women used to reach items on high shelves.\n\nJeth saw her intention and said, \"Don't bother with that; I'll get it,\" and came to stand behind her, reaching over her platinum head to rummage for the new can. His body touched hers. Her whole being tensed at his proximity, and for a few insane seconds she absurdly imagined that his lips had brushed the top of her hair. It seemed an age before he moved away. \"I've got it,\" he said at last. \"Come outside. I have your wine ready.\"\n\nThey sat sipping their drinks beside the pool and watched the last of the September sunsets hover near the horizon. Cara was convinced that nowhere in the world were there more dramatically beautiful sunsets than in West Texas: \"To make up for the fact that we don't have much else in the way of nature to brag about,\" Leon had said to her on the roundup.\n\n\"This is such an ideal place for parties, Mr. Langston,\" she said, indicating the spacious deck and pool. \"Do you ever use it for that?\" There had been no guests in the house since her arrival.\n\nShe could tell from the way Jeth toyed with his drink and did not answer immediately that her question had touched sensitive ground. Without the slightest change in tone, he replied, \"I find my Dallas town house more suitable for entertaining.\"\n\nCara stared at him. His meaning was unmistakable. \"Because of me?\"\n\n\"Yes,\" he replied, meeting her eyes steadily. \"Because of you.\"\n\n\"But, Mr. Langston\u2014!\" She was genuinely distressed. \"I don't mean to deprive you of the use of your home. Of course not! Why, it isn't as though I would _crash_ your parties. Surely you don't think I would!\" She was agitated and embarrassed. The wine had turned to vinegar on her tongue.\n\n\"Miss Martin, let's not ruin a salvaged evening by breaking our truce. It may surprise you, but I credit you with a great deal more propriety than that. I'm sure you'd be more than willing to stay out of sight while I'm entertaining, like some unsuitable relative confined to the attic while everyone else is having a good old time in the drawing room below. No, thanks. That's not my style. I go to Dallas often anyway. It's just as easy to fulfill my social obligations there.\"\n\nHe spoke with finality, and Cara's thoughts flew to the newspaper picture of Sonya Jeffers. No doubt his fianc\u00e9e knew all of his friends and business associates; she probably made a splendid hostess. She was also probably very curious about the woman living in her future home. Cara would have been.\n\nJeth changed the subject by asking about Marblehead. She answered his specific questions about its history, then, without mentioning his name, found herself describing all the places that she had loved and shared with Ryan. She was oddly comforted by speaking of them to the brother who had loved him. She told Jeth about Marblehead Harbor and Devereux Beach and the waterfront with its never-ending variety of sights and sounds and smells. She had been talking for some time when she suddenly broke off, aware that she was monopolizing the conversation and that Jeth's thoughts seemed far away. _In Dallas!_ Cara thought in stricken dismay.\n\n\"Forgive me,\" she said quietly. \"I didn't realize I had become boring.\"\n\nJeth glanced at her quickly. \"Nonsense. You know that you could never be boring. I was simply completely transported to Devereux Beach, that's all\u2014with you and Ryan.\"\n\nSo he had known, of course, of whom she was speaking. A sudden remark trembled on her lips, unspoken. She had almost said, _I wish you could have been there with us._\n\nJeth asked suddenly, \"Do you miss Boston very much?\"\n\n\"Not as much as I thought I would,\" Cara answered truthfully. In astonishment it occurred to her that she did not miss Boston at all.\n\n\"You must find West Texas vastly different.\"\n\n\"Not all that different, Mr. Langston,\" Cara replied. \"Perhaps because I grew up on the edge of the Atlantic, I am accustomed to vastness and space and uncluttered horizons.\"\n\n\"Do...you like anything about this part of the country?\"\n\nCara laughed. The wine had made her slightly reckless. \"If I said that I like everything about it, you would probably interpret that as meaning that I intend to stay and claim Ryan's share of the land in order to live here the rest of my life!\" When he looked startled, she said with gentle assurance, \"I have promised to return it to you, Mr. Langston, and so I will. But to answer your question, yes, I like West Texas. I like the clear, clean air and dry, honest heat. I even like the wind, which blows endlessly like it's searching for a home. And I like the land itself because it's uncompromising and hard, like you, Mr. Langston. However, when I went to plant my garden, I found that, given attention to its needs, the land can be very giving, very loving...\"\n\n\"Like me?\" Jeth asked cryptically, the gray eyes intent upon her face.\n\n\"Oh, that I wouldn't know.\" Cara felt her cheeks grow hot. The wine had gone to her head and she had said too much. She should never drink. It was obvious that she couldn't handle alcohol. \"Do you suppose we might put the steaks on? I'm getting a little tipsy.\"\n\nLater, when the evening was over, Jeth did not offer to escort her up the stairs to her room, and she thought she should be grateful for this unexpected consideration. How awkward to be taken to her bedroom door when he knew full well that she felt the physical vibrations between them\u2014sexual tensions that had increased as they began to play chess. Chess, she decided as the game wore on, was not a game to be played between a man and woman physically drawn to the other. Every move became fraught with a double meaning, and Cara grew more and more uncomfortable as Jeth's aggressive moves began to place her queen in hopeless jeopardy.\n\n\"Leaving before the game is through?\" he asked with cool mockery when she remarked at the lateness of the hour and asked to be excused. He could not have known how close his remark came to the truth or how deeply it pierced. Her year at La Tierra was now half over. She had not needed the carefully marked calendar she had discarded long ago to remind her of the rapidly passing days. Yes, she would be leaving before the game was through.\n\n\"Perhaps another time,\" she said, giving him a polite smile and searching with her toes for the high-heeled sandals she had slipped off beneath the game table. The sumptuous fur rug had been too tempting for her stockinged feet to resist. Unhurriedly, Jeth placed his cigar in the ashtray and stood up, his fit, powerful body emanating a physical magnetism that stopped her heart. As she watched him wide-eyed, the rancher came around the table and gently drew her by the wrists to her stockinged feet. Conscious of the disparity in their heights, Cara tried not to tense as she felt Jeth's smooth, dry fingertips, his thumbs still in control of her wrists, slide sensuously to nestle in her palms. She was too inexperienced to know if the action was deliberately provocative. All she knew was that his touch sent fire through her and that she could hardly breathe as he lifted her hands for his examination.\n\n\"I am glad to see that you managed to salvage these. Now they look as they always should.\"\n\n\"Well\u2014yes\u2014\" Cara was flustered and could not meet his eyes. She wondered if she should mention the box of gloves\u2014she had sent a brusque thank-you by way of Fiona\u2014but her pride and the shallow capacity of her lungs kept her silent.\n\n\"Do you think these hands can learn to hold and shoot a rifle?\"\n\nThe question was so unexpected that Cara's glance shot up, and her lips parted in surprise. Jeth's eyes dropped to their moist softness, and Cara instinctively pulled at her hands. The rancher's thumbs pressed deeper, and she allowed them to remain in his. \"Tomorrow morning after breakfast I'm taking you out on the range for some target practice. If you're going to ride Lady the far distances you do, you should take a rifle along and know how to use it. It's a practice of the ranch that I rigidly enforce so don't argue about it. You never know what you can run into out there, especially with winter coming and the coyotes hungry.\" He released her hands and with easy grace reached down the other side of the chair where he had been sitting. When he straightened up, the slim blue sandals dangled from his fingers. \"Were you looking for these?\" he asked with a wry lift of his brows.\n\nCara reached for them, and a little shock passed through her as he held them a fraction of a moment longer than necessary before yielding them to her. \"Good night, Mr. Langston,\" she said in a voice less firm than she would have liked. Then she fled the room before she could be compelled to stay.\n\nThe next day Jeth drove Cara in the jeep out to a remote section of the ranch to give her brusque lessons in aiming and firing a .30-30 rifle. Every nerve in her was alert to the nearness of his body as he positioned the stock of the gun into the small of her shoulder and held her steady while she fired. He seemed unaffected by the closeness of her head or of his arm unavoidably pressing her breast during the demonstration. Cara was so intensely aware of him that she had difficulty concentrating.\n\nDuring the drive back to the house, Jeth flicked a glance over the silk blouse and tailored slacks that she had chosen for the outing and remarked, \"If you plan to be here during the winter, you'll need some new ranch clothes. Tomorrow take the Continental and go see Miss Emma again.\"\n\n\"You'll probably think me a coward, Mr. Langston, but I'd rather wear the rags I have than have another encounter with Miss Emma. Besides, I don't have any money.\"\n\nThe Texan looked at her in surprise. \"What about Ryan's money?\"\n\n\"That's just it\u2014it's Ryan's money.\"\n\n\"Don't feel guilty about spending it now. You've earned it. If you're disinclined to spend it, consider it payment for your work while you've been at La Tierra. You've certainly earned more than your room and board. Marfa isn't the only place around here to buy clothes. You can go a few miles farther the other way to Alpine and shop. No one will know who you are if I'm not with you.\"\n\n\"Who will clean the stables?\" she asked, more for his reaction than anything. A warm little glow had begun in the center of her heart.\n\n\"No one as well as you,\" he answered, surprising her still further, even though his mouth remained stern and his eyes on the road straight ahead. \"However, I told you that you'd keep that job until I needed you more somewhere else. I need you in the study.\"\n\nInstantly on guard, she faced him. \"Doing what?\"\n\nJeth laughed shortly. \"Why, Miss Martin, what a suspicious mind you have! I want to take advantage of your skills as a librarian, not you. My library is in chaos. It's time the books and papers were put in some kind of order. Will you please see to it for the next few weeks?\"\n\nIt was a command couched in a request, Cara knew, but how much nicer to tell her like that than in the high-handed fashion he usually used with her. \"What about the fields? Am I still to work out there?\"\n\n\"Pepe will be putting them to bed for the winter. He won't need you for that. Your talents will be put to better use in the study. You'll begin Tuesday. Take tomorrow off and take your time looking around Alpine. There's a museum there that might interest you and a fairly good restaurant where you can get a decent lunch. Write me a check for the amount you think you'll need, and I'll leave you cash for it on the hall table. Also a map of the town and my keys to the car.\"\n\nHe thinks of everything, Cara thought, happy for the opportunity to have a change in her routine. She had only been off the ranch one time with Jeth, and tomorrow would be an especially nice time to get away: it was her birthday.\n\nThe fall roundup was in progress and Jeth Langston had been gone from the ranch over three weeks. Cara thought about him constantly as she indexed and catalogued the valuable collection of books in his study. Her suggestion that Ryan's books from upstairs also be included had earned her a silent look of gratitude from the rancher that had warmed her heart for days. \"How ridiculous!\" she chided herself. \"After all, I'm doing _him_ a favor, not the other way around!\" But she spent hours lost in the scrapbooks and photograph albums depicting the Langston family and the history of the ranch. By the time she had indexed them with the other memorabilia and documents, she felt intimately knowledgeable about every Langston who had ever been, including Jeth. He was impossible to imagine as an infant, but Cara found that indeed he had been one, and, from the photographs, held lovingly and often on his beautiful mother's lap.\n\nTouching Jeth through the photographs, learning about him in the articles and clippings she read, made her miss him terribly, with a craving that gnawed at her heart and violated her sleep. She longed for him to return to the ranch, if only briefly, as he had during the spring roundup, leaving Jim in charge. Just a glimpse of his tall figure striding toward the house from the landing strip would be enough. She could content herself with that.\n\nCara was puzzled that she had not seen Jim since their broken date. The foreman had been in the mountains with the roundup for the remuda when she returned from her shopping trip to Alpine, but she thought it strange that he hadn't sought her out to offer an explanation before he left for the October cattle drive.\n\nShe was sitting in the living room playing the Steinway when she sensed Jeth's presence. Her fingers stilled over the keys, her shoulders tensed in anticipation of her joy before she turned on the bench to find him watching her, the black Stetson, now returned for the winter, pushed back on his dark head. Quietly she pulled the cover down over the keyboard. \"Hello,\" she said, turning back to him. The interrupted notes of \"Clair de Lune\" hung in the air as they stared at each other.\n\nSlowly Jeth said, \"I haven't heard Debussy played like that since...well, in a long time,\" he amended. \"How have you been?\"\n\nLonely, Cara wanted to say, but she spoke calmly, giving him a slight smile. \"Busy. The library is finished.\"\n\n\"I'll go up and change and then you can show it to me. We'll have a drink together.\" He did not wait for her to answer but left the room, the welcome sound of his black boots striking the tiled floor.\n\nWhile Jeth changed, Cara decided to run out to the Feed-trough to see Leon. She had missed the dear old fellow. Like Jim, she had not seen him since her return to the ranch from her day's outing, not having wanted to interfere with his preparations to ready the chuckwagon for the roundup.\n\nA half hour after speaking with Leon, she stormed across the ranch yard into the house to Jeth Langston's study. A small balled fist rapped sharply on the door, and when Jeth called, \"Come in,\" Cara opened the door, not bothering to close it after her, and marched up to the rancher with blazing eyes and heaving chest. \"You are insufferable!\" she announced to the startled Texan. \"How could you fire Jim Foster just because he asked me out!\"\n\nJeth regarded her without speaking, all expression in the gray eyes slowly fading. Then he calmly returned to the task of pouring their drinks. \"Here,\" he said, handing her a glass of wine. \"Maybe that will settle you down.\" His eyes fell to the low opening of her blouse, then traveled back to her face. \"You still have a tan,\" he observed, \"and your hair is still sun streaked. What have you been doing to get so much sun?\"\n\nStruggling to gain control of her temper, Cara set the glass of wine down untasted. \"I've been helping Pepe,\" she answered. \"With so many men gone on the roundup, he needed help. This Indian summer has kept everything out there growing and productive. Now, Mr. Langston\u2014\"\n\n\"I didn't tell you to work out there,\" he interrupted. \"You take too much on yourself, Miss Martin.\"\n\n\"Mr. Langston, don't change the subject. How could you fire Jim because of me? He'd been with you for years and was an excellent foreman. No wonder you didn't get away from the roundup like you did in the spring...\" Cara bit her lip. She hadn't meant to say that.\n\nJeth's brows raised. \"So you noticed?\" He took a sip of his drink, considering her over its rim. \"Jim meant to weave his way into your affections, Miss Martin. I don't mean to shatter any illusions you might have concerning his feelings for you, but you could have been as plain as a fence post, and he would have done the same. He had in mind to convince you not to sell your share of La Tierra to me; then he meant to put himself in charge of running one-half of my ranch.\"\n\nCara knew Jeth spoke the truth. It was all as clear to her now as the straight nose on his hard, handsome face. There had been something basically self-serving in the foreman's character, an opportunistic streak that she had dimly suspected. But what really hurt was to know that Jeth thought Jim could have succeeded. That was why he had kept her busy the evening Jim was fired. That was why he had taken her out on the range all the next day, had sent her to Alpine to shop and sightsee the day after.\n\nAnother thought struck her. Jeth had never been jealous of Jim at all! He had only been fearful that his foreman would gain an inside track on her affections, a possibility that might have cost him half of his beloved La Tierra.\n\nHer fists still balled, Cara wanted to strike at the ruthless face that she loved with all of her heart and soul. \"You didn't have to pretend that you wanted me to have some new clothes, Mr. Langston, in order to prevent Jim from seeing me after you fired him. I wouldn't have turned over Ryan's share of the ranch to him, no matter how much you'd like to believe otherwise.\" Tears stung her eyes. \"I don't feel like showing you the library tonight. If you will excuse me\u2014\"\n\nShe was halfway to the door when Jeth's words stopped her. \"I sent you to Alpine because it was your birthday.\"\n\nCara was sure her feelings were expressed in the tensing of her shoulders, the halting of her footsteps. Because she knew her face would betray her, she did not turn around. \"How did you know?\"\n\n\"I know everything about you, Miss Martin. The detective, remember?\"\n\n\"Oh, yes.\" All he needed to know of her, he was saying, could be reduced to a few pages in a file folder.\n\nThe rancher had come up behind her. \"Turn around, Miss Martin.\" When she did, he saw the sheen of tears in her eyes. \"Are those for Jim?\"\n\nLet him think so, she thought. \"I feel responsible. If I hadn't been here, then this wouldn't have happened.\"\n\n\"Many things would not have happened if you had not been here, Miss Martin. Believe me, Jim is a minor casualty.\"\n\nThe tears dried in Cara's eyes. She understood what he was implying. He hated her very presence on La Tierra. Well, he needn't worry that she would impose herself on him in the future. She would stay completely out of his way. He would not set eyes on her again, not if she could help it. On the day her promise to Ryan was fulfilled, she would quietly disappear. He wouldn't even know what had happened to her, nor would he care, for on that day the papers would arrive releasing her claim to the ranch, and he would be free to marry.\n\n\"I dislike you intensely, Mr. Langston,\" Cara said bitterly. At the moment, it was the truth.\n\n\"I am aware of that, Miss Martin. It's such a shame.\"\n\nWithout a word, she turned and left him standing in the middle of the paneled room, a tall, forceful figure who gazed after her long after she was gone.\n\nAs winter approached, the shorter days made it more difficult for Cara to avoid the owner of La Tierra Conquistada. Twilight came early and fell fast, halting the ranch activities that kept Jeth out of the house. Having no assigned duties and finding solace in work, Cara volunteered her now-coveted services where they were needed. She helped Homer in the stables and Leon in the Feedtrough, ignoring Jeth when he happened to appear unexpectedly. For convenience, she had to bring Lady back to the smaller stable for the winter, where Jeth's bay was stalled, and resigned herself to the anguish she felt when their paths crossed there.\n\nStill, because he was essentially a man of routine, she was able to chart his comings and goings with some accuracy, and the two of them rarely met. For Cara, the long hours before bedtime were the hardest to fill. Occasionally she watched television with Fiona in the housekeeper's suite of rooms off the kitchen. In her own room she studied Spanish, which she was now able to speak with increasing fluency. She wrote her once-a-week letter to Harold St. Clair and read the best sellers and classics she got from the traveling bookmobile that stopped at the ranch every Tuesday.\n\nCara looked forward to the arrival of the bookmobile each week. She had become friendly with Honoria Sanchez, the gentle Mexican woman who was its driver. Honoria was also a librarian, and Cara enjoyed their professional chats.\n\nThanksgiving came and went and La Tierra began to gear itself for the Christmas holidays.\n\n\"They won't be nearly as exciting as in years past,\" Fiona grumbled in the kitchen one morning. \"Se\u00f1or Ryan is gone and El Patr\u00f3n will spend the holidays in Dallas.\"\n\nWith the Jeffers, Cara conjectured, and why not? They must be like family to him, and with this the first Christmas without Ryan...Sympathy for Jeth lay in her heart for days before she found the nerve to go to his study one evening after dinner.\n\nHe had expected Fiona, as was evident by the surprised lift of his brows, the unblinking stare with which he regarded her entrance into his sanctuary. \"Why, Miss Martin\u2014\" He spoke ironically. \"To what do I owe the unexpected pleasure of this visit?\" He rose languidly from the wingback chair to greet her, but not before his posture had suggested to Cara that he had been deeply sunk in his thoughts, his gaze lost in the fire that burned brightly in the limestone fireplace.\n\nCara's hands fidgeted at her sides. \"Uh, Mr. Langston, I\u2014I would like to discuss with you your plans for Christmas. Or rather, that is... _my_ plans for Christmas.\"\n\n\"Sit down, Miss Martin,\" Jeth invited, indicating the other chair next to the fireplace. \"You have plans for Christmas? I had thought you would be staying here.\"\n\n\"Yes, well, you see, that's what I want to talk to you about.\" She was acutely embarrassed. She had to moisten her lips to go on, a gesture that brought the lids half down over Jeth's eyes. \"Fiona has told me that ordinarily when...Ryan was alive, you stayed home for Christmas and that the ranch hosted many festivities and parties. This year I understand that you are going to Dallas to be with your...fianc\u00e9e's family\u2014\"\n\n\"My fianc\u00e9e?\" Jeth raised up in his chair, his expression instantly alert. \"Who told you about my fianc\u00e9e?\"\n\nSo it is true, Cara thought, a hand squeezing her heart. \"I read of your engagement in the Dallas _Morning News_ ,\" she answered, amazed to hear the steadiness of her voice. How is it possible that the dead can speak? \"I'm afraid you're leaving because of me, that you'd be entertaining if I weren't here. If I go away for several weeks, your normal holiday activities won't have to be interrupted. Mr. Langston\u2014\" Cara raised an imploring hand when she saw Jeth about to interrupt. \"I would like to do this for Ryan's sake. I can't bear for his brother to have to go somewhere else to spend Christmas because of me.\"\n\n\"Where will you go?\" Jeth asked noncommittally.\n\n\"I have several places,\" she answered swiftly. \"That doesn't have to be a concern of yours.\"\n\n\"Miss Martin,\" Jeth said on a sigh, \"I happen to know that you have nowhere to go. You have no money to get there even if you did, unless, of course, you use Ryan's money, which you won't do, not if I have learned anything about you. I appreciate your consideration for my feelings, but you can be assured that I would not allow you to run me out of my home. I prefer to be in Dallas this year for Christmas. There are too many memories here of...what should have been.\"\n\nHe had turned his dark head away from her to resume his contemplation of the fire. It was a sign of dismissal, Cara knew, and she ought to get up and go. But his last remark held her sadly enthralled, like the hum of music when the final notes are played. The two of them were what should have been, she was thinking\u2014not lovers, perhaps, but at least the best of friends. They had so many experiences in common. They had both been deprived of their parents at a sensitive time in their lives. They had each known the loss of worked-for dreams, deferred forever because of family obligations. And they had loved and lost in common a fine human being. Yet here they were, each sailing alone in his own ship on a sea of grief when they might have made the journey together, for a year at least.\n\nCara got up to go, and Jeth rose also. \"Very well, Mr. Langston.\" She held out her hand politely. \"If I do not see you before you leave, I hope your holidays will be pleasant.\"\n\nHis firm hand closed around hers. \"When did you read of my engagement, Miss Martin?\"\n\n\"Sometime last summer, Mr. Langston.\"\n\n\"I see. Happy holidays to you, too, Miss Martin.\"\n\nCara withdrew her hand and walked quickly from the room.\nChapter Twelve\n\nJeth was gone from La Tierra until the middle of an icy January. Somehow Cara got through Christmas Day, the ultimate emotional moment coming when she opened Jeth's gift to her. A week earlier, a tall Christmas tree had been erected before the window next to the Steinway, and she and Fiona had decorated it with the hallowed ornaments that had adorned La Tierra's Christmas trees for a century. But on Christmas morning, she and the housekeeper made their way to the headquarters building where another tree shone cozily in the window and around which La Tierra's resident cowhands gathered for the traditional opening of their presents.\n\nBefore Jeth's departure for Dallas, Cara had boldly slipped her present to him into his leather valise, which had been lying open on his bed. It was the foot-long piece of oak that she and Ryan had found their last day on the beach. She'd sent the piece away to be trimmed and set with the gemlike bits of bottle glass into the brand of La Tierra Conquistada.\n\nShe'd not expected to be remembered by the rancher, and so on Christmas morning when Leon, playing Santa Claus, handed her the small, exquisitely wrapped package with her name written in bold black ink on the white envelope tucked beneath the ribbon, her heart had all but stopped. She extracted the envelope first. Inside was a simple note: \"I trust you will wear this with your seagull as a symbol of another land you have conquered. Merry Christmas.\" It was signed with the single initial, _J._ The gift had been a small gold drop in the design of a prairie falcon, exquisitely made. Cara had pressed the trinket to her breast and reread the brief note dozens of times.\n\nShe was not aware that the owner of La Tierra was home until she saw the big gray Continental parked in the garage. Her pulse rate quickened as she ran the rest of the distance to the house. Thank heavens he was home! She had just come from the bunkhouse where she had been summoned by the new foreman to look at Leon, who had taken to his bed with a high fever. Cara diagnosed a severe case of influenza and told the foreman to call a doctor.\n\nWhen she entered the back door, frozen to the marrow in her jean jacket, she expected to find Jeth sitting at the table with the housekeeper, visiting over coffee. Neither was in the kitchen and Cara, knowing that Fiona had not been well either, felt a sense of alarm. \"Fiona!\" she called sharply.\n\n\"Out here!\" came the familiar voice, and Cara ran to the passage off the kitchen that led to the housekeeper's quarters. Jeth was carrying Fiona down the hall as if she weighed no more than a doll. \"Call a doctor!\" he ordered over his shoulder. \"She's burning up with fever.\"\n\n\"One is on the way,\" Cara said. \"Leon is sick, too.\"\n\nAn influenza epidemic swept through the ranks of La Tierra's personnel as icy winds tore across the bleak plains, sending gray tumbleweeds scuttling over the unimpeded distances like uprooted ghosts. Jeth agreed with Cara's suggestion to house the sick men in Ryan's room to prevent the virus from spreading to the skeleton crew remaining in the bunkhouse. The entire household staff succumbed to the rampant virus, which left the caring and feeding of the dozen sick men in Ryan's room to her. Also, of her own volition, she helped Toby in the Feedtrough to cook meals for the rest of the crew who now had the added burden of herding as many cattle as they could find into the deep draws for protection against freezing temperatures. They were especially worried about the cows, swollen now with their unborn calves to be delivered in early spring.\n\n\"Get away from me, child,\" Leon attempted to dissuade Cara when she came near to minister to him. \"I don't want you gettin' what I got.\"\n\n\"Hush, Leon,\" Cara told him gently, \"and eat some of this soup for me.\"\n\nCara and Jeth reached the same conclusion about the housekeeper and cook three days after the pair had taken to their sickbeds. Raking fingers distractedly through his dark hair, the gray eyes flinty with worry, Jeth admitted, \"I'll have hell to pay for this when they're well, but I'm calling an ambulance to take them to the hospital in Alpine. They're too old to fight this by themselves, and I'm not taking a chance of losing them. They're all I have.\"\n\nCara stared wordlessly at him, fear clutching at her throat, and Jeth saw something in her eyes that made him reach out suddenly and clasp the back of her neck, giving it a gentle shake. \"Hey, now, don't look like that. They'll be okay, little girl. Just see that you don't get sick, too. Understand?\"\n\nShe nodded, and Jeth released her as suddenly as he had touched her. Pulling on gloves, positioning the black Stetson, he said abruptly, \"I'll be out on the north range if anybody needs me. Call that ambulance, will you, and don't listen to any protests from those two.\"\n\nThe epidemic lasted three weeks. Cara's days were filled with washing and changing sheets, trooping up and down stairs with meals, fruit juices, and aspirin, interspersed with countless trips across the ranch yard to help Toby feed the tired cowhands three meals a day.\n\nIn the evening, though, fresh and relaxed from a fragrant bath and wearing one of the many soft wool dresses whose colors highlighted her beauty, Cara met Jeth in his study. They sat across from each other before a blazing fire and sipped their drinks while the January winds howled and fretted outside, seeking entrance into their haven of warmth and safety. Above them over the mantel, prominently displayed, was her Christmas present to Jeth, the glow of the pav\u00e9 setting bringing Ryan into the room. They had thanked each other simply for their gifts, and Cara never removed the gold chain that held the prairie falcon.\n\nThey talked of many subjects, and Cara discovered Jeth to be a wonderful conversationalist, his dry humor sometimes making her double over her glass of wine in a gale of laughter. She listened, fascinated, to his stories about the ranch and Texas and was surprised to learn that his knowledge of music equaled Ryan's.\n\nThey dined late, going into the kitchen to serve their plates from the reheated supper that Cara had brought from the Feedtrough earlier. Their meals were eaten in the small dining alcove, accompanied by a newly opened bottle of wine over which they lingered until it was gone. Then together they cleared away the table and washed the dishes.\n\nCara began to play the Steinway for Jeth when they were into the second frenzied week of fighting the epidemic and the freezing temperatures. She was standing before a window in the living room looking out at the storm-blackened night. Jeth had not yet come down, or so she thought, until she felt his hands on her shoulders. \"Relax,\" he said on a husky note as she stiffened. \"These delicate shoulders have been carrying too many burdens lately.\" Slowly he began to massage her tense neck muscles until she was drowning in a delicious, euphoric bliss. Bending his dark head close to her ear, he whispered, \"After dinner, play for us, Cara. We both need it.\"\n\nAnd so, with Jeth comfortable with cigar and snifter of brandy in the chair where he had heard her play \"Full Moon and Empty Arms,\" Cara played for them the music that had once been her life. Now their evenings ended with the chords of the Steinway trembling throughout the house. Afterward, Jeth would lead her to the foot of the stairs to say good night before returning to his study. It was always a slightly tense moment that neither prolonged. They seemed to share a tacit understanding about the folly of climbing the stairs together.\n\nJeth drove her several times to the hospital to visit Fiona and Leon, whose return to health could be measured by the degree of their indignation at having to remain until fully recovered. Driving back with Jeth on the last visit before their release, Cara was tensely aware of every movement from the man at her side. She was certain that, given the right provocation, she might burst apart inside. February was a week gone. She wanted desperately to clutch the days and hold them fast, for each one gone brought her closer to the last. The ranch was beginning to return to normal. The men had been released from the converted sick ward, and the household staff had returned. Already Jeth was busy making arrangements for the spring roundup.\n\n\"Why so glum?\" Jeth asked. \"Fiona and Leon are going to be home in a few days.\" Home. He made it sound like a place they shared, would always share. A sob rose in her throat, and she blinked back the unshed tears and looked out across the moonlit landscape that stretched to infinity. She sensed Jeth turning to her in puzzled inquiry. \"Cara?\" One of the strong hands left the steering wheel and covered hers in an inquiring squeeze. \"What's wrong? And don't say _nothing_ ,\" he told her firmly, before she did.\n\n\"Just a little tired, I guess,\" she answered.\n\n\"And...homesick?\"\n\nIn the darkness, she smiled ironically to herself. \"Yes, you could say that, too.\"\n\nThere was a small disappointed silence from Jeth. Cara could feel it. Then he said, in a voice that was suddenly hard, \"Forget about going back to Boston before the estate is settled, Cara. I want you here where I can keep an eye on you. I couldn't let you go anywhere where you might be prevailed upon to change your mind about selling to me.\"\n\nCara straightened in her seat and withdrew her hand. \"You mean you still think I would?\" She could hardly keep the appalled tremor out of her voice.\n\n\"Miss Martin, you are an enigma to me. You are the only woman I have ever met whose motives I cannot figure out. How do I know these last weeks have not been a ruse to get me to trust you, to let you leave without signing the papers so that when you are out of my...jurisdiction, you could run to the highest bidder? You've had plenty, you know, buyers who have tried to see you, who have called and written. They have all been intercepted and given...discouraging replies. Harold St. Clair's letters have been passed on to you intact, but none of the others. No one gets my land but me, Miss Martin. I am the only one bidding.\"\n\nSlowly, a coldness crawling like a snake through her body, Cara asked numbly, \"You...opened my mail...intercepted my calls? You had no right to do that, Mr. Langston.\"\n\n\"I had every right. You should know that I take any rights I choose on La Tierra.\"\n\n\"I could have _escaped_ the day I went to Alpine!\"\n\n\"Oh, Miss Martin, in _my_ car? I would have reported it stolen, and you wouldn't have gotten out of the county.\"\n\n\"You\u2014you led me to believe you let me use your car because it was my birthday...\"\n\n\"And so I did. I was tired of seeing you in those butchered rags. However, that's not to say that I trusted you to come back. You were monitored the entire day.\"\n\n\"Are you saying that I\u2014that I'm a _prisoner_ at La Tierra?\"\n\n\"Yes, Miss Martin. You have been since the day you set foot on it.\"\n\nCara contemplated opening the car door and running across the plains\u2014anywhere to escape that smooth voice that masked such an iron ruthlessness. \"I hate you,\" she spoke in a childish whisper through the choking disappointment that welled in her throat. \"I hate you so much.\"\n\n\"I have always suspected it, Miss Martin, even in your friendliest and nicest moments.\"\n\nCara turned violently away from him, burrowing her head in the upturned fur collar of her coat. A wave of desolation washed over her. She had thought they could at least part friends, that her last month on the ranch would be as wonderful as the past three weeks had been, providing her with memories that she could hold in her heart and return to, time and time again, like seashells that recapture the sound of the sea. Now she knew that Jeth Langston had never begun to care for her, not even a little. His suspicions had not been allayed. They had simply been set aside while he made use of her as a nurse and cook and housemaid. He spoke of ruses. What kind of ruse had _he_ used these past weeks in order to enlist her aid in helping him run the ranch, paying her off with the pleasure of his company, and worst of all, allowing her to enjoy a side of him she never dreamed existed. Weren't they ruses? Weren't they the cruelest ruses of all?\n\nHer head came out of its collar burrow. Cara regarded the lean, wolflike profile disdainfully. Because she had been cruelly dealt with, she would be cruel. \"You have names for all of us, Mr. Langston,\" she began, feeling her lip curl. \"All of us who are self-serving. What do you call a man who tries to seduce one woman while being engaged to another?\"\n\nCara had the satisfaction of seeing the muscle tighten along his jawline, but she was little comforted. \"I would call that man a fool, Miss Martin, who, fortunately, saw his mistake before it was too late.\"\n\nShe had no answer for that. Indeed, she had no answers for anything, and probably never would again.\n\nThey met on the stairs early the next morning as Jeth was returning to his room before leaving the house. \"Oh, Miss Martin,\" he hailed her in a voice that had returned to its former dispassion, \"a temporary housekeeper will be here this morning. No need for you to concern yourself with Fiona's activities. And I have a new man out at the Feedtrough to take over for Leon. You're relieved of those duties, too.\"\n\nCara gave him a long, level look. \"You are saying, then, that my services are not wanted.\"\n\n\"Not _needed_ , Miss Martin. Enjoy your leisure. That's what you came here for.\"\n\nFrom the cool measure they took of each other, Cara was thinking that those intimate, friendly evenings before the fire might never have been. The memory of them would be an exquisite torture, far more effective than his unabashed hostility would have been.\n\n\"And one other thing, Miss Martin,\" Jeth concluded formally. \"I am afraid I will have to forego the pleasure of your company in the coming evenings. With the roundup coming, I'll be working late.\" He nodded dismissively and went on up the stairs. Cara's eyes followed him. She wanted to shout, \"Who says the pleasure of my company would have been offered?\"\n\nFor Cara, who had so much time on her hands, February passed surprisingly quickly. March brought a succession of unprecedented mild days, deceptive in their gentleness, with clouds scampering across the blue skies like fluffy lambs at play. Cara covered her garden with hay to prevent a sudden thaw, which would render her lilies and irises vulnerable to the inevitable spring freezes still to come. \"Better to stay in deep freeze,\" she muttered grimly to the frozen ground, remembering how she had suffered under the thawing warmth of Jeth's attention.\n\nThe daffodils were up, and she collected golden masses of them for the house and cemetery. One afternoon as she laid an arrangement of them on Ryan's grave, she fell to her knees and began to cry uncontrollably, her shoulders heaving from the fury of her grief. \"Why, Ryan, why? Why did I have to come here? Was it to fall in love with him? You knew he would despise me. Why did you make me come, Ryan?\"\n\nA sudden noise made her lift her head and listen, and she heard the clop-clop of a horse's hooves cantering away. Horror-stricken, she stood up to observe Jeth Langston astride his bay, heading toward the ranch. The implacable back, the indomitable shoulders, the hard set of the black Stetson told her nothing. Had her words carried on the clear, dry air? Had he overheard her crying, or had he, seeing Lady tethered below, decided to bypass the cemetery for another day? Her shoulders slumped in despair. Would March twentieth never come?\n\nShe began to plan her escape. She would have to elude Fiona, whom she was sure now, in spite of the woman's growing affection for her, had been Jeth's watchdog. Cara would have to find a way to get out of the house and off the ranch without arousing suspicion. An idea, simple and foolproof, presented itself to her. She wrote for air and bus schedules, which, when they arrived, she was able to intercept without either Fiona's or Jeth's knowledge.\n\nLike one who knows her days are numbered, she observed the world of La Tierra with a sharper awareness, committing to memory all of the sights and sounds that she would never know again. Leon caught her staring at him on a morning when spring was still pretending to have arrived. \"What ya lookin' at me like that for?\" he snorted affectionately. \"Ya got a funny look on yore face, like yore sad about somethin'. Pretty girl like you shouldn't ever be sad, leastways not 'cause of an ol' geezer like me. Stop worryin' that I ain't well, 'cause I am!\"\n\nShe haunted Fiona in the kitchen until the housekeeper accused her of being underfoot. \"Go for a walk!\" she finally advised in despair. \"You're as jumpy as a rabbit in a gunnysack!\"\n\nWith one day remaining, Cara could not resist playing the Steinway once more. She chose \"MacArthur Park.\" Her heart sang the lyrics while her fingers played the melody. _After all the loves of my life...after all the loves of my life...you'll still be the one..._\n\n\"Very well done,\" Jeth said gravely from the doorway of his study when she had finished. \"You played that with great feeling.\" Cara froze on the bench. She hadn't known he was in his study at this hour of the day. She'd thought he would be at the roundup of the remuda. It was time for the spring cattle drive again.\n\nShe turned to him with a face as smooth and cold as marble. \"Thank you,\" she said tonelessly, and walked from the room.\n\nOnce outside, she ran to the stable to saddle Lady. Her heart was beating frenziedly. Don't love anything out there, Harold St. Clair had warned her\u2014no man, woman, or child; no horse or dog\u2014not even an armadillo. How she wished she had taken his advice! Cara kept the gun Jeth had taught her how to use in the tack room, and with hurried, frantic motions, she sheathed it in the saddle scabbard. Once mounted, she urged Lady into a fast sprint even before the gentle mare was out of the corral. Cara thought she heard someone call to her, but it was only the wind that blew in her ears, she decided, only the homeless wind that knew her and called her by name.\n\nSometime later, Cara reined Lady to a halt, her attention caught by a flock of buzzards, the prairie's precursors of death, circling lazily in the distance. Some hapless animal is down, thought Cara, and kneed Lady into a lope. She had never had to put an animal out of its misery on the range, but she knew that it was a law among cattlemen to do so rather than let the sharp-beaked buzzards tear at the soft undersides and eyes of their still-living prey.\n\nShe found the object of the buzzards' interest a few minutes later. It wore the brand of La Tierra on its flank and lay in golden ruin at the bottom of a ravine, a blond-maned palomino whose attempted leap across the wide chasm had resulted in a fall that had snapped both forelegs. Cara saw at once there was nothing to be done. The legs lay at a crazy angle, and the weak rise and fall of the exposed side suggested that the end was very near.\n\nShe dismounted and removed the gun from its scabbard. As she descended the rocky incline, a shocking thought struck like a serpent inside her brain. She told herself not to be ridiculous, that there were dozens of palomino stallions roaming the range at La Tierra, and that the horse in the ravine was not the broken magnificence that had once been Texas Star.\n\nThe palomino's glazed eyes were open and watched her approach with a flicker of welcome in the brown depths. Cara's resolve faltered, and she knelt down and stroked the rough, dry coat. \"I'm sorry, boy,\" she spoke softly. \"I'm sorry it has to be like this.\" She had only to back away now, aim the rifle, pull the trigger, and be gone. It had to be done and delaying accomplished nothing for either of them. But the palomino gave a soft whinny and tried to nuzzle her hand, and a spasm of pity moved within her chest. \"Don't\u2014don't make it harder. It will all be over in a second. You won't feel a thing.\" And then, because she had to know, Cara brushed back the hair at the base of the mane. There, pigmented into the hide, was a perfectly formed, five-pronged white star.\n\nThe cry she hurled toward the heavens startled the predators flying overhead, but only momentarily. The rush of their wings came nearer Cara's head. She roused herself with an effort and hurled a rock at the assemblage. Then, stepping back from the horse, she released the lever, sliding the bullet into the rifle's chamber. The palomino's ears perked at the loud click, as if he remembered the sound from the days when he had carried his master across the plains. Cara raised the .30-30 to her shoulder, and quickly, before tears could distort her aim, centered the sights on the white forehead and fired.\n\nThe report carried across the prairie and sent the scavengers flapping skyward in raucous number. The recoil slammed into Cara's shoulder and stunned her cheek, but she was beyond the impact of pain. Only Bill's words from the roundup found their way into the void of her mind: \"I figure the boss thinks that as long as Texas roams La Tierra, a part of Ryan does, too.\"\n\nDeep in shock, Cara was barely aware of the gun slipping from her fingers, of the black-vested figure stepping in front of her frozen vision. From far away came the sound of her toneless voice. \"I shot Texas Star, Mr. Langston. I shot Ryan's horse.\"\n\n\"It's all right, _querida_ , Cara. You did what you had to do. You have always done what you had to do.\"\n\nLike a robot she let herself be led out of the ravine and made to sit on the ground beside the big bay waiting for his master. Later, she did not remember being lifted onto the roomy saddle of Dancer, nor recall Jeth mounting behind her, cradling her in the safety of his arms. She only remembered, coming from somewhere, an acrid stench of mesquite smoke. It brought a sudden, quite total darkness, and she fainted.\n\nWhen Cara awoke, she thought at first she had been asleep in the undulating cabin of a ship. She lay still and blinked. The soft, lapping waves receded, and her surroundings came into focus. She was in her room at La Tierra. Moonlight, cold and pale, filtered through the open blinds and across the blanket in which she was cocooned. A fire was crackling in the fireplace, throwing dancing shadows on the wall, mocking the fierce wind that whipped around the corners of the balcony.\n\nFor a merciful moment, Cara's mind was empty of all thought. Then, like the return of sensation after a stunning blow, memory of the afternoon flooded back to her.\n\n\"No!\" Her denial was strangled as she fought to sit up and free her arms and legs from the entangling blanket. She must meet the returning tide on her feet\u2014she could drown lying here like this.\n\nInstantly a tall figure rose catlike from one of the fireside chairs and was at the bed as she sat up. \"Easy, Cara, easy,\" Jeth Langston spoke soothingly. \"The first few minutes will be the hardest.\"\n\nWith a total and utter sense of loss, a privation so great that she thought she would rather die than be denied this man for whom her soul craved, the realization came to her that this was to be her last night at La Tierra. After tomorrow she would never see him again. After him there would be no more lovers, no more deaths\u2014only hers. _After all the loves of my life..._\n\nCara tried to say his name, but a yearning, so intense that she feared it would rupture her chest, made it impossible to speak. Instead, she lay back down, turned away from him, and began to sob.\n\nWhen the tears were spent, the bed beneath her face was soaked and she had another of Jeth Langston's large handkerchiefs clutched in her hand. \"That must have been coming for some time,\" he remarked, when Cara turned in surprise to find him still in the room.\n\nConscious of how puffy her eyes must look, how red her nose, she swung her legs to the floor. The bedside clock read nine o'clock. \"You've been very kind to bother with me, Mr. Langston. I'm all right. Really. This is roundup time. You've plenty to do without having to bother about me.\"\n\nJeth had pulled one of the wingback chairs up so that he could prop his booted feet on the bed. Without removing them, he said, \"So Ryan told you about Texas Star, did he?\" When Cara wearily nodded, he said simply, \"He was in love with you, Cara. Don't you know that by now?\"\n\nShe raised her head, comprehension breaking over her face like a quiet sunrise. She regarded the rancher sadly. Yes, Ryan had loved her, not as the friend she had thought, but as deeply, as passionately, as she loved his brother. How naive she had been not to have known. Then why had he sent her here to be abused by the brother he also had loved?\n\n\"Are you hungry?\" he asked unexpectedly. \"I had Fiona keep you something warm in the oven. I can bring it up to you.\"\n\n\"No, I couldn't eat anything,\" Cara answered. \"I'd rather just have a bath.\"\n\n\"Have one, then. I'll see you in about thirty minutes with something that will make you sleep. My mother used to make it for Ryan and me when we had a bad night coming up.\"\n\nCara watched him walk to the door. As he opened it, she said softly, \"You've had lots of those, haven't you, Mr. Langston? Bad nights, I mean.\"\n\nThe rancher paused, hand on the doorknob. His expression was oddly tender. \"No more than you, little girl. Thirty minutes.\"\n\nWhen Jeth returned, Cara was sitting listlessly before the fire in a white, long-sleeved granny gown that covered her from neck to ankles. A shawl of pink flannel was over her shoulders, and fluffy white house shoes peeped from beneath the hem of her gown. The light from the fire played on her blond hair and in the dusky violet depths of her eyes.\n\n\"You look like a little girl,\" Jeth commented, handing her a hot mug of dark liquid. \"All you need is a teddy bear to complete the picture of scrubbed innocence.\"\n\n\"But we both know how misleading that would be, don't we?\" Cara said cynically as she took the mug.\n\n\"Do we?\" he said, with the still expression she could never read. Then he changed the subject abruptly. \"There's a freeze expected by morning. Will your garden be all right?\"\n\n\"Yes. I knew a freeze would come, so I covered it with hay. Weather is like people, always vacillating between hot and cold. Tell Fiona\u2014\" She stopped and caught her lip between her teeth. She had almost said, _Tell Fiona to remove the hay when the weather warms so that the iris bulbs can feel the sun._\n\nJeth asked, \"Tell Fiona what, Cara?\"\n\n\"Nothing. I don't know what I was about to say. I'm feeling rather groggy. This cocoa has alcohol in it.\"\n\n\"Brandy. Drink it up. It's better than a sedative.\"\n\nJeth had brought for himself a glass of bourbon, and they drank in silence, Cara thinking of tomorrow and Jeth's reaction when his lawyers called to inform him that Ryan's share of La Tierra had been restored to him free and clear. She had never intended to sell the land back to Jeth. Harold's letter, which she had received yesterday, had assured her that the appropriate papers would be in the hands of Jeth's attorneys tomorrow, the first day of spring.\n\n\"This is very good,\" Cara said. \"It goes down like warm fingers soothing away the hurt. Your mother must have been a wonderful woman.\"\n\n\"I've only known one other like her.\"\n\n\"The woman you're going to marry, of course.\"\n\n\"Yes, the woman I'm going to marry. She is the most courageous person I think I've ever known, and I admire and love her with all of my heart.\"\n\n\"Lucky her,\" said Cara flatly, setting her finished drink down and getting up suddenly. \"I think I'll say good night now, Mr. Langston\u2014\" The room began to spin like a kaleidoscope, and she thrust out her hands to steady herself. Her last fully conscious thought was of Jeth rising to catch them. After that she descended into a blissful oblivion in which she was buoyed up by something strong and swift that bore her away to a place of softness and warmth. In the dreamy depths in which she floated, she could feel Jeth's mouth, as soft as a whisper, against her lips. Over and over his lips found hers, and once she thought she felt the wetness of tears upon her cheeks, but they could only have been hers.\nChapter Thirteen\n\nCara awoke the next morning instantly alert. Rigidly she forced from her mind the events of yesterday, which she recalled quite clearly, telling herself that she had to concentrate on the day at hand. She was fuzzy only about what had happened when she tried to stand up last night. Jeth had been talking about his fianc\u00e9e, she remembered. Obviously she had fainted again, and he'd been obliged to put her to bed. The man would probably heave an enormous sigh of relief when she was gone!\n\nFiona looked at her sharply when she entered the kitchen but made no comment about Texas Star. The fact that she prepared Cara's favorite omelet, heaping it high with fresh tomatoes and peppers and the special picante sauce she loved, spoke more than words of her concern about Cara's ordeal.\n\nWhen she had eaten, Cara said casually, \"The bookmobile will be here in about an hour. May I have a paper sack to carry my books in? After it leaves, I think I'll go for a ride on Lady. She got shortchanged yesterday.\"\n\nFiona nodded and went to the cupboard to get a brown paper sack. \"You should rest today, se\u00f1orita. You look too pale.\"\n\nCara thanked her and went upstairs to pack what little she planned to take with her to Boston. The paper sack, of course, was for her few essentials. She would leave all the clothes Ryan had bought her, including the sable-lined raincoat. They had been meant, she knew now, to impress his brother, and had failed miserably. She touched the twin charms at her throat. The gift from Jeth she would keep; she could not have parted with it. She opened the bureau drawer that contained the three clean handkerchiefs of Jeth's she'd never returned. The one from last night she had washed and dried in the bathroom. It was ready to be folded with the others. After she had packed the paper sack, placing on top of her things the library books she meant to return when she escaped in the bookmobile, she went along to Jeth's room with the handkerchiefs. Without glancing around his quarters, she placed them on his bureau and hurriedly left.\n\nCara left the sack of \"books\" on the hall table, then went out to the stable to see Lady. The mare, too, was part of the plan. She intended to turn Lady loose without being saddled. If Jeth came looking for Cara, finding Lady gone, he would assume she was out on a ride. He wouldn't become suspicious until nightfall. By that time she would already be in Boston, having taken a bus from Alpine to Midland Air Terminal where she would catch a plane.\n\nAs Cara made her way to the stable, she noticed a number of ranch hands ringing one of the large corrals used to pen the remuda. There was no one to see her hide the saddle or turn Lady loose in the pasture adjoining the stable. Her heart was heavy as she started back across the ranch yard to wait for the bookmobile, which would soon be arriving. Homer Pritchard saw her and called, \"Come see what the boss caught yesterday, Miss Martin! If he ain't a sight to behold!\"\n\nThe group of men parted respectfully as she approached, and Cara gasped at what she saw. Inside the corral, bucking and snorting in derision of his captors, was the last unconquered challenger on La Tierra Conquistada.\n\n\"Devil's Own!\" she cried with such familiarity that the men turned in surprise and looked at her. The horse heard her and stood still. He gazed in her direction, ears pointed alertly.\n\n\"You two know each other?\" Homer queried in surprise.\n\n\"Indeed we do.\" She turned angrily and addressed the men draped along the corral fence. \"Don't you men have something better to do than to stand around gaping at that animal? Get off the fence!\" she snapped. \"Stop staring at him!\"\n\n\"Cara, what are you doing?\"\n\nRecognizing the familiar voice, Cara wheeled to face the owner of La Tierra Conquistada, her eyes dilating in their fury.\n\n\"What do you intend to do with that horse?\" she demanded tightly, vaguely conscious that every eye was on them and that the great black stallion was standing motionless in the center of the corral.\n\n\"Why do you want to know? What business is it of yours, little girl?\"\n\n\"Don't call me _little girl_! I'm not a little girl! And I'm making that horse my business. What do you intend to do with him? Break him? For what purpose? You have hundreds of horses at your beck and call. Why do you need him?\"\n\nJeth studied her closely. \"Why is that horse so special to you, Cara? Why do you care so much what happens to him?\"\n\n\"Listen to me, Jeth. Once that horse feels your saddle on his back, your bit under his tongue, he'll never be the same again. You'll brand his flank, but a horse like that...you'll be branding his heart. You may turn him loose when the roundup is over, but he would never be free again. If you can't love him, don't tame him. Isn't that what you once said to Ryan? And you could never love that horse. He's been too much of a thorn in your side.\"\n\nJeth smiled slightly. \"You don't think I could love something that has been...a thorn in my side?\"\n\nCara shook her head.\n\n\"Well.\" Jeth's voice was tender. Unexpectedly, he reached out and brushed a silken strand of hair out of her eyes. \"We'll talk about it tonight. Go back to the house now.\"\n\n\"It will be too late then.\"\n\n\"No. I won't let it be too late.\" An employee had come up from the headquarters building. \"Boss, there's a call for you from Dallas. Sounds pretty urgent.\"\n\nJeth gave the girl a soft glance. \"We'll talk tonight.\"\n\nWith a hollow ache Cara watched him walk away from her and disappear inside the headquarters building. The men began to disperse, and Cara turned back to the corral. Calmly, without hurry, she lifted the wire hoop that secured the gate, swung it wide, and stood waiting for Devil's Own to register her invitation. In less than a minute the ears flattened, the tail arched, and the startled shouts of the men were too late to deter the thundering hooves. With tail high and mane flowing, Devil's Own streaked through the open gate, past her to freedom, deflecting with ease the hastily thrown ropes of the few men who had gotten them into the air.\n\nCara had a glimpse of Homer's ashen face before she tore off across the yard to the house. Fiona was in the kitchen, unaware of what had transpired. \"I hear the bookmobile coming, se\u00f1orita,\" she said.\n\nHer escape was easier than she had ever imagined. She had simply asked Honoria for a ride into Alpine, and the young woman, glad of Cara's company, had eagerly granted her request. By early evening, Cara was landing at Logan International Airport in Boston where Harold St. Clair met her.\n\n\"My God!\" was all he could say at first when she disembarked carrying her paper sack. She was almost unrecognizable in the blue jeans and flannel-lined jean jacket, her short hair the color of platinum.\n\n\"Hello, Harold.\" She smiled and convinced him that she was really Cara. \"Thank you for meeting me and for everything else you've done. My apartment really is available to stay in tonight?\"\n\n\"Yes. The tenants left last week. Uh, is that your only luggage?\"\n\n\"I'm afraid so. I left everything else at La Tierra.\"\n\n\"So it would seem. That's why we've all been awfully worried about you.\"\n\n\"Who is _we_?\"\n\n\"Why, myself of course, and Jeth Langston, not to mention the whole ranch staff. When it was discovered that you were missing, Jeth Langston checked your room. None of your clothes were gone, and he thought something had happened to you. Apparently he must have turned the ranch and the whole county upside down looking for you, then something made him think that you'd taken a powder.\"\n\nHe found the handkerchiefs, Cara thought.\n\n\"He checked the bus company and found out that you bought a ticket to the airport. In the meantime, his lawyers called me to find out if I knew your whereabouts. I told them I was expecting you late this afternoon.\"\n\n\"Had they spoken already to Mr. Langston? Did he know La Tierra is his again?\"\n\n\"He knew, all right. They got in touch with him this morning.\"\n\nCara remembered the phone call that had interrupted her final conversation with the rancher. She suddenly felt drained. She was sorry to have worried Fiona or Leon unduly. \"Could you get in touch with him tonight, Harold, and tell him I'm here? I made some friends at La Tierra I wouldn't want to be worried about me.\"\n\n\"I'd be happy to, except that Jeth Langston seems to have disappeared, too. His lawyers can't find him, and his housekeeper has no idea where he's gone.\"\n\n\"Disappeared?\" Cara was aghast. \"With the spring roundup at hand? He probably went to Dallas to be with his fianc\u00e9e.\" _To tell her the good news_ , Cara could have added. _Now they can begin planning their wedding._\n\nHarold's face appeared troubled as he bit his lip nervously. \"Cara...there's something I must tell you\u2014\"\n\nCara's heart felt an apprehensive chill. \"What is it?\"\n\n\"A few days ago, as instructed, I mailed to Jeth Langston a registered letter that Ryan wrote shortly after he altered his will. It was to arrive on the twentieth of March.\"\n\n\"Did it?\"\n\n\"I don't know. I mentioned this to the Langston attorneys, who said, according to the housekeeper, the mail had arrived while Jeth was in the house. Whether the letter was among the correspondence, she didn't know. She had no idea whether he had signed for it. But he left shortly thereafter. As for his being in Dallas, he's not. His attorneys have checked. I just finished speaking with them before I picked you up. They thought perhaps he might come here.\"\n\n\"To Boston? What on earth for? Jeth Langston didn't come to Boston when Ryan was alive. There would certainly be no reason for him to come here now. The letter probably upset him. He's gone off somewhere to be alone.\"\n\nAt her apartment, Harold insisted on taking her out to dinner, at least a hamburger, he suggested, marking the clothes she was wearing and doubting whether the paper bag contained anything suitable for something more lavish. He thought of the red dinner dress with regret. Cara was not to try and put him off. She should eat, and in a couple of hours he would be back to take her out for a bite. In the meantime, while she rested, he would go back to the office and put in a call to La Tierra. As he was leaving, Cara said warmly, \"Thank you for everything, Harold, especially for advancing the money I needed. You know I'll pay back every cent.\"\n\n\"Cara, you don't owe me or the firm one nickel. Ryan took care of any and all expenses that you could possibly have incurred this year, which weren't many.\" The lawyer gave her a diffident smile. \"If I may say so, Cara, I am so glad you're back. I've been counting the days until you were. See you in a little while.\"\n\nWhen he had gone, Cara took her curling iron from the paper sack. She really should try to do something to look less like a waif, she decided, looking at her straight hair and unenhanced face in the mirror. How far away and long ago was that curly-haired, golden-skinned young woman who had once been reflected in her mirror. She had been so striking in her blue party dress and high-heeled shoes. Quite possibly, Cara would never see the likes of her again.\n\nAs she bathed, she wondered about the letter Ryan had written to be delivered on March twentieth. Very probably it explained to Jeth why she had come to live for a year on La Tierra Conquistada. It would have been like Ryan to clear her name with his brother, to end that chapter without a question mark remaining. But as for her, she would never be able to understand why Ryan had sent her there in the first place. There was a remote possibility that somehow Ryan had thought Jeth would come to care for her\u2014though why, when he had known his brother had a perfectly suitable fianc\u00e9e waiting in the wings? Surely Ryan would have known that Jeth would have found someone like her, a believed whore...intolerable. Perhaps the letter explained his reason for sending her there. Cara would never know.\n\nShe was ready by the time Harold's knock came at the door. She had put on makeup and her hair had been washed and softly curled about her face. Too bad about the flannel shirt and jeans, she thought, not really caring. Tomorrow she would get out of storage the dull, meager collection of clothes she had left behind a year ago. It was a good thing that she'd not discarded the old, brown \"monk's cassock.\" Her jean jacket was not suitable for the rigors of a New England spring.\n\nCara did not even make the effort to smile as she went to the door. Her heart was too full of Jeth, of the yawning emptiness of a future without him. She threw the door open wide as a recompense.\n\n\"Hello, Cara,\" said Jeth Langston grimly from the doorway. Over his arm was draped the sable-lined raincoat.\n\nFor a paralyzed few seconds, Cara thought she was hallucinating. Jeth Langston could not possibly be standing at her door. But then the man in the fleece-lined jacket moved, stepped forward over the threshold, the gray eyes beneath the fawn Stetson never leaving her face, and the small, shabby room was all at once filled with the man's very real and dangerous presence.\n\nCara backed away, her mouth and eyes round Os of amazement. \"Mr. Langston, what are you doing here? Surely you didn't come to\u2014to get even with me for setting Devil's Own free?\" In her alarm, it was the only reason she could think of.\n\n\"He's part of the reason I'm here, yes,\" Jeth answered, kicking the door softly shut and advancing toward her.\n\n\"You\u2014you monster! Can't you bear to lose anything you set your sights on!\"\n\n\"No, Miss Martin, I can't bear to lose anything I set my sights on.\"\n\n\"Well, content yourself with the knowledge that there's always another roundup. There will always be another opportunity to get your rope around that poor animal's neck!\" She had retreated as far as she could go, furious with herself for letting him frighten her so. This was her apartment, her town, her land! How dare he come here and try to intimidate her!\n\n\"That's as may be, but there won't ever be another Cara Martin.\"\n\n\"What is that supposed to mean?\" And why had he brought that coat? she wondered irrelevantly, looking up at the unforgettable face. He stood very near to her, and she saw that he didn't look nearly as ferocious as she had first thought. Jeth tossed the coat on a nearby chair.\n\n\"You'll need that,\" he said, pushing back his Stetson. As she waited for him to explain, he very calmly slipped his arm around her waist and drew her to him to kiss her.\n\n\"Mr. Langston, why are you here?\" Cara sighed in resignation after the long, deep kiss. Naturally she had responded in spite of herself, as he had known she would, from the small, self-satisfied smile playing about his lips. She had been too surprised to struggle, and after a while she hadn't wanted to. \"If you have any ideas about making hay with me, you'd best forget it. Harold St. Clair will be here any minute.\"\n\n\"No, he won't. I spoke to him a short while ago. Fortunately for me, he was working late. I managed to convince him that we wouldn't need his company tonight. He's a nice fellow. I'm afraid I've misjudged him, as I misjudged you, Cara.\"\n\n\"So that's why you came all this way,\" Cara said, the light dawning. \"To tell me you had misjudged me.\" And maybe to get in a little dallying to boot, she thought scornfully, noting that the lean, wolflike face showed not the slightest trace of remorse. \"Harold told me he had sent you a letter written by Ryan shortly after his will was altered. I am assuming you got it. It must have explained everything.\"\n\nJeth's brows rose innocently. \"You mean this?\" He reached inside his jacket pocket for a long envelope. An arm still around her, he yielded enough space for her to examine the envelope. She recognized Ryan's handwriting, then with a little catch of breath saw that the envelope was still sealed.\n\n\"But you haven't read it!\" she exclaimed, looking up at him in puzzlement.\n\n\"I didn't have to. I know what it says. In a little while we'll read it together. I have an idea it's to both of us.\"\n\n\"Jeth, what are you saying?\" Cara asked in confusion.\n\nJeth led her to the couch in front of the fireplace and sat her down. \"We need a fire in here,\" he said, laying wood in the grate. When a crackling blaze was going, he took off his jacket and hat and joined her on the couch. Resting an arm on the back of it behind her, he searched Cara's face with a baffling scrutiny. \"You still haven't figured it all out, have you? You still don't know why Ryan sent you to live on La Tierra?\"\n\nCara stared at him. \"How do you know he sent me? You haven't read the letter.\"\n\n\"Didn't he, Cara?\"\n\nCara hesitated, then nodded slowly. The year was over now. Her promise had been kept. There was no reason now for Jeth not to know the truth.\n\nThe rancher reached for her hand. \"He asked you to go, didn't he? Probably as he was dying, he got you to promise that you would live there for a year after his death, from the first day of spring to the next. Don't you know why?\"\n\n\"Well...I have thought that maybe...he wanted us to\u2014to care for each other. That's insane, of course. He knew you could never care for a woman you thought to be a\u2014a whore, and especially since you were engaged\u2014\"\n\n\"Cara\u2014\" Jeth pressed a kiss on a soft wave at her temple. \"He knew us both so well. Even though he knew he was sending you into a year of hell, he knew us well enough to know that we'd come through intact.\"\n\n\"Speak for yourself, Jeth Langston,\" Cara said bitterly.\n\n\"You aren't intact?\" With a sharp glance, Cara saw Jeth's brows raise in irony, his lips twist in amusement. \"Why not?\"\n\nTo hell with my pride! Cara thought stormily. What comfort is it to me now, anyway? She looked Jeth full in the face, obstinately. \"Because I did come to care for you, Mr. Langston. Lord knows why. You are the most overbearing, arrogant, intimidating\u2014 _man_ I'll ever know, but as you say, Ryan knew us both better than we knew ourselves. The caring became\u2014well, here is a collector's item for _your_ vanity! The caring became love. I am now, _Mr. Langston_ , in love with you\u2014and that should be punishment enough for all the trouble I've caused. Don't think, however, that's the reason I restored the land to you. I planned to, anyway, from the very beginning. It wasn't mine. Ryan should never have left it to me. I can't imagine why he did.\"\n\n\"Because it was the only way he could set us up to fall in love. If he hadn't left you the land, how could he have gotten you on La Tierra? When would our paths have ever crossed?\"\n\nCara blinked at him, trying to keep a tight lid on the hope that was trying to bubble up inside her. \"But you're not saying that you\u2014love me?\" she whispered.\n\n\"Of course I am. I told you last night after I brought you your cocoa.\"\n\n\"But you were speaking of your fianc\u00e9e!\"\n\n\"I was speaking of you, _querida_ , though you obviously didn't realize it. Of the woman I am going to marry.\"\n\n\"What about Sonya Jeffers?\" she cried. _Querida!_ He called her _querida_ , sweetheart!\n\n\"I haven't been engaged to Sonya Jeffers since I left you in your sickbed to go to Dallas to break our engagement. At the time, I still hadn't figured out all the puzzle, but I wasn't going to marry any other woman feeling the way I did about you.\"\n\n\"Jeth...\" Cara's arms flew around his neck. Tears shone in her eyes. \"That's what you meant that night in the car when we were on our way back from the hospital. I asked you what you thought of a man who tried to seduce one woman when he was engaged to another, and you said...you said...\"\n\n\"I remember what I said,\" Jeth said with a wry smile. \"And naturally you took the opposite of my meaning.\"\n\n\"But you had just told me I'd been a prisoner at La Tierra!\"\n\n\"That was a desperate move to keep you there. You'd just said that you were homesick for Boston, and I was scared that you'd leave before I could undo all the harm I'd done. By then I was head over heels in love with you, and I had to keep you there to somehow salvage the damage. I should have known I couldn't scare you into staying.\"\n\n\"It wasn't Boston I was homesick for. I was sick about having to leave La Tierra, which by then was home to me. But you became so cold to me after that, Jeth. I thought...I thought...you'd just been kind to me during a time when there was no one else to help you. When I was no longer of use to you, I thought you had rejected me.\"\n\n\"Oh, honey\u2014\" Jeth pulled her tightly to him as if he were afraid she might disappear into thin air. \"That was sheer self-preservation. You told me that night you hated me. I believed you. I couldn't imagine how you could even like me after all I'd done to you, much less love me. Then I began seeing some signs, and remembering others, that told me otherwise.\"\n\nCara said in a muffled voice against his chest, \"Probably one of them was that time you heard me crying my heart out in the cemetery. I knew you had heard me.\"\n\n\"There was a message in your torment, Cara. You left me messages all year, had I chosen to read them\u2014like that discarded calender of yours that I found a few months after you'd been at La Tierra. You had marked the days very black at first, with heavy crisscrosses that reflected your anger and desire for the year to be over. Then gradually the marks had become fainter, almost reluctant. After a while, you hadn't marked the days at all. Then you threw away the calendar.\n\n\"But the final proof I had of your feelings was the despair I heard in the way you played 'MacArthur Park.' I'd just brought Devil's Own down from the mountains. That's why I was in the study at that time of day. And then this morning when you set Devil's Own free after giving me that little speech about him, when you got so hopping mad at me for capturing him, I knew then, lovely lady, that I had you, that you were mine. You identified with that horse. You wanted him to be free, unbranded, as you never would be again. I intended telling you all of this tonight when I proposed\u2014\"\n\nWide-eyed, Cara looked up to interrupt him. \"Do you still plan to?\"\n\n\"Try to stop me. Try to say no.\"\n\n\"You kissed me while I was sleeping last night, didn't you?\" And cried, too, thought Cara. Imagine: Jeth Langston crying!\n\n\"Yes, just like I'm looking forward to doing when you are sleeping, and I want to wake you to\u2014\" His arms tightened. \"I have loved you, Cara Martin, since the day I first saw you in my attorney's office. You were exactly\u2014the very image of the woman I had always dreamed of\u2014the woman I wanted to marry. I thought I _was_ dreaming. After I'd held you, kissed you, touched you, you were like a fever in my blood. It nearly destroyed me to think that, as much as I loved Ryan, he had gotten to you first.\"\n\n\"But he didn't!\" Cara protested.\n\n\"Shh, I know that now, sweetheart, though it took me longer than it should have to realize it. Once I did, everything else fell into place. Even as late as the evening we were driving back from the hospital, I still couldn't believe you hadn't been Ryan's mistress. By then, I didn't care anymore. I just knew that I loved you and that after you there would never be anyone else for me.\"\n\n\"When,\" Cara asked, \"did you believe the truth about me?\"\n\n\"I'm ashamed to say, honey, that it was only yesterday when you were playing 'MacArthur Park.' I was sitting in the study thinking how everything I felt about you was in that song. Then it hit me like the kick of a mule that you were playing it to express how _you_ felt about _me_! In that moment, I think, I saw the whole picture clearly. All along there had been two conflicting points that I could never reconcile: Ryan, I knew, would never have loved a tramp\u2014not enough to leave her half of La Tierra. But Ryan, I also knew, could never have kept his hands off you. Being a man myself, I just couldn't let go of that idea. But then when I looked at that part of the equation from the viewpoint of a _brother_ , then I knew what Ryan had done...\"\n\nCara lifted her head to look at him inquiringly. \"What exactly _had_ he done?\" she asked softly.\n\n\"Shortly after he found out he was dying, he met you, Cara\u2014\"\n\n\"In the library,\" she offered, \"a little over two years ago.\"\n\n\"He saw in you the woman he knew I'd always wanted, needed, would especially need after he was gone. He decided you would be his gift to me, his final gift of love, to ease the pain of his death.\"\n\n\"Of course...\" Cara whispered sadly, thinking of all those times she had wondered at their relationship, all the conversations in which Ryan had tried to defend and explain his older brother to her.\n\n\"So he spent his remaining time setting the scene, so to speak. He bought you beautiful clothes to make you even more appealing to me. He arranged a will that would force us to be together and keep us together in spite of the obstacles. And then he extracted that promise from you that would finish setting us up\u2014\"\n\n\"He told me I'd need courage,\" Cara murmured.\n\n\"Which you had, honey, plenty of it. God, when I think of what I did to you\u2014how I made you suffer! But if you hadn't, Cara, I would have been without some other pieces to the puzzle. Why, I kept asking myself, would a girl like you take such abuse to simply stay at La Tierra for room and board? Why didn't you ever throw your weight around as half owner? Why didn't you ever ask to see the land, the oil, the water rights you'd inherited? Why, I asked, if you hadn't been Ryan's mistress, did he leave you half of La Tierra?\n\n\"By the time the roundup was over, I had just about decided that Ryan and you had made some sort of contract. I was willing to believe that you were telling me the truth about your never having been lovers. You were so damned innocent looking, and I was already so gone over you, and I wanted so desperately to believe you. Then you threw me that curve that day in the garden\u2014\"\n\n\"I had to, Jeth,\" Cara interrupted. \"I had to make you not want me; otherwise, you'd have made love to me. According to the newspaper article, you were just waiting for the estate to be settled to be married. I thought you only wanted to make love to me so that you could get me to sign the papers so you could marry Sonya. I knew that I'd sign those papers. And then I couldn't have stayed at La Tierra. I couldn't have fulfilled my promise to Ryan.\"\n\nJeth let out a short, incredulous whistle through his teeth. \"We called an awful lot of the shots wrong, didn't we? Still, I should have figured out the whole puzzle at Christmas when you told me that during the summer you'd read of my engagement to Sonya. I spent most of the holidays searching through the summer editions of the Dallas _Morning News_ trying to find that damned article. When I did, I pinpointed its release to the day that you put on that very convincing performance for me in your garden. I hoped, of course, that it had been more than hurt pride that had made you behave that way\u2014confess to being Ryan's mistress. But by that time, I'd abused you so...I didn't think I had a snowball's chance in hell of ever winning you again.\"\n\nCara reached up to stroke his hard cheek. \"Sonya was one of the casualties you were referring to when you told me that Jim had been a minor one, wasn't she?\"\n\nJeth nodded. \"I'm afraid so. She's a fine woman. I've known her family all of my life, and I've always known how Sonya felt about me, how she was hoping someday to become mistress of La Tierra. When Ryan died, I was suddenly so\u2014so alone. She was there. Someone I could trust, someone I could take for granted...If you hadn't come along, I would have married her. Ryan knew that, too, of course. He knew I would have turned to her for solace.\"\n\nCara buried her face in Jeth's shoulder, feeling a surge of sympathy for the girl who had come so close to marrying the man of her dreams only to lose him in the eleventh hour. \"You know now,\" she said gently after a while, \"why Ryan didn't tell you he was dying, why he didn't go home to be with you in his last days...\"\n\n\"Yes, honey, I know,\" Jeth said against her hair. \"I would have found out about the will and taken steps to contest it. And then I would never have met you, never known you, never loved you. Ryan was so convinced of your integrity that he knew I'd get the land back anyhow.\" His voice broke suddenly. \"But, Cara, he fell in love with you himself along the way. What that must have been like for him, knowing that he could never have you, never touch you\u2014\"\n\n\"Let's read his letter,\" Cara suggested, her eyes growing moist, and reached for the long envelope in Jeth's jacket pocket. She also slipped out another of the white monogrammed handkerchiefs.\n\n\"I come with a lifetime supply of those.\" Jeth grinned. \"Not, I hope, that you'll have many occasions to use them.\"\n\nRyan spoke to them, Cara decided as Jeth read, not from the grave, but from wherever it was that he watched over them, had watched over them from the beginning. As Jeth had suspected, the letter was addressed to both of them. Tears spilled down her cheeks when Jeth's voice broke, and she stole her hand into his to comfort him. In the ironical, lighthearted banter that she remembered, Ryan explained how he had set in motion the events, when he knew he was dying, that would lead the two of them to love. \"Sonya's a good girl,\" Ryan wrote, \"but marriage with her would never fulfill you, Jeth. The girl beside you is the only one for you.\" The letter closed with declarations of love for both of them and the dry wish that someday there might be a noisy assortment of little Langstons to make the rooms of the big house \"less like those of a mausoleum.\"\n\n\"Quite a man, wasn't he?\" Jeth said when he had finished the letter.\n\n\"Quite a man,\" Cara agreed. \"One to name our first son after.\"\n\n\"Cara\u2014\" Jeth's eyes were shining. \"I love you with all my heart. The losses are over now. The tears are done. Ryan can rest in peace.\" And he lowered his head to claim the lips of the woman he loved.\n\n# About the Author\n\nLeila Meacham is a writer and former teacher who lives in San Antonio, Texas. She is the bestselling author of the novels _Roses_ , _Tumbleweeds_ , _Somerset_ , __ and _Titans_. For more information, you can visit LeilaMeacham.com.\n\n# Also by Leila Meacham\n\n_Roses_\n\n_Tumbleweeds_\n\n_Somerset_\n\n_Titans_\n\n# [ACCLAIM FOR THE NOVELS OF  \nLEILA MEACHAM](TOC.xhtml)\n\n## TITANS\n\n\"The novel has it all: a wide cast of characters, pitch-perfect period detail, romance, plenty of drama, and skeletons in the closet (literally). Saga fans will be swooning.\"\n\n\u2014 _Booklist_ (starred review)\n\n\"It has everything any reader could want in a book...epic storytelling that plunges the reader headfirst into the plot...[Meacham] is a titan herself.\"\n\n\u2014 _Huffington Post_\n\n\"Emotionally resounding...Texas has never seemed grander...Meacham's easy-to-read prose helps to maintain a pace that you won't be able to quit, pushing through from chapter to chapter to find the next important nugget of this dramatic family tale. It is best savored over a great steak with a glass of wine and evenings to yourself.\"\n\n\u2014BookReporter\n\n## SOMERSET\n\n\"Bestselling author Meacham is back with a prequel to _Roses_ that stands on its own as a sweeping historical saga, spanning the nineteenth century...[Fans] and new readers alike will find themselves absorbed in the family saga that Meacham has proven\u2014once again\u2014talented in telling.\"\n\n\u2014 _Publishers Weekly_ (starred review)\n\n\"Entertaining...Meacham skillfully weaves colorful history into her lively tale... _Somerset_ has its charms.\"\n\n\u2014 _Dallas Morning News_\n\n\"Slavery, westward expansion, abolition, the Civil War, love, marriage, friendship, tragedy, and triumph\u2014all the ingredients (and much more) that made so many love _Roses_ so much\u2014are here in abundance.\"\n\n\u2014 _San Antonio Express-News_\n\n\"A story you do not want to miss...[Recommended] to readers of Kathryn Stockett's _The Help_ or Margaret Mitchell's _Gone with the Wind_. _Somerset_ has everything a compelling historical epic calls for: love and war, friendship and betrayal, opportunity and loss, and everything in between.\"\n\n\u2014 _BookPage_\n\n\"4\u00bd stars! This prequel to _Roses_ is as addictive as any soap opera...As sprawling and big as Texas itself, Meacham's epic saga is perfect for readers who long for the 'big books' of the past. There are enough adventure, tears, and laughter alongside colorful history to keep readers engrossed and satisfied.\"\n\n\u2014 _RT Book Reviews_\n\n## TUMBLEWEEDS\n\n\"[An] expansive generational saga...Fans of _Friday Night Lights_ will enjoy a return to the land where high school football boys are kings.\"\n\n\u2014 _Chicago Tribune_\n\n\"Meacham scores a touchdown...You will laugh, cry, and cheer to a plot so thick and a conclusion so surprising, it will leave you wishing for more. Yes, Meacham is really that good. And _Tumbleweeds_ is more than entertaining, it's addictive.\"\n\n\u2014Examiner.com\n\n\"If you're going to a beach this summer, or better yet, a windswept prairie, this is definitely a book you'll want to pack.\"\n\n\u2014 _Times Leader_ (Wilkes-Barre, PA)\n\n\"[A] sprawling novel as large as Texas itself.\"\n\n\u2014 _Library Journal_\n\n\"Once again, Meacham has proven to be a master storyteller...The pages fly by as the reader becomes engrossed in the tale.\"\n\n\u2014 _Lubbock Avalanche-Journal_ (TX)\n\n## ROSES\n\n\"Like _Gone with the Wind_ , as gloriously entertaining as it is vast... _Roses_ transports.\"\n\n\u2014 _People_\n\n\"Meacham's sweeping, century-encompassing, multigenerational epic is reminiscent of the film _Giant_ , and as large, romantic, and American a tale as Texas itself.\"\n\n\u2014 _Booklist_\n\n\"Enthralling.\"\n\n\u2014 _Better Homes and Gardens_\n\n\"The story of East Texas families in the kind of dynastic gymnastics we all know and love.\"\n\n\u2014Liz Smith\n\n\"Larger-than-life protagonists and a fast-paced, engaging plot...Meacham has succeeded in creating an indelible heroine.\"\n\n\u2014 _Dallas Morning News_\n\n\"[An] enthralling stunner, a good, old-fashioned read.\"\n\n\u2014 _Publishers Weekly_\n\n\"A thrilling journey...a treasure...a must-read. Warning: Once you begin reading, you won't be able to put the book down.\"\n\n\u2014Examiner.com\n\n\"[A] sprawling novel of passion and revenge. Highly recommended...It's been almost thirty years since the heyday of giant epics in the grand tradition of Edna Ferber and Barbara Taylor Bradford, but Meacham's debut might bring them back.\"\n\n\u2014 _Library Journal_ (starred review)\n\n\"A high-end _Thorn Birds_.\"\n\n\u2014TheDailyBeast.com\n\n\"I ate this multigenerational tale of two families warring it up across Texas history with the same alacrity with which I would gobble chocolate.\"\n\n\u2014Joshilyn Jackson, _New York Times_ bestselling author of  \n _gods in Alabama_ and _Backseat Saints_\n\n\"A Southern epic in the most cinematic sense\u2014plot-heavy and historical, filled with archaic Southern dialect and formality, with love, marriage, war, and death over three generations.\"\n\n\u2014Caroline Dworin, \"The Book Bench,\" NewYorker.com\n\n\"This sweeping epic of love, sacrifice, and struggle reads like _Gone with the Wind_ with all the passions and family politics of the South.\"\n\n\u2014 _Midwest Book Review_\n\n\"The kind of book you can lose yourself in, from beginning to end.\"\n\n\u2014 _Huffington Post_\n\n\"Fast-paced and full of passions...This panoramic drama proves evocative and lush. The plot is intricate and gives back as much as the reader can take...Stunning and original, _Roses_ is a must-read.\"\n\n\u2014TheReviewBroads.com\n\n\"May herald the overdue return of those delicious doorstop epics from such writers as Barbara Taylor Bradford and Colleen McCullough...a refreshingly nostalgic bouquet of family angst, undying love, and 'if only's.\"\n\n\u2014 _Publishers Weekly_\n\n\"Superbly written...a rating of ten out of ten. I simply loved this book.\"\n\n\u2014 _A Novel Menagerie_\nIn the sweeping tradition of the _New York Times_ bestselling _Roses_ , Leila Meacham delivers another grand yet intimate novel set against the rich backdrop of early twentieth-century Texas. In the midst of this transformative time in Southern history, two unforgettable characters emerge and find their fates irrevocably intertwined as they love, lose, and betray.\n\n## Turn the page for a preview of\n\n# TITANS\n\n## On sale now.\nPrologue\n\nFrom a chair beside her bed, Leon Holloway leaned in close to his wife's wan face. She lay exhausted under clean sheets, eyes tightly closed, her hair brushed and face washed after nine harrowing hours of giving birth.\n\n\"Millicent, do you want to see the twins now? They need to be nursed,\" Leon said softly, stroking his wife's forehead.\n\n\"Only one,\" she said without opening her eyes. \"Bring me only one. I couldn't abide two. You choose. Let the midwife take the other and give it to that do-gooder doctor of hers. He'll find it a good home.\"\n\n\"Millicent\u2014\" Leon drew back sharply. \"You can't mean that.\"\n\n\"I do, Leon. I can bear the curse of one, but not two. Do what I say, or so help me, I'll drown them both.\"\n\n\"Millicent, honey... it's too early. You'll change your mind.\"\n\n\"Do what I say, Leon. I mean it.\"\n\nLeon rose heavily. His wife's eyes were still closed, her lips tightly sealed. She had the bitterness in her to do as she threatened, he knew. He left the bedroom to go downstairs to the kitchen where the midwife had cleaned and wrapped the crying twins.\n\n\"They need to be fed,\" she said, her tone accusatory. \"The idea of a new mother wanting to get herself cleaned up before tending to the stomachs of her babies! I never heard of such a thing. I've a mind to put 'em to my own nipples, Mr. Holloway, if you'd take no offense at it. Lord knows I've got plenty of milk to spare.\"\n\n\"No offense taken, Mrs. Mahoney,\" Leon said, \"and... I'd be obliged if you _would_ wet-nurse one of them. My wife says she can feed only one mouth.\"\n\nMrs. Mahoney's face tightened with contempt. She was of Irish descent and her full, lactating breasts spoke of the recent delivery of her third child. She did not like the haughty, reddish-gold-haired woman upstairs who put such stock in her beauty. She would have liked to express to the missy's husband what she thought of his wife's cold, heartless attitude toward the birth of her newborns, unexpected though the second one was, but the concern of the moment was the feeding of the child. She began to unbutton the bodice of her dress. \"I will, Mr. Holloway. Which one?\"\n\nLeon squeezed shut his eyes and turned his back to her. He could not bear to look upon the tragedy of choosing which twin to feast at the breast of its mother while allocating the other to the milk of a stranger. \"Rearrange their order or leave them the same,\" he ordered the midwife. \"I'll point to the one you're to take.\"\n\nHe heard the midwife follow his instructions, then pointed a finger over his shoulder. When he turned around again, he saw that the one taken was the last born, the one for whom he'd hurriedly found a holey sheet to serve as a bed and covering. Quickly, Leon scooped up the infant left. His sister was already suckling hungrily at her first meal. \"I'll be back, Mrs. Mahoney. Please don't leave. You and I must talk.\"\nChapter One\n\nBarrows homestead near Gainesville, Texas, 1900\n\nOn the day Nathan Holloway's life changed forever, his morning began like any other. Zak, the German shepherd he'd rescued and raised from a pup, licked a warm tongue over his face. Nathan wiped at the wet wake-up call and pushed him away. \"Aw, Zak,\" he said, but in a whisper so as not to awaken his younger brother, sleeping in his own bed across the room. Sunrise was still an hour away, and the room was dark and cold. Nathan shivered in his night shift. He had left his underwear, shirt, and trousers on a nearby chair for quiet and easy slipping into as he did every night before climbing into bed. Randolph still had another hour's sleep coming to him, and there would be hell to pay if Nathan disturbed his brother.\n\nSocks and boots in hand and with the dog following, Nathan let himself out into the hall and sat on a bench to pull them on. The smell of bacon and onions frying drifted up from the kitchen. Nothing better for breakfast than bacon and onions on a cold morning with a day of work ahead, Nathan thought. Zak, attentive to his master's every move and thought, wagged his tail in agreement. Nathan chuckled softly and gave the animal's neck a quick, rough rub. There would be potatoes and hot biscuits with butter and jam, too.\n\nHis mother was at the stove, turning bacon. She was already dressed, hair in its neat bun, a fresh apron around her trim form. \"G'morning, Mother,\" Nathan said sleepily, passing by her to hurry outdoors to the privy. Except for his sister, the princess, even in winter, the menfolks were discouraged from using the chamber pot in the morning. They had to head to the outhouse. Afterward, Nathan would wash in the mudroom off the kitchen where it was warm and the water was still hot in the pitcher.\n\n\"Did you wake your brother?\" his mother said without turning around.\n\n\"No, ma'am. He's still sleeping.\"\n\n\"He's got that big test today. You better not have awakened him.\"\n\n\"No, ma'am. Dad about?\"\n\n\"He's seeing to more firewood.\"\n\nAs Nathan quickly buttoned into his jacket, his father came into the back door with an armload of the sawtooth oak they'd cut and stacked high in the fall. \"Mornin', son. Sleep all right?\"\n\n\"Yessir.\"\n\n\"Good boy. Full day ahead.\"\n\n\"Yessir.\"\n\nIt was their usual exchange. All days were full since Nathan had completed his schooling two years ago. A Saturday of chores awaited him every weekday, not that he minded. He liked farmwork, being outside, alone most days, just him and the sky and the land and the animals. Nathan took the lit lantern his father handed him and picked up a much-washed flour sack containing a milk bucket and towel. Zak followed him to the outhouse and did his business in the dark perimeter of the woods while Nathan did his, then Nathan and the dog went to the barn to attend to his before-breakfast chores, the light from the lantern leading the way.\n\nDaisy, the cow, mooed an agitated greeting from her stall. \"Hey, old girl,\" Nathan said. \"We'll have you taken care of in a minute.\" Before grabbing a stool and opening the stall gate, Nathan shone the light around the barn to make sure no unwanted visitor had taken shelter during the cold March night. It was not unheard of to find a vagrant in the hayloft or, in warmer weather, to discover a snake curled in a corner. Once a hostile, wounded fox had taken refuge in the toolshed.\n\nSatisfied that none had invaded, Nathan hung up the lantern and opened the stall gate. Daisy ambled out and went directly to her feed trough, where she would eat her breakfast while Nathan milked. He first brushed the cow's sides of hair and dirt that might fall into the milk, then removed the bucket from the sack and began to clean her teats with the towel. Finally he stuck the bucket under the cow's bulging udder, Zak sitting expectantly beside him, alert for the first squirt of warm milk to relieve the cow's discomfort.\n\nDaisy allowed only Nathan to milk her. She refused to cooperate with any other member of the family. Nathan would press his hand to her right flank, and the cow would obligingly move her leg back for him to set to his task. With his father and siblings, she'd keep her feet planted, and one of them would have to force her leg back while she bawled and trembled and waggled her head, no matter that her udder was being emptied. \"You alone got the touch,\" his father would say to him.\n\nThat was all right by Nathan and with his brother and sister as well, two and three years behind him, respectively. They got to sleep later and did not have to hike to the barn in inclement weather before the sun was up, but Nathan liked this time alone. The scents of hay and the warmth of the animals, especially in winter, set him at ease for the day.\n\nThe milk collected, Nathan put the lid on the bucket and set it high out of Zak's reach while he fed and watered the horses and led the cow to the pasture gate to turn her out for grazing. The sun was rising, casting a golden glow over the brown acres of the Barrows homestead that would soon be awash with the first growth of spring wheat. It was still referred to as the Barrows farm, named for the line of men to whom it had been handed down since 1840. Liam Barrows, his mother's father, was the last heir to bear the name. Liam's two sons had died before they could inherit, and the land had gone to his daughter, Millicent Holloway. Nathan was aware that someday the place would belong to him. His younger brother, Randolph, was destined for bigger and better things, he being the smarter, and his sister, Lily, would marry, she being beautiful and already sought after by sons of the well-to-do in Gainesville and Montague and Denton, even from towns across the border in the Indian Territory. \"I won't be living out my life in a calico dress and kitchen apron\" was a statement the family often heard from his sister, the princess.\n\nThat was all fine by Nathan, too. He got along well with his siblings, but he was not one of them. His brother and sister were close, almost like twins. They had the same dreams\u2014to be rich and become somebody\u2014and were focused on the same goal: to get off the farm. At nearly twenty, Nathan had already decided that to be rich was to be happy where you were, doing the things you liked, and wanting for nothing more.\n\nSo it was that that morning, when he left the barn with milk bucket in hand, his thoughts were on nothing more than the hot onions and bacon and buttered biscuits that awaited him before he set out to repair the fence in the south pasture after breakfast. His family was already taking their seats at the table when he entered the kitchen. Like always, his siblings took chairs that flanked his mother's place at one end of the table while he seated himself next to his father's at the other. The family arrangement had been such as long as Nathan could remember: Randolph and Lily and his mother in one group, he and his father in another. Like a lot of things, it was something he'd been aware of but never noticed until the stranger appeared in the late afternoon.\nChapter Two\n\nThe sun was behind him and sinking fast when Nathan stowed hammer and saw and nails and started homeward, carrying his toolbox and lunch pail. The sandwiches his mother had prepared with the extra bacon and onions and packed in the pail with pickles, tomato, and boiled egg had long disappeared, and he was hungry for his supper. It would be waiting when he returned, but it would be a while before he sat down to the evening meal. He had Daisy to milk. His siblings would have fed the horses and pigs and chickens before sundown, so he'd have only the cow to tend before he washed up and joined the family at the table.\n\nIt was always something he looked forward to, going home at the end of the day. His mother was a fine cook and served rib-sticking fare, and he enjoyed the conversation round the table and the company of his family before going to bed. Soon, his siblings would be gone. Randolph, a high school senior, seventeen, had already been accepted at Columbia University in New York City to begin his studies, aiming for law school after college. His sister, sixteen, would no doubt be married within a year or two. How the evenings would trip along when they were gone, he didn't know. Nathan didn't contribute much to the gatherings. Like his father, his thoughts on things were seldom asked and almost never offered. He was merely a quiet listener, a fourth at cards and board games (his mother did not play), and a dependable source to bring in extra wood, stoke the fire, and replenish cups of cocoa. Still, he felt a part of the family scene if for the most part ignored, like the indispensable clock over the mantel in the kitchen.\n\nZak trotted alongside him unless distracted by a covey of doves to flush, a rabbit to chase. Nathan drew in a deep breath of the cold late-March air, never fresher than at dusk when the day had lost its sun and the wind had subsided, and expelled it with a sense of satisfaction. He'd had a productive day. His father would be pleased that he'd been able to repair the whole south fence and that the expense of extra lumber had been justified. Sometimes they disagreed on what needed to be done for the amount of the expenditure, but his father always listened to his son's judgment and often let him have his way. More times than not, Nathan had heard his father say to his mother, \"The boy's got a head for what's essential for the outlay, that's for sure.\" His mother rarely answered unless it was to give a little sniff or utter a _humph_ , but Nathan understood her reticence was to prevent him from getting a big head.\n\nAs if his head would ever swell over anything, he thought, especially when compared to his brother and sister. Nathan considered that everything about him\u2014when he considered himself at all\u2014was as ordinary as a loaf of bread. Except for his height and strong build and odd shade of blue-green eyes, nothing about him was of any remarkable notice. Sometimes, a little ruefully, he thought that when it came to him, he'd stood somewhere in the middle of the line when the good Lord passed out exceptional intellects, talent and abilities, personalities, and looks while Randolph and Lily had been at the head of it. He accepted his lot without rancor, for what good was a handsome face and winning personality for growing wheat and running a farm?\n\nNathan was a good thirty yards from the first outbuildings before he noticed a coach and team of two horses tied to the hitching post in front of the white wood-framed house of his home. He could not place the pair of handsome Thoroughbreds and expensive Concord. No one that he knew in Gainesville owned horses and carriage of such distinction. He guessed the owner was a rich new suitor of Lily's who'd ridden up from Denton or from Montague across the county line. She'd met several such swains a couple of months ago when the wealthiest woman in town, his mother's godmother, had hosted a little coming-out party for his sister. Nathan puzzled why he'd shown up to court her during the school week at this late hour of the day. His father wouldn't like that, not that he'd have much say in it. When it came to his sister, his mother had the last word, and she encouraged Lily's rich suitors.\n\nNathan had turned toward the barn when a head appeared above a window of the coach. It belonged to a middle-aged man who, upon seeing Nathan, quickly opened the door and hopped out. \"I say there, me young man!\" he called to Nathan. \"Are ye the lad we've come to see?\"\n\nAn Irishman, sure enough, and obviously the driver of the carriage, Nathan thought. He automatically glanced behind him as though half expecting the man to have addressed someone else. Turning back his gaze, he called, \"Me?\"\n\n\"Yes, you.\"\n\n\"I'm sure not.\"\n\n\"If ye are, ye'd best go inside. He doesn't like to be kept waiting.\"\n\n\"Who doesn't like to be kept waiting?\"\n\n\"Me employer, Mr. Trevor Waverling.\"\n\n\"Never heard of him.\" Nathan headed for the barn.\n\n\"Wait! Wait!\" the man cried, scrambling after him. \"Ye must go inside, lad. Mr. Waverling won't leave until ye do.\" The driver had caught up with Nathan. \"I'm cold and... me backside's shakin' hands with me belly. I ain't eaten since breakfast,\" he whined.\n\nDespite the man's desperation and his natty cutaway coat, striped trousers, and stiff top hat befitting the driver of such a distinctive conveyance, Nathan thought him comical. He was not of particularly short stature, but his legs were not long enough for the rest of him. His rotund stomach seemed to rest on their trunks, no space between, and his ears and Irish red hair stuck out widely beneath the hat like a platform for a stovepipe. He reminded Nathan of a circus clown he'd once seen.\n\n\"Well, that's too bad,\" Nathan said. \"I've got to milk the cow.\" He hurried on, curious of who Mr. Waverling was and the reason he wished to see him. If so, his father would have sent his farmhand to get him, and he must tend to Daisy.\n\nThe driver ran back to the house and Nathan hurried to the barn. Before he reached it, he heard Randolph giving Daisy a smack. \"Stay still, damn you!\"\n\n\"What are you doing?\" Nathan exclaimed from the open door, surprised to see Randolph and Lily attempting to milk Daisy.\n\n\"What does it look like?\" Randolph snapped.\n\n\"Get away from her,\" Nathan ordered. \"That's my job.\"\n\n\"Let him do it,\" Lily pleaded. \"I can't keep holding her leg back.\"\n\n\"We can't,\" Randolph said. \"Dad said to send him to the house the minute he showed up.\"\n\nHis siblings often discussed him in the third person in his presence. Playing cards and board games, they'd talk about him as if he weren't sitting across the table from them. \"Wonder what card he has,\" they'd say to each other. \"Do you suppose he'll get my king?\"\n\n\"Both of you get away from her,\" Nathan commanded. \"I'm not going anywhere until I milk Daisy. Easy, old girl,\" he said, running a hand over the cow's quivering flanks. \"Nathan is here.\"\n\nDaisy let out a long bawl, and his brother and sister backed away. When it came to farm matters, after their father, Nathan had the top say.\n\n\"Who is Mr. Waverling, and why does he want to see me?\" Nathan asked.\n\nBrother and sister looked at each other. \"We don't know,\" they both piped together, Lily adding, \"But he's rich.\"\n\n\"We were sent out of the house when the man showed up,\" Randolph said, \"but Mother and Dad and the man are having a shouting match over you.\"\n\n\"Me?\" Nathan pulled Daisy's teats, taken aback. Who would have a shouting match over him? \"That's all you know?\" he asked. Zak had come to take his position at his knee and was rewarded with a long arc of milk into his mouth.\n\n\"That's all we know, but we think... we think he's come to take you away, Nathan,\" Lily said. Small, dainty, she came behind her older brother and put her arms around him, leaning into his back protectively. \"I'm worried,\" she said in a small voice.\n\n\"Me, too,\" Randolph chimed in. \"Are you in trouble? You haven't done anything bad, have you, Nathan?\"\n\n\"Not that I know of,\" Nathan said. Take him away? What was this?\n\n\"What a silly thing to ask, Randolph,\" Lily scolded. \"Nathan never does anything bad.\"\n\n\"I know that, but I had to ask,\" her brother said. \"It's just that the man is important. Mother nearly collapsed when she saw him. Daddy took charge and sent us out of the house immediately. Do you have any idea who he is?\"\n\n\"None,\" Nathan said, puzzled. \"Why should I?\"\n\n\"I don't know. He seemed to know about you. And you look like him... a little.\"\n\nAnother presence had entered the barn. They all turned to see their father standing in the doorway. He cleared his throat. \"Nathan,\" he said, his voice heavy with sadness, \"when the milkin's done, you better come to the house. Randolph, you and Lily stay here.\"\n\n\"But I have homework,\" Randolph protested.\n\n\"It can wait,\" Leon said as he turned to go. \"Drink the milk for your supper.\"\n\nThe milking completed and Daisy back in her stall, Nathan left the barn, followed by the anxious gazes of his brother and sister. Dusk had completely fallen, cold and biting. His father had stopped halfway to the house to wait for him. Nathan noticed the circus clown had scrambled back into the carriage. \"What's going on, Dad?\" he said.\n\nHis father suddenly bent forward and pressed his hands to his face.\n\n\"Dad! What in blazes\u2014?\" Was his father crying? \"What's the matter? What's happened?\"\n\nA tall figure stepped out of the house onto the porch. He paused, then came down the steps toward them, the light from the house at his back. He was richly dressed in an overcoat of fine wool and carried himself with an air of authority. He was a handsome man in a lean, wolfish sort of way, in his forties, Nathan guessed. \"I am what's happened,\" he said.\n\nNathan looked him up and down. \"Who are you?\" he demanded, the question bored into the man's sea-green eyes, so like his own. He would not have dared, but he wanted to put his arm protectively around his father's bent shoulders.\n\n\"I am your father,\" the man said.\n\n### Thank you for buying this ebook, published by Hachette Digital.\n\nTo receive special offers, bonus content, and news about our latest ebooks and apps, sign up for our newsletters.\n\nSign Up\n\nOr visit us at hachettebookgroup.com\/newsletters\n\n## Contents\n\n  1. Cover\n  2. Title Page\n  3. Welcome\n  4. Dedication\n  5. A Letter to My Friends, Fans, Readers of My Later-in-Life Novels, and Newcomers to the Books of Leila Meacham\n  6. Chapter One\n  7. Chapter Two\n  8. Chapter Three\n  9. Chapter Four\n  10. Chapter Five\n  11. Chapter Six\n  12. Chapter Seven\n  13. Chapter Eight\n  14. Chapter Nine\n  15. Chapter Ten\n  16. Chapter Eleven\n  17. Chapter Twelve\n  18. Chapter Thirteen\n  19. About the Author\n  20. Also by Leila Meacham\n  21. Acclaim for the Novels of Leila Meacham\n  22. An Excerpt from Titans\n  23. The Books of Leila Meacham\n  24. Newsletters\n  25. Copyright\n\n##  Navigation\n\n  1.  Begin Reading \n  2.  Table of Contents\n\nThis book is a work of fiction. Names, characters, places, and incidents are the product of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is coincidental.\n\nCopyright \u00a9 1984 by Leila J. Meacham  \nAuthor letter copyright \u00a9 2016 by Leila Meacham  \nExcerpt from _Titans_ copyright \u00a9 2016 by Leila Meacham\n\nCover design by Laura Klynestra  \nCover photo by Laura Stolfi\/Stocksy  \nCover copyright \u00a9 2016 by Hachette Book Group, Inc.\n\nHachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.\n\nThe scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbg.com. Thank you for your support of the author's rights.\n\nGrand Central Publishing  \nHachette Book Group  \n1290 Avenue of the Americas  \nNew York, NY 10104  \ngrandcentralpublishing.com  \ntwitter.com\/grandcentralpub\n\nOriginally published in hardcover in 1984 by the Walker Publishing Company, Inc., New York, New York.\n\nFirst Grand Central Publishing Edition: July 2016\n\nGrand Central Publishing is a division of Hachette Book Group, Inc.  \nThe Grand Central Publishing name and logo are trademarks of Hachette Book Group, Inc.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to www.hachettespeakersbureau.com or call (866) 376-6591.\n\nPermission to quote lines from \"MacArthur Park\" by Jimmy Webb graciously granted by Warner Bros. Music. \u00a9 1968 Canopy Music, all rights reserved, used by permission of Warner Bros. Music.\n\nLibrary of Congress Control Number: 2016934480\n\nISBNs: 978-1-4555-4130-0 (trade pbk.), 978-1-4555-4133-1 (library edition hardcover), 978-1-4555-4131-7 (ebook)\n\nE3-20160531-NF-DA\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTo\n\nTHE CHAIRMAN AND GOVERNORS OF THE\n\nGLASGOW SCHOOL OF ART,\n\nIN WHOSE SCHOOL I HAVE BEEN LECTURER\n\nON ARTISTIC ANATOMY\n\nFOR A NUMBER OF YEARS;\n\nTO THE HEADMASTER MR. FRANCIS H. NEWBERY,\n\nAND TO THE STUDENTS,\n\nTHIS BOOK IS RESPECTFULLY DEDICATED\n\nBY THEIR OBEDIENT SERVANT,\n\nJAMES M. DUNLOP.\n\nGLASGOW. 1899\nBibliographical Note\n\nThis Dover edition, first published in 2007, is an unabridged republication of the revised second edition of Anatomical Diagrams for the Use of Art Students, by James M. Dunlop, originally published by George Bell and Sons, London, in 1904.\n\nThe plates, which appear in color in the original edition, have been reproduced in black and white for the Dover edition.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nDunlop, James M.\n\nAnatomical diagrams for art students \/ James M. Dunlop.\n\np. cm.\n\n\"This Dover edition, first published in 2007. is an unabridged republication of the revised second edition of Anatomical diagrams for the use of art students, by James M. Dunlop. originally published by George Bell and Sons. London, in 1904. The plates, which appear in color in the original edition, have been reproduced in black and white for the Dover edition.\"\n\n9780486149479\n\n1. Anatomy, Artistic. I. Dunlop, James M., Anatomical diagrams for the use of art students. II. Title.\n\nNC760.D85 2007   \n743.4'9\u2014dc22\n\n2006102439\n\nManufactured in the United States of America Dover Publications. Inc., 31 East 2nd Street, Mineola, N.Y 11501\n\n# PREFACE.\n\nSCIENCE AND ART are indeed sisters, but they are very different in their tastes, and it is no easy task to cultivate with advantage the favour of both. Artistic Anatomy is in its nature a scientific pursuit, dealing partly in explicit observation of details of form, partly in the explanation of the causes producing them; while the details themselves are among those with which the followers of Art require to be familiar; and are sometimes of little apparent scientific importance save from an artistic point of view. In these circumstances it is little to be wondered at that this department of knowledge has not been more fully explored.\n\nProperly conceived of, Artistic Anatomy undertakes the systematic study of the particulars of superficial form, the accurate description of them one by one, and the investigation of the structural and functional causes on which they depend.\n\nAmong the phenomena to be considered, the proportions of the great divisions of the body one to another claim an important place, and have justly received attention from remote times. Rules have been laid down by which an ideal standard has been sought to be fixed, the deviations produced by age and sex being taken into account; and while such standards are more or less artificial, and not to be too slavishly followed to the extent of an unnatural uniformity, they certainly are invaluable as expressing a mean which cannot be deviated from to more than a limited extent without transgressing the laws of nature and producing deformity.\n\nEach part of the body has also its particular proportions, and the study of proportions passes gradually into that of details of shape. All these details are capable of being taken one by one and systematically described. But this cannot be done either accurately or instructively without reference to the subjacent structures on which they depend, and the actions governing the conditions of such structures.\n\nSubcutaneous prominences of bone afford so many constant points in the surface of the figure, while the softer subcutaneous tissues sometimes occur in masses of such firmness as to be but little affected by change of attitude, and in other instances are flaccid, pendulous, wrinkled or stretched. But the muscles and their tendons produce the greatest variations of local form in different persons and in different attitudes; muscular substance swelling when in action, while tendons are incapable alike of swelling and of altering their total length, but may stand out when they are tightened over the concavity formed by the bending of a joint. Also lines of attachment to subcutaneous bone, themselves incapable of change of form, may in different circumstances be prominent or sunk according to the degree of swelling of the muscles around. Besides all this it must be noted that muscular contractions cause, especially in the face, lines, elevations, and depressions, not corresponding to the shapes of the muscles, but produced by the displacement of skin and subcutaneous fat, as illustrated by the elevation of the cheek and lower eyelid in laughter, and by the formation at the same time of the lines called crows' toes, and it does seem possible that a more careful analysis than has been attempted of the lines and displacements occurring in different expressions might yield better results than are to be obtained from such works as those of Le Brun, Sir Charles Bell, Piderit and Darwin, however valuable these may be. It may also be mentioned that considerations in connection with balance, respiration, mental capacity and race fall within the scope of Artistic Anatomy.\n\nIf these views are allowed to be correct, it will be admitted that the field of Artistic Anatomy has never been covered; and if this task be ever undertaken it must be for its own sake, aiming at independent completeness, and not at mere assistance to Artists. Much will thus be brought to light, in all probability now unsuspected, and Art and Philosophy will both be gainers.\n\nWhile, however, Art is one thing and Artistic Anatomy quite another, and while it is to be acknowledged that beautiful representations may be achieved without any anatomical knowledge, this only shows how much can be done by practised observation led on by intuitive appreciation which, often unconsciously, guides the mind to the accomplishment of its aims. But such success is neither easy nor to be depended on, and the general average thus obtainable cannot be expected to be so good as would be obtained if observation were assisted by acquaintance with the meaning of the shapes observed. The greatest masters, including notably Michelangelo, Leonardo da Vinci and Raphael, have found that to give intelligence to their efforts at representation, and enable them to understand the indispensable relations of parts it was necessary to call in the aid of dissection. For the eye, though often, even when well trained, at fault, especially when invention is brought into play, is yet subtle to detect instinctively the unsatisfactoriness of error.\n\nIt seems sometimes to be supposed that Artistic Anatomy is merely Anatomy made easy for Artists by omitting explicit details and all mention of internal organs,\u2014superficial Anatomy in both senses of the word. But what is superficial in the sense of being slovenly is of little use to any one. The Professional Anatomist addressing his discourse to Artists, and desiring to give them the information for which they crave, cannot help seeing at once that there is much internal structure which can have no possible bearing on Art, but he will fail altogether in his purpose if he does not note that the artist seeks for direction with regard to details which are often of small interest to the surgeon, and have received little attention from Anatomists.\n\nTwo of the sets of considerations most important to the Artist will easily be seen to be, one, the part played by the skeleton in determining the external form, and another, the precise extent and attachments of superficial muscles, together with the disposition of muscular fibre and tendon in individual muscles. It is principally to these two considerations that Mr. Dunlop directs attention in the following pages, appealing to the eye, instead of depending on description; and it appears to me that the method which he has selected, and the manner in which he has carried it out, provide for the Art Student a singularly compendious and desirable book, easily consulted, and occupying ground which has not hitherto been taken up. It is not the whole subject of Artistic Anatomy, but only one department of it which is here dealt with. The facts taught are brought out with diagrammatic simplicity and precision which cannot fail to bring them clearly and prominently before the student, thus giving him immense assistance. I have pleasure therefore in anticipating for this useful work a great success.\n\nJOHN CLELAND.\n\n# Table of Contents\n\nTitle Page  \nDedication  \nCopyright Page  \nPREFACE.  \nSIDE VIEW OF THE HEAD AND NECK.  \nFRONT VIEW OF THE HEAD AND NECK.  \nSIDE VIEW OF TRUNK.  \nFRONT VIEW OF THE TRUNK.  \nBACK VIEW OF THE TRUNK.  \nOUTER VIEW OF THE LOWER LIMB.  \nFRONT VIEW OF THE LOWER LIMB.  \nINNER VIEW OF THE LOWER LIMB.  \nBACK VIEW OF THE LOWER LIMB.  \nOUTER VIEW OF THE UPPER LIMB.  \nFRONT VIEW OF THE UPPER LIMB.  \nINNER VIEW OF THE UPPER LIMB.  \nBACK VIEW OF THE UPPER LIMB.  \nTHE BONES IN RELATION TO THE OUTLINE OF FIGURE FRONT VIEW  \nCONSTRUCTION LINES OF THE STANDING FIGURE. FRONT VIEW.  \nTHE BONES IN RELATION TO OUTLINE OF FIGURE. BACK VIEW.  \nCONSTRUCTION LINES OF THE STANDING FIGURE. BACK VIEW.  \nTHE BONES IN RELATION TO OUTLINE OF FIGURE. SIDE VIEW.  \nPARTS OF THE BONES WHICH DIRECTLY AFFECT THE SURFACE FORM\u2014SIDE VIEW.  \nTHE BONES AS IN ACTION\u2014FRONT VIEW.  \nTHE MUSCLES AS IN ACTION\u2014FRONT VIEW.  \nTHE BONES AS IN ACTION.\u2014BACK VIEW.  \nTHE MUSCLES AS IN ACTION.\u2014BACK VIEW.  \nTHE BONES AS IN ACTION.\u2014SIDE VIEW.  \nTHE MUSCLES AS IN ACTION.\u2014SIDE VIEW.  \nDETAILS OF THE FACE.  \nMUSCLES OF THE HEAD.  \nMUSCLES OF THE NECK..  \nMUSCLES OF THE TRUNK.  \nMUSCLES OF THE THIGH\u2014FRONT VIEW.  \nMUSCLES OF THE BUTTOCK AND HIP.  \nMUSCLES OF THE THIGH\u2014BACK VIEW.  \nOUTER VIEW OF THE FOOT.  \nMUSCLES OF THE ARM.\u2014FRONT VIEW.  \nMUSCLES OF THE ARM.\u2014BACK VIEW.  \nMUSCLES OF FRONT AND INNER SIDE OF FOREARM. THE PRONATOR AND FLEXOR GROUP.  \nMUSCLES OF BACK AND OUTER SIDE OF FOREARM THE SUPINATOR AND EXTENSOR GROUP.  \nBONES OF THE HAND.\u2014PALMAR VIEW.  \nMUSCLES OF THE HAND.\u2014PALMAR VIEW.  \nBONES OF THE HAND.\u2014BACK VIEW.  \nSURFACE FORMS OF THE LIMBS.  \nDIAGRAMMATIC NOTES OF VARIOUS BONES  \nSURFACE FORMS OF THE TRUNK.  \nTHREE-QUARTER FRONT VIEW OF SKELETON.  \nTHREE QUARTER BACK VIEW OF SKELETON.  \nDIAGRAMS WITH NOTES ON PROPORTION.\n\n# SIDE VIEW OF THE HEAD AND NECK.\n\n# BONES.\n\n# MUSCLES.\n\n# FRONT VIEW OF THE HEAD AND NECK.\n\n# BONES\n\n# MUSCLES.\n\n# SIDE VIEW OF TRUNK.\n\n# BONES.\n\n# MUSCLES.\n\n# FRONT VIEW OF THE TRUNK.\n\n# BONES.\n\n# MUSCLES.\n\n# BACK VIEW OF THE TRUNK.\n\n# BONES.\n\n# MUSCLES.\n\n# OUTER VIEW OF THE LOWER LIMB.\n\n# BONES.\n\n# MUSCLES.\n\n# FRONT VIEW OF THE LOWER LIMB.\n\n# BONES.\n\n# MUSCLES.\n\n# INNER VIEW OF THE LOWER LIMB.\n\n# BONES.\n\n# MUSCLES\n\n# BACK VIEW OF THE LOWER LIMB.\n\n# BONES.\n\n# MUSCLES.\n\n# OUTER VIEW OF THE UPPER LIMB.\n\n# BONES.\n\n# MUSCLES.\n\n# FRONT VIEW OF THE UPPER LIMB.\n\n# BONES.\n\n# MUSCLES.\n\n# INNER VIEW OF THE UPPER LIMB.\n\n# BONES.\n\n# MUSCLES.\n\n# BACK VIEW OF THE UPPER LIMB.\n\n# BONES.\n\n# MUSCLES.\n\n# THE BONES IN RELATION TO THE OUTLINE OF FIGURE FRONT VIEW\n\n# CONSTRUCTION LINES OF THE STANDING FIGURE. FRONT VIEW.\n\nFig. 1 Fig. 2\n\nIn the above diagrams, Fig. 1 shows the leading constructive lines when the figure is standing upright and resting on both legs. In this position the line passing through the shoulders, and that drawn across the antr. supr. spines of the Pelvis, are both obviously at right angles to the vertical axis of the body.\n\nIn Figure 2 the weight of the body is carried mainly on one leg, and in this position the axial line of the body becomes a curve, but still the line drawn through the shoulders and that through the Pelvis may be regarded as at right angles to this imaginary curve. Observe that the Pelvis is higher on the side which supports the figure, and also that the hip makes a sharper angle on that side, and further, note the slope of the standing leg with the ground, necessary for the balance of the figure.\n\nThe axial line of the body carried down through the standing leg gives here a line of double curvature, which is the first line to be drawn in suggesting the pose.\n\n# THE BONES IN RELATION TO OUTLINE OF FIGURE. BACK VIEW.\n\n# CONSTRUCTION LINES OF THE STANDING FIGURE. BACK VIEW.\n\nThe above diagrams show the leading constructive lines of the back view of the similar positions to those already shown in the front view. The greater length of the trunk as viewed from behind will be observed on comparison with the front view. The lines across the back are drawn through the shoulders, the lower end of the thorax, the iliac crests, and the gluteal fold.\n\n# THE BONES IN RELATION TO OUTLINE OF FIGURE. SIDE VIEW.\n\n# PARTS OF THE BONES WHICH DIRECTLY AFFECT THE SURFACE FORM\u2014SIDE VIEW.\n\n# THE BONES AS IN ACTION\u2014FRONT VIEW.\n\n# THE MUSCLES AS IN ACTION\u2014FRONT VIEW.\n\n# THE BONES AS IN ACTION.\u2014BACK VIEW.\n\n# THE MUSCLES AS IN ACTION.\u2014BACK VIEW.\n\n# THE BONES AS IN ACTION.\u2014SIDE VIEW.\n\n# THE MUSCLES AS IN ACTION.\u2014SIDE VIEW.\n\n# DETAILS OF THE FACE.\n\n# MUSCLES OF THE HEAD.\n\n# MUSCLES OF THE NECK..\n\n# MUSCLES OF THE TRUNK.\n\n# MUSCLES OF THE THIGH\u2014FRONT VIEW.\n\n# MUSCLES OF THE BUTTOCK AND HIP.\n\n# MUSCLES OF THE THIGH\u2014BACK VIEW.\n\n# OUTER VIEW OF THE FOOT.\n\n# INNER VIEW OF THE FOOT.\n\n# MUSCLES OF THE ARM.\u2014FRONT VIEW.\n\n# MUSCLES OF THE ARM.\u2014BACK VIEW.\n\n# MUSCLES OF FRONT AND INNER SIDE OF FOREARM. THE PRONATOR AND FLEXOR GROUP.\n\n# MUSCLES OF BACK AND OUTER SIDE OF FOREARM THE SUPINATOR AND EXTENSOR GROUP.\n\n# BONES OF THE HAND.\u2014PALMAR VIEW.\n\n# MUSCLES OF THE HAND.\u2014PALMAR VIEW.\n\n# BONES OF THE HAND.\u2014BACK VIEW.\n\n# SURFACE FORMS OF THE LIMBS.\n\n# BONES OF THE UPPER AND LOWER LIMBS.\n\n# DIAGRAMMATIC NOTES OF VARIOUS BONES\n\n# SURFACE FORMS OF THE TRUNK.\n\n# THREE-QUARTER FRONT VIEW OF SKELETON.\n\n# THREE QUARTER BACK VIEW OF SKELETON.\n\n# DIAGRAMS WITH NOTES ON PROPORTION.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by The Online Distributed Proofreading Team at\nhttp:\/\/www.pgdp.net (This file was produced from images\ngenerously made available by The Internet Archive)\n\n\n\n\n\n\n\n\n\n\n                            RED WAGON STORIES\n                                   OR\n                        TALES TOLD UNDER THE TENT\n\n                                   BY\n                               WELLS HAWKS\n\n                             [Illustration]\n\n                           I. & M. OTTENHEIMER\n                               PUBLISHERS\n                      NO. 321 WEST BALTIMORE STREET\n                             BALTIMORE, MD.\n\n                             Cover Design by\n                             J. R. CROSSLEY.\n\n                            Copyrighted 1904.\n                          I. & M. OTTENHEIMER.\n                             BALTIMORE, MD.\n\n\n\n\nBetween the shows there were seven of the circus outfit who would sit\naround the ring bank and on the carpet pads just to talk. Here are some\nof the tales told under the big round top when the tent was empty.\n\nAnd to those happy days of bread and preserves, when we bare-footed kids\nsneaked out of the backyard gate to the circus lot and led the spotted\nponies to water, these little yarns are affectionately dedicated:--\n\n\n\n\nCONTENTS.\n\n\n                                                             PAGE.\n\n    THE PRESS AGENT'S STORY                                     7\n\n    THE OLD GRAFTER'S LAMENT                                   14\n\n    THE BILL POSTER'S VISIT                                    21\n\n    THE CANDY BUTCHER'S DREAM OF LOVE                          30\n\n    THE BOSS CANVASMAN'S YARN                                  33\n\n    THE SIDE SHOW SPIELER SPEAKS                               48\n\n    THE BAND MASTER'S SOLO                                     54\n\n    THE CANDY BUTCHER TALKS ABOUT A LOVE AFFAIR AND HIS\n      ENCOUNTER WITH THE BUCKWHEAT MAN                         59\n\n    THE CONCERT MANAGER GETS REMINISCENT                       70\n\n    THE HANDS AT THE WINDOW                                    75\n\n    THE CONCERT MANAGER TELLS THE BOYS AN ELEPHANT STORY       83\n\n\n\n\nTHE PRESS AGENT'S STORY.\n\n\nThe Press Agent of the Big Show had formerly been dramatic editor of\nthe leading daily in Council Bluffs. It was his star boast that he was\nthe only critic in the Middle West that ever had the nerve to roast Joe\nJefferson, and he said he did it in the interest of art.\n\n\"Art,\" says he, \"must be preserved, an' the only way to do it is by\nknockin'.\"\n\nThe Press Agent wore his hair long, had a smooth face, and looked like\na police reporter out on a three-column story with the facts coming in\nslowly. He hadn't much baggage, but he always carried about a ream of\nadjective hit paper, two lead pencils, and a pass-pad. No man ever heard\nhim talk without wondering what kind of stuff he beat out on a typewriter.\n\nThe saw dust spreader was smoothing out the ring for the night acts\nand the rest of the gang were sitting around roasting the route when\nthe Press Agent came through the red curtains at the dressing tent\nentrance picking his teeth with a straw. He sat down on the box where the\nGreaser Knife Thrower kept his keen steels, and filling his pipe waited\nfor a break in the conversation. Then he asked the gasoline man for a\nmatch. After he got the fire he saw there were no words loose from the\nring-bankers, so he starts his skein.\n\n\"Well, lads, we hit 'em up hard at the mat today, 12,000 on the blue\nboards an' the ticket wagon window down before the harness is on for\nthe entree. S'pose them laddy-bucks in No. 2 car will say it was a good\nbillin', but I'm tellin' you people that this is a readin' community, an'\nit was the press work that had the coin hittin' the window this date, an'\nthat's no cold cream con, either. The Gov'nor knows it, for he gives me\na good word an' a back pat jus' as the parade was startin' for the main\nhighway.\n\n\"I'm given youse the real word, an' it's this--when you can get 'em\nreadin' about the Big Show you've already got 'em feelin' for change to\nbuy, an' that's as true as ticker talk. The old man sees in the paper\nthat the Big Show will soon be on the lot, an' when he gets home to daily\nbread he tells it to the old woman; the kids get next and there's no let\nup on papa 'till he promises to buy in for the whole family. An' workin'\none is workin' all--that's my motto. It's the press work that gets 'em\ntalkin', an' it's the talkin' that'll make 'em give up even when wheat is\ndown to 48 an' interest on mortgages is starin' 'em in the face. Get the\npaper talk an' the money is so sure that you can be plannin' new acts for\nnext season before the first pasteboard hits the bottom of the red box on\nthe gate.\n\n\"But, say, it ain't no children's game to get this paper talk. The good\nold days when you could blow into the newspaper offices with a loud vest\nand a tiger claw hangin' on your watch guard is done. Them times the old\nagent would lay down a cigar on the editor's desk, spread a lot of salve\nabout the greatest yet and the only one in captivity story, and then work\nthe gag 'write me somethin', old man.' But them days is strictly past.\nIt's a new make up now, an' a new line of talk that wins 'em. You want\nto enter quiet like just as if you were one of them Sunday school boys\nwith a write-up on a rally in the church basement. The editor gives you\nthe size-up for this, an' when you says 'I'm ahead of the Big Show comin'\n25th and 26th,' he's so surprised that he's glad to see you, an' it's\nonce aroun' the track before the bunch sees the flag that he asks you out\nto drink before you spring your pass-pad. And, if you don't believe me,\nask soft talking Jim Jay Brady and have it passed off for gospel.\n\n\"It's the approach that makes the center shot this new century. Go in\neasy, be skimp with your talk, don't spread the salve too thick, an' give\n'em clean copy--that's the game; be you ahead of Henry Irving with ten\ncarloads of stuff, a dinky little farce comedy with a society dame doin'\nthe lead, a melodrama with a real convict a-cracking the safe, or one\nof them Broadway big ones--no matter, it's the same, an' what goes for\nthem goes for the Big Show, whether you've got 68 cars on the sidin', or\nyou have slipped in after night with rubber boots on--and that's no Tody\nHamilton catch line.\n\n\"But you don't want to be too certain; you can get your chances in this\nline just as easy as in the shootin' gallery when its bullets against\nclay pipes. Some of the boys that handles the copy for the Eastern press\ncan put up a frost that would keep Chicago beef around the world in a\nsailin' ship. But you can melt 'em if you make good. Remember hittin'\nBoston las' season an' runnin' up against one of these heady boys with a\nfoldin' forehead. I give it to 'em easy, an' when I says circus he looks\nat me through his windows an' says so haughty:\n\n\"'Ah, the circus! Quite a diverting entertainment. Originated with the\nGreeks.'\n\n\"Now wouldn't that make you itch? Me mind gets to chasin' 'roun' for a\nproper come-back, an' I tries to recollect the names of some of them old\nguys what went paddlin' 'roun' in a sheet an' sandals spittin' out wise\nwords that no one has forgot. An' mem'ry lands me at the right dock, so I\nhan's this to the college boy:\n\n\"'Yes,' sez I, 'I believe it was Aristophanes who wrote an epic on the\ncircus to be read at one of Nero's spring openin's.'\n\n\"The words is hardly out of me mouth when he gives me one of those\nlooks that would have made Peary thought he had found the pole. So I\nlays me copy on the desk and gives five bells to back water an' I'm in\nthe elevator. An' so help me Bob, I hardly reaches the pavement before\nI sees sheets of paper flutterin' in the air an' me copy falls on the\nasphalt. The college boy had lifted it. Now you would think that it was a\nsheet that roasted us. Not at all, not at all. He gives us a two-column\nwrite-up and three half tones that had me up to the bar an' dizzy for a\nday. The worst will fool you.\n\n\"An', say, that reminds me. Remember when we was playin' the week's run\nin Chicago las' July? Well, I'd been skatin' roun' to the papers an'\nbuyin' drinks for the press boys, an' it was joggin' along to three on\nthe dials when I remembers I have a bed at the hotel. It was out near the\nlot, an' I starts out to walk. I'm crossin' the railroad tracks when a\nweary steps out an' asks me for a match. I gives up, when another comes\ninto the talk an' says, 'Give us money.' Say, I didn't have but thirty\ncents, an' I gave up. But the highwaymen thought I was lyin', an' was\ngoing to tap me when I says, 'Now, boys, let's argue this out.' So I\ntakes them two Jesse Jameses under a lamp post an' gives them a josh talk\non the Big Show that has 'em serfy.\"\n\n\"Well,\" said the Boss Canvasman, who was always interested when there was\nany fight talk, \"what happened, what happened?\"\n\n\"What happened?\" says the Press Agent. \"Hear me! I takes out me pass-pad\nan' writes passes for them robbers until a policeman comes, when I turns\n'em over. An' that ain't all; I gets a column story in each of the\nafternoon sheets on how the Press Agent of the Big Show captures two bold\nboys, an' the Gov'nor gives me the good word and a double X, an' I says\nthankee an' repeats me motto, 'An' workin' one is workin' all,' to every\nbarkeeper that was sellin' after midnight that evenin'.\"\n\n\n\n\nTHE OLD GRAFTER'S LAMENT.\n\n\nThe Old Grafter had corns on his knuckles from holding greenbacks between\nhis fingers.\n\nHe looked a trifle seedy about the costume, but his moustache was\nwaxed--the moustache, too, was dyed and you saw the reason when he took\nhis hat off. The Old Grafter wore a celluloid collar and a polka dotted\ndickey, and when his vest was opened it showed up the shyness of his\nlinen.\n\nThe Concert Manager was springing gossip about the principal clown who\nwas having trouble with his wife who did the iron jaw swing. He saw the\nOld Grafter coming across the ring and he stopped, for it was pretty well\nknown that old-timer wouldn't stand for scandal. The Old Grafter bit off\nenough tobacco from the canvasman's plug to make a comfortable quid and\nthen sat down on the snake box. He was looking sad and there was silence.\nPresently he sent a splash of juice up against the center pole and after\nshifting the quid he opened up.\n\n\"Say fellers, I've been cuttin' the cards since John Robinson had money\nin tent shows an' I've come to the verdict, it's this--when you've got\nthe green in your pocket an' the suckers is tipped off they'll crowd you\nas thick as flies on the popcorn pile, but when there ain't no coin to\njingle you kin get so lonesome that you'll go to bed with a hot water\nbot'l for company.\"\n\nThis bit of wisdom impressed the gang, for no one spoke, and the Old\nGrafter threw his reversing bar and chinned out this--\n\n\"The old days is gone an' they's left the circus graft on a weedy sidin'\nwith no roun' trips back to the lan' of promise. Them was the one ring\ndays an' in them times there was allus fodder for the hogs. Today it's\none ring, two rings, three rings and a stage--the biggest tent on earth,\nbut for the grafter--nothin', nothin'. Me, what use to turn the shank of\nthe week with a bigger wad than the principal bareback gets, me makes\ntoday a dirty twenty on percentage an' sellin' reserved seats. I'm\nashamed to look the old days in the face. Why say in them days many a\ntime the proprietor of the Big Show was touchin' the grafter for cash\nwhen business was bad an' today so diff'rent, so diff'rent--if I gets\ninto a poker play on the train an' the ante's a nickel I've got to reach\ntwice to find the coin. If I'd had the good sense what's in Bill McGinnis\nhead I'd a bought a little road tavern like he did twenty years ago an'\nI'd a-had a bank book roostin' back of the bar. But I thinks there's\nstill somethin' doin' my end an' I waits an' loses--and what do I get--a\ncouple of treasuries and some change at the pay off durin' the season\nwith crackers and cheese for me an' the old woman in the winter. It's the\ndiff'rence 'tween horse radish an' saw dust an' its got me slippin' back.\n\n\"I'll tell you fellers somethin' 'bout the old days. 'Twas 'bout '76 an'\nwe was graftin' with a one ring outfit. We struck good crops and sunny\nweather in the one nighters in the Ohio valley. The farmers had money an'\nthere was peaches in the orchard for every boy with the troup that had a\nbag of tricks. Everybody was standin' in on the graft an' we had a fixer\ntwo days ahead so there'd be no call. We was carryin' a car with the lay\nout an' four tin horns that was science on faro and turnin' the wheel.\nThe big game was invited to the car an' there was allus a set out an'\nsumthin' to drink. The little fish was worked on the lot an' there was\ndays, many days when the graft was mor'n the ticket wagon count up, an'\nthe rake off was loafin' 'bout par, continuous. Good days them, me boys,\nfor ev'ry body from the boss of the outfit down to the stake driver.\nMoney was comin' easy an' when there was any protestin' on the part of\nthe patrons an' it got to fists, or gun play we passed along the Hey Rube\nan' there was Gettysburg till mornin' if they was lookin' for battle.\n\n\"The best burg we hit was a lit'l settlement where we had a two mile\nhaul up the pike from track to lot. Everything was ripe for graftin' an'\nwe was ready for harvest. Seems like a reform committee had to hit down\nall the games and the folks was hungry for gamlin'. The posters in No. 1\ncar piped us off on conditions an' it was said that them paste spreaders\ntraveled off with a roll from stud polker in the car after the bills was\non the stands.\n\n\"They wuz on the lots waitin' for us when the boss landed to lay off the\npitch for the round top, we wuz only usin' one then an' had no an'mals to\nspeak of. The fakirs got in the game early an' transparent cards from gay\nParee was the first bait and bitin' was good. 'Fore the parade started\nall hands was busy on the lot takin' care of the games an' say the\nfarmers had it with 'em in rolls. The foxy boy in the ticket wagon has\nall his bad coin ready and the constable with the badge has been fixed\nwith a ten to see there's no argument when short change is handed out.\n\n\"Oh! we worked systematic them days.\n\n\"Well, say, before the band had struck up the grand march for the entree\ngold bricks wuz sellin' like cod fish cakes at a <DW65> camp meetin' an'\nthe boys what was workin' the shells had to lay off to get the stiffness\nout'er their fingers.\n\n\"I hates to tell it, I hates to tell it in the days there's nothing'\ndoin'.\n\n\"You see I was cappin' for the boss of the show an' say that day keeps\nme busier than a man drivin' sheep. The outfit was gettin' thirty-five\nper cent. of the graft an' if the partic'lar grafter who was gettin' the\ncoin failed to come up we se'ed that he was prop'ly turned over to the\noff'cers of the law an' we did the prosecutin' on the groun' that we was\nrunnin' a strictly moral show. Say, while I was watchin', a farmer with\na bunch of weeds under his chin an' a face like a quince comes up to me\nan' makes a holler. Somebody had touched him for his wad 'fore he could\nget to the games an' he was dead sore. I se'ed that he was goin' to make\ntrouble so I remembers that his wagon is standin' in the dirt road by the\nlot. I gives a stakeman the tip he kicks the off bay in the flanks an'\nthere's a runaway. The corn cutter chases after his team an' forgits that\nhe ever had a roll.\n\n\"An' say at night down in the car the air was hot. The tin horns was busy\nand coin was droppin' like rotten apples in a mill race. The boys what\nwas dealin' the faro had monkeyed with the deck an' it was far to the bad\nfor the spenders. 'Bout time to start haulin' for the cars two burly boys\nbegins to talk fight an' it looks like Hey Rube all aroun'. One sticks\nhis knuckles into me face an' I says to him sort'er fierce like.\n\n\"Say, young fellow, if youse lookin' for fight I'll git one of the boys\nto stick his teeth in your neck an' you'll change your mind.\n\n\"There was no gun playin' but there was a lot of chinnin' and cussin' but\nwe finally gets the tin horns out an' starts 'em up the road to the first\nsection. The gang is hot after us an' there was only one thing saved us.\nJes' as the crowd was closin' in I sees the tiger den with the two blacks\npullin' it comin' over the hill. I chases forward to the trainer an' when\nthe cage gits up close we jes' shoves them two tin horn dealers in the\nden with the tigers an' saves their lives.\"\n\n\"Never could make a return date there, could you, Bill?\" asked the Boss\nCanvasman as he made one of the spotted coach dogs take a jump through\nhis hands.\n\n\"Return date, well I should reckon. Went back there in two months an'\nstill foun' 'em ripe. But there was only one way to do it. The Boss had\nto paint over all the cars an' wagons an' change the name of the show.\nThe hay boys thought it was a new outfit.\"\n\n\n\n\nTHE BILL POSTER'S VISIT.\n\n\nThe Bill Poster was a stranger to most every one in the outfit. He\ntraveled a month ahead of the show on Advertising Car No. 2, and while\nhe hung around during the run in New York he never got well enough\nacquainted to mix in with the ring bank squatters. While the outfit had\ngreat respect for him, and especially for his work, he wasn't generally\nunderstood, and this did not keep his popularity up to par. He always\nseemed solid with the business staff and called the assistant treasurer\nby his first name, and these two conditions were known to the sawdust\nboys. They always took off their hats to anybody on the staff, and the\nman in the ticket wagon was only known at the pay-off. Then, too, he\ndressed well and wore a diamond, that is, a real one, and altogether his\nfinancial condition was too good.\n\nBut for some reason or other the Bill Poster did happen back on the show\none afternoon. Just after the matinee, when the gang sat down on the\nbank for the spiel, he was seen walking across the track, and the boys at\nonce began to speculate on what brought the paste spreader back to home.\nSome of them thought it was for a call-down, and the concert manager\ndeclared that was the cause, for, said he, \"I never seen a town billed\nso rotten as this un.\" But the gasoline man, who was a close observer,\nthought different. He knew that there was a little fairy working in the\nFall of Rome ballet that was sweet on the paste boy, and he put the rest\nof the crowd wise before the conversation got too far from the shore.\n\n\"Cert'nly,\" he said. \"Didn't I see the guy in his plaid rags ev'ry night\nwhen he was playin' the Garden, gittin' the little lady at the dressin'\nroom door and blowin' her off to butter cakes an' coffee before she\nchases to the bridge an' home.\"\n\nThe gasoline man was getting real gabby on the love affair, when the Bill\nPoster came through the red curtains over the dressing tent entrance\nand walked across the ring to where the gang sat in the shadows. He had\na sassy little \"Howdy\" for everybody and then passed around a box of\nTurkish cigarettes. Everybody passed it up, and the Boss Canvasman bit\noff a two-by-three chew from his plug and looked sour.\n\n\"Slipped back to see the Boss,\" said the Bill Poster, as he lighted one\nof the Turkish boys with a match he took out of a sterling box that had a\nbeer ad. on it. \"Ain't no secret; I want a transfer. I'm good an' tired\nof the slow work on No. 2 car, an' so I gets a day off an' runs back to\nsee if the Boss won't put me with the opposition crew.\"\n\nThe gang was silent. Nobody had asked for the why, so nobody commented\non it. The fact that he was going to have nerve enough to ask to get in\nthe opposition crew filled the concert manager with disgust, for he knew\nsomething about bill posting, and also knew that it took a triple-plated\ncrackerjack to hold a place with this crowd of rush pasters of a\nthree-ring outfit.\n\n\"There's nothin' to it,\" continued the Bill Poster, \"gettin' into a town\nwhere ev'rything is dead ready, all the boards up, and nothing to do but\npaste. I want a little excitement. I allus gets it in the winter, when\nI'm billin' a hall show. Many a time I've laid me bundle of lithos under\na doorstep to punch some guy who was tearin' down my stuff in saloons\nwhere I'd spent up me money, and then hangin' his stuff in the window. I\ntell you the opposition crew is the crowd to have the ginger. When your\ncar is hangin' up on a grassy sidin' an' you gits a wire that the other\nshow is routin' three days ahead of your own bookin's, it makes you jump.\nThe boss wires the head of the gang to jump for the town and beat 'em\nup. Beat 'em out, but on the level, legitimate--but beat 'em up. Don't\ntear down none of their billin', but kill it if you have to buy the side\nof the Presbyterian meetin' house to git a showin' for them nine-\ntwenty-eight sheet stands.\"\n\nAs far as the gang on the bank was concerned, the Bill Poster was talking\nGreek, and he had 'em wingin'. The Concert Manager thought he was \"next,\"\nbut his coupling broke before his understanding left the city limits.\nJust then the Press Agent of the Big Show happened in and the talk hadn't\ngone three lengths before the Bill Poster and the newspaper man crossed\nbayonets. Both were doing the publicity gag, and both had a well set and\nriveted idea that each one and not the other was bringing the people into\nthe tent and giving the show a good gate to send back on the statement to\nthe high hat boys in the city who were doing the financing.\n\n\"Let me tell you something,\" said the Press Agent, as serious as if he\nwas arguing to get a half column write-up on fourteen dollars' worth of\nadvertising in the only daily in the town. \"Let me tell you. These days\nthe people who are spending money for amusements reads the papers, and\nit's the paper talk that lands the coin at the window. I know what I'm\ntalkin' about. Bill posting is all right, but it's the newspaper work\nthat does the real singin'.\"\n\n\"Come off!\" said the Bill Poster. \"You're only pluggin' your own job. You\ndon't mean to tell me that the boss of this outfit would keep all the\nprintin' shops in Cincinnatty goin' night an' day to git out the wall\nstuff if they didn't think it was some good. An' say, they wouldn't be\nrunnin' three billin' cars ahead of this here show if there wasn't some\ncome-back to the money they was blowin'. Why, say, what do you think they\nare? Your press work is all right, an' my bill postin' is all right, an'\nyou've got to have both.\"\n\n\"Well, maybe you're right,\" said the Press Agent; \"I guess they use the\nbilling to emphasize my work.\"\n\n\"I don't know so sure what you means, partner,\" said the Bill Poster,\n\"but the Boss of our car figgers it out this way: He says that the\nreadin' in the papers about the big show makes 'em look at the pictures\non the wall. And, says he, the pictures on the wall makes 'em read what\nis in the papers. An', say, he's been pastin' since the John Robinson\ndays.\"\n\n\"Guess he's right,\" said the Press Agent.\n\nThis last statement hit the gang as real good sense, and they half agreed\nthat the Bill Poster knew something about his business.\n\n\"I tell you, boys,\" continued the Bill Poster, as he took a seat on the\nsawdust pile and lighted another one of the Sultan's dreams, \"in me dull\nmoments, when we is travelin' an' there's nothin' to do but layin' out\npaper an' gittin' the buckets ready, I figgers it out this way: You can\ngit 'em with the paper talk all right; but there's one thing you can do\nwith good bill postin' and litho work, an' it's this, you can't make 'em\nread the papers, but bless me, you can make 'em see bill postin'. Say,\nme an' the gang I work with in New York have sniped the subway fence so\nhard with red-on-yellows that you would think there was nothing else\non Broadway. Did you see 'em? Well, you bet. There was so much color\nstickin' along the ditch that it hurt your eyes when you rode by in\na car. That's what I claims for proper billin'. You can git it where\nthey've got to see it.\n\n\"Say, to prove what I says is right, I'll tell you a little experience I\nhas. I was doin' the litho work for a cheap price house that was playin'\nthe old favorites with a stock. They puts on 'The White Squadron.' The\nboss comes to me and says, 'Look here, Jim, I wants you to do your best\nwith this piece; its costin' us a lot of money to get it on, and we wants\nto get it back. There's a diamond stud coming to you if you gets what\nI calls a good showin'.' Say, I would 'a' done it anyhow for them kind\nwords, but I says I'll git that diamond if I puts bills all over the\ntrees in Central Park an' goes up in stripes for ten years for doin' it.\nI was thinkin' all the time some new gag to work, when one mornin' comin'\ndown I reads that there's a yacht race in Harlem river that afternoon.\nYou know, boys, 'The White Squadron' is one of them naval pieces, an' has\na lot of ships in it. Well, the Sunday before I'd pasted up a lot of one\nan' a half sheet boards with type an' litho stuff, an' I has it loaded\nin a wagon ready to git out on the street some night and sit the boards\nin doorways. But no, says I. Me partner an' I drives the team out to the\nHarlem River bridge. The river is so thick with tugs and launches full of\npeople to see the boat race that you can hardly find the water. We waits\nuntil the race starts, an' then we clumps them boards into the river,\ncarefully like, so they will fall with the picture side up. They hits\nthe current and starts floatin' down. They all seems to cling together,\nand make a big raft, an' all you can see is 'White Squadron.' Everybody\non the bridge and the boats is a readin' and laughin', and we knows it's\nthe showin' of our lives. And say, the boards keeps on driftin' wid the\ncurrent an' gits so thick that when the guys in the paper boats hits that\npart of the river they gits stuck and the race has to be called off.\"\n\n\"Well,\" said the Press Agent, coolly, \"did the show do any business?\"\n\n\"Business!\" replies the Bill Poster; \"you'd a thought there was a fire in\nthe neighborhood ev'ry night, the crowd was there so thick fightin' to\ngit in. An' say, the guys what got broke up in the boat race is so stuck\non the joke that they gives a theatre party an' the papers is full of it.\"\n\n\"Yes,\" said the Press Agent, \"and it took the press agent to get that in\nthe papers.\"\n\n\"But the bill posters got 'em in the house,\" retorted the Bill Poster,\nwith the air of a man who had knocked the local welterweight over the\nropes.\n\n\n\n\nTHE CANDY BUTCHER'S DREAM OF LOVE.\n\n\nIt was generally conceded that the Candy Butcher was the handsomest man\nin the outfit. To be sure, the gent who did the sixty-one horse act in\nRing Three was a Charlie boy for good looks, but it was only when he was\nin the red coat and working. When he left the dressing tent and went red\nlight hunting in a one night stand he looked like a canvasman on a visit\nhome to his people, but he was a hot card when he had the dicer on in\nthe horse act. It was so different with the Candy Butcher--he was always\ndressed up and he never looked like he felt it.\n\nNo one ever saw the Candy Butcher wear a coat, but his checked trousers\nwere always creased whether the big show was playing a one night in\nKeokuk or doing a run in the Madison Square Garden over in the hot city.\nAnd he always wore a vest, but it was never buttoned and there was a red\nstriped shirt with one of those Montana boys screwed in the bosom right\nunder the dickey dot of a bow. The vest was something to speak about--it\nhad the band wagon way to the bad on the distribution of colors and\nlooked like page 89 in a wall paper drummer's sample book. There was a\nshiny chain with an elk's tooth and a tiger's claw and in one vest pocket\nwas a date book with a tooth brush and a blue pencil doing a duet on the\nother side of the rainbow. The Candy Butcher always wore pink underwear\nand had his sleeves rolls up to his elbows. And you don't want to forget\nthe little miniature of \"blondy\" that he had pinned right over his blood\npump.\n\nAnd as a matter of detail the Candy Butcher always had this get up the\nwhole season and no grease spots ever scored--he was just the same\nwhether it was back of the tub and the peanuts giving the \"five tonight\ngood people\" gag, selling concert tickets during the run-offs in the\nhippodrome or Sunday afternoon in the ladies' coach, section three,\ntelling the big blonde who did the cloud swing in the round top rigging\nthat of all the girls she was the onliest.\n\nThe gang was sitting on a carpet roll under the big top when the Candy\nButcher came across the track and sat down on the ring bank. He was\nlooking sad, but his pants were creased and the Montana boy was shining\nlike a frozen hunk of Kennebec water. He rolled himself a cigarette and\nthe gang being silent he edged in with this bunch of talk:\n\n\"I'm bluer this evenin' than the paste boards they're passin' out of the\nticket wagon an' if it wasn't for gettin' the dock at the pay off I'd be\nup against some boozery workin' the syphon like an engine at a tenement\nfire. I ain't got no life in me 't all an' I won't have until we leaves\nthe east an' strikes the west country. B'lieve me, me boys, the east is\nall right for business, for I can pass out the sour juice at five a throw\nright here as well as any where, but it's myself an' not the place. It\nwas too much of a feather bed las' season an' I was fool enough not to\nremember that I had to wake up. There ain't no use talkin', whenever a\nguy gets a good dream in this here life some sucker has got to give him\nthe alarm clock finish an' he wakes up with a yell.\n\n\"Say, I can call the turn on the folks on the blue boards an' have 'em\nall drinkin' lemon juice and shellin' peanuts an' I likes to do it, but\nme heart ain't in the work, this season and that's no lithograph josh\neither. I'll tell you and some of youse may give me the grin, but it's\nten to one you've had the soft spot yourselves, so I ain't a-carin'.\nRemember the Congress of Nations gag they was workin' las' season? Well\nright back of me lay out there was a lot of maidens that was doin' the\ngypsy village and fakin' a lot of beads and fortune tellin'. There was\none little fairy in the outfit that had me dead, an' I don't mind tellin'\nyou that she had me soft from the start. She wasn't none of these city\npick ups but a nice little gal that talked quiet and minded her own. She\ndidn't mix with the rest of the push an' we got thick the first day the\ncanvass went up. She tells me her story confidential like an' I give her\nme sympathy, for her people was dead agin her for troopin'. You see she\nhad been working in a New York hash house, where they had Bible talk on\nthe wall an' where they gave a splash of beans and a draw for a dime.\nShe gets tired an' a guy what has been eatin' at the place gives her\na job in a boardin' house waitin' on the table. Here she meets the man\nwhat has the Congress get up to put on an' he tells her gilt edge stories\nabout the circus business and to cut the talk down she joins out with\nthe show. Well, say, she was the real thing to me. In two months she had\nme stoppin' the booze an sendin' money home to the folks, an' it was a\ncenter shot to get that out of me. She was allus lookin' for a chance to\ndo me somethin' kind an' one day she did a little turn that I wont forget\nlong after I've past the old man' home.\n\n\"We had struck a rough run of one nighters in Ohio and was looking for\nbright things across the river in the West Virginney townships. I had to\ndo the German Emperor in the parade an' when we got back to the lot I\nbegins to get me stand ready for the sale. I'd packed up careful on the\nlast stand and thought it was sunny for the next. But when I got me chest\nopen I finds the citric acid jug missing, and the floaters I'd saved to\nthrow on the top of the tub was gone too. I had a cussin' spell for a\nbrief and then I goes on a still hunt for lemons--the real yellow. But\nbless me I couldn't find one in the village an' there was nothin' doin'\nwith the barkeeper what had 'em. Comin' back I see's a <DW55> doin' the\nshaker across from the lot. He has 'bout a dozen lemons and I offers him\na good price, but the brown boy wouldn't sell an' I was sorer than a\ndoped lion. I goes into the tent and meets Maggie, that was the name of\nme fairy, an' she was sewin' silvers on her little coat. She sees me sad\nlike an' I unloads me woes. The gal didn't say much, but she rubs her\nhand across me frowner an' sez, never mind, John, I'll help ye out, an'\ngoes 'way. Say, youse may give me the laugh, but durn me if that lass\ndidn't come back in 'bout an hour carryin' a bucket an' I mos' had a fit\nwhen I see's it full of gooseberries? What's the game? sez I.\n\n\"Watch me?\" sez Maggie, \"an' I'll keep you in the business.\"\n\n\"Durn me, boys, if that little maiden didn't mash them berries to a pulp,\nstrain 'em through the Hoochie Coochie gal's veil and have the tub full\nof sour juice in seven minutes. I pours in the water, finds me floaters\nand puts them on the ice bank an' before the gang is passin' once\n'roun' I'm sellin' the juice as if lemons was growin' on locust trees.\nGooseberries too an' the yaps couldn't git enough of it. It was better\nthan any graft ever in the one ring days an' the little gal had done it\nall. Ain't no use tellin' you that I gives her a new shirt waist an' she\ngives me a squeeze that makes me top spin like a merry-go-round.\n\n\"An' say I fixed that <DW55> that wouldn't give up. I tipped off one of the\ndrivers and when the first pole wagon leaves the lot with the eight grays\na pullin' it, the leads shies into the shaker stand an' gives it the\napple cart finish.\n\n\"But the little fairy I lost her an' that's why I'm sad. It was this\nway. The gal what did the twistin' for the Turks had the fever an' they\nshipped her home. The guy what had the Congress comes to Maggie, gives\nher the jersey and the gauzy pants and sez she must do the part. Maggie\nkicked an' said she was engaged for the gypsie village. The guy says \"not\nat all\" and Maggie pulls off the spangles an' goes home. An' say I ain't\nbeen right since, an' some days I feels like playin' quits myself.\"\n\nThe gang looked at the Candy Butcher consolingly, but no one spoke.\n\n\"The las' I hears from her,\" she said, \"she had gone back to waitin'.\"\nShe's slingin' hash in a Brooklyn Caf', but I loves her just as hard.\n\n\n\n\nTHE BOSS CANVASMAN'S YARN.\n\n\nThe Boss Canvasman was always sad. He never talked--he just chewed his\ntobacco and worked. Like the Candy Butcher, he never wore a coat, but he\ncut the pink underwear the Lemonade Boy flashed when he had his sleeves\nrolled up at the tub. Of course, the Boss had a coat, one that had run\nthrough a dozen seasons, but he always kept it strapped down under the\ndriver's cushion on the pole wagon. Whenever he did use it, the coat was\ndoing duty for a pillow when the last section was late pulling out, and\nhe was sleeping on a gondola with the wall poles for a mat.\n\nThe Boss Canvasman's pants were ancient history, and his vest was always\nopen. He wore one of those motorman watches, with a shoestring for a\nchain. He never looked at the watch except at night, for when it was\ndaytime he could pull off the hour on the second by the slant of the\nshadows across the big \"top.\" The Boss never wore a collar. On Sunday he\nwould put a gold button in the shirt band, lean disconsolately against\nthe tongue of the pole wagon, and feel uncomfortable because he was\ndressed up.\n\nThere was no coin and jewel flash about the Boss Canvasman. But he did\nwear a rusty button in the lapel of his vest--one of those G. A. R.\nthings. Across his face there was a long red scar, and sometimes when\nhe had been drinking he talked about the first Ohio Cavalry--Gettysburg\nwas the answer. He feared nobody nor anything. He had no friends, except\nprobably the Stock Boss, and there was a tie there, because the two had\ndone the wagon show long years back before three rings were dreamed of\nand farmers were living on their own hog meat and were happy. If he ever\ndid talk, it was when something went wrong, and then his line of words\nwere unfit for publication--even in a Chicago weekly.\n\nIt looked like a squall just as the matinee was breaking, and the boys\nat the cages were hurrying the people along to get the tent clear before\nthe water fell. The Boss Canvasman was hard at it getting the guys tight\nand throwing in cinders around the big poles, where the dirt was soft. He\nwas taking no chances on a blow. He had been mixed up in several of those\nwind things down in Texas, where a cyclone struck the lot, and all that\nwas left when the sun got back was the ticket wagon and the elephants\nthat were chained to earth. He knew his game.\n\nAfter the usual beef stew and the splash of beans had been put away with\na cup of black in the meal tent, the gang gathered about the rink bank\nfor a little rest. The Sawdust Spreader and the Gasoline Man were talking\nscandal, as usual. This time it was the Snake Charmer, who mixed it too\nstrong with the bottles on the last stand, lost the keys to her snake-box\nand two boas and a black boy starved to death before the feeders could\nget the rabbits under the fangs.\n\nThe gang sat down in silence until the Concert Manager cut in with some\nweather talk. It looked stormy, and as it was the last night of the stand\nand a long haul to the cars, everybody was feeling a little sore. With a\nstorm coming up, the tent to watch, and then the haul and an eight-hour\njump with a hustle for the march in spangles down the highway the next\nmorning. It had everybody grouchy and thinking about hall shows with a\nroof and a stove.\n\n\"Youse is doing a lot of guff about a rain,\" cut in the Side Show\nSpieler, as he polished up his shiny brim on the corner of the leaping\ntick, \"but youse kin stay inside when its droppin', but for me--the open\nand still the same old gab for the dimes.\"\n\nThe Boss Canvasman came up along the ring bank, and without noticing the\ncrowd on the sawdust began to jamb down the cinders around the net-pole\nwith the heel of his boot. From the distance came a low rumble of\nthunder. The gang looked at each other. Everybody in the outfit feared a\nstorm; not so much for the storm itself, but for the effect it had on the\nBoss Canvasman. He never talked, but when the basses under the hills were\ngrowling and the lightning was doing a fancy jig against the blue, he let\nloose a vocabulary that would put a canalboat captain to the blush of\nshame and send a sea-soaked old jib reefer to flight as a down and out\ncusser beyond appeal, cards torn up, tables turned over and police at the\ndoor. He used the same two-em words, but the way they hit the air would\nhave made holes in a battleship.\n\nThe broadcloth boys, who came over in the sailing ships and scared the\nIndians into religion by telling them how warm it was in perdition,\ncouldn't touch the Boss Canvasman for a scare when he got on the same\ntopic. He had a way of saying \"hell!\" that made you want to turn in a\nfire alarm just for personal comfort.\n\nPresently he came across to the gang, and to the surprise of all the\nrink-bank squatters loosened up.\n\n\"Talkin' 'bout workin' in the rain, is you?\" he said, with a sneer, and\na cross-hook glance at the Side Show Spieler. \"You've got no kick. Say,\nyou'll have your head on some Mamie's shoulder in the last day coach, up\nan' away, while me an' my gang is still workin' on this lot gittin' this\nround top on the wagon without streakin' her wid mud.\"\n\nThere was not a reply, for the boys knew he was right. The Side Show\nSpieler hung around a bit, and, with a typhoid smile, remarked that he\nguessed he'd bow hisself out, and more than that, he did.\n\n\"Speakin' about storms,\" said the Boss Canvasman, as he tied a long,\nrunning knot in the guy that held the triple bars, \"I guess you fellows\n'ceptin' some of the old boys, dunno what it is for a rain an' a blow. I\ndun bin circusin' it for forty year, and, say, I've met some blows an'\nlightnin' that no sailor chap ain't hit, I don't care how often he's bin\n'round the Horn.\"\n\nYou couldn't have looked into the face of that old fellow without\nbelieving every word. He was burned brown by every clime, creased and\nseamed by every frost, and parched and dryed by every wind, and his hands\nfor knots and gnarls had an old oak twice around the track and then past\nthe judges and turf writers, all off and back to the street cars.\n\n\"Say, I'm going to tell you fellers sumthin',\" said the Boss, as he sat\ndown and began twisting together a piece of rope that was getting to look\nlike a lion's tail. The gang was startled, for in the memory of none\nthere never lodged the fact that the Boss Canvasman had been seen to sit\ndown as long as the pole was standing. But he did, and what's more, he\nreeled off a yarn.\n\n\"Jes, 'fore the war broke out,\" he said, \"I had enlisted in the First\nOhio, I was workin' down the Valley of Virginia wid a little wagon show\ndoin' the same kind of work I'se doin' now, 'cept it was nothin' but play\nto this. Funny, too, for six months afore I was down that same valley wid\nthe cavalry cuttin' into them rebs, an' I don't mind telling you they was\na cuttin' us, too, wid Mosby in the woods an' 'Stonewall' Jackson, God\nknows where.\n\n\"Well, we had been doin' a day an' a night stand in one of the little\ntowns, an' had a fourteen mile haul down the pike for the next. We was\nhopin' for a moon, not so much to strike by, but for the drive, for\nthe people hadn't got the roads to right, an' they was still full of\nartillery ruts an' wagon train wrecks. But we pulled off the show, an'\nbefore the last bareback was on I has the menagerie canvas on the wagons,\nall stakes up and the dens down the road, with the boys leadin' the\nelephants and the two camels over the hills. It had been squally like,\nall night, but I had the round top tightened up so hard that youse could\nhave walked the ropes an' there was no danger.\n\n\"But jes' as Dutch Andy was playin' his last piece there was a bust\nof wind an' a flash of lightnin', an' she began to come down in solid\nsheets. We gets the people out and gets to work on the tent wall. This\npeels off in a jiffy, an' the rain lets up. Then down wid the big top an'\non the wagon. But sumthin' catches in the riggin' on the main pole, an' I\nsees I has to send me helper up on a climb to get her clear. Everything\nwas gone but the pole wagon and a few side show things, an' the 'bus we\nbosses rode in, with four grays pullin' it. Afore I sends my helper, Jim,\na fine boy, what had been a sailor, up the pole, I send four men out to\nhold her up by hangin' on the long rope to the far stake. Jim skins up to\nthe top an' gits her loose, but before he kin git down the gang holdin'\nthe long guy loses their holt, an' the pole falls.\n\n\"Well, we picks up Jim, an' he is pretty bad. Ribs in, an' a lot of cuts.\nWe tried ev'ry house aroun', but no doctor, though there was one good old\nlady who gave us some arnicy and strips of bandage she said she'd kept\nto use on her husband when he got shot up in the Saturday night fights\n'bout the tavern. So we piles poor Jim into the 'bus, and drives off\neasy, while we walks along quiet like an' sore. Poor Jim, he jes' groans\nan' talks 'bout doin' his best, an' I keeps givin' him liquor to make him\nforget it.\n\n\"But it was all over for Jim, an' we jes pullin' out of a clump of woods\ndown by a river when we sees he's dead. There was no use carryin' his\nbody long, an' he didn't have no people to ship it to, so we decided to\ngive him a decent burial. Two of the stakemen digs the place, an' we lays\npoor Jim away under a willow tree. Jes' then one of the boys speaks up\nan' sez:\n\n\"'Say, boys, it don't seem right to plant Jim without sayin' sumthin'.'\n\n\"But there wasn't a mother's son in the crowd knew what to say, though\nthey is all on to what the fellow means. We waits awhile an' I sez:\n\n\"'Well, there might be a little singin'.'\n\n\"An' I wishes that I had the principal clown there, for he was good on\nsad songs, 'specially if he'd been boozin' a little.\n\n\"'Might git the band,' sez one of the boys.\n\n\"'But the band is way ahead.' We is all studyin' like, when over the\nhills comes the whistling wagon.\n\n\"'Here is the calliope,' sez one of the stakemen.\n\n\"So we stops it, and 'Reddy' Cavenaugh, who played the whistles, besides\ndoublin' for Peter the Great in the street parade, sez he has enough\nsteam on to play a little. We backs the calliope around, an' three of the\nboys holds the hosses. Then Reddy played soft like, jes' as soft as he\ncould, on the whistles, an' we all lifts our hats.\"\n\n\"What did he play?\" asked the Candy Butcher, as he wiped away a tear with\nhis red cuff.\n\n\"Well,\" said the Boss Canvasman, \"he only knowed two tunes, 'When Johnny\nComes Marchin' Home' and 'The Blue Danube,' but we planted Jim to both.\"\n\n\n\n\nTHE SIDE SHOW SPIELER SPEAKS.\n\n\nThe Side Show Spieler was a tall dark man with a sad face. He was clean\nshaven, wore his hair slightly long and looked like William Jennings\nBryan, after the vote was in and counted and the telegraph operators had\ngone home. He had a deep baritone voice and a vocabulary that was always\nIt. The Spieler carried more education than any man on the pay-roll and\nit was said that he was the only man in the outfit that could read the\nLatin names on the animal cages. It was generally supposed that he was\none of the better days' boys, but he never told the story of his past\nlife to any of the gang.\n\nIt had started to rain just after the afternoon performance, and as it\nwas the night to strike and haul, the six squatters on the ring bank\nwere silent and sore. The Spieler came in from the menagerie and joined\nthe layout. He never sat down, so he stood for a while in front of the\nothers. No one spoke and he let a little conversation hit the air.\n\n\"What's the matter with you fellows? Sore 'cause it is raining! Don't see\nwhy, you all are under cover. I've got to stand out there in front of the\ntent and talk for dimes.\"\n\n\"Yes,\" said the Boss Canvasman, \"and I've got this tent to roll up an'\nload.\n\n\"Well be happy, be happy,\" said the Spieler. Then after a pause. \"Say,\nyou fellows can help me out a little. The Boss gives me a talk last\nnight, and says while the spiel for the little show is all right and good\nhe wants a new one for the big stands we strikes next week. I've been\ndigging up the old talk I used to tear off on the Midway at the Chicago\nShow and I've about studied her out, if youse don't care I'll just unroll\nhere an' see if its the proper josh.\"\n\nThere was no objection, so the Spieler mounts one of the red painted\nstools, the object holders stand on for the little lady to jump the\nbanners. Then he serves his spiel:\n\n\"And now good people if you will kindly give me your attention for a few\nmoments I will explain to you the great congress of freaks, oddities of\nmother nature and strange and curious collection of wonders shown in the\ntent. Remember you have plenty of time, the Big Show does not commence\nfor fully harf an hour. Surely you will not leave the lot until you\nhave seen all--all good people--all provided for you in this monster\nentertainment, this caravan of canvas covered world sought wonders. Come\na little closer. Please. Thank you. First, let me call your attention to\na remarkable group of reindeer. We have not one--three--five--or six of\nthese specimens of the animal kingdom, but a whole herd of them--a herd\nof them--a herd of reindeer from the land of the midnight sun, where\nthere is but one night, one day--reindeers, my friends, from the icy\nmountains of far away Norway the greatest group ever exhibited in any\ncolossal enterprise that has ever been organized by mortal man.\n\n\"Next you will find the ostrich farm. This strange bird that furnishes\nplumage for me ladies' bonnet, and that comes from the sunny sands of\nAfrica. And remember the cool of the evening is the best time to see the\nostriches, for it is then you may notice their marked pe-cu-li-ar-i-ties.\nListen, good people, reindeer and ostriches--reindeer from the frozen\nnorth--ostriches from torrid Africa--specimens from each zone, the most\nastounding representation of nature's wonderland ever shown. Reindeer and\nostriches--as the poet says:\n\n    \"From Greenland's icy mountains\n    From Afric's coral strand\n    Where them crystal waters\n    Run down the heathen land.\n\n\"And all for a dime, ten cents, will you hesitate? but wait, good people,\nthat is not all. The wonder of wonders is yet to come--Bobo--he eats 'em\nalive, he eats 'em alive. You must see Bobo. This strange and curious\nspecimen of humanity who exists upon poisonous reptiles, captured in the\njungles of the Tasmanian blue gum tree and brought to civilized America,\nhe still lives on snakes--Bobo, the snake eater--Bobo, he eats 'em alive,\nhe eats 'em alive.\n\n\"One moment, good people, one moment--this is not all. Listen--Wild\nRose--the half girl and half dog. This remarkable freak of nature that\nhas puzzled the scientists of two continents. Queen Mary, the largest\nfat woman ever shown under canvas or in hall of curios, the marvelous\nSamson, the giant of today, who bears upon his breast great rocks to be\nbroken with a sledge, and last but not least--Professor Corello and his\ntroupe of performing roaches, the only attempt ever made to develop the\nhitherto unknown powers of these insects. The greatest, most interesting\nand educating avalanche of remarkable freaks and strange and curious\npeople ever shown. And all for a dime, two nickels, good people--a dime,\nbut a dime. The performance is about to begin--one dime--the sight\nof an invested fortune, the greatest stroke of genius of the modern\nshowman--yours for a dime.\"\n\nThe Spieler took a long breath and then looked at his audience.\n\n\"How's that,\" he asked, \"how's that for a furnace talk?\"\n\n\"It's all right, Cap,\" said the Concert Manager; \"it'll bring 'em.\"\n\n\"Say,\" said the Boss Canvasman, \"how do you keep that voice of yours\nshoutin' all the time?\"\n\n\"Boozin', boozin' up,\" said the Spieler, \"boozin' up.\"\n\n\n\n\nTHE BAND MASTER'S SOLO\n\n\nThe leader of the Big Show's band wasn't much on technique, but if\nthere were any notes coming to an E-flat cornet that he had overlooked\nsomething was wrong with the whole theory of music.\n\nThe way he could blow melody out of that piece of polished brass was\nsomething that the rest of the outfit never understood. He was a little\nfellow with a very small moustache that ran largely to waxed ends. He\nalways wore a blue uniform and a cap, and he looked like a messenger boy.\nThe twenty-eight \"star soloists\" that he directed possessed more wind\nthan a Western cyclone and an Eastern typhoon blown into one, for twice a\nday they played from one-thirty until the last race in the hippodrome was\noff, and this was no five finger exercise.\n\nThe gang was rather talkative when the leader came across from the band\nstand, so he sat down on the corner of the elevated stage and hummed to\nhimself. Presently the chorus cut out and he soloed thusly:\n\n\"I ain't no Sousa, boys, an' there ain't no brass hangin' to my pea\njacket, but say, if there's any leader that can get more noise out of\nthem 28 than I can, I'll eat every bit of sawdust under the tent an' say\nthankee when I'm done.\"\n\nNobody disputed this distinction and the leader continued to cadenza:\n\n\"It ain't no snap tossin' off melody for a show like this. When I'm out\nwith the minstrels in the winter the game's easy, but the snap is nothin'\nbut blow, an' you've got a lot of crazy ones in the ring here to take\ncues from. An' talkin' about them 28 of mine, there ain't no show band in\nthe country that can beat 'em switchin'.\"\n\n\"Say, you know the night we opened in the Garden? Well, we was playin'\n'The Holy City' for the guy in spangles what rolls hissef up the spiral.\nThe music plot was 'Holy City' to the top, a little of the shiver while\nhe was makin' the last turn, an' then a lot of brass an' bing-bing when\nhe makes the rush to the ring. Well, the boys were playin' the 'Holy\nCity' fine and daisy when the equestrian director comes across the track\nan' whispers:\n\n\"'Here's Dewey comin' up by the reserved section.'\n\n\"So I knows he wants somethin' appropriate, an' I gives the signal for\n'Here Comes a Sailor.' Well, them twenty-eight switches like a limited\non a clear track an' the crowd on the boards goes wild. But the guy in\nthe tin ball, he's been kneelin' it up to 'The Holy City,' an' when the\nmusic changes to swift he can't work his knees fas' enough an' he lets go\nan' nearly breaks his back. He calls me a Dutch somethin', I didn't jes'\ncatch, an' it costs him 25 fine off the pay sheet.\n\n\"An' speakin' about noise, fifteen year ago I leaves home, where I was\nworkin' in a harness factory and leadin' the Silver Cornet Band in the\nevenin', an' goes on the road with a medicine show. We has one of them\nlong-haired boys doin' the fake dentist an' pullin' teeth without pain\nwhile his wife does the female doctor an' sells pills. We six brass has\nto play when Doc an' his wife is workin', an' in the mornin' go back of\nthe stage an' roll pills an' put 'em in fancy boxes what Doc sells with\nthe packages of Australian gold pens, the little joker transparent cards\nan' the South American Cyroola Corn Cure what he gives away to each an'\nevery purchaser of Dr. Sorino's Death Delayin' Pellets.\n\n\"Well, the game was to git some <DW53> in the crowd to come up on the stage\nan' have his tooth pulled for nothin' an' without pain. Doc gets the moke\nin the chair an' makes his spiel 'bout the great pain killer he has an'\nsays it won't hurt the boy on the velvet. The band was all brass except\nCooney Watson, who was playin' a kettle drum an' workin' the bass and\ncymbals with a pedal. While Doc was gittin' the forceps on the tooth we\nplayed soft an' quiet like an' as soon as he gives the jerk we lets loose\nwith a march an' you can't hear <DW65> man holler to save your life. It\nwas great, an' it worked the countries all the times. Cooney would make\nyou think there was a thunder storm comin' up the way he beat them drums.\n\n\"But poor Cooney. Doc picks up six Indians to make the show stronger an'\nintroduce his famous Indian bitters. The red boys had a new moon an' asks\nCooney to loan 'em the drum to do the Tom Tom. Cooney says no an' the\nBig Chief gets good an' sore, but says nothin'. The next day we has a\nparade an' we brass is on top of a wagon with Doc's ads, painted on the\nside. The Indians is ridin' along behind us. Well, say, we had hardly\nhit the main street when the Indian what was sore on the drummer throws\nhis lariat and lassos poor Cooney off the wagon, drums an' all, into the\nmiddle of a bunch of cows what was gettin' weighed. He was pretty bad, so\nwe shipped him home.\"\n\n\"That ain't Cooney beatin' the drum with us, is it?\" asks the Boss\nCanvasman as he tied a long running knot in the guy rope to the net under\nthe swings for the brother act.\n\n\"No indeed,\" says the Leader, \"Cooney never joins out again. The las' I\nseen of him he was workin' at his trade out in Indianny--he was paintin'\nthe roof of the courthouse when we had the parade.\"\n\n\n\n\nTHE CANDY BUTCHER TALKS ABOUT A LOVE AFFAIR AND HIS ENCOUNTER WITH THE\nBUCKWHEAT MAN.\n\n\nThe Candy Butcher of the Big Show looked like a cut-out in a Sunday\nsupplement. He was the best dressed man in the outfit, and no matter what\nhe was doing and where he was doing it he always looked fixed up, and he\nfelt it.\n\nHis pants were always creased whether the show was doing a run in the\nlarge city or playing the one-nighters on a single-track jerk-water\nbeyond the Wabash. He never wore his coat when working, and his loud\nlinen would have stopped a limited with one flash from the tower. He\nwas there with the pink underwear, and his stockings had more kinds of\ncolor in them than the side of the band wagon when the season was new.\nThe Candy Butcher was always dressed, and when he got behind the counter\nto pull off the \"Five tonight, good people!\" gag he would have made\nthe window of an East Side gents' furnishing store drop the curtain.\nThe Candy Butcher didn't mix in much with the men in the outfit. He had\na chemical moustache that he zephyred with a velvet voice, and he was\nalways aces with the ladies. When Section One was pulling out for a long\nSunday jump, be sure of Him for the day coach with the girls. He was good\nat that, and, while he didn't always make a landing, he managed generally\nto get his bowline fast to the pier before the current caught him.\n\nHe always wore his coat in the meal tent, but he took it off right after\nsupper and carried it on his arm. The make-up didn't miss the ulster\nmuch, for he had on a vest that was three strikes and out for rainbow\ncolors--one of those rum omelette tinted things that a Philadelphia\nbutton buyer puts on for Saturday night when he's waiting at the stage\ndoor for some spotlight Sadie. He was there with the cheap tailors, all\nright.\n\nThe squatters on the ring bank were just settling for the afternoon gab\nwhile the equestrian director, sore because he couldn't get away to keep\na date, was rearranging the horse acts with a piece of a pencil on the\nback of the night's card. The Candy Butcher came through a crevice in\nthe tent and stopped to talk to the Saw Dust Spreader, who was standing\nbehind the wardrobe basket pulling on his plush pants.\n\n\"What you dressin' for?\" said the Candy Butcher.\n\n\"Oh, they're gettin' cheap,\" said the Saw Dust Spreader. \"I've got to\ndouble for an object holder, an' I'm up for the leaps right after the\nentree.\"\n\n\"What do you care,\" said the Candy Butcher, \"long as peppermint is\nstriped?\"\n\nThen he laughed at his own little trade journal joke. He was full of\nthose. He was always reading song books and joke budgets when waiting to\nget up on the blue boards to sell tickets for the concert after the show.\nHe came across the track and joined the gang.\n\n\"Gee!\" said the Side Show Spieler, who was always good on the opening\nline, \"youse dressed up for fair tonight! Looks like youse goin' to a\nbirthday party.\"\n\n\"Not for me,\" replied the Candy Butcher, as he put another piece of gum\nunder the mustache. \"Cut out the parties for Willie in the Summer an'\nafter this in the winter no more front parlor talks for me after ma is\nin bed an' the old man is out in the cold switchin' down in the railroad\nyard.\"\n\n\"Sore again,\" said the Concert Manager; \"you've always got your kick\ncomin' on sumthin'.\"\n\n\"Well, I wouldn't,\" said the Candy Butcher, \"but you fellows is allus\nrunnin' me in when it comes to any girl talk. Say, I ain't the masher of\nthis outfit, take it from me. If I could do the look killin' an' have 'em\nrunnin' me like the boy what does the principal bareback--the popcorn and\nthe soft drinks--well, no more for Jamesie.\"\n\nThe gang sort of warmed up to this talk, so he let her out another notch\nand began.\n\n\"Say,\" he continued, \"youse fellows is allus lookin' for sumthin' soft.\nWell, say, I've got a game for this season that will kill 'em. No more\nkicks over the lemonade tub for me. I've got 'em all skinned on the sour\njuice game. Say, you may not know it, but it's a losin' game when you'r\nrunnin' to one-nighters and sellin' lemon juice. It looks good to see me\nhollerin' over the chunk of ice an' takin' in the half dimes while I'm\npassin' out the cold drink an' the peanuts; but, say, you never thought\nhow many of them nicks it takes to buy a box of lemons. All right, all\nright in the city, bo', for the yellow boys, but when you are run out in\nthe meadows it's diff'rent, diff'rent.\n\n\"So, says I, during the winter when I'm managin' me penny arcade wid the\ntalkin' machines, says, 'll get up a scheme that will make the lemons\nback to the shady groves for you, no more fore me. An', say, I gits me\nthink tank on it and I invents sumthin'. Say, you'll laugh, but I pulled\nit off this afternoon, an', say, they all fell for once.\"\n\n\"What is it, Bill?\" asked the Old Grafter, always ready for a new shot at\nthe purse.\n\n\"'What is it?' Well, say! You know how them guys is stringin' you about\nfake lemonade, 'cause you ain't got no yellow slices floatin' on the top\nof the tub. Well, what is you goin' to do when lemons is 45 per, an'\neven the barkeeps is usin' the acid for the sour. Well, me, I just has\na dozen lemon slices made out of celluloid, an', say, Bill, you can't\ntell 'em from the real, 'pon my word, boy, when they is floatin' aroun'\nin the tub after I has poured in me water, me citric acid and me sugar.\nWhy, say, it looks like one of them things at a Fresh Air give-out, where\neverything is dun on the level, 'cause the reporters is watchin'. I jes'\nworks me celluloid slices on the stan', an' when biz is dun wipes 'em\noff, puts 'em in the box, an' theyse jes' as good as new, an' there's\nmore comin' to the Dime Savin' for Will, an' that's no song book wit.\"\n\nThe crowd eyed him in silence with that awe that meets a pack of\nundergraduates when they first gaze upon the man who discovered some new\nchemical analysis they never expect to understand.\n\nThe Saw Dust Spreader joined the crowd just then looking like a cheap\nleading man in a ten and twenty \"Carmen,\" with the red pants and the\nlittle coat. It was he for gossip, so he broke in.\n\n\"Yes, you're pretty good,\" he said to the Candy Butcher, remembering the\nlaugh he got when he came across the tan bark. \"But, say, where was you\nall last week? No lyin' now, Willie, 'cause I'm on, dead on.\"\n\n\"Well, I dunno that it's any secret,\" said the Candy Butcher. \"I dun me\nduty an' I suffered for it.\"\n\nThe gang looked like a listening party, so he began to reel:\n\n\"Say, it's pretty tough when a fellow starts out to do the right thing by\na little lady and gets the flag. It jes' shows that whenever you gits to\ndreamin' good somebody is goin' to give you an alarm clock finish an' let\nyou wake up with a shriek. Remember two seasons ago, when we was workin'\nthe Congress of Nations gag in the manager's tent? Well, me it is who\nmeets a little lady who is doin' the bead stitchin' in the gypsy village.\nShe's a pretty little thing an' quietlike. Well, she seems lonesome like\nan' one wet night I carries her across the lot when the mud is up to your\nknees. She seems to like it an' we has a long talk in the car.\n\n\"It seems that she used to work in a bean place where she is called\nNumber 8. I thinks that is funny, so I allus calls her Number 8.\n\n\"Seems like her folks was sore on her for troopin', an' she comes to me\nfor sympathy. Well, she had me stoppin' the booze before we was two weeks\nout, an' I was gettin' quiet in me gab and cuttin' down on the swear\ntalk. She tells me the way she gets into the circus is that a big guy\nwhat was engagin' people for the Congress used to eat his butter cakes at\nher table, an' he keeps on tellin' her what a fine life it is to be an\nactress, an', as he has been readin' about it in books, she throws up the\nwaitin' job and joins out. The big guy gets half of her first week for\nthe gettin' her the job.\n\n\"But it seems that when Number 8 pulls out of the bean place that she\nbreaks the heart of the guy with the white cap that cooks the buckwheats\nin the window. He's been sweet on her from the first day she hollered\nthe hot cakes an' he pulls 'em off the griddle an' looks into her eyes.\nWell, this guy takes a solemn oath that he'll kill the bloke that makes\nhis Mamie give up waitin' an' go troopin'. He never gets next to the big\njay what gave her the job, but when he was doin' the big city he sees me\nchasin' her home every night. He gives me a look I don't like, an' I asks\nNumber 8 what it means.\n\n\"'Oh, don't mind him' she says, 'he used to belong to my euchre.'\n\n\"Now, if I'd been wise I'd a-known Number 8 was connin', for what did\nthat little fairy know about euchre parties? I knows now that she was\npullin' off some speech she heard in the theatre where the lady shoots\nthe dook across the card table because he brings the coachman into the\nparlor to get a drink while the other dook is sayin' his good-by speech.\nI is too sweet on the fairy then to know that she's connin' me.\n\n\"Well, jes' the last week we was playin' New York, who does I meet in the\npark lookin' at the fish but Number 8. She looks so sweet an' nice that\nit all comes back, an' me up an' speaks. She says sorter haughty:\n\n\"'Who are you, sir? Whom are you addressin'?\n\n\"Say, that hit me like a blast, when all of a sudden I get a welt across\nthe head wid sumthin' that is iron. Say, I falls, but is up, and though\nthe knock has given me the blood, I see that it's the guy what cooks the\nbuckwheats in the window who was tryin' to do the killin'. An' say, he\nhas run out of his bean place an' hit me wid the cake turner. I grasps\nhim, an' it's catch-as-catch-can, an' Number 8 screamin' on the bench. I\ngives him a couple of good ones when in a jiffy it's rainin' plates an'\ncoffee cups, an' I'm gettin, 'em on the face. Say, his whole gang from\nthe bean shop was out in their white coats, throwin' the crockery an' me\ngettin' it.\n\n\"I knows the show is shut up an' help is a long way off. Somebody yells,\n'Lick the brute,' an' I gets another pie plate in the eye.\n\n\"'What's he done?' says a cab driver.\n\n\"'He insulted me wife,' said the buckwheat cake man.\n\n\"I tried to explain, but he gives me another one with the cake-turner an'\nI'm on the asphalt.\n\n\"I'se gettin' it good, an' I sees I must get help or cut the season for\nthe city ward. So I yells 'Hey, Rube!'\n\n\"Well, what do you think? There was a couple of old tramps a-sleepin' on\na bench, an' when they hears me scream, me on me back wid the buckwheat\nman sittin' on me, I sees 'em move, I yells it again, an' one of them\nwearies says:\n\n\"'That sounds familiar-like to me.'\n\n\"'Hey, Rube!' I gives it again, an', say, they gets in, an' they put that\nwaiter gang into a pile that looks like a hash brown in a spill.\"\n\n\"Well, what happened to you?\" said the Canvasman, who was always there\nfor the battle tales.\n\n\"Me, say, I gets it all. A couple of dinnys pulls me to a box, an' in the\ncell all night for me. An', say, if it hadn't been for James A., Lord\nbless his soul, me to the island for winter quarters an' in stripes.\"\n\n\"But what becomes of the fairy?\" says the Saw Dust Spreader, who always\nlikes to know the finish.\n\n\"What becomes of her?\" says the Candy Butcher, feeling a couple of scars.\n\"I hears it all later. She had married the guy an' moves over to Jersey.\nHe's keepin' a saloon, an' she's cookin' the oysters. He gives one with\nevery drink.\"\n\n\n\n\nTHE CONCERT MANAGER GETS REMINISCENT.\n\n\nThe manager of the concert company looked like a Methodist minister\ngoing to see the bishop. He wore a high silk hat and a last-winter's\ndouble-breasted coat. Whenever he talked he held the hat in one hand and\nrubbed it with the other so that it looked like a clipped yearling at a\ncountry run off. His voice was a deep bass, and when he did the \"remember\ngood people the Big Show is not yet 'arf over\" from the elevated stage\nyou could gamble on the words hitting every ear.\n\nHe was sitting on the ring bank looking over his touch list when the\nconversation grew wavy and then dropped to a hush.\n\n\"This here tent business is pullin' down to a theatrical man,\" he said\nas he lighted a choked stogie. \"Give me the hall show every time for the\ncush and comfort an' I'll be easy an' shippin' in money to the backer\nif the bookin's is good. The rep shows is the thing me boys if you can\ndeliver, an' you can strike territory that ain't been ploughed to death\nby a lot of yellows.\n\n\"Had out a rep company that was a winner. We was playin' 'East Lynne' and\ndoin' it good with six people and a band on the balcony at 7 to 8. The\nway we threw them dramatic chunks into the ten, twenty and thirts was\nsumthin' remark'ble. We wasn't connin' neither, but givin' 'em a show\nthat had 'em weepin' from ring up to las' curt'n. Say, I had a leadin'\nlady that was the genuine. She had been up three times before the school\ncommissioners for declaimin' an' her old man thought she was a Mary\nAnderson. We joshed him along on the Mary Anderson gag an' the old guy\nchecked in with a five hundred for a starter to get the fit up and the\ngal's costumes. Say, she was a blonde with a figure that set the town\nhall tonight people on the road to ruin with all brakes off. The leadin'\nman was a cuff juggler and he wouldn't settle down, but he doubled in\nprops an' was all right. The heavy was one of those chesty boys who was\nalles givin' me the jab 'when I was with Booth.' He started out all\nright, all right in the first act, but he died out before the curt'n got\ndown; the old man was pretty rotten, thank you, but the way he could play\nan E flat cornet on the balcony was sumthin' strictly proper. I'm jes'\ntellin' youse what you can do with a lot of bum players, if you've got\nthe goods, an' youse gets the bookin's. I was workin' the crowd on a $300\nsalary an' playin' up into the gross on $750 a week an' livin' like the\nman what owns his lay out. But I let go.\n\n\"You see some of the managers down on the coal oil circuit in Central\nPennsylvania got the vaudeville bug and was yellin' for specialties.\nSo I gets the soubrette to do a rag time stunt between the second an'\nthird, an' the first night the gal'ry window jumps nine and a half to\nthe good. I says that's what they wants an' I keeps the specialty in for\ngood. But the Lady Isabel of the push was getting artistic an' she says\nno to the specialty. I says yes, an' her old man comes on an' says Mary\nAnderson didn't have no gal singin' and showin' her legs in her show, so\nme an' the old guy plays quits. Well, it was gettin' warm, so I picks\nup me little soubrette, gets a privilege at a fair an' starts in to do\nthe black tent. We had a little round top, blacker on the inside than a\nBow'ry alley. The game was to get the yaps inside, all lights out, flash\nthe calcium, an' then do the floatin' illusion. The little gal would\nfloat roun' the tent an' hand me out roses, and the gang would go daffy.\nYou see she was rigged up in one of these white gowns an' was chasin'\nround in a back flap stickin' her head and body through wherever I had a\nslit. But I has a good lime light man an' the payin's never coupled to\nthe con.\n\n\"It was good for thirty a day and the privilege was cheap, but say, the\nfinish was tragic. You see the gal had run off from home, where she was\nmakin' three dollars spinnin' yarn in a mill an' payin' her people two\nfifty board. She gets stuck on the show business an' goes out with a rep,\nwhere I picks her up. Well, it seems that her old man gets sort o' dippy\n'cause he didn' do the right thing by the little one an' started out to\nfin' her. Somebody tells the old boy she is dead an' he falls down for a\nwhile. But he gets up and goes wanderin' 'bout to all the shows lookin'\nfor the gal. Well, he gets into my show one day an' when we flashes the\nillusion there's a yell an' the old one says, 'me daughter, me daughter,'\nand the gal flops an' breaks up the show. She gets sorry an' goes home\nwith the gray hair an' I loses the graft and strikes this.\"\n\nThe Boss Canvasman started in to do a little cussin' because the round\ntop over the stage was sagging and he broke up the talk.\n\nBut the Press Agent wants the finish of the yarn, and he speaks up:\n\n\"Well, Pop, what became of the gal?\"\n\n\"Oh,\" says Pop, \"the old man goes under the ground an' the jig stepper\ngoes back to the business. Last season she was doublin' with the iron\nchested man doin' a singin' specialty in the side show. But they's both\nout now. The iron chested man is yellin' the stations on the Ninth Avenue\nL, and the Mamie girl is makin' ten a week posin' for chromos that you\nwouldn't hang over the thermometer, s' help me.\"\n\n\n\n\nTHE HANDS AT THE WINDOW.\n\n\nThe man who sold the red paste-boards at the ticket wagon always dressed\nbetter than the man who owned the circus. He ran largely to striped vests\nand wore shirts that were of the sassiest shade of pink. Like the Candy\nButcher, he never wore his coat when he was working, but the vest and\nthe trousers made him a swell enough picture. The rest of the outfit\nregarded him with the same awe that fills the spirit of a man earning\ntwelve dollars a week who accidentally runs into J. Pierpont Morgan. To\nbe sure, the Ticket Seller never had as much cash as the Wall Street man,\nbut he made a bigger flash. He always carried a roll, and as his accounts\nnever failed to balance and he stood good with the Boss, it was generally\npresumed that the gold boys he carried in his upper vest pocket were his\nown.\n\nThe gang that talked on the ring bank between the afternoon and the night\nshow had great respect for this member of the outfit. He had a line of\ntalk that sounded big, and that was always inspiring to the crowd. He\nknew just what the Big Show was taking in, and he liked to talk about it.\nHe talked figures, and the ring bank crowd only knew it by the thousands\nof people that roosted on the blue boards when the bill was being run\noff under the main top. In the winter months, when the Big Show had gone\nto quarters, the Ticket Man managed a burlesque house in Chicago, and it\nwas said that he wore so many diamonds on the pink bosom that it wasn't\nnecessary to turn on the lobby lights when the night audience was coming\nin. He was a loud boy, all right, but he backed the noise with coin. When\nthe Ticket Man wasn't talking about money, he let loose a bunch of racing\ntalk that would have feazed the principal writer on a daily turf sheet.\nHe used words that seldom got farther than the paddock, and he picked\nwinners like a diamond expert getting out the real shines from a pile of\nglass.\n\nAnd with both lines of talk he had 'em all faded when it came to the\ncon. He had a line of explanations that would make any man think he\nthrew down a two-spot when he knew that he walked up to the window with a\ndouble X.\n\n\"Lemme tell you fellows sumthin'\" he said, as he came in after supper in\nthe meal tent and sat down for a smoke. \"You'se can always hear a lot of\nknockin' for the boys what sells in the wagon. Now take it from me, don't\nyouse believe a word of it till youse gets the other side. You hear a lot\nof hollerin' that ev'rybody is gettin' done, and that the boys in the\nwagon is buyin' gov'ment bonds and furnishing flats on the short change\ngraft. But it ain't so--not altogether. It was in the old days, when I\nstarted in the business, but it ain't now. Graftin' is dead, take it from\nme.\"\n\n\"Yes,\" said the Old Grafter, with a deep sigh, \"you're right, Bill,\ngraftin' is dead, an' it was a sorry day for me when it died.\"\n\n\"You're right, Jim; half way right, but it won't go these days. But\nwhat I'm tellin' you is right; when you is sellin' hard tickets to a\nlong line that is rushin' to get to the tent, youse can't go out and\nhave a personal interview with every man that runs away an' leaves his\nchange on the window. An' say, whose goin' to take chances givin' it\nup when he does come back? Not me, not me; I see me little bank-roll in\nthe Dime Savin's lookin' like a busted baloon on the end of the season.\nSay, don't make no mistake, ev'ry hand at the window is a hand ag'in'\nyou. I know what I'm talkin' about. There ain't a man in the country who\ndon't think he's doin' a smart trick when he beats a circus man out of\nmoney. They'll all do you when they can. Mark my word. I've been passing\nout hard tickets with this outfit for nine years an' long before that\nwith a fly-by-night, an' I knows the game--ev'ry hand that comes up to\nthat window will do you if it can. I've had too many Reubs hand me a\ntwo-dollar bill an' say it was a tenner.\"\n\n\"What do you do then?\" asked the Concert Manager.\n\n\"Do?\" said the Ticket Man. \"Do? Why, I'd just look Reuben in the eye an'\nsay Brush on, haysy, you ain't got a ten; wait till you sell your wheat\nan' come back an' then I'll gamble with you.\"\n\n\"It's all right now, dead square an' on the level, but youse know what\nit was in the graftin' days,\" he continued, as he wiped the dust off\nhis patent leathers with a horse plume. \"We turned the season them days\nwith a bunch of money that was all our own, and nobody kickin'. I was\nout with a little graftin' show that had more gamblin' on one lot than a\ncounty fair, an', say, I wasn't takin' no chances with them grafters an'\ntinhorns! I knowed the countries would buy their tickets before they went\nat the games, an' say, I wasn't overlookin' my bit.\n\n\"Me an' the fixer stood pat. He travelled three days ahead, an' had it\nall squared with the Mayor of the burg and the police so the games could\ngo on an' no kick. Well, he would get hold of a smart lookin' constable,\ntake him aroun' back of the courthouse and give him a talk. The constable\nlikes it bein' taken into the confidence of a showman, an' lets ev'rybody\nsee it, 'cause he's proud.\n\n\"Well, the fixer sez, 'You're a smart lookin' young man; you do me a turn\nwhen the show comes, an' if the Boss likes you, he may take you along for\nChief Detective!'\n\n\"Say, that hits the Reub so hard, that Chief Detective talk, and for\nthree days he's seein' visions of hisself flashin' his tin all along\nthe line. Well, he reports to me, and I gives him instructions and a\nten-dollar bill, with promise of a five after the show if he does his\nwork. I posts him in front of the window, and has him fixed to butt in\nwhen there's any kick, an' say, 'No arguments; keep movin', gentlemen!'\n\n\"An' he does it good, and I throws the short change, an' with the band a\ngoin' in the tent and the crowd crazy to get at the games, I picks out a\nlittle bag of coin before the tinhorns lands on the roll. An' me helper,\nwho was a bit of a mechanic, he takes the sill of the window in the wagon\nand tilts it with the slant to the inside. A guy comes along and throws\nup a dollar bill, and says give me one. With one hand I throw down the\nticket, an' with the other throw up two quarters, or a quarter two tens\nand a five. They hit the glass, an' say, with that slant some of it was\nsure to come back, an' Mr. Man is in too big a hurry for the Big Show\nto notice, an' the constable, who's to get the five, keeps 'em movin'.\nAt night the constable says he wants to have a private conversation an'\ntakes me off.\n\n\"'Look a here,' sez he, 'that agent ahead of the circus sez if I could do\nthe work I might be made chief detective.'\n\n\"Not a word, keep quiet, says I. You'll spoil it.\n\n\"The constable don't just get next, but it soun's good. 'See that man\nover there?' sez I. Constable looks aroun' an' sees the big boy that\nallus stood by the gate an' who wasn't no more of an officer than me.\n\n\"Well, sez I, that man has been watchin' you. He's Mr. Pinkerton himself,\nand as soon as he makes out his official report to Superintendent Byrnes\nyou gets the job. He thanks me an' I give him the fivespot. He goes home\nan' tells ev'rybody for the names I spring was so good he didn't see\nthrough the con. But, of course, I've got to give the fixer a bit. An'\nsay, that's the trouble with graftin'. You've got to fix too many people\nto keep from gettin' a holler. But none of it now. I stick to what I\nsays, that ev'ry hand that comes up to the window is a hand ag'in' you,\nand will do you if it can. You're goin' to get short change, all right,\nbut don't do no old-fashioned graftin'. I tell you there's only one way\nto get it honest, an' if the band is playin' while you're sellin', it\nwill keep your line movin' at the window, an' it's a poor evenin' you\ncan't pick up fifteen at least that was left and nobody hurt. Of course,\nI calls them back--but I never could holler loud.\"\n\nThe Ticket Man took out a solid gold watch, with a stop movement, and his\ninitials in diamonds on the back.\n\n\"Well, guess I better get at it,\" and he left and went to the ticket\nwagon, where on the inside he sat on a little stool with the tickets in\none hand and the change on the other side.\n\nThen the hands began to come up and he was at work again.\n\n\n\n\nTHE CONCERT MANAGER TELLS THE BOYS AN ELEPHANT YARN.\n\n\nThe concert manager looked like a corn doctor who had dressed up to go\nto the City Hall for his license. He always wore a long black coat, and\nwhen a healthy ray of sunshine hit it square in the back it threw off a\nglare like a locomotive reflector. He had a smooth face, wore his hair\nlong, and it was his proudest boast that once down in Texarkana he had\nbeen asked if he was William Jennings Bryan. As the question was put by a\nswarthy Democrat who wanted to buy \"sumthin',\" he said he was, and they\nboth liquored up the rest of the afternoon, and no discoveries.\n\nThe Concert Manager had sifted the English language down to about nine\nfive-syllable words, and the rest of his talk was only ordinary. His\nvoice was his big hit. He could stand on the elevated stage between Ring\nOne and Ring Two, and no matter what was going on hit every ear drum\nwith his gag about \"directly after the performance,\" \"you need not leave\nyour seats,\" and that \"the gentlemanly ushers would soon pass along\nselling tickets.\" His voice got him all of the announcing, and he had a\nshape that always cast him for some big guy with fake whiskers when they\nwere doing anything spectacular for the entree.\n\nHe loved to address the multitude. Let the gang sit down on the ring\nbank for a chat before the night show, and it was he to cut in with a\ntalk--something he knew all about and which had the rest of the boys\nwinging before he hit the end of the first chapter. He had more different\nkinds of talk than any man in the outfit, and he always yearned to be\nhanding it out. In the winter he ran a hall show with a company that\nwould take a fall out of anything from \"Hamlet\" to a melodrama with three\nbig effects and a wholesale killing in each act. The gang was telling\nhard luck stories when he came across the track, and he was ready with\none at the first cue.\n\n\"'Say, boys,' he said with one of those back platform gestures, 'I'll\ntell you the star hard luck story--say, the best, and no appeal. Me, a\nlong time ago playing the little towns in Ohio with a merry-go-round.\nThink of it, yes, but don't laugh, for it was a good game in them days\nand I was turnin' coin. We put up on a lot for four days for a county\nfair, an' business is so good that I stops sleepin' back of the tent an'\ngoes to the inn. Well, say, the barkeeper of that tavern was a pass fiend\nfor fair, an' he keeps strikin' me for free rides on the wooden horses. I\nlaughs him off, but he keeps at it. So one day I gets mad an' throws him\nhard. He looks at me over the black bottle an' sez: 'Say, you give a free\nride ev'ry time they gets the brass ring, don't ye?'\n\n\"I says 'yes,' an' he shuts up like a law abider, an' that ends it. But\nwhat do you suppose that glass rubber does? Say, he comes out to the\nlot with his gang, an' while he wasn't lookin' he pours a hole bottle\nof gildin' fluid into the ring slide, an' say, everybody on the lot was\ngettin' gold rings an' ridin' free all night. It hit me so hard I had\nto fold up and to the wagons an' do the sneak without settlin' me last\npayment on the privilege to the fair folks.\"\n\nThis seemed to please the ring bank crowd and the Concert Manager was\nfull of loose gab. Finally the talk drifted to animals, and then to\nelephants, and up speaks the Concert Manager.\n\n\"I never comes through the animal tent and sees the elephant herd,\" he\nsays, \"but it has me rememberin' sumthin' that happens when I'm with the\nJohn Robinson show doin' the talkin' from the ring an' makin' me extra\nwith a couple of blackface turns, a singin' act an' a knockabout after\nthe Big Show.\n\n\"Say, for real heart them elephants has got all humanity beat to a\nstandstill. Seems to me that the worst weakness that is in the breast of\nman is the feelin' that pulls him away from another man when they ought\nto be stickin' together. Say, it ain't many a man that remembers a good\nturn, and kind acts is forgotten as quick as money that's owin'. But\nthem big boys with the trunks, say, they ain't forgettin' nothin'. Youse\ncan do a good turn for a man an' he'll throw you the next day, but the\nelephant ain't forgettin' the one what has done him sumthin' good, be he\nman or beast.\n\n\"Well, in them days we was carryin' six big fellows, an' part of my game\nwas to let their backs out to the storekeepers for banners. I'd lay it\nout with some drygoods dealer that was enterprisin' enough to do some\nreal sensational advertisin'. Then we'd have the banners painted, swing\n'em over the backs, an' Mr. Mann has a good showin' when we makes the\nparade down the street. I was payin' the trainer a little extry to see\nthat the banners got on, and, say, that boy for liquor was the original\nreservoir. It seems that the money I'm givin' him allus buys a drink. He\nkeeps it up right along an' gits many a warnin'. There wasn't a guy with\nthe outfit that could handle the beasts like he could, an' the Boss was\nallus afeerd sumthin' would happen.\n\n\"And it did by and by. The trainer don't pay attention to the warnin',\nand one matinee he queers the act. He gets his elephants all mixed up\nin his ring, an' there comes near bein' a breakaway but for a lot of\nspearin'. The Boss sees it all, an' he gets the trainer out without any\ntalk. The boys gets the elephants back to their stable, but ev'ry one is\nlooking for trouble.\n\n\"Well, say, the Big Show is started an' about halfway through when the\nBoss Canvasman is ready to strike the animal tent an' make for the cars.\nThe horses is put to the dens, an' they is all soon on the road, with the\nwalkin' stock followin'. Then the boys gets to the elephants an' tries\nto start 'em, but not a one will move. They gives them the spear, but\nnot a move. They gets to hollerin' an' more spearin', but they might as\nwell tried to move mountains. The Boss come in, an' he tries it, but no\ngo. They moves a lot of hay down the lot, thinkin' it will make 'em look\nfor another feedin', but no go. Then one of the trainers what had been\nhelpin' the head trainer stan's out an' calls 'em by name. They flaps\ntheir big ears, but not a move. It's time for the section to be pullin'\nout, an' we all hands is up against it.\n\n\"Well, I sez to myself, here's your chance to be smart an' redeem it all\nfor Jed,--the trainer what was fired for drinkin'. I hustles over to the\ntavern an' finds Jed puttin' away the juice like he was loadin' a train.\nI gives him a long talk an' tells him about the elephants. He calls 'em\nhis darlings an' wants to buy more drink. I begs him to come over an'\nhelp us out, but he's sore on the Boss an' won't make a move. After\nawhile I gets him to thinkin', an' he sez he'll do it. Jed was pretty\nfull an' was staggerin', but I gets him to the lot where ev'ry animal man\nin the outfit is a spearin' them beasts, while the Boss is cussin', an'\nthe whole town roostin' on the sidewalk an' givin' us the laugh. I gets\nJed in an' right in front of the Boss.\n\n\"The Boss is still sore, and sez, 'Take that loafer away.'\n\n\"'Wait,' sez I.\n\n\"Then afore there could be any argument Jed staggers out in front of the\nherd, and, leaning heavy-like on the bale of hay, begins callin':\n\n\"'King,' he yells.\n\n\"The biggest bull in the herd lifts his ear, then holds up his trunk an'\ngives a shriek that scares the crowd white.\n\n\"'Stop that spearin'!' sez Jed, an' every animal man gits out of the way.\n\n\"'King!' yells Jed again, an' with that he walks over an' touches the\nelephant on the trunk. The big bull wheels around and starts away.\n\n\"'Sam, Dick, Boss, Frank!' keeps yellin' Jed, an' the whole herd is\nturnin' an' takin' the road to the depot.\n\n\"The crowd on the lot is cheerin', an' the animal men walk along by the\nbeasts, but there's no more spearin'. Jed jes' leans against the hay an'\nkeeps callin'. He finishes, an' as the last elephant comes along he loops\ndown with his trunk, an' pickin' Jed up sits him on his trunk an' walks\naway proud like jes, as he does in Ring Three at the finish of the act.\n\n\"Say, the Boss is almost wild about it, an' the whole town follows the\nherd to the train. The Boss had Jed put in the sleeper, but Jed won't\nhave it, and crawls out to sleep in the elephant car, where he was most\nat home.\n\n\"Say, that's what I call havin' a heart. Them elephants knew that voice.\nJed had been leadin' 'em up hill and down hill, from sea to sea, an' they\nwasn't goin' to have it any other way.\"\n\n\"Well, did they take him back for good?\" asked the Saw Dust Spreader.\n\n\"Take him back,\" said the Concert Manager. \"Well, I should say, an' the\nBoss gives him a gold watch with elephant engravin' on it, with a fancy\nband with some wordin' about bein' a faithful servant.\n\n\"An', say, Jed was sober after that durin' the season, but when we got\ninto quarters the tavern for Jed all the time.\"\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Red Wagon Stories, by Wells Hawks\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# Some Tame Gazelle\n\n## A Novel\n\n### Barbara Pym\n\n# Contents\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\n\nChapter Sixteen\n\nChapter Seventeen\n\nChapter Eighteen\n\nChapter Nineteen\n\nChapter Twenty\n\nChapter Twenty-One\n\nChapter Twenty-Two\nSome tame gazelle, or some gentle dove:\n\nSomething to love, oh, something to love!\n\n\u2014THOMAS HAYNES BAYLY\n\n# Chapter One\n\nTHE NEW CURATE SEEMED quite a nice young man, but what a pity it was that his combinations showed, tucked carelessly into his socks, when he sat down. Belinda had noticed it when they had met him for the first time at the vicarage last week and had felt quite embarrassed. Perhaps Harriet could say something to him about it. Her blunt jolly manner could carry off these little awkwardnesses much better than Belinda's timidity. Of course he might think it none of their business, as indeed it was not, but Belinda rather doubted whether he thought at all, if one were to judge by the quality of his first sermon.\n\n'If only we could get back some of the fervour and eloquence of the seventeenth century in the pulpit today,' she had said to her sister Harriet, a plump elegant spinster in the middle fifties.\n\n'Oh, we don't want that kind of thing _here'_ Harriet had said in her downright way, for she had long ago given up all intellectual pursuits, while Belinda, who had never been considered the clever one, still retained some smattering of the culture acquired in her college days. Even now a light would shine in her mild greenish eyes, so decorously hidden behind horn-rimmed spectacles, at the mention of Young's _Night Thoughts_ or the dear Earl of Rochester's _Poems on Several Occasions._\n\nNeither she nor Harriet had ever married, but Harriet was making her usual fuss over the new curate and was obviously prepared to be quite as silly over him as she had been over his predecessors. She was especially given to cherishing _young_ clergymen, and her frequent excursions to the curates' lodgings had often given rise to talk, for people did like a bit of gossip, especially about a respectable spinster and church worker like Miss Harriet Bede. There was naturally nothing scandalous about these visits, as she always took with her a newly baked cake, some fresh eggs or fruit\u2014for the poor young men always looked half starved\u2014or even a hand-knitted pullover or pair of socks, begun by her in a burst of enthusiasm and usually finished, more soberly, by Belinda. And then of course she would ask them to supper.\n\nWas it tonight he was coming? Belinda wondered vaguely. It must be tonight, she decided, catching sight of a bowl of exceptionally fine pears on the little table by the window, and expensive bought chrysanthemums in the vases when there were perfectly good Michaelmas daisies in the garden. Dear Harriet, she wasn't really extravagant, only rather too lavish in her hospitality. The Reverend Edgar Donne was surely a simple young man and would not expect much. Naturally one did not think of the clergy as expecting anything in the way of material luxuries... Belinda paused, for she was remembering the vicar, Archdeacon Hoccleve, and how one couldn't really say that about him. But then dear Henry was different, in some ways not like a clergyman at all. For although Belinda had loved him faithfully for over thirty years, she sometimes had to admit that he had very few of the obvious virtues that one somehow expected of one's parish priest. His letter in this month's parish magazine, announcing the arrival of the new curate, had a peevish and condescending tone that a stranger might have thought not quite the thing for an archdeacon. But the village was used to it.\n\n'The Reverend Edgar Donne\u2014the name is of course pronounced _Dunne\u2014_ will be with us by the time you read these words', he wrote. 'Nobody will be more glad to welcome him than I myself, for whom these last few weeks have been more trying than any of you can possibly imagine. Without a curate it has been impossible for me to take the holiday I so badly need and I have been forced to cancel some of the services because I have not felt equal to taking them, as the ready help I looked for from fellow priests in neighbouring parishes has not been forthcoming....'\n\nOf course that was a dig at the Reverend Edward Plowman, who disliked the Archdeacon so much, and as he had quarrelled with Canon Glover what could he expect? thought Belinda, almost wishing that she were Deaconess Bede and could enter the pulpit herself. But even a deaconess was not permitted to celebrate Holy Communion\u2014it was of course the _early_ services which had been cancelled\u2014whereas in the Nonconformist churches, she believed, women ministers had equal status with men....\n\n'B' _linda!'_ Harriet's impatient voice interrupted her thoughts, 'it's nearly seven and Mr. Donne will soon be here.' Harriet appeared in the doorway, wearing only a celanese vest and knickers, as if her actual presence in the room would make Belinda realize more fully how late it was.\n\n'Why, Harriet, the curtains aren't drawn,' exclaimed Belinda in an agitated tone. 'Anybody might see into the room! And you know I never take as long to get ready as you do.'\n\n'All the same Mr. Donne will probably be punctual,' said Harriet, 'and it would be terrible if neither of us was ready. I've borrowed your lace scarf, as I must have something to cover up the neck of my green frock. Perhaps it would have been better if I hadn't tried to alter it to a Vee.'\n\n'Yes, dear.' Belinda spoke rather absently, for by now she was occupied with the problem of what _she_ should wear. She hoped that Harriet had not also borrowed her black velvet bridge coat, as she wanted it herself on these late September evenings. But then Harriet was probably too stout for it, although she liked her clothes to fit tightly and always wore an elastic roll-on corset.\n\nIn her room Belinda took out her blue marocain, a rather dim dress of the kind known as 'semi-evening'. Quite good enough for the curate, she decided, though if the Archdeacon had been coming as well she would probably have worn her velvet. She did hope that Harriet wouldn't put on a lot of lipstick, it was so unsuitable....\n\nAt that moment there was a ring at the bell and an agitated call from Harriet.\n\n'Belinda, you go! I haven't finished doing my hair.'\n\n'But surely Emily will go?' said Belinda. She was wondering whether to wear her little seed-pearl brooch or not.\n\n'No, Emily can't go. She's putting the sauce on the chicken.'\n\nBelinda hurried downstairs without the little brooch. She felt flustered and incomplete.\n\nThe figure on the doorstep might have been any of the other curates, except that Mr. Donne favoured a rather unfashionably high clerical collar. He doesn't remember me, thought Belinda, as she replied to his rather puzzled greeting.\n\n'This _is_ Miss Bede's house?' he asked, hesitating on the threshold.\n\n'Yes, I am Miss Bede,' said Belinda with simple dignity, 'but I expect you know my sister better.'\n\n'Ah, you must be Miss _Belinda_ Bede,' he announced, triumphant at having placed her. 'I've heard a lot about you from the Archdeacon.'\n\n'Oh, really? What did he say?' Belinda tried not to sound too coy and eager.\n\n'He\u2014er\u2014said you did a lot of good work in the parish,' replied the curate primly.\n\n'Oh...' Belinda could not help feeling disappointed. It made her sound almost unpleasant. If that was what he had really said, of course. It didn't sound at all like the Archdeacon, who never said the sort of things clergymen ought to say. It was so odd to think of him as being a clergyman at all... Belinda's thoughts slipped back to her college days when they had been students together. _Most_ odd... and yet there was no sadness or bitterness in her mind as she thought of him. It was obvious that poor Agatha had a very difficult time with him, although by her scheming she had made him an archdeacon. Their cook had told the Bedes' Emily who had told Harriet that the Archdeacon was very difficult to get up in the mornings, and of course one knew that he always made his curates do the early services which was really rather slack, because it wasn't as if he were very old or weak in health. And yet he had such charm, even now....\n\nThe curate coughed nervously and ventured a remark about the weather.\n\n'Yes, I love September,' agreed Belinda, guilty at having let her thoughts wander from her guest. 'Michaelmas daisies and blackberries and comforting things like fires in the evening again and knitting.'\n\n'Ah, knitting,' he smiled, and Belinda could see him glancing round the room as if he already expected to see the beginnings of a pullover for himself. But all that Belinda's cretonne work bag contained was a pink lacy-looking garment, a winter vest for herself. It was so annoying of Miss Jenner not to have any more 'Perliknit' left. She had had to buy a slightly thicker wool of a rather brighter pink to finish it off.\n\nFortunately at this moment, for the conversational going was heavy, a firm step was heard on the stairs and Harriet came into the room, radiant in flowered voile. Tropical flowers rioted over her plump body. The background was the green of the jungle, the blossoms were crimson and mauve, of an unknown species. Harriet was still attractive in a fat Teutonic way. She did not wear her pince-nez when curates came to supper.\n\nThe curate sprang up eagerly and seemed suddenly to lose some of his shyness.\n\n'Good evening, Mr. Donne,' said Harriet, 'I'm afraid I haven't my sister's punctual ways, but I'm sure she has been entertaining you better than I could have done. I had a classical education and it isn't a very good training for scintillating conversation.' She sat down rather heavily on the sofa beside him. 'Now we must not forget that the name is pronounced _Dunne,'_ she declared roguishly.\n\n'Well, actually, as a matter of fact...' the curate looked embarrassed, 'I don't pronounce it that way. I can't imagine why the Archdeacon thought I did.'\n\n'He was of course thinking of the seventeenth-century poet of that name,' said Belinda stoutly. The truth was, of course, that dear Henry could never resist a literary allusion and was delighted, in the way that children and scholars sometimes are, if it was one that the majority of his parishioners did not understand.\n\n'He will have to put a correction in the magazine next month,' chortled Harriet. 'I should like to see the Archdeacon having to climb down.'\n\n'It makes one feel quite odd to have one's name mispronounced or misspelt,' said Belinda evenly. 'Almost like a different person.'\n\n'Oh, yes,' agreed Harriet, 'like Gorringe's catalogue.'\n\nThe curate looked politely interested but puzzled.\n\n'You see,' Harriet explained, 'they once sent me a catalogue addressed to Miss _Bode,_ and somehow I'm so lazy that I never bothered to correct it. So now I have a dual personality. I always feel Miss Bode is my dowdy self, rather a frumpish old thing.'\n\n'She must certainly be most unlike Miss _Bede,'_ blurted out Mr. Donne with surprising gallantry.\n\nHarriet protested amid delighted giggles. Belinda felt rather left out and found her eyes fixed on the curate's combinations, which still showed. Surely it was much too warm for such garments, unless perhaps he wore them all the year round?\n\nDuring the short silence which followed, the tinkling of a cowbell was heard. The sisters had brought it back from a holiday in Switzerland and it was now used as a gong.\n\n'Ah, dinner,' said Harriet. 'Come, Mr. Donne, you shall take me in,' she added with mock solemnity.\n\nMr. Donne was quite equal to the occasion, for he had all the qualifications of a typical curate. Indeed, his maternal grandfather had been a bishop.\n\nIn the dining-room Harriet sat at one end of the table and Belinda at the other, with the curate in the middle. Harriet carved the boiled chicken smothered in white sauce very capably. She gave the curate all the best white meat.\n\nWere all new curates everywhere always given boiled chicken when they came to supper for the first time? Belinda wondered. It was certainly an established ritual at their house and it seemed somehow right for a new curate. The coldness, the whiteness, the muffling with sauce, perhaps even the sharpness added by the slices of lemon, there was something appropriate here, even if Belinda could not see exactly what it was.\n\n'I called at the vicarage on the way here,' said the curate. 'Mrs. Hoccleve very kindly promised me some apples.'\n\nHarriet looked rather annoyed. 'Their apples haven't done at all well this year,' she said, 'and I always think those red ones are rather tasteless. You must take some of our Cox's Oranges with you when you go.'\n\nThe curate murmured grateful thanks.\n\n'How is Mrs. Hoccleve's rheumatism?' asked Belinda.\n\n'Not very much better,' he replied. 'I hear she is going to Karlsbad in October. Apparently the waters there are very good.'\n\n'Nettles are an excellent thing, I believe,' said Harriet.\n\n'Indeed?' Mr. Donne looked so interested that he must have found it quite a strain. 'How should they be used?'\n\n'Oh, I don't really know,' Harriet beamed. 'Just nettles. Boiled, perhaps. People will try all sorts of odd remedies,' she added, with the complacency of one who is perfectly healthy.\n\n'Poor Agatha,' murmured Belinda, although she could not really feel very sympathetic.\n\nThere was a slight lapse in the conversation.\n\n'I hear you are a rowing man,' said Belinda, with what she felt was rather forced enthusiasm.\n\n'Oh, how splendid!' Harriet was of course delighted, as she would have been with any piece of information. 'I can just imagine you stroking an eight.'\n\n'Well, actually, I haven't done any for some time, but I used to be very keen.' The curate looked down at his chicken bone as if he would like to take it up in his fingers and gnaw it. He was not very well fed at his lodgings and the evening meal was particularly scrappy.\n\nHarriet picked up her bone and began to eat it in her fingers. She beamed on Mr. Donne and said brightly, 'Like Queen Victoria, you know, so much more sensible and convenient.'\n\nHe followed her example eagerly. Belinda looked on with some distaste. If only Harriet could see how foolish she looked. The white sauce was beginning to smear itself on her face.\n\n'I expect you are quite bewildered meeting so many new people,' she said, leading the conversation back into suitable channels.\n\n'Yes, in a way I am, but I find it fairly easy to remember them so far. I came across Miss Liversidge this afternoon in the village and have persuaded her to address a meeting of the Mothers' Union. She seems to have had a great many interesting experiences.'\n\nBelinda smiled. The idea of Edith Liversidge addressing the Mothers' Union amused her. One never knew what she might say to them and she would hardly set them a good example of tidiness. Dear Edith, she was always such a mess.\n\n'She's a kind of decayed gentlewoman,' said Harriet comfortably, helping the curate to trifle.\n\n'Oh _no,_ Harriet,' Belinda protested. Nobody could call Edith decayed and sometimes one almost forgot that she was a gentlewoman, with her cropped grey hair, her shabby clothes which weren't even the legendary 'good tweeds' of her kind and her blunt, almost rough, way of speaking. 'Miss Liversidge is really splendid,' she declared and then wondered why one always said that Edith was 'splendid'. It was probably because she hadn't very much money, was tough and wiry, dug vigorously in her garden and kept goats. Also, she had travelled abroad a good deal and had done some relief work after the 1914 war among refugees in the Balkans. Work of rather an unpleasant nature too, something to do with sanitation. Belinda hoped that Harriet wouldn't mention it in front of Mr. Donne. 'Of course she has made a home for poor Miss Aspinall, who's a kind of relation,' she said hastily. 'I always thinks it's very unselfish to have a comparative stranger to live with you when you've been used to living alone.'\n\n'Ah, well, we ought to share what we have with others,' said Mr. Donne with rather disagreeable unctuousness.\n\n'Oh, Mr. Donne, I can't imagine you sharing your home with Connie Aspinall,' Harriet burst out, 'she's so dreary.'\n\nMr. Donne smiled. 'Well, perhaps I didn't mean to be taken quite literally,' he said.\n\n'Now she's a decayed gentlewoman if you like,' said Harriet. 'She can talk of nothing but the days when she used to be companion to a lady in Belgrave Square who was a kind of relation of one of Queen Alexandra's Ladies-in-Waiting.'\n\n'She plays the harp very beautifully,' murmured Belinda weakly, for poor Connie was really rather uninteresting and it was hard to think of anything nice to say about her.\n\n'Let's have coffee in the drawing-room,' said Harriet rather grandly. At one time she had wanted to call it the lounge, but Belinda would not hear of it. She had finally won her point by reminding Harriet of how much their dear mother would have disliked it.\n\nIn the drawing-room they arranged themselves as before, Harriet on the sofa with the curate and Belinda in one of the armchairs. Belinda took out her knitting and went on doing it rather self-consciously. It was beginning to look so very much like an undergarment for herself. The curate's combinations must be 'Meridian', she thought. It was nice and warm for pyjamas, too, in fact Harriet herself wore it in the winter. The close fabric fitted her plump body like a woolly skin.\n\nWhile they were drinking their coffee, Harriet went to the little table by the window and took up the bowl of pears which Belinda had noticed earlier in the evening.\n\n'Now you must have a pear,' she insisted. 'Do you know, when we were children our mother used to say that we could never keep fruit on the sideboard.'\n\nBelinda would have liked to add that they couldn't now, and that it was only because they had been having the curate to supper that there had been anything more than a withered apple or orange in the bowl this evening. Harriet's appetite was just as rapacious in her fifties as it had been in her teens.\n\nThe curate helped himself to a pear and began to peel it. He seemed to be getting rather sticky and there was some giggling and interchange of large handkerchiefs between him and Harriet.\n\nBelinda went on quietly with her knitting. The evening promised to be just like so many other evenings when other curates had come to supper. There was something almost frightening and at the same time comforting about the sameness of it all. It was odd that Harriet should always have been so fond of curates. They were so immature and always made the same kind of conversation. Now the Archdeacon was altogether different. One never knew what he might say, except that it was certain to be something unexpected and provocative. Besides, it was really more suitable to lavish one's affection on somebody of a riper age, as it was obviously natural that one should lavish it on somebody. Indeed, one of Belinda's favourite quotations, taken from the works of a minor English poet, was\n\n_Some tame gazelle, or some gentle dove:_\n\n_Something to love, oh, something to love!_\n\nBelinda, having loved the Archdeacon when she was twenty and not having found anyone to replace him since, had naturally got into the habit of loving him, though with the years her passion had mellowed into a comfortable feeling, more like the cosiness of a winter evening by the fire than the uncertain rapture of a spring morning.\n\nHarriet's tittering laugh disturbed Belinda's quiet thoughts. 'Oh, Mr. Donne, I'm not quite as stupid as you think! I used to know some Latin. _Ah quotiens illum doluit properare Calypso,'_ she retorted, flinging at him triumphantly the last remnants of her classical education.\n\nCan she be hinting at me to go? he wondered, but then decided that she had probably long ago forgotten the meaning of the line. All the same it was getting late. He mustn't outstay his welcome and the elder Miss Bede had yawned once or twice, although she stifled it very politely.\n\nDespite protests from Harriet, they were soon in the hall and the curate was putting on his overcoat. Harriet was fussing round him like a motherly hen.\n\n'Why, of course, it's the garden party tomorrow,' said Belinda, suddenly feeling very tired. 'There will be such a lot to do.'\n\nThe curate sighed with an affectation of weariness. 'I shall be almost glad when it is over,' he said. 'These functions are always very tiring for us.'\n\nHarriet smiled understandingly, as if including herself in the select brotherhood of the clergy. 'Never mind,' she said, 'there will be the coconut shies. I always love them. And you'll get a good tea. _I_ am in charge of the tea garden.'\n\n'Oh, _well,_ Miss Bede...' the curate moved towards the front door and Belinda was able to slip quietly into the background. She went into the drawing-room and began to tidy it, plumping up the cushions and removing the remains of the pears they had eaten. She put her knitting into its cretonne bag and took the parish magazine to read in bed. There was a nice new serial in it, all about a drunken organist and a young bank clerk, who was also a lay reader and had been wrongfully accused of embezzlement. And of course the Archdeacon's letter was always worth a second reading.\n\n# Chapter Two\n\nALTHOUGH THE MISSES BEDE had a maid they were both quite domesticated and helped her in various small ways, clearing away the breakfast things, dusting their own bedrooms and doing a little cooking when they felt like it. On this particular morning, however, which was the day of the vicarage garden party, Belinda decided that she could miss doing her room with a clear conscience, as there were so many more important things to be done. It was unlikely that Miss Liversidge would be visiting them and putting them to shame by writing 'E. Liversidge' with her finger, as she had once done when Emily had neglected to dust the piano. Typical of Edith, of course, going straight to the point with no beating about the bush. Not that she could talk either, with dog's hairs all over the carpet and the washing-up left overnight.\n\nThis morning, as she went about humming _God moves in a mysterious way,_ Belinda wondered what to do first. She had to arrange for some deck-chairs they had promised to be taken over to the vicarage. The cake she had made to be raffled\u2014the Archdeacon was broad-minded and didn't disapprove of such things\u2014must be finished off with its mauve paper frill. The seams of Harriet's cr\u00eape de Chine dress had to be let out, as Harriet seemed to have grown stouter since she had last worn it. Perhaps that was the most important thing of all, for Harriet intended to wear it that afternoon.\n\nWhile she was sewing, Belinda began to wonder what everyone would be wearing at the garden party. Agatha Hoccleve would of course wear a nice suitable dress, but nothing extreme or daring. As the wife of an archdeacon she always had very _good_ clothes, which seemed somehow to emphasize the fact that her father had been a bishop. Then there was Edith Liversidge, who would look odd in the familiar old-fashioned grey costume, whose unfashionably narrow shoulders combined with Edith's broad hips made her look rather like a lighthouse. Her relation, Miss Aspinall, would wear a fluttering blue or grey dress with a great many scarves and draperies, and she would, as always, carry that mysterious little beaded bag without which she was never seen anywhere. Undoubtedly the most magnificent person there would be Lady Clara Boulding, who was to perform the opening ceremony. It was of course fitting that this should be so, as she was the daughter of an earl and the widow of their former Member of Parliament, an excellent man in his way, although he had never been known to speak in the House except on one occasion, when he had asked if a window might be opened or shut.\n\nBy now Belinda had tacked the seams of the dress and was fitting it on her sister, who twitched about impatiently, while Belinda ran round her with her mouth full of pins.\n\nHarriet was having one of her tirades against the Archdeacon.\n\n'All that nonsense in the parish magazine about him needing a holiday,' she stormed. 'If that's so, why doesn't he go to Karlsbad with Agatha? Unless she wants a holiday away from him\u2014you could hardly blame her if she preferred to go alone. _I_ certainly would.'\n\n'But surely Agatha isn't going to Karlsbad _alone?'_ asked Belinda eagerly.\n\n'Well, their Florrie told Emily that she and cook aren't looking forward to managing the Archdeacon by themselves, so it looks as if he isn't going with her. I think it would be nicer if he went too, then we might have a good sermon for a change. I never heard anything so depressing in my life as that horrid thing he read last Sunday\u2014all about worms, and such stilted language. Edith Liversidge walked out in the middle, and'\u2014Harriet chortled at the memory of it\u2014'one of the churchwardens ran after her with a glass of water, thinking she felt faint or something.'\n\n'But Harriet,' said Belinda gently, 'Henry was reading a passage from _Urn Burial,_ I thought he read it magnificently,' she sighed. Of course the real truth of the matter was that poor Henry was too lazy to write sermons of his own and somehow one didn't think of him as being clever in a theological kind of way. That is, no scholarly study of any of St. Paul's Epistles had as yet appeared under Archdeacon Hoccleve's name, although he had once remarked to Belinda that he thought the Apocalyptic literature remarkably fine.\n\nHarriet continued her tirade. 'If it weren't so far to walk,' she said, 'I should certainly go to Edward Plowman's church; he does at least preach good homely sermons that everyone can understand. He works systematically through the Ten Commandments and the Beatitudes, I believe; much the most sensible thing to do. Besides, he's such a nice man.'\n\n'But Harriet,' said Belinda anxiously, 'he _is_ rather high. He wears a biretta and has incense in the church. It's all so\u2014well\u2014 _Romish.'_ Broad-minded as she was, Belinda was unable to keep a note of horror out of her voice.\n\nHarriet became defiant. 'Edward Plowman is such a fine-looking man, too,' she declared. 'Like Cardinal Newman.'\n\n'Oh, no, Harriet,' protested Belinda. 'Cardinal Newman had a much bigger nose. And besides, he really did go _over,_ you know, and I'm sure Edward Plowman would never do that.'\n\n'Oh, then I must have been thinking of somebody else,' said Harriet vaguely. 'Anyway,' she went on, 'Mr. Donne could certainly preach better sermons than the Archdeacon, I'm sure.' She pulled the dress off over her head. 'You needn't bother to oversew the seams\u2014they won't show.'\n\nWhen Belinda had finished the sewing she decided that she had better go over to the vicarage to see if she could help Agatha in any way. It did not take her long to reach the vicarage gate, as it was very near her own house. When they had finally decided to spend their old age together, Harriet had insisted that they should be well in touch with the affairs of the parish. Belinda had not felt so strongly about it, although when Archdeacon Hoccleve had been made vicar she was naturally glad that their house was so near his. She imagined friendly poppings in and out but somehow, dear Henry not being quite like other clergymen, it hadn't worked out like that. And then of course there was Agatha. It was difficult to be completely informal with her, either because of her father having been a bishop or for some more subtle reason, Belinda had never been quite sure which.\n\nShe walked up the vicarage drive. The Archdeacon had a hankering after the picturesque and would have liked a ha-ha, a ruined temple, grottoes, waterfalls and gloomily overhanging trees. He fancied himself to be rather like one of those eighteenth-century clergymen suffering from the spleen, but Agatha was a practical woman, who liked neat borders and smooth lawns, flowers in the front garden and vegetables at the back. So the vicarage garden, as Belinda saw it on this September morning, was admirably suited to a garden party but there were no grottoes.\n\nBelinda walked up to the front door, but before she had time to ring the bell Agatha appeared, carrying many bundles of brightly coloured paper. She was wearing a plain but well-cut dress of striped Macclesfield silk and looked rather harassed.\n\n'How are you, Agatha?' asked Belinda. 'I've come to see if there's anything I can do to help and I must see the Archdeacon about the Sunday School children's recitations.'\n\n'Henry is having a bath,' said Agatha shortly.\n\nSurely rather late? thought Belinda. It was past eleven and oughtn't an archdeacon to rise earlier than that?\n\n'So of course you can't see him now,' continued Agatha in the same tone of voice, which implied that she had the privilege not allowed to Belinda of seeing an archdeacon in his bath. 'You will have to wait,' she concluded, with a note of something like triumph in her voice.\n\n'Why, of course,' said Belinda meekly. Agatha always seemed to be most formidable in the mornings. In the evenings she was often quite affable and would talk about begonias and the best way to pickle walnuts.\n\n'You seem very busy,' said Belinda, moving towards Agatha as if to help her. 'Can't I do something while I'm waiting for the Archdeacon?'\n\nAgatha nodded reluctantly. 'I was going to arrange the garden-produce stall,' she said, thinking that Belinda Bede was rather a nuisance although she no doubt meant well. 'You might help me to pin the coloured paper round it. I thought green and orange and perhaps red would show off the vegetables rather nicely.'\n\n'What lovely marrows!' exclaimed Belinda, catching sight of them among a heap of miscellaneous garden produce. They were gleaming yellow and dark green, with pale stripes. Surely the poor soil of the vicarage garden could not have produced such beauties?\n\n'Yes, they are fine,' agreed Agatha. 'They are from Count Bianco's garden. He brought them round himself early this morning.'\n\n'Poor old Count Bianco,' said Belinda gently. Ricardo Bianco was an Italian count, who for some unexplained reason had settled in the village many years ago. He was a gentle melancholy man, beloved by everyone for his generosity and courtly manners and he had admired Harriet Bede for more years than could now be remembered. He had the habit of asking her to marry him every now and then, and Harriet, although she always refused him, was really very fond of him and often asked his advice about her gardening problems. Gardening and his childhood in Naples were his chief topics of conversation, though he would occasionally enjoy a melancholy talk about his old friend John Akenside, who had been killed in a riot in Prague, when he had just been sitting at an open-air caf\u00e9 taking a glass of wine, as was his custom in the evening, doing no harm to anybody. 'Ricardo is so _devoted_ to Harriet,' said Belinda, giving the words a full meaning which was not lost on Agatha Hoccleve.\n\nAgatha went rather pink and said angrily, 'Count Bianco comes of a very old Italian family. I always think he and Lady Clara Boulding would be very suited to each other, but of course her father's earldom was only a nineteenth-century creation,' she mused.\n\nBelinda was rather annoyed at this. 'I don't think Lady Clara and Ricardo would be at all suited to each other,' she said, repeating his Christian name with triumph. 'Harriet and Ricardo have a great many tastes in common, especially gardening. Why, whenever he comes to our house he nearly always brings with him some roots or seeds...' here Belinda broke off, aware that this sounded rather ridiculous, but Agatha did not seem to have noticed. She was just opening her mouth to say something else, when their attention was diverted by somebody calling out in a loud voice.\n\nBelinda recognized the voice as that of the Archdeacon. He was leaning out of one of the upper windows, calling to Agatha, and he sounded very peevish. Belinda thought he looked so handsome in his dark green dressing-gown with his hair all ruffled. The years had dealt kindly with him and he had grown neither bald nor fat. It was Agatha who seemed to have suffered most. Her pointed face had lost the elfin charm which had delighted many and now looked drawn and harassed. She had rheumatism too, but Belinda realized that she would have to have something out of self-defence and perhaps with the passing of the years it had become a reality. One never knew.\n\nThe voice went on calling. It seemed that the moths had got into the Archdeacon's grey suit and why had Agatha been so grossly neglectful as to let this happen? The tirade was audible to anyone in the garden or in the road beyond.\n\nBelinda turned away from the window and began to hang festoons of green paper along the top of the stall. The gardener, who was weeding one of the flower beds near by, also turned away. He could not bear the Venerable Hoccleve, as the servants called him. He was a bit mad in his opinion, wanting yew trees on the lawn and something he called a ha-ha, which no gardener had ever heard of.\n\nEventually Agatha returned to the business of decorating, looking extremely annoyed, but not mentioning the incident. She began to take down all Belinda's decorations and arrange them another way. Belinda thought it better to say nothing, so they went on with their work in silence. At last Belinda, who felt rather uncomfortable, drew Agatha's attention to the arrangement of the marrows.\n\n'I think they would look rather effective in a kind of pyramid, if it could be managed,' she suggested, thinking to herself that it would obviously be better if Agatha were to humour dear Henry a little more. But of course Belinda could hardly give an archdeacon's wife a few hints on how to manage her husband.\n\nAt that moment one of the marrows fell over and the pyramid had to be rebuilt. While they were doing this, the Archdeacon came out on to the lawn.\n\n'Good morning,' he said, ignoring his wife. 'I see that I have kept you waiting, but so many annoying things have happened that it was quite impossible to be ready any sooner.' He darted a quick, angry glance at Agatha.\n\nBelinda spoke hastily in order to change the subject.\n\n'I've brought a list of the recitations the children have learnt, so you can choose which ones you think best from it,' she said, knowing perfectly well that he would find fault with the pieces and ask why they had not been taught more Middle English lyrics or passages from Gower and Chaucer.\n\nHe smiled with an affectation of weariness and then sighed. 'Ah, yes. There is so much to be done before this afternoon. I haven't been able to sleep for thinking about it. Nobody can possibly know how much I have to do,' he went on, with another meaning glance at Agatha.\n\n'Perhaps if you had got up earlier, Henry,' she said sharply. 'Florrie called you at eight. I was up at _seven.'_\n\nThe Archdeacon laughed and began to pace about the lawn with his hands in his pockets. Belinda was embarrassed and began to walk slowly towards the house. Eventually the Archdeacon followed. They walked together into his study. He was smiling to himself in a sardonic way that Belinda found very disconcerting. It was unsuitable for a clergyman to look sardonic. Perhaps Harriet was right to prefer the more conventional Mr. Plowman and Mr. Donne.\n\nThere was an awkward silence and to break it Belinda descended weakly to flattery.\n\n'I did enjoy hearing you read _Urn Burial_ last Sunday,' she said. 'It is so _very_ fine and you read it so well.'\n\n'Ah, yes. Do you remember when I used to read Milton to you?' he said, his thoughts going back to the days when Belinda's frank adoration had been so flattering. By this time he had forgotten how bored he had been by her constancy. Agatha never asked him to read aloud to her when they were alone together in the evenings. 'Do you remember the magnificent opening lines of _Samson Agonistes?'_ he asked, warming to his subject and looking dangerously on the brink of reminding her of them. Indeed, the first words were already out of his mouth when Belinda interrupted him, and directed his attention to the matter in hand. There was of course nothing she would have liked better than to hear dear Henry reciting Milton, but somehow with Agatha outside and so much to be done it didn't seem quite the thing. Also, it was the morning and it seemed a little odd to be thinking about poetry before luncheon.\n\n'Now,' she said, 'what about these recitations? I have a list of the ones they know, so I think perhaps you'd better let me choose the most suitable ones. I know you must be very busy with other things,' she added soothingly, 'and even though Mr. Donne can help you, I know that you like to see to everything personally.'\n\n'I doubt whether our friend Donne will be much help,' said the Archdeacon. 'His sermons are very poor. He and Edward Plowman are about a match for each other.'\n\n'Oh, but one must be tolerant,' said Belinda, 'and many people prefer a simple sermon. I've heard people say that Edward Plowman is considered quite a saint in his parish.'\n\nThe Archdeacon laughed rather bitterly. 'Do you wonder when his parish consists almost entirely of doting spinsters?' he said. It was one of the Archdeacon's grievances that people never made a fuss of him as they did of Father Plowman or of the younger curates, although he pretended to despise such adulation. And then too, Lady Clara Boulding, whose country seat lay midway between the two villages, had chosen to attend Plowman's church rather than his. The Archdeacon could not help feeling bitter about this, for although Belinda might put a pound note into the collection bag on Easter Sunday, it was hardly the same as Lady Clara's five or ten. In these days of poverty the spirit in which it was given counted for very little.\n\n'You need not make fun of doting spinsters,' said Belinda, roused by his mockery. 'After all, it isn't always our fault...' she stopped in confusion, fearing that he might make some sarcastic retort.\n\n'No, women like to have something to dote on,' he said mildly enough, 'I have noticed that. And we in the Church are usually the victims.'\n\n'We are _all_ in the Church,' said Belinda gently. 'I think I should go out into the garden again and help Agatha. There must still be a great deal to do.'\n\nOut in the garden, Agatha, surrounded now by several willing helpers, for she was popular among the church workers because of her distinguished ecclesiastical connections, had finished decorating and arranging the garden-produce stall.\n\nThe vicarage garden was beginning to look like a fairground. Stalls, coconut shies, bran-tubs and even a fortuneteller's booth had taken root on the lawn. The Archdeacon always hated this annual garden party and tried to have as little to do with it as possible, although he had to put in an appearance to fawn on the more distinguished visitors. There was always a possibility that Lady Clara Boulding might decide to come to his church, which was really nearer if one walked across the fields, although it was difficult to imagine anyone as impressive as Lady Clara doing that.\n\n'It looks as if everything is finished,' said Belinda. 'I don't feel as if I have done my share.'\n\n'You have put up with my ill-humour for ten minutes,' said the Archdeacon, 'which is more than anyone else could have done.'\n\nBelinda flushed with embarrassment and secret pleasure. She felt herself to be somehow exalted above the groups of busy women, who had been arranging pyramids of apples, filling bran-tubs and decorating stalls with coloured paper. Once, she knew, she _had_ been different, and perhaps after all the years had left her with a little of that difference. Perhaps she was still an original shining like a comet, mingling no water with her wine. But only very occasionally, mostly she was just like everyone else, rather less efficient, if anything. Even her paper decorations had been taken down and rearranged. There was nothing of her handiwork left on the garden-produce stall.\n\n'Why, look,' she exclaimed, unable to deal with the Archdeacon's curious compliment, 'there's Edith Liversidge. Whatever is she doing?' For Miss Liversidge, looking even more dishevelled than usual, was pushing her way through a thick clump of rhododendrons on the opposite side of the lawn.\n\n'Oh, Archdeacon,' she called in her rough, mannish voice, 'there you are! I've been looking everywhere for you.'\n\n'Well, Miss Liversidge, I hardly see why you should have expected to find me in the rhododendrons,' he said.\n\n'Oh, that's the treasure hunt,' she explained. 'I've just been arranging some of the clues. We shall have everybody tied up in knots this afternoon.'\n\n'That will certainly be diverting,' said the Archdeacon politely, 'but I had imagined it was only for the younger people.'\n\n'Oh, nonsense, everyone will be encouraged to join in. Now, what I really wanted to see you about was the cloakroom arrangements. Lavatories, you know. What has been done?' Edith rapped out the question with brusque efficiency.\n\nBelinda turned away in embarrassment. Surely Edith could have asked Agatha and need not have troubled the Archdeacon with such an unsuitable thing? But he appeared to be enjoying the conversation and entered into the discussion with grave courtesy.\n\n'I cannot really say. I had imagined that people would use their own discretion,' he ventured.\n\n'Children are not noted for their discretion,' said Edith bluntly, 'and even grown-ups aren't angels.'\n\nThe Archdeacon smiled. 'No, not even the higher orders of the clergy would claim to be quite that. Perhaps you can help us, Miss Liversidge, we all know your experience in these matters.'\n\n'Yes, but of course it wasn't at all the same thing in the Balkans after the war,' said Edith, perhaps unnecessarily. 'Still, I have been thinking things over. We must have clear notices put up. I've got Mr. Matthews from the Art School at work on them now.'\n\n'Poor Matthews, a prostitution of his talents, I feel,' said the Archdeacon. 'I think Gothic lettering would be most suitable. What is your opinion, Miss Bede?'\n\nPoor Belinda, confused at being drawn into the conversation, could only murmur that in her opinion the largest and clearest kind of lettering would obviously be the best.\n\nMiss Liversidge looked from one to the other impatiently. 'I thought the ladies should use the ground-floor cloakroom and the gentlemen the place behind the toolshed.'\n\n'The Place Behind the Toolshed, what a sinister sound that has,' mused the Archdeacon. 'I'm sure your arrangements will be admirable, Miss Liversidge, though perhaps hardly necessary.'\n\n'We shall see about that,' she said in a dark tone, and then stumped off in search of her relative, Miss Aspinall, calling her as if she were a dog, 'Connie! Connie! Come along! Time to go home to lunch.'\n\nMiss Aspinall, who had been enjoying a snobbish little talk with Agatha, hurried after her. She could never keep pace with Edith and was always a few steps behind her.\n\n'I think Edith Liversidge is really disgusting,' said Harriet indignantly. 'Mr. Donne and I could overhear what she was saying from the tea garden. He seemed most embarrassed.'\n\n'Ah, what it is to be young,' sighed the Archdeacon. 'Or perhaps he is what the higher orders of the clergy would not claim to be. One never knows.'\n\n'He is an excellent preacher,' said Harriet stoutly, if irrelevantly, 'and he seems to have the coconuts _very_ well organized. Now Mr. Donne,' she called, bringing him into the group, 'don't forget that you promised to let me win a coconut.'\n\n'Ah, Miss Bede, I'm sure your skill will win the biggest one of all,' said Mr. Donne gallantly.\n\nAt this point Belinda thought it would be as well if they went home to luncheon. They would need to reserve all their strength for the afternoon, she explained.\n\n# Chapter Three\n\n'DON'T YOU THINK YOU would be more comfortable in low-heeled shoes, dear?' suggested Belinda tentatively. 'One's feet always get so tired standing about.' She glanced down at her own\u2014long, English gentlewoman's feet she always thought them, sensibly clad in shoes that were rather too heavy for the printed cr\u00eape de Chine dress and coatee she was wearing.\n\nHarriet glanced down too. 'I always think low heels are so dowdy,' she said. 'Besides, high heels are definitely the fashion now.'\n\n'Yes, I suppose they are,' agreed Belinda, for Harriet always knew things like that. And yet, she thought, at our age, surely all that was necessary was to dress suitably and if possible in good taste, without really thinking of fashion? With the years one ought to have grown beyond such thoughts but somehow one never did, and Belinda set out for the afternoon conscious that she was wearing dowdy shoes.\n\nAs they walked to the vicarage, Belinda regulating her normally brisk step in consideration for Harriet's high heels, they were overtaken by Count Bianco, who was escorting Miss Liversidge and Miss Aspinall, both of whom were dressed exactly as Belinda had anticipated. Count Bianco wore a light grey suit and a panama hat. He carried a stick and grey gloves and there was a fine rose in his buttonhole. As they came together he gave them a courtly bow, which, from anybody else might have seemed exaggerated.\n\n'How charming you are looking, Miss Harriet,' he said, 'and you also, Miss Belinda,' he added, as a courteous afterthought. 'Poor old John Akenside,' he went on meditatively, 'how he loved the hot weather.'\n\n'Nonsense, Ricardo,' said Edith Liversidge, 'he always went as red as a lobster in the sun.' She had known the Count's friend in what she called her Balkans Days and it was rumoured that he had been very fond of her, but had been too shy to declare himself. It seemed odd to think that anyone could have loved Edith, who seemed a person to inspire fear and respect rather than any more tender emotion, but, as Belinda had once suggested, perhaps the unpleasant nature of her work in the Balkans had hardened her and she had once been more lovable.\n\nThey walked on to the lawn where a group of people had assembled. Belinda could see the Archdeacon standing at the top of the front-door steps, against a background of Victorian stained glass, the vicarage being built in the Gothic style. She thought he looked splendid, and somehow the glass set off his good looks.\n\nLady Clara Boulding was to open the garden party officially at half-past two and as she had now arrived, there seemed no reason why she should not get on with it at once. But the crowd was obviously waiting for something. Agatha Hoccleve, who was standing by her husband, nudged him and said in an agitated and audible whisper, 'Henry, a _prayer.'_\n\nThe Archdeacon started. He had been wondering whether Lady Clara would give some definite contribution to the church-roof fund as well as buying things at the stalls. He cleared his throat.\n\n'Let us ask for God's blessing on our endeavours,' he said, in a loud voice which quite startled some people.\n\nBelinda looked down at the grass and then at Agatha's neat su\u00e8de shoes, so much more suited to the occasion than her own.\n\nThe Archdeacon began to recite a prayer. _O Lord God, who seest that we put not our trust in anything that we do, mercifully grant that by Thy Power we may be defended against all adversity...._\n\nHarriet looked at Belinda and frowned. The Archdeacon always chose such unsuitable prayers. _Prevent us O Lord in all our doings,_ was the obviously correct one for such an occasion. These little departures from convention always annoyed her.\n\nBelinda, on the other hand, was thinking loyally, what an excellent choice! It strikes just the right note of humility. When Henry prays for defence from adversity, he must mean too much confidence in our own powers. One knew that pride often came before a fall. Or perhaps he was not referring to the garden party specifically, but taking in the larger sphere of life outside it... here Belinda's thoughts became confused and a doubt crept into her mind, which was quickly and loyally pushed back. For it could not be that dear Henry had just said the first prayer that came into his head....\n\nThere was a short pause. Count Bianco replaced his panama hat and everyone began to move, relieved to be normal once more.\n\nBut they were not to be released yet. Lady Clara enjoyed opening garden parties and bazaars. Indeed, apart from attending memorial services in fashionable London churches, it was her chief recreation. She stood on a grassy bank, slightly raised above the rest of the crowd. She was still a handsome woman, and if her speech contained rather too much of her late husband's meaningless parliamentary phraseology, her voice was nevertheless pleasant and soothing. Miss Aspinall, who had detached herself from Miss Liversidge in order to be among the foremost of the little group who would go round the stalls with Lady Clara, was listening with a pathetically eager expression on her thin face. Nobody knew how much Edith got on her nerves and how different it all was from the days when she had been companion to Lady Grudge in Belgrave Square. Treated like one of the family, _such_ kindness... Connie's eyes filled with tears and she had to turn away.\n\nAt last Lady Clara stepped down from her grassy platform and made her way towards the stalls, accompanied by Agatha and the Archdeacon, who had a particularly ingratiating smile on his face. At a respectful distance behind them came Miss Aspinall with a group of lady helpers, who were hurrying to get to their places at the stalls. Lady Clara's progress was slow and stately but profitable. She bought some jam, two marrows, half a dozen lavender sachets, a tea cosy, a pair of bed socks, some paper spills in a fancy case and an embroidered _Radio Times_ cover.\n\nBelinda, now busy at the garden-produce stall, was wondering whether she ought to wrap Lady Clara's marrows up, and if so what was the best way of doing it. They had only newspapers for wrapping, so she chose _The Times_ as being the most suitable and made them up into a rather clumsy parcel. Lady Clara's chauffeur was to collect them later.\n\nNot long after this Agatha came back to the stall and began to fluster the helpers by rearranging them and collecting all the money together into one tin, so that they were all tumbling over each other to get change instead of each one having her own little pile.\n\n'What's this?' asked Agatha sharply, pointing to the Times-shrouded parcel which Belinda had put into a corner.\n\n'Oh, that's Lady Clara's marrows,' Belinda explained.\n\n'Wrapped in newspaper?' Agatha's tone was expressive. 'I'm afraid that won't do at all.' She produced some blue tissue paper from a secret hiding-place and began to undo Belinda's parcel.\n\n'Oh, dear, I'm so sorry, I didn't know there was any other paper,' said Belinda in confusion. 'I saw them lying there and I thought perhaps they ought to be wrapped up and put aside in case anybody sold them by mistake.'\n\n'I don't think anybody would be so stupid as to do that,' said Agatha evenly. 'They were the two finest marrows on the stall, I chose them myself.'\n\n'Oh, well...' Belinda gave a weak little laugh. All this fuss about two marrows. But it might go deeper than that, although it did not do to think so.\n\n'Perhaps you would like to go and have tea,' said Agatha, who was having difficulty with the bulk of the marrows and the fragility of the tissue paper and did not want Belinda to see. 'We may as well go in turns.'\n\n'Well, yes, if it isn't too early,' said Belinda.\n\n'Oh, no, Lady Clara is already having hers. She has gone with Count Bianco.' Agatha stood up and reached for a ball of string.\n\nBelinda felt herself hurrying away, routed was perhaps the word, Agatha triumphant. It was a pity they sometimes had these little skirmishes, especially when Agatha was so often triumphant. All over two marrows, even if they were the finest on the stall.\n\nBelinda looked around to see if she could find Harriet. She felt that she wanted to tell somebody about the marrows and perhaps laugh over them. Harriet's healthy indignation would do her as much good as a cup of tea, she thought. But Harriet was nowhere to be seen. And where was the Archdeacon? It would be just like him to retire to the house and have a bath. But he had already had one today, as Belinda knew, so she guessed that he was probably attending on some of the more distinguished visitors.\n\nAs she entered the tea garden she saw Harriet sitting at a table with the curate. Harriet was handing him a plate of cakes and urging him in her penetrating voice to try one of the pink ones which she had made specially for him. Perhaps it will be better if I don't disturb them, thought Belinda, turning round to look for a vacant place at one of the other tables. And then she came face to face with the Archdeacon. He also wanted his tea, and as they had so often had tea together in the past what could be more natural than that they should have it together this afternoon?\n\nThey sat down at a table for two. Belinda began to be assailed by various doubts. What would people think to see her having tea with the Archdeacon while his wife was still working tirelessly at the garden-produce stall? It was a pity really to worry about what people thought, but, Belinda flattered herself, she wasn't entirely old and unattractive, even in her sensible shoes, and she still had the marrows on her conscience, although she did not feel that she could tell the Archdeacon about them.\n\nHe on the other hand had no such scruples. Belinda began to wish that he wouldn't talk so loudly, for although she knew that it was only one of his little oddnesses to complain about his wife, other people might not realize this. So she put two lumps of sugar into his tea, and tried as tactfully as she could to change the subject of the conversation.\n\n'But Agatha has been so busy arranging things for the garden party and the concert tonight,' said Belinda in a low voice. 'She can't see to everything at once.' Raising her voice, she went on, 'Speaking of the concert reminds me that Harriet is still undecided as to what she is going to play. Of course she has a large repertoire, but one must choose something suitable and not too long...' Belinda babbled on. '... she's very anxious to play a Brahms intermezzo, but it may be a little heavy for a village concert. I thought perhaps some Mendelssohn, some of the _Songs without Words_ are so charming...' She looked at the Archdeacon anxiously, to see if he had yet forgotten Agatha's negligence in letting the moths get into his grey suit.\n\nHis face betrayed that he had not. In fact all the bright conversation about. the concert seemed to have been wasted on him. 'I don't think you'd have done that,' he said thoughtfully, gazing at a piece of bread and butter on his plate.\n\nBelinda saw that it was no good trying to change the subject yet. He must be humoured out of it. She seemed to be having a difficult afternoon altogether, what with the episode of the marrows and now having to humour the Archdeacon. Archdeacons ought not to need humouring, she told herself angrily. Supposing Henry were a bishop, could one still expect no improvement?\n\n'What are you smiling at?' asked the Archdeacon crossly. 'People look very foolish smiling at nothing.'\n\n'I wasn't smiling at nothing,' retorted Belinda. 'I was wondering if you'd still make such a fuss about unimportant trifles if you were a bishop.'\n\n'Unimportant trifles! The only good suit I have ruined, and you call it an unimportant trifle.'\n\n'We are supposed not to take heed of what we shall wear,' said Belinda unconvincingly.\n\n'My dear Belinda, we are not in the Garden of Eden. That is no solution to the problem. We may as well face the facts. Agatha ought not to have let the moth get into that suit. It was her duty to see that they didn't. I am sure that you would have seen that it was put away with moth balls...' the Archdeacon's voice had now grown so loud that people at the other tables were beginning to look at them with interest and amusement. Belinda felt most embarrassed.\n\n'It would have smelt of camphor then and you would probably have disliked that,' she said, almost in a whisper.\n\nThe Archdeacon gave a shout of laughter at this. Suddenly he was in a good temper again, fell on a plate of cakes and began to eat ravenously. 'I was too busy to have any luncheon,' he explained. 'So many tiresome things to do.'\n\n'It will be nice for you to go away for a holiday,' ventured Belinda.\n\nThe Archdeacon sighed heavily. 'Ah, if only I could.'\n\n'But now that Mr. Donne is here, surely it can be managed?'\n\n'One cannot leave the flock without a shepherd,' said the Archdeacon in a mocking tone.\n\n'But he said\u2014I mean we heard\u2014that Agatha was going to Karlsbad in October,' said Belinda, urged on by curiosity. 'Surely you will be going too?'\n\n'Alas, no.' The Archdeacon finished the last cake. 'And even if I were, it would hardly be a holiday for me.'\n\nBelinda could think of no reply to make to this and none seemed to be expected. She could neither agree nor protest, she felt, but did what seemed to her the best she could by getting up from the table and saying that she really must get back to the stall. 'I must go and relieve Agatha,' she said. 'I see she hasn't been for her tea yet.'\n\n'It will please her not to have any,' said the Archdeacon. 'I wonder that you have had any. I thought women enjoyed missing their meals and making martyrs of themselves.'\n\n'We may do it, but I think we can leave the enjoyment of it to the men,' said Belinda, pleased at having thought of an answer. But Henry was really too bad, there was no knowing what he might say next. And he was _not_ going to Karlsbad... She hoped nobody had overheard their conversation. It had really been most unsuitable, but somehow she felt better for it and had almost forgotten the episode of the marrows.\n\nBack at the garden-produce stall, Belinda saw Agatha, looking rather tired and flustered, bundling what remained of the flowers, fruit and vegetables on to the front of the stall.\n\n'Oh, there you are,' she exclaimed, making Belinda feel that she had been away too long.\n\n'Yes, you must be longing for your tea, but surely you could have gone before now? Wouldn't Miss Liversidge or Miss Aspinall have taken charge of the stall?' said Belinda, doing the best she could.\n\n'Oh, well, I may as well go now,' said Agatha grudgingly, 'we seem to have taken quite a lot of money.'\n\n'Yes, and the tea garden has been crowded. The Archdeacon was still there when I left,' Belinda added, thinking that this might encourage Agatha.\n\n'I have had no luncheon,' she said. 'I shall really be glad of a cup of tea.'\n\n'Oh, dear, I wish I'd known that, then you could have gone first,' said Belinda. Had there been no luncheon at all at the vicarage today? Surely a bad arrangement, or had the Archdeacon and his wife wished to outdo each other in self-denial?\n\n'Well, Belinda, I expect _you_ enjoyed your tea,' said Harriet, advancing towards the stall.\n\nBelinda was thankful that Agatha was out of hearing. 'Yes, I thought the cakes were lovely,' she said.\n\n'You and the Archdeacon looked so cosy. Having a nice conversation about moth balls, too, most domestic. What a _pity_ it is about Agatha. They have really nothing in common.'\n\n'Oh, Harriet, you're quite wrong,' said Belinda stoutly. 'Agatha is a most intelligent woman. She knows a great deal about medieval English literature. And then there's _palaeography,'_ she continued, as if her emphatic tone would explain its importance in the married life of Agatha and the Archdeacon.\n\n'Oh, yes, that's about apes, isn't it?' said Harriet, losing interest in the subject. 'Do you think a fur cape would be too hot for the concert this evening? The gold lam\u00e9 jacket doesn't really go so well with my blue as the white fur. Besides, it's rather severe and needs something to soften it.'\n\nEventually Harriet decided to wear the white fur cape, and everyone agreed that she looked very handsome, although one of the more spiteful Sunday School teachers whispered to her friend that she suspected it was not real ermine, but only shaved coney.\n\nAt the beginning of the concert, the Archdeacon, looking very striking in clerical evening dress, made a charming little speech. He seemed to have recovered completely from his bad temper of the afternoon, because they had made a splendid lot of money at the garden party and there was a good attendance at the concert. He beamed on the crowd of zealous church workers, as he praised their untiring efforts, and they in their turn were so greatly carried away by his charm that they forgot all his annoying oddities and began to think themselves fortunate to have such a distinguished-looking vicar.\n\nBelinda was sitting by Count Bianco. She had seldom seen him so animated. He did not refer even once to the sad death of his friend John Akenside. After Harriet had played her Brahms intermezzo, he declared in an enthusiastic mixture of English and Italian that for him everything would be an anticlimax after this.\n\nBelinda found herself thinking, as she often did, that it would be an excellent thing if Harriet would marry the Count. He was wealthy and he had a beautiful house and garden: and, moreover, as Agatha had remarked that morning, he came of a very old Italian family. Belinda was sure that he would have no objection to Harriet making cakes and other dainties for the curates. He was such a kind-hearted man.\n\nIn the meantime a child was reciting, rather too fast, but Belinda caught one or two lines.\n\n_In dingles deep and mountains hoar_\n\n_they combatted the tusky boar._\n\nShe tried to remember why the Archdeacon had been anxious to include this, for it was not a particularly suitable poem. Then she realized that it was in order that he might explain to an audience not really interested in such linguistic niceties, the history of the rare word _dingle._ How it is first known in the twelfth or thirteenth century in a work called _Sawles Warde:_ then it is revived by the Elizabethans, who gave it to Milton\u2014you remember it in _Comus,_ of course....\n\nThe Reverend Edward Plowman, sitting in the front row by Agatha, listened to the explanations jealously. How like Hoccleve to show off his knowledge on such an unsuitable occasion! Father Plowman, as he was called by his devoted parishioners, was not a clever man. He had failed to take Honours in Theology, but he worked hard in his parish and the elaborate ritual of his services was ample compensation for the intellectual poverty of his sermons. He was greatly beloved by his flock and one Christmas he had received so many pairs of hand-worked slippers that he gave the Archdeacon a pair. The gift was accepted rather grudgingly, especially as they were a size too small. This evening Father Plowman was not wearing his usual costume of cassock and biretta and his evening dress was less well cut than the Archdeacon's. He shifted uneasily, reflecting that even the best seats were hard. But soon there would be an interval. Would there be refreshments? he wondered. He tried to remember whether they had had refreshments at the last concert. He could hardly ask Mrs. Hoccleve. These recitations were really rather heavy going, though this was better. _Time wasted is existence, used is life_... one might almost use a line like that for a text. He began to meditate on the theme, although he did not really approve of these literary sermons. Still, he had no doubt that he could do them as well as Hoccleve.\n\nIn the interval, during which he enjoyed some excellent coffee and cakes, Father Plowman talked to the elder Miss Bede about the death-watch beetle and gave her a short dissertation on, its habits. This put Belinda into an elegiac mood and somehow prepared her for the next item on the programme, which was a harp solo by Miss Aspinall. There was certainly something elegiac about poor Connie. Her thin, useless hands, her fluttering grey dress\u2014surely a cast-off from Belgrave Square?\u2014even the instrument itself with its Victorian association, made Belinda think of past glories, of more elegant gatherings than this one, at which Connie might have played. The little beaded bag came with her on to the platform and she took out of it a little lace-edged handkerchief, on which she wiped her hands before she started to play. What she played Belinda hardly knew, but it had a melancholy air and the applause which greeted it was restrained though sincere. The village people thought poor Miss Aspinall was not quite right in the head and considered it very clever of her to be able to play at all.\n\nCertainly their most noisy enthusiasm was reserved for the curate, who appeared last on the programme and had a sensational success. Belinda read in her programme that he was to sing _Believe me if all those endearing young charms_ and _The Lost Chord,_ both very suitable songs and particular favourites of hers. Like all sentimental people she cherished the idea of loving a dear ruin, and found her eyes filling with tears as he sang the affecting words. Count Bianco too was very much moved, except that he thought of himself as the ruin, perhaps being loved by Harriet.\n\nBelinda could not see the Archdeacon very well, but she could not help feeling that he was a little displeased at the tumultuous applause which greeted Mr. Donne's songs. Some of the rougher members of the audience, accommodated on benches at the back of the hall, were even stamping their feet and whistling. There was no doubt of his success and popularity.\n\n'The Archdeacon was looking quite annoyed,' said Harriet with satisfaction, as they drank their Ovaltine before going to bed that evening. 'Imagine _him_ singing, though. I think Agatha was pleased at Mr. Donne's success, she would do anything to disagree with the Archdeacon.'\n\nBelinda was too tired to argue. 'I thought poor Connie played very nicely,' she said.\n\nOh, yes,' agreed Harriet, 'and you should have heard Edith talking. She seems to think the whole success of the garden party was due to her arrangements. I didn't see anybody disappearing behind the toolshed though, did you, Belinda? At least, nobody we know, that is.'\n\n'No, Harriet, I did not,' said Belinda in a weary but firm tone. 'After all, most of us were there only two or three hours.'\n\n'Yes, I suppose that's really nothing,' said Harriet yawning.\n\n# Chapter Four\n\n'I THOUGHT WE MIGHT have a cauliflower cheese for lunch today,' Harriet announced at breakfast one morning. 'We shall only need a light meal as we are having the duck this evening.'\n\n'Oh, of course, Mr. Donne is coming again, isn't he,' said Belinda. 'I think perhaps it's a mistake to ask him _too_ often, you know. It seems no time since he was last here.'\n\n'Why, Belinda, it's nearly three weeks,' said Harriet indignantly.\n\n'Yes, I suppose it must be. How quickly the time goes.' Belinda began piling up the plates, scraping fish bones from one to another. Suddenly she stopped, and an expression almost of horror came over her face. 'But Harriet,' she said, 'Miss Prior is coming today. Had you forgotten that?'\n\n'Yes, I had, but I don't see what difference it makes,' said Harriet. 'It's rather a good thing, really. She'll be able to patch that chair cover that's getting so worn, and perhaps start on my new velvet dress.'\n\n'But, Harriet, we _can't_ give her only cauliflower cheese,' went on Belinda with unusual persistence. 'You know how she enjoys her meals and we always give her meat of some kind.'\n\n'You surely aren't suggesting that we should have the duck for lunch, are you?' asked Harriet with a note of challenge in her voice.\n\n'Well, I don't know, really...' Belinda hesitated. She was a little afraid of her sister sometimes. 'Would it matter if we gave Mr. Donne cauliflower cheese? I'm sure he wouldn't mind. We could have some soup and a fairly substantial sweet, and with coffee afterwards it would be quite a nice little meal. I'm sure I would think it very nice. After all, when we had supper with Edith Liversidge on Friday we only had baked beans and _no_ sweet, as far as I remember, just some coffee and biscuits...' poor Belinda floundered on, disconcerted by Harriet's stony silence.\n\n'Miss Prior will just have to put up with cauliflower cheese,' said Harriet firmly. 'If you expect Mr. Donne to, why shouldn't she?'\n\n'Oh, dear, I can't explain exactly. We always seem to have this argument every time she comes,' said Belinda. 'But one feels that perhaps Miss Prior's whole life is just a putting up with second best all the time. And then she's so easily offended. I suppose it's cowardly of me, but I do hate any kind of _atmosphere.'_\n\nThe trouble was that Miss Prior wasn't entirely the meek person one expected a little sewing woman to be. Belinda had two feelings about her\u2014Pity and Fear, like Aristotle's _Poetics,_ she thought confusedly. She was so very nearly a gentlewoman in some ways that one felt that she might even turn out to be related to a clergyman or something like that. She could never have her meals with Emily in the kitchen, nor would she presume to take them with Belinda and Harriet. They must be taken in to her on a tray. She was so touchy, so conscious of her position, so quick to detect the slightest suspicion of patronage. One had to be _very_ careful with Miss Prior.\n\nShe arrived at ten o'clock punctually, a little dried-up woman of uncertain age, with a brisk, birdlike manner and brown, darting eyes. Her dress was drab and dateless, but immaculate, and she wore what appeared to be rather a good cameo brooch at the neck.\n\nShe did her work in the little morning-room on the ground floor, where Belinda usually did the flowers. This was generally tidied beforehand, but today, on showing her in, Belinda noticed to her dismay that it had not been done. Everything looked dusty, there were bits of cotton on the carpet, and worst of all, two vases of dead chrysanthemums. Their stems showed black and slimy in the yellow water.\n\n'Good morning, Miss Prior,' Belinda's tone was bright and welcoming, but she looked a little harassed, wondering why Emily had not dusted the room and how the dead flowers could have been forgotten. She had taken them out of the drawing-room hastily the day before, when Lady Clara Boulding had been seen coming to the door.\n\n'I think I would like the window open,' declared Miss Prior. 'I'm afraid the smell of chrysanthemums always upsets me.'\n\n'I'm so sorry...' Belinda moved in the direction of the window.\n\n'Oh, don't trouble, Miss Bede, I can manage perfectly well.' To Belinda's anxious eyes Miss Prior looked unusually fragile as she lifted the heavy frame.\n\n'There...' Miss Prior took a deep breath of the sharp October air, then glanced round the room, alert and birdlike. 'And perhaps I could have a duster?'\n\n'Oh, _dear_...' Belinda was almost speechless with confusion. 'I'm afraid Emily doesn't seem to have done it this morning'\u2014that was obvious\u2014'and I know how particular you are.'\n\nMiss Prior smiled. 'Well, I wouldn't like to say that, Miss Bede. I can hardly afford to be in _my_ position, but I think we all work better in bright, clean surroundings, don't you?'\n\n'Yes, of course, I'm _so_ sorry....'\n\n'That's quite all right, Miss Bede. You will make me think I am being a nuisance.' Miss Prior went to the sewing machine and drew off the cover with a brisk movement. 'Now, what is it to be this morning?'\n\n'I think the chair covers are the most important, and then there are the new bathroom curtains and some sheets to be put sides to middle...' Belinda went on rapidly, as if speaking quickly would make the work seem less, 'and my sister was wondering whether you could start on a dress for her out of some brown velvet she's got. Perhaps you could begin to cut it out?'\n\n'Very good, Miss Bede, I will commence with the chair covers,' said Miss Prior.\n\nBelinda was saved any further explanations by the appearance of Harriet, who strode into the room with a bundle of silky velvet in her arms.\n\n'Good morning, Miss Prior,' she said. 'Now I want you to start on this dress first of all, if you will.'\n\nBelinda waited rather fearfully, but she need not have been afraid, for Miss Prior seemed much meeker with Harriet and began to admire the stuff and ask her what kind of style she had in mind.\n\n'Oh, I've bought a _Vogue_ pattern,' said Harriet, 'size 38, so you can just follow that.'\n\nMiss Prior darted a doubtful upward glance at the bulk of Harriet towering over her. 'I wonder if it's going to be big enough on the _hips?'_ she ventured. 'That's where you usually need it, isn't it?' She took out her familiar tape-measure which was in a little case shaped like a frog. 'Lady Boulding on the bust, Miss Harriet on the hips, that's what I always say to myself,' she chanted brightly.\n\nBelinda laughed. 'Don't you say anything for me, Miss Prior?'\n\n'Oh, well, Miss Bede, you never wear very _fitting_ dresses, do you, if you see what I mean? A few inches here or there doesn't make much difference.'\n\n'No, I suppose it doesn't,' said Belinda, depressed by this picture of herself in shapeless, unfashionable garments.\n\n'Now, Mrs. Hoccleve,' went on Miss Prior, turning the knife in the wound, 'she _has_ got some lovely things. Not that I make much for her, you know, except a few summer dresses, like I do for Lady Clara. But her clothes are from the _best_ houses. She's just got a lovely navy two-piece with a lemon blouse...' Miss Prior's voice trailed off into a kind of rapture.\n\nIt isn't right, thought Belinda indignantly, for a clergyman's wife to get her clothes from the best houses. She ought to be a comfortable, shabby sort of person, in an old tweed coat and skirt or a sagging stockinette jumper suit. Her hats should be shapeless and of no particular style and colour. Like my old gardening hat.\n\n'Would you like a cup of tea, Miss Prior?' she said aloud. 'I am just going to make some.'\n\n'Thank you, Miss Bede, I was just thinking I should like a cup. I find I can start my work better after a cup of tea. There's quite a nip in the air these mornings and I had to breakfast earlier than usual today.'\n\nMiss Prior's tone was uncomplaining, even bright, but Belinda felt she could bear no more and hurried out of the room to put the kettle on.\n\n'I don't think today is going very well,' she remarked to Harriet, when they were alone drinking their own tea. 'First of all the room wasn't dusted and I'd forgotten to throw away those dead chrysanthemums, and then Miss Prior made me feel as if I really ought to have offered tea sooner. I wonder whether she would have preferred cocoa?' Belinda looked worried. 'It's more sustaining if one has had an early breakfast. Still, she is having a piece of cake with her tea.'\n\n'Oh, and there was no paper in the downstairs lavatory,' chortled Harriet. 'She came to me just now, _so_ confidential. I couldn't think what she was going to say.'\n\n'Oh, dear,' sighed Belinda, 'I meant to get some more toilet rolls yesterday.'\n\n'I just gave her an old _Church Times,'_ said Harriet airily.\n\n'Oh, Harriet, I wish you hadn't done that. I feel Miss Prior is the kind of person who wouldn't like to use the _Church Times._ And I'm still not quite happy about giving her cauliflower cheese.'\n\nBelinda continued to be anxious about Miss Prior's lunch, even though she herself supervised the laying of the tray which Emily was to take in to her.\n\n'It certainly _looked_ very nice,' she said, helping herself to a rather smaller portion of the cauliflower cheese than she had given to Miss Prior, 'and I asked Emily to be sure to wash the cauliflower _very_ well and to put plenty of cheese in it.'\n\nHarriet helped herself liberally and ate with enjoyment.\n\n'I think the damson flan will make quite a good contrast;' went on Belinda. 'I know she will like that.'\n\n'Oh, I do wish you'd stop worrying about Miss Prior,' said Harriet in exasperation.\n\n'I think I will just look in and see if she is all right,' said Belinda when she had finished her meal. 'I expect she would like some coffee.'\n\nBelinda hesitated for a moment outside the door of the morning-room. She could hear the whirr of the sewing machine inside, but Miss Prior never spent very long over her meals. She did not like to be seen in the act of eating or drinking, it seemed to make her more conscious of her position.\n\nBelinda opened the door and went in. Her heart sank when she saw the tray on the little table, for although everything else had been eaten, the cauliflower cheese had been pushed to one side of the plate and was almost untouched.\n\n'Oh, dear, I'm afraid you haven't enjoyed your lunch, Miss Prior,' said Belinda, who now felt near to tears. 'Don't you like cauliflower cheese?'\n\n'Oh, yes, Miss Bede, I do sometimes,' said Miss Prior in an offhand tone, not looking up from her work.\n\nBelinda went on standing in the doorway watching Miss Prior negotiating an awkward bit of chair cover. Then she looked again at the tray, wondering what she could say next. And then, in a flash, she realized what it was. It was almost a relief to know, to see it there, the long, greyish caterpillar. Dead now, of course, but unmistakable. It needed a modern poet to put this into words. Eliot, perhaps.\n\nBelinda burst into a torrent of apologies. How careless of Emily not to wash the cauliflower more thoroughly! How unfortunate that it should have been Miss Prior who had got the caterpillar!\n\n'I'm afraid I didn't feel like going on with it after that,' said Miss Prior almost smugly.\n\n'No, of course not, I quite understand,' said Belinda. But she did not really understand. If this had happened to her in somebody else's house she would have pretended she hadn't seen it and gone on eating. It might have required courage, but she would have done it. 'You must be hungry still,' she went on, 'perhaps you would like a poached egg, or two poached eggs? Emily could easily do them.'\n\nMiss Prior gave a little laugh. 'Well, no, Miss Bede, thank you all the same. It would seem funny to have a meal the wrong way round like that, wouldn't it? You wouldn't fancy that yourself now, would you?'\n\n'If I were really hungry I don't think I should mind,' said Belinda bravely. 'But you must have an egg with your tea. Perhaps that will make up for it.'\n\n'Thank you, Miss Bede, that would be very nice. I am going to the vicarage tomorrow,' Miss Prior went on conversationally, putting down her work for a moment. 'Mrs. Hoccleve wants one or two things done before she goes away.'\n\n'I'm sure you wouldn't get a caterpillar in your cauliflower cheese there,' said Belinda lightly.\n\nMiss Prior made a noise like a snort. 'It might be about all I _would_ get,' she said. _'Very_ poor meals there.' She lowered her voice, 'Between ourselves, Miss Bede, Mrs. Hoccleve doesn't keep a good table. At least, _I_ never see any proof of it. An old dried-up scrap of cheese or a bit of cottage pie, _no_ sweet, sometimes. I've heard the maids say so, too, you know how these things get about. Scarcely any meat except at the week-end, the Sunday roast, you know. You always have such nice meals, Miss Bede, and you give me just the same as you have yourselves, I know that. After all, it might just as easily have been you or Miss Harriet that got the unwelcome visitor today,' she concluded with a little giggle.\n\nBelinda's eyes filled with tears and she experienced one of those sudden moments of joy that sometimes come to us in the middle of an ordinary day. Her heart like a singing bird, and all because Agatha didn't keep as good a table as she did and Miss Prior had forgiven her for the caterpillar, and the afternoon sun streaming in through the window over it all. 'You're doing that chair cover _beautifully,_ Miss Prior,' she said warmly, 'and how well you've got on with Miss Harriet's dress.'\n\n'It really does pay to show her a little consideration,' she said to Harriet afterwards. 'I'm sure she works better when one does. Besides, I'm afraid she's sometimes made to feel inferior, poor soul. Next time she comes we'll have something really nice for lunch, perhaps even a chicken,' she mused.\n\nBut Harriet's thoughts were already with Mr. Donne and the duck they were to have that evening. Could they perhaps have something original served with it, like the orange salad they had had at Count Bianco's? One wanted to give people really interesting food.\n\n'It was a good idea to give Miss Prior an egg for her tea,' went on Belinda, 'I think it made up for the caterpillar.'\n\n'Of course, Mr. Donne _may_ prefer apple sauce, though it would be more ordinary,' said Harriet thoughtfully.\n\nSo the sisters continued antiphonally, each busy with her own line of thought. But at last they found themselves in agreement on the subject of Harriet's brown velvet dress. It was going to be very successful and the new bracelet-length sleeves were most becoming. 'Not even Agatha has a dress with the new sleeves,' said Harriet proudly.\n\nBelinda felt a little depressed at being reminded of Agatha and her clothes from the 'best houses', but soon brightened up when she remembered Miss Prior's remarks about her poor table.\n\n'Do you know, Harriet,' she said, 'Miss Prior told me that the only time they have meat at the vicarage is at the week-end? I can't believe that, really, and yet I've always had a suspicion that Agatha was just the tiniest bit mean.'\n\n'Well, it's a good thing Mr. Donne doesn't have to live there,' said Harriet stoutly.\n\n# Chapter Five\n\nONE AFTERNOON HARRIET SET out for the curate's lodgings, carrying a large basket. Besides a cake and some apple jelly, she was taking some very special late plums which she had been guarding jealously for the last few weeks. She hurried along, hoping that she would not meet anybody on the way, as she and Belinda were going to tea with Count Bianco and she had not much time. She therefore felt very annoyed when she saw the Archdeacon coming towards her, and would have hurried on, had they not come face to face on the pavement.\n\nThe Archdeacon had been visiting a rich parishioner, who was thought to be dying. The poor were much too frightened of their vicar to regard him as being of any possible comfort to the sick, but the Archdeacon liked to think of himself as fulfilling some of the duties of a parish priest and there was something about a deathbed that appealed to his sense of the dramatic. He had also taken the opportunity of visiting the workhouse that afternoon and was altogether in a pleasant state of melancholy.\n\n'When I visit these simple people,' he said affectedly with his head on one side, 'I am reminded of Gray's _Elegy.'_ He began to quote:\n\n_Far from the madding crowd's ignoble strife_\n\n_Their sober wishes never learn'd to stray;_\n\n_Along the cool sequestered vale of life_\n\n_They kept the noiseless tenor of their way._\n\n'Oh, _quite'_ agreed Harriet, annoyed at being delayed. 'One of the finest poems in our language,' she pronounced hoping that there the matter would end.\n\nBut the Archdeacon had had a tiring afternoon and was in no hurry to return to his good works.\n\n'Indeed it is,' he agreed. 'Johnson's criticism of it is so apt, as you will remember\u2014\"Sentiments to which every bosom returns an echo.\"'\n\nBut Harriet could stay no longer. 'Oh, yes, like the Apes of Brazil,' she remarked, and moved off, leaving the Archdeacon puzzling over what she meant. He thought it unlikely that it would be a literary quotation and yet it seemed somehow Elizabethan. Perhaps Belinda would know it. She often wasted her time reading things that nobody else would dream of reading.\n\nHarriet walked away in the opposite direction. It would be very annoying if the curate was out, although she supposed she could always leave the things in his sitting-room with a note. His lodgings were situated in what Harriet considered one of the more sordid streets of the village, a row of late Victorian red brick villas called Jubilee Terrace. Every window had its lace curtains and she imagined that she detected stealthy movements behind them as she walked along to the house where the curate lodged. Well, let them watch her and gossip about it too if they liked, she thought stoutly, it would do some of them good to realize that charity began at home. That Miss Beard, a Sunday School teacher too, pretending to be watering the ferns in the front room\u2014Harriet called out 'Good afternoon!' to her, and was pleased to see her scuttle furtively back into the shelter of the lace curtains.\n\nThe curate's house was called 'Marazion'. Harriet walked up the little path bordered with shells and rang the front-door bell. Mr. Donne himself answered it. He expressed himself delighted to see her and was quite overwhelmed with gratitude when she presented her gifts.\n\nHe gazed at the jelly in wonder as if he had never seen anything like it before, but then recovered sufficiently to show Harriet into his sitting-room. This was quite a nice room, not as meanly furnished as Harriet could have wished, though Belinda was relieved that they did not have to provide the curate with furniture as well as food.\n\nHarriet looked eagerly round, searching for those personal touches that make a room interesting. Other curates had lodged here, but this was the first time Harriet had visited Mr. Donne and there were bound to be differences of detail. The first thing she noticed was a large oar, fastened on the wall over the mantelpiece, with photographs of rowing groups hanging underneath it.\n\n'Ah, now which one is you?' she asked, going up to one of the groups and peering at it. They all looked so alike but at last she discovered Mr. Donne and he pointed out to her his best friend who was a curate in London and another who had been called to the Mission Field.\n\nOh, dear,' she said, her face clouding, 'I hope you will not go overseas. I mean, you aren't _commanded_ to go by your bishop, are you?'\n\n'Oh no, it is a personal matter. The call comes from within, as it were,' explained Mr. Donne, rather red in the face.\n\nHarriet seemed satisfied with this explanation and turned her attention to his books, which were not a particularly original selection. Shakespeare, some standard theological works, a few paper-backed detective novels and the _Oxford Book of English Verse._\n\n'I do hope your electric light is good,' she said anxiously. 'You know how harmful it is to read in a bad light, like Milton or whoever it was, although of course he went blind, so perhaps his eyes were weak to begin with. Oh, good, you have a reading-lamp\u2014I was thinking that I could let you have one that we don't use very much.'\n\nMr. Donne thanked her for the kind thought. 'As a matter of fact,' he assured her, 'I don't work very much in the evenings, except when I'm preparing a sermon. The Boys' Club and the Scouts take up most of my time.'\n\nScouts seemed to remind Harriet that she had thought of knitting him a pair of socks or stockings.\n\n'That's very kind of you,' he said, 'I wear them out terribly quickly and can never have enough. I've just had two pairs sent me today, so I shall be quite well off.'\n\nHarriet bristled with indignation. 'Oh? Who made those for you? Your mother or an aunt perhaps?' These occurred to her as the only people who could legitimately be allowed to knit socks for a curate whom she regarded as her property.\n\n'No, as a matter of fact, it was a relation of Mrs. Hoccleve's, she was up at the University when I was, at least she was doing research. She's a kind of female Don.'\n\n'Oh, I see.' Harriet was a little pacified, but the whole thing was unsatisfactory and needed to be looked into. It was not somehow natural for a female Don to knit for a curate, especially as she sounded to be quite a young woman.\n\n'Her name's Olivia Berridge and she's awfully nice,' said the curate in a kind of burst.\n\n'Well, I shall have to be going now,' said Harriet, putting on her gloves. 'My sister and I are going to tea with Count Bianco.'\n\nThe curate thanked her once more for her gifts and came to the front gate with her. Nevertheless Harriet could not help feeling that the visit had been disappointing. This Olivia Berridge knitting socks for him, that was the trouble. Harriet wondered if there could be anything more between them than that. Of course there had been no photographs other than the rowing groups in his sitting-room, though she had not been able to see his bedroom. There might well be a photograph of Miss Berridge there. And yet it seemed that she must be several years older than he was, so perhaps there was nothing in it after all. Mr. Donne was so _very_ young, not more than twenty-three or four, and there was really nobody here in whom he could confide. Agatha was so unsympathetic and the Archdeacon was definitely peculiar, perhaps even the tiniest bit _mental._ He had that odd sloping forehead which was supposed to be a sign of mental deficiency. Harriet had read about it in _Harmsworth's Encyclopaedia._ She let her thoughts wander at random on this interesting point, but by the time she reached home the worry of Olivia Berridge was again uppermost and as soon as she saw Belinda she asked her what she knew about 'a relation of Agatha Hoccleve's doing research at the University'.\n\nBelinda frowned. 'I believe she has a niece who does research in Middle English,' she said. 'Something to do with _The Owl and the Nightingale,_ I think, but I'm not sure what aspect. Of course there is still much that is obscure in that poem and several disputed readings....'\n\nHarriet interrupted her impatiently. 'Oh, I dare say,' she said, 'but how old is this girl?'\n\n'Oh, I don't think she's very young,' said Belinda, 'at least, she's about thirty, I think, which is young really, isn't it?'\n\nHarriet seemed satisfied and hurried away to change for tea at Count Bianco's. She came downstairs again looking very elegant in a green suit with a cape trimmed with monkey fur. She had decided to break in her new python-skin shoes.\n\nAfter some delay, because Harriet couldn't walk very fast in her new shoes, they arrived at Count Bianco's house.\n\nThey had tea almost at once. The Count had ordered his cook to make all Harriet's favourite cakes and there were four different kinds of jam.\n\nBelinda enjoyed her tea quietly while Harriet and the Count talked. Every time they visited Ricardo's house, Belinda was struck by the excellence of everything in it. The house itself was an interesting continental-looking building with a tower at one end and balconies and window-boxes, filled at all times of the year with suitable flowers. And the garden was delightful, with its perfectly tended herbaceous borders and rockeries, a grove of lime trees and some fine Lombardy poplars. The joys of the vegetable garden, too, were considerable. Belinda wondered how anybody could remain unmoved at the sight of the lovely marrows, and the magnificent pears, carefully tied up in little cotton bags, so that they should not fall before they were ripe or be eaten by the birds. At the bottom of the vegetable garden was a meadow, which Ricardo had planted with such of his native Italian flowers as could be induced to grow in the less sunny English climate. This part of the garden was his especial delight, and on fine evenings he would sit for hours in a deck-chair reading Tacitus or Dante, or brooding over the letters of his friend John Akenside.\n\nThis afternoon he was anxious that they should see a fine show of Michaelmas daisies, eight different varieties, each one a different colour and one a particularly rare one, which he had brought back from the south of France, when he was there in the spring. And he was thinking of having a pond made, for water-lilies and goldfish, and where did Harriet think would be the best place to have it?\n\n'Oh, Ricardo, how lovely!' said Harriet, in raptures at the thought of the pond. 'Will you swim in it? If it had a nice concrete bottom it would be quite clean, and so romantic to swim in the moonlight with the fishes.'\n\nBelinda shivered. The fishes would be so cold and slimy and besides, Ricardo didn't do romantic things like that. 'Leigh Hunt writes rather charmingly about a fish', she said aloud, _'Legless, unloving, infamously chaste';_ she paused. Perhaps it was hardly suitable, really, and she was a little ashamed of having quoted it, but these little remembered scraps of culture had a way of coming out unexpectedly.\n\n'Swans would be nice,' Harriet went on, 'Or would they eat the fishes?'\n\nRicardo was uncertain, but said that he had thought of getting some peacocks, they would look so effective on the terrace.\n\nHarriet agreed that they would, and they moved off together, leaving Belinda bending over the Michaelmas daisies. She did not want to listen to another proposal of marriage and probably a refusal as well. It was some months since Ricardo had last proposed to Harriet and Belinda could feel that another offer was due.\n\nWhen they came back to the house she could tell that he had once more been disappointed. It seemed a suitable time to talk about Ricardo's old friend John Akenside, and how he must miss him, even after all these years.\n\nYes, indeed, Ricardo wondered at times whether his own end was near. Would Belinda come into his study and see the photograph which he meant to have as the frontispiece to his long-awaited edition of the letters of John Akenside?\n\nThey went into the house, leaving Harriet to collect some plants from the gardener.\n\nRicardo's study gave the impression that he was a very studious and learned man. The walls were lined with very dull-looking books and the large desk covered with papers and letters written in faded ink. His task of collecting and editing the letters of his friend took up most of his time now, and it was doubtful whether he read anything but a few lines of Dante or a sentence of Tacitus.\n\nA large photograph of John Akenside stood in a prominent position on the desk, showing him in some central European court dress. He looked uncomfortable in the white uniform and faintly ridiculous, like something out of a musical comedy. Perhaps his collar was crooked or the row of medals too ostentatious to be quite convincing, for there was something indefinably wrong about it, which marred the grandeur of the whole effect. Belinda never minded laughing at this photograph, because she felt that somehow John would understand. Indeed, as she looked at the face, she thought she detected a twinkle in the eyes, which seemed to look slyly round the corners of the rimless glasses, and the mouth was curled into a half smile, self-conscious, but at the same time a little defiant.\n\nTogether Belinda and Ricardo studied the portrait and for some minutes neither spoke. Then Ricardo said rather sadly, 'I think he would have wished the world to see this one.'\n\nBelinda agreed. 'Yes, perhaps you're right. And yet he never sought worldly glory, did he...' she mused sentimentally. 'He was always so humble\u2014hardly downtrodden,' she added hastily, remembering his rather shambling gait and his fingers stained with red ink, 'but I think he never realized his unique gifts, from which we, who were privileged to be his friends, received so much benefit...' Belinda stopped, rather tied up in this sentence of appreciation. She felt as if she were writing his obituary notice in _The Times._ Being with Ricardo often made her talk like this. Of course, she thought, I believe John Akenside had a finger in nearly every European political pie at the time of his death, and yet one had never been sure what it was that he actually _did._\n\nRicardo continued the funeral oration. 'At the beginning of my edition of the letters,' he said, 'which will also contain a short biographical memoir, I am going to quote that beautiful opening sentence from the _Agricola_ of Tacitus\u2014 _Clarorum virorum facta moresque posteris tradere, antiquitus usitatum, ne nostris quidem temporibus quamquam incuriosa suorum aetas omisit..._ ' he recited solemnly in his quaint Italian pronunciation.\n\nBelinda waited patiently while Ricardo finished the complicated Latin sentence. She had never been much of a classical scholar.\n\nAfter he had finished there was a long silence. At last Belinda noticed that it was nearly seven o'clock and at that moment Harriet came into the room, bustling and cheerful, and carrying a basket full of rare rock plants. Her appearance seemed to cheer Ricardo and they began to talk about the care of woolly-leaved alpine plants, a conversation in which Belinda was glad to join.\n\n# Chapter Six\n\nON THE WAY HOME Belinda decided to call at the vicarage to see Agatha. Harriet, calculating that the curate would probably not be there, went home to make some sardine eggs for supper, as it was Emily's evening out.\n\nBelinda found Agatha sitting in the drawing-room, mending the Archdeacon's socks. It was a gloomy, un-homely kind of room, though Belinda could never quite decide why she did not like it. The electric light was too dim or the chair covers too drab or perhaps it was just that Agatha was there and that behind her was a bookcase, where, behind glass, was a complete set of the publications of the Early English Text Society. Belinda noticed rather sadly that the crimson socks Henry had bought in Vienna were not among the pile on Agatha's sewing table. But how stupid of me, she thought, socks don't last thirty years, and how could a man in Henry's position wear crimson socks! The ones Belinda noticed tonight were of the most sober archidiaconal colours. She wanted to say, 'Oh, Agatha, let me help you,' but thought better of it. Agatha might consider it a reflection on her darning, and certainly would not care to be reminded that Belinda had darned socks for Henry before she had ever set eyes on him. 'Never interfere between husband and wife', as Belinda remembered her dear mother telling her, and one could not be too careful, even about an apparently trivial thing like a sock.\n\nAgatha seemed pleased to see her and they began to talk of parish matters. There would be a great deal to do when Agatha went away, indeed, she was beginning to wonder whether it would be possible to go after all.\n\nBelinda reflected on the truth of the saying that husbands and wives grow to be like each other, for it might almost have been Henry talking. So badly in need of a holiday, and yet who could be left in charge of the Mothers' Union and there was that rather delicate affair of the altar brasses and the unpleasantness between Miss Jenner and Miss Beard... listening to her Belinda began to feel very gloomy indeed. It seemed almost as if Agatha had decided not to go.\n\n'I'm sure I should be very willing to do what I could,' she said doubtfully, aware that she was not a mother and was far too much of a moral coward to deal satisfactorily with even the slightest unpleasantness. 'I often go into Miss Jenner's shop to buy knitting wool,' she added, 'perhaps I could say a word....'\n\nTo Belinda's surprise, Agatha seemed grateful for her offer of help and they found themselves talking about Mr. Donne and what a great asset he was to the parish, after which it was the most natural thing in the world for Belinda to ask after the Archdeacon.\n\nAgatha smiled indulgently. He was well, considering everything, she said.\n\nConsidering what? Belinda wondered, and ventured to remark that men were really much more difficult to please than women, who bore their burdens without complaining.\n\nAgatha nodded and sighed. There was a short pause during which Agatha seemed to be intent on finding a piece of wool to match the sock she was mending. Belinda took up the _Church Times_ and began glancing idly through the advertisements. A priest's cloak for sale, 44-inch chest\u2014clerical evening dress, tall, slim build, never worn\u2014she paused, wondering what story, sad or dramatic, lay behind those words. She had just turned to the back pages and was wondering whether Harriet would care to spend part of their summer holiday at a Bright Christian Guest House at Bognor, when the door opened and the Archdeacon came in.\n\nHe kissed Agatha in a hasty, husbandly way, which rather surprised Belinda, who had not thought that any outward signs of affection ever passed between them. Perhaps it distressed her a little, too, but he seemed so genuinely pleased to see her that she soon recovered and was listening happily to his account of how he had spent the afternoon, visiting a deathbed and then going on to see the old people in the workhouse.\n\n'These humble people remind me of Gray's _Elegy,'_ he said affectedly with his head on one side.\n\nNeither Belinda nor Agatha had heard his conversation with Harriet, so that they listened with respectful interest while he quoted the appropriate verse. Nor were they in a hurry to be gone, as Harriet had been, and so did not say 'Oh, _quite'_ when he had finished but enlarged intelligently on the charming theme. Agatha Was reminded of _Piers Plowman,_ Belinda of the poetry of Crabbe, which she could not remember very exactly, but she felt she had to be reminded of something out of self-defence, for Agatha had got a First and knew all about _Piers Plowman._ Indeed, she seemed about to quote from it and would probably have done so had not the Archdeacon suddenly been reminded of Wordsworth and some suitable lines in _The Excursion._ Then he began to read from _The Prelude._ Belinda thought Agatha looked rather bored and fidgety, but she herself was delighted and lived happily in the past until the entry of Mr. Donne brought her back into the present.\n\n'Your sister brought me some delicious plums this afternoon,' he said, addressing Belinda, 'and some homemade cake and jelly. I'm afraid I'm getting quite spoilt.'\n\nThe Archdeacon looked envious. The plums in their garden hadn't done particularly well this year and Agatha was always too busy with parochial work to make jelly or cakes or even to ask the cook to make them.\n\n'Ah, well, you won't always be a curate,' said Belinda indulgently.\n\n'That doesn't follow at all,' said the Archdeacon. 'Look at Plowman and all the gifts he gets. I suppose it has something to do with celibacy.'\n\nAgatha smiled complacently. 'Well, dear, people know that you are not in need of these things,' she said.\n\n'I will bring you some of the plums tomorrow,' said the curate nobly.\n\n'Now, you see, I have given Donne a chance to be unselfish,' said the Archdeacon, 'so good comes out of everything.'\n\nBelinda was silent, wondering if by any chance there were any plums left and whether she would have the courage to bring the Archdeacon a pot of the blackberry jelly which she herself had made a week or two ago. Perhaps when Agatha went away... a cake, too, perhaps with coffee icing and filling and chopped nuts on the top, or a really rich fruit cake....\n\n'We really must do something about the Harvest Festival,' said the Archdeacon wearily. 'I suppose I shall have to get Plowman to preach. A pity Canon Harvey is such a difficult man, he's really a better preacher.'\n\n'Ah, yes,' Agatha nodded sympathetically.\n\nBelinda did not join in the conversation. She remembered that the Archdeacon and Canon Harvey had a long-standing quarrel about the use of _Songs of Praise_ in church, which the latter considered 'savoured of Pantheism in many instances'. They had had a heated correspondence about it in the local paper.\n\n'Of course Plowman knows a good deal about the technical side of farming,' said the Archdeacon. 'That is some advantage.'\n\n'Yes, indeed,' said the curate earnestly. 'He believes that digging is a kind of worship and that we get nearer to God by digging. At least, it may not be exactly that,' he stammered in confusion, 'but of course one does feel that the countryman is nearer God, in a way.'\n\n'Nature's cathedral,' retorted the Archdeacon scornfully. 'One sees what you mean, of course.'\n\n'I think Mr. Donne was remembering the Latin _colere,_ which has the double meaning of dig and worship, as in cult and agriculture,' said Agatha helpfully. 'You explained it so well in your sermon about the spiritual meaning of harvest time,' she added, turning to her husband. 'It would be nice to hear that again.'\n\nThe Archdeacon nodded and looked pleased. 'Yes, I think that may be the solution,' he said. 'I felt at the time that it was perhaps too subtle for some of the congregation.'\n\nBelinda was silent with admiration. What a splendid wife Agatha was! She could never have dealt with him half so cleverly herself, she thought humbly. She remembered the sermon, of course, but it had been so obscure, that even she had been forced to abandon all efforts to understand it.\n\n'Oh, Mrs. Hoccleve,' burst out the curate eagerly, 'I nearly forgot to tell you, I had a pair of hand-knitted socks from Olivia Berridge. Wasn't it nice of her?'\n\n'Yes, she mentioned you the last time I heard from her,' said Agatha thoughtfully, and then began to explain to Belinda that Olivia Berridge was a niece of hers whom Mr. Donne had met when he was an undergraduate.\n\n'We both used to sing in the Bach Choir,' explained the curate, making the acquaintance sound respectable, even dull, Belinda thought.\n\n'She's a very clever girl,' Agatha went on, 'and she's doing some really excellent work on certain doubtful readings in _The Owl and The Nightingale.'_ She sighed, and looked down at the sock she was mending. 'I envy her that opportunity.'\n\n'Well, my dear, there is no reason why you shouldn't get down to something like that yourself,' said the Archdeacon. 'I am sure you have more time to spare than I have.'\n\n'I do so admire people who do obscure research,' said Belinda. 'I'm sure I wish I could.'\n\n'Of course I have done a good deal of work on Middle English texts myself in the past,' said Agatha, smiling.\n\n'Now, Agatha, Belinda does not wish to be forced to admire you,' said the Archdeacon. 'After all, academic research is not everything. We must remember George Herbert's lines:\n\n_A servant with this clause_\n\n_Makes drudgery divine,_\n\n_Who sweeps a room as for Thy laws,_\n\n_Makes that and the action fine._\n\n'Yes, they are comforting,' Belinda agreed. 'And yet,' she went on unhappily, 'I don't sweep rooms, Emily does that. The things I do seem rather useless, but I suppose it could be applied to any action of everyday life, really.'\n\n'Oh, certainly, Miss Bede,' said Mr. Donne, with curately heartiness. 'We cannot all have the same gifts,' he added, with what Belinda felt was an insufferably patronizing air.\n\n'Olivia is a very forceful young woman,' said the Archdeacon, 'and rather a bluestocking in appearance. What do you think, Donne?'\n\n'Well, I can't say that I've really noticed,' said the curate. 'I mean, it's what a person _is_ that matters most, isn't it?'\n\n'Ah, yes, the clergy at any rate should feel that,' said the Archdeacon sardonically. 'It might be an idea for one of your sermons, Donne. You could take the lilies of the field text and work it out quite simply. I'm not sure that I won't take it myself, though. It might be a way of reaching the evening congregations, they like something of that kind. Never waste your erudite quotations on them, they don't appreciate or understand them.'\n\nThe curate murmured something about not really knowing any erudite quotations, at which the Archdeacon nodded and looked satisfied.\n\nAgatha rolled up the last pair of socks, and there was a pleasant silence, during which Belinda became rather sentimental as she contemplated the cosy domestic scene. Agatha, surrounded by the socks and her affectionate husband, dear Agatha, almost; it was very seldom that Belinda was able to think of her like that. We really ought to love one another, she thought warmly, it was a pity it was often so difficult. But as she walked home, her thoughts took a more definite and interesting turn. She began to wonder if perhaps Mr. Donne loved Olivia Berridge. By the time she had reached her own house, however, she had decided that the whole idea was so upsetting that it could not possibly be so. In any case, he would not have the chance of seeing her very often, and a few pairs of socks through the post could not really do very much.\n\n# Chapter Seven\n\nWHEN THE DAY CAME for Agatha to go away, Belinda and Harriet watched her departure out of Belinda's bedroom window. From here there was an excellent view of the vicarage drive and gate. Belinda had brought some brass with her to clean and in the intervals when she stopped her vigorous rubbing to look out of the window, was careful to display the duster in her hand. Harriet stared out quite unashamedly, with nothing in her hand to excuse her presence there. She even had a pair of binoculars, which she was now trying to focus.\n\nThe sisters had said goodbye to Agatha the day before. Belinda was sure that she would rather be alone on her last morning to say goodbye to her husband, and there were always so many last-minute things to see to that the presence of strangers could be nothing but a hindrance, she thought. She really felt quite unhappy to think of Agatha and the Archdeacon being parted, for the cosy domestic scene which she had witnessed on her last visit to the vicarage had made a deep impression, and she felt that she ought to keep reminding herself of it. Of course they did have their little differences, there was no denying that, but it was equally certain that they were devoted to each other and that Agatha was an admirable wife.\n\nBelinda and Harriet had been at their posts by the window for about ten minutes before there was any sign of life at the vicarage. Harriet had suggested that they should be there early, as, according to her calculations, Agatha would have to start for the station at least twenty minutes before the train went at half-past eleven. To watch anyone coming or going in the village was a real delight to them, so that they had looked forward to this morning with an almost childish excitement. And yet it was understandable, for there were so many interesting things about a departure, if one could watch it without any feeling of sorrow or regret. What would Agatha wear? Would she have a great deal of luggage or just a suitcase and a hat-box? Would the Archdeacon go with her to the station in the taxi, or would he be too busy to spare the time? If he did _not_ go to the station would he kiss Agatha goodbye before she got into the taxi, or would he already have done that in the house? Belinda and Harriet were busy discussing these interesting questions when Harriet gave a little cry of pleasure and amusement.\n\n'Oh, look,' she exclaimed, 'the curate in his shirt-sleeves!'\n\nBelinda looked. It was indeed the curate, wearing no coat and carrying a large round hat-box. As far as she could see he looked flushed and dishevelled.\n\n'I do hope they didn't make him carry the trunk downstairs,' she said, peering anxiously through the field-glasses. 'He looks rather tired.'\n\nThe next person to appear in the drive was the Archdeacon. He was carrying a suitcase and looking round him uncertainly, as if he did not know what to do with it. But at this moment a taxi appeared, so he advanced towards it with a threatening air.\n\n'That old car of Palmer's!' exclaimed Harriet in disgust. 'All the stuffing's coming out of the seats! I suppose the Archdeacon was too mean to order Haines.'\n\n'Oh, Harriet, I'm sure it wasn't that,' said Belinda loyally. 'Probably Haines was engaged for this morning, and anyway, I don't think Palmer is any cheaper.'\n\n'They've got plenty of time,' said Harriet, looking at her watch, 'but I expect the Archdeacon wants to make quite sure she doesn't miss the train. I expect they'll be glad to get away from each other for a bit,' she added.\n\nBelinda was about to contradict her sister and remind her of what a devoted couple the Hoccleves were, when Agatha herself appeared, carrying a fur coat over her arm and a small dressing-case.\n\n'Oh, that's the case with gold fittings, isn't it?' said Harriet. 'I always think it must be so heavy, though. I don't like her hat very much, it makes her face look too sharp.'\n\nBelinda suddenly felt that there was something indecent about their curiosity and turned away to clean the brass candlesticks on the mantelpiece. But nothing would move Harriet from the window. She kept up a flow of comments on Agatha's clothes, the behaviour of the Archdeacon and the curate. Belinda only hoped nobody could see her, with the field-glasses glued to her eyes. It would look so bad, somehow, though she did not doubt that others in the village were doing exactly the same thing.\n\nBelinda went downstairs, humming _God moves in a mysterious way,_ and telling herself that it was not right that she should feel relieved because Agatha was going away. Of course she was glad that Agatha was to have a well-deserved holiday and the waters would undoubtedly help her rheumatism, so there was room for gladness, but she ought not to have to tell herself this after the first thought that came into her mind had been how nice it would be to be able to ask Henry in to tea or supper without having to ask Agatha as well.\n\nShe went into the kitchen with a rather firmer step than usual and quite startled Emily, who was reading the _Daily Minor_ over her mid-morning cup of tea.\n\n'Oh, Emily, I hoped you would have got on to the silver by now,' she said. 'Miss Harriet and I have done the bedrooms'\u2014she paused guiltily\u2014'and I think I will see to the lunch myself. I am going to make a risotto out of the chicken that was left over.'\n\n'Yes, Miss Bede.' Emily began to assemble the materials for silver cleaning. 'I see Mrs. Hoccleve's gone,' she remarked.\n\n'Oh, yes, it was today she was going,' said Belinda casually.\n\n'I hope she won't come to any harm, you never know with foreigners, do you?' said Emily.\n\n'An English gentlewoman can never come to any harm,' said Belinda, more to herself than to Emily.\n\n'But you do hear of people having nasty things happen to them,' persisted Emily. 'I've read it in the papers. But of course Mrs. Hoccleve's elderly, really, isn't she, so it's different?'\n\nBelinda was silent. She felt she could hardly agree that Agatha was elderly when she herself was a year older and thought of herself as only middle-aged. And yet, middle-aged or elderly, what was the difference really? _Calm of mind, all passion spent..._ she had known that before she was thirty. 'Don't waste the Silvo like that, Emily,' she said with unaccustomed sharpness, 'you won't get a better polish. It's the rubbing that does it.'\n\nThe sound of heavy footsteps on the stairs told Belinda that Harriet had finished her business there, and as the kettle was boiling, she made a pot of tea and took it into the dining-room.\n\n'I think I shall see if I can alter my black coat and make the sleeves like Agatha's,' Harriet was saying, half to herself. 'Do you think there is anything to let out on the seams?'\n\n'Your coat is so nice as it is,' said Belinda doubtfully, for she had had experience of Harriet's attempts at alteration. 'Altering a coat is so much more difficult than a dress.'\n\n'Yes,' agreed Harriet gravely, 'I think you're right. I might buy some of that leopard-skin trimming though and put it on the cuffs and pockets. That would be a change, and sleeves are going to be _important_ this winter, I believe.'\n\n'Have they got Agatha away safely?' asked Belinda casually.\n\n'Oh, yes,' said Harriet, in a more cheerful tone of voice. 'Mr. Donne went in the taxi with her. I suppose he would see her off at the station. And do you _know,'_ she leaned forward eagerly, 'the Archdeacon didn't even kiss her. He just waved his hand, like this.' Harriet gave a rather improbable imitation of how the Archdeacon had said goodbye to his wife.\n\n'I expect they said their real goodbyes in the house,' said Belinda. 'After all it's rather upsetting, isn't it, a parting like that?'\n\n'The Archdeacon didn't look in the least bit upset,' said Harriet. 'After the taxi had gone he stood in the drive grinning and rubbing his hands, looking as pleased as Punch.'\n\n'Oh, no, Harriet, I can't believe that,' said Belinda, and so, comfortably arguing, they drank their tea and were just finishing it when there was a cry from Harriet, who pointed in the direction of the window.\n\n'Look,' she cried, for she had been so absorbed in her task of 'strengthening' a pair of corsets with elastic thread that she had not noticed the Archdeacon creeping up the drive. Neither had Belinda, but she was less observant and sharp.\n\n'I thought I would take you by surprise,' he said. 'I am glad to find you both engaged in the trivial round, the common task.'\n\nBelinda was too agitated to think of any clever reply, while Harriet was bundling the corsets under a cushion in one of the armchairs. Belinda noticed to her horror that they were imperfectly hidden and planted herself firmly in front of the chair. It was too bad of Harriet to make these little embarrassments. The two cats were curled up in the basket-chair on the other side of the fire, so it was quite a problem to know where to seat the Archdeacon. But Harriet recovered her composure more quickly than Belinda, turned out the cats with a quick movement and offered him the chair.\n\n'I'm afraid we have annoyed them,' said the Archdeacon, 'they are looking positively baleful. And yet I feel that I need rest more than they do.' He sighed and stretched out his hands to the warmth of the fire.\n\n'We always call them the brethren dwelling in unity,' said Harriet. _'Behold how good and joyful a thing it is, brethren, to dwell together in unity,'_ she quoted, as if by way of explanation. 'The psalm, you know....'\n\n'Of course he knows,' said Belinda rather sharply, and yet it was odd how one sometimes felt that he might _not._ She began to wonder why he had come; it was unusual for him to call in the morning.\n\n'I expect you know Agatha has just gone,' said the Archdeacon, in answer to her thoughts. 'Such a business getting her to the station, I really feel quite exhausted. These departures are always more tiring for those who are left behind.'\n\n'Oh, dear, we should have offered you some tea,' said Belinda reflecting that it was in fact Mr. Donne who had gone to the station with Agatha. 'We had ours some time ago so it won't be very nice. I'll get Emily to make you some more.'\n\n'Well, that is kind of you, but I had some refreshment at the vicarage,' said the Archdeacon. 'I really felt justified in having something.'\n\nBelinda nodded sympathetically, but she could see Harriet looking scornful and so began talking quickly about the Harvest Festival and the decorations which were to be done the next day.\n\n'We must have more corn this year,' said the Archdeacon. 'Corn is an essential part of harvest, perhaps the most important part of all.'\n\n'Ah, yes, bread is the staff of life,' said Harriet solemnly. 'But we mustn't forget the other fruits of the earth. Ricardo Bianco has some very fine marrows and pumpkins, and bigger things really show up better.'\n\n'The church always looks very nice,' said Belinda, fearing they were going to have an argument.\n\n'Yes, there are always plenty of willing helpers,' said the Archdeacon complacently.\n\n'I do hope there won't be any unpleasantness this year.' said Belinda, her face clouding. 'Last year there was the embarrassment of Miss Prior, if you remember.'\n\n'The Embarrassment of Miss Prior,' said the Archdeacon, savouring the words. 'It sounds almost naughty, but I fear it was not. I cannot recall the circumstances.'\n\n'Oh, I remember,' said Harriet. 'When Miss Prior came to decorate it was found that somebody else had already done the lectern and she's always done it for the last twenty years or more.'\n\n'Yes, poor little soul,' said Belinda reminiscently, 'she was rather late. She had been finishing some curtains for Lady Clara Boulding\u2014you know, those heavy maroon velvet ones in her morning-room\u2014and she was nearly crying. She does so enjoy doing the lectern and making a bunch of grapes hang down from the bird's mouth. Of course the only disadvantage is that they do distract the Sunday School children's attention so; last year they were very much inclined to giggle\u2014Miss Jenner and Miss Smiley had a very difficult time with them.'\n\n'If only they would try to teach them that it is perfectly right and fitting that we should bring the fruits of the earth into God's House at Harvest Time,' said the Archdeacon rather peevishly.\n\n'But children don't understand things like that,' said Belinda, 'and in any case young people are so prone to giggle. I can remember I was.'\n\nHarriet chortled reminiscently at some schoolgirl joke, but would not reveal it when asked.\n\nEventually the Archdeacon stood up to go and Belinda was about to hurry to the kitchen to start preparing the risotto, when Harriet pointed towards the Archdeacon's left foot and exclaimed loudly, 'Oh, you've got a hole in your sock!'\n\n'Damn,' said the Archdeacon firmly and unmistakably. 'I suppose it was too much to hope that my clothes would be left in order.'\n\n'I expect Agatha doesn't like darning,' said Harriet tactlessly. 'I'm not at all fond of it myself, so I can sympathize.'\n\n'Oh, but a sock is liable to go into a hole at any time,' said Belinda hastily. 'It doesn't look a very big one. Perhaps it could be cobbled together...' she was already rummaging in her work basket for some wool of the right shade. 'I'm afraid this grey is rather too light,' she said, 'but I don't think it will show very much.'\n\n'Oh, that doesn't matter,' said the Archdeacon impatiently. 'What a fuss it all is over such a trifling matter.'\n\nBelinda smiled as she threaded her needle. Dear Henry, he was so inconsistent, but perhaps a hole in a sock was hardly as important as moths in a suit. 'I think it would be best if you put your foot up on this little chair,' she said, 'then I can get at your heel to mend the sock.'\n\nThe Archdeacon submitted himself to her ministrations with rather an ill will, and there was one anxious moment when Belinda inadvertently pricked him with the needle and it seemed as if he would lose his temper.\n\nHarriet did her best to divert him with conversation and eventually he recovered his good humour and began to ask her the origin of her elusive quotation about the Apes of Brazil. He thought that it might be Elizabethan, it reminded him of that poem with the lines about making Tullia's ape a marmoset and Leda's goose a swan.\n\n'I don't remember anything about the Apes of Brazil,' said Belinda anxiously, for the darning of the sock was an all-engrossing occupation.\n\n'Do you mean what I said that afternoon we met in the village?' asked Harriet. 'That's not a quotation, that's natural history.' She laughed delightedly.\n\nThe Archdeacon seemed surprised and Harriet began to explain.\n\n'It's quite simple, really,' she said. 'When the Apes of Brazil beat their chests with their hands or paws, or whatever apes have, you can hear the sound two miles away.'\n\n'Oh, Harriet,' said Belinda, as if reproving a child, 'surely not two _miles?_ You must be mistaken.'\n\n'Two miles,' said Harriet firmly. 'Father Plowman told me.'\n\nThe Archdeacon laughed scornfully at this.\n\n'It was at Lady Clara Boulding's house,' said Harriet indignantly. 'We were having a most interesting conversation, I can't remember now what it was about.'\n\n'I cannot imagine what the subject of it can have been,' said the Archdeacon, 'and I did not know that Plowman had ever been in Brazil.'\n\n'You said something about sentiments to which every bosom returns an echo,' said Harriet, 'so I naturally thought of the Apes of Brazil.'\n\n'I think the minds of the metaphysical poets must have worked something like that,' said Belinda thoughtfully. 'Donne and Abraham Cowley, perhaps.'\n\n'Cowley was a very stupid man,' said the Archdeacon shortly. 'I cannot understand the revival of interest in his works.'\n\n'I think the hole is mended now,' said Belinda. 'It doesn't look so bad now; of course the wool _is_ just a little too light.'\n\n'My dear Belinda, you have done it quite exquisitely,' said the Archdeacon. 'I must take care to be passing your house every time I have a hole in my sock.'\n\nBelinda smiled and went quite pink with pleasure and confusion. She went with him to the front door and then returned to the dining-room where Harriet had collapsed heavily into a chair and was fanning herself with the parish magazine.\n\n'Thank goodness, he's gone,' she said. 'I really don't know how Agatha manages to put up with him all the time. No wonder she's gone away.'\n\n'Harriet, _do_ speak more quietly,' said Belinda in an agitated whisper, for Emily had just come into the room to lay the table. 'I must go and start the risotto,' she said and went into the kitchen, where she walked aimlessly about in circles trying to assemble all the ingredients she needed. For somehow it was difficult to concentrate. The mending of the sock had been an upsetting and unnerving experience, and even when she had made the risotto she did not feel any pleasure at the thought of eating it.\n\n'Nearly twenty-past one!' said Harriet, as they sat down to their meal. 'The Archdeacon has delayed everything. I suppose he imagined Emily would be cooking.'\n\n'I don't suppose he thought about it at all, men don't as a rule,' said Belinda, 'they just expect meals to appear on the table and they do.'\n\n'Of course Emily usually does cook,' went on Harriet, 'it's only that she can't manage foreign dishes.' She took a liberal second helping of risotto. 'This is really delicious.'\n\n'It was Ricardo's recipe,' said Belinda absently.\n\n'We really must go and get some more blackberries soon,' said Harriet. 'Although in October the devil will be in them. You know what the country people say.'\n\nBelinda smiled.\n\n'Mr. Donne is very fond of blackberry jelly,' said Harriet. 'Apparently he very much enjoyed the apple jelly I took him. He said he really preferred it for breakfast\u2014instead of marmalade, you know.'\n\n'I wonder what it would be like to be turned into a pillar of salt?' said Belinda surprisingly, in a far-away voice.\n\n'Belinda!' Harriet exclaimed in astonishment. 'Whatever made you think of that? Potiphar's wife, wasn't it, in the Old Testament somewhere?'\n\n'I think it was Lot's wife,' said Belinda, 'but I can't remember why. I should imagine it would be very restful,' she went on, 'to have no feelings or emotions. Or perhaps,' she continued thoughtfully, 'it would have been simpler to have been born like Milton's first wife, an image of earth and phlegm.'\n\n'Oh, Belinda, don't be disgusting!' said Harriet briskly. 'And do pass the cheese. You are hopelessly inattentive. When Mr. Donne was here the other night you never passed him anything. If it hadn't been for me he would have _starved.'_\n\nBelinda came back to everyday life again. How many curates would starve and die were it not for the Harriets of this world, she thought. 'I'm sorry, dear,' she said. 'I must try not to be so absent-minded. Today has been rather trying, hasn't it really\u2014too much happening.'\n\n'Yes,' agreed Harriet. 'Agatha going and the Archdeacon coming. Who knows what he may be up to now that she's gone?'\n\n'Oh, Harriet, I wish you wouldn't talk like that,' said Belinda. 'It's really most unsuitable. And besides,' she went on, half to herself, 'what could he be up to when you come to think of it?' Her voice trailed off rather sadly, but she rose from the table briskly enough and spent the afternoon doing some useful work in the garden.\n\n# Chapter Eight\n\nTHE NEXT DAY BELINDA had a letter from Dr. Nicholas Parnell, a friend of her undergraduate days and the Librarian of her old University Library. He wrote of the successful tour which Mr. Mold, the deputy Librarian, had made in Africa. 'He has penetrated the thickest jungles,' wrote Dr Parnell, 'where no white man, and certainly no deputy Librarian, has ever set foot before. The native chiefs have been remarkably generous with their gifts and Mold has collected some five thousand pounds, much of it in the form of precious stones and other rareties. I suspect that a great many of them have not the slightest idea to what they are contributing, but, where Ignorance is bliss...'\n\nBelinda sighed. Dear Nicholas was really quite obsessed with the Library and its extensions. She wished he would remember that the two things which bound them together were the memory of their undergraduate days and our greater English poets. She turned to the end of the letter, where she found more cheering news. The Librarian thought he might be able to come and spend a few days with the Archdeacon while Agatha was away. Perhaps Mr. Mold would come too. 'The Library can safely be left in charge of old Mr. Lydgate,' he concluded. 'He is a little wandering now and is continually worrying about and pronunciation of the Russian \"l\". However, his duties will be light.'\n\nHow nice it would be to see dear Nicholas again, thought Belinda, eating her scrambled egg and feeling happy and proud that she, a middle-aged country spinster, should number famous librarians among her friends. At least, the Library was famous, she emended. Dear Nicholas had rather sunk into obscurity since his scholarly publications of twenty years ago, and now that he had definitely abandoned all intellectual pursuits, she assumed that no more in that line was to be expected from him. Still, _Floreat Bibliotheca,_ and she was sure that under his guidance it would. And, what was perhaps even more important, the Library would be adequately heated and the material comfort of the readers considered. For who can produce a really scholarly work when he is sitting shivering in a too heavy overcoat, struggling all the time against the temptation to go out and get himself a warming cup of coffee?\n\nThe same afternoon Belinda went into the village to do a little shopping. She had to give an order at the grocer's and the butcher's, and, if there was time, she would go and choose some wool to make Ricardo Bianco a nice warm pair of socks. She wondered if he had tried taking calcium tablets for his chilblains; they were supposed to be very good.\n\nShe entered the wool shop, kept by Miss Jenner, who was also a Sunday School teacher. She always liked going to Miss Jenner's as the attractive display of different wools fired her imagination. Harriet would look splendid in a jumper of that coral pink. It would be a good idea for a Christmas present, although it was impossible to keep anything secret from Harriet owing to her insatiable curiosity. And here was an admirable clerical grey. Such nice soft wool too... would she ever dare to knit a pullover for the Archdeacon? It would have to be done surreptitiously and before Agatha came back. She might send it anonymously, or give it to him casually, as if it had been left over from the Christmas charity parcel. Surely that would be quite seemly, unless of course it might appear rather ill-mannered?\n\n'This is a lovely clerical grey,' said Miss Jenner, as if sensing her thoughts. 'I've sold quite a lot of this to various ladies round here\u2014especially in Father Plowman's parish. I was saying to the traveller only the other day that I knew this would be a popular line. He even suggested I might knit _him_ a pullover'\u2014she laughed shrilly\u2014'the idea of it!'\n\nBelinda smiled. She could well imagine the scene. Miss Jenner was so silly with the travellers that it was quite embarrassing to be in the shop when one of them arrived. Still, poor thing, Belinda thought, the warm tide of easy sentimentality rising up within her, it was probably the only bit of excitement in her drab life. She was getting on now, and with her sharp, foxy face and prominent teeth had obviously never been pretty. Living over the shop with her old mother must be very dull. And perhaps we are all silly over something or somebody without knowing it; perhaps her own behaviour with the Archdeacon was no less silly than Miss Jenner's with the travellers. It was rather a disquieting thought, especially when Miss Jenner, with a smirk on her face, began to tell her that eight ounces was the amount of wool that ladies usually bought.\n\n'It will go very well with my Harris tweed costume,' said Belinda firmly. 'I think I will have _nine_ ounces, in case I decide to make long sleeves.' After all, she _might_ make a jumper for herself, now that she came to think of it she was certain that she would, either that or something else equally safe and dull. When we grow older we lack the fine courage of youth, and even an ordinary task like making a pullover for somebody we love or used to love seems too dangerous to be undertaken. Then Agatha might get to hear of it; that was something else to be considered. Her long, thin fingers might pick at it critically and detect a mistake in the ribbing at the Vee neck; there was often some difficulty there. Agatha was not much of a knitter herself, but she would have an unfailing eye for Belinda's little mistakes. And then the pullover might be too small, or the neck opening too tight, so that he wouldn't be able to get his head through it. Belinda went hot and cold, imagining her humiliation. She would have to practise on Harriet, whose head was fully as big as the Archdeacon's. And yet, in a way, it would be better if Harriet didn't know about it, she might so easily blurt out something... Obviously the enterprise was too fraught with dangers to be attempted and Belinda determined to think no more about it. _God moves in a mysterious way,_ she thought, without irreverence. It was wonderful how He did, even in small things. No doubt she would know what to do with the wool as time went on.\n\nThis afternoon Belinda had naturally hoped that she might meet the Archdeacon, but it was now nearly tea-time, and although she had been through the main street and all the most likely side streets, Fate had not brought them together. She decided that there was nothing for it but to go home; after all, there would be many more opportunities.\n\nBut when she had got as far as the church, she saw a familiar figure wandering about among the tomb-stones, with his hands clasped behind his back and an expression of melancholy on his face. It was, of course, the Archdeacon. But what was he doing in the churchyard when it was nearly tea-time? Belinda wondered. This would hardly be a suitable time to interrupt his meditations by telling him that she had had a letter from Nicholas Parnell and that she did hope they would both come to supper when he came to stay. She began to walk rather more slowly, uncertain what to do. She looked in her shopping basket to see if she had forgotten anything. She remembered now that the careful list she had made was lying on top of the bureau in the dining-room, so she could hardly expect to check things very satisfactorily. There was no reason why she should not hurry home to tea.\n\n'These yew trees are remarkably fine,' said a voice quite close to her, 'they must be hundreds of years old.'\n\nBelinda looked up from her basket. The Archdeacon had now come to the wall.\n\n'Oh, good afternoon,' she said, hoping that he had not noticed her obvious reluctance to go home. 'You quite startled me. I didn't see you,' she added, hoping that she might be forgiven or at least not found out, in this obvious lie.\n\nThe Archdeacon smiled. 'I was thinking out my sermon for Sunday,' he said. 'I find the atmosphere so helpful. Looking at these tombs, I am reminded of my own mortality.'\n\nBelinda contemplated a design of cherubs' heads with a worn inscription underneath it. 'Yes, indeed,' she said, hoping that the gentle melancholy of her tone would make amends for her trite reply.\n\n'I have lately been reading Young's _Night Thoughts,'_ went on the Archdeacon, in his pulpit voice. 'There are some magnificent lines in it that I had forgotten.'\n\nBelinda waited. She doubted now whether it would be possible to be back for tea at four o'clock. She could hardly break away when the Archdeacon was about to deliver an address on the mortality of man.\n\nHe began to quote:\n\n_We take no note of time_\n\n_But from its loss. To give it then a tongue_\n\n_Is wise in man. As if an angel spoke,_\n\n_I feel the solemn sound. If heard aright_\n\n_It is the knell of my departed hours..._\n\n'I thought of those lines when I heard the clock strike just now,' he explained.\n\n'It must be wonderful, and unusual too, to think of time like that,' said Belinda shyly, realizing that when she heard the clock strike her thoughts were on a much lower level. She suspected that even dear Henry was guilty of more mundane thoughts occasionally. At four o'clock in the afternoon, surely the most saintly person would think rather of tea than of his departed hours? She stood silent, looking into her basket.\n\n'Not that Young was a great theologian, or even a great poet,' the Archdeacon went on hastily. 'Much of the _Night Thoughts_ consists of platitudes expressed in that over-elaborate and turgid style, which the minor eighteenth-century poets mistakenly associated with Milton.'\n\n'Oh, yes, the style is certainly rather flowery,' said Belinda, doing the best she could, for she was beginning to be uneasily conscious of Harriet waiting for her tea, the hot scones getting cold and Miss Beard, that excellent church worker and indefatigable gossip, passing by on the other side of the road.\n\n'That may be, but I do find in it a little of the wonder and awe which is generally supposed to be absent from the literature of that age.' The Archdeacon stood looking at Belinda with his head on one side, as if he expected her to agree with him.\n\nBut Belinda said nothing, for she was thinking how handsome he still was. His long pointed nose only added to the general distinction of his features. There was quite a long pause until the clock struck a quarter-past four.\n\n'Tea,' said the Archdeacon, suddenly human once more. 'I'm all by myself,' he said rather pathetically. 'Won't you come and share my solitary meal? I don't know if there will be any cake,' he added doubtfully.\n\nBelinda started. 'Oh, _no,'_ she said, drawing back a little, and then remembering her manners, she added: 'Thank you very much but Harriet will be expecting me.' She did not dare to invite him to share their undoubtedly more appetizing meal and almost smiled when she pictured what Harriet's reaction would be were she to bring him home unexpectedly. All the same it would have been very nice to have had tea with him, she thought regretfully, quite like old times. Perhaps he would ask her again, though it was the kind of spontaneous invitation that comes perhaps only once in a lifetime. 'You must come to tea with us some time,' she said, doing her best to assume a light, social manner. 'I will ask Harriet what is the best day, though,' she added hastily, 'I expect you are very busy.'\n\n'Ah, yes,' he sighed, 'nobody can possibly know how busy.'\n\n'Then I mustn't keep you any longer,' said Belinda, moving away.\n\n'Well, the tombs are always with us,' he replied enigmatically, raising his hat with a sweeping gesture.\n\nBelinda could think of nothing to say to this, so she smiled and walked home very quickly. As she had expected, Harriet was waiting impatiently in the drawing-room. The tea was already in, and the hot scones stood in a little covered dish in the fireplace.\n\n'Oh, Belinda, when _will_ you learn to be punctual,' she said, in a despairing voice.\n\n'I'm so sorry, dear,' said Belinda humbly. 'I should have been here by four, but I met the Archdeacon.' She looked about her rather helplessly for a place to put her coat. 'I'm sorry you waited tea for me.'\n\n'Well, I was rather hungry,' said Harriet nobly, 'but having to wait will make me enjoy it all the more. What meat did you order?'\n\n'Mutton,' said Belinda absently.\n\n'But we haven't any red-currant jelly,' said Harriet. 'One of us will have to go out tomorrow morning and get some. Mutton's so uninteresting without it.'\n\nBelinda sat down by the fire and began to pour out the tea.\n\n'Where did you see the Archdeacon?' asked Harriet.\n\n'In the churchyard,' said Belinda. 'He was walking about among the tombs.'\n\nHarriet snorted.\n\n'But, Harriet,' Belinda leaned forward eagerly, 'he asked me to go to tea with him, but of course I couldn't very well have gone.'\n\n'I don't see why not,' said Harriet. 'I can't believe you didn't want to.'\n\n'No, it wasn't exactly that,' said Belinda slowly. 'I didn't really mind one way or the other,' she lied, 'but I knew you would be expecting me back and I thought you might wonder where I was. And then Florrie and the cook might have thought it funny if I went there the minute Agatha was out of the house. You know how servants gossip, especially in a small place like this. I don't want to be silly in any way, of course there would have been nothing _in_ it, but I decided it would be better if I didn't go.' She put the rest of her scone into her mouth with an air of finality.\n\nHarriet was obviously disappointed. 'I do wish you'd gone,' she lamented. 'So little of interest happens here and one may as well make the most of life. Besides, dear,' she added gently, 'I don't think anybody would be likely to gossip about you in that old tweed coat.'\n\n'No, you're quite right. I suppose it will have to go to Mrs. Ramage next time she comes.' She got up and rang the bell for Emily to clear away the tea things. When she was going out with the tray, Emily turned to Harriet rather nervously and said, 'Excuse me, m'm, but would you mind if I just slip out to the post?'\n\nOh, no, Emily,' said Harriet firmly, 'there's no need for that. I shall be writing some letters myself, so I can take yours as well. There is plenty for you to do here.'\n\nEmily went out of the room with a sulky expression on her face, and was heard to bang the tray rather heavily on the table in the passage.\n\n'She only wants to go and gossip with the vicarage Florrie,' said Harriet, triumphant at having frustrated her. 'And we can't have that, can we?' she said turning to Belinda for support.\n\nBut Belinda was not listening. She was wondering what they would have talked about if she _had_ gone to tea, or rather what Henry would have talked about. It had started to rain outside, and the soft patter of the rain in the leaves, combined with the rapidly falling darkness, made her feel pleasantly melancholy. She wondered if Henry were looking at the twilight, missing Agatha, she thought dutifully, or even regretting that she had not stayed to tea. It _would_ have been nice to go... Belinda put down her knitting and sat dreaming. Of course there was a certain pleasure in not doing something; it was impossible that one's high expectations should be disappointed by the reality. To Belinda's imaginative but contented mind this seemed a happy state, with no emptiness or bitterness about it. She was fortunate in needing very little to make her happy.\n\nShe was still sitting idly with her knitting in her lap, when the front door-bell rang, and Miss Liversidge and Miss Aspinall were shown into the room.\n\n'We were just passing and thought we'd drop in,' Edith explained.\n\nThey stood in the doorway, a tall drooping figure and a short stout one, both wearing mackintoshes, and that wet-weather headgear so unbecoming to middle-aged ladies and so incongruously known as a 'pixie hood'.\n\n'Do take off your wet things,' said Belinda rousing herself.\n\n'You had better stay to supper,' said Harriet rather too bluntly. 'It won't be very much but we shall be having it soon.'\n\nWhy yes, it will be a good chance to repay the baked beans, thought Belinda. She wondered whether they ought perhaps to open a tin of tongue and get Emily to make a potato salad. Or would a macaroni cheese be better? With some bottled fruit and coffee to follow that should really be enough.\n\n'I think I'll just go and tell Emily about supper,' she said.\n\n'Oh, please don't trouble to make any difference for us,' said Connie. 'Bread and cheese or whatever you're having will do for us, won't it, Edith?'\n\nEdith gave a short bark of laughter. 'Well, I must say that I should like to feel an effort was being made, even if only a small one,' she said in a jocular tone. 'I think we all like to feel that.'\n\n'But we only came to see you,' said Connie. Her eyes brightened a little and she said in a low voice, 'We think we have a piece of news.'\n\n'News? What kind of news?' asked Harriet rather sharply.\n\n'We have heard that Mr. Donne is engaged,' said Edith, in a loud triumphant tone.\n\n'To a niece of Mrs. Hoccleve's, a Miss Berry,' chimed in Connie.\n\n'Miss Berridge, I think, if it's the niece who's doing research,' said Belinda, looking rather fearfully at her sister.\n\n'Oh, I don't think that can be true,' declared Harriet indignantly. 'She has made him a pair of socks, but I don't think there is anything more than that between them.'\n\n'Miss Prior told us,' persisted Edith, 'and she is usually very accurate. She has been a good deal at the vicarage lately, getting Mrs. Hoccleve's clothes ready to go away. She may very well have heard something.'\n\n'But Miss Berridge is some years older than Mr. Donne,' said Harriet, equally persistent. 'It would be a most unsuitable marriage. Besides,' she added, her tone taking on a note of disgust, 'she's doing some research or something like that, isn't she, Belinda.'\n\n'Yes, on some doubtful reading in _The Owl and the Nightingale._ It doesn't seem a very good training for a wife,' said Belinda uncertainly, thinking of Agatha and her inability to darn. 'Still, if she has knitted him a pair of socks perhaps she is not entirely lacking in the feminine arts.'\n\nEdith gave a snort. 'I believe some of these old poems are very _coarse,_ so she may not be such a blue stocking as we think.'\n\nThere was a short silence during which the front-door bell rang again and Mr. Donne was shown into the room carrying a bundle of parish magazines.\n\n'Miss Jenner couldn't manage to deliver them this month,' he explained, 'so I am doing it.'\n\n'Just the person we wanted to see,' said Harriet. 'Now, _you_ can surely tell us. Is it true that you are engaged to be married?' The words rang out as a challenge.\n\n'I\u2014engaged?' Mr. Donne made a kind of bleating noise and a movement with his arms which scattered the parish magazines all over the floor. 'It's certainly the first I've heard of it,' he went on, recovering something of his usual manner. 'Who is the fortunate lady?'\n\n'Miss Berridge,' said Edith Liversidge firmly.\n\n'Miss Berridge?' he echoed in a puzzled tone. 'Well, of course, she's a very good sort, and I like her very much...' he hesitated, perhaps feeling that he was being ungallant.\n\n'But you think of her more as an elder sister, I expect,' prompted Harriet with determination.\n\n'Well, yes, I suppose I do,' he agreed gratefully. 'Anyway she's much too clever to look at anyone like me.'\n\n'Is she beautiful?' persisted Edith.\n\n'Well, not exactly _beautiful,'_ he said, looking embarrassed, 'but very nice and so kind.'\n\n_Ah, had she been more beauteous and less kind,_\n\n_She might have found me of another mind._\n\nthought Belinda, but decided it might be better not to quote the lines.\n\n'Well, that's that,' said Harriet. 'There is no truth in the rumour. Isn't it amazing how people will gossip?'\n\n'I never thought there was,' said Connie to Belinda. 'I think Mr. Donne will marry some pretty _young_ thing.' She sighed and her eyes bulged sentimentally.\n\n'I may not get married at all,' said Mr. Donne almost defiantly. 'Many clergymen do not.'\n\n'No, a single curate is in many ways more suitable,' said Belinda thoughtfully. 'More in the tradition, if you see what I mean. And then of course there's the celibacy of the clergy isn't there?' she added quickly.\n\n'Is there?' said Edith scornfully. 'I thought St. Paul said it was better to marry than burn.'\n\n'Well, it is hardly a question of that,' said Belinda in a confused way. 'I mean, of burning. One would hardly expect it to be.' She felt rather annoyed with Edith, who must surely know less than anybody about what St. Paul had said, for introducing this unsuitable aspect of the question.\n\nFortunately, Harriet, who had disappeared from the room while she was speaking, now came back with the news that supper was ready.\n\n'You will stay, won't you, Mr. Donne?' she asked, turning to him with a beaming smile. 'I'm afraid it won't be much of a meal...' she waved her hands deprecatingly.\n\nEdith Liversidge moved into the dining-room with a confident step. They would all benefit from Mr. Donne's presence, she knew, and noted with sardonic approval that there was a large bowl of fruit salad on the table and a jug of cream as well as a choice of cold meats.\n\nOh dear, thought Belinda, recognizing tomorrow's luncheon, surely the tin of tongue would have been enough?\n\n'Let's all have a glass of sherry,' said Harriet, going over to the sideboard, where a decanter and glasses had been set out on a tray. 'After all, we _might_ have been going to drink to Mr. Donne's engagement.'\n\n# Chapter Nine\n\n'I SUPPOSE THEY REALLY _have_ come,' said Harriet I doubtfully. 'Emily is usually quite accurate in her information and she had this from the vicarage Florrie, who ought to know if anyone does. She told her that two gentlemen had arrived to stay at the vicarage last night, but of course we have no proof that it is Dr. Parnell and Mr. Mold. It might be two clergymen coming to see the Archdeacon about something.'\n\n'Yes, I suppose it might be,' said Belinda, 'but somehow clergymen _don't_ come to see him about things, do they? I don't know why.'\n\n'They came by night,' declared Harriet, 'like Nicodemus. Isn't Mr. Mold called Nicodemus?'\n\n'Oh, _no,_ Harriet, his name is Nathaniel.'\n\n'Nathaniel Mold,' said Harriet, trying it. 'Nat Mold. I think that sounds rather common, doesn't it?'\n\n'Well, we shall just call him Mr. Mold,' said Belinda, 'so I don't think we need worry. I believe Nicholas always calls him Nathaniel. He hates abbreviations.' She got up from the table and went to the window. 'It seems quite a nice morning after all that heavy rain,' she said. 'I think I shall go out into the village a little later on. I expect Nicholas will be taking a stroll and I am so looking forward to seeing him. Perhaps we shall meet.'\n\n'I won't come with you,' said Harriet nobly. 'After all, he is really your friend, not mine, and I expect you will have a lot to talk about.' Privately, Harriet thought him rather a boring little man, but she hoped for great things from Mr. Mold, who was reputed to be something of a 'one for the ladies'. This piece of information had also been gleaned from the vicarage Florrie, but Harriet had thought it wiser not to tell her sister. She wondered how Florrie, a plain, lumpish girl, had managed to find it out in so short a time.\n\nBelinda was fortunate enough to come face to face with Dr. Parnell before she had gone very far, and as they were just outside the Old Refectory, a tea shop run by gentlewomen, it seemed a good idea to go inside and have a cup of coffee. Dear Nicholas looked rather cold and peevish, she thought, wondering if he had had an adequate breakfast at the vicarage.\n\n'I don't suppose you are really in need of anything,' she said, as they sat down, 'but morning coffee is a pleasant, idle habit, I always think.'\n\n'Good morning, Miss Bede.' Mrs. Wilton, a pleasant-faced woman with rather prominent teeth, and wearing a smock patterned with a herbaceous border, stood before them. She stared at Dr. Parnell with frank interest and then at Belinda. Nicholas Parnell was small and bearded and did not somehow look the kind of person one would marry, Belinda realized. All the same, she felt proud of his distinction and could not resist introducing him to Mrs. Wilton, who was, after all, a canon's widow.\n\n'Oh, the _Library,'_ said Mrs. Wilton in a reverent tone. 'My husband used to read there when he was an undergraduate. I've heard so much about it.'\n\n'Of course we have central heating there now,' said Dr. Parnell. 'There have been great improvements in the last ten years or so. We also have a Ladies' Cloakroom in the main building now,' he added, his voice rising to a clear, ringing tone. 'That is a very great convenience.' He chuckled into his beard as Mrs. Wilton went away to fetch their coffee. 'I do not approve of this hushed and reverent attitude towards our great Library. After all, it is a place for human beings, isn't it?'\n\n'Yes, I suppose it is,' said Belinda doubtfully, for she was remembering some of the strange people who used to work there in her undergraduate days, many of whom could hardly have been called human beings if one were to judge by their looks.\n\n'These are excellent cakes,' said Dr. Parnell, eating heartily, 'although I had such a late breakfast that I can hardly do them justice. I must say I was surprised that dear Henry was not up before me. I had quite expected that there would be a Daily Celebration. Now that I come to think of it, I distinctly remember seeing 'D' against the church in _Mowbray's Guide._ I hope I shall not have to write and correct them.'\n\n'Oh, no,' said Belinda, always anxious to defend the Archdeacon. 'There is always a Daily Celebration but I expect Mr. Donne\u2014he's the curate\u2014would be taking it. Probably Henry thought it would be more courteous to breakfast with you on your first morning here.'\n\n'Ah, Belinda, I see you have not changed. We did not breakfast until half-past nine, so your argument falls to pieces. I left poor Henry in the churchyard, as I came out just now. He said the tombs put him in mind of his own mortality.'\n\n'And did he quote Young's _Night Thoughts_ to you?' asked Belinda, suddenly disloyal.\n\n'Indeed, he did. I left him because he was so tiresomely melancholy. And then he has been trying to make me subscribe to some fund for the church roof,' said Dr. Parnell.\n\n... _but perforated sore,_\n\n_And drill'd in holes the solid oak is found_\n\n_By worms voracious, eating through and through...._\n\nhe quoted solemnly, so that Belinda could hardly help smiling, although she knew it was very naughty of her. As they walked out of the Old Refectory towards the church she tried to remember what it was that Father Plowman had told her about the death-watch beetle and its habits, as if to make amends for her lapse. But before she had got very far, they had reached the churchyard wall and Belinda could see that the Archdeacon was sitting in his favourite seat under the yew trees. She felt a faint irritation to see him sitting there in the middle of the morning when so many people, women mostly, were going about their household duties and shopping. She supposed that men would be working too, but somehow their work seemed less important and exhausting.\n\n'What is he doing in the churchyard, I wonder?' she asked Dr. Parnell, but she did not really expect him to be able to tell her. The Archdeacon's affected eighteenth-century melancholy failed to charm her this morning.\n\n'I think he's meditating on his sermon for Sunday morning,' said Dr. Parnell. 'I understand that it is to be something rather out of the ordinary.'\n\nWhen the Archdeacon saw them he smiled benevolently, but at the same time condescendingly. It was as if he were letting them see how fortunate they were to be able to stroll in the village on a fine October morning, while he was condemned to sit among the tombs thinking out his sermon.\n\n'Isn't that seat rather _damp?'_ inquired Belinda sharply. 'We had some very heavy rain during the night, and you know how easily you catch cold.' She felt that as Agatha was so many miles away she was justified in adopting this almost wifely tone towards him.\n\nHe looked up irritably; Belinda had spoilt the romance of his environment. It was just the kind of remark that Agatha would make and, now that he came to think of it, he supposed the seat _was_ rather damp. He felt a distinct chill striking up through his bones and began to wonder if he were perhaps catching cold. He would never have noticed it if Belinda had not put the idea into his head. He rose rather ungraciously and came towards them.\n\n'It seems impossible to find peace and quiet anywhere,' he remarked. 'I had settled down in my study after breakfast when the girl came in with the vacuum cleaner and drove me into the churchyard. Now I am interrupted again.'\n\nBelinda smiled at this picture. 'I'm sorry if we have disturbed you,' she said. 'I think we should really have walked past if you had not got up and come to us.'\n\n'That would have been most unfriendly,' said the Archdeacon unreasonably. 'Besides, it is not every day that we have visitors. We should really make some effort to entertain them.'\n\n'Belinda has been doing her best,' said Dr. Parnell. 'She has given me an excellent cup of coffee and introduced me to a charming lady who showed great reverence when the Library was mentioned. It is really rather gratifying. I should be delighted to show her round,' he added. 'She would find every convenience. The next thing will be to have some kind of a restaurant where readers can take luncheon or tea together. Do you know,'\u2014he tapped his walking stick on the ground\u2014'I have had to have notices printed requesting readers not to _eat_ in the Library? One would hardly have thought it possible.'\n\nDuring this time an idea had been taking shape in Belinda's mind, and it was one which she knew her sister would approve. The talk about eating had made her think how nice it would be if they had a little supper party on Sunday evening. So, with unusual boldness, she issued the invitation, though she realized that her own rather timid way did not compare with Harriet's careless joviality. 'If you have no other engagement on Sunday evening,' she began, 'I was wondering if perhaps... I mean, would you care to come to supper at our house after Evensong? And Mr. Mold too, of course.'\n\n'That would be delightful,' said Dr. Parnell. 'One feels somehow that Sunday evening should be spent _away_ from a vicarage if at all possible.'\n\n'Sunday is always a heavy day for me,' said the Archdeacon 'and this Sunday will be particularly so. I intend to preach myself both morning and evening. These people are so sunk in lethargy that they do not know their own wickedness.'\n\nBelinda looked a little startled. 'I know,' she said inadequately. 'I mean, one is.' All the same it was uncomfortable to be reminded of one's sinfulness in the middle of a bright morning.\n\n'Sloth and lethargy,' said Dr. Parnell, with relish. 'But I take it you will accept Belinda's invitation, I know Nathaniel will want to.'\n\n'I shall come if I possibly can,' said the Archdeacon, passing his hand over his eyes with a gesture of weariness, 'but it may be that I shall be completely exhausted by the evening.'\n\n'But you will need a meal,' said Dr. Parnell, 'and I expect Belinda will want to know the numbers. It makes some difference with the catering, the arrangement of the table and that kind of thing.'\n\n'Ah, yes, I do not understand these mysteries,' said the Archdeacon. 'I think you can take it that I shall come,' he added, turning to Belinda with almost a smile.\n\n'You would hardly believe what I found Henry doing when we arrived last night,' said Dr. Parnell, in an easy, conversational tone.\n\nBelinda, who was of a credulous nature, refrained from making a guess.\n\n'Playing Patience on the floor of his study,' he went on. 'A complicated variety called Double Emperor.'\n\n'Patience is a very intelligent relaxation,' said Belinda, her usual loyalty coming to the rescue. 'You don't realize how hard Henry works. I mean,' she added obscurely, 'there are things to do in a country parish that people don't know about unless they live in one. Your work in the Library has its fixed hours, but a clergyman is at everybody's beck and call.' Of course, she reflected sadly, people would never dare to trouble the Archdeacon with their worries; they would go hurrying along Jubilee Terrace to Mr. Donne. Still, the smile that Henry gave her made her realize that being a little untruthful sometimes had its compensations.\n\nThe church clock struck half-past twelve.\n\n'Ah, lunch-time,' said the Archdeacon, and the party broke up to return to their respective houses.\n\nWhen Belinda got home she found Harriet in a state of great excitement.\n\n'Oh, Belinda,' she said, in a loud voice, 'he really is _charming.'_\n\nAs Emily was at this moment bringing in the meat, Belinda waited until they were settled at the table before she made any further inquiries.\n\n'Harriet, I wish you _wouldn't_ talk in front of Emily,' she began, but her own curiosity prevented her from saying any more. 'Who's charming?' she asked.\n\n'Why, Mr. Mold,' declared Harriet with enthusiasm. 'I saw him this morning.' Should she tell Belinda that she had seen him coming out of the Crownwheel and Pinion? she wondered. Better not, perhaps, and yet it would spoil the story to leave out such a piece of information.\n\n'But, Harriet, how could you have seen him?' asked Belinda rather impatiently. 'I understood from Nicholas that he was tired and was spending the morning in bed.'\n\n'Well, he must have got up because I saw him in the street,' said Harriet defiantly. She wished Belinda would not always behave quite so much like an elder sister. She decided that she would not tell her story in full. 'I spoke to him,' she declared.\n\nBelinda was incredulous. 'But, Harriet, you don't _know_ him,' she said.\n\n'Oh, of course he didn't realize who I was,' she explained. 'I met him coming out of the Crownwheel and Pinion, and he asked me the way to the Post Office; and as I happened to be going along to buy some stamps, we walked there together.' She paused, triumphant.\n\nBelinda put down her knife and fork in astonishment. The Crownwheel and Pinion in the morning! Surely Harriet had been mistaken? It sounded as if she had been 'picked up' by some commercial traveller. Most distasteful.\n\n'I don't think it can have been Mr. Mold,' she declared, looking very worried. 'After all, I've only met him once many years ago and you've never met him. I don't think it _can_ have been him,' she repeated, with a puzzled frown on her face.\n\n'It _was_ Mr. Mold,' said Harriet patiently. 'He said he was a stranger here, and that he had arrived last night and was staying at the vicarage.'\n\n'Oh, well, if he said that...' Belinda had to admit that it probably had been Mr. Mold. But for a deputy librarian to go to the Crownwheel and Pinion in the morning... surely it was unthinkable! And yet perhaps it was not so surprising, when one came to consider it, for after all Mr. Mold was not _quite_... He had started his career in the Library as a boy fetching books for readers, and although one didn't want to be snobbish and his ability had undoubtedly brought him to a distinguished position, it was certainly true that lack of breeding showed itself. Belinda could not help wishing that it had not been Harriet who had seen Mr. Mold. She would be sure to tell people and the whole situation was so embarrassing. She wondered if Nicholas knew, because really he was to blame for bringing such a man to the village.\n\n'Of course,' she said, more to herself than to Harriet, 'he may have felt ill or something. One must be careful not to judge people too hardly and, I dare say that in a town there is really no harm in a man going into a public house for a pint of beer in the morning, but these things _are_ regarded rather differently in a village and I should have thought he would have realized that.'\n\n'He certainly didn't look ill,' said Harriet, 'in fact quite the contrary. Rather a rosy complexion really and a well-built figure, not _fat,_ of course... his suit was very well cut, a dark blue with a narrow stripe and a maroon tie. He didn't look at all _flashy,_ though.'\n\n'Oh, no,' said Belinda, 'one would hardly expect an official of one of the greatest libraries in England to look flashy.'\n\n'And he had the most delightful manners,' Harriet went on. 'He didn't try to take advantage of me in any way,' she explained.\n\nHad she not thought it would be rather indelicate, Belinda would have laughed at this remark. The idea of anybody taking advantage of a respectable spinster, plumply attractive it must be admitted, in the main street of a respectable village in daylight, struck her as being rather ridiculous. But she thought it wiser not to let Harriet see that she was amused. Instead, she went on to tell her how she had invited the Archdeacon and his visitors to supper on Sunday evening.\n\nHarriet was delighted. She enjoyed entertaining and often complained that they did not do enough. 'I will see if Mr. Donne is free,' she said. 'I expect he would like to come.'\n\n'Yes, if you like,' said Belinda doubtfully, 'but I had thought it would be nice to ask Ricardo, then we shall all be more of an age, as it were.'\n\n'Oh, but I think we need youth, and Mr. Donne is so amusing,' persisted Harriet.\n\n'We shall be rather short of women of course,' said Belinda. 'I suppose we could ask Edith and Connie. I have a feeling Edith and Nicholas would get on rather well together. They are both interested in the same kind of thing.'\n\n'What, in lavatories?' asked Harriet bluntly.\n\nBelinda, who had been going to say 'conveniences', was forced to agree that this was what she had in mind, and told Harriet about his pride in the Ladies' Cloakroom which had recently been added to the Library.\n\n'I should have thought he had better things to think about,' retorted Harriet, 'and we certainly don't want to encourage Edith. Mr. Donne was so embarrassed when she was talking like that to the Archdeacon on the morning of the garden party. And then poor Connie is so dreary, isn't she? Does it really matter if we don't have equal numbers? After all I can manage Ricardo and Mr. Donne and you can have the Archdeacon and Dr. Parnell.'\n\n'Well, we shall have to think about it,' said Belinda. 'After all, Edith and Connie are always free and don't mind being asked at the last minute. The Archdeacon is preaching rather a special sermon on Sunday morning,' she added, getting up from the table, 'and he said he was preaching in the evening too and will be very tired. So I should like the supper to be particularly nice.'\n\n'Oh, of course,' Harriet agreed, 'but whatever we give him will be better than what he would get at the vicarage. We must be careful not to have the same as we had the last time Mr. Donne was here.'\n\n'I know Henry is fond of chicken,' said Belinda thoughtfully. 'Perhaps that would be the best.'\n\n'I really must look it up in my diary,' said Harriet, 'but I _think_ we had chicken the last time Mr. Donne was here.'\n\n# Chapter Ten\n\nIT WAS ON THE next Sunday morning that the Archdeacon preached his famous sermon on the Judgment Day.\n\nThe day had begun as other Sundays did. After breakfast Belinda had consulted with Emily about the roast beef, and together they had decided what time it ought to be put into the oven and how long it ought to stay there. The vegetables\u2014celery and roast potatoes\u2014were agreed upon, and the pudding\u2014a plum tart\u2014chosen. In addition, the chickens for the supper party were to be put on to boil and Emily was to start making the trifle if she had time. The jellies had been made on Saturday night and were now setting in the cool of the cellar. Belinda had suggested that they might have a lighter luncheon than usual, as there was so much to do, but Harriet was not going to be cheated of her Sunday roast, and had managed to persuade her sister that there would be plenty of time to get things ready in the afternoon and early evening. If was of course out of the question that either of them should attend Evensong.\n\nAt half-past ten Harriet began to prepare herself for church. This morning she was taking particular care with her appearance. On ordinary Sundays she had to look nicer than Agatha, as well as wearing something that would cause Count Bianco to burst into ecstatic compliments, and she liked the curate to see that his generation still had something to learn from hers in matters of elegance and good taste. But this Sunday was a particularly important one, for Dr. Parnell and Mr. Mold would be among the congregation and it was most important that she should make a good impression. She could not help regretting that when she had met Mr. Mold in the village and directed him to the Post Office, she had been wearing rather a countrified tweed coat, as was perhaps only to be expected on a weekday morning in a country village. This morning she was determined to make amends for this. Mr. Mold would hardly recognize the plump woman he had met outside the Crownwheel and Pinion in the elegant creature he was to see this morning. Once or twice, though, she felt a twinge of anxiety. Supposing he were not there?\n\n'Belinda,' she called down the stairs, 'Mr. Mold isn't a Roman Catholic or a Methodist, is he?'\n\n'No, I don't _think_ so,' Belinda called back, wondering why her sister should want to know. Perhaps Harriet had some doctrinal difficulty to be solved, although she had never before betrayed any interest in that direction. The Church of England had been good enough for a long line of dear curates; it would have been presumptuous of her to attempt to go further than that.\n\n'What I meant, was, will he be in church this morning?' Harriet explained.\n\n'Oh, surely,' said Belinda. 'I expect everyone will want to hear the special sermon.'\n\nHarriet snorted, as if expressing her contempt for anyone who would go to church to hear the Archdeacon preach.\n\n'Henry is very particular about the observance of Sunday,' Belinda went on. 'I'm sure he wouldn't like anyone staying at the vicarage not to attend Divine Service.'\n\nHarriet, who had got to the stage of arranging the veil on her hat, was too preoccupied to make any answer, but she could not help wondering if Belinda had forgotten one occasion when the Archdeacon himself had not been in church, and had later been seen in the vicarage garden, obviously in excellent health.\n\nBy ten minutes to eleven Harriet was ready, and waiting impatiently in the hall.\n\n'Belinda!' she called in an agitated voice. 'If you don't hurry up somebody might take our pew.'\n\nBelinda reflected unhappily that the church was never likely to be full enough for that to happen, unless there was a bishop or somebody very special preaching, like that time when, she could only imagine through some mistake, they had had a handsome Brother from a religious community, obviously intended for Father Plowman's church. She looked quickly in her bag to see if she had a half-crown and a clean handkerchief, picked up her prayer book, and hurried downstairs.\n\nBy this time Harriet was half way out of the gate. Belinda received her scolding meekly and was still silent when Harriet, quite kindly of course, began to criticize the clothes she was wearing.\n\n'You ought to have tied your scarf in a bow,' she said, 'it's much smarter, and you know that hats are turning _up_ at the back this winter, don't you?'\n\n'Yes, dear,' said Belinda. 'I like yours very much, but I don't think I could wear one like that myself.'\n\n'Oh, it's quite easy,' said Harriet airily, tipping her hat forward to an angle which Belinda considered a little too rakish for church, 'but you'd have to have your hair curled up at the back,' she added.\n\n'Yes, I know,' said Belinda hopelessly, looking at Harriet's carefully arranged ringlets. But I doubt whether Henry would like me any better, she thought.\n\nThis worldly conversation had carried them almost to the vicarage. There was as yet no sign of the important visitors. Harriet looked in at the gate rather anxiously; of course it was not quite five minutes to eleven, there was plenty of time for them to appear. It would perhaps be better to be settled in their own pew before they arrived. She hurried Belinda into the church.\n\nWhen they were inside Belinda knelt down hastily to say a prayer, but Harriet waited until she had arranged her bag and umbrella, removed her gloves and loosened her silver-fox fur. The next moment Belinda found herself being nudged by her sister, who whispered rather loudly, 'Here they come, they're going to sit by Ricardo.'\n\nWhen they were sitting down again, Harriet assured Belinda triumphantly that she had not been mistaken, it _had_ been Mr. Mold whom she had seen in the village. It was rather difficult to study them at all intently, because they were sitting behind Harriet and Belinda, but it was possible to do it not too obviously by putting your umbrella in the stand behind and taking some time in doing it. When Harriet had gone through this process, she was able to inform Belinda that Dr. Parnell was wearing a dark tie and that Mr. Mold had on the same suit she had seen him in before.\n\nThe service began quite uneventfully with one of the usual morning hymns, _New every morning is the love._ As they sang, Belinda noticed that the Archdeacon was not joining with them, but looking rather sternly round the church. As she did not want to catch his eye, Belinda looked down at her prayer book and concentrated on Keble's fine lines\n\n_Through sleep and darkness safely brought, Restored to life, and power and thought._\n\nNot that she ever thought of herself as having much _power,_ but she was certainly alive and might be considered capable of a small amount of thought. She could at least thank God for that. The curate was joining heartily in the singing and Belinda hoped he was saving enough voice to read the lessons.\n\nObviously the Archdeacon was out to impress his visitors, for the _Te Deum_ and the _Benedictus_ were sung to elaborate, unfamiliar settings, which the congregation could not attempt and which seemed rather beyond the choir at some points. The Archdeacon himself read the first lesson and the curate the second. The Archdeacon also intoned many of the prayers and his voice went up and down in the oddest way. Of course the voice should go up or down, Belinda couldn't quite remember which, at the end of a line, but there seemed to be something wrong somewhere and so much disturbance was caused among the choir boys that Mr. Gibson, the organist, had to hurry out of his place to control them.\n\nBelinda thought that as the Archdeacon was going to preach, he was perhaps doing too much of the service himself, and what with the curious intoning and the curate's church voice, which was like nothing so much as a bleating sheep's, it was difficult for Belinda to keep from smiling. And even she was forced to admit to herself that they were getting a little too much for their money, when she realized that they were going to have the Litany.\n\nJust before he went to the Litany desk, the Archdeacon glanced round the congregation with what appeared to be a look of malicious amusement on his face. At least, that was how it must have seemed to most people, but perhaps it could hardly have been amusement. Indulgence for his sinful flock was more likely and certainly more fitting. Everyone knelt down rather angrily. They had had the Litany last Sunday and the Archdeacon never made any attempt to shorten it. As he could not sing, he made up for it by making his voice heard as much as possible in other ways.\n\nBelinda was trying hard to concentrate on her sins, but somehow the atmosphere was not very suitable this morning and she was at last forced to give it up. Staring at the Archdeacon's back, she reflected that he was still very handsome. Perhaps he would read aloud to them when he came to supper tonight, though, as she would be the only person who wanted to listen, it might be rather difficult to arrange. Harriet could play the piano and the curate might be asked to sing, but the main entertainment of the evening would be the conversation. Dear Nicholas was so delightfully witty and Mr. Mold would no doubt be able to tell them many interesting things about the Library. By the time the Archdeacon had ascended the pulpit steps, Belinda had forgotten all about the special sermon, and settled herself comfortably in her pew, as did the rest of the congregation, having just sung with great vigour that the world was very evil.\n\nThe text was given out, quite a usual one from the Revelation. _And I saw a new heaven and a new earth, for the first heaven and the first earth were passed away._\n\nHarriet looked at her watch. She supposed they would have to endure the Archdeacon for at least twenty minutes, possibly twenty-five minutes or even half an hour. She sighed and tried to listen to what he was saying. It was some consolation that he was preaching a sermon of his own composition instead of one of those tedious literary things that Belinda said he read so magnificently.\n\n'We are apt to accept this vision of the new heaven and the new earth with too much complacency,' he declared.\n\nOh, well, thought Harriet, clergymen are always saying things like that.\n\n'But do we realize all that must happen before we can hope to share in this bliss? If indeed, we are found worthy. I say again, do we realize? Have we any idea at all?' The Archdeacon paused impressively and peered at his congregation; a harmless enough collection of people\u2014old Mrs. Prior and her daughter, Miss Jenner, Miss Beard and Miss Smiley in front with the children, ever watchful to frown on giggles or fidgets\u2014the Bank Manager, who sometimes read the lessons\u2014the Misses Bede and the guests from the vicarage\u2014Count Bianco\u2014Miss Liversidge and Miss Aspinall\u2014of course they did not realize but he was going to tell them. 'The _Judgment Day'_ he almost shouted, so loudly that Harriet had to take out her handkerchief to stifle her inappropriate amusement, and old Mrs. Prior let out a kind of moan. 'That day may be soon,' he went on, 'it may even be _tomorrow.'_\n\nThe congregation shifted awkwardly in their seats. It was uncomfortable to be reminded that the Judgment Day might be tomorrow.\n\n_'Dies Irae,'_ he continued, lingering on the words with enjoyment. Belinda saw Edith Liversidge purse her lips disapprovingly at this Romish expression. 'Day of Wrath,' he translated. 'And what a terrible day that will be!'\n\nThe congregation, still rather uneasy and disturbed, reminded themselves that of course such a thing couldn't _really_ happen. Why, scientists told us that it would take millions of years for the sun to move sufficiently far away from the earth for life to become extinct. At least it was perhaps not exactly that, but something very like it. They knew enough to realize that the Archdeacon was being ridiculous and that the Judgment Day could not possibly be tomorrow. When the first uncomfortable shock had passed they were able to laugh at themselves. How could they have been so silly as to be alarmed!\n\nBut even as they were thinking thus, the relentless voice from the pulpit was pouring scorn on those scientists who thought they knew how the world had begun and how it would end. How _could_ they know? These matters were incomprehensible mysteries known to God alone. The Judgment Day was as likely to be tomorrow as at any time in the far distant future. The world was indeed very evil, as they had just been singing in that fine hymn translated from the Latin, the times were waxing late. All through our literature poets had been haunted by the idea of the Last Day and what it would be like....\n\nThe congregation suddenly relaxed. It was just going to be one of the Archdeacon's usual sermons after all. There had been no need for those uncomfortable fears. They settled down again, now completely reassured, and prepared themselves for a long string of quotations, joined together by a few explanations from the Archdeacon.\n\nHe began at the seventeenth century. Belinda reflected that if he had gone back any further, the sermon would have assumed Elizabethan proportions. As it was, it promised to be longer than usual. She listened admiringly. The Archdeacon was quoting Thomas Flatman's lines written in 1659, to show how poets of the latter half of that century had imagined that the Judgment Day was near.\n\n_'Tis not far off; methinks I see_\n\n_Among the stars some dimmer be;_\n\n_Some tremble as their lamps did fear_\n\n_A neighbouring extinguisher...._\n\nAnd curiously enough one of the oldest inhabitants of the parish had remarked to him only the other day that the stars did not seem to be as bright as they were when he was a boy. It was very significant. The Archdeacon liked the sound of his own voice and so did Belinda, and she was delighted to hear him read about thirty more lines of Flatman's poem.\n\nThose of the congregation who were still listening\u2014Harriet's attention had long since wandered\u2014smiled complacently. That had been in 1659, they thought, and nothing had come of this man's noticing that some of the stars were dimmer. Why even the Archdeacon himself was forced to admit it! 1659. 1660. What had happened in 1660? His hearers resented this history lesson. The Restoration. Everyone knew that. But here was the Archdeacon trying to tell them that the Restoration was itself a kind of Judgment Day.\n\nBelinda tried hard to follow, but she found this point rather obscure. She was frowning slightly with the effort of concentration. Harriet was looking at the curate, but he had sunk so deeply into his stall that very little of him was visible. By looking out of the corner of her left eye and turning her head slightly, she could see Dr. Parnell and Mr. Mold. Mr. Mold was looking at his watch and Dr. Parnell appeared to be smiling at some private joke. Count Bianco, sitting in front of Dr. Parnell, had long ago given up any attempt to follow the sermon. A Roman Catholic by upbringing, he still found the service confusing and only attended the Archdeacon's church because he felt it might bring him nearer to Harriet. This morning she had looked in his direction; she had distinctly turned her head. Could it be that she was looking at _him?_\n\nWhen the Archdeacon reached the eighteenth century, the going was a little easier. Several people smiled at the lines he quoted from Blair's poem _The Grave:_\n\n_When the dread trumpet sounds, the slumb'ring dust,_\n\n_Not unattentive to the call shall wake,_\n\n_And ev'ry joint possess its proper place,_\n\n_With a new elegance of form unknown_\n\n_To its first state. Nor shall the conscious soul_\n\n_Mistake his partner...._\n\nBelinda liked this very much, but she was uneasily conscious that the Archdeacon had already been preaching for nearly half an hour, and she began to worry about the beef. It would be roasted to a cinder by now, unless Emily had had the sense to turn down the oven. Harriet did so like it underdone, and they were usually well out of church and sitting down to their meal by half-past twelve. And Henry had only got as far as the eighteenth century without yet having mentioned Edward Young, who was sure to be brought into the sermon somehow. She had rather lost the thread of what he was saying now, but suddenly felt herself on safer ground when she heard him mention the _Night Thoughts._ He seemed to be implying that each person listening to him this morning was little better than the unknown Lorenzo, for whose edification the poem had been written. Even Belinda thought the Archdeacon was going a little too far when he likened his congregation to such as _'call aloud for ev'ry bauble drivel'd o'er by sense'._ Whatever it might mean it certainly sounded abusive. He concluded his reading from Young by flinging a challenge at them.\n\n_... Say dreamers of gay dreams,_\n\n_How will you weather an eternal night,_\n\n_Where such expedients fail?_\n\nHe paused dramatically and the sermon was at an end. There was quite a stir in the congregation, for some of them had been dreaming gay dreams most of the morning, although many of them had given the sermon a chance, and had only allowed their thoughts to wander when it had passed beyond their comprehension. They now fidgeted angrily in bags and pockets for their collect-money. One or two even let the plate pass them, waving it on with an angry gesture.\n\nBelinda soon recovered from her first feeling of shocked surprise. Of course dear Henry had not really meant to insult them. He had obviously been carried away by the fine poetry, and naturally he must have meant to include himself among those he condemned. It had really been one of the finest sermons she had ever heard him preach, she told herself loyally, even if the ending had been rather sudden and unusual. It didn't do people any harm to hear the truth occasionally. We were all inclined to get too complacent sometimes. She thought rather vaguely of great preachers like Savonarola, Donne and John Wesley. No doubt they had not spared the feelings of their hearers either, but as she was unable to think of anything that any one of them had said, she could not be absolutely sure. As they were singing the last hymn _Ye servants of the Lord,_ Belinda tried to think of some intelligent criticisms, for she did not want her praise of dear Henry to be lacking in discernment. He might welcome intelligent criticism, she thought, knowing perfectly well that he would not. Perhaps there had been rather too many literary quotations, and she had the feeling that it was not quite the thing to read bits of Restoration drama in church... but it had certainly been a fine and unusual sermon. She could not help wondering whether he would continue it this evening, going through the Victorians and the modern poets and so bringing it up to date. But that would hardly be suitable for the evening congregation, who, as he had admitted himself, liked simpler stuff.\n\nAs they came out of church they passed the time of day with Dr. Parnell and Mr. Mold, but Belinda hurried Harriet away before they could get involved in conversation. Mr. Mold's manner seemed very free and he had looked almost as if he were going to _wink_ at Harriet. Ricardo, who was hovering hopefully by a tomb-stone, saw her whisked away before he could do more than bow and say good morning. But he comforted himself with the prospect of seeing her that evening.\n\n'I was sorry not to stop and talk to Ricardo,' said Belinda, 'but we are so late as it is.'\n\n'Yes,' said Harriet, 'I expect many people's Sunday dinner will be ruined. I wonder what they are having at the vicarage?'\n\n'I think they are having duck,' said Belinda. 'At least, I saw one in Hartnell's on Saturday which was labelled for the vicarage. And of course,' she said thoughtfully, as she watched her sister carve the over-cooked beef, 'duck needs to be _very_ well done, doesn't it? It can't really be cooked too much.'\n\n# Chapter Eleven\n\nTHE GUESTS WERE DUE to arrive at about eight o'clock, by which time Evensong would be over. Both Belinda and Harriet had of course been much too busy with their preparations to attend it. Belinda had felt very much tempted, indeed, the thought of missing one of the Archdeacon's sermons was almost unbearable, but she consoled herself with the reflection that looking after his material welfare was just as important as her own spiritual welfare, if such it could be called, and that she was making the sacrifice in a good cause.\n\nA very nice supper had been prepared. It had to be so, for not only must the Archdeacon be pleased, but Harriet had thought the curate needed feeding up as he had been looking especially thin and pale lately. She could only hope it was nothing to do with that Miss Berridge. There were to be cold chickens with ham and tongue and various salads, followed by trifles, jellies, fruit and Stilton cheese. An extra leaf had been put in the dining-room table as, much against Harriet's will, Belinda had decided to invite Miss Liversidge and Miss Aspinall. She really thought that five men and two women was a little disproportionate and such a party might give rise to talk. Emily would think it so funny. It was rather an undertaking to have seven people to supper, but as most of the food was cold and could be prepared well in advance there was no reason why everything should not go very well.\n\nHarriet was a little inclined to worry about what they should drink. Mr. Mold would be used to living in style, she thought, and would surely expect whisky or gin.\n\n'But we have a very good sherry,' said Belinda. 'I am sure that is quite correct, and there will be the hock and afterwards port.' Whisky was to Belinda more a medicine than a drink, something one took for a cold with hot milk or lemon. It was not at all suitable for a Sunday evening supper party at which there were to be clergymen and ladies present.\n\n'We must be sure that the hock is _chilled_ enough,' said Harriet. 'Not iced, of course that would be a serious error; I shouldn't like Mr. Mold to think that we didn't know about wine.'\n\n'Nicholas is a great connoisseur,' said Belinda. 'It seems right that a librarian should be, I think. Good wine and old books seem to go together.'\n\n'And of course the Archdeacon likes a drop,' said Harriet rather vulgarly. 'We shall have to watch him. I don't suppose Agatha lets him have much. Good heavens, it's nearly seven o'clock. We must go and change.'\n\nHarriet was determined that this evening should see the climax of her elegance and only lamented the fact that she could not wear full evening dress. Still, her new brown velvet would be magnificent with her gold necklace and long ear-rings. Belinda had decided to wear her blue chiffon. Henry had once said that he liked her in pale colours, and although that had been over thirty years ago it was possible that he still might. Her crystal beads and ear-rings went quite well with it, and when she had put on a little rouge the whole effect was rather pleasing. She went into the bathroom where Harriet was splashing about in the bath like a plump porpoise. Her curls were protected by a round cap of green oilskin and the room was filled with the exotic scent of bath salts.\n\nHarriet looked at Belinda critically. 'Yes, you look very nice,' she said, 'but I think I should use some more lipstick if I were you. Artificial light is apt to make one look paler.'\n\n'Oh, no, Harriet, I don't think I can use any more,' said Belinda. 'I shouldn't really feel natural if I did. _Thou art not fair for all thy red and white,'_ she quoted vaguely, leaving Harriet to wallow in her bath.\n\nAt five minutes to eight Belinda was downstairs in the drawing-room, waiting for somebody to arrive. She was sitting in a chair by the fire with a book on her knee, which she was not reading. It was something by an old friend of Harriet's\u2014a former curate\u2014Theodore Grote, now Bishop of Mbawawa in Africa. Dear Theo, he had certainly done splendid work among the natives, at least, that was what everyone said, although nobody seemed to know exactly what it was that he had done. Certainly they still _looked_ very heathen, grinning away in their leafy dress. But perhaps that was before he converted them. She opened the book with a view to finding out, but she could not settle to reading and walked restlessly round the room, moving the flowers, rearranging the cushions and altering the position of chairs. She brought out a photograph of Nicholas Parnell in his academic robes and put it on the mantelpiece; she also displayed on a small table a little pamphlet he had written about central heating in libraries. It was prettily bound and had a picture of a phoenix on the cover with a Greek inscription underneath it. Something about Prometheus, Harriet had said, for Belinda was like Shakespeare in having little Latin and no Greek.\n\nAt eight o'clock the bell rang. Men's voices were heard in the hall and a minute or two later Emily showed the Archdeacon, Dr. Parnell, Mr. Mold and the curate into the room. Belinda shook hands with them rather formally. The Archdeacon advanced towards an armchair by the fire and sank down into it rather dramatically, as if exhausted. Dr. Parnell took up his own pamphlet and said that he was glad to find somebody who had cut its beautiful pages.\n\n'I'm afraid you're hardly a best seller,' said Mr. Mold jovially. 'Nor even as much ordered in the Library as Rochester's poems,' he added.\n\nBelinda frowned and looked embarrassed when the curate asked, with his usual eager interest, what poems those were.\n\n'I am afraid they are rather _naughty'_ said Dr. Parnell. 'We have had to lock them away in a special place, together with other books of a similar nature. All the same, they are quite often asked for by our readers.'\n\n'Oh, well, I suppose people have to study them,' said Belinda, handing round cigarettes and wondering how she could change the subject.\n\n'We should not like to think that they ordered them for any other reason,' said Dr. Parnell, chuckling and rubbing his hands in front of the fire.\n\nBelinda was greatly relieved when Miss Liversidge and Miss Aspinall arrived and she was able to introduce them. She was so afraid that Nicholas and Edith would discover their common interest in sanitary arrangements too soon, that she resolutely kept them apart. It was all very difficult and she wished Harriet would hurry up and come in. Only the curate was making what Belinda considered to be suitable conversation for the awkward interval between arriving and sitting down to eat.\n\n'Do you know,' he was saying eagerly, 'there was quite a _nip_ in the air this evening? I shouldn't be surprised if we had frost.'\n\n'And the later the frost the harder the winter,' said a cheerful voice in the background. 'I do hope you're all wearing warm underclothes.'\n\nIt was Harriet, ready far sooner than Belinda had hoped was possible, looking splendid too, in her brown velvet and gold ornaments.\n\nThe curate laughed heartily and assured her that he had his on.\n\nHarriet was in excellent form and soon had everybody laughing, except for the Archdeacon, who went on reclining in the armchair, not speaking.\n\n'Well, what did you preach about this evening?' asked Harriet. 'I haven't heard any comments on the sermon yet.'\n\nThe Archdeacon roused himself. 'It was a continuation of this morning's, brought up to date as it were,' he explained.\n\n'Oh, it was beautiful,' gushed Connie Aspinall, 'I did so enjoy it.'\n\nThe Archdeacon looked pleased. 'I had feared it might be rather too obscure,' he said. 'Eliot is not an easy poet.'\n\nBelinda gasped. Eliot! And for the evening congregation! But it must have been magnificent to hear him reading Eliot. 'Perhaps you will give the sermon again,' she suggested timidly, 'for the benefit of those who were not at Evensong.'\n\n'Perhaps I will,' said the Archdeacon, 'perhaps I will.' He paused. 'I have been considering giving a course of sermons on Dante,' he mused. 'Not of course in the original\u2014Carey's translation, perhaps.'\n\n'It would be a fine and unusual subject,' said Belinda doubtfully.\n\n'Well, you can count me out,' said Edith bluntly. 'I couldn't make much of the sermon this morning. Too full of quotations, like _Hamlet.'_ She gave her short barking laugh.\n\n'I think people prefer the more obvious aspects of the Christian teaching,' said Nicholas regretfully. 'I mean, I am afraid that they do. Simple sentiments in intelligible prose. A great pity really. I can see how it limits the scope of the more enterprising clergy.'\n\n_'Sentiments to which every bosom returns an echo,'_ said the Archdeacon. 'Yes, one appreciates that, and yet, why shouldn't Eliot express those sentiments?'\n\n'Do the bosoms of people nowadays return any echo?' said Mr. Mold. 'One wonders really.'\n\n'Well, if they do, it certainly isn't as loud as the echo made by the Apes of Brazil,' chortled Harriet, and in response to a very pressing appeal by Mr. Mold, she began to explain it all over again.\n\n'Can the sound really be heard two _miles_ away?' said Connie, whose voice held just the same note of rapt awe as when she had praised the Archdeacon's sermon. 'Isn't that wonderful?'\n\nThe Archdeacon looked rather annoyed. 'Oh, it is nothing very unusual,' he said shortly, so that Belinda began to wonder whether he was about to embark on some tale of his own. But he lapsed into silence again, so that she was forced to continue the conversation with a bright and rather insincere remark about Agatha, and what a pity it was that she was not with them that evening.\n\n'I had a postcard from her yesterday,' said the Archdeacon. 'She says she is getting on quite well speaking Anglo-Saxon and Old High German in the shops.'\n\n'Ah, well, Agatha is so clever,' said Belinda, without bitterness. 'I'm afraid I've forgotten all my Anglo-Saxon. But surely the vocabulary is rather limited?'\n\n'Yes, quite ridiculous,' said Edith shortly. 'Like trying to talk Latin in Italy.'\n\n'Why, Ricardo isn't here yet,' said Belinda. 'What do you say to a glass of sherry while we are waiting?'\n\n'I say yes,' said Mr. Mold promptly.\n\n'Yes _please,_ Nathaniel,' Dr. Parnell reminded him. 'We should not like it to be thought that an official of our great Library was lacking in manners.'\n\n'I feel that _I_ have been lacking in manners for not offering it sooner,' said Belinda quite sincerely, thus taking upon herself the blame for all the little frictions of the evening. But it was so obvious that women should take the blame, it was both the better and the easier part, and just as she was pouring out the sherry, Ricardo arrived, with such profuse and gallant apologies for being late that everyone was put into a good humour. He was a little encumbered by a magnificent pot of chrysanthemums, which he presented with a ceremonious gesture.\n\nShortly after this they went into supper, Edith and Harriet followed by Mr. Mold and the curate, making for the dining-room with what Belinda considered indecent haste. But even those who followed more slowly moved with confident anticipation. Belinda had taken care to arrange the table so that Harriet should sit between Ricardo and Mr. Mold, when she might see how superior dear Ricardo was. Belinda herself sat by the Archdeacon and Dr. Parnell, while Miss Liversidge and Miss Aspinall were fitted in where there happened to be spaces.\n\n'Why, the table is groaning!' exclaimed Dr. Parnell. 'I like that expression so much, but one is hardly ever justified in using it or even expecting it. Certainly not at Sunday supper.'\n\nBelinda looked pleased. 'I hope everything will be nice,' she said. 'I never see why Sunday supper should be the dreary meal it usually is. I mean,' she added hastily, remembering that they had had just such a dreary Sunday supper at the vicarage a few weeks ago, 'that it can sometimes be.'\n\n'We always have cold meat with beetroot and no potatoes,' said the Archdeacon, as if reading her thoughts. 'Nothing could be more unappetizing.'\n\n'Of course the servants are often out on Sunday evening,' said the curate, 'and one likes to feel that they are having an easier time.'\n\n'I don't feel that at all,' said the Archdeacon. 'I like to feel that somebody has taken a little trouble in preparing a meal for me. I think we deserve it after the labour of Sunday.'\n\n'Especially after that magnificent sermon this morning,' murmured Belinda loyally, giving him some of the best of the chicken.\n\n'All that reading must be very tiring,' said Dr. Parnell spitefully.\n\n'It was some very fine poetry,' said Ricardo vaguely, for he had not grasped much of the sermon apart from the bare fact that it was supposed to be about the Judgment Day.\n\n'We certainly thought it magnificent, didn't we, Harriet?' said Belinda, turning towards her sister rather urgently. It was tactless of Harriet not to have made any comment. But then she herself was rather to blame for introducing the subject when they had already discussed it once. It was a pity that her loyalty had got the better of her.\n\n'Oh, splendid,' said Harriet, rather too enthusiastically. 'But of course I'm _not_ a theologian,' she added, with a brilliant smile.\n\nMr. Mold laughed at this and so did Dr. Parnell, but the Archdeacon looked rather annoyed.\n\n'Few of us are true theologians,' said the curate sententiously. 'But after all, the real knowledge comes from within and not from books.'\n\nBelinda looked at him rather apprehensively. She hoped the wine wasn't beginning to have its effect on him. White wine wasn't really intoxicating and he had only had one glass of sherry.\n\n'That is rather a disconcerting thought,' said Dr. Parnell. 'You are preaching sedition. I maintain that the true knowledge comes from _books._ It would be a poor prospect for me and for Nathaniel if everybody thought as you do.'\n\n'Ah, but it is true what Mr. Donne says,' said Ricardo thoughtfully. 'My dear friend John Akenside used to say that he learned more about the political situation in central Europe in those quiet moments with a glass of wine at a caf\u00e9 table than by all his talks with Pribitchevitch's brother.'\n\n'Oh, yes, I can believe that,' said Edith Liversidge, 'John liked his glass of wine.'\n\n'But surely the intrigues of Balkan politics can hardly be compared with the true knowledge that comes from within?' Dr. Parnell protested. 'I hardly think it is the same thing.'\n\nBelinda too, thought Ricardo's remarks hardly relevant, but one could not argue the point. Perhaps it was a mistake to have any kind of serious conversation when eating, or even anywhere at all in mixed company. Men took themselves so seriously and seemed to insist on arguing even the most trivial points. So, at the risk of seeming frivolous, she turned the conversation to something lighter.\n\n'I can never think of Belgrade without thinking of the public baths,' she said.\n\nMr. Mold looked across the table expectantly. Perhaps this would be more amusing than the knowledge that comes from within.\n\n'Why does it remind you of public baths?' said the Archdeacon. 'It seems most unlikely.'\n\n'Oh, well,' said Belinda, who hadn't really a story to tell, 'I once heard somebody describing them and I thought it was rather funny. A lot of old men all swimming about in a pool of hot water,' she concluded weakly, hoping that somebody would laugh.\n\nMost of them did, especially the Archdeacon, who seemed to be in a good temper again.\n\n'I never heard of there being any hot springs in Belgrade,' said Ricardo seriously.\n\n'Oh, I expect the water was artificially heated,' explained the curate, turning earnestly to Belinda for further information on this interesting point.\n\nBelinda felt rather flustered at the interest which everyone was taking in her silly little story. 'I don't really know,' she said. 'I've never been to Belgrade myself, and even if I had I don't suppose I should have visited the public baths.'\n\n'Not the ones with the old men in them, we hope,' said Mr. Mold, with almost a wink.\n\nBelinda was rather taken aback. She didn't think she liked Mr. Mold very much. Of course one didn't want to be snobbish, but it really was true that low origin always betrayed itself somewhere.\n\n'Oh, Belinda never remembers where she's been,' said Harriet, hardly improving the situation. 'Now, Mr. Mold, do have some more trifle,' she said, favouring him with a brilliant smile.\n\n'Perhaps I may have some too,' said the Archdeacon. 'It really is delicious. I don't know when I've tasted anything so good.'\n\nBut Belinda hardly noticed his praise. She was thinking indignantly that Nicholas had always encouraged Mr. Mold too much as a boy, although one would have thought that moving in a cultured intellectual society would have cured him of any tendency to make jokes not quite in the best of taste. And yet, she thought doubtfully, the Library, great though it was, did not always attract to it cultured and intellectual persons. Nicholas himself, obsessed with central heating and conveniences, was perhaps not the best influence for a weak character like Mr. Mold. Belinda began to wish that she were in Karlsbad with dear Agatha, helping her to get cured of her rheumatism. She imagined herself in the pump-room, if there was one, drinking unpleasant but salutary waters, and making conversation with elderly people. Perhaps taking a gentle walk in the cool of the evening with an old clergyman or a retired general....\n\nAll around her the conversation buzzed pleasantly. Mr. Mold's little lapse was quite forgotten, if indeed it had ever been noticed by anyone except Belinda. Harriet was asking Ricardo if it was true that the fleas on the Lido were so wonderful. She had heard that they bounced balls and drew little golden carriages.\n\n'Indeed, they are,' said Ricardo gravely. 'I have seen them myself.'\n\nThe curate leaned forward eagerly. 'It is wonderful what things animals and even insects can be made to do if they are trained with kindness,' he said, his face aglow with interest.\n\nEveryone agreed with this very just remark. Dr. Parnell even went so far as to observe that it was also true of people.\n\n'I should love to go to Venice,' said Belinda 'I think there is something very special about Italy. It is so rich in literary associations.'\n\n'Ah...' Ricardo put down his spoon and was obviously on the point of bursting into a flow of Dante, but the Archdeacon was too quick for him and got in first with Byron.\n\n_I stood in Venice on the Bridge of Sighs_\n\n_A palace and a prison on each hand:_\n\nhe quoted, and remarked that he had thought of visiting Italy in the spring, but that of course it was almost impossible for him to take a holiday however much he might need it.\n\n'Oh, but I don't see why you shouldn't,' said Harriet, loudly and tactlessly. 'It isn't as if you had a frightful lot to do, and I'm sure Mr. Donne could manage perfectly well when you were away.'\n\nThere was an uncomfortable silence.\n\n'You know, you really should take a holiday,' said Edith Liversidge. 'A change does everyone good. Everyone would benefit.'\n\n'You would feel you were doing good to others as well as yourself,' said Dr. Parnell, 'so you would have a double satisfaction.'\n\n'I doubt whether I could allow myself that luxury,' said the Archdeacon quite good humouredly. 'Of course a change can sometimes be a good thing. I have often wondered whether I ought to have a town parish. There is more scope for preaching.'\n\n'Oh, yes,' Connie Aspinall broke in eagerly. 'I remember when Canon Kendrick was rector of St. Ermin's there wasn't a vacant seat at Evensong\u2014you had to be there half an hour before it started. He said some very shocking things.'\n\n'Well, of course people like that,' said Dr. Parnell. 'Kendrick was a contemporary of mine. He got a very poor degree, I believe, but he found out what his line was and made a success of it.'\n\n'What a cynical way to talk,' said Harriet indignantly. 'He was probably a very sincere man.'\n\n'Oh, he was,' said Connie, 'but I suppose he thought it his duty to say those things, unpleasant though they were.'\n\n'Yes, there is evil even in Belgravia,' said Dr. Parnell.\n\n'There is evil everywhere,' said the curate.\n\nBelinda looked around her uncomfortably as if expecting to find the devil sitting at the table, but by the time they were drinking their coffee in the drawing-room that solemn subject had somehow been forgotten and Ricardo, talking about the picturesque costume of the ancient Etruscans, was the centre of the little group. Belinda marvelled at the way conversation rushed from one subject to another with such bewildering speed, but decided that this was one which could give offence to nobody and was therefore to be encouraged. So she appeared interested although she knew nothing whatever about the ancient Etruscans. It was hardly the kind of thing one would know about, unless one had had the advantage of a classical education, as Nicholas and Ricardo had.\n\n'Was their dress anything like that of the Germanic tribes?' she asked. 'I mean the ones Tacitus described in his _Germania,'_ she added vaguely, for it was a long time since she had taken her first University examination in which it had been one of the set books.\n\nRicardo paused and looked thoughtful, but before he had time to answer, Harriet let out a cry of joy, as if she had suddenly come upon something which she had long ago given up for lost.\n\n_Locupletissimi veste distinguuntur, non fluitante_...\n\nshe paused appealingly, and waved her plump hands about, searching for the rest of the quotation. While she did so, Belinda and Dr. Parnell began to laugh, the curate and Miss Aspinall looked amazed and expectant, while the Archdeacon smiled a little doubtfully, for he was not a very good Latinist, nor had he known his Tacitus very well as an undergraduate. Mr. Mold looked frankly bored, although he could not help thinking that Harriet looked very attractive, waving her hands about in the air. Ricardo was frowning, but only for a moment. How terrible it would be if he were to fail her! He cleared his throat...\n\n_non fluitante sed stricto et singulos artus exprimente,_\n\nHe recited.\n\nReally, reflected Belinda, Ricardo's faculty for quoting Tacitus is quite frightening!\n\n'I suppose _veste_ means vest,' said the curate earnestly, with an expression of painful intelligence on his face.\n\n'Hardly in the modern sense, perhaps,' said Ricardo thoughtfully, 'although it was a tight-fitting garment, as you hear from the description.'\n\n'More like men's long pants,' said Edith Liversidge bluntly. 'They must have looked rather comic.'\n\n'Yes, it is strange that the rich men should have been distinguished by the wearing of underclothes,' said Belinda thinking that the conversation was getting more than usually silly. She was herself smiling, as she could not help thinking of the curate's combinations. It was a good thing, she felt, when Ricardo suggested that Harriet might honour them by playing something on the piano.\n\n'How very talented your sister must be,' said Mr. Mold, who was sitting on the sofa by Belinda. 'She would be an asset to any household,' he declared pompously.\n\nBelinda tried to think of something to say which would put him off, but could think of nothing without being disloyal to her sister. Certainly she looked very splendid sitting at the piano; it was not surprising that he should admire her.\n\nThe first chords of _The Harmonious Blacksmith_ jangled forth. Belinda wondered why Harriet had chosen this particular piece, and began to be a little anxious about the later variations, which she knew were rather tricky. But Harriet avoided this difficulty by playing only the first two variations.\n\n'I don't want to bore you with the whole lot,' she said, and broke into a gay Chopin mazurka.\n\nThere was now an atmosphere of peace and contentment in the room. Everyone had eaten well, there was a good fire and comfortable chairs. The Archdeacon, in the best chair, was nodding now. Miss Aspinall had found a polite listener in the curate, who was asking her just the kind of questions she liked about the past glories of her life in Belgrave Square. Dr. Parnell and Miss Liversidge were talking, but in a low voice, about the 'improvements' in the Library and what further ones could conveniently be made. Ricardo and Mr. Mold were both admiring Harriet and vying for her attention. Only Belinda was unoccupied, but she was quite happy in the knowledge that the party had really been quite successful. Of course, if the Archdeacon had not been asleep, she could have had some conversation with him, but it was nice to know that he felt really at home, and she would not for the world have had him any different.\n\n# Chapter Twelve\n\nTHE NEXT DAY HARRIET could talk of nothing but Mr. Mold. At breakfast she declared that he was remarkably young-looking for his age.\n\n'I suppose he must be in the early fifties, but that's really the Prime of Life, isn't it?' she said to Belinda, who had not so far contributed anything to her sister's eulogy apart from the observation that he certainly had rather a high colour. Harriet repeated 'the Prime of Life', and went on eating her sausage.\n\n'Yes, I suppose he is,' agreed Belinda, but rather doubtfully, for she was not really sure what the Prime of Life was. She had always thought that her own prime was twenty-five, so that by her reckoning Mr. Mold must be nearly thirty years past it. 'Personally he isn't a type that appeals to me very much,' she added, remembering the joke about the public baths in Belgrade.\n\n'Oh, I know he isn't always quoting Gray's _Elegy,'_ said Harriet pointedly, 'but he's so amusing, such a Man of the World,' she added naively. 'I wonder how long he will be staying?'\n\n'I expect he will have to get back to the Library soon,' said Belinda. 'Old Mr. Lydgate is in charge now, but I should think he is hardly up to the work really, though,' she added irrelevantly, 'he had some very interesting experiences in Ethiopia.'\n\nWith that she went upstairs, thinking that it would really be a good thing when both the Librarian and Mr. Mold went back. Although it was nice seeing different people, especially if they were old friends as dear Nicholas was, Belinda found it rather unsettling. The effort of trying to talk to so many people last night and keep them at peace with each other had quite exhausted her. But there was some satisfaction mixed with her tiredness, for she felt it had been quite a successful party. Edith and Connie had obviously enjoyed both the food and the company, and as inviting them had been something in the nature of a duty, one could feel special satisfaction there. Nicholas and Edith had had a very full conversation about conveniences and he had invited her to come for a personally conducted tour of the Library, where her advice on certain points would be much valued. The curate, Count Bianco and Mr. Mold had seemed quite happy, though perhaps the word was scarcely applicable to the Count, who preferred his gentle state of melancholy which must have been enriched by Harriet's attentions to Mr. Mold and the curate. Belinda's only fear was that the Archdeacon had been bored, though she had decided that his going to sleep showed rather that he felt at home in her house and she was determined to go on thinking so. As she dusted her dressing-table, she broke into Addison's noble hymn, _The spacious firmament on high._\n\n_In Reason's ear they all rejoice..._ How admirable that was! Belinda began to think rather confusedly about the eighteenth century, and what in her undergraduate essays she had called its 'Rationalism'. Had not her favourite, Young, said something about his heart becoming the convert of his head? How useful that must have been! Belinda began to look back on her own life and came to the regretful conclusion that she had admired the great eighteenth-century poet without really taking his advice. She comforted herself by reflecting that it was now too late to do anything about it, but as she opened a drawer she came upon some skeins of grey wool, the wool she had bought to knit the Archdeacon a pullover. She knew now that she would never do it. She would make a jumper for herself, safe, dull and rather too thick. Surely this was proof that her heart had now become the convert of her head? Or was it just fear of Agatha?\n\nShe shut the drawer and turned her attention to other work, preparing to live this day as if her last. As it was a nice bright morning, she felt that it would be a good opportunity to do some gardening, and later, if she had time, she might write a letter to poor Agatha, who was probably feeling rather lonely all by herself in Karlsbad. It was quite a luxury to be able to think of her as 'poor Agatha'; it showed that absence could do more than just make the heart grow fonder.\n\nBelinda went downstairs and put on her goloshes and an old mackintosh. She decided to put some bulbs in the beds in the front garden and then move round to the back. If people came to the door it was more likely that they would come later, and by then she would be out of sight. She began to plant tulip bulbs in between the wall-flower plants. They would make a pretty show in the late spring. She noticed how splendidly the aubretias had done; they were spreading so much that they would soon have to be divided. Belinda remembered when she had put them in as little cuttings. They had had a particularly hard winter that year, so that she had been afraid the frost would kill them. But they had all lived and flourished. How wonderful it was, when one came to think of it, what a lot of hardships plants could stand! And people too. Here Belinda realised how well her own heart, broken at twenty-five, had mended with the passing of the years. Perhaps the slave had grown to love its chains, or whatever it was that the dear Earl of Rochester had said on that subject. Belinda was sure that our greater English poets had written much about unhappy lovers _not_ dying of grief, although it was of course more romantic when they did. But there was always hope springing eternal in the human breast, which kept one alive, often unhappily... it would be an interesting subject on which to read a paper to the Literary Society, which the Archdeacon was always threatening to start in the village. Belinda began to collect material in her mind and then imagined the typical audience of clergy and female church workers, most of them unmarried. Perhaps after all it would hardly be suitable. She must consider, too, what was fitting to her _own_ years and position.\n\nBy this time she had planted most of the bulbs, and her back was aching. She stood up to rest herself, and looked idly over the wall. The road was deserted and there was no sign of life at the vicarage, but of course it was barely half-past ten, and in the village the best people did not appear till later, when they would start out to do their shopping or to meet a friend at the Old Refectory for coffee. Belinda leaned her arms on the wall, apparently lost in thought. It did not occur to her that she would look odd to anyone passing by. Absent-mindedly she scraped the moss off the wall with a trowel. She would rest for a few minutes and then put some scyllas in the rockery. It was nice to think that she had the whole morning before her. She must go into the cellar and see if the bulbs she had planted in bowls were showing any shoots yet. It would be fatal if they were left in the darkness too long....\n\nShe looked up from the moss and glanced in the direction of the vicarage, to see if anything had happened since she last looked. In the next instant she knew that a great deal had happened. Mr. Mold had come out of the gate and was walking rapidly towards her.\n\nHe must be coming _here,_ thought Belinda, in a flurry of agitation. For some seconds she wondered whether she ought to go and warn Harriet, but even now she could hear his step on the pavement. How quickly he walked! It would be as much as Belinda could do to hide herself before he came through the gate and walked up the drive.\n\nShe looked around frantically. There was no time to run back to the house, as he would see her, and even if he did not recognize her in her gardening clothes it would look so conspicuous. So Belinda concealed herself as best she could behind a large rhododendron bush, which grew on one side of the little drive leading up to the front door. She was fully aware how foolish she would feel if she were discovered in this undignified crouching position, but she could not imagine that Mr. Mold would take the trouble to penetrate the thickness of the bush before he rang the bell and announced himself. She wished he would hurry, for it was very uncomfortable behind the bush and rather dirty. Also, Belinda felt like laughing and it would indeed be terrible if Mr. Mold were startled on the doorstep by a sudden burst of laughter coming out of a bush.\n\nThe crunching of the gravel told Belinda that her ordeal was nearly over. From her hiding-place she could observe him quite well, and she noticed that he looked unusually smart. He seemed to be dressed all in grey and carried very new gloves and a walking-stick. Belinda could not see whether he had a flower in his buttonhole or not, but she thought it was not unlikely.\n\nWhen he was safely in the house, Belinda took the opportunity to run as fast as she could into the back garden, where she arrived rather out of breath. She sat down on an upturned box in the toolshed and began to consider the situation.\n\nWhat could bring him to their house so early in the day? It seemed unlikely that he was the kind of person who would call to thank them for the supper party. It was more probable that he had come to demand a subscription for the Library extensions. It might be that he was calling on several people for this purpose and had come to their house first because it was nearest. Belinda hoped Harriet was not going to be disappointed. She seemed to have taken quite a fancy to Mr. Mold, and it would be so unfortunate if she got any _ideas_ about him. For Belinda was sure that if Mr. Mold ever did decide to marry he would choose for his bride some pretty, helpless young woman, perhaps a reader in the Library, who asked him in appealing tones where she could find the _Dictionary of National Biography._ In any case, he was certainly not good enough for Harriet, who would soon tire of his florid complexion and facetious humour. Also, he was not really a gentleman; that seemed to matter a great deal. All the same, it would be interesting to know why he had called.\n\nHarriet, who had been sitting over the fire in the dining-room at her usual task of 'strengthening corsets' with elastic thread, had not been so slow in finding an answer to this important question. It was quite obvious that he had come to see _her._ She had no time to go into this question more deeply, for almost immediately after she had heard the front-door bell, Emily had come into the dining-room and announced Mr. Nathaniel Mold.\n\n'Tell him I shall be with him in a minute,' said Harriet, rolling up the corsets and putting them under a cushion. It would never do for him to see her with her face all flushed and shining from sitting over the fire. 'Oh, and Emily, I hope you have switched on the electric fire in the drawing-room?'\n\nFive minutes later she walked down the stairs looking considerably more elegant, her face rather heavily powdered and her hair neatly arranged. There was no need to hurry, she decided, as she paused for a moment in the hall to take a final look at herself in the mirror.\n\nShe opened the drawing-room door quietly. Mr. Mold was standing with his back to her. At Harriet's entry he turned round, rather startled. He was holding in his hand a copy of _Stitchcraft,_ in which he had been reading how to make a table runner. It is always difficult to know how one ought to be occupied when waiting for a lady in her drawing-room, and he had resisted the temptation to probe into the pigeon holes of the large desk, which stood invitingly open. _Stitchcraft_ was dull but safe, he felt.\n\n'Good morning,' said Harriet, advancing towards him with hands outstretched in welcome and a brilliant smile on her face. 'I'm so sorry to have kept you waiting. I'm afraid you didn't find anything very interesting to read, but it must be a change for you to look at a frivolous feminine paper.'\n\n'A very pleasant one,' he said gallantly, 'but of course we do take all these papers in the Library. I was in charge of cataloguing them at one time. I learnt quite a lot about needlework and beauty culture.'\n\n'How wonderful\u2014to think that these papers are preserved,' said Harriet, laughing. 'But I suppose you would be much too important to have anything to do with them now.'\n\n'Oh, no!' Mr. Mold smiled and laughed and looked generally rather coltish.\n\n'Do sit down,' said Harriet, sinking into the softness of the sofa. 'It's very nice of you to have called,' she went on, hoping that he would soon give her some clue as to why he had come.\n\nShe was even more handsome in daylight than she had been in the evening, he decided, which was indeed very surprising. He had almost expected to be disappointed at their second meeting and had planned an alternate course of action should this happen.\n\n'It is a very great pleasure to see you again,' he said rather stiffly. 'I felt I wanted to call and thank you and your sister for the very delightful party last night. I enjoyed it immensely.'\n\n'Oh, I'm afraid my sister is out in the garden,' said Harriet, half rising, 'but I'm sure she would like to see you.'\n\n'Oh, well perhaps you could convey my thanks to her. I expect she is busy and I shouldn't like to bother her,' said Mr. Mold quickly. He had not been at all taken with the sister, and the last thing he wanted was to have to sit making conversation with her.\n\nThere was quite a long pause. Mr. Mold began to feel rather uncomfortable. This was not at all his usual style. Perhaps it would have been better if he'd had a whisky before he came out, though half-past ten was a little early even for him. Still, it might have given him courage, though he could not help feeling that he might be more successful if there were a certain diffidence or nervousness about his bearing. He could not draw upon his experience in such matters because he had never before proposed marriage to anyone. His intrigues had been mostly with the kind of women who would hardly make suitable wives for the deputy Librarian of one of England's greatest libraries; nor had they ever been considered as such.\n\nWhat _is_ the matter with him? Harriet was wondering. He was not at all like his usual self, in fact he seemed quite nervous, almost like poor Ricardo when he was about to propose to her. She determined to put him at his ease, so she said in a light joking way, 'Now, I do hope you haven't come to say goodbye. It will be very naughty of you to run off and leave us so soon.' She found this way of talking very good with curates and it certainly seemed to make Mr. Mold less shy.\n\n'Unfortunately my work demands that I should go back this afternoon, or tomorrow at the latest,' he said, gaining courage from her manner. 'But before I went I hoped I should be able to have a talk with you.' He looked at her plump, handsome profile expectantly.\n\nVery much what I expected, thought Harriet complacently, but she was pleased and flattered to discover that she had been right. It was gratifying to feel that such a Man of the World as Mr. Mold obviously was should want to marry her. But should she accept him? When it came to the point, Harriet found herself surprisingly undecided, considering how well she had spoken of Mr. Mold to Belinda. She began to see that there were many reasons why she should refuse his offer when it came. To begin with she had known him for such a short time; indeed, this morning was only their third meeting. Harriet was not the kind of person to believe with Marlowe that\n\n_Where both deliberate, the love is slight:_\n\n_Whoever loved, that loved not at first sight?_\n\nObviously that was quite ridiculous. How could one possibly know all the things that had to be known about a person at first sight? Belinda had said she believed Mr. Mold had a very nice house, but then poor Belinda was so vague, and for all that the house might be semidetached and not at all in an advantageous position. If Mr. Mold were very much in love with her it might be unkind to hurt his feelings\u2014Harriet did not stop to consider how many times she must have hurt the feelings of her faithful admirer Count Bianco\u2014but a smart and floridly handsome admirer in the Prime of Life would be much more acceptable to her than a husband of the same description. In her girlhood imaginings Harriet had always visualized a tall, pale man for her husband, hence her partiality for the clergy. People of Mr. Mold's type could never look well in a pulpit. And finally, who would change a comfortable life of spinsterhood in a country parish, which always had its pale curate to be cherished, for the unknown trials of matrimony? Harriet remembered Belinda once saying something about people preferring to bear those ills they had, rather than flying to others that they knew not of, or something like that. It had been quite one of Belinda's most sensible observations.\n\nThus Harriet's mind was practically made up to refuse Mr. Mold's offer when it came. In the meantime, she waited for him to declare himself. He was nearly as slow as poor Ricardo, who always took so long to come to the point that Harriet sometimes found herself helping him out.\n\nThere was a pause. Harriet sighed. Perhaps even Mr. Mold needed to be helped a little; she had thought he might be better at coming to the point than Ricardo.\n\n'How I envy you living in that lovely town,' she said, looking at him rather intensely. 'Your house is in the Woodbury road, isn't it? I always think that's the very nicest part.'\n\n'Yes, it is pleasant. My house is on a corner, so it has a rather larger garden than the others.' Again there was a short pause, then Mr. Mold burst out with rather forced joviality, 'You know, I feel that you and I have so much in common....'\n\nHarriet said nothing. She was going rapidly over her own interests and comparing them with those that Mr. Mold might be supposed to have. A certain standard of living, comfort, good food, all these they might share, but as before her mind went back to what was undoubtedly her greatest interest\u2014curates. Perhaps she did not define it in that one simple word, but the idea was there, and with it the suspicion that Mr. Mold was the kind of person who was not entirely at his ease with the clergy.\n\nEncouraged by her silence Mr. Mold went on: 'What I mean to say is, that I think we should be very happy if we married. My house is large and comfortable and my financial position is sound... and,' he added, rather as an afterthought, 'I loved you the moment I saw you.'\n\nHarriet almost laughed when she remembered their first meeting in the village, when she had been wearing that awful old tweed coat, too! It was really amazing how blind love made people. Nevertheless, she was disappointed. Proposals from Ricardo several times a year had accustomed her to passionate pleadings, interspersed with fine phrases from the greater Italian poets. Besides, Ricardo never proposed sitting down. Always standing or even kneeling, indeed, his courtly manners had often caused Harriet some amusement. Compared with Ricardo, Mr. Mold sounded so prosaic and casual. He didn't sound as if he really _cared_ at all. She glanced at him hastily; little beads of sweat were glistening on his forehead and his face was crimson. Harriet could not help remembering that Ricardo always looked pale, and although these differences were rather trivial, they seemed somehow to add themselves to the list of reasons why she should not accept Mr. Mold's proposal.\n\n'Dear Mr. Mold,' she began, not quite as certain of herself as usual, for she was not yet used to rejecting _him,_ and did not know how he would take it. 'It is really charming of you to say such kind things, and I am deeply honoured by your proposal, but I feel I cannot accept it. It would not be fair to _you,'_ she added hastily, not wishing to appear unkind.\n\nMr. Mold looked genuinely disappointed. 'Of course I know this must be a shock to you,' he ventured. 'Perhaps you would like to wait a few days and decide after thinking it over?'\n\nBut Harriet didn't think she would like to do that. Thinking things over was so tiring, and really there was nothing to think about. The more she considered it, the less attractive the prospect of this marriage seemed to be. He had been so jolly last night, that was what she had liked. Perhaps that was because he had been a little drunk? And somehow he didn't look so handsome at close quarters. Was it possible that he was just _past_ the Prime of Life? she wondered.\n\nSo she smiled at him very charmingly and repeated that although she was flattered and deeply touched by his proposal she thought it would be kinder to give him his answer now.\n\n'I'm afraid my sister and I are _very_ confirmed spinsters,' she added, in a lighter vein.\n\nMr. Mold felt like saying that he had not intended to marry her sister as well, for he was now annoyed rather than hurt at her refusal, and did not consider that she had sufficiently realized the compliment he had paid her in asking her to be his wife. He muttered something about it being a great pity, and then Harriet said she hoped that he would have a pleasant journey back; the afternoon train was a very convenient one and she believed there was a restaurant car on it. Dr. Parnell would be staying a little longer, perhaps? It was such a real pleasure for them to see visitors as they lived such uneventful lives in this quiet village. She did hope that Mr. Mold would come and see them again next time he was in the neighbourhood.\n\nAs he stood on the front doorstep, Mr. Mold extended a cordial invitation to her to come and visit him some time. 'You'll always find me in the Library,' he added jovially, almost his old self again.\n\n'Reading _Stitchcraft,_ I suppose,' said Harriet, on a teasing note.\n\nAs he went out of the gate, he even waved one of his new gloves at her. Perhaps after all the Librarian was right when he said that marriage was a tiresome business and that he and Mold were lucky not to have been caught. He looked at his watch. There would be plenty of time for a chat with the landlord of the Crownwheel and Pinion before lunch. Marriage might put a stop to all that kind of thing.\n\nWhile Mr. Mold's proposal was being rejected in the drawing-room, Belinda was in the dining-room, writing a letter to Agatha. 'We have had remarkably mild weather lately,' she wrote, 'and I have been able to do a lot of gardening, in fact I have just been putting in the last of the bulbs. I have noticed your pink chrysanthemums showing buds, which is very early for them, isn't it?\n\n'The Archdeacon preached a very fine sermon on Sunday, about the Judgment Day. We were all very much impressed by it. You will be glad to hear that he is looking well and has a good appetite.'\n\nHere Belinda paused and laid down her pen. Was this last sentence perhaps a little presumptuous? Ought an archdeacon to be looking well and eating with a good appetite when his wife was away? And ought Belinda to write as if she knew about his appetite?\n\nShe turned to the letter again and added 'as far as I know' to the sentence about the appetite.\n\n'It was so nice to see Nicholas Parnell again, and I think he enjoys coming here for a quiet holiday. He brought the deputy Librarian, Mr. Mold, with him. I don't know whether you have met him? Personally, his type does not appeal to me very much. He is supposed to be a great ladies' man, and is too fond of making jokes not always in the best of taste. Harriet saw him coming out of the Crownwheel and Pinion in the morning, which I thought a pity.'\n\nHere Belinda laid down her pen again. Was she being quite fair to Mr. Mold? She had allowed herself to get so carried away by her own feelings about him that she had rather forgotten she was writing to Agatha, in whom she did not normally confide.\n\n'Still, I daresay he is a very nice man,' she went on, 'when one really knows him.'\n\nThis last sentence reminded Belinda that he had now been closeted in the drawing-room with Harriet for some considerable time. Belinda had not yet been able to decide why he had come, indeed, she had rather forgotten about the whole thing. Nothing was further from her mind than a proposal of marriage, and had she known what was going on, she would probably have rushed into the drawing-room, even if she had still been wearing her old gardening mackintosh and goloshes, and tried her best to stop it, for one was never quite sure what Harriet would do. Especially after her apparent admiration of Mr. Mold and her continual harping on the Prime of Life. Belinda went so far as to go into the hall, but could not bring herself to listen at the drawing-room door. From where she stood she could hear a low murmur of voices. It was no use being impatient, and the last thing she wanted was to see Mr. Mold herself, so she went back to her letter. Writing to Agatha was not easy, more of a duty than a pleasure, but Belinda felt that she might like to hear some of the details of the parish life which the Archdeacon probably would not give her, so she wrote about the autumn leaves and berries they had used to decorate the church, the organist's illness and Miss Smiley's brave attempt to play at Evensong, the success of the Scouts' Jumble Sale and other homely matters.\n\nAt last she heard the sound of a door opening, then conversation and laughter. Harriet and Mr. Mold had come out of the drawing-room. Belinda waited until she judged him safely out of the front door and then went eagerly into the hall to hear the result of his visit.\n\nShe found Harriet standing in front of the mirror, rubbing her hands together and looking pleased with herself. Her face was rather red and she looked more elegant than was usual at such an early hour of the day.\n\n'Well,' she said, with a hint of triumph in her voice, 'that's that.'\n\n'Yes,' said Belinda, 'but what? I hope you didn't promise him anything for the Library Extension Fund. There are far more deserving causes in the parish.'\n\n'But, Belinda, surely you guessed why he had come?' said Harriet patiently, for really her sister was very stupid. 'He came to ask me to marry him,' she declared, smiling.\n\n'Oh, _Harriet...'_ Belinda was quite speechless. She might have known that something dreadful like this would happen. As if he would bother to come and ask for a subscription to the Library funds! Her supposition seemed very vain and feeble now. Still, as Belinda would not have to live with them, perhaps she need not see very much of her over-jovial brother-in-law\u2014that would be some consolation, though it would hardly make up for the loss of her sister. Of course, she supposed, she could always have a companion to live with her, some deserving poor relation like Connie Aspinall, or she might advertise in the _Church Times;_ somebody with literary interests and fond of gardening, a churchwoman, of course. Belinda shuddered as she thought of the applications and the task of interviewing them; she was sure she would never have the strength to reject anyone, however unsuitable. Perhaps, after all, it would be better to live alone.\n\n'Of course, I couldn't accept him,' said Harriet, rather loudly, for she had expected Belinda to show real interest, instead of just standing and staring at the floor.\n\nThe look of relief that brightened Belinda's face was pathetic in its intensity.\n\n'Oh, _Harriet...'_ again she was speechless. However could she have thought for a moment that her sister would do such a thing?\n\n'Indeed I couldn't,' said Harriet calmly. 'Why I hardly know him, and you remember what Shakespeare said about when lovely woman stoops to folly...' she made a significant gesture with her hand.\n\nBelinda frowned. 'I don't think it was _Shakespeare,_ dear,' she said absently. 'I must ask Henry. I have an idea it may be Pope.' But what did it matter? Belinda was so overcome with joy and relief at Harriet's news that she kissed her impulsively and suggested that they should have some meringues for tea, as Harriet was so fond of them.\n\nTogether they went into the dining-room, where Harriet with many ludicrous and exaggerated imitations, gave a demonstration of how Mr. Mold had proposed to her.\n\n'Oh, Harriet, you mustn't be so unkind!' protested Belinda, in the intervals of laughing, for her sister was really much funnier than Mr. Mold could possibly have been. They laughed even more when the corsets were discovered under a cushion.\n\n'Just imagine if Emily had brought him in here and he had discovered them while he was waiting. Or if the Archdeacon had when he came the other day,' chortled Harriet.\n\n'Oh, Harriet,' said Belinda faintly. There was a vulgar, music-hall touch about it all that one could associate with Mr. Mold but hardly with the Archdeacon.\n\n'I expect he's consoling himself in the Crownwheel and Pinion,' said Harriet, 'so we needn't really pity him.'\n\nShe was perfectly right; so much so that, when he arrived at the vicarage rather late for lunch, Dr. Parnell was constrained to whisper to his friend the Archdeacon, 'I fear poor Nathaniel is not entirely sober.'\n\n# Chapter Thirteen\n\nAS MR. MOLD SETTLED himself comfortably in his first-class corner seat he decided that he had probably had a lucky escape. And indeed, he reflected, _Love is only one of many passions and it has no great influence on the sum of life,_ as the Librarian was so fond of quoting.\n\nA few days later Belinda and Harriet were invited to tea at the vicarage. It was hardly surprising that Mr. Mold's proposal, which appeared to be known to the Archdeacon and Dr. Parnell, should be the chief topic of conversation.\n\nDr. Parnell was inclined to think it a pity that Harriet had refused his colleague, for although he had always been of the opinion that it must be very tiresome to be married, he did not deny that it was an interesting state. Indeed, he often regretted that the Archdeacon was the only one of his friends who had a wife. As a young man Dr. Parnell had looked forward to the time when Belinda would come to him for advice on the trials of matrimony. In those days he had hoped that she might marry the Archdeacon, and was almost as disappointed as she had been at her failure to captivate him. He had never liked Agatha, but he could not help admiring her skill, and when by her powers her husband was raised to the dignity of archdeacon, Mr. Parnell, as he then was, had aptly remarked that Henry was indeed fortunate in having won the love of a good woman. Nevertheless, he considered himself almost equally fortunate in _not_ having done so, and often used to remark to John Akenside that he did not think poor Henry was quite as _free_ as he had been.\n\nBut there was no denying that Harriet and Mr. Mold would have made an admirable couple. They had both reached an age when temperament and character were settled, and instead of one dominating the other they would have been able to live in comfortable harmony. Besides, there would be plenty of money, so that if there had been love, which Dr. Parnell rather doubted, it would have been less likely to fly out of the window, as he had been told it did when poverty came in at the door.\n\nSitting round the fire in the Archdeacon's study, they considered the problem.\n\n'Of course I never advise anyone to enter into that state without long and careful thought,' said Dr. Parnell, 'but I should be the last to admit impediments to the marriage of true minds, and it seems to me that you and Nathaniel have a great many tastes in common.'\n\nHarriet denied this indignantly: perhaps she was still thinking of curates. 'The only thing we have in common is a love of good food,' she said, thinking that Dr. Parnell was being more than usually interfering. 'I could never marry Mr. Mold.'\n\n'But surely liking the same things for dinner is one of the deepest and most lasting things you could possibly have in common with anyone,' argued Dr. Parnell. 'After all, the emotions of the heart are very transitory, or so I believe; I should think it makes one much happier to be well-fed than well-loved.'\n\nBelinda did not trouble to contradict this statement, romantic and sentimental though she was. She was feeling much too happy and peaceful to indulge in any argument. For here she was sitting on the sofa with the person she had loved well and faithfully for thirty years, and whom she still saw as the beautiful young man he had been then, although he was now married and an archdeacon. And as if this were not enough, had she not just escaped having a brother-in-law who was not really a gentleman, and made jokes not always in the best of taste? When one reached middle age it was even more true that all change is of itself an evil and ought not to be hazarded but for evident advantage. She smiled at Dr. Parnell indulgently, but said nothing. The Archdeacon in his turn smiled affectionately at her, and thought what a nice peaceful creature she was, so different from his own admirable wife, with her busy schemes for his preferment.\n\nDr. Parnell was still regretting Harriet's hasty action, and suggested that she might write Mr. Mold a letter giving him some _hope,_ for he had heard that even hope was better than nothing.\n\nBut Harriet, who knew she was being teased, merely listened with a smile on her face and said with dignity that she believed she could do a great deal better for herself. She looked at the three of them rather mysteriously, and Belinda wondered whether she could be making plans to captivate Dr. Parnell.\n\n'You would have kept poor Nathaniel out of mischief,' he said, still harping on the same subject.\n\n'I daresay,' remarked Harriet, 'and I expect he needs it. Do you know,' she leaned forward confidentially, 'I believe he _drinks...'_ she said, pronouncing this last word in a suitably hushed whisper.\n\n'Oh, Harriet,' protested Belinda, for she could now afford to feel kindly towards Mr. Mold, 'I don't think you should say that. We all like to take something occasionally, a drink can be a great comfort at times.'\n\n'I am glad to hear that you are so broad-minded,' said the Archdeacon. 'I remember Agatha being quite shocked when I said something of the kind to the Mothers' Union once.'\n\n'Well, I suppose it is a dangerous thing to say,' said Dr. Parnell. 'They might abuse the comfort of drink.'\n\n'Whereas _we_ know how to be moderate,' said Harriet primly.\n\n'I cannot imagine Agatha taking too much,' said Dr. Parnell. He chuckled. 'No, I'm afraid I can't.'\n\nBelinda gave him a shocked glance. 'Have you heard from Agatha again?' she asked the Archdeacon brightly.\n\n'Yes, as a matter of fact I had a letter by the lunch-time post,' he said. 'You can read it if you want to,' he added, taking a letter out of his pocket and handing it to her.\n\nBelinda took the letter rather gingerly, thinking it odd that he should hand it to her so willingly. But when she came to read Agatha's neat handwriting, she saw that the letter contained nothing private. It seemed to be a long list of things he must not forget to do. It was admirably practical, but unromantic. And yet, after so many years of being married to a charming but difficult man like the Archdeacon, perhaps it was rather too much to expect that Agatha should dwell on the desolation of life without him. All the same, Belinda could not help remembering her own letters, and she was sure that even now she could have found something a little more _tender_ to write about than Florrie's and cook's wages and the Mothers' Union tea. She was just going to hand the letter back when she noticed that there was a postscript over the page.\n\n'I forgot to tell you that among the people staying here is the Bishop of Mbawawa. I believe the Bedes know him. He is a delightful man, so friendly, and he tells many interesting stories about the splendid work he has been doing among the natives. I am trying to persuade him to come home with me, as I am sure everyone would be interested to meet him.'\n\nBelinda stopped short in amazement as she read these words. 'Harriet,' she said, _'who_ do you think is there?'\n\nHarriet, who was quietly enjoying a substantial tea, looked up and asked who was where.\n\n'In Karlsbad,' said Belinda.\n\n'I don't know,' she said, not very interested. 'It's the sort of place where King Edward VII might be, only of course it could hardly be him.'\n\n'It's an old friend of yours,' said Belinda.\n\n'Is he an old friend?' asked the Archdeacon.\n\n'I should like to number bishops among my friends,' said Dr. Parnell.\n\nHarriet seemed to brighten up at this. 'Bishops? Well, of course I know quite a number,' she mused. This was not really surprising, for after all every bishop has once been a curate. 'It couldn't be Willie Amery, I suppose or Oliver Opobo and Calabar\u2014isn't that a lovely title?\u2014no, he's in Nigeria, I believe. Of course it might be Theo Grote, Theodore Mbawawa, as he signs himself,' she smiled to herself. _'That_ would be the nicest of all.'\n\n'Yes, Harriet, it's Theo Grote,' said Belinda. 'I knew you would be interested to hear that.'\n\n'Oh _ho,'_ said Dr. Parnell, seeing that Harriet had gone quite pink in the face. 'I believe we are going to see some old broth being warmed up. I like to see that.'\n\n'Agatha talks of bringing him to stay here,' said the Archdeacon distastefully. He disliked other members of his calling.\n\nClean sheets on the spare bed and a tin of biscuits on the little table in case he should feel hungry in the night, thought Belinda irrelevantly.\n\n'He must be about fifty-seven or fifty-eight,' said Harriet, who seemed to have been doing a little calculation. 'It _will_ be nice to see dear Theo again.'\n\n'On the threshold of sixty,' mused Dr. Parnell. 'That's a good age for a man to marry. He needs a woman to help him into his grave.'\n\n'But that's just the Prime of Life,' said Harriet indignantly. 'I'm sure we shan't find Theo at all _doddery.'_\n\nBelinda began to suspect that Harriet regarded the Bishop as a possible husband. She had certainly been very much in love with him when she was a schoolgirl and he a willowy curate in the early twenties. Belinda had often thought that the reason why Harriet made so much of Mr. Donne was because he reminded her of dear Theo Grote. And then Belinda had often heard her say that a bishop needed a wife to help him with certain intimate problems in his diocese, things which a woman could deal with better than a man. It seemed a little hard, Belinda thought, that this new menace should appear, just when she was so relieved at having escaped Mr. Mold, but she would just have to leave Harriet to her schemes. Belinda trembled for the unmarried Bishop of Mbawawa if he did not feel inclined to enter into that blessed state.\n\n'Shall I read aloud to you?' suggested the Archdeacon hopefully. He went over to the bookshelves and invited requests for what anyone would like. But Dr. Parnell suddenly got up from his chair and announced that he thought he had better do his packing. It was so tiresome to have a rush at the last minute. Harriet, too, had suddenly remembered that she was to deliver some parish magazines and was already half way out of the door, thanking the Archdeacon for a delightful tea party, and inviting him to drop in at four o'clock any afternoon. She and Dr. Parnell hurried out of the room together, the latter remarking that he wondered Henry could spare the time from his parochial duties to listen to the sound of his own voice.\n\n'We should all have time to improve our minds,' said Belinda smiling happily.\n\nThe Archdeacon turned towards her with a volume of Spenser in his hand. 'I think it would be pleasant to have something from the _Faerie Queene,'_ he declared.\n\nThe clock struck half-past five. Belinda settled herself comfortably in her chair. She felt rather drowsy and the _Faerie Queene_ was such a soothing poem. It just went on and on.\n\nAt six o'clock the Archdeacon suggested a little Wordsworth. Belinda agreed that this would be very nice. She had always been so fond of _The Prelude._\n\nAt half-past six Belinda began to murmur something about being sure that she was disturbing the Archdeacon, who must have a great deal to do.\n\nWell, yes, he supposed that she _was_ disturbing him really, but it was very pleasant to be disturbed occasionally, especially when there were so many tiresome things to do.\n\n'Do you know,' he said suddenly, with the air of one who has made an important discovery, 'this reminds me of the old days. I used to read aloud to you then. Does it remind you?'\n\nBelinda was speechless, as she considered this proof of man's oddness. Whatever did he imagine that it reminded her of? 'Oh, yes, it's quite like the old days,' she said at last, and then tried to think of something more intelligent to continue the conversation.\n\nSilences were awkward things, especially when one's mind was only too apt to wander back into the past and remember it so vividly that it became more real than the present. Unless she fixed her attention on something definite, she might find herself saying the wrong thing. Her eyes lighted on a set of Bible commentaries. Well, nobody could expect her to talk about them. She must try again. The mantelpiece is dusty, she thought. Florrie needs keeping up to the mark, and I don't believe she's used the Hoover on this carpet since Agatha went away. Agatha. There was something definite. There was nothing vague or nebulous about an archdeacon's wife, even when she wasn't there. I loved you more than Agatha did, thought Belinda, but all I can do now is to keep silent. I can't even speak to Florrie about the dusty mantelpiece, because it's nothing to do with me. It never was and it never will be.\n\n'Florrie never bothers to dust my study when Agatha's away,' said the Archdeacon, seeing where Belinda was looking.\n\n'No, things always go wrong in a house when there's no woman at the head of things,' agreed Belinda. 'I mean, it's different when Agatha's away.'\n\nThe Archdeacon sighed. 'Yes, it is different,' he agreed. 'But there it is. We can't alter things, can we?'\n\nBelinda did not know what to say to this, as she was not quite sure what he meant. She was just wondering what would happen if she led the conversation round to more personal things than dusty mantelpieces, when the door opened and in came Dr. Parnell, complaining that he was hungry and asking if they were never going to have anything to eat.\n\n'Why, yes, it must be nearly supper-time,' said Belinda, starting to put on her gloves. 'I must go.'\n\n'Oh, but I insist that you stay,' protested the Archdeacon.\n\n'I really couldn't,' said Belinda mechanically. 'Harriet will be expecting me.'\n\n'Please, dear Belinda,' he said coaxingly. 'You know I asked you to tea the other day and you wouldn't come. The least you can do is to stay now. For the sake of old times,' he added, with uncharacteristic heartiness.\n\n'Really, Henry, I think you might have put it better,' said Dr. Parnell. 'I should hardly imagine that poor Belinda can really wish to be reminded of old times.'\n\nBut Belinda only smiled. 'All right, I will stay,' she said.\n\n'I shall never understand women,' said Dr. Parnell complacently.\n\n# Chapter Fourteen\n\nBELINDA ARRIVED HOME THAT evening feeling very happy. It had been so nice having supper at the vicarage without the restraining presence of Agatha, the efficient wife and good philologist. During those few hours Belinda had almost imagined herself back in her youth. As she had listened to the Archdeacon giving a short dissertation on the Beast Fable in the Middle Ages, she had found herself looking at her watch and thinking that she would have to be back at her college by half-past ten if she did not want the Principal to ask any awkward questions. When the Archdeacon had gone on to discuss the sources of his Judgment Day sermon, she had realized regretfully that the Principal of whom she had stood in awe had been in her grave at least ten years.\n\nIt was not until she reached her own front door that Belinda began to feel a little uneasy, and wonder if it had been quite the thing for her to spend a whole evening at the vicarage without Agatha or Harriet to chaperone her. For although Dr. Parnell had been there, he wasn't quite the same as some respectable middle-aged woman. And yet why should not she be allowed her occasional joys, such very mild ones, which were mostly remembrance of things past?\n\nSound of hearty laughter came from the drawing-room. The curate was there. Perhaps he had dropped in after Evensong, as Harriet had so often told him to, and had stayed to supper. Belinda was glad that he was there, as his presence would save her a little from Harriet's ruthless cross-examination, which was bound to come sooner or later.\n\nBelinda went upstairs to take off her hat and coat and then into the middle of the cheerful noise.\n\nHarriet and the curate were sitting on the sofa, deeply engrossed in a book. The curate leaped up and expressed himself delighted to see her. Belinda thought it a little unnecessary of him to welcome her to her own house, but she said nothing.\n\n'I suppose you've had supper?' asked Harriet.\n\n'Yes,' said Belinda, 'I stayed at the vicarage. I hope you didn't wait for me?'\n\n'Oh, no,' said Harriet, 'in fact I didn't expect you back even as soon as _this.'_\n\nBelinda laughed rather uncomfortably. 'What are you reading?' she asked, hoping to change the subject.\n\n'We were reading Catullus. I really don't know how we got on to it,' said Harriet merrily. 'Mr. Donne's so good at Latin but of course it's quite thirty years since I read a _word_ of it.'\n\n'Oh, come,' said the curate playfully. 'I can't believe that.'\n\nBelinda took up her knitting. She remembered Dr. Parnell saying that he thought Catullus rather too indelicate for a young girl to read. If this were so, for Belinda's scanty knowledge of Latin would not enable her to find out for herself, how much more indelicate must the great Roman poet be for a young curate! 'There is a pretty translation of one of his poems by Thomas Campion,' she said vaguely, 'but I suppose it's not like reading the original.'\n\n'No, my friend Olivia Berridge always says that. You remember perhaps, she's Mrs. Hoccleve's niece,' explained the curate.\n\n'Oh, yes,' said Harriet, 'she knows Anglo-Saxon and things like that, doesn't she? And of course she made you those socks. I thought the toes were not very well grafted, in that grey pair you showed me. It's quite easy to do, really. You just say knit and slip off, purl and keep on\u2014or it may be the other way round.'\n\nThe curate looked mystified. 'She's very clever,' he said. 'I expect she knows that.'\n\nHarriet looked a little annoyed and the conversation flagged. Mr. Donne got up to go.\n\n'Perhaps you would like to borrow some books?' suggested Belinda, seeing that he was looking at the shelves. It was a little difficult to guide his choice, but eventually he went away with some thrillers and the selected poems of the Earl of Rochester, a volume of which Belinda was particularly fond. It had been given to her by Dr. Parnell on her twenty-first birthday. Belinda felt that it would not be likely to harm Mr. Donne's morals, as it professed to be _a collection of such pieces only, as may be received in a vertuous court, and may not unbecome the Cabinet of the Severest Matron._\n\nHe had hardly gone out of the front door, when Harriet turned eagerly to Belinda and said, 'Now tell me _all,_ about it.'\n\nBelinda looked up from her knitting rather startled. 'But, Harriet,' she protested, 'there's really nothing to tell. Henry read aloud to me and then we talked a bit and then he persuaded me to stay to supper, which I did. But I don't know whether I ought to have done that,' she added rather unhappily. 'I mean, I shouldn't like Agatha to think...'\n\n'No, no, of course not,' said Harriet soothingly. 'Now wouldn't you like a nice cup of Ovaltine?' she said, fussing round Belinda like a motherly hen.\n\n'Well, I don't know, I think I would,' said Belinda. Perhaps a nourishing milky drink was needed to bring her down to earth but it seemed an unromantic end to the evening.\n\nHarriet had already gone into the kitchen and soon returned with the Ovaltine and a selection of biscuits and cakes.\n\n'Now,' she said, as if speaking to an invalid, 'drink it up while it's hot and don't try to talk till you've finished. There'll be plenty of time for you to tell me all about it.'\n\n'But what is there to tell?' protested Belinda, rousing herself. 'I've told you what we did.'\n\nHarriet chose a chocolate biscuit. 'I do believe the Archdeacon has been asking you to elope with him!' she declared triumphantly.\n\n'Oh, Harriet, how dreadful you are!' said Belinda, unable to help laughing at this monstrous suggestion. 'As if a clergyman, let alone an archdeacon, would do a thing like that.'\n\n'Then he's been telling you that he's very fond of you, and hinting that he wishes he'd married you instead of Agatha,' went on Harriet, gallantly persevering.\n\n'Well, hardly that,' ventured Belinda, growing a little more confidential, for the Ovaltine had loosened her tongue. 'I mean, it's a bit late for anything like that, isn't it? Henry is always loyal to Agatha and feels quite _differently_ about her,' she added hastily, in case her sister should take her up wrongly.\n\nHarriet agreed with this ambiguous statement. 'Yes, I'm sure he does,' she said, 'but there's no reason why he shouldn't be fond of you as well. Clergymen are always saying that we should love one another.'\n\n'Oh, _Harriet,'_ protested Belinda, rather shocked, 'you know quite well that isn't at all the same kind of thing. But of course Henry and I have always been friends and I hope we always will be.'\n\nHarriet sighed. Poor Belinda was so unworldly, so sentimental.\n\n'Of course he is fond of you,' she declared boldly. 'Anybody can tell that by the way he keeps smiling at you, when he thinks nobody's looking.'\n\nBelinda had always thought that they were smiles of pity rather than of love. But hadn't one of our greater English poets said something about Pity being akin to Love? Or had she made it up herself? A vague recollection of Aristotle's _Poetics_ came into her mind. But that was Pity and Fear, rather like her feeling for Miss Prior, not at all the same thing....\n\n'I'm sure everyone knows,' persisted Harriet, nothing daunted by her sister's unwillingness to confide in her and determined to make something interesting out of Belinda's evening at the vicarage.\n\n'Knows what?' asked Belinda, rather startled.\n\n'Why, that you love each other,' beamed Harriet, as if she were giving her blessing to a young couple, instead of making rather a scandalous suggestion about a married archdeacon and a respectable spinster.\n\nBelinda was now rather agitated and could not think of anything to say.\n\n'Don't deny that he's making the most of Agatha's absence,' Harriet went on, 'and anyway everybody knows that you knew him long before Agatha did.'\n\n'I don't see what that has to do with it,' said Belinda. 'One can hardly claim people on that basis.'\n\n'But didn't he say _anything?_ Surely you weren't just reading poetry _all_ the time?'\n\n'No, not all the time.' Belinda smiled as she remembered their conversation. 'We talked about the dust on the mantelpiece.'\n\n'Oh, Belinda, surely...' Harriet searched vainly in the tin for another chocolate biscuit.\n\n'And I said that of course things did go wrong when there was no woman at the head of things,' Belinda went on. 'I mean, servants neglect their duty and that sort of thing. I think I said, \"It's different when Agatha's away\", or something like that.'\n\n'Oh, Belinda, wasn't that rather obvious?'\n\nBelinda looked startled. 'Oh, I meant things in the house, naturally. He surely couldn't have taken it any other way?'\n\n'Well, what did he say to that?' went on Harriet relentlessly. 'Do you remember?'\n\n'Oh, yes. I always remember everything Henry says.' She smiled. 'Thirty years of it. It's a pity I don't remember other things so well. He said, \"Yes, it is different, but there it is. We can't alter things, can we?\"'\n\nHarriet let out a cry of joy. 'Go on,' she said. 'What happened next?'\n\n'Oh, then Nicholas came in and we had supper.' Belinda paused. 'Well, dinner really, because there was soup, though I _think_ it was tinned. Still, it was very nice, mushroom or something. It had little bits of things in it.'\n\n'Oh, Belinda, I don't want to hear about the soup,' said Harriet. How sickening that Nicholas should have come in just at that moment. Just like him,' she grumbled. 'To think of it, the moment you've been waiting for for thirty years!' She paused dramatically.\n\n'But you know I've never expected anything,' protested Belinda. 'I've no right to.'\n\n'But don't you see what he meant when he said that about not being able to alter things?' said Harriet. 'He meant he'd rather have you than Agatha, only of course he couldn't put it quite as crudely as that.'\n\n'We never mentioned ourselves,' said Belinda hopelessly. 'We were talking about his study not being dusted.'\n\n'Now if only he were a _widower,'_ mused Harriet.\n\n'But he isn't,' said Belinda stoutly.\n\n'No, and Agatha's very tough in spite of her rheumatism,' lamented Harriet. 'And soon he won't even be a grass widower because she's coming back.'\n\n'Yes, she's coming back,' said Belinda, even more stoutly. How odd if Henry were a widower, she thought suddenly. How embarrassing, really. It would be like going back thirty years. Or wouldn't it? Belinda soon saw that it wouldn't. For she was now a contented spinster and her love was like a warm, comfortable garment, bedsocks, perhaps, or even woollen combinations; certainly something without glamour or romance. All the same, it was rather nice to. think that Henry _might_ prefer her to Agatha, although she knew perfectly well that he didn't. It was one of the advantages of being the one he hadn't married that one could be in a position to imagine such things.\n\nBelinda gave a contented sigh. It had been such a lovely evening. Just one evening like that every thirty years or so. It might not seem much to other people, but it was really all one needed to be happy. But Harriet was saying something, so she could not indulge in such thoughts for long.\n\n'I don't like the way Mr. Donne keeps mentioning that Olivia Berridge,' she said. 'Although he did say that he was definitely _not_ engaged to her, one never knows.'\n\n'He doesn't sound as if he were in love with her,' said Belinda doubtfully. 'But of course Miss Berridge may have made up her mind to marry him. She sounds a bit like Agatha,' she added.\n\n'But Mr. Donne is nothing like the Archdeacon,' said Harriet indignantly.\n\n'No, I don't think anybody is quite like the Archdeacon,' said Belinda quietly.\n\n# Chapter Fifteen\n\nDURING THE NEXT WEEK or so the village began to look forward to the homecoming of the Archdeacon's right hand, as many of the church workers called Mrs. Hoccleve. They were conventional enough to use this expression, which they had often heard, without troubling to ask themselves whether it could really be applied here. It was a known thing that the wives of clergy often were their right hands, and even if the Hoccleves were sometimes rather snappy with each other there was no doubt that she was the power behind him. The news that she was to be accompanied by the Bishop of Mbawawa had spread rapidly, and little groups of eager Sunday School teachers could be seen talking about it. Some had heard that he was black, a real African bishop, but Harriet soon put them right on this point and achieved a new importance through having known him as a curate. Belinda did not share in this glory, for after so many years she found it difficult to remember which of the many curates her sister had cherished was Theodore Grote.\n\n'He was thin and dark, wasn't he?' she said anxiously. 'But somehow I can't see his face.' There was a blank above the clerical collar, as it were, for so many had been thin and dark.\n\n'Oh, Belinda, you _must_ remember him at that Whist Drive,' said Harriet smiling tenderly, for she had no difficulty in recalling him as one of the most sought-after curates in the history of the Church of England. In his heyday there had been quite a procession of doting women towards his lodgings, carrying cooked pheasants and chickens, iced cakes, even jellies in basins... Harriet sighed over her reminiscences and then remarked in a regretful tone of voice that she did not think Belinda had really made the most of Agatha's absence.\n\nBelinda could not but agree, for Harriet was perfectly right. And yet, how could one make the most of the absence of an archdeacon's wife. It was a thing no truly respectable spinster could or would do. She pointed this out to Harriet, who refused with characteristic obstinacy to understand and merely remarked that we none of us got any younger.\n\nAh yes, thought Belinda, as she turned to knitting the dull grey jumper that might have been a pullover for the Archdeacon, that was it. She thought about it so seldom but now she became melancholy at the realization that the fine madness of her youth had gone. She was no longer an original shining like a comet, indeed, it would have been unsuitable if she had been. _Change and decoy in all around I see... All, all are gone, the old familiar faces..._ Dear Nicholas was back in the Library, John Akenside was in heaven, while his earthly remains rested in an English cemetery in the Balkans, and if Harriet married Theodore Mbawawa, even she would be gone... Who was there apart from the forbidden Archdeacon? One's women friends, of course, people like Edith Liversidge and Connie Aspinall, but they were a cold comfort. Belinda grew even more melancholy, and then she remembered Count Bianco. There was always Ricardo. Perhaps they could read Dante together and find some consolation in the great Italian poet. She did not think she would be equal to reading Tacitus.\n\nIn the meantime Belinda had promised to go to the station with the Archdeacon to meet Agatha and the Bishop. On this day she was classed with Agatha's nearest and dearest in a way which seemed to her rather ironical. Who but a man could be so lacking in finer feelings as to think of such a thing? she wondered. But of course she said she would go.\n\n'After all, you both love the Archdeacon,' Harriet had explained, and Belinda supposed that it was true, though one could hardly admit it even to oneself. Possibly, thought Belinda, I love him even more than Agatha does, but my feeling may be the stronger for not having married him.\n\nAs they waited on the station, Belinda decided that her sister had been wise to stay at home, for it was bitterly cold. Harriet liked her comfort and had decided that she would appear to better advantage in a less bleak setting.\n\n'The Bishop will surely find our climate very different from that of the tropics,' remarked the curate, as they were stamping their feet on the platform to keep warm.\n\nBelinda thought this remark to be so obvious as not to require an answer, so she turned to the Archdeacon and said that she thought his watch must be fast, as they seemed to have been waiting a long time.\n\n'Oh, no, but the train is late,' he said, with a superior smile. So Milton's Adam must have smiled on Eve. He was not pleased at the prospect of having to entertain the Bishop for an indefinite length of time, but nevertheless he was looking forward to it with a kind of grim relish. He remembered certain minor discomforts about the spare room at the vicarage as he stood there on the cold platform. It was a gloomy room with a northerly aspect and a tall, dark monkey-puzzle growing close to the window, which looked out on to an old potting-shed, full of flower-pots and dried-up roots and bulbs. And in addition, although the Archdeacon had not personally made the bed, he knew that there were sides-to-middle sheets on it, for Florrie had come into his study that morning, very agitated because all the whole sheets were still at the laundry. The Archdeacon was delighted. He seemed to remember also that the mattress was a particularly lumpy one, worn into uncomfortable bumps and hollows by a variety of visiting clergy, and that the bedside lamp did not work. All the same he had taken pleasure in making a suitable selection of books for the bedside table\u2014a volume of Tillotson's sermons, Klaeber's edition of _Beowulf,_ the Poems of Mrs. Hemans, an old Icelandic grammar, and, as a concession to the Bishop's connection with Africa, a particularly dull anthropological work, which had been included with some other books he had bought at a sale. The Bishop would naturally want thrillers\u2014the clergy always did, he found\u2014but he was keeping his own supply locked up in his study.\n\nBelinda saw him smiling to himself and wondered whether it could be because Agatha was coming home. Naturally it must be, as she knew he was not pleased at the prospect of the Bishop. Prospect of bishops, she thought, liking the phrase, but at that moment the curate espied the train coming round the bend into the station.\n\n'There they are!' he shouted, rushing towards a first-class carriage where Agatha's face had appeared at the window.\n\n'I thought you would be travelling at the rear. I didn't look for you at the front,' said the Archdeacon rather reproachfully.\n\nHe kissed his wife, not very affectionately, Belinda thought, but she had kept herself rather in the background, waiting for this reunion of nearest and dearest to be over. There was a forced smile on her face to be used when needed. She looked about her and saw emerging from the railway carriage what she imagined must be the Bishop of Mbawawa. He too was looking as if he did not quite know what to do with himself.\n\nBelinda moved towards him and introduced herself. 'I don't suppose you remember me,' she said, smiling rather awkwardly. Nor did she remember him, if it came to that, for she could have sworn that she had never seen him in all her life. Could a beautiful curate have grown into this tall, stringy-looking man, with a yellow, leathery complexion? His expression reminded Belinda of a sheep more than anything; his face was long, his forehead domed and his head bald. He was even rather toothy, a thing that Harriet abhorred. Could it be the same person? she wondered.\n\n'But I certainly do remember you,' he was saying. 'You knitted me a beautiful scarf when I was a curate.'\n\nBelinda was decidedly taken aback. She had always thought it rather wonderful that she had never done anything of the kind. He must have been thinking of Harriet or another of his many admirers.\n\n'Oh, did I?' she said vaguely, which helped neither of them, and caused the Bishop to be assailed with doubts.\n\n'You did such splendid work with the Guild of St Agnes,' he went on, less certainly.\n\nBelinda, who had never heard of any such guild, felt rather foolish. 'I daresay you remember my sister better,' she ventured, 'Harriet, you know.'\n\n'Ah, _yes.'_ His face cleared. 'I shall look forward to renewing my acquaintance with her,' he said politely. 'She was always interested in missions, if I remember rightly.'\n\n'Oh, yes, she was,' said Belinda, thinking that she might as well agree for a change. 'She is looking forward to seeing you so much.'\n\nThe Bishop was just saying something about the pleasure of meeting old friends and looked as if he might almost be about to quote some Mbawawa proverb, when Agatha came over to them and Belinda found herself shaking hands cordially and telling her how well she looked.\n\n'You're looking _splendid,'_ said Belinda, and indeed Agatha was quite fat in the face and seemed in very good spirits. Belinda was shocked to find herself wondering whether a month's absence from her husband could have anything to do with it.\n\nThe curate was now bundling them all into a taxi, saying that he would walk as he had to call and see one of the church wardens. So Belinda found herself sitting by Agatha, while the Bishop and the Archdeacon squatted rather incongruously on the little folding seats.\n\n'I don't think Henry is looking very well,' said Agatha, with something of her old sharpness, so that Belinda felt that it was her fault.\n\n'Oh, I think he's quite well really,' said Belinda quickly. 'A little tired, perhaps, but then he's had so much to do while you've been away.'\n\nThe Archdeacon was unable to resist joining in this interesting conversation, so he rather rudely interrupted the Bishop, who was telling him something about the organization of his diocese, to remark that he was really quite exhausted and did not know how he was going to get through the rest of the winter.\n\n'Oh, I expect you'll manage all right,' said Agatha lightly. 'I'm sure Bishop Grote will be only too glad to help with some of the services.'\n\n'Indeed I shall,' said the Bishop, 'and I daresay my own experience in organizing a large diocese will be of use to you. My African priests are dear, good fellows, but they sometimes need a helping hand.'\n\nBelinda did not dare to look at the Archdeacon's face, but she could feel the blackness of his look and she wondered if her own indignation on his account could be seen in her face. To class an English archdeacon with African priests! Surely that was going too far? Not, she hastened to assure herself, because they were Africans: she was certain that, as the Bishop had said, they were dear, good fellows, but she was surprised that he should have so little sense of what was fitting to the occasion.\n\nThe remainder of the drive was taken in silence. Fortunately it was not far, but it was long enough for Belinda to realize that the Bishop and the Archdeacon had taken an instant dislike to each other. This was in some ways rather unlucky, as it was essential to the success of Harriet's plans that the Bishop should stay some time, and if the Archdeacon were really rude to him, as he might very well be, there was no knowing what might happen. Of course Harriet might not feel so enthusiastic after she had seen the Bishop. Belinda hoped that this might be so, as she did not think she would like him as a brother-in-law and she certainly did not want her sister to leave her and go to Africa.\n\nThey reached the vicarage and Belinda hovered uncertainly by the front door.\n\n'Belinda will stay to tea, of course,' said the Archdeacon quickly.\n\n'Oh, yes, certainly,' said Agatha. 'We want to hear all the parish news.'\n\n'I should like a little moral support,' said the Archdeacon in a low voice. 'I am not sure that tea will be enough.'\n\nBelinda walked quickly into the hall after Agatha. The Bishop followed them, but the Archdeacon stayed behind to supervise the unloading of the luggage.\n\n'I hope it isn't inconvenient,' said Belinda in her usual apologetic manner.\n\n'Not at all,' said Agatha coolly. 'One extra for tea is no trouble at all\u2014I never find that it is.'\n\n'Oh, no, nor do I,' said Belinda, floundering deeper. She was quite grateful to the Bishop for turning the conversation to the Mbawawa hunting customs, by which it appeared that when an animal was killed unlimited hospitality was extended to all neighbouring tribes.\n\n'What a nice custom,' said Belinda inadequately, 'but I imagine they would have to have more than one animal, wouldn't they?'\n\n'Oh, it is a ritual eating,' the Bishop explained. 'The meat is not actually consumed.'\n\n'Well, I hope our tea will be a little more satisfying,' said Agatha, smiling indulgently at the Bishop. 'Where is Harriet, by the way?' she asked Belinda, in a sharper tone.\n\n'She couldn't come to the station,' said Belinda evasively. After all it was none of Agatha's business where Harriet was. She could hardly have told the assembled company that Harriet preferred to wait for some more elegant occasion before renewing her acquaintance with the Bishop. She had visions of herself advancing towards him graciously in her brown velvet or wine cr\u00eape de Chine, and such clothes would hardly have been possible at the station on a cold winter afternoon.\n\n'I hope she is quite well?' said Agatha politely.\n\n'Yes, thank you,' said Belinda. 'She was doing something else, as a matter of fact.'\n\n'With Ricardo, perhaps?' suggested the Archdeacon helpfully.\n\n'Is she married then?' asked the Bishop. 'I did not know.'\n\n'Oh, no,' said Belinda confusedly. 'Ricardo isn't her _husband._ He is an Italian count,' she added quickly, as if that did away with any possibility of a misunderstanding. 'One of our oldest friends here.'\n\n'Of course everybody in the village knows him,' said Agatha. 'He is a charming man, most friendly. I ought not to gossip, I know, but I don't think anybody would be surprised if he were to marry Lady Clara Boulding, the widow of our former Member of Parliament,' she explained for the Bishop's benefit.\n\n'There is no reason why you shouldn't gossip, my dear,' said the Archdeacon, 'but I think a great many people would be surprised if things happened as you suggested. I had an idea that Ricardo's fancy lay in quite another direction.'\n\nAt this the Bishop directed a rather coy glance towards Belinda, who began talking quickly about missions and the very interesting preacher they had had one Sunday evening, from the diocese of Ndikpo, or some such name.\n\n'Ah, yes, Ndikpo,' the Bishop shook his head. 'The labourers are indeed few in that field. They have no African priests there.'\n\n'I suppose some parts are more backward than others,' Belinda ventured. 'I mean it takes some time before the natives are ready to be ordained.'\n\n'Ah, yes, yes. Time and Money,' the Bishop nodded again and then asked whether the Archdeacon gave out collecting boxes for Missions to his parishioners.\n\n'No,' replied the Archdeacon shortly. 'We have a collection at Mattins every now and then.'\n\n'But surely it would be more lucrative if people took collecting boxes?' suggested the Bishop. 'You would find that the spirit of friendly rivalry would increase the amount considerably. You might publish the results in the parish magazine when the boxes were opened. I have always found that the best way.'\n\n'It wouldn't work here,' said the Archdeacon emphatically.\n\n'I think it's an excellent idea,' said Agatha. 'Miss Smiley does it for the Zenana Mission, you know.'\n\n'Does she?' said the Archdeacon. 'I really could not undertake to do anything about it myself, I am much too busy, as you will realize when you have been here a little longer, my dear Bishop.'\n\nBelinda could only wonder how he was to be made to realize this, but, loyal as ever, she agreed that the Archdeacon was much too busy and, much to her own surprise and dismay, heard herself offering to take on the organization and distribution of the boxes.\n\n'I knew it!' said the Bishop. 'As soon as I saw Miss Bede, I said to myself \"here is one of _us\u2014_ that splendid work for the Guild of St. Agnes\"\u2014do you know, Archdeacon, she even denied that she had been connected with it?'\n\nBelinda was by now covered with confusion and began to wonder whether she had indeed worked splendidly for the Guild of St. Agnes; after all, her memory was not always completely reliable. Perhaps she had also knitted the Bishop a beautiful scarf. She felt that it was time to be going.\n\n'Oh, Belinda is a very excellent person altogether,' said the Archdeacon with casual charm. 'I don't know what we should do without her.'\n\nBelinda reflected a little bitterly on these words as she walked home, but the sight of Ricardo and Edith Liversidge, deep in conversation, soon turned her thoughts to other matters. She could guess what they were talking about and was not surprised to hear Ricardo saying that he had hoped the tombstone would be of white marble rather than commonplace grey stone.\n\n'Well, as long as he's dead I don't see that it matters what his tombstone is like,' said Edith very sensibly, but with a lack of feeling regrettable in one who had lost her love so tragically. 'I'm sure John wouldn't mind,' she went on jovially. 'He was never all that particular about appearances.'\n\n'A mind and spirit as great as his needed no outward decoration,' said Ricardo solemnly.\n\nBelinda waited until he had gone before interrupting with an invitation to Edith to come in for a cup of tea.\n\n'I've had mine at the vicarage,' she explained, 'but I expect Harriet will still be having hers.'\n\n'Well, if it isn't a nuisance,' said Edith, with unusual consideration.\n\n'Oh, one extra for tea is no trouble at all,' said Belinda gaily. 'I never find that it is.'\n\n'It might quite easily be,' said Edith. 'If there were only a small piece of cake left, for instance.'\n\nBelinda was confident that the tea table would be well stocked and was quite ready to forget that she had had her own tea at the vicarage. After the little strains and awkwardnesses there she felt that she deserved a second tea and was able to do full justice to the potato cakes and her favourite Belgian buns.\n\nAt tea they were all very gay, in the way that happy, unmarried ladies of middle age often are. Naturally they talked about the Bishop. Try as she would, Belinda could not give a flattering description. After several attempts to soften the blow, she burst out, 'Well, Harriet, there's no getting away from it, he reminds me of a _sheep!'_\n\n'But surely a very handsome sheep?' Harriet protested. 'Of course I haven't seen him for many years, but people don't alter all that much, and he was such an exceptionally good-looking curate.'\n\n'Are you sure you're thinking of the same one?' suggested Belinda timidly. 'You've known so _many._...'\n\nBut Harriet indignantly denied the possibility of such a mistake. 'And anyway,' she went on stoutly, to justify herself in case of a possible disappointment, 'you can't judge a person by his face.'\n\n'No, of course not,' Belinda agreed. 'I'm sure he's an excellent man,' she said doubtfully. 'Have you ever heard of the Guild of St Agnes, by the way?'\n\n'Oh, I don't think so,' said Harriet. 'There's the Society of St. Monica, but that's for widows, I believe.'\n\n'You mean you have to be a widow?' said Belinda. 'I think it could hardly be that.'\n\n'I know,' said Edith suddenly, 'it does work among Fallen Women\u2014Connie's patroness was the President I believe. They used to have teas and sales of work in the house in Belgrave Square.'\n\n'So that's it,' said Belinda, who hardly knew what advice she could give to a Fallen Woman, let alone what kind of splendid work she could have done. She explained the connection and they all laughed very heartily.\n\nWhen they had finished tea Edith suddenly began doing a Balkan folk dance which encouraged Harriet to give a very ludicrous imitation of Mr. Mold's proposal. But as Belinda laughed she found herself almost wishing that Harriet were even now Mrs. Nathaniel Mold. Then at least there could be no danger of having the Bishop of Mbawawa for a brother-in-law.\n\n# Chapter Sixteen\n\nIT WAS THE MORNING after the Bishop's arrival and there was a feeling of suppressed excitement in the air. At the Misses Bede's house the morning passed in the usual way until just before luncheon, when the front-door bell rang. Before Belinda and Harriet could begin to guess who it was, Emily had announced the Archdeacon.\n\nIt was at once evident that he was in a good temper, which Belinda thought rather surprising, although there was a certain relish in disliking somebody, she supposed, which might account for it.\n\n'I hope the Bishop is well?' she ventured.\n\n'Oh, tolerably well, I think,' said the Archdeacon, rubbing his hands in front of the fire. 'I believe he did not sleep very well, but our spare bed is notoriously uncomfortable.'\n\n'We could easily have him here,' said Harriet, 'our spare bed has a new mattress and is really most comfortable. I have tried it myself.'\n\n'That is kind of you,' said the Archdeacon, 'but the clergy are used to discomfort. They even enjoy it, you know.'\n\nBelinda looked at him doubtfully, but he appeared to be quite serious.\n\n'I hope you are writing to poor Nathaniel Mold,' he said, seeing that Harriet was seated at the writing-desk.\n\n'Oh, no,' said Harriet, thinking it rather interfering of the Archdeacon. 'That is all finished. I was writing to Gorringes' for a new winter dressing-gown.'\n\n'Well, they say that a bird in the hand is worth two in the bush,' declared the Archdeacon.\n\n'I'm not sure that I understand your meaning there,' said Harriet coyly. 'What do you mean by two in the bush?'\n\n'Why, Ricardo and the Bishop,' said the Archdeacon slyly.\n\nBelinda felt inclined to add that poor Ricardo was almost as good as in the hand, but she said nothing as she thought the conversation rather unbecoming. It would have been quite another matter if Edith or Connie had been there instead of the Archdeacon, but for somebody of Harriet's age to discuss her suitors with the vicar of the parish seemed to Belinda hardly the thing.\n\n'But I haven't seen the Bishop for over thirty years,' protested Harriet, enjoying herself very much.\n\n'Then you will see him tonight,' said the Archdeacon.\n\n_'Tonight?'_ echoed the sisters incredulously, as if it were the most unlikely thing in the world.\n\n'Yes, I came to see you about it. The Bishop is giving a lantern lecture with slides, and I wanted to know if you would be good enough to work the lantern,' said the Archdeacon, turning to Harriet.\n\n'Why, I should love to,' said Harriet, for she had an unexpected genius for working the lantern and had done it for many years now.\n\n'It should be an unusual experience,' said the Archdeacon, 'to renew your acquaintance with the Bishop over a slide put in upside down.'\n\nHarriet went off into a peal of delighted laughter.\n\n'What's the lecture to be about?' asked Belinda, thinking that somebody ought to show an intelligent interest in it.\n\n'Oh, his natives I believe,' said the Archdeacon rather scornfully. 'Songs and dances and that kind of thing!'\n\n'It should be very interesting,' ventured Belinda.\n\n'And amusing too,' said the Archdeacon. 'The Bishop was practising this morning.'\n\n'The songs or the dances?' asked Harriet.\n\n'Oh, the songs, as far as I could hear. I daresay it will not be possible to demonstrate the dances.'\n\n'I hope _not,'_ said Belinda rather indignantly, for from what one heard about these native dances it did not seem as if they were the sort of thing that could properly be performed in a parish hall.\n\nMoved by a sudden impulse of friendliness, Harriet asked the Archdeacon if he would stay to luncheon. 'We're having pheasant,' she added temptingly.\n\n'I'm so sorry,' he said, 'but I'm afraid Agatha will be expecting me. Otherwise nothing could have given me greater pleasure.'\n\n'Yes, that's the worst of having a wife,' said Harriet jovially.\n\n'It is really much wiser for a man to stay single,' said the Archdeacon, 'and then it doesn't matter if he's late for lunch.'\n\nAfter he had gone Harriet remarked that if he had been single _now,_ he might have discovered that there were even greater advantages, but she soon changed the subject, and began asking Belinda's opinion about her hair. Should she leave the back in a neat roll or comb it out into fluffy curls?\n\nBelinda gave some sort of an answer, as she realized that Harriet was determined to have the fluffy curls, and wondered whether she herself should wear her blue marocain or an old wool dress. The parish hall was inclined to be draughty and she had no particular wish to impress the Bishop. On the other hand, the Archdeacon would certainly be there and she did not wish to appear dowdy before him. It was a difficult problem. Harriet had already decided that she would wear her brown velvet, and possibly her fur cape, though working the lantern she would probably be warm enough without it.\n\nEmily was also going to the lecture with the vicarage Florrie, who had given her a most glowing account of the Bishop. He had apparently given her some very pretty African beads and a wooden comb, carved by one of his native converts.\n\n'Putting silly ideas into her head,' Harriet had said to Belinda after hearing this. 'Theo ought to be careful,' she said ominously.\n\nThe lecture was to begin at eight o'clock, but Harriet insisted that they should be in good time as she had all sorts of things to do in connection with the lantern, which was inclined to be temperamental.\n\n'And you will want to get a good seat,' she said.\n\n'Oh, not particularly,' said Belinda. 'I don't suppose the hall will be very full and all the chairs are equally hard.'\n\nThe problem of where to sit was settled by their meeting Miss Liversidge and Miss Aspinall at the door of the hall.\n\n'Let's go somewhere at the back, where we can have a good laugh,' said Edith.\n\nBelinda agreed that she would also like to sit somewhere at the back, although she did not give any such crude reason for her preference. Poor Miss Aspinall would have liked to sit nearer the front, in case Lady Clara Boulding should be there, but she knew it was no use saying so, and sat meekly on Belinda's other side, glancing hopefully back at the door when anybody came in. Harriet was looking very important, perched up on a table, manipulating the lantern and trying out some specimen slides. Canterbury cathedral, a field with cows, and the head and shoulders of a bearded clergyman followed one another in quick succession: the lantern was obviously working well.\n\nThe hall began to fill up until there were very few vacant seats and the Bishop could be seen threading his way among the chairs towards Harriet, carrying a box of lantern slides. Belinda craned forward eagerly, yet as unconcernedly as she could, to witness their reunion after so many years. Edith Liversidge did the same and even went so far as to stand up for a better view.\n\n'How many years did you say it was?' she inquired.\n\n'I can't remember exactly, but I think it must be nearly thirty,' said Belinda in a more subdued tone of voice, for she did not want Miss Beard and Miss Smiley, who were sitting in front of them with a group of fellow teachers, to hear all their conversation. Things half heard were apt to be wickedly exaggerated and Miss Beard, in spite of being an excellent Sunday school teacher, was very much inclined to gossip.\n\n'What a long time!' breathed Connie. 'There is something very wonderful in meeting a friend again after many years.'\n\n'That rather depends,' said Edith brusquely. 'I can think of some I'd much rather not meet.'\n\n'I suppose in that case you would hardly call them friends,' said Belinda. 'Although one doesn't really know what a person is going to be like after thirty years.'\n\nHarriet's position on the table made it necessary for the Bishop to gaze up at her. She bent graciously and extended her hand as if to take his, but received instead the box of lantern slides. Belinda was indignant. How rude and casual of him! she thought. How like a bishop! she went on and then stopped, realizing the injustice of this generalization. For she was certain that Willie Amery or Oliver Opobo and Calabar would not have behaved like this. Theodore Grote was cold, a cold fish as she remembered their dear mother calling him. _Legless, unloving, infamously chaste,_ she thought detachedly, remembering Ricardo's gold fish, and was then ashamed of herself for thinking of it. There could be no excuse, for Leigh Hunt was not even one of our greater poets. Still, there _was_ something fishlike about Bishop Grote. Fish and sheep. Was that possible?\n\n'I do wish I knew what they were saying,' said Connie, 'though of course it's the most unpardonable curiosity. Meetings like this ought to be really _sacred.'_\n\n'He's obviously just saying something about the slides,' said Edith. 'Connie is much too romantic. I suppose she thinks he ought to be quoting poetry.'\n\n'Well, he might,' said Belinda, 'if he were that sort of a person, which I doubt. He didn't even shake hands, otherwise he might have quoted that nice line of Cleveland's, where he describes a lady's hand _tender as 'twere a jelly gloved..._ I always like that, but somehow it doesn't apply to peoples' hands now.'\n\nEdith looked down complacently at her own fingers, gnarled and stained. 'Not in the country,' she said, 'though Connie's always fussing about hers, rubbing them with lotion and all that sort of nonsense. I always tell her that nobody's likely to want to hold her hand now, so why bother.'\n\nBelinda thought this rather unkind and sympathized with Connie. It wasn't exactly that one hoped to have one's hand held....\n\n'Look, there's the Archdeacon and Father Plowman,' said Connie. 'I suppose it must be going to begin.'\n\n'I imagine one can smoke here?' said Edith, producing a squashed paper packet of Woodbines and offering it to Belinda.\n\n'No, thank you,' said Belinda. She felt that it would be unbecoming for her to smoke, though it seemed right that Edith should do so. Anything that she did seemed to be in character. Her appearance tonight in a homespun skirt with white blouse and Albanian embroidered waistcoat made Belinda feel dowdy and insignificant, one of the many thousand respectable middle-aged spinsters, the backbones or busybodies of countless parishes throughout the country.\n\nThe Archdeacon had mounted the platform and was introducing the Bishop in a short and almost gracious speech.\n\n'I, for one, am eagerly looking forward to hearing more about this fascinating country and its people,' he said. 'Many of us will envy Bishop Grote his unique opportunities. It may even be that I too shall feel the urge to labour in a foreign field,' he concluded, with what Belinda could only think was sarcasm, for nothing more unlikely could be imagined.\n\nThe audience settled down on the hard chairs. Belinda noticed that Agatha was wearing a becoming new dress, dark green, with little pleated ruffles at the shoulders and neck. From the best houses, she thought, with sad resignation.\n\n'The climate of Mbawawa is temperate and the soil very fertile,' began the Bishop, waving his pointer vaguely in the air.\n\nThe first slide appeared. It showed a seascape with some kind of tropical palms in the foreground. Belinda had seen the same type of picture on the covers of dance tunes about the South Sea Islands.\n\n'When I say temperate,' went on the Bishop, 'I dare say many of you might find it rather hot.' He paused and tapped his pointer vigorously on the floor.\n\nThere appeared in rapid succession several pictures of handsome natives, dressed in bunches of leaves and garlands of flowers. Some members of the audience were inclined to giggle at these, but the Bishop hastily explained that the pictures were of the natives as they _used_ to be.\n\n'We have since introduced a form of European dress which is far more in keeping with Christian ideas of morality,' he said. Another slide followed, showing the natives clad in this way. 'I should like to add here,' he went on, 'that we are often very much in need of garments for our people and should welcome gifts of clothing or material\u2014light cotton materials, of course, nothing elaborate or costly.'\n\nIt would be typical of the perfidy of human nature, thought Belinda indignantly, if the church workers fell so much in love with the Bishop that they forgot about all the other more deserving charities such as the Clothe-Our-Children League and the Society for helping the Poor in Pimlico to which they were accustomed to contribute. She could already notice in the half darkness the beaming looks of approval on their faces, as they nodded and smiled to each other, planning working parties and schemes to raise money. Of course the Mbawawa _were_ a deserving cause, she supposed, but were they not happier in their leaves and flowers? Naturally one wished them to have the benefits of Christianity; it was rather difficult to see where one should draw the line. They could hardly appear at a service in a dress of leaves, she reflected, when she herself felt that a short-sleeved dress was unsuitable. But need they wear those shapeless cotton garments? Perhaps the architecture of the church had something to do with it: one's style of dress ought to be somehow in keeping... her thoughts wandered on against a background of bleating Bishop's voice. He had somehow got on to the subject of music.\n\n'The language is well suited to singing,' he declared. 'It is soft and pleasing, vowels and liquid sounds predominating. You may be interested to hear that the alphabet contains only eighteen letters,' he went on, 'and I think that if you saw it written you would hardly call it an alphabet at all. Such an odd collection of letters with long tails and squiggles! You see, the Mbawawa had never written their language down until a few years ago, when missionaries attempted it. Then some clever people in London, experts in African languages, made up this alphabet, and I think nobody was more surprised than the Mbawawa themselves!' The Bishop laughed heartily and wiped his brow. 'But I haven't come here tonight to tell you about the alphabet. I think we can safely leave that to the clever people in London,' he added, with what Belinda felt was insufferable patronage, considering the distinction of his audience, which contained at least four University graduates, five, if one counted Father Plowman's failed B.A.\n\n'Their chief musical instrument is the Mhamha, M-H-A-M-H-A,' he spelled the word out and one of the Sunday school teachers could be seen fumbling in her handbag for pencil and paper. There now followed another slide of grinning natives holding musical instruments.\n\n'I dare say some of you would like to hear what the language sounds like,' said the Bishop, 'so I am going to sing a few verses of a song which the Mbawawa adapt to many occasions, birth, marriage, death, all the great events in this mortal life have their own form of it.' He paused, as if wondering which was most suited to a gathering in a parish hall. 'Let me try and give you a gay marriage song,' he said. 'Imagine yourselves taking part in a Mbawawa wedding.'\n\n'I do not feel myself equal to that,' whispered Edith to Belinda. 'Death would have been a better choice, or even birth.'\n\nThe voice of the Bishop rang out through the hall in song. Many handkerchiefs were taken out hastily, especially among the younger members of the audience, for the noise which filled the hall was quite unexpected. Even Belinda, who had heard the Bishop sing as a curate, was a little unprepared. And yet perhaps the Mbawawa _did_ have voices like that and it was wrong to feel that one wanted to laugh. Belinda glanced at Harriet to see how she was reacting. As far as it was possible to see, she was displaying remarkable self-control, for she was very prone to giggle, and appeared to be gazing at the Bishop with rapt attention. Most of the audience were stirring uneasily. Even Agatha was smiling a little, but she managed to make it look as if she were not really amused, but pleased and approving, which was quite another matter.\n\nAt last the noise stopped, and some people relieved their feelings by clapping.\n\n'The song has many more verses,' explained the Bishop. 'Indeed, if the singer is particularly gifted he can go on almost indefinitely; I have known the marriage song go on all night, but I fear I should find the hall empty if I attempted that.'\n\nDuring the laughter which followed he tapped his pointer on the floor and another slide clicked into place.\n\n'Now this is another characteristic musical instrument. It is called the Hmwoq, spelled H-M-W-O-Q.'\n\nEveryone looked with interest at the curiously shaped object which had now appeared on the screen. It was certainly a very peculiar shape and there was more giggling from the back of the hall. It could hardly be what it seemed to be, thought Belinda doubtfully, though one knew that among primitive peoples one might find almost anything. The anthropologist who went among them must go with an open mind....\n\nThe Bishop turned towards the screen and prodded it uncertainly. Then he advanced towards the edge of the platform and said in a loud clear voice, 'I think that slide is upside down.'\n\nEveryone turned to look at Harriet, who was not in the least embarrassed at having such attention drawn to her. Indeed, Belinda could not be absolutely sure that her sister had not purposely put the slide in upside down.\n\n'I am _so_ sorry, My Lord Bishop,' Harriet's voice rang clearly through the hall. 'How _stupid_ of me,' she said, smiling most charmingly into the darkness.\n\nThe Bishop responded graciously enough by saying that he feared _he_ was too stupid to explain the picture unless it were the right way up, and his explanation was very confused even when the slide was correctly shown. Leaving it rather hurriedly, he produced a large sea shell from an inner pocket and applied it to his lips.\n\nBy this time his hearers knew more or less what to expect, so that they were able to bear the strange sounds which came out of the shell with more composure. The noise seemed to be a hollower and more resonant version of the Bishop's own singing voice.\n\n'Wonderful how he does it, isn't it?' whispered Father Plowman to Agatha Hoccleve, who could not but agree that it was indeed most wonderful.\n\n'This instrument is used particularly in agricultural rites,' explained the Bishop, 'where the ceremony of propitiating the earth goddess is carried out.'\n\n'Phallic,' murmured Edith, nodding her head. 'Quite the usual thing.'\n\nFortunately the Sunday school teachers did not know the word, thought Belinda, or they would most certainly have turned round. It was rather like Edith to show off her smattering of anthropological knowledge, she felt, particularly if it were something rather embarrassing.\n\nAfter the music came more slides of wedding and funeral scenes, and finally one of the Bishop himself in gaiters and leafy garlands, at which everyone clapped vigorously. It was a relief to be able to let off steam, for much laughter had been bottled up. But the climax came when he turned his back on the audience, fumbled in a suitcase and reappeared facing them in a huge painted wooden mask, with hinged beak, large round eyes and hanging raffia mane, which completely covered his head and shoulders. This brought the house down and there was laughter and clapping from the front seats, stamping and whistling from the back benches.\n\nAll that followed was inevitably an anticlimax. The Bishop went on to give a list of rather stray facts which he might have got out of the _Encyclopaedia Britannica._ He mentioned that they tattooed little, that the native chiefs sometimes weighed as much as one hundred and eighty pounds, and that although infanticide was prevalent, cannibalism was almost unknown. They lived chiefly on yams and millet, but rats and mice were also eaten. Beer was brewed from guinea corn; fire was made from a paste of salt, pepper and lizard dung.\n\n'And in conclusion,' he added, 'for the benefit of any anthropologists who may be listening to me, I may as well state that the basic social unit is an exogamous patrilineal kindred or extended family or even clan. I don't think we need worry overmuch about that.'\n\nHearty laughter greeted these remarks. There was no anthropologist in the audience.\n\nFather Plowman mounted the platform and began to propose a vote of thanks.\n\n'I for one shall never forget this fascinating lecture,' he said. 'I am sure that after tonight there will be many who will be eager to visit this beautiful country and see all these wonders for themselves.'\n\n'Putting ideas into their heads,' muttered Edith, and she was not far wrong, for one of the Sunday school teachers was even at that moment toying with the idea of asking the Bishop whether he could find a place for her in his Mission, and even Miss Aspinall was wondering whether it might not be possible to go out there to teach the gentler arts.\n\n'Truly the wonders of this world are without number. Let us thank God for His goodness to us,' concluded Father Plowman, and everyone agreed that it was a most fitting end to the evening.\n\nIt was a little spoilt by the Archdeacon rising to his feet and saying that he was sure everyone would wish him to thank Miss Harriet Bede for her admirable working of the lantern, without which the lecture would not have been half so enjoyable. Of course it was right that she should be thanked, but several people felt that Father Plowman's words should have been the last. Only Belinda was pleased, both because of his acknowledgment of her sister and because no evening was complete for her which did not include a few words from the Archdeacon.\n\n'So like him, that kind thought,' she said to Edith, 'remembering Harriet when the Bishop never said a word, nor Father Plowman for that matter. I knew the Archdeacon wouldn't forget.'\n\n'Oh, I expect he just wanted to be different,' said Edith, struggling into her mannish navy blue overcoat. 'What happens now?'\n\n'I think we go home,' said Belinda, 'but I dare say Harriet will go and have refreshments at the vicarage with the Bishop. They'll probably have coffee and sandwiches or something light.'\n\n'Oh, I hate standing about balancing a cup and plate and making conversation,' said Edith. 'Come along, Connie,' she called, turning round, 'we're going home.'\n\nBut Connie, with a hasty gathering up of bits and pieces and a fluttering of grey draperies, had hurried towards the front of the hall, where she could be seen among the little cluster of people waiting to shake hands with the Bishop.\n\n'Don't make her come away,' pleaded Belinda. 'She would probably like to go to the vicarage with the others.'\n\n'Well, come and take pot luck with me,' said Edith roughly. 'Just coffee and baked beans\u2014you know our kind of supper.'\n\n'That will be lovely,' murmured Belinda.\n\nAt the door of Edith's cottage a big, shaggy dog came bounding towards them, his muddy paws scrabbling against their coats and stockings, and inside the living-room, for it could hardly be called a drawing-room, everything was so primitive and comfortless that Belinda felt really sympathetic towards poor Connie. After Belgrave Square too... Her harp, shrouded in a holland cover, seemed out of place in the untidy room with its smell of dog and cigarette smoke.\n\nBelinda stood uncertainly on the threshold of the little kitchen, watching Edith cutting bread and scooping the beans out of their tin into a saucepan.\n\n'Hand me that ash tray, will you?' said Edith, but not before Belinda had seen a grey wedge of ash drop into the beans. 'Drat it,' she said. 'Too late. Hope you don't mind?'\n\n'Of course not,' said Belinda nobly, remembering Miss Prior and the caterpillar. Perhaps there was something after all in being a gentlewoman.\n\n# Chapter Seventeen\n\nWHEN BELINDA AWOKE NEXT morning, she decided that she did not feel very well. She was not sure whether this was because of the ash in the baked beans, the half-empty bottle of Empire port that Edith had found in the back of a cupboard or the damp walk home, in rather thin shoes. She was inclined to think it must be the last, for what else could have given her such an unromantic, snivelling cold?\n\n'Oh, dear,' said Harriet, sitting down heavily on Belinda's bed, 'the Bishop was coming to tea and I suppose I shall have to put him off if you're going to be ill.'\n\n'Why?' asked Belinda stupidly.\n\n'Well, really, what would people think?'\n\n'They needn't know I'm in bed, and after all, it's only a matter of time,' said Belinda, who was in no mood to humour her sister's coy scruples.\n\n'Yes, perhaps it is,' agreed Harriet, but rather doubtfully. 'He asked particularly if you would be here, though.'\n\n'Did he? Well, we certainly can't have tea in my bedroom,' said Belinda plaintively.\n\n'No, of course not,' Harriet agreed. 'Now are you sure you couldn't fancy a little sausage?' she said brightly. 'Emily will have cooked enough for both of us.'\n\nBelinda did not think she fancied anything at all, but was persuaded to try some weak tea and a piece of toast. And would Harriet be very kind and bring her the _Oxford Book of Victorian Verse?_ She might feel like reading later on.\n\nHarriet went downstairs and came back with a tray and the book.\n\n'Isn't it rather heavy to read in bed?' she ventured. 'I've brought you something smaller as well. Here's the Fourth Book of Virgil. I know you like the part about Dido and Aeneas. It's such a nice thin little book.'\n\n'Oh, Harriet, how kind. But it's all in Latin, and you know I can't read it.'\n\n'Never mind, dear,' said Harriet soothingly. 'I shouldn't read at all, if I were you. Just try and rest.'\n\n'I can't think how I caught this cold,' said Belinda.\n\n'I'll go and get you some whisky from the Crownwheel and Pinion,' declared Harriet. 'I shall go as soon as it's open.'\n\n'Oh, Harriet, don't go _there,'_ said Belinda, rather concerned. 'I'm sure you could get some at Abbot's, and anyway I don't think I really need it. If I stay in bed and keep warm I'm sure to be better in a day or two. Hot lemon is really a much nicer drink.'\n\n'You never know when you may need whisky,' said Harriet mysteriously. 'It's just as well to have it in the house.'\n\n'I seem to remember a recipe in _Tried Favourites\u2014_ a sort of substitute for whisky,' said Belinda. 'I dare say it would be quite easy to make.'\n\n'I think our guests would hardly thank us if we offered them that,' said Harriet.\n\n'Our guests?' Belinda sank back weakly on to her pillows, unable to face the idea of guests who needed to be entertained with whisky. 'I think I'll just rest until lunch-time,' she said. 'I dare say I shan't read after all.'\n\nSo Harriet left her and went out to do the shopping. She met several people and told each one about her sister's indisposition, making little or much of it according to the status of her hearer. To the Archdeacon she gave the most exact details, thinking that somehow he ought to be possessed of all the facts.\n\n'She had weak tea and dry toast for breakfast,' said Harriet confidentially, 'and then she asked for the _Oxford Book of Victorian Verse.'_\n\n'She called for madder music and for stronger wine,' said the Archdeacon, but Harriet was not familiar with our great Victorian poets and so the quotation passed over her head.\n\nShe pointed out rather sharply that strong wine was the last thing that should be given to an invalid, although a little brandy might be helpful in cases of biliousness.\n\n'But of course Belinda isn't bilious,' she said hastily. 'Nothing like that.'\n\n'Poor Belinda, I am really extremely sorry. Do tell her how very sorry I am. I only wish I could go and see her.'\n\n'Oh, she's not at all seriously ill,' said Harriet. 'Just a little chill. I'm sure it would alarm people if you were seen going to the house. People always think the worst when they see a clergyman.'\n\n'Dear me, I hardly know how to take that,' said the Archdeacon. 'I should have liked to think that we brought comfort to the sick.'\n\n'Oh, well, I suppose you do, in a way,' said Harriet, who was finding it difficult to convey that it all depended on the clergyman.\n\n'I must look out some books for her to read,' said the Archdeacon.\n\n'Thank you very much, but she really has plenty to read.'\n\n'All the same, there might be something she'd like,' persisted the Archdeacon. 'I sometimes wish that I could afford to be ill so that I could read some of the things I normally never have time for.'\n\nHarriet looked contemptuous but said nothing. 'I must be going now,' she said at last. 'I still have quite a lot of shopping to do.'\n\nOn her return she found that Belinda had been to sleep and felt a little better.\n\n'I saw the Archdeacon,' said Harriet triumphantly. 'He seemed quite concerned to hear that you were ill and almost suggested coming to see you, but I soon nipped that in the bud.'\n\nBelinda gathered her faded pink bed-jacket more closely round her shoulders. 'Oh, no, I couldn't have him coming to see me,' she said. 'Not without warning, anyway.'\n\n'Well, of course,' said Harriet pompously, 'it is, or should be, customary for a clergyman to visit the sick in his parish. But perhaps that's only for the poor people really, to see if they have all they want and so on.'\n\n'Yes, I suppose I have everything I want,' said Belinda rather sadly.\n\n'Naturally if you were seriously ill or dying it would be another matter,' went on Harriet reassuringly.\n\n'But I'm not,' said Belinda regretfully, thinking of Henry reading _Samson Agonistes_ to her on her death-bed.\n\nAfter lunch she settled down to her own thoughts. Harriet had brought up a light novel from the circulating library and this lay with the _Oxford Book of Victorian Verse_ on the eiderdown. But Belinda did not feel like reading. She was quite enjoying her illness now that she felt a little better and could allow her thoughts to wander at random in the past and future without the consciousness that she ought to be more profitably employed. She had no doubt that there would soon be another proposal of marriage in the drawing-room, perhaps even this afternoon, although she judged the Bishop to be a more prudent man than Mr. Mold. He had certainly not behaved very cordially to Harriet at the lecture, but Belinda was sure that he would not be able to hold out long against her charms. Nor had Harriet seemed as enthusiastic as might have been expected. Could it be that she had found him less attractive than she anticipated, or was it the very depth of her feeling that kept her from speaking of it? Belinda puzzled over this for some time and then fell to thinking of her own life.\n\nThere was very little new to be said or thought about it, she decided. She had loved dear Henry for so many years now that she no longer thought of her love as a hopeless passion. Indeed, Belinda felt that no spinster of her age and respectability could possibly have such a thing for an archdeacon. The fierce flame had died down, but the fire was still glowing brightly.\n\n_My very ashes in their urn,_\n\n_Shall like a hallowed lamp for ever burn...._\n\nHow much more one appreciated our great literature if one loved, thought Belinda, especially if the love were unrequited! She touched the books affectionately but made no effort to read either of them. As Harriet had said, the _Oxford Book of Victorian Verse_ was rather heavy to hold, and many of the poems in it were uncomfortably sentimental for afternoon reading.\n\nSuddenly there was a noise in the hall. Belinda sat up in bed and listened. At first she thought it must be the Bishop arriving rather too early for tea, but then she realized that it was a woman's voice. It sounded almost like Agatha's. She crouched under the bedclothes and began to wonder whether she had a temperature after all and ought not to see people. It was surely not normal to have a sudden longing to hide under the bedclothes when one heard the vicar's wife in the hall, even if one did love her husband better than she did?\n\nBelinda sat up bravely and took out her hand-mirror. She knew that she looked most unattractive and thought what a good thing it was that she was not seriously ill or dying. Her hair was out of curl, her cheeks were pale and her nose needed powdering. She would not have liked Henry to see her like this, even on her death-bed.\n\nAgatha was all too soon in the room, saying, 'Poor Belinda, I was so sorry when Henry told me you were ill. I thought I'd come and see how you were.'\n\nBelinda, who was trying to smuggle the hand-mirror out of sight, murmured that it was very kind of Agatha and that she was feeling much better.\n\n'And now,' said Agatha, rather too briskly, 'what has been the matter with you?'\n\n'Oh, I think I must have caught a slight chill,' said Belinda vaguely. 'Perhaps I was sitting in a draught at the Bishop's lecture,' she ventured, feeling ashamed of not knowing exactly what was the matter with her and why. 'The lecture was most interesting, wasn't it?'\n\n'Oh, yes, fascinating,' said Agatha. 'The Bishop is such a dear man, so kind and amusing.'\n\nHe must be especially pleasant to Agatha, thought Belinda, who had not formed at all that impression of him. Indeed Agatha was quite animated when she spoke of him and even looked a little flushed.\n\n'He asked where you were last night after the lecture,' she went on. 'We had quite a little gathering at the vicarage, you know.'\n\n'I thought you would probably have enough people there without me,' said Belinda weakly, feeling as she so often did with Agatha that she had somehow done the wrong thing. 'I went to supper with Edith Liversidge. I'm surprised that he should have remembered me at all.'\n\n'Well, of course it is a bishop's duty to remember people,' said Agatha. 'My father had a wonderful memory for names and faces.'\n\n'Yes, I suppose they meet so many people,' said Belinda, feeling rather damped. Not that she wanted the Bishop to remember her particularly, but it was like Agatha to take away any illusions she might have cherished.\n\n'It's some years since you last met, isn't it?' said Agatha conversationally.\n\n'Oh, yes, about thirty years, I think. We none of us grow any younger, do we? _Timor mortis conturbat me,'_ murmured Belinda, staring straight in front of her.\n\nAgatha looked at her sharply. Sometimes she wondered whether Belinda was quite all there. She said such odd things.\n\nThere was a short pause, but before it had time to become awkward a hearty voice was heard outside the door and Edith Liversidge strode into the room, followed by Connie Aspinall. Their arms were full of books and parcels.\n\n'I must be going now,' said Agatha, who disliked Edith. 'Too many visitors at once will tire you.'\n\nBelinda thanked her for her kindness, but was quite relieved to be left alone with Edith and Connie.\n\n'We've brought you some books,' said Edith. 'And Connie's made you a sponge cake. You know I'm no hand at that kind of thing.'\n\n'Oh, how kind....'\n\n'I thought you might like to see some old copies of _The Gentlewoman,'_ said Connie. 'There's a picture of Lady Grudge's daughter in one' of them.'\n\n'How interesting, I shall look forward to reading them,' said Belinda. 'You must point her out to me.'\n\nAt this moment the front-door bell rang. 'That must be the Bishop arriving,' said Belinda.\n\n'The Bishop?' asked Edith, rather surprised.\n\n'Yes, he is coming to tea today.'\n\n'Does he know you're ill or is he expecting to see both of you?'\n\n'Oh, I'm supposed to be there too, but I suddenly woke up ill,' said Belinda pathetically. 'So Harriet will have to entertain him alone.'\n\n'Alone?' said Edith. 'I don't think he'll like that.'\n\n'Why ever not?' asked Belinda, rather worried. Life was quite difficult enough without Edith making disturbing suggestions. Harriet was going to marry the Bishop and Belinda would be left in her old age to die a lonely death, or with nobody but a paid companion to cheer her last hours. Surely that was enough? She had been trying to prepare herself for the worst and did not wish to be unsettled.\n\nMeanwhile Edith expounded her ideas of what a bishop would think quite proper. 'I don't think Harriet will get him,' she said bluntly.\n\nBelinda had been thinking the same thing not so long ago, but now she was inclined to disagree. After all, dear Edith had had little or no experience of bishops, although poor John had been a very good man in his way.\n\n'I think he has successfully avoided so many women in his life that not even Harriet will be able to catch him,' she went on. 'I know his sort.'\n\nBelinda thought this rather a vulgar way of putting it, though it could hardly be denied that it was what Harriet intended to do.\n\n'I believe women can do almost anything if they are really desperate,' she ventured. 'In one of Lyly's plays, _Endimion,_ I think....'\n\n'But I don't think Harriet is really desperate,' Edith interrupted. 'Do you?'\n\n'I really couldn't say,' said Belinda plaintively. 'I think I should like a drink of barley water. It's on the little table.'\n\n'Oh, let me get it for you,' said Connie, coming forward with a glass and a copy of _The Gentlewoman_ in her hand. 'This is Lady Joan Grudge,' she said eagerly, indicating a group of people enjoying a joke at a race meeting.\n\n'Yes, she's very pretty, isn't she,' murmured Belinda, and was then informed that she had been looking at the wrong girl, for nobody could call Lady Joan pretty. 'Such a nice expression,' she emended. 'She looks very jolly.'\n\n'Yes, she is a _very_ sweet girl....'\n\nDownstairs Harriet had just made the discovery that Bishop Grote never ate anything for his tea.\n\nNow this was exceedingly awkward, for how can any real contact be established between two persons when one is eating and the other merely watching? For some minutes Harriet did not know what to do. Her recollections of the Bishop as a curate had included cream buns and hot buttered toast, with licking of fingers. Eventually she had to resort to a kind of arch scolding, which was really more suitable for very young curates than for bishops.\n\n'Now, that's _naughty_ of you,' she said. 'I expected you to have a really good appetite. I shall be _much_ too embarrassed to eat alone,' she added, liberally spreading a piece of buttered toast with strawberry jam.\n\n'I am really very sorry,' said the Bishop complacently, but with no intention of changing his habits even to be polite to his hostess. 'But your sister will be eating something, won't she?' he inquired, looking anxiously towards the door for a sign of the elder Miss Bede.\n\n'Oh, I forgot to tell you,' said Harriet airily, 'poor Belinda is in bed today. She isn't well.'\n\nThe Bishop started and half rose from his chair. 'Nothing infectious or contagious, I hope?' he asked, rather too eagerly.\n\n'No, no,' Harriet smiled reassuringly, but at the same time a little dangerously, so that the Bishop knew that there was as yet no possibility of escape. 'Just a slight chill,' she explained, 'but one can't be too careful.'\n\nThe Bishop murmured some words of regret but he found it difficult to keep a note of displeasure out of his voice. So might he have rebuked a rebellious African who appeared at Divine Service in his flowery but inadequate native costume. To begin with, he was not sure that he believed this story about Belinda's illness. People didn't suddenly become ill like this, he told himself angrily, and Belinda had seemed quite well at the lecture last night. If she were really indisposed why hadn't Harriet written or sent a message, changing the invitation to another day? She could so easily have done this. That nice maid who had answered the door would have been only too willing to take a message to the vicarage. It was only a few minutes away. Besides, he wanted to see Belinda. He thought her much nicer than Harriet and she had knitted him such a beautiful scarf when he was a curate, however much she might deny it. Harriet must indeed be heartless to leave her sister lying alone and ill upstairs while she entertained guests\u2014the Bishop's indignation had got the better of his accuracy\u2014in the drawing-room.\n\nHarriet interrupted his thoughts by asking if his tea were too strong.\n\n'Oh, no, it is very nice, thank you,' he said quite civilly.\n\n'Perhaps you would like some more sugar in it?' she persisted.\n\n'No, thank you. I never take more than one lump.'\n\nThere was a pause while Harriet, who was finding dear Theo not at all as she had imagined, racked her brains for something to say. Surely he had lost some of his charm of manner? she asked herself anxiously. It went without saying that he had once had charm of manner, but what had happened to it now? He did not appear to be enjoying himself at all and was behaving almost as if this visit were a duty rather than a pleasure. She stretched forward and helped herself to another piece of buttered toast. And how extremely irritating this not eating was. It was impolite, too, most impolite.\n\n'I suppose it was in Africa that you got into the habit of not eating any tea?' she asked brightly.\n\n'Oh, no, it was when I was a minor canon,' he replied seriously. 'I found that it interfered with Evensong.'\n\nHarriet burst into a peal of laughter. She thought this very funny and stored it up to tell Belinda. But the Bishop's sheep's face hardly altered its expression.\n\n'A minor canon,' she giggled. 'Now when you were a deacon I seem to remember you eating crumpets for tea,' she said, trying to bring back to him the remembrance that he had once been a typically charming curate with endearing human weaknesses. _'Sic transit gloria mundi,'_ she added.\n\n'I beg your pardon?' The Bishop's voice held a note of surprise. He thought Harriet an extremely silly woman and was wondering how soon he could decently get away. It was in vain that Harriet asked him intelligent questions about the flora and fauna of the Mbawawa country and tried to draw him out on the missionaries' attitude towards polygamy. He seemed disinclined for conversation and at five o'clock got up to go.\n\nAs they went out into the hall, Miss Liversidge and Miss Aspinall came down the stairs from Belinda's room.\n\n'Poor Belinda,' said Connie, 'I think she seems rather low. I must say I thought her looking _not at all well.'_\n\n'Rubbish,' said Edith. 'It's only a chill. She'll be up and about in a day or two.'\n\n'I am indeed sorry to hear this,' said the Bishop. 'Will you give her my very kindest regards and good wishes for a speedy recovery? In my diocese we have a special song for such an occasion. It is almost entirely on one note.'\n\nThe three ladies looked up expectantly.\n\n'Perhaps I had better not sing it now,' said the Bishop. 'It might disturb Miss Belinda. The African temperament is not quite like ours.'\n\n'Oh, Belinda, he's so stupid and _dull!'_ Harriet burst out, when the visitors had gone. 'He sent you his kindest regards. I don't think he really believed you were ill until he saw Edith and Connie.'\n\n'I'm sorry he was such a disappointment,' said Belinda, 'but perhaps he will improve on further acquaintance. He is really quite the opposite to Mr. Mold, isn't he? I mean, perhaps he doesn't have all his goods in the shop window.'\n\nHarriet laughed scornfully and became absorbed in looking at _The Gentlewoman._\n\n'It was kind of Connie to bring these,' said Belinda. 'It makes one feel so secure to look at a paper like this.' She pointed out Lady Joan Grudge, enjoying a joke with friends at a race meeting, a group at a dance held in Eaton Square for somebody's debutante daughter, a party of titled people at a night club and other comforting unrealities. Lulled in security and contentment, they passed the next half-hour very pleasantly until there was a ring at the front door and the sisters started up in agitation.\n\n'Oh, dear, I wonder who that is?' said Harriet, hastily squeezing her feet into the elegant shoes which she had kicked off after the Bishop had gone.\n\n'I really don't think I can do with any more visitors tonight,' said Belinda feebly. 'What is it, Emily?' she asked, as the maid appeared in the doorway.\n\nFor answer Emily thrust forward a large bundle, shrouded in many sheets of blue tissue paper. 'Flowers for the invalid, Miss Belinda,' she said brightly, in a nurse's tone.\n\nHarriet rushed forward. Nobody ever sent _Belinda_ flowers, but the florist's label was clearly addressed to Miss Belinda Bede. Harriet unwrapped the tissue paper to reveal a dozen beautiful chrysanthemums, bronze and white.\n\nBelinda's heart leapt. They were from Henry. Harriet had seen him that morning, so he knew she was ill, but in any case Agatha would probably have told him.\n\n'There doesn't seem to be a card with them,' said Harriet, fumbling with maddening deliberation. 'Oh, yes, here we are.' She tossed the little envelope over to Belinda, who tore it open eagerly.\n\nWhen one has reached Belinda's age, and even before, one takes these small disappointments calmly. Of course the flowers were not from the Archdeacon, how could they have been? It would have been most unsuitable, unless, of course, Agatha had joined in the gift, Belinda told herself, as she struggled to decipher the unfamiliar handwriting.\n\n' _With best wishes for a speedy recovery\u2014Theodore Mbawawa.'_ she read. 'Oh, Harriet, from the Bishop!' she sank back weakly on to her pillows. 'I really don't think I can bear any more today. Theodore Mbawawa... doesn't that sound odd... I suppose it's what he calls himself....'\n\n# Chapter Eighteen\n\nBELINDA DID NOT KEEP to her bed for very long, and was soon about again. The Bishop stayed on at the vicarage, though it was evident that he and the Archdeacon disliked each other. Agatha, however, seemed to prefer his company to that of her husband, and Belinda could not help noticing the way she beamed at him when beaming was certainly not one of her normal expressions. Perhaps she felt naturally more at ease with bishops, as her father had been one, and it may have been a disappointment to her that her husband was only an archdeacon. Certainly she and Bishop Grote made a very suitable couple, if only because there was something slightly unpleasant about each of them. Belinda began to weave a little fantasy in which they somehow 'came together' and the Archdeacon was left alone and in need of comfort. How this was to come about she did not know, as divorce was against her principles and the Archdeacon's too, she imagined, and she would hardly have wished the Archdeacon to be removed by death and so put beyond the reach of her comfort. It was somehow out of the question, even in a fantasy, that Agatha should die. People like Agatha didn't die. It might of course be discovered that the marriage of Henry and Agatha had not been legal, but that happened only in the novels of Mrs. Henry Wood. And supposing Henry were to be left alone and apparently in need of comfort, and did not turn to Belinda? How was that to be borne? It might well be that he would find Agatha's absence comfort enough. When she got to this point, Belinda was firm with herself and set to work helping Emily with the mincemeat and Christmas puddings, for it was already December and it was even rumoured that the Bishop was to spend Christmas at the vicarage.\n\nSoon Christmas cards began to arrive and every post brought something.\n\n_'A Happy, Holy Christmass,'_ said Harriet, reading from Father Plowman's card. 'Very nice wording, and such a pretty picture of the Nativity. I think Christ _mass_ is rather nice, Belinda. The Archdeacon's card is _very_ ordinary, just _A Happy Christmas and a Bright New Year, from Agatha and Henry Hoccleve,_ and no picture at all.'\n\n'I expect Agatha had them done,' said Belinda. 'I imagine Henry would have wanted some quotation, but it was probably cheaper without. Of course Christ _mass_ is rather High, isn't it, I mean, the use of the word _mass._ It always looks like a mis-spelling to me.'\n\n'I wonder what Mr. Donne's will be like,' said Harriet anxiously. 'I hope he will remember us.'\n\n'Oh, surely,' said Belinda, indignant not so much for herself as for her sister and the many delicacies she had prepared for him. 'He has been here so much.'\n\n'He might even send a calendar,' mused Harriet. 'That's one degree better than a card.'\n\n'Yes, one of those with a quotation from Shakespeare or a Great Thought for every day. I always think it's nice to have one in some convenient place so that you can read it at the beginning of the day. And yet the thoughts they choose are often so depressing, aren't they, as if Great Thinkers were never cheerful.'\n\n'Well, Mr. Donne's calendar certainly won't go _there,'_ said Harriet, bristling. 'It shall go in the dining-room, where we can read it at breakfast.'\n\n'Yes, dear, that's what I mean.'\n\nBut Mr. Donne upset their plans by calling round in person with his present, or rather presents, an expensive looking box of chocolates with a coloured picture of Hampton Court on the lid which the sisters felt he could ill afford, and a photograph of himself which he gave to them rather shyly, obviously embarrassed by Harriet's cries of joy.\n\n'Oh, how _lovely!_ And how good of you! Belinda, isn't it _good_ of Mr. Donne?' She thrust the photograph at Belinda, who was rather at a loss, as it looked so exactly like any of the other photographs of curates in Harriet's collection upstairs that she could hardly think what to say.\n\n'The lighting is very good,' she ventured, noticing it on his nose and clerical collar. 'So often a photograph is spoiled by bad lighting. You look very serious,' she added, with what was for her a forced note of playfulness. 'Almost as if you were thinking out a sermon.'\n\n'Oh, we shall have you for tomorrow morning, shan't we,' said Harriet, for the Archdeacon, contrary to his normal practice, had been trying out a course of sermons on the evening congregation, sermons written in so-called 'simple language' and full of sentiments to which every bosom might be expected to return an echo, though he had not, of course, mentioned Harriet's Apes of Brazil. Some of his hearers had found the sermons almost too simple and were even beginning to wonder whether the Archdeacon himself were not returning to his second childhood.\n\n'Oh, no, Bishop Grote is to preach on Sunday morning,' said the curate.\n\n'Well, I suppose he must keep his hand in,' said Belinda. 'I expect it will be about Christmas as it's so near.'\n\n'Yes, on Tuesday,' said the curate. 'I can hardly believe it myself, the weather's so mild.'\n\n'They say a green Christmas means a full churchyard,' declared Harriet with satisfaction. 'I dare say some old people will be taken.'\n\n'Taken?' The curate looked puzzled. 'Ah, yes, I see. I suppose we must expect that.'\n\nThey were silent for a moment, until Belinda, not liking to see his young face clouded over, said, 'I really can't think of any old people who are likely to die at the moment. Besides, it's the weather _after_ Christmas that we have to fear, isn't it?'\n\n'Oh, yes, if it's mild at Christmas it will be cold afterwards,' agreed Harriet. 'It makes one feel very anxious.'\n\nBelinda, looking at Harriet's sturdy figure, could hardly help smiling.\n\nWhen Sunday came it occurred to Belinda that perhaps the Bishop had his uses after all. For when the Archdeacon came to give out the notices of the Christmas Services it appeared that Bishop Grote and Mr. Donne were to take the seven and eight o'clock Celebrations of Holy Communion, while the Archdeacon himself was to preach at Mattins and conduct the Celebration afterwards at twelve, for the benefit of the elderly and lazy. He took the opportunity to say a few words of warning to those who intended to go to Midnight Mass at Father Plowman's church, dwelling darkly on the dangers they might meet there and pronouncing the word _Rome_ with such horrifying emphasis that many of his hearers were quite alarmed, and those who had thought of doing such a thing began to tell themselves that perhaps the parish church was more convenient after all.\n\nThe Bishop's sermon, when it came, was not particularly suited to the season, being very much like his lecture suitably adapted for the pulpit. He had chosen for his text a verse from the psalms, _In them hath he set a tabernacle for the sun, which cometh forth as a bridegroom out of his chamber, and rejoiceth as a giant to run his course._\n\nBelinda hoped that Harriet would not be upset by the reference to bridegrooms, but she appeared to be quite unmoved and it was evident that she had very sensibly put away any hopes she might once have had. She even whispered to Belinda that he certainly wasn't the preacher he used to be, though he still had that same way of gripping the edge of the pulpit when he wished to emphasize a point.\n\nThe text seemed to have little reference to the sermon, although the more intelligent of the congregation saw it as referring to the Bishop himself. He was the giant and his course was the Mission Field. Belinda noticed, however, that when he prayed for his flock he gave the impression that they were so entirely heathen that she began to wonder whether dear Theo had done such wonderful work among them after all. What if the whole of his life had been so taken up with avoiding designing spinsters and widows that no other work had been possible? It was an interesting idea and one which she was able to follow up that evening, when she and Harriet were invited to supper at the vicarage.\n\nBelinda was not sure why they had been asked, but it seemed as if Agatha had decided to dispose of several people to whom she owed invitations, for the company included, besides themselves, the Bishop, Father Plowman, Mr. Donne and Miss Aspinall, who had been asked at the last minute instead of Lady Clara Boulding, who had suddenly decided to spend Christmas in Switzerland with her married daughter. Miss Aspinall was radiant, or as near it as she could be, glittering with beads and chains and agreeing rapturously with everything that everybody said. This was rather difficult with four clergymen present, as, with the exception of the curate who hardly ventured an opinion on anything, they tended to disagree with each other wherever they could.\n\nIt was such a pity, Belinda reflected, that clergymen were so apt to bring out the worst in each other, especially with the season of Peace and Goodwill so near. As a species they did not _get on,_ and being in a small country village made things even more difficult. These embarrassments would not arise in London where the clergy kept themselves to themselves in their own little sets, High, Broad and Low, as it were. It was so odd to hear Father Plowman calling the curate Father Donne, though the curate himself did not appear to think it so. On the contrary, he had that evening preached a most successful sermon in Father Plowman's church on the text _We heard of the same at Ephrata and found it in the wood,_ and had been very much impressed by the elaborate service. He would discuss it with Olivia Berridge some time; she was always so sensible and would be sure to give him good advice. He would be seeing her in the New Year as he had been invited to stay for a few days with the chaplain of his old college, in whose rowing he still took a very keen interest. When there was a suitable pause in the conversation, he ventured to mention this visit.\n\n'Oh, if you should see Mr. Mold, do give him my very kindest regards,' said Harriet, fingering her long rope of cultured pearls.\n\n'Do you think that is wise?' said the Archdeacon. 'Even kindest regards are a poor substitute for the deeper feelings. I hear that the poor fellow is in quite a bad way as it is.'\n\nThe Bishop looked a little alarmed and Agatha, frowning at her husband, hastened to turn the conversation to Olivia, and how glad she would be to see Mr. Donne. 'She is generally up during the vacation, you know,' she explained. 'She does a good deal of reading then.'\n\nMr. Donne looked rather embarrassed. 'Oh, yes, it will be jolly to see Olivia again,' he said heartily. 'I expect we shall go for some walks together. She's very keen on walking.'\n\n'Has she made you any more socks?' asked Belinda innocently.\n\n'Yes, indeed, and a pullover too,' said Mr. Donne. 'She's really awfully good.'\n\n'Well, I hope she knows how to graft a toe by now,' said Harriet bluntly. 'Belinda could show her.'\n\n'Olivia is a very clever girl,' said Agatha. 'I'm sure she is quite equal to it.'\n\n'I should hardly call her a girl,' said the Archdeacon spitefully. 'But I suppose women like to think of themselves as girls long after they are thirty.'\n\n'Oh, Olivia is only thirty-one or two,' said Agatha impatiently, 'and her work on _The Owl and the Nightingale_ has really been a most substantial contribution to Middle English studies.'\n\n'All the same, it is important to know how to graft a toe,' persisted Harriet. 'What is it, Belinda, knit and slip off, then purl and keep on? I never can remember.'\n\nJust as Belinda was thinking of a tactful answer, the Bishop broke in, saying with a reminiscent sigh, 'Ah, the socks I had knitted for me when I was a curate!'\n\n'I know,' agreed Father Plowman, 'some small, some large, some short, some long, but all acceptable because of the goodwill that inspired the knitters.'\n\n'I should have thought a sock was very little use unless it was the right size,' said the Archdeacon sourly.\n\nWhen she heard this, Belinda was thankful that she had decided against knitting him a pullover and went cold with horror at the thought of what she had escaped. For there would surely have been something wrong with it. She attended to her soup, straight out of a tin with no subtle additions, she decided. Perhaps only one tin among so many, watered down or with potato water added. It certainly had very little taste.\n\n'What delicious soup, Mrs. Hoccleve,' said Miss Aspinall timidly. 'Such a delicate flavour.'\n\n'It reminds me of our native fermented porridge,' said the Bishop. 'The flavour is somewhat similar.'\n\n'Oh, how _interesting,'_ said Connie. 'How is it made?'\n\n'My dear Bishop, I hope you will remember that we are at the dinner table and spare us a detailed description,' broke in the Archdeacon.\n\n'Yes, I suppose these natives are very disgusting,' said Harriet complacently. 'It is better not to know too much about them.'\n\n'Many of them will be celebrating the festival of Christmas on Tuesday, just as we shall be doing,' said the Bishop on a faint note of reproach. 'Perhaps it will not be exactly the same in detail, but their feelings will be as ours.'\n\n'I suppose it is because of your work there that they will be able to,' said the curate.\n\nThe Bishop smiled and was about to answer when the Archdeacon gave a short bark of laughter and exclaimed, 'Ah, no, that's where you're wrong. The Romans were there first. Father Vigilio of the Padua Fathers, I believe.'\n\n'Yes, certainly, but I had the honour of starting the first Church of England Mission among the Mbawawa,' said the Bishop, 'though the Roman Catholics _were_ there before me.'\n\n'What a shame,' said Harriet indignantly, but Belinda felt that her wrath was directed not so much towards the Church of Rome as the rather dry-looking rissoles, cabbage and boiled potatoes which were now set before them. _Rissoles!_ Belinda could imagine her sister's disgusted comments later. At least one would have expected a bird of some kind, especially when there was a bishop present, when indeed all the gentlemen were in Holy Orders.\n\n'I suppose the African's leaning towards ritual would make him a ready convert to Roman Catholicism,' Belinda ventured. 'I mean, one knows their love of bright, gaudy things,' she added rather unfortunately. 'The Church of England might seem rather plain to them.'\n\n'Bright and gaudy?' said Father Plowman, on a pained note. 'Oh, Miss Bede, surely you cannot mean that?'\n\n'I'm sorry,' said Belinda, in confusion, 'naturally I didn't mean to imply...'\n\n'Well, Plowman is still with us, you know,' said the Archdeacon almost jovially. 'I don't think he need take your remarks so personally.'\n\nBelinda chewed her stringy cabbage and listened gratefully to dear Henry talking about Frazer and _The Golden Bough,_ which he thought remarkably fine.\n\n'At one time I had the idea of giving a course of sermons based on it,' he said, 'but I came to the conclusion, regretfully I must admit, that with a congregation of limited intelligence it would be _too dangerous.'_\n\nBelinda liked the sound of this and could almost have imagined them all back in Victorian days, when a father might forbid a book 'inimical to the faith of the day' to be read in his house.\n\n'How debased anthropology has become since Frazer's day,' sighed the Bishop, 'a mere matter of genealogies, meaningless definitions and jargon, _words, words, words,_ as Hamlet has it; lineage, sib, kindred, extended family, ramage\u2014one doesn't know where one is. Even the good old term _clan_ is suspect.'\n\n'What is a sib?' asked Harriet. 'It sounds a nice, friendly kind of thing, or it might be something to eat, a biscuit, perhaps.'\n\nThe Bishop shook his head and said nothing, either because he did not deign to be associated with present-day anthropological terminology or because he did not really know what a sib was.\n\nThe Archdeacon recalled the Anglo-Saxon meaning of the word, and talked for some minutes about the double meaning of peace and relationship, but Harriet had lost interest and soon they were all in the drawing-room, drinking coffee made with coffee essence. When the gentlemen joined them it was suggested that Harriet should play the piano and she gave a showy performance of Manuel de Falla's _Pantomime._ Then the Bishop sang an unaccompanied Mbawawa Christmas carol, which everyone agreed was very moving. When he had finished, Father Plowman suggested with admirable good manners that the Archdeacon should read aloud to them.\n\nThe Archdeacon was so surprised at this that for some minutes he could not even think of anything to read.\n\n'Let it be something that all can understand,' suggested Father Plowman, thinking of an occasion when the Archdeacon had insisted on reading Chaucer's _Canterbury Tales_ with an attempt at the original pronunciation.\n\nThere was a pause, nobody liking or perhaps wishing to make any suggestion, until Miss Aspinall timidly ventured the observation that Keats had written some very lovely poems. She was, of course, remembering Lady Grudge's 'evenings' in Belgrave Square, when Canon Kendrick used to read aloud to them.\n\n'Ah, yes, we will have _Hyperion,'_ said the Archdeacon. _'Remarkably_ fine.'\n\nThere was a murmur of assent, during which Harriet could be heard asking the curate if _Hyperion_ were a very _long_ poem; she had quite forgotten.\n\nBelinda turned to the Bishop and made a chatty remark about always having liked the lines about _Sorrow more beautiful than Beauty's self._\n\nThe Bishop nodded and gave her what Belinda thought was rather an intimate smile. 'I am sure that any poem _you_ admire must be very fine,' he said in a low voice.\n\nBelinda was so startled that she wondered whether she could have heard correctly. 'I'm afraid I like what I remember from my student days,' she said. 'I hardly ever read anything new.' _Hyperion_ had no memories for her, as the Archdeacon had never read it to her then, so that she was able to listen to it quite dispassionately and join with the polite murmurs that followed his performance.\n\n'And yet I think I prefer the earlier Keats,' she said rather boldly, 'I was always very fond of _Isabella_ when I was a young girl.'\n\nThe Archdeacon smiled indulgently and Agatha said quite kindly, 'Well, of course, _Isabella_ is rather a young girl's poem, isn't it?'\n\n'Oh, yes, completely,' agreed Belinda. 'It is many years since I read it.' It would indeed be an ominous sign if she felt drawn to it at her time of life, she felt.\n\n'What a fine poem Young's _Night Thoughts_ is,' said the Bishop. 'I have been reading it every night myself. I have a most interesting collection of books in my room,' he went on. 'There is an Icelandic grammar among them and I have been comparing that language with the Mbawawa.'\n\n'But do you find any similarity?' asked Agatha doubtfully.\n\n'Oh, none whatever,' said the Bishop almost gaily, 'but it is a fascinating study, _fascinating...'_ his voice trailed off on a bleating note.\n\n'I am surprised and gratified that you find the books interesting,' said the Archdeacon. 'I made the selection myself, but I had no idea of your tastes.'\n\nThe evening ended with a song from the curate. Harriet, who accompanied him, was anxious that he should try an Elizabethan love song, and after a rather faltering beginning he sang quite charmingly, Belinda thought, but without much conviction.\n\n_Love is a fancie,_\n\n_Love is a frenzie,_\n\n_Let not a toy then breed thee such annoy...._\n\nPerhaps there was no frenzy in his feeling for Miss Berridge, and love was hardly a _toy._ Surely Count Bianco's affection for Harriet could not be so described, or Belinda's for the Archdeacon? And yet tonight she had the feeling that there might be some truth in what the poet said. It was excellent advice to those of riper years, especially when the imagination became too active. That intimate note in the Bishop's voice, for example, and the way he had seemed to look at her during the reading of the poem. It might just as easily have been Connie Aspinall he was looking at. Belinda had been forced to mention the fact that the chrysanthemums he had sent her were still lasting very well. She almost wished that they might die, and noticed with relief when she got home that some of the foliage was tinged with brown. Suddenly she took them out of their vase and, although it was dark, went out with them to the dustbin. They _were_ dead really and one did not like to feel that flowers from the wrong person might be everlasting.\n\n# Chapter Nineteen\n\n'I MUST GO AND see Ricardo,' declared Belinda, one morning early in the New Year. 'Edith tells me that he has a slight attack of gout which keeps him in the house. It's rather difficult to know what to take him, though.'\n\n'Yes, you have to be very careful with gout.' said Harriet. 'No beef or strawberries or port wine. Do you think I ought to go as well?'\n\n'That would certainly do him more good than anything, but you mustn't come with me. I'm sure he'd prefer to see you _alone.'_\n\n'Yes, I really will go,' said Harriet. 'You may tell him to expect me,' she added graciously.\n\nNow that Harriet's plans about the Bishop were clearly not likely to come to anything, Belinda was determined to bring Count Bianco and her sister together as much as possible. She felt this to be her duty, and although she was not particularly anxious that Harriet should marry and leave her alone, she thought that if a marriage had been arranged in heaven she would prefer Ricardo to be the happy man. He was devoted to Harriet and they had many tastes in common: he came of an ancient Italian family and was very comfortable financially. The only thing that might possibly be against him was that he was not in the Church, but even this was not as great a drawback as might at first appear, for would there not always be tender curates in need of sympathetic attention and perfectly baked cakes?\n\nSo Belinda reasoned within herself as she walked up the drive to Ricardo's house. She walked slowly, for she was thinking rather sentimentally of how Ricardo had loved her sister well and faithfully for many years. Surely he deserved some reward for his constancy? She herself had loved the Archdeacon even longer, but naturally there was no hope of any reward for her now, at least not in _this_ world, she reflected piously, and we are given to understand that we shall be purged of all earthly passions in that _other_ life.\n\nThe Count was in and would be delighted to see her. Belinda had been careful to announce herself as Miss _Belinda_ Bede, with special emphasis on the Christian name, for she did not want Ricardo to expect Harriet and then be disappointed.\n\nHe was in the library, reading a little here and there in his many books. His gouty foot was bound up and rested on a low stool. Beside him on a little table was a pile of letters, which Belinda guessed to be those of his friend, the late John Akenside. There was also a Serbo-Croatian dictionary and the works of Alfred, Lord Tennyson.\n\nThe Count greeted Belinda with a sad smile. 'It is indeed kind of you to call,' he said, attempting to get up, but Belinda put her hand on his arm and said how sorry she had been to hear about his gout.\n\n'It is an inconvenience,' he said, 'but I am accustomed to it.'\n\nTouched by his patience and resignation, Belinda wondered how she could show her sympathy. She found it a little difficult to make conversation with Ricardo at the best of times, and could do no more than touch on various matters of general interest. It was inevitable that they should find themselves talking about the Bishop, who showed no signs of moving from the vicarage, where he had now been for nearly two months.\n\n'I hear that he is to be married soon,' said Ricardo, in a calm, patient tone.\n\n'Oh, surely not!' exclaimed Belinda, wondering how it was possible that Ricardo should come out with a piece of news that she and Harriet knew nothing of. 'We haven't heard anything, and I can't really imagine that anybody would want to marry him.'\n\n'I heard that your sister was to marry him,' said Ricardo pathetically.\n\nBelinda now laughed aloud for joy, all the more because it might so nearly have happened. In fact, she told herself soberly, there was still time; but she could at least reassure Ricardo.\n\n'It certainly isn't true at the moment,' she said, 'and I think it most unlikely that it ever will be. Wherever did you hear such a thing?'\n\nRicardo could not remember exactly; perhaps his manservant had heard it somewhere, or it may have been the Archdeacon who had told him when he called a few days ago. Yes, he was sure now that it must have been the Archdeacon. He had seemed quite certain that he was not misinformed.\n\nThe wicked _liar,_ thought Belinda angrily. An archdeacon making mischief and spreading false rumours, that was what it amounted to. Although, she told herself hastily, it was possible that Ricardo had misunderstood him, had read too much into a hint or taken a joke too seriously.\n\n'There is no truth in it whatever,' she declared positively, hoping as she did so that the Bishop was not at this moment in their drawing-room asking Harriet to be his wife. 'Harriet does not really care for him at all,' she went on boldly.\n\nRicardo smiled and looked almost happy, but then his face clouded as he asked if the Bishop were still at the vicarage?\n\n'Yes, he is still there,' said Belinda, 'but I do not think he will stay much longer. He will have to be getting back to his diocese.'\n\n'Then there is still time,' said Ricardo despondently. 'Even now he may be asking her.'\n\nBelinda shifted uneasily in her chair. Of course one never knew for certain what Harriet might be up to, or the Bishop, for that matter. She was grateful when Ricardo's manservant appeared with sherry and biscuits on a silver tray.\n\n'Have you been working on the letters this morning?' she asked, indicating the pile on the table.\n\n'Yes, I have been reading them before you came. How wise he was! He knew what would happen there; no man understood the Balkan mind as he did.'\n\n'No, I'm sure they didn't,' said Belinda inadequately, for she was never quite clear as to what _had_ happened there except that poor John had been killed in a riot.\n\nThere was a silence, during which Belinda racked her brains for something intelligent to say. But she was too late to stop Ricardo from getting back on to the subject of Harriet.\n\n'It is many days since I have seen your sister,' he said. 'It may be that there is something she does not wish to tell me.'\n\nThe warmth of the room and the unaccustomed effect of sherry in the morning were beginning to make Belinda feel a little vague and carefree, in the mood to make rash promises.\n\n'Harriet is coming to see you very soon,' she said. 'I can promise you that.'\n\n'She will never marry me, she does not love me,' said Ricardo as if speaking his thoughts aloud.\n\n'Now, Ricardo, you mustn't lose hope,' said Belinda comfortably. 'I know she is fond of you and even if she will not love you, always remember'\u2014her eyes lighted on the works of Alfred, Lord Tennyson\u2014'that it is better to have loved and lost than never to have loved at all. I always think those lines are such a great comfort; so many of us have loved and lost.' She frowned: nobody wanted to be one of many, and she did not like this picture of herself, only one of a great crowd of dreary women. Perhaps Tennyson was rather hackneyed after all.\n\nBut Ricardo did not appear to think so. 'You are so kind and understanding,' he said. 'I feel that there is a great bond between us.'\n\nBelinda did not quite know what to say, so she merely smiled and said that she was sure that some day everything would be just as Ricardo wished.\n\n'Then I shall ask her again,' he declared, fired with fresh courage and looking as if he were about to quote Dante or Tacitus at any moment; probably the former, Belinda thought, for it seemed unlikely that there would be anything suitable in Tacitus.\n\n'Yes, Ricardo, do,' she said, 'but not yet. Wait until the spring, when the daffodils are out in your meadow.'\n\n'If I am spared till then,' said Ricardo sadly, looking down at his bandaged foot. He then went on to talk of the fine new bulbs that he had planted in the meadow and to calculate when they would be at their best.\n\nBelinda left the house feeling that she had done good, and with a picture of daffodils and scyllas in her mind. She saw Harriet, the radiant Countess, picking grapes in the conservatory, adorning the head of Ricardo's dinner table, opening a garden f\u00eate or bazaar. But all this was in the distant future. For the present Belinda was glad that she had been able to cheer Ricardo and to give him a little hope. What a good thing it was that hope sprang eternal in the human breast! What would she herself have done without hope? Even if nothing came of it, she thought obscurely, for she could not have said exactly what it was that she hoped for _now._ It would be enough if things could return to normal and be as they were before Mr. Mold and the Bishop had appeared in the village. They could get on very well without them.\n\nBelinda took out her shopping list and stopped for a moment, deep in contemplation of it. Coffee, rice, steel wool, kitchen soap, written in her own hand and then, in Harriet's, tinned peaches, sponge cakes, sherry _(not_ cooking), set of no. 14 knitting needles _(steel)_... Belinda frowned. They had plenty of knitting needles of all sizes and did they really _need_ sherry or peaches or sponge cakes? Perhaps Mr. Donne was coming to supper again.\n\n'Good morning!' said a bright, cheerful voice, which Belinda did not at first recognize as Agatha's, calling to her from the other side of the narrow street.\n\n'Good morning,' Belinda called back, and was just moving on when she saw that Agatha was hurrying across to speak to her.\n\n'Isn't it a lovely morning?' she said, beaming with such unusual good humour that Belinda stared at her quite curiously, wondering what could have happened to bring about this change.\n\n'Yes, isn't it. Really quite mild,' murmured Belinda expectantly.\n\n'I have some great news for you,' said Agatha, smiling.\n\n'News? For me?' All kinds of wild ideas rushed through Belinda's head, most of which she rejected hastily. Henry had been made a dean or a bishop, that was it. It seemed unlikely, in a way, and yet what else could it be?\n\n'I had a letter from my niece Olivia this morning,' went on Agatha. 'She and Edgar are to be married\u2014quite soon.' She paused and peered so intently and beamingly at Belinda that the latter drew back, a little embarrassed.\n\n'Edgar?' said Belinda stupidly. 'Do I know him?'\n\n'Why, of course you do! Our curate, Mr. Donne,' said Agatha with some of her usual impatience which made Belinda feel more at home. 'Such a suitable thing altogether, I've been hoping all along that it would happen like this.'\n\n'But isn't she a lot older than he is,' said Belinda tactlessly.\n\n'Oh, well, a year or two, but Mr. Donne needs an older woman. Besides, he's rather shy and an older woman can often help things along, you know.'\n\n'How do you mean?' asked Belinda. 'I suppose if young people want to marry, they will. I mean if they both do.'\n\n'Ah, yes,' said Agatha, 'but there is often a natural hesitation on the part of the man, especially if he feels, as I know Edgar does, that a woman is far superior to him intellectually.'\n\nAnd older too, thought Belinda perversely. 'Yes, I suppose a young man might well hesitate in those circumstances,' she said aloud.\n\n'He who hesitates is lost,' said Agatha briskly. 'I told Edgar that and I dropped a hint to Olivia.'\n\n'Oh, did _she_ propose to _him?'_ asked Belinda in a loud, interested tone. 'I've often wondered if it was done very much. I suppose it must be done a good deal more than one realizes.'\n\n'Oh, yes,' said Agatha casually, 'it is not at all unusual. Men are understandably shy about offering what seems to them very little and when a woman realizes this she is perfectly justified in helping him on a bit, as it were.'\n\nAt this moment an idea came into Belinda's head. At first it seemed fantastic, then quite likely, and finally almost a certainty. Agatha had proposed to Henry. Why had this never occurred to her before? And now that it had, what was the use of it? Belinda could not answer this, but she knew that she could put it away in her mind and take it out again when she was feeling in need of comfort.\n\n'Yes, I suppose it can happen like that,' she agreed calmly. 'There is no reason why it shouldn't. And yet,' she ventured, 'I don't think I should ever feel certain enough to take on that responsibility myself. I know men have to take it, but supposing one met somebody else afterwards...' she stopped in confusion.\n\n'Ah, yes,' Agatha's face seemed to soften for a moment, 'That can happen too. One wonders how often it _does_ happen when one knows that it _can.'_\n\nBelinda hurried home in a great turmoil. So many exciting things to tell Harriet and somehow the curate's engagement seemed to be the least exciting of them all. Nevertheless, she could not help wondering how her sister would take the news. Not that one could say it had really been a 'disappointment' to Harriet in the usual sense, but what would she do without a curate to dote upon? It was unlikely that Miss Berridge\u2014perhaps they would soon be calling her Olivia\u2014would approve of anybody else doting on her husband, for Harriet would not like it to be suggested that she was too old and unattractive for there to be any danger, which led Belinda to speculate upon the age at which a single woman could safely have a curate to live with her without fear of scandal. She feared that whatever the age might be, seventy-five or even eighty, it would be many years before Harriet would attain it. What a solution it would be! Belinda sighed as she walked through the gate, fearful of what might happen.\n\nBut Harriet had already heard the news and although it was obvious that she was rather upset, her attitude was rather one of indignation and pity for Mr. Donne.\n\n_'Poor_ young man,' she said, 'I could hardly believe it when I heard the news. Of course it's obvious that she's been after him for a long time. I expect _she_ proposed to _him.'_\n\n'Why, yes,' said Belinda eagerly, 'Agatha as good as told me so. And I think _she_ proposed to Henry, and now she finds that she prefers the Bishop. At least,' she added, feeling that she had gone rather too far, 'she might not necessarily have meant that, but she did hint at it.'\n\n'I can quite imagine it,' said Harriet. 'If only _you_ could have thought of proposing, Belinda.'\n\n'It wouldn't have occurred to me, I'm afraid. And think how dreadful it would be to be refused. I sometimes wonder how men can bear it, though they usually go off and ask somebody else, don't they, all except Ricardo, that is. But I think it's much better not to have asked, not to know definitely that one wasn't wanted,' said Belinda hastily. 'I always feel that a man should do his own proposing.'\n\n'Yes, and then he can be blamed for the results,' said Harriet stoutly.\n\n'Perhaps clergymen feel that they ought not to ask people to marry them,' said Belinda. 'The celibacy of the clergy, you know,' she added vaguely.\n\n'It's much better for a curate not to marry. Just imagine, a _married_ curate,' said Harriet in disgust.\n\n'Ricardo has given me the recipe for ravioli,' said Belinda. 'He seemed rather low, I thought, and I promised that you would go up and see him.'\n\n'Oh, yes, I really ought to go,' said Harriet, 'perhaps this afternoon,' she added, brightening up a little, as if the thought of the Count's admiration would do something to make up for the prospect of a married curate. 'No, this afternoon won't do. I shall go tomorrow after I've had my hair done.'\n\n'All right, but let it be soon,' said Belinda. 'It would cheer him up so much. I think I shall make some ravioli for supper; it seems quite easy.'\n\n'Yes, that would be nice. We have some cold meat, haven't we, for the filling?' Harriet suddenly chuckled. 'I wonder if Agatha _does_ prefer the Bishop to Henry,' she said. 'How ironical life is; he sent you those flowers and you weren't at all pleased.'\n\nBelinda looked startled, almost as if she expected to see the chrysanthemums still there. But the place where they had stood was reassuringly filled with dried Cape gooseberries and honesty.\n\n# Chapter Twenty\n\nBELINDA ALWAYS LIKED WORKING in the kitchen when Emily was not there and was glad that she had decided to make the ravioli on her afternoon out. Emily always seemed so critical, though generally in a silent way which was far more unnerving than if she had put it into words. Belinda could feel her scornful, pitying glances as she creamed butter and sugar or rubbed fat into flour. For this reason she usually chose some foreign dish of which Emily would be unlikely to have knowledge.\n\nThis afternoon she felt a great sense of freedom and spread the things around her in a most wanton manner, though the recipe did not need complicated ingredients. The secret seemed to lie in the kneading or rolling, which was to be carried out for a full half-hour or until the paste was quite smooth and 'of the consistency of the finest chamois leather', as the Count's translation of the Italian read.\n\nWhen Belinda had been kneading and rolling for about ten minutes she felt she must rest. It was exhausting work, and the paste was nowhere near the desired consistency yet. It was sticky, full of little lumps and greyish looking\u2014not at all like any kind of chamois leather.\n\nHarriet was bustling in and out of the kitchen as she was expecting a visit from Mrs. Ramage, the wardrobe woman. She had spread practically the whole of Belinda's wardrobe out on the floor, and was quite ruthless in brushing aside Belinda's feeble protest on seeing a nearly new green cr\u00eape afternoon frock among the things to be sold.\n\n'Oh, but Harriet, I rather like that dress,' said Belinda, 'and there's still a lot of wear in it. I'm sure Miss Prior could bring it up to date in some way, if it needs it. Perhaps a little lace collar or a contrasting jabot,' she suggested uncertainly.\n\n'Neither lace collars nor jabots are being worn at the moment,' said Harriet firmly, 'and I've always thought it rather a trying shade of green. It makes you look yellow.'\n\nBelinda paused in her kneading, remembering the many times she had worn the dress. Had she always looked yellow in it? It was a disturbing thought. 'I suppose that old tweed coat _is_ past wearing,' she went on sadly, 'but I've always liked it so much.'\n\n'It's no use being sentimental about things,' said Harriet. 'You shouldn't keep a clutter of clothes you never wear just because you once liked them.'\n\nBelinda made no comment on this, for she was thinking that Harriet's words might be applied to more serious things than clothes. If only one could clear out one's mind and heart as ruthlessly as one did one's wardrobe....\n\n'I shall see Mrs. Ramage in the dining-room,' declared Harriet. 'I shall not take more than two or three things in at once. I shall start by asking \u00a35 for your green dress.'\n\n'But I believe it hardly cost that when it was new,' protested Belinda. 'What a good thing _you_ are seeing her,' she added, thinking also that it was just as well that Harriet had something to take her mind off Mr. Donne's engagement. 'I'm afraid I never have the courage to ask a big price but just agree to what she offers.'\n\nHarriet snorted. 'She'd probably offer a pound for the lot if you asked her.' The front-door bell rang. 'There, that must be her now.' Harriet strode out with a purposeful step, carrying Belinda's old tweed coat and an old jumper suit of her own over her arm. She would lead up to the green cr\u00eape dress artistically and not bring it out until the last moment.\n\nBelinda returned to her kneading and rolling. The paste still did not seem quite right. Perhaps it was too sticky. She sprinkled more flour on the board and on her hands and went on rather grimly. Her back was aching a little now and she was startled when the front-door bell rang again, and stood for a moment undecided what to do. It was no use expecting Harriet to answer it and she herself with her floury hands and generally dishevelled appearance was really in no fit state to go. But of course it probably wouldn't be anybody who mattered. It certainly wouldn't be the Archdeacon at ten to three in the afternoon.\n\nThe bell rang again, a long ring, as if it had been firmly pressed. Belinda wiped her floury hands on her apron and hurried into the hall.\n\nA man's figure showed through the frosted glass panel of the front door. A tall figure, but definitely not the Archdeacon's. It was probably a man selling something. A suitcase would be opened on the doorstep, full of combs, cards of safety pins and darning wool, packets of needles... Still, such things were useful, one could always do with them, thought Belinda opening the door, for she felt much too sorry for the men not to buy something.\n\n'Ah, Miss Bede, good afternoon.' It was an unctuous voice, a clergyman's voice, a Bishop's voice. Why was it that they were so unmistakable? Only the Archdeacon's was different.\n\n'Oh, dear, Bishop Grote.' Theodore Mbawawa. Belinda rubbed her hands vigorously on her grey tweed skirt\u2014for her apron was already too floury to be of much use\u2014and backed into the hall. 'I'm afraid I'm hardly in a fit state to receive visitors, but do come in.' She edged towards the drawing-room door and put out a still floury hand to open it.\n\nBut the Bishop was almost too quick for her. His hand reached the knob simultaneously with hers. For one panic-stricken moment she even imagined that it lingered for a fraction of a second, but then dismissed the unworthy thought almost before it had time to register in her mind. She was in an agitated state, and she had read somewhere that in any case middle-aged spinsters were apt to imagine things of this kind....\n\nInside the drawing-room Belinda stood uncertainly, while the Bishop advanced towards the fireplace, where a fire was laid but not yet lit. He made a remark about the weather, observing that it was a raw and chilly afternoon. In his diocese, he added, they would be enjoying some of the hottest weather now.\n\n'It must be a lovely climate,' said Belinda, fumbling in the Toby jug on the mantelpiece for a box of matches. 'I'm so sorry about there being no fire. We usually light it just before tea.' She wondered if she could perhaps offer him a cup now. It was certainly a little early, but it would at least fill in the time until Harriet had finished with Mrs. Ramage and would also give Belinda an opportunity to slip away and make herself more presentable. 'I will go and tell my sister that you are here,' she said crouching over the fire and setting a match to a corner of the _Church Times._ 'She will be so pleased to see you.'\n\nThe Bishop held up his hand. 'No, please, Miss Bede. It is _you_ I have come to see.'\n\nBelinda stood up. 'Oh?' Whatever could he want? _'Me?_ Please sit down, won't you? I must apologize again for my untidiness, but I was doing something in the kitchen.'\n\nThe Bishop smiled. 'And doing it admirably, I'm sure.'\n\nBelinda smiled uneasily. She began to wonder whether she had thanked him enough for the flowers, though that was some weeks ago now, and was just going to make a remark about them when he began to speak in a hurrying way, as if he were not quite sure of himself.\n\n'Miss Bede, I am sure you must have realized\u2014have noticed, that is\u2014my preference for you above all the other ladies of the village,' he said, and peered at her so intently that Belinda\u2014they were sitting together on the sofa\u2014drew back, considerably alarmed.\n\n'No, I don't think I have,' she said anxiously. 'In any case you can hardly know me very well or you would realize that there is nothing very special about me.'\n\n'Ah, well, one hardly looks for beauty at our time of life,' he said, with a return of some of his usual complacency. _'She is not fair to outward view_... how does Wordsworth put it?'\n\n'Not _Wordsworth,'_ said Belinda automatically. 'Coleridge, Hartley Coleridge, I think.' She felt rather annoyed. Not even a middle-aged spinster likes to be told in so many words that she is not fair to outward view. Besides, she felt that the Bishop had taken an unfair advantage of her, calling on Emily's afternoon off, when she had had no opportunity to tidy herself. 'Although I am not beautiful myself and never have been,' she went on, 'I must confess that I like to see beauty in other people.'\n\n'You mean beauty of character, ah, yes. That is something we all like to see.'\n\n'No, I mean beauty of person,' said Belinda obstinately.\n\nThe Bishop smiled. 'Then perhaps you will not be so ready to accept what I have to offer,' he said, though it was obvious that he really thought quite otherwise.\n\n'Offer?' said Belinda in a startled tone. 'I don't think I understand.' The man on the doorstep opening his suitcase was simpler and less alarming than this. She hardly dared to let herself guess what the Bishop meant; it was too fantastic and terrible to be thought of.\n\n'Perhaps you are not accustomed to receiving such offers?' he went on. 'Or perhaps it is some time since you last had one? After all, this is a quiet country village; it is unlikely that you would meet many strangers.'\n\n'That may be,' said Belinda feeling very angry, 'but I think I can say that I have had my share, in the past, that is. Naturally not lately,' she fumbled, her natural honesty getting the better of her.\n\n'I think I had better speak more plainly,' the Bishop went on. 'I am asking you to marry me.'\n\nThere was a short but awkward silence, and then Belinda heard herself stammering out the first words that came into her head, 'Oh, but I _couldn't....'_\n\n'My dear, you are equal to being the wife of a bishop,' he said kindly, making a movement towards her. 'You need have no fears on that account. When I was a younger man I held views about the celibacy of the clergy, young curates often do, you know,' he smiled indulgently, 'it is a kind of protection, if you see what I mean. But a man does need a helpmeet, you remember in _Paradise Lost....'_\n\nBelinda interrupted him with a startled exclamation. _'Paradise Lost!'_ she echoed in horror. _'Milton....'_\n\n'I think when one has reached er\u2014riper years,' the Bishop continued, 'things are different, aren't they?'\n\nA man needs a woman to help him into his grave, thought Belinda, remembering a remark Dr. Parnell had made. Well, there would be plenty who would be willing to do that.\n\n'I'm afraid I can't marry you,' she said, looking down at her floury hands. 'I don't love you.'\n\n'But you respect and like me,' said the Bishop, as if that went without saying. 'We need not speak of love\u2014one would hardly expect that now.'\n\n'No,' said Belinda miserably, 'I suppose one would not _expect_ it. But you see,' she went on, 'I did love somebody once and perhaps I still do.'\n\n'Ah...' the Bishop shook his head, 'he died, perhaps? A very sad thing.'\n\nThey were both silent. He died, yes, it was better that the Bishop should think that, it sounded more suitable; there was even something a little noble about it. _She never married..._ Belinda began to see herself as a romantically tragic figure.\n\n'Of course, as Lord Byron says,' began the Bishop, and then paused.\n\nCould Lord Byron have said anything at all suitable? Belinda wondered. _When we two parted in silence and tears?_ Possibly, though the poem was not really applicable. 'Do tell me,' she said, her literary curiosity driving other thoughts from her mind. 'What _did_ Lord Byron say?'\n\nBut the Bishop was standing up now and saying that he did not think he would be able to stay for tea, although Belinda was not conscious of having offered it. 'I think it is perhaps a little early for tea, Miss Bede, and I have still another call to make.'\n\n'Oh, I expect you will get tea there,' said Belinda in a full, relieved tone. 'Now that I come to think of it, we have only very little cake, just a small piece of gingerbread, I believe. When one has guests one likes to have rather more than that to offer them.' She frowned, wishing she had not used the word 'offer', but the Bishop did not seem to be at all upset, or even, indeed, to have noticed. 'I'm so sorry,' said Belinda ambiguously. 'I am really most honoured that you should have felt... but I'm sure you will understand how it is.'\n\n'Do not give it another thought, Miss Bede,' he said briskly, 'I assure you that _I_ shall not. After all, we must remember that _God moves in a mysterious way, His wonders to perform.'_\n\n'Yes, certainly,' agreed Belinda, feeling a little annoyed that he should quote her favourite hymn. But perhaps it was presumptuous to suppose that God would be more likely to reveal His ways to her than to the Bishop. She did not quite see how the lines applied here, no doubt he had something else in mind. Perhaps he would come another day and ask Harriet? At all events he was not going to give her refusal another thought, so he could not care very much. It was not very flattering to her, though she supposed that as she was not fair to outward view she could hardly expect anything else.\n\nIt was not until they were in the hall that she realized that she had been offered and refused something that Agatha wanted, or that she may have wanted, for the hint she had given had been very slight. She wondered if the Bishop had any idea of it.\n\n'It is nice that you have been able to stay so long here,' she said, with unaccustomed guile. 'I expect the Archdeacon and Mrs. Hoccleve will miss you when you go back.'\n\n'Yes, I think I can say that they will.' The Bishop smiled to himself. 'I have been able to give the Archdeacon a few tips, although a small country parish hardly presents the same problems as a large African diocese.'\n\n'No, of course not,' said Belinda. 'Nobody would imagine that it did.'\n\n'Mrs. Hoccleve has been most kind in helping me to buy various things that I need to take back to Africa with me. She has also knitted me some socks.'\n\n'Oh, how kind!' exclaimed Belinda. 'There is nothing like hand-knitted socks.'\n\n'No, indeed, there isn't. Particularly when they are not quite long enough in the foot.' The Bishop laughed with a silly, bleating noise. 'Quite between ourselves, of course,' he added.\n\n'Of course,' repeated Belinda, closing the front door behind him. She felt that she could almost love Agatha as a sister now. The pullover that she might have made for the Archdeacon would surely have been wrong somewhere, but as it had never even been started, it lacked the pathos of the socks not quite long enough in the foot. To think of Agatha as pathetic was something so new that Belinda had to sit down on a chair in the hall, quite overcome by the sensation. She began to find ways of making things better and more bearable. Agatha couldn't really have meant that she cared for the Bishop; nobody could love a man like that. She almost longed to see Agatha and to be crushed by one of her sharp retorts, to know that she was still the same.\n\nAt last she remembered the ravioli, and was almost glad of an excuse to stop thinking about these disturbing matters. She paused for a moment by the looking-glass and studied her wispy hair, flushed face smeared with flour and faded blue overall. Looking like that one could not feel even a romantic figure whose lover had died.\n\nThe sound of raised, almost angry, voices came from behind the closed door of the dining-room. It was a clash of wills between Harriet and Mrs. Ramage, but Harriet would win in the end. It was known that the Misses Bede had 'good' things\u2014though hardly of the same standard as Mrs. Hoccleve\u2014and Mrs. Ramage would be unwilling to leave without buying them.\n\nBelinda went quietly back to the kitchen and sat down. She wished Harriet would come, so that she could tell her all about it. After all, she supposed, it was something to have been considered worthy to be the wife of a bishop, even if only a colonial one. There was something rather sad about the kitchen now. It was beginning to get dark, and the greyish mass of dough on the table reminded Belinda of the unfinished ravioli. Twenty minutes more kneading, and perhaps it would be of the consistency of the finest chamois leather.\n\n_The trivial round, the common task\u2014_ did it furnish _quite_ all we needed to ask? Had Keble _really_ understood? Sometimes one almost doubted it. Belinda imagined him writing the lines in a Gothic study, panelled in pitch-pine and well dusted that morning by an efficient servant. Not at all the same thing as standing at the sink with aching back and hands plunged into the washing-up water.\n\n'Three pounds, fifteen and six!' Harriet came triumphant into the kitchen, waving the notes in her hand. 'She was pleased with your green dress, but she wondered how you could ever have worn it. \"Not at all Miss Bede's colour\", she said.'\n\n'No, I begin to wonder now myself how I could ever have worn it,' said Belinda. 'Perhaps it is hardly surprising that Bishop Grote does not think me fair to outward view, though I think I was wearing my blue marocain that evening at the vicarage, and I always think I look quite nice in that.'\n\n'Oh, was it Theo who called just now?' asked Harriet. 'What did he want?'\n\n'He wanted me to be his wife,' said Belinda, enjoying the dramatic simplicity of her announcement.\n\n_'No!'_ Harriet's surprise was a little uncomplimentary, but her joy and relief at having her sister spared to her more than made up for it. 'What a pity you and Agatha can't change, though,' she lamented. 'But of course he can't really care for her very much or he wouldn't have asked you, would he?'\n\n'I don't know,' said Belinda, who was beginning to think that she did not understand anything any more. 'Anyway I don't suppose Agatha really cares for him. I ought not to have told you what she said.' She felt that she could not tell even Harriet about the socks and was glad when she left the subject and came out with a piece of news of her own. Mrs. Ramage, in the intervals of bargaining, had told her that she had heard that Mr. Donne had been offered a 'post' at his old University\u2014chaplain in the college or something like that.\n\n'How suitable,' said Belinda, 'but of course it will mean him leaving here, won't it?'\n\n'Oh, yes, of course,' said Harriet casually. 'But don't you see, we shall get a new curate? The Archdeacon will never be able to manage by himself.'\n\n'No, of course not,' agreed Belinda fervently, 'he couldn't possibly manage by himself. He will certainly have to get a new curate.'\n\n'This is really a place for a young man,' said Harriet.\n\n'Well, I don't know. A young man might want more scope, a more active parish with young people. Something in the East End of London, perhaps,' Belinda suggested. 'I should think this curacy might very well suit an older man.'\n\n'Oh, I can't imagine that,' said Harriet in disgust. 'And anyway, curates are nearly always young.'\n\n'Not necessarily,' said Belinda, feeling that she ought to help her sister to face up to the problem from every possible angle. 'Sometimes a man in middle life suddenly feels called upon to take Holy Orders. I always feel it must be so awkward and upsetting for his family.'\n\n'Oh, dear,' Harriet's face clouded. 'I do hope it won't be anyone like that.'\n\n'I'm not saying it will be, but it _could_ be,' said Belinda. 'I think the Archdeacon would prefer a young man, though.'\n\n'Yes, working with the Archdeacon must be a great experience,' said Harriet obscurely. 'A young man of good family, just ordained, that's what we really want. Do you suppose the Archdeacon will advertise in the _Church Times?'_\n\n'He could hardly advertise for somebody of good family,' said Belinda smiling.\n\n'They sometimes say \"genuine Catholic\" or \"prayer-book Catholic\",' mused Harriet, 'but of course we should hardly want that here.'\n\n'Oh, Harriet, _look!'_ Belinda held up the sheet of ravioli she had been rolling.\n\n'But, Belinda, it's just like a piece of leather. I'm sure that can't be right,' protested Harriet.\n\n'It is,' said Belinda joyfully, 'it's even finer than the finest chamois leather.'\n\n# Chapter Twenty-One\n\nTHE NEXT FEW WEEKS were entirely taken up with preparations for the curate's wedding. It may be said without exaggeration that it was the only topic of conversation in the village at this time, and many a church worker's dingy life had been brightened by the silver and white invitation card, which was prominently displayed on many mantelpieces. The marriage of their dear Mr. Donne was something in which all could share, for had he not at some time or another been to meals at all their houses? Many were the chickens which had been stuffed and roasted or boiled and smothered in white sauce in his honour.\n\nOne afternoon, a few days before the wedding, there was a gathering at the parish hall, the object of which was to make a presentation to Mr. Donne and his fianc\u00e9e. Miss Berridge was to stay at the vicarage and be married from there, as her parents were dead and Mrs. Hoccleve, perhaps because of her niece's substantial contribution to Middle English studies, had always been particularly fond of her.\n\nThere was great excitement in the hall for nobody had yet seen Miss Berridge, though Edith Liversidge had caught a glimpse of her coming from the station the previous evening. As it had been a very dark night she had not been able to give a satisfactory description, except to say that she had been wearing a fur coat, a dark, rather bushy fur, musquash, she thought, and that she was very tall, perhaps taller than Mr. Donne, who was only of middle height for a man.\n\n'Taller than he is,' said Harriet in a disgusted tone, as they sat waiting for the proceedings to begin. 'What a pity! I always think it looks so bad.'\n\n'Yes,' agreed Belinda, 'but of course it can't be helped. I mean, there are other things more important.'\n\n'Tall women always droop,' said Edith sharply. 'I'm always telling Connie to hold herself up. It never had any effect until Bishop Grote said something about liking tall women\u2014she's been better since then.'\n\n'Did he say that?' asked Belinda, interested. 'I didn't think he really minded what people looked like\u2014or expected much from people of our age, anyway.'\n\n'Bishops ought not to mind or expect,' said Edith, 'but I suppose they're human. Besides, don't forget all those native women...' she paused darkly. 'He was describing their costume or lack of it when he came to tea with us the other day. Poor Connie was quite embarrassed.'\n\n'Well, he's gone now,' said Belinda comfortably, 'and he'll probably be quite glad to be back among those dear, good fellows.'\n\n'Agatha was sorry to see him go,' said Harriet. 'That was obvious. _Ah, quotiens illum doluit properare Calypso\u2014_ Ah, how many times did Calypso grieve at his hastening.'\n\n'Connie was sorry too,' said Edith. 'If it hadn't been that she was due for her annual visit to Belgrave Square she would have been very low.'\n\n'I do hope she will be back in time for Mr. Donne's presentation,' said Harriet, 'she wouldn't want to miss his speech, I'm sure.'\n\n'Her train is due in at a quarter to three, so I suppose she should be here for most of it\u2014in at the death, you know.'\n\n'Poor Mr. Donne,' sighed Harriet, 'one almost feels that it _is_ a death.'\n\nAt this moment there was a stir among the rows of waiting women, and the door on to the platform opened. Agatha Hoccleve, in a black tailored costume of good cut, came in, followed by a tall, pleasant-looking woman in the early or middle thirties, in a blue tweed costume of rather less good cut. She had a pale, rather long face and wore spectacles. Her hair was neatly arranged at the back of her head, though this was rather difficult to see as her navy felt hat was pulled down at a sensible angle.\n\nLong, English gentlewoman's feet, thought Belinda, noticing her shoes and good, heavy silk stockings.\n\nMiss Berridge was followed by Mr. Donne, looking rather sheepish, Father Plowman, whose parish had wished to join in the presentation, and, finally, the Archdeacon, smiling sardonically and bearing in his arms a large square object shrouded in a cloth which he placed on a small table at one side of the platform.\n\n'She _is_ taller than he is,' whispered Harriet, 'and she looks much older. What a pity! She's rather plain, too, isn't she? Why doesn't she use lipstick?'\n\n'Ladies and Gentlemen...' Agatha Hoccleve's clear voice rang out. She was a confident public speaker and this afternoon's audience of parish women with a few churchwardens and choirmen held no terrors for her. 'It gives me great pleasure to introduce my niece, Miss Berridge, soon to be Mrs. Donne, to you all. I only wish she were going to stay longer with us, but we must not\u2014I am sure we do not\u2014grudge her to Mr. Donne, and I am equally sure that he'\u2014she gave a sideways glance to where the curate was sitting looking down at his shoes\u2014'will not grudge _us_ the opportunity of getting to know her while she is here. Of course _I_ know her already,' she added with a little laugh, and sat down amid the mild clapping which followed.\n\nBy this time the Archdeacon had got up and moved over to the side of the platform where the shrouded object stood and seemed about to uncover it, but he was prevented by an agitated gesture from Father Plowman. The situation was saved by Agatha stepping forward and saying in a loud voice, 'Before the Archdeacon makes the presentation, I think Mr. Plowman has something to say.'\n\n_'Father_ Plowman,' giggled Harriet.\n\n'Yes, indeed, I have,' said that clergyman, with a grateful glance at Agatha and a rather baleful one at the Archdeacon. 'My parishioners and I felt that we could not let this opportunity pass without adding our good wishes and our widow's mite, as it were, towards this gift. I think we shall not soon forget Father Donne's gifts to _us._ I mean,' he added, sensing a faint bewilderment among his hearers, 'his Sunday evening sermons. I can see him now, walking across the fields in the evening sunshine, his cassock and surplice over his arm...' Mr. Donne himself now looked a little startled as Father Plowman's church was some seven or eight miles away and he had always gone over on his bicycle... 'pausing perhaps to drink in the beauty of our old church seen in that gracious evening light, pondering his message to us that evening, the _gift_ he was bringing us.' Father Plowman paused, a little overcome by his eloquence. 'May he, with the help of Miss Berridge, go on from strength to strength, as I am sure he will. _From glory to glory advancing, we praise Thee, O Lord.'_ He bowed his head. 'And now a prayer, _Prevent us O Lord in all our doings...'_\n\nThe obvious prayer, of course, thought Belinda, who had noted with anxiety the expression of irritation on the Archdeacon's face. Perhaps Father Plowman ought not to have said a prayer at all\u2014it should have been left to the last speaker\u2014the Archdeacon\u2014who was now advancing once more towards the shrouded object.\n\nAs if in deliberate contrast, he adopted a more ominous tone, and dwelt, not so much on the parish's loss, though he did mention that Mr. Donne's going would mean a great deal of extra work for _him,_ as on the difficulties that Mr. Donne might be expected to encounter in his new life. 'The University is a stony and barren soil,' he declared in a warning tone, 'one might almost say that the labourers are too many for the scanty harvest that is to be reaped there. The undergraduates are as in Anthony \u00e0 Wood's day, much given over to drinking and gaming and vain brutish pleasures.' He looked as if he were warming to his subject, and Belinda began to fear that he might quote other and more unsuitable passages from that crabbed antiquary, but after a short pause he left the subject and went on. 'It is to be hoped that Mr. Donne may succeed where others have failed. Indeed, with the help of Miss Berridge'\u2014he gave her a most charming smile\u2014'one feels that he may. I am certain too that with her considerable linguistic gifts she will be a great help to him if ever he feels called upon to labour in the Mission Field.'\n\nThere was a visible stir in the audience at these words and some indignant whispering.\n\n_From Greenland's icy mountains_\n\n_From India's coral strand...._\n\n'It is not so very long since we were singing Bishop Heber's fine hymn in _our_ fine old church.' The Archdeacon paused as if to let the significance of his words sink in and then began to fumble with the cloth which shrouded the square object.\n\n'Isn't it the table cloth out of the morning-room at the vicarage?' whispered Harriet. 'How unsuitable!'\n\n_We take no note of time but from its loss._\n\n_To give it then a tongue is wise in man...._\n\nThe Archdeacon paused\u2014'Edward Young, the eighteenth-century poet and divine wrote those words nearly two hundred years ago. Not a _great_ poet, you may say, no, one would hardly call him that, but I think his words are still true today. That is why we are giving this chiming clock to Mr. Donne and Miss Berridge as a wedding present.' He flung aside the cloth with a dramatic gesture that Belinda thought very fine, if a little too theatrical for the occasion. 'May it do something to ease the burdens they will be called upon to bear in their new life.'\n\n'I should think Mr. Donne's burdens will be infinitely lighter in his new life than they have been here,' murmured Edith, 'but I suppose the Archdeacon cannot bear to think of anybody without some kind of burden.'\n\n'Well, we know that life is never without them,' said Belinda loyally. 'It is perhaps just as well to remember that. And even if we appear to have none we really _ought_ to have...' her voice trailed off obscurely, for Miss Berridge had come forward and was making a speech of thanks. Her voice was clear and ringing, as if she were used to giving lectures or addressing meetings. What an excellent clergyman's wife she would make with this splendid gift!\n\n'Edgar and I are simply delighted...' there was comfort in the words, as if she were protecting Mr. Donne in a sensible tweed coat or even woollen underwear. It was obvious that she would take care of him, not letting him cast a clout too soon. She would probably help with his sermons too, and embellish them with quotations rarer than her husband, with his Third Class in Theology, could be expected to know. A helpmeet indeed.\n\n'Rather _toothy_ when she smiles, isn't she,' whispered Harriet. 'I wonder what _he_ will say.'\n\nMr. Donne's speech was very short. 'Olivia has said exactly what I would have said,' he began. Here again the use of Christian names gave a cosy, intimate feeling. Agatha and Father Plowman were smiling and even the Archdeacon was looking benevolent.\n\nMr. Donne concluded on a serious note. 'I think we shall both remember what the Archdeacon has said about the burdens we may have to bear in our new life. I hope we may not be found wanting at the testing time. And now, let us pray....'\n\nBelinda wondered whether he would be able to think of another suitable prayer when Father Plowman had rather unfairly used the obvious one already. But she had to admit that his choice was an admirable one. _Lord, we pray Thee that Thy grace may always prevent and follow us, and make us continually to be given to all good works..._ She bowed her head and could see out of the corner of her eye Miss Prior and Miss Jenner, creeping in through a side door carrying a tea-urn. When they realized that a prayer was being said, they stood stiffly with the urn, like children playing a game of 'statues'.\n\nThe prayer ended, and after a decent pause Miss Berridge and Mr. Donne\u2014or Olivia and Edgar as they had now become in the minds of their hearers\u2014came down from the platform and moved among the audience, shaking hands and chatting.\n\nBelinda found herself talking to Miss Berridge and offering her a cake from a plate which seemed to have got into her hand. She felt a glow of warm friendliness towards her, perhaps because of her rather plain, good-humoured face, her sensible felt hat, her not particularly well-cut tweed suit and her low-heeled shoes. Nothing from the 'best houses' here\u2014all was as it should be in a clergyman's wife.\n\n'Where will you live?' Belinda asked. 'I suppose suitable accommodation is provided for married chaplains even in a place that is in other ways as old-fashioned as our University.'\n\n'Oh, we've already taken a very nice house, rather Gothic in style, but I think it will be comfortable and it has a large garden,' said Olivia. 'And Edgar will have rooms in college as well, of course. I hope you and your sister will come and see us when we are settled in. Edgar tells me you have been so good to him.'\n\nBelinda managed to stop herself saying, 'Oh, it was really Harriet, my sister, she dotes on curates,' and asked instead whether it was decided who should come in place of Mr. Donne. 'Will he be a young man or an older one who needs the quiet and country air of a little place like this?' she asked.\n\n'Both, in a way,' said Olivia. 'He _is_ young, but he has recently been ill\u2014I think you'll like him very much. I believe he's a Balliol man and he's certainly very handsome, dark and rather Italian-looking, really. Edgar looks quite plain beside him.' She laughed affectionately. 'Anyway, you'll see him at our wedding.'\n\n'How splendid! My sister will be delighted,' said Belinda with unguarded enthusiasm. 'She has such a respect for Balliol men,' she added hastily.\n\n'Yes, it still maintains its great tradition of scholarship,' agreed Olivia. 'I had an idea that Archdeacon Hoccleve was a Balliol man.'\n\n'No, he isn't, as a matter of fact,' said Belinda, 'but he is a very fine preacher. His knowledge of English literature is quite remarkable for a clergyman.'\n\n'His sermons are full of quotations,' said Harriet bluntly. 'I consider Mr. Donne a _much_ better preacher.'\n\n'I think that English Literature and Theology can be very happily combined,' said Olivia gracefully. 'I daresay I shall find myself encouraging Edgar to write more literary sermons.'\n\n'Ah, yes,' said her fianc\u00e9, 'I expect Olivia will help me to outdo even the Archdeacon with obscure quotations from the _Ormulum.'_\n\n'Whatever is that?' asked Harriet. 'It sounds very learned.'\n\n'A kind of moral treatise, I believe.'\n\n'Oh, well, I hope you won't listen to her,' said Harriet. 'I suppose people liked things like that in the old days.'\n\n'And I hope that we in these days may still be said to like \"things like that\",' said Father Plowman, smiling indulgently at Harriet, as if she were a naughty child.\n\n'Perhaps we would like them better if we could understand them,' said Belinda. 'I mean the language, of course.' She spoke rather hastily as she could see the Archdeacon approaching and thought it might be as well to change the subject before he came. 'I wish somebody would have another cake,' she said, offering the plate which she was still holding. Harriet took it from her and began to encourage Mr. Donne to eat a cake with pink icing, much to the amusement of Olivia, who did not appear to mind Harriet's proprietary attitude in the very least.\n\nBelinda, feeling that she had monopolized the young couple for quite long enough, withdrew quietly and found herself near the tea-urn over which Miss Prior and Miss Jenner were presiding.\n\n'Would you like another cup, Miss Bede?' asked Miss Prior. 'I know you're one for tea, like I am.'\n\n'Yes, please, I would,' said Belinda, feeling this to be a comfortable classification. 'I'm sure we need plenty of tea after all this excitement.'\n\n'We certainly do,' agreed Miss Prior. 'Miss Aspinall, too! _That_ news was a surprise, I can tell you. A shock, you might almost say.'\n\n'Oh, I always thought he had his eye on her,' said Miss Jenner in her shrill, arch voice. 'I said to mother only the other day that something would come of it.'\n\n'Come of it?' asked Belinda, looking over to where, as she now realized, Connie Aspinall, fresh from Belgravia, was standing in the centre of a little crowd. She looked almost happy and was talking with unusual animation. Even Belgrave Square could hardly have made such a change in her. 'I don't quite understand,' Belinda went on, 'has something happened to Miss Aspinall? Is _she_ getting married as well?'\n\n'Didn't you know?' Miss Jenner almost shrieked. 'Miss Aspinall's got engaged to Bishop Grote. You remember, Miss Bede, he was staying at the vicarage not so long ago.'\n\nBelinda had taken a large gulp of tea and narrowly escaped choking, but she was able to indicate that she did remember Bishop Grote. 'But this is amazing, wonderful,' she emended, 'news. I must go and congratulate her immediately.' She put down her cup and made her way over to Miss Aspinall's corner.\n\n'My _dear_ Connie,' she said, 'I've just heard your wonderful news. I _am_ glad.'\n\n'Yes, I'm so happy,' said Connie. 'Theodore told me that as soon as he came here he felt that he was destined to find happiness, and that when he saw me he knew _it was to be'_ She gave a mysterious emphasis to these last words.\n\nBelinda was silent for a moment in thankfulness and wonder. She did not now resent the Bishop's quoting of her favourite hymn _God moves in a mysterious way._ It was manifest that He did.\n\n'But how did you come to meet again?' she asked, curiosity taking the place of wonder. 'Did he visit you in Belgrave Square?'\n\n'No, we met one afternoon in the Army and Navy Stores,' said Connie. 'That was the wonderful part of it. I had gone into the garden-furniture department to buy a trowel and I was standing looking at a sweet little bird bath\u2014you know they have such pretty carved stone ones\u2014when I heard footsteps behind me\u2014the place was _quite_ deserted\u2014I happened to look round and there was Theodore.'\n\n'How extraordinary!' said Belinda, imagining the shock she herself would have felt.\n\n'Yes, it seemed he had lost his way. He wanted to buy a new tin trunk to take back to Africa and found himself in the garden-furniture department by mistake. The man he asked had misdirected him or the department had been moved, I believe.'\n\n'How extraordinary life is,' Belinda interposed. 'To think that the moving of a department, if it _had_ been moved....'\n\n'And so then he asked me to go and have tea with him and luckily there was a Fuller's teashop quite near and he proposed to me over tea.'\n\n'Did you have that lovely walnut cake?' asked Harriet, who had now joined them.\n\n'I don't suppose Connie noticed what she was eating,' said Belinda. 'My dear, I'm so glad for you. I hope you will be very happy. I'm _sure_ you will,' she added, feeling that she may have sounded doubtful.\n\n'Well, I think we shall be,' said Connie. 'At first I naturally felt rather nervous\u2014the importance of the position, you see\u2014but Theodore assured me that I was quite equal to being a bishop's wife.'\n\n'Yes, I am _sure_ he did,' said Belinda, remembering. 'Are you to be married here or in London?'\n\n'Oh, from Belgrave Square. Lady Grudge insisted on it. She's been so _very_ kind and is most interested in the splendid work Theodore's been doing.'\n\n'I feel that Connie's cup is almost too full,' said Belinda, as she and Harriet walked home together, 'but I'm very glad she's so happy. Perhaps they really do love each other, though he may have told her not to expect that. But perhaps it was only _I_ who was not to expect love,' she added rather sadly.\n\n'And being married from Belgrave Square!' said Harriet, who was in high spirits. 'One feels that is almost the best thing about it. _Hymen Io!'_ she chanted, producing a most appropriate left-over fragment of her classical education. 'That is suitable for weddings isn't it?'\n\n'I suppose so,' said Belinda, who had not had the same advantages. 'Of course there are many beautiful Epithalamia in the English language, but one feels somehow that they are more for _young_ people.'\n\nBut Harriet had lost interest in Connie and the Bishop. She was remembering Olivia Berridge's description of the new curate. 'Balliol and rather Italian-looking,' she breathed, 'and just recovered from an illness. Oh, Belinda, he will need such special care!' Later that evening she could be seen studying a book of Invalid Cookery, and was quite annoyed when Belinda pointed out that he would probably be eating with a normal appetite by the time he came to them.\n\n# Chapter Twenty-Two\n\nFORTUNATELY THE WEDDING DAY was fine, one of those unexpected spring days that come too soon and deceive one into thinking that the winter is over. Belinda had great difficulty in restraining Harriet from casting off her woollen vest, and was only successful after she had pointed out, with great seriousness, that the curate would surely not be so foolish as to leave off his winter combinations in February.\n\n'Oh, I suppose Miss Berridge would see to that,' said Harriet sulkily. 'It's quite obvious that she's going to fuss over him and turn him into a molly-coddle.'\n\nBelinda could think of no answer to make to this, but could only wonder at the shortness of Harriet's memory. It seemed such a little time since she had thought of nothing but knitting and cooking special dishes for Mr. Donne.\n\n_By many deeds of shame_\n\n_We learn that love grows cold,_\n\nthought Belinda. Did the author of that fine hymn realize how often his lines had been frivolously applied? She turned to her prayer-book and saw that he had died in 1905: so he was beyond the realization. No doubt we are more frivolous now than we were then, she thought, feeling in her jewel-box for her little seed-pearl brooch. The neck of her new dress\u2014a soft shade of blue that could not possibly make her look yellow\u2014needed something to relieve its severity. Miss Prior had really made it very well; it even fitted quite closely as a result of Belinda's timid and carefully phrased request for something a little less shapeless than usual. Harriet had ordered a creation from Gorringe's which Belinda considered an unnecessary expense. She had not, however, mentioned this to her sister as she realized that the loss of a dear curate was something of an occasion and this was, as far as she could remember, the first time one had been lost by marriage. Usually it was the Mission Field or the East End of London that claimed them, or, more rarely, a comfortable living in the gift of a distant relative....\n\n'Belinda! Are you ready?' Harriet stood in the room, magnificent in furs and veiled hat.\n\n'Oh, yes, I just have to put on my hat and coat. There! You _do_ look nice, Harriet. I wonder if a veil would suit me? One feels that it's _kinder_ somehow\u2014not that your complexion needs to be hidden, of course, but I sometimes think that for _me..._ and yet, I don't know...' she broke off, wondering whether it would really make any difference now if she were to appear with her face softened by clouds of veiling. Henry knew the face underneath and would not be deceived and that was all that really mattered.\n\n'Yes, I think it's one of my most successful hats,' said Harriet. 'Yours is nice too,' she added kindly. 'I think I should wear it just a _little_ further back, though.'\n\nBelinda adjusted it without looking in the mirror and they went downstairs.\n\n'It's really a perfect day,' said Harriet, 'amazingly mild.'\n\n'But I'm glad you kept on your vest,' said Belinda, 'it would have been most unwise to leave it off. It's certainly mild compared with what it has been, but it may be colder tonight.'\n\nAs they passed the vicarage she drew Harriet's attention to an almond tree in full blossom.\n\n'Look, the almond tree's out,' she said. 'I always feel it looks lovelier every year, so beautiful that one can hardly bear it.'\n\nHarriet looked at her sister apprehensively. She hoped Belinda wasn't going to cry too much at the wedding. They did not have many weddings in the village and it would hardly be surprising if Belinda began to distress herself by remembering that if Fate had willed otherwise she herself might have married the Archdeacon thirty years ago. She was relieved when Belinda went on to talk about ordinary things, whether it would be wise to take more than one glass of champagne at the reception, always assuming it were offered, what Agatha would be wearing and whether the new curate would be at the church.\n\nThe sisters arrived in good time so that they could choose their place carefully. Harriet looked round quite unashamedly to watch people coming into the church. Belinda occasionally turned her head, but most of the time she sat quietly with her hands folded in her lap, though she could not help showing her interest and even whispering to Harriet when Agatha came in, very elegant in dark red, with a fur coat and wide-brimmed hat. A group of nondescript-looking women\u2014possibly relatives or friends of the bride, fellow workers in the field of Middle English studies\u2014came behind Agatha and they sat together in one of the front pews.\n\nDr. Nicholas Parnell had unfortunately not been able to come down for the wedding as he had to welcome to the Library some distinguished Russian visitors who were bringing with them a number of interesting relics. Mr. Mold had recently left for China, where he was to make a tour of libraries and a special study of the heating systems employed in them. There was, however, to be one distinguished visitor, the University Professor of Middle English, who was to give the bride away.\n\n'Perhaps he wanted Miss Berridge too,' Harriet had suggested, 'but hadn't the courage to speak.'\n\n'Oh, surely not,' said Belinda, 'or how could he bear to come to the wedding?' She could certainly not have attended Agatha and Henry's.\n\nIn the meantime the church was filling up. Count Bianco, whose gout was well enough now to allow him to go out, had chosen his seat carefully so that he could have a good view of Harriet. Edith Liversidge came in alone, for Connie Aspinall was in Belgravia, preparing for her own wedding in a few weeks time. Edith had honoured the occasion by wearing her best purple dress, and a kind of toque on her head, a little like a tea-cosy but dignified for all that. How dear John would have loved to see her, thought Belinda sadly. Being in church often made her think of John and his tragic death in Prague. They said it was a golden city, but how much more golden must it be from its associations with him.\n\nShe began to study the white and silver leaflet, announcing the marriage of Olivia Mary Berridge with Edgar Bernard Amberley Donne. The young couple had not chosen _God moves in a mysterious way_ among the hymns, indeed, Belinda had hardly hoped for that. Their selection, except for a Middle English lyric to a setting by a relative of the bride's, was conventional. _Gracious Spirit, Holy Ghost_ and _The voice that breathed o'er Eden_ were very suitable. It was true that the latter was often sung to the tune of _Brief life is here our portion,_ but it did no harm to be reminded of that. Brief life _was_ here our portion... Belinda's eyes strayed to the rows of memorial tablets to past vicars. She could hardly bear to think that one day Henry's would be among them and that she might be there to see it.\n\nAt this moment there was a stir of excitement as the curate appeared with his best man, a stocky, red-haired young clergyman, a typical rowing man, as Harriet whispered to Belinda.\n\nShortly afterwards the bride entered on the arm of the Professor of Middle English, a tall, thin man, ill at ease in his formal clothes. Belinda wondered whether Henry and Agatha were remembering their own wedding day. She had heard that people did on these occasions unless they were in a position to look forward rather than back. Count Bianco might be looking forward, but more quietly and with less rapture than a younger man. Was there anybody in the church without some romantic thought? Possibly Miss Prior, sitting with her old mother, was more interested in the bride's dress which, Belinda noticed with approval, was not white but sapphire-blue velvet, the kind of thing that would 'come in' afterwards and could if necessary be dyed and worn for years. She really looked very nice, almost pretty and not as tall as she had seemed to be at the presentation.\n\n'If I'm ever married I shall certainly have a _fully_ choral ceremony,' said Harriet enthusiastically as they filed out of the church. 'Or is there a special one for those of riper years?'\n\nBelinda said nothing because she had been crying a little and could not trust herself to speak yet.\n\n'No, that's only baptism,' said Edith cheerfully. 'I believe Connie and the Bishop are having quite an elaborate affair. You will be getting your invitations soon. She's having a dress made at Marshall's\u2014embossed chenille velvet\u2014though goodness knows what use that will be in the tropics,' she snorted.\n\n'No, it doesn't seem as if it would be much use,' Belinda agreed. 'But of course he won't be Bishop of Mbawawa all his life. I suppose he may retire and write a book about his experiences. They often do, don't they?'\n\n'With cassock and surplice in Mbawawa-land,' retorted Edith. 'Yes, one knows the kind of thing only too well.'\n\n'What a beautiful day,' said a gentle voice behind them. 'It is quite like Naples.'\n\n'My dear Ricardo,' said Edith, looking round, 'where in Naples would you see such a very odd-looking crowd of people as are now coming out of this church?'\n\n'I don't see the new curate among them,' said Harriet, looking worried. 'It can't be the best man, can it? Nobody would call him Italian-looking. Ricardo, Italians don't ever have red hair, do they?'\n\nThe Count considered the question seriously. 'Red hair is certainly not unknown in Italy,' he pronounced at length, 'but I do not think you would say that it is characteristic of the Italians. In the north of course it is sometimes found, but,' he paused impressively, 'I do not remember that I ever saw a red-haired man in Naples.'\n\n'Oh, that's good,' said Harriet perfunctorily, 'it can't be him, then. But I did hope he was going to be at the wedding.'\n\n'I daresay there was some difficulty about the trains,' suggested Belinda. 'We shall probably find that he is at the reception.'\n\nThis was to be held at the parish hall. The curate had invited all the village, so that the vicarage drawing-room would hardly have contained them. The bride and bridegroom were in the doorway to shake hands with the guests as they filed in.\n\nBelinda murmured what she hoped was a suitable greeting and passed on but she heard Harriet laughing loudly with Mr. Donne, so that she wondered rather fearfully what her sister could have said. She found herself standing by Agatha and confronted by a tray of glasses of champagne which one of the hired waiters was holding in front of her.\n\n'I must just have a sip, to drink their health, you know,' she said apologetically, feeling Agatha's eyes on her.\n\n'Oh, certainly,' said Agatha. 'It is quite the thing.'\n\n'I'm not really very used to drinking champagne,' Belinda admitted.\n\n'Aren't you?' Agatha gave a little social laugh, which would normally have crushed Belinda and made her feel very gauche and inferior. But today she did not mind. She was almost glad to be able to see Agatha as her old self again. The socks not quite long enough in the foot, which the Bishop had so unkindly mentioned, had been worrying Belinda. She had suddenly seen Agatha as pathetic and the picture was disturbing. Now she knew that there could never be anything pathetic about Agatha. Poised and well-dressed, used to drinking champagne, the daughter of a bishop and the wife of an archdeacon\u2014that was Agatha Hoccleve. It was Belinda Bede who was the pathetic one and it was so much easier to bear the burden of one's own pathos than that of somebody else. Indeed, perhaps the very recognition of it in oneself meant that it didn't really exist. Belinda took a rather large sip of champagne and looked round the hall with renewed courage.\n\nMost of the guests were known to her, although some people from neighbouring villages had been invited. She noticed Father Plowman, near the food and with a well-filled glass. Lady Clara Boulding, putting a small _vol-au-vent_ whole into her mouth, and a group of Sunday School teachers, Miss Beard, Miss Smiley and Miss Jenner, standing in a corner by themselves and looking suspiciously into their glasses. Agatha had left her now and was greeting some of the guests, so Belinda moved over to where Miss Prior and her mother were standing.\n\n'Very nice, isn't it?' she said inadequately, nodding and smiling in their direction.\n\n'Oh, yes, it's quite nice,' said Mrs. Prior. 'It's nice to see everyone enjoying themselves. I like to see that.'\n\n'Mother was saying she wished there was a cup of tea,' said Miss Prior in a low voice, 'but we'll have one when we get home. You see, Miss Bede, we're not really used to drinking champagne. It's different for you of course.'\n\n'Well, I don't often have it,' Belinda admitted, feeling that she must stand midway between Agatha and the Priors in this matter, 'but of course we all want to drink the health of the bride and bridegroom in it.'\n\n'Yes, of course,' Miss Prior agreed. 'She looks very sweet, doesn't she? That _Vogue_ pattern makes up well in velvet, it was in the December book. Your dress has turned out quite nicely, too, Miss Bede. I had such trouble with the sleeves, you wouldn't believe it. I found I'd put them in the wrong way round!' Miss Prior laughed rather shrilly and took a gulp of champagne.\n\n'Are they the right way round now?' asked Belinda.\n\n'Oh, of course, Miss Bede. I took them out again. I couldn't let you wear a dress with the sleeves in the wrong way round.'\n\n'No,' said Belinda, feeling all the same that this kind of thing might very well happen to her. 'Mrs. Hoccleve's dress is very smart, isn't it?' she ventured, feeling that it was not at all the thing to discuss the guests' clothes with Miss Prior but being unable to resist the temptation. Perhaps, she thought, Miss Prior's profession will excuse me.\n\n'Smart, yes, that's what I would call it too,' said Miss Prior. 'But red's not her colour. The material's good, I can see that, but you take a look at the seams inside\u2014you won't find them finished off like mine are.'\n\n'No, I dare say not,' said Belinda, realizing that she would certainly never have the opportunity of examining the dress so minutely, but feeling, none the less, that Agatha's splendour was considerably diminished.\n\n'Oh, look, the Archdeacon's going to make a speech,' said Miss Prior. 'Pray silence for the Venerable Hoccleve,' she giggled.\n\nThe champagne was having a different effect on Belinda, who was now gazing very sentimentally at the Archdeacon, thinking how nice he looked and what a clever speech he was making, not at all obvious or vulgar as wedding speeches so often were. She had really no idea what he was talking about, but there were a great many quotations in it, including one from Spenser which really seemed to be quite appropriate, something about love being a celestial harmony of likely hearts\n\n_Which join together in sweet sympathy,_\n\n_To work each others' joy and true content...._\n\nHe was not saying anything about Burdens or sudden calls to the Mission Field.\n\nThe curate replied very nicely after the health had been drunk and was followed by the Professor of Middle English, who made an unintelligible but obviously clever little speech about _The Owl and the Nightingale,_ embellished with quotations from that poem. Agatha and Olivia were smiling knowledgeably at each other, and Belinda turned away to meet Harriet, who was moving towards her through the crowd. Her face radiated joy and happiness. How nice it is that Harriet is entering so whole-heartedly into their feelings, thought Belinda, for she had been so afraid that her sister might be made unhappy by the curate's marriage and departure.\n\n'The third from the left,' whispered Harriet eagerly.\n\nBelinda looked about her, rather puzzled. Then she saw what her sister meant, for in a corner she saw five curates, all young and all pale and thin, with the exception of one, who was tall and muscular and a former Rugby Blue, as she afterwards learned.\n\nThe third from the left. How convenient of the curates to arrange themselves so that Belinda could so easily pick out Harriet's choice. He was dark and rather Italian-looking, paler and more hollow-cheeked than the others. Now Belinda understood her sister's joy and suddenly she realized that she too was happier than she had been for a long time.\n\nFor now everything would be as it had been before those two disturbing characters Mr. Mold and Bishop Grote appeared in the village. In the future Belinda would continue to find such consolation as she needed in our greater English poets, when she was not gardening or making vests for the poor in Pimlico.\n\nHarriet would accept the attentions of Count Bianco and listen patiently and kindly to his regular proposals of marriage. Belinda did not go any further than this in her plans for the future: she could only be grateful that their lives were to be so little changed. It was true that the curate on whom Harriet had lavished so much care and affection was now a married man and lost to them, but another had come in his place, so like, that they would hardly realize the difference, except that he was rather Italian-looking and had had a nervous breakdown.\n\n_Then she fretted, ah, she fretted,_\n\n_But 'ere six months had gone past,_\n\n_She had got another poodle dog_\n\n_Exactly like the last...._\n\nthought Belinda frivolously, but the old song had come into her head and seemed appropriate. Some tame gazelle or some gentle dove or even a poodle dog\u2014something to _love,_ that was the point.\n\n'I think I'll ask him after the reception, although it's rather soon. But we do want to make him welcome _at once,_ don't we?' Harriet was speaking eagerly to her sister.\n\nBelinda smiled. 'Of course, dear.' Asking the new curate to supper seemed a particularly happy thought.\n\n'I knew you would agree,' said Harriet, making boldly for the curates' corner.\n\nBelinda was looking round the room to see if she could find some sympathetic person to whom she could say that Dr. Johnson had been so right when he had said that all change is of itself an evil, when she saw Harriet approaching with the new curate.\n\nShe smiled and shook hands with him, but before either of them could utter a suitable platitude, Harriet had burst in with the news that the young man was coming to supper with them on his first Sunday evening in the village, which would be in about a fortnight's time.\n\n'He says he is fond of boiled chicken,' she added.\n\nBelinda laughed awkwardly and hoped that the new curate would not be embarrassed by Harriet's behaviour.\n\nBut he seemed completely at ease as Harriet confided to him that she always liked to eat chicken bones in her fingers.\n\n'Like dear Queen Victoria used to,' she sighed.\n\n'Now, I'm sure _you_ don't remember Queen Victoria,' he said gallantly.\n\n'We older people remember a great deal more than you think,' said Harriet coyly.\n\n'Oh, _come,_ now,' laughed the curate, and although his voice was rather weak as a result of his long illness, Belinda was overjoyed to hear that it had the authentic ring.\nAll rights reserved, including without limitation the right to reproduce this book or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.\n\nCopyright \u00a9 1950, 1983 Barbara Pym\n\nCover design by Jason Gabbert\n\n978-1-4532-7963-2\n\nThis edition published in 2013 by Open Road Integrated Media, Inc.\n\n180 Maiden Lane\n\nNew York, NY 10038\n\nwww.openroadmedia.com\n\n# **BARBARA PYM**\n\n**FROM OPEN ROAD MEDIA**\n\n**Find a full list of our authors and**\n\n**titles atwww.openroadmedia.com**\n\n**FOLLOW US:**\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n#    \nJOSEPH HELLER\n\n# Now and Then\n\n_From Coney Island to Here_\n\nJoseph Heller was born and raised in Brooklyn.  \nHe lives in East Hampton, New York.\n\n**_Also by_ JOSEPH HELLER**\n\n_Catch-22_\n\n_We Bombed in New Haven_ (play)\n\n_Something Happened_\n\n_Good as Gold_\n\n_No Laughing Matter_  \n(with Speed Vogel)\n\n_God Knows_\n\n_Picture This_\n\n_Closing Time_\n\nVINTAGE BOOKS EDITION\n\n_Copyright \u00a9 1998 by Skimton, Inc._\n\nAll rights reserved under International and Pan-American Copyright Conventions. Published in the United States by Vintage Books, a division of Random House, Inc., New York, and simultaneously in Canada by Random House of Canada Limited, Toronto. Originally published in hardcover in the United States by Alfred A. Knopf, Inc., New York, in 1998.\n\nThe Library of Congress has cataloged the Knopf edition as follows:  \nHeller, Joseph.  \nNow and then: from Coney Island to here \/ by Joseph Heller.  \np. cm.  \n1. Heller, Joseph\u2014Biography. 2. Novelists, American\u201420th century\u2014  \nBiography.  \nI. Title.  \nPS3558.E476Z468 1998  \n813\u203254\u2014dc21 97-49658\n\neISBN: 978-0-307-76696-0\n\n_Author photograph: Jerry Bauer_  \n _Title page photograph courtesy of Carol Schops_\n\nwww.randomhouse.com\n\nv3.1\n_For my sister, Sylvia_\n\n# Contents\n\n_Cover_\n\n_About the Author_\n\n_Other Books by This Author_\n\n_Title page_\n\n_Copyright_\n\n_Dedication_\n\n_Acknowledgments_\n\nChapter 1 - The Gold Ring\n\nChapter 2 - Coney\n\nChapter 3 - Sea Gate\n\nChapter 4 - Work\n\nChapter 5 - On and On\n\nChapter 6 - And On and On\n\nChapter 7 - And On and On and On\n\nChapter 8 - Peace\n\nChapter 9 - Psychiatry\n\nChapter 10 - Danny the Bull\n\n# ACKNOWLEDGMENTS\n\n_The author would like to thank Donald Kaplan,  \nJudy Walsh and the staff of the Brooklyn Public Library,  \nand the entire staff of the East Hampton Public Library._\n\n# 1  \nThe Gold Ring\n\nTHE GOLD RING on the carousels was made of brass. Even as kids in Coney Island we didn't believe it was the real thing. By the time we'd grown old enough to ride the outside horses and lunge out sideways to grasp the metal rings that swung toward us for the final few rotations, the carousel was no longer enchanting and we had no deep desire for the free ride that the last, lucky gold one awarded. By then we had nickels enough to go around again if we wanted to, but we tended to spend them on attractions that were higher and faster, more spectacular\u2014roller coasters\u2014and, for fun, the electric bump-cars.\n\nWe were luckier with the staying power of our craving for things like pretzels, potato chips, jelly doughnuts, and chocolate bars. Mark Twain is said to have remarked that by the time we're tall enough to reach the jar of jam on the high shelf of the cupboard, we find that we've lost our taste for jam. No such rueful fate struck me or my friends or any in the small family of four of which I was the youngest, not with edibles like halvah or salted peanuts, ice cream, kosher corned beef and hot dogs, or even salami sandwiches. When we found ourselves with enough cash to obtain as much of these delicacies as we wanted, we still had a hearty appetite for them, and we tended to indulge ourselves, and still do, by eating as much as, sometimes more, much more, than we truly did, and do, want.\n\nOf late, my best defense against corpulence has been to keep out of the house supplies of things to eat that reason cautions I shouldn't be stuffing into myself. Pistachio nuts, for instance, whether in petite jars or five-pound bags, have a feeble chance of extended survival once I discover them close at hand. If there's ice cream in the freezer, I feel a commanding moral responsibility to move it out of the house as quickly as I'm able to swallow as much as is there. Lately, I've discovered that salty pretzels go very well with just about any dessert I am likely to have at home. They're also good by themselves. If, before going to bed, I happen to remember we have sliced turkey breast in the refrigerator, the odds are heavy that I'll put some in my mouth as I find my way into the bathroom to brush my teeth\u2014on a couple of crackers, of course, or half a flat of pita bread, with salt and mustard.\n\nBut it's ice cream that still tastes most wonderful and is richest in evocative associations extending backward in time almost to the formation of memory itself, to brands and items long extinct\u2014to Dixie cups, for example, with their prized photographs of cowboy movie stars on the inside of the lid under a transparent waxed-paper seal. Like the evocations of the cookie to Proust, a meditation on ice cream soon takes me back to the age of eight or nine and into a family setting in which a small container is shared with bliss by the four of us, a mother, a sister, a brother, and me. (I was by many years the youngest.) In summer, ice cream was everywhere. In autumn, though, after the change back from daylight saving time, and even in winter on a black night after dinner and before bedtime, the idea of ice cream might be voiced, taking on a sacramental meaning to our small family in our small apartment\u2014four rooms, looking out on West 31st Street near Surf Avenue in Coney Island. My mother\u2014finally relaxing in front of the radio with the rest of us, after shopping for dinner, preparing dinner, serving dinner, and cleaning up in the kitchen after dinner\u2014might say with her Jewish intonations that she would certainly very much welcome the taste of a little ice cream. We had no refrigerator then, no freezer\u2014no family living in our apartment building did\u2014and there would be no ice cream in the house. At that late hour only the soda fountain in a drugstore two blocks away was open. I was the one who would volunteer to go. I would be given a dime to bring back a container. The flavor of our unanimous choice in those years was called Golden Glow. It's hard today to believe that just half a pint of bulk ice cream could have been so satisfying to the four of us, but that's all, as I recall, that a dime paid for. More than a dime for ice cream they couldn't bring themselves to spend. We were prudent with money because we didn't have much, but I, the \"baby\" in the family, was never allowed to feel that.\n\nMY SISTER, SYLVIA, was seven years older than I. My brother, Lee, originally Eli, born in Russia and brought to this country at the age of six, was seven years older than she was. In reality, they were only my half brother and half sister, the children of my father and his first wife, who had died. My mother was therefore a stepmother to them. They were, I realize only now, technically orphans, and although they never said so, they must have felt at least a little like orphans. I was, then, the baby in the family, treated by everyone, in effect, like an only child, which in some ways I was.\n\nI had no inkling of these family relationships until I was well into my teens, and I was shocked nearly speechless when confronted by the discovery, which unfolded at my brother's wedding. There, my mother's role in the procession down the aisle was to walk behind him, alone, and I listened dumbfounded to the officiating rabbi praise her so generously for the loving care she had given the groom, the son who was not biologically her own, and the daughter, too. I felt victimized, disgraced. My response to rage then, as it chiefly is still, was to break off speaking to the person offending me. I stopped speaking to my sister one time when she took up cigarettes and another time when she bleached her hair. And this time I may have fallen silent with all three of them, possibly with so deep and vindictive a hurt that I would obstinately refuse to tell any of them the cause, for that would be speaking. My oldest friend in the world, Marvin Winkler, with whom as an infant I had often been lodged together in a playpen, was amazed not long ago when I recalled the incident and related my stunned reaction. He was puzzled by my surprise, for he had been informed of the relationship by his mother when he was still a child and cautioned against hurting my feelings by bringing it up. My sister, too, was taken aback to read my account of this event in a biographical piece about me for which she had also been interviewed. She, along with my mother and my brother, had simply assumed, she claimed, that I'd known all along. It was not a scandal, not even a secret. They didn't talk about it because there was no need to.\n\nOn the other hand, I know it is true that neither my brother nor my sister ever said anything at all to me about my father, least of all about an earlier marriage, until out of curiosity as an adult I began to inquire. And only once did my mother talk to me about him, volunteering the information that my father could eat a whole chocolate cake at a single sitting\u2014as a delivery driver for a wholesale bakery firm, cakes were easy for him to obtain\u2014and that before he went into the hospital with a bleeding gastric ulcer, his stool was as black as coal. She told me this as an admonition against my own appetite, for I could always eat as much cake as was given me, and still want more. I discovered early in life that shelled walnuts in a bowl of raisins make a lovely late-evening treat for uninterrupted munching till bedtime. I once overheard my mother recounting that when I was a nursing infant, she had to tear me away from the breast, for I would never finish, and I can believe her.\n\nDIFFERENT AS EACH OF US was from the others, and however much there was inside us that we didn't want to talk about, we were, and functioned as, a close family. It was not in our nature to complain, quarrel, demand, or gossip. I remember a number of tantrums I discharged as a juvenile, but I believe that from the time I was an adolescent, I had acquired the family's stoic, resigned, and undemanding nature and didn't complain, quarrel, or gossip, either. My mother, finding herself after six years of marriage a widow with three children, two of them not naturally her own, had brought us up as the mother of us all, and my brother and sister had related to her as such. My brother, I learned, had at one time saved up money enough to buy her a better radio. The gift was poignantly apt, for my mother loved melodic music, Puccini arias in particular. And in those days of radio entertainment, amid the glorious profusion of comedy and variety shows, there was an opulence of weekly broadcasts featuring short selections from opera and other light classical pieces.\n\nAlthough the period was one of severe economic depression, everybody sooner or later seemed able to find employment. By the time I started school, my brother, fourteen years older than I, was already working at a brokerage in Wall Street as a customer's man, a job for which he was to prove ill suited by his modest manner and unaggressive temperament, and by a disposition to be continually obliging to others. My sister, after graduating from high school and encountering a number of startling rebuffs at employment agencies, would eventually find herself starting at R. H. Macy's department store, where she would remain for something like forty years. And I at sixteen, after classes at high school, would be cutting a dashing figure in a khaki quasi-military uniform as a part-time messenger boy delivering telegrams for Western Union, in office buildings in Manhattan weekdays, on a bicycle weekends to residences in Brooklyn, and mostly exulting in my duties with a sense of adventure and attentive curiosity. My mother, who had been a seamstress and something of a dressmaker before her marriage, worked steadily at home at her sewing machine mending garments for neighbors and doing alterations for a cousin, Sadie Pacon, who owned a dress shop on Gravesend Neck Road in lower Flatbush nearby. She also found steady work at home from laundries, turning frayed collars on men's shirts so that, from the outside and at the neck at least, the shirts looked almost good as new. Some weekend nights, mainly Saturdays, my brother would find extra income filling in at a catering hall for some banquet, dressing up appropriately for his suave role in the cloakroom or as one of the hall's hosts greeting and directing arriving guests. Neighbors outside the apartment building seeing him depart in a tuxedo or a natty, double-breasted, summer sport jacket might remark that he was off on a heavy date. But I knew he was going to work.\n\nMy mother liked to read. In Europe, her family had been bookbinders. From the Coney Island public library I would bring her Yiddish translations of novels. She enjoyed Tolstoi, especially _Anna Karenina_ , with which she had long been familiar, and thought Dostoyevsky was crazy. Her brother (my Uncle Sam) late in life was employed repairing books for the library of Brandeis University, where his son, Harry Stein, a varsity player in James Madison High School in Brooklyn and City College in New York, was one of the athletic coaches.\n\nWhen my mother noticed an apple turning soft she would briskly plan a noodle pudding that made use of it rather than have it go to waste. She took threadbare bed-sheets to her scissors and her sewing machine and converted them into window curtains. (Often, as her eyesight weakened with age, I would help with the sewing by threading her needle. Today I would not be able to.) If brother and sister had quarreled, she wouldn't let them go to bed until they had talked it out and made peace. She was never more happy than when a friend from the old country appeared unexpectedly. The visit to Coney Island of a Mrs. Rosen filled her with joy.\n\nMy sister's clothing as a teenager in high school, she recalls, came to her mainly as hand-me-downs from an older first cousin, who, because she was already at work in a business with her father, favored dresses of dark color, which my mother altered to fit. She also remembers feeling endlessly self-conscious, because she wore dresses rather than the skirts with blouses or sweaters fashionable among her schoolmates. Widowed twice, she is esteemed by a stepdaughter from her first marriage and by the three children from her second, all of them grown now.\n\nFor a little while, far back before the age of sixteen, out of eagerness more than need, I hawked newspapers in the early evening, peddling the next morning's editions the night before, crying, _\" 'Merican, News_ , and _Mirror_ , morning pape'!\" The _American_ , a Hearst newspaper more reactionary than his _Mirror_ , was thicker than the others, cost more, and had few readers in the Italian and Jewish neighborhoods I traversed in my desperate hurry to keep ahead of other newsboys who were trying desperately to keep ahead of me. I soon dropped that heavier paper from my inventory, and my sales cry became \"Hey, get your morning _News_ and _Mirror_ , morning pape'!\" At best this was very small stuff, and I was lucky if I earned as much as a dollar. I bought the papers for a penny and a half, I think I remember, and sold them for two cents each, hoping for an occasional tip of a penny or two. People who wanted both might give me a nickel. I bought the papers from trucks near the subway station at Stillwell Avenue, usually eating a frankfurter while I waited for them to arrive, and wended my way home on a route along the boardwalk and the populous avenues of Surf and Mermaid in the hope that my last batch would be entirely gone by the time I got there. If not, a humorous, audacious plea to neighbors sitting outside their houses might do the trick. \"Extra, Extra! Hitler dies... his mustache!\" was one shouted ploy to command immediate attention. Any papers that remained I might give away to mothers and fathers there who were my favorites.\n\nA wondrous incident remains indelibly alive\u2014it occurred only one time. At a boardwalk Irish bar one night, a thin man sitting alone at a table beckoned to me and reached out for one of my papers. As I stood there and waited in suspense, he opened it from the back, studying the racing results a few moments, and then returned it. And then he gave me a dime and wanted no newspaper. I was in heaven, strolling on air as I went back outside. I was in love with a world that had such humans in it.\n\nFrom repeated personal experience I've learned that few pleasures are so thoroughly reinforcing to the spirit as the arrival of unexpected money. Not long ago, I was amazed to receive in the mail, with no advance word, a royalty check from my literary agent for almost $18,000, representing cumulative royalties over the years from the publication of one of my novels in East Germany. I had long since forgotten that I'd even had a novel published there, and for the next hour or so I found myself grinning and humming a happy show tune or two. But I doubt very much that the sensation of well-being I experienced from this relatively huge windfall surpassed the joy brought me by that ten-cent gift I once received as a newsboy. It feels now as though that dime meant enormously more to me, and perhaps it did.\n\nEach of these benign surprises affecting me was a rapture someone born rich is not likely ever to enjoy. There are some pleasures money can't buy.\n\nOUR STREET AND neighborhood felt safe, insular, and secure, to a child an ethnic stronghold. Just about all the parents were immigrants, the majority from Eastern Europe. There was nothing and no one to fear physically.\n\nI did not really know who Hitler was. Our elders did, but I doubt that even they, or anyone, could have guessed at the evil immensity of the brutal destruction he was going to set loose. I remember a day: It is late afternoon, and I can hear again the newspaper vendors tearing through our street with their high-pitched cries and their \"extra\" editions of the papers with glaring headlines announcing that Hindenburg had resigned as president of Germany to make way for Hitler as chancellor. They knew they would sell every copy that day, whereas I didn't know who Hindenburg was. Until much later, I thought he was a zeppelin.\n\nThere was just about no fear of violence in that part of Coney Island where I lived and grew up. And there was practically no crime, unless one considers the peddling of ice cream or soft drinks on the beach a criminal activity\u2014the police did, but we didn't\u2014or later on, approaching the years of the war and to a broader extent afterward, the possession and smoking of marijuana. I didn't consider using marijuana a crime, either, although I never wanted to become \"schooled\" to be what was called a \"viper.\" I doubt that my brother or sister even knew about the increasing prevalence of this activity, and my mother would have been of a disapproving mind toward anything thought wrong. Then after the war came heroin.\n\nIn the nineteen years I lived on that street before going into the army\u2014except for two months away as a blacksmith's helper in a southern navy yard, I lived nowhere else\u2014I never heard of a rape, an assault, or an armed robbery in our neighborhood. I do recall having heard of a bookmaker shot to death in a poolroom, but that was on West 25th Street, six blocks away, on the very border between the Jewish and Italian sections. It wasn't really in our neighborhood, although we were eventually using that spot as a meeting place and the most convenient site for placing our bets on professional baseball games. It was run by a short, squat man called Sammy the Pig. The poolroom nearer my home was owned and run by a man we knew as Weepy. Nicknames were legion in Coney Island. _Foofson_ is the Yiddish word for the number fifteen. Sammy Foofsen, an older youth, was already Sammy Fifteen by the time I heard of him. A little while later, even his nickname acquired a nickname: He was referred to as Six-Six-Three. These mutations in title appeared to arise and take hold quite naturally, as though by spontaneous generation; no one person originated them. A guy called Chicago was Chi, then Shy. Mursh the Cripple had a leg in a clumsy metal brace from polio. In time, Mursh the Cripple matured into Mursh the Hopper. He was older, too\u2014one of the first guys we knew in Coney Island to own a car; it was an automobile with just hand controls. And a prank of his was to sit in the space beside the driver's seat, staring straight ahead with a placid smile and, with no person behind the wheel, operate the car with manual movements below the dashboard that weren't visible to disbelieving other drivers or pedestrians, who, startled, usually looked twice. My friend Marvin Winkler was Beansy for some reason, which seems almost preferable to Marvin. Murray Rabinowitz was Rup since childhood, in honor of his early surgery for a hernia, known then as a \"rupture.\" Danny Rosoff was Danny the Count. Squeezy and Frenchy were two girls\u2014I don't know how they came to merit such racy monikers. Danny the Bull, another fellow my age, was given that name before the war because of his broad chest and muscular shoulders. After the war, he was soon rendered frail and woeful by an addiction to heroin. His father, a barber named Max, gave him a cruel new title, derogating him to customers in his shop as My Son the Junkie.\n\nTHE LONE ACT of violence affecting me was one I unconsciously took pains not to find out much about until I was well into my thirties, and that was the death of my father in a hospital\u2014from internal bleeding after surgery for a condition that today I imagine would likely be considered almost routine, his bleeding ulcer. I prefer to believe that these days, with such protective facilities as intensive-care units and surgical recovery rooms standard, the falling open of his stitches would have been detected in time and that he would have survived.\n\nI was just past five. I knew he was gone\u2014I don't remember that I was ever told, I just knew\u2014but I had no notion of the cause. For thirty years I didn't care to ask. I didn't dare? It wasn't a secret. I didn't know, I eventually recognized in one startling moment\u2014and it came to me in an illuminating burst of understanding\u2014because I had never wanted to ask; they didn't tell me, I guess, because with the passing of years they simply assumed I knew. It was not a gratifying subject for family discussion. The impact upon my brother and sister\u2014losing both parents in so short a time and finding themselves in the care of a stepmother they had known only six years\u2014would have been enormous. But not only did we not complain much in my family, we didn't talk much about anything deeply felt. We didn't ask for much, either. Only one time ever was I present when one of us shed a tear, at the funeral of a spouse. And that's about all it was, a tear.\n\nAbout my father, I simply lost interest in him after he was gone. Later on\u2014outside the family, of course\u2014it became a conceit of personality to assert impertinently when the subject arose that I never missed him, and I don't believe I did. I didn't have that tender admiration for a large and generous nature that Nabokov expresses for his father. I hardly knew mine at all. If anything, the passing away of Mr. Isaac Daniel Heller was for me more a matter of embarrassment than anything else. At the beginning of each term in elementary school, it was the procedure of the teacher to call the children one by one to the desk in front of the classroom to obtain information that included the father's name and perhaps his occupation. Someone, teacher or sister, taught me the word _deceased._ I said it always in a repentant murmur brought low by a burden of shame and humiliation, and I hoped none of my classmates would hear.\n\nI did not feel I missed him. In later years, I might even wisecrack irreverently when hearing someone talk of an appalling home life caused by a father that I was fortunate mine was gone before he could inflict much damage. But I was biting my fingernails at the age of seven. And except for two hospital confinements very much later, during which my anxieties were focused on inescapable concerns, I have gone on biting them and I still do. In my late teens and afterward, I was suffering headaches before I understood what headaches were. Earlier, when someone spoke of a headache, I had no grasp of what was meant; once I did get the idea, I realized that I had been having them all along, usually on weekend mornings and afternoons. They've stopped. I studied my face in the mirror excessively. I had overheard that I was a good-looking child, mostly from repeated comments by my sister's girlfriends, and I wanted to reassure myself that I had not changed for the worse. I suffered boils more often than what was considered normal and had a school year of warts: There were seventeen on my hands and arms, including one large flat one on an elbow that didn't disintegrate with the others and had to be burned off by Dr. Abe Levine (who charged two dollars for the office visit and the treatment). I had a crazy fear I was going bald, because the hairline at my forehead was so much higher than at my ears. I did what I conceived I could to \"train\" my hair, and I moved the part from one side to the other, and then to the center and then off again to the side, and then tried it without a part, plastered flat with a stiffening hair tonic, and then with waves. I did other little things I still wouldn't want to tell anybody. I measured my height by the level of my eyes and therefore believed dismally that I was half a head shorter than I indeed was\u2014that I was much shorter than boys who were shorter than I. It was vanity, of course, but I'm inclined now to believe that it was a very anxious vanity founded on a wobbly formation of what R. D. Laing might have called \"ontological insecurity.\"\n\nI was prone to fantasizing, daydreaming\u2014and to good effect: I brought my rather overworking imagination into the classroom in book reports and compositions, and in this schoolwork I excelled. I can think of these examples: For a book report on _Tom Sawyer_ in one of the earliest grades I donned the mantle of Tom Sawyer tasked with the assignment of writing a book report on the work in which he is the principal personality; in the seventh grade, when I was about twelve, for an autobiographical written assignment supposedly by a subject in history, I became, in a first-person account\u2014not Abraham Lincoln, which would have been ordinary, others would have chosen him\u2014but the _metal_ in the gun that was used to shoot him. I was born, I remember, in a mine in Chile, in a shovelful of iron ore. Papers like these were read aloud as outstanding, which was the reward I aimed for. I very strongly did want to excel and be noticed.\n\nAnd one time I used some savings to surprise my brother with a gift for Father's Day. I bought him a carton of cigarettes, Camels, and it cost a dollar then. And I still am unable to decide, aware now of unconscious motivation, whether the deed was one of sincere gratitude and affection or merely a stunt contrived to excite comment and win me some complimentary attention. Or both. Another time, nearing the age of sixty, while experimenting with psychoanalysis as a remedy for alleviating stressful moods in the throes of a rancorous divorce, I had a coherent dream in which my brother, Lee, who then was living and in good health, was dead and resurrected from the grave to rejoin the family in the old apartment house in Coney Island, where my mother, who had long since died, was alive and active. On my back on the analyst's couch, I protested astutely that I probably wouldn't have had that dream of my father-as-brother the night before were I not in this treatment and if I hadn't known I would be coming to see him that morning. And he, chuckling in his upholstered armchair, didn't disagree, and conceded that the psychoanalytical process was indeed to a large extent \"neurogenic,\" a word I had not heard before and have not heard since.\n\nNOT LONG AFTER my father died, we moved from our old apartment into one of equal size directly across the street, and that is the one I think of as the home in which I grew from childhood. It had four rooms. All of the apartments seemed to have four rooms, one a full kitchen, one a living room that routinely also functioned as somebody's sleeping place. In spring, with us as with others, the ROOM FOR RENT sign went up, the notice went out to neighbors, and we took in a boarder for the summer, usually a single adult or an adult couple with friends or relatives nearby. None of us, not my brother or sister or I, could recall with certainty how we disposed of ourselves for the period in which we gave up that room. My sister does remember a situation in our first apartment with two parents and a child boarding in one room, while one of us slept in the kitchen. She can't remember which one of us it was.\n\nIt was to one of these boarders that I owe my first exposure to classical music. He politely requested permission to tune in to station WNYC on our radio when it wasn't in use (in use by me, usually, listening\u2014to the despair of my mother\u2014to big-band swing). WNYC was a classical music station. I didn't have to listen when he played the music he preferred, but I did have to hear. And I was overwhelmed when I recognized in these classical performances so many melodies already familiar to me as popular songs from _Your Hit Parade_ and other radio programs: \"Our Love,\" \"I'm Always Chasing Rainbows,\" \"You Are [perhaps \"This Is\"] My Song of Love,\" \"The Lamp Is Low,\" and a host, more than one host, of others (\"The Isle of May,\" \"April Showers,\" \"No Star Is Lost in My Blue Heaven,\" Tchaikovsky again, and what appeared to be a genuine Negro spiritual and was, in fact, astonishingly, the largo from Dvo\u0159\u00e1k's _New World_ symphony, \"Going Home\"). And it was while I was listening alertly to discover more of these that other strains I hadn't heard before were played and repeated and became recognizable and appreciated.\n\nWhen families moved from one flat in Coney Island to another, it was mainly to take advantage of the usual \"concession\"\u2014a rent-free first month or two given to the new tenant. When they moved away from Coney Island into areas of loftier economic standing, it was because they could at last afford to. The clearest memory I preserve of our earlier apartment was of my brother, after all our belongings had been transferred, going back with me and bringing a broom to sweep out, still one more time, the corners of all the rooms. He didn't want the new tenants, he explained, to think that the ones preceding them, us, were slobs. He would retain this conscientious and considerate trait, this regard for comportment, till the end of his life.\n\nIn the new apartment, after being put to bed, I would frequently and furtively imagine, or hope, that a policeman was stationed at each end of our segment of the dark block, keeping vigilant watch on my building and window to safeguard me from\u2014I cannot imagine what. Except for the threat of true poverty, against which I was protectively kept ignorant, there were no dangers. Both inside and outside the house we were safe. There were no kidnappings or burglaries, and always in decent weather there were scores of kids on the street to play with, and grown-ups to watch and keep an eye on us. From the earliest grades on, we were encouraged, boys and girls, to walk on our own to the grade school close to half a mile away, crossing streets. We walked home for lunch at noon, walked back to school for the afternoon session, returned on foot when classes ended at three, without fear, without harm. Except in summer, there wasn't much danger from automobiles; this was a low-income neighborhood during the Great Depression, and there weren't many automobiles. From about the age of nine or ten, we could play in the street at night or roam the boardwalk until we sagged with sleepiness or someone came to fetch us.\n\nThe two apartments I spoke of were in buildings on a side street that ran between Surf Avenue, the thoroughfare that was closest to the boardwalk and seashore, and, receding from Surf Avenue in a direction opposite the beach, the tracks of a trolley line running in a straight sand-and-stony trench we called Railroad Avenue. Once across these tracks on my walk to school, I might be joined by my friend Albie Covelman, a classmate from the second half of first grade on, or, a few years later, by Seymour Ostrow, who had moved to our block from Sea Gate, a sequestered, private settlement a few streets past the other side of the school, after his own father died. At Mermaid Avenue, we would join a thickening pedestrian flow of other friends and acquaintances from school who were coming from homes a few streets past my own. At the school, waiting on line as instructed or romping about the spacious yard, there would be many more boys and girls from kindergarten classes through eighth grade. They had come from all directions. I must note that I remember no blacks in my elementary school, and none in my high school, not even on the football or basketball teams. The high school, several miles away, drew its students from a number of surrounding areas of diverse populations, but not yet so diverse as to include \"Negroes.\" I recall no discussions of the subject before World War II.\n\nThe streets in Coney Island were divided into three residential segments, sometimes four, by Surf Avenue, by the trolley line, and by Mermaid Avenue, which had the most stores. Some of the streets had apartment houses between the beach and Surf Avenue, others didn't. On one whole block of prime property between the boardwalk and Surf Avenue and open only in summertime was a Catholic orphanage for boys, who came, I think, for short stays. With fair hair and milky, freckled skin they looked sickly to us when we passed and stared; and we, with our suntans and dark complexions, must have seemed peculiar and exotic to them. The avenues paralleled each other on linear courses from the high fences and guarded entrances of Sea Gate at one end of Coney Island to the final station of the BMT subway train routes in the amusement section in the middle, and each of the three segments between them was thickly populated, each block of each street a close-knit miniature village in its own right. Surf Avenue, the largest of them, ran past the train stop into and through the amusement area, past the games and rides and other sidewalk attractions; a Surf Avenue trolley line that began outside Sea Gate ran well past Coney Island and came to its last stop at Sheepshead Bay.\n\nDespite its bustle, solid population, and long eminence, Coney Island is not impressive in size and not really an island but part of a Brooklyn sand spit hardly more than half a mile wide, with the ocean on one side and a bay, Gravesend Bay, bordering the other. In that small span of street in which we lived, surely no more than two hundred yards between Surf Avenue and the Norton's Point trolley tracks, there were four apartment houses of three or four stories each and a sprinkling of smaller dwellings for one or two families, and there were always enough other boys of the right age to play our street games with, the older boys training us to catch and throw a ball when they needed more players to complete a team. Girls, too. My sister, near eighty now and living in Florida, is still friendly with several women she met on the block in those days. And there were always parents, mostly smiling mothers, happily overseeing us from the windowsills of their apartments as we played ball or tag or roller-skated. When weather and domestic chores permitted, they sat outdoors on chairs carried downstairs and placed alongside each other in front of the apartment houses on both sides of the street, conversing with each other in Yiddish, with us in English, or in a patois of both.\n\nRegularly on pleasant afternoons as they sat there, a Mrs. Shatzkin who lived near us would trudge by, carrying in a sack of black oilcloth a large covered kettle holding homemade potato knishes\u2014thick, circular, good-sized dumplings of mashed potatoes richly seasoned with pepper and browned onions in pockets of a special dough of her own invention which were deep-fried in vegetable oil\u2014freshly made and still warm, and from which she strived to eke out her living. She made another kind, too, with a filling of a buckwheat grain called kasha. Both kinds were unusually good, and sold for a nickel apiece. My mother was among those who relished them, and would call me to her to share one with me. A Mrs. Gelber, who lived around the corner on Surf Avenue, made in her kitchen what we called \"jelly apples\" and carried them around to sell. They cost a nickel, too. Soon she was making them in quantity to supply food stands in the amusement area. And soon Mrs. Shatzkin had leased a large storefront on the corner, with a staff of middle-aged women\u2014probably relatives, because they all resembled her\u2014making her knishes in full view and selling them to customers on the street; then she expanded into a much larger outlet on the more crowded boardwalk and supplied food stands in other locations, too. At least four generations of Shatzkins, all with the same florid faces, stippled with large brown and gray freckles, earned their keep from the knishes originated by the maternal founder of that thrifty dynasty.\n\nOther women came by frequently with trays of sesame candies and the like, which sold for two pennies apiece. Some early evenings in spring, summer, and fall, a hearty black woman would appear in the street and in the courtyards of the summer bungalow colonies\u2014a street singer singing Yiddish songs with accurate Yiddish pronunciation. If we were upstairs, my mother invariably would wrap a few pennies in a scrap of newspaper and toss them down from the window, as she did with others who arrived to serenade us with accordions or violins. My mother had the same keen appreciation for food that I am blessed with, though not nearly my capacity. She was particularly fond of certain salty, smoky flavors. Toward the final years of her life she surreptitiously came upon bacon and instantly found the taste delicious. Who wouldn't? She adored smoked whitefish, too.\n\nSo do I. I look like her, though I'm taller. High forehead that is broad and flat, large face with prominent cheekbones, a determined, sometimes surly mouth. Not long ago, my friend Marvin, who lives a distance away, exclaimed on the resemblance after we hadn't seen each other for a couple of years. My sister and brother grew more and more to look like each other as they aged. Their wide eyes were a clear, bright blue. Mine, like my mother's, are brown.\n\nALL OF US in Coney Island did whatever we legally could to earn what money we needed, and somehow everyone I know of managed to fare pretty well. The fathers all worked, as did the older brothers and sisters as soon as they had liberated themselves from high school by graduating. Almost none of us then thought seriously about going to college or wanted to go, or could have afforded to. (My brother did want to go, would have given anything to have been able to, my sister, Sylvia, tells me.) For me, years before the time for decision, Lee was sending away for catalogs and applications to colleges such as MIT, Oberlin, Harvard, Yale, and others of that elevated level for which I could not possibly have paid and to which I could not then conceivably have gained admission. At New York University after the war, when I was selected in my junior year for membership in Phi Beta Kappa\u2014as much for success with the publication of a few short stories as for classroom distinction\u2014I didn't know what Phi Beta Kappa was. But Lee did and was flushed with pleasure in the pride he took in me. \"Some Phi Beta Kappa,\" was his happy comment on that ignorance of mine. But for him it was already too late for college, and I couldn't be sure whether a longing for a university education was another painful disappointment he suffered or merely a whimsical reverie in which he indulged. He owned a tennis racquet, which he kept in a wooden press in a closet and which I never saw him use. I don't know where that came from, if not from a few summers when he was away as a camp counselor. He had a Red Cross badge for lifesaving which he earned in those days. At times as he aged he could be arbitrary, critical, and argumentative over very small matters\u2014what bread was best for what kind of sandwich. But he was instinctively obliging and well-mannered, especially with strangers.\n\nFor a while, a very short while, when I was just eighteen, in a concession to respectable conformity I did accede to the plan of applying for admission to night school at Brooklyn College. My high-school transcript was better than average, and I was accepted. However, I decided impulsively one evening before term began, while sleepily registering for classes, that I much preferred my nighttime social life, unremarkable as that was, and I dropped out before attending. When the time arrived that I was able to go, felt realistically that I _could_ go, to college, I wanted to and did. I was twenty-two when I started, and I enrolled like a million or two other recent American war veterans, with the federal government paying a monthly subsistence allowance and just about all my educational costs.\n\nMost of the fathers I knew of had salaried jobs in Manhattan in what was called \"the garment center,\" performing one specialized function or another in the sequence essential to the manufacture of clothing. None advanced to salesmen. But a number of friends in the expanding circle I acquired as I grew up came from families with small businesses of their own right there in Coney Island, mostly stores: Louie Kessler, Solly Mirror, and Lenny Karafiol owned bakeries. Esther Dessick's family owned the fish store\u2014none of these retail shops on Mermaid Avenue filled premises roomy enough to be called markets. Lily Dashevsky came from fruits and vegetables, Murray Singer's father was the butcher. Louie Berkman's father owned a lucrative junkyard. Marvin Winkler's father prospered as a bookmaker and eventually was able to move his family into the richer milieu of Sea Gate. In the flat just below our apartment lived Irving Kaiser, a year younger than I but a friend from early childhood who was one of the three people I grew up with who were killed in the war, and just below him on street level was the tailor shop owned by his father. And the father of Danny the Bull was a barber.\n\nLOOKING BACK, I find it something of a miracle that from such a beginning the four of us in my family separately and independently eventually found ourselves through most of our lives with enough money to satisfy our needs and our material wants. Our expectations, while varying considerably, were disciplined. We did not want what we could not hope to have, and we were not made bitter or envious by knowing of people who had much more. The occasional neighborhood Communist proselytizer got nowhere with us. Neither, I must record, did the dedicated anti-Communist ideologue, not then or later. We worked at what we could because we never doubted we had to work, and we felt fortunate indeed that we could find work.\n\nIn time, both my brother and sister with their spouses were able to retire comfortably to modest dwellings in West Palm Beach, Florida, with savings and pension benefits from companies with which each had been working many years, Sylvia from her employment with Macy's as a minor executive, a sales supervisor in one department or another, my brother as head of the mail room at MCA, the thriving talent agency that eventually acquired the Universal Pictures motion-picture studio. People I've run into who worked with my sister invariably speak of her with tremendous affection. Men I've met who worked for my brother describe him as a gracious, generous, helpful, and kind boss. It has proved impossible for me to give either one of them anything. Once I tried to buy my brother a better car than the one he was driving in Florida, and was not allowed to.\n\nRight or wrong, I'm convinced that no one in the family lied to the others about anything, or to anyone else. If they did, it would only be to say they were feeling all right when they weren't, or that there was nothing they needed when there might be something they did.\n\nFollowing the success of my first novel, published when I was thirty-eight, there was a considerable financial return from my second, _Something Happened_ , and I was able to retire from my own salaried position, which was with the English Department of the City College of New York, where I had spent the previous four years. I went on writing, of course, but I thought of this work think of it now, at this very moment!\u2014not so much as working for money but as a challenging and eternally and increasingly harrowing (and remunerative) application of the mind to leisure, in much the same way I imagine a rock climber or mountain hiker contends with his pastime or an amateur bridge player or golfer tussles with the frustrations and adversities of his particular obsessive recreation. I make a living from mine.\n\nELSEWHERE DURING MY CHILDHOOD, in the world and in this country, there was turmoil. There were violent strikes. Blacks were lynched, and photographs of these heartless crimes seemed invariably to picture among the lynchers men wearing the American Legion field caps of veterans of World War I. Labor organizers in rural communities were beaten, jailed. A bonus army of veterans descended on Washington, D.C., with demands for a promised payment and was forcibly dispersed by military units commanded by officers soon to be honored in World War II. In the weekly newsreels in movie theaters we watched infuriated dairy farmers spill into streams milk they couldn't sell at a profit; countless people who would have wanted the milk lacked money to buy it. There were no jobs for huge and growing numbers. For the evicted and other homeless there were shantytowns across the country called Hoovervilles, so named for the sitting president, Herbert Hoover. Men stood on breadlines, families were fed in soup kitchens. Banks failed.\n\nSomehow, we, on that minute parcel of seashore at the lower tip of Brooklyn that was home to separate but overlapping neighborhoods mainly of Jewish and Italian families, managed to escape the worst of the consequences of the stock-market crash and the Depression. And so, it seemed, did most of the nearby districts of Brooklyn, one of five boroughs in New York City, which, with approximately two million residents, had as it does now a population greater than all but the few largest cities in the country.\n\nI don't know how that happened or why we were so lucky.\n\nI do know that the prevailing sympathy in our locality was always automatically for all strikers, and for the poor and the downtrodden. This meant we were not Republicans. Chanting with others \"Hoover, Hoover, rah, rah, rah\/Put him in the ashcan, ha, ha, ha!\" in a torchlight parade on election eve in November 1932 was my first political action, taken at the age of nine years, six months.\n\nTHE HEALTH OF the four of us was generally very good. I remember no illness of significance striking any of us until well past the age of fifty. It would have been hard to tell if our health hadn't been good, since no one in our family made a fashion of ailments or talked much about the distress they brought. My brother, even in his last years, even with one excruciating foot infection that swelled the tissues until the flesh seemed ready to explode, would consistently attempt to minimize his condition; he put off going to a doctor _for anything_ until further postponement was no longer humanly, humanely, possible, until symptoms or someone _forced_ him to go.\n\nLate in my mother's life, however, there was one accident of permanent and drastic consequence. While I was away in preflight training at aviation cadet school in Santa Ana, California, she, at home alone, tumbled from a stool while hanging kitchen curtains and broke a hip. She dragged herself to the kitchen window and, spying Jeannie Goldman, a girl my age who lived in a facing apartment just across the street, shouted out to her for help. Jeannie rushed upstairs\u2014apartment doors then were never locked in the daytime, there was no need\u2014and telephoned for the ambulance.\n\nPerhaps it was that crawl across the floor that did it: Emergency treatment and surgery as a ward patient in Coney Island Hospital left her with one leg shorter than the other, and she was reduced to walking in one heavier, built-up shoe and with a limp and a cane.\n\nShe went on doing everything in the house she had done before, including shopping on Mermaid Avenue and climbing the stairs to our apartment two floors above street level, but the cane and the observable limp were humiliations over which she never stopped grieving for as long as she lived. She had my vanity, or I inherited hers. If ever she needed assistance for anything unusual or just wanted to talk, she would bang with her cane on the floor, and Mrs. Rose Kaiser would come up from below to accommodate her. When Mrs. Kaiser, by then also a widow, died and unknown new tenants moved in below, my mother's spirit seemed to ebb, and she lapsed more and more into that hopeless, emotionless state of resignation we recognize so often in the aged. But from her, there was never a plea or complaint. In my fifties I grew friendly with the novelist James Jones, whose mother was alive and ailing, and one time in answer to his question I told him that both my parents had long since passed on. Grimly, he responded: \"You're lucky.\" Grimly, I understood him.\n\nBUT FARTHER BACK, in better days, I can find a deeply satisfying episode that bonded us closely to each other; this was the time when I, so much younger than the others, was nevertheless able to cooperate with Sylvia and Lee to help teach my mother sufficient English to qualify her to vote.\n\nShe voted for Roosevelt, of course, for Franklin Delano Roosevelt, as did everyone else in the neighborhood, if not, as it felt to us, the world. And then, for the office of New York State governor and later of U.S. senator, for Herbert Lehman, who possessed the double virtue of being not only a New Deal Democrat but also a Jew. These candidates always won. I couldn't vote until the election of 1944, while still overseas in the service, and I voted for Roosevelt by absentee ballot. He won again. I wasn't wounded in the war. I wasn't killed, either, or taken prisoner, and those were the good old, gone-but-not-forgotten days when the candidates I favored most had the best chance of election. That never happens anymore. Even now, though I know this is an illusion, it feels improbable to me that there should have been any such curious things back then as Jewish Republicans, and perhaps there weren't any. The brass ring seemed good as gold.\n\n# 2  \nConey\n\nCONEY ISLAND, with its beaches, crowds, commotion, and couple of hundred entertainments, has always been magical to children and a gaudy magnet for adults. People came from everywhere. Early in this century, even Sigmund Freud dropped in for a look on his trip to the United States; the Russian author Maxim Gorky was a sightseer, too. The milling crowds through the 1930s included soldiers on leave and sailors in port, crewmen in the American and foreign merchant marine. Whole families, sometimes clans of extended families, would journey from Manhattan and the Bronx and other parts of Brooklyn to spend the day and early evening. Those who eschewed the lockers and other facilities of the bathhouses would make camp on blankets under the boardwalk, changing in and out of bathing suits, eating from tubs of cooked foods prepared at home before setting out. The place was better known than we who lived there realized and had been a famous, and notorious, playland and summer resort far longer than we could appreciate\u2014at its most wicked when its first recreational community was established at Norton's Point just after the Civil War at the western tip of what later came to be called Sea Gate. Visitors came by ferry. So infamous for vice did Norton's Point grow\u2014for pickpockets, prostitutes, gamblers, hoodlums\u2014that numbers of tough and cunning rogues with schemes of their own began moving eastward for safety into what eventually grew to become Coney Island proper.\n\nThe founding of Luna Park and George C. Tilyou's Steeplechase Park dates back to the last years of the 1890s. Both had been long established and were already in decline by the time I grew aware of them. Although I didn't know it until I began this history, at one period there were three prominent horse-racing tracks in the area: one in Sheepshead Bay close by, another in the district called Gravesend, to which all of Coney Island is joined; and another in Brighton Beach, closer still.\n\nNotable stake races of the present day like the Suburban and Futurity Handicaps were features of the season and were better known nationally than the Kentucky Derby; the Preakness for its first fifteen years was in Coney Island too. Prizefighting also: The first contest in New York State for the world heavyweight title took place at the Coney Island Athletic Club (in 1899, between Bob Fitzsimmons and James Jeffries. Jeffries triumphed, though a three-to-one underdog). The racing season brought socialites with names like Whitney, Vanderbilt, and Belmont into this home area of mine, as well as big spenders of lesser pedigree. The socially correct visitors preferred the sedate environment of Manhattan Beach and Oriental Beach at the eastern end of what was then called the Island. But the trainers and jockeys as well as the touts, gamblers, and other raffish drifters who followed the horses, and the prizefights, too, also came in thick crowds and brought business and action into the lively center, to the hotels and eating places and to the beer gardens with their singing waiters and female entertainers, to flirtatious beer-hall waitresses hustling customers to drink more, and to freelance prostitutes as well, of whom, I now learn, there were continually growing numbers. A New York State antibetting law of 1909 put an end to Coney Island as a center of horse racing in the East. All three tracks were gone by the time I came along, and the only horseflesh we knew belonged to the iceman, the milkman, and the man at the reins of the Brighton Laundry wagon.\n\nSurviving into my time was the spacious, elegant German beer garden Feltman's, a famous meeting place for fun-loving celebrities of the Gay Nineties, but it was on the way out by the time I discovered it. It was the original Mr. Felt-man himself who is credited with inventing the frankfurter, a small boiled sausage that he served on a roll. Also in operation in my time as a Coney Island amusement was an exhibition of premature babies in incubators. I swear to God it's true! I now find out that the facility was the enterprise of a European physician of some international reputation, staffed with nurses, containing up-to-date equipment, and devoting more effort to preserving the lives of these infants than hospitals at the time were inclined to do. Nonetheless, it does seem grotesque.\n\nCONEY ISLAND WAS generously extolled in hit songs that are still sung: in Rodgers and Hart's \"Manhattan\" and \"The Lady Is a Tramp,\" and in Cole Porter's \"You're the Top,\" in which a night at Coney is up there with the Colosseum and the Louvre museum, the Tower of Pisa and the Mona Lisa. In film, too, the place was celebrated. Year after year, those weekly newsreels typical of the 1930s never failed to highlight the opening of the season each Decoration Day weekend with a shot of a flat-bottomed boat accelerating down the watery slope of the Shoot-the-Chutes in Luna Park, and another of one of the circular rides in operation in Steeplechase Park, or of a woman's skirt blown above her head by an unexpected blast of air from a hidden jet in the floor. In the movie _Manhattan Melodrama_ of the early thirties, characters played by William Powell and Myrna Loy go to Coney Island on a date. In _The Devil and Miss Jones_ , Robert Cummings and Jean Arthur join the Coney Island crowds for pleasure on a day off from work, accompanied by Spring Byington and Charles Coburn. William Powell and Myrna Loy and Robert Cummings and Jean Arthur might all well have gone there as their natural selves, too, for celebrities were continually reported by people who knew people who swore they had spotted them.\n\nCelebrity sightings were most common after the Labor Day holiday during a series of nightly parades called Mardi Gras, which were contrived to signal the close of the season on the one hand while, on the other, prolonging it by several days. Oddly, this postseason celebration also claimed the high moral purpose of providing money to rebuild the Coney Island Rescue Home, a shelter for wayward girls that had burned down. Predictably, the effect of those parades was to attract hopeful floods of yet more wayward girls to the area.\n\nIn the 1920s, lawmakers finally began bringing about reforms that eventually transformed Coney Island's character. These changes were accelerated by improvements in electric trolley-car service and in the extension of the subway system into the Island itself, right to a busy crossroad on Surf Avenue. For a fare of only a nickel, rapid transport brought in multitudes of people seeking safe and inexpensive varieties of recreation, and by the thirties, my era, the sinful prestige of the notorious beach resort was unearned and all but gone.\n\nAt the Mardi Gras parades from then on, for those of us quickly bored by the marchers, the greatest joy consisted of nothing more exciting than throwing handfuls of confetti into the faces of girls, who appeared flattered by these attentions. It was not much fun. In rowdier days, I've read, rougher sports would stuff the confetti down and up their dresses and into their mouths. That doesn't seem like much fun either.\n\nNoticeable in almost all the sophisticated tributes to Coney Island I've cited is a more than faint aura of the patronizing, of people going slumming for fun. They are making a fashion of the unfashionable, in much the same jaunty spirit in which chic and adventurous New Yorkers would safely voyage up to Harlem before the war, my war, for the music and the food (mainly spareribs) and also to buy marijuana. I don't know where those Coney Island addicts on the needle like Danny the Bull, Raymie Glickman, Philly Penner, George Weiss, and others went for their heroin after the war, for I had married shortly after my discharge and moved from Brooklyn, and that was not something I could have found out from my mother.\n\nIt didn't take long for me or any of the others migrating into the broader world to learn that one widely used way of deprecating a place was, and still is, to scoff that it is just like Coney Island. And, for the record, no one I've known from Coney has ever referred to Coney Island by just that abbreviated term \"Coney.\"\n\nBY 1920, soon after the BMT subway company extended four of its lines into Coney Island, huge masses of people were arriving\u2014first, tens of thousands on summer weekends, then hundreds of thousands, finally a million and more. Most came now from the working classes and the newly emerging lower-middle class. Many belonged to immigrant groups recently settled in the East, especially from Eastern Europe (Jews) and Southern Europe (Italians). The Irish were already there, as were the Germans and Scandinavians. Witness the names of enterprises already established: Hahn's Baths, Scoville's, Feltman's, Stauch's, Paddy Shea's Saloon, Shannon's. A man named Handwerker, first name Nathan (a nephew went to high school with me), had fortuitously opened a small hot-dog stand on Surf Avenue just across the street from the corner on which the terminal of the new subway station arose, selling his savory product from a griddle for only five cents, in contrast to Feltman's costlier boiled ones for ten cents. Feltman may have invented the hot dog, but Nathan's perfected it, and soon customers stood in packs five deep, clamoring to be served.\n\nProbably it was about then, with the subway now granting convenient access to just about anywhere else in the city, that the wood bungalows, brick homes, and apartment houses I knew began to be built and immigrant families to move in. As early as 1921, a progressive reformer, Bruce Bliven, was complaining in the progressive _New Republic_ about a new population that was displacing \"native American Stock.\" He had noticed that just about all of the people he saw on the Coney Island beach now seemed to have \"black hair.\"\n\nEVEN AT THIS LATE DATE, people I meet with a large stock of memories of visits to Coney Island still express surprise upon hearing that I grew up there, that families lived there, and still do, and that children were brought up there, and still are.\n\nThe single image they retain is of a gigantic, sprawling, fenced-in amusement area with an abundance of games and rides, of sideshows and food stands, that is closed down and locked up at the end of each season until the following spring. In fact, though, the amusement area was perhaps only fifteen blocks long, on a strip only one block deep from Surf Avenue to the boardwalk, in a seaside community that is about two and a half miles long and about half a mile wide and which, by contemporary suburban standards, was densely populated with year-round residents. Even back in 1929, when I entered kindergarten, there were sufficient families with young children living in Coney Island to overcrowd the two elementary schools there\u2014and each of these stood five stories high and a full block wide. One was in the Jewish area, the other in the Italian, but this division into Jewish and Italian was anything but absolute. There were Bartolinis, Palumbos, Salimeris, and a Charlie Anderson in my school, P.S. 188, and a Klineline (German) and Bannon on my street, and there were Mandels, Goldbergs, and Kesslers in the other school, P.S. 80, which rose near both the Catholic church and a synagogue of yellow brick on Mermaid Avenue that was perhaps the largest of the several in Coney Island. A Julia Revelli was in my class all the way through elementary school.\n\nThe second apartment house in which we lived was owned for a few years after we moved in by the Provenzano family. They dwelled on the second-floor front, opposite the Kaiser family across the large landing, which sometimes served as a play area. Tony Provenzano was a year older than I, proportionately larger, and probably stronger. I never had to find out. Tony was disconcertingly rigid in his unwillingness to cooperate in automatically allowing me to take the lead in deciding the games to be played or to assume the captainship of the team, as all the others of our approximate age group in the street were inclined to do. There was in the Provenzano apartment a player piano with a wealth of piano rolls. I was endlessly fascinated and would listen as long as they would let me. Tony had a large collection of lead soldiers, too, the first I'd seen, and we played with them also, although there isn't much one can do with lead soldiers except stand them up and look at them and try, without much gratification, to play at war. Across that first-floor hall in Irving Kaiser's apartment was an early phonograph\u2014a Victrola, as all phonographs were then called generically. It was wound up and powered by a hand crank. The favorite record, preferred even to a couple by Enrico Caruso, was one called \"Cohen on the Telephone,\" a comic monologue by a man with a Yiddish accent attempting with ineffectual desperation to place a phone call. The records, like those in our cellar clubs much later, were made of shellac; they were played with a metal needle, and they shattered if dropped. There was a complete set of volumes in the Kaiser apartment called _The Book of Knowledge_ that held our interest for hours. (For a short while, the Kaisers had a summer boarder who also held my interest: She was a snappy young lady from Pennsylvania who walked about freely in a half slip and brassiere, and I would drop downstairs and hang around casually with Irving in the hope of again seeing her half undressed. In a week or so she had a man of middle years as a gentleman caller, a man who owned a concession in the amusement center. And a week or so after that, she was removed by the police.) _The Book of Knowledge_ contained not only simple information about almost everything in the world; it also provided instructions for making and doing things. Howie and Henny Ehrenman, brothers in an apartment house on Surf Avenue just around the corner, still remember that I ruined their mother's best frying pan one afternoon, demonstrating how to make perfect butterscotch from _The Book of Knowledge._ They remember I ruined the butterscotch, too.\n\nWhen Nettie, one of the two Provenzano daughters (the other was Rosemarie), had her church wedding, the Hellers attended as a family, possibly excepting my brother, who tried to avoid all such ceremonial events as assiduously as I attempt to avoid them now. That was the first time any of us had been inside a Christian church of any kind. I wouldn't bet that any of us felt we had a right to be there.\n\nAfter the Provenzano family surrendered ownership of the apartment house, they moved down into the Italian section of Coney Island, on Surf Avenue just about opposite one of the several stretches occupied by Steeplechase Park. I went at least once to visit them in their new quarters. The next time I passed that way, Tony had a food stand in the building and we waved. The last time I passed, the food stand was gone. So was Steeplechase Park, all but the framework of the defunct Parachute Jump, which still towers as a mournful landmark, presumably because the cost of demolishing it would be higher than anyone might wish to spend to be rid of it. And Irving Kaiser, as I've said, was killed in the war, in Italy. If, as I've heard, his life was ended by an artillery shell, it could be that he never knew what hit him. In that one respect, just that one, it might be said he was lucky.\n\nToday, the Italian neighborhood remains; the Italian restaurants best known then, Gargulio's and Carolina, are still functioning. But the Jewish section, right up to the barrier gates of Sea Gate\u2014which are guarded more vigilantly today than ever before by a private security force\u2014has given way almost wholly to a population mainly Hispanic and African American. There are fewer stores; possibly no supermarkets; and, I am given to believe from newspaper reports, a higher crime rate than in my day. When last I looked, the storefronts on Mermaid Avenue still stood, but most were boarded up at the windows and had nothing going on inside. I wondered then what residents did for a convenient drugstore, tailor, or shoe-repair shop. People in Sea Gate today must drive a good distance to a mall for all their marketing. At Surf Avenue and West 31st Street, where my two old apartment houses had been, two tall modern units of a housing project stand.\n\nConey Island is, of course, in lower Brooklyn, at the lowest tip, lying laterally along the southern front of that borough like an inverted anvil facing the Atlantic Ocean just outside the mouth of New York Harbor. It is attached to Brooklyn at the Gravesend section and runs from Norton's Point at the western end of Sea Gate, which it includes, eastward in a descending order of numbered streets that now contain the city's Aquarium at West 8th Street and end at Ocean Parkway. There Coney Island terminates at the beginning of a contiguous locality called Brighton and\/or Brighton Beach. Past Brighton is Sheepshead Bay, thriving in past days with seafood restaurants and flotillas of public, party-fishing boats. And on the south side of the bay along that same unbroken stretch of shorefront lies the smaller community of Manhattan Beach. Manhattan Beach was once prized for its formal hotels and upper-class residents and vacationers, who came mainly for the horse races in Sheepshead Bay. In a later life it had a pair of fancy casinos; these were not gambling places but dance halls presenting famous bands and singers of the time. Brighton Beach was known for its handball players and handball competitions. Manhattan Beach was considered the most well-to-do section; Sea Gate was next; then Brighton. Coney Island was the lowest in income level but by far the best known and most exciting. People from the other sections frequently came to ours to dance at our social clubs and ride the rides. We did not often go to theirs.\n\nWhile certainly not a slum, Coney Island was in my time a depressed area for its year-round dwellers, except for Sea Gate, and this is even truer today than in the past. It appears to me today a rather squalid scene, and it must have seemed so back then to others better off than we who weren't living there. The older my mother became, the more she detested the place for its harsh and rowdy intrusions, especially in summer, when the streets were constantly filled with people and noise, often late into the night. Screened apartment windows were open at night, for this was before the days when air conditioners were ubiquitous. My mother liked the bathing and, before the accident of the broken hip, would go to the beach on sunny days, taking along an empty milk bottle to fill with ocean water for washing the sand from her feet when she returned. But she was distressed and angered by the vulgar turmoil and ceaseless, turbulent motion, the sheer volume and activity, of the crowds everywhere. Bitterly she would denounce our part of the world as a _\"chozzer_ mart,\" a pig market. I might feel secretly chastened when I heard her, but I was also secretly unsympathetic, for I was part of the restless commotion and, once old enough for membership in one of the local social clubs, thoughtlessly helped contribute to the din and disorder of the street with our late-night banter and our club room's outdoor loudspeakers blasting away with \"jump\" music from the unrelenting record players within.\n\nWe were children from poor families, but didn't know it. I don't think I have ever in my life thought of myself as underprivileged, as unfairly deprived of something I might reasonably wish to own and didn't. Although incomes were low, everyone's father did seem to have a job, and later everyone's older brother and sister; finally, we, too, were out of school and working. It was a blessing of our childhood to be oblivious of our low economic state and of how others might regard us. We had our beach and our boardwalk, our safe streets, the food and clothing we needed, and I don't believe the circumstance of moderate poverty was too upsetting to our parents either. Nearly all were immigrants and living on a roughly equal level. This was the nature of life; they had learned that in Europe. It was not stylish to bemoan. They expected life to be hard, and most were living better than they had been able to in the Old World. I doubt that many had known the luxury of running water, central heating, and indoor plumbing before they arrived in America. Until my mother and my brother, the latter a child of six, set out on the sea journey to New York from somewhere in western Russia\u2014they sailed separately on different voyages, for my father's first wife was still alive, and he and my mother had not yet met\u2014neither had seen an orange. My brother was offered his first one on the steamship carrying him across the Atlantic. Because it was an object alien to him, he refused it. The next day, after hearing from others how wonderful the flavor was, he was eager to try, but the supply was exhausted; there were no more.\n\nThat was life, too.\n\nThe elders in our families worked hard, both the men and the women, and did not often quarrel. Quarreling was not how things got settled. They didn't drink to excess and seldom divorced, and, unless they were more artfully circumspect than my own generation, they didn't fool with the wives or husbands of their friends and neighbors.\n\nI once heard George Mandel, an early friend from Coney Island and the first of the people I knew to succeed in becoming a novelist, present in a television interview a description of Coney Island I wouldn't try to improve upon: If a person did have to grow up in a slum (he used that word _slum_ for comic exaggeration) he could imagine no better one.\n\nTHERE WERE apartment houses on just about every side street in my section of Coney Island and on every block of the avenues they crossed. All of the larger ones were brick buildings with three or four landings of apartments above the ground floor, which might contain a row of stores like Mr. Kaiser's tailor shop downstairs and, for a while as in mine, a corner grocery store, later remodeled into another couple of apartments. On every street were narrower, lower houses made for the most part of dull yellow brick, with porches up a stoop of stairs and basements below street level, and these could be dwelling places for two or three families, though many were owned and occupied by single families. Of the larger apartment houses, none but a few of the newer ones provided elevators. These were self-service, and for a while were baffling contraptions of great interest as well as places for restful, occasionally mischievous, play on rainy days. Pressing all the buttons to slow the passage of the elevator and irritate the next passengers was one of our tricks. A scene from this time I still preserve in my memory is of aging men and women (eventually the old men and women were our fathers and mothers, and my own mother was among them) laboring up the steep flights of stairs, the women usually with arms filled with brown bags from the grocer or butcher. Anyone in our family going downstairs from our second-floor domicile was reminded to \"take down the garbage\" to the covered metal pails always waiting in the alleyway of our house. There were wooden houses, too, some with small yards in front and back, meager gardens, with an architecture of clapboard walls and sloping, shingled rooftops instead of the flat tops with the clotheslines and radio antennas of our building.\n\nScattered everywhere about the Island, mostly in neighborhoods near the beach, were complexes of bungalows and frame buildings called \"villas\" or \"courts\" or \"esplanades,\" which might all have been there before we were. Certainly, they were already in place when I came along to notice them, and were still there after the war when I married and moved away to attend college. These wooden structures of various sizes would lie silent and shut for nine months of the year and then teem with families who rented the cramped bedrooms and housekeeping units for the summer and came crowding into them excitedly with their bedding and baby carriages\u2014almost all of them arriving, it seemed, in the same few days. They would pour in from other parts of Brooklyn such as Williamsburg and East New York, and from Manhattan, the Bronx, and New Jersey, to pass the next few months in a bustling proximity to each other that would have been intolerable to people more cultivated and less sociable, and likely _was_ intolerable to the generation of young they reared. The summer renters would for the most part spend their months in these quarters happily, for the same families returned with bedding and growing children to the same rudimentary \"villas,\" \"courts,\" and \"esplanades\" year after year. Some of the boys came to be summer friends; their return was awaited and they grew acceptable enough to play in our card games for pennies or soda-bottle tops, or to qualify for a position on the Surf Avenue punchball team of West 31st Street, especially when we were short of players.\n\nA small summer bungalow structure of two stories with a center courtyard stood on my side of the street between my apartment house and a lesser one at the corner of the Railroad Avenue trolley stop. This other apartment house had a candy store on street level, in which Mr. Moses and his wife and daughter worked; they lived in an apartment in the back. He sold newspapers, too, both Yiddish and English, and for a long time the telephone in the public telephone booth inside his store was the only one on the street and was used by every family, both for making calls and for receiving them. In those days, an unseen telephone operator was still on duty at what was called the switchboard, a woman who said \"Number please?\" Any of us hurrying to summon anyone to the candy store to receive a phone call could expect a two-cent tip, the going rate. My sister recalls now that any boy calling her for a date had to do all his courting and preliminary kidding around on the telephone in Mr. Moses's candy store.\n\nAcross the street from my house was an expansive facility called the West End Villa\u2014a row of one-family bungalows. Off-season, we would play our games of tag on the porches and over the railings. Behind these, inside a development on two levels similar to those in today's motels, were strips of adjoining summer apartments, and it was due to one of these, probably, that my brother met the girl who would in time become his wife. Perhaps he didn't actually meet her there, but her family rented one of these apartments summer after summer, and I was familiar with her parents and her brother, Ben, before they became his in-laws.\n\nHer name when we met her was Perle; then, for some reason, for one or two seasons she changed it to Lynn. There was something of a fashion then among young women\u2014though in that period we said girls\u2014for changing names. I remember seeing a girl named Gertrude whom I'd known all through primary school suddenly pointed out to me as Gail when she showed up as a potential date for one of the fellows in the \"senior\" group of a social club of which I was already, and only, a junior member. She was then in my French class in high school. The club was in a store with a back room on a side street two blocks away from my house and was called Club Alteo. What was unusual about the Alteo was that it was already second generation; many of the members, then in their twenties, were the younger brothers of the original organizers, who, maturing into full-fledged adults, had outgrown the social and athletic activities of the numerous \"SACs\"\u2014social and athletic clubs\u2014always flourishing in Coney Island and in Brighton Beach, too. I was one of a group of neighborhood boys in our mid-teens already organized into the Alteo Juniors. We were permitted to use the club room for our weekly meetings and as a place to hang out during the day and into the early evening to listen to music, listen to the tall stories told by the older guys, and learn how to dance the lindy hop. We learned as well to distinguish between the very good music of bands such as Jimmy Lunceford's, Duke Ellington's, and Count Basie's, and the more popular \"commercial\" stuff of Glenn Miller and Tommy Dorsey, with Benny Goodman preferable to the latter two but not nearly as enthralling as the best, who without exception were black. In the evenings, when their own social activities got going, our elders showed us the door, outside which we often loitered awhile on the sidewalk just to watch.\n\nGertrude, now Gail, was brought to the club room by an older, pretty sister, and Gail was wearing lipstick and mascara, done up in a way none of us had seen her before. In short order she became the girlfriend of twenty-two-or-three-year-old Murray Beckerman. It was conceded by his peers that he was the best lindy hopper in the club and, along with a nonmember named Curly Klein, good enough to take the floor without shame at the Savoy Ballroom in Harlem. In all other ways he seemed unremarkable to us teenagers when compared to the George Mandels, Roy Roycemans, and Danny Rosoffs who were also members: George and Roy could talk of Beethoven as well as Basie; and it was from the lips of Danny the Count that I first heard the names Hemingway, Fitzgerald, Faulkner, Dos Passos, all uttered without a given name, as though they had long been familiars of his. He spoke to me about them because he knew I was already writing short stories (short stories that were spectacularly immature and inept). I confess that even back then I began to wonder whether Danny had actually read these authors or had merely read _about_ them. I wonder still.\n\nNeither Gertrude nor I could ever be at ease with each other after exchanging glances in the club room. She was Gail there and Gertrude in the French class. But she, with a nubile bosom dauntlessly accentuated by the tight sweater and uplifting undergarment cups then in style, could note with confidence that she was attractive to older men in their twenties; whereas we, poor things at fifteen, didn't know of any girls our own age who might be attracted to us. Young people then were slower to mature socially than they seem to be today; at least that was true of many of the teenage boys in Coney Island.\n\nIn a little while, my sister-in-law, Lee's wife, reverted to her original name of Perle, to the confusion of my mother. (I don't know what eventually happened to Gertrude\/Gail.) When I got married, which I did at twenty-two, in October of 1945, it was to a young woman named Shirley. As my mother aged, the distinction between Perlie and Shirley was not always easy for her to preserve. With increasing frequency, and with exasperation, too, for she was always sufficiently alert to recognize her slip as she was making it, she would use one name when calling or referring to the other, much the same way that I now, as I age, occasionally blunder with my second wife by addressing her by the name of my first. This is not a sage thing for a husband to do.\n\nJUST ABOUT AS NUMEROUS as the bungalow dwellings for summer renters were the bathhouses for people who came just for the day. There were bathhouses down the length of Coney Island, almost all of them between Surf Avenue and the boardwalk and beach, and on just about every street. Most of them were simple facilities with lockers and showers, such as Hahn's Baths just down the block from where we lived, and Ward's baths across the street facing it. The strongest appeal they exercised was their immediate access to the beach and the ocean. My sister remembers working at Hahn's one summer as the locker girl in the female section; while still a schoolgirl, she worked other summers at one of the many soft-ice-cream stands on the boardwalk\u2014we called it \"frozen custard\"\u2014and intermittently as a \"shill\" at one of the games of skill or chance, pretending to be playing as a paying customer in order to draw others to witness how she fared and thus be enticed to play, too. There was little need for kitchen services in any of these humble bathhouses, for an abundance of food stands on and underneath the boardwalk provided almost everything that might be craved in the way of fast food and soft drink.\n\nHere and there, extending right up to the boardwalk, were a number of elaborate swimming establishments featuring outdoor pools with water slides and diving boards, together with a few simple sports activities like Ping-Pong, shuffleboard, basketball shooting, and, highly favored by those who knew how to use them, punching bags. Raven Hall (named after one of the prominent early developers, Ravenhall, I've just found out), which was far down in the amusement area and functioning long before I was, seemed by far the largest. Later on, Steeplechase augmented its myriad features by installing a pool that might have rivaled Raven Hall in size. Washington Baths seemed to have the most loyal following: People I've met through the years say they bought season tickets summer after summer. McLaughlin Baths, right in our neighborhood, drew its patrons from the Scandinavian, mainly Norwegian, and Irish populations in Bay Ridge, an area distinguished for a brawling toughness and the predictable anti-Semitism then generally common in America among groups that were not Jewish. The younger, high-spirited, rougher ones at McLaughlin's took occasional pleasure in loudly mocking the Jews on the beach by whom they found themselves surrounded, and once or twice each summer there was a report of some fracas brewing that, as far as I know, never did erupt into anything of large consequence. The war seemed to put an end to that kind of local friction.\n\nThe swimming pools at most of these establishments were located right up against the boardwalk and always lured spectators from the throngs of curious strollers, who paused to watch through the openings in the enclosing fences. These were, after all, the only swimming pools ever seen by any of us except on the screens of movie theaters. It was good to have proof they were real. The swan dive and the quick jackknife from the low and medium diving boards were the favorites; perhaps they were also the easiest. Likewise, the men adept with the rhythms and variations of the punching bag happily worked up sweats performing for admiring viewers assembled on the boardwalk. On the beach, athletes who stood on their heads balanced on the abdomen and knees of supine girlfriends were so prevalent as to hardly command more than a glance. A disappearing species of beach gymnast was the athlete who stood on his hands and walked about a bit in jerky strides. I haven't seen any of these clumsy jokers perform that stunt in years, and I hope I never see one again. Pinochle games played by elderly Jewish men with vigorous gestures and impassioned discussions were more exciting spectator sports as we tried to learn the subtleties of the game and figure out what these characters from the Old World were gloating and disputing about. Portable radios made their appearance with huge batteries in large cases that seemed to weigh half a hundred pounds. Ukulele and banjo players abounded. It was on the beach in Coney Island that I first heard western ballads such as \"Clementine\" and \"Red River Valley.\"\n\nBACK IN THE EARLY THIRTIES, some shrewd speculators decided to erect a luxury hotel right at the boardwalk on West 29th Street, the Half Moon Hotel, and soon afterward they were tormented by shadows of insolvency. The Half Moon, so christened after Henry Hudson's vessel, remains the most majestic building in Coney Island, and I think I remember watching it going up from the window of my apartment. I knew nothing about Henry Hudson and his fatal voyage then, but I would have thought someone more knowledgeable might have proposed a name auguring better prospects for the venture than the end met by that seafaring explorer.\n\nIn retrospect, it's hard to guess what in the world it was that led those crafty real-estate operators to suppose that, with the charisma of Coney Island already fading among the swells, people with enough money to go somewhere else would choose to come to Coney Island. The Half Moon Hotel achieved a brief flare of notoriety in about 1940 with the death of one of the few figures acquiring some measure of fame through association with Coney Island, a man named Abe Reles. Although he never lived in Coney Island, Abe Reles, who hailed from Brownsville, a fiercer area of Brooklyn distinguished for its gangsters and a prizefighter named Bummy Davis, was secretly sequestered in the hotel in protective custody as a material witness while waiting to give crucial testimony at the trial of some men involved with a gang known as Murder, Incorporated. Reles broke into public notice with a fatal splash when he either fell, jumped, or was thrown from the window of his hotel room before he could testify. At the time very few people other than the police authorities knew he was there, and although it was never proved, the rumor existed, and still is believed, that he was helped out the window by the very policemen to whose protective custody he had been entrusted. Eventually, after the war, during which it was utilized by either the coast guard or the navy, the hotel became a Hebrew home for the aged, and this metamorphosis, though somewhat somber, was an apt symbol for the old and faltering Island itself, which had certainly seen more vital days. My mother in her last years spent some time there as a resident, following emergency surgery for a strangulated hernia from which she never regained her strength. The surgery also revealed that she had become diabetic. No doubt she would have greatly preferred living with one of us, provided her presence was welcomed warmly and no source of conflict; a cynical realist, she knew that wasn't possible and didn't expect it to happen, and, to my knowledge, she didn't ask. Consistent with modern convention, none of the three of us, all married by then and living in confining apartments, felt we could have her. The subject never arose, but I would guess that each of us secretly suffered at least some remorse. In our family, we did not often talk about sad things.\n\nEVEN BEFORE WORLD WAR II, the Island as diversion and playground had been fading in verve and enterprise, its amusement area persistently shrinking. And since the end of that war, in 1945, with the exception of a few government-financed public housing projects, there has not (to my knowledge) been a single new residential dwelling of size constructed. The amusement area, once as up-to-date as any in the world and with a reputation for being just about the best, has not provided a spectacular new attraction since the Parachute Jump from the 1939\u201340 New York World's Fair was installed by Steeplechase.\n\nConey Island still presents a boardwalk that seemed then, and probably still does, the longest, widest, and most splendid boardwalk in the whole world. It has a wide beach of fine sand its entire length, a beach that continues well beyond Coney Island into Brighton Beach and Manhattan Beach. One has only to stumble with shock and lacerated arches upon the shorefronts of Nice or the English Brighton\u2014and remember with a sense of affront that these are also called beaches\u2014to begin to fully appreciate the spacious shorefront of Coney Island as a truly distinguished natural treasure.\n\nAnd Coney Island also had for us those long, paved streets, almost entirely free of motor traffic except in summer, that were better playgrounds than any recreation engineer has ever devised.\n\nOur favorite street game in spring and summer was punchball, a variation on conventional baseball played with a springy rubber ball\u2014the \"Spaldeen\" was best, but it was costlier than the others. The ball was propelled from home plate by a fist rather than by the shaft of a broomstick or any other kind of bat, hence the name \"punchball.\" In fall, we played hockey on roller skates and also a game with a football called association, involving passing, dodging, and tagging, with no tackling or blocking. In this I excelled as a receiver. Then, once we learned how, we played real football on the sand beach. The first time we played a team on hard ground, and that was in Sea Gate, I was astounded at just how hard the ground was when you fell and struck it. We wore no padding or helmets. Earlier, when we were still more juvenile than adolescent, there were games like \"hango-seek\" (hide-and-go-seek), tag, and follow-the-leader on the beach and around the empty summer bungalows, with me almost always the leader. For follow-the-leader on the boardwalk summer evenings, I originated a variation based on guile and impishness rather than agility. I would excuse myself with winning politeness to people standing in conversation, step between them, and move on. The second fellow was obliged to do the same. With the interruption from the third, suspicion was aroused, and by the fourth or fifth, the victims broke off to stare and caught on, usually with good-humored surprise. The longer the line, the broader the laughter. In barrooms, waiters and bartenders didn't tolerate us with such good humor. I, the leader (in follow-the-leader, one almost always remained the leader; if I fell in jumping from one porch railing to another, I insisted the others would have to fall, too) would enter innocently, make my way around as though in search of someone, and exit through another door, while the second young fellow was already coming in as though looking for me, and after him a third, innocently trailing both of us. It was the fourth or fifth who would be roared at and shooed out. And the one after him would hesitate to go forward and unhappily have to move to the end of the line, while, I of course, was always spared.\n\nA typical Saturday morning in spring and summer found us playing a punchball game against a team from another block\u2014which more often than not we won\u2014then coming home triumphantly for a well-earned glass of cold milk and hungering for lunch. Saturday early afternoon saw us going to the movies with a dime for admission and a nickel more for candy to see a double feature (two movies), the latest episode in a cowboy or Tarzan serial, a newsreel, and a Mickey Mouse or Betty Boop cartoon or travelogue thrown in, and then walking home afterward in serious discussion of plans for the rest of that day. Sunday usually was simpler, with most of the morning spent eating breakfast and perusing the colored funny papers and the advertisements of toys we'd like someday to own, such as two-wheel bicycles, and trips we'd dream of someday being able to take. From the beginning of time we had two large movie palaces to attend\u2014the RKO Tilyou and Loew's Coney Island. Then, on an empty lot on Surf Avenue around my corner one block away, a new one, the Surf Theater, was constructed, and we had three. The opening of the Surf Theater was an event I remember. The picture was _One Night of Love_ and starred the opera singer Grace Moore, who was killed in an airplane crash not many years later. Much of the music, including the title song, was based on melodies of Puccini, and my mother went, too.\n\nTHE AMUSEMENT AREA of Coney Island\u2014that stretch encompassing the rides and the games, the food stands and the penny arcades\u2014was of greatest interest to us in the cooler days of spring and late summer, or to break the routine even on hot days when one of us would get hold of some free passes to Luna Park or Steeplechase. These were the only two true amusement parks then existing, each charging an admission price and incorporating attractions exclusively its own, in name if not necessarily in originality: The slow boat ride through darkness was in one of the parks called the Tunnel of Love and in the other (I believe) the Red Mill. All the individual attractions required a specific ticket of admission or a separate fee, and at Steeplechase everyone had to pay extra for the Parachute Jump. Stauche's indoor arena, one of the original large restaurants early in the century, was still standing, but now it was a place where prizefights were occasionally held and where marathon dance contests were for a while in vogue. For half a mile between Surf Avenue and the boardwalk, past Steeplechase almost to Feltman's, there was (and is) a pedestrian thoroughfare called the Bowery, the name appropriated, of course, from that earlier boulevard of fabled excess in lower Manhattan. Almost every inch of it was consecrated to some kind of attraction, and even the cross streets were crammed with sideshows, games, rides, and food stands, with the cries of barkers and ticket sellers, the sputtering of engines and the rolling turmoil of wheels, and the maniacal bedlam of amplified mechanical laughter crashing incessantly from the various spook rides. And famous Feltman's was still in business, though inexorably expiring, still large and garish but waning from neglect now that Diamond Jim Brady and other bigwigs from the Gay Nineties had themselves passed on.\n\nAt one point on the Coney Island boardwalk, a Howard Johnson's ice cream stand suddenly surfaced, with many very bright lights and twenty-six different flavors. Having overcome our surprise, we stared with wonder at this oddity and scanned the names of the flavors, musing on why, with vanilla, chocolate, and strawberry standard, and burnt almond and butter pecan available, and then pistachio for the plain novelty of color rather than for any particular excellence of taste, there was a need for so many more flavors. And who in hell was Howard Johnson, of whom we had never heard? On the boardwalk we preferred our soft ice cream, our frozen custard, to the true ice cream we normally enjoyed at home or in our street, and Howard Johnson's was soon gone. Twice there were bold ventures with outdoor bowling alleys on the boardwalk. These were exotic gambles, too. Bowling then was still _goyisheh naches_ , (Gentile pleasure), and these didn't last long, either.\n\nWe learned early on that a boiled frankfurter anywhere is not as good as one broiled on a grill to the point of splitting\u2014you only had to ingest a boiled one in a stadium at a professional baseball game to know you were tasting only hot water and mustard; that the Wonder Wheel with its rolling, swaying gondolas, which is one of the two mechanical attractions from our antiquity still in operation, was obviously superior to George C. Tilyou's Ferris wheel in Steeplechase Park, but that both were for sightseeing squares or for adults with children who were squares and still too young to be exposed to the terror of anything but height; that the Cyclone, which is the second old-timer still extant, was far and away the best of all roller coasters; and that it was futile to search anywhere in the universe for a tastier potato knish than Shatzkin's when they were still made by hand by old women who were relatives or friends of the family. (Although I still run into mulish people, hogs, who didn't mind the thicker, yellow Gabila's, and to people brought up in adjacent Brighton Beach who still salivate at the mention of Mrs. Stahl's, the queen of potato knishes there.)\n\nAll of this was practical, worldly knowledge that taught us always to look for fair value for money. We also learned at an early age a fact of capitalism that directed us toward the antithetical principle that it is usually impossible to obtain fair value. The difference, to Aristotle as well as Karl Marx, is known as profit. We learned this first of all from the Coney Island barkers who offered to guess your weight, guess your age, guess your name or occupation, the country you came from or the date you were born, guess anything at all about you including the color of your underwear, for a dime, a quarter, a half dollar, or a dollar, the prize at stake improving with the increase in the amount bet. The fact was that the barker could never lose. He knew no more about the tricks of this trade than you do, but he always came out ahead, right or wrong, because the customer could never win. Phrased more nicely, while the customer might walk away with a prize, the proprietor could never lose, because the prize at stake invariably cost him less, considerably less, than the patron had spent to win it. Of course, the customer also walked away with the satisfying emotional reward of having outsmarted the expert, while the proprietor in turn enjoyed the emotional reward of knowing he had just pocketed another piece of small change as profit.\n\nBut honest to God, a kid could win a whole coconut for a penny at the penny-pitch game and, once having acquired the know-how, knock a hole through the soft spot in the head of the shell with a hammer and nail to get at the milk, then smash the whole coconut into fragments to be distributed liberally to everyone around. The trick to winning a coconut was not to toss each penny at one of the metal button targets that would set off a ringing to announce a winner but to aim it into a patch of losing pennies already lying there and set several of them vibrating down the incline onto one of the buttons.\n\nThe catch to winning a coconut was that nobody really enjoyed eating coconut or drinking coconut milk. But we all had to pretend we did in order to savor the victory.\n\nTHERE WAS A COMPETITION between Luna Park and Steeplechase that dated back to the founding of both at the turn of the century. I forget the names of the two men involved with the spectacular opening of Luna Park a few years after Steeplechase, but Steeplechase the Funny Place was solely the creation of an individual named George C. Tilyou. Tilyou was a born entrepreneur. As a boy, he began accumulating capital by selling vials of Coney Island sand and bottles of Coney Island seawater to wide-eyed visitors. Even people from places with beaches of their own returned home with precious souvenirs of Coney Island sand and ocean water. His idea for a Ferris wheel, the first in Coney Island, came from one he saw while on his honeymoon at the Exposition in Chicago in 1893. He undertook construction of a copy half its size and ballyhooed it from the day he broke ground as the largest in the world. When the first of his amusement parks burned to the ground in one of Coney Island's periodic fires, he promptly began recovering his losses by charging people ten cents to enter and see the smoking ruins. \"If Paris is France,\" he is reported saying, \"then Coney Island between June and September is the world.\" This was not wholly true. Tilyou died in 1914, nine years before I was born, and is buried in Green-Wood Cemetery in Brooklyn, and all that any of us knew about him was that his name was on Steeplechase Park and on one of our two capacious movie theaters, and that there was, or had been, a family home on Surf Avenue across the street from the amusement park. It was a wooden affair of a couple of floors set back from the street, with that family name, TILYOU, carved on the vertical face of a stone step at the bottom of a small entranceway with the letters already sunk almost halfway into the ground at the edge of the sidewalk. We would note the marker and the house on the way to his movie theater. It didn't occur to us then that he was already dead. It didn't cross our minds that he had once been alive. We never thought of him at all as somebody human, or thought of his name as any more of a family name than the equally improbable one of Loew.\n\nOf the two amusement parks, there seemed to be a near unanimous preference for Luna Park, and we were inclined to be contemptuous of anyone, usually someone from someplace else, who raved about Steeplechase. The most we could feel about Steeplechase was that it was \"all right\"\u2014we couldn't honestly say, \"It stinks.\" Once we got inside Luna Park, there seemed to be a good deal more open space in the middle in which to meander and more interesting things to look at. The architecture was a fantastic, almost nightmarish corruption of the Moorish and the Byzantine in circus-clown colors of chalk white and cherry red with ornamental stripes of black and bright green on minarets, spires, and onion domes. Every few hours Luna Park offered a free circus in a small tented ring to everyone already inside the park. Its commencement was heralded by a short parade of a few listless elephants and other performers to the beat of a drummer and the notes of a trumpet and trombone. I remember a high-diving act outside in public view, I'll take my oath I do\u2014someone diving from a tall platform into a thimbleful of water; and I'll swear to this one, too: a man or a woman shot from a cannon into a net a good distance away, maybe four or five times daily.\n\nLuna Park began with the Shoot-the-Chutes, which was a pretty good ride to go on and to watch, a flat-bottomed boat sliding down a high incline slippery with water into a bouncing splash in a large pool at the bottom and coasting to a stop. The best part, sadly, was waiting for your turn. It also had the Mile Sky Chaser, the highest roller coaster in Coney Island. God, it was high. And something to stare up at from the ground, that long pull of the cars to the top of the first descent. With wonder and dread, I mustered enough nerve to approach it as a customer for the first time at the age of eleven or twelve. It was taken for granted that as a step toward maturity every boy would sooner or later have to ride the Mile Sky Chaser, as older ones already had; this was another in the progression of rites of initiation. I contemplated this looming challenge with the pragmatic and stoic frame of mind that has stood me in good stead in other perilous situations since then and with which I long ago came to face the necessary act of dying itself: If others could go through it and survive, I could, too.\n\nAs it turned out, the Mile Sky Chaser was a cinch\u2014after the first time. Though high, it was tame, not nearly as fast or as twisting and jolting as the Tornado or the Thunderbolt, or, grandest of all, the Cyclone, which was reserved by everyone for last, the consummation of the roller-coaster agenda. I worked up to the big event of the Cyclone over one or two summers by growing used to and scornful of the lesser rides. The easiest of all the roller coasters was the rather childish one in Steeplechase. It was nothing much, and not intended to be in that rather mannerly playground, and except for a series of rapid bumps at the conclusion\u2014they were bumps, not dips\u2014it provided no surprises. Luna Park had a better, minor one, called the Dragon's Gorge, which, inventively, was partly indoors and partly in darkness. I'm not sure how much later I mastered the Tornado and the Thunderbolt; these were speedy and rattling variations of each other, with women screaming in true or pretended fear and men with their mouths gaping, pretending to grin. But first came the day of my induction to the Mile Sky Chaser, accompanied for morale, as was usual, by a couple of friends who had already been and guaranteed that it was nothing.\n\nAs it turned out, it was not only a cinch, it soon proved a bore. After experimenting over time with a seat at the very front, the back, the middle, I could anticipate accurately every dive and turn of the Mile Sky Chaser with my eyes closed better than, years later, I was ever able to read an aerial map in the air corps with my eyes open, or am able even now to read a road map. As a matter of fact, we often would take this whole ride, and others, with our eyes closed in attempts, mainly futile, to amplify our sensations of thrill and fright. Soon the only excitement the Mile Sky Chaser yielded was the chance to display to marveling strangers how coolly we braved its apparent perils.\n\nNone of us, then or afterward, was ever able to be that casual about the Cyclone or to chance the same liberties. I was discharged from the army in May of 1945\u2014the war against Japan was still on when our government began demobilizing\u2014and found myself back home during a spell of beautiful weather in spring with little idea of what to do with myself during the day. My cellar clubs had passed away from attrition. In an ardor of liberation, I went hurrying down Surf Avenue in civilian pants and my army field jacket, looking for any old friend to rejoice with. Through a coincidence of good luck emanating from bad, I was among the first to be released from the service under the point system newly in effect. The bad luck was that I had lost most of my pocket money in a dice game or card game on a base in San Angelo, Texas, and thus happened to be on hand that weekend when an order arrived to discharge a certain number of officers. That last part was the good luck. Back home, it seemed that just about everyone I knew from the neighborhood was still away. Suddenly, on Surf Avenue, I spied Davey Goldsmith, whom I hadn't seen in three years, home on furlough and doing the same thing I was, looking for someone to pal around with. We embraced each other jubilantly and then hurried, almost running, down into the amusement area in search of old times, our receding childhood. I was better at schoolwork than Davey, but he was smarter and more self-assured than I was in just about all other ways. (The last time I saw him he was prospering as the owner of a division of a hatband manufacturing company for which he had worked since finishing high school.) At Nathan's Famous, we gulped down some hot dogs and a couple of bags of their unrivaled fried potatoes, and that indeed was exactly like old times. It was _better_ than old times, because we now had cash in our pockets for everything we might feasibly want. (As an officer with flight pay on combat duty overseas, I had been able to send home plenty.) When we went on the Parachute Jump, I was tense again for just the few seconds of suspense that preceded the unexceptional drop along protective guide wires. But coasting down, I had to wonder why anyone would want to ride it a second time. The Cyclone, it turned out, hadn't diminished with age\u2014its age or mine\u2014or with familiarity. We rode it once. I don't know what Davey was doing, but I was holding on for dear life through the racing plunges and veers, and I tottered off with a thumping ache in my head and a wrenched neck. And all at once I understood without needing to put anything into words that this was a part of my life that was definitely over. After sixty missions overseas, I was now selective in my adventures, and I had no doubt that I would never want to ride that or any other roller coaster again. Davey Goldsmith might love to, but Joey Heller wouldn't. I was twenty-two and would soon be engaged to be married and I was too old. And like someone very much older, for I have crossed similar thresholds of loss since, I felt with sadness that something dear was behind me forever, but I also felt that loss with tremendous relief. It is often pleasing to be free of even good things, and childhood is one of them. Youth is another.\n\nFollowing the Cyclone, Davey guided me by trolley car to a fantastic clam bar in a Sheepshead Bay restaurant called Lundy's to teach me how to eat raw clams on the half shell. I had never before even looked a clam in the eye. I didn't think it could be done. Davey showed me it could, and I've been eating them ever since.\n\nWHILE STEEPLECHASE, \"the funny place,\" was also always the lesser place, it was not all that bad. Much of it was indoors under a vast roofed pavilion, away from the sun, shielded from rain. For ten or fifteen cents you could purchase entry to the park but not admittance to any of the rides, and many wise people did just that. There were food stands inside\u2014cotton candy, frankfurters, ice cream\u2014and games of skill and chance, like the penny-pitch for the coconuts. There were people to watch, and of best value were those people who had bought tickets to all the rides and used up very few of them; if asked, they might give you the rest of their tickets as they were exiting. There was, without fee, a theater with comfortable seating for about three hundred, on whose stage unsuspecting patrons who had just taken the Steeplechase ride and were exiting on foot suddenly found themselves in the spotlight as spectacles before an audience. They would have to cross that stage to escape, braving the pranks of two or three jesters in wait for them, one a stunted clown; the second costumed as a farmer, or a New Yorker's stereotyped image of a hick farmer; and the third as a rangy cowboy. There were several shaky vertical columns of painted barrels strung along ropes, and these might abruptly begin to move, as would various shuttling sections of the floor. Also waiting to surprise them were mild electric shocks and upward gusts of air to raise the skirts of women and blow off men's hats. Those were the days in which men still wore hats, even to amusement parks in summer\u2014straw hats and panama hats. One of the costumed figures sported a long wooden clapper with which he threatened to whack fannies, another brandished a metal wand at the end of a wire cord connected to an electrical outlet, openly threatening shocks. People coming upon them onstage without warning and with an audience watching must surely have been stricken, if only for a second, with the bewildering terror that they had somehow dropped into a chamber containing some of the milder tortures of the first level of hell.\n\nFor twenty-five cents in Steeplechase you could buy a circular pink ticket with numbers 1 to 25 that entitled you to twenty-five attractions. The tickets came on a loose white string that went about the wearer's neck to protect against loss. For fifty cents you bought a blue ticket that allowed you thirty-one attractions, the added six being the premium rides that were judged the best of the lot. Our practice was to get inside any way we could and then to politely ask well-dressed ladies and gentlemen moving in the direction of the exits if they were leaving and if so, could we have the remainder of their tickets. There was little reason to refuse us. Many of the elderly would use hardly any of their tickets, sometimes riding only on the elegant carousel\u2014created originally for some German king by a master bridge-builder in Leipzig\u2014on which they could smoothly circle for several minutes seated on the benches in one of the ornate gondolas. And in this way we could accumulate enough tickets to go on any ride as many times as we wanted, with the result that we soon didn't want to go on any.\n\nLUNA PARK, as I've said, seemed to us much the better of the two competitors. Yet Luna Park closed first, a few years after the war ended, following a few fires and some desperate and futile attempts to arrest the critical decrease in customers. A housing project of complex design built by private developers for people of middle income now stands in its place.\n\nSteeplechase held on gamely into the sixties before it gave up the battle and closed shop.\n\nA few years before that, I went there one afternoon with my friends George Mandel and Mario Puzo on what proved to be a final trip. Mario had by then published two novels, _The Dark Arena_ and _The Fortunate Pilgrim_ , but not yet _The Godfather._ George had published _Flee the Angry Strangers_ and _The Wax Boom._ My _Catch-22_ had appeared shortly before, in 1961. I had met Mario years earlier through George. It was a lazy, drowsy day, and the three of us had come from the city with our children, I with my two, George with his two, and Mario with one or two of his five.\n\nThe very qualities that had disappointed us in the past made Steeplechase now ideal for languid fathers in their forties there with young children. It was clean, it was orderly, it was safe. While the children chased about in gawking exploration and enjoyed themselves first on one ride that moved slowly around in a circle, then on another that did exactly the same thing, the three of us could rest calmly on a bench and talk quietly about such things as publishers, book advertising, advances, and royalties, and that lousy Book-of-the-Month Club that had paid no attention to any of us. Luckily for us, the kids didn't want to go on many rides that would have necessitated our going with them, and we weren't really eager to go on any. Coming in, Mario, who was not from Coney Island and was rather portly, had chanced the Magic Barrel. He sank down slowly to the revolving floor and was unable to right himself, presenting a ludicrous picture as he rolled around there helplessly for a minute, laughing, with an unlighted long, long cigar in his hand, until finally the attendant in crimson coat and green jockey cap walked in and helped him through. After that he wanted no more. The place was very still and rather empty that bright afternoon. After a couple of pleasant hours, we prepared to leave, and a thought about the passing of generations occurred to me as we walked toward the exit: It struck me that if a kid like the one I used to be approached and asked for our tickets, he would have gotten from us three blue ones that were just about complete, except for the number-one ride punched for Mario. The next time I visited Coney Island, Steeplechase was no longer there, and only the red skeleton of the old Parachute Jump marked the spot, like a funereal obelisk.\n\nEACH YEAR AFTER LABOR DAY, the traditional end of the summer season at most American resorts, there was that week of parades on Surf Avenue quaintly mistitled Mardi Gras by promoters who didn't know or didn't care that the title was traditionally connected with spring and Lent. If you hate parades as much as I did then and still do, you would have loved to hate the ones at the Coney Island Mardi Gras, for they represented everything about parades that is sordid, tawdry, and synthetic. They were long and they were tiresome. They were intended to draw crowds to the Island for that extension of the summer's business, and this they did. Celebrity attendance was most prevalent during Mardi Gras, and the city's daily newspapers, which then numbered about seven, often printed smiling shots of someone from the movies who had been snapped there the day before as though having a good time.\n\nNegative as we felt, we nevertheless would go to the Mardi Gras just about every night, more for the rides and the food than the parades. Merely mixing with the crowds was diverting. Every night it was just about the same parade. One might be called \"Firemen's Night,\" another \"Policemen's Night,\" and so forth, but the floats were mostly identical, the same marching bands with honking brass and clarinets would go clumping by, and there was a goodly sprinkling of the ladies' auxiliaries to almost every group that marched. (Just about everything that marched had a ladies' auxiliary, and I'm still not sure what they are.) And we always had a nickel for at least one hot dog. The droves of people moving on the sidewalk in both directions and for the most part paying little heed to the marchers were potentially more interesting than the parades. On Saturday afternoon there was a different kind of procession called the \"Baby Parade,\" an utterly grotesque competition for which mothers prepared their helpless offspring for weeks in advance and paraded them along in strollers past judges in the passionate hope of winning distinction and a prize for their efforts and their tot. I doubt that the first prize was a scholarship to West Point.\n\nOne night I did see Mayor La Guardia ride by in a car in the Mardi Gras. He was wearing a brown fedora with the rolled brim of a cowboy hat. I confess I was thrilled. I was young and I was thrilled and I'm glad, for our vision of Fiorello H. La Guardia as a figure of exceptional integrity for one holding political office endures. Except for FDR, I cannot think of many since then I've felt worth a second look.\n\nBack when the area's permanent residential and seasonal population was almost entirely white, it was standard campaign procedure for political candidates for local or national office to make a stop in Coney Island. Then as now, the purpose for a politician was less to hobnob with voters than to be photographed with a frankfurter while on a leisurely stroll along the boardwalk, always smiling as though merry, introducing himself to people easily excited by the prospect of seeing their picture in a newspaper. Even before I turned impenitently cynical about all American contenders for elective position, I began to feel that this was too much of a condescending insult to impose on a low-income neighborhood already distressed with signs of unmistakable and irreversible decline. One of the more offensive images I still hold in memory is a newspaper picture of Nelson Rockefeller, campaigning either for himself or in support of some other Republican, sinking his good-natured teeth into a Coney Island hot dog. And one of the more ridiculous memories is of Henry Cabot Lodge, patrician scion of the impeccable New England line of Lodges, lending himself to that same demeaning ritual\u2014demeaning to himself and demeaning to the local electorate he was humoring. Because of his innate good taste, one\u2014this one\u2014can still take sadistic pleasure from the thought of the piercing abhorrence he no doubt was suffering and pray, with abiding malice, that some of the mustard dripped off his hot dog and ran down into his shirtsleeve. I remember remarking at the time that it was sufficient to have people like these in public office without having to put up with their fellowship as well, and that they wouldn't be walking through Coney Island smiling if they had to live there.\n\nTHE YEARS OF MY CHILDHOOD were the years of the Great Depression, although in childhood we didn't have a firm idea of everything that term signified. In our wonderful streets, we could play our games. Also wonderful was that our street was right outside the doors and windows of our apartment houses. We could see in an instant which of our friends were already out there and what kind of game was about to start. If in doubt, we could call down and ask. Possibly the most valuable of our resources were all the other children. Almost always there were enough children nearby of all ages to organize any kind of activity and field any kind of team.\n\nThis, then, was the block\u2014a section of a street between avenues that teemed with all the companionship we young ones needed, both boys and girls, and in the beginning our interests were centered almost exclusively on the block (not the whole street) that was our own. Our closest friends in school might be from other blocks on other streets, but outside school we didn't have much to do with them unless it was to challenge _their_ blocks to a game of punchball or hockey. When a family moved from one street to another, the children almost invariably attempted at first to maintain affiliation with the old block and resisted assimilation into the new. Occasionally, two blocks would decide to have a fight; when that happened, we would mobilize our respective forces a safe distance apart, throw stones at each other for a few minutes, and then break off to return to our customary diversions without antagonism, which probably hadn't existed in the first place.\n\nThe beach as much as the street exerted a strong influence on our daily activities. In summer we were there much of every warm day. What better place to be? In autumn we played football there, which meant we were spoiled and tenderized by the soft sand and unconditioned for the real game when the time came to play on standard terrain. I take it as a cause for some negative pride that not a single one of the fellows I grew up with ever amounted to much as an athlete, or tried to.\n\nThere was a rhythm to our sequence of play activities that was both seasonal and intuitive. One mild day in early spring the sun would be out and suddenly the kids on every block were punching grounders to each other in the middle of the street, and the punchball season was awakening. The girls were skipping rope on the sidewalk or playing hopscotch on numbered grids marked out in white chalk. There would come a certain moment after Labor Day when every boy seemed to understand that summer was definitely on the way out and it was time to start tossing footballs. The same was true of hockey\u2014sooner or later everybody roller-skated: All over the Island, on virtually the same day on every block, the flocks of flying footballs would thin and the grinding of steel-wheel roller skates would be heard in the land. At the beach in the summer, of course, there was the swimming, along with the occasional games of tag in the water and fetching things up from the bottom and the sunbathing out at the end of the mossy rock breakwater, over which we trained ourselves early to sprint with eye-catching agility across the irregular crags of massive stones. At night the busier streets were much too cluttered for games, and we would drift to the boardwalk to roam for hours, enviously watching the gaudy, kaleidoscopic world of grown-ups, until our big brothers or sisters were sent to find us amid the sauntering throngs and pull us home to sleep.\n\nConsidering how loosely we were guarded, the late hours we kept, and the vast influx of strangers into the Island each summer, it's surprising how little harm befell us. Where we lived, as I've stated, there was practically no crime and few serious accidents. I can't, for example, recall hearing of a single murder (other than the one occurring long before, outside a poolroom) in Coney Island up until the time I left to enlist in the army. I was not aware of any incident of child molestation or domestic abuse of any kind in any combination. One time, a strange male attempted to lure into his automobile Norma Goldman, who lived across the street with her family in the apartment we had formerly occupied; the street buzzed with that one for a day or so, and the net effect was an emphatic caution impressed upon all of us, boys and girls, against ever going off with a stranger. While still in my earliest grades of elementary school, a classmate who lived in my building, Jackie Keshner, was struck by a car at the corner and was out of school for a while with a leg in a cast; I would spend part of most afternoons keeping him up to date on our schoolwork, particularly on geography, which proved the easiest and most rewarding subject to transmit. In geography, I remember, we were gravely taught that bread is \"the staff of life.\" We weren't taught what the phrase means, and I still don't know. (Sadly, I sense that geography as a subject for early study is probably obsolete everywhere. It surely is needed: While planning a family trip to Italy in the summer of 1966, one of my two children, a middle-class teenager midway through a costly private school in Manhattan, wanted to know why, instead of journeying by steamship, we couldn't travel there by car.) Robbery? I heard of just one: An armored car at the Rubel Ice Company, off a good distance from where we lived, was held up by armed robbers who made good their getaway across Gravesend Bay by speedboat. But that still sounds like an industrial crime and, like most others of that class, had nothing to do with us. An interesting sidelight in our chronicle, though irrelevant, is that the Schechter chicken company, also a distance away, which supplied the freshly killed chickens that are still unmatched for flavor and texture by anything frozen or refrigerated for transport, was the small concern whose complaint against prevailing federal rules was the basis for the U. S. Supreme Court decision declaring the New Deal's NRA unconstitutional. The gangster Abe Reles, the Rubel Ice Company, and Schechter's chickens\u2014the historical high points that brought newspaper fame to our realm at that time.\n\nOne summer when I was just into my teens, a younger boy from a family in my building drowned. A nonswimmer, he had ventured, it was reported, out along the safety ropes into water over his head and was swept away when his hold was broken by a wave unexpectedly large. I knew his sister, a girl two years ahead of me in school; before the season had passed, he seemed to me almost forgotten. He had another sister substantially older, mature. She was a slight, black-haired girl with dimples, and I remember her face more distinctly than the others, although I don't recall that we ever exchanged so much as a nod of recognition. She had a boyfriend, and because of her boyfriend I was too bashful to meet her eye. His name was Jack, and at the beach the two of them were always lying entwined in each other's arms on a blanket or the sand, dozing, talking, kissing, under the boardwalk mostly, separating themselves from others in her family group, which also was not the custom. Strolling there and back, they always held hands or walked with their arms around each other. In the Club Alteo was a boy named Herbie with a girlfriend named Teddy, and these two were always to be seen there lying on a couch, in the back room or the open one. And next door to my own Club Highlight later on, there was a member of the Club Amo Pharmacy named Arnie with a girlfriend called Bobbie, and these two also were always either dancing sensuously cheek-to-cheek or lying together in a sweethearts' embrace in full view on one of the couches. I have no idea what happened to any of this trio of ardent and loving couples. Did their unembarrassed affections endure? Did they marry and remain constant? If they're alive, they are older than I am and they are strangers I'd rather not meet. I can picture them only as young and still interlocked in their devoted snuggles and, with much misgiving, I succumb to temptation to repeat some stanzas from A. E. Housman:\n\nWith rue my heart is laden  \nFor golden friends I had,  \nFor many a rose-lipt maiden,  \nAnd many a lightfoot lad.\n\nBy brooks too broad for leaping  \nThe lightfoot lads are laid;  \nThe rose-lipt girls are sleeping  \nIn fields where roses fade.\n\nRhymed lines about death or the lachrymose march of time are about as far as I ever was able to get in a struggle to like poetry, and I long ago gave up trying. Besides, of those three couples I've cited, just one of the performers, the girl Bobbie, had golden hair, while another's was honey brown. I cannot swear that the boys were lightfooted. I didn't often see them standing.\n\nALTHOUGH I DID NOT at the time fully appreciate the degree of danger, at least three times before leaving home I came close to dying young, to being killed accidentally: once, for a kite; again when swimming out too far; and a third, by an automobile while being chased by my brother one night after refusing to come upstairs to bed when called from a window by my mother. I was swift on sneakers and dodged away from him in play, not fear, and I was still giggling to myself like a dimwit when the car squealed to a stop in the nick of time, as they say, and I found myself in sudden headlights, resting against its grille. I had turned the flight from Lee into a game; I had done the same thing several years before on the day of my father's funeral. Then, older boys on the block, who well knew what was taking place that day after I had succeeded in convincing myself I didn't, had to chase me over the railings of the porches and surround me in order to take me in hand and conduct me into the automobile carrying us to the chapel, which I remember not at all and from which, because I was just past five, I possibly might have been kept away, and then to the cemetery, which I recall only vaguely. In the military, I came close innumerable times to dying young, too, but didn't appreciate that either until I saw blood pouring from a man wounded in my plane. I expect my war experience in this regard corresponds to that of every infantryman, marine, paratrooper, etc., going into modern battle for the first time. I, luckily, was spared that dreadful recognition until I was far along in my tour of duty, on my thirty-seventh mission. But after that, I was in a state close to panic as we took off from the landing strip at the start of every one of those missions remaining.\n\nAs for the kite: Older young men at the beach with unlimited funds would sometimes send batches of kites aloft on strings tied to each other, with the result that the central cord eventually developed quite a strong pull, and the ancillary ones, too. Watching these grown-ups by myself one time, I saw one string of kites tear loose and drift inland. I moved up the street after them and saw them descending limply above my apartment house. I bounded up the staircases to the roof. There I found that the end of the string to which they had been attached had caught on a radio aerial wire strung a couple of feet out beyond the low protective brick parapet around the roof of our building. The closest one was dangling there by its cloth tail, but it wasn't close enough; it was out of reach. I couldn't see that at first. I set myself on that low wall of old stone and began carefully stretching out toward it as far as I could, inching myself out nearer and nearer the edge (while my mother sat downstairs talking to some neighbors in front of the house; what a ghastly surprise she might have received!). I still couldn't take hold of the string or the tail of the kite. Then I glanced down along the side of the building and saw myself perched very high up above our alley and the garbage cans below. I felt no fear. I went on leaning out farther, as though I hadn't caught on that I was putting myself at precarious risk. A moment's reflection could have reminded me that had I asked my mother, sister, or brother for the nickel or dime for a kite of my own, I probably would have had it; if not, I could have pilfered still another coin from the secret cache of my sister's I had discovered. But I wanted _that_ kite. It was a wish grown into a lust. I kept stretching and craving, craving and stretching. And then, for whatever obscure reason, I stopped; in an instant I decided to give up. I turned yellow, as they say, and, thank God, I simply quit. I straightened up carefully and inched back from the edge as cautiously as I had moved out, more frightened of a mistake in retreating than I had been in exposing myself to danger just a few seconds before.\n\nI feel now that I risked injury, could have been paralyzed or perhaps even killed, every time we played tag in an empty bathhouse and I vaulted from the top of one row of empty lockers to the top of another. And I _know_ now that I was almost killed every time I swam out to the bell buoy off our block, even though nothing really threatened me the several times I did.\n\nThe summer would begin officially for us, I suppose, on that day in late June we called \"promotion,\" when we would come running jubilantly home on that last day of school, waving our final report cards, me with my A in classwork and B+ in deportment, calling out to everyone we flew by that we had been promoted. \"Over the ocean\/tomorrow's promotion\" was a refrain we chanted. Another was \"No more classes\/no more books\/no more teachers'\/dirty looks.\" By that day, we were already brown enough from the sun to be envied by every pale working adult and summer renter, for we would have been swimming in the ocean and capering in our bathing trunks for more than a month, since May. The Coney Island beach, then as now, was packed inhumanely on Saturdays and Sundays, but that didn't bother us as children, for we were almost never in want of a place to sit. Instead, we were always in and out of the water, jumping off the wooden jetty leading to that breakwater of large rocks or skipping nimbly out to the end of the rocks to rest or watch the strange adults who fished for hours with sinkers, reel, and rod and never seemed to catch a damned thing except an orange crab. We could catch crabs at will by cracking open a mussel torn from the rocks and lowering it into the water on a string through one of the openings. But once we had the crabs, we didn't know what to do with them. Only when an Italian family moved onto the block across the street did I discover with amazement and a momentary nausea that mussels and crabs could be eaten.\n\n\"Where are you going with those?\" I inquired of Dolly Partini one day, when I came upon her returning from the beach with a pailful of black mussels.\n\nIt was many, many years before I could muster the courage to try one, and today I well up with optimistic cheer whenever I spot them on the menu in a new restaurant, offered either in a white Normandy broth or in red marinara.\n\nWe learned to swim in shallow water by the age of seven or eight. The waves in Coney Island are seldom very high, and even when rough are never very rough, in comparison with other Atlantic beaches. The shore is shielded in part on one side by Sandy Hook projecting out from New Jersey and by Rockaway Beach jutting out from Queens on the other, and isn't fully exposed to the sea. From the day we found nerve enough to try the \"dead man's float\" and discovered we could remain bobbing for more than a stroke or two, we began preparing ourselves for the swim to \"the third pole.\" This was not really to a pole but to the heavy rope connecting two of the poles farthest out and boxing in the protected swimming area at its outermost limit. The distance of each of the parallel lines of upright, sturdy wooden poles extending into the water from the first one, at the shore, to the farthest, third pole was at most between forty and fifty yards. At low tide we could walk more than halfway and, when we dared, pull ourselves out along the ropes for most of the rest. At the second and third poles, where the thick ropes of hemp were fastened above the waterline, we would have to let go and swim past the pole and take hold of the rope where it dipped down again. But the day we did get out to the third pole for the first time\u2014no matter how\u2014was a day on which we had accomplished a noble and heroic feat. To be able to swim there regularly back and forth without fear was to possess definite status in the young male community. After that, the only sea challenge left was the bell buoy, rocking and ringing about half a mile out, perhaps a quarter of a mile, maybe five hundred yards, past the rope at the third pole.\n\nThe buoy was of vivid red and bobbed and clanged so unremittingly that we soon turned deaf to the sound, as we did to the reverberations of the electric trolley cars rolling by on Surf Avenue practically in the apartment next door and on Railroad Avenue not much farther away. Relatives who came to spend a night for the first time would tremble in shock at the volume of the various noises they had suffered and to which our whole family had by then grown selectively deaf. All of us in our innocence and ignorance spoke freely of that bell buoy as a \"bellboy.\" Not until after I had my degree in English from New York University and a master's degree from Columbia and after a year more in English at Oxford on a Fulbright scholarship and two years teaching English at Pennsylvania State University, then a college, were my eyes opened upon meeting my friend Marvin Green, who then was selling artwork for slide shows at a time when I was commissioning some as a writer in the promotion department of _Time_ magazine and who had loved sailing since a child: The letters \"buoy\" were not pronounced \"boy\" as in _buoyant_ but more properly \"boo-ey,\" as in no other English word I know of. I laughed scornfully the first time I heard him say it and thought he was crazy. Of course I rushed to a dictionary to check. It still sounds better my way.\n\nThe swim to the bell buoy was not especially dangerous if, first, one knew how to swim, and, next, knew how to go about reaching it. There was really no way to work oneself up to the attempt through preparation. A day would simply arrive when one of us who hadn't yet been there felt certain enough that he could make it, and he would just tag along tensely with a group of other boys who had been to the bell buoy and survived. Valiant as we might prove, and young, we were nevertheless too craven and too maturely wise ever to try it alone.\n\nBlack-topped harbor poles far out in the water tilting left or right would tell us which way the ocean was flowing. We would move three or four blocks up or down the beach in order to have the tide carry us toward, rather than away from, the small, floating buoy with a circular platform for a rim that was our destination. The tide was usually powerful that far out; and if we had ever miscalculated and missed the bell buoy, I think we would have drowned when exhausted. We were too young and not nearly hardy enough to make the trip out there and back without a rest.\n\nStarting out, we would swim in a group directly to the safety rope at the third pole and there dawdle awhile to catch our breath, build up morale, and replenish what little energy we had expended. Then, with a unanimous nod, we would start swimming, not toward the buoy but heading straight out, banking on the flow of the tide to convey us horizontally toward the floating object that was our target. We would move leisurely, alternating a dog paddle with a comfortable variation of the sidestroke. We would talk in fragments of conversation as we progressed\u2014conversation that had no purpose other than to tacitly supply mutual reassurance\u2014and we would make sure we kept close together as we advanced outward. Gradually, we would find ourselves nearing the bell buoy. It was not a strenuous feat, really; it was more a matter of patience than endurance\u2014cool patience. You simply kept paddling calmly and talking calmly, and after a while you could begin making out the markings on the red paint and were nearly there. The only moments of anxiety might arise when you looked back at the dwindling, now miniature shore and realized\u2014if you had a haunted imagination like mine\u2014how very far away help was. Suddenly, everything normally large there was so _small!_ (Today, a mere revisit in memory to that Lilliputian picture is sufficient to chill.) But, of course, help of a sort was always near in the friends we were swimming with. And all the while, the tide was bearing us closer. When we found ourselves only fifteen or twenty yards away, we would turn over dynamically into the fashionable Australian crawl, as we then called it, and swim the rest of the way rapidly, to arrive in a racing splash. Once there, we would haul ourselves aboard onto our feet and, seesawing in tandem from one side to the other, clang that bell triumphantly to capture the attention of everyone on the beach. It was a revelation to observe the pressure with which the tide foamed against the base.\n\nThe swim back never had that keen thrill of drama that is often striking to a young boy taking a manful chance, mainly, I suppose, because each minute brought us closer to safety rather than away from it. And this time our target was a few miles of beach that would have been impossible to overshoot. One time, though, one boy in a group of four, tired badly and exclaimed that he didn't feel he could make it back. Without urgency, without even a sense of danger materializing, each of us simply took hold of a piece of him and cooperated in hauling him gently in, close enough to the rope at the third pole for him to manage the last few yards alone. It was not an occasion for exultation. I don't think we felt then that we had just saved a life; we didn't really believe he'd been in danger of dying.\n\nBut I don't recall that I ever set out on a swim to the bell buoy after that. I doubt that he did either. His name was Irving Kaiser, the same Irving Kaiser who lived in the apartment directly below ours\u2014a year younger than I, he was thinner, too, and I was thin\u2014and the same one who was killed by an artillery shell in Italy six or seven years later. I am more saddened by his death as I write about him now than I believe I let myself be at the time the news reached me, and even when I was back living in Coney Island and seeing his mother frequently. I believe I had by then already trained myself defensively to stifle painful emotion. I am walking proof of at least part of Freud's theories of repression and the domain of the unconscious, and perhaps, in writing this way here and in other things I've published, of denial and sublimation, too.\n\nToday, I wouldn't try that swim for a million dollars, tax-free, although I don't doubt I could make it there and back. It's this haunted imagination that's mine still.\n\nMy brother had no recollection at all of the episode of the chase and the car when I brought it up some years ago during one of our infrequent spells of hesitant family reminiscence. With the benign revisions made by time in his perceptions, to his ungrudging mind I had always been \"a good kid\" and, though past sixty, still was. He was proud of me and proud of his son, Paul, who was doing well first as a talent agent and later in television production. Together, we had been fulfilling the dreams of accomplishment that he had no doubt nourished for himself.\n\nBut I remember vividly. It was nighttime and dark, and when I saw Lee emerge from the entrance to our building to take possession of me and lead me upstairs to bed or the bathtub, I instantly and recklessly transformed my disobedience into the street game of a pursuit. He was faster, being so much older, but I was swift on sneakers and tricky, and I dodged and faked and led him in an exasperating race back and forth from one side of the street to the other, laughing uncontrollably each time I twisted away to elude him again. And suddenly there were headlights blinding me and a tortured screech of automobile wheels braking, and my merriment ended with both of us stretched out on the bumper of a car that had squealed to a stop against us just one moment before.\n\nI'll mention one other occasion when, speaking metaphorically now, I thought I'd come close to putting two of us in the family away for good, one of them me. This was a prank that none of us ever forgot, but even today it makes a lesser impression on me than it did on the others.\n\nRising on the sidewalk outside our building, near the curb, was a telephone pole, with spikes sunk into opposite sides on escalating levels to serve as steps for the serviceman to climb to the seat at the repair box about two stories up. They weren't difficult to mount once you attained the starting point at the bottom, a sturdy wedge of wood hammered into the pole for a first foothold about three feet from the ground. I found out how easy it was one day when, given a boost to that bottom wedge by a friend, I began going up the spikes, hand over hand and leg after leg. At the top, when I was at the seat, I discovered with happy surprise that I was gazing right into the open window of our kitchen. And I saw my mother there, busy at the stove inside just a few feet away.\n\n\"Ma, can I have a glass of milk?\" I asked in my most innocent voice.\n\nThe look of bewildered amazement and then stark horror that struck her face when she turned and saw me poised in air just outside the window was so extreme that I feared I, too, might die of fright at just that moment. It could have been about then that my mother made for the first time the remark about my idea of humor that she would repeat many times afterward in maternal surprise and tribute, though often with a dismayed shake of the head:\n\n\"You've [He's] got a twisted brain.\"\n\nOne time I saw on her face a look of outrage and astonishment even more distraught, when, while glaring squarely into her eyes during one of my occasional infantile outbursts, I called her a bastard.\n\nMy belief, erroneous, was that inasmuch as her command of English was poor, she had not heard that dirty street word before and would have no idea what it meant, no more than I accurately knew then what it meant. At once I saw with terror that I was mistaken. She gasped with incredulity, and staggered back a step. And I knew in instant reflex that I never wanted to see her again with such an expression of deep hurt. I prayed she would never tell anyone.\n\nONE OTHER TIME about then I had a secret I wanted to guard. Prowling about the kitchen by myself, with a rapacious appetite starting to agitate me as always, I found in the cupboard a bulb of garlic with several of the cloves already broken loose. I thought surely that if I ate one or two, nobody would know. They soon knew. Everyone knew. For the next few days, people even half a block away knew.\n\n# 3  \nSea Gate\n\nSEA GATE, on the western edge of a sliver of Brooklyn oceanfront scraped up and discarded by the receding glacier of the last ice age, remains historic in these annals for two later and more striking occurrences: a fire and a feel\u2014and for an alleged business connection with a dignitary named Alfred E. Smith, a noteworthy summer presence there in the third decade of this century. An earlier celebrity dweller in Sea Gate was one John Y. McKane, political boss and self-appointed police chief on the Island, who saw to it that all things there ran in the peaceful mode he fancied until he himself was scrutinized by a higher authority, indicted, tried, and imprisoned.\n\nThe rumor of the involvement of Alfred E. Smith in the development of Sea Gate might be groundless, but he himself was real. Al Smith, formerly the very popular Democratic governor of New York, became the first Catholic candidate\u2014he was defeated\u2014for the presidency. A rarity in office for his unwillingness, I've read, to make use of his public service for his private enrichment, he was later made use of for his prestige and contacts by others more farseeing who were absorbed in the profitable formation of various real-estate properties. One of these was the Empire State Building. Another undertaking vaguely ascribed to him in local folklore was the formation of the \"Sea Gate Association\" for the purchase and consolidation of land on the western rim of Coney Island and the conversion of that tidy, choice tract into an elite residential seaside community poking unassumingly out into the entrance to New York Harbor, with the Atlantic Ocean on one side and the tranquil inlet of a bay washing the other. It was a community of homeowners forbidding admittance to nonresidents, with its own uniformed police force guarding the gates at each of the two entrances and a tall chain-link barricade rising on its boundary between the limits of its private property and the rest of Coney Island.\n\nSea Gate begins where the Coney Island boardwalk and avenues end. The chain-link fence extends all the way down across the sand into the water's edge at both low tide and high, a rampart manifestly differentiating those inside from those outside. No doubt it extends that far in violation of federal law, which conventionally holds beaches to be a public part of the national seashore. In summertime, about a mile and a half away at the opposite end of Coney Island, beat the heart of the respectably notorious amusement section, with its billions of electric lights at Luna Park and Steeplechase and hundreds of other attractions. The newer community of Sea Gate reposed as far from the hurly-burly of the crowds and their seasonal pleasures and as far from the immigrant subcultures evolving in between as it was possible to be without backpedaling away into the water itself.\n\nThere, as in other places we know of, the organization of an \"association\" suggests the dictates of proprietary covenants among the earliest settlers to restrict the ownership and rental of properties to those exactingly approved by the governing body already in place. Probably Jews were not admitted at first, or Italians either. The fence spanning the beach and avenues, and the Sea Gate policemen at the gates, were of course intended to keep curious wayfarers out rather than insiders in. They contributed, as well, to a constant and none-too-subtle semiotic reminder of class distinction. A ban against stores of every kind, apart from aesthetic considerations, implies additionally a rather frigid determination of the early settlers to distance themselves from just about everything else in Coney Island and to persist there independent of the service facilities to be found in the community outside. There were still no stores in Sea Gate, not one, when I, like Columbus in the West Indies, or stout Cortez on his peak in Dari\u00e9n, grew old enough to \"discover\" this new world, and the inconveniences to families I came to know were many and often strenuous. God knows what these early inhabitants did for their headaches and haircuts. Aspirin could be toted in a pocket, but pockets couldn't hold barbers, and the inhabitants of Sea Gate certainly wouldn't have elected to wander deep into Coney Island to have their hair trimmed by Max the Barber, the father of Danny the Bull. They must have counted on conveying many of life's necessities with them when they came there, for summers or as year-round residents, and they didn't come by subway and trolley car. Before the construction of superhighways and the Brooklyn\u2014Battery Tunnel, the journey by automobile from Manhattan and places still farther would not have been swift.\n\nTheir aversion to collegial fellowship with those outside the fence is readily explained by the reputation borne, and earned, by Coney Island as \"the playground of the poor.\" The poor are despised even by their brothers, teaches the Bible in more than one place, and circumstantial confirmation of this truth can be found in the tendency of just about all of us to move out of Coney Island as soon as we could manage to. In general, though, we didn't move far. And we continued to return for our diversions more often than we went to any other place and to focus our social life on our friendships with those still living there and on the remaining social clubs with their phonographs, couches, back rooms, and card tables.\n\n\u2022 \u2022 \u2022\n\nTHE FIRE I SAW inside Sea Gate was my first fire, apart from those regulated small ones, with broken fruit crates for fuel, strung out like signals along the Coney Island beach Tuesday nights in summer acknowledgment of the fireworks shot aloft from the boat anchored off the end of Steeplechase Pier.\n\nAt these Coney Island \"barnfires\" of ours, we ate charred, sandy potatoes, roasted in wood flames and not fit for human consumption, and sizzling burnt marshmallows that blistered the tongue. We called the potatoes \"mickeys,\" although we didn't know why. We cleared our debris from the beach after we had put out our fire and left the spot clean for the next day. We were good little boys when we were good. We were afraid of our teachers, although we liked them all, dreaded displeasing them, and were petrified by the mere possibility of getting into trouble in school.\n\nThe blaze inside Sea Gate was a daytime affair, and the real thing.\n\nIt was big.\n\nAlong with others on our block, I stood on the corner of Surf Avenue and gazed with awe at the broad flush of orange fire and billows of dense smoke, first black, then white, that dominated the horizon of the squat skyline something like half a mile away. What was burning? It was the yacht club, people who were older and wiser kept telling each other. I had not heard those words before. It was, I have since ascertained, the Atlantic Yacht Club, and it was described as \"swank.\" Others are better qualified than I to itemize the traits of what that word signifies in depiction of a yacht club. I didn't even know then what a yacht club was.\n\nNow that I do have some idea of what a yacht club is, I return to the memory of the fire and the presence of a yacht club with a pleased bemusement that never diminishes.\n\nWhat a yacht club was doing in Coney Island at that time remains a charming mystery I would not choose to unravel by reductive research. My surmises are more intriguing. Certainly, Coney Island was at least as appropriate a location for a yacht club as the building between Fifth and Sixth Avenues on West 44th Street in Manhattan currently occupied by the New York Yacht Club. For one thing, it was closer to water. At the back of Coney Island is Gravesend Bay, which, though rather narrow, is nevertheless a body of water. In grade school there were those who bragged they had walked across the bay on the few days it was frozen in winter. Beyond the bay curves the western shoreline of Brooklyn proper, providing one of the contours shaping the enormous, magnificent, and now largely inactive New York seaport. And beyond that, on clear days, one can easily pinpoint the gray structures of certain of Manhattan's skyscrapers.\n\nIn miles therefore, or perhaps \"knots,\" the trip from the city to Sea Gate is shorter by sea than by land, and we know that those oddballs who do enjoy boating, whether by engine or sail, are usually in no hurry anyway and haven't much else they want to do with their time.\n\nBack then there was a company, the Iron Steamboat Company, that ran a regular schedule of shuttle trips from the Battery in lowest Manhattan to the Steeplechase Pier, transporting visitors from the city directly into the amusement area, and these ships seemed to perform prosperously, for they went on operating until Coney Island changed with the war and its alluring mystique of novelty and glamour began irretrievably to wane. From vantage points on our boardwalk and beach we could watch these white side-wheeler steamboats coming and going, and easily spot, as well, the freighters and the giant, international ocean liners, trying to identify the liners by the number of smokestacks as they steamed slowly in and out before us. A luxury cruise ship, the _Morro Castle_ , went up in flames one day not far offshore; that would have been my second Coney Island fire, but I was elsewhere when it happened. However, I did go more than once to view the devastated hulk after it had been towed into Gravesend Bay (by name a fitting resting place, it might at first seem, for Gravesend Bay is compounded of grim English words whose portentous message I did not appreciate until I began writing about it; but, alas, I now know that the term arises in nonphilosophical innocence from Dutch words meaning nothing more sinister than \"lord's beach\"). The journey by water to Sea Gate from some waterway inland was neither treacherous nor long, and it is easy to picture boatmen, and boatwomen, from docking places in the Hudson and East Rivers and even from Westchester, Connecticut, and the shores of Long Island lifting anchor to set course for Coney Island and the Atlantic Yacht Club in Sea Gate.\n\nThey had to bring their meals with them or eat in the dining room at the yacht club, if the yacht club had one, for together with the absence of stores, there were no restaurants in Sea Gate. At that time, probably, Jews were not accepted for membership at the club. But Jews were not sailors then, either, not my Brooklyn Jews, so there was no insult to anything but pride. In those preassimilation days, they did not go golfing, play tennis, or ski. And they rarely divorced; if any wanted to, they would have had to learn how.\n\nBy the time I found myself old enough to wander about Sea Gate, through the courtesy of one or another school acquaintance who lived there, there was no trace of the swanky yacht club I'd seen burning away or of any other. It had been there, it was gone. The salient landmark now was Lindbergh Park, memorable still for the airplane pilot whose name honors it and, to me, for the milestone event of my first feel. The park is a tiny clearing\u2014the land area of all Sea Gate is small\u2014that served as one of several gathering places for green adolescents of both sexes for giggling hijinks on balmy nights. Word went out from other fellows in my Boy Scout troop that two girls there whom I recognized from grade school would permit you to squeeze their breasts if you waited politely in line with the other boys for your turn in the kissing games. One of the girls was petite and pretty. The second was larger and jolly. I waited my turn and drew the buxom one. The gossip proved true. I felt a female bosom, and I learned something\u2014I learned something fast. I learned that once you had a breast in your hand, there wasn't much you could do with it. Not until the initiating days of my cellar clubs not long after did I begin to comprehend that this first liberty beyond kissing was a step, a passport of sorts, that might, or might not, be a prelude to an advancing stage of intimacies more disorderly.\n\nBy that time in Sea Gate, the founding fathers, along with the yacht club, had been removed from the scene by the inexorable stresses of economic and social change and replaced by a different order of landowner with a less demanding ethos. Two-family houses of red brick had sprung up for rental on the idle lots between the capacious wooden summer mansions traditional at seashores. Some years ago, several of these mansions were put to use by the city as dwellings for people on welfare, to the alarm and chagrin of that generation of residents, and this practice may still be in force.\n\nWith the change in population, there was no longer much difference in race, creed, or place of national origin between those living inside the gates and those outside. There did exist a presumed financial disparity, but this was something that in childhood we were slow to comprehend and which was not necessarily accurate. The girls and boys of Sea Gate went to the same public school we did and usually had a longer walk to get to it; there was no talk then of any such thing as a private school. They went to the same parties at graduation and to the same movie theaters for Saturday matinees, to the same secondary school by the same trolley cars or school buses. Those fellows who did join social clubs later came outside to join one of ours. Girls I remember\u2014Hannah Tansman and Gladys Simon\u2014did come to classes in grade school wearing fur coats in winter, but memory hints that they were both, perhaps, from families in the fur business. A blond cutie of nine or ten named June Owitz always seemed to be wearing a bright sweater, possibly cashmere. But if there ever was coolness or snobbishness, I didn't notice it, and no one I knew, inside or outside Sea Gate, ever spoke of it. But parents are parents, and in my maturity, I have the feeling that had I been a parent with a daughter living in Sea Gate, I might not have approved of someone like me.\n\nWhen I entered secondary school, the Coney Island celebrities on the Abraham Lincoln High School football team were largely from Sea Gate: Eddie Mann, Phil Metling, Len Finkleman. When I left, they were from Coney Island streets like my own: Herb Poplinger, Red Goldstein, Richie Wertheim, and others. On my first trip to Lincoln High on a school bus, I had the football lineman Len Finkleman pointed out to me, and with a pang gave up all dreams of even trying out for the football team. He was two years ahead of me in classes and appeared two hundred pounds ahead of me in the evolutionary progression of bone and muscle. In due course I turned out for fencing lessons instead and made the team, but lost interest before even one interscholastic match.\n\nI confess that I loved school, loved both grade school and high school. I loved the vacations and the end of vacations. The thought of playing hookey in grade school hardly ever crossed my mind\u2014I wouldn't have been able to think of a more enjoyable way to spend the time. On days when I was absent with a fever, I would watch the clock until Irving Kaiser from the apartment downstairs and Ira Lopata across the street were due to get home in order to converse with them from my window. Twice in high school I was a truant. I calculated with remorse afterward that I hadn't had as good a time at the stage show and film at New York's Paramount Theater or in Weepy's poolroom as I would have had in the classes, cafeteria, and sweetshop at Lincoln. I had no pleasure in disobedience for its own sake, and I take none now.\n\nI liked school because I was good at it. I took readily to reading, writing, and arithmetic. I even liked the homework, was stimulated to accomplishment by the challenge of each demanding obligation. I liked very much having something to do. I am still less comfortable with unlimited leisure than with the organizing and worrisome feeling that I have something I must get done, like finishing this book, or even this paragraph.\n\nThe elementary school year and classes then were divided into halves, into an A section and a B. In my first grade, IAI, Miss Wolfe one day held up a flash card bearing the word KING. It was our first exposure to an _-ing_ word. I concentrated a moment and raised my hand\u2014I was the first and only child to do so\u2014and when called upon said, \"King.\" Don't ask me how I knew. A day or so later, I was \"skipped\" half a year, transferred from class IAI into Miss Leiberman's \"fast\" IBI. There I found myself among a group of students with whom I was to remain all the way until graduation, and then through high school as well. We, the I class, regardless of grade, were always the \"bright\" class, and we advanced together with very few changes in our membership, most of which resulted from new families moving into the district and old ones moving away. Two (2) was the slow class, and 3, 4, and 5 were in between. I was bright, and so were Ruth Gerstein, Hannah Tansman, Gladys Simon, Albie Covelman, Eugene Dolgin, Seymour Ostrow, Phyllis Ritterman, and maybe as many as a dozen others I could name if I put my mind to it. As early as the fourth or fifth grade, Phyllis Ritterman had made it known that she wished to grow up to be a novelist\u2014this before the rest of us knew what a novelist was. She had inscribed as much with her signature in my souvenir graduation album. Almost fifty years after graduation from this primary school, I was invited to speak at a college in Arizona or Texas\u2014I forget exactly where. While the audience was dispersing afterward, a woman facing me squarely as she sat down beside me challenged: \"You don't know who I am, do you?\" It required no more than a second to summon up the answer. \"Sure I do. You're Phyllis Ritterman, aren't you?\" From that encounter came my formulation of a theory I find to be generally true: Time may age a physiognomy but doesn't otherwise alter it beyond recognition. (On the chance you may care, I'll contribute the unrelated opinion that psychotherapy doesn't change much either\u2014not character, personality, feelings, or even, in the long run, behavior. I speak here from both personal experience and observation.)\n\nI avidly enjoyed our sessions in the early grades in rapid arithmetic; I naturally would, for these were competitions in accuracy and speed, and I was one of the two, three, or four who were always out in front, and I embraced the chance to show my stuff. In algebra and plane geometry later on, with some tips on attitude from my brother, Lee, I was a whiz kid of sorts in the introductory levels, but not, I discovered with resignation, beyond those. My brother was addicted to mystery riddles. I liked to grapple with them, too. In the Lee Heller fabrication, X was not merely an unknown quantity: X was a slippery party to be tracked down and pinned by the pertinent factual evidence contained in the enclosing data. Included one time on an examination in the 7A or 7B algebra class of Mr. T. D. Bartells was a vexing problem of a peculiarly complicated trickiness. \"Only one person in the class got that one right,\" revealed Mr. Bartells, and then looked my way. \"And it had to be you,\" he added, and threw a stick of chalk at me.\n\nThrowing chalk was his mischievous procedure for keeping control of his mischievous students, of whom I inveterately was one, wisecracking, talking, interrupting, and concocting an occasional practical joke\u2014innocent, I was maliciously inclined to believe\u2014on a fellow student; for example, if the correct answer was two and he didn't know it, I might pretend to help him with the whisper \"forty-two hundred and eighty-nine.\" And surely Mr. Bartells would guess what had happened. I had a million of them. We had been tittering about Mr. Bartells as we advanced each year toward the grades he taught, for T. D. Bartells was known secretly in a punning play on his name as \"Titty Bottles.\" I would be most surprised if he didn't know that, too.\n\nI was never in real trouble in school. The few times a parent was demanded, my sister came. My mother would have been frightened, and I would have felt disgraced by her broken English. My sister recalls intervening only once, and that was over an incident I have forgotten entirely. My offense was not misconduct but boredom. I would clearly be musing on other things in apparent distraction and not be paying rapt attention, but each time the teacher sought to trap me with a sudden question, I frustrated her with the correct answer. I would be gazing out the window, at the wall, down at my hands or feet; sometimes I appeared to be talking to myself, mouthing words. If so, they were probably sentences I was testing for the trick ending of a short story I was already contemplating writing someday, or possibly a phrase with which to begin or conclude my next book report.\n\nVery early, I now choose to think, I was exhibiting a pragmatic fixation on making use of my fantasizing and putting my daydreams to work. The early ambition to become a writer of fiction was one of them.\n\nI was generally at home with the logical and at sea with the theoretical. (I coped with matters beyond my comprehension by taking for granted the truth of what I was told and attempting to go on from there.) In biology in early high school\u2014another triumph\u2014the teacher, in discussing bacteria, raised the question of how we could tell that the process of pasteurization did not eliminate all the bacterial life in a bottle of milk. The answer appeared to me so obvious that at first I didn't even raise my hand. When no one else in the class could give it, I did.\n\nBut I lacked innate efficiency when it came to the abstract. My brain went dull with quadratic equations, negative numbers, the multiplication of fractions, trigonometry, beginning calculus. In chemistry, which should have been easy, I often found myself in difficulty with the mathematics of chemical reactions. It was along about this time that certain of my close friends (Marvin Winkler among them) threw in the towel in the struggle with such useless malarkey and transferred out of Lincoln into vocational school. I got by because my memory was reliable and I was able to duplicate through rote the procedures I was supposed to follow. But a logarithm remained unfathomable, and I still haven't solved the eternal mystery of pi (or even understood what the mystery is).\n\nEnglish classes, on the other hand, _were_ as easy as pie. I had a surer grasp of grammar when I was learning it than I have now since I've been writing professionally and since my experience of teaching freshman English on a college level. Correcting freshman compositions, with that terrorizing responsibility of always appearing infallible, did much to undermine my assurance about syntax and spelling. So much uncertainty generated by looking things up to make absolutely sure eventually generated long-lived doubts that still obtain.\n\nReading was exciting, engrossing, and I was doing a lot of it at home on my own. An older brother and sister already working at jobs in Manhattan\u2014it was not a short commute\u2014brought a variety of magazines into the apartment which expanded my mind in a widening radius of common knowledge. There were books, too, from commercial circulating libraries, as contrasted with public libraries\u2014current novels, mysteries, best-sellers that were easy to read, which was not in all respects a very good thing to grow used to. It took a long while afterward for me to relate with patience to literary works in which something more than rapid plots with lots of dialogue furnished the essential components. Magazines like _Collier's_ and _Liberty_ , which dependably published works of popular fiction, invariably contained in each issue a \"short, short story\" of a thousand words or less with a trick ending, and the prospect of writing these successfully stimulated my imagination and ambition at an early age.\n\nWhen I was nine or ten, a visiting cousin on my father's side, Nat Siegel, who might already have completed the course work to become an accountant, brought me the gift of a children's version of the _Iliad._ It was the first work I'd read that truly fired me with excitement and emotion. I could hardly cease reading and rereading it. I couldn't forget it, and I have not forgotten it. It was not only a simplified version but also a narrative version that carried the action to a conclusive finish well beyond the confines of the original. Not until I was almost through high school did I hear that the _Iliad_ was not a fascinating small novel for children but a large epic poem. And not until I read it somewhere along the line in college did the knowledge sink in that the subject of the work is the wrath of Achilles and not the exhausting war against Troy. My disillusionment and surprise were not easily overcome. So deeply affected was I by my first exposure to the work in that small volume that I still find it difficult to accept that in Homer's version there is no Trojan horse, no death of Achilles, no final victory and sack of the city. It may say something about my frame of mind that I preferred Hector to Achilles and felt more compassion for Priam than for any of the heroic Greeks, and that in Shakespeare, I liked Hotspur better than I did the prince, and still do, more even than Falstaff\u2014and I like to think that _everyone_ of right mind feels the same way.\n\nI wrote my next book report on the _Iliad_ , to my delight and to that of my teacher. I decorated the cover with tracings of drawings from the pages of the thin volume. And the month after that, I wrote another, completely different book report on the same work, submitting it with other illustrations. As gently as possible, the teacher took me aside to stress something I had missed: that reading rather than writing was the objective of those assignments and that more than one report on a book was not the practice. It could have been about then that I moved on to _Tom Sawyer, a_ work that didn't really please me at all.\n\nThe tendency to be unaware of matters that should be obvious has stayed with me. Going for a master's degree in American literature at Columbia University after the war, I worked for more than a year on a thesis on the Pulitzer Prize plays, a trivial, unfruitful subject that I'm surprised now was approved, without bothering to find out what a thesis was supposed to be. Not until long afterward, when I was asked in conversation what the thesis of my thesis was, did I realize I'd had none. It was accepted anyway. Some time after that, while occupied with writing the latter portions of _Catch-22_ , I applied for a job of minor executive status in the advertising-promotion department of _McCall's_ , then a women's service magazine with a huge circulation, and was required to submit to a battery of psychological tests that included the Rorschach. I found myself dilating at almost inexhaustible length on each of the whole blots on each of the cards, less volubly on the color ones, barely remarking on any of the details. I scarcely noticed them; I could see only one big picture. If God is in the details, as more than one contemporary authority continues to maintain, I think I've passed Him by in everyone's work but my own. I think it was with intuitive good sense that when I finally applied myself seriously to the writing of fiction, I skirted the requirements of convincing detail fundamental to that genre called realism.\n\nI have one more classroom achievement I'm eager to boast of. In one of my schools, either primary or secondary, we read _Treasure Island_ and were assigned to bury a treasure of our own somewhere on a map entirely of our own devising. In my map\u2014inventively, I must say\u2014I marked the various sites with the names of well-known people of the time. I had a Rudy Valley, a W. C. Hill (a well-known radio commentator), a meadow called W. C. Fields, and a forest called Lefty Grove (a famous baseball pitcher on the Boston Red Sox). I would have had a Veronica Lake, if it hadn't been too soon. Certainly I had a Grace or Victor Moore, and other geographical features with place names of that kind. A landmark was a racetrack called Steeplechase. Outside a woods called Helen Twelvetrees (a former screen actress), my treasure was buried in a meadow named Luna Park. It was all in the region of Pancreas, and on the Isle of Langerhans, one in that body of pancreatic cells whose name I had come upon adventitiously in some mystery novel brought into the house shortly before by my brother or my sister. You entered the territory through some mouth or other\u2014I might have used \"gorge,\" too, but I doubt I was that clever\u2014and made your way downward through the alimentary canal. The teacher\u2014named Miss Perks, I remember, a substitute who came for a day and remained for the term\u2014was greatly impressed. I had guessed she would be.\n\nHIGH SCHOOL AND PUBERTY brought significant changes\u2014to our attitudes, responses, and social behavior\u2014and so, as might be guessed, did Hitler, Mussolini, and World War II. We reeled into adolescence with different increments of biological experience and velocity, the girls, naturally, maturing sooner and evidencing signs more prominently: A girl named Sonia showed the fullest bosom in the seventh or eighth grade, whereas a Sea Gate girl named Ruth, whom I had a crush on, slender, dark, and slightly taller, was showing hardly anything at all. But all of us who went on to Abraham Lincoln High School rather than to Brooklyn Tech or to a vocational school began our secondary school education on the same day.\n\nThe school was some distance away, and for the first time in our young lives we found ourselves associating closely in classrooms with strangers from other parts of the world, even if those other parts were only contiguous parts of Brooklyn. There were the boys and girls from that second, Italian section of Coney Island, whom we now came to know much better; along with them were pupils from those bordering territories that at first were as alien to us as Alaska would have been. The pupils from Brighton close by, who were Jewish, too, were most like us\u2014the girls on balance a bit better groomed, but the boys no less rowdy and ill-mannered than we were, better at sports because better coached. Naked swimming in the school pool one gym period a week was an indefensible oppression. The only guys who didn't seem to mind were those with very big dicks. The school was just over three miles from where we lived, too far to walk if we didn't want to, and, except for a few revitalizing afternoons each spring, we didn't want to. Our Surf Avenue trolley line bore us into the amusement area, turned up to Neptune Avenue, and dropped us at a corner almost at the school itself. Those from Sea Gate had a longer way to go, for they first had to make their way out of their enclave on foot or by internal jitney. For a few more nickels a week we could journey back and forth by a private bus that departed each morning from a spot on Mermaid Avenue and returned to that same place each afternoon. These buses were individual personal enterprises; the school, though far away, took no responsibility for getting us there. When last I looked, a neighborhood acquaintance from one of the social clubs, Novack by name, was making a good adult living as owner of one, perhaps more than one, of these school buses.\n\nThe school, though large and relatively new, was already overcrowded, and for the first year after enrolling, all of us in the entering class were obliged to attend classes in an annex. The Lincoln Annex occupied the top two floors of a typical elementary public school of red brick with white stone trim apart from Abraham Lincoln's main grounds. It was packed with students\u2014boys and girls, young men and women, from a strange and somewhat haphazard assortment of ethnic and occupational backgrounds we hadn't encountered before. Taken all together, we were a motley crew. My recollection is that all of us were white. The school was in an unfamiliar Italian residential neighborhood, and we moved about in it cautiously. Fortunately, we didn't have to move far: The Italian grocery store supplying our first hero sandwiches for a nickel with Italian salami, baloney, or ham on Italian bread was just across the street, and the tracks for the streamlined trolley to which we transferred from our own were just a couple of blocks away.\n\nIn the high-school annex and afterward in the main building, we attended classes with students we hadn't, with perhaps a couple of chance exceptions, known before and who hadn't to any fuller extent known each other. Along with the increase in the number of new student-acquaintances, our choice of elective courses further contributed to the diffusion of our Coney Island social sets into the general school population. For a language to study, many of us first chose Spanish in the belief that it would prove easier than French or German; two of us, one the son of the local pharmacist, picked Latin in hopes of becoming a physician, and he eventually did. The boys, with very few exceptions, decided on an academic curriculum, which was meant to guide them toward a college education almost none then desired; the girls, though with many more exceptions, did, like my sister, Sylvia, select a commercial course directing them toward inevitable, traditional jobs as secretaries and bookkeepers. I made room in my program for a class in typing. Although I had no typewriter closer than an old one in Irving Kaiser's apartment downstairs, I felt in my bones that if ever I was to succeed as a hard-boiled news reporter or novelist or playwright\u2014and I expected that to be soon\u2014I might succeed more quickly if I knew how to type. I was one of a handful of boys in a room full of girls I hadn't previously met. I lusted for several who fancied tight skirts and sweaters (I'm afraid I still do, and for those now often pictured in advertisements in slinky shifts) and was in love with none.\n\nIt surprises me today to contemplate my complete lack of interest then in working on the school newspaper, the _Lincoln Log_ , or contributing to the literary magazine, _Cargoes._ Seymour Ostrow, who also was thinking of becoming a hard-nosed journalist and who later, after the war, became a softhearted criminal lawyer instead, succeeded in becoming a reporter for the _Lincoln Log;_ I didn't even try. The few friends I made who also read books and were intensely occupied with the literary magazine spoke effortlessly to each other of writers such as Chaucer, Keats, and Yeats; these were names I don't think I'd heard before. Yet I was preoccupied always, although without remarkable productivity or technical expertise, with the writing of short stories and sarcastic, humorous (I thought) nonfiction pieces. These were works I didn't want anyone connected with the literary magazine to know about. Mr. Grumet encouraged such creative efforts in his advanced English course, and with his assignments I was able to hold my own. My favorite author for a time was not Chaucer, Keats, or Yeats but Damon Runyon, of whom not many people, if they know of him at all, are apt to think highly any longer. My favorite humorists later on (recommended to me by Danny the Count as \"Benchley\" and \"Wodehouse,\" as though every lad in the world routinely knew their Christian names) became Robert Benchley and P. G. Wodehouse, and these were quantum leaps forward to nearly inexhaustible delights. My favorite source for the best in contemporary American fiction was _Collier's_ magazine\u2014until a friend of my sister's, who, upon learning from her that I was interested in writing and reading, presented me with a hardbound copy of Irwin Shaw's collection _Sailor off the Bremen_ , and then my ambitions turned more serious, and my assumptions of brilliant and immediate fame were moderated accordingly. It must have been about then, after Irwin Shaw, that I began my never-ending tussles with the texts in the magazine _The New Yorker_ , where his fiction often appeared.\n\nNotwithstanding a fear of failure at the literary magazine and my facade of indifferent reluctance, in 1939, when I was sixteen, still a schoolboy, and Russia invaded Finland, I wrote a short, short story about that war and a young, heroic Finnish soldier defending his post with ingenious tricks I've forgotten against Red forces overwhelming in mass. I showed this effort to nobody. After hours of hard labor, and with copious erasures necessitated by the constant growth of my vocabulary, I got the story down on white paper on the old typewriter in Irving Kaiser's house. I thought it was great. In fact, I not only wrote the story, I submitted it\u2014I mailed it away\u2014for publication in _Collier's, Liberty_ , and the New York _Daily News_ , our local newspaper of huge circulation that then also published fiction regularly.\n\nIt was not accepted.\n\nThose were my first rejection slips.\n\nNo, I don't still have them.\n\nNor have I kept my first letter or two of acceptance. The first of those came six years later, for a short work written in my tent in Corsica and also tapped out on a borrowed typewriter, this one belonging to a recent replacement pilot who was out flying combat missions while I, my missions completed, sat safely on my ass and wrote stories on his keyboard. The next was for a freshman theme at the University of Southern California\u2014but those can wait until I'm at least out of high school, can't they?\n\nI CAN DIVULGE NOW that during both my early schools, elementary and secondary, hardly a year went by in which I did not for at least part of a term have a secret and serious, nonsexual crush on one girl or another: June, Ruth, Hannah, Gladys, Mimi, Naomi, another Ruth\u2014who can remember them all? Each had what I idealized as an irresistibly pretty face. I used to wonder with dumbfounded amazement why all the boys were not in love with the one I was in love with at the time. Usually, my infatuation was with a girl who sat alongside me at a desk between me and the teacher, so that I could stare at her dreamily, in profile, to my heart's fulfillment. On days these girls were late or absent, I might sink into a miserable dread of disappointment and rejection. These loves were romantic, not carnal, and it was fortunate, so far as I know, that they weren't returned. Otherwise I might have undergone a brief early marriage and a quick early divorce\u2014instead of the long marriage I enjoyed, one of thirty-five years, and the long, unhappy, embittering divorce, which took more than three years to formalize. Luckily, my schoolboy crushes were over by the time I entered college; by then I was already married and spared those heartaches.\n\nOn Fridays, I acknowledge now with some pride and shame, with that indelicate pride _in_ shame come upon too often in confessional autobiographical pages like this one, I wore to school for a while my official Boy Scout uniform. Troop meetings throughout the various neighborhoods were held Friday evenings, and girls who were Girl Scouts came to classes that day in their uniforms of green, too. It was the fashion. My membership in the Boy Scouts was for me a freak occurrence; mostly till then I had obstinately resisted participation in organizations and organized group activities, and I still do. In grammar school I wouldn't once join the GO (our \"General Organization\" to help support student activities) and perversely refused to go along on voluntary excursions to zoos and botanical gardens; in high school I passed up the class trip to the 1939\u201440 World's Fair, and when my team won the intramural softball tournament after a somewhat spectacular outfield running catch by me for the final out with the winning runs on base, I neglected to drop by the athletic office to pick up my PSAL pin, for I knew I would never wear it. I have no idea what happened to my Phi Beta Kappa key or, for that matter, to my Air Medal or bombardier's wings. I never voted in a school election. I don't vote now. I haven't gone to a school or air force reunion and did not attend my college graduation ceremony. For a couple of fruitless years I was enrolled as a voter in the Democratic Party under the gullible impression that I could influence the choice of candidates, an American delusion still besetting millions.\n\nThose two were about my only lapses. But the Boy Scouts for us was more social than spiritual, and I was soon having fun at their games as a patrol leader. In high school, an Italian girl with the shortened name of Mimi was a Girl Scout. She was different from the taciturn others in my galaxy of stately belles. She was of pale complexion, and she was talkative, witty, and forward, likable rather than beautiful. Direct. We kidded with each other a lot, and we were able to make each other laugh. One Friday, both of us in uniform, she suggested that we plan an overnight hike with just a friend of hers and another boy. Because I didn't quickly grasp what she was suggesting, I declined with some faltering evasions. If I had known for sure what she had in mind, I probably would have declined more abruptly. I was backward that way, she was not. Her warmth toward me seemed to lessen. She knew more than I did, and I was too innocent to learn.\n\nALMOST WITHOUT PERCEIVING IT, we were growing up. And many of the resulting transmutations were pervasive and subtle.\n\nEarlier, in grade school, a difference of a year or half a year between us meant merely a difference in class level and mattered not at all in the hierarchy of the block and the street; now, the difference in age brought a difference in school locations\u2014with different classmates and acquaintances, different loyalties, different teachers, homework, enthusiasms. Old relationships loosened as new friendships formed. Those left behind had to feel left out of much that was going on. Friends made friends with people from different parts of Coney Island, and they became our friends, too. Boys from different streets soon were coming together to organize themselves into new and better punchball and football teams and grew as interested in girls as in athletics. More and more we tended to hang out with guys who were friendly with the girls we knew; girls, too, became more interested in themselves as girls, and in boys, too. Jeannie Goldman from across the street joined a social club with girls from down the Island who had gone to the public school in the Italian section. She and I never did get to mingle socially, to the presumed regret of her mother, who'd been fond of me through my childhood, and mine, who'd always liked Jeannie, and later esteemed her all the more after the accident of the broken hip. Geraldine Scharf on the next block joined a girls' club, too, and I never found myself with her at a party again. The boys these girls found were generally older. Our girls were generally a year or so younger. I can remember no strong social ties with Irving Kaiser after I moved into high school one year before he did, although I continued to use his typewriter and we continued to know each other until we each followed our separate destinies into military service.\n\nWhen, while still in high school, a bunch of us finally got together and chipped in for rent, furniture, and phonograph for a social club of our own\u2014my first social club, Club Hilight, was located in the cellar of a two-family house\u2014just about all the members were on the same grade and age level, and close to half were fellows from down the Island I hadn't known in elementary school. And the girls didn't have to be from our neighborhood, either. On weekend evenings they came from Brighton, Flatbush, East New York, and elsewhere and were flattered by invitations to drop down to our club room to dance and neck. In the summer they'd arrive on the Island from everywhere. I believe it close to a fact that until I went into the army, I never took a girl out on a date, and not many of the other fellows did either. It is still a magic of biology to me how the sexually advanced of both genders managed from earliest years to sense each other out, to recognize and acknowledge each other's mutuality of procreative desires and intuitively come together for joint satisfaction. Cliques established themselves on that single basis of sexual headway: Periodically, a special allegiance would form between two or three males who had latched on to a girl agreeable to putting out for all of them in sequence on the same occasion, and they, even when with the rest of us, would cluster together as though in conspiracy and talk\u2014not only about what they were up to with her, but also about everything else\u2014furtively. The dedicated marijuana smokers would cling together, too, as did the heroin users later on. To the disappointment of those who were still too young or otherwise excluded from one coterie or another, the old ties of the block and the street were melting away.\n\nAnd suddenly there were those boys and girls we'd known forever who belonged to no group at all: the boy without personality, the girl who wasn't attractive and couldn't live comfortably with that knowledge, the odd ones, the quiet ones, the nervous ones, even the studious ones, the crippled ones who couldn't play ball or dance the lindy hop. Mursh the Hopper. The stutterer. These were the friendless ones who didn't fit and automatically were being left out of everything, without pity, without attention. It has not been much different since.\n\nLife was turning real in ways we could no longer ignore. In 1939 Germany invaded Poland from the west, Russia invaded from the east, and the big war began. In 1940 our Selective Service Act went into effect, and young men in the neighborhood began to fade from sight and return on furlough in military uniform. George Mandel, a Club Alteo senior while I was still a junior, was called up early, and his gorgeous \"wheels\" disappeared with him. As an illustrator, originator, and writer in the burgeoning field of comic books, he'd been earning very good money while still very young, and he began to show up in a latest-model blue convertible with automatic top and hydromatic transmission, the first we had seen and the niftiest car we had ever been permitted to caress with our fingertips. Henny Ehrenman, closer to me in age though older, also soon went, to an airfield in Colorado to which I was later assigned for a while. Home on furlough, he described only a good time in the western saloons of Denver with women more permissive than any we had known, or known of. None who were drafted or enlisted seemed to mind. At seventeen I didn't think I would ever have to go. Some of the draft-eligible ones, while they waited, found defense work doing things with a metal alloy called monel; not long after, as I was waiting in turn, I had a couple of defense jobs myself. Others bided their time in a factory manufacturing a fabric called chenille. But a cousin of mine who was only slightly older was killed in a motor accident on an army air base. And after Pearl Harbor they were shipping men overseas. After Pearl Harbor, I smugly took for granted, along with a few million other innocents, that once \"we\" were in it, the war would soon be over, before I could be called up. \"Those Japs must be crazy,\" I remember saying to the friends I was with on the afternoon of Pearl Harbor day. \"We'll wipe them right out.\"\n\nIt was known even before I went overseas that Abie Ehrenreich, an aerial gunner my age who probably had enlisted immediately for him to be in battle so soon, had been in a plane shot down over North Africa and was already a prisoner of war. George Mandel went into Europe with the infantry, succumbing to the allure of the action when he had the option of remaining Stateside as a member of a permanent party unit in an instructional capacity. He will not disagree when I say he should have known better. In Holland he was shot in the leg in an ambush. Recuperating in a hospital when hearing of the Battle of the Bulge, he checked himself out to rejoin his unit and shortly afterward was hit in the head by a sniper. This was a wound whose numerous distressing effects have never entirely left him. General George C. Patton would have been proud of him. My George will not be pleased to hear me say that.\n\nLife was real, life was earnest, and it was turning harsh and earnest for several of us in other ways as well. One boy we had gone to grade school with would finally die of the \"weak heart\" that had compelled his exclusion from all physical-education classes; I remember another from the Alteo seniors who passed away early from Hodgkin's disease, the first time we'd heard of it (a closer friend, Lou Berkman, was fifty-two when he finally gave in to that one); a friend from the Boy Scouts died of leukemia. Neighborhood youths would grow up unaccountably mean and vicious and begin hanging around with toughs from other neighborhoods. Izzy Nish could be wicked. Smokey was sweet-natured but could be dangerous. It was best at all times to keep far away from Louie Schwartz, an inordinately pugnacious young prizefighter. The legend had it that when he moved far upward in class for a bout with \"raging bull\" Jake LaMotta, even his mother went to _shul_ with hundreds of others in Coney Island to pray that LaMotta would not knock him out quickly but carry him the distance and pound him mercilessly. Single girls would surprisingly turn up pregnant when there still was massive odium attached to that condition and force boys several years older into unwanted, short marriages to effect legitimacy. One girl of sultry, exquisite beauty and a year younger than I, was so obviously pregnant by the time of her high-school graduation that the administration was of two minds about permitting her to take part in commencement exercises. The kid from Brighton we had teased as a sissy would indeed turn out to be homosexual, before we even understood what that was. Many of us were already working after high-school classes at jobs we liked only a little\u2014I, an exception, got a kick out of mine. Others were dropping out of school as soon as the law allowed, at age sixteen, to go to work at jobs they didn't like at all. We could stay up late now, almost all night if we wished, and in summer in the amusement area we could stay up late enough to view the overflowing trash baskets as the crowds thinned, to see the litter of watermelon rinds and chewed corncobs on the ground amid the ruins of hot-dog rolls, and to comprehend that Coney Island is a rather unclean place when the ticket booths close and the lights go out.\n\nThe aromatic foods that had been fried and grilled turned greasy. In the early hours of the next day the odors in the street already signalled decay. Even the fresh breezes from the sea, which had awakened keen appetites earlier and stimulated the other senses, could no longer bear away those repellent effluvia of garbage. We had already realized that in winter Coney Island was in the main a lonely, dark, and windy place for people grown too old for homework, roller skating, or playing tag. In winter, marijuana smokers I recognized would often huddle for warmth and each other's company in the lobby of my apartment house, which had no doorman, of course, and was open to anyone who wished to walk in. I knew the smell. I knew them by name. Nobody chased them away.\n\n# 4  \nWork\n\nOF THE VARIOUS JOBS I held before going into the army, just one, that of a uniformed Western Union messenger, was exotic by Coney Island standards. My post as a blacksmith's helper was the more surprising, but by the time I became one, at least a dozen other Coney Islanders had gone south to work in the Norfolk Navy Yard in Portsmouth, Virginia. Yet not until I tumbled into this new situation did any of us appreciate that there were such things as blacksmiths still in existence except at a horse track, let alone in an up-to-date, state-of-the-art, vast American navy yard. Others from Coney Island, already working in the Norfolk Navy Yard, were by then in jobs equally remote and unforeseen\u2014helpers to steamfitters, machinists, sheet-metal workers\u2014in an arcane profusion of factory hangars spreading just about everywhere on the level land except the dry docks, and there painters and riggers were toiling continually at the vessels. An \"E for Efficiency\" banner floated above the main entrance to the navy yard. This had been earned even before my arrival.\n\nWe were housed in individual rooms in single-story barracks, in a stretched-out, labyrinthine sprawl of clapboard bungalows that at times seemed infinite. A Coney Island man slightly older than the rest of us, who was already a licensed plumber, drove down in his auto. Better paid than we were and more experienced in such matters as mating, he lost little time finding a girlfriend, a cheerful, plump, blond young woman from one of the soda shops in town. He conveyed her into his quarters in our barrack for a little while one evening, and the very next day was directed to move out. A friend I made from a different part of Brooklyn also fell into some kind of trouble over a girl. Late one afternoon after work, the police came and took him away for questioning; the next day all his things were gone, and so was he, back home, without a parting word to anyone. Our friendship had been founded largely on our corresponding developing appreciation for classical music. On calm nights in Virginia we could tease in music on our little radios even from station WQXR in New York. Novices as listeners, we both took indignant and vociferous exception to more experienced critics who extolled Bach and Mozart at the expense of tuneful romantic compositions like those of Tchaikovsky and C\u00e9sar Franck, which we then preferred. I would guess that his tastes, too, have broadened by now.\n\nI have to emphasize that it was not insurance against the military draft that drew us southward to war work in the navy yard in Virginia, which afforded no deferments to mere helpers. After Pearl Harbor, almost no one I knew of wanted to evade service. Even pudgy Marvin Winkler took himself into the marines as soon as he was old enough to be vulnerable to conscription. (He married _before_ he was old enough, he reminded me recently; because he was under twenty-one, his mother needed to accompany the couple to the marriage bureau at the city hall and with her signature and person provide consent to the union. No such parental authorization was required for the bride.) Nor was it patriotism, either, that led us to Virginia. Neither outside the army nor in do I recall hearing patriotic statements from anyone but our official propaganda sources, and these were issued to us as lessons in \"orientation.\" Rather, we were drawn almost feverishly to our jobs in defense work by the chance of a colossal elevation in pay over what we had been accustomed to making elsewhere: At the navy yard, we would earn a salary of _a dollar an hour_ for an eight-hour day in a forty-hour week. On top of that was the boost to time-and-a-half for overtime on weekends: twelve dollars a day for each Saturday and Sunday if we chose to work, adding up to a total of sixty-four dollars for a seven-day week! In our jobs in the city, we had been lucky to make as much as twenty dollars a week; rarely did any of us make more. Thirty was a fortune, and a fortune befalling only those lucky few with uncles or older brothers or brothers-in-law already prospering at something and in a position to hire.\n\nIn Virginia, naturally, we chose, starting out, to work weekends, for that was where the big money lay. The toil was physical, and that was new. A snapshot of me I sent home after just a few of those seven-day workweeks was (with some puckish intention) pathetic. The photo caused my whole family to wish I would give the job up and come home even sooner than I finally did. In the picture, I'm just back from another day at the navy yard and even skinnier than I was when I departed from home. In my unpressed, oversized work shirt, trousers, and metal-lined factory cap picked out with a superabundance of caution by my brother, Lee\u2014both of us had absolutely no experience with this breed of outfit\u2014I contrived for the camera to look cadaverous, hollow-eyed, and anguished. With a hint of an elfish leer, I was affecting the remorseless exhaustion of a creature on its last legs. The pose was not altogether fake. Work in the blacksmith shop was truly tiring. The heat was high, the manual labor usually hard, and the smudged sweat soiling my shirt and staining my face in the photograph was the real thing.\n\nNot long ago, my sister was reminded of another, much earlier small drama about my clothes, of which I have no recollection. This had to do with a neatly packed suitcase and a two-week trip from home to a country summer camp to which, startled, I found myself being sent away. The camp operated as a charity for underprivileged city children, and I qualified. Probably it was Lee who learned about it and applied on my behalf. I remember the camp well enough but nothing about the suitcase. What amazed and tickled the others in the family was that upon my return after two weeks away, my suitcase was packed precisely in as good order as when I had carried it off. I simply hadn't thought to unpack it. From the top I had taken my toothbrush and comb and a change of clothes for sports the first day, and I had managed with just those and my travel outfit for the two full weeks. My explanation was that, after all, no one had told me to unpack.\n\nAnd no doubt it was so much easier to leave the suitcase alone. All my life it's been the same\u2014been so much easier to let other people attend to all such duties for me. In my book _Closing Time_ I say of a character, Yossarian, that he couldn't learn to make a bed and would sooner starve than cook. _That_ is autobiography.\n\nThis summer camp, supposedly a treat, was experienced by me as a lonely and puzzling ordeal pockmarked by ceaseless physical inconvenience: The cot was not my own bed; I had to make it up myself; we got busy each day earlier than I wanted to; the lake was cold and less buoyant than the ocean I was used to; the footing was slimy; there were insects I didn't know about; and I was scared of bees. The ecology of the woods was mysterious and unappealing. Was I glad to get back!\n\nCoincidentally, as I learned much later, Mario Puzo was following a parallel course at approximately the same age, but with a reaction opposite to mine. The camp he attended was a different charity, sponsored by the _New York Herald Tribune_ Fresh Air Fund. And he reviews his junket into the woods as perhaps the best time in his whole life\u2014until, I surmise, the success of _The Godfather_ was assured (his third novel, by the way). But Mario escaped to country camp from the stifling cauldron and molten cobblestone streets of Hell's Kitchen in Manhattan.\n\nI was from Coney Island.\n\nFLY FORWARD THIRTY YEARS for a possible link between that trip to camp, my first from home, and the leaden gloom with which I have been filled every time I've prepared as an adult for a trip away. There's nothing specific that might account for these emanations of despair. But I do hate packing, the mere thought of it; I loathe unpacking. Weeks before a trip, I find myself brooding in torment over which suit to take, the right shoes to bring for each pair of trousers, the correct tie to match each shirt and jacket, the appropriate shirts. I determine not to take too much and always do. My piercing anxiety when I leave home is that the automobile chauffeuring me to the airport or railroad station will forget to come or break down along the way, that I will misplace or lose my ticket or passport. As a result, I'm usually there at the airport at least an hour sooner than anyone wants me to be, including me. I'm usually prompter than prompt for all appointments. Once aboard a plane and in secure possession of the seat reserved for me, all dread dissipates, and I feel I've already triumphed; whatever venture I am about is already a success. I have no fear of the journey itself, although I will try to make it a point\u2014in a superstition rejected in childhood and thoughtfully reclaimed as a young adult in wartime\u2014to inconspicuously cross the fingers of both hands on landings and takeoffs. I grew very much into the habit of doing this in the air corps as my combat duties progressed toward fulfillment, and I did so in dead earnest on my last flight to Naples for the steamship taking me back. (No one knows I cross my fingers, and I would sooner embrace the tortures of hell than have a single soul find out.)\n\nI suffer none of these forebodings coming home, except perhaps when landing in a thunderstorm, when I am most likely to remember the precaution. But I do hate to unpack, to separate the things to be laundered from those for the cleaners, the medicines from the toiletries, the magazines from the real mail, the real mail from the junk mail. I proceed reluctantly with all this in petite stages spanning at least a couple of days. And I frequently yearn for a mother or an older brother or sister to do it all for me.\n\nPEOPLE IN CONEY ISLAND with work in Manhattan have to travel far to get to it. Not as far as Norfolk, Virginia, but farther than you or I would want to travel today, unless we had to. Then we had to.\n\nFirst, for about fifteen or twenty minutes, there was the Norton's Point trolley ride to the Stillwell Avenue subway station, the trolley stopping at every corner coming and going during the morning and evening rush hours. People from Sea Gate had to spend another fifteen or twenty minutes just to get to the trolley stop when setting out. Obtaining seats on the subway train in the morning was not a difficulty for us: The last stop on the line coming out, we were the starting point going back and the first ones to board. People coming home who caught the train at Times Square, the start and the finish for the three express lines running back and forth between Coney Island and the city, enjoyed the same benefit at the end of the workday, but the crush of passengers was great and you had to step lively to be among the first to sprint furiously into the vacant carriage nearest you and dash for a choice seat, facing forward and next to the window. Just one station down from Times Square, at 34th Street\u2014Herald Square, where Sylvia got on and off for her job at Macy's department store for nearly forty years, finding a seat of any kind was already a stroke of luck. The seats were usually completely taken by the time the doors closed there and the train was on its way toward the third express stop downtown, 14th Street\u2014Union Square. Those who had no seat when the train left 14th Street were condemned to stand in uncomfortable close quarters\u2014uncomfortable especially for women\u2014until the train had rumbled across the bridge joining Manhattan to Brooklyn and was rolling deep into different residential areas served by the separate subway lines. Then, people who did have seats began vacating them to dismount. The subway rides in rush hour between Times Square and Coney Island took almost an hour\u2014the Sea Beach line was most direct and lasted a few minutes less, but we tended to go home on the first train to arrive\u2014and were seldom the happiest parts of anyone's workday. It seemed to me then a perpetual tragedy and a scandalous economic outrage\u2014and this idea was not original with me, I knew\u2014that to have to travel so long just to get to work, or even, for that matter, to ever have to go to work at all, was a dismal and unjust fate for enlightened mankind to suffer.\n\nIn the mid-1950s, when at _Time_ magazine, I discovered without commiserating and with some surprise that many even of my outwardly affluent colleagues faced a journey of some length to their place of labor. I was past the age of thirty but still callow when I began my advertising sales promotion work there, writing copy and producing related visual aids to assist others in the selling of advertising pages. Soon I made the acquaintance of a family member of one of the cofounders of _Time_ magazine, the initial venture in what had flourished to become the universally respected publishing megalopolis by which we both were employed. A young man named Britten Hadden, who died very young of a streptococcus infection, had been a partner of the young Henry R. Luce in the creation of _Time._ I was mildly bewildered to find that my contemporary Hadden, Peter, had his home by preference outside the city, in Long Island, in a place called Locust Valley, whence he undertook to travel daily to his work as a space salesman at _Time_ , going by automobile to his train station, by train to the terminal in Manhattan, and by taxicab from Pennsylvania Station to the Time-Life Building (which then rose facing the ice-skating rink in Rockefeller Center), submitting himself with free will to a daily commute whose total time was probably close to the one from Coney Island to Times Square. I wondered why he did that. I learned also from our Research Department, which relied to a heavy extent on statistical information released by the government's Census Bureau and several allied agencies, that there was an established group in their taxonomy known as a Class A suburb, and that Locust Valley was definitely one of them. I, together with my wife and small family (a daughter, Erica, and a son, Ted), was already comfortably ensconced in Manhattan and could get to work easily by subway or on foot, in ten minutes one way and twenty the other. I ultimately accepted the fact that many corporate executives on very high income levels, there and in other companies, _chose_ to spend a considerable allotment of their daily hours traveling back and forth to work rather than live in the city. The publisher of _Time_ then, James Linen, was among them; so was just about every other person of exalted position there whom I can think of now. I thought it bizarre that people of such means should live where they did when they could easily have afforded to live where I did, in an apartment on the Upper West Side of Manhattan, and get to the office or back as quickly as I could. I felt none of the pity for them I had once experienced for my Coney Island workers of old, and for myself. In fact, I could only think they were nuts.\n\nI soon found one other respect in which I thought them berserk\u2014an obsessive and near-addictive passion for golf. The buoyant enslavement to golf-playing was epidemic in the business departments but mercifully sparing of all us writers and the commercial artists in the Promotion Department (perhaps because we couldn't afford it; it was bruited about, maybe even accurately, that the company picked up all the costs of golf club membership and related expenses for the space salesmen), and it affected me then as a token of some psychological disorder. Today, of course, everyone knows that men drawn to golf are unhappy at home and have difficulty bonding with women.\n\nTHE MEMORY I RETAIN most strongly of Pete Hadden hasn't to do with advertising work, mine or his, but with a perfect baseball game pitched by a Yankee named Don Larsen during a World Series\u2014no hits, no errors, no bases on balls, twenty-seven batters faced, twenty-seven batters retired. Such a feat was without precedent in a World Series game\u2014it has not been repeated\u2014and it has been extremely rare anywhere else. In many of the Time Inc. offices each fall during the World Series there were personal table radios brought in and installed for the duration, to enable people to go on listening to every game at work when they could no longer do so at the bars in the nearby restaurants in which they had spent their long lunches. About half an hour after that perfect game had concluded, I recall watching Pete, for whom I was at that time preparing a project, drift as though stunned into the open office in which we in the sales-presentation department worked and draw near me with a reverent expression of awe. \"Today,\" he said, in a hushed voice, \"we saw history being made.\"\n\nIt appears to me today in a sentimental mirage that, at _Time_ magazine in that golden age of the 1950s, and at _Life_ , too, and maybe even at the arid business publication _Fortune_ , the World Series no sooner ended in early fall of each year than the Christmas office parties at once began. Or perhaps it was the commencement of Thanksgiving festivities. It doesn't matter. No holiday was needed to replenish the _joie de vivre_ that prevailed during business hours in those corridors in those days. There were always sufficient birthdays, anniversaries, retirements, transfers, and whatnots to suffice as a good reason for a party somewhere on our floor. The liquor would flow, the canap\u00e9s would appear, the socializing would spill over after business hours into small groups in one nearby bar or another. Small wonder we were often reluctant to hurry home. The women at work there were lively, educated, and bright; the only sad faces normally to be found were those on people with a hangover from excessive drinking the night before, or that same day at lunch, or on some man at last drearily aware, in a mood of inescapable self-discovery, that he did in fact have a serious drinking problem and was in truth an alcoholic. It was a rumor then, possibly no more than a rumor, that the company maintained an ongoing arrangement with the Payne-Whitney Clinic at New York Hospital for the discreet admission and treatment of important employees in that predicament as well as of those with various other types of serious depression. There was another rumor that when the company first instituted its most liberal medical benefits program, one that in its introductory phases provided reimbursement for psychiatric care, it transpired that more than 10 percent of the total number of our freewheeling, easy-living Time Inc. employees around the world were receiving one form of psychotherapy or another. The Time corporation was a prodigal and indulgent workplace in which people often broke down (and one of the people teetering nervously on the brink was the character Bob Slocum in that excellent novel _Something Happened)._\n\nAn old friend from _Time_ , Gerald Broidy, helpfully recounted to me not long ago a business occasion of that period in which he was peripherally involved\u2014organizing a dinner meeting at which a number of highest top-management executives of Time Inc. met with a number of management officials of equal status at the _New York Times._ The dinner was held in the New York apartment of one of the dominant figures at Time, whose wife was a member of the family owning the _Times._ Broidy did not attend. Of course, drinks were served beforehand. Early the very next morning, he learned by telephone from his exasperated counterpart at the newspaper that one of the imbibing nobles from Time Inc. passed out as soon as the meal began and slept soundly through the remainder of the meeting with his head on the table, to the surprise and consternation, I'd guess, only of those dignitaries from the _Times._\n\n\"Which one could it have been?\" the man at the newspaper indignantly demanded.\n\nAnd Broidy, with the habitual laughter that is still his, answered, \"Any one of them!\"\n\nI believe that it was for a sales call by Pete Hadden that I inventively blended my recent literary education with the immediate business need and devised an opening for his easel presentation that was soon in demand by other space salesmen and is remembered to this day as something of a classic. I forget which product or company was the prospective customer for new or increased advertising, but on the opening board of the easel demonstration, I offered an enlarged copy of the Tenniel illustration for _Through the Looking-Glass_ showing the Red Queen skimming over the ground at top speed with Alice in tow, and with her explanation in a caption: \"Now, here, you see, it takes all the running you can do, to keep in the same place.\"\n\nOther salesmen were soon requesting this picture as an introduction for their own sales calls. From me personally there followed new written presentations with this opening, one attempting to persuade the Simmons mattress company to forget consumer publications for a while and employ our newsmagazine to sell mattresses to the hotel and motel owners who subscribed in disproportionate numbers, and another to convince the space buyers for H. J. Heinz to use _Time_ to advertise their individual and miniature customer servings of ketchup to the proprietors of diners, coffee shops, and hotels.\n\nIt was through such inspired ideas that I received a $1,000 raise at the end of each of my first two years, climbing from my starting salary of $9,000 to $11,000 (if memory serves me correctly). After my third year I received an explanation but no increase: There was a company-wide salary freeze imposed as a consequence of the starting-up costs of the new magazine _Sports Illustrated._ I did then what I had done twice before since coming back to New York City from my teaching position as instructor at Pennsylvania State College. I purchased a new hat\u2014a gray fedora with a dark band; bought a new white-on-white shirt with French cuffs; dug out my cuff links; and roamed about secretly, exploring for a better-paying job. I soon found one doing the same work at Look magazine at an annual salary of $13,000. A friend from _Time_ , Arky Gonzalez, who earlier had left for more money at _Reader's Digest_ , cautioned that things were very different in the rest of the business world from what they were at _Time_ , and that I might soon miss being there.\n\nI still do.\n\nAt one of the annual sales conventions, to which I was taken to assist with the projection equipment for the assortment of slide shows presented, I witnessed Henry Luce himself sternly notify his audience that, to his mind, the business of the Time-Life corporation was publishing and that he was not going to fold _Sports Illustrated_ merely to assuage the doubts of the investors on Wall Street or allay internal concerns about salary increases and the size of continuing company contributions to the employee pension plan.\n\nAt the festive Time Inc. conferences I attended at deluxe resorts in locations like Florida, Nassau, and Bermuda, on a table immediately inside the entrance to the dining room for breakfast each morning were huge glass vats of Bloody Mary and Brandy Alexander mixes standing ready for all arrivals; and at the golf course every afternoon, I was told, always close to the tee at each hole was a large drum of ice with chilled bottles of beer to fortify the thirsty Time Inc. sportsmen as they braved the challenge of the next fairway. A young space salesman with a name good enough for John Cheever or F. Scott Fitzgerald, Seth Bidwell, was head of an out-of-town office largely, I had heard, because he was a three-handicap on the links and would cut an estimable figure with the customers he played with. Another space salesman with a great name was Royal Peterson II; slender, tall, courteous, clad always in suits well chosen and well tailored, he fit his name to perfection. Yet another salesman, I'd been told, had thrown up into the swimming pool at a sales convention I did not attend. With the company so paternalistic, he probably wasn't discharged; more likely, the accident was overlooked, or else he was eased sideways into a new position that precluded his appearance at future conferences. Notwithstanding all the foregoing, the company kept growing in one way or another during the three years I was there\u2014 _Time_ proudly hailed the milestone of reaching two million in paid circulation\u2014and continued growing hugely during the forty years that followed, despite my departure. It merged ultimately, against my wishes, with Warner Communications, formerly Warner Brothers, a motion-picture entity psychedelically resplendent in its own size and reputation.\n\nBut sometimes nothing fails like success. The organization multiplied in value and deteriorated in worth. More money flowed in from television shows and ancillary facilities than from publications. The magazines are today without special dignity or significance, even among other magazines, and I suspect it can't presently be as enjoyable working on them for merger-minded businesspeople whose bristling work ethic, one doubtless beloved by institutional investors, can credibly be refined to automatically trigger the feisty comeback \"They're not here to enjoy themselves, they're here to work.\"\n\nIf there are colorful personalities left in management, they are colorful invisibly.\n\nIn what seemed a disgracefully rapid sequence not long ago, perhaps the same week, maybe in the same breath, the news-conscious controllers of the company owning the news-sensitive magazine _Time_ announced in public forum a single payment of some $70 million to a single executive, one acquired in the merger, and the discharge, for motives purely of economy, of some two hundred employees. Such are the generous freedoms afforded by American free enterprise. And such is the character of the enterprise.\n\nAN ADVANTAGE I ENJOYED commuting from Brooklyn to my earliest job in the city with Western Union was that I started for work directly after my last class at Abraham Lincoln High School. At the end of my schoolday I would board the subway at the Ocean Parkway station of the Brighton Line, and since I was already past the trolley ride from Coney Island and the first two train stops, the time of the trip into the city was shortened by some thirty minutes. And in the early afternoon, when I left school for work, the trains were not crowded and I could pick my seat.\n\nAt the first office to which I was assigned, I stored my uniform in a locker in a central changing room for a number of satellite offices in the business districts of downtown Manhattan radiating out from Union Square and from the Flatiron Building on 23rd Street, where Broadway crosses Fifth Avenue. When I was transferred uptown, the uniform hung in a small closet in a single, small office in what was then the General Motors Building, at 57th Street and Broadway. Finally, I kept the uniform at home after I was shifted to an office in Brooklyn not far away in lower Flatbush and worked only weekends. At long last I could sport my dashing Western Union messenger garb right in Coney Island as I rode on my bicycle back and forth between home and office, sometimes even delivering telegrams right in the neighborhood. This never happened, but a hovering dread I forged in my imagination dealt with the risk that sooner or later I would have to deliver to a family I knew a yellow envelope stamped with two red stars. In the days of the telegram, before the facsimile machine and the ubiquitous telephone rendered such swift communications all but obsolete, the two-starred message brought news of tragedy.\n\nIn short order I was able to, and I did, ride a bike just about everywhere without using my hands, even turning corners into streets broad enough to encompass a sweeping arc. My bicycle came as a present from my Aunt Esther as soon as she learned I needed one to keep my messenger's job. The bike emerged from the stock in the toy store owned by her and my Uncle Julius in Williamsburg, which itself was a long haul away. I traveled there by train. I don't remember on whose borrowed bike I had first learned to ride and whose I had used to begin at the new place, but by the time my Aunt Esther came through, I could ride competently, and no one lacked faith that I could accomplish the long ride back to Coney Island unharmed. Aunt Esther, whose personality seemed the dominant one in her family, mapped instructions directing me to Kings Highway, which turned out to be a busier and farther-reaching thoroughfare than I or anyone else but Aunt Esther and Uncle Julius had dreamed it was. They owned an automobile and drove when they visited us. A daughter, Janet, was studying medicine, a stark anomaly then for a woman, and a son, Philip, was soon a lawyer, later a judge. They were relations on my father's side but remained close to us after he died. All of the cousins on both my mother's and father's side were considerably older than I but not older than my brother and sister. That could be a reason I haven't remained close. My brother was distant from them by temperament; my cousin, Nat Siegel, also on my father's side, may already have been an accountant when he gave me the _Iliad._ His older brother, Morris, was already a physician, and that could have been reason enough for Lee to remain aloof. Sylvia, on the other hand, was warm and convivial as always, even to participating with them in a cousins' club when there were still enough of them around. Lee seemed always more outgoing to people he knew slightly than to those he knew well. An unusually modest man, at nineteen or twenty he took off from home without warning one day and made his way as a hobo to California and back over the summer. He didn't perceive anything extraordinary enough in that feat even to mention it to his son, Paul, and seemed genuinely surprised that both of us were impressed when I finally brought it up, after Paul was already grown and married. Eventually my mother confided to me with some bitterness that she had been terribly embarrassed in the past by Lee's habit of remaining in his room when relatives were present. Aunts and uncles conversed with my mother in Yiddish when they visited. Only now do I understand that when they spoke of \"Itchy,\" which they frequently did, they were talking of my father, Itchy being the diminutive of the name Yitzak, which is Yiddish for Isaac.\n\nThe day I got my bike was the day Lou Gehrig retired from baseball, for I remember listening in Aunt Esther's house on the radio to his tearful farewell as I ate the lunch she had prepared for me. Pedaling away on my new bicycle in a spirit of vigorous exuberance, I began the long ride back through mysterious urban regions I had never been in before. Brooklyn is a vast realm, and the trip seemed an unending adventure through fresh surroundings then presumed to be safe. I drank it all in eagerly. At last, after half an hour or longer and with a pervading stir of relief, I recognized ahead the elevated train station I knew at a familiar strip of Kings Highway, a congested shopping area with all kinds of stores and a Dubrow's cafeteria opening at dawn and serving terrific food into the early hours of the next day. Shortly after, as forecast, I arrived at Ocean Parkway. In ten minutes or so I rolled past my high school. In familiar regions now, cycling the final thirty minutes down the broad boulevard into Coney Island and then home was a triumph.\n\nI have remarked before on a congenital unconsciousness of mine about details: Not until recently did it penetrate my mind that Kings Highway in Brooklyn is really _the Kings_ highway and that the adjoining borough of Queens is really the Queen's. I don't know which British king and queen they memorialize and don't ache to find out (you can look that one up yourself, if you care more than I do). I would take an offhand shot at William and Mary, because they alone of the kingdom's royal couples seem to go together (Victoria's Albert was a mere consort), and I would probably be wrong.\n\nI do know that in addition to the bicycle, my Aunt Esther gave me a dollar and, as usual, instructions for my mother to feed me more red meat. Either I seemed anemic to Aunt Esther or she was infused with a fear of anemia and convinced that a diet rich in beef would always defeat it. Aunt Esther never failed to give me a one-dollar bill whenever she saw me. At the cemetery, when my father was buried, I remember she gave me a dollar. I remember little more. It was the first dollar I ever had, and I don't know what I did with it. (At five years old, I couldn't do much of anything.) Others, Freud tells me, often unknowingly associate currency with gold and gold with excrement, and vice versa, whether they want to or not. I associate money with life, and an absence of money with death. I can't help it. I, too, can try a guess at the reason. But I still can't help it.\n\n# 5  \nOn and On\n\nAT THE AGE OF SIXTEEN I was old enough to obtain my \"working papers\" and seek legitimate employment.\n\nFederal legislation governing child labor enacted not long before proscribed as unlawful the official hiring of anyone younger than that in just about any formal capacity. (Shirley Temple, I must suppose, was a cheeky and heartwarming exception.)\n\nAt sixteen one also was allowed to quit school, but I never wanted to do anything like that, and neither did any of my friends. Quitting high school would have been a shame to the family and a scandal in the neighborhood. As it happened, not until I was past twenty-eight and face-to-face with the forbidding requirement of a Ph.D. if I was to continue teaching did I finally back timidly away from further institutional education and take steps to try my hand in the real world. A doctorate in literature is not a simple commitment. By then, I had been married for six years, had one small child, and had served two rather tiring years at very low pay as an instructor in the Department of English Composition at Pennsylvania State College\u2014twelve classroom hours weekly with four sections of Expository Writing, Comp. 5, each of first-year students who, with only a few exceptions, were no more interested in learning the practice of expository writing than I was in teaching it. By then, too, considerably encouraged by the publication in national magazines of a few short stories of mine that were written and had appeared while I was still a college undergraduate at NYU, I had a stronger ambition to write literature than to go on studying it. (I must note here, in an unconvincing pretense of humility, that in 1948, when most of these few stories appeared, Norman Mailer published _The Naked and the Dead;_ Gore Vidal, younger than both of us, had given us his war novel _Williwaw_ and that year _The City and the Pillar;_ and Truman Capote, also younger, had attracted national attention with his _Other Voices, Other Rooms.)_\n\nBut back in May of 1939, while still attending high school (and possibly already typing preposterously and zestfully away in Irving Kaiser's apartment downstairs on my short story, which was aimed first at the New York _Daily News_ , about my youthful Finnish soldier ingeniously and heroically thwarting the invasion of his native land by Soviet forces), I turned sixteen and set aside my creative passions to rush off enthusiastically to apply for my working papers.\n\nEither Sylvia or Lee had a friend who managed a Western Union telegraph office in the Bensonhurst part of Brooklyn, which is close to Coney Island. In that era just about every residential district had a Western Union office. Through that contact, I secured an interview with a Mr. Shotter in personnel at the Western Union headquarters building at 60 Hudson Street in Manhattan. And through him I was hired as a part-time messenger to work afternoons on weekdays after school and half a day Saturdays when needed. Since the five-day workweek was not then standard, I was wanted most Saturdays to help service the many businesses that stayed open. From a Western Union supply department, equivalent to the kind of quartermaster outlet I was to encounter later in the military, I drew my jaunty uniform. The vision has grown indistinct with the passage of time, but I believe it included brown leather puttees of a type recognizable to us from photos of cavalry officers in World War I and the Spanish-American War. (I am almost positive I wore leather puttees at some stage of my life, and I know it wasn't in World War II and not in the Boy Scouts.) A minimum-wage law was already in effect, and I believe I have it right that the minimum at which I started was twenty-five cents an hour, which increased to forty cents while I was still with Western Union. Working, on average, three to four hours each afternoon and four more on Saturday, but not on business holidays, my take-home pay ranged from five to six dollars a week. When I was let go after more than a year, I took my direction from some older fellows and followed the trail they had blazed to the unemployment-insurance office. To my great surprise, I was eligible. To my greater surprise, I received a hefty, guaranteed six dollars a week, the minimum from that source stipulated by legislation, for thirteen weeks. Here was as much or more than I had been earning by working, and without the unavoidable expense of twenty cents a day for trolley and subway fare.\n\nOh, what a wonderful world indeed this was that had such blessings in it, I felt like singing out loud each time the postman delivered my six-dollar check\u2014but for only those thirteen weeks.\n\nBy agreement at home, whatever I earned was my own for allowance. Lee and Sylvia were both already out of high school and working in the city. And at that time my take-home pay of six bucks a week was plenty.\n\nThe assumption in the family when the opportunity for employment arose was that I would be hired for the Bensonhurst office, and it could have been about then that I began training myself furiously on borrowed bicycles. Instead, I was placed in an office in the city, one on East 17th Street just west of Fourth Avenue, a few blocks from Union Square, where I had never been, and a few blocks down from the Flatiron Building, formerly an architectural highlight in the city, of which I had never heard. Manhattan was very different in that period (so was I), and so was the area fronting Union Square. With such bustling department stores as S. Klein's, Ohrbach's, Mays, it was a shopping district second only to the one with Macy's and Gimbel's twenty streets uptown, which in turn was second (if two \"seconds\" aren't unacceptable) only to the elegant department stores on Fifth Avenue in the fifties. Today, all these stores on 14th Street at Union Square have closed and haven't been replaced. Union Square hasn't nearly the same character without them, and without the fiery labor-movement rallies of the past. But Macy's is different now, too, without Gimbel's just down the street. Fifth Avenue isn't nearly the same today, either, and neither is the Champs-\u00c9lys\u00e9es or, as I found out with sharp pangs of disappointment on a recent vacation trip, the good old Via Veneto in Rome, on which I had spent so many happy, roistering hours during my military service overseas. The spirit of pleasure in which I had luxuriated there during the war and a few times after is still unmistakably the ambience of the city, but it has moved from the Via Veneto to other districts, and I haven't had time to learn where they are.\n\nI believe it was on street level in the Flatiron Building itself that a large, central Western Union office was situated, with a locker room providing space and facilities for forty or fifty of us to change into our work clothes and back at the staggered hours dictated by our different work schedules. I smoked my first cigarettes there\u2014Spuds, then Kools. Prior to my starting work, I hadn't troubled myself to conceive that there might be such things as telegraph offices in the business districts and that there were floods of messengers pacing back and forth on telegraph business. In movies they always pedaled bikes. On the day I began, I walked from school to the subway station as rehearsed, checked in and changed at the one office and reported for work at the other, and my seasoning in the city commenced.\n\nI should explain for out-of-towners that when New Yorkers speak of the city, they often imperiously mean only Manhattan.\n\nI learned a good deal very quickly. I learned first, as already noted, that telegrams were delivered on foot, that those delivered to offices required no signature of receipt, and that we not only delivered telegrams but were just as often summoned by electrical signal and code to specific reception rooms in office buildings to pick up messages typed on yellow forms to be carried back for transmission by teletype to addressees elsewhere. For those who don't know and do care, I will explain that the words of a telegram came out of a teletype machine on unrolling paper strands about half an inch broad; these were pasted onto yellow forms that were folded into envelopes by the operators and handed to the people working the counter, who assembled them in groups by destination and then handed them for delivery to the next one of us in line on the bench or at the counter. I detected in due course that, as a consequence of our guaranteed hourly minimum wage, there was no longer any real incentive to hurry, unlike the way it worked in the previous epoch spoken about by old-timers, during which they were paid by the number of items handled\u2014piecework, so to speak\u2014and might earn more, if lucky, or might earn less.\n\nI learned, with disgusting effort, to overcome the revolting tastes of cigarette smoking inevitably afflicting the amateur and to acquire the habit. (Sixteen years later, past thirty, I kicked the habit in a squall of pique when the price of a pack rose another penny to twenty-six cents the same week the first of the lung-cancer reports appeared.) I learned that Fifth Avenue neatly divides the East Side of Manhattan from the West Side, a handy detail that once learned is never forgotten, and that the streets and avenues are nicely numbered in consecutive order, until the numbers decline to zero, where streets begin carrying names and the trials to memory begin. Years later, I stumbled across an exception to the methodical numbering of streets, a tangled and astonishing intersection in Greenwich Village where West 4th and West 10th Streets cross as a result of convoluted determinants that, once understood, are forgotten instantly. I never learned what most of the offices I regularly went into and out of produced to justify the cost of their maintenance. A carpet company was a carpet company, but many of the others were confusing. The American Woolen Building, about the biggest and busiest in our sector, was an enigma to me for the longest time, for it was impossible to believe that all the companies I called on there were in some way connected with the woolen textile business.\n\nOne time on service at another office for two days I discovered the Washington Market, of which I had never heard, situated then on Manhattan's Lower West Side and the transfer center for most of the fruits and vegetables shipped into the city. Thus began my first contemplations of the nearly countless transactions implicated in the movement of vegetables and fruits from the far-flung acres in which they had been nurtured into Lily Dashevsky's family's fruit store on Mermaid Avenue in Coney Island, where my mother could purchase what she needed to bring into our apartment for us. Another time I worked a bit farther uptown in a quarter alongside the Hudson River in which there were establishments committed, believe it or not, to the slaughter of pigs. I did not believe it, even while there. This was as startling to me as it was to everyone I told, but I know it was true. The odors were thick and repugnant, the workers looked grim as jailers, I soon comprehended what the squealing noises deep inside the buildings were about, and I was thankful I didn't have to work there a second day.\n\nMy closest acquaintance in this first Western Union office was a short and slightly older part-timer who lived in the Bay Ridge section of Brooklyn, where Jews did not live and would not have lived comfortably if they had to. Through him, one sweltering summer afternoon after work, I saw how a nervy, experienced kid his age could go confidently into a saloon and come out with a cold and sweating silver pitcher of beer and a couple of glasses. That day I learned also that few things are as lusciously refreshing as ice-cold beer on a sweltering summer afternoon after half a day's Western Union work.\n\nRiding back into Brooklyn with him on the West End subway line, the only line carrying people to Bay Ridge, I would often witness a mass cultural phenomenon that calls into question the complacent dogma with which we mislead ourselves into believing that we arrive at our biases rationally and independently. At a major express stop, all the people traveling into Bay Ridge and environs left the train for a shuttle extension, and all the copies of the afternoon Republican newspapers, principally the _World-Telegram_ and _Journal-American_ , got off the train with them; while the liberal, Democratic newspapers like the _New York Post_ , and later on a newcomer called _PM_ , continued onward with their readers into the Jewish regions of Borough Park, Bensonhurst, and Coney Island.\n\nExcitements and surprises in the telegram-delivery business were few. With one part of my brain I played at walking about my routes at an unvarying pace that would carry me from street to street without halting for a traffic light, adroitly varying my sequence of calls when I beheld the tide of traffic turning against me, and I became quite good at that. There was a thrill one lucky afternoon when, passing the Players Club on Gramercy Park South with a telegram for delivery nearby, I spied and instantly recognized the figure descending the outdoor steps and moving across the sidewalk to hail a taxi: it was Charles Coburn, a character actor familiar on sight from movies like _The Devil and Miss Jones._ There was no mistaking him. The office buzzed with excitement when I hurried back with the news, and I couldn't wait for the day to end and the trip home to Coney Island to be over to tell everyone in my family and on the street that I had seen Charles Coburn in the flesh. I prayed they would believe me.\n\nAnother day, an older, full-time messenger\u2014older meant that he was out of high school, about twenty or twenty-one, and would start worrying about military conscription after the Selective Service Act was passed in 1940\u2014returned to home base from a call to an apartment near Gramercy Park with the juicy tale confided to the rest of us that he had been admitted by a man and woman and been requested by the man to engage in sex with the woman. He revealed nothing more, other than that, of course, the woman was good-looking. It mattered not one whit to me whether he was speaking the truth: In my per-fervid, young imagination, here was another glittering reward lying in wait for me as I continued into my maturity.\n\nI am still waiting.\n\nSix years farther on into my maturity, as a sophomore of twenty-three in my fiction-writing course at NYU, I made use of the incident in a short story called \"World Full of Great Cities\" that never attained publication in a periodical but has several times been printed in anthologies. The last time, I recall, it was taken by a Japanese publisher for publication in English for pupils in Japan studying American ways and literature. For that story, I fortunately hit upon a point of view that proved apt. By then I was familiar with Sherwood Anderson's \"I Want to Know Why\" and Ernest Hemingway's \"My Old Man,\" and I was deliberately imitating them in kind: a young man's discovery of something disagreeable in the adult world which he had not expected. There was this difference, though: In those works, the reader knows what the uncorrupted and inexperienced young narrator does not. In mine, the reader has no more insight into motivations than I, the narrator-as-Western-Union-boy, had when I first heard of the incident, had when I wrote the story, or have to this day.\n\nTHE UPTOWN OFFICE to which I eventually was transferred lay on a low floor of the General Motors Building, occupying the whole Broadway block of West 57th and 58th Streets, near the active traffic crossings of Columbus Circle, and allowed the three of us there to window-watch the systematic creation of a tall apartment building in the days when construction workers still used red-hot rivets tossed by tongs from the small furnaces ascending punctually with the scaffoldings, glowing bolts that were caught by others in metal cones for hammering into the junctures of the orange-crusted girders hoisted by cranes. The apartment building was at 240 Central Park South, and I know people who are comfortably quartered there today. The trip from Brooklyn to this new place of work was somewhat longer, two local stations past the last express stop at Times Square. But riding home, I was always sure of a seat.\n\nThe new office was small, a single room, and served only companies located in the same building, an arrangement chiefly sustained, I judge now, by the volume of business generated in the few uppermost floors entirely occupied by divisions of the General Motors Corporation. Here, too, I worked only half days, changing after school into and out of my uniform in the gentlemen's lavatory on the landing outside our office. I can't remember if I worked Saturdays as well.\n\nThree of us proved enough to handle the work. In charge of the office was Miss McCormack, a blond, pretty woman of small stature, who was enduring repeated problems with a boyfriend about whom Tom Fitzgerald, who'd been there with her awhile full-time, teased her with a friendly, gentle irony. She was the sole operator of the teletype machine, processing incoming and outgoing messages. There was no real need for her at the counter; Tom and I could by turn easily take charge of the telegrams and electrical requests for pickups and organize the routes through the different floors of the building ourselves. Tom Fitzgerald was a tall, equable, taciturn young man of upright posture who moved calmly and was wise, very wise, beyond his years, which were about twenty-one. Soon, certain events outside the office, for which he bore no responsibility, would be causing him to worry about the draft, for the Selective Service Act of 1940 mandated the registration for active duty of all males between the ages of twenty-one and twenty-six. And before the attack at Pearl Harbor, people did worry about the draft. If they worried after Pearl Harbor, they did so covertly, and males of eligible age still in civilian dress were prompt with apologetic explanations they frequently felt obliged to give.\n\nI'm sure I was there through that summer of 1940, because I easily recall the torrents of telegrams to multiple addressees gushing out from the executive floors of the General Motors Corporation to delegates at the Republican National Convention championing the selection of Wendell Willkie as the candidate to run against FDR for the office of president. Willkie won the nomination and lost the election. Originally from Indiana, Willkie, like Roosevelt, favored support to Britain in the war and opposition to Hitler; in addition, Willkie accepted many of the New Deal programs and therefore had little ammunition to employ against Roosevelt other than the two-term tradition going back to George Washington. That was an election in which employees in Woolworth's stores were compelled by their employers to wear WILLKIE FOR PRESIDENT buttons. He lost anyway, and I was not surprised. A devoted reader by then of liberal newspapers, I couldn't see why anyone would resist Roosevelt for a third term, and later, a fourth. I couldn't believe then, because at the time I hadn't heard of any, that there were such things as human beings of good intention who were Republicans, and I'm still not sure. By now I have the same doubts about Democrats.\n\nThe ages of eligibility for the draft expanded progressively in both directions, upward and downward, as our immersion in combat grew greater, and when the government, in 1942, announced the lowering of the draft age to nineteen after I had already passed eighteen, I and a few friends from Coney Island concluded that the time was ripe to enlist. A persuasive enticement was the promise of picking the branch and the division of the service we preferred. It was a promise kept for those of us choosing the U.S. Army Air Corps; no one I knew personally who chose the air corps was subject to any staggering shock of the kind suffered later by such as Kurt Vonnegut and Mel Brooks, who both were wrenched from one military specialty, for which they were in training, and thrust into the infantry for shipment overseas as combat replacements, for which duty they received no training.\n\nIn this new office with Miss McCormack and Tom Fitzgerald the work was more interesting to me because there was so much less of it. I had time for my homework, for extended conversation (which remains a favorite personal recreation), and, by copying Tom, for reaping the fruits of a surprising preoccupation with calligraphy. Working like an artisan of old, Tom was fashioning an elaborate handwriting for himself. Seated beside him as he concentrated, I awaited his decisions on each capital and lowercase letter in our alphabet and labored intently at duplicating his choices when he made his final judgments. By the time I was transferred into the Brooklyn office the following spring, his handwriting had become my handwriting. In time I let myself forsake it, embarrassed by the flamboyant cursives implying deliberate effort and suggesting affectation, and today from Tom Fitzgerald's new handwriting of 1940 I preserve only the long looping sweep under the J in my signature and modified vestiges of the same below my y's, g's, and j's, and in my capital H.\n\nI don't know what happened to Tom in the military or afterward, or to his handwriting, either. He had a Jewish girlfriend he alluded to occasionally. He was friendly with a young, mild-mannered, redheaded, Irish elevator operator who lived near him in the Bronx, and I was friendly with him, too. I was on good terms with all the elevator operators\u2014they recognized me as a fellow workingman. A way to remain in their good graces, I had learned very quickly at the first job after one gruff rebuke, was always to walk down to the lobby from the second floor rather than force an elevator to stop for you there, and always to use the staircase in either direction for a single floor. Today, from the social vantage point I enjoy, and probably from yours, too, it seems almost antediluvian that well-brought-up young white Americans would settle in contentedly to such occupations as full-time elevator operator or messenger; but that was then, and the social world you and I know today is only one aspect of the real world, a single facet, and the rest of today's real world is one more reality from which we, with our privileges, strive to remain safely insulated, and of which we definitely wish to keep ourselves securely ignorant.\n\nI FOUND OUR NEW SETTING in the city more congenial than my first. On the thoroughfare just behind our building, on Eighth Avenue, was a spacious Horn & Hardart Automat. The Automats were favorite eating places of mine then and\u2014a year later\u2014even more so, when I found myself by pure coincidence back in a more exalted job in the same building, with my preferred meal their succulent chopped-sirloin plate, which included any two vegetables and a roll and butter for twenty-five cents. Their baked beans were an exceptional treat, too, as all alive today who have eaten them will attest, and so were their mashed potatoes (I wish I had a plateful of that chopped sirloin with vegetables here right now). I also liked their orange and lemon glac\u00e9 cakes for a nickel as dessert. A Nedick's was close by, too, for a hot dog and orange drink now and then, and on the corner of West 57th Street was a Chock Full O' Nuts sparkling with cleanliness, at which for a nickel you could enjoy a nutted cream-cheese sandwich on raisin bread. This was not especially filling, but would serve until mealtime and in that respect was more satisfying than a hot dog, whose savory juices and textures always left me wanting another one, and still do.\n\nInside the General Motors Building was an office run by two youngish men manufacturing formulas for soft-ice-cream machines, and as often as not, when I stopped in with a telegram, they asked for my judgment of a new alteration in the recipe for chocolate or banana which they were seeking to improve. Other times when on that floor I would pop in anyway to offer my help. Another small office, one that functioned largely by telegram, was occupied by a well-mannered and well-dressed horse-race tout. A few times every week he would address the same message recommending the same horse to a few hundred subscribers. I assumed he performed reasonably well, since he stayed in business, but I was too young for bookmakers, and my twenty-five or forty cents an hour didn't yield much room for error. One other venture stood out prominently, occupying virtually an entire floor of the building and called the Manhattan Mutual Automobile Casualty Company. There, a very beautiful, dark, buxom, married, mature woman known as Miss Peck or Miss Beck never failed to greet me with a smile and thank me each time I stopped in on telegram business, and I looked forward to calling there just to experience her welcome and to hear her speak to me again with such genuine, feminine warmth.\n\nWhen about a year later, through statistical chances long enough for a whopping prize in a state lottery, I reported to Manhattan Mutual from an employment agency to work full-time at the more distinguished job of file clerk, she didn't know me and didn't recall ever having laid eyes on me before.\n\nIt must have been the debonair figure I'd cut in my messenger's uniform that had caused her to notice me before.\n\nClothes do make the man.\n\n# 6  \nAnd On and On\n\nTHE TRAFFIC LIGHTS on Ocean Parkway were staggered, our first encounter with this stratagem for allowing automobiles to proceed steadily at a restricted rate of speed. Separating the six or eight traffic lanes were two attractive pedestrian islands. Benches sat every twenty yards or so beneath the foliage of modest shade trees, and these trees and benches and tended pavement walks ran from the lower end of the parkway in Brighton to its farthermost point at one of the grand entrances to Prospect Park. The sylvan aspect of Ocean Parkway, already established by the time I first saw it from my bicycle, likely was a halcyon dividend from one of the public works programs of FDR's New Deal. Ocean Parkway was perhaps the most handsome of the numerous major traffic routes in the borough, if not in all the city, and possibly it still is. A lane for bicycles ran the length of the pedestrian walks from beginning to end, and I was soon making use of it for my telegram business. Stops were dictated only by traffic lights wherever avenues crossed the parkway, and these were mostly a good way apart. To help cope with the predestined boredom that has proved twin to the confidence arising with my successful performance in most jobs I've held\u2014in just about all of them, propelled by anxiety going in, I've done my best work at the beginning\u2014I set my mind to work constructing a diverting challenge: riding the full stretch of Ocean Parkway between the office and home without having to stop for a single red light. That proved easy; much of the time I did better than the mass of automobiles, whose restless drivers were habitually tempted to speed faster than the staggered lights programmed them to do. Because that challenge, once mastered, grew boring in turn, I introduced the daring innovation of performing the same unbroken bicycle journey without using my hands. Frequently I had to veer into the questionable tactic of decreasing speed to allow a traffic light ahead to change before I got to it and could burst safely through. That was permissible under the rules; I made the rules. Soon I was maestro of that undertaking, too, and capering on my wheels, I basked in the admiring scrutiny of the women on the benches\u2014the mothers and grandmothers with children in strollers and baby carriages\u2014as I show-boated past them, my arms swinging at my sides or, in a more peremptory bid for attention, clasping my hands across the crown of my Western Union messenger's cap, arms akimbo.\n\n\"Look, Ma, no hands!\" I shouted silently, heaping laurels on myself as I sped along.\n\nEven after all this while, Brooklyn remains to me a vast, unexplored territory, now enormously more complex culturally than it was, with the surging and discordant infusion of Asian, Russian, African-American, Arabic, and other peoples, together with Hispanic, brown and black, from Latin America and the Caribbean. I am still awed by newspaper accounts of violent happenings in neighborhoods I've been hearing about for a lifetime and am still unable to pinpoint on a map: Red Hook, Bath Beach, Gowanus, Greenpoint, Sunset Park, East New York, and even, to underline the point, Midwood, which when I was in high school and at the Western Union office just off Kings Highway, was close at hand\u2014but where? I delivered telegrams in Midwood, but with the confused notion that I was still in lower Flatbush, and for all I know now I could have been correct.\n\nWhen Lee married, he and Perle lived first in an area called Crown Heights. With my mother and sister I visited there several times but have only a nebulous idea of its location. I do know, from the newspapers, that Crown Heights at present is home to a large black community and a large Hasidic Jewish community, who live side by side antagonistically. Certainly, my worries about these two communities would be different, but one group would be no less intimidating to me than the other, and I'm glad I live here now and not there. Take Bushwick. If I were to trouble myself at long last to find out just where the Bushwick section of Brooklyn is, I don't doubt that I'd succeed. But I also don't doubt that I'd soon forget.\n\nI DON'T KNOW what we did at my new Western Union office just off Kings Highway when it rained\u2014I have a dim impression of olive slickers issued by the company, but I wouldn't bet my life on it, or yours\u2014and I can't recall bicycling at work in winter. I have no bad memories of this last, brief tour of Western Union duty. I have no memories at all of the people there.\n\nFrom this office I would blithely go wheeling off with my telegrams into familiar neighborhoods, even on occasion my own, as well as areas surrounding the office and my high school that were totally new to me, places with names like Flatlands and Canarsie. Small discoveries still titillate me. I knew that Bedford Avenue was beyond the right-field fence of Ebbets Field, the stadium of the Brooklyn Dodgers, and was the place where baseballs struck over that fence for home runs presumably fell. Yet I was really surprised to find Bedford Avenue extending all the way down into residential lower Flatbush bearing its own name, instead of its equivalent, East 25th Street. Coney Island Avenue, so denominated despite its running into the eastern end of Brighton and nowhere near Coney Island, was East 11th Street, and Ocean Parkway I was finally able to fix into place as East 6th Street. Till then I knew only the streets of Coney Island, which were \"Wests,\" and these went reticulating away backward in a contrary direction, my own.\n\nThe longest avenue in Brooklyn is probably Flatbush Avenue, extending from the Manhattan Bridge, which indeed connects with Manhattan, down in a straight diagonal through the borough almost to the seashore across the border of Queens. I discovered the long, lower part of Flatbush Avenue on my bicycle; the upper half I traveled ten years later when I lived in Manhattan and drove across the bridge into Brooklyn for visits to Sylvia, Lee, and my mother. Sylvia and Lee, each married, then lived close to each other, near the subway stop at Kings Highway. This was near the Western Union office in which, while finishing high school, I had worked ten years earlier. I liked best of all, after my first odyssey along Flatbush Avenue, the long routes out, sometimes all the way into the distant settlement of Gerritsen Beach, a small cluster of one-family houses standing along canals that connected the waters of whatever inlet comes in from the sea or bay there. I hadn't heard of Gerritsen Beach before and I haven't heard of it since. There the names on the telegrams were not Jewish and not Italian and neither were the faces on the people, who tended to be fair-haired. I never failed to experience a thrill each time I pedaled past the military air-base installation of Floyd Bennett Field, at a time, in early spring of 1941, when nobody I knew had ever been up in an airplane or ever expected to be.\n\nAnd just three years later, a mere three, in early spring of 1944, I was able to thrill again to Floyd Bennett Field shortly before going overseas, when I landed there in a plane with a flight of B-25S on a training flight from our base in Columbia, South Carolina. (The revisionary idea arises that it could have been Mitchell Field on Long Island we flew to, but the time elapsed doesn't change, and the point is the same.)\n\nOnce on the ground we had some time, and I led two surly bombardier acquaintances of mine (I couldn't call them buddies, for we weren't that close), Bowers and Bailey, who'd come in on other planes, into the city for a meal at the famous Lindy's restaurant, then on Broadway and about 50th Street. I had been there before on furlough (and as something of a glorified, though anonymous, serviceman celebrity) right after receiving my bombardier's wings and commission as a second lieutenant. They were both from Chicago\u2014along with the _Chicago Tribune_ , their favorite newspaper, they were fervid isolationists and hated FDR, and England, too\u2014and hadn't seen anything like Lindy's before, and they weren't entirely at home with its raucous splendor or mainly Jewish menu. (I forget what they ordered, but my habitual meal there in daytime was a chopped-liver-and-smoked-turkey sandwich on rye bread, along with a sour pickle and sour tomato or two, and strawberry cheesecake for dessert.) They had both read Damon Runyon, though, and one of them commented that all the men there (including me?) fit the description of Runyon's character Harry the Horse, by which he meant, I was aware, that they all looked Jewish, and of course we did.\n\nIn the Italian homes at which I stopped with my telegrams while still with Western Union as I waited for an adult to acknowledge receipt with a signature or to return with a tip for me, I inevitably encountered a strong, pungent odor unmistakably different from the one in my own apartment, although of that one, through natural adjustment, I was scarcely conscious. Only lately have I come to realize that the special pervasive aroma of which I speak is Italian and that it was ingrained in wallpaper, rugs, plaster, and upholstery impregnated for years with the steaming fragrances of olive oil, garlic, and tomatoes.\n\nIn those Western Union days I often received a tip in return for my telegram. On birthdays and Mother's Day there was frequently more than one telegram for a single person, and so, too, at the catering halls for weddings. The amount of the tip was most often a nickel, occasionally a dime. My biggest windfall could not have been foretold. Late one afternoon outside a house on Bedford Avenue, returning to my bike after a delivery, I was spotted by a frantic young man who pounced at me as though I were a compassionate angel appearing in answer to a desperate prayer. If he gave me a dollar, he wanted to know, would I go into another house close by and sing \"Happy Birthday\"\u2014that is, deliver, as a freelance independent contractor, what was officially called a singing telegram\u2014to a girl named Phyllis at a party inside? I said sure, and I did ring the bell, did enter when bade, did advance inside toward a partying group, and did, standing perfectly stationary with shoulders squared, sing \"Happy Birthday\" to Phyllis, from Eddie. My audience was thrilled and applauded. Returning outside, I signaled to Eddie that the job was done and that his way was clear. I abstained from confiding that an older woman there, likely a mother or an aunt, had tipped me another quarter for my singing telegram.\n\nI found singing \"Happy Birthday\" easy work\u2014lucrative, too\u2014and regretted that I wasn't able to make a career of it.\n\nIn late spring of 1941, with FDR securely back in office, I was scheduled to finish at Abraham Lincoln High School. I had terminated my work with Western Union by the day of graduation. If there was such thing as a prom then, I didn't hear of it; if I had, I wouldn't have gone. To this day, I am leery of group participation and all formalized group activities, and every time I relent I have cause to repent. In protest against the Vietnam War, I went on a march in New York, one in which I was first grouped with other literary people and then ranked in alphabetical order. As a result, I found myself shoulder to shoulder with two literary critics who had already made known their disapproval of me and could correctly adduce, from that circumstance if none other, that I didn't think much of them, either. I was wary of other marches after that and attended only one more, that largest one in Washington, D.C., to which I went in a plane with a group of friends. Erica, my daughter, was there, too, traveling with a busload of high-school classmates. Also on that march was a group of young Catholic college students, all of them females, and the accompanying Sisters from Marymount College in Westchester, where I went to speak about my work as an author the following week, and they were victims of tear-gassing while waiting innocently at a curb for their chartered bus to arrive. I did not even think of going to my college commencement exercises. I did go to my high-school graduation; my mother, Lee, and Sylvia went proudly, too. We had an early dinner together at a restaurant, and after that I took the subway with some friends to a jazz club on 52nd Street to listen to Billie Holiday. We had our own social club by then and knew how exceptional she was.\n\nMy first club, Club Hilight, began in a cellar and then moved to a more palatial home, in a storefront on Surf Avenue, just a block and a half away from my house. One day my brother dropped in unexpectedly to summon me home. I was smoking a cigarette. I had kept my smoking secret from the family, and I couldn't say now which of us was the more flustered. I was caught; he had caught me. Each of us was supposed to do something. Neither seemed sure what that something was.\n\n\"How long have you been doing that?\" Lee finally asked disapprovingly, when we were walking outside together.\n\nI had crushed out the butt. Both he and Sylvia smoked, and I could have retorted by asking how old he was when he began, but I can't be glib when it counts. With a most innocent voice and a most guileless face I can tell convincingly the most far-fetched lie to the credulous, but only when the point is humorous with nothing real at stake. With pressure on, I stammer, I'm dumb.\n\n\"As long as you're doing it, you might as well do it at home,\" he decided. \"You should try not to do anything outside the house that you wouldn't want us to know about.\" And after a pause, he added: \"Here, have one of mine.\"\n\nThis was censure that had a positive effect, and a year afterward helped buttress an inhibition already planted in my conscience, one that I credit with vitiating all the narcotic effects of smoking marijuana, apart from a false appetite. Smoking marijuana the several times I did left me with a sore throat and feeling very hungry. Smoking marijuana was something I definitely would not have wanted my family to know about. I couldn't get the least bit high the times I tried, not then or later, a little while after the war, when I puffed only to accommodate close friends while in their company (I didn't want to abuse their friendship as a \"drag\" or \"bringdown.\") From lessons learned afterward through that eventual, and inconclusive, couple of years in psychoanalysis, I surmise that I couldn't get high because for reasons nestled deep in my unconscious I simply didn't really want to. In truth, I didn't want to be smoking pot at all.\n\nAs for the graduation itself, the festivities had nothing special to do with me. I was not valedictorian (thanks be to God!), won no prizes, and was not singled out by name for any honors other than my diploma. Probably, though, I was the sole possessor of a singular distinction that passed unrecognized. It's more likely than not that I was the only male member of that graduating class named Joey who had received, or was then still receiving, unemployment insurance, raking in my beefy six dollars a week.\n\nThere was nothing more by way of celebration than I have related, no thought of anything even remotely like a bonus or vacation. I don't think anyone I knew of then\u2014not Lee, not Sylvia\u2014had ever gone anywhere on vacation. On Saturday morning, I rested. On Sunday, with Lee and Sylvia counseling over my shoulders, I read the want ads, and on Monday morning, I set out by subway to call on employment agencies and seek my fortune in the city.\n\nMY FIRST EMPLOYMENT AGENCY, the Wall Street Employment Bureau, with a downtown address in the Manhattan financial district on an oblique side street named Beaver Street (it fascinates me that such details take enduring root in the memory), vectored me right back uptown to the General Motors Building for my first interview and onto the payroll of that Manhattan Mutual Automobile Casualty Company, whose comely, maternal, and properly married, dark-haired Miss Beck (or Peck) was unable to believe she had seen me before and signified with a shrug and an unconcerned dip of the head that it hardly made a difference. A new and about-to-be more appropriate crush at the reception desk up front conducted me deep into the back part of the office, where I had never been before. I was spoken to first by Miss Sullivan, secretary to one of the higher executives; she questioned me in kindly fashion, mainly, I judge now, to verify whether I could hear and understand English. After that, her boss needed no longer than five minutes to approve my appearance, I suppose, and I was taken on right then and there to work as a clerk in the file room. My starting salary was sixty dollars a month, which seemed okay. I was elated to have any job at all. I gathered fairly soon that the same wage would be my final salary, too, as long as I remained a file clerk. Had I thought about it, I would have sensed even then that there wasn't much else at Manhattan Mutual that I could quickly learn to do capably, or would want to.\n\nAt the time, age eighteen, I doubt I had any inclination to think so systematically. Like my friends of the same age, I was all but oblivious of the future. None of us knew seriously what we wanted to do or would like to become. Descriptive terms for jobs \"with a chance for advancement\" or \"maybe a good future\" meant little to us, for we didn't plan; we weren't in a position to. The war was continuing, the draft was a stimulating prospect, and we took what we could get. If one of us ever did receive a promotion, bettering himself with a move from one menial level to one less lowly, we construed the event not as one of destiny or reward but as an astonishing turn of plain good luck. I don't remember how much of my sixty dollars a month I gave to the family to contribute to the general upkeep, but I did give some, whatever amount was requested by my brother or sister. They wouldn't have demanded much.\n\nThe file room itself was a spacious cage of chain-wire fence that rose from floor to ceiling and projected into the center of that part of the office floor, a fence through whose square gaps those of us inside could see outside, and people outside could hear and observe us. Because we could communicate freely by voice through the barrier, it was seldom necessary for others to enter our cage, although they often did stroll in.\n\nI learned quickly what the company did: It issued liability insurance to taxis, to fleets, and to independent owners, for claims resulting from accidents, claims for PD's (property damage) and PI's (personal injury), and to limousines as well. I learned quickly, too, and impressed people back home with the knowledge, that at that time the license plates of all taxis in the city began with an \"O\" and those of the private car services with the letter \"Z.\" Each time a cab or one of our insured limousines was involved in an accident\u2014and in New York City at that busy period when cruising taxis were numerous in all five boroughs, there were collisions of varying degrees of seriousness every day and night\u2014a report was made, and a file containing the report was opened and then filed in one of our numerous file cabinets, to be retrieved and filed again each time a document was added. What we did almost exclusively\u2014and there was a bunch of us to do it\u2014was locate and deliver the file to the right desk each time it was requested and return it to be filed again when it was done with. After a case was closed, usually through negotiation, the file traveled downstairs to a storeroom for dead records and into a place in one of the walls of cabinets there. The file cabinets upstairs were metal; these were of paperboard. After a suitable period, these records were transferred to a rented space in a Manhattan warehouse for dead records that were even deader. One of these dead records, too, would come back to life occasionally, and one of us would be dispatched to the dreary catacombs of the warehouse to fetch it.\n\nThere were five of us doing this work full-time, and I became friendliest with Stanley Levy, some two years older than I, who lived far out in Brooklyn near Coney Island, and had, with his black hair and cleft chin, a perceptible resemblance to Cary Grant, and, I conclude from people who knew Cary Grant, was much the wittier. Lou, who'd been there longest, was a slim, friendly Italian from the Bronx; Jerry was Jewish and lived in the Bronx also. Ralph was boss of us all, a spry, springy, hurrying, short young man still under thirty with waxen, wavy blond hair. Ralph, if one went by appearances, always appeared to be doing at least twice the work of any of us, of _all_ of us combined\u2014for he sped about to and fro at least twice as fast, always with a sheet of notepaper in his hand, and then one day he was abruptly let go. We were not told why. That slip of paper, Stanley Levy joked after his exit, never had anything written on it.\n\nEntering the file room as boss of us all came Miss Dunbar from another, distant part of the floor, from around a dogleg where the adjusters, legal, and sales people were quartered and tended to keep largely to themselves. Miss Dunbar was a woman of formidable build of about fifty, with a knowing, disdainful look and a derisive half smile. Things were going to be different from that moment on, she firmly let us know when she entered to take charge, and necessary changes would be made. None of us knew what she was talking about; we had no idea what had gone wrong. And, in effect, things were in no way different. We went about our work as before, untroubled and in good spirits, kidding with those girls who kidded back and with the men who encouraged banter\u2014George Schwartz, Mr. Spiese, Harriet Jackman, Miss Beck, Virginia, Eunice. Miss Dunbar's sardonic half smile and knowing look proved to be no more than that: It was just her look, and the only change occurred when Lou's number came up in the draft and he was forced to depart for military service. There was a farewell Italian dinner in his Bronx home for the other three of us in the friendliest atmosphere (a hell of a long subway ride for Stanley Levy and me), prepared and superintended by a beaming, affectionate mother, affectionate to all of us. This was my first Italian home-cooked meal, and I rashly misjudged. Too polite back then to inquire and make sure, I took the meat balls following the soup and spaghetti to be the main course and prudently (better safe than sorry) helped myself to one, maybe two or three extra; and I thereafter had difficulty with the chicken and vegetables and the sausage stuffing that followed before the fruit and cheese and cake for dessert.\n\nThe storage room for dead records one floor below us, with its couple of old desks end to end and several discarded swivel chairs, was anything but dreary. For us it was a kind of playground. There was one key, and the file room had it. I ate lunch there with the others whenever I brought sandwiches from home, which was often. My mother would prepare and pack my favorites, along with an apple, orange, or banana: On two seeded rolls, always two and always rolls with poppy seeds, there could be canned salmon with grated onion and sliced tomato or kosher salami or baloney, with plenty of mustard. Another treat of fractionally my own design (I liked to think) was sliced chicken\u2014white meat vigorously preferred, then as today\u2014with lettuce and\u2014this was the magic\u2014mayonnaise lathered on the inside of one of the halves of roll and ketchup soaked into the other. I was on the very threshold of Russian dressing and didn't know it.\n\nThere were occasional rubber-band and paper-clip fights in the storeroom or dice games for pennies and nickels or \"underleg.\" (In \"underleg,\" the player swiftly skidded a coin beneath each shoe, so swiftly that neither player could note which side was up. The second player then slid coins of equal value underneath after them. If the coins matched, the matching player won. When they didn't, the first player did. Most times, they split.) You could kill time down there comfortably alone while pretending to still be searching for a file you had already found and go on with the morning's crossword puzzle or complete the entry for the contest in one of the two tabloid newspapers to pick the most winners of the coming Saturday's college football games. Now and then I would choose a file at random to search inside for matters of special interest. I couldn't find any. The storeroom was also a secret trysting place for hasty sexual encounters, for more than one of us, as two of us inadvertently discovered the day I left. So were the two flights of stairs between the floors for even swifter ones. I learned much about office romances from my special friend Virginia, a cheerful, humorous flirt, and exceedingly pretty, too. She was four years older than I and had been through college. Once we were friends, she confided freely about dates with one lawyer there and one adjuster, each of whom threatened to put her out of his car if she didn't prove more accommodating than she chose to be. She lived in New Jersey and did not say she walked back. She also had a sweet and sad nonphysical relationship with a quiet, almost elderly married man, which was confined to a drink after work now and then and an occasional dinner. Nothing could come of this, and neither of them wished that anything more would.\n\nI went to my first opera from the office one Saturday afternoon, a matinee of _Carmen_ at the old Metropolitan Opera House downtown somewhere in the West Thirties. I went alone. My choice was a good one. Through weekly and nightly radio broadcasts presenting short sections of classical-music favorites, I had grown familiar with some of the score\u2014the Toreador melody, the \"Habanera,\" the spirited overture (two of them were already popular in our streets through corruption by doggerel lyrics: \"Toreador\/Don't spit on the floor\/Use the cuspidor\/That's what it's for\" and \"I lost my shirt\/I want my shirt\"\u2014I've forgotten the rest; this last was taught me by Lee, from whom I had also learned all of the stanzas of \"Ivan Tzivitsky Tzizar\"). At the performance of _Carmen_ , I was exhilarated by the dances, the acting, charmed by the amazing entrance of real horses drawing coaches out onto an indoor stage in the colorful procession introducing the final act. I told everybody\u2014real, live horses!\n\nThen I pushed my luck. As part of a naive and eventually futile agenda to initiate my friend Lou Berkman into the peculiar bliss of musical appreciation, I purchased two tickets to my second opera, an evening performance, and brought him along as my treat. Lou was already working in his father's junk shop in Brooklyn, an arena of enterprise far more lucrative than the name suggests, and one that supported two or three of the Berkman families rather well for as long as was necessary. After work, he came uptown by subway to my office and kidded with my perky friend Virginia at the receptionist's desk until I was ready to leave. He paid for my dinner of chopped sirloin, mashed potatoes, and baked beans at the Automat across the avenue, and we proceeded downtown by subway. My tickets were, of course, the cheapest, and we finally found our seats very high up along the side and in the topmost row, from which point we could not see perhaps a third of the stage. My choice this time was unfortunate: _Tannh\u00fcuser._ I was misled by faith and inexperience: The melody of the \"Pilgrim's Chorus\" had been appropriated for our Thanksgiving song in music class in elementary school; it is also the melody with which the overture begins, and I naively thought that such recognition would make it easier to connect with the rest of the musical score. For those who may not know Wagner or may not know _Tannh\u00e4user_ , there is a large difference between that one and _Carmen_ , a large difference and a lo-o-ong one. Because of that evening, Lou never attended another opera as long as he lived, and I have never wanted to hear _Tannh\u00e4user_ again.\n\nBack at work, with the other Lou gone from the file room and Miss Dunbar, unlike mercurial Ralph, forever stationary and watchful, the atmosphere inside our workplace suffered a gradual shrinking of levity. Lou's replacement did not mesh perfectly with the well-knit camaraderie in which the rest of us had previously flourished. And when Stanley Levy's number came up in the draft, too, and he gave notice of his intention to leave, I gave notice with him. We left the same afternoon. Stanley would be going into the army, I would be going to the navy yard in Virginia. On the day of our departure, there emerged through an amusing coincidence the secret that more than one of us had been making use of the storeroom for dead records on the floor below for speedy, romantic alliances during office hours. There was only that one key to the storeroom, and two of us wanted it. Ever pliable, I deferred to seniority and collected my own farewell cuddle on the staircase halfway down, where once again I could not obtain all that my heart wished for in the way of an amatory send-off. Virginia, as customary, wore a good deal of bright red lipstick, and we couldn't even kiss passionately without smearing both our mouths with inculpating evidence. I was deeply in love with her, which was good. I was no longer in love with her the minute after we parted, and that was better.\n\n# 7  \nAnd On and On and On\n\nIN A FAR SECTION of the long shed of our blacksmith shop, into which I never had occasion to go, a huge sea anchor under construction curved a pointed shank into the air. I couldn't imagine how they were building it. Periodically, I would wheel a barrel of small objects over to one of the circular pools for \"tempering,\" and I would gaze with some wonderment at the thick, colossal arm ascending from the ground higher than everything around it. I puzzled even then over the mystery of the massive links of chain needed to anchor the anchor, how and where these were fabricated and joined. The drop forges in my section were mammoth structures, and these produced only such small items as bolts, rivets, and buckles. I still don't know how these links of chain were made and have stopped wondering. Today I worry instead about tiny jewelry chains, how in the world the minuscule filaments are assembled into those flexible segments.\n\nWe swallowed salt tablets all day long. For our health. The population at that time was being urged to consume a lot of salt on steamy days to guard against heat exhaustion. Because of the many burning furnaces in the shop, this seemed unusually important for us. The furnaces were mainly small and stood apart from each other. The cylinders for dispensing the salt tablets were hung like light fixtures all about, on walls and posts. It must have been at least as hot and hard and uncomfortable for the blacksmiths as it was for me, because they worked closer to the fires longer and did the turning and pounding and trimming of the hot metal on the anvils and the dies at the drop forges. My job consisted mainly of carrying one at a time the glowing metal rods from the furnace to the forge or the anvil\u2014I used tongs, of course, and wore the bulky, fire-resistant gloves Lee had picked out for me\u2014each time the man I was delegated to work with that day signalled for another unit, then clearing out of his way the one he had newly shaped. But the blacksmiths were paid much more money than I was, and they were accustomed by life to arduous physical labor. Unlike me, they accepted hard work as natural.\n\nA surprising number of them were from rural North Carolina, the northern border of which wasn't that far away; and a good many of the rest had rambled down to avail themselves of the temporary opportunity of defense work, for as long as that opportunity lasted, from agrarian towns in Virginia. Where else could a worthy blacksmith earn a decent living in peacetime?\n\nI had been prepared by northern friends for the various oddities of speech I swiftly encountered. The most singular of these, unlike any of the exaggerated travesties I had heard in movie theaters, flowed from those who regularly transmuted the familiar (familiar to me) _ou_ vowel sound (as in \"out\") into something else\u2014into _oo_ \u2014so that a mouse became a moose (I never thought to probe how a moose was pronounced locally); and so, for example, the sentence \"There's a mouse running about the house\" became \"There's a moose running aboot the hoose.\" I was much too politic and sensitive ever to joke with the men about this, or even to indicate that I'd heard anything humorously atypical. I have almost always been able (with a sympathetic respect that might be either wisdom or sublimation of cowardice) to at least perceive the basis of the other person's point of view, and to see myself as others might see me. To my mother, and Lee and Sylvia, too, to everyone in our family, it would have been a profound character failing ever to deprecate anyone just for being different or handicapped. Bigotry was a heinous offense.\n\nBecause we tried to see ourselves as others might see us, we tended to be a self-conscious bunch. A household anecdote epitomizing that point related to the Jewish holy day of Yom Kippur. My mother, who in general was nonobservant, preferred that I go to one of the two synagogues on our street late in the afternoon to say the _Kaddish_ for my dead father, more, I'd guess, to keep up traditional good appearances than from a belief that a prayer from me would be of much help to either my father or the Lord. I attempted to do as she wished for as long as I lived at home, which was until I went into the army. She never insisted; that was not her nature. But in early September of 1949, when I and my first wife, Shirley, were preparing to sail to England for an academic year on one of the earliest of the Fulbright scholarships, she did rouse herself to exhort. By then she had long since broken her hip and was limping about with a cane; that accident befell her while I was still in training as a bombardier. With a diffident urgency that obviously was embarrassing to her, she called both of us aside, a calendar in one hand and in the other a scrap of paper on which the date of that year's approaching Yom Kippur was already noted. We must remember, she emphasized to us, to be sure to find a temple in whatever city we were in when that day came and attend at least part of the services. \"Else,\" she explained in her faulty English, \"the people will think you're a 'Comminist.' \" It happened that we were in Paris on Yom Kippur, my wife and I, age twenty-five and twenty-six, respectively, and in Paris for the first time, and our attempt to scour the city for a synagogue was not wholehearted. We didn't stumble upon one; yet I don't suppose that many of the people in Paris assumed we were \"Comminists.\" And it is insulting to God, I suspect, to imagine He cares whether I pray to Him or even knows where or who I am.\n\nOn emergency leave after my mother's accident, I was appreciative of the benevolence with which the various bureaucratic desks of the army joined to secure my furlough in what seemed a matter of minutes, certainly less than an hour. I was midway through my preflight training at the Santa Ana Army Air Base in California, a huge installation in which thousands, perhaps tens of thousands, of aviation cadets were received, examined, evaluated, and assigned to classes for preflight training as pilots, bombardiers, and navigators. The telegram had been sent by my sister. My orderly room referred me to the office of the chaplain, which on the spot arranged a loan to me from the Red Cross to pay for the railroad tickets procured for me by the transportation office, which also awarded me a priority rating for the train.\n\nEntering the hospital in Brooklyn by myself some five days later, I had no idea what I would find. For reasons I don't understand and never expect to, I had constructed the bizarre scenario that I might not recognize my mother and feared that my failure to do so might sink her into deep despair. A couple of dozen beds in the open women's ward of Coney Island Hospital stood before me. Facing the entrance when I stepped in was a bed holding a white-haired woman about my mother's age whose attention I captured instantly. She rose on an elbow to observe me more intently. I stared right back with the tentative beginnings of a smile. Her gaze remained fixed on me and I started across to her. I hugged her gently while kissing her once or twice and sat down. I was appalled that she didn't seem to recognize me or respond appropriately to my name. This was worse than I had imagined. It required a few more uncomfortable minutes of awkward talk for both of us to realize that we had never set eyes on each other before. I glanced about wretchedly. At the far end of the ward I then clearly spied my mother practically levitating out of her bed, plaster cast and all, and waving wildly in furious and frustrated exasperation to attract my attention. She looked exactly as I remembered, and she told me yet again that I had a twisted brain. She said it one more time when I revealed that I would soon be flying regularly in an airplane. It went without saying that she feared I would be killed in a crash.\n\nAfter reading this, anyone who has recently read _Catch-22_ for the third or fourth time might be struck by the parallel between the account of my mother I've just given and an episode in the novel in which Yossarian is visited in a hospital bed by a family of tearful strangers, but I don't remember that I consciously had the former in mind when I was devising the latter.\n\nI HAD AN UNCOMPLICATED TIME with my Southerners in the blacksmith shop. It was easy for me to reason from the outset that my own Brooklyn manner of speaking would appear equally quaint and deformed to them, as it well might to citizens of countless other regions all across the continent, and doubtless still does. We seemed eccentric to each other.\n\nTo all the white men I worked with then, far back in 1942, the Negroes, blacks, or African Americans also working there, however one chooses to designate them, were held in such irrelevant regard and were fixed so solidly in caste that they were never even spoken about, and those there as helpers in the blacksmith shop were never addressed by anyone but me except in terse reference to work. The one or two in my section I most frequently joined with to cooperate on something and with whom\u2014with what I now properly characterize as no more than a patronizing affectation of acceptance\u2014I tried to establish friendly conversational relationships, intelligently withdrew and kept their distance. What surprised me more than the total absence of relationships between whites and blacks was the apparently uniform virulence with which Protestants regarded Catholics. I first learned about this from a dark and roly-poly Irish blacksmith I worked with often, who'd been reared in Northern Ireland, exposed to the enmity of the Protestant \"Orangemen\" there\u2014it was the first time I'd heard that word _Orangemen_ \u2014and was now in the Norfolk Navy Yard in Portsmouth, Virginia, disapproved of and shunned by the others. Once enlightened by him, I plainly saw how he was mistrusted by his coworkers and coolly boycotted.\n\nThe blacks there were disdained and beneath notice, and the Catholics were detested with a touchy hostility. But a Jewish New Yorker like myself was a novel rarity. The local men had never seen one before, and I was the only one in the shop. And I had never been with Southerners. Thus there existed between us a mutual curiosity to be indulged.\n\nAt lunchtime and coffee breaks when we chatted in our group, others might drift close to gawk at me and listen with friendly smirks to our conversations. We always got along just fine\u2014almost always\u2014with much courtesy and consideration expressed by them to me about the mechanical mysteries and dangers of this work that to me was entirely new. As the first duty of my first day there, I was directed by the foreman to move a barrel of bolts to the grinding wheels to mill the surplus fringes of metal smoothly from the heads. I looked at the barrel and I looked at the hand truck, and I didn't know what to do. I knew it wouldn't work, but I decided to try: While the foreman and the assistant foreman observed, I squatted, wrapped my arms around the barrel as far as they would go, and tried to lift. The assistant foreman, a man named Beeman, who rarely smiled at me or anyone else, gestured to me to step aside and demonstrated how simple it was to slide the bottom ledge of the hand truck under the barrel and leverage the full weight back onto the wheels for rolling. I got the hang of that one quickly. He led me to the electrical grinding wheels to make certain I knew how to don goggles and to show me what they wanted done. I proved I could learn that one quickly, too.\n\nA single exception to the harmonious relationships I otherwise enjoyed occurred during one of our daily lunchtime breaks when the general discussion turned to religion and I chose to volunteer the information that Jesus was in fact Jewish and of presumed Jewish parentage. The immediate and united stiffening of the entire circle of white faces was an instantaneous warning that they had never been told this before and did not want to be told it now, or ever. Even my closest pals bristled. And I discreetly chose not to push the point. The symbolic rubric INRI on icons of the crucifixion would not have sufficed to convince them; and besides, I admit, I didn't myself know back then that these letters represented in Latin an abbreviation of \"Jesus of Nazareth, King of the Jews.\"\n\nYears later, in 1962, after the publication of _Catch-22_ , I met Mel Brooks on the summer vacation resort of Fire Island, and I heard him joke, in the hoarse shout that is his second nature, that every Jew should have a big Gentile for a friend. My special friend in our group in the blacksmith shop was the biggest one there, a lumbering, easygoing fellow and the son of the foreman of the shop as well. No one would have dared harass me with him as my protector, but even without him I would have been untouched by any kind of bullying or more subtle abuse. There was no Jew-baiting all the weeks I was there, although there was plenty of raillery about my being a New Yorker and my quaint locutions and barbaric pronunciations. And I took care not to make too much of their being rustic Southerners.\n\nOne man I worked with often, from a farming community in North Carolina, could count and measure quantities with vertical lines in sets of five, the fifth line drawn sideways to cross through the other four, but he had never been taught the basics of division and multiplication. It was one of my functions to work out for him the number of long bars to be requisitioned for a fixed number of small units to be made from them. But he could do the work: With chalk he would measure and mark the lines for the three-foot lengths to be cut from each rod of perhaps thirty feet. And he would arrange them in his furnace and then pound and fashion them into whatever shape they were supposed to take. One day when I was occupied at the cutting machine and at the same time computing amounts with him while he measured and marked, I luckily turned my eyes back to the machine and just did manage to spy the shearing blade descending on my fingers as they fed in the metal. I pulled my hand back in sudden horror, saving my fingers from dismemberment at the last moment. Never again did I shift my gaze to speak to him or anyone else while at that cutting machine, and to this day I count that near tragedy among the most chilling experiences of my life, more frightening than anything that occurred on my second mission to Avignon or any of the other dangerous things that happened to me in wartime. Perhaps, as is classic in nightmares, what I tremble at most in the memory of the shearing machine is a calamity that never occurred, while everything that happened in combat did. I grade in retrospect those missing fingers I didn't lose right up there with the several times I swam out intrepidly when younger to the Coney Island bell buoy, when I didn't drown, and now am in recurrent fear that I might have.\n\nA jollier episode than that near disaster at the shearing machine involved a working friendship with a young master blacksmith from nearby North Carolina. After a few weeks\u2014after, I conclude, he finally decided he approved of me\u2014he invited me, more than once, to drive out with him on a weekend day to the town he lived in, where, he promised, he would introduce me to some girls who would get a kick out of meeting me and would let me have from them as much as I wanted of anything I wanted.\n\nThe temptation was strong. The decisive drawback was stronger: I, fortunately, and unfortunately, too, always look ahead with caution to calculate consequences, even when drunk, and I could clearly see that I would have to give up a day of overtime, more likely two days if I slept over. Whereupon a dispiriting pall settled over my lubricious visions at the thought that afterward I would find myself again doing as much work for eight dollars a day as I could have been doing for twelve.\n\nThere was a second factor, a rather droll one, that perhaps also helped me overcome desire. In the country argot of his home territory, the common vulgar term for the primary female genital was the same as the one commonly employed elsewhere for the male sex organ\u2014as I heard with some sense of shock each time he applied it to _them\u2014_ and that took a good bit of getting used to. The possibility arises only as I write this that he might have been directing me to males, to boys and men, but I strongly doubt that, for we were all talking as usual about sexual congress with women. And my day and night fantasies about women converged always on what he called a different thing and we called pussy.\n\nAnother source of practically daily amusement to my coworkers sprang from the work shoes picked out for me in Manhattan by my cautious brother, Lee. Ever concerned for my safety and everyone else's, Lee sought and found safety work shoes reinforced inside with domed steel shells to protect the toes and the front of the foot against crushing injury by impact from falling weights. Neither of us had the tiniest idea what kind of massive objects in the navy yard I'd be in danger from, and factory accidents do happen. In consequence, the shoes I wore to work were abnormally large and long and very heavy.\n\nAs it turned out, there was a tenet of folklore in my Virginia blacksmith shop holding that the size of a man's foot was directly indicative of the mass and range of his penis. None of my coworkers had ever seen, or even heard of, shoes with interior reinforcing structures like mine, and my feet appeared to be the largest, longest, thickest, bulkiest feet that any of them had ever laid eyes on. Much, therefore, was expected of me. It could be that those reinforced work shoes were the reason my colleague had wanted to bring me to his hometown to display to the women he knew there. The continual teasing I was subjected to in the form of mock praise and admiration was baffling to cope with. I was most loath to disappoint them. Secretly, I schemed to use the urinals in the bathroom only when that chamber was vacant, an attempt requiring more luck and craft than I could dependably command, for in the government navy yard, as in all other workplaces in the world, the rest rooms serve not only as toilets but also as sheltered spaces in which to hang out for short respites of recuperation from the unavoidable monotony of work. Rather than risk eventual discovery and bring down upon my head all the taunting, crude, homespun derision that was sure to follow\u2014as though I were the one responsible for exaggerated expectations\u2014I soon put myself to the expense of purchasing an ordinary pair of shoes and threw my armored ones away. To those at the fires who noticed the change and commented, I feigned a casual shrug and explained: \"Circumcised. You know that.\"\n\nIn actual practice, there appeared to be no greater chance of injury working there than I would today face trying to prepare a hot meal for myself in my kitchen. The work was pretty much as I've said, tending the furnace and bearing bars of glowing metal from the fire to the true artisan; transferring weights from place to place on a hand truck or occasionally by an overhead traveling hoist that traversed the length of the shop on a track near a sidewall and functioned at the touch of button controls like those found today on a remote TV control; cutting metal into specific lengths; grinding bolts into level surfaces at the top, and threading them, too. At the threading machines, almost as a frolic, I devised a speed system of synchronized alternation with the man on the second unit alongside me that allowed us together to thread the bolts at a breathtaking pace unknown before in that shop, a rate that drew admirers and invoked murmurs of amused approbation, even from tight-mouthed Mr. Beeman. But that was diverting only sporadically, for no one was in a rush for the barrels of threaded bolts and we had no true need to hurry, other than to distract ourselves and others.\n\nNow and then I had to work with the heavy sledgehammer. I was not good at that. Luckily, I didn't have to heft it all the way up over my shoulder for maximum power but could let gravity supply most of the needed force as I let it fall with a bang atop some rounding or rimming tool already held in place, Here, the hazard was more to the man I was working for than to me. Apart from that lapse at the shearing machine, I ran no more risk in the blacksmith shop than I had braved as a file clerk or Western Union boy. As a Western Union boy, good God, I could have been struck by a car!\n\nMore powerful to me in memory than anything but that shearing machine is a sinister sense of obscure and unnameable risks I might have been running on the train and boat trips I took to get to the navy yard. I am sensitive to such journeys now, when crimes against innocents seem endemic in our national life, although I doubt I was haunted by the threat of such perils back then. If I were, I might not have made the trip, not by myself. I doubt the family would have wanted me to.\n\nThe journey took all of one day and the morning of the next. I traveled alone in this first train trip from home, and I remember being silent and solitary for just about all of it. It must have been as forlorn and solemn an expedition for me as the bus ride away to summer camp. I spoke to no one, and no one spoke to me. I'm not good at starting conversations with strangers, and I've never picked up a girl unless the initiating remarks were made by someone else. I was a prisoner of the clock, helpless, a grieving slave to the slow passage of time. I was cut off from everyone. I may by then have already been afflicting myself involuntarily with the gloomy Freudian construct that people who went away from home sometimes did not return.\n\nThe train ride finally terminated for me in a place in Virginia called Cape Charles. There, in compliance with an itinerary of instructions laid out meticulously on paper for me by Lee, I boarded a ferry for a water ride that, to my surprise, consumed another few hours. The length of the boat ride and the enormous expanse of the bay, Chesapeake Bay, were astounding to me\u2014till then, my only ferry rides had taken no more than fifteen minutes, from the tip of Manhattan to Staten Island. Daylight was ending by the time I reached Norfolk. Another trip by ferry would bring me to Portsmouth and the navy yard. However, it had not been thought advisable that I arrive in Portsmouth at night to make my way around a strange city after dark.\n\nOn arrival, therefore, I checked into the first inexpensive hotel I could find. It was in appearance... unpretentious. The rosy neon lights from a coffee-shop restaurant across the street bled reflectively through my window into the room. I had an uneventful dinner alone there at the counter. I seemed to be the only guest in the hotel. The desk clerk was a skinny, bleary-eyed man of about thirty with a red patch of a birthmark on one side of his chin. I had done a good deal of reading in American fiction by then, and at dinner I had decided that if he proposed sending a woman up to my room, I would brace myself to say yes, but I didn't know how to ask for one. (Now I think I would know how but doubt I'll ever want to.) Waking up early the next morning, I was jittery. I still caught no sign of any other guests, and I was surprised and relieved when I found I had made my way safely out of the hotel with my suitcases and my life.\n\nThe ferry ride to Portsmouth was short. On getting off the boat, I asked questions at the dock and proceeded several miles by bus to the office in the bungalow complex in which I would reside, to check in and register for a room, and when finally I spotted some familiar Coney Island faces, I felt myself securely at home.\n\nUpon reporting for work my first morning I learned not only what to do with a hand truck, a grinding wheel, and my barrel of iron bolts too heavy for human arms to hoist, but also how to clock in when arriving. Both the friendly foreman and the uncommunicative assistant foreman were waiting just inside the entrance to the shop, and not only for me. They took notice of every worker coming in, and with eagle eyes kept watch on a large, framed board inside a cabinet against the wall bearing round brass checks arranged in numerical order in rows on separate small rings. I was assigned one of these. Every man checking in on our day shift had, upon entering, to turn his brass check over from left to right to mark his presence. One or both of these supervisors was on station there before eight every morning to make certain that no one man turned over the checks for anyone other than himself. When the work whistle confirmed the end of our day shift, the procedure was renewed in reverse, to verify that someone who'd reported in for the day's work hadn't escaped in the interval. And in the morning, as soon as the work whistle blew to herald the formal beginning of our shift, a hinged glass door was swung closed over the whole display of this version of a time clock and was locked until quitting time.\n\nAnyone not on the premises on time had forfeited not just the hourly fraction of his lateness; he was barred for the remainder of the day, no matter his readiness to work, his skill, or his excuse. It seemed, therefore, that no acute shortage of labor cried out for remedy. It appeared as well that there was no great rush for any of the products that flowed from our shop. We all kept busy faithfully at what we were supposed to be doing, but the rhythm was methodical rather than hurried. When we worked up a sweat it was more from the temperature than the rapidity of the work. People took days off whenever they wanted to. Apart from the occasional assault against boredom at the threading machines by me and a partner, no one ever sped to break records. Periodically, though, a wave of advance warning would circulate lazily through the shop and awake ripples of disturbance vaguely similar to those in an ant colony or a hive of bees when intruded upon: Government inspectors would soon be coming through.\n\nIn reaction, the fires in all the furnaces would be turned up, and in each of them rods and strips would be inserted for heating. The drop forges would pump and gasp and pound. Grinding wheels would be turned on and threading machines would thread, even when there was next to nothing to thread. The tapping of hammers would be heard in the land, and all of us were so abnormally immersed in our labors as not even to see the high-ranking naval officers and ill-at-ease white-collar civilians passing through with perfunctory yet discerning pauses, as though they understood what they were looking at and what we were doing and why.\n\nThey didn't.\n\nWe didn't either.\n\nWe worked seven days a week and, except for our licensed plumber with the automobile and his girlfriend from the soda shop and for my acquaintance from a different part of Brooklyn whose unrevealed social activities had occasioned a visit from the police, there was not much for any of us in the way of entertainment. I can't remember going to even one movie during the seven or eight continuous weeks I worked without a day off. After we had showered and rested a little at the end of our day shift, it was time for dinner. To be on the job at eight every morning dictated retiring early, and we were doing that seven days a week. Just outside the entrance to the navy yard, in the diner in which we ate breakfast, we were surprised at first to see that certain kinds of liquor were offered, and more surprised to behold it being consumed at that hour, as though liquor for breakfast were commonplace.\n\nWe ate most of our dinners in the boardinghouse of a rather easygoing, voluptuous woman of some years with a shy daughter of about twenty who assisted her in the serving and on whom she kept a vigilant, puritanical eye that did not harmonize with the subtle and perhaps unintended coquettishness in her own sensual bearing. The meals were economical and good, basic: chicken, pork, beef, lamb chops, stews\u2014no fish, thank goodness. The single strange element in her cuisine was something they called corn bread and was not the corn bread we Jewish New Yorkers knew was corn bread. Our corn bread was a plump, hardy, firm, yeasty, East European loaf, and their corn bread was cake. By now, though, I've grown used to their corn bread and wouldn't mind having some right now, with a little raspberry or strawberry jam to go with it. On Passover, all the Jewish laborers in Portsmouth were invited to traditional holiday fare in a large hall rented by members of the Jewish community. It was a happy and novel occasion for us, with much greatly welcomed hospitality, and what we commented most about was the southern pronunciation of both the English and the Yiddish (and the Hebrew, too, in the compressed observances).\n\nIn Norfolk, we'd been told, there was a \"cathouse\" at the address 30 Bank Street. I don't think any of us flirted seriously with the thought of going there; if others went and talked about it, they didn't talk about it to me. Again, a powerful force toward abstinence was monetary, the sobering, unspoken awareness that we would have to atone later with approximately three full days at hard labor for a fling of half an hour in a lewd debauchery that might or might not prove euphorically uplifting. Or worse, might prove ruinously addictive.\n\nAt length, I finally did treat myself to a spree of a day off\u2014two full days, I think; a weekend. I forget what I did with it\u2014it could have been a drive to New York packed into the speeding automobile of one of us at a unit cost lower than the normal commercial fare; the memory is a blur. If so, I don't know what I did there. I can remember (or think I do) the address of a bordello in Norfolk I never went to, but nothing of how I passed the time back home in Coney Island or somewhere else after something like _fifty-six_ consecutive days of formidable labor. But... whatever I did do spoiled me, broke my spirit, sapped my willpower, put an end to my career as a factory worker. Going back to that work even after so short a furlough grew unthinkable, and proved with the passage of the first few days an agony of phobic anguish such that only a man of a steelier character than mine could subdue.\n\nI surrendered with hardly a fight.\n\nI gave notice.\n\nI packed up and left.\n\nBoy\u2014was I happy!\n\nON MY WAY HOME from Virginia, I stopped in Washington, D.C., for one night and two days to visit with my Coney Island friend Marty Kapp. Marty was from 23rd Street, down in the Italian section. We hadn't met until high school, and he became close friends with the rest of us by joining the first of our social clubs. In our bunch he was something of an anomaly\u2014even his stance was upright and he didn't, like the rest of us, slouch along like a cool hipster when he walked. He didn't twirl a length of key chain or even peg his pants. I believe he was one of the rare ones among us\u2014Sy Ostrow was another\u2014who went directly from high school into college, to Brooklyn College, which was free and near to us. It was from Marty's father, an electrician or plumber, that I obtained the obligatory letter of recommendation for the job at the navy yard attesting to my mechanical experience (which I didn't have) acquired working for him (which I had never done). It was not, alas, the only time in my life I cut corners, as the charitable young lady from California who surreptitiously helped me meet the foreign-language requirement for my master's degree in English at Columbia University can warrant, if she's alive and remembers (if she's alive, she remembers). I'm sorry now I did that. It was my malfeasance then that compelled me later, with logical good conscience, to show mercy to two students I caught cheating when I was teaching college in Pennsylvania.\n\nWhile I had been laboring away at the navy yard, Marty was working in Washington in some governmental capacity, either in a civil-service position or perhaps already enrolled in the navy's V-12 program, an agenda of specialized military education in which he was to spend the duration of the war. He found a bed for me in the communal apartment he was sharing with a lively group of others about the same age as the two of us.\n\nThe wartime capital city seemed afire with sunshine and energy that weekend. I spent much of my first day, a Saturday on which Marty was occupied responsibly with whatever were his duties, roaming about goggle-eyed in a state of orthodox reverence. I was appropriately awed by the Washington Monument and the sleek, domed Jefferson Memorial, and deeply moved by the solemnity of the seated figure in the Lincoln shrine. I was as spontaneously thrilled by all I saw of official Washington as any uncritical first-time visitor from the hinterlands (i.e., from anywhere outside the District of Columbia) possessing just a superficial and idealized knowledge of American history and a tangential, headline-founded awareness of current events. Even schoolboys now know they can't rely on schoolbooks and official public statements for any honest grasp of the historical truth. In Lincoln's Gettysburg Address, for example, once you're past the simple arithmetic at the beginning of the leading sentence, there's not much left that can be empirically verified. It's a wonderful speech, one of the best on record, certainly, and it's mostly a lot of baloney. So much for speeches!\n\nOn Sunday, Marty and I attended an outdoor symphonic concert, my first concert. I remember the featured work but not the performers; it was the Tchaikovsky Violin Concerto, which I hadn't heard before. I have a good ear for music and a magnetic memory for musical phrases. A day or two after I was back in Coney Island, I was fortunate enough to catch the same concerto on the radio. I stopped everything to concentrate, listening intently from beginning to end, and had the feeling when it was over that I had known that music all my life. To this day I can recall and hum the major melodic passages in the openings of all three movements. Also today, the violin concerto of Tchaikovsky, once the most beloved and moving of composers to me, is just about the only work of his I can stand.\n\nBACK HOME IN CONEY ISLAND I was nineteen and at loose ends, carefree, but in the state of boyish lunacy and na\u00efvet\u00e9 still widespread among late adolescents, I'd bet. Though carefree, I was nonetheless in a developmental quandary. Others around me were likewise adrift. Bobby Magrill, the pharmacist's son, was set on becoming a physician. George Mandel, our comic-books cartoonist who was making $300 or $400 a week when I was making $15, already in the army, where he would suffer his serious head wound in the Battle of the Bulge, had attended lessons at the Art Students League. These were exceptions. The rest of us were without concrete direction or motivation.\n\nFor us, the military draft looming somewhere ahead was the only certainty in a future that was in all other respects murky. It was hard to give serious consideration to what in later life we intended to do, _could_ do. Danny the Count and Sheiky Silverman sold costume jewelry as peddlers outside the busy entrances of the subway station at Union Square. Sheiky also peddled ice cream on the Coney Island beach that summer, 1942, when he wasn't leading the policemen pursuing him in an agile chase through the crowds in the sand. Louie Berkman was in his father's junk shop. We found jobs for each other. Marvin Winkler worked a short while stringing pearls in a factory where Murray \"Rup\" Rabinowitz already had some standing; later, Marvin did shipping in the garment center for our Coney Island friend Jackie Sachs.\n\nI looked around for what work I could find and took what work I could get. Davey Goldsmith needed assistance as a shipping clerk during a business rush at the U.S. Hatband Company, and I pitched in with him for a while at $21 a week, learning how to unfold and seal flat corrugated boxes. Now and then, when the need arises, I wish I remembered how. Macy's, through Sylvia, hired me for a couple of weeks to count stock in an inventory period. Someone introduced me to a job in a millinery factory in Manhattan to pack hats for shipping and to keep the floors clean. Until then, I\u2014a fledgling, aspiring author\u2014hadn't bothered to find out what \"millinery\" was; I'd imagined it was something like lingerie. The millinery we made was of straw, and the scraps of straw nicked my fingers when I did my best to keep the floors clean. Before the first week was out, one of the scowling elderly partners growled at me not to come in any longer. He was telling me I was fired. Lee was no longer a customer's man on Wall Street but somehow the supervisor in a small factory doing subcontracting work for a larger defense manufacturer. I went to work for Lee on a drill press. I adjusted a template over a square of something that was not wood and brought the spinning auger down into each opening in the dotted pattern of the several holes. Doing just this eight hours a day, four hours at a clip, was nerve-racking. Lee was not hurt or surprised when I confessed I couldn't stand it. Along with so many other Americans of my generation in an era of economic depression that had not truly passed, I'd been biding my time, waiting in numb hope for some unknown, defining reality finally to pop up that would clarify the course I should follow, wind me up, and start me on my way.\n\nWe had not thought it would be a war.\n\nThe announcement by the government that nineteen-year-olds would soon be called up for military service supplied the impetus. We saw no incentive in avoiding the draft. A group of us enlisted\u2014we had nothing better to do. And we were further motivated by the opportunity to choose the branch of service we preferred.\n\nThe morning I left for the army, I had orders to report to a waiting room at the Pennsylvania Railroad Station for transportation to the area's military reception center at Camp Upton, Long Island. My mother and Sylvia walked with me to the trolley stop outside Mr. Moses's candy store at Railroad Avenue. We talked routinely as we waited; I promised I would write often and telephone from Long Island if I could. We hugged formally and kissed cheeks automatically as the trolley car drew near and slowed to a stop. Later, many years later, I was profoundly shaken (and tried not to show it) when my sister happened to disclose that my mother dissolved into tears and collapsed with weeping as soon as the trolley car carried me off on my merry way. Sobbing violently, my mother could barely keep standing, and Sylvia strained to hold her erect as they struggled back to the apartment house.\n\nDope that I was, I didn't see that there was anything to cry about.\n\nMy mother never mentioned the occasion to me, and I never brought it up with her. Our family tendency to keep disturbing emotions to ourselves has lasted as long as we have.\n\nTHE AMIABLE PSYCHOLOGIST Erik Erikson writes somewhere _(Childhood and Society?)_ of a \"Moratorium\" that emerges in the lives of most Western young people between the end of adolescence and the onset of maturity, during which the individual doesn't truly know what he or she is or where he wants to go, doesn't indeed truly know _who_ he is, or what he or she should decide to become. It is a season of baffled uncertainty over identity and can lead to grave mistakes. (Witness the skinheads in England, the neo-Nazis in Germany, the neophyte Nazis in Moscow. What is the average starting age of the cocaine or heroin addict? The average mugger?) To the extent Erickson is right, and for those of us Americans in that stage who weren't harmed physically or damaged emotionally by military service, the war came along at just the right time. (For Europeans, there is never a right time for a European war.) It put an end to our confusion and ambivalence, took most powers of decision out of our hands, and swept us into a national endeavor considered admirable and just.\n\nAnd remunerative, too. My total income upon entering the air force as a private was as much as I'd been able to command outside, and as an officer on flight status was greater than I was able to earn afterward when starting out in my civilian pursuits, first as a college instructor at Pennsylvania State College ($3,000 a year) and next as an advertising copywriter at a small New York agency called the Merrill Anderson Company ($60 a week, $260 per month). But at the Merrill Anderson Company, as an unexpected employee benefit, I drank my first Gibsons with a copy chief named Gert Conroy and learned to love extra-dry martinis in a chilled glass with a twist of lemon peel. (Also, one morning I arrived at work with my pastry and container of coffee and a mind brimming with ideas, and immediately in longhand put down on a pad the first chapter of an intended novel, a chapter that, after expansion and revision, was published in 1955 in the periodical _New World Writing_ #7 as the opening of a novel in progress, a work to be titled... _Catch-18._ I was at _Time_ when a contract for the publication of the novel was offered on the basis of 250 pages and working at _McCall's_ magazine when the manuscript was completed at last in January 1961 and published in October. My salary at _McCall's_ was good, I had two children, and I was obliged to wait almost a full year until the motion picture rights were sold before I had confidence enough to leave.)\n\nAfter the war, Marty Kapp continued what technical education he'd begun in the navy V-12 program and graduated as a soil engineer (yet another thing I'd not heard of before). For all his career he worked as a soil engineer with the Port Authority of New York\u2014on airfields and buildings, I know, and perhaps on bridges and tunnels, too\u2014and had risen to some kind of executive status before he died. He did well enough to die on a golf course. Davey Goldsmith went back to work at the hatband company, prospered there, and eventually inherited a division of that company. I was able to go to college. Lou Berkman left the junk shop to start a plumbing supply company in Middletown, New York, and looked into real estate with the profits from that successful venture. Marvin Winkler passed up college, too, and was soon in the photographic film business (his first in a sequence of modest enterprises), converting surplus air corps aerial film into color film for home use. Albie Covelman went to Bucknell and was soon an executive with a coffee-equipment supply company. Sy Ostrow, who was taught Russian in the service so that he could function as an interpreter, returned to college and, with pained resignation, saw realistically that he had no better alternative than to study law. Stanley Levy, from the file room at the automobile casualty company, became an insurance broker and handled what little business I could give him when we at last caught up with each other. Like Marty Kapp, Stanley died on a golf course. It's my opinion they died of boredom.\n\nI was mustered out of the service even before the war was over, demobilized under a point system put into effect as soon as Germany capitulated at the beginning of May in 1945.\n\nBut even before the war ended with the surrender of Japan in August, I felt myself walking around on easy street, in a state of fine rapture. What more could I ask for? I was in love and engaged to be married (to the same young woman I was in love with). I was twenty-two years old. I would be entering college as a freshman at the University of Southern California, with tuition and related costs paid for by the government. And I'd just had a short story accepted for publication.\n\n# 8  \nPeace\n\nIN DECEMBER OF 1944, when I had completed the tour of duty then defined at sixty missions and was awaiting orders for transportation back to the States, two chaste beginners, both lieutenants freshly shipped overseas, moved into my tent. They replaced a pair who had finished their tours and already left, and one of these new men brought into the tent with him the uncommon personal asset of a portable typewriter. Don't ask me why\u2014I've lost the explanation, though it seems likely he was nurturing the same authorial aspirations I was, and Tom Sloan also.\n\nTom Sloan, a lead bombardier, who was from Philadelphia, and Hall A. Moody, a wing bombardier from Mississippi, were among the friends in Corsica I was closest to. Tom was a conscientious, quick, and capable lead bombardier in whose formation I found it almost a pleasure to fly. Like Moody, I was merely a wing bombardier and operated only a toggle switch, compressing it in the speediest fraction of the instant after I saw the bombs starting downward out of the open bomb-bay doors in the lead plane. Once overseas, I never made use of the bombsight on which I had been trained, or even had one with me in the nose of my plane. I had a .50-caliber machine gun mounted on a ball swivel instead, and fortunately never had to make use of that implement either. Four of the wing bombardiers in the six planes in each of our formations carried no bombsights but single machine guns instead.\n\nTom was only a few years older than I, no older than twenty-five, I'm sure, but he was already married and the father of an infant child he'd seen not more than a few times. He was resolutely intent on surviving to rejoin the family he missed so greatly, and he was increasingly and visibly perturbed that he might not succeed. It's a joy to me now to report that he did. Shipping orders for the two of us arrived the same day; not till they were in hand did we truly feel saved.\n\nHall A. Moody, my age or even younger, was married also, and I relate with pride, not scorn, that neither he nor Tom Sloan ever exhibited even the slightest interest in sex with another woman, not on rest leaves in Rome and not in Sicily, Cairo, or Alexandria. I, unmarried and in depth of experience still almost a virgin, was the boyish and ravenous satyr. Moody was in several ways an innocent: He had never before _even heard_ of oral sex, in either direction. He had not heard of it by either the genteel Latinate names we all now know or the vulgar familiar ones we now all use, and the mere suggestion of any such distorted practice was to him an outrageous, debasing horror he declined to envision and caused him to redden and clench his fists in passionate indignation. When Tom and I left together for the States, Moody was still on combat status and I don't know what became of him, but his name doesn't appear on the list of fatalities in our squadron history, and I assume he survived. And Tom Sloan's infant child, the one he missed so dearly and painfully and reticently when I knew him, was about one year old then and now would be past fifty.\n\nWith travel orders in hand, Tom and I were flown from Corsica\u2014not to Rome, which I would have favored for a final, farewell, youthful revel\u2014but to Naples on the first leg of our journey homeward. Given the choice of returning by air or sea, I expressed unequivocal preference for the sea, because it had become my furtive and sacred resolve never to go up in an airplane again. (In fact, once back safely, I declined to fly again, as soldier or civilian, for something like the next seventeen years, traveling instead by train to company conventions when at _Time_ and _McCall's_ and by cruise steamship when the meetings were in Bermuda and Nassau. Not until the stupefying boredom of overnight rail travel ultimately outweighed all fears of dying did I weaken, and by then there were jets.)\n\nAfter a week or so in Naples, I sailed with a few thousand other servicemen on a converted and highly modern troopship, not long previously the S.S. _America_ , I believe I remember, formerly the premier luxury liner of the American passenger fleet. There were six officers in my cabin on two tiers of bunk beds, sleeping, napping, reading, eating, talking, because there wasn't much else to do in the ten days or so we were afloat. None of us had known any of the others before. I cannot imagine now how I spent so much time doing essentially nothing but sleeping, napping, reading, eating, and talking, and I know I wouldn't ever want to have to do it again. We sailed without convoy or other naval escort. Our cabin wasn't first class, because we weren't on deck; but it wasn't steerage either, for we did have a porthole.\n\nWe were heading for Boston, but didn't know our destination until we were already in port. From Boston we were transported by rail to Atlantic City for routine processing and reassignment, and from there, on furlough, I traveled home to Coney Island and was triumphantly back with my family as something of a glamorous war hero. I never thought to ask what they worried about and said about me while they waited for my letters, and I still don't know.\n\nAmong the first things I did during my medical examination in Atlantic City was ask to be taken off flying status. The doctor was surprised: He found it odd that, after surviving combat duty, I should want to give up half my base pay to avoid only four hours a month of flight time in the States\u2014there seemed little chance of my being sent overseas again. With the shameless knowledge that I was lying, I pleaded that I was afraid to fly, claiming that the dread I might have was keeping me awake: The mere remembrance of the gasoline fumes inside a plane was sickening to me. By that time there were already thousands of airmen like me who had completed combat duty and were back dillydallying in the States while the government tried to decide what to do with us, and he graciously granted my request. And the lie I thought I was telling him turned out to be true, for by then, I realized, I had genuinely grown terrified of flying.\n\nIN BEING ASSIGNED to my tent, the two new replacements, both of them pilots, lucked into one of the most comfortable and best-appointed pyramid canvas abodes in the squadron. We not only had a gasoline-dripping homemade stove for warmth in winter but a splendid fireplace as well. On the wall of the room in my home in East Hampton in which I am arranging these words\u2014rather, rearranging them\u2014is mounted a framed and glass-encased photograph taken inside our tent, lifted from an elegant, bound volume of a squadron history prepared after the war by our squadron public-relations officer, an assiduous captain named Everett Thomas. The page is captioned \"The Holidays Corsica 1944\" and the photograph highlights a large kitchen knife, a cake, and the five of us then dwelling inside the tent. In the background is a spacious fireplace with blazing flames, topped by a broad wood-beamed mantel fabricated from salvaged railroad ties. We are all posing rather obviously for what professes to be a candid photograph.\n\nA young pilot named Bob Vertrees is shown at a table with a long kitchen knife, and he is slicing into what looks like a Christmas fruitcake that presumably has reached him from home. The two newcomers, seated also, are looking on politely. One gazes from the side; the other, possessor of a sparse mustache and the typewriter, practically faces the camera, but his eyes are properly averted toward the cake. Looking on almost impassively between them is a rather short, high-cheeked pilot named Edward Ritter, short but not so short as to have fallen short of the minimum physical requirements for pilot training. Of those shown, Ritter, arriving overseas after me, has been there in the tent with me longest. I am bending forward in a chair at the right, almost in profile, casually smoking a cigarette and clad in my officer's cap with the wire frame removed, a liberty of fashion conferred on air corps personnel as a distinctive privilege, and a leather flight jacket bearing a round patch of the squadron's insignia, a brazen depiction against a dark background of a slim, well-busted naked female wielding a thunderbolt while her windblown hair streams behind her and she sits suggestively straddling a long, projecting bomb that free-falls forward and downward. On a low table just behind me is the portable typewriter, which I was already, with the owner's permission, making much use of, for letters and for manuscripts. Behind that, at the rear of the tent, are the cozy household flames in the fireplace and the broad mantel of sturdy timber, which had been carpeted for the season with a winding garland of Christmas foliage. And on the wall above the fireplace are a number of large, glossy photographs of beautiful women. They are not exactly of the cheesecake kind, but the one immediately behind my head is of an attractive female reclining on a divan in a becoming dress and a full, silken bosom of a quality that still never fails to provoke in me an intuitive reflection of desire and speculation. Vertrees is grinning as he cuts the cake and both of the new men are smiling, too, while Ritter and I look on, absorbed and content. I've been overseas longer than the others, but we're all about the same age. We're kids not long past twenty, and we look like kids who are only past twenty.\n\nIn December 1944 I was twenty-one and a half, and it is hard now, it boggles the mind now, to believe that a young kid like Ritter, whom I'd known the longest and who was somewhat stumpy in physique, was ever, let alone routinely as an occupational specialty, permitted to fly as a pilot at the controls of a twin-engined Mitchell medium-sized bomber that carried a bomb load of four thousand pounds (eight five-hundred-pounders or four of a thousand pounds each) and five other human beings!\n\nHow in hell did he learn to do that?\n\nI, who didn't apply for my first driver's license until I was twenty-eight, find it difficult to envision even now that a kind and unaggressive boy, so young, could learn to fly a bomber. And there were others in the squadron of slighter build and even fewer years who were pilots, too.\n\nA little while before, Vertrees, who was a pilot also, had been wounded in the hand by a chip of flak, injured so trivially that he was out of action only a few days and returned to combat duty in almost no time.\n\nRitter's biography overseas was perplexing. Taciturn, good-natured, soft-spoken, even shy, from Kentucky, he was something of a tireless wonder as a handyman, one with unlimited patience who took pleasure in making and fixing things. The construction of the fireplace was entirely of his devising, if I recall accurately. We have our place in the squadron history book only by virtue of that fireplace. He erected and maintained the stove, too, although by winter every tent contained one like it, a continuously feeding gasoline unit that drew fuel from a can outside and dripped it onto sand in a drum that stood near the center pole and helped warm the interior. We also used the stove to heat our supplementary snacks at night from the boxes of K or C rations that were in abundant supply and the hot water with which we shaved and washed outside in the morning. A flak helmet was our basin. The metal fins on the upright, inverted skeleton of a frame in which unfused bombs were shipped was our washstand.\n\nIn combat, Ritter as pilot had ditched safely once into the Mediterranean waters coming back from a mission on which his engine had been shot out, and all the men with him had made it from the plane into the life rafts without injury and had been rescued by air-sea surface craft before nightfall. I remember awaiting their return. Another time, he flew back with one engine feathered and landed safely, and I believe there was one other time he crash-landed without harm to anyone aboard, either at a different landing strip closer to where his crippled plane was in difficulty or at our own, code-named Genoa. (\"Hello, Genoa, hello, Genoa\" was a pilot's saluting radio call to the control tower of our field at Alisan, Corsica, and the title of a story I once thought I might write and don't believe I ever did.) Hall A. Moody was aboard with Ritter on one of those near disasters, and for that reason, both were awarded a recreational trip to Egypt. Sloan and I went along because we had completed our sixty missions and there wasn't much else for the squadron to do with us as we shambled around waiting for our transfer orders. Also, I suspect there was a forlorn, covert command dream that, refreshed upon our return and intoxicated with gratitude, we would beg to be allowed to fly more missions than the sixty we already had. Neither one of us volunteered to do that.\n\nRemarkably, through all his unlucky series of mishaps the pilot Ritter remained imperviously phlegmatic, demonstrating no symptoms of fear or growing nervousness, even blushing with a chuckle and a smile whenever I gagged around about him as a jinx, and it was on these qualities of his, his patient genius for building and fixing things and these recurring close calls in aerial combat, only on these, that I fashioned the character of Orr in _Catch-22._ (I don't know if he's aware of that. I don't know if he's even read the book, for I've never been in touch with him or almost any of the others.)\n\nIn a nearby tent just across a railroad ditch in disuse was the tent of a friend, Francis Yohannon, and it was from him that I nine years later derived the unconventional name for the heretical Yossarian. The rest of Yossarian is the incarnation of a wish. In Yohannon's tent also lived the pilot Joe Chrenko, a pilot I was especially friendly with, who later, in several skimpy ways, served as the basis for the character Hungry Joe in _Catch-22._ In that tent with them was a pet dog Yohannon had purchased in Rome, a lovable, tawny cocker spaniel he bought while others were purchasing and smuggling back contraband Italian pistols, Berettas. In my novel I turned the dog into a cat to protect its identity.\n\nRome was a wonderful city to go to (and still is), and, given his astonishing achievement, even a limited cast of characters ought to include a tribute to our able squadron executive officer, Major Cover (Major de Coverley in _Catch-22)._ Now, I still don't know what an executive officer is or what one does, but whatever ours did, he did exceedingly well. A few weeks after I arrived overseas for combat in early May of 1944 as a replacement bombardier, the Germans retreated north from the open city of Rome (there was no connection between these events, I'm sure). The first American soldiers were in Rome on the morning of June 4, and close on their heels, perhaps even beating them into the city, sped our congenial executive officer, Major Cover, to rent two apartments there for use by the officers and enlisted men in our squadron: For officers there was an ornate four-bedroom suite with much marble and many mirrors in the reputable residential district of the Via Nomentana, with a separate bedroom with double bed for each officer, and a maid to see that the rooms were tidied; for the enlisted men, who came in larger bunches, there was an extended arrangement of rooms on two floors somewhere up past the top of the Via Veneto, with cooks and maids, and with female friends of the cooks and maids who liked to hang out there with the enlisted men just for sport.\n\nHardly did we learn that the city of Rome was ours than officers and enlisted men who'd been in combat longer were returning starry-eyed from enchanted rest leaves there, speaking, rhapsodically and disbelievingly, of restaurants, nightclubs, dance halls, and girls, girls, girls\u2014girls in their summer dresses strolling with smiles on the Via Veneto. And these fellow fliers taught us the most valuable Italian phrase for just about everything we might want when our turn to go there on a rest leave came:\n\n_\"Quanto costa?\"_\n\nAnd thanks to Major Cover and the apartments, we were frequently on leave there.\n\nRITTER HAD ARRIVED in my tent early on in my tour of duty to occupy the cot vacated by a bombardier from Oklahoma named Pinkard, shot down and killed on a mission to the railway bridge north of the city of Ferrara in the Po Valley. There were several of these missions to Ferrara shortly after I arrived, and the target was a much more terrifying one than I was able to grasp in my absence of experience and my idiotic faith in my own divine invulnerability. On another mission to Ferrara, one I don't think I was on, a radio gunner I didn't know was pierced through the middle by a wallop of flak\u2014it was always flak that destroyed, for there were no German fighter planes attacking us the whole time I was there\u2014and he died, moaning, I was told, that he was cold. For my episodes of Snowden in the novel, I fused the knowledge of that tragedy with the panicked copilot and the thigh wound to the top turret gunner in my own plane on our second mission to Avignon. The rest of the details are all pretty much as I related, except that I did not, like Yossarian, discard my uniform or sit naked in a tree, and I was not given a medal while undressed. Back over water on the way home from that mission, a plane with a bombardier friend of mine named Wohlstein in the crew (he could drive, and with a car from the motor pool we would go exploring together through the tiny Corsican mountain villages on our eastern side of the island), pulled out of formation, toppled into a spin, and corkscrewed down to a fatal crash into the water that killed all five men aboard. The pilot in his plane that day was Earl C. Moon, who was copilot in the crew with which I flew in a B-25 across the Atlantic from South Carolina to Algeria and then to Corsica. I didn't witness that crash or know about it until we were back on the ground, for I was entirely occupied ministering to the thigh wound of my top-turret gunner, with bandages, sulfanilamide, and morphine, and with sickly attempts at solicitous and reassuring platitudes while we flew on and on. The aerial gunner and I had not known each other before. When I went to visit him in the hospital the next day, he must have been given blood transfusions, for his Mediterranean color was back, and he was in ebullient spirits. We greeted each other as the closest of pals and never saw each other again.\n\nEXCEPT FOR THE INSERTION into the novel of a radio gunner who is mortally wounded, the incidents I actually experienced in my plane on this second mission to Avignon were very much like those I related in fictional form. A copilot panicked and I thought I was doomed. By that time I had learned through experience that this war was perilous and that they were trying to kill me. The earlier cluster of missions I had flown to Ferrara when first overseas had assumed in my memory the character of a fantasy nightmare from which I had luckily escaped without harm in my trusting innocence, like an ingenuous kid in a Grimm fairy tale. And I also knew from the serious tone in the briefing room and from an earlier mission to Avignon that this target was a dangerous one.\n\nAll four squadrons in the group were involved, flying into southern France in a single large bunch, then separating near the city to simultaneously attack three separate targets that were several miles apart. My plane was in the last of the three elements turning in, and as we neared our IP, the initial point from which we would begin our bomb run, I looked off into the distance to see what was taking place with the other formations. The instant I looked, I glimpsed far off amid black bursts of flak a plane in formation with an orange glow of fire on its wing. And the instant I spied the fire, I saw the wing break off and the plane nose over and fall straight down, like a boulder\u2014rotating slowly with its remaining wing, but straight down. There was no possibility of parachutes. Then we made the turn toward our target and we were in it ourselves.\n\nThe very first bursts of flak aimed at us were at an accurate height, and that was a deadly sign. We could hear the explosions. I have since read of the tactic developed by the Germans of sending a monitor plane out to fly alongside our bombers and radio our exact altitude and speed to the antiaircraft batteries below, and it's possible they were doing it that day. Soberly and tensely, I did what I had to\u2014we all did. When I observed the bomb-bay doors of the lead bombardier opening, I opened mine; when I saw his bombs begin to go, I toggled away mine; when the indicator on my dial registered that all our bombs were away, I announced on the intercom that our bombs were away. When a gunner in the rear looking down into the bomb-bay announced that the bomb-bay was clear, I flipped the switch that closed the doors. And then our whole formation of six planes wrenched away upward at full throttle into a steep and twisting climb. And then the bottom of the plane just seemed to drop out: we were falling, and I found myself pinned helplessly to the top of the bombardier's compartment, with my flak helmet squeezed against the ceiling. What I did not know (it was reconstructed for me later) was that one of the two men at the controls, the copilot, gripped by the sudden fear that our plane was about to stall, seized the controls to push them forward and plunged us into a sharp descent, a dive, that brought us back down into the level of the flak.\n\nI had no power to move, not even a finger. And I believed with all my heart and quaking soul that my life was ending and that we were going down, like the plane on fire I had witnessed plummeting only a few minutes before. I had no time for anything but terror. And then just as suddenly\u2014I think I would have screamed had I been able to\u2014we leveled out and began to climb away again from the flak bursts, and now I was flattened against the floor, trying frantically to grasp something to hold on to when there was nothing. And in another few seconds we were clear and edging back into formation with the rest of the planes. But as I regained balance and my ability to move, I heard in the ears of my headphones the most unnatural and sinister of sounds: silence, dead silence. And I was petrified again. Then I recognized, dangling loosely before me, the jack to my headset. It had torn free from the outlet. When I plugged myself back in, a shrill bedlam of voices was clamoring in my ears, with a wail over all the rest repeating on the intercom that the bombardier wasn't answering. \"The bombardier doesn't answer!\" \"I'm the bombardier,\" I broke in immediately. \"And I'm all right.\" \"Then go back and help him, help the gunner. He's hurt.\"\n\nIt was our top gunner who was wounded, and his station was in the front section of the plane just behind the pilot's flight deck. But so deeply, over time, have the passages in the novel entrenched themselves in me that I am tempted even now to think of the wounded man as the radio gunner in the rear. Our gunner was right there on the floor in front of me when I moved back through the crawlway from my bombardier's compartment, and so was the large oval wound in his thigh where a piece of flak\u2014a small one, judging from the entrance site on the inside\u2014had blasted all the way through. I saw the open flesh with shock. I had no choice but to do what I had to do next. Overcoming a tremendous wave of nausea and revulsion that was close to paralyzing, I delicately touched the torn and bleeding leg, and after the first touch, I was able to proceed with composure.\n\nAlthough there was a lot of blood puddling about, I could tell from my Boy Scout days\u2014I had earned a merit badge in first aid\u2014that no artery was punctured and thus there was no need for a tourniquet. I followed the obvious procedure. With supplies from the first-aid kit, I heavily salted the whole open wound with sulfanilamide powder. I opened and applied a sterile compress, maybe two\u2014enough to close and to cover everything injured. Then I bandaged him carefully. I did the same with the small hole on the inside of his thigh. When he exclaimed that his leg was starting to hurt him, I gave him a shot of morphine\u2014I may have given him two if the first didn't serve quickly enough to soothe us both. When he said he was starting to feel cold, I told him we would soon be back on the field and he was going to be all right. Truthfully, I hadn't the slightest idea where we were, for my attention had been totally concentrated on him.\n\nWith a wounded man on board, we were given priority in landing. The flight surgeon and his medical assistants and an ambulance were waiting to the side at the end of the runway. They took him off my hands. I might have seemed a hero and been treated as something of a small hero for a short while, but I didn't feel like one. They were trying to kill me, and I wanted to go home. That they were trying to kill all of us each time we went up was no consolation. They were trying to kill _me._\n\nI was frightened on every mission after that one, even the certified milk runs. It could have been about then that I began crossing my fingers each time we took off and saying in silence a little prayer. It was my sneaky ritual.\n\nTHAT WAS A MISSION on August 15th. A week before, I had seen a plane shot down just after the bomb run on the first of our squadron's missions to the bridges of Avignon. I was in the leading flight and when I looked back to see how the others were doing, I saw one plane pulling up above and away from the others, a wing on fire beneath a tremendous, soaring plume of orange flame. I saw a parachute billow open, then another, then one more before the plane began spiraling downward, and that was all. Dick Hirsch, a fellow from Chicago I'd trained with in bombardiering school in Victorville, California, was in one of those chutes. He came down in a field in Provence, was picked up by the underground, hidden, clothed in civilian garb, and smuggled back through the battle lines into the Allied part of Italy. Once back in the squadron he was rotated home without delay, to his surprise and ours, it being the policy never to expose to recapture anyone who had once had contact with the underground. The pilot killed in his plane was a blond fellow from upstate New York named James Burrhus, and I knew him, too, having flown missions with him. The copilot was a younger kid named Alvin Yellon, recently arrived, and I found out more about him fifty years later when I received a letter from his brother asking whether the mission I had described in my writing could have been the one on which his brother had met his death. It was.\n\nIt was all very much more dangerous, I realize now, than I had let myself recognize at the time and since. I believe I've read that more than 10 percent of the three hundred thousand American World War II dead were air corps crewmen.\n\nAnd for those on the ground on the Western Front in Europe that Christmas season, it was even colder and more lethal. I remember seeing in _Stars and Stripes_ , the official military newspaper, photographs of bundled-up Americans plodding their bleak way in desperate winter combat clothing through drifts of snow taller than they were. They looked like foreigners in a different, distant war. Theirs _was_ a different war. As a consequence of the colossal reciprocal miscalculations by both the Allied and the German high commands, miscalculations historically and endemically inherent in high commands, there ensued that Christmas the disasters of what we've learned to call the Battle of the Bulge. Our generals believed that the Germans could not and would not attack through the Ardennes Forest, and maintained defenses there that were thin and inexperienced; the Germans believed they could indeed break through and that the course of the war would be altered to their significant advantage if they did.\n\nBoth were wrong.\n\nThe Germans did break through. But the course of the war was altered most significantly, I'd bet, only for the thousands on both sides who were killed, wounded, or imprisoned in that devastating misadventure by aggressive Germans and complacent Americans. My Coney Island friend George Mandel received his head wound from a sniper there, the bullet penetrating his helmet but going into the brain only so far, and my contemporary friend and unintentional contemporary literary rival for attention, Kurt Vonnegut, was among those taken prisoner. Officers at the top, of course, always know much more than those they command at the bottom, but, until it's all over, there's never a way to know if they've known enough.\n\nWhile combat service in the air corps wasn't altogether the cushy sinecure some might suppose, we did enjoy, together with decent food, enclosed latrines, and heat in our tents, at least one other precious benefit that American ground forces until the Vietnam War did not: a specified tour of duty, after which we were detached from combat, replaced, and rotated home.\n\nI have often wondered whether there were many combat men in our ground forces coming ashore on D-Day who made it through the following eleven months till May and the end of the war with Germany without getting killed, wounded, or taken prisoner.\n\nI wouldn't be surprised to hear there was not even one.\n\nAmong the last several missions in my own tour of duty was an especially theatrical one in which I still take both a military and a civilian pride, the civilian pride bred of my sole assertion of leadership and authority as an officer. The assignment that morning was a hurried one. The destination was the large Italian seaport of La Spezia. The target was an Italian cruiser reportedly being towed out into a deep channel of the harbor by the Germans, to be scuttled there as an obstacle to approaching Allied ground forces pressing steadily north. I was relieved to discover myself assigned to one of the planes in a chaff element. The chaff element was a flight of three planes that led all the others in over the target, with their rear gunners jettisoning open bales of aluminized \"chaff from the two side gun ports to confound the radar below that was directing the enemy's antiaircraft fire. We carried no bombs. Because we carried no bombs, we could go zigzagging in at top speed and vary our altitudes. Because we carried no bombs, I shrewdly deduced that there was no need for a bombardier. Therefore, after priming and test-firing the machine gun in the bombardier's compartment in the nose of the plane, I resolved to sit that mission out\u2014literally.\n\nThe flight deck of the B-25, that part of the plane occupied by pilot and copilot, had armor plate on the floor and the backrests of both their seats. On that kind of mission in that point in my career, I would take two flak suits with me, one to wear and one to shield as much of the rest of me as I could curl up and cover. That day I took up front with me to the machine gun only the one I wore. After preparing the machine gun to fire and crawling back through the narrow tunnel connecting the nose of the plane with the rest, I retrieved the second flak suit, mounted to the flight deck, and sat down on the floor right behind the pilot, with my back against the back of his chair. I had armor plate below and armor plate at my back, a flak helmet on my head, and an extra flak suit to cover my groin and legs. And as an additional precaution, I sat with my parachute pack already hooked to my parachute harness. This time I had my parachute with me. Normally, the parachute pack was too bulky to fit through the crawlspace to the bombardier's compartment and had to be left behind; there was no escape hatch up front, anyway. I was the veteran of that crew, and the two newer pilots didn't know what to make of me. \"It's okay,\" I assured them crisply. \"Let me know if German fighters show up and I'll go back.\"\n\nWhen I looked behind us after we had flown through the flak at La Spezia and turned off, I was greatly satisfied with myself and all that I saw, and with all the others as well. We were unharmed; the turbulent oceans of dozens and dozens of smutty black clouds from the countless flak bursts were diffused all over the sky at different heights. The other flights were coming through without apparent damage. And down below I could watch the bombs from one cascade after another exploding directly on the ship that was our target.\n\nAnd soon after that, I was finished. I was alive and I was well. In that photograph I spoke of earlier of the five of us in the tent exists a huge and invisible divide between me and the others. I was through and they, particularly the two newcomers, now faced more missions to fly to survive, because the official number for those still on combat status, at least in our bomber group, had lately been raised.\n\nHow did they feel about that?\n\nAs casually oblivious and indifferent, I'd guess, as I had felt in midcourse when the number had been raised first from fifty to fifty-five and then to sixty. For them, it was up to seventy.\n\nWHILE MY COMRADES-IN-ARMS were off flying missions, I hunkered in my tent and played with the typewriter, naturally preferring to do that when alone. Our medium bombers were of medium range, our bomber group a tactical one, striking almost exclusively at rail and highway bridges. The flight time of missions averaged about three hours, though sometimes they were closer to two. It didn't take long to go from Corsica past Elba into central Italy and turn around and come back. I have a log before me that marks the time of missions to Poggibonsi, Pietrasanta, and Orvieto at exactly two hours, to Parma at two hours and ten minutes, to Ferrara, farther north in the Po Valley, at three hours and ten minutes, to Avignon, much farther away in France in the Rhone Valley, at four hours and thirty-five minutes. Sometimes we\u2014now they\u2014flew two missions the same day.\n\nOn the morning of August 15, 1944, the day of the invasion of southern France, we\u2014I, too, back then\u2014bombed gun emplacements on a beach near Marseilles shortly after daybreak (the cannons turned out to be wooden dummies), and in the afternoon we went on that second mission to Avignon, all four squadrons separating over three targets to divide the antiaircraft fire. We divided the antiaircraft fire, but nevertheless all three units suffered casualties.\n\nAll that day of the invasion when we weren't ourselves in the air, we watched huge armadas of heavy bombers passing high overhead, higher than we ever flew, from their bases near Naples\u2014hundreds and hundreds of them in their somber, purring formations, more planes on our side than I could believe existed in the entire world. All the advantages in the war now seemed unmistakably ours. But in late December came the German offensive in the Battle of the Bulge, and it took five more months after that for Germany finally to capitulate to us and to the USSR, in the east.\n\nTHE SUBJECT MATTER of all the short works with which I was now busying myself and had been busying myself even earlier was not founded on experiences of my own. I hadn't had any then that seemed worth translating into fiction. I borrowed the action and the settings from the works of other writers, who may\u2014I didn't consider the possibility then\u2014in turn have been borrowing from the works of still others. These experiences, which I as author dealt with knowledgeably, were vicarious and entirely literary, gleaned from wanderings as a reader, and they ranged from the picturesque whimsies of William Saroyan to the hard-nosed, sexist attitudes, particularly toward women and marriage, of Hemingway and Irwin Shaw, embodying as well implicit assessments of materialism, wealth, Babbittry, and ideals of masculinity and male decency that I ingenuously accepted as irreducibly pure and nullifying all others.\n\nIn the immense replacement depot in Constantine, Algeria, where I spent a few weeks with the crewmen with whom I had flown overseas in our small B-25 before being assigned to Corsica, my primary inspiration as a neophyte writer was Saroyan. He appealed to my taste and seemed easy to emulate and well worth copying. (The stories that seemed easiest to emulate and most worth copying were short ones with few descriptive passages written in literary vocabulary and with a large proportion of vernacular dialogue.) In one of his collections was a story titled (I am working here from memory) \"Did You Ever Fall in Love with a Midget Weighing Thirty-eight Pounds?\" In one of the stories from my Algerian period (working still from memory) was a young man in New York romantically involved with a girl who walked around on her hands. I have no recollection of my title. (I can imagine now that I imagined then the title \"Did You Ever Fall in Love with a Girl Who Walked Around on Her Hands?\") Where she walked on her hands and where the story went I have also mercifully forgotten.\n\nBy then I was familiar with most of the work of Hemingway, Irwin Shaw, and Jerome Weidman. Jerome Weidman's prewar collection of short stories from _The New Yorker_ called _The Horse That Could Whistle Dixie_ was another favorite of mine, as were the two novels of his I'd read (which were brought into the apartment by Sylvia or Lee from the circulating library in Magrill's drugstore on the corner of Mermaid Avenue). These were I _Can Get It for You Wholesale_ and its sequel, _What's in It for Me?_ , and I thought them marvelous, as I did Budd Schulberg's _What Makes Sammy Run?_ The Studs Lonigan trilogy of James T. Farrell was another sophisticated favorite of mine and everybody else's, mainly for the realistic action and realistic words. James Joyce's _Ulysses_ floated briefly into the Coney Island apartment also, no doubt borne there, I recognize now, by the notoriety of the court victory over its banning. _Ulysses_ sailed back to Magrill's drugstore unread by any of us, although I still can tingle with the _frisson_ of astonishment I enjoyed coming upon two forbidden words in the very early pages, one of which describes the green color of the sea, the other the gray, sunken state of the world. John O'Hara was known to me also.\n\nI had a large assortment of other reading to draw upon for stimulation while overseas. Professor Matthew Bruccoli at the University of South Carolina reminded me just a while ago of those plentiful Armed Services (or Armed Forces) Editions, which I had forgotten about. Here\u2014there\u2014was a government publishing venture of scope without precedent and one that probably hasn't been and won't be repeated: editions of more than twelve hundred works in printings of fifty thousand copies each, distributed without charge to American servicemen everywhere. Bruccoli, an insatiable collector, has assembled more than a thousand of these titles. He wants them all and won't ever get them, for I doubt that even he can now locate those missing two hundred volumes.\n\nIn one of these paperbound books, a collection of short stories, I found a work by Stephen Crane, \"The Open Boat,\" and in this tale of shipwrecked sailors adrift in a lifeboat is a line of dialogue repeated by a man at the oars like a Wagnerian _leitmotif_ (although I did not learn about Wagnerian _leitmotifs_ until later): \"Spell me, Billy.\" (I'm not absolutely sure of the name Billy. I no longer have the Armed Forces, or Armed Services, Edition, and Bruccoli might not have it among his thousand, either.)\n\nFrom this sonorous reiterated chord, which I pondered in my tent, and perhaps from a one-act play of Saroyan's called _Hello, Out There_ that I might already have read, could have sprung the notion of the short story to be titled \"Hello, Genoa, Hello, Genoa,\" which would be related entirely in brisk intercom radio dialogue between a bomber pilot, or a few of them, and the control tower at the air base in Corsica. It is (even to this day, I feel) an engaging notion with a beguiling title, and I never tried writing it (or have no recollection that I did).\n\nInstead, I wrote a short story in a day or two that I called \"I Don't Love You Anymore.\" It is about two thousand words long, and like everything from my Algerian and Corsican periods, it is based on things I knew nothing about except from my sifting around in the works of other writers. Consequently, it reflects the style and point of view of many of the malign and histrionic fictions by American male authors of that time and ours: A married, worldly-wise serviceman, with whom we are intended to sympathize, feels temporarily, upon finding himself back home, and offering no specific complaint, that he no longer wants his marriage and no longer loves his wife. (Whatever that last was meant to mean, I truly had no idea. It was a convention.)\n\nI don't know how it happened, but after I was back in the States and just out of the service, the story was submitted as an unsolicited manuscript and accepted for publication in _Story_ , a periodical then publishing fiction only and held in very high esteem. By pure luck in timing, the war in Europe being just over, it chanced that the magazine was devoting an entire issue to fiction by men and women from the services.\n\nI was back in Coney Island living in the apartment on West 31st Street when the note of acceptance arrived with the mailman. Overnight, I was, or felt myself, a local celebrity\u2014I saw to that by showing the letter to everyone I knew\u2014and I became talked about, too, in a small social enclave on Riverside Drive in Manhattan, where I was already going steady, keeping company, with a girl I had met not long before and to whom I soon would give an engagement ring and would marry the afternoon of October 2, 1945, just before boarding a Pullman train to California to begin college in Los Angeles.\n\nThe acceptance of my story, of course, impressed everybody, although no one I knew of except Danny the Count had every heard of _Story_ magazine, and probably it was from Danny the Count that I first heard about it. Possibly it was at his suggestion that I made the submission. The payment I received for \"I Don't Love You Any more\" was\u2014I had no idea what to expect\u2014twenty-five dollars.\n\nI was soon calculating deliriously. I had spent only two days on the story at a time when I was distracted by the war. Once I applied myself industriously, I figured, I could easily manage to complete four stories a week, approximately sixteen a month, and, by selling them for twenty-five dollars each, I would soon find myself with earnings almost as large as what I had been raking in as a first lieutenant on combat duty overseas.\n\nI had great expectations.\n\nHere was a beginning, a first convincing demonstration of a promising future, and even my prospective in-laws were proudly persuaded. Soon after the wedding and the train trip to Los Angeles, I derived a second and even greater indication of potential triumph from my first-year course in freshman English.\n\nGETTING GOOD GRADES in classes at the University of Southern California in 1945 proved ridiculously easy (Art Buchwald, a student there at the same time, I later learned, conceivably may have floundered in the classroom, but I did not). It was finding an apartment that was impossibly hard.\n\nAfter luxuriating for a week or more at the Ambassador Hotel for the second part of our honeymoon\u2014it was one of L.A.'s better-known hotels (it was the hotel in whose kitchen Robert Kennedy was shot to death twenty-three years later during his California presidential-primary campaign)\u2014we settled into a large bedroom in a small rooming house downtown on South Figueroa Street, between Washington Boulevard and somewhere else. The costs of the stay at the Ambassador Hotel were met from the cash gifts collected at the wedding, by far the most substantial portion of which flowed into our pockets from my wife's side of the family. The initial segment of the honeymoon was spent in a drawing room of a train going overnight from New York to Chicago and another drawing room of a different train going from Chicago to Los Angeles for three more nights. It was not much fun. We read a lot. The observation car and the club car bedazzled only briefly. A Pullman berth in a drawing room of the _Twentieth-Century Limited_ and then on the _Chief_ or the _Super Chief_ from Chicago to Los Angeles is not ideal as a nuptial couch. There was a beauty shop on board, and my wife, Shirley, went once to have her hair and nails done.\n\nThe rooming house into which we moved was obtained through a housing-locating service at the university. It was the property and dwelling of an elderly, shrinking, white-haired couple named Mr. and Mrs. Hunter. A trolley line just outside bore me to the college in one direction and in the other trundled into Pershing Square in downtown Los Angeles and to the pickup and drop-off points of buses going to whichever of the racetracks was in session. There we could find Chinese restaurants and Mexican food, too.\n\nIn the household and marriage of our landlords, Mrs. Hunter was beyond doubt the energetic party. Ghostly, tidy Mr. Hunter, in his light-blue or light-gray cardigan sweater, with an ineffaceable and cryptic smile on his trim, pale face, would drift by wordlessly like some harmless spirit whenever our paths crossed during the academic year we lived in his house. In sharp contrast, Mrs. Hunter, when I was away mornings at school, would corner my wife and try to convert her to Christianity, specifically into a California sect to which, she confided with great secrecy, she had already deeded the house and everything else she owned. Some kind of Jewish evangelist was already in the city at the time\u2014formerly a rabbi, she boasted, and of late a proselytizer for Christ\u2014and she thought we would profit greatly by listening to his radio broadcasts and attending his services. On the other hand, she also confided to Shirley that her husband, Mr. Hunter, was not, or had never been, much good at servicing her sexually and that there were, or had been, periods in her life when her cravings had grown so desperate that she wanted to, or in fact did, seat herself on a cake of ice to cool them. They had never brought forth children.\n\nWe had no cooking facilities, and what limited kitchen privileges were offered (refrigeration was about the extent of them) we hesitated to make use of for fear of being trapped in peculiar conversations. On our corner of Washington Boulevard was a good Greek coffee shop, and we ate most of our dinners there and in a few other neighborhood places. On weekend splurges we indulged ourselves in publicized glamorous haunts like the Brown Derby or Romanoff's in hopes of catching a glimpse of some movie personality, any movie personality. (We scored only one, Rosalind Russell, and I would never have recognized her because of her tiny chin, but my Shirley did.) That part of Washington Boulevard near which we lived was also the center of the city's mortuary-monument industry.\n\nGiven my wife's rather privileged and sheltered upbringing, I am incredulous now that she succeeded in spending a year with me under such conditions or even agreed to try, and that her mother, Dottie, permitted her to. In 1949, four years later, when I received my Fulbright scholarship to Oxford, her mother's abrupt response to her daughter's exciting news was, I was told:\n\n\"You're not going to England!\"\n\nWe went.\n\nIn California, as a married freshman student without friends or family, I found it hard to meet people, and we enjoyed very little in the way of a social life. There was much larking around on campus, especially between the fraternities and the sororities, but we had no part in that. I imagine that an even more confounding time was suffered by the younger male students entering the university, any university, directly from high school. They were very young by comparison to us and they were rendered mute and invisible, were overshadowed by the numbers of veterans crowding into the classrooms, almost all of them my own age, twenty-two, and over. We were older, bigger, and better read and we had experienced more and were zealous to learn (or else we were football players).\n\nIt must have seemed a preposterous hope for any of those younger students to succeed in qualifying for collegiate athletics. The football teams consisted mainly of older young men who had been playing on college teams before the war and had spent the years since then growing huger in muscle and bone mass and, often, honing their pigskin skills on one of the service teams. They moved lethargically through classes on a preplanned schedule to conserve their eligibility until they were most needed. The year I attended Southern California, the football team went to the Rose Bowl, with a backfield that was an average age, I'd guess, of twenty-four. I believe the star was twenty-six. The opposing team was Alabama, whose All-American player, Harry Gilmer, had been something of a standout, I think, even before the war. As a student I received two tickets to the big game. Back then, I already detested organized cheering sections and sneered at things like college spirit and cheer-leading squads. Because Shirley had given up trying to grow interested in football and was apathetic about the Rose Bowl, I sold my two tickets for twenty-five dollars each to a couple from Alabama, and we spent that day, and the money, at one of the local horse tracks, Santa Anita or Hollywood Park.\n\nThe motives for my decision to go to the University of Southern California remain opaque, but they doubtless included the indispensable one that I was accepted there. I don't doubt that they were also evasive in purpose, intended to delay, to buy time. I didn't want\u2014I felt myself much too young\u2014to have to decide right away what I was going to do for the rest of my life. Once afforded the opportunity to attend college, I decided to make use of it. Going to college was easier and more appealing than going to work and certainly, then, more respected. And what work could I have found that would not have been a blow to the spirit after my jubilant homecoming from the war?\n\nBut I didn't know where I ought to try to go. Places on the altitude of Harvard and Yale didn't really exist for me\u2014they were beyond both my grasp and my reach, outside my world and outside my imagining. What would the two of us do with ourselves at Harvard or Yale even if I had applied and been admitted? Who would talk to us? Any concept of higher education as high education was alien to me. I wanted to find out; I wasn't sure I wanted to learn. Anything in the nature of a scholarly love of knowledge was soon cooling realistically with my growing awareness that I would never make the effort to acquire the Greek, Latin, and German referred to so copiously in the illuminating works of literary explication I was guided to. At the same time, the idea was growing stronger that I should strive to make a name for myself as an author of fiction rather than as a critic or some other kind of intellectual. My hyperactive enthusiasm to learn more about just about everything existed principally for my own blissful satisfaction, and still does. (I have given up on Wittgenstein and Sartre and philosophy, but in recent years I have been burrowing into neo-Darwinism, which is congenial to my religious skepticism, and writhing in an exasperating quandary over quantum mechanics, which, to my mind, remains impossible even to define, let alone comprehend.)\n\nI had been stationed twice in lower California during the war and had enjoyed both interludes there. In fact, I pretty much enjoyed everywhere I'd been in the army\u2014Corsica, too, apart from the perils; even San Angelo, Texas, where I was sent for several months after Europe with almost nothing to do almost all the time. I am generally not a hard person to please (although people close to me might find that hard to believe, and distant acquaintances impossible). At the air base in Santa Ana, when I was there for classification as an aviation cadet and preflight training, there were the calm recreations in the town itself on weekends off and the longer, fatiguing overnight excursions by bus from the air base to Los Angeles. At bombardier training school in Victorville in the desert, we could treat ourselves when we desired to the fairly riotous Saturday nights off at Balboa Beach, with its flashy pier and dance halls and bars, and the billions of young girls and women who were also drawn there with the goal of carousing around. For me as for many others, lower California emitted a cinematic charisma as a place of beautiful excitements and charming opulence and opportunity\u2014it still does\u2014and no one expressed reservations about our decision to go there. And, of course, the moderate California weather was always seductive.\n\nThis was in stark contrast to the horrible, deeply depressing, incapacitating winter milieu into which I was harshly plunged on my furlough after I'd returned by steamship to the States from Corsica in January and found myself back in Coney Island. I soon knew without need to put anything into words that, given a choice, I would not want to live there again. Coney Island had long before lost its magic for me and for most of the rest of us as an amusement area, and there was nothing but Nathan's open in winter, anyway. It was black and it was bleak. There was no one around and there was nothing going on. There were no social clubs. I had no girlfriend and didn't know any girls I felt I might want to see intimately, or who might want to see me. In Coney Island that winter there was no place but the movies to go to on a date; other places were too far away, and I didn't have a car. If I'd had a car, I wouldn't have known how to drive it. I had nothing to do.\n\nI missed the army, where there was always something to do, even when there was nothing.\n\nIn a day or two, sister Sylvia and brother Lee took note of my pathetic inertia, and it was Lee who, typically, came up with a suggestion that was not as implausibly far-fetched as it had sounded to all three of us at first mention. Sylvia and I couldn't have thought of it in a million years, but Lee, the selfless wishful thinker, came up with the proposal that marvelously did the trick: Why didn't I take myself off for a week or so to a long-established resort hotel, not far off in the Catskills, which was relatively fashionable to New Yorkers and known to be popular with people who could afford it, and of which I, in my parochial simplicity, had never heard?\n\nGrossinger's.\n\nWe knew I had the money: Allotments from my air corps pay, which we all thought mighty substantial, had been coming to them directly from the start, to be banked in my name. As a serviceman back from overseas I would probably be showered with attention and courtesies. It sounded okay to me. Lee made the reservations and arranged for the van pickup for the trip from Brooklyn to Liberty, New York, and the plan turned out to be a rejuvenating solution not only for the rest of my furlough but also for the shaping of much of my life ahead.\n\nI had never been on ice skates before, but found I could skate capably after the first few minutes, and it was indeed a dashing figure I cut as I circled the rink in my green winter flight jacket with its fur collar and silver first lieutenant's bar. For the evenings I had my officer's dress with my wings and campaign ribbons for overseas service, the only impressive one among them, I knew, the Air Medal with a large number of oak-leaf clusters (which came as routine with the completion of every fixed number of missions and not, as others might suppose, for the accomplishment of anything particularly valorous). I had my gleaming Mediterranean suntan, my health, my youth, my good looks, and my modesty. I was Jewish. I could project self-confidence. I was respectful and polite, and I was made much of by the parents there, especially those with single daughters, and by single women unaccompanied by parents. Even I knew it was a sure thing I would win the weekly dance contest on Saturday night, even though I was not then, am not now, and have not ever been anything much as a dancer. As fortune would have it\u2014the kind of storybook good fortune one used to expect to find in romance magazines\u2014I met the girl there, that week in January, whom I would marry in October and remain married to for thirty-five years before we separated.\n\nWhen I returned to the city we were already going steady.\n\nWe dated when I commuted into New York weekends from Atlantic City until I was shipped to San Angelo, Texas. While I was there we exchanged letters. We saw each other again, in New Jersey and New York, when I was transferred in May to Fort Dix to be processed for discharge. And we went on seeing each other when I was out. It was Shirley's mother who took the initiative when I was alone with her one afternoon in her living room and supplied the impetus I hadn't found the power to generate on my own.\n\n\"Barney thinks,\" said Dottie, with the devious premeditation that was second nature to her and occasionally endearing\u2014it was likely that Barney, the husband, had no inkling of what she was up to\u2014\"that it's because you don't have the money that you don't give her a ring.\"\n\nI had never heard such language before.\n\n\"Give her a ring?\" I inquired, puzzled, for the possibility of my doing any such incomprehensible thing hadn't once entered my mind. \"What for?\"\n\n\"To get engaged to be married.\"\n\nI had money enough for the ring. She made the purchase and billed me for just $500. Friends and families on both sides were delighted with the match. I was a young and unformed twenty-two and a half; the bride had just passed twenty-one. Almost every fellow I'd grown up with in Coney Island was getting married at about that time: Marvin Winkler and Evelyn, Davey Goldsmith and Estelle, Lou Berkman and Marion, Marty Kapp and Sylvia, Sy Ostrow and Judy, and others. (As far as I know, mine and only one other of these early marriages would end in divorce.)\n\nTHE MARRIAGE SEEMS to have been a good neurotic fit, observed the psychiatrist at my introductory consultation with him in which I tried to present fairly all the imperfections in me and in my wife underlying the growing estrangement and irreconcilable disputes that had been arising uncontrollably.\n\nIt surely must have been a good neurotic fit, to endure for thirty-five years. Why then did we eventually separate and divorce? It was a good fit, but, well... I doubt either of us could have said. I don't believe either of us wanted to.\n\n(I found out in ensuing sessions of psychoanalysis that there was nothing generically denigrating about my new mentor's use of that term _neurotic._ \"All of us are neurotic,\" he erupted one day, after I'd strayed again from my free-associating into what he called my habit of intellectualizing. \"Only psychotics are not neurotic\")\n\nMOST WEEKDAY MORNINGS at Mr. and Mrs. Hunter's I spent at school. The thought of cutting classes didn't seriously occur to me in college any more than it had in high school. The temptation to do so, had it arisen, would have tormented me. I enjoyed the work and the classroom competition.\n\nAfternoons and evenings we often roamed about the city, sightseeing. After all, this was Hollywood, the land of Mickey Rooney and Judy Garland, Van Johnson, Dorothy Lamour. Since we seldom had to be anywhere at an appointed hour, we had time to travel by bus. We looked at the La Brea tarpits and the house of Hearst's mistress, Marion Davies's house at Santa Monica, at the names immortalized in the pavement outside Grauman's Chinese Theater. We went to Tijuana one weekend and even ate a taco each from a street vendor without falling ill. Shirley and I caught our first sight of shish kebab grilling on a skewer in the neon-glaring window of a restaurant on a pulsating street near the UCLA campus early one evening, and it became a reliable weekly meal, with rice and salad, and a side order of french-fried potatoes, for under two dollars. Our year in Los Angeles was educational: Back in New York, we were the only people we knew ever to have heard of shish kebab, and we boosted it heartily. But grilled lamb is not particularly succulent, and we soon advanced to dishes like beef bourguignonne, coq au vin, cappelletti bolognese, shrimp creole, lobsters broiled or steamed\u2014who can remember them all?\n\nAlso in the afternoons and most evenings I did my reading and writing for school and wrote and rewrote my stories and light nonfiction pieces that I mailed away for publication to one magazine after another; they all came back. I had a portable typewriter of my own now, purchased from Macy's through Sylvia at her employee's discount\u2014in the days before discount stores, when a discount was still a discount. Having heard that there existed such a thing as a writer's keyboard, I had ordered it with the idea that my written work would flow that much more swiftly. It had such trivial modifications as quotation marks and question mark in lower case. It made no difference. Typing a little more rapidly only meant that I would have to begin retyping pages, paragraphs, even sentences that much sooner. Both of us read a great deal for pleasure. Shirley read _Wuthering Heights_ and was astounded that it diverged so much from the movie. I read everything I believed to be instructive. On our table radio we listened to the one classical-music station in Los Angeles and to the radio-network comedy programs that were already our favorites. Fred Allen was available and Jack Benny was still funny.\n\nTwo authors for whom I found myself with a new and fervent enthusiasm were Aldous Huxley, for reasons I now can no longer begin to guess, and H. L. Mencken, whose merits as a thinker and writer are brilliantly obvious to this day: a blunt and mordant dislike for every kind of hokum, a command of American English that is eloquently and indisputably his own, a voice whose surface aspects others might seek to copy but who has not, to my mind, been surpassed in individuality and substance.\n\nI was then among the susceptible many trying to copy him, believing I was the only one. Words like \"clodpate\" and \"bastinado\" began showing up regularly in my writer's vocabulary for the scornfully superior humorous pieces I was dashing off at least once a week, interspersed with the short stories I was cranking out at just about the same rate. In the college library I discovered one work of his published in 1919 or so, which, I am ashamed now to say, I responded to with laughing admiration, virtual applause, and which, if in print today and he still among the quick (as he might say, and did), would see him in the stocks for the rest of his natural life, with no fair-minded person dissenting. The volume, a slim one, is titled _In Defense of Women._ The ostensible (and disingenuous) argument is that women in general are the more intelligent members of the human species, and men, in consequence, comparatively stupid. The proof? Women induce men to marry them, to live with them, and to tolerate their presence socially. I am embarrassed to confess that more than Mencken's vocabulary found its way into my literary thinking. It was the fashion, the convention of the time, to present women in a stereotyped way as targets to be patronizingly derided. I am thankful now that none of these efforts of mine found its way into print or even, to my knowledge, still exists.\n\nI had applied to Southern California with a major in journalism, thinking, without thinking clearly, that there'd be a lot of writing and reading involved and that newspaper work by its nature would be invariably arresting, if not always electrifying. While I was registering and noted the required courses, such as copyediting, rewriting, proofreading, and layout, I switched like a fugitive to a major in English and with a deep breath relaxed and blessed my luck. I had a deep feeling of relief that I'd escaped, by the skin of my teeth, an irreparable and very great disaster.\n\nThat semester, as afterward, I enrolled in more classes than the stipulated minimum. With all costs met by the government, why not? By attending college summer sessions, too, I managed to complete the four-year curriculum in three years. I finished my science requirements that first year with overlapping courses in bacteriology, botany, and zoology. These were areas of study totally new to me and entirely fascinating. I would have liked an introductory course to physics, too (I still would), but couldn't find the space to fit one in (and still can't). A survey course in the history of civilization\u2014two semesters, I believe\u2014didn't have a minute in it or demand attention to even one item that didn't thrill me. Flashbacks to my elation upon hearing for the first time about the Indus, the Tigris, the Euphrates, the Ganges are still vivid\u2014what resonant sounds these marvelous words discharge\u2014and Praxiteles and Pericles, too. The work in English, freshman composition, was grist for my mill: Punctuation and syntax were easy, the written assignments valued opportunities to show my stuff.\n\nI tried artfully in all my written work to address my subject with a paper that would also, with ingenious alteration, be suitable for publication in one of the popular magazines I knew about. Thus, when research for a theme on paleontology brought me into contact with paragraphs on geological periods with names like Mesozoic and Jurassic, terms I could retain by rote but not aspire to understand, I first met the assignment adequately and next converted the composition into a humorous spoof, presented (I hoped) with the abrasive superiority of an H. L. Mencken, of the deliberate use of unintelligible jargon affected by pedagogues (another word I lifted from Mencken). The new piece, once typed, went out everywhere and went nowhere. It was perhaps not as humorous as I thought.\n\nThe short stories I wrote at that time tended to be plotted extravagantly and often to be resolved miraculously by some kind of ironic divine intervention on the side of the virtuous and oppressed. In one I remember, a French farmer in Provence who has secretly collaborated with the German occupiers watches in speechless horror as every one of his ten sons is picked out in a random selection from the village people for reprisal executions. In another, a visitor in search of a persecuted figure in the Christian rural South who has mysteriously disappeared turns out to be Jesus or some kind of avenging angel bent on retribution for his murder. It gives me pleasure to say that these stories went nowhere either.\n\nIn an elective course in contemporary theater (I had thoughts of becoming a playwright, too. It seemed easier. There were fewer words) was a blond, outgoing girl named Mary Alden, who early on took a testy dislike to me, which she made little effort to mask. She was my own age, perhaps slightly older, with a couple of years' experience working in the theater. The two of us dominated classroom discussion, and we were on opposite sides of just about every question. She was galled by my contradictions. Having recently come from New York, I had the advantage of referring to people active there in show business of whom she had not yet heard and the teacher had. She found my presumption unforgivable. I was sorry that Mary Alden didn't like me, for I liked her. She held as sacred what I took for sport.\n\nUnfortunately, that has happened with me several times since\u2014with a graduate philosophy major named Norwood Hanson on the steamship carrying a large contingent of Fulbright scholars to England; with a faculty colleague at Penn State named Gordon Smith, when we visited the hallowed battlefield and cemetery sites at Gettysburg; and more than once with a fellow student at NYU named David Krause, but David and I remained friends through college and afterward. The other two refused to speak to me ever again.\n\nMy mother could have told them I had a twisted brain.\n\nFinding in my first term in college that I could attain top grades in just about all my courses with little strain, I devised my program for the next semester with veteran savoir faire, coming up with a schedule designed to release me from classes two or three afternoons a week early enough to join my wife downtown and catch the bus that would get us to the racetrack in time to bet the daily double. (This was childish fancy, for neither of us had cashed a ticket on the daily double the several times we had been to the races, and I have never cashed one since.) Dismounting from the trolley car with schoolbooks still in hand after one of these half days on campus, I was surprised by the sight of Shirley waving with one arm and hastening toward me with a face rosy with joy. In her hand was an opened letter that had come in the mail that morning after I had left the rooming house. It was a note of acceptance from _Esquire_ magazine for a nonfiction piece I had written originally as a theme in my freshman English class. Assigned the subject of describing some kind of method or device, I had invented a series of sure-fire systems, all of them foolish, for always winning money at the racetrack.\n\nAnd with the note had come a check for $200!\n\nI was now, at twenty-two, a certified young author of fiction and, with _Beating the Bangtails_ , of nonfiction also (an illustrious distinction in 1946, two years before Vidal, Capote, Mailer, and others came rolling onto the national scene and made me feel backward, aged, envious, and derelict).\n\nI was not, however, a certified handicapper of horses, and most of the $200 I earned from my freshman theme on how to win money betting on horses was lost at the racetrack the next half-dozen times we went.\n\nTHE VEXATIOUS PROBLEM of obtaining a fitting place to live, which had obstructed the conjugal dreams of us newly-weds in Los Angeles, was brushed away with dispatch in New York by another exercise of the untiring enterprise of Dottie, wielded with Barney's usual reserved and benevolent approval. Friends of theirs were residents and owners of a modernized, narrow, five-story elevator apartment building at a wonderful location in Manhattan, on West 76th Street just off Central Park West, around the corner from the New-York Historical Society and just a few steps from the American Museum of Natural History. When a one-room studio apartment facing front on the first floor became vacant, we moved in.\n\nWe were figuratively just a hop, skip, and jump from a couple of subway lines and bus stops and from all kinds of shopping. One subway carried me expeditiously downtown and back to NYU for the next two years; the other rode me uptown to Columbia University for the year of my master's degree. The amount of room at our disposal wasn't notably greater than what was ours in the California rooming house, but here a roll-out sofa bed permitted us to better utilize what space we had. And here we had a compact kitchen area that conveniently held a table for four. We could entertain, and we did. Among the people we had over for dinner once or twice was Maurice \"Buck\" Baudin, Jr., the instructor in the fiction-writing course in which I and some new friends at NYU were doing so well. Buck was a mere five years older than I, but even after I had moved on from NYU I couldn't with composure bring myself to call him anything but \"Mister.\" The furnishing of the apartment was handled by Dottie and Shirley and was done tastefully and rationally and, I imagine, inexpensively, by their standards. Lee's wife, Perlie, had looked into such things as interior decorating and gushed in praise at the sensible arrangement and the authentic look of several of the almost-genuine antique pieces. Dottie was pleased to have her daughter and her writer\/college-student son-in-law installed in an apartment she would be able to display to her friends with pride.\n\nAnd so, for that matter, was I.\n\nMy former mother-in-law, Dorothy Held, was one of those cultivated, modern women who knew the difference between turquoise and aquamarine. On the other hand, she also knew the difference between sirloin steak and top sirloin, prime rib and top round, and that only first-cut brisket was suitable for a good pot roast.\n\nMy favorite steak before I met her, after I'd left home and discovered it, was filet mignon, always well done. Not until I'd met Dottie's daughter and was eating at her table was I introduced to the joys of rare beef, to the rare sirloin steak, right up to the bone, and to the heavenly and substantial charm of a standing roast of rare prime rib of beef with a crust liberally seasoned with garlic, salt, and paprika. Shirley's family was much better off than my own, with experience vastly more cosmopolitan: They even read the _New York Times._ And her parents found delight in watching me eat\u2014always a second helping, often a third. They were generous in other ways, too. They were well off but not wealthy. Barney\u2014Bernard Held\u2014was partners with a large company in a much smaller firm manufacturing dresses. He apparently made a very good income during the war and for many years after, but the Helds tended, with Dottie providing the impetus, to spend just about everything coming in and to greatly enjoy whatever they spent it on\u2014including us. They spent much on us, and it was largely with their encouragement and help that I was able to spend almost the next six years of my life, and the first six years of my marriage, going to college.\n\nA gratifying event in our marriage was the birthday party my wife made for my sister, Sylvia. Well into the marriage, after some twenty-five years, Shirley was horrified to hear one day that my sister had never in her life been given a birthday party, and resolved to make her one. I had never seen Sylvia so filled with effusive happiness as she was at our house the next time her birthday rolled around. Tears of laughter streamed down her cheeks, and her voice soon was hoarse with her whoops of hilarious reminiscence. With Lee it was the same; he was talking and chortling and wiping his eyes, too. I had never before heard either of them, alone or together, in such exuberant discourse. I listened. I learned. Among the memories Sylvia recalled with merriment was of her first year out of high school when, at age seventeen or eighteen, she would journey into the city with others to search for a job. This would have been in about 1934, when jobs were not easy to find. She and her female friends would chip in for a pack of cigarettes and pool their money for variety in their lunches in a cafeteria. And at the employment agencies, she remembered, they often were spared long waits in vain by someone appearing from inside to announce that they might as well leave: There were no jobs for Jews that day.\n\nThat she could recall this with mirth at her birthday party was testimony to the astonishing good fortune we had all been enjoying since.\n\nOnce, in Finland on a book tour, I met a local writer, Danny Katz, a Finnish humorist whose Jewish parents, like mine, had emigrated from Russia. \"They weren't too smart,\" he muttered, chuckling. \"They settled here.\"\n\nOurs were smarter. They had settled in New York.\n\nAT THE END OF OUR first year back in New York, a full apartment just above us with a complete kitchen and large living room and bedroom opened up and was offered to us. Into it we promptly moved, and there we lived until the fall of 1949, when we sailed to England. I've lost track of the numbers, but I'm certain we were helped significantly with the rent and other expenses, for I very strongly doubt that my monthly subsistence allowance from the government while I continued at school would have enabled us to live on a scale that, then as now, seemed lavish. (I did for a while work part-time after school in the circulation department of a magazine called _American Home_ , filing and finding the address plates of subscribers, but the hourly income and take-home pay from that would not have been gigantic.) To friends from Coney Island, almost all of whom by then had moved out of Coney Island, and to other friends I made at NYU, our standard of living was kingly.\n\nA commodious, well-furnished apartment in a well-maintained building in a prime neighborhood in Manhattan was as precious then as it is today. We held parties and dinners there often, and it was a good meeting place before going out with friends from locations less felicitous. On balmy nights we served drinks on the roof, where there were comfortable chairs and well-tended plants. In summer, Dottie and Barney always took a country house somewhere or booked space in a luxury hotel for the season\u2014I remember the Lido in Lido Beach, Long Island, and the Forest House at Lake Mahopac, New York\u2014and we would go there on weekends and during other breaks in my summer-school sessions.\n\nAll in all, we were living the life of Riley.\n\nAnd it wasn't costing me much.\n\nGetting high grades at New York University required much more application than had my freshman year at Southern Cal, but I managed in almost all my courses. The competition was tougher. Lacking a campus and a campus social life and no longer an active institution in intercollegiate sports, the Washington Square College of the university was something in the nature of a commuting college. And students who traveled to classes, many from homes quite distant, weren't there to fool around. Especially not the war veterans. Two friends I made in philosophy courses, Edward Blaustein and a student named Kahn, received Rhodes scholarships at a time when few, very few, were awarded to Jews. Eddie Blaustein was later a professor of law, then president of Bennington College (the first Jewish president, he said, of a non-Jewish college), then president of Rutgers University. I don't know what happened to Kahn, but I don't doubt that he, too, did all right. The main lesson I learned in philosophy was to react with skepticism and treat with Socratic malice all emphatic ideological beliefs, especially those of my favorite philosophy teachers.\n\nIn the fiction-writing courses of Maurice Baudin was David Krause, who journeyed from faraway New Jersey, journeyed through rainstorms and blizzards\u2014he would, in a figure of speech first coined by Mario Puzo in connection with himself, sooner eat a broom than miss a day of class. He got straight A's, too, and was a professor of Irish and English literature at Brown University all his adult life until his retirement not long ago. David wrote perfect short stories but couldn't be persuaded to submit them for publication because he didn't think they were worthy. And we had Alex Austin, a meager, short fellow who seldom raised his voice above a whisper, even when reading aloud in the classroom. By the time he enrolled in the class, more than two hundred poems and short stories by Alex had been published in \"little\" magazines most of us hadn't even heard of. It was an essential part of his daily regimen, like brushing teeth, to write at least one short story every afternoon\u2014he would sit down at his typewriter devotionally, often without a thought in mind, and simply begin typing. He had novel-length manuscripts, too, and Baudin was reduced to imploring negotiations with him to limit the number he handed in.\n\nUnlike David Krause, who was a genuine idealist (he saw teaching as a calling; I went into it because I thought it would be easier than a business office), I mailed out everything, worthy or not. When I learned that Buck Baudin was selling stories to _Good Housekeeping_ and receiving $1,500 apiece for them, I appended that magazine and all other women's magazines accepting fiction to my list of prospective benefactors. Hardly a workday went by that didn't find the postman delivering to 20 West 76th Street at least a few of my stamped, self-addressed manila envelopes returning manuscripts of mine with rejection slips. _The New Yorker_ then was as admirably efficient in editorial procedure as it was superb in editorial content. I used to joke\u2014and it wasn't much of an exaggeration\u2014that a story I would mail to _The New Yorker_ in the morning would be back with its concise, slighting rejection slip in the afternoon mail that same day.\n\nSometime during my second or third semester with Baudin\u2014I took a third semester without credit because I wanted to keep working with him\u2014he chose four short stories of mine and delivered them for consideration to his literary agent. The reader's report from there said that none in the group was suitable for publication. Of the four, three were subsequently accepted by magazines when submitted by me as unsolicited manuscripts.\n\nMuch later, when I taught for four years at the City College of New York between the publication of my first novel, _Catch-22_ , and the completion of my second, _Something Happened_ , I several times attempted the same thing with my own literary agent for those students whose work seemed especially promising, and was rewarded with the same negative result.\n\nA comment I always seemed to get back from Baudin with each short story, regardless of other notes of praise or criticism, was that I was taking too long to begin, dawdling at the opening, as though hesitant to get going and move forward into what I had in mind. It's a quirk of mine, perhaps a psychological flaw, that has lasted. I have cogitated over it in the closest secrecy, secrecy until now, as a quality that might credibly be described as anal retentive. From the manuscript of _Catch-22_ , after it had been accepted, I deleted of my own volition something like 50 of the first 250 pages without loss of incident. (The editor of these very words, Robert Gottlieb, who was also the editor of _Catch-22_ , snickers at the words \"of my own volition.\" Okay, it was at his suggestion that I did what I could to move into the middle much sooner.)\n\nIn the margin of one story of mine in college, \"Castle of Snow,\" Baudin asked why I simply didn't begin at the top of page four and instructed me to start there when I read it aloud to the class. It went well that way, to my grateful astonishment, and I began on my original page four when I typed a clean copy.\n\nMailed out by me to the \"Fiction Editor\" of the _Atlantic Monthly_ with that revision, it was rejected, but with a personal letter from a woman there who signed her name and seemed to be implying that with several small alterations it might be accepted if resubmitted. (I wish she had suggested one more, for the choice of Chaucer as the favorite author of an East European immigrant forced to sell his books jars the teeth now with its blatant improbability.) I made the changes, amending what was amiss, providing what was lacking (I welcomed text suggestions then, and I've welcomed them since), mailed the revised manuscript directly to her, and the story was accepted for publication as an \"Atlantic First.\" Coincidentally, in the issue of the magazine in which it finally appeared was a story by the equally young James Jones, also presented as an Atlantic First. (Neither of us was fortunate enough to win the semiannual bonus prize for the best of the Atlantic Firsts in that period.) I was paid $250. With a contact now at the magazine, it was not surprising that sooner or later I would place another work there, and I did. For this second one I received just $200, since it wasn't as long as the first one (and not as good, or even much good at all).\n\nI was a star in the class, and I would have been a celebrity on campus if the Washington Square College of New York University had a campus.\n\nMuch the same process was simultaneously going on between me and a benevolent editor at _Esquire_ magazine who finally signed his name George Wiswell in elaboration of the initials to the penciled notes of encouragement he had been adding to the formal rejection slips. Eventually he rejected a story regretfully, lamenting some defect in motivation or characterization and virtually pledging that he would recommend it for publication if that fault could be remedied. I made the attempt, the story was accepted, and for this one I received $300. Sometime later they took another one and paid me less, because the second one was shorter than the first (and not as good, or even much good at all). As I advanced in college, I acquired standards and learned to be more critical, and before I finished I also learned that, apart from their being mine, there wasn't much distinctive about all but two or three of the stories I was writing at this time. I now wanted to be new, in the way that I thought, as I discovered them, Nabokov, Celine, Faulkner, and Waugh were new\u2014not necessarily different, but new. Original.\n\nI mention as a public service the amounts I received in payment for these stories, for people generally have a misinformed notion of the scale of remuneration of successful American authors of fiction, and of freelance writers of any kind. In talking to groups of aspiring authors, I have frequently emphasized, particularly when the audience was made up of young people, that unless they had money, would inherit money, or would marry money, they were going to have to work at something else for quite a while, even if everything they wrote from that moment on found publication. Irwin Shaw would have agreed. And so would John Cheever. I used to boast that I could reel off the names of a dozen or more American authors of distinguished national reputation who weren't earning enough from their writing to ignore other sources of income and devote themselves wholly to their work, and probably I still can. Irwin Shaw could give in an instant the paltry total of dollars he had received for all the short stories he had ever published in _The New Yorker._\n\nA flattering consequence of my collegiate fame was that I was prevailed upon against my better judgment to submit a short story to the college literary magazine just getting started, and I did. For those who care, it was called \"Lot's Wife\" and was about\u2014oh, never mind that!\u2014the wife was icily indifferent and petty in regard to the victim of an auto accident, the man thoroughly sympathetic but no match for her.\n\nOf more interest is the unflattering aftermath of that charitable gesture, which was a patronizing review of the story in the college newspaper by a fellow student with literary ambitions who condescended to find it pallid, riddled with faults, lacking any compensating merit.\n\nThat should have steeled me against unkind critiques in the future, but nothing does.\n\nHowever, at just about that time I was ecstatic to learn that one of my stories, that same \"Castle of Snow,\" had been selected for inclusion in the Martha Foley annual anthology of best short stories.\n\n_That_ was ameliorating! There was no doubt I was on my way.\n\nAnd only twelve years later, after working two years at Penn State, one year at an advertising agency, a year at the Army-Air Force Post Exchange System (from which job I believe I was about to be fired), one year in the Advertising Department of the Remington Rand Corporation (which manufactured office systems and equipment and not, as I had feared, rifles), three years at _Time_ , one year at _Look_ , and three years at _McCall's_ , I finished my first novel and saw it published.\n\nAS I'VE SAID AND REPEAT, I wrote the first chapter in longhand one morning in 1953, hunched over my desk at the advertising agency (from ideas and words that had leaped into my mind only the night before); the chapter was published in the quarterly _New World Writing_ #7 in 1955 under the title \"Catch-18.\" (I received twenty-five dollars. That same issue carried a chapter from Jack Kerouac's _On the Road_ , under a pseudonym.) In 1957, while working at _Time_ , when the novel was only about half done, I received a contract for its publication from Bob Gottlieb at Simon & Schuster (providing for a guaranteed advance of $1,500, half on signing the contract, half on acceptance of the completed manuscript). The novel was completed in early 1961, when I was working for _McCall's_ , and published in October of that year under the title _Catch-22._ (What you may have heard was true: The number in the title was changed to skirt a conflict that same season with the Leon Uris novel _Mila 18.)_\n\nThe novel was not the instant success many people assume it was, not at all on the scale of such immediate national acclaim as greeted the first novels of Norman Mailer, James Jones, and others. It was not a best-seller and it won no prizes. There were reviews that were good, a good many that were mixed, and there were reviews that were bad, very bad, almost venomously spiteful, one might be tempted to say (and I am the one that might say it). In the Sunday _New York Times Book Review_ , for example, in a slender notice located so far back that the only people apt to notice it were those friends and relations awaiting it, the reviewer declared that the \"novel gasps for want of craft and sensibility,\" \"is repetitious and monotonous,\" \"fails,\" \"is an emotional hodgepodge,\" and is no novel; and in _The New Yorker_ , the critic decided that _Catch-22_ \"doesn't even seem to have been written; instead it gives the impression of having been shouted onto paper,\" \"what remains is a debris of sour jokes,\" and in the end \"Heller wallows in his own laughter and finally drowns in it.\" (I am tempted to drown in my own gloating laughter even as I set this down. What restrains me is the knowledge that the lashings still smart, even after so many years, and if I ever pretend to be a jolly good sport about them, as I am doing right now, I am only pretending.)\n\nOn a brighter side came generous statements of cheerful approval from a large range of prominent writers\u2014a gratifying and encouraging surprise. Gradually, as reviews continued to appear for weeks and even months after publication, requests for interviews began to arrive, for radio and television too (and all were accepted eagerly). After a first printing of about seven thousand copies, modest additional printings were distributed throughout the year as interest in the work continued and built. I believe there were fourteen or fifteen printings, and sales of the novel exceeded thirty thousand copies before the mass-market paperback edition was released.\n\nMy young daughter, Erica, and my younger son, Ted, were mildly baffled to find their father a sudden object of attraction and the recipient of compliments from people we hadn't known. Frequently they were titillated (in short time they found themselves driven to a studio by Tony Curtis himself in his Rolls Royce, to witness the filming of an episode of a popular TV series), but the effects were not always salutary. As the two of us were walking along the beach at Fire Island that first summer, my son of six belligerently thrust himself between me and a stranger who had come up to praise me and asserted stridently, \"I'm the one who wrote the book!\" (Probably he won't remember that, but I do.)\n\nFrom then on, I was something of a celebrity presence in the household, and that is never, or hardly ever, an entirely good thing. It would have been witless of me to attempt to ward off these flattering acknowledgments, and hypocritical to pretend I did anything other than lap them up.\n\nIn England, where the novel was published several months later, the history was different. By that time, the London literary world was already largely familiar with the work through word of mouth about the American edition and its growing reputation, and its appearance there was treated as something of an event. Best-seller lists were new and rudimentary in England back then, but _Catch-22_ was quickly at the head of them. News of this success abroad made its way back to New York and helped keep up the momentum of _Catch-22's_ expanding recognition. In the late summer of 1962, that same Sunday _New York Times Book Review_ reported that the underground book New Yorkers seemed to be talking about most was _Catch-22._ The novel was probably more heavily advertised that year than any other, but it was still, to the _Times_ , \"underground.\"\n\nAnd right after that came September and the paperback edition, and a surge in popular appeal that seemed to take the people at the publisher, Dell, completely by surprise, despite their elaborate promotion and distribution strategies. It seemed for a while that the people there could not fully bring themselves to believe the sales figures and that they would never catch up. Still, by the end of 1963 there had been eleven printings. _Catch_ was, I believe, the best-selling paperback book that year, with more than two million copies sold.\n\nSadly, those numbers did not translate into the wealth they might suggest. (The cover prices of these paperbacks was seventy-five cents, the average royalty was somewhere between 6 and 10 percent, and it was divided equally between the original publisher and the author\u2014netting me, I estimate, an unassuming three or four cents a copy.) It was not until the motion picture rights were sold about a year after publication that I felt confident enough, and substantial enough, to leave my job at _McCall's_ magazine, where I was in charge of a department of three others turning out advertising\/sales presentations, and then I did so cautiously on an unpaid leave of absence for one year. The money from the movie sale totaled $100,000, less 15 percent in commissions divided between agent and lawyer. At my request, payments would be spread over four years, allowing me more than sufficient time to throw myself wholeheartedly into my new career as a novelist and complete another work\u2014I thought.\n\nBut not until twelve years later, in the fall of 1974, a mere thirteen years after _Catch-22_ (four of those years spent working in the English department at the City College of New York and another teaching year divided between Yale and the University of Pennsylvania), was my second novel, _Something Happened_ , completed and readied for publication.\n\n_Something Happened_ was much more successful in its initial edition than _Catch-22_ (Irwin Shaw, who had read an advance copy, took me aside at a small gathering to tell me softly, \"I think it's a masterpiece.\" By now I feel free to say I agreed), and since its publication in 1974, I haven't had to work for money at anything I didn't want to work at or associate much with people I don't like. And I won't.\n\nI don't give presents anymore, either, and I no longer observe holidays. I hardly ever hurry.\n\n# 9  \nPsychiatry\n\nTHE FIRST TIME I met my father face-to-face to talk to him, so to speak, was in the office of a psychoanalyst sometime in 1979, when I was already fifty-six years old. My father had been dead for more than fifty of those years.\n\nFor as long as I could recall, I had been the periodic victim of a dream of escalating nightmarish dimensions. It would recur, unchanged, every now and then when I was sleeping at home in my own bed or away from home in hotel rooms, or, almost guaranteed, within the first few nights in strange beds in rented summer places. The dream was always the same in action and emotion. It would always unfold in the identical place, in that same bedroom in the same Coney Island apartment in which I had lived until I was past nineteen and left for the army. I would be in bed in the dream, sleeping, just as I was in bed sleeping when I was having the dream. Unexpectedly, a man would begin coming into the apartment, entering not criminally through the window in my room or through any other in the apartment, but at the front door. His face was in shadows. I would know right off that he was there, and in my dream I would come awake. I was frightened from the beginning and would grow increasingly frightened as I visualized him making his way toward me without hesitation, moving noiselessly past one room after the other along the hallway that ran through the apartment from the front entrance to the last room at the very end, in which I had been lying in my bed, and in fact still was, although in the dream it was always the bed I had slept in while I still lived there. My fear would quickly heighten to sheer horror as I watched myself sitting up in bed and saw the man draw closer and closer. In unbearable dread and panic, I was compelled to cry out for help, and the words would swell in my paralyzed throat and take form as gibberish, they would jam and choke me, I would strain and struggle with all my force, which wasn't much, to shriek out, pleading for safety and survival. And eventually, of course, before he was in my room, before I could distinguish any features of the face that was always cloaked in darkness, before whatever menacing outcome I'd been anticipating with such heart-stopping terror could take place, I would ride to my own rescue by coming awake\u2014as always happens with nightmares, Freud says somewhere.\n\nOr I would be nudged awake with displeasure and rebuked in bad temper by whoever was in the room with me and had been wrested from sleep by my unintelligible gasps and wordless groans.\n\nWhile recalling and then reminiscing out loud about this dream, with my psychiatrist (psychoanalyst) out of sight behind me as I lay on the couch in his office, I was stricken into silence for a moment by the stunning recognition that I had ceased having this dream, that I hadn't once suffered it for something like the two years past, not since I had entered into the patient-analyst relationship with him. He said nothing. I wondered why it had departed, wondered where that familiar, extremely harrowing, yet completely harmless dream of mine had gone, and as I turned my head to gaze at him, he saw me settle into one of those deliberate, stubborn quiets he'd learned by then meant that I was insisting on some kind of explanation.\n\n\"You don't need that dream anymore,\" he commented at last, with palpable reluctance. \"You have me here now.\"\n\nI grasped his interpretation with instant clarity and gratification, perceiving its rightness at once.\n\n\"Then why,\" I asked, and it may have seemed I was doubting him, \"was there fear?\"\n\n\"If that wish of yours were to come true,\" he answered, \"wouldn't you be afraid?\"\n\nI AM NOT GIVING his name, because I am relying on evocations, and he might not be pleased to see me make public reconstructions that are not accurate, or even some that are. At our second consultation, prior to our formally beginning therapy, he appeared somewhat put out when I responded negatively to his guess that in the interval I had reviewed his credentials. The idea of doing so hadn't crossed my mind; I wouldn't have known how, and I would have felt like a sneak if I had. (On the other hand, he took for granted that I had done so and might have judged me neurotic for _not_ doing so.) He was, I subsequently found out, though younger than I, a man of distinguished reputation in his field, greatly respected in New York and elsewhere, and he seemed to want me to know that. He was familiar with my reputation\u2014he was one of those readers I've heard praise _Catch-22_ for having had, they felt, a wholesome impact on their thinking when young\u2014so we started out with some kind of equivalence in mutual regard, and it didn't decline.\n\nAs fully as I could, I filled him in on everything I felt relevant, doing that with what I thought was commendable objectivity. My language probably was laced with technical terms standard in psychology, because that's the way I am (I suppose there was at least a little bit of showing off). I had read a lot of Freud by then and also had read much about him, and, like all imaginative Americans of my generation who knew about him, had been mesmerized (a term technically medical in etymology) by the egocentric implications and promises of psychoanalysis. In the course of my program for therapy, which continued with sporadic interruptions into a third year, I confessed willingly to everything I could remember and guessed at everything I couldn't\u2014well, almost everything. (I didn't tell my analyst that when I was young and afraid that my hair would wave or curl, I would sometimes go to sleep wearing one of my sister's hair nets, or that one time when my eyes felt sore\u2014they often still do\u2014I conjured up, entirely on my own, the reasonable notion of alleviating the irritation with a few drops of olive oil in each eye for lubrication, and consequently saw the world through a blur for the rest of the week. There were a couple of other secrets on that mortifying infantile plane which I wouldn't want him to know\u2014or you, either.)\n\nI was flattered and quick to agree when he said that any therapy less than analysis would be kid stuff for me. This was in the dark ages before Prozac and other mood elevators and stabilizers were at hand and before psychoanalysis, and Freud himself, had coursed inevitably into the buffeting gales of criticism, denial, and opposition by which they are mauled today.\n\nHe expressed the apprehension that I might exhibit a tendency to \"intellectualize\" and thereby obstruct the process. I told him not to worry about that, although, if pressed, I would have had to reveal that I wasn't really sure what either one of us was talking about.\n\nWe decided on four sessions weekly when possible; it often was not possible. There were the ordinary disruptions caused by things such as colds and intestinal flu. More unusual and, in time, more tacitly amusing to both of us, I think, was that frequently he and I would each be tempted and succumb to attractive invitations to travel. He went once to China, I went once to a writers' conference at Berkeley, and we both acquiesced to frequent requests to speak at universities and such. Also, I was usually away for the whole summer with my wife, no matter the intermittent marital discord\u2014he would be away in August, anyway\u2014and I was not going to give up my July out of the city merely to preserve my sanity or attain the peace of mind I was paying for.\n\nI told him frankly that among my motives for coming to see him (truthfully, I could think of no other real reason) was the tactical wish to have a psychiatric medical authority of my own to quote in comeback during domestic arguments\u2014even to misquote, by attributing to him statements that had not been made. He laughed and was not surprised.\n\nHe wanted me to understand that he could be of no help to me with my current emotional problems, a demurral I quickly professed to comprehend. He did believe, however, that a program of analysis might prove of some prophylactic value in warding off that late-life depression that eventually creeps up on so many of the aging and cloistered.\n\nHe was mistaken on that first point. He helped me enormously in the matter of day-to-day embroilments at home and elsewhere by being someone impartial and intelligent I could talk to with absolute freedom and who could point out obvious aspects and approaches in each crisis that I could not rationally spot for myself.\n\nAs for warding off potential depression, well, because of him or not, here I am in late life, and I'm no more depressed, despondent, dour, morose, cynical, bitter, drowsy, lonely, angry, and all the rest than I've been all along. If I'm gripped at times by the somber realization that I have nothing to do that I would enjoy doing or that not much merry is going on, it's because I don't and there isn't. I feel in pretty good shape as long as I have work I want to do. I know things are going well for me if my first thoughts upon waking up each day are related to what I'm writing. Today, my frame of mind is very good, for I awoke this morning with the idea of putting in right here these two sentences I am putting in. It's when I feel I can't find anything I want to do that I'm inclined to sleep a lot and to imagine I'm being infected by the brooding and miasmic lethargy of what the French call a _cafard_ and we used to call \"the blues.\" It grows harder and harder to meet new people you'd like to become close friends with (and the odds are great that you're going to find that out).\n\nHe bound me to this: I was not to make fundamental changes in my life once we began\u2014not in my marriage, my work, or other areas\u2014without discussion with him. So obvious was the reasoning underlying this condition that I agreed without hesitation.\n\nAnd by the end of the summer of 1981, I broke that promise over and over without contrition and without even recognizing I was doing so. I changed accountants, I changed book publishers and instituted a lawsuit against one of them, I left my literary agent, I changed lawyers and then changed lawyers again, I moved out of the marital apartment not once but twice\u2014once in summer, when he was away, and once, this time permanently, during Christmas, when he also was away\u2014and finally, perhaps as a grand and auspicious climax, I unilaterally and abruptly broke off my treatments with him.\n\nThat sudden termination followed a disappointing return to the city after a summer away alone, half in Aspen, where I soon made friends, and half in Santa Fe, where I already had some, when I lost all patience and felt myself running out of money. Expenses were building, income was not. A lawyer had counseled me to just relax all summer and assured me that all disagreements relating to the divorce would be peacefully settled by the time I returned. None was. And I had books to finish as well.\n\nAnd in mid-December of that year I was felled by a rare, paralyzing neurological disease called Guillain-Barr\u00e9 syndrome, whose etiology remains mysterious, and I was pretty much out of action for the next nine months. Stress? Maybe.\n\nFrom the beginning of the analysis, I tried my best to be a very good patient, one effortlessly surpassing the average, dreaming the dreams I was supposed to, retrieving the memories I guessed he wanted me to retrieve, unlocking all the unmentionable buried wishes I might have surreptitiously repressed, in order thereby to uncover the sources of the doubts, fears, drives, and inhibitions with which I was in blind conflict (although I might not know it). But I couldn't get far. My theory now about psychoanalytical theory is that corrective therapy demands unwavering concentration by a patient of intelligence with a clear and untroubled head who is not in need of it. My head was a swarming fun house of lawsuits, quarrels, work, and too many other distracting entanglements to count. Every day, something new might take dominion in my mind, if not from life then in the concentrated ruminations for my writing. While searching for myself as a fatherless Coney Island child, I had one new novel, _Good as Gold_ , published, and another, _God Knows_ , was almost halfway completed before I was through. And of course we talked about those and my two previous novels, about their content and what they might reveal about me, and about the little dramas playing out continually around each of them. (In one appalling mental lapse, I failed to notice that the surname of the reviewer of _Good as Gold_ in the Sunday _New York Times_ \u2014a good review, but not a rave\u2014was the same as that of my analyst, even to the deviant spelling. They were distant relations, it turned out, but that had no effect on either the book review or the extent of my cure.)\n\nI would imagine that in my eagerness to make an excellent impression as a patient, I was not alone among male subjects in a sophisticated hurry to plead guilty to such likely flaws as an Oedipus complex, castration fears, impotence fears, performance fears, ill will, ambivalent emotions toward everyone close to me, latent homosexuality, rage, shame, lustful thoughts, unconscious hatreds and unaccountable losses of confidence, vague perceptions of confrontational inadequacy and abstract dangers, and disguised, subterranean volcanoes of murderous aggression. Had I known then of narcissism as a specific pathology, I might have thrown that in, too. All that serious stuff was easy. It was the simple, unknown stuff that was hard and that eventually proved unconquerable: to go beyond facile academic logic and probe deep into memory for an understanding of feelings and events that might not even be there or ever have occurred, not even in fantasies.\n\nI was blocked often (or thought I was) and freely confessed to that failing even before I could be charged with it. \"Resistance\" was another of the terms I thought to volunteer. (And I'll freely confess right here that the more than ordinary difficulties I am having with these passages might be arising afresh from \"resistance\"\u2014from what some might infer in this place as a psychological block and in other sections of this book dismiss as mere writer's block.)\n\nI have always been a fertile dreamer. Now I was giving value to each dream as a treasure of significance. I was intrigued by the way my dreams began to center on our sessions, even to the physical location of his office, and on what I took to be the elusive essence of my monologues. I was beguiled by the frequency with which such details as the number on his office door began appearing in different dream contexts and, when awake, by the cunning and cute lapses in memory, the endearing tricks of the mind that seemed to fit perfectly into Freudian theory. The most dramatic of these occurred on the last session of one season. I was coasting along in the kind of voluble free-association we both believed in when he interrupted brusquely to demand, \"Don't you know that you were twenty minutes late today?\"\n\nNo, I had not known.\n\nDuring one of the periods of separation preceding my divorce when I was living alone in a sublet furnished apartment, I intrepidly flew off for an unlikely weekend rendezvous, a blind date with a woman I hadn't met or known about before (my most dangerous mission, marveled my friends) and whom nobody I knew had heard of either. (The farcical means by which this unconventional tryst chanced to come about might appear in a sequel, should I ever decide to write one\u2014and I probably never will.) Having arrived, I was at first afflicted by an inability to function sexually, I had to confess to him awkwardly when we resumed on my return (not the only time that had happened to me in my lifetime, I was constrained to admit to him\u2014and to you).\n\n\"I wished you were there with me,\" I said as a joke.\n\n\"I was,\" he said laconically.\n\n\"Then why didn't you help?\" I said.\n\n\"You wanted me to help you do something you didn't really want to do?\" he demanded.\n\nFrom that exchange came much wisdom about the thousand and one other things I often thought I wanted to do but truly preferred not to, and I ceased my regretful wishful thinking about scores of them. I'd never really wanted to live in a house in Tuscany or the French Riviera or have Elizabeth Taylor and Marilyn Monroe in love with me, and I didn't really covet for myself the bolder public life lived by Norman Mailer, although there was much there to envy. People with choices generally do what they want to do and have no real choice but to be what they already are, and I think even my former mentor would agree. I don't actually want to go to the White House for dinner or receive an award of some kind from whichever transient happens to be living there, although I _would_ like to get my hands on a Nobel Prize or two.\n\nGod knows I deserve it.\n\nOur dialogues were not always so one-sided. From me he first learned the word _catamite._ I was surprised that he didn't know it and he was surprised that I did. I'd found it first long before in a book on forensic medicine that had somehow fallen into my hands in my tent overseas and had taken for granted that it was a technical medical term known universally. And from me came a comment about trying to change neurotic behavior which brought a quick guffaw from him. \"It's like asking a hunchback to stand up straight,\" I said.\n\nHe liked that one and said he might use it often. I gave him license, since that clever phrase had originated with a friend, Speed Vogel, of whom and with whom I have elsewhere written. _(No Laughing Matter_ , if you're genuinely interested.)\n\nThen there were the dreams.\n\nOn days we were scheduled, I usually awoke in a state of high expectancy, especially if I came alive with a sumptuous dream, outlining to myself what purportedly would be spontaneous later, and it dawns on me now that the subject matter of the analysis had in large part become the analysis itself.\n\n\"Oh, boy! Have I got a dream for you!\" I might say to myself proudly, when I awoke from one with a complete plot, a story line fraught with apparent meaning and latent meaning and replete with the right kind of symbols\u2014symbols of sex and death anyone could recognize: steeples, purses, tearful partings. There was a week in which I heard three jokes all dealing with a shipwrecked man releasing a female genie from a bottle and being granted a wish that was sexual and had been misunderstood. While asleep, I conjured up a fourth that was as good as the others for a joke that was conjured up while asleep. Naturally, I began our day's session with that. Fortunately, he hadn't heard any of the others, and I was able to dispose of much of my fifty minutes with jokes and laughter before I settled down to whatever was that day's most disturbing problem, one that, while perhaps not a choice lode for self-discovery, was uppermost in my mind and couldn't possibly be avoided if I was going to express myself with candor, either by free-associating or in blunt complaint and appeal. (I remember two of the jokes\u2014one involved a large man with a little head walking into a bar, the other a man walking into a bar with a smirking ten-inch mannequin perched on his shoulder. I've forgotten the third one and have no memory of the one I dreamed, either because it wasn't as comical as we thought or because I am again prey to an unconscious and deliberate memory lapse.)\n\nTHERE IS A RELUCTANCE to proceed, and it is formidable. It is more than a writer's block. I had forgotten completely, in another memory lapse, that the core of this chapter was to be not my escapades in psychoanalysis at the time of the breakup of my marriage but the death of my father when I was a small boy and the traumatic effects, as all assume, it must certainly have exercised in shaping me. (Rich material, certainly, for the penultimate chapter, and I'm glad I'll never have to review this book\u2014or even write the jacket copy for a book with a passage about how difficult it was to write this passage.) There is a lack of enthusiasm for racing ahead, most rare for me this close to the ending of a book, and it is more likely due to the tender subject matter than to boredom or fatigue or any two-or three-day virus.\n\nNearly twenty-five years earlier, I'd had my one other personal adventure with psychological investigation, and I talked about that with the psychiatrist often and vividly. Certain impressions left from that encounter have remained ineradicable. In 1958, when I was thirty-five and leaving a job at one magazine for a better one at _McCall's_ magazine as head of a small department ancillary to advertising, industrial psychology was much in vogue\u2014at least at McCall's Corporation. I was required to submit to a battery of tests for psychological evaluation before my new appointment could be confirmed. The examinations took up most of one day and perhaps the better part of another and were conducted in mid-Manhattan in the offices of a partnership formed by two professors on the faculty of Columbia University. The procedures were congenial. Most of the tests were new to me, and a couple of them I hadn't heard of before, such as sentence completion, which was both perplexing to me in the range of theoretical possibilities of interpretation and amusing as a guessing game, and the Thematic Apperception Test, the TAT, which, for me anyway, was at times especially challenging and at other times profoundly affecting. The Rorschach test was an immediate catalyst, and I pricked up my senses with the alertness and gusto of, as they say, a cavalry horse at the first notes of the bugle when I saw it coming and we were about to begin.\n\nI charged into the first card. And from then on I talked and talked endlessly about the assorted forces I could easily imagine at work in the individual inkblots in the succession of cards that followed, offering no original responses about any, I'm sure, but dissertating on each until out of breath and the examiner gave evidence of restlessness and we turned to the next. I went gabbing and gabbing away\u2014blithely, inexhaustibly\u2014until I collided with card number 4 or 5, the first of the ink blots in color, and was at once catapulted into a state of startled confusion and silence. My loquacity was annihilated. I was numb. That first color card is labeled the \"color shock\" card, I learned speedily from a technical book I obtained as soon as I could at the local library, and, for me at least, the card was named fittingly. (For others, too, I must conclude, or where would that title have come from?)\n\nFrom that point on I proceeded gingerly, far less sure of myself. It wasn't long before then that I had at last taken the trouble to learn from my sister the cause of my father's death. Applying for my first life insurance policy\u2014I had my wife and two children\u2014I drew a blank when I came to the request on the insurance application for the cause of his death. I was jolted with surprise, and disappointed in myself, too, for I divined that the reason I didn't know how he had died was that I had never wanted to find out. The cause of death, as I've said, was internal bleeding following surgery for a stomach ulcer, and I was mistaken earlier when I assumed he had been a ward patient at Coney Island Hospital. I learned just recently from Sylvia that the hospital was Columbia Presbyterian Hospital in Manhattan, a long, long ride by subway from Coney Island, and it fell upon Sylvia, then just thirteen, to escort my mother to the hospital when they received word of my father's death. One takes the BMT express subway to its final stop at Times Square and then switches to a line of the IRT. But two IRT lines are there: One continues up through Manhattan to the station stop for the hospital; the other diverges into the Bronx and ends somewhere before the border of Westchester County, which is outside the city. Sylvia had never been that far from home. It was her macabre luck to choose the wrong line, and the mournful subway journey of bereaved mother and teenage daughter to the hospital for the certification of death took two hours longer than it should have.\n\nThe predominant color on those Rorschach inkblots that do have color is, I remember (or think I remember), red, in splotches of different densities and shapes. And on that first card, after I recovered from my shock, and on the color cards unveiled subsequently, I could truly see only blood, blood, and amputated or excised bodily organs, human organs. Here and there when faced with green I tried to fake a flower or a fruit, but my heart wasn't in it, and my soul couldn't follow through. So self-conscious and embarrassed was I by the consistency of my gruesome replies that I felt forced to call attention, with a nervous laugh, to my awareness of their morbid uniformity. (Just as with the black-and-white blots, I had a lesser dilemma with genital symbols: I didn't want the examiner to think I feared seeing any; I didn't want him to think I was seeing too many.)\n\nAfter the rite of taking the tests, there was an extended conversational interview. The mood was relaxed. In the course of discussion, I naturally mentioned that I was about halfway through writing a novel for which I already had been given a publishing contract. What was it about? That question still makes me squirm (and I'm still tremendously glad I never had to write a review of _Catch-22_ , either\u2014or its jacket copy). It was set toward the end of World War II\u2014I could be straightforward about that much\u2014and concerned men in an American bomber group on an island off the coast of Italy. And\u2014I swear!\u2014it was only then, in trying to talk about it coherently, that I realized for the first time how extensively I was focusing on the grim details of human mortality, on disease, accidents, grotesque mutilations. I was again awash in the reds and pinks of the Rorschach color cards, in blood, in the deaths of such characters as Kid Sampson and Snowden and even with my colorless Soldier in White. Although I had from the start, from the second chapter on, been dutifully following a disciplined outline, I hadn't perceived till then how much material of a gory nature was embodied in its fulfillment. There was certainly an awful lot for a novel that has since been described by many as among the funniest they've read.\n\nIt has struck me since\u2014it couldn't have done so then\u2014that in _Catch-22_ and in all my subsequent novels, and also in my one play, the resolution at the end of what narrative there is evolves from the death of someone in the chapter just before, and it is always the death of someone other than the main character. In _Catch-22_ , the death of Snowden, who has perished four hundred pages earlier\u2014in fact, before the present action of the novel has even begun\u2014is written about fully only in the chapter before the final one. And in my most recent novel, _Closing Time_ , it is the death of Lew as narrated by his wife that precedes the surrealistic catastrophes at the climax, though Yossarian goes on living. And even in the novel before that one, _Picture This_ , by which time I'd grown cautiously aware of that undeviating pattern and tried to make an effort to break it, the death of Socrates in the _Crito_ as related formally by Plato (to me in translation, of course) is almost against my will the precursor of the final few pages. And\u2014lo and behold\u2014here I am, much to my surprise, doing exactly the same thing with this book! I still haven't broken the pattern. I don't know why this should be so\u2014it reflects no aesthetic or philosophical principle. I would speculate that the individual personality, no matter how protean in creativity, has a character all its own and can construct flexibly only within the parameters of its own nature. The novels of other writers, too, like the mature compositions of classical composers, tend to be more alike than we wish to own.\n\nI've come to recognize something else. Where I stress tender relationships, they are usually most tender between a father and a child, as in _Something Happened_ , and also in _God Knows_ , where King David's most wrenching recollections are of the death of his son Absalom and of the newborn child who is the result of his transgression with Bathsheba.\n\nThe Thematic Apperception Test for the job at _McCall's_ provided me with one indelible prototype that still touches me poignantly whenever it comes to mind. In the TAT, one is asked to comment subjectively about simple drawings that are objectively neutral, to project onto the picture a personal vision of what is taking place, to create a text. It was not that easy, and I would bet that there are many people who at times can't bring themselves to fantasize anything at all. The picture I remember, and it's the lone one I recall, is still with me, together with the emotions it aroused at the time, indissolubly joined.\n\nA woman with white hair is in a room with a grown youth with dark hair. That's all. But to me it was a scene of pain, of miserable, unwanted separation. There are no emotions on the faces pictured in the drawing, but I could sense the sadness of a mother and a son. The boy is going away and they are speechless with sorrow. Another person looking at that same drawing, I've since realized, could easily be projecting a scene of triumph and elation\u2014could imagine, for instance, that a boy away from home at college finds pleasant lodgings with a woman who is happy to have him. But I didn't. (I could venture now that the absence of a father in the picture could be the key to my response to the scene, but I didn't think of that then and I would once more be casting about for a textbook psychoanalytical explanation for the grimness I imposed on the scene, and still do. Certainly, in the picture there was no father, only the mother\u2014or rather just a woman.)\n\nI didn't know him. He isn't there. I would not recognize him. I have only a handful of impressions of him\u2014there is only one snapshot, a photograph in which he is younger than I am now but will always seem older\u2014and the authenticity of a couple of my memories is questionable. Allowed in the driver's seat of an auto of his or his bakery delivery van I am encouraged to press the automatic starter and pretend I'm driving at each grinding lurch. He drives me to the Coney Island Hospital to have my tonsils removed. (And that wasn't fun. Lying there alone afterward, I would have given my life for a drink of water to assuage the maddening thirst.) He is playing with me on the sidewalk with an airplane with a wind-up, rubber-band propeller he has bought for me, when Lee comes walking home from the trolley stop after his runaway summer as a hobo, and there is a joyful reunion. (\"When you come from California,\" my mother would repeat for years afterward, \"you've _got_ to take a bath.\") Sleeping in the same bedroom with my mother and father and at least one time being taken into bed with them\u2014make what you will out of that, but I don't doubt it's true. He commands me back from an open window when he spies me making my way out it toward the fire escape. On the day of his funeral, I wouldn't get in the car for the trip to the funeral parlor or the cemetery, and when older boys on the block came after me, I turned the chase into a game.\n\nSylvia was not at ease answering questions about our father, and I didn't try to extract much. She remembers sitting by the window at night doing her homework by the lamplight from outside, but she doesn't attribute the absence of lighting indoors to any cruel stinginess\u2014none of the families we knew liked the electric company\u2014and after all of us had been driven to wear eyeglasses, she no longer blamed anyone for her defective eyesight. She retains a memory of herself and my mother in the terrible week of mourning that followed the funeral. It was summer and hot. Someone had dressed her, a child of thirteen, in black garments, but my mother advised her to change into something lighter and go outside in the street, where it might be cooler. (I wasn't around for that. I don't know where I was that week, but I doubt it was there. I would remember. In almost three years of therapy I could get so far and no farther.) He was always good to me, Sylvia says\u2014all of them were. His first wife was ill and died and not much later he was ill, too, and he was not always kind to Lee. The youngest in a family of several boys who had emigrated from Eastern Europe to the New York area, he was the least prosperous and the most helpful and cooperative, and was that way with my mother's relatives, too.\n\nLee talked about him to me one time only and with a mixture of deep and conflicting emotions. His eyes were misty and he kept trying to smile. That conversation took place under unhappy circumstances. In New York from West Palm Beach on a family visit, his wife, Perlie, was troubled by shoulder pain and asked us\u2014me and my first wife, Shirley\u2014to recommend a physician, preferably an orthopedic specialist. The doctor she found took X rays and found advanced lung cancer. During the first program of chemotherapy at Mount Sinai Hospital in New York, Lee and I had dinner alone a few nights in succession at an Italian restaurant just down the block from my apartment\u2014Tony's, on West 79th Street. It was the most time we had ever spent alone together and the longest we'd ever talked. Both of us drank whiskey.\n\nHe wouldn't acknowledge himself as a runaway from home that long-ago summer. He had only gone to New Jersey to apply for a job, and when it was clear he wouldn't get it, he had simply decided on the spot to keep going west. He sent postcards home every week to let the family know he was well. His most aching childhood recollection was of riding beside my father winter mornings in a horse-drawn delivery van in upper Manhattan, when they lived there before I was born. On icy uphill streets, my father would have to use the whip on the horse, and people on the street would berate them both for their cruelty to the animal. My father was a good and kind man, Lee said, although he would use his belt to beat Lee for any kind of misbehavior. That sounded brutal to me. \"There were lots of Jewish criminals around and he didn't want me to turn out bad.\" Then he did have a good reason, I said in a conciliatory tone. \"Sure. But I was just a little kid. How bad could I have been to deserve that?\" Then he wasn't always so good and gentle, was he? \"He was always very good. His wife was sick and died and he was sick, too.\"\n\nLee did not turn out bad. None of us did. At the age of eighty-four, a widower, he died on the living-room floor of his small condominium in West Palm Beach in a way we would all choose to go, if we were eighty-four and could force ourselves to take leave of life then. Back from a vacation cruise with a group of men and women he'd grown friendly with, he died instantly from a heart attack in the midst of optimistically planning things for the immediate future: He had borrowed some books from the local library that day and had just returned from the supermarket with some staples with which to restock his kitchen.\n\nOne more report, from an uncle by marriage in my first wife's family, who'd been a young boy on the block with us when my father was still alive. He described my father to me as cheerful, jolly, friendly, helpful\u2014and so here I am again, as I mentioned earlier, with another book whose penultimate chapter deals with death, the death of someone other than the principal character in it.\n\nTHE LAST TIME I was with my father or had anything to do with him was more than sixty-five years ago, on the day he was buried in a cemetery somewhere in Long Island or New Jersey. Sylvia might remember where but would be uncomfortable if I asked her, and knowing where would make no difference anyway. Until now I haven't even thought of asking. I've never grappled much with the idea of trying to find out more about him. I prefer not to. I still prefer not to. And knowing more would make no difference now, either. I know him by his absence. Until just now, if I thought about the day of his funeral at all, I was disposed to remember it as one of the more rewarding in my young life. I sat waiting placidly on a stone bench on a stone patio with a railing, just outside the main office at the entrance to the cemetery grounds. It was summer, I had been dressed neatly, and there was sunshine. People milling about fussed over me. Some handed me coins\u2014dimes, an occasional quarter. My Aunt Esther gave me a whole dollar. I felt rich.\n\n# 10  \nDanny the Bull\n\nDANNY THE BULL murdered his mother.\n\nYou couldn't ask for a better opening line than that one for a final chapter, could you?\n\nIt had something to do with heroin, money for heroin. Either she gave him money, five dollars for a fix, or she didn't, or she had been doing both, giving him when she had it and he demanded it, denying him when she didn't. Not like his father, Max the Barber, who would make a contemptuous, bitter show of disavowal whenever his son came abjectly into the barber shop to implore, who was an awful man (in the judgment of my friend Marvin Winkler, who knew them better than I), a kind of insolent \"wise guy,\" abrasive with unfunny remarks. I know the type, and so do you. His father would make sarcastic fun of Danny before others in the barber shop, renouncing him as \"my son, the junkie,\" and boasting with smug pride that he would not give him a dime.\n\nDanny the Bull's mother and father lived apart; they were no longer married. He lived with his mother down around West 23rd Street, near the big yellow-brick synagogue and the big red-brick Catholic church, where the Jewish sector merged with the Italian. Max the Barber is still the only man I've heard of living in Coney Island at that time who was divorced. He had married again. His second wife was a local woman who also had been married earlier and who brought into this marriage children of her own with the last name of Glickman. In due course, Max the Barber's stepson, Raymie Glickman, fell dead in the street from an overdose of heroin. And Leona, Raymie's wife, also died of a heroin overdose. Max the Barber, the father of Danny the Bull, though a rasping, unfunny wise guy, did have cause for discontent.\n\nSoon after the war, with the social clubs finished and school days past, our gathering places were reduced to a very few for those still living in the Island and for those many more who had moved elsewhere but returned regularly for visits with family and friends. I was already in the city and would travel down from Manhattan.\n\nWe were young adults now. On Mermaid Avenue near Happy's Luncheonette was the popular daytime hangout for males\u2014the poolroom belonging to Sammy the Pig. There we could hobnob with lanky Pinya, a rabbi's son with the mien of a hungry friar who was our local bookmaker; with Smokey Bleeker, a well-built, friendly sort of tough guy from an older generation\u2014he was in his thirties; and with Sammy the Pig himself, short and stubby in a porcine way, who didn't gab much and wasn't much fun. In the rear of the poolroom near a large, chalk-lined blackboard mounted to the side was a Western Union ticker-tape machine that typed out up-to-the-minute data on the progress of sporting events nationwide. In season came the inning-by-inning progress of all major-league baseball games. Streetwise sophisticates understood that an unusual lag in a ticker report usually indicated something happening in that half inning of the game; new bets might be made in the delay, odds might change. Bewildering to me still is how some of the guys there who had barely staggered through arithmetic classes for their elementary-school diplomas could now calculate fractional odds and multiple payoffs on parlays with a dazzling rapidity and exactitude that left very little room for betterment by even the swiftest of today's digital computers.\n\nOne of these savants had been a fellow student on my own grade-school level at P.S. 188 with the last name Silverman. For reasons known only to himself, he had changed his name to Ershky Jones, and we now called him that. To most of us, the game of checkers was a matter of haphazard luck. But Ershky saw much more than that; somehow he had penetrated to essentials. He won every time against every player, with none of us ever able to spot the things he saw that we didn't.\n\nHe could also run up walls. He would run a few feet up the perpendicular walls of apartment houses and rooms, turn, and come back down. He would practice running up walls and down while conversing cynically about things like baseball, politics, and the antiunion and anti-Democratic Party biases of the Hearst newspapers and the New York _Daily News._ He was better than anyone else we knew at running up walls. None of the rest of us had the will to try it more than once.\n\nSmokey Bleeker was almost always around, too, at Happy's Luncheonette or in Sammy the Pig's poolroom, grinning, joking, affable, and crassly unschooled in speech, even in comparison to the Brooklyn articulations of Ershky Jones and me. Loose-limbed, muscular, and agile, Smokey always had coarse, black stubble shadowing his countenance, yet always looked as though he had recently shaved. He had been brought up someplace funny and far away and had once worked on a farm\u2014in Pennsylvania, Poland, or both. By the time I met him he had been in prison and had fought in the prize ring. He and another older neighborhood hard guy named Izzy Nish, this one wiry, had formerly buddied around together in a number of shady, daring ventures, one of which had brought a carload or two of equally dauntless Italians with knives charging down upon them in some other part of Brooklyn. Izzy Nish managed to flee, leaving Smokey behind to be carved up in a doorway. He had narrowly survived. Now the two did not speak and took care not to cross paths. They favored different poolrooms. Smokey's face was gashed with scars, and a circular part of the top of his nose had been sliced out into a hollow. That might be the explanation for his perpetual, neat stubble of beard.\n\nOne year Smokey Bleeker enjoyed a season of celebrity because he had a hand, a hand with fingers clenched into a fist, in the genesis of a black eye of a luminous immensity unlike any that people around Happy's or Sammy the Pig's had ever conceived possible before. The black eye was not sported by Smokey himself, naturally, but by a burly fellow of my own generation with a short, thick neck who had been nicknamed\u2014perfectly, and not by me\u2014Quasimodo. Quasi (we sounded the opening consonant like a k rather than a _kw)_ was a sullen person quick to take offense and accustomed to having his own way. He generally did have his own way, for most of us kept out of his way. After once catching sight of his black eye, we could easily understand why a black eye is traditionally known as a \"shiner.\" Quasi's shiner was of a solar brilliance, an injury mending itself over time in prismatic hues from purple to blue to green to yellow to tangerine, and it glowed like a beacon and seemed to polish the ground in front of him as he walked. People (people like me, who deliberately loitered on Happy's corner in hopes of glimpsing for the first time the shiner I'd already heard of) came there, came to Coney Island, just to see the fabled shiner of Quasi's with their own eyes. Some talk about it still, as I'm doing now, and Marvin Winkler does, too.\n\nThat gargantuan black eye came into being as the immediate aftermath of a brief sidewalk fistfight outside the poolroom between Quasi and Smokey Bleeker. I don't have all the details of the provocation. They had something to do with Quasimodo's walking up to where Smokey was seated listening to the radio at the front of the poolroom and changing the station without a word for permission or apology. Smokey, offended, protested that Quasi's action was \"rude.\" (Smokey's employment of that word _rude_ was an astounding incongruity for more than one reason, but people on the spot swore it was the one he spoke.) Quasimodo invited him to step outside to settle the dispute if there was objection. Smokey stepped. Outside the poolroom, Quasi, overlooking Smokey's experience in the prize ring, ducked right down to bull forward and seize him around the waist, lift him, and throw or wrestle him to the ground. Smokey, remembering he'd been a boxer and impervious to physical fear, unleashed an uppercut timed to perfection as Quasi's head came down, and that was the beginning and the end of the fisticuffs.\n\nThe ambulance from Coney Island Hospital pulled up outside Sammy the Pig's poolroom just in time to save Quasimodo from suffocating, for the gigantic swelling around his nose and mouth was choking his windpipe off from the rest of him.\n\nOf Izzy Nish, Smokey's erstwhile confederate in a miscellany of misdeeds that ultimately landed Smokey in the hospital with knife wounds and perhaps also in prison, I know less. He was not one to hang around with nothing to do and was not inclined to exchange pleasantries with younger kids like me. He was married to the sister of a Coney Island friend by the time he was pointed out to me and probably was employed on a standard work schedule. My occasional random sightings of Izzy Nish all occurred in our other Coney Island poolroom, Weepy's, where a largely different group passed the hours at different times of the day. His taut, tawny-colored face also was marked with a scar or two, but what charisma he possessed lay in his pool playing. He was superb. An expert pocket-billiards player is a miracle to behold to someone incapable like me, and Izzy Nish was among the best I've ever watched. Shooting pool, he was a stolid avatar of that self-possessed, mute, concentrated proficiency that is second nature to the shark and mystifying testimony to the maze of intricate nuances inherent in so pacific a competition. For those who don't know, foresight in billiards directs the player not to the billiard ball that is safest to sink but to the one that will leave him and the white cue ball in the best position to sink the next, and the next, and the next. And finally the last, which should be left in a position enabling you to sink it forcefully at the end of a rack while enabling you to break apart the new rack with your cue ball in the same shot, and thereby continue. It was like chamber music watching Izzy move smoothly and wordlessly through one rack of billiard balls, two racks, and sometimes three.\n\nWeepy himself, Murray by name, the owner of this poolroom, was a family man with daughters. Unlike Sammy's, Weepy's had no bookmaker and no ticker machine. There were salami sandwiches on a roll for a nickel each and chocolate-flavored drinks in cold bottles. Women were welcome at Weepy's but were rarely present.\n\nBlackie the Pinochle Player, in a more retiring way, was another local personality. He was a champ at cards, though not at picking horses. I'd laid eyes on him only a few times, never at play, but I'd heard, more than once, the anecdotal account of one tormenting game he'd played at high stakes and the renowned remark he'd made immediately after it. When the bidding was over and he was about to begin play, he calculated with distress that he had drastically overbid. In pinochle, unlike bridge, the player in command has no partner and doesn't play with the support of a dummy hand; he faces two opponents united against him. Blackie, caught between folding the hand and paying up or playing the hand and paying double if he lost (quadruple, for his trump would have had to be spades), reckoned despondently that he needed to take every one of the fifteen tricks, every point left in the deck, to make his bid. He chose to play. He played slowly, and slowly he made all the tricks. Closing in toward the end, when he could guess with assurance where the rest of the cards lay and realized that he was certain to make the hand, his muscles went slack and he drooped with the strain. Looking down with an unhappy sigh and shaking his head with dismay, as though he had lost, not won, he moaned out loud:\n\n\"I played the game all wrong.\"\n\nHAPPY'S LUNCHEONETTE, on Mermaid Avenue a few blocks from Weepy's poolroom, was a lively, clean, well-lighted place, a safe and convenient setting in which to meet with our wives if it was that kind of evening or to hang around in with just the fellows. There were no longer many other spots. It opened early and went on serving very late. Lounging around doing nothing before we went where we were going or after we had been, we could easily put away as a snack a tuna sandwich or two on white toast with a leaf of lettuce inside and a slender length of pickle alongside. Before the war, tuna salad hadn't been heard of. Now a simple tuna sandwich went down as easily as a potato chip or a salted peanut. I told you we liked to eat, all of us, and those of us I still claim as friends still do. We've never regretted a mouthful. For a true hunger, Happy's pork chops were good, and his hamburgers, too. Our drinks of preference were chocolate egg creams or malted milkshakes of any flavor, or in winter a hot chocolate. A taste for coffee takes time to mature.\n\nHappy's, together with the street corner outside, was also a clustering spot for the neighborhood's many surreptitious \"vipers,\" a commissary of sorts where the dedicated marijuana smokers could buy and sell and borrow and swap, and a couple of years later on, when they evolved as a subculture, for the junkies to drift into and linger idly in their auras of furtive and suspicious confidences.\n\nIt was around the corner from Happy's and down the side street one night that observers caught their first sight of heroin in use. A friendly generous pusher showed up to coach a bunch of those there, at no cost, how to mainline heroin. Of the dozen or so willing to accept the injection directly into the vein, every one was quickly retching violently and vomiting into the street. You would have bet that not a single one of them would ever want to go near the stuff again. And you would have lost. All did and all were soon addicted.\n\nOne more watering place for spring, summer, and autumn socializing was a roomy square bar on the boardwalk in our Jewish neighborhood. A bar was an uncommon place for us then, but this one fronted wide onto the boardwalk, projecting the airy character of a resort, and was run by someone we knew. The atmosphere was of a spacious caf\u00e9, a cabaret without live entertainment. Women came there, too, with escorts or with each other. Everyone there knew everyone else, at least by sight. I had long since been living away from Coney Island, and I was led to this place by Marvin Winkler after one of my visits to my mother. It was there that I bumped into Howie Ehrenman from West 31st Street whom I hadn't seen in so long a time, ever since we had departed separately for the war. He, with his brother Henny and the family, now lived somewhere else in Brooklyn. Howie had grown taller, and was quarreling confidently with an adoring girlfriend at the bar. She was a stunning beauty\u2014an Italian girl with lush, olive features and blond hair that glittered\u2014and I thought her ravishing. I haven't seen him since. Marvin directed my attention to a fresh-faced, pretty girl of a different complexion, peaches and cream, who sat facing us at the opposite side of the rectangular bar, pert, laughing, gleaming. She looked, said Marvin, \"like a bottle of milk,\" didn't she\u2014that figure of speech was his\u2014and you would never guess, would you, he confided, that she was being treated for gonorrhea. My instant reaction was to scan enviously the male faces in the room\u2014I was younger, of course, and a romantic even then\u2014for sight of the likely guys, the lucky, likely ones, who might have contracted that disease from that cheery doll. She did indeed look as clean and pasteurized and homogenized as a wholesome bottle of milk. Ever since I heard that simile from Marvin I have been waiting for a chance to use it in my writing, and now I have.\n\nLIKE SOME I KNEW in the neighborhood and others I heard about, Danny the Bull had acted more than once to commit himself voluntarily for treatment in the federal drug rehabilitation facility in Lexington, Kentucky. A quirk in the law then (perhaps still) left a person free to use drugs but decreed it criminal to possess them. In seeking help, one could admit to use and addiction without incurring arrest, and many, perhaps all, of these desperate Coney Islanders did.\n\nAt Lexington, Kentucky, the treatment was \"cold turkey\"\u2014immediate, total withdrawal. It was from those coming back that we first learned the term. This was before the development of methadone and similar alleviating pharmaceutical substitutions, and going cold turkey on heroin was an excruciating ordeal, or at least was said to be by any who had endured it. Leaving Lexington at the completion of a program of treatment, the subject could, if he (or she) wished, obtain from an orderly in an underhanded way a list of sources of supply for the illegal drug of choice on the route home from the institution. What made the outlook for almost all of them dismal was that when the time arrived to go back home, they had no place to go _but_ back home.\n\nFor a very short period I was acquainted with a minister involved in drug rehabilitation work for the East Harlem Protestant parish in New York. The population of his parish was predominantly Hispanic, mostly Puerto Rican. He was a ruddy, buoyant man of a likable disposition, but his pessimism about his work was total: He felt the futility of bringing the weak-willed, damaged addict back into the same situation from which he had come and expecting him to overcome the numerous psychological, social, and environmental influences that had lured him into his addiction to begin with.\n\nA couple of us were witness to a striking and depressing illustration of this fragile despair. We met Danny the Bull by chance one day outside Happy's while he was killing time and waiting, maybe, for another look at Quasi's black eye. The Bull had just come back from another stay in Lexington. He was animated, convinced, firm, almost chipper. Suddenly, he stiffened; then he was quivering, and looking elsewhere. His trembling appearance now was one of agonized indecision and terror. We gazed in the direction he was staring. On the sidewalk across Mermaid Avenue stood an inert, huddling figure one of us recognized as another neighborhood junkie, always a likely source of supply and information about supply. Danny the Bull held out against temptation for not even one whole minute, withdrew from us, and crossed over again.\n\nIn California there was Synanon House, an organization much publicized for successful work with addicts. So celebrated was its reputation for dramatic successes that a film director for whom I was doing some rewriting on a slight sex comedy _(Sex and the Single Girl_ , and the title is more risqu\u00e9 than the content) assigned a researcher to Synanon to give him material for a feature film to be set there that would have a happy ending and great commercial possibilities. And for all I know, he made it. Danny the Bull knew of the place, too. With a friend who was also a junkie, he flew out to Santa Monica to apply for admission. The verdict of doom handed down to him there was that they might succeed in helping his friend but they could do nothing for him. How they could tell I don't know. Danny came back alone to Coney Island and into the apartment on Mermaid Avenue in which he lived with his mother. A short while later, he burst through a door and choked her to death with his bare hands as she lay in her bathtub.\n\nHe insisted afterward in an incoherent explanation of his rage that he had done it because it was all her fault, that she had kept him a junkie by giving him money, and that if she had acted as his father, Max the Barber, had done and given him nothing, he probably would have been cured. He was consigned to a hospital for the criminally insane and if he's still alive he may still be there.\n\nAbout the thunderstruck, horrified, immigrant Jewish woman who was his mother, I would submit that she had not the smallest idea of what was happening to her as she was being strangled to death by her son in her bathtub, or why. And I assume she had been just as helplessly dumbfounded by the addiction and the miserable, weird changes it had produced in her young boy's behavior about which she understood nothing\u2014unlike mothers today, and laudable public officials, who know a great deal about the drug problem, and still can do nothing.\n\nAnother neighborhood heroin junkie with whom I had been much better acquainted was a kid named Solly. We had been through grade school together and sometimes played in the same games on the same team. I don't remember what happened to him in high school, but I doubt that he finished. He had been to Lexington, too. He was like Mark Twain, who found it easy to give up smoking, having done it so many times; Solly had kicked the heroin habit more than once. Marvin Winkler pitched in to assist him after the last attempt. He provided Solly with steady, regular employment in a pastry-baking venture newly begun in New Jersey. Solly strived happily and industriously and took pride in his work. He kept clean. \"I make the best honey-glazed doughnuts in the world,\" he would boast with sincerity. \"People come from all over New Jersey and Connecticut to eat my honey-glazed.\" Soon he had a girlfriend, and the last I heard was that they were talking of marriage. Then Marvin had to close the business to move to California. We don't know what happened to Solly after that.\n\nI can say with some pride that not one of the friends in the three distinct social groups I ran around with in Coney Island (in general, the people from one didn't cotton to those in another) became an addict or thought for a second of ever yielding to anything so self-destructive or in fact to anything over which they had no control. (As with LSD much later on, there was no longing in any of us for that rarefied state of mental elevation about which users bragged.) My conviction is strong that those who did give way to drug addiction were individuals without any compelling attributes of personality.\n\nSPOOKY WEINER was not a junkie and not likely ever to become one. However, through peculiar physiological attunements of unknown origin, he ripened for a while, strangely, into the local authority on gage, or pot (by later generations called \"grass\"), and was the regnant arbiter on \"garbage.\" He had a gift, an unerring flair, for the detection of inferior or adulterated seeds of marijuana. This genius did not pass unnoticed by older hands, members of the bygone Alteo Seniors social club with whom we were still in contact. They would take him along on drives into the city to Harlem or Greenwich Village for major purchases. There, in murky hallways, the suspense would thicken as Spooky sampled and tested. Others in the close crowd of prospective buyers and hopeful sellers would cease to breathe as Spooky inhaled deeply from a stick, or reefer (later a \"joint\"), rolled by his own hands. They waited on tenterhooks for the effect to register and for the squeezed-out verdict to emerge from a throat disinclined yet to exhale. As often as not, the verdict would come with that one word: \"garbage.\" When the verdict was garbage, the transaction was canceled and they would have to search about for a better connection with a better grade of hemp, or gage, or pot, or grass.\n\nSpooky's famous sensitivity to garbage eventually bore him into a scrape with the law that many of us in whom the incident lives, including himself, still find humorous. There came a Saturday night when I was scheduled to meet with him early the next day at my apartment in the city to aid in the launching of a business enterprise of his having to do with the early-morning home delivery of Sunday morning breakfasts of bagels, lox (smoked salmon), and cream cheese. I had, as a charitable service, written the advertising copy for the direct-mail solicitations already distributed by him with a bold headline of his own creation. (That headline read: SLEEP LATER SUNDAY MORNINGS! The company name, with offices on Surf Avenue in Coney Island, was Greenacre Farms, Inc.) Early that Saturday evening, I received a telephone call from Danny the Count in Coney Island advising me with some glee that the appointment for the next morning was off; Spooky wouldn't be there.\n\n\"Why not?\" I responded, and I was already chortling along with the Count in anticipation of his reply, for I was not unaware of the mishaps with which Spooky's business career had been continually and supernaturally blotched. \"Where is he?\"\n\n\"He's in jail,\" said Danny, with a burst of a bronchial laugh that sent me off laughing, too.\n\n\"What happened?\"\n\nWhat happened was that Spooky, with his exceptional devotion to the good in marijuana and his sensitive antennae for \"garbage,\" had been delegated to go into the city to buy a large quantity of rolled reefers for which four or five others had chipped in. The purchase completed, he returned to Coney Island to Weepy's poolroom to await the arrival of his partners. Hardly had he seated himself on the brim of a pool table in the rear than he spied a posse of plainclothes policemen flooding in from the front. In those primeval times in Coney Island, plainclothes police officers in warm weather could be easily marked by two distinguishing traits: by the shirts worn outside the trousers to conceal the tools of the trade at the belt, and by the Celtic-pink and Saxon-white skin of their faces, which were definitively Gentile. They were there looking for someone else, but Spooky couldn't know that. Deftly, he flipped his packet to the floor into the shadows near a wall, praying no one would see.\n\n\"Pick up the package, sonny,\" said the officer who saw. \"Or I'll peel that mustache off your face.\"\n\nSpooky accompanied them to jail with his mustache on.\n\nThe fix went in, the judgment was mild, the arresting officer couldn't be altogether positive that he actually had seen the suspect toss the goods away.\n\nSometime later, Spooky found himself filling out an official application for something like the renewal of a driver's license, and he was faced with the query of whether he had ever been arrested. Fearing a penalty of fine and imprisonment, he decided to make a clean breast of things and answered yes. The follow-up question, though, required him to state the cause. Here he allowed a spasm of humor to override his prudence. With a lightened heart, he wrote:\n\n\"Failure to dispose of garbage.\"\n\nOUR PUNCH BALL GAMES were over. There was no longer amusement for us in the amusement area. We were moving away to live in other places; no one I know of who lived there has moved back or would want to. We were used to the rides and the games and we went there now only to shepherd others who had never been or, lacking any better diversion, to eat. When Nathan's, which was open winters, too, introduced pea soup, it was the best pea soup in the world, singularly so on frosty, stygian nights, and when they began selling pizzas, the pizza was the equal of the best in the world. (As with Solly's honey-glazed doughnuts, people who knew came from far and wide to savor Nathan's delectables and were never disappointed.)\n\nLuna Park had already declined into an unruly and disintegrating relic and was gone; Steeplechase was ailing, sluggishly deteriorating, although we couldn't see that. Observing the gaiety of people less jaded than ourselves cheerfully moving about inside brought a complacent satisfaction. What we definitely were not used to and quickly embraced were the advantageous circumstances our American society was affording us and into which we were advancing with eagerness.\n\nMy trips into Brooklyn after our return from California were irregular. We went as a married couple, or I went alone. Often we journeyed to Brooklyn but didn't go to Coney Island, and much of what I learned I heard from others, mostly from Marvin, who remains the oldest of my very close friends (although we haven't seen each other now in two or three years), and also from Louie Berkman and, news about a different set of people, from Davey Goldsmith, Danny the Count, Harold Bloom, and others. Bulletins of another character came to me from my mother, who remained in the Coney Island apartment until her intestinal surgery rendered her too dependent and impaired to continue living alone, and from Sylvia and Lee, who had their separate apartments nearby. As a married couple, my wife, Shirley, and I socialized most agreeably; we had Marvin and Evelyn, Lou Berkman and Marion, Dave Goldsmith and Estelle, later George Mandel and Miki; still living in Coney Island was Danny the Count, always funny; Evelyn's girlfriend Maxine, married to a former prizefighter; and Maxine's limber older sister, June, an excellent lindy-hop dancer (a very infrequent thing in a white woman) who fell in love with Danny the Count and in order to marry him divorced a husband to whom she had been married early. We mixed easily and we normally could meet as conveniently in the city as in Coney Island.\n\nIn Brooklyn when we had no special plans, where we went and what we did often depended on who had the car. One night Heshie Bodner had the only car, and an older eminence of considerable substance and brawn named Scarface Louie decided that he and his large dog had a number of business calls to make in other parts of Brooklyn and lower Manhattan.\n\n\"Whose station wagon is that outside?\" Scarface Louie inquired, leaning into Happy's.\n\nHeshy made believe he didn't hear.\n\n\"Heshy's,\" a younger innocent named Spotty Dave volunteered, helpfully pointing.\n\nAnd Heshy for the next several hours was off on a disquieting and perilous journey.\n\nBy occupation, Scarface Louie had something to do with the union workers in taverns and small restaurants; either he ran the labor unions or he busted them. That evening he needed meetings with people outside one barroom after another, while Heshy and the dog were ordered to wait in the car. The conversations were quick and quiet but vehement, too. Once or twice they picked up someone else at one place and dropped him off at another. When another rider was in the car, the big dog sat on the front seat beside Heshy. Outside one dark saloon a man was slapped around on the sidewalk, with Scarface Louie looking on collectedly, making gestures of approval. There was even a robbery before they were through. At a gas station where the attendant was less than courteous, Louie and a friend decided they might as well take the cash. Heshy and the dog didn't step from the car during the long trip except for Heshie and the dog to pee against the tire now and then. The dog was a Great Dane.\n\n\"Thanks, Heshy,\" said Scarface Louie when he was dropped off at Happy's. \"We'll do it again soon.\"\n\nTHE MOVEMENT OUT OF Coney Island after the war wasn't then, not at first, an incidence of white flight before an influx of people who were of darker skin and of even poorer economic station. That came later (and continues everywhere, and the apprehensiveness and prejudice at work appear to include social and economic anxiety along with the racial). Rather, the migration away was an optimistic drive toward betterment in response to the multitude of captivating opportunities that flowered in the wake of our victory in the war. Among the richest and most sensible and far-reaching was the G.I. Bill of Rights: Along with the obvious good of making available a higher education to those like myself who'd never hoped realistically to acquire one, it detained several million of us from pouring all together into the workforce and inciting unrest by over-flooding it. Conjoined with this powerful impulse toward upward mobility was the allure of the suburbs to many in the overcrowded cities (to many in the same suburbs and outer districts the longing was to transfer into the overcrowded heart of the cities) and the easy fulfillment of a newly awakened aspiration of families to own their own homes. Shortly after the war came the first Levittown, out in Long Island, in what was then open farmland, followed soon by scores of similar construction developments, advertising attractive houses at fabulously reasonable prices, with paltry down payments, low interest rates, and government financing through G.I. loans. Willie Siegel from my own street in Coney Island (already as a young adult called Bill Siegel, just as I was no longer Joey but Joe) was the first contemporary I knew of to buy his own house in a suburb and perhaps the first to make that quantum growth in maturity of wanting one.\n\nI was gone from Coney Island permanently with my marriage in October of 1945, five months following my discharge from the army. After the year in England and the two years of teaching in Pennsylvania, we settled into Manhattan in an impressive, rent-controlled apartment building with elevators, doormen, and elevator operators, and I went on living in that building for the next twenty-eight years, until the final separation leading to the eventual divorce.\n\nDavey Goldsmith had soon moved away into an apartment in Brighton, still close enough to all he knew; Marty Kapp was first in Riverdale, just north of Manhattan, next in New Jersey; Lou Berkman started his plumbing-supply business in Middletown, New York. All of us married, all of us in our twenties. Lou's family, like mine, was still in Brooklyn, and we would see each other there as well as in Manhattan. Marvin Winkler had moved with his family from Sea Gate in Coney Island to Ocean Parkway even before the war. Albie Covelman moved away with his family then, too, to somewhere close in Bensonhurst, I believe. It was just after a four-handed pinochle game at Albie's house one lazy Sunday afternoon in December that we heard the news of the attack at Pearl Harbor and declared categorically that the Japs had gone crazy and would be slapped down in a week or so. That Hitler then declared war on us before we did on him was a momentous event that seemed to elude my notice. (Had Hitler not blundered that way, there might have unfolded in the European Theater an alternate war in which...?)\n\nGeorge Mandel had also earlier moved with his family into a different part of Brooklyn. After the war, while I was still at NYU, he already had a large, bohemian, bachelor loft in Manhattan, where he could paint and write. He was near Greenwich Village and near my school, and I would drop by there often, as would Danny the Count and Marvin Winkler. All of us were ecstatically astonished when the paperback reprint rights to his first novel, _Flee the Angry Strangers_ , published in 1953, were sold for a price of $25,000! His share of half that seemed a fortune. And $12,500 then _was indeed_ a fortune to someone living largely on disability payments for his war wound, and to someone like me, who had worked as a college instructor and next as an advertising copywriter for the same small salary of sixty dollars a week. I thought I, too, might try a novel of my own one day when I felt myself competent to complete one. As it turned out, I didn't feel competent to begin one until I was already past thirty.\n\nI DON'T GO THERE anymore. From my home in East Hampton to Coney Island is a long drive. I wouldn't want to go even if I lived closer. My friends are scattered. Their parents are dead. The only people I know who still live there are Sandy Kern, the widow of Ira, a boy who was a classmate in grade school and high school, and Frances Goodman, a friend of my sister's of her own generation. But they both live in Sea Gate, behind the fences and the private security forces. (And I've just learned that Sandy has moved, too.) The beach is near, the ocean is there, the weather is more clement in winter and summer than the extremes in the rest of the city, but they are there for the soothing feelings of safety and peace of mind that a homogeneous haven provides as much as for anything else.\n\nThere is not much of a Jewish neighborhood remaining outside the gates. Twenty years back, sometime around 1978, I made a trip back into the residential area of Coney Island for my novel _Good as Gold_ , just to note the changes. That Marvin and I met in a bar on Mermaid Avenue in itself signified a large change\u2014a change in us that we met in a bar, and a change in our old neighborhood of Coney Island, in which a bar, that particular bar, was the only place left for maturing people in the old crowd still living there to gather feeling safe. Smokey was there that evening, and that was the last time I saw him. He was getting old, he admitted, grinning and laughing as always, and he told me how he had found that out. That summer he'd been peddling ice cream on the beach, and he and an Italian kid in his twenties from down the Island had crossed into what each wanted to think of as a monopoly domain. Neither liked the competition. And the other, like Quasimodo in a different time and place, invited Smokey to settle it with fists under the boardwalk if he had a mind to. \"And,\" said Smokey, putting his head back with a grin and basking in the tale and the memory, \"he beat the hell out of me\u2014so easily.\" Smokey hadn't been able to see a single punch coming, he related to me, practically boasting. \"And I'm the guy that used to chase away all the other peddlers!\" And that's how Smokey could tell he was getting old. (I included his account of that incident, name and all, in _Good as Gold_ , as I also used Sylvia's birthday party, and the torturing wounds\u2014not scars, unhealing _wounds_ \u2014of Lee's lifelong ambivalent love for our father.)\n\nI also went back for an afternoon to look around, and afterward I recorded these impressions of mine of Coney Island in that work of fiction:\n\nFour springy, dark-skinned bloods in sneakers were coming his way, and he [Gold] knew in a paralyzing flash of intuition that it was ending for him right then and there, with a knife puncture in the heart....\n\nThey passed without bothering him, deciding to let him live. His time had not yet come.\n\nGold had noted earlier all the boarded-up, ruined shops on the three major lateral avenues of Coney Island and wondered where all the people went now to buy food, have their suits and dresses mended and dry cleaned, their shoes and radios fixed, and their medical prescriptions filled. In his rented car, he drove alone one more time the desolate length of Mermaid Avenue to the high chain-link fence of the private residential area of Sea Gate, where owners of the larger houses were now accepting welfare families, turned left toward the beach and boardwalk, and made his way back slowly along Surf Avenue. He did not see a drugstore. Behind the guarded barriers of Sea Gate, which once grandly sported a yacht club and was restricted to well-off Christians, younger Jewish families now congregated for safety and sent their children to whatever private schools they could. Elderly men and women, as always, probably still crept forth from secret places each morning and prowled the streets and boardwalk for patches of warming sunlight, conversing in Yiddish, and Raymie Rubin's mother had been killed one day on her return. [A true event: Raymie Miller's mother had been murdered by an intruder in her ground-floor apartment in the building on West 31st in which we had first lived.] Gold did not pass a single Jewish delicatessen. There was no longer a movie house operating in Coney Island: drugs, violence, and vandalism had closed both garish, overtowering theaters years before. The brick apartment house in which he had spent his whole childhood and nearly all his adolescence had been razed: on the site stood something newer and uglier that did not seem a nourishing improvement for the Puerto Rican families there now.\n\nThe newer, high-rise apartment building in the public-housing project there doubtless had elevators, better plumbing, and central air conditioning. I preferred my old apartment house, with its window in the bathroom and a kitchen roomy enough to hold a table at which a family of four could sit comfortably for meals.\n\nFarther down the Island, the Italian section appears compressed but pretty much intact, with the landmark restaurants of Gargulio's and Carolina still functioning. And the Steeplechase Pier, with Steeplechase long gone, was recently officially christened with the name Auletta Pier, to honor a citizen prominent in that section\u2014also, incidentally, the father of the journalist Ken Auletta.\n\nBut when last I was there, the fishermen out on the pier with lines down for crabs and their bobs and sinkers out wishfully for whatever small fish might be hooked were almost entirely Hispanic, as were the women and men with small boys and girls romping delightedly around the pier in the sunshine. All were overjoyed to see me, and the group with me, for I was there with a British film crew for a television show of which they all now felt themselves selected to be part.\n\nI HAD BEEN BACK only a few times. There was the occasion in the sixties mentioned previously when George Mandel and Mario Puzo and I relaxed in Steeplechase and found pleasure in watching our small children scampering about the grounds. And there was an afternoon and that one evening for on-site research sometime about 1978, when I spoke with Smokey in the bar.\n\nEarly in 1982, while a patient in a rehabilitation hospital, the Rusk Institute in New York, recuperating from the muscular weakness of my Guillain-Barr\u00e9 syndrome, I went again. Then I was the accomplished native guide on an abbreviated autobiographical tour with Mary Kay Fish, a physical therapist from upstate New York who had never been to Coney Island; Valerie Jean Humphries, a nurse; and Jerry McQueen, my friend with the car who was then a homicide detective with the city's police force, and a very good one. Valerie Humphries was one of the covey of nurses I fell in love with during a confinement lasting almost six months in two hospitals. We were married in 1987, and she still seems to like me.\n\nSince then I've gone back to Coney Island only three times, and always with television film crews from Europe who wanted to shoot footage for their programs of the place from which I had come, especially since it was a place with a world reputation and so tellingly visual.\n\nThe first group was from England, the next from Germany. This last time I was again with a production group from England. As before, all of the Europeans had heard of Coney Island, and their faces were alive with expectation as we arrived and made ready to disembark from the van and the car. Doubtless they had all been to playlands more modern and luxurious in a variety of cities and countries. But this was Coney Island, and to their minds Coney Island was a luminous myth of which they had long heard and which emitted the animating mystique of legend.\n\nTheir enthusiasm was infectious; I turned eager, too, and proprietary. And at Nathan's, the rallying place I had selected, I charged with more ravenous hunger than the others to the griddle for hot dogs and to the next post for the fried potatoes (with extra salt, please). They progressed to the food counters with the timidity of reverence. The food at Nathan's has never failed to impress my Europeans, and _most especially_ those from Britain, where the bland sausages they put up with have been the topic of caustic jokes since the days of rationing in the last world war.\n\nThis was in May 1994, and the experience was again exhilarating, an adventure for all of us. A complex of apartment houses rose where Luna Park had stood, and the former location of Steeplechase was a vacant area of several blocks, but they didn't seem to mind.\n\nThe defunct Parachute Jump was to them an Eiffel Tower of reddened steel.\n\nIt was midweek, a schoolday in early spring. There were no large crowds. The only high attractions left in operation in Coney Island were the Wonder Wheel and the Cyclone. They hardly noticed the difference.\n\nThey had never, it seemed, laid eyes on such a boardwalk, such a broad, long, sand beach, so sweeping a seascape of the ocean to the horizon at the vast opening into New York Harbor. Unpopulated, the spectacle was serenely perfect, sublime, eternal.\n\nOn Surf Avenue, where games and rides and confectioners and shows had once raucously functioned, one alongside the other almost to infinity, the numbers were cramped into a contained area, and most of the stalls were now the spilling retail counters of a flea market. That was okay with them, too.\n\nThey didn't object. Of the mechanical attractions in operation, a greater proportion than ever before were miniaturized rides for children, but the Europeans with me, of course, didn't miss the ones that were gone. And they didn't care when I spoke of the changes. For them, it was a rapture to be there\u2014if only for that first time.\n\nWe\u2014Valerie and I\u2014rode the Wonder Wheel with them while they taped footage from our swaying gondola on high. But the Wonder Wheel is tame. The Cyclone is not. Jittery beforehand with trepidation, a soundman and a young British woman who was assistant to the producer determined together to brave it; they were shaking when they staggered off afterward and wobbled back to the rest of us, pale with incredulity and jabbering with exultation from the thrilling ordeal. Although it was a schoolday, the rides for the children were full. Buses from a number of schools were parked on the side streets. The children had been brought from their separate schools for a day's outing in spring. They varied widely in their distinct groupings, black, brown, white, Asian\u2014one cluster obviously was from an Orthodox Jewish institution\u2014and their gleeful shrieks of laughter echoed everywhere as they tore around in the orbits of the different mechanisms, definitely having fun (as all children everywhere should always be allowed to do).\n\nI may be kidding myself, but I think I had more.\n\nIRWIN SHAW IS GONE and is very sorely missed by all those who knew him and gloried in his large mind and boisterous generosity of spirit. James Jones is gone, too. Both already were famous novelists when _Catch-22_ came out, and\u2014unacquainted with its author\u2014praised it before publication and helped lift it off to a fortunate start. And we had become friends. Mario Puzo's legs are troubling him. Mine are still weak from that neuropathy of fifteen years ago (and from those fifteen years of aging, too). Speed Vogel has become susceptible to respiratory infections and takes Coumadin for blood clots. Julie Green takes Coumadin now, too, and so does David Goodman and my sister, Sylvia. Lou Berkman did die of Hodgkin's disease, as related in _Closing Time._ My first wife, Shirley, is gone, too\u2014lung cancer, like my sister-in-law, Perlie. George Mandel is still actively around, good-humored as always and growing at least as hard of hearing as the rest of us. Marvin Winkler hangs on, though gorgeously overweight in face and form, as does his wife, Evelyn, who works in California as a teacher of preschool children, and he marvels each time we talk by telephone that the three of us have lasted this long and seem to be mostly in good health. On the first day of May every year he telephones from California to wish me happy birthday. Every February I try to remember to telephone him on his birthday, which falls late in the month. (In Eugene, Oregon, there is a man I've never met or spoken to named Bradford Willett. For something like thirty-five years now, starting soon after the publication of _Catch-22_ , he has sent me a birthday card. Through several changes of address they have managed to catch up with me. Probably, he was in his twenties when he began, and I am delighted and touched deeply to receive them.)\n\nIt is sometimes hard now to look back, to take stock, and to realize, to truly believe, that I have been to college\u2014that was not a common thing in my youth, as I've said, and movies about college life were very popular, and were always comedy romances; to know further that I have even taught at colleges\u2014daydreaming of that would have been a lunatic fantasy at the beginning; and that, further yet, I studied on a scholarship at Oxford University in England for one year, at St. Catherine's College, and that after a term there in 1991 as a Christiensen Fellow, I was appointed an Honorary Visiting Fellow.\n\nJust think: I am an Honorary Visiting Fellow of St. Catherine's College of Oxford University!\n\nI have much to be pleased with, including myself, and I am. I have wanted to succeed, and I have. I look younger than my years, much younger to people who are young, and I am in reasonably good health. My appetite continues hearty, and is complemented by a sterling digestive system that almost never lets me down. I still have most of a good head of hair and probably I have sufficient income and money to go on living as well as I want to, with enough left over, I feel, to please my few heirs (Valerie, sister Sylvia, children Erica and Ted). I am in love with my wife, still find other women appealing, enjoy a good many close friendships, and I have just finished writing this book.\n\nIt will take about a year to be published, and I expect much of what I've just said to still be true when it is.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**Begin Reading**\n\nTable of Contents\n\nNewsletters\n\nCopyright Page\n\nHachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.\n\nThe scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.\n_For Des_\n\n# Prologue\n\nThe story of their lives was all there.\n\nSandwiched between two battered green leather covers, on a hundred pages of charcoal paper, a bulging patchwork of newsprint, starting on the first page with the engagement notice Deirdre had snipped out of _The Irish Times_ six decades ago, the paper tobacco-colored now with age, the typeface a faintly quaint relic of a bygone age.\n\n#### Mr. M. MacEntee\u2014Miss D. O'Sullivan\n\nThe engagement is announced between Manus, only son of David and Eleanor MacEntee of Kildangan, Co. Kildare, and Deirdre, youngest daughter of Eamonn and Mary O'Sullivan of Ennis, Co. Clare.\n\nThat was the first time Deirdre's name had ever appeared in print. A brief mention, and one that for most of the women of her generation would be the only newsworthy event of their lives. An event as frivolous and short-lived as a single firework on a summer's night, falling somewhere between the announcement of their birth\u2014if indeed anyone had bothered to announce their birth\u2014and that of their death, however many decades later. In between the two, a farmhouse somewhere along the road. A husband, rising up out of the bed before dawn and not returning until after dusk. A clatter of children and a couple of hospital stays. A new hat for each special occasion.\n\nSo it was for Deirdre's sisters, but not for Deirdre. Deirdre's life was always going to be different, something her father seemed to see in her from the start. \"The world is full of boring people,\" he used to say to her, and Deirdre's heart would squirm with the secret pleasure of knowing that he was referring to her sisters.\n\nThe youngest of five girls, Deirdre was separated by six years from the next sister up. And while the other four were one prettier than the other, with their neat little noses and their dancing brown eyes, Deirdre always stood apart from them. With her long-boned face and her wide full mouth\u2014a mouth with hardly a dip in the bow of the upper lip\u2014she was unusual-looking, or so her father kept telling her. \"I'll let you in on a secret,\" he said to her, when she was no more than six or seven. \"You're not a pretty little thing, like your sisters. You're something far better than that. You're _unusual_ -looking.\"\n\n\"You must promise me you'll never be afraid to be different,\" he told her, on more than one occasion. And because Deirdre loved her father more than anyone, because she never for one moment doubted the wisdom of what he told her, she made it her life's work to fulfill that promise. When she left home for Dublin at just seventeen, it was her father who walked her to the train. He found her a carriage with a nun sitting in it, and before he left her there he presented her with the scrapbook as a parting gift. \"All you have to do now is fill it,\" he said to her, his voice brisk and brusque. \"Not a bother to you, Deirdre O'Sullivan.\" As the train pulled away, Deirdre looked back to see him standing on the platform, his arm raised in a rigid salute and tears raining down his cheeks.\n\nThat was the morning after the old Abbey Theater burned down. The smell of it in the air as Deirdre's train pulled into Heuston station, the news on everyone's lips. The Abbey, gone up in flames! Deirdre rushed up the quays and stood with the crowd gathered to peer at the smoldering remains of the national theater. As the Dublin fire brigade hosed down the rubble, as the players and stagehands did everything they could to salvage precious relics from the wreckage, Deirdre looked on in despair, imagining that she had arrived in the capital too late to fulfill her dream of becoming an Abbey actress. In fact the fire would prove to be her opportunity.\n\nAs she stood on the street outside, a man in shirtsleeves leaned out of an upstairs window and started flinging costumes out on to the street. Deirdre was among the onlookers who helped salvage them, spending the rest of the afternoon pegging wet gowns and uniforms on to a clothesline that had been strung across the empty auditorium. Before the day was out, she had the stage manager on a promise to get her an audition for the Abbey School of Acting; on the day of the audition, he was the one who schooled her in the flowery Irish she would need to impress the theater's director.\n\nIt was as a student of the Abbey school that Deirdre secured her first walk-on part just two months later, playing a nurse in the production of _The Silver Tassie_ that marked the start of the theater's long exile on Pearse Street. By the following spring she had her first speaking part, as Nellie the Post in _The Righteous are Bold_ , a play that scandalized her mother as much as it delighted her father. When the notice appeared in the following day's _Irish Independent_ , Deirdre's father sent the hotel porter out for five copies, one of which Deirdre pasted into her scrapbook, using her fountain pen to draw a wide loop of violet ink around the salient sentence:\n\nDeirdre O'Sullivan delivered a vivid performance as Nellie the Post...\n\nDeirdre leaned in to squint at the small print, but she couldn't make out the letters. Blurry discs across her eyes; the first one had appeared a year ago in the corner of her left eye and slipped across her eyeball. A partial eclipse that affected both eyes now, so that everything was steeped in sepia, as if the whole world was aging with her.\n\n\"You have cataracts,\" the doctor had said, when finally she was persuaded to see one. Turning off his miner's flash-light, he wheeled backward on his chair with something like glee. \"Nothing whatsoever to worry about,\" he said. \"It's extremely common at your age. I'm going to refer you to an eye surgeon.\"\n\nSpinning his chair round to face his desk, he dropped his chin to his chest and started tapping on his keyboard. Using his middle finger in a stabbing motion, as a pianist would to hit a final, definitive note, he set the printer in motion. It spat out a page, which he handed over to Deirdre with deadly gravity.\n\n\"You know what this is?\" Deirdre said to him, once she was standing.\n\nHe looked up at her expectantly, waiting for her to answer her own question.\n\n\"This,\" she said, waving the piece of paper at him. \"This is the thin end of the wedge.\"\n\nThe paper she was holding in her hand was a ticket to everything she had always feared. A summons that sooner or later would see her sitting in an ugly armchair in a triple-glazed sunroom in an overheated nursing home, surrounded by other old ladies, and nothing to separate her from them, nothing to make her different. The possibility that her entire personality, and the life she had built on the strength of it, the prospect that all of it might turn out to be nothing more than a notion she had taken upon herself, this was something Deirdre could not bear to contemplate.\n\nAs soon as she was out on the street, she slipped the referral letter into the nearest bin. Giddy as a schoolgirl, she sailed along the pavement, holding her umbrella out imperiously to bring an approaching bus to a stop. She climbed on board, brandishing her free travel pass like a police badge, aware of a mischievous sense of her own power. Not since she had boarded the bus from Ennis to Dublin all those years ago, with her money sewn into her knickers for fear of the criminals she might encounter in the city, not since then had she had such a drunken sense of freedom. The fact that she had found a way to escape the ordinary, at this late stage of her life, it was almost like being immortal.\n\nShe was only a week in Dublin when she met Manus. Six weeks later they were engaged, and by the following summer they were married. A photograph of their wedding appeared in the _Irish Independent'_ s Wedding Bells column; looking at the clipping now, Deirdre had to bite her bottom lip to stop herself from smiling. Manus looked so camp, it was ridiculous. \"How did you not know he was gay?\" Deirdre's daughters had asked her many times over the years. \"Darlings,\" she had told them, \"if I'd known what a homosexual was, I might have recognized him as one. But we neither of us would have had a notion what the word meant. You have to understand how innocent we were. Nobody ever told us anything.\"\n\nManus was a Protestant, which was scandal enough in those days. Of course, he had to convert to marry her. In a gesture of reciprocity, their wedding reception was held not in her hometown but in the long-overgrown walled garden of the estate his father managed in County Kildare. To the horror of Deirdre's mother, there were no chairs, only rugs laid out on the grass for the guests to sprawl on. When Manus left Deirdre for another man\u2014twenty-eight years and three children later\u2014Deirdre's mother claimed not to be in the least bit surprised. After what she'd seen of their wedding day, she had long been prepared for the worst.\n\nWhen Manus left Deirdre, it was a point of pride with her not to react the way everyone expected her to. Everyone expected her to be heartbroken, and humiliated, and of course she was both of those things, but she was damned if she was going to show it. \"Manus and I were always more like friends than lovers,\" she told everyone she met, aware that she needed to subvert the unspoken suggestion of some sexual inadequacy on her part that might have provoked her husband to seek his thrills elsewhere. \"All is fair in love and war,\" she told other people, hinting at a broader dimension to her marriage than might have been outwardly apparent. \"It's a great relief to me,\" she said, \"not to have to put up with him stealing my face creams anymore.\"\n\nSuch was the force of Deirdre's ability to create a part for herself, such was the gusto with which she reprised the role year after year, and decade after decade, that she nearly came to believe it herself. What she almost entirely succeeded in obfuscating, what not even her children guessed at, was the fact that she had never not been in love with him. She had spent her entire life loving a man who could not love her. It was too big a thing to contemplate.\n\nOh, Manus, Manus, Manus. Every cutting of his was pasted into her scrapbook with such naive pride. Reviews of his novel, and feature pieces written with all the genteel reverence of another age. A black-and-white photograph of Manus sitting at his desk in this very study at the top of the house, with the light from the window falling in on him. In the photograph he has an upright typewriter before him, a sheet of paper primed to receive his next novel, a novel that would never come to be written.\n\nDeirdre was sitting at that same desk now, with the same wintry light falling in through the window, only less of it. In the fifty years that had passed since that photograph was taken, the wisteria she had planted as a young bride had stealthily climbed three floors, immersing the back of the house in a deep green sea of leaves, so that even on the top floor you had the sense of being in an underwater cave. A green gloom over everything, it had the advantage of disguising the dust. The patches of dry rot climbing down out of the ceiling cornices. The bucket in the corner to catch the drip from the leaking roof. Deirdre had long ago given up pouring money into that bloody roof.\n\nShe turned the pages of the scrapbook impatiently, passing over articles so well known to her that she didn't need to read them; they were like familiar fields flashing by the window of a train. The piece about Manus from _The New York Times_. The photograph of them both posing on the tarmac at Idlewild, he wearing dark glasses and she a ruby-red turban to match her coat; they looked for all the world like movie stars, which was what they almost were.\n\nTurning the next page, she came upon a publicity photograph of herself, and stopped. A photograph of her standing alone onstage in a flowing gown, her chin lifted slightly to display her long, pale neck to advantage. Her eyes were open wide and artfully vacant, as if she were just an empty vessel aching to channel all the great parts that still lay in store for her. The beauty of her, undeniable by then. Not a small-town beauty like her sisters, but a beauty capable of commanding the national stage. Capable perhaps, even, of conquering the world.\n\nHad she known then that she stood balanced on the brink? Review after review, pasted lovingly into her scrapbook, attesting to her triumphant position as the queen of the Dublin stage. There was talk of movie parts, perhaps a move to Broadway, until the beginning of the end was announced in the form of a tiny patch of newsprint no bigger than a postage stamp; it was attached to the page by a piece of tape as dry and yellow as a flake of old skin.\n\n**MacEntee** \u20144 July 1960, at Stella Maris Nursing Home. To Deirdre, wife of Manus MacEntee, a daughter, Alma. _Deo gratias_.\n\nAlma, who arrived in a flurry of numbers. Talk of inches, then ounces. Weeks, then months. All Deirdre wanted to know was when she could go back to work. \"See that scar on my head?\" Alma would joke in later years, pointing to a chickenpox dent on her left temple. \"That's the mark my mother's heel made when she was climbing over me to get back on to the stage.\" And for a year or two it seemed that motherhood had done nothing to slow Deirdre down. It was only when she fell pregnant again, and sick this time with a morning sickness that lasted all day, only then were her ambitions finally derailed. Reluctantly, she passed up on the part of Brigid in _Shadow and Substance_ (a part she was a little long in the tooth for, truth be told) and took to the bed to await her deliverance.\n\nDeirdre suffered a collapsed lung giving birth to Acushla, and it took her a long time to recover, and although Manus did insert a birth notice, nobody thought to save the paper when it appeared, so no record exists of Acushla's birth, much to Deirdre's regret. There is no record of her father's death either, an event that occurred on the same day Acushla was born. Deirdre was too weak to attend his funeral. She cried for a year afterward, two years maybe; she began to think she would cry forever. \"Why didn't you ask anyone for help?\" asked Acushla once. \"Why didn't you go to the doctor?\" Oh, it was hard to explain. You didn't ask for help back then. It just wasn't something you did. \"What got you out of it?\" asked Acushla, even though she already knew the answer. \"Your brother did,\" said Deirdre with a smile. \"It was your brother got me out of it.\"\n\n**MacEntee** \u2014to Manus and Deirdre (n\u00e9e O'Sullivan), a son, Macdara.\n\nMacdara, who is Deirdre's favorite, according to the girls. As evidence, they cite the fact that the PIN for Deirdre's bank card is the year of Macdara's birth. \"Don't be ridiculous,\" Deirdre says. \"I love all my children equally.\" Even though that's not strictly true. She loves all her children in equal measure, certainly, but not all at the same time, and not in the same manner.\n\nHer firstborn Deirdre loves with a mixture of awe and pride. Alma, who arrived in the world fully in charge of herself; no sooner did she open her eyes than she took command of her surroundings, a command she never once let go. Sometimes it seems to Deirdre that she loves Alma precisely because Alma never asks for her love, and never indicates that she needs it.\n\nAcushla, on the other hand, was born crying out for attention. A fretful baby, she would wake in the middle of the night in need of comfort, a comfort Deirdre did not have it in her to give. Many was the night Deirdre spent walking Acushla around the house in the dead of night, her own helpless tears pooling with her daughter's to form a puddle of sticky grief in the hollow of her collarbone. The legacy of that awful time is a tenderness between Deirdre and Acushla, a tenderness verging on pity for the misery they shared. It is nothing more or less than another strain of love.\n\nBy comparison, Deirdre's love for Macdara was always miraculously straightforward. From the moment she set eyes on him in the hospital, she loved him with a love that was as uncomplicated as it was unexpected. By then, she had ceased to hope for anything out of motherhood, but with Macdara's arrival, after a birth as easy as a bump in the road, she was transformed overnight into an earth mother. Macdara was breast-fed from the start, where the girls had been bottle-fed. Where the girls had been left with nannies, Macdara went everywhere Deirdre went, nestled to her chest in a sling, and at night he crept into the bed between his parents, something the girls would never have been allowed to do.\n\nAfter Macdara was born, Deirdre did not even contemplate returning to work, determined not to miss a moment of her belated joy in motherhood. It was only when he was dispatched to boarding school that she relaunched her career, enjoying a brief but glorious three years back in the footlights. She was just preparing to go on tour with _The Gigli Concert_ when Macdara's crisis came, a crisis that occurred three weeks into his scholarship to the University of Provence (for some reason never entirely understood by Deirdre, the crisis was triggered by a picture of a hermaphrodite he saw in a book). Macdara's breakdown was so severe, and the challenge of helping him back to full health so enormous, that Deirdre dropped out of the tour, an act of career sabotage from which there would be no way back. Thirty years later, mother and son are still shuffling around the same space, brushing up against each other's foibles with a mutual acceptance and an ever-increasing endearment that baffles everyone but themselves. \"You're enabling him,\" Alma said to her once. \"We enable each other,\" Deirdre replied, cutting her off at the pass.\n\n\"Mum!\"\n\nIt was Macdara's voice, calling up to her. \"Do you want tea?\"\n\nBy the sound of it, he had climbed as far as the landing.\n\n\"Please, darling!\" answered Deirdre, rising out of her chair to shout her answer, only to feel the familiar lick of pain racing down her right leg. \"Sciatica,\" the doctor had said. \"Again, perfectly normal.\" The bladder incontinence Deirdre did not deign to mention to him, certain in the knowledge that this too would be written off as a normal consequence of aging.\n\nLowering herself carefully back down into her chair, Deirdre began to speed through the scrapbook, looking for Macdara's only item of news, after his birth. She skimmed over Alma's first article for the college newspaper. Alma's first article for a national newspaper. A front-page photograph of Alma's wedding to Michael Collins (much was made of the fact that the groom was the youngest sitting deputy in the nation's parliament. Much too of the name he bore, a name passed down to him by his father despite the irony of naming your son after the dead hero of the opposition\u2014oh how they laughed at the notion of a Fianna F\u00e1il Michael Collins!). The pages that followed were plastered with coverage of Mick's early political career. His media appearances fast became so numerous that Deirdre had been forced to stop saving them. She took to keeping only those clippings that featured Alma. Alma photographed side by side with Mick at a state dinner. Alma pictured at the edge of a media scrum, microphone in hand. Alma in a full-page interview about the new generation of women making a name for themselves in television.\n\nThere was the odd article about Acushla too. A morsel in a gossip column about her engagement to Mick's twin brother Liam. A picture of her wearing a large hat to Ladies' Day at the Spring Show. An appearance in a list of Ireland's best-dressed women. The only time Macdara's name made it into any form of publication was an entry in his school annual noting that he had triumphed in the spellathon. Deirdre had cut the piece out and dutifully pasted it into her scrapbook, confident that it would be no more than a prelude to greater things. In retrospect, it might have been better to leave it out.\n\nHearing Macdara's tread on the landing, she flipped the page.\n\n\"Here's your tea,\" he said, coming into the room without knocking. Hunched, the way very tall people often are, he bore the mug of tea forward with all the practiced care of a butler. He was wearing a freshly ironed shirt, as always, and one of the hand-knitted ties he had learned to make in occupational therapy\u2014he'd been making them for people as Christmas presents ever since. A V-necked lambswool sweater and camel-colored cords, and on his feet a pair of Ugg boots. Macdara was very prone to cold feet.\n\n\"Scrapbooking?\" he asked her, as he set the mug down on the worn leather surface of the desk.\n\n\"Yes,\" she said, wafting her hand over the open pages. \"I was just going to paste in an article on your sister but I got carried away.\"\n\nShe glanced down at the page in front of her.\n\n\"My mother's death notice,\" she said. And there was silence for a moment as they both stared down at it. Macdara was standing on the far side of the desk, so he had to twist his neck so as not to be looking at it upside down.\n\n**O'Sullivan** , Mary. In her 95th year. Widow of the late Eamonn...\n\n\"When was that?\" asked Macdara, looking up at her. His eyes gray and guileless.\n\n\"Nineteen eighty-four.\"\n\nHe was in France when she died. Both of them know this but neither of them mentions it.\n\n\"Thirty years ago,\" he says, with surprise.\n\n\"Yes,\" she replies, but she's not sure whether he's referring to his grandmother's death or the ill-fated stay in France. Or indeed both.\n\n\"Well,\" he says, cocking his head as if he's about to say something else. He drifts toward the door with his head still tilted, and the conversation is left hanging. The next thing he's gone, and whatever it was that he was going to say, it remains unsaid.\n\nOnce Macdara is gone, the room seems emptier than it was before he entered it. Deirdre sits for a moment searching the air for some visible change, but there is nothing but a faint hum.\n\nShe makes a conscious effort to summon a memory of her mother. A desperate desire in her, to remember her mother as a young woman\u2014all that comes to mind are frozen images from photographs. The living, breathing memory is that of a bag of bones in a bed. A pair of angry staring eyes. A smell that remains on Deirdre's breath\u2014despite all the money they spent on the nursing home, there was always that greenhouse smell of human rot. The memory of it horrifies Deirdre. It frightens her, and try as she might, she cannot supplant it. The fact is that her mother lived too long. She outlived even herself.\n\n... deeply regretted by her daughters Margaret (Peggy), Mary, Catherine (Kitty), Maeve and Deirdre.\n\nAll of them still alive. Peggy must be ninety-three by now, and Mary ninety-one. Kitty on the verge of ninety. Even Maeve must be eighty-six, which makes Deirdre eighty at her next birthday. A fact that has been passing through her head with increasing regularity since the start of the year. The number is puzzling to her; it seems incorrect.\n\n\"I don't fancy the idea of it,\" she confided to her ex-husband recently, thinking that he of all people would understand.\n\nInfuriatingly, Manus just laughed.\n\n\"I've always been younger than everyone else,\" Deirdre went on, anxious to make herself understood. \"I can't see myself as an old person. I don't even _like_ old people.\"\n\n\"Nonsense,\" he said. \"You have the makings of a magnificent old lady. You were born to be an old lady.\"\n\nOh it annoyed her no end when he said that. It trespassed upon her vanity.\n\n\"You're going to live to be a hundred,\" he said. And she became aware of a great and petulant desire to prove him wrong.\n\n\"Let's face it,\" he told her, with a tease in his voice. \"You come from a long line of survivors.\"\n\n\"Oh, Manus,\" she said. \"That's the problem. It fills me with dread, the thought of getting old.\"\n\n\"Indeed,\" he said, with a grim chuckle. \"And yet you must admit, it's better than the alternative.\"\n\nDeirdre had smiled at him, and nodded, but in her head she had started churning a slow-moving thought. Turning over the words he had used until she had extracted the meaning out of them. What had caught her attention was not her ex-husband's cheerful fatalism in the face of the aging process. What had taken hold of her was a notion that he had unintentionally planted in her head, the notion that there was an alternative.\n\n\"You can't kill yourself just because you've developed cataracts,\" said her doctor. \"Sure, you're as fit as a fiddle. You're in better nick than I am.\"\n\n\"Don't even _think_ about patronizing me,\" she said. \"I've given this a lot of thought. I'm going to be eighty in October and I feel I've had a good inning. I've decided to quit while I'm ahead.\"\n\n\"I wonder could you be suffering from depression?\" he asked her, peering at her more closely.\n\n\"Do I look like I'm suffering from depression?\"\n\n\"You're not suffering from a terminal illness,\" he said, ticking another box in his head.\n\n\"Of course I am! I'm suffering from the onset of old age, which in all cases that I know of has proved to be fatal.\"\n\nHe didn't even smile.\n\n\"You do know it's illegal for me to help you?\"\n\n\"I'm not asking you to help me. I'm simply asking you to provide me with my medical records and confirm that I'm of sound mind.\"\n\n\"Oh, this is all most unusual.\"\n\n\"Exactly,\" said Deirdre, happily.\n\n\"I would strongly urge you to inform your family of your plans.\"\n\n\"Of course,\" she said, although of course she had absolutely no intention of telling them. That was all part of the fun, the surprise they would get when she sprang it on them. Manus in particular she delighted in surprising. She couldn't help wishing she would be around to see his face when he heard. It was Manus, more than anyone, who would appreciate the supreme stylishness of this thing she was planning to do.\n\n\"I wonder have you stopped to consider the effect this will have on your children?\" asked the doctor.\n\nDeirdre looked at him, wondering how on earth he could be so maddeningly stupid.\n\n\"Don't you see? I'll be doing them a favor.\"\n\nAnd really, that did seem to Deirdre to be the case. By releasing them from any duty of care to her as she got older, by dividing between them the not insignificant sum of money she had scrimped and saved over the years before it was swallowed up by doctors' bills and nursing home charges, it did seem to Deirdre that she would be doing her family a huge favor.\n\nOnce the decision was made, there was a great falling away of her fears. The things that had worried her before\u2014the cataracts and the sciatica, the dilapidated state of her house\u2014these seemed no more than mere irritations to her now. The clouds she had seen gathering\u2014the cancers and the dementia, the incontinence and the immobility\u2014they had miraculously cleared, and for the moment at least, Deirdre saw nothing but blue sky above her. A frothy sweetness to everything, it was as if her life was made of nothing but sherbet. She found herself prone to unexpected bursts of happiness, singing as she went about the house, songs she had not sung since her girlhood.\n\nWhen the old house creaked at night, Deirdre rolled over in the bed and went straight back to sleep. When winter storms tore a long portion of guttering from the back wall, she picked it up and tossed it into the flower bed. When the weathermen predicted flooding as a consequence of global warming, and wet summers in perpetuity, she didn't pay a blind bit of attention, because she wasn't going to be there to worry about it.\n\n\"I've decided to throw a party for my birthday,\" she told her children. \"I thought it would be nice to gather us all together under the one roof.\"\n\n\"Why would that be nice?\" asked Alma, with a disdainful expression that was most typical of her.\n\n\"We could combine it with your fiftieth if you like?\" Deirdre suggested to Acushla, in a moment of rather rash generosity.\n\n\"Why on earth would I want to go advertising the fact that I'm turning fifty?\" asked Acushla, appalled.\n\n\"It's going to be a proper party,\" said Deirdre doggedly. \"We'll have champagne, and canap\u00e9s, and printed invitations.\"\n\n\"It sounds more like a wedding,\" said Macdara, in a morose tone.\n\nAnd Deirdre giggled, giddy as a young bride. Because Macdara was right! This was not at all unlike a wedding, what with its focus on the fripperies rather than the momentous step into the unknown that would follow.\n\nDeirdre had an image in mind of her party, an image that drew on a painting she had always loved. The painting was by Sir John Lavery, and it had been hanging for many years in the National Gallery; for many years Deirdre had been in the habit of dropping into the gallery to visit it, as you might visit an old friend. The subject of the painting was Lady Lavery, and she was depicted seated in a vast and grand room, wearing long, rich robes of satin and silk. On her head she wore an elaborate feathered turban, and her daughters and a dog were arrayed around her, with a servant bearing a tray aloft in the background. In the painting, Lady Lavery's long, pale face was luminous, her posture serene, her eyes sadly knowing.\n\nDeirdre fancies that she used to look like Lady Lavery, a resemblance that was pointed out to her once by a drunk at a party. \"Look,\" said the drunk, \"it's the lady on the face of the old pound note!\" And while Deirdre had brushed off his ramblings with a self-deprecating laugh, while she had pretended to be embarrassed by him, really she was flattered. Ever since then it has formed a private but important part of her image of herself, this notion of a resemblance to Lady Lavery, so that when she thinks back on herself as a young woman, it is Lady Lavery's face that comes to mind, clearer than any memory of her own.\n\nIn imagining her party, Deirdre has an image in her mind of that particular painting. She visualizes herself seated in similarly theatrical attire in her own living room, with her family arrayed around her. In planning the party, she is constructing a subtle _tableau vivant_ in her head. It occurs to her fleetingly that the desire to create this tableau might be the only reason for having the party. If she were to dredge her soul, she might find this to be true. But Deirdre has been playing a part for so many years that she has long since given up on trying to find the place where her own personality ends and the pretending begins.\n\n**In Her Own Words:**\n\n**Alma MacEntee**\n\n**in conversation with Dara Lynch**\n\n**What's your favorite film?** _La Dolce Vita._\n\n**What's your favorite book?** _Love in a Cold Climate_ , by Nancy Mitford. She makes my family seem almost normal, by comparison.\n\n**What part of your body do you like best?** My breasts. If I say so myself, I've great tits.\n\n**Part of the body that you like least?** My teeth. I come from a generation before perfect teeth.\n\n**What is the trait you most deplore in yourself?** Slovenliness.\n\n**What is the trait you most deplore in others?** Humorlessness.\n\n**Which living person do you most admire, and why?** My father, because he has never lost his sense of innocence.\n\n**What is your greatest regret?** I don't believe in regrets.\n\n**What's your favorite smell?** The smell of rashers being cooked for me.\n\n**Who's your favorite fictional hero\/heroine?** Scarlett O'Hara.\n\n**And your hero\/heroine in real life?** Hillary Clinton.\n\n**Have you ever said \"I love you\" and not meant it?** No.\n\n**What is your most treasured possession?** My waistline.\n\n**What beauty product could you not live without?** My factor 50 sunscreen.\n\n**Who would you choose to play you in the film of your life?** Julianne Moore.\n\n**Who's your ideal dinner guest?** My ex-husband (I'd poison him).\n\n**Motto?** Be yourself; everyone else is already taken.\n\n# Alma\n\nAlma MacEntee. Five syllables that, when spoken out loud (as they often are), tumble into each other with an energy that seems to come from within. Almamacentee. A bubbling stream of a word that, in the time it takes for it to travel through the locks and sluice gates of your brain, calls to mind a slew of images, slabs of information, half-remembered snippets of gossip. A gash of red lipstick, a helmet of unashamedly red hair. That hair that seems to get redder all the time, heightening the effect of the china-white skin and the always-amused eyes. In your head you can hear that oh-so-familiar voice, with its almost inappropriate suggestion of intimacy.\n\nAlma MacEntee. In the less than three seconds it takes for you to hear her name, you have her whole history in your head, and with it your own. You remember her beginnings, how she spilled out of the continuity studio and shouldered her way through the dandruff-smattered suits on election night, an oversized microphone in her hand, the questions she asked famous now for their fearlessness. In her next incarnation she was standing on a small, round podium in a pool of artificial light, a dark, murmuring crowd at her feet. The silver gown she wore fell from her hips like an oil spill and her hair was styled into two shining chestnut wings that swept off her painted face. \"And now for the results of the Belgian jury,\" she said. \" _Royaume-Uni, dix points_. United Kingdom, ten points. _Irlande, douze points_. Ireland twelve points.\" As the crowd went wild, Alma MacEntee allowed herself a smile. Of course, nowadays her eyes wrinkle when she smiles, which is why she is careful not to do it so often. Nowadays she is more often seen with her eyebrows disdainfully raised, her elbow resting on the studio table as if in preparation for a bout of arm-wrestling, one manicured fingernail poised to signal an interruption to this or that politician.\n\nShe was married to a politician, once. You remember their whirlwind romance. You remember the front-page photos of their wedding and the kiss for the cameras. You remember the divorce, back when divorce was a scandal. (Was it even legal to get divorced back then? It seems that Alma MacEntee was a divorcee before divorce existed.) She was the lover of a prominent businessman, the lover of a famous actor, the lover of a newspaper owner, long before the word \"lover\" was ever spoken out loud. She was a woman before her time. And while all the Marys and Anns of the small screen were spoken of with their surnames attached, to distinguish them from all the other Marys and Anns out there in the world, while the Eileens and Sharons were sometimes confused with or mistaken for other Eileens and Sharons, Alma MacEntee required only four letters to define her.\n\nAlma.\n\nOn the first day of April\u2014a day she always considered the first day of spring, treating as a bad joke the notion that spring arrived in these parts in February, or even March\u2014Alma woke at seven, as she always did, to the sound of the radio. She lay in her vast empty bed, as was her practice, while she listened to the headlines, suppressing a growl as she heard the morning newscaster mispronounce Barack Obama's name again. No matter how many times Alma lobbed angry rants into General Mail, no matter how many times she raised the issue with the subs, the newsreaders persisted in mispronouncing his name, placing the stress on the first syllable instead of the second. Was the man himself not authority enough on the pronunciation of his own name? Was his wife not to be trusted to get it right? All you had to do was listen to them, for Christ's sake. Swinging her feet out of bed, Alma fought the urge to ring in.\n\n\"I'm a crank,\" she said, even though there was no one there to hear her. She nodded, as always, in agreement with herself. \"I know. I'm a crank.\"\n\nPulling her black velvet robe off the back of the bedroom door, she made her way down into the kitchen, where another radio was tuned in to the same station. She turned on the coffee machine, and while she waited for it to heat up, she stared out of the window into the small back garden, where her daughter's bicycle lay abandoned against the wall, a garland of plastic flowers wound around the handlebars and a threadbare tarpaulin of cobwebs hanging over it. Across the skyline the neighbor's washing swayed in the breeze, underpants and shirts hanging upside down like a row of dead cats strung up by their front paws.\n\nAlma turned away from the window to make her coffee. In her mind she was already running through her agenda for the day, careful not to get anything out of sequence. The hairdresser's first. Then the dry cleaner's. And she had better stop by the paper for a minute on her way in to work, just to cut off any trouble at the pass.\n\nFor that, she'd be needing some war paint.\n\n\"Well,\" said Jim, when she walked into his office without knocking.\n\nHe was sitting at his desk with his jacket off, wearing a lemon-yellow shirt (cotton-polyester mix, Alma would be willing to bet) with a Cross pen in the breast pocket. A cheap red tie curled up in his in-tray like a sleeping snake.\n\n\"I got your message,\" said Alma.\n\nJim propped his feet up on the desk. He leaned back in his chair and intertwined his hands behind his head, revealing the underarms of his shirt, dark and wet.\n\nAlma's expression did not waver.\n\nShe advanced into the room, walking from her hips. She ignored the two chairs that sat facing the desk, weaving her way round to sit on the edge of it instead. She had one bum cheek on the surface, one in mid-air, her pencil skirt riding high above her knees. She wrapped her left ankle around her right leg, balancing herself on a single three-inch heel. (Men like Jim aren't used to having beautiful women flirting with them. This was something that Alma understood. Flirt with a man like Jim, and he's yours for life.)\n\n\"I'm listening,\" he said, his eyes bulging as he looked at her. Alma allowed the faintest touch of a smile to tug at one corner of her lips.\n\n\"I thought it was funny. Go on, Jim. Admit it. You thought it was funny too.\"\n\nHe sighed.\n\n\"People seem to have taken offense.\"\n\n\"Oh,\" she said. \"People.\"\n\nEyebrows raised, she waited for him to say more.\n\n\"The earthworm analogy, in particular, seems to have upset them.\"\n\nShe shrugged.\n\n\"I thought the earthworm thing was quite mild, actually. I've said way worse stuff than that in my time.\"\n\nAnd she had. There was the piece about Ladies' Day at the races (Crufts for humans), the one about West End musicals (strictly special needs, she had suggested, and was forced to apologize a week later, to special needs people and West End musicals both. It was the first of many apologies).\n\n\"Personally, I think the obesity piece was my most offensive to date.\"\n\nHe put his hand over his face.\n\n\"Don't remind me.\"\n\nShe had him now. She knew she had him. But he still had to put up the pretense of a fight.\n\n\"Alma,\" he said, throwing his arms out wide. \"You know as well as I do. I pay you to be provocative. People read your column _because_ it's provocative. But there's a line there, and we need to stay on the right side of it.\"\n\n\"So what are you saying? The earthworm thing was the wrong side of the line?\"\n\n\"Forty-six e-mails, twelve letters and nineteen phone calls.\"\n\nAlma planted one hand on the surface of the desk. The other hand she propped on her hip, and like a teapot, she tipped herself forward.\n\n\"Jim,\" she said, in her deepest, throatiest voice. \"People have no sense of humor anymore.\"\n\n\"These are serious times. Call it evolution.\"\n\n\"In that case, we're the dinosaurs, darling.\"\n\nHe laughed. It was the use of the word \"we\" that was warming him up. The thought of him and her, wrapped up together in that one little word.\n\n\"We're not extinct yet,\" he said, his raw, doughy face flushed with his own fumbling flirtation.\n\n\"No, darling,\" said Alma, narrowing her eyes at him. \"We're not extinct yet. And we must not go down without a fight.\"\n\nThe phone on Jim's desk rang and he reached out to pick it up, winking at her with his left eye, while his greedy little right eye slobbered all over her. She hopped down off the desk and sashayed toward the door, enjoying the knowledge that he was watching her arse as she went. At the door, she turned to blow him a kiss.\n\nHis secretary was sitting outside, watching her with heavy eyes.\n\n\"You know yourself,\" said Alma, pausing to lay a hand on the secretary's desk. \"A girl has to make a living.\"\n\nEight hours later (after three more cups of coffee, two interminable editorial meetings and a marathon session in makeup), she was sitting in the studio. Her hair was sprayed to the texture of dry bark, and she was wearing a creamy white jacket that set off her coloring perfectly. A skimpy black tank top revealed her milky cleavage. Thirty seconds to air and she took a cosmetic purse out from under the table, opened it up and started applying a fresh layer of lipstick.\n\n\"Jesus,\" said the program editor, in the control room. \"I wish she wouldn't do that.\"\n\n\"I wouldn't mind,\" said the studio director, \"but that shade of lipstick she insists on wearing? It's completely inappropriate for a news program.\"\n\n\"I'll tell you what. Why don't you be the one to tell her?\"\n\nSniggers from the shadows, interrupted by Alma's voice, amplified through the gallery.\n\n\"Where are we going first?\"\n\nThe program editor went into a spasm, grappling at the heap of papers in front of him.\n\n\"Jesus! Does she not have the running order?\"\n\nThis tendency of Alma's to wander into the studio without her notes was the stuff of legend. Legendary also was her ability to manage without them. She had been known to present an entire program without so much as glancing down at her scripts. She had sailed through the first segment of an election special once with her constituency guide sitting on top of the coffee machine in the corridor; when at last the ad break arrived and a broadcast assistant dashed in with it, it was only to be stared at by Alma as if to say, what would I need that for? She had winged her way through the peace process, presenting programs late into the night from outside Hillsborough Castle, or Stormont, without so much as a handwritten lead-in. She had done outside broadcasts from rain-sodden piers and muddy fields, or on bridges over flooded rivers, with no more equipment than a powder puff and a golf umbrella. When the autocue went down on budget night 2010, she had not even raised an eyebrow, because she had all the figures in her head.\n\nSo if Alma was a picture of calm fifteen seconds to air and flying the studio blind, then this should have been no surprise to anyone. What was surprising was the panic in the gallery. The gallery had never learned not to panic. The program editor was standing up, gripping his skull with his hands to contain his stress. A broadcast assistant was standing at the studio door with Alma's scripts, her little heart pounding with fear as she waited for the opportunity to slip inside.\n\nThe director leaned in to the talkback.\n\n\"Straight to Government Buildings,\" she said, holding the button down. \"You should have Simon McFeeley in vision.\"\n\n\"Simon!\" said Alma, as if she had just bumped into him on the street. \"How are you, my darling?\"\n\nHer voice shattered the sanctity of the empty studio.\n\n\"Hi, Alma,\" came Simon's voice in return, delayed by a two-second lag.\n\n\"Ten seconds to air,\" said the broadcast coordinator.\n\nAlma gave her hair one last little pat. She dropped her chin a little, to disguise the wrinkles on her neck, as the broadcast coordinator began the countdown. The sting began to roll, graphics tumbling through splintered space. Alma appeared on the monitor, face full square to the camera, blue eyes twinkling. Her perfect symmetry lent her a luminescence onscreen that was not apparent in the flesh. She looked, quite simply, magnificent.\n\n\"Good evening,\" she said. \"And welcome to _Headline_. Coming up on tonight's program, the latest on the talks between the Troika and the Department of Finance. A rude awakening for the drinks industry. And the abolition of the Senate: we debate the pros and cons. But first I'm joined from Government Buildings by our political editor Simon McFeeley...\"\n\n\"Sweet Jesus,\" said the program editor, leaning back into his chair. \"That woman puts years on me.\"\n\nThe lights were still on in the stadium when Alma's taxi swung into the square.\n\nBloody waste of electricity, she thought, wondering to herself was this the seed of a column. Not enough, she decided, and let the thought sink to the bottom of her mind.\n\nThe taxi driver slowed to a crawl, bending low over the steering wheel as he peered up at the stadium. The place was still relatively new; it had been named after the insurance company who sponsored it, and while at first the city's residents had vowed never to use this new name, it was slowly slipping into the lexicon in much the same way as the stadium itself had ingratiated itself into the city's affections. From a distance, it had a curiously transparent appearance, like a great glass bubble on the skyline, but up close and hovering above the forty dark houses of the square, it looked like a spaceship. A great glowing ship of steel and glass.\n\nThe taxi driver was gaping out of the window at it. An Indian man, or Pakistani perhaps, he had a laminated picture of his children on the dashboard.\n\n\"Anywhere here is fine,\" said Alma, impatient to get out of the cab.\n\nThe driver stopped at the corner of the square, but he continued to look up at the stadium. \"You are very lucky,\" he said. \"You are so lucky, to live beside this magnificent stadium. I would be very happy to live beside this stadium.\"\n\nIt was something that Alma found impossible to understand, the devotion of sports fans to this secular temple. From far and wide they came, posing for photographs in front of it. (\"It's a _football stadium_ ,\" Alma had written in one of her columns. \"It's not the Taj Mahal.\")\n\n\"I'll sign for that,\" she said, desperate to be out of the cab. The arches of her feet were aching and her mouth was dry. She was longing to take off her shoes. Longing for a drink, and the ritual post-program cigarette; it was the only one she allowed herself these days.\n\nThe driver turned round to face her. \"On my day off, I am going to come back here. I am going to bring my son to see your gorgeous stadium.\"\n\nThe word \"gorgeous,\" spoken in a Dublin accent.\n\n\"It's not really my stadium,\" said Alma, leaning into the gap between the seats. \"Now if you don't mind, I'll sign your docket for you.\"\n\n\"Of course,\" said the driver. \"Sorry.\" He rooted about in the glove compartment until he found the docket book. Rooted again until he found a pen. Alma signed her name, adding a hefty tip as she always did.\n\nAs she climbed the steps to her house, the square behind her was in shadow. A small enclave of forty red-brick Victorian villas, at one time these would have been solid working-class homes, but then people like Alma started moving in, attracted by the relatively reasonable prices and the proximity to town. Barely twenty minutes' walk to Grafton Street, Alma didn't even need to keep a car, choosing to make liberal use of taxis with impunity instead.\n\nShe raised her knee to her chest to prop up her handbag while she fished around for her front-door keys. By the time she was finally turning the key in the lock, the taxi was sliding out of the square. She pushed open the front door and tossed her keys on to the table inside. She stepped out of her shoes, and with her stockinged foot she gave the door a shove to close it, but the shove didn't take. It didn't take because there was someone standing behind her, holding the door open with his boot.\n\nHORROR ATTACK ON JOURNO was the banner headline in the evening paper the next day. BRAVE ALMA REFUSED TO GIVE THIEVES HER RINGS said a smaller headline on the inside page. And in even smaller print there followed the gory detail: TV STAR LOSES TWO FINGERS IN HORROR ATTACK.\n\n\"It's the familiarity with which they use your first name,\" said her mother. \"That's what I'd have a problem with.\"\n\nAlma's mother had a peculiar talent for tangents. She could discover tangents that no one else knew were there, seizing on them with a single-minded zeal, as if she alone had the power to see through to the heart of the matter.\n\n\"Whatever about the fingers,\" she said. \"The fingers you can live without. It's the familiarity, don't you see? They're on first-name terms with you. That's what I'd have a problem with.\"\n\nAlma looked to her brother first, then to her father for help.\n\nMacdara was standing over by the window. With his eyes roaming the ceiling, it was hard to know if he'd heard. Macdara was often adrift like this. No matter what room he was in, he seemed always to be looking for a way out.\n\nAlma's father was sitting to the left of her bed in a padded chair that doubled as a commode. He was wearing his customary daytime outfit. A nautical blazer, shiny from over-wear, with a handmade shirt that was thirty years old and scuffed at the cuffs and collar. A pair of white cotton trousers that rode high enough on his ankles to reveal a natty pair of turquoise socks. With his shock of white hair, he looked like an aging Captain Sensible: all he was missing was the parrot on his shoulder. He was sitting bolt upright in the chair, but his head had drooped to one side. A lick of hair had fallen across his forehead and his eyes were closed. Poor old boy, Alma thought. He had been the first to arrive. Woken by the early-morning call, he had burst into the hospital room barely twenty minutes later, unshaven and wild-eyed, the nurse trailing after him starstruck. If this was a test of love, then Alma's father had won it hands down.\n\nIt had taken her mother a good hour to arrive (even though her house was nearer, as the crow flies), and when she did come she was dressed for a performance. Her long gray hair was swept up off her face in great swirls that came together in a huge bird's nest on top of her head, and she was wearing her trademark black riding skirt with a purple silk blouse buttoned high up her neck. Thrown over it all was a black wool cape. She looked like she was going to a Bloomsday event.\n\n\"Darling,\" she said, as she bent over her daughter's bed.\n\nAlma found herself drowning in Yardley's Lily of the Valley. So her mother had even taken the time to douse herself in perfume.\n\n_Maman, nul points_.\n\n\"Who would you like me to ring?\" the nurse had asked Alma, as soon as she came out of the anesthetic. \"There was no ICE contact on your phone. We didn't know who to ring.\"\n\nThirst was Alma's first thought. Her mouth was so dry she couldn't swallow. Her next thought was that she was cold. A terrible cold, as if she'd been refrigerated from the inside out.\n\n\"Cold,\" she said. \"Very cold.\"\n\nThe nurse came and covered her with some extra blankets. She gave her a sponge stick to suck, which relieved the parched mouth but not the cracked lips. As her physical discomfort crashed over her like a wave, Alma became aware of her right hand, bulkier than it should be and swaddled tight by her side.\n\n\"How did I get here?\" she asked, noticing the curtained cubicle she found herself in. The large empty window to her left, with the daylight just starting to leach through the clouds. From somewhere outside the cubicle she became aware of noises. A clattering of tin; by the sound of it, someone was shaking a sack full of saucepans. The nurse was checking a bag of fluid that was hanging by Alma's bed. She bent down to write something on a chart, answering Alma's question without looking at her.\n\n\"You were brought in by ambulance.\"\n\nAs soon as she said it, Alma remembered. She remembered the ambulance men in their astronauts' boots traipsing across the bloodstained floor of her kitchen. She remembered worrying that they'd walk the blood into the hall carpet on their way back out. \"Darling,\" one of them had said to her, once they'd strapped her into the wheelchair. \"Now, darling,\" he had said as they lifted the chair down the front steps. Alma remembered being touched by the way he said it, as if she was his daughter, or his sweetheart even.\n\n\"Who called them?\" she asked.\n\nBut the nurse didn't answer. She was copying numbers down on to the chart from a digital monitor by the bedside.\n\n\"Any pain?\" she asked, and she paused to scrutinize Alma's face.\n\n\"No. No pain.\"\n\n\"It was Mr. Maguire who operated on you,\" said the nurse. \"He's gone home to get a few hours' sleep, but when he gets back he'll come and talk to you. In the meantime, is there anyone you'd like us to ring?\"\n\nThe first person who came to mind was Mick. Mick was the one to ring, naturally. Why did they even have to ask? She was just about to tell them when she remembered that she was no longer married to him. Hadn't been married to him for fifteen years. As she retraced the steps in her mind, it was with a vast sense of loss. She missed him, all of a sudden, as she had never missed him before. She could have cried with the pain of missing him.\n\nThe next person she thought of was Nora, but as soon as she thought of her daughter, she thought also of her absence. Pointless to even try and contact her.\n\n\"My mum,\" she said to the nurse. \"In my phone you'll find numbers for my mum and dad.\"\n\nThinking, Great! Fifty years old and my next of kin are my mum and dad.\n\nSo here they were, Alma's parents. Sitting on opposite sides of her bed, like an old married couple settled into their armchairs around the fire on a winter's evening. (Macdara had disappeared without anyone noticing.)\n\n\"Is he asleep?\" asked Alma's mother, nodding toward Alma's father.\n\n\"Who?\" said her father, snapping himself awake. \"Not at all. I was just dozing.\"\n\n\"It's the narcolepsy,\" said her mother, speaking to Alma. \"It's genetic. His father had it before him.\"\n\n\"Nonsense,\" said Alma's father, yawning. \"You're just jealous of my ability to fall asleep at will.\"\n\nAlma's mother sat up even straighter in her chair and without comment began to nudge any stray hairs away from her forehead, using spidery fingers. That famous hair; it lent her an air of theatricality that was by no means accidental. She liked to think she still belonged to the theater, even though it was thirty years since she'd been on a stage.\n\n_Mad old bag_ was the general impression she gave, whereas Alma's father was a _wonderful eccentric_. An inequity that was apparent to nobody more than to Alma's mother herself. \"Your father,\" she would say, \"seems to have achieved a kind of cult status in his old age. But I, on the other hand, remain a mere curiosity.\" And while all of his eccentricities seemed effortless (the disco dancing and the eye makeup, the clapped-out old Jag); while his multiplying oddities seemed more an unpeeling of his personality than an adornment to it, hers had a somewhat forced air about them, as if she was just hashing over a part she'd been playing for years.\n\n\"Oh,\" said Alma's father. \"I forgot. I have something for you from Sam.\" Shifting his weight, he began to feel around in the pockets of his jacket. \"I know I put it in here somewhere.\"\n\nAt last he located what he was looking for, tucked inside his breast pocket. A small, much-folded parcel made out of a single sheet of paper, which he passed across to Alma. Painstakingly using her left hand, she unfolded it and revealed a perfectly round coin of golden foiled paper; painted on the surface of it, in black ink, was the silhouette of a small bird in flight.\n\n\"Oh,\" she said, clamping it to her heart. \"I love it. Tell Sam I said thank you.\"\n\nSam, who you couldn't help but love, even though Sam, just twenty-four years old at the time and fresh out of Tangier, was the reason why Alma's parents' marriage had broken down. (\"Oh, that's rubbish, darling,\" Alma's mother had said once. \"It wasn't the affair with a younger man that ruined our marriage. It was your father's vulgar addiction to Hellmann's mayonnaise.\")\n\n\"How is he?\" Alma's mother asked now, with genuine concern.\n\nCheerfully, Alma's father bobbed his head.\n\n\"Oh, he's much the same. Although he's not happy with the new regime I have him on. I'm feeding him mackerel three times a day. Apparently it has miraculous healing powers.\"\n\n\"You make him sound like a sea lion,\" said Alma's mother, her eyes wide and startled. \"Of course the fish oil seems to be the cure for everything nowadays. That and the blueberries.\" She pronounced the word with great suspicion, as if it was a trick.\n\n\"Oh, yes. Sam and I are all for the blueberries. We eat them for breakfast, with granola.\"\n\n\"It's well for you,\" said Alma's mother, \"that you can afford to live on blueberries.\"\n\nNever did she let an opportunity go by to remind him of her impecunity. Alma laid her head back against her pillow and closed her eyes as they continued to bat back and forth across her bed.\n\n\"I'm afraid Sam's diet isn't the problem. It's the climate. He wasn't bred for this ghastly climate.\"\n\n\"We were none of us bred for this climate. If we were, we'd have fins.\"\n\n\"What was that Seamus Heaney poem?\"\n\nAnd so their conversation meandered from superfoods to Seamus Heaney. From Seamus Heaney to Yeats and from Yeats to book tokens and on to Greene's bookshop and the dapper little man who used to deliver their books for them, what was his name?\n\nAs Alma listened to them, it seemed to her that their conversation was like a deck of cards that had been well and truly shuffled and then dealt out, each card appearing at random but still familiar. The whole time she lay there listening to them, the one thing they did not discuss was the thing that had happened to Alma.\n\n\"Remember that scene in _Reservoir Dogs_?\"\n\nAlma's visitors nodded nervously.\n\n\"Well, it wasn't a bit like that.\"\n\nShe was sitting up against a bank of pillows, a black satin kimono jacket draped elegantly over her shoulders and the wounded hand lying discreetly by her side. She had managed to style her hair with some hand cream, which, incredibly, someone had brought her as a gift, but in the absence of her regular, ruthless blow-dry, her hair was taking its first tentative steps back to nature. Thicker and frizzier, it had the effect of softening her face. She had contrived somehow to apply some makeup, using her left hand and without the use of a mirror. The action of painting on her eyebrows, for so long a reflex, now had to be performed with careful deliberation. Even the application of lipstick, which with her right hand she could have done in her sleep, felt weirdly new. It was as if she'd been turned inside out.\n\nAlma's visitors were gathered around the bed, sitting in an assortment of ugly hospital chairs filched from other, less visited rooms. The air was thick with the fug of tiger lilies. The windowsills, the ledges, the surface of the bedside locker, all crammed with expensive floral arrangements. What with the heat in the room and all that foliage, it was like being in one of the greenhouses at the Botanic Gardens.\n\n\"Get a bottle of Prosecco out of the fridge,\" Alma told her visitors when they arrived, gesturing with her good hand toward the small fridge in the corner of the room. \"I've decided that Prosecco is going to play a key role in my recovery. Declan, you can do the honors.\"\n\nDeclan, who had started out in the newsroom on the same day as Alma. She remembers him, straight out of _The Tuam Herald_ , with his cowlick and his confirmation suit. He was head of news now, and therefore Alma's boss. Were she and Declan friends? She thought not. And yet he had come to visit her, all the same, out of duty most likely. And Pat Deacon and Mary O'Malley had volunteered, or more likely been volunteered. A reluctant delegation, they bore with them bags of small gifts sent by those lucky enough to be spared.\n\nDeclan poured the Prosecco into four ribbed plastic cups that he found in a stack on top of the fridge. Awkwardly, they all toasted Alma, keeping their eyes on her face so as not to look at the hand. They sipped at the Prosecco, with the plastic cups buckling in their hands, each of them secretly determined to leave their drink unfinished (Declan because he was driving, Pat because she had to chair the afternoon news conference and Mary because she had to read the two o'clock bulletin).\n\nThe door opened. A head poked around it, a head that for some reason always called to Alma's mind a raw chicken, or a lanky turkey. Free range, with that yellow tinge.\n\n\"You all know Jim,\" said Alma. \"Long-suffering editor of my newspaper column.\"\n\nJim was holding a bunch of pre-packed flowers, clearly from the Spar. (\"Follow the crowd,\" the nurse had said, when he asked her which room.)\n\n\"Hiya, Jim,\" said the others, shifting in their chairs to acknowledge his arrival.\n\n\"Well,\" said Jim, looking around for somewhere to put the flowers.\n\n\"Just put them in the sink,\" said Alma. \"They've run out of vases.\"\n\nJim settled himself against the scorching-hot radiator as Alma continued her story.\n\n\"I'm afraid I was unlucky in my attacker. I'm not sure he'd severed many fingers before. He made quite a meal of it. Funny, the bone was the easy bit. I suppose when you think about it, it makes sense. You know when you chop up a chicken? It's easy enough to cut through the bone, if you have a sharp knife. The skin is the tricky bit. There's always a bit of skin that just won't break.\"\n\nA horrified silence in the room.\n\n\"Oh, dear,\" said Alma. \"Is this hard to listen to? I'm so sorry.\"\n\nShe took a sip of her Prosecco, and flashed them a smile.\n\n\"Actually, I'm not a bit sorry. Pity about you.\"\n\nThey set down their plastic cups, glanced at each other to synchronize their departure and, like a flurry of startled birds, took off. Declan made the first move, but Mary and Pat jumped up after him and they walked out together, taking grateful gulps of the cold air outside. Conscious, each of them, of their ten movable fingers.\n\n\"She's got balls,\" said Pat. \"You have to say that for her.\"\n\n\"Ah, she's some woman,\" said Declan. \"She'd be great in a war.\"\n\n\"Nonsense,\" said Mary. \"All that bravado, it's the drugs talking. She must be up to her eyeballs in morphine. I hate to think how she'll feel when it wears off.\"\n\n\"God help her,\" said Pat. \"What a horrible thing to happen.\"\n\nThe morphine was heaven for the first twenty-four hours.\n\nAfter that, Alma began to feel imprisoned by it. She was struggling to distinguish between day and night. She was too high and she wanted to climb down. Whereas before she had thrilled in this soaring pain-free elation, now she wanted rid of it. She needed to face up to reality, whatever reality was. When her surgeon came by on his rounds, she asked him to take her bandages off. She wanted to see the damage to her hand.\n\n\"Are you sure?\"\n\nAlma was sitting on top of the covers, with her white La Perla nightdress pulled up high on her thigh to reveal two long and shapely legs. She was confident he would not notice the stubble on her calves, the plum-colored polish on her toenails that was starting to chip.\n\n\"Come on, Luke,\" she said. \"I'm a big girl. How bad can it be?\" (She had met him at a dinner party once and was determined to continue their relationship on the same footing, as a social acquaintance rather than a patient.)\n\nShe shifted herself up in the bed and he sat down on the edge of it. He looked her in the eye and Alma felt something unfurl inside her, something she recognized belatedly as a rustle of panic. She forced herself not to blink.\n\n\"Okay,\" he said, and without turning he spoke to the nurse who was hovering behind him. \"Jean, can you get me some scissors, please. I'm going to change Miss MacEntee's dressings for her.\"\n\n\"Careful,\" croaked Alma. \"That could be misinterpreted.\"\n\nEven to her own ears, the attempt at flirting fell flat.\n\nThe bandages came off easily, exposing first of all an entirely normal-looking thumb. Next to emerge was a relatively unscathed index finger; the only evidence of any injury was the crust of dried blood that had settled into the wrinkles on the finger, like the sediment left at the bottom of a glass of red wine. The fingernail had black blood caked into the cuticle, as if she had been digging in soil. Carefully, he peeled back the gauze to reveal what was left of the middle finger. It had been severed just above the first knuckle and the stump had been sewn up with what looked like barbed wire, the rounded tip of it blackened, as if the fingertip had been burned off. The ring finger, too, was blackened and criss-crossed with barbed wire. It had been severed just below the knuckle.\n\nAlma noticed that she was breathing out through her mouth. In through the nose, out through the mouth, the way they teach you in pre-natal classes.\n\n\"I warned you it wouldn't be pretty.\"\n\nShe nodded.\n\nHe removed the last bit of bandaging, and a perfectly intact, precious little finger emerged. Alma felt a rush of pity for it.\n\n\"Poor little finger. He's been left out on a bit of a limb, hasn't he?\"\n\nThe joke fell out of her, mis-spaced. Misjudged. Maguire didn't even smile.\n\n\"We had a job tidying it up,\" he said. \"Most of the amputations we see are power-tool-related, which makes them fairly straightforward, but yours was a bit trickier than that. We were three hours in surgery trying to reattach the fingers.\"\n\nHe reminded Alma of a mechanic discussing a tricky problem with an engine. She gave up any hope of flirting with him.\n\n\"You're as well off, actually,\" he was saying. \"Reattached fingers tend to be a bit troublesome. \"Believe it or not, you're as well off without them.\"\n\nIt was not true that Alma had refused to give her attackers the rings. She would have given them the rings in two seconds if she'd only been able to get them off her fingers. She'd have thrown the bloody things at them, and good riddance to them. They were a relic of her marriage, those rings. An ugly, oversized ruby that came down from Mick's grandmother, and an ostentatious clutch of diamonds he'd bought her with some cash he didn't want the tax man to get his hands on. When they'd split up, she had transferred the rings to her right hand rather than give them back to him. If she'd been wearing them on her left hand, her attackers would have chopped those fingers off instead and Alma would have been left with the use of her right hand. She'd have been able to put her makeup on properly. She'd have been able to write, for God's sake. In a way, she decided, this was all Mick's fault.\n\nTwo days since the accident\u2014why did everyone insist on calling it an accident? Two days since the _attack_ , and not a squeak out of him. He must have heard about it by now. He was in Brussels, for God's sake, not Ulan Bator. There was no way he wouldn't have heard. More than a year since she had spoken to him, but she imagined that maybe now, because of this, he would break the silence. What was she expecting, flowers? Who was she kidding? Even at the best of times Mick had never been a great man for the flowers. The first time he took her out to dinner, he ducked under the table to look at her legs. That was the most romantic thing he ever did.\n\nEven when Nora was born, there were no roses. No \"push gift\"\u2014isn't that what they call it now? A revolting notion, and anyway, Nora was born by C-section because she was lying breech. Alma had banned Mick from the operating theater, mortified by the notion of him seeing her with her belly sliced open. Mick had been relieved that she had taken the decision out of his hands. He waited in the pub around the corner until they sent for him, the phone ringing behind the bar; he said it was the only time he ever left a pint half drunk. Alma remembers how the midwife handed Nora to him, an awkward maneuver of the elbows, as if between them they were trying to make a cat's cradle. Mick bent down low over the baby and whispered something in her ear, something Alma couldn't hear. It was the start of a long father\u2013daughter conversation, one that might on the face of it have seemed to be driven by their differences. To Alma's eyes, they were more alike than either of them realized. The same stubbornness in each of them, even if it found a different expression.\n\nShe was always between them, Nora. Driving them apart, keeping them together. Hard to know which. For as long as Nora was a child, they were attached by their mutual attachment to her. And even after they separated, even after Mick went to Brussels and the house they had shared was sold, after Alma and Nora had moved into their new little house and the furniture was divided up, even after all of that, they managed to cooperate in a semi-civilized fashion on the parenting of Nora. But now that Nora was grown-up, now that there were no more school reports to discuss, no parent\u2013teacher meetings at which to present a ham-fisted united front, no graduation ceremonies to attend (one of them on either side of her, with Nora's arms binding them from behind, like a tight chain), now that Nora had drifted out beyond all the milestones, into the murky world she had chosen to inhabit, there was no reason for Mick and Alma to ever speak.\n\nThinking about Nora gave Alma an uneasy feeling, but she could not have said why. Her daughter was occupying her mind with an insistence that was almost physical. Not the Nora of recent times, but the Nora of long ago. Nora at twelve years of age, with her school kilt and her embarrassing budding breasts, and her big sad eyes. Had Alma noticed at the time that Nora was sad? It seemed to her now that her daughter had needed her then, and that she had not been there for her.\n\nOh, she'd been there, of course, she'd been there. She had driven Nora to school every morning and she had come home from work every evening, and they had sat down together to eat whatever meal the child-minder had prepared for them. The child-minder was like an aunt to Nora, she was like a granny to her\u2014Nora was better off with the child-minder, that's what Alma used to tell herself. Anyone could see that Alma wasn't cut out for the job.\n\n\"How do you manage as a single mother?\" she was often asked, in interviews for the lifestyle pages of the papers. And Alma had a stable of stock phrases that she rolled out in reply. \"Oh, I have lots of help,\" she would say, or \"It's a team effort.\" \"It takes a whole village to raise a child,\" she would say. \"I have the greatest admiration for anyone who manages to stay home with their children. I know I couldn't do it.\"\n\nAs she lay in her hospital bed, it occurred to Alma that what she was feeling was guilt. A guilt she had never before stopped for even a moment to entertain, it had her in its grip now and it wouldn't let go of her. With the light outside her window hovering between night and day and the hospital room cast in a bruised blue shadow, with her own voice ringing phrases from the past in her ears, phrases that ran loose in her head, banging the walls as they went and raising echoes to the rafters, Alma realized with great sorrow that she was alone in her hour of need, with no one to blame but herself. She could not call on her daughter now, no matter how much she might feel in need of her. She could not call on her because she had long ago forfeited the right to do so.\n\n\"I've been trying to track her down,\" said Connie, falling into the wing-backed chair beside Alma's bed, with her oversized handbag at her feet and her suede wedges turned inward at the toes. A green felt cloche hat sat on top of her beautiful Modigliani head. The hat's wisp of green netting hovered over her fabulous eyebrows. \"I think she might be on a ship, headed for Gaza. That was her plan, the last time I spoke to her.\"\n\n\"Oh, Jesus,\" said Alma. In her mind she was seeing a headline sequence. A ship at night, with a helicopter hovering overhead. Camouflaged figures wearing night-vision goggles, dropping down on to the deck. The crackle and flash of gunfire. Grenades, louder and brighter again. How many people were killed? Eight? Ten? What were the chances of it happening again?\n\n\"I think they may have sailed from Cyprus, or Greece. Nobody seems to know.\"\n\nAlma pictured a big top-heavy boat, bobbing about on the waves with no radar, no communications, no defenses.\n\n\"I've been trying all the numbers I have for her,\" said Connie. \"Her Irish mobile seems to have been disconnected. The last number she gave me was for a Greek mobile, but I've been ringing and ringing it and I keep getting some guy who doesn't speak any English. I'll try e-mailing her this evening. She may be checking her e-mails.\"\n\n\"Actually,\" said Alma. \"I'd prefer if you didn't.\"\n\nConnie turned her mouth upside down, as she always did when she was dubious about something.\n\n\"I don't see any point in telling her, Connie. If she's on a boat in the middle of nowhere, there's no point in her being told.\"\n\nConnie tilted her head and drew a breath in. She was about to speak when Alma stopped her.\n\n\"Connie, you're going to have to leave this to me. Nora is not to be told.\"\n\nConnie pursed her lips and knitted her eyebrows at Alma, to show that she was unhappy. Like a mime artist, her every feeling played itself out on her face.\n\nConnie of the stolen name, a name Alma had hoarded since childhood, stashing it in a secret place in her heart for the daughter she would one day have (at no point did Alma ever consider the possibility that she would have a son). The only person she confided the name to was her sister, and her sister stole it from her. When Acushla's daughter was born, three months to the day before Alma's, Acushla named her Constance. Furious and hurt, feeling betrayed and blindsided, Alma was forced to go back to the drawing board, and the name she came up with was Nora. In time her sister was to commit worse crimes against her (one crime in particular was far, far worse), but it was the theft of the name that rankled still, long after everything else had ceased to matter.\n\nIt seemed to Alma now that the theft of the name was a crime of lasting consequence. With that name went everything she would have wanted for her daughter, as if the name carried within itself the blueprint of a life. For it is Connie now, who is constant. Connie with her house and her husband and her two little boys. It is Connie who is here at Alma's bedside, while her own daughter is nowhere to be found. It is Connie who is even now unloading bottles of Prosecco from her voluminous handbag and stashing them in the fridge. Connie whom Alma loves, with an easy camaraderie that she has never been able to muster for so much as a single moment with Nora.\n\n\"Has my mum been to see you?\"\n\n\"Oh, yes,\" said Alma. \"She came yesterday. Briefly.\"\n\nAlma knew that Connie had a great desire\u2014a pathetic, childish desire\u2014for her mother and her aunt to be friends again.\n\n\"She brought that orchid,\" said Alma, gesturing with her freshly bandaged hand toward the window, where a spindly black stalk threw out perfect little bursts of magenta flowers.\n\nIt was so typical of Acushla to bring an orchid. She had brought a scented candle too, and a tray of out-of-season peaches, all from Marks & Spencer. Dressed in soft shell-pinks and mauves, she had perched on the edge of the visitor's chair, with her knees falling to the side and her handbag in her lap, like a woman in a waiting room. The space between the chair and the bed may as well have been a vast canyon; as the two sisters spoke, they were both afraid of falling into it. The air between them swirling with all the things they couldn't say to each other.\n\n\"The orchid shouldn't be in the window,\" said Connie, and she went to move it. \"Where will I put it?\"\n\nFor a moment Alma was tempted to tell her to take it home. But she stopped herself.\n\n\"Here,\" she said, clearing a space on her bedside locker. \"Put it here. It's so beautiful, it deserves pride of place.\"\n\nPleased, Connie set the orchid down on the surface of the locker.\n\n\"Well,\" she said. \"I'm afraid I have to dash. I've to collect the gruesome twosome from Mum's.\"\n\n\"All right, doll. Thanks for the supplies.\"\n\n\"No worries. I'll be back tomorrow. Let me know if you need anything.\"\n\nOn her way out of the door, she had to sidestep to avoid bumping into a small Oriental man walking behind an enormous bunch of roses. A jumble of different colors, there were yellow roses, and pink roses, and roses as red as cherries; peach-colored roses, and roses the color of Christmas clementines. The man deposited them on the table tray that hovered over Alma's bed and Alma foraged among the stems with her awkward left hand until she unearthed the florist's envelope. Ripping it open with her teeth, she found a small card bearing a four-line message, typed out in tiny letters.\n\n_You must turn your mournful ditty_\n\n_To a merry measure._\n\n_I will never come for pity,_\n\n_I will come for pleasure._\n\nAlma let her hand drop down on to her lap, with the florist's card still in it. She knew who the flowers were from, even though there was no name. She recognized in the gesture, three days late and already overblown, all the hallmarks of the film director she had been sleeping with, on and off, for the past few months. The very absence of his name, that was a vanity. The extravagance of the bouquet, that was another one\u2014he would have wanted to be certain that his flowers dwarfed everyone else's. The use of the secondhand sliver of Shelley from _Out of Africa_ , that was a vanity heaped on vanities. So she was Karen Blixen to his Denys Finch Hatton, was that what he was suggesting? Or was she Meryl Streep to his Robert Redford? The old Alma would have loved the comparison. The old Alma would have respected the lack of sentiment. But the last few days had changed her, and this was a new Alma, one who was saddened by the flowers, and made lonely by them. She would have preferred that he come and see her, instead of sending her more bloody flowers.\n\nFor the first time in as long as she could remember, for the first time since she was a kid, perhaps, Alma laid her head down and cried wet tears of self-pity that left skid marks of black mascara across the starched white surface of the hospital pillow.\n\nAfter three days in the hospital, Alma was told she could go home. Three days that had been provided to her more as a favor than a necessity\u2014the private wing of the hospital was half empty, so nobody was in a rush to send a fully insured patient home.\n\n\"I'm not sure I'm ready to go,\" she said, panicked at the thought of it.\n\n\"I can't justify keeping you here,\" said Maguire, sitting down companionably on the edge of her bed. \"Your hand is healing well. You've seen the physio, and the pain specialist. I'll see you in a week's time, in my private clinic. In the meantime, I'd recommend the liberal use of painkillers, whenever you need them.\"\n\n\"Couldn't I stay another day or two? I'd like to get some value out of that bloody insurance I've been paying all these years.\"\n\n\"Alma,\" he said, dipping his balding head and looking her sternly in the eye. \"You need to go home.\"\n\nAfter he was gone, she turned her head and stared out of the window. From six floors up, she had a view of the city below her. She could pick out the landmarks of her life, from the water tower at the college where she had once been a student to the transmission mast at the television station where she had spent her whole career. For all its familiarity, there was something about the scene that wasn't quite right. As if there was a kaleidoscope inside her head, one that since the attack had been twisted ever so slightly by an unseen hand, altering the patterns and colors that up until now Alma had been accustomed to seeing. In some weird way that she couldn't yet describe, the entire landscape of her life had become strangely unfamiliar to her.\n\nAfter the attack, all Alma's old columns came back to haunt her. There was the one about the Neighborhood Watch scheme (a license for men with small penises to exploit their vigilante tendencies). The one about Meals on Wheels (shoot me if I'm ever forced to eat an overcooked stew out of a Tupperware container). The demands of friendship (what's so awful about solitude?). After the attack, Alma was afraid to be alone after dark in her own home, but the men on her square were so well informed about her views that they resisted the temptation to check on her, for fear of causing offense. Alma found herself dependent on ready meals because none of the neighboring women dared to darken her door with a casserole. She was desperate for company, but most of her friends and colleagues assumed she'd prefer to be left alone while she recuperated. She found herself stewing in an isolation that was entirely of her own making.\n\nOn her first full day home, she forced herself up to Sandymount village, on the pretext of buying some firelighters. Although it was well into April by now, the temperatures were still low and Alma felt in need of the comfort of an open fire. As she made her way up Sandymount Road, she was as wobbly as a newborn foal. The air that swirled around her was strangely threatening, every sound sharply heightened. When she saw an old lady bearing down on her, bent almost double under the weight of two supermarket bags, she felt a curious stirring of fear at the prospect of an encounter. The old lady drew level with Alma, and paused for a moment.\n\n\"Good to see you out and about,\" she said approvingly, straightening her head but not her body.\n\n\"Thank you,\" said Alma, buoyed up by a pride that was out of all proportion, like a child who has been praised by a teacher.\n\n\"Now, my love,\" said the weary-looking lady behind the counter in the caf\u00e9, when Alma went in on a whim to order a cup of coffee. \"You sit down there and I'll bring it over to you.\" Alma unwound her scarf from her neck and slipped out of the sleeves of her coat, letting the warm clouds of steam from the coffee machine wrap themselves around her as she sat and waited.\n\nThere was a young woman sitting at the table next to her, with a baby asleep in her lap. The young woman was holding her cup of coffee out to one side to avoid any danger of spilling it on the baby, and she was staring into space, her eyes dull with a tiredness the coffee would not relieve. The baby was sleeping with his arms thrown out either side of him, back blissfully arched, belly rising and falling with his breath. Alma was reminded of when Nora was a baby. Nora used to fall asleep just like that, with her arms stretched out either side of her and her head thrown back, mouth open. For a moment it seemed to Alma that the baby might have been Nora, so deeply was her memory resurrected. She found herself overcome by nostalgia, a yearning in her for the sheer ordinariness of being a young mother again, until it occurred to her that she had never been an ordinary young mother. Determined to be extraordinary, she had gone back to work when Nora was three weeks old, with barely a thought for what she was leaving behind.\n\n\"Best time in your life,\" she heard someone say, and looking up she realized that the weary-looking waitress had arrived with her coffee. She too was gazing at the baby.\n\n\"Best time in your life,\" she repeated wistfully. \"When your children are small.\"\n\nAlma stared at her.\n\n\"You don't feel it going, do you?\" said the woman.\n\n\"No,\" said Alma. And it seemed to her all of a sudden that it was the truest thing. The truest and saddest thing.\n\nThe woman put a plate down in front of her.\n\n\"I thought you might like a scone, on the house. I've sliced it for you, to make it easier.\"\n\nAlma was about to decline the offer\u2014normally she would never eat a scone with her coffee\u2014but the comforting smell of the baking soda made her tummy hollow out with hunger, so instead she said thank you.\n\n\"You're very kind,\" she said, blinking her eyes several times in quick succession to stop them from welling up. The kindness of people, that was something Alma had never noticed before. She was profoundly affected by it.\n\n\"I've decided to give up my column,\" she told Jim that evening. She had phoned him rather than going in to see him. The thought of walking through that newsroom filled her with dread. She'd have sooner walked through a snakepit.\n\n\"Now don't rush into any decisions,\" he said. \"You're only just home. It may take you a while to get your strength back.\"\n\n\"It has nothing to do with my strength,\" said Alma. \"Physically, I feel fine. It's just that I can't bring myself to say anything nasty about anyone anymore. I feel like going to live in a bloody Amish community.\"\n\nHow could she explain it to anyone? In order to say all the mean and amusing things that she was expected to say in her column, it was necessary for her to feel that she was different to other people. But since the attack, it had begun to dawn on her that there was nothing different about her at all. She was just the same as everybody else.\n\n\"Come on,\" said Jim. \"Give it time. You'll soon be back to your wicked old ways.\"\n\n\"I wish I could be so sure,\" said Alma. \"Things were much easier for me before I discovered I was human.\"\n\nAfter the attack, or the incident as her mother insisted on calling it, Alma took six weeks off work. (Whenever she felt guilty about it, she reminded herself of that curtailed maternity leave.) Dawn broke earlier every morning, and the evenings were getting longer, but to Alma's surprise, she found that she had no trouble filling her time. She would rise early, making herself a mug of fresh coffee as usual, but instead of standing by the kitchen counter with the radio on as she drank it, she took to sitting beside the French windows in the quiet to savor it. The quiet was new to her and she found it surprisingly pleasant, the absence of all that noise.\n\nSummer had arrived, and with it a burst of color. After the longest winter anyone could remember\u2014a winter that ran right through the month of April and into May, so that people still had their heating on and their fires lit throughout the May bank holiday weekend\u2014the weather changed overnight. The sudden rise in temperatures caused all the summer flowers to bloom at once so that in the space of a single week you had the first tulips bursting open, along with the lilac and the early roses.\n\nIn Alma's garden, two little robins were building a nest where two walls met under the protection of a small weeping cherry tree. Alma found that she liked to sit and watch as the she-robin fussed about at the nest. The he-robin would make occasional trips from the nest to the wall, or from the nest to the top of the shed, returning with scraps of ivy in his beak, or small twigs, which he handed over to the she-robin. The division of labor, while unequal, seemed harmonious. There seemed to be no question between them but that she was in charge.\n\n\"Fair play to you,\" said Alma, raising her mug to them. \"You've got it all ironed out.\"\n\nAfter she'd watched the robins for a while, after she'd drunk a mug or two of coffee, Alma would go about opening the mountain of mail the postman delivered every morning. Day after day she received cards from complete strangers, cards that told of people's hardships and wished her fortitude in dealing with her own. There were letters from people she came across in the course of her work; the politicians in particular emerged as very diligent letter-writers. Some typed their letters, while others handwrote them; it was the fact that they had taken the time to write at all that Alma found so moving. It turned out they were all human beings after all.\n\nSome days Macdara came for lunch, bringing with him curious delicacies he'd unearthed in Lidl or Aldi, or in the Polish section of the local supermarket, delicacies like venison sausage or smoked eel. He would deposit them without explanation on the kitchen counter and Alma would unwrap them with all the discomfort of a cat owner who's been honored with a present of a dead mouse. But where the old Alma would have expressed her disgust in lurid language, the new Alma accepted these offerings in a spirit of quiet martyrdom, rather than risk hurting her brother's fragile feelings.\n\nHe and Alma would sit on either side of her impractical little kitchen table, with their strange lunch set out on a platter between them, and they would talk about small things. A sense of something delicate and tentative unfurling between them, something that, given time, might slowly take root in this space provided by their mutual unemployment. For the first time in a long time, Alma asked herself no questions about the mystery that was her brother's life. For years she had bombarded herself and anyone who would listen with angry queries about him. (What on earth does he do all day? How does he survive? Where's he going to live if we ever have to sell Mum's house? I hope he doesn't think we're going to look after him when she's gone?) Now, in her isolation, she found herself accepting her brother for what he was. A sweet if somewhat unexplained presence.\n\nAcushla came by with lunch other days, bringing enough expensive salads to feed an army. She never could stop; she was always rushing off somewhere else, with some other good deed to do. Alma tried to get her to take some of the salads away with her again, but Acushla wouldn't hear of it and Alma didn't have the energy to be arguing with her.\n\nHer mother was the other extreme. Bringing a leftover piece of salmon for the two of them to share, or a chicken leg that had been languishing at the bottom of her fridge, she would breeze in without warning, like a gust of wind that comes in through a briefly opened door, jumping up again just as suddenly to go. In her wake she always left a lingering smell of Elizabeth Arden face powder and mothballs, a smell that reminded Alma acutely of her childhood.\n\nAlma's father could not visit her for long, because of Sam, but once or twice a week, following a touchingly formal invitation, she would venture out to see them in their \"little shoebox\" (Alma's father refused to use the word \"apartment,\" on the basis that it was vulgar\u2014when his imagination failed him, he would revert to calling his place \"the flat\"). \"Would you join us for a spot of lunch in our little glass eyrie?\" he would say, checking always with Sam before he gave Alma a time. Alma was shocked to find how grateful she was for these invitations. During the daytime particularly, it was good to get out. Any invitations for the evenings she made a rule of turning down.\n\nShe would eat her evening meal at the kitchen table with a novel spread open on a wooden Prop-It, which allowed her to read while she ate. She would drink one or two glasses of red wine with her meal, so that by the time the light had begun to drain out of the garden she would be feeling mildly sedated. She would draw all the curtains in the front room tight against the night and put the chain across the front door, locking the door carefully with the new triple lock system she'd installed after the attack. Then, taking her mobile phone with her, and the rest of the bottle of wine, she would move upstairs, using the sound of the TV to drown out the nighttime noises in the house. All the footfalls and the tappings, the human and inhuman sounds that filtered through the walls from the houses on either side of her. Alma had never noticed them before. Now they scared the life out of her. She took to leaving the TV on all night.\n\nSix weeks Alma was off work, and in all that time she did not go to the hairdresser. She did not go for her customary weekly pedicure\u2014the manicures, for obvious reasons, she also canceled, allowing Connie to cut the nails of what fingers she had left. She watched with morbid fascination as, slowly but steadily, her toenail polish grew out with her toenails. Her hair lost its chestnut sheen as the color bled out of it and she studied her roots every day in the mirror, fascinated to learn\u2014for she had long ago forgotten\u2014what her natural hair color was. By the end of the six weeks she had a clump of dull brown hair, every third strand of it silver, and at the crown of her head an inch of brave new growth as shocking and white as snow. She studied this new growth in the mirror every day with a protective pride. She was like a man who, for the first time in his life, decides to grow a beard. She was fascinated by this as-yet-unexplored aspect of herself.\n\n\"Good God!\" said her hairdresser, when at last she appeared. \"You've gone snow white!\"\n\nHe scrabbled with his fingers in Alma's hair, looking up at the mirror every so often to gawp at her. He couldn't disguise his excitement.\n\n\"It must have happened overnight!\"\n\n\"Is it any surprise?\"\n\nTenderly, he cupped her head with his hands. \"Poor baby,\" he said, talking to her through the mirror. \"But don't worry. We'll come up with a plan of attack.\"\n\nAlma shook her head.\n\n\"No plan of attack. We're going with it.\"\n\nHis eyes flew wide.\n\n\"No!\"\n\nShe smiled. For the first time in weeks, she was enjoying herself. Stirrings of freedom moving through her. (\"A woman's hair is her crowning glory,\" Alma's mother used to always say. It gave Alma great pleasure to think what her mother would say when she saw this.)\n\n\"We're going for a crew cut,\" she said, addressing herself in the mirror. \"And the color stays the way it is.\"\n\nLooking at herself in the mirror, there was no doubt in her mind but that she wanted to do this. She could not go back into the world looking the same as she used to. What had happened to her had changed her so profoundly that she needed her appearance to reflect that change.\n\n\"Okay,\" said the hairdresser, looking at her with new-found respect. \"Okay. You go, girl!\"\n\nSlowly, and with great aplomb, he pulled the scissors out of his belt. With a bullfighter's flourish, he draped a plastic cape around her shoulders. \"Let's do it,\" he said, looking up at her one last time, just to check that she wasn't having second thoughts. \"Okay,\" he said, and he began to snip. Snip, snip, snip. Snip, snip, snip. \"Omigod,\" he said. \"I can't believe we're doing this.\"\n\nAnd when it was done, \"Omigod. You look STUNNING.\"\n\n\"Oh. My. God,\" said Nora. \"What happened to your hair?\"\n\n\"Nora! Where on earth are you? We've been worried about you.\"\n\nOn the screen, Nora was all distorted. Her face was too small, her eyes too big. She looked like a woodland animal who has stumbled upon a secret camera and is inspecting it, oblivious to its purpose.\n\n\"Nora!\" said Alma. \"Can you hear me?\"\n\nNora's face froze, disappeared, and then appeared again in a slightly different position.\n\n\"What happened to your hair?\"\n\n\"Oh, I had a bit of an accident.\"\n\nThere she was again, using that word.\n\n\"What kind of an accident?\" asked Nora. \"An accident to your _hair_?\"\n\n\"No,\" said Alma quietly. \"No, no. An accident to my hand.\"\n\nNora seemed confused.\n\n\"But you're all right?\" she asked.\n\nAnd Alma wanted to say, No! No, she wanted to say. No, I am not all right! But watching Nora's waiting face, she could not bring herself to do it.\n\n\"I'm grand,\" she said. \"You know me. I'm pretty tough.\"\n\nLike a dieter who refuses a chocolate they have been offered, or a teetotaler turning down a drink, Alma had a deeply satisfying sense of her own self-restraint. A sense of pride that she had not told her daughter what had happened. A sense of regret, too, that the moment was gone and could not be brought back.\n\nNora nodded, relieved.\n\n\"You look good,\" she said. \"The hair suits you.\"\n\nIt was hard to tell in the darkness, but Nora looked suntanned.\n\n\"Where are you?\" asked Alma. \"You look like you've been out in the sun.\"\n\nNora tilted her head to one side and her hair fell across her forehead. Using a gesture of her father's, she pushed it away.\n\n\"I can't tell you where I am,\" she said. \"But don't worry, I'm perfectly safe.\"\n\n\"Well when are you coming home?\"\n\n\"Soon,\" said Nora. \"I just have to do something first, and then I'll be home.\"\n\nAlma had a sudden urge to cry. She felt a swell of tears rising up in her like vomit. She had to gulp to force them back down her throat.\n\n\"Nora!\" she said, unable to keep the panic out of her voice. \"You mind yourself, all right? I want you to promise me that you'll mind yourself.\"\n\n\"Don't worry,\" said Nora. \"I'm fine. Look, I'll talk to you soon. Give my love to everyone.\"\n\nAnd Alma was about to send a message of love to Nora\u2014it was in her mind to tell Nora that she loved her, but the connection was already lost.\n\n\"Alma!\" said her boss. \"Jesus.\"\n\nShe kept forgetting what a shock it was to people to see her without her hair. Without her hair she looked like a different person. Her skin tone had changed, that was the first thing. Whereas before it had had a pale, ethereal quality in relation to her hair color, now it looked positively pink. Her eyes were brighter, their blue more intense, and she had taken to wearing less makeup. The effect of this was to make her look younger. Of her old look, the only thing she had retained was the red lipstick.\n\n\"Alma,\" said Declan, jumping up out of his chair and sticking his hand out to grasp hers before realizing his mistake and whipping it away. (\"Use the hand as normally as possible,\" the doctor had said when he took her bandages off, but Alma found this was easier said than done.)\n\n\"Alma. You look great!\"\n\nHe leaned forward to kiss her. His mind would be whirring, she knew. Wondering could he put her back on TV looking like this? Wondering could he stop her going back on TV, or would that constitute discrimination? Poor Declan\u2014since he took on this job he'd had an employment appeals tribunal always biting at his heels. The old Alma would have reveled in his misery.\n\n\"Welcome back,\" he said, with his head stretched back on his neck as he opened his arms to her. \"It's great to have you back.\"\n\nShe allowed him to hug her, hoping he wouldn't notice that she was shaking.\n\n\"Declan,\" she said, as she sat down. \"Here's the thing. I can't go back to being on the telly. I want a move to radio.\"\n\nFair play to him, he made a decent fist of disguising his relief.\n\n\"I don't think you should make any rash decisions,\" he said. \"Why don't you settle back in first? Take a few days to catch up.\"\n\nShe cut him off with a shake of her head.\n\n\"I don't want to be in front of the camera anymore,\" she said, trying to be as direct with him as she could without giving too much of herself away. \"The telly's a young person's game, you know that, Declan. I've held the line for long enough, but it's a losing battle and I don't have the energy to fight it anymore.\"\n\nHe looked at her with a puzzled expression on his face.\n\n\"Are you trying to tell me that the telly's no place for an older woman? You're the very last person I would have expected to say that.\"\n\nShe held his gaze and smiled.\n\n\"Yeah, well, vanity prevails,\" she said. \"I don't want the nation to watch me aging under those studio lights.\" (When the truth was that she was afraid the nation would see something else\u2014something she would not be capable of disguising for the cameras. She was afraid she would not be able to conceal her poor, bruised soul.)\n\n\"Look, Declan. The truth is, I'm too fragile for the telly. What happened to me is written all over my face. I can't hide it.\" She could hear the wobble in her own voice and it terrified her. She uncrossed her legs, leaning forward over her knees and fixing her eyes on his. \"Believe me, you don't want me on the telly. I'm like Princess bloody Diana. Too much empathy. It's not pretty.\"\n\nThis was supposed to make him laugh, but it didn't. The expression on his face was one that Alma didn't recognize at first, so unfamiliar was it to her. It was the one thing she could not bear, to be pitied. She reached down deep inside herself and tried to access something of her old, lost bravado.\n\n\"Come on, Declan, it's not like anyone died. I'll only be moving across to the other side of the newsroom.\"\n\nHe nodded. His expression funereal.\n\n\"Sure,\" he said. \"It just feels like the end of an era.\"\n\n\"Ah, now. Let's not get too dramatic. Why don't you buy me a cup of coffee in the canteen and we'll come up with a plan for the next stage of my career.\"\n\nHe stood up, fumbling his coins around in his pocket. Already his mind had turned to plugging holes in his rosters.\n\n\"Actually, it so happens that I'm short a newsreader for the radio bulletins this afternoon. But of course, if it's too soon...\"\n\nVery quickly she made the calculation that she would be better off getting it over with as soon as possible.\n\n\"Why not?\" she said. \"I'll have to get back on the horse sometime.\"\n\nBut no sooner had she said it than the panic began to rise in her. A lump of nausea formed in the back of her throat and she swallowed it back down, moving ahead of Declan toward the door of his office. Swinging her handbag over her shoulder, she planted one high-heeled foot in front of the other with exaggerated precision. She let her hips swing from side to side, attempting to muster a display of her old swagger. Without her hair to toss around, it just didn't feel the same.\n\nAll morning a steady stream of people made their way over to Alma's desk to welcome her back. Tentative some of them, intrepid others, they all avoided looking at her hand, admiring her hair instead, with varying degrees of conviction. Mary O'Malley arrived into the office with a small bunch of pink roses she'd picked from her own garden, their stems wrapped in damp kitchen paper. She settled them in front of Alma's computer, using an empty smoothie bottle as a makeshift vase. One of the young radio reporters brought her back a Kit Kat from his coffee break, laying it down beside her keyboard without a word as he passed. And Ray O'Donnell\u2014fellow news anchor and longtime nemesis of Alma's\u2014came in the door backward, carrying a wet umbrella and a laptop, and the first thing he did was walk over and kiss her on the top of her head.\n\nBefore the attack, none of the young reporters would have dared to approach Alma, no colleague would have dreamed of kissing her. There could have been no clearer indication of how it had changed her, not only in her own eyes but in the eyes of everyone else too. She felt miserably undeserving of people's kindnesses, like someone who has arrived without a present at a party but is made to feel welcome nonetheless. In the gentle attentions of her colleagues Alma saw only reflections of her own unworthiness. It was a relief to her when the time came round for the afternoon news conference.\n\nShe arrived just as the conference was starting, making for the last empty chair at the far side of the table. As she sat down, the news editor pushed a news list toward her. Was she imagining it or did he look at her nervously? She glanced around the table but everyone had their heads buried in the list.\n\n\"Okay,\" he said. \"We've got unemployment figures out today. The Minister is available. Likewise all the usual suspects.\"\n\nAlma kept her eyes on his face. Once or twice, as he rattled through the list (ward closures in one of the capital's main hospitals, a gangland murder, another rise in the jobless figures), he looked up at her, as if in anticipation of something\u2014but what?\n\n\"Brussels,\" he said, reading the next slug off his list. \"This is a story about our European Commissioner, Michael Collins.\"\n\nThere was a shift in the air pressure inside the room. All those lungs full of carbon dioxide that everyone was afraid to breathe out. A current of excitement like the atmosphere inside a classroom when the monotony of the day is about to be broken by a pre-planned prank.\n\nValiantly, the news editor plowed on.\n\n\"Someone has posted a video on YouTube that appears to show the Commissioner stealing a pepper grinder from a restaurant table in Brussels.\"\n\nOh, the pleasure in the room; it was like a sudden release of gas. Legs were uncrossed, spines unfurled, shoulders rolled back. An atmosphere of lip-smacking malice was unleashed.\n\n\"Hang on now,\" said the news editor, with his hand up to still the room. \"We need to think this one through carefully.\"\n\nAll eyes on him.\n\n\"What's to think through?\" asked one of the program editors.\n\n\"Well, we need to be fair about this. We need to check the authenticity of the footage. Make sure it wasn't a setup.\"\n\n\"Where did it come from?\"\n\n\"Someone with a mobile phone.\"\n\n\"And we're sure it's him?\"\n\n\"Oh, it's him all right.\"\n\n\"Well feck it, I don't know about anyone else, but I'm running it.\"\n\nDespite a few minor quibbles, there was general agreement. What with everything else that was going on, everyone agreed that the nation could do with a bit of light relief.\n\nAlma watched the footage, for the umpteenth time, on the nine o'clock news. She was tucked up in bed, the dregs of a glass of wine beside her on the bedside table. Fair play to Ray O'Donnell, he managed to read the intro with a straight face.\n\n\"Ireland's European Commissioner, Michael Collins, finds himself at the center of a media storm tonight following the publication on a video-sharing website of mobile phone footage that appears to show him removing a pepper grinder from a table at a Brussels restaurant.\"\n\nThe footage showed a pavement caf\u00e9. White tablecloths, with wicker chairs under blood-red umbrellas. Mick was sitting alone at a table with a coffee cup in front of him. He was wearing a dark-colored overcoat, a yellow wool scarf doubled round his neck. (That's what a decade in Brussels does to a man, thought Alma. Before he went to Brussels, Mick would never in a million years have been seen dead in yellow.) The footage showed the waiter approaching the table with the bill on a small steel tray. Mick dug into his pocket and handed him some notes. The waiter dipped into his apron and gave Mick some change, then moved off to another table and Mick was alone again. He leaned forward, elbow on the table, and lifted the pepper grinder an inch off the surface, using his hand as an upside-down claw. He turned the head of the grinder around once, twice, three times, with his free hand cupped underneath it to catch the pepper grounds, which he examined closely. He looked up briefly to check if anyone was watching. Then, with a shifting of the shoulders, one up and the other down, as if they were scales, he slipped the pepper grinder into his pocket.\n\nAlma watched as the TV showed exterior shots of Mick's apartment, with the curtains drawn. They showed the gate to an underground garage, resolutely closed. They showed a close-up of a doorbell, and next to it a small typewritten label: _Monsieur M. Collins_.\n\n\"Every attempt has been made to contact Mr. Collins,\" went the voice-over. \"But so far all our efforts have been unsuccessful. The Commissioner's whereabouts are this evening unknown.\"\n\nAlma reached for the remote control and turned the television off. The quiet that followed had a sound of its own, which she found herself listening to. It was the sound of her house making its presence known to her. The sound of her own blood pounding in her ears. As she lay there listening, she heard another sound, one that it took her a moment to recognize. It was the sound of her doorbell, winding its way through the dark and empty rooms downstairs.\n\n# WIKIPEDIA\n\n**Michael \"Mick\" Collins** (born 9 March 1956) is the European Commissioner for Internal Market and Services and a former Irish government minister. He was first elected to D\u00e1il Eireann in February 1982 and held his seat in Tipperary North until 2002, when he was nominated by Taoiseach Bertie Ahern to serve as Ireland's European Commissioner. A brother of the politician and former minister Liam Collins, he was married to the Irish television presenter Alma MacEntee, with whom he has a daughter, Nora. They separated in 1998 and subsequently divorced. In April 2013 he became a YouTube sensation when mobile phone footage showed him removing a pepper grinder from a table in a Brussels restaurant. The footage went viral and within twenty-four hours had been viewed by more than twenty million people.\n\n#### Contents\n\n1. Early and personal life\n\n2. Early political career\n\n3. Period in government\n\n4. Brussels\n\n5. Pepper grinder incident\n\n# Mick\n\nIt was hard to know which of them was the more shocked.\n\nShe was sitting on the edge of her armchair, her velvet robe bound tightly across her chest, her bare feet balancing on the points of her toes. Mick kept looking from her face to her hair, her hair to her face. The face, he knew. Even without her makeup, he knew her face. But the hair belonged to someone else, and he couldn't get used to it. On the doorstep he had wondered for a second had he arrived at the wrong house. Until she said, \"Mick! Jesus. What are you doing here?\"\n\nHer face was thrown into shadow by the glare of the lamp she'd turned on in the corner of the room. She had offered to light the fire and he'd said, \"Ah, no, sure it's not that cold. I'd kill for a glass of wine, though.\" She had handed him the bottle first, then the opener.\n\n\"So,\" she said to him, once they'd settled themselves on either side of the barren fireplace. \"Why here?\"\n\nHe'd been asking himself the same question.\n\n\"The truth?\"\n\nEven as he said it, he was wondering did he have it in him to be truthful. He could tell her that she was the first person he'd thought of, except that wasn't true. He had thought of his brother first, and then he had thought better of it. His brother would not welcome him arriving on his doorstep.\n\n\"The truth is there was no one else.\"\n\nShe came down off the points of her toes, crossed one leg over the other and sat back into the armchair.\n\n\"I'm not sure whether I should be amused by that, or annoyed.\"\n\nMick leaned forward, resting his forearms on his knees, so that he was looking up at her. He adopted his most earnest expression.\n\n\"You were the only one I could trust. I figured you wouldn't have the heart to turn me away.\"\n\n\"Trust,\" she repeated, following the word up with an inward hiss of her breath. He liked the way she did that, the sharp intake of breath, as if she was taking a drag on one of the cigarettes she had long ago given up.\n\n\"You invited me in,\" he said. \"You could have turned me away.\"\n\nShe let a reluctant laugh out of her, a laugh like a yelp.\n\n\"And miss out on hearing the story from the horse's mouth? Are you mad? You knew my curiosity would get the better of me. You old bollocks.\"\n\nHe laughed himself now. Leaned his head back against the headrest, content just to watch her. She had drawn her legs up on to her chair, her bare feet tucked in under her, like a praying saint. She reached her right hand out for her wineglass, gripping it by the stem with her thumb and her index finger.\n\n\"Does that cause you pain?\"\n\nShe hesitated, and he found himself wondering why. The question wasn't difficult. Maybe she was thinking that he didn't deserve an answer. Maybe she was thinking, Fuck you, it's a bit late now to be asking.\n\nShe transferred the wineglass to her left hand and held the right hand out in front of her, studying it. From where he was sitting, it looked like she had her two middle fingers clenched in toward her palm\u2014that old trick you play on a child to let on that you're missing a finger.\n\n\"The pain I can live with,\" she said, in a voice that sounded a little distracted. \"The pain can be solved, with enough painkillers. The look of it is a bit unsightly, but I'm getting used to that too.\"\n\nShe turned the hand over and looked at it from the other side, as if she was seeing it for the first time.\n\n\"It's the fear,\" she said. When she looked up, her eyes were surprisingly unguarded. \"The fear is very bad.\"\n\n\"I'm so sorry,\" he said. And really, he meant it. He felt an awful swell of guilt rise up in him as he contemplated the ordeal she had been through. He had heard about it, of course, he had heard all about it. But he had not thought about it, not beyond his initial revulsion. He had not thought about the effect it would have on her, imagining that she would weather it with her usual resilience, that she would scorn his sympathy. But a thing like that would change you, how could it not? What was wrong with him that he hadn't been in touch with her?\n\n\"Did they catch them yet?\" he asked. \"Any arrests? Any idea who they were?\"\n\nShe shook her head. Over and over again she shook her head at the questions he asked of her, but she did not make so much as a single sound.\n\n\"I'm so sorry this happened to you,\" he said.\n\nShe continued to shake her head impatiently as she tried to explain.\n\n\"Do you know, I never used to notice the sound of sirens before? When I was in the hospital, I started hearing ambulance sirens day and night. And I thought it was just because I was in hospital that I was hearing them. But I still hear them now; there seem to be sirens going off all the time. And every time I hear one, I can't help thinking about the poor person inside. I'm like one of those old ladies who bless themselves whenever an ambulance goes by.\"\n\nShe was looking not directly at him but somewhere off to the side of him. Her attention seemed to be focused on something inside herself, and it was only when she had it in her sights that she turned her eyes on him.\n\n\"You see, you become aware of all the bad things that are happening out there, and the worst of it is that you know for a fact that the bad thing can happen to you. I will never again be able to live under the illusion that the bad thing won't happen to me.\"\n\nHe stared at her, struggling to take in this new version of the woman he'd married. In his mind he had a vision of her on the night they met. Twenty-one years old and precociously fearless, in her knee-high boots and her white skirt suit, an old cravat of her father's knotted tightly around her neck. She had interviewed him at the count center, the two of them standing high above the crowds on a platform of bare boards, supported by builders' scaffolding. Once the camera lights had been turned off he had leaned in and whispered in her ear, \"Let's pray the cameraman shot us from the waist up. We don't want the whole nation knowing that the new TD for Tipperary North had a boner for his interviewer.\" She'd thrown her head back and laughed out loud, a shameless laugh that convinced him then and there that she was the woman for him.\n\n\"Jesus,\" he said, looking at her now. \"I'd string them up for doing this to you. I swear to God, Alma, I'd hold them down and cut off their balls, one by one.\"\n\nShe smiled and rolled her eyes at him.\n\n\"Thanks, Mick, but I suspect you're in enough trouble as it is.\"\n\nHe nodded. Allowed her the opening.\n\n\"So? What have you got to say for yourself?\"\n\n\"I don't know. What do you want to call it? A moment of madness?\"\n\n\"Ah, now. What is it you earn? Two hundred grand? Three hundred grand? What possesses someone who earns three hundred grand a year to steal a five-euro pepper grinder? That's what all the papers are going to be asking. You might want to come up with a half-decent answer.\"\n\nHe adopted his most reasonable voice.\n\n\"It's not that easy to find a decent pepper grinder. The one I have in the flat is broken. I keep meaning to buy a new one but I never seem to have the time to get to the shops.\"\n\nShe was shaking her head in disbelief.\n\n\"You're going to have to come up with something a bit better than that, Mick.\"\n\nWith the knuckle of his index finger he began to worry at his lip. Nudging the lip with his knuckle, it was a tic of his. An indication that he was rattled.\n\n\"I know,\" he said. \"I know.\"\n\n\"They're going to have a field day.\"\n\nHe squirmed in his chair. He put his hands over his face and began to stretch the skin away from his forehead, as if his face was a latex mask that he could pull off. He was talking through the gap between his hands.\n\n\"You might try not to sound quite so pleased.\"\n\n\"It's not that I'm pleased. I feel sorry for you, Mick, just as you feel sorry for me. So here we are, feeling sorry for each other. Who would have thought?\"\n\nHe took a long slurp of his wine, resting the glass on his knee while he studied her. \"You're looking good, girl. Despite everything, you look good.\"\n\nAnd she did. To him, she looked bloody good.\n\n\"You're a dirty liar. I look old, is what I look. I look like a washed-up old wreck.\"\n\n\"You're barely fifty, Alma, for God's sake. You're looking pretty good for fifty.\"\n\n\"Fifty-three.\"\n\n\"Ah, come on. You were always able to take a compliment. Don't stop now.\"\n\nHe reached out and picked the bottle of wine up off the cold hearth. Poured the last of it out between their two glasses.\n\n\"What about Nora?\" she asked him. \"Have you heard from her?\"\n\n\"Not a whisper. And you?\"\n\n\"The last time I spoke to her, she wouldn't tell me where she was,\" said Alma.\n\n\"Oh, I know where she is all right. I make it my business to know where she is. She's in Egypt. Having failed to get into Gaza by sea, she's now trying to tunnel her way in through Egypt.\"\n\n\"Don't they bomb those tunnels?\"\n\nHe shrugged.\n\n\"From time to time.\"\n\nHe did not mention that the tunnels had been bombed as recently as December. He did not mention that the tunnels collapse, too. That the plywood that holds up the walls sometimes comes crashing down in a rain of rocks and rubble and shoddy building materials and that people get buried alive under there and that they die that way, unless someone can dig them out in time. None of this did he mention to Alma, even though he had made it his business to know it.\n\n\"I take it she doesn't know what happened to you.\"\n\nAgain, she shook her head.\n\n\"We're not in the habit of telling each other much.\"\n\n\"No,\" he said, by way of agreement, and they lapsed into silence for a moment.\n\n\"Do you think she's planning on ever forgiving us?\" he asked suddenly.\n\n\"For what?\"\n\n\"For fucking up her life. Isn't that what this is all about? She's punishing us for what we did to her.\"\n\n\"I'm relieved to hear there's two of us in it,\" she said. \"I always thought it was just me she was punishing.\"\n\nShe sighed, and he saw something he had never seen in her before. A weariness of spirit; it was the first time he had ever known Alma to be tired.\n\n\"I miss her,\" she said simply.\n\n\"Yeah,\" he said. \"So do I.\"\n\nShe rose out of her chair. \"Well, her room's empty. I'm sure she won't mind you sleeping there. Top of the stairs, it's the door straight ahead of you. Bathroom's on the landing. There's a spare toothbrush in the cupboard under the sink. Don't even think about using mine.\" (That was always a bone of contention during their marriage, her suspicion that he used her toothbrush.)\n\nShe yanked her robe even tighter across her chest in a matronly display of modesty. What did she think he was going to do, try and jump her?\n\n\"All right,\" she said. \"I'm off to bed. Good night, Mick.\"\n\nHe stood up. Leaned in and gave her a peck on the cheek, careful not to lay a hand on her.\n\n\"Good night,\" he said. \"And Alma... thanks for letting me stay.\"\n\nShe turned out into the hall and, using the banisters to support her, climbed the stairs. The smell of her perfume stayed in the room behind her\u2014or was it just the memory of her perfume?\n\nMick sank back down into the armchair. He let himself fall against the back of the seat and closed his eyes. It was a long time before he opened them again, and when he did, he realized that his fingers and toes were numb. Barely able to summon the energy to move, he hauled himself up out of the chair and crept upstairs, anxious not to disturb her.\n\nHe woke early, not knowing where he was. The bed was harder than the one he was used to, the pillows softer, and he wondered for a moment was he in a hotel? So many hotels in his life. Hotels and planes, hotels and trains; sometimes he had to check his location on his BlackBerry just to remind himself where he was.\n\nOh, fuck. Now that he remembered, he wished he hadn't.\n\nHe turned over and buried his face in the pillow, enjoying the sensation of smothering himself. His airways blocked, his eyelids forced shut, he listened to the sound of his own struggling breath, heard from inside his own head. At last, gasping for air, he flipped himself over so that he was lying on his side, facing the window.\n\nThe curtains were open and the day was dawning a pale, powdery pink. From this angle, the stadium was in profile. The great western rim of it like the edge of a gladiator's helmet, its glass tiles angled to reflect the sky. Seen from this perspective, it was a beautiful sight.\n\nOutside the window, a telephone wire bounced up and down in the breeze. A row of small birds gathered along it, one of them tilted this way, one tilted that way. Every so often one of the birds would fly away and then return to the wire. Watching them, it seemed to Mick that they were holding a wordless conversation, a dialogue they were acting out without sound. From behind the stadium a train appeared. It crept along like a mouse scurrying under a giant's sleeping head. A moment later it had disappeared, leaving a new stillness in its wake.\n\nMick rolled over on to his back and studied the posters on the wall in front of him. There was one of _The MotorcycleDiaries_, Che Guevara riding pillion with his arms stretched out either side of him. A poster of _Alive_ , which was Nora's favorite film when she was a kid. She must have watched that film a hundred times, and forced Mick to watch it with her. Staring at the poster, all Mick could think was that he missed his daughter. He missed her not as she was now, but as she was then. He missed all the moments that he had missed of her life, this room of hers that he had never even seen, and this view from the window that was her view every morning when she woke up. He missed all the things that he had missed and that could not now be brought back, no matter how much he missed them.\n\nThe smell of coffee brought him downstairs. Alma was standing by the open French doors, coffee cup in hand, facing out into the back garden. She gave no indication that she'd heard him come into the room\u2014for a moment he wondered had she not heard him\u2014but then she started talking, without so much as turning her head.\n\n\"I've a robins' nest in my garden,\" she said, in a faraway voice. \"Yesterday there were only three eggs in it but today there's a fourth. If you stick your head out the bathroom window you can see them. I don't like to go out there in case I disturb them.\"\n\nHe pulled out a chair and sat himself down at the kitchen table. He was wearing trousers and a shirt, no sweater, no socks. The tiles were cold on his bare feet and he nestled the sole of one foot against the other for warmth.\n\n\"Good morning,\" she said, turning to face him and speaking in an entirely different tone of voice.\n\n\"Good morning,\" he said.\n\n\"I'm afraid there's no waitress service in this house. Coffee's on the hob. Bread in the bread bin. Butter and jam on the counter.\"\n\nHe smiled in recognition of an old pattern. Any tenderness that ever existed between them by night, it would have vanished by daylight. He stood up, poured himself a cup of coffee, and sat back down at the table.\n\n\"I've some light reading for you,\" she said, throwing a copy of _The Irish Times_ down in front of him. \"Bottom of the front page.\"\n\nHe scanned the headline and let his head flop down on to his chest.\n\n\"Listen, you don't have a BlackBerry charger, do you? My phone's gone dead.\"\n\n\"Probably just as well. You might want to think twice about turning it on.\"\n\n\"Sure, but I need to ring my office.\"\n\n\"Here.\"\n\nShe passed him her own phone. \"Now,\" she said, carrying her cup and saucer over to the sink. \"I need to get into the shower.\"\n\nHe made himself two slices of toast that he slathered with apricot jam, the sound of the power shower in the bathroom upstairs like the roar of an aircraft taking off overhead. He had to wait for the noise to stop before he rang his special adviser. It was only when he went to dial the number that he realized he didn't know it. Didn't know his office number either, didn't know any of the numbers. He had them all in his BlackBerry.\n\n\"Fuck,\" he said, trying to remember how people used to find phone numbers before they had mobile phones, trying to remember how he used to do things before he had advisers. He looked around the room and spotted an iPad on the kitchen counter. He powered it up and with a fumbling forefinger stabbed the icon for the Internet connection. It took him ten minutes to navigate his way through the Commission website to a number for his own office. Another five to negotiate the call answering system. Noises upstairs as Alma moved from room to room.\n\n\"Jesus,\" he muttered. \"How the hell are people supposed to contact us?\" Until it occurred to him. Sure, that was the whole point.\n\n\" _Bonjour_ ,\" said a woman's voice at last. \" _Bureau du Commissaire Europ\u00e9en pour le march\u00e9 int\u00e9rieur et les services_.\"\n\nIt was a point of pride with Mick never to speak French. Fucking Frogs; it was bad enough to have to march to their tune without speaking their bloody language too. At least the Germans spoke English.\n\n\"Good morning,\" he said, with all the authority he could muster. \"This is Commissioner Collins speaking. I need to speak to Feargal McCarthy immediately.\"\n\n\" _Un instant_.\"\n\nShe put the call through straightaway. That in itself was suspicious. It was as if they were waiting for him to call in. He imagined the whole office falling silent, everyone frozen in motion as Feargal picked up the phone.\n\n\"Mick! Where are you?\"\n\n\"Never mind that. Now, what's the score?\"\n\n\"Well. We're under siege. We've had every media organization in the world on to us already this morning. We've had requests for interviews from every TV and radio station on the planet. We've got our own government looking for an explanation. And we've got the Commission president breathing down our necks.\"\n\nNasal at the best of times, at moments of excitement Feargal's sinuses tended to seize up on him and his voice took on a distinctly adenoidal pitch. Mick moved the phone away from his ear just a fraction to lessen the impact. The difference this made was surprising and extremely pleasant. Rather than coming out of the handset loud and clear, as if he was in the room with him, Feargal's voice now seemed much further away and therefore less real.\n\n\"We're putting out a holding statement. You know, it was all a misunderstanding, et cetera et cetera...\"\n\n\"Good,\" said Mick. \"Good man.\"\n\n\"You've no public engagements today or tomorrow, so that buys us a bit of time.\"\n\n\"Good stuff.\"\n\n\"If we could meet up maybe, somewhere other than the office\u2014\"\n\n\"Ah, feck it, I think the battery on my phone's about to go...\"\n\n\"Is there another number\u2014\"\n\n\"What's that? No, can't hear you. I think the phone must be running out of juice. I'll tell you what. I'll try you again later. In the meantime, just keep on doing what you're doing, good man.\"\n\n\"But wait\u2014\"\n\nTaking the phone away from his ear altogether and using his thumb as you would to flick a switch, Mick closed the call down. When he looked up, Alma was standing in the kitchen doorway in a belted black dress. With her high heels on, and her red lipstick, she looked almost like her old self.\n\n\"This thing seems to be gathering a bit of momentum,\" Mick told her. \"It might be as well if I lay low for a day or two. How would you feel about harboring a fugitive?\"\n\nKnowing her well enough to know that the idea would appeal to her.\n\nAs soon as the door was closed behind her, as soon as he'd seen her taxi reverse out of the square (waiting a few seconds in case she had to come back for something), Mick set about checking out the house. He patrolled the front room first, taking in objects both familiar and unfamiliar. The print above the fireplace in the dining room, that was one of Sam's. Mick recognized the motifs immediately\u2014spindly minarets against a midnight-blue skyline, and the outline of a bird in flight etched out in Sam's signature gold leaf.\n\nHe wandered into the dining room, taking stock of the highly polished antique mahogany table and its matching mahogany chairs, their seats upholstered in a deep pink velvet. He wondered who had chosen the table and chairs. Alma had never shown any interest in antiques when he was married to her. Could it have been one of her many lovers? That actor fella she was seeing at one stage struck Mick as a suitable candidate, pretentious fucker that he was, and him out of a butcher's shop in Mullingar. Mick had met him a few years back, at an arse-licking session in the embassy, and he'd made a point of establishing a bond with him as a fellow Midlands man. Your man couldn't get away from him fast enough. Oh, he had Mullingar well and truly in the rear-view mirror, that fucker. And Alma too, by the sound of it. Last Mick heard, he was gallivanting around with some young one out of an Australian soap opera.\n\nMick swept forward to study the photographs on the mantelpiece. There was one of Nora as a baby, looking exactly like herself, her eyes wide with indignation. There was a picture of Alma's mother on the Abbey stage, a spotlight pooling at her feet. There was a formal portrait of her father at his writing desk, the same portrait that had appeared on the back of his book. At the edge of the mantelpiece was an old snapshot of Alma herself. It showed her sitting on a low stone wall, wearing a white sundress and a pair of huge sunglasses. Mick identified it as one of their honeymoon photographs, and he was as certain as he could be that it had originally been a photograph of the two of them, but now it showed only Alma. Looking closer, he saw the evidence. A blue knee jutting into the picture from the right-hand side. She must have cut him out of the picture and had it reframed.\n\nHe headed upstairs, bypassing the bathroom and making straight for her bedroom. The bed was unmade, a pile of pillows on one side, dented still by her head. There were clothes heaped on the ottoman, clothes piled up on the armchair by the window, high-heeled shoes scattered around the floor. It had always been a mystery to him how someone so particular about some things could be so slovenly about others. (\"But I _like_ being a slob,\" she had told him once.) Before he left the room, he had a peek into her underwear drawer. He had to know did she still wear the same satin underwear. He was thrilled to find that she did.\n\nHe stripped off on the landing, taking a visceral pleasure in wandering naked through his ex-wife's house. Leaving his clothes in a puddle on the carpet, he headed to the bathroom and ran the shower, waiting until it was steaming hot before he stepped in. Afterward, he helped himself to a clean white towel that he found on the heated towel rail. He collected his clothes from the landing, putting the underpants on inside out for the sake of hygiene. He put on the same shirt he'd been wearing yesterday, and the same trousers, for want of anything else to wear. On his way downstairs he stopped off at the bathroom again and gave himself a generous all-over spraying with a deodorant he found on the bathroom counter. An old trick, from his student days, it almost made him feel young again.\n\nHe made himself a mug of tea and settled down in front of the TV in the living room, feet on the coffee table, mug on the arm of the sofa. With some difficulty he managed to steer the channel to Sky News. They were showing pictures from Syria, but the crawl at the bottom of the screen made reference to him: _Spokesperson for Commissioner Collins says pepper grinder incident was \"a misunderstanding\"_...\n\nAs Mick watched, the newsreader appeared full screen.\n\n\"You're watching Sky News,\" she said. \"Coming up this hour, new footage shows the European Commissioner Michael Collins stealing a _pen_ from the Russian president Vladimir Putin.\"\n\n\"Oh, feck it,\" said Mick, sitting up so fast that his tea slopped all over the arm of the couch.\n\nOn the screen, he saw himself in Moscow last year, Putin to the right of him. The Russian president was talking and Mick was listening, or pretending to listen, to the translator in his ear. As he watched, he saw himself reach out and lift Putin's pen off the table in front of him. He turned the pen around in his fingers and then, unmistakably, he could be seen tucking it into his breast pocket.\n\n\"Oh, fuck, fuck, fuck!\"\n\nHe clutched his head with his hands and, sinking back into the couch, let his legs float upward. Like an upended beetle, he grappled at the air with his feet.\n\n\"Oh, fuck, fuck, fuck, fuck, fuck,\" he said. Then he righted himself and went looking for his phone, which he had plugged into Alma's charger.\n\n\"It's a free-for-all,\" squealed Feargal. \"The things they're tweeting about you, you would not believe. Former employees, ex-girlfriends, they're all coming out of the woodwork. I'll tell you one thing, you've really managed to make some enemies over the years. It looks like this is payback time.\"\n\nMick listened, the sound of static filling the air inside his head. He was having trouble concentrating.\n\n\"So, what do we do now?\"\n\n\"Well, wherever you are, we think it's best if you lie low for a day or two. Where are we now? Thursday. The hope is this will all have blown over by next week.\"\n\n\"Any word from on high?\"\n\n\"They're suggesting the best way out of this may be a treatment program.\"\n\n\"Treatment for what?\"\n\n\"Well,\" said Feargal, \"according to the psychiatrist who's just been interviewed on BBC 24, it's a classic case of kleptomania. What did she say? I'm paraphrasing here, so bear with me. The irrational urge to steal things regardless of economic necessity belongs on the spectrum of obsessive\u2014compulsive disorder.\"\n\n\"For fuck's sake,\" said Mick. \"Did it never occur to them that I just fancied the bastard's pen?\"\n\nHe spent the rest of the afternoon avoiding the news channels. He hopped around the lower reaches of the remote control, finding nothing but cookery programs. Who would have thought there was so much cookery on TV? You could spend your whole afternoon just watching people cook. He imagined a whole continent, morbidly obese and fed on fast food, sitting on their couches and watching cookery programs on daytime TV.\n\nHe discovered that his favorite was a program called _Come Dine with Me_. He watched with perverse fascination as five strangers cooked for each other in their own homes, night after night, competing for a prize of a thousand pounds. A thousand pounds? he thought, puzzled. Why would anyone want to expose themselves and their pathetic homes for a paltry thousand pounds? He watched, struggling to understand, as they snooped around each other's houses, making fun of each other's possessions for the cameras. He watched, appalled, as they cooked meal after gruesome meal for each other. He found himself marvelling at the awfulness of people. The niceness of them, too.\n\nHe had the dinner ready for her when she got home, a menu constructed on the basis of what he had managed to find in her kitchen.\n\n\"This is all very domestic,\" she said, as she deposited a large paper bag on the kitchen counter. He peered into it and saw multiple packs of black socks, white underwear, white T-shirts. He was overcome by nostalgia. She used to always shop for his clothes when they were married.\n\n\"Thank you,\" he said, touched.\n\nShe ignored him.\n\n\"So, they're staking out your apartment. They've someone camped outside your office. They're even keeping an eye on your brother's house.\"\n\nMick chuckled.\n\n\"I'd be tempted to pitch up there just for the hell of it.\"\n\nHe was standing at the cooker. A tea towel slung over his left shoulder, he was putting the finishing touches to the meal he'd prepared for her with some prawns he'd salvaged from the bottom of her freezer.\n\n\"So, when did you learn to cook?\" she asked. She was standing against the kitchen counter, holding the glass of wine he'd poured for her.\n\n\"When you threw me out. It was either cook or starve. Has Nora not told you about my culinary prowess?\"\n\n\"Nora tells me nothing, about you or anything else. I always assumed you lived on restaurant food in Brussels. I imagined you surviving on truffles.\"\n\nHe laughed.\n\n\"Yeah, there's a bit of truffling in Brussels all right. Brussels is all about the truffles and the mussels. Bit of a culture shock for a country lad like me.\"\n\nWhen he met her, he had never even tasted an avocado.\n\n\"So, what are they saying?\"\n\n\"Jesus, Mick, you've no idea. It's taken on a life of its own.\"\n\nHe upended a pot of boiled rice into a colander, and the steam rushed up at him.\n\n\"Come on. Tell me the worst.\"\n\n\"Oh, it's open season. Anyone who ever worked for you, they're spilling the beans. Some press officer with the Commission says you made an inappropriate comment about her legs. The waitress in the caf\u00e9 near your office, she says you used to hassle her to go out with you AND you never left her a tip. Oh, and what about the secretary who used to pick up your dry-cleaning for you? She says you didn't pay her back.\"\n\nHe had two plates out on the kitchen counter and he was serving the rice up in little mounds, the way he'd seen it done on the telly.\n\n\"The dirty cow!\" he said. \"Once, maybe twice, I forgot to pay her back. But don't you worry. She used to tape the receipts on to my desk, to remind me. I used to buy her chocolates from Pierre Marcolini, for Christ's sake. I did everything to cozy up to that bloody woman, bar riding her.\"\n\nHe paused in ladling out the prawns to look up at her.\n\n\"Ah, don't look at me like that. You know what I mean.\"\n\nHe carried their plates over to the table, pulling out a chair for her before he sat down himself.\n\n\"Prawns flash-fried with butter and brandy,\" he announced.\n\nTentatively, she speared a prawn with her fork and put it in her mouth.\n\n\"I don't know why you're looking so nervous. It's not like I'm about to poison you.\"\n\nShe smiled.\n\n\"Not bad for someone who used to eat his dinner in the middle of the day.\"\n\nThat was something she had teased him about when they were first married.\n\n\"Tomorrow's going to be more challenging,\" he said. \"Tomorrow I'm down to frozen peas and four eggs.\"\n\n\"I suppose you want me to do some shopping?\"\n\n\"Ah, no. That would spoil the fun. I'm quite enjoying this. It's like that program on the telly. What's it called? _Ready Steady Cook_.\"\n\n\"It's like the diary of Anne Frank, that's what it is.\"\n\nHe raised his glass to her.\n\n\"And you, my girl, are the heroine of the Resistance.\"\n\n\"If they knew you were here,\" said Alma, shaking her head at the thought of it. \"If they knew you were here, Jesus Christ, Mick. I'd be hung out to dry.\"\n\nFor two days, Mick was happy enough to hang around the house. Not for forty years had he had a pajama day, not since he was a child and his school had been closed down for a week because of a burst pipe. He lay on the couch for hours on end, eating Alma's biscuits and drinking tea as he watched back-to-back editions of _Come Dine with Me_. In his mind, the notion that this was a welcome, albeit temporary, reprieve from the world.\n\nOn day three, everything changed. He woke, and before he even opened his eyes, he knew that the buoyancy of the past two days was gone. It was as if the world had turned while he was asleep and now everything was cast in a different light. Outside the window the telephone wire hung in the air, motionless and bereft of birds. The sky was a dense flat white, the tiles on the edge of the stadium sullen and unreflective. Mick lay in the bed, steeped in gloom.\n\nWhereas initially he had chosen to view his predicament as a comical one (the adventures of a sane man in a crazy world), now it began to take on a less cheerful complexion. What at first had seemed a bit of an escapade\u2014a lark, an adventure\u2014now began to take on the appearance of a crisis. His career, everything he had worked so hard to achieve, it was all hanging in the balance. And all because of a fucking pepper grinder. He found himself raging against the injustice of it.\n\nSwinging his legs out of the bed, he reached for his phone and put a call through to Feargal.\n\n\"Okay, Feargal,\" he said, \"I've had enough of this. I'm coming back.\"\n\n\"Oh, I don't know if that's a good idea, Mick.\" Feargal's voice squeaked with protest.\n\n\"Hold on a second. Don't you think this whole thing is getting a bit out of proportion? I filched a fucking pepper grinder. I'm happy to pay for it. The Putin thing, that was just a mistake. I didn't realize what I was doing. What would I want with Putin's fucking pen anyway? Jesus, Feargal, what's the world coming to when a man gets subjected to a public bloody flogging just because he steals a bloody pen? It's not like I'm Dominique Strauss-Kahn, for fuck's sake. It's not like I'm fucking Berlusconi. All I did was pocket a bloody biro.\"\n\nSilence on the other end of the line.\n\n\"There's a bit more to it than that, I'm afraid.\"\n\n\"What do you mean?\"\n\n\"This doesn't reflect well on the Commission, Mick. Surely you can see that. What with everything else that's going on\u2014people going hungry in Greece, half of Spain out of work\u2014the Commission needs to be seen to take this seriously.\"\n\n\"What are you trying to tell me, Feargal?\"\n\n\"They're putting you on paid leave while they decide what to do with you.\"\n\n\"What!\" He couldn't believe what he was hearing. \"You know what this is, Feargal? It's a fucking witch hunt!\"\n\n\"Mick,\" said Feargal. \"Can you please stop bombarding me with expletives. I've had about enough of listening to your foul language.\"\n\nAnd that was the moment. When Feargal McCarthy\u2014Feargal fucking McCarthy, whom Mick had hired as a spotty kid straight out of UCD\u2014when Feargal fucking McCarthy pulled the PC card on him, that's when Mick knew it was all over.\n\nHe ventured out wearing his overcoat and his dark glasses, a woolly hat of Nora's pulled down low over his ears. The first person he met was Alma's next-door neighbor, bald as an egg, in a bulging diamond-pattern jumper.\n\n\"Howrya,\" said the neighbor, coming out of his house.\n\n\"Howrya,\" said Mick, wondering had the neighbor recognized him.\n\n\"Grand day,\" said the neighbor. And he pulled a rolled-up towel out from under his sweater.\n\n\"Thinking of a swim?\" asked Mick.\n\nThe neighbor stood for a moment looking up at the sky. Then he turned to Mick.\n\n\"That's what I was thinking. High tide's about now.\"\n\n\"Looks like we're in for some rain.\"\n\n\"Ah, sure that won't hurt me if I'm wet already.\"\n\nThe neighbor turned and looked at the upstairs window of his house.\n\n\"Don't go telling on me now. If she found out, she'd kill me.\"\n\nSure, who would I tell? wondered Mick, walking with the neighbor toward the corner.\n\n\"Your secret's safe with me,\" he said, as he and the neighbor parted ways. \"Enjoy your swim now.\"\n\nThe neighbor put his index finger to his lips to reiterate the need for discretion. And Mick thought, hang on a minute, who's the fugitive here?\n\nIn the Spar, he filled a basket with groceries. He scanned the front pages of the papers, afraid to bend down to study them properly in case he aroused any suspicion. With his sunglasses on he could hardly see, but he was afraid to take them off for fear of being recognized. He placed himself in line for the checkout, looking straight ahead of him. When his turn came, he marched forward and hefted his basket up on to the counter.\n\n\"Hiya,\" he said to the young man at the till. A stocky, mahogany-skinned man, he was Malaysian by the looks of him, or maybe even Mauritian. Mick prided himself on being able to guess these things.\n\n\"Where is it you're from,\" he asked. \"Malaysia?\"\n\n\"Me?\" said the young man, surprised. \"I'm from Brazil.\" He jerked his head to the man working at the next till along, \"He's from Malaysia,\" he said, laughing as if it was all a big joke.\n\n\"Good stuff,\" said Mick, wondering to himself when it had come to pass that the only people doing an honest day's work in Ireland were the foreigners.\n\n\"Do you have a loyalty card?\" asked the young man.\n\n\"What?\" said Mick. \"A loyalty card? No. I don't have a loyalty card.\"\n\n\"Would you like one?\"\n\n\"No. No, you're all right, I don't need a loyalty card.\"\n\n\"You should get one,\" said the young man, his brown eyes full of concern. \"It doesn't take long. You fill out this form.\"\n\nAlready he was producing the form from under the counter.\n\n\"Honestly,\" said Mick, waving it away. \"I don't need a loyalty card. I'm not here for long.\"\n\n\"Okay,\" said the young man, putting the form back under the counter. He looked disconsolate, as if he'd just failed to save a soul. Even as he was scanning Mick's groceries, he was ruminating over it. \"You can save a lot of money if you have a loyalty card.\"\n\nMick emerged from the Spar with a plastic bag weighing down each arm. The rain had started and he had no raincoat. No umbrella either, and no means of holding one. There was nothing to save him from getting a wetting. He lingered in the doorway, looking hopelessly out at the rain, and it was then that he spotted Alma's father's car. It was definitely Manus's car, a powder-blue 1962 Mark 2 Jag\u2014sure, who else would drive a car like that? It was stopped at the traffic lights, with Manus's mad head in proud silhouette above the steering wheel. Mick stepped out on to the street, ready to wave him down, a sudden rush of affection unfolding inside him for his former father-in-law.\n\nThe lights changed and the car began to move, prowling through the intersection.\n\n\"Manus!\" called Mick, arm in the air.\n\nThe car cruised toward him, and Mick had to jump back on to the pavement to avoid being mowed down. Alma's father was in profile now, detached from the world like an old sea captain at the wheel of a great ship out on the open sea. Oblivious to the furious wake he was generating, he splattered Mick with mud as he passed.\n\n\"Manus,\" called Mick again, setting his shopping bags down so that he could wave with both hands.\n\nThe car moved slowly past the Spar, blithely occupying the center of the road so that other cars were forced to pull in just to let it pass.\n\n\"Manus,\" called Mick, a third time. Even though by now it was clear that Manus hadn't seen him.\n\nMajestically, the car moved off down the avenue, cresting the bridge and disappearing the other side of it. Mick picked up his shopping bags again and continued on his journey, feeling sadly invisible.\n\n\"Are you the fella from the Corporation?\" asked a very old woman he met at the corner of the square. She was wearing a transparent plastic scarf knotted under her chin, and a large transparent raincoat, even though the rain had stopped and the sun was out again.\n\nMick put his shopping bags down and pointed at himself.\n\n\"Who, me?\"\n\n\"Yes, you. The Corporation said they were going to send someone round to look at the drains. The drains is all blocked, I've been trying for weeks to get them to send someone round.\"\n\n\"No,\" said Mick, sorry to disappoint her. \"No, I'm afraid I'm not the man from the Corporation.\"\n\nThe woman took a step toward him.\n\n\"Only, there's more rain forecast and I'm worried those drains is going to flood. They don't clean out the drains,\" she said, confiding in him now. \"Years ago, there used to be always fellas out cleaning the drains, but nowadays you never see them.\"\n\nMick could feel dried mud splatters caking on the skin of his ankles. His clothes were damp and hung heavy on him but still he had a desire to linger. A desire for human contact.\n\n\"How long are you living on the square?\" he asked her.\n\n\"Oh, I'm here all my life.\"\n\nShe turned and pointed to a house behind her. Mick wasn't sure if she meant the one with the yellow door or the one with the white door.\n\n\"I was born in that house,\" she said. \"My mother had six of us in that house and then she died. My father only lasted three months after her. It was my grandfather who reared us.\"\n\nMick found himself surprisingly affected by her story.\n\n\"How very sad,\" he said.\n\n\"Ah, sure that was the way of it in those days. Times was harder then. People had to manage as best they could.\"\n\n\"Yes,\" said Mick thoughtfully, and his own misfortunes suddenly seemed very feeble to him.\n\n\"Well,\" said the woman, \"we live in hope.\"\n\n\"Of what?\" asked Mick, thinking for a moment that she had some answer to offer him, something that would show him the way out of the predicament he was in.\n\nShe looked at him like he was simple.\n\n\"We live in hope,\" she said, \"of the man coming about the drains before the place gets flooded.\"\n\n\"Oh, that,\" said Mick, bending to pick up his bags. \"Yes, indeed. We live in hope.\"\n\nWith the shopping bags weighing him down, he made his way along the last stretch of the square.\n\n\"You wouldn't do me a favor?\" asked the bald neighbor. He was standing out in front of his house. \"You wouldn't take me towel? Stick it on a radiator for me. I'll get it back off you later.\"\n\n\"Sure,\" said Mick, putting his shopping bags down again so he could take the towel. \"Afraid you're going to be busted?\" he asked, weighing the wet towel in his hand as if he were trying to guess how heavy it was.\n\n\"Something like that,\" said the neighbor. \"It's the cancer,\" he said, by way of explanation. \"I'm in the middle of me chemo. If the wife knew I'd been in for a swim, she'd kill me.\"\n\nMick nodded, not sure what to say.\n\n\"I'll throw it in the dryer,\" he said. \"Come and get it whenever you want.\"\n\n\"Nice one,\" said the neighbor. And he slipped into his house, pulling the door shut quietly behind him.\n\nMick took Alma's spare keys out of his pocket and tackled her new triple-lock system. Two Chubbs, one Yale, all of them put in the day she'd come home from the hospital, she'd told him. Just as he was losing his patience, the door gave in.\n\nHe unpacked the shopping, humming away to himself happily. He was thinking about the people he'd met, and the feeling it gave him to talk to them. He liked talking to people. That was something he'd missed, all the years he'd been living in Brussels. It only occurred to him now how much he'd missed it. Now that he was back among his own people.\n\n\"I met some of your neighbors,\" said Mick, once they were sitting down to dinner.\n\nRisotto with baby peas; before adding the stock to the rice, he had thrown in a generous glass of a fine Spanish sherry he'd found in Alma's drinks cabinet. The result was sublime, a dish that was more than the sum of its parts. The peas as sweet as gumdrops, the stock neither liquid nor solid, and the rice offering just the right resistance on the teeth.\n\n\"Don't tell me you've been drawing them down on me,\" she said. \"I've spent years studiously avoiding them.\"\n\nRain falling on the skylight, like handfuls of small pebbles striking the glass. It had been raining for hours.\n\n\"Well, I met the man next door,\" said Mick. \"The man with the cancer.\"\n\n\"I didn't know he had cancer.\"\n\n\"He's doing chemotherapy. He still swims in the sea, though. He doesn't want his wife to find out.\"\n\nAlma sighed.\n\n\"That pair,\" she said. \"They drive me mad. He insists on sticking a parking cone outside his house so no one will take his space. On Sunday mornings they cook cabbage and the smell of it seeps into my house through the fireplace.\"\n\n\"There's an old lady across the square. I got talking to her too.\"\n\n\"Not Dolores?\"\n\n\"I didn't get her name. Nice old lady...\"\n\n\"Invisible hairnet, lipstick on the teeth?\"\n\n\"As I said, a nice old lady. She's concerned about the drains.\"\n\nAlma smiled. Very slowly she began to shake her head from side to side.\n\n\"You're unbelievable! You're canvassing my bloody neighbors! Where are we? Three years from the next election? So what are you thinking? Time to reinvent yourself?\"\n\nShe was still shaking her head.\n\n\"I know exactly what you're thinking. You're thinking you'll snatch that seat right out from under your brother's nose.\"\n\nHe stared at her, shocked that she had guessed what he was thinking before he had even formulated it himself. His brother's seat, lost at the last election, it would be ripe for the picking next time round. The thought had been percolating at the back of his head. Not even a thought yet, more like mild vapors, conspiring to take ahold of him.\n\n\"Ah, come on,\" he said. And he lowered his face to look down into his risotto. \"Come on,\" he repeated, digging around in the rapidly congealing rice with his fork. \"What do you take me for?\"\n\nWhen he looked up at her again she was still staring at him. She had one eyebrow raised, her eyes twinkling at him and her lips clamped shut to suppress a smile. She saw right through him\u2014always had done\u2014and yet she liked him all the same. That was the thing about himself and Alma, he thought, they had always liked each other.\n\n\"What happened to us?\" he asked her. \"We were so good together. We brought out the best in each other.\"\n\nShe snapped her head to the side, looking out into the garden. In her throat she sounded a dry little laugh, or an expression of indignation, he didn't know which.\n\n\"You know what happened, Mick.\"\n\nBut did he? It seemed to him that they had never talked about it. Not then, and not since. Not properly.\n\n\"The thing with Acushla,\" he said. \"I still don't know if you believe me. You can ask me a hundred times and I'd give you the same answer over and over again. Nothing. Happened. Between. Me. And. Acushla.\"\n\n\"It wasn't just that,\" she said, looking down at her plate now. \"It was all the others, too.\"\n\n\"They didn't matter.\" Was there a way of saying this without making it sound like a broken record? He tried to put extra feeling into it, tried to make every word ring true. \"Not a single one of them mattered to me, Alma.\"\n\nShe looked up at him as she answered in a throwaway voice.\n\n\"They mattered to me.\"\n\nSuddenly he felt very, very bad. Not just because it had cost him his marriage, not just because it had cost him his home and his daughter, but because he had hurt her.\n\n\"I didn't take it seriously enough,\" he said, and it was a revelation to himself as much as it was to her. \"I took it for granted, what we had. I was a gobshite.\"\n\n\"In fairness,\" she said, softening, \"we neither of us did. We neither of us knew what it was we had.\"\n\nAnd it was typical of her, thought Mick, to be so just.\n\nHe dropped his shoulders forward, lowering his head so that he was looking up at her. Eyes narrowed, he ventured to ask, \"Do you think you'd ever be able to give me another chance?\"\n\nShe grimaced, the way you do when you're breaking bad news to someone.\n\n\"I don't know, Mick. The honest answer to that is, I simply do not know.\"\n\nThe flood came at nightfall.\n\nA wet, dark, silent thing, it slithered out from under the gates of the stadium and coiled itself around the gutters, expanding until it had filled the streets right up to the top of the curbs. Noiselessly, it spilled over, flooding the pavements. It seeped into the grass and began to rise like a sponge cake, the water levels steadily climbing the inside walls of the square. It crept up the pathways of the houses, slipping under their front doors without anyone noticing. Its stealth was a living, breathing thing, silent in the night.\n\nThe first Alma and Mick knew of it was when the neighbor banged on their door. Two dull thuds that bore the sound of a fist rather than a row of knuckles. Alma and Mick were waiting for the start of the nine o'clock news, when they heard it.\n\n\"Jesus,\" said Mick. \"What the hell was that?\"\n\nHe looked over at Alma. She had not moved from where she was sitting. Hands in her lap, head held high, she had closed her eyes, as if by doing so she could make herself disappear. For a second, Mick had a memory of Nora as a child. Whenever they played hide-and-seek, Nora would put her hands over her eyes, imagining that if she couldn't see him, then he couldn't see her either.\n\n\"Stay here,\" he said. \"I'll get it.\"\n\nWhen he opened the door, the flood poured in. The neighbor was standing outside in his pajamas, his wellies deep in the water. The street outside was moving but the sky was still. The rain had stopped, leaving behind an eerie silence.\n\n\"What is it?\" asked Alma, coming up behind Mick.\n\n\"It's the river,\" said the neighbor. \"The river's gone and burst its banks.\"\n\nBehind him Mick could make out dark figures wading through the water with buckets and flashlights. On the doorsteps, backlit by their brightly painted front halls, children stood in their pajamas watching. Above them loomed the stadium, dimly lit from within. It looked sheepish, as if it was trying not to draw attention to itself.\n\n\"Hold on a second,\" said Mick. \"I'm coming.\"\n\nHe reached back into the hall for a raincoat, grabbing a bright yellow mac of Nora's that was hanging on the coatrack. He just about managed to squeeze himself into it. There were no wellies in his size so he took his shoes off, rolled his trousers up over his knees and stepped out into the floodwaters in his bare feet. He was thinking about broken glass. About rats and Weil's disease. Underfoot, he could feel sharp stones. Uneven tarmac. A manhole cover.\n\n\"You stay here,\" he shouted back to Alma. \"Get a brush and try to sweep the water out. I'll find out if there's anything we can do to stop it rising any farther.\"\n\nTogether with the neighbor, Mick waded out into the middle of the street, up to his knees in the cold black water.\n\n\"Where does the water drain to?\" he asked the neighbor.\n\n\"It doesn't. The drains are all blocked up.\"\n\n\"All right. We'll need to open a manhole, so. Has anyone got a crowbar?\"\n\n\"I do, in the house. I'll go and get it.\"\n\n\"Wait,\" said Mick. \"I don't know your name.\"\n\n\"Maurice,\" said the neighbor. \"My name's Maurice.\"\n\nHe held out his hand to shake Mick's.\n\n\"Maurice, I'm not sure whether you should be out here. I'm worried you might catch something.\"\n\n\"Ah, well. It's a bit late for that,\" said Maurice. \"And anyway, nothing this exciting has happened round here for years. I'm not about to miss it now.\"\n\nMaurice waded back into his house, his pajama bottoms sagging under an inadequate drawstring. A moment later he returned with a crowbar.\n\n\"I came across the manhole cover here somewhere,\" said Mick, walking around in the pitch-black water. He was scoping out the ground with the soles of his feet. Every so often he would touch on something slimy, or something that moved. At one point something scurried between his feet and he scrambled, comically, to lift his legs off the ground, hopping first on one foot and then the other. At last his right toe made contact with the dimpled metal surface of the manhole cover.\n\n\"Here it is!\" he shouted. \"Here it is, Maurice, I have it!\"\n\n\"Good man,\" said Maurice, and he waded over to Mick with the crowbar. Taking it from him, Mick blind-guided it in through the opening on the cover of the manhole and leaned on it with all his weight. The cover didn't budge.\n\n\"Give us a hand here.\"\n\nMaurice came around behind him and together they grabbed on to the crowbar.\n\n\"All right, are you ready? One, two, three.\"\n\nThe two of them leaned down on the crowbar and gradually, with great reluctance, the manhole cover lifted. Carefully, Mick hefted it to one side, and he and Maurice stood peering down into the black flood waters. In both of their minds was the possibility that whatever was down there might make its way up\u2014a geyser of raw sewage\u2014to add to their troubles.\n\nAs they gazed into the impenetrable depths, slowly but steadily the manhole began to draw on the flood water. From all around them it was being sucked downward until, unbelievably, the water levels began to fall.\n\n\"Sweet Jesus,\" said Mick. \"It's working.\"\n\n\"I don't believe it,\" said Maurice, looking up at him. His strangely hairless face was shining in the moonlight. \"We did it,\" he said, and he slapped Mick on the back.\n\nThe water was hovering around Mick's ankles, and still falling. A festival air about the place, all around the square people were squelching about in the puddles. Children had ventured out, dressing gowns lapping around the rims of their wellies. The old lady Mick had befriended came out of her house in a quilted robe. Plastic bags tied around her feet, she began handing out mugs of tea that she poured from a giant flask. Someone else passed around biscuits and they all stood about in the dull glow of the stadium, enjoying a neighborly midnight feast. And into this happy atmosphere, into this almost magical gathering of people brought together by their combined efforts to avert disaster, arrived Mick's brother. With his rain jacket open at the neck to reveal a shirt and tie, and his suit trousers tucked into his wellies, he peered down into the open manhole, just as the last slops of the floodwaters were being swallowed up.\n\n\"You might want to put the cover back on that,\" he said. \"That's a danger to small children the way it is.\"\n\n\"You should have seen your face.\" Even now Mick was heaving with childish laughter; he had to clutch a hand to his ribs to contain it. \"Fuck me,\" he said. \"It was worth anything to see your face.\"\n\n\"You looked like Paddington Bear in that raincoat,\" said Liam drily. He picked up his water and sipped at it.\n\n\"Would you not have a pint?\" Mick had asked him.\n\nThe two of them were standing at the bar; Liam was looking anxiously around him, alert to the potential damage of being seen in his brother's company. Three years out from the next election, but already he was shaping up for it. Even after the beating he'd taken, even after the ritual humiliation of his recent defeat, he couldn't wait to get back in the ring. Mick could see it in his eyes.\n\n\"Ah, have a pint with me,\" said Mick.\n\n\"I'm happy with the Ballygowan,\" said Liam. \"I'm watching my glucose levels. You know yourself.\"\n\nMick looked at him. Noticed that he was thinner. He looked older, too. His cheeks slack, like an empty hammock, his complexion gray in the yellow light of the pub.\n\n\"What prompted this?\"\n\n\"Ah, I've been seeing this dietician. She has me off gluten. Dairy, you know, the usual.\"\n\n\"And when you say _seeing_?\"\n\n\"Jesus, Mick. Is that all you think about? I'm seeing her for my cholesterol levels. I'm trying to get the cholesterol down. Our father died of a heart attack at fifty, or had you forgotten that? It's time we started minding ourselves.\"\n\n\"Well in case you hadn't heard, Liam, my cholesterol is the least of my fucking problems.\"\n\nSo there they were, back to square one. Fighting each other for space in the back of the car. Measuring their MiWadi glasses against each other, using an upended ruler to mark out the slightest discrepancy in the levels. Setting the alarm to be the first to wake on Christmas morning so as to plunder the other one's stocking. From the moment they arrived into the world, seventeen minutes apart and roaring the both of them, from that moment on they had been out to get each other.\n\nThey collected their drinks from the bar and carried them over to a table in the corner.\n\n\"So, are you not going to ask?\"\n\nLiam shrugged. Took another sip of his water.\n\n\"I don't have to ask. I saw it on the telly.\"\n\n\"Oh, and that's it?\"\n\n\"What more could there be?\"\n\n\"Well, you might let me try and explain.\"\n\nLiam dipped two fingers into his water glass and drew out an ice cube. Putting it into his mouth, he crunched on it.\n\n\"I don't need to listen to you explain. I'm your brother, Mick. I know you. You're the guy who used to steal robins' eggs out of their nests just for the crack. You're the guy who used to nick bottles of milk off old ladies' doorsteps. You're the fucker who robbed my hurling medal, for God's sake. You robbed it out of my school bag and you let Tommy Mangan take the blame.\"\n\nMick let a deep breath out of him.\n\n\"I did not rob that medal.\"\n\nAnd in so far as he knew, it was true. He had no memory of ever robbing that bloody medal.\n\nLiam was staring at him, staring right into his eyes.\n\n\"I'd still love to know what you did with it. You must have hidden it away somewhere, somewhere you knew I'd never find it. It had my name engraved on the back of it, so you couldn't even go showing it off to anyone. Why would you bother? That's what I never understood. Was it just for the pleasure of knowing that you had something that belonged to me?\"\n\nListening to him, Mick found himself wondering was it possible that he _had_ robbed the medal? Was it possible that he had robbed it and didn't remember?\n\n\"This is ridiculous,\" he said. \"I don't even remember the bloody thing. It seems only to exist in your memory.\"\n\n\"You see? You're still denying it.\"\n\nMick picked up his pint and took a gulp, trying to keep a lid on his rage. It was ridiculous for them to be fighting over this. Jaysus, it was embarrassing at their age.\n\n\"Would you listen to us,\" he said. \"And us in our fifties.\"\n\n\"Yeah,\" said Liam, but he was determined not to let it go. \"Fifty-seven years old, but some of us still have some growing up to do.\"\n\nAnd it was a mark perhaps of the doldrums that Mick found himself in that he let his brother have the last word.\n\n\"Well,\" he said. \"None of this has turned out exactly how we planned it, has it?\"\n\nLiam shrugged, his face grim as he answered.\n\n\"We've been thrown a few curveballs all right.\"\n\n\"Ah, come on. We dropped the fucking ball, Liam. We stood on the shagging thing and burst it. We took it and stuck a pin in it.\"\n\nLiam's eyes took on an opaque quality.\n\n\"It's been a tough time, I'm not denying that. It's been a tough time for the whole country. But we need to look on our present troubles as an opportunity. There's a great opportunity now for a fresh start.\"\n\n\"Jaysus,\" said Mick, with a bark. \"You're starting to sound like a fucking politician.\"\n\nLiam turned his face to look at his brother head-on. Defensive now. Accusatory.\n\n\"What else am I going to do, Mick? What else _can_ I do? Head up a charity? Go and write a crime novel? Politics is my trade, Mick. It's what I do. It's the only thing I know how to do. I don't see any other options out there for me. There isn't exactly a queue at my door for my services. I can hardly retire at fifty-seven. I have to find some way of filling the next ten years.\"\n\nThere was panic in his voice and Mick almost felt sorry for him.\n\n\"I hear you,\" he said. \"Sure, aren't we in the same boat, the two of us? Two washed-up old farts.\"\n\nBut Liam didn't like that. He didn't like to think their predicaments were in any way comparable.\n\n\"Speak for yourself,\" he said, the fucker.\n\nMick let it go.\n\n\"So, what's your plan?\" asked Liam.\n\n\"I don't really have one,\" said Mick. \"Sure, my term was up soon anyway. They were only dying for an excuse to get rid of me.\"\n\nLiam weighed that up.\n\n\"Yeah,\" he said. \"But at least while you were in Brussels you were still inside the tent, instead of outside it pissing in.\"\n\nMick laughed, a laugh that stayed trapped inside his chest.\n\n\"Have you not heard?\" he said. \"The tent has been blown away in the storm. There is no fucking tent!\"\n\n\"So, we pitch a new one,\" said Liam, with grim determination.\n\nWhere was he getting this shit from? Some counselor? Some overpriced life coach? Some change-management guru that he'd signed up to?\n\n\"Do you know what, Liam? I think I'd prefer to sleep out in the open.\"\n\nMick picked up his pint and drained it, his eyes on his brother the whole time. He beckoned to the barman for another.\n\n\"Tell me,\" said Liam, as they waited for the pint to arrive. \"How's Alma doing?\"\n\nMick dropped his head to the side. Thought about it for a minute.\n\n\"Physically,\" he said, \"she's grand. She went and chopped all her hair off, but otherwise she's grand. Mentally, though, I think she's very fragile.\"\n\nLiam raised his eyebrows.\n\n\"That's not a word I'd ever have associated with Alma.\"\n\nMick laughed.\n\n\"Fierce, yes. Fearless. Ferocious. All the F words. But not fragile.\"\n\n\"Yes, it was always Acushla was the fragile one.\"\n\nAnd they were both of them silent for a moment, thinking not of Acushla as she was now, but Acushla as she was when they first met her.\n\nMick remembers her in her pale pink bridesmaid's dress and her gold strappy sandals walking across the lawn at his wedding. She was holding her sandals by their straps in one hand, the way a poacher carries a brace of game birds. Later, when she was dancing barefoot, he saw that the soles of her feet were stained green by the grass. He saw his brother watching her and he knew instantly what was going to happen. Two brothers marrying two sisters: it seemed like it was meant to be.\n\n\"How is she?\" asked Mick. \"Acushla.\"\n\n\"How's Acushla?\" repeated Liam. \"The truth is, Mick, I'd be the last person to know. We live in the same house but we hardly have a word to say to each other.\"\n\nMick looked at his brother as if he was only seeing him for the first time.\n\n\"I'm sorry to hear that,\" he said. \"I really am.\"\n\nAnd it was true, he _was_ sorry. He was surprised at how sorry he was. He had assumed that Liam and Acushla were happy. Despite everything, he had imagined their marriage to be a success, holding it up against the very public failure of his own. To find out now that it was not what he had imagined it to be, there was no satisfaction in it for him. He felt only sorrow.\n\nAs his brother looked back at him, Mick made a conscious effort to hold his gaze. Eyeball to eyeball for five seconds. Eyes open wider. Six, seven, eight, nine, ten. Just like that game they used to play when they were kids: the first to blink would lose. Staring into Liam's steady blue eyes, it seemed to Mick that he could see his brother's feelings moving like dancers behind a screen. Resentment, self-pity, suspicion.\n\nLiam blinked and Mick spat out a laugh in triumph.\n\n\"You fucker,\" said Liam, picking up his water glass.\n\nIn the far corner of the pub, unbeknownst to either of them, a man was holding his phone up in the air to take a photograph of them. Bending down low over his lap, the man posted the photograph on Twitter, wondering with a happy burp of warm hops how long it would take someone to pick up on it.\n\n# Acushla MacEntee At Home\n\nHer name is synonymous with understated glamour, so it's no surprise to find Acushla MacEntee Collins immaculately clothed in a gray silk shirt-dress and simple gold jewelry on a Monday morning in early summer. Welcoming Style magazine into her home in Dublin 4, the 45-year-old wife of the former Minister for Justice says the key to her look is simplicity. \"I think the most important thing is to dress for yourself. At my age, I don't care what anyone else thinks. I'm very comfortable in my own skin.\"\n\nAcushla has a grown-up daughter and two small grandsons but she hardly looks old enough to be a mother, let alone a grandmother. So what's her secret? \"If you're asking me whether I've used Botox, then the answer is no,\" she replies, with disarming candor. \"I'm afraid I'm too squeamish to allow anyone near me with a needle. I'm terrified of going under the knife. So the only alternative is to grow old gracefully. I try to drink a lot of water, and I stay out of the sun. Apart from that, it's all down to genes. I have my mother to thank for that. She's about to turn eighty and she still has the complexion of a teenager.\"\n\nOf course, Acushla's mother is none other than Deirdre O'Sullivan, star of the Abbey stage and a legendary figure in Irish theater. \"From an early age, my mother instilled in us the importance of presenting your best face to the world. We were always taught to walk with our heads held high and our shoulders well back. You might be wearing rags, but if you walk like a queen, then that's what the world will take you for.\"\n\nIt's salutary advice for the times we live in. And Acushla MacEntee is no stranger to hardship. Two years ago, she was the wife of a senior member of the Cabinet. Now her husband is fighting to win back the seat he lost in the last election. \"Politics is such a tough game,\" she says, \"and it's very hard watching someone you love take a knock like that. But Liam is a remarkably resilient person. I'm so proud of the way he has picked himself up and started again. He really cares about the future of this country and he wants to be a part of it.\"\n\nFor her part, Acushla is content to support him. \"I've always been more comfortable playing the supporting role than the lead. Even when my sister and I would put on little plays as children, she always took center stage and I was left with all the bit parts! Since I married Liam I've been very happy to let him occupy the limelight. My job is in the wings, but it's an important job, the job of wife and mother. Now I'm a grandmother as well, so I find I've plenty to keep me occupied!\"\n\n# Acushla\n\nAcushla stood at the kitchen window and watched as Connie unloaded her grandsons from the back of the car. She was short of breath, her heart beating too fast in her chest, like a distressed bird. Instinct and counter-instinct battling it out with each other. She was looking forward to the morning ahead of her\u2014she always looked forward to these Friday mornings with the boys\u2014just as a part of her was dreading it. Already she was anticipating the droop of their heavy little heads over the kitchen table. Their pudgy fists clutching the crayons, and the frantic windshield-wiper motion of their hands across the page. The voracious appetite for fresh paper.\n\nShe had it all ready for them\u2014a neat stack of foolscap pages in the center of the table, and on top of it, acting as a paperweight, the tub of crayons. The Ribena was already mixed, the plastic beakers standing by on the kitchen counter to receive it. The boys would knock the beakers over, of course. They would spill the Ribena all over their artwork and Acushla would be forced to hang the pages out like wet laundry, draping them over the backs of chairs and radiators. Later, the dry drawings would flutter to the floor, enjoying a moment of flight before coming to land under the kitchen dresser, or the Aga, from where Acushla would gather them up and stash them outside in the green bin.\n\nOh, she was tired just thinking about it. She was tired and apprehensive at the same time. As she watched the boys tripping along the gravel path toward the kitchen steps, her heart was in her mouth. The way they walked, with the weight of their bodies thrown out over their feet, it was inevitable that one of them would fall. And sure enough, as Acushla watched, Oscar went tumbling down. Connie swept up behind him with Ernie hanging on to her right hand, a huge shoulder bag acting as a counterbalance on her other side. With a scoop of her free hand she set Oscar on his feet, and with the same tipping motion he began to tumble forward again, tripping over his own toes. Acushla watched him spill down the stairs and wondered should she have taken a whole Valium this morning instead of a half?\n\nShe opened the kitchen door and stood aside as Oscar stormed in, his brother following fast behind him. They kicked their shoes off and together they made straight for the far side of the kitchen, pulling a stool alongside the counter so they could climb up to the biscuit tin.\n\n\"Boys!\" said Connie. \"Are you not going to say hello to your granny?\"\n\nThey turned around, just.\n\n\"Hi, Ganny.\"\n\nA biscuit in each hand, they scrambled back down, leaving the lid off the tin. Helplessly, Acushla stood and watched them. She'd be the whole afternoon cleaning up.\n\n\"Oscar needs to take his antibiotics,\" said Connie. \"And I'll leave you some Calpol, in case the fever comes back.\"\n\nHer daughter was rooting around in her handbag, the loose waves of her hair falling down over her long, thin face. She was wearing a vintage raincoat, open on to a short green dress that had the look of a charity shop about it. The matching green leather gloves she wore made it difficult for her to negotiate her way around the inside of the handbag. Impatiently, she pulled one glove off with her teeth and dipped her naked hand back inside. She unpacked half a dozen loose nappies on to the table, along with a blister pack of Calpol fastmelts and a small brown bottle of medicine.\n\n\"How much Calpol should I give him?\" asked Acushla anxiously. You'd think she would know this. She was a mother herself; surely she would know this? When she became a grandmother, it was as if everything she had once known about babies had been wiped clean and she was forced to start over again. She was as nervous minding Connie's boys as she would have been minding a complete stranger's children. More so, maybe. She was terrified she would let something happen to them.\n\n\"Give him five mil of the antibiotics,\" Connie was saying. \"And one fastmelt every four hours.\"\n\nAcushla was sorry now that she'd taken the Valium. She was struggling to absorb Connie's instructions. A breeze blowing through her head, it picked up the words and swirled them round inside her skull.\n\n\"Shit,\" said Connie. \"Look at the time.\"\n\n\"Where do you have to be?\"\n\n\"Oh, I've to do a feature piece for the _Indo_ about this abortion thing. I've found a woman who's willing to talk about it. They need the piece by tomorrow, so I'll have to write it up tonight when the boys are in bed. These people think that freelancers have no lives...\"\n\nConnie was still talking, but Acushla's head had emptied of everything but that one word. A word that was on the radio every time she turned it on these past few weeks. On the front pages of all the papers, again. On her daughter's lips, even now, as she rooted around in her handbag, her voice muffled by a tumble of hair.\n\n\"This woman was raped, but she had to go to England earlier this year for an abortion because she couldn't get one here. I swear to God, there are times when I think this country is still in the Dark Ages.\"\n\nAcushla didn't answer her, but Connie didn't seem to notice. She had her head in her handbag. A reporter's notebook coming out on to the table, and a hooded fleece. A freezer bag bursting with Lego.\n\n\"Finally,\" said Connie, pulling out a glossy magazine, its spine doubled back on itself to reveal a photograph of her mother sitting elegantly balanced on the arm of the drawing-room couch.\n\n\"I thought you might want a spare copy. For Grandmother's scrapbook.\" (From the moment Connie had arrived in the world, it had been made clear to her that her grandmother's title was not to be abbreviated. Under no circumstances was anyone to call her \"Granny,\" or \"Nana.\" Nothing short of \"Grandmother\" would do. Whereas Connie's mother delighted in being \"Ganny\" to her boys.)\n\nAcushla made an effort to smile.\n\n\"The photo's nice,\" said Connie, leaning in for a second to plant a kiss on her mother's cheek before rushing for the door. She paused and looked back, with a wink. \"I have to say, you're looking pretty good. For _forty-five_.\"\n\nAs her daughter swept past, Acushla swayed, and for a moment she feared she might topple over.\n\n\"You're an angel, Mum. I don't know where I'd be without you.\"\n\nHer head disappeared past the kitchen window. Looking up at the clock on the wall above, Acushla saw that it was only two minutes past ten. That left two hours and fifty-eight minutes before Connie would be back. It seemed like an awful long time.\n\nBy eleven o'clock, they were in Accident and Emergency.\n\n\"He's got a piece of Lego stuck up his nose,\" Acushla told the nurse. \"I don't know how he managed it. I was with him the whole time.\" She was desperate not to appear negligent.\n\n\"Let's have a look,\" said the nurse, dipping down to see.\n\nObligingly, Oscar tipped his head back for her. This was not the first time he had stuck things up his nose, so he knew the drill.\n\n\"Well, I can see it up there,\" said the nurse. \"But we're going to have to wait for the doctor to take it out.\"\n\nOscar whirled off to join his brother, who was pulling a stream of tickets out of the dispenser.\n\n\"Lads,\" growled the nurse, her index finger raised.\n\nAcushla kneaded the handbag on her lap.\n\n\"How long do you think the wait will be?\"\n\nThe nurse shrugged, unsympathetic.\n\n\"There's a few people ahead of you. You're looking at an hour at the very least.\"\n\nWispy thoughts floated unbidden through Acushla's mind. The thought of the amount of money she'd raised for this hospital over the years. The thought of the fund-raising dinner that was taking place this very night. She had the hospital CEO's number in her phone. In a matter of hours he would be sitting across the table from her; she could ring him right now if she wanted to. The thought roamed around her head like an unwanted guest. Of course she would not ring him! Not in a million years would she ring him about something so trivial. But it would make a good story for later, the story of her morning in Accident and Emergency. She pictured herself regaling the table with it. It gave her great buoyancy to think that she had this story to tell.\n\nShe picked herself up and moved toward the row of plastic seats in the waiting area. The boys were kneeling on the ground, making imaginary pancakes on a brightly colored plastic stove. Acushla picked up a tattered copy of _Hello!_ magazine and sat down to read about Kate Middleton's pregnancy, but the words kept jumping around on the page. Pressing in around her, the presence of the other people in the waiting room.\n\nThere was a fevered baby screaming. A toddler pedaling a plastic tractor up and down the corridor in a frenzy. There was a little girl in pink pajamas curled up on her mother's lap, her eyes silently roaming the room. The mother looked young and poor. She kept petting the little girl's head, planting kiss after kiss on her threadbare skull. Acushla felt guilty looking at them. With her designer handbag in her lap, and her expensive coat folded inside-out across her knees\u2014with her boisterously healthy boys and their frivolous medical emergency\u2014she felt overprivileged and unworthy by comparison. She vowed to cede her place in the queue to the woman if her ticket happened to be called first.\n\nShe thought about ringing Connie, but, heroically, she didn't. Every five minutes she thought about ringing Connie, but she resisted the temptation each time, proud of herself every moment that went by for holding the line. Anticipating already the pride she would take in telling her daughter there had been no need to ring.\n\n\"Oh, we managed just fine,\" she would say. \"Didn't we, boys? We managed just grand.\"\n\nFrom her handbag she produced two bags of popcorn to keep them going, and dipping into the inside pocket she fished out a Valium for herself. As she broke the little yellow pill in half, she saw that the polish on one of her fingernails had chipped away. She felt a rising sense of despair at the thought that she would have to do her nails again before the dinner tonight. She still had her dress to collect from the dry-cleaner's, and she had promised to stop by the hotel to check the flowers. The goodie bags would need to be stashed under the chairs, the chocolate favors distributed around the tabletops. What had seemed on paper a simple list of chores now began to take on a nightmare complexion. Too late, Acushla realized that she had taken on too much, again.\n\nThe dinner was in aid of the children's hospital, one of half a dozen charities that Acushla patronized. The venue was a five-star hotel barely a mile from her home. A brand-new, purpose-built palace, its completion had coincided with the high-water mark in the country's fortunes. No sooner was the red carpet rolled out, no sooner was the first Rolls-Royce pulling up outside than the nation's finances were in free fall.\n\n\"We'll have a job getting money out of people tonight,\" said one of Acushla's fellow committee members as they moved around the tables checking place names. An angular beauty by the name of Laura, this woman had been a model in her past life. Small-town stuff, standing around Stephen's Green in a skimpy bikini, covered in fake tan and goose bumps. She'd gone on to become the wife of a well-known cardiologist. The mother of a child with cancer, she was a tireless fund-raiser for the children's hospital. Acushla couldn't help but feel inferior to her, as if her suffering had earned her extra points in the game of life.\n\n\"You look gorgeous,\" said Acushla, surveying the simple satin gown, the effortlessly twined hair, the barely there makeup.\n\n\"So do you,\" said Laura. \"But then you always look lovely.\"\n\n\"Oh, I don't know about that,\" said Acushla breathlessly. \"I had to throw myself together. You know yourself\u2014one of those days.\"\n\nShe had not had time to put rollers in her hair and she had been forced to patch her chipped nails rather than redoing them from scratch, little chinks in her armor that left her feeling horribly vulnerable.\n\n\"That's a great dress,\" said Laura, as they moved on to the next table.\n\n\"Well, the sleeves do wonders for the bingo wings,\" said Acushla, with deliberate self-deprecation.\n\n\"Oh, get away out of that,\" said Laura. \"You don't have a pick on you.\"\n\nWas Acushla imagining it, or was Laura's tone a little impatient? Irritated, even?\n\n\"Your sister was very good to honor her commitment to us. Everyone would have understood if she hadn't felt up to it\u2014under the circumstances.\"\n\n\"Oh, Alma's a real trouper,\" said Acushla. But the words sounded a big hollow clang in the air. She tried them again, like a showjumper coming around for a fresh attempt at a fence. \"She's an absolute trouper,\" she said. \"She wouldn't dream of letting us down.\"\n\nAs soon as Alma's name was announced, there was an explosion of applause. Heads turned to see, arms draped over the backs of chairs. As she stood up and weaved her way toward the stage, a low outbreak of commentary rose from the tables.\n\n\"Fair play to her,\" said the man on Acushla's right. \"That's courage for you.\"\n\n\"Oh, she's got mettle,\" said someone else. \"I'll say that for her.\"\n\nAcushla twisted from the waist, clapping with what she hoped looked like genuine affection.\n\nAlma reached the podium and paused, expertly adjusting the microphone while she waited for the applause to die down. She was wearing a simple black tuxedo suit with a pair of ankle-strap stilettos. The double-breasted suit jacket was fastened by a single button, and she was wearing nothing under it.\n\n\"She looks ay-may-zing,\" mouthed the woman to Acushla's right.\n\nAcushla gave a nodding smile, crinkling her eyes in agreement as her sister's voice, familiar as a shiver, slunk around the ballroom.\n\n\"You're too kind,\" said Alma into the microphone, quelling the last of the applause with the power of that beautiful voice. A voice like honey in the sun, as one columnist had famously observed; it was a voice with the power to make a man lie down at her feet and lick them, he had written. For years Alma had kept that clipping tacked up on the door of one of her kitchen cupboards, along with a letter from another fan who wrote to tell her that he liked to close his eyes and jerk off to the sound of her reading the news. She took great pride in that letter, much to the horror of her sister.\n\n\"There is no more worthy cause,\" said Alma, her voice all the more powerful for its unsentimental tone, \"no battle more worth the fighting than the battle to provide state-of-the-art health care to our sickest children. Among us tonight are some of the doctors and nurses who work at the coalface of our hospitals, as well as the dedicated volunteers who support them in their mission. Truly, this is the work of the angels.\"\n\nAlready the waiters were going round the tables topping up people's wineglasses, the committee members rising out of their chairs and gathering at the foot of the stage, ready to bear the auction items around the room.\n\n\"Never has the funding of our hospitals been a more critical issue,\" said Alma. \"Never has the task of fund-raising been more arduous. But rest assured, never has your generosity been more gratefully appreciated.\"\n\nBy the end of the night the auction had fetched forty-nine thousand euro. Which was good money, everybody said; it was a great result for the times that were in it. But it was a far cry from the night that someone paid a hundred and fifty grand for a round of golf with P\u00e1draig Harrington.\n\n\"I have one remaining item to auction,\" said Alma, leaning in to the microphone in conspiratorial fashion. She took something out of her jacket pocket, a small, transparent object that she placed on the podium in front of her.\n\n\"For those of you who can't see, this is an unremarkable but rather infamous glass pepper grinder. Now, can I start the bidding at a hundred?\"\n\n\"That was a bit of a showstopper,\" said Acushla, when she ran into her sister in the powder room.\n\nAlma was sitting in front of the mirror touching up her foundation. Acushla slid on to the seat next to her and took out her own makeup bag, delving into it for her concealer.\n\n\"Sorry,\" said Alma unapologetically, as she dabbed at the side of her nose with the index finger of her maimed right hand. She was looking at Acushla through the mirror. \"I'm afraid I couldn't resist it.\"\n\nAcushla avoided looking at her sister's reflection. She had always been squeamish, and the sight of her sister's wounded hand was more than she could stomach. It seemed to her a little inconsiderate of Alma not to be more discreet in her use of it. She could easily have used her left hand to do her makeup.\n\n\"I don't know why you're apologizing,\" she said coldly. \"It brought the house down. It'll be the talk of the town tomorrow.\"\n\nSitting poker straight on the upholstered stool, she sucked her cheeks in and swept a slick of fresh blusher across her cheekbones, the reflection of her sister's face hovering at the edge of her peripheral vision.\n\n\"Are you annoyed with me?\" asked Alma, brandishing her uncapped lipstick in midair. She appeared bemused by the idea.\n\n\"Why would I be annoyed with you?\" said Acushla, still without making eye contact. She was shaping her eyebrows with her index finger now, licking the finger and smoothing. \"No,\" she went on, in a tinny voice. \"It's just it would have been nice to know that Mick was staying with you. I seem to be the only person in the room who didn't know, that's all.\"\n\n\"Sure it's all over Twitter,\" said Alma, her eyebrows hoisted high. \"And anyway, I assumed your husband would have told you.\"\n\nAcushla felt as if she'd been backed into a corner.\n\n\"Yeah, well, I haven't seen him,\" she said, aware of what this revealed about her marriage. \"We're both so busy these days, we hardly ever see each other.\"\n\nShe zipped up her makeup bag and combed her hair back with her fingers. She looked across at her sister's reflection in the mirror. Alma was looking back at her, with no effort to disguise her humor now.\n\n\"Don't tell me you're jealous. I swear to God, I think you're jealous that he came to me rather than to you.\"\n\n\"Don't be ridiculous,\" said Acushla, stretching out her spine.\n\n\"After all these years,\" said Alma, \"you're still trying to steal my husband!\"\n\n\"Oh, for God's sake,\" said Acushla, and for want of anything else to say, she stood up and made for the door.\n\nOutside in the corridor, a photographer stopped them and asked them to pose for a picture. Without a word, the two sisters draped their arms around each other's waists and inclined the crowns of their heads a little toward each other, to catch a flattering angle. Too late, it occurred to Acushla that she'd been caught on the wrong side of the picture. Her left-hand side had always been her bad side, but she couldn't swap places now. As the flash erupted, she was drowning in the certainty that her sister had stolen a march on her, again.\n\n\"Poor Acushla,\" Acushla's mother liked to say. \"She spent her entire childhood trying to catch up with Alma. She used to follow her around like a lost puppy. Oh, it was painful to watch.\"\n\nAnd it's true, Acushla's earliest memories are all of Alma moving ahead of her like a figure in a dream. Alma in the playground, making for the empty swing. Alma on the street, chasing after the ice-cream van. Alma on the beach, heading for the waterline. In her memory, Acushla is always running after Alma, calling out to her but never catching up.\n\nWhen Alma started school, Acushla took it so badly that her mother had to buy her a school bag too, and a lunch box just like Alma's. When Alma started swimming lessons, Acushla was so determined to prove she could swim that she jumped into the deep end and nearly drowned. When Alma started wearing a bra, Acushla stuffed socks inside her vest to make it look like she was wearing one too, and when Alma got her period, Acushla lay in bed and willed hers to come, to no avail. When Acushla's own milestones did eventually come around, in their own good time, they brought her no satisfaction, coming so long after her sister's as to have no value to her at all.\n\nAcushla was fourteen when she faced up to the fact that she was never going to catch up with her sister. No matter how hard she tried, Alma was always going to be smarter than her at her schoolwork, better at sports, more popular with the boys. And it occurred to Acushla how futile it was for her to continue trying to beat her sister at her own game. Instead of focusing on Alma's strengths, Acushla adjusted her sights, forcing herself to study her sister's weaknesses instead. In everything that her sister was not, she carved out a whole new personality for herself.\n\nThe new Acushla would be gentle where Alma was rough. Meticulous where Alma was careless. Thoughtful toward people where Alma was casual. With absolute determination, Acushla sat down and designed a new style of handwriting for herself. She acquired a watch and took pride in her newfound punctuality; she even started making lists, to overcome her tendency toward forgetfulness. She remembered people's birthdays, and made cards for them, something Alma would never have dreamed of doing. When Alma was drawn into the debating club at school, Acushla volunteered for the benevolent society. While Alma was launching a school newspaper, Acushla was preparing to accompany a group of handicapped people on a pilgrimage to Lourdes. In her final year of school, Acushla was chosen to be head girl, and there was an added satisfaction to her achievement because Alma had never even made prefect.\n\nThat was the year their father left home, moving in with his as-yet-nameless Moroccan lover in a scandal that was passed around the country and parsed with glee, bringing a splash of much-needed color to the otherwise drab winter of 1980. Their brother was sent off to boarding school in a disastrously misguided attempt to insulate him from the jibes of his contemporaries. And while Macdara had done nothing but irritate Acushla when he was in the house, once he was gone she found that she missed him. She missed the physicality of their horseplay. A bump of the shoulders as they passed each other on the stairs, a shin kick under the table at dinner. When Macdara was living at home, he and Acushla had played at hating each other, but without that game to play she was strangely bereft. She missed him most at breakfast time, when they would set up the box of cornflakes as a wall between them so they wouldn't have to look at each other; with Macdara gone, there was no need for the box of cornflakes.\n\nAlma was in college by then and seldom surfaced until midday. Acushla's mother took her breakfast in bed, to recover from the exertions of the night before. She had chosen that winter to relaunch her stage career, with a triumphant run as Grace in Brian Friel's _Faith Healer_ at the Abbey\u2014a part that one critic said was made for her. She dined out most nights after the show, with the result that Acushla ate alone every evening as well as every morning, with the exception of Sunday.\n\nShe was supposed to be studying for her Leaving Cert. Everyone had assumed that she would go to college, perhaps to study Arts, but Acushla was brewing a different plan, one that she had stored up secretly and nurtured during those long winter evenings with the food of her family's neglect. When she should have been memorizing the life cycle of the liver fluke, or committing Cordelia's words to heart, she was filling out application forms for Aer Lingus, and rehearsing for her interview.\n\n\"You're going to be an _air hostess_ ,\" her mother repeated, when Acushla made the announcement. \"What do you mean you're going to be an _air hostess_?\"\n\nThey were all of them gathered round the dining table to celebrate Macdara's sixteenth birthday. It was the first time he'd been allowed home from school for the weekend. The first time they'd all been together since their father had left. Sam had been included in the party, purely it seemed to Acushla because it gave her mother such pleasure to think of the fuss it would cause when people heard that she'd invited her husband's gay lover to dinner.\n\n\"I've been offered a job by Aer Lingus,\" announced Acushla. \"As soon as I've finished my Leaving Cert, I'll be starting my training.\"\n\n\"You must like to travel,\" said Sam, agreeably.\n\nDeirdre gave him a withering look.\n\nOh, what a scene there was that night. One parent a draper's daughter from Ennis, and the other the son of a penniless estate manager from County Kildare, they had their snobberies in common and they were agreed on one thing. No daughter of theirs was going to become an air hostess. They would have preferred her to announce that she was joining the circus, or the chorus line at the Folies-Berg\u00e8re; they would have seen the glamour in that. They wouldn't have minded her falling pregnant outside of marriage, or running off with some dissolute poet, if only he had an ounce of talent. But for her to become an air hostess? God, no. There was absolutely no question of it.\n\nBy the time September came round, Acushla was safely enrolled in First Arts in UCD. She staggered through two years of college, each summer sacrificed to the misery of repeats, until a reprieve came her way in the form of Liam Collins. They were married the Christmas of Acushla's final year and Connie was born the first week in May, providing Acushla with the perfect excuse not to sit her finals. And while the lack of any letters after her name never caused her a moment's regret, while it never once occurred to her to go back and finish her degree, she always felt a pang of something not unlike pain whenever she caught a glimpse of an Aer Lingus uniform. The sight of that petrol-blue coat, with the leather gloves and the sensible low-heeled shoes, never failed to stir something in her. It seemed to Acushla that there was another life that she might have lived, a life beyond that uniform. She could imagine that there existed a house somewhere full of rowdy children and a handsome pilot husband. A laughing version of herself dressed carelessly in a pair of jeans and a sweater, with her hair tumbling down her back in effortless, unkempt curls. It seemed to Acushla that all of this could have existed for her, if only she had managed to defy her parents' prejudices.\n\nThe house was in darkness when she got home, the windows staring out at her, large and black and empty. She went into the kitchen and switched on the kettle to make herself a cup of valerian tea, lining up her vitamins on the counter while she was waiting. Two multivitamins for her general health, extra calcium for her bones, and Siberian ginseng for the stress. She washed it all down with a double dose of soluble vitamin C. The fizz of the orange flavoring gave her an instant feeling of well-being, like the comfort a child takes in a sucky sweet.\n\nAt least she has her health, that's what she tells herself. She has friends who've had chronic back trouble, friends who suffer from diabetes. Friends who've had emergency hysterectomies, and double mastectomies\u2014their bodies papered over again but still bearing the scars. Alone among the women she knows, Acushla remains miraculously unscathed. Her belly is flat and trim, her breasts still small and upright. Her nipples in particular are triumphantly youthful, suckled as they were by just the one baby. Such scars as Acushla bears are all on the inside, and invisible to anyone but herself.\n\n\"Would you describe what you feel as guilt?\" a therapist asked her once.\n\nAnd Acushla has been thinking about that ever since. Can you feel guilty and still have no regrets? Can you be remorseful even though you wouldn't have done anything differently?\n\n\"I would suggest that what you're feeling is grief,\" suggested another therapist. \"Grief that has never had a chance to find its expression.\"\n\n\"Betrayal,\" said another. \"You were betrayed by the very person you should have been able to trust.\"\n\nAnd of course they were right. They were all right, every last one of them. Acushla does feel betrayed. She feels sad and guilty; she feels full up with grief. But it's anger that she feels more than anything. Anger like the stagnant water in a pond, or the toxic liquid that's left behind in an old mine tailings dump. For twenty years it has been seeping silently out into her life. Poisoning her slowly, without anyone noticing.\n\nOf the five people who know, it was her mother she told first. Her sister, who took care of all the arrangements. Her husband, who imposed a rigid code of silence on the whole thing, as if the silence were paramount and not the thing itself. Her father was away at the time and only found out after the fact; it was Acushla who told him, spilling the story out of her in spite of her husband's directive. Over and over again Liam said that no one must know and Acushla never thought to defy him. Not at the time, and not since\u2014only once in all the years that followed did she ever tell another soul what had happened to her, and that was when she told Mick. Mick is the fifth person who knows\u2014something that in Acushla's mind creates a bond between them that can never be broken. Mick's kindness to her, and the sweet, brief friendship that grew out of it, this is a holy thing to Acushla. A memory that she treasures, despite all the trouble it brought down on them. Mick salvaged her faith in men.\n\nIt was three male doctors who broke the news to her\u2014all of them middle-aged, all of them wearing pinstripe suits. They leaned away from her as if it was radioactive, this thing that she had inside her. One doctor was sitting, the other two standing. Arms behind their backs, they kept looking at each other rather than at her. Talking in sentences that trailed off in the middle. The words they used seemed to form some kind of a code, but Acushla was not able to make out what it was they were trying to say. They seemed to be circling something, but before they got to the center of it, the consultation was over. They sprang to action, one of them jumping up to grab her coat for her as his colleague dashed to hold the door open. Out in the corridor a nurse slipped her a torn-off piece of paper with an English telephone number written on it in blue biro. \"I could be fired for doing this,\" she whispered.\n\nWhen Acushla told her husband, he said the same thing.\n\n\"I could lose my job,\" he said. \"I'd have to give up my place in Cabinet. I'd never be reelected.\"\n\nHis career was riding on Acushla's silence. A silence that she maintained through election after election. Through abortion referendum after abortion referendum\u2014referenda that convulsed the country for weeks and months on end, filling the streets with protesters and the airwaves with rage and outrage. Acushla took to turning the radio off during those times. She started switching the TV channel away from the news. She avoided buying a newspaper or answering the door, rushing away from shop counters without her change and standing back from the other mothers outside the school gates so as to stop herself coming into contact with this one word that could not be spoken.\n\nA word that in Acushla's head reverberates even now with hard consonants, bouncing off the walls of her skull like a small ball bearing, rolling around in her head with no way of escaping. A word that is never spoken between her and Liam\u2014if ever they refer to it, which they seldom do, they speak of when she \"lost the baby.\" The other word is deployed only in the context of the legislation that Liam voted against time and time again over the years, in line with his party whip. In line with the views of the majority of voters, views that were carefully calibrated in poll after poll, while what happened to Acushla was submerged in a dense solution of secrecy. Even now that Liam's career is in tatters; now that his reputation is on the floor with the nation's finances and any hope of ever salvaging either of them hanging by a frayed thread\u2014even now the secrecy remains.\n\nAcushla was sitting at her dressing table when she heard him come in. The silver raw silk curtains were only partially drawn, and outside, the stadium occupied the night like a huge paper lantern. Stars speckled the sky above it, surprisingly bright for the city; their presence made Acushla feel very small.\n\nShe sat and listened as her husband closed the front door behind him. She heard the sound of a light switch being turned on, and a pause while he checked the post on the hall table. Then his footsteps moved across the wooden floor, and she heard creaking treads as he climbed the stairs.\n\n\"You're up,\" he said as he came into the room, his shoes moving noiselessly over the thick-pile carpet.\n\nShe didn't answer. She removed her pearl earrings, dropping them one by one into the crystal ashtray that she kept on her dressing table for this exact purpose.\n\nLiam crossed the room behind her, the mirror reflecting only the middle section of him, from the shoulders to the knees. Acushla gave no indication that she had seen him.\n\nHe removed his jacket and draped it carefully over the garment butler. His trousers he took off and folded neatly along their crease, sliding them legs first into the trouser press and letting the waist fall over the edge. With his hand cupped in the manner of a man rolling a pair of dice, he dropped his cuff links into a small silver bowl on his dressing table and rolled his shirt into a ball, stashing it in the laundry basket beside the window, along with his socks.\n\n\"Jesus,\" he said, \"that was a long day.\"\n\nAcushla watched him in the mirror as he moved across to the bed and slipped his hand under the pillows, where he found a pair of pajamas that had been neatly folded and stowed there for him. He put them on, taking the time to fasten all the buttons. He got into bed and plugged his phone into the charger before switching on the lamp and taking his reading glasses out of the case that he kept on his bedside table. Putting on the glasses, he reached out for a copy of a satirical magazine he subscribed to. He didn't seem to have noticed that she wasn't speaking to him.\n\n\"I saw Alma tonight.\"\n\n\"Oh?\"\n\n\"She was doing the auction for us.\"\n\n\"Oh, right.\"\n\n\"I gather Mick's staying with her.\"\n\nNo answer.\n\n\"I hear you met up with him for a pint.\"\n\nAt last he lowered his magazine and looked up at her. Their eyes met through the mirror, and his reflection started talking to hers.\n\n\"Do you know what kills me? He's wandering around like he doesn't have a care in the world. He doesn't seem to have any idea of the impact this has had on the party. It's the last bloody thing we need.\"\n\nA high-pitched wobble in his voice, as if his brother had given him a shove and he was struggling to retain his balance.\n\n\"I wouldn't mind, but we were just starting to see a bit of a bounce in the polls. We were just starting to alter people's perceptions of us. We'll be back to square one again after this.\"\n\nWhen she didn't say anything, he sighed and raised his magazine again, leaving their conversation hanging in the air.\n\nAcushla stood and turned off the overhead light, then hung her robe on the hook on the back of the door. Wrapping her nightdress around her legs, she climbed modestly into the bed, staying as close to the edge as she could. Without saying good night to him, without saying anything to him at all, she turned to lie facing the door. She reached a hand out for the small plastic pill jar that she kept on her bedside table; taking the lid off it, she tipped out a Valium, popped it into her mouth and washed it down with a swig of water. As she waited for the tranquilizer to take ahold of her, it seemed to her that she could not survive another second of the silent scream that was her marriage.\n\n\"Why did you marry him in the first place?\"\n\nThat's what a therapist had asked her once. And Acushla had paused on the question, pretending to puzzle over it. She had sighed, and fiddled with her wedding ring, and said it was hard to remember.\n\n\"I think I was trying to get one up on my sister.\"\n\n\"Seriously?\" asked the therapist.\n\n\"Oh, I don't know,\" she said wistfully. \"We were young. We hardly knew each other. We neither of us had a clue what we were getting ourselves into.\"\n\nIt was at Mick and Alma's wedding that she met him. Six foot two and lean as a greyhound, with his hair falling down over his eyes and his face tanned the color of a ripe red apple from the hay harvesting, he was the image of his brother, only better-looking. As soon as she caught sight of him, Acushla became acutely aware of him, and everyone else in the room disappeared. A feeling more like danger than love, although at the time she did not distinguish between the two.\n\n\"Watch out for that fella,\" Mick warned her. \"He's a cool customer.\" To Acushla's ears it sounded like he was giving her his blessing.\n\nAcushla was the bridesmaid and Liam the best man. Liam was supposed to ask her up to dance, but he didn't. He stood against the bar instead, smoking cigarette after cigarette and watching her on the dance floor. When she drifted out into the garden, she could feel his eyes on her back and when she came back into the room there he was, still watching her. By dawn the next day, his face drained pale and deadly serious from the night's drinking, he had announced his intention to marry her.\n\n\"We'll see about that,\" she said, and she whirled away from him, determined to test the measure of his love. The measure of her own she did not think to take, so giddy was she with the notion of being loved.\n\nHe started phoning the house every day; when Acushla came on the phone she found he had little to say to her, but somehow this only made his persistence all the more touching. \"It's him,\" her mother would say, with her hand over the mouthpiece, and her eyes thrown up to heaven. \"The poor eejit.\"\n\n\"He's out there again,\" she would say, looking out the front window to where Liam had taken to standing under the branches of the acacia tree, lighting one cigarette off another as he waited for Acushla to come out and give him her answer. Crouched on the threadbare rug in her mother's darkened bedroom, peering out over the base of the window frame, Acushla could see the burning tip of his cigarette glowing orange in the shadows of the garden.\n\n\"Away with you,\" her mother would shout, leaning out of her bedroom window with her hair falling down over the front of her nightgown like some middle-aged, never-rescued Rapunzel. \"Away with you now,\" she would say, as Acushla peered out from behind her shoulder. \"This is no time to come calling on anyone, sure it's after midnight.\"\n\nThrough the gap in the curtains her mother would monitor his eventual retreat with all the satisfaction of a conquering general. It played to Deirdre's sense of vanity for her daughters to drive young men to lose the run of themselves. It played to her insatiable appetite for drama.\n\n\"It all sounds very romantic,\" ventured one therapist.\n\n\"Oh, I don't know if I'd describe it as romantic,\" said Acushla, racking her brains for a more accurate word. \"It was thrilling all right. It was very flattering. But I don't think I'd call it romantic.\"\n\n\"I was starting to feel like the boring one in the family,\" she told another therapist. \"There was my dad, shacked up with a Moroccan man half his age. My mother was in the Trocadero half the night being serenaded by opera singers. My sister was just married and fast becoming a TV star. And I was still stuck in college. I was impatient for something to happen to me.\"\n\n\"And that's when he came along?\"\n\n\"That's when he came along.\"\n\nIn a way, it was funny.\n\n\"He thought he was getting a younger version of my sister. And I thought I was getting a more handsome version of his brother. And by the time we both realized our mistake, it was too late.\"\n\n\"Well,\" said her father. \"Never a dull moment.\"\n\nHe was studying the newspaper Acushla had brought him. The photograph on the front page was a picture of Alma standing at the auction podium brandishing the famous pepper grinder. A wing of white hair fell down over her face; her neck was as long and pale as a swan's.\n\n**STOLEN PEPPER MILL  \nSELLS FOR THOUSANDS  \nAT CHARITY AUCTION.**\n\nHer father chuckled.\n\n\"I'll say one thing for her. She certainly knows how to throw the cat among the pigeons.\"\n\nHe was fumbling with the coffeepot. He insisted on using one of those tin Italian espresso makers that you have to assemble and place on the gas ring. The Nespresso machine that Acushla had bought him for Christmas was sitting gathering dust on the kitchen counter.\n\n\"I hear Mick's staying with her?\" said Acushla.\n\n\"So I hear.\"\n\nAcushla took the pastries she'd brought out of their paper bag and arranged them on a plate. This was their regular Saturday-morning ritual. He made the coffee; she brought the pastries.\n\n\"Naturally, she wouldn't admit it, but I gather she's quite pleased to have him back.\"\n\nAcushla studied the things on his shelf. There was a disturbing number of jars of Branston Pickle. Various chutneys and unopened pots of English mustard. He seemed to be stockpiling them, but for what?\n\n\"She has him in Nora's room,\" he said, as the coffee began to hiss and splutter on the cooker. \"But I predict it's only a matter of time before he makes the journey across the landing.\"\n\nAcushla became aware of a pain in her side. Clamping her hand to her belly, she made a pretense of studying the collage on the kitchen wall. A patchwork of postcards and photographs from all the places they had been over the years, it told the story of a life of travel and culture, a life of love. There were opera tickets from Verona and La Scala and matchbooks from Raffles in Singapore and the Algonquin Hotel in New York. There was a cartoon of Sam sketched in around the margins of a Los Angeles restaurant menu by David Hockney and a snatch of verse by Seamus Heaney written in seeping fountain pen on to a white paper napkin and dedicated to Manus. Here they were standing on the terrace of the Peggy Guggenheim museum in Venice, both of them wearing black polo necks and dark glasses, while another photograph pictured them on a Tokyo street in a snowstorm. From the evidence on the wall, it seemed they had been to every country in the world. The only place they had not been to was Morocco. Once Sam made the decision to move in with Acushla's father, the country of his birth was the one place in the world he could never visit.\n\n\"How's Sam?\" Acushla asked, glancing toward the open kitchen door.\n\n\"Oh, you know Sam. He's as sweet as ever, thank God. He may be losing his marbles, but he still has his lovely temperament. Now. I'll carry the tray, if you'll follow with the pot of coffee.\"\n\nAcushla trailed after him into the living room, her head struggling to keep up with her heart. She was overwhelmed, as she often was, by the richness of her father's life. The love that was vested in every word he spoke about Sam. The tenderness, even the tragedy of it, was all thick and rich. Every time she visited her father and Sam in their home, Acushla felt like a lonely little cold-water fish who has ventured into the tropics. She didn't know whether to be pleased for them that they had found this happiness, or sad for herself that she hadn't.\n\n\"Sam,\" said her father, as he set the tray down on the dining table. \"Look who's here. It's Acushla.\"\n\nSam was sitting over by the vast glass wall of the apartment, his armchair angled to face out, a pack of Marlboro Red and an overflowing ashtray on an occasional table beside him.\n\n\"Oh,\" he said, getting to his feet. \"Acushla.\" As if he had just heard her name for the first time.\n\nShe came forward to kiss him, first on one cheek, then on the other.\n\n\"It's almost warm enough to sit out on the balcony,\" said her father. \"But I'm afraid it's not safe. Those bloody seagulls seem to think this is a cliff. You're taking your life in your hands venturing out there. Poor Sam got a nasty cut on his head the other day when one of them made a dive for him.\"\n\n\"Isn't there anything you can do about it?\"\n\n\"Short of shooting the bastards?\"\n\n\"I was thinking more along the lines of an awning, or a sun umbrella. I could get one for you, if you like.\"\n\n\"Your sister suggested we hang up some old CDs. Apparently the seagulls are repelled by their own reflection.\"\n\n\"When was Alma here?\" asked Acushla, trying to keep her voice breezy and light.\n\n\"Yesterday,\" said her father. \"She brought Michael with her.\"\n\nThere was no getting away from it, she had that stitch in her side again.\n\n\"I always said those two would get back together,\" said her father, oblivious to the pain he was causing her. \"They're much too well suited to be apart.\"\n\nAnd Acushla said nothing, even though she had thought many times over the years that it was Liam and Alma who would have been better suited to each other. It was Liam and Alma who should have been married, they would have been perfect for each other. And Acushla would have been perfect for Mick.\n\nThe amazing thing was how two men who had emerged from the same womb, at the same time, could be so very, very different.\n\nThe cartoonists got it straightaway\u2014the difference between them\u2014and they delighted in exaggerating it. Whereas Mick was always depicted as fatter and more jovial than he really was, Liam was leaner and more menacing. Mick was the bon viveur, always holding a bottle of wine and a fat cigar. Liam was the worrier, with his pinched face poring over the latest polls. \"Mick Collins is too likable to be party leader,\" wrote one newspaper pundit at the time of a party heave, \"and Liam Collins isn't likable enough.\"\n\nPoor Liam! The moment that line was printed, it came to define him. No matter what he did after that, no matter how hard he worked, and what political points he scored, he was never able to throw off the notion that there was something lacking in him, some essence of charm that was a prerequisite for the top job and that was inexplicably absent in him. When the final meltdown came, on the night the electorate rose out of its stupor and punched the party of government full square in the face, Liam Collins earned himself another epithet, one he would never live down. He showed himself to be a poor loser.\n\nThe night Liam lost his seat, he stormed out of the count center without speaking a word to the waiting media. Without shaking the hand of the young woman who had displaced him. Without stopping to accept the commiserations of his election workers, or to thank them for their efforts. He climbed into a waiting car and went straight home, where he walked wordlessly past Acushla and straight up the stairs to bed. The next morning he wouldn't get up. It was three weeks into his collapse before she managed to persuade him to see a doctor.\n\nShe drove him to the appointment, and waited for him while he went in. On the way home, she left him sitting in the car while she filled the prescription the doctor had given him. Incomprehension in his eyes, and desperation with it, he was more like a sick animal than a man. And heartbreaking as it was to see him like that, Acushla found herself hoping that this illness was the chance she had been waiting for, the chance at last of a fresh start.\n\nThere was one evening in particular when they sat up late and talked. They opened a bottle of wine, and before they knew it, they had finished it. They opened another, and for the first time Liam talked to her about his humiliation. He told her about the panic attacks, how for days after the election he had woken in the dead of night with the weight of a concrete block on his chest. The first time it happened, he thought he was having a heart attack.\n\n\"Why didn't you tell me?\" she asked, her head inclined toward his in sympathy, her hand on his freckled, feathery arm.\n\n\"Oh, I don't know. I suppose I didn't want to admit to you that I was scared.\"\n\n\"Oh, Liam...\" she said. And it seemed to her in that moment that something was opening up for them, a tiny crack through which, in time, the love might pour.\n\nHe cried that night, big wet baby tears that slid out of him without a word. She cradled his head to her chest and stroked his hair. She kissed him on the forehead, and for the first time in a long time she felt strong. For the first time in a long time they made love, and after they were finished, they wrapped themselves up face to face in each other's arms, like they used to do when they were first married. And Acushla remembered something that she had long ago forgotten, something that had existed between them before everything went wrong. She remembered that they were once allies.\n\nWhen she woke the next day, she was curled in toward the center of the bed, with her face up against her husband's back. She slid in closer to him, snaking her hand around his waist and nestling the fronts of her knees into the backs of his. It seemed to her that morning, as she breathed in the familiar smell of his skin, that the past twenty years had been left behind them and they could pick up again from when last they had a chance of happiness.\n\nThat evening Acushla prepared fillet steaks for them, as a treat. She pulled the cork on a bottle of wine, leaving it on the kitchen counter to breathe. She freshened up her makeup and reapplied her lipstick, and when Liam came in the door from his counseling session, she turned expecting a kiss.\n\nHe walked straight through the kitchen and into the living room, bending down to pick up the remote control off the coffee table.\n\n\"There's a poll out,\" he said.\n\nHe stood in front of the TV, pointing the remote at the screen.\n\n\"Jesus. It's even worse than I thought.\"\n\nThe lowest poll ratings since the foundation of the party, the future of the very organization was now in doubt. The penalty to be exacted for their role in the country's economic collapse, this ruthless mob vengeance.\n\n\"Like it was all our fault,\" he growled.\n\n\"Here,\" said Acushla, moving toward him with a glass of wine. \"Have a drink.\"\n\nHe turned and looked at her as if she was a stranger.\n\n\"You know I'm not meant to be drinking, with all the medication I'm on.\"\n\n\"Oh. I just thought...\"\n\nHis jaw was clenched as he spoke to her, his eyes cold.\n\n\"I shouldn't have drunk that wine last night. I can't afford to be doing that, in my situation. I can't afford to let my guard down.\"\n\nSo it was a mistake, that moment of weakness. It was a regret to him how he had unburdened himself to her. That brief oasis of tenderness they had happened upon, that little patch of warmth in the cold waters of their marriage, he was determined not to venture into it again.\n\nThe next thing he had himself going to a dietician who took him off gluten. He took up Pilates on the advice of Connie, and meditation on the advice of the Pilates teacher. He gave up coffee and tea, and slowly, over the course of the next year, he constructed a new narrative for himself. A narrative of recovery and rebirth. Through sacrifice and self-discipline Liam would build himself up again. He would become a tougher and wiser person and, like the country itself, he would live to fight another day. What saddened Acushla most, what she could not bear to watch, was the complete success with which he managed to rehabilitate himself. By the end of the year he had made a full recovery; if anything, he was harsher than he had been before, and more brittle. He had the added arrogance, too, of a man who has battled his demons and won, where another less capable creature might have lost. To her infinite sorrow Acushla was forced to accept the fact that the man who had hauled himself out of the wreckage of her husband's career was not going to turn out to be a nicer man than the one he had been before.\n\nAcushla was coming out of the nail salon when she met Macdara.\n\nHands held out either side of her, with her fingers splayed to keep her newly painted nails clear of danger, she was just wondering how she'd get her car keys out of her handbag when she saw him. Unaware that he was being observed, Macdara was waiting at the pedestrian crossing, dressed in his usual gentlemanly fashion, in corduroy trousers with a slightly-too-small tweed jacket and a wide wool scarf wrapped several times around his neck. There was an innocence about him as he raised his face to check for the green man. Something of the little boy he had once been. Acushla was overcome by a wave of tenderness toward him. A wave of love for this brother whose world had drifted so far away from hers.\n\n\"Acushla!\" he said, when he reached her.\n\n\"Macdara.\"\n\nShe would have liked to hug him but she didn't want to smudge her nails, so instead she reached up to kiss him, bumping her cheek against his clean-shaven chin. Her lips brushed against the soft-coiled wool of his scarf. A smell of extra-strong mints off him, he had a brown paper bag tucked under his arm in a way that suggested something clandestine. Briefly, Acushla thought of pornographic magazines, of discount double packs of chocolate biscuits, before her imagination ran aground.\n\n\"I was planning on taking the bus,\" said Macdara. \"But if you have your car with you, I might take a lift.\"\n\nThis curious directness of his, it was a result of his spending so much time alone. Acushla knew this, but even so it struck her as odd.\n\n\"Would you like me to drop you home?\"\n\nMacdara paused, studying her with his steady eyes.\n\n\"Are you not going down to the court hearing?\"\n\nAcushla began to grow concerned for her brother's state of mind.\n\n\"What court hearing?\"\n\n\"Haven't you heard?\" he said, speaking with great gentleness. \"They've arrested someone for the attack on Alma.\"\n\n\"Really?\" said Acushla, struggling to regain her grip on the conversation. \"That's great news.\"\n\n\"He's going to be charged this afternoon.\"\n\n\"But it's Saturday...\"\n\n\"There's a special sitting.\"\n\n\"I had no idea.\" (Later she would find a clutch of missed calls on her phone, and a message on her answering machine from her father, but for now Acushla was under the impression that she had not been informed.)\n\n\"The court appearance is at two,\" said Macdara, slowly stretching out his left arm to check his watch. Everything Macdara did, he did slowly. \"If you're happy to give me a lift, we can go down there together.\"\n\n\"Oh, I don't know,\" said Acushla, shy all of a sudden at the thought of going, and a little petulant at not being told. \"Do you think she'd want us there?\"\n\nIn her mind she was still nursing the sting of last night's brush with her sister. The offer Acushla had made her of a lift home, Alma had refused it out of hand. Acushla couldn't help thinking of all those salads she'd bought Alma when she was off work\u2014salads Alma had invariably tried to return to her as she was leaving. The orchid she'd bought her\u2014the color chosen so carefully to cheer her sister up\u2014Alma had dumped it in her dark downstairs loo. And back, and back, through all the little hurts her sister had inflicted on her over the years, not a one of them ever forgotten.\n\n\"I'm not sure she'd want me there,\" she said, wary of incurring yet another wound.\n\n\"Of course she would!\" said Macdara.\n\nAnd he spoke with such emphasis, and such conviction, that Acushla couldn't but be drawn into the beautiful orbit of his innocence.\n\nThey arrived late because of the traffic and got caught up in a one-way system that brought them in a wide loop around the Four Courts; it was only when they'd finally found a parking space on the quays that they realized they should have been in the new courts building, and by the time they walked up there it was all over.\n\nRain falling down out of a strangely cloud-free sky, they arrived to find everyone gathered in a small circle out on the street. The circle wobbled on sight of them, with Mick shuffling in closer to Alma, and Deirdre stepping back to make space for Macdara. Acushla squeezed in under her father's umbrella, his free arm pulling her in so close that she could smell Pears soap off him, and burned toast. A dry gloom prevailed under the canopy of the umbrellas.\n\n\"Well,\" said Deirdre, with a grim expression.\n\n\"Well?\" said Macdara, his transparent eyes roving from one to the other of them. \"What happened?\"\n\n\"He got bail.\"\n\nAcushla ventured to look across at Alma. Alma raised her eyebrows and glanced around at them all. Everyone seemed to be waiting for her to speak.\n\n\"Well, I don't know about the rest of you,\" she said, using her left hand to fasten the buttons of her raincoat all the way up to her neckline. \"But I could do with a drink.\"\n\nThe circle splintered as the family drifted in twos and threes across the slick wet street toward the nearest pub. Mick yanked the door open, standing out on the pavement to let them all through. First Deirdre, stamping her boots and shaking out the dripping hem of her cape, followed by Macdara, who bent his head down low, like someone entering a cave. Then Alma, on three-inch heels, with her raincoat tightly belted in the manner of a spy, or a stripper. Manus stumbled after her, managing to right himself again just inside the door. Acushla was left until last, with Mick holding the door open for her. Placing the palm of his hand briefly on the small of her back, he ushered her into the pub, and Acushla closed her eyes briefly as a memory ran through her. A shiver of a memory, it served to remind her how much she had once desired him.\n\nNothing happened between them.\n\nShe had assured Alma of this, at the time and many times since, indignantly at first, then with impatience and finally tears. Nothing had ever happened between them, that was the honest-to-God truth; they just had lunch together. A lunch that came about entirely by accident\u2014Acushla had been in town shopping, and she was on her way back to fetch her car when she ran into an anti-abortion demonstration. Old men bearing placards with photographs of dead fetuses. Young women brandishing buggies with healthy babies in them. She found herself trapped among them and she panicked. Holding her hands up in front of her, she began to push her way out, her eyes squinting to block out the images, her head shaking the sounds away. She fell out of the crowd and straight into the path of her brother-in-law, who was slipping out the side gate of the D\u00e1il. He took one glance at her and grabbed her by the arm, steering her up the street and in through the doors of the Shelbourne Hotel.\n\nThe air inside was a comforting hush. The pile of the carpet a thick bed of grass under her feet. Acknowledging the doorman with a nod, Mick guided Acushla to a low table in the corner of the lounge. The trees of Stephen's Green filled the enormous window. A piano in the corner, its notes little more than a watery trickle.\n\n\"The baby had no brain,\" she said. \"Did you ever hear of such a thing?\"\n\nHis face crumpled, as if she'd kicked him.\n\n\"Ah, Acushla. He never told me.\"\n\n\"And Alma didn't tell you either?\"\n\nShe had always assumed that Alma would have told him.\n\nHe shook his head.\n\n\"When did this happen?\" he asked her.\n\n\"Last year.\"\n\nHe nodded slowly and she could see that his mind was working on multiple tracks, as always. The personal and the political.\n\n\"Was it Liam who went to England with you?\"\n\nShe paused, ashamed of the answer. Not so much for Liam as for herself.\n\n\"He let you go alone, the fucker!\"\n\n\"My mum came with me,\" said Acushla quietly. \"But that's not the point. The point is that this happens to people. That's what you need to understand. This happens to people you know.\"\n\n\"Can you tell me about it?\"\n\nLeaning forward in his chair, he made a steeple of his fingers and held it to his lips, and Acushla had the impression that never in her whole life had anyone ever listened to her so intently. It gave her great strength to be listened to like that. It gave her eloquence.\n\n\"Thank you for telling me,\" he said when she was finished. \"I'm very grateful to you for telling me that story.\"\n\nAs they were leaving, he placed his hand on the small of her back and guided her out through the revolving door. A gesture that flooded her with longing. She had a great desire to be close to this man who had shown her such kindness. (\"He has the Mickey gene,\" Liam always said. \"He never met a woman he didn't want to sleep with.\") At that moment Acushla found herself imagining what it would be like to be one of those women.\n\nOut on the street, Mick drew her into a fierce embrace. Once he'd released her, he gripped her upper arms with his big bear hands, as if he was about to pick her up.\n\n\"You know where I am,\" he said with great seriousness. \"If ever you need anything, you know where I am.\"\n\nThen he took her face in his two hands and bent down to kiss her. A kiss that was aimed at her forehead perhaps, or the bridge of her nose; it was Acushla who angled her head back in order to receive it on the lips. The kiss lasted only a second or two, no more than that, but it was long enough all the same for each of them to be aware of a terrifying and thrilling possibility. The possibility of something more.\n\nThe kiss was witnessed by Alma as she passed by the hotel in a taxi. She told the taxi driver to go around the block but by the time they had passed the hotel again, her husband and her sister were gone. Nothing that Acushla said to Alma afterward succeeded in convincing her that the encounter she had seen was entirely innocent.\n\n\"I saw you!\" she said. \"I saw you with my own two eyes.\"\n\n\"You won't tell Liam, will you?\" said Acushla. An unfortunate reaction, and one that seemed to imply guilt where there was none. From that moment on, it was a losing battle. \"It's not my fault if you can't trust your husband,\" she said to Alma, unwisely choosing to mount an offensive instead of defending herself. \"Oh, for God's sake,\" she said when it came up again. \"How many times do we have to go over this? Nothing happened.\" Until the time came when she was reduced to tears of frustration at being so endlessly misunderstood.\n\n\"He was just being nice to me,\" she said plaintively. \"Nobody in my whole life has ever been that nice to me.\"\n\nInside the pub, Mick ordered whiskeys for Deirdre, Manus and Alma. A pint for himself, and a white wine for Acushla. For Macdara, he called up a ginger ale. (Macdara always was a useless drinker.)\n\n\"Let me give you a hand,\" said Acushla, while the others made for a corner seat. Coats and raincoats stacked behind their heads, they settled on to the plush velvet banquette. The feeling of a railway coach about it. Something of the rain was still present in the air inside the pub.\n\n\"Well, kid,\" said Mick, as they waited for the barman to assemble the drinks. \"You look a million dollars.\" (From the night he first met her, in her mother's kitchen, he had always called her \"kid.\")\n\nShe looked down at herself, as if to check, and saw the camel cashmere coat she was wearing. The hem of her burned-orange silk skirt peeping out from under it. She had one tan leather boot planted up on the brass foot-rail under the bar, the other on the red tiled floor.\n\n\"You don't look too bad yourself,\" she said, tilting her face up to his. \"Despite everything.\"\n\nHe laughed.\n\n\"Ah, you know me. I thrive on adversity.\"\n\n\"Seriously,\" she said, lowering her voice. \"Are you all right?\"\n\n\"Sure,\" he answered lightly. \"I'm having a grand time. The life of leisure suits me nicely.\"\n\nAcushla nodded, not sure whether to believe him or not.\n\n\"What about yourself, kid?\"\n\n\"Oh, you know. I'm grand.\"\n\nShe caught his eye, hoping he might see just by looking at her that she was lying. But he was distracted. He was pulling banknotes out of his sagging trouser pocket to pay for the drinks. Reaching along the bar, he corralled the three glasses of whiskey, one finger inside the rim of each, so that he could pick them up together. With the other hand he grabbed Macdara's ginger ale.\n\n\"Just take your own glass of wine,\" he said. \"I'll come back for the pint.\"\n\nApproaching the table, Acushla realized that something was wrong. For a moment she wondered was it something she had done, but then she saw that they were all looking off to one side, their eyes rooted to the spot where two men had taken up position on high stools at the bar.\n\n\"What's going on?\" she whispered, as she slipped on to a low seat beside Macdara.\n\nHer father was the first to speak.\n\n\"It seems your sister's assailant is celebrating his freedom by treating himself to a drink.\"\n\nAcushla turned her head briefly to the side. One of the men had his back to her, but she could see the other clearly. In the second or two that she had him in her sights, she formed an impression of a round white face. A pair of restless little eyes.\n\n\"It's unacceptable that he can just walk in here,\" Manus was saying. \"It's simply unacceptable.\"\n\nWith his knees all knobbly inside his thin cotton trousers and his knotty old hands lying clenched on his thighs, he looked very old all of a sudden. Old and vulnerable; a gust of wind would be enough to topple him over. Deirdre was sitting next to him, bullish by comparison. She was looking in the direction of the two men. She had her eye teeth gritted on one side, as if she was biting off a loose thread.\n\n\"I've a mind to go over there and have a word with them,\" she said, in a loud voice.\n\nYou and whose army? thought Acushla, wondering even as she did so where she had heard that phrase before.\n\n\"Perhaps we had better leave,\" said Manus tentatively. The electric blue of his eyes sparking with alarm.\n\n\"The fuck we will,\" said Mick. \"If anyone's going to leave, it's him.\"\n\n\"We can't just sit here and ignore him,\" said Acushla, horrified at the thought.\n\nBefore anyone knew what was happening, Alma had picked up her whiskey glass and drained it in one go. She took her raincoat in her poor right hand, and slid out along the banquette. Raising herself out of the seat and on to her heels, she walked slowly and deliberately toward the bar. The two men turned on their stools to face her, their backs rounded, knees splayed. Alma drew level with them and stopped. Ignoring one, she stared the other straight in the face. Standing four feet away from him for one second. Two. Three. Then, without a word, she walked on toward the door. Her attacker turned back to his companion and muttered something inaudible, his head bent down over his pint.\n\nIn the stunned seconds that followed, Acushla jumped up and ran for the door.\n\nThere was no sign of her sister on the footpath outside. Acushla looked left and right, twice, before she spotted her on the far side of the road. Alma was leaning back against the river wall, three lanes of traffic passing between them. Acushla stepped into it, giving the cars no choice but to stop.\n\n\"Alma,\" she said, when she got to the other side. \"Are you all right?\"\n\nAlma's face had drained of all color. The bones of her jaw were rigid, and her mouth was slightly open, her breath coming out in shallow gasps. She had her raincoat draped over her shoulders and Acushla saw that great shudders were passing through her, as if she had just been rescued from a cold sea.\n\n\"Oh, Alma,\" said Acushla, in surprise more than anything, and she opened her arms up wide. Her sister stepped toward her, and Acushla wrapped her in tight and whispered into her hair. \"Alma, sweetheart. It's going to be all right. I promise you\u2014it's going to be all right.\"\n\nAs the others spilled out of the pub behind them, their familiar voices bubbling above the sound of the traffic, Acushla held her sister in her arms. Aware of Alma's birdlike bones under her clothes. Her sister's skull against her cheekbone. A stray strand of her hair streaking her lips. Alma's heart was thumping against Acushla's chest. Her breath hot and wet against her neck. Her perfume raising dust-cloud memories in Acushla's mind of the perfume they used to make as children by crushing rose petals and water in an empty jam jar. Not since they were children had they been this close.\n\nAt dinner that night, Liam sat, as he always did, at the head of the long dining table. A table they had bought early on in their marriage in the expectation of having a clatter of children. Acushla sat as always to his right, facing the empty chair Connie had once occupied. When Connie left home, Acushla had suggested to Liam that he take Connie's place, so they would be looking across the table at each other as they ate instead of sitting at right angles to each other, with Acushla staring at the wall. But Liam dismissed the suggestion out of hand, determined to retain his position at the head of the table. And while Acushla never mentioned it again, it never ceased to bother her. Every time she sat down, it was with a stirring of anger that she was forced to sit to one side of him at the table while he occupied the head. Day by day that anger gathered, like drops of rain falling one by one into a bucket until suddenly the bucket was full to the brim and threatening to spill over. Sometimes Acushla feels like she's waiting for it to spill over.\n\n\"I presume you heard what happened in court today,\" she said.\n\n\"I heard he got bail.\"\n\nHe had his fork in his right hand, using it as a shovel to scoop up a load of mashed potato. Acushla couldn't bear it when he did that; in her mother's house it would never have been considered acceptable to take your fork in your right hand.\n\n\"He came into the pub while we were there,\" she told him.\n\n\"Really?\" he asked, looking up at her now.\n\n\"How can it be,\" she said, \"that he can just walk into the same pub as her like that? Surely that shouldn't be allowed.\"\n\nHe raised one shoulder and dropped the other as he weighed it up.\n\n\"I suppose if he was allowed out on bail, without restrictions...\"\n\n\"But how can it be,\" she said, \"that she has to go on with her life now, knowing that he's out there.\"\n\n\"Well hang on, he hasn't been found guilty of anything\u2014\"\n\n\"I swear to God,\" said Acushla, without allowing him to finish, \"this country does my head in sometimes.\"\n\nHe sighed.\n\n\"We do have a justice system.\" In his voice was that air of weary exasperation he liked to use with her, as if he was tired of always having to explain things to her. \"It may not work perfectly, but\u2014\"\n\n\"Nothing ever changes,\" she said, and her voice was shrill now with her own barely understood rage. \"Nobody in this bloody country ever does anything to bring about change.\"\n\n\"What are you talking about?\" he asked, his impatience getting the better of him.\n\n\"Oh, forget it,\" she said, shaking her head.\n\nHe placed his knife and fork together and stood up, his chair making a loud, angry sound as it scraped against the kitchen tiles. Without saying another word, he went into the living room, where he picked up the remote control, turned on the TV and flopped down on the couch. Through the open double doors Acushla could see the large flat-screen, with the graphics whirling for the nine o'clock news. As she picked up his plate, she heard the newsreader starting in on the headlines: \"Protests in Dublin city center as the D\u00e1il prepares to hear submissions in relation to abortion legislation.\"\n\nWith the plate in one hand and his dirty cutlery in the other, Acushla turned to face the TV. On the screen she saw shots of people bearing placards\u2014some printed, others homemade. A close-up of a young woman with WOMEN'S RIGHTS NOW written in blue marker on the skin of her cheek. Another woman had an empty buggy with her, a placard propped where a baby would normally be. The reporter's voice ran over the pictures: \"... more than twenty years after the constitutional referendum on the X case, and Ireland still has no legislation on the circumstances under which doctors are permitted to terminate a pregnancy.\"\n\nAcushla took a few steps forward, so that she was standing on the threshold between the kitchen and the living room. Liam was slumped on the couch with the remote control on his knee, his expression perfectly impassive. When the story came to an end, and the newsreader introduced a report from Syria, he didn't even blink. He remained motionless on the couch, the images on the screen reflected in his stubbornly opaque eyes.\n\nFor a second Acushla contemplated flinging the plate she was holding to the ground. She imagined it crashing against the kitchen tiles, a projectile of china splinters flying through the air. She imagined Liam turning toward her in shock. \"What the bejeezus?\" he would say.\n\nHer fingers tightened on the steak knife she was holding, and for one terrifying moment she contemplated stabbing him with it. She imagined herself closing in on him, her hand raised high over her head. By the time Liam realized what was happening, it would be too late. She would have brought the knife down on him in a single stab, driving it into the soft spot between his shoulder blades. She had seen a TV chef kill a lobster that way once.\n\nThis is how murders happen, she thought, as her thoughts spun out of control. Like a washing machine with its transit bolts loose about to enter the spin cycle, or a space shuttle malfunctioning as it left the launch pad, something seemed to have come unstuck inside Acushla's head, some small thing that nevertheless had the potential to cause a catastrophe.\n\nShe turned and walked back into the kitchen, where she forced her fingers to release the knife into the dishwater. She let the plate she was holding slide into the limp suds, drying her hands on a tea towel that she found on the kitchen counter. As she climbed the two long flights of stairs to her bedroom, she felt that her skull might explode, splattering blood and brain matter all over her spotless paintwork.\n\nReaching her bedroom, she walked to her dressing table. She sat down in front of the mirror, but for once she did not look into it. She did not unscrew the lid of her cleanser, as she normally would, to go about the task of removing her makeup. She did not brush her hair, or change her diamond earrings for sleepers. She did not reach for the little jar of Valium she kept hidden in her cosmetic bag. Instead she opened the right-hand drawer and took out a pad of ivory writing paper. Locating a biro at the back of the drawer, she began to draft a letter. Imagining for a moment that it would be a difficult letter to write, she was surprised to find how easily the words came tumbling out of her. It was almost as if she had it written already, inside her head.\n\n_Dear Sir,_\n\n_Like many women in Ireland, I have been taking a personal interest in this week's Oireachtas committee hearings on abortion._\n\n_Twenty years ago, I was forced to travel to England for an abortion because I could not get one at home. The baby I was carrying had no chance of surviving outside the womb, but my doctors said there was nothing they could do to help me, so I went to England._\n\n_I was heartbroken to lose my baby and I felt very let down by the doctors who were supposed to be looking after me. I was also angry with the politicians who were responsible for making the laws that turned women like me into common criminals. Twenty years later and nothing has changed._\n\n_Shame on them all._\n\n_Yours sincerely,_\n\n_Acushla Collins_\n\n# Connie\n\n\"Oh, Mum,\" said Connie. She was about to ask if it was true. It was on the tip of her tongue to ask, but she stopped herself just in time. By the silence in her grandmother's kitchen she knew, by the look on her mother's face, and the loopy convent-girl handwriting. Connie had only ever seen her mum's handwriting on shopping lists before, or on birthday cards, never in a letter, and it was the spelling mistakes that broke her heart. She had always suspected that her mother was dyslexic, a suspicion she had never had the courage to voice for fear of hurting Acushla's feelings. It reared its head again now as she looked down at the letter in her hand.\n\n\"Oh, Mum,\" she said, grasping at a small handful of words among the millions and millions that were at her disposal. There was a vast world of words out there, and Connie wanted to pick the exact right ones. It seemed tremendously important that she say the right thing.\n\nShe looked down at the letter again, not because she had failed to understand it the first time, but because she needed to buy herself some time. She leaned in over the table, tilting the page backward to catch the light, but the words scrambled in front of her eyes, and the longer she stared at it, the less sense she was able to make of it.\n\n\"It's a good letter,\" she said, looking up again. \"It's a really powerful letter.\"\n\nThe kitchen was lit by a single pendant light that hung down low over the large round table. Encircled by a black linen shade, the bulb threw a small spotlight over the table, leaving the rest of the room in darkness. In the corners, piles of books rose from the floor in spindly towers, while spider plants spilled down from the side tables and cobwebs older than Connie formed a canopy overhead.\n\nConnie's mother, her grandmother and her aunt were sitting opposite her, their faces pale. Three versions of the same face, carved in white stone out of the darkness; Connie had never noticed before how alike they all were. As they waited for her to say something, she had the sense that she was at a job interview. A formality hanging in the air, she found herself suddenly deprived of her fluency.\n\n\"I never knew,\" she said. But even as she said it, the memories were raining down on her. Little morsels of memories, like feathers, or snowflakes falling down out of the skies, they settled into the empty spaces in the story of her mother's life. Spaces Connie had never even noticed were there, until now.\n\nA memory of the day Ernie was born. Connie was sitting up in her hospital bed and her mother was leaning in over the bedside cot where Ernie was swaddled in a blue blanket. Acushla was wearing an oversized pair of sunglasses, which she did not remove, and behind the sunglasses Connie could see that her eyes were swimming with tears. At the time Connie had thought her a little overwrought by the experience of becoming a grandmother.\n\n\"You've been carrying this around with you all these years,\" she said, as the understanding of it continued to fall down on her. \"All these years, you've been carrying this around without telling anyone.\"\n\nHer mother nodded, and Connie watched in amazement as Alma rubbed her back in a gesture of support. Self-conscious as a mourner at a funeral, Connie reached across for her mum's hand. She took the brittle fingers in her own and held on to them, so that they were chained together across the table. She squeezed them tight, by way of apology, as she remembered with a flush of shame how hard she had fought her mother's love when she was a child.\n\nWhen Connie was a child, it was her mother's watery voice that woke her for school every morning. Her mother's anxious face was the first thing she saw at the school gates every afternoon. Her mother's fussing hands helped her into her pajamas at night, and brushed her hair, and with every gesture her mother poured out of her a love that was too weighty for one little girl to bear. Love enough for two children, it seemed to Connie now, and it grieved her to think how little of that love she had returned.\n\n\"I'm so sorry,\" she said. Her mother's fingers were grasping hers so tightly that they might have been bound together by a thin, taut piece of wire.\n\n\"Sure, you weren't to know,\" said her mum, mildly.\n\nBut that wasn't entirely true either. As a child, Connie was aware that her mother was sad, but she never stopped to wonder why. No more than she stopped to wonder why her grandmother was so eccentric. Or why her aunt Alma was so harsh. That was all just part of the landscape of her childhood, along with the endless catfights between her mother and her aunt, fights that saw Connie forever being thrown into the back of the car to be barrelled away from Alma's house as her mother babbled a litany of petty grievances at the rain-streaming windshield and other cars beeped at them and her mother started crying, again. Those arguments were always followed by reconciliations, which were followed by more arguments and more reconciliations, until the last, big argument, one as deadly serious as the others had been frivolous. (\"I knew you were a fool,\" said Alma. \"I knew you were a jealous little cow, but I never knew you were a treacherous bitch.\") Connie never did find out what that argument was about, but those words still play out in her head, all these years later. Looking at her mother now, it occurred to Connie to wonder what else there was about her that she didn't know.\n\n\"So, what happens now?\" she asked, remembering the letter she was holding in her hand.\n\n\"Well, someone will need to type it up,\" said Connie's grandmother, rising to her feet with a scrape of her chair and moving off into the gloam of the kitchen. A cave-like place of long-exhausted bulbs and unwashed dishes, empty jam jars cluttered the counters along with small heaps of unopened bills and hoarded egg boxes awaiting a new incarnation.\n\n\"I could do it,\" said Connie, turning to her mum and speaking very deliberately. \"I could type it up tomorrow.\"\n\n\"Excellent,\" said Deirdre, her huge voice welling up out of the corner. She was fussing around in a bottom cupboard, withdrawing ancient bottles of gin, and vermouth and cr\u00e8me de menthe. When she straightened up, she was breathless, a bottle of sherry in her hand, grasped by the neck.\n\n\"You might want to correct the spellings while you're at it. Your mother always was the most hopeless speller.\"\n\nShe deposited the bottle of sherry on the table with a thump and Connie stared at her, wondering were these little cruelties of hers deliberate, or were they unconscious? Did she even think about these things before she said them? If anything, it seemed to Connie recently that her grandmother was becoming more sweeping in her pronouncements, and more cavalier toward other people's feelings. She reminded her of a big old jellyfish, wafting her way through life, oblivious to the pain that was caused to others by the tentacles that came trailing in her wake.\n\n\"Macdara was the best speller of the lot of you,\" she was saying. \"Macdara won first prize in the spelling competition at school, did you know that? 'Trepidatious' was the word that sealed his victory, as I remember.\"\n\n\"How could we forget?\" said Alma, inclining her face toward her sister with a slow roll of the eyes.\n\nLining four crystal glasses up on the table, Deirdre began to pour a small measure of sherry into each. The glasses were so dusty they were almost opaque, the cuts in the crystal as black as soot. Connie took a corner of her scarf and gave the rim of her glass a discreet wipe before she lifted it to her lips. Her grandmother was struggling to untie a small plastic supermarket bag, her fingers wobbling at the knot. Eventually she got it loose to reveal an already opened pack of ginger nut biscuits. Connie stared at the pack suspiciously, thinking, Expiry date? She reached a hand out and took one, dunking it into the puddle-colored drink. The sherry tasted moldy, or was it the biscuit? She nibbled at it anyway.\n\n\"What about Dad?\" she asked, all of a sudden. \"Where's Dad in all of this?\"\n\nShe looked from one to the other of them. Saw the answer on their faces.\n\n\"So he doesn't know about the letter?\"\n\n\"This is women's business,\" said her grandmother. \"Your father gave up the right long ago to have a say in any of this.\"\n\nAnd if there was a militant edge to her voice, if there was a touch of bitterness, who could blame her? This was a woman whose husband had left her for a Moroccan man half her age. If Alma was a bit tough, maybe you'd have to be tough if you discovered that your husband was sleeping with every female reporter on the island. If Connie's mother was fragile, and prone to sorrow, well, it was very clear to Connie now why that might be. All of a sudden _everything_ seemed very clear to Connie.\n\n\" _Sl\u00e1inte_ ,\" said her grandmother, lifting her small glass and puckering her lips to sip from it. The rest of them lifted their glasses briefly in response, and as they did so, Connie experienced a shudder of vertigo. A sense of the future and the past cohabiting with the present, as if the scene in front of her was only the most minute and inadequate manifestation of a vast, non-linear reality, one that her brain could only catch the tiniest glimpse of. It was only then that it occurred to her\u2014amid the creeping spider plants and the dusty books, with her grandmother's distinct smell of never-washed lambswool and chemist-shop perfume hanging in the stale air\u2014it was then that it occurred to Connie that it was not a secret, this thing that her mother had told her. It was not a secret because at some level she had known it all along.\n\nShe remembers it.\n\nA smattering of sequential happenings, like numbered dots of ink on a page. It's only now that she realizes what it is that she remembers, now that she can join the dots.\n\nShe remembers her mother going to London. An unusual trip\u2014even at the time she would have known that it was out of the ordinary for her mother and her grandmother to go to London to see a show. Her father was busy with work so she was sent to stay with her aunt Alma, and her mother forgot to pack a toothbrush for her so Alma had to take her down to the chemist's to buy a new one. Later that day, or maybe it was another day, Alma took Connie and Nora on a carriage ride around Stephen's Green, and afterward she treated them to lunch in Captain America's. Connie ordered a banana split that came with a purple paper parasol stuck into it, and a long-handled spoon to eat it with. She still remembers the foamy taste of the canned whipped cream and the nibs of roasted hazelnuts that were sprinkled on top. She remembers the sense that Alma was spoiling her; she remembers not understanding why.\n\nIt was probably only a day or two that Connie stayed with Alma, but in her memory it seems like a week, a month, a year. When eventually Alma brought her home, it was to her grandmother's house rather than to her own, and her mother was upstairs, resting after the journey. Of course, in retrospect this made no sense. The flight from London was less than an hour, an anomaly that Connie would not have noticed at the time. She would have been distracted by the presents they brought back. She remembers her grandmother opening a suitcase out on the kitchen floor. In it was a Hamleys bag, and out of the Hamleys bag came two baby dolls, one for Connie and one for Nora. Connie remembers Alma's reaction, heard but not understood.\n\n\"Jesus, Mother! Isn't that just a little bit inappropriate?\"\n\n\"Don't be ridiculous,\" said Deirdre, handing out the dolls.\n\nNora's doll was dressed in a pink Babygro and Connie's doll was dressed in a blue one, which was fine by Connie because at the time blue was her favorite color. Connie laid her doll down on the floor and went about the work of undressing it. She had difficulty getting the poor doll's arms out of its sleeves\u2014the arm was fused into a rigid bend at the elbow, so you had to pull the fabric of the suit to breaking point to get it free\u2014but once that was done, all you had to do was pull the Babygro down over the doll's belly and yank it free of the legs. Under the Babygro, Connie discovered, the doll was wearing a cloth nappy, and under the nappy, to her horror, was a small plastic penis. She dropped the doll on to the tiles as if it had caught fire.\n\nAlma let out a squawk.\n\n\"Oh, sweet Jesus,\" she said, dry, rasping laughs coming out of her. Her hand was clasped to her chest and her eyes were streaming silent cartoon tears, tears that had been dyed bright blue by her mascara. (At that time Alma was a great woman for the blue mascara.)\n\nConnie remembers wailing.\n\n\"Would you stop that caterwauling!\" said her grandmother, moving across the kitchen with murderous intent. \"This can easily be resolved.\"\n\nShe came back with a bread knife.\n\n\"Pass me that doll.\"\n\nAlma was retching with laughter. Tears leaving wet blue runs on her white skin, like streaks of toilet cleaner on the enamel of a lavatory bowl.\n\n\"No,\" she was saying. \"Please, no. This is too much.\"\n\nConnie remembers picking the doll up by the arm and passing it up to her grandmother. Her wailing had stopped and in its place was a sticky, breathless curiosity.\n\nHer grandmother laid the doll down on the kitchen table and, using a rapid sawing motion, she removed the offending penis, slipping it into the pocket of her voluminous skirt. She handed the doll back to Connie.\n\n\"Now,\" she said. \"Problem solved.\"\n\nConnie shook her head.\n\n\"I don't want it.\"\n\n\"Don't be ridiculous. It's exactly the same now as the other doll.\"\n\nAgain, Connie shook her head.\n\n\"I don't want it.\"\n\nAnd so they stood facing each other in the kitchen, Connie aged seven, and her grandmother aged fifty-seven, and one of them more determined than the other. They would have been standing there all night if the standoff had not been resolved by Nora. Six years old and already a peacemaker, she came forward with her own doll and offered it to Connie.\n\nAnd so it came to pass that Connie got the girl doll, with her pale pink Babygro and her discreetly featureless plastic bum. Nora took the poor butchered boy doll, with his blue Babygro and underneath it a tiny rough-edged crater where his little plastic penis should have been. But life would have its revenge on Connie, by visiting on her two boy babies, one after the other, as living reincarnations of that poor rejected boy doll.\n\nConnie turned her key carefully in the front door, anxious not to make the slightest noise that might wake the boys. She slipped her feet out of her shoes and picked them up, using her fingers as a prong. With her huge fake fur muffling her movements, she swept through the living room and into the small kitchen at the back of the house. Moonlight was streaming in the back window, throwing a melancholy glow over the dinner dishes that were stacked in the sink where she had left them. A red light flashed on the washing machine where a load had finished but not been unpacked. A carton of milk stood on the counter with a spill leading away from it like a snail trail. The kettle was off its moorings and a bottle cap lay where it had fallen on the kitchen floor. Connie sighed, wondering for the hundredth time how it was possible for anyone to create such a mess heating up a single baby's bottle.\n\nShe crept up the stairs, stopping on the landing to peek into the boys' room, where, by the blue light of the nightlight, she saw Oscar asleep in his cot. With his knees pulled up under his chest and his bum in the air, he looked like a Muslim at prayer. Ernie was splayed out in his little bed with the covers thrown off him and his dinosaur pajamas all a-tangle. His hair was matted with sweat and in his right fist he was clutching a small plastic figurine he had found in the playground\u2014a miniature Woody from _Toy Story_. He had refused to let it out of his hand ever since. Connie stood for a moment in the doorway and watched her sons sleeping. She would have liked to kiss them, but she was afraid of waking them. Softly, she stepped back out on to the landing, leaving the door of their room ajar.\n\nThe light in her bedroom was on but Emmet was fast asleep, an open copy of a book about Lance Armstrong slumped facedown on his chest\u2014Connie had given it to him for Christmas but he never seemed to be able to get past the first page without passing out. She plucked the book off him, closing it and sliding it on to his bedside table. She shrugged her coat off and then her cardigan, climbing out of her leggings and tossing them on to the chair in the corner. Without even bothering to put on a nightdress, she climbed into the warm bed in her T-shirt and knickers, pulling the bobbin out of her hair and dropping it on the floor before reaching out to turn off the light.\n\nShe had only been out of the house for an hour or two, but she felt as if she had returned home from a long journey. The echoes of the evening's revelations lapping over her like waves, a sense of clouds and sky over her head, a feeling of water moving all about her, she could have been lying on a darkened beach, or in the bottom of a boat, instead of in her own bed.\n\nShe curled over on to her side for comfort, and Emmet fell in behind her, nuzzling his chin into the curve of her collarbone. Connie was reminded of those fortune-telling fish that you hold in the palm of your hand. The way they curl up is supposed to tell you something about yourself. She lay there trying to remember what it was those fish were supposed to tell you, but she was so tired that she fell asleep without ever remembering.\n\nAt some point during the night, Connie wakes to the sound of a small boy calling out in the dark. She cocks her ears and listens out for which of them it might be, identifying the cry as Ernie's. Glancing over briefly at her husband, who is fast asleep (or pretending to be), she leaps out of the bed, anxious to get to the child before he goes and wakes his brother.\n\nEver since the boys arrived\u2014one after the other in the space of less than two years\u2014sleep has become the dominating force in Connie's house. Sleep, and the lack of it, is what governs Connie and Emmet's relationship. It has become the currency of their marriage\u2014a small stack of casino chips that they barter with each other, exchanging an extra hour in bed for a sexual favor or a night on the town. Like water that has become dangerously scarce, what little sleep that is available is measured out between them with meticulous precision, each of them guarding their own share with a primitive ferocity.\n\nIn the four years since they became parents, there has been only a handful of occasions when they have both slept through the night. Between the colic and the teething, between the nightmares and the head colds and the barking dogs and the mating cats and the car alarms going off out on the street, there is seldom a night that one or the other of the boys doesn't wake up. And while at first Connie and Emmet took it in turns to get up\u2014in the beginning, Emmet even helped with the night feeds\u2014now that he is trying to finish his book, on top of teaching full-time, Connie feels in all conscience that he needs his night's sleep. So, tired as she is, when she hears Ernie calling out for her in the dark, Connie hauls herself out of bed and staggers into his room.\n\nThe room is sealed off from the world by blackout curtains, the night-light throwing outsize fish shapes across the walls. Ernie's eyes are wide open, his pupils pinned. \"There was a wolf,\" he says. In Ernie's dreams, there is always a wolf.\n\n\"There aren't any wolves in Ireland,\" whispers Connie, settling herself down on the bare floorboards beside his bed. Her back against the clammy wall, she slides her hand under the duvet to reassure her child.\n\n\"It's only a dream,\" she says, closing her eyes and using her own breathing to still his, her own hand to cool his; whether it works or not she doesn't know, because she falls asleep before him.\n\nSome time later\u2014Connie has no way of knowing how long, but it's still dark outside\u2014she wakes to find herself sprawled on the floor. A crick in her neck; her shoulder hurts where it's been resting on the wooden boards. Her nose and toes are frozen to snapping point. Terrified of waking the boys, she crawls out of the room on her hands and knees, aware even as she is doing so of the absurdity of the situation she finds herself in. Nothing matters so long as she makes it back to her own bed without waking them.\n\n\"Mum.\"\n\nShe keeps her eyes closed for a few seconds, maybe even for as long as a minute, as some hopeful part of her brain clings to the possibility that he might go away again if she doesn't open them.\n\n\"Mummy,\" he says. \"Mummy. Wake up.\"\n\nHe's pulling at her arm now, using all the weight of his stocky little body to haul on her, as if he were attempting to pull a large boat out of the water.\n\n\"Stop it,\" she says, opening her eyes to glare at him. \"Get off me.\" In her irritation, she sounds like a small child herself.\n\n\"Mum, I'm hungry.\"\n\n\"I'm coming.\" she says. \"You go on down and turn on the telly.\"\n\nAs she passes out of the room, she looks back and sees that her husband is curled up on his side. His face buried in a mound of pillows, he is sound asleep. In Connie's head, a stopwatch begins to count the seconds that he remains asleep while she herself is awake. She pads down the steep stairs to the living room, awarding herself a point for every creaking step. With every toy she stoops to pick up, with every movement of her exhausted limbs, her sense of martyrdom grows. When she steps on a small piece of Lego, the pain only serves as proof of her misery.\n\n\"Fuck, fuck, fuck,\" she says as she rubs the ball of her foot. The boys' eyes stare at her sideways from the couch. Another second and their eyes slide back to the TV.\n\nMoving across the room, Connie trips over a phone charger and curses again. She bends down and yanks the charger out of the wall. A temper raging in her, she wants to throw it across the room. Instead she slams it down on the desk, shocked to find herself up against the limits of her own personality. Forbearance, patience, calm\u2014Connie understands these things now. She understands why they are virtues.\n\nIn the kitchen, she heats some porridge and serves it up to the boys in front of the TV. She cleans off the kitchen counter and sweeps the kitchen floor. In her mind she's totting up all the jobs she's done in the house already today, measuring her tally of household chores against Emmet's. Every time she puts out a garbage bag, every time she bends over to pick up a pair of dirty underpants or wipe the pee stains off the toilet seat, the resentment sloshes around in her heart. Even as she's hating him, she hates this in herself. Never in her wildest dreams did Connie think she would ever turn into this kind of woman.\n\nConnie, who always _preferred_ the company of men to that of women. She couldn't wait to get out of school, smothered by that all-girl environment. It was only when she got to college that she found herself in her element. As a student of history and politics, she was one of the few women in a class full of earnest young men, and she reveled in being one of the lads while delighting at the same time in not being one of them. She made a point of sitting in the front row of the lecture hall, taking copious notes in green ink, before swinging her neat little bum back up the auditorium steps, pretending not to notice the effect she was having on the boys. Her shining hour came during a presentation on the causes of the American Civil War, when she set out a complicated analogy likening the countdown to the conflict to a game of strip poker between North and South. And if the mention of strip poker by a twenty-two-year-old Connie\u2014wearing her tight black polo neck and her tartan miniskirt\u2014if the mere mention of the word was enough to cause a rise in the temperature of the room, and a few coughs and splutters from the shyer boys, well that was something Connie would have taken no heed of. At that time she was of the view that there was nothing to separate the men and women of her generation.\n\n\"The only difference between the men and women of our generation is that the men pee standing up and the women pee sitting down,\" she argued at a college debate (even though she and a girlfriend had once managed to subvert this rule at an outdoor concert where the queue for the ladies' Portaloo was unmercifully long). Speaking in support of the motion that \"Society no longer has a need for feminism,\" Connie declared that the women of her generation only had themselves to blame if they failed to take up their rightful place in society. Afterward, in the college bar, between shots of tequila and the flagrant chain-smoking of Marlboro Lights (when Connie looks back on her college days she sees herself always in a cloud of smoke), she made the same case to her class tutor. While her tutor slid his hand up her skirt, Connie told him how future generations would have no need for special courses on women's history because women would be out there making history for themselves, an argument that seemed only to heighten the tutor's admiration for her, driving his hand farther up her thigh and his tongue into her ear.\n\nThe boy wonder of the history department, he was just back from Oxford with his newly earned PhD (a study of the Irish chaplains who served in the First World War). With his never-brushed hair and his murky green eyes\u2014eyes the color of an empty wine bottle with the light shining through\u2014Emmet brought a touch of rock-star glamour into the fossilized atmosphere of the history department. Heavily made-up girls lined the front rows of all his lectures; the same girls were to be found leaning against the wall outside his office, or sitting cross-legged on the floor waiting for him to come out, while back in his rooms Connie lay naked in his bed. That was where she spent most of her final year, smoking cigarettes and reading novels and listening to Emmet's music collection, when she should have been in the library. When her final results came through, she had earned herself not the first she had once hoped for, but a bare 2.1. But by then she and Emmet had moved in together, taking a one-year lease on a small flat above a Lebanese restaurant on Camden Street, and the loss of that elusive first seemed a small price to pay for love.\n\nOh, Connie, she says to herself now, as she stuffs another load of his dirty socks and knickers into the washing machine, slamming the door so hard that for a minute she's afraid she's broken the catch. Connie, Connie, Connie. How could you have let this happen to you?\n\n\"Hey,\" he says, when he comes down the stairs just after nine.\n\nFreshly showered and wearing blue jeans and a white shirt, with a pea-green lambswool sweater his mother gave him for his birthday, he smells of peppermint toothpaste and suds, his hair wet and dark. He comes into the kitchen and picks an apple out of the fruit bowl for his breakfast. He looks so handsome, and so clean, that Connie can't help but fancy him. In her oversized cardigan and her grubby T-shirt, she feels like a slob. Hair unbrushed, mascara down around her cheekbones, her hands are in the sudsy water, fondling last night's dishes.\n\n\"Well?\" he says, teasing her with his eyes.\n\n\"Well,\" she says, determined to match him monosyllable for monosyllable.\n\nHe comes up behind her and slips a hand under her cardigan, taking hold of her right bum cheek and squeezing it. He snakes the other hand round her waist, sliding it up the front of her chest and taking her right breast lightly in the palm of his hand, with his fingers splayed on either side of her nipple, and his lips worrying the back of her neck. Hands submerged in the warm, wet sink, she's shackled by him. She closes her eyes for a second, breathing in the smooth, clean feel of him.\n\n\"How about a ride?\" he whispers into her ear.\n\n\"How about an elbow in the goolies,\" she says, pushing him away from her with her hip as she takes another plate out of the sink and stows it on the drying rack. Drip, drip, drip. She hears him laughing as he moves out into the living room. He's gathering up his papers and stuffing them into his laptop bag, in preparation for the day at his mother's house. His mother has created a special workspace for him in her dining room. She stocks up on the coffee he drinks and the biscuits he likes, all in the service of the great book. A book he needs to write if he's ever going to get tenure\u2014if he's ever going to make professor, which is what Emmet fully intends to be.\n\n\"You could always take the day off,\" says Connie, leaning her head back and throwing her voice after him. \"It is a Sunday, after all.\"\n\nOf all the days that she spends alone, it's the Sundays that she finds the hardest.\n\n\"Come on, Connie. You know I can't.\"\n\nAnd of course, she does know. She knows that his deadline is only six months away. That the book has to be ready for next year's centenary of the start of the war. That it would have been written years ago if it wasn't for her and the boys...\n\n\"I know,\" she says. \"I know.\"\n\nHe hovers in the kitchen doorway.\n\n\"So, what was the drama?\" He has his laptop bag in one hand, his bicycle helmet in the other. \"Last night, what was the big drama?\"\n\n\"Oh,\" says Connie. \"It's a long story. I'll tell you later.\"\n\nAs he leaves, she hears him call out a goodbye to the boys. She hears the door close behind him and is overcome by despair. The day ahead of her seems impossibly long. The task of filling it impossibly arduous, a task made all the more difficult by the boys' racing reserves of energy. By her own epic exhaustion.\n\nAn exhaustion that lies in wait for her every morning when she wakes up; it follows her around all day, hanging like lead weights off the hem of her clothes, constantly trying to pull her down. She recharges herself with coffee, enjoying a brief burst of energy before she goes flat again. She eats, and experiences a short sugar burst, which dips again after half an hour, leaving her more tired than she was before. Like a woman in danger of being washed away by raging flood waters, she grabs on to any piece of driftwood she can find to steady herself. A mug of tea, a bar of chocolate, a sneaky cigarette. Anything that might give her the strength to muddle through another hour of another interminable day.\n\nKeep swimming, she tells herself, channeling Dory in _Finding Nemo_. Keep swimming, keep swimming. Sometimes it seems like all she has in her head these days are cartoons.\n\nShe was folding the washing when her phone rang. Her mother's number on the screen\u2014as soon as Connie saw it, she wondered had her mother changed her mind. Too late if so, the e-mail was already gone\u2014Connie had typed it up and sent it off ten minutes ago.\n\n\"So, they rang me.\"\n\n\"Okay,\" said Connie. \"We were expecting that. What did they say?\"\n\n\"It was a woman. She wanted to know if the letter was genuine.\"\n\n\"Well, that's fair enough. Did she ask you anything else?\"\n\n\"She asked me if your father knew about it.\" (Always \"your father,\" never Liam.)\n\n\"And what did you say?\"\n\n\"I said I would prefer if he wasn't told.\"\n\nConnie had a vision of the news conference, and the discussion that would inevitably take place.\n\n\"Was that it?\"\n\n\"No. Someone else rang me back afterward.\"\n\n\"Probably a reporter.\"\n\nAcushla's voice broke through with a question, as tentative as a puff of smoke.\n\n\"Do you think they'll print it?\"\n\nConnie paused, not quite sure how to answer. A confession from the wife of a former minister that she had traveled to England for an abortion, in contravention of the protection accorded to the unborn by the constitution? _Of course_ they would print it. Just try and stop them. She began to wonder did her mum have any idea what she was getting herself into?\n\n\"Oh, they'll print it all right,\" she said gently. \"There's no question, Mum, but they'll print it.\" (They would run a news story too, most likely on the front page, but Connie thought it better not to mention that yet.) \"You're not getting cold feet, are you?\"\n\n\"No,\" said her mum, with too much emphasis in her voice. She sounded like she was trying to convince herself. \"No, no. I want to do this.\"\n\nConnie wedged her mobile in between her shoulder and her ear so that she could get on with folding the dry clothes while she was talking.\n\n\"Okay,\" she said. \"Because it's not too late to pull out if you don't want to.\"\n\nShe was holding a pair of Ernie's trousers up against her belly, running the palm of her hand over them to flatten out the kinks. She folded them over once, twice, and set them down on the kitchen table, taking care to keep them well clear of the jam spill she had failed until now to notice.\n\n\"No, I do want to,\" said her mother. Her voice had a disjointed quality to it, which led Connie to think she'd taken a Valium. \"It's just... I don't know. It's just that I'm nervous, I suppose.\"\n\nConnie plucked a T-shirt of Emmet's off the clothes horse as she listened. She pinned it to her chest with her chin. Taking the sleeves, she doubled them toward the center. Folded the T-shirt over twice and set it down on the table.\n\n\"I don't know how people will react,\" her mother was saying. \"I'm a bit nervous, I suppose, about people's reactions.\"\n\nConnie reached deep inside herself, trying to find the energy to give her mother the encouragement she needed.\n\n\"I think you'll be surprised,\" she said, \"how sympathetic people will be. You'll get great support from people once they know.\"\n\nShe picked up a pair of Emmet's boxer shorts, fastening the single button on the crotch before folding them over.\n\n\"When are they going to publish it?\"\n\n\"They're talking about tomorrow. Tomorrow is the day.\"\n\n\"What about Dad? Are you worried about how he's going to react when it's published?\"\n\n\"No!\" said her mum defiantly, her voice skittering off the word. \"No, I know how he'll react. He'll hit the roof.\"\n\nAnd whereas before it had seemed to Connie that there was fear in her mother's voice, and trepidation, whereas a moment ago she had sensed a pitiful anxiety about how people would react, the mention of her father seemed to have invoked a spirit of brinksmanship in her. It occurred to Connie that perhaps her mother's newfound activism was not about women's rights, or women's freedoms and men's role in suppressing them. Perhaps it was just about one man and one woman, and the endlessly unhappy dance they were leading each other, a dance that would continue, most likely, long after this storm had passed. The thought of it made Connie feel sad and faintly sick.\n\nAfter her mother had gone off the line, she stood in the kitchen for a moment, with the phone cradled in the crook of her neck. She wondered what to do with the stray sock she was holding. Three times already this sock had gone through the wash, and still Connie couldn't find its partner. Holding it with the tips of two fingers as if it were a dead mouse, she dropped it into the bin.\n\nOne thing about living with three men is the amount of socks they have. Another thing is the amount of shoes. You'd swear you were living with a family of octopuses. Octopi? Whatever. Connie seems to spend her life putting away shoes.\n\nEverywhere she looks there are runners strewn about the floor. A pair of small red wellies abandoned inside the front door. A single sandal peeping out from between the cushions of the couch. There are slippers in the log basket and Crocs in the toy trunk; a pair of Emmet's leather brogues sitting neatly side by side on the floor at the foot of the armchair, as if the person who was wearing them had spontaneously combusted, leaving only their shoes behind them.\n\nIn a frenzy, Connie goes around the house piling shoes high in her arms like firewood. When she's gathered them all up, she finds there's nowhere to put them. She stands in the middle of her tiny house, trying to create by sheer force of desire some new storage space where none exists. Walking over to the playpen, she bends down, releases her arms and a cascade of shoes tumbles out.\n\nThe idea of the playpen was that the boys would play happily in it and Connie would sit at the table and do some work, but it has turned out the other way around. Often of a morning Connie will climb into the playpen with her laptop while the boys have the run of the house. Her latest innovation is to upend a box of cornflakes on to the kitchen tiles and let them turn the cornflakes into a quarry, running their diggers through them and shunting them around the floor, grinding the cereal into dust with the knees of their corduroy trousers. The cost of a box of cornflakes is less than two euro. The clean-up time, ten minutes of vigorous sweeping or five minutes with the hoover. The time bought, as much as an hour. If she's focused about it, she can get her blog written in the space of an hour.\n\nConnie writes a weekly blog, which she distributes to her modest Twitter following. She produces the occasional feature article for one of the national newspapers, where the editor is someone Alma knows. She conducts potted interviews by e-mail, quizzing minor celebrities about the silliest of things (What items are always to be found in your fridge? What's your favorite smell?). And while it's not exactly the career she once had in mind for herself\u2014a career with a regular paycheck, and colleagues, and lunch\u2014the cost of childcare makes it unfeasible for her to work outside the home. With a house in negative equity to the tune of two hundred thousand euro, and four people to feed on a junior lecturer's salary, Connie's career is a luxury that, for the moment at least, they cannot afford.\n\nOut in front of the house there's a stagnant river, and between the house and the river there's a green area, and it's there that Connie brings the boys to burn up some time. With a mug of tea in her hand and the sun on her face, she watches them at play, a warm weight settling over her like a mosquito net, creating a thin veil between her and the world.\n\nA dog emerges from one of the houses and the boys make a beeline for it.\n\n\"Wait, guys. Be careful,\" says Connie. But her voice sounds thin and unconvincing, like Willy Wonka. Stop. Don't. Come back.\n\nAlready the boys are throwing a stick for the dog. The dog fetches the stick and comes back with it and Ernie tries to wrestle it from his mouth while Oscar tries to mount the animal like a horse. Connie calls out to him.\n\n\"Oscar. Don't torment that poor creature.\"\n\nBut her words might as well be thoughts for all the impact they have. They might as well be speech bubbles, floating away in the air. Above the river wall the seagulls are wheeling, letting out heartbroken cries as they plummet. A woman on the bridge is throwing bread for them, and above the woman's head sits the stadium, like a big hair dryer hood. \"The bedpan\" the locals call it. But then the locals have a name for everything.\n\n\"Nice day for it,\" says Connie's neighbor, stepping out of her house and yanking her shopping cart over the threshold after her. A birdlike woman with twig-thin legs that end in big bird feet, she is well into her eighties but still as fit as a fiddle.\n\n\"Amazing,\" says Connie, smiling up at her.\n\n\"They're saying we're in for a good summer,\" says the neighbor.\n\n\"We don't get good summers.\"\n\n\"I'm telling you. The fella was on the radio this morning. He says the good weather's here till September.\"\n\n\"I'll believe it when I see it,\" says Connie. \"After everything we've suffered these last few years, a proper summer is exactly what we need, seeing as none of us can afford to leave the country.\"\n\nFive years since Connie has had a holiday, and she finds herself craving the heat.\n\n\"Well in anyway,\" says the neighbor, tucking her chin into her neck, \"it'll save on the gas bills.\"\n\nThe neighbor trundles off down the road, dragging her cart after her, and Connie closes her eyes to savor the sunshine. With her eyes shut, the light becomes liquid, pooling around her. From what seems like very far away, she hears wailing. Is it Ernie? she wonders. Or Oscar? It takes her a second to decide. She opens her eyes to see her youngest falling across the grass toward her.\n\nHe tumbles into her arms, his skull as hard as a nut under her chin, his tears big wet smears on the skin of her hand. Wrapping her arms around his squidgy belly, Connie feels so full of love for him that she could burst. She pulls him into her, burying her face in the hollow at the back of his neck and kissing him there over and over again. It's only when he's had a fall that she gets to kiss him like this.\n\n\"My baby,\" she says, breathing in the powdery smell behind his ear. The word sends a jolt through him. He stiffens and fights to get free of her.\n\nConnie wrinkles her nose.\n\n\"Ah, not again,\" she says, bending down to inspect the soles of his shoes.\n\n\"Shit,\" she says, without a trace of irony. \"Shit, shit, shit, shit, shit.\"\n\nShe prizes his left shoe off him and carries it into the front garden. Picking up a toothbrush that she keeps on the windowsill for this very purpose, and holding the sole of the shoe under the outside tap, she begins to work the dog shit out of the grooves of the runners. Her movements are driven by rage. Rage against the faceless dog owners who don't pick up after their dogs; rage against Dublin City Council, who don't enforce the fines against the dog owners who don't pick up after their dogs; rage against her husband, who at this moment is sitting at his desk in his mother's house eating chocolate biscuits. Fucking asshole, she says to herself, as she scrubs the side of the shoe and holds it under the tap, turning her face away to avoid the fine spray of shitty water that bounces back off its surface. Asshole, she says again, addressing her husband in her mind. For some reason, it seems clear to her that this is all his fault.\n\nA beep sounded loud and close and Connie looked up to see her mum's car rolling up to the door.\n\n\"I thought we'd get some coffees and go to the playground,\" she said through the open car window. \"I need to find some way of putting in the day.\"\n\nNot for the first time, Connie was grateful for her company.\n\n\"Hang on,\" she said, dashing into the house to grab her handbag. Keys, spare nappies, baby wipes... She pulled the door behind her. Already the boys were scrambling into the back of their granny's car, clambering over the large overnight bag that was there. Opening the passenger door, Connie had to move a leather vanity case that occupied the front footwell along with several pairs of shoes.\n\n\"Why am I getting a _Thelma and Louise_ feeling about this?\" She looked sideways at her mum and saw her eyelashes wobbling away behind her sunglasses.\n\n\"I thought it might be best if I stayed with your grandmother for a few days.\"\n\nShe was looking up at the rearview mirror as she pulled out. Her nails were displayed like pale pink shells along the steering wheel; the large dress ring she was wearing on her right hand threw off shards of colored light.\n\n\"Is that Grandmother's ring?\" asked Connie, even though there was no doubt in her mind that it was.\n\n\"Oh,\" said her mum, still looking into the mirror as she answered. \"She gave it to me. She can't wear it anymore\u2014her fingers are too puffy.\"\n\n\"Right,\" said Connie, thinking uneasily of the picture her grandmother had given her the last time she visited\u2014a print by a well-known Irish artist. Connie knew it to be of some value so she had tried to refuse it, but her grandmother had insisted. \"I _want_ you to have it,\" she had said, in a high, airy voice. \"Who knows who might get their hands on it after I'm gone.\"\n\n\"You don't think she's losing her marbles, do you? It's very unlike her to be giving stuff away.\"\n\nConnie's grandmother had always guarded her few precious possessions so fiercely. Only once had Connie and Nora ever been allowed to play with her jewelry, and even then she had hovered over them making sure nothing went astray.\n\n\"I suppose she's just getting old,\" said Acushla, with a shrug in her voice.\n\nFrom the back of the car Connie could hear Ernie counting cars. Oscar was pretending to count too, repeating the numbers he'd heard coming out of his brother's mouth but in a random sequence. Connie closed her eyes to quell the sense she had of some nebulous dread.\n\nThe playground was swirling with children when they arrived. Connie opened the gate, and the boys disappeared in among them. She bolted the gate behind her and followed her mum over to a picnic table.\n\n\"So,\" she said. \"I suppose there's nothing for us to do now but wait.\"\n\nHer mother sat with her shoulders slumped forward, her hands hugging her coffee cup. An oyster-colored pashmina was draped elegantly over her shoulders. Her eyes were blinking shadows behind her sunglasses.\n\n\"I feel a bit bad,\" she said. \"Sneaking out on your father like that.\"\n\n\"Oh, come on, Mum. It's no more than he deserves.\"\n\nThe more Connie thought about it, the more it seemed to her it was the very least he deserved.\n\nHer mum sighed. \"I suppose.\"\n\n\"I feel like such an idiot,\" said Connie, \"for taking his side all those years.\"\n\nShe cringes with shame as she thinks of it, what a daddy's girl she was.\n\n\"You adored your dad,\" said her mum with a weak smile. \"It was never my desire to come between the two of you.\"\n\n\"I know,\" said Connie. When the truth was that her mother never could have come between them, even if she had wanted to. There was nobody could have come between Connie and her dad.\n\nIn her mother's presence Connie was a serious child, imperious and bossy, but as soon as her father came in the door, he brought a sea breeze with him, whipping Connie into peaks of giddy froth. The less time he spent at home, the more Connie delighted in his presence. On the many, many nights when he was out at some constituency event or other, Connie would insist on waiting downstairs for him, falling asleep on the couch in the breakfast room while her mother moved about the kitchen completing her chores. The next thing Connie knew, her father would be scooping her up in his arms and carrying her up the stairs. As he slid her into her own bed, she would curl up between the cold sheets like a little guard dog who was finally off duty now that her master was home.\n\nIn later years her dad used to take her on his rounds of the doorsteps on Saturday mornings. She would hand out election leaflets, and nod and smile as he promised to help someone skip a hospital waiting list, or dodge a prison sentence by grace of his intervention. The evident devotion of his daughter was a subtle vote-getter, and something that would not have been lost on him, not for one moment. When he was appointed to Cabinet for the third time, it was Connie who accompanied him to the start of the new D\u00e1il term. She got a new dress for the occasion and a new pair of patent pumps, and her picture was in all the papers the next day. (Her mother was at home in bed with a sore throat\u2014an excuse that in retrospect seemed a bit thin to Connie, but then in retrospect there were so many things that suddenly made sense.)\n\n\"Oh, God,\" she said. \"I was such a little misogynist!\"\n\nRemembering how she loved being with her father. How she preferred her grandfather's company to her grandmother's. It was Connie's grandfather who taught her to play drafts, challenging her to epic tournaments that swallowed up entire rainy afternoons, while Sam sat by the window painting. He always had a stash of chocolate in the top left-hand drawer of his desk, and it was he who taught her to dip the end of a Time Out into her mug and use it as a straw. (\"I don't normally subscribe to these new-fangled confections,\" he would say, \"but I have to make an exception for Time Out. It's a magnificent addition to the repertoire.\") Connie has such happy memories of those lovely, languorous afternoons at her grandfather's dining table, with a large Latin dictionary wedged in under her bum, and her shoes hanging like lead weights off her dangling legs, with her grandfather's old-time music swirling in the air and his smirking blue eyes watching her as she huffed him. A sense of comradeship between them, as if they were on the same side in the game of life.\n\nAnd they _were_ on the same side, that was the truth of it. Before she even knew there were sides, Connie had sided with her grandfather. Treacherous little girl that she was, she had sided with her father. She had sided with the entire male population over the female. Because women, it seemed to Connie, were always complaining about something. They were always bickering with each other, or bitching, and if they weren't crying about something they were mounting a silent protest, one that only they knew the logic of. And it seemed to Connie that there was some disposition to unhappiness in the women of her family, some determination to be unhappy that must be located somewhere on the second X chromosome. The men seemed so much happier than the women, and so much more fun to be around.\n\n\"Mum\u2014\" she started to say, but she was interrupted by the sound of a child calling out to her.\n\n\"Mum!\" It was Ernie, from the top of the slide. He was standing gripping the handrails with his little fists. His solemn face searching out hers for approbation. Connie gave him a circus smile and a thumbs-up.\n\n\"Ah, God,\" she said as she turned back to her mum. \"Wouldn't that melt your heart.\"\n\n\"He's a darling,\" said her mum, and it seemed to Connie that in their love for the boys they were in perfect tune with each other. All the little things that mattered to Connie\u2014the visceral fear of a fever, the momentousness of a newly glimpsed tooth, the joy in kissing the velvety sole of a baby's foot\u2014her mother understood all these things. Her mother supported her, slipping her small gifts of cash to pay for makeup or a haircut. She helped her with babysitting, taking up the slack wordlessly whenever Connie needed her, whereas her father seemed to be talking to her in another register, about all the things they used to talk about before she had babies, things that were so remote to Connie now that she could barely hear him...\n\n\"You know you've been a great mum to me,\" she said, leaning forward to search out her mother's eyes behind her sunglasses. \"You do know that, don't you? Nobody could have been a better mum.\"\n\nHer mother tilted her head to one side and sighed.\n\n\"I do my best,\" she said. \"Just like you do, darling. You try to be the best mother you can be...\"\n\nAnd Connie was about to answer her when something occurred to her.\n\n\"Hang on,\" she said. \"Where's Oscar gone?\"\n\nA moment of playground panic; a second went by, and then another second, before she located him, waiting for a little girl to vacate a swing. He had his hand down the front of his trousers to fondle himself. (\"Captain Elastic\" Emmet liked to call him, because he seemed never to tire of stretching his little penis out like bubble gum, to see how far it would go.)\n\n\"Oh, Jesus,\" said Connie, covering her mouth with her hand. \"Would you look at my son.\"\n\nHer mum took in the sight of Oscar, and bit her lip and smiled. As she turned back to Connie, the smile turned into a laugh, tears forming in her eyes as she shook her head helplessly from side to side. Connie began to laugh with her, the two of them shaking with speechless laughter so that at first neither of them noticed that Connie's phone was ringing. It was hopping up and down on the surface of the picnic table. The screen displayed a photograph of Connie's father's face. A red receiver icon and a green receiver icon, inviting her to accept or reject the call. Still breathless from laughing, Connie reached out to reject it.\n\nIt was early afternoon by the time they got back to the house.\n\n\"Can we bake?\" asked the boys as soon as they were in the door.\n\n\"Jesus, lads. We're only just back from the park. Would you not think of having a nap?\"\n\nTheir two little faces turned up to hers in expectation of disappointment. For weeks she'd been promising them they could bake. For weeks she'd been postponing the promise. Maybe later, she would say, maybe tomorrow. Hoping they would forget about it, but of course they didn't.\n\n\"You promised us we could bake.\"\n\n\"All right,\" she said wearily. \"Let's bake.\"\n\nSo here they were, standing precariously on a pair of kitchen chairs, with adult-sized aprons tied twice around their bellies and wooden spoons in their hands. Flour in their hair, and in their eyebrows and their eyelashes, flour in every crevice of their clothing. With their cheeks flaming and their eyes wide and serious from the task at hand, they were blithely unaware of the spectacle they presented.\n\nConnie was just about to put the tray of cookies into the oven when she saw a sudden movement in the corner of the room. She froze with the tray in her hand. She saw, but did not react. For a second, maybe two, she stood and stared at the rat. The rat sat and stared back. Connie dropped the cookie tray and screamed.\n\nIn one swooping movement she hooked a child under each arm and climbed up on to the couch. She managed to reach for her mobile, which was on the table. Stabbed at Emmet's number; the phone rang three times and went through to voice mail. She rang his number again, and again the phone went through to voice mail. Again and again she rang, picturing the tally of missed calls mounting on the face of his phone, a measure of her growing hysteria. Seven calls she made, and still the rat sat in the corner of the kitchen, picking away at a stray crumb of cookie dough.\n\nConnie decided to make a run for it. She hefted a child on to each hip and made for the door, her bare feet barely touching the floor, as if she was running through a pit of flames. She slammed the front door behind her and threw the boys into the back of her little car, climbing into the front seat herself and starting it up. The soles of her feet naked on the pedals.\n\nOn the way to Emmet's mother's house, she kept trying his number. With each unanswered call she chalked up another black mark against him. By the time she had pulled up outside the house, she had scored twenty-seven unanswered calls. She rang on the side door, but again she got no answer. She bent down and retrieved the spare key from under the mat, letting herself into the dark hall. Turning the boys loose on their granny's fridge, she made her way up the stairs to the first floor, driven now by fury. The interconnecting reception rooms were deserted, sunlight streaming in through the back window, throwing a lopsided rectangle of light across the pale carpet. The dining room table was piled high with Emmet's books, but there were no signs of recent industry. No coffee cups, no loose notes, no uncapped pens. With something approaching relish, Connie crept up the stairs to the second floor. Past the framed portrait of Emmet in his graduation robes, his parents standing proudly on either side of him. Past a mosaic of family wedding photographs. She took in her own with a single withering glance as she charged by.\n\nShe found him in his old room. The bedroom door was ajar, and through the opening Connie could see the bottom half of his body lying on top of the single bed in all his clothes. She could hear the television, with a sound coming out of it like a swarm of angry bees; she recognized it as the drone of racing cars looping a track. Gingerly, she stepped around the open door and saw that he was fast asleep, the remote control lying on his belly and his hand placed protectively over it.\n\n\"At least he was on his own,\" said her friend Orla, palm clamped to her chest with relief. \"I thought you were going to tell me there was another woman in the bed with him.\"\n\nA pale dream of a girl, Orla's skin was stretched so thinly over the fine bones of her face as to be almost transparent. Delicate colors moved beneath the surface. At one time a coral-pink flush would show, high up on her cheekbones; another time it was the blue veins you would notice, marbling her eyelids and her temples, and again, the purple shadows under her eyes.\n\n\"But that's the worst of it!\" said Connie, putting her daiquiri glass down on the low table for a moment and looking around at her friends. (It seemed clear to her that the only thing to do under the circumstances was to get drunk.) \"I think I'd have preferred to find him in bed with another woman. For all the action he gets these days, I'd have _understood_ if he was sleeping with another woman. I'd have understood if he was sleeping with another _man_. I swear to God, I'm so knackered, I wouldn't care. It's the fact that he was sleeping alone, that's what I can't forgive. The thought of all that delicious sleep. I don't see how I'm ever going to forgive him.\"\n\n\"Hard to see,\" said Rachel, shaking her head. The only one of them who was still single, Rachel could be relied upon to take a hard line against male transgressions.\n\n\"I'd kill him,\" said Trish. \"If Ross did that to me, I'd murder him.\" (Ross who can't keep his hands off other women when he's had a few drinks. Ross who once asked Connie did she like it up the arse. I'd want to kill him too, thought Connie. If I was married to Ross, I'd happily kill him.)\n\n\"The only problem with killing him,\" said Orla, in her reedy voice, \"is that you'd be left minding the boys on your own, which would be a bit of an own goal.\"\n\n\"Oh, I've no intention of killing him,\" said Connie, with all the command of a hanging judge. \"The punishment I have in mind is much slower and more painful. I intend to make him pay me back for all the sleep he's been stealing from me.\"\n\nBending down to pucker her lips over the tip of her straw, she drained the last of her frozen daiquiri in one long slurp, sending a glacier at breakneck speed through the cavities of her skull. Even as she was gasping with the pain, she was raising her hand to order another round.\n\n\"There should be some kind of warning,\" she said, after they'd moved on to the next bar. They were standing outside on the street, smoking Rachel's cigarettes, even though the rest of them were supposed to have given up. \"They should have a public health campaign,\" she went on as she lit up. \"People should be warned about the dangers of marriage.\"\n\n\"It's not possible for men and women to live together happily,\" she said to Rachel in the nightclub. The others seemed to have disappeared, leaving just the two of them slumped side-by-side on a velvet couch. \"I don't know a single couple who manage to cohabit happily.\"\n\nShe was stunned by the clarity of this thought.\n\n\"I swear to God, I can't think of anyone I know who has a happy marriage!\"(Her grandfather was the only person Connie could think of who was in a happy relationship, which said something in and of itself about relations between men and women.)\n\n\"What about your parents?\" asked Rachel, confused. \"I thought your parents were still married.\"\n\n\"Ah,\" said Connie, thinking of the reams of newsprint that must even now be reeling off the great printing presses in the west of the city. Corralling her slurred thoughts into line, she made a mental note to buy the newspaper on her way home. She imagined how she would wobble into the Spar while her taxi meter ticked away outside. She would squint at the small print in the dark backseat of the taxi, not sure if it was the darkness or the drink that was preventing her from reading it. Already she could imagine exactly what it would say.\n\n**FORMER  \nMINISTER'S  \nWIFE  \nBREAKS  \nSILENCE  \nON  \nABORTION**\n\n# Liam\n\nLiam endured a night riddled with dreams.\n\nNormally he tended to dream only when he was traveling, or when he was sick, but for some reason on this occasion he was assailed by dreams all night long. He dreamed first of his father. A strangely prosaic dream whereby he and his father were tending a bonfire in an oil barrel in the corner of the garden. (Back then you could still light a bonfire, something Liam's own government would later move to ban, relegating one of the great pleasures of his youth to the realm of nostalgia.) Often of a summer evening Liam's father used to make a bonfire for all the packaging waste from the kitchen, tossing it on to the flames with all the absorption of a child playing with matches. In the dream, Liam and his father stood admiring the curious colors thrown up by the melting plastic, and Liam could feel his face being roasted by the flames while at his back the chill crept in from the darkening woods around him. There was no smell in the dream, only swirls of thick smoke distorting the air and the sense of his father standing beside him, solid as a tree, with his boots planted firmly in the gravel.\n\nWithout warning, the dream morphed into a nightmare. Liam was turning the compost heap over with a pitchfork when he came across his father's body. A flesh-covered hand in the early stages of decomposition, like the body of the farmyard dog that Liam had encountered in the compost heap one spring morning, warm and steamy after a whole winter of being dead.\n\n\"Jesus!\"\n\nHe sat up in the bed, choking on his own breath. Eyes wide and staring, his brain tumbling through time. He turned toward his wife for ballast, only to find the bed empty beside him. He checked his watch and saw that it was almost three, which was late for Acushla to be home but not so late as to worry him. He rolled the duvet round himself and curled into the comfort of it, falling straight back into a tunnel of dreams that were all about Acushla. When he woke again, it was just before six and her side of the bed was still empty.\n\nHis first thought was that she must have had an accident and that he should ring the Guards. He reached for his phone and was about to dial 999, but first he found himself rehearsing what he would say to them. He would be forced to identify himself, and to explain that his wife had not come home last night, and there would follow all kinds of intrusive questions and innuendo as they tried to come up with sordid scenarios to explain her absence. Putting the phone down again, he leaned over and turned on the radio, hoping to catch the news. If there had been a fatal road accident in the city during the night, or some other catastrophe that might explain why she had not come home, it would surely be on the news. He sat up in bed as he listened to the six o'clock news.\n\nThe first headline was about a tornado in Oklahoma. The second concerned an ongoing scandal about penalty points, and the third referred to the wife of a former minister who had admitted to having had an abortion. It took Liam a moment to realize that the former minister they were referring to was him.\n\nLiam's phone started ringing as the newscaster was reading out the weather forecast (another day of sunshine, announced with a mixture of disbelief and delight). He eyed the phone, identifying without answering them calls from every political correspondent in the country. Calls from the news desks of all the major newspapers, and all the broadcasters. He sat on the edge of his bed, elbows on his bony knees and bare feet planted in the carpet, with his phone in his hand. The minutes slipped by, the phone rang and rang and still Liam sat in his pajamas on the edge of his bed. Like someone who finds themselves at the bottom of a deep well without a rope, or far out at sea without an oar, he allowed himself the luxury of a period of inaction to wallow in his despair. A despair that seemed so absolute that everything that went before it now paled in comparison. The loss of his seat, even that now seemed little more than a common or garden crisis, and one that could have been overcome. But this... it was hard to see how there was any way back from this.\n\nNot since he was a child, sitting cross-legged on the bare boards of their bedroom floor as his brother beat him at Snakes and Ladders\u2014not since then had he experienced such an abject feeling of defeat. It was all he could do not to cry.\n\nHis press officer phoned at thirteen minutes past seven. He must have delayed ringing until he'd listened to _It Says In The Papers._ With a sinking heart, Liam accepted the call.\n\n\"Jack,\" he said.\n\n\"All right,\" said Jack, without any preamble. \" _The Irish Times_ is leading with it and the _Indo_ have it shoehorned into their later edition. RT\u00c9, Newstalk, Today FM\u2014you name it, they're all over it. And I just got a call from the BBC, which means it's about to travel across the water.\"\n\nLiam could hear the bounce in Jack's voice. After two years in the wilderness, his phone was hopping again. He was back in business.\n\n\"I didn't have a clue,\" said Liam. \"I'm as surprised as you are.\"\n\n\"I figured as much.\"\n\nThat was Jack for you, he always had to be ahead of the curve. God forbid anyone would steal a march on him.\n\n\"It was an ambush,\" said Liam, resting his forehead on his hand. \"A fucking ambush is what it was.\"\n\nAnd the fact that Liam found himself using the F word, that was an indication of how rattled he was. In a world where profanity sometimes seemed to be a language in and of itself, Liam had always prided himself on keeping his vocabulary clean. Often in the privacy of his own head he would unleash a string of obscenities, but seldom if ever did he allow them to escape out of him.\n\n\"I hate to be the one to ask you this,\" said Jack, with most uncharacteristic hesitancy. \"Don't take any offense now, but do you think she's left you?\"\n\nLiam raked a hand through his hair.\n\n\"That would appear to be the case,\" he said, hearing the note of self-pity that he had tried but failed to keep out of his voice.\n\n\"Do you think she's likely to do any interviews?\" persisted Jack. \"I have a horrible vision of her turning up on _The Late Late Show_ on Friday night.\"\n\n\"How would I know?\" said Liam, and this time his voice came out more like a bark. A narky, yappy little bark. \"I don't know, Jack,\" he said, in a whine now. \"All these things you're asking me, I don't have the answers to any of them.\"\n\nHe clenched his head with his hands, squeezing the bones of his skull tightly between his fingers, as if it was a football he was trying to burst. His mind was empty of everything but the fact that his wife had left him, taking his career with her.\n\n\"Fuck, fuck, fuck, fuck, fuck,\" he muttered to himself. It was only when he heard Jack's breath on the other end of the phone that he realized he had spoken out loud.\n\nShe had no understanding of politics, that was the problem. She was ignorant not just of the nuts and bolts of political life, but of the impulses that drove it. The sense of a big picture, the call to public service, all of that was incomprehensible to Acushla. She was like a peasant living in a medieval village. Her life revolved around a small circle of relationships that were confined within her familial walls. The world outside might as well not have existed, so little did it concern her. She was a creature of the microcosm.\n\nWhen Liam first ran for public office, in a constituency that was hers by birth rather than his own, his wife was no help to him. She seemed to have no knowledge of the terrain, no insight into local issues, and no contacts worth speaking of. She resisted any suggestions that she take part in the canvass, on the basis that she wouldn't have a clue what to say. On the one occasion that she agreed to hand out election leaflets outside Sunday Mass, she found the experience humiliating and resisted ever doing it again. Absorbed at that time in the potty training of Connie, and in painting the kitchen cupboards a sunflower yellow\u2014a task she insisted on undertaking two weeks before the election, despite Liam's pleas for her to hold off\u2014Acushla made it clear from the start that his political career was nothing to do with her, and if at first she did nothing to hinder it, she did nothing to help either.\n\nAlma was a mighty help, enlisting at his kitchen table amid the paint fumes and the endless cups of instant coffee. Together they worked their way through the electoral register, the way you might work your way through an old orchard, trying to shake every last vote out of it. Alma had a masterful command of the facts and figures, spitting out strings of numbers representing first-preference votes from elections before she was even born, recalling the patterns of transfers and the vagaries of each electoral area; she remembered exactly how many ballots had been cast in each election, and how each of those ballots had played out for each candidate along the magnificent if somewhat eccentric path of the single transferable vote. Late into the night she would wax lyrical about the beauty of the PR system. The wonders of the tally. The drama of the five-day count. Many years later, when Liam became Minister for the Environment and was foisted with the unenviable task of introducing electronic voting to a country enamored of the paper ballot\u2014a task that would turn into a career-defining fiasco\u2014it was Alma who warned him against it.\n\n\"You're making a mistake,\" she said. \"You'll ruin all the fun\u2014surely you of all people would prefer a paper ballot to a bloody machine?\"\n\n\"You might as well ask the lobster whether he prefers a slow death or a fast one,\" he said dolefully.\n\nShe laughed that laugh of hers that was like a pigeon cooing, and often afterward she quoted it back to him. \"How's life in the lobster tank?\" she would ask whenever she ran into him around government buildings. When Liam lost his seat, Alma sent a courier round with a live lobster packaged in a Styrofoam box, thick elastic bands wound around its claws. At the time, Liam failed to see the funny side of it.\n\n\"Well,\" said Mick, with a chuckle in his voice. \"Never a dull moment.\"\n\nInstantly Liam was sorry he'd come. He was tempted to turn away again rather than suffer his brother laughing at him. It was infuriating to Liam how Mick managed to find the humor in everything.\n\nHis brother's text had caught him at a weak moment. He was standing in his kitchen in the aftermath of the crisis meeting Jack had pulled together. Empty mugs strewn about the table, and half a digestive biscuit languishing on a plate. His phone had for some reason stopped ringing, and while the endless noise had jangled his nerves, the absence of it was almost worse. The sense of something happening behind his back, that was something Liam had always hated. He was standing staring at the phone, waiting for it to ring\u2014he was almost willing it to ring\u2014when he saw a text message come in from Mick.\n\nI'm here if you need me. M.\n\nHe was so touched by his brother's gesture of support that he could have cried. Acting on impulse, he grabbed his car keys and drove straight over to Alma's. A house he'd only ever visited on the canvass\u2014standing on the doorstep shooting the breeze with Alma about the latest polls. He had never actually been inside.\n\n\"Come on in,\" said Mick, and reluctantly Liam stepped into the blood-red hall. Framed watercolors of roses on the walls, and on the floor a pale pink carpet. He had the uncomfortable feeling that he was venturing into a strictly female domain.\n\n\"Shhh,\" said Mick, finger to his lips. He led the way down a short flight of steps into the kitchen. \"Alma's asleep upstairs.\"\n\n\"Is she all right?\"\n\nMick closed the kitchen door behind them before he answered.\n\n\"She's a bit shook. Your man being let out on bail, it was the last thing she needed. She hasn't been sleeping well since.\"\n\nLiam nodded in reply, but he was having trouble concentrating on what Mick was saying, so preoccupied was he by his own troubles.\n\n\"What can I offer you?\" Mick was asking. \"Coffee? Tea?\"\n\nBarefoot and wearing nothing but a pair of jeans and an open-necked shirt, he looked mightily at home.\n\n\"I'm all right,\" said Liam, shaking his head. With his hands in his pockets, and his fingers stroking the inside seams, he walked to the glass-paned back door and peered out into the garden. He turned to see Mick watching him.\n\n\"You could at least say something,\" he said, angry all of a sudden with his brother, although he hardly knew why.\n\n\"What do you want me to say?\"\n\n\"I dunno. That you're surprised, maybe? That you're shocked? Say anything, but don't just stand there looking at me like that.\"\n\n\"I'm not going to pretend to you that I'm surprised,\" said Mick. \"I knew about it already. I've known for a long time.\"\n\n\"Of course. Alma would have told you.\"\n\nMick shook his head.\n\n\"Actually, it was Acushla told me. A very long time ago, Acushla told me about it.\"\n\nLiam couldn't believe what he was hearing. A landslide started in his mind, the ground under him falling away.\n\n\"When was this?\" he asked, grappling for some facts to steady himself.\n\n\"About a year after it happened. I bumped into her in town. The poor kid was upset. She needed someone to talk to.\"\n\nLiam narrowed his eyes as all kinds of images unfurled in his brain. He imagined Acushla in a flood of tears, with Mick bending to comfort her. The next thing he had an image of his wife in his brother's arms. He forced his eyes tight shut, his paranoia spinning out of control. He didn't know what to be thinking.\n\n\"What are you doing here?\" he muttered under his breath. (That's what his counselor had told him to say to the negative thoughts. Ask them what they're doing there, said the counselor, and tell them to go away.) \"What are you doing here?\" he said again, spitting the words out through gritted teeth. His attention had turned backward into his own head, and he was only barely aware of his brother standing in front of him.\n\n\"Jesus,\" said Mick, throwing up his hands in a gesture of surrender. \"Don't blame the messenger.\"\n\nLiam stared at him, trying to remember what else the counselor had told him. Don't get yourself into negative spirals. Don't go over to the dark side.\n\n\"I'm sorry,\" he said, swallowing hard and drawing his hand down over his face, as if he was holding a hot cloth that he could use to wipe away his thoughts and start afresh. When he took his hand away, he saw that Mick was looking at him with alarm.\n\n\"Sorry, Mick. I'm a bit upset.\"\n\nMick nodded, and this time Liam was glad he didn't say anything.\n\n\"It's been a shitty time,\" he said, hearing the whine come through in his voice again. How he hated that whine; no matter what he did, he couldn't keep it out of his voice. \"It's been such a shitty, shitty time,\" he said. \"Do you know, Mick? There are times when I despair of things ever getting better.\"\n\n\"It's the anger,\" he said. \"I've never come across anything like it.\"\n\nThe two of them were sitting in his car outside Alma's house. Mick with his towel and togs rolled up on his lap. Liam with his car keys primed to start the engine. (\"Would you not come for a swim?\" Mick had asked, offering him the loan of a pair of togs. \"You're all right,\" said Liam. \"But I'll drive you out there if you like.\")\n\n\"Nice jammer,\" said Mick, as Liam started up the engine. \"What is it?\"\n\n\"A Skoda,\" said Liam ruefully. \"Not that that stopped them from having a go at me.\"\n\n**COLLINS SPLASHES OUT ON FLASH NEW CAR** was the headline in one of the tabloids. As if he had any choice but to go out and buy a car. What did they expect him to do, walk everywhere? Gone was the infamous \"car with a star\"; the Garda driver was also, obviously, gone. Liam was forced to dig out his old driving license, which had lapsed in the years since he had become a minister. He had to resit the test, a humiliation the tabloids devoured with glee, mocking up pictures of him wearing L plates plastered to his chest and a dunce's cap on his head, under the headline **COLLINS GOES BACK TO SCHOOL**.\n\n\"The anger that's out there,\" he said, as he reversed carefully out of the square. \"You've no idea, Mick, what it's been like. It's like they take pleasure in abusing us. The last time I went to a match in the Aviva, I had a pint thrown over me.\" (Liam was careful always to use the new name of the stadium, in deference to its corporate sponsor.)\n\n\"I suppose you can understand it,\" Mick was saying. \"There's a lot of people out there having a tough time.\"\n\n\"Yeah, well that's fair enough, but they act like they had no hand or part in it! The very people who were buying those houses they couldn't afford, the same people who were buying brand-new cars on the never-never, and apartments in Bulgaria, they're the ones who are shouting the loudest!\"\n\nLiam could hear his own voice, thin and high-pitched with indignation.\n\n\"Yeah, well,\" said Mick. \"It's always easier to blame someone else for your troubles, isn't it?\"\n\nHe was bent over in the seat, rummaging around in the glove compartment. Liam was about to tell him to take his nose out of his bloody glove compartment when Mick straightened up, brandishing a CD case.\n\n\"Dire Straits,\" he said, and his shoulders were wobbling with laughter as he opened up the case and took the disc out. He shook his head and let out a long sigh of pleasure as he slid the CD into the player. \"Oh, Liam. That could not be more fucking appropriate.\"\n\nLiam steered the car through a roundabout. Taking the two o'clock exit into an industrial wasteland, he peered out at a row of travelers' caravans parked on the verge. A fire burning on the ground. A child wearing nothing but a sagging disposable nappy. Further along the road they drove past a scrap heap, old cars piled high and ready for flattening. A repository for shipping containers; they were stacked one on top of the other like giant Lego bricks. Round the next corner they came upon the sewage plant and then the electricity generating station, and on the other side of the road, the flat blue sea.\n\n\"I have to hand it to you,\" said Liam. \"You sure know how to pick a scenic spot for your swim.\"\n\nMick laughed.\n\n\"I wouldn't mind, but it's your constituency. You're supposed to love every square inch of it.\"\n\nThe fucking cheek of him! This was the guy who had stolen their father's seat out from under Liam's nose. A seat that should by rights have been Liam's. After all, it was Liam who had taken a course in auctioneering with a view to taking over his father's practice in town. It was Liam who took the bus home from college every weekend to help out on the farm. Liam who took an interest in the local news, while Mick rarely bothered to make the trip down from Dublin. When their father died\u2014of a heart attack so sudden that he fell and cracked his head on the tap in the farmyard\u2014it was Liam who found him. There was so much blood on the ground that Liam's first thought was that he'd been stabbed; it was only when the doctor arrived that they discovered the truth. Throughout the three days of the funeral, Liam was so shocked that he hardly knew where he was. And by the time his father was buried and the grave covered over, Mick had it all sorted that he'd be the one to defend the seat in the by-election, on the basis that he was the firstborn, and that was the end of that.\n\nMick bought himself a house in town and pretended to be living there, not that anybody was fooled. Especially after he married Alma, there was hardly a weekend a month that they spent there. He retained the seat all the same, buoyed up by his youth and the rumored promise of a junior ministry, rumors that he and Alma had themselves started. During his first term, he managed to secure a brand-new public swimming pool for the constituency, and an investor to come in and put up a gleaming new shopping center on the outskirts of town, sinking its foundations right on top of an ancient burial site despite the objections of a handful of local poets and tree huggers. Once the tills started ringing in the shopping center, Mick's seat was safe for as long as he wanted it.\n\nBy that time Liam was living in Dublin and the farm was leased out to a neighbor. When eventually their mother died, they sold the house, and the farm along with it. And while Liam did nothing to save the farm\u2014he did think about it at the time but ruled it out as impracticable\u2014he mourns the loss of it still. He imagines himself walking out of a summer evening to count the cattle, with the darkness gathering in the trees and the river like a strip of tinfoil in the moonlight. He sees himself turning to walk back across the damp grass to the house, pausing at the stile to smoke the last cigarette of the day; the vision is so real to him that it's almost as if some shadowy version of himself is still there, living the life that he always assumed was his to live. The life his father and grandfather lived before him. In many ways it is more real to Liam than the life he has ended up with, a life that was never of his own choosing.\n\nAs the long, slow notes of an electric guitar sounded the start of \"Brothers in Arms\"\u2014a track Liam had always loved, never mind his brother's derision\u2014he found himself enveloped in a fog of homesickness. He was homesick for the land. Homesick for the fields, and the trees. For the one-word exchanges that said more than a thousand when you moved among people who spoke the same language as you. He was homesick for home, which was always and forever the place where he was born. No matter how long he lived in the city, Liam had never been able to call it his own.\n\nLiam parked the car up at the base of the South Wall and watched as his brother set off for his swim. He was heading for the Half Moon swimming club\u2014a whitewashed area halfway along the seawall, with iron steps leading down into the water. Normally it was only the regulars who swam there, but the sunshine had brought out the fair-weather swimmers. Young men taking running jumps off the seawall, legs pedaling for a moment in the air before they hit the water.\n\nMick negotiated the uneven surface of the wall in a gentle waddle, his shoulders tilting from side to side as he went, his head bobbing up and down like a nodding dog. Towel under one arm, he raised the other arm in greeting to everyone he passed. Not a care in the world; it was maddening to Liam to watch how he had reinvented himself.\n\nIf you'd been offering odds a few weeks ago on the chances of Mick turning himself around after the pepper grinder thing, you'd have been looking at 100\/1, maybe even 1,000\/1. And yet Liam had opened his paper the other day to see that Mick had signed himself up for the summer season of _Celebrity MasterChef_. He was pictured wearing a white apron and brandishing a wooden spoon at the launch of the RT\u00c9 summer schedule. Liam could just see it now, how his brother would charm his way back into public life, landing like a bruised plum into the soft heart of the nation's affections.\n\n\"Don't bother waiting for me,\" Mick had said. \"I'll find my own way back.\"\n\nBut instead of driving away, Liam sat on in his car. There were half a dozen others lined up on either side of him in the car park, all with their engines turned off. Some of the drivers had their doors open, with their legs stretched out to catch the sun. Summer sounds in the air. From one car he could hear a ribbon of jazz. From behind him, the hum of an ice-cream van, and the patter of children's voices. From out of the babble, one sound isolated itself so that all the others disappeared. Opening his eyes, Liam saw that what he was hearing was the radio, streaming through the open window of the car next to him. He leaned forward to turn the dial on his own radio, and there she was. The sound of her voice familiar but different, like the sound of your own voice played back to you. Liam could tell by listening to her that she was nervous.\n\n\"We're talking about twenty years ago,\" she was saying. \"There was no Internet back then. I had no way of finding out any information, no way of getting in touch with other women who'd had similar experiences.\"\n\nHer voice, as fragile as a tune played on a tin whistle.\n\n\"I was utterly alone,\" she was saying. \"As far as I knew, I was the only person this had ever happened to.\"\n\nThere were long, empty spaces between her words and Liam found himself listening to her as he had never listened before. Listening to her as a stranger would, with nothing to stand in the way of his sympathies.\n\n\"When I came back, I couldn't tell anyone about it, which made it all the harder.\"\n\nThe interviewer's voice broke in on tiptoe.\n\n\"Can I ask you where your husband was when all of this was happening?\"\n\nAnother pause, and it seemed to Liam that the world turned on her answer.\n\n\"Well,\" she said at last. \"I can't really speak for him. I can only speak for myself.\"\n\n\"But he knew about it,\" said the interviewer, pressing her ever so gently. \"Your husband knew at the time that you'd had an abortion?\"\n\n\"Oh, yes,\" said Acushla. \"He knew. Of course he knew.\"\n\n\"And yet he subsequently opposed the legislation to introduce abortion information in this country, in line with the 1992 referendum.\"\n\n\"That's right.\"\n\n\"That must have been very difficult for you,\" said the interviewer.\n\n\"It was,\" said Acushla. \"It was absolutely devastating.\"\n\nThe presenter allowed a few seconds to go by before breathing in heavily, for effect.\n\n\"Acushla Collins. Thank you so much for joining us.\"\n\nLiam closed his eyes and let out the breath he was holding. His shoulders collapsed and his head fell down on to his chest. Like an inflatable doll with the air gone out of it, there was nothing inside of him to hold him up. Nothing whatsoever in his head for a minute, perhaps even two\u2014for the entire duration of an ad break, and then a piece of music, Liam thought of nothing at all. It was only when he heard the presenter preparing to read out the comments of the listeners that he came back to his senses.\n\nHe sat up in his seat, afraid for Acushla now. He of all people knew the bile that was out there. The unholy vitriol this interview was liable to bring down on her. He was bracing himself for it, but to his amazement the comments were overwhelmingly positive. There were one or two, of course, who condemned her for the action she had taken\u2014one listener in particular said that she would have God to answer to for her actions\u2014but by and large the response was sympathetic. Caller after caller praising her for her courage, and sharing similar stories. As Liam listened to them, he was aware of the tide of public opinion slipping out from under him. He felt stranded and alone.\n\nAnd what he wanted to say\u2014what he wanted people to understand\u2014was something he had tried to explain to Acushla a number of times over the years, something she never seemed to be able to grasp. What he wanted people to know was that it was his sorrow too, the fact that they lost that baby. It was his sorrow too.\n\nLiam always wanted a big family. As a young man, he always pictured himself surrounded by multiple offspring, an older version of himself stalking the fields with his sons and walking his daughters up the aisle of the small church where he himself had been married. In those visions he wears his age on the surface, like the lead actor in a family saga who has had talcum powder combed through his hair in the makeup department, and skin-colored putty applied to his face to denote the passage of time, even though the audience knows that he's still a young man underneath. His wife is in the garden gathering flowers for the table, and she too has talcum powder in her hair and a layer of putty on her face, but underneath everyone can see that she's still the same girl he fell in love with all those years ago.\n\nWhen Liam married Acushla, he was blinded by the beauty of her. Blindsided into believing that she fitted the part, he wrote her into the story of his life, ignoring his mother's running commentary in the background. (His mother made no secret of the fact that she would have preferred him to marry a local girl.) \"She's very pretty,\" his mother said, when Liam first brought her home. And Liam knew what she meant by that; she meant that Acushla was too pretty. \"She's a jittery little thing,\" said his mother, again implying that this quality was enough to make her unsuitable for the life Liam had in store for her. \"I can't see her getting the hang of it,\" by which she meant that Acushla would never master the uneven temperatures in the kitchen range, or learn how to set the fire at the back of the grate so as not to smoke out the front room, or how to drape her clothes over the upstairs banisters so as they wouldn't be damp when she went to get into them in the morning. It had taken Liam's mother years to get the hang of all these things, and it wasn't that she doubted Acushla's ability to do the same; she doubted her willingness. And she'd have been right.\n\n\"I can't seem to get warm,\" whispered Acushla, that first night she stayed in Liam's mother's house. Wearing the diamond he had bought her on the ring finger of her left hand (Mick had already given their grandmother's old ruby ring to Alma), Acushla was curled up in a shiver between the cave-damp sheets in the single bed in the guest room. Liam crept across the bare creaking boards and she scooted over in the bed to make room for him; he had to put his fingers in her mouth as he made love to her, to stop her from giggling. To Liam's surprise, Acushla was a very noisy lover.\n\nThat night was the making of Connie, a fact that was not lost for one moment on Liam's mother. \"She's fierce healthy-looking, for a premature baby,\" his mother remarked, when Connie landed in a Dublin maternity hospital barely six months into their marriage. Acushla had insisted on having her baby in Dublin, and when she left the hospital three days later, it was to her own mother's house she went, ostensibly for a few weeks; those weeks turned into months as Liam drove up and down every day to tend the farm. Eventually he found a fellow to take care of the day-to-day running of it and he rented a house in Dublin, leaving the drawings for the new bungalow he had planned to build gathering dust on the desk in his father's study.\n\nThere was no bungalow built. No more babies born, although there were two miscarriages within the space of as many years, by which time Liam had managed to capture the fourth seat in Dublin southeast on the basis of a bare four thousand first-preference votes\u2014it was the transfers that clinched it for him. And if Acushla's last, disastrous miscarriage was not his finest hour as a husband (always in his mind he thought of it as a miscarriage, unable to bring himself to use the word \"abortion\"), it wasn't fair to say that it was his fault either. There were rumblings of another election when it happened, and if there was one word you didn't want to encounter on the doorsteps it was the word \"abortion.\"\n\nAfterward, they went for marriage counseling, attending a number of tortuous hour-long sessions in a windowless room at the top of a Georgian building that housed an Indian restaurant in its basement\u2014forever afterward Liam would associate this phase of his marriage with the smell of tikka masala. The therapist Acushla had chosen was a middle-aged woman with undyed hair. She always wore the same smock top and the same fat-person jeans. The same pair of lace-up leather sneakers. As soon as Liam saw her, he knew he didn't stand a chance.\n\n\"Your wife,\" she said to him, using the word as a stick to beat him with. \"Your wife\u2014and I'm using her words, not mine\u2014feels that her needs are being sacrificed on the altar of your political career.\"\n\nLiam nearly choked with indignation when she said that. He crossed and uncrossed his legs in an attempt to disguise his impatience. The therapist was watching his every move with a knowing look in her eyes, as if she knew exactly what he was thinking. As if she knew exactly what he was going to say before he even opened his mouth.\n\n\"What my wife fails to understand,\" he said, looking unblinkingly at the therapist as he tried to ignore the sound of Acushla's sobbing, \"what she seems determined not to understand, is that I am not a free agent. I'm a member of a government that has been elected by the people to do a job.\" (He was Minister for Defense at the time, a fact he assumed the therapist was well aware of.) \"I have a most sacred job to do, and while I understand that Acushla and I have had our personal difficulties recently, I am nonetheless obliged to rise above them and look at the bigger picture.\"\n\nThe therapist dipped her head a little and rested her chin on her knuckles, without ever taking her eyes off Liam. Not since he was seven, and he and Mick had been called into the local priest's office to answer for the theft of some candles from the shrine in the church (which of course Mick was guilty of), not since then had Liam had the sense that someone could see right through him, right through to the wrinkled little walnut of guilt at the center of him, a guilt he had no sense of ever having warranted. How to explain to this woman the injustices that had assailed him his whole life?\n\n\"I'm just doing my best here,\" he said, appealing to her for some sympathy. But he should have known better.\n\n\"Has it ever occurred to you that you might be abdicating responsibility for the personal, and using your political duties as an excuse? It sounds to me like your public persona provides you with a pretext to distance yourself from what's happening at home.\"\n\n\"Ah, now,\" he said, taking ahold of the armrests of his chair and raising himself up tall. \"I'm not going to sit here and listen to you filling my wife's head with those kinds of ideas. I won't have you handing her ammunition to use against me. I won't stand for it.\"\n\nThe therapist looked up at him with a vicious glint in her eyes.\n\n\"Oh, really?\" she said. \"You and whose army?\"\n\nThey were all against him.\n\nThat's how it seemed to Liam now. Starting with his own mother, who had always sided with Mick. His twin brother, who had always delighted in putting one over on him. Even his wife never once took his side in anything. It seemed to Liam that the whole world was against him, and through no fault of his own.\n\nHe was always a good son. A good brother. A good husband. A good public representative, but still and all the voters had deserted him. After twenty-five years treading the footpaths of their bloody constituency, trampling in their dog shit and picking up their discarded chips bags, they had cast him out. A quarter of a century, during which time there wasn't a single night of the week but he was attending the removal of some constituent who had died, or a residents' association meeting. Not a Saturday morning but he was at a school fair, his arms full of cling-filmed banana bread and freezer-bagged fairy cakes, and the ear chewed off him about the need for a new classroom, or an extra resource teacher. After twenty-five years of this slavish community service they had dumped him, and for what?\n\nAs Liam drove back past the power station, past the sewage works and the scrap metal yard and the abandoned shipping containers, he began to talk to the radio, mounting a calmly reasoned defense of his record in government; it was a defense he had rolled out a hundred times in the past few years in interviews with print and broadcast media, but he continued to pursue it, because in his mind every word of it was true.\n\n\"We brought about the greatest period of prosperity this country has ever known,\" he said, cocking his head to one side as he spoke in his most patient and reasonable voice. \"We tripled child benefit. We banned smoking in public places. We brought peace to the island.\"\n\n\"Well that's all very well,\" the interviewer would say. \"But you also brought the country to ruin. You beggared the banks. You sold our sovereignty down the river. Generations to come will end up having to pay for the mess you made of things.\"\n\n\"No, S\u00e9an,\" said Liam, furrowing his brow as he spoke to the windshield. \"You see, that's where you're wrong. Yes, we suffered an economic crisis, yes, the effects of it have been the cause of much hardship for a great number of households, but the Irish people are nothing if not fair, and it's simply not fair to place the blame at the door of one party. What happened was not the fault of any one government. What happened here was the result of a global economic crisis...\"\n\nSo engrossed was Liam in the argument he was making, so determined was he to finally convince his imaginary audience of the reasonableness of what he was saying, that he failed to observe the yield sign at the approach to the roundabout. He failed to check the traffic coming from the right, and sailed straight into the path of a Louth-registered BMW. The BMW swerved to avoid him, letting out an angry beep. Heel of the hand on the wheel, Liam beeped back at him, three long defensive wails. As he drove on, eyeing the BMW driver warily in his rearview mirror while at the same time checking ahead of him in case a Garda had witnessed the incident, it seemed to him that he didn't have a friend in the world.\n\nIt was only when he stopped at the traffic lights on the Beach Road and indicated to turn left for home that he thought of one last port of call. Indicating right, he pulled on the wheel in preparation for a last-minute change of lane.\n\n\"Dad,\" said Connie, when she answered the door.\n\nShe was wearing dark glasses and her face was pinched and pale, her lips set in a grim straight line. Liam's hopes of a fair hearing plummeted at the sight of her.\n\nShe took a step back, pulling the door in on herself. The hallway was so small that she had to press back against the coatrack to let him through. A huge flower arrangement was balanced precariously on a side table at the bottom of the stairs; the vase wobbled precariously as Liam brushed by it and Connie went to steady it.\n\n\"Nice flowers,\" he said. And as a joke, \"What did he do?\"\n\n\"Don't go there,\" she said, in an ominous voice.\n\nLiam ducked his head to pass into the living area, plastic toys crunching under his shoes as he went.\n\n\"Boys,\" said Connie. \"Are you not going to say hello to your grandfather?\"\n\nThe boys were playing on the floor behind the couch, blankets and quilts draped over the dining chairs to form a den. One by one they emerged on their hands and knees.\n\n\"Hiya, lads,\" said Liam. He bent down from the waist to greet them, shaking hands with them as if he was on the campaign trail. They stared up at him, not quite sure what to make of him. He had his hands in the pockets of his sports jacket, and nervously he jangled the coins that were in there. The boys waited to see if this was a prelude to a gift. A conjuror's trick that would end in him producing a coin from his pocket, or better still a chocolate bar. Once they realized it wasn't going to happen, they ducked back into their den.\n\n\"Will you have tea?\" asked Connie, going to put the kettle on without waiting for an answer.\n\n\"Sure,\" he said, and he followed her into the kitchen, his hands still in his pockets. With the chairs all in use for the den, there was nowhere for him to sit. He paced down as far as the back wall and turned round.\n\n\"It's a bit cramped for you all here, isn't it?\"\n\nConnie stared at him with a puzzled look on her face.\n\n\"What do you mean?\"\n\nShe had pushed her sunglasses back on her head and Liam could see that her eyes were washed-out-looking. He found himself wondering had she been crying, and if so why.\n\n\"You must be very tight for space. The four of you.\"\n\nConnie shrugged.\n\n\"People raised ten kids in these houses. I think we can manage with two.\"\n\nHe turned to peer out into the backyard, where a pair of garden chairs occupied the small patio area, leaving barely enough room for the boys' trikes and scooters.\n\n\"It would be nice for the lads to have a garden,\" he said, craning his head up to search out the sky.\n\n\"We do have a garden.\" She was lining a pair of mugs up on the kitchen counter, and dropping tea bags into them. \"It's small, but it's still a garden.\"\n\nLooking out the window, Liam saw where Connie had installed creepers along the side fences. She'd planted shrubs and a maple tree, with a bird feeder hanging from its branches.\n\n\"I reckon the market might be bottoming out,\" he said, turning away from the window. \"It mightn't be the worst time for you to think about trading up.\"\n\n\"Hang on,\" said Connie, standing in the middle of the kitchen with a milk carton in her hand and an expression of disbelief on her face. \"Is that what you came here to talk to me about?\"\n\nHe paused, not sure what to say.\n\n\"You're unbelievable.\"\n\nThe way she looked at him, they might as well have been strangers.\n\n\"Mum's left you. She's camped out in Grandmother's house, doing back-to-back interviews about the abortion you somehow neglected to tell anyone about. The nation's up in arms at the thought of your blatant hypocrisy, and you're standing here talking to me about property prices. Are you completely mad?\"\n\nListening to her, it did seem to Liam that he wasn't handling this particularly well.\n\n\"I think I'm in shock. I wasn't expecting any of this, Connie. I had no idea it was coming. Your mother didn't think to warn me about it.\"\n\n\"You still don't get it, do you?\"\n\nShe put the milk carton down on the kitchen counter and took a step toward him. Her face was all screwed up as she spoke, as if she'd been sucking a lemon.\n\n\"It's always someone else's fault, isn't it? That's the thing with you. Everything that happens is someone else's fault.\"\n\n\"Ah, now\u2014\" he started to say, but Connie wouldn't allow him to speak.\n\n\"I don't want to hear it,\" she said, throwing her hands up in the air. \"I've no interest in hearing it. Don't you see? There's nothing you can say that will make it okay.\"\n\nLiam stared at her, struggling to understand what she was saying, this daughter of his who had once been such a great little friend to him. When did Connie stop being his friend? he wondered. It seemed to him that everything had changed between them once she started having babies.\n\n\"Come on, Connie. Don't you go turning against me now. You and me were always such great pals.\"\n\nConnie shuddered, and closed her eyes. When she opened them again he could see that she was full of tears, her mouth open like a bird so she could breathe.\n\n\"Yes,\" she said sadly, nodding her head and wiping her nose with the back of her hand. \"We were great pals. But I'm not taking sides with you against my mother anymore. I can't do it.\"\n\n\"So everyone's against me.\"\n\n\"It's not that everyone's against you,\" said Connie, and there was a whistle of urgency in her voice now, like the sound the old kettle used to make on the Aga in Liam's mother's kitchen when it was gone past the boil. \"It's just that you're in the wrong and you won't admit it. If you could only acknowledge it, then maybe there might be some hope for you.\"\n\nShe turned away from him and placed her hands on the kitchen counter, letting her head hang down on to her chest, like an athlete resting after a long run. Their two mugs of tea sat untouched on the counter beside her. Liam stood for a long time and watched her, his head slowly churning all the things she'd said.\n\n\"So what should I do?\" he asked her at last.\n\n\"What should you do?\" she repeated, turning her face to look at him.\n\nHe nodded, and she stared at him.\n\n\"I don't think I've ever heard you ask anyone for advice before.\"\n\nAnd come to think of it, Liam hadn't either\u2014he couldn't remember ever seeking anything other than professional advice, in the sanctity of a consulting room.\n\n\"Would you believe me when I tell you I'm asking for it now?\"\n\nConnie paused, and it seemed to him that she was weighing up whether to help him or not.\n\n\"One word. That's all I have for you.\"\n\n\"Okay.\"\n\n\"You might want to try saying you're sorry. Say it like you mean it. If you say it often enough, people might even start to believe you.\"\n\n\"Sorry,\" she repeated as she pushed him out the front door. With the flat of her hand on his back, she pounded him a few times with what could almost have passed for affection.\n\nSo, here he is again outside Acushla's mother's house.\n\nHe's standing underneath the acacia tree in the front garden, where he spent many a night when he was courting her, and the only thing that's missing is the cigarette. He finds himself wishing he never gave them up.\n\nHe bends to pick a stone up from the gravel, tossing it against the windowpane, but the noise it makes is hardly audible and there's no response. He takes another stone, tossing it harder this time, but this one bounces off the wooden window frame. The third stone he throws strikes the center of the bottom pane, shattering it instantly.\n\n\"Oh, shit!\" Liam doubles up, chest to his knees with his hands clutching his head as two startled faces appear in the top half of the window.\n\n\"Oh, shit,\" he says with a lurch of despair, as he comes to terms with the fact that things are still getting worse instead of better.\n\n\"Sorry,\" he calls out as the front door opens and Acushla and her mother appear in the doorway. Wearing long nightdresses and backlit by the bright hall, they have the appearance of two welcoming saints with the light of heaven shining behind them. Liam experiences a strange sensation of weightlessness, as if he has died and is about to enter the afterlife.\n\n\"I'm really, really sorry,\" he says, taking the steps two at a time so that before he knows it he is standing before them. Acushla's expression is stony but Liam has the impression that Deirdre is trying to suppress a smile.\n\n\"Can I come in?\" he asks, appealing to his mother-in-law rather than his wife.\n\n\"I suppose you had better,\" says Deirdre with great gravitas, pulling on the heavy old door to admit him. Acushla turns without so much as a glance at him and leads the way down the back stairs to the kitchen. Deirdre rolls her eyes and follows her daughter down, leaving Liam to bring up the rear. Past the empty spaces on the walls, spaces Deirdre has preserved for thirty years so that everyone can see where her husband removed those paintings that were his on the day he left her. \"Excuse the cold,\" she likes to tell her visitors, \"but it's not worth the cost of heating the whole house for me _alone_.\" (Whereas Acushla's father's flat is always as warm as the toast he likes to burn and scrape before he eats it.)\n\n\"Tell her I'm sorry,\" says Liam, addressing himself to Deirdre rather than Acushla.\n\nAcushla won't even look at him, so Deirdre has agreed to act as a conduit between them. With the three of them sitting round the large table, the low-hanging light casting shadows on their faces from above, the scene has all the appearance of a seance, except that the communion is between the living rather than the dead.\n\n\"Tell her I'll do anything to get her back. Anything she wants, I'll do it.\"\n\nDeirdre repeats this to her daughter and Acushla listens, her eyes resting with great composure on her mother's face. Liam can see that she's thinking about it.\n\n\"You heard him,\" says Deirdre. \"He says he'll do anything.\"\n\n\"Tell him I only have one condition,\" says Acushla, after a pause.\n\n\"No problem,\" he says straight away, addressing his wife's impassive profile.\n\nHe turns to Deirdre. \"Whatever it is, tell her I'll do it.\"\n\nDeirdre waits, alert as a croupier, for Acushla to show her hand.\n\n\"I want him to give up his place at the head of the table.\"\n\nWith the crown of her head high and proud, and her hands still in her lap, she enunciates every word with gentle precision. \"Every day for the rest of his life, I want him to sit across from me at the table.\"\n\nDeirdre blinks at her as if to say, _What?_ Is that all you're going to ask for?\n\n\"That's all I want,\" says Acushla, and she turns her head slowly to face Liam. Liam looks her in the eyes, and in them he sees something that he never knew was there. All these years, and never did he have the faintest inkling that Acushla had in her the stuff of political genius.\n\n\"You're on,\" he says, nodding slowly and looking at her with a new respect.\n\nThere were dozens of screaming teens gathered outside TV reception when they arrived.\n\n\"One Direction are on after you,\" explained the young woman with the clipboard who emerged to greet them. She led them up the stairs and ushered them into the green room, offering them tea or coffee, which they both declined. She guided them toward a sofa, waiting for them to sit down before she spun away again. Never had Liam been so aware of them as a couple as he was now, sitting nervously upright on the unyielding sofa like a pair of immigrants; all they had with them was their story.\n\n\"Liam,\" said the producer in greeting, \"Acushla, thanks very much for coming in to us.\"\n\n\"Thanks for having us,\" said Acushla, her face tilted upward in a practiced smile.\n\n\"Now,\" said the producer, his head spinning round as he looked for the young woman with the clipboard. \"We need to get Liam and Acushla down to makeup.\"\n\nThe way he said it, Liam had the feeling that he was talking in code. The young woman came and led them away down breeze-block corridors, up bare concrete stairwells and down more concrete stairwells until they arrived in the makeup department. They were put sitting side by side facing a huge mirror, the two of them draped in hairdresser's capes.\n\n\"You've gorgeous skin,\" said the woman who was doing Acushla's makeup. \"If only everyone had your skin, my job would be a lot easier.\"\n\nLiam could see that Acushla was pleased. She was shining with all the attention she was receiving. A divine grace settling over her like a bridal veil, he was reminded of the day he married her. He remembered how she had appeared in the doorway of the church on her father's arm. With her head bare and her golden hair adorned only by a single white camellia, she made her way down the aisle with glorious ease, her head bowed with pleasure under the weight of all the admiring attention. Suddenly Liam felt very sad at his failure down the years to give Acushla the attention she needed. It seemed to him now that it would have cost him nothing to give her a bit of attention.\n\n\"Nervous?\" he asked her, talking through the mirror.\n\n\"No,\" she said, with a smile. And considering this was her first time on TV, her level of calm was incomprehensible to him. He still developed a pit of fear in his stomach every time he prepared to go on air. Just as bad now, if not worse than the first time, despite all the media training he'd had over the years.\n\n\"If you're going to apologize, the trick is to do it properly,\" said the communications strategist he had hired to prepare him for this interview. \"No excuses, no whining. Just say that you were wrong, that you regret the way you handled it, and that you wish you'd done it differently.\"\n\nLiam was still repeating those instructions to himself when the host popped his head in to makeup to greet them.\n\n\"Liam,\" he said, pumping Liam's hand up and down. \"Good to see you.\"\n\n\"So brave of you,\" he said to Acushla, leaning down to bump his freshly made-up cheek against hers. \"Don't worry, I'll be gentle with you.\" And with a slow wink he was gone.\n\nAs Liam and Acushla waited hand in hand in the wings (make sure you walk onstage holding hands, the communications strategist had said), their host was straightening his tie and firing a quip into the waiting studio audience to warm them up. The newsreader on the nine o'clock news was wrapping up the bulletin with one late-breaking headline. Liam could see her face on the backstage monitor, he could see her lips moving and the measured movements of her head, but the volume on the monitor was turned off so he couldn't hear what it was that she was saying.\n\n\"Two minutes to air,\" said someone as the ads started rolling. Liam took a deep breath and squeezed Acushla's hand tighter while they waited for their cue.\n\n**Breaking news...** Three Irish human-rights activists are to be deported from Israel. Tarquin Kilpedder (73), Emer Barry (37) and Nora MacEntee Collins (28) were arrested in the Gaza Strip on Wednesday morning and taken to Givon Prison in the Israeli city of Ramla, where they've been held ever since. The three activists were in the area as volunteers with an international organization that provides solidarity to fishermen and farmers in Gaza.\n\n# Nora\n\nThe first thing Nora saw when the electric doors slid back was her parents, standing side by side at the welcome rail, like something that happens to you in a dream, when people from different parts of your life converge on you without explanation\u2014you might find your dentist and your first boyfriend together in your mother's kitchen, or your grandmother and your yoga teacher sitting in the front seat of a car while you sit in the back. Her brain gave a lurch as she tried to construct a narrative that would explain their presence here together. Her mother stepped out from behind the railing, bizarrely colorless with her undyed hair. She looked like an unpainted plaster model of herself.\n\n\"Nora,\" she said, looking down at Nora's feet. \"What in God's name happened to your shoes?\"\n\nThree days in an Israeli jail and the only thing her mother was worried about was her shoes. Nora looked down and saw the unpainted toenails, the smear of toe jam in the curve between each toe. The patches of raw skin where her feet had repeatedly burned and peeled while she was working in the fields.\n\n\"They never gave them back to me,\" she said, watching for her mother's reaction. \"They knew we'd look like nutters without our shoes.\"\n\n\"Well,\" said her mum, coming forward to hug her. Nora expected to be released from the hug immediately, but instead her mother hung on to her. For what seemed a very long time she did not let her go, and Nora was forced to settle into the hug. With her cheek lying against her mother's soft cashmere shoulder, and her eyes open on to the bright colors of the juice bar at the arrivals gate, she had a sense of what it must have been like to be a baby.\n\n\"It's good to have you home,\" said her mum, and Nora could have sworn that she had tears in her eyes.\n\n\"Nora,\" said her dad, coming forward to lift the backpack off her shoulder. He gave her an awkward squeeze, the backpack bumping against the both of them. \"Is that all the luggage you've got?\"\n\nA white light assaulted her, and a fraction of a second later there was the sound of a camera shutter. Her dad whirled round with the deep swing of a shot-putter, to confront the photographers.\n\n\"Now, lads,\" he said, putting up a stop hand.\n\nNora's mum put an arm around her and went to bundle her past the cameras.\n\n\"Wait a minute,\" said Nora, slipping out from under her arm. \"I haven't done anything wrong.\"\n\nShe turned toward the snappers, peeling off her sweater as she did so, and veterans of the media though they were, neither of her parents had the presence of mind to step out of the shot. The next day the papers would all publish different variations of the same photograph. A picture of Nora standing proud in the arrivals hall at Dublin airport, wearing a Free Gaza T-shirt, with her parents lurking behind her like common criminals.\n\nThe house was smaller than she remembered it, and more cluttered. The smell inside impossible to conjure up until you stepped back into it, but once you did, it was hard to imagine how you could ever have forgotten it. A smell of lingering peat dust and her mother's old-rose bath oil, with a base note of cheese that's been left for too long in the fridge.\n\nNora stood in the hall. Hemmed in between the bulging coatrack and the Mrs. Delany prints on the wall, she felt as if she was caught between two worlds. The world she was about to enter seemed unreal to her, or maybe she was unreal to it. A distance between them, like an encounter with a friend you haven't seen for a long time. She was reluctant to move any deeper into the house.\n\nShe was aware of a terrible sense of loss, but what it was that had been lost she could not have said. She felt as if she had left her body behind her in that hot little cell in Givon Prison. The memory of the dead air was still on her skin, the struggle to breathe in enough oxygen. \"Breathe shallow,\" Emer had said to her. \"Try to breathe shallow.\" For three days she and Emer had lain bunk over bunk, silent sometimes and sometimes talking, banging on the steel door of the cell for water, shouting out for Tarquin and waiting to hear Tarquin shout back. She could still hear the sound of his jaunty old voice, heard through the walls like a snatch of dialogue from an opera, in all its exaggerated bravado. Nora missed Tarquin already. She missed Emer too, and the simple meals they had been served, on blue plastic trays, of rice and cucumber and tomato. (Nora was a vegetarian and could not eat the chicken her captors offered her.) She missed her prison cell, she realized with surprise. While she was there, she had thought of nothing but the desire to get out. Now, given the choice, she would almost have gone back.\n\nThere were five plump silver fish lined up on a chopping board on the kitchen counter. Five dead eyes staring up at the kitchen ceiling. The presence of five raw fish in her mother's kitchen was only slightly more bizarre than the presence of Nora's father, a presence that had still only partially been explained.\n\n\" _Celebrity MasterChef_ ,\" said Nora again, not sure she had understood him correctly.\n\nShe was looking suspiciously at the fish.\n\n\"What are they?\"\n\n\"Sea bass. According to the fella in the fish shop, they're a bugger to bone.\"\n\nWith a tea towel slung over his shoulder and a grubby apron strung under the overhang of his belly, he was brandishing a long, thin filleting knife.\n\n\"The skills test,\" he said, eyeing the fish dolefully. \"That's the thing I'm most scared of. You wouldn't know what they'd throw at you.\"\n\n\"Will you've tea, Mick?\" asked her mum. She was taking china cups out of a cupboard underneath the counter and setting them on the table.\n\n\"Pour it out for me,\" he said, \"and I'll drink it while I'm doing this.\"\n\nNora was struck by the normality of their conversation. The way they spoke to each other was entirely normal; that was what was so weird about it.\n\nShe looked around the kitchen and saw all the familiar things on the walls\u2014the Edward Hopper print her mother had bought in MoMA, the Clinton\/Gore election poster she'd brought back from Washington, with a Hillary flyer wedged in between the glass and the frame. The china cups they were drinking from were part of the set Alma had been given as a bride\u2014as a child, Nora had broken one of them and hidden it in the bin rather than confess; as far as she knew, her mother had never noticed.\n\nNora sipped at her tea. Jet lag swirling around her, she still had the noise of the plane's engines in her ears. The chill of its air-conditioning system was still in her, and the smell of airline meals heating in the galley. A yearning for home that had not been satisfied by her return.\n\n\"Tell me about Acushla,\" she said, in an effort to anchor herself. Her voice sounded a false note, and she hardly recognized it.\n\n\"Oh, yes,\" said her mother. \"You missed all that.\"\n\nThere was something different about her mother but Nora couldn't put her finger on it. Something strangely rigid in her demeanor. Even the way she was sitting, perched like a schoolgirl with her hands tucked under her thighs, it was not like her.\n\n\"I've just been reading about it on the plane.\"\n\n\"Yes,\" said her father, lifting a fish spine by the tail and flicking it into the bin. \"I'm afraid she's usurped my place as the country's media darling.\"\n\n\"She's shown great courage,\" said Nora's mum solemnly, and Nora looked at her closely, not sure if she was being sarcastic or not.\n\n\"Unbelievable,\" said Nora, shaking her head.\n\nFor years Nora had despaired of her family's cynicism. For years she had tried to stir them to action, urging them to join her campaign against the use of Shannon airport by the U.S. military. (\"And watch all the American multinationals pull out of Ireland?\" said her father.) She had tried to sign them up to petitions in support of Falun Gong. (\"I would,\" said her grandfather, \"but I'd be afraid it might prevent me getting a visa for China.\") To join a vigil for the release of Aung San Suu Kyi. (\"How does she always manage to look so _elegant_?\" said Acushla.) That her meek and pampered aunt should become the figurehead of a protest movement, this seemed to Nora a most unlikely turn of events.\n\n\"How's Liam?\" she asked.\n\n\"Oh, he's trotting around after her like a wee lamb,\" said her father, with no attempt to disguise his pleasure. \"She has him by the bollocks, and she knows it. Now,\" he said, straightening up and looking at the clock. \"When would you ladies like to eat?\"\n\n\"Why don't you have a bath?\" said Nora's mum. \"I'll run you a nice hot bath and then we'll all sit down to dinner.\"\n\nUnder cover of the sound of gushing water, Nora slipped up to her old bedroom and closed the door quietly behind her. Like a guest in a stranger's house, she was desperate for a moment on her own. She sat down on her bed, pulling her knees up to her chest and resting her chin on them. Taking stock of her belongings, she searched for a pattern that would allow her to stitch herself back together again. The dream catcher in the window. The patchwork quilt on the bed. The snow globes she had collected as a child as mementoes of her parents' travels. The inspirational verses she had copied out and pinned on to her bulletin board. _Do not go where the path may lead, go instead where there is no path, and leave a trail_ (Ralph Waldo Emerson). _We must not fight injustice, we must make it invisible_ (Gandhi). Above her desk there was a framed portrait she had painted of herself in junior school, with oversized round eyes in a big round head. As she looked at it now, it seemed to her that she had captured something of herself in that self-portrait, an aloneness that she had not been aware of at the time.\n\nNora was fourteen when her parents split up, and at first she didn't notice that anything was wrong, because she was so used to her dad being away. It was the height of the peace process and her dad was Junior Minister at the Department of Foreign Affairs, so he spent more time in Belfast than he did at home. (\"Up there riding anything in a skirt,\" her mother told her later, and all Nora could think was, What kind of a person tells her daughter that?) Nora used to see her dad every night on TV. Walking into Stormont buildings, walking out of Stormont buildings. Standing outside Stormont buildings with a ragbag of other politicians and officials. To Nora they had the look of a band of puppets about them. Some muppetry quality in the strange assortment of shapes and sizes, all clumped together. Nora's dad was easy to pick out because he stood head and shoulders above everyone else.\n\nThe peace talks regularly went through the night, and often they went through the weekend, so it was no surprise to anyone if Nora's dad wasn't home much. But even after the peace process had ended, after the historic conclusion, for which the nuns led tentative prayers of thanks at Nora's school assembly\u2014after George Mitchell had flown home to New York and Tony Blair had gone triumphantly back to London, and the hotel rooms of Belfast had emptied themselves of reporters and camera crews, and the news bulletins had gone back to reporting other things that were happening in other places\u2014even then there was no sign of Nora's dad. It was only when she went through his wardrobe and discovered that his clothes were gone, only then did Nora figure out that he wasn't coming back. The final confirmation came not from either of her parents, but from a tabloid newspaper that Nora caught sight of one day in the Spar. She was on her way home from school and she had stopped to treat herself to a bag of popcorn with some coins she had found in her pocket when her parents' wedding photo caught her eye, looking out at her from the front page of one of the papers. \"The Good Friday Disagreement\" the paper had dubbed the split.\n\nNobody thought to tell Nora. There was never a day when her parents sat her down and explained to her what was happening (\"We both still love you... It's not your fault\"). At no point did either of them outline a plan to her (\"You'll spend weekdays with your mother and weekends with your father\"), let alone ask her how she felt about it. She was just another thing to be divided between them, at their convenience.\n\nAs Nora lingered in the slowly cooling bath her mother had run for her, with the slither of rose oil on her limbs and a scented steam misting up the windows and mirrors, she felt as if nothing had changed. Her parents had not asked her permission before they broke up her family; it seemed equally unfair to her now that they had not asked her permission before putting it back together again.\n\nHer mother's voice trilled up the stairs calling her down to dinner and Nora closed her eyes. She slid her bum along the oily base of the bath, pushing her knees into a deep bend. She arched her spine and craned her neck backward, allowing her head to sink down below the surface, where no further sounds could reach her. Just the sound of her own breath, amplified by the water and eerily alien.\n\n\"Habeas corpus,\" said her father, continuing a conversation that had begun in the car. \"I think we definitely have them on habeas corpus.\"\n\nHe was hunched over the kitchen counter, putting the finishing touches to his potato dauphinoise. Blowtorch in hand, he was carefully scorching the scales of scalloped potato that formed the surface.\n\n\"I still haven't worked out if you're pulling my leg,\" said Nora, turning to her mother for help.\n\n\"Oh, it's no joke,\" said her mum, looking up for a moment from setting the table.\n\n\"Were you even informed as to what they were charging you with?\" asked Nora's father. Poised on his toes like a jewel thief, he settled a perfect square of creamy caramelized potato into the center of each plate.\n\n\"You're missing the point,\" said Nora as she watched him.\n\nWith great care he leaned a naked white fillet of fish up against each potato square. Armed with a squidgy plastic bottle, he began to pipe small blisters of red sauce around the edges of the plates, dropping a sprinkling of crispy bacon crumbs on top of each blister.\n\n\"No bacon for me,\" said Nora, holding up her hand to stop him, but she was too late. She had not eaten meat since she was fourteen, but her dad seemed incapable of retaining this simple fact about her. The last time she visited him in Brussels, he suggested they go out for a steak.\n\n\"Okay,\" he said, throwing his hands up to beat an imaginary clock. \"That's it.\"\n\nBefore serving up the plates, he insisted on photographing them from every angle, muttering to himself as he went.\n\n\"They're going to get me on the presentation. I still haven't nailed it on the presentation.\"\n\nNora carried the first two plates over to the table and he followed with the last one.\n\n\"Well,\" he said, raising his glass of wine as he fell into his chair. His forehead shining with sweat, his face flushed from his efforts. \"To the two most important women in my life; it's good to have us all round the same table again.\"\n\nNora looked across nervously at her mother, expecting a snide remark, but instead her mum smiled, her eyes moving from Nora's dad to Nora before settling back on him. And again Nora saw something curiously unsure in her demeanor, something she had never seen in her before.\n\n\"They refused to allow you to make a phone call,\" her dad was saying. \"That's another grounds for complaint.\"\n\n\"You're missing the point,\" said Nora again, turning to face him. \"This isn't about me. This is about the people of Gaza. They're the ones who are being denied their rights, on a daily basis. They're the ones who are living in an open prison. That's the thing that matters here, not what happened to me.\"\n\nHow many times had they had this conversation? Or another one just like it. It was a well-worn path they liked to tramp along together.\n\n\"It's _wrong_ , what's happening in Gaza.\" She put her fork down and fixed her eyes on her father.\n\nHe looked at her as if he could barely contain his amusement.\n\n\"You agree with me,\" she said. \"I know you do! When it comes down to it, you believe in all the same things I do. But you keep coming up with excuses not to do anything about it.\"\n\n\"You're right,\" he said, with a maddening smirk, and she was reminded of the way he used to tickle her when she was a child. On weekend mornings, when she came into their bed in her pajamas, he used to catch her by the legs and tickle the soles of her feet. The more she wriggled and squealed, the harder he tickled her; he wouldn't give up until she begged for mercy.\n\n\"I'll tell you what,\" he said, serious all of a sudden. \"Once I've been crowned _Celebrity MasterChef Ireland_ , I might consider lending my support to one of your crusades. I could go back to Gaza with you, we could do a Gaza version of _I'm a Celebrity... Get Me Out of Here_.\"\n\nNora stared at him, not sure whether to laugh or cry.\n\n\"Ah, Nora,\" he said, choking on his own suppressed laughter. \"I'm sorry, I just get such a kick out of winding you up...\"\n\n\"I can't deal with him,\" said Nora, turning to her mother for support. \"How can you deal with a person who refuses to take anything seriously?\"\n\nHer mother seemed not to have been following the conversation. She was holding her fork with her left hand and her right hand was lying limp on the table, curled into a loose fist.\n\n\"Mum,\" said Nora. \"What happened to your _fingers_?\"\n\n\"I can't believe you didn't tell me.\"\n\nThey had moved into the living room after dinner, taking their wineglasses with them. The evening was too warm for a fire, warm enough even for Nora and her mother to remain bare-shouldered and barefooted. The sash window was open as far as it would go, a deepening blue sky outside. The stadium had absorbed all the glory of the summer night into its fish-scale skin.\n\nNora and her mother were curled up in opposite corners of the huge three-seater couch. Nora's dad was in the kitchen, practicing how to debone a chicken. (\"Is there no cleaver in this house?\" he had asked, before realizing his mistake.) Nora closed her eyes, unable to shake off the image of a huge square blade. She hugged one of the couch cushions to her chest so as to still her shuddering.\n\n\"I can't believe I didn't _notice_.\"\n\nNow that she had noticed, it was impossible to take her eyes off them. Those poor maimed fingers.\n\n\"Why didn't you tell me?\" she asked quietly, even though she knew the answer. Maybe if she had been in the habit of getting in touch more often, maybe then she would have been told.\n\nHer mother's eyelids fluttered momentarily, untold clouds passing behind them as she let the question go unanswered.\n\nNora sighed from deep down in her chest as it dawned on her that everything that had passed between them over the years had been leading to this moment. Every instance of neglect on the part of the mother (deliberate or accidental, it didn't make a difference, not at the time). Every act of revenge on the part of the daughter (noticed or unnoticed by her mother). Every step they took away from each other had in fact been bringing them closer to this. A moment that was always going to come. Perhaps not necessarily in this room, on this couch\u2014it could just as easily have been in a hospital ward with one or the other of them coming round to consciousness again after a road accident. It could have been at a funeral, the two of them standing over someone's grave. It could have played out in a hospital mortuary, with one of them laid out on the slab\u2014but it was always going to come. Nora experienced a great wave of relief that it had not come too late, this moment of reckoning.\n\n\"I would have come back. If I'd known, I would have come back.\"\n\n\"I know. That's why I couldn't tell you.\"\n\nHer mother's stockinged feet were knuckled up under Nora's bum, her mother's toes wriggling endlessly. Nora would have liked to change her position, to get away from the toes, but she did not want to appear to shy away from this awkward intimacy.\n\n\"It must have been terrifying,\" she said, treading oh-so-gently. She was trying to feel her way into the story, like someone venturing into a dark room, afraid of knocking something over.\n\nHer mother frowned the question away.\n\n\"I didn't think they were going to hurt me. I thought they'd just take whatever they could find and go. It never occurred to me that they would be prepared to hurt me.\"\n\nNora squeezed her eyes half shut and covered her mouth with her hand, unable to contain the images she was seeing in her mind.\n\n\"I can't bear to think about it. It's horrific.\"\n\nHer mother answered her in a matter-of-fact voice.\n\n\"These things happen to people,\" she said. \"And we are people.\"\n\n\"Oh, Mum,\" said Nora, and the way she said it, it sounded like she was pleading with her. Tears in her eyes, she didn't trust her voice to speak, so instead she reached out for her mother's hand. Cradling it between her own, she began to stroke it very gently, taking care not to avoid touching the poor wounded fingers.\n\nThere was a question Nora wanted to ask, but she didn't know whether to voice it or not.\n\n\"Mum,\" she said carefully. \"The people who did this to you. Did they catch them?\"\n\nHer mum reached out for her glass of wine.\n\n\"Oh, yes,\" she said, and her voice had changed. The tone of it was more brittle now, and more brusque. \"Yes, they've charged one of them. But the trial won't be until next year. He's been released on bail in the meantime. He's probably out there right now, breaking into someone else's house.\"\n\n\"Will you have to give evidence?\" asked Nora, as her father appeared in the doorway, his apron still on and smeared now with chicken guts. He placed the palm of his hand on the crown of her mother's head. Resting it there with great solemnity, as a priest might when giving a blessing.\n\n\"I thought we said we'd cross that bridge when we came to it?\"\n\nNora's mother stretched her neck back so that she was gazing up at him, and the look she gave him was one of such trust and dependence that Nora couldn't help but be touched.\n\n\"It's good to have you home,\" he said to Nora, using the word without a hint of self-consciousness. She thought of what she would have given over the years for the three of them to be together here like this, in quiet harmony. Now that it was happening, it was hard to know what to make of it.\n\nOn her way up to bed, Nora stopped to peek into her mother's bedroom and saw, with one slow sweep, her father's shoes under an armchair, one of his shirts hanging off the curtain rail, and a copy of _The Economist_ on what used to be the unoccupied side of the bed.\n\nAs a teenager, Nora was a spy. A dedicated listener at doors. A rifler of drawers. She always made it her business to know what man her mother was seeing. What woman her father was sleeping with. In her travels between her parents' separate residences, she was ever alert to the appearance of a new toothbrush beside the sink in her mother's house, or a man's sock behind the bedroom door. She took note of any face creams she found in her father's bathroom, or herb teas in his cupboards, using every detail to construct a picture a criminal profiler would have been proud of. And while she never stopped to consider the ethics of what she was doing while her parents were philandering with other people, now that it was the two of them together, she had an acute awareness of their privacy, and a desire not to violate it.\n\nWaking up on her first morning home, she lay in bed and listened as her father pottered about downstairs. She heard the stairs creaking as he carried a tray upstairs for her mother. She heard the springs groan as he climbed back into bed beside her. The sound of their voices through the wall; she couldn't make out what it was they were saying, just the sweet trickle of their conversation, like a nearby stream. Listening to it made her feel lonely all over again.\n\n\"Well,\" her father said, when finally she came downstairs. \"The dead have arisen.\"\n\nHe was standing at the cooker stirring a pot of custard for his ice cream. Barefoot and unshaven, he was wearing the peony-print kimono Nora had given her mum for her fiftieth birthday. When he turned toward her, Nora saw that the kimono was unbelted and hanging open on to a T-shirt and a flabby pair of boxer shorts. She was assailed by a mental image of her father's penis under there, sticky with sex.\n\n\"Here,\" he said, holding out a spoon to her. \"Taste this and tell me what you think. I've infused it with rosemary.\"\n\nNora turned her face away in disgust.\n\n\"There's no doubt about it,\" she told Connie. \"They're definitely sleeping together.\"\n\nShe had arrived at her cousin's door without warning. Walking across the bridge in the cool of the morning, she had skipped down the short flight of steps to the green area in front of Connie's house. A terrace of tiny red-brick houses, like something you might see in an English soap opera; every evening on the dot of half seven the _Coronation Street_ signature tune rang out down the row, followed by the _EastEnders_ sig at eight.\n\nNora peered through the front window, expecting to see the boys tumbling toward the door, but all she saw was Connie coming down the stairs. Long legs first, then a sage-green baby-doll nightie trimmed with black lace, until finally Connie's startled face appeared. She lit up when she saw her cousin.\n\n\"Nora!\" she squealed, throwing open the door.\n\n\"Did I wake you?\"\n\nConnie flung her arms around her. \"When did you get back?\"\n\n\"Only last night. You're the first person I've visited.\"\n\nNora stepped inside the door.\n\n\"Where's Emmet?\" she asked, looking around the empty room. \"Where are the boys?\"\n\n\"They've gone to the playground,\" said Connie. \"It's all part of a new regime. Every Sunday morning, come hell or high water, he has to take them to the playground.\"\n\n\"Oh?\"\n\n\"This is what I do while they're gone,\" said Connie, holding up a copy of the IKEA catalogue with a TV smile. \"I lie in bed reading the IKEA catalogue. That's how sad my life is...\"\n\nShe stopped herself and stood staring at Nora as if she'd only just noticed her standing there.\n\n\"I can't believe you're here! Come on into the kitchen. I'll make you coffee. I'm so happy to see you!\"\n\nThey settled on low stools in Connie's kitchen, surrounded by the smell of washing powder and soggy cereal. The sound of the tumble dryer stopping and then starting up again in the opposite direction. Connie had tucked the hem of her negligible nightie between her thighs and crossed them, to hide her knickers. Her hair was falling in loose spirals from the crown of her tiny head, and the shadows round her eyes were darkened by her day-old eye makeup. It seemed to Nora that Connie was getting more beautiful, if anything. She could never but be aware of Connie's beauty, with a tribal brew of jealousy and pride in equal measure.\n\n\"So,\" said Nora. \"It seems that my parents are sleeping together again.\"\n\nConnie wrinkled her nose.\n\n\"Can we be sure about this?\"\n\n\"There's no doubt about it. They're definitely sleeping together. Last night I heard sounds coming from their bedroom. At one stage I heard her give a kind of a squeal.\"\n\n\"Oh, Jesus,\" said Connie, baring her teeth.\n\n\"I can't get my head around it. They're getting on so _well_. It's like the divorce never happened. It's really weird.\"\n\n\"I don't find that weird at all. Your parents always got along well together. Way better than mine did. You got the feeling that they actually _liked_ each other.\"\n\nNora nodded as she acknowledged the truth of this.\n\n\"Your house was such a happy house,\" said Connie. \"Before your parents split up, it was a really happy house. Whereas the atmosphere in _my_ house\u2014do you remember? It was like Berlin before the wall came down.\"\n\n\"Well, now we know why.\"\n\n\"Yes,\" said Connie with a sigh. \"Now we know why.\"\n\nThey stared into each other's eyes, and it was as if the music had stopped and the parcel of family history they had been passing back and forth between them was being opened. One layer at a time, they were moving closer and closer to the nugget of truth that lay at the center of their lives.\n\n\"He used to bully her,\" said Connie, in a voice that was little more than a whisper. \"Do you remember? He used to bully her in front of us.\"\n\n\"I know,\" said Nora. \"I remember.\"\n\n\"He was a pig to her, and I did nothing to stop it. I feel so bad about that now. Especially now that we know what she was going through. I feel so bad for taking his side.\"\n\n\"Ah, Connie, you were just a kid.\"\n\n\"Still.\"\n\n\"She seems to be getting her revenge on him now.\"\n\nConnie smiled.\n\n\"I _almost_ feel sorry for him. He had to beg her to take him back.\"\n\n\"Do you think they still love each other?\"\n\n\"I don't know. I don't think love has much to do with it. I've reached the conclusion that relationships between men and women are all about power. It's all about the balance of power, and if you can't get that right, then it's a fight to the death.\"\n\nAnd it seemed to Nora that Connie wasn't talking about her parents anymore\u2014it seemed to Nora that she was talking about herself.\n\n\"Is everything okay between you and Emmet?\"\n\nConnie bent her head and looked down into her lap. She sighed, and for what seemed like a long time, she didn't speak.\n\n\"You don't have to talk about it if you don't want to.\"\n\n\"It's not that. Part of me does want to talk about it. I'm just afraid that if I start, I won't know where to stop.\"\n\nShe looked up at Nora and smiled an apologetic little smile, and Nora didn't know what to say to her. What could Nora say to Connie about husbands, and babies? Nora, who had never even had a proper boyfriend.\n\n\"Is it that bad?\"\n\n\"Oh, I don't know. I'm so tired I can't tell. We're still talking to each other, which is something, I suppose. We're still sleeping together, whenever we have the energy. But basically it's more like a war than a marriage.\"\n\n\"For all I know, that's normal.\"\n\nConnie burst out laughing.\n\n\"Oh, Nora. I missed you!\"\n\n\"I missed you too,\" said Nora, even though strictly speaking that wasn't true. When Nora was away, she missed her creature comforts all right. She missed her Barry's Tea and her Walnut Whips and she missed the salmon avocado rolls she and Connie craved from the sushi place on Capel Street, but curiously enough, she did not find herself missing people. She didn't miss her grandparents, even though she adored them both. She didn't miss Connie's boys, or Connie herself, even though Connie was the closest thing she had to a sister. Of all the people in the world, it was Connie who Nora loved best, and yet she did not miss her, not even for a moment, when she was away. Like a hot air balloon, the stays that tethered her to her family had to be severed every time she went away. And it was only when she returned that she felt herself being reattached.\n\n\"I wish you wouldn't keep disappearing off like that,\" said Connie, sticking her bottom lip out in a show of petulance. \"I don't like you being away so much.\"\n\n\"I always come back.\"\n\n\"Yeah, but not for long. I bet you're planning your next escape already.\"\n\nNora decided not to mention the e-mail she'd received from Amnesty last night\u2014an e-mail about the human-rights situation in Syria, it had her thinking of her next move already.\n\n\"My boys hardly recognize you, Nora. Every time you come back, I have to explain to them who you are.\"\n\nHow to say this without hurting Connie's feelings?\n\n\"Don't take this the wrong way, Connie, but sometimes I feel like there isn't much for me here.\"\n\n\"But this is your _home._ \"\n\nThat word again. Nora kept hearing it, and every time she heard it, she wondered what it meant.\n\n\"This is where your family is.\"\n\nAnd of course Nora couldn't say it, but that was the very thing that kept her away. The fact that they all lived within a square mile of each other, it was claustrophobic to Nora. She wondered how they could bear it.\n\n\"I know, Connie. I know, but...\"\n\nHow to explain it? The feeling she had when she was away of being free. Whereas here she felt so _un_ free. Unfree of her parents and her past, unfree of her feelings, feelings she had been trying for so long to get away from.\n\n\"Nora,\" said Connie, looking at her with the kindest of expressions in her beautiful eyes. \"Whether you like it or not, this is always going to be your home.\"\n\nNora's mother was not doing well.\n\nAt first she had seemed to everyone to be coping magnificently with what had happened to her. Like a building that has withstood the force of an earthquake, or a tree left standing after a storm, everyone was so relieved that she had survived it, they just assumed she would go on surviving it, day after day, into the future. But the cracks began to show over time, and the more time went by, the more she seemed to be struggling.\n\nNora would wake in the night to hear her padding about the house. The next thing, she would hear the sound of Eckhart Tolle's voice filtering through the walls; it reminded her of a Speak & Spell, with the cadences in all the wrong places. (It was Acushla who had suggested that Alma listen to _The Power of Now_ \u2014in a bizarre role reversal, Acushla was now giving Alma lessons in survival.) In the mornings Alma was groggy with the effects of what Nora suspected were sleeping pills, and in the evenings she was wakeful and anxious. She was getting headaches too, with increasing regularity. Incapacitating migraines that left her incapable of working for days at a time. Nora never remembered her mother suffering from migraines before.\n\n\"How are you doing?\" Nora asked, as she leaned into the darkened bedroom. The curtains were scrunched tight against the bright evening, the daylight outside rendering them luminously transparent. Her mum was lying fully clothed on top of the covers, a silk scarf draped over her eyes as a blindfold.\n\n\"I'm okay,\" she said, speaking in a tremulous whisper, as if there was a sleeping baby in the room and she was terrified of waking it.\n\nCarefully Nora perched on the edge of the bed, laying her hand on her mother's thigh.\n\n\"Can I get you anything?\" she whispered.\n\nHer mum shook her head, moving with the utmost care. The sense of some explosive substance that would detonate if disturbed.\n\n\"I brought you some water,\" whispered Nora. She held the glass up to her mother's lips for her to take a sip.\n\n\"Delicious,\" came the whisper. She mimed a thank-you that was audible only as a series of little puffs of breath. Nora took the glass away and set it down on the bedside table, careful not to make a sound as she was doing it. She let her eyes wander over the room, falling on shapes that were strangely unfamiliar in the shadows. A large mirror that gave the illusion of a door opening on to another room. A white sequined evening dress hanging from a cupboard door in a plastic dry-cleaner's sheath; it looked to Nora like a ghost about to rise up for a haunting. A black satin nightdress coiled on the floor; in the darkness it could have been a crouching cat.\n\nShe listened to her mum's breathing, shallow and deliberate, and she tried to follow it with her own breath. From outside the room came the sound of an ice-cream van. The jingle stopped abruptly and then started again just as suddenly. Nora listened as the sound of it receded into the distance, and when it was gone, she sat on in the silence. There was something restful about sitting in the stillness of her mother's illness. It was like being in the presence of a big drugged bear.\n\nThree days Nora had been home, and in all of that time nobody had asked her a single question about the month she had spent in Gaza. Oh, her father had asked her about the conditions inside the jail, and Connie wanted to know whether she'd met a man while she was away (which she hadn't), but nobody had expressed the remotest interest in the work she'd been doing, not until she visited Macdara.\n\n\"Now,\" he said. \"I want to hear all about it.\"\n\nHe was smiling at her with his eyes, letting his gaze rest on her the way very, very old people do, even though Macdara was only in his late forties. Macdara's ways were at odds with his appearance, which was curiously youthful. (\"What do you expect from someone who has nothing to do all day but take long baths?\" Nora had once heard Acushla say. \"No wonder he has no lines. He has nothing in the world to be worrying about.\")\n\nNora and Macdara sat at a wonky wrought-iron table that had been set into a small sunny patch in the corner of Nora's grandmother's garden. A garden that her grandmother kept untended as a testament to her husband's abandonment of her. From the ravening creepers to the perennially unpruned trees, everything pointed to the absence of a man in the house.\n\nWhen Nora and Connie were children, they used to spend hours acting out elaborate games in this garden. They played Sleeping Beauty here, with Connie the sleeping princess and Nora the valiant prince cutting his way through the brambles to reach her. They played Snow White, using Macdara's studio as the cottage of the Seven Dwarfs. They played Hansel and Gretel, with their grandmother cast (unbeknownst to her) in the role of the witch. For Nora, the garden still held the echoes of all those childhood games.\n\nShe had come to visit her grandmother, but found only Macdara at home.\n\n\"She's gone to the charity shop,\" he told her. \"For some reason she's taken it upon herself to bring all her old clothes to the charity shop.\"\n\n\"Really?\" said Nora, struggling to take in this information. Never once in her whole life did she ever remember her grandmother throwing anything out. (She's a Womble, Nora's grandfather used to say, forgetting that Nora was too young to remember the Wombles.)\n\n\"She's taken to decluttering all of a sudden,\" said Macdara. \"I've no idea what prompted it, but it's most out of character.\"\n\nHe led Nora outside, offering her a glass of lemon barley water, which he fetched from the gloom of the studio. They sat side by side, with their chairs turned to face the sun, sipping at the cloudy drink. Nora closed her eyes and allowed the silence to swarm round her as she waited for Macdara to say something. You wouldn't know what Macdara might say. He might just as easily say nothing.\n\nShe set her glass down on the slanting surface of the table and stretched her legs out in front of her to catch the sun. She slipped her feet out of her flip-flops, enjoying the wormy squelch of soil between the blades of uncut grass. She was aware of the sound of a train scuttling along the track behind the garden, aware of the cloying scent of the honeysuckle that had been allowed to run wild along the walls. She was aware of Macdara's presence, as tenuous as the company of a wild hedgehog, or a hare that had ventured into the garden. You were afraid to move in case you might scare him off.\n\n\"Lemon barley water is a good thing,\" he said eventually. \"I'm not sure whether people appreciate what a good thing lemon barley water is.\"\n\n\"It's delicious,\" said Nora. \"It reminds me of being a child.\"\n\nMacdara looked at her curiously.\n\n\"I keep forgetting,\" he said, \"that our little Nora's all grown up.\"\n\nAnd it might have sounded creepy coming from anyone else. If you didn't know Macdara, you might think this was a creepy thing to say. If you listened to Nora's mother talking about him, you might assume that he was handicapped perhaps, or an invalid of some sort. If you saw him in the midst of his family, silent and watchful where everyone else was so noisy, you would know there was something not quite right about him. But Nora had grown up with a child's view of her uncle, and that view of him had never left her. She saw the slight figure bulked out by cardigan on top of cardigan, turning from his desk as she and Connie pushed open the door of the studio. She saw the childlike man who would gracefully receive the tea they'd made for him from weeds they'd found in the garden. She saw the solitary audience member, sitting on a chair in the studio while she and Connie acted out their shambolic plays for him.\n\n\"What was it like?\" he asked her. \"What was it like, this place you've just been to?\"\n\nNora paused before she answered. There was something about Macdara that made you want to choose your words very carefully when you spoke to him. You had the sense that you were holding an instrument in your hands and that Macdara was listening out for the sound you would make with it; you had the feeling that he would know if you sounded an untrue note.\n\n\"It's a sad place,\" she said. \"I think it's the saddest place I've ever been.\"\n\n\"I'd like to hear about it,\" he said.\n\nAnd there was something wistful in his voice, this middle-aged man who had only once in his life been out of Ireland. He had been to France, many years ago, on a scholarship that was supposed to last a year but ended after just a week. A trip that was the butt forever afterward of many a family joke. It was Alma who suggested he write an account of it entitled \"A Week in Provence.\" Macdara smiled with good grace every time that joke was rolled out, but he never ventured off the island again. Talking to him about her travels, Nora felt like an ancient adventurer who brings back tales of distant lands, along with a trunk full of strange trophies, with which to mesmerize the people at home.\n\n\"It should be a beautiful place,\" she said. \"That's what's so sad about it. At one time it must have been the most beautiful place, this ancient city beside the sea. But there's almost nothing left of it now, only rubble and sand.\"\n\nMacdara had his head inclined toward her, his pale eyes endlessly patient.\n\n\"I can just picture it. You describe it well.\"\n\n\"The people have no way of making a living. They try to fish, but their fishing boats are taken away from them. They try to farm, but the snipers shoot at them while they're working in the fields. Even the olive trees,\" she said, and in her own voice she could hear a shrill note of indignation. \"The settlers break the branches off the olive trees to spoil the harvest.\"\n\n\"But that's terrible!\" said Macdara. \"That's just terrible.\" He was shaking his head as if he found it hard to believe what she was telling him.\n\n\"Yes,\" she said. \"It is terrible.\"\n\nAnd she wondered, if Macdara could see this so clearly, why did everyone else find it so hard to see how terrible it was?\n\nWhen the Garda came to the door, Nora's first thought was that it must be something to do with her deportation from Gaza.\n\nA Friday morning, her dad had left at the crack of dawn for another day of filming. Acushla had just arrived for coffee, along with Connie and the boys, bringing with them a box of small cakes. Instead of leading them straight into the kitchen, Nora's mum took the boys out into the back garden to show them the nest of newly hatched robins in the crook of the wall. The boys were more interested in the cakes than they were in the robins, but they were too polite to say so.\n\n\"Do you know a funny thing?\" said Alma, as she stepped back into the kitchen, closing the glass door carefully behind her. \"I've spent my entire adult life talking about politics and the economy and the state of the nation\u2014but I find that none of that's of any interest to me anymore.\"\n\nConnie was setting out the cups, but she paused and looked up at Alma, waiting to hear what her aunt was going to say.\n\nAlma was still standing at the kitchen door. She was gazing into midair as she spoke, while the rest of them stopped what they were doing to listen. Nora had a queer sense that they were all of them actors on a stage, waiting for Alma to deliver a key soliloquy before the action could resume.\n\n\"The thing is, I find I've no interest whatsoever in what's happening in Iraq, or Afghanistan,\" Alma was saying, as she walked over to the fridge and took out a carton of milk. \"I couldn't tell you who's in power in Israel. Or what's happening in Libya. It's like I've no space in my mind for any of those things anymore.\"\n\n\"Welcome to my world,\" said Connie with a snort. \"What you're describing is the state of my brain ever since I had children.\"\n\nAlma filled the milk jug and paused with the carton still in her hand.\n\n\"The only thing that interests me now is when those little robins are going to take flight. It's the only thing that seems real to me.\"\n\nShe had her head tilted to one side. A wistful note in her voice, as if she regretted the insight.\n\n\"I know it sounds mad, but I could quite happily sit here all day waiting for those little birds to fly away.\"\n\nAcushla stared at her, and Nora could see that she was still adjusting to this new incarnation of her sister.\n\n\"When can we have the cake?\" asked Ernie.\n\nHis brother was beside him, springing up and down on the spot.\n\n\"When can we have the cake? When can we have the cake? When can we have the cake?\"\n\n\"Now,\" said Acushla, coming to attention. \"Now is the time for the cake. Come and help me put them out on the plate.\"\n\nThe kettle boiled and switched itself off and Nora poured some water into the coffeepot to warm it. She was just emptying the water out into the sink when the doorbell rang.\n\n\"It's probably Grandmother,\" she told Connie. \"Would you mind getting it?\"\n\nBut instead of her grandmother, it was a female Garda who stepped into the kitchen (Nora's dad would have called her a ban-Garda). She had her hat in her hand, and her hair rolled into a neat little bun at the nape of her neck.\n\n\"Sorry to disturb you,\" she said, stepping forward tentatively with one navy-trousered leg. \"I can see that you're in the middle of something.\"\n\n\"Not at all,\" said Acushla. Best girl in the class, she was already extending an arm to usher the Garda forward. \"Will you have some coffee?\"\n\nThe little boys stood staring at the Garda, the cake for the moment forgotten.\n\n\"Honestly, I'm grand,\" said the Garda, looking at Acushla only briefly before turning her attention to Alma. \"I just wanted to have a quick word with Alma. There's something we need to bring to your attention.\"\n\nThe movement of air in the room was suspended as they all waited for the Garda to explain.\n\n\"A woman was killed in a house in Blessington last night. The state pathologist is only arriving at the scene now, so we've no official comment as of yet. But we believe she was killed by an intruder.\"\n\nNora became aware of her own breathing, hard and slow. Connie's breathing, and Acushla's breathing, and the little boys' and her mother's\u2014it seemed to Nora that they were all of them doing nothing but breathing. Breathing the Garda's words in.\n\n\"We've arrested a man and we're holding him in Blessington Garda station. But we thought you should know, it's the same man who's been charged with the attack on you.\"\n\nIn the pause that followed, the sound of the doorbell rang out, much louder than it had been before. Connie slid sideways out of the room, and a moment later she returned with her grandmother following behind her. Hair falling loose over her shoulders, Deirdre was wearing a magenta beret and holding a bunch of lavender that had clearly been stolen from someone's garden along the way.\n\n\"Did I miss something?\" she asked, in her huge billowing voice.\n\nBriefly the Garda explained the sequence of events again, while Nora watched her mother's face.\n\n\"Well,\" said Deirdre, addressing herself to Alma with her startled eyes wide and bright. \"It would appear that you had a rather lucky escape.\"\n\nAnd really, it did seem to them all that this was the case.\n\nThe crisis came the next morning.\n\nNora slipped downstairs to get herself a glass of water and found her mother sitting in the chair by the back door. Her mouth was turned upside down in a carnival mask of a sad face. Her eyes, dripping heartbroken tears.\n\n\"Mum!\" said Nora. \"What's wrong?\" She sank to her knees on the cold tiles, one careful hand on her mother's arm. Through the glass door, the sky was a cocktail of oranges and reds, but the garden was still steeped in darkness.\n\n\"What is it?\" asked Nora, but instead of replying, her mother just stared at her. In her eyes Nora saw a dull glint of catastrophe, as if she had caught sight of something so terrible that she could not put it into words.\n\n\"Oh, Mum. There's no need to be afraid. He's locked up now. He won't be allowed out again for a very long time.\"\n\nHer mother shook her head as the tears slid out of her.\n\n\"It's over, Mum. I know it doesn't feel like it, but it's over now.\" (They had been through all this last night. Time and time again they had talked it through. How the murder charge would take precedence. How Alma's case would be adjourned. It might never even come to court.)\n\n\"It's not that,\" said Alma, shaking her head vehemently. But what it was, she could not bring herself to say.\n\nNora sat with her and waited, looking at the lurid sky while her mother cried herself out.\n\nIt was the baby robins, she explained eventually. A cat perhaps, or a fox\u2014who knows what it might have been, but they were all gone. All that was left of them was some blood and feathers, and she couldn't bear it. Of all the things that had happened, this was the one thing she could not bear. The poor little baby robins.\n\n\"Right,\" said Nora's father, his hair sticking up in a tuft at the back of his head, his eyes wild with incomprehension. He had come down to find Nora and Alma wrapped up together, one seated in a chair and one on the floor, the mother's head resting on the daughter's.\n\n\"I don't understand,\" he whispered to Nora when she came back down after putting her mother to bed. He was assembling a breakfast tray for her, but he was so addled that he couldn't get the sequencing right and the toast popped before he had even turned on the kettle. \"Did something else happen?\"\n\n\"Yes,\" said Nora. \"Something else happened.\"\n\nHer father stared at her in genuine confusion and not for the first time, Nora found herself wondering how her parents had managed to live for so long in the world, while learning so very little.\n\nNora runs.\n\nWaking before everyone else, she slips out of the house. She runs to the lighthouse and back, a route that takes her on a path along the edge of the strand, and past the electricity plant, and out on to the seawall. She runs the surface of the wall, aware even as she's doing so that she's taking a risk with her ankles on the uneven surface. At the end of the wall a handful of Poles and Lithuanians are to be found fishing from the rocks behind the lighthouse\u2014before they came, nobody ever fished there. Nora turns and runs back without stopping, past the morning strollers and the dog walkers; on her right, the car ferry keeps pace with her as it plows its way into the heart of the city.\n\nShe is halfway back down the wall one morning when she sees her dad sauntering toward her, a pair of Ray-Bans on him and his towel under his arm. It's his walk that she recognizes first\u2014the ungainly waddle of him.\n\n\"Good morning!\" he calls out, once they're within earshot of each other. Nora raises her hand to greet him, but it's a few long seconds before they draw level.\n\n\"Well,\" she says. \"No filming today?\"\n\n\"Not today,\" he says cheerfully. Not a word has he said to them about the outcome of the contest, but there is no concealing the fact that he's at liberty again after only two days. Nora and Alma have drawn their own conclusions.\n\n\"If you'll wait for me, I'll treat you to a cup of coffee afterward,\" he says with a nod toward the coffee van at the base of the wall.\n\nNora sits and waits for him while he swims; once he's dressed again, they walk the last section of the wall together.\n\n\"What'll you have?\" he asks her, and Nora can't help but notice that he doesn't know how she takes her coffee.\n\n\"A latte,\" she says. \"Full-fat milk.\"\n\nAs he queues for the coffees, Nora takes a seat on a low wall in the sunshine. She closes her eyes and hears, but does not see, two women sitting down beside her. They start up a conversation, and it takes a moment for Nora to realize that they're talking about her dad.\n\n\"Is that your man?\" says one woman. \"What's his name, Collins.\"\n\n\"The fella who had the wife had the abortion?\"\n\n\"No, no. That's the brother you're talking about. This one's the fella who was out in Brussels. The one with the pepper grinder.\"\n\n\"Oh, that's right,\" says the other lady with a chuckle.\n\nNora opened one eye and saw her dad standing patiently in the coffee queue, his coins clutched in his hand like a child about to spend his pocket money. She felt a rush of affection for him, and a desire to protect him.\n\n\"He'd be on a good pension,\" said one of the women.\n\n\"Ah, he would be, yeah. He'd be on a grand pension.\"\n\n\"I wouldn't be worrying about him.\"\n\n\"No, he won't starve, that's for sure.\"\n\nNora smiled and closed her eyes again, surprised by the comfort it gave her to find herself among people who know who you are.\n\n\"What about this weather?\" said one of the women.\n\n\"It's miraculous,\" said the other. \"It's nothing short of bloody miraculous.\"\n\nThe weather was all anyone was talking about. Weather the country had not seen the like of in a generation, day after day broke to clear blue skies, and no end to it in sight. All round the coast, people took to the beaches, while in the midlands they headed for the lakes and rivers. In the cities, children in discount wet suits leapt joyously from buildings and cranes into any patch of water they could find. On farms and building sites, workingmen stripped off their shirts as telephones rang out in empty offices. The city parks turned yellow and dry as savannah, the citizens lazing on the parched grass so tanned and happy as to be barely recognizable even to themselves. A smell of barbecued meat hung on the air; hose bans were put in place and reservoirs ran dry. A summer with the power to change everything: after year upon year of austerity, it was just what the country needed.\n\nTo celebrate this miraculous summer, it was decided that a party should be held in the square. The party was scheduled to coincide with the screening of the first episode of _Celebrity MasterChef._ Nothing would do the neighbors but to have a ritual screening. Colored bunting was strung from the lampposts. Barbecues were borrowed and kegs of beer bought. They'd even erected a gazebo, out of old habit, in case of rain. Maurice the neighbor rigged some electrical cable out through the open door of his house to where a large screen had been set up in the center of the square. (\"Say nothing,\" he said. \"Me son nicked it from his office.\")\n\nBy the time the program went on air, they were all of them stuffed with charred sausages and shop-bought coleslaw. Half drunk on draft lager and warm white wine. While the kids played hide-and-seek among the bushes (Connie's boys included), the adults drew their chairs in close to the bonfire someone had lit in an empty-bellied barbecue. There was a huge cheer as the title sequence appeared on-screen, showing Mick standing in line with the other contestants. \"I'm just hoping not to make a complete eejit of myself,\" he said in the soundbite, and again everyone cheered.\n\nConnie gripped Nora's arm and whispered something into her ear, but what it was Nora couldn't hear. \"Have some more wine,\" whispered Maurice, leaning across to top up their glasses. His hair had started to grow back, giving him the appearance of a fledgling chick. Nora looked over and saw her parents sitting huddled side by side on foldout chairs in the grass, both of them swathed in picnic rugs. Her dad had one arm around her mum's shoulders and with the other he was smoking a fat cigar. Nora couldn't help but think how content they both looked.\n\nShe turned to say this to Connie, but Connie had turned toward Emmet, who was sitting on the far side of her. Her face nestled in the crook of his neck, she had her fingers tucked inside the waistband of his jeans. Nora looked about her and saw children sitting in parents' laps. She saw teenagers flirting awkwardly with each other as they rolled a sneaky joint. Even the old lady from across the square had a small dog on her knee. It seemed to Nora that she was the only one who was alone.\n\nShe leaned her head back to search out the sky, but what she saw instead was the stadium. Lit up from within, its great steel girders looked as fragile as balsa wood, its glass panels as thin as rice paper. In the sky above it, birds were flying. Dozens of seabirds, picked out by the stadium lights; they glowed like paper lanterns. As she watched the magical flight of those beautiful birds of light, it seemed to Nora that everything was possible, even for her.\n\n\"Come for tea,\" her grandfather had said. But it wasn't her grandfather who answered the door. It was a man Nora had never seen before. Slender as a girl, he had tight brown curls and brown eyes like conkers. He was wearing a Che Guevara T-shirt and a pair of crumpled linen trousers, and Nora noticed that he was in his bare feet. For a moment she wondered had she rung on the wrong door.\n\nFrom somewhere inside the apartment came the sound of her grandfather's voice.\n\n\"Nora! You haven't met Nando.\"\n\nHer grandfather stepped into the corridor. Backlit by the flood of daylight from the living room, he looked like he was standing onstage at the start of _The X Factor_.\n\n\"Nando's from Argentina,\" he said, as if that explained everything.\n\n\"Oh!\" said Nora in surprise. The only Nando she'd ever come across before was the Nando from _Alive_. Nando, the hero who walked across the mountains for help.\n\n\"Nando's a reiki healer,\" Manus went on to say. \"We're hoping that reiki might help to ease some of Sam's symptoms.\"\n\nNora stepped forward to plant a kiss on her grandfather's scratchy old cheek.\n\n\"Forgive my apparel, but I have a bit of a cold.\" He was dressed in an Aran sweater and Thai fisherman's pants, a pair of knitted slipper socks on his feet. He had a blue Tuareg scarf wrapped around his neck and head, leaving only his endlessly startling blue eyes uncovered.\n\n\"You look like Lawrence of Arabia,\" said Nora.\n\n\"You wouldn't be the first person to remark on the resemblance,\" he said, with some delight. \"Have I ever told you the story about the lady who came up to me in a supermarket in Los Angeles?\"\n\n\"The lady who thought you were Peter O'Toole?\" Nora glanced behind her to see if Nando was still there, but he had disappeared.\n\n\"Oh, I did tell you.\" He sounded a little disappointed that he wasn't going to get to tell the story again.\n\nHe stepped into the kitchen and Nora followed him.\n\n\"We'll have tea,\" he announced, and moved across to the sink to fill the kettle from the tap. He set it into its cradle and switched it on. As the kettle began to creak, Nora was listening out for other sounds, in other rooms. A feeling like being in a field at night, she was aware of the presence of other living creatures around her, heard but not seen.\n\n\"I want to hear all about your trip,\" announced her grandfather, turning his magnificent eyes on her. \"But let's wait until we're sitting down. I want Sam to hear it too.\"\n\nHe pulled at the coils of his scarf to loosen it a little about the chin. Small beads of perspiration were gathering along his forehead.\n\n\"Is that not a bit hot?\" asked Nora.\n\n\"On the contrary. It's actually very comforting. Plus I'm anxious not to pass my cold on to anyone. In particular I don't want Sam to get it. Although I must say, his immune system is in better shape than my own. He's as healthy as the proverbial butcher's dog.\"\n\nAs her grandfather equipped a tray with mugs and milk and sugar and fig rolls, Nora studied the various newspaper clippings he had pinned to the doors of his kitchen cupboards with thumbtacks. A letter someone had written to the paper calling for a nationwide campaign against stickers being placed on apples. A list of all the foods you should eat to counter memory loss. A review of a biography of Patrick Leigh Fermor.\n\n\"I was never in Israel myself,\" he said, \"or Palestine for that matter. But I was in Beirut, you know, in 1964. Beirut was quite something.\"\n\nThe kettle had boiled while he was talking.\n\n\"Wait, now,\" he said, as if she had been the one speaking. \"I have to concentrate on the task at hand. I'm like my car\u2014I can only do one thing at a time.\" (Driving anywhere with Nora's grandfather was a hair-raising experience\u2014if it was raining, you had to choose between using the indicators or the windshield wipers.)\n\nMoving with painstaking concentration, he poured water into the teapot, swirling it around before tipping it out again into the sink. With all the care of a scientist stooped over a laboratory bench, he transferred a heaped teaspoonful of Earl Grey tea leaves from a tin on the kitchen counter into the heated pot. Then another spoonful, and a final one before pouring in the boiling water. The smell of orange peel filled the kitchen.\n\n\"Now,\" he said, closing the lid. \"I'll carry the tea if you follow with the tray.\"\n\n\"Sam!\" said her grandfather. \"Nando! Tea is served.\"\n\nWith the great glass windows magnifying the heat of the afternoon sun, it was as hot as a boiler room, but Manus didn't seem to notice. Nora set the tray down on the table and slipped her cardigan off. Her arms were pale where she had kept them covered all the time she was in Gaza. Only her hands and face were tanned.\n\n\"Sam,\" said her grandfather, looking up at the door. \"Come and say hello to Nora.\"\n\n\"Nora,\" said Sam, delight dawning on him with the memory of her name. He walked toward her with his hands down by his sides. Behind him, Nora could see Nando hanging back. She was acutely aware of him watching her as she greeted Sam.\n\n\"Oh, Sam,\" she said as she hugged him to her. \"I'm so happy to see you.\"\n\n\"Nora,\" he said again as they pulled apart to face each other. He was working his lips over her name.\n\nNora smiled into his dark-lashed eyes, lashes that were so black they always looked wet. Eyes the color of coffee beans; at one time there had always been a sheen on them, a glinting hint of his buoyant good humor. Now there was only a worried stillness. As he chewed at his bottom lip, Nora saw that his teeth were nicotine-stained. His face was unshaven, but he was wearing a freshly ironed shirt and a pair of clean jeans, evidence of her grandfather's devoted but imperfect care.\n\n\"Come on,\" said Manus. \"Let's have the tea.\"\n\nNora and Sam sat down side by side and Nando took a seat opposite them. Nora tried not to look at him too much. She tried not to look for too long, but a jigsaw was forming in her head all the same. A series of fragments that came together to form a composite. The sugary brown eyes, the hair that curled behind his ears. He wore a signet ring on his right hand. A frayed leather bracelet around his slim wrist.\n\n\"How's your father taking his disappointment?\" Manus asked. He was holding the teapot protectively in his hands but he had paused in the pouring of it while he asked the question.\n\n\"Oh,\" said Nora. \"He's blaming the sea urchin. Apparently it's all the sea urchin's fault.\"\n\nHer grandfather resumed his tea-pouring with a smile.\n\n\"Have a fig roll,\" he said. \"You haven't lived until you've tried a fig roll.\" The way he said it, in his huge old voice, he made it sound like a voice-over on an ad. (How _do_ they get the figs into the fig rolls?)\n\n\"Whereabouts in Argentina are you from?\" Nora asked Nando. She was thinking, Buenos Aires. She was picturing darkly lit caf\u00e9s with white tablecloths, and people dancing the tango.\n\n\"I'm from Baroliche,\" he said. \"It's in the south.\" And she had the feeling that he wanted her to ask him another question. She had the feeling that she could ask him anything and he would answer her. There was something wide open about him, something that made you think of deep oceans, big skies.\n\n\"Nando is a ski instructor,\" said Manus. \"In another life.\"\n\nSo that was it. Altitude. Snow.\n\n\"Is that what brings you here? The summer?\"\n\n\"Well actually, it's winter now in Argentina.\"\n\n\"Of course,\" said Nora, feeling like an idiot. She flipped the world over in her head. Tried to imagine skiers traversing upside-down mountains, inverted penguins strutting around on upended icebergs and ships sailing along an upside-down sea. She thought of snow, and found herself wishing she could scoop up a handful of it and bury her face in it. She was so hot! Her cheeks felt flushed, her armpits sticky, her dress was clinging to her thighs. She would have liked to slip out to the bathroom and wash her face. Put on some deodorant, slap some powder on her nose. Instead of which she picked a magazine up off the table and began to fan herself with it in the hope that she could regain her cool.\n\n\"Now,\" said her grandfather. \"You promised you'd tell us all about your travels.\"\n\n# RELATIVE VALUES\n\nIn today's _Sunday Times_ , the writer Manus MacEntee and his granddaughter, the human-rights activist Nora MacEntee Collins, talk about passion and pride and the joys of an unconventional family.\n\n## _Manus_\n\nNora is the quintessential Scorpio. She's brave and calm but under the surface there's plenty of passion bubbling away. Ever since she was a little girl, she always had a built-in radar for injustice of any kind.\n\nNora was born during the era of glasnost and perestroika. It was the time of the famine in Ethiopia. A time for brave souls, and Nora is without a doubt one of them. It was a good day for the world the day Nora was born. Sometimes the world manages to deliver of itself precisely the kind of people it most needs.\n\nI tried to teach Nora to play drafts when she was a little girl but she was absolutely hopeless at it. No matter what I said to her, I could not induce her to huff me. But that's Nora for you, she's not capable of taking anything if it's at a cost to someone else.\n\nNora's an only child and I think her parents' divorce hit her very hard. For years she lived this nomadic existence, moving back and forth between two different homes in two different countries. What impressed me most was how she managed to remain loyal to both her parents throughout that time. She never once took sides, which says a lot for her strength of character.\n\nI'm so proud of the work that Nora does. She's a warrior for justice and I can't think of a better way to use the life you've been given. If everyone was like Nora, we would have something close to heaven on earth.\n\nMy dearest wish for her is that she will meet someone who deserves her. Nora is such a very fine person, it will be hard for her to find someone to match her, but when she does find such a person (and I do believe she will), he or she will be a very, very lucky sod.\n\n## _Nora_\n\nI love the look on people's faces when I introduce my grandfather to them. I tend to forget that other people have grandfathers who spend their days on the golf course, or doing the crossword. My grandfather is more likely to spend the day on a sunbed and the night out partying. I have friends who live in the flat below Grandad's, and whenever they have a party, he appears in his dressing gown in the small hours and joins in the dancing.\n\nI remember one time when I was a kid, Grandad picked me up from school in his Jag, and all the other kids in my class thought he was a rock star. It was like having one of the Rolling Stones as your grandfather.\n\nWhen I was born, Grandad and Sam had already been living together for years. It never occurred to me that there was anything strange about your grandfather living with another man. A boy in school told me once that my grandad was queer, but I didn't know what that meant so I took it as a compliment.\n\nThe most important thing Grandad taught me was the importance of kindness. He always says that the only thing that matters, in the end of the day, is how we've treated one another. I try to remember that.\n\nI'm convinced that Grandad is going to live forever. He seems to get younger and younger as time goes on. If you asked me his true age, I'd say he's about seventeen. Forever seventeen.\n\nWhen I was a little girl, Grandad used to read the death notices out of the paper to me\u2014he said there was a story in every one. When his death notice eventually comes to be written, it will make some story.\n\n# Manus\n\nIt's true that Manus is a great fan of the death notices. They're the first thing he turns to every morning when he sits down with his newspaper, a pot of tea in front of him and a single slice of toast (burned and then scraped, just the way he likes it) with a spoonful of his famous homemade marmalade. Manus makes marmalade every year from tins of canned oranges that he orders on the Internet. His marmalade is rough-cut and heavily spiced, in deliberate defiance of the wimpish excuse for a thing that you find in the shops these days; it was the disappointment of factory-made marmalade that forced him to start making his own. A joyous ritual that he carries out once a year to a soundtrack of the Andrews Sisters, or Bing Crosby, singing along in his tuneless voice as he stews the fruit in a huge catering pan that he keeps for this express purpose. Using the empty Hellmann's mayonnaise jars that he hoards all year-round, and the jam jars his ex-wife stockpiles for him, he makes gifts of his marmalade to everyone he knows for Christmas, holding back half a dozen for himself to see him through the year.\n\nHe and Sam are in the habit of having two breakfasts. The first, which they consume in bed with the radio on, consists of small earthenware cups of black coffee and little bowls of granola heaped high with Greek-style yogurt and pebbled with blueberries. It's Sam's job to make this breakfast, a task he carries out with great seriousness but increasingly erratic timing. Some days it could be eight in the morning by the time he wakes Manus with the tray. Other days it might be five or six.\n\nOn this particular morning, Manus was woken at three by noises in the kitchen and the smell of coffee brewing. By the time Sam appeared, heartbreakingly boyish-looking in nothing but his pajama bottoms, Manus was sitting up valiantly in the bed with his pillows stacked behind him and the clock radio turned to face the wall so that Sam would not see his mistake. There was chamber music playing instead of the morning news program, but Sam didn't seem to notice. There was so much now, that Sam didn't notice. Invisible shutters falling down around his mind, blocking out bits of the world without him even knowing it.\n\nSide by side and with hardly a word spoken between them, they ate their breakfast in the dead of night. Through the gap in the curtains Manus could see the sky outside, an unfathomable black. He was aware of a longing in him for the cold discomfort of early-morning flights, overnight train journeys, late nights that turn into early mornings. Somewhere on their travels they had come across a bakery that served coffee and hot buns to the revelers coming out of the nightclubs.\n\n\"Where was that bakery?\" he asked Sam. \"The bakery with the hot buns.\"\n\nAnd Sam seemed to know just what he was thinking, as if the thoughts flowed freely between their two minds.\n\n\"Barcelona,\" he said. \"They were cinnamon buns.\"\n\nSometimes Sam remembers things that Manus doesn't. Things from long ago, they are as real to Sam as things from yesterday. More so, perhaps.\n\nManus had the smell of cinnamon in his nostrils now, and the taste of treacly coffee in his throat. A memory of those black-tobacco cigarettes with no filter, their butts littering the ground. From the radio, the announcer's voice sidled in among his memories like smoke.\n\n\"Well, that was Schubert's famous string quintet for two violins, viola and two cellos. That piece of music, written just before the composer's death at the age of thirty-two...\"\n\nOnce they had finished their pre-dawn breakfast, Manus shifted his empty bowl and cup on to Sam's tray and stacked it over his, allowing Sam space to get out of the bed. Sam picked up the trays and carried them out to the kitchen while Manus slipped down under the covers again. He closed his eyes in the hope of going back to sleep, but the coffee had done its work and his mind was too alert. He lay in bed and listened to the strange, sucked-up silence of the night. He heard the soft, padding sound of Sam, rummaging around in the living room like a fox, then the sound of a lighter sparking up a cigarette. Now Manus knew for certain that he would not be able to go back to sleep, for fear that Sam would leave the lighted cigarette on the arm of the couch, for fear that it would fall down into the couch cushions and smolder. Your sense of smell fails to function when you're asleep\u2014Manus had read that in the paper recently, and ever since he has feared a fire in the night, imagining himself slipping into a smoke-induced coma as the flames lick at the foot of his bed.\n\n\"Burned alive,\" his death notice would read, which would at least be some consolation. For years he and Sam had made a play of coming up with ever more sensational story lines for their death notices, in keeping with the colorful ends they delighted in envisaging for themselves. \"Eaten by a crocodile,\" Sam might say as he and Manus made their way along a rickety rope bridge above a flooded river in Cambodia. And Manus would add cheerfully, \"As he would have wanted.\"\n\nIt was what each of them wanted, to go out with a bang. A plane crash would fit the bill, they were agreed upon that. And there was a time when they traveled so much that it seemed almost a likelihood. Another distinct possibility was a bus crash\u2014those ravines in Peru, they were almost begging you to take the plunge. Malaria would be glamorous, or even dengue fever. (\"Succumbed to a fever,\" the death notice would read, with deliberate poetic flourish.) Kidnapping was always in the running, especially when you insisted on taking a road trip through Colombia in the 1990s. \"Under no circumstances is anyone to pay a ransom for us,\" Manus had told his family before they left. \"If you do, I'll haunt you.\" Of course there was always an outside possibility that they would be taken by cannibals and made a human sacrifice of\u2014why else would anyone undertake to go on a walking tour of Papua New Guinea but out of a desire to be eaten? (\"Slowly, in Papua New Guinea,\" the death notice would read.)\n\nSlowly. That was what they had feared the most. Your faculties taking leave of you one by one, leaving only layers of indignity in their place. A humbling that you had no choice but to succumb to. And of course they had always assumed that it would be Manus who would suffer this humiliating decline, with Sam as the witness. Never once did it occur to them that it would be the other way round. Manus smiled at the thought that life was playing this little joke on them. For all the times they had tempted fate with their talk of crocodiles and ravines, for all their bravado in the face of death, life was determined to have the last laugh.\n\nManus fears death now, not for and of itself, but because he fears what will become of Sam when he's gone. A quarter of a century between them\u2014it is too much to hope for that Sam will go before him. Sam is only fifty-eight. Manus is eighty-two. When first they met, those numbers seemed like mere diversions\u2014they were nothing but fireworks lighting up the sky, talking points for other people to ooh and aah over, but now... now the numbers have taken on a deadly menace. No matter how he goes at them, Manus cannot make of them anything but what they are. A formula for disaster. An equation that cannot be resolved. He retreats into binaries, taking the days as they come, one by one.\n\nNever does Manus let a day go by without reading the death notices. As soon as he has their second breakfast laid out on the table, as soon as he has his toast buttered and his tea primed with milk, he opens the paper to the inside back page and, with an almost sensual pleasure, begins to read.\n\nWorking from top to bottom, he scans the columns, searching out certain key words, like \"tragically,\" or \"suddenly,\" for closer reading. He makes a point of skimming over any notices that allude to grandchildren, or great-grandchildren, as being too dull to bother with. Similarly the deaths of religious do not merit his attention, unless there's a mention of time spent on the missions. He has a particular distaste for the self-satisfied death notice, with its reference to a career spent in the higher echelons of the civil service perhaps, or an association with a particular golf or sailing club. Snippets of verse he finds vulgar in the extreme, likewise epitaphs in the Irish language (the deceased probably didn't speak a word of Irish). Manus respects brevity in a death notice, with a particular eye to the intriguing detail, like the mention this morning of a solicitor's firm in Mallow, with a telephone number supplied for \"any inquiries.\"\n\n\"What on earth could that mean?\" he asks aloud, looking up at Sam, without really expecting him to answer.\n\nSam is lost in the task of assembling his breakfast. Like all Moroccans, Sam is a hopeless sugar addict. Even now he is dumping spoonful upon spoonful of sugar into his tea; if Manus didn't stop him, he would empty the whole bowl into his cup. He pours a long lick of honey on to his bread, concentrating so intently that he is not aware of Manus watching him. He picks up the bread and eats it, humming to himself as he chews. What it is that he's humming Manus cannot make out.\n\n\"What are you humming, Sam?\" he asks, speaking with great gentleness so as to mask his irritation. The humming has a bad effect on him, so he tries to divert Sam. Even humming along with him is better than listening to Sam humming to himself. Anything is better than listening to the humming.\n\nSam looks up, confused.\n\n\"You were humming something. I was trying to make out what it was.\"\n\nSam looks back at him with desperately serious eyes.\n\n\"It sounded to me like it might have been the Ink Spots. Am I right?\"\n\nManus begins to hum, and after a bar or two, Sam joins in. The words singing themselves in their heads. Would you look at us, thinks Manus. Two old queens sitting here over our breakfast, humming away like half-wits. If anyone could see us! There's an absurdity to it, he thinks; it's like something out of Beckett. But there's a certain beauty to it too, with the smell of burned toast hanging in the air, and the sunlight tumbling in on them through the giant window. What there is, despite everything, is love.\n\nSam was the most beautiful creature he ever saw.\n\nThe first time Manus set eyes on him, he was walking down Grafton Street as Manus was walking up. A bitterly cold day at the hard end of winter; there was a frost on the ground even though it was past midday, but Sam was wearing no coat. His head was bare and he had no scarf or gloves on him, just an Aran sweater and a pair of threadbare tweed trousers. He had a pair of huge black dogs with him and they were straining on their leashes, pulling him along as if he were a chariot. His chin was raised and his head leaning back on his neck, and the way he walked, it seemed to Manus that his feet hardly touched the ground. Manus was just approaching the corner at Weirs when Sam swept past him. He turned to watch in his wake, arrested by love as by a thunderclap; he had no idea if he would ever see him again. All he knew was that he would never forget him.\n\nAt that time Manus was forty-eight. He had been married for nearly thirty years to a woman he liked very much and with whom he had three wonderful children. He had achieved literary superstardom with a novel that was generally considered to be the finest work of Irish fiction since _Ulysses_ (although nobody suggested it was in quite the same league), and this astounding and unexpected critical success had been crowned with a most gratifying commercial triumph when his book was made into a movie, starring Peter Finch and James Mason. On the day the check for the movie rights arrived, Manus went out and bought himself a brand-new Jaguar saloon. The car he chose had pale blue paintwork, with leather seats and a walnut dashboard, and Manus had been driving it ever since, with a fierce, undying loyalty that only increased with age. The older the car got, the more alarming the smells and sounds that it emitted, the more loyal Manus became to it. They were growing old together, Manus and his car. Both of them creaking and groaning and farting and belching (although the burning smell of oil that the car emitted was a thing of fragrant beauty compared to the smells that escaped from Manus these days). No matter how much trouble it gave him, Manus would not give up on that car, no more than it would give up on him. What passed between them was nothing more or less than love.\n\nIn the years that followed the release of his book, and the film of the book, Manus rested on his laurels, eking out a comfortable living from the largesse that had so unexpectedly dropped into his life\u2014a largesse that grew miraculously over the years as the result of a series of canny investments. And while he did on occasion make further attempts at writing, mapping out an idea for a novel or scribbling down a single passage of prose, he always returned to the conclusion that with that one novel, he had reached the limit of his abilities. For the most part, he was content with what he had achieved.\n\nDublin at that time was a place of empty spaces. The great Georgian squares that had once been the light and life of the city lay dark and derelict, their roofs stoved in so as to escape paying rates. Anyone who could get out of the city did get out, moving their families to the newly built housing estates in the suburbs, where shopping centers began to spring up, and schools and playgrounds, and if they weren't beautiful they were at least warm and clean, and nobody who made the move regretted it. Those few people who chose to stay behind had the city to themselves, and it took on the atmosphere of a small town, or a village, so intimate was the sense of community among those who remained. Manus couldn't walk down Grafton Street without bumping into a dozen people he knew, and at night he and his friends huddled in the snugs of a handful of pubs in the dark backstreets, pubs that felt more like the front parlors of people's homes than they did public houses.\n\nIt was only when they had children that Manus and Deirdre began to entertain at home. This was the mid-sixties, and people in Dublin had begun to drink wine. Often of a Friday Deirdre would make a big stew, and Manus would buy in a case of cheap Italian red, and people they knew and people they didn't know would come to eat and drink and, mainly, talk.\n\nIt was at one of these Friday gatherings that Sam appeared, several weeks after Manus had first sighted him. He arrived in the company of an obscure scion of the Guinness family whose dogs he had been employed to walk. The moment Manus saw him standing there by the fire\u2014wearing the same Aran sweater and the same paint-splattered tweed trousers he had been wearing that day on Grafton Street\u2014as soon as Manus saw him he realized that he had known all along that he would come, and that now he had come, they were both of them on a trajectory that could not be altered, or avoided, any more than it could be regretted.\n\nNow in his late fifties, Sam still has the jet-black hair he had as a young man, but it's coursed with white, as if he had dipped his fingers in wet paint before running his hand through it. He still has the bockety teeth he had then, teeth that stand apart from each other like tilting dominoes. Stained with age and tobacco, they lend him a perverse attraction, hinting at something dangerous and disreputable in his past. The hazelnut eyes are streaked now with confusion in place of the seductive humor that once danced across them like the flames and silhouettes of a James Bond title sequence.\n\nWhen the doorbell rang, Sam was sitting at the window. With the iPad on his knee, he was working at the screen with a stylus, looking up every so often at the vast view. He was looking not at the clusters of red-brick houses down below him, nor at the strip of sea in the distance, with the sails of the yachts out of Dun Laoghaire splintering the blurry horizon line; he did not see the church steeples peeping up like weeds above the trees, nor the office blocks with the workers moving about within their windows. He didn't even glance at the stadium, slouched in the landscape in the form of a huge sleeping S\u2014all he had eyes for was the old Victorian gasometer with its red wrought-iron pillars and its pointed peaks. When Sam paints the gasometer, what appears on his screen is a series of minarets, set against the clear blue sky of another place, at another time. For it is the minarets of Tangier that always appear in Sam's paintings, against ever-changing skies. The skyline of home, always present in his memory.\n\nHe never went back after he met Manus. There was an arranged marriage waiting for him at home, and a place in the family business, and he was afraid that if he did go home they would not allow him to leave again, afraid that something might have reached them of the life he was living. If that was the case, it would not have been beyond them to force him to stay. He caught sight of the city of his childhood only once in all the years, from a promontory beyond Tarifa as he and Manus were driving across the south of Spain. They got out of the car and stood and stared over the straits to where the sunlight flashed off the tips of the minarets, and they sat there until nightfall so that Sam could see the lights of Tangier in the dark. Ever since that night, Sam has been painting the same sight over and over again, in every light. Manus suspects it will be the one thing that remains when everything else is gone.\n\nThe doorbell rang while Manus was preparing the lunch, mashing smoked mackerel with mayonnaise and lemon juice to form a p\u00e2t\u00e9, which he planned to serve with slivers of brown soda bread and unsalted butter. (Manus minds his own health as well as Sam's, because he knows that he must stay alive as long as is humanly possible in order to care for Sam. It is the centerpiece of his existence, this need to stay alive.) He dropped the fork he'd been using into the sink and wiped his hands on a tea towel to rid them of lemon juice before making his way down the corridor to the intercom.\n\n\"I've a parcel for you,\" said the voice at the other end.\n\n\"Can't you leave it downstairs?\"\n\n\"It's registered, you'll have to sign for it.\"\n\n\"All right. I'll be down in a minute.\"\n\nIt was probably a book to be reviewed. _The Irish Times_ was in the habit of sending him books, and occasionally he obliged, just to keep his hand in. There again, it could be a gift. Acushla was in the habit of ordering gifts for him on the Internet. Thoughtful things for the winter, like flannel pajamas, or cashmere socks. Yes, it was probably a present from Acushla. By the time the lift reached the ground floor, Manus was as excited as a child at Christmas at the thought of what the parcel might hold.\n\nWhen he saw the size of it, when he took it from the postman and felt the weight of it, it was clear to him that this was not a pair of cashmere socks, or a book for that matter. Unless it was an encyclopedia, and encyclopedias didn't exist anymore, did they? Not since Wikipedia. This bloody thing weighed a ton and it was as hard and unyielding as a slab of stone. Manus signed the postman's console and lugged the parcel back to the lift, holding it down by his knees with two hands, like a builder shifting a paving block.\n\n\"What the devil?\" he asked himself as he heaved it down the corridor and in through the door of the apartment.\n\n\"A parcel,\" he told Sam, hoisting it on to the table. He was in the habit of speaking to Sam as you might speak to a dog or to a baby, without ever expecting an answer.\n\nHe examined the parcel, looking for any indication of who might have sent it, but there was none. Just Manus's own name and address, handwritten in bold capitals with a black Magic Marker. He turned the parcel upside down, looking for a return address, but all there was, in the top left-hand corner, was the number of a PO box in Dublin.\n\n\"Let's see now,\" he said as he tugged at the plastic wrapping the parcel was encased in, trying to rip a hole in it. The plastic stretched and thinned where he tugged at it but it would not rupture. He was forced to go into the kitchen to fetch a knife, making a stab in the parcel as if it were flesh and pulling the skin away in one long strip. What this revealed was a stack of loose-leaf low-grade printer paper, as thick as an old Dublin phone book; it was held together in the old-fashioned way with a Bridget's cross of twine. On the title page were two sparse lines of printed script, in 12-point Times New Roman: _Untitled, A Novel_.\n\nAfter he and Sam had finished their lunch, Manus cleared the table, leaving the dishes in the sink water to soak, a slovenly habit he had adopted recently in order to lessen the monotony of the task (all he had to do later was take them out of the water and prop them on the dish rack to dry). Afterward, he set Sam up in front of the TV and put on a DVD for him, selecting at random an episode of _The Waltons_ from the box set that Acushla had ordered for him over the Internet. Sam found _The Waltons_ soothing. Something about the pace of it, and the tranquil setting, seemed to instill a peace in him. When the end of each episode came, he would join in the voices bidding good night to each other as the lights went out around the old wooden house.\n\n\"Good night, Elizabeth. Good night, Mary Ellen.\"\n\nManus would join in too, although it was not clear to him whether Sam knew they were just playing along with a TV program or whether he was under the illusion that they too were tucked up under the eaves of that darkening house as the crickets cheeped and the wind chimes sounded and the signature tune swung up out of the darkness with its cheery trumpet and its rolling guitar, sounding its way into yet another episode. Sam was capable of watching three or four of them, back to back, of an afternoon. And Manus would take the opportunity to retreat to the far corner of the living room with a book, with the soft tones of John Boy's reminiscences adding a nostalgic undertow to whatever it was that he happened to be reading.\n\nWith Sam absorbed in the soft glow from the television, Manus slipped the twine binding off the manuscript and eased the great hunk of paper out of it. Picking the title page off with a licked finger and laying it to one side, he leaned down over the opening paragraph, with only the faintest stirring of curiosity. He was fully expecting to be disappointed. He had been sent so many books over the years, embarking on each one with great hope only to be let down. But he only had to read the first line to know that this one was different. By the end of the first page, he knew that this was the book he had been waiting for, the book that had existed in his imagination ever since the day he had packed his own novel into an envelope and sent it off. Always he had known that there was a book like this out there, and it was this knowledge that had stopped him from ever attempting to write again himself. Because unless you could write something as good as this, there was no point in even bothering.\n\nWhat this writer had achieved was something Manus had lusted after and longed for all his life. Not as a writer but as a reader; he had begun to wonder did it even exist, other than in his own longings. A book that rips a seam through the earth, throwing up out of the torn ground the beating wings of the human soul. What this writer had achieved was nothing other than the sublime.\n\nWhen Nora came by the next morning, she found him sitting at the dining table. Upside down in front of him, in a higgledy-piggledy pile, lay the seven hundred and eighty-seven pages he had spent the night reading. Seven hundred and eighty-seven pages and not a page too long! Not a single word wasted! Manus had spent the morning trying to compose his response. He had started draft after draft that he discarded in disgust, wasting page upon page of expensive headed paper before finally happening upon a formula of words that did some justice to the masterpiece he had just read. Posterity clanging in his ears, he was as certain as he could be that whatever he chose to say about this book would live on long after he was dead.\n\nHe was just signing his name to the letter when the doorbell rang. He looked round at the scrunched balls of paper that littered the surface of the table, at the empty coffee cups and the biscuit crumbs, at the Anglepoise lamp, still burning although it was nearly eleven. He folded his letter in three and slipped it into an envelope. Hastily, he inscribed the address of the post office box on the front of the envelope before rushing to open the door.\n\n\"Hi, Grandad.\"\n\nNora appeared to be dressed for a jog, in a T-shirt and leggings and a pair of battered old runners. Her hair was freshly washed and smelling of lemongrass. Her lips glistened with a slick of pink gloss. She had a pair of rolled-up yoga mats under her arm, and the sight of the yoga mats reminded Manus that she had promised to take him through a meditation practice. Nora was convinced that meditation would help him cope with the stresses of caring for Sam, even though Manus had tried to assure her that he didn't find it stressful caring for Sam.\n\n\"Hi, Sam,\" she said, raising her hand toward the window, where Sam was sitting in a chair in his pajamas, gazing out at the view. For once there was no cigarette in his hand.\n\n\"He hasn't had a cigarette all morning,\" whispered Manus. \"Normally by this hour he would have smoked half a dozen. Say nothing, but I think he may have forgotten that he's a smoker.\"\n\nNora was looking at him strangely.\n\n\"You look like you've been up all night!\"\n\n\"Perhaps,\" he said vaguely, looking into her rather than at her. \"You see, I've happened upon something quite extraordinary.\"\n\nShe looked about her, as if the thing he spoke of might be an object, or an animal that was lurking under the table.\n\n\"What is it?\"\n\nHe paused before he answered.\n\n\"A book,\" he said at last. The word came out all consonants, like a single note sounded with a guitar pick. \"A most extraordinary book.\"\n\nNora looked back at him with her big owl eyes.\n\n\"Did _you_ write this book?\"\n\n\"Lord, no! That's the whole point. Someone else did.\"\n\nNora was still staring at him.\n\n\"I don't understand.\"\n\n\"This,\" he said, patting the pile of pages on the table. \"This is the proof I've been waiting for.\"\n\nNora's eyes were wider now, if that were possible.\n\n\"Proof of what?\"\n\n\"Proof of everything!\" he said. \"Proof that I was right not to write a dozen mediocre novels. Proof that we are human, and that someone else has witnessed the lives we have lived. This book,\" he said, pausing for effect. \"This book is proof that we existed!\" (A phrase Manus had seen carved into the wooden wall of a reading nook in Shakespeare and Company in Paris; it had come back to him repeatedly as he was reading this book.)\n\nNora was smiling.\n\n\"That's perfect!\" she said, as if he had just pronounced the magic word. \"That's what meditation's all about. It's all about cultivating a deeper awareness of our own existence.\"\n\nHer smile was gently indulgent, like the smile of a good fairy. Not for the first time, Manus had the sense that Nora was actually much, much older than him, that she had lived for centuries and centuries and had a clear sight of things that he could only catch the most fleeting glimpses of.\n\n\"Now, are you ready?\" she asked.\n\nAlready she was rolling the mats out on the floor.\n\n\"I thought we'd start with some breathing exercises,\" she said, sitting down on the edge of a chair and propping her ankle on her knee as she removed one sneaker and the sock underneath it to reveal a broad, flat foot with blue-painted toenails. Swapping sides, she removed the other sneaker and the other sock and set both feet on the floor, her hands on her knees; the hands too had stubby blue nails.\n\n\"You might want to start by lying down.\"\n\n\"Easier said than done,\" he said as he looked at the ground. It seemed like a long way down.\n\nWith considerable creaking, he managed to lower himself slowly until he was crouched with his knees on the floor, the knuckles of his hands out in front of him to steady him, like a sprinter about to take off. Snatching his hands up off the ground, he allowed himself to fall to one side, landing heavily on one hip. From there it was an easy rollout to a horizontal position. The feeling of the hard floor under his back was surprisingly pleasant; it reminded him of his youth.\n\n\"I'm going to give you an eye pillow,\" said Nora. He had a sense of her hovering above him, and next thing he felt a soft weight on his eyelids, and a smell of dusty lavender.\n\n\"Now, I want you to start by taking a few really deep breaths. In through the nose and out through the mouth. In through the nose, out through the mouth...\"\n\nHe heard her voice moving away from him, and for the first time in a very long time, he forced himself to empty his mind of everything but the most basic mechanics of life. In through the nose, out through the mouth. In through the nose, out through the mouth...\n\nHe was aware of thoughts darting out like little fish from behind the rocks in his brain. Something Sam had said in the middle of the night, coming back to him now. The drip from the kitchen tap\u2014he must find someone to fix it. The book he had just read was a noise in his head, like the roar of a nearby waterfall.\n\n\"If your mind wanders, just bring it back to your breathing.\"\n\nBut he had lost the rhythm of it. The breath was stuck in his chest and he had to release it in a clumsy stumble out through his mouth. This breathing thing, it was trickier than it seemed. He made a fresh attempt, harnessing all his powers of concentration to control it.\n\nIn through the nose, out through the mouth, he chanted silently to himself, thinking ruefully of old dogs and new tricks as his thoughts marched off in a new direction. He made a halfhearted attempt to catch them, like a man chasing along a city pavement after a scrap of paper or a sweet wrapper that has been picked up by the wind and is dancing away from him, always just out of his reach.\n\nHe left Nora and Nando minding Sam while he went out to post his letter.\n\nNando had arrived early for his session. Nora seemed in no hurry to leave. When Manus left, they were sitting over a pot of tea while Nando told her all about his plans to convert an old shipping container into a home. Any other girl would have run a mile at the thought of such a plan, but much to Manus's amusement, Nora seemed charmed by it.\n\nThe meditation had rendered Manus curiously elated. He floated his car to a stop in a loading bay outside the post office and clambered out. He wasn't in the habit of parking his car so much as abandoning it anywhere it would fit. Generally a bus stop or a taxi rank, as the average parking space would not suffice.\n\nHe was inside the post office for five minutes perhaps, increasingly aware of an uncomfortably full bladder as a woman at the top of the queue arranged for a number of enormous parcels to be weighed and franked for posting to Latvia. The man ahead of him was buying savings stamps with the contents of his coin jar, and Manus waited with growing exasperation as the single post office worker counted out the coins. At last it was his turn. He approached the hatch and purchased a book of ten local stamps, fixing one carefully on to the front of the envelope before he moved away from the counter to hover in front of the postbox. With all the gravitas of a man voting in a historic election, he deposited the letter into the mouth of the box, listening out for the small thump from inside as it ended its fall.\n\nBuoyed up by the safe execution of his errand, he stepped out of the post office into the startling sunshine, only to find that his car had been clamped. The clampers were still on-site, a black man and a white man wearing matching uniforms. The white man was already climbing back into the truck, while the black man was straightening up after applying the clamp to the Jaguar's wheel. Poor sod, thought Manus, imagine the abuse he gets. He determined to be especially courteous to him.\n\n\"Gentlemen,\" said Manus, stepping forward with his hands up, to indicate that he intended them no harm. \"If you would be so good as to wait a moment until I've paid my fine, then you'll be free to take that contraption away with you again.\"\n\nIt was his full bladder he was thinking of, and the need to conclude this matter expeditiously. Taking his credit card out of his wallet, he handed it to the black man. The man had shiny ebony skin and a large, wide forehead. His bloodshot eyes were dull with the expectation of abuse.\n\n\"I wonder would you be so good as to do the honors for me?\"\n\nThe man furrowed his forehead, not sure what to make of what he was hearing.\n\n\"I don't carry a mobile phone,\" explained Manus apologetically. \"Generally your colleagues are kind enough to make the call for me.\"\n\nManus was on first-name terms with most of the city's clampers, and by and large they allowed him free range of the parking spots in town. This pair he had not encountered before. This pair must be new.\n\n\"It's all right,\" he said, nodding encouragingly. \"Just dial it in.\"\n\nShoulder up to his ear to hold the phone, the clamper put the call through. As he was reading off the credit card numbers, he kept glancing up at Manus, afraid that he was falling victim to some scam.\n\n\"Marvelous,\" said Manus, when the clamper came off the phone and returned the credit card. Once he had removed the clamp from the wheel, Manus handed him a ten-euro note, folded over lengthways so that it formed a taper.\n\n\"You're crazy,\" said the clamper, staring at the banknote he was being offered as if it were a lit fuse. \"You're completely crazy.\"\n\nHe was still shaking his head from side to side as he walked back to the truck. Manus pulled away from the curb, giving the clampers a long, low honk of his horn. The clampers beeped back, two jaunty beeps. That was another team that would never bother him again. For eighty euro, plus the ten-euro tip, he would enjoy free parking for the foreseeable future. For all of his eccentricities, Manus could be surprisingly sane when it came to money.\n\nOn the way home, he stopped to drop his already read newspaper in to his ex-wife and to use her lavatory. Deirdre claimed she couldn't afford a newspaper of her own, so Manus was in the habit of dropping his around to her once he'd finished with it. Even though the alimony he paid her was more than sufficient to finance the purchase of her own paper, it would have been less fun that way. It gave Deirdre such satisfaction to live off his hand-me-downs.\n\n\"Do you have time for a cup of tea?\" she asked him, without waiting for an answer. \"Here,\" she said, fanning a pile of what appeared to be takeaway brochures out on the table in front of him as he sat down. \"Have a look at those.\"\n\n\"What is it I'm looking at?\"\n\nHe opened one brochure to find lists of canap\u00e9s. According to the lists, you could have ham-hock terrine and tuna sushi rolls and smoked salmon blinis with horseradish cr\u00e8me fraiche. You could have mini-beefburgers and wild mushroom arancini and miniature onion and gruy\u00e8re tarts, all at what seemed to Manus an eye-watering price. And the most surprising part of the whole thing was that of all the people in all the world, his ex-wife seemed to him the very least likely to splash out on a raft of wildly expensive canap\u00e9s. There was something about this that didn't add up.\n\n\"They're from the caterers,\" she said, with a girlish lilt to her voice. \"For my party.\"\n\nLike a bell ringing in a far-off room, Manus had a faint recollection of this topic being raised before.\n\n\"Ah,\" he said, \"your party.\" Hoping this would prompt her to offer up some more clues. But she was busy ferrying china cups and saucers to the table. Then a matching milk jug and sugar bowl, and a shop-bought Swiss roll. There was something faintly comical about the way she went about assembling her quaint little tea party. She reminded him of a character out of Beatrix Potter\u2014an upright cat, or a hedgehog dressed in human clothes. With her great white eyebrows spilling down over her eternally surprised eyes and her two loops of white hair hanging like long ears on either side of her face (there were a few white hairs visible on her upper lip that he couldn't help but notice), she could even have been a very large old rabbit.\n\nShe set the teapot down on the table and sat down heavily in front of it. Lifting the lid, she gave the tea a stir with the handle of a knife.\n\nManus turned his cup right side up to reveal layers of tannin build-up on the tender white china inside. The tea set had belonged to his mother, but he had left it behind when he went, thinking it petty to claim it. Clearly, it hadn't been washed since. Deirdre did not believe in washing teacups, or teapots (or salad bowls, for that matter, or omelet pans). She believed it destroyed the flavor to wash them. She was adamant on wiping only.\n\n\"Remind me,\" asked Manus. \"When is your party?\"\n\n\"Well, my birthday's on the Friday,\" she said. \"So I thought we'd have the party on the Saturday.\"\n\nShe had never been a birthday person before. All the time he'd been married to her, she had refused to celebrate her birthday. In an attempt to overcome this prohibition, Manus had once made the mistake of throwing a surprise party for her. She had turned on her heel and walked back out of the room, furious that her friends had seen her without her makeup.\n\n\"You never cease to surprise me,\" he said. (She never ceased to _amuse_ him, with her childlike passions.) \"Why now?\"\n\nShe looked at him for a long time before she answered, her hands cupping the hot teapot as if it were a crystal ball.\n\n\"I'm going to be eighty. And the closer I get to it, the lighter I feel in my heart. For the first time in my life, I have a great desire to celebrate.\"\n\nNever had it occurred to Manus to envy his wife, but it occurred to him now. It occurred to him that she had nothing in the world to be worrying about. Her work on this earth was done.\n\n\"I've spent so many years worrying about Macdara. I've worried myself sick about Acushla, because she's so soft. I've even worried about Alma\u2014her hardness was such a worry to me. But a funny thing happened to me recently. I realized they're not my responsibility anymore. Their lives are their own to live and their troubles are their own, and there's nothing more I can do.\"\n\nManus didn't like to tell her this, but he had had the same thought many, many years ago.\n\n\"Their lives will go on,\" he said, \"long after you and I are gone.\"\n\n\"Yes,\" she said, \"I can see that now. Funny how I could never see that before.\"\n\n\"Let me offer you another comforting thought,\" he said. \"In a hundred years' time, nobody will even remember who any of us were.\"\n\nHe knew this would amuse her, and it did. This was the secret of their friendship, it seemed to him. The fact that they were amused by the same things.\n\n\"They'll remember you,\" she said, \"because of your book. Whereas my art was, by its very nature, transient.\"\n\nHe shrugged. He had never placed the same value on his book as other people had. He couldn't really see what all the fuss was about, couldn't see what made it stand out from anything else. The book he had read last night, now that was a different matter. The shadow of it stretched across his day like the shadow of a mighty mountain. He kept going back to it in his mind...\n\nDeirdre lifted the teapot and Manus held his cup out to her to receive the tea. He poured himself some milk and gave it a stir, leaning down to slurp the first mouthful, even though he knew this would annoy her. The annoyances were a part of it, this imperfect intimacy that existed between them. Like fond siblings, they were familiar with each other's every foible.\n\n\"After everything that's happened,\" she said, \"I feel it's an opportune time for a gathering of the tribe.\"\n\n\"What exactly do you have in mind?\" he asked, with some suspicion.\n\n\"Just a little family get-together. I thought it would be nice to gather the four generations together under the one roof. A meeting of the clan.\"\n\nThey lined up in front of Manus's eyes, two by two, as if they were boarding the Ark. He pictured himself and Sam coming up the front steps, then Alma and Mick, arm in arm and laughing, followed by Nora, who he saw walking alongside Nando. Now Connie and Emmet and the two boys, and after them, Acushla and Liam, side by side but not touching. At the back of the line was Macdara, all on his own. Why did the thought of it give him a shivery sense of foreboding? He felt as if someone was walking on his grave.\n\n\"We'll open up the double rooms,\" Deirdre was saying, her eyes sparkling with an intent that was almost sinister. \"We'll have champagne first, and canap\u00e9s, and then everybody will have to perform a party piece, even the little boys.\"\n\nManus found his suspicions growing, a feeling of unease that he could not explain. For a moment he wondered if perhaps Deirdre was planning some kind of ritual massacre? He had a vision of his entire family lying in lifeless puddles on the floor of her drawing room, while a toxic gas hovered in the air and she stood by the fireplace and sang \"The Last Rose of Summer.\" That had always been her party piece. Even now he could hear her singing it in her huge voice, a voice that could fill every inch of any room, no matter how large.\n\n\"Why am I getting a bad feeling about this?\" he asked her.\n\nShe stared at him with pantomime innocence, her spiky eyelashes blinking away at him.\n\n\"Why would you say that?\"\n\n\"Because I know you,\" he growled. \"And I know when you're up to something.\"\n\n\"I am not!\"\n\n\"Oh, yes you are.\"\n\n\"Oh, no I'm not.\"\n\n\"Oh, yes you are,\" he said, determined to have the last word. He knew her well enough to know that she was up to something. But what it might be he could not for the life of him have guessed.\n\nThey were always great friends.\n\nThey were friends before ever they were married and they remained friends afterward, strange as that might seem to some. \"Isn't it funny the way we've ended up being such good friends?\" she said to him once. It was the day of Connie's wedding and he had insisted on walking her out of the church behind the bride and groom, leaving Sam to trail behind with Macdara. \"Well, lass,\" he had said, holding out his arm for her to take. \"We should present a united front as grandparents.\" When the truth was that he wanted to save her the indignity of walking down the aisle alone. \"You're a good friend to me,\" she said to him later that evening, when he sat down beside her to take a rest from the dancing. \"You might even be my best friend,\" she added with mild disgust. \"God help us all.\"\n\nHe was twenty when he met her. A medical school dropout, he had managed to find a job with an advertising agency to tide him over until he fulfilled his dream of becoming a writer, a dream he never doubted would come to pass. Deirdre was seventeen and fresh out of Ennis. With her elocution-lesson accent and her thick black hair worn loose down her back, she was on the run from her mother's small-town ambitions for her, ambitions she was hell-bent on frustrating. They met in McDaids on a wet summer Saturday and Manus offered to buy her a drink. \"I'll have a brandy,\" she said boldly. He used to tease her about that afterward. Who drinks brandy on a summer's afternoon? Only a girl just up from the country!\n\n\"Your mother and I always had great affection for one another,\" Manus told Acushla when she came to him once looking for answers. She was wondering, not for the first time, whether to leave her husband. \"The secret to my relationship with your mother is that we never took each other too seriously,\" he told her. \"No matter what happened, we were always able to laugh about it.\" Looking at Acushla, he could see that laughter was the last thing on her mind. You would think, by looking at her, that she had forgotten how to laugh.\n\n\"Macushla,\" he had said, resting his hand on her thin arm. \"Macushla, my blue-eyed Macushla.\" (It was Manus who had the naming of her, and he had named her with that song in mind, a song he sang to her often when she was a child.) \"Macushla,\" he said. \"Macushla,\" and with his words he was deliberately evoking that beautiful old melody in both of their heads. \"Do you think you might try, for a change, not to take everything so very seriously?\"\n\nShe had smiled at him then, through her tears, and she had promised him that she would try. But Manus didn't hold out much hope for her. He had seen too much of life to think that there was any hope of a person changing their colors once they were cast. Acushla had been born with a deadly appetite for battle, a battle that she had waged first on her sister and then on her husband. Now at least it seemed to Manus that she had found a fight that was worthy of her. It was highly amusing to Manus, if not to Acushla, to think of her as a feminist icon. He was very much cheered not just by his younger daughter's belated blossoming, but by the notion that life still had it within its power to spring him the odd surprise.\n\nIt was one of the more unwelcome things about getting older, the sense of certainty that had begun to plague him. The stories of people's lives seemed at times so obvious to him, it was as if the whole world was a book that he had reached the end of, while everybody else was still reading. A sad and lonely feeling; there was nothing that gave him more pleasure than encountering a last-minute plot twist.\n\nIn Alma and Mick's wounded reunion Manus saw a beautiful logic, and it seemed to him that in his failure to foresee it, he had neglected to factor in the virulent strain of monogamy that affected all the women of his ex-wife's lineage. In Acushla's reconciliation with Liam, he recognized the same fatal strain of loyalty, one that he suspected Connie had inherited as well.\n\n\"I'm afraid we're one-man women,\" Deirdre said to him once, stating it as a sad fact, when he had asked her whether she ever saw herself marrying again. He was reminded of a story she used to tell the children when they were small. A story from her own childhood, about a swan whose mate was shot by a poacher; the widowed swan spent the rest of her days going up and down the river alone. \"Swans mate for life, you see,\" Deirdre would say. \"If a swan loses its mate, it will live out its days alone rather than ever mate again.\" Only now does Manus see that in leaving Deirdre, he doomed her to spending the rest of her life alone. A fate she bore with silent dignity, just like the swan in her story. It seems to Manus very sad that he never saw that until now.\n\nOf all the surprises that lie in wait around the corner, Manus prays that one of them is reserved for Macdara, but he fears that this is not the case. He sees Macdara living out his life alone, and he understands that Macdara's self-imposed celibacy is the result of his own highly publicized sexuality. If Manus has one regret, it is the hurt that he caused to Macdara. That's the one thing he would like to be able to go back and change.\n\n\"It wasn't you that put Macdara off sex,\" Deirdre said to him recently. \"It was all that rugby.\"\n\nThey were walking along a country lane, the hedgerows spilling over with life and color. Ahead of them the coffin of an old friend was being borne along the road from the church to the graveyard by a slow-moving hearse. The friend had been a contemporary of Deirdre's at the Abbey. He had stood best man to Manus at their wedding. Now he was dead and they were walking after his coffin.\n\n\"Poor Macdara,\" said Deirdre. \"He never recovered from being forced to play all that rugby.\"\n\nManus smiled at the comfort his ex-wife never ceased to find in these tangents of hers, tangents that for all their whimsy still contained somewhere within them the seeds of some grain of truth. It was true that Macdara had been traumatized by the smelly, sweaty-fleshed contact of the school scrum; it had produced in him a distaste for physical intimacy of any kind.\n\n\"Do you think he's lonely?\" asked Manus, looking down at his shoes as he walked, one foot appearing in his line of vision, then the other. He was trying to keep step with Deirdre, walking deliberately slowly in order to do so.\n\n\"Oh, no more than any of us,\" she said. \"Perhaps even less. Don't forget, Macdara chooses to be alone.\"\n\nAs they took their places round the open grave, a very old priest recited an inaudible prayer.\n\n\"Hypocrisy of the highest order,\" hissed Deirdre into Manus's ear. \"Sure, he always held that the notion of God was a nonsense.\"\n\nManus turned his face in order to whisper back.\n\n\"Hypocrisy, my dear, is the medium in which we move.\"\n\nA non-believer himself, Manus was nonetheless a great fan of Christian pageantry. In particular he was a sucker for a rousing Christian hymn. There was nothing quite so comforting to Manus as a good Protestant hymn.\n\n\"Well, lass,\" he said, as they began to move away from the grave. He held his arm out for her to fall in step with him, noticing with surprise that she had tears in her eyes. For a moment he thought that perhaps it was the sun that had made her eyes water, but when she spoke, her voice was affected too.\n\n\"Do you know what just occurred to me,\" she said, reaching out to grab his arm as she stumbled on the uneven ground.\n\n\"What?\"\n\nShe gazed up at him with a fierce look on her face. \"Now that Donal is gone, you're the only one left who remembers me for the girl I once was. And I'm the only one who remembers what you were like when you were young.\"\n\nThe funerals were coming hard and fast. Hardly a day went by but Manus spotted someone he knew in the death notices. Today it was Freddy Noble whose death was being announced. Beloved husband of Margaret, said the death notice. Deeply regretted by his devoted daughters Michaela and Ursula, his sons-in-law Arthur and Alan, granddaughters Isobel, Amy and Emily, and his many nieces and nephews. _Ar dheis D\u00e9 go raibh a anam d\u00edlis_.\n\nManus laughed.\n\n\"No mention of poor Patricia,\" he said to Sam.\n\n\"Patricia,\" said Sam. He appeared to be giving the name great thought. \"Patricia and her cassoulet.\"\n\n\"Forty years,\" said Manus. \"Freddy's been living with Patricia for forty years, and not a mention of her in the death notice.\"\n\nAs he spread a thin layer of marmalade on his toast, he was smiling to himself at the endlessly bizarre turns of human behavior. He was wondering to himself, fleetingly, whether Deirdre might expect a mention in his own death notice. Surely not, he thought, although technically she was still his wife. Manus had no doubt that Deirdre would outlive him. He was always telling her she would live to be a hundred, and for some reason this seemed to irritate her. She took it as an insult, which was why he persisted in mentioning it.\n\nManus would have liked to go to Freddy's funeral. He and Freddy had been in boarding school together, and when Manus sold his book, it was Freddy who had looked the contracts over for him. Often over the years he and Sam had been invited to eat with Freddy and Patricia in their little house down by the Dodder. (Manus and Sam could of course be counted upon not to be judgmental about Freddy's domestic arrangements.) Patricia was a terrific cook and a priceless mimic, and Manus had many a happy memory of nights around their table, replete with Patricia's famous cassoulet. Yes, Manus would have liked to be at Freddy's funeral, if only to see how the wife and daughters would handle the presence of Patricia. That would certainly be worth seeing, only Manus had to bring Sam to a doctor's appointment.\n\n\"Sam,\" he said. \"We should start getting ready. We have to be at Dr. Boylan's by eleven.\" Manus had been careful not to mention the appointment until now, aware that it would only rattle Sam to know of it too far in advance.\n\n\"Dr. Boylan,\" said Sam, flustered already. \"What time do we have to be there?\"\n\nHe rose from the table, unaware that his shirt was still open, the buttons unbuttoned and his white vest visible underneath.\n\n\"Eleven,\" said Manus. \"Don't worry, we have plenty of time.\"\n\nHe came around the table and drew Sam's shirt closed. Fastening the buttons, he took care to keep his hand movements as brusque and businesslike as possible. Sam had developed a tendency to mistake some of Manus's more innocent ministrations for sexual advances, with the result that Manus was wary of touching him. He was afraid to put his hand on Sam's arm, lest Sam take this as a come-on, afraid to stroke Sam's hair lest this should lead Sam to stroke him back. Sam's illness seemed to have had the effect of stoking his sexual appetite and freeing him of his inhibitions, whereas on Manus it had the opposite effect. Manus didn't like making love to Sam anymore, or being made love to by him. It didn't feel right, not when he couldn't be sure that Sam knew what he was doing. This wasn't something Manus felt he could talk to anyone about. Just another of the many little lonelinesses he had acquired since Sam first started to get sick.\n\n\"It's hot out there,\" said Manus. \"You'll hardly need a jacket.\"\n\nSelecting for himself a battered panama hat from the shelf in the cupboard, he set it on his head and checked for his keys and his wallet before ushering Sam out into the hall and closing the door behind them.\n\nAs they waited for the lift, Sam was anxious, impatiently watching the numbers on the display above the door.\n\n\"What time do we have to be there?\" he asked.\n\n\"Eleven,\" said Manus. \"We've loads of time.\"\n\n\"We've loads of time,\" said Sam, as if Manus was the one who needed convincing.\n\nWhen the lift arrived, there was someone already inside. A very obese young man, Sam and Manus only just about managed to squash in beside him. The lift doors closed and Manus noticed that Sam was humming the Oompa Loompa tune from _Charlie and the Chocolate Factory_ ; he prayed the young man would not recognize it. Painstakingly slowly they moved down through the six floors, arriving at last on the ground floor. The lift doors opened and Manus and Sam were forced to step out into the lobby to allow the young man to emerge. They got back in for the last leg of their journey, down to the basement garage. Sam was still humming as they stepped out of the lift and walked toward the car.\n\n\"Now,\" said Manus, opening the passenger door for Sam. He waited until Sam was safely in, then went round and climbed into the driver's seat. He started the engine, checking the petrol gauge to see if they would need to stop. The needle showed the tank a quarter full. That would get them to the doctor's surgery and back, if not much farther. Turning the radio on to a classical music station, Manus swung out of his parking place (the spaces on either side of him were empty, because no one could fit their car in beside his for fear of scraping it). He put his foot to the floor so as to make it up the ramp, and an unholy noise erupted from the back of the car, as if tin cans had been attached to the bumper.\n\n\"Damn it,\" said Manus, and he stopped the car at the top of the ramp, wrenching the handbrake up and hauling himself out. He went round the back of the car, stooping to look underneath it.\n\nThe fender was hanging off. It was clinging by one dull and neglected chrome arm on to the round rump of the car. When Manus touched it, it clattered to the ground, skittering away down the ramp. He inched down the slope to retrieve it, moving sideways and with great care, like a skier caught in deep snow. As he bent down to pick up the fender, he heard a noise that surprised him, a noise like something snapping. The last thing he saw was his car rolling backward down the ramp toward him. The last thing he heard was the orchestra of St. Martin-in-the-Fields, conducted by Sir Neville Marriner, with their beautiful rendition of \"Sheep May Safely Graze.\" The last thing he thought as he lay on the ground with the weight of his own car rolling over him was how curious it was that he of all people should be brought back into the fold in this way at the end.\n\n**MacEntee** , Manus (Dublin, and formerly of Kildangan, Co. Kildare)\u201426 June 2013 (tragically), in his 83rd year. Deeply regretted by his partner Sam, his wife of many years Deirdre (n\u00e9e O'Sullivan), his daughters Alma and Acushla and his son Macdara, his sons-in-law Michael and Liam Collins, his granddaughters Constance and Nora, his grandson-in-law Emmet and his great-grandsons Ernest and Oscar. Requiem Mass at Star of the Sea Church in Sandymount on Friday at 11:00, followed by a cremation service at Mount Jerome Cemetery.\n\n# Macdara\n\n\"I always thought he wanted to be buried at sea,\" said Acushla.\n\n\"You can't be serious,\" said Alma.\n\n\"That's what he always said he wanted,\" said Acushla, her words rising up and down the scale like notes on a piano.\n\n\"He said to it me too,\" said Deirdre. \"Many a time he told me he'd like a sea burial.\"\n\nThey were waiting for the undertaker, the four of them sitting in the hush of the waiting room\u2014they had turned down the offer of a home visit in favor of neutral ground.\n\n\"I did hear him mention it once or twice,\" said Alma, \"but I always assumed it was a joke. He used to say he'd like to be eaten by a crocodile, but I think it's safe to assume that was a joke too.\"\n\nNobody replied. Alma looked from one to the other of them, her eyes wide and unblinking as if to say, _surely_ you agree with me?\n\nMacdara coughed, to break the silence.\n\n\"I don't remember him ever showing any interest in sailing,\" said Acushla. \"Does anyone remember him ever showing any interest in sailing?\"\n\n\"No,\" said Deirdre quickly. \"Just sailors.\"\n\nMacdara leaned forward.\n\n\"If it's what he wanted, we could always ask.\"\n\nThe sound of his own voice was a surprise to him as much as it was to anyone else.\n\n\"We could always ask them,\" he said tentatively, \"whether it's possible?\"\n\nAs soon as he'd finished speaking, he sank farther into his chair, pulling his chin down on to his chest and his shoulders up to his ears. Alma leaned her head back against the wall, closing her eyes with a sigh, while Acushla propped an elbow on an elegantly crossed knee, wobbling her head encouragingly. His mother gave him a discreet wink, as if they were bridge partners and she was communicating something to him about their next move. Macdara was just on the point of venturing to speak again when the undertaker came into the room. He advanced with his arm extended to greet the grieving widow, who was already rising from her chair to accept his condolences.\n\n\"She's describing herself as his _wife_ ,\" Alma hissed as they were leaving. Her mother had disappeared back into the dark recesses of the funeral home in search of a bathroom.\n\n\"Well, technically she _is_ his wife,\" said Acushla, moving through the door Macdara was holding open for her. \"They were never divorced.\"\n\nThe three siblings formed a loose circle out on the pavement, assailed by a barrage of light and heat, along with the smell of chips from the takeaway next door.\n\n\"Sweet Jesus,\" said Alma. \"This presents us with a bit of a diplomatic situation.\"\n\n\"What do we do?\" asked Acushla, reaching up with a perfectly manicured hand to pull her sunglasses down over her eyes. \"You don't think she'll want to sit in the front pew in the church, do you?\"\n\n\"I don't see any indication of her taking a backseat. As far as she's concerned, she's the grieving widow.\"\n\n\"So, where does that leave poor Sam?\"\n\n\"Right where she always wanted him,\" said Alma. \"At the mercy of her generosity.\"\n\n\"Oh, pray,\" said Acushla. \"Pray she doesn't use this as an opportunity to get her revenge on Sam. I wouldn't put it past her to pull rank on him now that she can.\"\n\n\"I'm afraid we'll just have to make sure that doesn't happen,\" said Alma.\n\nAnd they all fell silent again as they contemplated the rocky path that lay ahead of them. Macdara turned to see the undertaker ushering their mother out the door of the funeral home. Her hair had come loose of its moorings and her cape hung lopsided off her stooped shoulders. As she squinted into the glare of the sunshine, Macdara was struck for the first time by how old she had become. She had gone and got old without him even noticing.\n\nIt was always his mother Macdara was closest to. Ever since he was small, his mother had been his silent ally, the champion of his every idiosyncrasy. There was no aberration Macdara could adopt, no whim of his that his mother would not indulge. From his refusal ever to eat meat that had any bones in it, or fruit that had any seeds in it (apples, pears, grapes\u2014Macdara had an unnatural fear of their offspring growing inside his belly), to his insistence on wearing wellies every day to school, his mother bent to every turn of his personality. Macdara's clothes had to be washed a dozen times before he wore them because he hated the feel of new fabric. He had to have flannel sheets because he found cotton too scratchy. His shoes had to be weathered by the shoemaker, and his hair had to be cut at home because he would not go to the barber's. On the rare occasions that Deirdre resolved to trim his fingernails, she would always have to ring the neighbors first, to warn them not to call the Guards when they heard the screams. When Macdara was sick, as he often was, his mother never forced him to go to school, allowing him to spend the day in her bed instead, reading detective stories. Some days he was allowed to stay in her bed even when he wasn't sick. Some days he was allowed to stay in her bed just because it was raining, or because it was cold, or because he didn't want to go to school.\n\n\"It's very hard for Macdara to go to school,\" his mother used to say, a phrase that his sisters pounced on and repeated ad nauseam. \"It's very hard for Macdara to go to school,\" they would mimic, bobbling their sleek little heads from side to side as they performed the imitation. And Macdara would skulk away, withdrawing further and further into his own silent world, a world that only his mother seemed to understand. Because the truth was that it _was_ very hard for Macdara to go to school. There were days when it was hard for Macdara to go out of the house. He was like a soft-shell crab, or a turtle without a shell. He seemed to have been born without any of the body armor that other people use to survive in the world.\n\nMacdara's father was baffled by him. With his boyish enthusiasm for life and his eternally optimistic view of human nature, Manus was at a loss to understand Macdara's fears, fears that were born not of experience but of some pathological sensitivity his father was not wired to comprehend. But while another man might have forced him to engage in blood sports and eat spare ribs with his bare hands, while another man would no doubt have argued that Macdara's weaknesses were all due to his mother's indulgence and that it was his duty as a father to provide a counterweight, Manus was happy to let Macdara off. Like a dog and cat who are forced to share the same house, Manus and Macdara managed to live alongside each other quite happily for years, while ignoring each other almost entirely. When his father moved out, Macdara was surprised by the size of the hole he left in the house, a hole that was filled by the sound of his mother shouting, and Alma slamming doors and Acushla's heartbroken sobbing.\n\nMacdara had pitied his sisters then, because their loss was so much greater than his. And he pitied them again now, because they had lost the parent they loved more. For his part, Macdara couldn't help but be relieved that it was his father and not his mother who had been the first to go.\n\nIt was his sisters who told him.\n\nThey appeared at the studio door, Alma with her head hanging down and Acushla hovering behind her. Macdara noticed straightaway how pale they both looked.\n\nAlma took a deep breath in through her mouth. She had a strange half-smile on her face, more of a grimace than a smile, and her eyes were all red.\n\n\"Macdara,\" she said. \"I'm afraid Dad's had an accident.\"\n\n\"Oh, dear,\" said Macdara. And he pictured the scene. He saw his father's car standing at a jaunty angle in the middle of a major intersection with smoke coming out of all its orifices as a distraught woman motorist inspected the damage to her new Mercedes. \"I hope there isn't too much damage.\"\n\nHis sisters looked at each other and it occurred to Macdara that they did seem afraid. They seemed afraid of _him_.\n\n\"Why are you _afraid_?\" he asked, seizing on that one word Alma had used, over all others. A word with the implication of something that was still to come, something that could even now be prevented. \"You both look like you've seen a ghost.\"\n\nNeither of them spoke, and all of a sudden Macdara too was very afraid.\n\nHe held up his right index finger.\n\n\"Don't tell me,\" he said, taking sudden command of the situation. \"Whatever it is, this thing that you were about to tell me. I don't want you to say it.\"\n\nHe turned away from them then, and walked back into the studio, leaving his sisters standing in the open doorway. Into his bedroom he went, and sat down on the end of the bed, with his hands cupping his knees. His mind was blank but his breath was coming hard and fast. He concentrated all of his attention on getting his breath back.\n\nAfter they'd been to the undertaker's, they returned to Macdara's studio to draft the death notice. Leaning their upper bodies in through the open door with their feet still planted outside in the garden, they called to Macdara's mind a pair of explorers surveying the interior walls of a cave.\n\n\"Come on in,\" he said to them, gathering his papers up off the table and removing stray clothes from the chairs. \"Sit down. Make yourselves at home.\"\n\nHe filled a jug of water from the tap and poured some lemon barley cordial into it. He set three tumblers on the table, and out of the kitchen press he took a pack of Nice biscuits, arranging a generous supply of them in a wide fan on a willow-pattern plate.\n\n\"Why do I get the feeling we're in an Enid Blyton novel?\" said Alma, slipping her shoes off as she sprawled in a chair. \"I feel like I'm one of the Famous Five.\"\n\n\"The Secret Seven,\" said Macdara, and despite the circumstances, he was aware of a feeling of great pride bubbling up in him, pride at having his sisters as guests here in his little house. \"I think the Secret Seven are the ones you're thinking of,\" he said happily. \"They were the ones who had a den.\" (When Macdara was a kid, he used to read those books over and over again\u2014it was the Five Find-Outers and Dog who were his favorites.)\n\n\"I was a Nancy Drew girl myself,\" said Alma, and to Macdara's great pleasure, she reached out for a biscuit and began to nibble at it. Her face was puffy with what Macdara recognized as the effects of medication. Her figure plumper as a result of all the gourmet meals Mick had been feeding her. A softness to her that had not been there before. For the first time in his life, Macdara was not scared of her.\n\nDoyenne of a hundred fund-raising committees, Acushla took a pad of paper out of her handbag and a freshly sharpened pencil. Flipping the cover of the pad back on itself and positioning the pencil expectantly over the blank page, she looked up at the others.\n\n\"Okay, where do we start?\"\n\n\"MacEntee,\" said Alma, settling back into her chair and closing her eyes as she began to dictate the copy. \"Manus. In his eighty-third year. As the result of a tragic accident...\"\n\nAcushla's head was bowed down sideways over her page, the tip of her tongue probing the corner of her mouth. The sound of her pencil scratching away at the surface of the paper.\n\n\"Deeply regretted by...\"\n\nMacdara waited to see what Alma would say next.\n\n\"... his loving partner Sam.\"\n\n_Scratch scratch scratch_ went Acushla's pencil.\n\n\"His ex-wife Deirdre...\"\n\nAcushla looked up.\n\n\"She's not going to like that.\"\n\nAlma threw her hands out in exasperation. Her poor fingers made Macdara wince. He couldn't look at them without feeling pain.\n\n\"We can't say 'wife,' as if they were still married,\" said Alma.\n\n\"We can't say 'ex-wife' either,\" said Acushla back to her.\n\n\"I've a suggestion,\" said Macdara.\n\nThey both turned to look at him, as if they had only just become aware of his presence.\n\n\"How about 'wife of many years'?\"\n\nAcushla froze as she considered it. Alma dipped her head like a conductor guiding an orchestra into a quiet section of a symphony.\n\n\"Do you know,\" she said, raising her head again. \"That could work.\"\n\n\"It's brilliant,\" said Acushla, scratching away furiously at the page. \"Absolutely brilliant.\"\n\n\"It's beautiful,\" said Alma. \"Although since we're being such sticklers for detail, I'm not sure it's entirely accurate to use the word 'tragedy.' It seems to me that 'a tragedy in his eighty-third year' is a bit of a contradiction in terms.\"\n\nMacdara brought his shoulders together to make himself smaller, and leaned in over the table.\n\n\"But it _is_ tragic,\" he said, fixing upon it as a point of fact.\n\nAlma raised her eyebrows as she waited for him to explain.\n\n\"By its very definition it's a tragedy. The death of a man because of a fatal flaw is defined as a tragedy.\"\n\n\"I don't understand,\" said Alma. \"What's the fatal flaw?\"\n\n\"Don't you see? That stupid car was the fatal flaw.\"\n\nAlma smiled and nodded. Something dawning in her eyes, something almost like respect.\n\n\"So we're sticking with 'tragedy'?\" asked Acushla, anxious to move on.\n\n\"Absolutely,\" said Alma. \"A tragedy it is.\"\n\nA tragedy was what it felt like to Macdara at times. The enormity of it taking him by the throat and tightening its grip on him until his eyes were watering and he could hardly breathe. It was in those moments that it seemed to Macdara that the world had stopped its turning, and it was hard for him to imagine that any other human being had ever been bereaved quite like he had been bereaved. That anybody else's death could ever have had quite such a profound effect on the physical world, altering the taste of all the foods in it, distorting the look of a room when he walked into it, and bending the scale of the streets and the buildings outside so that everything he looked at appeared strangely but inexplicably wrong. Only the sounds of the world were unchanged, the noise of the traffic and the exploding roar of the crowd from the stadium, the babble of the football fans gathered outside the pub after the match had finished. Macdara felt like shouting at them as he walked past, \"What are you laughing about? Did you not hear? My father _died_.\"\n\nAt other moments, it seemed to him the most ordinary thing in the world that his father should have died. Doesn't everybody die, after all? Why should his father be any different? In those moments it seemed to Macdara that it was a mistake to make too much of a fuss of this one ordinary old death. It seemed unseemly to make too much of it.\n\n\"At my age you don't get sad when someone old dies,\" said his mother. \"At my age you see it as a cause for celebration when someone of Manus's age dies.\"\n\nThey were all sitting around his mother's table, eating an Irish stew that a neighbor had dropped in. Nora had heated it up while Connie set the table. It was only when he began eating that Macdara discovered how hungry he was.\n\n\"You don't really mean that,\" said Acushla, looking at their mother with brimming eyes. Connie too had tears in her eyes. The expression on Nora's face was more like disappointment. Even Alma appeared shocked.\n\n\"Manus lived a long and rich life,\" said Deirdre, speaking with slow emphasis. \"He had the good fortune to be whisked away before falling prey to decrepitude. Something you should all get down on your knees and give thanks for, since you're the ones who would have ended up looking after him.\"\n\nBlinking her dry eyes, she stared brazenly back at them all, and for the first time in his life, Macdara saw something in her that he did not like. A ruthlessness that he'd never seen before. He found himself wondering when it had crept in.\n\nMacdara's father's death was reported in all the papers, as you might expect, and the TV news even ran a pre-prepared obituary. How long had it lain in wait? Macdara wondered, before experiencing a pang of something not unlike jealousy at the thought that there would be no obituaries to mark the passing of his own outwardly unremarkable life. Unless? Unless. Macdara still harbored an intermittent but ardent hope that he might yet achieve something remarkable with his life.\n\nIt was in this spirit of hope that he had sent his book off to his father. Taking every care to preserve his anonymity, he had printed the manuscript off and parceled it up, despite the clamor of voices in his head telling him he was a fool. He had walked to the post office, tormented by those voices. Even as he was standing in the queue, they would not fall silent.\n\nHe's going to hate it. This rubbish? Sure it's hardly even literate... Self-indulgent crap... What kind of delusional freak would think this worth publishing? What a shameful waste of paper and ink... You'd have been better off using the paper to wipe your arse... And to think, you could have been out doing something useful with your time, instead of writing this pathetic crap. Pathetic, pathetic, pathetic.\n\nTen, twelve times a night Macdara woke in the grip of a sickening panic, remembering with a physical lurch of his heart his wildly reckless act. What on earth had he been thinking, sending the book to his father? He would roll over on to his tummy in agony, burying his face in the pillow to smother the thought. He would contemplate for one wonderful moment the possibility that he had only dreamed that he had posted it, abandoning that fantasy with a despair that was almost hysterical. He would bang his forehead off the pillow over and over again, punishing himself for his own foolishness. Eventually he would turn over on to his back again, talking himself back to sleep with a flock of familiar reassurances, like sheep you might count.\n\nAt least he had not put his name to the book, that's what he told himself. There was no way his father could have known that he was the author. His father probably hadn't even had time to read the book before he died. If there was any comfort to be found for Macdara in Manus's sudden death, it was in the realization that in all likelihood he must have died before he got a chance to read his book.\n\nThey decided to wake him in the funeral home. It was not the custom to have a wake at home anymore, not in the city at least.\n\n\"That's a country thing,\" said Alma.\n\n\"I don't like the idea of having the body in the house,\" said Acushla. \"It gives me the creeps.\"\n\n\"Think of all the tea we'd have to make,\" said Deirdre. \"We'd have to give people _sandwiches_.\"\n\nMacdara agreed that on balance it was best to wake his father in the funeral home.\n\nIn death, his father looked like somebody else. A generic dead person, all barely covered bones and waxy skin. With his eyelids closed forever on his magnificent eyes, he could have been anybody.\n\n\"Who _is_ that?\" asked Ernie, eyes wide with awe as he was led up to the coffin.\n\n\"That's your great-grandfather,\" whispered Connie.\n\nMacdara could hear Ernie sucking this information into his chest in a raspy asthmatic wheeze.\n\n\"Here,\" said Connie, rooting around in her handbag, \"have a puff of your inhaler.\"\n\nOscar was up on tiptoes, trying to see over the rim of the coffin. Connie pushed him and his brother toward their grandmother and stood for a moment by herself at the head of the coffin. Macdara noticed her hands fluttering loose by her sides. It occurred to him that she was wondering whether she dared touch the corpse or not. As he watched, she placed a hand on the edge of the coffin for a moment before snatching it away.\n\n\"I can't believe he's gone,\" she said, sitting down beside Macdara. He could feel the swell of her sorrow rising and falling in her. She allowed herself to tilt a little to the side, so that her shoulder rested against his. He reached out a hand and patted her on the arm. He couldn't think of a thing to say.\n\n\"It's okay,\" said Connie. \"There's nothing you can say.\"\n\nOn the far side of the coffin, Connie's children had settled themselves on chairs between their grandparents, while Deirdre sat alone at the end of the row. Acushla was feeding the boys wine gums, dispensing the sweets straight out of her handbag one by one, while Liam tapped at the face of his phone. It occurred to Macdara that Connie's children would forever after associate the taste of wine gums with their great-grandfather's funeral. Those were the kind of things that children remembered.\n\nThe door of the room opened and out of the patch of daylight Mick and Alma materialized. Macdara noticed how Mick moved with practiced ease, working his way along the row of chairs to greet Acushla first, then Liam and finally Deirdre. Coming round the base of the coffin, he gave Connie a kiss on the cheek.\n\n\"Macdara,\" he said. \"I'm so sorry.\" He took Macdara's hand and grasped the flesh of his upper arm.\n\n\"It's all right,\" said Macdara, sitting back down again too quickly. No sooner was he sitting down than the door opened again and Nora came in. She had Sam with her, along with some other fellow Macdara did not recognize. He felt a stirring of anxiety at the presence of this stranger.\n\n\"Sam,\" he said, getting up and moving toward the door. He shook Sam's hand, taking care not to stand too close to him. He was worried, as always, about his breath. Worried there might be a smell off it, he liked to keep his distance from people in case this was so. (In school once, someone had made a comment about his breath, and he had been worrying about it ever since, despite his obsessive use of mouthwash and mints.)\n\n\"Sam,\" said Mick, closing in from one side. \"I'm so sorry.\" He pulled Sam into a half-hug, gripping his arm with one hand while the other went round Sam's shoulder. Sam was wearing a highly unsuitable burgundy velvet smoking jacket and a silk cravat. On his bottom half, a pair of jeans and runners. Acushla hovered beside Mick, waiting for her turn with him.\n\n\"Sam,\" she said gently. \"Do you want to come over and see Dad? He looks so nice, I think you should see him.\"\n\nSam allowed himself to be led over to the coffin. All eyes on him, he stood looking down at the face of the man he had lived with for more than three decades. Acushla stepped back a few paces to give him space.\n\n\"Do you think he knows what's happening?\" whispered Connie.\n\n\"Maybe it's just as well if he doesn't,\" Macdara whispered back.\n\n\"I'm not sure it's fair,\" he heard Alma say, her voice coming from somewhere behind him. \"It's not fair to have him here.\"\n\nMick's voice came in reply to her, somewhere above a whisper.\n\n\"Poor bugger. He looks like he doesn't have a clue what's going on.\"\n\nAnd still Sam stood beside the coffin. The twin candles that flanked it flickered and puffed breaths of white smoke out into the dark air. A door opened and closed. A child tumbled off a chair. Macdara watched as Sam reached his hand into the coffin and fumbled with the clothes Manus was wearing. It looked like he was trying to undo the buttons of his jacket.\n\n\"Oh, sweet Jesus,\" Macdara heard Alma mutter. \"What's he doing?\"\n\nThey all watched as Sam slipped his hand inside Manus's jacket, resting his palm on the place where his lover's heart had ceased beating.\n\nMacdara experienced a terrible rush of sorrow. For a moment he couldn't bear the weight of it and he let his head fall down on to his chest. When he looked up again, Nora and the stranger were on either side of Sam. Putting an arm gently around his shoulders, Nora led him away, guiding him past Acushla and Liam and the little boys, past Deirdre (who reached out and grasped his wrist without rising from her chair) and around the foot of the coffin to a chair at the end of the opposite row. He was sitting directly across from Deirdre now, with the coffin in between them, a discreet white satin sheet obscuring Manus's legs and feet. Was his father wearing shoes under there? wondered Macdara. Or was he just in his socks, or even his bare feet? Macdara's mind roamed these corridors with curious abandon, like a dog off its leash.\n\nThat was a good analogy, he thought, and he would have liked to write it down. But he had no notebook with him, no pen. No way of taking notes. Instead he sat studying the faces of his family lined up opposite him. He listened to their voices swirling around him. With his niece's melon-scented perfume fighting against the heady smell of candle wax, and his grand-nephews aiming surreptitious kicks at each other's shins while their grandmother turned to talk to her husband\u2014with his mother and Alma sitting side by side now and absentmindedly holding each other's hand, and the curly-haired stranger bending down to say something to Sam\u2014Macdara found his head flooded with thoughts. Thoughts that swooped down on him like birds gathering in a great oak tree. Words, sentences, whole paragraphs of prose settling around him like crows at dusk. He had a desperate urge to hold on to them, to grab them and stuff them into his pockets, these maddening little morsels of inspiration. And it was then that it came to him. As he sat by his father's coffin, wishing uselessly for a scrap of paper, or a sharp object that he could use to scratch words into the skin of his hand, it popped into his head. A sequence of words as beautifully jarring as it was perfect. As soon as it formed in his mind, he knew that he had found the title for his book. He knew also, with a crashing sense of defeat, that he would have forgotten it again by the time he had left this room.\n\nYou always think you'll remember these things, but you never do.\n\nThe letter arrived the following morning.\n\nIt was the day of the funeral, and Macdara recognized his father's handwriting immediately. He recognized the slanted, spiky script, executed as always in black ink with a thin-nibbed fountain pen. For one mad moment Macdara thought that his father was communicating with him from beyond the grave.\n\nCradling the letter in his open palms the way you might hold a wounded bird, he walked it over to the kitchen table and propped it up against a box of Alpen, taking a few steps back from the table to gain some distance from it. He needed time to think.\n\nThe address that had been handwritten by his father on the front of the envelope was the address of the PO box Macdara had hired at the local sorting office, at a cost of four hundred and seventy euro. The scheme for the hire of the PO box had been brewing in his mind for months. He had looked at it from every angle and it seemed to him that it was the only practical way to resolve the impasse in his head. The book was finished, had been finished for months, but there was no way on this earth that Macdara could even contemplate sending it to anyone until his father had read it. His father's opinion was the only opinion that mattered to him. His father's scorn was the only thing he feared. Because Macdara had written the book with only one reader in mind, and that reader was his father. It seemed to Macdara that there was something mystical about the way the letter had arrived on this of all days. He decided that he would shower first, and then he would get dressed, and only then would he open it.\n\nThe whole time he was in the shower, he was thinking about the letter. His mind like a wobbly weighing scale, one minute he was high on the hope that his father had liked the book, the next minute he had tipped over into the despairing certainty that he had hated it. As he dressed himself in his only suit\u2014a suit he'd had for more than thirty years\u2014it seemed to Macdara that he would be taking a great risk in opening the letter. If it was bad (as of course it would be), if it was as bad as he feared it would be, it would sully the day of his father's funeral. And Macdara wanted to keep the day of his father's funeral free of all worldly things.\n\nHe decided not to open the letter.\n\nThe book was three decades in the writing.\n\nYou do hear about this, don't you? You hear about people for whom it took thirty years to write a book. Macdara comes across these stories from time to time and he finds them comforting. What he hates hearing are stories of prolific writers, people who turn out a book every year for forty years. In Macdara's case, the writing is painfully slow, every sentence something to be worked and reworked, until eventually he is happy with it. Macdara has often spent a week on one sentence, a month on a single paragraph. And of course there have been mishaps. At one point a decade's work was wiped out when the old laptop he was using died\u2014it took him another year to recreate what was lost, and even then he couldn't be sure it was quite as good. Other times it seems to him that perhaps the rewrite was better, and he wonders should he destroy all his work and reconstruct it from memory, the way his father burns his toast and scrapes it?\n\nThe desire to write has been in him forever. He remembers riding around in the back of his father's car\u2014he was six years old, or maybe seven\u2014and there were trees going by the window, and Macdara was trying to find the right combination of words to describe them. Not to describe _what_ they were, that was something that didn't need describing. He yearned to describe _how_ they were. Like the pilot of a plane who traces words across the sky with puffs of smoke, or a graffiti artist armed with a fresh can of spray paint, all of Macdara's earliest instincts told him to carve out a place for himself in the world using nothing but words.\n\nThe first person to encourage him was his secondary-school English teacher. Macdara had landed into the school in fourth year, a refugee from the wreckage of his parents' marriage. It was thought that a bit of distance might put him out of harm's way, but the school was only a hundred miles from Dublin, and news traveled fast. By the time Macdara had unpacked his suitcase and stowed it under his bed, his story had done the rounds of the school, and by some unquestioned homophobic logic that was traded around the locker rooms along with ten-packs of Carrolls cigarettes, he had been labeled a bender. His English teacher encouraged him to pour what he was feeling out on to the page, which he did.\n\n\"Macdara is a great man for the use of the mixed metaphor,\" said the English teacher, closing in on Macdara's father at the parents' day at the end of the year.\n\nThe English teacher was most interested in Macdara's father. In particular there was the matter of a creative writing competition, which he was hoping Macdara's father would agree to judge.\n\n\"Macdara is very _earnest_ in his writing,\" he said, with a nudge in his voice, one eyebrow raised to suggest a shared joke. \"Sometimes a little _too_ earnest,\" he said. \"I'm trying to introduce the concept of creative restraint.\"\n\nMacdara's father fastened his famous blue eyes on the teacher and stared at him glacially as the mechanics of the proposed creative writing competition were explained to him. To Macdara's infinite relief, his father had dressed for minimum effect, in an almost-normal single-breasted tweed suit and a silk cravat, worn inside his open shirt.\n\n\"I wouldn't give you much for our smarmy friend,\" he said to Macdara afterward.\n\n\"He's an idiot,\" said Macdara, using great heat to disguise his colossal sense of betrayal. \"The guy's nothing but an idiot.\"\n\nHe was mortified that his work had been ridiculed. Sick with shame that his most solemn efforts had been made fun of. It would be more than three decades before Macdara dared to show his work to another human being, and even then it was under a thick cloak of anonymity.\n\n\"The word 'paradise' comes from the Persian word for garden,\" said the priest. \"In ancient Persia the greatest honor the king could bestow on one of his subjects was to invite them to walk with him in his garden in the cool of the evening.\n\n\"This is how we imagine Manus now,\" said the priest. \"Wandering the gardens of heaven, deep in conversation with the Lord. And Lord knows, they must have _plenty_ to talk about.\"\n\nLaughter broke out in the church, along with a gurgle of delight at the subversive nature of the joke. The priest was an amateur poet and a fellow Kildare man. An old friend of Manus's, at the end of the Mass he led his coffin out of the church with great pomp, holding the huge gilded cross high for all to see. Propping up the front right-hand side of the coffin with his left shoulder, Macdara fixed his eyes on the shuffling ruffle at the bottom of the priest's vestments, so as to avoid looking the mourners in the face as he passed. There was something touching and sad about the priest's sensible rubber-soled shoes, walking along under all those ruffles. Macdara always found people's shoes very touching and sad.\n\nFrom the balcony, the soloist was singing \"Nearer, My God, to Thee,\" which made Macdara even sadder.\n\nOutside the church, the sun fell down on them in warm yellow flakes. The little boys ran wild loops of the churchyard, like horses turned loose in a paddock, while the adults in the family were islanded by well-wishers. Macdara shook hands until his arm ached, and eventually the crowd began to thin. He could see his mother now, arching like a pergola over another old lady. His sisters, heads held high and laughing; they were both of them dressed up to the nines. He could see Sam being led to a car by Nando. (Macdara had been introduced to Nando outside the funeral home, and had judged him on the face of it to be a most trustworthy person.)\n\nHis father was to be cremated right after the funeral. The sea burial had turned out to be too complicated. To carry out a burial at sea, you had to have a special license from the Department of the Marine. You had to modify the coffin, by inserting a zinc lining and drilling holes into it, otherwise it would only float; there was even a question of weighing the body down. All things considered, it had been decided that it would be wiser to have Manus cremated, and to scatter his ashes at sea.\n\nMacdara sat beside his mother in the crematorium. Sam was with Nora and Nando, while Acushla and Alma sat side by side at the edge of the pew. As the curtains closed in on the coffin, they clung to each other, sobbing audibly. Macdara found their sudden outpouring of grief puzzling. It seemed like only a moment ago that they had been laughing. From some unseen speaker high up on the wall came the sound of music. Louis Armstrong, singing the mourners out into the sunshine.\n\nAfter the cremation, they all went to the pub. Acushla had wanted to put on a reception for everyone in her home and Alma had argued for a hotel, but Deirdre had insisted on going to the pub. She had insisted on the pub where she and Manus had met all those years ago, and where he and his friends had held court on so many a night. Nothing would do Deirdre but to have the reception upstairs in McDaids.\n\nThere were trays of sandwiches laid out on the bar when they arrived. Yellow light coming through the stained-glass windows. On the walls, black-and-white pictures of Manus and his cohort as young men. Macdara ordered a pint for himself, and one for Sam and Nora. Nando would only agree to having a half.\n\n\"I'm a useless drinker,\" said Nando apologetically.\n\n\"An excellent quality in a person,\" said Macdara. \"I wholeheartedly approve.\"\n\nHe raised his glass, making sure to catch Nando's eye before drinking from it\u2014someone had told him that it was deeply offensive to other cultures not to make eye contact while toasting. Macdara felt shy in the presence of Nando, as he often was in the company of anyone other than family. He began to long for an opportunity to escape the circle he found himself in, an opportunity that came soon enough with the arrival of a steady trickle of his father's various acquaintances; in twos and threes they came, clattering up the rickety stairs. From literary compatriots to country relations, from genuine friends to phoney hangers-on, they kept coming until the little room was heaving, the noise gathering like fumes under the glossy ceiling as the extractor fan in the wall whirred away inadequately. The temperature in the room was such that the men began shedding their suit jackets and ties, and opening up the buttons of their shirts. The women bared their tanned arms, giving the gathering a most un-funereal aspect. After three pints, Macdara could feel himself getting drunk, but that didn't stop him going to the bar to order a fourth. It wasn't often he had the opportunity to get drunk.\n\nHe found himself standing facing Acushla. Her face flushed from the heat, she wore her cardigan draped elegantly over the shoulders of her sleeveless black dress, and she was holding her glass of white wine out in front of her like a model displaying some kind of kitchen product in an advertisement. Her husband was standing beside her with his arm wound around the small of her back. Liam had his face turned to one side while he talked to someone behind him.\n\n\"You know that Daddy loved you,\" Acushla was saying, her voice pitched higher than usual.\n\n\"Of course,\" said Macdara, stumbling forward as he leaned in toward her. She put the palm of her free hand flat on his chest to stop his fall, leaning in to shout into his ear above the noise of the room. \"I know you weren't as close to him as me and Alma were, but that doesn't mean he didn't love you.\"\n\nAnd Macdara was about to tell her that he had never imagined this to be the case when he stopped himself. Acushla's reassurances had raised a question in his head, a question that had never existed before. He frowned as he contemplated it, swaying a little on the spot where he stood. Stepping to the side, he set his half-drunk pint down on the bar and made for the loo, aware only of a desire to get away from the noise and the people so that he could think.\n\nThe men's room was empty, but even so Macdara made for a cubicle. He had never been able to bring himself to use a urinal in a public toilet, preferring to lock himself into a cubicle so that he could sit down. Letting his suit trousers fall toward the wet tiled floor, he sat on the toilet, half closing his eyes to bring the graffiti on the back of the cubicle door into focus. There was a burst of noise as the door outside opened and closed again. In the conversation that followed, he identified his brother-in-law's voice, along with another voice that he didn't recognize. Desperately, Macdara strained to restrain his bladder, unable to bear the indignity of being heard to urinate.\n\n\"Well, we won't see his like again,\" Mick was saying, with drunken emphasis.\n\n\"You said it,\" said the other man. \"They broke the mold when they made old Manus.\"\n\nMacdara heard a stream of piss, and then another. He listened out for the sound of the taps being turned on for them to wash their hands, and the warm-air hand drier being used to dry them, but those sounds never came. Instead he heard the door hinge opening, and a burst of noise from outside as they left. Only when the door had closed behind them again did Macdara finally let go of his bladder. He washed his hands carefully outside before emerging furtively from the men's room. Grabbing his suit jacket from the hook in the hallway, he left the pub without saying good night to anyone.\n\nThe letter was waiting for him where he'd left it. Like a small stone that finds its way into your sock, or a scratch on the lens of your sunglasses, he had been acutely aware of it all day. He had been forced to apply great concentration to the ignoring of it. Now here it was, sitting patiently on the table. Another moment and he would be sitting down, reading its contents.\n\nHe took two steps back from the table and turned toward the fridge, postponing the decision momentarily while he searched for something to eat. There were two cold sausages that he had stowed earlier on the top shelf; they were glued to the china saucer by a pool of their own congealed fat. Macdara withdrew the saucer from the fridge and put it to rest on the kitchen counter while he dropped two slices of bread into the toaster. He peeled the sausages off the plate and split them each lengthways with a knife, revealing the globby sausagemeat inside. When the toast popped, he buttered it generously. He tiled one slice of toast with the cold sausages, using the other slice as a lid. Taking the sandwich in both hands, he began to eat it, marveling at the deliciousness of this thing that he had made. For a moment there was nothing in his mind, only the deliciousness of the cold sausage sandwich. But then the letter caught his eye. He realized that he did not want to open it.\n\nAs long as the letter went unopened, Macdara's world remained unchanged. But once he had opened it, once he had read its contents, he would be brought down one of two paths. Either his father had disliked the book (and the likelihood of this was very much uppermost in Macdara's mind), in which case Macdara would have to live forever with his father's disapprobation, a burden that would weigh heavily on a father-son relationship that had up until now managed to remain miraculously unconfrontational. In the unlikely event that his father had _liked_ the book (and it occurred to Macdara that this outcome was almost as terrifying as the prospect of his disliking it), well, that brought with it a whole new set of problems. If his father had liked the book, then there was never any reason for Macdara to have feared his scorn, and no need to have kept his writing a secret from him. If his father turned out to have liked the book, then Macdara would see that he had squandered the opportunity of a friendship with Manus, one based on mutual respect and the pursuit of a shared craft, rather than the pact of mutual misunderstanding they had observed with each other since Macdara was a child.\n\nIt occurred to Macdara now, as he licked the last bit of melted butter out of the corner of his mouth and burped a dry bubble of sausage-scented Guinness fumes out of his belly, it occurred to him that there was nothing to be gained by opening the letter. By not opening the letter, he could maintain forever the equilibrium that had existed between him and his father. An equilibrium that had lasted until the final whistle (or the final gasp of a dodgy handbrake). There was no point in revisiting it now. Revisiting it would only bring trouble.\n\nWith this thought in mind, Macdara took his father's letter and tore it in two. Observing glimpses of black ink where the envelope and the letter inside it had been rent, he resisted the temptation to peek, tearing those halves in two again, and those halves in two, and those halves in two, until all that remained were small flakes of writing paper and small flakes of envelope, like homemade confetti. Macdara swept them off the table into his cupped hand and dumped them into the trash can.\n\nAs he climbed into bed in his vest and underpants, having given himself a special amnesty from flossing and brushing his teeth, his head was spinning from the drink, his stomach unsure about the sausage sandwich, but his mind was untroubled by worries of any kind. He was asleep within seconds.\n\nThe letter!\n\nOh, Jesus, the letter. Macdara sprang out of bed and lurched through the kitchen, headfirst into the bin. Fingers scrabbling through butter-smeared paper and coffee grounds and melon pips, he began fishing out the scraps. There were fewer of them than you would imagine, and they had found their way farther down into the bin than he would have thought possible. He had to turn the contents out on to a newspaper on the floor, pawing his way through a soggy potpourri of eggshells and breadcrumbs, a melon half-shell with its reduced sticker still clinging to it, a toilet roll tube that he had neglected to recycle. Somewhere amid all this waste were his father's last words to him.\n\nMacdara laid the fragments out on the kitchen counter like pieces of a jigsaw. But what he had before him was more of a mosaic. Tiny shards of paper stained with ink that was no longer black but all the colors of an oil spill. Whatever it was that was wet in the bin, it had made of his father's script a watery batik. As his hangover swarmed noisily around his head, as the fact of his father's death uncoiled itself inside his belly, it became clear to Macdara that it was not going to be possible to reconstruct the letter.\n\nHe closed his eyes and let the tears slowly squeeze their way out of him. Tears that had been years in the forming. It was a surprise to Macdara what a relief it was to let them finally seep through.\n\nIt was a few weeks before they got round to going through his father's things.\n\nIn that time, it had been agreed that Nando would sleep in the apartment at night, so that Sam would not be on his own. The meager resources of the state would be called upon to provide carers for Sam during the day, with the family filling any remaining gaps between them. On the whole, it seemed to Macdara a most satisfactory resolution of the rather precarious situation Sam had been left in by Manus's death.\n\nSam seemed largely unaware of Manus's absence, and when on occasion he did comment on it, he seemed to absorb without emotion the explanation that Manus was dead. He showed no signs of being distressed, expressing only vague regret, as if it were the death of someone he hardly knew. \"Oh, dear,\" he would say, whenever he was reminded of it. \"Oh, dear,\" he would mutter, with mild surprise. \"That's very sad.\" It was only when Acushla went to clear out some of her father's things, only then did Sam become upset.\n\n\"We'll have to do it behind his back,\" she told Macdara. \"The next time the carer takes him out, we can all go round. It will be nice for us all to do it together.\"\n\nMacdara arrived first, followed within the space of a few seconds by his sisters. Wearing their sunglasses on their heads and carrying empty wine boxes and rolls of trash bags, they looked like aging film stars who had been sentenced to community service. One silver-haired, the other crowned a pale wheaten gold.\n\n\"Why don't we make a start on the clothes,\" said Acushla, tearing off a bin bag and handing it to Alma. \"Three piles,\" she said, pointing out spaces on the bedroom floor with a manicured fingertip. \"One for the bin. One for the charity shop. One for things we might want to keep.\"\n\nAlma nodded, and it occurred to Macdara how odd it was to hear Acushla telling Alma what to do. Stranger still to see Alma doing it.\n\nWhile his sisters worked their way through his father's wardrobe, Macdara made a start on his desk. Into a trash bag he tossed all the old matchbooks and swizzle sticks that he found in the middle drawer, all the paper clips and leaking batteries and dried-up pens. Checkbook stubs going back decades and business cards belonging to publishers long swallowed up by conglomerates, all of them went into the bin too. His father's much-stamped old passports he put aside as a memento, along with a box of slides that appeared to have been taken at somebody's wedding. Macdara held one of them up to the light and saw a clutch of women wearing sixties-style clothing; the images were too indistinct for him to recognize any of them.\n\nIn the side drawers, Macdara found his father's correspondence. Folders of documents relating to bank accounts and insurance policies. His father had filed and labeled them all meticulously, for fear of anything falling by the wayside. The documentation relating to his work was limited to a single drawer containing a sheaf of annual statements from his publisher. (The manuscript of his father's book, and most of his personal correspondence, had long since been donated to a university archive.) The previous year's royalties amounted to less than a hundred euro, something Macdara found very poignant. But his sadness was banished in an instant by his discovery in one of the drawers of a Time Out bar. Macdara laughed out loud at this joke from beyond the grave.\n\nIn the left-hand bottom drawer, he found what he was looking for. His own unbound manuscript, held loosely together by its original twine binding. On the first page, his father had inscribed in slanting capital letters his suggestion for a title. Turning the pages, Macdara could see that his father had peppered the margins with handwritten notes. As he breathlessly leafed through them, he was aware of the invisible dust of his father's final movements falling out from between the pages.\n\nBefore he left\u2014with the manuscript of his book carefully stashed in a supermarket carrier bag and his father's passports tucked away in the pocket of his jacket\u2014Macdara helped himself to a small selection of his father's more flamboyant clothing. Now that his father was gone, it seemed important to Macdara that his heirs adopt a little of his eccentricity. Otherwise the family would only sink back into the ordinariness whence they came, something Macdara was determined not to allow to happen.\n\n**_You are hereby invited to join_**\n\n**_Deirdre O'Sullivan_**\n\n**_in celebrating her 80th_**\n\n**_birthday at her home on_**\n\n**_Saturday 5 October at 6:00._**\n\n**_Strictly no presents._**\n\n**_Each guest is asked to_**\n\n**_prepare a short party piece._**\n\n**_R.S.V.P._**\n\n# Deirdre\n\nDeirdre woke on the morning of her party and noticed, for the first time in months, a chill in the air. A chill that registered first on the tip of her nose, and then in the cool intake of breath coming into contact with her warm lungs. A newborn feeling, brought on by the change of season. She felt nostalgic already for the summer that was gone. Eager at the same time for the joys of winter, she found herself thinking with longing of log fires and beef stew. Freezing-cold water straight from the tap. She was surprised how strong a pull was exerted on her by these temporal things.\n\nShe hopped out of bed and put on her robe\u2014a sensible purple fleece Acushla had bought for her in Marks & Spencer, it was undeniably comfortable, and Deirdre found herself wearing it more and more recently, despite the loss of self incurred. To her horror, comfort had begun to be more important to her than style, which was exactly what she had always feared from old age. She belted the robe loosely around her waist and slid her feet into her Chinese slippers\u2014slippers Manus had brought back for her from Hong Kong a hundred years ago. They were duck-egg blue with brocade birds on them, and faded though they were, Deirdre loved them.\n\nStopping off briefly at the lavatory on the return, Deirdre proceeded down two flights of stairs, alighting in her vast bare entrance hall just as a slab of ruby light from the stained-glass fanlight over the door fell in a diagonal across the bare floorboards, spotlighting the two wooden crates of champagne that were stacked against the wall. Veuve Clicquot, Deirdre had chosen, with a flush of pleasure at the thought of how Manus would have appreciated the reference to her widowhood. But of course Manus wasn't here to appreciate it, something Deirdre kept having to remind herself of, over and over again.\n\nIt wasn't that she had forgotten that he was dead. Only in her dreams was she under any illusion that he was still alive; during her waking hours she was fully aware of his absence. It was the nature of that absence that she was struggling with. The obstinately finite nature of it. It seemed unreasonable to Deirdre that death should claim Manus so absolutely. That his absence could not be alleviated by the occasional visit, the way someone living down the country might pop up to Dublin for a day, dropping in for a quick cup of tea before disappearing again. That he might not pass by in his car one afternoon with a beep and a flutter of the hand. In particular, it seemed unreasonable to Deirdre that he could not make even a brief appearance at her party. \"For One Night Only,\" as the ad in the newspaper might say of an old-time crooner coming out of retirement. She found herself frustrated by the fact that this was not going to be possible.\n\nDeirdre very much wanted Manus to be at her party. His presence had been her central motivation for having the party in the first place. In conceiving it, she had imagined herself putting on a final display of bravura, largely for his benefit. One last magnificent performance that he would chuckle over afterward, wondering at her capacity to surprise him even after all these years. With a crushing sense of anticlimax, Deirdre realized that she was throwing a party for someone who wasn't going to be there.\n\nIt was not part of Deirdre's plan that Manus would die before her. She had her letter to him written already, a letter that was to be given to him after her death, and in that letter she had said the one thing she had never said to him before, the thing she had been holding out on saying to him all these years. It was her last card, and one that she had delighted in hoarding, but she had always planned to lay it facedown on the table when the time came. The fact that she forgave him.\n\nWhen her children came to break the news to her that Manus had died, Deirdre was sitting in an old wing-backed chair in the breakfast room at the back of the house. A tray in her lap, she was busy licking the envelopes for her invitations. The dry taste of glue on her tongue\u2014a taste that reminded her of the Holy Communion wafers she had not received since she was a girl\u2014she was just licking the last of them when something made her look up at the window.\n\nThrough the wobbly glass she saw her three children making their way up the garden toward the house. Solemn-faced and single file, they weaved along the overgrown path, ducking their heads every so often to avoid the overhanging roses. Deirdre put her tray aside and stood to receive them. When they told her what had happened, explaining it to her several times before she understood what it was they were saying, she sank back down into her chair. She looked up at her children's faces, seeing the same expression of alarm in all their eyes.\n\n\"I don't believe it,\" she said in a tone of genuine wonderment. \"He's gone and done it to me again.\"\n\nWhen Manus left Deirdre, she was in her late forties. Menopausal and wretchedly mercurial with it, she was just struggling to come to terms with the loss of her youth when she was faced with the loss of her husband too. An event that gutted the household like a fire. In the aftermath of it, Deirdre was forced into survival mode, for the sake of the children. That first night without him, they sat in front of the TV in the breakfast room and ate greasy chips straight out of their wrappers. A treat that Deirdre had thought up because she couldn't bear the thought of them all sitting round the kitchen table without him, it failed miserably to alleviate the gloomy pall of his absence.\n\nThe children were all in shock, of course, although it manifested itself in different ways. Whereas Alma took to a round of aggressive serial dating, punishing any man who dared come near her for her father's act of faithlessness, Acushla cried herself to sleep every night, soaking her pillowcase with bitterly disenchanted tears. And while Macdara gave no obvious indication of having been affected by his father's defection, Deirdre would come down in the mornings to find the washing machine mid-cycle and his sheets inside. Macdara was fourteen years old at the time, and he had stopped wetting the bed when he was seven; the return of the habit was a clear sign of his distress. When Deirdre found a vile little note in the pocket of his school trousers\u2014a note that had been composed by a number of his classmates, judging by the varied handwriting\u2014she decided to send him off to boarding school, a decision she was forced to make alone because Manus and Sam had embarked on an extended tour of South America. From then on, Manus seemed always to be away whenever there was cause for either crisis or celebration.\n\nIt was while Manus was in India that Alma announced that she was getting married; he was in Indonesia when Acushla got engaged. When Acushla dropped out of college, he was on a road trip across Spain, and when Macdara had to be got home from France, Manus was on a remote Greek island and could not be contacted. When Deirdre accompanied Acushla to England to have the abortion, Manus was in Japan. When Mick and Alma broke up, he was wintering in South Africa. All the time Manus was gallivanting about the world, sending swashbuckling postcards from ever-more-glamorous locations, Deirdre was manning the home front. Like the survivor of a nuclear disaster, she was foraging about in the supermarket shelves for food that was past its sell-by date, dressed in secondhand clothes and Alma's cast-off Doc Marten boots. Although in fairness Manus never failed to meet her alimony payments, Deirdre never stopped worrying about money.\n\nAnd whereas Deirdre was forced to live out the rest of her life in the dark, creaking old house they had bought when they were first married, Manus settled like a soap bubble into his weightless glass apartment above the tree line. The lightness of his life\u2014the gliding, ethereal quality to him as he got older\u2014it seemed to Deirdre in such sharp contrast to the merciless gravity that weighed her down. Stocky as a stevedore, with arms and legs as thick as tree trunks, she was the product of all the years of heavy lifting she had done in his absence. What she wanted now more than anything was something of the lightness he had so effortlessly achieved. And just when she had found a way of achieving that, just when she had come up with a way, at last, of setting herself free, Manus went and pulled the rug out from under her again. In dying before Deirdre ever had the chance to implement her plan, he had stolen a march on her, and here she was, for the second time, left with the task of carrying her children through their grieving.\n\nDeirdre saw no reason to cancel the party. Acushla wanted to call it off, for fear of what people would think, and Alma suggested postponing it, at least until after Christmas. Even Macdara was in two minds.\n\n\"I don't know,\" he said, when he came into the kitchen with his invitation in his hand. (Deirdre had held off on posting them for a few weeks, in order to give things a chance to settle.) \"It's hard to see how we'll be in the mood.\" He turned the gilt-edged card over to study the back of it, even though there was nothing written there.\n\n\"On the other hand,\" ventured Deirdre, \"maybe it's just what we all need, to cheer us up.\"\n\nMacdara frowned, looking at the ceiling as if he was trying to add something up in his head. Ever since Manus had died, Deirdre had noticed this curious focus in him; his mind seemed very busy all the time, but elsewhere. She had the feeling that every word she said to him went unheard, like snow falling in the darkness. She tried speaking very softly, to get his attention.\n\n\"If anything, your father's death underlines the need for the rest of us to live for the day.\"\n\nLike a child who is jealous of a sibling's birthday, Deirdre was anxious for the attention to shift to her. For weeks they had spoken of nothing but Manus. And while at first Deirdre had borne the weight of their grief with good grace, the protracted cycle of mourning had started to irritate her. Acushla in particular had not stopped crying since Manus died. They only had to mention his name and she was off again. It seemed to Deirdre a profoundly unoriginal response to what was essentially a perfectly natural occurrence.\n\n\"At my age, you don't get sad when someone dies,\" she said, repeating once again what she had said to everyone at the funeral. \"When someone our age dies with all their faculties intact, it's a cause for celebration.\"\n\nHow many times over the course of those first few days, had she delivered those lines with her customary swagger to a mildly shocked audience. And while it was all part of a script she had spent a lifetime honing, to her own ears it had started to sound somewhat unconvincing. Like a priest saying prayers to a God he had come to doubt the existence of, or a mathematician following the rules of a formula he had come to question, Deirdre could not escape the unsettling feeling that she was talking nonsense.\n\n\"Everyone has to die sometime,\" she said as they left the church after the month's mind for Manus. They were all standing in the churchyard in the evening sunshine, and it seemed to Deirdre an opportune time to inject a little buoyancy into the proceedings. But no sooner had she said it than Acushla's eyes began to well up again. Her sister took her arm to steady her.\n\n\"Mother,\" said Alma, fixing her eyes on Deirdre. \"That's enough.\"\n\nDeirdre looked around at her family, seeking support from some quarter, but none of them would meet her eye. Connie had bent down to tuck Ernie's shirt into his trousers and Nora had turned to see where Nando had got to with Sam. Liam and Mick had their heads down, muttering something to each other. Even Macdara would not look at her, choosing to stare at his shoes instead. Deirdre found herself marooned by her own doing.\n\nIn the aftermath of Manus's death, Deirdre was forced to consider the possibility that she had underestimated the resilience of her family. She had counted on them taking his death in their stride, treating it with a minimum of sentimentality and a heavy dose of black humor. What she had not counted on was their grief, a grief that was as profound as it was genuine; the effect it had on her was discombobulating. And while she never once considered the possibility that she too was grieving, she found herself looking out the kitchen window every so often, half expecting to see his car sliding up to the pavement. A dozen times every morning she found herself putting on the kettle and warming the teapot for a ritual that no longer took place. Over and over again she emptied the water out of the teapot without ever having made the tea, and still she could not admit, even to herself, that she was missing him.\n\nDeirdre was going to invite some friends to her party, but then she realized she didn't have any. She had managed to divest herself of them all, semi-deliberately, as the years went by. She had come to the conclusion that it was not possible to maintain multiple friendships; it required far too much effort. It required you to do things that you didn't want to do, and to go to things you didn't want to go to, like other people's drinks parties, and their children's weddings. Like their dreaded retirement dinners, or the launch parties for their ghastly self-published memoirs. \"I can't imagine anything I'd like less,\" she began saying, because she had decided that she was too old for making excuses. \"If you don't mind me saying so, I'd prefer to have my toenails pulled out,\" she would say, in response to an invitation to a golden wedding anniversary, or a ladies' lunch, thinking that surely people would be amused by her honesty and appreciative of it. But they weren't a bit amused. It turned out that people were very easily offended, something that surprised Deirdre enormously. It baffled her to see how thin-skinned people were. You'd think they'd be a bit more hardy.\n\n\"I know this isn't your kind of thing,\" said Acushla when she stopped by one day with the newspaper. (Ever thoughtful, Acushla had taken upon herself her father's habit of bringing Deirdre the paper.) \"You don't have to do it if you don't want to,\" she went on, and Deirdre couldn't help but be irritated by her overly apologetic tone. \"It's just this thing I've been invited to. They give out awards to women who've done something during the year.\"\n\n\"Are you telling me that you've won one of these awards?\"\n\nIt was as Deirdre had feared.\n\n\"There's a dinner,\" said Acushla. \"I can always bring Liam, but I thought I'd ask you first, on the off chance that you might like to be there.\"\n\nDeirdre was just about to make her excuses when something in Acushla's expression gave her pause for thought.\n\n\"You don't have to answer now,\" said Acushla, cutting her off before she had a chance to say anything. \"Why don't you think about it and come back to me closer to the time?\"\n\nAnd Deirdre nodded, even though she knew with absolute certainty that she would never in a million years feel like getting all dressed up on a winter's evening only to sit through three courses of mediocre food in a hotel ballroom with a gaggle of silly women when she could be at home in her warm bed with a TV box set.\n\nIt was Alma who was responsible for getting her hooked on the box sets, with a gift of _Downton Abbey_ that Deirdre found indecently enjoyable (even though Maggie Smith's performance made her ill with envy).\n\n\"I had lunch with her the other day,\" said Mich\u00e9al MacAoda, when Deirdre bumped into him on Baggot Street. She was on her way to the liquor store to order the champagne for her party, and thankfully she had dressed for the occasion, choosing to wear a wine-colored silk blouse, pinned at the neck with the antique cameo brooch her mother had given her as a wedding present. She wore a long tweed skirt with deep pockets, and instead of her usual lace-up leather ankle boots, she had on a pair of soft brown leather brogues that she had long ago stolen from Manus, she and he having precisely the same-sized feet. Her suede gloves were one yellow and one brown, their partners lost long ago; it seemed to Deirdre they provided exactly the right note of eccentricity. When she saw Mich\u00e9al walking toward her on Baggot Street, she realized that she had been expecting to meet someone.\n\nHe stopped a few feet away from her and pointed at her. Deirdre was reminded of the famous poster of Kitchener: WANTS YOU!\n\n\"Deirdre O'Sullivan,\" he said.\n\n\"Mich\u00e9al MacAoda.\" She stepped forward to kiss him on both cheeks, a sense of her own long-lost glamour returning to her as a result of this encounter. It was Mich\u00e9al who had directed her in _The Cherry Orchard_ at the Gate, and at the wrap party he had made a fumbling pass at her. A pass that Deirdre had gently deflected, the memory of it was nonetheless fond to her.\n\n\"Isn't that funny,\" he said, gazing at her. \"I was only thinking about you the other day.\"\n\n\"Oh?\" said Deirdre, opening her eyes wide and plumping out her mouth.\n\n\"Yes,\" he said. \"It was after I met Maggie Smith. I had lunch with her the other day, and she planted the idea in my head of putting together a stage production of _Downton._ \"\n\nAnd of course, Deirdre knew what he was going to say next.\n\n\"I thought of you immediately.\"\n\n\"Oh, Mich\u00e9al. Don't be ridiculous. I haven't set foot on the stage for thirty years!\"\n\n\"Nonsense, Deirdre. You could do it in your sleep. The more I think about it, there's no one else who _could_ do it. It's got your name written all over it.\"\n\n\"All I can promise you is that I'll think about it,\" was what she said as they parted.\n\n\"You'll do more than think about it,\" he said, holding his finger up in the air in a parting shot.\n\nAs Deirdre sailed across Baggot Street, she was aware of an irresistible energy rising in her, an energy that signalled the advent of a new and most inconvenient lease of life.\n\nThe storm came on the afternoon of the party, with heavy rains and a high wind that sounded like a swarm of police sirens converging on some daring heist, or a dreadful car accident occasioned by the rain. Listening to the wail being carried on the air, Deirdre found it hard to believe that it was only the sound of the wind.\n\n\"Would you listen to that?\" she said. \"It sounds like the end of the world.\"\n\nShe had been planning on draping a string of candles in jam jars along the railings that bordered the front steps, but clearly that wasn't going to be possible now.\n\n\"I wonder would we put them inside? We could set them up inside the fanlight\u2014that way people would be able to see them from the outside when they arrive.\"\n\nMacdara came and stood beside her, looking up at where she was indicating.\n\n\"You'll need to use the stepladder,\" she said.\n\nAgain he hesitated, waiting for her to say more.\n\n\"It's in the cupboard under the stairs.\"\n\nAnd where any of the other members of the family would have been exasperated with Macdara for not knowing this, where the girls would have wondered whether he kept anything in his head but pure air, where his father (when he was alive) would have marveled at his ability to move through the world without accumulating any knowledge of the nuts and bolts of life, Deirdre did not entertain any of these thoughts. She had long ago accepted Macdara for what he was, and expected nothing else of him.\n\nDeirdre had arranged for Macdara to receive her guests for her. Not because she wanted to make a grand entrance, but because she very much wanted to watch them arrive without them being able to see her.\n\nShe put the finishing touches to her outfit, using her cameo brooch to fasten the long purple velvet cloak she had chosen to wear over her silk evening dress (a cloak she had worn when she played Gertrude in a 1983 production of _Hamlet_ on the Abbey stage; she had liked it so much that she couldn't resist filching it). She adjusted her satin headband so that the feather she'd attached to it rose dramatically from just behind her left ear. Then, turning out the light in her bedroom, she pulled a low upholstered chair up to the window and sat down. With the curtains partly drawn, she peered down into the empty street, waiting for her family to arrive.\n\nIt had stopped raining momentarily, but there was still a lot of water on the ground. A taxi swept to a stop at the gate, spreading great wings of water out on either side of it; it was lucky there was no one walking along the pavement or they would have been soaked. Mick appeared on the far side of the taxi and, coming around behind it, opened the door for Alma. She stuck one foot out first, letting it dangle rather inelegantly above the pavement. Then, getting ahold of herself, she managed to rise quite gracefully out of the taxi, with the help of the arm he offered her. She winked at him mischievously, and muttered something to him. They brought the best out in each other, those two, no question about it. As they climbed the front steps of the house, they were wobbling with laughter, but what it was they were laughing about Deirdre could not discern. She watched from above as the front door was opened from within and a handful of light was thrown over them. From below, she heard Macdara's voice.\n\n\"The animals came in two by two, hurrah, hurrah.\"\n\n\"Ah, Noah,\" said Alma, picking up on the joke.\n\n\"What a God-awful night,\" said Mick, stepping inside. The front door closed behind them, cutting off the sound of their voices. Deirdre sat very still for a moment, wondering would she be able to hear them downstairs now, but she couldn't. All she could hear was the low, buzzing static of the air in her room, and the sound of her own breathing, slightly faster and shallower than usual because of her excitement. She rolled her shoulders in their sockets, one by one, moving her head in a slow circle on her neck, with her eyes closed, to ease up the tension. These were things she had learned to do before going on stage.\n\nShe heard a sound outside and opened her eyes again just in time to see Nora and Nando coming through the gate. Nando had a guitar case slung across his back and Nora was carrying a wet paper bag in her arms, which Deirdre identified instantly as a gift. (It was just like Nora to flout the no-gift directive, even though she was the one who could least afford it.) They had no coats on them, no hats or scarves, no umbrella. Their hair was plastered to their heads and their faces were damp with rain, as if they'd been crying happy tears. As they flopped up the steps, Deirdre was thinking how nice it was to see Nora so happy. How odd it was that it should be this man\u2014a man who had landed in their lives from the other side of the world\u2014to finally tether her to home.\n\nLiam and Acushla were coming down the street. It had started raining again and Liam was holding a large black umbrella over their heads, Acushla struggling to keep up with him in her high heels. Oh, wouldn't you think she'd wear more sensible shoes on a night like this? But then Acushla never had an ounce of common sense. Poor Acushla.\n\nWhen Deirdre saw Connie and Emmet and the boys come traipsing up the front steps, she rose out of her chair. Gripping her lower back with her hands for support, she straightened up, allowing herself one last glance in the mirror before she swept down the stairs with the train of her cloak slithering step by step after her.\n\nWhen Deirdre entered the room, she had a chance to take a quick mental snapshot of her family before they turned to greet her.\n\nMacdara was standing in the center of the room, holding a bottle of champagne in his raised hand. Connie stood before him with her glass held out to receive it. She was balancing on a pair of platform wedges, with one little boy wrapped around her ankle. His brother was on all fours, disappearing behind the curtains. Liam stood peering out of the window, as if he was waiting for someone to arrive, while Nando was moving round the walls, studying the gallery of framed photographs, with Nora at his side as a guide. He had his arm snaked across the small of her back and she had hers stretched across his, like a pair of figure skaters about to execute some complex move. Emmet was hovering over by the table where the food was laid out. He had helped himself to a canap\u00e9, and as he chewed the half he had already bitten off, he was squinting at what remained in his hand, trying to figure out what it was. Over by the fireplace, Alma had settled herself into an armchair with her skirt pulled up over her knees to show off her legs and her face raised as she listened to something Mick was saying. He was standing with his back to the roaring fire, his hands splayed behind him to warm them while he talked to his wife. Acushla stood before them with a plate of canap\u00e9s, bending from the waist in the manner of an air hostess to offer one to Alma. Oh, why on earth didn't we just let her be an air hostess? thought Deirdre in a flash of clarity. It seemed so obvious to her now that Acushla was born to be an air hostess.\n\n\"Mum!\" said Acushla when she caught sight of Deirdre.\n\n\"Grandmother!\" said Connie, turning round. She had to drag her foot along the carpet with the child still clinging to her ankle.\n\nDeirdre felt time speeding up as they all converged on her.\n\n\"Deirdre,\" said Mick, advancing with his arms held wide. \"You look magnificent!\"\n\n\"Happy birthday, old girl,\" whispered Macdara as he brushed past her.\n\n\"Happy birthday, Gran-Mum,\" said the little boys, and Deirdre found herself wondering was that the moment when she had first loosened her grip on her own life, when she had allowed herself to be called Gran-Mum.\n\n\"Happy birthday, Grandmother,\" said Nora, coming forward to kiss her.\n\n\"Where's Sam?\" asked Deirdre. She had just noticed that Sam wasn't there.\n\n\"Oh, we decided it wasn't fair to bring him,\" said Nora. \"He's not really able to handle crowds, he gets stressed. He's much happier at home with the carer.\"\n\nFor a moment Deirdre was disappointed that Sam wasn't there, until it occurred to her that there had been a time when she and Sam could have been friends\u2014at any time over the years she could have made friends with him\u2014but that time had now come and gone. She had a sense of her life being carried rapidly downstream, of things and people being swept into the past, and herself along with them. A great desire in her to grab on to something. Anything, to still the passage of time.\n\nIt was Alma who volunteered to kick off the party pieces. She got to her feet in front of the fire, stepping out of her shoes so that she was standing in her stockings. Without her heels she was smaller, but somehow all the more powerful for it. She pushed the huge satin wrap she wore to the very edges of her shoulders and with her eyes closed she began to sing \"Down by the Salley Gardens.\" Alma was no singer, but she had chosen her piece well, rendering it so low and so slow that it was the words you heard, and not the singing of them. By the final verse she had abandoned any pretense of singing, devoting to those beautiful lines the full glory of her speaking voice.\n\nBefore the applause had even died down, Acushla was on her feet. Her choice of a Patsy Cline song was unfortunate. Too raw, too exposed, too high on the scale. As she sang, she couldn't quite hit all the notes. But this imperfect performance brought out only tenderness in Deirdre. She was aware of a fierce pride, verging almost on admiration, for this daughter of hers whose desires always ran so far ahead of her. Deirdre wondered was it too late for her to tell Acushla that she was proud of her. For one dreadful moment it seemed to her that she had left it too late, until she remembered the shy invitation to the Women of the Year awards\u2014an invitation Deirdre had actively considered turning down. She remembered it now with relief and vowed to accept it.\n\nWhen Acushla finished singing, she looked to Deirdre for approval and Deirdre saw that her daughter needed her still. She needed her now perhaps more than ever. She needed her mother to witness her triumph.\n\nAfter a short break to fill up people's glasses, Nando played an Argentinian folk song on the guitar. (\"He _would_ play the guitar,\" Deirdre heard Mick mutter to Alma.) Next up was Emmet, who told a long and inappropriate joke about a necrophiliac Frenchman. Alma flared her nostrils and narrowed her eyes at him. Acushla looked nervously at Deirdre to see if she was offended, which she wasn't, and Deirdre saw Nora glance pityingly at Connie, who sat ruffling Oscar's hair with her fingers as if she hadn't heard.\n\nFor his party piece, Oscar stood in the center of the room and turned his tongue upside down in his mouth. He was made to go round the room repeating the trick up close for all his older relatives. Then it was announced that Ernie would recite a poem, dedicated to his great-grandmother.\n\n\"MASHED POTATO, A LOVE POEM,\" he said, speaking in a loud monotone. \"BY SIDNEY HODDES.\"\n\n\"Stand out into the middle of the room,\" hissed Connie, giving him a little shove with her fingertips as if he were a small toy boat that she was pushing out toward the middle of a pond.\n\n\"Turn around to face Gran-Mum,\" shouted Acushla.\n\nErnie repositioned himself to face his great-grandmother, his face flushed as he resumed his recitation.\n\n\"Start again,\" said Alma. \"Start from the top.\"\n\n\"Nice and slow,\" said Acushla.\n\n\"Would you leave the poor child alone,\" said Liam. \"He's doing grand. Go on, Ernie. You're doing grand.\"\n\nErnie hesitated, looking up at the ceiling with immense seriousness, and it occurred to Deirdre that she would very much like to see this little boy become a man. It would be most interesting to see what kind of a man he would make, this aptly named earnest little boy. Only after he had finished reciting his poem did he realize that he'd forgotten a couple of lines in the middle, but by then everyone was clapping so hard that he retreated gratefully to his spot at his mother's feet.\n\n\"Who's next?\" asked Acushla, looking round the room. \"Macdara?\"\n\nReluctantly Macdara unfurled himself from the shadows. He took a piece of paper from the inside pocket of his jacket and unfolded it.\n\n\"This is something I wrote for you,\" he said, without looking up at Deirdre.\n\nThe room fell into silence. Falling with a bump, as everyone became still.\n\n\"Shush,\" said Connie to Oscar. \"Shush now.\"\n\nMacdara prepared to read, with all of them looking at him expectantly, quizzically even; they were not sure what was coming. Once he had started reading, Deirdre noticed that their heads began to bow, as if they were in church. Their faces took on soulful expressions while they listened, and Deirdre saw Alma place a hand on Mick's knee and Mick cover it with one of his. Liam sank down on to the arm of Acushla's chair, and Acushla sagged a little toward him until her shoulder was resting against his arm. Connie propped her chin on top of Oscar's head and absentmindedly kissed him over and over again. Nora and Nando sat back to back on the floor, each of them looking up at the ceiling, while Emmet watched Macdara with great intent. When the reading was finished, Deirdre understood that Macdara had become a different person in all of their eyes, and that this was his present to her, the answer to a question that she alone had never asked of him.\n\nMacdara's head drooped and his face flushed bright red amid the explosion of compliments that followed his reading.\n\n\"Mick,\" said Acushla, clapping her hands together like a giddy child. \"We haven't had Mick yet.\"\n\n\"No better man,\" said Liam with a dash of malice as his brother got to his feet. (Liam alone had refused to perform a party piece, showing a determination that Deirdre couldn't help but admire. Deirdre had always had a soft spot for Liam. Of her two sons-in-law, he was the one she preferred, perhaps because nobody else did.)\n\nMick set his wineglass down on a side table and coughed to clear his throat. He lurched his shoulders back and smoothed his hands over his belly.\n\n\"Help me out here, will you, son?\" he said to Nando, and with a nudge from Nora, Nando took up his guitar. He sat with his hand poised at the strings, his face alert as he waited to hear what it was that Mick was going to sing. Everyone else knew, of course. There was only one song that Mick had ever learned to sing, and that was \"Fernando.\" Before he even began, they were all smiling in anticipation, waiting for that familiar first line. And what Mick lacked in skill, he made up for in passion. He even had the Swedish accent to go with it. Hand on his heart like an opera singer, he held the other hand out to Alma as he sang. She was curled into her chair by the fire as she listened to him. A dreamy look on her face, and a memory there of the young girl she had been when first she met him. When Mick came to the chorus, they all sang along. And silly as it was, Deirdre felt like they were singing the story of their own lives, and each other's lives, in all their glorious imperfection. Mick's party piece was such a success that when it was finished, they launched into another round of the chorus, just for the joy of it.\n\n\"Folly that!\" said Mick triumphantly as he sat down. And for a moment it seemed that nobody would. But then Nora rose off the floor, dragging Connie with her by the arm. The boys were thrown off her in a tumble on to the carpet.\n\nOf the whole lot of them, Connie was by far and away the best singer, but to Deirdre's infinite irritation she never seemed to know the words of a single song. (Deirdre had told her more than once that it was a sinful waste of her talent not to learn the words of a few songs.) Nora, on the other hand, always knew the words but couldn't sing to save her life. The two of them together, Deirdre dreaded to think.\n\n\"Don't look so worried,\" said Nora. \"We've been practicing.\"\n\nShe nodded at Nando, who sounded a few notes on his guitar. The two girls stood, shifty and solemn, with their backs to the fire. As Connie's steel-fired voice lit into the first line of their serenade, Deirdre recognized with a smile what it was that they'd chosen, and she realized to her amazement that Connie had even taken the trouble to learn the words. Nora made a husky attempt at balancing the duet, but when the chorus came round, even she couldn't dampen the sound of Connie's voice, which had the power to fill a cathedral to the rafters. The song they had chosen was an anthem to love, and shamefully sentimental as it was, it did seem to Deirdre, as she listened to her granddaughters singing in glorious disharmony, that she was surrounded by love. The love of her family, there was no getting away from it. There was no getting away from the love she felt for them. And the result of all this love was that it would not be possible for her to leave them. Reluctantly Deirdre reached the conclusion that had always been lying in wait for her\u2014the conclusion that for her, at least, there was to be no easy way out.\n\nWith weary resignation, she saw that she would be forced to make that appointment to have her cataracts seen to. She would have to see someone about her hip too, perhaps even submit to surgery to fix it. She would force herself to do her pelvic-floor exercises again, in order to control her bladder. She would get a man in to look at the leaking roof, and the guttering at the back of the house, maybe even get the garden cut back. She would carry on patching herself and her surroundings as best she could, until they could be patched no more. This would be her gift to her family, a gift they would never even be aware of receiving.\n\n\"My party piece is a joke,\" she announced, springing it on them with relish. They would have been expecting her to produce a song. Taking advantage of the element of surprise, she launched straight into her piece at a nice jaunty pace, as befitted her choice of material.\n\n\"Now, Paddy Englishman, Paddy Irishman and Paddy Scotsman were on an expedition to chart a previously unexplored section of a notoriously dangerous river, a river that runs through the most impenetrable rain forest in the world, when they were captured by cannibals.\"\n\nSilence in the room; all Deirdre could hear was the sound of two logs collapsing into each other in the grate. The sound of Oscar's heavy breathing, and Acushla delicately clearing her throat.\n\n\"The cannibals tied the three men to a stake and announced their intention to skin them alive, make canoes out of their skins, and eat their flesh.\"\n\nErnie and Oscar's eyes widened and their mouths fell open. Oscar shunted himself back a few inches, so that he was tucked in between his mother's legs. Ernie put up his hand, as if he were in class.\n\n\"Wait,\" he said. \"How did Paddy Englishman, Paddy Irishman and Paddy Scotsman understand what the cannibals were saying to them?\"\n\n\"Good question!\" said Deirdre. \"The thing is, they had a translator.\"\n\nErnie seemed to be satisfied with that.\n\n\"'We're going to give you one last request, before we kill you,' said the chief cannibal. 'I'll have a cup of tea,' said Paddy Englishman. So the cannibals gave him a cup of tea, and then they skinned him alive, made a canoe out of his skin, and roasted his flesh on the fire.\n\n\"'What's your last request?' they asked Paddy Scotsman. 'I'll have a whisky,' he said. So they gave him a whisky, and they skinned him alive, made a canoe out of his skin, and roasted his flesh on the fire.\n\n\"So now it was Paddy Irishman's turn. 'What's your last request?' asked the chief cannibal. 'I'll have a fork,' said Paddy Irishman. Well, the cannibals were a bit confused at that, but since it was his last request, they gave him a fork. And do you know what he said?\"\n\nDeirdre paused now in the telling. She surveyed the faces of her family. Then, taking the imaginary fork in her fist, she stabbed herself vigorously in the chest, puncturing herself over and over again as she rolled out her punch line.\n\n\"'You're not makin' no fuckin' canoe out of me.'\"\n\nNever before had Deirdre spoken that word out loud, and never again would she. But on this occasion, no other word would do. She had rehearsed the piece out loud, using all manner of lesser expletives, but none of them had quite the impact she desired. She was amazed by the illicit thrill it gave her to use that dreadful, vulgar word. She was gratified by the stunned reaction of her family. For a full second they stared at her in amazement, then, under cover of the raucous applause that followed (with Emmet wolf-whistling through his fingers and Connie shouting, \"Way to go, old girl\"), Deirdre raised her glass in the air.\n\n\"To Manus,\" she said solemnly, dedicating to him the swan song she had planned for herself.\n\nThe room fell silent as they all raised their glasses. In the aftermath of the toast, Deirdre looked out beyond the circle her family formed in front of the fire, through the empty dining room and out the back window to where the night sky lay curiously empty. She could just about make out the stadium, barely visible in the darkness. With all but its most essential lights extinguished, it looked like a great ocean liner out at sea. Deirdre felt all the wistfulness of someone standing on the shore, knowing that the ship she had seen in the night would be gone in the morning, but she would still be here.\n\n**Proof That We Existed,** by Macdara MacEntee (Little, Brown \u20ac14.99)\n\nThis magnificent debut novel, an epic study of the human spirit, took more than three decades to write and runs to nearly eight hundred pages. Its publication comes less than a year after the death of the author's father, Manus, and while the timing is undoubtedly poignant, the scale of Macdara MacEntee's achievement is sure to secure a place for father and son among the great dynasties of the literary world.\n\n# Epilogue\n\nOf all the things that happened that year, it was the summer that would be most vividly remembered. Long after people had forgotten all about the horsemeat scandal, and the countrywide flooding, long after the abortion controversy had merged in people's memories with abortion controversies of other years, they would remember that glorious summer. When a popular radio program marked the turn of the year by conducting a poll of their listeners to identify the highs and lows of the twelve months just gone, it was not the country's exit from the IMF bailout that prevailed. It was not the election of a new pope, nor the visit to Dublin by Michelle Obama. The highlight of the year, as voted by radio listeners, was the long, hot summer.\n\nOf course for Manus's family, that summer would forever be associated with his death. Whenever anyone mentioned the long, hot summer, they would think not of children leaping joyously from piers, or of young people languishing on their backs in parks. Their thoughts turned instead to the sea of sunglasses that surrounded them in the churchyard on the day of his funeral. The smell of sunscreen as the mourners leaned in close to kiss them. The taste of the ice-creams they had eaten on their way to the crematorium. (Deirdre had insisted on stopping the funeral cort\u00e8ge to treat the children to an ice-cream, a move that took the undertakers by surprise but would no doubt have delighted Manus.)\n\nThey scattered his ashes the following spring, on a dry day at the end of March. They timed the trip for high tide, gathering at midday at a small harbor to the south of the city, where Liam had arranged for the use of a boat. As soon as they arrived, Liam climbed into the back and busied himself with the outboard engine, with Emmet to assist him. Mick stood in the well of the boat, holding a hand out to help the ladies as one by one they hopped down from the pier. Acushla settled the little boys in the bow, while Connie checked the fastenings on their life jackets. Alma and Deirdre took refuge on a bench running along the far side, Alma rooting in her handbag for her sunglasses, while Deirdre struggled to contain her hair in the wind. Macdara stood for a moment, one hand raised to hold his father's fedora to his head, as he enjoyed the sensation of the sea shifting under his feet. The taste of the salt in the air.\n\n\"Who are we waiting for?\" shouted Liam from the back of the boat. He had the engine started and was keen to get going.\n\n\"Just Sam and Nora and Nando,\" shouted Acushla. \"But we can't go without them. They've got the ashes.\"\n\nNo sooner had she said it than they appeared, rushing along the pier carrying rugs and bags and rain gear.\n\n\"Sorry we're late,\" said Nora, as she stood at the edge and looked down at them.\n\nShe reached her hand out to Nando and he handed her a cooler bag. She passed it down to Mick.\n\n\"Now,\" she said, \"watch out for Grandad.\"\n\nShe dropped a small rucksack down into the boat. Mick caught it with both hands, hugging it to his chest like a rugby ball.\n\n\"Jesus,\" he said. \"It wouldn't do to drop that.\" Carefully, he stowed it under a bench, with his raincoat wrapped around it to protect it from the spray.\n\n\"Take care now,\" he said, sticking a hand out to help Nora down. As they took their seats along the near side, Nando looked around him eagerly, as if he found them all delightful. It seemed to Macdara that it gave the family an added gloss to be seen through the eyes of this new person in their midst. It made them all seem shiny and new.\n\n\"Macdara,\" Acushla called out. \"Do you want to sit here?\" Already she was moving her handbag on to her lap and sliding her bum along the bench, shunting Alma and her mother further up to make space for him.\n\n\"I'm grand, thanks,\" said Macdara, standing with his feet apart to steady himself as the boat picked up speed. He was enjoying the feel of the wind in his face. The treat of being out at sea. Ahead of them, the chimneys at Poolbeg rose out of a low band of sea mist. A string of white sails appeared as bunting on the skyline. The sunlight fell on the water like splatters of rain.\n\nThe little boys were kneeling in the bow, each of them with an arm draped over the side, their fingers trailing the jade-green water. Connie was kneeling between them, with a firm grip on each of their life jackets. The boat turned as it left the protection of the harbor, smacking the waves sideways on now, so that great slaps of spray poured over the side. The little boys squealed with delight. Nando had taken an expensive-looking camera out of his bag and he had the lens trained on Connie and the lads, taking photograph after photograph of them.\n\nThe boat climbed the waves, and fell. Climbed, and fell, and Macdara allowed himself to drop down on to a seat beside Mick before he was knocked down. He took his father's hat off before it was blown away. With Dalkey Island ahead of them, Liam guided the boat between the island and the shore. Halfway through the Sound, he let the engine cut out and they all became aware of the seagulls, their screams like a hundred creaking farm gates.\n\n\"How's this for the spot?\" Liam shouted out.\n\nMick stood and looked around him. He saw the island, pockmarked by ruined fortifications and rocks and some scraggly goats. A seal popped its head up and disappeared down again under the waves. A slender yacht cut through the water of the Sound like a knife. A yacht with a black sail, it created an atmosphere of great and solemn melancholy, like something out of an Arthurian legend.\n\n\"Seems like as good a place as any,\" he said, looking to Alma for approval. She shrugged, as if to say, Sure. Mick looked to Sam, to check with him, and Sam looked back without giving any indication whether he understood what was going on. Nora put her arm around Sam's back and drew him in toward her, so that their two heads were almost touching. What she said to him could not be heard by the others because her words were carried away by the wind, but they saw that Sam was nodding.\n\n\"Let's do it,\" she said. And she reached her arm out for her dad to pass her the rucksack. Opening up the zip and taking out a small cardboard carton, she paused and looked around her.\n\n\"Who wants to do the honors?\"\n\nNobody answered.\n\n\"Okay, I'll do it,\" said Nora. \"If that's okay with everyone?\"\n\nShe got to her feet.\n\n\"Don't anyone else stand up,\" shouted Mick, \"or the boat will tip over!\"\n\nNora propped one knee up on the bench and leaned out over the edge of the boat.\n\n\"Which way is the wind blowing?\" she shouted, looking back toward Liam.\n\n\"That way,\" roared Liam, throwing his arm out the side of the boat, in the manner of an airport groundsman guiding an incoming plane toward its gate.\n\n\"Okay,\" said Nora. \"Here goes.\"\n\nShe opened the carton and for a moment nothing happened. Holding it from the bottom with both hands, she pitched it and a small cloud of dust billowed out. Like a swarm of bees suddenly released from a hive, it was picked up by the wind before spreading out and falling down into the sea. Nora gave the box one last shake, just to make sure there was nothing left in it.\n\n\"That's it,\" she said, turning back into the boat. She turned the carton upside down to demonstrate that it was empty.\n\nA seagull overhead let out a single high-pitched shriek. A moment later the rest of the flock followed with a spiral of raucous laughter.\n\n\"Well done,\" said Macdara. \"Well done, Nora.\"\n\nAnd he started clapping. Whether it was for his father or for Nora, he wasn't sure, but everyone else joined in, even the little boys.\n\n\"It's more like coral,\" said Nora with surprise. \"It looks more like coral than ashes.\"\n\nAnd it seemed to Macdara that it was just like Nora to inject the moment with a little poetry.\n\n\"I think the occasion calls for a song,\" said Deirdre, and she got to her feet. Moving unsteadily toward the center of the boat, she swung the loose end of her cape over her shoulder, in preparation for her performance. Fiddling for a moment with the hairpins that so inadequately held her hair up, she closed her eyes as she began to sing. The song she had chosen was a sea shanty, and a great favorite of Manus's. As she sang it, her voice was thrown about by the wind, which gave it an added bravery, and an added beauty.\n\n_Dress me up in me oilskins and jumper,_\n\n_No more on the docks I'll be seen,_\n\n_Just tell me old shipmates, I'm takin' a trip, mates,_\n\n_And I'll see you some day in Fiddlers' Green._\n\nAlma and Acushla were huddled together along one side of the boat, each of them staring down at the wet, shifting floor as they listened to their mother sing. Nora and Nando were sitting opposite them, Nora with her cheek resting on Nando's woolly shoulder and her eyes on her grandmother's face. Macdara and Mick sat facing Connie and the boys, and for once the boys were still, their small, grave faces framed by their huge life jackets as they listened to the singing. Connie's eyes shone with tears that she made no attempt to wipe away. Liam was sitting at the back of the boat, with his arm on the tiller to steady it and his face turned out to sea. Emmet was standing up at the back, a black shadow against the blue sky.\n\nAs she reached the final verse, Deirdre stretched a hand out toward Sam. Taking his hand in hers, she held on to it tight while she sang, her other hand stretched out to hold Mick's. For a moment Macdara was afraid that Mick would attempt to further the chain by grabbing his hand, but to his relief he didn't. Macdara was content to sit beside Mick, but not touching. When Deirdre got to the final chorus, they all joined in, and it seemed to Macdara that he had never known a moment of such perfect contentment as this, with the boat bobbing gently on the waves and his whole family in it, singing in unison.\n\n# ACKNOWLEDGMENTS\n\nThe events and characters in this book are entirely fictional, but I have consulted many people in the course of my research. For their expertise on vintage cars, I am grateful to Donal Byrne and Donal Morrissy. Neal Massey advised me on the ins and outs of a burial at sea. Aengas Mac Grianna was kind enough to talk to me about his experiences on _Celebrity MasterChef Ireland,_ while Mair\u00e9ad Delaney's vast knowledge of the Abbey Theater's history was a great resource\u2014to her and all the staff behind the scenes at the Abbey, my sincere thanks. To the late Garry Henderson, I owe the Hellmann's mayonnaise story as well as the beautiful \"Fiddler's Green,\" a verse of which is included by kind permission of John Conolly. For those readers who have a day-to-day memory of the summer of 2013, I confess that I have bent the weather in places to suit my purposes. The bit about the long, hot summer, I did not make up.\n\nMy agent Marianne Gunn O'Connor has been a stalwart support, as has her sub-agent Vicki Satlow. My former colleagues in RT\u00c9 have been unfailingly kind. Paul Durcan stepped forward with encouragement when I most needed it. Margaret Daly and Cormac Kinsella offered their wisdom and expertise most generously. To all of them, a warm thank-you.\n\nMy editors Rebecca Saunders and Helen Atsma worked hard to make this book the best book it could be. I am hugely grateful to both of them. Thanks also to Kati Nicholl, Joanna Smyth and Jane Selley for their valuable input. It has been my pleasure to work with Manpreet Grewal, Tamsyn Berryman, Maddie West and Stephanie Melrose from Little, Brown in London. Breda Purdue and Jim Binchy along with Siobh\u00e1n Tierney and all the team at Hachette in Dublin have been, as always, wonderful.\n\nMary Reynolds and Karen Coleman Muldowney helped in different but important ways. So too did Ray Murphy and Margaret Dunne. Hilary McGouran and Valerie Bistany read the book at various stages in its development, and their suggestions were enormously helpful. \u00c1ine Lawlor was a valuable early reader as were Sara Burke and Aoife Kavanagh. Finally and most importantly, I am grateful for the support of my family in encouraging me on this path I've chosen. I want to thank my brother Kevin and my sister Meg for their faith in me, and their love. My father Des, for his constant friendship and wise counsel; he also bore with good grace the theft of some of his gentle eccentricities. For Mark and Lucy and Clara, there are not enough words.\n\n# _Also by this author_\n\nTHIS IS HOW IT ENDS\n\n### Thank you for buying this ebook, published by Hachette Digital.\n\nTo receive special offers, bonus content, and news about our latest ebooks and apps, sign up for our newsletters.\n\nSign Up\n\nOr visit us at hachettebookgroup.com\/newsletters\n\n# Contents\n\n  1. Cover\n  2. Title Page\n  3. Welcome\n  4. Dedication\n  5. Prologue\n  6. Alma\n  7. Mick\n  8. Acushla\n  9. Connie\n  10. Liam\n  11. Nora\n  12. Manus\n  13. Macdara\n  14. Deirdre\n  15. Epilogue\n  16. Acknowledgments\n  17. Also by this Author\n  18. Newsletters\n  19. Copyright\n\n# Navigation\n\n  1. Begin Reading\n  2. Table of Contents\n\n# Copyright\n\nThis book is a work of fiction. Names, characters, places, and incidents are the product of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is coincidental.\n\nCopyright \u00a9 2015, 2016 by Kathleen MacMahon\n\nCover design by Lisa Honerkamp\n\nPainting by Francis Livingston\n\nCover copyright \u00a9 2016 by Hachette Book Group, Inc.\n\nAll rights reserved. In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher constitute unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nGrand Central Publishing\n\nHachette Book Group\n\n1290 Avenue of the Americas, New York, NY 10104\n\nhachettebookgroup.com\n\ntwitter.com\/grandcentralpub\n\nFirst ebook edition: July 2016\n\nGrand Central Publishing is a division of Hachette Book Group, Inc.\n\nThe Grand Central Publishing name and logo is a trademark of Hachette Book Group, Inc.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to www.hachettespeakersbureau.com or call (866) 376-6591.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nISBN 978-1-4555-1133-4\n\nE3-20160602-JV-PC\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Juliet Sutherland, Renald Levesque and the\nOnline Distributed Proofreaders Europe at\nhttp:\/\/dp.rastko.net\n\n\n\n\n\n\n\n\nCHRISTIANITY\n\nAND\n\nGREEK PHILOSOPHY;\n\nOR, THE RELATION BETWEEN\nSPONTANEOUS AND REFLECTIVE THOUGHT IN GREECE\nAND THE POSITIVE TEACHING OF\nCHRIST AND HIS APOSTLES.\n\n\nBY B.F. COCKER, D.D.,\n\nPROFESSOR OF MORAL AND MENTAL PHILOSOPHY IN THE UNIVERSITY OF MICHIGAN\n\n\"Plato made me know the true God, Jesus Christ showed me the way to\nhim.\"\n                                                        ST. AUGUSTINE\n\n\n\n\nNEW YORK: CARLTON & LANAHAN.\nSAN FRANCISCO: E. THOMAS.\nCINCINNATI: HITCHCOCK & WALDEN.\n\n1870.\n\n\n\nEntered, according to Act of Congress, in the year 1870, by HARPER &\nBROTHERS, in the Clerk's Office of the District Court of the United\nStates for the Southern District of New York.\n\n\n\n\nTO\n\nD.D. WHEDON, D.D.,\n\nMY EARLIEST LITERARY FRIEND, WHOSE VIGOROUS WRITINGS HAVE\nSTIMULATED MY INQUIRIES, WHOSE COUNSELS HAVE GUIDED\nMY STUDIES, AND WHOSE KIND AND GENEROUS WORDS\nHAVE ENCOURAGED ME TO PERSEVERANCE\nAMID NUMEROUS DIFFICULTIES,\nI DEDICATE THIS VOLUME AS A TOKEN OF MY MORE THAN ORDINARY AFFECTION\n\n_THE AUTHOR_.\n\n\n\n\nPREFACE.\n\n\nIn preparing the present volume, the writer has been actuated by a\nconscientious desire to deepen and vivify our faith in the Christian\nsystem of truth, by showing that it does not rest _solely_ on a special\nclass of facts, but upon all the facts of nature and humanity; that its\nauthority does not repose _alone_ on the peculiar and supernatural\nevents which transpired in Palestine, but also on the still broader\nfoundations of the ideas and laws of the reason, and the common wants\nand instinctive yearnings of the human heart. It is his conviction that\nthe course and constitution of nature, the whole current of history, and\nthe entire development of human thought in the ages anterior to the\nadvent of the Redeemer centre in, and can only be interpreted by, the\npurpose of redemption.\n\nThe method hitherto most prevalent, of treating the history of human\nthought as a series of isolated, disconnected, and lawless movements,\nwithout unity and purpose; and the practice of denouncing the religions\nand philosophies of the ancient world as inventions of satanic mischief,\nor as the capricious and wicked efforts of humanity to relegate itself\nfrom the bonds of allegiance to the One Supreme Lord and Lawgiver, have,\nin his judgment, been prejudicial to the interests of all truth, and\nespecially injurious to the cause of Christianity. They betray an utter\ninsensibility to the grand unities of nature and of thought, and a\nstrange forgetfulness of that universal Providence which comprehends all\nnature and all history, and is yet so minute in its regards that it\nnumbers the hairs on every human head, and takes note of every sparrow's\nfall, A juster method will lead us to regard the entire history of human\nthought as a development towards a specific end, and the providence of\nGod as an all-embracing plan, which sweeps over all ages and all\nnations, and which, in its final consummation, will, through Christ,\n\"gather together all things in one, both things which are in heaven and\nthings which are on earth.\"\n\nThe central and unifying thought of this volume is _that the necessary\nideas and laws of the reason, and the native instincts of the human\nheart, originally implanted by God, are the primal and germinal forces\nof history; and that these have been developed under conditions which\nwere first ordained, and have been continually supervised by the\nprovidence of God_. God is the Father of humanity, and he is also the\nGuide and Educator of our race. As \"the offspring of God,\" humanity is\nnot a bare, indeterminate potentiality, but a living energy, an active\nreason, having definite qualities, and inheriting fundamental principles\nand necessary ideas which constitute it \"the image and likeness of God.\"\nAnd though it has suffered a moral lapse, and, in the exercise of its\nfreedom, has become alienated from the life of God, yet God has never\nabandoned the human race. He still \"magnifies man, and sets his heart\nupon him.\" \"He visits him every morning, and tries him every moment.\"\n\"The inspiration of the Almighty still gives him understanding.\" The\nillumination of the Divine Logos still \"teacheth man knowledge.\" The\nSpirit of God still comes near to and touches with strong emotion every\nhuman heart. \"God has never left himself without a witness\" in any\nnation, or in any age. The providence of God has always guided the\ndispersions and migrations of the families of the earth, and presided\nover and directed the education of the race. \"He has foreordained the\ntimes of each nation's existence, and fixed the geographical boundaries\nof their habitations, _in order that they should seek the Lord_, and\nfeel after and find Him who is not far from any one of us.\" The\nreligions of the ancient world were the painful effort of the human\nspirit to return to its true rest and centre--the struggle to \"find Him\"\nwho is so intimately near to every human heart, and who has never ceased\nto be the want of the human race. The philosophies of the ancient world\nwere the earnest effort of human reason to reconcile the finite and the\ninfinite, the human and the Divine, the subject and God. An overruling\nProvidence, which makes even the wrath of man to praise Him, took up all\nthese sincere, though often mistaken, efforts into his own plan, and\nmade them sub-serve the purpose of redemption. They aided in developing\namong the nations \"the desire of salvation,\" and in preparing the world\nfor the advent of the Son of God. The entire course and history of\nDivine providence, in every nation, and in every age, has been directed\ntowards the one grand purpose of \"reconciling all things to Himself.\"\nChristianity, as a comprehensive scheme of reconciliation, embracing\n\"all things,\" can not, therefore, be properly studied apart from the\nages of earnest thought, of profound inquiry, and of intense religious\nfeeling which preceded it. To despise the religions of the ancient\nworld, to sneer at the efforts and achievements of the old philosophers,\nor even to cut them off in thought from all relation to the plans and\nmovements of that Providence which has cared for, and watched over, and\npitied, and guided all the nations of the earth, is to refuse to\ncomprehend Christianity itself.\n\nThe author is not indifferent to the possibility that his purpose may be\nmisconceived. The effort may be regarded by many conscientious and\nesteemed theologians with suspicion and mistrust. They can not easily\nemancipate themselves from the ancient prejudice against speculative\nthought. Philosophy has always been regarded by them as antagonistic to\nChristian faith. They are inspired by a commendable zeal for the honor\nof dogmatic theology. Every essay towards a profounder conviction, a\nbroader faith in the unity of all truth, is branded with the opprobrious\nname of \"rationalism.\" Let us not be terrified by a harmless word.\nSurely religion and right reason must be found in harmony. The author\nbelieves, with Bacon, that \"the foundation of all religion is right\nreason.\" The abnegation of reason is not the evidence of faith, but the\nconfession of despair. Sustained by these convictions, he submits this\nhumble contribution to theological science to the thoughtful\nconsideration of all lovers of Truth, and of Christ, the fountain of\nTruth. He can sincerely ask upon it the blessing of Him in whose fear it\nhas been written, and whose cause it is the purpose of his life to\nserve.\n\nThe second series, on \"Christianity and Modern Thought,\" is in an\nadvanced state of preparation for the press.\n\n      NOTE.--It has been the aim of the writer, as far as the\n      nature of the subject would permit, to adapt this work to\n      general readers. The references to classic authors are,\n      therefore, in all cases made to accessible English\n      translations (in Bohn's Classical Library); such changes,\n      however, have been made in the rendering as shall present\n      the doctrine of the writers in a clearer and more forcible\n      manner. For valuable services rendered in this department of\n      the work, by Martin L. D'Ooge, M. A., Acting Professor of\n      Greek Language and Literature in the University of Michigan,\n      the author would here express his grateful acknowledgment.\n\n\n\n\nCONTENTS.\n\n\n     CHAPTER I.\n     ATHENS, AND THE MEN OF ATHENS.\n\n     CHAPTER II.\n     THE PHILOSOPHY OF RELIGION.\n\n     CHAPTER III.\n     THE RELIGION OF THE ATHENIANS.\n\n     CHAPTER IV.\n     THE RELIGION OF THE ATHENIANS: ITS MYTHOLOGICAL AND SYMBOLICAL\n     ASPECTS.\n\n     CHAPTER V.\n     THE UNKNOWN GOD.\n\n     CHAPTER VI.\n     THE UNKNOWN GOD (_continued_).\n       IS GOD COGNIZABLE BY REASON?\n\n     CHAPTER VII.\n     THE UNKNOWN GOD (_continued_).\n       IS GOD COGNIZABLE BY REASON? (_continued_).\n\n     CHAPTER VIII.\n     THE PHILOSOPHERS OF ATHENS.\n       PRE-SOCRATIC SCHOOL.\n     _Sensational_: THALES--ANAXIMENES--HERACLITUS--ANAXIMANDER\n     LEOCIPPUS--DEMOCRITUS.\n\n     CHAPTER IX.\n     THE PHILOSOPHERS OF ATHENS (_continued_).\n       PRE-SOCRATIC SCHOOL (_continued_)\n     _Idealist_: Pythagoras--Xenophanes--Parmenides--Zeno. _Natural\n     Realist_: Anaxagoras.\n       THE SOCRATIC SCHOOL.\n       Socrates.\n\n     CHAPTER X\n     THE PHILOSOPHERS OF ATHENS (_continued_).\n       THE SOCRATIC SCHOOL (_continued_).\n       Plato.\n\n     CHAPTER XI.\n     THE PHILOSOPHERS OF ATHENS (_continued_).\n       THE SOCRATIC SCHOOL (_continued_).\n       Plato.\n\n     CHAPTER XII.\n\n     THE PHILOSOPHERS OF ATHENS (_continued_).\n       THE SOCRATIC SCHOOL (_continued_).\n       Aristotle.\n\n     CHAPTER XIII.\n     THE PHILOSOPHERS OF ATHENS (_continued_).\n       POST-SOCRATIC SCHOOL.\n       Epicurus and Zeno.\n\n     CHAPTER XIV.\n     The Propaedeutic Office of Greek Philosophy.\n\n     CHAPTER XV.\n     The Propaedeutic Office of Greek Philosophy (_continued_).\n\n\n\n\n\"_Ye men of Athens_, all things which I behold bear witness to your\ncarefulness in religion; for, as I passed through your city and beheld\nthe objects of your worship, I found amongst them an altar with this\ninscription, TO THE UNKNOWN GOD; whom, therefore, ye worship, though ye\nknow; Him not, Him declare I unto you. God who made the world and all\nthings therein, seeing He is Lord of heaven and earth, dwelleth not in\ntemples made with hands; neither is He served by the hands of men, as\nthough he needed any thing; for He giveth unto all life, and breath, and\nall things. And He made of one blood all the nations of mankind to dwell\nupon the face of the whole earth; and ordained to each the appointed\nseasons of their existence, and the bounds of their habitation, that\nthey should seek God, if haply they might feel after Him and find Him,\nthough he be not far from every one of us: for in Him we live, and move,\nand have our being; as certain of your own poets have said, _For we are\nalso His offspring_. Forasmuch, then, as we are the offspring of God, we\nought not to think that the Godhead is like unto gold, or silver, or\nstone, graven by the art and device of man. Howbeit, those past times of\nignorance God hath overlooked; but now He commandeth all men everywhere\nto repent, because He hath appointed a day wherein He will judge the\nworld in righteousness by that Man whom He hath ordained; whereof He\nhath given assurance unto all, in that He hath raised Him from the\ndead.\"--Acts xvii. 22-31.\n\n\n\n\nCHRISTIANITY\nAND\nGREEK PHILOSOPHY\n\n\n\n\nCHAPTER I.\n\nATHENS, AND THE MEN OF ATHENS.\n\n\n\"Is it not worth while, for the sake of the history of men and nations,\nto study the surface of the globe in its relation to the inhabitants\nthereof?\"--Goethe.\n\nThere is no event recorded in the annals of the early church so replete\nwith interest to the Christian student, or which takes so deep a hold on\nthe imagination, and the sympathies of him who is at all familiar with\nthe history of Ancient Greece, as the one recited above. Here we see the\nApostle Paul standing on the Areopagus at Athens, surrounded by the\ntemples, statues, and altars, which Grecian art had consecrated to Pagan\nworship, and proclaiming to the inquisitive Athenians, \"the strangers\"\nwho had come to Athens for business or for pleasure, and the\nphilosophers and students of the Lyceum, the Academy, the Stoa, and the\nGarden, \"_the unknown God_.\"\n\nWhether we dwell in our imagination on the artistic grandeur and\nimposing magnificence of the city in which Paul found himself a solitary\nstranger, or recall the illustrious names which by their achievements in\narts and philosophy have shed around the city of Athens an immortal\nglory,--or whether, fixing our attention on the lonely wanderer amid the\nporticoes, and groves, and temples of this classic city, we attempt to\nconceive the emotion which stirred his heart as he beheld it \"wholly\ngiven to idolatry;\" or whether we contrast the sublime, majestic theism\nproclaimed by Paul with the degrading polytheism and degenerate\nphilosophy which then prevailed in Athens, or consider the prudent and\nsagacious manner in which the apostle conducts his argument in view of\nthe religious opinions and prejudices of his audience, we can not but\nfeel that this event is fraught with lessons of instruction to the\nChurch in every age.\n\nThat the objects which met the eye of Paul on every hand, and the\nopinions he heard everywhere expressed in Athens, must have exerted a\npowerful influence upon the current of his thoughts, as well as upon the\nstate of his emotions, is a legitimate and natural presumption. Not only\nwas \"his spirit stirred within him\"--his heart deeply moved and agitated\nwhen he saw the city wholly given to idolatry--but his thoughtful,\nphilosophic mind would be engaged in pondering those deeply interesting\nquestions which underlie the whole system of Grecian polytheism. The\ncircumstances of the hour would, no doubt, in a large degree determine\nthe line of argument, the form of his discourse, and the peculiarities\nof his phraseology. The more vividly, therefore, we can represent the\nscenes and realize the surrounding incidents; the more thoroughly we can\nenter into sympathy with the modes of thought and feeling peculiar to\nthe Athenians; the more perfectly we can comprehend the spirit and\ntendency of the age; the more immediate our acquaintance with the\nreligious opinions and philosophical ideas then prevalent in Athens, the\nmore perfect will be our comprehension of the apostle's argument, the\ndeeper our interest in his theme. Some preliminary notices of Athens and\n\"the Men of Athens\" will therefore be appropriate as introductory to a\nseries of discourses on Paul's sermon on Mars' Hill.\n\nThe peculiar connection that subsists between Geography and History,\nbetween a people and the country they inhabit, will justify the\nextension of our survey beyond the mere topography of Athens. The people\nof the entire province of Attica were called Athenians (_Athenaioi_) in\ntheir relation to the state, and Attics _(Attikoi_) in regard to their\nmanners, customs, and dialect.[1] The climate and the scenery, the forms\nof contour and relief, the geographical position and relations of\nAttica, and, indeed, of the whole peninsula of Greece, must be taken\ninto our account if we would form a comprehensive judgment of the\ncharacter of the Athenian people.\n\nThe soil on which a people dwell, the air they breathe, the mountains\nand seas by which they are surrounded, the skies that overshadow\nthem,--all these exert a powerful influence on their pursuits, their\nhabits, their institutions, their sentiments, and their ideas. So that\ncould we clearly group, and fully grasp all the characteristics of a\nregion--its position, configuration, climate, scenery, and natural\nproducts, we could, with tolerable accuracy, determine what are the\ncharacteristics of the people who inhabit it. A comprehensive knowledge\nof the physical geography of any country will therefore aid us\nmaterially in elucidating the natural history, and, to some extent, the\nmoral history of its population. \"History does not stand _outside_ of\nnature, but in her very heart, so that the historian only grasps a\npeople's character with true precision when he keeps in full view its\ngeographical position, and the influences which its surroundings have\nwrought upon it.\"[2]\n\n[Footnote 1: Niebuhr's \"Lectures on Ethnography and Geography,\" p. 91.]\n\n[Footnote 2: Ritter's \"Geographical Studies,\" p. 34.]\n\nIt is, however, of the utmost consequence the reader should understand\nthat there are two widely different methods of treating this deeply\ninteresting subject--methods which proceed on fundamentally opposite\nviews of man and of nature. One method is that pursued by Buckle in his\n\"History of Civilization in England.\" The tendency of his work is the\nassertion of the supremacy of material conditions over the development\nof human history, and indeed of every individual mind. Here man is\npurely passive in the hands of nature. Exterior conditions are the\nchief, if not the _only_ causes of man's intellectual and social\ndevelopment. So that, such a climate and soil, such aspects of nature\nand local circumstances being given, such a nation necessarily\nfollows.[3] The other method is that of Carl Ritter, Arnold Guyot, and\nCousin.[4] These take account of the freedom of the human will, and the\npower of man to control and modify the forces of nature. They also take\naccount of the original constitution of man, and the primitive type of\nnations; and they allow for results arising from the mutual conflict of\ngeographical conditions. And they, especially, recognize the agency of a\nDivine Providence controlling those forces in nature by which the\nconfiguration of the earth's surface is determined, and the distribution\nof its oceans, continents, and islands is secured; and a providence,\nalso, directing the dispersions and migrations of nations--determining\nthe times of each nation's existence, and fixing the geographical bounds\nof their habitation, all in view of the _moral_ history and spiritual\ndevelopment of the race,--\"that they may feel after, and find the living\nGod.\" The relation of man and nature is not, in their estimation, a\nrelation of cause and effect. It is a relation of adjustment, of\nharmony, and of reciprocal action and reaction. \"Man is not\"--says\nCousin--\"an effect, and nature the cause, but there is between man and\nnature a manifest harmony of general laws.\"... \"Man and nature are two\ngreat effects which, coming from the same cause, bear the same\ncharacteristics; so that the earth, and he who inhabits it, man and\nnature, are in perfect harmony.\"[5] God has created both man and the\nuniverse, and he has established between them a striking harmony. The\nearth was made for man; not simply to supply his physical wants, but\nalso to minister to his intellectual and moral development. The earth is\nnot a mere dwelling-place of nations, but a school-house, in which God\nhimself is superintending the education of the race. Hence we must not\nonly study the _events_ of history in their chronological order, but we\nmust study the earth itself as the _theatre_ of history. A knowledge of\nall the circumstances, both physical and moral, in the midst of which\nevents take place, is absolutely necessary to a right judgment of the\nevents themselves. And we can only elucidate properly the character of\nthe actors by a careful study of all their geographical and ethological\nconditions.\n\n[Footnote 3: See chap. ii. \"History of Civilization.\"]\n\n[Footnote 4: Ritter's \"Geographical Studies;\" Guyot's \"Earth and Man;\"\nCousin's \"History of Philosophy,\" lec. vii., viii., ix.]\n\n[Footnote 5: Lectures, vol. i. pp. 162, 169.]\n\nIt will be readily perceived that, in attempting to estimate the\ninfluence which exterior conditions exert in the determination of\nnational character, we encounter peculiar difficulties. We can not in\nthese studies expect the precision and accuracy which is attained in the\nmathematical, or the purely physical sciences. We possess no control\nover the \"materiel\" of our inquiry; we have no power of placing it in\nnew conditions, and submitting it to the test of new experiments, as in\nthe physical sciences. National character is a _complex_ result--a\nproduct of the action and reaction of primary and secondary causes. It\nis a conjoint effect of the action of the primitive elements and laws\noriginally implanted in humanity by the Creator, of the free causality\nand self-determining power of man, and of all the conditions, permanent\nand accidental, within which the national life has been developed. And\nin cases where _physical_ and _moral_ causes are blended, and\nreciprocally conditioned and modified in their operation;--where primary\nresults undergo endless modifications from the influence of surrounding\ncircumstances, and the reaction of social and political\ninstitutions;--and where each individual of the great aggregate wields a\ncausal power that obeys no specific law, and by his own inherent power\nsets in motion new trains of causes which can not be reduced to\nstatistics, we grant that we are in possession of no instrument of exact\nanalysis by which the complex phenomena of national character may be\nreduced to primitive elements. All that we can hope is, to ascertain, by\npsychological analysis, what are the fundamental ideas and laws of\nhumanity; to grasp the exterior conditions which are, on all hands,\nrecognized as exerting a powerful influence upon national character; to\nwatch, under these lights, the manifestations of human nature on the\ntheatre of history, and then apply the principles of a sound historic\ncriticism to the recorded opinions of contemporaneous historians and\ntheir immediate successors. In this manner we may expect, at least, to\napproximate to a true judgment of history.\n\nThere are unquestionably fundamental powers and laws in human nature\nwhich have their development in the course of history. There are certain\nprimitive ideas, imbedded in the constitution of each individual mind,\nwhich are revealed in the universal consciousness of our race, under the\nconditions of experience--the exterior conditions of physical nature and\nhuman society. Such are the ideas of cause and substance; of unity and\ninfinity, which govern all the processes of discursive thought, and lead\nus to the recognition of Being _in se_;--such the ideas of right, of\nduty, of accountability, and of retribution, which regulate all the\nconceptions we form of our relations to all other moral beings, and\nconstitute _morality_;--such the ideas of order, of proportion, and of\nharmony, which preside in the realms of art, and constitute the\nbeau-ideal of _esthetics_;--such the ideas of God, the soul, and\nimmortality, which rule in the domains of _religion_, and determine man\na religious being. These constitute the identity of human nature under\nall circumstances; these characterize humanity in all conditions. Like\npermanent germs in vegetable life, always producing the same species of\nplants; or like fundamental types in the animal kingdom, securing the\nsame homologous structures in all classes and orders; so these\nfundamental ideas in human nature constitute its sameness and unity,\nunder all the varying conditions of life and society. The acorn must\nproduce an oak, and nothing else. The grain of wheat must always produce\nits kind. The offspring of man must always bear his image, and always\nexhibit the same fundamental characteristics, not only in his corporeal\nnature, but also in his mental constitution.\n\nBut the germination of every seed depends on conditions _ab extra_, and\nall germs are modified, in their development, by geographical and\nclimatal surroundings. The development of the acorn into a mature and\nperfect oak greatly depends on the exterior conditions of soil, and\nmoisture, light, and heat. By these it may be rendered luxuriant in its\ngrowth, or it may be stunted in its growth. It may barely exist under\none class of conditions, or it may perish under another. The Brassica\noleracea, in its native habitat on the shore of the sea, is a bitter\nplant with wavy sea-green leaves; in the cultivated garden it is the\ncauliflower. The single rose, under altered conditions, becomes a double\nrose; and creepers rear their stalks and stand erect. Plants, which in a\ncold climate are annuals, become perennial when transported to the\ntorrid zone.[6] And so human nature, fundamentally the same under all\ncircumstances, may be greatly modified, both physically and mentally, by\ngeographical, social, and political conditions. The corporeal nature of\nman--his complexion, his physiognomy, his stature; the intellectual\nnature of man--his religious, ethical, and esthetical ideas are all\nmodified by his surroundings. These modifications, of which all men\ndwelling in the same geographical regions, and under the same social and\npolitical institutions, partake, constitute the _individuality_ of\nnations. Thus, whilst there is a fundamental basis of unity in the\ncorporeal and spiritual nature of man, the causes of diversity are to be\nsought in the circumstances in which tribes and nations are placed in\nthe overruling providence of God.\n\n[Footnote 6: See Carpenter's \"Compar. Physiology,\" p. 625; Lyell's\n\"Principles of Geology,\" pp. 588, 589.]\n\nThe power which man exerts over material conditions, by virtue of his\nintelligence and freedom, is also an important element which, in these\nstudies, we should not depreciate or ignore. We must accept, with all\nits consequences, the dictum of universal consciousness that man is\n_free_. He is not absolutely subject to, and moulded by nature. He has\nthe power to control the circumstances by which he is surrounded--to\noriginate new social and physical conditions--to determine his own\nindividual and responsible character--and he can wield a mighty\ninfluence over the character of his fellow-men. Individual men, as\nLycurgus, Solon, Pericles, Alexander, Caesar, and Napoleon have left the\nimpress of their own mind and character upon the political institutions\nof nations, and, in indirect manner, upon the character of succeeding\ngenerations of men. Homer, Plato, Cicero, Bacon, Kant, Locke, Newton,\nShakspeare, Milton have left a deep and permanent impression upon the\nforms of thought and speech, the language and literature, the science\nand philosophy of nations. And inasmuch as a nation is the aggregate of\nindividual beings endowed with spontaneity and freedom, we must grant\nthat exterior conditions are not omnipotent in the formation of national\ncharacter. Still the free causality of man is exercised within a narrow\nfield. \"There is a strictly necessitative limitation drawing an\nimpassable boundary-line around the area of volitional freedom.\" The\nhuman will \"however subjectively free\" is often \"objectively unfree;\"\nthus a large \"uniformity of volitions\" is the natural consequence.[7]\nThe child born in the heart of China, whilst he may, in his personal\nfreedom, develop such traits of character as constitute his\nindividuality, must necessarily be conformed in his language, habits,\nmodes of thought, and religious sentiments to the spirit of his country\nand age. We no more expect a development of Christian thought and\ncharacter in the centre of Africa, unvisited by Christian teaching, than\nwe expect to find the climate and vegetation of New England. And we no\nmore expect that a New England child shall be a Mohammedan, a Parsee, or\na Buddhist, than that he shall have an Oriental physiognomy, and speak\nan Oriental language. Indeed it is impossible for a man to exist in\nhuman society without partaking in the spirit and manners of his country\nand his age. Thus all the individuals of a nation represent, in a\ngreater or less degree, the spirit of the nation. They who do this most\nperfectly are the _great_ men of that nation, because they are at once\nboth the product and the impersonation of their country and their age.\n\"We allow ourselves to think of Shakspeare, or of Raphael, or of Phidias\nas having accomplished their work by the power of their individual\ngenius, but greatness like theirs is never more than the highest degree\nof perfection which prevails widely around it, and forms the environment\nin which it grows. No such single mind in single contact with the facts\nof nature could have created a Pallas, a Madonna, or a Lear; such vast\nconceptions are the growth of ages, the creation of a nation's spirit;\nand the artist and poet, filled full with the power of that spirit, but\ngave it form, and nothing but form. Nor would the form itself have been\nattained by any isolated talent. No genius can dispense with\nexperience.... Noble conceptions already existing, and a noble school of\nexecution which will launch mind and hand upon their true courses, are\nindispensable to transcendent excellence. Shakspeare's plays were as\nmuch the offspring of the long generations who had pioneered the road\nfor him, as the discoveries of Newton were the offspring of those of\nCopernicus.\"[8] The principles here enounced apply with equal force to\nphilosophers and men of science. The philosophy of Plato was but the\nripened fruit of the pregnant thoughts and seminal utterances of his\npredecessors,--Socrates, Anaxagoras, and Pythagoras; whilst all of them\ndo but represent the general tendency and spirit of their country and\ntheir times. The principles of Lord Bacon's \"Instauratio Magna\" were\nincipient in the \"Opus Majus\" of Roger Bacon, the Franciscan friar. The\nsixteenth century matured the thought of the thirteenth century. The\ninductive method in scientific inquiry was immanent in the British mind,\nand the latter Bacon only gave to it a permanent form. It is true that\ngreat men have occasionally appeared on the stage of history who, like\nthe reformers Luther and Wesley, have seemed to be in conflict with the\nprevailing spirit of their age and nation, but these men were the\ncreations of a providence--that providence which, from time to time, has\n_supernaturally_ interposed in the moral history of our race by\ncorrective and remedial measures. These men were inspired and led by a\nspirit which descended from on high. And yet even they had their\nprecursors and harbingers. Wyckliffe and John Huss, and Jerome of Prague\nare but the representatives of numbers whose names do not grace the\nhistoric page, who pioneered the way for Luther and the Reformation. And\nno one can read the history of that great movement of the sixteenth\ncentury without being persuaded there were thousands of Luther's\npredecessors and contemporaries who, like Staupitz and Erasmus, lamented\nthe corruptions of the Church of Rome, and only needed the heroic\ncourage of Luther to make them reformers also. Whilst, therefore, we\nrecognize a free causal power in man, by which he determines his\nindividual and responsible character, we are compelled to recognize the\ngeneral law, that national character is mainly the result of those\ngeographical and ethological, and political and religious conditions in\nwhich the nations have been placed in the providence of God.\n\n[Footnote 7: See Dr. Wheedon's \"Freedom of the Will,\" pp. 164, 165.]\n\n[Footnote 8: Froude, \"Hist. of England,\" pp. 73, 74.]\n\nNations, like persons, have an _Individuality_. They present certain\ncharacteristic marks which constitute their proper identity, and\nseparate them from the surrounding nations of the earth; such, for\nexample, as complexion, physiognomy, language, pursuits, customs,\ninstitutions, sentiments, ideas. The individuality of a nation is\ndetermined mainly from _without_, and not, like human individuality,\nfrom within. The laws of a man's personal character have their home in\nthe soul; and the peculiarities and habits, and that conduct of life,\nwhich constitute his responsible character are, in a great degree, the\nconsequence of his own free choice. But dwelling, as he does, in\nsociety, where he is continually influenced by the example and opinions\nof his neighbors; subject, as he is, to the ceaseless influence of\nclimate, scenery, and other terrestrial conditions, the characteristics\nwhich result from these relations, and which are common to all who dwell\nin the same regions, and under the same institutions, constitute a\nnational individuality. Individual character is _variable_ under the\nsame general conditions, national character is _uniform_, because it\nresults from causes which operate alike upon all individuals.\n\nNow, that man's complexion, his pursuits, his habits, his ideas are\ngreatly modified by his geographical surroundings, is the most obvious\nof truths. No one doubts that the complexion of man is greatly affected\nby climatic conditions. The appearance, habits, pursuits of the man who\nlives within the tropics must, necessarily, differ from those of the man\nwho dwells within the temperate zone. No one expects that the dweller on\nthe mountain will have the same characteristics as the man who resides\non the plains; or that he whose home is in the interior of a continent\nwill have the same habits as the man whose home is on the islands of the\nsea. The denizen of the primeval forest will most naturally become a\nhuntsman. The dweller on the extended plain, or fertile mountain <DW72>,\nwill lead a pastoral, or an agricultural life. Those who live on the\nmargin of great rivers, or the borders of the sea, will \"do business on\nthe great waters.\" Commerce and navigation will be their chief pursuits.\nThe people whose home is on the margin of the lake, or bay, or inland\nsea, or the thickly studded archipelago, are mostly fishermen. And then\nit is a no less obvious truth that men's pursuits exert a moulding\ninfluence on their habits, their forms of speech, their sentiments, and\ntheir ideas. Let any one take pains to observe the peculiarities which\ncharacterize the huntsman, the shepherd, the agriculturist, or the\nfisherman, and he will be convinced that their occupations stamp the\nwhole of their thoughts and feelings; color all their conceptions of\nthings outside their own peculiar field; direct their simple philosophy\nof life; and give a tone, even, to their religious emotions.\n\nThe general aspects of nature, the climate and the scenery, exert an\nappreciable and an acknowledged influence on the _mental_\ncharacteristics of a people. The sprightliness and vivacity of the\nFrank, the impetuosity of the Arab, the immobility of the Russ, the\nrugged sternness of the Scot, the repose and dreaminess of the Hindoo\nare largely due to the country in which they dwell, the air they\nbreathe, the food they eat, and the landscapes and skies they daily look\nupon. The nomadic Arab is not only indebted to the country in which he\ndwells for his habit of hunting for daily food, but for that love of a\nfree, untrammelled life, and for those soaring dreams of fancy in which\nhe so ardently delights. Not only is the Swiss determined by the\npeculiarities of his geographical position to lead a pastoral life, but\nthe climate, and mountain scenery, and bracing atmosphere inspire him\nwith the love of liberty. The reserved and meditative Hindoo, accustomed\nto the profuse luxuriance of nature, borrows the fantastic ideas of his\nmythology from plants, and flowers, and trees. The vastness and infinite\ndiversity of nature, the colossal magnitude of all the forms of animal\nand vegetable life, the broad and massive features of the landscape, the\naspects of beauty and of terror which surround him, and daily pour their\nsilent influences upon his soul, give vividness, grotesqueness, even, to\nhis imagination, and repress his active powers. His mental character\nbears a peculiar and obvious relation to his geographical\nsurroundings.[9]\n\n[Footnote 9: Ritter, \"Geograph. Studies,\" p. 287.]\n\nThe influence of external nature on the imagination--the _creative_\nfaculty in man--is obvious and remarkable. It reveals itself in all the\nproductions of man--his architecture, his sculpture, his painting, and\nhis poetry. Oriental architecture is characterized by the boldness and\nmassiveness of all its parts, and the monotonous uniformity of all its\nfeatures. This is but the expression, in a material form, of that\nshadowy feeling of infinity, and unity, and immobility which an unbroken\ncontinent of vast deserts and continuous lofty mountain chains would\nnaturally inspire. The simple grandeur and perfect harmony and graceful\nblending of light and shade so peculiar to Grecian architecture are the\nproduct of a country whose area is diversified by the harmonious\nblending of land and water, mountain and plain, all bathed in purest\nlight, and canopied with skies of serenest blue. And they are also the\nproduct of a country where man is released from the imprisonment within\nthe magic circle of surrounding nature, and made conscious of his power\nand freedom. In Grecian architecture, therefore, there is less of the\nmassiveness and immobility of nature, and more of the grace and dignity\nof man. It adds to the idea of permanence a _vital_ expression. \"The\nDoric column,\" says Vitruvius, \"has the proportion, strength, and beauty\nof man.\" The Gothic architecture had its birthplace among a people who\nhad lived and worshipped for ages amidst the dense forests of the north,\nand was no doubt an imitation of the interlacing of the overshadowing\ntrees. The clustered shaft, and lancet arch, and flowing tracery,\nreflect the impression which the surrounding scenery had woven into the\ntexture of the Teutonic mind.\n\nThe history of painting and of sculpture will also show that the varied\n\"styles of art\" are largely the result of the aspects which external\nnature presented to the eye of man. Oriental sculpture, like its\narchitecture, was characterized by massiveness of form and tranquillity\nof expression; and its painting was, at best, but  sculpture. The\nmost striking objects are colossal figures, in which the human form is\nstrangely combined with the brute, as in the winged bulls of Nineveh and\nthe sphinxes of Egypt. Man is regarded simply as a part of nature, he\ndoes not rise above the plane of animal life. The soul has its\nimmortality only in an eternal metempsychosis--a cycle of life which\nsweeps through all the brute creation. But in Grecian sculpture we have\nless of nature, more of man; less of massiveness, more of grace and\nelegance; less repose, and more of action. Now the connection between\nthese styles of art, and the countries in which they were developed, is\nat once suggested to the thoughtful mind.\n\nAnd then, finally, the literature of a people equally reveals the\nimpress of surrounding cosmical conditions. \"The poems of Ossian are but\nthe echo of the wild, rough, cloudy highlands of his Scottish home.\" The\nforest songs of the wild Indian, the <DW64>'s plaintive melodies in the\nrice-fields of Carolina, the refrains in which the hunter of Kamtchatka\nrelates his adventures with the polar bear, and in which the South Sea\nIslander celebrates his feats and dangers on the deep, all betoken the\ninfluence which the scenes of daily life exert upon the thoughts and\nfeelings of our race. \"To what an extent nature can express herself in,\nand modify the culture of the individual, as well as of an entire\npeople, can be seen on Ionian soil in the verse of Homer, which, called\nforth under the most favorable sky, and on the most luxuriant shore of\nthe Grecian archipelago, not only charms us to-day, but bearing this\nimpress, has determined what shall be the classic form throughout all\ncoming time.\"[10]\n\n[Footnote 10: See Ritter, pp. 288, 289. Poetic art has unquestionably\nits _geographical_ distributions like the fauna and flora of the globe.\n\"If you love the images, not merely of a rich, but of a luxuriant fancy;\nif you are pleased with the most daring flights; if you would see a\npoetic creation full of wonders, then turn your eye to the poetry of the\n_orient_, where all forms appear in purple; where each flower glows like\nthe morning ray resting on the earth. But if, on the contrary, you\nprefer depth of thought, and earnestness of reflection; if you delight\nin the colossal, yet pale forms, which float about in mist, and whisper\nof the mysteries of the spirit-land, and of the vanity of all things,\nexcept honor, then I must point you to the hoary _north_.... Or if you\nsympathize with that deep feeling, that longing of the soul, which does\nnot linger on the earth, but evermore looks up to the azure tent of the\nstars, where happiness dwells, where the unquiet of the beating heart is\nstill, then you must resort to the romantic poetry of the\n_west_.\"--\"_Study of Greek Literature_,\" Bishop Esaias Tegner, p. 38.]\n\nIn seeking, therefore, to determine correctly what are the\ncharacteristics of a nation, we must endeavor to trace how far the\nphysical constitution of that people, their temperament, their habits,\ntheir sentiments, and their ideas have been formed, or modified, under\nthe surrounding geographical conditions, which, as we have seen, greatly\ndetermine a nation's individuality. Guided by these lights, let us\napproach the study of \"_the men of Athens_.\"\n\n_Attica_, of which Athens was the capital, and whose entire populations\nwere called \"Athenians,\" was the most important of all the Hellenic\nstates. It is a triangular peninsula, the base of which is defined by\nthe high mountain ranges of Cithaeron and Parnes, whilst the two other\nsides are washed by the sea, having their vertex at the promontory of\nSunium, or Cape Colonna. The prolongation of the south-western line\ntowards the north until it reaches the base at the foot of Mount\nCithaeron, served as the line of demarkation between the Athenian\nterritory and the State of Megara. Thus Attica may be generally\ndescribed as bounded on the north-east by the channel of the Negropont;\non the south-west by the gulf of AEgina and part of Megara; and on the\nnorth-west by the territory which formed the ancient Boeotia, including\nwithin its limits an area of about 750 miles.[11]\n\nHills of inferior elevation connect the mountain ranges of Cithaeron and\nParnes with the mountainous surface of the south-east of the peninsula.\nThese hills, commencing with the promontory of Sunium itself, which\nforms the vertex of the triangle, rise gradually on the south-east to\nthe round summit of Hymettus, and onward to the higher peak of\nPentelicus, near Marathon, on the east. The rest of Attica is all a\nplain, one reach of which comes down to the sea on the south, at the\nvery base of Hymettus. Here, about five miles from the shore, an abrupt\nrock rises from the plain, about 200 feet high, bordered on the south by\nlower eminences. That rock is the Acropolis. Those lower eminences are\nthe Areopagus, the Pmyx, and the Museum. In the valley formed by these\nfour hills we have the Agora, and the varied undulations of these hills\ndetermine the features of the city of Athens.[12]\n\n[Footnote 11: See art. \"Attica,\" _Encyc. Brit._]\n\n[Footnote 12: See Conybeare and Howson's \"Life and Epistles of St.\nPaul,\" vol. i. p. 346.]\n\nNearly all writers on the topography of Athens derive their materials\nfrom Pausanias, who visited the city in the early part of the second\ncentury, and whose \"Itinerary of Greece\" is still extant.[13] He entered\nthe city by the Peiraic gate, the same gate at which Paul entered some\nsixty years before. We shall place ourselves under his guidance, and, so\nfar as we are able, follow the same course, supplying some omissions, as\nwe go along, from other sources. On entering the city, the first\nbuilding which arrested the attention of Pausanias was the Pompeium, so\ncalled because it was the depository of the sacred vessels, and also of\nthe garments used in the annual procession in honor of Athena (Minerva),\nthe tutelary deity of Athens, from whom the city derived its name. Near\nthis edifice stood a temple of Demeter (Ceres), containing statues of\nthat goddess, of her daughter Persephone, and of Iacchus, all executed\nby Praxiteles; and beyond were several porticoes leading from the city\ngates to the outer Ceramicus, while the intervening space was occupied\nby various temples, the Gymnasium of Hermes, and the house of Polytion,\nthe most magnificent private residence in Athens.\n\n[Footnote 13: The account here given of the topography of Athens is\nderived mainly from the article on \"Athens\" in the _Encyc. Brit._]\n\nThere were two places in Athens known by the name of Ceramicus, one\nwithout the walls, forming part of the suburbs; and the other within the\nwalls, embracing a very important section of the city. The outer\nCeramicus was covered with the sepulchres of the Athenians who had been\nslain in battle, and buried at the public expense; it communicated with\nthe inner Ceramicus by the gate Dipylum. The Ceramicus within the city\nprobably included the Agora, the Stoa Basileios, and the Stoa Poecile,\nbesides various other temples and public buildings.\n\nHaving fairly passed the city gates, a long street is before us with a\ncolonnade or cloister on either hand; and at the end of this street, by\nturning to the left, we might go through the whole Ceramicus to the open\ncountry, and the groves of the Academy. But we turn to the right, and\nenter the Agora,--the market-place, as it is called in the English\ntranslation of the sacred narrative.\n\nWe are not, however, to conceive of the market-place at Athens as\nbearing any resemblance to the bare, undecorated spaces appropriated to\nbusiness in our modern towns; but rather as a magnificent public square,\nclosed in by grand historic buildings, of the highest style of\narchitecture; planted with palm-trees in graceful distribution, and\nadorned with statues of the great men of Athens and the deified heroes\nof her mythology, from the hands of the immortal masters of the plastic\nart. This \"market-place\" was the great centre of the public life of the\nAthenians,--the meeting-place of poets, orators, statesmen, warriors,\nand philosophers,--a grand resort for leisure, for conversation, for\nbusiness, and for news. Standing in the Agora, and looking towards the\nsouth, is the _Museum,_ so called because it was believed that _Musaeus_,\nthe father of poetry, was buried there. Towards the north-west is the\n_Pnyx,_ a sloping hill, partially levelled into an open area for\npolitical assemblies. To the north is seen the craggy eminence of the\n_Areopagus_, and on the north-east is the _Acropolis_ towering high\nabove the scene, \"the crown and glory of the whole.\"\n\nThe most important buildings of the Agora are the Porticoes or\ncloisters, the most remarkable of which are the Stoa Basileios, or\nPortico of the king; the Stoa Eleutherius, or Portico of the Jupiter of\nFreedom; and the Stoa Poecile, or Painted Porch. These Porticoes were\ncovered walks, the roof being supported by columns, at least on one\nside, and by solid masonry on the other. Such shaded walks are almost\nindispensable in the south of Europe, where the people live much in the\nopen air, and they afford a grateful protection from the heat of the\nsun, as well as a shelter from the rain. Seats were also provided where\nthe loungers might rest, and the philosophers and rhetoricians sit down\nfor intellectual conversation. The \"Stoic\" school of philosophy derived\nits name from the circumstance that its founder, Zeno, used to meet and\nconverse with his disciples under one of these porticoes,--the Stoa\nPoecile. These porticoes were not only built in the most magnificent\nstyle of architecture, but adorned with paintings and statuary by the\nbest masters. On the roof of the Stoa Basileios were statues of Theseus\nand the Day. In front of the Stoa Eleutherius was placed the divinity to\nwhom it was dedicated; and within were allegorical paintings,\ncelebrating the rise of \"the fierce democracy.\" The Stoa Poecile derived\nits name from the celebrated paintings which adorned its walls, and\nwhich were almost exclusively devoted to the representation of national\nsubjects, as the contest of Theseus with the Amazons, the more glorious\nstruggle at Marathon, and the other achievements of the Athenians; here\nalso were suspended the shields of the Scionaeans of Thrace, together\nwith those of the Lacedemonians, taken at the island of Sphacteria.\n\nIt is beyond our purpose to describe all the public edifices,--the\ntemples, gymnasia, and theatres which crowd the Ceramic area, and that\nportion of the city lying to the west and south of the Acropolis. Our\nobject is, if possible, to convey to the reader some conception of the\nancient splendor and magnificence of Athens; to revive the scenes amidst\nwhich the Athenians daily moved, and which may be presumed to have\nexerted a powerful influence upon the manners, the taste, the habits of\nthought, and the entire character of the Athenian people. To secure this\nobject we need only direct attention to the Acropolis, which was crowded\nwith the monuments of Athenian glory, and exhibited an amazing\nconcentration of all that was most perfect in art, unsurpassed in\nexcellence, and unrivalled in richness and splendor. It was \"the\npeerless gem of Greece, the glory and pride of art, the wonder and envy\nof the world.\"\n\nThe western side of the Acropolis, which furnished the only access to\nthe summit of the hill, was about 168 feet in breadth; an opening so\nnarrow that, to the artists of Pericles, it appeared practicable to fill\nup the space with a single building, which, in serving the purpose of a\ngateway to the Acropolis, should also contribute to adorn, as well as\nfortify the citadel. This work, the greatest achievement of civil\narchitecture in Athens, which rivalled the Parthenon in felicity of\nexecution, and surpassed it in boldness and originality of design,\nconsisted of a grand central colonnade closed by projecting wings. This\nincomparable edifice, built of Pentelic marble, received the name of\nPropylaea from its forming the vestibule to the five-fold gates by which\nthe citadel was entered. In front of the right wing there stood a small\nIonic temple of pure white marble, dedicated to Nike Apteros (Wingless\nVictory).\n\nA gigantic flight of steps conducted from the five-fold gates to the\nplatform of the Acropolis, which was, in fact, one vast composition of\narchitecture and sculpture dedicated to the national glory. Here stood\nthe Parthenon, or temple of the Virgin Goddess, the glorious temple\nwhich rose in the proudest period of Athenian history to the honor of\nMinerva, and which ages have only partially effaced. This magnificent\ntemple, \"by its united excellences of materials, design, and decoration,\ninternal as well as external, has been universally considered the most\nperfect which human genius ever planned and executed. Its dimensions\nwere sufficiently large to produce an impression of grandeur and\nsublimity, which was not disturbed by any obtrusive subdivision of\nparts; and, whether viewed at a small or greater distance, there was\nnothing to divert the mind of the spectator from contemplating the unity\nas well as majesty of mass and outline; circumstances which form the\nfirst and most remarkable characteristic of every Greek temple erected\nduring the purer ages of Grecian taste and genius.\"[14]\n\n[Footnote 14: Leake's \"Topography of Athens,\" p. 209 et seq.]\n\nIt would be impossible to convey any just and adequate conception of the\nartistic decorations of this wonderful edifice. The two pediments of the\ntemple were decorated with magnificent compositions of statuary, each\nconsisting of about twenty entire figures of colossal size; the one on\nthe western pediment representing the birth of Minerva, and the other,\non the eastern pediment, the contest between that goddess and Neptune\nfor the possession of Attica. Under the outer cornice were ninety-two\ngroups, raised in high relief from tablets about four feet square,\nrepresenting the victories achieved by her companions. Round the inner\nfrieze was presented the procession of the Parthenon on the grand\nquinquennial festival of the Panathenaea. The procession is represented\nas advancing in two parallel columns from west to east; one proceeding\nalong the northern, the other along the southern side of the temple;\npart facing inward after turning the angle of the eastern front, and\npart meeting towards the centre of that front.\n\nThe statue of the virgin goddess, the work of Phidias, stood in the\neastern chamber of the cella, and was composed of ivory and gold. It had\nbut one rival in the world, the Jupiter Olympus of the same famous\nartist. On the summit or apex of the helmet was placed a sphinx, with\ngriffins on either side. The figure of the goddess was represented in an\nerect martial attitude, and clothed in a robe reaching to the feet. On\nthe breast was a head of Medusa, wrought in ivory, and a figure of\nVictory about four cubits high. The goddess held a spear in her hand,\nand an aegis lay at her feet, while on her right, and near the spear, was\na figure of a serpent, believed to represent that of Erichthonius.\n\nAccording to Pliny, the entire height of the statue was twenty-six\ncubits (about forty feet), and the artist, Phidias, had ingeniously\ncontrived that the gold with which the statue was encrusted might be\nremoved at pleasure. The battle of the Centaurs and Lapithae was carved\nupon the sandals; the battle of the Amazons was represented on the aegis\nwhich lay at her feet, and on the pedestal was sculptured the birth of\nPandora.\n\nThe temple of Erechtheus, the most ancient structure in Athens, stood on\nthe northern side of the Acropolis. The statue of Zeus Polieus stood\nbetween the Propylaea and the Parthenon. The brazen colossus of Minerva,\ncast from the spoils of Marathon, appears to have occupied the space\nbetween the Erechtheium and the Propylaea, near the Pelasgic or northern\nwall. This statue of the tutelary divinity of Athens and Attica rose in\ngigantic proportions above all the buildings of the Acropolis, the\nflashing of whose helmet plumes met the sailor's eye as he approached\nfrom the Sunian promontory. And the remaining space of the wide area was\nliterally crowded with statuary, amongst which were Theseus contending\nwith the Minotaur; Hercules strangling the serpents; the Earth imploring\nshowers from Jupiter; and Minerva causing the olive to sprout, while\nNeptune raises the waves. After these works of art, it is needless to\nspeak of others. It may be sufficient to state that Pausanias mentions\nby name towards three hundred remarkable statues which adorned this part\nof the city even after it had been robbed and despoiled by its several\nconquerors.\n\nThe Areopagus, or hill of Ares (Mars), so called, it is said, in\nconsequence of that god having been the first person tried there for the\ncrime of murder, was, beyond all doubt, the rocky height which is\nseparated from the western end of the Acropolis by a hollow, forming a\ncommunication between the northern and southern divisions of the city.\nThe court of the Areopagus was simply an open space on the highest\nsummit of the hill, the judges sitting in the open air, on rude seats of\nstone, hewn out of the solid rock. Near to the spot on which the court\nwas held was the sanctuary of the Furies, the avenging deities of\nGrecian mythology, whose presence gave additional solemnity to the\nscene. The place and the court were regarded by the people with\nsuperstitious reverence.\n\nThis completes, our survey of the principal buildings, monuments, and\nlocalities within the city of Athens. We do not imagine we have\nsucceeded in conveying any adequate idea of the ancient splendor and\nglory of this city, which was not only the capital of Attica, but also\n\n     \"The eye of Greece, mother of art and eloquence.\"\n\nWe trust, however, that we have contributed somewhat towards awakening\nin the reader's mind a deeper interest in these classic scenes, and\nenabling him to appreciate, more vividly, the allusions we may hereafter\nmake to them.\n\nThe mere dry recital of geographical details, and topographical notices\nis, however, of little interest in itself, and by itself. A tract of\ncountry derives its chief interest from its historic _associations_--its\nimmediate relations to man. The events which have transpired therein,\nthe noble or ignoble deeds, the grand achievements, or the great\ndisasters of which it has been the theatre, these constitute the living\nheart of its geography. Palestine has been rendered forever memorable,\nnot by any remarkable peculiarities in its climate or scenery, but by\nthe fact that it was the home of God's ancient people--the Hebrews and\nstill more, because the ardent imagination of the modern traveller still\nsees upon its mountains and plains the lingering footprints of the Son\nof God. And so Attica will always be regarded as a classic land, because\nit was the theatre of the most illustrious period of ancient\nhistory--_the period of youthful vigor in the life of humanity, when\nviewed as a grand organic whole_.\n\nHere on a narrow spot of less superficies than the little State of Rhode\nIsland there flourished a republic which, in the grandeur of her\nmilitary and naval achievements, at Marathon, Thermopylae, Plataea, and\nSalamis, in the sublime creations of her painters, sculptors, and\narchitects, and the unrivalled productions of her poets, orators, and\nphilosophers, has left a lingering glory on the historic page, which\ntwenty centuries have not been able to eclipse or dim. The names of\nSolon and Pericles; of Socrates, Plato, and Aristotle; of Isocrates and\nDemosthenes; of Myron, Phidias, and Praxiteles; of Herodotus, Xenophon,\nand Thucydides; of Sophocles and Euripides, have shed an undying lustre\non Athens and Attica.\n\nHow much of this universal renown, this imperishable glory attained by\nthe Athenian people, is to be ascribed to their geographical position\nand surroundings, and to the elastic, bracing air, the enchanting\nscenery, the glorious skies, which poured their daily inspiration on the\nAthenian mind, is a problem we may scarcely hope to solve.\n\nOf this, at least, we may be sure, that all these geographical and\ncosmical conditions were ordained by God, and ordained, also, for some\nnoble and worthy end. That God, \"the Father of all the families of the\nearth,\" cared for the Athenian people as much as for Jewish and\nChristian nations, we can not doubt. That they were the subjects of a\nProvidence, and that, in God's great plan of human history, they had an\nimportant part to fulfill, we must believe. That God \"determined the\ntime of each nation's existence, and fixed the geographical bounds of\nits habitation,\" is affirmed by Paul. And that the _specific_ end for\nwhich the nation had its existence was fulfilled, we have the fullest\nconfidence. _So far, therefore, as we can trace the relation that\nsubsists between the geographical position and surroundings of that\nnation, and its national characteristics and actual history, so far are\nwe able to solve the problem of its destiny; and by so much do we\nenlarge our comprehension of the plan of God in the history of our\nrace_.\n\nThe geographical position of Greece was favorable to the freest\ncommercial and maritime intercourse with the great historic\nnations--those nations most advanced in science, literature, and art.\nBounded on the west by the Adriatic and Ionian seas, by the\nMediterranean on the south, and on the east by the AEgean Sea, her\npopulations enjoyed a free intercommunication with the Egyptians,\nHebrews, Persians, Phoenicians, Romans, and Carthaginians. This\npeculiarity in the geographical position of the Grecian peninsula could\nnot fail to awaken in its people a taste for navigation, and lead them\nto active commercial intercourse with foreign nations.[15] The boundless\noceans on the south and east, the almost impassable mountains on the\nwest and north of Asia, presented insurmountable obstacles to commercial\nintercourse. But the extended border-lands and narrow inland seas of\nSouthern Europe allured man, in presence of their opposite shores, to\nthe perpetual exchange of his productions. An arm of the sea is not a\nbarrier, but rather a tie between the nations. Appearing to separate, it\nin reality draws them together without confounding them.[16] On such a\ntheatre we may expect that commerce will be developed on an extensive\nscale.[17] And, along with commerce, there will be increased activity in\nall departments of productive industry, and an enlarged diffusion of\nknowledge. \"Commerce,\" says Ritter, \"is the great mover and combiner of\nthe world's activities.\" And it also furnishes the channels through\nwhich flow the world's ideas. Commerce, both in a material and moral\npoint of view, is the life of nations. Along with the ivory and ebony,\nthe fabrics and purple dyes, the wines and spices of the Syrian\nmerchant, there flowed into Greece the science of numbers and of\nnavigation, and the art of alphabetical writing from Phoenicia. Along\nwith the fine wheat, and embroidered linen, and riches of the farther\nIndias which came from Egypt, there came, also, into Greece some\nknowledge of the sciences of astronomy and geometry, of architecture and\nmechanics, of medicine and chemistry; together with the mystic wisdom of\nthe distant Orient. The scattered rays of light which gleamed in the\neastern skies were thus converged in Greece, as on a focal point, to be\nrendered more brilliant by contact with the powerful Grecian intellect,\nand then diffused throughout the western world. Thus intercourse with\nsurrounding nations, by commerce and travel, contact therewith by\nimmigrations and colonizations, even collisions and invasions also,\nbecame, in the hands of a presiding Providence, the means of diffusing\nknowledge, of quickening and enlarging the active powers of man, and\nthus, ultimately, of a higher civilization.\n\n[Footnote 15: Humboldt's \"Cosmos,\" vol. ii. p. 143.]\n\n[Footnote 16: Cousin, vol. i. pp. 169, 170.]\n\n[Footnote 17: The advantageous situation of Britain for commerce, and\nthe nature of the climate have powerfully contributed to the perfection\nof industry among her population. Had she occupied a central, internal\nstation, like that of Switzerland, the facilities of her people for\ndealing with others being so much the less, their progress would have\nbeen comparatively slow, and, instead of being highly improved, their\nmanufactures would have been still in infancy. But being surrounded on\nall sides by the sea, that \"great highway of nations,\" they have been\nable to maintain an intercourse with the most remote as well as the\nnearest countries, to supply them on the easiest terms with their\nmanufactures, and to profit by the peculiar products and capacities of\nproduction possessed by other nations. To the geographical position and\nclimate of Great Britain, her people are mainly indebted for their\nposition as the first commercial nation on earth.--See art.\n\"Manufactrues,\" p. 277, _Encyc. Brit_.]\n\nThen further, the peculiar configuration of Greece, the wonderful\ncomplexity of its coast-line, its peninsular forms, the number of its\nislands, and the singular distribution of its mountains, all seem to\nmark it as the theatre of activity, of movement, of individuality, and\nof freedom. An extensive continent, unbroken by lakes and inland seas,\nas Asia, where vast deserts and high mountain chains separate the\npopulations, is the seat of immobility.[18] Commerce is limited to the\nbare necessities of life, and there are no inducements to movement, to\ntravel, and to enterprise. There are no conditions prompting man to\nattempt the conquest of nature. Society is therefore stationary as in\nChina and India. Enfolded and imprisoned within the overpowering\nvastness and illimitable sweep of nature, man is almost unconscious of\nhis freedom and his personality. He surrenders himself to the disposal\nof a mysterious \"_fate_\" and yields readily to the despotic sway of\nsuperhuman powers. The State is consequently the reign of a single\ndespotic will. The laws of the Medes and Persians are unalterable. But\nin Greece we have extended border-lands on the coast of navigable seas;\npeninsulas elaborately articulated, and easy of access. We have\nmountains sufficiently elevated to shade the land and diversify the\nscenery, and yet of such a form as not to impede communication. They are\nusually placed neither in parallel chains nor in massive groups, but are\nso disposed as to inclose extensive tracts of land admirably adapted to\nbecome the seats of small and independent communities, separated by\nnatural boundaries, sometimes impossible to overleap. The face of the\ninterior country,--its forms of relief, seemed as though Providence\ndesigned, from the beginning, to keep its populations socially and\npolitically disunited. These difficulties of internal transit by land\nwere, however, counteracted by the large proportion of coast, and the\naccessibility of the country by sea. The promontories and indentations\nin the line of the Grecian coast are hardly less remarkable than the\npeculiar elevations and depressions of the surface. \"The shape of\nPeloponnesus, with its three southern gulfs, the Argolic, Laconian, and\nMessenian, was compared by the ancient geographers to the leaf of a\nplane-tree: the Pagasaean gulf on the eastern side of Greece, and the\nAmbrakian gulf on the western, with their narrow entrances and\nconsiderable area, are equivalent to internal lakes: Xenophon boasts of\nthe double sea which embraces so large a portion of Attica; Ephorus, of\nthe triple sea by which Boeotia was accessible from west, north, and\nsouth--the Euboean strait, opening a long line of country on both sides\nto coasting navigation. But the most important of all Grecian gulfs are\nthe Corinthian and Saronic, washing the northern and north-eastern\nshores of Peloponnesus, and separated by the narrow barrier of the\nIsthmus of Corinth. The former, especially, lays open AEtolia, Phokis,\nand Boeotia, as the whole northern coast of Peloponnesus, to water\napproach.... It will thus appear that there was no part of Greece proper\nwhich could be considered as out of the reach of the sea, whilst most\nparts of it were easy of access. The sea was thus the sole channel for\ntransmitting improvements and ideas as well as for maintaining\nsympathies\" between the Hellenic tribes.[19] The sea is not only the\ngrand highway of commercial intercourse, but the empire of movement, of\nprogress, and of freedom. Here man is set free from the bondage imposed\nby the overpowering magnitude and vastness of continental and oceanic\nforms. The boisterous and, apparently, lawless winds are made to obey\nhis will. He mounts the sea as on a fiery steed and \"lays his hand upon\nher mane.\" And whilst thus he succeeds, in any measure, to triumph over\nnature, he wakes to conscious power and freedom. It is in this region of\ncontact and commingling of sea and land where man attains the highest\nsuperiority. Refreshing our historic recollections, and casting our eyes\nupon the map of the world, we can not fail to see that all the most\nhighly civilized nations have lived, or still live, on the margin of the\nsea.\n\n[Footnote 18: Cousin, vol. i. pp. 151, 170.]\n\n[Footnote 19: Grote's \"Hist, of Greece,\" vol. ii. pp. 221, 225.]\n\nThe peculiar configuration of the territory of Greece, its forms of\nrelief, \"so like, in many respects, to Switzerland,\" could not fail to\nexert a powerful influence on the character and destiny of its people.\nIts inclosing mountains materially increased their defensive power, and,\nat the same time, inspired them with the love of liberty. Those\nmountains, as we have seen, so unique in their distribution, were\nnatural barriers against the invasion of foreign nations, and they\nrendered each separate community secure against the encroachments of the\nrest. The pass of Thermopylae, between Thessaly and Phocis, that of\nCithaeron, between Boeotia and Attica; and the mountain ranges of Oneion\nand Geraneia, along the Isthmus of Corinth, were positions which could\nbe defended against any force of invaders. This signal peculiarity in\nthe forms of relief protected each section of the Greeks from being\nconquered, and at the same time maintained their separate autonomy. The\nseparate states of Greece lived, as it were, in the presence of each\nother, and at the same time resisted all influences and all efforts\ntowards a coalescence with each other, until the time of Alexander.\nTheir country, a word of indefinite meaning to the Asiatic, conveyed to\nthem as definite an idea as that of their own homes. Its whole\nlandscape, with all its historic associations, its glorious monuments of\nheroic deeds, were perpetually present to their eyes. Thus their\npatriotism, concentrated within a narrow sphere, and kept alive by the\nsense of their individual importance, their democratic spirit, and their\nstruggles with surrounding communities to maintain their independence,\nbecame a strong and ruling passion. Their geographical surroundings had,\ntherefore, a powerful influence upon their political institutions.\nConquest, which forces nations of different habits, characters, and\nlanguages into unity, is at last the parent of degrading servitude.\nThese nations are only held together, as in the Roman empire, by the\niron hand of military power. The despot, surrounded by a foreign\nsoldiery, appears in the conquered provinces, simply to enforce tribute,\nand compel obedience to his arbitrary will. But the small Greek\ncommunities, protected by the barriers of their seas and gulfs and\nmountains, escaped, for centuries, this evil destiny. The people, united\nby identity of language and manners and religion, by common interest and\nfacile intercommunication, could readily combine to resist the invasions\nof foreign nations, as well as the encroachments of their own rulers.\nAnd they were able to easily model their own government according to\ntheir own necessities and circumstances and common interests, and to\nmake the end for which it existed the sole measure of the powers it was\npermitted to wield.[20]\n\n[Footnote 20: _Encyc. Brit_, art. \"Greece.\"]\n\nThe soil of Attica was not the most favorable to agricultural pursuits.\nIn many places it was stony and uneven, and a considerable proportion\nwas bare rock, on which nothing could be grown. Not half the surface was\ncapable of cultivation. In this respect it may be fitly compared to some\nof the New England States. The light, dry soil produced excellent\nbarley, but not enough of wheat for their own consumption. Demosthenes\ninforms us that Athens brought every year, from Byzantium, four hundred\nthousand _medimni_ of wheat. The alluvial plains, under industrious\ncultivation, would furnish a frugal subsistence for a large population,\nand the mildness of the climate allowed all the more valuable products\nto ripen early, and go out of season last. Such conditions, of course,\nwould furnish motives for skill and industry, and demand of the people\nfrugal and temperate habits. The luxuriance of a tropical climate tends\nto improvidence and indolence. Where nature pours her fullness into the\nlap of ease, forethought and providence are little needed. There is none\nof that struggle for existence which awakens sagacity, and calls into\nexercise the active powers of man. But in a country where nature only\nyields her fruits as the reward of toil, and yet enough to the\nintelligent culture of the soil, there habits of patient industry must\nbe formed. The alternations of summer and winter excite to forethought\nand providence, and the comparative poverty of the soil will prompt to\nfrugality. Man naturally aspires to improve his condition by all the\nmeans within his power. He becomes a careful observer of nature, he\ntreasures up the results of observation, he compares one fact with\nanother and notes their relations, and he makes new experiments to test\nhis conclusions, and thus he awakes to the vigorous exercise of all his\npowers. These physical conditions must develop a hardy, vigorous,\nprudent, and temperate race; and such, unquestionably, were the Greeks.\n\"Theophrastus, and other authors, amply attest the observant and\nindustrious agriculture prevalent in Greece. The culture of the vine and\nolive appears to have been particularly elaborate and the many different\naccidents of soil, level, and exposure which were to be found, afforded\nto observant planters materials for study and comparison.\"[21] The\nGreeks were frugal in their habits and simple in their modes of life.\nThe barley loaf seems to have been more generally eaten than the wheaten\nloaf; this, with salt fish and vegetables, was the common food of the\npopulation. Economy in domestic life was universal. In their manners,\ntheir dress, their private dwellings, they were little disposed to\nostentation or display.\n\n[Footnote 21: Grote, \"Hist. of Greece,\" vol. ii. p. 230.]\n\nThe climate of Attica is what, in physical geography, would be called\n_maritime_. \"Here are allied the continental vigor and oceanic softness,\nin a fortunate union, mutually tempering each other.\"[22] The climate of\nthe whole peninsula of Greece seems to be distinguished from that of\nSpain and Italy, by having more of the character of an inland region.\nThe diversity of local temperature is greater; the extremes of summer\nand winter more severe. In Arcadia the snow has been found eighteen\ninches thick in January, with the thermometer at 16 deg. Fahrenheit, and it\nsometimes lies on the ground for six weeks. The summits of the central\nchains of Pindus and most of the Albanian mountains are covered with\nsnow from the beginning of November to the end of March. In Attica,\nwhich, being freely exposed to the sea, has in some measure an insular\nclimate, the winter sets in about the beginning of January. About the\nmiddle of that month the snow begins to fall, but seldom remains upon\nthe plain for more than a few days, though it lies on the summit of the\nmountain for a month.[23] And then, whilst Boeotia, which joins to\nAttica, is higher and colder, and often covered with dense fogs, Attica\nis remarkable for the wonderful transparency, dryness, and elasticity of\nits atmosphere. All these climatal conditions exerted, no doubt, a\nmodifying influence upon the character of the inhabitants.[24] In a\ntropical climate man is enfeebled by excessive heat. His natural\ntendency is to inaction and repose. His life is passed in a \"strenuous\nidleness.\" His intellectual, his reflective faculties are overmastered\nby his physical instincts. Passion, sentiment, imagination prevail over\nthe sober exercises of his reasoning powers. Poetry universally\npredominates over philosophy. The whole character of Oriental language,\nreligion, literature is intensely imaginative. In the frozen regions of\nthe frigid zone, where a perpetual winter reigns, and where lichens and\nmosses are the only forms of vegetable life, man is condemned to the\nlife of a huntsman, and depends mainly for his subsistence on the\nprecarious chances of the chase. He is consequently nomadic in his\nhabits, and barbarous withal. His whole life is spent in the bare\nprocess of procuring a living. He consumes a large amount of oleaginous\nfood, and breathes a damp heavy atmosphere, and is, consequently, of a\ndull phlegmatic temperament. Notwithstanding his uncertain supplies of\nfood, he is recklessly improvident, and indifferent to all the lessons\nof experience. Intellectual pursuits are all precluded. There is no\nmotive, no opportunity, and indeed no disposition for mental culture.\nBut in a temperate climate man is stimulated to high mental activity.\nThe alternations of heat and cold, of summer and of winter, an elastic,\nfresh, and bracing atmosphere, a diversity in the aspects of nature,\nthese develop a vivacity of temperament, a quickness of sensibility as\nwell as apprehension, and a versatility of feeling as well as genius.\nHistory marks out the temperate zone as the seat of the refined and\ncultivated nations.\n\n[Footnote 22: Guyot, \"Earth and Man,\" p. 181.]\n\n[Footnote 23: _Encyc. Brit._, art. \"Greece.\"]\n\n[Footnote 24: The influence of climatic conditions did not escape the\nattention of the Greeks. Herodotus, Hippocrates, and Aristotle speak of\nthe climate of Asia as more enervating than that of Greece. They\nregarded the changeful character and diversity of local temperature in\nGreece as highly stimulating to the energies of the populations. The\nmarked contrast between the Athenians and the Boeotians was supposed to\nbe represented in the light and heavy atmosphere which they respectively\nbreathed.--_Grote_, vol. ii. pp. 232-3.]\n\nThe natural scenery of Greece was of unrivalled grandeur--surpassing\nItaly, perhaps every country in the world. It combined in the highest\ndegree every feature essential to the highest beauty of a landscape\nexcept, perhaps, large rivers. But this was more than compensated for by\nthe proximity of the sea, which, by its numerous arms, seemed to embrace\nthe land on nearly every side. Its mountains, encircled with zones of\nwood, and capped with snow, though much lower than the Alps, are as\nimposing by the suddenness of their elevation--\"pillars of heaven, the\nfosterers of enduring snows.\"[25] Rich sheltered plains lie at their\nfeet, covered with an unequally woven mantle of trees, and shrubs, and\nflowers,--\"the verdant gloom of the thickly-mantling ivy, the narcissus\nsteeped in heavenly dew, the golden-beaming crocus, the hardy and\never-fresh-sprouting olive-tree,\"[26] and the luxuriant palm, which\nnourishes amid its branches the grape swelling with juice. But it is the\ncombination of these features, in the most diversified manner, with\nbeautiful inland bays and seas, broken by headlands, inclosed by\nmountains, and studded with islands of every form and magnitude, which\ngives to the scenery of Greece its proud pre-eminence. \"Greek scenery,\"\nsays Humboldt, \"presents the peculiar charm of an intimate blending of\nsea and land, of shores adorned with vegetation, or picturesquely girt\nwith rocks gleaming in the light of aerial tints, and an ocean beautiful\nin the play of the ever-changing brightness of its deep-toned wave.\"[27]\nAnd over all the serene, deep azure skies, occasionally veiled by light\nfleecy clouds, with vapory purple mists resting on the distant mountain\ntops. This glorious scenery of Greece is evermore the admiration of the\nmodern traveller. \"In wandering about Athens on a sunny day in March,\nwhen the asphodels are blooming on Colones, when the immortal mountains\nare folded in a transparent haze, and the AEgean slumbers afar among his\nisles,\" he is reminded of the lines of Byron penned amid these scenes--\n\n[Footnote 25: Pindar.]\n\n[Footnote 26: Sophocles, \"oedipus at Colonna.\"]\n\n[Footnote 27: \"Cosmos,\" vol. ii. p. 25.]\n\n     \"Yet are thy skies as blue, thy crags as wild;\n     Sweet are thy groves, and verdant are thy fields,\n     Thine olives ripe as when Minerva smiled,\n     And still his honeyed wealth Hymettus yields;\n     There the blithe bee his fragrant fortress builds,\n     The freeborn wanderer of the mountain air;\n     Apollo still thy long, long summer gilds,\n     Still in his beams Mendeli's marbles glare;\n     Art, Glory, Freedom fail, but nature still is fair.\"[28]\n\n[Footnote 28: Canto ii., v. lxxxvi., \"Childe Harold.\"]\n\nThe effect of this scenery upon the character, the imagination, the\ntaste of the Athenians must have been immense. Under the influence of\nsuch sublime objects, the human mind becomes gifted as with inspiration,\nand is by nature filled with poetic images. \"Greece became the\nbirth-place of taste, of art, and eloquence, the chosen sanctuary of the\nmuses, the prototype of all that is graceful, and dignified, and grand\nin sentiment and action.\"\n\nAnd now, if we have succeeded in clearly presenting and properly\ngrouping the facts, and in estimating the influence of geographical\nposition and surroundings on national character, we have secured the\nnatural _criteria_ by which we examine, and even correct the portraiture\nof the Athenian character usually presented by the historian.\n\nThe character of the Athenians has been sketched by Plutarch[29] with\nconsiderable minuteness, and his representations have been permitted,\nuntil of late years, to pass unchallenged. He has described them as at\nonce passionate and placable, easily moved to anger, and as easily\nappeased; fond of pleasantry and repartee, and heartily enjoying a\nlaugh; pleased to hear themselves praised, and yet not annoyed by\ncriticism and censure; naturally generous towards those who were poor\nand in humble circumstances, and humane even towards their enemies;\njealous of their liberties, and keeping even their rulers in awe. In\nregard to their intellectual traits, he affirms their minds were not\nformed for laborious research, and though they seized a subject as it\nwere by intuition, yet wanted patience and perseverance for a thorough\nexamination of all its bearings. \"An observation,\" says the writer of\nthe article on \"_Attica_,\" in the Encyclopaedia Britannica, \"more\nsuperficial in itself, and arguing a greater ignorance of the Athenians,\ncan not easily be imagined.\" Plutarch lived more than three hundred\nyears after the palmy days of the Athenian Demos had passed away. He was\na Boeotian by birth, not an Attic, and more of a Roman than a Greek in\nall his sympathies. We are tempted to regard him as writing under the\ninfluence of prejudice, if not of envy. He was scarcely reliable as a\nbiographer, and as materials for history his \"Parallel Lives\" have been\npronounced \"not altogether trustworthy.\"[30]\n\n[Footnote 29: \"De Praecept.\"]\n\n[Footnote 30: _Encyc. of Biography_, art. \"Plutarch.\"]\n\nThat the Athenians were remarkable for the ardor and vivacity of their\ntemperament,--that they were liable to sudden gusts of passion,--that\nthey were inconstant in their affections, intolerant of dictation,\nimpatient of control, and hasty to resent every assumption of\nsuperiority,--that they were pleased with flattery, and too ready to\nlend a willing ear to the adulation of the demagogue,--and that they\nwere impetuous and brave, yet liable to be excessively elated by\nsuccess, and depressed by misfortune, we may readily believe, because\nsuch traits of character are in perfect harmony with all the facts and\nconclusions already presented. Such characteristics were the natural\nproduct of the warm and genial sunlight, the elastic bracing air, the\nethereal skies, the glorious mountain scenery, and the elaborate\nblending of sea and land, so peculiar to Greece and the whole of\nSouthern Europe.[31] These characteristics were shared in a greater or\nless degree by all the nations of Southern Europe in ancient times, and\nthey are still distinctive traits in the Frenchman, the Italian, and the\nmodern Greek.[32]\n\n[Footnote 31: \"As the skies of Hellas surpassed nearly all other\nclimates in brightness and elasticity, so, also, had nature dealt most\nlovingly with the inhabitants of this land. Throughout the whole being\nof the Greek there reigned supreme a quick susceptibility, out of which\nsprang a gladsome serenity of temper, and a keen enjoyment of life;\nacute sense, and nimbleness of apprehension; a guileless and child-like\nfeeling, full of trust and faith, combined with prudence and forecast.\nThese peculiarities lay so deeply imbedded in the inmost nature of the\nGreeks that no revolutions of time and circumstances have yet been able\nto destroy them; nay, it may be asserted that even now, after centuries\nof degradation, they have not been wholly extinguished in the\ninhabitants of ancient Hellas.\"--\"_Education of the Moral Sentiment\namongst the Ancient Greeks_.\" By FREDERICK JACOBS, p. 320.]\n\n[Footnote 32: These are described by the modern historian and traveller\nas lively, versatile, and witty. \"The love of liberty and independence\ndoes not seem to be rooted out of the national character by centuries of\nsubjugation. They love to command; but though they are loyal to a good\ngovernment, they are apt readily to rise when their rights and liberties\nare infringed. As there is little love of obedience among them, so\nneither is there any toleration of aristocratic pretensions.\"--_Encyc.\nBrit._, art. \"Greece.\"]\n\nThe consciousness of power, the feeling of independence, the ardent love\nof freedom induced in the Athenian mind by the objective freedom of\nmovement which his geographical position afforded, and that\nsubordination and subserviency of physical nature to man so peculiar to\nGreece, determined the democratic character of all their political\ninstitutions. And these institutions reacted upon the character of the\npeople and intensified their love of liberty. This passionate love of\npersonal freedom, amounting almost to disease, excited them to a\nconstant and almost distressing vigilance. And it is not to be wondered\nat if it displayed itself in an extreme jealousy of their rulers, an\nincessant supervision and criticism of all their proceedings, and an\nintense and passionate hatred of tyrants and of tyranny. The popular\nlegislator or the successful soldier might dare to encroach upon their\nliberties in the moment when the nation was intoxicated and dazzled with\ntheir genius, their prowess, and success; but a sudden revulsion of\npopular feeling, and an explosion of popular indignation, would overturn\nthe one, and ostracism expel the other. Thus while inconstancy, and\nturbulence, and faction seem to have been inseparable from the\ndemocratic spirit, the Athenians were certainly constant in their love\nof liberty, faithful in their affection for their country,[33] and\ninvariable in their sympathy and admiration for that genius which shed\nglory upon their native land. And then they were ever ready to repair\nthe errors, and make amends for the injustice committed under the\ninfluence of passionate excitement, or the headlong impetuosity of their\ntoo ardent temperament. The history of Greece supplies numerous\nillustrations of this spirit. The sentence of death which had been\nhastily passed on the inhabitants of Mytilene was, on sober reflection,\nrevoked the following day. The immediate repentance and general sorrow\nwhich followed the condemnation of the ten generals, as also of\nSocrates, are notable instances.\n\n[Footnote 33: When immense bribes were offered by the king of Persia to\ninduce the Athenians to detach themselves from the alliance with the\nrest of the Hellenic States, she answered by the mouth of Aristides\n\"that it was impossible for all the gold in the world to tempt the\nRepublic of Athens, or prevail with it to sell its liberty and that of\nGreece!\"]\n\nIn their private life the Athenians were courteous, generous, and\nhumane. Whilst bold and free in the expression of their opinions, they\npaid the greatest attention to rules of politeness, and were nicely\ndelicate on points of decorum. They had a natural sense of what was\nbecoming and appropriate, and an innate aversion to all extravagance. A\ngraceful demeanor and a quiet dignity were distinguishing traits of\nAthenian character. They were temperate and frugal[34] in their habits,\nand little addicted to ostentation and display. Even after their\nvictories had brought them into contact with Oriental luxury and\nextravagance, and their wealth enabled them to rival, in costliness and\nsplendor, the nations they had conquered, they still maintained a\nrepublican simplicity. The private dwellings of the principal citizens\nwere small, and usually built of clay; their interior embellishments\nalso were insignificant--the house of Polytion alone formed an\nexception.[35] All their sumptuousness and magnificence were reserved\nfor and lavished on their public edifices and monuments of art, which\nmade Athens the pride of Greece and the wonder of the world.\nIntellectually, the Athenians were remarkable for their quickness of\napprehension, their nice and delicate perception, their intuitional\npower, and their versatile genius. Nor were they at all incapable of\npursuing laborious researches, or wanting in persevering application and\nindustry, notwithstanding Plutarch's assertion to the contrary. The\ncircumstances of every-day life in Attica, the conditions which\nsurrounded the Athenian from childhood to age, were such as to call for\nthe exercise of these qualities of mind in the highest degree. Habits of\npatient industry were induced in the Athenian character by the poverty\nand comparative barrenness of the soil, demanding greater exertion to\nsupply their natural wants. And an annual period of dormancy, though\nunaccompanied by the rigors of a northern winter, called for prudence in\nhusbanding, and forethought and skill in endeavoring to increase their\nnatural resources. The aspects of nature were less massive and\nawe-inspiring, her features more subdued, and her areas more\ncircumscribed and broken, inviting and emboldening man to attempt her\nconquest. The whole tendency of natural phenomena in Greece was to\nrestrain the imagination, and discipline the observing and reasoning\nfaculties in man. Thus was man inspired with confidence in his own\nresources, and allured to cherish an inquisitive, analytic, and\nscientific spirit. \"The French, in point of national character, hold\nnearly the same relative place amongst the nations of Europe that the\nAthenians held amongst the States of Ancient Greece.\" And whilst it is\nadmitted the French are quick, sprightly, vivacious, perhaps sometimes\nlight even to frivolity, it must be conceded they have cultivated the\nnatural and exact sciences with a patience, and perseverance, and\nsuccess unsurpassed by any of the nations of Europe. And so the\nAthenians were the Frenchmen of Greece. Whilst they spent their \"leisure\ntime\"[36] in the place of public resort, the porticoes and groves,\n\"hearing and telling the latest news\" (no undignified or improper mode\nof recreation in a city where newspapers were unknown), whilst they are\ncondemned as \"garrulous,\" \"frivolous,\" \"full of curiosity,\" and\n\"restlessly fond of novelties,\" we must insist that a love of study, of\npatient thought and profound research, was congenial to their natural\ntemperament, and that an inquisitive and analytic spirit, as well as a\ntaste for subtile and abstract speculation, were inherent in the\nnational character. The affluence, and fullness, and flexibility, and\nsculpture-like finish of the language of the Attics, which leaves far\nbehind not only the languages of antiquity, but also the most cultivated\nof modern times, is an enduring monument of the patient industry of the\nAthenians.[37] Language is unquestionably the highest creation of\nreason, and in the language of a nation we can see reflected as in a\nmirror the amount of culture to which it has attained. The rare balance\nof the imagination and the reasoning powers, in which the perfection of\nthe human intellect is regarded as consisting, the exact correspondence\nbetween the thought and the expression, \"the free music of prosaic\nnumbers in the most diversified forms of style,\" the calmness, and\nperspicuity, and order, even in the stormiest moments of inspiration,\nrevealed in every department of Greek literature, were not a mere happy\nstroke of chance, but a product of unwearied effort--and effort too\nwhich was directed by the criteria which reason supplied. The plastic\nart of Greece, which after the lapse of ages still stands forth in\nunrivalled beauty, so that, in presence of the eternal models it\ncreated, the modern artist feels the painful lack of progress was not a\nspontaneous outburst of genius, but the result of intense application\nand unwearied discipline. The achievements of the philosophic spirit,\nthe ethical and political systems of the Academy, the Lyceum, the Stoa,\nand the Garden, the anticipations, scattered here and there like\nprophetic hints, of some of the profoundest discoveries of \"inductive\nscience\" in more modern days,--all these are an enduring protest against\nthe strange misrepresentations of Plutarch.\n\n[Footnote 34: These are still characteristics of the Greeks. \"They are\nan exceedingly temperate people; drunkenness is a vice remarkably rare\namongst them; their food also is spare and simple; even the richest are\ncontent with a dish of vegetables for each meal, and the poor with a\nhandful of olives or a piece of salt fish.... All other pleasures are\nindulged with similar propriety; their passions are moderate, and\ninsanity is almost unknown amongst them.\"--_Encyc. Brit._, art.\n\"Greece.\"]\n\n[Footnote 35: Niebuhr's Lectures, vol. i. p. 101.]\n\n[Footnote 36: Eukaireo corresponds exactly to the Latin _vacare_, \"to be\nat leisure.\"]\n\n[Footnote 37: Frederick Jacobs, on \"Study of Classic Antiquity,\" p. 57.]\n\nIn Athens there existed a providential collocation of the most favorable\nconditions in which humanity can be placed for securing its highest\nnatural development. Athenian civilization is the solution, on the\ntheatre of history, of the problem--What degree of perfection can\nhumanity, under the most favorable conditions, attain, without the\nsupernatural light, and guidance, and grace of Christianity?[38] \"Like\ntheir own goddess Athene the people of Athens seem to spring full-armed\ninto the arena of history, and we look in vain to Egypt, Syria, and\nIndia, for more than a few seeds that burst into such marvellous growth\non the soil of Attica.\"[39]\n\n[Footnote 38: It has been asserted by some theological writers, Watson\nfor example, that no society of civilized men has been, or can be\nconstituted without the aid of a religion directly communicated by\nrevelation, and transmitted by oral tradition;--\"that it is possible to\nraise a body of men into that degree of civil improvement which would\nexcite the passion for philosophic investigation, without the aid of\nreligion... can have no proof, and is contradicted by every fact and\nanalogy with which we are acquainted.\" (_Institutes_, vol. i. p. 271;\nsee also Archbishop Whately, \"Dissertation,\" etc., vol. i. _Encyc.\nBrit._, p. 449-455).\n\nThe fallacy of the reasoning by which this doctrine is sought to be\nsustained is found in the assumption \"that to all our race the existence\nof a First Cause is a question of philosophy,\" and that the idea of God\nlies at the end of \"a gradual process of inquiry\" and induction, for\nwhich a high degree of \"scientific culture\" is needed. Whereas the idea\nof a First Cause lies at the beginning, not at the end of philosophy;\nand philosophy is simply the analysis of our natural consciousness of\nGod, and the presentation of the idea in a logical form. Faith in the\nexistence of God is not the result of a conscious process of reflection;\nit is the spontaneous and instinctive logic of the human mind, which, in\nview of phenomena presented to sense, by a necessary law of thought\nimmediately and intuitively affirms a personal Power, an intelligent\nMind as the author. In this regard, there is no difference between men\nexcept the clearness with which they apprehend, and the logical account\nthey can render to themselves, of this instinctive belief. Spontaneous\nintuition, says Cousin, is the genius of all men; reflection the genius\nof few men. \"But Leibnitz had no more confidence in the principle of\ncausality, and even in his favorite principle of sufficient reason, than\nthe most ignorant of men;\" the latter have this principle within them,\nas a law of thought, controlling their conception of the universe, and\ndoing this almost unconsciously; the former, by an analysis of thought,\nsucceeded in defining and formulating the ideas and laws which\nnecessitate the cognition of a God. The function of philosophy is simply\nto transform alethes doxa into itisteme--right opinion into science,--to\nelucidate and logically present the immanent thought which lies in the\nuniversal consciousness of man.\n\nThat the possession of the idea of God is essential to the social and\nmoral elevation of man,--that is, to the civilization of our race, is\nmost cheerfully conceded. That humanity has an end and destination which\ncan only be secured by the true knowledge of God, and by a participation\nof the nature of God, is equally the doctrine of Plato and of Christ.\nNow, if humanity has a special end and destination, it must have some\ninstinctive tendings, some spermatic ideas, some original forces or\nlaws, which determine it towards that end. All development supposes some\noriginal elements to be unfolded or developed. Civilization is but the\ndevelopment of humanity according to its primal idea and law, and under\nthe best exterior conditions. That the original elements of humanity\nwere unfolded in some noble degree under the influence of philosophy is\nclear from the history of Greece; there the most favorable natural\nconditions for that development existed, and Christianity alone was\nneeded to crown the result with ideal perfection.]\n\n[Footnote 39: Max Muller, \"Science of Language,\" p. 404, 2d series.]\n\nHere the most perfect ideals of beauty and excellence in physical\ndevelopment, in manners, in plastic art, in literary creations, were\nrealized. The songs of Homer, the dialogues of Plato, the speeches of\nDemosthenes, and the statues of Phidias, if not unrivalled, are at least\nunsurpassed by any thing that has been achieved by their successors.\nLiterature in its most flourishing periods has rekindled its torch at\nher altars, and art has looked back to the age of Pericles for her\npurest models. Here the ideas of personal liberty, of individual rights,\nof freedom in thought and action, had a wonderful expansion. Here the\nlasting foundations of the principal arts and sciences were laid, and in\nsome of them triumphs were achieved which have not been eclipsed. Here\nthe sun of human reason attained a meridian splendor, and illuminated\nevery field in the domain of moral truth. And here humanity reached the\nhighest degree of civilization of which it is capable under purely\n_natural_ conditions.\n\nAnd now, the question with which we are more immediately concerned is,\nwhat were the specific and valuable results attained by the Athenian\nmind in _religion_ and _philosophy_, the two momenta of the human mind?\nThis will be the subject of discussion in subsequent chapters.\n\nThe order in which the discussion shall proceed is determined for us by\nthe natural development of thought. The two fundamental momenta of\nthought and its development are spontaneity and reflection, and the two\nessential forms they assume are religion and philosophy. In the natural\norder of thought spontaneity is first, and reflection succeeds\nspontaneous thought. And so religion is first developed, and\nsubsequently comes philosophy. As religion supposes spontaneous\nintuition, so philosophy has religion for its basis, but upon this basis\nit is developed in an original manner. \"Turn your attention to history,\nthat living image of thought: everywhere you perceive religions and\nphilosophies: everywhere you see them produced in an invariable order.\nEverywhere religion appears with new societies, and everywhere, just so\nfar as societies advance, from religion springs philosophy.\"[40] This\nwas pre-eminently the case in Athens, and we shall therefore direct our\nattention first to the Religion of the Athenians.\n\n[Footnote 40: Cousin, \"Hist. of Philos.,\" vol. i. p. 302.]\n\n\n\n\nCHAPTER II.\n\nTHE PHILOSOPHY OF RELIGION.\n\n\n      All things which I behold bear witness to your carefulness\n      in religion deisidaimonesterois.--ST. PAUL.\n\nAs a prelude and preparation for the study of the religion of the\nAthenians, it may be well to consider religion in its more abstract and\nuniversal form; and inquire in what does religion essentially consist;\nhow far is it grounded in the nature of man; and especially, what is\nthere in the mental constitution of man, or in his exterior conditions,\nwhich determines him to a mode of life which may be denominated\n_religious?_ As a preliminary inquiry, this may materially aid us in\nunderstanding the nature, and estimating the value of the religious\nconceptions and sentiments which were developed by the Greek mind.\n\nReligion, in its most generic conception, may be defined as a form of\nthought, feeling, and action, which has the _Divine_ for its object,\nbasis, and end. Or, in other words, it is a mode of life determined by\nthe recognition of some relation to, and consciousness of dependence\nupon, a _Supreme Being_. This general conception of religion underlies\nall the specific forms of religion which have appeared in the world,\nwhether heathen, Jewish, Mohammedan, or Christian.\n\nThat a religious destination appertains to man as man, whether he has\nbeen raised to a full religious consciousness, or is simply considered\nas capable of being so raised, can not be denied. In all ages man has\nrevealed an instinctive tendency, or natural aptitude for religion, and\nhe has developed feelings and emotions which have always characterized\nhim as a religious being. Religious ideas and sentiments have prevailed\namong all nations, and have exerted a powerful influence on the entire\ncourse of human history. Religious worship, addressed to a Supreme Being\nbelieved to control the destiny of man, has been coeval and coextensive\nwith the race. Every nation has had its mythology, and each mythologic\nsystem has been simply an effort of humanity to realize and embody in\nsome visible form the relations in which it feels itself to be connected\nwith an external, overshadowing, and all-controlling Power and Presence.\nThe voice of all ancient, and all contemporaneous history, clearly\nattests that the _religious principle_ is deeply seated in the nature of\nman; and that it has occupied the thought, and stirred the feelings of\nevery rational man, in every age. It has interwoven itself with the\nentire framework of human society, and ramified into all the relations\nof human life. By its agency, nations have been revolutionized, and\nempires have been overthrown; and it has formed a mighty element in all\nthe changes which have marked the history of man.\n\nThis universality of religious sentiment and religious worship must be\nconceded as a fact of human nature, and, as a universal fact, it demands\nan explanation. Every event must have a cause. Every phenomenon must\nhave its ground, and reason, and law. The facts of religious history,\nthe past and present religious phenomena of the world can be no\nexception to this fundamental principle; they press their imperious\ndemand to be studied and explained, as much as the phenomena of the\nmaterial or the events of the moral world. The phenomena of religion,\nbeing universally revealed wherever man is found, must be grounded in\nsome universal principle, on some original law, which is connate with,\nand natural to man. At any rate, there must be something in the nature\nof man, or in the exterior conditions of humanity, which invariably\nleads man to worship, and which determines him, as by the force of an\noriginal instinct, or an outward, conditioning necessity, to recognize\nand bow down before a Superior Power. The full recognition and adequate\nexplanation of the facts of religious history will constitute a\n_philosophy of religion_.\n\nThe hypotheses which have been offered in explanation of the religious\nphenomena of the world are widely divergent, and most of them are, in\nour judgment, eminently inadequate and unsatisfactory. The following\nenumeration may be regarded as embracing all that are deemed worthy of\nconsideration.\n\nI. The phenomenon of religion had its origin in SUPERSTITION, that is,\nin a _fear_ of invisible and supernatural powers, generated by ignorance\nof nature.\n\nII. The phenomenon of religion is part of that PROCESS or EVOLUTION OF\nTHE ABSOLUTE (i.e., the Deity), which gradually unfolding itself in\nnature, mind, history, and _religion_, attains to perfect\nself-consciousness in philosophy.\n\nIII. The phenomenon of religion has its foundation in FEELING--_the\nfeeling of dependence and of obligation_; and that to which the mind, by\nspontaneous intuition or instinctive faith, traces this dependence and\nobligation we call God.\n\nIV. The phenomenon of religion had its outbirth in the spontaneous\napperceptions of REASON, that is, the necessary _a priori ideas of the\nInfinite, the Perfect, the Unconditioned Cause, the Eternal Being_,\nwhich are evoked into consciousness in presence of the changeful and\ncontingent phenomena of the world.\n\nV. The phenomenon of religion had its origin in EXTERNAL REVELATION, to\nwhich _reason_ is related as a purely passive organ, and _heathenism_ as\na feeble relic.\n\nAs a philosophy of religion--an attempt to supply the rationale of the\nreligious phenomena of the world, the first hypothesis is a skeptical\nphilosophy, which necessarily leads to _Atheism_. The second is an\nidealistic philosophy (absolute idealism), which inevitably lands in\n_Pantheism_. The third is an intuitional or \"faith-philosophy,\" which\nfinally ends in _Mysticism_. The fourth is a rationalistic or\n\"spiritualistic\" philosophy, which yields pure _Theism_. The last is an\nempirical philosophy, which derives all religion from instruction, and\nculminates in _Dogmatic Theology_.\n\nIn view of these diverse and conflicting theories, the question which\nnow presents itself for our consideration is,--does any one of these\nhypotheses meet and satisfy the demands of the problem? does it fully\naccount for and adequately explain all the facts of religious history?\nThe answer to this question must not be hastily or dogmatically given.\nThe arbitrary rejection of any theory that may be offered, without a\nfair and candid examination, will leave our minds in uncertainty and\ndoubt as to the validity of our own position. A blind faith is only one\nremove from a pusillanimous skepticism. We can not render our own\nposition secure except by comprehending, assaulting, and capturing the\nposition of our foe. It is, therefore, due to ourselves and to the cause\nof truth, that we shall examine the evidence upon which each separate\ntheory is based, and the arguments which are marshalled in its support,\nbefore we pronounce it inadequate and unphilosophical. Such a criticism\nof opposite theories will prepare the way for the presentation of a\nphilosophy of religion which we flatter ourselves will be found most in\nharmony with all the facts of the case.\n\nI. _It is affirmed that the religious phenomena of the world had their\norigin in_ SUPERSTITION, _that is, in a fear of unseen and supernatural\npowers, generated from ignorance of nature_.\n\nThis explanation was first offered by Epicurus. He felt that the\nuniversality of the religious sentiment is a fact which demands a cause;\nand he found it, or presumed he found it not in a spiritual God, which\nhe claims can not exist, nor in corporeal god which no one has seen, but\nin \"phantoms of the mind generated by fear.\" When man has been unable to\nexplain any natural phenomenon, to assign a cause within the sphere of\nnature, he has had recourse to supernatural powers, or living\npersonalities behind nature, which move and control nature in an\narbitrary and capricious manner. These imaginary powers are supposed to\nbe continually interfering in the affairs of individuals and nations.\nThey bestow blessings or inflict calamities. They reward virtue and\npunish vice. They are, therefore, the objects of \"sacred awe\" and\n\"superstitious fear.\"\n\n                           Whate'er in heaven,\n     In earth, man sees mysterious, shakes his mind\n     With sacred awe o'erwhelms him, and his soul\n     Bows to the dust; the cause of things conceal\n     Once from his vision, instant to the gods\n     All empire he transfers, all rule supreme,\n     And doubtful whence they spring, with headlong haste\n     Calls them the workmanship of power divine.\n     For he who, justly, deems the Immortals live\n     Safe, and at ease, yet fluctuates in his mind\n     How things are swayed; how, chiefly, those discerned\n     In heaven sublime--to SUPERSTITION back\n     Lapses, and fears a tyrant host, and then\n     Conceives, dull reasoner, they can all things do,\n     While yet himself nor knows what may be done,\n     Nor what may never, nature powers defined\n     Stamping on all, and bounds that none can pass:\n     Hence wide, and wider errs he as he walks.[41]\n\n[Footnote 41: Lucretius, \"De Natura Rerum,\" book vi. vs. 50-70.]\n\nIn order to rid men of all superstitious fear, and, consequently, of all\nreligion, Epicurus endeavors to show that \"nature\" alone is adequate to\nthe production of all things, and there is no need to drag in a \"divine\npower\" to explain the phenomena of the world.\n\nThis theory has been wrought into a somewhat plausible form by the\nbrilliant and imposing generalizations of Aug. Comte. The religious\nphenomena of the world are simply one stage in the necessary development\nof mind, whether in the individual or the race. He claims to have been\nthe first to discover the great law of the three successive stages or\nphases of human evolution. That law is thus enounced. Both in the\nindividual mind, and in the history of humanity, thought, in dealing\nwith its problems, passes, of necessity, through, first, a\n_Theological_, second, a _Metaphysical_, and finally reaches a third, or\n_Positive_ stage.\n\nIn attempting an explanation of the universe, human thought, in its\nearliest stages of development, resorts to the idea of living personal\nagents enshrined in and moving every object, whether organic or\ninorganic, natural or artificial. In an advanced stage, it conceives a\nnumber of personal beings distinct from, and superior to nature, which\npreside over the different provinces of nature--the sea, the air, the\nwinds, the rivers, the heavenly bodies, and assume the guardianship of\nindividuals, tribes, and nations. As a further, and still higher stage,\nit asserts the unity of the Supreme Power which moves and vitalizes the\nuniverse, and guides and governs in the affairs of men and nations. The\n_Theological_ stage is thus subdivided into three epochs, and\nrepresented as commencing in _Fetichism_, then advancing to\n_Polytheism_, and, finally, consummating in _Monotheism_.\n\nThe next stage, the _Metaphysical_, is a transitional stage, in which\nman substitutes abstract entities, as substance, force, Being _in se_,\nthe Infinite, the Absolute, in the place of theological conceptions.\nDuring this period all theological opinions undergo a process of\ndisintegration, and lose their hold on the mind of man. Metaphysical\nspeculation is a powerful solvent, which decomposes and dissipates\ntheology.\n\nIt is only in the last--the _Positive_ stage--that man becomes willing\nto relinquish all theological ideas and metaphysical notions, and\nconfine his attention to the study of phenomena in their relation to\ntime and space; discarding all inquiries as to causes, whether efficient\nor final, and denying the existence of all entities and powers beyond\nnature.\n\nThe first stage, in its religious phase, is _Theistic_, the second is\n_Pantheistic_, the last is _Atheistic_.\n\nThe proofs offered by Comte in support of this theory are derived:\n\nI. _From Cerebral Organization_. There are three grand divisions of the\nBrain, the Medulla Oblongata, the Cerebellum, and the Cerebrum; the\nfirst represents the merely animal instincts the second, the more\nelevated sentiments, the third, the intellectual powers. Human nature\nmust, therefore, both in the individual and in the race, be developed in\nthe following order: (1.) in animal instincts; (2.) in social affections\nand communal tendencies; (3.) in intellectual pursuits. Infant life is a\nmerely animal existence, shared in common with the brute; in childhood\nthe individual being realizes his relation to external nature and human\nsociety; in youth and manhood he compares, generalizes, and classifies\nthe objects of knowledge, and attains to science. And so the infancy of\nour race was a mere animal or savage state, the childhood of our race\nthe organization of society, the youth and manhood of our race the\ndevelopment of science.\n\nNow, without offering any opinion as to the merits of the phrenological\ntheories of Gall and Spurzheim, we may ask, what relation has this order\nto the law of development presented by Comte? Is there any imaginable\nconnection between animal propensities and theological ideas; between\nsocial affections and metaphysical speculations? Are not the\nintellectual powers as much concerned with theological ideas and\nmetaphysical speculations as with positive science? And is it not more\nprobable, more in accordance with facts, that all the powers of the\nmind, instinct, feeling, and thought, enter into action simultaneously,\nand condition each other? The very first act of perception, the first\ndistinct cognition of an object, involves _thought_ as much as the last\ngeneralization of science. We know nothing of _mind_ except as the\ndevelopment of thought, and the first unfolding, even of the infant\nmind, reveals an intellectual act, a discrimination between a self and\nan object which is not self, and a recognition of resemblance, or\ndifference between _this_ object and _that_. And what does Positive\nscience, in its most mature and perfect form, claim to do more than \"to\nstudy actual phenomena in their orders of resemblance, coexistence, and\nsuccession.\"\n\nCerebral organization may furnish plausible analogies in favor of some\ntheory of human development, but certainly not the one proposed by Aug.\nComte. The attempt, however, to construct a chart of human history on\nsuch an _a priori_ method,--to construct an ideal framework into which\nhuman nature must necessarily grow, is a violation of the first and most\nfundamental principle of the Positive science, which demands that we\nshall confine ourselves strictly to the study of actual phenomena in\ntheir orders of resemblance, coexistence, and succession. The history of\nthe human race must be based on facts, not on hypotheses, and the facts\nmust be ascertained by the study of ancient records and existing\nmonuments of the past. Mere plausible analogies and _a priori_ theories\nbased upon them, are only fitted to mislead the mind; they insert a\nprism between the perceiving mind and the course of events which\ndecomposes the pure white light of fact, and throws a false light over\nthe entire field of history.\n\n2. _The second order of proof is attempted to be drawn from the\nanalogies of individual experience_.\n\nIt is claimed that the history of the race is the same as that of each\nindividual mind; and it is affirmed that man is _religious_ in infancy,\n_metaphysical_ in youth, and _positive_, that is, scientific without\nbeing religious, in mature manhood; the history of the race must\ntherefore have followed the same order.\n\nWe are under no necessity of denying that there is some analogy between\nthe development of mind in the individual man, and in humanity as a\nwhole, in order to refute the theory of Comte. Still, it must not be\noverlooked that the development of mind, in all cases and in all ages,\nis materially affected by exterior conditions. The influence of\ngeographical and climatic conditions, of social and national\ninstitutions, and especially of education, however difficult to be\nestimated, can not be utterly disregarded. And whether all these\ninfluences have not been controlled, and collocated, and adjusted by a\nSupreme Mind in the education of humanity, is also a question which can\nnot be pushed aside as of no consequence. Now, unless it can be shown\nthat the same outward conditions which have accompanied the individual\nand modified his mental development, have been repealed in the history\nof the race, and repeated in the same order of succession, the argument\nhas no value.\n\nBut, even supposing it could be shown that the development of mind in\nhumanity has followed the same order as that of the individual, we\nconfidently affirm that Comte has not given the true history of the\ndevelopment of the individual mind. The account he has given may perhaps\nbe the history of his own mental progress, but it certainly is not the\nhistory of every individual mind, nor indeed, of a majority even, of\neducated minds that have arrived at maturity. It would be much more in\nharmony with facts to say childhood is the period of pure receptivity,\nyouth of doubt and skepticism, and maturity of well-grounded and\nrational belief. In the ripeness and maturity of the nineteenth century\nthe number of scientific men of the Comtean model is exceedingly small\ncompared with the number of religious men. There are minds in every part\nof Europe and America as thoroughly scientific as that of Comte, and as\ndeeply imbued with the spirit of the Inductive Philosophy, which are not\nconscious of any discordance between the facts of science and the\nfundamental principles of theology. It may be that, in his own immediate\ncircle at Paris there may be a tendency to Atheism, but certainly no\nsuch tendency exists in the most scientific minds of Europe and America.\nThe faith of Bacon, and Newton, and Boyle, of Descartes, Leibnitz, and\nPascal, in regard to the fundamental principles of theology, is still\nthe faith of Sedgwick, Whewell, Herschel, Brewster, Owen, Agassiz,\nSilliman, Mitchell, Hitchcock, Dana, and, indeed of the leading\nscientific minds of the world--the men who, as Comte would say, \"belong\nto the elite of humanity.\" The mature mind, whether of the individual or\nthe race, is not Atheistical.\n\n3. _The third proof is drawn from a survey of the history of certain\nportions of our race._\n\nComte is far from being assured that the progress of humanity, under the\noperation of his grand law of development, has been uniform and\ninvariable. The majority of the human race, the vast populations of\nIndia, China, and Japan, have remained stationary; they are still in the\nTheological stage, and consequently furnish no evidence in support of\nhis theory. For this reason he confines himself to the \"elite\" or\nadvance-guard of humanity, and in this way makes the history of humanity\na very \"abstract history\" indeed. Starting with Greece as the\nrepresentative of ancient civilization, passing thence to Roman\ncivilization, and onward to Western Europe, he attempts to show that the\nactual progress of humanity has been, on the whole, in conformity with\nhis law. To secure, however, even this semblance of harmony between the\nfacts of history and his hypothetical law, he has to treat the facts\nvery much as Procrustes treated his victims,--he must stretch some, and\nmutilate others, so as to make their forms fit the iron bed. The natural\norganization of European civilization is distorted and torn asunder. \"As\nthe third or positive stage had accomplished its advent in his own\nperson, it was necessary to find the metaphysical period just before;\nand so the whole life of the Reformed Christianity, in embryo and in\nmanifest existence, is stripped of its garb of _faith_, and turned out\nof view as a naked metaphysical phenomenon. But metaphysics, again, have\nto be ushered in by theology; and of the three stages of theology\nMonotheism is the last, necessarily following on Polytheism, as that,\nagain, on Fetichism. There is nothing for it, therefore, but to let the\nmediaeval Catholic Christianity stand as the world's first monotheism,\nand to treat it as the legitimate offspring and necessary development of\nthe Greek and Roman polytheism. This, accordingly, Comte actually does.\nProtestantism he illegitimates, and outlaws from religion altogether,\nand the genuine Christianity he fathers upon the faith of Homer and the\nScipios! Once or twice, indeed, it seems to cross him that there was\nsuch a people as the Hebrews, and that they were not the polytheists\nthey ought to have been. He sees the fact, but pushes it out of his way\nwith the remark that the Jewish monotheism was 'premature.'\"[42]\n\n[Footnote 42: Martineau's Essays, pp. 61, 62.]\n\nThe signal defect of Comte's historical survey, however, is, that it\nfurnishes no evidence of the general prevalence of Fetichism in\nprimitive times. The writings of Moses are certainly entitled to as much\nconsideration and credence as the writings of Berosus, Manetho, and\nHerodotus; and, it will not be denied, they teach that the faith of the\nearliest families and races of men was _monotheistic_. The early Vedas,\nthe Institutes of Menu, the writings of Confucius, the Zendavesta, all\nbear testimony that the ancient faith of India, China, and Persia, was,\nat any rate, pantheistic; and learned and trustworthy critics, Asiatic\nas well as European, confidently affirm that the ground of the\nBrahminical, Buddhist, and Parsist faith is _monotheistic_; and that\n_one_ Being is assumed, in the earliest books, to be the origin of all\nthings.[43] Without evidence, Comte assumes that the savage state is the\noriginal condition of man; and instead of going to Asia, the cradle of\nthe race, for some light as to the early condition and opinions of the\nremotest families of men, he turns to Africa, the _soudan_ of the earth,\nfor his illustration of the habit of man, in the infancy of our race, to\nendow every object in nature, whether organic or inorganic, with life\nand intelligence. The theory of a primitive state of ignorance and\nbarbarism is a mere assumption--an hypothesis in conflict with the\ntraditionary legends of all nations, the earliest records of our race,\nand the unanimous voice of antiquity, which attest the general belief in\na primitive state of light and innocence.\n\n[Footnote 43: \"The Religions of the World in their Relation to\nChristianity\" (Maurice, ch. ii., iii., iv.).]\n\nThe three stages of development which Comte describes as necessarily\nsuccessive, have, for centuries past, been simultaneous. The\ntheological, the metaphysical, and the scientific elements coexist now,\nand there is no real, radical, or necessary conflict between them.\nTheological and metaphysical ideas hold their ground as securely under\nthe influence of enlarged scientific discovery as before; and there is\nno reason to suppose they ever had more power over the mind of man than\nthey have to-day. The notion that God is dethroned by the wonderful\ndiscoveries of modern science, and theology is dead, is the dream of the\n\"_profond orage cerebral_\" which interrupted the course of Comte's\nlectures in 1826. As easily may the hand of Positivism arrest the course\nof the sun, as prevent the instinctive thought of human reason\nrecognizing and affirming the existence of a God. And so long as ever\nthe human mind is governed by necessary laws of thought, so long will it\nseek...\n\n[Transcriber's note: In the original document, page 64 is a duplication\nof page 63. The real page 64 seems to be missing.]\n\n....eur, and consequently to develop its true philosophy. Its\nfundamental error is the assumption that all our knowledge is confined\nto the observation and classification of sensible phenomena--that is, to\nchanges perceptible by the senses. Psychology, based, as it is, upon\nself-observation and self-reflection, is a \"mere illusion; and logic and\nethics, so far as they are built upon it as their foundation, are\naltogether baseless.\" Spiritual entities, forces, causes, efficient or\nfinal, are unknown and unknowable; all inquiry regarding them must be\ninhibited, \"for Theology is inevitable if we permit the inquiry into\ncauses at all.\"\n\nII. The second hypothesis offered in explanation of the facts of\nreligious history is, _that religion is part of that_ PROCESS OR\nEVOLUTION OF THE ABSOLUTE (_i.e._, the Deity) _which, gradually\nunfolding itself in nature, mind, history, and religion, attains to the\nfullest self-consciousness in philosophy_.\n\nThis is the theory of Hegel, in whose system of philosophy the\nsubjective idealism of Kant culminates in the doctrine of \"_Absolute\nIdentity_.\" Its fundamental position is that thought and being, subject\nand object, the perceiving mind and the thing perceived, are ultimately\nand essentially _one_, and that the only actual reality is that which\nresults from their mutual relation. The outward thing is nothing, the\ninward perception is nothing, for neither could exist alone; the only\nreality is the relation, or rather synthesis of the two; the essence or\nnature of being in itself accordingly consists in the coexistence of two\ncontrarieties. Ideas, arising from the union or synthesis of two\nopposites, are therefore the _concrete realities_ of Hegel; and the\n_process_ of the evolution of ideas, in the human mind, is the process\nof all existence--_the Absolute Idea_.\n\n_The Absolute_(die Idee) thus forms the beginning, middle, and end of\nthe system of Hegel. It is the one infinite existence or thought, of\nwhich nature, mind, history, religion, and philosophy, are the\nmanifestation. \"The absolute is, with him, not the infinite _substance_,\nas with Spinoza; nor the infinite _subject_, as with Fichte; nor the\ninfinite _mind_, as with Schelling; it is a perpetual _process_, an\neternal thinking, without beginning and without end.\"[44] This _living,\neternal process of absolute existence is the God of Hegel_.\n\nIt will thus be seen that the _Absolute_ is, with Hegel, the sum of all\nactual and possible existence; \"nothing is true and real except so far\nas it forms an element of the Absolute Spirit.\"[45] \"What kind of an\nAbsolute Being,\" he asks, \"is that which does not contain in itself all\nthat is actual, even evil included?\"[46] The Absolute, therefore, in\nHegel's conception, does not allow of any existence out of itself. It is\nthe _unity_ of the finite and the infinite, the eternal and the\ntemporal, the ideal and the real, the subject and the object. And it is\nnot only the unity of these opposites so as to exclude all difference,\nbut it contains in itself, all the differences and opposites as elements\nof its being; otherwise the distinctions would stand over against\nabsolute as a limit, and the absolute would cease to be absolute.\n\nGod is, therefore, according to Hegel, \"no motionless, eternally\nself-identical and unchangeable being, but a living, eternal _process_\nof absolute self-existence. This process consists in the eternal\nself-distinction, or antithesis, and equally self-reconciliation or\nsynthesis of those opposites which enter, as necessary elements, into\nthe constitution of the Divine Being. This _self-evolution_, whereby the\nabsolute enters into antithesis, and returns to itself again, is the\neternal _self-actualization_ of its being, and which at once constitutes\nthe beginning, middle, and end, as in the circle, where the beginning is\nat the same time the end, and the end the beginning.\"[47]\n\n[Footnote 44: Morell, \"Hist, of Philos., p. 461.\"]\n\n[Footnote 45: \"Philos. of Religion,\" p. 204.]\n\n[Footnote 46: Ibid., chap. xi. p. 24.]\n\n[Footnote 47: Herzog's _Real-Encyc._, art. \"Hegelian Philos.,\" by\nUlrici.]\n\nThe whole philosophy of Hegel consists in the development of this idea\nof God by means of his, so-called, dialectic method, which reflects the\nobjective life-process of the Absolute, and is, in fact, identical with\nit; for God, says he, \"is only the Absolute Intelligence in so far as he\nknows himself to be the Absolute Intelligence, _and this he knows only\nin science_ [dialectics], _and this knowledge alone constitutes his true\nexistence._\"[48] This life-process of the Absolute has three \"moments.\"\nIt may be considered as the idea _in itself_--bare, naked, undetermined,\nunconscious idea; as the idea _out of itself_, in its objective form, or\nin its differentiation; and, finally, as the idea _in itself_, and _for\nitself_, in its regressive or reflective form. This movement of thought\ngives, _first_, bare, naked, indeterminate thought, or thought in the\nmere antithesis of Being and non-Being; _secondly_, thought\nexternalizing itself in nature; and, _thirdly_, thought returning to\nitself, and knowing itself in mind, or consciousness. Philosophy has,\naccordingly, three corresponding divisions:--1. LOGIC, which here is\nidentical with metaphysics; 2. PHILOSOPHY OF NATURE; 3. PHILOSOPHY OF\nMIND.\n\n[Footnote 48: \"Hist, of Philos.,\" iii. p. 399.]\n\nIt is beyond our design to present an expanded view of the entire\nphilosophy of Hegel. But as he has given to the world a _new_ logic, it\nmay be needful to glance at its general features as a help to the\ncomprehension of his philosophy of religion. The fundamental law of his\nlogic is the _identity of contraries or contradictions_. All thought is\na synthesis of contraries or opposites. This antithesis not only exists\nin all ideas, but constitutes them. In every idea we form, there must be\n_two_ things opposed and distinguished, in order to afford a clear\nconception. Light can not be conceived but as the opposite of darkness;\ngood can not be thought except in opposition to evil. All life, all\nreality is thus, essentially, the union of two elements, which,\ntogether, are mutually opposed to, and yet imply each other.\n\nThe identity of Being and Nothing is one of the consequences of this\nlaw.\n\n1. _The Absolute is the Being_ (das Absolute ist das Seyn), and \"the\nBeing\" is here, according to Hegel, bare, naked, abstract,\nundistinguished, indeterminate, unconscious idea.\n\n2. _The Absolute is the Nothing_ (das Absolute ist das Nichts). \"Pure\nbeing is pure abstraction, and consequently the absolute-negative, which\nin like manner, directly taken, is _nothing_.\" Being and Nothing are the\npositive and negative poles of the Idea, that is, the Absolute. They\nboth alike exist, they are both pure abstractions, both absolutely\nunconditioned, without attributes, and without consciousness. Hence\nfollows the conclusion--\n\n3. _Being and Nothing are identical_ (das Seyn und das Nichts ist\ndasselbe), Being is non-Being. Non-Being _is_ Being--the\nAnders-seyn--which becomes _as_ Being to the Seyn. Nothing is, in some\nsense, an actual thing.\n\n_Being_ and _Nothing_ are thus the two elements which enter into the one\nAbsolute Idea as contradictories, and both together combine to form a\ncomplete notion of bare production, or the _becoming_ of something out\nof nothing,--the unfolding of real existence in its lowest form, that\nis, of _nature_.\n\nThe \"_Philosophy of Nature_\" exhibits a series of necessary movements\nwhich carry the idea forward in the ascending scale of sensible\nexistence. The laws of mechanics, chemistry, and physiology are resolved\ninto a series of oppositions. But the law which governs this development\nrequires the self-reconciliation of these opposites. The idea,\ntherefore, which in nature was unconscious and ignorant of itself,\nreturns upon itself, and becomes conscious of itself, that is, becomes\n_mind_. The science of the regression or self-reflection of the idea, is\nthe \"_Philosophy of Mind_.\"\n\nThe \"_Philosophy of Mind_\" is subdivided by Hegel into three parts.\nThere is, first, the subjective or individual mind (_psychology_); then\nthe objective or universal mind, as represented in society, the state,\nand in history (_ethics, political philosophy,_ or _jurisprudence_, and\n_philosophy of history_); and, finally, the union of the subjective and\nobjective mind, or _the absolute mind_. This last manifests itself again\nunder three forms, representing the three degrees of the\nself-consciousness of the Spirit, as the eternal truth. These are,\nfirst, _art_, or the representation of beauty (aesthetics); secondly,\n_religion_, in the general acceptation of the term (philosophy of\nreligion); and, thirdly, _philosophy_ itself, as the purest and most\nperfect form of the scientific knowledge of truth. All historical\nreligions, the Oriental, the Jewish, the Greek, the Roman, and the\nChristian, are _the successive stages in the development or\nself-actualization of God_.[49]\n\nIt is unnecessary to indicate to the reader that the philosophy of Hegel\nis essentially pantheistic. \"God is not a _person_, but personality\nitself, _i.e._, the universal personality, which realizes itself in\nevery human consciousness, as so many separate thoughts of one eternal\nmind. The idea we form of the absolute is, to Hegel, the absolute\nitself, its essential existence being identical with our conception of\nit. Apart from, and out of the world, there is no God; and so also,\napart from the universal consciousness of man, there is no Divine\nconsciousness or personality.\"[50]\n\n[Footnote 49: See art. \"Hegelian Philosophy,\" in Herzog's _Real-Encyc._,\nfrom whence our materials are chiefly drawn.]\n\n[Footnote 50: Morell, \"Hist. of Philos.,\" p. 473.]\n\nThis whole conception of religion, however, is false, and conflicts with\nthe actual facts of man's religious nature and religious history. If the\nword \"religion\" has any meaning at all, it is \"a mode of life determined\nby the consciousness of dependence upon, and obligation to God.\" It is\nreverence for, gratitude to, and worship of God as a being distinct from\nhumanity. But in the philosophy of Hegel religion is a part of God--a\nstage in the development or self-actualization of God. Viewed under one\naspect, religion is the self-adoration of God--the worship of God by\nGod; under another aspect it is the worship of humanity, since God only\nbecomes conscious of himself in humanity. The fundamental fallacy is\nthat upon which his entire method proceeds, viz., \"the identity of\nsubject and object, being and thought.\" Against this false position the\nconsciousness of each individual man, and the universal consciousness of\nour race, as revealed in history, alike protest. If thought and being\nare identical, then whatever is true of ideas is also true of objects,\nand then, as Kant had before remarked, there is no difference between\n_thinking_ we possess a hundred dollars, and actually _possessing_ them.\nSuch absurdities may be rendered plausible by a logic which asserts the\n\"identity of contradictions,\" but against such logic common sense\nrebels. \"The law of non-contradiction\" has been accepted by all\nlogicians, from the days of Aristotle, as a fundamental law of thought.\n\"Whatever is contradictory is unthinkable. A=not A=O, or A--A=O.\"[51]\nNon-existence can not exist. Being can not be nothing.\n\n[Footnote 51: Hamilton's Logic, p. 58.]\n\nIII. The third hypothesis affirms _that the phenomenon of religion has\nits foundation in_ FEELING--_the feeling of dependence and of\nobligation_; and that to which the mind, by spontaneous intuition of\ninstinctive faith, traces that dependence and obligation we call God.\n\nThis, with some slight modification in each case, consequent upon the\ndifferences in their philosophic systems, is the theory of Jacobi,\nSchleiermacher, Nitzsch, Mansel, and probably Hamilton. Its fundamental\nposition is, that we can not gain truth with absolute certainty either\nfrom sense or reason, and, consequently, the only valid source of real\nknowledge is _feeling--faith, intuition_, or, as it is called by some,\n_inspiration_.\n\nThere have been those, in all ages, who have made all knowledge of\ninvisible, supersensuous, divine things, to rest upon an internal\n_feeling_, or immediate, inward vision. The Oriental Mystics, the\nNeo-Platonists, the Mystics of the Greek and Latin Church, the German\nMystics of the 14th century, the Theosophists of the Reformation, the\nQuietists of France, the Quakers, have all appealed to some _special_\nfaculty, distinct from the understanding and reason, for the immediate\ncognition of invisible and spiritual existences. By some, that special\nfaculty was regarded as an \"interior eye\" which was illuminated by the\n\"Universal Light;\" by others, as a peculiar sensibility of the soul--a\n_feeling_ in whose perfect calm and utter quiescence the Divinity was\nmirrored; or which, in an ecstatic state, rose to a communion with, and\nfinal absorption in the Infinite.\n\nJacobi was the first, in modern times, to give the \"faith-philosophy,\"\nas it is now designated, a definite form. He assumes the position that\nall knowledge, of whatever kind, must ultimately rest upon intuition or\nfaith. As it regards sensible objects, the understanding finds the\nimpression from which all our knowledge of the external flows, ready\nformed. The process of sensation is a mystery; we know nothing of it\nuntil it is past, and the feeling it produces is present. Our knowledge\nof matter, therefore, rests upon faith in these intuitions. We can not\ndoubt that the feeling has an objective cause. In every act of\nperception there is something actual and present, which can not be\nreferred to a mere subjective law of thought. We are also conscious of\nanother class of feelings which correlate us with a supersensuous world,\nand these feelings, also, must have their cause in some objective\nreality. Just as sensation gives us an immediate knowledge of an\nexternal world, so there is an internal sense which gives us an\nimmediate knowledge of a spiritual world--God, the soul, freedom,\nimmortality. Our knowledge of the invisible world, like our knowledge of\nthe visible world, is grounded upon faith in our intuitions. All\nphilosophic knowledge is thus based upon _belief_, which Jacobi regards\nas a fact of our inward sensibility--a sort of knowledge produced by an\nimmediate _feeling_ of the soul--a direct apprehension, without proof,\nof the True, the Supersensuous, the Eternal.\n\nJacobi prepared the way for, and was soon eclipsed by the deservedly\ngreater name of Schleiermacher. His fundamental position was that truth\nin Theology could not be obtained by reason, but by a feeling,\n_insight_, or intuition, which in its lowest form he called\n_God-consciousness_, and in its highest form, _Christian-consciousness._\nThe God-consciousness, in its original form, is the _feeling of\ndependence_ on the Infinite. The Christian consciousness is the perfect\nunion of the human consciousness with the Divine, through the mediation\nof Christ, or what we would call a Christian experience of communion\nwith God.\n\nRightly to understand the position of Schleiermacher we must take\naccount of his doctrine of _self_-consciousness. \"In all\nself-consciousness,\" says he, \"there are two elements, a Being ein Seyn,\nand a Somehow-having-become (Irgendweigewordenseyn). The last, however,\npresupposes, for every self-consciousness, besides the ego, yet\nsomething else from whence the certainty of the same\n[self-consciousness] exists, and without which self-consciousness would\nnot be just this.\"[52] Every determinate mode of the sensibility\nsupposes an _object_, and a _relation_ between the subject and the\nobject, the subjective feeling deriving its determinations from the\nobject. External sensation, the feeling, say of extension and\nresistance, gives world-consciousness. Internal sensation, the _feeling\nof dependence_, gives God consciousness. And it is only by the presence\nof world consciousness and God-consciousness that self consciousness can\nbe what it is.\n\nWe have, then, in our self-consciousness a _feeling of direct\ndependence_, and that to which our minds instinctively trace that\ndependence we call God. \"By means of the religious feeling, the Primal\nCause is revealed in us, as in perception, the things external, are\nrevealed in us.\"[53] The _felt_, therefore, is not only the first\nreligious sense, but the ruling, abiding, and perfect form of the\nreligious spirit; whatever lays any claim to religion must maintain its\nground and principle in _feeling_, upon which it depends for its\ndevelopment; and the sum-total of the forces constituting religious\nlife, inasmuch as it is a _life_, is based upon immediate\nself-consciousness.[54]\n\n[Footnote 52: Glaubenslehre, ch. i. Sec. 4.]\n\n[Footnote 53: Dialectic, p. 430.]\n\n[Footnote 54: Nitzsch, \"System of Doctrine,\" p. 23.]\n\nThe doctrine of Schleiermacher is somewhat modified by Mansel, in his\n\"_Limits of Religious Thought_.\" He maintains, with Schleiermacher, that\nreligion is grounded in _feeling_, and that the _felt_ is the first\nintimation or presentiment of the Divine. Man \"_feels_ within him the\nconsciousness of a Supreme Being, and the instinct to worship, before he\ncan argue from effects to causes, or estimate the traces of wisdom and\nbenevolence scattered through the creation.\"[55] He also agrees with\nSchleiermacher in regarding the _feeling of dependence_ as _a_ state of\nthe sensibility, out of which reflection builds up the edifice of\nReligious Consciousness, but he does not, with Schleiermacher, regard it\nas pre-eminently _the_ basis of religious consciousness. \"The mere\nconsciousness of dependence does not, of itself, exhibit the character\nof the Being on whom we depend. It is as consistent with superstition as\nwith religion; with the belief in a malevolent, as in a benevolent\nDeity.\"[56] To the feeling of dependence he has added the _consciousness\nof moral obligation_, which he imagines supplies the deficiency. By this\nconsciousness of moral obligation \"we are compelled to assume the\nexistence of a moral Deity, and to regard the absolute standard of right\nand wrong as constituted by the nature of that Deity.\"[57] \"To these two\nfacts of the inner consciousness the feeling of dependence, and\nconsciousness of moral obligation may be traced, as to their sources,\nthe two great outward acts by which religion, in its various forms, has\nbeen manifested among men--_Prayer_, by which they seek to win God's\nblessing upon the future, and _Expiation_, by which they strive to atone\nfor the offenses of the past. The feeling of dependence is the instinct\nwhich urges us to pray. It is the feeling that our existence and welfare\nare in the hands of a superior power; not an inexorable fate, not an\nimmutable law; but a Being having at least so far the attribute of\npersonality that he can show favor or severity to those who are\ndependent upon Him, and can be regarded by them with feelings of hope\nand fear, and reverence and gratitude.\"[58] The feeling of moral\nobligation--\"the law written in the heart\"--leads man to recognize a\nLawgiver. \"Man can be a law unto himself only on the supposition that he\nreflects in himself the law of God.\"[59] The conclusion from the whole\nis, there must be an _object_ answering to this consciousness: there\nmust be a God to explain these facts of the soul.\n\n[Footnote 55: Mansel, \"Limits of Religious Thought,\" p. 115.]\n\n[Footnote 56: Id., ib., p. 120.]\n\n[Footnote 57: Id., ib., p. 122.]\n\n[Footnote 58: Id., ib., pp. 119, 120.]\n\n[Footnote 59: Id., ib., p. 122.]\n\nThis \"philosophy of feeling,\" or of faith generated by feeling, has an\ninterest and a significance which has not been adequately recognized by\nwriters on natural theology. Feeling, sentiment, enthusiasm, have always\nplayed an important part in the history of religion. Indeed it must be\nconceded that religion is a _right state of feeling towards\nGod_--religion is _piety_. A philosophy of the religious emotion is,\ntherefore, demanded in order to the full interpretation of the religious\nphenomena of the world.\n\nBut the notion that internal feeling, a peculiar determination of the\nsensibility, is the source of religious ideas:--that God can be known\nimmediately by feeling without the mediation of the truth that manifests\nGod; that he can be _felt_ as the qualities of matter can be felt; and\nthat this affection of the inward sense can reveal the character and\nperfections of God, is an unphilosophical and groundless assumption. To\nassert, with Nitzsch, that \"feeling has reason, and is reason, and that\nthe sensible and felt God-consciousness generates out of itself\nfundamental conceptions,\" is to confound the most fundamental\npsychological distinctions, and arbitrarily bend the recognized\nclassifications of mental science to the necessities of a theory.\nIndeed, we are informed that it is \"by means of an _independent_\npsychology, and conformably to it,\" that Schleiermacher illustrates his\n\"philosophy of feeling.\"[60] But all psychology must be based upon the\nobservation and classification of mental phenomena, as revealed in\nconsciousness, and not constructed in an \"independent\" and a priori\nmethod. The most careful psychological analysis has resolved the whole\ncomplex phenomena of mind into thought, feeling, and volition.[61] These\norders of phenomena are radically and essentially distinct. They differ\nnot simply in degree but in kind, and it is only by an utter disregard\nof the facts of consciousness that they can be confounded. Feeling is\nnot reason, nor can it by any logical dexterity be transformed into\nreason.\n\n[Footnote 60: Nitzsch, \"System of Doctrine,\" p. 21.]\n\n[Footnote 61: Kant, \"Critique of Judg.,\" ch. xxii.; Cousin, \"Hist, of\nPhilos.,\" vol. ii. p. 399; Hamilton, vol. i. p. 183, Eng. ed.]\n\nThe question as to the relative order of cognition and feeling, that is,\nas to whether feeling is the first or original form of the religious\nconsciousness, or whether feeling be not consequent upon some idea or\ncognition of God, is one which can not be determined on empirical\ngrounds. We are precluded from all scrutiny of the incipient stages of\nmental development in the individual mind and in collective humanity. If\nwe attempt to trace the early history of the soul, its beginnings are\nlost in a period of blank unconsciousness, beyond all scrutiny of memory\nor imagination. If we attempt the inquiry on the wider field of\nuniversal consciousness, the first unfoldings of mind in humanity are\nlost in the border-land of mystery, of which history furnishes no\nauthentic records. All dogmatic affirmation must, therefore, be\nunjustifiable. The assertion that religious feeling precedes all\ncognition,--that \"the consciousness of dependence on a Supreme Being,\nand the instinct of worship\" are developed _first_ in the mind, before\nthe reason is exercised, is utterly groundless. The more probable\ndoctrine is that all the primary faculties enter into spontaneous action\n_simultaneously_--the reason with the senses, the feelings with the\nreason, the judgment with both the senses and the reason, and that from\ntheir primary and simultaneous action arises the complex result, called\nconsciousness, or conjoint knowledge.[62] There can be no clear and\ndistinct consciousness without the cognition of a _self_ and a\n_not-self_ in mutual relation and opposition. Now the knowledge of the\nself--the personal ego--is an intuition of reason; the knowledge of the\nnot-self is an intuition of sense. All knowledge is possible only under\ncondition of plurality, difference, and relation.[63] Now the judgment\nis \"the Faculty of Relations,\" or of comparison; and the affirmation\n\"_this_ is not _that_\" is an act of judgment; to know is, consequently,\nto judge.[64] Self-consciousness must, therefore, be regarded as a\nsynthesis of sense, reason, and judgment, and not a mere self-feeling\n(coenaesthesis).\n\n[Footnote 62: Cousin, \"Hist. of Philos.,\" vol. i. p. 357; vol. ii. p.\n337.]\n\n[Footnote 63: Id., ib., vol. i. p. 88.]\n\n[Footnote 64: Hamilton, \"Metaphys.,\" p. 277]\n\nA profound analysis will further lead to the conclusion that if ideas of\nreason are not chronologically antecedent to sensation, they are, at\nleast, the logical antecedents of all cognition. The mere feeling of\nresistance can not give the notion of without the a priori idea of\nspace. The feeling of movement of change, can not give the cognition of\nevent without the rational idea of time or duration. Simple\nconsciousness can not generate the idea of personality, or selfhood,\nwithout the rational idea of identity or unity. And so the mere \"feeling\nof dependence,\" of finiteness and imperfection, can not give the idea of\nGod, without the rational a priori idea of the Infinite, the Perfect,\nthe Unconditioned Cause. Sensation is not knowledge, and never can\nbecome knowledge, without the intervention of reason, and a concentrated\nself feeling can not rise essentially above animal life until it has,\nthrough the mediation of reason, attained the idea of the existence of a\nSupreme Being ruling over nature and man.\n\nMere feeling is essentially blind. In its _pathological_ form, it may\nindicate a want, and even develop an unconscious appetency, but it can\nnot, itself, reveal an _object_, any more than the feeling of hunger can\nreveal the actual presence, or determine the character and fitness, of\nany food. An undefinable fear, a mysterious presentiment, an instinctive\nyearning, a hunger of the soul, these are all irrational emotions which\ncan never rise to the dignity of knowledge. An object must be conjured\nby the imagination, or conceived by the understanding, or intuitively\napprehended by the reason, before the feeling can have any significance.\n\nRegarded in its _moral_ form, as \"the feeling of obligation,\" it can\nhave no real meaning unless a \"law of duty\" be known and recognized.\nFeeling, alone, can not reveal what duty is. When that which is right,\nand just, and good is revealed to the mind, then the sense of obligation\nmay urge man to the performance of duty. But the right, the just, the\ngood, are ideas which are apprehended by the reason, and, consequently,\nour moral sentiments are the result of the harmonious and living\nrelation between the reason and the sensibilities.\n\nMr. Mansel asserts the inadequacy of Schleiermacher's \"feeling of\ndependence\" to reveal the character of the Being on whom we depend. He\nhas therefore supplemented his doctrine by the \"feeling of moral\nobligation,\" which he thinks \"compels us to _assume_ the existence of a\nmoral Deity.\" We think his \"fact of religious intuition\" is as\ninadequate as Schleiermacher's to explain the whole phenomena of\nreligion. In neither instance does feeling supply the actual knowledge\nof God. The feeling of dependence may indicate that there is a Power or\nBeing upon whom we depend for existence and well-being, and which Power\nor Being \"we call God.\" The feeling of obligation certainly indicates\nthe existence of a Being to whom we are accountable, and which Being Mr.\nMansel calls a \"moral Deity.\" But in both instances the character, and\neven the existence of God is \"_assumed_\" and we are entitled to ask on\nwhat ground it is assumed. It will not be asserted that feeling alone\ngenerates the idea, or that the feeling is transformed into idea without\nthe intervention of thought and reflection. Is there, then, a _logical_\nconnection between the feeling of dependence and of obligation, and the\nidea of the Uncreated Mind, the Infinite First Cause, the Righteous\nGovernor of the world. Or is there a fixed and changeless co-relation\nbetween _the feeling_ and the _idea_, so that when the feeling is\npresent, the idea also necessarily arises in the mind? This latter\nopinion seems to be the doctrine of Mansel. We accept it as the\nstatement of a fact of consciousness, but we can not regard it as an\naccount of the genesis of the idea of God in the human mind. The idea of\nGod as the First Cause, the Infinite Mind, the Perfect Being, the\npersonal Lord and Lawgiver, the creator, sustainer, and ruler of the\nworld, is not a simple, primitive intuition of the mind. It is\nmanifestly a complex, concrete idea, and, as such, can not be developed\nin consciousness, by the operation of a single faculty of the mind, in a\nsimple, undivided act. It originates in the spontaneous operation of the\nwhole mind. It is a necessary deduction from the facts of the universe,\nand the primitive intuitions of the reason,--a logical inference from\nthe facts of sense, consciousness, and reason. A philosophy of religion\nwhich regards the feelings as supreme, and which brands the decisions of\nreason as uncertain, and well-nigh valueless, necessarily degenerates\ninto mysticism--a mysticism \"which pretends to elevate man directly to\nGod, and does not see that, in depriving reason of its power, it really\ndeprives man of that which enables him to know God, and puts him in a\njust communication with God by the intermediary of eternal and infinite\ntruth.\"[65]\n\n[Footnote 65: Cousin, \"True, Beautiful, and Good,\" p. 110.]\n\nThe religious sentiments in all minds, and in all ages, have resulted\nfrom the union of _thought_ and _feeling_--the living and harmonious\nrelation of reason and sensibility; and a philosophy which disregards\neither is inadequate to the explanation of the phenomena.\n\nIV. The fourth hypothesis is, _that religion has had its outbirth in the\nspontaneous apperceptions of_ REASON; that is, in the necessary, a\npriori ideas of the infinite, the perfect, the unconditioned Cause, the\nEternal Being, which are evoked into consciousness in presence of the\nchangeful, contingent phenomena of the world.\n\nThis will at once be recognized by the intelligent reader as the\ndoctrine of Cousin, by whom _pure reason_ is regarded as the grand\nfaculty or organ of religion.\n\nReligion, in the estimation of Cousin, is grounded on _cognition_ rather\nthan upon feeling. It is the knowledge of God, and the knowledge of duty\nin its relation to God and to human happiness; and as reason is the\ngeneral faculty of all knowing, it must be the faculty of religion. \"In\nits most elevated point of view, religion is the relation of absolute\ntruth to absolute Being,\" and as absolute truth is apprehended by the\nreason alone, reason \"is the veridical and religious part of the nature\nof man.\"[66] By \"reason,\" however, as we shall see presently, Cousin\ndoes not mean the discursive or reflective reason, but the spontaneous\nor intuitive reason. That act of the mind by which we attain to\nreligious knowledge is not a _process of reasoning_, but a pure\nappreciation, an instinctive and involuntary movement of the soul.\n\n[Footnote 66: Henry's Cousin, p. 510.]\n\nThe especial function of reason, therefore, is to reveal to us the\ninvisible, the supersensuous, the Divine. \"It was bestowed upon us for\nthis very purpose of going, without any circuit of reasoning, from the\nvisible to the invisible, from the finite to the infinite, from the\nimperfect to the perfect, and from necessary and eternal truths, to the\neternal and necessary principle\" that is God.[67] Reason is thus, as it\nwere, the bridge between consciousness and being; it rests, at the same\ntime, on both; it descends from God, and approaches man; it makes its\nappearance in consciousness as a guest which brings intelligence of\nanother world of real Being which lies beyond the world of sense.\n\nReason does not, however, attain to the Absolute Being directly and\nimmediately, without any intervening medium. To assert this would be to\nfall into the error of Plotinus, and the Alexandrian Mystics. Reason is\nthe offspring of God, a ray of the Eternal Reason, but it is not to be\nidentified with God. Reason attains to the Absolute Being indirectly,\nand by the interposition of truth. Absolute truth is an attribute and a\nmanifestation of God. \"Truth is incomprehensible without God, and God is\nincomprehensible without truth. Truth is placed between human\nintelligence, and the supreme intelligence as a kind of mediator.\"[68]\nIncapable of contemplating God face to face, reason adores God in the\ntruth which represents and manifests Him.\n\n[Footnote 67: Cousin, \"True, Beautiful, and Good,\" p. 103.]\n\n[Footnote 68: Id., ib., p. 99.]\n\nAbsolute truth is thus a revelation of God, made by God to the reason of\nman, and as it is a light which illuminates every man, and is\nperpetually perceived by all men, it is a universal and perpetual\nrevelation of God to man. The mind of man is \"the offspring of God,\"\nand, as such, must have some resemblance to, and some correlation with\nGod. Now that which constitutes the image of God in man must be found in\nthe reason which is correlated with, and capable of perceiving the truth\nwhich manifests God, just as the eye is correlated to the light which\nmanifests the external world. Absolute truth is, therefore, the sole\nmedium of bringing the human mind into communion with God; and human\nreason, in becoming united to absolute truth, becomes united to God in\nhis manifestation in spirit and in truth. The supreme law, and highest\ndestination of man, is to become united to God by seeking a full\nconsciousness of, and loving and practising the Truth.[69]\n\n[Footnote 69: Henry's Cousin, p. 511, 512.]\n\nIt will at once be obvious that the grand crucial questions by which\nthis philosophy of religion is to be tested are--\n\n1st. _How will Cousin prove to us that human reason is in possession of\nuniversal and necessary principles or absolute truths?_ and,\n\n2d. _How are these principles shown to be absolute? how far do these\nprinciples of reason possess absolute authority?_\n\nThe answer of Cousin to the first question is that we prove reason to be\nin possession of universal and necessary principles by the analysis of\nthe contents of consciousness, that is, by psychological analysis. The\nphenomena of consciousness, in their primitive condition, are\nnecessarily complex, concrete, and particular. All our primary ideas are\ncomplex ideas, for the evident reason that all, or nearly all, our\nfaculties enter at once into exercise; their simultaneous action giving\nus, at the same time, a certain number of ideas connected with each\nother, and forming a whole. For example, the idea of the exterior world,\nwhich is given us so quickly, is a complex idea, which contains a number\nof ideas. There is the idea of the secondary qualities of exterior\nobjects; there is the idea of the primary qualities; there is the idea\nof the permanent reality of something to which you refer these\nqualities, to wit, matter; there is the idea of space which contains\nbodies; there is the idea of time in which movements are effected. All\nthese ideas are acquired simultaneously, or nearly simultaneously, and\ntogether form one complex idea.\n\nThe application of analysis to this complex phenomenon clearly reveals\nthat there are simple ideas, beliefs, principles in the mind which can\nnot have been derived from sense and experience, which sense and\nexperience do not account for, and which are the suggestions of reason\nalone: the idea of the _Infinite_, the _Perfect_, the _Eternal_; the\ntrue, the beautiful, the good; the principle of causality, of substance,\nof unity, of intentionality; the principle of duty, of obligation, of\naccountability, of retribution. These principles, in their natural and\nregular development, carry us beyond the limits of consciousness, and\nreveal to us a world of real being beyond the world of sense. They carry\nus up to an absolute Being, the fountain of all existence--a living,\npersonal, righteous God--the author, the sustainer, and ruler of the\nuniverse.\n\nThe proof that these principles are absolute, and possessed of absolute\nauthority, is drawn, first, from the _impersonality of reason_, or,\nrather, the impersonality of the ideas, principles, or truths of reason.\n\nIt is not we who create these ideas, neither can we change them at our\npleasure. We are conscious that the will, in all its various efforts, is\nenstamped with the impress of our personality. Our volitions are our\nown. So, also, our desires are our own, our emotions are our own. But\nthis is not the same with our rational ideas or principles. The ideas of\nsubstance, of cause, of unity, of intentionality do not belong to one\nperson any more than to another; they belong to mind as mind, they are\nrevealed in the universal intelligence of the race. Absolute truth has\nno element of personality about it. Man may say \"my reason,\" but give\nhim credit for never having dared to say \"_my_ truth.\" So far from\nrational ideas being individual, their peculiar characteristic is that\nthey are opposed to individuality, that is, they are universal and\nnecessary. Instead of being circumscribed within the limits of\nexperience, they surpass and govern it; they are universal in the midst\nof particular phenomena; necessary, although mingled with things\ncontingent; and absolute, even when appearing within us the relative and\nfinite beings that we are.[70] Necessary, universal, absolute truth is a\ndirect emanation from God. \"Such being the case, the decision of reason\nwithin its own peculiar province possesses an authority almost divine.\nIf we are led astray by it, we must be led astray by a light from\nheaven.\"[71]\n\n[Footnote 70: Cousin, \"True, Beautiful, and Good,\" p. 40.]\n\n[Footnote 71: Id., \"Lectures,\" vol. ii. p. 32.]\n\nThe second proof is derived from _the distinction between the\nspontaneous and reflective movements of reason_.\n\nReflection is voluntary, spontaneity is involuntary; reflection is\npersonal, spontaneity is impersonal; reflection is analytic, spontaneity\nis synthetic; reflection begins with doubt, spontaneity with\naffirmation; reflection belongs to certain ones, spontaneity belongs to\nall; reflection produces science, spontaneity gives truth. Reflection is\na process, more or less tardy, in the individual and in the race. It\nsometimes engenders error and skepticism, sometimes convictions that,\nfrom being rational, are only the more profound. It constructs systems,\nit creates artificial logic, and all those formulas which we now use by\nthe force of habit, as if they were natural to us. But spontaneous\nintuition is the true logic of nature,--instant, direct, and infallible.\nIt is a primitive affirmation which implies no negation, and therefore\nyields positive knowledge. To reflect is to return to that which was. It\nis, by the aid of memory, to return to the past, and to render it\npresent to the eye of consciousness. Reflection, therefore, creates\nnothing; it supposes an anterior operation of the mind in which there\nnecessarily must be as many terms as are discovered by reflection.\nBefore all reflection there comes spontaneity--a spontaneity of the\nintellect, which seizes truth at once, without traversing doubt and\nerror. \"We thus attain to a judgment free from all reflection, to an\naffirmation without any mixture of negation, to an immediate intuition,\nthe legitimate daughter of the natural energy of thought, like the\ninspiration of the poet, the instinct of the hero, the enthusiasm of the\nprophet.\" Such is the first act of knowing, and in this first act the\nmind passes from _idea to being_ without ever suspecting the depth of\nthe chasm it has passed. It passes by means of the power which is in it,\nand is not astonished at what it has done. It is subsequently astonished\nwhen by reflection it returns to the analysis of the results, and, by\nthe aid of the liberty with which it is endowed, to do the opposite of\nwhat it has done, to deny what it has affirmed. \"Hence comes the strife\nbetween sophism and common sense, between false science and natural\ntruth, between good and bad philosophy, both of which come from free\nreflection.\"[72]\n\nIt is this spontaneity of thought which gives birth to _religion._ The\ninstinctive thought which darts through the world, even to God, is\nnatural religion. \"All thought implies a spontaneous faith in God, and\nthere is no such thing as natural atheism. Doubt and skepticism may\nmingle with reflective thought, but beneath reflection there is still\nspontaneity. When the scholar has denied the existence of God, listen to\nthe man, interrogate him, take him unawares, and you will see that all\nhis words envelop the idea of God, and that faith in God is, without his\nrecognition, at the bottom, in his heart.\"[73]\n\nReligion, then, in the system of Cousin, does not begin with reflection,\nwith science, but with _faith_. There is, however, this difference to be\nnoted between the theory of the \"faith-philosophers\" (Jacobi,\nSchleiermacher, etc.) and the theory of Cousin. With them, faith is\ngrounded on sensation or _feeling_; with him, it is grounded on\n_reason_. \"Faith, whatever may be its form, whatever may be its object,\ncommon or sublime, can be nothing else than the _consent of reason_.\nThat is the foundation of faith.\"[74]\n\n[Footnote 72: Cousin, \"True, Beautiful, and Good,\" p. 106.]\n\n[Footnote 73: \"Hist. of Philos.,\" vol. i. p. 137.]\n\n[Footnote 74: Ibid., vol. i. p. 90.]\n\nReligion is, therefore, with Cousin, at bottom, pure Theism. He thinks,\nhowever, that \"true theism is not a dead religion that forgets precisely\nthe fundamental attributes of God.\" It recognizes God as creator,\npreserver, and governor; it celebrates a providence; it adores a\nperfect, holy, righteous, benevolent God. It holds the principle of\nduty, of obligation, of moral desert. It not only perceives the divine\ncharacter, but feels its relation to God. The revelation of the\nInfinite, by reason, moves the feelings, and passes into sentiment,\nproducing reverence, and love, and gratitude. And it creates worship,\nwhich recalls man to God a thousand times more forcibly than the order,\nharmony, and beauty of the universe can do.\n\nThe spontaneous action of reason, in its greatest energy, is\n_inspiration_. \"Inspiration, daughter of the soul and heaven, speaks\nfrom on high with an absolute authority. It commands faith; so all its\nwords are hymns, and its natural language is poetry.\" \"Thus, in the\ncradle of civilization, he who possessed in a higher degree than his\nfellows the gift of inspiration, passed for the confidant and the\ninterpreter of God. He is so for others, because he is so for himself;\nand he is so, in fact, in a philosophic sense. Behold the sacred origin\nof prophecies, of pontificates, and of modes of worship.\"[75]\n\n[Footnote 75: \"Hist. of Philos.,\" vol. i. p. 129.]\n\nAs an account of the genesis of the idea of God in the human\nintelligence, the doctrine of Cousin must be regarded as eminently\nlogical, adequate, and satisfactory. As a theory of the origin of\nreligion, as a philosophy which shall explain all the phenomena of\nreligion, it must be pronounced defective, and, in some of its aspects,\nerroneous.\n\nFirst, it does not take proper account of that _living force_ which has\nin all ages developed so much energy, and wrought such vast results in\nthe history of religion, viz., the _power of the heart_. Cousin\ndiscourses eloquently on the spontaneous, instinctive movements of the\nreason, but he overlooks, in a great measure, the instinctive movements\nof the heart. He does not duly estimate the feeling of reverence and awe\nwhich rises spontaneously in presence of the vastness and grandeur of\nthe universe, and of the power and glory of which the created universe\nis a symbol and shadow. He disregards that sense of an overshadowing\nPresence which, at least in seasons of tenderness and deep sensibility,\nseems to compass us about, and lay its hand upon us. He scarcely\nrecognizes the deep consciousness of imperfection and weakness, and\nutter dependence, which prompts man to seek for and implore the aid of a\nSuperior Being; and, above all, he takes no proper account of the sense\nof guilt and the conscious need of expiation. His theory, therefore, can\nnot adequately explain the universal prevalence of sacrifices, penances,\nand prayers. In short, it does not meet and answer to the deep longings\nof the human heart, the wants, sufferings, fears, and hopes of man.\n\nCousin claims that the universal reason of man is illuminated by the\nlight of God. It is quite pertinent to ask, Why may not the universal\nheart of humanity be touched and moved by the spirit of God? If the\nideas of reason be a revelation from God, may not the instinctive\nfeelings of the heart be an inspiration of God? May not God come near to\nthe heart of man and awaken a mysterious presentiment of an invisible\nPresence, and an instinctive longing to come nearer to Him? May he not\ndraw men towards himself by sweet, persuasive influences, and raise man\nto a conscious fellowship? Is not God indeed the _great want_ of the\nhuman heart?\n\nSecondly, Cousin does not give due importance to the influence of\nrevealed truth as given in the sacred Scriptures, and of the positive\ninstitutions of religion, as a divine economy, supernaturally originated\nin the world. He grants, indeed, that \"a primitive revelation throws\nlight upon the cradle of human civilization,\" and that \"all antique\ntraditions refer to an age in which man, at his departure from the hand\nof God, received from him immediately all lights, and all truths.\"[76]\nHe also believes that \"the Mosaic religion, by its developments, is\nmingled with the history of all the surrounding people of Egypt, of\nAssyria, of Persia, and of Greece and Rome.\"[77] Christianity, however,\nis regarded as \"the summing and crown of the two great religious systems\nwhich reigned by turn in the East and in Greece\"--the maturity of\nEthnicism and Judaism; a development rather than a new creation. The\nexplanation which he offers of the phenomena of inspiration opens the\ndoor to religious skepticism. Those who were termed seers, prophets,\ninspired teachers of ancient times, were simply men who resigned\nthemselves wholly to their intellectual instincts, and thus gazed upon\ntruth in its pure and perfect form. They did not reason, they did not\nreflect, they made no pretensions to philosophy they received truth\nspontaneously as it flowed in upon them from heaven.[78] This immediate\nreception of Divine light was nothing more than the _natural_ play of\nspontaneous reason nothing more than what has existed to a greater or\nless degree in every man of great genius; nothing more than may now\nexist in any mind which resigns itself to its own unreflective\napperceptions. Thus revelation, in its proper sense, loses all its\npeculiar value, and Christianity is robbed of its pre-eminent authority.\nThe extremes of Mysticism and Rationalism here meet on the same ground,\nand Plotinus and Cousin are at one.\n\n[Footnote 76: \"Hist. of Philos.,\" vol. i. p. 148.]\n\n[Footnote 77: Ibid., vol. i. p. 216.]\n\n[Footnote 78: Morell, \"Hist. of Philos.,\" p. 661.]\n\nV. The fifth hypothesis offered in explanation of the religious\nphenomena of the world is that they had their origin _in_ EXTERNAL\nREVELATION, _to which reason is related as a purely passive organ, and\nEthnicism as a feeble relic_.\n\nThis is the theory of the school of \"dogmatic theologians,\" of which the\nablest and most familiar presentation is found in the \"Theological\nInstitutes\" of R. Watson.[79] He claims that all our religious knowledge\nis derived from _oral revelation alone_, and that all the forms of\nreligion and modes of worship which have prevailed in the heathen world\nhave been perversions and corruptions of the one true religion first\ntaught to the earliest families of men by God himself. All the ideas of\nGod, duty, immortality, and future retribution which are now possessed,\nor have ever been possessed by the heathen nations, are only broken and\nscattered rays of the primitive traditions descending from the family of\nNoah, and revived by subsequent intercourses with the Hebrew race; and\nall the modes of religious worship--prayers, lustrations,\nsacrifices--that have obtained in the world, are but feeble relics,\nfaint reminiscences of the primitive worship divinely instituted among\nthe first families of men. \"The first man received the knowledge of God\nby sensible converse with him, and that doctrine was transmitted, with\nthe confirmation of successive manifestations, to the early ancestors of\nall nations.\"[80] This belief in the existence of a Supreme Being was\npreserved among the Jews by continual manifestations of the presence of\nJehovah. \"The intercourses between the Jews and the states of Syria and\nBabylon, on the one hand, and Egypt on the other, powers which rose to\ngreat eminence and influence in the ancient world, was maintained for\nages. Their frequent dispersions and captivities would tend to preserve\nin part, and in part to revive, the knowledge of the once common and\nuniversal faith.\"[81] And the Greek sages who resorted for instruction\nto the Chaldean philosophic schools derived from thence their knowledge\nof the theological system of the Jews.[82] Among the heathen nations\nthis primitive revelation was corrupted by philosophic speculation, as\nin India and China, Greece and Rome; and in some cases it was entirely\nobliterated by ignorance, superstition, and vice, as among the\nHottentots of Africa and the aboriginal tribes of New South Wales, who\n\"have no idea of one Supreme Creator.\"[83]\n\n[Footnote 79: We might have referred the reader to Ellis's \"Knowledge of\nDivine Things from Revelation, not from Reason or Nature;\" Leland's\n\"Necessity of Revelation;\" and Horsley's \"Dissertations,\" etc.; but as\nwe are not aware of their having been reprinted in this country, we\nselect the \"Institutes\" of Watson as the best presentation of the views\nof \"the dogmatic theologians\" accessible to American readers.]\n\n[Footnote 80: Watson, \"Theol. Inst,\" vol. i. p. 270.]\n\n[Footnote 81: Id. ib., vol. i. p. 31.]\n\n[Footnote 82: See ch. v. and vi., \"On the Origin of those Truths which\nare found in the Writings and Religious Systems of the Heathen.\"]\n\n[Footnote 83: Ibid., vol. i. p. 274.]\n\nThe same course of reasoning is pursued in regard to the idea of duty,\nand the knowledge of right and wrong. \"A direct communication of the\nDivine Will was made to the primogenitors of our race,\" and to this\nsource _alone_ we are indebted for all correct ideas of right and wrong.\n\"Whatever is found pure in morals, in ancient or modern writers, may be\ntraced to _indirect_ revelation.\"[84] Verbal instruction--tradition or\nscripture--thus becomes the source of all our moral ideas. The doctrine\nof immortality, and of a future retribution,[85] the practice of\nsacrifice--precatory and expiatory, are also ascribed to the same\nsource.[86] Thus the only medium by which religious truth can possibly\nbecome known to the masses of mankind is _tradition_. The ultimate\nfoundation on which the religious faith and the religious practices of\nuniversal humanity have rested, with the exception of the Jews, and the\nfavored few to whom the Gospel has come, is uncertain, precarious, and\neasily corrupted tradition.\n\n[Footnote 84: Watson, \"Theol. Inst.,\" vol. ii. p. 470.]\n\n[Footnote 85: Id. ib., vol. i. p. 11.]\n\n[Footnote 86: Id. ib., vol. i. p. 26.]\n\nThe improbability, inadequacy, and incompleteness of this theory will be\nobvious from the following considerations:\n\n1. It is highly improbable that truths so important and vital to man, so\nessential to the well-being of the human race, so necessary to the\nperfect development of humanity as are the ideas of God, duty, and\nimmortality, should rest on so precarious and uncertain a basis as\ntradition is admitted, even by Mr. Watson, to be.\n\nThe human mind needs the idea of God to satisfy its deep moral\nnecessities, and to harmonize all its powers. The perfection of humanity\ncan never be secured, the destination of humanity can never be achieved,\nthe purpose of God in the existence of humanity can never be\naccomplished, without the idea of God, and of the relation of man to\nGod, being present to the human mind. Society needs the idea of a\nSupreme Ruler as the foundation of law and government, and as the basis\nof social order. Without it, these can not be, or be conserved.\nIntellectual creatureship, social order, human progress, are\ninconceivable and impossible without the idea of God, and of\naccountability to God. Now that truths so fundamental should, to the\nmasses of men, rest on tradition _alone_, is incredible. Is there no\nknown and accessible God to the outlying millions of our race who, in\nconsequence of the circumstances of birth and education, which are\nbeyond their control, have had no access to an oral revelation, and\namong whom the dim shadowy rays of an ancient tradition have long ago\nexpired? Are the eight hundred millions of our race upon whom the light\nof Christianity has not shone unvisited by the common Father of our\nrace? Has the universal Father left his \"own offspring\" without a single\nnative power of recognizing the existence of the Divine Parent, and\nabandoned them to solitary and dreary orphanage? Could not he who gave\nto matter its properties and laws,--the properties and laws through\nwhose operation he is working out his own purposes in the realm of\nnature,--could not he have also given to mind ideas and principles\nwhich, logically developed, would lead to recognition of a God, and of\nour duty to God, and, by these ideas and principles, have wrought out\nhis sublime purposes in the realm of mind? Could not he who gave to man\nthe appetency for food, and implanted in his nature the social instincts\nto preserve his physical being, have implanted in his heart a \"feeling\nafter God,\" and an instinct to worship God in order to the conservation\nof his spiritual being? How otherwise can we affirm the responsibility\nand accountability of all the race before God? Those theologians who are\nso earnest in the assertion that God has not endowed man with the native\npower of attaining the knowledge of God can not, on any principle of\nequity, show how the heathen are \"without excuse\" when, in involuntary\nignorance of God, they \"worship the creature instead of the Creator,\"\nand violate a law of duty of which they have no possible means to attain\nthe barest knowledge.\n\n2. This theory is utterly inadequate to the explanation of the\n_universality_ of religious rites, and especially of religious ideas.\n\nTake, for example, the idea of God. As a matter of fact we affirm, in\nopposition to Watson, the universality of this idea. The idea of God is\nconnatural to the human mind. Wherever human reason has had its normal\nand healthy development[87], this idea has arisen spontaneously and\nnecessarily. There has not been found a race of men who were utterly\ndestitute of some knowledge of a Supreme Being. All the instances\nalleged have, on further and more accurate inquiry, been found\nincorrect. The tendency of the last century, arbitrarily to quadrate all\nthe facts of religious history with the prevalent sensational\nphilosophy, had its influence upon the minds of the first missionaries\nto India, China, Africa, Australia, and the islands of the Pacific. They\n_expected_ to find that the heathen had no knowledge of a Supreme Being,\nand before they had mastered the idioms of their language, or become\nfamiliar with their mythological and cosmological systems, they reported\nthem as _utterly ignorant of God_, destitute of the idea and even the\nname of a Supreme Being. These mistaken and hasty conclusions have,\nhowever, been corrected by a more intimate acquaintance with the people,\ntheir languages and religions. Even in the absence of any better\ninformation, we should be constrained to doubt the accuracy of the\nauthorities quoted by Mr. Watson in relation to Hindooism, when by one\n(Ward) we are told that the Hindoo \"believes in a God destitute of\n_intelligence_\" and by another (Moore) that \"Brahm is the one eternal\n_Mind_, the self-existent, incomprehensible Spirit\". Learned and\ntrustworthy critics, Asiatic as well as European, however, confidently\naffirm that \"the ground of the Brahminical faith is Monotheistic;\" it\nrecognizes \"an Absolute and Supreme Being\" as the source of all that\nexists.[88] Eugene Burnouf, M. Barthelemy Saint-Hilaire, Koeppen, and\nindeed nearly all who have written on the subject of Buddhism, have\nshown that the metaphysical doctrines of Buddha were borrowed from the\nearlier systems of the Brahminic philosophy. \"Buddha.\" we are told, is\n\"_pure intelligence_\" \"_clear light_\", \"_perfect wisdom_;\" the same as\nBrahm. This is surely Theism in its highest conception.[89] In regard to\nthe peoples of South Africa, Dr. Livingstone assures us \"there is no\nneed for beginning to tell even the most degraded of these people of the\nexistence of a God, or of a future state--the facts being universally\nadmitted.... On questioning intelligent men among the Backwains as to\ntheir former knowledge of good and evil, of God, and of a future state,\nthey have scouted the idea of any of them ever having been without a\ntolerably clear conception on all these subjects.\"[90] And so far from\nthe New Hollanders having no idea of a Supreme Being, we are assured by\nE. Stone Parker, the protector of the aborigines of New Holland, they\nhave a clear and well-defined idea of a \"_Great Spirit_,\" the maker of\nall things.\n\n[Footnote 87: Watson, \"Theol. Inst.,\" vol. i. p. 46.]\n\n[Footnote 88: Maurice, \"Religions of the World,\" p. 59: _Edin.\nReview_,1862, art \"Recent Researches on Buddhism.\" See also Mueller's\n\"Chips from a German Workshop,\" vol. i. ch. i. to vi.]\n\n[Footnote 89: \"It has been said that Buddha and Kapila were both\natheists, and that Buddha borrowed his atheism from Kapila. But atheism\nis an indefinite term, and may mean very different things. In one sense\nevery Indian philosopher was an atheist, for they all perceived that the\ngods of the populace could not claim the attributes that belong to a\nSupreme Being. But all the important philosophical systems of the\nBrahmans admit, in some form or another, the existence of an Absolute\nand Supreme Being, the source of all that exists, or seems to\nexist.\"--Mueller, \"Chips from a German Workshop,\" vol. i. pp. 224,5.\n\nBuddha, which means \"intelligence,\" \"clear light,\" \"perfect wisdom,\" was\nnot only the name of the founder of the religion of Eastern Asia, but\nAdi Buddha was the name of the Absolute, Eternal Intelligence.--Maurice,\n\"Religions of the World,\" p. 102.]\n\n[Footnote 90: \"Missionary Travels and Researches in South Africa,\" p.\n158.]\n\nNow had the idea of God rested _solely_ on tradition, it were the most\nnatural probability that it might be lost, nay, _must_ be lost, amongst\nthose races of men who were geographically and chronologically far\nremoved from the primitive cradle of humanity in the East. The people\nwho, in their migrations, had wandered to the remotest parts of the\nearth, and had become isolated from the rest of mankind, might, after\nthe lapse of ages, be expected to lose the idea of God, if it were not a\nspontaneous and native intuition of the mind,--a necessity of thought. A\nfact of history must be presumed to stick to the mind with much greater\ntenacity than a purely rational idea which has no visible symbol in the\nsensible world, and yet, even in regard to the events of history, the\npersistence and pertinacity of tradition is exceedingly feeble. The\nSouth Sea Islanders know not from whence, or at what time, their\nancestors came. There are monuments in Tonga and Fiji of which the\npresent inhabitants can give no account. How, then, can a pure, abstract\nidea which can have no sensible representation, no visible image, retain\nits hold upon the memory of humanity for thousands of years? The Fijian\nmay not remember whence his immediate ancestors came, but he knows that\nthe race came originally from the hands of the Creator. He can not tell\nwho built the monuments of solid masonry which are found in his\nisland-home, but he can tell who reared the everlasting hills and built\nthe universe. He may not know who reigned in Vewa a hundred years ago,\nbut he knows who now reigns, and has always reigned, over the whole\nearth. \"The idea of a God is familiar to the Fijian, and the existence\nof an invisible superhuman power controlling and influencing nature, and\nall earthly things, is fully recognized by him.\"[91] The idea of God is\na common fact of human consciousness, and tradition alone is manifestly\ninadequate to account for its _universality_.\n\n[Footnote 91: \"Fiji and the Fijians,\" p. 215.]\n\n3. A verbal revelation would be inadequate to convey the knowledge of\nGod to an intelligence \"_purely passive_\" and utterly unfurnished with\nany _a priori_ ideas or necessary laws of cognition and thought.\n\nOf course it is not denied that important verbal communications relating\nto the character of God, and the duties we owe to God, were given to the\nfirst human pair, more clear and definite, it may be, than any knowledge\nattained by Socrates and Plato through their dialectic processes, and\nthat these oral revelations were successively repeated and enlarged to\nthe patriarchs and prophets of the Old Testament church. And\nfurthermore, that some rays of light proceeding from this pure fountain\nof truth were diffused, and are still lingering among the heathen\nnations, we have no desire, and no need to deny.\n\nAll this, however, supposes, at least, a natural power and aptitude for\nthe knowledge of God, and some configuration and correlation of the\nhuman intelligence to the Divine. \"We have no knowledge of a dynamic\ninfluence, spiritual or natural, without a dynamic reaction.\" Matter can\nnot be moved and controlled by forces and laws, unless it have\nproperties which correlate it with those forces and laws. And mind can\nnot be determined from without to any specific form of cognition, unless\nit have active powers of apprehension and conception which are governed\nby uniform laws. The \"material\" of thought may be supplied from without,\nbut the \"form\" is determined by the necessary laws of our inward being.\nAll our cognition of the external world is conditioned by the _a priori_\nideas of time and space, and all our thinking is governed by the\nprinciples of causality and substance, and the law of \"sufficient\nreason.\" The mind itself supplies an element of knowledge in all our\ncognitions. Man can not be taught the knowledge of God if he be not\nnaturally possessed of a presentiment, or an apperception of a God, as\nthe cause and reason of the universe. \"If education be not already\npreceded by an innate consciousness of God, as an operative\npredisposition, there would be nothing for education and culture to act\nupon.\"[92] A mere verbal revelation can not communicate the knowledge of\nGod, if man have not already the idea of a God in his mind. A name is a\nmere empty sign, a meaningless symbol, without a mental image of the\nobject which it represents, or an innate perception, or an abstract\nconception of the mind, of which the word is the sign. The mental image\nor the abstract conception must, therefore, precede the name; cognition\nmust be anterior to, and give the meaning of language.[93] The child\nknows a thing even before it can speak its name. And, universally, we\nmust know the _thing_ in itself, or image it by analogies and\nresemblances to some other thing we do know, before the name can have\nany meaning for us. As to purely rational ideas and abstract\nconceptions,--as space, cause, the infinite, the perfect,--language can\nnever convey these to the mind, nor can the mind ever attain them by\nexperience if they are not an original, connate part of our mental\nequipment and furniture. The mere verbal affirmation \"there is a God\"\nmade to one who has no idea of a God, would be meaningless and\nunintelligible. What notion can a man form of \"the First Cause\" if the\nprinciple of causality is not inherent in his mind? What conception can\nhe form of \"the Infinite Mind\" if the infinite be not a primitive\nintuition? How can he conceive of \"a Righteous Governor\" if he have no\nidea of right, no sense of obligation, no apprehension of a retribution?\nWords are empty sounds without ideas, and God is a mere name if the mind\nhas no apperception of a God.\n\n[Footnote 92: Nitzsch, \"System of Christian Doctrine,\" p. 10.]\n\n[Footnote 93: \"Ideas must pre-exist their sensible signs.\" See De\nBoismont on \"Hallucination,\" etc., p. iii.]\n\nIt may be affirmed that, preceding or accompanying the announcement of\nthe Divine Name, there was given to the first human pair, and to the\nearly fathers of our race, some visible manifestation of the presence of\nGod, and some supernatural display of divine power. What, then, was the\ncharacter of these early manifestations, and were they adequate to\nconvey the proper idea of God? Did God first reveal himself in human\nform, and if so, how could their conception of God advance beyond a rude\nanthropomorphism? Did he reveal his presence in a vast columnar cloud or\na pillar of fire? How could such an image convey any conception of the\nintelligence, the omnipresence, the eternity of God? Nay, can the\ninfinite and eternal Mind be represented by any visible manifestation?\nCan the human mind conceive an image of God? The knowledge of God, it is\nclear, can not be conveyed by any sensible sign or symbol if man has no\nprior rational idea of God as the Infinite and the Perfect Being.\n\nIf the facts of order, and design, and special adaptation which crowd\nthe universe, and the _a priori_ ideas of an unconditioned Cause and an\ninfinite Intelligence which arise in the mind in presence of these\nfacts, are inadequate to produce the logical conviction that it is the\nwork of an intelligent mind, how can any preternatural display of\n_power_ produce a rational conviction that God exists? \"If the universe\ncould come by chance or fate, surely all the lesser phenomena, termed\nmiraculous, might occur so too.\"[94] If we find ourselves standing amid\nan eternal series of events, may not miracles be a part of that series?\nOr if all things are the result of necessary and unchangeable laws, may\nnot miracles also result from some natural or psychological law of which\nwe are yet in ignorance? Let it be granted that man is _not_ so\nconstituted that, by the necessary laws of his intelligence, he must\naffirm that facts of order having a commencement in time prove mind; let\nit be granted that man has _no_ intuitive belief in the Infinite and\nPerfect--in short, no idea of God; how, then, could a marvellous display\nof _power_, a new, peculiar, and startling phenomenon which even seemed\nto transcend nature, prove to him the existence of an infinite\n_intelligence_--a personal God? The proof would be simply inadequate,\nbecause not the right kind of proof. Power does not indicate\nintelligence, force does not imply personality.\n\n[Footnote 94: Morell, \"Hist. of Philos.\" p. 737.]\n\nMiracles, in short, were never intended to prove the existence of God.\nThe foundation of this truth had already been laid in the constitution\nand laws of the human mind, and miracles were designed to convince us\nthat He of whose existence we had a prior certainty, spoke to us by His\nMessenger, and in this way attested his credentials. To the man who has\na rational belief in the existence of God this evidence of a divine\nmission is at once appropriate and conclusive. \"Master, we know thou art\na teacher sent from God; for no man can do the works which thou doest,\nexcept God be with him.\" The Christian missionary does not commence his\ninstruction to the heathen, who have an imperfect, or even erroneous\nconception of \"the Great Spirit,\" by narrating the miracles of Christ,\nor quoting the testimony of the Divine Book he carries along with him.\nHe points to the heavens and the earth, and says, \"There is a Being who\nmade all these things, and Jehovah is his name; I have come to you with\na message from Him!\" Or he need scarce do even so much; for already the\nheathen, in view of the order and beauty which pervades the universe,\nhas been constrained, by the laws of his own intelligence, to believe in\nand offer worship to the \"Agnostos Theos\"--the unseen and\nincomprehensible God; and pointing to their altars, he may announce with\nPaul, \"this God _whom ye worship_, though ignorantly, him declare I unto\nyou!\"\n\nThe results of our study of the various hypotheses which have been\noffered in explanation of the religious phenomena of the world may be\nsummed up as follows: The first and second theories we have rejected as\nutterly false. Instead of being faithful to and adequately explaining\nthe facts, they pervert, and maltreat, and distort the facts of\nreligious history. The last three each contain a precious element of\ntruth which must not be undervalued, and which can not be omitted in an\nexplanation which can be pronounced complete. Each theory, taken by\nitself, is incomplete and inadequate. The third hypothesis overrates\n_feeling_; the fourth, _reason_; the fifth, _verbal instruction_. The\nfirst extreme is Mysticism, the second is Rationalism, the last is\nDogmatism. Reason, feeling, and faith in testimony must be combined, and\nmutually condition each other. No purely rationalistic hypothesis will\nmeet and satisfy the wants and yearnings of the heart. No theory based\non feeling alone can satisfy the demands of the human intellect. And,\nfinally, an hypothesis which bases all religion upon historical\ntestimony and outward fact, and despises and tramples upon the\nintuitions of the reason and the instincts of the heart can never\ncommand the general faith of mankind. Religion embraces and\nconditionates the whole sphere of life--thought, feeling, faith, and\naction; it must therefore be grounded in the entire spiritual nature of\nman.\n\nOur criticism of opposite theories has thus prepared the way for, and\nobviated the necessity of an extended discussion of the hypothesis we\nnow advance.\n\n_The universal phenomenon of religion has originated in the a priori\napperceptions of reason, and the natural instinctive feelings of the\nheart, which, from age to age, have been vitalized, unfolded, and\nperfected by supernatural communications and testamentary revelations_.\n\nThere are universal facts of religious history which can only be\nexplained on the first principle of this hypothesis; there are special\nfacts which can only be explained on the latter principle. The universal\nprevalence of the idea of God, and the feeling of obligation to obey and\nworship God, belong to the first order of facts; the general prevalence\nof expiatory sacrifices, of the rite of circumcision, and the observance\nof sacred and holy days, belong to the latter. To the last class of\nfacts the observance of the Christian Sabbath, and the rites of Baptism\nand the Lord's Supper may be added.\n\nThe history of all religions clearly attests that there are two orders\nof principles--the _natural_ and the _positive_, and, in some measure,\ntwo authorities of religious life which are intimately related without\nnegativing each other. The characteristic of the natural is that it is\n_intrinsic_, of the positive, that it is _extrinsic_. In all ages men\nhave sought the authority of the positive in that which is immediately\n_beyond_ and above man--in some \"voice of the Divinity\" toning down the\nstream of ages, or speaking through a prophet or oracle, or written in\nsome inspired and sacred book. They have sought for the authority of the\nnatural in that which is immediately _within_ man--the voice of the\nDivinity speaking in the conscience and heart of man. A careful study of\nthe history of religion will show a reciprocal relation between the two,\nand indicate their common source.\n\nWe expect to find that our hypothesis will be abundantly sustained by\nthe study of the _Religion of the Athenians_.\n\n\n\n\nCHAPTER III.\n\nTHE RELIGION OF THE ATHENIANS.\n\n\n\"All things which I behold bear witness to your carefulness in religion\n(deisidaimonesterous). For as I passed through your city, and beheld the\nobjects of your worship, I found amongst them an altar with this\ninscription--'TO THE UNKNOWN GOD.' Whom therefore ye worship....\"--ST.\nPAUL.\n\nThrough one of those remarkable counter-strokes of Divine Providence by\nwhich the evil designs of men are overruled, and made to subserve the\npurposes of God, the Apostle Paul was brought to Athens. He walked\nbeneath its stately porticoes, he entered its solemn temples, he stood\nbefore its glorious statuary, he viewed its beautiful altars--all\ndevoted to pagan worship. And \"his spirit was stirred within him,\" he\nwas moved with indignation \"when he saw the city full of images of the\ngods.\"[95] At the very entrance of the city he met the evidence of this\npeculiar tendency of the Athenians to multiply the objects of their\ndevotion; for here at the gateway stands an image of Neptune, seated on\nhorseback, and brandishing the trident. Passing through the gate, his\nattention would be immediately arrested by the sculptured forms of\nMinerva, Jupiter, Apollo, Mercury, and the Muses, standing near a\nsanctuary of Bacchus. A long street is now before him, with temples,\nstatues, and altars crowded on either hand. Walking to the end of this\nstreet, and turning to the right, he entered the Agora, a public square\nsurrounded with porticoes and temples, which were adorned with statuary\nand paintings in honor of the gods of Grecian mythology. Amid the\nplane-trees planted by the hand of Cimon are the statues of the deified\nheroes of Athens, Hercules and Theseus, and the whole series of the\nEponymi, together with the memorials of the older divinities; Mercuries\nwhich gave the name to the streets on which they were placed; statues\ndedicated to Apollo as patron of the city and her deliverer from the\nplague; and in the centre of all the altar of the Twelve Gods.\n\n[Footnote 95: Lange's Commentary, Acts xvii. 16.]\n\nStanding in the market-place, and looking up to the Areopagus, Paul\nwould see the temple of Mars, from whom the hill derived its name. And\nturning toward the Acropolis, he would behold, closing the long\nperspective, a series of little sanctuaries on the very ledges of the\nrocks, shrines of Bacchus and AEsculapius, Venus, Earth, and Ceres,\nending with the lovely form of the Temple of Unwinged Victory, which\nglittered in front of the Propylaea.\n\nIf the apostle entered the \"fivefold gates,\" and ascended the flight of\nstone steps to the platform of the Acropolis, he would find the whole\narea one grand composition of architecture and statuary dedicated to the\nworship of the gods. Here stood the Parthenon, the Virgin House, the\nglorious temple which was erected during the proudest days of Athenian\nglory, an entire offering to Minerva, the tutelary divinity of Athens.\nWithin was the colossal statue of the goddess wrought in ivory and gold.\nOutside the temple there stood another statue of Minerva, cast from the\nbrazen spoils of Marathon; and near by yet another brazen Pallas, which\nwas called by pre-eminence \"the Beautiful.\"\n\nIndeed, to whatever part of Athens the apostle wandered, he would meet\nthe evidences of their \"carefulness in religion,\" for every public place\nand every public building was a sanctuary of some god. The Metroum, or\nrecord-house, was a temple to the mother of the gods. The council-house\nheld statues of Apollo and Jupiter, with an altar to Vesta. The theatre\nat the base of the Acropolis was consecrated to Bacchus. The Pnyx was\ndedicated to Jupiter on high. And as if, in this direction, the Attic\nimagination knew no bounds, abstractions were deified; altars were\nerected to Fame, to Energy, to Modesty, and even to Pity, and these\nabstractions were honored and worshipped as gods.\n\nThe impression made upon the mind of Paul was, that the city was\nliterally \"full of idols,\" or images of the gods. This impression is\nsustained by the testimony of numerous Greek and Roman writers.\nPausanias declares that Athens \"had more images than all the rest of\nGreece;\" and Petronius, the Roman satirist, says, \"it was easier to find\na god in Athens than a man.\"[96]\n\n[Footnote 96: See Conybeare and Howson's \"Life and Epistles of St.\nPaul;\" also, art. \"Athens,\" in _Encyclopaedia Britannica_, whence our\naccount of the \"sacred objects\" in Athens is chiefly gathered.]\n\nNo wonder, then, that as Paul wandered amid these scenes \"his spirit was\nstirred in him.\" He burned with holy zeal to maintain the honor of the\ntrue and only God, whom now he saw dishonored on every side. He was\nfilled with compassion for those Athenians who, notwithstanding their\nintellectual greatness, had changed the glory of God into an image made\nin the likeness of corruptible man, and who really worshipped the\ncreature _more_ than the Creator. The images intended to symbolize the\ninvisible perfections of God were usurping the place of God, and\nreceiving the worship due alone to him. We may presume the apostle was\nnot insensible to the beauties of Grecian art. The sublime architecture\nof the Propylaea and the Parthenon, the magnificent sculpture of Phidias\nand Praxiteles, could not fail to excite his wonder. But he remembered\nthat those superb temples and this glorious statuary were the creation\nof the pagan spirit, and devoted to polytheistic worship. The glory of\nthe supreme God was obscured by all this symbolism. The creatures formed\nby God, the symbols of his power and presence in nature, the ministers\nof his providence and moral government, were receiving the honor due to\nhim. Over all this scene of material beauty and aesthetic perfection\nthere rose in dark and hideous proportions the errors and delusions and\nsins against the living God which Polytheism nurtured, and unable any\nlonger to restrain himself, he commenced to \"reason\" with the crowds of\nAthenians who stood beneath the shadows of the plane-trees, or lounged\nbeneath the porticoes that surrounded the Agora. Among these groups of\nidlers were mingled the disciples of Zeno and Epicurus, who\n\"encountered\" Paul. The nature of these \"disputations\" may be easily\nconjectured, The opinions of these philosophers are even now familiarly\nknown: they are, in one form or another, current in the literature of\nmodern times. Materialism and Pantheism still \"encounter\" Christianity.\nThe apostle asserted the personal being and spirituality of one supreme\nand only God, who has in divers ways revealed himself to man, and\ntherefore may be \"known.\" He proclaimed that Jesus is the fullest and\nmost perfect revelation of God--the _only_ \"manifestation of God in the\nflesh.\" He pointed to his \"resurrection\" as the proof of his superhuman\ncharacter and mission to the world. Some of his hearers were disposed to\ntreat him with contempt; they represented him as an ignorant \"babbler,\"\nwho had picked up a few scraps of learning, and who now sought to palm\nthem off as a \"new\" philosophy. But most of them regarded him with that\npeculiar Attic curiosity which was always anxious to be hearing some\n\"new thing.\" So they led him away from the tumult of the market-place to\nthe top of Mars' Hill, where, in its serene atmosphere, they might hear\nhim more carefully, and said, \"May we hear what this new doctrine is\nwhereof thou speakest?\"\n\nSurrounded by these men of thoughtful, philosophic mind--men who had\ndeeply pondered the great problem of existence, who had earnestly\ninquired after the \"first principles of things;\" men who had reasoned\nhigh of creation, fate, and providence; of right and wrong; of\nconscience, law, and retribution; and had formed strong and decided\nopinions on all these questions--he delivered his discourse on the\n_being_, the _providence_, the _spirituality_, and the _moral\ngovernment_ of God.\n\nThis grand theme was suggested by an inscription he had observed on one\nof the altars of the city, which was dedicated \"To the Unknown God.\" \"Ye\nmen of Athens! every thing which I behold bears witness to your\n_carefulness in religion_. For as I passed by and beheld your sacred\nobjects I found an altar with this inscription, 'To the Unknown God;'\nwhom, therefore, ye worship, though ye know him not [adequately], Him\ndeclare I unto you.\" Starting from this point, the manifest carefulness\nof the Athenians in religion, and accepting this inscription as the\nevidence that they had some presentiment, some native intuition, some\ndim conception of the one true and living God, he strives to lead them\nto a deeper knowledge of Him. It is here conceded by the apostle that\nthe Athenians were a _religious people_. The observations he had made\nduring his short stay in Athens enabled him to bear witness that the\nAthenians were \"a God-fearing people,\"[97] and he felt that fairness and\ncandor demanded that this trait should receive from him an ample\nrecognition and a full acknowledgment. Accordingly he commences by\nsaying in gentle terms, well fitted to conciliate his audience, \"All\nthings which I behold bear witness to your carefulness in religion.\" I\nrecognize you as most devout; ye appear to me to be a God-fearing\npeople,[98] for as I passed by and beheld your sacred objects I found an\naltar with this inscription, \"To the Unknown God,\" whom therefore ye\nworship.\n\n[Footnote 97: Lange's Commentary, _in loco_.]\n\n[Footnote 98: \"Os before deisid.--so imports. I recognize you as\nsuch.\"--Lange's Commentary.]\n\nThe assertion that the Athenians were \"a religious people\" will, to many\nof our readers, appear a strange and startling utterance, which has in\nit more of novelty than truth. Nay, some will be shocked to hear the\nApostle Paul described as complimenting these Athenians--these pagan\nworshippers--on their \"carefulness in religion.\" We have been so long\naccustomed to use the word \"heathen\" as an opprobrious\nepithet--expressing, indeed, the utmost extremes of ignorance, and\nbarbarism, and cruelty, that it has become difficult for us to believe\nthat in a heathen there can be any good.\n\nFrom our childhood we have read in our English Bibles, Ye men of Athens,\nI perceive in all things ye are _too superstitious_ and we can scarcely\ntolerate another version, even if it can be shown that it approaches\nnearer to the actual language employed by Paul. We must, therefore, ask\nthe patience and candor of the reader, while we endeavor to show, on the\nauthority of Paul's words, that the Athenians were a \"religious people,\"\nand that all our notions to the contrary are founded on prejudice and\nmisapprehension.\n\nFirst, then, let us commence even with our English version: \"Ye men of\nAthens, I perceive that in all things ye are _too superstitious_.\" And\nwhat now is the meaning of the word \"superstition?\" It is true, we now\nuse it only in an evil sense, to express a belief in the agency of\ninvisible, capricious, malignant powers, which fills the mind with fear\nand terror, and sees in every unexplained phenomenon of nature an omen,\nor prognostic, of some future evil. But this is not its proper and\noriginal meaning. Superstition is from the Latin _superstitio_, which\nmeans a superabundance of religion,[99] an extreme exactitude in\nreligious observance. And this is precisely the sense in which the\ncorresponding Greek term is used by the Apostle Paul. Deisidaimonia\nproperly means \"reverence for the gods.\" \"It is used,\" says Barnes, \"in\nthe classic writers, in a good sense, to denote piety towards the gods,\nor suitable fear and reverence for them.\" \"The word,\" says Lechler, \"is,\nwithout doubt, to be understood here in a good sense; although it seems\nto have been intentionally chosen, in order to indicate the conception\nof _fear_(deido), which predominated in the religion of the apostle's\nhearers.\"[100] This reading is sustained by the ablest critics and\nscholars of modern times. Bengel reads the sentence, \"I perceive that ye\nare _very religious_\"[101] Cudworth translates it thus: \"Ye are every\nway _more than ordinarily religious.\"[102]_ Conybeare and Howson read\nthe text as we have already given it, \"All things which I behold bear\nwitness to your _carefulness in religion_.\"[103] Lechler reads \"very\ndevout;\"[104] Alford, \"carrying your _religious reverence very\nfar_;\"[105] and Albert Barnes,[106] \"I perceive ye are greatly devoted\nto _reverence for religion_.\"[107] Whoever, therefore, will give\nattention to the actual words of the apostle, and search for their real\nmeaning, must be convinced he opens his address by complimenting the\nAthenians on their being more than ordinarily religious.\n\n[Footnote 99: Nitzsch, \"System of Christ. Doctrine,\" p. 33.]\n\n[Footnote 100: Lange's Commentary, _in loco_.]\n\n[Footnote 101: \"Gnomon of the New Testament.\"]\n\n[Footnote 102: \"Intellectual System,\" vol. i. p. 626.]\n\n[Footnote 103: \"Life and Epistles of St. Paul,\" vol. i. p. 378.]\n\n[Footnote 104: Lange's Commentary.]\n\n[Footnote 105: Greek Test.]\n\n[Footnote 106: Notes on Acts.]\n\n[Footnote 107: Also Clarke's Comment., _in loco_.]\n\nNor are we for a moment to suppose the apostle is here dealing in hollow\ncompliments, or having recourse to a \"pious fraud.\" Such a course would\nhave been altogether out of character with Paul, and to suppose him\ncapable of pursuing such a course is to do him great injustice. If \"to\nthe Jews he became as a Jew,\" it was because he recognized in Judaism\nthe same fundamental truths which underlie the Christian system. And if\nhere he seems to become, in any sense, at one with \"heathenism,\" that he\nmight gain the heathen to the faith of Christ, it was because he found\nin heathenism some elements of truth akin to Christianity, and a state\nof feeling favorable to an inquiry into the truths he had to present. He\nbeheld in Athens an altar reared to the God _he_ worshipped, and it\nafforded him some pleasure to find that God was not totally forgotten,\nand his worship totally neglected, by the Athenians. The God whom they\nknew imperfectly, \"_Him_\" said he, \"I declare unto you;\" I now desire to\nmake him more fully known. The worship of \"the Unknown God\" was a\nrecognition of the being of a God whose nature transcends all human\nthought, a God who is ineffable; who, as Plato said, \"is hard to be\ndiscovered, and having discovered him, to make him known to all,\nimpossible.\"[108] It is the confession of a _want_ of knowledge, the\nexpression of a _desire_ to know, the acknowledgment of the _duty_ of\nworshipping him. Underlying all the forms of idol-worship the eye of\nPaul recognized an influential Theism. Deep down in the pagan heart he\ndiscovered a \"feeling after God\"--a yearning for a deeper knowledge of\nthe \"unknown,\" the invisible, the incomprehensible, which he could not\ndespise or disregard. The mysterious _sentiments_ of fear, of reverence,\nof conscious dependence on a supernatural power and presence\novershadowing man, which were expressed in the symbolism of the \"sacred\nobjects\" which Paul saw everywhere in Athens, commanded his respect. And\nhe alludes to their \"devotions,\" not in the language of reproach or\ncensure, but as furnishing to his own mind the evidence of the strength\nof their _religious instincts_, and the proof of the existence in their\nhearts of that _native apprehension_ of the supernatural, the divine,\nwhich dwells alike in all human souls.\n\n[Footnote 108: Timaeus, ch. ix.]\n\nThe case of the Athenians has, therefore, a peculiar interest to every\nthoughtful mind. It confirms the belief that religion is a necessity to\nevery human mind, a want of every human heart.[109] Without religion,\nthe nature of man can never be properly developed; the noblest part of\nman--the divine, the spiritual element which dwells in man, as \"the\noffspring of God\"--must remain utterly dwarfed. The spirit, the personal\nbeing, the rational nature, is religious, and Atheism is the vain and\nthe wicked attempt to be something less than man. If the spiritual\nnature of man has its normal and healthy development, he must become a\nworshipper. This is attested by the universal history of man. We look\ndown the long-drawn aisles of antiquity, and everywhere we behold the\nsmoking altar, the ascending incense, the prostrate form, the attitude\nof devotion. Athens, with her four thousand deities--Rome, with her\ncrowded Pantheon of gods--Egypt, with her degrading\nsuperstitions--Hindostan, with her horrid and revolting rites--all\nattest that the religious principle is deeply seated in the nature of\nman. And we are sure religion can never be robbed of her supremacy, she\ncan never be dethroned in the hearts of men. It were easier to satisfy\nthe cravings of hunger by logical syllogisms, than to satisfy the\nyearnings of the human heart without religion. The attempt of Xerxes to\nbind the rushing floods of the Hellespont in chains was not more futile\nnor more impotent than the attempt of skepticism to repress the\nuniversal tendency to worship, so peculiar and so natural to man in\nevery age and clime.\n\n[Footnote 109: The indispensable necessity for a religion of some kind\nto satisfy the emotional nature of man is tacitly confessed by the\natheist Comte in the publication of his \"Catechism of Positive\nReligion.\"]\n\nThe unwillingness of many to recognize a religious element in the\nAthenian mind is further accounted for by their misconception of the\nmeaning of the word \"religion.\" We are all too much accustomed to regard\nreligion as a mere system of dogmatic teaching. We use the terms\n\"Christian religion,\" \"Jewish religion,\" \"Mohammedan religion,\" as\ncomprehending simply the characteristic doctrines by which each is\ndistinguished; whereas religion is a mode of thought, and feeling, and\naction, determined by the consciousness of our relation to and our\ndependence upon God. It does not appropriate to itself any specific\ndepartment of our mental powers and susceptibilities, but it conditions\nthe entire functions and circle of our spiritual life. It is not simply\na mode of conceiving God in thought, nor simply a mode of venerating God\nin the affections, nor yet simply a mode of worshipping God in outward\nand formal acts, but it comprehends the whole. Religion (_religere_,\nrespect, awe, reverence) regulates our thoughts, feelings, and acts\ntowards God. \"It is a reference and a relationship of our finite\nconsciousness to the Creator and Sustainer and Governor of the\nuniverse.\" It is such a consciousness of the Divine as shall awaken in\nthe heart of man the sentiments of reverence, fear, and gratitude\ntowards God; such a sense of dependence as shall prompt man to pray, and\nlead him to perform external acts of worship.\n\nReligion does not, therefore, consist exclusively in knowledge, however\ncorrect; and yet it must be preceded and accompanied by some intuitive\ncognition of a Supreme Being, and some conception of him as a free moral\npersonality. But the religious sentiments, which belong rather to the\nheart than to the understanding of man--the consciousness of dependence,\nthe sense of obligation, the feeling of reverence, the instinct to pray,\nthe appetency to worship--these may all exist and be largely developed\nin a human mind even when, as in the case of the Athenians, there is a\nvery imperfect knowledge of the real character of God.\n\nRegarding this, then, as the generic conception of religion, namely,\n_that it is a mode of thought and feeling and action determined by our\nconsciousness of dependence on a Supreme Being_, we claim that the\napostle was perfectly right in complimenting the Athenians on their\n\"more than ordinary religiousness,\" for,\n\n1. They had, in some degree at least, that faith in the being and\nprovidence of God which precedes and accompanies all religion.\n\nThey had erected an altar to the unseen, the unsearchable, the\nincomprehensible, the unknown God. And this \"unknown God\" whom the\nAthenians \"worshipped\" was the true God, the God whom Paul worshipped,\nand whom he desired more fully to reveal to them; \"_Him_ declare I unto\nyou.\" The Athenians had, therefore, some knowledge of the true God, some\ndim recognition, at least, of his being, and some conception, however\nimperfect, of his character. The Deity to whom the Athenians reared this\naltar is called \"the unknown God,\" because he is unseen by all human\neyes and incomprehensible to human thought. There is a sense in which to\nPaul, as well as to the Athenians--to the Christian as well as to the\npagan--to the philosopher as well as to the peasant--God is \"_the\nunknown_,\" and in which he must forever remain the incomprehensible.\nThis has been confessed by all thoughtful minds in every age. It was\nconfessed by Plato. To his mind God is \"the ineffable,\" the unspeakable.\nZophar, the friend of Job, asks, \"Canst thou by searching find out God?\nCanst thou find out the Almighty to perfection?\" This knowledge is \"high\nas heaven; what canst thou do? deeper than hell; what canst thou know?\"\nDoes not Wesley teach us to sing,\n\n                 \"Hail, Father, whose creating call\n                    Unnumbered worlds attend;\n                  Jehovah, comprehending all,\n                    Whom none can comprehend?\"\n\nTo his mind, as well as to the mind of the Athenian, God was \"the great\nunseen, unknown.\" \"Beyond the universe and man,\" says Cousin, \"there\nremains in God something unknown, impenetrable, incomprehensible. Hence,\nin the immeasurable spaces of the universe, and beneath all the\nprofundities of the human soul, God escapes us in this inexhaustible\ninfinitude, whence he is able to draw without limit new worlds, new\nbeings, new manifestations. God is therefore to us\n_incomprehensible_.\"[110] And without making ourselves in the least\nresponsible for Hamilton's \"negative\" doctrine of the Infinite, or even\nresponsible for the full import of his words, we may quote his\nremarkable utterances on this subject: \"The Divinity is in part\nconcealed and in part revealed. He is at once known and unknown. But the\nlast and highest consecration of all true religion must be an altar 'to\nthe unknown God.' In this consummation nature and religion, Paganism and\nChristianity, are at one.\"[111]\n\n[Footnote 110: \"Lectures,\" vol. i. p. 104.]\n\n[Footnote 111: \"Discussions on Philosophy,\" p. 23.]\n\nWhen, therefore, the apostle affirms that while the Athenians worshipped\nthe God whom he proclaimed they \"knew him not,\" we can not understand\nhim as saying they were destitute of all faith in the being of God, and\nof all ideas of his real character. Because for him to have asserted\nthey had _no_ knowledge of God would not only have been contrary to all\nthe facts of the case, but also an utter contradiction of all his\nsettled convictions and his recorded opinions. There is not in modern\ntimes a more earnest asserter of the doctrine that the human mind has an\nintuitive cognition of God, and that the external world reveals God to\nman. There is a passage in his letter to the Romans which is justly\nentitled to stand at the head of all discourses on \"natural theology,\"\nRom. i. 19-21. Speaking of the heathen world, who had not been favored,\nas the Jews, with a verbal revelation, he says, \"That which may be known\nof God is manifest _in_ them,\" that is, in the constitution and laws of\ntheir spiritual nature, \"for God hath showed it unto them\" in the voice\nof reason and of conscience, so that in the instincts of our hearts, in\nthe elements of our moral nature, in the ideas and laws of our reason,\nwe are taught the being of a God. These are the subjective teachings of\nthe human soul.\n\nNot only is the being of God revealed to man in the constitution and\nlaws of his rational and moral nature, but God is also manifested to us\nobjectively in the realm of things around us; therefore Paul adds, \"The\ninvisible things of him, even his eternal power and Godhead, from the\ncreation are clearly seen, being understood by the things that are\nmade.\" The world of sense, therefore, discloses the being and\nperfections of God. The invisible attributes of God are made apparent by\nthe things that are visible. Forth out of nature, as the product of the\nDivine Mind, the supernatural shines. The forces, laws, and harmonies of\nthe universe are indices of the presence of a presiding and informing\nIntelligence. The creation itself is an example of God's coming forth\nout of the mysterious depths of his own eternal and invisible being, and\nmaking himself apparent to man. There, on the pages of the volume of\nnature, we may read, in the marvellous language of symbol, the grand\nconceptions, the glorious thoughts, the ideals of beauty which dwell in\nthe uncreated Mind, These two sources of knowledge--the subjective\nteachings of God in the human soul, and the objective manifestations of\nGod in the visible universe--harmonize, and, together, fill up the\ncomplement of our natural idea of God. They are two hemispheres of\nthought, which together form one full-orbed fountain of light, and ought\nnever to be separated in our philosophy. And, inasmuch as this divine\nlight shines on all human minds, and these works of God are seen by all\nhuman eyes, the apostle argues that the heathen world \"is without\nexcuse, because, knowing God (gnontes ton Theon) they did not glorify\nhim as God, neither were thankful; but in their reasonings they went\nastray after vanities, and their hearts, being void of wisdom, were\nfilled with darkness. Calling themselves wise, they were turned into\nfools, and changed the glory of the imperishable God for idols graven in\nthe likeness of perishable man, or of birds, and beasts, and creeping\nthings,...and they bartered the truth of God for lies, and reverenced\nand worshipped the things made rather than the Maker, who is blessed\nforever. Amen.\"[112]\n\n[Footnote 112: Rom. i. 21-25, Conybeare and Howson's translation.]\n\nThe brief and elliptical report of Paul's address on Mars' Hill must\ntherefore, in all fairness, be interpreted in the light of his more\ncarefully elaborated statements in the Epistle to the Romans. And when\nPaul intimates that the Athenians \"knew not God,\" we can not understand\nhim as saying they had _no_ knowledge, but that their knowledge was\nimperfect. They did not know God as Creator, Father, and Ruler; above\nall, they did not know him as a pardoning God and a sanctifying Spirit.\nThey had not that knowledge of God which purifies the heart, and changes\nthe character, and gives its possessor eternal life.\n\nThe apostle clearly and unequivocally recognizes this truth, that the\nidea of God is connatural to the human mind; that in fact there is not\nto be found a race of men upon the face of the globe utterly destitute\nof some idea of a Supreme Being. Wherever human reason has had its\nnormal and healthful development, it has spontaneously and necessarily\nled the human mind to the recognition of a God. The Athenians were no\nexception to this general law. They believed in the existence of one\nsupreme and eternal Mind, invisible, incomprehensible, infeffable--\"the\nunknown God.\"\n\n2. The Athenians had also that consciousness of dependence upon God\nwhich is the foundation of all the primary religious emotions.\n\nWhen the apostle affirmed that \"in God we live, and move, and have our\nbeing,\" he uttered the sentiments of many, if not all, of his hearers,\nand in support of that affirmation he could quote the words of their own\npoets, for we are also his offspring; [113] and, as his offspring, we\nhave a derived and a dependent being. Indeed, this consciousness of\ndependence is analogous to the feeling which is awakened in the heart of\na child when its parent is first manifested to its opening mind as the\ngiver of those things which it immediately needs, as its continual\nprotector, and as the preserver of its life. The moment a man becomes\nconscious of his own personality, that moment he becomes conscious of\nsome relation to another personality, to which he is subject, and on\nwhich he depends.[114]\n\n[Footnote 113:\n\n           \"Jove's presence fills all space, upholds this ball;\n            All need his aid; his power sustains us all,\n              _For we his offspring are_.\"\n                            Aratus, \"The Phaenomena,\" book v. p. 5.\n\nAratus was a poet of Cilicia, Paul's native province. He flourished B.C.\n277.\n\n       \"Great and divine Father, whose names are many,\n        But who art one and the same unchangeable, almighty power;\n        O thou supreme Author of nature!\n        That governest by a single unerring law!\n                      Hail King!\n        For thou art able, to enforce obedience from all frail mortals,\n        _Because we are all thine offspring,_\n        The image and the echo only of thy eternal voice.\"\n                                        Cleanthes, \"Hymn to Jupiter.\"\n\nCleanthes was the pupil of Zeno, and his successor as chief of the Stoic\nphilosophers.]\n\n[Footnote 114: \"As soon as a man becomes conscious of himself, as soon\nas he perceives himself as distinct from other persons and things, he at\nthe same moment becomes conscious of a higher self, a higher power,\nwithout which he feels that neither he nor any thing else would have any\nlife or reality. We are so fashioned that as soon as we awake we feel on\nall sides our dependence on something else; and all nations join in some\nway or another in the words of the Psalmist, 'It is He that made us, not\nwe ourselves.' This is the first _sense_ of the Godhead, the _sensus\nnuminis_, as it has well been called; for it is a _sensus_, an immediate\nperception, not the result of reasoning or generalization, but an\nintuition as irresistible as the impressions of our senses.... This\n_sensus numinis_, or, as we may call it in more homely language,\n_faith_, is the source of all religion; it is that without which no\nreligion, whether true or false, is possible.\"--Max Mueller, \"Science of\nLanguage,\" Second Series, p. 455.]\n\nA little reflection will convince us that this is the necessary order in\nwhich human consciousness is developed.\n\nThere are at least two fundamental and radical tendencies in human\npersonality, namely, to _know_ and to _act_. If we would conceive of\nthem as they exist in the innermost sphere of selfhood, we must\ndistinguish the first as _self-consciousness_, and the second as\n_self-determination_. These are unquestionably the two factors of human\npersonality.\n\nIf we consider the first of these factors more closely, we shall\ndiscover that self-consciousness exists under limitations and\nconditions. Man can not become clearly conscious of _self_ without\ndistinguishing himself from the outer world of sensation, nor without\ndistinguishing self and the world from another being upon whom they\ndepend as the ultimate substance and cause. Mere _coenoeesthesis_ is not\nconsciousness. Common feeling is unquestionably found among the lowest\nforms of animal life, the protozoa; but it can never rise to a clear\nconsciousness of personality until it can distinguish itself from\nsensation, and acquire a presentiment of a divine power, on which self\nand the outer world depend. The _Ego_ does not exist for itself, can not\nperceive itself, but by distinguishing itself from the ceaseless flow\nand change of sensation, and by this act of distinguishing, the _Ego_\ntakes place in consciousness. And the _Ego_ can not perceive itself, nor\ncognize sensation as a state or affection of the _Ego_ except by the\nintervention of the reason, which supplies the two great fundamental\nlaws of causality and substance. The facts of consciousness thus\ncomprehend three elements--self, nature, and God. The determinate being,\nthe _Ego_, is never an absolutely independent being, but is always in\nsome way or other codetermined by another; it can not, therefore, be an\nabsolutely original and independent, but must in some way or another be\na _derived_ and _conditioned_ existence.\n\nNow that which limits and conditions human self-consciousness can not be\nmere _nature_, because nature can not give what it does not possess; it\ncan not produce what is _toto genere_ different from itself.\nSelf-consciousness can not arise out of unconsciousness. This new\nbeginning is beyond the power of nature. Personal power, the creative\nprinciple of all new beginnings, is alone adequate to its production.\nIf, then, self-consciousness exists in man, it necessarily presupposes\nan absolutely _original_, therefore _unconditioned, self-consciousness_.\nHuman self-consciousness, in its temporal actualization, of course\npresupposes a nature-basis upon which it elevates itself; but it is only\npossible on the ground that an eternal self-conscious Mind ordained and\nrules over all the processes of nature, and implants the divine spark of\nthe personal spirit with the corporeal frame, to realize itself in the\nlight-flame of human self-consciousness. The original light of the\ndivine self-consciousness is eternally and absolutely first and before\nall. \"Thus, in the depths of our own self-consciousness, as its\nconcealed background, the God-consciousness reveals itself to us. This\ndescent into our inmost being is at the same time an ascent to God.\nEvery deep reflection on ourselves breaks through the mere crust of\nworld-consciousness, which separates us from the inmost truth of our\nexistence, and leads us up to Him in whom we live and move and\nare.\"[115]\n\n[Footnote 115: Mueller, \"Christian Doctrine of Sin,\" vol. i. p. 81.]\n\nSelf-determination, equally with self-consciousness, exists in us under\nmanifold _limitations_. Self-determination is limited by physical,\ncorporeal, and mental conditions, so that there is \"an impassable\nboundary line drawn around the area of volitional freedom.\" But the most\nfundamental and original limitation is that of _duty_. The\nself-determining power of man is not only circumscribed by necessary\nconditions, but also by the _moral law_ in the consciousness of man.\nSelf-determination alone does not suffice for the full conception of\nresponsible freedom; it only becomes, _will_, properly by its being an\nintelligent and conscious determination; that is, the rational subject\nis able previously to recognize \"the right,\" and present before his mind\nthat which he _ought_ to do, that which he is morally bound to realize\nand actualize by his own self-determination and choice. Accordingly we\nfind in our inmost being a _sense of obligation_ to obey the moral law\nas revealed in the conscience. As we can not become conscious of self\nwithout also becoming conscious of God, so we can not become properly\nconscious of self-determination until we have recognized in the\nconscience a law for the movements of the will.\n\nNow this moral law, as revealed in the conscience, is not a mere\nautonomy--a simple subjective law having no relation to a personal\nlawgiver out of and above man. Every admonition of conscience directly\nexcites the consciousness of a God to whom man is accountable. The\nuniversal consciousness of our race, as revealed in history, has always\nassociated the phenomena of conscience with the idea of a personal Power\nabove man, to whom he is subject and upon whom he depends. In every age,\nthe voice of conscience has been regarded as the voice of God, so that\nwhen it has filled man with guilty apprehensions, he has had recourse to\nsacrifices, and penances, and prayers to expatiate his wrath.\n\nIt is clear, then, that if man has _duties_ there must he a\nself-conscious Will by whom these duties are imposed, for only a real\nwill can be legislative. If man has a _sense of obligation_, there must\nbe a supreme authority by which he is obliged. If he is _responsible_,\nthere must be a being to whom he is accountable.[116] It can not be said\nthat he is accountable to himself, for by that supposition the idea of\nduty is obliterated, and \"right\" becomes identical with mere interest or\npleasure. It can not be said that he is simply responsible to\nsociety--to mere conventions of human opinions and human\ngovernments--for then \"_right_\" becomes a mere creature of human\nlegislation, and \"_justice_\" is nothing but the arbitrary will of the\nstrong who tyrannize over the weak. Might constitutes right. Against\nsuch hypotheses the human mind, however, instinctively revolts. Mankind\nfeel, universally, that there is an authority beyond all human\ngovernments, and a higher law above all human laws, from whence all\ntheir powers are derived. That higher law is the Law of God, that\nsupreme authority is the God of Justice. To this eternally just God,\ninnocence, under oppression and wrong, has made its proud appeal, like\nthat of Prometheus to the elements, to the witnessing clouds, to coming\nages, and has been sustained and comforted. And to that higher law the\nweak have confidently appealed against the unrighteous enactments of the\nstrong, and have finally conquered. The last and inmost ground of all\nobligation is thus the conscious relation of the moral creature to God.\nThe sense of absolute dependence upon a Supreme Being compels man, even\nwhile conscious of subjective freedom, to recognize at the same time his\nobligation to determine himself in harmony with the will of Him \"in whom\nwe live, and move, and are.\"\n\n[Footnote 116: \"The thought of God will wake up a terrible monitor whose\nname is Judge.\"--Kant.]\n\nThis feeling of dependence, and this consequent sense of obligation, lie\nat the very foundation of all religion. They lead the mind towards God,\nand anchor it in the Divine. They prompt man to pray, and inspire him\nwith an instinctive confidence in the efficacy of prayer. So that prayer\nis natural to man, and necessary to man. Never yet has the traveller\nfound a people on earth without prayer. Races of men have been found\nwithout houses, without raiment, without arts and sciences, but never\nwithout prayer any more than without speech. Plutarch wrote, eighteen\ncenturies ago, If you go through all the world, you may find cities\nwithout walls, without letters, without rulers, without money, without\ntheatres, but never without temples and gods, or without _prayers_,\noaths, prophecies, and sacrifices, used to obtain blessings and\nbenefits, or to avert curses and calamities.[117] The naturalness of\nprayer is admitted even by the modern unbeliever. Gerrit Smith says,\n\"Let us who believe that the religion of reason calls for the religion\nof nature, remember that the flow of prayer is just as natural as the\nflow of water; the prayerless man has become an unnatural man.\"[118] Is\nman in sorrow or in danger, his most natural and spontaneous refuge is\nin prayer. The suffering, bewildered, terror-stricken soul turns towards\nGod. \"Nature in an agony is no atheist; the soul that knows not where to\nfly, flies to God.\" And in the hour of deliverance and joy, a feeling of\ngratitude pervades the soul--and gratitude, too, not to some blind\nnature-force, to some unconscious and impersonal power, but gratitude to\nGod. The soul's natural and appropriate language in the hour of\ndeliverance is thanksgiving and praise.\n\n[Footnote 117: \"Against Kalotes,\" ch. xxxi.]\n\n[Footnote 118: \"Religion of Reason.\"]\n\nThis universal tendency to recognize a superior Power upon whom we are\ndependent, and by whose hand our well-being and our destinies are\nabsolutely controlled, has revealed itself even amid the most\ncomplicated forms of polytheistic worship. Amid the even and undisturbed\nflow of every-day life they might be satisfied with the worship of\nsubordinate deities, but in the midst of sudden and unexpected\ncalamities, and of terrible catastrophes, then they cried to the Supreme\nGod.[119] \"When alarmed by an earthquake,\" says Aulus Gellius, \"the\nancient Romans were accustomed to pray, not to some one of the gods\nindividually, but to God in general, _as to the Unknown_.\"[120]\n\n[Footnote 119: \"At critical moments, when the deepest feelings of the\nhuman heart are stirred, the old Greeks and Romans seem suddenly to have\ndropped all mythological ideas, and to have fallen back on the universal\nlanguage of true religion.\"--Max Mueller, \"Science of Language.\" p. 436.]\n\n[Footnote 120: Tholuck, \"Nature and Influence of Heathenism,\" p. 23.]\n\n\"Thus also Minutius Felix says, 'When they stretch out their hands to\nheaven they mention only God; and these forms of speech, _He is great_,\nand _God is true_, and _If God grant_(which are the natural language of\nthe vulgar), are a plain confession of the truth of Christianity.' And\nalso Lactantius testifies, 'When they swear, and when they wish, and\nwhen they give thanks, they name not many gods, but God only; the truth,\nby a secret force of nature, thus breaking forth from them whether they\nwill or no;' and again he says, 'They fly to God; aid is desired of God;\nthey pray that God would help them; and when one is reduced to extreme\nnecessity, he begs for God's sake, and by his divine power alone\nimplores the mercy of men.'\"[121] The account which is given by Diogenes\nLaertius[122] of the erection of altars bearing the inscription \"to the\nunknown God,\" clearly shows that they had their origin in this general\nsentiment of dependence on a higher Power. \"The Athenians being\nafflicted with pestilence invited Epimenides to lustrate their city. The\nmethod adopted by him was to carry several sheep to the Areopagus,\nwhence they were left to wander as they pleased, under the observation\nof persons sent to attend them. As each sheep lay down it was sacrificed\nto _the propitious God_. By this ceremony it is said the city was\nrelieved; but as it was still unknown what deity was propitious, an\naltar was erected _to the unknown God_ on every spot where a sheep had\nbeen sacrificed.\"[123]\n\n[Footnote 121: Cudworth, vol. i. p. 300.]\n\n[Footnote 122: \"Lives of Philosophers,\" book i., Epimenides.]\n\n[Footnote 123: See Townsend's \"Chronological Arrangement of New\nTestament,\" note 19, part xii.; Doddridge's \"Exposition;\" and Barnes's\n\"Notes on Acts.\"]\n\n\"The unknown God\" was their deliverer from the plague. And the erection\nof an altar to him was a confession of their absolute dependence upon\nhim, of their obligation to worship him, as well as of their need of a\ndeeper knowledge of him. The gods who were known and named were not able\nto deliver them in times of calamity, and they were compelled to look\nbeyond the existing forms of Grecian mythology for relief. Beyond all\nthe gods of the Olympus there was \"one God over all,\" the Father of gods\nand men, the Creator of all the subordinate local deities, upon whom\neven these created gods were dependent, upon whom man was absolutely\ndependent, and therefore in times of deepest need, of severest\nsuffering, of extremest peril, then they cried to the living, supreme,\neternal God.[124]\n\n[Footnote 124: \"The men and women of the Iliad and Odyssey are\nhabitually religious. The language of religion is often on their\ntongues, as it is ever on the lips of every body in the East at this\nday. The thought of the gods, and of their providence and government of\nthe world, is a familiar thought. They seem to have an abiding\nconviction of their _dependence_ on the gods. The results of all actions\ndepend on the will of the gods; _it lies on their knees_ (Theon ev\ngounasi keitai, Od. i. 267), is the often repeated and significant\nexpression of their feeling of dependence.\"--Tyler, \"Theology of Greek\nPoets,\" p. 165.]\n\n3. The Athenians developed in a high degree those religious emotions\nwhich always accompany the consciousness of dependence on a Supreme\nBeing.\n\nThe first emotional element of all religion is _fear_. This is\nunquestionably true, whether religion be considered from a Christian or\na heathen stand-point. \"The _fear_ of the Lord is the beginning of\nwisdom.\" Associated with, perhaps preceding, all definite ideas of God,\nthere exists in the human mind certain feelings of _awe_, and\n_reverence_, and _fear_ which arise spontaneously in presence of the\nvastness, and grandeur, and magnificence of the universe, and of the\npower and glory of which the created universe is but the symbol and\nshadow. There is the felt apprehension that, beyond and back of the\nvisible and the tangible, there is a _personal, living Power_, which is\nthe foundation of all, and which fashions all, and fills all with its\nlight and life; that \"the universe is the living vesture in which the\nInvisible has robed his mysterious loveliness.\" There is the feeling of\nan _overshadowing Presence_ which \"compasseth man behind and before, and\nlays its hand upon him.\"\n\nThis wonderful presentiment of an invisible power and presence pervading\nand informing all nature is beautifully described by Wordsworth in his\nhistory of the development of the Scottish herdsman's mind:\n\n     So the foundations of his mind were laid\n     In such communion, not from terror free.\n     While yet a child, and long before his time,\n     Had he perceived the presence and the power\n     Of greatness; and deep feelings had impressed\n     So vividly great objects, that they lay\n     Upon his mind like substances, whose presence\n     Perplexed the bodily sense.\n                        ... In the after-day\n     Of boyhood, many an hour in caves forlorn,\n     And 'mid the hollow depths of naked crags,\n     He sat, and even in their fixed lineaments,\n     Or from the power of a peculiar eye,\n     Or by creative feeling overborne,\n     Or by predominance of thought oppressed,\n     Even in their fixed and steady lineaments\n     He traced an ebbing and a flowing mind....\n     Such was the Boy,--but for the growing Youth,\n     What soul was his, when, from the naked top\n     Of some bold headland, he beheld the sun\n     Rise up, and bathe the world in light! He looked:\n     Ocean and earth, the solid frame of earth\n     And ocean's liquid mass, in gladness lay\n     Beneath him; far and wide the clouds were touched.\n     And in their silent faces could he read\n     Unutterable love. Sound needed none,\n     Nor any voice of joy; his spirit drank\n     The spectacle: sensation, soul, and form\n     All melted into him; they swallowed up\n     His animal being; in them did he live,\n     And by them did he live; they were his life,\n     In such access of mind, in such high hour\n     Of visitation from the living God.[125]\n\nBut it may be said this is all mere poetry; to which we answer, in the\nwords of Aristotle, \"Poetry is a thing more philosophical and weightier\nthan history.\"[126] The true poet is the interpreter of nature. His soul\nis in the fullest sympathy with the grand ideas which nature symbolizes,\nand he \"deciphers the universe as the autobiography of the Infinite\nSpirit.\" Spontaneous feeling is a kind of inspiration.\n\nIt is true that all minds may not be developed in precisely the same\nmanner as Wordsworth's herdsman's, because the development of every\nindividual mind is modified in some measure by exterior conditions. Men\nmay contemplate nature from different points of view. Some may be\nimpressed with one aspect of nature, some with another. But none will\nfail to recognize a mysterious _presence_ and invisible _power_ beneath\nall the fleeting and changeful phenomena of the universe. \"And sometimes\nthere are moments of tenderness, of sorrow, and of vague mystery which\nbring the feeling of the Infinite Presence close to the human\nheart.\"[127]\n\n[Footnote 125: \"The Wanderer.\"]\n\n[Footnote 126: Poet, ch. ix.]\n\n[Footnote 127: Robertson.]\n\nNow we hold that _this feeling and sentiment of the Divine_--the\nsupernatural--exists in every mind. It may be, it undoubtedly is,\nsomewhat modified in its manifestations by the circumstances in which\nmen are placed, and the degree of culture they have enjoyed. The African\nFetichist, in his moral and intellectual debasement, conceives a\nsupernatural power enshrined in every object of nature. The rude Fijian\nregards with dread, and even terror, the Being who darts the lightnings\nand wields the thunderbolts. The Indian \"sees God in clouds, and hears\nhim in the wind.\" The Scottish \"herdsman\" on the lonely mountain-top\n\"feels the presence and the power of greatness,\" and \"in its fixed and\nsteady lineaments he sees an ebbing and a flowing mind.\" The\nphilosopher[128] lifts his eyes to \"the starry heavens\" in all the depth\nof their concave, and with all their constellations of glory moving on\nin solemn grandeur, and, to his mind, these immeasurable regions seem\n\"filled with the splendors of the Deity, and crowded with the monuments\nof his power;\" or he turns his eye to \"the Moral Law within,\" and he\nhears the voice of an intelligent and a righteous God. In all these\ncases we have a revelation of the sentiment of the Divine, which dwells\nalike in all human minds. In the Athenians this sentiment was developed\nin a high degree. The serene heaven which Greece enjoyed, and which was\nthe best-loved roof of its inhabitants, the brilliant sun, the mountain\nscenery of unsurpassed grandeur, the deep blue sea, an image of the\ninfinite, these poured all their fullness on the Athenian mind, and\nfurnished the most favorable conditions for the development of the\nreligious sentiments. The people of Athens spent most of their time in\nthe open air in communion with nature, and in the cheerful and temperate\nenjoyment of existence. To recognize the Deity in the living powers of\nnature, and especially in man, as the highest sensible manifestation of\nthe Divine, was the peculiar prerogative of the Grecian mind. And here\nin Athens, art also vied with nature to deepen the religious sentiments.\nIt raised the mind to ideal conceptions of a beauty and a sublimity\nwhich transcended all mere nature-forms, and by images, of supernatural\ngrandeur and loveliness presented to the Athenians symbolic\nrepresentations of the separate attributes and operations of the\ninvisible God. The plastic art of Greece was designed to express\nreligious ideas, and was consecrated by religious feeling. Thus the\nfacts of the case are strikingly in harmony with the words of the\nApostle: \"All things which I behold bear witness to your carefulness in\nreligion,\" your \"reverence for the Deity,\" your \"fear of God.\"[129] \"The\nsacred objects\" in Athens, and especially \"the altar to the Unknown\nGod,\" were all regarded by Paul as evidences of their instinctive faith\nin the invisible, the supernatural, the divine.\n\n[Footnote 128: Kant, in \"Critique of Practical Reason.\"]\n\n[Footnote 129: See Parkhurst's Lexicon, under _Deisidaimonia_, which\nSuidas explains by eulabeia peri to Theion--_reverence for the Divine_,\nand Hesychius by Phubutheia--_fear of God_. Also, Josephus, Antiq., book\nx. ch. iii, Sec. 2: \"Manasseh, after his repentance and reformation, strove\nto behave himself (te deisidaimonia chrestheia) in the _most religious\nmanner_ towards God.\" Also see A. Clarke on Acts xvii.]\n\nAlong with this sentiment of the Divine there is also associated, in all\nhuman minds, an _instinctive yearning_ after the Invisible; not a mere\nfeeling of curiosity to pierce the mystery of being and of life, but\nwhat Paul designates \"a feeling after God,\" which prompts man to seek\nafter a deeper knowledge, and a more immediate consciousness. To attain\nthis deeper knowledge--this more conscious realization of the being and\nthe presence of God, has been the effort of all philosophy and all\nreligion in all ages. The Hindoo Yogis proposes to withdraw into his\ninmost self, and by a complete suspension of all his active powers to\nbecome absorbed and swallowed up in the Infinite.[130] Plato and his\nfollowers sought by an immediate abstraction to apprehend \"the\nunchangeable and permanent Being,\" and, by a loving contemplation, to\nbecome \"assimilated to the Deity,\" and in this way to attain the\nimmediate consciousness of God. The Neo-Platonic mystic sought by\nasceticism and self-mortification to prepare himself for divine\ncommunings. He would contemplate the divine perfections in himself; and\nin an _ecstatic_ state, wherein all individuality vanishes, he would\nrealize a union, or identity, with the Divine Essence.[131] While the\nuniversal Church of God, indeed, has in her purest days always taught\nthat man may, by inward purity and a believing love, be rendered capable\nof spiritually apprehending, and consciously feeling, the presence of\nGod. Some may be disposed to pronounce this as all mere mysticism. We\nanswer, The living internal energy of religion is always _mystical_, it\nis grounded in _feeling_--a \"_sensus numinis_\" common to humanity. It is\nthe mysterious sentiment of the Divine; it is the prolepsis of the human\nspirit reaching out towards the Infinite; the living susceptibility of\nour spiritual nature stretching after the powers and influences of the\nhigher world. It is upon this inner instinct of the supernatural that\nall religion rests. I do not say every religious idea, but whatever is\npositive, practical, powerful, durable, and popular. Everywhere, in all\nclimates, in all epochs of history, and in all degrees of civilization,\nman is animated by the sentiment--I would rather say, the\npresentiment--that the world in which he lives, the order of things in\nthe midst of which he moves, the facts which regularly and constantly\nsucceed each other, are not _all_. In vain he daily makes discoveries\nand conquests in this vast universe; in vain he observes and learnedly\nverifies the general laws which govern it; _his thought is not inclosed\nin the world surrendered to his science_; the spectacle of it does not\nsuffice his soul, it is raised beyond it; it searches after and catches\nglimpses of something beyond it; it aspires higher both for the universe\nand itself; it aims at another destiny, another master.\n\n[Footnote 130: Vaughan, \"Hours with the Mystics,\" vol. i. p. 44.]\n\n[Footnote 131: Id. ib., vol. i. p. 65.]\n\n      \"'Par dela tous ces cieux le Dieu des cieux reside.'\"[132]\n\nSo Voltaire has said, and the God who is beyond the skies is not nature\npersonified, but a supernatural Personality. It is to this highest\nPersonality that all religions address themselves. It is to bring man\ninto communion with Him that they exist.[133]\n\n[Footnote 132: \"Beyond all these heavens the God of the heavens\nresides.\"]\n\n[Footnote 133: Guizot, \"L'Eglise et la Societe Chretiennes\" en 1861.]\n\n4. The Athenians had that deep consciousness of sin and guilt, and of\nconsequent liability to punishment, which confesses the need of\nexpiation by piacular sacrifices.\n\nEvery man feels himself to be an accountable being, and he is conscious\nthat in wrong-doing he is deserving of blame and of punishment. Deep\nwithin the soul of the transgressor is the consciousness that he is a\nguilty man, and he is haunted with the perpetual apprehension of a\nretribution which, like the spectre of evil omen, crosses his every\npath, and meets him at every turn.\n\n     \"'Tis guilt alone,\n       Like brain-sick frenzy in its feverish mode,\n       Fills the light air with visionary terrors,\n       And shapeless forms of fear.\"\n\nMan does not possess this consciousness of guilt so much as it holds\npossession of him. It pursues the fugitive from justice, and it lays\nhold on the man who has resisted or escaped the hand of the executioner.\nThe sense of guilt is a power over and above man; a power so wonderful\nthat it often compels the most reckless criminal to deliver himself up,\nwith the confession of his deed, to the sword of justice, when a\nfalsehood would have easily protected him. Man is only able by\npersevering, ever-repeated efforts at self-induration, against the\nremonstrances of conscience, to withdraw himself from its power. His\nsuccess is, however, but very partial; for sometimes, in the moments of\nhis greatest security, the reproaches of conscience break in upon him\nlike a flood, and sweep away all his refuge of lies. \"The evil\nconscience is the divine bond which binds the created spirit, even in\ndeep apostasy, to its Original. In the consciousness of guilt there is\nrevealed the essential relation of our spirit to God, although\nmisunderstood by man until he has something higher than his evil\nconscience. The trouble and anguish which the remonstrances of this\nconsciousness excite--the inward unrest which sometimes seizes the slave\nof sin--are proofs that he has not quite broken away from God.\"[134]\n\n[Footnote 134: Mueller, \"Christian Doctrine of Sin,\" vol. i. pp. 225,\n226.]\n\nIn Grecian mythology there was a very distinct recognition of the power\nof conscience, and a reference of its authority to the Divinity,\ntogether with the idea of retribution. Nemesis was regarded as the\nimpersonation of the upbraidings of conscience, of the natural dread of\npunishment that springs up in the human heart after the commission of\nsin. And as the feeling of remorse may be considered as the consequence\nof the displeasure and vengeance of an offended God, Nemesis came to be\nregarded as the goddess of retribution, relentlessly pursuing the guilty\nuntil she has driven them into irretrievable woe and ruin. The Erinyes\nor Eumenides are the deities whose business it is to punish, in hades,\nthe crimes committed upon earth. When an aggravated crime has excited\ntheir displeasure they manifest their greatest power in the disquietude\nof conscience.\n\nAlong with this deep consciousness of guilt, and this fear of\nretribution which haunts the guilty mind, there has also rested upon the\nheart of universal humanity a deep and abiding conviction that\n_something must be done to expiate the guilt of sin_--some restitution\nmust be made, some suffering must be endured,[135] some sacrifice\noffered to atone for past misdeeds. Hence it is that men in all ages\nhave had recourse to penances and prayers, to self-inflicted tortures\nand costly sacrifices to appease a righteous anger which their sins had\nexcited, and avert an impending punishment. That sacrifice to atone for\nsin has prevailed universally--that it has been practised \"_sem-per,\nubique, et ab omnibus,_\" always, in all places, and by all men--will not\nbe denied by the candid and competent inquirer. The evidence which has\nbeen collected from ancient history by Grotius and Magee, and the\nadditional evidence from contemporaneous history, which is being now\nfurnished by the researches of ethnologists and Christian missionaries,\nis conclusive. No intelligent man can doubt the fact. Sacrificial\nofferings have prevailed in every nation and in every age. \"Almost the\nentire worship of the pagan nations consisted in rites of deprecation.\nFear of the Divine displeasure seems to have been the leading feature of\ntheir religious impressions; and in the diversity, the costliness, the\ncruelty of their sacrifices they sought to appease gods to whose wrath\nthey felt themselves exposed, from a consciousness of sin, unrelieved by\nany information as to the means of escaping its effects.\"[136]\n\n[Footnote 135: Punishment is the penalty due to sin; or, to use the\nfavorite expression of Homer, not unusual in the Scriptures also, it is\nthe payment of a debt incurred by sin. When he is punished, the criminal\nis said to pay off or pay back (apotinein) his crimes; in other words,\nto expiate or atone for them (Iliad, iv. 161,162),\n\n      syn te megalo apetisan syn sphesin kephalesi gynaixi te kai\n      tekeessin.\n\nthat is, they shall pay off, pay back, atone, etc., for their treachery\nwith a great price, with their lives, and their wives and\nchildren.--Tyler, \"Theology of Greek Poets,\" p. 194.]\n\n[Footnote 136: Magee, \"On the Atonement,\" No. V. p. 30.]\n\nIt must be known to every one at all acquainted with Greek mythology\nthat the idea of _expiation_--atonement--was a fundamental idea of their\nreligion. Independent of any historical research, a very slight glance\nat the Greek and Roman classics, especially the poets, who were the\ntheologians of that age, can leave little doubt upon this head.[137]\nTheir language everywhere announces the notion of _propitiation_, and,\nparticularly the Latin, furnishes the terms which are still employed in\ntheology. We need only mention the words ilasmos, ilaskomai, lytron,\nperipsema, as examples from the Greek, and _placare, propitiare,\nexpiare, piaculum_, from the Latin. All these indicate that the notion\nof expiation was interwoven into the very modes of thought and framework\nof the language of the ancient Greeks.\n\n[Footnote 137: In Homer the doctrine is expressly taught that the gods\nmay, and sometimes do, remit the penalty, when duly propitiated by\nprayers and sacrifices accompanied by suitable reparations (\"Iliad,\" ix.\n497 sqq.). \"We have a practical illustration of this doctrine in the\nfirst book of the Iliad, where Apollo averts the pestilence from the\narmy, when the daughter of his priest is returned without ransom, and a\n_sacrifice_ (elatombe) is sent to the altar of the god at sacred\nChrysa.... Apollo hearkens to the intercession of his priest, accepts\nthe sacred hecatomb, is delighted with the accompanying songs and\nlibations, and sends back the embassy with a favoring breeze, and a\nfavorable answer to the army, who meanwhile had been _purifying_\n(apelymainonto) themselves, and offering unblemished hecatombs of bulls\nand goats on the shore of the sea which washes the place of their\nencampment.\"\n\n\"The object of the propitiatory embassy to Apollo is thus stated by\nUlysses: Agamemnon, king of men, has sent me to bring back thy daughter\nChryses, and to offer a sacred hecatomb for (yper) the Greeks, that we\nmay _propitiate_ (ilasomestha) the king, who now sends woes and many\ngroans upon the Argives\" (442 sqq.).--Tyler, \"Theology of Greek Poets,\"\npp. 196, 197.]\n\nWe do not deem it needful to discuss at length the question which has\nbeen so earnestly debated among theologians, as to whether the idea of\nexpiation be a primitive and necessary idea of the human mind, or\nwhether the practice of piacular sacrifices came into the post-diluvian\nworld with Noah, as a positive institution of a primitive religion then\nfirst directly instituted by God. On either hypothesis the practice of\nexpiatory rites derives its authority from God; in the latter case, by\nan outward and verbal revelation, in the former by an inward and\nintuitive revelation.\n\nThis much, however, must be conceded on all hands, that there are\ncertain fundamental intuitions, universal and necessary, which underlie\nthe almost universal practice of expiatory sacrifice, namely, _the\nuniversal consciousness of guilt, and the universal conviction that\nsomething must be done to expiate guilt_, to compensate for wrong, and\nto atone for past misdeeds. But _how_ that expiation can be effected,\nhow that atonement can be made, is a question which reason does not seem\ncompetent to answer. That personal sin can be atoned for by vicarious\nsuffering, that national guilt can be expiated and national punishment\naverted by animal sacrifices, or even by human sacrifices, is repugnant\nto rather than conformable with natural reason. There exists no\ndiscernible connection between the one and the other. We may suppose\nthat eucharistic, penitential, and even deprecatory sacrifices may have\noriginated in the light of nature and reason, but we are unable to\naccount for the practice of piacular sacrifices for substitutional\natonement, on the same principle. The ethical principle, that one's own\nsins are not transferable either in their guilt or punishment, is so\nobviously just that we feel it must have been as clear to the mind of\nthe Greek who brought his victim to be offered to Zeus, as it is to the\nphilosophic mind of to-day.[138] The knowledge that the Divine\ndispleasure can be averted by sacrifice is not, by Plato, grounded upon\nany intuition of reason, as is the existence of God, the idea of the\ntrue, the just, and good, but on \"tradition,\"[139] and the\n\"interpretations\" of Apollo. \"To the Delphian Apollo there remains the\ngreatest, noblest, and most important of legal institutions--the\nerection of temples, sacrifices, and other services to the gods,... and\nwhat other services should be gone through with a view to their\n_propitiation_. Such things as these, indeed, _we neither know\nourselves, nor in founding the State would we intrust them to others_,\nif we be wise;... the god of the country is the natural interpreter to\nall men about such matters.\"[140]\n\n[Footnote 138: \"He that hath done the deed, to suffer for it--thus cries\na proverb thrice hallowed by age.\"--AEschylus, \"Choeph,\" 311.]\n\n[Footnote 139: \"Laws,\" book vi. ch. xv.]\n\n[Footnote 140: \"Republic,\" book iv. ch. v.]\n\nThe origin of expiatory sacrifices can not, we think, be explained\nexcept on the principle of a primitive revelation and a positive\nappointment of God. They can not be understood except as a\ndivinely-appointed symbolism, in which there is exhibited a confession\nof personal guilt and desert of punishment; an intimation and a hope\nthat God will be propitious and merciful; and a typical promise and\nprophecy of a future Redeemer from sin, who shall \"put away sin by the\nsacrifice of himself.\" This sacred rite was instituted in connection\nwith the _protevangelium_ given to our first parents; it was diffused\namong the nations by tradition, and has been kept alive as a general,\nand, indeed, almost universal observance, by that deep sense of sin, and\nconsciousness of guilt, and personal urgency of the need of a\nreconciliation, which are so clearly displayed in Grecian mythology.\n\nThe legitimate inference we find ourselves entitled to draw from the\nwords of Paul, when fairly interpreted in the light of the past\nreligious history of the world, is, that the Athenians were a religious\npeople; that is, _they were, however unknowing, believers in and\nworshippers of the One Supreme God_.\n\n\n\n\nCHAPTER IV.\n\nTHE RELIGION OF THE ATHENIANS: ITS MYTHOLOGICAL AND SYMBOLICAL ASPECTS.\n\n\n\"That there is one Supreme Deity, both philosophers and poets, and even\nthe vulgar worshippers of the gods themselves frequently acknowledge;\nwhich because the assertors of gods well understood, they affirm these\ngods of theirs to preside over the several parts of the world, yet so\nthat there is only one chief governor. Whence it follows, that all their\nother gods can be no other than ministers and officers which one\ngreatest God, who is omnipotent, hath variously appointed, and\nconstituted so as to serve his command.\"--LACTANTIUS.\n\nThe conclusion reached in the previous chapter that the Athenians were\nbelievers in and worshippers of the One Supreme God, has been challenged\nwith some considerable show of reason and force, on the ground that they\nwere _Polytheists_ and _Idolaters_.\n\nAn objection which presents itself so immediately on the very face of\nthe sacred narrative, and which is sustained by the unanimous voice of\nhistory, is entitled to the fullest consideration. And as the interests\nof truth are infinitely more precious than the maintenance of any\ntheory, however plausible, we are constrained to accord to this\nobjection the fullest weight, and give to it the most impartial\nconsideration. We can not do otherwise than at once admit that the\nAthenians were _Polytheists_--they worshipped \"many gods\" besides \"the\nunknown God.\" It is equally true that they were _Idolaters_--they\nworshipped images or statues of the gods, which images were also, by an\neasy metonymy, called \"gods.\"\n\nBut surely no one supposes that this is all that can be said upon the\nsubject, and that, after such admissions, the discussion must be closed.\nOn the contrary, we have, as yet, scarce caught a glimpse of the real\ncharacter and genius of Grecian polytheistic worship, and we have not\nmade the first approach towards a philosophy of Grecian mythology.\n\nThe assumption that the heathen regarded the images \"graven by art and\ndevice of man\" as the real creators of the world and man, or as having\nany control over the destinies of men, sinks at once under the weight of\nits own absurdity. Such hypothesis is repudiated with scorn and\nindignation by the heathens themselves. Cotta, in _Cicero_, declares\nexplicitly: \"though it be common and familiar language amongst us to\ncall corn Ceres, and wine Bacchus, yet who can think any one so mad as\nto take that to be really a god that he feeds upon?\"[141] And _Plutarch_\ncondemns the whole practice of giving the names of gods and goddesses to\ninanimate objects, as absurd, impious, and atheistical: \"they who give\nthe names of gods to senseless matter and inanimate things, and such as\nare destroyed by men in the using, beget most wicked and atheistical\nopinions in the minds of men, since it can not be conceived how these\nthings should be gods, for nothing that is inanimate is a god.\"[142] And\nso also the Hindoo, the Buddhist, the American Indian, the Fijian of\nto-day, repel the notion that their visible images are real gods, or\nthat they worship them instead of the unseen God.\n\n[Footnote 141: Cudworth's \"Intell. System,\" vol. ii. p. 257, Eng. ed.]\n\n[Footnote 142: Quoted in Cudworth's \"Intell. System,\" vol. ii. p. 258,\nEng. ed.]\n\nAnd furthermore, that even the invisible divinities which these images\nwere designed to represent, were each independent, self-existent beings,\nand that the stories which are told concerning them by Homer and Hesiod\nwere received in a literal sense, is equally improbable. The earliest\nphilosophers knew as well as we know, that the Deity, in order to be\nDeity, must be either _perfect_ or nothing--that he must be _one_, not\nmany--without parts and passions; and they were scandalized and shocked\nby the religious fables of the ancient mythology as much as we are.\n_Xenophanes_, who lived, as we know, before Pythagoras, accuses Homer\nand Hesiod of having ascribed to the gods every thing that is\ndisgraceful amongst men, as stealing, adultery, and deceit. He remarks\n\"that men seem to have created their gods, and to have given them their\nown mind, and voice, and figure.\" He himself declares that \"God is\n_one,_ the greatest amongst gods and men, neither in form nor in thought\nlike unto men.\" He calls the battles of the Titans and the Giants, and\nthe Centaurs, \"the inventions of former generations,\" and he demands\nthat God shall be praised in holy songs and nobler strains.[143]\nDiogenes Laertius relates the following of _Pythagoras_, \"that when he\ndescended to the shades below, he saw the soul of Hesiod bound to a\npillar of brass and gnashing his teeth; and that of Homer, as suspended\non a tree, and surrounded by serpents; as a punishment for the things\nthey had said of the gods.\"[144] These poets, who had corrupted\ntheology, _Plato_ proposes to exclude from his ideal Republic; or if\npermitted at all, they must be subjected to a rigid expurgation. \"We\nshall,\" says he, \"have to repudiate a large part of those fables which\nare now in vogue; and, especially, of what I call the greater\nfables,--the stories which Hesiod and Homer tell us. In these stories\nthere is a fault which deserves the gravest condemnation; namely, when\nan author gives a _bad representation of gods and heroes_. We must\ncondemn such a poet, as we should condemn a painter, whose pictures bear\nno resemblance to the objects which he tries to imitate. For instance,\nthe poet Hesiod related an ugly story when he told how Uranus acted, and\nhow Kronos had his revenge upon him. They are offensive stories, and\nmust not be repeated in our cities. Not yet is it proper to say, in any\ncase,--what is indeed untrue--that gods wage war against gods, and\nintrigue and fight among themselves. Stories like the chaining of Juno\nby her son Vulcan, and the flinging of Vulcan out of heaven for trying\nto take his mother's part when his father was beating her, and all other\nbattles of the gods which are found in Homer, must be refused admission\ninto our state, _whether they are allegorical or not_. For a child can\nnot discriminate between what is allegorical and what is not; and\nwhatever is adopted, as a matter of belief, in childhood, has a tendency\nto become fixed and indelible; and therefore we ought to esteem it as of\nthe greatest importance that the fables which children first hear should\nbe adapted, as far as possible, to promote virtue.\"[145]\n\n[Footnote 143: Max Muller, \"Science of Language,\" pp. 405, 406.]\n\n[Footnote 144: \"Lives,\" bk. viii. ch. xix. p. 347.]\n\n[Footnote 145: \"Republic,\" bk. ii. ch. xvii.]\n\nIf, then, poetic and allegorical representations of divine things are to\nbe permitted in the ideal republic, then the founders of the state are\nto prescribe \"the moulds in which the poets are to cast their fictions.\"\n\n\"Now what are these moulds to be in the case of _Theology?_ They may be\ndescribed as follows: It is right always to represent God as he really\nis, whether the poet describe him in an epic, or a lyric, or a dramatic\npoem. Now God is, beyond all else, _good in reality_, and therefore so\nto be represented. But nothing that is good is hurtful. That which is\ngood hurts not; does no evil; is the cause of no evil. That which is\ngood is beneficial; is the cause of good. And, therefore, that which is\ngood is not the cause of _all_ which is and happens, but only of that\nwhich is as it should be.... The good things we must ascribe to God,\nwhilst we must seek elsewhere, and not in him, the causes of evil\nthings.\"\n\nWe must, then, express our disapprobation of Homer, or any other poet,\nwho is guilty of such a foolish blunder as to tell us (Iliad, xxiv. 660)\nthat:\n\n     'Fast by the threshold of Jove's court are placed\n     Two casks--one stored with evil, one with good:'\n\nand that he for whom the Thunderer mingles both--\n\n     'He leads a life checkered with good and ill.'\n\nBut as for the man to whom he gives the bitter cup unmixed--\n\n                                         'He walks\n     The blessed earth unbless'd, go where he will.'\n\nAnd if any one asserts that the violation of oaths and treaties by the\nact of Pandarus was brought about by Athene and Zeus (Iliad, ii. 60), we\nshould refuse our approbation. Nor can we allow it to be said that the\nstrife and trial of strength between tween the gods (Iliad, xx.) was\ninstigated by Themis and Zeus.... Such language can not be used without\nirreverence; it is both injurious to us, and contradictory in\nitself.[146]\n\nInasmuch as God is perfect to the utmost in beauty and goodness, _he\nabides ever the same_, and without any variation in his form. Then let\nno poet tell us that (Odyss. xvii. 582)\n\n                   'In similitude of strangers oft\n     The Gods, who can with ease all shapes assume,\n     Repair to populous cities.'\n\nAnd let no one slander Proteus and Thetis, or introduce in tragedies, or\nany other poems, Hera transformed into the guise of a princess\ncollecting\n\n     'Alms for the life-giving children of Inachus, river of Argos,'\n\nnot to mention many other falsehoods which we must interdict.[147]\n\n\"When a poet holds such language concerning the gods, we shall be angry\nwith him, and refuse him a chorus. Neither shall we allow our teachers\nto use his writings for the instruction of the young, if we would have\nour guards grow up to be as god-like and god-fearing as it is possible\nfor men to be.\"[148]\n\nWe are thus constrained by the statements of the heathens themselves, as\nwell as by the dictates of common sense, to look beyond the external\ndrapery and the material forms of Polytheism for some deeper and truer\nmeaning that shall be more in harmony with the facts of the universal\nreligious consciousness of our race. The religion of ancient Greece\nconsisted in something more than the fables of Jupiter and Juno, of\nApollo and Minerva, of Venus and Bacchus. \"Through the rank and\npoisonous vegetation of mythic phraseology, we may always catch a\nglimpse of an original stem round which it creeps and winds itself, and\nwithout which it can not enjoy that parasitical existence which has been\nmistaken for independent vitality.\"[149]\n\n[Footnote 146: \"Republic,\" bk. ii. ch. xix.]\n\n[Footnote 147: \"Republic,\" bk. ii. ch. xx. Much more to the same effect\nmay be seen in ch. ii.]\n\n[Footnote 148: \"Republic,\" bk. ii. ch. xxi.]\n\n[Footnote 149: Max Mueller, \"Science of Language,\" 2d series, p. 433.]\n\nIt is an obvious truth, attested by the voice of universal consciousness\nas revealed in history, that the human mind can never rest satisfied\nwithin the sphere of sensible phenomena. Man is impelled by an inward\nnecessity to pass, in thought, beyond the boundary-line of sense, and\ninquire after causes and entities which his reason assures him must lie\nbeneath all sensible appearances. He must and will interpret nature\naccording to the forms of his own personality, or according to the\nfundamental ideas of his own reason. In the childlike subjectivity of\nthe undisciplined mind he will either transfer to nature the phenomena\nof his own personality, regarding the world as a living organism which\nhas within it an informing soul, and thus attain a _pantheistic_\nconception of the universe; or else he will fix upon some extraordinary\nand inexplicable phenomenon of nature, and, investing it with\n_super-natural_ significance, will rise from thence to a religious and\n_theocratic_ conception of nature as a whole. An intelligence--a mind\n_within_ nature, and inseparable from nature, or else _above_ nature and\ngoverning nature, is, for man, an inevitable thought.\n\nIt is equally obvious that humanity can never relegate itself from a\nsupernatural origin, neither can it ever absolve itself from a permanent\ncorrelation with the Divine. Man feels within him an instinctive\nnobility. He did not arise out of the bosom of nature; in some\nmysterious way he has descended from an eternal mind, he is \"the\noffspring of God.\" And furthermore, a theocratic conception of nature,\nassociated with a pre-eminent regard for certain apparently supernatural\nexperiences in the history of humanity, becomes the foundation of\ngovernments, of civil authority, and of laws. Society can not be founded\nwithout the aid of the Deity, and a commonwealth can only be organized\nby Divine interposition. \"A Ceres must appear and sow the fields with\ncorn.\" And a Numa or a Lycurgus must be heralded by the oracle as\n\n     \"Dear to Jove, and all who sit in the halls of the Olympus.\"\n\nHe must be a \"descendant of Zeus,\" appointed by the gods to rule, and\none who will \"prove himself a god.\" These divinely-appointed rulers were\nregarded as the ministers of God, the visible representatives of the\nunseen Power which really governs all. The divine government must also\nhave its invisible agents--its Nemesis, and Themis, and Dike, the\nministers of law, of justice, and of retribution; and its Jupiter, and\nJuno, and Neptune, and Pluto, ruling, with delegated powers, in the\nheavens, the air, the sea, and the nethermost regions. So that, in fact,\nthere exists no nation, no commonwealth, no history without a Theophany,\nand along with it certain sacred legends detailing the origin of the\npeople, the government, the country itself, and the world at large. This\nis especially true of India, Egypt, Greece, and Rome. Their primitive\nhistory is eminently _mythological_.\n\nGrecian polytheism can not be otherwise regarded than as a\npoetico-historical religion of _myth_ and _symbol_ which is under-laid\nby a natural Theism; a parasitical growth which winds itself around the\noriginal stem of instinctive faith in a supernatural Power and Presence\nwhich pervades the universe. The myths are oral traditions, floating\ndown from that dim; twilight of _poetic_ history, which separates real\nhistory, with its fixed chronology, from the unmeasured and unrecorded\neternity--faint echoes from that mystic border-land which divides the\nnatural from the supernatural, and in which they seem to have been\nmarvellously commingled. They are the lingering memories of those\nmanifestations of God to men, in which he or his celestial ministers\ncame into visible intercourse with our race; the reality of which is\nattested by sacred history. In all these myths there is a theogonic and\ncosmogonic element. They tell of the generation of the celestial and\naerial divinities--the subordinate agents and ministers of the Divine\ngovernment. They attempt an explanation of the genesis of the visible\nuniverse, the origin of humanity, and the development of human society.\nIn the presence of history, the substance of these myths is preserved by\n_symbols_, that is, by means of natural or artificial, real or striking\nobjects, which, by some analogy or arbitrary association, shall suggest\nthe _idea_ to the mind. These symbols were designed to represent the\ninvisible attributes and operations of the Deity; the powers that\nvitalize nature, that control the elements, that preside over cities,\nthat protect the nations: indeed, all the agencies of the physical and\nmoral government of God. Beneath all the pagan legends of gods, and\nunderlying all the elaborate mechanism of pagan worship, there are\nunquestionably philosophical ideas, and theological conceptions, and\nreligious sentiments, which give as meaning, and even a mournful\ngrandeur to the whole.\n\nWhilst the pagan polytheistic worship is, under one aspect, to be\nregarded as a departure from God, inasmuch as it takes away the honor\ndue to God alone, and transfers it to the creature; still, under another\naspect, we can not fail to recognize in it the effort of the human mind\nto fill up the chasm that seemed, to the undisciplined mind, to separate\nGod and man--and to bridge the gulf between the visible and the\ninvisible, the finite and the infinite. It was unquestionably an attempt\nto bring God nearer to the sense and comprehension of man. It had its\norigin in that instinctive yearning after the supernatural, the Divine,\nwhich dwells in all human hearts, and which has revealed itself in all\nphilosophies, mysticisms, and religions.[150] This longing was\nstimulated by the contemplation of the living beauty and grandeur of the\nvisible universe, which, to the lively fancy and deep feeling of the\nGreeks, seemed as the living vesture of the Infinite Mind,--the temple\nof the eternal Deity. In this visible universe the Divinity was partly\nrevealed, and partly concealed. The unity of the all-pervading\nIntelligence was veiled beneath an apparent diversity of power, and a\nmanifoldness of operations. They caught some glimpses of this universal\npresence in nature, but were more immediately and vividly impressed by\nthe several manifestations of the divine perfections and divine\noperations, as so many separate rays of the Divinity, or so many\nsubordinate agents and functionaries employed to execute the will and\ncarry out the purposes of the Supreme Mind.[151] That unseen,\nincomprehensible Power and Presence was perceived in the sublimity of\nthe deep blue sky, the energy of the vitalizing sun, the surging of the\nsea, the rushing wind, the roaring thunder, the ripening corn, and the\nclustering vine. To these separate manifestations of the Deity they gave\n_personal names_, as Jupiter to the heavens, Juno to the air, Neptune to\nthe sea, Ceres to the corn, and Bacchus to the vine. These personals\ndenoted, not the things themselves, but the invisible, divine powers\nsupposed to preside over those several departments of nature. By a kind\nof prosopopoeia \"they spake of the things in nature, and parts of the\nworld, as persons--and consequently as so many gods and goddesses--yet\nso as the intelligent might easily understand their meaning, _that these\nwere in reality nothing else but so many names and notions of that one\nNumen,--divine force and power which runs through all the world,\nmultiformly displaying itself._\"[152] \"Their various deities were but\ndifferent names, different conceptions, of that Incomprehensible Being\nwhich no _thought_ can reach, and no _language_ express.\"[153] Having\ngiven to these several manifestations of the Divinity personal names,\nthey now sought to represent them to the eye of sense by _visible\nforms_, as the symbols or images of the perfections of the unseen, the\nincomprehensible, the unknown God. And as the Greeks regarded man as the\nfirst and noblest among the phenomena of nature, they selected the human\nform as the highest sensible manifestation of God, the purest symbol of\nthe Divinity. Grecian polytheism was thus a species of _mythical\nanthropomorphism_.\n\n[Footnote 150: The original constitution of man is such that he \"seeks\nafter\" God Acts xvii. 27. \"All men yearn after the gods\" (Homer,\n\"Odyss.\" iii. 48).]\n\n[Footnote 151: \"Heathenism springs directly from this, that the mind\nlays undue stress upon the bare letter in the book of creation; that it\nseparates and individualizes its objects as far as possible; that it\nplaces the sense of the individual part, in opposition to the sense of\nthe whole,--to the _analogia fidei_ or _spiritus_ which alone gives\nunity to the book of nature, while it dilutes and renders as transitory\nas possible the sense of the universal in the whole.... And as it laid\ngreat stress upon the letter in the book of nature, it fell into\npolytheism. The particular symbol of the divine, or of the Godhead,\nbecame a myth of some special deity.\"--Lange's \"Bible-work,\" Genesis, p.\n23.]\n\n[Footnote 152: Cudworth, \"Intellect. System,\" vol. i. p. 308.]\n\n[Footnote 153: Max Mueller, \"Science of Language,\" p. 431.]\n\nA philosophy of Grecian mythology, such as we have outlined in the\npreceding paragraphs, is, in our judgment, perfectly consistent with the\nviews announced by Paul in his address to the Athenians. He intimates\nthat the Athenians \"thought that the Godhead was _like unto_ (e nai\nomoion)--to be imaged or represented by human art--by gold, and silver,\nand precious stone graven by art, and device of man;\" that is, they\nthought the perfections of God could be represented to the eye by an\nimage, or symbol. The views of Paul are still more articulately\nexpressed in Romans, i. 23, 25: \"They changed the glory of the\nincorruptible God into the _similitude of an image_ of corruptible\nman,.... and they worshipped and served the thing made, para--_rather_\nthan, or _more_ than the Creator.\" Here, then, the apostle intimates,\nfirst, that the heathen _knew_ God,[154] and that they worshipped God.\nThey worshipped the creature besides or even more than God, but still\nthey also worshipped God. And, secondly, they represented the\nperfections of God by an image, and under this, as a \"_likeness_\" or\nsymbol, they indirectly worshipped God. Their religious system was,\nthen, even to the eye of Paul, a _symbolic_ worship--that is, the\nobjects of their devotion were the _omoiomata_--the similitudes, the\nlikenesses, the images of the perfections of the invisible God.\n\n[Footnote 154: Verse 21.]\n\nIt is at once conceded by us, that the \"sensus numinis,\" the natural\nintuition of a Supreme Mind, whose power and presence are revealed in\nnature, can not maintain itself, as an influential, and vivifying, and\nregulative belief amongst men, without the continual supernatural\ninterposition of God; that is, without a succession of Divine\nrevelations. And further, we grant that, instead of this symbolic mode\nof worship deepening and vitalizing the sense of God as a living power\nand presence, there is great danger that the symbol shall at length\nunconsciously take the place of God, and be worshipped instead of Him.\nFrom the purest form of symbolism which prevailed in the earliest ages,\nthere may be an inevitable descent to the rudest form of false worship,\nwith its accompanying darkness, and abominations, and crimes; but, at\nthe same time, let us do justice to the religions of the ancient\nworld--the childhood stammerings of religious life--which were something\nmore than the inventions of designing men, or the mere creations of\nhuman fancy; they were, in the words of Paul, \"a _seeking after God_, if\nhaply they might feel after him, and find him, who is not far from any\none of us.\" It can not be denied that the more thoughtful and\nintelligent Greeks regarded the visible objects of their devotion as\nmere symbols of the perfections and operations of the unseen God, and of\nthe invisible powers and subordinate agencies which are employed by him\nin his providential and moral government of the world. And whatever\nthere was of misapprehension and of \"ignorance\" in the popular mind, we\nhave the assurance of Paul that it was \"_overlooked_\" by God.\n\nThe views here presented will, we venture to believe, be found most in\nharmony with a true philosophy of the human mind; with the religious\nphenomena of the world; and, as we shall subsequently see, with the\nwritings of those poets and philosophers who may be fairly regarded as\nrepresenting the sentiments and opinions of the ancient world. At the\nsame time, we have no desire to conceal the fact that this whole\nquestion as to the origin, and character, and philosophy of the\nmythology and symbolism of the religions of the ancient world has been a\nsubject of earnest controversy from Patristic times down to the present\nhour, and that even to-day there exists a wide diversity of opinion\namong philosophers, as well as theologians.\n\nThe principal theories offered may be classed as the _ethical_, the\n_physical_, and the _historical_, according to the different objects the\nframers of the myths are supposed to have had in view. Some have\nregarded the myths as invented by the priests and wise men of old for\nthe improvement and government of society, as designed to give authority\nto laws, and maintain social order.[155] Others have regarded them as\nintended to be allegorical interpretations of physical phenomena--the\npoetic embodiment of the natural philosophy of the primitive races of\nmen;[156] whilst others have looked upon them as historical legends,\nhaving a substratum of fact, and, when stripped of the supernatural and\nmiraculous drapery which accompanies fable, as containing the history of\nprimitive times.[157] Some of the latter class have imagined they could\nrecognize in Grecian mythology traces of sacred personages, as well as\nprofane; in fact, a dimmed image of the patriarchal traditions which are\npreserved in the Old Testament scriptures.[158]\n\nIt is beyond our design to discuss all the various theories presented,\nor even to give a history of opinions entertained.[159] We are fully\nconvinced that the hypothesis we have presented in the preceding pages,\nviz., _that Grecian mythology was a grand symbolic representation of the\nDivine as manifested in nature and providence_, is the only hypothesis\nwhich meets and harmonizes all the facts of the case. This is the theory\nof Plato, of Cudworth, Baumgarten, Max Mueller, and many other\ndistinguished scholars.\n\n[Footnote 155: Empedocles, Metrodorus.]\n\n[Footnote 156: Aristotle.]\n\n[Footnote 157: Hecataeus, Herodotus, some of the early Fathers, Niebuhr,\nJ.H. Voss, Arnold.]\n\n[Footnote 158: Bochart, G.J. Vossius, Faber, Gladstone.]\n\n[Footnote 159: To the English reader who desires an extended and\naccurate acquaintance with the classic and patristic literature of this\ndeeply interesting subject, we commend the careful study of Cudworth's\n\"Intellectual System of the Universe,\" especially ch. iv. The style of\nCudworth is perplexingly involved, and his great work is unmethodical in\nits arrangement and discussion. Nevertheless, the patient and\npersevering student will be amply rewarded for his pains. A work of more\nprofound research into the doctrine of antiquity concerning God, and\ninto the real import of the religious systems of the ancient world, is,\nprobably, not extant in any language.]\n\nThere are two fundamental propositions laid down by Cudworth which\nconstitute the basis of this hypothesis.\n\n1. _No well-authenticated instance can be furnished from among the Greek\nPolytheists of one who taught the existence of a multiplicity of\nindependenty uncreated, self-existent deities; they almost universally_\n_believed in the existence of_ ONE SUPREME, UNCREATED, ETERNAL GOD,\n\"_The Maker of all things_\"--\"_the Father of gods and men_,\"--\"_the\nsole Monarch and Ruler of the world_.\"\n\n2. _The Greek Polytheists taught a plurality of_\"GENERATED DEITIES,\"\n_who owe their existence to the power and will of the Supreme God, who\nare by Him invested with delegated powers, and who, as the agents of his\nuniversal providence, preside over different departments of the created\nuniverse_.\n\nThe evidence presented by Cudworth in support of his theses is so varied\nand so voluminous, that it defies all attempts at condensation. His\nvolumes exhibit an extent of reading, of patient research, and of varied\nlearning, which is truly amazing. The discussion of these propositions\ninvolves, in fact, nothing less than a complete and exhaustive survey of\nthe entire field of ancient literature, a careful study of the Greek and\nLatin poets, of the Oriental, Greek, and Alexandrian philosophers, and a\nreview of the statements and criticisms of Rabbinical and Patristic\nwriters in regard to the religions of the pagan world. An adequate\nconception of the varied and weighty evidence which is collected by our\nauthor from these fields, in support of his views, could only be\nconveyed by transcribing to our pages the larger portion of his\nmemorable _fourth_ chapter. But inasmuch as Grecian polytheism is, in\nfact, the culmination of all the mythological systems of the ancient\nworld, the fully-developed flower and ripened fruit of the cosmical and\ntheological conceptions of the childhood-condition of humanity, we\npropose to epitomize the results of his inquiry as to the _theological_,\nopinions of the Greeks, supplying additional confirmation of his views\nfrom other sources.\n\nAnd first, he proves most conclusively that Orpheus, Homer, and\nHesiod,[160] who are usually designated \"the theologians\" of Greece, but\nwho were in fact the depravers and corrupters of pagan theology, do not\nteach the existence of a multitude of _unmade, self-existent, and\nindependent deities_. Even they believed in the existence of _one_\nuncreated and eternal mind, _one Supreme God_, anterior and superior to\nall the gods of their mythology. They had some intuition, some\napperception of the _Divine_, even before they had attached to it a\nsacred name. The gods of their mythology had all, save one, a temporal\norigin; they were generated of Chaos and Night, by an active principle\ncalled _Love_. \"One might suspect,\" says Aristotle, that Hesiod, and if\nthere be any other who made _love_ or _desire_ a principle of things,\naimed at these very things (viz., the designation of the efficient cause\nof the world); for Parmenides, describing the generation of the\nuniverse, says:\n\n     'First of all the gods planned he _love_;'\n\nand further, Hesiod:\n\n     'First of all was Chaos, afterwards Earth,\n     With her spacious bosom,\n     And _Love_, who is pre-eminent among all the immortals;'\n\nas intimating here that in entities there should exist some _cause_ that\nwill impart motion, and hold bodies in union together. But how, in\nregard to these, one ought to distribute them, as to the order of\npriority, can be decided afterwards.[161]\n\n[Footnote 160: We do not concern ourselves with the chronological\nantecedence of these ancient Greek poets. It is of little consequence to\nus whether Homer preceded Orpheus, or Orpheus Homer. They were not the\nreal creators of the mythology of ancient Greece. The myths were a\nspontaneous growth of the earliest human thought even before the\nseparation of the Aryan family into its varied branches.\n\nThe study of Comparative Mythology, as well as of Comparative Language,\nassures us that the myths had an origin much earlier than the times of\nHomer and Orpheus. They floated down from ages on the tide of oral\ntradition before they were systematized, embellished, and committed to\nwriting by Homer, and Orpheus, and Hesiod. And between the systems of\nthese three poets a perceptible difference is recognizable, which\nreflects the changes that verbal recitations necessarily and\nimperceptibly undergo.]\n\n[Footnote 161: \"Metaphysics,\" bk. i. ch. iv.]\n\nNow whether this \"first principle,\" called \"_Love_,\" \"the cause of\nmotion and of union\" in the universe, was regarded as a personal Being,\nand whether, as the ancient scholiast taught, Hesiod's love was \"the\nheavenly Love, which is also God, that other love that was born of Venus\nbeing junior,\" is just now of no moment to the argument. The more\nimportant inference is, that amongst the gods of Pagan theology but\n_one_ is self-existent, or else none are. Because the Hesiodian gods,\nwhich are, in fact, all the gods of the Greek mythology, \"were either\nall of them derived from chaos, love itself likewise being generated out\nof it; or else love was supposed to be distinct from chaos, and the\nactive principle of the universe, from whence, together with chaos, all\nthe theogony and cosmogony was derived.\"[162] Hence it is evident the\npoets did not teach the existence of a multiplicity of unmade,\nself-existent, independent deities.\n\n[Footnote 162: \"Cudworth,\" vol. i. p. 287.]\n\nThe careful reader of Cudworth will also learn another truth of the\nutmost importance in this connection, viz., _that the theogony of the\nGreek poets was, in fact, a cosmogony_, the generation of the gods\nbeing, in reality, the generation of the heavens, the sun, the moon, the\nstars, and all the various powers and phenomena of nature. This is dimly\nshadowed forth in the very names which are given to some of these\ndivinities. Thus Helios is the sun, Selena is the moon, Zeus the\nsky--the deep blue heaven, Eos the dawn, and Erse the dew. It is\nrendered still more evident by the opening lines of Hesiod's\n\"Theogonia,\" in which he invokes the muses:\n\n     \"Hail ye daughters of Jupiter! Grant a delightsome song.\n     Tell of the race of immortal gods, always existing,\n     Who are the offspring of the earth, of the starry sky,\n     And of the gloomy night, whom also the ocean nourisheth.\n     Tell how the gods and the earth at first were made,\n     And the rivers, and the mighty deep, boiling with waves,\n     And the glowing stars, and the broad heavens above,\n     And the gods, givers of good, born of these.\"\n\nWhere we see plainly that the generation of the gods is the generation\nof the earth, the heaven, the stars, the seas, the rivers, and other\nthings produced by them. \"But immediately after invocation of the Muses\nthe poet begins with Chaos, and Tartara, and Love, as the first\nprinciples, and then proceeds to the production of the earth and of\nnight out of chaos; of the ether and of day, from night; of the starry\nheavens, mountains, and seas. All which generation of gods is really\nnothing but a poetic description of the cosmogonia; as through the\nsequel of the poem all seems to be physiology veiled under fiction and\nallegory.... Hesiod's gods are thus not only the animated parts of the\nworld, but also the other things of nature personified and deified, or\nabusively called gods and goddesses.\"[163] The same is true both of the\nOrphic and Homeric gods. \"Their generation of the gods is the same with\nthe generation or creation of the world, both of them having, in all\nprobability, derived it from the Mosaic cabala, or tradition.\"[164]\n\nBut in spite of all this mythological obscuration, the belief in one\nSupreme God is here and there most clearly recognizable. \"That Zeus was\noriginally to the Greeks the Supreme God, the true God--nay, at some\ntime their only God--can be perceived in spite of the haze which\nmythology has raised around his name.\"[165] True, they sometimes used\nthe word \"Zeus\" in a physical sense to denote the deep expanse of\nheaven, and sometimes in a historic sense, to designate a hero or\ndeified man said to have been born in Crete. It is also true that the\nHomeric Zeus is full of contradictions. He is \"all-seeing,\" yet he is\ncheated; he is \"omnipotent,\" yet he is defied; he is \"eternal,\" yet he\nhas a father; he is \"just,\" yet he is guilty of crime. Now, as Mueller\nvery justly remarks, these contradictions may teach us a lesson. If all\nthe conceptions of Zeus had sprung from one origin, these contradictions\ncould not have existed. If Zeus had simply and only meant the Supreme\nGod, he could not have been the son of Kronos (Time). If, on the other\nhand, Zeus had been a mere mythological personage, as Eos, the dawn, and\nHelios, the sun, he could never have been addressed as he is addressed\nin the famous prayer of Achilles (Iliad, bk. xxi.).[166]\n\n[Footnote 163: Cudworth, vol. i. pp. 321, 332.]\n\n[Footnote 164: Id., ib., vol. i. p. 478.]\n\n[Footnote 165: Max Mueller, \"Science of Language,\" p. 457.]\n\n[Footnote 166: Id., ib., p. 458.]\n\nIn Homer there is a perpetual blending of the natural and the\nsupernatural, the human and divine. The _Iliad_ is an incongruous medley\nof theology, physics, and history. In its gorgeous scenic\nrepresentations, nature, humanity, and deity are mingled in inextricable\nconfusion. The gods are sometimes supernatural and superhuman\npersonages; sometimes the things and powers of nature personified; and\nsometimes they are deified men. And yet there are passages, even in\nHomer, which clearly distinguish Zeus from all the other divinities, and\nmark him out as the Supreme. He is \"the highest, first of Gods\" (bk.\nxix. 284); \"most great, most glorious Jove\" (bk. ii. 474). He is \"the\nuniversal Lord\" (bk. xi. 229); \"of mortals and immortals king supreme,\"\n(bk. xii. 263); \"over all the immortal gods he reigns in unapproached\npre-eminence of power\" (bk. xv. 125). He is \"the King of kings\" (bk.\nviii. 35), whose \"will is sovereign\" (bk. iv. 65), and his \"power\ninvincible\" (bk. viii. 35). He is the \"eternal Father\" (bk. viii. 77).\nHe \"excels in wisdom gods and men; all human things from him proceed\"\n(bk. xiii. 708-10); \"the Lord of counsel\" (bk. i. 208), \"the all-seeing\nJove\" (bk. xiii. 824). Indeed the mere expression \"Father of gods and\nmen\" (bk. i. 639), so often applied to Zeus, and him _alone_, is proof\nsufficient that, in spite of all the legendary stories of gods and\nheroes, the idea of Zeus as the Supreme God, the maker of the world, the\nFather of gods and men, the monarch and ruler of the world, was not\nobliterated from the Greek mind.[167]\n\n[Footnote 167: \"In the order of legendary chronology Zeus comes after\nKronos and Uranos, but in the order of Grecian conception Zeus is the\nprominent person, and Kronos and Uranos are inferior and introductory\nprecursors, set up in order to be overthrown, and to serve as mementos\nof the powers of their conqueror. To Homer and Hesiod, as well as to the\nGreeks universally, Zeus is the great, the predominant God, 'the Father\nof gods and men,' whose power none of the gods can hope to resist, or\neven deliberately think of questioning. All the other gods have their\nspecific potency, and peculiar sphere of action and duty, with which\nZeus does not usually interfere; but it is he who maintains the\nlineaments of a providential government, as well over the phenomena of\nOlympus as over the earth.\"--Grote, \"Hist. of Greece,\" vol. i. p. 3.\n\nZeus is not only lord of heaven but likewise the ruler of the lower\nworld, and the master of the sea.--Welcher, \"Griechische Goetterlehre,\"\nvol. i. p. 164. The Zeus of the Greek poets is unquestionably the god of\nwhom Paul declared: In him we live and move, and have our being, as\ncertain of your own poets have also said--\n\n     \"'For we are his offspring.'\"\n\nNow whether this be a quotation from Aratus or Cleanthes, the language\nof the poets is, \"We are the offspring of Zeus;\" consequently the Zeus\nof the poets and the God of Christianity are the same God.\n\n\"The father of gods and men in Homer is, of course, the Universal Father\nof the Scriptures.\"--Tyler, \"Theology of Greek Poets,\" p. 171.]\n\n\"When Homer introduces Eumaios, the swineherd, speaking of this life and\nthe higher powers that rule it, he knows only of just gods 'who hate\ncruel deeds, but honor justice and the righteous works of men' (Od. xiv.\n83). His whole life is built up on a complete trust in the divine\ngovernment of the world without any artificial helps, as the Erinys, the\nNemesis, or Moira. 'Eat,' says the swineherd, 'and enjoy what is here,\nfor _God_[168] will grant one thing, but another he will refuse,\nwhatever he will in his mind, for he can do all things' (Od. xiv. 444;\nx. 306). This surely is religion, and it is religion untainted by\nmythology. Again, the prayer of the female slave, grinding corn in the\nhouse of Ulysses is religious in the truest sense--'Father Zeus, thou\nwho rulest over gods and men, surely thou hast just thundered in the\nstarry sky, and there is no cloud anywhere. Thou showest this as a sign\nto some one. Fulfill now, even to me, miserable wretch, the prayer which\nI now offer'\" (Od. xx. 141-150).[169]\n\n[Footnote 168: No sound reason can be assigned for translating _Theos_\nby \"_a_ god\" as some have proposed, rather than \"_God_.\" But even if it\nwere translated \"a god,\" this god must certainly be understood as Zeus.\nPlato tells us that Zeus is the most appropriate name for God. \"For in\nreality the name Zeus is, as it were, a sentence; and persons dividing\nit in two parts, some of us make use of one part, and some of another;\nfor some call him Zen, and some Dis. But these parts, collected together\ninto one, exhibit the nature of the God;... for there is no one who is\nmore the cause of living, both to us and everything else, than he who is\nthe ruler and king of all. It follows, therefore, that this god is\nrightly named, through whom _life_ is present in all living\nbeings.\"--Cratylus, Sec. 28.\n\nTheos was usually employed, says Cudworth, to designate _God_ by way of\npre-eminence, Theoi to designate inferior divinities.]\n\n[Footnote 169: Mueller, \"Science of Language,\" p. 434.]\n\nThe Greek tragedians were the great religious instructors of the\nAthenian people. \"Greek tragedy grew up in connection with religious\nworship, and constituted not only a popular but a sacred element in the\nfestivals of the gods.... In short, strange as it may sound to modern\nears, the Greek stage was, more nearly than any thing else, the Greek\npulpit.[170] With a priesthood that offered sacrifice, but did not\npreach, with few books of any kind, the people were, in a great measure,\ndependent on oral instruction for knowledge; and as they learned their\nrights and duties as citizens from their orators, so they hung on the\nlips of the 'lofty, grave tragedians' for instruction touching their\norigin, duty, and destiny as mortal and immortal beings.... Greek\ntragedy is essentially didactic, ethical, mythological, and\nreligious.\"[171]\n\n[Footnote 170: Pulpitum, a stage.]\n\n[Footnote 171: Tyler, \"Theology of Greek Poets,\" pp. 205, 206.]\n\nNow it is unquestionable that, with the tragedians, Zeus is the Supreme\nGod. AEschylus is pre-eminently the theological poet of Greece. The great\nproblems which lie at the foundation of religious faith and practice are\nthe main staple of nearly all his tragedies. Homer, Hesiod, the sacred\npoets, had looked at these questions in their purely poetic aspects. The\nsubsequent philosophers, Plato and Aristotle, developed them more fully\nby their didactic method. AEschylus stands on the dividing-line between\nthem, no less poetic than the former, scarcely less philosophical than\nthe latter, but more intensely practical, personal, and _theological_\nthan either. The character of the Supreme Divinity, as represented in\nhis tragedies, approaches more nearly to the Christian idea of God. He\nis the Universal Father--Father of gods and men; the Universal Cause\n(panaitios, Agamem. 1485); the All-seer and All-doer (pantopies,\npanergetes, ibid, and Sup. 139); the All-wise and All-controlling\n(pankrates, Sup. 813); the Just and the Executor of justice (dikephoros,\nAgamem. 525); true and incapable of falsehood (Prom. 1031);\n\n     pseudegorein gar ouk epistatai stoma\n     to dion, alla pan epos telei,--\n\nholy (agnos, Sup. 650); merciful (preumenes, ibid. 139); the God\nespecially of the suppliant and the stranger (Supplices, passim); the\nmost high and perfect One (teleion upsiston, Eumen. 28); King of kings,\nof the happy, most happy, of the perfect, most perfect power, blessed\nZeus (Sup. 522).[172] Such are some of the titles by which Zeus is most\nfrequently addressed; such the attributes commonly ascribed to him in\nAEschylus.\n\nSophocles was the great master who carried Greek tragedy to its highest\nperfection. Only seven out of more than a hundred of his tragedies have\ncome down to us. There are passages cited by Justin Martyr, Clemens\nAlexandrinus, and others which are not found in those tragedies now\nextant. The most famous and extensively quoted passage is given by\nCudworth.[173]\n\n     Eis tais aletheiaisin, eis estin Theos,\n     Os ouranon t' eteuxe kai gaian makran,\n     Poniou te karapon oidma, kanemon bian, k. t. l.[174]\n\nThis \"one only God\" is Zeus, who is the God of justice, and reigns\nsupreme:\n\n     \"Still in yon starry heaven supreme,\n     Jove, all-beholding, all-directing, dwells--\n     To him commit thy vengeance.\"--\"Electra,\" p. 174 sqq.\n\nThis description of the unsleeping, undecaying power and dominion of\nZeus is worthy of some Hebrew prophet--\n\n     \"Spurning the power of age, enthroned in might,\n     Thou dwell'st mid heaven's broad light;\n     This was in ages past thy firm decree,\n     Is now, and shall forever be:\n     That none of mortal race on earth shall know\n     A life of joy serene, a course unmarked by woe.\"\n\n     \"Antigone,\" pp. 606-614.[175]\n\n[Footnote 172: Tyler, \"Theology of Greek Poets,\" pp. 213, 214.]\n\n[Footnote 173: \"Intellectual Syst.,\" vol. i. p. 483.]\n\n[Footnote 174: \"There is, in truth, one only God, who made heaven and\nearth, the sea, air, and winds,\" etc.]\n\n[Footnote 175: \"Theology of Greek Poets,\" p. 322.]\n\nWhether we regard the poets as the principal theological teachers of the\nancient Greeks, or as the compilers, systematizers, and artistic\nembellishers of the theological traditions and myths which were afloat\nin the primitive Hellenic families, we can not resist the conclusion\nthat, for the masses of the people Zeus was the Supreme God, \"the God of\ngods\" as Plato calls him. Whilst all other deities in Greece are more or\nless local and tribal gods, Zeus was known in every village and to every\nclan. \"He is at home on Ida,[176] on Olympus, at Dodona.[177] While\nPoseidon drew to himself the AEolian family, Apollo the Dorian, Athene\nthe Ionian, there was one powerful God for all the sons of\nHellen--Dorians, AEolians, Ionians, Achaeans, viz., the Panhellenic\nZeus.\"[178] Zeus was the name invoked in their solemn nuncupations of\nvows--\n\n     \"O Zeus, father, O Zeus, king.\"\n\nIn moments of deepest sorrow, of immediate urgency and need, of greatest\nstress and danger, they had recourse to Zeus.\n\n     \"Courage, courage, my child!\n     There is still in heaven the great Zeus;\n     He watches over all things, and he rules.\n     Commit thy exceeding bitter griefs to him,\n     And be not angry against thine enemies,\n     Nor forget them.\"[179]\n\n[Footnote 176: \"Iliad,\" bk. iii. 324.]\n\n[Footnote 177: Bk. xvi. 268.]\n\n[Footnote 178: Mueller, p. 452.]\n\n[Footnote 179: Sophocles, \"Electra,\" v. 188.]\n\nHe was supplicated, as the God who reigns on high, in the prayer of the\nAthenian--\n\n     \"Rain, rain, O dear Zeus, on the land of the Athenians and on their\n     fields.\"\n\nIt has been urged that, as Zeus means the sky, therefore he is no more\nthan the deep concave of heaven personified and deified, and that\nconsequently Zeus is not the true, the only God. This argument is only\nequalled in feebleness by that of the materialist, who argues that\n\"spiritus\" means simply breath, therefore the breath is the soul. Even\nif the Greeks remembered that, originally, Zeus meant the sky, that\nwould have no more perplexed their minds than the remembrance that\n\"thymos\"--mind--meant originally blast. \"The fathers of Greek theology\ngave to that Supreme Intelligence, which they instinctively recognized\nas above and ruling over the universe, the name of Zeus; but in doing\nso, they knew well that by Zeus they meant more than the sky. The\nunfathomable depth, the everlasting calm of the ethereal sky was to\ntheir minds an image of that Infinite Presence which overshadows all,\nand looks down on all. As the question perpetually recurred to their\nminds, 'Where is he who abideth forever?' they lifted up their eyes, and\nsaw, as they thought, beyond sun, and moon, and stars, and all which\nchanges, and will change, the clear blue sky, the boundless firmament of\nheaven. That never changed, that was always the same. The clouds and\nstorms rolled far below it, and all the bustle of this noisy world; but\nthere the sky was still, as bright and calm as ever. The Almighty Father\nmust be there, unchangeable in the unchangeable heaven; bright, and\npure, and boundless like the heavens, and like the heavens, too, afar\noff.\"[180] So they named him after the sky, _Zeus_, the God who lives in\nthe clear heaven--the heavenly Father.\n\n[Footnote 180: Kingsley, \"Good News from God,\" p. 237, Am. ed.]\n\nThe high and brilliant sky has, in many languages and many religions,\nbeen regarded as the dwelling-place of God. Indeed, to all of us in\nChristian times \"God is above;\" he is \"the God of heaven;\" \"his throne\nis in the heavens;\" \"he reigns on high.\" Now, without doing any violence\nto thought, the name of the abode might be transferred to him who dwells\nin heaven. So that in our own language \"heaven\" may still be used as a\nsynonym for \"God.\" The prodigal son is still represented as saying, I\nhave sinned against \"_heaven_.\" And a Christian poet has taught us to\nsing--\n\n     \"High _heaven_, that heard my solemn vow,\n     That vow renewed shall daily hear,\" etc.\n\nWhenever, therefore, we find the name of heaven thus used to designate\nalso the Deity, we must bear in mind that those by whom it was\noriginally employed were simply transferring that name from an object\nvisible to the eye of sense to another object perceived by the eye of\nreason. They who at first called God \"_Heaven_\" had some conception\nwithin them they wished to name--the growing image of a God, and they\nfixed upon the vastest, grandest, purest object in nature, the deep blue\nconcave of heaven, overshadowing all, and embracing all, as the symbol\nof the Deity. Those who at a later period called heaven \"_God_\" had\nforgotten that they were predicating of heaven something more which was\nvastly higher than the heaven.[181]\n\n[Footnote 181: See \"Science of Language,\" p. 457.]\n\nNotwithstanding, then, that the instinctive, native faith of humanity in\nthe existence of one supreme God was overlaid and almost buried beneath\nthe rank and luxuriant vegetation of Grecian mythology, we can still\ncatch glimpses here and there of the solid trunk of native faith, around\nwhich this parasitic growth of fancy is entwined. Above all the\nphantasmata of gods and goddesses who descended to the plains of Troy,\nand mingled in the din and strife of battle, we can recognize an\novershadowing, all-embracing Power and Providence that dwells on high,\nwhich never descends into the battle-field, and is never seen by mortal\neyes--_the Universal King and Father,--the \"God of gods_.\"\n\nBesides the direct evidence, which is furnished by the poets and\nmythologists, of the presence of this universal faith in \"_the heavenly\nFather_,\" there is also a large amount of collateral testimony that this\nidea of one Supreme God was generally entertained by the Greek pagans,\nwhether learned or unlearned.[182] Dio Chrysostomus says that \"all the\npoets call the first and greatest God the Father, universally, of all\nrational kind, as also the King thereof. Agreeably with which doctrine\nof the poets do mankind erect altars to Jupiter-King (Dios Basileos) and\nhesitate not to call him Father in their devotions\" (Orat. xxxvi.). And\nMaximus Tyrius declares that both the learned and the unlearned\nthroughout the pagan world universally agree in this; that there is one\nSupreme God, the Father of gods and men. \"If,\" says he, \"there were a\nmeeting called of all the several trades and professions,... and all\nwere required to declare their sense concerning God, do you think that\nthe painter would say one thing, the sculptor another, the poet another,\nand the philosopher another? No; nor the Scythian neither, nor the\nGreek, nor the hyperborean. In regard to other things, we find men\nspeaking discordantly one to another, all men, as it were, differing\nfrom all men... Nevertheless, on this subject, you may find universally\nthroughout the world one agreeing law and opinion; _that there is one\nGod, the King and Father of all, and many gods, the sons of God,\nco-reigners together with God_\"(Diss. i. p. 450).\n\n[Footnote 182: Cudworth, vol. i. pp. 593, 594.]\n\nFrom the poets we now pass to the philosophers. The former we have\nregarded as reflecting the traditional beliefs of the unreasoning\nmultitude. The philosophers unquestionably represent the reflective\nspirit, the speculative thought, of the educated classes of Greek\nsociety. Turning to the writings of the philosophers, we may therefore\nreasonably expect that, instead of the dim, undefined, and nebulous form\nin which the religious sentiment revealed itself amongst the\nunreflecting portions of the Greek populations, we shall find their\ntheological ideas distinctly and articulately expressed, and that we\nshall consequently be able to determine their religious opinions with\nconsiderable accuracy.\n\nNow that Thales, Pythagoras, Xenophanes, Anaxagoras, Empedocles,\nSocrates, Plato, and Aristotle were all believers in the existence of\none supreme, uncreated, eternal God, has been, we think, clearly shown\nby Cudworth.[183]\n\n[Footnote 183: Vol. i. pp. 491-554.]\n\nIn subsequent chapters on \"_the Philosophers of Athens_,\" we shall enter\nmore fully into the discussion of this question. Meantime we assume\nthat, with few exceptions, the Greek philosophers were \"genuine\nTheists.\"\n\nThe point, however, with which we are now concerned is, _that whilst\nthey believed in one supreme, uncreated, eternal God, they at the same\ntime recognized the existence of a plurality of generated deities who\nowe their existence to the power and will of the Supreme God, and who,\nas the agents and ministers of His universal providence, preside over\ndifferent departments of the created universe_. They are at once\nMonotheists and Polytheists--believers in \"one God\" and \"many gods.\"\nThis is a peculiarity, an anomaly which challenges our attention, and\ndemands an explanation, if we would vindicate for these philosophers a\nrational Theism.\n\nNow that there can be but one infinite and absolutely perfect Being--one\nsupreme, uncreated, eternal God--is self-evident; therefore a\nmultiplicity of such gods is a contradiction and an impossibility. The\nearly philosophers knew this as well as the modern. The Deity, in order\nto be Deity, must be one and not many: must be perfect or nothing. If,\ntherefore, we would do justice to these old Greeks, we must inquire what\nexplanations they have offered in regard to \"the many gods\" of which\nthey speak. We must ascertain whether they regarded these \"gods\" as\ncreated or uncreated beings, dependent or independent, temporal or\neternal We must inquire in what sense the term \"god\" is applied to these\nlesser divinities,--whether it is not applied in an accommodated and\ntherefore allowable sense, as in the sacred Scriptures it is applied to\nkings and magistrates, and those who are appointed by God as the\nteachers and rulers of men. \"_They are called gods_ to whom the word of\nGod came.\"[184] And if it shall be found that all the gods of which they\nspeak, save _one_, are \"generated deities\"--dependent beings--creatures\nand subjects of the one eternal King and Father, and that the name of\n\"god\" is applied to them in an accommodated sense, then we have\nvindicated for the old Greek philosophers a consistent and rational\nTheism. In what relation, then, do the philosophers place \"_the gods_\"\nto the one Supreme Being?\n\n_Thales_, one of the most ancient of the Greek philosophers, taught the\nexistence of a plurality of gods, as is evident from that saying of his,\npreserved by Diogenes Laertius, \"The world has life, and is full of\ngods.\"[185] At the same time he asserts his belief in one supreme,\nuncreated Deity; \"God is the oldest of all things, because he is unmade,\nor ungenerated.\"[186] All the other gods must therefore have been\n\"generated deities,\" since there is but one unmade God, one only that\nhad \"no beginning.\"[187]\n\n[Footnote 184: See John x. 35.]\n\n[Footnote 185: \"Lives,\" bk. i.; see also Aristotle's \"De Anima,\" bk. i.\nch. viii. panta Thion plere.]\n\n[Footnote 186: \"Lives,\" bk. i.]\n\n[Footnote 187: \"Lives,\" bk. i.]\n\n_Xenophanes_ was also an assertor of many gods, and one God; but his one\nGod is unquestionably supreme. \"There is one God, the greatest amongst\ngods and men;\" or, \"God is one, the greatest amongst gods and men.\"[188]\n\n_Empedocles_ also believed in one Supreme God, who \"is wholly and\nperfectly mind, ineffable, holy, with rapid and swift-glancing thought\npervading the whole world,\" and from whom all things else are\nderived,--\"all things that are upon the earth, and in the air and water,\nmay be truly called the works of God, who ruleth over the world, out of\nwhom, according to Empedocles, proceed all things, plants, men, beasts,\nand _gods_.\"[189] The minor deities are therefore _made_ by God. It will\nnot be denied that _Socrates_ was a devout and earnest Theist. He taught\nthat \"there is a Being whose eye pierces throughout all nature, and\nwhose ear is open to every sound; extending through all time, extended\nto all places; and whose bounty and care can know no other bounds than\nthose fixed by his own creation.\"[190] And yet he also recognized the\nexistence of a plurality of gods, and in his last moments expressed his\nbelief that \"it is lawful and right to pray to the gods that his\ndeparture hence may be happy.\"[191] We see, however, in his words\naddressed to Euthydemus, a marked distinction between these subordinate\ndeities and \"Him who raised this whole universe, and still upholds the\nmighty frame, who perfected every part of it in beauty and in goodness,\nsuffering none of these parts to decay through age, but renewing them\ndaily with unfading vigor;... even he, _the Supreme God_, still holds\nhimself invisible, and it is only in his works that we are capable of\nadmiring him.\"[192]\n\n[Footnote 188: Clem. Alex., \"Stromat.\" bk. v.]\n\n[Footnote 189: Aristotle, \"De Mundo,\" ch. vi.]\n\n[Footnote 190: Xenophon's \"Memorabilia,\" i. 4.]\n\n[Footnote 191: \"Phaedo,\" Sec. 152.]\n\n[Footnote 192: \"Memorabilia,\" iv. 3.]\n\nIt were needless to attempt the proof that _Plato_ believed in one\nSupreme God, and _only_ one. This one Being is, with him, \"the first\nGod;\" \"the greatest of the gods;\" \"the God over all;\" \"the sole\nPrinciple of the universe.\" He is \"the Immutable;\" \"the All-perfect;\"\n\"the eternal Being.\" He is \"the Architect of the world; \"the Maker of\nthe universe; the Father of gods and men; the sovereign Mind which\norders all things, and passes through all things; the sole Monarch and\nRuler of the world.[193]\n\nAnd yet remarkable as these expressions are, sounding, as they do, so\nlike the language of inspiration,[194] there can be no doubt that Plato\nwas also a sincere believer in a plurality of gods, of which, indeed,\nany one may assure himself by reading the _tenth_ book of \"the Laws.\"\n\n[Footnote 193: See chap. xi.]\n\n[Footnote 194: Some writers have supposed that Plato must have had\naccess through some medium to \"the Oracles of God.\" See Butler, vol. ii.\np. 41.]\n\nAnd, now that we have in Plato the culmination of Grecian speculative\nthought, we may learn from him the mature and final judgment of the\nancients in regard to the gods of pagan mythology. We open the _Timaeus_,\nand here we find his views most definitely expressed. After giving an\naccount of the \"generation\" of the sun, and moon, and planets, which are\nby him designated as \"visible gods,\" he then proceeds \"to speak\nconcerning the other divinities:\" \"We must on this subject assent to\nthose who in former times have spoken thereon; who were, as they said,\nthe offspring of the gods, and who doubtless were well acquainted with\ntheir own ancestors..... Let then the genealogy of the gods be, and be\nacknowledged to be, that which they deliver. Of Earth and Heaven the\nchildren were Oceanus and Tethys; and of these the children were\nPhorcys, and Kronos, and Rhea, and all that followed these; and from\nthese were born Zeus and Hera, and those who are regarded as brothers\nand sisters of these, and others their offspring.\n\n\"When, then, _all the gods were brought into existence_, both those\nwhich move around in manifest courses [the stars and planets], and those\nwhich appear when it pleases them [the mythological deities], the\nCreator of the Universe thus addressed them: 'Gods, and sons of gods, of\nwhom I am the father and the author, produced by me, ye are\nindestructible because I will.... Now inasmuch as you have been\n_generated_, you are hence _not_ immortal, nor wholly indissoluble; yet\nyou shall never be dissolved nor become subject to the fatality of\ndeath, because _so I have willed_.... Learn, therefore, my commands.\nThree races of mortals yet remain to be created. Unless these be\ncreated, the universe will be imperfect, for it will not contain within\nit every kind of animal.... In order that these mortal creatures may be,\nand that this world may be really a cosmos, do you apply yourselves to\nthe creation of animals, imitating the exercises of my power in\n_creating_ you.'\"[195]\n\n[Footnote 195: \"Timaeus,\" ch. xv.]\n\nHere, then, we see that Plato carefully distinguishes between the sole\nEternal Author of the universe, on one hand, and the \"souls,\" vital and\nintelligent, which he attaches to the heavenly orbs, and diffuses\nthrough all nature, on the other. These subordinate powers or agents are\nall created, \"_generated_ deities,\" who owe their continued existence to\nthe _will_ of God; and though intrusted with a sort of deputed creation,\nand a subsequent direction and government of created things, they are\nstill only the _servants_ and the _deputies_ of the Supreme Creator, and\nDirector, and Ruler of all things. These subordinate agents and\nministers employed in the creation and providential government of the\nworld appear, in the estimation of Plato, to have been needed--\n\n1. _To satisfy the demands of the popular faith_, which presented its\nfacts to be explained no less than those of external nature. Plato had\nevidently a great veneration for antiquity, a peculiar regard for\n\"tradition venerable through ancient report,\" and \"doctrines hoary with\nyears.\"[196] He aspired after supernatural light and guidance; he longed\nfor some intercourse with, some communication from, the Deity. And\nwhilst he found many things in the ancient legends which revolted his\nmoral sense, and which his reason rejected, yet the sentiment and the\nlesson which pervades the whole of Grecian mythology, viz., that the\ngods are in ceaseless intercourse with the human race, and if men will\ndo right the gods will protect and help them, was one which commended\nitself to his heart.\n\n[Footnote 196: Ibid., ch. v.]\n\n2. These intermediate agents seem to have been demanded to _satisfy the\ndisposition and tendency which has revealed itself in all systems, of\ninterposing some scale of ascent between the material creation and the\ninfinite Creator_.\n\nThe mechanical theory of the universe has interposed its long series of\nsecondary causes--the qualities, properties, laws, forces of nature; the\nvital theory which attaches a separate \"soul\" to the various parts of\nnature as the cause and intelligent director of its movements. Of these\n\"souls\" or gods, there were different orders and degrees--deified men or\nheroes, aerial, terrestrial, and celestial divinities, ascending from\nnature up to God. And this tendency to supply some scale of ascent\ntowards the Deity, or at least to people the vast territory which seems\nto swell between the world and God, finds some countenance in \"the\nangels and archangels,\" \"the thrones, and dominions, and principalities,\nand powers\" of the Christian scriptures.[197]\n\n3. These inferior ministers also seemed to Plato to _increase the\nstately grandeur and imperial majesty of the Divine government._ They\nswell the retinue of the Deity in his grand \"circuit through the highest\narch of heaven.\"[198] They wait to execute the Divine commands. They are\nthe agents of Divine providence, \"the messengers of God\" to men.\n\n[Footnote 197: \"The gods of the Platonic system answer, in office and\nconception, to the angels of Christian Theology.\"--Butler, vol. i. p.\n225.]\n\n[Footnote 198: \"Phaedrus,\" Sec. 56,7.]\n\n4. And, finally, the host of inferior deities interposed between the\nmaterial sensible world and God seemed to Plato as _needful in order to\nexplain the apparent defects and disorders of sublunary affairs_. Plato\nwas jealous of the Divine honor. \"All good must be ascribed to God, and\nnothing but good. We must find evil, disorder, suffering, in some other\ncause.\"[199] He therefore commits to the junior deities the task of\ncreating animals, and of forming \"the mortal part of man,\" because the\nmortal part is \"possessed of certain dire and necessary passions.\"[200]\n\n[Footnote 199: \"Republic,\" bk. ii. p.18.]\n\n[Footnote 200: \"Timaeus,\" xliv.]\n\nAristotle seems to have regarded the popular polytheism of Greece as a\nperverted relic of a deeper and purer \"Theology\" which he conceives to\nhave been, in all probability, perfected in the distant past, and then\ncomparatively lost. He says--\"The tradition has come down from very\nancient times, being left in a mythical garb to succeeding generations,\nthat these (the heavenly bodies) are gods, and that the Divinity\n_encompasses the whole of nature_. There have been made, however, to\nthese certain fabulous additions for the purpose of winning the belief\nof the multitude, and thus securing their obedience to the laws, and\ntheir co-operation towards advancing the general welfare of the state.\nThese additions have been to the effect that these gods were of the same\nform as men, and even that some of them were in appearance similar to\ncertain others amongst the rest of the animal creation. The wise course,\nhowever, would be for the philosopher to disengage from these traditions\nthe false element, and to embrace that which is true; and the truth lies\nin that portion of this ancient doctrine which regards the first and\ndeepest ground of all existence to be the _Divine_, and this he may\nregard as a divine utterance. In all probability, every art, and\nscience, and philosophy has been over and over again discovered to the\nfarthest extent possible, and then again lost; and we may conceive these\nopinions to have been preserved to us as a sort of fragment of these\nlost philosophers. We see, then, to some extent the relation of the\npopular belief to these ancient opinions.\"[201] This conception of a\ndeep Divine ground of all existence (for the immateriality and unity of\nwhich he elsewhere earnestly contends)[202] is thus regarded by\nAristotle as underlying the popular polytheism of Greece.\n\n[Footnote 201: \"Metaph.,\" xi. 8.]\n\n[Footnote 202: Bk. xi. ch. ii. Sec. 4.]\n\nThe views of the educated and philosophic mind of Greece in regard to\nthe mythological deities may, in conclusion, be thus briefly stated--\n\nI. _They are all created beings_--\"GENERATED DEITIES,\" _who are\ndependent on, and subject to, the will of one supreme God_.\n\nII. _They are the_ AGENTS _employed by God in the creation of, at least\nsome parts of, the universe, and in the movement and direction of the\nentire cosmos; and they are also the_ MINISTERS _and_ MESSENGERS _of\nthat universal providence which he exercises over the human race_.\n\nThese subordinate deities are, 1. the greater parts of the visible\nmundane system animated by intelligent souls, and called \"_sensible\ngods_\"--the sun, the moon, the stars, and even the earth itself, and\nknown by the names Helios, Selena, Kronos, Hermes, etc.\n\n2. Some are _invisible powers_, having peculiar offices and functions\nand presiding over special places provinces and departments of the\nuniverse;--one ruling in the heavens (Zeus), another in the air (Juno),\nanother in the sea (Neptune), another in the subterranean regions\n(Pluto); one god presiding over learning and wisdom (Minerva), another\nover poetry, music, and religion (Apollo), another over justice and\npolitical order (Themis), another over war (Mars), another over corn\n(Ceres), and another the vine (Bacchus).\n\n3. Others, again, are _ethereal_ and _aerial_ beings, who have the\nguardianship of individual persons and things, and are called _demons,\ngenii_, and _lares_; superior indeed to men, but inferior to the gods\nabove named.\n\n\"Wherefore, since there were no other gods among the Pagans besides\nthose above enumerated, unless their images, statues, and symbols should\nbe accounted such (because they were also sometimes abusively called\n'gods'), which could not be supposed by them to have been unmade or\nwithout beginning, they being the workmanship of their own hands, we\nconclude, universally, that all that multiplicity of Pagan gods which\nmake so great a show and noise was really either nothing but several\nnames and notions of one supreme Deity, according to his different\nmanifestations, gifts, and effects upon the world personated, or else\nmany inferior understanding beings, generated or created by one supreme:\nso that one unmade, self-existent Deity, and no more, was acknowledged\nby the more intelligent Pagans, and, consequently, the Pagan Polytheism\n(or idolatry) consisted not in worshipping a multiplicity of unmade\nminds, deities, and creators, self-existent from eternity, and\nindependent upon one Supreme, but in mingling and blending some way or\nother, unduly, creature-worship with the worship of the Creator.\"[203]\n\n[Footnote 203: Cudworth, \"Intellectual System,\" vol. i. p. 311.]\n\nThat the heathen regard the one Supreme Being as the first and chief\nobject of worship is evident from the apologies which they offered for\nworshipping, besides Him, many inferior divinities.\n\n1. They claimed to worship them _only_ as inferior beings, and that\ntherefore they were not guilty of giving them that honor which belonged\nto the Supreme. They claimed to worship the supreme God incomparably\nabove all. 2. That this honor which is bestowed upon the inferior\ndivinities does ultimately redound to the supreme God, and aggrandize\nhis state and majesty, they being all his ministers and attendants. 3.\nThat as demons are mediators between the celestial gods and men, so\nthose celestial gods are also mediators between men and the supreme God,\nand, as it were, convenient steps by which we ought with reverence to\napproach him. 4. That demons or angels being appointed to preside over\nkingdoms, cities, and persons, and being many ways benefactors to us,\nthanks ought to be returned to them by sacrifice. 5. Lastly, that it can\nnot be thought that the Supreme Being will envy those inferior beings\nthat worship or honor which is bestowed upon them; nor suspect that any\nof these inferior deities will factiously go about to set up themselves\nagainst the Supreme God.\n\nThe Pagans, furthermore, apologized for worshipping God in images,\nstatues, and symbols, on the ground that these were only schetically\nworshipped by them, the honor passing from them to the prototype. And\nsince we live in bodies, and can scarcely, conceive of any thing without\nhaving some image or phantasm, we may therefore be indulged in this\ninfirmity of human nature (at least in the vulgar) to worship God under\na corporeal image, as a means of preventing men from falling into\nAtheism.\n\nTo the Christian conscience the above reasons assigned furnish no real\njustification of Polytheism and Idolatry; but they are certainly a tacit\nconfession of their belief in the one Supreme God, and their conviction\nthat, notwithstanding their idolatry, He only ought to be worshipped.\nThe heathen polytheists are therefore justly condemned in Scripture, and\npronounced to be \"_inexcusable_.\" They had the knowledge of the true\nGod--\" they _knew God_\" and yet \"they glorified him not as God.\" \"They\nchanged the glory of the incorruptible God into a likeness of\ncorruptible man.\" And, finally, they ended in \"worshipping and serving\nthe creature _more_ than the Creator.\"[204]\n\n[Footnote 204: Romans i. 21, 25.]\n\nIt can not, then, with justice be denied that the Athenians had some\nknowledge of the true God, and some just and worthy conceptions of his\ncharacter. It is equally certain that a powerful and influential\nreligious sentiment pervaded the Athenian mind. Their extreme\n\"carefulness in religion\" must be conceded by us, and, in some sense,\ncommended by us, as it was by Paul in his address on Mars' Hill. At the\nsame time it must also be admitted and deplored that the purer theology\nof primitive times was corrupted by offensive legends, and encrusted by\npolluting myths, though not utterly defaced.[205] The Homeric gods were\nfor the most part idealized, human personalities, with all the passions\nand weaknesses of humanity. They had their favorites and their enemies;\nsometimes they fought in one camp, sometimes in another. They were\nsusceptible of hatred, jealousy, sensual passion. It would be strange\nindeed if their worshippers were not like unto them. The conduct of the\nHomeric heroes was, however, better than their creed. And there is this\nstrange incongruity and inconsistency in the conduct of the Homeric\ngods,--they punish mortals for crimes of which they themselves are\nguilty, and reward virtues in men which they do not themselves always\npractise. \"They punish with especial severity social and political\ncrimes, such as perjury (Iliad, iii. 279), oppression of the poor (Od.\nxvii. 475), and unjust judgment in courts of justice (Iliad, xvi. 386).\"\nJupiter is the god of justice, and of the domestic hearth; he is the\nprotector of the exile, the avenger of the poor, and the vigilant\nguardian of hospitality. \"And with all the imperfections of society,\ngovernment, and religion, the poem presents a remarkable picture of\nprimitive simplicity, chastity, justice, and practical piety, under the\nthree-fold influence of moral feeling, mutual respect, and fear of the\ndivine displeasure; such, at least, are the motives to which Telemachus\nmakes his appeal when he endeavors to rouse the assembled people of\nIthaca to the performance of their duty (Od. ii. 64).\"[206]\n\n[Footnote 205: \"There was always a double current of religious ideas in\nGreece; one spiritualist, the other tainted with impure\nlegends.\"--Pressense.]\n\n[Footnote 206: Tyler, \"Theology of Greek Poets,\" pp. 167, 168;\nPressense, \"Religion before Christ,\" p. 77.]\n\nThe influence of the religious dramas of AEschylus and Sophocles on the\nAthenian mind must not be overlooked. No writer of pagan antiquity made\nthe voice of conscience speak with the same power and authority that\nAEschylus did. \"Crime,\" he says, \"never dies without posterity.\" \"Blood\nthat has been shed congeals on the ground, crying out for an avenger.\"\nThe old poet made himself the echo of what he called \"the lyreless hymn\nof the Furies,\" who, with him, represented severe Justice striking the\nguilty when his hour comes, and giving warning beforehand by the terrors\nwhich haunt him. His dramas are characterized by deep religious feeling.\nReverence for the gods, the recognition of an inflexible moral order,\nresignation to the decisions of Heaven, an abiding presentiment of a\nfuture state of reward and punishment, are strikingly predominant.\n\nWhilst AEschylus reveals to us the sombre, terror-stricken side of\nconscience, Sophocles shows us the divine and luminous side. No one has\never spoken with nobler eloquence than he of moral obligation--of this\nimmortal, inflexible law, in which dwells a God that never grows old--\n\n        \"Oh be the lot forever mine\n            Unsullied to maintain,\n         In act and word, with awe divine,\n            What potent laws ordain.\n\n     \"Laws spring from purer realms above:\n      Their father is the Olympian Jove.\n      Ne'er shall oblivion veil their front sublime,\n      Th' indwelling god is great, nor fears the wastes of time.\"[207]\n\nThe religious inspiration that animates Sophocles breaks out with\nincomparable beauty in the last words of oedipus, when the old banished\nking sees through the darkness of death a mysterious light dawn, which\nillumines his blind eyes, and which brings to him the assurance of a\nblessed immortality.[208]\n\n[Footnote 207: \"oedipus Tyran.,\" pp. 863-872.]\n\n[Footnote 208: Pressense, \"Religion before Christ,\" pp. 85-87.]\n\nSuch a theology could not have been utterly powerless. The influence of\ntruth, in every measure and degree, must be salutary, and especially of\ntruth in relation to God, to duty, and to immortality. The religion of\nthe Athenians must have had some wholesome and conserving influence of\nthe social and political life of Athens.[209] Those who resign the\ngovernment of this lower world almost exclusively to Satan, may see, in\nthe religion of the Greeks, a simple creation of Satanic powers. But he\nwho believes that the entire progress of humanity has been under the\ncontrol and direction of a benignant Providence, must suppose that, in\nthe purposes of God, even Ethnicism has fulfilled some end, or it would\nnot have been permitted to live. God has \"_never left himself without a\nwitness_\" in any nation under heaven. And some preparatory office has\nbeen fulfilled by Heathenism which, at least, repealed the _want_, and\nprepared the mind for, the advent of Christianity.\n\n[Footnote 209: The practice, so common with some theological writers, of\ndrawing dark pictures of heathenism, in which not one luminous spot is\nvisible, in order to exalt the revelations given to the Jews, is\nexceedingly unfortunate, and highly reprehensible. It is unfortunate,\nbecause the skeptical scholar knows that there were some elements of\ntruth and excellence, and even of grandeur, in the religion and\ncivilization of the republics of Greece and Rome; and it is\nreprehensible, because it is a one-sided and unjust procedure, in so far\nas it withholds part of the truth. This species of argument is a\ntwo-edged sword which cuts both ways. The prevalence of murder, and\nslavery, and treachery, and polygamy, in Greece and Rome, is no more a\nproof that \"the religions of the pagan nations were destructive of\nmorality\" (Watson, vol. i. p. 59), than the polygamy of the Hebrews, the\nfalsehoods and impositions of Mediaeval Christianity, the persecutions\nand martyrdoms of Catholic Christianity, the oppressions and wrongs of\nChristian England, and the slavery of Protestant America, are proofs\nthat the Christian religion is \"destructive of morality.\" What a fearful\npicture of the history of Christian nations might be drawn to-day, if\nall the lines of light, and goodness, and charity were left out, and the\ncrimes, and wrongs, and cruelties of the Christian nations were alone\nexhibited!\n\nHow much more convincing a proof of the truth of Christianity to find in\nthe religions of the ancient world a latent sympathy with, and an\nunconscious preparation for, the religion of Christ. \"The history of\nreligions of human origin is the most striking evidence of the agreement\nof revealed religion with the soul of man--for each of these forms of\nworship is the expression of the wants of conscience, its eternal thirst\nfor pardon and restoration--rather let us say, its thirst for\nGod.\"--Pressense, p. 6.]\n\nThe religion of the Athenians was unable to deliver them from the guilt\nof sin, redeem them from its power, and make them pure and holy. It gave\nthe Athenian no victory over himself, and, practically, brought him no\nnearer to the living God. But it awakened and educated the conscience,\nit developed more fully the sense of sin and guilt, and it made man\nconscious of his inability to save himself from sin and guilt; and \"the\nday that humanity awakens to the want of something more than mere\nembellishment and culture, that day it feels the need of being saved and\nrestored from the consequences of sin\" by a higher power. AEsthetic taste\nhad found its fullest gratification in Athens; poetry, sculpture,\narchitecture, had been carried to the highest perfection; a noble\ncivilization had been reached; but \"the need of something deeper and\ntruer was written on the very stones.\" The highest consummation of\nPaganism was an altar to \"the unknown God,\" the knowledge of whom it\nneeded, as the source of purity and peace.\n\nThe strength and the weakness of Grecian mythology consisted in the\ncontradictory character of its divinities. It was a strange blending of\nthe natural and the supernatural, the human and the divine. Zeus, the\neternal Father,--the immortal King, whose will is sovereign, and whose\npower is invincible,--the All-seeing Jove, has some of the weaknesses\nand passions of humanity. God and man are thus, in some mysterious way,\nunited. And here that deepest longing of the human heart is met--the\nunconquerable desire to bring God nearer to the human apprehension, and\ncloser to the human heart. Hence the hold which Polytheism had upon the\nGrecian mind. But in this human aspect was also found its weakness, for\nwhen philosophic thought is brought into contact with, and permitted\ncritically to test mythology, it dethrones the false gods. The age of\nspontaneous religious sentiment must necessarily be succeeded by the age\nof reflective thought. Popular theological faiths must be placed in the\nhot crucible of dialectic analysis, that the false and the frivolous may\nbe separated from the pure and the true. The reason of man demands to be\nsatisfied, as well as the heart. Faith in God must have a logical basis,\nit must be grounded on demonstration and proof. Or, at any rate, the\nquestion must be answered, _whether God is cognizable by human reason_?\nIf this can be achieved, then a deeper foundation is laid in the mind of\nhumanity, upon which Christianity can rear its higher and nobler truths.\n\n\n\n\nCHAPTER V.\n\nTHE UNKNOWN GOD.\n\n\n\"As I passed by, and beheld your sacred objects, I found an altar with\nthis inscription, _To the Unknown God_.\"--ST. PAUL.\n\n\"That which can be _known_ of God is manifested in their hearts, God\nhimself having shown it to them\" [the heathen nations].--ST. PAUL.\n\nHaving now reached our first landing-place, from whence we may survey\nthe fields that we have traversed, it may be well to set down in\ndefinite propositions the results we have attained. We may then carry\nthem forward, as torches, to illuminate the path of future and still\nprofounder inquiries.\n\nThe principles we have assumed as the only adequate and legitimate\ninterpretation of the facts of religious history, and which an extended\nstudy of the most fully-developed religious system of the ancient world\nconfirms, may be thus announced:\n\nI. A religious nature and destination appertain to man, so that the\npurposes of his existence and the perfection of his being can only be\nsecured in and through religion.\n\nII. The idea of God as the unconditioned Cause, the infinite Mind, the\npersonal Lord and Lawgiver, and the consciousness of dependence upon and\nobligation to God, are the fundamental principles of all religion.\n\nIII. Inasmuch as man is a religious being, the instincts and emotions of\nhis nature constraining him to worship, there must also be implanted in\nhis rational nature some original _a priori_ ideas or laws of thought\nwhich furnish the necessary cognition of the object of worship; that is,\nsome native, spontaneous cognition of God.\n\nA mere blind impulse would not be adequate to guide man to the true end\nand perfection of his being without rational ideas; a tendency or\nappetency, without a revealed object, would be the mockery and misery of\nhis nature--an \"ignis fatuus\" perpetually alluring and forever deceiving\nman.\n\nThat man has a native, spontaneous apperception of a God, in the true\nimport of that sacred name, has been denied by men of totally opposite\nschools and tendencies of thought--by the Idealist and the Materialist;\nby the Theologian and the Atheist. Though differing essentially in their\ngeneral principles and method, they are agreed in asserting that God is\nabsolutely \"_the unknown_;\" and that, so far as reason and logic are\nconcerned, man can not attain to any knowledge of the first principles\nand causes of the universe, and, consequently, can not determine whether\nthe first principle or principles be intelligent or unintelligent,\npersonal or impersonal, finite or infinite, one or many righteous or\nnon-righteous, evil or good.\n\nThe various opponents of the doctrine that God can be cognized by human\nreason may be classified as follows: I. _Those who assert that all human\nknowledge is necessarily confined to the observation and classification\nof phenomena in their orders of co-existence, succession, and\nresemblance_. Man has no faculty for cognizing substances, causes,\nforces, reasons, first principles--no power by which he can _know_ God.\nThis class may be again subdivided into--\n\n1. Those who limit all knowledge to the observation and classification\nof _mental_ phenomena (_e. g_., Idealists like J. S. Mill).\n\n2. Those who limit all knowledge to the observation and classification\nof _material_ phenomena (_e. g_., Materialists like Comte).\n\nII. _The second class comprises all who admit that philosophic knowledge\nis the knowledge of effects as dependent on causes, and of qualities as\ninherent in substances; but at the same time assert that \"all knowledge\nis of the phenomenal_.\" Philosophy can never attain to a positive\nknowledge of the First Cause. Of existence, absolutely and in itself, we\nknow nothing. The infinite can not by us be comprehended, conceived, or\nthought. _Faith_ is the organ by which we apprehend what is beyond\nknowledge. We believe in the existence of God, but we can not _know_\nGod. This class, also, may be again subdivided into--\n\n1. Those who affirm that our idea of the Infinite First Cause is\ngrounded on an _intuitional_ or subjective faith, necessitated by an\n\"impotence of thought\"--that is, by a mental inability to conceive an\nabsolute limitation or an infinite illimitation, an absolute\ncommencement or an infinite non-commencement. Both contradictory\nopposites are equally incomprehensible and inconceivable to us; and yet,\nthough unable to view either as possible, we are forced by a higher\nlaw--the \"Law of Excluded Middle\"--to admit that one, and only one, is\nnecessary (_e. g_., Hamilton and Mansel).\n\n2. Those who assert that our idea of God rests solely on an _historical_\nor objective faith in testimony--the testimony of Scripture, which\nassures us that, in the course of history, God has manifested his\nexistence in an objective manner to the senses, and given verbal\ncommunications of his character and will to men; human reason being\nutterly incapacitated by the fall, and the consequent depravity of man,\nto attain any knowledge of the unity, spirituality, and righteousness of\nGod (_e. g_., Watson, and Dogmatic Theologians generally).\n\nIt will thus be manifest that the great question, the central and vital\nquestion which demands a thorough and searching consideration, is the\nfollowing, to wit: _Is God cognizable by human reason_? Can man attain\nto a positive cognition of God--can he _know_ God; or is all our\nsupposed knowledge \"a learned ignorance,\"[210] an unreasoning faith? We\nventure to answer this question in the affirmative. Human reason is now\nadequate to the cognition of God; it is able, with the fullest\nconfidence, to affirm the being of a God, and, in some degree, to\ndetermine his character. The parties and schools above referred to\nanswer this question in the negative form. Whether Theologians or\nAtheists, they are singularly agreed in denying to human reason all\npossibility of _knowing_ God.\n\n[Footnote 210: Hamilton's \"Philosophy,\" p. 512.]\n\nBefore entering upon the discussion of the negative positions enumerated\nin the above classification, it may be important we should state our own\nposition explicitly, and exhibit what we regard as the true doctrine of\nthe genesis of the idea of God in the human intelligence. The real\nquestion at issue will then stand out in clear relief, and precision\nwill be given to the entire discussion.\n\n(i.) _We hold that the idea of God is a common phenomenon of the\nuniversal human intelligence_. It is found in all minds where reason has\nhad its normal and healthy development; and no race of men has ever been\nfound utterly destitute of the idea of God. The proof of this position\nhas already been furnished in chap, ii.,[211] and needs not be re-stated\nhere. We have simply to remark that the appeal which is made by Locke\nand others of the sensational school to the experiences of infants,\nidiots, the deaf and dumb, or, indeed, any cases wherein the proper\nconditions for the normal development of reason are wanting, are utterly\nirrelevant to the question. The acorn contains within itself the\nrudimental germ of the future oak, but its mature and perfect\ndevelopment depends on the exterior conditions of moisture, light, and\nheat. By these exterior conditions it may be rendered luxuriant in its\ngrowth, or it may be stunted in its growth. It may barely exist under\none class of conditions; it may be distorted and perverted, or it may\nperish utterly under another. And so in the idiotic mind the ideas of\nreason may be wanting, or they may be imprisoned by impervious walls of\ncerebral malformation. In the infant mind the development of reason is\nyet in an incipient stage. The idea of God is immanent to the infant\nthought, but the infant thought is not yet matured. The deaf and dumb\nare certainly not in that full and normal correlation to the world of\nsense which is a necessary condition of the development of reason.\nLanguage, the great vehiculum and instrument of thought, is wanting, and\nreason can not develop itself without words. \"Words without thought are\ndead sounds, _thoughts without words are nothing_. The word is the\nthought incarnate.\"[212] Under proper and normal conditions, the idea of\nGod is the natural and necessary form in which human thought must be\ndeveloped. And, with these explanations, we repeat our affirmation that\nthe idea of God is a common phenomenon of the universal human\nintelligence.\n\n[Footnote 211: Pp. 89,90.]\n\n[Footnote 212: Mueller, \"Science of Language,\" p. 384.]\n\n(ii.) _We do not hold that the idea of God, in its completeness, is a\nsimple, direct, and immediate intuition of the reason alone, independent\nof all experience, and all knowledge of the external world_. The idea of\nGod is a complex idea, and not a simple idea. The affirmation, \"God\nexists,\" is a _synthetic_ and _primitive_ judgment spontaneously\ndeveloped in the mind, and developed, too, independent of all reflective\nreasoning. It is a necessary deduction from the facts of the outer world\nof nature and the primary intuitions of the inner world of reason--a\nlogical deduction from the self-evident truths given in sense,\nconsciousness, and reason. \"We do not _perceive_ God, but we _conceive_\nHim upon the faith of this admirable world exposed to view, and upon the\nother world, more admirable still, which we bear in ourselves.\"[213]\nTherefore we do not say that man is born with an \"innate idea\" of God,\nnor with the definite proposition, \"there is a God,\" written upon his\nsoul; but we do say that the mind is pregnant with certain natural\nprinciples, and governed, in its development, by certain necessary laws\nof thought, which determine it, by a _spontaneous logic_, to affirm the\nbeing of a God; and, furthermore, that this judgment may be called\n_innate_ in the sense, that it is the primitive, universal, and\nnecessary development of the human understanding which \"is innate to\nitself and equal to itself in all men.\"[214]\n\n[Footnote 213: Cousin, \"True, Beautiful and Good,\" p.102.]\n\n[Footnote 214: Leibnitz.]\n\nAs the vital and rudimentary germ of the oak is contained in the acorn;\nas it is quickened and excited to activity by the external conditions of\nmoisture, light, and heat, and is fully de developed under the fixed and\ndeterminative laws of vegetable life--so the germs of the idea of God\nare present in the human mind as the intuitions of pure reason\n(_Rational Psychology_); these intuitions are excited to energy by our\nexperiential and historical knowledge of the facts and laws of the\nuniverse (_Phenomenology_); and these facts and intuitions are developed\ninto form by the necessary laws of the intellect (_Nomology_, or\n_Primordial Logic_).\n\nThe _logical demonstration_ of the being of God commences with the\nanalysis of thought. It asks, What are the ideas which exist in the\nhuman intelligence? What are their actual characteristics, and what\ntheir primitive characteristics? What is their origin, and what their\nvalidity? Having, by this process, found that some of our ideas are\nsubjective, and some objective that some are derived from experience,\nand that some can not be derived from experience, but are inherent in\nthe very constitution of the mind itself, as _a priori_ ideas of reason;\nthat these are characterized as self-evident, universal, and necessary\nand that, as laws of thought, they govern the mind in all its\nconceptions of the universe; it has formulated these necessary\njudgments, and presented them as distinct and articulate propositions.\nThese _a priori_, necessary judgments constitute the major premise of\nthe Theistic syllogism, and, in view of the facts of the universe,\nnecessitate the affirmation of the existence of a God as the only valid\nexplanation of the facts.\n\nThe _natural_ or _chronological order_ in which the idea of God is\ndeveloped in the human intelligence, is the reverse process of the\nscientific or logical order, in which the demonstration of the being of\nGod is presented by philosophy; the latter is _reflective_ and\n_analytic_, the former is _spontaneous_ and _synthetic._ The natural\norder commences with the knowledge of the facts of the universe,\nmaterial and mental, as revealed by sensation and experience. In\npresence of these facts of the universe, the _a priori_ ideas of power,\ncause, reason, and end are evoked into consciousness with greater or\nless distinctness; and the judgment, by a natural and spontaneous logic,\nfree from all reflection, and consequently from all possibility of\nerror, affirms a necessary relation between the facts of experience and\nthe _a priori_ ideas of the reason. The result of this involuntary and\nalmost unconscious process of thought is that natural cognition of a God\nfound, with greater or less clearness and definiteness, in all rational\nminds. The _a posteriori_, or empirical knowledge of the phenomena of\nthe universe, in their relations to time and space, constitute the minor\npremise of the Theistic syllogism.\n\nThe Theistic argument is, therefore, necessarily composed of both\nexperiential and _a priori_ elements. An _a posteriori_ element exists\nas a condition of the logical demonstration The rational _a priori_\nelement is, however, the logical basis, the only valid foundation of the\nTheistic demonstration. The facts of the universe alone would never lead\nman to the recognition of a God, if the reason, in presence of these\nfacts, did not enounce certain necessary and universal principles which\nare the logical antecedents, and adequate explanation of the facts. Of\nwhat use would it be to point to the events and changes of the material\nuniverse as proofs of the existence of a _First Cause,_ unless we take\naccount of the universal and necessary truth that \"every change must\nhave an efficient cause;\" that all phenomena are an indication of\n_power_; and that \"there is an ultimate and sufficient reason why all\nthings exist, and are as they are, and not otherwise.\" There would be no\nlogical force in enumerating the facts of order and special adaptation\nwhich literally crowd the universe, as proofs of the existence of an\n_Intelligent Creator_, if the mind did not affirm the necessary\nprinciple that \"facts of order, having a commencement in time, suppose\nmind as their source and exponent.\" There is no logical conclusiveness\nin the assertion of Paley, \"that _experience_ teaches us that a designer\nmust be a person,\" because, as Hume justly remarks, our \"experience\" is\nnarrowed down to a mere point, \"and can not be a rule for a universe;\"\nbut there is an infinitude of force in that dictum of reason, that\n\"intelligence, self-consciousness, and self-determination necessarily\nconstitute personality.\" A multiplicity of different effects, of which\nexperience does not always reveal the connection, would not conduct to a\nsingle cause and to _one_ God, but rather to a plurality of causes and a\nplurality of gods, did not reason teach us that \"all plurality implies\nan ultimate indivisible unity,\" and therefore there must be a _First\nCause_ of all causes, a _First Principle_ of all principles, _the\nSubstance_ of all substances, _the Being_ of all beings--_a God_ \"of\nwhom, in whom, and to whom are all things\" (panta ek tou Theou, en to\nTheo eis ton Theon).\n\nThe conclusion, therefore, is, that, as the idea of God is a complex\nidea, so there are necessarily a number of simple _a priori_ principles,\nand a variety of experiential facts conspiring to its development in the\nhuman intelligence.\n\n(iii.) _The universe presents to the human mind an aggregation and\nhistory of phenomena which demands the idea of a God--a self-existent,\nintelligent, personal, righteous First Cause--as its adequate\nexplanation._\n\nThe attempt of Positivism to confine all human knowledge to the\nobservation and classification of phenomena, and arrest and foreclose\nall inquiry as to causes, efficient, final, and ultimate, is simply\nfutile and absurd. It were just as easy to arrest the course of the sun\nin mid-heaven as to prevent the human mind from seeking to pass beyond\nphenomena, and ascertain the ground, and reason, and cause of all\nphenomena. The history of speculative thought clearly attests that, in\nall ages, the inquiry after the Ultimate Cause and Reason of all\nexistence--the arche, or First Principle of all things--has been the\ninevitable and necessary tendency of the human mind; to resist which,\nskepticism and positivism have been utterly impotent. The first\nphilosophers, of the Ionian school, had just as strong a faith in the\nexistence of a Supreme Reality--an Ultimate Cause--as Leibnitz and\nCousin. But when, by reflective thought, they attempted to render an\naccount to themselves of this instinctive faith, they imagined that its\nobject must be in some way appreciable to sense, and they sought it in\nsome physical element, or under some visible and tangible shrine. Still,\nhowever imperfect and inadequate the method, and however unsatisfactory\nthe results, humanity has never lost its positive and ineradicable\nconfidence that the problem of existence could be solved. The resistless\ntide of spontaneous and necessary thought has always borne the race\nonward towards the recognition of a great First Cause; and though\nphilosophy may have erred, again and again, in tracing the logical order\nof this inevitable thought, and exhibiting the necessary nexus between\nthe premises and conclusion, yet the human mind has never wavered in the\nconfidence which it has reposed in the natural logic of thought, and man\nhas never ceased to believe in a God.\n\nWe readily grant that all our empirical knowledge is confined to\nphenomena in their orders of co-existence, succession, and resemblance.\n\"To our objective perception and comparison nothing is given but\nqualities and changes; to our inductive generalization nothing but the\nshifting and grouping of these in time and space.\" Were it, however, our\nimmediate concern to discuss the question, we could easily show that\nsensationalism has never succeeded in tracing the genetic origin of our\nideas of space and time to observation and experience; and, without the\n_a priori_ idea of _space_, as the place of bodies, and of _time_, as\nthe condition of succession, we can not conceive of phenomena at all.\nIf, therefore, we know any thing beyond phenomena and their mutual\nrelations; if we have any cognition of realities underlying phenomena,\nand of the relations of phenomena to their objective ground, it must be\ngiven by some faculty distinct from sense-perception, and in some\nprocess distinct from inductive generalization. The knowledge of real\nBeing and real Power, of an ultimate Reason and a personal Will, is\nderived from the apperception of pure reason, which affirms the\nnecessary existence of a Supreme Reality--an Uncreated Being beyond all\nphenomena, which is the ground and reason of the existence--the\ncontemporaneousness and succession--the likeness and unlikeness, of all\nphenomena.\n\nThe immediate presentation of phenomena to sensation is the _occasion_\nof the development in consciousness of these _a priori_ ideas of reason:\nthe possession of these ideas or the immanence of these ideas, in the\nhuman intellect, constitutes the original _power_ to know external\nphenomena. The ideas of space, time, power, law, reason, and end, are\nthe logical antecedents of the ideas of body, succession, event,\nconsecution, order, and adaptation. The latter can not be conceived as\ndistinct notions without the former. The former will not be revealed in\nthought without the presentation to sense, of resistance, movement,\nchange, uniformity, etc. All actual knowledge must, therefore, be\nimpure; that is, it must involve both _a priori_ and _a posteriori_\nelements; and between these elements there must be a necessary relation.\n\nThis necessary relation between the _a priori_ and _a posteriori_\nelements of knowledge is not a mere subjective law of thought. It is\nboth a law of thought and a law of things. Between the _a posteriori_\nfacts of the universe and the _a priori_ ideas of the reason there is an\nabsolute nexus, a universal and necessary correlation; so that the\ncognition of the latter is possible only on the cognition of the former;\nand the objective existence of the realities, represented by the ideas\nof reason, is the condition, _sine qua non_, of the existence of the\nphenomena presented to sense. If, in one indivisible act of\nconsciousness, we immediately perceive extended matter exterior to our\npercipient mind, then Extension exists objectively; and if Extension\nexists objectively, then Space, its _conditio sine qua non_, also exists\nobjectively. And if a definite body reveals to us the _Space_ in which\nit is contained, if a succession of pulsations or movements exhibit the\nuniform _Time_ beneath, so do the changeful phenomena of the universe\ndemand a living _Power_ behind, and the existing order and regular\nevolution of the universe presuppose _Thought_--prevision, and\npredetermination, by an intelligent mind.\n\nIf, then, the universe is a created effect, it must furnish some\nindications of the character of its cause. If, as Plato taught, the\nworld is a \"created image\" of the eternal archetypes which dwell in the\nuncreated Mind, and if the subjective ideas which dwell in the human\nreason, as the offspring of God, are \"copies\" of the ideas of the\nInfinite Reason--if the universe be \"the autobiography of the Infinite\nSpirit which has also repeated itself in miniature within our finite\nspirit,\" then may we decipher its symbols, and read its lessons straight\noff. Then every approach towards a scientific comprehension and\ngeneralization of the facts of the universe must carry us upward towards\nthe higher realities of reason. The more we can understand of Nature--of\nher comprehensive laws, of her archetypal forms, of her far-reaching\nplan spread through the almost infinite ages, and stretching through\nillimitable space--the more do we comprehend the divine Thought. The\ninductive generalization of science gradually _ascends_ towards the\nuniversal; the pure, essential, _a priori_ reason, with its universal\nand necessary ideas, _descends_ from above to meet it. The general\nconceptions of science are thus a kind of _ideoe umbratiles_--shadowy\nassimilations to those immutable ideas which dwell in essential reason,\nas possessed by the Supreme Intelligence, and which are participated in\nby rational man as the offspring and image of God.\n\nWithout making any pretension to profound scientific accuracy, we offer\nthe following tentative classification of the facts of the universe,\nmaterial and mental, which may be regarded as hints and adumbrations of\nthe ultimate ground, and reason, and cause, of the universe. We shall\nventure to classify these facts as indicative of some fundamental\nrelation; (i.) to Permanent Being or Reality; (ii.) to Reason and\nThought; (iii.) to Moral Ideas and Ends.\n\n(i.) _Facts of the universe which indicate some fundamental relation to\nPermanent Being or Reality_.\n\n1. _Qualitative_ Phenomena (properties, attributes, qualities)--the\npredicates of a _subject_; which phenomena, being characterized by\nlikeness and unlikeness, are capable of comparison and classification,\nand thus of revealing something as to the nature of the _subject_.\n\n2. _Dynamical_ Phenomena (protension, movement, succession)--events\ntranspiring in _time_, having beginning, succession, and end, which\npresent themselves to us as the expression of _power_, and throw back\ntheir distinctive characteristics on their _dynamic_ source.\n\n3. _Quantitative_ Phenomena (totality, multiplicity, relative unity)--a\nmultiplicity of objects having relative and composite unity, which\nsuggests some relation to an absolute and indivisible _unity_.\n\n4. _Statical_ Phenomena (extension, magnitude, divisibility)--bodies\nco-existing in _space_ which are limited, conditioned, relative,\ndependent, and indicate some relation to that which is self-existent,\nunconditioned, and absolute.\n\n(ii.) _Facts of the universe which indicate some fundamental relation to\nReason or Thought_.\n\n1. _Numerical and Geometrical Proportion_.--Definite proportion of\nelements (Chemistry), symmetrical arrangement of parts\n(Crystallography), numerical and geometrical relation of the forms and\nmovements of the heavenly bodies (Spherical Astronomy), all of which are\ncapable of exact mathematical expression.\n\n2. _Archetypal Forms_.--The uniform succession of new existences, and\nthe progressive evolution of new orders and species, conformable to\nfixed and definite ideal archetypes, the indication of a comprehensive\n_plan_(Morphological Botany, Comparative Anatomy).\n\n3. _Teleology of Organs_.--The adaptation of organs to the fulfillment\nof special functions, indicating _design_(Comparative Physiology).\n\n4. _Combination of Homotypes and Analogues_.--Diversified homologous\nforms made to fulfill analogous functions, or special purposes fulfilled\nwhilst maintaining a general plan, indicating _choice_ and\n_alternativity_.\n\n(iii.) _Facts of the universe which indicate some fundamental relation\nto Moral Ideas and Ends_.\n\n1. _Ethical Distinctions_.--The universal tendency to discriminate\nbetween voluntary acts as right or wrong, indicating some relation to an\n_immutable moral standard of right_.\n\n2. _Sense of Obligation_.--The universal consciousness of dependence and\nobligation, indicating some relation to Supreme _Power_, an Absolute\n_Authority_.\n\n3. _Feeling of Responsibility_.--The universal consciousness of\nliability to be required to give account for, and endure the\nconsequences of our action, indicating some relation to a Supreme\n_Judge_.\n\n4. _Retributive Issues_.--The pleasure and pain resulting from moral\naction in this life, and the universal anticipation of pleasure or pain\nin the future, as the consequence of present conduct, indicate an\n_absolute Justice_ ruling the world and man.\n\nNow, if the universe be a _created effect_, it must, in some degree at\nleast, reveal the character of its Author and cause. We are entitled to\nregard it as a created symbol and image of the Deity; it must bear the\nimpress of his _power_; it must reveal his infinite _presence_; it must\nexpress his _thoughts_; it must embody and realize his _ideals_, so far,\nat least, as material symbols will permit. Just as we see the power and\nthought of man revealed in his works, his energy and skill, his ideal\nand his taste expressed in his mechanical, artistic, and literary\ncreations, so we may see the mind and character of God displayed in his\nworks. The skill and contrivance of Watts, and Fulton, and Stephenson\nwere exhibited in their mechanical productions. The pure, the intense,\nthe visionary impersonation of the soul which the artist had conjured in\nhis own imagination was wrought out in Psyche. The colossal grandeur of\nMichael Angelo's ideals, the ethereal and saintly elegance of Raphael's\nwere realized upon the canvas. So he who is familiar with the ideal of\nthe sculptor or the painter can identify his creations even when the\nauthor's name is not affixed. And so the \"eternal Power\" of God is\n\"clearly seen\" in the mighty orbs which float in the illimitable space.\nThe vastness of the universe shadows forth the infinity of God. The\nindivisible unity of space and the ideal unity of the universe reflect\nthe unity of God. The material forms around us are symbols of divine\nideas, and the successive history of the universe is an expression of\nthe divine thought; whilst the ethical ideas and sentiments inherent in\nthe human mind are a reflection of the moral character of God.\n\nThe reader can not have failed to observe the form in which the Theistic\nargument is stated; \"_if_ the finite universe is a created effect, it\nmust reveal something as to the nature of its cause: _if_ the existing\norder and arrangement of the universe had a commencement in time, it\nmust have an ultimate and adequate cause.\" The question, therefore,\npresents itself in a definite form: \"_Is the universe finite or\ninfinite; had the order of the universe a beginning, or is it eternal_?\"\n\nIt will be seen at a glance that this is the central and vital question\nin the Theistic argument. If the order and arrangement of the universe\nis _eternal_, then that order is an inherent law of nature, and, as\neternal, does not imply a cause _ab extra:_ if it is not eternal, then\nthe ultimate cause of that order must be a power above and beyond\nnature. In the former case the minor premise of the Theistic syllogism\nis utterly invalidated; in the latter case it is abundantly sustained.\n\nSome Theistic writers--as Descartes, Pascal, Leibnitz, and Saisset--have\nmade the fatal admission that the universe is, in some sense, _infinite_\nand _eternal_. In making this admission they have unwittingly\nsurrendered the citadel of strength, and deprived the argument by which\nthey would prove the being of a God of all its logical force. That\nargument is thus presented by Saisset: \"The finite supposes the\ninfinite. Extension supposes first space, then immensity: duration\nsupposes first time, then eternity. A sudden and irresistible judgment\nrefers this to the necessary, infinite, perfect being.\"[215] But if \"the\nworld is infinite and eternal,\"[216] may not nature, or the totality of\nall existence (to pan), be the necessary, infinite, and perfect Being?\nAn infinite and eternal universe has the reason of its existence in\nitself, and the existence of such a universe can never prove to us the\nexistence of an infinite and eternal God.\n\n[Footnote 215: \"Modern Pantheism,\" vol. ii. p. 205.]\n\n[Footnote 216: Ibid, p. 123.]\n\nA closer examination of the statements and reasonings of Descartes,\nPascal, and Leibnitz, as furnished by Saisset, will show that these\ndistinguished mathematicans were misled by the false notion of\n\"_mathematical_ infinitude.\" Their infinite universe, after all, is not\nan \"absolute,\" but a \"relative\" infinite; that is, the indefinite. \"The\nuniverse must extend _indefinitely_ in time and space, in the infinite\ngreatness, and in the infinite littleness of its parts--in the infinite\nvariety of its species, of its forms, and of its degrees of existence.\nThe finite can not express the infinite but by being _multiplied_\ninfinitely. The finite, so far as it is finite, is not in any reasonable\nrelation, or in any intelligible proportion to the infinite. But the\nfinite, as _multiplied_ infinitely,[217] ages upon ages, spaces upon\nspaces, stars beyond stars, worlds beyond worlds, is a true expression\nof the Infinite Being. Does it follow, because the universe has no\nlimits,--that it must therefore be eternal, immense, infinite as God\nhimself? No; that is but a vain scruple, which springs from the\nimagination, and not from the reason. The imagination is always\nconfounding what reason should ever distinguish, eternity and time,\nimmensity and space, _relative_ infinity and _absolute_ infinity. The\nCreator alone is eternal, immense, absolutely infinite.\"[218]\n\n[Footnote 217: \"The infinite is distinct from the finite, and\nconsequently from the multiplication of the finite by itself; that is,\nfrom the _indefinite_. That which is not infinite, added as many times\nas you please to itself, will not become infinite.\"--Cousin, \"Hist, of\nPhilos.,\" vol. ii. p. 231.]\n\n[Footnote 218: Saisset, \"Modern Pantheism,\" vol. ii. pp. 127, 128.]\n\nThe introduction of the idea of \"the mathematical infinite\" into\nmetaphysical speculation, especially by Kant and Hamilton, with the\ndesign, it would seem, of transforming the idea of infinity into a\nsensuous conception, has generated innumerable paralogisms which\ndisfigure the pages of their philosophical writings. This procedure is\ngrounded in the common fallacy of supposing that _infinity_ and\n_quantity_ are compatible attributes, and susceptible of mathematical\nsynthesis. This insidious and plausible error is ably refuted by a\nwriter in the \"North American Review.\"[219] We can not do better than\ntransfer his argument to our pages in an abridged form.\n\n[Footnote 219: \"The Conditioned and the Unconditioned,\" No. CCV. art.\niii. (1864).]\n\nMathematics is conversant with quantities and quantitative relations.\nThe conception of quantity, therefore, if rigorously analyzed, will\nindicate _a priori_ the natural and impassable boundaries of the\nscience; while a subsequent examination of the quantities called\ninfinite in the mathematical sense, and of the algebraic symbol of\ninfinity, will be seen to verify the results of this _a priori_\nanalysis.\n\nQuantity is that attribute of things in virtue of which they are\nsusceptible of exact mensuration. The question _how much_, or _how many_\n(_quantus_), implies the answer, _so much_, or _so many_ (_tantus_); but\nthe answer is possible only through reference to some standard of\nmagnitude or multitude arbitrarily assumed. Every object, therefore, of\nwhich quantity, in the mathematical sense, is predicable, must be by its\nessential nature _mensurable._ Now mensurability implies the existence\nof actual, definite limits, since without them there could be no fixed\nrelation between the given object and the standard of measurement, and,\nconsequently, no possibility of exact mensuration. In fact, since\nquantification is the object of all mathematical operations, mathematics\nmay be not inaptly defined as _the science of the determinations of\nlimits_. It is evident, therefore, that the terms _quantity_ and\n_finitude_ express the same attribute, namely, _limitation_--the former\nrelatively, the latter absolutely; for quantity is limitation considered\nwith relation to some standard of measurement, and finitude is\nlimitation considered simply in itself. The sphere of quantity,\ntherefore, is absolutely identical with the sphere of the finite; and\nthe phrase _infinite quantity_, if strictly construed, is a\ncontradiction in terms.\n\nThe result thus attained by considering abstract quantity is\ncorroborated by considering concrete and discrete quantities. Such\nexpressions as _infinite sphere, radius, parallelogram, line,_ and so\nforth, are self-contradictory. A sphere is limited by its own periphery,\nand a radius by the centre and circumference of its circle. A\nparallelogram of infinite altitude is impossible, because the limit of\nits altitude is assigned in the side which must be parallel to its base\nin order to constitute it a parallelogram. In brief, all figuration is\nlimitation. The contradiction in the term _infinite line_ is not quite\nso obvious, but can readily be made apparent. Objectively, a line is\nonly the termination of a surface, and a surface the termination of a\nsolid; hence a line can not exist apart from an extended quantity, nor\nan infinite line apart from an infinite quantity. But as this term has\njust been shown to be self-contradictory, an infinite line can not exist\nobjectively at all. Again, every line is extension in one dimension;\nhence a mathematical quantity, hence mensurable, hence finite; you must\ntherefore, deny that a line is a quantity, or else affirm that it is\nfinite.\n\nThe same conclusion is forced upon us, if from geometry we turn to\narithmetic. The phrases _infinite number, infinite series, infinite\nprocess_, and so forth, are all contradictory when literally construed.\nNumber is a relation among separate unities or integers, which,\nconsidered objectively as independent of our cognitive powers, must\nconstitute an exact sum; and this exactitude, or synthetic totality, is\nlimitation. If considered subjectively in the mode of its cognition, a\nnumber is infinite only in the sense that it is beyond the power of our\nimagination or conception, which is an abuse of the term. In either case\nthe totality is fixed; that is, finite. So, too, of _series_ and\n_process_. Since every series involves a succession of terms or numbers,\nand every process a succession of steps or stages, the notion of series\nand process plainly involves that of _number_, and must be rigorously\ndissociated from the idea of infinity. At any one step, at any one term,\nthe number attained is determinate, hence finite. The fact that, by the\nlaw of the series or of the process, _we_ may continue the operation _as\nlong as we please_, does not justify the application of the term\ninfinite to the operation itself; if any thing is infinite, it is the\nwill which continues the operation, which is absurd if said of human\nwills.\n\nConsequently, the attribute of infinity is not predicable either of\n'diminution without limit,' 'augmentation without limit,' or 'endless\napproximation to a fixed limit,' for these mathematical processes\ncontinue only as we continue them, consist of steps successively\naccomplished, and are limited by the very fact of this serial\nincompletion.\n\n\"We can not forbear pointing out an important application of these\nresults to the Critical Philosophy. Kant bases each of his famous four\nantinomies on the demand of pure reason for unconditioned totality in a\nregressive series of conditions. This, he says, must be realized either\nin an absolute first of the series, conditioning all the other members,\nbut itself unconditioned, or else in the absolute infinity of the series\nwithout a first; but reason is utterly unable, on account of mutual\ncontradiction, to decide in which of the two alternatives the\nunconditioned is found. By the principles we have laid down, however,\nthe problem is solved. The absolute infinity of a series is a\ncontradiction _in adjecto_. As every number, although immeasurably and\ninconceivably great, is impossible unless _unity_ is given as its basis,\nso every series, being itself a number, is impossible unless a _first\nterm_ is given as a commencement. Through a first term alone is the\nunconditioned possible; that is, if it does not exist in a first term,\nit can not exist at all; of the two alternatives, therefore, one\naltogether disappears, and reason is freed from the dilemma of a\ncompulsory yet impossible decision. Even if it should be allowed that\nthe series has no first term, but has originated _ab aeterno_, it must\nalways at each instant have a _last term_; the series, as a whole, can\nnot be infinite, and hence can not, as Kant claims it can, realize in\nits wholeness unconditioned totality. Since countless terms forever\nremain unreached, the series is forever limited by them. Kant himself\nadmits that it _can never be completed_, and is only potentially\ninfinite; actually, therefore, by his own admission, it is finite. But a\nlast term implies a first, as absolutely as one end of a string implies\nthe other; the only possibility of an unconditioned lies in Kant's first\nalternative, and if, as he maintains Reason must demand it, she can not\nhesitate in her decisions. That _number is a limitation_ is no new\ntruth, and that every series involves number is self-evident; and it is\nsurprising that so radical a criticism on Kant's system should never\nhave suggested itself to his opponents. Even the so-called _moments_ of\ntime can not be regarded as constituting a real series, for a series can\nnot be real except through its divisibility into members whereas time is\nindivisible, and its partition into moments is a conventional fiction.\nExterior limitability and interior divisibility result equally from the\npossibility of discontinuity. Exterior illimitability and interior\nindivisibility are simple phases of the same attribute of _necessary\ncontinuity_ contemplated under different aspects. From this principle\nflows another upon which it is impossible to lay too much stress,\nnamely; _illimitability and indivisibility, infinity and unity,\nreciprocally necessitate each other_. Hence the Quantitative Infinites\nmust be also Units, and the division of space and time, implying\nabsolute contradiction, is not even cogitable as an hypothesis.[220]\n\n\"The word _infinite_, therefore, in mathematical usage, as applied to\n_process_ and to _quantity_, has a two-fold signification. An infinite\nprocess is one which we can continue _as long as we please_, but which\nexists solely in our continuance of it.[221] An infinite quantity is one\nwhich exceeds our powers of mensuration or of conception, but which,\nnevertheless, has bounds and limits in itself.[222] Hence the\npossibility of relation among infinite quantities, and of different\norders of infinities. If the words _infinite, infinity, infinitesimal_,\nshould be banished from mathematical treatises and replaced by the words\n_indefinite, indefinity,_ and _indefinitesimal_, mathematics would\nsuffer no loss, while, by removing a perpetual source of confusion,\nmetaphysics would get great gain.\"\n\n[Footnote 220: By the application of these principles the writer in the\n\"North American Review\" completely dissolves the antinomies by which\nHamilton seeks to sustain his \"Philosophy of the Conditioned.\" See\n\"North American Review,\" 1864, pp. 432-437.]\n\n[Footnote 221: De Morgan, \"Diff. and Integ. Calc.\" p. 9.]\n\n[Footnote 222: Id., ib., p. 25.]\n\nThe above must be regarded as a complete refutation of the position\ntaken by _Hume_, to wit, that the idea of nature eternally existing in a\nstate of order, without a cause other than the eternally inherent laws\nof nature, is no more self-contradictory than the idea of an\neternally-existing and infinite mind, who originated this order--a God\nexisting without a cause. The eternal and infinite Mind is indivisible\nand illimitable; nature, in its totality, as well as in its individual\nparts, has interior divisibility, and exterior limitability. The\ninfinity of God is not a _quantitative_, but a _qualitative_ infinity.\nThe miscalled eternity and infinity of nature is an _indefinite_\nextension and protension in time and space, and, as _quantitative_, must\nnecessarily be limited and measurable, therefore _finite_.\n\nThe universe of sense-perception and sensuous imagination is a\nphenomenal universe, a genesis, a perpetual becoming, an entrance into\nexistence, and an exit thence; the Theist is, therefore, perfectly\njustified in regarding it as disqualified for _self-existence_, and in\npassing behind it for the Supreme Entity that needs no cause. Phenomena\ndemand causation, entities dispense with it. No one asks for a cause of\nthe _space_ which contains the universe, or of the Eternity on the bosom\nof which it floats. Everywhere the line is necessarily drawn upon the\nsame principle; that entities _may_ have self-existence, phenomena\n_must_ have a cause.[223]\n\n[Footnote 223: \"Science, Nescience, and Faith,\" in Martineau's \"Essays,\"\np. 206.]\n\nIV. _Psychological analysis clearly attests that in the phenomena of\nconsciousness there are found elements or principles which, in their\nregular and normal development, transcend the limits of consciousness,\nand attain to the knowledge of Absolute Being, Absolute Reason, Absolute\nGood_, i.e., GOD.\n\nThe analysis of thought clearly reveals that the mind of man is in\npossession of ideas, notions, beliefs, principles (as _e.g._, the idea\nof space, duration, cause, substance, unity, infinity), which are not\nderived from sensation and experience, and which can not be drawn out of\nsensation and experience by any process of generalization. These ideas\nhave this incontestable peculiarity, as distinguished from all the\nphenomena of sensation, that, whilst the latter are particular,\ncontingent, and relative, the former are _universal_, _necessary_, and\n_absolute_. As an example, and a proof of the reality and validity of\nthis distinction, take the ideas of _body_ and of _space_, the former\nunquestionably derived from experience, the latter supplied by reason\nalone. \"I ask you, can not you conceive this book to be destroyed?\nWithout doubt you can. And can not you conceive the whole world to be\ndestroyed, and no matter whatever in existence? You can. For you,\nconstituted as you are, the supposition of the non-existence of bodies\nimplies no contradiction. And what do we call the idea of a thing which\nwe can conceive of as non-existing? We call it a _contingent_ and\n_relative_ idea. But if you can conceive this book to be destroyed, all\nbodies destroyed, can you suppose space to be destroyed? You can not. It\nis in the power of man's thought to conceive the non-existence of\nbodies; it is not in the power of man's thought to conceive the\nnon-existence of space. The idea of space is thus a _necessary_ and\n_absolute_ idea.\"[224]\n\n[Footnote 224: Cousin's \"Hist. of Philos.,\" vol. ii. p. 214.]\n\nTake, again, the ideas of _event_ and _cause_. The idea of an event is a\n_contingent_ idea; it is the idea of something which might or might not\nhave happened. There is no impossibility or contradiction in either\nsupposition. The idea of cause is a _necessary_ idea. An event being\ngiven, the idea of cause is necessarily implied. An uncaused event is an\nimpossible conception. The idea of cause is also a _universal_ idea\nextending to all events, actual or conceivable, and affirmed by all\nminds. It is a rational fact, attested by universal consciousness, that\nwe can not think of an event transpiring without a cause; of a thing\nbeing the author of its own existence; of something generated by and out\nof nothing. _Ex nihilo nihil_ is a universal law of thought and of\nthings. This universal \"law of causality\" is clearly distinguishable\nfrom a _general_ truth reached by induction. For example, it is a very\ngeneral truth that, during twenty-four hours, day is succeeded by night.\nBut this is not a necessary truth, neither is it a universal truth. It\ndoes not extend to all known lands, as, for example, to Nova Zembla. It\ndoes not hold true of the other planets. Nor does it extend to all\npossible lands. We can easily conceive of lands plunged in eternal\nnight, or rolling in eternal day. With another system of worlds, one can\nconceive other physics, but one can not conceive other metaphysics. It\nis impossible to imagine a world in which the law of causality does not\nreign. Here, then, we have one absolute principle (among others which\nmay be enumerated), the existence and reality of which is revealed, not\nby sensation, but by reason--a principle which transcends the limits of\nexperience, and which, in its regular and logical development, attains\nthe knowledge of the Absolute Cause--the First Cause of all causes--God.\n\nThus it is evident that the human mind is in possession of two distinct\norders of primitive cognitions,--one, contingent, relative, and\nphenomenal; the other universal, necessary, and absolute. These two\ndistinct orders of cognition presuppose the existence in man of two\ndistinct faculties or organs of knowledge--_sensation_, external and\ninternal, which perceives the contingent, relative, and phenomenal, and\n_reason_, which apprehends the universal, necessary, and absolute. The\nknowledge which is derived from sensation and experience is called\n_empirical_ knowledge, or knowledge _a posteriori_, because subsequent\nto, and consequent upon, the exercise of the faculties of observation.\nThe knowledge derived from reason is called _transcendental_ knowledge,\nor knowledge _a priori_, because it furnishes laws to, and governs the\nexercise of the faculties of observation and thought, and is not the\nresult of their exercise. The sensibility brings the mind into relation\nwith the _physical_ world, the reason puts mind in communication with\nthe _intelligible_ world--the sphere of _a priori_ principles, of\nnecessary and absolute truths, which depend upon neither the world nor\nthe conscious self, and which reveal to man the existence of the soul,\nnature, and God. Every distinct fact of consciousness is thus at once\n_psychological_ and _ontological_, and contains these three fundamental\nideas, which we can not go beyond, or cancel by any possible\nanalysis--the _soul_, with its faculties; _matter_, with its qualities;\n_God_, with his perfections.\n\nWe do not profess to be able to give a clear explication and complete\nenumeration of all the ideas of reason, and of the necessary and\nuniversal principles or axioms which are grounded on these ideas. This\nis still the grand desideratum of metaphysical science. Its achievement\nwill give us a primordial logic, which shall be as exact in its\nprocedure and as certain in its conclusions as the mathematical\nsciences. Meantime, it may be affirmed that philosophic analysis, in the\nperson of Plato, Aristotle, Kant, and Cousin, has succeeded in\ndisengaging such _a priori_ ideas, and formulating such principles and\nlaws of thought, as lead infallibly to the cognition of the _Absolute\nBeing_, the _Absolute Reason_, the _Absolute Good_, that is, GOD.\n\nIt would carry us too far beyond our present design were we to exhibit,\nin each instance, the process of _immediate abstraction_ by which the\ncontingent and relative element of knowledge is eliminated, and the\nnecessary and absolute principle is disengaged. We shall simply state\nthe method, and show its application by a single illustration.\n\nThere are unquestionably _two_ sorts of abstraction: 1. \"_Comparative_\nabstraction, operating upon several real objects, and seizing their\nresemblances in order to form an abstract idea, which is collective and\nmediate; collective, because different individuals concur in its\nformation; mediate, because it requires several intermediate\noperations.\" This is the method of the physical sciences, which\ncomprises comparison, abstraction, and generalization. The result in\nthis process is the attainment of a _general_ truth. 2. \"_Immediate_\nabstraction, not comparative; operating not upon several concretes, but\nupon a single one, eliminating and neglecting its individual and\nvariable part, and disengaging the absolute part, which it raises at\nonce to its pure form.\" The parts to be eliminated in a concrete\ncognition are, first, the quality of the object, and the circumstances\nunder which the absolute unfolds itself; and secondly, the quality of\nthe subject, which perceives but does not constitute it. The phenomena\nof the me and the not-me being eliminated, the absolute remains. This is\nthe process of rational psychology, and the result obtained is a\n_universal_ and _necessary_ truth.\n\n\"Let us take, as an example, the principle of cause. To be able to say\nthat the event I see must have a cause, it is not indispensable to have\nseen several events succeed each other. The principle which compels me\nto pronounce this judgment is already complete in the first as in the\nlast event; it can not change in respect to its object, it can not\nchange in itself; it neither increases nor decreases with the greater or\nless number of applications. The only difference that it is subject to\nin regard to us is that we apply it, whether we remark it or not,\nwhether we disengage it or not from its particular application. The\nquestion is not to eliminate the particularity of the phenomenon wherein\nit appears to us, whether it be the fall of a leaf or the murder of a\nman, in order immediately to conceive, in a general and abstract manner,\nthe necessity of a cause for every event that begins to exist. Here it\nis not because I am the same, or have been affected in the same manner\nin several different cases, that I have come to this general and\nabstract conception. A leaf falls; at the same moment I think, I\nbelieve, I declare that this falling of the leaf must have a cause. A\nman has been killed; at the same instant I believe, I proclaim that this\ndeath must have a cause. Each one of these facts contains particular and\nvariable circumstances, and something universal and necessary, to wit,\nboth of them can not but have a cause. Now I am perfectly able to\ndisengage the universal from the particular in regard to the first fact\nas well as in regard to the second fact, for the universal is in the\nfirst quite as well as in the second. In fact, if the principle of\ncausality is not universal in the first fact, neither will it be in the\nsecond, nor in the third, nor in the thousandth; for a thousandth is not\nnearer than the first to the infinite--to absolute universality. It is\nthe same, and still more evidently, with _necessity_. Pay particular\nattention to this point; if necessity is not in the first fact, it can\nnot be in any; for necessity can not be formed little by little, and by\nsuccessive increments. If, on the first murder I see, I do not exclaim\nthat this murder had necessarily a cause, at the thousandth murder,\nalthough it shall be proved that all the others had causes, I shall have\nthe right to think that this murder has, very probably, also a cause,\nbut I shall never have the right to say that it _necessarily_ had a\ncause. But when universality and necessity are already in a single case,\nthat case is sufficient to entitle me to deduce them from it,\"[225] and\nwe may add, also, to affirm them of every other event that may\ntranspire.\n\n[Footnote 225: Cousin, \"True, Beautiful, and Good,\" pp. 57, 58.]\n\nThe following _schema_ will exhibit the generally accepted results of\nthis method of analysis applied to the phenomena of thought:\n\n(i.) _Universal and necessary principles, or primitive judgments from\nwhence is derived the cognition of Absolute Being_.\n\n1. _The principle of Substance_; thus enounced--\"every quality supposes\na _subject_ or real being.\"\n\n2. _The principle of Causality_; \"every thing that begins to be supposes\na _power_ adequate to its production, _i.e._, an efficient cause.\"\n\n3. _The principle of Unity_; \"all differentiation and plurality supposes\nan incomposite unity; all diversity, an ultimate and indivisible\nidentity.\"\n\n4. _The principle of the Unconditioned_; \"the finite supposes the\ninfinite, the dependent supposes the self-existent, the temporal\nsupposes the eternal.\"\n\n(ii.) _Universal and necessary principles, or primitive judgments, from\nwhich is derived the cognition of the Absolute Reason_.\n\n1. _The principle of Ideality_; thus enounced, \"facts of order--definite\nproportion, symmetrical arrangement, numerical relation, geometrical\nform--having a commencement in time, present themselves to us as the\nexpression of _Ideas_, and refer us to _Mind_ as their analogon, and\nexponent, and source.\"\n\n2. _The principle of Consecution_; \"the uniform succession and\nprogressive evolution of new existences, according to fixed definite\narchetypes, suppose a unity of _thought_--a comprehensive _plan_\nembracing all existence.\"\n\n3. _The principle of Intentionality or Final Cause_; \"every means\nsupposes an _end_ contemplated, and a choice and adaptation of means to\nsecure the _end_.\"\n\n4. _The principle of Personality_; \"intelligent purpose and voluntary\nchoice imply a personal agent.\"\n\n(iii.) _Universal and necessary principles, or primitive judgments, from\nwhence is derived the cognition of the Absolute Good_.\n\n1. _The principle of Moral Law_; thus enounced, \"the action of a\nvoluntary agent necessarily characterized as _right_ or _wrong_,\nsupposes an immutable and universal standard of right--an absolute moral\nLaw.\"\n\n2. _The principle of Moral Obligation_; \"the feeling of obligation to\nobey a law of duty supposes a _Lawgiver_ by whose authority we are\nobliged.\"\n\n3. _The principle of Moral Desert_; \"the feeling of personal\naccountability and of moral desert supposes a _judge_ to whom we must\ngive account, and who shall determine our award.\"\n\n4. _The pnnciple of Retribution_; \"retributive issues in this life, and\nthe existence in all minds of an impersonal justice which demands that,\nin the final issue, every being shall receive his just deserts, suppose\na being of _absolute justice_ who shall render to every man according to\nhis works.\"\n\nA more profound and exhaustive analysis may perhaps resolve all these\nprimitive judgments into one universal principle or law, which Leibnitz\nhas designated \"_The principle or law of sufficient reason_,\" and which\nis thus enounced--there must be an ultimate and sufficient reason why\nany thing exists, and why it is, rather than otherwise; that is, if any\nthing begins to be, something else must be supposed as the adequate\nground, and reason, and cause of its existence; or again, to state the\nlaw in view of our present discussion, \"_if the finite universe, with\nits existing order and arrangement, had a beginning, there must be an\nultimate and sufficient reason why it exists, and why it is as it is,\nrather than otherwise_.\" In view of one particular class of phenomena,\nor special order of facts, this \"principle of sufficient reason\" may be\nvaried in the form of its statement, and denominated \"the principle of\nsubstance,\" \"the principle of causality,\" \"the principle of\nintentionality,\" etc.; and, it may be, these are but specific judgments\nunder the one fundamental and generic law of thought which constitutes\nthe _major_ premise of every Theistic syllogism.\n\nThese fundamental principles, primitive judgments, axioms, or necessary\nand determinate forms of thought, exist potentially or germinally in all\nhuman minds; they are spontaneously developed in presence of the\nphenomena of the universe, material and mental; they govern the original\nmovement of the mind, even when not appearing in consciousness in their\npure and abstract form; and they compel us to affirm _a permanent being_\nor _reality_ behind all phenomena--a _power_ adequate to the production\nof change, back of all events; a _personal Mind_, as the explanation of\nall the facts of order, and uniform succession, and regular evolution;\nand a _personal Lawgiver_ and _Righteous Judge_ as the ultimate ground\nand reason of all the phenomena of the moral world; in short, to affirm\n_an Unconditioned Cause of all finite and secondary causes; a First\nPrinciple of all principles; an Ultimate Reason of all reasons; an\nimmutable Uncreated Justice, the living light of conscience; a King\nimmortal, eternal, invisible, the only wise God, the ruler of the world\nand man_.\n\nOur position, then, is, that the idea of God is revealed to man in the\nnatural and spontaneous development of his intelligence, and that the\nexistence of a Supreme Reality corresponding to, and represented by this\nidea, is rationally and logically demonstrable, and therefore justly\nentitled to take rank as part of our legitimate, valid, and positive\n_knowledge_.\n\nAnd now from this position, which we regard as impregnable, we shall be\nprepared more deliberately and intelligibly to contemplate the various\nassaults which are openly or covertly made upon the doctrine that _God\nis cognizable by human reason_.\n\n\n\n\nCHAPTER VI.\n\nTHE UNKNOWN GOD (_continued_).\n\nIS GOD COGNIZABLE BY REASON?\n\n\n\"The abnegation of reason is not the evidence of faith, but the\nconfession of despair.\"--LIGHTFOOT.\n\nAt the outset of this inquiry we attempted a hasty grouping of the\nvarious parties and schools which are arrayed against the doctrine that\nGod is cognizable by human reason, and in general terms we sought to\nindicate the ground they occupy.\n\nViewed from a philosophical stand-point, we found one party marshalled\nunder the standard of Idealism; another of Materialism and, again,\nanother of Natural Realism. Regarded in their theological aspects, some\nare positive Atheists; others, strange to say, are earnest Theists;\nwhilst others occupy a position of mere Indifferentism. Yet,\nnotwithstanding the remarkable diversity, and even antagonism of their\nphilosophical and theological opinions, they are all agreed in denying\nto reason any valid cognition of God.\n\nThe survey of Natural Theism we have completed in the previous chapter\nwill enable us still further to indicate the exact points against which\ntheir attacks are directed, and also to estimate the character and force\nof the weapons employed. With or without design, they are, each in their\nway, assailing one or other of the principles upon which we rest our\ndemonstration of the being of God. As we proceed, we shall find that\nMill and the Constructive Idealists are really engaged in undermining\n\"the _principle of substance_;\" their doctrine is a virtual denial of\nall objective realities answering to our subjective ideas of matter,\nmind, and God. The assaults of Comte and the Materialists of his school\nare mainly directed against \"_the principle of causality_\" and \"_the\nprinciple of intentionality_;\" they would deny to man all knowledge of\ncauses, efficient and final. The attacks of Hamilton and his school are\ndirected against \"the _principle of the unconditioned_,\" his philosophy\nof the conditioned is a plausible attempt to deprive man of all power to\nthink the Infinite and Perfect, to conceive the Unconditioned and\nUltimate Cause; whilst the Dogmatic Theologians are borrowing, and\nrecklessly brandishing, the weapons of all these antagonists, and, in\naddition to all this, are endeavoring to show the insufficiency of \"_the\nprinciple of unity_\" and the weakness and invalidity of \"the _moral\nprinciples_,\" which are regarded by us as relating man to a Moral\nPersonality, and as indicating to him the existence of a righteous God,\nthe ruler of the world. It is necessary, therefore, that we should\nconcentrate our attention yet more specifically on these separate lines\nof attack, and attempt a minuter examination of the positions assumed by\neach, and of the arguments by which they are seeking, directly or\nindirectly, to invalidate the fundamental principles of Natural Theism.\n\n(i.) _We commence with the Idealistic School_, of which John Stuart Mill\nmust be regarded as the ablest living representative.\n\nThe doctrine of this school is that all our knowledge is necessarily\nconfined to _mental_ phenomena; that is, \"to _feelings_ or states of\nconsciousness,\" and \"the succession and co-existence, the likeness and\nunlikeness between these feelings or states of consciousness.\"[226] All\nour general notions, all our abstract ideas, are generated out of these\nfeelings[227] by \"_inseparable association_,\" which registers their\ninter-relations of recurrence, co-existence, and resemblance. The\nresults of this inseparable association constitute at once the sum total\nand the absolute limit of all possible cognition.\n\n[Footnote 226: J. S. Mill, \"Logic,\" vol. i. p. 83 (English edition).]\n\n[Footnote 227: In the language of Mill, every thing of which we are\nconscious is called \"feeling.\" \"Feeling, in the proper sense of the\nterm, is a genus of which Sensation, Emotion, and Thought are the\nsubordinate species.\"--\"Logic,\" bk. i. ch. iii. Sec. 3.]\n\nIt is admitted by Mill that one _apparent_ element in this total result\nis the general conviction that our own existence is really distinct from\nthe external world, and that the personal _ego_ has an essential\nidentity distinct from the fleeting phenomena of sensation. But this\npersuasion is treated by him as a mere illusion--a leap beyond the\noriginal datum for which we have no authority. Of a real substance or\nsubstratum called Mind, of a real substance or substratum called Matter,\nunderlying the series of feelings--\"the thread of consciousness\"--we do\nknow and can know nothing; and in affirming the existence of such\nsubstrata we are making a supposition we can not possibly verify. The\nultimate datum of speculative philosophy is not \"_I think_,\" but simply\n\"_Thoughts or feelings are_.\" The belief in a permanent subject or\nsubstance, called matter, as the ground and plexus of physical\nphenomena, and of a permanent subject or substance, called mind, as the\nground and plexus of mental phenomena, is not a primitive and original\nintuition of reason. It is simply through the action of the principle of\nassociation among the ultimate phenomena, called feelings, that this\n(erroneous) separation of the phenomena into two orders or\naggregates--one called mind or self; the other matter, or not\nself--takes place; and without this curdling or associating process no\nsuch notion or belief could have been generated. \"The principle of\nsubstance,\" as an ultimate law of thought, is, therefore, to be regarded\nas a transcendental dream.\n\nBut now that the notion of _mind_ or _self_, and of _matter_ or not\n_self_, do exist as common convictions of our race, what is philosophy\nto make of them? After a great many qualifications and explanations, Mr.\nMill has, in his \"Logic,\" summed up his doctrine of Constructive\nIdealism in the following words: \"As body is the mysterious _something_\nwhich excites the mind to feel, so mind is the mysterious _something_\nwhich feels and thinks.\"[228] But what is this \"mysterious something?\"\nIs it a reality, an entity, a subject; or is it a shadow, an illusion, a\ndream? In his \"Examination of Sir Wm. Hamilton's Philosophy,\" where it\nmay be presumed, we have his maturest opinions, Mr. Mill, in still more\nabstract and idealistic phraseology, attempts an answer. Here he defines\nmatter as \"_a permanent possibility of sensation_,\"[229] and mind as \"_a\npermanent possibility of feeling_.\"[230] And \"the belief in these\npermanent possibilities,\" he assures us, \"includes all that is essential\nor characteristic in the belief in substance.\"[231] \"If I am asked,\"\nsays he, \"whether I believe in matter, I ask whether the questioner\naccepts this definition of it. If he does, I believe in matter: and so\ndo all Berkeleians. In any other sense than this, I do not. But I affirm\nwith confidence that this conception of matter includes the whole\nmeaning attached to it by the common world, apart from philosophical,\nand sometimes from theological theories. The reliance of mankind on the\nreal existence of visible and tangible objects, means reliance on the\nreality and permanence of possibilities of visual and tactual\nsensations, when no sensations are actually experienced.\"[232]\n\"Sensations,\" however, let it be borne in mind, are but a subordinate\nspecies of the genus feeling.[233] They are \"states of\nconsciousness\"--phenomena of mind, not of matter; and we are still\nwithin the impassable boundary of ideal phenomena; we have yet no\ncognition of an external world. The sole cosmical conception, for us, is\nstill a succession of sensations, or states of consciousness. This is\nthe one phenomenon which we can not transcend in knowledge, do what we\nwill; all else is hypothesis and illusion. The _non-ego_, after all,\nthen, may be but a mode in which the mind represents to itself the\npossible modifications of the _ego_.\n\n[Footnote 228: \"Logic,\" bk. i, ch. iii. Sec. 8.]\n\n[Footnote 229: \"Examination of Sir Wm. Hamilton's Philosophy,\" vol. i.\np. 243.]\n\n[Footnote 230: Ibid., vol. i. p. 253.]\n\n[Footnote 231: Ibid., vol. i. p. 246.]\n\n[Footnote 232: Ibid., vol. i. pp. 243, 244.]\n\n[Footnote 233: \"Logic,\" bk. i. ch. iii. Sec. 3.]\n\nAnd now that matter, as a real existence, has disappeared under Mr.\nMill's analysis, what shall be said of mind or self? Is there any\npermanent subject or real entity underlying the phenomena of feeling? In\nfeeling, is there a personal self that feels, thinks, and wills? It\nwould seem not. Mind, as well as matter, resolves itself into a \"series\nof feelings,\" varying and fugitive from moment to moment, in a sea of\npossibilities of feeling. \"My mind,\" says Mill, \"is but a series of\nfeelings, or, as it has been called, a thread of consciousness, however\nsupplemented by believed possibilities of consciousness, which are not,\nthough they might be, realized.\"[234]\n\n[Footnote 234: \"Examination of Sir Wm. Hamilton's Philosophy,\" vol. i.\np. 254.]\n\nThe ultimate fact of the phenomenal world, then, in the philosophy of\nMill, is neither matter nor mind, but feelings or states of\nconsciousness associated together by the relations, amongst themselves,\nof recurrence, co-existence, and resemblance. The existence of self,\nexcept as \"a series of feelings;\" the existence of any thing other than\nself, except as a feigned unknown cause of sensation, is rigorously\ndenied. Mr. Mill does not content himself with saying that we are\nignorant of the _nature_ of matter and mind, but he asserts we are\nignorant of the _existence_ of matter and mind as real entities.\n\nThe bearing of this doctrine of Idealism upon Theism and Theology will\nbe instantly apparent to the reader. If I am necessarily ignorant of the\nexistence of the external world, and of the personal _ego_, or real\nself, I must be equally ignorant of the existence of God. If one is a\nmere supposition, an illusion, so the other must be. Mr. Mill, however,\nis one of those courteous and affable writers who are always conscious,\nas it were, of the presence of their readers, and extremely careful not\nto shock their feelings or prejudices; besides, he has too much\nconscious self-respect to avow himself an atheist. As a speculative\nphilosopher, he would rather regard Theism and Theology as \"open\nquestions,\" and he satisfies himself with saying, if you believe in the\nexistence of God, or in Christianity, I do not interfere with you. \"As a\ntheory,\" he tells us that his doctrine leaves the evidence of the\nexistence of God exactly as it was before. Supposing me to believe that\nthe Divine mind is simply the series of the Divine thoughts and feelings\nprolonged through eternity, that would be, at any rate, believing God's\nexistence to be _as real as my own_[235]. And as for evidence, the\nargument of Paley's 'Natural Theology,' or, for that matter, of his\n'Evidences of Christianity,' would stand exactly as it does.\n\nThe design argument is drawn from the analogies of human experience.\nFrom the relation which human works bear to human thoughts and feelings,\nit infers a corresponding relation between works more or less similar,\nbut superhuman, and superhuman thoughts and feelings. _If_ it prove\nthese, nobody but a metaphysician needs care whether or not it proves a\nmysterious _substratum_ for them.[236] The argument from design, it\nseems to us, however, would have no validity if there be no external\nworld offering marks of design. If the external world is only a mode of\nfeeling, a series of mental states, then our notion of the Divine\nExistence may be only \"an association of feelings\"--a mode of Self. And\nif we have no positive knowledge of a real self as existing, and God's\nexistence is no more \"real than our own,\" then the Divine existence\nstands on a very dubious and uncertain foundation. It can have no very\nsecure hold upon the human mind, and certainly has no claim to be\nregarded as a fundamental and necessary belief. That it has a very\nprecarious hold upon the mind of Mr. Mill, is evident from the following\npassage in his article on \"_Later Speculations of A. Comte_.\"[237] \"We\nventure to think that a religion may exist without a belief in a God,\nand that a religion without a God may be, even to Christians, an\ninstructive and profitable object of contemplation.\"\n\nAnd now let us close Mr. Mill's book, and, introverting our mental gaze,\ninterrogate _consciousness_, the verdict of which, even Mr. Mill assures\nus, is admitted on all hands to be a decision without appeal.[238]\n\n[Footnote 235: \"Examination of Sir Wm. Hamilton's Philosophy,\" vol. i.\np. 254.]\n\n[Footnote 236: \"Examination of Sir Wm. Hamilton's Philosophy,\" vol. i.\np. 259.]\n\n[Footnote 237: Westminster Review, July, 1835 (American edition), p. 3.]\n\n[Footnote 238: \"Examination of Sir Wm. Hamilton's Philosophy,\" vol. i.\np. 161.]\n\n1. We have an ineradicable, and, as it would seem, an intuitive faith in\nthe real existence of an external world distinct from our sensations,\nand also of a personal self, which we call \"I,\" \"myself,\" as distinct\nfrom \"my sensations,\" and \"my feelings.\" We find, also, that this is\nconfessedly the common belief of mankind. There have been a few\nphilosophers who have affected to treat this belief as a \"mere\nprejudice,\" an \"illusion;\" but they have never been able, practically,\nso to regard and treat it. Their language, just as plainly as the\nlanguage of the common people, betrays their instinctive faith in an\nouter world, and proves their utter inability to emancipate themselves\nfrom this \"prejudice,\" if such it may please them to call it. In view of\nthis acknowledged fact, we ask--Does the term \"_permanent possibility of\nsensations_\" exhaust all that is contained in this conception of an\nexternal world? This evening I _remember_ that at noonday I beheld the\nsun, and experienced a sensation of warmth whilst exposing myself to his\nrays; and I _expect_ that to-morrow, under the same conditions, I shall\nexperience the same sensations. I now _remember_ that last evening I\nextinguished my light and attempted to leave my study, but, coming in\ncontact with the closed door, experienced a sense of resistance to my\nmuscular effort, by a solid and extended body exterior to myself; and I\n_expect_ that this evening, under the same circumstances, I shall\nexperience the same sensations. Now, does a belief in \"a permanent\npossibility of sensations\" explain all these experiences? does it\naccount for that immediate knowledge of an _external_ object which I had\non looking at the sun, or that presentative knowledge of _resistance_\nand _extension_, and of an extended, resisting _substance_, I had when\nin contact with the door of my study? Mr. Mill very confidently affirms\nthat this belief includes all; and this phrase expresses all the meaning\nattached to extended \"matter\" and resisting \"substance\" by the common\nworld.[239] We as confidently affirm that it does no such thing; and as\n\"the common world\" must be supposed to understand the language of\nconsciousness as well as the philosopher, we are perfectly willing to\nleave the decision of that question to the common consciousness of our\nrace. If all men do not believe in a permanent _reality_--a substance\nwhich is external to themselves, a substance which offers resistance to\ntheir muscular effort, and which produces in them the sensations of\nsolidity, extension, resistance, etc.--they believe nothing and know\nnothing at all about the matter.\n\n[Footnote 239: \"Examination of Sir Wm. Hamilton's Philosophy,\" vol. i.\np. 243.]\n\nStill less does the phrase \"_a permanent possibility of feelings\"_\nexhaust all our conception of a personal self. Recurring to the\nexperiences of yesterday, I _remember_ the feelings I experienced on\nbeholding the sun, and also on pressing against the closed door, and I\nconfidently _expect_ the recurrence, under the same circumstances, of\nthe same feelings. Does the belief in \"a permanent possibility of\nfeelings\" explain the act of memory by which I recall the past event,\nand the act of prevision by which I anticipate the recurrence of the\nlike experience in the future? Who or what is the \"I\" that remembers and\nthe \"I\" that anticipates? The \"ego,\" the personal mind, is, according to\nMill, a mere \"series of feelings,\" or, more correctly, a flash of\n\"_present_ feelings\" on \"a background of possibilities of present\nfeelings.\"[240] If, then, there be no permanent substance or reality\nwhich is the subject of the present feeling, which receives and retains\nthe impress of the past feeling, and which anticipates the recurrence of\nlike feelings in the future, how can the _past_ be recalled, how\ndistinguished from the present? and how, without a knowledge of the past\nas distinguished from the present, can the _future_ be forecast? Mr.\nMill feels the pressure of this difficulty, and frankly acknowledges it.\nHe admits that, on the hypothesis that mind is simply \"a series of\nfeelings,\" the phenomena of memory and expectation are \"inexplicable\"\nand \"incomprehensible.\"[241] He is, therefore, under the necessity of\ncompleting his definition of mind by adding that it is a series of\nfeelings which \"_is aware of itself as a series_;\" and, still further,\nof supplementing this definition by the conjecture that \"_something\nwhich has ceased to exist, or is not yet in existence, can still, in a\nmanner, be present_.\"[242] Now he who can understand how a series of\nfeelings can flow on in time, and from moment to moment drop out of the\npresent into non-existence, and yet be _present_ and _conscious of\nitself as a series_, may be accorded the honor of understanding Mr.\nMill's definition of mind or self, and may be permitted to rank himself\nas a distinguished disciple of the Idealist school; for ourselves, we\nacknowledge we are destitute of the capacity to do the one, and of all\nambition to be the other. And he who can conceive how the _past_ feeling\nof yesterday and the _possible_ feeling of to-morrow can be in any\nmanner _present_ to-day; or, in other words, how any thing which has\nceased to exist, or which never had an existence, can _now_ exist, may\nbe permitted to believe that a thing can be and not be at the same\nmoment, that a part is greater than the whole, and that two and two make\nfive; but we are not ashamed to confess our inability to believe a\ncontradiction. To our understanding, \"possibilities of feeling\" are not\nactualities. They may or may not be realized, and until realized in\nconsciousness, they have no real being. If there be no other background\nof mental phenomena save mere \"possibilities of feeling,\" then present\nfeelings are the only existences, the only reality, and a loss of\nimmediate consciousness, as in narcosis and coma, is the loss of all\npersonality, all self-hood, and of all real being.\n\n[Footnote 240: \"Exam. of Hamilton,\" vol. i. p. 260.]\n\n[Footnote 241: Ibid, p. 262.]\n\n[Footnote 242: Ibid.]\n\n2. What, then, is the verdict of consciousness as to the existence of a\npermanent substance, an abiding existence which is the subject of all\nthe varying phenomena? Of what are we really conscious when we say \"I\nthink,\" \"I feel,\" \"I will?\" Are we simply conscious of thought, feeling,\nand volition, or of a self, a person, which thinks, feels, and wills?\nThe man who honestly and unreservedly accepts the testimony of\nconsciousness in all its integrity must answer at once, _we have an\nimmediate consciousness, not merely of the phenomena of mind, but of a\npersonal self as passively or actively related to the phenomena_. We are\nconscious not merely of the act of volition, but of a self, a power,\nproducing the volition. We are conscious not merely of feeling, but of a\nbeing who is the subject of the feeling. We are conscious not simply of\nthought, but of a real entity that thinks. \"It is clearly a flat\ncontradiction to maintain that I am not immediately conscious of myself,\nbut only of my sensations or volitions. Who, then, is that _I_ that is\nconscious, and how can I be conscious of such states as _mine?_\"[243]\n\n[Footnote 243: Mansel, \"Prolegomena Logica,\" p. 122, and note E, p.\n281.]\n\nThe testimony of consciousness, then, is indubitable that we have a\ndirect, immediate cognition of _self_--I know myself as a distinctly\nexisting being. This permanent self, to which I refer the earlier and\nlater stages of consciousness, the past as well as the present feeling,\nand which I know abides the same under all phenomenal changes,\nconstitutes my personal identity. It is this abiding self which unites\nthe past and the present, and, from the present stretches onward to the\nfuture. We know self immediately, as existing, as in active operation,\nand as having permanence--or, in other words, as a \"_substance_.\" This\none immediately presented substance, myself, may be regarded as\nfurnishing a positive basis for that other notion of substance, which is\nrepresentatively thought, as the subject of all sensible qualities.\n\n3. We may now inquire what is the testimony of consciousness as to the\nexistence of the extra-mental world? Are we conscious of perceiving\nexternal objects immediately and in themselves, or only mediately\nthrough some vicarious image or representative idea to which we\nfictitiously ascribe an objective reality?\n\nThe answer of common sense is that we are immediately conscious, in\nperception, of an _ego_ and a _non-ego_ known together, and known in\ncontrast to each other; we are conscious of a perceiving subject, and of\nan external reality, as the object perceived.[244] To state this\ndoctrine of natural realism still more explicitly we add, that we are\nconscious of the immediate perception of certain essential attributes of\nmatter objectively existing. Of these primary qualities, which are\nimmediately perceived as real and objectively existing, we mention\n_extension_ in space and _resistance_ to muscular effort, with which is\nindissolubly associated the idea of _externality_. It is true that\nextension and resistance are only qualities, but it is equally true that\nthey are qualities of something, and of something which is external to\nourselves. Let any one attempt to conceive of extension without\nsomething which is extended, or of resistance apart from something which\noffers resistance, and he will be convinced that we can never know\nqualities without knowing substance, just as we can not know substance\nwithout knowing qualities. This, indeed, is admitted by Mr. Mill.[245]\nAnd if this be admitted, it must certainly be absurd to speak of\nsubstance as something \"unknown.\" Substance is known just as much as\nquality is known, no less and no more.\n\n[Footnote 244: Hamilton, \"Lectures,\" vol. 1. p. 288.]\n\n[Footnote 245: \"Logic,\" bk. i. ch. iii. Sec. 6.]\n\nWe remark, in conclusion, that if the testimony of consciousness is not\naccepted in all its integrity, we are necessarily involved in the\nNihilism of Hume and Fichte; the phenomena of mind and matter are, on\nanalysis, resolved into an absolute nothingness--\"a play of phantasms in\na void.\"[246]\n\n(ii.) We turn, secondly, to the _Materialistic School_ as represented by\nAug. Comte.\n\nThe doctrine of this school is that all knowledge is limited to\n_material_ phenomena--that is, to appearances _perceptible to sense_. We\ndo not know the essence of any object, nor the real mode of procedure of\nany event, but simply its relations to other events, as similar or\ndissimilar, co-existent or successive. These relations are constant;\nunder the same conditions, they are always the same. The constant\nresemblances which link phenomena together, and the constant sequences\nwhich unite them, as antecedent and consequent, are termed _laws_. The\nlaws of phenomena are all we know respecting them. Their essential\nnature and their ultimate causes, _efficient_ or _final_, are unknown\nand inscrutable to us.[247]\n\n[Footnote 246: Masson, \"Recent British Philos.,\" p. 62.]\n\n[Footnote 247: See art. \"Positive Philos. of A. Comte,\" _Westminster\nReview_, April, 1865, p. 162, Am. ed.]\n\nIt is not our intention to review the system of philosophy propounded by\nAug. Comte; we are now chiefly concerned with his denial of all\ncausation.\n\n1. _As to Efficient Causes_.--Had Comte contented himself with the\nassertion that causes lie beyond the field of sensible observation, and\nthat inductive science can not carry us beyond the relations of\nco-existence and succession among phenomena, he would have stated an\nimportant truth, but certainly not a new truth. It had already been\nannounced by distinguished mental philosophers, as, for example, M. de\nBiran and Victor Cousin.[248] The senses give us only the succession of\none phenomenon to another. I hold a piece of wax to the fire and it\nmelts. Here my senses inform me of two successive phenomena--the\nproximity of fire and the melting of wax. It is now agreed among all\nschools of philosophy that this is all the knowledge the senses can\npossibly supply. The observation of a great number of like cases assures\nus that this relation is uniform. The highest scientific generalization\ndoes not carry us one step beyond this fact. Induction, therefore, gives\nus no access to causes beyond phenomena. Still, this does not justify\nComte in the assertion that causes are to us absolutely _unknown_. The\nquestion would still arise whether we have not some faculty of\nknowledge, distinct from sensation, which is adequate to furnish a valid\ncognition of cause. It does not by any means follow that, because the\nidea of causation is not given as a \"physical quaesitum\" at the end of a\nprocess of scientific generalization, it should not be a \"metaphysical\ndatum\" posited at the very beginning of scientific inquiry, as the\nindispensable condition of our being able to cognize phenomena at all,\nand as the law under which all thought, and all conception of the system\nof nature, is alone possible.\n\n[Footnote 248: \"It is now universally admitted that we have no\nperception of the causal nexus in the material world.\"--Hamilton,\n\"Discussions,\" p. 522.]\n\nNow we affirm that the human mind has just as direct, immediate, and\npositive knowledge of _cause_ as it has of _effect._ The idea of cause,\nthe intuition _power_, is given in the immediate consciousness of _mind\nas determining its own_ operations. Our first, and, in fact, our only\npresentation of power or cause, is that of _self as willing_. In every\nact of volition I am fully conscious that it is in my power to form a\nresolution or to refrain from it, to determine on this course of action\nor that; and this constitutes the immediate presentative knowledge of\npower.[249] The will is a power, a power in action, a productive power,\nand, consequently, a cause. This doctrine is stated with remarkable\nclearness and accuracy by Cousin: \"If we seek the notion of cause in the\naction of one ball upon another, as was previously done by Hume, or in\nthe action of the hand upon the ball, or the primary muscles upon the\nextremities, or even in the action of the will upon the muscles, as was\ndone by M. Maine de Biran, we shall find it in none of these cases, not\neven in the last; for it is possible there should be a paralysis of the\nmuscles which deprives the will of power over them, makes it\nunproductive, incapable of being a cause, and, consequently, of\nsuggesting the notion of one. But what no paralysis can prevent is the\naction of the will upon itself, the production of a resolution; that is\nto say, the act of causation entirely mental, the primitive type of all\ncausality, of which all external movements are only symbols more or less\nimperfect. The first cause for us, is, therefore, the _will_, of which\nthe first effect is volition. This is at once the highest and the purest\nsource of the notion of cause, which thus becomes identical with that of\npersonality. And it is the taking possession, so to speak, of the cause,\nas revealed in will and personality, which is the condition for us of\nthe ulterior or simultaneous conception of external, impersonal\ncauses.\"[250]\n\n[Footnote 249: \"It is our _immediate consciousness of effort_, when we\nexert force to put matter in motion, or to oppose and neutralize force,\nwhich gives us this internal conviction of _power_ and _causation_, so\nfar as it refers to the material world, and compels us to believe that\nwhenever we see material objects put in motion from a state of rest, or\ndeflected from their rectilinear paths and changed in their velocities\nif already in motion, it is in consequence of such an _effort_ somehow\nexerted.\"--Herschel's \"Outlines of Astronomy,\" p. 234; see Mansel's\n\"Prolegomena,\" p. 133.]\n\n[Footnote 250: \"Philosophical Fragments,\" Preface to first edition.]\n\nThus much for the origin of the idea of cause. We have the same direct\nintuitive knowledge of cause that we have of effect; but we have not yet\nrendered a full and adequate account of the _principle of causality_. We\nhave simply attained the notion of our personal causality, and we can\nnot arbitrarily substitute our personal causality for all the causes of\nthe universe, and erect our own experience as a law of the entire\nuniverse. We have, however, already seen (Chap. V.) that the belief in\nexterior causation is _necessary_ and _universal_. When a change takes\nplace, when a new phenomenon presents itself to our senses, we can not\navoid the conviction that it must have a cause. We can not even express\nin language the relations of phenomena in time and space, without\nspeaking of causes. And there is not a rational being on the face of the\nglobe--a child, a savage, or a philosopher--who does not instinctively\nand spontaneously affirm that every movement, every change, every new\nexistence, _must_ have a cause. Now what account can philosophy render\nof this universal belief? One answer, and only one, is possible. The\n_reason_ of man (that power of which Comte takes no account) is in fixed\nand changeless relation to the principle of causation, just as _sense_\nis in fixed and changeless relation to exterior phenomena, so that we\ncan not know the external world, can not think or speak of phenomenal\nexistence, except as _effects_. In the expressive and forcible language\nof Jas. Martineau: \"By an irresistible law of thought _all phenomena\npresent themselves to us as the expression of power_, and refer us to a\ncausal ground whence they issue. This dynamic source we neither see, nor\nhear, nor feel; it is given in _thought_, supplied by the spontaneous\nactivity of mind as the correlative prefix to the phenomena\nobserved.\"[251] Unless, then, we are prepared to deny the validity of\nall our rational intuitions, we can not avoid accepting \"this subjective\npostulate as a valid law for objective nature.\" If the intuitions of our\nreason are pronounced deceptive and mendacious, so also must the\nintuitions of the senses be pronounced illusory and false. Our whole\nintellectual constitution is built up on false and erroneous principles,\nand all knowledge of whatever kind must perish by \"the contagion of\nuncertainty.\"\n\n[Footnote 251: \"Essays,\" p. 47.]\n\nComte, however, is determined to treat the idea of causation as an\nillusion, whether under its psychological form, as _will_, or under its\nscientific form, as _force_. He feels that Theology is inevitable if we\npermit the inquiry into causes;[252] and he is more anxious that\ntheology should perish than that truth should prevail. The human will\nmust, therefore, be robbed of all semblance of freedom, lest it should\nsuggest the idea of a Supreme Will governing nature; and human action,\nlike all other phenomena, must be reduced to uniform and necessary law.\nAll feelings, ideas, and principles guaranteed to us by consciousness\nare to be cast out of the account. Psychology, resting on\nself-observation, is pronounced a delusion. The immediate consciousness\nof freedom is a dream. Such a procedure, to say the least of it, is\nhighly unphilosophical; to say the truth about it, it is obviously\ndishonest. Every fact of human nature, just as much as every fact of\nphysical nature, must be accepted in all its integrity, or all must be\nalike rejected. The phenomena of mind can no more be disregarded than\nthe phenomena of matter. Rational intuitions, necessary and universal\nbeliefs, can no more be ignored than the uniform facts of\nsense-perception, without rendering a system of knowledge necessarily\nincomplete, and a system of truth utterly impossible. Every one truth is\nconnected with every other truth in the universe. And yet Comte demands\nthat a large class of facts, the most immediate and direct of all our\ncognitions, shall be rejected because they are not in harmony with the\nfundamental assumption of the positive philosophy that all knowledge is\nconfined to _phenomena perceptible to sense_. Now it were just as easy\nto cast the Alps into the Mediterranean as to obliterate from the human\nintelligence the primary cognitions of immediate consciousness, or to\nrelegate the human reason from the necessary laws of thought. Comte\nhimself can not emancipate his own mind from a belief in the validity of\nthe testimony of consciousness. How can he know himself as distinct from\nnature, as a living person, as the same being he was ten years ago, or\neven yesterday, except by an appeal to consciousness? Despite his\nearnestly-avowed opinions as to the inutility and fallaciousness of all\npsychological inquiries, he is compelled to admit that \"the phenomena of\nlife\" are \"_known by immediate consciousness_.\"[253] Now the knowledge\nof our personal freedom rests on precisely the same grounds as the\nknowledge of our personal existence. The same \"immediate consciousness\"\nwhich attests that I exist, attests also, with equal distinctness and\ndirectness, that I am self-determined and free.\n\n[Footnote 252: \"The _inevitable tendency_ of our intelligence is towards\na philosophy radically theological, so often as we seek to penetrate, on\nwhatever pretext, into the intimate nature of phenomena\" (vol. iv. p.\n664).]\n\n[Footnote 253: \"Positive Philos,\" vol. ii. p. 648.]\n\nIn common with most atheistical writers, Comte is involved in the fatal\ncontradiction of at one time assuming, and at another of denying the\nfreedom of the will, to serve the exigencies of his theory. To prove\nthat the order of the universe can not be the product of a Supreme\nIntelligence, he assumes that the products of mind must be characterized\nby freedom and variety--the phenomena of mind must not be subject to\nuniform and necessary laws; and inasmuch as the phenomena presented by\nexternal nature are subject to uniform and changeless laws, they can not\nbe the product of mind. \"Look at the whole frame of things,\" says he;\n\"how can it be the product of mind--of a supernatural Will? Is it not\nsubject to regular laws, and do we not actually obtain _prevision_ of\nits phenomena? If it were the product of mind, its order would be\nvariable and free.\" Here, then, it is admitted that _freedom is an\nessential characteristic of mind_. And this admission is no doubt a\nthoughtless, unconscious betrayal of the innate belief of all minds in\nthe freedom of the will. But when Comte comes to deal with this freedom\nas an objective question of philosophy, when he directs his attention to\nthe only will of which we have a direct and immediate knowledge, he\ndenies freedom and variety, and asserts in the most arbitrary manner\nthat the movements of the mind, like all the phenomena of nature, must\nbe subject to uniform, changeless, and necessary laws. And if we have\nnot yet been able to reduce the movements of mind, like the movements of\nthe planets, to statistics, and have not already obtained accurate\nprevision of its successions or sequences as we have of physical\nphenomena, it is simply the consequence of our inattention to, or\nignorance of, all the facts. We answer, there are no facts so directly\nand intuitively known as the facts of consciousness; and, therefore, an\nargument based upon our supposed ignorance of these facts is not likely\nto have much weight against our immediate consciousness of personal\nfreedom. There is not any thing we know so immediately, so certainly, so\npositively, as this fact--_we are free_.\n\nThe word \"force,\" representing as it does a subtile menial conception,\nand not a phenomenon of sense, must also be banished from the domains of\nPositive Science as an intruder, lest its presence should lend any\ncountenance to the idea of causation. \"Forces in mechanics are only\n_movements_, produced, or tending to be produced.\" In order to \"cancel\naltogether the old metaphysical notion of force,\" another form of\nexpression is demanded. It is claimed that all we do know or can\npossibly know is the successions of phenomena in time. What, then, is\nthe term which henceforth, in our dynamics, shall take the place of\n\"force?\" Is it \"Time-succession?\" Then let any one attempt to express\nthe various forms and intensities of movement and change presented to\nthe senses (as _e.g._, the phenomena of heat, electricity, galvanism,\nmagnetism, muscular and nervous action, etc.) in terms of\nTime-succession, and he will at once become conscious of the utter\nhopelessness of physics, without the hyperphysical idea of force, to\nrender itself intelligible.[254] What account can be rendered of\nplanetary motion if the terms \"centrifugal force\" and \"centripetal\nforce\" are abandoned? \"From the two great conditions of every Newtonian\nsolution, viz., projectile impulse and centripetal tendency, eject the\nidea of _force_, and what remains? The entire conception is simply made\nup of this, and has not the faintest existence without it. It is useless\nto give it notice to quit, and pretend that it is gone when you have\nonly put a new name upon the door. We must not call it 'attraction,'\nlest there should seem to be a _power_ within; we are to speak of it\nonly as 'gravitation,' because that is only 'weight,' which is nothing\nbut a 'fact,' as if it were not a fact that holds a power, a true\ndynamic affair, which no imagination can chop into incoherent\nsuccessions.[255] Nor is the evasion more successful when we try the\nphrase, 'tendency of bodies to mutual approach.' The approach itself may\nbe called a phenomenon; but the 'tendency' is no phenomenon, and can not\nbe attributed by us to the bodies without regarding them as the\nresidence of force. And what are we to say of the _projectile impulse_\nin the case of the planets? Is that also a phenomenon? Who witnessed and\nreported it? Is it not evident that the whole scheme of physical\nastronomy is a resolution of observed facts into dynamic equivalents,\nand that the hypothesis posits for its calculations not phenomena, but\nproper forces? Its logic is this: _If_ an impulse of certain intensity\nwere given, and _if_ such and such mutual attractions were constantly\npresent, then the sort of motions which we observe in the bodies of our\nsystem _would follow_. So, however, they also would _if_ willed by an\nOmnipotent Intelligence.\"[256] It is thus clearly evident that human\nscience is unable to offer any explanation of the existing order of the\nuniverse except in terms expressive of Power or Force; that, in fact,\nall explanations are utterly unintelligible without the idea of\ncausation. The language of universal rational intuition is, \"all\nphenomena are the expression of power;\" the language of science is,\n\"every law implies a force.\"\n\n[Footnote 254: See Grote's \"Essay on Correlation of Physical Forces,\"\npp. 18-20; and Martineau's \"Essays,\" p. 135.]\n\n[Footnote 255: \"Gravity is a real _power_ of whose agency we have daily\nexperience.\"--Herschel, \"Outlines of Astronomy,\" p. 236.]\n\n[Footnote 256: Martineau's \"Essays,\" p. 56.]\n\nIt is furthermore worthy of being noted that, in the modern doctrine of\nthe Correlation and Conservation of Forces, science is inevitably\napproaching the idea that all kinds of force are but forms or\nmanifestations of some _one_ central force issuing from some _one_\nfountain-head of power. Dr. Carpenter, perhaps the greatest living\nphysiologist, teaches that \"the form of force _which may be taken as the\ntype of all the rest_\" is the consciousness of living effort in\nvolition.[257] All force, then, is of one type, and that type is mind;\nin its last analysis external causation may be resolved into Divine\nenergy. Sir John Herschel does not hesitate to say that \"it is\nreasonable to regard the force of gravitation as the direct or indirect\nresult of a consciousness or will exerted somewhere.\"[258] The humble\nChristian may, therefore, feel himself amply justified in still\nbelieving that \"power belongs to God;\" that it is through the Divine\nenergy \"all things are, and are upheld;\" and that \"in God we live, and\nmove, and have our being;\" he is the Great First Cause, the\nFountain-head of all power.\n\n[Footnote 257: \"Human Physiology,\" p. 542.]\n\n[Footnote 258: \"Outlines of Astronomy,\" p. 234.]\n\n2. _As to Final Causes_--that is, reasons, purposes, or ends _for_ which\nthings exist--these, we are told by Comte, are all \"disproved\" by\nPositive Science, which rigidly limits us to \"the history of _what is_,\"\nand forbids all inquiry into reasons _why it is_. The question whether\nthere be any intelligent purpose in the order and arrangement of the\nuniverse, is not a subject of scientific inquiry at all; and whenever it\nhas been permitted to obtrude itself, it has thrown a false light over\nthe facts, and led the inquirer astray.\n\nThe discoveries of modern astronomy are specially instanced by Comte as\ncompletely overthrowing the notion of any conscious design or\nintelligent purpose in the universe. The order and stability of the\nsolar system are found to be the _necessary_ consequences of\ngravitation, and are adequately explained without any reference to\npurposes or ends to be fulfilled in the disposition and arrangement of\nthe heavenly bodies. \"With persons unused to the study of the celestial\nbodies, though very likely informed on other parts of natural\nphilosophy, astronomy has still the reputation of being a science\neminently religious, as if the famous words, 'The heavens declare the\nglory of God, had lost none of their truth... No science has given more\nterrible shocks to the doctrine of _final causes_ than astronomy.[259]\nThe simple knowledge of the movement of the earth must have destroyed\nthe original and real foundation of this doctrine--the idea of the\nuniverse subordinated to the earth, and consequently to man. Besides,\nthe accurate exploration of the solar system could not fail to dispel\nthat blind and unlimited admiration which the general order of nature\ninspires, by showing in the most sensible manner, and in a great number\nof different respects, that the orbs were certainly not disposed in the\nmost advantageous manner, and that science permits us easily to conceive\na better arrangement, by the development of true celestial mechanism,\nsince Newton. All the theological philosophy, even the most perfect, has\nbeen henceforth deprived of its principal intellectual function, the\nmost regular order being thus consigned as necessarily established and\nmaintained in our world, and even in the whole universe, _by the simple\nmutual gravity of its several parts_.\"[260]\n\nThe task of \"conceiving a better arrangement\" of the celestial orbs, and\nimproving the system of the universe generally, we shall leave to those\nwho imagine themselves possessed of that omniscience which comprehends\nall the facts and relations of the actual universe, and foreknows all\nthe details and relations of all possible universes so accurately as to\nbe able to pronounce upon their relative \"advantages.\" The arrogance of\nthese critics is certainly in startling and ludicrous contrast with the\naffected modesty which, on other occasions, restrains them from\n\"imputing any intentions to nature.\" It is quite enough for our purpose\nto know that the tracing of evidences of _design_ in those parts of\nnature accessible to our observation is an essentially different thing\nfrom the construction of a scheme of _optimism_ on _a priori_ grounds\nwhich shall embrace a universe the larger portion of which is virtually\nbeyond the field of observation. We are conscious of possessing some\nrational data and some mental equipment for the former task, but for the\nlatter we feel utterly incompetent.[261]\n\n[Footnote 259: In a foot-note Comte adds: \"Nowadays, to minds\nfamiliarized betimes with the true astronomical philosophy, the heavens\ndeclare no other glory than that of Hipparchus, Kepler, Newton, and all\nthose who have contributed to the ascertainment of their laws.\" It seems\nremarkable that the great men who _ascertained_ these laws did not see\nthat the saying of the Psalmist was emptied of all meaning by their\ndiscoveries. No persons seem to have been more willing than these very\nmen named to ascribe all the glory to Him who _established_ these laws.\nKepler says: \"The astronomer, to whom God has given to see more clearly\nwith his inward eye, from what he has discovered, both can and will\nglorify God;\" and Newton says: \"This beautiful system of sun, planets,\ncomets could have its origin in no other way than by the purpose and\ncommand of an intelligent and powerful Being. We admire him on account\nof his perfections, we venerate and worship him on account of his\ngovernment.\"--Whewell's \"Astronomy and Physics,\" pp. 197, 198.]\n\n[Footnote 260: \"Positive Philosophy,\" vol. ii. pp. 36-38; Tulloch,\n\"Theism,\" p. 115.]\n\n[Footnote 261: Chalmers's \"Institutes of Theology,\" vol. i. pp. 117,\n118.]\n\nThe only plausible argument in the above quotation from Comte is, that\nthe whole phenomena of the solar system are adequately explained by the\nlaw of gravitation, without the intervention of any intelligent purpose.\nLet it be borne in mind that it is a fundamental principle of the\nPositive philosophy that all human knowledge is necessarily confined to\nphenomena _perceptible to sense_, and that the fast and highest\nachievement of human science is to observe and record \"the invariable\nrelations of resemblance and succession among phenomena.\" We can not\npossibly know any thing of even the existence of \"causes\" or \"forces\"\nlying back of phenomena, nor of \"reasons\" or \"purposes\" determining the\nrelations of phenomena. The \"law of gravitation\" must, therefore, be\nsimply the statement of a fact, the expression of an observed order of\nphenomena. But the simple statement of a fact is no _explanation_ of the\nfact. The formal expression of an observed order of succession among\nphenomena is no _explanation_ of that order. For what do we mean by an\nexplanation? Is it not a \"making plain\" to the understanding? It is, in\nshort, a complete answer to the questions _how_ is it so? and _why_ is\nit so? Now, if Comte denies to himself and to us all knowledge of\nefficient and final causation, if we are in utter ignorance of \"forces\"\noperating in nature, and of \"reasons\" for which things exist in nature,\nhe can not answer either question, and consequently nothing is\nexplained.\n\nPractically, however, Comte regards gravitation as a force. The order of\nthe solar system has been established and is still maintained by the\nmutual gravity of its several parts. We shall not stop here to note the\ninconsistency of his denying to us the knowledge of, even the existence\nof, force, and yet at the same time assuming to treat gravitation as a\nforce really adequate to the explanation of the _how_ and _why_ of the\nphenomena of the universe, without any reference to a supernatural will\nor an intelligent mind. The question with which we are immediately\nconcerned is whether gravitation _alone_ is adequate to the explanation\nof the phenomena of the heavens? A review _in extenso_ of Comte's answer\nto this question would lead us into all the inextricable mazes of the\nnebular hypothesis, and involve us in a more extended discussion than\nour space permits and our limited scientific knowledge justifies. For\nthe masses of the people the whole question of cosmical development\nresolves itself into \"a balancing of authorities;\" they are not in a\nposition to verify the reasonings for and against this theory by actual\nobservation of astral phenomena, and the application of mathematical\ncalculus; they are, therefore, guided by balancing in their own minds\nthe statements of the distinguished astronomers who, by the united\nsuffrages of the scientific world, are regarded as \"authorities.\" For\nus, at present, it is enough that the nebular hypothesis is rejected by\nsome of the greatest astronomers that have lived. We need only mention\nthe names of Sir William Herschel, Sir John Herschel, Prof. Nichol, Earl\nRosse, Sir David Brewster, and Prof. Whewell.\n\nBut if we grant that the nebular hypothesis is entitled to take rank as\nan established theory of the development of the solar system, it by no\nmeans proves that the solar system was formed without the intervention\nof intelligence and design. On this point we shall content ourselves\nwith quoting the words of one whose encyclopaedian knowledge was\nconfessedly equal to that of Comte, and who in candor and accuracy was\ncertainly his superior. Prof. Whewell, in his \"Astronomy and Physics,\"\nsays: \"This hypothesis by no means proves that the solar system was\nformed without the intervention of intelligence and design. It only\ntransfers our view of the skill exercised and the means employed to\nanother part of the work; for how came the sun and its atmosphere to\nhave such materials, such motions, such a constitution, and these\nconsequences followed from their primordial condition? How came the\nparent vapor thus to be capable of coherence, separation, contraction,\nsolidification? How came the laws of its motion, attraction, repulsion,\ncondensation, to be so fixed as to lead to a beautiful and harmonious\nsystem in the end? How came it to be neither too fluid nor too\ntenacious, to contract neither too quickly nor too slowly for the\nsuccessive formation of the several planetary bodies? How came that\nsubstance, which at one time was a luminous vapor, to be at a subsequent\nperiod solids and fluids of many various kinds? What but design and\nintelligence prepared and tempered this previously-existing element, so\nthat it should, by its natural changes, produce such an orderly\nsystem\"?[262] \"_The laws of motion alone will not produce the regularity\nwhich we admire in the motion of the heavenly bodies_. There must be an\noriginal adjustment of the system on which these laws are to act; a\nselection of the arbitrary quantities which they are to involve; a\nprimitive cause which shall dispose the elements in due relation to each\nother, in order that regular recurrence may accompany constant change,\nand that perpetual motion may be combined with perpetual\nstability.\"[263]\n\n[Footnote 262: \"Astronomy and Physics,\" p. 109.]\n\n[Footnote 263: Chalmers's \"Institutes of Theology,\" vol. i. p. 119.]\n\nThe harmony of the solar system in all its phenomena does not depend\nupon the operation of any _one_ law, but from the special adjustment of\nseveral laws. There are certain agents operating throughout the entire\nsystem which have different properties, and which require special\nadjustment to each other, in order to their beneficial operation. 1st.\nThere is _Gravitation,_ prevailing apparently through all space. But it\ndoes not prevail alone. It is a force whose function is to balance other\nforces of which we know little, except that these, again, are needed to\nbalance the force of gravitation. Each force, if left to itself, would\nbe the destruction of the universe. Were it not for the force of\ngravitation, the centrifugal forces which impel the planets would fling\nthem off into space. Were it not for these centrifugal forces, the force\nof gravitation would dash them against the sun. The ultimate fact of\nastronomical science, therefore, is not the law of gravitation, but the\n_adjustment_ between this law and other laws, so as to produce and\nmaintain the existing order.[264] 2d. There is _Light_, flowing from\nnumberless luminaries; and _Heat_, radiating everywhere from the warmer\nto the colder regions; and there are a number of adjustments needed in\norder to the beneficial operation of these agents. Suppose we grant that\nby merely mechanical causes the sun became the centre of our system, how\ndid it become also the _source of its vivifying influences_? \"How was\nthe fire deposited on this hearth? How was the candle placed on this\ncandlestick?\" 3d. There is an all-pervading _Ether_, through which light\nis transmitted, which offers resistance to the movement of the planetary\nand cometary bodies, and tends to a dissipation of mechanical energy,\nand which needs to be counter-balanced by well-adjusted arrangements to\nsecure the stability of the solar system. All this balancing of opposite\nproperties and forces carries our minds upward towards Him who holds the\nbalances in his hands, and to a Supreme Intelligence on whose\nadjustments and collocations the harmony and stability of the universe\ndepends.[265]\n\n[Footnote 264: Duke of Argyll, \"Reign of Law,\" pp. 91, 92.]\n\n[Footnote 265: M'Cosh, \"Typical Forms and Special Ends,\" ch. xiii.]\n\nThe recognition of all teleology of organs in vegetable and animal\nphysiology is also persistently repudiated by this school. When Cuvier\nspeaks of the combination of organs in such order as to adapt the animal\nto the part which it has to play in nature, Geoffroy Saint Hilaire\nreplies, \"I know nothing of animals which have to play a part in\nnature.\" \"I have read, concerning fishes, that, because they live in a\nmedium which resists more than air, their motive forces are calculated\nso as to give them the power of progression under these circumstances.\nBy this mode of reasoning, you would say of a man who makes use of\ncrutches, that he was originally destined to the misfortune of having a\nleg paralyzed or amputated.[266] \"With a modesty which savors of\naffectation, he says, \"I ascribe no intentions to God, for I mistrust\nthe feeble powers of my reason. I observe facts merely, and go no\nfarther. I only pretend to the character of the historian of _what is_.\"\n\"I can not make Nature an intelligent being who does nothing in vain,\nwho acts by the shortest mode, who does all for the best.\"[267] All the\nsupposed consorting of means to ends which has hitherto been regarded as\nevidencing Intelligence is simply the result of \"the elective affinities\nof organic elements\" and \"the differentiation of organs\" consequent\nmainly upon exterior conditions. \"_Functions are a result, not an end_.\nThe animal undergoes the kind of life that his organs impose, and\nsubmits to the imperfections of his organization. The naturalist studies\nthe play of his apparatus, and if he has the right of admiring most of\nits parts, he has likewise that of showing the imperfection of other\nparts, and the practical uselessness of those which fulfill no\nfunctions.\"[268] And it is further claimed that there are a great many\nstructures which are clearly useless; that is, they fulfill no purpose\nat all. Thus there are monkeys, which have no thumbs for use, but only\nrudimental thumb-bones hid beneath the skin; the wingless bird of New\nZealand (Apteryx) has wing-bones similarly developed, which serve no\npurpose; young whalebone whales are born with teeth that never cut the\ngums, and are afterwards absorbed; and some sheep have horns turned\nabout their ears which fulfill no end. And inasmuch as there are some\norganisms in nature which serve no purpose of utility, it is argued\nthere is no design in nature; things are _used_ because there are\nantecedent conditions favorable for _use_, but that use is not the _end_\nfor which the organ exists. The true naturalist will never say, \"Birds\nhave wings given them _in order_ to fly;\" he will rather say, \"Birds fly\n_because_ they have wings.\" The doctrine of final causes must,\ntherefore, be abandoned.\n\n[Footnote 266: Whewell, \"History of Inductive Sciences,\" vol. ii. p.\n486.]\n\n[Footnote 267: Id., ib., vol. ii. p. 490.]\n\n[Footnote 268: Martin's \"Organic Unity in Animals and Vegetables,\" in M.\nQ. Review, January, 1863.]\n\nIt is hardly worth while to reply to the lame argument of Geoffroy,\nwhich needs a \"crutch\" for its support. The very illustration,\nundignified and irrelevant as it is, tells altogether against its\nauthor. For, first, the crutch is certainly a _contrivance_ designed for\nlocomotion; secondly, the length and strength and lightness of the\ncrutch are all matters of calculation and _adjustment_; and, thirdly,\nall the adaptations of the crutch are well-considered, in order to\nenable the lame man to walk; the function of the crutch is the final\ncause of its creation. This crutch is clearly out of place in Geoffroy's\nargument, and utterly breaks down. It is in its place in the\nteleological argument, and stands well, though it may not behave as well\nas the living limb. The understanding of a child can perceive that the\ndesign-argument does not assert that men were intended to have amputated\nlimbs, but that crutches are designed for those whose limbs are\nparalyzed or amputated.\n\nThe existence of useless members, of rudimentary and abortive limbs,\ndoes seem, at first sight, to be unfavorable to the idea of supremacy of\npurpose and all-pervading design. It should be remarked, however, that\nthis is an argument based upon our ignorance, and not upon our\nknowledge. It does not by any means follow that because we have\ndiscovered no reasons for their existence, therefore there are no\nreasons. Science, in enlarging its conquests of nature, is perpetually\ndiscovering the usefulness of arrangements of which our fathers were\nignorant, and the reasons of things which to their minds, were\nconcealed; and it ill becomes the men who so far \"mistrust their own\nfeeble powers\" as to be afraid of ascribing any intention to God or\nnature, to dogmatically affirm there is no purpose in the existence of\nany thing. And then we may ask, what right have these men to set up the\nidea of \"utility\" as the only standard to which the Creator must\nconform? How came they to know that God is a mere \"utilitarian;\" or, if\nthey do not believe in God, that nature is a miserable \"Benthamite?\" Why\nmay not the idea of beauty, of symmetry, of order, be a standard for the\nuniverse, as much as the idea of utility, or mere subordination to some\npractical end? May not conformity to one grand and comprehensive plan,\nsweeping over all nature, be perfectly compatible with the adaptation of\nindividual existences to the fulfillment of special ends? In civil\narchitecture we have conformity to a general plan; we have embellishment\nand ornament, and we have adaptation to a special purpose, all combined;\nwhy may not these all be combined in the architecture of the universe?\nThe presence of any one of these is sufficient to prove design, for mere\nornament or beauty is itself a purpose, an object, and an end. The\nconcurrence of all these is an overwhelming evidence of design. Wherever\nfound, they are universally recognized as the product of intelligence;\nthey address themselves at once to the intelligence of man, and they\nplace him in immediate relation to and in deepest sympathy with the\nIntelligence which gave them birth. He that formed the eye of man to\nsee, and the heart of man to admire beauty, shall He not delight in it?\nHe that gave the hand of man its cunning to create beauty, shall He not\nhimself work for it? And if man can and does combine both \"ornament\" and\n\"use\" in one and the same implement or machine, why should not the\nCreator of the world do the same? \"When the savage carves the handle of\nhis war-club, the immediate purpose of his carving is to give his own\nhand a firmer hold. But any shapeless scratches would be enough for\nthis. When he carves it in an elaborate pattern, he does so for the love\nof ornament, and to satisfy the sense of beauty.\" And so \"the harmonies,\non which all beauty depends, are so connected in nature that _use_ and\n_ornament_ may often both arise out of the same conditions.\"[269]\n\n[Footnote 269: Duke of Argyll, \"Reign of Law,\" p. 203.]\n\nThe \"true naturalist,\" therefore, recognizes two great principles\npervading the universe--_a principle of order_--a unity of plan, and _a\nprinciple of special adaptation_, by which each object, though\nconstructed upon a general plan, is at the same time accommodated to the\nplace it has to occupy and the purpose it has to serve. In other words,\nthere is _homology of structure_ and _analogy of function_, conformity\nto _archetypal forms_ and _Teleology of organs_, in wonderful\ncombination. Now, in the Materialistic school, it has been the prevalent\npractice to set up the unity of plan in animal structures, in opposition\nto the principle of Final Causes: Morphology has been opposed to\nTeleology. But in nature there is no such opposition; on the contrary,\nthere is a beautiful co-ordination. The same bones, in different\nanimals, are made subservient to the widest possible diversity of\nfunctions. The same limbs are converted into fins, paddles, wings, legs,\nand arms. \"No comparative anatomist has the slightest hesitation in\nadmitting that the pectoral fin of a fish, the wing of a bird, the\npaddle of the dolphin, the fore-leg of a deer, the wing of a bat, and\nthe arm of a man, are the same organs, notwithstanding that their forms\nare so varied, and the uses to which they are applied so unlike each\nother.\"[270] All these are homologous in structure--they are formed\nafter an ideal archetype or model, but that model or type is variously\nmodified to adapt the animal to the sphere of life in which it is\ndestined to move, and the organ itself to the functions it has to\nperform, whether swimming, flying, walking, or burrowing, or that varied\nmanipulation of which the human hand is capable. These varied\nmodifications of the vertebrated type, for special purposes, are\nunmistakable examples of final causation. Whilst the silent members, the\nrudimental limbs instanced by Oken, Martins, and others--as fulfilling\nno purpose, and serving no end, exist in conformity to an ideal\narchetype on which the bony skeletons of all vertebrated animals are\nformed,[271] and which has never been departed from since time began.\nThis type, or model, or plan, is, however, itself an evidence of\n_design_ as much as the plan of a house. For to what standard are we\nreferring when we say that two limbs are morphologically the same? Is it\nnot an _ideal_ plan, a _mental_ pattern, a metaphysical conception? Now\nan _ideal_ implies a mind which preconceived the idea, and in which\nalone it really exists. It is only as \"an _order of Divine thought_\"\nthat the doctrine of animal homologies is at all intelligible; and\nHomology is, therefore, the science which traces the outward embodiment\nof a Divine Idea.[272] The principle of intentionality or final\ncausation, then, is not in any sense invalidated by the discovery of \"a\nunity of plan\" sweeping through the entire universe.\n\n[Footnote 270: Carpenter's \"Comparative Physiology,\" p. 37.]\n\n[Footnote 271: Agassiz, \"Essay on Classification,\" p. 10.]\n\n[Footnote 272: Whewell's \"History of Inductive Sciences,\" vol. i. p.\n644; \"The Reign of Law,\" p. 208; Agassiz, \"Essay on Classification,\" pp.\n9-11.]\n\nWe conclude that we are justly entitled to regard \"the principle of\nintentionality\" as a primary and necessary law of thought, under which\nwe can not avoid conceiving and describing the facts of the\nuniverse--_the special adaptation of means to ends necessarily implies\nmind_. Whenever and wherever we observe the adaptation of an organism to\nthe fulfillment of a special end, we can not avoid conceiving of that\n_end_ as foreseen and premeditated, the _means_ as selected and adjusted\nwith a view to that end, and creative energy put forth to secure the\nend--all which is the work of intelligence and will.[273] And we can not\ndescribe these facts of nature, so as to render that account\nintelligible to other minds, without using such terms as \"contrivance,\"\n\"purpose,\" \"adaptation,\" \"design.\" A striking illustration of this may\nbe found in Darwin's volume \"On the Fertilization of Orchids.\" We select\nfrom his volume with all the more pleasure because he is one of the\nwriters who enjoins \"caution in ascribing intentions to nature.\" In one\nsentence he says: \"The _Labellum_ is developed into a long nectary, _in\norder_ to attract _Lepidoptera_; and we shall presently give reasons for\nsuspecting the nectar is _purposely_ so lodged that it can be sucked\nonly slowly, _in order_ to give time for the curious chemical quality of\nthe viscid matter settling hard and dry\" (p. 29). Of one particular\nstructure he says: \"This _contrivance_ of the guiding ridges may be\ncompared to the little instrument sometimes used for guiding a thread\ninto the eye of a needle.\" The notion that every organism has a use or\npurpose seems to have guided him in his discoveries. \"The strange\nposition of the _Labellum_, perched on the summit of the column, ought\nto have shown me that here was the place for experiment. I ought to have\nscorned the notion that the _Labellum_ was thus placed _for no good\npurpose_. I neglected this plain guide, and for a long time completely\nfailed to understand the flower\" (p. 262).[274]\n\n[Footnote 273: Carpenter's \"Principles of Comparative Physiology,\" p.\n723.]\n\n[Footnote 274: Edinburgh Review, October, 1862; article, \"The\nSupernatural.\"]\n\nSo that the assumption of final causes has not, as Bacon affirms, \"led\nmen astray\" and \"prejudiced further discovery;\" on the contrary, it has\nhad a large share in every discovery in anatomy and physiology, zoology\nand botany. The use of every organ has been discovered by starting from\nthe assumption _that it must have some use_. The belief in a creative\npurpose led Harvey to discover the circulation of the blood. He says:\n\"When I took notice that the valves in the veins of so many parts of the\nbody were so placed that they gave a free passage to the blood towards\nthe heart, but opposed the passage of the venal blood the contrary way,\nI was incited to imagine that so provident a cause as Nature has not\nplaced so many valves _without design_, and no design seemed more\nprobable than the circulation of the blood.\"[275] The wonderful\ndiscoveries in Zoology which have immortalized the name of Cuvier were\nmade under the guidance of this principle. He proceeds on the\nsupposition not only that animal forms have _some_ plan, _some_ purpose,\nbut that they have an intelligible plan, a discoverable purpose. At the\noutset of his \"_Regne Animal_\" he says: \"Zoology has a principle of\nreasoning which is peculiar to it, and which it employs to advantage on\nmany occasions; that is, the principle of the conditions of existence,\ncommonly called final causes.\"[276] The application of this principle\nenabled him to understand and arrange the structures of animals with\nastonishing clearness and completeness of order; and to restore the\nforms of extinct animals which are found in the rocks, in a manner which\nexcited universal admiration, and has commanded universal assent.\nIndeed, as Professor Whewell remarks, at the conclusion of his \"History\nof the Inductive Sciences,\" \"those who have been discoverers in science\nhave generally had minds, the disposition of which was to believe in an\n_intelligent Maker_ of the universe, and that the scientific\nspeculations which produced an opposite tendency were generally those\nwhich, though they might deal familiarly with known physical truths, and\nconjecture boldly with regard to unknown, do not add to the number of\nsolid generalizations.\"[277]\n\n[Footnote 275: \"History of Inductive Science,\" vol. ii. p. 449.]\n\n[Footnote 276: \"History of Inductive Science,\" vol. ii. p. 2, Eng. ed.]\n\n[Footnote 277: Ibid., vol. ii. p. 491. A list of the \"great discoverers\"\nis given in his \"Astronomy and Physics,\" bk. iii. ch. v.]\n\n\n\n\nCHAPTER VII.\n\nTHE UNKNOWN GOD (_continued_).\n\nIS GOD COGNIZABLE BY REASON? (_continued_).\n\n\n      \"The faith which can not stand unless buttressed by\n      contradictions is built upon the sand. The profoundest faith\n      is faith in the unity of truth. If there is found any\n      conflict in the results of a right reason, no appeal to\n      practical interests, or traditionary authority, or\n      intuitional or theological faith, can stay the flood of\n      skepticism.\"--ABBOT.\n\nIn the previous chapter we have considered the answers to this question\nwhich are given by the Idealistic and Materialistic schools; it devolves\nupon us now to review (iii.) the position of the school of _Natural\nRealism_ or _Natural Dualism_, at the head of which stands Sir William\nHamilton.\n\nIt is admitted by this school that philosophic knowledge is \"the\nknowledge of effects as dependent on their causes,\"[278] and \"of\nqualities as inherent in substances.\"[279]\n\n[Footnote 278: \"Lectures on Metaphysics,\" vol. i. p. 58.]\n\n[Footnote 279: Ibid., vol. i. p. 138.]\n\n1. _As to Events and Causes_.--\"Events do not occur isolated, apart, by\nthemselves; they occur and are conceived by us only in connection. Our\nobservation affords us no example of a phenomenon which is not an\neffect; nay, our thought can not even realize to itself the possibility\nof a phenomenon without a cause. By the necessity we are under of\nthinking some cause for every phenomenon, and by our original ignorance\nof what particular causes belong to what particular effects, it is\nrendered impossible for us to acquiesce in the mere knowledge of the\nfact of the phenomenon; on the contrary, we are determined, we are\nnecessitated to regard each phenomenon as _only partially known until we\ndiscover the causes_ on which it depends for its existence.[280]\nPhilosophic knowledge is thus, in its widest acceptation, the knowledge\nof effects as dependent on causes. Now what does this imply? In the\nfirst place, as every cause to which we can ascend is only an effect, it\nfollows that it is the scope, that is, the aim, of philosophy to trace\nup the series of effects and causes until we arrive at _causes which are\nnot in themselves effects_,\"[281]--that is, to ultimate and final\ncauses. And then, finally, \"Philosophy, as the knowledge of effects in\ntheir causes, necessarily tends, not towards a plurality of ultimate or\nfinal causes, but towards _one_ alone.\"[282]\n\n[Footnote 280: Ibid., vol. i. p. 56.]\n\n[Footnote 281: \"Lectures on Metaphysics,\" vol. i. p. 58.]\n\n[Footnote 282: Ibid., vol. i. p. 60.]\n\n2. _As to Qualities and Substance, or Phenomena and Reality_.--As\nphenomena appear only in conjunction, we are compelled, by the\nconstitution of our nature, to think them conjoined in and by something;\nand as they are phenomena, we can not think them phenomena of nothing,\nbut must regard them as properties or qualities of something.[283] Now\nthat which manifests its qualities--in other words, that in which the\nappearing causes inhere, that to which they belong--is called their\n_subject_, or _substance_, or _substratum_.[284] The subject of one\ngrand series of phenomena (as, _e.g._, extension, solidity, figure,\netc.) is called _matter_, or _material substance_. The subject of the\nother grand series of phenomena (as, _e.g._, thought, feeling, volition,\netc.) is termed _mind_, or _mental substance_. We may, therefore, lay it\ndown as an undisputed truth that consciousness gives, as an ultimate\nfact, a primitive duality--a knowledge of the _ego_ in relation and\ncontrast to the _non-ego_, and a knowledge of the _non-ego_ in relation\nand contrast to the _ego_[285] Natural Dualism thus establishes the\nexistence of two worlds of _mind_ and _matter_ on the immediate\nknowledge we possess of both series of phenomena; whilst the Cosmothetic\nIdealists discredit the veracity of consciousness as to our immediate\nknowledge of material phenomena, and, consequently, our _immediate\nknowledge of the existence of matter_.[286]\n\n[Footnote 283: Ibid., vol. i. p. 137.]\n\n[Footnote 284: Ibid., vol. i. p. 137.]\n\n[Footnote 285: Ibid., vol. i. p. 292.]\n\n[Footnote 286: Ibid., vol. i. pp. 292, 295.]\n\nThe obvious doctrine of the above quotations is, that we have an\nimmediate knowledge of the \"_existence_ of matter\" as well as of \"the\n_phenomena_ of matter;\" that is, we know \"_substance_\" as immediately\nand directly as we know \"_qualities_.\" Phenomena are known only as\ninherent in substance; substance is known only as manifesting its\nqualities. We never know qualities without knowing substance, and we can\nnever know substance without knowing qualities. Both are known in one\nconcrete act; substance is known quite as much as quality; quality is\nknown no more than substance. That we have a direct, immediate,\npresentative \"face to face\" knowledge of matter and mind in every act of\nconsciousness is asserted again and again by Hamilton, in his\n\"Philosophy of Perception.\"[287] In the course of the discussion he\nstarts the question, \"_Is the knowledge of mind and matter equally\nimmediate?_\" His answer to this question may be condensed in the\nfollowing sentences. In regard to the immediate knowledge of _mind_\nthere is no difficulty; it is admitted to be direct and immediate. The\nproblem, therefore, exclusively regards the intuitive perception of the\nqualities of _matter_. Now, says Hamilton, \"if we interrogate\nconsciousness concerning the point in question, the response is\ncategorical and clear. In the simplest act of perception I am conscious\nof _myself_ as a perceiving _subject_, and of an external _reality_ as\nthe object perceived; and I am conscious of both existences in the same\nindivisible amount of intuition.\"[288] Again he says, \"I have frequently\nasserted that in perception we are conscious of the external object,\nimmediately and _in itself_.\" \"If, then, the veracity of consciousness\nbe unconditionally admitted--_if the intuitive knowledge of matter and\nmind_, and the consequent reality of their antithesis, be taken as\ntruths,\" the doctrine of Natural Realism is established, and, \"without\nany hypothesis or demonstration, the _reality of mind_ and the _reality\nof matter_.\"[289]\n\n[Footnote 287: Philosophy of Sir William Hamilton, part ii.]\n\n[Footnote 288: Ibid., p. 181.]\n\n[Footnote 289: Ibid., pp. 34, 182.]\n\nNow, after these explicit statements that we have an intuitive knowledge\nof matter and mind--a direct and immediate consciousness of self as a\nreal, \"self-subsisting entity,\" and a knowledge of \"an external reality,\nimmediately and _in itself_,\" it seems unaccountably strange that\nHamilton should assert \"_that all human knowledge, consequently all\nhuman philosophy, is only of the Relative or Phenomenal_;\"[290] and that\n\"_of existence absolutely and in itself we know nothing_.\"[291] Whilst\nteaching that the proper sphere and aim of philosophy is to trace\nsecondary causes up to ultimate or first causes, and that it\n_necessarily tends_ towards one First and Ultimate Cause, he at the same\ntime asserts that \"first causes do not lie within the reach of\nphilosophy,\"[292] and that it can never attain to the knowledge of the\nFirst Cause.[293] \"The Infinite God can not, by us, be comprehended,\nconceived, or thought.\"[294] God, as First Cause, as infinite, as\nunconditioned, as eternal, is to us absolutely \"_The Unknown_.\" The\nscience of Real Being--of Being _in se_--of self-subsisting entities, is\ndeclared to be impossible. All science is only of the phenomenal, the\nconditioned, the relative. Ontology is a delusive dream. Thus, after\npages of explanations and qualifications, of affirmations and denials,\nwe find Hamilton virtually assuming the same position as Comte and\nMill--_all human knowledge is necessarily confined to phenomena_.\n\n[Footnote 290: \"Lectures on Metaphysics,\" vol. i. p. 136]\n\n[Footnote 291: Ibid., vol. i. p. 138.]\n\n[Footnote 292: Ibid., vol. i. p. 58.]\n\n[Footnote 293: Ibid., vol. i. p. 60.]\n\n[Footnote 294: Ibid., vol. ii. p. 375.]\n\nIt has been supposed that the chief glory of Sir William Hamilton rested\nupon his able exposition and defense of the doctrine of Natural Realism.\nThere are, however, indications in his writings that he regarded \"the\nPhilosophy of the Conditioned\" as his grand achievement. The Law of the\nConditioned had \"not been generalized by any previous philosopher;\" and,\nin laying down that law, he felt that he had made a new and important\ncontribution to speculative thought.\n\nThe principles upon which this philosophy is based are:\n\n1. _The Relativity of all Human Knowledge._--Existence is not cognized\nabsolutely and in itself, but only under special modes which are related\nto our faculties, and, in fact, determined by these faculties\nthemselves. All knowledge, therefore, is _relative_--that is, it is of\nphenomena only, and of phenomena \"under modifications determined by our\nown faculties.\" Now, as the Absolute is that which exists out of all\nrelation either to phenomena or to our faculties of knowledge, it can\nnot possibly be _known_.\n\n2. _The Conditionality of all Thinking_.--Thought necessarily supposes\nconditions. \"To think is to condition; and conditional limitation is the\nfundamental law of the possibility of thought. As the eagle can not\nout-soar the atmosphere in which he floats, and by which alone he is\nsupported, so the mind can not transcend the sphere of limitation within\nand through which the possibility of thought is realized. Thought is\nonly of the conditioned, because, as we have said, to think is to\ncondition.\"[295] Now the Infinite is the unlimited, the unconditioned,\nand as such can not possibly be _thought_.\n\n3. _The notion of the Infinite--the Absolute, as entertained by man, is\na mere \"negation of thought._\"--By this Hamilton does not mean that the\nidea of the Infinite is a negative idea. \"The Infinite and the Absolute\nare _only_ the names of two counter _imbecilities_ of the human\nmind\"[296]--that is, a mental inability to conceive an absolute\nlimitation, or an infinite illimitation; an absolute commencement, or an\ninfinite non-commencement. In other words, of the absolute and infinite\nwe have no conception at all, and, consequently, no knowledge.[297]\n\nThe grand law which Hamilton generalizes from the above is, \"_that the\nconceivable is in every relation bounded by the inconceivable_.\" Or,\nagain, \"The conditioned or the thinkable lies between two extremes or\npoles; and these extremes or poles are each of them unconditioned, each\nof them inconceivable, each of them exclusive or contradictory of the\nother.\"[298] This is the celebrated \"Law of the Conditioned.\"\n\n[Footnote 295: \"Discussions,\" p. 21.]\n\n[Footnote 296: Ibid., p. 28.]\n\n[Footnote 297: \"Lectures on Metaphysics,\" vol. ii. pp. 368, 373.]\n\n[Footnote 298: Ibid., vol. ii. p. 373.]\n\nIn attempting a brief criticism of \"the Philosophy of the Conditioned,\"\nwe may commence by inquiring:\n\nI. _What is the real import and significance of the doctrine \"that all\nhuman knowledge is only of the relative or phenomenal_?\"\n\nHamilton calls this \"the great axiom\" of philosophy. That we may\ndistinctly comprehend its meaning, and understand its bearing on the\nsubject under discussion, we must ascertain the sense in which he uses\nthe words \"_phenomenal_\" and \"_relative._\" The importance of an exact\nterminology is fully appreciated by our author; and accordingly, in\nthree Lectures (VIII., IX., X.), he has given a full explication of the\nterms most commonly employed in philosophic discussions. Here the word\n\"_phenomenon_\" is set down as the necessary \"_correlative_\" of the word\n\"_subject_\" or \"_substance_.\" \"These terms can not be explained apart,\nfor each is correlative of the other, each can be comprehended only in\nand through its correlative. The term '_subject_' is used to denote the\nunknown (?) basis which lies under the various _phenomena_ or properties\nof which we become aware, whether in our external or internal\nexperience.\"[299] \"The term '_relative_' is _opposed_ to the term\n'_absolute_;' therefore, in saying that we know only the relative, I\nvirtually assert that we know nothing absolutely, that is, _in and for\nitself, and without relation to us and our faculties_.\"[300] Now, in the\nphilosophy of Sir William Hamilton, \"the absolute\" is defined as \"that\nwhich is aloof from relation\"--\"that which is out of all relation.\"[301]\nThe _absolute_ can not, therefore, be \"_the correlative_\" of the\nconditioned--can not stand in any relation to the phenomenal. The\n_subject,_ however, is the necessary correlative of the phenomenal, and,\nconsequently, the subject and the absolute are not identical.\nFurthermore, Hamilton tells us the subject _may be comprehended_ in and\nthrough its correlative--the phenomenon; but the absolute, being aloof\nfrom all relation, can not be comprehended or conceived at all. \"The\nsubject\" and \"the absolute\" are, therefore, not synonymous terms; and,\nif they are not synonymous, then their antithetical terms, \"phenomenal\"\nand \"relative,\" can not be synonymous.\n\n[Footnote 299: \"Lectures on Metaphysics,\" vol. i. p. 148.]\n\n[Footnote 300: Ibid., vol. i. p. 137.]\n\n[Footnote 301: \"Discussions,\" p. 21.]\n\nIt is manifest, however, that Hamilton does employ these terms as\nsynonymous, and this we apprehend is the first false step in his\nphilosophy of the conditioned. \"All our knowledge is of the relative\n_or_ phenomenal.\" Throughout the whole of Lectures VIII. and IX., in\nwhich he explains the doctrine of the relativity of human knowledge,\nthese terms are used as precisely analogous. Now, in opposition to this,\nwe maintain that the relative is not always the phenomenal. A thing may\nbe \"in relation\" and yet not be a phenomenon. \"The subject or substance\"\nmay be, and really is, on the admission of Hamilton himself,\n_correlated_ to the phenomenon. The ego, \"the conscious _subject_\"[302]\nas a \"_self-subsisting entity_\" is necessarily related to the phenomena\nof thought, feeling, etc.; but no one would repudiate the idea that the\nconscious subject is a mere phenomenon, or \"series of phenomena,\" with\nmore indignation than Hamilton. Notwithstanding the contradictory\nassertion, \"that the _subject_ is unknown,\" he still teaches, with equal\npositiveness, \"that in every act of perception I am conscious of self,\nas a perceiving _subject.\"_ And still more explicitly he says: \"As\nclearly as I am conscious of existing, so clearly am I conscious, at\nevery moment of my existence, that the conscious Ego is not itself a\nmere modification [a phenomenon], nor a series of modifications\n[phenomena], but that it is itself different from all its modifications,\nand a _self-subsisting entity_.\"[303] Again: \"Thought is possible only\nin and through the consciousness of Self. The Self, the I, is recognized\nin every act of intelligence as the _subject_ to which the act belongs.\nIt is I that perceive, I that imagine, I that remember, etc.; these\nspecial modes are all only the phenomena of the I.\"[304] We are,\ntherefore, conscious of the _subject_ in the most immediate, and direct,\nand intuitive manner, and the subject of which we are conscious can not\nbe \"_unknown_.\" We regret that so distinguished a philosophy should deal\nin such palpable contradictions; but it is the inevitable consequence of\nviolating that fundamental principle of philosophy on which Hamilton so\nfrequently and earnestly insists, viz., \"that the testimony of\nconsciousness must be accepted in all its integrity\".\n\n[Footnote 302: Philosophy of Sir William Hamilton (edited by O.W.\nWight), p. 181.]\n\n[Footnote 303: \"Lectures on Metaphysics,\" vol. i. p. 373.]\n\n[Footnote 304: Ibid., vol. i. p. 166.]\n\nIt is thus obvious that, with proper qualifications, we may admit _the\nrelativity of human knowledge_, and yet at the same time reject the\ndoctrine of Hamilton, _that all human knowledge is only of the\nphenomenal_.\n\n\"The relativity of human knowledge,\" like most other phrases into which\nthe word \"relative\" enters, is vague, and admits of a variety of\nmeanings. If by this phrase is meant \"that we can not know objects\nexcept as related to our faculties, or as our faculties are related to\nthem,\" we accept the statement, but regard it as a mere truism leading\nto no consequences, and hardly worth stating in words. It is simply\nanother way of saying that, in order to an object's being known, it must\ncome within the range of our intellectual vision, and that we can only\nknow as much as we are capable of knowing. Or, if by this phrase is\nmeant \"that we can only know things by and through the phenomena they\npresent,\" we admit this also, for we can no more know substances apart\nfrom their properties, than we can know qualities apart from the\nsubstances in which they inhere. Substances can be known only in and\nthrough their phenomena. Take away the properties, and the thing has no\nlonger any existence. Eliminate extension, form, density, etc., from\nmatter, and what have you left? \"The thing in itself,\" apart from its\nqualities, is nothing. Or, again, if by the relativity of knowledge is\nmeant \"that all consciousness, all thought are relative,\" we accept this\nstatement also. To conceive, to reflect, to know, is to deal with\ndifference and relation; the relation of subject and object; the\nrelation of objects among themselves; the relation of phenomena to\nreality, of becoming to being. The reason of man is unquestionably\ncorrelated to that which is beyond phenomena; it is able to apprehend\nthe necessary relation between phenomena and being, extension and space,\nsuccession and time, event and cause, the finite and the infinite. We\nmay thus admit the _relative character of human thought_, and at the\nsame time deny that it is an ontological disqualification.[305]\n\nIt is not, however, in any of these precise forms that Hamilton holds\nthe doctrine of the relativity of knowledge. He assumes a middle place\nbetween Reid and Kant, and endeavors to blend the subjective idealism of\nthe latter with the realism of the former. \"He identifies the\n_phenomenon_ of the German with the _quality_ of the British\nphilosophy,\"[306] and asserts, as a regulative law of thought, that the\nquality implies the substance, and the phenomenon the noumenon, but\nmakes the substratum or noumenon (the object in itself) unknown and\nunknowable. The \"phenomenon\" of Kant was, however, something essentially\ndifferent from the \"quality\" of Reid. In the philosophy of Kant,\n_phenomenon_ means an object as we envisage or represent it to\nourselves, in opposition to the _noumenon_, or a thing as it is in\nitself. The phenomenon is composed, in part, of subjective elements\nsupplied by the mind itself; as regards intuition, the forms of space\nand time; as regards thought, the categories of Quantity, Quality,\nRelation, and Modality. To perceive a thing in itself would be to\nperceive it neither in space nor in time. To think a thing in itself\nwould be not to think it under any of the categories. The phenomenal is\nthus the product of the inherent laws of our own constitution, and, as\nsuch, is the sum and limit of all our knowledge.[307]\n\n[Footnote 305: Martineau's \"Essays,\" p. 234.]\n\n[Footnote 306: M'Cosh's \"Defense of Fundamental Truth,\" p. 106.]\n\n[Footnote 307: Mansel's \"Lectures on the Philosophy of Kant,\" pp. 21,\n22.]\n\nThis, in its main features, is evidently the doctrine propounded by\nHamilton. The special modes in which existence is cognizable\" are\npresented to, and known by, the mind _under modifications determined by\nthe faculties themselves_.\"[308] This doctrine he illustrates by the\nfollowing supposition: \"Suppose the total object of consciousness in\nperception is=12; and suppose that the external reality contributes 6,\nthe material sense 3, and the mind 3; this may enable you to form some\nrude conjecture of the nature of the object of perception.\"[309] The\nconclusion at which Hamilton arrives, therefore, is that things are not\nknown to us as they exist, but simply as they appear, and as our minds\nare capable of perceiving them.\n\n[Footnote 308: Hamilton's \"Lectures on Metaphysics,\" vol. i. p. 148.]\n\n[Footnote 309: Hamilton's \"Lectures on Metaphysics,\" vol. ii. p. 129;\nand also vol. i. p. 147.]\n\nLet us test the validity of this majestic deliverance. No man is\njustified in making this assertion unless, 1. He knows things as they\nexist; 2. He knows things not only as they exist but as they appear; 3.\nHe is able to compare things as they exist with the same things as they\nappear. Now, inasmuch as Sir William Hamilton affirms we do not know\nthings as they exist, but only as they appear, how can he know that\nthere is any difference between things as they exist and as they appear?\nWhat is this \"_thing in itself_\" about which Hamilton has so much to\nsay, and yet about which he professes to know nothing? We readily\nunderstand what is meant by the _thing_; it is the object as existing--a\nsubstance manifesting certain characteristic qualities. But what is\nmeant by _in itself_? There can be no _in itself_ besides or beyond the\n_thing_. If Hamilton means that \"the thing itself\" is the thing apart\nfrom all relation, and devoid of all properties or qualities, we do not\nacknowledge any such thing. A thing apart from all relation, and devoid\nof all qualities, is simply pure nothing, if such a solecism may be\npermitted. With such a definition of Being _in se_, the logic of Hegel\nis invincible, \"Being and Nothing are identical.\"\n\nAnd now, if \"the thing in itself\" be, as Hamilton says it is, absolutely\n_unknown_, how can he affirm or deny any thing in regard to it? By what\nright does he prejudge a hidden reality, and give or refuse its\npredicates; as, for example, that it is conditioned or unconditioned, in\nrelation or aloof from relation, finite or infinite? Is it not plain\nthat, in declaring a thing in its inmost nature or essence to be\ninscrutable, it is assumed to be partially _known_? And it is obvious,\nnotwithstanding some unguarded expressions to the contrary, that\nHamilton does regard \"the thing in itself\" as partially known. \"The\nexternal reality\" is, at least, six elements out of twelve in the \"total\nobject of consciousness.\"[310] The primary qualities of matter are known\nas in the things themselves; \"they develop themselves with rigid\nnecessity out of the simple datum of _substance occupying space_.\"[311]\n\"The Primary Qualities are apprehended as they are in bodies\"--\"they are\nthe attributes of _body as body_,\" and as such \"are known immediately in\nthemselves,\"[312] as well as mediately by their effects upon us. So that\nwe not only know by direct consciousness certain properties of things as\nthey exist in things themselves, but we can also deduce them in an _a\npriori_ manner. \"The bare notion of matter being given, the Primary\nQualities may be deduced _a priori_; they being, in fact, only\nevolutions of the conditions which that notion necessarily implies.\" If,\nthen, we know the qualities of things as \"in the things themselves,\"\n\"the things themselves\" must also be, at least, partially known; and\nHamilton can not consistently assert the relativity of _all_ knowledge.\nEven if it be granted that our cognitions of objects are only in part\ndependent on the objects themselves, and in part on elements superadded\nby our organism, or by our minds, it can not warrant the assertion that\nall our knowledge, but only the part so added, is relative. \"The\nadmixture of the relative element not only does not take away the\nabsolute character of the remainder, but does not even (if our author is\nright) prevent us from recognizing it. The confusion, according to him,\nis not inextricable. It is for us 'to analyze and distinguish what\nelements,' in an 'act of knowledge,' are contributed by the object, and\nwhat by the organs or by the mind.\"[313]\n\n[Footnote 310: \"Lectures on Metaphysics,\" vol. ii. p. 129.]\n\n[Footnote 311: Philosophy of Sir Wm. Hamilton, p. 357]\n\n[Footnote 312: Ibid., pp. 377, 378.]\n\n[Footnote 313: Mill's \"Examination of Hamilton's Philosophy,\" vol. i. p.\n44.]\n\nAdmitting the relative character of human thought as a psychological\nfact, Mr. Martineau has conclusively shown that this law, instead of\nvisiting us with disability to transcend phenomena, _operates as a\nrevelation of what exists beyond_. \"The finite body cut out before our\nvisual perception, or embraced by the hands, lies as an island in the\nemptiness around, and without comparative reference to this can not be\nrepresented: the same experience which gives us the definite object\ngives us also the infinite space; and both terms--the limited appearance\nand the unlimited ground--are apprehended with equal certitude and\nclearness, and furnished with names equally susceptible of distinct use\nin predication and reasoning. The transient successions, for instance,\nthe strokes of a clock, which we count, present themselves to us as\ndotted out upon a line of permanent duration; of which, without them, we\nshould have no apprehension, but which as their condition, is\nunreservedly known.\"[314]\n\n\"What we have said with regard to space and time applies equally to the\ncase of Causation. Here, too, the finite offered to perception\nintroduces to an Infinite supplied by thought. As a definite body\nreveals also the space around, and an interrupted succession exhibits\nthe uniform time beneath, so does the passing phenomenon demand for\nitself a power beneath. The space, and time, and power, not being part\nof the thing perceived, but its conditions, are guaranteed to us,\ntherefore, on the warrant, not of sense, but of intellect.\"[315]\n\n\"We conclude, then, on reviewing these examples of Space, and Time, and\nCausation, that ontological ideas introducing us to certain fixed\nentities belong no less to our knowledge than scientific ideas of\nphenomenal disposition and succession.\"[316] In these instances of\nrelation between a phenomenon given in perception and an entity as a\nlogical condition, the correlatives are on a perfect equality of\nintellectual validity, and the relative character of human thought is\nnot an ontological disqualification, but a cognitive power.\n\n[Footnote 314: \"Essays,\" pp. 193,194.]\n\n[Footnote 315: Ibid., p. 197.]\n\n[Footnote 316: Ibid., p. 195.]\n\nThere is a thread of fallacy running through the whole of Hamilton's\nreasonings, consequent upon a false definition of the Absolute at the\noutset. The Absolute is defined as _that which exists in and by itself,\naloof from and out of all relation_. An absolute, as thus defined, does\nnot and can not exist; it is a pure abstraction, and, in fact, a pure\nnon-entity. \"The Absolute expresses perfect independence both in being\nand in action, and is applicable to God as self-existent.\"[317] It may\nmean the absence of all _necessary_ relation, but it does not mean the\nabsence of _all_ relation. If God can not _voluntarily_ call a finite\nexistence into being, and thus stand in the relation of cause, He is\ncertainly under the severest limitation. But surely that is not a limit\nwhich substitutes choice for necessity. To be unable to know God out of\nall relation--that is, apart from his attributes, apart from his created\nuniverse, is not felt by us to be any privation at all. A God without\nattributes, and out of all relations, is for us no God at all. God as a\nbeing of unlimited perfection, as infinitely wise and good, as the\nunconditioned cause of all finite being, and, consequently, as\nvoluntarily related to nature and humanity, we can and do know; this is\nthe living and true God. The God of a false philosophy is not the true\nGod; the pure abstractions of Hegel and Hamilton are negations, and not\nrealities.\n\n2. We proceed to consider the second fundamental principle of Hamilton's\nphilosophy of the conditioned, viz., that \"conditional limitation is the\nfundamental law of the possibility of thought,\" and that thought\nnecessarily imposes conditions on its object.\n\n\"Thought,\" says Hamilton, \"can not transcend consciousness:\nconsciousness is only possible under the antithesis of a subject and an\nobject known only in correlation, and _mutually limiting each\nother_\"[318] Thought necessarily supposes conditions; \"to think is\nsimply to condition,\" that is, to predicate limits; and as the infinite\nis the unlimited, it can not be thought. The very attempt to think the\ninfinite renders it finite; therefore there can be no infinite _in\nthought_, and, consequently, the infinite can not be known.\n\n[Footnote 317: Calderwood's \"Philosophy of the Infinite,\" p. 179.]\n\n[Footnote 318: \"Discussions,\" p. 21.]\n\nIf by \"the infinite in thought\" is here meant the infinite compassed or\ncontained in thought, we readily grant that the finite can not contain\nthe infinite; it is a simple truism which no one has ever been so\nfoolish as to deny. Even Cousin is not so unwise as to assert the\nabsolutely comprehensibility of God. \"In order absolutely to comprehend\nthe Infinite, it is necessary to have an infinite power of\ncomprehension, and this is not granted to us.\"[319] A finite mind can\nnot have \"an infinite thought.\" But it by no means follows that, because\nwe can not have infinite thought, we can have no clear and definite\nthought of or concerning the Infinite. We have a precise and definite\nidea of infinitude; we can define the idea; we can set it apart without\ndanger of being confounded with another, and we can reason concerning\nit. There is nothing we more certainly and intuitively know than that\nspace is infinite, and yet we can not comprehend or grasp within the\ncompass of our thought the infinite space. We can not form an _image_ of\ninfinite space, can not traverse it in perception, or represent it by\nany combination of numbers; but we can have the _thought_ of it as an\nidea of Reason, and can argue concerning it with precision and\naccuracy.[320] Hamilton has an idea of the Infinite; he defines it; he\nreasons concerning it; he says \"we must believe in the infinity of God.\"\nBut how can he define the Infinite unless he possesses some knowledge,\nhowever limited, of the infinite Being? How can he believe in the\ninfinity of God if he has no definite idea of infinitude? He can not\nreason about, can not affirm or deny any thing concerning, that of which\nhe knows absolutely nothing.\n\n[Footnote 319: \"Lectures on History of Philosophy,\" vol. i. p. 104.]\n\n[Footnote 320: \"To form an _image_ of any infinitude--be it of time or\nspace [or power]; to go mentally through it by successive steps of\nrepresentation--is indeed impossible; not less so than to traverse it in\nour finite perception and experience. But to have the _thought_ of it as\nan idea of the reason, not of the phantasy, and assign that thought a\nconstituent place in valid beliefs and consistent reasonings, appears to\nus as not only possible, but inevitable.\"--Martineau's \"Essays,\" p.\n205.]\n\nThe grand logical barrier which Hamilton perpetually interposes to all\npossible cognition of God _as infinite_ is, that to think is to\ncondition--to limit; and as the Infinite is the unconditioned, the\nunlimited, therefore \"the Infinite can not be _thought_.\" We grant at\nonce that all human thought is limited and finite, but, at the same\ntime, we emphatically deny that the limitation of our thought imposes\nany conditions or limits upon the object of thought. No such affirmation\ncan be consistently made, except on the Hegelian hypothesis that\n\"Thought and Being are identical;\" and this is a maxim which Hamilton\nhimself repudiates. Our thought does not create, neither does it impose\nconditions upon, any thing.\n\nThere is a lurking sophism in the whole phraseology of Hamilton in\nregard to this subject. He is perpetually talking about \"thinking a\nthing\"--\"thinking the Infinite.\" Now we do not think a thing, but we\nthink _of_ or _concerning_ a thing. We do not think a man, neither does\nour thought impose any conditions upon the man, so that he must be as\nour thought conceives or represents him; but our thought is of the man,\nconcerning or about the man, and is only so far true and valid as it\nconforms to the objective reality. And so we do not \"think the\nInfinite;\" that is, our thought neither contains nor conditions the\nInfinite Being, but our thoughts are _about_ the Infinite One; and if we\ndo not think of Him as a being of infinite perfection, our thought is\nneither worthy, nor just, nor true.[321]\n\n[Footnote 321: Calderwood's \"Philosophy of the Infinite,\" pp. 255, 256.]\n\nBut we are told the law of all thought and of all being is\ndetermination; consequently, negation of some quality or some\npotentiality; whereas the Infinite is \"_the One and the All_\" (ti En kai\nPyn),[322] or, as Dr. Mansel, the disciple and annotator of Hamilton,\naffirms, \"the sum of all reality,\" and \"the sum of all possible modes of\nbeing.\"[323] The Infinite, as thus defined, must include in itself all\nbeing, and all modes of being, actual and possible, not even excepting\nevil. And this, let it be observed, Dr. Mansel has the hardihood to\naffirm. \"If the Absolute and the Infinite is an object of human\nconception at all, this, and none other, is the conception\nrequired.\"[324] \"The Infinite Whole,\" as thus defined, can not be\nthought, and therefore it is argued the Infinite God can not be known.\nSuch a doctrine shocks our moral sense, and we shrink from the thought\nof an Infinite which includes evil. There is certainly a moral\nimpropriety, if not a logical impossibility, in such a conception of\nGod.\n\n[Footnote 322: Hamilton's \"Lectures on Metaphysics,\" Appendix, vol. ii.\np. 531.]\n\n[Footnote 323: \"Limits of Religious Thought,\" p. 76.]\n\n[Footnote 324: Ibid.]\n\nThe fallacy of this reasoning consists in confounding a _supposed_\nQuantitative Infinite with _the_ Qualitative Infinite--the totality of\nexistence with the infinitely perfect One. \"Qualitative infinity is a\nsecondary predicate; that is, the attribute of an attribute, and is\nexpressed by the adverb _infinitely_ rather than the adjective\n_infinite_. For instance, it is a strict use of language to say, that\nspace is infinite, but it is an elliptical use of language to say, God\nis infinite. Precision of language would require us to say, God is\ninfinitely good, wise, and great; or God is good, and his goodness is\ninfinite. The distinction may seem trivial, but it is based upon an\nimportant difference between the infinity of space and time on the one\nhand, and the infinity of God on the other. Neither philosophy nor\ntheology can afford to disregard the difference. Quantitative Infinity\nis illimitation by _quantity_. Qualitative Infinity is illimitation by\n_degree_. Quantity and degree alike imply finitude, and are categories\nof the finite alone. The danger of arguing from the former kind of\ninfinitude to the latter can not be overstated. God alone possesses\nQualitative Infinity, which is strictly synonymous with _absolute\nperfection_; and the neglect of the distinction between this and\nQuantitative Infinity, leads irresistibly to pantheistic and\nmaterialistic notions. Spinozism is possible only by the elevation of\n'infinite extension' to the dignity of a divine attribute. Dr. Samuel\nClarke's identification of God's immensity with space has been shown by\nMartin to ultimate in Pantheism. From ratiocinations concerning the\nincomprehensibility of infinite space and time, Hamilton and Mansel pass\nat once to conclusions concerning the incomprehensibility of God. The\ninconsequence of all such arguments is absolute; and if philosophy\ntolerates the transference of spatial or temporal analogies to the\nnature of God, she must reconcile herself to the negation of his\npersonality and spirituality.\"[325] An Infinite Being, quite remote from\nthe notion of _quantity_, may and does exist; which, on the one hand,\ndoes not include finite existence, and, on the other hand, does not\nrender the finite impossible to thought. Without contradiction they may\ncoexist, and be correlated.\n\nThe thought will have already suggested itself to the mind of the reader\nthat for Hamilton to assert that the Infinite, as thus defined (the One\nand the All), is absolutely unknown, is certainly the greatest\nabsurdity, for in that case nothing can be known. This Infinite must be\nat least partially known, or all human knowledge is reduced to zero. To\nthe all-inclusive Infinite every thing affirmative belongs, not only to\nbe, but to be known. To claim it for being, yet deny it to thought, is\nthus impossible. The Infinite, which includes all real existence, is\ncertainly possible to cognition.\n\nThe whole argument as regards the conditionating nature of all thought\nis condensed into four words by Spinoza--\"_Omnis determinatio est\nnegatio_;\" all determination is negation. Nothing can be more arbitrary\nor more fallacious than this principle. It arises from the confusion of\ntwo things essentially different--_the limits of a being_, and _its\ndeterminate and distinguishing characteristics_. The limit of a being is\nits imperfection; the determination of a being is its perfection. The\nless a thing is determined, the more it sinks in the scale of being; the\nmost determinate being is the most perfect being. \"In this sense God is\nthe only being absolutely determined. For there must be something\nindetermined in all finite beings, since they have all imperfect powers\nwhich tend towards their development after an indefinite manner. God\nalone, the complete Being in whom all powers are actualized, escapes by\nHis own perfection from all progress, and development, and\nindetermination.\"[326]\n\n[Footnote 325: North American Review, October, 1864, article, \"The\nConditioned and the Unconditioned,\" pp. 422, 423. See also Young's\n\"Province of Reason,\" p. 72; and Calderwood's \"Philosophy of the\nInfinite,\" p. 183.]\n\n[Footnote 326: Saisset, \"Modern Pantheism,\" vol. ii. p. 71.]\n\nAll real being must be determined; only pure Nothing can be\nundetermined. _Determination_ is, however, one thing; and _limitation_\nis essentially another thing. \"Even space and time, though cognized\nsolely by negative characteristics, are determined in so far as\ndifferentiated from the existences they contain; but this\ndifferentiation involves no limitation of their infinity.\" If all\ndistinction is determination, and if all determination is negation, that\nis (as here used), limitation, then the infinite, as distinguished from\nthe finite, loses its own infinity, and either becomes identical with\nthe finite, or else vanishes into pure nothing. If Hamilton will persist\nin affirming that all determination is limitation, he has no other\nalternatives but to accept the doctrine of Absolute Nihilism, or of\nAbsolute Identity. If the Absolute is the indeterminate--that is, no\nattributes, no consciousness, no relations--it is pure non-being. If the\nInfinite is \"the One and All,\" then there is but one substance, one\nabsolute entity.\n\nHerbert Spencer professes to be carrying out, a step farther, the\ndoctrine put into shape by Hamilton and Mansel, viz., \"the philosophy of\nthe Unconditioned.\" In other words, he carries that doctrine forward to\nits rigidly logical consequences, and utters the last word which\nHamilton and Mansel dare not utter--\"Apprehensible by us there is no\nGod.\" The Ultimate Reality is absolutely unknown; it can not be\napprehended by the human intellect, and it can not present itself to the\nintellect at all. This Ultimate Reality can not be _intelligent_,\nbecause to think is to condition, and the Absolute is the unconditioned;\ncan not be _conscious_, because all consciousness is of plurality and\ndifference, and the Absolute is one; can not be _personal_, because\npersonality is determination or limitation, and the Infinite is the\nillimitable. It is \"audacious,\" \"irreverent,\" \"impious,\" to apply any of\nthese predicates to it; to regard it as Mind, or speak of it as\nRighteous.[327] The ultimate goal of the philosophy of the Unconditioned\nis a purely subjective Atheism.\n\n[Footnote 327: \"First Principles,\" pp. 111, 112.]\n\nAnd yet of this Primary Existence--inscrutable, and absolutely\nunknown--Spencer knows something; knows as much as he pleases to know.\nHe knows that this \"ultimate of ultimates is _Force_,\"[328] an\n\"_Omnipresent Power_,\"[329] is \"_One_\" and \"_Eternal_.\"[330] He knows\nalso that it can not be intelligent, self-conscious, and a\npersonality.[331] This is a great deal to affirm and deny of an\nexistence \"absolutely unknown.\" May we not be permitted to affirm of\nthis hidden and unknown something that it is _conscious Mind_,\nespecially as Mind is admitted to be the only analogon of Power; and\n\"the _force_ by which we produce change, and which serves to symbolize\nthe causes of changes in general, is the final disclosure of\nanalysis.\"[332]\n\n[Footnote 328: \"First Principles,\" p. 235.]\n\n[Footnote 329: Ibid., p. 99.]\n\n[Footnote 330: Ibid., p. 81.]\n\n[Footnote 331: Ibid., pp. 108-112.]\n\n[Footnote 332: Ibid., p. 235.]\n\n3. We advance to the review of the third fundamental principle of\nHamilton's philosophy of the Unconditioned, viz., that the terms\ninfinite and absolute are names for a \"mere negation of thought\"--a\n\"mental impotence\" to think, or, in other words, the absence of all the\nconditions under which thought is possible.\n\nThis principle is based upon a distinction between \"positive\" and\n\"negative\" thought, which is made with an air of wonderful precision and\naccuracy in \"the Alphabet of Human Thought.\"[333] \"Thinking is\n_positive_ when existence is predicated of an object.\" \"Thinking is\n_negative_ when existence is not attributed to an object.\" \"Negative\nthinking,\" therefore, is not the thinking of an object as devoid of this\nor that particular attribute, but as devoid of all attributes, and thus\nof all existence; that is, it is \"the negation of all\nthought\"--_nothing_. \"When we think a thing, that is done by conceiving\nit as possessed of certain modes of being or qualities, _and the sum of\nthese qualities constitutes its concept or notion_.\" \"When we perform an\nact of negative thought, this is done by thinking _something_ as _not_\nexisting in this or that determinate mode; and when we think it as\nexisting in no determinate mode, _we cease to think at all--it becomes a\nnothing_.\"[334] Now the Infinite, according to Hamilton, can not be\nthought in any determinate mode; therefore we do not think it at all,\nand therefore it is for us \"a logical Non-entity.\"\n\n[Footnote 333: \"Discussions,\" Appendix I. p. 567.]\n\n[Footnote 334: \"Logic,\" pp. 54, 55.]\n\nIt is barely conceivable that Hamilton might imagine himself possessed\nof this singular power of \"performing an act of negative thought\"--that\nis, of thinking and not thinking at once, or of \"thinking something\"\nthat \"becomes nothing;\" we are not conscious of any such power. To think\nwithout an object of thought, or to think of something without any\nqualities, or to think \"something\" which in the act of thought melts\naway into \"nothing,\" is an absurdity and a contradiction. We can not\nthink about nothing. All thought must have an object, and every object\nmust have some predicate. Even space has some predicates--as\nreceptivity, unity, and infinity. Thought can only be realized by\nthinking something existing, and existing in a determinate manner; and\nwhen we cease to think something having predicates, we cease to think at\nall. This is emphatically asserted by Hamilton himself.[335] \"Negative\nthinking\" is, therefore, a meaningless phrase, a contradiction in terms;\nit is no thought at all. We are cautioned, however, against regarding\n\"the negation of thought\" as \"a negation of all mental ability.\" It is,\nwe are told, \"an attempt to think, and a failure in the attempt.\" An\nattempt to think about _what_? Surely it must be about some object, and\nan object which is _known_ by some sign, else there can be no thought.\nLet any one make the attempt to think without something to think about,\nand he will find that both the process and the result are blank\nnothingness. All thought, therefore, as Calderwood has amply shown, is,\nmust be, _positive_. \"Thought is nothing else than the comparison of\nobjects known; and as knowledge is always positive, so must our thought\nbe. All knowledge implies an object _known_; and so all thought involves\nan object about which we think, and must, therefore, be positive--that\nis, it must embrace within itself the conception of certain qualities as\nbelonging to the object.\"[336]\n\n[Footnote 335: \"Logic,\" p. 55.]\n\n[Footnote 336: \"Philosophy of the Infinite,\" p. 272.]\n\nThe conclusion of Hamilton's reasoning in regard to \"negative thinking\"\nis, that we can form no notion of the Infinite Being. We have no\npositive idea of such a Being. We can think of him only by \"the thinking\naway of every characteristic\" which can be conceived, and thus \"ceasing\nto think at all.\" We can only form a \"negative concept,\" which, we are\ntold, \"is in fact no concept at all.\" We can form only a \"negative\nnotion,\" which, we are informed, \"is only the negation of a notion.\"\nThis is the impenetrable abyss of total gloom and emptiness into which\nthe philosophy of the conditions leads us at last.[337]\n\n[Footnote 337: Whilst Spencer accepts the general doctrine of Hamilton,\nthat the Ultimate Reality is inscrutable, he argues earnestly against\nhis assertion that the Absolute is a \"mere negation of thought.\"\n\n\"Every one of the arguments by which the relativity of our knowledge is\ndemonstrated distinctly postulates the _positive existence_ of something\nbeyond the relative. To say we can not know the Absolute is, by\nimplication, to affirm there _is_ an Absolute. In the very denial of our\npower to learn _what_ the Absolute is, there lies hidden the assumption\n_that_ it is; and the making of this assumption proves that the Absolute\nhas been present to the mind, not as nothing, but as _something_. And so\nwith every step in the reasoning by which the doctrine is upheld, the\nNoumenon, everywhere named as the antithesis of the Phenomenon, is\nthroughout thought as actuality. It is rigorously impossible to conceive\nthat our knowledge is a knowledge of appearances only, without, at the\nsame time, conceiving a Reality of which these are appearances, for\nappearances without reality are unthinkable.\n\n\"Truly to represent or realize in thought any one of the propositions of\nwhich the argument consists, the unconditioned must be represented as\n_positive_, and not negative. How, then, can it be a legitimate\nconclusion from the argument that our consciousness of it is negative?\nAn argument, the very construction of which assigns to a certain term a\ncertain meaning, but which ends in showing that this term has no\nmeaning, is simply an elaborate suicide. Clearly, then, the very\ndemonstration that a definite consciousness [comprehension] of the\nAbsolute is impossible, unavoidably presupposes an indefinite\nconsciousness of it [an apprehension].\"--\"First Principles,\" p. 88.]\n\nStill we have the word _infinite_, and we have _the notion_ which the\nword expresses. This, at least, is spared to us by Sir William Hamilton.\nHe who says we have no such notion asks the question _how we have it?_\nHere it may be asked, how have we, then, the word infinite? How have we\nthe notion which this word expresses? The answer to this question is\ncontained in the distinction of positive and negative thought.\n\nWe have a positive concept of a thing when we think of it by the\nqualities of which it is the complement. But as the attribution of\nqualities is an affirmation, as affirmation and negation are relatives,\nand as relatives _are known only in and through each other_, we can not,\ntherefore, have a _consciousness_ of the affirmation of any quality\nwithout having, at the same time, the _correlative consciousness_ of its\nnegation. Now the one consciousness is a positive, the other\nconsciousness is a negative notion; and as all language is the reflex of\nthought, the positive and negative notions are expressed by positive and\nnegative names. Thus it is with the Infinite.[338] Now let us carefully\nscrutinize the above deliverance. We are told that \"relatives are known\nonly in and through each other;\" that is, such relatives as _finite_ and\n_infinite_ are known necessarily in the same act of thought. The\nknowledge of one is as necessary as the knowledge of the other. We can\nnot have a consciousness of the one without the correlative\nconsciousness of the other. \"For,\" says Hamilton, \"a relation is, in\ntruth, a thought, one and indivisible; and while the thinking a relation\n_necessarily involves the thought of its two terms,_, so it is, with\nequal necessity, itself involved in the thought of either.\" If, then, we\nare _conscious_ of the two terms of the relation in the same \"one and\nindivisible\" mental act--if we can not have \"the consciousness of the\none without the consciousness of the other\"--if space and position, time\nand succession, substance and quality, infinite and finite, are given to\nus in pairs, then 'the _knowledge of one is as necessary as the\nknowledge of the other,_' and they must stand or fall together. The\nfinite is known no more positively than the infinite; the infinite is\nknown as positively as the finite. The one can not be taken and the\nother left. The infinite, discharged from all relation to the finite,\ncould never come into apprehension; and the finite, discharged of all\nrelation to the infinite, is incognizable too. \"There can be no\nobjection to call the one 'positive' and the other 'negative,' provided\nit be understood that _each_ is so with regard to the other, and that\nthe relation is convertible; the finite, for instance, being the\nnegative of the infinite, not less than the infinite of the\nfinite.\"[339]\n\n[Footnote 338: _Logic,_ p. 73.]\n\n[Footnote 339: Martineau's \"Essays,\" p. 237.]\n\nTo say that the finite is comprehensible in and by itself, and the\ninfinite is incomprehensible in and by itself, is to make an assertion\nutterly at variance both with psychology and logic. The finite is no\nmore comprehensible _in itself_ than the infinite. \"Relatives are known\nonly in and through each other.\"[340] \"The conception of one term of a\nrelation necessarily implies that of the other, it being the very nature\nof a relative to be thinkable only through the conjunct thought of its\ncorrelative.\" We comprehend nothing more completely than the infinite;\n\"for the idea of illimitation is as clear, precise, and intelligible as\nthe idea of limitability, which is its basis. The propositions \"A is X\"\n\"A is not X,\" are equally comprehensible; the conceptions A and X are in\nboth cases positive data of experience, while the affirmation and\nnegation consist solely in the copulative or disjunctive nature of the\npredication. Consequently, if X is comprehensible, so is not--X; if the\nfinite is comprehensible, so is the infinite.\"[341]\n\nWhilst denying that the infinite can by us be _known_, Hamilton tells us\nhe is \"far from denying that it is, must, and ought to be\n_believed_.\"[342] \"We must believe in the infinity of God.\"\n\"Faith--belief--is the organ by which we apprehend what is beyond\nknowledge.\"[343] We heartily assent to the doctrine that the Infinite\nBeing is the object of faith, but we earnestly deny that the Infinite\nBeing is not an object of knowledge. May not knowledge be grounded upon\nfaith, and does not faith imply knowledge? Can we not obtain knowledge\nthrough faith? Is not the belief in the Infinite Being implied in our\nknowledge of finite existence? If so, then God as the infinite and\nperfect, God as the unconditioned Cause, is not absolutely \"the\nunknown.\"\n\n[Footnote 340: Hamilton's \"Logic,\" p. 73.]\n\n[Footnote 341: North American Review, October, 1864, article\n\"Conditioned and the Unconditioned,\" pp. 441, 442.]\n\n[Footnote 342: Letter to Calderwood, Appendix, vol. ii. p. 530.]\n\n[Footnote 343: \"Lectures on Metaphysics,\" vol. ii. p. 374.]\n\nA full exposition of Sir William Hamilton's views of _Faith_ in its\nconnection with Philosophy would have been deeply interesting to us, and\nit would have filled up a gap in the interpretation of his system. The\nquestion naturally presents itself, how would he have discriminated\nbetween faith and knowledge, so as to assign to each its province? If\nour notion of the Infinite Being rests entirely upon faith, then upon\nwhat ultimate ground does faith itself rest? On the authority of\nScripture, of the Church, or of reason? The only explicit statement of\nhis view which has fallen in our way is a note in his edition of\nReid.[344] \"We _know_ what rests upon reason; we _believe_ what rests\nupon authority. But reason itself must rest at last upon authority; for\nthe original data of reason do not rest upon reason, but are necessarily\naccepted by reason on the authority of what is beyond itself. These data\nare, therefore, in rigid propriety, Beliefs or Trusts. Thus it is that,\nin the last resort, we must, per force, philosophically admit that\nbelief is the primary condition of reason, and not reason the ultimate\nground of belief.\"\n\n[Footnote 344: P. 760; also Philosophy of Sir William Hamilton, p. 61.]\n\nHere we have, first, an attempted distinction between faith and\nknowledge. \"We _know_ what rests upon reason;\" that is, whatever we\nobtain by deduction or induction, whatever is capable of explication and\nproof, is _knowledge_. \"We _believe_ what rests upon authority;\" that\nis, whatever we obtain by intellectual intuition or pure apperception,\nand is incapable of explication and of proof, is \"a _belief or trust_.\"\nThese instinctive beliefs, which are, as it were, the first principles\nupon which all knowledge rests, are, however, indiscriminately called by\nHamilton \"cognitions,\" \"beliefs,\" \"judgments.\" He declares most\nexplicitly \"that the principles of our knowledge must themselves be\n_knowledges_;\"[345] and these first principles, which are \"the primary\ncondition of reason,\" are elsewhere called \"_a priori cognitions_;\" also\n\"native, pure, or transcendental _knowledge_,\" in contradistinction to\n\"_a posteriori cognitions_,\" or that knowledge which is obtained in the\nexercise of reason.[346] All this confusion results from an attempt to\nput asunder what God has joined together. As Clemens of Alexandria has\nsaid, \"Neither is faith without knowledge, nor knowledge without faith.\"\nAll faith implies knowledge, and all knowledge implies faith. They are\nmingled in the one operation of the human mind, by which we apprehend\nfirst principles or ultimate truths. These have their light and dark\nside, as Hamilton has remarked. They afford enough light to show _that_\nthey are and must be, and thus communicate knowledge; they furnish no\nlight to show _how_ they are and _why_ they are, and under that aspect\ndemand the exercise of faith. There must, therefore, first be something\n_known_ before there can be any _faith_.[347]\n\n[Footnote 345: Ibid., p. 69.]\n\n[Footnote 346: \"Lectures on Metaphysics,\" vol. ii. p. 26.]\n\n[Footnote 347: M'Cosh, \"Intuitions,\" pp. 197, 198; Calderwood,\n\"Philosophy of the Infinite,\" p. 24.]\n\nAnd now we seem to have penetrated to the centre of Hamilton's\nphilosophy, and the vital point may be touched by one crucial question,\n_Upon what ultimate ground does faith itself rest?_ Hamilton says, \"we\nbelieve what rests upon _authority_.\" But what is that authority? I. It\nis not the authority of Divine Revelation, because beliefs are called\n\"instinctive,\" \"native,\" \"innate,\" \"common,\" \"catholic,\"[348] all which\nterms seem to indicate that this \"authority\" lies within the sphere of\nthe human mind; at any rate, this faith does not rest on the authority\nof Scripture. Neither is it the authority of Reason. \"The original data\nof reason [the first principles of knowledge] do not rest upon the\nauthority of reason, but _on the authority of what is beyond itself_.\"\nThe question thus recurs, what is this ultimate ground beyond reason\nupon which faith rests? Does it rest upon any thing, or nothing?\n\n[Footnote 348: Philosophy of Sir Wm. Hamilton, pp. 68, 69.]\n\nThe answer to this question is given in the so-called \"Law of the\nConditioned,\" which is thus laid down: \"_All that is conceivable in\nthought lies between two extremes, which, as contradictory of each\nother, can not both be true, but of which, as mutual contradictories,\none must_.\" For example, we conceive _space_, but we can not conceive it\nas absolutely bounded or infinitely unbounded. We can conceive _time_,\nbut we can not conceive it as having an absolute commencement or an\ninfinite non-commencement. We can conceive of _degree_, but we can not\nconceive it as absolutely limited or as infinitely unlimited. We can\nconceive of _existence_, but not as an absolute part or an infinite\nwhole. Therefore, \"the Conditioned is that which is alone conceivable or\ncogitable; the Unconditioned, that which is inconceivable or\nincogitable. The conditioned, or the thinkable, lies between two\nextremes or poles; and each of these extremes or poles are\nunconditioned, each of them inconceivable, each of them exclusive or\ncontradictory of the other. Of these two repugnant opposites, the one is\nthat of Unconditional or Absolute Limitation; the other that of\nUnconditional or Infinite Illimitation, or, more simply, the Absolute\nand the Infinite; the term _absolute_ expressing that which is finished\nor complete, the term _infinite_ that which can not be terminated or\nconcluded.\"[349]\n\n\"The conditioned is the mean between two extremes--two inconditionates,\nexclusive of each other, neither of which _can be conceived as\npossible_, but of which, on the principle of contradiction, and excluded\nmiddle, _one must be admitted as necessary_. We are thus warned from\nrecognizing the domain of our knowledge as necessarily co-extensive with\nthe horizon of our faith. And by a _wonderful revelation_, we are thus,\nin the very consciousness of our inability to conceive aught above the\nrelative and the finite, _inspired with a belief in_ the existence of\nsomething unconditioned beyond the sphere of all comprehensible\nreality.\"[350] Here, then, we have found the ultimate ground of our\nfaith in the Infinite God. It is built upon a \"mental imbecility,\" and\nbuttressed up by \"contradictions!\"[351]\n\n[Footnote 349: \"Lectures on Metaphysics,\" vol. ii. pp. 368, 374. With\nHamilton, the Unconditioned is a genus, of which the Infinite and\nAbsolute are species.]\n\n[Footnote 350: \"Discussions on Philosophy,\" p. 22.]\n\n[Footnote 351: The warmest admirers of Sir William Hamilton hesitate to\napply the doctrine of the unconditioned to Cause and Free-will. See\n\"Mansel's Prolegom.,\" Note C, p. 265.]\n\nSuch a faith, however, is built upon the clouds, and the whole structure\nof this philosophy is \"a castle in the air\"--an attempt to organize\nNescience into Science, and evoke something out of nothing. To pretend\nto believe in that respecting which I can form no notion is in reality\nnot to believe at all. The nature which compels me to believe in the\nInfinite must supply me some object upon which my belief can take hold.\nWe can not believe in contradictions. Our faith must be a rational\nbelief--a faith in the ultimate harmony and unity of all truth, in the\nveracity and integrity of human reason as the organ of truth; and, above\nall, a faith in the veracity of God, who is the author and illuminator\nof our mental constitution. \"We can not suppose that we are created\ncapable of intelligence in order to be made victims of delusion--that\nGod is a deceiver, and the root of our nature a lie.\"[352] We close our\nreview of Hamilton by remarking:\n\n[Footnote 352: Philosophy of Sir William Hamilton, p. 21.]\n\n1. \"The Law of the Conditioned,\" as enounced by Hamilton, is\ncontradictory. It predicates contradiction of two extremes, which are\nasserted to be equally incomprehensible and incognizable. If they are\nutterly incognizable, how does Hamilton _know_ that they are\ncontradictory? The mutual _relation_ of two objects is said to be known,\nbut the objects themselves are absolutely unknown. But how can we know\nany relation except by an act of comparison, and how can we compare two\nobjects so as to affirm their relation, if the objects are absolutely\nunknown? \"The Infinite is defined as Unconditional Illimitation; the\nAbsolute as Conditional Limitation. Yet almost in the same breath we are\ntold that each is utterly inconceivable, each the mere negation of\nthought. On the one hand, we are told they _differ_; on the other, we\nare told they do _not differ_. Now which does Hamilton mean? If he\ninsist upon the definitions as yielding a ground of conceivable\ndifference, he must abandon the inconceivability; but if he insist upon\nthe inconceivability, he must abandon the definition as sheer verbiage,\ndevoid of all conceivable meaning. There is no possible escape from this\ndilemma. Further, two negations can never contradict; for contradiction\nis the asserting and the denying of the same proposition; two denials\ncan not conflict. If Illimitation is negative, Limitation, its\ncontradictory, is positive, whether conditional or unconditional. In\nbrief, if the Infinite and Absolute are wholly incomprehensible, they\nare not distinguishable; but if they are distinguishable, they are not\nwholly incomprehensible. If they are indistinguishable, they are to us\nidentical; and identity precludes contradiction. But if they are\ndistinguishable, distinction is made by difference, which involves\npositive cognition; hence one, at least, must be conceivable. It\nfollows, therefore, by inexorable logic, that either the contradiction\nor the inconceivability must be abandoned.\"[353]\n\n[Footnote 353: North American Review, October, 1864, pp. 407, 408.]\n\n2. \"The Law of the Conditioned,\" as a ground of faith in the Infinite\nBeing, is utterly void, meaningless, and ineffectual. Let us re-state it\nin Hamilton's own words: \"The conditioned is the _mean_ between two\nextremes, two inconditionates exclusive of each other, neither of which\n_can be conceived as possible_, but of which, on the principle of\nContradiction and Excluded Middle, _one must be admitted as necessary_.\"\nIt is scarcely needful to explain to the intelligent reader the above\nlogical principles; that they may, however, be clearly before the mind\nin this connection, we state that the principle of Contradiction is\nthis: \"A thing can not at the same time be and not be; _A is_, _A is\nnot_, are propositions which can not both be true at once.\" The\nprinciple of Excluded Middle is this: \"A thing either is or is not--_A\neither is or is not B_; there is no _medium_.\"[354] Now, to mention the\nlaw of Excluded Middle and two contradictories with a _mean_ between\nthem, in the same sentence, is really astounding. \"If the two\ncontradictory extremes are equally incogitable, yet include a cogitable\nmean, why insist upon the necessity of accepting either extreme? This\nnecessity of accepting one of the contradictories is wholly based upon\nthe supposed impossibility of a _mean_; if a mean exists, _that_ may be\ntrue, and both contradictories together false. But if a mean between two\ncontradictories be both impossible and absurd, Hamilton's 'conditioned'\nentirely vanishes.\"[355] If both contradictories are equally unknown and\nequally unthinkable, we can not discover _why_, on his principles, we\nare bound to believe _either_.\n\n[Footnote 354: Hamilton's \"Logic,\" pp. 58, 59; \"Metaphysics,\" vol. ii.\np. 368.]\n\n[Footnote 355: North British Review, October, 1864, pp. 415, 416.]\n\n3. The whole of this confusion in thought and expression results from\nthe habit of confounding the sensuous imagination with the non-sensuous\nreason, and the consequent co-ordination of an imageable conception with\nan abstract idea. The objects of sense and the sensuous imagination may\nbe characterized as extension, limitation, figure, position, etc.; the\nobjects of the non-sensuous reason may be characterized as universality,\neternity, infinity. I can form an _image_ of an extended and figured\nobject, but I can not form an _image_ of space, time, or God; neither,\nindeed, can I form an image of Goodness, Justice, or Truth. But I can\nhave a clear and precise idea of space, and time, and God, as I can of\nJustice, Goodness, and Truth. There are many things which I can most\nsurely _know_ that I can not possibly _comprehend_, if to comprehend is\nto form a mental image of a thing. There is nothing which I more\ncertainly know than that space is infinite, and eternity unbeginning and\nendless; but I can not comprehend the infinity of space or the\nillimitability of eternity. I know that God is, that he is a being of\ninfinite perfection, but I can not throw my thoughts around and\ncomprehend the infinity of God.\n\n(iv.) We come, lastly, to consider the position of the _Dogmatic\nTheologians_.[356] In their zeal to demonstrate the necessity of Divine\nRevelation, and to vindicate for it the honor of supplying to us all our\nknowledge of God, they assail every fundamental principle of reason,\noften by the very weapons which are supplied by an Atheistical\nphilosophy. As a succinct presentation of the views of this school, we\nselect the \"_Theological Institutes_\" of R. Watson.\n\n[Footnote 356: Ellis, Leland, Locke, and Horsley, whose writings are\nextensively quoted in Watson's \"Institutes of Theology\" (reprinted by\nCarlton & Lanahan, New York).]\n\n1st. The invalidity of \"_the principle of causality_\" is asserted by\nthis author. \"We allow that the argument which proves that the _effects_\nwith which we are surrounded have been _caused_, and thus leads us up\nthrough a chain of subordinate causes to one First Cause, has a\nsimplicity, an obviousness, and a force which, when we are previously\nfurnished with the idea of God, makes it, at first sight, difficult to\nconceive that men, under any degree of cultivation, should be inadequate\nto it; yet if ever the human mind commenced such an inquiry at all, it\nis highly probable that it would rest in the notion of an _eternal\nsuccession of causes and effects_, rather than acquire the ideas of\ncreation, in the proper sense, and of a Supreme Creator.\"[357] \"We feel\nthat our reason rests with full satisfaction in the doctrine that all\nthings are created by one eternal and self-existent Being; but the Greek\nphilosophers held that matter was eternally co-existent with God. This\nwas the opinion of Plato, who has been called the Moses of\nphilosophy.\"[358]\n\nFor a defense of \"the principle of causality\" we must refer the reader\nto our remarks on the philosophy of Comte. We shall now only remark on\none or two peculiarities in the above statement which betray an utter\nmisapprehension of the nature of the argument. We need scarcely direct\nattention to the unfortunate and, indeed, absurd phrase, \"an eternal\nsuccession of causes and effects.\" An \"eternal succession\" is a\n_contradictio in adjecto_, and as such inconceivable and unthinkable. No\nhuman mind can \"rest\" in any such thing, because an eternal succession\nis no rest at all. All \"succession\" is finite and temporal, capable of\nnumeration, and therefore can not be eternal.[359] Again, in attaining\nthe conception of a First Cause the human mind does not pass up \"through\na chain of subordinate causes,\" either definite or indefinite, \"to one\nFirst Cause.\"\n\n[Footnote 357: Watson's \"Institutes of Theology,\" vol. i. p. 273.]\n\n[Footnote 358: Id., ib., vol. i. p. 21.]\n\n[Footnote 359: See _ante_, pp. 181, 182, ch. v.]\n\nLet us re-state the principle of causality as a universal and necessary\nlaw of thought. \"_All phenomena present themselves to us as the\nexpression of_ POWER, and refer us to a causal ground whence they\nissue.\" That \"power\" is intuitively and spontaneously apprehended by the\nhuman mind as Supreme and Ultimate--\"the causal ground\" is a personal\nGod. All the phenomena of nature present themselves to us as \"effects,\"\nand we know nothing of \"subordinate causes\" except as modes of the\nDivine Efficiency.[360] The principle of causality compels us to think\ncausation behind nature, and under causation to think of Volition.\n\"Other forces we have no sort of ground for believing; or, except by\nartifices of abstraction, even power of conceiving. The dynamic idea is\neither this or nothing; and the logical alternative assuredly is that\nnature is either a mere Time-march of phenomena or an expression of\nMind.\"[361] The true doctrine of philosophy, of science, and of\nrevelation is not simply that God did create \"in the beginning,\" but\nthat he still creates. All the operations of Nature are the operations\nof the Divine Mind. \"Thou takest away their breath, they die, and return\nto their dust. Thou sendest forth thy spirit, they are created; and thou\nrenewest the face of the earth.\"[362]\n\n[Footnote 360: The modern doctrine of the Correlation and Homogenity of\nall Forces clearly proves that they are not many, but _one_--\"a dynamic\nself-identity masked by transmigration.\"--Martineau's \"Essays,\" pp.\n134-144.]\n\n[Footnote 361: Martineau's \"Essays,\" pp. 140, 141.]\n\n[Footnote 362: Psalm civ.]\n\nThe assertion that Plato taught \"the eternity of matter,\" and that\nconsequently he did not arrive at the idea of a Supreme and Ultimate\nCause, is incapable of proof. The term yle=matter does not occur in the\nwritings of Plato, or, indeed, of any of his predecessors, and is\npeculiarly Aristotelian. The ground of the world of sense is called by\nPlato \"the receptacle\" (ypodoche), \"the nurse\" (tithene) of all that is\nproduced, and was apparently identified, in his mind, with _pure\nspace_--a logical rather than a physical entity--the mere negative\ncondition and medium of Divine manifestation. He never regards it as a\n\"cause,\" or ascribes to it any efficiency. We grant that he places this\nvery indefinite something (opoionoun ti) out of the sphere of temporal\norigination; but it must be borne in mind that he speaks of \"creation in\neternity\" as well as of \"creation in time;\" and of time itself, though\ncreated, as \"an eternal image of the generating Father.\"[363] This one\nthing, at any rate, can not be denied, that Plato recognizes creation in\nits fullest sense as the act of God.\n\nThe admission that something has always existed besides the Deity, as a\nmere logical condition of the exercise of divine power (_e.g._, space),\nwould not invalidate the argument for the existence of God. The proof of\nthe Divine Existence, as Chalmers has shown, does not rest on the\nexistence of matter, but on the orderly arrangement of matter; and the\ngrand question of Theism is not whether the _matter of the world_, but\nwhether the _present order of the world_ had a commencement.[364]\n\n2d. Doubt is cast by our author upon the validity of \"_the principle of\nthe Unconditioned or the Infinite_.\" \"Supposing it were conceded that\nsome faint glimmering of this great truth [the existence of a First\nCause] might, by induction, have been discovered by contemplative minds,\nby what means could they have _demonstrated_ to themselves that he is\neternal, self-existent, immortal, and independent?\"[365] \"Between things\nvisible and invisible, time and eternity, beings finite and beings\ninfinite, objects of sense and objects of faith, _the connection is not\nperceptible_ to human observation. Though we push our researches,\ntherefore, to the extreme point whither the light of nature can carry\nus, they will in the end be abruptly terminated, and we must stop short\nat an immeasurable distance between the creature and the Creator.\"[366]\n\n[Footnote 363: Plato, \"Timaeus,\" Sec. xiv.]\n\n[Footnote 364: Chalmers's \"Natural Theology,\" bk. i. ch. v.; also\nMahan's \"Natural Theology,\" pp. 21-23.]\n\n[Footnote 365: Watson's \"Institutes of Theol.,\" vol. i. p. 274.]\n\n[Footnote 366: Id., ib., vol. i. p. 273.]\n\nTo this assertion that the connection of things visible and things\ninvisible, finite and infinite, objects of sense and objects of faith,\nis utterly imperceptible to human thought, we might reply by quoting the\nwords of that Sacred Book whose supreme authority our author is seeking,\nby this argument, to establish. \"The _invisible_ things of God, even his\neternal power and god-head, from the creation, are clearly _seen_, being\n_understood by the things which are made_.\" We may also point to the\nfact that in every age and in every land the human mind has\nspontaneously and instinctively recognized the existence of an invisible\nPower and Presence pervading nature and controlling the destinies of\nman, and that religious worship--prayer, and praise, and\nsacrifice--offered to that unseen yet omnipresent Power is an universal\nfact of human nature. The recognition of an _immediate_ and a\n_necessary_ \"connection\" between the visible and the invisible, the\nobjects of sense and the objects of faith, is one of the most obvious\nfacts of consciousness--of universal consciousness as revealed in\nhistory, and of individual consciousness as developed in every rational\nmind.\n\nThat this connection is \"not perceptible to human observation,\" if by\nthis our author means \"not perceptible to sense,\" we readily admit. No\none ever asserted it was perceptible to human observation. We say that\nthis connection is perceptible to human _reason_, and is revealed in\nevery attempt to think about, and seek an explanation of, the phenomenal\nworld. The Phenomenal and the Real, Genesis and Being, Space and\nExtension, Succession and Duration, Time and Eternity, the Finite and\nthe Infinite, are correlatives which are given in one and the same\nindivisible act of thought. \"The conception of one term of a relation\nnecessarily implies that of the other; it being the very nature of a\ncorrelative to be thinkable only through the conjunct thought of its\ncorrelative; for a relation is, in truth, a thought one and indivisible;\nand whilst the thinking of one relation necessarily involves the thought\nof its two terms, so it is, with equal necessity, itself involved in the\nthought of either.\"[367] Finite, dependent, contingent, temporal\nexistence, therefore, necessarily supposes infinite, self-existent,\nindependent, eternal Being; the Conditioned and Relative implies the\nUnconditioned and Absolute--one is known only in and through the other.\nBut inasmuch as the unconditioned is cognized solely _a priori_, and the\nconditioned solely _a posteriori_, the recognition by the human mind of\ntheir necessary correlation becomes the bridge whereby the chasm between\nthe subjective and the objective may be spanned, and whereby Thought may\nbe brought face to face with Existence.\n\n[Footnote 367: Hamilton's \"Metaphysics,\" vol. ii. pp. 536, 537.]\n\nThe reverence which, from boyhood, we have entertained for the\ndistinguished author of the \"Institutes\" restrains us from speaking in\nadequate terms of reprobation of the statement that \"the _First Cause_\"\nmay be known, and yet not conceived \"as eternal, self-existent,\nimmortal, and independent\". Surely that which is the ground and reason\nof all existence must have the ground and reason of its own existence in\nitself. That which is _first_ in the order of existence, and in the\nlogical order of thought, can have nothing prior to itself. If the\nsupposed First Cause is not necessarily self-existent and independent,\nit is not the _first_; if it has a dependent existence, there must be a\nprior being on which it depends. If the First Cause is not eternal, then\nprior to this Ultimate Cause there was nothingness and vacuity, and pure\nnothing, by its own act, became something. But \"_Ex nihilo nihil_\" is a\nuniversal law of thought. To ask the question whether the First Cause be\nself-existent and eternal, is, in effect, to ask the question \"who made\nGod?\" and this is not the question of an adult theologian, but of a\nlittle child. Surely Mr. Watson must have penned the above passage\nwithout any reflection on its real import[368].\n\n[Footnote 368: In an article on \"the Impending Revolution in Anglo-Saxon\nTheology\" Methodist Quarterly Review, (July, 1863), Dr. Warren seems to\ntake it for granted that the \"aiteological\" and \"teleological\" arguments\nfor the existence of God are utterly invalidated by the Dynamical theory\nof matter. \"Once admit that _real power_ can and does reside in matter,\nand all these reasonings fail. If inherent forces of matter are\ncompetent to the production of all the innumerable miracles of movement\nin the natural world, what is there in the natural world which they can\nnot produce. If all _the exertions of power_ in the universe can be\naccounted for without resort to something back of, and superior to,\nnature, what is there which can force the mind to such a resort?\" (p.\n463). \"Having granted that _power_, or _self-activity_, is a natural\nattribute of all matter, what right have we to deny it _intelligence_?\"\n(p. 465). \"_Self-moving matter must have thought and design_\" (p. 469).\n\nIt is not our intention to offer an extended criticism of the above\npositions in this note. We shall discuss \"the Dynamical theory\" more\nfully in a subsequent work. If the theory apparently accepted by Dr.\nWarren be true, that \"_the ultimate atoms of matter are as uniformly\nefficient as minds_, and that we have the same ground to regard the\nforce exerted by the one _innate_ and _natural_ as that exerted by the\nother\" (p. 464), then we grant that the conclusions of Dr. Warren, as\nabove stated, are unavoidable. We proceed one step farther, and boldly\nassert that the existence of God is, on this hypothesis, incapable of\nproof, and the only logical position Dr. Warren can occupy is that of\nspiritualistic Pantheism.\n\nDr. Warren asserts that \"the Dynamical theory of matter\" is now\ngenerally accepted by \"Anglo-Saxon _naturalists_.\" \"One can scarcely\nopen a scientific treatise without observing the altered stand-point\"\n(p. 160). We confess that we are disappointed with Dr. Warren's\ntreatment of this simple question of fact. On so fundamental an issue,\nthe Doctor ought to have given the name of at least _one_ \"naturalist\"\nwho asserts that \"the ultimate atoms of matter are as uniformly\nefficient as minds.\" Leibnitz, Morrell, Ulrici, Hickok, the authorities\nquoted by him, are metaphysicians and idealists of the extremest school.\nAt present we shall, therefore, content ourselves with a general denial\nof this wholesale statement of Dr. Warren; and we shall sustain that\ndenial by a selection from the many authorities we shall hereafter\npresent. \"No particle of matter possesses within itself the power of\nchanging its existing state of motion or of rest. Matter has no\nspontaneous power either of rest or motion, but is equally susceptible\nto each as it may be acted on by _external_ causes\" (Silliman's\n\"Principles of Physics,\" p. 13). The above proposition is \"a truth on\nwhich the whole science of mechanical philosophy ultimately depends\"\n(Encyclopaedia Britannica, art. \"Dynamics,\" vol. viii. p. 326). \"A\nmaterial substance existing alone in the universe could not produce any\neffects. There is not, so far as we know, a self-acting material\nsubstance in the universe\" (M'Cosh, \"Divine Government, Physical and\nMoral,\" p. 78). \"Perhaps the only true indication of matter is\n_inertia_.\" \"The cause of gravitation is _not resident_ in the particles\nof matter merely,\" but also \"_in all space_\" (Dr. Faraday on\n\"Conservation of Force,\" in \"Correlation and Conservation of Force.\" (p.\n368). He also quotes with approbation the words of Newton, \"That gravity\nshould be innate, inherent, and essential to matter, is so great an\nabsurdity, that I believe no man who has in philosophic matters a\ncompetent faculty of thinking can ever fall into it\" p. 368). \"The\n'force of gravity' is an improper expression\" (p. 340). \"Forces are\ntransformable, indestructible, and, _in contradistinction from matter_,\nimponderable\" (p. 346). \"The first cause of things is Deity\" (Dr. Mayer,\nin \"Correlation and Conservation of Force,\" p. 341). \"Although the word\n_cause_ may be used in a secondary and subordinate sense, as meaning\nantecedent forces, yet in an abstract sense it is totally inapplicable;\nwe can not predicate of any physical agent that it is abstractedly the\ncause of another\" (p. 15). \"Causation is the _will_,\" \"creation is the\nact, of God\" Grove on \"Correlation of Physical Forces,\" (p. 199).\n\"Between gravity and motion it is impossible to establish the equation\nrequired for a rightly-conceived _causal_ relation\" (\"Correlation and\nConservation of Force,\" p. 253). See also Herschel's \"Outlines of\nAstronomy,\" p. 234.\n\nIt certainly must have required a wonderful effort of imagination on the\npart of Dr. Warren to transform \"weight\" and \"density,\" mere passive\naffections of matter, into self-activity, intelligence, thought, and\ndesign. Weight or density are merely relative terms. Supposing one\nparticle or mass of matter to exist alone, and there can be no\nattractive or gravitating force. There must be a cause of gravity which\nis distinct from matter.]\n\n3d. The validity of \"_the principle of unity_\" is also discredited by\nWatson. \"If, however, it were conceded that some glimmerings of this\ngreat truth, the existence of a First Cause, might, by induction, have\nbeen discovered, by what means could they have demonstrated to\nthemselves that the great collection of bodies which we call the world\nhad but _one_ Creator.\"[369]\n\n[Footnote 369: \"Institutes of Theology,\" vol. i. p. 275.]\n\nWe might answer directly, and at once, that the oneness or unity of God\nis necessarily contained in \"the very notion of a First Cause\"--a\n_first_ cause is not many causes, but _one_. By a First Cause we do not,\nhowever, understand the first of a numerical series, but an arche--a\nprinciple, itself unbeginning, which is the source of all beginning. Our\ncategorical answer, therefore, must be that the unity of God is a\nsublime deliverance of reason--God is one God. It is a first principle\nof reason that all differentiation and plurality supposes an incomposite\nunity, all diversity implies an indivisible identity. The sensuous\nperception of a plurality of parts supposes the rational idea of an\nabsolute unity, which has no parts, as its necessary correlative. For\nexample, extension is a congeries of indefinitesimal parts; the\ncontinuity of matter, as _empirically_ known by us, is never absolute.\nSpace is absolutely continuous, incapable of division into integral\nparts, illimitable, and, as _rationally_ known by us, an absolute unity.\nThe cognition of limited extension, which is the subject of quantitative\nmeasurement, involves the conception of unlimited space, which is the\nnegation of all plurality and complexity of parts. And so the cognition\nof a phenomenal universe in which we see only difference, plurality, and\nchange, implies the existence of a Being who is absolutely unchangeable,\nidentical, and one.\n\nThis law of thought lies at the basis of that universal desire of unity,\nand that universal effort to reduce all our knowledge to unity, which\nhas revealed itself in the history of philosophy, and also of inductive\nscience. \"Reason, intellect, nous, concatenating thoughts and objects\ninto system, and tending upward from particular facts to general laws,\nfrom general laws to universal principles, is never satisfied in its\nascent till it comprehends all laws in a single formula, and consummates\nall conditional knowledge in the unity of unconditional existence.\" \"The\nhistory of philosophy is only the history of this tendency, and\nphilosophers have borne ample testimony to its reality. 'The mind,' says\nAnaxagoras, 'only knows when it subdues its objects, when it reduces the\nmany to the one.' 'The end of philosophy,' says Plato, 'is the intuition\nof unity.' 'All knowledge,' say the Platonists, 'is the gathering up\ninto one, and the indivisible apprehension of this unity by the knowing\nmind.'\"[370]\n\n[Footnote 370: Hamilton's \"Metaphysics,\" vol. i. pp. 68, 69.]\n\nThis law has been the guiding principle of the Inductive Sciences, and\nhas led to some of its most important discoveries. The unity which has\nbeen attained in physical science is not, however, the absolute unity of\na material substratum, but a unity of _Will_ and of _Thought_. The late\ndiscovery of the monogenesis, reciprocal convertibility, and\nindestructibility of all Forces in nature, leads us upward towards the\nrecognition of one Omnipresent and Omnipotent Will, which, like a mighty\ntide, sweeps through the universe and effects all its changes. The\nuniversal prevalence of the same physical laws and numerical relations\nthroughout all space, and of the same archetypal forms and teleology of\norgans throughout all past time, reveals to us a Unity of Thought which\ngrasps the entire details of the universe in one comprehensive\nplan.[371] The positive _a priori_ intuitions of reason and the _a\nposteriori_ inductions of science equally attest _that God is one_.\n\n[Footnote 371: We refer with pleasure to the articles of Dr. Winchell,\nin the North-western Christian Advocate, in which the _a posteriori_\nproof of \"the Unity of God\" is forcibly exhibited, and take occasion to\nexpress the hope they will soon be presented to the public in a more\npermanent form.]\n\n4th. By denying that man has any intuitive cognitions of right and\nwrong, or any native and original feeling of obligation, Mr. Watson\ninvalidates \"the moral argument\" for the existence of a Righteous God.\n\n\"As far as man's reason has applied itself to the discovery of truth or\n_duty_ it has generally gone astray.\"[372] \"Questions of morals do not,\nfor the most part, lie level to the minds of the populace.\"[373] \"Their\nconclusions have no _authority_, and place them under no\n_obligation_.\"[374] And, indeed, man without a revelation \"is without\n_moral control_, without _principles of justice_, except such as may be\nslowly elaborated from those relations which concern the grosser\ninterests of life, without _conscience_, without hope or fear in another\nlife.\"[375]\n\n[Footnote 372: \"Institutes of Theology,\" vol. ii. p. 470.]\n\n[Footnote 373: Ibid., vol. i. p. 15.]\n\n[Footnote 374: Ibid., vol. i. p. 228.]\n\n[Footnote 375: Ibid., vol. ii. p. 271.]\n\nNow we shall not occupy our space in the elaboration of the proposition\nthat the universal consciousness of our race, as revealed in human\nhistory, languages, legislations, and sentiments, bears testimony to the\nfact that the ideas of right, duty, and responsibility are native to the\nhuman mind; we shall simply make our appeal to those Sacred Writings\nwhose verdict must be final with all theologians. That the fundamental\nprinciples of the moral law do exist, subjectively, in all human minds\nis distinctly affirmed by Paul, in a passage which deserves to be\nregarded as the chief corner-stone of moral science. \"The Gentiles\n(ephne, heathen), which have not the written law, do by the guidance of\nnature (reason or conscience) the works enjoined by the revealed law;\nthese, having no written law, are a law unto themselves; who show\nplainly the works of the law written on their hearts, their conscience\nbearing witness, and also their reasonings one with another, when they\naccuse, or else excuse, each other.\"[376] To deny this is to relegate\nthe heathen from all responsibility. For Mr. Watson admits \"that the\nwill of a superior is not in justice binding unless it be in some mode\nsufficiently declared.\" Now in the righteous adjudgments of revelation\nthe heathen are \"without excuse.\" The will of God must, therefore, be\n\"sufficiently declared\" to constitute them accountable. Who will presume\nto say that the shadowy, uncertain, variable, easily and unavoidably\ncorrupted medium of tradition running through forty muddy centuries is a\n\"sufficient declaration of the will of God?\" The law is \"written on the\nheart\" of every man, or all men are not accountable.\n\n[Footnote 376: Romans, ch. ii. ver. 14-15.]\n\nNow this \"law written within the heart\" immediately and naturally\nsuggests the idea of a Lawgiver who is over us. This felt presence of\nConscience, approving or condemning our conduct, suggests, as with the\nspeed of the lightning-flash, the notion of a Judge who will finally\ncall us to account. This \"accusing or excusing of each other,\" this\nrecognition of good or ill desert, points us to, and constrains us to\nrecognize, a future Retribution; so that some hope or fear of another\nlife has been in all ages a universal phenomenon of humanity.\n\nIt is affirmed, however, that whilst this capacity to know God may have\nbeen an original endowment of human nature, yet, in consequence of the\nfall, \"the understanding and reason are weakened by the deterioration of\nhis whole intellectual nature.\"[377] \"Without some degree of education,\nman is _wholly_ the creature of appetite. Labor, feasting, and sleeping\ndivide his time, and wholly occupy his thoughts.\"[378]\n\n[Footnote 377: \"Institutes of Theology,\" vol. i. p. 15.]\n\n[Footnote 378: Ibid., vol. i. p. 271.]\n\nWe reverently and believingly accept the teaching of Scripture as to the\ndepravity of man. We acknowledge that \"the understanding is darkened\" by\nsin. At the same time, we earnestly maintain that the Scriptures do not\nteach that the fundamental laws of mind, the first principles of reason,\nare utterly traversed and obliterated by sin, so that man is not able to\nrecognize the existence of God, and feel his obligation to Him. \"_Though\nthey_(the heathen) _knew God_ (dioti gnontes), they did not glorify him\nas God, neither were thankful, but became vain in their imagination, and\ntheir foolish hearts were darkened. They changed the truth of God into a\nlie, and worshipped and served the creature _more_ than the Creator.\"\n\"And as they did not _approve of holding God with acknowledgment_, God\ndelivered them over to an unapproving mind, to work those things which\nare not suitable.\" After drawing a fearful picture of the darkness and\ndepravity of the heathen, the Apostle adds, \"Who, _though they_ KNOW\n_the law of God_, that they who practise such things are worthy of\ndeath, not only do them, but even are well pleased with those who\npractise them.\"[379] The obvious and direct teaching of this passage is\nthat the heathen, in the midst of their depravity and idolatry, are not\nutterly ignorant of God; \"they _know_ God\"--\"they _know_ the law of God\n\"--\"they worship Him,\" though they worship the creature _more than_ Him.\nThey know God, and are unwilling to \"acknowledge God.\" \"They know the\nrighteousness of God,\" and are \"haters of God\" on account of his purity;\nand their worshipping of idols does not proceed from ignorance of God,\nfrom an intellectual inability to know God, but from \"corruption of\nheart,\" and a voluntary choice of, and a \"pleasure\" in, the sinful\npractices accompanying idol worship. Therefore, argues the Apostle, they\nare \"without excuse.\" The whole drift and aim of the argument of Paul\nis, not to show that the heathen were, by their depravity, incapacitated\nto know God, but that because they knew God and knew his righteous law,\ntherefore their depravity and licentiousness was \"inexcusable.\"\n\n[Footnote 379: Romans, ch. i. ver. 23-32.]\n\nWe conclude our review of opposing schools by the re-affirmation of our\nposition, _that God is cognizable by human reason._ The human mind,\nunder the guidance of necessary laws of thought, is able, from the facts\nof the universe, to affirm the existence of God, and to attain some\nvalid knowledge of his character and will. Every attempt to solve the\ngreat problem of existence, to offer an explanation of the phenomenal\nworld, or to explore the fundamental idea of reason, when fairly and\nfully conducted, has resulted in the recognition of a Supreme\n_Intelligence_, a personal _Mind_ and _Will_, as the ground, and reason,\nand cause of all existence. A survey of the history of Greek Philosophy\nwill abundantly sustain this position, and to this we shall, in\nsubsequent chapters, invite the reader's attention.\n\n\n\n\nCHAPTER VIII.\n\nTHE PHILOSOPHERS OF ATHENS.\n\nPRE-SOCRATIC SCHOOL.\n\nSENSATIONAL:\nTHALES--ANAXIMENES--HERACLITUS--ANAXIMANDER--LEUCIPPUS--DEMOCRITUS.\n\n\n      \"Then certain philosophers of the Epicureans and of the\n      Stoics encountered Paul.\"--Acts xvii. 18.\n\n      \"Plato affirms that this is the most just cause of the\n      creation of the world, that works which are good should be\n      wrought by the God who is good; whether he had read these\n      things in the Bible, or whether by his penetrating genius he\n      beheld _the invisible things of God as understood by the\n      things which are made_\"--ST. AUGUSTINE, \"De Civ. Dei,\" lib.\n      xi. ch. 21.\n\nOf all the monuments of the greatness of Athens which have survived the\nchanges and the wastes of time, the most perfect and the most enduring\nis her philosophy. The Propylaea, the Parthenon, and the Erechtheum,\nthose peerless gems of Grecian architecture, are now in ruins. The\nmagnificent sculpture of Phidias, which adorned the pediment, and outer\ncornice, and inner frieze of these temples, and the unrivalled statuary\nof gods and heroes which crowded the platform of the Acropolis, making\nit an earthly Olympus, are now no more, save a few broken fragments\nwhich have been carried to other lands, and, in their exile, tell the\nmournful story of the departed grandeur of their ancient home. The\nbrazen statue of Minerva, cast from the spoils of Marathon, which rose\nin giant grandeur above the buildings of the Acropolis, and the flashing\nof whose helmet plumes was seen by the mariner as soon as he had rounded\nthe Sunian promontory; and that other brazen Pallas, called, by\npre-eminence, \"the Beautiful;\" and the enormous Colossus of ivory and of\ngold, \"the Immortal Maid\"--the protecting goddess of the\nParthenon--these have perished. But whilst the fingers of time have\ncrumbled the Pentelic marble, and the glorious statuary has been broken\nto pieces by vandal hands, and the gold and brass have been melted in\nthe crucibles of needy monarchs and converted into vulgar money, the\nphilosophic _thought_ of Athens, which culminated in the dialectic of\nPlato, still survives. Not one of all the vessels, freighted with\nimmortal thought, which Plato launched upon the stream of time, has\nfoundered. And after the vast critical movement of European thought\nduring the past two centuries, in which all philosophic systems have\nbeen subjected to the severest scrutiny, the _method_ of Plato still\npreserves, if not its exclusive authority unquestioned, at least its\nintellectual pre-eminence unshaken. \"Platonism is immortal, because its\nprinciples are immortal in the human intellect and heart.\"[380]\n\n[Footnote 380: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n9.]\n\nPhilosophy is, then, the world-enduring monument of the greatness and\nthe glory of Athens. Whilst Greece will be forever memorable as \"the\ncountry of wisdom and of wise men,\" Athens will always be pre-eminently\nmemorable as the University of Greece. This was the home of Socrates,\nand Plato, and Aristotle--the three imperial names which, for twenty\ncenturies, reigned supreme in the world of philosophic thought. Here\nschools of philosophy were founded to which students were attracted from\nevery part of the civilized world, and by which an impulse and a\ndirection was given to human thought in every land and in every age.\nStanding on the Acropolis at Athens, and looking over the city and the\nopen country, the Apostle would see these _places_ which are inseparably\nassociated with the names of the men who have always been recognized as\nthe great teachers of the pagan world, and who have also exerted a\npowerful influence upon Christian minds of every age. \"In opposite\ndirections he would see the suburbs where Plato and Aristotle, the two\npupils of Socrates, held their illustrious schools. The streamless bed\nof the Ilissus passes between the Acropolis and Hymettus in a\nsouth-westerly direction, until it vanishes in the low ground which\nseparates the city from the Piraeus.\" Looking towards the upper part of\nthis channel, Paul would see gardens of plane-trees and thickets of\nangus-castus, \"with other torrent-loving shrubs of Greece.\" Near the\nbase of Lycabettus was a sacred inclosure which Pericles had ornamented\nwith fountains. Here stood a statue of Apollo Lycius, which gave the\nname to the _Lyceum_. Here, among the plane-trees, Aristotle _walked_,\nand, as he walked, taught his disciples. Hence the name Peripatetics\n(the Walkers), which has always designated the disciples of the\nStagirite philosopher.\n\nOn the opposite side of the city, the most beautiful of the Athenian\nsuburbs, we have the scene of Plato's teaching. Beyond the outer\nCeramicus, which was crowded with the sepulchres of those Athenians who\nhad fallen in battle, and were buried at the public expense, the eye of\nPaul would rest on the favored stream of the Cephisus, flowing towards\nthe west. On the banks of this stream the _Academy_ was situated. A\nwall, built at great expense by Hipparchus, surrounded it, and Cimon\nplanted long avenues of trees and erected fountains. Beneath the\nplane-trees which shaded the numerous walks there assembled the\nmaster-spirits of the age. This was the favorite resort of poets and\nphilosophers. Here the divine spirit of Plato poured forth its sublimest\nspeculations in streams of matchless eloquence; and here he founded a\nschool which was destined to exert a powerful and perennial influence on\nhuman minds and hearts in all coming time.\n\nLooking down from the Acropolis upon the Agora, Paul would distinguish a\ncloister or colonnade. This is the Stoa Poecile, or \"Painted Porch,\" so\ncalled because its walls were decorated with fresco paintings of the\nlegendary wars of Greece, and the more glorious struggle at Marathon. It\nwas here that Zeno first opened that celebrated school which thence\nreceived the name of _Stoic_. The site of the _garden_ where Epicurus\ntaught is now unknown. It was no doubt within the city walls, and not\nfar distant from the Agora. It was well known in the time of Cicero, who\nvisited Athens as a student little more than a century before the\nApostle. It could not have been forgotten in the time of Paul. In this\n\"tranquil garden,\" in the society of his friends, Epicurus passed a life\nof speculation and of pleasure. His disciples were called, after him,\nthe Epicureans.[381]\n\n[Footnote 381: See Conybeare and Howson's \"Life and Epistles of St.\nPaul,\" vol. i., Lewes's \"Biographical History of Philosophy;\" and\nEncyclopaedia Britannica, article, \"Athens,\" from whence our materials\nfor the description of these \"places\" are mainly derived.]\n\nHere, then, in Athens the Apostle was brought into immediate contact\nwith all the phases of philosophic thought which had appeared in the\nancient world. \"Amongst those who sauntered beneath the cool shadows of\nthe plane-trees in the Agora, and gathered in knots under the porticoes,\neagerly discussing the questions of the day, were the philosophers, in\nthe garb of their several sects, ready for any new question on which\nthey might exercise their subtlety or display their rhetoric.\" If there\nwere any in that motley group who cherished the principles and retained\nthe spirit of the true Platonic school, we may presume they felt an\ninward intellectual sympathy with the doctrine enounced by Paul. With\nPlato, \"philosophy was only another name for _religion_: philosophy is\nthe love of perfect Wisdom; perfect Wisdom and perfect Goodness are\nidentical: the perfect Good is God himself; philosophy is the love of\nGod.\"[382] He confessed the need of divine assistance to attain \"the\ngood,\" and of divine interposition to deliver men from moral ruin.[383]\nLike Socrates, he longed for a supernatural--a divine light to guide\nhim, and he acknowledged his need thereof continually.[384] He was one\nof those who, in heathen lands, waited for \"the desire of nations;\" and,\nhad he lived in Christian times, no doubt his \"spirit of faith\" would\nhave joyfully \"embraced the Saviour in all the completeness of his\nrevelation and advent.\"[385] And in so far as the spirit of Plato\nsurvived among his disciples, we may be sure they were not among the\nnumber who \"mocked,\" and ridiculed, and opposed the \"new doctrine\"\nproclaimed by Paul. It was \"the philosophers of the Epicureans and of\nthe Stoics who _encountered_ Paul.\" The leading tenets of both these\nsects were diametrically opposed to the doctrines of Christianity. The\nruling spirit of each was alien from the spirit of Christ. The haughty\n_pride_ of the Stoic, the Epicurean abandonment to _pleasure_, placed\nthem in direct antagonism to him who proclaimed the crucified and risen\nChrist to be \"_the wisdom_ of God.\"\n\n[Footnote 382: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n61.]\n\n[Footnote 383: \"Republic,\" bk. vi. ch. vi. vii.]\n\n[Footnote 384: Butler's \"Lectures,\" vol. i. p. 362.]\n\n[Footnote 385: Wheedon on \"The Will,\" p. 352; also Butler's \"Lectures,\"\nvol. ii. p. 252]\n\nIf, however, we would justly appreciate the relation of pagan philosophy\nto Christian truth, we must note that, when Paul arrived in Athens, the\nage of Athenian glory had passed away. Not only had her national\ngreatness waned, and her national spirit degenerated, but her\nintellectual power exhibited unmistakable signs of exhaustion, and\nweakness, and decay. If philosophy had borne any fruit, of course that\nfruit remained. If, in the palmy days of Athenian greatness, any field\nof human inquiry had been successfully explored; if human reason had\nachieved any conquests; if any thing true and good had been obtained,\nthat must endure as an heir-loom for all coming time; and if those\ncenturies of agonizing wrestlings with nature, and of ceaseless\nquestioning of the human heart, had yielded no results, then, at least,\nthe _lesson_ of their failure and defeat remained for the instruction of\nfuture generations. Either the problems they sought to solve were proved\nto be insoluble, or their methods of solution were found to be\ninadequate; for here the mightiest minds had grappled with the great\nproblems of being and of destiny. Here vigorous intellects had struggled\nto pierce the darkness which hangs alike over the beginning and the end\nof human existence. Here profoundly earnest men had questioned nature,\nreason, antiquity, oracles, in the hope they might learn something of\nthat invisible world of _real_ being which they instinctively felt must\nlie beneath the world of fleeting forms and ever-changing appearances.\nHere philosophy had directed her course towards every point in the\ncompass of thought, and touched every _accessible_ point. The sun of\nhuman reason had reached its zenith, and illuminated every field that\nlay within the reach of human ken. And this sublime era of Greek\nphilosophy is of inestimable value to us who live in Christian times,\nbecause _it is an exhaustive effort of human reason to solve the problem\nof being_, and in its history we have a record of the power and weakness\nof the human mind, at once on the grandest scale and in the fairest\ncharacters.[386]\n\n[Footnote 386: See article \"Philosophy,\" in Smith's \"Dictionary of the\nBible.\"]\n\nThese preliminary considerations will have prepared the way for, and\nawakened in our minds a profound interest in, the inquiry--1st. What\npermanent _results_ has Greek philosophy bequeathed to the world? 2d. In\nwhat manner did Greek philosophy fulfill for Christianity a\n_propoedeutic_ office?\n\nIt will at once be obvious, even to those who are least conversant with\nour theme, that it would be fruitless to attempt the answer to these\nimportant questions before we have made a careful survey of the entire\nhistory of philosophic thought in Greece. We must have a clear and\ndefinite conception of the problems they sought to solve, and we must\ncomprehend their methods of inquiry, before we can hope to appreciate\nthe results they reached, or determine whether they did arrive at any\ndefinite and valuable conclusions. It will, therefore, devolve upon us\nto present a brief and yet comprehensive epitome of the history of\nGrecian speculative thought.\n\n\"_Philosophy_,\" says Cousin, \"_is reflection_, and nothing else than\nreflection, in a vast form\"--\"Reflection elevated to the rank and\nauthority of a _method_.\" It is the mind looking back upon its own\nsensations, perceptions, cognitions, ideas, and from thence to the\n_causes_ of these sensations, cognitions, and ideas. It is thought\npassing beyond the simple perceptions of things, beyond the mere\nspontaneous operations of the mind in the cognition of things to seek\nthe _ground_, and _reason_, and _law_ of things. It is the effort of\nreason to solve the great problem of \"Being and Becoming,\" of appearance\nand reality, of the changeful and the permanent. Beneath the endless\ndiversity of the universe, of existence and action, there must be a\nprinciple of unity; below all fleeting appearances there must be a\npermanent substance; beyond this everlasting flow and change, this\nbeginning and ending of finite existence, there must be an _eternal\nbeing_, the source and cause of all we see and know, _What is that\nprinciple of unity, that permanent substance_, or principle, or being?\n\nThis fundamental question has assumed three separate forms or aspects in\nthe history of philosophy. These forms have been determined by the\nobjective phenomena which most immediately arrested and engaged the\nattention of men. If external nature has been the chief object of\nattention, then the problem of philosophy has been, _What is the\narche--_the beginning; what are the first principles_--the elements from\nwhich, the ideas or laws according to which, the efficient cause or\nenergy by which, and the reason or end for which the universe exists?_\nDuring this period reflective thought was a PHILOSOPHY OF NATURE. If the\nphenomena of mind--the opinions, beliefs, judgments of men--are the\nchief object of attention, then the problem of philosophy has been,\n_What are the fundamental Ideas which are unchangeable and permanent\namid all the diversities of human opinions, connecting appearance with\nreality, and constituting a ground of certain knowledge or absolute\ntruth?_ Reflective thought is now a PHILOSOPHY OF IDEAS. Then, lastly,\nif the practical activities of life and the means of well-being be the\ngrand object of attention, then the problem of philosophy has been,\n_What is the ultimate standard by which, amid all the diversities of\nhuman conduct, we may determine what is right and good in individual,\nsocial, and political life?_ And now reflective thought is a PHILOSOPHY\nOF LIFE. These are the grand problems with which philosophy has grappled\never since the dawn of reflection. They all appear in Greek philosophy,\nand have a marked chronology. As systems they succeed each other, just\nas rigorously as the phenomena of Greek civilization.\n\nThe Greek schools of philosophy have been classified from various points\nof view. In view of their geographical relations, they have been divided\ninto the _Ionian_, the _Italian_, the _Eleatic_, the _Athenian_, and the\n_Alexandrian_. In view of their prevailing spirit and tendency, they\nhave been classified by Cousin as the Sensational, the Idealistic, the\nSkeptical, and the Mystical. The most natural and obvious method is that\nwhich (regarding Socrates as the father of Greek philosophy in the\ntruest sense) arranges all schools from the Socratic stand point, and\ntherefore in the chronological order of development:\n\n     I. THE PRE-SOCRATIC SCHOOLS.\n     II. THE SOCRATIC SCHOOLS.\n     III. THE POST-SOCRATIC SCHOOLS.\n\nThe history of philosophy is thus divided into three grand epochs. The\nfirst reaching from Thales to the time of Socrates (B.C. 639-469): the\nsecond from the birth of Socrates to the death of Aristotle (B.C.\n469-322); the third from the death of Aristotle to the Christian era\n(B.C. 322, A.D. 1). Greek philosophy during the first period was almost\nexclusively a philosophy of nature; during the second period, a\nphilosophy of mind; during the last period, a philosophy of life.\nNature, man, and society complete the circle of thought. Successive\nsystems, of course, overlap each other, both in the order of time and as\nsubjects of human speculation; and the results of one epoch of thought\nare transmitted to and appropriated by another; but, in a general sense,\nthe order of succession has been very much as here indicated. Setting\naside minor schools and merely incidental discussions, and fixing our\nattention on the general aspects of each historic period, we shall\ndiscover that the first period was eminently _Physical_, the second\n_Psychological_, the last _Ethical_. Every stage of progress which\nreason, on _a priori_ grounds, would suggest as the natural order of\nthought, or of which the development of an individual mind would furnish\nan analogy, had a corresponding realization in the development of\nGrecian thought from the time of Thales to the Christian era. \"Thought,\"\nsays Cousin, \"in the first trial of its strength is drawn without.\" The\nfirst object which engages the attention of the child is the outer\nworld. He asks the \"_how_\" and \"_why_\" of all he sees. His reason urges\nhim to seek an explanation of the universe. So it was in the _childhood_\nof philosophy. The first essays of human thought were, almost without\nexception, discourses peri physeos (De rerum natura), of the nature of\nthings. Then the rebound of baffled reason from the impenetrable\nbulwarks of the universe drove the mind back upon itself. If the youth\ncan not interpret nature, he can at least \"know himself,\" and find\nwithin himself the ground and reason of all existence. There are\n\"_ideas_\" in the human mind which are copies of those \"_archetypal\nideas_\" which dwell in the Creative Mind, and after which the universe\nwas built. If by \"analysis\" and \"definition\" these universal notions can\nbe distinguished from that which is particular and contingent in the\naggregate of human knowledge, then so much of eternal truth has been\nattained. The achievements of philosophic thought in this direction,\nduring the Socratic age, have marked it as the most brilliant period in\nthe history of philosophy--the period of its _youthful_ vigor. Deeply\nimmersed in the practical concerns and conflicts of public life,\n_manhood_ is mainly occupied with questions of personal duty, and\nindividual and social well-being. And so, during the hopeless turmoil of\ncivil disturbance which marked the decline of national greatness in\nGrecian history, philosophy was chiefly occupied with questions of\npersonal interest and personal happiness. The poetic enthusiasm with\nwhich a nobler age had longed for _truth_, and sought it as the highest\ngood, has all disappeared, and now one sect seeks refuge from the storms\nand agitations of the age in Stoical indifference, the other in\nEpicurean effeminacy.\n\nIf now we have succeeded in presenting the real problem of philosophy,\nit will at once be obvious that the inquiry was not, in any proper\nsense, _theological_. Speculative thought, during the period we have\nmarked as the era of Greek philosophy, was not an inquiry concerning the\nexistence or nature of God, or concerning the relations of man to God,\nor the duties which man owes to God. These questions were all remitted\nto the _theologian_. There was a clear line of demarkation separating\nthe domains of religion and philosophy. Religion rested solely on\nauthority, and appealed to the instinctive faith of the human heart. She\npermitted no encroachment upon her settled usages, and no questioning of\nher ancient beliefs. Philosophy rested on reason alone. It was an\nindependent effort of thought to interpret nature, and attain the\nfundamental grounds of human knowledge--to find an arche--a first\nprinciple, which, being assumed, should furnish a rational explanation\nof all existence. If philosophy reach the conclusion that the arche was\nwater, or air, or fire, or a chaotic mixture of all the elements or\natoms, extended and self-moved, or monads, or to pan, or uncreated mind,\nand that conclusion harmonized with the ancient standards of religious\nfaith--well; if not, philosophy must present some method of\nconciliation. The conflicts of faith and reason; the stragglings of\ntraditional authority to maintain supremacy; the accommodations and\nconciliations attempted in those primitive times, would furnish a\nchapter of peculiar interest, could it now be written.\n\nThe poets who appeared in the dim twilight of Grecian\ncivilization--Orpheus, Musaeus, Homer, Hesiod--seem to have occupied the\nsame relation to the popular mind in Greece which the Bible now sustains\nto Christian communities.[387] Not that we regard them as standing on\nequal ground of authority, or in any sense a revelation. But, in the eye\nof the wondering Greek, they were invested with the highest sacredness\nand the supremest authority. The high poetic inspiration which pervaded\nthem was a supernatural gift. Their sublime utterances were accepted as\nproceeding from a divine afflatus. They were the product of an age in\nwhich it was believed by all that the gods assumed a human form,[388]\nand held a real intercourse with gifted men. This universal faith is\nregarded by some as being a relic of still more distant times, a faint\nremembrance of the glory of patriarchal days. The more natural opinion\nis, that it was begotten of that universal longing of the human heart\nfor some knowledge of that unseen world of real being, which man\ninstinctively felt must lie beyond the world of fleeting change and\ndelusive appearances. It was a prolepsis of the soul, reaching upward\ntowards its source and goal. The poet felt within him some native\naffinities therewith, and longed for some stirring breath of heaven to\nsweep the harp-strings of the soul. He invoked the inspiration of the\nGoddess of Song, and waited for, no doubt believed in, some \"deific\nimpulse\" descending on him. And the people eagerly accepted his\nutterance as the teaching of the gods. They were too eager for some\nknowledge from that unseen world to question their credentials. Orpheus,\nHesiod, Homer, were the theologoi--the theologians of that age.[389]\n\n[Footnote 387: \"Homer was, in a certain sense, the Bible of the\nGreeks.\"--Whewell, \"Platonic Dialogues,\" p. 283.]\n\n[Footnote 388: The universality of this belief is asserted by Cicero:\n\"Vetus opinio est, jam usque ab heroicis ducta temporibus, eaque et\npopuli Romani et omnium gentium firmata consensu, versari quandem inter\nhomines divinationem.\"--Cicero, \"De Divin.\" bk. i. ch. i.]\n\n[Footnote 389: Cicero.]\n\nThese ancient poems, then, were the public documents of the religion of\nGreece--the repositories of the national faith. And it is deserving of\nespecial note that the philosopher was just as anxious to sustain his\nspeculations by quoting the high traditional authority of the ancient\ntheologian, as the propounder of modern novelties is to sustain his\nnotions by the authority of the Sacred Scriptures. Numerous examples of\nthis solicitude will recur at once to the remembrance of the student of\nPlato. All encroachments of philosophy upon the domains of religion were\nwatched as jealously in Athens in the sixth century before Christ, as\nthe encroachments of science upon the fields of theology were watched in\nRome in the seventeenth century after Christ. The court of the Areopagus\nwas as earnest, though not as fanatical and cruel, in the defense of the\nancient faith, as the court of the Inquisition was in the defense of the\ndogmas of the Romish Church. The people, also, as \"the sacred wars\" of\nGreece attest, were ready quickly to repel every assault upon the\nmajesty of their religion. And so philosophy even had its martyrs. The\ntears of Pericles were needed to save Aspasia, because she was suspected\nof philosophy. But neither his eloquence nor his tears could save his\nfriend Anaxagoras, and he was ostracized. Aristotle had the greatest\ndifficulty to save his life. And Plato was twice imprisoned, and once\nsold into slavery.[390]\n\n[Footnote 390: Cousin's \"Lectures on the History of Philosophy,\" vol. i.\np. 305.]\n\nIt is unnecessary that we should, in this place, again attempt the\ndelineation of the theological opinions of the earlier periods of\nGrecian civilization. That the ancient Greeks believed in _one Supreme\nGod_ has been conclusively proved by Cudworth. The argument of his\nfourth chapter is incontrovertible.[391] However great the number of\n\"generated gods\" who crowded the Olympus, and composed the ghostly array\nof Greek mythology, they were all subordinate agents, \"demiurges,\"\nemployed in the framing of the world and all material things, or else\nthe ministers of the moral and providential government of the eis Theos\nagentos--the one uncreated God. Beneath, or beyond the whole system of\npagan polytheism, we recognize a faith in an _Uncreated Mind_, the\nSource of all the intelligence, and order, and harmony which pervades\nthe universe the Fountain of law and justice; the Ruler of the world;\nthe Avenger of injured innocence; and the final Judge of men. The\nimmortality of the soul and a state of future retribution were necessary\ncorollaries of this sublime faith. This primitive theology was\nunquestionably the people's faith; the faith, also, of the philosopher,\nin his inmost heart, however far he might wander in speculative thought.\nThe instinctive feeling of the human heart, the spontaneous intuitions\nof the human reason, have led man, in every age, to recognize a God. It\nis within the fields of speculative thought that skepticism has had its\nbirth. Any thing like atheism has only made its appearance amid the\nefforts of human reason to explain the universe. The native sentiments\nof the heart and the spontaneous movements of the reason have always\nbeen towards faith, that is, towards \"a religious movement of the\nsoul.\"[392] Unbridled speculative thought, which turns towards the outer\nworld alone, and disregards \"the voices of the soul,\" tends towards\n_doubt_ and irreligion. But, as Cousin has said, \"a complete\nextravagance, a total delusion (except in case of real derangement), is\nimpossible.\" \"Beneath reflection there is still spontaneity, when the\nscholar has denied the existence of a God; listen to the man,\ninterrogate him unawares, and you will see that all his words betray the\nidea of a God, and that faith in a God is, without his recognition, at\nthe bottom of his heart.\"[393]\n\n[Footnote 391: \"Intellectual System of the Universe;\" see also ch. iii.,\n\"On the Religion of the Athenians.\"]\n\n[Footnote 392: Cousin's \"Hist. of Philos.,\" vol. i. p. 22.]\n\n[Footnote 393: Id., ib., vol. i. p. 137.]\n\nLet us not, therefore, be too hasty in representing the early\nphilosophers as destitute of the idea of a God, because in the imperfect\nand fragmentary representations which are given us of the philosophical\nopinions of Thales, and Anaximenes, and Heraclitus, and Diogenes of\nApollonia, we find no explicit allusions to the _Uncreated Mind_ as the\nfirst principle and cause of all. A few sentences will comprehend the\nwhole of what remains of the opinions of the earliest philosophers, and\nthese were transmitted for ages by _oral_ tradition. To Plato and\nAristotle we are chiefly indebted for a stereotype of those scattered,\nfragmentary sentences which came to their hands through the dim and\ndistorting medium of more than two centuries. Surely no one imagines\nthese few sentences contain and sum up the results of a lifetime of\nearnest thought, or represent all the opinions and beliefs of the\nearliest philosophers! And should we find therein no recognition of a\npersonal God, would it not be most unfair and illogical to assert that\nthey were utterly ignorant of a God, or wickedly denied his being? If\nthey say \"there is no God,\" then they are foolish Atheists; if they are\nsilent on that subject, we have a right to assume they were Theists, for\nit is most natural to believe in God. And yet it has been quite\ncustomary for Christian teachers, after the manner of some Patristic\nwriters, to deny to those early sages the smallest glimpse of underived\nand independent knowledge of a Divine Being, in their zeal to assert for\nthe Sacred Scriptures the exclusive prerogative of revealing Him.\n\nNow in regard to the theological opinions of the Greek philosophers, we\nshall venture this general _lemma_--_the majority of them recognized an\n\"incorporeal substance\"_[394]_ an uncreated Intelligence, an ordering,\ngoverning Mind_. Leucippus, Democritus, and Epicurus, who were\nMaterialists, are perhaps the only exceptions. Many of them were\nPantheists, in the higher form of Pantheism, which, though it associates\nthe universe with its framer and mover, still makes \"the moving\nprinciple\" superior to that which is moved. The world was a living\norganism,\n\n     \"Whose body nature is, and God the soul.\"\n\nUnquestionably most on them recognized the existence of _two_ first\nprinciples, substances essentially distinct, which had co-existed from\neternity--an incorporeal Deity and matter.[395] We grant that the free\nproduction of a universe by a creative fiat--the calling of matter into\nbeing by a simple act of omnipotence--is not elementary to human reason.\nThe famous physical axiom of antiquity, \"_De nihilo nihil, in nihilum\nposse reverti\"_ under one aspect, may be regarded as the expression of\nthe universal consciousness of a mental inability to conceive a creation\nout of nothing, or an annihilation.[396] \"We can not conceive, either,\non the one hand, nothing becoming something, or something becoming\nnothing, on the other hand. When God is said to create the universe out\nof nothing, we think this by supposing that he evolves the universe out\nof himself; and in like manner, we conceive annihilation only by\nconceiving the Creator to withdraw his creation from actuality into\npower.\"[397] \"It is by _faith_ we understand the worlds were framed by\nthe _word of God_, so that things which are were not made from things\nwhich do appear\"--that is, from pre-existent matter.\n\n[Footnote 394: \"Ousian asomaton.\"--Plato.]\n\n[Footnote 395: Cudworth's \"Intellectual System,\" vol. i. p. 269.]\n\n[Footnote 396: Mansell's \"Limits of Religious Thought,\" p. 100.]\n\n[Footnote 397: Sir William Hamilton's \"Discussions on Philosophy,\" p.\n575.]\n\nThose writers[398] are, therefore, clearly in error who assert that the\nearliest question of Greek philosophy was, What is God? and that various\nand discordant answers were given, Thales saying, water is God,\nAnaximenes, air; Heraclitus, fire; Pythagoras, numbers; and so on. The\nidea of God is a native intuition of the mind. It springs up\nspontaneously from the depths of the human soul. The human mind\nnaturally recognizes God as an uncreated Mind, and recognizes itself as\n\"the offspring of God.\" And, therefore, it is simply impossible for it\nto acknowledge water, or air, or fire, or any material thing to be its\nGod. Now they who reject this fundamental principle evidently\nmisapprehend the real problem of early Grecian philosophic thought. The\nexternal world, the material universe, was the first object of their\ninquiry, and the method of their inquiry was, at the first stage, purely\nphysical. Every object of sense had a beginning and an end; it rose out\nof something, and it fell back into something. Beneath this ceaseless\nflow and change there must be some permanent principle. What is that\nstoicheon--that first element? The changes in the universe seem to obey\nsome principle of law--they have an orderly succession. What is that\nmorphe--that form, or ideal, or archetype, proper to each thing, and\naccording to which all things are produced? These changes must be\nproduced by some efficient cause, some power or being which is itself\nimmobile, and permanent, and eternal, and adequate to their production.\nWhat is that arche tes kineseos--that first principle of movement Then,\nlastly, there must be an end for which all things exist--a good reason\nwhy things are as they are, and not otherwise. What is that to ou eneken\nkai to agathon--that reason and good of all things? Now these are all\narchai or first principles of the universe. \"Common to all first\nprinciples,\" says Aristotle, \"is the being, the original, from which a\nthing is, or is produced, or is known.\"[399] First principles,\ntherefore, include both elements and causes, and, under certain aspects,\nelements are also causes, in so far as they are that without which a\nthing can not be produced. Hence that highest generalization by\nAristotle of all first principles; as--1. The Material Cause; 2. The\nFormal Cause; 3. The Efficient Cause; 4. The Final Cause. The grand\nsubject of inquiry in ancient philosophy was not alone what is the final\n_element_ from which all things have been produced? nor yet what is the\n_efficient cause_ of the movement and the order of the universe? _but\nwhat are those First Principles which, being assumed, shall furnish a\nrational explanation of all phenomena, of all becoming?_\n\n[Footnote 398: As the writer of the article \"Attica,\" in the\nEncyclopaedia Britannica.]\n\n[Footnote 399: \"Metaphysics,\" bk. iv. ch. i. p. 112 (Bohn's edition).]\n\nSo much being premised, we proceed to consider the efforts and the\nresults of philosophic thought in\n\n\n\nTHE PRE-SOCRATIC SCHOOLS.\n\n\"The first act in the drama of Grecian speculation was performed on the\nvaried theatre of the Grecian colonies--Asiatic, insular, and Italian,\nverging at length (in Anaxagoras) towards Athens.\" During the progress\nof this drama two distinct schools of philosophy were developed, having\ndistinct geographical provinces, one on the east, the other on the west,\nof the peninsula of Greece, and deriving their names from the localities\nin which they flourished. The earliest was the _Ionian;_ the latter was\nthe _Italian_ school.\n\nIt would be extremely difficult, at this remote period, to estimate the\ninfluence which geographical conditions and ethnical relations exerted\nin determining the course of philosophic thought in these schools.\nUnquestionably those conditions contributed somewhat towards fixing\ntheir individuality. At the same time, it must be granted that the\ndistinction in these two schools of philosophy is of a deeper character\nthan can be represented or explained by geographical surroundings; it is\na distinction reaching to the very foundation of their habits of\nthought. These schools represent two distinct aspects of philosophic\nthought, two distinct methods in which the human mind has essayed to\nsolve the problem of the universe.\n\nThe ante-Socratic schools were chiefly occupied with the study of\nexternal nature. \"Greek philosophy was, at its first appearance, a\nphilosophy of nature.\" It was an effort of the reason to reach a \"first\nprinciple\" which should explain the universe. This early attempt was\npurely speculative. It sought to interpret all phenomena by\n_hypotheses_, that is, by suppositions, more or less plausible,\nsuggested by physical analogies or by _a priori_ rational conceptions.\n\nNow there are two distinct aspects under which nature presents itself to\nthe observant mind. The first and most obvious is the _simple phenomena_\nas perceived by the senses. The second is the _relations_ of\n_phenomena_, cognized by the reason alone. Let phenomena, which are\nindeed the first objects of perception, continue to be the chief and\nalmost exclusive object of thought, and philosophy is on the highway of\npure physics. On the other hand, instead of stopping at phenomena, let\ntheir relations become the sole object of thought, and philosophy is now\non the road of purely mathematical or metaphysical abstraction. Thus two\nschools of philosophy are developed, the one SENSATIONAL, the other\nIDEALIST. Now these, it will be found, are the leading and\ncharacteristic tendencies of the two grand divisions of the pre-Socratic\nschools; the Ionian is _sensational_, the Italian is _idealist_.\n\nThese two schools have again been the subject of a further subdivision\nbased upon diverse habits of thought. The Ionian school sought to\nexplain the universe by _physical analogies._ Of these there are two\nclear and obvious divisions--analogies suggested by living organisms,\nand analogies suggested by mechanical arrangements. One class of\nphilosophers in the Ionian school laid hold on the first analogy. They\nregarded the world as a living being, spontaneously evolving itself--a\nvital organism whose successive developments and transformations\nconstitute all visible phenomena. A second class laid hold on the\nanalogy suggested by mechanical arrangements. For them the universe was\na grand superstructure, built up from elemental particles, arranged and\nunited by some ab-extra power or force, or else aggregated by some\ninherent mutual affinity. Thus we have two sects of the Ionian school;\nthe first, _Dynamical_ or vital; the second, _Mechanical_.[400]\n\n[Footnote 400: Ritter's \"Ancient Philosophy,\" vol. i. pp. 191, 192.]\n\nThe Italian school sought to explain the universe by rational\nconceptions and _a priori_ ideas. Now to those who seek, by simple\nreflection, to investigate the relations of the external world this\nmarked distinction will present itself: some are relations _between_\nsensible phenomena--relations of time, of place, of number, of\nproportion, and of harmony; others are relations _of_ phenomena to\nessential being--relations of qualities to substance, of becoming to\nbeing, of the finite to the infinite. The former constituted the field\nof Pythagorean the latter of Eleatic contemplation. The Pythagoreans\nsought to explain the universe by numbers, forms, and harmonies; the\nEleatics by the _a priori_ ideas of unity, substance, Being _in se_, the\nInfinite. Thus were constituted a _Mathematical_ and a _Metaphysical_\nsect in the Italian school. The pre-Socratic schools may, therefore, be\ntabulated in the following order:\n\n     I. IONIAN (Sensational), (1.) PHYSICAL {Dynamical or Vital.\n                                            {Mechanical.\n\n     II. Italian (Idealist), {(2.) MATHEMATICAL Pythagoreans.\n                             {(3.) METAPHYSICAL Eleatics.\n\nI. _The Ionian or Physical School._--We have premised that the\nphilosophers of this school attempted the explanation of the universe by\nphysical analogies.\n\nOne class of these early speculators, the _Dynamical_, or vital\ntheorists, proceeded on the supposition of a living energy infolded in\nnature, which in its spontaneous development continuously undergoes\nalteration both of quality and form. This imperfect analogy is the first\nhypothesis of childhood. The child personifies the stone that hurts him,\nand his first impulse is to resent the injury as though he imagined it\nto be endowed with consciousness, and to be acting with design. The\nchildhood of superstition (whose genius is multiplicity) personifies\neach individual existence--a rude Fetichism, which imagines a\nsupernatural power and presence enshrined in every object of nature, in\nevery plant, and stock, and stone. The childhood of philosophy (whose\ngenius is unity) personifies the universe. It regards the earth as one\nvast organism, animated by one soul, and this soul of the world as a\n\"created god.\"[401] The first efforts of philosophy were, therefore,\nsimply an attempt to explain the universe in harmony with the popular\ntheological beliefs. The cosmogonies of the early speculators in the\nIonian school were an elaboration of the ancient theogonies, but still\nan elaboration conducted under the guidance of that law of thought which\nconstrains man to seek for _unity_, and reduce the many to the one.\n\nTherefore, in attempting to construct a theory of the universe they\ncommenced by postulating an arche--a first principle or element out of\nwhich, by a _vital_ process, all else should be produced. \"Accordingly,\nwhatever seemed the most subtle or pliable, as well as _universal_\nelement in the mass of the visible world, was marked as the seminal\nprinciple whose successive developments and transformations produced all\nthe rest.\"[402] With this seminal principle the living, _animating_\nprinciple seems to have been associated--in some instances perhaps\nconfounded, and in most instances called by the same name. And having\npursued this analogy so far, we shall find the _most decided and\nconclusive_ evidence of a tendency to regard the soul of man as similar,\nin its nature, to the soul which animates the world.\n\n[Footnote 401: Plato's \"Laws,\" bk. x. ch. i.; \"Timaeus,\" ch. xii.]\n\n[Footnote 402: Butler's \"Lectures on Ancient Philosophy,\" vol. 1. p.\n292.]\n\n_Thales of Miletus_(B.C. 636-542) was the first to lead the way in the\nperilous inquiry after an arche, or first principle, which should\nfurnish a rational explanation of the universe. Following, as it would\nseem, the genealogy of Hesiod, he supposed _water_ to be the primal\nelement out of which all material things were produced. Aristotle\nsupposes he was impressed with this idea from observing that all things\nare nourished by moisture; warmth itself, he declared, proceeded from\nmoisture; the seeds of all things are moist; water, when condensed,\nbecomes earth.\n\nThus convinced of the universal presence of water, he declared it to be\nthe first principle of things.[403]\n\nAnd now, from this brief statement of the Thalean physics, are we to\nconclude that he recognized only a _material_ cause of the universe?\nSuch is the impression we receive from the reading of the First Book of\nAristotle's Metaphysics. His evident purpose is to prove that the first\nphilosophers of the Ionian school did not recognize an _efficient_\ncause. In his opinion, they were decidedly materialistic. Now to\nquestion the authority of Aristotle may appear to many an act of\npresumption. But Aristotle was not infallible; and nothing is more\ncertain than that in more than one instance he does great injustice to\nhis predecessors.[404] To him, unquestionably, belongs the honor of\nhaving made a complete and exhaustive classification of causes, but\nthere certainly does appear something more than vanity in the assumption\nthat he, of all the Greek philosophers, was the only one who recognized\nthem all. His sagacious classification was simply a resume of the labors\nof his predecessors. His \"principles\" or \"causes\" were incipient in the\nthought of the first speculators in philosophy. Their accurate\ndefinition and clearer presentation was the work of ages of analytic\nthought. The phrases \"efficient,\" \"formal,\" \"final\" cause, are, we\ngrant, peculiar to Aristotle; the ideas were equally the possession of\nhis predecessors.\n\n[Footnote 403: Aristotle's \"Metaphysics,\" bk. i. ch. iii.]\n\n[Footnote 404: Lewes's \"Biographical History of Philosophy,\" p. 77;\nCousin's \"The True, the Beautiful, and the Good,\" p. 77.]\n\nThe evidence, we think, is conclusive that, with this primal element\n(water), Thales associated a formative principle of motion; to the\n\"material\" he added the \"efficient\" cause. A strong presumption in favor\nof this opinion is grounded on the psychological views of Thales. The\nauthor of \"De Placitis Philosophorum\" associates him with Pythagoras and\nPlato, in teaching that the soul is incorporeal, making it naturally\nself-active, and an intelligent substance.[405] And it is admitted by\nAristotle (rather unwillingly, we grant, but his testimony is all the\nmore valuable on that account) that, in his time, the opinion that the\nsoul is a principle, aeikineton--ever moving, or essentially\nself-active, was currently ascribed to Thales. \"If we may rely on the\nnotices of Thales, he too would seem to have conceived the soul as a\n_moving principle_.\"[406] Extending this idea, that the soul is a moving\nprinciple, he held that all motion in the universe was due to the\npresence of a living soul. \"He is reported to have said that the\nloadstone possessed a soul because it could move iron.\"[407] And he\ntaught that \"the world itself is _animated_, and full of gods.\"[408]\n\"Some think that _soul_ and _life_ is mingled with the whole universe;\nand thence, perhaps, was that [opinion] of Thales that all things are\nfull of gods,\"[409] portions, as Aristotle said, of the universal soul.\nThese views are quite in harmony with the theology which makes the Deity\nthe moving energy of the universe--the energy which wrought the\nsuccessive transformations of the primitive aqueous element. They also\nfurnish a strong corroboration of the positive statement of\nCicero--\"Aquam, dixit Thales, esse initium rerum, Deum autem eam mentem\nquae ex aqua cuncta fingeret.\" Thales said that water is the first\nprinciple of things, but God was that mind which formed all things out\nof water;[410] as also that still more remarkable saying of Thales,\nrecorded by Diogenes Laertius; \"God is the most ancient of all things,\nfor he had no birth; the world is the most beautiful of all things, for\nit is the workmanship of God.\"[411] We are aware that some historians of\nphilosophy reject the statement of Cicero, because, say they, \"it does\nviolence to the chronology of speculation.\"[412] Following Hegel, they\nassert that Thales could have no conception of God as Intelligence,\nsince that is a conception of a more advanced philosophy. Such an\nopinion may be naturally expected from the philosopher who places God,\nnot at the commencement, but at the _end_ of things, God becoming\nconscious and intelligent in humanity. If, then, Hegel teaches that God\nhimself has had a progressive development, it is no wonder he should\nassert that the idea of God has also had an historic development, the\n_last_ term of which is an _intelligent God_. But he who believes that\nthe idea of God as the infinite and the perfect is native to the human\nmind, and that God stands at the beginning of the entire system of\nthings, will feel there is a strong _a priori_ ground for the belief\nthat Thales recognized the existence of an _intelligent God who\nfashioned the universe_.\n\n[Footnote 405: Cudworth's \"Intellectual System,\" vol. i. p. 71.]\n\n[Footnote 406: Aristotle, \"De Anima,\" i. 2, 17.]\n\n[Footnote 407: Id., ib., i. 2, 17.]\n\n[Footnote 408: Diogenes Laertius, \"Lives of the Philosophers,\" p. 18\n(Bohn's ed.).]\n\n[Footnote 409: Aristotle, \"De Anima,\" i. 17.]\n\n[Footnote 410: \"De Natura Deor.,\" bk. i. ch. x.]\n\n[Footnote 411: \"Lives,\" etc., p. 19.]\n\n[Footnote 412: Lewes's \"Hist. Philos.,\" p. 4.]\n\n_Anaximenes of Miletus_ (B.C. 529-480) we place next to Thales in the\nconsecutive history of thought. It has been usual to rank Anaximander\nnext to the founder of the Ionian School. The entire complexion of his\nsystem is, however, unlike that of a pupil of Thales. And we think a\ncareful consideration of his views will justify our placing him at the\nhead of the Mechanical or Atomic division of the Ionian school.\nAnaximenes is the historical successor of Thales; he was unquestionably\na vitalist. He took up the speculation where Thales had left it, and he\ncarried it a step forward in its development.[413]\n\nPursuing the same method as Thales, he was not, however, satisfied with\nthe conclusion he had reached. Water was not to Anaximenes the most\nsignificant, neither was it the most universal element. But air seemed\nuniversally present. \"The earth was a broad leaf resting upon it. All\nthings were produced from it; all things were resolved into it. When he\nbreathed he drew in a part of this universal life. All things are\nnourished by air.\"[414] Was not, therefore, _air_ the arche, or primal\nelement of things?\n\n[Footnote 413: Ritter's \"History of Ancient Philosophy,\" vol. i. p.\n203.]\n\n[Footnote 414: Lewes's \"Biographical History of Philosophy,\" p. 7.]\n\nThis brief notice of the physical speculations of Anaximenes is all that\nhas survived of his opinions. We search in vain for some intimations of\nhis theological views. On this merely negative ground, some writers have\nunjustly charged him with Atheism. Were we to venture a conjecture, we\nwould rather say that there are indications of a tendency to Pantheism\nin that form of it which associates God necessarily with the universe,\nbut does not utterly confound them. His fixing upon \"_air_\" as the\nprimal element, seems an effort to reconcile, in some apparently\nintermediate substance, the opposite qualities of corporeal and\nspiritual natures. Air is invisible, impalpable, all-penetrating, and\nyet in some manner appreciable to sense. May not the vital\ntransformations of this element have produced all the rest? The writer\nof the Article on Anaximenes in the Encyclopaedia Britannica tells us (on\nwhat ancient authorities he saith not) that \"he asserted this air was\nGod, since the divine power resides in it and agitates it.\"\n\nSome indications of the views of Anaximenes may perhaps be gathered from\nthe teachings of Diogenes of Apollonia (B.C. 520-490,) who was the\ndisciple, and is generally regarded as the commentator and expounder of\nthe views of Anaximenes. The air of Diogenes was a soul; therefore it\nwas _living_, and not only living, but conscious and _intelligent_. \"It\nknows much,\" says he; \"for without _reason_ it would be impossible for\nall to be arranged duly and proportionately; and whatever objects we\nconsider will be found to be so arranged and ordered in the best and\nmost beautiful manner.\"[415] Here we have a distinct recognition of the\nfundamental axiom that _mind is the only valid explanation of the order\nand harmony which pervades the universe_. With Diogenes the first\nprinciple is a \"divine air,\" which is vital, conscious, and intelligent,\nwhich spontaneously evolves itself, and which, by its ceaseless\ntransformations, produces all phenomena. The soul of man is a detached\nportion of this divine element; his body is developed or evolved\ntherefrom. The theology of Diogenes, and, as we believe, of his master,\nAnaximenes also, was a species of Materialistic Pantheism.\n\n[Footnote 415: Lewes's \"Biographical History of Philosophy,\" p. 8;\nRitter's \"History of Ancient Philosophy,\" vol. i. p. 214.]\n\n_Heraclitus of Ephesus_(B.C. 503-420) comes next in the order of\nspeculative thought. In his philosophy, _fire_ is the arche, or first\nprinciple; but not fire in the usual acceptation of that term. The\nHeraclitean \"fire\" is not flame, which is only an intensity of fire, but\na warm, dry vapor--an _ether_, which may be illustrated, perhaps, by the\n\"caloric\" of modern chemistry. This \"_ether_\" was the primal element out\nof which the universe was formed; it was also a vital power or principle\nwhich animated the universe, and, in fact, the _cause_ of all its\nsuccessive phenomenal changes. \"The world,\" he said, \"was neither made\nby the gods nor men, and it was, and is, and ever shall be, an\n_ever-living fire_, in due proportion self-enkindled, and in due measure\nself-extinguished.\"[416] The universe is thus reduced to \"an eternal\nfire,\" whose ceaseless energy is manifested openly in the work of\ndissolution, and yet secretly, but universally, in the work of\nrenovation. The phenomena of the universe are explained by Heraclitus as\n\"the concurrence of opposite tendencies and efforts in the motions of\nthis ever-living fire, out of which results the most beautiful harmony.\nThis harmony of the world is one of conflicting impulses, like the lyre\nand the bow. The strife between opposite tendencies is the parent of all\nthings. All life is change, and change is strife.\"[417]\n\n[Footnote 416: Ritter's \"History of Ancient Philosophy,\" vol. i. p.\n235.]\n\n[Footnote 417: Lewes's \"Biographical History of Philosophy,\" p. 70;\nRitter's \"History of Ancient Philosophy,\" vol. i. p. 244.]\n\nHeraclitus was the first to proclaim the doctrine of the perpetual\nfluxion of the universe (to reon, to gignomenon--Unrest and\nDevelopment), the endless changes of matter, and the mutability and\nperishability of all individual things. This restless, changing flow of\nthings, which never _are_, but always are _becoming_, he pronounced to\nbe the _One_ and the _All_.\n\nFrom this statement of the physical theory of Heraclitus we might\nnaturally infer that he was a Hylopathean Atheist. Such an hypothesis\nwould not, however, be truthful or legitimate. On a more careful\nexamination, his system will be found to stand half-way between the\nmaterialistic and the spiritual conception of the Author of the\nuniverse, and marks, indeed, a transition from the one to the other.\nHeraclitus unquestionably held that all substance is material, for a\nphilosopher who proclaims, as he did, that the senses are the only\nsource of knowledge, must necessarily attach himself to a material\nelement as the primary one. And yet he seems to have _spiritualized_\nmatter. \"The moving unit of Heraclitus--the Becoming--is as immaterial\nas the resting unit of the Eleatics--the Being.\"[418] The Heraclitean\n\"_fire_\" is endowed with _spiritual_ attributes. \"Aristotle calls it\npsyche--soul, and says that it is asomatotaton, or absolutely\nincorporeal (\"De Anima,\" i. 2. 16). It is, in effect, the common ground\nof the phenomena both of mind and matter it is not only the animating,\nbut also the intelligent and regulating principle of the universe; the\nZynos Logos, or universal Word or Reason, which it behooves all men to\nfollow.\"[419] The psychology of Heraclitus throws additional light upon\nhis theological opinions. With him human intelligence is a detached\nportion of the Universal Reason. \"Inhaling,\" said he, \"through the\nbreath the Universal Ether, which is Divine Reason, we become\nconscious.\" The errors and imperfections of humanity are consequently to\nbe ascribed to a deficiency of the Divine Reason in man. Whilst,\ntherefore, the theory of Heraclitus seems to materialize mind, it may,\nwith equal fairness, be said to spiritualize matter.\n\n[Footnote 418: Zeller's \"History of Greek Philosophy,\" vol. i. p. 57.]\n\n[Footnote 419: Butler's \"Lectures,\" vol. i. p. 297, note.]\n\nThe general inference, therefore, from all that remains of the doctrine\nof Heraclitus is that he was a Materialistic Pantheist. His God was a\nliving, rational, intelligent Ether--a soul pervading the universe. The\nform of the universe, its ever-changing phenomena, were a necessary\nemanation from, or a perpetual transformation of, this universal soul.\n\nWith Heraclitus we close our survey of that sect of the physical school\nwhich regarded the world as a living organism.\n\nThe second subdivision of the physical school, _the Mechanical_ or\n_Atomist theorists_, attempted the explanation of the universe by\nanalogies derived from mechanical collocations, arrangements, and\nmovements. The universe was regarded by them as a vast superstructure\nbuilt up from elemental particles, aggregated by some inherent force or\nmutual affinity.\n\n_Anaximander of Miletus_ (born B.C. 610) we place at the head of the\nMechanical sect of the Ionian school; first, on the authority of\nAristotle, who intimates that the philosophic dogmata of Anaximander\n\"resemble those of Democritus,\" who was certainly an Atomist; and,\nsecondly, because we can clearly trace a genetic connection between the\nopinions of Democritus and Leucippus and those of Anaximander.\n\nThe arche, or first principle of Anaximander, was to apeiron, _the\nboundless, the illimitable, the infinite_. Some historians of philosophy\nhave imagined that the infinite of Anaximander was the \"unlimited all,\"\nand have therefore placed him at the head of the Italian or \"idealistic\nschool.\" These writers are manifestly in error. Anaximander was\nunquestionably a sensationalist. Whatever his \"infinite\" may be found to\nbe, one thing is clear, it was not a \"metaphysical infinite\"--it did not\ninclude infinite power, much less infinite mind.\n\nThe testimony of Aristotle is conclusive that by \"the infinite\"\nAnaximander understood the multitude of primary, material particles. He\ncalls it \"a migma, or mixture of elements.\"[420] It was, in fact, a\n_chaos_--an original state in which the primary elements existed in a\nchaotic combination without _limitation_ or division. He assumed a\ncertain \"_prima materia_,\" which was neither air, nor water, nor fire,\nbut a \"mixture\" of all, to be the first principle of the universe. The\naccount of the opinions of Anaximander which is given by Plutarch (\"De\nPlacita,\" etc.) is a further confirmation of our interpretation of his\ninfinite. \"Anaximander, the Milesian, affirmed the infinite to be the\nfirst principle, and that all things are generated out of it, and\ncorrupted again into it. _His infinite is nothing else but matter._\"\n\"Whence,\" says Cudworth, \"we conclude that Anaximander's infinite was\nnothing else but an infinite chaos of matter, in which were actually or\npotentially contained all manner of qualities, by the fortuitous\nsecretion and segregation of which he supposed infinite worlds to be\nsuccessively generated and corrupted. So that we may easily guess whence\nLeucippus and Democritus had their infinite worlds, and perceive how\nnear akin these two Atheistic hypotheses were.\"[421] The reader, whose\ncuriosity may lead him to consult the authorities collected by Cudworth\n(pp. 185-188), will find in the doctrine of Anaximander a rude\nanticipation of the modern theories of \"spontaneous generation\" and \"the\ntransmutation of species.\" In the fragments of Anaximander that remain\nwe find no recognition of an ordering Mind, and his philosophy is the\ndawn of a Materialistic school.\n\n[Footnote 420: Aristotle's \"Metaphysics,\" bk. xi. ch. ii.]\n\n[Footnote 421: Cudworth's \"Intellectual System,\" vol. i. pp. 186, 187.]\n\n_Leucippus of Miletus_ (B.C. 500-400) appears, in the order of\nspeculation, as the successor of Anaximander. _Atoms_ and _space_ are,\nin his philosophy, the archai, or first principles of all things.\n\"Leucippus (and his companion, Democritus) assert that the plenum and\nthe vacuum [_i.e._, body and space] are the first principles, whereof\none is the Ens, the other Non-ens; the differences of the body, which\nare only figure, order, and position, are the causes of all\nothers.\"[422]\n\n[Footnote 422: Aristotle's \"Metaphysics,\" p. 21 (Bohn's edition).]\n\nHe also taught that the elements, and the worlds derived from them, are\n_infinite_. He describes the manner in which the worlds are produced as\nfollows: \"Many bodies of various kinds and shapes are borne by\namputation from the infinite [_i.e._, the chaotic migma of Anaximander]\ninto a vast vacuum, and then they, being collected together, produce a\nvortex; according to which, they, dashing against each other, and\nwhirling about in every direction, are separated in such a way that like\nattaches itself to like; bodies are thus, without ceasing, united\naccording to the impulse given by the vortex, and in this way the earth\nwas produced.\"[423] Thus, through a boundless void, atoms infinite in\nnumber and endlessly diversified in form are eternally wandering; and,\nby their aggregation, infinite worlds are successively produced. These\natoms are governed in their movements by a dark negation of\nintelligence, designated \"Fate,\" and all traces of a Supreme Mind\ndisappear in his philosophy. It is a system of pure materialism, which,\nin fact, is Atheism.\n\n[Footnote 423: Diogenes Laertius, \"Lives,\" p. 389.]\n\n_Democritus of Abdera_ (B.C. 460-357), the companion of Leucippus, also\ntaught \"that _atoms_ and the _vacuum_ were the beginning of the\nuniverse.\"[424] These atoms, he taught, were infinite in number,\nhomogeneous, extended, and possessed of those primary qualities of\nmatter which are necessarily involved in extension in space--as size,\nfigure, situation, divisibility, and mobility. From the combination of\nthese atoms all other existences are produced; fire, air, earth, and\nwater; sun, moon, and stars; plants, animals, and men; the soul itself\nis an aggregation of round, moving atoms. And \"motion, which is the\ncause of the production of every thing, he calls _necessity_.\"[425]\nAtoms are thus the only real existences; these, without any pre-existent\nmind, or intelligence, were the original of all things.\n\n[Footnote 424: Diogenes Laertius, \"Lives,\" p. 395.]\n\n[Footnote 425: Id, ib., p. 394.]\n\nThe psychological opinions of Democritus were as decidedly materialistic\nas his physical theories. All knowledge is derived from sensation. It is\nonly by material impact that we can know the external world, and every\nsense is, in reality, a kind of touch. Material images are being\ncontinually thrown off from the surface of external objects which come\ninto actual contact with the organs of sense. The primary qualities of\nmatter, that is, those which are involved in extension in space, are the\nonly objects of real knowledge; the secondary qualities of matter, as\nsoftness, hardness, sweetness, bitterness, and the like, are but\nmodifications of the human sensibilities. \"The sweet exists only in\nform--the bitter in form, hot in form, color in form; but in causal\nreality only atoms and space exist. The sensible things which are\nsupposed by opinion to exist have no real existence, but atoms and space\nalone exist.\"[426]\n\n[Footnote 426: Lewes's \"Biographical History of Philosophy,\" p. 96. The\nwords of Democritus, as reported by Sextus Empiricus.]\n\nThus by Democritus was laid the basis of a system of absolute\nmaterialism, which was elaborated and completed by Epicurus, and has\nbeen transmitted to our times. It has undergone some slight\nmodifications, adapting it to the progress of physical science; but it\nis to-day substantially the theory of Democritus. In Democritus we have\nthe culmination of the mechanical theory of the Ionian or Physical\nschool. In physics and psychology it terminated in pure materialism. In\ntheology it ends in positive Atheism.\n\nThe fundamental error of all the philosophers of the physical school was\nthe assumption, tacitly or avowedly, that sense-perception is the only\nsource of knowledge. This was the fruitful source of all their erroneous\nconclusions, the parent of all their materialistic tendencies. This led\nthem continually to seek an arche, or first principle of the universe,\nwhich should, under some form, be appreciable to _sense_; and\nconsequently the course of thought tended naturally towards materialism.\n\nThales was unquestionably a dualist. Instructed by traditional\nintimations, or more probably guided by the spontaneous apperceptions of\nreason, he recognized, with more or less distinctness, an incorporeal\nDeity as the moving, animating, and organizing cause of the universe.\nThe idea of God is a truth so self-evident as to need no demonstration.\nThe human mind does not attain to the idea of a God as the last\nconsequence of a series of antecedent principles. It comes at once, by\nan inherent and necessary movement of thought, to the recognition of God\nas the First Principle of all principles. But when, instead of\nhearkening to the simple and spontaneous intuitions of the mind, man\nturns to the world of sense, and loses himself in discursive thought,\nthe conviction of a personal God becomes obscured. Then, amid the\nendlessly diversified phenomena of the universe, he seeks for a cause or\norigin which in some form shall be appreciable to sense. The mere study\nof material phenomena, scientifically or unscientifically conducted,\nwill never yield the sense of the living God. Nature must be\ninterpreted, can only be interpreted in the light of certain _a priori_\nprinciples of reason, or we can never \"ascend from nature up to nature's\nGod.\" Within the circle of mere sense-perception, the dim and\nundeveloped consciousness of God will be confounded with the universe.\nThus, in Anaximenes, God is partially confounded with \"air,\" which\nbecomes a symbol; then a vehicle of the informing mind; and the result\nis a semi-pantheism. In Heraclitus, the \"ether\" is, at first, a\nsemi-symbol of the Deity; at length, God is utterly confounded with this\nether, or \"rational fire,\" and the result is a definite _materialistic\npantheism_. And, finally, when this feeling or dim consciousness of God,\nwhich dwells in all human souls, is not only disregarded, but pronounced\nto be an illusion--a phantasy; when all the analogies which intelligence\nsuggests are disregarded, and a purely mechanical theory of the universe\nis adopted, the result is the utter negation of an Intelligent Cause,\nthat is, _absolute Atheism_, as in Leucippus and Democritus.\n\n\n\n\nCHAPTER IX.\n\nTHE PHILOSOPHERS OF ATHENS _(continued_).\n\nPRE-SOCRATIC SCHOOL _(continued_).\n\nIDEALIST: PYTHAGORAS--XENOPHANES--PARMENIDES--ZENO. NATURAL REALIST:\nANAXAGORAS.\n\n\nSOCRATIC SCHOOL.\n\nSOCRATES.\n\n\nIn the previous chapter we commenced our inquiry with the assumption\nthat, in the absence of the true inductive method of philosophy which\nobserves, and classifies, and generalizes facts, and thence attains a\ngeneral principle or law, two only methods were possible to the early\nspeculators who sought an explanation of the universe--1st, That of\nreasoning from physical analogies; or, 2d, That of deduction from\nrational conceptions, or _a priori_ ideas.\n\nAccordingly we found that one class of speculators fixed their attention\nsolely on the mere phenomena of nature, and endeavored, amid sensible\nthings, to find a _single_ element which, being more subtile, and\npliable, and universally diffused, could be regarded as the ground and\noriginal of all the rest, and from which, by a vital transformation, or\nby a mechanical combination and arrangement of parts, all the rest\nshould be evolved. The other class passed beyond the simple phenomena,\nand considered only the abstract _relations_ of phenomena among\nthemselves, or the relations of phenomena to the necessary and universal\nideas of the reason, and supposed that, in these relations, they had\nfound an explanation of the universe. The former was the Ionian or\nSensation school; the latter was the Italian or Idealist school.\n\nWe have traced the method according to which the Ionian school\nproceeded, and estimated the results attained. We now come to consider\nthe method and results of\n\nTHE ITALIAN OR IDEALIST SCHOOL.\n\nThis school we have found to be naturally subdivided into--1st, The\n_Mathematical_ sect, which attempted the explanation of the universe by\nthe abstract conceptions of number, proportion, order, and harmony; and,\n2d, The _Metaphysical_ school, which attempted the interpretation of the\nuniverse according to the _a priori_ ideas of unity, of Being _in se_,\nof the Infinite, and the Absolute.\n\n_Pythagoras of Samos_(born B.C. 605) was the founder of the Mathematical\nschool.\n\nWe are conscious of the difficulties which are to be encountered by the\nstudent who seeks to attain a definite comprehension of the real\nopinions of Pythagoras. The genuineness of many of those writings which\nwere once supposed to represent his views, is now questioned. \"Modern\ncriticism has clearly shown that the works ascribed to Timaeus and\nArchytas are spurious; and the treatise of Ocellus Lucanus on 'The\nNature of the All' can not have been written by a Pythagorean.\"[427] The\nonly writers who can be regarded as at all reliable are Plato and\nAristotle; and the opinions they represent are not so much those of\nPythagoras as \"the Pythagoreans.\" This is at once accounted for by the\nfact that Pythagoras taught in secret, and did not commit his opinions\nto writing. His disciples, therefore, represent the _tendency_ rather\nthan the actual tenets of his system; these were no doubt modified by\nthe mental habits and tastes of his successors.\n\n[Footnote 427: Lewes's \"Biographical History of Philosophy,\" p. 24.]\n\nWe may safely assume that the proposition from which Pythagoras started\nwas the fundamental idea of all Greek speculation--_that beneath the\nfleeting forms and successive changes of the universe there is some\npermanent principle of unity_[428] The Ionian school sought that\nprinciple in some common physical element; Pythagoras sought, not for\n\"elements,\" but for \"relations,\" and through these relations for\nultimate laws indicating primal forces.\n\n[Footnote 428: See Plato, \"Timaeus,\" ch. ix. p. 331 (Bohn's edition);\nAristotle's \"Metaphysics,\" bk. v. ch. iii.]\n\nAristotle affirms that Pythagoras taught \"that _numbers_ are the first\nprinciples of all entities,\" and, \"as it were, a _material_ cause of\nthings,\"[429] or, in other words, \"that numbers are substances that\ninvolve a separate subsistence, and are primary causes of\nentities.\"[430]\n\n[Footnote 429: Aristotle's \"Metaphysics,\" bk. i. ch. v.]\n\n[Footnote 430: Id., ib., bk. xii. ch. vi.]\n\nAre we then required to accept the dictum of Aristotle as final and\ndecisive? Did Pythagoras really teach that numbers are real\nentities--the _substance_ and cause of all other existences? The reader\nmay be aware that this is a point upon which the historians of\nphilosophy are not agreed. Ritter is decidedly of opinion that the\nPythagorean formula \"can only be taken symbolically.\"[431] Lewes insists\nit must be understood literally.[432] On a careful review of all the\narguments, we are constrained to regard the conclusion of Ritter as most\nreasonable. The hypothesis \"that numbers are real entities\" does\nviolence to every principle of common sense. This alone constitutes a\nstrong _a priori_ presumption that Pythagoras did not entertain so\nglaring an absurdity. The man who contributed so much towards perfecting\nthe mathematical sciences, who played so conspicuous a part in the\ndevelopment of ancient philosophy, and who exerted so powerful a\ndetermining influence on the entire current of speculative thought, did\nnot obtain his ascendency over the intellectual manhood of Greece by the\nutterance of such enigmas. And further, in interpreting the philosophic\nopinions of the ancients, we must be guided by this fundamental\ncanon--\"The human mind has, under the necessary operation of its own\nlaws, been compelled to entertain the same fundamental ideas, and the\nhuman heart to cherish the same feelings in all ages.\" Now if a careful\nphilosophic criticism can not render the _reported_ opinions of an\nancient teacher into the universal language of the reason and heart of\nhumanity, we must conclude either that his opinions were misunderstood\nand misrepresented by some of his successors, or else that he stands in\nutter isolation, both from the present and the past. His doctrine has,\nthen, no relation to the successions of thought, and no place in the\nhistory of philosophy. Nay, more, such a doctrine has in it no element\nof vitality, no germ of eternal truth, and must speedily perish. Now it\nis well known that the teaching of Pythagoras awakened the deepest\nintellectual sympathy of his age; that his doctrine exerted a powerful\ninfluence on the mind of Plato, and, through him, upon succeeding ages;\nand that, in some of its aspects, it now survives, and is more\ninfluential to-day than in any previous age; but this element of\nimmutable and eternal truth was certainly not contained in the inane and\nempty formula, \"that numbers are real existences, the causes of all\nother existences!\" If the fame of Pythagoras had rested on such \"airy\nnothings,\" it would have melted away before the time of Plato.\n\n[Footnote 431: \"History of Ancient Philosophy,\" vol. i. p. 359.]\n\n[Footnote 432: \"Biographical History of Philosophy,\" p. 38.]\n\nWe grant there is considerable force in the argument of Lewes. He urges,\nwith some pertinence, the unquestionable fact that Aristotle asserts,\nagain and again, that the Pythagoreans taught \"that numbers are the\nprinciples and substance of things as well as the causes of their\nmodifications;\" and he argues that we are not justified in rejecting the\nauthority of Aristotle, unless better evidence can be produced.\n\nSo far, however, as the authority of Aristotle is concerned, even Lewes\nhimself charges him, in more than one instance, with strangely\nmisrepresenting the opinions of his predecessors.[433] Aristotle is\nevidently wanting in that impartiality which ought to characterize the\nhistorian of philosophy, and, sometimes, we are compelled to question\nhis integrity. Indeed, throughout his \"Metaphysics\" he exhibits the\negotism and vanity of one who imagines that he alone, of all men, has\nthe full vision of the truth. In Books I. and XII. he uniformly\nassociates the \"_numbers_\" of Pythagoras with the \"_forms_\" and\n\"_ideas_\" of Plato. He asserts that Plato identifies \"forms\" and\n\"numbers,\" and regards them as real entities--substances, and causes of\nall other things. \"_Forms are numbers_[434]... so Plato affirmed,\nsimilar with the Pythagoreans; and the dogma that numbers are causes to\nother things--of their substance-_he, in like manner, asserted with\nthem_.\"[435] And then, finally, he employs the _same_ arguments in\nrefuting the doctrines of both.\n\n[Footnote 433: \"Aristotle uniformly speaks disparagingly of Anaxagoras\"\n(Lewes's \"Biographical History of Philosophy\"). He represents him as\nemploying mind (nous) simply as \"a _machine_\" for the production of the\nworld;--\"when he finds himself in perplexity as to the cause of its\nbeing necessarily an orderly system, he then drags it (mind) in by force\nto his assistance\" \"Metaphysics,\" (bk. i. ch. iv.). But he is evidently\ninconsistent with himself, for in \"De Anima\" (bk. i. ch. ii.) he tells\nus that \"Anaxagoras saith that mind is at once a _cause of motion_ in\nthe whole universe, and also of _well_ and _fit_.\" We may further ask,\nis not the idea of fitness--of the good and the befitting--the final\ncause, even according to Aristotle?\n\nHe also totally misrepresents Plato's doctrine of \"Ideas.\" \"Plato's\nIdeas,\" he says, \"are substantial existences--real beings\"\n(\"Metaphysics,\" bk. i. ch. ix.). Whereas, as we shall subsequently show,\n\"they are objects of pure conception for human reason, and they are\nattributes of the Divine Reason. It is there they substantially exist.\"\n(Cousin, \"History of Philosophy,\" vol. i. p. 415). It is also pertinent\nto inquire, what is the difference between the \"formal cause\" of\nAristotle and the archetypal ideas of Plato? and is not Plato's to\nagathon the \"final cause?\" Yet Aristotle is forever congratulating\nhimself that he alone has properly treated the \"formal\" and the \"final\ncause!\"]\n\n[Footnote 434: This, however, was not the doctrine of Plato. He does not\nsay \"forms are numbers.\" He says: \"God formed things as they first arose\naccording to forms _and_ numbers.\" See \"Timaeus,\" ch. xiv. and xxvii.]\n\n[Footnote 435: Aristotle's \"Metaphysics,\" bk. i. ch. vi.]\n\nNow the writings of Plato are all extant to-day, and accessible in an\nexcellent English translation to any of our readers. Cousin has\nshown,[436] most conclusively (and we can verify his conclusions for\nourselves), that Aristotle has totally misrepresented Plato. And if, in\nthe same connection, and in the course of the same argument, and in\nregard to the same subjects, he misrepresents Plato, it is most probable\nhe also misrepresents Pythagoras.\n\n[Footnote 436: \"The True, the Beautiful, and the Good,\" pp. 77-81.]\n\nIt is, however, a matter of the deepest interest for us to find the\nevidence gleaming out here and there, on the pages of Aristotle, that he\nhad some knowledge of the fact that the Pythagorean numbers were\nregarded as _symbols_. The \"numbers\" of Pythagoras are, in the mind of\nAristotle, clearly identified with the \"forms\" of Plato. Now, in Chapter\nVI. of the First Book he says that Plato taught that these \"forms\" were\nparadeigmata--models, patterns, exemplars after which created things\nwere framed. The numbers of Pythagoras, then, are also models and\nexemplars. This also is admitted by Aristotle. The Pythagoreans indeed\naffirm that entities subsist by an _imitation_ (mimesis) of\nnumbers.[437] Now if ideas, forms, numbers, were the models or paradigms\nafter which \"the Operator\" formed all things, surely it can not be\nlogical to say they were the \"material\" out of which all things were\nframed, much less the \"efficient cause\" of things. The most legitimate\nconclusion we can draw, even from the statements of Aristotle, is that\nthe Pythagoreans regarded numbers as the best expression or\nrepresentation of those laws of proportion, and order, and harmony,\nwhich seemed, to their eyes, to pervade the universe. Their doctrine was\na faint glimpse of that grand discovery of modern science--that all the\nhigher laws of nature assume the form of a precise quantitative\nstatement.\n\n[Footnote 437: Aristotle's \"Metaphysics,\" bk. i. ch. vi.]\n\nThe fact seems to be this, the Pythagoreans busied themselves chiefly\nwith what Aristotle designates \"the _formal_ cause,\" and gave little\nattention to the inquiry concerning \"the _material_ cause.\" This is\nadmitted by Aristotle. Concerning fire, or earth, or the other bodies of\nsuch kind, they have declared nothing whatsoever, inasmuch as affirming,\nin my opinion, nothing that is peculiar concerning _sensible_\nnatures.[438] They looked, as we have previously remarked, to the\nrelations of phenomena, and having discovered certain \"numerical\nsimilitudes,\" they imagined they had attained an universal principle, or\nlaw. \"If all the essential properties and attributes of things were\nfully represented by the relations of numbers, the philosophy which\nsupplied such an explanation of the universe might well be excused from\nexplaining, also, that existence of objects, which is distinct from the\nexistence of all their qualities and properties. The Pythagorean\ndoctrine of numbers might have been combined with the doctrine of atoms,\nand the combination might have led to results worthy of notice. But, so\nfar as we are aware, no such combination was attempted, and perhaps we\nof the present day are only just beginning to perceive, through the\ndisclosures of chemistry and crystallography, the importance of such an\ninquiry.\"[439]\n\n[Footnote 4398: Id., ib., bk. i. ch. ix.]\n\n[Footnote 439: Whewell's \"History of Inductive Sciences,\" vol. i. p.\n78.]\n\nThese preliminary considerations will have cleared and prepared the way\nfor a fuller presentation of the philosophic system of Pythagoras. The\nmost comprehensive and satisfactory exposition of his \"method\" is that\ngiven by Wm Archer Butler in his \"_Lectures on Ancient Philosophy_,\" and\nwe feel we can not do better than condense his pages.[440]\n\n[Footnote 440: Lecture VI. vol. i.]\n\nPythagoras had long devoted his intellectual adoration to the lofty idea\nof _order_, which seemed to reveal itself to his mind, as the presiding\ngenius of the serene and silent world. He had, from his youth, dwelt\nwith delight upon the eternal relations of space, and determinate form,\nand number, in which the very idea of _proportion_ seems to find its\nfirst and immediate development, and without the latter of which\n(number), all proportion is absolutely inconceivable. To this ardent\ngenius, whose inventive energies were daily adding new and surprising\ncontributions to the sum of discoverable relations, it at length began\nto appear as if the whole secret of the universe was hidden in these\nmysterious correspondences.\n\nIn making this extensive generalization, Pythagoras may, on his known\nprinciples, be supposed to have reasoned as follows: The mind of man\nperceives the relations of an eternal _order_ in the proportions of\nspace, and form, and number. That mind is, no doubt, a portion of the\nsoul which animates and governs the universe; for on what other\nsupposition shall we account for its internal principle of activity--the\nvery principle which characterizes the prime mover, and can scarce be\nascribed to an inferior nature? And on what other supposition are we to\nexplain the identity which subsists between the principles of order,\nauthenticated by the reason and the facts of order which are found to\nexist in the forms and multiplicities around us, and independent of us?\nCan this sameness be other than the sameness of the internal and\nexternal principles of a common nature? The proportions of the universe\ninhere in its divine soul; they are indeed its very essence, or at\nleast, its attributes. The ideas or principles of Order which are\nimplanted in the human reason, must inhere in the Divine Reason, and\nmust be reflected in the visible world, which is its product. Man, then,\ncan boldly affirm the necessary harmony of the world, because he has in\nhis own mind a revelation which declares that the world, in its real\nstructure, must be the image and copy of that divine _proportion_ which\nhe inwardly adores.[441]\n\n[Footnote 441: It is an opinion which goes as far back as the time of\nPlato, and even Pythagoras, and has ever since been widely entertained,\nthat beauty of _form_ consists in some sort of _proportion_ or _harmony_\nwhich may admit of a mathematical expression; and later and more\nscientific research is altogether in its favor. It is now established\nthat complementary colors, that is, colors which when combined make up\nthe full beam, are felt to be beautiful when seen simultaneously; that\nis, the mind is made to delight in the unities of nature. At the basis\nof music there are certain fixed ratios; and in poetry, of every\ndescription, there are measures, and correspondencies. Pythagoras has\noften been ridiculed for his doctrine of \"the music of the spheres;\" and\nprobably his doctrine was somewhat fanciful, but later science shows\nthat there is a harmony in all nature--in its forms, in its forces, and\nin its motions. The highest unorganized and all organized objects take\ndefinite forms which are regulated by mathematical laws. The forces of\nnature can be estimated in numbers, and light and heat go in\nundulations, whilst the movements of the great bodies in nature admit of\na precise quantitative expression. The harmonies of nature in respect of\ncolor, of number, of form, and of time are forcibly exhibited in\n\"Typical Forms and Special Ends in Creation,\" by M'Cosh.]\n\nAgain, the world is assuredly _perfect_, as being the sensible image and\ncopy of the Divinity, the outward and multiple development of the\nEternal Unity. It must, therefore, when thoroughly known and properly\ninterpreted, answer to all which we can conceive as perfect; that is, it\nmust be regulated by laws, of which we have the highest principles in\nthose first and elementary properties of numbers which stand next to\n_unity_. \"The world is then, through all its departments, _a living\narithmetic in its development, a realized geometry in its repose_.\" It\nis a kosmos (for the word is purely Pythagorean)--the expression of\n_harmony_, the manifestation, to sense, of everlasting _order_.\n\nThough Pythagoras found in geometry the fitting initiative for abstract\nspeculation, it is remarkable that he himself preferred to constitute\nthe science of Numbers as the true representative of the laws of the\nuniverse. The reason appears to be this: that though geometry speaks\nindeed of eternal truths, yet when the notion of symmetry and proportion\nis introduced, it is often necessary to insist, in preference, upon the\nproperties of numbers. Hence, though the universe displays the geometry\nof its Constructor or Animator, yet nature was eminently defined as the\nmimesis ton arithmon--the imitation of numbers.\n\nThe key to all the Pythagorean dogmas, then, seems to be the general\nformula of _unity in multiplicity_:--unity either evolving itself into\nmultiplicity, or unity discovered as pervading multiplicity. The\nprinciple of all things, the same principle which in this philosophy, as\nin others, was customarily called _Deity_, is the primitive unit from\nwhich all proceeds in the accordant relations of the universal scheme.\nInto the sensible world of multitude, the all-pervading Unity has\ninfused his own ineffable nature; he has impressed his own image upon\nthat world which is to represent him in the sphere of sense and man.\nWhat, then, is that which is at once single and multiple, identical and\ndiversified--which we perceive as the combination of a thousand\nelements, yet as the expression of a single spirit--which is a chaos to\nthe sense, a cosmos to the reason? What is it but\nharmony--proportion--the one governing the many, the many lost in the\none? The world is therefore a _harmony_ in innumerable degrees, from the\nmost complicated to the most simple: it is now a Triad, combining the\nMonad and the Duad, and partaking of the nature of both; now a Tetrad,\nthe form of perfection; now a Decad, which, in combining the four\nformer, involves, in its mystic nature, all the possible accordances of\nthe universe.[442]\n\nThe psychology of the Pythagoreans was greatly modified by their\nphysical, and still more, by their moral tenets. The soul was arithmos\neauton kinon--a self-moving number or Monad, the copy (as we have seen)\nof that Infinite Monad which unfolds from its own incomprehensible\nessence all the relations of the universe. This soul has three elements,\nReason (nous), Intelligence (phren), and Passion (Thymos). The two last,\nman has in common with brutes, the first is his grand and peculiar\ncharacteristic. It has, hence, been argued that Pythagoras could not\nhave held the doctrine of \"transmigration.\" This clear separation of man\nfrom the brute, by this signal endowment of reason, which is\nsempiternal, seems a refutation of those who charge him with the\ndoctrine.\n\nIn the department of morals, the legislator of Crotona found his\nappropriate sphere. In his use of numerical notation, moral good was\nessential unity--evil, essential plurality and division. In the fixed\ntruths of mathematical abstractions he found the exemplars of social and\npersonal virtue. The rule or law of all morality is resemblance to God;\nthat is, the return of number to its root, to unity,[443] and virtue is\nthus a harmony.\n\n[Footnote 442: That is, 1+2+3+4=10. There are intimations that the\nPythagoreans regarded the Monad as God, the Duad as matter, the Triad as\nthe complex phenomena of the world, the Tetrad as the completeness of\nall its relations, the Decad as the cosmos, or harmonious whole.]\n\n[Footnote 443: Aristotle, \"Nichomachian Ethics,\" bk. i. ch. vi.]\n\nThus have we, in Pythagoras, the dawn of an _Idealist_ school; for\nmathematics are founded upon abstractions, and there is consequently an\nintimate connection between mathematics and idealism. The relations of\nspace, and number, and determinate form, are, like the relations of\ncause and effect, of phenomena and substance, perceptible _only in\nthought_; and the mind which has been disciplined to abstract thought by\nthe study of mathematics, is prepared and disposed for purely\nmetaphysical studies. \"The looking into mathematical learning is a kind\nof prelude to the contemplation of real being.\"[444] Therefore Plato\ninscribed over the door of his academy, \"Let none but Geometricians\nenter here.\" To the mind thus disciplined in abstract thinking, the\nconceptions and ideas of reason have equal authority, sometimes even\nsuperior authority, to the perceptions of sense.\n\n[Footnote 444: Alcinous, \"Introduction to the Doctrines of Plato,\" ch.\nvii.]\n\nNow if the testimony of both reason and sense, as given in\nconsciousness, is accepted as of equal authority, and each faculty is\nregarded as, within its own sphere, a source of real, valid knowledge,\nthen a consistent and harmonious system of _Natural Realism_ or _Natural\nDualism_ will be the result. If the testimony of sense is questioned and\ndistrusted, and the mind is denied any immediate knowledge of the\nsensible world, and yet the existence of an external world is maintained\nby various hypotheses and reasonings, the consequence will be a species\nof _Hypothetical Dualism_ or _Cosmothetic Idealism_. But if the\naffirmations of reason, as to the unity of the cosmos, are alone\naccepted, and the evidence of the senses, as to the variety and\nmultiplicity of the world, is entirely disregarded, then we have a\nsystem of _Absolute Idealism_. Pythagoras regarded the harmony which\npervades the diversified phenomena of the outer world as a manifestation\nof the unity of its eternal principle, or as the perpetual evolution of\nthat unity, and the consequent _tendency_ of his system was to\ndepreciate the _sensible_. Following out this tendency, the Eleatics\nfirst neglected, and finally denied the variety of the universe--denied\nthe real existence of the external world, and asserted an absolute\n_metaphysical_ unity.\n\n_Xenophanes of Colophon_, in Ionia (B.C. 616-516), was the founder of\nthis celebrated school of Elea. He left Ionia, and arrived in Italy\nabout the same time as Pythagoras, bringing with him to Italy his Ionian\ntendencies; he there amalgamated them with Pythagorean speculations.\n\nPythagoras had succeeded in fixing the attention of his countrymen on\nthe harmony which pervades the material world, and had taught them to\nregard that harmony as the manifestation of the intelligence, and unity,\nand perfection of its eternal principle. Struck with this idea of\nharmony and of unity, Xenophanes, who was a poet, a rhapsodist, and\ntherefore by native tendency, rather than by intellectual discipline, an\nIdealist, begins already to attach more importance to _unity_ than\nmultiplicity in his philosophy of nature. He regards the testimony of\nreason as of more authority than the testimony of sense; \"and he holds\nbadly enough the balance between the unity of the Pythagoreans and the\nvariety which Heraclitus and the Ionians had alone considered.\"[445]\n\nWe are not, however, to suppose that Xenophanes denied entirely the\nexistence of _plurality_. \"The great Rhapsodist of Truth\" was guided by\nthe spontaneous intuitions of his mind (which seemed to partake of the\ncharacter of an inspiration), to a clearer vision of the truth than were\nhis successors of the same school by their discursive reasonings. \"The\nOne\" of Xenophanes was clearly distinguished from the outward universe\n(ta polla) on the one hand, and from the \"_non-ens_\" on the other. It\nwas his disciple, Parmenides, who imagined the logical necessity of\nidentifying plurality with the \"_non-ens_\" and thus denying all\nimmediate cognition of the phenomenal world. The compactness and logical\ncoherence of the system of Parmenides seems to have had a peculiar charm\nfor the Grecian mind, and to have diverted the eyes of antiquity from\nthe views of the more earnest and devout Xenophanes, whose opinions were\ntoo often confounded with those of his successors of the Eleatic school.\n\"Accordingly we find that Xenophanes has obtained credit for much that\nis, exclusively, the property of Parmenides and Zeno, in particular for\ndenying plurality, and for identifying God with the universe.\"[446]\n\n[Footnote 445: Cousin, \"History of Philosophy,\" vol. i. p. 440.]\n\n[Footnote 446: See note by editor, W.H. Thompson, M.A., on pages 331,\n332 of Butler's \"Lectures,\" vol. i. His authorities are \"Fragments of\nXenophanes\" and the treatise \"De Melisso, Xenophane, et Gorgia,\" by\nAristotle.]\n\nIn theology, Xenophanes was unquestionably a _Theist_. He had a profound\nand earnest conviction of the existence of a God, and he ridiculed with\nsarcastic force, the anthropomorphic absurdities of the popular\nreligion. This one God, he taught, was self-existent, eternal, and\ninfinite; supreme in power, in goodness, and intelligence.[447] These\ncharacteristics are ascribed to the Deity in the sublime words with\nwhich he opens his philosophic poem--\n\n     \"There is one God, of all beings, divine and human, the greatest:\n     Neither in body alike unto mortals, neither in mind.\"\n\nHe has no parts, no organs, as men have, being\n\n     \"All sight, all ear, all intelligence;\n     Wholly exempt from toil, he sways all things by _thought_ and\n                                                           _will_.\"[448]\n\nXenophanes also taught that God is \"uncreated\" or \"uncaused,\" and that\nhe is \"excellent\" as well as \"all-powerful.\"[449] And yet, regardless of\nthese explicit utterances, Lewes cautions his readers against supposing\nthat, by the \"one God,\" Xenophanes meant a Personal God; and he asserts\nthat his Monotheism was Pantheism. A doctrine, however, which ascribes\nto the Divine Being moral as well as intellectual supremacy, which\nacknowledges an outward world distinct from Him, and which represents\nHim as causing the changes in that universe by the acts of an\nintelligent volition, can only by a strange perversion of language be\ncalled pantheism.\n\n[Footnote 447: Lewes's \"Biographical History of Philosophy,\" p. 38;\nRitter's \"History of Ancient Philosophy,\" vol. i. pp. 428, 429.]\n\n[Footnote 448: Ritter's \"History of Ancient Philosophy,\" vol. i. pp.\n432, 434.]\n\n[Footnote 449: Butler's \"Lectures,\" vol. i. p. 331, note; Ritter's\n\"History of Ancient Philosophy,\" vol. i. p. 428.]\n\n_Parmenides of Elea_ (born B.C. 536) was the philosopher who framed the\npsychological opinions of the Idealist school into a precise and\ncomprehensive system. He was the first carefully to distinguish between\n_Truth_ (aletheian) and _Opinion_ (doxan)--between ideas obtained\nthrough the reason and the simple perceptions of sense. Assuming that\nreason and sense are the only sources of knowledge, he held that they\nfurnish the mind with two distinct classes of cognitions--one variable,\nfleeting, and uncertain; the other immutable, necessary, and eternal.\nSense is dependent on the variable organization of the individual, and\ntherefore its evidence is changeable, uncertain, and nothing but a mere\n\"_seeming_.\" Reason is the same in all individuals, and therefore its\nevidence is constant, real, and true. Philosophy is, therefore, divided\ninto two branches--_Physics_ and _Metaphysics_; one, a science of\nabsolute knowledge; the other, a science of mere appearances. The first\nscience, Physics, is pronounced illusory and uncertain; the latter,\nMetaphysics, is infallible and immutable.[450]\n\nProceeding on these principles, he rejects the dualistic system of the\nuniverse, and boldly declared that all essences are fundamentally\n_one_--that, in fact, there is no real plurality, and that all the\ndiversity which \"appears\" is merely presented under a peculiar aesthetic\nor sensible law. The senses, it is true, teach us that there are \"many\nthings,\" but reason affirms that, at bottom, there exists only \"the\none.\" Whatever, therefore, manifests itself in the field of sense is\nmerely illusory--the mental representation of a phenomenal world, which\nto experience seems diversified, but which reason can not possibly admit\nto be other than \"immovable\" and \"one.\" There is but one Being in the\nuniverse, eternal, immovable, absolute; and of this unconditioned being\nall phenomenal existences, whether material or mental, are but the\nattributes and modes. Hence the two great maxims of the Eleatic school,\nderived from Parmenides--ta panta en, \"_The All is One_\" and to auto\nnoein te kai einai (Idem est cogitare atque esse), \"_Thought and Being\nare identical._\" The last remarkable dictum is the fundamental principle\nof the modern pantheistic doctrine of \"absolute identity\" as taught by\nSchelling and Hegel.[451]\n\n[Footnote 450: Ritter's \"History of Ancient Philosophy,\" vol. i. pp.\n447, 451.]\n\n[Footnote 451: Id., ib., vol. i. pp. 450, 455.]\n\nLewes asserts that \"Parmenides did not, with Xenophanes, call 'the One'\nGod; he called it Being.[452] In support of this statement he, however,\ncites no ancient authorities. We are therefore justified in rejecting\nhis opinion, and receiving the testimony of Simplicius, \"the only\nauthority for the fragments of the Eleatics,\"[453] and who had a copy of\nthe philosophic poems of Parmenides. He assures us that Parmenides and\nXenophanes \"affirmed that '_the One,_' or unity, was the first Principle\nof all,....they meaning by this One _that highest or supreme God_, as\nbeing the cause of unity to all things.... It remaineth, therefore, that\nthat _Intelligence_ which is the cause of all things, and therefore of\nmind and understanding also, in which all things are comprehended in\nunity, was Parmenides' one Ens or Being.[454] Parmenides was, therefore,\na spiritualistic or idealistic Pantheist.\n\n_Zeno of Elea_ (born B.C. 500) was the logician of the Eleatic school.\nHe was, says Diogenes Laertius, \"the inventor of Dialectics.\"[455] Logic\nhenceforth becomes the organon[456]--organon of the Eleatics.\n\n[Footnote 452: \"Biographical History of Philosophy,\" p. 50.]\n\n[Footnote 453: Encyclopaedia Britannica, article \"Simplicius.\"]\n\n[Footnote 454: Cudworth's \"Intellectual System,\" vol. i. p. 511.]\n\n[Footnote 455: \"Lives,\" p. 387 (Bohn's edition).]\n\n[Footnote 456: Plato in \"Parmen.\"]\n\nThis organon, however, Zeno used very imperfectly. In his hands it was\nsimply the \"reductio ad absurdum\" of opposing opinions as the means of\nsustaining the tenets of his own sect. Parmenides had asserted, on _a\npriori_ grounds, the existence of \"the One.\" Zeno would prove by his\ndialectic the non-existence of \"the many.\" His grand position was that\nall phenomena, all that appears to sense, is but a _modification_ of the\nabsolute One. And he displays a vast amount of dialectic subtilty in the\neffort to prove that all \"appearances\" are unreal, and that all movement\nand change is a mere \"seeming\"--not a reality. What men call motion is\nonly a name given to a series of conditions, each of which, considered\nseparately, is rest. \"Rest is force resistant; motion is force\ntriumphant.\"[457] The famous puzzle of \"Achilles and the Tortoise,\" by\nwhich he endeavored to prove the unreality of motion, has been rendered\nfamiliar to the English reader.[458]\n\n[Footnote 457: Lewes's \"Biographical History of Philosophy,\" p. 60.]\n\n[Footnote 458: Ritter's \"History of Ancient Philosophy,\" vol. i. pp.\n475, 476.]\n\nAristotle assures us that Zeno, \"by his one Ens, which neither was moved\nnor movable, meaneth God.\" And he also informs us that \"Zeno endeavored\nto demonstrate that there is but one God, from the idea which all men\nhave of him, as that which is the best, supremest, most powerful of all,\nor an absolutely perfect being\" (\"De Xenophane, Zenone, et\nGorgia\").[459]\n\nWith Zeno we close our survey of the second grand line of independent\ninquiry by which philosophy sought to solve the problem of the universe.\nThe reader will be struck with the resemblance which subsists between\nthe history of its development and that of the modern Idealist school.\nPythagoras was the Descartes, Parmenides the Spinoza, and Zeno the Hegel\nof the Italian school.\n\nIn this survey of the speculations of the pre-Socratic schools of\nphilosophy, we have followed the course of two opposite streams of\nthought which had their common origin in one fundamental principle or\nlaw of the human mind--the _intuition of unity_--\"or the desire to\ncomprehend all the facts of the universe in a single formula, and\nconsummate all conditional knowledge in the unity of unconditioned\nexistence.\" The history of this tendency is, in fact, the history of all\nphilosophy. \"The end of all philosophy,\" says Plato, \"is the intuition\nof unity.\" \"All knowledge,\" said the Platonists, \"is the gathering up\ninto one.\"[460]\n\n[Footnote 459: Cudworth's \"Intellectual System,\" vol. i. p. 518.]\n\n[Footnote 460: Hamilton's \"Metaphysics,\" vol. i. pp. 67-70 (English\nedition).]\n\nStarting from this fundamental idea, _that, beneath the endless flux and\nchange of the visible universe, there must be a permanent principle of\nunity_, we have seen developed two opposite schools of speculative\nthought. As the traveller, standing on the ridges of the Andes, may see\nthe head-waters of the great South American rivers mingling in one, so\nthe student of philosophy, standing on the elevated plane of analytic\nthought, may discover, in this fundamental principle, the common source\nof the two great systems of speculative thought which divided the\nancient world. Here are the head-waters of the sensational and the\nidealist schools. The Ionian school started its course of inquiry in the\ndirection of _sense_; it occupied itself solely with the phenomena of\nthe external world, and it sought this principle of unity in a\n_physical_ element. The Italian school started its course of inquiry in\nthe direction of _reason_; it occupied itself chiefly with rational\nconceptions or _a priori_ ideas, and it sought this principle of unity\nin purely _metaphysical_ being. And just as the Amazon and La Plata\nsweep on, in opposite directions, until they reach the extremities of\nthe continent, so these two opposite streams of thought rush onward, by\nthe force of a logical necessity, until they terminate in the two\nUnitarian systems of _Absolute Materialism_ and _Absolute Idealism_,\nand, in their theological aspects, in a pantheism which, on the one\nhand, identifies God with matter, or, on the other hand, swallows up the\nuniverse in God.\n\nThe radical error of both these systems is at once apparent. The\ntestimony of the primary faculties of the mind was not regarded as each,\nwithin its sphere, final and decisive. The duality of consciousness was\nnot accepted in all its integrity; one school rejected the testimony of\nreason, the other denied the veracity of the senses, and both prepared\nthe way for the _skepticism_ of the Sophists.\n\nWe must not, however, lose sight of the fact that there were some\nphilosophers of the pre-Socratic school, as Anaxagoras and Empedocles,\nwho recognized the partial and exclusive character of both these\nsystems, and sought, by a method which Cousin would designate as\nEclecticism, to combine the element of truth contained in each.\n\n_Anaxagoras of Clazomencoe_ (B.C. 500-428) added to the Ionian\nphilosophy of a material element or elements the Italian idea of a\n_spirit_ distinct from, and independent of the world, which has within\nitself the principle of a spontaneous activity--Nous autocrates, and\nwhich is the first cause of motion in the universe--arche tes\nkineseos.[461]\n\n[Footnote 461: Cousin, \"History of Philosophy,\" vol. i. p. 411.]\n\nIn his physical theory, Anaxagoras was an Atomist. Instead of one\nelement, he declared that the elements or first principles were\nnumerous, or even infinite. No point in space is unoccupied by these\natoms, which are infinitely divisible. He imagined that, in nature,\nthere are as many kinds of principles as there are species of compound\nbodies, and that the peculiar form of the primary particles of which any\nbody is composed is the same with the qualities of the compound body\nitself. This was the celebrated doctrine of _Homoeomeria_, of which\nLucretius furnishes a luminous account in his philosophic poem \"De\nNatura Rerum\"--\n\n                        \"That bone from bones\n     Minute, and embryon; nerve from nerves arise;\n     And blood from blood, by countless drops increased.\n     Gold, too, from golden atoms, earths concrete,\n     From earths extreme; from fiery matters, fire;\n     And lymph from limpen dews. And thus throughout\n     From primal kinds that kinds perpetual spring.\"[462]\n\nThese primary particles were regarded by Anaxagoras as eternal; because\nhe held the dogma, peculiar to all the Ionians, that nothing can be\nreally created or annihilated (de nihilo nihil, in nihilum nil posse\nreverti). But he saw, nevertheless, that the simple existence of\n\"_inert_\" matter, even from eternity, could not explain the motion and\nthe harmony of the material world. Hence he saw the necessity of another\npower--_the power of Intelligence_. \"All things were in chaos; then came\nIntelligence and introduced Order.\"[463]\n\nAnaxagoras, unlike the pantheistic speculators of the Ionian school,\nrigidly separated the Supreme Intelligence from the material universe.\nThe Nous of Anaxagoras is a principle, infinite, independent\n(autocrates), omnipresent (en panti pantos moioa enon), the subtilest\nand purest of things (lepitotaton panion chrematon kaikai katharotaton);\nand incapable of mixture with aught besides; it is also omniscient\n(panta egno), and unchangeable (pas omoios esti).--Simplicius, in\n\"Arist. Phys.\" i. 33.[464]\n\n[Footnote 462: Good's translation, bk. i. p. 325.]\n\n[Footnote 463: Diogenes Laertius, \"Lives,\" p. 59.]\n\n[Footnote 464: Butler's \"Lectures on Philosophy,\" vol. i. p. 305, note.]\n\nThus did Anaxagoras bridge the chasm between the Ionian and the Italian\nschools. He accepted both doctrines with some modifications. He believed\nin the real existence of the phenomenal world, and he also believed in\nthe real existence of \"The Infinite Mind,\" whose Intelligence and\nOmnipotence were manifested in the laws and relations which pervade the\nworld. He proclaimed the existence of the Infinite Intelligence (\"the\nONE\"), who was the Architect and Governor of the Infinite Matter (\"the\nMANY\").\n\nOn the question as to the origin and certainty of human knowledge,\nAnaxagoras differed both from the Ionians and the Eleatics. Neither the\nsense alone, nor the reason alone, were for him a ground of certitude.\nHe held that reason (logos) was the regulative faculty of the mind, as\nthe Nous, or Supreme Intelligence, was the regulative power of the\nuniverse. And he admitted that the senses were veracious in their\nreports; but they reported only in regard to phenomena. The senses,\nthen, perceive _phenomena_, but it is the reason alone which recognizes\n_noumena_, that is, the reason perceives being in and through phenomena,\nsubstance in and through qualities; an anticipation of the fundamental\nprinciple of modern psychology--\"_that every power or substance in\nexistence is knowable to us, so far only, as we know its phenomena_.\"\nThus, again, does he bridge the chasm that separates between the\nSensationalist and the Idealist. The Ionians relied solely on the\nintuitions of sense; the Eleatics accepted only the apperceptions of\npure reason; he accepted the testimony of both, and in the synthesis of\nsubject and object--the union of an element supplied by sensation, and\nan element supplied by reason, he found real, certain knowledge.\n\nThe harmony which the doctrine of Anaxagoras introduced into the\nphilosophy of Athens, soon attracted attention and multiplied disciples.\nHe was teaching when Socrates arrived in Athens, and the latter attended\nhis school. The influence which the doctrine of Anaxagoras exerted upon\nthe mind of Socrates (leading him to recognize Intelligence as the cause\nof order and special adaptation in the universe),[465] and also upon the\ncourse of philosophy in the Socratic schools, is the most enduring\nmemorial of his name.[466]\n\n[Footnote 465: \"Phaedo,\" Sec. 105.]\n\n[Footnote 466: Aristotle's \"Metaphysics,\" bk. i. ch. iii.]\n\nWe have devoted a much larger space than we originally designed to the\nante-Socratic schools--quite out of proportion, indeed, with that we\nshall be able to appropriate to their successors. But inasmuch as all\nthe great primary problems of thought, which are subsequently discussed\nby Plato and Aristotle, were started, and received, at least, typical\nanswers in those schools, we can not hope to understand Plato, or\nAristotle, or even Epicurus, or Zeno of Cittium, unless we have first\nmastered the doctrines of Heraclitus, Pythagoras, Parmenides, and\nAnaxagoras.[467] The attention we have bestowed on these early thinkers\nwill, therefore, have been a valuable preparatory discipline for the\nstudy of\n\nII. THE SOCRATIC SCHOOL.\n\nThe first cycle of philosophy was now complete. That form of Grecian\nspeculative thought which, during the first period of its development,\nwas a philosophy of nature, had reached its maturity; it had sought \"the\nfirst principles of all things\" in the study of external nature, and had\nsignally failed. In this pursuit of first principles as the basis of a\ntrue and certain knowledge of the system of the universe, the two\nleading schools had been carried to opposite poles of thought. One had\nasserted that _experience_ alone, the other, that _reason_ alone was the\nsole criterion of truth. As the last consequence of this imperfect\nmethod, Leucippus had denied the existence of \"the one,\" and Zeno had\ndenied the existence of \"the many.\" The Ionian school, in Democritus,\nhad landed in Materialism; the Italian, in Parmenides, had ended in\nPantheism; and, as the necessary result of this partial and defective\nmethod of inquiry, which ended in doubt and contradiction, a spirit of\ngeneral skepticism was generated in the Athenian mind. If doubt be cast\nupon the veracity of the primary cognitive faculties of the mind, the\nflood-gates of universal skepticism are opened. If the senses are\npronounced to be mendacious and illusory in their reports regarding\nexternal phenomena, and if the intuitions of the reason, in regard to\nthe ground and cause of phenomena, are delusive, then we have no ground\nof certitude. If one faculty is unveracious and unreliable, how can we\ndetermine that the other is not equally so? There is, then, no such\nthing as universal and necessary truth. Truth is variable and uncertain,\nas the variable opinion of each individual.\n\n[Footnote 467: Maurice's \"Ancient Philosophy,\" p. 114; Butler's\n\"Lectures on Ancient Philosophy,\" vol. ii. pp. 87, 88.]\n\nThe Sophists, who belonged to no particular school, laid hold on the\nelements of skepticism contained in both the pre-Socratic schools of\nphilosophy, and they declared that \"the sophia\" was not only\nunattainable, but that no relative degree of it was possible for the\nhuman faculties.[468] Protagoras of Abdera accepted the doctrine of\nHeraclitus, that thought is identical with sensation, and limited by it;\nhe therefore declared that there is no criterion of truth, and _Man is\nthe measure of all things_.[469] Sextus Empiricus gives the\npsychological opinions of Protagoras with remarkable explicitness.\n\"Matter is in a perpetual flux, whilst it undergoes augmentations and\nlosses; the senses also are modified according to the age and\ndisposition of the body. He said, also, that the reason of all phenomena\nresides in matter as substrata, so that matter, in itself, might be\nwhatever it appeared to each. But men have different perceptions at\ndifferent periods, according to the changes in the things perceived....\nMan is, therefore, the criterion of that which exists; all that is\nperceived by him exists; _that which is perceived by no man does not\nexist_.\"[470] These conclusions were rigidly and fearlessly applied to\nethics and political science. If there is no Eternal Truth, there can be\nno Immutable Right. The distinction of right and wrong is solely a\nmatter of human opinion and conventional usage.[471] \"That which\n_appears_ just and honorable to each city, is so for _that city_, so\nlong as the opinion prevails.\"[472]\n\n[Footnote 468: Encyclopaedia Britannica, article \"Sophist.\"]\n\n[Footnote 469: Plato's \"Theaetetus\" (anthropos--\"the individual is the\nmeasure of all things\"), vol. i. p. 381 (Bohn's edition).]\n\n[Footnote 470: Lewes's \"Biographical History of Philosophy,\" p. 117.]\n\n[Footnote 471: \"Gorgias,\" Sec. 85-89.]\n\n[Footnote 472: Plato's \"Theaetetus,\" Sec. 65-75.]\n\nThere were others who laid hold on the weapons which Zeno had prepared\nto their hands. He had asserted that all the objects of sense were mere\nphantoms--delusive and transitory. By the subtilties of dialectic\nquibbling, he had attempted to prove that \"change\" meant \"permanence,\"\nand \"motion\" meant \"rest.\"[473] Words may, therefore, have the most\nopposite and contradictory meanings; and all language and all opinion\nmay, by such a process, be rendered uncertain. One opinion is,\nconsequently, for the individual, just as good as another; and all\nopinions are equally true and untrue. It was nevertheless desirable, for\nthe good of society, that there should be some agreement, and that, for\na time at least, certain opinions should prevail; and if philosophy had\nfailed to secure this agreement, rhetoric, at least, was effectual; and,\nwith the Sophist, rhetoric was \"the art of making the worst appear the\nbetter reason.\" All wisdom was now confined to a species of \"word\njugglery,\" which in Athens was dignified as \"the art of disputation.\"\n\n[Footnote 473: \"And do we not know that the Eleatic Palamedes (Zeno)\nspoke by art in such a manner that the same things appeared to be\nsimilar and dissimilar, one and many, at rest and in\nmotion?\"--\"Phaedrus,\" Sec. 97.]\n\nSOCRATES (B.C. 469-399), the grand central figure in the group of\nancient philosophers, arrived in Athens in the midst of this general\nskepticism. He had an invincible faith in truth. \"He made her the\nmistress of his soul, and with patient labor, and unwearied energy, did\nhis great and noble soul toil after perfect communion with her.\" He was\ndisappointed and dissatisfied with the results that had been reached by\nthe methods of his predecessors, and he was convinced that by these\nmethods the problem of the universe could not be solved. He therefore\nturned away from physical inquiries, and devoted his whole attention to\nthe study of the human mind, its fundamental beliefs, ideas, and laws.\nIf he can not penetrate the mysteries of the outer world, he will turn\nhis attention to the world within. He will \"know himself,\" and find\nwithin himself the reason, and ground, and law of all existence. There\nhe discovered certain truths which can not possibly be questioned. He\nfelt he had within his own heart a faithful monitor--a _conscience_,\nwhich he regarded as the voice of God.[474] He believed \"he had a divine\nteacher with him at all times. Though he did not possess wisdom, this\nteacher could put him on the road to seek it, could preserve him from\ndelusions which might turn him out of the way, could keep his mind fixed\nupon the end for which he ought to act and live.\"[475] In himself,\ntherefore, he sought that ground of certitude which should save him from\nthe prevailing skepticism of his times. The Delphic inscription, Gnoti\nseauton, \"_know thyself_\" becomes henceforth the fundamental maxim of\nphilosophy.\n\n[Footnote 474: The Daemon of Socrates has been the subject of much\ndiscussion among learned men. The notion, once generally received, that\nhis _daimon_ was \"a familiar genius,\" is now regarded as an exploded\nerror. \"Nowhere does Socrates, in Plato or Xenophon, speak of _a_ genius\nor demon, but always of a _doemoniac something (to daimonion_, or\n_daimonin ti_), or of a _sign_, a _voice_, a _divine sign_, a _divine\nvoice_\" (Lewes's \"Biographical History of Philosophy,\" p. 166).\n\"Socrates always speaks of a _divine or supernatural somewhat_ ('divinum\nquiddam,' as Cicero has it), the nature of which he does not attempt to\ndivine, and to which he never attributes personality\" (Butler's\n\"Lectures on Ancient Philosophy,\" vol. i. p. 357). The scholar need not\nto be informed that _to daimonion_, in classic literature, means the\ndivine Essence (Lat. _numen_), to which are attributed events beyond\nman's power, yet not to be assigned to any special god.]\n\n[Footnote 475: Maurice's \"Ancient Philosophy,\" p. 124.]\n\nTruth has a rational, _a priori_ foundation in the constitution of the\nhuman mind. There are _ideas_ connatural to the human reason which are\nthe copies of those archetypal ideas which belong to the Eternal Reason.\nThe grand problem of philosophy, therefore, now is--_What are these\nfundamental_ IDEAS _which are unchangeable and permanent, amid all the\ndiversifies of human opinion, connecting appearance with reality, and\nconstituting a ground of certain knowledge or absolute truth_? Socrates\nmay not have held the doctrine of ideas as exhibited by Plato, but he\ncertainly believed that there were germs of truth latent in the human\nmind--principles which governed, unconsciously, the processes of\nthought, and that these could be developed by reflection and by\nquestioning. These were embryonate in the womb of reason, coming to the\nbirth, but needing the \"maieutic\" or \"obstetric\" art, that they might be\nbrought forth.[476] He would, therefore, become the accoucheur of ideas,\nand deliver minds of that secret truth which lay in their mental\nconstitution. And thus _Psychology_ becomes the basis of all legitimate\nmetaphysics.\n\n[Footnote 476: Plato's \"Theaetetus,\" Sec. 22.]\n\nBy the general consent of antiquity, as well as by the concurrent\njudgment of all modern historians of philosophy, Socrates is regarded as\nhaving effected a complete revolution in philosophic thought, and, by\nuniversal consent, he is placed at the commencement of a new era in\nphilosophy. Schleiermacher has said, \"the service which Socrates\nrendered to philosophy consisted not so much in the truths arrived at\n_as in the_ METHOD _by which truth is sought_.\" As Bacon inaugurated a\nnew method in physical inquiry, so Socrates inaugurated a new method in\nmetaphysical inquiry.\n\nWhat, then, was this _new method_? It was no other than the _inductive_\nmethod applied to the facts of consciousness. This method is thus\ndefined by Aristotle: \"Induction is the process from particulars to\ngenerals;\" that is, it is the process of discovering laws from facts,\ncauses from effects, being from phenomena. But how is this process of\ninduction conducted? By observing and enumerating the real facts which\nare presented in consciousness, by noting their relations of resemblance\nor difference, and by classifying these facts by the aid of these\nrelations. In other words, it is _analysis_ applied to the phenomena of\nmind.[477] Now Socrates gave this method of psychological analysis to\nGreek philosophy. There are two things of which Socrates must justly be\nregarded as the author,--the _inductive reasoning_ and _abstract\ndefinition_.[478] We readily grant that Socrates employed this method\nimperfectly, for methods are the last things perfected in science; but\nstill, the Socratic movement was a vast movement in the right direction.\n\n[Footnote 477: Cousin's \"Lectures on the History of Philosophy,\" vol. i.\np. 30.]\n\n[Footnote 478: Aristotle's \"Metaphysics,\" vol. xii. ch. iv. p. 359\n(Bohn's edition).]\n\nIn what are usually regarded as the purely Socratic dialogues,[479]\nPlato evidently designs to exhibit this method of Socrates. They proceed\ncontinually on the firm conviction that there is a standard or criterion\nof truth in the reason of man, and that, by _reflection_, man can\napprehend and recognize the truth. To awaken this power of reflection;\nto compel men to analyze their language and their thoughts; to lead them\nfrom the particular and the contingent, to the universal and the\nnecessary; and to teach them to test their opinions by the inward\nstandard of truth, was the aim of Socrates. These dialogues are a\npicture of the conversations of Socrates. They are literally an\neducation of the thinking faculty. Their purpose is to discipline men to\nthink for themselves, rather than to furnish opinions for them. In many\nof these dialogues Socrates affirms nothing. After producing many\narguments, and examining a question on all sides, he leaves it\nundetermined. At the close of the dialogue he is as far from a\ndeclaration of opinions as at the commencement. His grand effort, like\nthat of Bacon's, is to furnish men a correct method of inquiry, rather\nthan to apply that method and give them results.\n\n[Footnote 479: \"Laches,\" \"Charmides,\" \"Lysis,\" \"The Rivals,\" \"First and\nSecond Alcibiades,\" \"Theages,\" \"Clitophon.\" See Whewell's translation,\nvol. i.]\n\nWe must not, however, from thence conclude that Socrates did not himself\nattain any definite conclusions, or reach any specific and valuable\nresults. When, in reply to his friends who reported the answer of the\noracle of Delphi, that \"Socrates was the wisest of men,\" he said, \"he\nsupposed the oracle declared him wise _because he knew nothing_,\" he did\nnot mean that true knowledge was unattainable, for his whole life had\nbeen spent in efforts to attain it. He simply indicates the disposition\nof mind which is most befitting and most helpful to the seeker after\ntruth. He must be conscious of his own ignorance. He must not exalt\nhimself. He must not put his own conceits in the way of the thing he\nwould know. He must have an open eye, a single purpose, an honest mind,\nto prepare him to receive light when it comes. And that there is light,\nthat there is a source whence light comes, he avowed in every word and\nact.\n\nSocrates unquestionably believed in one Supreme God, the immaterial,\ninfinite Governor of all. He cherished that instinctive, spontaneous\nfaith in God and his Providence which is the universal faith of the\nhuman heart. He saw this faith revealed in the religious sentiments of\nall nations, and in the tendency to worship so universally\ncharacteristic of humanity.[480] He appealed to the consciousness of\nabsolute dependence--the persuasion, wrought by God in the minds of all\nmen, that \"He is able to make men happy or miserable,\" and the\nconsequent sense of obligation which teaches man he ought to obey God.\nAnd he regarded with some degree of affectionate tenderness the common\nsentiment of his countrymen, that the Divine Government was conducted\nthrough the ministry of subordinate deities or generated gods. But he\nsought earnestly to prevent the presence of these subordinate agents\nfrom intercepting the clear view of the Supreme God.\n\nThe faith of Socrates was not, however, grounded on mere feeling and\nsentiment. He endeavored to place the knowledge of God on a rational\nbasis. We can not read the arguments he employed without being convinced\nthat he anticipated all the subsequent writers on Natural Theology in\nhis treatment of the argument from _special ends_ or _final causes_. We\nventure to abridge the account which is given by Xenophon of the\nconversation with Aristodemus:[481]\n\n[Footnote 480: \"Memorabilia,\" bk. i. ch. iv. Sec. 16.]\n\n[Footnote 481: Ibid., bk. i. ch. iv.]\n\n\"I will now relate the manner in which I once heard Socrates discoursing\nwith Aristodemus concerning the Deity; for, observing that he never\nprayed nor sacrificed to the gods, but, on the contrary, ridiculed those\nwho did, he said to him:\n\n\"'Tell me, Aristodemus, is there any man you admire on account of his\nmerits? Aristodemus having answered, 'Many,--'Name some of them, I pray\nyou,' said Socrates. 'I admire,' said Aristodemus, 'Homer for his Epic\npoetry, Melanippides for his dithyrambics, Sophocles for his tragedy,\nPolycletus for statuary, and Zeuxis for painting.'\n\n\"'But which seemed to you most worthy of admiration, Aristodemus--the\nartist who forms images void of motion and intelligence, or one who has\nskill to produce animals that are endued, not only with activity, but\nunderstanding?'\n\n\"'The latter, there can be no doubt,' replied Aristodemus, 'provided the\nproduction was not the effect of chance, but of wisdom and contrivance.'\n\n\"'But since there are many things, some of which we can easily see the\nuse of, while we can not say of others to what purpose they are\nproduced, which of these, Aristodemus, do you suppose the work of\nwisdom?'\n\n\"'It would seem the most reasonable to affirm it of those whose fitness\nand utility are so evidently apparent,' answered Aristodemus.\n\n\"'But it is evidently apparent that He who, at the beginning, made man,\nendued him with senses because they were good for him; eyes wherewith to\nbehold what is visible, and ears to hear whatever was heard; for, say,\nAristodemus, to what purpose should odor be prepared, if the sense of\nsmelling had been denied or why the distinction of bitter or sweet, of\nsavory or unsavory, unless a palate had been likewise given,\nconveniently placed to arbitrate between them and proclaim the\ndifference? Is not that Providence, Aristodemus, in a most eminent\nmanner conspicuous, which, because the eye of a man is so delicate in\nits contexture, hath therefore prepared eyelids like doors whereby to\nsecure it, which extend of themselves whenever it is needful, and again\nclose when sleep approaches? Are not these eyelids provided, as it were,\nwith a fence on the edge of them to keep off the wind and guard the eye?\nEven the eyebrow itself is not without its office, but, as a penthouse,\nis prepared to turn off the sweat, which falling from the forehead might\nenter and annoy that no less tender than astonishing part of us. Is it\nnot to be admired that the ears should take in sounds of every sort, and\nyet are not too much filled with them? That the fore teeth of the animal\nshould be formed in such a manner as is evidently best for cutting, and\nthose on the side for grinding it to pieces? That the mouth, through\nwhich this food is conveyed, should be placed so near the nose and eyes\nas to prevent the passing unnoticed whatever is unfit for\nnourishment?... And canst thou still doubt, Aristodemus, whether a\n_disposition of parts like this should be a work of chance, or of wisdom\nand contrivance_?'\n\n\"'I have no longer any doubt,' replied Aristodemus; 'and, indeed, the\nmore I consider it, the more evident it appears to me that man must be\nthe masterpiece of some great Artificer, carrying along with it infinite\nmarks of the love and favor of Him who hath thus formed it.'\n\n\"'But, further (unless thou desirest to ask me questions), seeing,\nAristodemus, thou thyself art conscious of reason and intelligence,\nsupposest thou there is no intelligence elsewhere? Thou knowest thy body\nto be a small part of that wide-extended earth thou everywhere\nbeholdest; the moisture contained in it thou also knowest to be a\nportion of that mighty mass of waters whereof seas themselves are but a\npart, while the rest of the elements contribute out of their abundance\nto thy formation. It is the _soul_, then, alone, that intellectual part\nof us, which is come to thee by some lucky chance, from I know not\nwhere. If so, there is no intelligence elsewhere; and we must be forced\nto confess that this stupendous universe, with all the various bodies\ncontained therein--equally amazing, whether we consider their magnitude\nor number, whatever their use, whatever their order--all have been\nproduced by chance, not by intelligence!'\n\n\"'It is with difficulty that I can suppose otherwise,' returned\nAristodemus; 'for I behold none of those gods whom you speak of as\nframing and governing the world; whereas I see the artists when at their\nwork here among us.'\n\n\"'Neither yet seest thou thy soul, Aristodemus, which, however, most\nassuredly governs thy body; although it may well seem, by thy manner of\ntalking, that it is chance and not reason which governs thee.'\n\n\"'I do not despise the gods,' said Aristodemus; 'on the contrary, I\nconceive so highly of their excellency, as to suppose they stand in no\nneed of me or of my services.'\n\n\"'Thou mistakest the matter,' Aristodemus, 'the great magnificence they\nhave shown in their care of thee, so much the more honor and service\nthou owest them.'\n\n\"'Be assured,' said Aristodemus, 'if I once could persuade myself the\ngods take care of man, I should want no monitor to remind me of my\nduty.'\n\n\"'And canst thou doubt, Aristodemus, if the gods take care of man? Hath\nnot the glorious privilege of walking upright been alone bestowed on\nhim, whereby he may with the better advantage survey what is around him,\ncontemplate with more ease those splendid objects which are above, and\navoid the numerous ills and inconveniences which would otherwise befall\nhim? Other animals, indeed, they have provided with feet; but to man\nthey have also given hands, with which he can form many things for use,\nand make himself happier than creatures of any other kind. A tongue hath\nbeen bestowed on every other animal; but what animal, except man, hath\nthe power of forming words with it whereby to explain his thoughts and\nmake them intelligible to others? But it is not with respect to the body\nalone that the gods have shown themselves bountiful to man. Their most\nexcellent gift is that of a soul they have infused into him, which so\nfar surpasses what is elsewhere to be found; for by what animal except\nman is even the existence of the gods discovered, who have produced and\nstill uphold in such regular order this beautiful and stupendous frame\nof the universe? What other creature is to be found that can serve and\nadore them?... In thee, Aristodemus, has been joined to a wonderful soul\na body no less wonderful; and sayest thou, after this, the gods take no\nthought for me? What wouldst thou, then, more to convince thee of their\ncare?'\n\n\"'I would they should send and inform me,' said Aristodemus, 'what\nthings I ought or ought not to do, in like manner as thou sayest they\nfrequently do to thee.'\"\n\nIn reply, Socrates shows that the revelations of God which are made in\nnature, in history, in consciousness, and by oracles, are made _for_ all\nmen and _to_ all men. He then concludes with these remarkable words:\n\"As, therefore, amongst men we make best trial of the affection and\ngratitude of our neighbor by showing him kindness, and make discovery of\nhis wisdom by consulting him in our distress, do thou, in like manner,\nbehave towards the gods; and if thou wouldst experience what their\nwisdom and their love, render thyself deserving of some of those divine\nsecrets which may not be penetrated by man, and are imparted to those\nalone who consult, who adore, and who obey the Deity. Then shalt thou,\nmy Aristodemus, understand _there is a Being whose eye passes through\nall nature, and whose ear is open to every sound; extended to all\nplaces, extending through all time; and whose bounty and care can know\nno other bounds than those fixed by his own creation_\".[482]\n\n[Footnote 482: Lewes's translation, in \"Biog. History of Philosophy,\"\npp. 160-165.]\n\nSocrates was no less earnest in his belief in the immortality of the\nsoul, and a state of future retribution. He had reverently listened to\nthe intuitions of his own soul--the instinctive longings and aspirations\nof his own heart, as a revelation from God. He felt that all the powers\nand susceptibilities of his inward nature were in conscious adaptation\nto the idea of immortality, and that its realization was the appropriate\ndestiny of man. He was convinced that a future life was needed to avenge\nthe wrongs and reverse the unjust judgments of the present life;[483]\nneeded that virtue may receive its meet reward, and the course of\nProvidence may have its amplest vindication. He saw this faith reflected\nin the universal convictions of mankind, and the \"common traditions\" of\nall ages.[484] No one refers more frequently than Socrates to the grand\nold mythologic stories which express this faith; to Minos, and\nRhadamanthus, and AEacus, and Triptolemus, who are \"real judges,\" and\nwho, in \"the Place of Departed Spirits, administer _justice_.\"[485] He\nbelieved that in that future state the pursuit of wisdom would be his\nchief employment, and he anticipated the pleasure of mingling in the\nsociety of the wise, and good, and great of every age.\n\n[Footnote 483: \"Apology,\" Sec. 32, p. 329 (Whewell's edition).]\n\n[Footnote 484: Ibid.]\n\n[Footnote 485: \"Apology,\" p. 330.]\n\nWhilst, then, Socrates was not the first to teach the doctrine of\nimmortality, because no one could be said to have first _discovered_ it\nany more than to have first discovered the existence of a God, he was\ncertainly the first to place it upon a philosophic basis. The Phaedo\npresents the doctrine and the _reasoning_ by which Socrates had elevated\nhis mind above the fear of death. Some of the arguments may be purely\nPlatonic, the argument especially grounded on \"ideas;\" still, as a\nwhole, it must be regarded as a tolerably correct presentation of the\nmanner in which Socrates would prove the immortality of the soul.\n\nIn _Ethics_, Socrates was pre-eminently himself. The systematic\nresolution of the whole theory of society into the elementary principle\nof natural law, was peculiar to him. _Justice_ was the cardinal\nprinciple which must lie at the foundation of all good government. The\nword sophia--_wisdom_--included all excellency in personal morals,\nwhether as manifested (reflectively) in the conduct of one's self, or\n(socially) towards others. And _Happiness_, in its purity and\nperfection, can only be found in virtuous action.[486]\n\n[Footnote 486: Butler's \"Lectures on Ancient Philosophy,\" vol. i. pp.\n360, 361.]\n\nSocrates left nothing behind him that could with propriety be called a\n_school_. His chief glory is that he inaugurated a new _method_ of\ninquiry, which, in Plato and Aristotle, we shall see applied. He gave a\nnew and vital impulse to human thought, which endured for ages; \"and he\nleft, as an inheritance for humanity, the example of a heroic life\ndevoted wholly to the pursuit of truth, and crowned with martyrdom.\"\n\n\n\n\nCHAPTER X.\n\nTHE PHILOSOPHERS OF ATHENS (_continued_).\n\nTHE SOCRATIC SCHOOL (_continued_).\n\nPLATO.\n\n\nWe have seen that the advent of Socrates marks a new era in the history\nof speculative thought. Greek philosophy, which at first was a\nphilosophy of nature, now changes its direction, its character, and its\nmethod, and becomes a philosophy of mind. This, of course, does not mean\nthat now it had mind alone for its object; on the contrary, it tended,\nas indeed philosophy must always tend, to the conception of a rational\nideal or _intellectual system of the universe_. It started from the\nphenomena of mind, began with the study of human thought, and it made\nthe knowledge of mind, of its ideas and laws, the basis of a higher\nphilosophy, which should interpret all nature. In other words, it\nproceeded from psychology, through dialectics, to ontology.[487]\n\n[Footnote 487: Cousin's \"Lectures on the History of Philosophy,\" vol. i.\np. 413.]\n\nThis new movement we have designated in general terms as the _Socratic\nSchool_. Not that we are to suppose that, in any technical sense,\nSocrates founded _a_ school. The Academy, the Lyceum, the Stoa, and the\nGarden, were each the chosen resort of distinct philosophic sects, the\nlocality of separate schools; but Athens itself, the whole city, was the\nscene of the studies, the conversations, and the labors of Socrates. He\nwandered through the streets absorbed in thought. Sometimes he stood\nstill for hours lost in profoundest meditation; at other times he might\nbe seen in the market-place, surrounded by a crowd of Athenians, eagerly\ndiscussing the great questions of the day.\n\nSocrates, then, was not, in the usual sense of the word, a teacher. He\nis not to be found in the Stoa or the Grove, with official aspect,\nexpounding a system of doctrine. He is \"the garrulous oddity\" of the\nstreets, putting the most searching and perplexing questions to every\nbystander, and making every man conscious of his ignorance. He delivered\nno lectures; he simply talked. He wrote no books; he only argued: and\nwhat is usually styled his school must be understood as embracing those\nwho attended him in public as listeners and admirers, and who caught his\nspirit, adopted his philosophic _method_, and, in after life, elaborated\nand systematized the ideas they had gathered from him.\n\nAmong the regular or the occasional hearers of Socrates were many who\nwere little addicted to philosophic speculation. Some were warriors, as\nNicias and Laches; some statesmen, as Critias and Critobulus; some were\npoliticians, in the worst sense of that word, as Glaucon; and some were\nyoung men of fashion, as Euthydemus and Alcibiades. These were all alike\ndelighted with his inimitable irony, his versatility of genius, his\ncharming modes of conversation, his adroitness of reply; and they were\ncompelled to confess the wisdom and justness of his opinions, and to\nadmire the purity and goodness of his life. The magic power which he\nwielded, even over men of dissolute character, is strikingly depicted by\nAlcibiades in his speech at \"the Banquet.\"[488] Of these listeners,\nhowever, we can not now speak. Our business is with those only who\nimbibed his philosophic spirit, and became the future teachers of\nphilosophy. And even of those who, as Euclid of Megara, and Antisthenes\nthe Cynic, and Aristippus of Cyrenaica, borrowed somewhat from the\ndialectic of Socrates, we shall say nothing. They left no lasting\nimpression upon the current of philosophic thought, because their\nsystems were too partial, and narrow, and fragmentary. It is in Plato\nand Aristotle that the true development of the Socratic philosophy is to\nbe sought, and in Plato chiefly, as the disciple and friend of Socrates.\n\n[Footnote 488: \"Banquet,\" Secs. 39, 40.]\n\nPlato (B.C. 430-347) was pre-eminently the pupil of Socrates. He came to\nSocrates when he was but twenty years of age, and remained with him to\nthe day of his death.\n\nDiogenes Laertius reports the story of Socrates having dreamed he found\nan unfledged cygnet on his knee. In a few moments it became winged and\nflew away, uttering a sweet sound. The next day a young man came to him\nwho was said to reckon Solon among his near ancestors, and who looked,\nthrough him, to Codrus and the god Poseidon. That young man was Plato,\nand Socrates pronounced him to be the bird he had seen in his\ndream.[489]\n\n[Footnote 489: Diogenes Laertius, \"Lives of the Philosophers,\" bk. iii.\nch. vii.]\n\nSome have supposed that this old tradition intimates that Plato departed\nfrom the method of his master--he became fledged and flew away into the\nair. But we know that Plato did not desert his master whilst he was\nliving, and there is no evidence that he abandoned his method after he\nwas dead. He was the best expounder and the most rigid observer of the\nSocratic \"organon.\" The influence of Socrates upon the philosophy of\nPlato is everywhere discernible. Plato had been taught by Socrates, that\nbeyond the world of sense there is a world of eternal truth, seen by the\neye of reason alone. He had also learned from him that the eye of reason\nis purified and strengthened by _reflection_, and that to reflect is to\nobserve, and analyze, and define, and classify the facts of\nconsciousness. Self-reflection, then, he had been taught to regard as\nthe key of real knowledge. By a completer induction, a more careful and\nexact analysis, and a more accurate definition, he carried this\nphilosophic method forward towards maturity. He sought to solve the\nproblem of _being_ by the principles revealed in his own consciousness,\nand in the _ultimate ideas of the reason_ to find the foundation of all\nreal knowledge, of all truth, and of all certitude.\n\nPlato was admirably fitted for these sublime investigations by the\npossession of those moral qualities which were so prominent in the\ncharacter of his master. He had that same deep seriousness of spirit,\nthat earnestness and rectitude of purpose, that longing after truth,\nthat inward sympathy with, and reverence for justice, and purity, and\ngoodness, which dwelt in the heart of Socrates, and which constrained\nhim to believe in their reality and permanence. He could not endure the\nthought that all ideas of right were arbitrary and factitious, that all\nknowledge was unreal, that truth was a delusion, and certainty a dream.\nThe world of sense might be fleeting and delusive, but the voice of\nreason and conscience would not mislead the upright man. The opinions of\nindividual men might vary, but the universal consciousness of the race\ncould not prevaricate. However conflicting the opinions of men\nconcerning beautiful things, right actions, and good sentiments, Plato\nwas persuaded there are ideas of Order, and Right, and Good, which are\nuniversal, unchangeable, and eternal. Untruth, injustice, and wrong may\nendure for a day or two, perhaps for a century or two, but they can not\nalways last; they must perish. The _just_ thing and the _true_ thing are\nthe only enduring things; these are eternal. Plato had a sublime\nconviction that his mission was to draw the Athenian mind away from the\nfleeting, the transitory, and the uncertain, and lead them to the\ncontemplation of an Eternal Truth, an Eternal Justice, an Eternal\nBeauty, all proceeding from and united in an Eternal Being--the ultimate\nagathon--_the Supremely Good_. The knowledge of this \"Supreme Good\" he\nregarded as the highest science.[490]\n\n[Footnote 490: \"Republic,\" bk. vi. ch. xvi. p. 193.]\n\nAdded to these moral qualifications, Plato had the further qualification\nof a comprehensive knowledge of all that had been achieved by his\npredecessors. In this regard he had enjoyed advantages superior to those\nof Socrates. Socrates was deficient in erudition, properly so called. He\nhad studied men rather than books. His wisdom consisted in an extensive\n_observation_, the results of which he had generalized with more or less\naccuracy. A complete philosophic method demands not only a knowledge of\ncontemporaneous opinions and modes of thought, but also a knowledge of\nthe succession and development of thought in past ages. Its instrument\nis not simply psychological analysis, but also historical analysis as a\ncounterproof.[491] And this erudition Plato supplied. He studied\ncarefully the doctrines of the Ionian, Italian, and Eleatic schools.\nCratylus gave him special instruction in the theories of\nHeraclitus.[492] He secured an intimate acquaintance with the lofty\nspeculations of Pythagoras, under Archytas of Tarentum, and in the\nwritings of Philolaus, whose books he is said to have purchased. He\nstudied the principles of Parmenides under Hermogenes,[493] and he more\nthan once speaks of Parmenides in terms of admiration, as one whom he\nhad early learned to reverence.[494] He studied mathematics under\nTheodoras, the most eminent geometrician of his day. He travelled in\nSouthern Italy, in Sicily, and, in search of a deeper wisdom, he pursued\nhis course to Egypt.[495] Enriched by the fruits of all previous\nspeculations, he returned to Athens, and devoted the remainder of his\nlife to the development of a comprehensive system \"which was to combine,\nto conciliate, and to supersede them all.\"[496] The knowledge he had\nderived from travel, from books, from oral instruction, he fused and\nblended with his own speculations, whilst the Socratic spirit mellowed\nthe whole, and gave to it a unity and scientific completeness which has\nexcited the admiration and wonder of succeeding ages.[497]\n\n[Footnote 491: Cousin's \"Lectures on the History of Philosophy,\" vol. i.\np. 31.]\n\n[Footnote 492: Aristotle's \"Metaphysics,\" bk. i. ch. vi.]\n\n[Footnote 493: Diogenes Laertius, \"Lives of the Philosophers,\" bk. iii.\nch. viii. p. 115.]\n\n[Footnote 494: See especially \"Theaetetus,\" Sec. 101.]\n\n[Footnote 495: Ritter's \"History of Ancient Philosophy,\" vol. ii. p.\n147.]\n\n[Footnote 496: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n22.]\n\n[Footnote 497: Encyclopaedia Britannica, article \"Plato.\"]\n\nThe question as to _the nature, the sources, and the validity of human\nknowledge_ had attracted general attention previous to the time of\nSocrates and Plato. As the results of this protracted controversy, the\nopinions of philosophers had finally crystallized in two well-defined\nand opposite theories of knowledge.\n\n1. That which reduced all knowledge to the accidental and passively\nreceptive quality of the organs of sense and which asserted, as its\nfundamental maxim, that \"_Science consists in_\naisthesis--_sensation_.\"[498]\n\nThis doctrine had its foundation in the physical philosophy of\nHeraclitus. He had taught that all things are in a perpetual flux and\nchange. \"Motion gives the appearance of existence and of generation.\"\n\"Nothing _is_, but is always a _becoming\"_[499] Material substances are\nperpetually losing their identity, and there is no permanent essence or\nbeing to be found. Hence Protagoras inferred that truth must vary with\nthe ever-varying sensations of the individual. \"Man (the individual) is\nthe measure of all things.\" Knowledge is a purely relative thing, and\nevery man's opinion is truth for him.[500] The law of right, as\nexemplified in the dominion of a party, is the law of the strongest;\nfluctuating with the accidents of power, and never attaining a permanent\nbeing. \"Whatever a city enacts as appearing just to itself, this also is\njust to the city that enacts it, so long as it continues in force.\"[501]\n\"The just, then, is nothing else but that which is expedient for the\nstrongest.\"[502]\n\n[Footnote 498: \"Theaetetus,\" Sec. 23.]\n\n[Footnote 499: Ibid., Secs. 25, 26.]\n\n[Footnote 500: Ibid., Secs. 39, 87.]\n\n[Footnote 501: Ibid., Sec. 87.]\n\n[Footnote 502: \"Republic,\" bk. i. ch. xii.]\n\n2. The second theory is that which denies the existence (except as\nphantasms, images, or mere illusions of the mind) of the whole of\nsensible phenomena, and refers all knowledge to the _rational\napperception of unity_ (to en) _or the One_.\n\nThis was the doctrine of the later Eleatics. The world of sense was, to\nParmenides and Zeno, a blank negation, the _non ens_. The identity of\nthought and existence was the fundamental principle of their philosophy.\n\n      \"Thought is the same thing as the cause of thought; For\n      without the thing in which it is announced, You can not find\n      the thought; for there is nothing, nor shall be, Except the\n      existing.\"[503]\n\n[Footnote 503: Parmenides, quoted in Lewes's \"Biog. History of\nPhilosophy,\" p. 54.]\n\nThis theory, therefore, denied to man any valid knowledge of the\nexternal world.\n\nIt will at once be apparent to the intelligent reader that the direct\nand natural result of both these theories[504] of knowledge was a\ntendency to universal skepticism. A spirit of utter indifference to\ntruth and righteousness was the prevailing spirit of Athenian society.\nThat spirit is strikingly exhibited in the speech of Callicles, \"the\nshrewd man of the world,\" in \"Gorgias\" (Sec.85, 86). Is this new to our\nears?\" My dear Socrates, you talk of _law_. Now the laws, in my\njudgment, are just the work of the weakest and most numerous; in framing\nthem they never thought but of themselves and their own interests; they\nnever approve or censure except in reference to _this._ Hence it is that\nthe cant arises that tyranny is improper and unjust, and to struggle for\neminence, guilt. Unable to rise themselves, of course they would wish to\npreach liberty and equality. But nature proclaims the law of the\nstronger.... We surround our children from their infancy with\npreposterous prejudices about liberty and justice. The man of sense\ntramples on such impositions, and shows what Nature's justice is.... I\nconfess, Socrates, philosophy is a highly amusing study--in moderation,\nand for boys. But protracted too long, it becomes a perfect plague. Your\nphilosopher is a complete novice in the life _comme il faut_.... I like\nvery well to see a child babble and stammer; there is even a grace about\nit when it becomes his age. But to see a man continue the prattle of the\nchild, is absurd. Just so with your philosophy.\" The consequence of this\nprevalent spirit of universal skepticism was a general laxity of morals.\nThe Aleibiades, of the \"_Symposium_,\" is the ideal representative of the\nyoung aristocracy of Athens. Such was the condition of society\ngenerally, and such the degeneracy of even the Government itself, that\nPlato impressively declares \"that God alone could save the young men of\nhis age from ruin.\"[505]\n\n[Footnote 504: Between these two extreme theories there were offered\ntwo, apparently less extravagant, accounts of the nature and limits of\nhuman knowledge--one declaring that \"_Science_(real knowledge) _consists\nin right opinion_\" (doxa alethes), but having no further basis in the\nreason of man (\"Theaestetus,\" Sec. 108); and the other affirming that\n\"_Science is right opinion with logical explication or definition_\"\n(meta loxou, \"Theaetetus,\" Sec. 139). A close examination will, however,\nconvince us that these are but modifications of the sensational theory.\nThe latter forcibly remind us of the system of Locke, who adds\n\"reflection\" to \"sensation,\" but still maintains that all on \"simple\nideas\" are obtained from without, and that these are the only material\nupon which reflection can be exercised. Thus the human mind has no\ncriterion of truth within itself, no elements of knowledge which are\nconnatural and inborn.]\n\n[Footnote 505: \"Republic,\" bk. vi. ch. vii.]\n\nTherefore the grand, the vital, the most urgent question for his times,\nas indeed for all times, was, _What is Truth? What is Right_? In the\nmidst of all this variableness and uncertainty of human opinion, is\nthere no ground of certainty? Amid all the fluctuations and changes\naround us and within us, is there nothing that is immutable and\npermanent? Have we no ultimate standard of Right? Is there no criterion\nof Truth? Plato believed most confidently there was such a criterion and\nstandard. He had learned from Socrates, his master, to cherish an\nunwavering faith in the existence of an Eternal Truth, an Eternal Order,\nan Eternal Good, the knowledge of which is essential to the perfection\nand happiness of man, and which knowledge must therefore be presumed to\nbe attainable by man. Henceforth, therefore, the ceaseless effort of\nPlato's life is to attain a standard (kriterion)[506]--a CRITERION OF\nTRUTH.\n\n[Footnote 506: \"Theaetetus,\" Sec. 89.]\n\nAt the outset of his philosophic studies, Plato had derived from\nSocrates an important principle, which became the guide of all his\nsubsequent inquiries. He had learned from him that the criterion of\ntruth must be no longer sought amid the ever-changing phenomena of the\n\"sensible world.\" This had been attempted by the philosophers of the\nIonian school, and ended in failure and defeat. It must therefore be\nsought in the metaphenomenal--the \"intelligible world;\" that is, it must\nbe sought in the apperceptions of the reason, and not in opinions\nfounded on sensation. In other words, he must look _within_. Here, by\nreflection, he could recognize, dimly and imperfectly at first, but\nincreasing gradually in clearness and distinctness, two classes of\ncognitions, having essentially distinct and opposite characteristics. He\nfound one class that was complex (synkegumenon), changeable (thateron),\ncontingent and relative (ta pros ti schesin echonta); the other, simple\n(kexorismenon), unchangeable (akineton), constant (tauton), permanent\n(to on aei), and absolute (anypotheton = aploun). One class that may be\nquestioned, the other admitting of no question, because self-evident and\nnecessary, and therefore compelling belief. One class grounded on\nsense-perception, the other conceived by reason alone. But whilst the\nreason recognizes, it does not create them. They are not particular and\nindividual, but universal. They belong not to the man, but to the race.\n\nHe found, then, that there are in all minds certain \"principles\" which\nare fundamental--principles which lie at the basis of all our cognitions\nof the objective world, and which, as \"mental laws,\" determine all our\nforms of thought; and principles, too, which have this marvellous and\nundeniable character, that they are encountered in the most common\nexperiences, and, at the same time, instead of being circumscribed\nwithin the limits of experience, transcend and govern it--principles\nwhich are _universal_ in the midst of particular phenomena--_necessary,_\nthough mingled with things contingent--to our eyes _infinite_ and\n_absolute_, even when appearing in us the relative and finite beings\nthat we are.[507] These first or fundamental principles Plato called\nIDEAS (ideai).\n\n[Footnote 507: Cousin's \"The True, the Beautiful, and the Good,\" p. 40.]\n\nIn attempting to present to the reader an adequate representation of the\nPlatonic Ideas, we shall be under the necessity of anticipating some of\nthe results of his Dialectical method before we have expounded that\nmethod. And, further, in order that it may be properly appreciated by\nthe modern student, we shall avail ourselves of the lights which modern\npsychology, faithful to the method of Plato, has thrown upon the\nsubject. Whilst, however, we admit that modern psychology has succeeded\nin giving more definiteness and precision to the \"doctrine of Ideas,\" we\nshall find that all that is fundamentally valuable and true was present\nto the mind of Plato. Whatever superiority the \"Spiritual\" philosophy of\nto-day may have over the philosophy of past ages, it has attained that\nsuperiority by its adherence to the principles and method of Plato.\n\nIn order to the completeness of our preliminary exposition of the\nPlatonic doctrine of Ideas, we shall conditionally assume, as a natural\nand legitimate hypothesis, the doctrine so earnestly asserted by Plato,\nthat the visible universe, at least in its present form, is an _effect_\nwhich must have had a _cause_,[508] and that the Order, and Beauty, and\nExcellence of the universe are the result of the presence and operation\nof a \"regulating Intelligence\"--a _Supreme Mind_.[509] Now that,\nanterior to the creation of the universe, there must have existed in the\nEternal Mind certain fundamental principles of Order, Right, and Good,\nwill not be denied. Every conceivable _form_, every possible _relation_,\nevery principle of _right_, must have been eternally present to the\nDivine thought. As pure intelligence, the Deity must have always been\nself-conscious--must have known himself as substance and cause, as the\nInfinite and Perfect. If then the Divine Energy is put forth in creative\nacts, that energy must obey those eternal principles of Order, Right,\nand Good. If the Deity operate at all, he must operate rightly, wisely,\nand well. The created universe must be an _image_, in the sphere of\nsense, of the ideas which inhere in the reason of the great First Cause.\n\n[Footnote 508: \"Timaeus,\" ch. ix.]\n\n[Footnote 509: \"Phaedo,\" Sec. 105.]\n\n\"Let us declare,\" says Plato, \"with what _motive_ the Creator hath\nformed nature and the universe. He was _good_, and in the good no manner\nof envy can, on any subject, possibly subsist. Exempt from envy, he had\nwished that all things should, as far as possible, _resemble\nhimself_.... It was not, and is not to be allowed for the Supremely Good\nto do any thing except what is most _excellent_ (kalliston)--most\n_fair_, most _beautiful_.\"[510] Therefore, argues Plato, \"inasmuch as\nthe world is the most beautiful of things, and its artificer the best of\ncauses, it is evident that the Creator and Father of the universe looked\nto the _Eternal Model_(paradeigma), pattern, or plan,\"[511] which lay in\nhis own mind. And thus this one, only-generated universe, is the _image_\n(eikon) of that God who is the object of the intellect, the greatest,\nthe best, and the most perfect Being.[512]\n\n[Footnote 510: \"Timaeus,\" ch. x.]\n\n[Footnote 511: Ibid., ch. ix.]\n\n[Footnote 512: \"Timaeus,\" ch. lxxiii.]\n\nAnd then, furthermore, if this Supreme Intelligence, this Eternal Mind,\nshall create another _mind_, it must, in a still higher degree, resemble\nhim. Inasmuch as it is a rational nature, it must, in a peculiar sense,\npartake of the Divine characteristics. \"The soul,\" says Plato, \"is that\nwhich most partakes of the _Divine_\"[513] The soul must, therefore, have\nnative _ideas_ and sentiments which correlate it with the Divine\noriginal. The ideas of substance and cause, of unity and identity, of\nthe infinite and perfect, must be mirrored there. As it is the\n\"offspring of God,\"[514] it must bear some traces and lineaments of its\nDivine parentage. That soul must be configured and correlated to those\nprinciples of Order, Right, and Good which dwell in the Eternal Mind.\nAnd because it has within itself the same ideas and laws, according to\nwhich the great Architect built the universe, therefore it is capable of\nknowing, and, in some degree, of comprehending, the intellectual system\nof the universe. It apprehends the external world by a light which the\nreason supplies. It interprets nature according to principles and laws\nwhich God has inwrought within the very essence of the soul. \"That which\nimparts truth to knowable things, and gives the knower his power of\nknowing truth, is the _idea of the good_, and you are to conceive of\nthis as the source of knowledge and of truth.\"[515]\n\n[Footnote 513: \"Laws,\" bk. v. ch. i.]\n\n[Footnote 514: Ibid., bk. x.]\n\n[Footnote 515: \"Republic,\" bk. vi. ch. xviii.]\n\nAnd now we are prepared to form a clear conception of the Platonic\ndoctrine of Ideas. Viewed in their relation to the Eternal Reason, as\ngiving the primordial thought and law of all being, these principles are\nsimply eide auta kath auta--_ideas in themselves_--the essential\nqualities or attributes of Him who is the supreme and ultimate Cause of\nall existence. When regarded as before the Divine imagination, giving\ndefinite forms and relations, they are the tupoi, the paradeigmata--_the\ntypes_, _models, patterns, ideals_ according to which the universe was\nfashioned. Contemplated in their actual embodiment in the laws, and\ntypical forms of the material world, they are eikones--_images_ of the\neternal perfections of God. The world of sense pictures the world of\nreason by a participation (methexis) of the ideas. And viewed as\ninterwoven in the very texture and framework of the soul, they are\nomoiomata--copies of the Divine Ideas which are the primordial laws of\nknowing, thinking, and reasoning. Ideas are thus the nexus of relation\nbetween God and the visible universe, and between the human and the\nDivine reason.[516] There is something divine in the world, and in the\nhuman soul, namely, _the eternal laws and reasons of things_, mingled\nwith the endless diversity and change of sensible phenomena. These ideas\nare \"the light of the intelligible world;\" they render the invisible\nworld of real Being perceptible to the reason of man. \"Light is the\noffspring of the Good, which the Good has produced in his own likeness.\nLight in the visible world is what the _idea of the Good_ is in the\nintelligible world. And this offspring of the Good--light--has the same\nrelation to vision and visible things which the Good has to intellect\nand intelligible things.\"[517]\n\n[Footnote 516: \"Now, Idea is, as regards God, a mental operation by him\n(the notions of God, eternal and perfect in themselves); as regards us,\nthe first things perceptible by mind; as regards Matter, a standard; but\nas regards the world, perceptible by sense, a pattern; but as considered\nwith reference to itself, an existence.\"--Alcinous, \"Introduction to the\nDoctrines of Plato,\" p. 261.\n\n\"What general notions are to our minds, he (Plato) held, ideas are to\nthe Supreme Reason (nous basileus); they are the eternal thoughts of the\nDivine Intellect, and we attain truth when our thoughts conform with\nHis--when our general notions are in conformity with the\nideas.\"--Thompson, \"Laws of Thought,\" p. 119.]\n\n[Footnote 517: \"Republic,\" bk. vi. ch. xix.]\n\n_Science_ is, then, according to Plato, _the knowledge of universal,\nnecessary, unchangeable, and eternal ideas_. The simple cognition of the\nconcrete phenomena of the universe is not regarded by him as _real_\nknowledge. \"Science, or real knowledge, belongs to _Being_, and\nignorance to _non_-Being.\" Whilst that which is conversant only \"with\nthat which partakes of both--of being and non-being--and which can not\nbe said either to be or not to be\"--that which is perpetually\n\"becoming,\" but never \"really is,\" is \"simply _opinion_, and not real\nknowledge.\"[518] And those only are \"philosophers\" who have a knowledge\nof the _really-existing_, in opposition to the mere seeming; of the\n_always-existing_, in opposition to the transitory; and of that which\nexists _permanently_, in opposition to that which waxes and wanes--is\ndeveloped and destroyed alternately. \"Those who recognize many beautiful\nthings, but who can not see the Beautiful itself, and can not even\nfollow those who would lead them to it, they _opine_, but do not _know_.\nAnd the same may be said of those who recognize right actions, but do\nnot recognize an absolute righteousness. And so of other ideas. But they\nwho look at these ideas--permanent and unchangeable ideas--these men\n_really know_.\"[519] Those are the true philosophers alone who love the\nsight of truth, and who have attained to the vision of the eternal\norder, and righteousness, and beauty, and goodness in the Eternal Being.\nAnd the means by which the soul is raised to this vision of real Being\n(to ontos on) is THE SCIENCE OF REAL KNOWLEDGE.\n\nPlato, in the \"Theaetetus,\" puts this question by the interlocutor\nSocrates, \"What is Science (Episteme) or positive knowledge?\"[520]\nTheaetetus essays a variety of answers, such as, \"Science is sensation,\"\n\"Science is right judgment or opinion,\" \"Science is right opinion with\nlogical definition.\" These, in the estimation of the Platonic Socrates,\nare all unsatisfactory and inadequate. But after you have toiled to the\nend of this remarkable discussion, in which Socrates demolishes all the\nthen received theories of knowledge, he gives you no answer of his own.\nHe abruptly closes the discussion by naively remarking that, at any\nrate, Theaetetus will learn that he does not understand the subject; and\nthe ground is now cleared for an original investigation.\n\n[Footnote 518: \"Republic,\" bk. v. ch. xx.]\n\n[Footnote 519: Ibid., bk. v. ch. xxii.]\n\n[Footnote 520: \"Theaetetus,\" Sec. 10.]\n\nThis investigation is resumed in the \"Republic.\" This greatest work of\nPlato's was designed not only to exhibit a scheme of Polity, and present\na system of Ethics, but also, at least in its digressions, to propound a\nsystem of Metaphysics more complete and solid than had yet appeared. The\ndiscussion as to the _powers_ or _faculties_ by which we obtain\nknowledge, the _method_ or _process_ by which real knowledge is\nattained, and the ultimate _objects_ or _ontological grounds_ of all\nreal knowledge, commences at Sec. 18, book v., and extends to the end of\nbook vii.\n\nThat we may reach a comprehensive view of this \"sublimest of sciences,\"\nwe shall find it necessary to consider--\n\n1st. _What are the powers or faculties by which we obtain knowledge, and\nwhat are the limits and degrees of human knowledge?_\n\n2d. _What is the method in which, or the processes and laws according to\nwhich, the mind operates in obtaining knowledge?_\n\n3d. _What are the ultimate results attained by this method? what are the\nobjective and ontological grounds of all real knowledge?_\n\nThe answer to the first question will give the PLATONIC PSYCHOLOGY; the\nanswer to the second will exhibit the PLATONIC DIALECTIC; the answer to\nthe last will reveal the PLATONIC ONTOLOGY.\n\nI. PLATONIC PSYCHOLOGY.\n\nEvery successful inquiry as to the reality and validity of human\nknowledge must commence by clearly determining, by rigid analysis, what\nare the actual phenomena presented in consciousness, what are the powers\nor faculties supposed by these phenomena, and what reliance are we to\nplace upon the testimony of these faculties? And, especially, if it be\nasserted that there is a science of absolute Reality, of ultimate and\nessential Being, then the most important and vital question is, By what\npower do we cognize real Being? through what faculty do we obtain the\nknowledge of that which absolutely _is_? If by sensation we only obtain\nthe knowledge of the fleeting and the transitory, \"_the becoming_\" how\ndo we attain to the knowledge of the unchangeable and permanent, \"the\n_Being_?\" Have we a faculty of universal, necessary, and eternal\nprinciples? Have we a faculty, an interior eye which beholds \"_the\nintelligible_,\" ideal, spiritual world, as the eye of sense beholds the\nvisible or \"_sensible world_?\"[521]\n\nPlato commences this inquiry by first defining his understanding of the\nword dynamis--_power_ or _faculty_. \"We will say _faculties_ (dynameis)\nare a certain kind of real existences by which we can do whatever we are\nable (_e.g._, to know), as there are powers by which every thing does\nwhat it does: the eye has a _power_ of seeing; the ear has a _power_ of\nhearing. But these powers (of which I now speak) have no color or figure\nto which I can so refer that I can distinguish one power from another.\n_In order to make such distinction, I must look at the power itself, and\nsee what it is, and what it does. In that way I discern the power of\neach thing, and that is the same power which produces the same effect,\nand that is a different power which produces a different effect_.\"[522]\nThat which is employed about, and accomplishes one and the same purpose,\nthis Plato calls a _faculty_.\n\n[Footnote 521: \"Republic,\" bk. vi. ch. xviii.]\n\n[Footnote 522: Ibid., bk. v. ch. xxi.]\n\nWe have seen that our first conceptions (_i.e._, first in the order of\ntime) are of the mingled, the concrete (to synkechymenon), \"the\nmultiplicity of things to which the multitude ascribe beauty, etc.[523]\nThe mind \"contemplates what is great and small, not as distinct from\neach other, but as confused.[524] Prior to the discipline of\n_reflection_, men are curious about mere sights and sounds, love\nbeautiful voices, beautiful colors, beautiful forms, but their\nintelligence can not see, can not embrace, the essential nature of the\nBeautiful itself.[525] Man's condition previous to the education of\nphilosophy is vividly presented in Plato's simile of the cave.[526] He\nbeholds only the images and shadows of the ectypal world, which are but\ndim and distant adumbrations of the real and archetypal world.\n\nPrimarily nothing is given in the abstract (to kegorismenon), but every\nthing in the concrete. The primary faculties of the mind enter into\naction spontaneously and simultaneously; all our primary notions are\nconsequently synthetic. When reflection is applied to this primary\ntotality of consciousness, that is, when we analyze our notions, we find\nthem composed of diverse and opposite elements, some of which are\nvariable, contingent, individual, and relative, others are permanent,\nunchangeable, universal, necessary, and absolute. Now these elements, so\ndiverse, so opposite, can not have been obtained from the same source;\nthey must be supplied by separate powers. \"Can any man with common sense\nreduce under one what _is infallible_, and what is _not\ninfallible?_\"[527] Can that which is \"_perpetually becoming_\" be\napprehended by the same faculty as that which \"_always is?_\"[528] Most\nassuredly not.\n\n[Footnote 523: Ibid., bk. v. ch. xxii.]\n\n[Footnote 524: Ibid., bk. vii. ch. viii.]\n\n[Footnote 525: Ibid., bk. v. ch. xx.]\n\n[Footnote 526: Ibid., bk. vii. ch. i., ii.]\n\n[Footnote 527: \"Republic,\" bk. v. ch. xxi.]\n\n[Footnote 528: Ibid., bk. v. ch. xxii.; also \"Timaeus,\" Sec. 9.]\n\nThese primitive intuitions--the simple perceptions of sense, and the _a\npriori_ intuitions of the reason, which constitute the elements of all\nour complex notions, have essentially _diverse objects_--the sensible or\nectypal world, seen by the eye and touched by the hand, which Plato\ncalls doxasten--_the subject of opinion_; and the noetic or archetypal\nworld, perceived by reason, and which he calls dianontiken--_the subject\nof rational intuition or science_. \"It is plain,\" therefore, argues\nPlato, \"that _opinion_ is a different thing from _science_. They must,\ntherefore, have a different _faculty_ in reference to a different\nobject--science as regards that which _is_, so as to know the nature of\nreal _being_--opinion as regards that which can not be said absolutely\nto be, or not to be. That which is known and that which is opined can\nnot possibly be the same,... since they are naturally faculties of\ndifferent things, and both of them are faculties--_opinion_ and\n_science_, and each of them different from the other.\"[529] Here then\nare two grand divisions of the mental powers--a faculty of apprehending\nuniversal and necessary Truth, of intuitively beholding absolute\nReality, and a faculty of perceiving sensible objects, and of judging\naccording to appearance.\n\n[Footnote 529: Ibid., bk. v. ch. xxi., xxii.]\n\nAccording to the scheme of Plato, these two general divisions of the\nmental powers are capable of a further subdivision. He says: Consider\nthat there are two kinds of things, the _intelligible_ and the\n_visible_; two different regions, the intelligible world and the\nsensible world. Now take a line divided into two equal segments to\nrepresent these two regions, and again divide each segment in the same\nratio--both that of the visible and that of the intelligible species.\nThe parts of each segment are to represent differences of clearness and\nindistinctness. In the visible world the parts are _things_ and\n_images_. By _images_ I mean shadows,[530] reflections in water and in\npolished bodies, and all such like representations; and by _things_ I\nmean that of which images are resemblances, as animals, plants, and\nthings made by man.\n\nYou allow that this difference corresponds to the difference of\n_knowledge_ and _opinion_; and the _opinionable_ is to the _knowable_ as\nthe _image_ to the _reality_.[531]\n\n[Footnote 530: As in the simile of the cave (\"Republic,\" bk. vii. ch. i.\nand ii.).]\n\n[Footnote 531: The analogy between the \"images produced by reflections\nin water and on polished surfaces\" and \"the images of external objects\nproduced in the mind by sensation\" is more fully presented in the\n\"Timaeus,\" ch. 19.\n\nThe eye is a light-bearer, \"made of that part of elemental fire which\ndoes not burn, but sheds a mild light, like the light of day.... When\nthe light of the day meets the light which beams from the eye, then\nlight meets like, and make a homogeneous body; the external light\nmeeting the internal light, in the direction in which the eye looks. And\nby this homogeneity like feels like; and if this beam touches any\nobject, or any object touches it, it transmits the motions through the\nbody to the soul, and produces that sensation which we call _seeing_....\nAnd if (in sleep) some of the strong motions remain in some part of the\nframe, they produce within us likenesses of external objects,... and\nthus give rise to dreams.... As to the images produced by mirrors and by\nsmooth surfaces, they are now easily explained, for all such phenomena\nresult from the mutual affinity of the external and internal fires. The\nlight that proceeds from the face (as an object of vision), and the\nlight that proceeds from the eye, become one continuous ray on the\nsmooth surface.\"]\n\nNow we have to divide the segment which represents intelligible things\nin this way: The one part represents the knowledge which the mind gets\nby using things as images--the other; that which it has by dealing with\nthe ideas themselves; the one part that which it gets by reasoning\ndownward from principles--the other, the principles themselves; the one\npart, truth which depends on hypotheses--the other, unhypothetical or\nabsolute truth.\n\nThus, to explain a problem in geometry, the geometers make certain\nhypotheses (namely, definitions and postulates) about numbers and\nangles, and the like, and reason from them--giving no reason for their\nassumptions, but taking them as evident to all; and, reasoning from\nthem, they prove the propositions which they have in view. And in such\nreasonings, they use visible figures or diagrams--to reason about a\nsquare, for instance, with its diagonals; but these reasonings are not\nreally about these visible figures, but about the mental figures, and\nwhich they conceive in thought.\n\nThe diagrams which they draw, being visible, are the images of thoughts\nwhich the geometer has in his mind, and these images he uses in his\nreasoning. There may be images of these images--shadows and reflections\nin water, as of other visible things; but still these diagrams are only\nimages of conceptions.\n\nThis, then, is _one_ kind of intelligible things: _conceptions_--for\ninstance, geometrical conceptions of figures. But in dealing with these\nthe mind depends upon assumptions, and does not ascend to first\nprinciples. It does not ascend above these assumptions, but uses images\nborrowed from a lower region (the visible world), these images being\nchosen so as to be as distinct as may be.\n\nNow the _other_ kind of intelligible things is this: that which the\n_Reason_ includes, in virtue of its power of reasoning, when it regards\nthe assumptions of the sciences as (what they are) assumptions only, and\nuses them as occasions and starting-points, that from these it may\nascend to the _Absolute_, which does not depend upon assumption, the\norigin of scientific truth.\n\n_The reason takes hold of this first principle of truth_, and availing\nitself of all the connections and relations of this principle, it\nproceeds to the conclusion--using no sensible image in doing this, but\ncontemplates the _idea alone_; and with these ideas the process begins,\ngoes on, and terminates.\n\n\"I apprehend,\" said Glaucon, \"but not very clearly, for the matter is\nsomewhat abstruse. _You wish to prove that the knowledge which by the\nreason, in an intuitive manner, we may acquire of real existence and\nintelligible things is of a higher degree of certainty than the\nknowledge which belongs to what are commonly called the Sciences_. Such\nsciences, you say, have certain assumptions for their basis; and these\nassumptions are by the student of such sciences apprehended not by\nsense, but by a mental operation--by conception.\n\n\"But inasmuch as such students ascend no higher than assumptions, and do\nnot go to the first principles of truth, they do not seem to have true\nknowledge, intellectual insight, intuitive reason, on the subjects of\ntheir reasonings, though the subjects are intelligible things. And you\ncall this habit and practice of the geometers and others by the name of\nJUDGMENT (dianoia), not reason, or insight, or intuition--taking\njudgment to be something between opinion, on the one side, and intuitive\nreason, on the other.\n\n\"You have explained it well,\" said I. \"And now consider these four kinds\nof things we have spoken of, as corresponding to four affections (or\nfaculties) of the mind. INTUITIVE REASON (noesis), the highest; JUDGMENT\n(dianoia)(or _discursive reason_), the next; the third, BELIEF (pistis);\nand the fourth, CONJECTURE, or _guess_ (eikasia); and arrange them in\norder, so that they may be held to have more or less certainty, as their\nobjects have more or less truth.\"[532] The completeness, and even\naccuracy of this classification of all the objects of human cognition,\nand of the corresponding mental powers, will be seen at once by studying\nthe diagram proposed by Plato, as figured on the opposite page.\n\n[Footnote 532: \"Republic,\" bk. vi. ch. xx. and xxi.]\n\nPLATONIC SCHEME OF THE OBJECTS OF COGNITION, AND THE RELATIVE MENTAL POWERS\n___________________________________________________________________________\n            |                                 |\n            | VISIBLE WORLD                   | INTELLIGIBLE WORLD\n            | (the object of Opinion--doxa).  |(the object of Knowledge or\n            |                                 | Science--ipytteme).\n            |_________________________________|____________________________\n            |                |                |              |\n            | Things.        | Images.        | Intuitions.  | Conceptions.\n____________|________________|________________|______________|_____________\n\nAnd may be thus further expanded:\n___________________________________________________________________________\n            |                                 |\n            | VISIBLE WORLD.                  |   INTELLIGIBLE WORLD.\n____________|_________________________________|____________________________\n            |                |                |             |\n            | Things         | Images         | Ideas       | Conceptions\nOBJECT      |                |                |             |\n            | zoa. k. t. l.  | icones.        | ideai.      | duenoemata.\n____________|________________|________________|_____________|______________\n            |                |                |             |\n            | Belief.        | Conjecture.    | Intuition.  |Demonstration.\nPROCESS     |                |                |             |\n            | piotis.        | eikasia.       | noesis.     | ipisieie.\n____________|________________|________________|_____________|______________\n            |                |                |             |\n            | SENSATION.     | PHANTASY.      | INTUITIVE   | DISCURSIVE\nFACULTY     |                |                |   REASON.   |    REASON.\n            | aisthesis.     | phantasia.     | nous.       | logos.\n____________|________________|________________|_____________|______________\n            |                |                |             |\nMODERN      | SENSE.         | IMAGINATION.   | REASON.     | JUDGMENT.\nNOMENCLATURE|Presentative    |Representative  |Regulative   | Logical\n            | Faculty.       | Faculty        | Faculty.    |    Faculty.\n____________|________________|________________|_____________|______________\n            |                                 |\n            | MEMORY.                         | REMINISCENCE\n            | mneme.                          |   anamesis.\n            | The Conservative Faculty--      | The Reproductive Faculty--\n            | \"the preserver of sensation\"    |\"the recollection of the\n            | (soteria aisin, seos.) [533]    | things which the soul\n            |                                 | saw (in Eternity) when\n            |                                 | journeying in the train of\n            |                                 | the Deity.\"[534]\n            |[Footnote 533: \"Philebus,\" Sec. 67] | [Footnote 534: Phaedrus,\n            |                                 |                     Sec. 62.]\n____________|_________________________________|____________________________\n\n\nThe foregoing diagram, borrowed from Whewell, with some modifications\nand additions we have ventured to make, exhibits a perfect view of the\nPlatonic scheme of the _cognitive powers_--the faculties by which the\nmind attains to different degrees of knowledge, \"having more or less\ncertainty, as their objects have more or less truth.\"[535]\n\n1st. SENSATION (aisthesis).--This term is employed by Plato to denote\nthe passive mental states or affections which are produced within us by\nexternal objects through the medium of the vital organization, and also\nthe cognition or vital perception or consciousness[536] which the mind\nhas of these mental states.\n\n2d. PHANTASY (phantasia).--This term is employed to describe the power\nwhich the mind possesses of imagining or representing whatever has once\nbeen the object of sensation. This may be done involuntarily as \"in\ndreams, disease, and hallucination,\"[537] or voluntarily, as in\nreminiscence. Phantasmata are the images, the life-pictures (zographena)\nof sensible things which are present to the mind, even when no external\nobject is present to the sense.\n\n[Footnote 535: \"Republic,\" bk. vii. ch. xix.]\n\n[Footnote 536: \"In Greek philosophy there was no term for\n'consciousness' until the decline of philosophy, and in the latter ages\nof the language. Plato and Aristotle, to say nothing of other\nphilosophers, had no special term to express the knowledge which the\nmind has of the operation of its own faculties, though this, of course,\nwas necessarily a frequent matter of consideration. Intellect was\nsupposed by them to be cognizant of its own operations.... In his\n'Theaetetus' Plato accords to sense the power of perceiving that it\nperceives.\"--Hamilton's \"Metaphysics,\" vol. i. p. 198 (Eng. ed.).]\n\n[Footnote 537: \"Theaetetus,\" Sec. 39.]\n\nThe conjoint action of these two powers results in what Plato calls\n_opinion_ (doxa). \"Opinion is the complication of memory and sensation.\nFor when we meet for the first time with a thing perceptible by a sense,\nand a sensation is produced by it, and from this sensation a memory, and\nwe subsequently meet again with the same thing perceived by a sense, we\ncombine the memory previously brought into action with the sensation\nproduced a second time, and we say within ourselves [this is] Socrates,\nor a horse, or fire, or whatever thing there may be of such a kind. Now\nthis is called _opinion_, through our combining the recollection brought\npreviously into action with the sensation recently produced. And when\nthese, placed along each other, agree, a true opinion is produced; but\nwhen they swerve from each other, a false one.\"[538] The dixa of Plato,\ntherefore answers to the experience, or the _empirical knowledge_ of\nmodern philosophy, which is concerned only with appearances (phenomena),\nand not with absolute realities, and can not be elevated to the dignity\nof _science_ or real knowledge.\n\nWe are not from hence to infer that Plato intended to deny all reality\nwhatever to the objects of sensible experience. These transitory\nphenomena were not real existences, but they were _images_ of real\nexistences. The world itself is but the image, in the sphere of sense,\nof those ideas of Order, and Proportion, and Harmony, which dwell in the\nDivine Intellect, and are mirrored in the soul of man. \"Time itself is a\nmoving image of Eternity.\"[539] But inasmuch as the immediate object of\nsense-perception is a representative image generated in the vital\norganism, and all empirical cognitions are mere \"conjectures\" (eikasiai)\nfounded on representative images, they need to be certified by a higher\nfaculty, which immediately apprehends real Being (to on). Of things, as\nthey are in themselves, the senses give us no knowledge; all that in\nsensation we are conscious of is certain affections of the mind\n(pathos); the existence of self, or the perceiving subject, and a\nsomething external to self, a perceived object, are revealed to us, not\nby the senses, but by the reason.\n\n[Footnote 538: Alcinous, \"Introduction to the Doctrine of Plato,\" p.\n247.]\n\n[Footnote 539: \"Timaeus,\" Sec. 14.]\n\n3d. JUDGMENT (dianoia, logos), _the Discursive Faculty, or the Faculty\nof Relations_.--According to Plato, this faculty proceeds on the\nassumption of certain principles as true, without inquiring into their\nvalidity, and reasons, by deduction, to the conclusions which\nnecessarily flow from these principles. These assumptions Plato calls\nhypotheses (ypotheseis). But by hypotheses he does not mean baseless\nassumptions--\"mere theories--\"but things self-evident and \"obvious to\nall;\"[540] as for example, the postulates and definitions of Geometry.\n\"After laying down hypotheses of the odd and even, and three kinds of\nangles [right, acute, and obtuse], and figures [as the triangle, square,\ncircle, and the like], he _proceeds on them as known, and gives no\nfurther reason about them_, and reasons downward from these\nprinciples,\"[541] affirming certain judgments as consequences deducible\ntherefrom.\n\n[Footnote 540: \"Republic,\" bk. vi. ch. xx.]\n\n[Footnote 541: Ibid., bk. vi. ch. xx.]\n\nAll judgments are therefore founded on _relations_. To judge is to\ncompare two terms. \"Every judgment has three parts: the subject, or\nnotion about which the judgment is; the predicate, or notion with which\nthe subject is compared; and the copula, or nexus, which expresses the\nconnection or relation between them.[542] Every act of affirmative\njudgment asserts the agreement of the predicate and subject; every act\nof negative judgment asserts the predicate and subject do not agree. All\njudgment is thus an attempt to reduce to unity two cognitions, and\nreasoning (logizesthai) is simply the extension of this process. When we\nlook at two straight lines of equal length, we do not merely think of\nthem separately as _this_ straight line, and _that_ straight line, but\nthey are immediately connected together by a comparison which takes\nplace in the mind. We perceive that these two lines are alike; they are\nof equal length, and they are both straight; and the connection which is\nperceived as existing between them is a _relation of sameness or\nidentity._[543] When we observe any change occurring in nature, as, for\nexample, the melting of wax in the presence of heat, the mind recognizes\na causal efficiency in the fire to produce that change, and the relation\nnow apprehended is a _relation of cause and effect_[544] But the\nfundamental principles, the necessary ideas which lie at the basis of\nall the judgments (as the ideas of space and time, of unity and\nidentity, of substance and cause, of the infinite and perfect) are not\ngiven by the judgment, but by the \"highest faculty\"--\"the _Intuitive\nReason_,[545] which is, for us, the source of all unhypothetical and\nabsolute knowledge.\n\n[Footnote 542: Thompson's \"Laws of Thought,\" p. 134.]\n\n[Footnote 543: \"Phaedo,\" Secs. 50-57, 62.]\n\n[Footnote 544: \"Timaeus,\" ch. ix.; \"Sophocles,\" Sec. 109.]\n\n[Footnote 545: \"Republic,\" bk. vi. ch. xxi.]\n\nThe knowledge, therefore, which is furnished by the Discursive Reason,\nPlato does not regard as \"real Science.\" \"It is something between\nOpinion on the one hand, and Intuition on the other.\"[546]\n\n[Footnote 546: Ibid., bk. vi. ch. xxi.]\n\n4th. REASON (nous)--_Intuitive Reason_, is the organ of self-evident,\nnecessary, and universal Truth. In an immediate, direct, and intuitive\nmanner, it takes hold on truth with absolute certainty. The reason,\nthrough the medium of _ideas_, holds communion with the world of real\nBeing. These ideas are the _light_ which reveals the world of unseen\nrealities, as the sun reveals the world of sensible forms. \"_The idea of\nthe good_ is the _sun_ of the Intelligible World; it sheds on objects\nthe light of truth, and gives to the soul that knows, the power of\nknowing.\"[547] Under this light, the eye of reason apprehends the\neternal world of being as truly, yes more truly, than the eye of sense\napprehends the world of phenomena. This power the rational soul\npossesses by virtue of its having a nature kindred, or even homogeneous\nwith the Divinity. It was \"generated by the Divine Father,\" and, like\nhim, it is in a certain sense \"_eternal_.\"[548] Not that we are to\nunderstand Plato as teaching that the rational soul had an independent\nand underived existence; it was created or \"generated\" in eternity,[549]\nand even now, in its incorporate state, is not amenable to the\nconditions of time and space, but, in a peculiar sense, dwells in\neternity; and therefore is capable of beholding eternal realities, and\ncoming into communion with absolute beauty, and goodness, and\ntruth--that is, with God, the _Absolute Being_.\n\n[Footnote 547: Ibid., bk. vi. ch. xix.; see also ch. xviii.]\n\n[Footnote 548: The reader must familiarize himself with the Platonic\nnotion of _\"eternity\" as a fixed state out of time existing\ncontemporaneous with one in time_, to appreciate the doctrine of Plato\nas stated above. If we regard his idea of eternity as merely an\nindefinite extension of time, with a past, a present, and a future, we\ncan offer no rational interpretation of his doctrine of the eternal\nnature of the rational essence of the soul. An eternal nature\n\"generated\" in a \"past\" or \"present\" time is a contradiction. But that\nwas not Plato's conception of \"eternity,\" as the reader will discover on\nperusing the \"Timaeus\" (ch. xiv.). \"God resolved to create a moving image\nof eternity, and out of that eternity which reposes in its own\n_unchangeable unity_ he framed an eternal image moving according to\nnumerical succession, which we call _Time_. Nothing can be more\ninaccurate than to apply the terms, _past, present, future_, to real\nBeing, which is immovable. Past and future are expressions only suitable\nto generation which proceeds through time.\" Time reposes on the bosom of\neternity, as all bodies are in space.]\n\n[Footnote 549: \"Timaeus,\" ch. xvi., and \"Phaedrus,\" where the soul is\npronounced arche de ageneton.]\n\nThus the soul (psyche) as a composite nature is on one side linked to\nthe eternal world, its essence being generated of that ineffable element\nwhich constitutes the real, the immutable, and the permanent. It is a\nbeam of the eternal Sun, a spark of the Divinity, an emanation from God.\nOn the other side it is linked to the phenomenal or sensible world, its\nemotive part[550] being formed of that which is relative and phenomenal.\nThe soul of man thus stands midway between the eternal and the\ncontingent, the real and the phenomenal, and as such, it is the mediator\nbetween, and the interpreter of, both.\n\n[Footnote 550: thymeides, the seat of the nobler--epithymetikon, the\nseat of the baser passions.]\n\nIn the allegory of the \"Chariot and Winged Steeds\"[551] Plato represents\nthe lower or inferior part of man's nature as dragging the soul down to\nthe earth, and subjecting it to the slavery and debasement of corporeal\nconditions. Out of these conditions there arise numerous evils that\ndisorder the mind and becloud the reason, for evil is inherent to the\ncondition of finite and multiform being into which we have \"fallen by\nour own fault.\" The present earthly life is a fall and a punishment. The\nsoul is now dwelling in \"the grave we call the body.\" In its incorporate\nstate, and previous to the discipline of education, the rational element\nis \"asleep.\" \"Life is more of a dream than a reality.\" Men are utterly\nthe slaves of sense, the sport of phantoms and illusions. We now\nresemble those \"captives chained in a subterraneous cave,\" so poetically\ndescribed in the seventh book of the \"Republic;\" their backs are turned\nto the light, and consequently they see but the shadows of the objects\nwhich pass behind them, and they \"attribute to these shadows a perfect\nreality.\" Their sojourn upon earth is thus a dark imprisonment in the\nbody, a dreamy exile from their proper home. \"Nevertheless these pale\nfugitive shadows suffice to revive in us the reminiscence of that higher\nworld we once inhabited, if we have not absolutely given the reins to\nthe impetuous untamed horse which in Platonic symbolism represents the\nemotive sensuous nature of man.\" The soul has some dim and shadowy\nrecollection of its ante-natal state of bliss, and some instinctive and\nproleptic yearnings for its return.\n\n[Footnote 551: \"Phaedrus,\" Sec. 54-62.]\n\n     \"Our birth is but a sleep and a forgetting;\n     The soul that rises with us, our life's star,\n       Has had elsewhere its setting,\n       And cometh from afar,\n       Not in entire forgetfulness,\n       And not in utter nakedness,\n     But trailing clouds of glory, do we come\n     From God, who is our home.\"[552]\n\n[Footnote 552: Wordsworth, \"Ode on the Intimations of Immortality,\" vol.\nv.]\n\nExiled from the true home of the spirit, imprisoned in the body,\ndisordered by passion, and beclouded by sense, the soul has yet longings\nafter that state of perfect knowledge, and purity, and bliss, in which\nit was first created. Its affinities are still on high. It yearns for a\nhigher and nobler form of life. It essays to rise, but its eye is\ndarkened by sense, its wings are besmeared by passion and lust; it is\n\"borne downward, until at length it falls upon and attaches itself to\nthat which is material and sensual,\" and it flounders and grovels still\namid the objects of sense.\n\nAnd now, with all that seriousness and earnestness of spirit which is\npeculiarly Christian, Plato asks how the soul may be delivered from the\nillusions of sense, the distempering influence of the body, and the\ndisturbances of passion, which becloud its vision of the real, the good,\nand the true?\n\nPlato believed and hoped this could be accomplished by _philosophy_.\nThis he regarded as a grand intellectual discipline for the purification\nof the soul. By this it was to be disenthralled from the bondage of\nsense[553] and raised into the empyrean of pure thought \"where truth and\nreality shine forth.\" All souls have the faculty of knowing, but it is\nonly by reflection, and self-knowledge, and intellectual discipline,\nthat the soul can be raised to the vision of eternal truth, goodness,\nand beauty--that is, to the vision of God. And this intellectual\ndiscipline was the _Platonic Dialectic_.\n\n[Footnote 553: Not, however, fully in this life. The consummation of the\nintellectual struggle into \"the intelligible world\" is death. The\nintellectual discipline was therefore melete thanatou, _a preparation\nfor death_.]\n\n\n\n\nCHAPTER XI.\n\nTHE PHILOSOPHERS OF ATHENS (_continued_.)\n\nTHE SOCRATIC SCHOOL (_continued_).\n\nPLATO.\n\n\nII. THE PLATONIC DIALECTIC.\n\nThe Platonic Dialectic is the Science of Eternal and Immutable\nPrinciples, and the _method_ (organon) by which these first principles\nare brought forward into the clear light of consciousness. The student\nof Plato will have discovered that he makes no distinction between logic\nand metaphysics. These are closely united in the one science to which he\ngives the name of \"_Dialectic_\" and which was at once the science of the\nideas and laws of the Reason, and of the mental process by which the\nknowledge of Real Being is attained, and a ground of absolute certainty\nis found. This science has, in modern times, been called _Primordial_ or\n_Transcendental Logic_.\n\nWe have seen that Plato taught that the human reason is originally in\npossession of fundamental and necessary ideas--the copies of the\narchetypal ideas which dwell in the eternal Reason; and that these ideas\nare the primordial laws of thought--that is, they are the laws under\nwhich we conceive of all objective things, and reason concerning all\nexistence. These ideas, he held, are not derived from sensation, neither\nare they generalizations from experience, but they are inborn and\nconnatural. And, further, he entertained the belief, more, however, as a\nreasonable hypothesis[554] than as a demonstrable truth, that these\nstandard principles were acquired by the soul in a pre-existent state in\nwhich it stood face to face with ideas of eternal order, beauty,\ngoodness, and truth.[555] \"Journeying with the Deity,\" the soul\ncontemplated justice, wisdom, science--not that science which is\nconcerned with change, and which appears under a different manifestation\nin different objects, which we choose to call beings; but such science\nas is in that which alone is indeed _being_.[556] Ideas, therefore,\nbelong to, and inhere in, that portion of the soul which is properly\nousia--_essence_ or _being_; which had an existence anterior to time,\nand even now has no relation to time, because it is now in\neternity--that is, in a sphere of being to which past, present, and\nfuture can have no relation.[557]\n\n[Footnote 554: Within \"the eikoton mython idea--the category of\nprobability.\"--\"Phaedo.\"]\n\n[Footnote 555: \"Phaedo,\" Sec. 50-56.]\n\n[Footnote 556: \"Phaedrus,\" Sec. 58.]\n\n[Footnote 557: See note on p. 349.]\n\nAll knowledge of truth and reality is, therefore, according to Plato, a\nREMINISCENCE (anamnesis)--a recovery of partially forgotten ideas which\nthe soul possessed in another state of existence; and the _dialectic_ of\nPlato is simply the effort, by apt _interrogation_, to lead the mind to\n\"_recollect_\"[558] the truth which has been formerly perceived by it,\nand is even now in the memory though not in consciousness. An\nillustration of this method is attempted in the \"_Meno_\" where Plato\nintroduces Socrates as making an experiment on the mind of an uneducated\nperson. Socrates puts a series of questions to a slave of Meno, and at\nlength elicits from the youth a right enunciation of a geometrical\ntruth. Socrates then points triumphantly to this instance, and bids Meno\nobserve that he had not taught the youth any thing, but simply\ninterrogated him as to his opinions, whilst the youth had recalled the\nknowledge previously existing in his own mind.[559]\n\n[Footnote 558: \"To learn is to recover our own previous knowledge, and\nthis is properly to _recollect._\"--\"Phaedo\" Sec. 55.]\n\n[Footnote 559: \"Meno,\" Sec. 16-20. \"Now for a person to recover knowledge\nhimself through himself, is not this to _recollect_.\"]\n\nNow whilst we readily grant that the instance given in the \"_Meno_\" does\nnot sustain the inference of Plato that \"the boy\" had learnt these\ngeometrical truths \"in eternity,\" and that they had simply been brought\nforward into the view of his consciousness by the \"questioning\" of\nSocrates, yet it certainly does prove that _there are ideas or\nprinciples in the human reason which are not derived from without--which\nare anterior to all experience, and for the development of which,\nexperience furnishes the occasion, but is not the origin and source_. By\na kind of lofty inspiration, he caught sight of that most important\ndoctrine of modern philosophy, so clearly and logically presented by\nKant, _that the Reason is the source of a pure_ a priori _knowledge_--a\nknowledge native to, and potentially in the mind, antecedent to all\nexperience, and which is simply brought out into the field of\nconsciousness by experience conditions. Around this greatest of all\nmetaphysical truths Plato threw a gorgeous mythic dress, and presented\nit under the most picturesque imagery.[560] But, when divested of the\nrich coloring which the glowing imagination of Plato threw over it, it\nis but a vivid presentation of the cardinal truth that _there are ideas\nin the mind which have not been derived from without_, and which,\ntherefore, the mind brought with it into the present sphere of being.\nThe validity and value of this fundamental doctrine, even as presented\nby Plato, is unaffected by any speculations in which he may have\nindulged, as to the pre-existence of the soul. He simply regarded this\ndoctrine of pre-existence as highly probable--a plausible explanation of\nthe facts. That there are ideas, innate and connatural to the human\nmind, he clung to as the most vital, most precious, most certain of all\ntruths; and to lead man to the recognitions of these ideas, to bring\nthem within the field of consciousness, was, in his judgment, the great\nbusiness of philosophy.\n\nAnd this was the grand aim of his _Dialectic_--to elicit, to bring to\nlight the truths which are already in the mind--\"a maieusis\" a kind of\nintellectual midwifery[561]--a delivering of the mind of the ideas with\nwhich it was pregnant.\n\n[Footnote 560: As in the \"Phaedo,\" Secs. 48-57; \"Phaedrus,\" Secs. 52-64;\n\"Republic,\" bk. x.]\n\n[Footnote 561: \"Theaetetus,\" Secs. 17-20.]\n\nIt is thus, at first sight, obvious that it was a higher and more\ncomprehensive science than the art of deduction. For it was directed to\nthe discovery and establishment of First Principles. Its sole object was\nthe discovery of truth. His dialectic was an _analytical_ and _inductive\nmethod_. \"In Dialectic Science,\" says _Alcinous_, \"there is a dividing\nand a defining, and an analyzing, and, moreover, that which is inductive\nand syllogistic.\"[562] Even _Bacon_, who is usually styled \"the Father\nof the Inductive method,\" and who, too often, speaks disparagingly of\nPlato, is constrained to admit that he followed the inductive method.\n\"An induction such as will be of advantage for the invention and\ndemonstration of Arts and Sciences must distinguish the essential nature\nof things (naturam) by proper rejections and exclusions, and then after\nas many of these negatives as are sufficient, by comprising, above all\n(super), the positives. Up to this time this had not been done, nor even\nattempted, _except by Plato alone, who, in order to attain his\ndefinitions and ideas, has used, to a certain extent, the method of\nInduction_.\"[563]\n\n[Footnote 562: \"Introduction to the Doctrines of Plato,\" vol. vi. p.\n249. \"The Platonic Method was the method of induction.\"--Cousin's\n\"History of Philosophy,\" vol. i. p. 307.]\n\n[Footnote 563: \"Novum Organum,\" vol. i. p. 105.]\n\nThe process of investigation adopted by Plato thus corresponds with the\ninductive method of modern times, with this simple difference, that\nBacon conducted science into the world of _matter_, whilst Plato\ndirected it to the world of _mind_. The dialectic of Plato aimed at the\ndiscovery of the \"laws of thought;\" the modern inductive philosophy aims\nat the discovery of the \"laws of nature.\" The latter concerns itself\nchiefly with the inquiry after the \"causes\" of material phenomena; the\nformer concerned itself with the inquiry after the \"first principles\" of\nall knowledge and of all existence. Both processes are, therefore,\ncarried on by _interrogation_. The analysis which seeks for a law of\nnature proceeds by the interrogation of nature. The analysis of Plato\nproceeds by the interrogation of mind, in order to discover the\nfundamental _ideas_ which lie at the basis of all cognition, which\ndetermine all our processes of thought, and which, in their final\nanalysis, reveal the REAL BEING, which is the ground and explanation of\nall existence.\n\nNow the fact that such an inquiry has originated in the human mind, and\nthat it can not rest satisfied without some solution, is conclusive\nevidence that the mind has an instinctive belief, a proleptic\nanticipation, that such knowledge can be attained. There must\nunquestionably be some mental initiative which is the _motive_ and\n_guide_ to all philosophical inquiry. We must have some well-grounded\nconviction, some _a priori_ belief, some pre-cognition \"ad intentionem\nejus quod quaeritur,\"[564] which determines the direction of our\nthinking. The mind does not go to work aimlessly; it asks a specific\nquestion; it demands the \"_whence_\" and the \"_why_\" of that which is.\nNeither does it go to work unfurnished with any guiding principles. That\nwhich impels the mind to a determinate act of thinking is the possession\nof a _knowledge_ which is different from, and independent of, the\nprocess of thinking itself. \"A rational anticipation is, then, the\nground of the _prudens quaestio_--\"the forethought query, which, in\nfact, is the prior half of the knowledge sought.\"[565] If the mind\ninquire after \"laws,\" and \"causes,\" and \"reasons,\" and \"grounds,\"--the\nfirst principles of all knowledge and of all existence,--\"it must have\nthe _a priori_ ideas of \"law,\" and \"cause,\" and \"reason,\" and \"being _in\nse\"_ which, though dimly revealed to the mind previous to the discipline\nof reflection, are yet unconsciously governing its spontaneous modes of\nthought. The whole process of induction has, then, some rational ground\nto proceed upon--some principles deeper than science, and more certain\nthan demonstration, which reason contains within itself, and which\ninduction \"draws out\" into clearer light.\n\n[Footnote 564: Bacon.]\n\n[Footnote 565: Coleridge, vol. ii. p. 413.]\n\nNow this mental initiative of every process of induction is the\nintuitive and necessary conviction _that there must be a sufficient\nreason why every thing exists, and why it is as it is, and not\notherwise_;[566] or in other words, if any thing begins to be, some\nthing else must be supposed[567] as the ground, and reason, and cause,\nand law of its existence. This \"_law of sufficient_ (or _determinant)\nreason_\"[568] is the fundamental principle of all metaphysical inquiry.\nIt is contained, at least in a negative form, in that famous maxim of\nancient philosophy, \"_De nihilo nihil_\"--\"Adynaton ginestai ti ek\nmedenos prouparxontos.\" \"It is impossible for a real entity to be made\nor generated from nothing pre-existing;\" or in other words, \"nothing can\nbe made or produced without an efficient cause.\"[569] This principle is\nalso distinctly announced by Plato: \"Whatever is generated, is\nnecessarily generated from a certain aitian\"--_ground, reason_, or\n_cause_; \"for it is wholly impossible that any thing should be generated\nwithout a cause.\"[570]\n\n[Footnote 566: \"Phaedo,\" Sec. 103.]\n\n[Footnote 567: _Suppono_, to place under as a support, to take as a\nground.]\n\n[Footnote 568: This generic principle, viewed under different relations,\ngives--\n\n     1st. _The principle of Substance_--every quality supposes a subject\n       or real being.\n\n     2d. _The principle of Causality_--every thing which begins to be\n       must have a cause.\n\n     3d. _The principle of Law_--every phenomenon must obey some uniform\n       law.\n\n     4th. _The principle of Final Cause_--every means supposes an end,\n       every existence has a purpose or reason why.\n\n     5th. _The principle of Unity_--all plurality supposes a unity as\n       its basis and ground.]\n\n[Footnote 569: Cudworth's \"Intellectual System,\" vol. ii. p. 161.]\n\n[Footnote 570: \"Timaeus,\" ch. ix.]\n\nThe first business of Plato's dialectic is to demonstrate that the\nground and reason of all existence can not be found in the mere objects\nof sense, nor in any opinions or judgments founded upon sensation.\nPrinciples are only so far \"first principles\" as they are permanent and\nunchangeable, depending on neither time, nor place, nor circumstances.\nBut the objects of sense are in ceaseless flux and change; they are\n\"_always becoming_;\" they can not be said to have any \"_real being_.\"\nThey are not to-day what they were yesterday, and they will never again\nbe what they are now; consequently all opinions founded on mere\nphenomena are equally fluctuating and uncertain. Setting out, therefore,\nfrom the assumption of the fallaciousness of \"_opinion_\" it examined the\nvarious hypotheses which had been bequeathed by previous schools of\nphilosophy, or were now offered by contemporaneous speculators, and\nshowed they were utterly inadequate to the solution of the problem. This\nscrutiny consisted in searching for the ground of \"contradiction\"[571]\nwith regard to each opinion founded on sensation, and showing that\nopposite views were equally tenable. It inquired on what ground these\nopinions were maintained, and what consequences flowed therefrom, and it\nshowed that the grounds upon which \"opinion\" was founded, and the\nconclusions which were drawn from it, were contradictory, and\nconsequently untrue.[572] \"They,\" the Dialecticians, \"examined the\nopinions of men as if they were error; and bringing them together by a\nreasoning process to the same point, they placed them by the side of\neach other: and by so placing, they showed that _the opinions are at one\nand the same time contrary to themselves, about the same things, with\nreference to the same circumstances, and according to the same\npremises_.\"[573] And inasmuch as the same attribute can not, at the same\ntime, be affirmed and denied of the same subject,[574] therefore a thing\ncan not be at once \"changeable\" and \"unchangeable,\" \"movable\" and\n\"immovable,\" \"generated\" and \"eternal.\"[575] The objects of sense,\nhowever generalized and classified, can only give the contingent, the\nrelative, and the finite; therefore the permanent ground and sufficient\nreason of all phenomenal existence can not be found in opinions and\njudgments founded upon sensation.\n\n[Footnote 571: \"The Dialectitian is one who syllogistically infers the\ncontradictions implied in popular opinions.\"--Aristotle, \"Sophist,\" Secs.\n1, 2.]\n\n[Footnote 572: \"Republic,\" bk. vi. ch. xiii.]\n\n[Footnote 573: \"Sophist,\" Sec. 33; \"Republic,\" bk. iv. ch. xii.]\n\n[Footnote 574: See the \"Phaedo,\" Sec. 119, and \"Republic,\" bk. iv. ch.\nxiii., where the Law of Non-contradiction is announced.]\n\n[Footnote 575 \"Parmenides,\" Sec. 3.]\n\nThe dialectic process thus consisted almost entirely of\n_refutation_,[576] or what both he and Aristotle denominated _elenchus_\n(elenchos)--a process of reasoning by which the contradictory of a given\nproposition is inferred. \"When refutation had done its utmost, and all\nthe points of difficulty and objection had been fully brought out, the\ndialectic method had accomplished its purpose; and the affirmation which\nremained, after this discussion, might be regarded as setting forth the\ntruth of the question under consideration;\"[577] or in other words,\n_when a system of error is destroyed by refutation, the contradictory\nopposite principle, with its logical developments, must be accepted as\nan established truth_.\n\n[Footnote 576: Confutation is the greatest and chiefest of\npurification.--\"Sophist,\" Sec. 34.]\n\n[Footnote 577: Article \"Plato,\" Encyclopaedia Britannica.]\n\nBy the application of this method, Plato had not only exposed the\ninsufficiency and self-contradiction of all results obtained by a mere\n_a posteriori_ generalization of the simple facts of experience, but he\ndemonstrated, as a consequence, that we are in possession of some\nelements of knowledge which have not been derived from sensation; that\nthere are, in all minds, certain notions, principles, or ideas, which\nhave been furnished by a higher faculty than sense; and that these\nnotions, principles, or ideas, transcend the limits of experience, and\nreveal the knowledge of _real being_--to ontos on--_Being in se_.\n\nTo determine what these principles or ideas are, Plato now addresses\nhimself to the _analysis of thought_. \"It is the glory of Plato to have\nborne the light of analysis into the most obscure and inmost region; he\nsearched out what, in this totality which forms consciousness, is the\nprovince of reason; what comes from it, and not from the imagination and\nthe senses--from within, and not from without.\"[578] Now to analyze is\nto decompose, that is, to divide, and to define, in order to see better\nthat which really is. The chief logical instruments of the dialectic\nmethod are, therefore, _Division_ and _Definition_. \"The being able to\n_divide_ according to genera, and not to consider the same species as\ndifferent, nor a different as the same,\"[579] and \"to see under one\naspect, and bring together under one general idea, many things scattered\nin various places, that, by _defining_ each, a person may make it clear\nwhat the subject is,\" is, according to Plato, \"dialectical.\"[580]\n\n[Footnote 578: Cousin's \"Lectures on the History of Philosophy,\" vol. i.\np. 328.]\n\n[Footnote 579: \"Sophist,\" Sec. 83.]\n\n[Footnote 580: \"Phaedrus,\" Secs. 109, 111.]\n\nWe have already seen that, in his first efforts at applying reflection\nto the concrete phenomena of consciousness, Plato had recognized two\ndistinct classes of cognitions, marked by characteristics essentially\nopposite;--one of \"_sensible_\" objects having a definite outline, limit,\nand figure, and capable of being imaged and represented to the mind in a\ndeterminate form--the other of \"_intelligible_\" objects, which can not\nbe outlined or represented in the memory or the imagination by any\nfigures or images, and are, therefore, the objects of purely rational\nconception. He found, also, that we arrive at one class of cognitions\n\"_mediately_\" through images generated in the vital organism, or by some\ntestimony, definition, or explication of others; whilst we arrive at the\nother class \"_immediately_\" by simple intuition, or rational\napperception. The mind stands face to face with the object, and gazes\ndirectly upon it. The reality of that object is revealed in its own\nlight, and we find it impossible to refuse our assent--that is, it is\n_self-evident_. One class consisted of _contingent_ ideas--that is,\ntheir objects are conceived as existing, with the possibility, without\nany contradiction, of conceiving of their non-existence; the other\nconsisted of _necessary_ ideas--their objects are conceived as existing\nwith the absolute impossibility of conceiving of their non-existence.\nThus we can conceive of this book, this table, this earth, as not\nexisting, but we can not conceive the non-existence of space. We can\nconceive of succession in time as not existing, but we can not, in\nthought, annihilate duration. We can imagine this or that particular\nthing not to have been, but we can not conceive of the extinction of\nBeing in itself. He further observed, that one class of our cognitions\nare _conditional_ ideas; the existence of their objects is conceived\nonly on the supposition of some antecedent existence, as for example,\nthe idea of qualities, phenomena, events; whilst the other class of\ncognitions are _unconditional_ and _absolute_--we can conceive of their\nobjects as existing independently and unconditionally--existing whether\nany thing else does or does not exist, as space, duration, the infinite,\nBeing _in se_. And, finally, whilst some ideas appear in us as\n_particular_ and _individual_, determined and modified by our own\npersonality and liberty, there are others which are, in the fullest\nsense, _universal_. They are not the creations of our own minds, and\nthey can not be changed by our own volitions. They depend upon neither\ntimes, nor places, nor circumstances; they are common to all minds, in\nall times, and in all places. These ideas are the witnesses in our\ninmost being that there is something beyond us, and above us; and beyond\nand above all the contingent and fugitive phenomena around us. Beneath\nall changes there is a _permanent_ being. Beyond all finite and\nconditional existance there is something _unconditional_ and _absolute_.\nHaving determined that there are truths which are independent of our own\nminds--truths which are not individual, but universal--truths which\nwould be truths even if our minds did not perceive them, we are led\nonward to a _super-sensual_ and super-natural ground, on which they\nrest.\n\nTo reach this objective reality on which the ideas of reason repose, is\nthe grand effort of Plato's dialectic. He seeks, by a rigid analysis,\nclearly to _separate_, and accurately to _define_ the _a priori_\nconceptions of reason. And it was only when he had eliminated every\nelement which is particular, contingent, and relative, and had defined\nthe results in precise and accurate language, that he regarded the\nprocess as complete. The ideas which are self-evident, universal, and\nnecessary, were then clearly disengaged, and raised to their pure and\nabsolute form. \"You call the man dialectical who requires a reason of\nthe essence or being of each thing. As the dialectical man can define\nthe essence of every thing, so can he of the good. He can _define_ the\nidea of the good, _separating_ it from all others--follow it through all\nwindings, as in a battle, resolved to mark it, not according to opinion,\nbut according to science.\"[581]\n\n[Footnote 581: \"Republic,\" bk. vii. ch. xiv.]\n\n_Abstraction_ is thus the process, the instrument of the Platonic\ndialectic. It is important, however, that we should distinguish between\nthe method of _comparative_ abstraction, as employed in physical\ninquiry, and that _immediate_ abstraction, which is the special\ninstrument of philosophy. The former proceeds by comparison and\ngeneralization, the latter by simple separation. The one yields a\ncontingent general principle as the result of the comparison of a number\nof individual cases, the other gives an universal and necessary\nprinciple by the analysis of a single concrete fact. As an illustration\nwe may instance \"the principle of causality.\" To enable us to affirm\n\"that every event must have a cause,\" we do not need to compare and\ngeneralize a great number of events. \"The principle which compels us to\npronounce the judgment is already complete in the first as in the last\nevent; it can change in regard to its object, it can not change in\nitself; it neither increases nor decreases with the greater or less\nnumber of applications.\"[582] In the presence of a single event, the\nuniversality and necessity of this principle of causality is recognized\nwith just as much clearness and certainty as in the presence of a\nmillion events, however carefully generalized.\n\n[Footnote 582: Cousin's \"The True, the Beautiful, and the Good,\" pp. 57,\n58.]\n\nAbstraction, then, it will be seen, creates nothing; neither does it add\nany new element to the store of actual cognitions already possessed by\nall human minds. It simply brings forward into a clearer and more\ndefinite recognition, that which necessarily belongs to the mind as part\nof its latent furniture, and which, as a law of thought, has always\nunconsciously governed all its spontaneous movements. As a process of\nrational inquiry, it was needful to bring the mind into intelligible and\nconscious communion with the world of _Ideas_. These ideas are partially\nrevealed in the sensible world, all things being formed, as Plato\nbelieved, according to ideas as models and exemplars, of which sensible\nobjects are the copies. They are more fully manifested in the\nconstitution of the human mind which, by virtue of its kindred nature\nwith the original essence or being, must know them intuitively and\nimmediately. And they are brought out fully by the dialectic process,\nwhich disengages them from all that is individual and phenomenal, and\nsets them forth in their pure and absolute form.\n\nBut whilst Plato has certainly exhibited the true method of\ninvestigation by which the ideas of reason are to be separated from all\nconcrete phenomena and set clearly before the mind, he has not attempted\na complete enumeration of the ideas of reason; indeed, such an\nenumeration is still the grand desideratum of philosophy. We can not\nfail, however, in the careful study of his writings, to recognize the\ngrand Triad of Absolute Ideas--ideas which Cousin, after Plato, has so\nfully exhibited, viz., the _True_, the _Beautiful_, and the _Good_.\n\nPLATONIC SCHEME OF IDEAS\n\nI. _The idea of_ ABSOLUTE TRUTH or REALITY (to alethes--to on)--the\nground and efficient cause of all existence, and by participating in\nwhich all phenomenal existence has only so far a reality, sensible\nthings being merely shadows and resemblances of ideas. This idea is\ndeveloped in the human intelligence in its relation with the phenomenal\nworld; as,\n\n1. _The idea of_ SUBSTANCE (ousia)--the ground of all phenomena, \"the\nbeing or essence of all things,\" the permanent reality.--\"Timaeeus,\" ch.\nix. and xii.; \"Republic,\" bk. vii. ch. xiv.; \"Phaedo,\"Secs. 63-67, 73.\n\n2. _The idea of_ CAUSE (aitia)--the power or efficiency by which things\nthat \"become,\" or begin to be, are generated or produced.--\"Timaeus,\" ch.\nix.; \"Sophist,\" Sec. 109; \"Philebus,\" Secs. 45, 46.\n\n3. _The idea of_ IDENTITY (auto to ison)--that which \"does not change,\"\n\"is always the same, simple and uniform, incomposite and\nindissoluble,\"--that which constitutes personality or\nself-hood.--\"Phaedo,\" Secs. 61-75; \"Timaeus,\" ch. ix.; \"Republic,\" bk. ii.\nch. xix. and xx.\n\n4. _The idea of_ UNITY (to en)--one _mind_ or intelligence pervading the\nuniverse, the comprehensive conscious _thought_ or _plan_ which binds\nall parts of the universe in one great whole (to pan)--the principle of\n_order_.--\"Timaeus,\" ch. xi. and xv.; \"Republic,\" bk. vi. ch. xiii.;\n\"Philebus,\" Secs. 50-51.\n\n5. _The idea of the_ INFINITE (to apeiron)--that which is unlimited and\nunconditioned, \"has no parts, bounds, no beginning, nor middle, nor\nend.\"--\"Parmenides,\" Secs. 22, 23.\n\nII. _The idea of_ ABSOLUTE BEAUTY (to kalon)--the formal cause of the\nuniverse, and by participation in which all created things have only so\nfar a real beauty.--\"Timaeus,\" ch. xi, \"Greater Hippias,\" Secs. 17, 18;\n\"Republic,\" bk. v. ch. 22.\n\nThis idea is developed in the human intelligence in its relation to the\norganic world; as,\n\n      1. _The Idea of_ PROPORTION or SYMMETRY (symmetria)--the\n      proper relation of parts to an organic whole resulting in a\n      harmony (cosmos), and which relation admits of mathematical\n      expression.--\"Timaeus,\" ch. lxix.; \"Philebus,\" Sec. 155\n      (\"Timaeus,\" ch. xi. and xii., where the relation of numerical\n      proportions to material elements is expounded).\n\n      2. _The idea of_ DETERMINATE FORM (paradeigma\n      archetypos)--the eternal models or archetypes according to\n      which all things are framed, and which admit of geometrical\n      representation.--\"Timaeus,\" ch. ix.; \"Phaedo,\" Sec.112 (\"Timaeus,\"\n      ch. xxviii.-xxxi., where the relation of geometrical forms\n      to material elements is exhibited).\n\n      3. _The idea of_ RHYTHM (rythmos)--measured movement in time\n      and space, resulting in melody and grace.--\"Republic,\" bk.\n      iii. ch. xi. and xii.; \"Philebus,\" Sec. 21.\n\n      4. _The idea of_ FITNESS or ADAPTATION\n      (chresimon)--effectiveness to some purpose or end.--\"Greater\n      Hippias,\" Sec. 35.\n\n      5. _The idea of_ PERFECTION (teleiotes)--that which is\n      complete, \"a structure which is whole and finished--of whole\n      and perfect parts.\"--\"Timaeus,\" ch. xi., xii., and xliii.\n\n\nIII. _The idea of_ ABSOLUTE GOOD (to agathon)--the final _cause_ or\n_reason_ of all existence, the sun of the invisible world, that pours\nupon all things the revealing light of truth.\n\nThe first Good[583] (_summum bonum_) is God the highest, and Mind or\nIntelligence (nous), which renders man capable of knowing and resembling\nGod. The second flows from the first, and are virtues of mind. They are\ngood by a participation of the chief good, and constitute in man a\nlikeness or _resemblance_ to God.--\"Phaedo,\" Secs.110-114; \"Laws,\" bk. i.\nch. vi., bk. iv. ch. viii.; \"Theaetetus,\" Secs. 84, 85; \"Republic,\" bk. vi.\nch. xix., bk. vii. ch. iii., bk. x. ch. xii.[584]\n\n[Footnote 583: \"Let us declare, then, on what account the framing\nArtificer settled the formation of the universe. He was GOOD;\" and being\ngood, \"he desired that all things should as much as possible resemble\nhimself.\"--\"Timaeus,\" ch. x.]\n\n[Footnote 584: \"At the utmost bounds of the intellectual world is the\n_idea of the Good_, perceived with difficulty, but which, once seen,\nmakes itself known as the cause of all that is beautiful and good; which\nin the visible world produces light, and the orb that gives it; and\nwhich in the invisible world directly produces Truth and\nIntelligence.\"--\"Republic,\" bk. vii. ch. iii.]\n\nThis idea is developed in the human intelligence in its relation to the\nworld of moral order; as,\n\n1. _The idea of_ WISDOM or PRUDENCE (phronesis)--thoughtfulness,\nrightness of intention, following the guidance of reason, the right\ndirection of the energy or will.--\"Republic,\" bk. iv. ch. vii., bk. vi.\nch. ii.\n\n2. _The idea of_ COURAGE or FORTITUDE (andria)--zeal, energy, firmness\nin the maintenance of honor and right, virtuous indignation against\nwrong.--\"Republic,\" bk. iv. ch. viii.; \"Laches;\" \"Meno,\" Sec. 24.\n\n3. _The idea of_ SELF-CONTROL or TEMPERANCE\n(sophrosyne)--sound-mindedness, moderation, dignity.--\"Republic,\" bk.\niv. ch. ix.; \"Meno,\" Sec. 24; \"Phaedo,\" Sec. 35.\n\n4. _The idea of_ JUSTICE (dikalosyne)--the harmony or perfect\nproportional action of all the powers of the soul.--\"Republic,\" bk. i.\nch. vi., bk. iv. ch. x.-xii., bk. vi. ch. ii. and xvi.; \"Philebus,\" Sec.\n155; \"Phaedo,\" Sec. 54; \"Theaetetus,\" Secs. 84, 85.\n\nPlato's idea of Justice comprehends--\n\n(1) EQUITY (isotes)--the rendering to every man his due.--\"Republic,\"\nbk. i. ch. vi.\n\n(2.) VERACITY (alepheia)--the utterance of what is true.--\"Republic,\"\nbk. i. ch. v., bk. ii. ch. xx., bk. vi. ch. ii.\n\n(3.) FAITHFULNESS (pistotes)--the strict performance of a\ntrust.--\"Republic,\" bk. i. ch. v., bk. vi. ch. ii.\n\n(4.) USEFULNESS (opheltmon)--the answering of some valuable\nend.--\"Republic,\" bk. ii. ch. xviii., bk. iv. ch. xviii.; \"Meno,\" Sec. 22.\n\n(5.) BENEVOLENCE (eunoia)--seeking the well-being of\nothers.--\"Republic,\" bk. i. ch. xvii., bk. ii. ch. xviii.\n\n(6.) HOLINESS (osiotes)--purity of mind, piety.--\"Protagoras,\" Secs. 52-54;\n\"Phaedo,\" Sec. 32; \"Theaetetus,\" Sec. 84.\n\nThe final effort of Plato's Dialectic was to ascend from these ideas of\nAbsolute Truth, and Absolute Beauty, and Absolute Goodness to the\n_Absolute Being_, in whom they are all united, and from whom they all\nproceed. \"He who possesses the true love of science is naturally carried\nin his aspirations to the _real Being_; and his love, so far from\nsuffering itself to be retarded by the multitude of things whose reality\nis only apparent, knows no repose until it have arrived at union with\nthe _essence_ of each object, by the part of the soul which is akin to\nthe permanent and essential; so that this divine conjunction having\nproduced intelligence and truth, the knowledge of _being_ is won.\"[585]\n\n[Footnote 585: \"Republic,\" bk. vi. ch. v.]\n\nTo the mind of Plato, there was in every thing, even the smallest and\nmost insignificant of sensible objects, a _reality_ just in so far as it\nparticipates in some archetypal form or idea. These archetypal forms or\nideas are the \"_thoughts of God_\"[586]--they are the plan according to\nwhich he framed the universe. \"The Creator and Father of the universe\nlooked to an _eternal model_.... Being thus generated, the universe is\nframed according to principles that can be comprehended by reason and\nreflection.\"[587] Plato, also, regarded all individual conceptions of\nthe mind as hypothetical notions which have in them an _a priori_\nelement--an idea which is unchangeable, universal, and necessary. These\nunchangeable, universal, and necessary ideas are copies of the Divine\nIdeas, which are, for man, the primordial laws of all cognition, and all\nreasoning. They are possessed by the soul \"in virtue of its kindred\nnature to that which is permanent, unchangeable, and eternal.\" He also\nbelieved that every archetypal form, and every _a priori_ idea, has its\nground and root in a higher idea, which is _unhypothetical_ and\n_absolute_--an idea which needs no other supposition for its\nexplanation, and which is, itself, needful to the explanation of all\nexistence--even the idea of an _absolute_ and _perfect Being_, in whose\nmind the ideas of absolute truth, and beauty, and goodness inhere, and\nin whose eternity they can only be regarded as eternal.[588] Thus do the\n\"ideas of reason\" not only cast a bridge across the abyss that separates\nthe sensible and the ideal world, but they also carry us beyond the\nlimits of our personal consciousness, and discover to us a realm of real\nBeing, which is the foundation, and cause, and explanation of the\nphenomenal world that appears around us and within us.\n\n[Footnote 586: Alcinous, \"Doctrines of Plato,\" p. 262.]\n\n[Footnote 587: \"Timaeus,\" ch. ix.]\n\n[Footnote 588: Maurice's \"Ancient Philosophy,\" p. 149.]\n\nThis passage from psychology to ontology is not achieved _per saltum_,\nor effected by any arbitrary or unwarrantable assumption. There are\nprinciples revealed in the centre of our consciousness, whose regular\ndevelopment carry us beyond the limits of consciousness, and attain to\nthe knowledge of actual being. The absolute principles of _causality_\nand _substance_, of _intentionality_ and _unity_, unquestionably give us\nthe absolute Being. Indeed the absolute truth _that every idea supposes\na being in which it resides_, and which is but another form of the law\nor principle of substance, viz., _that every quality supposes a\nsubstance or being in which it inheres_, is adequate to carry us from\nIdea to Being. \"There is not a single cognition which does not suggest\nto us the notion of existence, and there is not an unconditional and\nabsolute truth which does not necessarily imply an absolute and\nunconditional Being.\"[589]\n\n[Footnote 589: Cousin's \"Elements of Psychology,\" p. 506.]\n\nThis, then, is the dialectic of Plato. Instead of losing himself amid\nthe endless variety of particular phenomena, he would search for\nprinciples and laws, and from thence ascend to the great Legislator, the\n_First Principle of all Principles_. Instead of stopping at the\nrelations of sensible objects to the general ideas with which they are\ncommingled, he will pass to their _eternal Paradigms_--from the just\nthing to the idea of absolute justice, from the particular good to the\nabsolute good, from beautiful things to the absolute beauty, and thence\nto the ultimate reality--_the absolute Being_. By the realization of the\nlower idea, embodied in the forms of the visible universe and in the\nnecessary laws of thought, he sought to rise to the higher idea, in its\npure and abstract form--the _Supreme Idea_, containing in itself all\nother ideas--the _One Intelligence_ which unites the universe in a\nharmonious whole. \"The Dialectic faculty proceeds from hypothesis to an\nunhypothetical principle.... It uses hypotheses as steps, and\nstarting-points, in order to proceed from thence to the _absolute_. The\nIntuitive Reason takes hold of the First Principle of the Universe, and\navails itself of all the connections and relations of that principle. It\nascends from idea to idea, until it has reached the Supreme Idea\"--the\n_Absolute Good_--that is, _God_.[590]\n\n[Footnote 590: \"Republic,\" bk. vi. ch. xx. and xxi.]\n\nWe are thus brought, in the course of our examination of the Platonic\nmethod, to the _results_ obtained by this method--or, in other words, to\n\nIII. THE PLATONIC ONTOLOGY.\n\nThe grand object of all philosophic inquiry in ancient Greece was to\nattain to the knowledge of real Being--that Being which is permanent,\nunchangeable, and eternal. It had proceeded on the intuitive conviction,\nthat beneath all the endless diversity of the universe there must be a\nprinciple of _unity_--below all fleeting appearances there must be a\npermanent _substance_--beyond all this everlasting flow and change, this\nbeginning and end of finite existence, there must be an eternal Being,\nwhich is the _cause_, and which contains, in itself, the _reason_ of the\norder, and harmony, and beauty, and excellency which pervades the\nuniverse. And it had perpetually asked what is this permanent,\nunchangeable, and eternal substance or being?\n\nPlato had assiduously labored at the solution of this problem. The\nobject of his dialectic was \"to lead upward the soul to the knowledge of\nreal being,\"[591] and the conclusions to which he attained may be summed\nup as follows:\n\n1st. _Beneath all_ SENSIBLE _phenomena there is an unchangeable\nsubject-matter, the mysterious substratum of the world of sense, which\nhe calls the receptacle (ipodoche) the nurse (tithene) of all that is\nproduced_.[592]\n\nIt is this \"substratum or physical groundwork\" which gives a reality and\ndefiniteness to the evanescent phantoms of sense, for, in their\nceaseless change, _they_ can not justify any title whatever. It alone\ncan be styled \"_this_\" or \"_that_\" (tode or touto); they rise no higher\nthan \"_of such kind_\" or \"_of what kind or quality\" (toiouton or\nopoionoun ti).[593] It is not earth, or air, or fire, or water, but \"an\ninvisible _species_ and formless universal receiver, which, in the most\nobscure way, receives the immanence of the intelligible.\"[594] And in\nrelation to the other two principles (_i.e._, ideas and objects of\nsense), \"it is _the mother_\" to the father and the offspring.[595] But\nperhaps the most remarkable passage is that in which he seems to\nidentify it with _pure space_, which, \"itself imperishable, furnishes a\n_seat_ (edran) to all that is produced, not apprehensible by direct\nperception, but caught by a certain spurious reasoning, scarcely\nadmissible, but which we see as in a dream; gaining it by that judgment\nwhich pronounces it necessary that all which is, be _somewhere_, and\noccupy a _certain space_.\"[596] This, it will be seen, approaches the\nCartesian doctrine, which resolves matter into _simple extension.[597]\n\n[Footnote 591: \"Republic,\" bk. vii. ch. xii. and xiii.]\n\n[Footnote 592: \"Timaeus,\" ch. xxii.]\n\n[Footnote 593: \"Timaeus,\" ch. xxiii.]\n\n[Footnote 594: Ibid., ch. xxiv.]\n\n[Footnote 595: Ibid., ch. xxiv.]\n\n[Footnote 596: Ibid., ch. xxvi.]\n\n[Footnote 597: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n171.]\n\nIt should, however, be distinctly noted that Plato does not use the word\nyle--matter. This term is first employed by Aristotle to express \"the\nsubstance which is the subject of all changes.\"[598] The subject or\nsubstratum of which Plato speaks, would seem to be rather a logical than\na material entity. It is the _condition or supposition_ necessary for\nthe production of a world of phenomena. It is thus the\n_transition-element_ between the real and the apparent, the eternal and\nthe contingent; and, lying thus on the border of both territories, we\nmust not be surprised that it can hardly be characterized by any\ndefinite attribute.[599] Still, this unknown recipient of forms or ideas\nhas a _reality_; it has \"an abiding nature,\" \"a constancy of existence;\"\nand we are forbidden to call it by any name denoting quality, but\npermitted to style it \"_this_\" and \"_that_\" (tode kai touto).[600]\nBeneath the perpetual changes of sensible phenomena there is, then, an\nunchangeable subject, which yet is neither the Deity, nor ideas, nor the\nsoul of man, which exists as the means and occasion of the manifestation\nof Divine Intelligence in the organization of the world.[601]\n\n[Footnote 598: \"Metaphysics,\" bk. vii. ch. i.]\n\n[Footnote 599: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n178.]\n\n[Footnote 600: \"Timaeus,\" ch. xxiii.]\n\n[Footnote 601: Ibid., ch. xiiii]\n\nThere has been much discussion as to whether Plato held that this\n\"_Receptacle_\" and \"_Nurse_\" of forms and ideas was eternal, or\ngenerated in time. Perhaps no one has more carefully studied the\nwritings of Plato than William Archer Butler, and his conclusions in\nregard to this subject are presented in the following words: \"As, on the\none hand, he maintained a strict system of dualism, and avoided, without\na single deviation, that seduction of pantheism to which so many\nabstract speculators of his own school have fallen victims; so, on the\nother hand, it appears to me that he did not scruple to place this\nprinciple, the opposite of the Divine intelligence, in a sphere\nindependent of temporal origination.... But we can scarcely enter into\nhis views, unless we ascertain his notions of the nature of _Time_\nitself. This was considered to have been created with the rest of the\nsensible world, to finish with it, if it ever finished--to be altogether\nrelated to this phenomenal scene.[602] 'The generating Father determined\nto create a moving image of eternity (aionos); and in disposing the\nheavens, he framed of this eternity, reposing in its own unchangeable\nunity, an eternal _image_, moving according to numerical succession,\nwhich he called _Time_. With the world arose days, nights, months,\nyears, which all had no previous existence. The past and future are but\nforms of time, which we most erroneously transfer to the eternal\nsubstance (aidion ousian); we say it was, and is, and will be, whereas\nwe can only fitly say _it is_. Past and future are appropriate to the\nsuccessive nature of generated beings, for they bespeak motion; but the\nBeing eternally and immovably the same is subject neither to youth nor\nage, nor to any accident of time; it neither was, nor hath been, nor\nwill be, which are the attributes of fleeting sense--the circumstances\nof time, imitating eternity in the shape of number and motion. Nor can\nany thing be more inaccurate than to apply the term _real being_ to\npast, or present, or future, or even to non-existence. Of this, however,\nwe can not now speak fully. _Time_, then, was formed with the heavens,\nthat, together created, they may together end, _if indeed an end be in\nthe purpose of the Creator_; and it is designed as closely as possible\nto resemble the eternal nature, its exemplar. The model exists through\nall eternity; the world has been, is, and will be through all\n_time_.'[603] In this ineffable eternity Plato places the Supreme Being,\nand the archetypal ideas of which the sensible world of time partakes.\nWhether he also includes under the same mode of existence the\n_subject-matter_ of the sensible world, it is not easy to pronounce; and\nit appears to me evident that he did not himself undertake to speak with\nassurance on this obscure problem.\"[604] The creation of matter \"out of\nnothing\" is an idea which, in all probability, did not occur to the mind\nof Plato. But that he regarded it as, in some sense, a _dependent_\nexistence--as existing, like time, by \"the purpose or will of the\nCreator\"--perhaps as an eternal \"generation\" from the \"eternal\nsubstance,\" is also highly probable; for in the last analysis he\nevidently desires to embrace all things in some ultimate _unity_--a\ntendency which it seems impossible for human reason to avoid.\n\n[Footnote 602: See _ante_, note 4, p. 349.]\n\n[Footnote 603: \"Timaeus,\" ch. xiv.]\n\n[Footnote 604: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n171-175.]\n\n2d. _Beneath all mental phenomena there is a permanent subject or\nsubstratum which he designates_ THE IDENTICAL (to auto)--_the rational\nelement of the soul--\"the principle of self-activity\" or\nself-determination_.[605]\n\nThere are three principles into which Plato analyzes the soul--the\nprinciple of the _Identical_, the _Diverse_, and the _Intermediate\nEssence_.[606] The first is indivisible and eternal, always existing in\n_sameness_, the very substance of _Intelligence_ itself, and of the same\nnature with the Divine.[607] The second is divisible and corporeal,\nanswering to our notion of the passive _sensibilities_, and placing the\nsoul in relation with the visible world. The third is an intermediate\nessence, partaking of the natures of both, and constituting a medium\nbetween the eternal and the mutable--the conscious _energy_ of the soul\ndeveloped in the contingent world of time. Thus the soul is, on one\nside, linked to the unchangeable and the eternal, being formed of that\nineffable element which constitutes the _real_ or _immutable Being_, and\non the other side, linked to the sensible and the contingent, being\nformed of that element which is purely _relative_ and _contingent_. This\nlast element of the soul is regarded by Plato as \"mortal\" and\n\"corruptible,\" the former element as \"immortal\" and \"indestructible,\"\nhaving its foundations laid in eternity.\n\n[Footnote 605: \"Laws,\" bk. x. ch. vi. and vii.; \"Phaedrus,\" Sec. 51; \"arche\nkineseos.\"]\n\n[Footnote 606: \"Timaeus,\" ch. xii.; tauton, thateron, and ousia or to\nsymmisgomenon.]\n\n[Footnote 607: \"Laws,\" bk. v. ch. i.]\n\nThis doctrine of the eternity of the free and rational element of the\nsoul must, of course, appear strange and even repulsive to those who are\nunacquainted with the Platonic notion of eternity as a fixed state out\nof time, which has no past, present, or future, and is simply that which\n\"always _is_\"--an everlasting _now_. The soul, in its elements of\nrationality and freedom, has existed anterior to time, because it now\nexists in eternity.[608] In its actual manifestations and personal\nhistory it is to be contemplated as a \"generated being,\" having a\ncommencement in time.\n\nNow, that the human soul, like the uncreated Deity, has always had a\ndistinct, conscious, personal, independent being, does not appear to be\nthe doctrine of Plato. He teaches, most distinctly, that the \"divine,\"\nthe immortal part, was created, or rather \"generated,\" in eternity. \"The\nDeity himself _formed the divine_, and he delivered over to his\ncelestial offspring [the subordinate and generated gods] the task of\n_forming the mortal_. These subordinate deities, copying the example of\ntheir parent, and receiving from his hands the _immortal principle_ of\nthe human soul, fashioned subsequently to this the mortal body, which\nthey consigned to the soul as a vehicle, and in which they placed\nanother kind of soul, mortal, the seat of violent and fatal\naffections.\"[609] He also regarded the soul as having a derived and\ndependent existence. He draws a marked distinction between the divine\nand human forms of the \"self-moving principle,\" and makes its\ncontinuance dependent upon the will and wisdom of the Almighty Disposer\nand Parent, of whom it is \"the first-born offspring.\"[610]\n\n[Footnote 608: See _ante_, note 4, p. 349, as to the Platonic notions of\n\"Time\" and \"Eternity.\"]\n\n[Footnote 609: \"Timaeus,\" ch. xliv.]\n\n[Footnote 610: See the elaborate exposition in \"Laws,\" bk. x. ch. xii.\nand xiii.]\n\nThat portion of the soul which Plato regarded as \"immortal\" and \"to be\nentitled divine,\" is thus the \"_offspring of God_\"--a ray of the\nDivinity \"generated\" by, or emanating from, the Deity. He seems to have\nconceived it as co-eternal with its ideal objects, in some mysterious\nultimate _unity_. \"The true foundation of the Platonic theory of the\nconstitution of the soul is this fundamental principle of his\nphilosophy--the _oneness of truth and knowledge_.[611] This led him\nnaturally to derive the _rational_ element of the soul (that element\nthat _knows_), that possesses the power of noesis from the _real_\nelement in things (the element that _is_)--the nooumenon; and in the\noriginal, the final, and, though imperfectly, the present state of that\nrational element, he, doubtless, conceived it united with its object in\nan eternal conjunction, or even identity. But though intelligence and\nits correlative intelligibles were and are thus combined, the soul is\n_more_ than pure intelligence; it possesses an element of personality\nand consciousness distinct to each individual, of which we have no\nreason to suppose, from any thing his writings contain, Plato ever meant\nto deprive it.\"[612] On the contrary, he not only regarded it as having\nnow, under temporal conditions, a distinct personal existence, but he\nalso claimed for it a conscious, personal existence after death. He is\nmost earnest, and unequivocal, and consistent in his assertion of the\ndoctrine of the immortality of the soul. The arguments which human\nreason can supply are exhibited with peculiar force and beauty in the\n\"Phaedo,\" the \"Phaedrus,\" and the tenth book of the \"Republic.\" The most\nimportant of these arguments may be presented in a few words.\n\n[Footnote 611: See Grant's \"Aristotle,\" vol. i. pp. 150, 151.]\n\n[Footnote 612: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n209, note.]\n\n1. _The soul is immortal, because it is incorporeal_. There are two\nkinds of existences, one compounded, the other simple; the former\nsubject to change, the latter unchangeable; one perceptible to sense,\nthe other comprehended by mind alone. The one is visible, the other is\ninvisible. When the soul employs the bodily senses, it wanders and is\nconfused; but when it abstracts itself from the body, it attains to\nknowledge which is stable, unchangeable, and immortal. The soul,\ntherefore, being uncompounded, incorporeal, invisible, must be\nindissoluble--that is to say, immortal.[613]\n\n[Footnote 613: \"Phaedo,\" Secs. 61-75.]\n\n2. _The soul is immortal, because it has an independent power of\nself-motion_--that is, it has self-activity and self-determination. No\narrangement of matter, no configuration of body, can be conceived as the\noriginator of free and voluntary movement.\n\nNow that which can not move itself, but derives its motion from\nsomething else, may cease to move, and perish. \"But that which is\nself-moved, never ceases to be active, and is also the cause of motion\nto all other things that are moved.\" And \"whatever is continually active\nis immortal.\" This \"self-activity is,\" says Plato, \"the very essence and\ntrue notion of the soul.\"[614] Being thus essentially _causative_, it\ntherefore partakes of the nature of a \"principle,\" and it is the nature\nof a principle to exclude its _contrary_. That which is essentially\nself-active can never cease to be active; that which is the cause of\nmotion and of change, can not be extinguished by the change called\ndeath.[615]\n\n3. _The soul is immortal, because it possesses universal, necessary, and\nabsolute ideas_, which transcend all material conditions, and bespeak an\norigin immeasurably above the body. No modifications of matter, however\nrefined, however elaborated, can give the Absolute, the Necessary, the\nEternal. But the soul has the ideas of absolute beauty, goodness,\nperfection, identity, and duration, and it possesses these ideas in\nvirtue of its having a nature which is one, simple, identical, and in\nsome sense, eternal.[616] If the soul can conceive an immortality, it\ncan not be less than immortal. If, by its very nature, \"it has hopes\nthat will not be bounded by the grave, and desires and longings that\ngrasp eternity,\" its nature and its destiny must correspond.\n\nIn the concluding sections of the \"Phaedo\" he urges the doctrine with\nearnestness and feeling as the grand motive to a virtuous life, for \"the\nreward is noble and the hope is great.\"[617] And in the \"Laws\" he\ninsists upon the doctrine of a future state, in which men are to be\nrewarded or punished as the most conclusive evidence that we are under\nthe moral government of God.[618]\n\n[Footnote 614: \"Phaedrus,\" Secs. 51-53.]\n\n[Footnote 615: \"Phaedo,\" Secs. 112-128.]\n\n[Footnote 616: Ibid., Secs. 48-57, 110-115.]\n\n[Footnote 617: Ibid., Secs. 129-145.]\n\n[Footnote 618: The doctrine of Metempsychosis, or transmigration of\nsouls, can scarcely be regarded as part of the philosophic system of\nPlato. He seems to have accepted it as a venerable tradition, coming\nwithin the range of probability, rather than as a philosophic truth, and\nit is always presented by him in a highly mythical dress. Now of these\nmythical representations he remarks in the \"Phaedo\" (Sec. 145) that \"no man\nin his senses would dream of insisting _that they correspond to the\nreality_, but that, the soul having been shown to be immortal, this, or\nsomething like this, is true of individual souls or their habitations.\"\nIf, as in the opinions of the ablest critics, \"the Laws\" is to be placed\namongst the last and maturest of Plato's writings, the evidence is\nconclusive that whatever may have been his earlier opinions, he did not\nentertain the doctrine of \"Metempsychosis\" in his riper years. But when,\non the one hand, the soul shall remain having an intercourse with divine\nvirtue, it becomes divine pre-eminently; and pre-eminently, after having\nbeen conveyed to a _place_ entirely holy, it is changed for the better;\nbut when it acts in a contrary manner, it has, under contrary\ncircumstances, placed its existence in some _unholy spot_.\n\n     _This is the judgment of the gods, who hold Olympus._\n\n\"O thou young man,\" [know] \"that the person who has become more wicked,\n_departs to the more wicked souls;_ but he who has become better, to the\nbetter both in life and in all deaths, to do and suffer what is fitting\nfor the like.\"--\"Laws,\" bk. x. ch. xii. and xiii.]\n\n4. _Beyond all finite existences and secondary causes, all laws, ideas,\nand principles, there is an_ INTELLIGENCE _or_ MIND, _the First\nPrinciple of all Principles, the Supreme Idea on which all other ideas\nare grounded; the Monarch and Lawgiver of the universe, the ultimate\nSubstance from which all other things derive their being and essence,\nthe First and efficient Cause of all the order, and harmony, and beauty,\nand excellency, and goodness, which pervades the universe, who is called\nby way of pre-eminence and excellence the Supreme Good_, THE GOD (o\nTheos), \"_the God over all_,\" (o epi pasi Theos).\n\n_This_ SUPREME MIND,[619] Plato taught, is incorporeal,[620]\nunchangeable,[621] infinite,[622] absolutely perfect,[623] essentially\ngood,[624] unoriginated,[625] and eternal.[626] He is \"the Father, and\nArchitect, and Maker of the Universe,\"[627] \"the efficient Cause of all\nthings.\"[628] \"the Monarch and Ruler of the world,\"[629] \"the sovereign\nMind that orders all things, and pervades all things,\"[630] \"the sole\nPrinciple of all things,\"[631] and \"the Measure of all things,\"[632] He\nis \"the Beginning of all truth,\"[633] \"the Fountain of all law and\njustice,\"[634] \"the Source of all order and beauty,\"[635] \"the Cause of\nall good;\"[636] in short, \"he is the Beginning, the Middle, and End of\nall things.\"[637]\n\n[Footnote 619: \"Phaedo,\" Secs. 105-107.]\n\n[Footnote 620: Diogenes Laertius, \"Lives,\" bk. iii. ch. 77.]\n\n[Footnote 621: \"Republic,\" bk. ii. ch. xix.; \"Timaeus,\" ch. ix.]\n\n[Footnote 622: \"Apeleius,\" bk. i. ch. v.]\n\n[Footnote 623: \"Republic,\" bk. ii. ch. xx.]\n\n[Footnote 624: \"Timaeus,\" ch. x.; \"Republic,\" bk. ii. ch. xviii.]\n\n[Footnote 625: \"Timaeus,\" ch. ix.-x.]\n\n[Footnote 626: Ibid., ch. xii.]\n\n[Footnote 627: Ibid., ch. ix.]\n\n[Footnote 628: \"Phaedo,\" Sec. 105.]\n\n[Footnote 629: \"Laws,\" bk. x. ch. xii.; \"Republic,\" bk. vii. ch. iii.;\n\"Philebus,\" Sec. 50.]\n\n[Footnote 630: \"Philebus,\" Sec.51.]\n\n[Footnote 631: \"Republic,\" bk. vi. ch. xix.]\n\n[Footnote 632: \"Laws,\" bk. iv. ch. viii.]\n\n[Footnote 633: \"Republic,\" bk. ii. ch. xxi.]\n\n[Footnote 634: \"Laws,\" bk. iv. ch. vii.]\n\n[Footnote 635: \"Philebus,\" Sec. 51; \"Timaeus,\" ch. x.]\n\n[Footnote 636: \"Republic,\" bk. ii. ch. xviii.; \"Timaeus,\" ch. x.]\n\n[Footnote 637: \"Laws,\" bk. iv, ch. vii.]\n\nBeyond the sensible world, Plato conceived another world of\nintelligibles or _ideas_. These ideas are not, however, distinct and\nindependent existences. \"What general notions are to our own minds,\nideas are to the Supreme Reason (nous basileus); they are the _eternal\nthoughts_ of the Divine Intellect.\"[638] Ideas are not substances, they\nare qualities, and there must, therefore, be some ultimate substance or\nbeing to whom, as attributes, they belong. \"It must not be believed, as\nhas been taught, that Plato gave to ideas a substantial existence. When\nthey are not objects of pure conception for human reason, they are\nattributes of the Divine Reason. It is there they substantially\nexist.\"[639] These eternal laws and reasons of things indicate to us the\ncharacter of that Supreme Essence of essences, the Being of beings. He\nis not the simple aggregate of all laws, but he is the Author, and\nSustainer, and Substance of all laws. At the utmost summit of the\nintellectual world of Ideas blazes, with an eternal splendor, the idea\nof the _Supreme Good_ from which all others emanate.[640] This Supreme\nGood is \"far beyond all existence in dignity and power, and it is that\nfrom which all things else derive their being and essence.\"[641] The\nSupreme Good is not the truth, nor the intelligence; \"it is the Father\nof it.\" In the same manner as the sun, which is the visible image of the\ngood, reigns over the world, in that it illumes and vivifies it; so the\nSupreme Good, of which the sun is only the work, reigns over the\nintelligible world, in that it gives birth to it by virtue of its\ninexhaustible fruitfulness.[642] _The Supreme Good is_ GOD _himself_,\nand he is designated \"the good\" because this term seems most fittingly\nto express his essential character and essence.[643] It is towards this\nsuperlative perfection that the reason lifts itself; it is towards this\ninfinite beauty the heart aspires. \"Marvellous Beauty!\" exclaims Plato;\n\"eternal, uncreated, imperishable beauty, free from increase and\ndiminution... beauty which has nothing sensible, nothing corporeal, as\nhands or face: which does not reside in any being different from itself,\nin the earth, or the heavens, or in any other thing, but which exists\n_eternally and absolutely in itself, and by itself;_ beauty of which\nevery other beauty partakes, without their birth or destruction bringing\nto it the least increase or diminution.\"[644] The absolute being--God,\nis the last reason, the ultimate foundation, the complete ideal of all\nbeauty. God is, _par excellent_, the Beautiful.\n\n[Footnote 638: Thompson's \"Laws of Thought,\" p. 119.]\n\n[Footnote 639: Cousin, \"Lectures on the History of Philosophy,\" vol. i.\np. 415. \"There is no quintessential metaphysics which can prevail\nagainst common sense, and if such be the Platonic theory of ideas,\nAristotle was right in opposing it. But such a theory is only a chimera\nwhich Aristotle created for the purpose of combating it.\"--\"The True,\nthe Beautiful, and the Good,\" p. 77.]\n\n[Footnote 640: \"Republic,\" bk. vii. ch. iii.]\n\n[Footnote 641: \"Ibid.,\" bk. vi. ch. xviii. and xix.]\n\n[Footnote 642: \"Republic,\" bk. vii. ch. iii.]\n\n[Footnote 642: Ritter's \"History of Ancient Philosophy,\" vol. ii. p.\n275.]\n\n[Footnote 644: \"Banquet,\" Sec. 35. See Cousin, \"The True, the Beautiful,\nand the Good,\" Lecture IV., also Lecture VII. pp. 150-153; Denis,\n\"Histoire des Theories et Idees Morales dans l'Antiquite,\" vol. i. p.\n149.]\n\nGod is therefore, with Plato, _the First Principle of all Principles;_\nthe Divine energy or power is the _efficient cause_, the Divine beauty\nthe _formal cause_, and the Divine goodness the _final cause_ of all\nexistence.\n\n_The eternal unity of the principles of Order, Goodness, and Truth, in\nan ultimate reality--the_ ETERNAL MIND, is thus the fundamental\nprinciple which pervades the whole of the Platonic philosophy. And now,\nhaving attained this sublime elevation, he looks down from thence upon\nthe _sensible, the phenomenal world_, and upon _the temporal life of\nman;_ and in the light of this great principle he attempts to explain\ntheir meaning and purpose. The results he attained in the former case\nconstitute the Platonic _Physics_, in the latter, the Platonic _Ethics_.\n\nI. PLATONIC PHYSICS.\n\nFirmly believing in the absolute excellence of the Deity, and regarding\nthe Divine Goodness as the Final Cause of the universe, he pronounces\nthe physical world to be an _image_ of the perfection of God.\nAnaxagoras, no doubt, prepared the way for this theory. Every one who\nhas read the \"Phaedo,\" will remember the remarkable passage in which\nSocrates gives utterance to the disappointment which he had experienced\nwhen expecting from physical science an explanation of the universe.\n\"When I was young,\" he said--\"it is not to be told how eager I was about\nphysical inquiries, and curious to know _how the universe came to be as\nit is_; and when I heard that Anaxagoras was teaching that all was\narranged by _mind_, I was delighted with the prospect of hearing such a\ndoctrine unfolded; I thought to myself, if he teaches that mind made\nevery thing to be as it is, he will explain _how it is_ BEST _for it to\nbe_, and show that so it is.\" But Anaxagoras, it appears, lost sight of\nthis principle, and descended to the explanation of the universe by\nmaterial causes. \"Great was my hope,\" says Socrates, \"and equally great\nmy disappointment.\"[645]\n\n[Footnote 645: \"Phaedo,\" Secs. 105, 106.]\n\nPlato accepted this suggestion of Anaxagoras with all his peculiar\nearnestness, and devoted himself to its fuller development. It were a\nvain and profitless theory, which, whilst it assumed the existence of a\nSupreme Mind, did not represent that mind as operating in the universe\nby _design_, and as exhibiting his intelligence, and justice, and\ngoodness, as well as his power, in every thing. If it be granted that\nthere is a Supreme Mind, then, argued Plato, he must be regarded as \"the\nmeasure of all things,\" and all things must have been framed according\nto a plan or \"model\" which that mind supplied. Intelligence must be\nregarded as having a _purpose_, and as working towards an _end_, for it\nis this alone which distinguishes reason from unreason, and mind from\nmere unintelligent force. The only proper model which could be presented\nto the Supreme Intelligence is \"the eternal and unchangeable model\"[646]\nwhich his own perfection supplies, \"for he is the most excellent of\ncauses.\"[647] Thus God is not simply the maker of the universe, but the\nmodel of the universe, because he designed that it should be an IMAGE,\nin the sphere of sense, of his own perfections--a revelation of his\neternal beauty, and wisdom, and goodness, and truth. \"God was _good_,\nand being good, he desired that the universe should, as far as possible,\n_resemble_ himself.... Desiring that all things should be _good_, and,\nas far as might be, nothing evil, he took the fluctuating mass of things\nvisible, which had been in orderless confusion, and reduced it to\n_order_, considering this to be the _better_ state. Now it was and is\nutterly impossible for the supremely good to form any thing except that\nwhich is _most excellent_ (kalliston--most fair, most beautiful\").[648]\nThe object at which the supreme mind aimed being that which is \"_best_,\"\nwe must, in tracing his operations in the universe, always look for\n\"_the best_\" in every thing.[649] Starting out thus, upon the assumption\nthat the goodness of God is the final cause of the universe, Plato\nevolved a system of _optimism_.\n\nThe physical system of Plato being thus intended to illustrate a\nprinciple of optimism, the following results may be expected:\n\n1. That it will mainly concern itself with _final causes_. The universe\nbeing regarded chiefly, as indeed it is, an indication of the Divine\nIntelligence--every phenomenon will be contemplated in that light.\nNature is the volume in which the Deity reveals his own perfections; it\nis therefore to be studied solely with this motive, that we may learn\nfrom thence the perfection of God. The _Timaeus_ is a series of ingenious\nhypotheses designed to deepen and vivify our sense of the harmony, and\nsymmetry, and beauty of the universe, and, as a consequence, of the\nwisdom, and excellence, and goodness, of its Author.[650]\n\n[Footnote 646: \"Timaeus,\" ch. ix.]\n\n[Footnote 647: Ibid.]\n\n[Footnote 648: Ibid., ch. x.]\n\n[Footnote 649: Ibid., ch. xix.]\n\n[Footnote 650: \"Being is related to Becoming (the Absolute to the\nContingent) as Truth is to Belief; consequently we must not marvel\nshould we find it impossible to arrive at any certain and conclusive\nresults in our speculations upon the creation of the visible universe\nand its authors; it should be enough for us if the account we have to\ngive be as probable as any other, remembering that we are but men, and\ntherefore bound to acquiesce in merely probable results, without looking\nfor a higher degree of certainty than the subject admits of\"--\"Timaeus,\"\nch. ix.]\n\nWhatever physical truths were within the author's reach, took their\nplace in the general array: the vacancies were filled up with the best\nsuppositions admitted by the limited science of the time.[651] And it is\nworthy of remark that, whilst proceeding by this \"high _a priori_ road,\"\nhe made some startling guesses at the truth, and anticipated some of the\ndiscoveries of the modern inductive method, which proceeds simply by the\nobservation, comparison, and generalization of facts. Of these prophetic\nanticipations we may instance that of the definite proportions of\nchemistry,[652] the geometrical forms of crystallography,[653] the\ndoctrine of complementary colors,[654] and that grand principle that all\nthe highest laws of nature assume the form of a precise quantitative\nstatement.[655]\n\n2. It may be expected that a system of physics raised on optimistic\nprinciples will be _mathematical_ rather than experimental. \"Intended to\nembody conceptions of proportion and harmony, it will have recourse to\nthat department of science which deals with the proportions in space and\nnumber. Such applications of mathematical truths, not being raised on\nascertained facts, can only accidentally represent the real laws of the\nphysical system; they will, however, vivify the student's apprehension\nof harmony in the same manner as a happy parable, though not founded in\nreal history, will enliven his perceptions of moral truth.\"[656]\n\n[Footnote 651: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n157.]\n\n[Footnote 652: \"Timaeus,\" ch. xxxi.]\n\n[Footnote 653: Ibid., ch. xxvii.]\n\n[Footnote 654: Ibid., ch. xlii.]\n\n[Footnote 655: \"It is Plato's merit to have discovered that the laws of\nthe physical universe are resolvable into numerical relations, and\ntherefore capable of being represented by mathematical\nformulae.\"--Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p. 163.]\n\n[Footnote 656: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n163.]\n\n3. Another peculiarity of such a system will be an impatience of every\nmerely _mechanical_ theory of the operations of nature.\n\n\"The psychology of Plato led him to recognize mind wherever there was\nmotion, and hence not only to require a Deity as first mover of the\nuniverse, but also to conceive the propriety of separate and subordinate\nagents attached to each of its parts, as principles of motion, no less\nthan intelligent directors. These agents were entitled '_gods_' by an\neasy figure, discernible even in the sacred language,[657] and which\nserved, besides, to accommodate philosophical hypotheses to the popular\nreligion. Plato, however, carefully distinguished between the sole,\nEternal Author of the Universe, on the one hand, and that 'soul,' vital\nand intelligent, which he attaches to the world, as well as the spheral\nintelligences, on the other. These 'subordinate deities,' though\nintrusted with a sort of deputed creation, were still only the deputies\nof the Supreme Framer and Director of all.\"[658] The \"gods\" of the\nPlatonic system are \"subordinate divinities,\" \"generated gods,\" brought\ninto existence by the will and wisdom of the Eternal Father and Maker of\nthe universe.[659] Even Jupiter, the governing divinity of the popular\nmythology, is a descendant from powers which are included in the\ncreation.[660] The offices they fulfill, and the relations they sustain\nto the Supreme Being, correspond to those of the \"angels\" of Christian\ntheology. They are the ministers of his providential government of the\nworld.[661]\n\n[Footnote 657: Psalm lxxxii. I; John x. 34.]\n\n[Footnote 658: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n164.]\n\n[Footnote 659: \"Timaeus,\" ch. xv.]\n\n[Footnote 660: Ibid.]\n\n[Footnote 661: \"Laws,\" bk. x.]\n\nThe application of this fundamental conception of the Platonic\nsystem--_the eternal unity of the principles of Order, Goodness, and\nTruth in an ultimate reality, the Eternal Mind_--to the elucidation of\nthe _temporal life_ of man, yields, as a result--\n\nII. THE PLATONIC ETHICS.\n\nBelieving firmly that there are unchangeable, necessary, and absolute\nprinciples, which are the perfections of the Eternal Mind, Plato must,\nof course, have been a believer in an _immutable morality_. He held that\nthere is a rightness, a justice, an equity, not arbitrarily constituted\nby the Divine will or legislation, but founded in the nature of God, and\ntherefore eternal. The independence of the principles of morality upon\nthe mere will of the Supreme Governor is proclaimed in all his\nwritings.[662] The Divine will is the fountain of efficiency, the Divine\nreason, the fountain of law. God is no more the creator of _virtue_ than\nhe is the creator of _truth_.\n\nAnd inasmuch as man is a partaker of the Divine essence, and as the\nideas which dwell in the human reason are \"copies\" of those which dwell\nin the Divine reason, man may rise to the apprehension and recognition\nof the immutable and eternal principles of righteousness, and \"by\ncommunion with that which is Divine, and subject to the law of order,\nmay become himself a subject of order, and divine, so far as it is\npossible for man.\"[663]\n\n[Footnote 662: In \"Euthyphron\" especially.]\n\n[Footnote 663: \"Republic,\" bk. vi. ch. xiii.]\n\nThe attainment of this consummation is the grand purpose of the Platonic\nphilosophy. Its ultimate object is \"_the purification of the soul_,\" and\nits pervading spirit is the aspiration after perfection. The whole\nsystem of Plato has therefore an eminently _ethical_ character. It is a\nspeculative philosophy directed to a practical purpose.\n\nPhilosophy is the _love of wisdom_. Now wisdom (sophia) is expressly\ndeclared by Plato to belong alone to the Supreme Divinity,[664] who\nalone can contemplate reality in a direct and immediate manner, and in\nwhom, as Plato seems often to intimate, knowledge and being coincide.\nPhilosophy is the aspiration of the soul after this wisdom, this perfect\nand immutable truth, and in its realization it is a union with the\nPerfect Wisdom through the medium of a divine affection, the _love_ of\nwhich Plato so often speaks. The eternal and unchangeable Essence which\nis the proper object of philosophy is also endowed with _moral_\nattributes. He is not only \"the Being,\" but \"the Good\" (to agathon), and\nall in the system of the universe which can be the object of rational\ncontemplation, is an emanation from that goodness. The love of truth is\ntherefore the love of God, and the love of Good is the love of truth.\nPhilosophy and morality are thus coincident. Philosophy is the love of\nPerfect Wisdom; Perfect Wisdom and Perfect Goodness are identical; the\nPerfect Good is God; philosophy is the \"_Love of God_.\"[665] Ethically\nviewed, it is this one motive of _love_ for the Supreme Wisdom and\nGoodness, predominating over and purifying and assimilating every desire\nof the soul, and governing every movement of the man, raising man to a\nparticipation of and communion with Divinity, and restoring him to \"the\n_likeness_ of God.\" \"This flight,\" says Plato, \"consists in resembling\nGod (omoiosis Theo), and this resemblance is the becoming just and holy\nwith wisdom.\"[666] \"This assimilation to God is the enfranchisement of\nthe divine element of the soul. To approach to God as the substance of\ntruth is _Science_; as the substance of goodness in truth is _Wisdom_,\nand as the substance of Beauty in goodness and truth is _Love_.\"[667]\n\nThe two great principles which can be clearly traced as pervading the\nethical system of Plato are--\n\n1. _That no man is willingly evil_.[668]\n\n2. _That every man is endued with the power of producing changes in his\nmoral character_[669]\n\n[Footnote 664: \"Phaedrus,\" Sec. 145.]\n\n[Footnote 665: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n61.]\n\n[Footnote 666: \"Theaetetus,\" Sec. 84.]\n\n[Footnote 667: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n277.]\n\n[Footnote 668: \"Timaesus,\" ch. xlviii.]\n\n[Footnote 669: \"Laws,\" bk. v. ch. i., bk. ix. ch. vi., bk. x. ch. xii.]\n\nThe first of these principles is the counterpart ethical expression of\nhis theory of _immutable Being_. The second is the counterpart of his\ntheory of phenomenal change, or _mere Becoming_.\n\nThe soul of man is framed after the pattern of the immutable ideas of\nthe _just_, and the _true_, and the _good_, which dwell in the Eternal\nMind--that is, it is made in the image of God. The soul in its ultimate\nessence is formed of \"the immutable\" and \"the permanent.\" The presence\nof the ideas of the just, and the true, and the good in the reason of\nman, constitute him a moral nature; and it is impossible that he can\ncease to be a moral being, for these ideas, having a permanent and\nimmutable being, can not be changed. All the passions and affections of\nthe soul are merely phenomenal. They belong to the mortal, the\ntransitory life of man; they are in endless flow and change, and they\nhave no permanent reality. As phenomena, they must, however, have some\nground; and Plato found that ground in the mysterious, instinctive\nlonging for the _good_ and the _true_ which dwells in the very essence\nof the soul. These are the realities after which it strives, even when\npursuing pleasure, and honor, and wealth, and fame. All the restlessness\nof human life is prompted by a longing for the _good_. But man does not\nclearly perceive what the _good_ really is. The rational element of the\nsoul has become clouded by passion and ignorance, and suffered an\neclipse of its powers. Still, man longs for the good, and bears witness,\nby his restlessness and disquietude, that he instinctively desires it,\nand that he can find no rest and no satisfaction in any thing apart from\nthe knowledge and the participation of the Supreme, the Absolute Good.\n\nThis, then, is the meaning of the oft-repeated assertion of Plato \"_that\nno man is willingly evil_;\" viz., that no man deliberately chooses evil\nas evil. And Plato is, at the same time, careful to guard the doctrine\nfrom misconception. He readily grants that acts of wrong are\ndistinguished as voluntary and involuntary, without which there could be\nneither merit nor demerit, reward nor punishment.[670] But still he\ninsists that no man chooses evil in and by itself. He may choose it\nvoluntarily as a means, but he does not choose it as an end. Every\nvolition, by its essential nature, pursues, at least, an _apparent_\ngood; because the end of volition is not the immediate act, but the\nobject for the sake of which the act is undertaken.[671]\n\n[Footnote 670: \"Laws,\" bk. ix. ch. vi.]\n\n[Footnote 671: \"Gorgias,\" Secs. 52, 53.]\n\nHow is it, then, it may be asked, that men become evil? The answer of\nPlato is, that the soul has in it a principle of change, in the power of\nregulating the desires--in indulging them to excess, or moderating them\naccording to the demands of reason. The circumstances in which the soul\nis placed, as connected with the sensible world by means of the body,\npresent an occasion for the exercise of that power, the end of this\ntemporal connection being to establish a state of moral discipline and\nprobation. The humors and distempers of the body likewise deprave,\ndisorder, and discompose the soul.[672] \"Pleasures and pains are unduly\nmagnified; the democracy of the passions prevails; and the ascendency of\nreason is cast down.\" Bad forms of civil government corrupt social\nmanners, evil education effects the ruin of the soul. Thus the soul is\nchanged--is fallen from what it was when first it came from the\nCreator's hand. But the eternal Ideas are not utterly effaced, the image\nof God is not entirely lost. The soul may yet be restored by remedial\nmeasures. It may be purified by knowledge, by truth, by expiations, by\nsufferings, and by prayers. The utmost, however, that man can hope to do\nin this life is insufficient to fully restore the image of God, and\ndeath must complete the final emancipation of the rational element from\nthe bondage of the flesh. Life is thus a discipline and a preparation\nfor another state of being, and death the final entrance there.[673]\n\n[Footnote 672: \"Gorgias,\" Secs. 74-76.]\n\n[Footnote 673: \"Phaedo,\" Secs. 130, 131.]\n\nIndependent of all other considerations, virtue is, therefore, to be\npursued as the true good of the soul. Wisdom, Fortitude, Temperance,\nJustice, the four cardinal virtues of the Platonic system, are to be\ncultivated as the means of securing the purification and perfection of\nthe inner man. And the ordinary pleasures, \"the lesser goods\" of life,\nare only to be so far pursued as they are subservient to, and compatible\nwith, the higher and holier duty of striving after \"the resemblance to\nGod.\"\n\n\n\n\nCHAPTER XII.\n\nTHE PHILOSOPHERS OF ATHENS (_continued_).\n\nTHE SOCRATIC SCHOOL (_continued_).\n\nARISTOTLE.\n\n\nAristotle was born at Stagira, a Greek colony of Thrace, B.C. 384. His\nfather, Nicomachus, was a physician in the Court of Amyntas II., King of\nMacedonia, and is reported to have written several works on Medicine and\nNatural History. From his father, Aristotle seems to have inherited a\nlove for the natural sciences, which was fostered by the circumstances\nwhich surrounded him in early life, and which exerted a determining\ninfluence upon the studies of his riper years.\n\nImpelled by an insatiate desire for knowledge, he, at seventeen years of\nage, repaired to Athens, the city of Plato and the university of the\nworld. Plato was then absent in Sicily; on his return Aristotle entered\nhis school, became an ardent student of philosophy, and remained until\nthe death of Plato, B.C. 348. He therefore listened to the instructions\nof Plato for twenty years.\n\nThe mental characteristics of the pupil and the teacher were strikingly\ndissimilar. Plato was poetic, ideal, and in some degree mystical.\nAristotle was prosaic, systematic, and practical. Plato was intuitive\nand synthetical. Aristotle was logical and analytical. It was therefore\nbut natural that, to the mind of Aristotle, there should appear\nsomething confused, irregular, and incomplete in the discourses of his\nmaster. There was a strange commingling of questions concerning the\ngrounds of morality, and statements concerning the nature of science; of\ninquiries concerning \"real being,\" and speculations on the ordering of a\nmodel Republic, in the same discourse. Ethics, politics, ontology, and\ntheology, are all comprised in his Dialectic, which is, in fact, the one\ngrand \"science of the idea of the good.\" Now to the mind of Aristotle it\nseemed better, and much more systematic, that these questions should be\nseparated, and referred to particular heads; and, above all, that they\nshould be thoroughly discussed in an exact and settled terminology. To\narrange and classify all the objects of knowledge, to discuss them\nsystematically and, as far as possible, exhaustively, was evidently the\nambition, perhaps also the special function, of Aristotle. He would\nsurvey the entire field of human knowledge; he would study nature as\nwell as humanity, matter as well as mind, language as well as thought;\nhe would define the proper limits of each department of study, and\npresent a regular statement of the facts and principles of each science.\nAnd, in fact, he was the first who really separated the different\nsciences and erected them into distinct systems, each resting upon its\nown proper principles. He distributed philosophy into three\nbranches:--(i.) _Theoretic_; (ii.) _Efficient_; (iii.) _Practical_. The\nTheoretic he divided into--1. _Physics_; _2. Mathematics_; 3.\n_Theology_, or the Prime Philosophy--the science known in modern times\nas Metaphysics. The Efficient embraces what we now term the arts,--1.\n_Logic; 2. Rhetoric_; 3. _Poetics_. The Practical comprises--_1.\nEthics_; 2. _Politics_. On all these subjects he wrote separate\ntreatises. Thus, whilst Plato is the genius of abstraction, Aristotle is\neminently the genius of classification.\n\nSuch being the mental characteristics of the two men--their type of mind\nso opposite--we are prepared to expect that, in pursuing his inquiries,\nAristotle would develop a different _Organon_ from that of Plato, and\nthat the teachings of Aristotle will give a new direction to philosophic\nthought.\n\nARISTOTELIAN ORGANON.\n\nPlato made use of psychological and logical analysis in order to draw\nfrom the depth of consciousness certain fundamental ideas which are\ninherent in the mind--born with it, and not derived from sense or\nexperience. These ideas he designates \"the intelligible species\" (ta\nnooumena gene) as opposed to \"the visible species\"--the objects of\nsense. Such ideas or principles being found, he uses them as\n\"starting-points\" from which he may pass beyond the sensible world and\nascend to \"the absolute,\" that is, to God.[674] Having thus, by\nimmediate abstraction, attained to universal and necessary ideas, he\ndescends to the outer world, and attempts by these ideas to construct an\nintellectual theory of the universe.[675]\n\nAristotle will reverse this process. He will commence with _sensation_,\nand proceed, by induction, from the known to the unknown.\n\nThe repetition of sensations produces _recollection_, recollection\n_experience_, and experience produces _science_.[676] \"Science and art\nresult unto men by means of experience....\" \"Art comes into being when,\nfrom a number of experiences, one universal opinion is evolved, which\nwill embrace all similar cases. For example, if you know that a certain\nremedy has cured Callias of a certain disease, and that the same remedy\nhas produced the same effect on Socrates and on several other persons,\nthat is _Experience_; but to know that a certain remedy will cure all\npersons attacked with that disease, is _Art_. Experience is a knowledge\nof individual things (ton kathekasta); art is that of universals (ton\nkatholou).\"[677]\n\n[Footnote 674: \"Republic,\" bk. vi. ch. xx.]\n\n[Footnote 675: \"Timaeus,\" ch. ix.]\n\n[Footnote 676: \"Metaphysics,\" bk. i. ch. i.]\n\n[Footnote 677: Ibid.]\n\nDisregarding the Platonic notion of the unity of all Being in the\nabsolute idea, he fixed his immediate attention on the manifoldness of\nthe phenomenal, and by a classification of all the objects of experience\nhe sought to attain to \"general notions.\" Concentrating all his\nattention on the individual, the contingent, the particular, he ascends,\nby induction, from the particular to the _general_; and then, by a\nstrange paralogism, \"the _universal_\" is confounded with \"the _general_\"\nor, by a species of logical sleight-of-hand, the general is transmuted\ninto the universal. Thus \"induction is the pathway from particulars to\nuniversals.\"[678] But how universal and necessary principles can be\nobtained by a generalization of limited experiences is not explained by\nAristotle. The experiences of a lifetime, the experiences of the whole\nrace, are finite and limited, and a generalization of these can only\ngive the finite, the limited, and at most, the general, but not the\nuniversal.\n\n[Footnote 679: \"Topics,\" bk. i. ch. xii.; \"Ethics,\" bk. vi. ch. iii.]\n\nAristotle admits, however, that there are ideas or principles in the\nmind which can not be explained by experience, and we are therefore\nentitled to an answer to the question--how are these obtained? \"Sensible\nexperience gives us what is _here_, _there_, _now_, in such and such a\nmanner, but it is impossible for it to give what is _everywhere_ and _at\nall times_.\"[680] He tells us further, that \"science is a conception of\nthe mind engaged in universals, and in those things which exist of\nnecessity, and since there are _principles of things demonstrable and of\nevery science_ (for science is joined with reason), it will be neither\nscience, nor art, nor prudence, which discovers the principles of\nscience;... it must therefore be (nous) pure intellect,\" or the\nintuitive reason.[681] He also characterizes these principles as\n_self-evident_. \"First truths are those which obtain belief, not through\nothers, but through themselves, as there is no necessity to investigate\nthe '_why_' in scientific principles, but each principle ought to be\ncredible by itself.\"[682] They are also _necessary_ and _eternal_.\n\"Demonstrative science is from necessary principles, and those which are\n_per se_ inherent, are necessarily so in things.\"[683] \"We have all a\nconception of that which can not subsist otherwise than it does.... The\nobject of science has a necessary existence, therefore it is _eternal_.\nFor those things which exist in themselves, by necessity, are all\neternal.\"[684] But whilst Aristotle admits that there are \"immutable and\nfirst principles,\"[685] which are not derived from sense and\nexperience--\"principles which are the foundation of all science and\ndemonstration, but which are themselves indemonstrable,\"[686] because\nself-evident, necessary, and eternal; yet he furnishes no proper account\nof their genesis and development in the human mind, neither does he\nattempt their enumeration. At one time he makes the intellect itself\ntheir source, at another he derives them from sense, experience, and\ninduction. This is the defect, if not the inconsistency, of his\nmethod.[687]\n\n[Footnote 680: \"Post. Analytic,\" bk. i. ch. xxxi.]\n\n[Footnote 681: \"Ethics,\" bk. vi. ch. vi.]\n\n[Footnote 682: \"Topics,\" bk. i. ch. i.]\n\n[Footnote 683: \"Post. Analytic,\" bk. i. ch. vi.]\n\n[Footnote 684: \"Ethics,\" bk. vi. ch. iii.]\n\n[Footnote 685: Ibid., bk. vi. ch. xi.]\n\n[Footnote 686: \"Post. Analytic,\" bk. i. ch. iii.]\n\n[Footnote 687: Hamilton attempts the following mode of reconciling the\ncontradictory positions of Aristotle:\n\n\"On the supposition of the mind virtually containing, antecedent to all\nexperience, certain universal principles of knowledge, in the form of\ncertain necessities of thinking; still it is only by repeated and\ncomparative experiments that we compass the certainty; on the one hand,\nthat such and such cognitions can not but be thought as necessary,\nnative generalities; and, on the other, that such and such cognitions\nmay or may not be thought, and are, therefore, as contingent, factitious\ngeneralizations. To this process of experiment, analysis, and\nclassification, through which we attain to a scientific knowledge of\nprinciples, it might be shown that Aristotle, not improperly, applies\nthe term _Induction_.\"--\"Philosophy,\" p. 88.]\n\nThe human mind, he tells us, has two kinds of intelligence--the\n_passive_ intelligence (nous pathetikos), which is the receptacle of\nforms (dectikon tou eidous); and the _active_ intelligence (nous\npoietikos), which impresses the seal of thought upon the data furnished\nby experience, and combines them into the unity of a single judgment,\nthus attaining \"general notions.\"[688] The passive intelligence (the\n\"external perception\" of modern psychology) perceives the individual\nforms which appear in the external world, and the active intelligence\n(the intellect proper) classifies and generalizes according to fixed\nlaws or principles inherent in itself; but of these fixed laws--prota\nnoemata--first thoughts, or _a priori_ ideas, he offers no proper\naccount; they are, at most, purely subjective. This, it would seem, was,\nin effect, a return to the doctrine of Protagoras and his school, \"that\nman--the individual--is the measure of all things.\" The aspects under\nwhich objects present themselves in consciousness, constitute our only\nground of knowledge; we have no direct, intuitive knowledge of Being _in\nse_. The noetic faculty is simply a _regulative_ faculty; it furnishes\nthe laws under which we compare and judge, but it does not supply any\noriginal elements of knowledge. Individual things are the only real\nentities,[689] and \"universals\" have no separate existence apart from\nindividuals in which they inhere as attributes or properties. They are\nconsequently pure mental conceptions, which are fixed and recalled by\ngeneral names. He thus substitutes a species of conceptual-nominalism in\nplace of the realism of Plato. It is true that \"real being\" (to on) is\nwith Aristotle a subject of metaphysical inquiry, but the proper, if not\nthe only subsistence, or ouaia, is the form or abstract nature of\nthings. \"The essence or very nature of a thing is inherent in the _form_\nand _energy_\"[690] The science of Metaphysics is strictly conversant\nabout these abstract intellectual forms just as Natural Philosophy is\nconversant about external objects, of which the senses give us\ninformation. Our knowledge of these intellectual forms is, however,\nfounded upon \"beliefs\" rather than upon immediate intuition, and the\nobjective certainty of science, upon the subjective necessity of\nbelieving, and not upon direct apperception.\n\n[Footnote 688: \"On the Soul,\" ch. vi.; \"Ethics,\" bk. vi. ch. i.]\n\n[Footnote 689: \"Metaphysics,\" bk. vi. ch. xiii.]\n\n[Footnote 690: Ibid., bk. vii. ch. iii.]\n\nThe points of contrast between the two methods may now be presented in a\nfew sentences. Plato held that all our cognitions are reducible to two\nelements--one derived from _sense_, the other from _pure reason_; one\nelement particular, contingent, and relative, the other universal,\nnecessary, and absolute. By an act of _immediate abstraction_ Plato will\neliminate the particular, contingent, and relative phenomena, and\ndisengage the universal, necessary, and absolute _ideas_ which underlie\nand determine all phenomena. These ideas are the thoughts of the Divine\nMind, according to which all particular and individual existences are\ngenerated, and, as divine thoughts, they are real and permanent\nexistences. Thus by a process of immediate abstraction, he will rise\nfrom particular and contingent phenomena to universal and necessary\nprinciples, and from these to the First Principle of all principles, the\nFirst Cause of all causes--that is, to _God_.\n\nAristotle, on the contrary, held that all of our knowledge begins with\n\"the singular,\" that is, with the particular and the relative, and is\nderived from sensation and experience. The \"sensible object,\" taken as\nit is without any sifting and probing, is the basis of science, and\nreason is simply the architect constructing science according to certain\n\"forms\" or laws inherent in mind. The object, then, of metaphysical\nscience is to investigate those \"universal notions\" under which the mind\nconceives of and represents to itself external objects, and speculates\nconcerning them. Aristotle, therefore, agrees with Plato in teaching\n\"that science can only be a science of universals,\"[691] and \"that\nsensation alone can not furnish us with scientific knowledge.\"[692] How,\nthen, does he propose to attain the knowledge of universal principles?\nHow will he perform that feat which he calls \"passing from the known to\nthe unknown?\" The answer is, by _comparative abstraction_. The universal\nbeing constituted by a relation of the object to the thinking subject,\nthat is, by a property recognized by the intelligence alone, in virtue\nof which it can be retained as an object of thought, and compared with\nother objects, he proposes to _compare, analyze, define,_ and _classify_\nthe primary cognitions, and thus evoke into energy, and clearly present\nthose principles or forms of the intelligence which he denominate\n\"universals.\" As yet, however, he has only attained to \"general\nnotions,\" which are purely subjective, that is, to logical definitions,\nand these logical definitions are subsequently elevated to the dignity\nof \"universal principles and causes\" by a species of philosophic\nlegerdemain. Philosophy is thus stripped of its metaphysical character,\nand assumes a strictly _logical_ aspect. The key of the Aristotelian\nmethod is therefore the\n\nARISTOTELIAN LOGIC.\n\nPure Logic is the science of the formal laws of thought. Its office is\nto ascertain the rules or conditions under which the mind, by its own\nconstitution, reasons and discourses. The office of Applied Logic--of\nlogic as an art--is \"to form and judge of conclusions, and, through\nconclusions, to establish proof. The conclusions, however, arise from\npropositions, and the propositions from conceptions.\" It is chiefly\nunder the latter aspect that logic is treated by Aristotle. According to\nthis natural point of view he has divided the contents of the logical\nand dialectic teaching in the different treatises of the _Organon_.\n\n[Footnote 691: \"Ethics,\" bk. vi. ch. vi.]\n\n[Footnote 692: \"Post. Analytic,\" bk. i. ch. xxxi.]\n\nThe first treatise is the \"_Categories_\" or \"Predicaments\"--a work which\ntreats of the universal determinations of Being. It is a classification\nof all our mental conceptions. As a matter of fact, the mind forms\nnotions or conceptions about those natures and essences of things which\npresent an outward image to the senses, or those, equally real, which\nutter themselves to the mind. These may be defined and classified; there\nmay be general conceptions to which all particular conceptions are\nreferable. This classification has been attempted by Aristotle, and as\nthe result we have the ten \"Categories\" of _Substance, Quantity,\nQuality, Relation, Time, Place, Position, Possession, Action, Passion_.\nHe does not pretend that this classification is complete, but he held\nthese \"Predicaments\" to be the most universal expressions for the\nvarious relations of things, under some one of which every thing might\nbe reduced.\n\nThe second treatise, \"_On Interpretation_,\" investigates language as the\nexpression of thought; and inasmuch as a true or false thought must be\nexpressed by the union or separation of a subject and a predicate, he\ndeems it necessary to discuss the parts of speech--the general term and\nthe verb--and the modes of affirmation and denial. In this treatise he\ndevelops the nature and limitations of propositions, the meaning of\ncontraries and contradictions, and the force of affirmations and denials\nin _possible, contingent_, and _necessary_ matter.\n\nThe third are the \"_Analytics_,\" which show how conclusions are to be\nreferred back to their principles, and arranged in the order of their\nprecedence.\n\nThe First or Prior Analytic presents the universal doctrine of the\nSyllogism, its principles and forms, and teaches how must reason, if we\nwould not violate the laws of our own mind. The theory of reasoning,\ngenerally, with a view to accurate demonstration, depends upon the\nconstruction of a perfect syllogism, which is defined as \"a discourse in\nwhich, certain things being laid down, something else different from the\npremises necessarily results, in consequence of their existence.\"[693]\nConclusions are, according to their own contents and end, either\n_Apodeictic_, which deal with necessary and demonstrable matter, or\n_Dialectic_, which deal with probable matter, or _Sophistical_, which\nare imperfect in matter or form, and announced, deceptively, as correct\nconclusions, when they are not. The doctrine of Apodeictic conclusions\nis given in the \"_Posterior Analytic_,\" that of Dialectic conclusions in\nthe \"_Topics_,\" and that of the Sophistical in the \"_Sophistical\nElenchi_.\"\n\nNow, if Logic is of any value as an instrument for the discovery of\ntruth, the attainment of certitude, it must teach us not only how to\ndeduce conclusions from premises, but it must certify to us the validity\nof the principles from whence we reason and this is attempted by\nAristotle in the Posterior Analytic. This treatise opens with the\nfollowing statement: All doctrine, and all intellectual discipline,\narises from a prior or pre-existent knowledge. This is evident, if we\nsurvey them all; for both mathematical sciences, and also each of the\narts, are obtained in this manner. The same holds true in the case of\nreasonings, whether through [deductive] _Syllogism_ or through\n_Induction_, for both accomplish the instruction they afford from\ninformation previously known--the former (syllogistic reasoning)\nreceiving it, as it were, from the traditions of the intelligent, the\nlatter (inductive reasoning) manifesting the universal through the light\nof the singular.[694] Induction and Syllogism are thus the grand\ninstruments of logic.[695]\n\n[Footnote 693: \"Prior Analytic,\" bk. i. ch. i.; \"Topics,\" bk. i. ch. i.]\n\n[Footnote 694: \"Post. Analytic,\" bk. i. ch. i.]\n\n[Footnote 695: \"We believe all things through syllogism, or from\ninduction.\"--\"Prior Analytic,\" bk. ii. ch. xxiii.]\n\nBoth these processes are based upon an _anterior_ knowledge.\nDemonstrative science must be from things true, first, immediate, more\nknown than, prior to, and the causes of, the conclusion, for thus there\nwill be the appropriate first principles of whatever is\ndemonstrated.[696] The first principles of demonstration, the material\nof thought, must, consequently, be supplied by some power or faculty of\nthe mind other than that which is engaged in generalization and\ndeductive reasoning. Whence, then, is this \"anterior knowledge\" derived,\nand what tests or criteria have we of its validity?\n\n1. In regard to deductive or syllogistic reasoning, the views of\nAristotle are very distinctly expressed.\n\nSyllogistic reasoning \"proceeds from generals to particulars.\"[697] The\ngeneral must therefore be supplied as the foundation of the deductive\nreasoning. Whence, then, is this knowledge of \"the general\" derived? The\nanswer of Aristotle is that the universal major proposition, out of\nwhich the conclusion of the syllogism is drawn, _is itself necessarily\nthe conclusion of a previous induction, and mediately or immediately an\ninference_--a collection from individual objects of sensation or of\nself-consciousness. \"Now,\" says he, \"demonstration is from universals,\nbut induction from particulars. It is impossible, however, to\ninvestigate universals except through induction, since things which are\nsaid to be from abstraction will be known only by induction.\"[698] It is\nthus clear that Aristotle makes _deduction necessarily dependent upon\ninduction_. He maintains that the highest or most universal principles\nwhich constitute the primary and immediate propositions of the former\nare furnished by the latter.\n\n[Footnote 696: \"Post. Analytic,\" bk. i. ch. ii.]\n\n[Footnote 697: Ibid., bk. i. ch. xviii.; \"Ethics,\" bk. vi. ch. iii.]\n\n[Footnote 698: \"Post. Analytic,\" bk. i. ch. xviii.]\n\n2. General principles being thus furnished by induction, we may now\ninquire whence, according to Aristotle, are the materials for induction\nderived? What is the character of that \"anterior knowledge\" which is the\nbasis of the inductive process?\n\nInduction, says Aristotle, is \"the progression from singulars to\nuniversals.\"[699] It is an illation of the universal from the singular\nas legitimated by the laws of thought. All knowledge, therefore, begins\nwith singulars--that is, with individual objects. And inasmuch as all\nknowledge begins with \"individual objects,\" and as the individual is\nconstantly regarded by Aristotle as the \"object of sense,\" it is claimed\nthat his doctrine is that all knowledge is derived from _sensation_, and\nthat science and art result to man (_solely_) by means of _experience._\nHe is thus placed at the head of the empirical school of philosophy, as\nPlato is placed at the head of the ideal school.\n\n[Footnote 699: \"Post. Analytic,\" bk. i. ch. xviii.]\n\nThis classification, however, is based upon a very superficial\nacquaintance with the philosophy of Aristotle as a whole. The practice,\nso commonly resorted to, of determining the character of the\nAristotelian philosophy by the light of one or two passages quoted from\nhis \"Metaphysics,\" is unjust both to Aristotle and to the history of\nphilosophic thought. We can not expect to attain a correct understanding\nof the views of Aristotle concerning the sources and grounds of all\nknowledge without some attention to his psychology. A careful study of\nhis writings will show that the terms \"sensation\" (aisthesis) and\n\"experience\" (empeira) are employed in a much more comprehensive sense\nthan is usual in modern philosophic writings.\n\n\"Sensation,\" in its lowest form, is defined by Aristotle as \"an\nexcitation of the soul through the body,\"[700] and, in its higher form,\nas the excitation of the soul by any object of knowledge. In this latter\nform it is used by him as synonymous with \"intuition,\" and embraces all\nimmediate intuitive perceptions, whether of sense, consciousness, or\nreason. \"The universe is derived from particulars, therefore we ought to\nhave a sensible perception (aisthesis) of these; and this is intellect\n(nous).\"[701] Intelligence proper, the faculty of first principles, is,\nin certain respects, a sense, because it is the source of a class of\ntruths which, like the perceptions of the senses, are immediately\nrevealed as facts to be received upon their own evidence. It thus\nanswers to the \"sensus communis\" of Cicero, and the \"Common Sense\" of\nthe Scottish school. Under this aspect, \"Sense is equal to or has the\nforce of Science.\"[702] The term \"Experience\" is also used to denote,\nnot merely the perception and remembrance of the impressions which\nexternal objects make upon the mind, but as co-extensive with the whole\ncontents of consciousness--all that the mind _does_ of its own native\nenergy, as well as all that it _suffers_ from without. It is evidently\nused in the Posterior Analytic (bk. ii. ch. xix.) to describe the whole\nprocess by which the knowledge of universals is obtained. \"From\nexperience, or from every universal remaining in the soul, the\nprinciples of art and science arise.\" The office of experience is \"to\nfurnish the principles of every science\"[703]--that is, to evoke them\ninto energy in the mind. 'Experience thus seems to be a thing almost\nsimilar to science and art.[704] In the most general sense, \"sensation\"\nwould thus appear to be the immediate perception or intuition of facts\nand principles, and \"experience\" the operation of the mind upon these\nfacts and principles, elaborating them into scientific form according to\nits own inherent laws. The \"experience\" of Aristotle is analogous to the\n\"reflection\" of Locke.\n\n[Footnote 700: \"De Somn.,\" bk. i.]\n\n[Footnote 701: \"Ethics,\" bk. vi. ch. xi.; see also ch. vi.]\n\n[Footnote 702: \"De Cen. Anim.\"]\n\n[Footnote 703: \"Prior Analytic,\" bk. i. ch. xix.]\n\n[Footnote 704: \"Metaphysics,\" bk. i. ch. i.]\n\nSo much being premised, we proceed to remark that there is a distinction\nperpetually recurring in the writings of Aristotle between the elements\nor first principles of knowledge which are \"clearest in their own\nnature\" and those which \"are clearest to our perception.\"[705] The\ncauses or principles of knowledge \"are _prior_ and _more known_ to us in\ntwo ways, for what is prior in nature is not the same as that which is\nprior to us, nor that which is more known (simply in itself) the same as\nthat which is more known to us. Now I call things prior and more known\nto us, those which are _nearer to sense_; and things prior and more\nknown simply in themselves, those which are _remote from sense_; and\nthose things are most remote which are especially _universal_, and those\nnearest which are _singular_; and these are mutually opposed.\"[706] Here\nwe have a distribution of the first or prior elements of knowledge into\ntwo fundamentally opposite classes.\n\n(i.) _The immediate or intuitive perceptions of sense,_\n\n(ii.) _The immediate or intuitive apperceptions of pure reason,_\n\n[Footnote 705: \"Ethics,\" bk. i. ch. iv.; \"Metaphysics,\" bk. ii. ch. i.;\n\"Rhetoric,\" bk. i. ch. ii.; \"Prior Analytic,\" bk. ii. ch. xxiii.]\n\n[Footnote 706: \"Post. Analytic,\" bk. i. ch. ii.]\n\nThe objects of sense-perception are external, individual, \"nearest to\nsense,\" and occasionally or contingently present to sense. The objects\nof the intellect are inward, universal, and the essential property of\nthe soul. They are \"remote from sense,\" \"prior by nature;\" they are\n\"forms\" essentially inherent in the soul previous to experience; and it\nis the office of experience to bring them forward into the light of\nconsciousness, or, in the language of Aristotle, \"to evoke them from\npotentiality into actuality.\" And further, from the \"prior\" and\nimmediate intuitions of sense and intellect, all our secondary, our\nscientific and practical knowledge is drawn by logical processes.\n\nThe Aristotelian distribution of the intellectual faculties corresponds\nfully to this division of the objects of knowledge. The human intellect\nis divided by Aristotle into,\n\n1. The Passive or Receptive Intellect (nous paphetikos).--Its office is\nthe reception of sensible impressions or images (Phantasmata) and their\nretention in the mind (myeme). These sensible forms or images are\nessentially immaterial. \"Each sensoriurn (aistheteron) is receptive of\nthe sensible quality _without the matter_, and hence when the sensibles\nthemselves are absent, sensations and phantasikos remain.\"[707]\n\n[Footnote 707: \"De Anima,\" bk. iii. ch. ii.]\n\n2. The Active or Creative Intellect (nous pointikos).--This is the power\nor faculty which, by its own inherent power, impresses \"form\" upon the\nmaterial of thought supplied by sense-perception, exactly as the First\nCause combines it, in the universe, with the recipient matter.\n\n\"It is necessary,\" says Aristotle, \"that these two modes should be\nopposed to each other, as matter is opposed to form, and to all that\ngives form. The receptive reason, which is as matter, becomes all things\nby receiving their forms. The creative reason gives existence to all\nthings, as light calls color into being. The creative reason transcends\nthe body, being capable of separation from it, and from all things; it\nis an everlasting existence, incapable of being mingled with matter, or\naffected by it; prior, and subsequent to the individual mind. The\nreceptive reason is necessary to individual thought, but it is\nperishable, and by its decay all memory, and therefore individuality, is\nlost to the higher and immortal reason.\"[708]\n\nThis \"Active or Creative Intellect\" is again further subdivided, by\nAristotle--\n\n1. The _Scientific_ (epistemonikon) part--the \"virtue,\" faculty, or\n\"habit of principles.\" He also designates it as the \"place of\nprinciples,\" and further defines it as the power \"which apprehends those\nexistences whose principles can not be otherwise than they are\"--that\nis, self-evident, immutable, and necessary truths[709]--the _intuitive\nreason_.\n\n2. The _Reasoning_ (logistikon) part--the power by which we draw\nconclusions from premises, and \"contemplate contingent matter\"[710]--the\n_discursive reason_.\n\nThe correlatives _noetic_ and _dianoetic_, says Hamilton, would afford\nthe best philosophic designation of these two faculties; the knowledge\nattained by the former is an \"intuitive principle\"--a truth at first\nhand; that obtained by the latter is a \"demonstrative proposition\"--a\ntruth at second hand.\n\nThe preceding notices of the psychology of Aristotle will aid us\nmaterially in interpreting his remarks \"_Upon the Method and Habits\nnecessary to the ascertainment of Principles_.\"[711]\n\n[Footnote 708: \"De Anima,\" bk. iii. ch. v.]\n\n[Footnote 709: \"Ethics,\" bk. vi. ch. i.]\n\n[Footnote 710: Ibid.]\n\n[Footnote 711: \"Post. Analytic,\" bk. ii, ch. xix., the concluding\nchapter of the Organon.]\n\n\"That it is impossible to have scientific knowledge through\ndemonstration without a knowledge of first immediate principles, has\nbeen elucidated before.\" This being established, he proceeds to explain\nhow that \"knowledge of first, immediate principles\" is developed in the\nmind.\n\n1. The knowledge of first principles is attained by the _intuition of\nsense_--the immediate perception of external objects, as the _exciting_\nor _occasional cause_ of their development in the mind.\n\n\"Now there appears inherent in all animals an innate power called\n_sensible perception_ (aisthesis); but sense being inherent, in some\nanimals a permanency of the sensible object is engendered, but in others\nit is not engendered. Those, therefore, wherein the sensible object does\nnot remain have no knowledge without sensible perception, but others,\nwhen they perceive, retain one certain thing in the soul,... with some,\n_reason_ is produced from the permanency (of the sensible impression),\n[as in man], but in others it is not [as in the brute]. From sense,\ntherefore, as we say, memory is produced, and from the repeated\nremembrance of the same thing we get experience.... From experience, or\n_from every universal remaining in the soul_--the one besides the many\nwhich in all of them is _one_ and the _same_--the principles of art and\nscience arise. If experience is conversant with generation, the\nprinciples of art; if with being, the principles of science.... Let us\nagain explain: When one thing without difference abides, there is then\nthe first universal (notion) [developed] in the soul; for the singular\nindeed is perceived by sense, _but sense is [also] of the\nuniversal_\"--that is, the universal is immanent in the sensible object\nas a property giving it \"form.\" \"It is manifest, then, that primary\nthings become necessarily known by induction, for thus sensible\nperception produces [develops or evokes] the _universal_.\" 2. The\nknowledge of first principles is attained by the _intuition of pure\nintellect_ (nous)--that is, \"_intellect itself is the principle of\nscience_\" or, in other words, intellect is the _efficient, essential\ncause_ of the knowledge of first principles.\n\n\"Of those habits which are about intellect by which we ascertain truth,\n_some[712] are always true_, but others[713] admit the false, as opinion\nand reasoning. But science and (pure) intellect are always true, and no\nother kind of knowledge, except intellect [intellectual intuition], is\nmore accurate than science. And since the principles of demonstration\nare more known, and all science is connected with reason, there could\nnot be a science of principles. But since nothing can be more true than\nscience, except intellect, intellect will belong to principles. From\nthese [considerations] it is evident that, as demonstration is not the\nprinciple of demonstration, so neither is science the principle of\nscience. If, then, we have no other true genus (of habit) besides\nscience, _intellect will be the principle of science_; it will also be\nthe principle (or cause of the knowledge) of the principle.\"\n\n[Footnote 712: The \"noetic.\"]\n\n[Footnote 713: The \"dianaetic.\"]\n\nThe doctrine of Aristotle regarding \"first principles\" may perhaps be\nsummed up as follows: All demonstrative science is based upon\n_universals_ \"prior in nature\"--that is, upon _a priori_, self-evident,\nnecessary, and immutable principles. Our knowledge of these \"first and\nimmediate principles\" is dependent primarily on _intellect_ (nous) or\nintuitive reason, and secondarily on sense, experience, and induction.\nPrior to experience, the intellect contains these principles in itself\npotentially, as \"forms,\" \"laws,\" \"habitudes,\" or \"predicaments\" of\nthought; but they can not be \"evoked into energy,\" can not be revealed\nin consciousness, except on condition of experience, and they can only\nbe scientifically developed by logical abstraction and definition. The\nultimate ground of all truth and certainty is thus a mode of our own\nmind, a subjective necessity of thinking, and truth is not in things,\nbut in our own minds.[714] \"Ultimate knowledge, as well as primary\nknowledge, the most perfect knowledge which the philosopher can attain,\nas well as the point from which he starts, is still a proposition. All\nknowledge seems to be included under two forms--knowledge _that_ it is\nso; knowledge _why_ it is so. Neither of these can, of course, include\nthe knowledge at which Plato is aiming--knowledge which is correlated\nwith Being--a knowledge, not _about_ things or persons, but _of_\nthem.\"[715]\n\n[Footnote 714: \"Metaphysics,\" bk. v. ch. iv.]\n\n[Footnote 715: Maurice's \"Ancient Philosophy,\" p. 190.]\n\nARISTOTELIAN THEOLOGY\n\nTheoretical philosophy, \"the science which has truth for its end,\" is\ndivided by Aristotle into Physics, Mathematics, and Theology, or the\nFirst Philosophy, now commonly known as \"Metaphysics,\" because it is\nbeyond or above physics, and is concerned with the primitive ground and\ncause of all things.[716]\n\nIn the former two we have now no immediate interest, but with Theology,\nas \"the science of the Divine,\"[717] the _First Moving Cause_, which is\nthe source of all other causes, and the original ground of all other\nthings, we are specially concerned, inasmuch as our object is to\ndetermine, if possible, whether Greek philosophy exerted any influence\nupon Christian thought, and has bequeathed any valuable results to the\nTheology of modern times.\n\n\"The Metaphysics\" of Aristotle opens by an enumeration of \"the\nprinciples or causes\"[718] into which all existences can be resolved by\nphilosophical analysis. This enumeration is at present to be regarded as\nprovisional, and in part hypothetical--a verbal generalization of the\ndifferent principles which seem to be demanded to explain the existence\nof a thing, or constitute it what it is. These he sets down as--\n\n[Footnote 716: \"Physics are concerned with things which have a principle\nof motion in themselves; mathematics speculate on permanent, but not\ntranscendental and self-existent things; and there is another science\nseparate from these two, which treats of that which is immutable and\ntranscendental, if indeed there exists such a substance, as we shall\nendeavor to show that there does. This transcendental and permanent\nsubstance, if it exist at all, must surely be the sphere of the\n_divine_--it must be the first and highest principle. Hence it follows\nthat there are three kinds of speculative science--Physics, Mathematics,\nand Theology.\"--\"Metaphysics,\" bk. x. ch. vii.]\n\n[Footnote 717: \"Metaphysics,\" bk. i. ch. ii.]\n\n[Footnote 718: Aition--cause--is here used by Aristotle in the sense of\n\"account of\" or \"reason why.\"]\n\n1. The Material Cause (ten ylen kai to ypokeimenon)--the matter and\nsubject--that _out of_ which a given thing has been originated. \"From\nthe analogy which this principle has to wood or stone, or any actual\nmatter out of which a work of nature or of art is produced, the name\n'material' is assigned to this class.\" It does not always necessarily\nmean \"matter\" in the now common use of the term, but \"antecedents--that\nis, principles whose inherence and priority is implied in any existing\nthing, as, for example, the premises of a syllogism, which are the\nmaterial cause of the conclusion.\"[719] With Aristotle there is,\ntherefore, \"matter as an object of sense,\" and \"matter as an object of\nthought.\"\n\n2. The Formal Cause (Ten ausian kai to ti einai)--the being or abstract\nessence of a thing--that primary nature on which all its properties\ndepend. To this Aristotle gave the name of eidos--the form or exemplar\n_according to_ which a thing is produced.\n\n3. The Moving or Efficient Cause (othen e arche tes kineseos)--the\norigin and principle of motion--that _by which_ a thing is produced.\n\n4. The Final Cause (to ou eneken kai to agathon)--the good end answered\nby the existence of any thing--that for the sake of which_ any thing is\nproduced--the eneka tou, or reason for it.[720] Thus, for instance, in a\nhouse, the wood out of which it is produced is the _matter (yle), the\nidea or conception according to which it is produced is _the form_\n(eidos morphe), the builder who erects the house is the _efficient_\ncause, and the reason for its production, or the end of its existence is\nthe _final_ cause.\n\n[Footnote 719: Encyclopaedia Britannica, article \"Aristotle;\" \"Post.\nAnalytic,\" bk. ii. ch. xi.]\n\n[Footnote 720: \"Metaphysics,\" bk. i. ch. iii.]\n\nCauses are, therefore, the elements into which the mind resolves its\nfirst rough conception of an object. That object is what it is, by\nreason of the matter out of which it sprang, the moving cause which gave\nit birth, the idea or form which it realizes, and the end or object\nwhich it attains. The knowledge of a thing implies knowing it from these\nfour points of view--that is, knowing its four causes or principles.\n\nThese four determinations of being are, on a further and closer\nanalysis, resolved into the fundamental antithesis of MATTER and FORM.\n\n\"All things that are produced,\" says Aristotle,[721] \"are produced from\nsomething (that is, from _matter_), by something (that is, _form_), and\nbecome something (the totality--to synolon);\" as, for example, a statue,\na plant, a man. To every subject there belongs, therefore, first,\n_matter_ (yle); secondly, _form_ (morphe). The synthesis of these two\nproduces and constitutes _substance_, or ousia. Matter and form are thus\nthe two grand causes or principles whence proceed all things. The\nformative cause is, at the same time, the moving cause and the final\ncause; for it is evidently the element of determination which impresses\nmovement upon matter whilst determining it; and it is also the end of\nbeing, since being only really exists when it has passed from an\nindeterminate to a determinate state.\n\n[Footnote 721: \"Metaphysics,\" bk. vi. ch. vii.]\n\nIn proof that the eidos or form is an _efficient_ principle operating in\nevery object, which makes it, to our conception, what it is, Aristotle\nbrings forward the subject of generation or production.[722] There are\nthree modes of production--natural, artificial, and automatic. In\nnatural production we discern at once a matter; indeed Nature, in the\nlargest sense, may be defined as \"that out of which things are\nproduced.\" Now the result formed out of this matter or nature is a given\nsubstance--a vegetable, a beast, or a man. But what is the _producing_\ncause in each case? Clearly something akin to the result. A man\ngenerates a man, a plant produces another plant like to itself. There\nis, therefore, implied in the resulting thing a _productive force_\ndistinct from matter, upon which it works. And this is the eidos, or\nform. Let us now consider artificial production. Here again the form is\nthe producing power. And this is in the soul. The art of the physician\nis the eidos, which produces actual health; the plan of the architect is\nthe conception, which produces an actual house. Here, however, a\ndistinction arises. In these artificial productions there is supposed a\nnoesis and a poiesis. The noesis is the previous conception which the\narchitect forms in his own mind; the poiesis is the actual creation of\nthe house out of the given matter. In this case the conception is the\nmoving cause of the production. The form of the statue in the mind of\nthe artist is the motive or cause of the movement by which the statue is\nproduced; and health must be in the thought of the physician before it\ncan become the moving cause of the healing art. Moreover, that which is\ntrue of artificial production or change is also true of spontaneous\nproduction. For example, a cure may take place by the application of\nwarmth, and this result is accomplished by means of friction. This\nwarmth in the body is either itself a portion of health, or something is\nconsequent upon it which is like itself, which is a portion of health.\nEvidently this implies the previous presence either of nature or of an\nartificer. It is also clearly evident that this kind of generating\ninfluence (the automatic) should combine with another. There must be a\nproductive power, and there must be something out of which it is\nproduced. In this case, then, there will be a yle and an eidos.[723]\n\n[Footnote 722: Ibid.]\n\n[Footnote 723: Maurice's \"Ancient Philosophy,\" pp. 205, 206.]\n\nFrom the above it appears that the _efficient_ cause is regarded by\nAristotle as identical with the _formal_ cause. So also the _final_\ncause--the end for the sake of which any thing exists--can hardly be\nseparated from the perfection of that thing, that is, from its\nconception or form. The desire for the end gives the first impulse of\nmotion; thus the final cause of any thing becomes identical with the\ngood of that thing. \"The moving cause of the house is the builder, but\nthe moving cause of the builder is the end to be attained--that is, the\nhouse.\" From such examples as these it would seem that the\ndeterminations of form and end are considered by Aristotle as one, in so\nfar as both are merged in the conception of _actuality_; for he regarded\nthe end of every thing to be its completed being--the perfect\nrealization of its idea or form. The only fundamental determinations,\ntherefore, which can not be wholly resolved into each other are _matter\nand form_.[724]\n\n[Footnote 724: Schwegler's \"History of Philosophy,\" pp. 120, 123.]\n\nThe opposition of matter and form, with Aristotle, corresponds to the\nopposition between the element of _generality_ and the element of\n_particularity_. Matter is indeterminate; form is determinate. Matter,\nabstracted from form, in thought, is entirely without predicate and\ndistinction; form is that which enters into the definition of every\nsubject, and without which it could not be defined. Matter is capable of\nthe widest diversity of forms, but is itself without form. Pure form is,\nin fact, that which is without matter, or, in other words, it is the\npure conception of being. Matter is the necessary condition of the\nexistence of a thing; form is the essence of each thing, that in virtue\nof which substance is possible, and without which it is inconceivable.\nOn the one side is passivity, possibility of existence, capacity of\naction; on the other side is activity, actuality, thought. The unity of\nthese two in the realm of determined being constitutes every individual\nsubstance. The relation of matter and form, logically apprehended, is\nthus the relation of POTENTIALITY and ACTUALITY.\n\nThis is a further and indeed a most important step in the Aristotelian\ntheology. Matter, as we have seen, after all, amounts to merely capacity\nfor action, and if we can not discover some productive power to develop\npotentiality into actuality, we look in vain for some explanation of the\nphenomena around us. The discovery, however, of energy (energeia), as a\nprinciple of this description, is precisely what we wanted, and a\nmomentary glance at the actual phenomena will show its perfect identity\nwith the eidos, or form.[725] \"For instance, what is a calm? It is\nevenness in the surface of the sea. Here the sea is the subject, that\nis, the matter in _capacity_, but the evenness is the _energy_ or\nactuality;... energy is thus as form.\"[726] The form (or idea) is thus\nan energy or actuality (energeia); the matter is a capacity or\npotentiality (dynamis), requiring the co-operation of the energy to\nproduce a result.\n\nThese terms, which are first employed by Aristotle in their\nphilosophical signification, are characteristic of his whole system. It\nis, therefore, important we should grasp their precise philosophical\nimport; and this can only be done by considering them in the strictest\nrelation to each other. It is in this relation they are defined by\nAristotle. \"Now energeia is the existence of a thing not in the sense of\nits potentially existing. The term _potentially_ we use, for instance,\nof the statue in the block, and of the half in the whole (since it may\nbe subtracted), and of a person knowing a thing, even when he is not\nthinking of it, but might be so; whereas energeia is the opposite. By\napplying the various instances our meaning will be plain, and one must\nnot seek a definition in each case, but rather grasp the conception of\nthe analogy as a whole,--that it is as that which builds to that which\nhas a capacity for building; as the waking to the sleeping; as that\nwhich sees to that which has sight, but whose eyes are closed; as the\ndefinite form to the shapeless matter; as the complete to the\nunaccomplished. In this contrast, let the energeia be set off as forming\nthe one side, and on the other let the potential stand. Things are said\nto be in energeia not always in like manner (except so far as there is\nan analogy, that as this thing is in this, and related to this, so is\nthat in that, or related to that); for sometimes it implies _motion_ as\nopposed to the _capacity of motion_, and sometimes _complete existence_\nopposed to _undeveloped matter_\".[727] As the term dynamis has the\ndouble meaning of \"_possibility of existence_\" as well as \"_capacity of\naction_\" so there is the double contrast of \"_action_\" as opposed to the\ncapacity of action; and \"_actual existence_\" opposed to possible\nexistence or potentiality. To express accurately this latter antithesis,\nAristotle introduced the term entelecheia[728]--entelechy, of which the\nmost natural account is that it is a compound of en telei echein--\"being\nin a state of perfection.\"[729] This term, however, rarely occurs in the\n\"Metaphysics,\" whilst energeia is everywhere employed, not only to\nexpress activity as opposed to passivity, but complete existence as\nopposed to undeveloped matter.\n\n[Footnote 725: \"That which Aristotle calls 'form' is not to be\nconfounded with what we may perhaps call shape [or figure]; a hand\nsevered from the arm, for instance, has still the outward shape of a\nhand, but, according to Aristotelian apprehension, it is only a hand now\nas to matter, and not as to form; an actual hand, a hand as to form, is\nonly that which can do the proper work of a hand.\"--Schwegler's \"History\nof Philosophy,\" p. 122.]\n\n[Footnote 726: \"Metaphysics,\" bk. vii. ch. ii.]\n\n[Footnote 727: \"Metaphysics,\" bk. viii. ch. vi.]\n\n[Footnote 728: \"Entelechy indicates the perfected act, the completely\nactual.\"--Schw.]\n\n[Footnote 729: Grant's Aristotle's \"Ethics,\" vol. i. p. 184.]\n\n\"In Physics dynamis answers to the necessary conditions for the\nexistence of any thing before that thing exists. It thus corresponds to\nyle, both to the prote yle--the first matter, or matter devoid of all\nqualities, which is capable of becoming any definite substance, as, for\nexample, marble; and also to the eschate yle--or matter capable of\nreceiving form, as marble the form of the statue.\" Marble then exists\npotentially in the simple elements before it is marble. The statue\nexists potentially in the marble before it is carved. All objects of\nthought exist, either purely in potentiality, or purely in actuality, or\nboth in potentiality and in actuality. This division makes an entire\nchain of all existence. At the one end is matter, the prote yle which\nhas a merely potential existence, which is necessary as a condition, but\nwhich having no form and no qualities, is totally incapable of being\nrealized by the mind. At the other end of the chain is pure form, which\nis not at all matter, the absolute and the unconditioned, the eternal\nsubstance and energy without matter (ousia aldios kai energeia aneu\ndynameos), who can not be thought as non-existing--the self-existent\nGod. Between these two extremes is the whole row of creatures, which out\nof potentiality evermore spring into actual being.[730]\n\n[Footnote 730: Id., ib., vol. i. p. 185.]\n\nThe relation of actuality to potentiality is the subject of an extended\nand elaborate discussion in book viii., the general results of which may\nbe summed up in the following propositions:\n\n1. _The relation of Actuality to Potentiality is as the Perfect to the\nImperfect_.--The progress from potentiality to actuality is motion or\nproduction (kinesis or genesis). But this motion is transitional, and in\nitself imperfect--it tends towards an end, but does not include the end\nin itself. But actuality, if it implies motion, has an end in itself and\nfor itself; it is a motion desirable for its own sake.[731] The relation\nof the potential to the actual Aristotle exhibits by the relation of the\nunfinished to the finished work, of the unemployed builder to the one at\nwork upon his building, of the seed-corn to the tree, of the man who has\nthe capacity to think, to the man actually engaged in thought.[732]\nPotentially the seed-corn is the tree, but the grown-up tree is the\nactuality; the potential philosopher is he who is not at this moment in\na philosophic condition; indeed, every thing is potential which\npossesses a principle of development, or of change. Actuality or\nentelechy, on the other hand, indicates the _perfect act_, the end\ngained, the completed actual; that activity in which the act and the\ncompleteness of the act fall together--as, for example, to see, to\nthink, where the acting and the completed act are one and the same.\n\n2. _The Relation of Actuality to Potentiality is a causal Relation_.--A\nthing which is endued with a simple capacity of being may nevertheless\nnot actually exist, and a thing may have a capacity of being and really\nexist. Since this is the case, there must ensue between non-being and\nreal being some such principle as _energy_, in order to account for the\ntransition or change.[733] Energy has here some analogy to motion,\nthough it must not be confounded with motion. Now you can not predicate\neither motion or energy of things which are not. The moment energy is\nadded to them they are. This transition from potentiality to actuality\nmust be through the medium of such principles as propension or _free\nwill_, because propension or free will possess in themselves the power\nof originating motion in other things.[734]\n\n[Footnote 731: \"Metaphysics,\" bk. viii. ch. vi.]\n\n[Footnote 732: Ibid., bk. viii. ch. vi.]\n\n[Footnote 733: Ibid., bk. viii. ch. iii.]\n\n[Footnote 734: Ibid., bk. viii. ch. v.]\n\n3. _The Relation of Actuality and Potentiality is a Relation of\nPriority_.--Actuality, says Aristotle, is prior to potentiality in the\norder of reason, in the order of substance, and also (though not\ninvariably) in the order of time. The first of all capacities is a\ncapacity of energizing or assuming a state of activity; for example, a\nman who has the capacity of building is one who is skilled in building,\nand thus able to use his energy in the art of building.[735] The primary\nenergizing power must precede that which receives the impression of it,\nForm being older than Matter. But if you take the case of any particular\nperson or thing, we say that its capacity of being that particular\nperson or thing precedes its being so actually. Yet, though this is the\ncase in each particular thing, there is always a foregone energy\npresumed in some other thing (as a prior seed, plant, man) to which it\nowes its existence. One pregnant thought presents itself in the course\nof the discussion which has a direct bearing upon our subject. [Dynamis]\nhas been previously defined as \"a principle of motion or change in\nanother thing in so far forth as it is another thing\"[736]--that is, it\nis fitted by nature to have motion imparted to it, and to communicate\nmotion to something else. But this motion wants a resting-place. There\ncan be no infinite regression of causes. There is some primary [dynamis]\npresupposed in all others, which is the beginning of change. This is\n[Greek: physis], or nature. But the first and original cause of all\nmotion and change still precedes and surpasses nature. The final cause\nof all potentiality is energy or _actuality_. The one proposed is prior\nto the means through which the end is accomplished. A process of\nactualization, a tendency towards completeness or perfection ([Greek:\ntelos]) presupposes an absolute actuality which is at once its beginning\nand end. \"One energy is invariably antecedent to another in time, up to\nthat which is primarily and eternally the Moving Cause.\"[737]\n\n[Footnote 735: \"Metaphysics,\" bk. viii. ch. viii.]\n\n[Footnote 736: Ibid., bk. iv. ch. xii.]\n\n[Footnote 737: Ibid., bk. viii. ch. viii.]\n\nAnd now having laid down these fundamental principles of metaphysical\nscience, as preparatory to Theology, Aristotle proceeds to establish the\nconception of the Absolute or Divine Spirit _as the eternal, immutable\nSubstance, the immaterial Energy, the unchangeable Form of Forms, the\nfirst moving Cause_.\n\nI. _The Ontological Form of Proof_.--It is necessary to conceive an\neternal and immutable substance--an actuality which is absolute and\nprior, both logically and chronologically, to all potentiality; for that\nwhich is potential is simply contingent, it may just as easily not be as\nbe; that which exists only in capacity is temporal and corruptible, and\nmay cease to be. Matter we know subsists merely in capacity and\npassivity, and without the operation of Energy,(energeia), or the\nformative cause, would be to us as non-entity. The phenomena of the\nworld exhibits to us the presence of Energy, and energy presupposes the\nexistence of an eternal substance. Furthermore, matter and potentiality\nare convertible terms, therefore the primal Energy or Actuality must be\n_immaterial_.[738]\n\n2. _The Cosmological Form of Proof_.--It is impossible that there should\nbe _motion_, genesis, or a chain of causes, except on the assumption of\na first Moving Cause, since that which exists only in capacity can not,\nof itself energize, and consequently without a principle of motion which\nis essentially active, we have only a principle of immobility. The\nprinciple \"ex nihilo nihil\" forbids us to assume that motion can arise\nout of immobility, being out of non-being. \"How can matter be put in\nmotion if nothing that subsists in energy exist, and is its cause?\" All\nbecoming, therefore, necessarily supposes that which has not become,\nthat which is eternally self-active as the principle and cause of all\nmotion. There is no refuge from the notion that all things are \"born of\nnight and nothingness\" except in this belief.[739]\n\n[Footnote 738: \"Metaphysics,\" bk. xi. ch. vi.]\n\n[Footnote 739: Ibid., bk. xi. ch. vii., viii.]\n\nThe existence of an eternal principle subsisting in energy is also\ndemanded to explain the _order_ of the world. \"For how, let me ask, will\nthere prevail _order_ on the supposition that there is no subsistence of\nthat which is eternal, and which involves a separable existence, and is\npermanent.\"[740] \"All things in nature are constituted in the best\npossible manner.\"[741] All things strive after \"the good.\" \"The\nappearance of ends and means in nature is a proof of design.\"[742] Now\nan end or final cause presupposes intelligence,--implies a _mind_ to see\nand desire it. That which is \"fair,\" \"beautiful,\" \"good,\" an \"object of\ndesire,\" can only be perceived by Mind. The \"final cause\" must therefore\nsubsist in that which is prior and immovable and eternal; and _Mind_ is\n\"that substance which subsists absolutely, and according to\nenergy.\"[743] \"The First Mover of all things, moves all things without\nbeing moved, being an eternal substance and energy; and he moves all\nthings as the object of reason and of desire, or love.\"[744]\n\n[Footnote 740: Ibid., bk. x. ch. ii.]\n\n[Footnote 741: \"Ethics,\" bk. i. ch. ix.]\n\n[Footnote 742: \"Nat. Ausc.,\" bk. ii. ch. viii.]\n\n[Footnote 743: \"Metaphysics,\" bk. xi. ch. vii.]\n\n[Footnote 744: Ibid.]\n\n3. _The Moral Form of Proof_.--So far as the relation of potentiality\nand actuality is identical with the relation of matter and form, the\nargument for the existence of God may be thus presented: The conception\nof an absolute matter without form, involves the supposition of an\nabsolute form without matter. And since the conception of form resolves\nitself into _motion_, _conception_, _purpose_ or _end_, so the Eternal\nOne is the absolute principle of motion (the proton kinoun), the\nabsolute conception or pure intelligence (the pure ti en einai), and the\nabsolute ground, reason, or end of all being. All the other predicates\nof the First Cause follow from the above principles with logical\nnecessity.\n\n(i.) _He is, of course, pure intellect_, because he is absolutely\nimmaterial and free from nature. He is active intelligence, because his\nessence is pure actuality. He is self-contemplating and self-conscious\nintelligence, because the divine thought can not attain its actuality in\nany thing extrinsic; it would depend on something else than self--some\npotential existence for its actualization. Hence the famous definition\nof the absolute as \"the thought of thought\" (noesis noeseos).[745] \"And\ntherefore the first and actual perception by mind of Mind itself, doth\nsubsist in this way throughout all eternity.\"[746]\n\n[Footnote 745: Schwegler's \"History of Philosophy,\" p. 125.]\n\n[Footnote 746: \"Metaphysics,\" bk. xi. ch. ix.]\n\n(ii.) _He is also essential life_. \"The principle of life is inherent in\nthe Deity, for the energy or active exercise of mind constitutes life,\nand God constitutes this energy; and essential energy belongs to God as\nhis best and everlasting life. Now our statement is this--that the Deity\nis a living being that is everlasting and most excellent in nature, so\nthat with the Deity life and duration are uninterrupted and eternal; for\nthis constitutes the essence of God.\"[747]\n\n(iii.) _Unity belongs to him_, since multiplicity implies matter; and\nthe highest idea or form of the world must be absolutely\nimmaterial.[748] The Divine nature is \"devoid of parts and indivisible,\nfor magnitude can not in any way involve this Divine nature; for God\nimparts motion through infinite duration, and nothing finite--as\nmagnitude is--can be possessed of an infinite capacity.\"[749]\n\n(iv.) _He is immovable and ever abideth the same_; since otherwise he\ncould not be the absolute mover, and the cause of all becoming, if he\nwere subject to change.[750] God is impassive and unalterable ([Greek:\napathes kai analloioton]); for all such notions as are involved in\npassion or alteration are outside the sphere of the Divine\nexistence.[751]\n\n[Footnote 747: \"Metaphysics,\" bk. xi. ch. vii.]\n\n[Footnote 748: Ibid.]\n\n[Footnote 749: Ibid.]\n\n[Footnote 750: Ibid., bk. xi. ch. viii.]\n\n[Footnote 751: Ibid., bk. xi. ch. vii.]\n\n(v.) _He is the ever-blessed God_.--\"The life of God is of a kind with\nthose highest moods which, with us, last a brief space, it being\nimpossible they should be permanent; whereas, with Him they are\npermanent, since His ever-present consciousness is pleasure itself. And\nit is because they are vivid states of consciousness, that waking, and\nperception, and thought, are the sweetest of all things. Now essential\nperception is the perception of that which is most excellent,... and the\nmind perceives itself by participating of its own object of perception;\nbut it is a sort of coalescence of both that, in the Divine Mind,\ncreates a regular identity between the two, so that with God both (the\nthinker and the thought, the subject and object) are the same. In\npossession of this prerogative, He subsists in the exercise of energy;\nand the contemplation of his own perfections is what, to God, must be\nmost agreeable and excellent. This condition of existence, after so\nexcellent a manner, is what is \"so astonishing to us when we examine\nGod's nature, and the more we do so the more wonderful that nature\nappears to us. The mood of the Divine existence is essential energy,\nand, as such, it is a life that is most excellent, blessed, and\neverlasting.[752]\n\n[Footnote 752: \"Metaphysics,\" bk. xi. ch. vii.]\n\nThe theology of Aristotle may be summed up in the following sentences\nselected from book xi. of his \"Metaphysics:\"\n\n\"This motionless cause of motion is a necessary being; and, by virtue of\nsuch necessity, is the all-perfect being. This all-pervading principle\npenetrates heaven and all nature. It eternally possesses perfect\nhappiness; and its happiness is in action. This primal mover is\nimmaterial; for its essence is in energy. It is pure thought--thought\nthinking itself--the thought of thought. The activity of pure\nintelligence--such is the perfect, eternal life of God. This primal\ncause of change, this absolute perfection, moves the world by the\nuniversal desire for the absolute good, by the attraction exercised upon\nit by the Eternal Mind--the serene energy of Divine Intelligence.\"\n\nIt can not be denied that, so far as it goes, this conception of the\nDeity is admirable, worthy, and just. Viewed from a Christian\nstand-point, we at once concede that it is essentially defective. There\nis no clear and distinct recognition of God as Creator and Governor of\nthe universe; he is chiefly regarded as the Life of the universe--the\nIntellect, the Energy--that which gives excellence, and perfection, and\ngladness to the whole system of things. The Theology of Aristotle is, in\nfact, metaphysical rather than practical. He does not contemplate the\nDeity as a moral Governor. Whilst Plato speaks of \"being made like God\nthrough becoming just and holy,\" Aristotle asserts that \"all moral\nvirtues are totally unworthy of being ascribed to God.\"[753] He is not\nthe God of providence. He dwells alone, supremely indifferent to human\ncares, and interests, and sorrows. He takes no cognizance of individual\nmen, and holds no intercourse with man. The God of Aristotle is not a\nbeing that meets and satisfies the wants of the human heart, however\nwell it may meet the demands of the reason.\n\n[Footnote 753: \"Ethics,\" bk. x. ch. viii.]\n\nMorality has no basis in the Divine nature, no eternal type in the\nperfections and government of God, and no supports and aids from above.\nThe theology of Aristotle foreshadows the character of the\n\nARISTOTELIAN ETHICS.\n\nWe do not find in Aristotle any distinct recognition of an eternal and\nimmutable morality, an absolute right, which has its foundation in the\nnature of God. Plato had taught that there was \"an absolute Good, above\nand beyond all existence in dignity and power;\" which is, in fact, \"the\ncause of all existence and all knowledge,\" and which is God; that all\nother things are good in proportion as they \"partake of this absolute\nGood;\" and that all men are so far good as they \"resemble God.\" But with\nthis position Aristotle joins issue. After stating the doctrine of Plato\nin the following words--\"Some have thought that, besides all these\nmanifold goods upon earth, there is some _absolute good_, which is the\ncause to all these of their being good\"--he proceeds to criticise that\nidea, and concludes his argument by saying--\"we must dismiss the idea at\npresent, for if there is any one good, universal and generic, or\ntranscendental and absolute, it obviously can never be realized nor\npossessed by man; whereas something of this latter kind is what we are\ninquiring after.\" He follows up these remarks by saying that \"Perhaps\nthe knowledge of the idea may be regarded by some as useful, as a\npattern (paradeigma) by which to judge of relative good.\" Against this\nhe argues that \"There is no trace of the arts making use of any such\nconception; the cobbler, the carpenter, the physician, and the general,\nall pursue their vocations without respect to the _absolute good_, nor\nis it easy to see how they would be benefited by apprehending it.\"[754]\nThe good after which Aristotle would inquire is, therefore, a _relative\ngood_, since the knowledge of the absolute good can not possibly be\nrealized.\n\n[Footnote 754: \"Ethics,\" bk. i. ch. vi.]\n\nInstead, therefore, of seeking to attain to \"a transcendental and\nabsolute good \"--a fundamental idea of right, which may be useful as a\nparadigm by which we may judge of relative good, he addresses himself\nsolely to the question, \"what is good for man\"--what is the good\nattainable in action? And having identified the Chief Good with the\nfinal and perfect end of all action, the great question of the _Ethics_\nis, \"_What is the end of human action?_\" (ti esti to ton prakton\nteloa).[755]\n\n[Footnote 755: \"Ethics,\" bk. i. ch. xiii.]\n\nNow an end or final cause implies an intelligence--implies a mind to\nperceive and desire it. This is distinctly recognized by Aristotle. The\nquestion, therefore, naturally arises--is that end fixed for man by a\nhigher intelligence, and does it exist for man both as an idea and as an\nideal? Can man, first, intellectually apprehend the idea, and then\nconsciously strive after its realization? Is it the duty of man to aim\nat fulfilling the purposes of his Creator? To this it may be answered\nthat Aristotle is not at all explicit as to God's moral government of\nthe world. \"Moral government,\" in the now common acceptation of the\nterm, has no place in the system of Aristotle, and the idea of \"duty\" is\nscarcely recognized. He considers \"the good\" chiefly in relation to the\nconstitution and natural condition of man. \"_It is_\" says he, \"_the end\ntowards which nature tends_.\" As physical things strive unconsciously\nafter the end of their existence, so man strives after the good\nattainable in life. Socrates had identified virtue and knowledge, he had\ntaught that \"virtue is a Science.\" Aristotle contended that virtue is an\nart, like music and architecture, which must be attained by exercise. It\nis not purely intellectual, it is the bloom of the physical, which has\nbecome ethical. As the flower of the field, obeying the laws of its\norganization, springs up, blooms, and attains its own peculiar\nperfection, so there is an instinctive desire (orexis) in the soul which\nat first unconsciously yearns after the good, and subsequently the good\nis sought with full moral intent and insight. Aristotle assumes that the\ndesires or instincts of man are so framed as to imply the existence of\nthis end (telos).[756] And he asserts that man can only realize it in\nthe sphere of his own proper functions, and in accordance with the laws\nof his own proper nature and its harmonious development.[757] It is not,\nthen, through instruction, or through the perfection of knowledge, that\nman is to attain the good, but through exercise and habit (ephos). By\npractice of moral acts we become virtuous, just as by practice of\nbuilding and of music, we become architects and musicians; for the\nhabit, which is the ground of moral character, is only a fruit of\noft-repeated moral acts. Hence it is by these three things--nature,\nhabit, reason--that men become good.\n\n[Footnote 756: Ibid, bk. i. ch. ii.]\n\n[Footnote 757: \"Ethics,\" bk. i. ch. vii.]\n\nAristotle's question, therefore, is, _What is the chief good for man as\nman_? not what is his chief good as a spiritual and an immortal being?\nor what is his chief good as a being related to and dependent upon God?\nAnd the conclusion at which he arrives is, that it is _the absolute\nsatisfaction of our whole nature_--that which men are agreed in calling\n_happiness_. This happiness, however, is not mere sensual pleasure. The\nbrute shares this in common with man, therefore it can not constitute\nthe happiness of man. Human happiness must express the completeness of\nrational existence. And inasmuch as intelligence is essential activity,\nas the soul is the _entelechy_ of the body, therefore the happiness of\nman can not consist in a mere passive condition. It must, therefore,\nconsist in _perfect activity_ in well-doing, and especially in\ncontemplative thought,[758] or as Aristotle defines it--\"_It is a\nperfect practical activity in a perfect life_.\"[759] His conception of\nthe chief good has thus two sides, one internal, that which exists in\nand for the consciousness--a \"complete and perfect life,\" the other\nexternal and practical. The latter, however, is a means to the former.\nThat complete and perfect life is the complete satisfaction and\nperfection of our rational nature. It is a state of peace which is the\ncrown of exertion. It is the realization of the divine in man, and\nconstitutes the absolute and all-sufficient happiness.[760] A good\naction is thus an End-in-itself (teleion telos) inasmuch as it secures\nthe _perfection_ of our nature; it is that for the sake of which our\nmoral faculties before existed, hence bringing an inward pleasure and\nsatisfaction with it; something in which the mind can rest and fully\nacquiesce; something which can be pronounced beautiful, fitting,\nhonorable, and perfect.\n\n[Footnote 758: \"If it be true to say that happiness consists in doing\nwell, a life of action must be best both for the state and the\nindividual. But we need not, as some do, suppose that a life of action\nimplies relation to others, or that those only are active thoughts which\nare concerned with the results of action; but far rather we must\nconsider those speculations and thoughts to be so which have their _end\nin themselves_, and which are for their own sake.\"--\"Politics,\" bk. vii.\nch. iii.]\n\n[Footnote 759: \"Ethics,\" bk. i. ch. x.]\n\n[Footnote 760: \"Ethics,\" bk. x. ch. viii.]\n\nFrom what has been already stated, it will be seen that the Aristotelian\nconception of _Virtue_ is not conformity to an absolute and immutable\nstandard of right. It is defined by him as _the observation of the right\nmean (mesotes) in action_--that is, the right mean relatively to\nourselves. \"Virtue is a habit deliberately choosing, existing as a mean\n(meson) which refers to us, and is defined by reason, and as a prudent\nman would define it; and it is a mean between two evils, the one\nconsisting in excess, the other in defect; and further, it is a mean, in\nthat one of these falls short of, and the other exceeds, what is right\nboth in passions and actions; and that virtue both finds and chooses the\nmean.\"[761] The perfection of an action thus consists in its containing\nthe right degree--the true mean between too much, and too little. The\nlaw of the mesotes is illustrated by the following examples: Man has a\nfixed relation to pleasure and pain. In relation to pain, the true mean\nis found in neither fearing it nor courting it, and this is _fortitude_.\nIn relation to pleasure, the true mean stands between greediness and\nindifference; this is _temperance_. The true mean between prodigality\nand narrowness is _liberality_; between simplicity and cunning is\n_prudence_; between suffering wrong and doing wrong is _justice_.\nExtending this law to certain qualifications of temper, speech, and\nmanners, you have the portrait of a graceful Grecian gentleman. Virtue\nis thus _proportion, grace, harmony, beauty in action_.\n\n[Footnote 761: Ibid, bk. ii. ch. vi.]\n\nIt will at once be seen that this classification has no stable\nfoundation. It furnishes no ultimate standard of right. The _mean_ is a\nwavering line. It differs under different circumstances and relations,\nand in different times and places. That mean which is sufficient for one\nindividual is insufficient for another. The virtue of a man, of a slave,\nand of a child, is respectively different. There are as many virtues as\nthere are circumstances in life; and as men are ever entering into new\nrelations, in which it is difficult to determine the correct method of\naction, the separate virtues can not be limited to any definite number.\n\nImperfect as the ethical system of Aristotle may appear to us who live\nin Christian times, it must be admitted that his writings abound with\njust and pure sentiments. His science of Ethics is a _discipline of\nhuman character in order to human happiness_. And whilst it must be\nadmitted that it is directed solely to the improvement of man in the\npresent life, he aims to build that improvement on pure and noble\nprinciples, and seeks to elevate man to the highest perfection of which\nhe could conceive. \"And no greater praise can be given to a work of\nheathen morality than to say, as may be said of the ethical writings of\nAristotle, that they contain nothing which a Christian may dispense\nwith, no precept of life which is not an element of Christian character;\nand that they only fail in elevating the heart and the mind to objects\nwhich it needed Divine Wisdom to reveal.\"[762]\n\n[Footnote 762: Encyclopaedia Britannica, article \"Aristotle.\"]\n\n\n\n\nCHAPTER XIII.\n\nTHE PHILOSOPHERS OF ATHENS _(continued)_\n\nPOST-SOCRATIC SCHOOL.\n\nEPICURUS AND ZENO.\n\n\nPhilosophy, after the time of Aristotle, takes a new direction. In the\npre-Socratic schools, we have seen it was mainly a philosophy of nature;\nin the Socratic school it was characterized as a philosophy of mind; and\nnow in the post-Socratic schools it becomes a philosophy of life--a\nmoral philosophy. Instead of aiming at the knowledge of real Being--of\nthe permanent, unchangeable, eternal principles which underlie all\nphenomena, it was now content to aim, chiefly, at individual happiness.\nThe primary question now discussed, as of the most vital importance, is,\nWhat is the ultimate standard by which, amid all the diversities of\nhuman conduct and opinion, we may determine what is right and good in\nindividual and social life?\n\nThis remarkable change in the course of philosophic inquiry was mainly\ndue--\n\n1st. _To the altered circumstances of the times_. An age of civil\ndisturbance and political intrigue succeeded the Alexandrian period. The\ndifferent states of Greece lost their independence, and became gradually\nsubject to a foreign yoke. Handed over from one domination to another,\nin the struggles of Alexander's lieutenants, they endeavored to\nreconquer their independence by forming themselves into confederations,\nbut were powerless to unite in the defense of a common cause. The Achaean\nand Etolian leagues were weakened by internal discords; and it was in\nvain that Sparta tried to recover her ancient liberties.\n\nDivided amongst themselves, the smaller states invoked the aid of\ndangerous allies--at one time appealing to Macedon, at another to Egypt.\nIn this way they prepared for the total ruin of Greek liberty, which was\ndestined to be extinguished by Rome.[763]\n\n[Footnote 763: Pressense, \"Religions before Christ,\" pp. 136-140.]\n\nDuring this period of hopeless turmoil and social disorder, all lofty\npursuits and all great principles were lost sight of and abandoned. The\nphilosophic movement followed the downward course of society, and men\nbecame chiefly concerned for their personal interest and safety. The\nwars of the Succession almost obliterated the idea of society, and\nphilosophy was mainly directed to the securing of personal happiness; it\nbecame, in fact, \"the art of making one's self happy.\" The sad reverses\nto which the Grecian mind had been subjected produced a feeling of\nexhaustion and indifference, which soon reflected itself in the\nphilosophy of the age.\n\n2d. In connection with the altered circumstances of the age, we must\nalso take account of _the apparent failure of the Socratic method to\nsolve the problem of Being_.\n\nThe teaching of Aristotle had fostered the suspicion that the dialectic\nmethod was a failure, and thus prepared the way for a return to\nsensualism. He had taught that individuals alone have a real existence,\nand that the \"essence\" of things is not to be sought in the elements of\nunity and generality, or in the _idea_, as Plato taught, but in the\nelements of diversity and speciality. And furthermore, in opposition to\nPlato, he had taught his disciples to attach themselves to sensation, as\nthe source of all knowledge. As the direct consequence of this teaching,\nwe find his immediate successors, Dicearchus and Straton, deliberately\nsetting aside \"the god of philosophy,\" affirming \"that a _divinity_ was\nunnecessary to the explanation of the existence and order of the\nuniverse.\" Stimulated by the social degeneracy of the times, the\ncharacteristic skepticism of the Greek intellect bursts forth anew. As\nthe skepticism of the Sophists marked the close of the first period of\nphilosophy, so the skepticism of Pyrrhonism marked the close of the\nsecond. The new skepticism arrayed Aristotle against Plato as the\nearlier skeptics arrayed atomism against the doctrine of the Eleatics.\nThey naturally said: \"We have been seeking a long time; what have we\ngained? Have we obtained any thing certain and determinate? Plato says\nwe have. But Aristotle and Plato do not agree. May not our opinion be as\ngood as theirs? What a diversity of opinions have been presented during\nthe past three hundred years! One may be as good as another, or they may\nbe all alike untrue!\" Timon and Pyrrhon declared that, of each thing, it\nmight be said to be, and not to be; and that, consequently, we should\ncease tormenting ourselves, and seek to obtain an _absolute calm_, which\nthey dignified with the name of _ataraxie_. Beholding the overthrow and\ndisgrace of their country, surrounded by examples of pusillanimity and\ncorruption, and infected with the spirit of the times themselves, they\nwrote this maxim: \"Nothing is infamous; nothing is in itself just; laws\nand customs alone constitute what is justice and what is iniquity.\"\nHaving reached this extreme, nothing can be too absurd, and they cap the\nclimax by saying, \"We assert nothing; no, not even that we assert\nnothing!\"\n\nAnd yet there must some function, undoubtedly, remain for the \"wise man\"\n(sophos).\n\nReason was given for some purpose. Philosophy must have some end. And\ninasmuch as it is not to determine speculative questions, it must be to\ndetermine practical questions. May it not teach men to _act_ rather than\nto _think_? The philosopher, the schools, the disciples, survive the\ndarkening flood of skepticism.\n\nThree centuries before Christ, the Peripatetic and Platonic schools are\nsucceeded by two other schools, which inherit their importance, and\nwhich, in other forms, and by an under-current, perpetuate the disputes\nof the Peripatetics and Platonists, namely, the Epicureans and Stoics.\nWith Aristotle and Plato, philosophy embraced in its circle nature,\nhumanity, and God; but now, in the systems of Epicurus and \"Zeno\", moral\nphilosophy is placed in the foreground, and assumes the chief, the\novershadowing pre-eminence. The conduct of life--morality--is now the\ngrand subject of inquiry, and the great theme of discourse.\n\nIn dealing with _morals_ two opposite methods of inquiry were possible:\n\n1. _To judge of the quality of actions by their_ RESULTS.\n\n2. _To search for the quality of actions in the actions them selves_.\n\nUtility, which in its last analysis is _Pleasure_, is the test of right,\nin the first method; an assumed or discovered _Law of Nature_, in the\nsecond. If the world were perfect, and the balance of the human\nfaculties undisturbed, it is evident that both systems would give\nidentical results. As it is, there is a tendency to error on each side,\nwhich is fully developed in the rival schools of the Epicureans and\nStoics, who practically divided the suffrages of the mass of educated\nmen until the coming of Christ.\n\nEPICUREANS.\n\nEpicurus was born B.C. 342, and died B.C. 270. He purchased a Garden\nwithin the city, and commenced, at thirty-six years of age, to teach\nphilosophy. The Platonists had their academic Grove: the Aristotelians\nwalked in the Lyceum: the Stoics occupied the Porch: the Epicureans had\ntheir Garden, where they lived a tranquil life, and seem to have had a\ncommunity of goods.\n\nThere is not one of all the various founders of the ancient\nphilosophical schools whose memory was cherished with so much veneration\nby his disciples as that of Epicurus. For several centuries after his\ndeath, his portrait was treated by them with all the honors of a sacred\nrelic: it was carried about with them in their journeys, it was hung up\nin their schools, it was preserved with reverence in their private\nchambers; his birthday was celebrated with sacrifices and other\nreligious observances, and a special festival in his honor was held\nevery month.\n\nSo much honor having been paid to the memory of Epicurus, we naturally\nexpect that his works would have been preserved with religious care. He\nwas one of the most prolific of the ancient Greek writers. Diogenes\ncalls him \"a most voluminous writer,\" and estimates the number of works\ncomposed by him at no less than three hundred, the principal of which he\nenumerates.[764] But out of all this prodigious collection, not a single\nbook has reached us in a complete, or at least an independent form.\nThree letters, which contain some outlines of his philosophy, are\npreserved by Diogenes, who has also embodied his \"Fundamental\nMaxims\"--forty-four propositions, containing a summary of his ethical\nsystem. These, with part of his work \"On Nature,\" found during the last\ncentury among the Greek MSS. recovered at Herculaneum, constitute all\nthat has survived the general wreck.\n\n[Footnote 764: Diogenes Laertius, \"Lives of the Philosophers,\" bk. x.\nch. xvi., xvii.]\n\nWe are thus left to depend mainly on his disciples and successors for\nany general account of his system. And of the earliest and most\nimmediate of these the writings have perished.[765] Our sole original\nauthority is Diogenes Laertius, who was unquestionably an Epicurean. The\nsketch of Epicurus which is given in his \"Lives\" is evidently a \"labor\nof love.\" Among all the systems of ancient philosophy described by him,\nthere is none of whose general character he has given so skillful and so\nelaborate an analysis. And even as regards the particulars of the\nsystem, nothing could be more complete than Laertius's account of his\nphysical speculations. Additional light is also furnished by the\nphilosophic poem of Lucretius \"On the Nature of Things,\" which was\nwritten to advocate the physical theory of Epicurus. These are the chief\nsources of our information.\n\n[Footnote 765: Some fragments of the writings of Metrodorus, Phaedrus,\nPolystratus, and Philodemus, have been found among the Herculanean\nPapyri, and published in Europe, which are said to throw some additional\nlight on the doctrines of Epicurus. See article on \"Herculanean Papyri,\"\nin Edinburgh Review, October, 1862.]\n\nIt is said of Epicurus that he loved to hearken to the stories of the\nindifference and apathy of Pyrrhon, and that, in these qualities, he\naspired to imitate him. But Epicurus was not, like Pyrrhon, a skeptic;\non the contrary, he was the most imperious dogmatist. No man ever showed\nso little respect for the opinions of his predecessors, or so much\nconfidence in his own. He was fond of boasting that he had made his own\nphilosophy--_he_ was a \"self-taught\" man! Now \"Epicurus might be\nperfectly honest in saying he had read very little, and had worked out\nthe conclusions in his own mind, but he was a copyist, nevertheless; few\nmen more entirely so.\"[766] His psychology was certainly borrowed from\nthe Ionian school. From thence he had derived his fundamental maxim,\nthat \"sensation is the source of all knowledge, and the standard of all\ntruth.\" His physics were copied from Democritus. With both, \"atoms are\nthe first principle of all things.\" And in Ethics he had learned from\nAristotle, that if an absolute good is not the end of a practical life,\n_happiness_ must be its end.[767] All that is fundamental in the system\nof Epicurus was borrowed from his predecessors, and there is little that\ncan be called new in his teaching.\n\n[Footnote 766: Maurice's \"Ancient Philosophy,\" p. 236.]\n\n[Footnote 767: \"Ethics,\" bk. i. ch. vi]\n\nThe grand object of philosophy, according to Epicurus, _is the\nattainment of a happy life_. \"Philosophy,\" says he, \"is the power by\nwhich reason conducts men to happiness.\" Truth is a merely relative\nthing, a variable quantity; and therefore the pursuit of truth for its\nown sake is superfluous and useless. There is no such thing as absolute,\nunchangeable right: no action is intrinsically right or wrong. \"We\nchoose the virtues, not on their own account, but for the sake of\npleasure, just as we seek the skill of the physician for the sake of\nhealth.\"[768] That which is nominally right in morals, that which is\nrelatively good in human conduct, is, therefore, to be determined by the\neffects upon ourselves; that which is agreeable--pleasurable, is right;\nthat which is disagreeable--painful, is wrong. \"The virtues are connate\nwith living pleasantly.\"[769] Pleasure (edone), then, is the great end\nto be sought in human action. \"Pleasure is the chief good, the beginning\nand end of living happily.\"[770]\n\n[Footnote 768: \"Fundamental Maxims,\" preserved in Diogenes Laertius,\n\"Lives of the Philosophers,\" bk. x. ch. xxx.]\n\n[Footnote 769: \"Epicurus to Menaeceus,\" in Diogenes Laertius, \"Lives of\nthe Philosophers,\" bk. x. ch. xxvii.]\n\n[Footnote 770: Id., ib.]\n\nThe proof which Epicurus offers in support of his doctrine, \"that\npleasure is the chief good,\" is truly characteristic. \"All animals from\nthe moment of their birth are delighted with pleasure and offended with\npain, by their natural instincts, and without the employment of reason.\nTherefore we, also, of our own inclination, flee from pain.\"[771] \"All\nmen like pleasure and dislike pain; they naturally shun the latter and\npursue the former.\" \"If happiness is present, we have every thing, and\nwhen it is absent, we do every thing with a view to possess it.\"[772]\nVirtue thus consists in man's doing deliberately what the animals do\ninstinctively--that is, choose pleasure and avoid pain.\n\n[Footnote 771: Diogenes Laertius, \"Lives of the Philosophers,\" bk. x.\nch. xxix.]\n\n[Footnote 772: Id., ib., bk. x. ch. xxvii.]\n\n\"Every kind of pleasure\" is, in the estimation of Epicurus, \"alike\ngood,\" and alike proper. \"If those things which make the pleasures of\ndebauched men put an end to the fears of the mind, and to those which\narise about the heavenly bodies [supernatural powers], and death and\npain,... we should have no pretense for blaming those who wholly devote\nthemselves to pleasure, and who never feel any pain, or grief (which is\nthe chief evil) from any quarter.\"[773] Whilst, however, all pleasures\nof the body, as well as the mind, are equal in dignity, and alike good,\nthey differ in intensity, in duration, and, especially, in their\nconsequences. He therefore divides pleasure into two classes; and in\nthis, as Cousin remarks, is found the only element of originality in his\nphilosophy. These two kinds of pleasure are:\n\n1. _The pleasure of movement, excitement, energy_ (edone en\nkinesei).[774] This is the most lively pleasure; it supposes the\ngreatest development of physical and mental power. \"Joy and cheerfulness\nare beheld in motion and energy.\" But it is not the most enduring\npleasure, and it is not the most perfect. It is accompanied by\nuneasiness; it \"brings with it many perturbations,\" and it yields some\nbitter fruits.\n\n[Footnote 773: \"Fundamental Maxims,\" No. 9, in Diogenes Laertius, \"Lives\nof the Philosophers,\" bk. x. ch. xxxi.]\n\n[Footnote 774: Diogenes Laertius, \"Lives of the Philosophers,\" bk. x.\nch. xxviii.]\n\n2. _The second kind of pleasure is the pleasure of repose, tranquillity,\nimpassibility_ (edone katastematike). This is a state, a \"condition,\"\nrather than a motion. It is \"the freedom of the body from pain, and the\nsoul from confusion.\"[775] This is perfect and unmixed happiness--the\nhappiness of God; and he who attains it \"will be like a god among men.\"\n\"The storm of the soul is at an end, and body and soul are perfected.\"\n\nNow, whilst \"no pleasure is intrinsically bad,\"[776] prudence\n(phronesis), or practical wisdom, would teach us to choose the highest\nand most perfect happiness. Morality is therefore the application of\nreason to the conduct of life, and virtue is wisdom. The office of\nreason is to \"determine our choices\"--to take account of the duration of\npleasures, to estimate their consequences, and to regard the happiness\nof a whole lifetime, and not the enjoyment of a single hour. Without\nwisdom men will choose the momentary excitements of passion, and follow\nafter agitating pleasures, which are succeeded by pain; they will\nconsequently lose \"tranquillity of mind.\" \"It is not possible,\" says\nEpicurus, \"to live pleasantly without living prudently and honorably and\njustly.\"[777] The difference, then, between the philosopher and the\nordinary man is this--that while both seek pleasure, the former knows\nhow to forego certain indulgences which cause pain and vexation\nhereafter, whereas the ordinary man seeks only immediate enjoyment.\nEpicurus does not dispense with virtue, but he simply employs it as a\nmeans to an end, namely, the securing of happiness.[778]\n\n[Footnote 775: Id., ib.]\n\n[Footnote 776: \"Fundamental Maxims,\" No. 7.]\n\n[Footnote 777: Ibid., No. 5.]\n\n[Footnote 778: Pressense, \"Religions before Christ,\" p. 141.]\n\nSocial morality is, like private morality, founded upon _utility._ As\nnothing is intrinsically right or wrong in private life, so nothing is\nintrinsically just or unjust in social life. \"Justice has no independent\nexistence: it results from mutual contracts, and establishes itself\nwherever there is a mutual engagement to guard against doing or\nsustaining any injury. Injustice is not intrinsically bad; it has this\ncharacter only because there is joined with it the fear of not escaping\nthose who are appointed to punish actions marked with this\ncharacter.\"[779] Society is thus a contract--an agreement to promote\neach other's happiness. And inasmuch as the happiness of the individual\ndepends in a great degree upon the general happiness, the essence of his\nethical system, in its political aspects, is contained in inculcating\n\"the greatest happiness of the greatest number.\"\n\nIf you ask Epicurus what a man shall do when it is clearly his immediate\ninterest to violate the social contract, he would answer, that if your\ngeneral interest is secured by always observing it, you must make\nmomentary sacrifices for the sake of future good. But \"when, in\nconsequence of new circumstances, a thing which has been pronounced just\ndoes not any longer appear to agree with utility, the thing which was\njust... ceases to be just the moment it ceases to be useful.\"[780] So\nthat self-interest is still the basis of all virtue. And if, by the\nperformance of duty, you are exposed to great suffering, and especially\nto death, you are perfectly justified in the violation of any and all\ncontracts. Such is the social morality of Epicurus.\n\nWith coarse and energetic minds the doctrine of Epicurus would\ninevitably lead to the grossest sensuality and crime; with men whose\ntemperament was more apathetic, or whose tastes were more pure, it would\ndevelop a refined selfishness--a perfect egoism, which Epicurus has\nadorned with the name \"tranquillity of mind--impassibility,\"\n(ataraxia).[781]\n\n[Footnote 779: \"Fundamental Maxims,\" Nos. 35, 36.]\n\n[Footnote 780: Ibid., No. 41.]\n\n[Footnote 781: It is scarcely necessary to discuss the question whether,\nby making pleasure the standard of right, Epicurus intended to encourage\nwhat is usually called sensuality. He earnestly protested against any\nsuch unfavorable interpretation of his doctrine:--\"When we say that\npleasure is a chief good, we are not speaking of the pleasures of the\ndebauched man, or those which lie in sensual enjoyment, as some think\nwho are ignorant, and who do not entertain our opinions, or else\ninterpret them perversely; but we mean the freedom of the body from\npain, and the soul from confusion\" (\"Epicurus to Menaeceus,\" in Diogenes\nLaertius, \"Lives,\" bk. x. ch. xxvii.). The most obvious tendency of this\ndoctrine is to extreme selfishness, rather than extreme sensuality--a\nselfishness which prefers one's own comfort and case to every other\nconsideration.\n\nAs to the personal character of Epicurus, opinions have been divided\nboth in ancient and modern times. By some the garden has been called a\n\"sty.\" Epicurus has been branded as a libertine, and the name\n\"Epicurean\" has, in almost all languages, become the synonym of\nsensualism. Diogenes Laertius repels all the imputations which are cast\nupon the moral character of his favorite author, and ascribes them to\nthe malignity and falsehood of the Stoics. \"The most modern criticism\nseems rather inclined to revert to the vulgar opinion respecting him,\nrejecting, certainly with good reason, the fanatical panegyrics of some\nFrench and English writers of the last century. Upon the whole, we are\ninclined to believe that Epicurus was an apathetic, decorous, formal\nman, who was able, without much difficulty, to cultivate a measured and\neven habit of mind, who may have occasionally indulged in sensual\ngratifications to prove that he thought them lawful, but who generally\npreferred, as a matter of taste, the exercises of the intellect to the\nmore violent forms of self-indulgence. And this life, it seems to us,\nwould be most consistent with his opinions. To avoid commotion, to make\nthe stream of life flow on as easily as possible, was clearly the aim of\nhis philosophy.\"--Maurice's \"Ancient Philosophy,\" p. 236.]\n\nTo secure this highest kind of happiness--this pure impassivity, it was\nnecessary to get rid of all superstitious fears of death, of\nsupernatural beings, and of a future retribution.[782] The chief causes\nof man's misery are his illusions, his superstitions, and his\nprejudices. \"That which principally contributes to trouble the spirit of\nmen, is the persuasion which they cherish that the stars are beings\nimperishable and happy (_i.e.,_ that they are gods), and that then our\nthoughts and actions are contrary to the will of those superior beings;\nthey also, being deluded by these fables, apprehend an eternity of\nevils, they fear the insensibility of death, as though that could affect\nthem....\" \"The real freedom from this kind of trouble consists in being\nemancipated from all these things.\"[783] And this emancipation is to be\nsecured by the study of philosophy--that is, of that philosophy which\nexplains every thing on natural or physical principles, and excludes all\nsupernatural powers.\n\n[Footnote 782: Lucretius, \"On the Nature of Things,\" bk. i. 1. 100-118.]\n\n[Footnote 783: Epicurus to Herodotus, in Diogenes Laertius, \"Lives of\nthe Philosophers,\" p. 453 (Bohn's edition).]\n\nThat ignorance which occasions man's misery is two-fold, (i.) _Ignorance\nof the external world, which leads to superstition._ All unexplained\nphenomena are ascribed to unseen, supernatural powers; often to\nmalignant powers, which take pleasure in tormenting man; sometimes to a\nSupreme and Righteous Power, which rewards and punishes men for their\ngood or evil conduct. Hence a knowledge of Physics, particularly the\nphysics which Democritus taught, was needful to deliver men from false\nhopes and false fears.[784] (ii.) _Ignorance of the nature of man, of\nhis faculties, powers, and the sources and limits of his knowledge_,\nfrom whence arise illusions, prejudices, and errors. Hence the need of\nPsychology to ascertain the real grounds of human knowledge, to explain\nthe origin of man's illusions, to exhibit the groundlessness of his\nfears, and lead him to a just conception of the nature and end of his\nexistence.\n\n[Footnote 784: \"The study of physics contributes more than any thing\nelse to the tranquillity and happiness of life.\"--Diogenes Laertius,\n\"Lives,\" bk. x. ch. xxiv. \"For thus it is that _fear_ restrains all men,\nbecause they observe many things effected on the earth and in heaven, of\nwhich effects they can by no means see the causes, and therefore think\nthat they are wrought by a _divine_ power. For which reasons, when we\nhave clearly seen that _nothing can be produced from nothing_, we shall\nhave a more accurate perception of that of which we are in search, and\nshall understand whence each individual thing is generated, and how all\nthings are done without the agency of the gods.\"--Lucretius, \"On the\nNature of Things,\" bk. i. l. 145-150.]\n\nPhysics and Psychology are thus the only studies which Epicurus would\ntolerate as \"conducive to the happiness of man.\" The pursuit of truth\nfor its own sake was useless. Dialectics, which distinguish the true\nfrom the false, the good from the bad, on _a priori_ grounds, must be\nbanished as an unnecessary toil, which yields no enjoyment. Theology\nmust be cancelled entirely, because it fosters superstitious fears. The\nidea of God's taking knowledge of, disapproving, condemning, punishing\nthe evil conduct of men, is an unpleasant thought. Physics and\nPsychology are the most useful, because the most \"agreeable,\" the most\n\"comfortable\" sciences.\n\nEPICUREAN PHYSICS.\n\nIn his physical theories Epicurus followed Leucippus and Democritus. He\nexpounds these theories in his letters to Herodotus and Pythocles, which\nare preserved in Diogenes Laertius.[785] We shall be guided mainly by\nhis own statements, and when his meaning is obscure, or his exposition\nis incomplete, we shall avail ourselves of the more elaborate statements\nof Lucretius,[786] who is uniformly faithful to the doctrine of\nEpicurus, and universally regarded as its best expounder.\n\nThe fundamental principle of his philosophy is the ancient maxim--\"_de\nnihilo nihil, in nihilum nil fosse reverti_;\" but instead of employing\nthis maxim in the sense in which it is used by Parmenides, Anaxagoras,\nEmpedocles, and others, to prove there must be something self-existent\nand eternal, or in other words, \"that nothing which once was not can\never of itself come into being,\" he uses it to disprove a divine\ncreation, and even presents the maxim in an altered form--viz., \"nothing\nis ever _divinely_ generated from nothing;\"[787] and he thence concludes\nthat the world was by no means made for us by _divine_ power.[788]\nNature is eternal. \"The universal whole always was such as it now is,\nand always will be such.\" \"The universe also is infinite, for that which\nis finite has a limit, but the universe has no limit.\"[789]\n\n[Footnote 785: \"Lives of the Philosophers,\" bk. x.]\n\n[Footnote 786: \"De Natura Rerum.\"]\n\n[Footnote 787: Lucretius, \"On the Nature of Things,\" bk. i.]\n\n[Footnote 788: Ibid.]\n\n[Footnote 789: Diogenes Laertius, \"Lives of the Philosophers,\" bk. x.\nch. xxiv.]\n\nThe two great principles of nature are a _vacuum_, and a _plenum._ The\nplenum is _body_, or tangible nature; the vacuum is _space_, or\nintangible nature. \"We know by the evidences of the senses (which are\nour only rule of reasoning) that _bodies_ have a real existence, and we\ninfer from the evidence of the senses that the vacuum has a real\nexistence; for if space have no real existence, there would be nothing\nin which bodies can move, as we see they really do move. Let us add to\nthis reflection that one can not conceive, either in virtue of\nperception, or of any analogy founded on perception, any general quality\npeculiar to all beings, which is not either an attribute, or an\naccident, of the body or of the vacuum.\"[790]\n\nOf bodies some are \"combinations\"--concrete bodies--and some are\nprimordial \"elements,\" out of which combinations are formed. These\nprimordial elements, out of which the universe is generated, are\n\"_atoms_\" (atomoi). These atoms are \"the first principles\" and \"seeds\"\nof all things.[791] They are \"_infinite_ in number,\" and, as their name\nimplies, they are \"_infrangible\" \"unchangeable_\" and\n\"_indestructible.\"_[792] Matter is, therefore, not infinitely divisible;\nthere must be a point at which division ends.[793]\n\nThe only qualities of atoms are _form_, _magnitude_, and _density._ All\nthe other sensible qualities of matter--the secondary qualities--as\ncolor, odor, sweetness, bitterness, etc.--are necessarily inherent in\nform. All secondary qualities are changeable, but the primary atoms are\nunchangeable; \"for in the dissolution of combined bodies there must be\nsomething _solid_ and _indestructible,_ of such a kind that it will not\nchange, either into what does not exist, or out of what does not exist,\nbut the change results from a simple displacement of parts, which is the\nmost usual case, or from an addition or subtraction of particles.\"[794]\n\n[Footnote 790: Diogenes Laertius, \"Lives of the Philosophers,\" bk. x.\nch. xxiv.]\n\n[Footnote 791: Id., ib., bk. x. ch. xxv.]\n\n[Footnote 792: Id., ib., bk. x. ch. xxiv.]\n\n[Footnote 793: Id., ib.; Lucretius, \"On the Nature of Things,\" bk. i. l.\n616-620.]\n\n[Footnote 794: Diogenes Laertius, \"Lives of the Philosophers,\" bk. x.\nch. xxiv.]\n\nThe atoms are not all of one _form_, but of different forms suited to\nthe production of different substances by combination; some are square,\nsome triangular, some smooth and spherical, some are hooked with points.\nThey are also diversified in _magnitude_ and _density_. The number of\noriginal forms is \"incalculably varied,\" but not infinite. \"Every\nvariety of forms contains an infinitude of atoms, but there is not, for\nthat reason, an infinitude of forms; it is only the number of them which\nis beyond computation.\"[795] To assert that atoms are of every kind of\nform, magnitude, and density, would be \"to contradict the phenomena;\n\"for experience teaches us that objects have a finite magnitude, and\nform necessarily supposes limitation.\n\n[Footnote 795: Id., ib.]\n\nA variety of these primordial forms enter into the composition of all\nsensible objects, because sensible objects possess different qualities,\nand these diversified qualities can only result from the combination of\ndifferent original forms. \"The earth has, in itself, primary atoms from\nwhich springs, rolling forth cool _water_, incessantly recruit the\nimmense sea; it has also atoms from which _fire_ arises.... Moreover,\nthe earth contains atoms from which it can raise up rich _corn_ and\ncheerful _groves_ for the tribes of men....\" So that \"no object in\nnature is constituted of one kind of elements, and whatever possesses in\nitself must numerous powers and energies, thus demonstrates that it\ncontains more numerous kinds of primary particles,\"[796] or primordial\n\"seeds of things.\"\n\n\"The atoms are in a continual state of _motion_\" and \"have moved with\n_equal rapidity_ from all eternity, since it is evident the vacuum can\noffer no resistance to the heaviest, any more than the lightest.\" The\nprimary and original movement of all atoms is _in straight lines, by\nvirtue of their own weight_. The vacuum separates all atoms one from\nanother, at greater or less distances, and they preserve their own\npeculiar motion in the densest substances.[797]\n\n[Footnote 796: Lucretius, \"On the Nature of Things,\" bk. i. l. 582-600.]\n\n[Footnote 797: Diogenes Laertius, \"Lives of the Philosophers,\" bk. x.\nch. xxiv.; Lucretius, \"On the Nature of Things,\" bk. i. l. 80-92.]\n\nAnd now the grand crucial question arises--_How do atoms combine so as\nto form concrete bodies?_ If they move in straight lines, and with equal\nrapidity from all eternity, then they can never unite so as to form\nconcrete substances. They can only coalesce by deviating from a straight\nline.[798] How are they made to deviate from a straight line? This\ndeviation must be introduced _arbitrarily_, or by some _external cause_.\nAnd inasmuch as Epicurus admits of no causes \"but space and matter,\" and\nrejects all divine or supernatural interposition, the _new_ movement\nmust be purely arbitrary. They deviate _spontaneously,_ and of their own\naccord. \"The system of nature immediately appears _as a free agent_,\nreleased from tyrant masters, to do every thing of itself spontaneously,\nwithout the help of the gods.\"[799] The manner in which Lucretius proves\nthis doctrine is a good example of the petitio principii. He assumes, in\nopposition to the whole spirit and tendency of the Epicurean philosophy,\nthat man has \"a free will,\" and then argues that if man who is nothing\nbut an aggregation of atoms, can \"turn aside and alter his own\nmovements,\" the primary elements, of which his soul is composed, must\nhave some original spontaneity. \"If all motion is connected and\ndependent, and a new movement perpetually arises from a former one in a\ncertain order, and if the primary elements do not produce any\ncommencement of motion by deviating from the straight line to break the\nlaws of fate, so that cause may not follow cause in infinite succession,\n_whence comes this freedom of will_ to all animals in the world? whence,\nI say, is this liberty of action wrested from the fates, by means of\nwhich we go wheresoever inclination leads each of us? whence is it that\nwe ourselves turn aside, and alter our motions, not at any fixed time,\nnor in any fixed part of space, but just as our own minds prompt?....\nWherefore we must necessarily confess that the same is the case with the\nseeds of matter, and there is some other cause besides strokes and\nweight [resistance and density] from which this power [of free movement]\nis innate in them, since we see that _nothing is produced from\nnothing_.\"[800] Besides form, extension, and density, Epicurus has found\nanother inherent or essential quality of matter or atoms, namely,\n\"_spontaneous\" motion._\n\n[Footnote 798: \"At some time, though at no fixed and determinate time,\nand at some point, though at no fixed and determinate point, they turn\naside from the right line, but only so far as you can call the least\npossible deviation.\"--Lucretius, \"On the Nature of Things,\" bk. ii. l.\n216-222.]\n\n[Footnote 799: Lucretius, \"On the Nature of Things\" bk. ii. 1.\n1092-1096.]\n\n[Footnote 800: Id., ib., bk. ii. l. 250-290.]\n\nBy a slight \"voluntary\" deflection from the straight line, atoms are now\nbrought into contact with each other; \"they strike against each other,\nand by the percussion new movements and new complications\narise\"--\"movements from high to low, from low to high, and horizontal\nmovements to and fro, in virtue of this reciprocal percussion.\" The\natoms \"jostling about, _of their own accord_, in infinite modes, were\noften brought together confusedly, irregularly, and to no purpose, but\nat length they _successfully coalesced_; at least, such of them as were\nthrown together suddenly became, in succession, the beginnings of great\nthings--as earth, and air, and sea, and heaven.\"[801]\n\n[Footnote 801: Lucretius, \"On the Nature of Things,\" bk. ii. l.\n1051-1065.]\n\nAnd now Lucretius shall describe the formation of the different parts of\nthe world according to the cosmogony of Epicurus. We quote from Good's\ntranslation:\n\n     But from this boundless mass of matter first\n     How heaven, and earth, and ocean, sun, and moon,\n     Rose in nice order, now the muse shall tell.\n     For never, doubtless, from result of thought,\n     Or mutual compact, could primordial seeds\n     First harmonize, or move with powers precise.\n     But countless crowds in countless manners urged,\n     From time eternal, by intrinsic weight\n     And ceaseless repercussion, to combine\n     In all the possibilities of forms,\n     Of actions, and connections, and exert\n     In every change some effort to create--\n     Reared the rude frame at length, abruptly reared,\n     Which, when once gendered, must the basis prove\n     Of things sublime; and whence eventual rose\n     Heaven, earth, and ocean, and the tribes of sense.\n\n     Yet now nor sun on fiery wheel was seen\n     Riding sublime, nor stars adorned the pole,\n     Nor heaven, nor earth, nor air, nor ocean lived,\n     Nor aught of prospect mortal sight surveyed;\n     But one vast chaos, boisterous and confused.\n     Yet order hence began; congenial parts\n     Parts joined congenial; and the rising world\n     Gradual evolved: its mighty members each\n     From each divided, and matured complete\n     From seeds appropriate; whose wild discortderst,\n     Reared by their strange diversities of form,\n     With ruthless war so broke their proper paths,\n     Their motions, intervals, conjunctions, weights,\n     And repercussions, nought of genial act\n     Till now could follow, nor the seeds themselves\n     E'en though conjoined in mutual bonds, co\n     Thus air, secreted, rose o'er laboring earth;\n     Secreted ocean flowed; and the pure fire,\n     Secreted too, toward ether sprang sublime.\n\n     But first the seeds terrene, since ponderous most\n     And most perplext, in close embraces clung,\n     And towards the centre conglobating sunk.\n     And as the bond grew firmer, ampler forth\n     Pressed they the fluid essences that reared\n     Sun, moon, and stars, and main, and heaven's high wall.\n     For those of atoms lighter far consist,\n     Subtiler, and more rotund than those of earth.\n     Whence, from the pores terrene, with foremost haste\n     Rushed the bright ether, towering high, and swift\n     Streams of fire attracting as it flowed.\n\n     Then mounted, next, the base of sun and moon,\n     'Twixt earth and ether, in the midway air\n     Rolling their orbs; for into neither these\n     Could blend harmonious, since too light with earth\n     To sink deprest, while yet too ponderous far\n     To fly with ether toward the realms extreme:\n     So 'twixt the two they hovered; _vital_ there\n     Moving forever, parts of the vast whole;\n     As move forever in the frame of man\n     Some active organs, while some oft repose.[802]\n\n[Footnote 802: Lucretius, \"On the Nature of Things,\" b. v. l. 431-498]\n\nAfter explaining the origin and causes of the varied celestial\nphenomena, he proceeds to give an account of the production of plants,\nanimals, and man:\n\n     Once more return we to the world's pure prime,\n     Her fields yet liquid, and the tribes survey\n     First she put forth, and trusted to the winds.\n\n     And first the race she reared of verdant herbs,\n     Glistening o'er every hill; the fields at large\n     Shone with the verdant tincture, and the trees\n     Felt the deep impulse, and with outstretched arms\n     Broke from their bonds rejoicing. As the down\n     Shoots from the winged nations, or from beasts\n     Bristles or hair, so poured the new-born earth\n     Plants, fruits, and herbage. Then, in order next,\n     Raised she the sentient tribes, in various modes,\n     By various powers distinguished: for not heaven\n     Down dropped them, nor from ocean's briny waves\n     Sprang they, terrestrial sole; whence, justly _Earth_\n     Claims the dear name of mother, since alone\n     Flowed from herself whate'er the sight surveys.\n\n     E'en now oft rears she many a sentient tribe\n     By showers and sunshine ushered into day.[803]\n     Whence less stupendous tribes should then have risen\n     More, and of ampler make, herself new-formed,\n     In flower of youth, and _Ether_ all mature.[804]\n\n     Of these birds first, of wing and plume diverse,\n     Broke their light shells in spring-time: as in spring\n     Still breaks the grasshopper his curious web,\n     And seeks, spontaneous, foods and vital air.\n\n     Then rushed the ranks of mortals; for the soil,\n     Exuberant then, with warmth and moisture teemed.\n     So, o'er each scene appropriate, myriad wombs\n     Shot, and expanded, to the genial sward\n     By fibres fixt; and as, in ripened hour,\n     Their liquid orbs the daring foetus broke\n     Of breath impatient, nature here transformed\n     Th' assenting earth, and taught her opening veins\n     With juice to flow lacteal; as the fair\n     Now with sweet milk o'erflows, whose raptured breast\n     First hails the stranger-babe, since all absorbed\n     Of nurture, to the genial tide converts.\n     Earth fed the nursling, the warm ether clothed,\n     And the soft downy grass his couch compressed.[805]\n\n[Footnote 803: The doctrine of \"spontaneous generations\" is still more\nexplicitly announced in book ii. \"Manifest appearances compel us to\nbelieve that animals, though possessed of sense, are generated from\nsenseless atoms. For you may observe living worms proceed from foul\ndung, when the earth, moistened with immoderate showers, has contracted\na kind of putrescence; and you may see all other things change\nthemselves, similarly, into other things.\"--Lucretius, \"On the Nature of\nThings,\" bk. i. l. 867-880.]\n\n[Footnote 804: Ether is the father, earth the mother of all organized\nbeing.--Id., ib., bk. i. l. 250-255.]\n\n[Footnote 805: Id., ib., bk. v. l. 795-836.]\n\nA state of pure savagism, or rather of mere animalism, was the primitive\ncondition of man. He wandered naked in the woods, feeding on acorns and\nwild fruits, and quenched his thirst at the \"echoing waterfalls,\" in\ncompany with the wild beast.\n\nThrough the remaining part of book v. Lucretius describes how speech was\ninvented; how society originated, and governments were instituted; how\ncivilization commenced; and how religion arose out of ignorance of\nnatural causes; how the arts of life were discovered, and how science\nsprang up. And all this, as he is careful to tell us, without any divine\ninstruction, or any assistance from the gods.\n\nSuch are the physical theories of the Epicureans. The primordial\nelements of matter are infinite, eternal, and self-moved. After ages\nupon ages of chaotic strife, the universe at length arose out of an\n_infinite_ number of atoms, and a _finite_ number of forms, by a\nfortuitous combination. Plants, animals, and man were spontaneously\ngenerated from ether and earth. Languages, society, governments, arts\nwere gradually developed. And all was achieved simply by blind,\nunconscious nature-forces, without any designing, presiding, and\ngoverning Intelligence--that is, without a God.\n\nThe evil genius which presided over the method of Epicurus, and\nperverted all his processes of thought, is clearly apparent. The end of\nhis philosophy was not the discovery of truth. He does not commence his\ninquiry into the principles or causes which are adequate to the\nexplanation of the universe, with an unprejudiced mind. He everywhere\ndevelops a malignant hostility to religion, and the avowed object of his\nphysical theories is to rid the human mind of all fear of supernatural\npowers--that is, of all fear of God.[806] \"The phenomena which men\nobserve to occur in the earth and the heavens, when, as often happens,\nthey are perplexed with fearful thoughts, overawe their minds with a\ndread of the gods, and humble and depress them to the earth. For\nignorance of natural causes obliges them to refer all things to the\npower of the divinities, and to resign the dominion of the world to\nthem; because of those effects they can by no means see the origin, and\naccordingly suppose that they are produced by divine influence.\"[807]\n\n[Footnote 806: \"Let us trample religion underfoot, that the victory\ngained over it may place us on an equality with heaven\" (book i.). See\nDiogenes Laertius, \"Lives of the Philosophers,\" bk. x. ch. xxiv. pp.\n453,454 (Bohn's edition); Lucretius, \"On the Nature of Things,\" bk. i.\nl. 54-120.]\n\n[Footnote 807: Lucretius, \"On the Nature of Things,\" bk. vi. l. 51-60.]\n\nTo \"expel these fancies from the mind\" as \"inconsistent with its\ntranquillity and opposed to human happiness,\" is the end, and, as\nLucretius believes, the glory of the Epicurean philosophy. To accomplish\nthis, God must be placed at an infinite distance from the universe, and\nmust be represented as indifferent to every thing that transpires within\nit. We \"must beware of making the Deity interpose here, for that Being\nwe ought to suppose _exempt from all occupation_, and perfectly\nhappy,\"[808]--that is, absolutely impassible. God did not make the\nworld, and he does not govern the world. There is no evidence of design\nor intelligence in its structure, and \"such is the faultiness with which\nit stands affected, that it can not be the work of a Divine power.\"[809]\n\nEpicurus is, then, an unmistakable Atheist. He did not admit a God in\nany rational sense. True, he _professed_ to believe in gods, but\nevidently in a very equivocal manner, and solely to escape the popular\ncondemnation. \"They are not pure spirits, for there is no spirit in the\natomic theory; they are not bodies, for where are the bodies that we may\ncall gods? In this embarrassment, Epicurus, compelled to acknowledge\nthat the human race believes in the existence of gods, addresses himself\nto an old theory of Democritus--that is, he appeals to dreams. As in\ndreams there are images that act upon and determine in us agreeable or\npainful sensations, without proceeding from exterior bodies, so the gods\nare images similar to those of dreams, but greater, having the human\nform; images which are not precisely bodies, and yet not deprived of\nmateriality which are whatever you please, but which, in short, must be\nadmitted, since the human race believes in gods, and since the\nuniversality of the religious sentiment is a fact which demands a\ncause.\"[810]\n\n[Footnote 808: Diogenes Laertius, \"Lives of the Philosophers,\" bk. x.\nch. xxv.; Lucretius, \"On the Nature of Things,\" bk. i. l. 55-60.]\n\n[Footnote 809: Lucretius, \"On the Nature of Things,\" bk. v. l. 195-200.]\n\n[Footnote 810: Cousin's \"Lectures on the History of Philosophy,\" vol. i.\np. 431.]\n\nIt is needless to offer any criticism on the reasoning of Epicurus. One\nfact will have obviously presented itself to the mind of the reflecting\nreader. He starts with atoms having form, magnitude, and density, and\nessays to construct a universe; but he is obliged to be continually\nintroducing, in addition, a \"_nameless something_\" which \"remains in\nsecret,\" to help him out in the explanation of the phenomena.[811] He\nmakes life to arise out of dead matter, sense out of senseless atoms,\nconsciousness out of unconsciousness, reason out of unreason, without an\nadequate cause, and thus violates the fundamental principle from which\nhe starts, \"_that nothing can arise from nothing_.\"\n\nEPICUREAN PSYCHOLOGY.\n\nIn the system of Epicurus, the soul is regarded as corporeal or\nmaterial, like the body; they form, together, one nature or substance.\nThe soul is composed of atoms exceedingly diminutive, smooth, and round,\nand connected with or diffused through the veins, viscera, and nerves.\nThe substance of the soul is not to be regarded as simple and\nuncompounded; its constituent parts are _aura_, heat, and air. These are\nnot sufficient, however, even in the judgment of Epicurus, to account\nfor _sensation_; they are not adequate to generate sensible motives such\nas revolve any thoughts in the mind. \"A certain fourth nature, or\nsubstance, must, therefore, necessarily be added to these, _that is\nwholly without a name_; it is a substance, however, than which nothing\nexists more active or more subtile, nor is any thing more essentially\ncomposed of small and smooth elementary particles; and it is this\nsubstance which first distributes sensible motions through the\nmembers.\"[812]\n\n[Footnote 811: As, _e.g._, Lucretius, \"On the Nature of Things,\" bk.\niii. l. 260-290.]\n\n[Footnote 812: Id., ib., bk. iii. l. 237-250.]\n\nEpicurus is at great pains to prove that the soul is material; and it\ncan not be denied that he marshals his arguments with great skill.\nModern materialism may have added additional illustrations, but it has\ncontributed no new lines of proof. The weapons are borrowed from the old\narsenal, and they are not wielded with any greater skill than they were\nby Epicurus himself, I. The soul and the body act and react upon each\nother; and mutual reaction can only take place between substances of\nsimilar nature. \"Such effects can only be produced by _touch_, and touch\ncan not take place without _body_.\"[813] 2. The mind is produced\ntogether with the body, it grows up along with it, and waxes old at the\nsame time with it.[814] 3. The mind is diseased along with the body, \"it\nloses its faculties by material causes, as intoxication, or by severe\nblows; and is sometimes, by a heavy lethargy, borne down into a deep\neternal sleep.\"[815] 4. The mind, like the body, is healed by medicines,\nwhich proves that it exists only as a mortal substance.[816] 5. The mind\ndoes not always, and at the same time, continue _entire_ and\n_unimpaired_, some faculties decay before the others, \"the substance of\nthe soul is therefore divided.\" On all these grounds the soul must be\ndeemed mortal; it is dissolved along with the body, and has no conscious\nexistence after death.\n\n[Footnote 813: Lucretius, \"On the Nature of Things,\" bk. iii. l.\n138-168.]\n\n[Footnote 814: Id., ib., bk. iii. l. 444-460.]\n\n[Footnote 815: Id., ib., bk. iii. l. 438-490.]\n\n[Footnote 816: Id., ib., bk. iii. l. 500-520.]\n\nSuch being the nature of the soul, inasmuch as it is material, all its\nknowledge must be derived from sensation. The famous doctrine of\nperception, as taught by Epicurus, is grounded upon this pre-supposition\nthat the soul is corporeal. \"The eidola aporroiai--_imagines, simulacra\nrerum, etc_., are, like pellicles, continually flying off from objects;\nand these material 'likenesses,' diffusing themselves everywhere in the\nair, are propelled to the perceptive organs.\" These images of things\ncoming in contact with the senses produce _sensation_ (aisthesis). A\nsensation may be considered either as regards its object, or as regards\nhim who experiences it. As regards him who experiences it, it is simply\na passive affection, an agreeable or disagreeable feeling, passion, or\nsentiment (to pathos). But along with sensation there is inseparably\nassociated some knowledge of the object which excites sensation; and it\nis for this reason that Epicurus marked the intimate relation of these\ntwo phenomena by giving them analogous names. Because the second\nphenomenon is joined to the first, he calls it\nepaisthesis--_perception_. It is sensation viewed especially in regard\nto its object--_representative sensation_, or the \"sensible idea\" of\nmodern philosophy. It is from perception that we draw our general ideas\nby a kind of prolepsis (prolepsis) an anticipation or laying hold by\nreason of that which is implied in sensation. Now all sensations are\nalike true in so far as they are sensations, and error arises from false\nreasoning about the testimony of sense. All knowledge is purely relative\nand contingent, and there is no such thing as necessary and absolute\ntruth.\n\nThe system of Epicurus is thus a system of pure materialism, but not a\nsystem of materialism drawn, as a logical consequence, from a careful\nand unprejudiced study of the whole phenomena of mind. His openly avowed\ndesign is to deliver men from the fear of death, and rid them of all\napprehension of a future retribution. \"Did men but know that there was a\nfixed limit to their woes, they would be able, in some measure, to defy\nthe religious fictions and menaces of the poets; but now, since we must\nfear eternal punishment at death, there is no mode, no means of\nresisting them.\"[817] To emancipate men from \"these terrors of the\nmind,\" they must be taught \"that the soul is mortal, and dissolves with\nthe body\"--that \"death is nothing to us, for that which is dissolved is\ndevoid of sensation, and that which is devoid of sensation is nothing to\nus.\"[818] Starting with the fixed determination to prove that\n\n     \"Death is nothing, and naught after death,\"\n\nhe will not permit any mental phenomena to suggest to him the idea of an\nincorporeal spiritual substance. Matter, under any form known to\nEpicurus, is confessedly insufficient to explain sensation and thought;\na \"nameless something\" must be _supposed_. But may not \"that principle\nwhich _lies entirely hid, and remains in secret_\"[819]--and about which\neven Epicurus does not know any thing--be a spiritual, an _immaterial_\nprinciple? For aught that he knows it may as properly be called\n\"_spirit_\" as matter. May not _sensation_ and _cognition_ be the result\nof the union of matter and spirit; and if so, may not their mutual\naffections, their common sympathies, be the necessary conditions of\nsensation and cognition in the present life? A reciprocal relation\nbetween body and mind appears in all mental phenomena. A certain\nproportion in this relation is called mental health. A deviation from it\nis termed disease. This proportion is by no means an equilibrium, but\nthe perfect adaptation of the body, without injury to its integrity, to\nthe purposes of the mind. And if this be so, all the arguments of\nmaterialism fall to the ground.\n\n[Footnote 817: Lucretius, \"On the Nature of Things,\" bk. i. l. 100-118.]\n\n[Footnote 818: Diogenes Laertius, Maxim 2, in \"Lives of the\nPhilosophers,\" bk. x. ch. xxxi.]\n\n[Footnote 819: Lucretius, \"On the Nature of Things,\" bk. iii. l.\n275-280.]\n\nThe concluding portion of the third book, in which Lucretius discourses\non _death_, is a mournful picture of the condition of the heathen mind\nbefore Christianity \"brought life and immortality fully to light.\" It\ncomes to us, like a voice from the grave of two thousand years, to prove\nthey were \"without hope.\" To be delivered from the fear of future\nretribution, they would sacrifice the hope of an immortal life. To\nextintinguish guilt they would annihilate the soul. The only way in\nwhich Lucretius can console man in prospect of death is, by reminding\nhim that he will _escape the ills of life_.\n\n     \"'But thy dear home shall never greet thee more!\n     No more the best of wives!--thy babes beloved,\n     Whose haste half-met thee, emulous to snatch\n     The dulcet kiss that roused thy secret soul,\n     Again shall never hasten!--nor thine arm,\n     With deeds heroic, guard thy country's weal!--\n     Oh mournful, mournful fate!' thy friends exclaim!\n     'One envious hour of these invalued joys\n     Robs thee forever!--But they add not here,\n     '_It robs thee, too, of all desire of joy_'--\n     A truth, once uttered, that the mind would free\n     From every dread and trouble. 'Thou art safe\n     The sleep of death protects thee, _and secures\n     From all the unnumbered woes of mortal life!_\n     While we, alas! the sacred urn around\n     That holds thine ashes, shall insatiate weep,\n     Nor time destroy the eternal grief we feel!'\n     What, then, has death, if death be mere repose,\n     And quiet only in a peaceful grave,--\n     What has it thus to mar this life of man?\"[820]\n\n[Footnote 820: Lucretius, \"On the Nature of Things,\" bk. iii. l.\n906-926.]\n\nThis is all the comfort that Epicureanism can offer; and if \"the wretch\nstill laments the approach of death,\" she addresses him \"with voice\nsevere\"--\n\n     \"Vile coward! dry thine eyes--\n     Hence with thy snivelling sorrows, and depart!\"\n\nIt is evident that such a system of philosophy outrages the purest and\nnoblest sentiments of humanity, and, in fact, condemns itself. It was\nborn of selfishness and social degeneracy, and could perpetuate itself\nonly in an age of corruption, because it inculcated the lawfulness of\nsensuality and the impunity of injustice. Its existence at this precise\nperiod in Grecian history forcibly illustrates the truth, that Atheism\nis a disease of the heart rather than the head. It seeks to set man free\nto follow his own inclinations, by ridding him of all faith in a\nDivinity and in an immortal life, and thus exonerating him from all\naccountability and all future retribution. But it failed to perceive\nthat, in the most effectual manner, it annihilated all real liberty, all\ntrue nobleness, and made of man an abject slave.\n\nSTOICISM.\n\nThe Stoical school was founded by Zeno of Citium, who flourished B.C.\n290. He taught in the Stoa Poecile, or Painted Porch; and his disciples\nthence derived the name of Stoics. Zeno was succeeded by Cleanthes (B.C.\n260); and Cleanthes by Chrysippus (B.C. 240), whose vigorous intellect\ngave unity and completeness to the Stoical philosophy. He is reported to\nhave said to Cleanthes,--\"Give me your doctrines, and I will find the\ndemonstrations.\"[821]\n\n[Footnote 821: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. vii.]\n\nNone of the writings of the early Stoics, save a \"Hymn to Jupiter,\" by\nCleanthes, have survived. We are chiefly indebted to Diogenes\nLaertius[822] and Cicero[823] for an insight into their system. The Hymn\nof Cleanthes sheds some light on their Theology, and their moral\nprinciples are exhibited in \"The Fragments\" of Epictetus, and \"The Life\nand Meditations\" of Marcus Aurelius.\n\n[Footnote 822: \"Lives of the Philosophers,\" bk. vii.]\n\n[Footnote 823: \"De Fm.,\" and \"De Natura Deorum.\"]\n\nThe philosophy of the Stoics, like that of the Epicureans, was mainly a\nphilosophy of life--that is, a _moral_ philosophy. The manner in which\nthey approached the study of morals, and the principles upon which they\ngrounded morality, were, however, essentially different.\n\nThe grand object of Epicurus was to make the current of life flow on as\ncomfortably as possible, without any distracting thoughts of the past or\nany disturbing visions of the future. He therefore starts with this\nfundamental principle, that the true philosophy of life is to enjoy\none's self--the aim of existence is to be happy. Whatever in a man's\nbeliefs or conduct tends to secure happiness is _right_; whatever\nawakens uneasiness, apprehension, or fear, is _wrong_. And inasmuch as\nthe idea of a Divine Creator and Governor of the universe, and the\nbelief in a future life and retribution, are uncomfortable thoughts,\nexciting superstitious fears, they ought to be rejected. The Physics and\nthe Psychology of Epicurus are thus the natural outgrowth of his\nMorality.\n\nZeno was evidently a more earnest, serious, and thoughtful man. He\ncherished a nobler ideal of life than to suppose \"man must do\nvoluntarily, what the brute does instinctively--eschew pain, and seek\npleasure.\" He therefore seeks to ascertain whether there be not some\n\"principle of nature,\" or some law of nature, which determines what is\nright in human action--whether there be not some light under which, on\ncontemplating an action, we may at once pronounce upon its intrinsic\n_rightness_, or otherwise. This he believes he has found in the\n_universal reason_ which fashioned, and permeates, and vivifies the\nuniverse, and is the light and life of the human soul. The chief good\nis, confessedly, to live according to nature; which is to live according\nto virtue, for nature leads us to that point.... For our individual\nnatures are all part of the universal nature; on which account, the\nchief good is to live in a manner corresponding to one's own nature, and\nto universal nature; doing none of those things which the common law of\nmankind (the universal conscience of our race) forbids. _That common law\nis identical with_ RIGHT REASON _which pervades every thing, being the\nsame with Jupiter_ (Zeus), _who is the regulator and chief manager of\nall existing things_.[824] The foundation of the ethical system of the\nStoics is thus laid in their philosophy of nature--their Physiology and\nPsychology. If, therefore, we would apprehend the logical connection and\nunity of Stoicism, we must follow their order of thought--that is, we\nmust commence with their\n\nPHYSIOLOGY.\n\nDiogenes Laertius tells us that the Stoics held \"that there are two\ngeneral principles in the universe--the _passive_ principle (to\nnaschon), which is matter, an existence without any distinctive quality,\nand the _active_ principle (to poioun), which is the reason existing in\nthe passive, that is to say, God. For that He, being eternal, and\nexisting throughout all matter, makes every thing.\"[825] This Divine\nReason, acting upon matter, originates the necessary and unchangeable\nlaws which govern matter--laws which the Stoics called logoi\nspermatikoi--generating reasons or causes of things. The laws of the\nworld are, like eternal reason, necessary and immutable; hence the\neimarmene--the _Destiny_ of the Stoics, which is also one of the names\nof the Deity.[826] But by Destiny the Stoics could not understand a\nblind unconscious necessity; it is rather the highest reason in the\nuniverse. \"Destiny (eimarmene) is a connected (eiromene) cause of\nthings, or the reason according to which the world is regulated.\"[827]\n\n[Footnote 824: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. liii.]\n\n[Footnote 825: Id., ib., bk. vii. ch. lxviii.]\n\n[Footnote 826: \"They teach that God is unity, and that he is called\nMind, and _Fate_, and Jupiter.\"--Id., ib., bk. vii. ch. lxviii.]\n\n[Footnote 827: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. lxxiv.]\n\nThese two principles are not, however, regarded by the Stoics as having\na distinct, separate, and independent existence. One is substance\n(ousia); the other is quality (poios). The primordial matter is the\npassive ground of all existence--the original substratum for the Divine\nactivity. The Divine Reason is the active or formative energy which\ndwells within, and is essentially united to, the primary substance. The\nStoics, therefore, regarded all existence as reducible, in its last\nanalysis, to _one substance_, which on the side of its passivity and\ncapacity of change, they called _hyle_ (yle);[828] and on the side of\nits changeless energy and immutable order, they called God. The\ncorporeal world--physical nature--is \"a peculiar manifestation\" of God,\ngenerated from his own substance, and, after certain periods, absorbed\nin himself. Thus God, considered in the evolution of His power, is\nnature. And nature, as attached to its immanent principle, is called\nGod.[829] The fundamental doctrine of the Stoics was a spiritual, ideal,\nintellectual pantheism, of which the proper formula is, _All things are\nGod, but God is not all things_.\n\n[Footnote 828: Or \"matter.\" A good deal of misapprehension has arisen\nfrom confounding the intellectual yle of Aristotle and the Stoics with\nthe gross physical \"matter\" of the modern physicist. By \"matter\" we now\nunderstand that which is corporeal, tangible, sensible; whereas by yle,\nAristotle and the Stoics (who borrowed the term from him) understood\nthat which is incorporeal, intangible, and inapprehensible to sense,--an\n\"unknown something\" which must necessarily be _supposed_ as the\ncondition of the existence of things. The _formal_ cause of Aristotle is\n\"the substance and essence\"--the primary nature of things, on which all\ntheir properties depend. The _material_ cause is \"the matter or subject\"\nthrough which the primary nature manifests itself. Unfortunately the\nterm \"material\" misleads the modern thinker. He is in danger of\nsupposing the _hyle_ of Aristotle to be something sensible and physical,\nwhereas it is an intellectual principle whose inherence is implied in\nany physical thing. It is something distinct from _body_, and has none\nof those properties we are now accustomed to ascribe to matter. Body,\ncorporeity, is the result of the union of \"hyle\" and \"form.\" Stobaeus\nthus expounds the doctrine of Aristotle: Form alone, separate from\nmatter (yle) is _incorporeal_; so matter alone, separated from form, is\nnot _body_. But there is need of the joint concurrence of both\nthese--matter and form--to make the substance of body. Every individual\nsubstance is thus a totality of matter and form--a sinolon.\n\nThe Stoics taught that God is _oneliness_ (Diogenes Laertius, \"Lives of\nthe Philosophers,\" bk. vii. ch. lxviii.); that he is _eternal_ and\n_immortal_ (bk. vii. ch. lxxii.); he could not, therefore, be corporeal,\nfor \"body _infinite, divisible,_ and _perishable_\" (bk. vii. ch.\nlxxvii.). \"All the parts of the world are perishable, for they change\none into another; therefore the world is perishable\" (bk. vii. ch.\nlxx.). The Deity is not, therefore, absolutely identified with the world\nby the Stoics. He permeates all things, creates and dissolves all\nthings, and is, therefore, _more_ than all things. The world is finite;\nGod is infinite.]\n\n[Footnote 829: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. lxx.]\n\nSchwegler affirms that, in physics, the Stoics, for the most part,\nfollowed Heraclitus, and especially \"carried out the proposition that\nnothing incorporeal exists; every thing is essentially _corporeal_.\" The\npantheism of Zeno is therefore \"_materialistic._\"[830] This is not a\njust representation of the views of the early Stoics, and can not be\nsustained by a fair interpretation of their teaching. They say that\nprinciples and elements differ from each other. Principles have no\ngeneration or beginning, and will have no end; but elements may be\ndestroyed. Also, that elements have bodies, and have forms, _but\nprinciples have no bodies, and no forms_.[831] Principles are,\ntherefore, _incorporeal._ Furthermore, Cicero tells us that they taught\nthat the universal harmony of the world resulted from all things being\n\"contained by one _Divine_ SPIRIT;\"[832] and also, that reason in man is\n\"nothing else but part of the _Divine_ SPIRIT merged into a human\nbody.\"[833] It thus seems evident that the Stoics made a distinction\nbetween corruptible _elements_ (fire, air, earth, water) and\nincorruptible _principles_, by which and out of which elements were\ngenerated, and also between corporeal and incorporeal substances.\n\n[Footnote 830: Schwegler's \"History of Philosophy,\" p. 140.]\n\n[Footnote 831: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. lxviii.]\n\n[Footnote 832: \"De Natura Deorum,\" bk. ii. ch. xiii.]\n\n[Footnote 833: Ibid, bk. ii. ch. xxxi.]\n\nOn a careful collation of the fragmentary remains of the early Stoics,\nwe fancy we catch glimpses of the theory held by some modern pantheists,\nthat the material elements, \"having body and form,\" are a vital\ntransformation of the Divine substance; and that the forces of\nnature--\"the generating causes or reasons of things\" (logoi\nspermatikoi)--are a conscious transmutation of the Divine energy. This\ntheory is more than hinted in the following passages, which we slightly\ntranspose from the order in which they stand in Diogenes Laertius,\nwithout altering their meaning. \"They teach that the Deity was in the\nbeginning by _himself_\".... that \"first of all, he made the four\nelements, fire, water, air, and earth.\" \"The fire is the highest, and\nthat is called aether, in which, first of all, the sphere was generated\nin which the fixed stars are set...; after that the air; then the water;\nand the sediment, as it were, of all, is the earth, which is placed in\nthe centre of the rest.\" \"He turned into water the whole substance which\npervaded the air; and as the seed is contained in the product, so, too,\nHe, being the seminal principle of the world, remained still in\nmoisture, making matter fit to be employed by himself in the production\nof things which were to come after.\"[834] The Deity thus draws the\nuniverse out of himself, transmuting the divine substance into body and\nform. \"God is a being of a certain quality, having for his peculiar\nmanifestation universal substance. He is a being imperishable, and who\nnever had any generation, being the maker of the arrangement and order\nthat we see; and who at certain periods of time _absorbs all substance\nin himself and then reproduces it from himself_.\"[835] And now, in the\nlast analysis, it would seem as though every thing is resolved into\n_force_. God and the world are _power, and its manifestation_, and these\nare ultimately one. \"This identification of God and the world, according\nto which the Stoics regarded the whole formation of the universe as but\na period in the development of God, renders their remaining doctrine\nconcerning the world very simple. Every thing in the world seemed to be\npermeated by the Divine life, and was regarded as the flowing out of\nthis most perfect life through certain channels, until it returns, in a\nnecessary circle, back to itself.\"[836]\n\n[Footnote 834: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. lxviii., lxix.]\n\n[Footnote 835: Id., ib., bk. vii. ch. lxx.]\n\n[Footnote 836: Schwegler's \"History of Philosophy,\" p. 141.]\n\nThe God of the Stoics is not, however, a mere principle of life\nvitalizing nature, but an _intelligent_ principle directing nature; and,\nabove all, a _moral_ principle, governing the human race. \"God is a\nliving being, immortal, rational, perfect, and intellectual in his\nhappiness, unsusceptible of any kind of evil; having a foreknowledge of\nthe world, and of all that is in the world.\"[837] He is also the\ngracious Providence which cares for the individual as well as for the\nwhole; and he is the author of that natural law which commands the good\nand prohibits the bad. \"He made men to this end that they might be\nhappy; as becomes his fatherly care of us, he placed our good and evil\nin those things which are in our own power.\"[838] The Providence and\nFatherhood of God are strikingly presented in the \"Hymn of Cleanthes\" to\nJupiter--\n\n[Footnote 837: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. lxxii.]\n\n[Footnote 838: Marcus Aurelius, bk. iii. ch. xxiv.]\n\n     Most glorious of the immortal Powers above!\n     O thou of many names! mysterious Jove:\n     For evermore almighty! Nature's source!\n     Thou governest all things in their order'd course!\n     All hail to thee! since, innocent of blame,\n     E'en mortal creatures may address thy name;\n     For all that breathe, and creep the lowly earth,\n     Echo thy being with reflected birth--\n     Thee will I sing, thy strength for aye resound:\n     The universe, that rolls this globe around,\n     Moves wheresoe'er thy plastic influence guides,\n     And, ductile, owns the god whose arm presides.\n     The lightnings are thy ministers of ire;\n     The double-forked and ever-living fire;\n     In thy unconquerable hands they glow,\n     And at the flash all nature quakes below.\n     Thus, thunder-armed, thou dost creation draw\n     To one immense, inevitable law:\n     And, with the various mass of breathing souls,\n     Thy power is mingled, and thy spirit rolls.\n     Dread genius of creation! all things bow\n     To thee: the universal monarch thou!\n\n     Nor aught is done without thy wise control,\n     On earth, or sea, or round the ethereal pole,\n     Save when the wicked, in their frenzy blind,\n     Act o'er the follies of a senseless mind,\n     Thou curb'st th' excess; confusion, to thy sight,\n     Moves regular; th' unlovely scene is bright.\n     Thy hand, educing good from evil, brings\n     To one apt harmony the strife of things.\n     One ever-during law still binds the whole,\n     Though shunned, resisted, by the sinner's soul.\n     Wretches! while still they course the glittering prize\n     The law of God eludes their ears and eyes.\n     Life, then, were virtue, did they thus obey;\n     But wide from life's chief good they headlong stray.\n     Now glory's arduous toils the breast inflame;\n     Now avarice thirsts, insensible of shame;\n     Now sloth unnerves them in voluptuous ease,\n     And the sweet pleasures of the body please.\n     With eager haste they rush the gulf within,\n     And their whole souls are centred in their sin.\n     But, oh, great Jove! by whom all good is given!\n     Dweller with lightnings and the clouds of heaven!\n     Save from their dreadful error lost mankind!\n     Father! disperse these shadows of the mind!\n     Give them thy pure and righteous law to know;\n     Wherewith thy justice governs all below.\n     Thus honored by the knowledge of thy way,\n     Shall men that honor to thyself repay;\n     And bid thy mighty works in praises ring,\n     As well befits a mortal's lips to sing:\n     More blest, nor men, nor heavenly powers can be,\n     Than when their songs are of thy law and thee.[839]\n\n[Footnote 839: Sir C. A. Elton's version, published in \"Specimens of\nAncient Poets,\" edited by William Peters, A. M., Christ Church, Oxford.]\n\nPSYCHOLOGY.\n\nAs in the world there are two principles, the passive and the active, so\nin the understanding there are two elements: a passive\nelement--_sensation_, and an active element--_reason_.\n\nAll knowledge commences with the phenomena of sensation (aisthesis).\nThis produces in the soul an image (phantasia), which corresponds to the\nexterior object, and which Chrysippus regarded as a modification of the\nmind (alloiosis).[840]\n\nAssociate with sensibility is thought--the faculty of general ideas--the\northos logos, or right reason, as the supreme power and the guiding\nlight of humanity. This active principle is of divine origin, \"a part or\nshred of the Divinity.\"\n\n[Footnote 840: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. xxxiv.]\n\nThis \"right reason,\" or \"common reason,\" is the source and criterion of\nall truth; \"for our individual natures are all parts of the universal\nnature,\" and, therefore, all the dictates of \"common reason\" are\n\"identical with that right reason which pervades every thing, being the\nsame with Jupiter, who is the regulator and chief manager of all\nthings.\"\n\nThe fundamental canon of the logic of the Stoics, therefore, was that\n\"what appears to all, that is to be believed, for it is apprehended by\nthe reason, which is common and Divine.\"\n\nIt is needless to remark that the Stoics were compelled by their\nphysiological theory to deny the proper immortality of the soul. Some of\nthem seem to have supposed that it might, for a season, survive the\ndeath of the body, but its ultimate destination was absorption into the\nDivine essence. It must return to its original source.\n\nETHICS.\n\nIf reason be the great organizing and controlling law of the universe,\nthen, to live conformable to reason is the great practical law of life.\nAccordingly, the fundamental ethical maxim of the Stoics is, \"Live\nconformably with nature--that is, with reason, or the will of the\nuniversal governor and manager of all things.\"[841] Thus the chief good\n(eudaimonia) is the conformity of man's actions to reason--that is, to\nthe will of God, \"for nothing is well done without a reference to\nGod.\"[842]\n\n[Footnote 841: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. liii.]\n\n[Footnote 842: Marcus Aurelius, bk. iii. Sec. II.]\n\nIt is obvious that this doctrine must lead to a social morality and a\njurisprudence the very opposite of the Epicurean. If we must do that\nwhich is good--that is, that which is reasonable, regardless of all\nconsequences, then it is not for the pleasurable or useful results which\nflow from it that justice should be practised, but because of its\nintrinsic excellence. Justice is constituted good, not by the law of\nman, but by the law of God. The highest pleasure is to do right; \"this\nvery thing is the virtue of the happy man, and the perfect happiness of\nlife, when every thing is done according to a harmony of the genius of\neach individual to the will of the Universal Governor and Manager of all\nthings.\"[843] Every thing which interferes with a purely rational\nexistence is to be eschewed; the pleasures and pains of the body are to\nbe despised. To triumph over emotion, over suffering, over passion; to\ngive the fullest ascendency to reason; to attain courage, moral energy,\nmagnanimity, constancy, was to realize true manhood, nay, \"to be\ngodlike; for they have something in them which is, as it were, a\ngod\"[844]\n\nThe sublime heroism of the Stoic school is well expressed in the manly\nprecept, \"Anechou\"--_sustine_--endure. \"Endure the sorrows engendered by\nthe bitter struggle between the passions support all the evils which\nfortune shall send thee--calumny, betrayal, poverty, exile, irons, death\nitself.\" In Epictetus and Marcus Aurelius this spirit seems to rise\nalmost to the grandeur of Christian resignation. \"Dare to lift up thine\neyes to God and say, 'Use me hereafter to whatsoever thou pleasest. I\nagree, and am of the same mind with thee, indifferent to all things.\nLead me whither thou pleasest. Let me act what part thou wilt, either of\na public or a private person, of a rich man or a beggar.'\"[845] \"Show\nthose qualities,\" says Marcus Aurelius, \"which God hath put in thy\npower--sincerity, gravity, endurance of labor, aversion to pleasure,\ncontentment with thy portion and with few things, benevolence,\nfrankness, no love of superfluity, freedom from trifling,\nmagnanimity.\"[846]\n\n[Footnote 843: Diogenes Laertius, \"Lives of the Philosophers,\" bk. vii.\nch. liii.]\n\n[Footnote 844: Id., ib., bk. vii. ch. xliv.]\n\n[Footnote 845: Arrian, \"Diss. Epict.,\" bk. ii. ch. xviii.]\n\n[Footnote 846: \"I read to-day part of the 'Meditations of Marcus\nAntonius' [Aurelius]. What a strange emperor! And what a strange\nheathen! Giving thanks to God for all the good things he enjoyed! In\nparticular for his good inspirations, and for twice revealing to him, in\ndreams, things wherby he was cured of (otherwise) incurable distempers.\nI make no doubt but this is one of the 'many' who shall come from the\neast and the west, and sit down with Abraham, Isaac, and Jacob,' while\nthe 'children of the kingdom'--nominal Christians--are 'shut\nout.'\"--Wesley's \"Journal,\" vol. i, p. 353.]\n\nAmid the fearful moral degeneracy of imperial Rome, Stoicism became the\nrefuge of all noble spirits. But, in spite of its severity, and its\napparent triumph over the feelings, it brought no real freedom and\npeace. \"Stoical morality, strictly speaking, is, at bottom, only a\nslavish morality, excellent in Epictetus; admirable still, but useless\nto the world, in Marcus Aurelius.\" Pride takes the place of real\ndisinterestedness. It stands alone in haughty grandeur and solitary\nisolation, tainted with an incurable egoism. Disheartened by its\nmetaphysical impotence, which robs God of all personality, and man of\nall hope of immortality; defeated in its struggle to obtain purity of\nsoul, it sinks into despair, and often terminates, as in the case of its\ntwo first leaders, Zeno and Cleanthes, and the two Romans, Cato and\nSeneca, in self-murder. \"Thus philosophy is only an apprenticeship of\ndeath, and not of life; it tends to death by its image, _apathy_ and\n_ataraxy._\"[847]\n\n[Footnote 847: Cousin's \"Lectures on the History of Philosophy,\" vol. i.\np. 439.]\n\n\n\n\nCHAPTER XIV.\n\nTHE PROPAEDEUTIC OFFICE OF GREEK PHILOSOPHY.\n\n\n\"Philosophy, before the coming of the Lord, was necessary to the Greeks\nfor righteousness, and it now proved useful for godliness, being in some\npart a preliminary discipline (propaideia tis ousa) for those who reap\nthe fruits of faith through demonstration. Perhaps we may say it was\ngiven to the Greeks with this special object; for philosophy was to the\nGreeks what the Law was to the Jews, 'a schoolmaster to bring them to\nChrist.'\"--CLEMENS ALEXANDRINUS.\n\nPhilosophy, says Cousin, is the effort of _reflection_--the attempt of\nthe human mind to develop in systematic and logical form that which has\ndimly revealed itself in the spontaneous thought of ages, and to account\nto itself in some manner for its native and instinctive beliefs. We may\nfurther add, it is the effort of the human mind to attain to truth and\ncertitude on purely rational grounds, uncontrolled by traditional\nauthorities. The sublime era of Greek philosophy was, in fact, an\nindependent effort of human reason to solve the great problems of\nexistence, of knowledge, and of duty. It was an attempt to explain the\nphenomenal history of the universe, to interpret the fundamental ideas\nand laws of human reason, to comprehend the utterances of conscience,\nand to ascertain what Ultimate and Supreme Reality underlies the world\nof phenomena, of thought, and of moral feeling.[848] And it is this\nwhich, for us, constitutes its especial value; that it was, as far as\npossible, a result of simple reason; or, if at any time Faith asserted\nits authority, the distinction is clearly marked: If this inquiry was\nfully, and honestly, and logically conducted, we are entitled to presume\nthat the results attain by this effort of speculative thought must\nharmonize with the positive utterances of the Divine Logos--the Eternal\nReason, whose revelations are embalmed and transmitted to us in the Word\nof God. If the great truth that man is \"the _offspring of God\"_ and as\nsuch \"_the image and glory of God_\" which is asserted, alike, by Paul\nand the poet-philosophers of Tarsus and Mysia, be admitted, then we may\nexpect that the reason of man shall have some correlation with the\nDivine reason. The mind of man is the _chef-d'oeuvre_ of Divine art. It\nis fashioned after the model which the Divine nature supplies. \"Let us\nmake man in _our_ image after _our_ likeness.\" That image consists in\nepignosis--_knowledge;_ dikalosyne--_justice_; and\nosiotes--_benevolence._ It is not merely the _capacity_ to know, to be\njust, and to be beneficent; it is _actual_ knowledge, justice, and\nbenevolence. It supposes, first, that the fundamental ideas of the true,\nthe just, and the good, are connate to the human mind; second, that the\nnative determination of the mind is towards the realization of these\nideas in every mental state and every form of human activity; third,\nthat there is a constitutional sympathy of reason with the ideas of\ntruth, and righteousness, and goodness, as they dwell in the reason of\nGod. And though man be now fallen, there is still within his heart some\nvestige of his primal nature. There is still a sense of the divine, a\nreligious aptitude, \"a feeling after God,\" and some longing to return to\nHim. There are still ideas in the reason, which, in their natural and\nlogical development compel him to recognize a God. There is within his\nconscience a sense of duty, of obligation, and accountability to a\nSuperior Power--\"a law of the mind,\" thought opposed and antagonized by\ndepraved passions and appetites--\"the law in the members.\" There is yet\na natural, constitutional sympathy of reason with the law of God--\"it\ndelights in that law,\" and consents \"that it is good,\" but it is\noverborne and obstructed by passion. Man, even as unregenerate, \"wills\nto do that which is good,\" but \"how to perform that which is good he\nfinds not,\" and in the agony of his soul he exclaims, \"Oh, wretched man\nthat I am, who shall deliver me!\"[849]\n\n[Footnote 848: Plato sought also to attain to the Ultimate Reality\nunderlying all aesthetic feeling--the Supreme Beauty as well as the\nSupreme Good.]\n\n[Footnote 849: Romans, ch. vii.]\n\nThe Author of nature is also the Author of revelation. The Eternal\nFather of the Eternal Son, who is the grand medium of all God's direct\ncommunications to our race--the revealer of God, is also \"the Father of\nthe spirits of all flesh.\" That divine inbreathing which first\nconstituted man \"a living soul\"--that \"inspiration of the Almighty\nwhich giveth man understanding,\" and still \"teacheth him knowledge,\"\nproceeds from the same Spirit as that which inspires the prophets and\nseers of the Old Testament Church, and the Apostles and teachers of the\nnew. That \"true light which lighteth every man that cometh into the\nworld\" shone on the mind of Anaxagoras, and Socrates, and Plato, as well\nas on the mind of Abraham and Rahab, Cornelius and the Syro-Phoenician\nwoman, and, in a higher form, and with a clearer and richer effulgence,\non the mind of Moses, Isaiah, Paul and John. It is not to be wondered\nat, then, if, in the teaching of Socrates and Plato, we should find a\nstriking _harmony_ of sentiment, and even form of expression, with some\nparts of the Christian revelation. No short-sighted jealousy ought to\nimpugn the honesty of our judgment, if, in the speculations of Plato, we\ncatch glimpses of a world of ideas not unlike that which Christianity\ndiscloses, and hear words not unfamiliar to those who spake as they were\nmoved by the Holy Ghost.\n\nIf, then, there exists some correlation between Divine and human reason,\nand if the light which illuminates all minds in Christian and in heathen\nlands is the _same_ \"true light,\" though differing in degrees of\nbrightness, it is most natural and reasonable to expect some connection\nand some correspondence between the discoveries of philosophy and the\nrevelations of the Sacred Oracles.\n\nAlthough Christianity is confessedly something which is above reason and\nnature--something communicated from above, and therefore in the fullest\nsense supernatural and superhuman, yet it must stand in _relation_ to\nreason and nature, and to their historic development; otherwise it could\nnot operate on man at all. \"We have no knowledge of a dynamic influence,\nspiritual or natural, without a dynamic reaction.\" Matter can only be\nmoved by forces, and according to laws, as it has properties which\ncorrelate it with these forces and laws. And mind can not be determined\nfrom without to any specific form of cognition, unless it have powers of\napprehension and conception which are governed by uniform laws. If man\nis to be instructed by a verbal revelation, he must, at least, be\ncapacitated for the reception of divine communication--must have a power\nof forming supersensuous conceptions, and there must be some original\ncommunity of thought and idea between the mind that teaches and the mind\nthat is taught. A revelation from an invisible God--a being \"whom no man\nhas ever seen or ever can see\" with the eye of sense--would have no\naffinity for, and no power to affect and enlighten, a being who had no\npresentiment of an invisible Power to which he is in some way related. A\nrevealed law promulgated from an unseen and utterly unknown Power would\nhave no constraining authority, if man had no idea of right, no sense of\nduty, no feeling of obligation to a Supreme Being. If, therefore,\nreligious instruction be not already preceded by an innate consciousness\nof God, and of obligation to God, as an operative predisposition, there\nwould be nothing for revelation to act upon. Some relation between the\nreason which planned the universe, and which has expressed its thoughts\nin the numerical relations and archetypal forms which are displayed\ntherein, and the reason of man, with its ideas of form and number,\nproportion and harmony, is necessarily supposed in the statement of Paul\nthat \"the invisible things of God from the creation are seen.\" Nature to\nus could be no symbol of the Divine Thought, if there were no\ncorrelation between the reason of man and the reason of God. All\nrevelation, indeed, supposes some community of nature, some affinities\nof thought, some correlation of ideas, between the mind communicating\nspiritual knowledge, and the mind to which the communication is made. In\napproaching man, it must traverse ground already occupied by man; it\nmust employ phrases already employed, and assume forms of thought\nalready familiar to man. It must address itself to some ideas,\nsentiments, and feelings already possessed by man. If religion is the\ngreat end and destination of man, then the nature of man must be\nconstituted for religion. Now religion, in its inmost nature, is a\ncommunion, a fellowship with God. But no creature can be brought into\nthis communion \"save one that is constitutionally related to God in\nterms that admit of correspondence.\" There must be intelligence offered\nto his intelligence, sentiment to his sentiment, reason to his reason,\nthought to his thought. There must be implanted in the human mind some\nfundamental ideas and determinations grounded upon this fact, that the\nreal end and destination of man is for religion, so that when that\nhigher sphere of life and action is presented to man, by an outward\nverbal revelation, there shall be a recognized harmony between the inner\nidea and determination, and the outer revelation. We can not doubt that\nsuch a relation between human nature and reason, and Christianity,\nexists. We see evidences of this in the perpetual strivings of humanity\nto attain to some fuller and clearer apprehension of that Supreme Power\nwhich is consciously near to human thought, and in the historic\ndevelopment of humanity towards those higher forms of thought and\nexistence which demand a revelation in order to their completion. This\noriginal capacity, and this historical development, have unquestionably\nprepared the way for the reception of Christianity.\n\nChristianity, then, must have some connection with the reason of man,\nand it must also have some relation to the progressive developments of\nhuman thought in the ages which preceded the advent of Christ.\nChristianity did not break suddenly upon the world as a new commencement\naltogether unconnected with the past, and wanting in all points of\nsympathy and contact with the then present. It proceeded along lines of\nthought which had been laid through ages of preparation; it clothed\nitself in forms of speech which had been moulded by centuries of\neducation, and it appropriated to itself a moral and intellectual\nculture which had been effected by long periods of severest discipline.\nIt was, in fact, the consummation of the whole moral and religious\nhistory of the world.\n\nA revelation of new truths, presented in entirely new forms of thought\nand speech, would have defeated its own ends, and, practically, would\nhave been no revelation at all. The divine light, in passing through\nsuch a medium, would have been darkened and obscured. The lens through\nwhich the heavenly rays are to be transmitted must first be prepared and\npolished. The intellectual eye itself must be gradually accustomed to\nthe light. Hence it is that all revelation has been _progressive_,\ncommencing, in the infancy of our race, with images and symbols\naddressed to sense, and advancing, with the education of the race, to\nabstract conceptions and spiritual ideas. The first communications to\nthe patriarchs were always accompanied by some external, sensible\nappearance; they were often made through some preternatural personage in\nhuman form. Subsequently, as human thought becomes assimilated to the\nDivine idea, God uses man as his organ, and communicates divine\nknowledge as an internal and spiritual gift. The theistic conception of\nthe earliest times was therefore more or less anthropomorphic, in the\nprophetic age it was unquestionably more spiritual. The education of\nHebraic, Mosaic, and prophetic ages had gradually developed a purer\ntheism, and prepared the Jewish mind for that sublime announcement of\nour Lord's--\"God is a spirit, and they that worship him must worship in\nspirit.\" For ages the Jews had worshipped in Samaria and Jerusalem, and\nthe inevitable tendency of thought was to localize the divine presence;\nbut the gradual withdrawment from these localities of all visible tokens\nof Jehovah's presence, prepared the way for the Saviour's explicit\ndeclaration that \"neither in this mountain of Samaria, nor yet at\nJerusalem, shall men worship the Father,\" to the exclusion of any other\nspot on earth; the real temple of the living God is now the heart of\nman. The _Holiness_ of God was an idea too lofty for human thought to\ngrasp at once. The light of God's ineffable purity was too bright and\ndazzling to burst at once on human eyes. Therefore it was gradually\ndisplayed. The election of a chosen seed in Abraham's race to a nearer\napproach to God than the rest of pagan humanity; the announcement of the\nDecalogue at Sinai amidst awe-inspiring wonders; the separation of a\nsingle tribe to the priestly office, who were dedicated to, and purified\nin an especial manner for the service of the tabernacle; the\nsanctification of the High-priest by sacrifice and lustration before he\ndared to enter \"the holiest place\"--the presence-chamber of Jehovah: and\nthen the direct and explicit teaching of the prophets--were all\nadvancing steps by which the Jewish mind was lifted up to the clearer\napprehension of the holiness of God, the impurity of man, the distance\nof man from God, and the need of Mediation.\n\nThe ideas of _Redemption_ and _Salvation_--of atonement, expiation,\npardon, adoption, and regeneration--are unique and _sui-generis_. Before\nthese conceptions could be presented in the fullness and maturity of the\nChristian system, there was needed the culture and education of the ages\nof Mosaic ritualism, with its sacrificial system, its rights of\npurification, its priestly absolution, and its family of God.[850]\nRedemption itself, as an economy, is a development, and has\nconsequently, a history--a history which had its commencement in the\nfirst Eden, and which shall have its consummation in the second Eden of\na regenerated world. It was germinally infolded in the first promise,\ngradually unfolded in successive types and prophecies, more fully\ndeveloped in the life, and sayings, and sufferings of the Son of God,\nand its ripened fruit is presented to the eye of faith in the closing\nscenic representations of the grand Apocalypse of John. \"Judaism was not\ngiven as a perfect religion. Whatever may have been its superiority over\nsurrounding forms of worship, it was, notwithstanding, a provisional\nform only. The consciousness that it was a preparatory, and not a\ndefinite dispensation, is evident throughout. It points to an end beyond\nitself, suggests a grander thought than any in itself; its glory\nprecisely consists in its constant looking forward to a glorious future\ndestined to surpass it.\"[851]\n\n[Footnote 850: Romans, IX 4-6.]\n\n[Footnote 851: Pressense, \"Religions before Christ,\" p. 202.]\n\nThus the determinations which, through Redemption, fall to the lot of\nhistory, as Nitzsch justly remarks, obey the emancipating law of\n_gradual progress_.[852] Christianity was preceded by ages of\npreparation, in which we have a gradual development of religious phrases\nand ideas, of forms of social life and intellectual culture, and of\nnational and political institutions most favorable to its advent and its\npromulgation; and \"in the fullness of time\"--the maturity and fitness of\nthe age--\"God sent his own Son into the world.\"\n\n[Footnote 852: \"System of Doctrine,\" p. 73.]\n\nThis work of preparation was not confined alone to Judaism. The divine\nplan of redemption comprehended all the race; its provisions are made in\nview of the wants of all the race; and we must therefore believe that\nthe entire history of the race, previous to the coming of the Redeemer,\nwas under a divine supervision, and directed towards the grand centre of\nour world's history. Greek philosophy and Grecian civilization must\ntherefore have a place in the divine plan of history, and they must\nstand in an important relation to Christianity. He who \"determined the\ntime of each nation's existence, and fixed the geographical boundaries\nof their habitation in order that they may seek the Lord,\" can not have\nbeen unmindful of the Greek nation, and of its grandest age of\nphilosophy. \"The Father of the spirits of all flesh\" could not be\nunconcerned in the moral and spiritual welfare of any of his children.\nHe was as deeply interested in the Athenian as in the Hebrew. He is the\nGod of the Gentile as well as the Jew. His tender mercies are over all\nhis works. If the Hebrew race was selected to be the agent of his\nprovidence in one special field, and if the Jewish theocracy was one\ngrand instrument of preparatory discipline, it was simply because,\nthrough these, God designed to bless all the nations of the earth. And\nsurely no one will presume to say that a civilization and an\nintellectual culture which was second only to the Hebrew, and, in some\nof its aspects, even in advance of the Hebrew, was not determined and\nsupervised by Divine Providence, and made subservient to the education\nand development of the whole race. The grand results of Hebrew\ncivilization were appropriated and assimilated by Christianity, and\nremain to this day. And no one can deny that the same is true of Greek\ncivilization. Through a kind of historic preparation the heathen world\nwas made ready for Christ, as a soil is prepared to receive the seed,\nand some precious fruits of knowledge, of truth, and of righteousness,\neven, were largely matured, which have been reaped, and appropriated,\nand vitalized by the heaven-descended life of Christianity.\n\nThe chief points of excellence in the civilization of the Greeks are\nstrikingly obvious, and may be readily presented. High perfection of the\nintellect and the imagination displaying itself in the various forms of\nart, poetry, literature, and philosophy. A wonderful freedom and\nactivity of body and of mind, developed in trade, and colonization, in\nmilitary achievement, and in subtile dialectics. A striking love of the\nbeautiful, revealing itself in their sculpture and architecture, in the\nfree music of prosaic numbers, and the graceful movement and measure of\ntheir poetry. A quickness of perception, a dignity of demeanor, a\nrefinement of taste, a delicacy of moral sense, and a high degree of\nreverence for the divine in nature and humanity. And, in general, a ripe\nand all-pervading culture, which has made Athens a synonym for all that\nis greatest and best in the genius of man; so that literature, in its\nmost flourishing periods has rekindled its torch at her altars, and art\nhas looked back to the age of Pericles for her purest models.[853] All\nthese enter into the very idea of Greek civilization. We can not resist\nthe conviction that, by a Divine Providence, it was made subservient to\nthe purpose of Redemption; it prepared the way for, and contributed to,\nthe spread of the Gospel.\n\n[Footnote 853: In Lord Brougham's celebrated letter to the father of the\nhistorian Macaulay in regard to the education of the latter, we read:\n\"If he would be a great orator, he must go at once to the fountain-head,\nand be familiar with every one of the great orations of Demosthenes....\nI know from experience that nothing is half so successful in these times\n(bad though they be) as what has been formed on the Greek models. I use\npoor illustrations in giving my own experience, but I do assure you that\nboth in courts and Parliament, and even to mobs, I have never made so\nmuch play (to use a very modern phrase) as when I was almost translating\nfrom the Greek. I composed the peroration of my speech for the Queen, in\nthe Lords, after reading and repeating Demosthenes for three or four\nweeks.\"]\n\nIts subserviency to this grand purpose is seen in the Greek tendency to\ntrade and colonization. Their mental activity was accompanied by great\nphysical freedom of movement. They displayed an inherent disposition to\nextensive emigration. \"Without aiming at universal conquest, they\ndeveloped (if we may use the word) a remarkable catholicity of\ncharacter, and a singular power of adaptation to those whom they called\nBarbarians. In this respect they were strongly contrasted with the\nEgyptians, whose immemorial civilization was confined to the long valley\nwhich extended from the cataracts to the mouth of the Nile. The Hellenic\ntribes, on the other hand, though they despised the foreigners, were\nnever unwilling to visit them and to cultivate their acquaintance. At\nthe earliest period at which history enables us to discover them, we see\nthem moving about in their ships on the shores and among the islands of\ntheir native seas; and, three or four centuries before the Christian\nera, Asia Minor, beyond which the Persians had not been permitted to\nadvance, was bordered by a fringe of Greek colonies; and lower Italy,\nwhen the Roman Republic was just becoming conscious of its strength, had\nreceived the name of Greece itself. To all these places they carried\ntheir arts and literature, their philosophy, their mythology, and their\namusements.... They were gradually taking the place of the Phoenicians\nin the empire of the Mediterranean. They were, indeed, less exclusively\nmercantile than those old discoverers. Their voyages were not so long.\nBut their influence on general civilization was greater and more\npermanent. The earliest ideas of scientific navigation and geography are\ndue to the Greeks. The later Greek travellers, Pausanias and Strabo, are\nour best sources of information on the topography of St. Paul's\njourneys.\n\n\"With this view of the Hellenic character before us, we are prepared to\nappreciate the vast results of Alexander's conquests. He took the meshes\nof the net of Greek civilization which were lying in disorder on the\nedge of the Asiatic shore, and spread them over all the countries he\ntraversed in his wonderful campaigns. The East and the West were\nsuddenly brought together. Separate tribes were united under a common\ngovernment. New cities were built as the centres of political life. New\nlines of communication were opened as the channels of commercial\nactivity. The new culture penetrated the mountain ranges of Pisidia and\nLycaonia. The Tigris and Euphrates became Greek rivers. The language of\nAthens was heard among the Jewish colonies of Babylonia, and a Grecian\nBabylon was built by the conqueror in Egypt, and called by his name.\n\n\"The empire of Alexander was divided, but the effects of his campaigns\nand policy did not cease. The influence of these fresh elements of\nsocial life was rather increased by being brought into independent\naction within the sphere of distinct kingdoms. Our attention is\nparticularly directed to two of the monarchical lines which descended\nfrom Alexander's generals--the Ptolemies, or the Greek kings of Egypt,\nand the Seleucidae, or the Greek kings of Syria. Their respective\ncapitals, Alexandria and Antioch, became the metropolitan centres of\ncommercial and civilized life in the East.\"[854] Antioch was for ages\nthe home of science and philosophy. Here the religious opinions of the\nEast and the West were blended and mutually modified. Here it was\ndiscovered by the heathen mind that a new religion had appeared, and a\nnew revelation had been given.[855] In Alexandria all nations were\ninvited to exchange their commodities and, with equal freedom, their\nopinions. The representatives of all religions met here. \"Beside the\nTemple of Jupiter there rose the white marble Temple of Serapis, and\nclose at hand stood the synagogue of the Jews.\" The Alexandrian library\ncontained all the treasures of ancient culture, and even a copy of the\nHebrew Scriptures.\n\n[Footnote 854: Conybeare and Howson, \"Life and Epistles of St. Paul,\"\nvol. i. pp. 8-10.]\n\n[Footnote 855: Acts, xi. 26.]\n\nThe spread of the Greek _language_ was one of the most important\nservices which the cities of Antioch and Alexandria rendered to\nChristianity. The Greek tongue is intimately connected with the whole\nsystem of Christian doctrine.\n\nThis language, which, in symmetry of structure, in flexibility and\ncompass of expression, in exactness and precision, in grace and\nelegance, exceeds every other language, became the language of theology.\nNext in importance to the inspiration which communicates the superhuman\nthought, must be the gradual development of the language in which the\nthought can clothe itself. That development by which the Greek language\nbecame the adequate vehicle of Divine thought, the perfect medium of the\nmature revelation of truth contained in the Christian Scriptures, must\nbe regarded as the subject of a Divine providence. Christianity waited\nfor that development, and it awaited Christianity. \"The Greek tongue\nbecame to the Christian more than it had been to the Roman or the Jew.\nThe mother-tongue of Ignatius at Antioch was that in which Philo\ncomposed his treatises at Alexandria, and which Cicero spoke at Athens.\nIt is difficult to state in a few words the important relation which\nAlexandria, more especially, was destined to bear to the whole Christian\nChurch.\" In that city, the Old Testament was translated into Greek;\nthere the writings of Plato were diligently studied; there Philo, the\nPlatonizing Jew, had sought to blend into one system the teachings of\nthe Old Testament theology and the dialectic speculations of Plato.\nNumenius learns of Philo, and Plotinus of Numenius, and the ecstasy of\nPlotinus is the development of Philo's intuitions. A _theological\nlanguage_ by this means was developed, rich in the phrases of various\nschools, and suited to convey the spiritual revelation of Christian\nideas to all the world. \"It was not an accident that the New Testament\nwas written in Greek, the language which can best express the highest\nthoughts and worthiest feelings of the intellect and heart, and which is\nadapted to be the instrument of education for all nations; nor was it an\naccident that the composition of these books and the promulgation of the\nGospels were delayed till the instruction of our Lord, and the writings\nof his Apostles could be expressed in the dialect [of Athens and] of\nAlexandria.\"[856] This must be ascribed to the foreordination of Him\nwho, in the history of nations and of civilizations, \"worketh all things\naccording to the counsel of his own will.\"\n\n[Footnote 856: Conybeare and Howson, \"Life and Epistles of St. Paul,\"\nvol. i. p. 10.]\n\nNow it is the doctrine of the best philologists that language is a\n_growth_. Gradually, and by combined efforts of successive generations,\nit has been brought to the perfection which we so much admire in the\nidioms of the Bible, the poetry of Homer, Dante, and Shakspeare, and the\nprose compositions of Demosthenes, Cicero, Johnson, and Macaulay. The\nmaterial or root-element of language may have been the product of mental\ninstinct, or perhaps the immediate gift of God by revelation; but the\nformal element must have been the creation of thought, and the result of\nrational combination. Language is really the incarnation of thought;\nconsequently the growth of a language, its affluence, comprehension, and\nfullness must depend on the vigor and activity of thought, and the\nacquisition of general ideas. Language is thus the best index of\nintellectual progress, the best standard of the intellectual attainment\nof an age or nation. The language of barbaric tribes is exceedingly\nsimple and meagre; the paucity of general terms clearly indicating the\nabsence of all attempts at classification and all speculative thought.\nWhilst the language of educated peoples is characterized by great\nfullness and affluence of terms, especially such as are expressive of\ngeneral notions and abstract ideas. All grammar, all philology, all\nscientific nomenclature are thus, in fact, _psychological deposits_,\nwhich register the progressive advancement of human thought and\nknowledge in the world of mind, as the geological strata bear testimony\nto the progressive development of the material world. \"Language,\" says\nTrench, \"is fossil poetry, fossil history,\" and, we will add, fossil\nphilosophy. Many a single word is a concentrated poem. The record of\ngreat social and national revolutions is embalmed in a single term.[857]\nAnd the history of an age of philosophic thought is sometimes condensed\nand deposited in one imperishable word.[858]\n\n[Footnote 857: See Trench \"On the Study of Words,\" p. 20, where the word\n\"frank\" is given as an illustration.]\n\n[Footnote 858: For example, the cosmos of the Pythagoreans, the eide of\nthe Platonists, and the ataraxia of the Stoics.]\n\nIf, then, language is the creation of thought, the sensible vesture with\nwhich it clothes itself, and becomes, as it were, incarnate--if the\nperfection and efficiency of language depends on the maturity and\nclearness of thought, we conclude that the wonderful adequacy and\nfitness of the Greek language to be the vehicle of the Divine thought,\nthe medium of the most perfect revelation of God to men, can only be\nexplained on the assumption that the ages of philosophic thought which,\nin Greece, preceded the advent of Christianity, were under the immediate\nsupervision of a providence, and, in some degree, illuminated by the\nSpirit of God.\n\nGreek philosophy must therefore have fulfilled a propaedeutic office for\nChristianity. \"As it had been intrusted to the Hebrews to preserve and\ntransmit the heaven-derived element of the Monotheistic religion, so it\nwas ordained that, among the Greeks, all seeds of human culture should\nunfold themselves in beautiful harmony, and then Christianity, taking up\nthe opposition between the divine and human, was to unite both in one,\nand show how it was necessary that both should co-operate to prepare for\nthe appearance of itself and the unfolding of what it contains.\"[859]\nDuring the period of Greek philosophy which preceded the coming of\nChrist, human reason, unfolding itself from beneath, had aspired after\nthat knowledge of divine things which is from above. It had felt within\nitself the deep-seated consciousness of God--the sporadic revelation of\nHim \"who is not far from any one of us\"--the immanent thought of that\nBeing \"in whom we live and move and are,\" and it had striven by analysis\nand definition to attain a more distinct and logical apprehension. The\nheart of man had been stirred with \"the feeling after God\"--the longing\nfor a clearer sense of the divine, and had struggled to attain, by\nabstraction or by ecstasy, a more immediate communion with God. Man had\nbeen conscious of an imperative obligation to conform to the will of the\ngreat Supreme, and he sought to interpret more clearly the utterances of\nconscience as to what duty was. He had felt the sense of sin and guilt,\nand had endeavored to appease his conscience by expiatory offerings, and\nto deliver himself from the power of sin by intellectual culture and\nmoral discipline. And surely no one, at all familiar with the history of\nthat interesting epoch in the development of humanity, will have the\nhardihood to assert that no steps were taken in the right direction, and\nno progress made towards the distant goal of human desire and hope. The\nlanguage, the philosophy, the ideals of moral beauty and excellence, the\nnoble lives and nobler utterances of the men who stand forth in history\nas the representatives of Greek civilization, all attest that their\nnoble aspiration and effort did not end in ignominious failure and utter\ndefeat. It is true they fell greatly beneath the realization of even\ntheir own moral ideals, and they became painfully conscious of their\nmoral weakness, as men do even in Christian times. They learned that,\nneither by intellectual abstraction, nor by ecstasy of feeling, could\nthey lift themselves to a living, conscious fellowship with God. The\nsense of guilt was unrelieved by expiations, penances, and prayers. And\nwhilst some cultivated a proud indifference, a Stoical apathy, and\nothers sank down to Epicurean ease and pleasure, there was a noble few\nwho longed and hoped with increasing ardor for a living Redeemer, a\npersonal Mediator, who should \"stand between God and man and lay his\nhand on both.\" Christ became in some dim consciousness \"the Desire of\nNations,\" and the Moral Law became even to the Greek as well as the Jew\n\"a school-master to lead them to Him.\"\n\n[Footnote 859: Neander's \"Church History,\" vol. i. p. 4.]\n\nThe arrival of Paul at Athens, in the close of this brilliant period of\nGreek philosophy, now assumes an aspect of deeper interest and\nprofounder significance. It was a grand climacteric in the life of\nhumanity--an epoch in the moral and religious history of the world. It\nmarked the consummation of a periodic dispensation, and it opened a new\nera in that wonderful progression through which an overruling Providence\nis carrying the human race. As the coming of the Son of God to Judea in\nthe ripeness of events--\"the fullness of time\"--was the consummation of\nthe Jewish dispensation, and the event for which the Jewish age had been\na preparatory discipline, so the coming of a Christian teacher to\nAthens, in the person of \"the Apostle of the Gentiles,\" was the\n_terminus ad quem_ towards which all the phases in the past history of\nphilosophic thought had looked, and for which they had prepared.\nChristianity was brought to Athens--brought into contact with Grecian\nphilosophy at the moment of its exhaustion--at the moment when, after\nages of unwearied effort, it had become conscious of its weakness, and\nits comparative failure, and had abandoned many questions in despair.\nGreek philosophy had therefore its place in the plan of Divine\nProvidence. It had a mission to the world; that mission was now\nfulfilled. If it had laid any foundation in the Athenian mind on which\nthe Christian system could plant its higher truths--if it had raised up\ninto the clearer light of consciousness any of those _ideas_ imbedded in\nthe human reason which are germane to Christian truth--if it had\nrevealed more fully the wants and instincts of the human heart, or if it\nhad attained the least knowledge of eternal truth and immutable right,\nupon this Christianity placed its _imprimatur_. And at those points\nwhere human reason had been made conscious of its own inefficiency, and\ncompelled to own its weakness and its failure, Christianity shed an\neffulgent and convincing light.\n\nTherefore the preparatory office of Greek religion and Greek philosophy\nis fully recognized by Paul in his address to the Athenians. He begins\nby saying that the observations he had made enabled him to bear witness\nthat the Athenians were indeed, in every respect, \"a God-fearing\npeople;\"--that the God whom they knew so imperfectly as to designate Him\n\"the Unknown,\" but whom \"they worshipped,\" was the God he worshipped,\nand would now more fully declare to them. He assures them that their\npast history, and their present geographical position, had been the\nobject of Divine foreknowledge and determination. \"He hath determined\nbeforehand the times of each nation's existence, and fixed the\ngeographical boundaries of their habitation,\" all with this specific\ndesign, that they might \"seek after,\" \"feel after,\" and \"find the Lord,\"\nwho had never been far from any one of them. He admits that their\npoet-philosophers had risen to a lofty apprehension of \"the Fatherhood\nof God,\" for they had taught that \"we are all his offspring;\" and he\nseems to have felt that in asserting the common brotherhood of our race,\nhe would strike a chord of sympathy in the loftiest school of Gentile\nphilosophy. He thus \"recognized the Spirit of God brooding over the face\nof heathenism, and fructifying the spiritual element in the heart even\nof the natural man. He feels that in these human principles there were\nsome faint adumbrations of the divine, and he looked for their firmer\ndelineation to the figure of that gracious Master, higher and holier\nthan man, whom he contemplated in his own imagination, and whom he was\nabout to present to them.\"[860]\n\n[Footnote 860: Merivale's \"Conversion of the Roman Empire,\" p. 78.]\n\nThis function of ancient philosophy is distinctly recognized by many of\nthe greatest of the Fathers, as Justin, Clement, Origen, Augustine, and\nTheodoret. Justin Martyr believed that a ray of the Divine Logos shone\non the mind of the heathen, and that the human soul instinctively turned\ntowards God as the plant turns towards the sun. \"Every race of men\nparticipated in the Word. And they who lived with the Word were\nChristians, even if they were held to be godless; as, for example, among\nthe Greeks, Socrates and Heraclitus, and those like them.\"[861] Clement\ntaught that \"philosophy, before the coming of the Lord, was necessary to\nthe Greeks for righteousness; and now it proved useful for godliness,\nbeing a sort of preliminary discipline for those who reap the fruits of\nfaith through demonstration.... Perhaps we may say that it was given to\nthe Greeks with this special object, for it brought the Greek nation to\nChrist as the Law brought the Hebrews.\"[862] \"Philosophy was given as a\npeculiar testament to the Greeks, as forming the basis of the Christian\nphilosophy.\"[863] Referring to the words of Paul, Origen says, the\ntruths which philosophers taught were from God, for \"God manifested\nthese to them, and all things that have been nobly said.\"[864] And\nAugustine, whilst deprecating the extravagant claims made for the great\nGentile teachers, allows \"that some of them made great discoveries, so\nfar as they received help from heaven; whilst they erred as far as they\nwere hindered by human frailty.\"[865] They had, as he elsewhere\nobserves, \"a distant vision of the truth, and learnt, from the teaching\nof nature, what prophets learnt from the spirit.\"[866] In addressing the\nGreeks, Theodoret says, \"Obey your own philosophers; let them be your\ninitiators; for they announced beforehand our doctrines.\" He held that\n\"in the depths of human nature there are characters inscribed by the\nhand of God.\" And that \"if the race of Abraham received the divine law,\nand the gift of prophecy, the God of the universe led other nations to\npiety by natural revelation, and the spectacle of nature.\"[867]\n\n[Footnote 861: \"First Apology,\" ch. xlvi.]\n\n[Footnote 862: \"Stromata,\" bk. i. ch. v.]\n\n[Footnote 863: \"Stromata,\" bk. vi. ch. viii.]\n\n[Footnote 864: \"Contra Celsum,\" bk. vi. ch. iii.]\n\n[Footnote 865: \"De Civitate Dei,\" bk. ii. ch. vii.]\n\n[Footnote 866: Sermon lxviii. 3.]\n\n[Footnote 867: See Smith's \"Bible Dictionary,\" article \"Philosophy;\"\nPressense, \"Religions before Christ,\" p. II; Butler's \"Lectures on\nAncient Philosophy,\" vol. ii. pp. 28-40.]\n\nIn attempting to account for this partial harmony between Philosophy and\nRevelation, we find the Patristic writers adopting different theories.\nThey are generally agreed in maintaining some original connection, but\nthey differ as to its immediate source. Some of them maintained that the\nancient philosophers derived their purest light from the fountain of\nDivine Revelation. The doctrines of the Old Testament Scriptures were\ntraditionally diffused throughout the West before the rise of\nphilosophic speculation. If the theistic conceptions of Plato are\nsuperior to those of Homer it is accounted for by his (hypothetical)\ntour of inquiry among the Hebrew nation, as well as his Egyptian\ninvestigations. Others maintained that the similarity of views on the\ncharacter of the Supreme Being and the ultimate destination of humanity\nwhich is found in the writings of Plato and the teachings of the Bible\nis the consequence of _immediate_ inspiration. Origen, Jerome, Eusebius,\nClement, do not hesitate to affirm that Christ himself revealed his own\nhigh prerogatives to the gifted Grecian. From this hypothesis, however,\nthe facts of the case compel them to make some abatements. In the\nmid-current of this divine revelation are found many acknowledged\nerrors, which it is impossible to ascribe to the celestial illuminator.\nPlato, then, was _partially_ inspired, and clouded the heavenly beam\nwith the remaining grossnesses of the natural sense.[868] Whilst a\nthird, and more reasonable, hypothesis was maintained by others. They\nregarded man as \"the offspring and image of the Deity,\" and maintained\nthere must be a correlation of the human and divine reason, and,\nconsequently, of all discovered truth to God. Therefore they expected to\nfind some traces of connection and correspondence between Divine and\nhuman thought, and some kindred ideas in Philosophy and Revelation.\n\"Ideas,\" says St. Augustine, \"are the primordial forms, as it were, the\nimmutable reason of things; they are not created, they are eternal, and\nalways the same: they are contained in the Divine intelligence and\nwithout being subject to birth and death, they are _types_ according to\nwhich is formed every thing that is born and dies.\" The copies of these\narchetypes are seen in nature, and are participated in by the reason of\nman; and there may therefore be some community of idea between man and\nGod, and some relation between Philosophy and Christianity.\n\n[Footnote 868: Butler's \"Lectures on Ancient Philosophy,\" vol. ii. p.\n41.]\n\nThe various attempts which have been made to trace the elevated theism\nand morality of Socrates and Plato to Jewish sources have signally\nfailed. Justin Martyr and Tertullian claim that the ancient philosophers\n\"borrowed from the Jewish prophets.\" Pythagoras and Plato are supposed\nto have travelled in the East in quest of knowledge.[869] The latter is\nimagined to have had access to an existing Greek version of the Old\nTestament in Egypt, and a strange oversight in chronology brings him\ninto personal intercourse with the prophet Jeremiah. A sober and\nenlightened criticism is compelled to pronounce all these statements as\nmere exaggerations of later times.[870] They are obviously mere\nsuppositions by which over-zealous Christians sought to maintain the\nsupremacy and authority of Scripture. The travels of Pythagoras are\naltogether mythical, the mere invention of Alexandrian writers, who\nbelieved that all wisdom flowed from the East.[871] That Plato visited\nEgypt at all, rests on the single authority of Strabo, who lived at\nleast four centuries after Plato; there is no trace in his own works of\nEgyptian research. His pretended travels in Phoenicia, where he gained\nfrom the Jews a knowledge of the true God, are more unreliable still.\nPlato lived in the fourth century before Christ (born B.C. 430), and\nthere is no good evidence of the existence of a Greek version of the Old\nTestament before that of \"the Seventy\" (Septuagint), made by order of\nPtolemy Philadelphus, B.C. 270. Jeremiah, the prophet of Israel, lived\ntwo centuries before Plato; consequently any personal intercourse\nbetween the two was simply impossible. Greek philosophy was\nunquestionably a development of Reason alone.[872]\n\n[Footnote 869: Mr. Watson adopts this hypothesis to account for the\ntheistic opinions of the ancient philosophers of Greece. See \"Institutes\nof Theology,\" vol. i. pp. 26-34.]\n\n[Footnote 870: Ritter's \"History of Ancient Philosophy,\" vol. ii. p.\n147.]\n\n[Footnote 871: Max Muller, \"Science of Language,\" p. 94.]\n\n[Footnote 872: See on this subject, Ritter's \"History of Ancient\nPhilosophy,\" vol. i. pp. 147, 148; Encyclopaedia Britannica, article\n\"Plato,\" vol. xvii. p. 787; Smith's \"Bible Dictionary,\" article\n\"Philosophy;\" and Thompson's \"Laws of Thought,\" p. 326.]\n\nSome of the ablest Christian scholars and divines of modern times, as\nCudworth, Neander, Trench, Pressense, Merivale, Schaff, after the most\ncareful and conscientious investigation, have come to this conclusion,\nthat Greek philosophy fulfilled a preparatory mission for Christianity.\nThe general conclusions they reached are forcibly presented in the words\nof Pressense:\n\nIt would be difficult to overstate the importance of Greek philosophy\nwhen viewed as a preparation to Christianity. Disinterested pursuit of\ntruth is always a great and noble task. The imperishable want of the\nhuman mind to go back to first principles, suffices to prove that this\nprinciple is divine. We may abuse speculation; we may turn it into one\nof the most powerful dissolvents of moral truths; and the defenders of\npositive creeds, alarmed by the attitude too often assumed by\nspeculation in the presence of religion, have condemned it as\nmischievous in itself, confounding in their unjust prejudice its use and\nits abuse. But, for all serious thinkers, philosophy is one of the\nhighest titles of nobility that humanity possesses: and when we consider\nits mission previous to Christianity, we feel convinced that it had its\nplace in the Divine plan. It was not religion in itself that philosophy,\nthrough its noblest representatives, combated, but polytheism. It\ndethroned the false gods. Adopting what was best in paganism, philosophy\nemployed it as an instrument to destroy paganism, and thus clear the way\nfor definite religion. Above all, it effectually contributed to purify\nthe idea of Divinity, though this purification was but an approximation.\nIf at times it caught glimpses of the highest spiritualism, yet it was\nunable to protect itself against the return and reaction of Oriental\ndualism. In spite of this imperfection, which in its way served the\ncause of Christianity by demonstrating the necessity of revelation, men\nlike Socrates and Plato fulfilled amongst their people a really sublime\nmission.\n\nThey were to the heathen world the great prophets of the human\nconscience, which woke up at their call. And the awakening of the moral\nsense was at once the glory and ruin of philosophy; for conscience, once\naroused, could only be satisfied by One greater than they, and must\nnecessarily reject all systems which proved themselves insufficient to\nrealize the moral idea they had evoked.\n\n\"But to perish thus, and for such a cause, is a high honor to a\nphilosophy. It was this made the philosophy of Greece, like the Hebrew\nlaws, though in an inferior sense, a schoolmaster that led to Jesus\nChrist, according to the expression of Clement of Alexandria. Viewed in\nthis light, it was a true gift of God, and had, too, the shadow of good\nthings to come, awakening the presentiment and desire of them, though it\ncould not communicate them. Nor can we conceive a better way to prepare\nfor the advent of Him who was to be 'the Desire of Nations' before\nbecoming their Saviour.\"[873]\n\n[Footnote 873: \"Religions before Christ,\" pp. 101, 102.]\n\nIn previous chapters we have endeavored to sketch the history of the\ndevelopment of metaphysical thought, of moral feeling and idea, and of\nreligious sentiment and want, which characterized Grecian civilization.\nIn now offering a brief _resume_ of the history of that development,\nwith the design of more fully exhibiting the preparatory office it\nfulfilled for Christianity, we shall assume that the mind of the reader\nhas already been furnished and disciplined by preparatory principles. He\ncan scarce have failed to recognize that this development obeyed a\n_general law_, however modified by exterior and geographical conditions;\nthe same law, in fact, which governs the development of all individual\nfinite minds, and which law may be formulated thus:--_All finite mind\ndevelops itself, first, in instinctive determinations and spontaneous\nfaiths; then in rising doubt, and earnest questioning, and ill-directed\ninquiry; and, finally, in systematic philosophic thought, and rational\nbelief_. These different stages succeed each other in the individual\nmind. There is, first, the simplicity and trust of childhood; secondly,\nthe undirected and unsettled force of youth; and, thirdly, the wisdom of\nmature age. And these different stages have also succeeded each other in\nthe universal mind of humanity. There has been, 1st. _The era of\nspontaneous beliefs_--of popular and semi-conscious theism, morality,\nand religion, 2d. _The transitional age_--the age of doubt, of inquiry,\nand of ill-directed mental effort, ending in fruitless sophism, or in\nskepticism. 3d. _The philosophic or conscious age_--the age of\nreflective consciousness, in which, by the analysis of thought, the\nfirst principles of knowledge are attained, the necessary laws of\nthought are discovered, and man arrives at positive convictions, and\nrational beliefs. In the history of Grecian civilization, the first is\nthe Homeric age; the second is the pre-Socratic age, ending with the\nSophists; and the third is the grand Socratic period. History is thus\nthe development of the fundamental elements of humanity, according to an\nestablished law, and under conditions which are ordained and supervised\nby the providence of God. \"The unity of civilization is in the unity of\nhuman nature; its varieties, in the variety of the elements of\nhumanity,\" which elements have been successively developed in the course\nof history. All that is fundamental in human nature passes into the\nmovement of civilization. \"I say all that is fundamental; for it is the\nexcellency of history to take out, and throw away all that is not\nnecessary and essential. That which is individual shines for a day, and\nis extinguished forever, or stops at biography.\" Nothing endures, except\nthat which is fundamental and true--that which is vital, and organizes\nitself, develops itself, and arrives at an historical existence.\n\"Therefore as human nature is the matter and basis of history, history\nis, so to speak, the judge of human nature, and historical analysis is\nthe counter-proof of psychological analysis.\"[874]\n\n[Footnote 874: Cousin's \"Lectures on the History of Philosophy,\" vol. i.\np. 31.] Nature, individual mind, and collective humanity, all obey the\nlaw of progressive development; otherwise there could be no history, for\nhistory is only of that which has movement and progress. Now, all\nprogress is from the indefinite to the definite, from the inorganic to\nthe organic and vital, from the instinctive to the rational, from a dim,\nnebulous self-feeling to a high reflective consciousness, from sensuous\nimages to abstract conceptions and spiritual ideas. This progressive\ndevelopment of nature and humanity has not been a series of creations\n_de novo_, without any relation, in matter or form, to that which\npreceded. All of the present was contained in embryonic infoldment in\nthe past, and the past has contributed its results to the present.[875]\nThe present, both in nature, and history, and civilization, is, so to\nspeak, the aggregate and sum-total of the past. As the natural history\nof the earth may now be read in the successive strata and deposits which\nform its crust, so the history of humanity may be read in the successive\ndeposits of thought and language, of philosophy and art, which register\nits gradual progression. As the paleontological remains imbedded in the\nrocks present a succession of organic types which gradually improve in\nform and function, from the first sea-weed to the palm-tree, and from\nthe protozoa to the highest vertebrate, so the history of ancient\nphilosophy presents a gradual progress in metaphysical, ethical, and\ntheistic conceptions, from the unreflective consciousness of the Homeric\nage, to the high reflective consciousness of the Platonic period. And as\nall the successive forms of life in pre-Adamic ages were a preparation\nfor and a prophecy of the coming of man, so the advancing forms of\nphilosophic thought, during the grand ages of Grecian civilization, were\na preparation and a prophecy of the coming of the Son of God.\n\n[Footnote 875: The writer would not be understood as favoring the idea\nthat this development is simply the result of \"natural law.\" The\nconnection between the past and the present is not a material, but a\n_mental_ connection. It is the bond of Creative Thought and Will giving\nto organic forces a foreseen direction towards the working out of a\ngrand plan. See Agassiz, \"Contributions to Natural History,\" vol. i. pp.\n9, 10; Duke of Argyll, \"Reign of Law,\" ch. v.]\n\nWe shall now endeavor to trace this process of gradual preparation for\nChristianity in the Greek mind--\n\n(i.) _In the field of_ THEISTIC _conceptions_.\n\n(ii.) _In the department of_ ETHICAL _ideas and principles_.\n\n(iii.) _In the region of_ RELIGIOUS _sentiment_.\n\nIn the field of theistic conception the propaedeutic office of Grecian\nphilosophy is seen--\n\nI. _In the release of the popular mind from Polytheistic notion, and the\npurifying and spiritualizing of the Theistic idea_.\n\nThe idea of a Supreme Power, a living Personality, energizing in nature,\nand presiding over the affairs of men, is not the product of philosophy.\nIt is the immanent, spontaneous thought of humanity. It has, therefore,\nexisted in all ages, and revealed itself in all minds, even when it has\nnot been presented to the understanding as a definite conception, and\nexpressed by human language in a logical form. It is the thought which\ninstinctively arises in the opening reason of childhood, as the dim and\nshadowy consciousness of a living mind behind all the movement and\nchange of the universe. Then comes the period of doubt, of anxious\nquestioning, and independent inquiry. The youth seeks to account to\nhimself for this peculiar sentiment. He turns his earnest gaze towards\nnature, and through this living vesture of the infinite he seeks to\ncatch some glimpses of the living Soul. In some fact appreciable to\nsense, in some phenomenon he can see, or hear, or touch, he would fain\ngrasp the cause and reason of all that is. But in this field of inquiry\nand by this method he finds only a \"receding God,\" who falls back as he\napproaches, and is ever still beyond; and he sinks down in exhaustion\nand feebleness, the victim of doubt, perhaps despair. Still the\nsentiment of the Divine remains, a living force, in the centre of his\nmoral being. He turns his scrutinizing gaze within, and by\nself-reflection seeks for some rational ground for his instinctive\nfaith. There he finds some convictions he can not doubt, some ideas he\ncan not call in question, some thoughts he is compelled to think, some\nnecessary and universal principles which in their natural and logical\ndevelopment ally him to an unseen world, and correlate and bind him fast\nto an invisible, but real God. The more his mind is disciplined by\nabstract thought, the clearer do these necessary and universal\nprinciples become, and the purer and more spiritual his ideas of God.\nGod is now for him the First Principle of all principles, the First\nTruth of all truths; the Eternal Reason, the Immutable Righteousness,\nthe Supreme Good. The normal and healthy development of reason, the\nmaturity of thought, conduct to the recognition of the true God.\n\nAnd so it has been in the universal consciousness of our race as\nrevealed in history. There was first a period of spontaneous and\nunreflective Theism, in which man felt the consciousness of God, but\ncould not or did not attempt a rational explanation of his instinctive\nfaith. He saw God in clouds and heard Him in the wind. His smile\nnourished the corn, and cheered the vine. The lightnings were the\nflashes of his vengeful ire, and the thunder was his angry voice. But\nthe unity of God was feebly grasped, the rays of the Divinity seemed\ndivided and scattered amidst the separate manifestations of power, and\nwisdom, and goodness, and retribution, which nature presented. Then\nplastic art, to aid and impress the imagination, created its symbols of\nthese separate powers and principles, chiefly in human form, and gods\nwere multiplied. But all this polytheism still rested on a dim\nmonotheistic background, and all the gods were subordinated to\nZeus--\"the Father of gods and men.\" Humanity had still the sense of the\ndependence of all finite being on one great fountain-head of\nIntelligence and Power, and all the \"generated gods\" were the subjects\nand ministers of that One Supreme. This was the childhood of humanity so\nvividly represented in Homeric poetry.\n\nThen came a period of incipient reflection, and speculative thought, in\nwhich the attention of man is drawn outward to the study of nature, of\nwhich he can yet only recognize himself as an integral part. He searches\nfor some arche--some first principle, appreciable to sense, which in its\nevolution shall furnish an explanation of the problem of existence. He\ntries the hypothesis of \"_water_\" then of \"_air_\" then of \"_fire_\" as\nthe primal element, which either is itself, or in some way infolds\nwithin itself an informing Soul, and out of which, by vital\ntransformation, all things else are produced. But here he failed to find\nan adequate explanation; his reason was not satisfied. Then he sought\nhis first principle in \"_numbers_\" as symbols, and, in some sense, as\nthe embodiment of the rational conceptions of order, proportion, and\nharmony,--God is the original _monas_--unity--One;--or else he sought it\nin purely abstract \"_ideas_\" as unity, infinity, identity, and all\nthings are the evolution of an eternal thought, one and identical, which\nis God. And here again he fails. Then he supposes an unlimited\n_migma_--a chaotic mixture of elements existing from eternity, which was\nseparated, combined, and organized by the energy of a Supreme Mind, the\n_nous_ of Anaxagoras. But he holds not firmly to this great principle;\n\"he recurs again to air, and ether, and water, as _causes_ for the\nordering of all things.\"[876] And after repeated attempts and failures,\nhe is disappointed in his inquiry, and falls a prey to doubt and\nskepticism. This was the early youth of our humanity, the period that\nopens with Thales and ends with the Sophists.\n\n[Footnote 876: Thus Socrates complains of Anaxagoras. See \"Phaedo,\" Sec.\n108.]\n\nThe problem of existence still waits for and demands a solution. The\nheart of man, also, still cries out for the living God. The Socratic\nmaxim, \"know thyself,\" introverts the mental gaze, and self-reflection\nnow becomes the method of philosophy. The Platonic analysis of thought\nreveals elements of knowledge which are not derived from the outer\nworld. There are universal and necessary principles revealed in\nconsciousness which, in their natural and logical development, transcend\nconsciousness, and furnish the cognition of a world of Real Being,\nbeyond the world of sense. There are absolute truths which bridge the\nchasm between the seen and the unseen, the fleeting and the permanent,\nthe finite and the infinite, the temporal and the eternal. There are\nnecessary laws of thought which are also found to be laws of things, and\nwhich correlate man to a living, personal, righteous Lord and Lawgiver.\nFrom absolute ideas Plato ascends to an _absolute Being_, the author of\nall finite existence. From absolute truths to an _absolute Reason_, the\nfoundation and essence of all truth. From the principle of immutable\nright to an _absolutely righteous Being_. From the necessary idea of the\ngood to a being of _absolute Goodness_--that is, to _God_. This is the\nmaturity of humanity, the ripening manhood of our race which was\nattained in the Socratic age.\n\nThe inevitable tendency of this effort of speculative thought, spread\nover ages, and of the intellectual culture which necessarily resulted,\nwas to undermine the old polytheistic religion, and to purify and\nelevate the theistic conception. The school of Elea rejected the gross\nanthropomorphism of the Homeric theology. Xenophanes, the founder of the\nschool, was a believer in\n\n      \"_ One God_, of all beings divine and human the greatest,\n      Neither in body alike unto mortals, neither in ideas.\"\n\nAnd he repels with indignation the anthropomorphic representations of\nthe Deity.\n\n\"But men foolishly think that gods are born as men are,\nAnd have, too, a dress like their own, and their voice, and their figure:\nBut if oxen and lions had hands like ours, and fingers,\nThen would horses like unto horses, and oxen to oxen,\nPaint and fashion their god-forms, and give to them bodies\nOf like shape to their own, as they themselves too are fashioned.\"[877]\n\nEmpedocles also wages uncompromising war against all representations of\nthe Deity in human form--\n\n      \"For neither with head adjusted to limbs, like the human,\n      Nor yet with two branches down from the shoulders outstretching,\n      Neither with feet, nor swift-moving limbs,....\n      He is, wholly and perfectly, _mind_, ineffable, holy,\n      With rapid and swift-glancing thought pervading the world.\"[878]\n\n[Footnote 877: Ritter's \"History of Ancient Philosophy,\" vol. i. pp.\n431, 432.]\n\n[Footnote 878: Ibid., vol. i. pp. 495, 496.]\n\nWhen speaking of the mythology of the older Greeks, Socrates maintains a\nbecoming prudence; he is evidently desirous to avoid every thing which\nwould tend to loosen the popular reverence for divine things.[879] But\nhe was opposed to all anthropomorphic conceptions of the Deity. His\nfundamental position was that the Deity is the Supreme Reason, which is\nto be honored by men as the source of all existence and the end of all\nhuman endeavor. Notwithstanding his recognition of a number of\nsubordinate divinities, he held that the Divine is one, because Reason\nis one. He taught that the Supreme Being is the immaterial, infinite\nGovernor of all;[880] that the world bears the stamp of his\nintelligence, and attests it by irrefragable evidence;[881] and that he\nis the author and vindicator of all moral laws.[882] So that, in\nreality, he did more to overthrow polytheism than any of his\npredecessors, and on that account was doomed to death.\n\n[Footnote 879: Xenophon, \"Memorabilia,\" bk. i. ch. iii. Sec. 3.]\n\n[Footnote 880: Id., ib., bk. i. ch. iv. Secs. 17, 18.]\n\n[Footnote 881: Id., ib., bk. i. ch. i. Sec. 19.]\n\n[Footnote 882: Ritter's \"History of Ancient Philosophy,\" vol. ii. p. 63;\nButler's \"Lectures on Ancient Philosophy,\" vol. i. p. 359.]\n\nIt was, however, the matured dialectic of Plato which gave the\ndeath-blow to polytheism. \"Plato, the poet-philosopher, sacrificed Homer\nhimself to monotheism. We may measure the energy of his conviction by\nthe greatness of the sacrifice. He could not pardon the syren whose\nsongs had fascinated Greece, the fresh brilliant poetry that had\ninspired its religion. He crowned it with flowers, but banished it,\nbecause it had lowered the religious ideal of conscience.\" He was\nsensible of the beauty of the Homeric fables, but he was also keenly\nalive to their religious falsehood, and therefore he excluded the poets\nfrom his ideal republic. In the education of youth, he would forbid\nparents and teachers repeating \"the stories which Hesiod and Homer and\nthe other poets told us.\" And after instancing a number of these stories\n\"which deserve the gravest condemnation,\" he enjoins that God must be\nrepresented as he is in reality. \"God,\" says he, \"is, beyond all else,\ngood in reality, and therefore so to be represented;\" \"he can not do\nevil, or be the cause of evil;\" \"he is of simple essence, and can not\nchange, or be the subject of change;\" \"there is no imperfection in the\nbeauty or goodness of God;\" \"he is a God of truth, and can not lie;\" \"he\nis a being of perfect simplicity and truth in deed and word.\"[883] The\nreader can not fail to recognize the close resemblance between the\nlanguage of Plato and the language of inspiration.\n\nThe theistic conception, in Plato, reaches the highest purity and\nspirituality. God is \"_the Supreme Mind_,\" \"incorporeal,\"\n\"unchangeable,\" \"infinite,\" \"absolutely perfect,\" \"essentially good,\"\n\"unoriginated and eternal.\" He is \"the Father and Maker of the world,\"\n\"the efficient Cause of all things,\" \"the Monarch and Ruler of the\nworld,\" \"the Sovereign Mind that orders all things,\" and \"pervades all\nthings.\" He is \"the sole principle of all things,\" \"the beginning of all\ntruth,\" \"the fountain of all law and justice,\" \"the source of all order\nand beauty;\" in short, He is \"the beginning, middle, and end of all\nthings.\"[884]\n\n[Footnote 883: \"Republic,\" bk. ii. Secs. 18-21.]\n\n[Footnote 884: See _ante_, ch. xi. pp. 377, 378, where the references to\nPlato's writings are given.]\n\nAristotle continued the work of undermining polytheism. He defines God\nas \"the Eternal Reason\"--the Supreme Mind. \"He is the immovable cause of\nall movement in the universe, the all-perfect principle. This principle\nor essence pervades all things. It eternally possesses perfect\nhappiness, and its happiness consists in energy. This primeval mover is\nimmaterial, for its essence is energy--it is pure thought, thought\nthinking itself--the thought of thought.\"[885] Polytheism is thus swept\naway from the higher regions of the intelligence. \"For several to\ncommand,\" says he, \"is not good, there should be but one chief. A\ntradition, handed down from the remotest antiguity, and transmitted\nunder the veil of fable, says that all the stars are gods, and that the\nDivinity embraces the whole of nature. And round this idea other\nmythical statements have been agglomerated, with a view to influencing\nthe vulgar, and for political and moral expediency; as for instance,\nthey feigned that these gods have human shape, and are like certain of\nthe animals; and other stories of the kind are added on. Now, if any one\nwill separate from all this the first point alone, namely, that they\nthought the first and deepest grounds of existence to be Divine, he may\nconsider it a divine utterance.\"[886] The popular polytheism, then, was\nbut a perverted fragment of a deeper and purer \"Theology.\" This passage\nis a sort of obituary of polytheism. The ancient glory of paganism had\npassed away. Philosophy had exploded the old theology. Man had learned\nenough to make him renounce the ancient religion, but not enough to\nfound a new faith that could satisfy both the intellect and the heart.\n\"Wherefore we are not to be surprised that the grand philosophic period\nshould be followed by one of incredulity and moral collapse,\ninaugurating the long and universal _decadence_ which was, perhaps, as\nnecessary to the work of preparation, as was the period of religious and\nphilosophic development.\"\n\n[Footnote 885: \"Metaphysics,\" bk. xii.]\n\n[Footnote 886: \"Metaphysics,\" bk. xi. ch. viii. Sec. 19.]\n\nThe preparatory office of Greek philosophy in the region of speculative\nthought is seen--\n\n2. _In the development of the Theistic argument in a logical\nform._--Every form of the theistic proof which is now employed by\nwriters on natural theology to demonstrate the being of God was\napprehended, and logically presented, by one or other of the ancient\nphilosophers, excepting, perhaps, the \"moral argument\" drawn from the\nfacts of conscience.\n\n(I.) _The_ AETIOLOGICAL _proof_, or the argument based upon the principle\nof causality, which may be presented in the following form:\n\n      All genesis or becoming supposes a permanent and uncaused\n      Being, adequate to the production of all phenomena.\n\n      The sensible universe is a perpetual genesis, a succession\n      of appearances: it is \"always becoming, and never really\n      is.\"\n\n      Therefore, it must have its cause and origin in a permanent\n      and unoriginated Being, adequate to its production.\n\nThe major premise of this syllogism is a fundamental principle of\nreason--a self-evident truth, an axiom of common sense, and as such has\nbeen recognized from the very dawn of philosophy. [Greek: Adounaton\nginesthai ti ek medenos prouparxonios]--_Ex nihilo nihil_--_Nothing\nwhich once was not, could ever of itself come into being_. Nothing can\nbe made or produced without an efficient cause, is the oldest maxim of\nphilosophy. It is true that this maxim was abusively employed by\nDemocritus and Epicurus to disprove a Divine creation of any thing out\nof nothing, yet the great body of ancient philosophers, as Pythagoras,\nXenophanes, Parmenides, Zeno, Anaxagoras, Empedocles, Plato, and\nAristotle, regarded it as the announcement of an universal conviction,\nthat nothing can be produced without an efficient cause;--order can not\nbe generated out of chaos, life out of dead matter, consciousness out of\nunconsciousness, reason out of unreason. A first principle of life, of\norder, of reason, must have existed anterior to all manifestions of\norder, of life, of intelligence, in the visible universe. It was clearly\nin this sense that Cicero understood this great maxim of the ancient\nphilosophers of Greece. With him \"_De nihilo nihil fit\"_ is equivalent\nto \"_Nihil sine causa_\"--nothing exists without a cause. This is\nunquestionably the form in which that fundamental law of thought is\nstated by Plato: \"Whatever is generated is necessarily generated from a\ncertain cause, for it is wholly impossible that any thing should be\ngenerated without a cause.\"[887] And the efficient cause is defined as\n\"a power whereby that which did not previously exist was afterwards made\nto be.\"[888] It is scarcely needful to remark that Aristotle, the\nscholar of Plato, frequently lays it down as a postulate of reason,\n\"that we admit nothing without a cause.\"[889] By an irresistible law of\nthought, \"_all phenomena present themselves to us as the expression of\npower_, and refer us to a causal ground whence they issue.\"\n\n[Footnote 887: \"Timaeus,\" ch. ix.; also \"Philebus,\" Sec. 45.]\n\n[Footnote 888: \"Sophist,\" Sec. 109.]\n\n[Footnote 889: \"Post. Analytic,\" bk. ii. ch. xvi.; \"Metaphysics,\" bk. i.\nch. i. Sec. 3.]\n\nThe major premise of this syllogism is a fact of observation.\n\nTo the eye of sense and sensible observation, to scientific induction\neven in its highest generalizations, the visible universe presents\nnothing but a history and aggregation of phenomena--a succession of\nappearances or effects having more or less resemblance. It is a\nceaseless flow and change, \"a generation and corruption,\" \"a becoming,\nbut never really _is_;\" it is never in two successive moments the\n_same_.[890] All our cognitions of sameness, uniformity, causal\nconnection, permanent Being, real Power, are purely rational conceptions\n_given in thought_, supplied by the spontaneous intuition of reason as\nthe correlative prefix to the phenomena observed.[891]\n\n[Footnote 890: \"Timaeus,\" ch. ix.]\n\n[Footnote 891: Ibid.]\n\nTherefore the ancient philosophers concluded justly, there must be\nsomething [Greek: agenneton]--something which was never generated,\nsomething [Greek: autophyes] and [Greek: authypostaton]--self-originated\nand self-existing, something [Greek: tauton] and [Greek:\naionion]--immutable and eternal, the object of rational\napperception--which is the real ground and efficient cause of all that\nappears.\n\n(2.) The COSMOLOGICAL proof, or the argument based upon the principle of\norder, and thus presented:\n\n      Order, proportion, harmony, are the product and expression\n      of Mind.\n\n      The created universe reveals order, proportion, and harmony.\n\n      Therefore, the created universe is the product of Mind.\n\nThe fundamental law of thought which underlies this mode of proof was\nclearly recognized by Pythagoras. All harmony and proportion and\nsymmetry is the result of _unity_ evolving itself in and pervading\n_multiplicity_. Mind or reason is unity and indivisibility; matter is\ndiverse and multiple. Mind is the determinating principle; matter is\nindeterminate and indefinite. Confused matter receives form, and\nproportion, and order, and symmetry, by the action and interpenetration\nof the spiritual and indivisible element. In presence of facts of order,\nthe human reason instinctively and necessarily affirms the presence and\naction of Mind.\n\n\"Pythagoras had long devoted his intellectual adoration to the lofty\nidea of Order. To his mind it seemed as the presiding genius of the\nserene and silent world. He had from his youth dwelt with delight upon\nthe eternal relations of space and number, in which the very idea of\nproportion seems to find its first and immediate development, until at\nlength it seemed as if the whole secret of the universe was hidden in\nthese mysterious correspondences. The world, in all its departments,\nmoral and material, is a living arithmetic in its development, a\nrealized geometry in its repose; it is a '_cosmos_' (for the word is\nPythagorean), the expression of harmony, the manifestation to sense of\neverlasting order; and the science of _numbers_ is the truest\nrepresentation of its eternal laws.\" Therefore, argued Pythagoras and\nthe Pythagoreans, as the reason of man can perceive the relations of an\neternal order in the proportions of extension and number, the laws of\nproportion, and symmetry, and harmony must inhere in a Divine reason, an\nintelligent soul, which moves and animates the universe. The harmonies\nof the world which address themselves to the human mind must be the\nproduct of a Divine mind. The world, in its real structure, must be the\nimage and copy of that divine proportion which the mind of man adores.\nIt is the sensible type of the Divinity, the outward and multiple\ndevelopment of the Eternal Unity, the Eternal One--that is, God.\n\nThe same argument is elaborated by Plato in his philosophy of beauty.\nGod is with him the last reason, the ultimate foundation, the perfect\nideal of all beauty--of all the order, proportion, harmony, sublimity,\nand excellence which reigns in the physical, the intellectual, and the\nmoral world. He is the \"Eternal Beauty, unbegotten and imperishable,\nexempt from all decay as well as increase--the perfect--the Divine\nBeauty\"[892] which is beheld by the pure mind in the celestial world.\n\n[Footnote 892: \"Banquet,\" Sec. 35.]\n\n(3.) The Teleological proof, or the argument based upon the principle of\nintentionality or Final Cause, and is presented in the following form:\n\n      The choice and adaptation of means to the accomplishment of\n      special ends supposes an intelligent purpose, a Designing\n      Mind.\n\n      In the universe we see such choice and adaptation of means\n      to ends.\n\n      Therefore, the universe is the product of an intelligent,\n      personal Cause.\n\nThis is peculiarly the Socratic proof. He recognized the necessity and\nthe irresistibility of the conviction that the choice and adaptation of\nmeans to ends is the effect of Purpose, the expression of Will.[893]\nThere is an obviousness and a directness in this mode of argument which\nis felt by every human mind. In the \"Memorabilia\" Xenophon has preserved\na conversation of Socrates with Aristodemus in which he develops this\nproof at great length. In reading the dialogue[894] in which Socrates\ninstances the adaptation of our organization to the external world, and\nthe examples of design in the human frame, we are forcibly reminded of\nthe chapters of Paley, Whewell, and M'Cosh. Well might Aristodemus\nexclaim: \"The more I consider it, the more it is evident to me that man\nmust be the masterpiece of some great Artificer, carrying along with it\ninfinite marks of the love and favor of Him who has thus formed it.\" The\nargument from Final Causes is pursued by Plato in the \"Timaeus;\" and in\nAristotle, God is the Final Cause of all things.[895]\n\n[Footnote 893: \"Canst thou doubt, Aristodemus, whether a disposition of\nparts like this (in the human body) should be the work of chance, or of\nwisdom and contrivance?\"--\"Memorabilia,\" bk. i. ch. iv.]\n\n[Footnote 894: \"Memorabilia,\" bk. i. ch. iv.]\n\n[Footnote 895: Aristotle clearly recognizes that an end or final cause\nimplies Intelligence. \"The appearance of ends and means is a proof of\nDesign.\"--\"Nat. Ausc.,\" bk. ii. ch. viii.]\n\n(4.) The Ontological or Ideological proof, or the argument grounded on\nnecessary and absolute ideas, which may be thrown into the following\nsyllogism:\n\n      Every attribute or quality implies a subject, and absolute\n      modes necessarily suppose an Absolute Being. Necessary and\n      absolute truths or ideas are revealed in human reason as\n      absolute modes.\n\n      Therefore universal, necessary, and absolute ideas are modes\n      of the absolute subject--that is, God, the foundation and\n      source of all truth.\n\nThis is the Platonic proof. Plato recognized the principle of substance\n([Greek: ousia ypokeimenon]), and therefore he proceeds in the \"Timaeus\"\nto inquire for the real ground of all existence; and in the \"Republic,\"\nfor the real ground of all truth and certitude.\n\nThe universe consists of two parts, permanent existences and transient\nphenomena--being and genesis; the one eternally constant, the other\nmutable and subject to change; the former apprehended by the reason, the\nlatter perceived by sense. For each of these there must be a principle,\nsubject, or substratum--a principle or subject-matter, which is the\nground or condition of the sensible world, and a principle or substance,\nwhich is the ground and reason of the intelligible world or world of\nideas. The subject-matter, or ground of the sensible world, is \"the\nreceptacle\" and \"nurse\" of forms, an \"invisible species and formless\nreceiver (which is not earth, or air, or fire, or water) which receives\nthe immanence of the intelligible.\"[896] The subject or ground of the\nintelligible world is that in which ideal forms, or eternal archetypes\ninhere, and which impresses form upon the transitional element, and\nfashions the world after its own eternal models. This eternal and\nimmutable substance is God, who created the universe as a copy of the\neternal archetypes--the everlasting thoughts which dwell in his infinite\nmind.\n\n[Footnote 896: \"Timaeus,\" ch. xxiv.]\n\nThese copies of the eternal archetypes or models are perceived by the\nreason of man in virtue of its participation in the Ultimate Reason. The\nreason of man is the organ of truth; by an innate and inalienable right,\nit grasps unseen and eternal realities. The essence of the soul is akin\nto that which is real, permanent, and eternal;--_It is the offspring and\nimage of God_; therefore it has a true communion with the realities of\nthings, by virtue of this kindred and homogeneous nature. It can,\ntherefore, ascend from the universal and necessary ideas, which are\napprehended by the reason, to the absolute and supreme Idea, which is\nthe attribute and perfection of God. When the human mind has\ncontemplated any object of beauty, any fact of order, proportion,\nharmony, and excellency, it may rise to the notion of a quality common\nto all objects of beauty--from a single beautiful body to two, from two\nto all others; from beautiful bodies to beautiful sentiments, from\nbeautiful sentiments to beautiful thoughts, until, from thought to\nthought, we arrive at the highest thought, which has no other object\nthan the perfect, absolute, _Divine Beauty_.[897] When a man has, from\nthe contemplation of instances of virtue, risen to the notion of a\nquality common to all these instances, this quality becomes the\nrepresentative of an ineffable something which, in the sphere of\nimmutable reality, answers to the conception in his soul. \"At the\nextreme limits of the intellectual world is the _Idea of the Good_,\nwhich is perceived with difficulty, but, in fine, can not be perceived\nwithout concluding that it is the source of all that is beautiful and\ngood; that in the visible world it produces light, and the star whence\nlight directly comes; that in the invisible world it directly produces\ntruth and intelligence.\"[898] This _absolute Good is God_.\n\n[Footnote 897: \"Banquet,\" Sec. 34.]\n\n[Footnote 898: \"Republic,\" bk. vii. ch. iii.]\n\nThe order in which these several methods of proof were developed, will\nat once present itself to the mind of the reader as the natural order of\nthought. The first and most obvious aspect which nature presents to the\nopening mind is that of movement and change--a succession of phenomena\nsuggesting the idea of _power_. Secondly, a closer attention reveals a\nresemblance of phenomena among themselves, a uniformity of nature--an\norder, proportion, and harmony pervading the _cosmos_, which suggest an\n_identity and unity of power and of reason_, pervading and controlling\nall things. Thirdly, a still closer inspection of nature reveals a\nwonderful adaptation of means to the fulfillment of special ends, of\norgans designed to fulfill specific functions, suggesting the idea of\n_purpose_, _contrivance_, and _choice_, and indicating that the power\nwhich moves and determines the universe is a _personal_, _thinking_, and\n_voluntary_ agent. And fourthly, a profounder study of the nature of\nthought, an analysis of personal consciousness, reveals that there are\nnecessary principles, ideas, and laws, which universally govern and\ndetermine thought to definite and immovable conceptions--as, for\nexample, the principles of causality, of substance, of identity or\nunity, of order, of intentionality; and that it is only under these laws\nthat we can conceive the universe. By the law of substance we are\ncompelled to regard these ideas, which are not only laws of thought but\nalso of things, as inherent in a subject, or Being, who made all things,\nand whose ideas are reflected in the reason of man. Thus from universal\nand necessary ideas we rise to the _absolute Idea_, from immutable\nprinciples to a _First Principle of all principles_, a _First Thought_\nof all thoughts--that is, to _God_. This is the history of the\ndevelopment of thought in the individual, and in the race--_cause_,\n_order_, _design_, _idea_, _being_, GOD.\n\n\n\n\nCHAPTER XV.\n\nTHE PROPAEDEUTIC OFFICE OF GREEK PHILOSOPHY _(continued)_.\n\n\n      \"If we regard this sublime philosophy as a preparation for\n      Christianity instead of seeking in it a substitute for the\n      Gospel, we shall not need to overstate its grandeur in order\n      to estimate its real value.\"--Pressense.\n\n      \"Plato made me to know the true God. Jesus Christ showed me\n      the way to Him.\"--St. Augustine.\n\nThe preparatory office of Grecian philosophy is also seen in _the\ndepartment of morals_.\n\nI. _In the awakening and enthronement of Conscience as a law of duty,\nand the elevation and purification of the Moral Idea_.\n\nThe same law of evolution, which we have seen governing the history of\nspeculative thought, may also be traced as determining the progress of\nethical inquiry. In this department there are successive stages marked,\nboth in the individual and the national mind. There is, first, the\nsimplicity and trust of childhood, submitting with unquestioning faith\nto prescribed and arbitrary laws; then the unsettled and ill-directed\nforce of youth, questioning the authority of laws, and asking reasons\nwhy this or that is obligatory; then the philosophic wisdom of riper\nyears, recognizing an inherent law of duty, which has an absolute\nrightness and an imperative obligation. There is first a dim and shadowy\napprehension of some lines of moral distinction, and some consciousness\nof obligation, but these rest mainly upon an outward law--the observed\npractice of others, or the command of the parent as, in some sense, the\ncommand of God. Then, to attain to personal convictions, man passes\nthrough a stage of doubt; he asks for a ground of obligation, for an\nauthority that shall approve itself to his own judgment and reason. At\nlast he arrives at some ultimate principles of right, some immutable\nstandard of duty; he recognizes an inward law of conscience, and it\nbecomes to him as the voice of God. He extends his analysis to history,\nand he finds that the universal conscience of the race has, in all ages,\nuttered the same behest. Should he live in Christian times, he discovers\na wondrous harmony between the voice of God within the heart, and the\nvoice of God within the pages of inspiration. And now the convention of\npublic opinion, and the laws of the state, are revered and upheld by\nhim, just so far as they bear the imprimatur of reason and of\nconscience--that is, of God.\n\nThis history of the normal development of the individual mind has its\ncounterpart in the history of humanity. There is (1.) _The age of\npopular and unconscious morality_; (2.) _The transitional, skeptical, or\nsophistical age_; and (3.) _The philosophic or conscious age of\nmorality_.[899] In the \"Republic\" of Plato, we have these three eras\nrepresented by different persons, through the course of the dialogue.\nThe question is started--what is Justice? and an answer is given from\nthe stand-point of popular morality, by Polemarchus, who quotes the\nwords of the poet Simonides,\n\n      \"To give to each his due is just;\"[900]\n\nthat is, justice is paying your debts. This doctrine being proved\ninadequate, an answer is given from the Sophistical point of view by\nThrasymachus, who defines justice as \"the advantage of the\nstrongest\"--that is, might is right, and right is might.[901] This\nanswer being sharply refuted, the way is opened for a more philosophic\naccount, which is gradually evolved in book iv., Glaucon and Adimantus\npersonifying the practical understanding, which is gradually brought\ninto harmony with philosophy, and Socrates the higher reason, as the\npurely philosophic conception. Justice is found to be the right\nproportion and harmonious development of all the elements of the soul,\nand the equal balance of all the interests of society, so as to secure a\nwell-regulated and harmonious whole.\n\n[Footnote 899: Grant's \"Aristotle's Ethics,\" vol. i. p. 46.]\n\n[Footnote 900: \"Republic,\" bk. i. Sec. 6.]\n\n[Footnote 901: Ibid., bk. i. Sec. 12.]\n\nThe era of _popular and unconscious morality_ is represented by the\ntimes of Homer, Hesiod, the Gnomic poets, and \"the Seven Wise Men of\nGreece.\"\n\nThis was an age of instinctive action, rather than reflection--of poetry\nand feeling, rather than analytic thought. The rules of life were\npresented in maxims and proverbs, which do not rise above prudential\ncounsels or empirical deductions. Morality was immediately associated\nwith the religion of the state, and the will of the gods was the highest\nlaw for men. \"Homer and Hesiod, and the Gnomic poets, constituted the\neducational course,\" to which may be added the saws and aphorisms of the\nSeven Wise Men, and we have before us the main sources of Greek views of\nduty. When the question was asked--\"What is right?\" the answer was given\nby a quotation from Homer, Hesiod, Simonides, and the like. The morality\nof Homer \"is concrete, not abstract; it expresses the conception of a\nheroic life, rather than a philosophic theory. It is mixed up with a\nreligion which really consists in a celebration of the beauty of nature,\nand in a deification of the strong and brilliant qualities of human\nnature. It is a morality uninfluenced by a regard for a future life. It\nclings with intense enjoyment and love to the present world, and the\nstate after death looms up in the distance as a cold and repugnant\nshadow. And yet it would often hold death preferable to disgrace. The\ndistinction between a noble and ignoble life is strongly marked in\nHomer, and yet a sense of right and wrong about particular actions seems\nfluctuating\" and confused.[902] A sensuous conception of happiness is\nthe chief good, and mere temporal advantage the principal reward of\nvirtue. We hear nothing of the approving smile of conscience, of inward\nself-satisfaction, and peace, and harmony, resulting from the practice\nof virtue. Justice, energy, temperance, chastity, are enjoined, because\nthey secure temporal good. And yet, with all this imperfection, the\npoets present \"a remarkable picture of primitive simplicity, chastity,\njustice, and practical piety, under the three-fold influence of right\nmoral feeling, mutual and fear of the divine displeasure.\"[903]\n\n[Footnote 902: Grant's \"Aristotle's Ethics,\" vol. i. p. 51.]\n\n[Footnote 903: Tyler, \"Theology of the Greek Poets,\" p. 167.]\n\nThe _transitional, skeptical_, or _sophistical era_ begins with\nProtagoras. Poetry and proverbs had ceased to satisfy the reason of man.\nThe awakening intellect had begun to call in question the old maxims and\n\"wise saws,\" to dispute the arbitrary authority of the poets, and even\nto arraign the institutions of society. It had already begun to seek for\nsome reasonable foundation of authority for the opinions, customs, laws,\nand institutions which had descended to them from the past, and to ask\nwhy men were obliged to do this or that? The question whether there is\nat bottom any real difference between truth and error, right and wrong,\nwas now fairly before the human mind. The ultimate standard of all truth\nand all right, was now the grand object of pursuit. These inquiries were\nnot, however, conducted by the Sophists with the best motives. They were\nnot always prompted by an earnest desire to know the truth, and an\nearnest purpose to embrace and do the right. They talked and argued for\nmere effect--to display their dialectic subtilty, or their rhetorical\npower. They taught virtue for mere emolument and pay. They delighted, as\nCicero tells us, to plead the opposite sides of a cause with equal\neffect. And they found exquisite pleasure in raising difficulties,\nmaintaining paradoxes, and passing off mere tricks of oratory for solid\nproofs. This is the uniform representation of the sophistical spirit\nwhich is given by all the best writers who lived nearest to their times,\nand who are, therefore, to be presumed to have known them best.\nGrote[904] has made an elaborate defense of the Sophists; he charges\nPlato with gross misrepresentation. His portraits of them are denounced\nas mere caricatures, prompted by a spirit of antagonism; all antiquity\nis presumed to have been misled by him. No one, however, can read\nGrant's \"Essay on the History of Moral Philosophy in Greece\"[905]\nwithout feeling that his vindication of Plato is complete and\nunanswerable: \"Plato never represents the Sophists as teaching a lax\nmorality to their disciples. He does not make sophistry to consist in\nholding wicked opinions; he represents them as only too orthodox in\ngeneral,[906] but capable of giving utterance to immoral paradoxes for\nthe sake of vanity. Sophistry rather tampers and trifles with the moral\nconvictions than directly attacks them.\" The Sophists were wanting in\ndeep conviction, in moral earnestness, in sincere love of truth, in\nreverence for goodness and purity, and therefore their trifling,\ninsincere, and paradoxical teaching was unfavorable to goodness of life.\nThe tendency of their method is forcibly depicted in the words of Plato:\n\"There are certain dogmas relating to what is _just_ and _good_ in which\nwe have been brought up from childhood--obeying and reverencing them.\nOther opinions recommending pleasure and license we resist, out of\nrespect for the old hereditary maxims. Well, then, a question comes up\nconcerning what is right? He gives some answer such as he has been\ntaught, and straightway is refuted. He tries again, and is again\nrefuted. And, when this has happened pretty often, he is reduced to the\nopinion that _nothing is either right or wrong_; and in the same way it\nhappens about the just and the good, and all that before we have held in\nreverence. On this, he naturally abandons his allegiance to the old\nprinciples and takes up with those he before resisted, and so, from\nbeing a good citizen, he becomes lawless.\"[907] And, in point of fact,\nthis was the theoretical landing-place of the Sophists. We do not say\nthey became practically \"lawless\" and antinomian, but they did arrive at\nthe settled opinion that right and wrong, truth and error, are solely\nmatter of private opinion and conventional usage. Man's own fluctuating\nopinion is the measure and standard of all things.[908] They who \"make\nthe laws, make them for their own advantage.\"[909] There is no such\nthing as Eternal Right. \"That which _appears_ just and honorable to each\ncity is so for that city, as long as the opinion prevails.\"[910]\n\n[Footnote 904: \"History of Greece.\"]\n\n[Footnote 905: Aristotle's \"Ethics,\" vol. i. ch. ii.]\n\n[Footnote 906: \"His teachings will be good counsels about a man's own\naffairs, how best to govern his family; and also about the affairs of\nthe state, how most ably to administer and speak of state\naffairs.\"--\"Protag.,\" Sec. 26.]\n\n[Footnote 907: \"Republic,\" bk. vii. ch. xvii.]\n\n[Footnote 908: \"Theaetetus,\" Sec. 23.]\n\n[Footnote 909: \"Gorgias,\" Secs. 85-89.]\n\n[Footnote 910: \"Theaetetus,\" Secs. 65-75.]\n\nThe age of the Sophists was a transitional period--a necessary, though,\nin itself considered, an unhappy stage in the progress of the human\nmind; but it opened the way for, _The Socratic, philosophic_, or\n_conscious age of morals_. It has been said that \"before Socrates there\nwas no morality in Greece, but only propriety of conduct.\" If by this is\nmeant that prior to Socrates men simply followed the maxims of \"the\nTheologians,\"[911] and obeyed the laws of the state, without reflection\nand inquiry as to the intrinsic character of the acts, and without any\nanalysis and exact definition, so as to attain to principles of ultimate\nand absolute right, it must be accepted as true--there was no philosophy\nof morals. Socrates is therefore justly regarded as \"the father of moral\nphilosophy.\" Aristotle says that he confined himself chiefly to ethical\ninquiries. He sought a determinate conception and an exact definition of\nvirtue. As Xenophon has said of him, \"he never ceased asking, What is\npiety? what is impiety? what is noble? what is base? what is just? what\nis unjust? what is temperance? what is madness?\"[912] And these\nquestions were not asked in the Sophistic spirit, as a dialectic\nexercise, or from idle curiosity. He was a perfect contrast to the\nSophists. They had slighted Truth, he made her the mistress of his soul.\nThey had turned away from her, he longed for more perfect communion with\nher. They had deserted her for money and renown, he was faithful to her\nin poverty.[913] He wanted to know what piety was, that he might be\npious. He desired to know what justice, temperance, nobility, courage\nwere, that he might cultivate and practise them. He wrote no books,\ndelivered no lectures; he instituted no school; he simply conversed in\nthe shop, the market-place, the banquet-hall, and the prison. This\nphilosophy was not so much a _doctrine_ as a _life_. \"What is remarkable\nin him is not the _system_ but the _man_. The memory he left behind him\namongst his disciples, though idealized--the affection, blended with\nreverence, which they never ceased to feel for his person, bear\ntestimony to the elevation of his character and his moral purity. We\nrecognize in him a Greek of Athens--one who had imbibed many dangerous\nerrors, and on whom the yoke of pagan custom still weighed; but his life\nwas nevertheless a noble life; and it is to calumny we must have\nrecourse if we are to tarnish its beauty by odious insinuations, as\nLucian did, and as has been too frequently done, after him, by\nunskillful defenders of Christianity,[914] who imagine it is the gainer\nby all that degrades human nature. Born in a humble position, destitute\nof all the temporal advantages which the Greeks so passionately loved,\nSocrates exerted a kingship over minds. His dominion was the more real\nfor being less apparent.... His power consisted of three things: his\ndevoted affection for his disciples, his disinterested love of truth,\nand the perfect harmony of his life and doctrine.... If he recommended\ntemperance and sobriety, he also set the example; poorly clad, satisfied\nwith little, he disdained all the delicacies of life. He possessed every\nspecies of courage. On the field of battle he was intrepid, and still\nmore intrepid when he resisted the caprices of the multitude who\ndemanded of him, when he was a senator, to commit the injustice of\nsummoning ten generals before the tribunals. He also infringed the\niniquitous orders of the thirty tyrants of Athens. The satires of\nAristophanes neither moved nor irritated him. The same dauntless\nfirmness he displayed when brought before his judges, charged with\nimpiety. 'If it is your wish to absolve me on condition that I\nhenceforth be silent, I reply I love and honor you, but I ought rather\nto obey the gods than you. Neither in the presence of judges nor of the\nenemy is it permitted me, or any other man, to use every sort of means\nto escape death. It is not death but crime that it is difficult to\navoid; crime moves faster than death. So I, old and heavy as I am, have\nallowed myself to be overtaken by death, while my accusers, light and\nvigorous, have allowed themselves to be overtaken by the light-footed\ncrime. I go, then, to suffer death; they to suffer shame and iniquity. I\nabide by my punishment, as they by theirs. All is according to order.'\nIt was the same fidelity to duty that made Socrates refuse to escape\nfrom prison, in order not to violate the laws of his country, to which,\neven though irritated, more respect is due than to a father. 'Let us\nwalk in the path,' he says 'that God has traced for us.' These last\nwords show the profound religious sentiment which animated Socrates....\nIt is impossible not to feel that there was something divine in such a\nlife crowned with such a death.\"[914]\n\n[Footnote 911: Homer, Hesiod, etc.]\n\n[Footnote 912: \"Memorabilia,\" bk. i. ch. i. p. 16.]\n\n[Footnote 913: Lewes's \"Biographical History of Philosophy,\" p. 122.]\n\n[Footnote 914: Watson's \"Institutes of Theology,\" vol. i. p. 374.]\n\n[Footnote 915: Pressense, \"Religions before Christ,\" pp. 109-111.]\n\nSocrates laid the foundation for conscious morality by placing the\nground of right and wrong in an eternal and unchangeable reason which\nilluminates the reason and conscience of every man. He often asserted\nthat morality is a science which can not be taught. It depends mainly\nupon principles which are discovered by an inward light. Accordingly he\nregarded it as the main business of education to \"draw out\" into the\nlight of consciousness the principles of right and justice which are\ninfolded within the conscience of man--to deliver the mind of the secret\ntruth which was striving towards the light of day. Therefore he called\nhis method the \"maieutic\" or \"obstetric\" art. He felt there was\nsomething divine in all men (answering to his _to daimonion_ or\n_daimonion ti_--a divine and supernatural something--a warning\n\"voice\"--a gnomic \"sign\"--a \"law of God written on the heart\"), which by\na system of skillful interrogations he sought to elicit, so that each\nmight hear for himself the voice of God, and, hearing, might obey. Thus\nwas he the \"great prophet of the human conscience,\" and a messenger of\nGod to the heathen world, to prepare the way of the Lord.\n\nThe morality of conscience was carried to its highest point by Plato.\nFrom the moment he became the disciple of Socrates he sympathized deeply\nwith the spirit and the method of his master. He had the same deep\nseriousness of spirit, that same earnestness of purpose, that same\ninward reverence for justice, and purity, and goodness, which dwelt in\nthe heart of Socrates. A naturally noble nature, he loved truth with all\nthe glow and fervor of his young heart. He felt that if any thing gave\nmeaning and value to life, it must be the contemplation of absolute\ntruth, absolute beauty, and absolute Good. This absolute Good is God,\nwho is the first principle of all ideas, the fountain of all the order\nand proportion and beauty of the universe, the source of all the good\nwhich exists in nature and in man. To practise goodness--to conform the\ncharacter to the eternal models of order, proportion, and excellence, is\nto resemble God. To aspire after perfection of moral being, to secure\nassimilation to God ([Greek: omoiosis Theo]) is the noble aspiration of\nPlato's soul.\n\nWhen we read the \"Gorgias,\" the \"Philebus,\" and especially the\n\"Republic,\" with what noble joy are we filled on hearing the voice of\nconscience, like a harp swept by a seraph's hand, uttering such\ndeep-toned melodies! How does he drown the clamors of passion, the\ncalculations of mere expediency, the sophism of mere personal interest\nand utility. If he calls us to witness the triumph of the wicked in the\nfirst part of the \"Republic,\" it is in order that we may at the end of\nthe book see the deceitfulness of their triumph. \"As to the wicked,\" he\nsays, \"I maintain that even if they succeed at first in concealing what\nthey are, most of them betray themselves at the end of their career.\nThey are covered with opprobrium, and present evils are nothing compared\nwith those that _await them in the other life_. As to the just man,\nwhether in sickness or in poverty, these imaginary evils will turn to\nhis advantage in this life, _and after his death_; because the\nprovidence of the gods is necessarily attentive to the interests of him\nwho labors to become just, and to attain, by the practice of virtue, to\nthe most perfect resemblance to God which is possible to man.\"[916] He\nrises above all \"greatest happiness principles,\" and asserts distinctly\nin the \"Gorgias\" that it is better to suffer wrong than to do\nwrong.[917] \"I maintain,\" says he, \"that what is most shameful is not to\nbe struck unjustly on the cheek, or to be wounded in the body; but that\nto strike and wound me unjustly, to rob me, or reduce me to slavery--to\ncommit, in a word, any kind of injustice towards me, or what is mine--is\na thing far worse and more odious for him who commits the injustice,\nthan for me who suffer it.\"[918] It is a great combat, he says, greater\nthan we think, that wherein the issue is whether we shall be virtuous or\nwicked. Neither glory, nor riches, nor dignities, nor poetry, deserves\nthat we should neglect justice for them. The moral idea in Plato has\nsuch intense truth and force, that it has at times a striking analogy\nwith the language of the Holy Scriptures.[919]\n\n[Footnote 916: \"Republic,\" bk. x. ch. xii.]\n\n[Footnote 917: \"Gorgias,\" Secs. 59-80.]\n\n[Footnote 918: Ibid., Sec. 137.]\n\n[Footnote 919: Pressense, \"Religions before Christ,\" p. 129.]\n\nThe obligation of moral rectitude is, by Plato, derived from the\nauthoritative utterances of conscience as the voice of God. We must do\nright because reason and conscience say it is right. In the \"Euthyphron\"\nhe maintains that the moral quality of actions is not dependent on the\narbitrary will of a Supreme Governor;--\"an act is not holy because the\ngods love it, but the gods love it because it is holy.\" The eternal law\nof right dwells in the Eternal Reason of God, the idea of right in all\nhuman minds is a ray of that Eternal Reason; and the requirement of the\ndivine law that we shall do right is, and must be, in harmony with both.\n\nThe present life is regarded by Plato as a state of probation and\ndiscipline, the future life as one of reward and punishment.[920]\n\n[Footnote 920: \"Republic,\" bk. x. ch. xv., xvi.; \"Laws,\" bk. x. ch.\nxiii.]\n\nPlato was thus to the heathen world \"the great apostle of the moral\nidea;\" he followed up and completed the work of Socrates. \"The voice of\nGod, that still found a profound echo in man's heart, possessed in him\nan organ to which all Greece gave ear; and the austere revelation of\nconscience this time embodied in language too harmonious not to entice\nby the beauty of form, a nation of artists, they received it. The tables\nof the eternal law, carved in purest marble and marvellously sculptured,\nwere read by them.\"\n\nIn Plato both the theistic conception and the moral idea seem to have\ntouched the zenith. The philosophy of Aristotle, considered as a whole,\nappears on one side to have passed the line of the great Hellenic\nperiod. If it did not inaugurate, it at least prepared the way for the\ndecline. It perfected logic, as the instrument of ratiocination, and\ngave it exactness and precision, Yet taken all in all, it was greatly\ninferior to its predecessor. From the moral point of view it is a\ndecided retrogression. The god of Aristotle is indifferent to virtue. He\nis pure thought rather than moral perfection. He takes no cognizance of\nman. Morality has no eternal basis, no divine type, and no future\nreward. Therefore Aristotle's philosophy had little power over the\nconscience and heart.\n\nDuring the grand Platonic period human reason made its loftiest flight,\nit rose aloft and soared towards heaven, but alas! its wings, like those\nof Icarus, melted in the sun and it fell to earth again. Instead of wax\nit needed the strong \"eagle pinions of faith\" which revelation only can\nsupply. The decadence is strongly marked both in the Epicurean and Stoic\nschools. They both express the feeling of exhaustion, disappointment,\nand despair. The popular theology had lost its hold upon the public\nmind. The gods no longer visited the earth. \"The mysterious voice which,\naccording to the poetic legend related by Plutarch, was heard out at\nsea--'Great Pan is dead'--rose up from every heart; the voice of an\nincredulous age proclaimed the coming end of paganism. The oracles were\ndumb.\" There was no vision in the land. All faith in a beneficent\noverruling Providence was lost, and the hope of immortality was\nwell-nigh gone. The doctrines of a resurrection and a judgment to come,\nwere objects of derisive mockery.[921] Philosophy directed her attention\nsolely to the problem of individual well-being on earth; it became\nsimply a philosophy of life, and not, as with Plato, \"a preparation for\ndeath.\" The grosser minds sought refuge in the doctrines of Epicurus.\nThey said, \"Pleasure is the chief good, the end of life is to enjoy\nyourself;\" to this end \"dismiss the fear of gods, and, above all, the\nfear of death.\" The nobler souls found an asylum with the Stoics. They\nsaid, \"Fata nos ducunt--The Fates lead us! Live conformable to reason.\nEndure and abstain!\" Notwithstanding numerous and serious errors, the\nethical system of the Stoics was wonderfully pure. This must be\nconfessed by any one who reads the \"Enchiridion\" of Epictetus, and the\n\"Meditations\" of Aurelius. \"The highest end of life is to contemplate\ntruth and to obey the Eternal Reason. God is to be reverenced above all\nthings, and universally submitted to. The noblest office of reason is to\nsubjugate passion and conduct to virtue. Virtue is the supreme good,\nwhich is to be pursued for its own sake, and not from fear or hope. That\nis sufficient for happiness which is seated only in the mind, and\ntherefore independent of external things. The consciousness of\nwell-doing is reward enough without the applause of others. And no fear\nof loss, or pain, or even death, must be suffered to turn us aside from\ntruth and virtue.\"[922]\n\n[Footnote 921: Acts xvii. 32.]\n\n[Footnote 922: Marcus Aurelius.]\n\nThe preparatory office of Christianity in the field of ethics is further\nseen,\n\nII. _In the fact that, by an experiment conducted on the largest scale,\nit demonstrated the insufficiency of reason to elaborate a perfect ideal\nof moral excellence, and develop the moral forces necessary to secure\nits realization_.\n\nWe have seen that the moral idea in Socrates, Plato, Epictetus, Marcus\nAurelius, and Seneca rose to a sublime height, and that, under its\ninfluence, they developed a noble and heroic character. At the same time\nit must be conceded that their ethical system was marked by signal\nblemishes and radical defects. After all its excellence, it did not give\nroundness, completeness, and symmetry to moral life. The elements which\nreally purify and ennoble man, and lend grace and beauty to life, were\nutterly wanting. Their systems were rather a discipline of the reason\nthan a culture of the heart. The reason held in check the lower passions\nand propensities of the nature but it did not evoke the softer, gentler,\npurer emotions of the soul. The cardinal virtues of the ancient ethical\nsystems are Prudence, Justice, Temperance, and Courage, all which are in\nthe last analysis reduced to Wisdom. Humility, Meekness, Forgiveness of\ninjuries, Love of even enemies, Universal Benevolence, Real\nPhilanthropy, the graces which give beauty to character and bless\nsociety, are scarcely known. It is true that in Epictetus and Seneca we\nhave some counsels to humility, to forbearance, and forgiveness; but it\nmust be borne in mind that Christianity was now in the air, exerting an\nindirect influence beyond the limits of the labors of the indefatigable\nmissionaries of the Cross.[923] By their predecessors, these qualities\nwere disparaged rather than upheld. Resentment of injuries was applauded\nas a virtue, and meekness was proclaimed a defect and a weakness. They\nknew nothing of a forgiving spirit, and were strangers to the charity\n\"which endureth all things, hopeth all things, and never fails.\" The\nenlarged philanthrophy which overleaps the bounds of kindred and\nnationality, and embraces a common humanity in its compassionate regards\nand benevolent efforts, was unknown. Socrates, the noblest of all the\nGrecians, was in no sense cosmopolitan in his feeling. His whole nature\nand character wore a Greek impress. He could scarce be tempted to go\nbeyond the gates of Athens, and his care was all for the Athenian\npeople. He could not conceive an universal philanthropy. Plato, in his\nsolicitude to reduce his ideal state to a harmonious whole, answering to\nhis idea of Justice, sacrificed the individual. He superseded private\nproperty, broke up the sacred relations of family and home, degraded\nwoman, and tolerated slavery. Selfishness was to be overcome, and\npolitical order maintained, by a rigid communism. To harmonize\nindividual rights and national interests, was the wisdom reserved for\nthe fishermen of Galilee. The whole method of Plato's \"Politeia,\"\nbreathes the spirit of legalism in all its severity, untempered by the\nspirit of Love. This was the living force which was wanting to give\nenergy to the ideals of the reason and conscience, to furnish high\nmotive to virtue, to prompt to deeds of heroic sacrifice and suffering\nfor the good of others; and this could not be inspired by philosophy,\nnor constrained by legislation. This love must descend from above. \"The\nPlatonic love\" was a mere intellectual appreciation of beauty, and\norder, and proportion, and excellence. It was not the love of man as the\noffspring and image of God, as the partaker of a common nature, and the\nheir of a common immortality. Such love was first revealed on earth by\nthe incarnate Son of God, and can only be attained by human hearts under\nthe inspiration of his teaching and life, and the renewing influence of\nthe Holy Spirit. \"Love is of God, and every one that loveth is born of\nGod and knoweth God.\" To \"love our neighbor as ourself\" is the golden\nprecept of the Son of God, who is incarnate Love. The equality of all\nmen as \"the offspring of God\" had been nominally recognized by the Stoic\nphilosophers; its realization had been rendered possible to the popular\nthought by Roman conquest, law, and jurisprudence; these had prepared\nthe way for its fullest announcement and practical recognition by the\nworld. At this providential juncture St. Paul appears on Mars' Hill, and\nin the presence of the assembled philosophers proclaims, \"_God hath made\nof one blood all nations of men_.\" A lofty ideal of moral excellence had\nbeen attained by Plato--the conception of a high and inflexible\nmorality, which contrasted most vividly with the depravity which\nprevailed in Athenian society. The education \"of the public assemblies,\nthe courts, the theatres, or wherever the multitude gathered\" was\nunfavorable to virtue. And the inadequacy of all mere human teaching to\nresist this current of evil, and save the young men of the age from\nruin, is touchingly and mournfully confessed by Plato. \"There is not,\nthere never was, there never will be a moral education possible that can\ncountervail the education of which these are the dispensers; that is,\n_human_ education: I except, with the proverb, that which is Divine.\nAnd, truly, any soul that in such governments escapes the common wreck,\ncan only escape _by the special favor of heaven.\"_[924] He affirms again\nand again that man can not by himself rise to purity and goodness.\n\"Virtue is not natural to man, neither is it to be learned, but it comes\nto us by a divine influence. Virtue is the gift of God in those who\npossess it.\"[925] That \"gift of God\" was about to be bestowed, in all\nits fullness of power and blessing, \"_through Jesus Christ our Lord_.\"\n\n[Footnote 923: Seneca lived in the second century; Epictetus, in the\nlatter part of the first century.]\n\n[Footnote 924: \"Republic,\" bk. vi. ch. vi., vii.]\n\n[Footnote 925: \"Meno;\" see conclusion.]\n\nIn the department of _religious feeling_ and _sentiment_, the\npropaedeutic office of Greek philosophy is seen, in general, in the\nrevealing of the immediate spiritual wants of the soul, and the distinct\npresentation of the problem which Christianity alone can solve.\n\nI. _It awakened in man the sense of distance and estrangement from God,\nand the need of a Mediator--\"a daysman betwixt us, that might lay his\nhand upon us both_\"[926]\n\n[Footnote 926: Job ix. 33.]\n\nDuring the period of unconscious and unreflective theism, the sentiment\nof the Divine was one of objective nearness and personal intimacy. The\ngods interposed directly in the affairs of men, and held frequent and\nfamiliar intercourse with our race. They descend to the battle-field of\nTroy, and mingle in the bloody strife. They grace the wedding-feast by\ntheir presence, and heighten the gladness with celestial music. They\nvisit the poor and the stranger, and sometimes clothe the old and\nshrivelled beggar with celestial beauty. They inspire their favorites\nwith strength and courage, and fill their mouths with wisdom and\neloquence. They manifest their presence by signs and wonders, by visions\nand dreams, by auguries and prophetic voices. But more frequently than\nall, they are seen in the ordinary phenomena of nature, the sunshine and\nstorm, the winds and tempests, the hail and rain. The natural is, in\nfact, the supernatural, and all the changes of nature are the movement\nand action of the Divine. The feeling of dependence is immediate and\nuniversal, and worship is the natural and spontaneous act of man.\n\nBut the period of reflection is inevitable. Man turns his inquiring gaze\ntowards nature and desires, by an imperfect effort of physical\ninduction, to reach \"the first principle and cause of things.\" Soon he\ndiscovers the prevalence of uniformity in nature, the actions of\nphysical properties and agencies, and he catches some glimpses of the\nreign of universal law. The natural tendency of this discovery is\nobvious in the weakening of his sense of dependence on the immediate\nagency of God. The Egyptians told Herodotus that, as their fields were\nregularly irrigated by the waters of the Nile, they were less dependent\non God than the Greeks, whose lands were watered by rains, and who must\nperish if Jupiter did not send them showers.[927] As man advances in the\nfield of mere physical inquiry, God recedes; from the region of\nexplained phenomena, he retires into the region of unexplained\nphenomena--the border-land of mystery. The gods are driven from the\nwoods and streams, the winds and waves. Neptune does not absolutely\ncontrol the seas, nor AEolus the winds. The Divine becomes, no more a\nphysical arche--a nature-power, but a Supreme Mind, an ineffable Spirit,\nan invisible God, the Supreme Essence of Essences, the Supreme Idea of\nIdeas (eidos auto kath auto) apprehended by human reason alone, but\nhaving an independent, eternal, substantial, personal being. Through the\ninstrumentality of Platonism, the idea of God becomes clearer and purer.\nMan had learned that communion with the Divinity was something more than\nan apotheosis of humanity, or a pantheistic absorption. He caught\nglimpses of a higher and holier union. He had surrendered the ideal of a\nnational communion with God, and of personal protection through a\nfederal religion, and now was thrown back upon himself to find some\nchannel of personal approach to God. But alas! he could not find it. A\nGod so vastly elevated beyond human comprehension, who could only be\napprehended by the most painful effort of abstract thought; a God so\ninfinitely removed from man by the purity and rectitude of his\ncharacter; a God who was all pure reason, seemed alien to all the\nyearnings and sympathies of the human heart; and such a God, dwelling in\npure light, seemed inapproachable and inacessible to man.[928] The\npurifying of the religious idea had evoked a new ideal, and this ideal\nwas painfully remote. By the energy of abstract thought man had striven\nto pierce the veil, and press into \"the Holy of Holies,\" to come into\nthe presence of God, and he had failed. And he had sought by moral\ndiscipline, by self-mortification, by inward purification, to raise\nhimself to that lofty plane of purity, where he might catch some\nglimpses of the vision of a holy God, and still he failed. Nay, more, he\nhad tried the power of prayer. Socrates, and Plato, and Cleanthes had\nbowed the knee and moved the lips in prayer. The emperor Aurelius, and\nthe slave Epictetus had prayed, and prayer, no doubt, intensified their\nlonging, and sharpened and agonized their desire, but it did not raise\nthem to a satisfying and holy _koinonia_ in the divine life. \"It seems\nto me\"--said Plato--as Homer says of Minerva, that she removed the mist\nfrom before the eyes of Diomede,\n\n      'That he might clearly see 'twixt Gods and men.'\n\nso must he, in the first place, remove from your soul the mist that now\ndwells there, and then apply those things through which you will be able\nto know[929] and rightly pray to God.\n\n[Footnote 927: Herodotus, vol. ii. bk. ii. ch. xiii. p. 14 (Rawlinson's\nedition).]\n\n[Footnote 928: \"To discover the Maker and Father of the universe is a\nhard task;.... to make him known to all is impossible.\"--\"Timaeus,\" ch.\nix.]\n\n[Footnote 929: \"Second Alcibiades,\" Sec. 23.]\n\nTo develop this innate desire and \"feeling after God\" was the grand\ndesign of providence in \"fixing the times\" of the Greek nation, and \"the\nboundaries of their habitation.\"[930] Man was brought, through a period\nof discipline, to feel his need of a personal relation to God. He was\nmade to long for a realizing sense of his presence--to desire above all\nthings a Father, a Counsellor, and a Friend--a living ear into which he\nmight groan his anguish, or hymn his joy; and a living heart that could\nbeat towards him in compassion, and prompt immediate succor and aid. The\nidea of a pure Spiritual Essence without form, and without emotion,\npervading all, and transcending all, is too vague and abstract to yield\nus comfort, and to exert over us any persuasive power. \"Our moral\nweakness shrinks from it in trembling awe. The heart can not feed on\nsublimities. We can not make a home of cold magnificence; we can not\ntake immensity by the hand.\"[931] Hence the need and the desire that God\nshall condescendingly approach to man, and by some manifestation of\nhimself in human form, and through the sensibilities of the human heart,\ncommend himself to the heart of man--in other words, the need of an\n_Incarnation_. Thus did the education of our race, by the dispensation\nof philosophy, prepare the way for him who was consciously or\nunconsciously \"_the Desire of Nations_,\" and the deepening earnestness\nand spiritual solicitude of the heathen world heralded the near approach\nof Him who was not only \"the Hope of Israel\" but \"the Saviour of the\nworld.\"\n\n[Footnote 930: Acts xvii. 26, 27.]\n\n[Footnote 931: Caird.]\n\nThe idea of an _Incarnation_ was not unfamiliar to human thought, it was\nno new or strange idea to the heathen mind. The numberless metamorphoses\nof Grecian mythology, the incarnations of Brahm, the avatars of Vishnu,\nand the human form of Krishna had naturalized the thought.[932] So that\nwhen the people of Lystra saw the apostles Paul and Barnabas exercising\nsupernatural powers of healing, they said, \"The gods have come down to\nus in the likeness of men!\" and they called Barnabas Jupiter, and Paul,\nMercurius. The idea in its more definite form may have been, and indeed\nwas, communicated to the world through the agency of the dispersed Jews.\nSo that Virgil, the Roman poet, who was contemporary with Christ, seems\nto re-echo the prophecy of Isaiah--\n\n     The last age decreed by the Fates is come,\n     And a new frame of all things does begin;\n     A holy progeny from heaven descends\n     Auspicious in his birth, which puts an end\n     To the iron age, and from which shall arise\n     A golden age, most glorious to behold.\n\n[Footnote 932: Young's \"Christ of History,\" p. 248.]\n\nII. _Finally, Greek philosophy prepared the way for Christianity by\nawakening and deepening the consciousness of guilt, and the desire for\nRedemption_.\n\nThe consciousness of sin, and the consequent need of expiation for sin,\nwere gradually unfolded in the Greek mind. The idea of sin was at first\nrevealed in a confused and indefinite feeling of some external,\nsupernatural, and bewildering influence which man can not successfully\nresist; but yet so in harmony with the sinner's inclination, that he can\nnot divest himself of all responsibility. \"Homer has no word answering\nin comprehensiveness or depth of meaning to the word _sin_, as it is\nused in the Bible..... The noun _amartia_ which is appropriated to\nexpress this idea in the Greek of the New Testament, does not occur in\nthe Homeric poems..... The word which is most frequently employed to\nexpress wrong-doing of every kind is _ate_, with its corresponding\nverb..... The radical signification of the word seems to be a\nbefooling--a depriving one of his senses and his reason, as by\nunseasonable sleep, and excess of wine, joined with the influence of\nevil companions, and the power of destiny, or the deity. Hence, the\nGreek imagination, which impersonated every great power, very naturally\nconceived of Ate as a person, a sort of omnipresent and universal cause\nof folly and sin, of mischief and misery, who, though the daughter of\nJupiter, yet once fooled or misled Jupiter himself, and thenceforth,\ncast down from heaven to earth, walks with light feet over the heads of\nmen, and makes all things go wrong. Hence, too, when men come to their\nsenses, and see what folly and wrong they have perpetrated, they cast\nthe blame on Ate, and so, ultimately, on Jupiter and the gods.\"[933]\n\n[Footnote 933: Tyler, \"Theology of the Greek Poets,\" pp. 174, 175.]\n\n     \"Oft hath this matter been by Greeks discussed,\n      And I their frequent censure have incurred:\n      Yet was not I the cause; but Jove, and Fate,\n      And gloomy Erinnys, who combined to throw\n      A strong delusion o'er my mind, that day\n      I robb'd Achilles of his lawful prize.\n      What could I do? a Goddess all o'erruled,\n      Daughter of Jove, dread Ate, baleful power\n      Misleading all; with light step she moves,\n      Not on the earth, but o'er the heads of men.\n      With blighting touch, and many hath caused to err.\"[934]\n\nAnd yet, though Agamemnon here attempts to shuffle off the guilt of his\ntransgression upon Ate, Jove, and Fate, yet at other times he confesses\nhis folly and wrong, and makes no attempt to cast the responsibility on\nthe gods.[935] Though misled by a \"baleful power,\" he was not compelled.\nThough tempted by an evil goddess, he yet followed his own sinful\npassions, and therefore he owns himself responsible.\n\nTo satisfy the demands of divine justice, to show its hatred of sin, and\nto deter others from transgression, sin is punished. Punishment is the\npenalty due to sin; in the language of Homer, it is the payment of a\ndebt incurred by sin. When the transgressor is punished he is said to\n\"pay off,\" or \"pay back\" his crimes; in other words, to expatiate or\natone for them.\n\n                             \"If not at once,\n     Yet soon or late will Jove assert their claim,\n     And heavy penalty the perjured pay\n     With their own blood, their children's, and their wives'.\"[936]\n\nAt the same time the belief is expressed that the gods may be, and often\nare, propitiated by prayers and sacrifices, and thus the penalty is\nremitted.\n\n     \"The Gods themselves, in virtue, honor, strength,\n      Excelling thee, may yet be mollified;\n      For they when mortals have transgressed, or fail'd\n      To do aright, by sacrifice and pray'r,\n      Libations and burnt-off'rings, may be sooth'd.\"[937]\n\n[Footnote 934: \"Iliad,\" bk. xix. l. 91-101 (Lord Derby's translation).]\n\n[Footnote 935: Ibid., bk. ix. l. 132-136.]\n\n[Footnote 936: Ibid., bk. iv. l. 185-188.]\n\n[Footnote 937: Ibid., bk. ix. l. 581-585.]\n\nPolytheism, then, as Dr. Schaff has remarked, had the voice of\nconscience, and a sense, however obscure, of sin. It felt the need of\nreconciliation with deity, and sought that reconciliation by prayer,\npenance, and sacrifice.[938]\n\nThe sense of the exceeding sinfulness of sin, and the absolute need of\nexpiation, is determined with increasing clearness and definiteness in\nthe tragic poets.\n\nThe first great law which the Tragedians recognize, as a law written on\nthe heart, is \"that the sinner must suffer for his sins.\" The connection\nbetween sin and suffering is constantly recognized as a natural and\nnecessary connection, like that between sowing and reaping.\n\n     A haughty spirit, blossoming, bears a crop\n     Of woe, and reaps a harvest of despair.[939]\n\n\"Lust and violence beget lust and violence, and vengeance too, at the\nappointed time.\"[940] \"Impiety multiplies and perpetuates itself.\"[941]\n\"The sinner pays the debt he contracted, ends the career that he\nbegins,\"[942] \"and drinks to the dregs the cup of cursing which he\nhimself had filled.\"[943] Conscience is the instrument in the hands of\nJustice and Vengeance by which the Most High inflicts punishment. The\nretributions of sin are \"wrought out by God.\"\n\nThe consequences of great crimes, especially in high places, extend to\nevery person and every thing connected with them. \"The country and the\ncountry's gods are polluted.\"[944] \"The army and the people share in the\ncurse.\"[945] \"The earth itself is polluted with the shedding of\nblood,\"[946] \"and even the innocent and the virtuous who share the\nenterprises of the wicked may be involved in their ruin, as the pious\nman must sink with the ungodly when he embarks in the same ship.\"[947]\n\n[Footnote 938: Tyler, \"Theology of the Greek Poets,\" p. 258.]\n\n[Footnote 938: AEschylus, \"Persae,\" l. 821.]\n\n[Footnote 940: \"Agamemnon,\" l. 763.]\n\n[Footnote 941: Ibid., l. 788.]\n\n[Footnote 942: Ibid., l. 1529.]\n\n[Footnote 943: Ibid., l. 1397.]\n\n[Footnote 944: Ibid., l. 1645.]\n\n[Footnote 945: \"Persae,\" _passim._]\n\n[Footnote 946: \"Sup.,\" 265.]\n\n[Footnote 947: \"Theb.,\" p. 602.]\n\nThe pollution and curse of sin, when once contracted by an individual,\nor entailed upon a family, will rest upon them and pursue them till the\npolluted individual or the hated and accursed race is extinct, unless in\nsome way the sin can be expiated, or some god interpose to arrest the\npenalty. The criminal must die by the hand of justice, and even in Hades\nvengeance will still pursue him.[948] Others may in time be washed away\nby ablutions, worn away by exile and pilgrimage, and expiated by\nofferings of blood.[949] But great crimes can not be washed away; \"For\nwhat expiation is there for blood when once it has fallen on the\nground.\"[950] Thus the law (_[Greek: nomos]_)--for so it is expressly\ncalled--as from an Attic Sinai, rolls its reverberating thunders, and\npronounces its curses upon sin, from act to act and from chorus to\nchorus of that grand trilogy--the \"Agamemnon,\" the \"Choephoroe,\" and the\n\"Eumenides.\"\n\n[Footnote 948: \"Sup.,\" l. 227.]\n\n[Footnote 949: \"Eum.,\" l. 445 seq.]\n\n[Footnote 950: \"Choeph.,\" l. 47.]\n\nBut after the law comes the gospel. First the controversy, then the\nreconciliation. A dim consciousness of sin and retribution as a fact,\nand of reconciliation as a _want_, seems to have revealed itself even in\nthe darkest periods of history. This consciousness underlies not a few\nof the Greek tragedies. \"The 'Prometheus Bound' was followed by the\n'Prometheus Unbound,' reconciled and restored through the intervention\nof Jove's son. The 'oedipus Tyrannus' of Sophocles was completed by the\n'oedipus Colonus,' where he dies in peace amid tokens of divine favor.\nAnd so the 'Agamemnon' and 'Choephoroe' reach their consummation only in\nthe 'Eumenides,' where the Erinyes themselves are appeased, and the\nFuries become the gracious ones. This is not, however, without a special\ndivine interposition, and then only after a severe struggle between the\npowers that cry for justice and those that plead for mercy.\"\n\nThe office and work which, in this trilogy, is assigned to Jove's son,\nApollo, must strike every reader as at least a remarkable resemblance,\nif not a foreshadowing of the Christian doctrine of _reconciliation_.\n\"This becomes yet more striking when we bring into view the relation in\nwhich this reconciling work stands to [Greek: Zeus Soter], Jupiter\nSaviour--[Greek: Zeus tritos], Jupiter the third, who, in connection\nwith Apollo and Athena, consummates the reconciliation. Not only is\nApollo a [Greek: Soter], a Saviour, who, having himself been exiled from\nheaven among men, will pity the poor and needy;[951] not only does\nAthena sympathize with the defendant at her tribunal, and, uniting the\noffice of advocate and judge, persuade the avenging deities to be\nappeased;[952] but Zeus is the beginning and end of the whole process.\nApollo appears as the advocate of Orestes only at her bidding;[953]\nAthena inclines to the side of the accused, as the offspring of the\nbrain of Zeus, and of like mind with him.\"[954] Orestes, after his\nacquittal, says that he obtained it\n\n     \"By means of Pallas and of Loxias\n      And the third Saviour who doth all things sway.\"[955]\n\nPlatonism reveals a still closer affinity with Christianity in its\ndoctrine of sin, and its sense of the need of salvation. Plato is\nsacredly jealous for the honor and purity of the divine character, and\nrejects with indignation every hypothesis which would make God the\nauthor of sin. \"God, inasmuch as he is good, can not be the cause of all\nthings, as the common doctrine represents him to be. On the contrary, he\nis the author of only a small part of human affairs; of the larger part\nhe is not the author; for our evil things far outnumber our good things.\nThe good things we must ascribe to God, whilst we must seek elsewhere,\nand not in him, the causes of evil.\"[956] The doctrine of the poets,\nwhich would in some way charge on the gods the errors of men, he sternly\nresists. We must express our disapprobation of Homer, or any other poet,\nif guilty of such foolish blunders about the gods as to tell us[957]\n\n     'Fast by the threshold of Jove's court are placed\n      Two casks, one stored with evil, one with good,'\n\nAnd that he for whom the Thunderer mingles both\n\n     'He leads a life checker'd with good and ill.'\n\n[Footnote 951: \"Sup.,\" l. 214.]\n\n[Footnote 952: \"Eum.,\" l. 970.]\n\n[Footnote 953: Ibid., l. 616.]\n\n[Footnote 954: Ibid., l. 664, 737.]\n\n[Footnote 955: Tyler's \"Theology of the Greek Poets,\" especially ch. v.,\nfrom which the above materials are drawn.]\n\n[Footnote 956: \"Republic,\" bk. ii. ch. xviii.]\n\n[Footnote 957: \"Iliad,\" xxiv., l. 660.]\n\nNor can we let our young people know that, in the words of AEschylus--\n\n     \"'When to destruction God will plague a house\n       He plants among the members guilt and sin.'\"[958]\n\nWhatever in the writings of Homer and the tragic poets give countenance\nto the notion that God is, in the remotest sense the author of sin, must\nbe expunged. Here is clearly a great advance in ethical conceptions.\n\nThe great defect in the ethical system of Plato was the identification\nof evil with the inferior or corporeal nature of man--\"the irascible and\nconcupiscible elements,\" fashioned by the junior divinities. The\nrational and immortal part of man's nature, which is derived immediately\nfrom God--the Supreme Good, naturally chooses the good as its supreme\nend and destination. Hence he adopted the Socratic maxim \"that no man is\nwillingly evil,\" that is, no man deliberately chooses evil as evil, but\nonly as a _seeming_ good--he does not choose evil as an end, though he\nmay choose it voluntarily as a means. Plato manifests great solicitude\nto guard this maxim from misconception and abuse. Man has, in his\njudgment, the power to act in harmony with his higher reason, or\ncontrary to reason; to obey the voice of conscience or the clamors of\npassion, and consequently he is the object of praise or blame, reward or\npunishment. \"When a man does not consider himself, but others, as the\ncause of his own sins,.... and even seeks to excuse himself from blame,\nhe dishonors and injures his own soul; so, also, when contrary to\nreason.... he indulges in pleasure, he dishonors it by filling it with\nvice and remorse.\"[959] The work and effort of life, the end of this\nprobationary economy, is to make reason triumphant over passion, and\ndiscipline ourselves to a purer and nobler life.\n\n[Footnote 958: \"Republic,\" bk. ii. ch, xviii., xix.]\n\n[Footnote 959: \"Laws,\" bk. v. ch. i.]\n\nThe obstacles to a virtuous life are, however, confessedly numberless,\nand, humanly speaking, insurmountable. To raise one's self above the\nclamor of passion, the power of evil, the bondage of the flesh, is\nacknowledged, in mournful language, to be a hopeless task. A cloud of\nsadness shades the brow of Plato as he contemplates the fallen state of\nman. In the \"Phaedrus\" he describes, in gorgeous imagery, the purity, and\nbeauty, and felicity of the soul in its anterior and primeval state,\nwhen, charioteering through the highest arch of heaven in company with\nthe Deity, it contemplated the divine justice and beauty; but \"this\nhappy life,\" says he, \"we forfeited by our transgression.\" Allured by\nstrange affections, our souls forgot the sacred things that we were made\nto contemplate and love--we _fell_. And now, in our fallen state, the\nsoul has lost its pristine beauty and excellence. It has become more\ndisfigured than was Glaucus, the seaman \"whose primitive form was not\nrecognizable, so disfigured had he become by his long dwelling in the\nsea.\"[960] To restore this lost image of the good,--to regain \"this\nprimitive form,\" is not the work of man, but God. Man can not save\nhimself. \"Virtue is not natural to man, neither is it to be learned, but\nit comes by a divine influence. _Virtue, is the gift of God_.\"[961] He\nneeds a discipline, \"an education which is divine.\" If he is saved from\nthe common wreck, it must be \"by the special favor of Heaven.\"[962] He\nmust be delivered from sin, if ever delivered, by the interposition of\nGod.\n\n[Footnote 960: \"Republic,\" bk. x. ch. xi.]\n\n[Footnote 961: \"Meno.\"]\n\n[Footnote 962: \"Republic,\" bk. vi. ch. vi., vii.]\n\nPlato was, in some way, able to discover the need of a Saviour, to\ndesire a Saviour, but he could not predict his appearing. Hints are\nobscurely given of a Conqueror of sin, an Assuager of pain, an Averter\nof evil in this life, and of the impending retributions of the future\nlife; but they are exceedingly indefinite and shadowy. In all instances\nthey are rather the language of _desire_, than of hope. Platonism\nawakened in the heart of humanity a consciousness of sin and a profound\nfeeling of want--the want of a Redeemer from sin, a spiritual, a divine\nRemedy for its moral malady--and it strove after some remedial power.\nBut it was equally conscious of failure and defeat. It could enlighten\nthe reason, but it could only act imperfectly on the will. Platonic was\na striking counterpart to Pauline experience prior to the apostle's\ndeliverance by the power and grace of Christ. It discovered that \"the\nLaw is holy, and the commandment is holy, and just, and good.\" It\nrecognized that \"it is spiritual, but man is carnal, the slave of sin.\"\nIt could say, \"What I do I approve not; for I do not what I would, but\nwhat I hate. But if my will [my better judgment] is against what I do, I\nconsent unto the Law that it is good. And now it is no more I that do\nit, but sin, that dwelleth in me. For I know that in me, that is, in my\nflesh, good abideth not, for to will is present with me, but the power\nto do the right is absent: the good that I would, I do not; but the evil\nthat I would not, that I do. I consent gladly to the law of God in my\ninner man ['the rational and immortal nature'[963]]; but I behold a law\nin my members ['the irascible and concupiscible nature'[964]] warring\nagainst the law of my mind (or reason), and bringing me into captivity\nto the law of sin which is in my members. _Oh wretched man that I am!\nwho shall deliver me from the body of this death_?\"[965] Paul was able\nto say, \"I thank God (that he hath now delivered me), through Jesus\nChrist our Lord!\" Platonism could only desire, and hope, and wait for\nthe coming of a Deliverer.\n\n[Footnote 963: Plato.]\n\n[Footnote 964: Ibid.]\n\n[Footnote 965: Romans, vii.]\n\nThis consciousness of the need of supernatural light and help, and this\naspiration after a light supernatural and divine, which Plato inherited\nfrom Socrates, constrained him to regard with toleration, and even\nreverence, every apparent approach, every pretension, even, to a divine\ninspiration and guidance in the age in which he lived. \"'The greatest\nblessings which men receive come through the operation of _phrensy_\n([Greek: mania]--inspired exaltation), when phrensy is the gift of God.\nThe prophetess of Delphi, and the priestess of Dodona, many are the\nbenefits which in their phrensies (moments of inspiration) they have\nbestowed upon Greece; but in their hours of self-possession, few or\nnone. And too long were it to speak of the Sibyl, and others, who,\ninspired and prophetic, have delivered utterances beneficial to the\nhearers. Indeed, this word phrenetic or maniac is no reproach; it is\nidentical with mantic--prophetic.[966] And often when diseases and\nplagues have fallen upon men for the sins of their forefathers, some\nphrensy too has broken forth, and in prophetic strain has pointed out a\nremedy, _showing how the sin might be expiated, and the gods appeased_\n(by prayers, and purifications, and atoning rites).... So many and yet\nmore great effects could I tell you of the phrensy which comes from the\ngods.\"[967] Some have discerned in all this merely the food for a feeble\nridicule. They regard these sentiments as simply an evidence of the\npower and prevalence of superstition clouding the loftiest intellects in\nancient times. By the more thoughtful and philosophic mind, however,\nthey will be accepted as an indication of the imperishable and universal\nfaith of humanity in a supernatural and supersensuous world, and in the\npossibility of some communication between heaven and earth.[968] And\nabove all, it is a conclusive proof that Plato believed that the\nknowledge of _salvation_--of a remedy for sin, a method of expiation for\nsin, a means of deliverance from the power and punishment of sin, must\nbe revealed from Heaven.\n\n[Footnote 966: [Greek: Mania], phrensy; _[Greek: pantis]_, a\nprophet--one who utters oracles in a state of divine phrensy; _[Greek:\npantike]_, the prophetic art.]\n\n[Footnote 967: \"Phaedrus,\" Sec. 47-50 (Whewell's translation).]\n\n[Footnote 968: \"_Vetus opinio est_, jam usque ab heroicis ducta\ntemporibus, eaque et populi Romani et _omnium gentium_ firmata consensu,\nversari quandem inter homines divinationem.\"--Cicero, \"De Divin.,\" i.\nI.]\n\nPaul, then, found, even in that focus of Paganism, the city of Athens,\nreligious aspirations tending towards Jesus Christ. A true philosophic\nmethod, notwithstanding its shortcomings and imperfections, concluded by\ndesiring and seeking \"the Unknown God,\" by demanding him from all forms\nof worship, from all schools of philosophy. The great work of\npreparation in the heathen world consisted in the developing of the\n_desire_ for salvation. It proved that God is the great want of every\nhuman soul; that there is a profound affinity between conscience and the\nliving God; and that Tertullian was right when he wrote the \"Testimonium\nAnimae naturaliter Christianae.\"[969] And when it was sufficiently\ndemonstrated that \"the world by philosophy knew not God (as a Redeeming\nGod and Saviour), then it pleased God by the foolishness of preaching to\nsave them that believe.\" This was all a dispensation of divine\nprovidence, which was determined by, or \"in, the wisdom of God.\"[970]\n\nThe history of the religions and philosophies of human origin thus\nbecomes to us a striking confirmation of the truth of Christianity. It\nshows there is a wondrous harmony between the instinctive wants and\nyearnings of the human heart, as well as the necessary ideas and laws of\nthe reason, and the fundamental principles of revealed religion. There\nis \"a law written on the heart\"--written by the finger of God, which\ncorresponds to the laws written by the same finger on \"tables of stone.\"\nThere are certain necessary and immutable principles and ideas infolded\nin the reason of man, which harmonize with the revelations of the\nEternal Logos in the written word.[971] There are instinctive longings,\nmysterious yearnings of the human heart, to which that unveiling of the\nheart of God which is made in the teaching and life of the incarnate God\nmost satisfyingly answers. Within the depths of the human spirit there\nis an \"oracle\" which responds to the voice of \"the living oracles of\nGod.\"\n\n[Footnote 969: Pressense, \"Religions before Christ\" (Introduction);\nNeander, \"Church History,\" vol. i. (Introduction).]\n\n[Footnote 970: I Corinthians, i. 21.]\n\n[Footnote 971: \"The surmise of Plato, that the world of appearance\nsubsists in and by a higher world of Divine Thought, is confirmed by\nChristianity when it tells us of a Divine subsistence--that Eternal Word\nby whom and in whom all things consist.\"--Vaughan, \"Hours with the\nMystics,\" vol. i. p. 213.]\n\nHere, then, are two distinct and independent revelations--the unwritten\nrevelation which God has made to all men in the constitution of the\nhuman mind, and the external written revelation which he has made in the\nperson and teaching of his Son. And these two are perfectly harmonious.\nWe have here two great volumes--the volume of conscience, and the volume\nof the New Testament. We open them, and find they announce the _same_\ntruths--one in dim outline, the other in a full portraiture. There are\nthe same fundamental principles underlying both revelations. They both\nbear the impress of _divinity_. The history of philosophy may have been\nmarked by many errors of interpretation; so, also, has the history of\ndogmatic theology. Men may have often misunderstood and misinterpreted\nthe dictates of conscience; so have theologians misunderstood and\nmisinterpreted the dictates of revelation. The perversions of conscience\nand reason have been plead in defense of error and sin; and so, for\nages, have the perversions of Scripture been urged in defense of\nslavery, oppression, falsehood, and wrong. Sometimes the misunderstood\nutterances of conscience, of philosophy, and of science have been\narrayed against the incorrect interpretations of the Word of God. But\nwhen both are better understood, and more justly conceived, they are\nfound in wondrous harmony. When the New Testament speaks to man of God,\nof duty, of immortality, and of retribution, man feels that its\nteachings \"commend themselves to his conscience\" and reason. When it\nspeaks to him of redemption, of salvation, of eternal life and\nblessedness, he feels that it meets and answers all the wants and\nlongings of his heart. Thus does Christianity throw light upon the\noriginal revelations of God in the human conscience, and answers all the\nyearnings of the human soul. So it is found in individual experiences,\nso it has been found in the history of humanity. As Leverrier and Adams\nwere enabled to affirm, from purely mathematical reasoning, that another\nplanet must exist beyond _Uranus_ which had never yet been seen by human\neyes, and then, afterwards, that affirmation was gloriously verified in\nthe discovery of _Neptune_ by the telescope of Galle; so the reasonings\nof ancient philosophy, based on certain necessary laws of mind, enabled\nman to affirm the existence of a God, of the soul, of a future\nretribution, and an eternal life beyond the grave; and, then,\nsubsequently, these were brought fully into light, and verified by the\nGospel.\n\nWe conclude in the words of Pressense: \"To isolate it from the past,\nwould be to refuse to comprehend the nature of Christianity itself, and\nthe extent of its triumphs. Although the Gospel is not, as has been\naffirmed, the product of anterior civilizations--a mere compound of\nGreek and Oriental elements--it is not the less certain that it brings\nto the human mind the satisfaction vainly sought by it in the East as in\nthe West. _Omnia subito_ is not its device, but that of the Gnostic\nheresy. Better to say, with Clement of Alexandria and Origen, that the\nnight of paganism had its stars to light it, but that they called to the\nMorning-star which stood over Bethlehem.\"\n\n\"If we regard philosophy as a preparation for Christianity, instead of\nseeking in it a substitute for the Gospel, we shall not need to\noverstate its grandeur in order to estimate its real value.\"\n\n\n\n\nCONTENTS.\n\n\nA.\n\n     Abstraction, comparative and immediate, 187-189; 362-364.\n\n     AEschylus, his conception of the Supreme Divinity, 146;\n       his recognition of human guilt, and need of expiation, 515-517.\n\n     AEtiological proof of the existence of God, 487-489.\n\n     Anaxagoras, an Eclectic, 311;\n       in his physical theory an Atomist, 312;\n       taught that the Order of the universe can only be explained by\n       Intelligence, 312;\n       his psychology, 313;\n       the teacher of Socrates, 313.\n\n     Anaximander, his first principle _the infinite_, 290;\n       his infinite a chaos of primary elements, 290.\n\n     Anaximenes, a vitalist, 286;\n       his first principle _air_, 287.\n\n     Aristotle, his opinion of the popular polytheism of Greece, 157;\n       his classification of causes, 280, 404, 405;\n       his misrepresentations of Pythagoras, 299;\n       his classification of the sciences, 389;\n       his Organon, 389-394;\n       his Logic, 394-403;\n       his Theology, 404-417;\n       his Ethics, 417-421;\n       his Categories, 395;\n       his logical treatises, 396;\n       on induction and deduction, 396-398;\n       his psychology, 398, 401;\n       on how the knowledge of first principles is attained, 394, 402,\n          403;\n       on Matter and Form, 405-408;\n       on Potentiality and Actuality, 408-412;\n       his proof of the Divine existence, 412-415;\n       on the chief good of man, 419, 420;\n       his doctrine of the Mean, 420, 421;\n       defect of his ethical system, 505.\n\n     [Archai], or first principles, the grand object of\n        investigation in Greek Philosophy, 271, 274, 279, 280.\n\n     Athenians, criticism on Plutarch's sketch of their character, 45;\n       their vivacity, 45;\n       love of freedom, 46--and of country, 46;\n       private life of, 47;\n       intellectual character of, 48;\n       inquisitive and analytic, 48;\n       rare combinations of imagination and reasoning powers, 49;\n       religion of, 98;\n       the Athenians a religious people, 102;\n       their faith in the being and providence of God, 107;\n       their consciousness of dependence on God, 110, 116;\n       their religious emotions, 117;\n       their deep consciousness of sin and guilt, 122-124;\n       their sense of the need of expiation, 124, 125;\n       their religion exerted some wholesome moral influence, 162, 163.\n\n     Athens, topography of 27;\n       the Agora, 28;\n       its porticoes, 29;\n       the Acropolis, 30;\n       its temples, 31;\n       the Areopagus, 33;\n       sacred objects in, 98, 99;\n       images of the gods, 99;\n       localities of schools of philosophy in, 266-268.\n\n     Attica, geographical boundaries of, 26;\n       a classic land, 34;\n       its geographical and cosmical conditions providentially ordained\n       for great moral   ends, 34, 35; soil of, not favorable to\n       agriculture, 40--necessitated industry and frugality, 41; the\n       climate of, 41--its influence on the mental character of the\n       people, 42.\n\n\nB.\n\n     Bacon, his assertion that the search after final causes had misled\n       scientific inquirers, 222.\n\n\nC.\n\n     Categories of Aristotle, 395.\n\n     Causality, principle of, 189;\n       assailed by the Materialists, 194--especially by Comte, 203-209;\n       the intuition of _power_ a fact of immediate consciousness, 204;\n       consciousness of _effort_ the type of all force, 211;\n       Aristotle on Causality, 413;\n       aetiological proof of existence of God, 487-489.\n\n     Cause, origin of the idea of, 204, 205.\n\n     Causes, Aristotle's classification of, 280, 404, 405.\n\n     Chief good of man, Aristotle on, 419, 420.\n\n     Cleanthes, his hymn to Jupiter, 452, 453.\n\n     Comte, his theory of the origin of religion, 57-65;\n       his doctrine that all knowledge is confined to material\n       phenomena, 203; denies all causation, both efficient and final,\n       203-214.\n\n     Conditioned, law of the, 227, 228;\n       is contradictory, 250;\n       as a ground of faith, meaningless and void, 251.\n\n     Cosmological proof of the existence of God, 489, 490.\n\n     Cousin, his theory that religion had its outbirth in the\n       spontaneous apperceptions\n       of reason, 78-84;\n       criticism thereon, 84-86.\n\n     Criterion of truth, Plato's search after, 333, 334.\n\n     Cudworth, his interpretation of Grecian mythology, 139, 143.\n\n     Cuvier, on final causes, 216, 222.\n\nD.\n\n     Darwin, his inability to explain the facts of nature without\n       recognizing design, 221, 222.\n\n     Democritus, taught that atoms and the vacuum are the beginning of\n       all things, 292; an absolute materialist, 293.\n\n     Dependence, consciousness of, the foundation of primary religious\n       emotions, 110-113.\n\n     Development, law of mental, 478; three successive stages clearly\n       marked, in the individual, 478--in the universal mind of\n       humanity, 479, 480;\n           (1) in the field of Theistic conceptions, 481-494;\n           (2) in the department of morals, 495-509;\n           (3) in the department of religious sentiment, 509-522.\n\n     Dialectic of Plato, 353-369.\n\n     Dogmatic Theologians, assert that all our knowledge of God is\n       derived from the teaching of the Scriptures, 86,167; cast doubt\n       upon the principle of causality, 253-255--upon the principle of\n       the unconditioned, 255-257--upon the principle of unity,\n       258-261--and upon the immutable principles of morality, 261-263.\n\n     Dynamical or Vital school of ancient philosophers, 282-289.\n\nE.\n\n     Eclecticism of Anaxagoras, 311.\n\n     Emotions, the religious, 117-122;\n       sentiment of the Divine exists in all minds, 119-121;\n       also instinctive yearning after the Invisible, 121, 122.\n\n     Empedocles, a believer in one Supreme God, 153.\n\n     Epicurus, his theory of the origin of religion, 56, 57;\n       his Ethics, 427-432;\n       his Physics, 433-438;\n       taught that pleasure is the chief end of life, 428--that\n       ignorance of nature is the sole cause of unhappiness, 432--that\n       Physics and Psychology are the only studies conducive to\n       happiness, 432--that the universe is eternal and infinite,\n       433--that concrete bodies are combinations of atoms, 434--that\n       atoms have spontaneity, 436, and some degree of freedom, 436,\n       437; the parts of the world self-formed, 437, 438; plants,\n       animals, and man are spontaneously generated, 438; a state of\n       savagism the primitive condition of man, 439; his Atheism, 441;\n       his Psychology, 442-444; the soul material and mortal, 445, 446.\n\n     Eternity, Platonic notion of, 349 (_note_), 372, 373.\n\n     Eternity of Matter, how taught by Plato, 371-373;\n       distinctly affirmed by Epicurus, 433.\n\n     Eternity of the Soul, Plato's doctrine of, 373-375.\n\n     Ethical ideas and principles, gradual development of, 495, 496;\n       (1) the age of popular and unconscious morals, 497, 498;\n       (2) the transitional or sophistical age, 498-500;\n       (3) the philosophic or conscious age, 500-506.\n\n     Ethics of Plato, 383-387, 502-505;\n       of Aristotle, 417-42l;\n       of Epicurus, 427-432;\n       of the Stoics, 454, 456.\n\n     Expiation for sin, the need of, 124;\n       universally acknowledged, 124--especially in Grecian mythology,\n       125--and in the language of Greece and Rome, 125.\n\nF.\n\n     Facts of the universe, classification of, 175-177.\n\n     Fathers, the early, recognized the propaedeutic office of Greek\n       philosophy, 473-475.\n\n     Feeling, theories which ground all religion on, 70-74;\n       its inadequacy, 74-78.\n\n     Final Causes, impossibility of interpreting nature without\n       recognizing, 221, 222;\n       the assumption of final causes a means of discovery, 222, 223;\n       Cuvier on, 216, 222;\n       argument of Socrates from, 320-324;\n       Plato on, 380-382;\n       Aristotle on, 405, 413, 414;\n       teleological proof of the existence of God, 490, 491.\n\n     Force, the idea of, rejected by Comte, 207.\n\n     Forces, all of one type, and that type mind, 211.\n\n     Freedom, human, 19;\n       exists under limitations, 20;\n       both admitted and denied by Comte, 208, 209;\n       of Will, as taught by Plato, 386, 387;\n       admitted by Epicurus, 486.\n\nG.\n\n     Geoffrey St. Hilaire, his pretense of not ascribing any intentions\n       to nature, 216, 217.\n\n     Geography and History, relations between, 14;\n       opposite theories concerning, 15;\n       theory of Buckle, 16--of Ritter, Guyot, and Coubin, 16;\n       the relation one of adjustment and harmony, 16.\n\n     God, universality of idea of, 89;\n       Athenians believed in one God, 107, 147, 148;\n       idea of God a common phenomenon of human intelligence, 168, 169;\n       the development of this idea dependent on experience conditions,\n       169-172; the phenomena of the universe demand a God for their\n       explanation, 172-175: there are principles revealed in\n       consciousness which necessitate the idea of God, 184-189; proofs\n       of the existence of God employed by Aristotle, 412-416--by\n       Socrates, 320-324; views of God entertained by the Stoics, 452,\n       453; logical proofs of the existence of God developed by Greek\n       philosophy, 487-494; gradual development of Theistic conception,\n       481-487.\n\n     Gods of Grecian Mythology, how regarded by the philosophers,\n       151-157; views of Plato regarding them, 383.\n\n     Great men, represent the spirit of their age, 20;\n       the creation of a providence interposing in history, 21.\n\n     Greece, its geographical relations favorable to free intercourse\n       with the great historic nations, 35--to commerce, 36--to the\n       diffusion of knowledge, 36--and to a high degree of civilization,\n       36; peculiar configuration of Greece conducive to activity and\n       freedom, 36-38--and independence, 38; natural scenery, 43--its\n       influence on imagination and taste, 44.\n\n     Greek Civilization, a preparation for Christianity, 465-468.\n\n     Greek Language, a providentially prepared vehicle for the perfect\n       revelation of Christianity, 468-470.\n\n     Greek Philosophy, first a philosophy of Nature, 271, 281, 282;\n       next a philosophy of Mind, 271, 316-318;\n       lastly a philosophy of Life, 271, 422;\n       prepared the way for Christianity, 457-522.\n\n     Greeks, the masses of the people believed in one Supreme God, 147,\n       148.\n\n     Guilt, consciousness of, a universal fact, 122, 123;\n       recognized in Grecian mythology, 123, 124;\n       awakened and deepened by philosophy, 513-518.\n\nH.\n\n     Hamilton, Sir W., teaches that philosophic knowledge is the\n       knowledge of effects as dependent on causes, 224, 225;\n       and of qualities as inherent in substances, 225, 226;\n       and yet asserts all human knowledge is necessarily confined to\n       phenomena, 227;\n       his doctrine of the relativity of all knowledge, 227, 229-236;\n       his philosophy of the conditioned, 228;\n       conditional limitation the law of all thought, 236-242;\n       the Infinite a mere negation of thought, 242-246;\n       asserts we must believe in the infinity of God, 246;\n       takes refuge in faith, 247;\n       faith grounded on the law of the conditioned, 243, 249--that is,\n       on contradiction, 249, 250.\n\n     Hegel, his philosophy of religion, 65-70.\n\n     Heraclitus, his first principle _ether_, 288;\n       change, the universal law of all existence, 288;\n       a Materialistic Pantheist, 289.\n\n     Hesiod, on the generation of the gods, 142.\n\n     Homer, his conception of Zeus, 144, 145.\n\n     Homeric doctrine of sin, 513,514.\n\n     Homeric theology, 143-145, 509, 510.\n\n     Humanity, fundamental ideas and laws of, 18;\n       developed and modified by exterior conditions, 19;\n       the most favorable conditions existed in Athens.\n\nI.\n\n     Idealism, furnishes no adequate explanation of the common belief\n       in an external world, 193,199--and of a personal self, 200-202;\n       Cosmothetic Idealism, 305;\n       absolute Idealism, 305.\n\n     Ideas, Platonic doctrine of, 334-337;\n       Platonic scheme of, 364-367.\n\n     Images of the gods, how regarded by Cicero, 129--by Plutarch, 129;\n       the heathens apologized for the use of images, 159.\n\n     Immortality of the soul, taught by Socrates, 324--and by Plato,\n       375, 376; denied by Epicurus, 444-446.\n\n     Incarnation, the idea of, not unfamiliar to heathen thought, 512.\n\n     Induction, the psychological method of Plato, 356, 357.\n\n     Induction and Deduction, Aristotle on, 397, 398.\n\n     Infinite, the, not a mere negation of thought, 242-244;\n       known as the necessary correlative of the finite, 245;\n       as comprehensible in itself, as the finite is comprehensible in\n       itself, 246;\n       in what sense known, 252.\n\n     Infinite Series, the phrase, when literally construed, a\n       contradiction, 181,182.\n\n     Infinity, qualitative and quantitative, 239;\n       qualitative infinity possessed by God alone, 184, 239.\n\n     Intentionality, principle of, 190;\n       denied by Materialists, 194;\n       a first law of thought, 221-223; recognized by Socrates, 320-324.\n\n     Ionian School of Philosophy, a physical and sensational school,\n       281; subdivided into Mechanical and Dynamical, 282, 283.\n\n     Italian School of Philosophy, an Idealist school, 281;\n       subdivided into the Mathematical and Metaphysical, 282, 296.\n\nJ.\n\n     Jacobi, his faith-philosophy, 71.\n\nK.\n\n     Knowledge, Hamilton's doctrine of relativity of, 229-236;\n       opposite theories of knowledge among ancient philosophers, 330,\n       331; the tendency of these theories, 332;\n       Plato's theory of, 333, 334;\n       Plato's science of real knowledge, 337, 338.\n\n\n     L.\n\n     Language, inadequate to convey the idea of God, 92-94;\n       Greek language the best medium for the Christian revelation,\n       468-470.\n\n     Leucippus, his first principles _atoms_ and _space_, 291;\n       a pure Materialist, 292.\n\n     Logic of Aristotle, 394-403.\n\n     Logical Treatises of Aristotle, 395, 396.\n\n     Lucretius, the expounder of the doctrines of Epicurus, 426,427;\n       his account of the origin of worlds, 437, 438;\n       of plants, animals, and man, 438.\n\nM.\n\n     Mansel, bases religion on feeling of dependence, 72--and sense of\n       obligation, 73.\n\n     Materialists deny the principle of causality, 194, 203--and of\n       intentionality or final cause, 211-225;\n       Anaximander, Leucippus, and Democritus belong to the\n       materialistic school, 286-293:\n       Epicurus a materialist, 442-446.\n\n     Mathematical Infinite, not absolute, 179, 180;\n       capable of exact measurement, therefore limited, 180;\n       infinite sphere, radius, line, etc., self-contradictory, 180,\n       181.\n\n     Matter, did Plato teach the eternity of? 371-373;\n       the doctrine of the Stoics concerning matter, 449 (_note_).\n\n     Matter and Form, Aristotle on, 405-408.\n\n     Mean, Aristotle's doctrine of the, 420.\n\n     Mediator, consciousness of the need of a, awakened by Greek\n       philosophy, 509-513.\n\n     Metaphysical thought, law of its development, 478-480;\n       three different stages in the individual mind, 478, 479;\n       and in the universal consciousness of our race, 479.\n\n     Metempsychosis regarded by Plato as a mere hypothesis, 376\n       (_note_).\n\n     Mill, J. S., his doctrine that all knowledge is confined to mental\n       phenomena, 193;\n       his definition of matter, 196;\n       his views of personal identity, 196, 197;\n       his theological opinions, 197.\n\n     Miracles, not designed to prove the existence of God, 95.\n\n     Moral principles, universal and immutable, which lead to the\n       recognition of a God, 190;\n       the Dogmatic Theologians seek to invalidate the argument\n       therefrom, 261-263.\n\n     Mystics, base all religious knowledge on internal feeling, 70.\n\n     Mythology, philosophy of Greek, 134-139;\n       Cudworth's interpretation of, 139-143;\n       recognized the consciousness of guilt and need of expiation,\n       123-125.\n\n N.\n\n     National Character, a complex result, 17;\n       conjoint effect of moral and physical influences, 17;\n       human freedom not to be disregarded in the study of, 20;\n       influence of geographical surroundings, 23--of climate and\n       natural scenery, on the pursuits and mental character of nations,\n       23--on creative art, 24--and literature of nations, 25.\n\n     Nations, individuality of, 22;\n       determined mainly from without, 22.\n\n     Natural Realism, 305;\n       Anaxagoras a natural realist, 311-313.\n\n     Nature, interpreted by man according to fundamental laws of his\n      reason, 133.\n\nO.\n\n     Obligation, the sense of, lies at the foundation of religion, 115.\n\n     Ontological proof of the existence of God, 491-493.\n\n     Ontology, of Plato, 369-379;\n       the subject-matter of the world of sense, 370-373;\n       the permanent substratum of mental phenomena, 373-376;\n       the first Principle of all principles--God, 377-379, 491-493.\n\n     Optimism of Plato, 382.\n\n     Order of the Universe, had it a beginning, or is it eternal?\n       178-184.\n\n     Order, principle of, pervades the universe, 220, 221;\n       recognized by Pythagoras, 301;\n       Cosmological proof of the existence of God, 489, 490.\n\n P.\n\n     Parmenides, his theory of knowledge, 307-308;\n      a spiritualistic Pantheist, 308, 309.\n\n     Paul, St., at Athens, 14;\n       his emotion when he saw the city full of idols, 100;\n       the subject of his discourse, 101;\n       brought into contact with all the phases of philosophic thought,\n       268, 269;\n       his arrival at Athens an epoch in the moral history of the world,\n       472;\n       he recognized the preparatory office of Greek philosophy, 473.\n\n     Philosophers of Athens, 101;\n       believed in one supreme, uncreated, eternal God, 151-157;\n       their views of the mythological deities, 158, 159;\n       their apologies for images and image-worship, 159, 160.\n\n     Philosophic Schools, classification of, 271-273;\n       Pre-Socratic 280-314;\n       Socratic, 314-421;\n       Post-Socratic, 422-456.\n\n     Philosophy, the world-enduring monument of the glory of Athens,\n       265, 260;\n       defined, 270, 271;\n       an inquiry after first causes and principles, 271, 457;\n       not in any proper sense a theological inquiry, 273-277, 279;\n       the love of wisdom, 384, 385.\n\n     Philosophy in its relation to Christianity, 268-270;\n       sympathy of Platonism, 268;\n       antagonism of Epicureanism and Stoicism, 269;\n       the Propaedeutic office of philosophy, 457-524--recognized by St.\n       Paul, 473--and many of the early Fathers, 473-475;\n       philosophy undermined Polytheism, and purified the Theistic idea,\n       481-487;\n       developed the Theistic argument in a logical form, 487-494;\n       it awakened Conscience and purified the Ethical idea, 495-506;\n       demonstrated the insufficiency of reason to elaborate a perfect\n       ideal of moral excellence, 506-509;\n       awakened in man the sense of distance from God, and the need of\n       a Mediator, 509-513;\n       deepened the consciousness of sin, and the desire for a Redeemer,\n       513-522;\n       the history of philosophy a confirmation of the truth of\n       Christianity, 522-524.\n\n     Philosophy of Religion, 53;\n       based on the correlation between Divine and human reason,\n       458-462.\n\n     Plato, condemns the poets for their unworthy representations of\n       the gods, 130-132;\n       his views of the gods of Grecian mythology, 154-157:\n       the sympathy of his philosophy with Christianity, 268:\n       followed the philosophic method of Socrates, 328;\n       his moral qualifications for the study of philosophy, 328, 329;\n       his literary qualifications, 329, 330;\n       his search after a criterion of truth, 333, 334;\n       his doctrine of Ideas, 334-337;\n       his science of real knowledge, 337, 338;\n       his answer to the question, What is Science? 338, 339;\n       his Psychology 339-352;\n       his scheme of the intellectual powers, 345;\n       on the nature of the soul, 350;\n       his dialectic, 353-369;\n       his grand scheme of ideas, 364-367;\n       his Ontology, 369-379;\n       on the creation of time, 372;\n       did he teach that matter is eternal? 371, 372;\n       on the eternity of the rational element of the soul, 373-375;\n       on the immortality of the soul, 375, 376;\n       on God as the First Principle of all principles, 377-379;\n       his Physics, 380-383;\n       his Ethics, 383-387, 502-505;\n       defects of his\n       ethical system, 518;\n       his philosophy not derived from Jewish sources, 476;\n       felt the need of a superhuman deliverer from sin and guilt,\n       519-521.\n\n     Plutarch, his sketches of Athenian character, 44;\n       criticism on, 45;\n       on the universality of prayer and sacrifice, 115.\n\n     Poets, the Greek, believed in the existence of one uncreated Mind,\n       141;\n       their theogony was a cosmogony, 142;\n       the theologians of Greece, 274, 275.\n\n     Polytheism, Greek, a poetico-historical religion of myth and\n       symbol, 134;\n       its immoralities, 160, 161;\n       undermined by Philosophy, 484-487.\n\n     Post-Socratic Schools, classification of, 425;\n       a philosophy of life, 422-424.\n\n     Potentiality and Actuality, Aristotle on, 408-412.\n\n     Prayer, natural to man, 115.\n\n     Preparation for Christianity, not confined to Judaism alone,\n       464, 465;\n       Greek civilization also prepared the way for Christ, 465-468;\n       Greek language a providential development as the vehicle of a\n       more perfect revelation, 468-470;\n       Greek philosophy fulfilled a propaedeutic office, 470-472.\n\n     Pre-Socratic Schools, classification of, 280-282; 295, 296.\n\n     Principles, _universal and necessary_, how attained by the method\n       of Plato, 361-364, 390;\n       how, by the method of Aristotle, 390-394, 402, 403.\n\n     Psychological analysis, logical demonstration of the existence of\n       God begins with, 170;\n       reveals principles which in their logical development attain to\n       the knowledge of God, 184-189.\n\n     Psychology of Heraclitus, 289;\n       of Pythagoras, 304;\n       of Parmenides, 307, 308;\n       of Anaxagoras, 313;\n       of Protagoras, 315;\n       of Socrates, 317, 318;\n       of Plato, 339-352;\n       of Aristotle, 392, 398-401;\n       of Epicurus, 442-444;\n       of the Stoics, 453, 454.\n\n     Pythagoras, his doctrine that numbers are the first principles of\n       things, 297;\n       how to be interpreted, 297-304;\n       misrepresented by Aristotle, 298-300;\n       psychology of, 304.\n\nR.\n\n     Reason, insufficiency of, to elaborate a perfect ideal of moral\n       excellence, 505-509.\n\n     Redemption, desire of, awakened and defined by Greek philosophy,\n       513-521.\n\n     Relativity of all knowledge, Hamilton's doctrine of, 229-236.\n\n     Religion, the philosophy of, 53;\n       defined 53, 106;\n       universality of religious phenomena, 54;\n       hypothesis offered in explanation of, 55;\n       hypothesis of Epicurus and Comte, 56-65--of Hegel, 65-70--of\n       Jacobi and Schleiermacher, 70-78--of Cousin, 78-86--of Dogmatic\n       Theologians, 86-96--author's theory, 96, 97;\n       religion of the Athenians, 98--its mythological and symbolic\n       aspects, 128--exerted some wholesome influences, 161-163.\n\n     Reminiscence, Plato on, 354, 355.\n\n     Revelation, progressive, 462-464;\n       harmony of the two revelations in the volume of conscience and\n       the volume of the New Testament, 522-524.\n\nS.\n\n     Sacrifice, universal prevalence of, 115, 124;\n       prompted by the universal consciousness of guilt, 126:\n       expiatory sacrifices grounded on a primitive revelation, 127.\n\n     Schleiermacher, his theory that all religion is grounded on the\n       feeling of absolute dependence, 71, 72.\n\n     Science, Plato's answer to the question, What is Science? 338, 339.\n\n     Self-determination, limited by idea of duty, 113;\n       implies accountability, 114;\n       recognizes a Lawgiver and Judge, 115.\n\n     Socrates, his desire for truth, 316;\n       his daemon, 317 _(note_);\n       his philosophic method, 318, 319;\n       a believer in one Supreme God, 320;\n       his argument for the existence of God from final causes, 320-324;\n       his belief in immortality and a future retribution, 324, 325;\n       his Ethics, 325;\n       the great prophet of the human conscience, 500-502.\n\n     Socratic School, 314.\n\n     Sophists, 315, 316;\n       their skeptical tendency, 315;\n       their defective ethics, 498, 499.\n\n     Sophocles, believed in one Supreme God, 147.\n\n     Soul, Plato on the nature of the, 350, 373;\n       eternity of the rational element, 373-375.\n\n     Spencer, H., carries the law of the Conditioned forward to its\n       logical consequences, Atheism, 241, 242.\n\n     Stoical School, 446;\n       its philosophy a moral philosophy, 447.\n\n     Stoics, their Physiology, 448-453; their\n       Psychology, 453, 454;\n       their Ethics, 454-456;\n       their Theology, 452,453.\n\n     Substance, principle of, 189;\n       Idealism seeks to undermine it, 193;\n       Reason affirms a permanent substance as the ground of all mental\n       phenomena, 201--and of the phenomena of the sensible world,\n       202, 203.\n\n     Sufficient Reason, law of, recognized by Plato, 359.\n\n     Superstition, meaning of the term as used by Paul, 103.\n\n T.\n\n     Teleological proof of the existence of God, 490, 491.\n\n     Thales, a believer in one uncreated God, 152;\n       his first principles, 283;\n       he regards _water_ as the material cause, 284;\n       and God as the efficient cause, 285.\n\n     Theistic argument, in its logical form, 487-494.\n\n     Theistic conception, gradual development of, 481-484,\n\n     Theological opinions of the early periods of Greek civilization,\n       150, 151; 276-278.\n\n     Theology of Aristotle, 404-417;\n       identical with Metaphysics, 404, 416.\n\n     Theology of the Greek poets, 143-151;\n       proposed reform of Poetry by Plato, 131, 132.\n\n     Thinking, conditionality of, 228;\n       in what sense to be understood, 237;\n       thought imposes no limits upon the object of thought, 237, 238.\n\n     Thought, negative and positive, 242, 243;\n       negative thought an impossibility, 243;\n       all thought must be positive, 243.\n\n     Time, Platonic notion of, 371, 372.\n\n     Tragedians, the Greek, were the public religious teachers of the\n       Athenians, 145;\n       their theology, 146, 147;\n       influence of the religious dramas on the Athenian mind, 161-163;\n       guiltiness of man, and need of reconciliation confessed by,\n       515-517.\n\nU.\n\n     Unconditioned, principle of, 189;\n       assailed by Hamilton, 194.\n\n     Unity of God, 259;\n       an affirmation of reason, 259-261;\n       Xenophanes taught the unity of God, 307--also Parmenides,\n       309--and Plato, 377--and Aristotle, 415.\n\n     Unity, principle of, 189;\n       attempt of Dogmatic Theologians to prove its insufficiency, 194,\n       258-261;\n       recognized by Pythagoras, 296;\n       his effort to reduce all the phenomena of nature to a Unity, 303,\n       304.\n\n     Universal and necessary Principles, classification of, 189, 190;\n       these the foundation of our cognition of a God, 191;\n       how attained according to Plato, 360-364;\n       how by the method of Aristotle, 390-394, 402, 403.\n\n     Universe, the, is it finite or infinite? 178-184;\n       Epicurus teaches that it is infinite, 433.\n\n     Unknown God, the true God, 104;\n       God not absolutely unknown, 107-110;\n       classification of opponents to the doctrine that God can be\n       cognized by reason, 166-168;\n       Idealist School of Mill, 194-203;\n       Materialistic School of Comte, 203-223;\n       Hamiltonian School, 224-252;\n       School of Dogmatic Theologians, 252-263.\n\nW.\n\n     Watson, Richard, represents the views of Dogmatic Theologians 86;\n       asserts that all our religious knowledge is derived from oral\n       revelation, 86-88, 167;\n       incompleteness and inadequacy of this theory, 88-96;\n       in vindicating for the Scriptures the honor of revealing all our\n       knowledge of God, he casts doubt upon the principle of Causality,\n       253-255--on the principle of the Unconditioned, 255-257--on the\n       principle of Unity, 258-261--and on the immutable principles of\n       Morality, 261-263.\n\n     Wordsworth, on the Sentiment of the Divine, 118.\n\nX.\n\n     Xenophanes, his attack on Polytheism, 130;\n       his faith in one God, 153, 306, 307.\n\nZ.\n\n     Zeno of Citium, the founder of the Stoical School, 446;\n       a Spiritualistic Pantheist, 450, 451.\n\n     Zeno of Elea, maintained the doctrine of Absolute Identity, 309.\n\n     Zeus, originally the Supreme and only God of the Greeks, 143;\n       the Homeric Zeus, the Supreme God, 144, 145.\n\n\nTHE END.\n\n\n\nVALUABLE STANDARD WORKS\nFOR PUBLIC AND PRIVATE LIBRARIES,\nPublished by HARPER & BROTHERS, New York.\n\n\n_For a full List of Books suitable for Libraries, see _Harper &\nBrother's Trade-List _and_ Catalogue, _which may be had gratuitously on\napplication to the Publishers personally, or by letter enclosing Five\nCents_.\n\n\nHARPER & BROTHERS _will send any of the following works by mail, postage\nprepaid, to any part of the United States, on receipt of the price_.\n\nMOTLEY'S DUTCH REPUBLIC. 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Arctic Researches and Life among the\nEsquimaux: being the Narrative of an Expedition in Search of Sir John\nFranklin, in the Years 1860, 1861, and 1862. By Charles Francis Hall.\nWith Maps and 100 Illustrations. The Illustrations are from Original\nDrawings by Charles Parsons, Henry L. Stephens, Solomon Eytinge, W.S.L.\nJewett, and Granville Perkins, after Sketches by Captain Hall. 8vo,\nCloth, $5 00.\n\nHALLAM'S CONSTITUTIONAL HISTORY OF ENGLAND, from the Accession of Henry\nVII. to the Death of George II. 8vo, Cloth, $2 00.\n\nHALLAM'S LITERATURE. Introduction to the Literature of Europe during the\nFifteenth, Sixteenth, and Seventeenth Centuries. By Henry Hallam. 2\nvols. 8vo, Cloth, $4 00.\n\nHALLAM'S MIDDLE AGES. State of Europe during the Middle Ages. By Henry\nHallam. 8vo, Cloth, $2 00.\n\nHILDRETH'S HISTORY OF THE UNITED STATES. First Series: From the First\nSettlement of the Country to the Adoption of the Federal Constitution.\nSecond Series: From the Adoption of the Federal Constitution to the End\nof the Sixteenth Congress. 6 vols., 8vo, Cloth, $18 00.\n\nJAY'S WORKS. Complete Works of Rev. William Jay: comprising his Sermons,\nFamily Discourses, Morning and Evening Exercises for every Day in the\nYear, Family Prayers, &c. Author's enlarged Edition, revised. 3 vols.,\n8vo, Cloth, $6 00.\n\nJOHNSON'S COMPLETE WORKS. The Works of Samuel Johnson, LL.D. With an\nEssay on his Life and Genius, by Arthur Murphy, Esq. Portrait of Johnson\n2 vols., 8vo, Cloth, $4 00.\n\nKINGLAKE'S CRIMEAN WAR. The Invasion of the Crimea, and an Account of\nits Progress down to the Death of Lord Raglan. By Alexander William\nKinglake. With Maps and Plans. Two Vols. ready. 12mo, Cloth, $2 00 per\nvol.\n\nKRUMMACHER'S DAVID, KING OF ISRAEL. David, the King of Israel: a\nPortrait drawn from Bible History and the Book of Psalms. By Frederick\nWilliam Krummacher, D.D., Author of \"Elijah the Tishbite,\" &c.\nTranslated under the express Sanction of the Author by the Rev. M.G.\nEaston, M.A. With a Letter from Dr. Krummacher to his American Readers,\nand a Portrait. 12mo, Cloth, $1 75.\n\nLAMB'S COMPLETE WORKS. The Works of Charles Lamb. Comprising his\nLetters, Poems, Essays of Elia. Essays upon Shakspeare, Hogarth, &c.,\nand a Sketch of his Life, with the Final Memorials, by T. Noon Talfourd.\nPortrait. 2 vols. 12mo, Cloth, $3 00.\n\n\n\n\n\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Christianity and Greek Philosophy, by\nBenjamin Franklin Cocker\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nEndorsements\n\n\"Need help with your marriage? In this insightful and solid book Rhonda Stoppe gives wives real help! She puts her finger on problems and attitudes that plague every married woman\u2014and then supplies answers, solutions, and counsel from God's Word.\"\n\nElizabeth George\n\nbestselling author, A Woman After God's Own Heart\u00ae\n\n\"A vast amount of the misery and brokenness people experience is the result of lies they believe. This is certainly true of marriage. Rhonda Stoppe debunks misleading myths with grace, humor, and candor and leaves you with hope that your marriage can become more than the stuff of your dreams: it can be the handiwork of God.\"\n\nRichard Blackaby, PhD\n\nauthor, Customized Parenting in a Trending World\n\n\"If you've ever been through a difficult season in marriage, you may have secretly wondered whether you'd be happier with someone else...In this terrific book, Rhonda walks through several critical areas of discontentment, busts the myths that are so easy to believe, and shows just how important it is to rely on God\u2014and how to do that\u2014in order to move from heartache to joy in your marriage. This is the book for every woman who wants to be content and happy in her marriage but may not know how.\"\n\nShaunti Feldhahn\n\nsocial researcher and bestselling author of For Women Only\n\n\"I just loved reading If My Husband Would Change, I'd Be Happy! Are you searching for the secret to a happy marriage? Stoppe not only delivers practical steps to finding your happily-ever-after, she shares a life-changing message of how to use a Christ-centered marriage for the glory of God.\"\n\nErica Galindo\n\nfounder, CEO, and editorial director SonomaChristianHome.com\n\n\"If My Husband Would Change, I'd Be Happy packs practical wisdom and many \"Aha!\" moments to shed light on the myths that sabotage relationships. Through storytelling, humor, and biblical insights, Rhonda Stoppe reveals where true joy originates and the secret of becoming your husband's best friend. BRAVO! Rhonda on such a well-written and much-needed message.\"\n\nJulie Gorman\n\nauthor of What I Wish My Mother Had Told Me About Men\n\n\"I loved this book. Read it only if you dare to be inspired to better your marriage. If My Husband Would Change I'd Be Happy is a must-read for anyone. Rhonda's love for others shines through in every page.\"\n\nCarmen Whittaker\n\nwife of Fermin Whittaker, executive director California Southern Baptist Convention\n\n\"If you're finding your \"happily ever after\" not turning out as expected and being shattered by your not-so-perfect Prince Charming, then Rhonda Stoppe is the fairy godmother who will point you in the right direction for a Happy Ending made in the kingdom of heaven!\"\n\nAngela Rose\n\nfounder, TheLaundryMoms.com\n\n\"Rhonda and Steve Stoppe have been long-distance pastors and mentors to my wife and I over the past 15 years. They are the most grounded-in-the-Bible people we have ever met. Because of that they have been an amazing example of how God intended a married couple to live as husband and wife, and mother and father.\"\n\nJosh Berry\n\npresident, LabeLive\n\n\"Rhonda Stoppe makes me want to be a better wife, a better mother, and a better grandmother. But most of all, Rhonda inspires me to be a better Christian through her own very real and passionate relationship with Jesus.\"\n\nJoy Lucius\n\njournalist, American Family Journal  \nauthor, The Dandelion Trial\n\n\"When my husband had an affair, I thought my marriage was over. But Rhonda and Steve Stoppe helped us begin the healing process in our marriage by covering my husband and me with the love of our Savior. Their personal mentorship has been instrumental in making our heavy circumstances less burdensome, and they provide godly marriage counsel that imparts hope for the future.\"\n\nAngie\n\nhumbled and blessed wife and student of the Word\n\nHARVEST HOUSE PUBLISHERS\n\nEUGENE, OREGON\nUnless otherwise indicated, all Scripture quotations are from the New King James Version\u00ae. Copyright \u00a9 1982 by Thomas Nelson, Inc. Used by permission. All rights reserved.\n\nVerses marked ESV are from The ESV\u00ae Bible (The Holy Bible, English Standard Version\u00ae), copyright \u00a9 2001 by Crossway, a publishing ministry of Good News Publishers. Used by permission. All rights reserved.\n\nVerses marked NASB are taken from the New American Standard Bible\u00ae, \u00a9 1960, 1962, 1963, 1968, 1971, 1972, 1973, 1975, 1977, 1995 by The Lockman Foundation. Used by permission. (www.Lockman.org)\n\nCover by Dugan Design Group, Bloomington, Minnesota\n\nCover illustration \u00a9 Big_Ryan \/ iStock\n\nBack cover photo \u00a9 Jackie Plaza Photography\n\nIF MY HUSBAND WOULD CHANGE, I'D BE HAPPY\n\nCopyright \u00a9 2015 Rhonda Stoppe\n\nPublished by Harvest House Publishers\n\nEugene, Oregon 97402\n\nwww.harvesthousepublishers.com\n\nISBN 978-0-7369-6286-5 (pbk.)\n\nISBN 978-0-7369-6287-2 (eBook)\n\nAll rights reserved. No part of this electronic publication may be reproduced, stored in a retrieval system, distributed, or transmitted in any form or by any means\u2014electronic, mechanical, digital, photocopy, recording, or any other\u2014without the prior written permission of the publisher. The authorized purchaser has been granted a nontransferable, nonexclusive, and noncommercial right to access and view this electronic publication, and purchaser agrees to do so only in accordance with the terms of use under which it was purchased or transmitted. Participation in or encouragement of piracy of copyrighted materials in violation of author's and publisher's rights is strictly prohibited.\nDedication\n\nTo my husband, Steve Stoppe, the love of my life for more than 30 years. By your example, you have taught me how to love selflessly and fervently. You are my Stoppe-Ever-After. I pray God gives us many more years to serve Him together in this mission:\n\nTo know Christ and make Him known.\n\nAnd I cannot wait to one day stand before Christ's Throne with you to celebrate with all those who have come to salvation through our ministry together!\n\nTo my children and their spouses:\n\nTony and Kylene\n\nJake and Meredith\n\nBrandon and Jessy\n\nEstevan and Kayla\n\nI am in awe of how your marriages reflect the love of Christ to a world in desperate need of a Savior.\n\nSoli Deo gloria\nAcknowledgments\n\nThank you Gayle, Molly, Joan, Elaine, Pam, Penny, and Marge for becoming the Titus 2 women in my life when I was a young bride. The way you love your husbands drew me to you, and the way you love Christ caused me to want to love Him as well. \"I thank my God upon every remembrance of you\" (Philippians 1:3).\n\nSpecial thanks to...\n\nMy husband\u2013\u2013Steve Stoppe, for agreeing to write \"From a Husband's Perspective\" in this book. As a pastor, counselor, and husband, your words are insightful and inspiring to me\u2013\u2013and any woman who takes them to heart.\n\nMy editor\u2013\u2013Steve Miller, whose insights and wise counsel have helped Steve and me author this marriage resource for a generation who is desperate to believe a happy marriage can last for a lifetime.\nContents\n\nEndorsements\n\nDedication\n\nAcknowledgments\n\n1. If He Would Change, I'd Be Happy\n\nHe Was Never Meant to Be Your Happily-Ever-After\n\n2. I Will Respect Him When He Earns My Respect\n\nThe Concept of Unconditional Respect\n\n3. I'm Falling Out of Love with Him\n\nStaying in Love Is All in Your Mind\n\n4. Our Kids Would Obey if He Were a Better Father\n\nMarriage Survival Guide for Parents\n\n5. I Would Be Happier Married to Someone Else\n\nThe Grass Is Not Greener on the Other Side of the Fence\n\n6. He Would Love Me More if I Were Prettier\n\nThe Secret to Keeping His Attention\n\n7. All He Wants Is Sex\n\nWhen You Long for Romance\n\n8. More Money Equals Less Stress\n\nGrow Rich in Ways You Never Imagined\n\n9. Every Couple Fights\n\nEight Steps to Making Peace\n\n10. Our Marriage Would Be Better if Bad Things Would Stop Happening\n\nThe Joy of the Lord Is Your Strength\n\n11. If Momma Ain't Happy, Ain't Nobody Happy\n\nIt's Not Your Husband's Job to Make You Happy\n\n12. Happily-Ever-After Is a Fairy Tale\n\nTen Keys to a More Fulfilling Marriage\n\nAppendix: How to Have a Relationship with Jesus\n\nNotes\n\nAbout the Publisher\n\nIf He Would Change, I'd Be Happy\n\nHe Was Never Meant to Be Your Happily-Ever-After\n\n######\n\nI love being in love\u2014don't you? From the depths of my soul I have had an adoration for my husband that has only grown deeper over the past 30-plus years that we have been married. Is this adoration a result of being married to a perfect man? Of course not\u2014even though as a young bride I was convinced all of my happiness would be realized on the day I said, \"I do\" because my husband had promised to be my happily-ever-after.\n\nOn the day of our wedding, I walked down the aisle clutching my father's arm because I was trembling. I could hardly believe the day I had dreamt of was finally upon us! I had spent six months planning our wedding, and by the time we were to say our vows all I could think of was that I would soon be Mrs. Steven W. Stoppe. (I remember writing my new name over and over again just to establish how I would sign it\u2014did you do that?)\n\nI was so nervous as all eyes turned toward me, the bride, who was wearing the biggest hat-veil thing anyone had ever seen! And then, when my eyes met Steve's, nothing else mattered. It took my breath away to see him looking so incredibly handsome in his white tuxedo. (Big hats for veils and white long-tailed tuxedos were in fashion in the 1980s\u2014so don't judge me!)\n\nAs our eyes met, I remember thinking, I cannot believe I am actually marrying this amazing man. I am going to be the best wife he could ever ask for. Did you feel that way when you got married?\n\nSteve and I wrote our own wedding vows\u2014a real challenge for my not-so-romantically inclined guy. But he was a good sport, and he wrote wonderful words promising to love me \"as long as God shall give me life\"\u2014as Steve so sweetly whispered into my ear that day. As a reminder of our promises to one another, the vows we wrote have hung on a wall in our home ever since that day.\n\nTo my delight, for our honeymoon, Steve planned a monthlong trip that included driving to see a number of national parks across the United States. What a thrill it was to have an entire month to enjoy ourselves as husband and wife! (Although I have to admit, the nights that we camped out were a bit challenging when I learned how important it was to be near a restroom after lovemaking\u2014can I get a witness?)\n\nSteve's Aha Moment\n\nI'm sure that any illusions of grandeur Steve may have had about me when we got married were dashed during the second week of our honeymoon, while we were in Yellowstone National Park.\n\nAfter a couple of weeks of travel and a lot of fast food, I had gotten pretty constipated\u2014I'm just keepin' it real as I write this. You can imagine how uncomfortable I was whenever it was time to be romantic. So Steve decided to help out his poor bound-up wife by giving me a couple of laxatives\u2014something I had never taken before. He assured me if I took the pills before bedtime, I would have a most satisfying bowel movement in the morning, and all would be well.\n\nAt Steve's prompting, I took two pills. The next morning, nothing happened. So we decided to just enjoy our day at Yellowstone and try again that night by taking some more laxatives.\n\nWhen we drove into the park, we went straight to see the geyser called Old Faithful. Both Steve and I were excited to watch the hot water erupt out from the crater in the ground. We were impressed by how high the water shot up into the sky\u2014so much so that Steve decided it would look even more impressive to view the spectacle from a higher vantage point.\n\nSo up a nearby mountain we hiked. Old Faithful spouts at regular intervals throughout the day, and we calculated that we would be able to reach the top of the mountain just in time to see the geyser shoot forth before dark.\n\nWell, I probably don't even have to tell you what happened next. As we hiked and I was getting exercise, my innards began to make the most horrible gurgling sounds. As a blushing bride, I attempted to keep my husband from hearing the atrocious noise. However, the more we hiked, the clearer it became to me that the little pills, along with my morning coffee, were beginning to do a mighty work in my intestines.\n\nShamefully, I had to tell Steve what was happening, but I assured him I would be able to make it to the top of the mountain in time to see Old Faithful in action. But a short time later, I began to realize not only would I not be able to make it to the top, I was also in danger of not making it back down in time to find a bathroom before I experienced my own geyser spouting off!\n\nPoor Steve\u2014I know he was torn over what to do. He really wanted to continue the hike. I am sure he secretly evaluated the possibility of letting me make my way back down to the bathroom by myself, but then realized this was one of those \"for better or worse\" moments in which he had promised to love me just two weeks before.\n\nIn the end, we hiked slowly down the mountain. I had to stop several times to compose myself before I could go on. And all the while, Steve was laughing hysterically. I am happy to report I did make it to a restroom in time!\n\nBy the time I got out, the sun was setting and the park was about to close. So Steve never did have an opportunity to hike back up that mountain.\n\nWhen the Honeymoon Is Over\n\nDating, courtship, and planning the wedding are all glorious experiences for most women. But after the honeymoon, when the wedding gifts are in their proper place and life begins to happen, often the glorious experiences fade into the endless routines of to-do lists, juggling finances, and learning to serve one another. Did this happen to you?\n\nIt wasn't long before my weekends became consumed with doing laundry and housework. Gone were the Saturdays before marriage, during which Steve and I would spend an entire day at a park lazing by a river, enjoying one another's company. Even as a newlywed, those carefree days already seemed like a distant memory as I washed the dinner dishes and imagined the river running down my kitchen sink.\n\nI remember one Saturday in particular. I was in the house, defrosting our freezer. (They don't even make refrigerators that don't self-defrost anymore, do they?) As I painstakingly chipped away big chunks of ice, I could hear Steve and his brother, Dan, laughing in the garage. Dan had come over to help Steve work on a project. I should have been grateful for the help, but I found myself resenting that they were having a grand old time together while I was stuck in the house thawing out that miserable refrigerator and doing yet another load of his dirty laundry. Steve was a carpenter in those days, so his clothes got exceptionally dirty.\n\nIt didn't take long before I was annoyed by how much work was involved with being a wife. Soon resentment began building in my heart toward Steve. Even though I had already seen marriages in my family fall apart from resentment, I found myself falling for the mistake of harboring wrong attitudes.\n\nFamily of Origin\n\nHow would you describe your parents' marriage? Were they head-over-heels for each other? Did you have a terrific role model from their marriage or other marriages in your family that were characterized by joy, laughter, and delight? I hope this was true for you. In my own upbringing, it was not. So when I thought about what I wanted in my marriage, I had a long list of what I did not want.\n\nHowever, I soon learned that making a list of what you do not want your marriage to become is not an effective way to move toward having the marriage of your dreams.\n\nHow did you come up with the ideals for your \"dream marriage\"? After the wedding, were you surprised to discover that many of your expectations for your marriage didn't come true? Did you assume the fun and carefree experiences you enjoyed while you were dating would continue into married life? I did.\n\nWhile life cannot always be one fun experience after another, you can definitely have a truly satisfying marriage with a love that grows deeper as time goes on. Key to making this happen is breaking free of the common myths wives believe\u2014myths that make us look for marital happiness in all the wrong places. And in the chapters ahead, we'll talk about the secret to building a marriage that brings the kind of lifelong fulfillment you desire and that others will want to emulate.\n\nExpect to Feel Betrayed\n\nIn the early days of our marriage, I found myself becoming less and less enamored with my husband because he did not measure up to my expectations. For example, Steve loved to make peanut-butter toast. And not just in the morning for breakfast. Steve would make peanut-butter toast several times throughout the day. Why I had not noticed this man's obsession with peanut-butter toast while we were dating, I'll never know.\n\nIt wasn't the peanut-butter toast that bothered me, but the crumbs that were left behind every time Steve made this concoction. I have this unexplainable abhorrence to finding crumbs on my kitchen counter and floor. Mind you, I am not a spotless housekeeper, but there's just something about crumbs that gets to me!\n\nThinking he was saving me the trouble of washing a plate, Steve would invariably make his toast directly on the countertop\u2014the countertop! This would leave so many crumbs it was almost unbearable to me.\n\nFor the first few months of our marriage, I just quietly wiped up the crumbs while uttering little manipulative\u2014okay, maybe even passive-aggressive\u2014comments about how much I despised crumbs. Then one day I walked into the kitchen and found the countertop covered in crumbs. I must have gasped audibly, because Steve came running into the kitchen to see what was the matter.\n\nI burst into tears, and explained to Steve how his leaving crumbs behind made me feel like he didn't respect all the work I did to keep the house nice. The poor guy\u2014he just stood there stunned that I would rant so much over peanut-butter toast and crumbs. And he wondered why I would feel so betrayed by something as simple as the fact that he left a few crumbs on the kitchen counter once in a while.\n\nThis story sounds funny now, but when we were first married, the crumb dilemma truly devastated me. Because that's when it began to dawn on me that my husband was not the perfect person I had imagined him to be.\n\nThe Danger of Unrealistic Expectations\n\nOne of the biggest threats to a happy marriage is when one or both parties have unrealistic expectations of each other. When those expectations are not realized, you might feel betrayed. And this is when you may begin to believe myths that lead you to have unrealistic or incorrect expectations that do harm to your relationship. In this book, we will shed light on those myths.\n\nWhen my expectations of Steve were not being met, I remember feeling betrayed because he had promised to always make me happy. How self-absorbed I was back then. God used my disillusionment to show me my selfish heart. Have you ever had expectations come crashing down around you when reality sets in? How did that experience make you feel? Are you in a similar situation right now? Or maybe you have experienced years of disappointment in your marriage. Whatever the case, let's talk for a moment about how disappointment turns to disillusionment.\n\nYou might feel betrayed when you come to realize the man you married is not the man you had perceived him to be. If you have been married for any amount of time, I am sure that by now you have your own secret list of things you wish you could change about your husband.\n\nI find it interesting that frequently, the very qualities a woman was attracted to while dating her man often become the rub in their relationship after they are married. For example:\n\nI could go on, but you get the picture. Have you considered your husband may have his own secret list of disappointments about you as well? Rather than dwelling on what you wish your husband would change, what if you were to make a list of how you have changed after marriage? And instead, work to be the woman your husband had hoped you would be\u2014the wife you meant to be\u2014on the day you said, \"I do.\"\n\nSeriously, stop for a moment and evaluate the type of wife you had hoped to be...and the kind of wife you actually are. When your husband looked at your beautiful face as you cascaded down the aisle, what kind of wife did he expect you would be? Have you measured up to your own expectations\u2014let alone his?\n\nI have taught this message at women's events. When I ask each woman to measure herself against the kind of wife she meant to be\u2014and the kind her husband had hoped she would be\u2014tears begin to fall. If we are honest with ourselves, I think most of us wives wish we could go back and change the way we have treated our husbands at certain times. Am I right?\n\nIn more than 30 years of ministry, Steve and I have listened to countless couples reveal how disappointed they were in the person whom they married. Whenever a wife can convince her husband to come in for biblical marriage counseling, she often secretly says to herself, \"Oh good. Now my husband is going to find out all the ways he needs to change to be a better husband\u2014so that I can be happy.\"\n\nCan I let you in on a little secret? Looking to your husband to make you happy is an unfair expectation. And no matter how \"perfect\" he is, he will never bring you true joy. Because the purpose for which you exist is not to find happiness in your marriage relationship\u2014contrary to every fairy tale you ever heard as a little girl.\n\nYou were created to delight in your Creator. God made you to long for intimacy with Him\u2014to delight in Him. So any other relationship that you pursue to fill the void only God can fill will always come up short. In the same way, you can never be your husband's source of true joy.\n\nI Needed Help!\n\nDo you know a godly couple who have been married for a long time and are still deeply in love? Doesn't your heart long to have a marriage like theirs? What's their secret?\n\nAs you observe such a couple, you may be tempted to say, \"Oh that wife is so lucky to be married to such a wonderful man. I wish my husband were more like him.\"\n\nUpon closer observation, however, you might be surprised to learn that the secret to their happy marriage isn't related to how \"ideal\" they are as spouses. Rather, it's because their relationship is grounded in a love that is deeper than their own love for each other. A marriage flourishes when both husband and wife love Christ more than any other person in life\u2014including one's own spouse.\n\nIn Mark 12:30, Jesus declared that the priority of life is to love God with all of your being\u2014all of it. Do you love God like that? As a young Christian, I would have answered, \"Yes, of course I love God!\" and pointed to my busyness in serving God as evidence of that love. However, every once in a while I would meet someone who straight-up loved Jesus. A person whose life wasn't about doing things for God; rather, they lived to love God so much that they couldn't help but love others. Have you ever met anyone like that?\n\nYou know what kind of people I am talking about. You have to spend only a few minutes with them to realize they have an authentic love deep in their heart for their Savior.\n\nThis kind of wholehearted love is available to anyone who has a relationship with God through His Son, Jesus. When you learn to devote your heart to loving the Lord, there will be a natural outpouring of God's love spilling out of your heart and onto those around you\u2014especially to your husband. (To learn more about a relationship with God through Jesus, please turn to the article in the appendix of this book, \"How to Have a Relationship with Jesus.\")\n\nIt all comes down to this: The key to having an all-out love for your husband and experiencing fulfillment in your marriage does not lie in how well your husband measures up to your expectations, but in how well you love God.\n\nIt is humanly impossible to love selflessly because we are all born with a sin nature that seeks our own good above anyone else's. The only people who are able to love the way Jesus intended are those who have a personal relationship with God through Christ, are filled with the Holy Spirit, and are pursuing a deeper love for the Lord. Because God provides His supernatural love to those who love Him, He offers hope for true love to anyone who would follow Christ. Romans 5:5 says, \"God's love has been poured into our hearts through the Holy Spirit who has been given to us\" (ESV). We will talk more about hope for a happy marriage in chapter 11.\n\nLoving your husband amounts to so much more than your emotions and feelings for him at any given moment. Love is a choice. And God's love gives you the ability to love your husband even when he doesn't measure up to your expectations. Listen to what the Bible says:\n\nAbove all these put on love, which binds everything together in perfect harmony...Above all, keep loving one another earnestly, since love covers a multitude of sins (Colossians 3:14; 1 Peter 4:8 ESV).\n\nDid you know God loves you in this way? Psalm 139:17-18 says, \"How precious also are Your thoughts to me, O God! How great is the sum of them! If I should count them, they would be more in number than the sand.\"\n\nAre you really taking in what this scripture says? The perfect Creator of heaven and earth makes it a point to think precious thoughts toward you\u2014you!\n\nLet's be honest: You and I both know that if God wanted to, He could write a long list of all our flaws and the ways we fail Him every day. Yet because of His great love, God says, \"I, even I am He who blots out your transgressions for My own sake, and I will not remember your sins\" (Isaiah 43:25).\n\nWhat kind of love is that? A love that chooses to forget your sins and focus on precious thoughts toward you. I know I do not deserve this kind of love. Do you?\n\nDo you see where I am going with this? If God loves you so overwhelmingly even though you don't measure up to His expectations, and you are called to love others as He loves, then you are to have that same kind of love for your husband.\n\nYour Marriage Is a Light\n\nJesus said, \"By this all will know that you are My disciples, if you have love for one another\" (John 13:35). Why do you think Satan works so hard to destroy Christian marriage relationships?\n\nYour genuine love for each other will be a light that tells your children\u2014and a watching world\u2014that knowing the Savior really does make a difference in your lives. Letting this light shine does not happen by accident. In fact, if you make marital love all about your feelings, you will certainly miss the opportunity to shine Christ's light.\n\nWhen life is hard, your hormones are acting up, the bills pile up, and the kids get sick\u2014this is when the light of God's kind of love has the potential to shine the brightest. Jesus said, \"Let your light so shine before men, that they may see your good works and glorify your Father in heaven\" (Matthew 5:16).\n\nAs a young bride, I was drawn to some happily married couples in our church because their love shone as a bright light to me. As I began to look to more mature couples who seemed to delight in one another, do you know what I found? An untapped resource of wisdom that was exactly what I needed to teach me how to have a happy marriage!\n\nTitus 2:4 instructs older women to teach the younger how to love their husbands. The Greek word translated \"love\" in this verse is phileo, which refers to a friendship love. And that is just what these older women taught me\u2014how to enjoy my husband for who he is, not who I wished he would be. They taught me how to become his closest and dearest friend.\n\nIn writing this book, it is my sincere desire to be a Titus 2 woman in your life. Because when you learn the secret of becoming your husband's closest friend, you will become your husband's greatest treasure\u2014and he will become yours as well.\n\nWhether you are newly married or you have been married for a number of years, it is never too late to transform your marriage by applying God's principles to your marriage relationship.\n\nOne Woman's Story\n\nDo you find yourself hoping that it's not too late to make your marriage into the kind of relationship you and your husband will thoroughly enjoy for the rest of your lives?\n\nI know a woman\u2014we will call her Lydia\u2014who has been in a difficult marriage for many years. Not long after she and her husband were married, Lydia realized the man her husband pretended to be and the man he turned out to be were vastly different. After a season of fighting and arguing, Lydia could see how the unrest in her home was hurting her children, stealing her joy, and flat-out making her a miserable person.\n\nSo, taking the advice of a Titus 2 woman in her life, Lydia determined to stop waiting for her husband to change so she could be happy. She stopped lamenting over how disappointed she was in him. And instead, she devoted herself to knowing and loving Christ through prayer, Bible study, and fellowship with Christians.\n\nIn her pursuit, Lydia has discovered the secret to true joy. Although her husband has changed very little over the years, not long ago Lydia joyfully said to me, \"I just love my husband. I know it is Christ in me that is loving him through me\u2014because I just love him!\"\n\nWhat an incredible legacy this woman is leaving for her children. Because of her example, she is raising some of the most merciful people I know.\n\nSo Much More Than Happily-Ever-After\n\nThe world is longing to see married people who grow more in love over time. When Christian marriages do not exude true love, it hurts the name of Christ. Do you realize God wants the love between you and your husband to be a testimony of His love to a watching world? Happy marriages are one of the greatest tools God uses to draw unbelievers toward Him. And the effect of that testimony begins in your home, especially to your children.\n\nWhile writing this book I laughed and cried over the love stories I have included. And the truths that I discovered as I researched and studied God's Word have forever changed the way I relate to my husband. I cannot wait to share these stories and insights with you!\n\nIf you're like most women, you are so busy you rarely read through an entire book. I ask you to make a commitment now to keep working your way through this book until you reach the end. Every chapter includes subheads that help break the chapter into smaller parts so you don't have to feel overwhelmed by trying to read an entire chapter in one sitting. Keep the book on your nightstand, read one or two subsections at a time, and keep moving forward at a pace that works for you. I know you'll be glad if you do this.\n\nAt the close of each chapter, Steve has written a section called \"From a Husband's Perspective.\" As you read what he says, you will not only glean a man's perspective, but also insights from the many years of biblical counseling he has done with husbands and wives.\n\nAt the end of each chapter you will find two sections called \"Thinking It Through\" and \"Living It Out.\" These will allow you to study and apply the truths you've learned. (These questions also work well in a group setting should you decide to lead or participate in a small group study through this book.)\n\nFinally, at the end of each chapter you will find a link to my website, NoRegretsWoman.com, where you will be able to watch a short video of Steve and me discussing the topics of each chapter, and\/or an audio link to a particular message or a Christian love song. All of these are intended to help you process what you have learned in the chapter.\n\nSo if you are ready, let's take this journey together. I expect we will become great friends as we shine truth on some of the myths wives believe and learn how to love our husbands in a way that brings glory to Christ. In so doing, you will discover how to build a no-regrets marriage, and more importantly, one day you will stand before the Lord in heaven and hear Him say, \"Well done, good servant!\"\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nPhoto credit: JPlazaPhotography\n\nI Will Respect Him When He Earns My Respect\n\nThe Concept of Unconditional Respect\n\n######\n\nMost likely you have heard the Bible story of the teenage boy who killed a giant with his slingshot. The Bible says David was young and attractive, so you can imagine how the ladies of Israel would have taken notice of this new hero on the scene.\n\nThe women were so taken by David that the song they sang about him quickly became the number-one hit song. As the girls were singing, \"Saul has slain his thousands, and David his ten thousands\" (1 Samuel 18:7), I can only imagine how they might have swooned at even a glimpse of this overnight sensation.\n\nOne woman in particular fell head over heels for David. Her name was Michal, the daughter of King Saul. When King Saul found out his daughter loved David, he sent word to him: \"You shall be my son-in-law today\" (verse 21).\n\nIn my opinion, what made David even more attractive was his response to King Saul: \"Does it seem to you a light thing to be a king's son-in-law, seeing I am a poor and lightly esteemed man?\" (verse 23). David's response reveals his humble heart. There is nothing more attractive than a handsome man filled with God-honoring humility\u2014don't you agree?\n\nWhat a lucky girl Michal was! Back then, the daughters of a king didn't have any say about who they would marry. In fact, Michal's sister Merab was supposed to marry David, but at the last minute, King Saul married her off to someone else instead.\n\nI wonder what the marriage of this young regal couple would have been like. I wish the Bible gave us some details about the wedding. Can't you just imagine how enthralled the women of the land would have been over the romantic event? Yet the Bible simply states, \"Then Saul gave [David] Michal his daughter as a wife\" (verse 27).\n\nWhat we do learn is that, after Michal and David were married, King Saul became increasingly jealous against David's popularity and sought to kill him. In 1 Samuel 19:11-14, we see Michal deceiving her own father to protect her husband's life.\n\nAs with any marriage, there are the challenges and stresses of life. So it was with David and Michal. Since King Saul was seeking to kill David, the latter had no choice but to go into hiding, leaving Michal to await his return.\n\nImagine how much heartbreak this brought to young David. His father-in-law, who was also the father of his best friend, Jonathan, hated him so much he wanted to see David dead. David wrote many psalms during this season of sorrow and struggle, revealing the depth of the heartache and loneliness he experienced.\n\nAfter many years of exile, the day finally came when David was anointed king of Israel. After his coronation, one of his actions was to have the Ark of the Covenant brought into the City of David. When the Ark finally arrived, we are told that, \"David danced before the LORD with all his might\" (2 Samuel 6:14).\n\nFinally the running was over, and the promise God had made so many years earlier had come to pass. David was the king of Israel. And in humble gratitude he danced, for all he was worth, upon the streets of the city! Wouldn't you love to have been there that day? Watching this godly young man worship the Lord in total abandon would have been glorious.\n\nAs David danced, the people celebrated and watched their new king in action. And David's wife Michal also watched\u2014from her window. Wouldn't you expect Michal to run into the streets and dance with her husband to celebrate all God had done to bring about this wonderful moment? I know that's where you would have found me\u2014how about you?\n\nBut not so with Michal. Second Samuel 6:16 says she \"despised [David] in her heart\" when she saw him dancing in front of the people. Why would she be so upset with her husband's display of joyful worship? One Bible teacher observes, \"Michal considered David's unbridled joyful dancing as conduct unbefitting for the dignity and gravity of a king.\"\n\nAfter the coronation parade, David went with the priest and the people to offer burnt and peace offerings to the Lord, so Michal had plenty of time to seethe over her perceived humiliation before David arrived home.\n\nIsn't it interesting that we don't find Michal at the tent when David and the priest were giving offerings to the Lord? And where was she when David distributed a celebration meal to the multitude? I'll tell you where she was\u2014at home preparing her argument for when David walked in the door. Have you ever done that? I have.\n\nYou know what I am talking about. Something gets you so upset that you are in no frame of mind to do anything else until you've poured out your anger on your husband. As a young bride, I can remember a time when I waited for Steve's arrival home so I could blast him as soon as he walked in the door. From the hurt look on his face I knew he felt wounded and confused by my betrayal. Even today I regret the words I said to him so many years ago.\n\nSecond Samuel 6:20 says after everyone went home, David \"returned to bless his household.\" The poor guy was completely unaware Michal was getting ready to pounce on him. In fact, she didn't even wait for David to come to her\u2014she went to meet him. Imagine the hug, kiss, and words of affirmation David expected to hear from Michal. Instead, what he heard was this: \"How glorious was the king of Israel today, uncovering himself today in the eyes of the maids of his servants, as one of the base fellows shamelessly uncovers himself!\" (2 Samuel 6:20).\n\nWhat a slap in the face for David. In one moment all the wonder and joy from his glorious day were dashed by his wife's disrespectful accusation. How did David respond?\n\nIt was before the LORD, who chose me instead of your father and all his house, to appoint me ruler over the people of the LORD, over Israel. Therefore I will play music before the LORD. And I will be even more undignified than this, and will be humble in my own sight. But as for the maidservants of whom you have spoken, by them I will be held in honor (2 Samuel 6:21-22).\n\nIt seems Michal was so concerned about how the women of the city would perceive her husband\u2014likely the very women who earlier had sung about David's accomplishments\u2014that she felt it was necessary to disrespect him in a demeaning, sarcastic manner.\n\nAs can be expected, Michal's hurtful words did not endear David to her. In fact, the Bible says after that encounter, \"Michal the daughter of Saul had no children to the day of her death\" (2 Samuel 6:23). Some Bible commentators say Michal may have had no children because God closed her womb, and others suggest the more popular belief that from that day forward, King David had no further relations with Michal. Either way, the most disgraceful experience for a woman in those days was to not bear children. So the result of Michal's disrespect was her own disgrace.\n\nA Man's Insight\n\nAfter writing this, I couldn't shake the feeling of sorrow I had for the way David and Michal's marriage was wounded by her hurtful words. What a sad ending to a story that started out like a fairy tale. I felt the need to seek insights from a man's perspective, so over our morning coffee I asked my husband, Steve, what a woman might learn from this biblical account.\n\nSteve had some great points that helped me to gain understanding from both David and Michal's perspectives, and I'd like to share them with you because there's a lot here that wives can apply to their marriages. Steve said, \"The issues between Michal and David are as follows:\n\n1. Michal failed to see the big picture.\n\n2. Michal loved herself and her reputation more than she loved her husband.\n\n3. Michal's response showed disrespect to her husband.\"\n\nLet's unpack these three points to see what we can learn from them.\n\nFailing to See the Big Picture\n\nThe first issue we want to look at is that Michal failed to see the big picture. She had fallen in love with the teenage warrior who had battled a giant. She would have heard the song of the Israelite women as they celebrated David's victory. To be married to such a courageous, attractive man would have been quite a conquest for any woman. Michal found herself to be that woman.\n\nBut when Saul began to pursue David, the latter had to flee for his life. Nowhere in Scripture do we find that Michal was on the run with her husband. So we can only wonder if she waited in her comfy palace for David's return, likely only hearing about her husband's status through various messengers.\n\nWhen David was finally able to return, he determined to honor the Lord by bringing the Ark of the Covenant back to Israel. However, because the Ark was not moved according to God's instructions, one of David's men died while the Ark was in transit.\n\nImagine how David probably felt when the man died. In fact, he may have felt responsible for his death. After all, as king, David should have known God's requirements for moving the Ark so he could guide and protect his people.\n\nWhen the Ark finally arrived in the City of David, what a glorious occasion it was for the young king. No wonder he danced in the streets of the city!\n\nAnd where was Michal during all this activity? In her home, watching the festivities through a window. She likely saw only a brief snapshot of her husband's actions, which happened to include his dancing in the streets. And because Michal was self-centered rather than God-centered, all she could think about was the reputation she had worked so hard to maintain as royalty. As Saul's daughter and David's wife, she must have thought she was someone pretty special.\n\nCan you think of a time when you failed to see the big picture of your husband's circumstance? Maybe he spent the day battling giants at work while you were at home holding down the fort. Husbands often feel their wives do not understand the battles they have to face at work on a daily basis. Have you ever considered what your husband deals with? Here are a few possibilities to ponder:\n\n\u2022 Holding fast to his integrity in a work environment that says, \"The ends justify the means.\"\n\n\u2022 Keeping his eyes from longing at other women.\n\n\u2022 Putting up with office politics.\n\n\u2022 Worrying that someone younger or smarter is after his job.\n\n\u2022 Trying to make enough money to make ends meet.\n\n\u2022 Battling physical and emotional fatigue.\n\nMany men spend long hours at their job. For some their work is labor intensive, for others it is emotionally exhausting. When Steve worked in construction, he would come home physically spent. I remember watching him jump into our swimming pool after a hot day of working in the sun. Steam would rise from the surface of the water when he got into the pool.\n\nAnd these days, as a pastor, my husband comes home emotionally tired. He has listened and talked to people all day long. So that I could encourage my husband, I had to learn from him what his day was really like and not what I perceived it to be. Then I could discern when to talk to him about important matters or encourage him upon his return home.\n\nHow often husbands are greeted by their wives with a list of complaints! For example, let's say your husband had an amazing week at work. Maybe he is in sales, and after many business meetings and hours of presentations he lands the biggest account ever. On Friday, the boss calls all the staff together to honor your husband's efforts and thank him for a job well done.\n\nAll during the drive home, your husband is singing his heart out worshipping the Lord for His favor. He stops to buy a bucket of chicken for dinner, and some flowers to give you when he arrives.\n\nAll the while, you are at home watching the clock. Where could he be? you think to yourself. He's an hour late.\n\nWhen your husband finally pulls into the driveway, you meet him at the door. In a condescending tone, you say, \"Where have you been? You are late.\"\n\nHe starts to explain, and when you see the bucket of chicken, you exclaim, \"Chicken, seriously? You bought chicken? You could have called, you know. I already made spaghetti. We are having spaghetti tonight. You can take that chicken to work with you for lunch...if you're not already scheduled to have some wonderful 'business lunches' next week, while I eat peanut butter and jelly with the kids, again.\"\n\nAs you walk back into the house, you growl over your shoulder, \"By the way, you forgot to put the garbage cans away last night. They're still out on the street. Can you go get them now, please? It so embarrasses me when you leave the cans out there. Our neighbors must think you are lazy.\"\n\nAfter bringing in the garbage cans, your husband places the bouquet of flowers on the kitchen table. To which you respond, \"Flowers? It's not my birthday. That was last month, when you forgot the day\u2014remember?\"\n\nI know you would never be the woman in this scenario, but maybe you can relate to a similar rendition of the story. Is it any wonder husbands arrive home from work, grab the remote, and sit in front of the television until it is time to go to sleep?\n\nWhen it comes to your husband, are you failing to see the big picture and note all he does to provide for your family? Are you preoccupied with his shortcomings, or do you honor his efforts and celebrate his successes and offer him unconditional respect at home in spite of his failures?\n\nWhen Your Reputation Becomes More Important\n\nThe next point we can glean from Michal and David's story is Michal loved herself and her reputation more than she loved her husband. Like the woman who was more concerned about what the neighbors thought when the garbage cans were left out too long, do you have times when you throw your husband under the bus in an attempt to save your own reputation?\n\nMy husband was a youth pastor for 18 years, and we both dearly loved all that came with that job. After that, Steve accepted a position as senior pastor of a church. Shepherding adults is certainly different from pastoring teens. I soon discovered women in the church would come to me, rather than my husband, when they had a complaint. Or they would say things to me like, \"Can you have your husband call my husband? He is really under a lot of stress at work and needs some encouragement.\"\n\nIt didn't take long for me to develop a habit of telling my husband, \"You need to call so-and-so because his wife said he is struggling.\"\n\nWhat I soon discovered was that Steve would become very quiet and withdrawn when I made these types of comments. One day, as I was asking him if he had yet made the encouraging phone call, I had an \"aha moment.\"\n\nWhen Steve offered no reason why he had not followed up with the man, I pressed him to do so, adding, \"I don't want Mrs. Smith to think I forgot to tell you about her husband.\"\n\nAs I was speaking, the Lord opened my understanding. By agreeing to deliver Mrs. Smith's message to my husband, I was now part of the equation. I didn't want to look bad and have her think I had forgotten to pass along her request. And I didn't want her to think\u2014or tell others\u2014that my husband was uncaring about their situation. I was concerned about my reputation\u2014and ours.\n\nI then asked Steve, \"Honey, when I say you need to call this person, are you interpreting that as me telling you how to do your job?\"\n\nI could see the relief come across Steve's face. The poor guy had been trying to endure my \"helpfulness\" with a good attitude. Yet each time I would press him to make a contact, all he could hear was, \"Hey buddy, I know you're the pastor, but I am here to help you do your job right. If you drop the ball not only do you look bad as a pastor, but I look bad to these women who are asking me to get you to call their husbands.\"\n\nIn a moment everything changed! I came to realize how much my husband valued my respect. And in this new, uncharted territory of Minister's Wife, I was coming across more concerned about my own reputation and how Steve would be perceived by the women of the church than I was about trusting in Steve's leadership as a pastor. (Kind of like Michal's response to David, huh?)\n\nI quickly asked Steve to forgive me for being disrespectful. I assured him that I trusted his leadership and knew he was seeking the Lord daily for direction. I promised Steve from that day forward, \"When women come to me with a request or question, I will tell them they need to talk directly to you\u2014the pastor.\"\n\nThat was 15 years ago. And I am so thankful the realization came early on so I did not develop a habit of disrespecting my husband under the impression I was helping him do his job better. Along with this adjustment, I also determined not to correct him in public, talk over him in a meeting, or undermine his authority behind his back. The women of our church have a great respect for my husband because he is a godly leader\u2014and I believe, in part, it is also because they have seen my example of respecting him as the spiritual leader of our body. (If you are married to a pastor and would like more insights on how to minister to your husband, you may want to visit NoRegretsWoman.com to download my ebook I Sleep with the Pastor.)\n\nThe Cost of Showing Disrespect\n\nThe third issue we see in Michal and David's story is Michal's response showed disrespect to her husband. When David heard his wife's harsh words, he was quick to remind her that his actions were not about her, but were between him and God. David said he had danced \"before the LORD.\" He then reminded Michal of all God had done up to that point to make him king. Then finally David said, \"And I will be even more undignified than this, and will be humble in my own sight\" (verse 22).\n\nDavid made sure to explain his motives to his wife. He said, \"Look at what God has done. My dancing was in worship to Him. I know my heart was in the right place, and I don't care what you or anyone else thinks.\"\n\nFrom that day on David and Michal's relationship was forever changed as he withdrew from her. If the Bible commentaries are correct in suggesting the reason Michal had no children was because David never again went to bed with her, then we can conclude that Michal's disrespect of her husband led to his loss of affection for her.\n\nLove and Respect\n\nWomen long to be loved by their husbands. In all the years my husband and I have done biblical marriage counseling, we have seen that wives are often plagued with the question, \"Does he love me as much as I love him?\"\n\nFor the most part, women ache to know their husbands love them unconditionally. God created women to have this longing, which is why He instructed men, \"Husbands, love your wives, just as Christ also loved the church and gave Himself for her\" (Ephesians 5:25).\n\nOver the years, I have learned to make myself vulnerable in asking Steve to help me feel loved by him. You may think this sounds needy\u2014okay, call me needy. But I know how much I value being loved by my man, and I am willing to ask him from time to time to show me his love.\n\nThe Bible calls husbands to \"live with your wives in an understanding way\" (1 Peter 3:7 NASB). But let's be honest, ladies\u2014we are a mystery. We aren't even sure what we want much of the time. Based on your hormones and many other variables, actions that say \"I love you\" today may not be what you need tomorrow. If your husband is going to live with you in an understanding way, it is your responsibility to gently coach him\u2014for the rest of your life\u2014how he can best show you his love.\n\nFor example, when I was younger I really appreciated my husband's compliments about my appearance. If he failed to notice when I made an extra effort to look good, rather than hinting or pouting until he noticed, I would say, \"Hey, did you notice my new dress\/hairstyle\/etc.?\" When he responded with a compliment, I did not allow myself to think, Oh yeah, now you say something, buddy. Maybe I'll stop making the effort, and then we'll see how long it takes you to notice. I'm sure you'll agree this sounds petty, but over the years that I have mentored women, I've seen this kind of attitude surface frequently.\n\nNow that I am older, I tell my husband how much more I need his kind affirmation. We joke about how merciful God is to cause our vision to diminish as we age because we see each other through lenses blurred that help to soften all our wrinkles.\n\nAnd Just as Much as You Need to Feel Loved...\n\nJust as deeply as wives long to be loved without condition, husbands desire to receive unconditional respect from their wives. Again, God knew women did not need to be instructed to love their man\u2014this comes naturally. But to respect him is another story. That's why the Bible instructs, \"Let the wife see that she respects her husband\" (Ephesians 5:33 ESV).\n\nWomen tend to nurture and mother the people they care about. But your husband does not need a mom. He wants a wife who believes in him, relies on him, and celebrates his accomplishments.\n\nAn interesting side note here if you happen to be a mother of sons: In my book Moms Raising Sons to Be Men, one insight I share about helping your sons grow into men is the need for you to discern when you should stop treating your sons as little boys and start showing them respect as men. If you have adolescent boys, you might want to visit my website at www.NoRegretsWoman.com\/media and listen to my radio message \"Hand Him His Manhood or He Will Fight You for It.\"\n\nWhen a man feels disrespected by his wife, he tends to pull away and not show her the love she craves. That's how David responded to Michal. And when a woman does not feel loved, she will respond by disrespecting her husband. In his book Love and Respect, Dr. Emerson Eggerichs calls this the \"Crazy Cycle.\" He says, \"The Love and Respect Connection is the key to any problem in a marriage...How the need for love and the need for respect play off of one another in a marriage has everything to do with the kind of marriage you will have.\"\n\nDr. Eggerichs encourages wives to write a letter to their husband about why they respect him. Here's how one woman responded\u2014and many others are learning the same:\n\nI am sad that I have been married twenty-two years and just now understand the respect message. I wrote my husband two letters about why I respected him. I am amazed at how it has softened him in his response to me. I have prayed for years that my husband would love me and speak my love language. But when I began to speak his language, then he responded with what I have wanted...This revelation...has changed everything in my marriage\u2014my approach, my response, my relationship to God and my husband. It was the missing piece.\n\nBut He Doesn't Deserve My Respect\n\nI completely understand a woman's resistance to showing respect to a man who has not earned it. But just as God instructs a husband to love his wife whether she earns his love or not, God commands a wife to show respect to her husband without condition. Let's consider Dr. Eggerichs's insight into this matter:\n\nA wife faces two choices. She can try to make personal adjustments and treat her husband respectfully according to what Scripture says, or she can continue with a sour look, and a negative, disrespectful attitude...To continue with disrespect only means shooting herself in both feet...\n\nLearning to show your husband respect is vital to a healthy marriage. My husband and I have watched failing marriages be turned around when a wife determines to obey God's mandate to show respect to her husband.\n\nAre you worried if you show respect your husband will \"get his way\" when conflict arises? Or are you afraid your respectful manner will lessen your chances of motivating your husband to change? Listen to one woman's response to this: \"If I step out in faith, claiming God's Word as the basis for my action, then I am trusting God to bring to pass what He said He would do. I can't go wrong with that! I've determined that is the path I am going to take no matter how unfamiliar it seems.\"\n\nAccording to God's Word, showing respect to your husband is not optional. As you spend time in the Bible, ask God to help you focus on and express to your husband what you respect about him. And when you do, don't be surprised if your husband responds in a more loving manner. Because your husband needs to be respected by you, when you bless him with honor, he will come to view you as a treasure. And you will become your husband's closest confidant, friend, and encourager. Your respect will motivate your husband to attempt feats he might otherwise only dream about\u2014because a man respected by his woman can accomplish great things!\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nTo be respected is probably one of the most important needs your husband has. In general, men crave respect. For example, with the promise of respect, gangs seduce young boys to do terrible acts. Countries go to war and bar fights break out all because some guy felt disrespected. Most men who have anger issues will admit their anger is triggered when they feel dishonored.\n\nYour husband longs to be respected by you. There is a good chance he married you because he found satisfaction in the way you showed him honor. So my question to you is this: How are you doing now?\n\nWhen we were newlyweds, Rhonda did not always recognize how important her respect was to me. When she talked to me as if she were my mother, I would inadvertently discount whatever she was saying. I didn't mean to disregard her; I think it was just a subconscious defense mechanism. Have you observed your husband shutting you down when you try to \"help\" him accomplish a task? He may be feeling disrespected.\n\nI remember when Rhonda had her \"aha moment\" and realized how her constant \"reminding\" me of my responsibilities was making me feel disrespected. Are your attempts to \"help\" your husband do better coming across as disrespectful? Sometimes letting your husband forget to follow through on a commitment is the wiser choice. Men learn from their mistakes, so letting your husband pay a late fee, run out of gas, or miss an appointment may do more to help him remember next time.\n\nWhen you ask your husband to take care of something for you, you would do well to realize men don't mean to put it out of their minds\u2014men compartmentalize. So when we direct our full attention to one task, we may forget or delay doing something else our wives have asked us to do.\n\nDon't interpret your husband's forgetfulness as him not caring enough about you to do what you have asked. Sometimes a husband will want to do a task his own way, but because he knows his wife will badger him until he does it her way, he will avoid the job altogether.\n\nAnd then sometimes we men just forget! Most men, when they realize they've let their wives down, become disappointed in themselves and even berate themselves in their thoughts. If you chastise your husband while he is processing his own disappointment in himself, don't be surprised if your husband responds with anger\u2014or as I did with Rhonda, shuts you down.\n\nAs Rhonda learned to let me off the hook when I didn't come through for her, I found myself trying harder. And because Rhonda worked to think honorable thoughts toward me\u2014even when I didn't do things the way she wanted me to\u2014she would respond to me in a respectful manner. This made me gravitate toward her even more. (In case you are wondering, your husband knows when you are disappointed in him, even if you don't say a word. You have to think honorably toward your husband in order to show him genuine respect.)\n\nWhen you learn to give your husband the respect he so desperately needs from you, you'll be blessing him with an incredible gift. Husbands hear their married friends complain all the time about how their wives dishonor them. What if your husband was one of the few who could say, \"Not my wife\u2014she is my greatest supporter\"? Can you imagine how the ability to say this would stir your husband's feelings of love toward you?\n\nAnd, when you honor your husband, you will also be walking in obedience to the Lord's command, \"Let the wife see that she respects her husband\" (Ephesians 5:33 ESV). Over many years of biblical counseling done with couples, I have seen marriages transformed when the wife learns to unconditionally respect her husband. God gave your husband a longing for respect, and it is the Lord who will bless you\u2014and your marriage\u2014when you learn to satisfy this desire. And don't be surprised if your respect kindles in your husband a deeper love for you and a bond of unity that will stand the tests of time.\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. How did you feel after reading about the way Michal treated David?\n\n2. Has God convicted you of any areas in which you are withholding respect for your husband? Or possibly about ways you are blatantly disrespecting him? Take some time to ask the Lord to forgive you and help you turn from your sin.\n\n3. Prayerfully consider and then write the qualities you respect about your husband.\n\n######\n\nLIVING IT OUT\n\n######\n\n1. Spend some time writing a letter to your husband. Tell him why you appreciate him and the qualities you respect about him. (If you're not a letter writer, text your husband what it is you respect about him.)\n\n2. Determine to be a wife who shows unconditional respect to your husband. If you need more help in this area, look for a godly. older woman who respects her husband and ask her to mentor you.\n\n3. One of the best ways to learn what respecting your husband looks like in everyday life is to fellowship with couples who have learned the secret of unconditional love and respect. Look for married couples in your church whom you would like to emulate and spend time with them.\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nI'm Falling Out of Love with Him\n\nStaying in Love Is All in Your Mind\n\n######\n\nIs this really my life?\" the young bride said through tears. Theresa couldn't believe she found herself \"falling out of love\" with the man she had vowed to love forever\u2014only 18 months previous.\n\nTheresa composed herself and told me the story of the whirlwind romance that led up to her marriage. As she shared with me how she and her husband met, I saw a sparkle in her eye and a gentle smile across her lips. When she talked about the long walks she and her fianc\u00e9 would take on the beach, holding hands and dreaming about how happy they would be as husband and wife, another tear trickled down her cheek.\n\nWhat had happened? Theresa couldn't point to any particular event that had caused her feelings for her husband to change. It had all happened gradually. \"Life just got in the way,\" was how she put it.\n\nWhat's the Key to Staying in Love?\n\nTheresa's story is not uncommon. Many couples find themselves in trouble when they wrongly make the tasks of everyday living their priority\u2014rather than nurturing their love for one another. So how can you cultivate a loving relationship with your husband that will stand the test of time?\n\nThe first insight into building a love that lasts is to take your focus off of how much you want to be loved by your husband. If you become obsessed with your longing to feel loved, you will become more preoccupied with self-satisfaction than with building a happy relationship. And this, in turn, will undermine the health of your marriage.\n\nYou may be surprised to learn the secret to loving your husband well lies in learning to love God deeply. Because when your love for the Lord is genuine, He gives you His supernatural ability to love others selflessly\u2014including your husband.\n\nThe marriages I most want to emulate are those of husbands and wives who have learned to love God so much that their passion for one another is almost supernatural. Don't you want a marriage like that?\n\nSo how can you learn to love God so deeply that it spills over into your marriage? Jesus said the greatest priority of life is to \"love the LORD your God with all your heart, with all your soul, with all your mind, and with all your strength\" (Mark 12:30).\n\nLet's take a closer look at Jesus' words, shall we? Notice how He said you are to love God: with all your heart, all your soul, all your mind, all your strength. It's an all-out love. It holds nothing back. And it involves every part of your being\u2014your emotions, your inner self, and your thoughts. This kind of love seeks to grow closer to God and know Him intimately. That's how you fall more and more in love with someone\u2014by getting to know them.\n\nGrowing more deeply in love with the Lord means spending time with Him. But first, for this to happen, you must have a relationship with God through Christ. If you have received Christ as your Savior and Lord, then you are a child of God and you have a relationship with Him. And if you haven't taken this step or you aren't sure whether you have, then I would encourage you to read \"How to Have a Relationship with Jesus\" in the appendix of this book (see page 211).\n\nGetting back to loving God\u2014here are specific steps you can take to grow in that love:\n\n\u2022 Devote yourself to discovering God's character qualities through Bible study. Read His Word and get to know Him better. The more you read the Bible, the more you'll learn God's desires for your life. You'll come to see life\u2014and your marriage relationship\u2014from His perspective.\n\n\u2022 Pray to Him daily. Communicate with and talk to Him. Love grows through frequent interaction.\n\n\u2022 Fellowship with other believers who have a genuine love for God. Let their love for the Lord serve as a contagious influence for you.\n\nI can say with confidence that pursuing intimacy with God transformed my marriage, and it can transform yours as well. That's because when your love for God is right, He will help you love your husband the way your heart longs to love him.\n\nHas all this talk caused you to consider how loving God more can help your marriage? Francis Chan, author of Crazy Love, says:\n\nThe solution [to loving God more] isn't to try harder, fail, and then make bigger promises, only to fail again. It does no good to muster up more love for God, to will yourself to love Him more. When loving Him becomes an obligation...we end up focusing even more on ourselves...The answer lies in letting Him change you...The fact is, I need God to help me love God. And if I need His help to love Him, a perfect being, I definitely need His help to love other, fault-filled humans...As we begin to focus more on Christ, loving Him and others becomes more natural.\n\nDid you notice that last line? As you \"focus more on Christ, loving Him and others becomes more natural.\"\n\nLoving God with all your heart, soul, mind, and strength will enable you to love your husband with a selfless love that does not diminish when times are hard. As you love God and yield to the Holy Spirit's working in you, you'll be empowered to love your husband with God's perfect love. Can you imagine how many marriages would be saved if Christians committed to this one principle of loving God so deeply that His love spilled over into their relationship with their spouse?\n\nWhen Love Starts to Fade\n\nAlong with growing your love for the Lord, here are three practical steps you can take when you notice that your love for your husband is fading:\n\nRepent\n\nEven though Steve and I have biblically counseled married couples for many years, it still surprises me when spouses are convinced that the trouble in their marriage is no fault of their own. When a wife has this outlook, she has usually become so focused on how her husband has not measured up to her expectations that she is unable to see her own contribution to the discord.\n\nMight that describe you? Asking God to help search your heart and make you aware of your sin is the first step toward repentance. To repent requires you to agree with God that your thoughts and attitudes are sinful. It is easy to make excuses or justify sinful acts, so take some time to be alone with the Lord and pray, \"Search me, O God, and know my heart; try me, and know my anxieties; and see if there is any wicked way in me, and lead me in the way everlasting\" (Psalm 139:23-24).\n\nIf the Holy Spirit reveals to you areas of dissatisfaction toward your husband, will you agree with God that your resentment is sin? And will you confess your sin?\n\nI know, I know\u2014I can hear you saying, \"But you don't know my husband. Why doesn't he have to repent?\" You're right\u2014I don't know your husband, but God does. And only God can do a work in your husband, not you.\n\nResentful thoughts and manipulative remarks will never accomplish good in your marriage. Instead, they will only drive a wedge between you and your love. Don't let that happen to the point that one day restoration may seem impossible.\n\nUnchecked resentment always leads to bitterness. Hebrews 12:15 warns that a root of bitterness will spring up trouble and defile many. During our nearly 20 years of work in youth ministry, my husband and I frequently saw the great damage bitterness can cause. We have watched children raised in Christian homes become rebellious and resentful toward the Lord because of the bitterness they witnessed in their parents' marriages. Don't believe your family will go unscathed if your disillusionment with your husband turns into resentment, and ultimately bitterness.\n\nThe apostle Paul, in his letter to the Galatians, warned, \"Do not be deceived; God is not mocked; for whatsoever a man sows, that he will also reap\" (Galatians 6:7). Consider the prophet Hosea's warning: \"They sow the wind, and reap the whirlwind\" (Hosea 8:7). Beware, because tiny seeds of resentment can sprout into a tornado of destruction that tears your family apart!\n\nIf God reveals to you areas of sin that you are harboring against your husband, ask Him to make you truly contrite over your sin. Then commit to daily asking God to make your heart tender toward your husband. To soften your heart is not beyond His power.\n\nThe next step to rekindling your love for your husband is...\n\nRemember\n\nTake a moment to think back to the way things were when you fell in love with your husband. You thought about him throughout the day. You looked him in the eye when he talked, and you listened intently to what he had to say. Remember how you would tell your girlfriends the qualities you loved about him?\n\nOne of my favorite dating memories took place when I was 15 years old. Steve and I were just starting to become interested in one another. One night I arrived at our school's gymnasium, where I would be cheerleading for our basketball team. When I walked into the gym, Steve had just dribbled a basketball down the court. As he came down from shooting a layup, we were face-to-face. It was a moment frozen in time. Our eyes met, he smiled, and then he ran back down the court to join his team. That magical moment is forever burned in my memory as the instant I fell in love with my husband.\n\nDo you remember a time when you couldn't wait for your husband's gaze to meet yours? A moment in particular that took your breath away? When you make a habit of remembering details of how you and your husband fell in love, you can rekindle feelings of adoration you may have forgotten. Looking at old pictures, reading old love notes, and just talking about past memories with your husband can have a wonderfully positive influence on your relationship.\n\nBelieve the Best About Your Husband\n\nRelationships in which people always believe the best about you are priceless, wouldn't you agree? Would your husband count you as one of those relationships? Does he have the confidence you will cover his mistakes with kindness? Or does he worry, \"I wonder how I will disappoint her today?\"\n\nWhen you think about your husband, do you tend to dwell on the things about him that disappoint you? Left unchecked, this practice can seriously undermine your love for your husband. The result will be dissatisfaction with your marriage. And over time, you may find yourself coming to believe the myth I am falling out of love with my husband.\n\nSo what can you do? How can you cultivate a new way of thinking about your husband?\n\nIt starts by making a deliberate decision to think on his good qualities and refuse to dwell on how he doesn't make you happy. In this way you can rekindle your affection for your husband and learn to delight in him. And what husband doesn't want to be enjoyed by his wife?\n\nWhat a gift you give your man when you determine to take pleasure in his good qualities and overlook his less-than-admirable ones. Wouldn't you want your husband to do this for you as well? Let this behavior begin with you.\n\nWhenever your husband does something that displeases you, determine that you will continue to think the best of him. Don't be quick to assign wrong motives to his actions. For example, when you go to the bathroom in the middle of the night and fall into the toilet because he left the seat up, don't angrily assume he doesn't care enough about you to put down the seat. No, simply realize that he forgot. And then choose to forgive him for his forgetfulness. Wouldn't you want your husband to forgive you when you inadvertently forget to do something he has asked you to do? Offer him the same grace you hope he will extend to you.\n\nThe Power of Right Thoughts\n\nA great way to develop a healthy thought life toward your husband is to follow the advice of the apostle Paul: \"Whatever is true, whatever is honorable, whatever is just, whatever is pure, whatever is lovely, whatever is commendable, if there is any excellence, if there is anything worthy of praise, think about these things\" (Philippians 4:8 ESV).\n\nAt this point you may be thinking, What difference will it make if I try to think on what is best about my husband? He will never be anything but negative and unappreciative.\n\nI know a woman who would beg to differ. Her name is Anne.\n\nAnne had been married for more than a decade to a man she had learned to \"tolerate\" as she put it. But after her pastor gave a sermon on Philippians 4:8, she determined she would make every effort to have only good and honorable thoughts about her husband, Ted.\n\nAt first Anne could hardly find any good thoughts to replace the negative ones. But with God's help and a resolve to heal her marriage, Anne disciplined herself to put out of her mind any negative thoughts she had. Instead, she tried to dwell on positive thoughts about Ted.\n\nEventually, Anne found that as she obeyed God's instruction to think what was best about Ted, Christ's peace began to wash over her mind. No longer did Anne find herself anxious, unhappy, or restless.\n\nAs Anne's thought life was being transformed, her attitudes and actions were also changing. Soon Anne was not only thinking well of her husband; she also made it a point to verbally affirm him as well.\n\nAfter experiencing so many years of Anne's sharp tongue and condescending tone, Ted was wary of her new demeanor. He had learned a long time ago to keep his mouth shut, watch TV until bedtime, and not cross his wife if he wanted a relaxing evening.\n\nOver time, Ted grew to trust that Anne's new manner was not a passing phase. He found himself looking forward to arriving home after work. He even started to linger in the kitchen after dinner to talk with Anne as she cleaned up the dinner dishes.\n\nOne day Ted told Anne, \"I'll do the dishes tonight, honey. You do so much for me and the kids. It's the least I can do.\" Anne just about fell out of her chair.\n\nIt has been more than ten years since Anne determined to think what was best about Ted. She will tell you that decision saved her marriage. And because of Anne's example, Ted has learned to do the same and dwell on her good qualities as well. Today their marriage is one that others desire to emulate.\n\nBut My Husband Doesn't Deserve It\n\nYou may be tempted to say, \"You don't know how my husband has disappointed me. He doesn't deserve for me to focus on his good qualities because they could never outweigh his bad ones.\"\n\nGod could make the same statement about you\u2014and me. Our feeble human attempts to do good will never outweigh our bad. The prophet Isaiah said, \"All our righteousnesses are like filthy rags\" (Isaiah 64:6). The original Hebrew text in this passage reveals that by \"filthy rags,\" Isaiah meant menstrual rags. Yuck! I think the prophet wanted to impress on us how little God values our religious practices apart from loving Him. This gives a real picture of how impossible it is for our good deeds to ever make up for our bad.\n\nAre You Merciful?\n\nGod's love is merciful. His mercy not only forgives our sins but covers our weaknesses and provides relief from penalty.\n\nJesus encouraged us to show this same mercy toward each other when He said, \"Blessed are the merciful for they shall obtain mercy.\" This means God will cover all your weaknesses with mercy when you are merciful to others.\n\nDoes the spiritual quality of mercy show in your attitude toward your spouse? \"Mercy does not hold a grudge, harbor resentment, capitalize on another's failure or weakness, or publicize another's sin.\" According to this list, would you be characterized by mercy? Or do you hold grudges and harbor resentment against your husband? If so, it's time to stop.\n\nDo you look for ways to come alongside your husband's weaknesses to be his helper, or do you watch for him to make mistakes so you can point them out, belittle him in front of the kids, or publicly ridicule him? Such actions are not merciful.\n\nIn case you may think showing mercy is merely optional, Jesus commanded His followers, \"Be merciful as your Father is merciful.\" Are you ready to ask God to help you be merciful toward your husband no matter what? In so doing, your attitude toward him is sure to reflect God's grace. And as you make it a habit to cover your husband's shortcomings with compassion, you might find him responding in kind.\n\nThe final step for relighting your love for your husband is...\n\nReturn\n\nBack when you and your husband were dating, what were some of the things you did to win his heart? How might you return to carrying out those actions?\n\nWhen Steve and I were dating, he drove a really cool car\u2014a Ford Mustang Mach 1. The car's muffler could be heard rumbling down the street from a block away. I remember choosing just the right outfit, putting on my makeup, and curling my hair in anticipation of our date. Inevitably, I would hear the deep rumbling of Steve's car as I was spraying the final mist of hairspray on my hair and checking my makeup one last time. My heart would skip a beat knowing that my love would soon arrive to pick me up for our date.\n\nWhen your husband is about to arrive home from work, do you anticipate his return? Do you put in the same effort to look nice as you did when you were dating? It's easy for wives to make sure they are looking their best whenever they go out in public, yet become lazy about their appearance at home.\n\nIn her book For Women Only, Shanti Feldhahn invited men to anonymously express some of their deepest desires for their marriage relationship. Listen to what one man said about what it means to him when his wife doesn't keep up her appearance: \"When you don't take care of yourself, I feel unvalued and unhappy.\"\n\nMen tell me there is a sort of secret affirmation they give one another when they see a man whose wife makes an effort to look pretty. I have heard men say, \"It's really not about if she is the perfect size.\" Rather they are affirmed when, through her appearance, the woman on their arm says to the world, \"I care about this man. I value him. I dress for him. I am his prize.\" Do you carry yourself as if you are your husband's prize?\n\nI remember the day I realized I had been letting myself go. Somewhere amidst my kids' toddler years, I looked in the mirror after my husband had arrived home from work and saw how bad I looked! I walked out of our bedroom and asked him, \"How long have I looked this bad?\" He just smiled and said, \"You're busy with the kids all day,\" then winked at me. The poor guy. Notice he did not say, \"Oh honey, you don't look bad.\" From that day onward I determined to prepare myself each day to look pretty for Steve's homecoming.\n\nWhat are some other ways you caught your husband's affections in the past that you could return to today? Here are some actions you might want to try:\n\n\u2022 When he talks to you, stop what you are doing and look at him.\n\n\u2022 Laugh out loud at his jokes\u2014even if you are so familiar with them the punch lines no longer take you by surprise.\n\n\u2022 Don't talk to him like you are his mother. (We will talk more about this later.)\n\n\u2022 Tell him what you admire about him\u2014often.\n\n\u2022 Thank him for working to help support the family. (And if he is out of work, find other accomplishments to praise\u2014even if it is something as simple as remembering to put down the toilet seat!)\n\n\u2022 Be his girlfriend. This means sit with him while he works on a project, go to the hardware store with him, go out with him to his favorite burger joint.\n\n\u2022 Find reasons to touch him. Scratch his back. Rub his neck. Hold his hand.\n\n\u2022 Have sex with him. (We will talk more about this in chapter 7.)\n\nThis list is certainly not exhaustive, but hopefully it will help get you started on thinking about the many ways you can fan the flames of love for your husband. And when you are trying to rekindle your love for him, remember you are not alone. God wants you to learn to love your husband with His merciful love. He will gladly help you. All you need to do is ask (see James 4:2).\n\nDid you know God is more interested in you having a loving marriage than you are? He is the One who can heal your marriage and make your love last a lifetime.\n\nBeginning with an authentic love for the Lord, God's love will spill out of your heart and into your marriage. When this happens, your marriage will become the most glorious of all your earthly relationships. And when you tap into this delightful resource, you will discover the secret to growing more in love with your husband with each passing year. In time, the love displayed in your marriage will be the kind others will want to emulate.\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nThis chapter talks about some expectations you may have for a loving marriage. You want your husband to be everything you imagined he would be, but let's face it\u2014we as men fall short. It hurts me to write this statement as much as it does for you to hear it.\n\nLike your husband, there are times when I am anything but attentive. However, Rhonda covers this with love\u2014Christ's love. Was this always the case? No. Over the years, she has learned to love Christ first, and her love for me spills out from her passion for Christ.\n\nAs a result, Rhonda has learned to think loving thoughts toward me, which is a great source of encouragement for me. Lest you think my wife is superhuman and you can never attain her level of spiritual maturity, let me say she isn't perfect in how she loves me. But Rhonda is always ready to confess and make right when she acts toward me in an unloving manner. This causes me to up my game and be the kind of husband God wants me to be. You may discover your husband will respond in the same way.\n\nI challenge you to apply the principles Rhonda laid out in this chapter and see if God can use your love and obedience for Him to influence a more loving response from your husband. If you don't see a positive result right away, don't give up. When you commit to living in obedience to God's plan for a loving marriage, He will bless you for your faithfulness.\n\nEven if your husband doesn't respond initially the way you would like, your commitment to loving God with your whole heart will be rewarded. The godly life you lead may eventually stir in your husband a desire to obey the Lord as well. And consider how your children will learn to love when they observe your unconditional love for their father\u2014no matter how he responds. By simply adjusting your mind to think well of your husband, you are laying up treasure in heaven. An eternal reward for temporal obedience sounds like a great layaway plan, don't you agree?\n\nIn the many years that I have biblically counseled married couples, I have seen marriages transformed when wives committed to thinking well of their husbands. If your husband is able to rest in knowing you think well of him, you will become his delight.\n\nWhen you become a joy to your husband, his love for you is sure to grow. And when you are thinking only what is good about your husband, your heart will be filled with love for him as well.\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. What part of this chapter did you relate to most? Why?\n\n2. With Philippians 4:8 as a guide, name two positive thoughts about your husband you will commit to dwelling on this week.\n\n######\n\nLIVING IT OUT\n\n######\n\n1. You fell in love with your husband because your mind instructed your heart how to feel. List three qualities that made you fall in love with him. Then tell your husband at least one characteristic you love about him.\n\n2. Briefly write out a \"magical moment\" you remember from the days you were dating your husband. What can you do this week to create a magical moment in your marriage? (See \"Create Magical Moments\" on the next page for some ideas.)\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nCreate Magical Moments\n\nA key part of becoming a couple who stays in love is committing yourselves to creating memorable encounters that celebrate your love. How can you create these magical moments? You don't have to plan an expensive date. On the contrary, frequently the most romantic interactions between couples are simple and occur in the course of everyday life.\n\nHere are four ways to put some magic back into your marriage:\n\n1. Make eye contact from across a crowded room, and give him a flirtatious smile or a wink.\n\n2. Come up behind him while he is talking to his friends and rub his back or his neck, or kiss his cheek. (Don't interrupt the conversation; just distract him a bit, and then be on your way.)\n\n3. Send him a loving or flirtatious text in the middle of the day. Tell him how much you cannot wait to see him when he gets home. Or tell him of the romantic evening you have planned for later. (Be careful of what you say in your text in case one day your 14-year-old picks up your phone and sees your flirtatious comments\u2014I speak from experience!)\n\n4. Be creative. Try to recall what you did to capture your husband's attention when you were dating. And then work to rekindle the magic like you did back then.\n\nOur Kids Would Obey if He Were a Better Father\n\nMarriage Survival Guide for Parents\n\n######\n\nAfter Steve and I had been married a couple of years, I began to get the baby bug. My girlfriend Beth had recently given birth to an adorable baby girl. Whenever Beth and her husband, Dave, would come over to our house, I would watch longingly as Beth nurtured her baby girl, Kristal.\n\nKristal was one of those babies who loved the baby swing. She would sit for hours in the swing with her pacifier in her mouth as Beth, Dave, Steve, and I would visit and play games late into the evenings. The joy I observed in our friends was so precious I was certain having a baby was \"just what our marriage needed.\"\n\nWhen I approached Steve about this, his practical response was, \"What\u2014now? We just bought this house. We are in the middle of a huge remodel and you want a baby?\"\n\nI knew Steve was right to question the rationale of wanting a baby at that time in our lives. We had agreed to move into the house we had purchased so we could use all our money to remodel it, rather than renting an apartment while the work was being done. And to top it off, my income was what we used to pay the house payment\u2014while Steve's income paid for all other bills and the materials needed for the remodel. Since Steve and I had already promised each other I would quit my job to be a stay-at-home mom once we had kids, I can see now how the thought of having children would have weighed heavily on Steve's already strained budget plans.\n\nHowever, at the time, all I could see was how badly I wanted to be a mother. And each time we enjoyed time with Beth and Dave, I was even more convinced our marital bliss would not be complete until we could procreate!\n\nAs the months passed and Steve watched Kristal become more than just a squirming little newborn, I saw a new interest spark in him. Steve loved playing with Kristal, and was overjoyed when he could get her to laugh out loud at his silliness.\n\nWhen Steve and I finally had a serious talk about having a baby, I assured him I most likely wouldn't get pregnant right away. Some of my girlfriends had taken up to a year to conceive. With a deep breath, and a prayer gently whispered, Steve agreed to \"try\" to have a baby.\n\nAbout a month later, Beth and Dave came to our house again for a visit. We ate our usual snacks\u2014including my husband's favorite potato chips and onion dip. (Oh how I miss those days when we were younger and could eat junk food and not feel crummy afterward.)\n\nThe next day while I was at work, I kept burping that nasty onion dip. And two days later, I was still burping. That's when I thought, I wonder if this is morning sickness?\n\nThirty years ago, when you wanted to find out if you were pregnant, you had to go to the doctor to take a pregnancy test. Today, you can simply run to the local pharmacy and buy an at-home pregnancy test.\n\nDuring my lunch hour I visited my doctor. After I returned to work, the doctor's office called with the message, \"Mrs. Stoppe, your test is positive.\" I wasn't sure exactly what that meant, so I asked, \"Positive what? Positive I'm pregnant, or positive I'm not?\" (At the time I was thinking, Steve will think it's a positive thing if my test turns out negative.)\n\nAs it turned out, I was pregnant. So on my way home from work I bought all the fixings for a nice dinner, including Cornish game hens and stuffing. I wanted to serve a special dinner when I announced to Steve he was going to be a daddy.\n\nWhen Steve got home, we sat down to a lovely candlelit table set with our wedding china. His eyes lit up when he saw the beautiful Cornish game hens come out of the oven. And then it happened\u2014as I carried the bowl with the stuffing in it to the table, Steve looked at the stuffing, turned up his nose, and said, \"Stuffing? I hate stuffing.\"\n\nWell, you can imagine the disappointment I felt in that moment! I burst into tears, and Steve was stunned.\n\nThen Steve pulled me onto his lap and said, \"It's all right; I'll eat it. I'm sure you did a good job of cooking it. I'm just not a fan of stuffing. Really, it's ok; look I'll eat some now,\" as he reached to take a huge bite.\n\nI stopped him before he could gag on the stuffing, which would have completely destroyed the evening. Then he wiped away my tears again and asked, \"Okay, now tell me what's really wrong.\"\n\nI apologized for overreacting to the situation, and explained how I had hoped to cook a special meal to celebrate a special occasion. A look of concern came across Steve's face\u2014at first he thought maybe he had forgotten our anniversary. I said, \"It's not our anniversary, or my birthday. But you are going to be a father.\"\n\nSteve's look of concern turned to fear. He said, \"Are you sure? You said it would take several months. How do you know?\"\n\nAfter I assured Steve the doctor had confirmed my suspicions, he held me tightly (looking back, I think he did that so I wouldn't see the terror on his face), but Steve's hug and gentle whisper, \"I'm gonna be a father...wow\" are sweet memories I will never forget\u2014in spite of the stuffing incident.\n\nOur first child, Meredith, was born on Christmas Eve (after 52 hours of labor\u2014can I hear you say, \"Wow!\"?). I was so happy to bring our little bundle of joy home and start living as a family that I left the hospital the very next day\u2014and besides, it was Christmas.\n\nAs time passed and we settled into our routines, Steve and I made an alarming discovery. Our baby, Meredith, was nothing like Kristal. Whereas Kristal would gladly accept a pacifier, Meredith would gag and spit it out. Whereas Kristal would sit in the baby swing for hours on end, Meredith would freak out and cry because the swing terrified her. And while Kristal would sleep soundly in her bed each evening, Meredith screamed wildly for hours on end. (Since I didn't know much about babies, it wasn't until Beth's mother came over for a visit that she informed me the reason Meredith pulled her knees up and screamed each night was because she had colic.)\n\nThis turned out to be the hardest year of our marriage. Up to this point, we had agreed on pretty much everything we did as a couple. But now disagreements were more frequent. For example, because of Meredith's constant crying I became frazzled, exhausted, and an emotional wreck. When I would ask Steve to take Meredith for a walk so she would feel better, he insisted she needed to \"cry it out\" in her crib. Since we didn't see eye-to-eye on how to deal with Meredith's crying, I worried, How many other things will we disagree on as we raise this child?\n\nThe irony of all this is I had originally anticipated only the bliss that having a baby would bring. I hadn't expected the difficulties and exhaustion as well.\n\nAfter three long months of constant crying, Meredith woke up one morning as a happy baby. No more tummy trouble, and from that day on, she has been an absolute delight. (Meredith\u2014if you're reading this, please know how very thankful we are for you, and in hindsight, we even thank the Lord for the work He did in us as you cried your little eyes out.)\n\nIron Sharpens Iron\n\nThey say hindsight is always 20\/20. When Steve and I were first married, I exhibited a bad attitude toward him when he messed up the house, forgot to do something, or said something that didn't sit well with me. Oh how I wish I could go back to those first months of marriage and relive them knowing what I know now. At any rate, the season I spent adjusting to married life was what God used to show me how selfish a person I was. Once I recognized that, the Lord was able to begin chiseling away at me and make me into a woman who reflected more accurately Christ's selfless love toward my husband. Then when I became a mother, once again the Lord showed me just how self-absorbed I was. Raising little ones is either gonna \"make yah or break yah\"\u2014isn't that true?\n\nProverbs 27:17 says, \"Iron sharpens iron, and one man sharpens another\" (ESV). Have you ever noticed what happens when iron is rubbed against iron? Sparks fly! And while the end result is both knives become sharper and better instruments, the sharpening process requires friction\u2014which produces sparks. In the same way, God uses the relationships in our lives to sharpen us. As we rub up against each other, the friction can make us better instruments for His use.\n\nRaising children is sure to invite friction into a marriage. Not only are parents passionate about their kids, they are emotionally vested as well\u2014even to the point of being unreasonable sometimes. Let me explain: In nearly 20 years of youth ministry work, Steve and I encountered many parents who were blinded by their passion for their kids. For example, moms will go to great lengths to defend their child if they feel a teacher is treating him or her unjustly\u2014even when it's clear the child is the one at fault.\n\nMost parents have strong opinions about how their kids should be disciplined, what they should be allowed to do, and so on. But when mom and dad disagree on these important issues, sparks will fly. If you and your husband disagree about a specific parenting issue and the friction causes the two of you to evaluate your motives and surrender to Christ's leading, then you will become sharpened instruments for the Master's use.\n\nOn the contrary, when conflict over your kids leads you to sinful actions, you can know your motivations are not in line with God's will. In the middle of parenting disagreements when you react sinfully to your husband, do you stop to evaluate why you are willing to sin in the situation? If not, you are sadly missing an opportunity God is providing for you\u2014and your husband\u2014to ask the Holy Spirit for help. Once you are willing to admit your own sinful motivations and bring your thoughts and actions in line with what the Bible teaches regarding your situation, you will be able to have a rational, respectful discussion with your husband.\n\nBut Girls Just Wanna Have Fun!\n\nWhen Meredith was about 13, Steve was the youth pastor at a church in Austin, Texas. Steve's rule had always been that our kids were not allowed to ride in a car with a driver who was under the age of 18.\n\nAs the daughter of the youth pastor, Meredith was friends with a lot of kids under the age of 18 who had driver's licenses. Whenever she would ask if she could ride with one of her teenage friends, Steve promptly told her, \"Are they eighteen or older? If not, then the answer is no.\"\n\nPoor Meredith. I felt so sorry for her. She was trying so hard to fit in with the older kids\u2014good kids. But whenever Steve said no, she felt like a little kid and was \"mortified with embarrassment\" (I believe that's how she defined it).\n\nI loved the kids in our youth group. They had so much fun together going bowling, boating, and such. So whenever Steve told Meredith she wasn't allowed to go with them unless the driver was at least 18 years old, I truly felt sorry for her.\n\nSince I tried very hard to never disagree with Steve's mandate in front of Meredith, she would have thought I was in complete agreement with her father's rule. But behind closed doors Steve and I had many discussions about how unfair I thought he was being to Meredith. On one occasion, I even remember crying as I told Steve he was \"responding out of an irrational fear of what might happen to Meredith.\"\n\nYou know how you know when you've pushed your husband too far? Well, with that comment, I knew I'd done just that! Steve's nostrils flared (if you knew anything about Steve, you would know this means he is really mad), and he responded, \"Fine. You let her do whatever you think is best\u2014if you are willing to live with the consequence if something happens to her.\"\n\nSeriously\u2014how was I supposed to respond to that statement? Truth be told, over the years that Steve had been a youth pastor, we had seen our fair share of inexperienced teenage drivers get into car accidents\u2014some of them fatal. I was more than aware of the statistics regarding teenage-driver accidents. I just wanted to believe nothing would happen to our daughter because these were good kids.\n\nAfter I spent some time pondering the matter, the Holy Spirit convicted me over the fact I hadn't been honoring my husband's wishes. In the end, I stood by Steve's decision to not allow Meredith to ride with inexperienced drivers. Meredith continued to be upset, embarrassed, and distant with us whenever the issue came up.\n\nOne night after we had been out as a family, we arrived home to find a car parked in front of our house. A father of one of our youth group teens walked up to Steve's car window and tearfully explained how one of the teenage girls, who had been driving with another friend in her car, had just been killed in an accident. I'll spare you the details, but suffice it to say, these were two good girls. There was no drinking involved, and they were not doing anything illegal. The accident was simply a result of the girl's inexperience as a driver.\n\nMeredith was devastated to hear her friend had died\u2014as were we. After the funeral, Meredith came to us to apologize for being angry about our \"stupid rule\" because now she realized we were just trying to protect her.\n\nThat experience made me realize how very much I needed my husband's wisdom and input when it came to decisions regarding our kids' well-being. I usually leaned toward letting the kids do what I thought would be \"fun,\" while Steve was less concerned about their having a good time and more interested in keeping them safe and molding their character to be Christlike.\n\nWhat if I had fought with Steve in front of Meredith? What if I had pushed to get Steve to change his mind and Meredith had been allowed to ride with her friends? Maybe nothing bad would have happened to her. However, any bad that would have occurred would have been the consequence of my insistence, which, in turn, could have damaged our marriage\u2014and ultimately, our family.\n\nWhen you and your husband do not work to present a united front to your children, they will inevitably look for the point of your contention and get the two of you bickering over who is right so they can go on their merry way and do whatever they please.\n\nTake a moment and evaluate the way you and your husband parent. Do you teach your kids to respect their father by how you treat him in their presence? Do you talk about your kids' father behind his back, or complain to your children when your husband does not measure up to your expectations? If you do not instill a healthy respect in your kids for their father, you will suffer the consequences when you need your husband's reinforcement during their preteen and teenage years.\n\nBy the time your children have reached their middle-school years, if you and your husband have not learned how to work together in raising your children for God's glory, you will certainly be faced with conflict in your marriage. During their preteen years kids usually start asking to participate in social activities that are less supervised by adults. Couple that with adolescent hormones, and the emotions of both child and parent can escalate rather quickly.\n\nSadly, in the 20 years Steve and I worked with teens, we witnessed many marriages damaged\u2014even destroyed\u2014as parents battled with their spouse over each and every decision pertaining to their kids.\n\nBut What if My Husband Is a Bad Father?\n\nIs your husband aloof about parenting? Or possibly too controlling? I know people who fall into each of these categories. So what can you do when your husband does not measure up to your expectations as a father? For insight, let's learn from a mom whose husband was extremely controlling.\n\nTina's husband, Bob, had always been a control freak. When they were dating, Tina found security in Bob's take-charge manner as he planned every detail of their dates to a tee. After they were married, however, Bob's strong tendencies to control were often a point of contention. To avoid Bob's barrage of questioning when he arrived home, Tina had learned to keep certain details of how she spent her days from her husband. As time went on, Tina learned Bob's desire to control was deeply seated in fear\u2014a fear of not being able to protect his family.\n\nWhen Bob and Tina's teenage son began to resist his father's strong hand of control, intense arguments between father and son broke loose in their home. Tina felt sorry for her son, and would often get into the middle of the arguments. Until one day Tina's son pulled her aside and asked her to please stay out of the fight. He explained to his mother that her stepping in actually made Bob angrier and less reasonable to deal with.\n\nAs a favor to her son, Tina stepped back\u2014way back\u2014whenever the two would go toe-to-toe. That is when Tina learned to pray like she had never prayed before. While her husband and son were having heated arguments, Tina would go to her prayer chair and intercede for both of them. First, Tina would ask the Lord to reveal any sin in her heart so that she might confess it and be cleansed so her prayers would be effective (see James 5:16). Then she prayed specifically for her husband's controlling ways not to push her son into rebellion. And finally, Tina prayed for her husband to be convicted of how his fear was driving his incessant need to control everyone\u2014and every event\u2014in their family.\n\nAnd do you know what happened? God did a work in Tina's heart. As she made a habit of meeting with the Lord and confessing her own sin in each matter, her prayers became an effective resource in resolving the conflict between her husband and son. Years have passed since those difficult days, and Tina's commitment to pray for rather than defend her son made a lasting impression on him as a man who now serves Christ. And God changed her husband, ever so slowly. Anyone who was watching how the Lord worked to resolve Bob's control issues would say the events God used were nothing short of miraculous!\n\nPrayer Is the Key\n\nYou see, when Tina agreed to get out of the way, God was right there to do His work. Do you have a husband who is controlling? Can you apply what you have learned from Tina's story to your own marriage? Prayer is the key. The problem is, most of us don't want to wait for God to do the work in the person we are praying for. Often when wives do not see an instant transformation in their husband, they look for a way out. They convince themselves that they and the kids would be better off not living under the difficult situation. However, you must remember that God frequently does His best work refining us when we are in the midst of difficult circumstances. (Please understand I am not referring to situations involving spousal abuse.)\n\nFor example, the value of persistent, passionate prayer is one of the shining virtues in the life of Hannah, who was the mother of Samuel. In the years before Hannah was able to bear children, she endured a troubled home life. Her husband, Elkanah, was a bigamist. Because Hannah had been unable to conceive, it is likely that in order to have children, Elkanah took a second wife\u2014Peninnah.\n\nCan you imagine the hurt and rivalry that Elkanah invited into his home by marrying another woman? The Bible says that Elkanah preferred Hannah over Peninnah because he loved her deeply (see 1 Samuel 1:4-5). So here you have the makings for some intense rivalry all under one roof. What was Elkanah thinking?\n\nWhat a mess these three had on their hands, don't you think? In the book Twelve Extraordinary Women we glean this insight: \"Hannah was in constant anguish because of her own infertility. She was further tormented by Peninnah's taunts. The burden and stress made life almost unbearable.\"\n\nAlthough Hannah and Elkanah's marriage was marred by tension, the two did love one another deeply. As Twelve Extraordinary Women says,\n\nHannah's love for her husband is the first key to understanding her profound influence as a mother. Contrary to popular opinion, the most important characteristic of a godly mother is not her relationship with her children. It is her love for her husband. The love between husband and wife is the real key to a thriving family...Furthermore, all parents need to heed this lesson: what you communicate to your children through your marital relationship will stay with them for the rest of their lives. By watching how mother and father treat one another, they will learn the most fundamental lessons of life\u2014love, self-sacrifice, integrity, virtue, sin, sympathy, compassion, understanding and forgiveness. Whatever you teach about those things, right or wrong, is planted deep within their hearts.\n\nEach year, Hannah and Elkanah traveled together to the temple to worship God. One year in particular, Hannah was so sorrowful over her situation she could not even eat. (Lucky girl\u2014whenever I am distraught, all I want to do is eat. What about you?) Despite her difficult situation, she never became embittered. Rather, she became a woman characterized by a steadfast prayer life. First Samuel 1:12 says Hannah \"continued praying before the LORD.\"\n\nWith a broken heart, Hannah was driven to her knees. Her trials were the very tool God used to make her a woman of intense prayer. And on this particular visit to the temple, here's what Hannah said: \"O LORD of hosts, if You will indeed look on the affliction of Your maidservant and remember me, and not forget Your maidservant, but will give Your maidservant a male child, then I will give him to the LORD all the days of his life\" (1 Samuel 1:11).\n\nAnd when God answered Hannah's prayer and gave her a son, her immediate response was to pray. First Samuel 2:1-10 records her words, and the prayer is a beautiful masterpiece of thanksgiving.\n\nSo Hannah's troubles taught her to pray with intense passion, and when God answered her, she continued to remain steadfast in prayer.\n\nHannah's difficult marriage was also the catalyst God used to cause her to dedicate Samuel to the Lord. Do you realize Samuel is one of the few men in the Bible for whom we cannot find any record of rebellion against God? What a legacy Hannah left in her son Samuel\u2014all because she turned to God in her trial and did not become bitter or look for a way to escape her difficult marriage.\n\nIn the same way God used conflict in Hannah's home to make her a woman of prayer, the Lord wants to mold you into a woman who prays at all times, including in difficult circumstances. And if you follow Hannah's example of keeping your heart pure before the Lord, your prayers will not be hindered. Your fervent petitions for your children will do more to prepare them for God's plan for their lives than anything you can do on your own, including rescuing them from a difficult situation they may be facing.\n\nTo Know Christ and Make Him Known\n\nIf you have a personal relationship with Christ, then you have been invited to be on a mission with God. And what is that mission? To put it simply: to know Christ and to make Him known.\n\nIn your marriage relationship, do you tend to focus on how you can have a better marriage? In parenting, is your aim to become a better parent? While these are worthwhile goals, if you make them the focus of your life, you will have missed your mission. Sadly, the Christian church today leans much more toward helping people build happy lives rather than missional lives.\n\nFor example, when you got married, was your goal to live happily ever after with your husband? Have you made it your life's goal to work hard so you can have a nice home in which to raise a family? Again, these are not necessarily bad goals. The problem lies in making these \"good\" goals idols in your life\u2014idols that take priority over God's mission for you to know Christ more intimately day by day through prayer, Bible study, and fellowship with other Christians.\n\nReading book after book on how to be a better wife, mother, or Christian yet neglecting to spend dedicated amounts of time with God and His Word is settling for far less than what the Lord has for you. For it is through time with Him and the Bible that you will learn the character of Christ. Your love for Him will grow only as you sit in His presence and get to know Him more intimately through the pages of Scripture.\n\nIf your mission is to know Christ and make Him known, you must devote yourself to knowing Him so well that you recognize His providential hand in your own life\u2014and that of your family.\n\nIn his book You and Me Forever, Francis Chan states, \"[Our kids] must see the Gospel brought to life when they observe our parenting. We strive to demonstrate a beautiful picture of Christ in hopes that they will find Him attractive and give their lives to knowing Him.\"\n\nWhen your life's purpose is to know your Savior more and more with each passing day, the natural outcome will be for you to think with a biblical worldview. When this happens, you will learn to see life's ups and downs as opportunities to make Christ known to those around you\u2014especially to your children. And the more time you spend with Jesus, the more His character will spill out of your obedient life and thus create in others a desire to know Him too.\n\nD.L. Moody, a prominent evangelist and minister of the nineteenth century, said, \"If we attempt to feed others we must first be fed ourselves.\" This means, as a mother, you must take the time to daily study God's Word so you are ready to teach to others the truths you learn\u2014including your kids. I don't think the Lord could make your assignment, as a mom, any clearer than what He laid out in Deuteronomy 6:4-9:\n\nHear, O Israel: The LORD our God, the LORD is one! You shall love the LORD your God with all your heart, with all your soul, and with all your strength. And these words which I command you today shall be in your heart. You shall teach them diligently to your children, and shall talk of them when you sit in your house, when you walk by the way, when you lie down, and when you rise up. You shall bind them as a sign on your hand, and they shall be as frontlets between your eyes. You shall write them on the doorposts of your house and on your gates.\n\nFor more insights into what this passage says about parenting, see pages 184-85 of my book Moms Raising Sons to Be Men. There, I share how you can help your kids develop a biblical worldview.\n\nYou Are an Ambassador for Christ\n\nThe apostle Paul often found himself in circumstances much worse than any you and I will likely ever face. And yet he never lost sight of his mission to proclaim Christ. Listen as Paul pleads with the believers in Ephesus\u2014he urged them to be\n\npraying at all times in the Spirit, with all prayer and supplication. To that end keep alert with all perseverance, making supplication for all the saints, and also for me, that words may be given to me in opening my mouth boldly to proclaim the mystery of the gospel, for which I am an ambassador in chains, that I may declare it boldly, as I ought to speak (Ephesians 6:18-20 ESV).\n\nIf you were in prison, would your prayer request be, \"Hey, guys, I'm in chains for telling people about Christ. Could you keep praying for me to have additional opportunities to speak the gospel more boldly?\"\n\nYou won't likely find yourself in chains anytime soon, but sometimes a difficult marriage can make you feel like you're in prison. What if God has you right where He wants you? What if, like in Paul's life, the Lord knew you would be most effective sharing with others\u2014including your children\u2014the hope of salvation because of your pain?\n\nLet me put it to you this way: If the only way God can bring your kids into the kingdom is by showing them how faith in Jesus is real through your struggle, is it worth it? My friend Tina thinks it is. She said, \"If my husband's controlling bent is what drove me to my knees and my son to Christ, it was all worth it!\" Wow. Isn't she right?\n\nFirst Peter 4:11 says, \"If anyone speaks, let him speak as the oracles of God. If anyone ministers, let him do it with the ability which God supplies, that in all things God may be glorified through Jesus Christ, to whom belong the glory and the dominion forever and ever. Amen.\"\n\nIn Christ, God has called you to be His mouthpiece to those whom He is drawing to Himself. As you grow to know God through Bible study, memorizing Scripture, and prayer, you will no doubt glorify Christ because the natural outpouring of one who knows Jesus intimately is a love for others. This love spills over first into your marriage, then to your children, and then into all the other relationships the Lord brings your way.\n\nSecond Corinthians 5:20 makes this profound statement: \"Now then, we are ambassadors for Christ, as though God were pleading through us: we implore you on Christ's behalf, be reconciled to God.\" How does realizing you are Christ's ambassador to reconcile others to Him influence the way you relate to your husband and your children? Maybe you should take a moment to dwell on the fact the Lord has called you to be His ambassador. Then ask God to give you the courage to set aside all other pursuits\u2014even personal happiness\u2014to become the woman He has ordained you to be in this generation.\n\nThe apostle John said, \"I have no greater joy than to hear that my children walk in truth\" (3 John 4). Because when your children surrender to the truth of the gospel, the Spirit will do the work that no parental behavioral modification can ever accomplish. It is the Holy Spirit who changes everything!\n\nRemember, the circumstance in which you now find yourself is just a season. Your life will soon be past. Psalm 102:11-12 says, \"My days are like a shadow that lengthens, and I wither away like grass. But You, O LORD, shall endure forever, and the remembrance of Your name to all generations.\" When you live with an eternal perspective, you will find the courage to follow Jesus and live with the knowledge that your obedience to Christ will result in eternal rewards, as well as God's greatest good in this life.\n\nYour kids' security lies in the health of your marriage relationship. When you live with your eyes focused on the mission God has called you to\u2014to know Christ and make Him known\u2014you will have learned the key to building a no-regrets marriage. When you determine to live in a manner that reflects a genuine love for Christ\u2014no matter how smooth or difficult your marriage relationship may be\u2014you will do far more to draw your kids to salvation than any words you could ever say. And isn't that your ultimate goal?\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nI have a friend named Ken who devoted his life to working hard\u2014and playing hard. His free time was spent on all sorts of recreational activities from motorcycle riding and sailing to muscle cars. Although these hobbies were not bad in and of themselves, the time Ken spent enjoying them consumed his weekends, and on most Sundays, kept him away from church. All the while Ken's wife, Dorinda, continued to take their children to church every week. And she quietly prayed for the Lord to convict her husband regarding his misguided pursuits.\n\nKen's preoccupation with self had taken from Dorinda a husband who was once devoted to Christ. And it pulled Ken far away from any kind of ministry for the Lord. After a number of years, the Spirit began to show Ken that he was wasting his life. One day, Ken reluctantly agreed to attend a particular Bible study our church was offering to our men's group.\n\nAs Ken studied how men in the Bible experienced God when they walked in obedience to Him, he came to understand that he would need to reevaluate the way he was living. After a season of wrestling with God over his newfound convictions, Ken realized the priority of his life needed to be loving God with all his heart, soul, mind, and strength (see Mark 12:30).\n\nIt was then that Ken said, \"I need to adjust my life to what I have been learning.\"\n\nAt last, Dorinda's prayers were answered! Ken was ready to make knowing Christ and making Him known the supreme priorities of his life.\n\nAs a result, Ken got rid of many of his toys and set aside some of his hobbies to become a faithful follower of Christ. Talk about a transformation! Dorinda was overjoyed as she witnessed her husband's newfound passion to serve the Lord.\n\nAfter a time, Ken has come to enjoy some of his hobbies again, but now he and Dorinda do them together with the purpose of telling others about Christ. They also teach weekly Bible studies and mentor men and women in their church. And the two grandsons they are raising in their home get to watch their grandparents' marriage reflect their all-out commitment to knowing Christ and making Christ known.\n\nPerhaps your husband doesn't do what Ken did, or maybe you are secretly rejoicing at the prospect of your husband giving up some of the things that take up his free time. That's not my point here.\n\nMy point is this: Dorinda's deepest longing was to have her husband in church with her on Sundays. So what did she do? Did she nag, manipulate, cry, or pout until her husband gave in to her demands? No. Rather, Dorinda prayed.\n\nIf you long for your husband to be a better spiritual leader so that your children might be more obedient, don't nag him until he agrees to change. Nagging never works with husbands. In fact, if ever Rhonda nags me, I tend to want to do the opposite of what she asks.\n\nInstead, pray diligently each day for the Lord to transform your husband\u2013\u2013and your children\u2014with the truth.\n\nWhatever stresses you may experience in your marriage (and when you are raising children together, there will be seasons of stress), remember that the answer is not to stand against one another in conflict, but to stand united in prayer. Pray for your husband, and pray for your kids. Because the Bible promises, \"The prayer of a righteous person has great power as it is working.\"\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. From this chapter fill in the blanks: God's mission for your life is for you to _________________ Christ and to ____________________________ known.\n\n2. Write out 2 Corinthians 5:20. As an ambassador for Christ, what appeal does God desire to make through you to your children?\n\n3. What would your children say is the priority of your life? Read Job 6:24 and then ask God to help you understand whether you may have gone astray. Ask Him to teach you how to adjust your life to what you have learned in this chapter.\n\n######\n\nLIVING IT OUT\n\n######\n\n1. Look up 1 Peter 4:11 (ESV) and fill in the missing words:\n\nWhoever speaks, as one who speaks__________________; whoever serves, as one who serves ___________________ \u2014in order that __________________________ God may be glorified through Jesus Christ. To him belong glory and dominion forever and ever. Amen.\n\n2. Describe two steps you will take to demonstrate to your children that the most important pursuit in life is not personal happiness, but to build a life that stores up eternal rewards as you seek to know Christ and make Him known.\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nI Would Be Happier Married to Someone Else\n\nThe Grass Is Not Greener on the Other Side of the Fence\n\n######\n\nMy dear friend Vi was married for 42 years to Curt, the love of her life. When I asked Vi to tell me about how she and Curt met, she said, \"We were in college, on a choir tour. I played the piano and he was going into music ministry, so we were a good fit.\" Vi jokingly added, \"Although, when you play the piano and you marry a minister, you're never quite sure if he married you for love or because he needed a pianist!\"\n\nThroughout their married life, Vi and Curt served the Lord in full-time ministry. Vi often referred to their marriage as a waltz through life divinely choreographed by the Lord. Their waltz came to an end when Curt was diagnosed with cancer at age 68. As his health failed, Vi never left his side.\n\nOn the last day of his life, Curt looked to his sweetheart and said, \"Vi, am I dying?\" To which Vi tearfully responded, \"Yes, dear\u2014you are dying.\"\n\nAnd then in the joyful spirit Curt so adored, Vi whispered, \"Curt, you are going home to see Jesus! What is the first thing you want to say to Him when you see His face?\"\n\nCurt closed his eyes and smiled as he considered the moment he would stand in the presence of the Lord. And then without hesitation, he looked into Vi's gentle eyes and said, \"I am going to thank Him for giving me you.\" (Read Vi's poem at the end of this chapter on pages 86-87.)\n\nI have told Vi and Curt's story a number of times, but I can never get through it without crying. Even now as I write this, tears are streaming down my face. We all long to have a marriage like Vi and Curt's, don't we? A love that lasts through the tests of time. A happily-ever-after story our grandkids will tell their children about long after we are gone.\n\nIn this day of quick divorces and remarriages, lifelong relationships like Vi and Curt's are all too few. Sadly, statistics generally show that around half of those who promise \"Till death do us part\" will never celebrate a fiftieth wedding anniversary, let alone a twenty-fifth.\n\nWhen Your Husband Lets You Down\n\nSo what's the problem? Why are people so quick to abandon their vows to seek happiness in the arms of another? When couples are blissfully engaged to be married, betrayal is the furthest thing from their minds. But after they tie the knot and the years go by, the grass starts to look greener elsewhere. How is it that so many marriages are ruined because the husband or wife falls in love with someone else? And what can you do to ensure this doesn't happen to your marriage?\n\nFirst, realize that women are not usually tempted to fantasize about leaving for another man unless they feel like their husband repeatedly does not measure up to their expectations. The danger lies in allowing your disappointment to cause you to think, I would be happier married to someone else.\n\nWhenever your husband lets you down, you most likely feel betrayed. When I use the word betrayed, I'm not talking about your husband having an affair with another woman, which is the ultimate betrayal. Rather, I am referring to the everyday disappointments that make you feel as though your husband is being disloyal to you. Some common examples of such betrayals are:\n\n\u2022 when your husband vents about you on the phone to his mother\n\n\u2022 when he spent an excessive amount of money on a frivolous item without telling you\n\n\u2022 when he looks too intently at another woman\n\n\u2022 when he looks at pornography\n\n\u2022 when he makes comments to embarrass or undermine you in front of others\n\nPreparing, in advance, to deal with these types of disloyalty will allow you to determine how you will handle feelings of betrayal before they happen. Now, I am not saying you should watch your husband in a way that suggests you don't trust him. Rather, settle in your mind how you plan to respond with forgiveness before an infraction occurs. And be ready to offer the same grace to your husband that you would want from him in the times that you disappoint him.\n\nIf you are not prepared with a response and forgiveness, then you are more likely to deal with your disappointment irrationally. And this is when many wives begin to believe the myth I would be happier married to someone else. Beware, for if you toy with this idea for too long, you will invite Satan to wreak havoc in your life. The devil knows that if he can get you to dream about a happier life with another man, he will have gained a foothold toward destroying your marriage. Don't give him that chance (see 2 Corinthians 2:11; Ephesians 4:27).\n\nIn the face of your husband's everyday betrayals, you have choices to make. You can either be ill-prepared for the letdown and withdraw emotionally, or you can choose to cover the betrayal with love\u2014God's love.\n\nHow you respond will either create an emotional distance in your relationship or deepen your love for one another. You will find strength and peace to cover your disappointment with grace when you fall to your knees and seek the One who will never betray you.\n\nThe Grass May Look Greener, but It's Full of Thorns\n\nI come from a long line of broken marriages. So as young bride, I wanted to learn the secret to a happy and lasting marriage. For help, I looked to a number of godly women in our church whose marriages I wanted to emulate. When you become friends with happily married couples, you can learn from their example how to have a joyful union. Can you think of at least one godly couple whom you would like to emulate? If not, pray for the Lord to lead you to some. (Joining a Bible study with some older women is a great place to start.)\n\nMost of the couples Steve and I spent time with were older than we were. And the wives in these marriages took seriously the instruction in Titus 2 for older women to teach younger women how to love their husbands. This made them great mentors for me. (And it's also the reason I am writing this book\u2014to pass on to you the godly principles they taught me so I can be a Titus 2 woman in your life. And then you, in turn, can teach these principles to the women God brings into your life. See how it works?)\n\nWhen I would ask these women, \"What is the key to a happy marriage?\" I was surprised at their answers. The overall message I gleaned from these Titus 2 women was this:\n\nThe real secret to a happy marriage is not in how much you love your husband, but how much you love Christ. God created us to worship Him. When you make it a priority to worship God through quiet time with the Lord in Bible study, prayer, repentance, and obedience to His will, you will find your joy, identity, and sense of well-being in your relationship with your Creator. When this happens, you will not feel the need to find your worth in your relationship with your spouse, and you will never be tempted to look to another man to fulfill you either.\n\nGod created you with a need to be loved and to feel significant. But He never intended for you to fulfill those desires through marriage\u2014or through any relationship with a person. Rather, God wants to fill the longings of your heart with Himself. The problem is that sin stole away mankind's desire for intimacy with the Creator. And now, because of sin in your heart\u2014and mine\u2014we focus on self and struggle with self-worship. In this state of self-love, you are susceptible to think I deserve to be happy and to believe Satan's lie I would be happier with someone else. In this vulnerable state, when one romantic relationship fails to make you feel complete, there is a temptation to replace it with another one.\n\nThe only way to guard against having a distorted sense of love and self-worth is to have a healthy personal relationship with Jesus. Again, that involves growing your love for Him through prayer, spending time in Bible study, and fellowshipping with other Christians. When you determine to find your joy in Christ, you will be set free from looking to others to fill the void only God can satisfy.\n\nSo once you decide to look to God as the sole source of true happiness, you are ready to apply the principles Steve and I learned from the happy couples who befriended us so many years ago.\n\nEight Insights We Learned from Happily Married Couples\n\nHere are eight practical ways you can cultivate a happy marriage:\n\n1. Have Realistic Expectations\n\nOne reason people become unhappy with their marriage is because the relationship doesn't turn out to be all they had expected. Did you think your husband would be the answer to all your hopes for happily-ever-after? If so, at some point after the honeymoon was over, you came to realize you had married a normal human being and not the Prince Charming you imagined him to be.\n\nMy \"aha moment\" came during our honeymoon when my brand-new husband proceeded to use the bathroom in front of me. I was shocked. I mean, I knew this big hunk of a man relieved himself, but it never occurred to me he would do it in my presence!\n\nThe sooner you realize you and your husband are both imperfect people, the better you will be prepared to cover with grace the times you let one another down.\n\n2. Your Husband Is Not Like You\n\nYou don't have to be married for too long to discover your husband is not like you. All too often, couples attempt to define unity in marriage as \"sameness.\" But unity isn't sameness.\n\nFor example, one way men and women differ from each other is the way they respond to discord. In conflict, a woman will generally pull away, secretly hoping her husband will come after her to show her how much he cares. But most men require time to process a heated conversation. They often need to distance themselves from the situation and think through what was said. This is why, after a disagreement, your husband may go out to the garage to work on a task. While he is contemplating his own feelings, trying to understand how you are feeling, and possibly looking for a way to resolve the conflict, it would be easy for you to interpret his pulling away as rejection or a lack of concern for you and your feelings.\n\nSo don't be quick to assume your husband's retreat means that he doesn't care. And learn how to give your husband the space he needs to come back and have a rational conversation\u2014when he is ready.\n\nRemember, unity in marriage does not mean you have to see eye-to-eye with your husband on every detail of life. In his book What Did You Expect? author Paul David Tripp says,\n\nUnity in marriage is not the result of sameness...God has designed that you will be married to someone different from you. Unity is, rather, the result of what husband and wife do in the face of inevitable differences...The more you look at your spouse and see the imprint of God's fingers...the more you will be able to resist the temptation to try to remake him in your own image...The more you see divine beauty...in the differences between you, the less you will be irritated by them.\n\nWhen you begin to celebrate God's imprint on your husband, you will be prepared to implement the next insight I learned from happy couples:\n\n3. Think the Best About Your Husband\n\nI know that in chapter 3 we already talked about dwelling on your husband's good qualities, but it bears repeating in this context. Remind yourself regularly of the qualities you love about your husband. And resist the temptation to compare him to the \"ideal husband\" you dream of having. Wouldn't you want your husband to do the same for you?\n\nEven wit deliberate effort toward resisting the temptation to compare, married couples will often lean toward viewing one another through a negative lens. If thinking the best about each other is not yet a habit within your marriage, someone has to take the first step. Let that someone be you.\n\nMany couples I have talked to will admit to having had a mediocre marriage\u2014or even a bad one\u2014until one of them determined to stop comparing their spouse to the person they wished he or she would be. In so doing, their newfound habit of thinking positive thoughts spilled over into affirming words and kind service to their spouse. More often than not, the actions of one spouse not only turned the marriage around, but in many instances even saved the marriage.\n\n4. Be Kind to One Another\n\nAre you kind? I don't mean are you nice to the mailman, or the bagger at the grocery store. It's easy to be kind to people you only see for a few minutes each day. But in general, do you have a kind disposition? Is kindness your default mode, or do you have to force yourself not to lash out when you are offended? Maybe a better question to ask would be this: Does your husband think you are kind?\n\nEphesians 4:32 says, \"Be kind one to another.\" This is a command, not a suggestion. And yet wouldn't you agree there are times that being kind is not the easiest response? And if you have little ones at home, lack of sleep alone can have a negative influence upon your attempts to remain kind. One marriage expert says women in their thirties (when most women have small children) go through what he refers to as \"the unfriendly years.\" Can I get a witness? As a stay-at-home mom, I recall being tired, overworked, and \"underpaid.\" Unfriendly would certainly define the way I sometimes treated my husband in those days. (I remember thinking, He gets to go to work every day in an air-conditioned office and have lunch with grown-ups, while I'm chasing after kids and up to my elbows in laundry.)\n\nHave you ever struggled with this type of thinking? Would you characterize yourself as unfriendly to your husband? If while reading this you find yourself resentful\u2014for whatever reason\u2014realize that you'll only end up hurting your marriage. Whatever the situation or your circumstances, if you make an extra effort to be kind to your husband now, you will enjoy the benefit of a happier marriage as time goes on.\n\nThe seeds of kindness Steve and I planted during the chaotic years of raising children have borne fruit, and today we find ourselves in the midst of a delightful empty-nest season. By contrast, I know many wives who were unkind to their husbands because they harbored an unforgiving attitude and resentment over their husband's lack of help when the kids were little, only to reap a broken marriage when the children grew up and left home.\n\n5. Refuse to Fantasize About Being Married to Someone Else\n\nA major threat to a happy marriage is the temptation to believe you married the wrong person.\n\nWatching soap operas or romantic movies can easily lead to a restless heart within women. After her own life fell apart, a friend of mine told me, \"Whenever you teach women, warn them not to watch soap operas. I used to watch them for hours and wish my life was as interesting as the lives of the people on the show. When the drama visited my own marriage, it nearly destroyed me. The lifestyle I thought would spark a new fire brought devastation and destruction.\"\n\nWhen times get hard\u2014and they will\u2014allowing yourself to daydream about what it would be like to escape the hardship will only invite trouble into your marriage. If you are looking up old boyfriends on the Internet, or dreaming about what life would be like if you were married to a different man, you are already in the process of undermining the foundation of your marriage. And when you let the ground under your marriage get shaky, it will one day crumble beneath your feet.\n\nIn the Sermon on the Mount, Jesus told the story of the foolish man who built his house upon the sand (Matthew 7:26-27). Building your hopes for a happy marriage with someone other than your husband is certainly foolish. And when the storms of life come, your house will come crashing down around you.\n\n6. Your Husband Is Not Your Enemy\n\nIn times of strife, remind yourself that your husband is not your enemy. Your real adversary is the devil, who is a roaring lion seeking to devour you\u2014and your marriage. When you realize Satan comes to steal, kill, and destroy, you will know that it's the tempter who is your enemy, and not your husband.\n\n7. Love Christ More Than You Love Your Husband\n\nI know we discussed this several times in this book, but I cannot stress enough this key principle: When you determine to grow more deeply in love with Christ, you will find your worth in your relationship to Him. When this happens, you will not look to your husband to meet the needs only God can fill. When you live to love Jesus, His love for your husband will spill out of your heart. And your heart will become joyfully satisfied with your husband.\n\nOne marriage counselor offers this insight: \"Love is being unwilling to ask your spouse to be the source of your identity, meaning and purpose, or inner sense of well-being, while refusing to be the source of his.\"\n\n8. Determine That Divorce Will Never Be an Option\n\nIf ever you think leaving your marriage is the answer to your problems, remind yourself: Wherever you go, there you are. This means whatever struggles you may be having, realize you are half of the problem. If you choose to leave a difficult marriage, you can be certain you'll take all your unresolved issues into your next relationship as well. And while we are on this subject, let's talk a bit more about the consequences of divorce.\n\nThe Effects of Divorce\n\nWhenever couples have built a distance between themselves through hurtful words, unwillingness to forgive, or neglect, they may start to believe divorce is the only answer. Sadly, we live in a culture that thinks the goal of life is happiness at all costs, so divorce has become a very common choice, even among Christians. However, you can be sure the happiness you are seeking will elude you as you face the devastating fallout that comes with a divorce.\n\nOver the course of our years in youth work, my husband and I observed again and again the stress that raising a teen can have upon a marriage (which helped give us wisdom for when we raised our own teens). Some of the parents we knew chose to abandon their marriage vows when the times got rough\u2014only to find themselves facing a whole new set of problems that come with raising teenagers in a broken home.\n\nThe children's pastor of our church told me, \"In almost 20 years of working with kids, I have observed that the most harmful decision a parent can make for their child is to get a divorce.\" (Please understand that this statement does not hold true if there is abuse in the home.)\n\nI have heard women say, \"Our kids will be happier if they are not exposed to our constant fighting.\" Sadly, after the divorce, I have heard these same women regretfully admit how deeply their children were wounded by the breakup. Here are some heartbreaking realities that children of divorce will likely face:\n\n\u2022 Mom and Dad are less focused on their children as they work to establish their new single life. Whether the parents are focused on career, dating, or dealing with their own hurt after the divorce, kids inevitably suffer the consequences.\n\n\u2022 One or both of the parents remarry, which introduces stepchildren or siblings from the new marriage into the family. When this happens, your kids will struggle with feeling overlooked or less valued by their biological parents. Then there is the all-too-common threat of kids being sexually abused by a stepparent or step-sibling.\n\n\u2022 Children who grow up in broken homes deal with deep-seated insecurity issues that often lead to them repeating the cycle of divorce in their own marriages.\n\nOver the many years I have mentored women, I've learned that one major contributor to a woman's distrust of her husband's loyalty is her own parents' divorce. So don't deceive yourself into believing your broken marriage won't have any effect on your children's marriages.\n\nWhen my parents divorced after 30 years of marriage, I was well into my twenties\u2014and happily married with two children of my own. Even though I was an adult, their divorce shook my security in ways I would never have dreamed. Without a doubt, my parents' divorce is one of the most grievous experiences I have ever endured\n\nTill Death Do Us Part\n\nCan you think of an older couple you know who has stuck it out through the bad times? A couple you would like to emulate? My husband's parents, Bill and Eleanore, were just such a couple. My mother-in-law was deeply in love with \"Willie,\" as she called him. And I was captivated by their adoration for each other.\n\nAt first glance, you would think they never had any struggles in their marriage. But the reality was that Bill and Eleanore had weathered a number of difficult storms in life. For example, Bill was deployed to Korea not long after their marriage. While he was away, Eleanore suffered alone through a tragic miscarriage. In the first five years of their marriage, the couple silently grieved over their inability to conceive. So you can imagine their relief\u2014and elation\u2014when Steve was born. And then three years later, God blessed them with another son, Daniel.\n\nWhen Steve and I were dating, he used to tell me his parents never fought. I didn't believe him. I would say, \"Every married couple fights. Your parents are just hiding it from you and your brother.\"\n\nBut through the years I came to discover that Steve's perception of his parents' relationship was spot-on. The two genuinely adored one another. When life had been hard, rather than looking for a \"better life\" with someone else, they pressed into their relationship with Christ\u2014and ultimately into one another.\n\nBill and Eleanore's devotion for each other shone the brightest when Eleanore was diagnosed with Alzheimer's disease. First she forgot how to play the piano. Playing music for her church had been her great delight. You can imagine her sorrow when she couldn't remember how to play her favorite hymns. Bill grieved quietly for his bride.\n\nFor a decade, Bill devoted himself to Eleanore's care. Many nights I would hear him sobbing on the front porch as my husband held his dad in his arms. After a number of years the stress of taking care of Eleanore was taking a toll on Bill's health, so we suggested he put her in an assisted-care facility. To which Bill replied, \"She is my sweetheart. I would never dream of leaving her care to someone else.\"\n\nTalk about love. My kids and I were privy to watching true love lived out through Bill and Eleanore. Even when \"Ellie\" (as Bill called her) didn't remember who Bill was, he continued to take care of her. He even went so far as to sell their home, quit his job, and move into a house on our ranch so he could care for his love full-time.\n\nThe love demonstrated by my in-laws has left a lasting impression upon me, their children, and grandchildren. Jesus said, \"Greater love has no one than this, than to lay down one's life for his friends\" (John 15:13). Steve's father certainly personified this kind of selfless love until the day his beloved Eleanore passed on into eternity.\n\nHave you ever considered the story you are writing with your life? This generation is desperate for love stories like those in this chapter. Won't you join the ranks of Vi and Curt, who waltzed in each other's arms until Curt's final breath, and Bill and Eleanore, whose undying love withstood a failing mind?\n\nWhen you learn that true joy and satisfaction is not found in a perfect marriage with a perfect person, but rather comes from your relationship with Christ, you will learn the secret to a happy marriage. I pray you will apply the biblical principles laid out in this book so you can build a no-regrets marriage. And through your example, may your children and grandchildren learn the secret to a happy marriage.\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nAfter reading about my parents, I had a hard time writing this conclusion to Rhonda's chapter. Remembering how much my mom and dad loved one another brings about emotions that are strangely sad, yet joyful. Growing up under my parents' example, I learned what God intended for unconditional love between a husband and wife to look like. Because my parents did not attempt to find their identity, acceptance, or value in each other, but rather in their relationship with Christ, they learned how to love one another selflessly. Even when my mom didn't remember who my dad was because of her Alzheimer's, my father continued to love and serve her\u2014all the while grieving the loss of who she had once been.\n\nFor you as a believer, finding your identity, acceptance, or value in your spouse\u2014or any other earthly relationship\u2014is always a dangerous path to trod. You will certainly be disappointed if you attempt to establish your worth based on your husband's view of you. Because at some point your husband will let you down, and your perceived security will be shattered. It is in this place of disappointment that a woman may be tempted to look outside of her marriage relationship to another man to find her worth.\n\nGalatians 2:20 says, \"I have been crucified with Christ; it is no longer I who live, but Christ lives in me; and the life which I now live in the flesh I live by faith in the Son of God, who loved me and gave Himself for me.\" As much as Rhonda loves, accepts, and affirms me, she can never do for me all that Christ has done. And the same is true for you and your spouse.\n\nInstead of looking to your husband to make you feel treasured, make it your goal to die to yourself daily. And live by faith in the Son of God, who loved you so much He gave His life so that you could be made alive in Christ. Learn to remind yourself daily, through Scripture, just how precious you are to God. When you do, you will let your husband off the hook for being the source of your self-worth, and guard yourself from the temptation to look elsewhere for happiness.\n\nMake no mistake\u2014I am not saying you shouldn't want your husband to say and do things that make you feel loved. First Peter 3:7 tells us Christian men to live with our wives according to understanding. That means God expects husbands to try to meet their wives' emotional needs. But know that we men often need help with understanding what you need from us. If your husband is to try to learn what actions he can take to affirm his love for you, he will need you to coach him. Because, left to ourselves, more often than not we guys will get it wrong. And trying to figure out what our wives need at any given moment can be a rather intimidating task for all men. So if you find your husband not doing enough to make you happy, don't ever believe the lie that another man would understand your needs better\u2014because it's just not true.\n\nIn a marriage, both the husband and the wife bear the responsibility when they wrongly expect to find their worth in the way their spouse treats them. Accepting the fact that the ultimate goal of your life is to bring glory and honor to God is the first step you can take toward building a marriage that stands the tests of time.\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. Which couple's story had the greatest impact on you, and why? What inspired or challenged you most? Then write down what type of story you would like your children and grandchildren to one day tell about your marriage. What is one major adjustment you can make in your marriage to help write that story?\n\n2. If your heart is broken over a difficult marriage, what encouragement can you draw from Psalm 34:18 and James 4:8?\n\n######\n\nLIVING IT OUT\n\n######\n\n1. If you have developed a habit of being unkind, realize that God can help you change. James 4:2 says, \"You do not have because you do not ask.\" So begin by seeking help from the Lord. Write out a prayer asking God to help you be kind to your husband.\n\n2. In a notebook, write out the eight principles we can learn from happily married couples found in this chapter. Next to each insight, make a note of how you will apply what you have learned to your own marriage. To help you build a no-regrets marriage, keep this list where you can refer to it often.\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nThe Waltz\n\n\"Will you come and dance with me?\" he asked me oh so tenderly,\n\n\"And waltz with me around the floor like no two ever danced before?\"\n\nI smiled at him; my heart skipped a beat and before I realized it, I was on my feet.\n\nAccepting his invitation seemed so right\u2014I was willing to dance all night.\n\nHe looked to the Maestro and said, \"Conductor, if you please.\"\n\nAnd at his direction, the orchestra played with the greatest of ease\n\nThe most beautiful waltz I'd ever heard\n\nAnd we danced and danced without saying a word.\n\nWe both knew when the music began that we were a part of a much larger plan,\n\nAnd this waltz that we started just minutes ago was the first of a lifetime of dances\n\nTo show that we were becoming partners for life;\n\nNot just on the dance floor, but as husband and wife.\n\nI followed his lead as the music played on and we moved 'cross the floor in complete unison.\n\nWhat freedom I felt with his hand in mine, keeping in step, stepping in time.\n\nThe future was bright for two people that night who waltzed as though dancing on air.\n\nLife was just grand while holding his hand and gone were all worries and cares.\n\nAs the music played on and the clocked ticked away, nighttime approached at the close of the day\n\nAnd the music took on a different mood; and we found ourselves with a new attitude.\n\nWe were tired, without a doubt, and found that even good things wear you out!\n\nBut when one would faint, the other was strong and together we faced each new song.\n\nThere were times, I'll admit when I wanted to quit, but how could I leave him alone?\n\nWe'd partnered for life and I was his wife, \"flesh of his flesh and bone of his bone.\"\n\nHe was my \"head\" and I, his \"helpmate,\" we've waltzed now for so many years.\n\nI've laughed when he laughed and cried when he cried, shedding innumerable tears.\n\nNow we've become one and though the dancing is done, we still make quite a smart pair.\n\nI'll never regret that day when we met and he asked me to spin 'round the floor.\n\nSometimes it seemed slow, the music you know, and other times it was so very fast.\n\nBut together we stayed whatever was played and looking back, I thank God for the past.\n\nOne, two, three\u2014one, two, three\u2014it's the rhythm of the waltz you see;\n\nOne Conductor, plus two partners, make a unit of three.\n\nThe Conductor selected the tunes and tempo for each melody.\n\nAnd side by side, we simply complied with the music He played faithfully.\n\n\u2014Vi Estel 2014\n\nHe Would Love Me More if I Were Prettier\n\nThe Secret to Keeping His Attention\n\n######\n\nGeorge M\u00fcller is well remembered as a man who rescued thousands of helpless orphaned children from the cruel streets of England. In his famous orphan houses in Bristol, he helped provide for their basic needs and gave them an education as well.\n\nAs a young man, George met the beautiful Ermegarde at a Bible study. He loved the way her curls danced around her face when she giggled. When George was convinced God was calling him to be a missionary, he shared his dream with Ermegarde\u2014whom he intended to marry. Ermegarde turned up her nose and said, \" 'I could never be a missionary. Missionaries are poor...Be a lawyer, or a doctor, and leave being a missionary for other people who don't have anything better to do!' With that she stood up and stomped out of the room.\"\n\nAfter agonizing over this for several weeks, George knew he had to end his relationship with Ermegarde. Only then would he be free to follow the Lord's leading.\n\nLater, George met Mary Groves, who was not at all like Ermegarde\u2014either in external beauty or inner attitudes. As one biography tells it:\n\nThe relationship blossomed, and George found himself in love with Mary. Such a feeling surprised him for more than one reason. First, Mary was eight years older than he was. And second, he had not been looking for or even considering a wife. As far as he was concerned, a wife would slow him down. What if God called him to go someplace strange or remote? Could he expect a wife to follow him? And would he feel as though marriage made him a prisoner?\n\nAlthough George may not have expected just any wife to follow him, there was something about Mary.\n\nSoon the two were married. Mary set up housekeeping, and a week later, all her belongings were in George's tiny home. When George saw Mary's fancy silver and china, he talked with her and asked her to sell all her treasures for their ministry. Mary responded, \"Do what you think is best...and may God help us both.\"\n\nA while later, when George and Mary were out walking, George said,\n\n\"Mary, thank you for selling the things. Now there is another matter we need to talk about...It's the pew rent. I can't see how we can follow Jesus' command to treat all men equally if we give rich people the best pews.\"\n\nMary said, \"But George, that's our only income...\"\n\nGeorge said, \"I know it's hard, but I think it is the right thing to do.\"\n\nMary said, \"Do what you think is best, George. I can trust God, just like you do.\"\n\nGeorge stopped and hugged his wife. Tears spilled down his cheeks. Mary had been right\u2014marriage had not become a prison for him. Instead, it had given him a partner in the faith.\n\nTogether, George and Mary chose to trust God fully to meet their needs, and to give all else to ministry. It's reported that over the course of his life, nearly 1.5 million pounds passed through George's hands to provide food, shelter, and clothing for the multitudes of orphaned children in his care. He died with very little money to his name because he gave it all to ministry.\n\nWhat's more, George and Mary determined to never ask any person for money to support their orphanages. They simply prayed with faith, and watched God provide for their every need\u2014sometimes at the very last possible moment. It is for this reason that George Muller is considered to be a man of incredible faith.\n\nWhen Mary M\u00fcller died, George preached a short message at her funeral and quoted Psalm 119:68 (KJV): \"Thou art good, and doest good.\" The funeral service was one of the largest Bristol had ever seen. Thousands of letters poured in from orphans whom Mary had nurtured as children.\n\nGeorge was comforted by the letters, but he missed Mary greatly. Though she may not have been pretty by the world's standards, George knew his wife was one of the most beautiful women this world had ever known. In her, he had found a good thing.\n\nHe Who Finds a Wife Finds a Good Thing\n\nAre you a good thing? When your husband thinks about you, does a smile come across his lips? Not because of your outward appearance, but because of your inner beauty\u2014\"the hidden person of the heart, with the incorruptible beauty of a gentle and quiet spirit, which is very precious in the sight of God\" (1 Peter 3:4)?\n\nThe world says a man has done well if his wife is lovely to look at. But the Bible says, \"Charm is deceitful and beauty is passing, but a woman who fears the LORD, she shall be praised\" (Proverbs 31:30).\n\nProverbs 18:22 says, \"He who finds a wife finds a good thing, and obtains favor from the LORD.\" George M\u00fcller certainly found a good thing in his beloved Mary. Fortunately, when faced with the opportunity to marry a beautiful woman with little concern for God's plans, M\u00fcller decided to end the relationship. How differently George M\u00fcller's legacy might have been had he married Ermegarde. Maybe he would have been a successful doctor or lawyer, but he never would have known the life God had intended for him.\n\nMary M\u00fcller's outer appearance did not keep her husband from adoring her. Her glorious inner beauty shone on her face\u2014and in her life. This type of beauty shines more brightly over the passing years, and it is one that graciously spills over onto those around her.\n\nBeauty for God's Purpose\n\nIn the Bible, we read of women whom God worked through on account of their beauty. Queen Esther is a perfect example. The Bible says Esther was fair and beautiful (Esther 2:7). God had a purpose in her beauty, and it wasn't because He favored her. No, the Lord used Esther's looks to attract the attention of the king of Persia, who chose Esther to become his queen. This put her in the perfect position to be God's tool to help spare the people of Israel from annihilation.\n\nWhether you are pretty in the world's estimation or you would consider yourself average, don't believe the myth My husband would love me more if I were prettier. The truth is, a woman holds her husband's attention captive because of her inner loveliness.\n\nSo Is It Wrong to Want Your Husband to Think You Are Beautiful?\n\nAlmost every married woman wants her husband to tell her she's pretty. The trouble begins when you believe you will feel more valued if your husband makes you feel beautiful.\n\nBut when you start to think external beauty is what gives you value, you fail to understand your worth to your Creator. And when this happens, you become vulnerable to looking for affirmation on a human level.\n\nAnytime Christians seek from another person what only God can give, they make the person an idol. In a marriage relationship, if you look to your husband to satisfy your desire to be treasured, he will ultimately let you down. Because God never intended for your spouse to fill the void only He can fill.\n\nGod created you to find your worth and purpose in your relationship with Him alone. Listen to how much God treasures you: \"In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation for our sins\" (1 John 4:10). And, \"See how great a love the Father has bestowed on us, that we would be called children of God; and such we are\" (1 John 3:1 NASB).\n\nA wise woman will learn to find her worth in her standing with Christ so she can delight in her husband's compliments from a pure heart. When the Lord is the delight of your heart, the kind words you receive from your husband will only add to your joy.\n\nSo is it wrong to want your husband to think you are beautiful? I don't think so. A biblical example of a husband captivated by his wife's beauty can be found in the Song of Solomon, where King Solomon refers to his beloved as \"fairest among women.\"\n\nWhile it is nice to receive such affirmation from your man, just make sure you are finding your worth in your vertical relationship with God, and not the horizontal relationship with your husband. And if it turns out you are married to a man who is not complimentary like King Solomon, by God's grace you can know your worth in Christ, and you can learn to cover your husband's shortcomings with God's love. First Peter 4:8 (NASB) says, \"Above all, keep fervent in your love for one another, because love covers a multitude of sins.\"\n\nWhat if He Never Tells Me I'm Pretty?\n\nWhat should you do if your husband constantly forgets to compliment you? I have heard women tell their husbands things like, \"Karen's husband is so sweet. He always tells her she's pretty.\" In saying this, wives expect their husbands to hear, \"I wish you would compliment me like Karen's husband compliments her.\"\n\nInstead, what their husbands hear is this: \"Wow, Karen's husband is a great guy. Why can't you be more like him?\"\n\nWhen your husband hears you comparing him to another man, he is not likely to try to become more like the man you have held up as an example. Rather, he is more likely to feel disrespected by you and shut down.\n\nIf you really want to help your husband understand how he can minister to you in this area, tell him. Prayerfully consider your words and your motivation before you have a heart-to-heart with him. Ask God for wisdom. James 1:5 says anyone who lacks wisdom can ask of God, who gives it liberally and generously. Have you ever asked God to give you His wisdom before you attempt to have sensitive or difficult conversations with your husband? Try it. Asking for God's wisdom will help you speak in a God-centered way rather than a self-centered one.\n\nAfter you talk to your husband, pray for God to help him understand your need. Let God do a work in his heart. Pray often, and don't expect your husband to change overnight. Be willing to remind him or even playfully nudge him to compliment you, if necessary.\n\nFirst Corinthians chapter 13 is called the \"Love Chapter\" of the Bible. The passage is filled with a wonderful list of the characteristics of genuine love. In verse 7 we read that love \"believes all things.\" That is, the most genuine kind of love believes the best about the other person.\n\nSo I ask you: Is your husband a good-willed man? Is it possible he is not intentionally overlooking opportunities to affirm your beauty? Are there things you forget to do for him? When that happens, do you hope he will believe the best about you? Can you apply this same kind of mercy to your husband if he neglects or forgets to compliment you?\n\nDon't Set Yourself Up for Disappointment\n\nIn her book What I Wish My Mother Had Told Me About Men, my friend Julie Gorman has this to say:\n\nDoesn't it feel good when a man affirms you? Of course! We all crave these things\u2014and that's what makes us vulnerable. As we search for significance and validation, God's word strongly commands, \"Do not put your trust in princes, in mortal men who cannot save\" (Psalm 146:3).\n\nEmbracing the fallacy that a man will validate our worth positions us for heartache and disappointment.\n\nWhy Do We Need to Hear We Are Beautiful?\n\nHave you ever considered what life must have been like for Adam and Eve before they fell into sin? Talk about a honeymoon! They would have found themselves in the most romantic resort, complete with an all-inclusive food menu. They had the whole garden to themselves, and Adam had eyes only for Eve.\n\nBefore the fall, Adam and Eve's unity with one another\u2014and their Creator\u2014would have been absolute paradise. Because they lived in joyful union with God, His character would have been reflected in all areas of their lives. Their love for one another would have been completely selfless and God-centered. They would have loved perfectly because they were a perfect reflection of the Creator's love.\n\nBut once sin entered into the world, not only was Adam and Eve's fellowship with God broken, but for the first time, their total unity with one another was violated as well. What a heartbreak this must have been!\n\nAfter they fell into sin, God questioned Adam about what he had done, and he blamed his wife. Oh wow! In that moment, Eve would have realized the honeymoon was definitely over. Can you imagine how hurt Eve must have been?\n\nFrom that day on, the heart of mankind became desperately sick and wicked. In her book Idols of the Heart, Elyse Fitzpatrick gives this insight:\n\nOur hearts, the fount from which all sin flows (Matthew 12:34), have ceased to be God-centered and have become self-centered. Rather than living to reflect God for His glory, man lives for his own glory, seeking happiness in his own reflection...Rather than desiring to enjoy the beauty and order of creation for God's glory, they deify outward appearances. They long for others to worship their beauty and creativity. They make a god of their home, clothing, car or anything that reflects their glory, beauty, or worth.\n\nElyse addresses a very real tendency we have to be self-focused rather than God-focused. When any desire becomes so important we would sin to get it, we can know that desire has become an idol.\n\nSo What Should I Do?\n\nBy now you may be asking, \"How can I balance being God-centered with wanting my husband to think I am pretty? And how can I discern if my desire is sinful?\"\n\nAsk yourself this question: If my husband does not make me feel beautiful, do I resent him? Resenting your husband is a sin. If his making you feel pretty is so important to you that you will sin to get it, then you can know you are idolizing your desire.\n\nThrough Bible study, the Holy Spirit can help you discern your motives. You will never see yourself more clearly than through the lens of Scripture. So you must be in the Word on a daily basis, asking God to help you become God-centered rather than self-centered. In this way you can discern your sin and confess it. As one Christian writer observed, \"It is hypocritical to pray for victory over our sins yet be careless in our intake of the Word of God.\"\n\nMaybe I Shouldn't Try So Hard\n\nNow, before you are tempted to think trying to be pretty for your husband is somehow \"unspiritual,\" I want to give you some insight into how a husband appreciates a wife who cares about how she looks.\n\nIn her book For Women Only, Shanti Feldhahn talks about an anonymous survey she conducted with more than one thousand men. Here's what men said about their wives making an effort to look pretty:\n\nIn a way this issue for men is like the romance issue for us. Maybe it shouldn't matter whether our husbands ever put one jot of effort into romancing us. But it does. We love him regardless, but it doesn't salve the empty wistfulness we feel or the pain we may suffer wondering why on earth our man doesn't see that this is so important to us.\n\nGuys feel the same way on the issue of our appearance\u2014or at least our effort. It is critical that we acknowledge that this male desire is both real and legitimate.\n\n[Another man said] \"I want to be proud of my wife. Every man has this innate competition with other men, and our wives are a part of that. Every man wants other men to think that he did well.\"\n\nNow hold on, before you translate the comments from those men to mean, \"We want you all to be skinny and look great in a bikini,\" listen to what a majority of men said:\n\n\"Sometimes I'll meet a man whose wife is overweight\u2014but she takes care of herself. She puts some effort into her appearance. She dresses neatly, or does her makeup and hair. If she is comfortable in her own skin and is confident, you don't notice the extra pounds. I look at that husband and think, He did well.\"\n\nThe Secret to Keeping His Attention\n\nWe all envy the couple who seem captivated by each other. You know who they are. They catch each other's gaze across the room, and give a flirtatious wink. Who is this wife who seems to hold her husband's attention in spite of children, financial difficulties, and those extra pounds she has held onto since the babies came?\n\nDon't you want to be that woman? What is her secret? How has this wife managed to keep her husband's attention, and what can you learn from her?\n\nFirst Timothy 2 encourages godly women to focus less on adorning themselves externally and instead, to live in a manner that professes godliness. God's secret to capturing your husband's affection for a lifetime is for you to be devoted to developing your inner beauty.\n\nAn Example of Inner Beauty\n\nA wonderful example of a woman who exuded beauty from within is Ruth. The Bible tells us Ruth lived in Moab. There, she was married to a Jewish man who evidently died at a relatively young age. When Ruth's widowed mother-in-law, Naomi, decided to return to her homeland of Israel, the recently widowed Ruth was determined to go with her. In doing this, Ruth professed her loyalty to the God of Israel and would not be turning back to Moab.\n\nAfter Naomi and Ruth arrived in Israel, they found themselves destitute. Since Naomi was too old to work, Ruth went out daily to glean the leftover grain in the fields. Through God's providence, Ruth ended up gathering grain in a field owned by Boaz, who graciously looked out for her safety and well-being. I especially love when Boaz told Ruth, \"Have I not commanded the young men not to touch you?\"\n\nImagine how frightened Ruth must have been to daily make herself vulnerable by gathering grain in a field where foreign men would likely have not treated her favorably. I get chills when I think about Boaz \"riding in on his horse,\" so to speak, and coming to her rescue. Now that's a knight-in-shining-armor story! So what made this woman from a foreign land so attractive to Boaz?\n\nIn my study Bible I found a list of character qualities that made Ruth not only lovely to look at, but inwardly radiant. I believe she personified the traits found in the excellent wife of Proverbs 31. Let's look at those qualities:\n\nDevoted to Her Family\n\nRuth was devoted to her family. She displayed her devotion to her mother-in-law when Naomi entreated her to return to her people. Ruth's response has become a popular quote for couples to cite in one form or another in their wedding ceremonies:\n\nEntreat me not to leave you, or to turn back from following after you; for wherever you go, I will go; and wherever you lodge, I will lodge; your people shall be my people, and your God, my God. Where you die, I will die, and there will I be buried. The LORD do so to me, and more also, if anything but death parts you and me.\n\nI am moved to tears when I read about how Ruth pledged lifelong devotion to her mother-in-law. Over the years, Steve and I have taught a six-week premarital course and counseled many engaged couples. The most important aspect of this course is helping couples understand that marriage is a covenant, not a contract.\n\nSteve explains, \"A contract is something that says, 'If you do this for me, I will do this for you.' \" He then goes on to say, \"A marriage covenant is an unconditionally binding promise between you and the Lord. You are not making a covenant with your betrothed, but with the Lord.\"\n\nDo you reflect this kind of devotion to your husband? If Ruth was this committed to her mother-in-law, I can only imagine the loyalty she would have displayed to Boaz when he became her husband. Do you think her devotion would have captured Boaz's attention? I do. And this kind of commitment will certainly capture the affections of your husband as well.\n\nOkay, time-out. Some of you may be saying, \"But you don't understand, Rhonda. I am married to a difficult man. Even if I were to be this committed to him, he wouldn't appreciate it. He's a jerk.\"\n\nScripture tells us of godly women who were married to jerks. For example, there is the story of Abigail and Nabal, which we find in 1 Samuel 25. Verse 3 says of them, \"She was a woman of good understanding and beautiful appearance; but the man was harsh and evil in his doings.\" Nabal, whose name means \"fool,\" acted terribly toward King David, and Abigail was quick to make amends for what her husband had done.\n\nAnd before the king of Persia took Esther as his queen, he had cruelly humiliated and banished his previous wife for simply not \"presenting herself\" at a party for all his drunken friends to ogle at (Esther 1:10-12,19).\n\nEsther's grace-filled life left a lasting legacy because her focus was not on the man to whom she was married. Rather, she was devoted to living in obedience to God's plan for her life. If you are married to an unreasonable man, have you considered that God may want to accomplish great feats through you as you keep your sights on Him and determine to grow in the grace and knowledge of our Lord?\n\nDelighted in Her Work\n\nThe next character quality that made Ruth beautiful is that she delighted in her work. Ruth did not shrink back when the only way she would be able to provide for herself and her mother-in-law was to tie up her skirts and gather grain in the fields. The excellent wife in Proverbs 31 can be found preparing meals, making investments, sewing clothes, and generally doing all those things that we say we want to do but never actually get around to accomplishing. Proverbs 31:27 says the virtuous woman's husband praises her because \"she watches over the ways of her household, and does not eat the bread of idleness.\"\n\nHow well do you delight in serving your husband\u2014and your family? Are you joyful when you clean the house, pick up the dry-cleaning, or make a meal? Or do you grumble when you have to clean\u2014as though it is some big surprise to you the house got dirty again? Are you constantly distracted by social media so you neglect your daily tasks? Can your husband and children tell that they are truly the priorities of your life, and that you take joy in caring for them?\n\nDependent upon God\n\nThe next shining quality Boaz recognized in Ruth was her dependence on God. Boaz observed it was \"the LORD God of Israel, under whose wings [Ruth had] come for refuge\" (Ruth 2:12). One great source of security for a godly man is to know his wife is seeking the Lord as she goes about her day. Whether you are working outside the home or managing your household, if your husband knows you are living in full dependence upon God, he can rest assured your choices will be honoring to the Lord. With that assurance, your husband's heart can safely trust you, as did the man married to the Proverbs 31 woman. And when a husband can trust in his wife, adoration is a natural response.\n\nC.H. Spurgeon's father, John, was an extremely busy man. He was often away from home, leaving the task of bringing up the family largely to his wife, Eliza.\n\nOne Sunday while on his way to church, John Spurgeon turned the carriage around and returned home out of concern for the spiritual well-being of his children. When he entered the house he heard the sound of his wife in earnest prayer. C.H. Spurgeon said, \"My father felt that he might safely go about his Master's business while his dear wife was caring so well for the spiritual interests of the children.\"\n\nWould your husband consider you a woman who depends upon God? If he were to come home in the middle of the day, like Spurgeon's father did, would he find you praying for your children, or pulling your hair out?\n\nLearning to daily read and apply truth from the Bible and pray throughout your day will help you develop a habit of depending on God. And when you do this...\n\n\u2022 Your husband can rest in your godly demeanor.\n\n\u2022 Your kids will find security in knowing their mommy regularly seeks wisdom from the Lord.\n\n\u2022 Your inner beauty will flourish, and your husband will be attracted to your gentle and quiet spirit, which is precious in God's sight (1 Peter 3:4).\n\nDedicated to Godly Speech\n\nThe next important quality that will make you the apple of your husband's eye is being dedicated to godly speech. We are told of the Proverbs 31 woman, \"She opens her mouth with wisdom, and on her tongue is the law of kindness\" (verse 26).\n\nDo you think before you talk? Or in the heat of the moment, do you just blurt out whatever comes to mind? There is nothing less attractive to a man than a woman with an unbridled tongue, a woman who spouts condescending remarks or venomous slurs.\n\nWhat are some ways you can be sure you are dedicating yourself to godly speech?\n\nDetermine to never make your husband feel inadequate or stupid in public or private. When he makes a mistake, or shares his thoughts, concerns, or ideas with you, be the one person he can count on to be supportive. When he talks about his ideas, don't make him feel inferior. If you do, he will likely stop sharing his dreams or confiding in you. You don't want that, do you?\n\nSpeak kind words. Ephesians 4:32 says, \"Be kind one to another.\" Does your husband cringe when he comes home from work because he wonders whether you are going to meet him with kindness or harshness? Decide today you will only speak kindness to him when he arrives home. If you do, he will likely come to think you are the most beautiful woman on earth. And he will certainly look forward to coming home to see you.\n\nAlways speak well of your husband. When your husband is not around, can he trust that what you say about him will be honorable? Do you affirm him in public and speak highly of his accomplishments, or does he worry you might embarrass him by belittling him or revealing one of his secrets?\n\nYour husband wants\u2014in fact, he needs\u2014you to be proud of him, and to always have his back. Proverbs 31:12 says an excellent wife \"does [her husband] good and not evil all the days of her life.\"\n\nIf you make a constant effort to be your husband's friend and confidant, he will view your worth as being \"far above rubies\" (Proverbs 31:10) because he will know he can trust you. And a man who treasures his wife will treat her like a treasure.\n\nPractical Ways to Keep His Attention\n\nAlong with developing your inner beauty, there are many ways to help your husband have eyes only for you. Here are a few ideas, and I'm sure you can think of more:\n\n1. Flirt with him. Catch his eye across a crowded room and give him a flirtatious wink. Whisper in his ear at the dinner table how you plan to enjoy his company later\u2014after the kids are put to bed.\n\n2. Present yourself so he will be proud of you. Dress to please him (Proverbs 31:22).\n\n3. Look joyfully toward the future. Rejoice in the times to come (Proverbs 31:25).\n\n4. Forgive him. And then don't keep a record of his wrongs. In the way you would hope your husband forgets about your past offenses, offer him the same grace.\n\n5. Have sex with your husband. Do you realize you are God's gift to him to satisfy his God-given sexual desires? We will discuss this further in chapter 7. But for now\u2014trust me\u2014men whose wives pursue them sexually know they are blessed. Your husband likely knows other men who complain about how disinterested in sex their wives are. If your man is one of the few married men whose wife not only enjoys sex with him, but actually pursues him, not only will you keep his attention, but he will slay dragons for you!\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nI have raised dogs all of my life. When I was a young man, German shepherds captured my attention. But after a less-than-favorable experience with one of my shepherds, I moved on to golden retrievers. Most retrievers are loyal, friendly, and loving dogs\u2014without the killer instincts of a German shepherd. And as puppies, golden retrievers are generally happy-go-lucky and clueless.\n\nMany of us men may try to come off like the German Shepherd, but truth be known, when it comes to knowing how to make our wives feel beautiful or special, many of us are more like clueless puppies.\n\nThrough the years, Rhonda has helped me understand how much she needs me to tell her she looks attractive. For me\u2014and many men I have talked to\u2014it is easy to develop an attitude that says, \"I told you you were pretty when I met you. If I change my mind, I'll let you know.\" But with my wife's sometimes not-so-gentle prompting, it didn't take me long to figure out \"That ain't how it's done!\"\n\nMy beloved has had to remind me numerous times over the years how much she values my words of affirmation. Even after more than 30 years of marriage, she still needs to know I find her attractive. Maybe you can relate?\n\nI'll let you in on a little secret: For some reason I am a little reticent to use the word pretty (a confession I have not even shared with Rhonda\u2014until now because she is editing this section for me\u2014thanks, Babe!). My hesitation to use that word probably comes from some regressed memory that would take a lot of therapy to get through. However, suffice it to say, pretty just doesn't feel like a manly word for me to use. I usually say something like, \"Baby, you look hot!\" But I digress.\n\nThe point is, don't feel hurt if you have to patiently remind your husband of your emotional need to hear him tell you you're attractive to him. We men really want to meet the needs of our wives, but sometimes we feel awkward, or simply forget how much you value our affirmation. If your hubby needs a little prodding, do it patiently and lovingly. Remind him of how much you want to be pretty for him, but you also need to hear from him when you put forth the effort. (If your husband is one who frequently tells you you're beautiful without being reminded, count your blessings and tell him you appreciate his sensitivity.)\n\nWhile we are on this subject, the next time you ask your husband's opinion about how you look in the outfit you are wearing, don't back him into a corner asking him questions like, \"Does my rear look big in these pants?\" Seriously\u2014how is a man supposed to answer that question? And, if you ask your man to choose between outfit A or outfit B, do him a favor and wear the one he chooses. If you're not willing to do so, you would be wise to not ask his opinion in the first place. (How would you feel if your husband gave you two options, and then promptly dismissed your choice?)\n\nHere is one more insight for you: When Rhonda and I are running late, the last thing I want to tell her is, \"Oh yes, Babe, the outfit you had on before looked way better on you. You go ahead and change. I'll just wait in the car watching the clock to calculate how fast I am going to have to drive to make it to the event on time.\" Get my point?\n\nAnd remember, rather than focus on developing external beauty that will not stand the test of time, devote yourself to cultivating the beauty that comes from within the heart. And, as Rhonda pointed out, the secret to capturing your husband's attention for a lifetime is in learning to find your worth in your relationship with Christ. When you spend your life developing your inner beauty and staying focused on the Lord, your husband's affection for you will grow as he observes the lovely woman of God you are becoming. The more consistently you pursue Christ, the more beautiful you will become to your husband, to others, and most importantly, to Christ.\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. Read 1 Timothy 2:9-10 and list five positive qualities that will help you develop your inner beauty. What practical steps will you take to develop these qualities?\n\n2. Read through Proverbs 31. From this passage, and in light of what you learned in this chapter, name two steps you will take to become a woman in whom your husband can trust.\n\n######\n\nLIVING IT OUT\n\n######\n\n1. What have you learned about where you should find your value? How can finding your worth in Christ help your marriage?\n\n2. Implement at least two of the \"The Secret to Keeping His Attention\" this week. Observe how your husband responds to your actions.\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nAll He Wants Is Sex\n\nWhen You Long for Romance\n\n######\n\nAs single adults, Tim and Karen met at church. Karen says, \"From the moment he walked in the door, Tim was the guy every single woman had their eye on.\"\n\nWith a twinkle in her eye, Karen told me how delighted she was the first time Tim's gaze met hers across the crowded sanctuary. And with that gaze, Tim had become the object of her affection.\n\nKaren said, \"Since Tim was a bit awkward in his pursuit, I helped him out by finding reasons to talk to or sit by him at church\u2014you know, to encourage him to pursue me.\"\n\nKaren smiled as she said, \"The day I watched Tim casually saunter across the room to take the seat next to me was the day all the single women knew, This guy's off the market.\"\n\nWhile they were dating, Karen remembers how difficult it had been for them to keep their hands off of each other. To keep their commitment to remain sexually pure until they were married, Tim and Karen determined not to spend time alone. This meant long talks at the coffee shop, walking hand-in-hand at the park, and lots of fun activities.\n\nKaren recalls, \"I seriously couldn't wait to give myself to Tim in our marriage bed. And since I was so sexually motivated before marriage, I was convinced I would enjoy sex with my husband.\"\n\nThe honeymoon did not disappoint Tim or Karen, and they thoroughly enjoyed sex for their first two years of marriage. But when Karen took on a job that required her to stand on her feet all day, and Tim's schedule brought him home late in the evening, their sex life took a backseat to everything else.\n\nKaren remembers thinking, I know we should have sex more often, but I'm just so tired. And Tim doesn't seem to mind\u2014he never says anything anyway.\n\nAs time passed, Tim and Karen found themselves becoming less and less intimate\u2014both in the bedroom and in the way they related to one another. Every night Tim came home and plopped down in front of the television, while Karen busied herself with social media.\n\nWhat Tim and Karen did not discuss was how unfulfilled and lonely they were feeling in their marriage. Whenever Tim approached Karen for sex, he felt as though she accommodated out of obligation, not because she wanted him sexually. And since Tim never seemed to pursue her romantically except when he wanted sex, Karen secretly resented Tim's advances.\n\nDoes Tim and Karen's story sound familiar? I wish I could say it's not the norm for married Christian couples, but sadly, this scenario is more common than you might think. Can you identify with Karen's attitude toward sex?\n\nWhen Steve and I were first married, sex was pretty amazing! The Christian books we had read to prepare us for the marriage bed really paid off. Learning one another's bodies was a delightful adventure we both enjoyed.\n\nAfter a couple of years of marriage, I became pregnant with our first child. I will never forget what a woman at work told me\u2014a woman whom I didn't know very well. She pulled me aside and said, \"Can I give you some advice? When you have your baby, don't make tending to the baby a priority over having sex with your husband. That's a mistake I made when I was a young mom\u2014a mistake you don't want to make.\"\n\nI remember thinking, How odd that this older woman would reveal to me such an intimate secret from her past. But I tucked her words away in the back of my mind.\n\nThe first time I heard our baby cry from her room while Steve and I were having sex, the older woman's words of advice rang in my ears.\n\nI knew Meredith was safe in her crib. So I made a decision to stay in the game, as it were, and not jump out of bed right that second. We took a moment to finish, then I quickly tended to Meredith. I later learned from my husband how much he appreciated what I had done. My actions told him, \"I value you. You are important to me too.\"\n\nWhat Does Sex Mean to Him?\n\nMost women understand men have a strong physical desire for sex. So why do wives make their husbands feel apologetic for wanting sex? I think one reason is because a women's need for sexual intimacy is emotionally driven\u2014we want to feel loved, desired, and beautiful. But when it seems as though a husband's desire for sex is a mere physical urge, it becomes easy to wrongly assume he is acting selfishly and resent it. But what gets overlooked is the fact that a husband's sense of well-being and confidence is very much wrapped up in the sexual intimacy he enjoys with his wife.\n\nFor example, as a woman, you likely find great fulfillment in your marriage relationship through conversation. So you might expect your husband to find satisfaction in this as well. But the truth is that while men can enjoy talking with their wives, most men do not find the same fulfillment in conversation as women do. Your husband's God-given need to connect with you physically means just as much to him as good communication means to you.\n\nNeither of you are wrong; you are just wired differently. By design, God made you to feel emotionally connected with your husband through conversation, and He made your husband to emotionally engage with you through sex.\n\nThe trouble comes when both husband and wife look past the other person's needs and refuse to give what the other one longs for, in hopes of coercing their spouse to meet their own need. This is always a recipe for disaster.\n\nMake no mistake\u2014refusing to satisfy to your husband's deepest need until he gives you the romance you desire will only serve to erode the loving environment you so desperately long for in your marriage.\n\nWhat do you suppose Jesus would advise wives to do when it comes to ministering to their husband's sexual needs? In Matthew 7:12, Jesus said, \"So whatever you wish that others would do to you, do also to them\" (ESV).\n\nOne key way to reflect God's perfect love to others is to treat them the way you want to be treated. In the same way you want your husband to learn how to meet your emotional need for intimacy and romance, God wants you to be willing to understand his emotional need for sex, and determine to satisfy his need\u2014whether or not he ever meets yours.\n\nThe secret to a happy marriage is to take your eyes off of yourself\u2014and your expectations\u2014and focus on following Christ's example of a humble servant when it comes to loving to your husband. Philippians 2:3-8 says:\n\nLet nothing be done through selfish ambition or conceit, but in lowliness of mind let each esteem others better than himself. Let each of you look out not only for his own interests, but also for the interests of others. Let this mind be in you which was also in Christ Jesus, who, being in the form of God, did not consider it robbery to be equal with God, but made Himself of no reputation, taking the form of a bondservant, and coming in the likeness of men. And being found in appearance as a man, He humbled Himself and became obedient to the point of death, even the death of the cross.\n\nJust as Jesus humbled Himself to serve God by serving others, when you humbly minister to your husband, even in the marriage bed, you are actually serving the Lord.\n\nSo why are wives so resistant to minister to their husband's need for sex? The most common reason is selfishness, plain and simple. Because of our sin nature, the basic problem all people have is a preoccupation with self. In short, every sin results from this preoccupation. (Yes, I just implied that not having sex with your husband is a sin that stems from selfishness.)\n\nWhen you are selfishly devoted to yourself rather than to God and others\u2014in this case, your husband\u2014you will resist giving of yourself to him selflessly. And without Christ's help, you can never reach a standard of selfless love on your own.\n\nNow, in defense of yourself, you might begin to rattle off a list of all the things I do for that man. I am sure you are a great wife who does many acts of service for her husband. But allow me to let you in on a little secret my husband shared with me years ago. As a rule, most men would forego a picture-perfect house\u2014or other things\u2014for great sex. Is it possible you may need to rethink your priorities?\n\nIf you're feeling a little convicted right now, you may be thinking, How can I become less self-focused? Even trying harder not to be self-focused can cause you to remain self-focused. Hebrews 12:2 says we are to be \"looking unto Jesus\"\u2014that is the only place we should fix our eyes.\n\nLooking back, as a young mother, I remember how my self-focus kept me from ministering to my husband's sexual needs. After a day filled with being climbed upon, nursed on, and touched by my kids, the idea of being touched in bed was something I had trouble wrapping my mind around. Maybe you can relate? But one day this thought occurred to me: At work, Steve is likely using most of the 5000 words an average man speaks in a day, and yet I still expect him to talk to me in the evening. I need to treat him the way I expect to be treated. Even though I have been touched all day, I need to joyfully make myself available to his touch.\n\nInspired by this new revelation, I wanted to become a wife who put Steve's needs above my own. And I discovered that the secret to becoming a selfless wife was found in daily Bible study and prayer. Because time in God's Word transformed me more into the woman God wanted me to be. As a result, my prayers and desires became others-focused rather than all-about-me. I also found that when I forsook time with the Lord, I became less interested in meeting my husband's needs and more focused on my own.\n\nAs you are sanctified by God's truth, repent of your selfishness, and pray for your marriage, the Holy Spirit will enable you to selflessly love your husband. You will want to see your husband's need for sex with Christ's compassion, and even want to fulfill his needs\u2014even if your husband isn't making an effort to meet your needs.\n\nOne Bible teacher says, \"[Selfless love] can only come from the indwelling Holy Spirit, whose first-fruit is love (Gal. 5:22). In Jesus Christ 'the love of God has been poured out within our hearts through the Holy Spirit who was given to us' (Rom. 5:5). Only Christ's own Spirit can empower us to love each other as He loves us (John 13:34).\"\n\nHe Just Wants You to Want Him\n\nTo help you gain insight into how much men want to be wanted by their wives, one survey revealed that 74 percent of men said they would not be satisfied sexually if their wives offered all the sex they wanted, but did so reluctantly or simply to accommodate their need.\n\nShaunti Feldhahn, author of For Women Only, shares this insight:\n\nAs much as men want sex, most of them would rather go out and clip the hedges in the freezing rain than make love with a wife who appears to be responding out of duty...If she's just responding because she has to, he's being rejected by his wife...\n\nConsider the painful words of this truly deprived husband:\n\n\"We've been married for a long time. I deeply regret and resent the lack of intimacy of nearly any kind for the duration of our marriage. I feel rejected, ineligible, insignificant, lonely, isolated and abandoned as a result. Not having the interaction I anticipated prior to marriage is like a treasure lost and irretrievable. It causes deep resentment and hurt within me. This in turn fosters anger and feelings of alienation.\"\n\nWhen you mistakenly view your husband's need for sex as some sort of primal urge to be satisfied from time to time, you are missing the true ministry God has given you to affirm your husband's deepest emotional needs through sex.\n\nDid you know that when you pursue your husband sexually, you have a profound influence on him in all areas of his life? Men tend to struggle with feelings of inadequacy and loneliness. When you find your husband sexually desirable, and he feels loved for who he is, then you fill him with a sense of strength, well-being, and confidence.\n\nMen are more confident and alive when they are enjoying a healthy sex life with their wives. \"One husband said, 'What happens in the bedroom really does affect how I feel the next day at the office.' Another wrote, 'Sex is a release of day-to-day pressures...and seems to make everything else better.' \"\n\nWhen your husband says he feels better after you have sex with him, you would be wise to understand he is not simply talking about the physical pleasure he experiences through love making. He may never be able to put into words the effect making love to you has on his emotional well-being. But it really does impact him in a big way. You just watch and see if the results aren't reflected in your husband's confidence and overall satisfaction with life.\n\nBut I Want Romance!\n\nWhen Steve and I were dating, he would drive 30 minutes across town during his lunch break just to drop off a bouquet of flowers. Because he had to get back to work before his lunch hour was over, he only had time to knock on the door, hand me the flowers, give me a kiss, and then jump back in his car.\n\nAs I watched Steve drive away, I would bury my face in the flowers and say, \"Oh, how romantic!\"\n\nAfter we were married, Steve often stopped by the flower shop on his way home from work to bring me lovely bouquets. When I became a stay-at-home mom, Steve continued the romantic practice of bringing me flowers. Only this time I did not say, \"Oh, how romantic!\" Rather, I said, \"Oh, how expensive!\"\n\nI made a big mistake when I said that. When my husband's romantic gesture was met with my practical this doesn't fit our stay-at-home-mom budget, I did not realize how my words discouraged him. He was attempting to keep the romance alive in our marriage by doing the one thing I had told him was romantic since our days of courtship. In one fell swoop, I had made him feel like he had failed in his attempt to be romantic, and made him feel bad we were on a tight budget.\n\nDon't mistakenly assume husbands don't care about making romantic gestures. A 2004 survey showed that 84 percent of men say they do want to be romantic, but most say they just don't know what romance looks like to their wives. So help your husband understand what is romantic to you.\n\nSteve and I had a discussion about romance before I sat down to write this chapter. He explained, \"We men really do want to be romantic, but for most of us this means getting out of our comfort zone. When we are afraid we won't measure up to our wife's expectations, it's tempting to just not try at all.\"\n\nSteve went on, \"I know that giving gifts is romantic, but I always put so much pressure on myself to think of romantic gifts to the point I end up at a loss.\"\n\nI then pointed out to Steve how I found it very romantic when he builds something for me. For example, I recently asked him to make a wardrobe mirror for me. He was not sure he agreed with my request to build the mirror's frame out of the reclaimed barn wood we had on our property, but he complied.\n\nWithin a few days, I had the most amazing wardrobe mirror, framed with shabby-chic barn wood, leaning up against the wall in my bedroom. I loved it! So much so I posted a picture of the mirror on Instagram. (Follow me @RhondaStoppe to see more #StoppeEverAfter posts.) This one post created a great deal of chatter among my Instagram followers about how much they wanted my husband to make a mirror for them as well. To which I replied, \"He just does this for me.\" To me that's romance!\n\nSo what says romance to you? You cannot very well expect your husband to know if you're not even sure of the answer yourself. So take some time to consider some of your husband's romantic gestures in the past. And then tell him how you found those actions romantic. For example, when our children were young, I made sure Steve knew the most romantic thing he could do for me was to clean up the dinner dishes and get the kids bathed and into bed\u2014while I took a hot bath. This romantic act was most certainly met with a grateful wife and some sweet lovemaking that evening.\n\nWhen your husband makes an effort to be romantic, help him know when he is on the right track. Remember, he is likely putting himself into a situation where he feels inadequate. This means your husband might think he is risking humiliation if he gets it wrong. He may even believe he will lose your respect if his attempt at romance fails. So if he ruins the dinner he was making, or\u2014as in my case\u2014he pays too much for flowers you can't afford, don't humiliate him.\n\nOne man said, \"If I make the effort to be romantic and she laughs at me, you can be sure I won't put myself in that vulnerable position again for a very long time.\"\n\nHow Does Your Husband Define Romance?\n\nHopefully this chapter is helping you understand that when your husband desires sex, he is not simply looking for a physical release. He likely longs for romance as well. So if most men truly are closet romantics, let's look at what speaks romance to your man.\n\nRemember when you were dating? How did you spend your time together as a couple? Did your husband sit across from you reading poetry or singing songs he wrote just for you as he gazed into your eyes? Probably not. (Or maybe he did, if you're married to an artsy kind of guy.) At any rate, I imagine your courtship hours were spent talking and playing together as a couple.\n\nIf you are like most women, the talking and listening you experienced from your husband-to-be filled your romance tank. By contrast, the times of playing together would likely have ranked number one on your husband's romance chart.\n\nWhich brings me to the first activity most men find romantic:\n\nPlay with your husband. What does he like to do? What activities did you enjoy doing together when you were dating? Did you hike, play golf, or go fishing? If joining your husband in such activities filled up your husband's romance tank then, most likely it will do the same today. (As long as you go along to enjoy his company, and not invade his solace with nagging or complaints about everyday-life issues he may be trying to escape through his playtime.)\n\nThe Day I Became a Biker Chick\n\nThe year our oldest daughter, Meredith, turned 18, my husband came home with a motorcycle. He wanted me to ride on the back with him while leaving our two younger kids at home with Meredith. Up to this point, I had been hesitant to ride with Steve. I thought, Who will take care of our kids if something happens to us?\n\nBut now that Meredith was 18, Steve thought it was a great time to buy a motorcycle. His logic was, \"If something happens to us, Meredith can take care of the kids.\"\n\nSeriously, Steve thought this made perfect sense. So I had a choice to make: I could succumb to my fear, or I could jump on the back of that bike, wrap my arms around my man, and ride off into the sunset.\n\nI chose the latter.\n\nAnd over the past decade, what adventures we have had on our motorcycle! Recently we rode the bike from Northern California to Seattle and back\u20141900 miles round-trip! This experience was pure romance for my husband. And I completely enjoyed the scenery as we rode the coastal highway. As for my romantic tank? Once we made it to Seattle, we caught a cruise ship to Alaska and had a delightful time of romance!\n\nJust like you enjoy time with your girlfriends, there are times your husband would prefer to do activities with his guy friends. But you may be surprised to learn that your husband might not always be looking for guys to do guy stuff with him. Rather. he might be hoping you will do guy stuff with him. And when you do, he is romanced.\n\nHere are two more activities that may fill your husband's need for romance:\n\nLet him pursue you. When you were dating your husband, part of the romance for him was in pursuing you. So devise ways to flirt with and entice him into pursuing you from time to time. When it comes to filling your husband's need for romance, you'll be surprised how far a little flirtation goes.\n\nHave sex with him. When your husband puts forth the effort to pursue you, he is really hoping the evening will end with him enjoying you sexually. And as we discussed earlier, his desire is for you to be looking forward to your time together as well.\n\nWhen you joyfully take your husband to bed, you not only satisfy his physical, God-given need for sex, you become the salve for his soul as well. When you make the effort to deeply engage with your husband through sex, you are saying to him, \"I love you. I want you. I am here for you. I believe in you.\" Is it any wonder why most men put sex as number one in their romance category?\n\nSecrets of Great Sex\n\nWhat if you don't enjoy sex? You're not the only woman who has wondered about this. While this book is not about how to have great sex, I have written an ebook titled A Christian Woman's Guide to Great Sex in Marriage. (Visit NoRegretsWoman.com to download your copy.)\n\nI will let you in on a little secret: The longer you take to enjoy foreplay in the marriage bed, the more your body will prepare you for an incredible sexual experience. While amidst the busyness of life quickies are an important way to connect with your husband sexually, do make time for some marathon sex once in a while.\n\nAs women, when we go without sexual satisfaction for a period of time, we tend to forget how much we like it, while the opposite is true of men. So taking the time to create great sexual encounters will make you want more of those experiences in the future. (And if you have trouble reaching orgasm through intercourse, know that around 75 percent of women have the same challenge. Read my ebook for insights about this issue.)\n\nIf you don't enjoy sex, perhaps it's because you had a bad sexual experiences in the past. In my case, I was molested when I was six years old. So early in our marriage, I had to learn how not to flashback to that experience while Steve and I were making love. By forgiving the man who violated me, talking to my husband about my struggle, and prayer together as a couple and in my own personal prayers\u2014often in the moment I was having the flash back\u2014I was able to conquer the destructive emotions and enjoy sex with my husband.\n\nIf you have been violated, or perhaps you feel shame over sexual encounters you had before marriage, you may want to seek out professional help from a biblical counselor.\n\nSatan loves secrets because they allow him to keep his grip on you. For me, my own abuse as a child played itself out in my early teens through inappropriate physical involvement with boys. It took me more than a decade to even say out loud what I had experienced as a child\u2014or realize how that experience had affected me. But when I made the effort to get godly counsel, my openness took away the shame and fear I had harbored for so many years. God delights in your marriage bed, and through godly counsel, you can be free of anything that would steal, kill, or destroy the good God planned for sex between you and your man.\n\nSo What's the Bottom Line?\n\nGreat sex doesn't happen by accident. Life is busy, so having sex with your husband can easily become a less-than-pressing issue for you. Yet it is critical to the health of your marriage that you schedule times in your week for sex\u2014and romance as well.\n\nWhen you romance your husband, cultivate passionate sexual experiences, and help your husband know how to fill up your romance tank, you will not only transform your sex life, but I believe your marriage will be fundamentally changed as well. You hold the key to building a romantic marriage. By applying the principles in this chapter, you can enjoy a passionate marriage that is deeply satisfying for both of you.\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nRhonda pretty much hit this spot-on regarding the frustration we men have in the romance department. Truth be known, I wasn't really that good at romance before we were married. When we were courting, Rhonda was just keen to look for, and encourage me, when I was romantic. Did you do this for your husband too? What if, as when you were dating your husband, you were to continue to watch for\u2014and celebrate\u2014even his simplest attempts to be romantic? If you do, you may find your husband trying harder to be romantic with you.\n\nSo what about sex? At the risk of being categorized as the guy who just wants sex from his wife \"because men are animals,\" I'll roll up my sleeves and try my best to tackle the topic of what sex means to a man. I feel like I am stepping out on a tightrope, but here goes...\n\nWhen a husband knows his wife desires him sexually, it gives him a special sense of empowerment. This feeling of empowerment, however, is not about \"conquering his woman,\" nor is it arrogance. Rather, it is a quiet confidence that comes over a man when he believes his wife actually wants to make love to him.\n\nI realize that for Christians, our confidence comes from Christ alone. But God also made men to need affirmation in the marriage bed. Have you considered how the Spirit may use you to bolster your husband's self-assurance? When you minister to your husband's deep need for sexual intimacy, you pour courage into his heart.\n\nIn this chapter, Rhonda addressed the sin of selfishness and how it enters this whole equation of intimacy. Let's look at what Philippians 2:3-4 (ESV) says: \"Count others more significant than yourselves. Let each of you look not only to his own interests, but also to the interests of others.\"\n\nIf God wants you to put the needs of others before your own\u2014with regard to every person you encounter\u2014how much more important is this attitude in the most intimate relationship you can have this side of glory?\n\nWhen my wife puts my needs above her own through intimacy, she helps me keep my focus on my Savior. She inspires me to continue in my labor of serving Him and brings a sense of fulfillment in the life God has given me.\n\nRhonda's selflessness empowers me to be effective for God's kingdom. And you may be delighted to find your husband energized to accomplish whatever God is calling him to do as well. In my experience, husbands who are sexually satisfied at home are deeply in love with their wives. I wholeheartedly agree with Rhonda's statement that \"when you find your husband sexually desirable, and he feels loved for who he is, then you fill him with a sense of strength, well-being, and confidence.\" And since selflessly having sex with your husband is truly serving Christ, you will be rewarded by our loving Savior as well!\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. Name one insight you learned regarding the emotional benefits a healthy sex life has upon your husband.\n\n2. Write out how understanding the way sex ministers to your husband's loneliness, insecurity, or desire to feel loved will influence the way you will offer (or even pursue) sexual experiences with your husband.\n\n######\n\nLIVING IT OUT\n\n######\n\nRead Philippians 2:1-16, and answer the following questions:\n\n1. From verses 2-4, list the instructions Paul gives to Christ's followers.\n\n2. How can you apply the truth of verses 3-4 to your marriage? More specifically, how should you apply this instruction to your sex life?\n\n3. According to verse 5, believers are supposed to \"have this mind among yourselves, which is yours in Christ Jesus\" (ESV). What is one Christlike quality you should apply to the way you think toward your husband and his need for sexual intimacy?\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nPhoto credit: Eric McFarland Photography\n\nIn prayer before son Brandon's wedding.\n\nMore Money Equals Less Stress\n\nGrow Rich in Ways You Never Imagined\n\n######\n\nFrom as far back as I can remember, my husband has wanted to be debt-free. When Steve and I were dating, he used to dream about owning\u2014and living off of\u2014his own land. Because I was in love with Steve, I was \"in love\" with whatever dreams he had for his future.\n\nWhen we got married, we lived in the San Francisco Bay area. Back in 1981, home prices were extremely high, and we quickly learned the only way we could afford our own home was to purchase a fixer-upper. So long before there were television shows about people who bought junky houses and turned them into fabulous dream homes, Steve and I charted into some unfamiliar territory. All our friends thought we were crazy as they watched us move into some of the most run-down properties to fix them up. But we rolled up our sleeves and worked hard, and transformed them into lovely homes. And just as soon as we would finish one house, Steve would come home with that little side grin of his and say, \"Baby, I found another fixer-upper.\" Which meant we would have to sell the house we had just finished and move into yet another construction site.\n\nOne day, Steve came home with a real estate ad for a ranch in the mountains east of San Jose, California. It was an 80-acre ranch at an extremely reasonable price. Steve said, \"If we sell this place [another house we had just finished], we can afford to pay cash for this ranch and finally be debt-free!\"\n\nI agreed to go with him to look at the ranch, which was at the end of an hour-long drive up a treacherous mountain road. To add to my anxiety, the road was icy the day we went to look at the property because it was snowing. At long last, we arrived at a charming, snow-covered, one-bedroom house. My heart melted as I saw my husband's eyes light up\u2014I knew my world was about to change once again.\n\nAnd change it did. We sold our home in the Bay Area and purchased the ranch. When the house closed escrow, we drove back up the mountain to take a look at our \"dream cottage.\" But this time when we arrived it looked more like a shack! The snow had melted, and what stood before us was not the adorable little cottage I had thought we were getting.\n\nI wanted to cry, but when Steve started to laugh, I took my cue from him and we both laughed hysterically for quite some time. Once we found the courage to go inside the house, we discovered someone had left the back door open, and there was evidence animals had been living in the house. (I'll spare you the details.)\n\nBy August we were living in our little house on a ranch. Since there was only one bedroom, we gave it to our two kids\u2014where they slept on a bunk bed because the room was too small to accommodate two separate beds. Steve and I slept on a sofa bed in the living room. We got to sleep in front of a nice wood-burning stove, so we convinced ourselves that was rather romantic. So romantic, in fact, that it wasn't long before we were surprised by our third pregnancy! I slept on the sofa bed through my entire pregnancy. (I'm pretty sure you gasped audibly after you read this.)\n\nEach day, Steve left for work at 5:30 in morning and would not return home until 7:00 at night. This meant long days of me being at home alone with the kids in a remote location. You can imagine how lonely this fast-paced city girl became.\n\nAnd did I mention we did not have power? When we closed escrow, I assumed I'd call the local power company and our electricity would be turned on. How very naive I had been. We ended up living for a year and a half on a generator\u2014yes, a generator. Later, the electric company finally strung power lines to our house with the help of a helicopter.\n\nI wish I could say I had a good attitude all through this transition period. But no\u2014I shed many tears. I remember crying to Steve, \"Please let me go to work and you stay home with the kids. I am living your dream. You get to go to town every day\u2014you're living my dream!\"\n\nThe poor guy would just stroke my hair and remind me we needed his salary if we were going to add on to the house and remain debt-free. I knew he was right, but that didn't make the situation any easier.\n\nEventually we realized we would need to attend a church closer to our home so our kids could make friends with other children who lived nearby (when I say nearby, I mean 45 minutes away from our house). We found a little church in a town called Patterson. The Sunday we joined, I fought back tears because I dearly loved our church family in San Jose. Leaving meant saying good-bye to precious friends and all the wonderful women who had been my mentors.\n\nAs life continued on the ranch, this city girl had some crazy encounters. For example, our neighbor's cows would regularly come over the hill and break our water lines\u2014leaving me with no water until Steve returned home that evening. I remember one day in particular\u2014when I was eight months pregnant\u2014during which I was standing on top of bales of hay and using a two-by-four to try to urge a cow to move away from our water lines.\n\nIn a moment of clarity, I realized how irrational my behavior was\u2014especially since I had left my two-year-old son alone in the house. I threw down the board, stomped my way back to the house, and shouted through tears, \"I don't care if the cows break the water lines. How is this my life? I was homecoming queen!\"\n\nI laugh now, but those years were certainly the trial by fire that the Lord used to purify my heart. It wasn't until all of life's everyday comforts were gone that I came to realize how much worth I had placed on them. Through this awareness, God began to teach me how to live what Jesus commanded in Matthew 6:20: \"Lay up for yourselves treasures in heaven, where neither moth nor rust destroys.\"\n\nAfter our third child was born, the company Steve worked for moved to Los Angeles, which now meant he was without a job. The small severance check he received allowed him to stay at home for the next several months to work full-time on an addition to our house. This also freed him up to serve as the youth pastor for our church in Patterson. Because we had no house payments, car payments, or even electricity bills, Steve could work for the church for no pay. It was great to discover how being debt-free enabled us to minister freely wherever God called us to serve.\n\nAfter five years, through the Lord's providence, we ended up moving to Austin, Texas, where God surprised us with an opportunity to plant a church. In just one summer, the youth group that met in our home grew to 200 teenagers. While watching our white carpet get dirtied and our furniture worn out, we celebrated as we saw the majority of those teens come to salvation in Christ. And it was through this outreach in our home the Lord brought our oldest son, Tony, into our family. (You can read the story of how Tony became our son in my book Moms Raising Sons to Be Men.)\n\nDuring the six years we ministered in Austin, our church back in Patterson contacted Steve several times, asking him to pray about coming back to be their senior pastor. Finally, Steve told me, \"I think the Lord wants us to move back to California.\" Though our hearts were in Texas and with all the people we were ministering to, I knew Steve was right. We could see, in different ways, that God was calling us back to California. Going back was simply an act of obedience to the Lord. Of course this meant Steve would take a substantial pay cut, and we had no idea how we would be able to make ends meet. But we had seen God provide before, and we knew He would do so again.\n\nWhile we were in Texas, we had let friends live in our ranch house in California. So when we returned, we were able to settle back into our home again. As Steve and I evaluated our circumstances, it dawned on us that the reason it was possible for us to resume serving the church in Patterson was because more than a decade ago, the Lord had placed upon Steve's heart a desire to live debt-free. God had worked all this out over a long period of time, and since our return to California 15 years ago, He has continued to faithfully provide for all our needs. That which He hasn't provided, we have come to realize we don't need. We love our precious church family deeply, and we've been blessed to see many people come to Christ. Truly, God has done \"exceeding abundantly above all that we ask or think, according to the power that works in us\" (Ephesians 3:20).\n\nWho Lives Like That?\n\nIf you want to discover how to grow rich in ways you never imagined, make friends with couples who have set their hearts upon eternity\u2014and learn from their example. To get you started, let me introduce you right now to two couples whose eternal perspective influences the way they live.\n\nFirst comes Francis and Lisa Chan. In his book Crazy Love, Francis offers this interesting observation about wealth:\n\nIf one hundred people represented the world's population, fifty-three of those would live on less than $2 a day. Do you realize that if you make $4,000 a month, you automatically make one hundred times more than the average person on this planet?\n\nWhich is more messed up\u2014that we have so much compared to everyone else, or that we don't think we are rich?...Whether we acknowledge our wealth or not, being rich is a serious disadvantage spiritually. As William Wilberforce once said, \"Prosperity hardens the heart.\"\n\nFrancis goes on to say, \"When I returned from my first trip to Africa, I felt strongly that we were to sell our house and move into something smaller, in order to give more away...We ended up moving into a house half the size of our previous home, and we haven't regretted it.\"\n\nBy way of background information, at one time Francis was the pastor of a large church in Simi Valley, California. He surprised his congregation when he decided to resign because things were going very well with the ministry. But both Francis and Lisa were convinced the Lord was calling them to minister somewhere else\u2014and most likely with fewer resources.\n\nPregnant with their fifth child at the time, Francis and Lisa Chan sold their house, having no idea where the Lord would lead. As they traveled through Asia they prayed, asking the Lord daily if this would be where He would have them stay and serve. Chan says of this journey,\n\nIt was an unforgettable time. I can't describe how good it feels to walk through the slums of India with your family, hold hands in prayer, and ask God if He wants you to stay and serve Him there. You would think being homeless and uncertain would be stressful, but this was one of the most peaceful times of my life.\n\nYou may be wondering if Lisa held the same convictions as her husband when they sold everything to follow God's leading. Listen to how Lisa instructed their children when the Chans made the decision to return to the States: \"We can't let people talk us out of things. Sometimes God convicts, but then we let people talk us out of it. We need to stick to our convictions.\"\n\nClearly, Lisa was as passionate about surrendering their comforts and possessions to the cause of Christ as her husband. Can you imagine how differently things would have turned out if Francis wanted to answer God's call but Lisa was too fearful to let go of the security of their home, friends, and possessions? But she\u2014and her husband\u2014had taken to heart what Jesus said in Matthew 6:19-21: \"Do not lay up for yourselves treasures on earth, where moth and rust destroy and where thieves break in and steal; but lay up for yourselves treasures in heaven...for where your treasure is, there your heart will be also.\"\n\nGod alone knows what awaits the Chans in eternity because they've chosen to pursue the things of heaven rather than earth. And He knows the treasures that could await you as well\u2014if you learn to desire Him more than anything else in life.\n\nMiddle-Aged Crazy\n\nGene and Diane have been good friends of our family for 25 years. On Sundays after church, when all our kids were little, Diane would invite our family over to their home to swim in their above-ground pool. She would say, \"I can't afford to make lunch for everyone, so bring stuff to make sandwiches and we'll all share!\"\n\nNowadays Gene and Diane are financially comfortable, middle-aged grandparents. Over the years, I have watched them share with others when they had little\u2014and when they have had much. Many people would think that by this time, with an empty nest, Gene and Diane have earned the right to enjoy the fruit of their hard labors. But recently, the Lord put on their hearts a desire to adopt three teenage girls from an orphanage in Bulgaria. As I write, they are at the orphanage visiting these girls for the first time.\n\nAs Gene and Diane gave forth, with an open hand, the little they had when they were young parents, the Lord blessed them with more. And because they have learned to be God's vessel to bless others, He is now blessing them with three new daughters from Bulgaria\u2014bringing even more joy into their lives!\n\nTurn Your Eyes Upon Jesus\n\nDo you feel uneasy after reading about Francis and Lisa, Gene and Diane, or even the adventures Steve and I had when we worked toward becoming debt-free? Realize that God's plans for you and your husband won't look like someone else's. The point of what I have shared is not to make you feel guilty when you look at how other people manage their finances. Nor is it to get you to imitate what others have done. Rather, learn to seek intimacy with Christ, and focus on eternity. Because the more time you spend with Jesus, the less you will obsess over having enough. And the more often you remind yourself that this earth is not your home, the less preoccupied you will be with earthly comforts and possessions. And a major blessing of having an eternal perspective is that there will be less strain on your marriage relationship with regard to finances.\n\nIn his book You and Me Forever, Francis Chan offers this insight:\n\nFew would deny that marriages are destroyed by selfishness. At times we all overvalue our own pursuits while ignoring the desires of God and others. But we can't cure our narcissism by trying to ignore ourselves. The solution is to stare at God. When we actually stare at Him, everything else fades to its proper place.\n\nAs you read this, is the Holy Spirit convicting you of being too focused on earthly treasures? Take some time to ask God about whether any of your marriage conflicts are a result of you and your spouse being distracted by worldly pursuits and possessions. If the Holy Spirit brings to mind anything God wants you to surrender, ask the Lord to help you let go of whatever it may be.\n\nAnd don't get discouraged or confused if, after you submit your possessions to Christ, you end up picking them back up again. The more you pursue intimacy with Christ and fix your eyes on Him, the more He will give you His perspective on finances and possessions. While you might still possess certain things, you'll live with the realization your true treasure is in heaven. In all of this I'm reminded of a song my mother-in-law often sang to us:\n\nTurn your eyes upon Jesus,\n\nLook full in His wonderful face,\n\nAnd the things of earth will grow strangely dim,\n\nIn the light of His glory and grace.\n\nBut We Have Real Money Problems\n\nOver the years, my husband and I have visited with a number of Christian couples whose marital strife stemmed from financial issues. Even when husband and wife are committed Christians, money troubles are difficult to handle. The struggles usually begin when they take their eyes off God as their provider and begin blaming one another for their circumstances. The common complaints we hear from them are \"My husband works too many hours,\" \"She spends too much money on frivolous things,\" \"We are in so much debt we can't breathe,\" and \"The kids are left with no supervision because we both work long hours.\"\n\nSomewhere along the way, these couples\u2014who've been blessed by God\u2014began to become preoccupied with their material goods (or their lack of them). Gradually their focus shifted from Christ, who is their provider, onto themselves. They became consumed with seeking fulfillment from what they had and were working to acquire more.\n\nMaybe you can identify with these couples. Has your focus turned from Christ to career? Have you mistakenly come to think more money equals security\u2014and ultimately a happier marriage? If so, it is time to rethink your priorities.\n\nIf you find yourselves in financial trouble, you may want to ask your pastor if there is an elder in your church who can help you learn some practical money management skills. Or you can visit Christian financial expert Dave Ramsey's website to download free budgeting tools and other helpful resources.\n\nAll of us will face financial concerns at one time or another. Earlier in our marriage, when Steve was a carpenter, sometimes he couldn't find work for months on end because of the rainy season. I can remember standing at the kitchen sink, praying, \"Lord, this is Your house. We will use it for ministry, but You need to make the payment on Your house!\"\n\nIt was during one of those hard times a Titus 2 woman reminded me of Jesus' advice to His followers: \"Seek first the kingdom of God and His righteousness, and all these things shall be added to you\" (Matthew 6:33).\n\nAuthor Jim George says that when it comes to priorities, \"you are to seek a life of spiritual growth and following after God's priorities for your life. Then, friend, God will provide for your family. That's God's promise! Seek God...and have everything! Seek the world...and lose everything (Luke 9:25).\"\n\nBut what if it seems as though God hasn't provided for your house payment? What if all you work for is lost? Did God not come through\u2014or is it possible He has a better plan? Read what God says in Isaiah 55:8-9:\n\nFor My thoughts are not your thoughts, nor are your ways My ways, declares the LORD.\n\nFor as the heavens are higher than the earth,\n\nSo are My ways higher than your ways\n\nAnd My thoughts than your thoughts (NASB).\n\nA Couple Who Lost Everything\n\nImagine if one day everything you and your husband had worked for was taken away because you were Christians. And not only did you lose your property and your home, but you were also deported. This is exactly what happened to a couple in the Bible\u2014a couple from Italy named Priscilla and Aquila (see Acts 18:1-3).\n\nThe first time we meet this dynamic duo is in the city of Corinth, where they worked as tentmakers. On a missionary journey to Corinth, the apostle Paul met this husband-and-wife ministry team. They ended up becoming close friends. In the New Testament you can find Priscilla and Aquila...\n\n\u2022 risking their lives for Paul's safety (Romans 16:3-4),\n\n\u2022 traveling to Syria with Paul on a missionary journey (Acts 18:18),\n\n\u2022 teaching \"the way of God more accurately...showing from the Scriptures that Jesus is the Christ\" (Acts 18:24-28), and\n\n\u2022 hosting a church in their house (1 Corinthians 16:19).\n\nI have often asked myself, Who are these people? They are amazing! I can't help but wonder if the apostle Paul was thinking of Priscilla and Aquila when he penned the statement, \"Godliness with contentment is great gain\" (1 Timothy 6:6). Don't you wish you could have been friends with Priscilla and Aquila? I do.\n\nThe people in the Bible faced many of the same difficulties we experience today. And in many cases, God didn't reveal to them any special insight about why they were going through the trials they endured. For example, when Priscilla and Aquila lost their home and were exiled, nowhere in Scripture do we read that God told them, \"Hey, I know you are losing your home and you'll have to flee the country, but don't worry. I'm going introduce you to this guy named Paul, and with him you are going to have some amazing adventures as you share the gospel.\"\n\nWhat if, when the couple had lost everything, Priscilla had become distraught with fear and worry? Or what if she had blamed Aquila for not figuring out a way to keep their home? Imagine the life of ministry Priscilla would have forfeited if she had spent the rest of her years lamenting over what was lost in Italy. Consumed with fear, worry, or bitterness, Priscilla would not have been a \"vessel of honor\" ready for the Master's use (see 1 Thessalonians 4:4).\n\nPriscilla and Aquila built a no-regrets life in their joyful service to Christ by encouraging the apostle Paul, influencing their generation with the gospel, and providing a place for Christians to meet together in their home in Corinth. But more than that, they took to heart Jesus' exhortation in Matthew 6:20 to lay up for themselves treasures in heaven.\n\nSure, they lost all their earthly possessions when they were forced to leave Italy. And take it from a woman who used to hold church in her home\u2014you pretty much hold your house with an open hand when you use it for ministry. But I guarantee you that Priscilla and Aquila are rejoicing in heaven today, with no regret over what following Christ cost them in the short time they were on earth.\n\nA Modern-Day Priscilla and Aquila\n\nOver the past several years, the economy has been rough in California. And we know a number of people who have lost their homes. One couple stands out above the others. Their names are Dale and Amy.\n\nDale and Amy purchased their home 25 years ago. Over the years, Amy has been a stay-at-home mom, written a Christian column for a newspaper, served as a substitute teacher in the public schools, and ministered to the women in her church. Dale has worked in the corporate world, and has also devoted himself to ministry in the church.\n\nAbout ten years ago, Dale and Amy's son\u2014who had been a prodigal\u2014wanted answers about life from the Bible. Dale and Amy offered to meet with their son and his friends once a week to answer their questions. Amy cooked a meal for anyone who would come\u2014and come they did.\n\nOne by one, the young adults who met in Dale and Amy's home surrendered their lives to Christ. Each week, Dale would teach the group. And week after week, more students would come. Soon there were college-aged men and women seated all over Dale and Amy's house, up the stairway, and on the floor.\n\nAs the years passed by and the young adults grew in their faith, many became active in ministries in their church. A good number of them married one another and established godly homes.\n\nWhile God was blessing Dale and Amy's ministry abundantly, the worsening economic climate put a strain on their investments. Soon it became apparent they would have no other choice but to move out of the house they loved so dearly\u2014the home in which they had raised all their children.\n\nAmy was sad to walk away from her home, but through her tears she continued to cling to the God she serves. Later, the Lord provided Dale and Amy with a quaint house in the country. Interestingly, Amy had always dreamed of living in the country.\n\nWhat's more, Dale and Amy's daughter and her family ended up moving into a house right next door to them. Now Amy spends her days enjoying her four grandchildren. And yes, Dale and Amy still minister to the young adults in their church.\n\nWhile Amy would tell you she wouldn't have expected change of this magnitude in this season of her life, she is thankful that the Lord knew their need\u2014even before they asked. And God has blessed Amy and Dale for keeping their eyes on Him even when life's circumstances seemed to disappoint.\n\nWhat About You?\n\nWhen God moved Steve and me to the country so many years ago, He did not tell us that one day Steve would pastor a small church and our debt-free lifestyle would provide a way for us to be in full-time ministry. In similar ways, all the couples mentioned in this chapter gave up earthly comforts and possessions in exchange for heaven's treasures:\n\n\u2022 Francis and Lisa Chan sold their home to follow the Lord when they didn't know where He would lead. Francis now pastors a church in San Francisco, California.\n\n\u2022 Gene and Diane are spending their retirement years caring for three adopted teenage daughters from Bulgaria.\n\n\u2022 Priscilla and Aquila were exiled from Italy. They had no idea the Lord would use them to minister to the apostle Paul and host a church in their house\u2014as well as be an example to Christians who continue to read about them 2000 years after their deaths.\n\n\u2022 Dale and Amy had a tremendous ministry going when they lost their home. God did not explain why He had allowed this to happen. Instead, He gave them His peace and joy, and reminded them of the home in heaven they would one day have forever. And because of this couple's selfless humility, God continues to use them today.\n\nWhat about you? What will history say about the way you've been handling your finances and possessions? Will your kids grow up remembering all the arguments their mommy and daddy had over money? Or will they learn, from your example, how to store up treasures in heaven?\n\nLet's Review\n\nTo discuss finances is always touchy, especially when money is the main cause of conflict between you and your husband. But to ignore these issues is to leave problems unresolved in your marriage. In review, I'd like to repeat a couple of key points from this chapter that I don't want you to miss:\n\n1. The more you keep your eyes fixed on Jesus, the less you will care about the temporal comforts and possessions so many married couples fight over. If you choose to live contentedly within your means and hold your possessions with an open hand, your marriage will be blessed because your focus is off earthly things and on Christ\u2014and His kingdom purposes.\n\n2. Realize that God's highest good for your marriage is not to make you happy by giving you everything your heart desires. Rather, God wants to make you holy through His Son so He can accomplish His perfect plan through you\u2014so you will enjoy His blessings for all eternity.\n\nAs a believer, the only place you will find true satisfaction is in God\u2014in His control over your life and His provision for your needs. When you remember He is the One who sees, hears, and meets the needs of His children, you can stop looking to your husband (or other earthly means) to make you feel secure through material gain. Through times of abundance and times of need, God is doing His work to mold you more and more into the image of His Son. When you commit to seeking first His kingdom and His righteousness, you will find contentment and peace in your marriage\u2014and life.\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nIt's not always possible for us to live debt-free\u2014I get that. So don't feel you need to do as Rhonda and I did, and go sell your house and move into a shack in the middle of nowhere. That is not the point of this chapter.\n\nIn Christian circles there is a tendency for us to ascribe absolute principles for non-absolute circumstances. For example, if a person feels led by the Spirit to simplify his lifestyle to the point of great sacrifice, it would be wrong for him to impose his conviction on every other Christian he met.\n\nWith that said, you should know 1 John 2:15 instructs all believers, \"Do not love the world or the things in the world. If anyone loves the world, the love of the Father is not in him.\" The best way to not set your heart on temporal things is to ask God to help you store up treasures in heaven\u2014treasures that will last.\n\nYou would be wise to ask God often to help you discern the distinction between what you want and what you need. Over and over I am amazed to see how faithful God is to the meet the needs of His children when they learn to keep the pursuit of Christ as their life's priority, rather than storing up earthly treasures.\n\nHebrews 12:2 says we are to look unto Jesus, the author and finisher of our faith. When you discipline yourself to focus on Christ and seek first the kingdom of God and His righteousness, you will come to realize what matters most are the treasures you store up in heaven. And when you live with an eternal perspective, you will also learn to recognize God's gracious provision in this life as well.\n\nI know a lot of people who, later in life, regret working to attain wealth while their marriages and their families crumbled around them. Sadly, their regret often comes too late to change the consequences of their choices.\n\nWhen you hold all you possess with an open hand, you will discover how God's blessings often come in nontangible ways\u2014such as peace in your home, inexplicable joy in life (even in times of hardship), and having children who follow Christ.\n\nI can say from experience that there is nothing I have worked to attain in this life that is more valuable to me than observing the faithfulness of my children as they grow up to serve our Savior\u2014nothing. God is truly faithful and so worthy of our praise!\n\nKeeping your focus on Christ and His Word is key to a marriage unencumbered by materialistic pursuits. Learn to weigh the motivation of your every decision by asking yourself these questions:\n\n\u2022 Is this desire self-serving?\n\n\u2022 Is this pursuit self-centered?\n\n\u2022 Will this activity promote my agenda or God's purpose?\n\n\u2022 Does my spouse feel pressure to perform because of my wants?\n\n\u2022 Do I truly desire this item, activity, or pursuit so that Christ may be exalted, His truth may be promoted, and others might be blessed when they see the love of God through me and my spouse?\n\n\u2022 Am I willing to sin to attain this desire? (If so, you can know your motivations are wrong.)\n\nGod may never ask you to sell certain belongings, or go into full-time ministry, but you can know He does want you to surrender all you possess to Him so He can bless others and reflect His glory through you. If you are consumed with career and cash, it's time you do some serious soul-searching. If you are a wife who learns to trust the Lord for provision rather than looking to your husband, then your spouse might find the freedom to step out in faith and serve the Lord in new ways.\n\nI cannot promise how the Lord will use you and your husband when you have a proper attitude toward earthly possessions and pursuits. But 1 Corinthians 2:9 promises you cannot even begin to imagine what God has planned for those who make loving Him their life's pursuit:\n\nAs it is written: \"Eye has not seen, nor ear heard, nor have entered into the heart of man the things which God has prepared for those who love Him\" (1 Corinthians 2:9).\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. What do your pursuits reveal about your goals in life? What truths stood out to you most in this chapter, and how can you apply them?\n\n2. Read Philippians 3:7-11, and write out the insights you glean from this passage.\n\n3. Read Luke 12:48, then take some time to prayerfully consider how God wants you to respond to what you have read.\n\n4. Ask God to show you how you can apply what you have learned in this chapter to any money-related conflicts in your marriage. Write your thoughts.\n\n######\n\nLIVING IT OUT\n\n######\n\n1. Name at least one insight you learned from this chapter that you can put into practice the next time you are tempted to disagree with your husband about finances.\n\n2. In this chapter, which couple's story resonated with you the most, and why? From their example, what will you apply to your own marriage?\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nEvery Couple Fights\n\nEight Steps to Making Peace\n\n######\n\nWhen Julie and Greg were married, neither of them anticipated the baggage that Julie was bringing into their relationship from her first marriage. Julie Gorman, author of What I Wish My Mother Had Told Me About Men, says, \"In an attempt to keep from being hurt, I added more and more conditions to my growing list of needs...Our marriage continued to weaken. Though we loved each other passionately, we also fought passionately. As our fights progressed, our Christian conduct regressed. Expletive adjectives assailed our once-redeemed vocabulary. Four-letter words became a common exchange...I was desperate, needy, and extremely smothering. His growing hostility culminated into an eruption.\n\n\"That's when it happened: the frightful night of painful revelation...the night that demanded my change...the night God got hold of my attention in order to set me free.\"\n\nIt wasn't until Greg stormed out of the house shouting, \"I love you, Julie\u2014but I can't live like this any longer!\" that Julie realized the fragile state of her marriage. And only then did Julie fall to the floor and cry out to God for His help.\n\nAnd God did help. Julie says, \"I'd love to say God healed our marriage instantaneously, but He didn't...it took at least a year for the Holy Spirit to overhaul our relationship and realign our thoughts with His. But that night...that fight...changed everything...And it provided a foundation that would heal our marriage.\"\n\nLooking back, Julie says, \"I was a control freak! I had constantly tried to align Greg to my endless conditions. I wanted\u2014no, needed\u2014him to function within my controlled environment. Any deviation threatened my security.\"\n\nSo Why Do You Fight?\n\nFortunately for Greg and Julie, their story ends happily. But after more than 30 years of mentoring women, I am sad to say I have seen too many couples run to divorce court when the \"final fight\" erupts between them.\n\nWhenever you and your husband have a disagreement, does it turn into a battle? Or maybe you're not a couple who resorts to name-calling or four-letter words, but what about the silent treatment? Couples who don't learn how to resolve conflict in a Christ-honoring way will tend to either lash out with angry words, or\u2014just as harmful\u2014ignore one another. Both kinds of responses hurt the relationship.\n\nSo why would two people who promised to love one another till their dying breath resort to fighting with one another whenever a disagreement surfaces? James 4:1-2 offers this insight: \"What causes quarrels and what causes fights among you? Is it not this, that your passions are at war within you? You desire and do not have, so you murder. You covet and cannot obtain, so you fight and quarrel\" (ESV).\n\nJames is saying that ultimately, selfish desires are the cause of quarrels. And, because one or both people in the conflict are focused on what they wrongly believe they must have to be happy, they will \"fight to the death\" to get what they want. And for some, this means the death of their marriage. Do you feel so strongly about getting your way that you will fight for it?\n\nYears ago, Steve and I were counseling a married couple who were constantly fighting. On more than one occasion, the police had been called to subdue the husband after he became extremely angry.\n\nWhen the husband visited alone with Steve, he said, \"I don't know why I get so angry. She just talks so fast, and comes at me with so many accusations. Before I can respond to one, she is on to the next.\" Finally, out of the frustration of not being able to get a word in edgewise, this husband had put his fist through the wall.\n\nAt the same time, when I met privately with the wife, she went through a long list of ways her husband had let her down. She shared with me that each time she began to tell her husband what he had done to once again not measure up to her expectations, his defenses went up, her voice got more shrill, and the two would jump headlong into another battle.\n\nIt was easy for Steve and me to see how both contributed to their volatile arguments, but since each was convinced they alone were right\u2014and their spouse was wrong\u2014they would not change the way they related to one another in times of conflict. Sadly, their hurtful words and destructive actions chipped away at their marriage until it was destroyed.\n\nBut How Do We Stop the Fighting?\n\nIf you and your husband have developed a habit of fighting with each other, or torturing one another with the silent treatment, know that bad habits do not end by merely wanting to stop them. As with anything in life, success comes from hard work. And that is true about healthy conflict resolution.\n\nIt is not enough to merely read the Bible and pray for a marriage free from conflict. Along with prayer, you must study the Scriptures and yield in obedience to what you learn. Listen to how desperately the psalmist yearned to keep God's statutes:\n\nOh, that my ways were directed to keep Your statutes! Then I would not be ashamed, when I look into all Your commandments...Your word I have hidden in my heart, that I might not sin against You...I have rejoiced in the way of Your testimonies, as much as in all riches. I will meditate on Your precepts, and contemplate Your ways. I will delight myself in your statutes; I will not forget Your word (Psalm 119:5-6,11,14-16).\n\nAre you truly desperate to keep God's statutes? If there is discord in your marriage, the key to change is for you to determine that obeying God's precepts is more important than having your own way.\n\n\"But how do I apply Scripture to my life?\" you ask. In his book The Pursuit of Holiness, Jerry Bridges offers these practical steps: \"As you read or study the Scriptures and then mediate on them during the day, ask yourself these three questions:\n\n\u2022 What does this passage teach concerning God's will for a holy life?\n\n\u2022 How does my life measure up to that Scripture?...Be specific; don't generalize.\n\n\u2022 What definite steps of action do I need to take to obey?\n\nBridges goes on to say, \"Avoid general commitments to obedience and instead aim for specific obedience...We deceive ourselves when we grow in knowledge of the truth without specifically responding to it (James 1:22).\"\n\nWhile it is glorious to delight in what Jesus did on the cross to save you, rejoicing in the gospel coupled with repentance and yielding yourself in obedience to God's Word is what will transform you into a wife who can resist the temptation to fight with your husband.\n\nAs you wrestle with old habits, don't get discouraged if you don't find instant victory. Any training requires hard work, and you can expect that at first you will experience some failures. But if you persevere in this process of studying what God's Word says about your sin, and prayerfully apply it to your life, you will gradually see progress. Eventually you will succeed more often than you fail, until one day heated conflicts with your spouse become a distant memory. And, when your passion for Christ overrides your passion to win an argument, you'll enjoy peace in your marriage relationship.\n\nSo the next time you look longingly at a genuinely happy couple and say, \"I'll bet they never fight. I wish our marriage was as happy as theirs,\" realize they didn't just get a happy marriage and you got a difficult one. No, you can be sure a happily married couple makes a determined effort to be happy and to resolve their conflicts in a way that does not tear down their marriage.\n\nEight Steps to Making Peace\n\nOnce you decide to apply biblical principles to how you handle conflict in your marriage, you will bring peace into your home. Here are eight practical steps to beginning the process:\n\n1. Admit You Have a Problem\n\nStop saying, \"I'm fine\" or \"Everything is fine\" when it isn't. For example, if you and your husband have occasional fights that result in hurtful words or actions, you have a problem. Or if your disagreements result in days or even weeks of the two of you not speaking to one another, you have a problem. And if your husband says whatever he thinks you want to hear to keep the peace, you have a problem.\n\nMaybe most of your fights occur at a certain time of the month, and when your hormones settle down, you tell yourself, The fight wasn't all that bad. He knows I didn't mean what I said. It was just my hormones. He just needs to understand I can't help myself. If so, you have a problem!\n\nAfter you admit you have a problem, the next step to peace is...\n\n2. Acknowledge Your Sinful Bent\n\nWe are all fallen creatures who are susceptible to sin. And in Genesis 3:16, God told Eve the consequences that all women would experience as a result of her disobedience: \"To the woman He said, \"I will greatly multiply your pain in childbirth; in pain you will bring forth children; yet your desire will be for your husband, and he will rule over you' \" (NASB).\n\nEver since Adam and Eve sinned in the garden, marriage has been plagued by two people who are bent on getting their own way. One Bible teacher observes,\n\nBecause of sin...husbands and wives will face struggles in their own relationship. Sin has turned the harmonious system of God-ordained roles into distasteful struggles of self-will...Husbands and wives will need God's help in getting along as a result. The woman's desire will be to lord it over her husband, but the husband will rule by divine design (Ephesians 5:22-25).\n\nAt the onset of a disagreement, are you willing to ask yourself if your bent toward trying to rule over your husband is at the root of the conflict? Will you then yield your self-will to God's plan for marriage, and ask Him to help you submit to the authority He has placed over you? Even if your husband is not acting in a respectful manner, out of obedience to God's command, you are to offer respect to your husband. One way to do this, for example, is to respond softly when your husband is harsh. Proverbs 15:1 says, \"A soft answer turns away wrath, but a harsh word stirs up anger\" (ESV).\n\nIn Genesis 3:16, when God promised women would have pain in childbirth, He wasn't kidding. With my first baby, I endured 52 hours of labor\u2014without any pain meds! So we all agree having a baby hurts.\n\nAlong with the pain in childbirth, women are plagued with monthly periods, cramps, and hormone issues. Let's visit the topic of hormones, shall we? After my third child was born, I was left with postpartum depression. And later, I was plagued with terrible PMS. So when my hormone levels dropped, I became weepy and agitated. I experienced horrible feelings of being out of control, along with bouts of anxiety.\n\nOne day when I was crying\u2014ranting, really\u2014to Steve about how hard my life was, I looked into his face to see him trying desperately to understand how he could make things better for me. In that moment, the Lord opened my eyes to see how much I was hurting my husband\u2014and our marriage. I decided right then I would never again allow myself to vent my frustrations to him while my hormones were affecting me. After that, for a few days each month I withdrew from my husband and kids\u2014to keep from treating them harshly. This also proved to be hurtful to my husband because my actions made him feel like I was rejecting him or giving him the silent treatment.\n\nWhen I tried to explain to Steve how hormonal changes in my body influenced the way I interacted with him and the kids, he just looked hurt and disappointed. Finally, I figured out an analogy I could use to help Steve grasp my situation. I said, \"Imagine if every time there's a full moon, you turn into a werewolf\u2014no matter how hard you try not to. So your only hope is to ask the people you dearly love to lock you up, and keep you locked up, until the full moon has passed so you don't attack or hurt anyone. That's what PMS is like for me. I can't stop it, I know it's coming, and the best thing I can hope for is to keep away from those whom I love until I am myself again.\"\n\nI wish you could have seen Steve's face. The werewolf analogy not only gave him a glimpse of the lack of control I feel when my hormones act up, but it also gave him a sense of how he could best help me get through such times\u2014by keeping the kids occupied, and giving me as much space as possible for those few days.\n\nBecause I was willing to admit I have a problem\u2014rather than blaming Steve for the way I was acting\u2014I helped my husband see how desperately I needed his help and understanding. (And now, as a middle-aged woman, I am having to ask Steve once again to understand not only the emotions that go along with menopause, but also the crazy hot flashes! If you are going through menopause, you may find encouragement and insight from my ebook The Midlife Wife. Visit NoRegretsWoman.com to download a copy.)\n\nOnce you have admitted you have a problem, and have acknowledged you have a sinful bent to be reckoned with, the next step to keeping peace in your relationship is...\n\n3. Refuse to Be Argumentative\n\nHave you ever been with a couple who, no matter what the husband says, the wife is ready to correct him? While the couple bickers back and forth, I find myself thinking, Who cares if the story your husband is telling happened on Tuesday or Wednesday? And why do you think we want to hear you argue with your husband about who said what to whom, and when?\n\nI don't know about you, but for me to spend an evening with a couple who quarrels is an exhausting experience. I cannot imagine how worn-out a person would feel if they lived in a constant state of conflict. Listen to what Proverbs 27:15-16 says about a quarrelsome wife: \"A continual dripping on a very rainy day and a contentious woman are alike; whoever restrains her restrains the wind, and grasps oil with his right hand.\"\n\nAnd consider these instructions from the apostle Paul: \"The Lord's servant must not be quarrelsome but kind to everyone, able to teach, patiently enduring evil...eager to maintain the unity of the Spirit in the bond of peace...If possible, so far as it depends on you, live peaceably with all\" (2 Timothy 2:24; Ephesians 4:3; Romans 12:18 ESV).\n\nBecause conflict steals our joy, causes anxiety, and robs us of our peace, immediately after Paul addressed the two women who were in conflict at the church in Philippi he said,\n\nRejoice in the Lord always; again I will say, rejoice. Let your reasonableness be known to everyone. The Lord is at hand; do not be anxious about anything, but in everything by prayer and supplication with thanksgiving let your requests be made known to God. And the peace of God, which surpasses all understanding, will guard your hearts and your minds in Christ Jesus.\n\nThe next time you feel like arguing with your husband, remember 1 Corinthians 13:4-5: Love does not insist on having its own way, nor is it irritable or resentful. By refusing to be argumentative, you will show Christ's love to your husband and initiate peace in your relationship. Which leads me to the next step to establishing peace in your relationship:\n\n4. Make Peace a Priority\n\nGet rid of whatever causes discord in your marriage. Whether you and your husband fight over finances, how to spend your free time, or how to discipline your children, it's time to STOP! No argument is worth winning when the love and unity of your marriage is at stake.\n\nI have heard couples bicker and get into full-blown arguments over the most insignificant issues\u2014all because they want what they want. Because of the stress involved in preparing for and going on a vacation, that's when some couples get into their biggest arguments. Others argue each month over their lack of money to cover their bills. If you are fighting over where or how to take a trip, don't take the trip. And if you have a monthly battle over not having enough money, look at how you can downsize or sell off what you don't need. And then realize that ultimately, God is your provider, not your husband.\n\nI remember one couple who came to Steve and me for advice. They were so strapped financially that both husband and wife had to work long hours each week to pay their bills. Which meant their three teenagers were left unsupervised at home late into the evenings. After the wife shared about their financial burdens, the lack of intimacy in their marriage, and concern over their children, I gently suggested they consider selling off some of their possessions and moving into a house they could more readily afford. That would then allow them to cut back on the number of hours they had to work.\n\nIn response, the woman explained how inconvenient it would be to move, how uncomfortable they would be in a smaller house, and how she worked hard to have nice things. Therefore, she was unwilling to make any changes. I said, \"If my children were at risk and my marriage was in trouble, I would sell all I had, move into an apartment, and work to bring healing to my family.\"\n\nSadly, she did not like my response. Within a year, she and her husband were divorced. They moved into separate apartments, and the kids ended up being less supervised than before.\n\nKing Solomon, the wisest man who ever lived, said, \"Better is a dinner of herbs where love is than a fattened ox and hatred with it.\" I agree with Solomon\u2014it's better to enjoy love and live on salad every night than to have prime rib all the time and be in a hateful marriage. I'll take lettuce and love over steak and snarls any day\u2014how about you?\n\nNow, you may be thinking, I hear what you are saying, Rhonda, but I am not the one who is argumentative. My husband picks at everything I do, and is constantly looking for a fight. If this is your situation, my heart goes out to you. I can only imagine how discouraged you might feel. And while I cannot change your husband, I know who can\u2014God.\n\nI have a dear friend whose harsh husband became more peaceable when she stopped fighting with him and trusted God to change him. And God can work wonders through your obedience as well.\n\nWhen you determine to practice righteous living no matter how your husband responds, God can bring His peace into your relationship. \"Peace cannot be divorced from holiness. 'Righteousness and peace have kissed each other' is the beautiful expression of Psalm 85:10. Where there is true peace, there is righteousness, holiness, and purity. May those things characterize you as you strive to be a peacemaker.\"\n\nSo what can you do as you wait on God to change your husband? In these final steps to making peace, here are a few suggestions from God's Word:\n\n5. Pray Without Ceasing\n\nIt is the effectual, fervent prayer of the righteous that avails much, so keep your heart pure before the Lord, and never stop praying for God to help your husband grow to be more like Christ (see 1 Thessalonians 5:17; James 5:16; 1 Peter 3:12). And if your husband is not a Christian, never give up praying for his salvation.\n\n6. Forgive Your Husband as Many Times as Necessary\n\nDon't keep a list of your husband's infractions to throw in his face the next time he lets you down. In Matthew 18:22, Jesus instructed us to forgive 70 times 7. His point? Be willing to forgive\u2014always. While our natural fleshly tendency toward withholding forgiveness makes this very difficult to do, remember that through Christ's strength, all things are possible (see Philippians 4:13; Matthew 19:26).\n\n7. Seek Godly Counselors\n\nLook for an older Christian woman whom you trust, and ask her to pray with you and teach you from the Bible how to love your husband. Read Christian books about marriage. And consider seeking advice from your pastor or a biblical counselor (see Titus 2:1-5).\n\n8. Learn to Be a Peacemaker\n\nJesus said, \"Blessed are the peacemakers.\" You can trust that God will bless you when you determine to be a wife who makes peace with her husband.\n\nWhat Is a Peacemaker and How Can I Become One?\n\nAs a middle child, I have many memories of being curled up on the couch with my fingers in my ears as I watched my siblings fight with one another. Because of this experience, I learned to shut down\u2014or flee\u2014whenever I was exposed to conflict. Since I would avoid conflict at all costs, I considered myself a peacekeeper. But I later learned that being a peacekeeper and a peacemaker are not the same thing.\n\nIn his book The Peacemaker, author Ken Sande says,\n\nPeacemakers are people who breathe grace. They draw continually on the goodness and power of Jesus Christ, and then they bring His love, mercy, forgiveness, strength, and wisdom to the conflicts of daily life. God delights to breathe His grace through peacemakers and use them to dissipate anger, improve understanding, promote justice, and encourage repentance and reconciliation...I have observed how even the most difficult...issues can be resolved constructively when even one...decides to breathe grace in the midst of conflict.\n\nWhen people are faced with conflict, they will usually respond in one of two ways:\n\n\u2022 Flee from the conflict\n\n\u2022 Attack the one with whom they are in conflict\n\nWhen you find yourself in conflict, which response is your natural tendency? The kind of family you grew up in may have a lot to do with how you respond when conflict arises. For example, if you come from a family where parents and siblings exhibited the attack response, you may have learned the same behavior. Or if you came from a family who \"stuffed\" their feelings to escape conflict, you may turn and run whenever an argument begins.\n\nCan you imagine how troubling it would be for a person who chooses to flee to have their spouse continue to attack them verbally while the person tries to avoid the conflict? The one in flight may be thinking, I would have to really despise someone to treat them like this. They must really hate me to say such hurtful things. (Take it from a person who flees\u2014whenever someone comes at me aggressively with hateful words, I instinctively think, They must hate me!)\n\nAt the same time, the spouse who is in attack mode may feel like the one who is trying to escape doesn't care enough to fight it out. Do you see how either response\u2014flee or attack\u2014can serve to undermine and eventually destroy harmony in a marriage?\n\nBiblical Conflict Resolution\n\nThe only way to build a marriage free of hurtful conflict is through biblical conflict resolution. Are you ready to roll up your sleeves and do the hard work of learning the right way to resolve conflict in your marriage? Here are some steps you can take to work through disputes with your spouse in a Christlike manner:\n\nAsk God to help you bring glory to Him by how you respond to conflict. When two imperfect people live together, there is bound to be conflict. A good marriage is defined by how you and your husband respond when you have disagreements.\n\nLooking for ways to reflect Christ's love to your husband when you see the beginning of a dispute will do more to strengthen your relationship with your husband than you can imagine. Many wives get so caught up in winning an argument at any cost they fail to see the long-term damage they are doing to their marriages\u2014and their families. How do you respond when you and your husband do not see eye-to-eye?\n\nIn the middle of an argument you can have a wonderful opportunity to reflect Christ's character by working to resolve the conflict in a way that honors your husband. When you determine to live in a manner that brings God glory, you give a correct estimation of His character to those who are watching how you live\u2014beginning with your children. And when your life reflects God's character, He can use your example to create in your children an appetite to know Christ. (On the contrary, any hypocrisy on your part could make your children reject your faith.)\n\nBack when I was involved in youth ministry, some of the most amazing kids I knew grew up in homes where they watched their godly mother respond in a Christlike manner to a harsh husband. Because these kids witnessed their mom's genuine faith, displayed through her difficult circumstances, they were drawn to a personal relationship with Christ as well.\n\nSometimes the best way to glorify God is to keep your tongue from evil (see Psalm 34:13). So the next time you and your husband begin to argue, ask God to help you stop yourself from saying words that will make matters worse. Psalm 139:4 says, \"There is not a word on my tongue, but behold, O LORD, You know it altogether.\" So in the time it takes a quarrelsome remark to get from your mind to your tongue, the Holy Spirit can\u2014and will\u2014remind you not to speak it. It is your job to ask Him for help, and then yield yourself in obedience to the Spirit's prompting. The more often you respond in obedience to God, the more He will transform you into a peacemaker who glorifies Him.\n\nThe next way you can respond with Christ's character is...\n\nTake responsibility for your own contribution to the conflict. When you have a disagreement with your husband, is blaming him for the problem your natural default mode? In an attempt to be helpful to your husband, do you tend to point out his flaws? You should know that regularly blaming your husband may make him feel attacked and will likely invite a counterattack\u2014or cause him to flee your presence.\n\nLearn from what Jesus said: \"Why do you look at the speck that is in your brother's eye, but do not notice the log that is in your own eye?\" Is it possible you've become so focused on your husband's shortcomings that you have failed to acknowledge your own? When you are willing to overlook your husband's offenses and honestly admit your own faults, you just might find he will begin to offer to you the same grace.\n\nDon't bring up your husband's past failures. In the heat of an argument, are you tempted to pull out the list of your husband's past infractions? For example, if he overspends on a particular item, causing you to have less money for groceries that month, do you immediately harp on every other time he has committed a similar offense? Reacting in this way will certainly build a wall of discord between the two of you.\n\nWhen we were newly married, Steve was invited by a man he worked with to invest in a business. The man told Steve his investment would pay out tenfold if he gave him a certain number of dollars. Steve and I talked about the investment and decided to take a chance by throwing our money into the venture. Not long after Steve wrote the initial investment check we learned there would be no payout\u2014and we would never see our money again. We were disappointed, but learned a valuable lesson that day.\n\nI hadn't given another thought to the bad investment until more than a decade later when my husband was teaching a marriage seminar. He recounted the story to the audience, and said, \"Men learn from their mistakes. And I learned a good lesson that day about get-rich-quick schemes.\"\n\nThen Steve addressed the wives in the group. He said, \"Rhonda has never again mentioned the loss of that money. Even though she could have thrown it in my face a number of times when we were struggling to make ends meet, she never did\u2014never. Ladies, I cannot tell you how much you will bless your husband if you forgive and forget when he makes mistakes.\n\n\"As a result of how Rhonda responded to me in that experience, my admiration, trust, and respect for her grew tremendously. And when you do the same, I am confident your husband's trust and respect will grow for you as well.\"\n\nWhenever your husband lets you down, trusting him to do better the next time communicates your commitment to reconciliation. Are you willing to do whatever it takes to daily communicate forgiveness and reconciliation to him?\n\nFreedom from Conflict\n\nMany wives will weep for a good marriage, but they will not roll up their sleeves to do what it takes to get there. When you draw on Christ's grace, follow His example, and put His teachings into practice, you can find freedom from impulsive, self-centered decisions that contribute to conflict. As you employ the aforementioned eight steps to making peace, and determine to stop fighting with your husband, loving conflict resolution is within your grasp. And when you become a peacemaker who lives to glorify God, your Christlike character will certainly bring peace to your marriage.\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nA woman once told me, \"If I just had a different husband, everything would be okay.\" Have you ever made this same statement or one like it? While you might think your life would be better if you had a different husband, if there is conflict in your marriage, odds are that he is only half the problem. And since part of the responsibility for conflict lies with you, if you were to divorce your husband, you would just be dragging yourself\u2014and your unresolved issues\u2014into yet another relationship.\n\nEven if you could find a man who would always let you have your way, if you've never dealt with your selfishness or any of the unbiblical ways you relate to others, it would only be a matter of time before the fighting started up again.\n\nIf you're like most married couples, the conflicts in your marriage relationship come as a result of disappointment. For example, when you have a specific expectation and your husband does not measure up to it, you will become irritated, hurt, or even angry. Maybe your husband refuses to squeeze the toothpaste from the bottom of the tube (which I admit is a pet peeve of mine), even though you have repeatedly asked him to change his behavior. So how do you respond? First you start to nag him about it. Then maybe you dwell on thoughts like, Doesn't he know how hard I work to keep the bathroom clean? He doesn't even respect all I do around here. (I know this sounds silly, but believe me, a majority of arguments in a marriage begin at this level of disappointment.)\n\nMaybe you don't lash out or say anything at all when you are disappointed. Instead, you just determine to keep your mouth shut\u2014all the while mentally rehearsing in your mind what you would like to say to your husband. Keeping your mouth shut \"to keep the peace\" without changing your negative thinking is like putting a Band-Aid on cancer and hoping it will go away. Obviously, just covering up the problem won't make it leave. You have to address the problem itself.\n\nIf you want to stop fighting with your husband, take a good, long look at yourself and bring the truth of God's Word into the ways you interact with your husband. Evaluate your contribution to your marriage conflicts, and confess your sins to God. Ask your husband to forgive you, and allow God's Word to transform you. Every one of these actions will go a long way toward bringing more peace into your marriage.\n\nLook again at the eight principles Rhonda laid out in this chapter. Make those steps the habits of your life. Developing new habits takes continued effort. When you fail, confess it to the Lord, then do the right thing the next time. And don't give up!\n\nHebrews 12:14 says, \"Pursue peace with all people.\" God has called His children to dwell in peace with one another. What more important place can you begin to live out peaceful relationships than in your own home? Discipline yourself to become a woman of peace, and soon you will find that resolving conflict God's way will become your passion. When this happens, your home will be a place where peaceful relationships are enjoyed. And without harsh conflict in the home, your children will feel more secure, and will come to learn the keys to biblical conflict resolution for their own marriages.\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. Write out and memorize Romans 12:18.\n\n2. When conflict arises, do you tend to flee or attack? How does your husband respond to conflict? From the eight steps to becoming a peacemaker, name two specific steps you will employ to bring peace into your marriage.\n\n3. With regard to the onset of an argument, what insight do you learn from Proverbs 15:1?\n\n######\n\nLIVING IT OUT\n\n######\n\n1. What does James 4:1-2 say is the reason for fighting and quarrels?\n\n2. Are you so passionate about getting your way that you will fight to get it? If so, name at least one way you will let God redirect your passions to glorify Him and bring peace to your marriage.\n\n3. If you grew up in a home where people fought\u2014or if your marriage has been characterized by strife\u2014the idea of conflict-free living may be unfamiliar to you. Don't give up. Your persistence will result in progress, and God will be glorified as peace reigns in your marriage. Take a moment now to pray and commit yourself to biblical conflict resolution.\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nOur Marriage Would Be Better if Bad Things Would Stop Happening\n\nThe Joy of the Lord Is Your Strength\n\n######\n\nSeveral years ago, Steve went on a dirt-bike camping retreat with some of the men in our church. Around four o'clock in the afternoon on the second day of their retreat, my phone rang. You know the call you get that causes your heart to sink? This was that kind of call. A friend on the other end of the line said Steve had been seriously injured in a motorcycle accident. I was told that Steve lay unable to move in a canyon where emergency personnel were having a hard time rescuing him. Rescuing him? I thought.\n\nI asked, \"How bad is his injury?\"\n\nMy friend didn't have much information except that Steve was hurt badly and was waiting to be rescued by helicopter. She also said that it had started to snow, so Steve was in the elements and going into shock as he waited. With the promise of a phone call when she knew more, I hung up the phone. I happened to be at my brother's house when I received this disturbing call, so we all stopped to pray for Steve and his impending rescue.\n\nHours went by\u2014and no phone call. Because there was no cell service where the men were camping, I was unable to reach any of them. Finally, about four hours later, I called my friend Denel, who said her husband, John, had just walked in the door from the camping trip. John informed us that a rescue team had just hiked Steve to the top of the canyon, where they were now airlifting him to the hospital. In the end, Steve had lain in the dirt and cold, and in excruciating pain, for seven hours while waiting to be rescued.\n\nAround nine o'clock that night I finally received a call telling me which hospital Steve would be flown to. My brother then drove me the 90-minute ride to Stanford Hospital.\n\nI was so relieved when I finally got to see Steve at the hospital that I immediately covered his face with my kisses\u2014and my tears. He was pale, covered with dirt, and still had gravel in his mouth, which I immediately began to scoop out from inside of his cheeks. It was then I learned that Steve had shattered his right hip and would require extensive surgery. I'll spare you all the gory details of what is involved in putting a person with a shattered hip socket into traction while they await surgery, but suffice it to say, our oldest daughter, Meredith, nearly passed out.\n\nOnce Steve was admitted to the hospital and given something to dull the pain, the severity of the incident began to dawn on me. The doctors were concerned his right leg would no longer be able to function, and words like wheelchair and prosthetics were being thrown out as they showed me X-rays of Steve's injury.\n\nMy husband was fragile\u2014fragile! I didn't know what to do with that. This was the man who would cut off the tip of his finger and then keep working (true story). Steve had always been the rock in our family. And no matter what went wrong, he would always say, \"It's going to be okay\"\u2014and we would believe him. Now who was going to say, \"it's going to be okay\"\u2014and make me believe it?\n\nFor nine days, Steve was in traction awaiting surgery. I never once went home. Over the course of those days, Steve had so many visitors the hospital staff was amazed at the party that was continually going on in his room. And the kids from our church made a mural to hang in Steve's room that told the story of his accident through their drawings\u2014complete with helicopters, motorcycles, and red blood spurting out of Pastor Steve's leg. Steve loved it!\n\nAnd, of course, our own kids were by our side through the whole ordeal. We laughed, prayed, and thoroughly enjoyed one another as we were gathered around Steve's traction-extended body. Our son Brandon turned 18 the night before Steve's surgery, so we all gathered up in Steve's room and sang happy birthday to Brandon, and then prayed for the Lord to guide the surgeon's hands as he repaired Steve's hip\u2014and make it so he would be able to walk again.\n\nSteve's surgery went well, in spite of the excruciating pain. The Lord graciously provided Steve with one of the best\u2014if not the best\u2014orthopedic surgeons in the nation. After placing a metal plate where Steve's hip socket had once been, the surgeon did an incredible job of repairing the damage that had been caused by the accident.\n\nAfter Steve was out of surgery, we all prayed together and thanked the Lord for sending us such an amazing physician. Then everyone went home\u2014with the exception of me and our friends John and Denel.\n\nI had encouraged John and Denel to go home too, but they had spent their own sleepless nights in the hospital when their daughter Cassie had had open-heart surgery, and they said they were staying. I gratefully accepted their offer. (Talk about living out 2 Corinthians 1:4\u2014they comforted us with the comfort by which they themselves had been comforted.)\n\nAnd If the Accident Wasn't Enough...\n\nAbout one-and-a-half hours after our two youngest kids had gone home, I received a phone call from our daughter Kayla, who was crying hysterically. She reported how icy and snowy it was up at our house. I immediately said, \"Oh, Lord\u2014no, not another accident!\"\n\nKayla calmed down enough to say, \"No, we are fine. It's the house. It's completely flooded with water!\"\n\nApparently during all the time we had been spending at the hospital, the water pipes in our house had slowly filled up with ice and finally burst. Because our wood-burning stove had not been lit for a number of days, the house had become very cold. And because Steve had just filled our 5000-gallon water tank before he left on his camping trip, the entire tank had emptied into our house when the pipes had burst.\n\nEven as I tell this story here, I have to stop and take a deep breath because of the enormity of all that was happening at the time. As I sat in the hallway of the hospital hearing the news about our flooded house, all I could think of was how relieved I was that the phone call had not been about Kayla and Brandon being in an accident. I kept telling Kayla, \"It's okay, honey. It's just stuff. It's going to be all right.\"\n\nThe nurses at the nurses' station (who had become good friends by now) were all watching me as I hung up the phone and dissolved into tears. Once I composed myself, I was able to smile and again repeat, \"It's just stuff.\" And to the watching nurses I was able to share how I had peace from knowing my God would work all things together for good (see Romans 8:28).\n\nI was so thankful John and Denel had stayed the night because they were able to encourage us while we were in shock over yet another catastrophe. It wasn't long before they even had us laughing at the \"comedy of errors\" the whole fiasco had been.\n\nAfter the kids' call, I called my brother, who immediately drove up to our house. Then I called our church family, who by the next morning were at our house removing all of the water and saturated flooring. All the while, they lovingly ministered to our kids.\n\nOnce we got home from the hospital, Steve's recovery was long and painful\u2014and I never left his side. I was determined not to leave him to feel lonely or discouraged. \"We're in this together, baby,\" I would tell him.\n\nAll through the long, sleepless nights and the sorrow I experienced while watching my sweet husband in so much pain, the Holy Spirit kept whispering to my heart, \"The joy of the LORD is your strength\" (Nehemiah 8:10). No, I did not hear an audible voice, but as loudly as I can describe to you as one's heart can hear God speaking, over and over again He reminded me the joy of the Lord is your strength. I shared this with no one, but pondered it in my heart during the darkest moments of those difficult days. Taking God's words seriously, I determined to fight for joy in spite of how difficult it was to watch my love suffer so. And all through that time we were living on concrete floors and sorting through possessions that had been ruined by the flood.\n\nThe Joy of the Lord Is Your Strength\n\nWhile Steve was recovering from surgery, I was very protective over him. Well-meaning people who wanted to visit could not know how little sleep Steve was getting, or how much pain he was enduring. One person in particular kept calling and asking to come up and visit with Steve. Several times I said no, but then Steve said, \"Ah, let him come up. It'll be okay.\"\n\nReluctantly I said to the caller, \"You can come up, but you can't stay long.\"\n\nWhen the man arrived, I left him alone with Steve to talk. By the time they were done, Steve had led the man to the Lord! (And here I was being a protective wife when God wanted to use my husband in his weakness.)\n\nThe first Sunday that Steve and I were able to attend church, we could hardly wait to see our sweet church family, who had so blessed us with prayer, food, and continuing on in the work of the ministry while we were away.\n\nAfter the service, the man who had accepted Christ in our home gave me an envelope to \"encourage me,\" he said. Inside was a lovely card with a beautiful scripture across the front. And when I opened the card, there it was\u2014an incredible love note to me from the Lord!\n\nUpon a small tile placed inside of the card was inscribed, The joy of the LORD is your strength\u2014Nehemiah 8:10.\n\nJust remembering that makes me cry! I hadn't told anyone about the Lord getting me through each day with those words from Nehemiah 8:10\u2014no one! And our heavenly Father, in His kindness, sent this new believer to me with a loving note from His Word\u2014the very words His Holy Spirit had been whispering to my heart. Grateful tears spilled from my eyes that day, just like they are now. In that moment, Psalm 139:17 washed over my heart and mind: \"How precious also are Your thoughts to me, O God! How great is the sum of them!\"\n\nThe same God who so gently walked us through our trials is the One who will walk with you through yours as well. How do you and your husband respond when you are going through difficult events? Are you drawn closer to one another as you look to Christ for answers? Or do you turn on one another and blame each other for the calamity?\n\nI have a friend whose son was hit by a car and killed. He told me that after the accident, a wedge grew between him and his wife. Both were grieving, and both were blaming themselves and each other for the accident. Sadly, over time, they got divorced.\n\nAre You Surprised by Trials?\n\nWhy is it some marriages fall apart when they go through hard times while others seem to grow stronger? Let's unpack this question, shall we?\n\nOver the years that I have been involved in ministry, the one common response I have observed in Christians when they encounter a trial is they are often taken by surprise. This includes individuals who know that bad things happen to good people (one even wrote a book about that). But when their own little world is visited by difficulties, they are not only shocked, but they are downright bewildered. And this is often when Christians begin to question God's love for them.\n\nThe apostle Peter warned that we should fully expect troubled times and warned, \"Beloved, do not be surprised at the fiery trial when it comes upon you to test you, as though something strange were happening to you\" (1 Peter 4:12 ESV).\n\nIn that verse, the word \"happening\" in the phrase \"as though something strange were happening to you\" means \"to fall by chance.\" One Bible teacher explains it this way: \"A Christian must not think that his persecution is something that happened accidentally. God allowed it and designed it for the believer's testing, purging and cleansing.\"\n\nBecause people tend to question God's goodness when trials come, it is important for you to understand how God allows\u2014and even orchestrates\u2014trials in the lives of His children to accomplish His purposes.\n\nIn Isaiah 55:8, God says, \"My thoughts are not your thoughts, nor are your ways My ways.\" When we learn to trust that God's ways are good regardless of how difficult our circumstances are, then our lives\u2014and marriages\u2014can grow stronger to reflect more clearly His glory to those around us. And if during your darkest hours you and your husband remind one another of God's goodness and encourage one another with Scripture, your hearts will move toward one another as you grow closer to Christ.\n\nFirst Peter 1:6 says, \"In this you greatly rejoice, though now for a little while, if need be, you have been grieved.\" In other words, that which brought grief to you can become a reason for rejoicing.\n\nSo wait a minute: Am I to believe that Steve's accident was not really an \"accident\" but something God allowed\u2014dare I say God deemed necessary\u2014for our spiritual growth?\n\nI believe the answer is yes! Sure, the pain my husband endured was terrible. And yes, the road to healing was long and difficult. However, there's no question that God accomplished good through what had happened. And we will never fully know all the good that God did until we stand in heaven, where God will reveal to each of us all the good He did on our behalf.\n\nAll through our trial, God continually gave us the peace to endure, and He provided for our every need. And when the house was flooded, our kids discovered a deep love and willing helpfulness from our church family. They showed us God's love not only by cleaning up the house but also by generously purchasing new flooring for our home.\n\nWhat's more, during Steve's time of recovery, the church was robbed and Steve's laptop\u2014with all his sermon notes\u2014was stolen. And the money the church had collected for our flooring turned up missing. (By this point, all we could do was laugh when yet another \"trial\" came into our already topsy-turvy lives.)\n\nThrough it all, the joy of the Lord truly was our strength. We and our children learned where the source of joy was during that time\u2014the Lord Himself. In the years since, we have watched our kids and their spouses fight for joy whenever trials have visited their lives. When Meredith and Jake gave birth to our new granddaughter, we were heartbroken at the news she had serious birth defects. Yet the Lord continues to be their strength and ours as we pray for our sweet little Ivy.\n\nLooking back at all the ways the Lord took care of us gives us more reason than ever to trust God in every circumstance.\n\nDon't Look at the Waves\n\nMost of us have heard the story of when Peter stepped out of the boat during a storm to walk on the water toward Jesus:\n\nPeter answered him, \"Lord, if it is you, command me to come to you on the water.\" He said, \"Come.\" So Peter got out of the boat and walked on the water and came to Jesus. But when he saw the wind, he was afraid, and beginning to sink he cried out, \"Lord, save me.\" Jesus immediately reached out his hand and took hold of him, saying to him, \"O you of little faith, why did you doubt?\" And when they got into the boat, the wind ceased. And those in the boat worshiped him, saying, \"Truly you are the Son of God\" (Matthew 14:28-33 ESV).\n\nPeter took a great step of faith when he stepped out onto the raging sea\u2014a faith I'm not too sure I would have had. How about you? Summoning his courage, he put one foot in front of the other as the waves raged around him. That had to be very intimidating\u2014even terrifying! But then he took his eyes off of Jesus and focused instead on the stormy wind and waters, and he began to sink. Before the waves could swallow Peter, Jesus reached out and pulled Peter to Himself\u2014and to safety.\n\nI know many sermons have been preached on the importance of keeping our eyes on Jesus when the storms of life come, but the message bears repeating. So let's look a little closer at this familiar story and see if we can gain a better understanding of how and why God permits trials in the lives of His followers. As we do so, I'd like for us to start by looking at the bigger picture of what happened in Matthew 14.\n\nFrom the Mountaintop\n\nRight before the disciples were caught in that terrible storm, they had just experienced the glorious miracle of Jesus feeding the 5000 with the loaves and fishes. Talk about a mountaintop experience! Top that off with the fact everyone who was fed witnessed the miracle too. That had to be incredible\u2014wouldn't you agree?\n\nRight after the 5000 were fed, Jesus made the disciples leave. Matthew 14:22 says, \"Immediately Jesus made His disciples get into the boat and go before Him to the other side, while He sent the multitudes away.\"\n\nJesus knew what was next\u2014a raging storm. And I wonder if the disciples were a little disappointed that they didn't have more time to bask in the glorious experience they had just been through.\n\nHave you ever found it hard to leave a place where you saw God's hand in action? For Steve and me, youth camp was one such place. There were many times when we saw the Spirit of God convict kids and draw them to surrender to Christ. At the end of each camp, we hated having to leave because we knew how the world would try to pull those kids away from their commitments to the Lord. Can you think of a time when you saw God do an amazing work\u2014and then you immediately had to go back to everyday life experiences? That's hard, isn't it?\n\nInto the Storm\n\nJesus did not join the disciples in the boat, He stayed behind to pray (Matthew 14:23). But He knew where the disciples would be. When we face trials, it can often feel like Jesus is far away. And yet Hebrews 13:5 promises that Jesus will never leave nor forsake us. Even though He may send us into a tumultuous sea, we can know Jesus is right there with us, and He is always praying for us as we toil in our specific situation (see Romans 8:34).\n\nMaybe you've had times when it seems as though Jesus doesn't respond immediately to your pleas for help. Remember when Mary and Martha sent for Jesus to come heal their sick brother? Jesus did not come right away. Instead, He waited two whole days. And He said to the disciples, \"This sickness is not unto death, but for the glory of God, that the Son of God might be glorified through it\" (John 11:4).\n\nI love how the apostle John assures the reader, \"Now Jesus loved Mary and Martha\" before he tells us that Jesus waited two more days before going to them. Don't you sometimes feel like God doesn't love you when He doesn't rescue you in a timely manner? Satan loves to whisper in your ear, \"If God really loved you, He wouldn't have allowed this to happen.\" Don't fall into the trap of believing this lie and walking away from the only One who can turn your trials into something beautiful for your good and His glory (Romans 8:28).\n\nFighting the Storm\n\nAfter fighting the storm late into the night, the disciples saw Jesus walking toward them on the water. But at first they didn't recognize Him. In fact, they thought He was a ghost and they cried out in fear (Matthew 14:25-26). Sometimes when we are in the midst of a terrible storm, we are so focused on fighting it that we don't recognize Jesus when He comes to us. And because the disciples were not expecting Jesus to rescue them, it may have never occurred to them to even watch for Him.\n\nIn the same way, do you find yourself working so hard to pay the bills, fight an illness, or heal your marriage that you are not even watching the horizon for your Savior to come and help? Maybe your eyes are fixed on your circumstances, and you are hoping that somehow, everything will work out. Are you placing your trust in yourself instead of the Lord?\n\nResting in God Himself\n\nWhen Jesus arrived at the boat, He told the terrified disciples, \"Take heart; it is I. Do not be afraid\" (Matthew 14:27 ESV). Notice Jesus didn't say, \"Don't be afraid; I am going to fix this situation.\" No, He simply said, \"Guys\u2014it's Me!\" When Jesus speaks, hearts are calmed and hope is renewed.\n\nWhenever you are going through a struggle, the best place to find courage is from our Lord Himself. Some people cry out, \"God, give me a sign\" or \"Show me what to do,\" when Jesus simply wants you to focus on Him\u2014not on how He is going to make everything better.\n\nWhen the apostle Paul struggled with an infirmity and asked God three times to heal it, God responded, \"My grace is sufficient for you, for My strength is made perfect in weakness\" (2 Corinthians 12:9). Then Paul said, \"Therefore I take pleasure in infirmities, in reproaches, in needs, in persecutions, in distresses, for Christ's sake. For when I am weak, then I am strong\" (verse 10).\n\nOnly God knows the reasons He has allowed the trials you experience. And the sooner you let go of trying to figure out how to escape the trial or how to fix the problem, the sooner your faith will grow and God's peace will rule in your heart. This goes for your marriage as well. When you stop looking to your husband to resolve the problem and let him off the hook, and he sees you walking in faith through the raging sea, peace will wash over your marriage in a beautiful surge, and you will find that the joy of the Lord is your strength.\n\nThere's one more observation I'd like to share from Matthew 14.\n\nStepping Out in Faith\n\nWhen Peter recognized that it was Jesus who was approaching the boat, he said, \"Lord, if it is You, command me to come to You on the water.\" And Jesus said, \"Come\" (Matthew 14:28-29). I love how Peter responded here. Poor Peter often said things that got him in trouble, and even today people talk about the times when he lacked faith. But here, Peter was the only disciple who stepped out in faith when Jesus called to him. All the other guys stayed in the boat. Oh sure, they watched Peter walk on the water. Maybe they were even cheering him on. But none of them had the courage to step out and join him.\n\nDo you have that kind of faith? When you fix your eyes on Jesus and remember that He is sovereignly in control of all of life's circumstances, you will not only weather the storms of life, but you may even find the courage to say, \"Bring it on, Jesus. Call me to step out in faith. My eyes are on You.\"\n\nYou Are on a Mission\n\nEvery follower of Christ is called to reach people for God in their generation. We are to make Christ known so that people might come to redemption. As a Christian, anything you do in your life should be filtered through this missional statement:\n\nTo know Christ and make Him known.\n\nWhen you and your husband learn to live with a mission perspective, you will stop looking to each other to fix a difficult situation and turn to God instead. And you will trust that whatever trials or blessings God allows to come your way are divinely orchestrated by Him so \"that you may be blameless and innocent, children of God without blemish in the midst of a crooked and twisted generation, among whom you shine as lights in the world\" (Philippians 2:15 ESV). This allows God to use whatever means necessary to shine His glory through your obedient life so that through your testimony, He can create an appetite in other people to know Christ.\n\nTo shine brightly means to use your blessings to bless others, and to always acknowledge that it is the Lord who provides\u2014rather than taking the glory for your accomplishments. And it means walking through painful circumstances with joy so that Christ's peace will be seen by everyone who is watching.\n\nAll through our marriage, Steve and I have experienced great blessings and deep sorrow. I am confident the Lord has more in store for us as we seek to serve Him and share the gospel with the people He brings across our paths. What about your marriage?\n\nI challenge you to stop here and take a moment to ponder the highs and lows of your married life. Have there been times when you felt alone in the boat on a raging sea? How did you respond? Did you blame your husband for not making enough money, or not being sympathetic enough when you were hurting? Or maybe you have not yet gone through a truly painful experience. If so, I hate to break it to you: you will. So rather than hope that will never happen, you would do well to begin growing your trust in the character of Christ through prayer and studying the Bible.\n\nIt is through Scripture that God has chosen to reveal His ways to us. So when you read God's Word, ask Him to help you know Him better and trust Him more. Then you will be better prepared to not only survive life's storms, but actually thrive in times of trouble.\n\nDetermine to live with a missional perspective. This means daily asking God to do those good works He planned to do through you before the foundation of this world (see Ephesians 1:3-6; 2:10). When you wake up each day, ask God to give you a passion to seek His face through His Word so you can learn to trust Him more.\n\nWill you courageously ask God to use every circumstance in your marriage to show others that a relationship with Christ is the only way to true happiness? Even if your husband does not keep his focus on Christ, when you do you can joyfully build a marriage that will grow stronger through life's blessings and sorrows\u2014and God will be glorified.\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nWhat Rhonda didn't tell you about my dirt-biking accident is how the accident came about.\n\nAn insight you may not know about men is that we have an uncanny ability to squeeze all the excitement we possibly can out of a day of fun with our buddies. The day of my accident was no exception. We were coming to the end of our day of riding into a canyon and jamming up the other side of the bowl on a moist shale hill while throwing a rooster tail of dirt and rock behind us. This spectacle looked awesome to those who were watching from the top of the canyon. (I know this information is likely neither here nor there to you, but your husband would probably appreciate it!)\n\nWith evening upon us, the inevitable words came: \"One more time!\" (I'm told it was me who shouted those words, but in my defense, I really cannot remember.) Within a few seconds I was on the ground in excruciating pain. During the seven hours I lay there with a shattered hip, many things went through my mind. Of course I questioned the wisdom of my zeal to take one more run at the hill, but mostly I was just wondering how I was ever going to get out of my predicament. And after enduring several hours of intense pain, I remember telling the Lord, \"It's really okay if You want to take me home now.\" Have you ever been in such great pain that death didn't look so bad? Scary, isn't it?\n\nWhen bad things happen, how do you respond? Most people work their way through a variety of responses, such as sadness, anger, fear, blaming others, blaming themselves, or even blaming God. I am amazed at how often people blame God for their struggles. Even when they can look back and see that the cause of their difficult circumstance was their own doing, they continue to blame God for the consequences of their poor choices.\n\nI have no doubt my poor judgment was the cause of my accident, but I am equally convinced the Lord allowed the trial to bring about His good in my life and the lives of those around me. Can you think of a time your poor choices brought about painful circumstances? In those times, how did you respond? Did the trial put a wedge between you and your husband, or did it draw you closer together as you looked to Christ for comfort?\n\nWhen you find yourself amidst an unexpected trial, is your automatic reflex to blame your husband? What would happen if, instead, you began to develop a habit of looking to Jesus? With your eyes fixed on Him\u2014rather than the trial\u2014you will find His peace and hope even if your circumstance doesn't change.\n\nWhen you learn to look for what good God might accomplish through the hardship you are facing, you will discover how to glorify Him in the trial. When you suffer, God can be glorified in a number of ways:\n\n\u2022 God will show Himself powerful on your behalf by rescuing you from the pain.\n\n\u2022 God will grow your trust in Him to prepare you for an even greater trial to come.\n\n\u2022 God will draw others to know Christ because of the faithfulness they observe in you.\n\n\u2022 Fellow believers will be encouraged as they observe your joyful trust and commitment to Christ.\n\n\u2022 The faith of your children can increase greatly when they see you and your husband determine to trust Christ through difficult times.\n\nJesus warned His disciples, \"These things I have spoken to you, that in Me you may have peace. In the world you will have tribulation; but be of good cheer, I have overcome the world\" (John 16:33). I love that Jesus gave His followers\u2014and us\u2014this assurance so that we can face life's trials with His peace.\n\nAnd Jesus' words should help you to realize that nothing surprises God. Every struggle, heartbreak, and disappointment is filtered through His loving hand. Ultimately, His purpose is to make Christ known through you. Remember that as a Christian, you are on a mission to shine brightly the hope of Christ to a world who desperately needs Him. Any blessings or struggles you experience are opportunities for the Lord to redeem the lost, encourage fellow believers, and mold you and your spouse more into the image of His Son. Living with this perspective will enable you to build a marriage that not only withstands the storms of life, but also holds out hope to those the Lord brings into your path.\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. Some of my favorite scriptures to recite when I am amidst a trial are listed below. To prepare yourself to focus on Christ during your next trial, write out and then memorize each one.\n\n2 Chronicles 20:12\n\nHebrews 12:2\n\n2 Corinthians 4:7-9\n\n2. Read Psalm 119:25-32, then make a list of actions you should take when your soul is \"covered with dust.\"\n\n######\n\nLIVING IT OUT\n\n######\n\n1. With pen in hand, list some of your favorite people in the Bible. Next to their names, write out a trial they endured and how God used that trial to show the world that He is the one true God. If you need some prompting, I have provided some names and passages below:\n\nDavid: 1 Samuel 17 (pay specific attention to verses 34-37)\n\nJoseph: Genesis 39\u201341 (notice Joseph's perspective in Genesis 50:20)\n\nHeroes of the Faith: Hebrews 11\n\nAfter you are done, spend some time in prayer. Ask God to help you live like these heroes of the faith. Pray that you will remember you are on a mission to hold out the hope of salvation to a generation in desperate need of Jesus.\n\n2. Why do you think looking to Jesus during a trial will help your marriage grow stronger? What good habits can you develop in preparation for the next time you find yourself facing one of life's storms? Write them.\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nPhoto credit: JPlazaPhotography\n\nIf Momma Ain't Happy, Ain't Nobody Happy\n\nIt's Not Your Husband's Job to Make You Happy\n\n######\n\nFrom the time Jim and Elizabeth George first met, they fell in love. Elizabeth says, \"Ours was truly love at first sight as we passed and smiled at each other regularly on our way to and from classes.\"\n\n\"It was a bright autumn day at the University of Oklahoma. As I hurried toward my first class after lunch, I noticed him again. He was smiling as he came my way...Well, evidently he noticed me, too, because soon [in November] a mutual friend set up a blind date for us.\"\n\nOn Valentine's Day, Jim asked Elizabeth to be his bride. And by June, 22-year-old Jim and 20-year-old Elizabeth were married. Of their courtship, Elizabeth says, \"There was never a dull moment, and fun was the center of all we did...Wow, what a whirlwind of excitement!\"\n\nOf their honeymoon, she adds, \"Truly, it seemed like we were standing on the threshold of a lifetime of joy, love, excitement, and passion...This is how our wonderful friendship and marriage all began.\"\n\nAfter the honeymoon, however, it didn't take long for reality to set in. Elizabeth was working full-time, and Jim worked long hours with a one-hour commute each way to his job. With life's demands weighing heavily on their marriage, Jim and Elizabeth focused on the work in front of them.\n\nElizabeth remembers, \"Things went well for a while. And then...both Jim and I would tell you that after eight years, things became awfully empty and got pretty rocky...\" She says,\n\nWe fumbled, we argued, and we let each other down. Because we didn't find fulfillment in our marriage we poured our lives into causes, friends, hobbies, and intellectual pursuits. Having two children also didn't fill the emptiness we each felt. Our married life droned on for eight frustrating years until, by an act of God's grace, we became a Christian family...Giving our lives to Jesus Christ made a tremendous difference inside our hearts.\n\nBecoming a Christian did not change overnight the habits that had been established in their marriage. As time went on and they learned to apply the Bible's truths to their relationship, theirs became a happy marriage. And they have since written a number of books in which they share what they have learned about God's design for marriage.\n\nIn her book A Wife After God's Own Heart, Elizabeth offers this valuable insight:\n\nYour commitment to follow God's plan for a wife makes a tremendous difference. How? It will make a difference in the atmosphere in your home, in your communication as a couple, in your heart as love for your husband blossoms and abounds, and in the way you treat him with great respect. It will also improve the climate of your marriage, paving the way for the two of you to dwell together in harmony.\n\nIn addition, Elizabeth strongly recommends that wives make sure to keep the fun in their marriage. One morning some years ago, Jim added the word fun to Elizabeth's daily planner to playfully remind her to include some downtime into their busy schedules. She says, \"If Jim and I aren't careful, we can give ourselves to all work and no play! So just as we learned to persevere at our work and in the upkeep of our home, we have learned (and are still learning!) to remember to have some fun along the way.\"\n\nCan You Relate?\n\nCan you relate to Jim and Elizabeth's experiences? I can. Before the wedding, did you dream of how much fun it would be to be married? Setting up house and caring for your man were likely welcome tasks on your list. But when the life you imagined doesn't happen, disappointment is sure to set in.\n\nIf you find yourself in the middle of an unfulfilling or even difficult marriage, do not despair. Jim and Elizabeth turned their marriage around, and with God's help, so can you. Let's find out what is involved in making that happen.\n\nDecide to Enjoy Your Spouse\n\nSome of the best marriages are enjoyed by couples who make the time and effort to playfully delight in one another. Right now, can you think of a couple whose joy is contagious? Don't you just love to spend time with them? Don't you long to be them? What can you do to become more like the couple you desire to emulate?\n\nFirst off, think back to a time when it was downright fun to spend time with your man. How did you enjoy one another back when you were dating? You didn't fold your arms, tap your foot, and say, \"Ok, I'm here. Now make me happy.\" No\u2014more than likely, you planned to have a good time simply because you were happy to be together with your love.\n\nWith all the distractions life throws at you each day, you have to decide daily to enjoy your spouse. And when you determine each day to celebrate the time you spend with your husband, you have taken an important step toward making your marriage a delight.\n\nI remember a brief conversation my parents had when I was 17 years old. My father, who had the day off and was working in the garage, came into the house and said to my mother, \"Hey, I need to run to the store to buy a ladder. Wanna come?\" Without even looking up from washing the dishes, my mom replied, \"Why would I want to go with you to buy a ladder? That's not fun.\"\n\nI watched my dad's face fall and shoulders shrug as he walked back out to the garage. My mom was completely oblivious to how her response had dashed my father's hopes for a fun trip to the store. However, the incident made an indelible impression on me.\n\nWhen you are looking for ways to enjoy your husband, even running the simplest of errands together can provide an opportunity for some fun. When our kids were young, Steve and I would wait until the little ones had gone to bed, and then leaving them in the care of their older sister, we would sneak off to a 24-hour home improvement store. Steve was doing yet another remodel of our home, so there were frequent purchases to be made. This meant we had little money to spend on dates, and less time than usual for Steve to devote to me.\n\nSo I had a choice to make. I could have insisted Steve do all the shopping himself and complained that he never had money or time to spend with me. Or I could go with him to the home improvement store and find ways to enjoy an otherwise arduous task. Remembering how my mother missed an opportunity to have fun with my dad, I chose to go with Steve.\n\nEven now I am smiling as I remember those late-night runs. We were surprised to discover the store played some pretty amazing music late at night. Steve and I actually danced in the aisles when our favorite '70s songs played. (He's probably going to grimace when he finds out I told you this.)\n\nWhen Momma Ain't Happy\n\nDo you ever feel like happy times are always just out of your grasp? No matter how much you try to plan time to enjoy one another, something always seems to hijack your plans? I know there are seasons in marriage when there is so much work to do that it seems impossible to make time for each other.\n\nWhen our children were young, there were days when I was simply exhausted. Our youngest daughter, Kayla, was plagued with ear infections that would inevitably flare up in the middle of the night. And after staying up through the night to comfort her, I could not go back to bed because our two-year-old son would wake up and need my attention. Have you had similar experiences?\n\nI remember how difficult those days of raising little ones were. I recall how Steve would attempt to bring a jovial atmosphere into our home only to have one of the kids spill their milk at the table, causing more work for me, as I so rudely chided. Even as my harsh words were leaving my lips, I would immediately regret them. My poor husband didn't stand a chance with my hormonally imbalanced, sleep-deprived disposition.\n\nAre you the mom of little ones? Or maybe you have teenagers in your home who are skilled at working Mom and Dad into an argument so they can get their way. Raising kids is both hard and wonderful\u2014exhausting and exhilarating.\n\nIn my own experience, when I allowed my circumstances to dictate my joy, my family was destined to ride with me on the roller coaster of my emotions. I did not become a joyful wife and mom until I learned the importance of spending time in prayer and daily Bible study. In other words, I needed to look to God. I could not expect Steve to give me a life free from difficulty so I would be happy. I could not ask him to do for me what only God can do.\n\nThe only way you can experience true joy\u2014no matter what your circumstances\u2014is by pursuing intimacy with Christ on a daily basis. As you determine to seek joy in your relationship with Him, you will discover that your happiness comes not from how well your day turns out, but from Jesus Himself.\n\nAs a young mom, when I determined to be daily cleansed by \"the washing of water by the word\" (Ephesians 5:26), I came to know a deep, abiding joy within my heart\u2014regardless of how the day unfolded. When you determine to know Christ through the pages of Scripture, then God's peace, wisdom, and joy will spill out of your life and into your home.\n\nYou may be tempted to believe you are too busy to practice the spiritual disciplines required to be a happy wife. Don't fall for that kind of thinking\u2014don't allow the urgent to take priority over the essential.\n\nWhile kids and their needs often present a sort of urgency, what they really need is a mom filled with God's wisdom and joy. Take this advice from an older woman: The season during which your kids need you will be over before you know it. Your husband, by contrast, needs you for a lifetime. Devote yourself to becoming a happy wife, and you will build a marriage both you and your husband will enjoy for the rest of your lives.\n\nCelebrating Your Anniversary at a Biker Bar?\n\nSome years ago when we were still raising our kids, Steve's parents lived with us on our ranch because his mom suffered from Alzheimer's disease. At this time, we were also remodeling our one-bedroom house. As our wedding anniversary approached, Steve and I talked about going out on a date to celebrate.\n\nSince Steve's father was dealing with his wife's illness, and we didn't have any babysitters living nearby, we resigned ourselves to not being able to go out. However, the more we acquiesced to the possibility of not celebrating, the more determined I became to figure out a way to have our date. In the end, we put our younger two children to bed, and left our 11-year-old daughter with the telephone in her hand on speed-dial to her grandfather, who lived next door.\n\nSteve and I drove five minutes up the road to a little biker bar\u2014the only eatery within an hour of our home out in the country. Sitting at the bar, we ordered burgers and fries. It didn't take long before we both were laughing long and hard at our circumstances. I remember saying, \"Here I am, a city girl, excited for an anniversary date with her man at a remote biker bar.\"\n\nBecause I chose not to sulk or complain about our circumstances, we had a wonderful evening. And do you know what I found out? First, the place we ate at makes an amazing burger\u2014and the fries were out of this world. Second, we were both reminded that the reason we got married was because we truly enjoyed spending time together\u2014no matter what we were doing. Even if it meant celebrating our anniversary by playing pinball at a biker bar in the middle of nowhere.\n\nHave you forgotten how to enjoy your husband? When your idea of a perfect evening cannot be met, do you sulk or complain? Have you noticed that when you grumble it drives your husband away from you? Complaining to your husband about not spending time with you is not going to make him want to spend time with you.\n\nThe Bible says, \"It is better to live in a corner of a roof than in a house shared with a contentious woman\" (Proverbs 21:9 NASB). Get the picture? Complaining about how disappointed you are when you can't have your way will not make your husband want to spend time with you. On the contrary, as this proverb points out, your contention will drive him away from you\u2014maybe to the garage, to longer hours at work, or even to the very corner of the housetop!\n\nHusbands are energized by a joyful wife. When you learn to become a person your husband enjoys spending time with, you may discover him looking for more opportunities to refresh himself with your company.\n\nThere Is Hope\n\nIf you are reading this and thinking, That's me! I am not a happy person. No wonder my husband doesn't enjoy spending time with me, well, there is hope! The first step to becoming a joyful wife is to realize you can change.\n\nWhen Elizabeth George's kids were little and her husband was busy working four jobs, she admits, \"At first I handled our new situation in the wrong way. I whined. When that didn't work, I cried. When that didn't work, I screamed. When that didn't work, I stomped and sulked, making good use of 'the cold shoulder.' What a brat I was!\"\n\nIn the same way God took Elizabeth from throwing tantrums to teaching other women how to be a wife after God's own heart, He can transform you as well.\n\nAre you ready to make some changes? To help make the following points easy to remember, let's use the word H-O-P-E as an acrostic:\n\nHate the sin.\n\nOpen your eyes.\n\nPray for God's help.\n\nEnjoy your husband.\n\nHate the Sin\n\nTake an honest look at the kind of wife you have become. Do you think your husband would say you are fun to be with? If your husband seems to look for reasons not to be with you, could it be because he doesn't enjoy your company? If so, it is time to face up to your part in what is lacking in your relationship. No matter how much you may be able to assign fault to your husband, will you courageously ask the Lord to help you see your sin in the situation? And then ask God to help you be ruthless with your sin so you hate it and its destructive consequences as much as God does? Colossians 3:5 says, \"Put to death...what is earthly in you\" (ESV).\n\nOpen Your Eyes\n\nStop placing all blame on your husband and take responsibility for what you can do to build a happy marriage. Jesus said, \"Why do you look at the speck in your brother's eye, but do not consider the plank in your own eye?...Hypocrite! First remove the plank from your own eye, and then you will see clearly to remove the speck from your brother's eye.\"\n\nHuman nature seeks to blame others for our unhappy circumstances. So if you are blaming your husband for the lack of pleasure you find in your marriage, you should know Jesus calls you to evaluate your own sin before you criticize your husband's shortcomings.\n\nOne Bible teacher says, \"Most people feel free to judge other people harshly because they erroneously think they are somehow superior...Other people are not under us, and to think so is to have the wrong view of them...The wretched and gross sin that is always blind to its own sinfulness is self-righteousness...The very nature of self-righteousness is to justify self and condemn others.\"\n\nDo you have a self-righteous attitude toward your husband? Have you developed a habit of blaming him for your unhappiness? Won't you open your eyes to your own sin? Once the plank is removed from your eye, then you can see yourself, God, and others more clearly. And only then will you be in the right mind-set to humbly take steps toward making your marriage a more enjoyable union.\n\nIn this age, happy marriages are rare. People often ask my husband and me, \"What's your secret to a happy marriage?\" This question provides us with a wonderful opportunity to tell them about Christ. Steve and I are careful to share how God is the source of our happiness, and through His Son, they too can experience true joy.\n\nOne of the most dynamic evangelistic resources you have at your disposal is a happy marriage that reflects the joy of the Lord. And the first place you as a couple should share the gospel is to your own children. Sadly, many children raised in Christian homes reject Christ because of their parents' unhappy marriages. When you and your husband are not in unison, not only will you lack joy, but others will be negatively influenced as well\u2014especially your children. On the contrary, when you and your husband are enjoying one another, your joy will spill over to all of your other relationships\u2014including those with your children. \"Your decision to live in obedience to God will reverberate righteousness in generations to follow...how glorious it will be for you to look back at your life and see the godly influence your actions had upon your children and grandchildren.\"\n\nPractical Advice to Becoming a Joyful Wife\n\nTo be happily married\u2014is this the longing of your heart? I know this was my desire as a young bride, but because I had no healthy marriages in my own family to learn from, I was unsure of how to make this happen.\n\nAs life's responsibilities pressed in, I realized I was not the fun-loving, joy-filled wife I had hoped to be. So I cried out to the Lord for help. He sent to me several older ladies in my church\u2014ladies who were Titus 2 women. And what did these godly women teach me? Were they quick to rattle off a list of things I needed to change? No. First, these wonderfully wise women befriended me.\n\nThen, over time, as these women became my friends, they were able to discern what I needed to learn from them so they could help me learn how to be a better wife. And their insights were life-changing. Here are some of my favorite pointers from my Titus 2 friends:\n\n\u2022 Take a nap when the baby sleeps so you are not too tired to stay up with your husband after the children go to bed. A well-rested wife is a joyful wife.\n\n\u2022 Keep your house clean, but don't obsess over making it so perfect your husband is not comfortable in his own home.\n\n\u2022 Never, ever raise your voice to your husband. Speak calmly and quietly when you are in disagreement because \"a soft answer turns away wrath, but a harsh word stirs up anger\" (Proverbs 15:1).\n\n\u2022 Always guard your husband's reputation. Never talk behind his back.\n\n\u2022 Never belittle or make fun of your husband.\n\n\u2022 Make time to enjoy each other.\n\n\u2022 Fight for joy, because \"the joy of the LORD is your strength\" (Nehemiah 8:10).\n\n\u2022 Pray for your husband.\n\n\u2022 Love God's Word and study it daily, and commit to applying its truths to your life. God's Word will be a lamp to your feet and a light to your path (see Psalm 119:105).\n\nThe last point is the most important one I learned\u2014to love God's Word. As I studied the Bible each week with these women and they helped me apply God's truth to my life, I began to be transformed by the renewing of my mind (Romans 12:2). My thinking became different. As a result, my marriage became different as well\u2014for the better. Life's experiences were being filtered through God's truth, and that truth was changing who I was.\n\nHappiness Is Contagious\n\nIn my book Moms Raising Sons to Be Men, I observed the following about these Titus 2 women:\n\nWhen I spent time with these women, I observed their peaceful responses to the chaos of life. They displayed a resolve to seek after the Lord in every situation. They were not just church ladies who did good things for God; their hearts reflected His heart. They were by no means perfect, but they were genuine.\n\nSteve and I chose to fellowship with these women and their husbands, and we were captivated by how much they delighted in one another. They made having fun with their spouses, and other happy couples, a priority. And their happiness was contagious. Our marriage was blessed as we took time to camp, water-ski, play games, and laugh out loud with these delightful couples.\n\nMy friendships with these women are among the greatest treasures of my life. For more than three decades, I have had the privilege of watching their marriages flourish and glorify Christ. And now, in this season of their lives, I am learning how a Titus 2 woman finds sweet joy even when her love goes home to be with the Lord.\n\nPray for God's Help\n\nAbout her own attempt to change her unlikable ways, Elizabeth George says she learned to pray at the first hint of frustration or self-pity. She also advises wives to pray three times a day for their husbands. Doing this will help you to draw closer to your husband and get to know him and his needs better, and most important of all, love him as God does.\n\nWhile there are many books written on the topic of prayer, might I suggest you just take time to pray? One tip I can share is that when I learned to talk out loud to the Lord, my prayer life became much more personal. Here is a prayer I have prayed for my husband for more than 30 years:\n\nLord, cause me to love my husband with Your selfless love. Help me not to keep a record of wrongs and to forgive quickly. Give me Your joy in my marriage, and Your peace in my home. Help my husband to have eyes for me only, and grow his love for me ever stronger by the power of Your Spirit.\n\nEnjoy Your Husband\n\nWhen your husband tells you about his day, or recounts a story, stop what you are doing and look him in the eye. Lean toward him as he talks. Smile, nod, and laugh when it's appropriate to do so. Don't make him work to get a response or chuckle out of you. Remember how any attempt he made to be funny while you were dating was met with your laughter? How about you bless him with that kind of attention again?\n\nDetermine to be his girlfriend. One way to do this is to simply sit with him while he works on a project. Not to criticize, or to add your two cents, but to enjoy watching him work. And if your husband asks you to go with him to the hardware store, drop everything and go with him. You never know\u2014you may find yourselves dancing in the aisle to music at midnight.\n\nRealizing your husband is not the source of your happiness and learning to find joy in your relationship with Christ is the key to a happy marriage. Elizabeth George determined to be a wife after God's own heart, and it transformed her marriage. When you do the same, there is H-O-P-E for your marriage as well.\n\nFROM A HUSBAND'S PERSPECTIVE\n\n######\n\nA Word from Steve\n\nIn this chapter Rhonda talked about how Elizabeth George grew to be a wife after God's own heart by determining to grow in her walk with Christ. So I thought it would be good to take a moment to share from the perspective of Elizabeth's husband, Jim. In his book A Husband After God's Own Heart, Jim says:\n\nI would like to report that my spiritual growth, which started when I was just six years old, was a magnificent upward spiral, and that it had few, if any valleys. But no. Sad to say, my spiritual growth in those early years was an up-and-down roller coaster. And the downward drop on that roller coaster continued into my early adult life and had a serious effect on my marriage...\n\nSpiritual growth is the key to all that is important in life. That's what Jesus meant 2000 years ago when He told a listening audience not to be anxious about life and living. He said, \"Do not worry, saying, 'What shall we eat?' or 'What shall we wear?' \" (Matthew 6:31). These things are definitely needful, but they are not what's really important. They are not your first priority. What's really important is your spiritual growth.\n\nIt's true that spiritual growth takes terrific effort. But, my friend, it's also true that the rewards are great...No matter how old you are or how long\u2014or short!\u2014you've been married, the day you accelerate your growth in the Lord is the day your marriage is positively impacted, improved, and strengthened\n\nAs I look back upon Jim and Elizabeth's story, I am encouraged to learn how God transformed both of them\u2014and their marriage. It happened because they devoted themselves to prayer, the study of God's Word, and living in obedience to what they learned from Scripture.\n\nRhonda and I have discovered that as well\u2014that the key to a happy marriage is found in spiritual growth. And I know the same will be true for you and your marriage.\n\nWhen I use the phrase spiritual growth, please understand I am not encouraging you to be more religious. Jesus urged His followers to love the Lord their God with all of their being (Mark 12:30). When your love for God becomes the single most important goal of your life, then prayer, reading your Bible, walking in obedience to His Word, and loving your husband will all become natural outpourings of your love for Christ.\n\nAs you devote yourself to spiritual growth, you will find lasting joy because you will learn to think biblically about your marriage relationship and life's circumstances.\n\nWill you commit to making whatever sacrifices are necessary to grow in your walk with Christ? The result will be for your blessing and God's glory, and your marriage will shine a bright light of hope to a generation who desperately needs to know that following Christ is the only answer to all of life's problems\u2014and the source of a happy marriage.\n\n######\n\nTHINKING IT THROUGH\n\n######\n\n1. Write out what hope you find from the following verses:\n\nPhilippians 2:13\n\n1 Peter 2:9\n\n2 Peter 1:3\n\n2. Are you devoted to spiritual growth? Take a few moments now to ask yourself these questions, and read the following scriptures:\n\n\u2022 How genuine is my love for God (Mark 12:30)?\n\n\u2022 Am I daily adjusting my life to the precepts I learn in Scripture (Psalm 119:1-8)?\n\n\u2022 Do I walk in obedience to Christ because of how much I love Him (John 3:36)?\n\n\u2022 Or do I practice a religion with no real love for Jesus (Mark 7:6)?\n\n\u2022 Am I in awe of Jesus as I discover His character through Bible study (Hebrews 1)?\n\n\u2022 What steps can I take to make growing in my love for Christ a top priority (Psalm 119:9-16)?\n\n######\n\nLIVING IT OUT\n\n######\n\n1. Would your husband say you are fun to be with? What are some ways you can make your times together more enjoyable for him?\n\n2. Have you been blaming your husband for your lack of joy? Who alone should be your source of joy?\n\n3. From what you've learned in this chapter, explain how you can become a joyful wife.\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nHappily-Ever-After Is a Fairy Tale\n\nTen Keys to a More Fulfilling Marriage\n\n######\n\nDo you ever wonder what Peter's wife was thinking on the day her husband came home to announce he would be leaving the family fishing business to follow Jesus? Would this woman, who is unnamed in Scripture, have questioned Peter's decision to forsake all he had worked for to follow the One he believed was the Messiah? The Bible doesn't give us any insight into how Peter's conversion affected his marriage, but you can be sure that when I get to heaven I plan to find Peter's wife and ask her for the details of their story.\n\nWhat very little we do know about Peter's wife includes the fact that Jesus healed her mother (Peter's mother-in-law) from a serious fever. Luke 4:39 tells us that \"He stood over her and rebuked the fever, and it left her. And immediately she arose and served them.\" We aren't told if Peter's wife witnessed this healing, but if she did, I would think this would have persuaded her to get behind her husband's decision to devote his life to following Jesus.\n\nAs we'll see in a moment, history reports that Peter's wife was a courageous follower of Jesus until her final breath. This dynamic couple must have been a powerful influence for the gospel at a time when Nero was persecuting Christians. At one point, when Peter was commanded to stop talking about Jesus, he responded, \"We cannot but speak the things which we have seen and heard.\"\n\nThe commitment of Peter and his wife to boldly proclaim what they witnessed of Jesus' life, death, burial, and glorious resurrection would eventually cost them their lives. Eusebius, a well-learned Roman historian who lived from about AD 260 to 340, made this observation about the final moments between the apostle Peter and his beloved wife:\n\nThe blessed Peter, seeing his own wife led away to execution, was delighted, on account of her calling and return to her country, and that he cried to her in a consolatory and encouraging voice, addressing her by name: \"Oh thou, remember the Lord!\" Such was the marriage of these blessed ones.\n\nCan you imagine the final moment between Peter and his wife? How their eyes must have communicated volumes to each other as she was marched toward her execution? What courage she must have received to hear her sweet husband proclaim, \"Remember the Lord!\" as she was escorted along the path to her death. Did Peter's words remind her that in a few moments the Lord would be waiting to receive her into His kingdom?\n\nHappily-Ever-After?\n\nPeter and his wife were a powerful testimony for Christ to their generation. Do you realize the Lord wants to use you and your mate as a testimony to people as well? Have you ever considered that God brought you together with your husband because He has a mission for the two of you to accomplish\u2014together?\n\nSadly, many Christian couples become so focused on themselves or their personal pursuits that they never reach their full potential in this life. Sure, they may work to raise good kids, pay their bills, and go to church, but in all of these pursuits it is easy to become preoccupied with the things of earth. Jesus wants us to stop focusing so much on our own comforts and happiness and instead, to store up treasures in heaven, \"where neither moth nor rust destroys.\"\n\nCan you imagine how glorious it will be to one day stand before the Lord shoulder to shoulder with the people you and your husband have led to Christ? I long for that day! I simply cannot wait to stand next to Steve and celebrate with him over the many people we will meet in heaven who came to know Jesus because God allowed us to share the gospel with them. I don't care if we never get to retire, buy a motor home, and travel the world. Nothing will compare to the day we realize how every sacrifice, every prayer, and every tear was worth all that God had planned to do through us before the foundation of this world (see Ephesians 1:4; 2:10). And God has plans for you and your husband as well.\n\nAuthor Francis Chan shares this about his wife, Lisa:\n\nIt is our mutual love for Jesus that binds us, and our love for His mission in particular. We both love helping people repent of their sin, turn to Jesus and be filled with the Spirit. I love watching her share her faith...This may sound weird, but watching her minister attracts me to her even more...We love being on mission together. In fact, it is the times when we neglect the mission and just focus on our own desires that conflict arises. Staying on the mission is what draws us closer together.\n\nLooking back on our years of marriage, Steve and I can attest to the truth of Chan's statement. When you and your husband learn to take your eyes off of yourselves to focus on Christ and the purpose for which He has brought you together\u2014to know Christ and make Him known\u2014you will experience a happily-ever-after marriage that will reap joyful rewards far beyond what this life has to offer.\n\nFor some, the story of Peter and his wife's final days would seem to end in tragedy. But for those who realize we are all sojourners in this life, their story really does end happily. For when they breathed their final breath, they found themselves together in the presence of the Lord rejoicing over the fruit of their labors\u2014fruit that will abide forever.\n\nTen Keys to a More Fulfilling Marriage\n\nThroughout this book we have visited a number of myths wives tend to believe about marriage. And the key truth affirmed in every chapter is that true happiness is not to be found in your relationship with your husband, but in your relationship with Christ. Let's bring it all together now and review ten key principles to a more fulfilling marriage:\n\n1. Your Husband Was Never Meant to Be Your Happily-Ever-After\n\nAsking your husband to be the source of your happiness is an unfair expectation. You were created to delight in Christ and to be consumed by your love for Him.\n\nWhen Christ invades your life, what spills over is a passion for Him and for His kingdom purposes...Your willingness to lay aside anything that besets your passionate pursuit of Christ and His leading will not only set an example for [others] to follow, but create an appetite in [them] to do the same...you must be set on fire by the single most glorious purpose of life\u2014to know Christ and joyfully exhibit His greatness in all areas of life\n\nWhen you resolve to pursue loving Christ with all of your being, you will find the secret to happiness lies in your relationship with God alone. Only then can you enjoy fellowship with your husband in a way that honors Christ and blesses your husband.\n\n2. Respecting Your Husband Will Inspire Him to Love You More\n\nRemember David and Michal's story? Michal failed to see the big picture when David danced through the streets of Jerusalem on the day the Ark of the Covenant was brought into Jerusalem. Because Michal loved herself and her reputation more than she loved her husband, she responded by chastising him for his actions. Michal's disrespectful attitude alienated her husband from her for the rest of her life.\n\nGod created your husband with a deep longing to be respected by you. Just as deeply as you long to be loved without condition, your husband desires to receive unconditional respect from you. Notice I said unconditional respect. This means you don't get to hold hostage your respect for your husband when you aren't happy with him. Ephesians 5:33 says, \"Let the wife see that she respects her husband\" (ESV). This is not a suggestion; this is the Lord's command to us as wives.\n\nWhen you believe in your husband, rely on him, and celebrate his accomplishments, you are meeting one of his deepest emotional needs. As much as you value your husband's efforts to treat you in a loving manner, he will be grateful for your effort to treat him with honor. And when you do, don't be surprised if your husband responds to you in a more loving manner.\n\nYour respect can motivate your husband to accomplish great achievements\u2014because a man who is honored by his wife can do great things!\n\n3. Staying in Love Is All About Your Love for God\n\nI've said this multiple times already, but I believe it bears repeating here: Any wife who displays godly character by loving her husband will tell you that the key to loving your husband does not lie in how well he measures up to your expectations, but in how well you love God.\n\nBefore you were married, when you fell in love with your man, you had positive and loving thoughts about him. In marriage, you must work to continue to think such thoughts about him. The Bible provides a wonderful formula that can be applied to how you think of your husband:\n\nWhatever is true, whatever is honorable, whatever is just, whatever is pure, whatever is lovely, whatever is commendable, if there is any excellence, if there is anything worthy of praise, think about these things. What you have learned and received and heard and seen in me\u2014practice these things, and the God of peace will be with you (Philippians 4:8-9 ESV).\n\nIf you have not made a habit of thinking the best about your husband, you will need to determine to take \"every thought into captivity to the obedience of Christ.\" With God's help, you can gain victory over negative thoughts about your husband and replace them with thoughts that are honorable, lovely, and commendable. In our many years of biblical marriage counseling, Steve and I have seen relationships transformed when wives committed to thinking well of their husbands.\n\nJesus said the ultimate priority of life is to love God with all of your being. When you do so, you will find yourself enabled to obey the second greatest commandment: \"Love your neighbor as yourself\" (Mark 12:31). In this case, your husband is your neighbor.\n\nI am confident that pursuing intimacy with God was the single most important influence in transforming my marriage, and that can be true for you too. When your love for God is right, He will help you to love your husband.\n\n4. Parenting as One Brings Unity into Your Marriage and Security to Your Kids\n\nYour children's security lies in the health of your marriage relationship. When you learn to live with your sights upon God's calling on your life\u2014to know Christ and make Him known\u2014this will influence how you live at home. God intends for you to live in a manner that draws your kids to Christ.\n\nRemember that your genuine love for the Lord\u2014no matter how happy or trying your marriage may be\u2014will do far more to draw your kids to Christ than any words you can ever say to them. Whatever trials you and your husband encounter, if your children see the two of you united in purpose to display Jesus' character in your home, they will experience security. Isn't that your desire?\n\nIf your husband is not a Christian, do not fret. God can shine brightly through a godly woman who determines to honor the Lord in how she relates to her unbelieving husband. Acts 16:1 tells us Timothy's mother, Eunice, had been married to a Greek man whom some Bible commentators say might not have been a believer. Listen to how Paul commends Eunice: \"I am reminded of your sincere faith, a faith that dwelt first in your grandmother Lois and your mother Eunice and now, I am sure, dwells in you as well.\" And because Timothy's mother taught him sound doctrine from his childhood, he was ready to receive the gospel when he heard it preached (see 2 Timothy 3:15).\n\nIn the same way, if your husband is not a believer\u2014or he's not a strong Christian\u2014your commitment to teach your children to love God's Word can prepare them to respond to the gospel.\n\nRemember, your kids will be most secure when they observe their parents united, so don't disagree with your husband in front of your children about certain rules or disciplines he may impose. Determine to bow together united in prayer, rather than stand in conflict with one another, because \"the effective, fervent prayer of a righteous man avails much\" (James 5:16).\n\n5. The Grass Is Not Greener on the Other Side of the Fence\n\nGod created you with a longing to feel loved and valued\u2014by Him. Problems in marriage begin when you look to your husband to find your worth. God wants to fill the longings of your heart with Himself. Only through knowing Christ and living in intimacy with Him will you discover fulfillment.\n\nBecause of sin, you and I struggle with self-worship. And when you are in a state of self-love, if you're not satisfied with how your husband treats you, you may fall for the myth that you would be happier with someone other than your husband. When you find yourself toying with that idea, you can know that Satan\u2014who comes to steal, kill, and destroy (see John 10:10)\u2014is seeking to ruin you and your family.\n\nWhen your marriage relationship fails to satisfy your longings, you would be wise to remember that no relationship can fill the void that only God Himself can fill. Realizing it is wrong to receive your sense of worth from your husband is the first step to setting him free from the burden of trying to give you what only God can give. And when you determine to find your worth in Christ, you will no longer need others to fill the void only Jesus can satisfy.\n\n6. The Secret to Keeping Your Husband's Attention Is Finding Your Worth in Christ\n\nWhat's the secret to keeping your husband's attention? While there are many points I could make to answer this question, I prefer to revisit a statement my husband, Steve, made in chapter 6 of this book:\n\nThe secret to capturing your husband's attention for a lifetime is in learning to find your worth in your relationship with Christ. When you spend your life developing your inner beauty and staying focused on the Lord, your husband's affection for you will grow as he observes the lovely woman of God you are becoming. The more consistently you pursue Christ, the more beautiful you will become to your husband, to others, and most importantly, to Christ.\n\n7. Pursuing Your Husband Sexually Will Fill Him with a Sense of Well-Being\n\nDon't make your husband apologize for wanting to have sex with you. Pursue him sexually, and you will have a profound influence upon him in all areas of his life. When you make your husband feel sexually desirable, he will feel loved for who he is. You will fill him with a sense of well-being, confidence, and overall satisfaction with life.\n\nGod has given you a ministry of affirming your husband's deepest emotional needs through sex. In the same way that you long for your husband to romance you with his words and acts of love, he desires to be romanced by you through sexual intimacy.\n\nWhen you happily take your husband to bed, you not only satisfy his God-given physical need for sex, but you bring healing to his weary soul as well. To learn more about this, download my ebook A Christian Woman's Guide to Great Sex in Marriage.\n\n8. Grow Rich in Ways You Never Imagined\n\nHave you become consumed with career and cash? Do you fall into the trap of believing that security for you and your family lies in how much you can attain in this life?\n\nRealize God's highest good for your marriage is not to give you everything your heart desires so you can be comfortable and happy. Rather, God wants to make you holy through His Son so you can live out His perfect plan for you\u2014so you will reap His blessing for all eternity.\n\nWhen Steve and I moved to our little house in the country so we could live debt-free, we had no idea how God was preparing us to serve Him in full-time ministry. The people we have led to Christ, taught and mentored in truth, and fellowshipped with in ministry are greater treasures than any possessions we may have given up. And the hope of one day, for all eternity, worshipping with those people around the throne of Christ is the greatest treasure we could ever have.\n\nIn the same way, the more you keep your eyes fixed upon Jesus, the less you will care about possessions or the earthly issues many couples fight over. Once you determine to daily seek first the kingdom of God and His righteousness (Matthew 6:33), you will find contentment and peace in your marriage\u2014as well as God's purpose for your life.\n\n9. Be a Peacemaker in Your Marriage Relationship\n\nThe only way to build a marriage free of hurtful discord is through biblical conflict resolution. By way of review from chapter 9, here are eight steps to making peace:\n\n\u2022 Admit you have a problem. Take an honest look at why you are in the conflict.\n\n\u2022 Acknowledge your sinful bent. Realize in your sin you will either want to win the argument at any cost, or flee the conflict and stuff your feelings of resentment.\n\n\u2022 Refuse to be argumentative. \"The Lord's servant must not be quarrelsome but kind to everyone, able to teach, patiently enduring evil...[Be] eager to maintain the unity of the Spirit in the bond of peace...If possible, so far as it depends on you, live peaceably with all\" (2 Timothy 2:24; Ephesians 4:3; Romans 12:18 ESV).\n\n\u2022 Make peace a priority. Learn to talk through a disagreement for the purpose of resolution. And determine to get rid of whatever causes discord in your marriage. No argument is worth winning when the love and unity of your marriage is at stake.\n\n\u2022 Pray without ceasing. When God's people pray with a pure heart, their prayers are powerful and effective. So before you pray, be sure to ask God to reveal any sins within your heart so that you may confess them to the Lord. And never stop praying for the Lord to work in you\u2014and your husband\u2014to make you more like Christ (see 1 Thessalonians 5:17; James 5:16; 1 Peter 3:12).\n\n\u2022 Forgive your husband as many times as necessary (Matthew 18:22).\n\n\u2022 Seek godly counsel. Search out godly women in your church who can mentor you (Titus 2:1-5).\n\n\u2022 Learn to be a peacemaker. Jesus said, \"Blessed are the peacemakers\" (Matthew 5:9).\n\nMake yourself so familiar with these eight steps that the next time you and your husband begin to argue, you can stop yourself from fighting and instead, take time to reflect on how you can show Christ's character. By working to resolve conflict in a way that honors your husband, you can begin to live in a manner that reflects the Lord's character to those who are watching how you live\u2014beginning with your children.\n\nWhen this happens, your home will be marked by peace. Your children will feel more secure, and the peacemaking habits you practice in your home will, in turn, train your kids how to make peace in their relationships\u2014including their future marriages.\n\n10. The Joy of the Lord Is Your Strength\n\nLife is filled with blessings and struggles. Learning to see each experience as an opportunity for the Lord to shine His light through you is the first step to realizing God has a purpose in whatever He allows to come your way\u2014even a difficult marriage.\n\nThe apostle Paul and Silas are great examples of people who determined to be joyful amidst a terrible situation. After they were beaten and imprisoned for preaching the gospel, we could easily understand if they had bitterly questioned God's love for them. But that wasn't their response. Instead, they chose to sing hymns of praise to God (Acts 16:25).\n\nAuthor Richard Blackaby observes,\n\nThis story teaches an important life lesson. Jesus promised that no one could take His joy away from believers (John 16:22); Paul and Silas proved this to be true. When these two men were unfairly abused, they had to make a choice. They could let resentment overtake their hearts, or they could allow Christ to fill them with His joy even in that awful situation. They chose joy.\n\nWhen you choose joy in each experience you encounter, you can become a vessel for the Lord to reach the lost and encourage others\u2014as well as mold you, your husband, and your children more into the image of Christ. When you live with this perspective, you will discover the secret to living above life's circumstances\u2014and the joy of the Lord will indeed be your strength.\n\nAs I write this chapter, Steve and I just received news from our youngest daughter, Kayla, and her husband, Estevan, that the baby she was carrying in her womb has died. Steve and I are in somber awe as we watch our daughter and her husband walk with God-centered joy through this grievous circumstance. This joy is shining brightly to their Christian friends as well as their nonbelieving friends. I cannot help but think how the past trials Kayla and Estevan have endured have taught them to choose joy no matter what happens. And this, in turn, will bring honor to God.\n\nOne Last Love Story\n\nCharles Haddon Spurgeon is a man whom I admire very much. God used him as an evangelist and a preacher to bring a great revival to England in the 1800s. Although his story is inspiring, my goal in talking about him is not to tell of his accomplishments, but rather to share with you the wonderful love story of Charles and his wife, Susannah.\n\nWhile Charles Spurgeon is widely recognized for the tremendous impact he had in his era, his wife, Susannah, had an invaluable place in the molding of her husband's character and ministry. We are told that \"he never could have been what he was without her...No two souls on earth from the first fair dawn were more perfectly adapted to each other than Charles and Susannah Spurgeon.\"\n\nTheir paths would likely never have crossed were it not for God's providence. Susannah was a refined city girl, and Charles came from the countryside. After her first encounter with Charles, the cultured city girl said, \"I was not at all fascinated by the young orator's eloquence, while his countrified manner and speech excited more regret than reverence.\"\n\nHowever, as the Lord divinely brought about more opportunities for interaction, Charles began to pursue Susannah. And two-and-a-half months later, he sent her a gift. It was a copy of John Bunyan's book Pilgrim's Progress, in which he inscribed:\n\nMiss Thompson\u2014with desires for her progress in the blessed pilgrimage\n\nFrom C.H. Spurgeon Ap. 20, 1854\n\nNot long after Charles gave Susannah the book, the two attended a grand gala event in London with some friends. They found opportunity to sit near one another, and also take a walk in a beautiful garden. It was at this time that Susannah realized her love for Charles: \"During that walk, on that memorable day in June, I believe God Himself united our hearts in indissoluble bonds of true affection...From that time our friendship grew apace, and quickly ripened into deepest love.\"\n\nWithin a few weeks, Charles pledged his love to Susannah and asked her to marry him. She describes the occasion like this:\n\n[He] told me how much he loved me...I trembled and was silent for joy and gladness...To me, it was a time as solemn as it was sweet, and with a great awe in my heart. I left my beloved, and hastening to the house and to an upper room, I knelt before God, and praised Him with happy tears, for His great mercy in giving me the love of so good a man.\n\nEven though the two were in love, early in their courtship they had to work through some difficulties. For example, on one occasion Charles invited Susannah to go with him to a place where he was speaking. Because of Charles's popularity, he was swarmed by people at the event and swept away from his love. Absentmindedly, Charles forgot about Susannah\u2014which wounded her deeply. Finally, in anger, Susannah took a cab home, where she spilled out her grief to her mother.\n\nSusannah's mother wisely explained to her daughter that she was going to marry a husband who was no ordinary man. She told Susannah that Charles's whole life was dedicated to God and His service, and if she was to be his wife, she must never hinder him by trying to put herself first in his heart. (A lesson this pastor's wife needs to be reminded of as well from time to time.)\n\nLater, Charles showed up at Susannah's home looking for her. He kindly listened to her as she spilled out how indignant she had felt earlier.\n\nOn January 8, 1856, Charles and Susannah were married. Afterward they enjoyed a ten-day honeymoon to Paris. (Now that sounds like a romantic honeymoon\u2014well done, Charles!)\n\nThat Susannah learned how to be the selfless wife of a man whose whole heart was devoted to serving Christ was what made her so beautiful\u2014and well suited\u2014for Charles. Their wonderful union shone brightly to all they would meet through their many years of serving the Lord together. Listen to Spurgeon's own words about his beloved:\n\nThough He who chose us all worlds before,\n\nMust reign in our hearts alone,\n\nWe fondly believe that we shall adore,\n\nTogether before His throne.\n\nFor nearly 40 years, Charles and Susannah served God side by side. Their pilgrimage took them through the heights of glory and the depths of sorrow and hardship. Through it all, their love for one another\u2014and the Lord\u2014grew to immeasurable proportions.\n\nThe Lord certainly answered the \"prayer\" Charles had written in the front of the book he had given Susannah when they were courting, for God had given Susannah \"progress in the blessed pilgrimage.\"\n\nAfter Charles died on January 31, 1892, Susannah was left alone to continue on in ministry. Although sorrow and loneliness were often evident in her words after her husband's death, she also wrote with a sense of glorious triumph that is known only by a woman who has walked near to the Savior:\n\nI have traveled far on life's journey, and having climbed one of the few remaining hills between earth and heaven, I stand awhile on this vantage ground and look back across the country through which the Lord had led me...\n\nI can see two pilgrims treading the highway of life together, hand in hand\u2014heart linked to heart. True, they have had rivers to ford, mountains to cross, fierce enemies to fight and many dangers to go through. But their Guide was watchful, their Deliverer unfailing, and of them it might truly be said, \"In all their affliction He was afflicted, and the Angel of His presence saved them; in His love and in His pity He redeemed them: and He bare them and carried them all the days of old.\"\n\nMostly they went on their way singing; and for one of them at least, there was no greater joy than to tell others of the grace and glory of the blessed King to whose land he was hastening. And while he thus spoke, the power of the Lord was seen and the angels rejoiced over repenting sinners.\n\nBut at last they came to a place on the road where two ways met. And here, amidst the terrors of a storm such as they had never before encountered, they parted company\u2014the one being caught up to the invisible glory, and the other, battered and bruised by the awful tempest, henceforth toiling along the road\u2014alone!\n\nBut the \"goodness and mercy\" which for so many years had followed the two travelers, did not leave the solitary one. Rather did the tenderness of the Lord \"lead on softly,\" and choose green pastures for the tired feet, and still waters for the solace and refreshment of His trembling child.\n\nHe gave, moreover, into her hands a solemn charge\u2014to help fellow pilgrims along the road, therewith filling her life with blessed interest, and healing her own deep sorrow by giving her power to relieve and comfort others.\n\nAnyone else need a tissue? I just love this kind of love\u2014don't you? Charles and Susannah were used mightily by God\u2014all because they lived in wholehearted devotion to Christ.\n\nWhile your husband might never be like Charles, still, your place as his wife is to determine to devote yourself to the Lord. That's essential no matter what your husband's spiritual state. When you live with a heart of surrender, God can work through you to accomplish incredible feats for His kingdom.\n\nD.L. Moody is a good example of how God can work through a person who is completely yielded to Him. Moody was a contemporary and friend of Spurgeon, and was moved to follow God in total abandon when he heard the British evangelist Henry Varley say, \"The world has yet to see what God can do with and for and through and in and by the man who is fully and wholly consecrated to Him.\" After hearing that, Moody thought to himself,\n\nHe said \"a man.\" He did not say a great man, nor a learned man, nor a rich man, nor a wise man, nor an eloquent man, nor a smart man, but simply \"a man.\" I am a man, and it lies with the man himself whether he will or will not make that entire and full consecration. I will try my uttermost to be that man.\n\n\"As [Moody] came to realize that it is God who does the actual work of ministry and that the most effective channel for ministry is a wholly surrendered life, he resolved more than ever to avail himself completely for the Lord's use.\"\n\nIn the same way, God will work effectively through you to transform you and your marriage for His purpose\u2014if you resolve to live wholly consecrated to Christ.\n\nParting Is Such Sweet Sorrow\n\nI can hardly believe we have come to the end of this book. I have thoroughly enjoyed walking through these pages with you\u2014so much so that I don't want for this to be my final admonition to you. On the other hand, after my having spent months on writing this book, Steve can't wait to have his wife back!\n\nI feel the need to encourage you not to simply set this book aside and chalk up another Christian self-help book to your reading list. If you determine to apply the truths you learned in this book, they will change you and your marriage because they are based on the life-transforming Word of God. So please keep this book nearby, review it from time to time, recommend it to your friends, host a book club, and then talk with each other about what you learned.\n\nAs we close, I want to emphasize once again what I believe is the key to building a no-regrets marriage: Jesus said the ultimate priorities of life are to love God with your whole being, and then to love others as you love yourself (Mark 12:30-31). I believe these are the most important admonitions I can leave with you.\n\nWhen your love for Christ becomes the single most important pursuit of your life, you will understand your worth in your relationship with Him. And then fellowshipping with Him through Bible study and prayer will become the longing of your heart. And when loving God becomes your passion, loving your husband for who he is\u2014and not who you want him to be\u2014will become a natural outpouring of your love for Christ.\n\nWill you commit to making whatever sacrifices are necessary to grow in your love for Christ? In this way, your marriage can become one others will want to emulate. And the result will be for your blessing and God's glory. Your marriage will shine a bright light of hope to a generation who desperately needs to know that Christ is the only answer to all of life's problems\u2014and the only source of a happy marriage.\n\n######\n\nTHINKING IT THROUGH\n\n######\n\nSpend some time praying over the \"Ten Keys to a More Fulfilling Marriage.\" List in a notebook what steps you feel most compelled to apply to your marriage. Write out some ways you can adjust your life to what you have learned.\n\n######\n\nLIVING IT OUT\n\n######\n\n1. Commit to reviewing what you've learned in this book by choosing to do one or more of the following:\n\n\u2022 Over the next 12 months, review one chapter each month and ask the Lord to show you how He would have you continue growing in your love for Him and your husband.\n\n\u2022 Using the principles in this book, mentor another Christian woman (or start a small group or online study group and go through the book together).\n\n\u2022 Use this book as a resource to share the gospel with a non-Christian woman in need of marriage help.\n\n\u2022 Buy a copy of this book for a Christian woman engaged to be married.\n\n2. Pray, pray, and pray some more for God to transform you through His Word. Ask Him to give you His perfect love for your husband, and pray for Him to make your marriage one that shines brightly the message that Christ is the answer to every need.\n\nVisit NoRegretsWoman.com to watch Steve and Rhonda's video link and\/or listen to their suggested audio link.\n\nPhoto credit: JPlazaPhotography\n\nPhoto credit: Eric McFarland Photography\nAppendix:\n\nHow to Have a Relationship with Jesus\n\n######\n\nWhat on earth could she possibly mean by a relationship with Jesus?\" you ask. I am so glad you want to know!\n\nDid you know that God created people so that He could have a relationship with them? When the Lord created Adam and Eve and put them in the Garden of Eden, He did not leave them there with a list of religious rituals to perform while He observed from afar. No, Genesis 3:8 says that God walked with Adam and Eve in the garden in the cool of the day. He spent time with them!\n\nYou have likely heard some form of the story of how God put a tree in the garden and commanded Adam and Eve not to eat of its fruit, or they would surely die (Genesis 2:17). Genesis chapter 3 records how one day Satan came and tempted Eve to partake of the forbidden fruit. Eve was deceived and seduced by Satan's lies and ate the fruit\u2014and of course Adam followed suit. In the moment that they disobeyed God's command not only did their bodies begin to die physically, but what's worse is that they died spiritually. Can you imagine how empty they must have felt when that happened?\n\nYou see, once Adam and Eve sinned, they had rejected God's rule and yielded themselves to Satan. And without someone to rescue them, they were without hope of ever being in right standing with God again. Because of their rebellion against God, they could no longer fellowship with Him, for God cannot allow sin in His presence. And unless God provided a way that Adam and Eve\u2014and by extension, all of mankind\u2014could have that relationship restored, they would forever be without hope. Every one of us was destined to spend eternity in hell, separated from God's presence.\n\nHowever, because of God's great love for His creation, He had planned a way to rescue us and bring us back to Himself (that's why we use the word salvation!).\n\nHave you ever wondered, Why did God put that tree in the garden anyway? I mean, if it hadn't been there, Adam and Eve would never have been tempted. That's a good question, and it's one I have pondered myself.\n\nI used to think that somehow Adam and Eve's sin caught God by surprise, and that the Trinity (God the Father, God the Son, and God the Holy Spirit) entered into a holy huddle to figure out Plan B for mankind's redemption (redemption is a big word that basically means \"buy back\"\u2014see Revelation 5:9).\n\nI have since learned that God knew that Adam and Eve would fall. Revelation 13:8 says that Jesus was \"slain from the foundation of the world.\" That means even before God created the world or people, He knew that all of us would need a Savior. And because of His great love for us, and His desire to have a people who would choose to love and serve Him, He put the tree in the garden to give Adam and Eve a choice. When they sinned (and He knew they would) God told them that He would offer up His Son to pay the price for their disobedience (see Romans 5:12-21).\n\nImagine\u2014God loved us so much that He sacrificed His only Son, that whoever believes in Him will not die but will live forever (John 3:16)! God says the very act of offering His greatest treasure, Jesus, was His way of showing you and me just how very much He loves us. \"God demonstrates His own love toward us, in that while we were still sinners, Christ died for us\" (Romans 5:8). What an amazing way for Him to show us how much He loves us, huh?\n\nSo what does it mean to believe in Him like John 3:16 says? Is it a mere mental assent to the truth that Jesus is fully God, and being fully God He took on the form of a man when He was born through a virgin? And that Jesus lived a sinless life, and willingly gave Himself up to die a cruel death on a cross, and then He victoriously rose from the dead\u2014so that His blood could wash away our sins and He could give us eternal life? While all of those statements are true, if you simply agree with the facts about Jesus, that does not mean you have a relationship with Him. In fact, James 2:19 says even the demons believe, and they tremble in fear because they know who Jesus is, and what He accomplished when He died for our sins.\n\nNo, having a relationship with Jesus is entering into a personal covenant (that's a big word that means \"vow\" or \"promise\") with Jesus. He wants us to make a lifelong commitment to Him\u2014but how?\n\nFirst, God wants you to repent of your sins (repent means to agree with God that you are a sinner in need of a Savior, and that you will turn away from your sins). The Bible says, \"All have sinned and fall short of the glory of God\" (Romans 3:23). Only the blood of Jesus can wash away your sins (Hebrews 9:14).\n\nI know it's easy to take offense when someone says, \"You're a sinner,\" but let's be honest: You and I both know that even though we try to do what's right, our natural instinct is to disobey God's laws.\n\nYou see, God gave us those laws not so that we could try to become sinless by doing all that they command, but to show us that we will never be able to measure up to the sinless life God requires of us to have a relationship with Him and enter into heaven when we die (see Galatians 2:16; 3:24).\n\nSo where does that leave us? If Galatians 2:16 says that no man is justified by the works of the law, then how can we possibly be restored to God and go to heaven? If God isn't making sure our good deeds outweigh our bad deeds by the time we die (a completely bogus concept not taught in Scripture), and if, as Romans 6:23 says, \"the wages of sin is death,\" how can we be rescued from judgment?\n\nI'm glad you asked! For the Bible also says, \"The gift of God is eternal life in Christ Jesus our Lord\" and that we are justified (made right) \"by faith in Jesus Christ\" (Romans 6:23; Galatians 2:16).\n\nThe Bible teaches that Jesus is not simply one of many ways to salvation; He is the only way. In John 14:6, Jesus said, \"I am the way, the truth, and the life. No one comes to the Father except through Me.\" Those are Jesus' words, not mine. The only way to an intimate relationship with God is through Jesus. It is only when you receive His free gift of salvation that Jesus' blood washes away all of your sins. God Himself says, \"Though your sins are like scarlet, they shall be as white as snow\" (Isaiah 1:18).\n\nThink of it\u2014God promises to wipe the slate completely clean! No matter how many bad decisions you have made up to this point, no matter how shameful your past, Jesus is offering you freedom from all of it! Freedom from shame and the bondage of sin.\n\nOnce Jesus washes away your sins, He promises never to throw them in your face again. The Bible says, \"As far as the east is from the west\" is how far God removes our sins from us (Psalm 103:12). (You do realize that east and west never meet, right? That means that in Christ, our sins are taken away forever!)\n\nBut you don't get to just say some magic words, \"I believe,\" and then go back to life as usual. Jesus says He wants you to surrender all that you are to Him. \"If you confess with your mouth the Lord Jesus and believe in your heart that God has raised Him from the dead, you will be saved\" (Romans 10:9-10).\n\nJesus doesn't ask you to simply add Him onto your life. He wants to be your life. And to anyone who becomes Jesus' follower, He promises that He will give you a new and pure heart. Second Corinthians 5:17 says, \"Old things have passed away; behold, all things have become new.\"\n\nBelieve me when I tell you that without a relationship with Jesus I was a selfish, arrogant, fearful, and materialistic woman. But when I accepted Jesus' free gift of salvation and surrendered my life to Him as my Lord, I was set free. I have never looked back! Jesus took the mess that I was and gave me a new heart. Through Jesus, God forgave all of my sins\u2014all of them! And when I said yes to entering into a relationship (there's that word again) with Jesus, He put within me His Holy Spirit. (So that's what was missing!) And God wants the same for you.\n\nWhen God fills you with His Spirit, life makes sense! In fact, it's the life you were born to live, in fellowship with your Creator. Nothing else in this life will ever satisfy your longing for Him\u2014nothing.\n\nIf you enter into a relationship with Jesus, you never have to worry about being \"good enough\" for God to love you or let you into heaven when you die. To those who are in Christ, God says He adopts us as His very own children. \"Behold what manner of love the Father has bestowed on us, that we should be called children of God!\" (1 John 3:1). Jesus says we can call God \"Abba! Father!\" (that means \"Daddy\"\u2014Romans 8:15). And God says His great love for us is perfect, immeasurable, and nothing we could ever do will make Him stop loving us! (see Romans 8:35-39). To top it off, God promises you will never be alone again. Jesus promises He will never leave you nor forsake you (Matthew 28:19-20; Hebrews 13:5). How awesome is that?\n\nAnd there's one more thing: If you decide to believe that Jesus died for you, and you choose to agree with God that you are in need of a Savior because of your sinful heart, and if you pray and submit to Jesus as the Lord of your life, then God's Spirit will fill your heart with His presence, peace, and purpose.\n\nWhen you receive Jesus' free gift of salvation, He promises to lead you, guide you, and accomplish great things for His kingdom through you for the rest of your life. Ephesians 2:8-10 says, \"By grace [that means you can't earn it] you have been saved through faith, and that not of yourselves; it is the gift of God, not of works, lest anyone should boast. For [you] are His workmanship, created in Christ Jesus for good works which God prepared beforehand.\" God has a plan for your life. Isn't that exciting?\n\nSo now you know what it means to have a relationship with Jesus. It is my prayer that the Holy Spirit is drawing you to Christ even at this moment, and that you will pray to receive Jesus as your Lord and Savior so that you can begin this wonderful journey of walking with Him for the rest of your life, and on into heaven in the next!\nNotes\n\nChapter 1\u2014If He Would Change, I'd Be Happy\n\n. Luke 19:17 (ESV).\n\nChapter 2\u2014I Will Respect Him When He Earns My Respect\n\n. The MacArthur Study Bible (Nashville, TN: Thomas Nelson, 1997), study note for 2 Samuel 6:16.\n\n. Dr. Emerson Eggerichs, Love and Respect (Nashville, TN: Thomas Nelson, 2004), 15.\n\n. Eggerichs, Love and Respect, 3-4.\n\n. Eggerichs, Love and Respect, 87-89.\n\n. Eggerichs, Love and Respect, 89.\n\nChapter 3\u2014I'm Falling Out of Love with Him\n\n. Francis Chan, Crazy Love (Colorado Springs, CO: David C. Cook, 2013), 103-4.\n\n. Matthew 5:7.\n\n. John MacArthur, Daily Readings from the Life of Christ (Chicago, IL: Moody, 2008), 72.\n\n. Luke 6:36.\n\n. Shanti Feldhahn, For Women Only (Sisters, OR: Multnomah, 2004), 166, 169.\n\nChapter 4\u2014Our Kids Would Obey If He Were a Better Father\n\n. John MacArthur, Twelve Extraordinary Women (Nashville, TN: Thomas Nelson, 2005), 92.\n\n. MacArthur, Twelve Extraordinary Women, 95.\n\n. Francis and Lisa Chan, You and Me Forever (San Francisco, CA: Claire Love Publishing, 2014), 163.\n\n. Steve Miller, D.L. Moody on Spiritual Leadership (Chicago, IL: Moody, 2004), 109, citing D.L. Moody, Great Joy (New York: Treat, 1877), 245.\n\n. The name of the Bible study that Steve referred to is Experiencing God by Henry Blackaby (Nashville: B&H, 1998).\n\n. James 5:16 (ESV).\n\nChapter 5\u2014I Would Be Happier Married to Someone Else\n\n. Paul David Tripp, What Did You Expect? (Wheaton, IL: Crossway, 2010), 210.\n\n. Dr. Emerson Eggerichs, Love & Respect Video Conference, Colorado Springs, Focus on the Family (2004).\n\n. See 1 Peter 5:8; John 10:10.\n\n. Tripp, What Did You Expect?, 196.\n\nChapter 6\u2014He Would Love Me More if I Were Prettier\n\n. Janet & Geoff Benge, George M\u00fcller, the Guardian of Bristol's Orphans (Seattle, WA: YWAM Publishing, 1999), 43.\n\n. Benge, George M\u00fcller, the Guardian of Bristol's Orphans, 75-76\n\n. Benge, George M\u00fcller, the Guardian of Bristol's Orphans, 79-80.\n\n. Song of Solomon 1:8.\n\n. Julie Gorman, What I Wish My Mother Had Told Me About Men (Franklin, TN: Authentic Publishers, 2013), 25.\n\n. Genesis 3:11-12.\n\n. Jeremiah 17:9.\n\n. Elyse Fitzpatrick, Idols of the Heart (Phillipsburg, NJ: P&R Publishing, 2001), 130, 132.\n\n. Hebrews 4:12.\n\n. Psalm 139:23-24.\n\n. Jerry Bridges, The Pursuit of Holiness (Colorado Springs, CO: NavPress, 1986), 78.\n\n. Shanti Feldhahn, For Women Only (Sisters, OR: Multnomah, 2004), 166, 169. Italics in original.\n\n. Ruth 2:9.\n\n. The MacArthur Study Bible (Nashville, TN: Thomas Nelson, 1997), 373.\n\n. Ruth 1:15-18; Proverbs 31:10-12,23.\n\n. Ruth 1:16-17.\n\n. 2 Peter 3:18.\n\n. Ruth 2:2,7,17,23; Proverbs 31:13-21,24,27.\n\n. Ruth 2:12.\n\n. Proverbs 31:11.\n\n. Rhonda Stoppe, Moms Raising Sons to Be Men (Eugene, OR: Harvest House, 2013), 180. Quote originally from Arnold Dallimore, Spurgeon, A New Biography (Carlisle, PA: The Banner of Truth Trust, 2005), 36.\n\n. Ruth 2:7; Proverbs 31:26.\n\n. 1 Corinthians 13:5.\n\nChapter 7\u2014All He Wants Is Sex\n\n. John MacArthur, Daily Readings from the Life of Christ (Chicago, IL: Moody, 2008), 184.\n\n. Shaunti Feldhahn, For Women Only (Sisters, OR: Multnomah, 2004), 94.\n\n. Feldhahn, For Women Only, 99, 101-2.\n\n. Feldhahn, For Women Only, 99.\n\n. Feldhahn, For Women Only, 139.\n\n. To find out more about biblical counseling and counselors near you, go to the website for the Association of Certified Biblical Counselors at <http:\/\/www.biblicalcounseling.com>.\n\nChapter 8\u2014More Money Equals Less Stress\n\n. Francis Chan, Crazy Love (Colorado Springs, CO: David C. Cook, 2013), 89-90.\n\n. Chan, Crazy Love, 133.\n\n. Chan, Crazy Love, 177.\n\n. Chan, Crazy Love, 181-82.\n\n. Francis and Lisa Chan, You and Me Forever (San Francisco, CA: Claire Love Publishing, 2014), 27.\n\n. Turn Your Eyes Upon Jesus, Words and Music by Helen H. Lemmel, 1864\u20131961.\n\n. Jim George, A Husband After God's Own Heart (Eugene, OR: Harvest House, 2004), 11.\n\n. To learn more about Dale and Amy's prodigal son\u2014and what to do if your child is wayward\u2014read my book Moms Raising Sons to Be Men (Eugene, OR: Harvest House, 2013).\n\nChapter 9\u2014Every Couple Fights\n\n. Julie Gorman, What I Wish My Mother Had Told Me About Men (Franklin, TN: Authentic Publishing, 2013), 63.\n\n. Gorman, What I Wish My Mother Had Told Me About Men, 65.\n\n. Gorman, What I Wish My Mother Had Told Me About Men, 120.\n\n. Jerry Bridges, The Pursuit of Holiness (Colorado Springs, CO: NavPress, 1986), 104.\n\n. The MacArthur Study Bible (Nashville, TN: Thomas Nelson, 1997), study note for Genesis 3:16.\n\n. If you suffer side effects from PMS, consider going to a doctor who specializes in hormonal issues.\n\n. Philippians 4:4-7 (ESV).\n\n. Proverbs 15:17 (ESV).\n\n. John MacArthur, Daily Readings from the Life of Christ (Chicago, IL: Moody, 2008), 78.\n\n. To find out more about biblical counseling and counselors near you, go to the website for the Association of Certified Biblical Counselors at <http:\/\/www.biblicalcounseling.com>.\n\n. Matthew 5:9.\n\n. Ken Sande, The Peacemaker (Grand Rapids, MI: Baker, 2004), 11.\n\n. Luke 6:41-42 (NASB).\n\nChapter 10\u2014Our Marriage Would Be Better if Bad Things Would Stop Happening\n\n. The MacArthur Study Bible (Nashville, TN: Thomas Nelson, 1997), study note for 1 Peter 4:12.\n\nChapter 11\u2014If Momma Ain't Happy, Ain't Nobody Happy\n\n. Elizabeth George, A Wife After God's Own Heart (Eugene, OR: Harvest House, 2004), 14.\n\n. Elizabeth George, A Woman After God's Own Heart (Eugene, OR: Harvest House, 1997), 57.\n\n. George, A Wife After God's Own Heart, 162, 14.\n\n. George, A Wife After God's Own Heart, 43, 162.\n\n. George, A Wife After God's Own Heart, 14.\n\n. George, A Woman After God's Own Heart, 58.\n\n. George, A Wife After God's Own Heart, 37.\n\n. George, A Wife After God's Own Heart, 171.\n\n. George, A Wife After God's Own Heart, 52.\n\n. Matthew 7:3-5.\n\n. John MacArthur, Daily Readings from the Life of Christ (Chicago, IL: Moody, 2008), 177-79.\n\n. Rhonda Stoppe, Moms Raising Sons to Be Men (Eugene, OR: Harvest House, 2013), 71.\n\n. I also share about this in Stoppe, Moms Raising Sons to Be Men, 15.\n\n. Stoppe, Moms Raising Sons to Be Men, 14.\n\n. Jim George, A Husband After God's Own Heart (Eugene, OR: Harvest House, 2004), 9-11.\n\n. George, A Husband After God's Own Heart, 10.\n\nChapter 12\u2014Happily-Ever-After Is a Fairy Tale\n\n. Acts 4:20.\n\n. Eusebius Pahmphilus, Eusebius' Ecclesiastical History (Grand Rapids, MI: Baker, 1974), 115-16.\n\n. Matthew 6:20.\n\n. Francis and Lisa Chan, You and Me Forever (San Francisco, CA: Claire Love Publishing, 2014), 112.\n\n. John 15:16.\n\n. Rhonda Stoppe, Moms Raising Sons to Be Men (Eugene, OR: Harvest House, 2013), 188.\n\n. 2 Corinthians 10:5.\n\n. Scripture doesn't specifically say if Timothy's father was or wasn't a believer. Rather, it points out that he was a Greek, whereas Timothy's mother was Jewish. Some commentators believe this points to Timothy having access to both Greek and Jewish cultures, while others think it's a clue that Timothy's father was an unbeliever.\n\n. 2 Timothy 1:5 (ESV).\n\n. Rhonda Stoppe, A Christian Woman's Guide to Great Sex in Marriage, www.NoRegretsWoman.com.\n\n. Richard Blackaby, Unlimiting God (Colorado Springs, CO: Multnomah, 2008), 131.\n\n. H.L. Wayland, as cited in Arnold Dallimore, Spurgeon: A New Biography (Carlisle, PA: The Banner of Truth Trust, 1987), 54.\n\n. Iain Murray, ed., C.H. Spurgeon Autobiography: The Early Years 1834\u20131859 (London: Banner of Truth, 1962), 280.\n\n. Iain Murray, as cited in Dallimore, Spurgeon: A New Biography, 56.\n\n. Iain Murray, as cited in Dallimore, Spurgeon: A New Biography, 57.\n\n. Dallimore, Spurgeon: A New Biography, 61 (emphasis added).\n\n. Susannah Spurgeon, Ten Years After (London: Passmore & Alabaster, 1895), vi-vii. Dallimore, Spurgeon: A New Biography, 150.\n\n. Steve Miller, D.L. Moody on Spiritual Leadership (Chicago, IL: Moody, 2004), 22.\n\n. Miller, D.L. Moody on Spiritual Leadership, 22.\nAbout the Publisher\n\n* * *\n\nTo learn more about Harvest House books and to read sample chapters, visit our website:\n\nwww.harvesthousepublishers.com\n\nHARVEST HOUSE PUBLISHERS\n\nEUGENE, OREGON\n\n* * *\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nThe dust of just beginning\n\n# **The dust of just beginning**\n\n**Don Kerr**\n\n(C) 2010 Don Kerr\n\nPublished by AU Press, Athabasca University  \n1200, 10011 -- 109 Street  \nEdmonton, AB T5J 3s8\n\n**Library and Archives Canada Cataloguing in Publication**\n\nKerr, Don  \nThe dust of just beginning \/ Don Kerr.\n\n(Mingling voices, ISSN 1917-9405)  \nPoems.  \nIssued also in an electronic format (978-1-897425-93-0).  \nISBN 978-1-897425-92-3\n\nI. Title.\n\nII. Series: Mingling voices\n\nPS8571.E71D88 2010 C811'.54 C2010-904774-5\n\nCover and book design by Natalie Olsen, Kisscut Design.  \nCover image by Don Proch, _Asessippi Valley_ (2006).  \nAuthor photo by Hans Dommasch.  \nPrinted and bound in Canada by Marquis Book Printing.\n\nPlease contact AU Press, Athabasca University at  \naupress@athabascau.ca for permissions and copyright  \ninformation.\n\nA volume in the Mingling Voices series:  \nISSN 1917-9405 (Print) ISSN 1917-9413 (Online)\n\n## **Contents**\n\nJUST BEGINNING\n\na height of prairies\n\nBilly\n\nthe lady gardener, Anne Szum\n\nthe voice of Anne\n\nsun\n\nthe jesus poems\n\nthe body poems\n\nthe waters of life\n\nwhat then is free?\n\nthe rote beads\n\nthe last day\n\nI told you I could drive\n\nIla\n\na memory\n\noh then\n\ngone large gone small\n\nthe dust of just beginning\n\nFOR IT'S LOVE\n\nlove poems\n\nmall\n\nbody poems II, the black poet\n\nJOURNEY MAN\n\na fearful carnality\n\nlost and found\n\nmy road\n\nthe first day\n\nwhere are we going next year?\n\nto the tax man\n\njourney man\n\nyou dream the poem\n\ntough terrain\n\nmind riding\n\nBlue River\n\nWE ARE THE ECHO\n\nkelly's alley\n\nhunting rabbits\n\non the line\n\nsummer south\n\nCousin Lynda\n\nwater boy\n\nwe are the echo\n\nthis is the day nothing happens\n\nbody idling over\n\nTO ESCAPE ONTO PAPER\n\ninto the woods\n\nword off\n\nvictim of story\n\nat the end of his fifth whiskey\n\nthe dead tired poem\n\nthe anonymous clouds\n\nnever start from here\n\nSamuel Beckett\n\nacknowledgement[s]\n\nabout the author\n\n## **JUST BEGINNING**\n### **a height of prairies**\n\na height of prairies over the river  \nsideways sun in the brown  \nstubble, twisted speech  \nof dead trees, the duplicitous  \nsun, but in decline, eyes  \nseeing for miles, all  \nat the same moment, elsewhere  \nthe room with the blinds drawn,  \nthe cancer advancing like traffic\n\nor the smell in the corridor  \nof cooking, a fire hose for decor,  \na life, long at its time, unmet  \nin words, a day in the  \nShirley apartments, the smell  \nof cooking, long demolished,  \nhome to a bowling alley,  \nautomobile garage, passing  \nunmarked\n\na day in Connaught  \nin Ladbroke Road\n\n### **Billy**\n\nlook, Billy dead, the city empties,  \nthe city of London empties,  \nif we don't enter we needn't  \nremember, nor see others  \nin his room, his things  \nscattered, his ghost thin  \nin my belly what is there to say  \nwithout the listening man?\n\nyour death Billy,  \nconfounding time, burying  \nyour friends, burying the stories,  \nplentiful as books in your room,  \nyour Vancouver, long buried,  \nlong demolished, sun the liar  \nsaying this is the last of all days, yet  \nwe bound for the next day, your music  \non the tape deck, we travel  \ntwo ways now, playing you  \none day at a time\n\nBilly at Notting Hill  \nat Ladbroke Road  \nat Gennaro's, at Prost's,  \ntalking the eyes out of a girl,  \ndancing near the wide Saskatchewan  \nin shoes shiny enough to show  \nthe mossy face, Billy leaving  \nChristmas at Finchley for the  \neight-mile walk or crying  \nin the crowded hospital at Paddington  \nand saying I don't usually cry,  \nthe pain swallowing Billy\n\n### **the lady gardener, Anne Szum**\n\nthe night edges over the house  \ninto the branches of the tree  \nthe branches of the dark green fir  \ninto the forest of the peony  \ninto the dirt under the peony\n\nin the dead centre of the day  \nin the mid of the mid day  \nthe sun like a perennial  \nthe bluebells happy in the  \nsweet breeze the lilies  \npointing skyward the raspberries  \nin spiky bud all wait oh they all wait  \nfor their first love the lady  \ngardener but she is unavoidably  \ndetained the faces of the pansies  \nthe petunias watch and wait but  \nshe is unavoidably  \ndetained\n\n### **the voice of Anne**\n\nif your words fall into her ear  \nand you are in the same room  \nthe one with the column of cds  \nthe heater that eats logs  \nthe walls insulated with books of poetry\n\nin that room, your words falling  \ninto that ear, need no other home  \nuntil she, the lady gardener,  \ndwells only in our rooms  \ncluttered as they are  \nwith all matter of the living  \nabsently watching cars  \npeople and the faintest  \nof faint snow falling  \nfrom a heaven grey as ghosts  \nor your eccentric angels falling  \ninto the pie-shaped lot  \non Connaught saying in their  \nodd way for heaven's sake  \npick the raspberries  \nred as thick blood  \nthe sparrow hopping about  \nlooking for the ear to pour  \nits song into while I, bereft,  \nfall back into the habit of books\n\nand she sits, makes tea,  \ntends the garden, reads,  \nin the voice of Anne,  \nall days in disarray\n\n### **sun**\n\nif there is no sun  \nand the sky draws down  \nyou walk through  \na veil of mist and are not  \nat home\n\nsun shone before you knew  \nsun and wherever you travel sun  \nis your home your dream  \nof Rome sun rising  \nand falling in Claude  \nof Lorraine the wide bay  \nor the bones leaning like flames  \ninto only sun\n\nwhen you see sun  \nrocking the lake, firing the woods  \nwith a latticework that greens  \nthe ferns, laying great hands  \non the hills, scouring the bare  \nvalleys and the small forests  \nof your arms and legs,  \nwould you praise god?  \nor let the mind go dead,  \nthe body drowsy bathing  \nin sun heat and day light\n\n### **the jesus poems**\n\n1.\n\njesus held me  \nin the grip of hell  \nmy grade three teacher  \nlit the fire  \nand only prayer  \nsaid over and over and over  \nmight keep me green and cool  \nterrified always to re-enter  \nthe brick prison of St. Joe's School  \nor to tell the terror  \nheld me in thrall\n\nbaseball was the way  \nand the light of day  \njesus never played baseball  \nmaybe umpire or scorekeeper  \nbut the soft bunt down third  \nthe leaping catch at second  \nto be lead-off batter  \nto wait in the on-deck circle  \nto do the chatter  \nto win at tough  \nSt. Mary's to forget the fire  \nconcentrating on the next pitch  \nwas the best way out of  \nall that other stuff\n\n2.\n\nThere is nothing I believe  \nwith mathematical precision,  \nno equation out of the self.  \nIf lonely enough or vanishing enough  \nwould Odetta effect a cure?  \nPatsy Cline or _Casablanca?_  \nYet there is the temptation,  \nthe nothing into everything,  \ntrue life in death, the miracle  \nof the cross, the Catholic  \ncalculating machine.\n\neasy enough to say  \nblack on a health day\n\n3.\n\nsweet church, its large  \nemptiness, candles burning  \nfor the dead, the boy counting  \nhow many souls could fit and fly  \nin the large auditorium of god,  \nquits at thousands, looks  \nat silence, the creak of a kneeler  \nechoes, he kneeling, hands clasped  \nholy, sinks his teeth into the  \nvarnished pew to leave his mark,  \nunder St. Joseph in the brown  \ncloak with a staff the candles  \nburning like souls, like hell,  \nlike purgatory and so beautiful,  \neyes caught, body gone quiet  \nhe crosses himself,  \nwalks down the stone steps  \ninto the wide street\n\n4.\n\nthis was the sermon  \nthat a great garden  \nour heart's desire  \ngreen and golden  \nwas surrounded by a high fence  \nwith a narrow entrance  \nand outside all was fury and fire  \nstorm and stormy plains  \nthe land of all fear\n\nI knew I would never  \ndiscover that narrow way  \nto the green garden  \nand day after day  \nI picked the deadhead poppies  \nin my mom's garden that they  \nwould flower orange and yellow  \nlike fire all the days  \nof summer\n\n5.\n\nIs it possible to be a fallen away  \nUnited Churcher? Like my dad.  \nWe discovered in the apple-box  \nbookshelves in the basement his prize  \nfor bible studies in grade six  \nin Nokomis. That wasn't  \nthe father we knew. He attended  \nCatholic mass with my mother  \non Easter Sunday and was reading  \na detective novel in the back pew  \nwhen the priest, confessions over,  \nasked him if it was the good book.  \nMy dad, halfway through  \na Rex Stout, grinned. He was  \nan accountant but not  \na Catholic. His balance sheet  \nwas numerical. \"Who made you?\"  \n\"God made me to... \" After church  \nwind in trees makes shadows.\n\n6.\n\nWhat was it you called me?  \ncatholic or socialist.  \nCall me something less  \nsomething you know little of.  \nCall me baroque.  \nFellini says labels  \nshould go on suitcases.\n\nThe old house on eleventh  \nwith so many gimcracks  \non the lawn folks could not  \nfathom the lady maker, mad  \nthey said, their clothes  \nshrinking. Or a church  \nI saw as a kid, basement  \nonly, or the one tree left  \nwhere we necked in the shadow  \nof long-gone Rosary Hall we edging  \nnearer to finale but hanging  \nfire anteroom to living room  \nand if you talk\n\ndon't focus\n\nokay\n\n7.\n\nfor fear, I moved silently  \nfor fear, I said little  \nfor fear, I kept to myself  \nfor fear, on my knees in prayer  \nfor fear, never entering  \nfor fear, never entering  \nfor fear, no life but fear\n\n### **the body poems**\n\n1.\n\nthe body is sick  \nthe head says  \nthe body is growing  \nmosquito bites  \nankle aches belly boils  \nand the like  \njust to spite mind  \nno choice but to live above that damn body  \nhead whose clarity reviles  \nthe accident of body  \nwants to be left alone  \nfeels trapped in  \nerrant & bloody sullied  \nstink of flesh and bone\n\n2.\n\nnearing the end, the body failing,  \nyou learn modesty in all desires,  \nexcept the desire for desire,  \na modest thirst, the rose hip  \nor the lemon or the mint tea,  \nsettling inoffensively in that body\n\noh that was\n\nin its time so arrogant,  \nso easy in its words  \na soft-shoe body  \ndancing body  \nblues body\n\nahh\n\n3.\n\nwhen I feel awful, so far at least,  \nit's provisional, like a hangover  \nthat has always its slow end  \nencoded in the very libations  \nyou drank, and at this moment  \non a cool morning in October  \nthe fumes from my coffee  \npour over this page like  \nclouds in fast motion, so  \nlight a grey, so tentative,  \nso provisional\n\n4.\n\nthe worm under the skin  \nleaches colour, rolling  \nin the sun, basking in dark,  \nthe colour of cement, pocked,  \nscarred with cracks, worn,  \nwalked on, the day on the move,  \ncars talking in their boring way,  \nthe sun blueing the sky, the worm  \ngrinning over its first coffee of the day\n\n5. a modest carnality\n\nwhen I ordered a grande au lait  \non Denman the girl asked,  \ntwo or three shots of espresso  \nand I said two, I wasn't man  \nenough for three and she said  \nshe was, the modest carnality,  \nin the swing-walk of the waitress,  \nin the hug of greeting, in the  \namazing summer legs of the  \nserver girl, in this  \nlight touch and that,  \nin the communion of smiles,  \nthe perfect stranger, the  \ntouch on the shoulder, the purring  \nof invisible antennae,  \nin for a penny\n\n6. the dance\n\nthose times ago the dance  \nenveloping us none other  \nthan dance hip check  \neye trap sun  \nrising in a series of steps  \ndown the railroad hotel  \nbody wakes in the shank  \nof next afternoon the sun  \nimprinting itself in the brain  \nfind a beer start a  \nslow dance waiting  \nfor the music of your lover  \nto take you over  \nand over again\n\n### **the waters of life**\n\nwhich then are the waters of life?  \nat Fishing Lake on quiet days  \nthe water lolling about  \nlike a lazy fish\n\nat Windermere the bonfire  \nof evening at the end  \nof the lapping day\n\nat English Bay the eye  \ndrawn to the line  \nwhere water ends  \nand sky begins\n\nwaves washing ashore  \nunder the still water sky\n\n### **what then is free?**\n\nlike water to find your own level  \nwithout guilt  \nto fade from view  \nwhile others talk  \nto arrive at each day  \nwithout a plan but with,  \nlet us say, poems to write  \nin the sun of morning  \nto have an appointment for lunch  \nyou want to keep\n\n### **the rote beads**\n\nHe knew he should be bereft  \nat the prayers for his mother  \nit was what he felt but  \nwhen the old priest did  \nthe stations of the rosary  \nthe rote beads he felt  \nonly anger. His mother  \nhad said the trouble  \nwith the Catholic seniors  \nresidence was everybody  \nwas so damned religious.  \nAge eating at her  \nnever got all the way.\n\n### **the last day**\n\nhe forgot one wound  \nin another  \none ghastly presence  \nreplaced by another  \na kind of motion disease  \nfrom body part to body  \npart, head to belly to  \nlimb, this day the last day  \nof the freshly dead  \nhearing all the words  \nthat lay you under  \non the last ride  \nthe cars smelling each other  \ndown memorial drive  \nunder the elm roof  \nshredding the sun\n\nthen the day too is done\n\n### **I told you I could drive**\n\nI've become a brandy drinker  \na swirl at the bottom of the fat-bellied  \nglass, with the elegant scent\n\non a particular day  \nof no particular sort  \nmy mother, having received  \nher first driver's licence,  \naged 72, drove to my father's grave  \nand said, \"See, Cam, I told you I could drive,\"  \nused one tank of gas and sold us the car\n\nit's not a bad brandy  \nI should be drinking  \nrye and ginger  \nmy father's and his brother's  \nand Canada's national drink\n\na toast to our fallen comrades\n\n### **Ila**\n\nIla who tended  \nour kids with verve,  \nleaving us now behind  \nthe wind beating us  \ndown swallowing all  \ntestimonials, chilling us  \nto the bone, driving us  \nto the warm cars,  \nyou betcha,  \nIla in the cold, cold  \nground, brother took the soil  \ntemperature, four inches  \ndown it was only thirty  \ncan't plant, yep, clouds  \nbundling over the April grey  \nstubble, cars gone, you  \nbetcha\n\n### **a memory**\n\na memory  \nrattling in the head  \nthe face of one long  \ndead in the dream  \nin the city you've never been  \nthe ambulance in its white coat  \nits cargo dying  \nin the dead of winter\n\nthe ghost in the belly\n\nsong by Mabel Mercer  \nby Oscar Brown  \nby Nina Simone  \nis then and then\n\noh then\n\n### **oh then**\n\nwhile journeys end  \nthey begin again  \nall finales provisional  \nthe meeting over the walking begins  \nthe walking over the bus begins it's  \nhardly news years ago there was  \none sunset I decided would never  \nend colours blurring the evening  \nby the river whatever evening  \nit was remains still colours blurring  \nall going down and every  \nsundown is a replay  \nof that evening antennae  \nquivering\n\noh then\n\n### **gone large gone small**\n\nis it rooms or parks  \nwe desire wombs or space  \nlow roofs or sky high  \nthe friend in the room  \nat Ladbroke has died  \nthe room at Connaught  \nthe room where the stories  \nwere stored gone dark  \nthe city empties  \nstreet by street  \ngone large gone small\n\n### **the dust of just beginning**\n\nthe trees by the river yellowing  \nthe day without breeze  \ngolden coins hovering  \nin the blue sun  \nthe car tailing through  \nparkland or prairie  \na chill in the air  \nfirst taste of winter  \nwhite and cold  \nis the taste of the first melt  \non the south side of stores  \non Broadway where the low rise  \nencourages spring arriving  \nstreet by street everywhere  \nthe scent of dust  \nslow stepping spring  \nin the nostril  \nthe dust of just  \nbeginning\n\n## **FOR IT 'S LOVE**\n### **love poems**\n\n1. for it's love\n\nfor it's love owns the body  \nmakes it bend like a tree  \nin the wind all one way\n\nfor it's love owns the heart  \nmakes it drown in the flood  \nin the wild tide of love\n\nfor it's love owns the head  \neyes look where they will  \nthere are no thoughts but love\n\nthe day going on forever  \nall parts caught by love  \nnowhere to turn but love\n\n2. remembrance of love\n\nremembrance of love  \nthe internal collapse as you  \nwalked in the door in any  \nroom at all in church  \nthe icon my eyes  \nprayed to oh then  \nheart full heart sick  \nthere outside my self  \nmy self was standing\n\na sweater of let us say pink  \nwhere breasts like birds  \nin nests entered my nervous  \nsystem and I was a goner\n\nhalf in love with loss\n\n3. music in the veins\n\nmusic in the veins  \nfeet a life of their own  \nit was nina simone  \noh flo flo flo me la and on  \nthe move till dawn  \nthe partner at arm's length  \nor close enough to trade  \nbody parts all the way down  \nthe mind on half pay  \nwaiting for nina to say  \nit was time to flo me la  \nin our juice-laden bodies\n\nand who among us in the tall cafe  \nhas swallowed a ghost today\n\n4. the nameless heart\n\nthe nameless heart  \nnamed heart drowns under  \nthe flood and reaches for dry  \nland receding faster than grasp\n\nthe heart in deep water  \npumping for all it's worth  \nits aorta and its long  \ntentacles like a winter  \nelm tree is desperate  \nfor the sign  \nfor the saving grace  \nfor the word from you oh lady  \nday to save this nameless  \nheart of mine\n\n5. love can be\n\nlove can be  \nso muted a solo  \nso sweet a duet  \ndaily the jazz trio  \nin the late-night lounge  \nthe talking going on  \nat the low tables  \nround as wafers  \nat odd times the bar  \ngone quiet the piano  \nso fluent in the night  \nor driving for miles  \non the ease of the wide  \nfour-lane highway joining  \nthe rv park on full hookup  \nfor the night\n\n### **mall**\n\nin the mall the family of man  \nhas gone forth and multiplied  \nand the cars of the family of man  \nhave gone forth and multiplied  \nbusy days in the  \nfabricated world\n\nenough of waiting, yes,  \nthe buying, selling, the walking  \nand waiting, cars in their carspots,  \nthe endless lines of desires,  \nthe feet dying, the sulking,  \nthe slow fire of anger,  \ntrying to stay sane,  \nthe sun on high and where  \nis a stranger to start  \nthe whole stupid bloody thing  \nall over again\n\n### **body poems II, the black poet**\n\n1. narcissus\n\ntranslation is hard  \nwalking is easy\n\nlove is a long line  \nand kissing is shorthand\n\nit's a sweet thing to say  \nI wrote poetry in Spain  \nI wish I could say that\n\nblack is the colour  \nof my true love's heart\n\nin the magic room  \nwe slept well  \nyou went bloody wild,  \nNarcissus,  \nbloody wild,  \nyou said  \nproud as hell\n\n2. kill speak\n\nkill speak, she said,  \nkill speak and take me to bed  \noh I'm too timid, I said,  \ntoo timid to take you to bed\n\nlife, she said, is so bloody bad  \nI had a devil for a dad  \nI'm too timid I said,  \ntoo timid to take you to bed\n\nI've a body a devil can love  \na body ripe for that guy up above  \nI've a mind that believes in God  \nand a body that tries to be good\n\nI've two breasts and a hungry cunt  \ncome aboard and join in the hunt  \nsweet Lord protect me I ask you please  \nsweet Lord before I fall to my knees\n\nkill speak me baby, she said,  \nkill speak and take me to bed  \nI never learned kill speak, he said,  \nshall I kill self to enter your bed?\n\nyes, she said, yes  \nwell, he said, hmm\n\n3. love is the knife\n\nlove is the knife  \nthat cuts to the bone  \noh a fine knife  \nan old lacerator  \nplunged into ugly love,  \nshe said, as if saying it  \nmade it more\n\n4. fantasies\n\nthe fantasies  \ngorge themselves on the barren  \nbed fantasy populates\n\nevery which way\n\nwhile plain day is a\n\nwindy blue\n\nthe first red finger of tulip  \nin the smelly mulch reborning  \nin the wrinkled leaves\n\nfirst splash\n\n5. the ease of wit\n\nremember the days  \nof the ease of wit  \nthe flowing in and out  \nfocus the faux pas,  \nsurface in lieu  \nof the dead serious,  \nthe art of the interrupt  \ndeath to tirade but  \nall honour to the solo  \nas for love give cole  \nporter the last aper\u00e7u\n\n## **JOURNEY MAN**\n### **a fearful carnality**\n\ntoday the clouds are aesthetic  \nwhite & thin, elegant &  \noptimistic yet dissolve  \ninto sun in a fearful  \ncarnality you walking or  \ndriving under them as if you  \nwere important your eyes  \ngorged the sun in that high-lit  \nscam dressing you down  \nlashing you to the day  \nsun-lashed back forty lashes  \nif you please to bring the  \ndead to light  \nbilly & anne kitty & cam  \nthe light so thick  \nby midday death  \nwas in decay\n\n### **lost and found**\n\nlost on this road  \nwe know like the back of our hand  \nunder clouds that cannot hold still  \ncan hold no shape at all  \nof course we are expected  \nthe scotch waiting in the cupboard hall  \nbut nobody knows where we are  \non the highway like any other  \nin the car that's any car  \non some wave or other on our way  \nto the old port of call\n\n### **my road**\n\nthis is my road  \nno one else wants it  \nthe clouds over my road  \nare first-rate clouds  \nthe fields by my road  \nare Olympic-sized fields  \nand I'm on the old way  \nto discover the exact feeling  \nof driving in the sweet spring  \non the great plains  \nthe road free of language  \nthe signs sparse  \nlife thin  \ncars fast\n\nthe high sun  \nin passing gear  \non my road\n\nthat runs and  \nruns\n\n### **the first day**\n\nIn the beginning there was delay  \nIn the beginning rain was already falling  \nIn the beginning the cell phone worked overtime  \nIn the beginning the last wash was done  \nthe last flowers planted  \nIn the beginning the travelling bags  \noverflowed with anticipation  \nand the dog left home in the company of a girl  \nworth wagging her tail over  \nIn the beginning the house resumed its silence  \nIn the beginning the van groaned  \nthe baby pushed all the buttons  \nand the clock quit  \nIn the beginning the rain clouds came in layers  \nalready in the coastal weather  \nwe were travelling to  \nIn the beginning the fields stretched forever  \nand the kids played pick-a-number  \nIn the beginning the black highway  \ndrew us onward  \nIn the beginning we were our destination  \nIn the middle a patch of blue sky appeared  \nWe were all reassured  \nIn the middle he saw the past  \nthe Scamp the Dart the Rambler the Chevrolet  \nHis eyes were the same  \nThey saw what was there  \nThe highways were wider smoother  \nIn the middle the van  \ndoes not break down  \nin the rubber-tire world  \nIn the middle he thought  \nof whiskey at the terminus  \nHe thought no further ahead than that  \nIn the middle the kids  \nran round the park in Vegreville  \nand invented happiness  \nIn the middle the sun came out  \nHe was sunloving and mindnumb  \nnow time the only time  \njust like last time  \nIn the middle the sun  \nrambled all over the place  \nand they sang Blue Skies  \nIn the middle the baby yelled \"cows cows\"  \nand became the cowboy  \nIn the middle he wanted that other time  \nHe wanted that other time  \nthe blue Chevrolet just before  \ndeath entered the world  \nOh he wanted that other time  \nToday he wanted today  \nalready lonely for today  \nand our vanguard group of seven  \nin the beginning the middle and the end\n\nIn the end they made four correct turns  \nIn the end they sat on the deck  \nin the full sun  \nNow only the earth moved  \nIn the end the barbecue worked on their behalf  \nIn the end they drove for more beer  \nThe evening was long at the end of June  \nIn the end the clouds massed, ran, thinned,  \ngrew ramparts, washboards, were white,  \nwere grey, were yellow with sundown,  \nstreaked with lightning, poured rain  \nIn the end the cards appeared  \nIn the end the television captured the children  \nIn the end our eyes collapsed and we dreamt  \ntravelling invisible roads  \nAt the end of the first day\n\n### **where are we going next year?**\n\nbad highway out of Richdale  \nmakes Saskatchewan drivers  \nfeel right at home\n\nthe Hanna escarpment says  \nthe days of the prairie  \nare numbered  \nthe valley of the Red Deer  \nis at hand  \nclouds in our heaven cool  \nwith mountain air  \nwe surround our old friend  \non two sides  \nand we are where we  \nwere meant to be\n\nGraham, give me my pillow.  \nIt's a chicken.  \nGive me my chicken.  \nIt's a turkey.  \nGraham!  \nYou have to pay fifteen, thirty,  \neighty thousand dollars.  \nBooger brain.  \nYou're a barney.  \nYou're a guinea pig.  \nI'm hungry.  \nWhere are we going next year?\n\n### **to the tax man**\n\nrolling down the railroad line  \na mile-long container train  \nsaving all that gasoline  \nspent on the four-lane highway  \nby trucks, cars, and us today  \nif I write enough to mean  \nwe'll deduct the gasoline  \ndon't need a regular metre  \nto deduct another litre  \nbut hell if I write enough  \nI'll take the motel cost off  \nhey, hey, this land is our land  \ntravelling on the tax man\n\n### **Journey man**\n\nI am a journey man  \non the low road high road  \nthe flags of the clouds  \nblowing upstream all a-flutter  \nmy heart the journey man  \non the known roads  \nfresh this morning as  \nnew-baked sun\n\n### **you dream the poem**\n\naround the corner the perfect  \nvalley  \nthe farmhouse with a silver roof  \nburning  \nthe easy highway the sun  \non your leg\n\nthe highway is the narrative cutting through  \nthe chaos of trees  \nfive days on the road in your own  \nhome  \neach highway a number each curve  \nnameless  \nto Kamloops or Cache Creek or the legendary  \nYakk  \ngoing the speed limit plus  \nfive  \nin a meeting in a basement room one window only  \nyou dream the poem of car  \nand road\n\n### **tough terrain**\n\nin the new town  \non the wrong road  \nthe women shopping  \nthe sky lowering a woman  \nwith white breasts drives  \noff in a four-wheel the day  \nwaiting for her on the edge  \nof age let the story begin  \nthe story of giving up  \nsmoking or beating cancer  \nor the story of love which  \ncarries her away like  \na ferocious four-wheel  \nin tough terrain\n\n### **mind riding**\n\nin the room and travelling too  \nyou look me in the eye and  \nsee nothing at all  \nthis random mind riding  \non highways of no man's devising  \nno man can map mind  \nwhich has itself forgotten  \nwhere it is so wandering it is  \nso absent as if all walls  \nwere open road the eye  \nclosed shall go  \nwhere it will\n\n### **Blue River**\n\n\"You're so old  \nyou'll forget  \nwhere you come from.\"  \nNope.  \nI come from youth.  \n\"You're so old  \nyou'll forget  \nwhere you're going to.\"  \nI'm going to Blue River  \nby brown cows and green trees  \nrock face and fast river.\n\n## **WE ARE THE ECHO**\n### **kelly 's alley**\n\nhe walked into the wrong alley  \nthat was kelly's alley  \ndaydreaming into danger  \nfor the day of the bully  \nwas at hand by the broken  \ngarage that smelled of poop  \neyes opening legs like jelly  \n\"you come back I'll break yr fuckin'  \nhead\" backing out in terror  \nend of the alley in sight tremor  \nof hope\n\n### **hunting rabbits**\n\nhunting rabbits with a .22  \npulling the barbed wire apart  \nto crawl through  \ninto the hilly bush  \nno crop no cow no farmhouse  \neyes alive watching for  \nmovement in the brush  \nbang, bang, got him!  \ngot him! said Ted,  \ndead rabbit brown  \ndark blood stain  \nlying in the spiky grass  \nour eyes alive watching again  \nfor movement in the low bush\n\n### **on the line**\n\nto work the line  \nis to live in the mind  \nthe body repeating itself  \nhand over hand over hand  \nthe mind on weekend\n\nto work the line  \nis to become the machine  \nfeed it with bottles at one end  \npack them out the other  \nat night grease the nipples  \nthe bottle-washing machine  \non general drone the labelling  \nmachine on steady clunk clank  \nwe the most quiet part of  \nthe machine each in his own  \nreverie of lawn or lake or love  \nall bodies equal  \non the line\n\n### **summer south**\n\npicking cherries in the Okanagan  \ntall pointed ladders lift you  \nto the long view, lake, hills,  \nroadway flowing with cars,  \na kind of power to be so tall  \nin the cave of a tree, clouds  \nstreaming over the brown hills,  \nyou pause, for the moment, the work  \non hold, of course it's summer,  \nsummer south, obstreperously  \nsummer, in a place you are  \nand aren't and you think  \nhave I given god the slip  \nperched high in the tree  \nthe purple cherries  \nbursting in the mouth\n\n### **Cousin Lynda**\n\nin Vernon Cousin Lynda was on the hunt  \nfor a hundred pounds of cucumbers  \na hundred pounds of tomatoes  \nfor dill pickles for relish  \npick them herself at 45\u00a2 a pound  \nto lay away the preserves  \npreserving herself publishing  \npickles I said, you're  \nyour mother and she said  \nyes\n\n### **water boy**\n\nthe body of the boy leaps  \ninto deep water dives  \nunder the body of water  \nfrog boy water boy  \ndanger boy no fears mother  \na blur way up there sound  \nechoes in the sweet  \ndeep water\n\n### **we are the echo**\n\nfaces echo  \nskip a father or so  \nthe nose of the clan ringrose  \nis on the rampage  \nor the black black hair  \nof the kerrs from god knows  \nwhere all the old photos  \nunnamed stiff with time  \nor I see in the bar  \nan echo of doug  \ndead a dozen years ago  \nbecause they were after him  \nthe gamblers paranoia awful  \nas dog piss in the snow  \nthe collie like the collie  \nwho whelped her all in the face  \nthe tail markings high and low  \nof the dead we are the echo\n\n### **this is the day nothing happens**\n\nthis is the day nothing happens  \na meeting to say we've come to the end  \nof the agenda for today soccer on tv  \nand hotspur won this is the day  \nnothing happens listening  \nfor the first time to a cd  \nby mark this is the day  \nnothing happens a son says  \nhow his interview went  \nexcellent a sister-in-law  \nhad half a lung removed  \nand is under morphine  \nthis is the day nothing happens  \nthis is the day nothing happens  \nto you\n\n### **body idling over**\n\nI bike to the coffee shop  \nsun stepping in the window  \nbody idling over  \nart on the wall  \nmusic on the sound system  \n\"Oh my, oh my, why must I explain?\"  \nthe dark tenor sings girl  \nin a ball cap walks past  \nthe coffee oasis  \nand there's that blue sky  \nburning and pine trees  \ntall as the school  \nnewspapers to tell of the  \nparent trap Cuba's secrets  \npoverty in Manila and then  \nI dream body idling over  \nto be on the road at Hope  \nwith all its intersections\n\n## **TO ESCAPE ONTO PAPER**\n### **into the woods**\n\ninto the woods walked the man  \ndedicated to ecology, and he walked in  \nwithout a pen, I said, without a pen!  \nincredulous that joy should have no record,  \nas if the self would boil and bubble  \nunrecorded, all of life clammering  \nto escape onto paper\n\n### **word off**\n\nshe walks in and desolates  \nthe day, a particular flavour  \nof face, a blonde face, greyish  \nhair, loose blue shirt,  \npressed jeans, wordless,  \nslender as a knife, eyes  \nlike weapons me  \nworking hard to word  \nher off\n\n### **victim of story**\n\nPhiloctetes remains true  \nto his pain, the festering  \nleg, for years abandoned  \non the rocky island, and no  \nwords of Odysseus the political,  \nno promise of fame, of victory,  \nthey are but words to the pain  \nwhich is his, which is  \nwho he has become, but myth  \nmore strong than self  \nand the god from the machine  \ndrove Philoctetes to Troy,  \nanother victim of story\n\n### **at the end of his fifth whiskey**\n\nat the end of his fifth whiskey  \nhe became certain he was certain  \nand those who spoke of the price  \nof gold of shares in Nortel of the  \nCayman Islands as grand tax haven were  \nhe was certain in his brilliant head  \ndead wrong and dead dead\n\n### **the dead tired poem**\n\non the day of the hangover  \non the day of the hollow chest  \non the day of the body in charge  \non the day of the rapid cloud  \non the day of the tossed trees  \nin full bloom  \non the day of the dead eye  \nbody working to hold its head up  \neveryone else exuding health  \nlike a plague of grin  \na day on which the first beer  \nlay on the horizon  \non the dead day of  \nthe dead tired poem\n\n### **the anonymous clouds**\n\nthere is no centre on  \na glum day the anonymous clouds  \nover the river whose cross-hatched  \nwaves cannot be named  \none from another  \nor the weeds the bush the trees\n\nsit on my bench for a minute  \nor by some other method  \nof measure shadows moving  \nin the short-hair grass\n\n### **never start from here**\n\nin the metropole  \nthe weight of opinion\n\nwe live in a thin country  \npoets slip out sideways\n\ntraffic is heavy and art  \nbrief poem inert  \nlives in an unmarked  \ngrave never start  \nfrom here\n\n### **Samuel Beckett**\n\nI sat next to Samuel Beckett  \nHe didn't say a word  \nI didn't say a word  \nHe said nothing  \nI said nothing  \nHe had a pint of Guinness  \nI had a gill of whiskey  \nHe read nothing  \nI read a script  \nI looked up  \nHe lifted his glass to his lips  \nI think it was Beckett  \nHis face looked like  \nan Ordinance Survey map  \nHe cleared his throat once  \nI drank  \nHe drank  \nshifted in his chair once  \nand it creaked  \nHe didn't say a word  \nIt must have been Beckett\n**\u00b6** This book is set in the Thesis Sans family,  \na typeface designed by Luc(as) de Groot in 1994.\n\n## **acknowledgement [s]**\n\nI would like to thank the third cloud  \non the left of the elm closest to the van,  \nthe cloud that is unravelling, the elm  \nthat twists in its own way.\n\nThanks also to the coffee house and bar  \nthat provide tables for a small fee,  \nand to the ideological drugs I've  \ntaken and the slow recovery from same.\n\nThanks to everyone who looks like  \nthey do, and to the labyrinth of  \nthe hawthorn, the garden, the  \nliving room and the bedroom.\n\nThanks for my friends, who read  \nearlier drafts, though  \nthe last draft is always best.\n\n[ PS. I thank all those who helped with these words:  \nfrom a Saskatoon Writer's Group, Anne Szumigalski,  \nJohn Clarke, David Carpenter, Dwayne Brenna; from  \nAU Press, Walter Hildebrandt, who asked for the  \nmanuscript, and Pamela MacFarland Holway,  \nwho fine tuned it. Ta. ]\n\n## **about the author**\n\nMax Frith is an author without  \na single publication, no novel, no  \nbook of poems, no play, but he has read  \nmany books and attends movies  \nregularly. He has entered  \nmany contests and writes at  \nodd moments in a school  \nscribbler. He won an eight-  \nminute poetry contest, and  \npursues women at a  \nrespectful distance.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nAlso by Teri Terry\n\nSLATED\n\nFRACTURED\n\nNANCY PAULSEN BOOKS\n\nPublished by the Penguin Group\n\nPenguin Group (USA) LLC\n\n375 Hudson Street, New York, NY 10014\n\nUSA | Canada | UK | Ireland | Australia\n\nNew Zealand | India | South Africa | China\n\npenguin.com\n\nA Penguin Random House Company\n\nCopyright \u00a9 2014 by Teri Terry.\n\nPenguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nTerry, Teri. Shattered \/ Teri Terry. pages cm.\u2014 (Slated trilogy ; 3)\n\nSummary: \"Sporting a new identity and desperate to fill in the blank spaces of her life pre-Slating, Kyla heads to a remote mountain town to try to reunite with the birth mother she was kidnapped from as a child. There she is hoping all the pieces of her life will come together and she can finally take charge of her own future\"\u2014Provided by publisher.\n\n[1. Memory\u2014Fiction. 2. Identity\u2014Fiction. 3. Government, Resistance to\u2014Fiction. 4. High schools\u2014Fiction. 5. Schools\u2014Fiction. 6. England\u2014Fiction. 7. Science fiction.] I. Title.\n\nPZ7.T2815Sh 2014 [Fic]\u2014dc23 2013041575\n\nISBN 978-1-101-59173-4\n\nThe publisher does not have any control over and does not assume any responsibility for third-party websites or their content.\n\nVersion_1\nFor my mother\n\n# Contents\n\nAlso by Teri Terry\n\nTitle Page\n\nCopyright\n\nDedication\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n\nChapter 41\n\nChapter 42\n\nEpilogue\n\nAcknowledgments\n1\n\nIT DOESN'T LOOK LIKE MUCH FROM THE OUTSIDE. BUT WHAT you get outside is often like that. People, especially, can be so different from what you can see that you'd never guess what goes on in their secret places. What they are capable of. In my case, what lurked within was so well hidden even I didn't know about it.\n\nAiden parks the car along the side of the run-down building. He glances at me. \"Don't look so scared, Kyla.\"\n\n\"I'm not,\" I start to object, but then I glance at the road, and all at once, I am. \"Lorders,\" I hiss, and scrunch down in the seat. A black van pulls in behind us, blocking us in. Leaden dread pools in my veins, holds me still and numb even as everything inside is screaming _run._ The fear takes me back: another time, another Lorder. Coulson. The gun in his hand, pointed at me, and then\u2014\n\n_Bang!_\n\nKatran's blood. A sea of hot red that covered us both, and took my friend away forever. So like my father's death years ago that it wrenched up that most buried memory. Both dead. Both my fault.\n\nAiden puts a hand over mine, one worried eye on the mirror and the van, one on me. A door opens; someone steps out. Not dressed in Lorder black? A slight figure, a woman, hat pulled down low to shield her face. She walks to the door of the building. It opens from the inside, and she disappears through it.\n\n\"Look at me, Kyla,\" Aiden says, his voice calm, reassuring, and I tear my eyes away from the van behind us. \"There is nothing to worry about; just don't draw their attention.\" He twists in the driver's seat, slips his arms around me and tries to pull me close, but I'm rigid with fear. \"Play along,\" he says, and I force my body to relax into his. He murmurs into my hair, \"Just giving them a reason why we were lingering. In case they're getting curious.\"\n\nI breathe in slow. _They're not after me._ They'll go away now. _They're not after me._ And then I'm clinging to Aiden, and his arms wrap around me even tighter. There are vehicle sounds behind; tires crunch on gravel. And keep going.\n\n\"They're gone,\" Aiden says, but he doesn't let go. And the relief is so strong that I sag against him, bury my face in his chest. His heart beats fast, drumming a _thump-thump_ of safety, warmth, and something _else._\n\nBut this is wrong. He's not Ben.\n\nMy fear is replaced by embarrassment, then anger: anger at myself. I pull away. How could I be such a total wimp and let them get to me like that? How could I cling to Aiden just because I was scared? And I remember what he said on the way earlier: Lorders come here. Lorders, government officials, and their families. People with money and power who can make others look away and keep quiet. That woman is probably a Lorder's wife. She is probably here for the same reason as me. I flush.\n\nAiden's blue eyes are warm, concerned. \"Are you sure you can pull this off, Kyla?\"\n\n\"Yes. Of course I can. And I thought you weren't supposed to call me that anymore.\"\n\n\"It'd be easier if you'd make up your mind what your new name is going to be.\"\n\nI don't say anything, because I sort of have, but I don't want to share it yet. I'm not sure he'll like it.\n\n\"Walk in like you own the place, and no one will look at you twice. It's all anonymous.\"\n\n\"Okay.\"\n\n\"Best get going before anyone else comes.\"\n\nMore Lorders?\n\nI open the car door, step out. It is cold, a gray January day. The chill is reason enough for the scarf wrapped around my head, obscuring an identity that will soon change. I square my shoulders, walk to the door. It opens, and I step inside.\n\nMy eyes widen; my feet almost falter until I remember: _Walk like I own the place._ This shiny place, with enormous, plush chairs, soft music, and a smiling nurse? A guard stands discrete in the corner. The woman we saw come out of the Lorder van moments ago is ensconced in a chair with a wineglass in one hand.\n\nThe nurse approaches, smiles. \"Welcome. Do you know your number?\"\n\n\"7162,\" I say, the number Aiden gave me earlier. Even though my name is best kept quiet, I'm not sure I like being known by a number, not after being Slated. Not after having a Levo around my wrist with my number carved into it, classifying me as a criminal for all to see. It's gone now; there are no visible marks left behind, but the scars remain.\n\nThe nurse checks a handheld screen, smiles again. \"Have a seat for a moment. Your IMET consultant will be with you very soon.\"\n\nI sit, startled when the chair moves, adjusts to my body. IMET: Image Enhancement Technology. Barely whispered about, hellishly expensive, and totally illegal. I'm here courtesy of favors owed to Aiden's organization, MIA. MIA may stand for Missing in Action, but it turns out they don't just find missing people and campaign for the truth about the Lorders to be revealed. Turns out they also sneak people out of the UK who need to disappear, and others in at the same time: IMET consultants who know a good black market opportunity when they see one.\n\nThe woman in the other chair turns toward me. She is attractive, fifty or so. If the rumors are true, she'll look twenty years younger before she leaves this place. There is an inquisitive glint to her eyes, a _What are you here for?_ look. I ignore her.\n\nA door opens, and footsteps approach. She starts to get up, but the steps continue past her, and a man stops in front of me. A doctor? But not like any doctor I've seen before. He is in scrubs, but they are a bright purple shimmery fabric. It matches his streaked hair and purple eyes to perfection, unnatural shimmer and all.\n\nHe holds out both hands, helps me up, and air-kisses my cheeks. \"Hello, darling. I'm Doc de Jour, but you can call me DJ. This way.\" His words are a lilting drawl, an unfamiliar accent: Irish?\n\nI follow him and suppress a smirk at the indignant look on the waiting woman's face. She must wonder who I am, why I take precedence. If she only knew.\n\n_If_ she knew, it'd go straight to her Lorder husband.\n\n\u2022 \u2022 \u2022\n\nDoc de Jour is disappointed in me. \"Are you _sure_ that is all you want done? Hair. In _brown._ \" He says it like brown hair is the ultimate crime of mediocrity. But blending in is what I need.\n\n\"Yes: brown.\"\n\nHe sighs. \"Such lovely hair you have, and so hard to match. Like sunshine on early daffodils 12. With highlights 9.\" He runs his fingers through it, a measuring look in his eyes, like he is copying it for his next patient. Then he studies my face. \"How about eye color?\"\n\n\"No. I like them green.\"\n\n\"They're distinctive; it's a risk,\" he says, and my eyes widen. What does he know? He winks. \"They _are_ an interesting shade. Almost apple-green 26, but more intense,\" he says, then spins the chair I'm sitting in around and looks me up and down. I squirm. \"Wouldn't you like to be taller?\"\n\nI raise an eyebrow. \"You can do that?\"\n\n\"Of course. It'll take a while, though.\"\n\nI bristle. \"What is wrong with my height?\"\n\n\"Nothing. If you don't mind jumping to see over things.\"\n\n\"Just the hair.\"\n\n\"Brown. You know IMET is accelerated gene-tech: It is permanent. Brown hair forever. It'll grow that way; no more blondy ever again, unless you come back to see me.\"\n\nHe hands me a mirror, and I look in it. So weird to think next time I do this I won't see the hair I've always had. The color is okay, I guess, but it is so fine\u2014I always wanted thicker hair. Like Amy's gorgeous dark hair, the first thing I'd noticed about my new sister when I was assigned to live with her family as a new Slated, just months ago. \"Wait a minute. I wonder if . . .\"\n\nHe leans down and stares into my eyes with his purple ones. They're hard not to look at. \"Yes?\"\n\n\"Can you make it longer? And thicker. Maybe . . . some streaks in it. Not anything weird: natural-looking.\"\n\nHe claps his hands. \"Consider it done.\"\n\n\u2022 \u2022 \u2022\n\nLater I'm told to lie back on a table that is like the chairs in the waiting area: It molds and grips my body. Flutters of panic fight to keep me awake. Is this how it was when I was Slated? Then, I had no choice; I saw the file photograph. I was tied to a table like any other criminal. The Lorders and their surgery stole my memories, put a chip in my brain that could've killed me before my Levo was taken off. This isn't the same. This is just hair. And it is my choice: I don't _have_ to do this.\n\nThere is faint music. Everything is misty and vague, and my eyes start to close.\n\nJust hair . . . but it is _the_ hair Ben slipped his fingers through when he kissed me.\n\nSince the Lorders took him away and erased his memory, he doesn't know who I am anymore. But what if he fights it, fights what the Lorders have done to him, and starts to remember? Starts to understand why I'm his _dream girl._ What then? He'll never find me if I look different.\n\nI swallow, struggle to form words, to tell them to stop, I changed my mind.\n\nBen . . .\n\nFaces blur in and out and vanish.\n\n\u2022 \u2022 \u2022\n\n_We run. Side by side in the night, but Ben's long legs beat out a slower measure than mine. It is raining, but we don't care. Up a dark hill now, he slips ahead; the narrow path cut into rock is running with water. Soon we're soaked and covered in mud. He's laughing when he reaches the top, and raises his hands to the sky as the rain pounds harder._\n\n_\"Ben!\" I reach up, slip my arms around him, and pull him under a tree, then burrow into his warmth._\n\n_But something isn't right._\n\n_\"Ben?\" I pull away a little, look into familiar eyes: brown like melted chocolate, shot through with warm glints. Puzzled eyes. \"What is it?\"_\n\n_He shakes his head, pushes me away. \"I don't understand.\"_\n\n_\"What?\"_\n\n_\"I thought I knew you, but I don't. Do I?\"_\n\n_\"It's me! It's . . .\" My voice trails away. I panic inside, casting about for a name, not just any name, but_ my _name. Who am I, really?_\n\n_He shakes his head, walks away. Runs up the path and is gone._\n\n_I sag against the tree. What now? Should I run after him, just so he can deny me again? Or go back the other way, alone._\n\n_The sky lights up: A blinding flash dazzles my eyes, shows the trees and pounding rain. Before darkness returns, a tremendous crash shakes into my bones._\n\n_While the rest of me whirls with pain at Ben's departure, some part of my brain processes: Standing under a tree in a thunderstorm is dangerous._\n\n_But who am I, really? Until I can answer, I don't know which way to go._\n2\n\nDAYS LATER, DJ HANDS ME A MIRROR FOR THE FIRST TIME. I stare, then reach out gingerly with my fingers. The hair\u2014my hair\u2014even _feels_ different, foreign. I don't look like me anymore. Of course, that is kind of the point. A rich brown it may be, but shimmering with golden highlights. They bring out the green of my eyes so much that I stare at them suspiciously, wondering if DJ had been unable to resist adding some enhancement to them as well, but decide they are still the eyes I was born with. My hair is not, not in any respect: It's silky, thick, halfway down my back. I wince as I turn my head: It's heavy, so much so it hurts. It'll take some getting used to.\n\n\"Your scalp will be tender for a while.\" DJ holds up a small bottle. \"Painkillers, no more than two a day for a week. So . . . ?\"\n\nI tear my eyes away from the mirror, and look up at him. \"So?\"\n\n\"Do you like what you see?\"\n\nI smile broadly. \"I like.\"\n\n\"One final touch is needed, I think.\" DJ places a finger on either side of my chin, tilts my face up and stares at my eyes. He stares long enough for it to be uncomfortable if it were anyone else, but somehow it isn't with him. It is like he is measuring, and assessing\u2014what? The skin, the bone structure supporting it, the tissues, almost as though if he stares long enough he can see the individual cells and the genes inside them. He nods to himself, then turns to a cabinet with many drawers; he opens one, then another, and draws something out, then holds it toward me. Something low-tech.\n\n\"Glasses? I don't need glasses.\"\n\n\"Trust me. Put them on,\" he says. I do, and look in the mirror. Startled, I gasp; look back at him, then again at the mirror.\n\nThe frames are a delicate silvery-gray metal and suit my face as if made for it, but that isn't what made me gasp: It's my eyes. The lenses are completely clear, yet somehow, I am changed. My eyes aren't green anymore. More a blue-gray. I turn my head side to side, take the glasses off, put them back on again. Study myself like looking at a stranger. This dark-haired girl is _other._ She looks older, too. No one would recognize her. Not just Ben; I could walk past Mum and Amy in the street, and they'd be none the wiser.\n\n\"That's amazing. You're amazing.\"\n\n\"Why, yes, I am.\" He smiles. \"And this technology\"\u2014he touches the glasses\u2014\"isn't known in the UK, at least not yet. So wearing them shouldn't arouse any suspicions.\" He spins my chair around so we are facing each other again. \"So. The green-eyed blond girl is gone, replaced by a more sophisticated version, one who can pass for the eighteen you need for ID and travel if necessary. What is next for you?\" I hesitate, and he laughs. \"Keep your secrets. I hope\u2014no, I am sure\u2014we will cross paths again.\"\n\n\"Thanks for everything.\"\n\nHe tilts his head, something in his eyes still measuring, assessing.\n\n\"What is it?\"\n\nHe shakes his head. \"Nothing, and everything. Time for you to go.\" He holds the door open. As I step through it, he adds, \"Tell Aiden I need to see him.\"\n\n\u2022 \u2022 \u2022\n\nLater that day I'm in a small room hidden in the back of a factory. A dark room where new identities are forged. New lives begin.\n\n\"Name?\" an unidentified man asks.\n\nThis is the moment. I'm not Lucy, the name given when I was born. I'm not Rain, the name I eventually chose for myself after I was taken by Nico and his Antigovernment Terrorists\u2014Free UK, as he called them\u2014and shaped to be their weapon against the Lorders. I'm not Kyla, the name picked for me at the hospital after I was caught and Slated for being an AGT terrorist.\n\nI am who I choose to be.\n\n\"Name?\" the question is repeated.\n\nI am none of them. I am all of them.\n\n\"Riley. Riley Kain,\" I answer, one name that combines them all.\n\nSoon I clutch a forged ID card in my hand: a dark-haired, gray-eyed eighteen-year-old cleared to travel and live her own life: Riley Kain.\n\nWhat life do I choose to live?\n3\n\nTHE BUS RATTLES DOWN CITY, THEN COUNTRY, ROADS. NO more hiding needed with my new ID and new look, and I'd insisted on traveling back from London by myself. But who could have known an AGT bomb would be found on one of the London trains today, the entire network shut down while they were all checked? So the bus was the only option. Every jar of the road reverberates through my sore head, and I have to hold my hands together to stop them from gathering my new hair up to support its weight.\n\nFields, farms, and villages rush past, become familiar. We're nearing the village I lived in with Mum and Amy, the one I left the day Nico and his remote-detonated AGT bomb nearly killed me. I ran away to hide at Mac's. Mac is a friend, yes, and one I trust, but he hasn't known me for long to take such a risk. He is the cousin of Amy's boyfriend, and somehow involved with Aiden and MIA. Without knowing or insisting on knowing all that happened\u2014what I've done, or why\u2014he and Aiden were there, offering help. A safe place to hide. A chance at a new life. The old one with Mum and Amy ended just a short time ago, but already it feels distant, another life slipping away.\n\nA long black car approaches from the other direction, a coffin in the back, and traffic slows to a crawl on both sides. A second black car follows the hearse. It has two occupants, arms linked: one young, with dark hair and skin, one older and pale. In a flash, they are gone. My eyes widen.\n\nThat was Mum and Amy.\n\n\u2022 \u2022 \u2022\n\nThe bus stops near the end of Mac's long lane, and I rush up it on foot. Most of me is grappling with what I saw: Whose funeral were they attending? A deep feeling of dread settles inside, while some other part of my mind is distracted, processing that the air and sky have that heavy chill about them that says _snow,_ but I've never seen snow, and wonder why I feel this sense of expectation. While there must have been snow when I was Lucy, a child growing up in the Lake District, her memories were Slated away.\n\nAnother bend and Mac's house appears: a lone building on a lonely lane. From this vantage point, a small sliver of white over the high back gate says a van is there: Aiden's?\n\nI'm expected. A curtain moves, and the door opens as I reach it: Mac.\n\n\"Wow. Is that really you, Kyla?\"\n\n\"It's Riley now,\" I say, going in and wincing as I take off my hat and scarf and chuck them on a chair.\n\nAiden is there now and sees my face. \"I told you I could have picked you up. Are you all right?\"\n\nI shrug and go past them to the computer down the hall. Skye, Ben's dog, tries to jump up and lick my face, but I give her a quick pat and push her back. Mac's computer is an illegal; it isn't government-monitored. I meant to do a general search for local news on the off chance that funeral was picked up, but something makes me go to MIA's website first.\n\nLucy Connor, missing from her home in Keswick since age ten. Recently reported found: I had pushed the button on the screen myself, hoping to find a way back to who I was all those years ago, through whoever reported me missing. Now clearly marked _Deceased._ I stare at the screen, unable to process the word.\n\nA hand touches my shoulder. \"You're looking well for a dead person. I like the new hair,\" Mac says.\n\nI turn; Aiden has also followed and stands next to him. There is something in his face. \"You _knew,_ \" I hiss.\n\nHe says nothing, and that says it all.\n\n\"Why deceased?\"\n\n\"You are. Officially,\" Aiden says. \"According to government records, you died when a bomb exploded at your assigned home. Lorders have reported you as dead.\"\n\n\"But there was no body. Lorders wouldn't be fooled. My bus went past a funeral procession on the way here; Mum and Amy were following the hearse. Was that _my_ funeral?\"\n\n\"I'm sorry. I didn't know it was today.\"\n\n\"But you _knew._ That they think I'm dead.\" I'm angry, but I'm also confused. \"Why would the Lorders say I died?\"\n\n\"Perhaps they don't want to admit they don't know what happened to you?\" Mac suggests.\n\n\"I don't understand why the Lorders would do that.\"\n\nAiden tilts his head to one side. He's not sure, either; the uncertainty is in his eyes. \"Perhaps they don't want to admit they failed,\" he says. Aiden had assumed the bomb at our house had been Lorder, as payback for my role in helping Ben cut off his Levo, and I never set him straight. He doesn't know the dangerous double game I'd played, for the Lorders and Nico's AGT. Guilt twists inside at secrets kept, for help repaid by silence. But he keeps his secrets, too.\n\nMy eyes fill with tears. \"I can't leave Mum and Amy thinking I died in that explosion. I can't.\"\n\nAiden sits next to me and takes my hands in his. \"You have to. It's better this way; they can't be made to tell what they don't know.\"\n\nI pull my hands away. \"No. _No._ I can't leave it like this. I didn't like it when I thought they thought I was missing, but this is far worse! I can't leave with them thinking I'm dead.\"\n\n\"You can't see them. They may be watched, in case you make contact; it's too dangerous,\" Aiden says.\n\n\"No one would recognize me anymore.\"\n\nAiden shakes his head. \"Think this through. You've got another life waiting for you in Keswick. Don't throw it away now.\"\n\n\"But Mum\u2014\"\n\n\"She wouldn't want you to take the risk,\" he says.\n\nAnd I fall silent. I know he's right. If I could take her aside and tell her the whole story and ask her what I should do, she'd say, _Stay safe._ My head throbs, and I twist my hair in my hands, flinch as it pulls, then hold it up. Who knew thick hair hurt so much? I ache to lie down, but all this needs to be dealt with now. Why did MIA list me as deceased because the Lorders said I'm dead?\n\n\"Are you okay?\" Mac asks.\n\nI shrug and flinch with that movement, also. \"There're some painkillers in my bag,\" I say, and Mac gets it for me, and a glass of water. I take one.\n\n\"You should rest,\" Aiden says.\n\n\"Not yet. You have to explain something to me first. Why did you put me as deceased on MIA? Do Lorders monitor it; did you do it for them?\"\n\nAiden and Mac exchange a glance. Mac answers. \"We don't _know_ they do; the links are hidden and changed frequently. But we can't make it too hard to get to, or it wouldn't be useful for those who need it. We assume Lorders monitor the website, and probably do so regularly.\"\n\n\"But what about when I reported myself found? Won't they know?\"\n\nAiden shakes his head. \"That doesn't appear anywhere on-screen; it notifies MIA. And as I've told you before, at length, only the individuals involved in a particular missing person case know about it, and only when they need to know. Listings get taken down when we judge it is safe to do so for all involved.\"\n\nI'd quizzed Aiden on this relentlessly already, on who knows where I am now and where I'm going. And I believe him when he says it is all on a need-to-know basis only; he still hasn't even told me who reported me missing. Though I guess it is my real mother, he won't say until he judges I _need to know._ He must've thought I was extra paranoid; he didn't know the real reason for all my questions. He didn't know about Nico's plant in MIA\u2014that I'd spotted one of MIA's drivers at the terrorist camp. I had to be sure he wouldn't know I'd reported myself found and tell Nico. I should warn Aiden about him, but how can I without telling him all the rest?\n\n\"But what happens generally when someone is found?\" I ask. \"If they're kids like me who were Slated, it'll never be safe for them to go back to their original lives. It's illegal.\"\n\n\"It doesn't usually happen like that,\" Aiden admits. \"Though sometimes people do get in touch secretly, but keep their separate lives.\"\n\n\"Sometimes. What happens most of the time when someone is found?\"\n\nAiden and Mac exchange a glance. Aiden answers. \"Usually when we find out what happened to somebody . . . it's too late.\"\n\n\"They're dead for real, you mean.\" He nods. \"But I'm different.\" Always back to _Kyla is different._\n\n\"But you're officially dead,\" Aiden says. \"You can't return to your life here. There are few choices. One is what you have chosen: to go back under a different identity; to find your past.\"\n\n\"I have to.\" I sigh. We've been over this before, but I never told Aiden the real reason. I never told him about my father's death, about his last words to me. _Never forget who you are!_ And I did forget. I have to find out who I was, for him.\n\n\"What is your new name again?\" Mac asks. I fish my ID out of my pocket and hand it to him. \"Riley Kain,\" he says. \"A little different, but I like it.\"\n\nAiden frowns. \"That sounds kind of close to Kyla, doesn't it?\"\n\n\"Not that close,\" I say. I knew he'd think that. If he knew my name with the AGT was Rain, he'd be really annoyed, but not many living know me by that name anymore.\n\n_Just Nico,_ a voice whispers inside. I push it away. That'd only matter if he came across my new name, and how could that ever happen? I'm not going anywhere near the AGT. This name lets me hang on to all the parts of myself. If I let go of them, what is left?\n\nMy head is fuzzy. I let Mac help me up and lead me to the sofa in the front room, and a blanket. He and Aiden are murmuring at the door.\n\nFor all my insisting it has to be done, that I have to find out who I was, I'm afraid. What will I find?\n\n\"Few choices?\" I say, Aiden's earlier words filtering through. \"What other choice is there?\"\n\nAiden steps back into the room, kneels next to me. Smooths my hair away from my face.\n\n\"You know, Kyla. You could tell your story for MIA, be one of our witnesses.\"\n\n\"Then run away again.\"\n\n\"I wouldn't put it that way. We'd hide you someplace safe, or you could leave completely while the evidence is being gathered. Until we are ready.\"\n\n\"To expose the Lorders to the world. To make the people bring the government crashing down.\"\n\n\"Yes.\"\n\nHe's a dreamer; the Lorders will never go quietly. If at all. But it is a good dream. I smile back at Aiden, and his lips quirk.\n\n\"You're nice on painkillers.\"\n\n\"Shut up.\"\n\n\"And your new hair is gorgeous.\"\n\n\"It hurts.\"\n\n\"Take another painkiller?\"\n\nI shake my head. \"Better not. Aiden, there are things I haven't told you.\"\n\n\"I know. Tell me when you're ready.\"\n\nAiden's eyes are warm, gentle. If he knew everything about me, all I've done, would they still smile at me this way? He is too trusting for this world; he has to know. I have to tell him.\n\nI sigh. \"There is one thing I have to tell you now, ready or not.\"\n\n\"What is that?\"\n\n\"Your driver. The one who came when we saw Ben running at that track. Don't trust him.\"\n\nAiden's face goes serious, withdrawn; thinking. \"That would explain a few things,\" he says finally. \"We'll look into it. But the curious thing is, how would you know anything about it?\"\n\nHow nice it would be to tell Aiden _everything._ To not carry the burden alone. But before I can form a sentence, he shakes his head.\n\n\"No; don't answer that question. Not while you're silly on painkillers. Tell me your secrets when you are sure you want to.\" He starts to stand, but my mind is drifting back to what he said before.\n\n\"Wait. What did you mean by I could _leave completely_?\"\n\n\"You could leave the country.\"\n\n\"I could?\"\n\n\"You know MIA helps people leave when it is too dangerous to stay. To slip out of the country, over the sea. To United Ireland, or beyond.\"\n\nUnited Ireland: a free place of whisper, not reality. Since they left the UK decades ago, their existence is never officially acknowledged. Would it be any better there than here?\n\nCould I do that, just leave it all behind? My eyes close. There is so much Aiden doesn't know. Things I didn't tell him. I told myself it was because knowledge is dangerous, that he is better off not knowing. But is that really the whole reason? An uncomfortable twist of my guts says there is more to it: more not wanting him to know the things I've done. To look at me without that warmth in his eyes. I have so few friends; I can't risk losing another.\n\nWilling or not to begin with, I really _was_ in the AGT. I really _was_ a terrorist. Even though I chose to turn my back on them and their methods in the end, how could I be a witness for MIA against the Lorders? I'm the poster child for why Slating is a good thing.\n\nOver the sea . . .\n\nTo what and to where? To the unknown.\n\nTo run away.\n\n\u2022 \u2022 \u2022\n\n_I trudge up the path. Up and up, as fast as short legs can go. Soon all the streets and buildings are gone from sight. All is still, quiet. Alone at last._\n\n_I'm nervous but remember the way, though I haven't come by myself before. The walk seems longer alone, and I'm relieved when I get to the gate._\n\n_There is an eerie low mist hugging the stones. They lumber, asleep, half hidden in white. There is sunshine above; the mountains are bright sentries all around their sleeping babies. I walk across the field, into the mist, and press my hands against a stone. The sun doesn't make it through the mist; they are cold and huge close up. But when you stand back and look at the mountains, the stones are small._\n\n_Children of the mountains, Daddy calls them, and so do I, though I know from school that the stone circle was put here at Castlerigg by men and Druids, not mountains. Thousands and thousands of years ago. I start on one side, touching each one and counting._\n\n_I'm more than halfway around when a voice calls out, \"I knew I'd find you here.\" Daddy._\n\n_I don't say anything; I keep counting the stones. The mountains had many children. I'm just one._\n\n_Daddy walks up to me. \"Number?\" he asks._\n\n_\"Twenty-four,\" I say, and he walks around with me, and I count out loud as we go._\n\n_\"Twenty-five.\"_\n\n_\"She's really worried.\"_\n\n_\"Twenty-six.\"_\n\n_\"She's scared something will happen to you if you are out of her sight.\"_\n\n_I sigh. \"Twenty-seven.\"_\n\n_\"I know she can be difficult.\"_\n\n_\"Twenty-eight.\"_\n\n_\"But she loves you.\"_\n\n_\"Twenty-nine.\"_\n\n_\"You shouldn't run away.\"_\n\n_\"But_ you _do sometimes. Thirty.\" We stop. \"And she makes me crazy.\"_\n\n_Daddy laughs. \"I'll let you in on a secret.\" He looks both ways. \"Sometimes, she makes me crazy, too. Let's go home and be crazy together.\"_\n\n_\"Finish first?\" I say._\n\n_\"Of course.\"_\n\n_We keep counting, both out loud now, until we get to forty._\n\n_\"Done,\" I say, and we walk across to the gate. I look back. The mist is starting to bleed away. The stone children will be happy when they wake up in the sunshine; they have one another to play with when we are gone._\n\n_Later, I promise never to run away again. But my fingers are crossed when I say it._\n4\n\nI WAKE EARLY, STIFF AND ALARMED AS I SEEM UNABLE TO move. Then I realize that Skye has climbed onto the sofa and is sprawled across my legs: a heavy golden retriever blanket, one disinclined to wake up and tricky to dislodge.\n\nI pad into the kitchen to make tea, and peer out the window. The world is dipped in frost and makes my hands itch for a pencil and sketch pad: Intricate white patterns trace fence and trees, decorate cars and parts of cars in Mac's backyard, one that is more workshop than garden. No snow, at least not yet, so I had that wrong. And best of all: no white van, so Aiden is gone. That'll make today's plan easier. Once I work out what it is.\n\nI find my sketch pad and settle back on the sofa with my tea and Skye, meaning to draw frost's delicate patterns, but instead a stone circle insists on being rendered. And a small blond girl\u2014me, perhaps eight years old?\u2014hands pressed against a stone. Was that dream a real place? Everything inside says _yes._ I might find it when I go to Keswick; I might touch each stone, and count the mountain's children once again. But he won't find me there, not this time. He is gone forever.\n\nDad died trying to rescue me from Nico and the AGT five years ago, but the memory is recent; it had been buried so deep, for so long, that when it finally came back, it felt like it just happened.\n\nWhy am I going back? Dad won't be there. I can't remember anyone else from that life. Was it my real mother I was running away from in that dream?\n\n_She loves you,_ he said. Fingers crossed or not, I promised not to run away again. It wasn't my choice when I left before, but now it is: I have to go back.\n\nBut I can't leave yet, not without saying good-bye. Not this time. I have to tell Mum and Amy what really happened.\n\nI'm pulling on boots when Mac finally emerges, bleary-eyed and yawning.\n\nHe raises an eyebrow. \"So, let me guess: You're going to walk Skye. Just a short jaunt around the fields and back.\"\n\n\"Sure. That's it.\" Skye's tail thumps on the ground with the word _walk._\n\n\"Where are you going?\"\n\n\"I think you know.\"\n\n\"Aiden won't be happy.\"\n\n\"But you know I have to do this.\"\n\nHe stares levelly back. \"I'm beginning to realize more and more that there are times when, no matter the risk, something must be done. Some things _must_ be said. Is this one of those times?\"\n\n\"Yes. I have to tell Mum. She's lost too many other people in her life.\" Mac, of all people, should understand: from the guilt he has lived with since his school bus was bombed over six years ago. For surviving, yes, but most of all, for not speaking out about other survivors, like Mum's son, Robert, who later disappeared and was Slated. Gone without a trace. Just like her parents, the first Lorder prime minister and his wife, both assassinated by an AGT bomb when she was younger than I am now. I can't leave her thinking the same happened to me.\n\nSkye slumps back down between us, evidently having worked out that the walk isn't happening, at least not with me.\n\n\"I'll take you later,\" Mac promises her, then turns back to me. \"I just happened to drive through your village the other day.\"\n\n\"You did?\"\n\n\"Your house is still uninhabitable from the fire that spread after the blast. No one is living there. Where would they be?\"\n\n\"Oh. I didn't think of that. They're probably staying at Aunt Stacey's.\" I frown. Aunt Stacey and Mum are close, and she _seems_ all right. But her brother is Mum's ex\u2014a Lorder. If Stacey sees me, would she keep it to herself? \"I know: I'll try Mum's work. She goes for a walk at lunch most days. I'll lurk about and see if I can catch her coming or going.\"\n\n\"Sounds a bit thin.\"\n\n\"It's the best I've got.\"\n\n\"Want me to drive you?\"\n\n\"No. I'm less conspicuous on my own.\" That is what I say out loud, but this is something I have to do alone. And despite my new hair, despite my new ID, going there is still risky. If anyone is actually watching for me, would they be fooled?\n\n\"Take my bicycle.\"\n\n\"Okay.\" I smile. \"Thanks.\"\n\n\"All right then. But be careful. And have some breakfast first.\"\n\n\u2022 \u2022 \u2022\n\nI'm too early for Mum's lunch break, and something makes me stop at the graveyard. I get off the bicycle and lean it against the crumbling stone wall. Frost outlines bare trees; headstones are traced in ghostly white. I step through the gate and start down the path, my breath a moving shroud around me in the cold air.\n\nIt is a small village church, and the newest grave isn't hard to find. There is no headstone yet, if there will indeed be one, but the ground is disturbed: a patch of brown in gray frost-tipped grass, with a cover of scattered flowers. Was some other unidentified girl buried here, or was the casket empty, perhaps weighted with stones so no one would notice?\n\nI kneel down, take off my gloves, and reach out tentative fingers to a frozen lily. Is its fragile beauty preserved by cold? No. A petal shatters at my touch.\n\n\"Hello,\" a voice says, piercing the quiet, and I jump. A voice I know.\n\nI stand, turn. Stare at her, unable to speak.\n\n\"Were you a friend of Kyla's?\" Mum asks.\n\n\"You don't know me?\"\n\nHer brows knit together. She looks older, though it hasn't been long since I last saw her. Her eyes are tired, red. \"Sorry, have we met?\"\n\nTears well up in my eyes. I take off my glasses, sweep my dark hair to one side, wincing a little as the extra weight still hurts. \"It's me. It's Kyla,\" I whisper.\n\nShe goes pale, shakes her head.\n\n\"Mum?\" I reach a hand toward her, but then she steps back, turns, and scans the churchyard and road beyond.\n\n\"Put those glasses back on,\" she says, and after I do, she links an arm in mine. She pulls me down the path at the back of the church, then out the gate into the woods behind it, walking fast. The path twists, then divides, and we take the less traveled branch.\n\nShe finally stops. Wheezing a little, she turns and looks at me.\n\n\"It really _is_ you. You're really okay.\"\n\nMy tears start again, and then hers. She pulls me in for a hug. We stand there for a long time, not moving, not speaking.\n\nShe finally pulls away. \"Your hair?\" She reaches out to touch it. \"IMET?\"\n\nI nod.\n\n\"How? No, don't answer! Is it . . .\" She hesitates. \"Is it Lorders?\"\n\nI shake my head. \"They don't know where I am. And it wasn't them who tried to kill me, but for some reason of their own they've said I died. I don't understand why.\"\n\n\"So it wasn't their bomb, then. David said it wasn't, but . . .\" And she shrugs, no need to fill in the sentence. She didn't believe him. Why would she believe her estranged husband, after all he put us through?\n\n\"No. It was AGT.\"\n\nShe pales. \"They're after you?\"\n\nI shrug. \"They think I betrayed them to the Lorders.\"\n\n\"Did you?\"\n\nI shake my head. \"Not on purpose. Lorders tracked me to them.\" I don't say the rest of it: that I went against Nico's plans. That I wasn't there with her and the rest of her family next to Prime Minister Gregory so Nico could detonate the bomb I unknowingly wore. That instead I left to bust out his prisoner, Dr. Lysander: my doctor. The one who invented Slating. If he finds out I'm still alive, Nico's desire for revenge would have nothing to do with logic; to him, it'd be personal.\n\n\"Then maybe it's a good thing the Lorders have said you are dead. Maybe the AGT will believe them.\" She reaches a hand to my cheek. \"I'm so glad you're all right, but you shouldn't have come here. It's too dangerous. And how'd you know where to find me? I didn't even know I was coming. I just went for a walk, and my feet brought me.\"\n\n\"I didn't. I thought you'd be at work; I was going to try there. I couldn't leave with you thinking I died.\"\n\nShe grips me in a fierce hug. \"Have you got somewhere safe to go?\"\n\n\"I think so. I'll try to get word to you later.\"\n\n\"Don't. It's safer that way.\"\n\n\"What about Amy? How is she?\"\n\n\"She's distraught. But I can't tell her about you. At least, not now.\"\n\nMy tears are starting again. Amy has been my big sister since I was assigned to her family after I was Slated. No matter that it has only been a matter of months; Amy'd never do anything to hurt me on purpose. But could she keep such a big secret from the world?\n\n\"She's safer if she doesn't know,\" Mum says. \"I'll look after her.\"\n\n\"I know. All right.\"\n\n\"Dr. Lysander called, sent flowers. She seemed genuinely distressed about you.\"\n\nAnother twist of pain inside. Dr. Lysander doesn't deserve to not know the truth, but there is no safe way to tell her.\n\nMum stares at me so long, it's like she is memorizing my face, then kisses my cheek. \"I better go. Wait awhile before you follow.\" She grips me tight one more time, then turns and half runs back down the path.\n\nI lean against a tree, hugging my arms around myself.\n\nSo much pain: hers, Amy's, mine. And that whole charade of a funeral. For what? Why did the Lorders pretend I died?\n\n\u2022 \u2022 \u2022\n\nAfter a while I trudge back through the woods. When I reach the church, I hang back by the gate, but there is no one in sight. I retrieve the bicycle and start the miles back to Mac's.\n\nBefore long, thick heavy white flakes fall from the sky, swirling gently about me. I hold my hands out to catch them as they fall; they settle on my hat, my hair, changing brown to white. Obscuring my disguise; obscuring all of me. Pedaling gets harder as the snow thickens on the ground, and after a while I get off and push the bike.\n\nWhen I finally get back to the house, I'm soaked and half frozen. A relieved Mac has me sit in front of the fire.\n\nSkye is glued to the window, her eyes darting with each snowflake. \"She looks a little freaked out by the weather,\" I say.\n\n\"That's nothing; when there is a thunderstorm, she shakes and hides under the bed. Speaking of hiding, Aiden called while you were gone.\"\n\n\"And?\"\n\n\"I told him you went for a walk.\"\n\nMac's face says it all. \"I gather he didn't believe you, and isn't happy about it.\"\n\n\"How'd you guess? So, did everything go well? Did you say what you had to say?\"\n\n\"Yes.\"\n\n\"Feeling ready to move on now?\"\n\n\"Can I warm up first?\"\n\n\"You've got until tomorrow morning. Aiden's coming at nine. Trains are back on, and tickets have been organized; there's a computer file for you to study tonight with details of your new life.\"\n\n\u2022 \u2022 \u2022\n\nThere is one more good-bye I need to make. Late that night, after Mac's gone to sleep, I stand on a chair in the kitchen and get the owl sculpture down from the top of the fridge. I put it on the table and run my fingers lightly across its beak, the outstretched wings. All made of scrap bits of metal, but in a brilliant synthesis: It looks and feels so real. Ben's mother made it, made it for me at his request from a drawing I did. It seems so long ago. Now she is dead, killed along with her husband by Lorders. Just for asking too many questions about what happened to Ben.\n\nI run my fingers along the back until I feel the faint edge of paper, grip it between two fingernails, and pull. I unfold the note that holds Ben's last words to me; his last words while he was still _my Ben._\n\n_Dear Kyla,_\n\n_If you have found this, it means things have gone very wrong. I'm sorry to cause you pain. But know that this was my decision, and mine alone. No one else is to blame._\n\n_Love, Ben_\n\nDespite his words, at the time I thought it was my fault that Ben wanted to cut off his Levo, and all that followed: his seizures, his mum telling me to leave. The Lorders taking him, my not knowing if he was alive or dead. Then he was found by MIA, changed somehow by the Lorders, so he didn't even know who I was. That last time I saw Ben I tried, I really tried, to get through to him, to tell him to resist the Lorders. There was a moment when I saw something in his eyes; I thought he believed me, that he understood. All I can do for Ben now is hope.\n\nAnd the other thing I worked out well after the fact was that Nico had been working on Ben to cut off his Levo, to try to provide the trauma that would trigger the return of my memories of being with Nico and the AGT. But even with that, it is still my fault. If it weren't for me, Nico wouldn't have had a reason to go near him, would he?\n\nI stare at the note in my hands. Should I take it with me? I'm tempted. But somehow it belongs where I found it the first time, where it has been hiding ever since. I fold it up, carefully slip it inside the owl, and put the owl back on top of Mac's fridge. He'll keep it safe.\n\nMaybe one day, Ben and I will come back for it. Together.\n5\n\nTHE NEXT MORNING, SNOW IS THICK ON THE GROUND; THE lane is impassable. After a call from Aiden, Mac says he'll walk down the lane with me to meet him on the main road.\n\nI hesitate by the door, reluctant to leave what I know for what I do not, a place where I feel safe . . . for what?\n\nMac meets my eye. \"You'll be back.\"\n\n\"Will I?\"\n\n\"Oh yes. Skye would be very upset if you didn't visit again.\" He opens the door, and Skye bounds out and off the step, then skids to a stop, startled as the snow comes almost all the way to her nose.\n\nI step out, gather some into my gloved hands, and hold it out for her to sniff. \"It's snow,\" I explain. I roll it into a ball and throw it. She jumps to chase it, leaping in and out of the snow instead of running through it, then looks very puzzled when the snowball is indistinguishable from the rest of the snow where it landed.\n\nMac laughs and insists on carrying my small duffel bag of belongings. We plunge down the lane, snow well past our knees.\n\n\"So,\" I say, \"did Aiden still sound annoyed?\"\n\n\"He is at me.\"\n\n\"Oh. Sorry.\"\n\nMac shrugs. \"He'll get over it. Once he sees that you're all right.\"\n\nWhen we reach the main road, thankfully plowed, Aiden's van is already there.\n\n\"Thanks for putting up with me. For everything,\" I say. Where would I be now without Mac's place to run to, to hide away in?\n\nMac gives me a hug and opens the van door, then holds Skye when she tries to jump into the van next to me. I wave through the window, blinking furiously, trying to keep myself together until they are gone from sight.\n\nAiden nods once when I say hello, then keeps his attention on the icy road, on keeping us on it. The silence is as frosty as the winter morning until he pulls in front of the train station.\n\n\"Aiden, I'm sorry. But I had to see Mum before I could go. Don't blame Mac; he couldn't have stopped me. Don't let us say good-bye like this.\"\n\nHe catches my hand in his. Face serious, his deep blue eyes stare into mine. \"Kyla, please be more careful in the future. Don't let anything slip. Your life, and those of others, depends on your not getting caught.\"\n\n\"Don't let anything slip\u2014like getting my name wrong?\"\n\n\"Exactly.\"\n\n\"Like you just did? I'm Riley now, remember?\"\n\nA trace of a smile crosses his face. He reaches into a folder, then hands me a plastic card. \"Here is your train ticket. Don't lose it.\"\n\nI roll my eyes, tuck it in my pocket. \"I'll try not to.\"\n\n\"Have you got your ID?\"\n\nI give him a look, but he doesn't relent. I sigh, fish into my bag, and hold out my new ID card so he can see it, then tuck it away again.\n\n\"And have you got your story straight from the file I sent? Tell me.\"\n\n\"I'm Riley Kain. I'm eighteen, and my birthday is September 17, 2036. I'm from Chelmsford, an only child. My parents are both schoolteachers. I'm going to Keswick and staying at some place for under-twenty-ones along Derwentwater, called Waterfall House for Girls, and I'm signing up for CAS: the Cumbrian Apprenticeship Scheme. Do I really have to do that, by the way?\"\n\n\"You can't just visit; you have to be there for a reason.\" He smiles properly this time, and the tight knot of tension inside me eases. \"I did consider a hospitality job, majoring in dish-washing; we've got a connection in a hotel there. So things could be worse.\"\n\n\"Thanks. But you haven't told me one kind of big bit of information.\"\n\n\"What is that?\"\n\n\"How do I find who reported me missing?\"\n\nHis lips quirk. \"I've told you before: It's need-to-know.\"\n\nI stare at him, indignant. \"Who needs to know more than I do now! Aren't you going to tell me?\"\n\n\"Can't I leave it as a surprise?\"\n\nI glare.\n\n\"Just kidding\u2014it'll be easy to find your mother, Stella Connor. She runs Waterfall House. She knows you're coming; she knows you are her missing daughter.\"\n\n_My mother._ My actual, real mother: the one who gave birth to me, not one assigned in a Lorder insta-family. She was the one who reported me missing, just like I thought. My mother . . . the one I can't remember.\n\nAiden squeezes my hand, as if he can see the thoughts that are keeping me from speaking. \"Get going. Don't look like security worries you, or they'll pay extra attention. Just sail through the gate like you haven't a care in the world.\"\n\n\"Okay,\" I manage to say. But I'm still sitting in the van, and Aiden is still holding my hand.\n\n\"Kyla\u2014Riley, I mean\u2014take care of yourself. You know what to do if you need help, if anything goes wrong?\"\n\nI nod. Aiden's file also mentioned a certain community notice board. One where a coded note will reach his contact.\n\n\"I hope this works out for you. I hope you find what you're looking for. But if you don't . . .\" His voice trails away. \"Anyhow, best get going.\" He still holds my hand in his, some emotion too raw and private in his eyes, but I can't look away. Seconds slow and stretch until finally he lets go.\n\nI clamber out of the van with my bag, shut the door, then turn and hold up my hand in a good-bye; a hand now empty and cold. Words are stuck in a throat that feels tight. Another friend I may never see again. I stare at him through the window, storing him up: the way he tilts his head to one side when he looks at me intently like he is now, the fiery glint of his red hair in the morning sun. Aiden has done so much for me, and all I do is make him worry and cause problems. None of this could have been easy to arrange, and I didn't even say a proper thank-you.\n\nBut like he can see what is inside me, Aiden nods his head. _It's okay. Go on,_ he mouths.\n\nI turn, square my shoulders, and walk away from his van to the front of the station. As I approach, the barriers open. Aiden's file said they detect tickets and ID anywhere on your person and operate automatically; they also scan for weapons. Guards in a booth glance my way and then back at their security screens. I'm through. An arrow coded to my ticket lights up at my feet, shows which way to go. I start to walk away from the barriers to the designated elevator, still thinking of all the things I should have said . . .\n\nDJ! What with first me being upset about my so-called funeral, then Aiden being angry I went to see Mum, I forgot all about the IMET doctor's message\u2014that he wants to see Aiden. I turn to look through the glass barriers, but Aiden's van is already disappearing from sight.\n\nToo late. I hope it wasn't important.\n6\n\nTHE ELEVATOR DROPS SWIFTLY, THEN OPENS TO AN UNDERground platform. The train is already there; once again an arrow at my feet is coded to my ticket and points the way to the right car of the train and then to my seat. Other passengers move around me, following arrows of their own.\n\nHave I ever been on a train before? If I have, I don't remember. I put my bag into the overhead, then second thoughts have me pull it back down to retrieve my ID and put it in my pocket with my ticket before shoving it back up again. I can't lose my ID. Unlike most people's, mine would be quite a bother to replace.\n\nThe train is about half full; no one sits next to me. I have a window seat, and when the train leaves moments later, a vid runs in the window: glorious countryside, or Antarctic glaciers, or a steamy jungle. All at the flick of a switch, and I can't stop myself from trying them all. I'm glad Aiden's file explained this, or I would have been both alarmed and baffled. After a while I notice almost no one else uses the window vid, and I turn mine off. I study fellow passengers instead.\n\nA few are younger and in jeans like me, perhaps students or off on apprenticeships, but most look like businesspeople. Both men and women in suits, much like my assigned dad wore when he was off supposedly installing and maintaining government computer systems. Though who knows what he really did for the Lorders? He traveled all over the country, or so he said; a nervous thought makes me check every passenger I can see to make sure he isn't here. He did travel by car to some places, and there were buses also for short journeys, like to London, but most long-distance vehicular travel is banned now: All must travel by environment-friendly high-speed train.\n\nThe minutes tick to an hour; the train stops several times at other underground stations. At one, a harassed-looking mother with a boy about four years old gets on, his small hand clenched tight in hers. They sit a few rows in front of me. Before long his head peeks over the seat, dark eyes staring at mine. I smile and he dips down. Seconds later his head bobs up again, giggling and flashing a crooked grin this time, until his mother makes him sit down. He squirms into her lap and her arms go around him.\n\nA mother holding her child close. Was that how things were with me and my mother? I blink hard, then stare at the window vid screen, as blank and dead as my memories of her. I close my eyes. Maybe when we see each other, it'll all come back, like I am ten years old again. Maybe we'll run to each other and she'll hold me, and I'll be _home._ I'll know who I was, who I am.\n\nMaybe, I won't.\n\nThere is a panic inside, one that says _run._ That not knowing may be better than knowing; that things will change, and change isn't always good. I'd been desperate before to know who I was, where I came from, why I was Slated. Finding out about Nico's AGT and their plans for me didn't make anything better, did it?\n\nSome part of me notices that while I've been thinking, the train has stopped. For much longer than at any other stop. I open my eyes; the doors are still closed. We're not at a station?\n\nI glance about at the other passengers, and it is tangible, the growing unease. What is happening? The woman and boy get out of their seats and walk to the connecting door to the next car at the front of ours. I've seen people go in and out of it, returning with steaming cups in their hands. But this time the door won't open. They go back to their seats.\n\nMoments later the locked door opens, and unease turns to dread. Lorders. Two of them, with steely, dead eyes. In black ops gear, vests. One has a weapon in hand, the other a small device. There is a train guard with them, a bead of sweat on his brow.\n\n\"Get your tickets and IDs out, folks,\" the train guard says, his voice not quite steady. And passengers shuffle, get cards out of bags and pockets. I get mine out, hand shaking. _Get a grip._ Aiden's notes said a ticket and ID check is common. That mine'll pass fine, to stay calm if it happens. But he never said anything about Lorders being involved.\n\nThe Lorder with the gun stays at the door; the other follows the guard. When they get to the first passenger, the guard scans his ticket and ID. Then the Lorder holds up the device he carries and orders the passenger to look inside it until it beeps; first with one eye, then the other.\n\nA portable retinal scanner?\n\nThis is not a standard check. Swirls of fear turn to panic. Glasses must come off to scan retinas; they'll see my eye color is masked. If only I'd let DJ _change_ them to gray permanently, not mask them; vanity to keep my eyes green might kill me. I could take them off before they get here, hope they don't notice, but then I panic further: Who knows if my retinal key will show up the wrong name, that of a dead girl, Kyla Davis? We had them done at school. And at the hospital. I glance back, but there are Lorders at the back door also. Blocking the way.\n\nNowhere to go. Trapped. As a Slated, seeking out my past life is completely illegal. Not to mention the IMET and traveling under a false identity. After everything, is this as far as I get? Keswick should be just minutes away now. Did my fake ID trigger some warning? Are they looking for me?\n\nThey get closer, row by row. The guard checks each ticket and ID; the Lorder operates the retinal scanner.\n\nSomething bumps my foot and I almost scream. I glance down; the small boy is crawling under the seats. Ahead they have reached his mother. Her face is beyond pale, more gray, and her shaking hand holds out her ID and ticket. The guard scans them; they pass. But the Lorder's lips curve in a small smile of satisfaction. He knows. He is certain he's found the one he looks for. _It's not me._ He holds up the retinal scanner to her eye. Instead of a beep, it buzzes. His smile widens.\n\nHis hand clamps on her shoulder, pulls her up. Pushes her into the aisle. \"Walk!\" he barks. They start toward the front of the car. There is a small cry behind. I don't dare turn, but she does, and her face crumples. Moments later one of the Lorders from the back of the car walks past, dragging the small boy along with him.\n\nThey disappear through the front connecting door. No one says anything; no one looks at anyone. I'm horrified, but also _relieved._ They weren't after me. Not this time. But if my seat had been before hers, and they'd scanned my retinas . . . I quake inside.\n\nAnd then I'm ashamed. What will happen to them now? I'll never know if she did anything bad enough to warrant being hauled off by the Lorders like that; I'll never know what happens to her, or her son. What if everyone in this car had said, together, _No, you can't take them._ Could we have stopped it?\n\nThe answer might have been yes, for a few minutes. But they'd have reinforcements at the next station; we'd all be arrested and taken away. We'd face the same fate as she will. Is that a good enough reason to say nothing?\n\nWhat if every person in the country said _no,_ all at once, like Aiden thinks they will if they know what really goes on. They can't arrest every single one of us.\n7\n\nI STEP OUT OF THE DIM STATION ELEVATOR INTO DAZZLING sunshine. Keswick sunshine. It is cold, crisp; the air is so chill that breathing it in almost makes me cough. No snow on the ground here today, but above? White-peaked fells. There is a prickle on the back of my neck, my spine, but not from the cold. It is a physical reaction to being in this place, to breathing this air. I stand stock-still, gaping up at the mountains, until a whisper of sanity draws me back to here and now. _Don't draw attention._ I force my eyes to drop and look around me.\n\nOnly a few other passengers have come off here, and they are walking swiftly away. There is a Lorder van parked next to the station, blocking the view of one of the elevators; are they taking their new prisoners from the train? I walk away from any watchful eyes. Adjusting my bag on my shoulder, I find and follow the town center sign Aiden's notes said would be there. There is no recognition inside me now, of this station, or where to go. I glance back, and over the archway containing the elevators and ticket office is carved _2050._ This station didn't exist when I lived here. It's new.\n\nTen minutes later I've reached the center of town, and the prickling feeling of wonder, of both knowing and not knowing this place, comes back. There is a crowded pedestrian area leading up to an ancient building with a sign that says MOOT HALL. Cobblestones crumble underfoot, with a vague sense they are smaller than they should be. Because I'm bigger now?\n\nI shake my head. Am I imagining things? There is no definite memory, only shadows that seem to mist away if I stare. Maybe it's just the _longing_ to know this place.\n\nOn arrival in Keswick, I'm to go to Waterfall House. And my _mother._ I swallow; the word sounds all wrong. The house is along the shores of Derwentwater, on almost the opposite side of the lake from Keswick. I memorized maps of how to get there: about three miles to walk on footpaths. Or there is a boat across the water. Or a bus on the road.\n\nWalking takes longest. Walk it is. Roads then paths lead out of the town center, past a ruined theater and down to the lake; paths wander through woods with views over the lake, then drop to cut down to the water. There is ice silvering out from its deep blue edges; the ground underfoot is hard frozen. There are people, some with dogs, ambling on paths in all directions, breath puffing out white around their faces. They dwindle away the farther I get from Keswick. Soon I'm alone.\n\nMy feet move slower and slower, head full of a peculiar madness. I want to laugh and cry at the same time. I want to touch every tree, every rock on the way. I want to know them, to take them into me so they cement out whispers of memory. My head feels full of fuzzy cotton confusion, of _wanting_ to remember being here before, but nothing is definite. It could just be the wanting that makes me feel this way, that makes my feet long to walk back and forth over the same places to make me remember them, if not from before, from now.\n\nI shake my head. Aiden told me she knows I'm coming; she'll wonder what has happened to me . . . again. I start walking at a proper pace. What could it have been like for her? For _my mother._ I say the words over and over inside my head, tasting them, but they still don't feel right, don't sound right. I'm _her daughter_ \u2014that feels weird, too. I disappeared when I was ten years old. Seven years ago. How do you get through something like that? And then her husband died, a few years after I vanished, when he tried to rescue me. My fault. She might blame me.\n\nAnd so my feet go faster and slower and faster again for the rest of the walk, as my thoughts tumble inside. When I finally see the house in the distance, my feet stop completely. From Aiden's file I know it used to be Lodore Falls Hotel; Waterfall House for Girls it is called now. The exterior covered in Lake District gray slate fits the foreground of lake below and woods rising behind, the snow-touched fells beyond. It is warm from this distance, like a soft-focus, dreamy castle, even though I know much of it was destroyed in the riots decades ago, then rebuilt with more concrete and less slate. I carry on. The closer I walk, the harsher it gets.\n\nWhen I finally reach the house, I hesitate at the door. This is it. Will she know me? Will I know her? Eagerness and fear war inside of me, laced with caution. As Aiden's notes pointed out, many girls live here. None of them can realize what we are to each other.\n\nDo I knock? Go in?\n\nAs if to answer my question, the door opens, and a girl steps out. She nods and keeps going. I walk through the door before it swings closed.\n\nThere are other girls in the entrance area. Two in chairs, chatting. A woman stands near a large desk. She is tall; long blond hair swept back, dark roots peeking through; thin, maybe forty years old. Neatly, _very_ neatly dressed. Even her buttons shine. Is it her? Nothing about her is familiar. I walk up to the desk.\n\n\"Yes?\" she says.\n\n\"Uh, hi. I'm Riley Kain. I think I'm staying here.\"\n\n\"You're _late._ I was about to send some of the girls out to look for you in case you got lost in the woods.\" Is it her, my mother? Her lips are pursed, words calm and clear, but her eyes sweep over me with longing and confusion. She expects me to be blond, to have green eyes. She doesn't know about the IMET?\n\nWith my back to the other girls, I take my glasses off, as if to rub my eyes. Green eyes. Hers widen slightly. I put them back on.\n\n\"Your ID?\" she says, and I take it out. She scans it into a netbook, hand shaking slightly. \"You are indeed staying with us, Riley. I'm Stella Connor. You can call me Stella.\"\n\nI stare back at her. _Stella Connor_ : Lucy Connor's mother. But nothing about her or the name is familiar, and bitter disappointment at the lack of memory gnaws inside.\n\n\"You've missed lunch, I'm afraid. Tea is here in the conservatory at four, and dinner in the hall at seven. Here is your list of rules.\" She hands me a substantial number of sheets stapled together, touching my hand as she does so. \"We'll talk tonight,\" she adds, her words such a quiet whisper that I'm not sure I heard or imagined them.\n\n\"Madison?\" she calls out, and one of the girls looks up. \"Can you show Riley to her room, please? The tower.\"\n\nThe girl bounces out of her chair: cute, with dark curly hair, not much taller than me, a mischievous glint to her eyes. She walks over. \"Sure thing, Mrs. C.\"\n\nStella's eyes narrow. Not happy with the \"Mrs. C\" thing.\n\n\"This way!\" Madison says, with a dramatic flourish. I follow her through a door, down halls to stairs. She looks back. \"Take her to the tower!\" she mimics, one finger pointing dramatically at the staircase, her voice so like Stella's that I have to laugh.\n\nAt the top of the stairs, Madison flings the door open. \"I can't believe she's put you in the tower. It's been empty for ages. She only let one person stay here awhile last year, and that was just because a bunch of rooms were wrecked in floods and all the other rooms were full, and as soon as one was empty she shifted her out.\"\n\n\"How many girls stay here?\" I ask as I walk in and put my bag on the bed.\n\n\"Not so many now. Including you, there are, I believe, seventeen of us. Everyone leaves Waterfall-Weirdo if they can get a place anywhere else.\"\n\n\"Why Weirdo?\"\n\n\"You met the Queen of the Weird downstairs, didn't you notice? Wait 'til you read the list of rules.\" She takes it from my hand and brandishes it before putting it on the desk by the bed. \"Break any of the rules at your peril,\" she says in her Stella-mimic voice, and I try not to smirk; that _is_ my mother she's making fun of. \"And then there is her family.\" She rolls her eyes.\n\nFamily? Do I have other family? \"Why? Who are they?\" I ask, trying not to look too curious.\n\n\"Her mother is the JCO for all of England. Not somebody you want to be in the same room with. Thankfully, she hardly ever visits.\"\n\nJCO? I stare at her in shock. I have a _grandmother._ And my grandmother is not only a Lorder, but a Juvenile Control Officer, and not only that, but for all of England? My mouth falls open.\n\nMadison doesn't seem to notice. \"What are you doing in Keswick, anyhow?\"\n\n\"I'm here for the apprenticeship scheme.\"\n\n\"CAS? That starts tomorrow, doesn't it?\"\n\nI nod. There was an outline of the scheme in Aiden's notes, and it was the reason for my hasty trip up here: to make it for the first day. \"What do you do?\"\n\n\"I'm working at Cora's Caf\u00e9. It's my day off today. Can't wait until I'm twenty-one next summer so I can get out of here. I'd just moved into this fab flat with three others when they brought in that latest stupid YP law two years ago, and we had to give it up.\"\n\nI look at her blankly.\n\n\"Don't you even know why you're staying here? JCO Young Persons Law 29(b).\" She stands up bolt-straight. \"Thou shalt live with either family or in approved structured accommodation with supervision until the age of twenty-one,\" she intones in a nasally voice, then pretends to strangle herself. \"What do they think we'll get up to? It's not like there is much of anything to _do_ in Keswick, even if we weren't stuck out here.\"\n\nMadison opens a door to show me my bathroom. \"You may be on your own in the tower, but at least you don't have to share your bathroom. Don't miss rule nine: No more than five minutes per shower. If you go over, she turns the hot water off for the whole house for a day. Somehow, she _always_ knows. She does randoms, too: walks the halls in the middle of the night at odd times to make sure you don't breach rules six or eleven.\"\n\n\"Thanks.\" I smile, look at her. _Please leave._ I need to be alone awhile.\n\nShe must see it on my face. \"You want me to go, right.\"\n\n\"Ah . . .\"\n\n\"No worries. See you at tea downstairs at four. _Don't_ be late: rule number two.\"\n\nAlone at last, I circle the room: a double bed, an empty wardrobe, a desk, and a chair. More wardrobes across the room\u2014locked. And a lot of empty space; it's a big room. Did this used to be Lucy's room\u2014my room\u2014is that why Stella keeps it empty? I shrug. No idea. Nothing in it feels familiar.\n\nI pull the curtains open wide. There are windows all around: lake on one side, woods on the other. Gorgeous views, and I close my eyes, try to imagine this room and me in it, younger, looking out the window with my dad, but I can't.\n\nThere is an odd noise at the door: scratching? A gray paw appears underneath. I open it.\n\nA gray cat looks up at me, then pushes past me through the open door. It takes a running leap at the bed and sits there daintily, washing one paw, her green eyes on me all the while.\n\nLucy's gray kitten, her tenth birthday present\u2014one of the very few memories I've had of being her since I was Slated. Is it . . . this cat?\n\nI walk over to the bed, sit on the other end cross-legged. \"Is it you?\" I whisper. She stalks across the bed, walks all around me in a circle as if checking me out thoroughly. I hold out one hand, and she rubs her chin against it. Soon I've coaxed her onto my lap; I stroke her and she curls up, purring.\n\nThe list of rules is next to me where Madison left it, and I pick it up and look at the first page. _Rule one: Be nice to Pounce (the cat)._\n\n\"Pounce?\" I say, and she stirs, looks at me with slit eyes, then pulls her paws tight around her head as if to say, _Be quiet; can't you see I'm sleeping?_ Pounce sounds to me the sort of name a ten-year-old would give a kitten.\n\nWell. Stella might be a little weird, but given what she puts as rule number one, maybe she and I will get along all right, after all.\n8\n\nI MAKE IT TO TEA AT EXACTLY ONE MINUTE TO FOUR, stomach rumbling. Madison and the girl I saw her with earlier are there, and two others; there is no sign of Stella, and I'm told the others are at work in various places around Keswick. There is a teapot and a plate of warm scones with jam we all swoop on with delight. They usually just get dry biscuits at tea, Madison tells me. Is this a special treat for me?\n\nAfter tea, they give me a quick tour of the place. There is a TV room with sofas and fireplaces, a library, and a dining room with one long table already set for dinner.\n\nI wander back to my room to unpack. When we assemble for dinner at seven, Madison pulls me into a seat next to hers. Soon all but two seats are taken. There is a sea of friendly, curious eyes, and names are called out, too many to remember at once. And it all seems . . . nice. Cozy. Not a place to try to get away from.\n\nStella walks in as a clock chimes seven, and chatter quiets down. She takes the empty chair at the end of the table. She looks at the other empty seat and frowns. \"Does anyone know where Ellie is?\" There is a murmur of _no,_ shaken heads.\n\n\"Maybe she's not hungry. Maybe she's not well. Maybe she found something better to do,\" Madison says, and the room falls silent.\n\nStella frowns. \"Then she should have sent word. Could someone check her room, please?\"\n\nAnother girl volunteers, and returns moments later. \"She's in her room. She fell asleep,\" she says, and I wonder: Why doesn't Ellie come along now?\n\nThe tension on Stella's face relaxes, and gradually everyone else does also. Serving dishes are passed around. I'm relieved I'm too many seats away to have to try to chat with Stella in front of everyone, but now and then can't stop my eyes glancing over, finding hers, then spinning away again. This is so _surreal_ : in a room having dinner with my actual real mother for the first time in seven years, yet we sit apart, not speaking. There is a part of me that wants to jump up and say, _Enough already!_ And another part happy to keep up the appearance of strangers, to hang back, to observe.\n\nWhen we're done, everyone starts leaving except two on dishes duty, stacking plates. The others are wandering out in twos and threes; some head to the TV room, some in other directions, and I stand, uncertain. Did Stella mean for us to talk now? But Madison links my arm in hers and draws me along with her; a few others follow us down a hall and up a few stairs to knock on a door. \"Come in,\" a voice calls from inside.\n\n\"Did you bring me anything?\" a girl asks, and is introduced as the sleepy Ellie. \"I'm starving!\"\n\nMadison and the others produce rolls and other bits pilfered from dinner.\n\n\"I don't understand\u2014why didn't you just come and eat with the rest of us?\" I ask. \"What was the point in sending someone to check on you, then leave you here?\"\n\nMadison rolls her eyes. \"You can't have dinner if you're late. Against Weirdo rule number three.\"\n\n\"Don't be so unkind. She's all right,\" Ellie says, and I'm relieved to hear someone stick up for her. But it doesn't seem to be the popular opinion.\n\n\"It's ridiculous making us account for every second of the day. We're not babies,\" another girl says.\n\n\"You know why, though,\" Ellie answers, and I get the sense that this is a conversation everyone has heard before.\n\nMadison scowls. \"Yeah, but how many years ago was that? Shouldn't she be over it by now?\"\n\n\"Over what?\" I ask. An uncomfortable feeling says I already know, but I shouldn't. Do I ask because it would be normal to ask, or do I need to hear it? Hear somebody else say things I know to be true, but can't remember.\n\n\"You don't get over things like that,\" Ellie says to Madison, shaking her head, then turns to me. \"Her daughter went missing. No one knows what happened to her. I think Stella is afraid of something happening to one of us; she's just looking out for us all.\"\n\n\u2022 \u2022 \u2022\n\nLate that night there is a faint knock on my door, and it opens. I sit up, heart pounding.\n\nHall light frames her: Stella.\n\nShe looks different, hair down, a long flannel robe wrapped tight around her; more soft and uncertain. Pounce pushes past her, runs across the room and jumps up onto my bed.\n\nStella pulls the chair next to the bed and sits in it. She grips my hand so tight, it starts to hurt.\n\n\"Lucy? Is it really you?\" she whispers. She reaches out her other hand, shaking, to my hair. \"What has happened to your beautiful hair?\"\n\n\"It's changed, permanently: IMET.\"\n\n\"We could dye it, I suppose.\"\n\n\"No. I'm trying to not be recognized.\"\n\n\"Oh. Of course.\" She sighs. \"I can always stop dyeing mine.\"\n\n\"Why? Do we need to match?\"\n\nShe starts, pulls her hand away. \"Not exactly. It's just that I didn't know you when you came in. I didn't know my own daughter. You didn't know me, either, did you?\"\n\nI hesitate, shake my head. She looks hurt. \"I'm sorry. You know I was Slated, don't you?\"\n\nShe nods. \"She told me.\"\n\n\"Who?\"\n\nShe looks away. \"I don't know. Whoever it was who told me you were finally coming home.\"\n\nIn MIA?\n\n\"Tell me your story, Lucy. Tell me everything you can about where you've been these seven years.\"\n\nI hold still a moment. I came here because I wanted to find out about my missing past, my years here; of course she wants the same in return, to know about the parts of my life she has missed since then. A fair exchange? But much of what has been my life these last years I don't want to say out loud. Some demons are best kept locked up, hidden away.\n\n\"Lucy?\"\n\n\"Could you not call me Lucy? It's just that it is dangerous. No one can know who I really am.\"\n\n\"No one can hear us now.\"\n\n\"But you might slip up when other people are around.\"\n\nShe half smiles. \"I'll try, Lu\u2014\" She jumps, guiltily. \"Riley,\" she says. \"What should you call me?\" Her eyes hunger, and I know what she wants to hear, but I can't bring myself to do it.\n\n\"I should call you what all the girls do, for the same reason: Stella.\"\n\nShe frowns, and sighs. \"Oh, all right. Tell me about your life, _Riley._ \"\n\nAnd I stare back at her. _Should_ I tell her everything, no matter whether I want to or not? Is it dangerous to know? \"I don't know everything. A lot of my memories are gone.\"\n\n\"What you do know, then.\"\n\n\"I think I was kidnapped when I was ten. I didn't understand why for a long time.\"\n\nHer lip curls. \"The AGT.\"\n\nMy eyes widen. She knows, or guesses? \"Yes, it was them. They had some sort of plan, to fracture my personality. So that when I was Slated some memories would survive.\"\n\nStella's face wars between sadness and horror. \"You must have been so scared.\"\n\nSo little memory of that time remains, but what does isn't good: late at night hearing a doctor's voice saying over and over again, _You have no family; they didn't want you; they gave you to us._ My eyes start to sting, and I blink. \"Are you sure you want to know?\" I ask. \"Everything? It isn't easy to talk about. It might be harder to hear.\"\n\nStella hesitates. \"Yes. Tell me,\" she says, and slips an arm across my shoulders, hesitant, and some of the resistance inside melts enough for me to lean into her a moment, and tell her the blackest early memory from _then._\n\nI hold my left hand up. \"They made me\u2014as Lucy\u2014be right-handed. Broke my left fingers so I had no choice.\" She cradles my hand in hers, staying silent. Nods once to say _go on,_ but doesn't press. But I can't bring myself to tell her the thing that happened that finally cemented the personality split: that Dad snatched me back from the AGT, that we nearly got away. But Nico caught us. The gun in Nico's hand. Does she know how Dad\u2014her husband\u2014died?\n\nI straighten up. \"Later on, they succeeded: I had a split personality. When I was left-handed, I trained with the AGT as one of them; now and then I was right-handed, and I was Lucy. When the Lorders caught and Slated me, the other part of me hid away and Lucy was dominant, so I was Slated as right-handed, and it was Lucy's memories that were Slated. The later memories I had with the AGT survived. Lucy's early life is gone.\"\n\n\"Why would they do such a thing?\"\n\n\"As far as I understand, it was all part of a scheme to show the Lorders that Slating could fail: that any Slated criminal could be violent, even though that was supposed to be impossible. That none were safe.\" I don't spell out what the consequences of Nico's plans would have been. With no way to tell which Slated might turn, what would the Lorders have done to all the Slateds? I shudder inside.\n\n\"But if you were Slated, why haven't you got a Levo?\"\n\nThis is venturing into no-go territory; it would be dangerous for her to know how I was caught between the violent plans of Nico's AGT and Lorder blackmail. How they tracked me to the AGT, and I thought Agent Coulson was going to kill me, but Katran\u2014terrorist, yes, but an old friend who really _cared_ about me\u2014raced to my rescue, and Coulson shot Katran point-blank in front of me. How holding Katran as he died made me finally remember my dad's death. Because of Dr. Lysander, the Lorders thought I'd done as they wanted; they let me go, removed my Levo.\n\n\"Lucy? Sorry, Riley, I mean. What happened to your Levo?\" Stella prompts, and I wonder how long I've been staring into space.\n\n\"It was cut off,\" I say. A small lie. The Lorder method of removal was gentle: a few buttons pushed on a machine, and it painlessly sprang away.\n\n\"I didn't think that was possible,\" she says.\n\n\"It is,\" I say, and this I say with truth. I cut Ben's Levo off with a grinder, didn't I? He survived. Barely, but he did; then the Lorders took him away.\n\n\"There is something I don't understand. If you were Slated as right-handed, how can your years here be gone? You were left-handed until you were ten. You _must_ remember!\" She says the words like if she wants it enough, it will be so.\n\n\"I don't understand all the neurology of it. It's like what hand was dominant was plastic; it could be bent and changed. I think doing that was part of how my personality was fractured.\"\n\n\"So young.\" She shakes her head. \"But some memories stayed with you after you were Slated?\"\n\n\"Not exactly. To begin with, I was just like any other Slated. I had this new family, and\u2014\"\n\n\"Were they nice?\"\n\n\"Mostly. Mum and my sister were, though Mum was difficult to work out at first.\"\n\nShe holds still. \"You called this other woman _Mum._ \"\n\n\"I was Slated. They told us to do that.\"\n\n\"Sorry. It doesn't matter. And then?\"\n\n\"I started to get memories back.\" I hold back _how._ She doesn't need to know that I was attacked, that fear and rage crashed through the boundaries and made Rain emerge: the half of me that was pure AGT, pure terrorist, under Nico's spell, and ready to do whatever he asked.\n\n\"So what do you remember?\"\n\nI shake my head. \"I'm sorry. The memories I have are from after I left here. With the AGT. Before then is the half that was Slated.\"\n\nShe looks back at me, eyes desperate and pleading. \"But do you remember anything about me? Do you remember anything from here, before, at all?\"\n\nSomething, I don't know what, makes me say _no._ Even though there are some little snippets that _have_ come back: this cat, now curled up between us. Playing chess with Dad, and the rook. Is it because, as she said, I was left-handed when I was little? If that is true, then more may come back. Or is it because these are things that Rain knew? The worst memory of all\u2014Dad's death\u2014was suppressed, buried so deep it didn't come back until Katran died.\n\n\"Lucy? Riley, I mean. What is it?\"\n\nI shake my head. Does she know how he died? Does she know it was my fault? I can't say it out loud. Not tonight.\n\nI look past her, at the bedroom we're in. \"Was this my room?\" I ask.\n\nShe shakes her head no, and I'm relieved. It seemed _so_ not my room. I had that right at least. \"I put you in here because it's away from the other girls. Easier for me to visit.\" She hesitates. \"It used to be my room. A long time ago.\"\n\n\"Tell me everything I can't remember,\" I say. \"Please. I want to know it all.\"\n\nShe seems to hesitate, then holds out her hand again. A small thing, yet somehow it is so hard for me to reach out and take hers, to hold a stranger's hand, when her eyes are so full of desperate _want._ I do, and she grips mine tight once again. She smiles. \"What do you want to know?\"\n\n\"Everything, from the beginning. Tell me about when I was born. Where was I born? Was . . .\" And I hesitate. I've been so reluctant to mention him that it is just penetrating now that Stella hasn't, either. \"Was my father there?\"\n\nShe shakes her head, lips in a thin line. \"He wasn't there. He rarely was for the hard bits.\"\n\nMy eyes widen, a retort working its way up, but I bite it back.\n\n\"But you, Lucy, were the most beautiful baby that ever drew breath.\" She smiles. \"I'll show you.\" She gets up and takes out keys from her robe pocket. She goes to one of the locked wardrobes. \"I put albums in here for you: photos, all sorts of things you can look through from before. There are eleven albums, one for each year. We'll start making another one now, won't we?\"\n\nShe extracts an album and brings it over, places it in my hands, and I eagerly turn the pages. Well, okay; I _was_ a pretty cute baby. There is shot after shot of my general chubby baby cuteness: in a cot holding out hands and laughing; giggling in the bath; covered in mushy food. Always smiling. Didn't I ever howl? A few have Stella in them, also: hair dark then, smiling in a way that goes into her eyes. And there are empty spaces here and there; someone is missing. Removed? \"Why aren't there any photos of my dad?\"\n\nShe snaps the album shut. \"That's enough for tonight. You need to get some sleep. You have an early start tomorrow, don't you?\" She slips the album back into the wardrobe, locks it again.\n\n\"Can I have a key?\"\n\nShe hesitates, then shakes her head. \"No. You need your rest. We'll look at them together, all right? Good night, Lucy.\" She goes out the door.\n\nWell.\n\n_The Queen of the Weird_ : I hear Madison's words echo in my head, then feel bad. That's not fair. She's had a terrible hand dealt to her, hasn't she? Having her only child vanish when she was ten, then back seven years later, Slated, with no memory of her. She obviously had issues with Dad, also. I need to work out what that is about, what I should or shouldn't tell her about him. I sigh. I'm gripped by a need to know all I can of him, all I've forgotten, and more. I wonder if there are photos of him anywhere?\n\nI slip Pounce off my knees, walk across the room to the wardrobe with the albums in it, and assess the lock. A few twists with a hairpin and the lock clicks: open sesame! A skill learned from Nico.\n\nInside, the wardrobe has clothes hanging on one side\u2014summer dresses, put away for the winter? And the other side is shelves. The first few have albums numbered one to eleven as she said. But if she took Dad out of album number one, chances are the same is true of them all. The shelves below contain things wrapped in tissue paper. Curious, I draw out one bundle, take it to the bed, and open the paper carefully. Inside are neatly folded children's clothes. A girl's. Mine?\n\nI hesitate. I am trespassing on Stella's memories, wrapped up and locked away, for how long? It feels wrong.\n\nBut her memories should be my memories. I hold up a small dress, sized for perhaps a nine- or ten-year-old. It is pink and ruffled, really cute; way too cute, in fact.\n\n_I hated dresses. Especially pink ones._\n\nI almost stagger, put the dress down on the bed.\n\n_She made me wear it._\n\nMy head is spinning; I feel ill. I don't want to see any more. I fold them back up in the tissue paper, as careful as I can with shaking hands. This isn't what I was looking for.\n\nDad. I want photos of Dad.\n\nI put the bundle back where it was. The lower shelves just contain more tissue-wrapped bundles of what feels like clothing. More memories preserved and locked away. I stand back.\n\nThere is a top shelf, too high for me to easily reach, and I drag the desk chair across the floor and stand on it. There is a plastic box, pushed back so I didn't see it from below. I pull it off the shelf, put it on the desk, and take off the lid. _Bingo._ Framed photos, ones she has put away, out of sight. There has to be one in here.\n\nBut instead there are photos of a woman, one I don't recognize. The ones on top look old, going by the clothes, the hairstyles. Farther down is one of the same woman with a little girl, one hand on her shoulder; another with the girl a few years older. I gasp as I realize the girl is a dark-haired young version of Stella. The woman must be her mother: my grandmother. The Lorder JCO.\n\nI peer closer at her face, but don't see it in her, the Lorder stare. There are more recent ones; she is older, hair swept up and silvery gray, but she looks good for whatever age she must be. Sixty-something at least? She is thin, dressed well in clothes that look expensive but not showy. A kind smile on her face. I hold up a portrait of her and stare at her eyes; for no reason I can identify, I shiver, and hurriedly put it back down.\n\nI continue through the box. At the bottom is one last frame, and I draw it out.\n\nA group shot from a wedding: happy couple in the middle, a couple next to the groom that are probably his parents, and next to the bride, my grandmother.\n\nIt is hard to recognize the bride as Stella. Not so much from the unwinding of years or the white dress, but the youthful joy of her smile. And next to her in some version of a suit is Dad. Younger than my dreams, my memories, but there is no mistaking him. I reach out a shaking hand to the frame, to touch him. But he isn't looking at the camera; he is gazing at Stella, with so much love on his face that it is hard to even look at him.\n\nWhat happened to them?\n\nI pack the photos back away as they were, put the box back on the shelf. Lock the wardrobe and switch off the light. There are more boxes up there, and another locked wardrobe next to the first, but that is enough for one night.\n\nIn bed, suddenly aware how cold I have become, I pull the covers up and cuddle Pounce. She stays, warm and purring, and reminds me of Sebastian. I feel a pang of homesickness, for Mum and Amy.\n\nStella I can't think of as _Mum,_ or even as _Mother._ At least, not yet.\n\nThe only photo of Dad I've found so far in wardrobe number one is the wedding photo. Did Stella destroy them all, but couldn't bring herself to get rid of that one?\n\nAnd Stella hides all signs of her mother away in a plastic box in a locked wardrobe. Why?\n\nI suppose her being a Lorder is a good enough reason.\n\n\u2022 \u2022 \u2022\n\n_We sneak to the back door._\n\n_Daddy grins, holds up one finger to his lips. \"Quiet now, Lucy; we're spies.\"_\n\n_\"On a secret mission?\" I whisper, pulling my coat on when he holds it out._\n\n_He nods and winks, and we slip under the windows along the back of the house. He looks back at me following. \"Hmmm . . . Wait here a second,\" he says. He retraces our steps and moments later comes back, holding my wellies in one hand._\n\n_I roll my eyes._\n\n_\"Put them on, Lucy. One less thing to get yelled at over.\" He winks again. I struggle out of my hated pink shoes, already a bit dirty from the great garden escape, and am about to toss them behind the bushes when Daddy grabs them and carefully places them on a windowsill._\n\n_\"They'll be able to follow our trail,\" I warn._\n\n_He shrugs. \"I'm pretty sure she'll know where we've gone, anyhow.\"_\n\n_\"So why be sneaky?\"_\n\n_\"We're spies, remember?\"_\n\n_\"But I'm not dressed like a spy.\" I frown and hold out the ridiculous pink skirt that sticks out under my coat and do a spin in my camo wellies._\n\n_He laughs and bows down low. \"You are, indeed, the perfect picture of a demented princess spy, Your Majesty. Come on; your official birthday spying chariot awaits.\" We start to walk toward the lake and the kayaks._\n\n_But then a door bangs above. A voice calls out, \"Get back in here at once; your grandmother is here.\"_\n\n_\"Busted,\" I say._\n\n_\"Better go back, Lucy.\"_\n\n_\"Why?\"_\n\n_\"She just wants to say happy birthday. Go on.\"_\n\n_I sigh and start trudging back to the house, feet like lead. When I reach the window with my waiting shoes, I turn around; Daddy is gone. A distant splash says my spy chariot has launched without me._\n\n_At the back door I take off my wellies and slip on the pink satin shoes. They're better for spying in, anyhow. Still in the game, I creep without a whisper of sound; not down the main hall, no. Spies go careful, quiet, down secret ways. I slip through Mummy's study and out the door hidden behind the curtains. Down the tiny hall that goes around the sitting room where I know they'll be._\n\n_One more step, then another . . ._\n\n_Their voices change from a murmur into words I can hear, then wish I hadn't._\n9\n\nMEOW? MEOOOOW.\n\nHmph? I open one eye. It's still dark, and Pounce is scratching at my bedroom door. I get up and open it for her. She disappears down the stairs.\n\nI squint at my watch: 5:20. Thanks for the early wake-up call, cat. I yawn and stretch, shivering as I shrug my robe on and pull it close. There's no way I'll get back to sleep now.\n\nThat dream was so _weird,_ yet somehow I know in my gut it was real. It happened. Was it that horrible pink dress that brought it back?\n\nIt was happy at first, off on an adventure with Dad; then . . . what? I overheard _something_ between Stella and her mother. Something upsetting. What was it?\n\nI head down the stairs and up the hall for a drink. As I go, motion sensor lights momentarily blind my eyes, illuminating patches of darkness, then go out just as the next takes over. I start down the wrong hall, uncertain of the way, and double back to find the reception area from yesterday with its tea things.\n\nWhile I wait for the kettle to boil, I turn the lights off again and wander to the windows that overlook the lake, but it is lost in inky darkness. A spy kayak: Are they still there? I smile to myself, then frown. Dad went off without me, left me to go back and face them alone. _Never there for the hard bits,_ isn't that what Stella said? No. That isn't fair. Trying to rescue me from Nico was a _very_ hard bit. Failing was the hardest of them all.\n\nThe room is suddenly cast in bright illumination. A girl yawns in the doorway, then jumps when she sees me: Madison.\n\n\"You don't strike me as the early morning type,\" I say.\n\n\"Who, me? No, to be honest. But it's a seven o'clock start at the caf\u00e9 to take care of all the early breakfast-eaters of Keswick. How about you?\" she says, both of us heading for the kettle.\n\n\"The CAS thing starts at eight.\"\n\n\"Lucky. Couldn't sleep?\" I shake my head. \"Nervous?\"\n\nI look at her quickly, then realize she means what I am officially here for: CAS. I've been so caught up in Stella and my missing past that I haven't thought about it at all. Another new place, new people, more not knowing what to do or say while trying to remember to answer to _Riley Kain_ and not to say anything she wouldn't. Suddenly it is all way too worrying. I sigh.\n\n\"Tell you what. Come with me on the six thirty bus, and I'll show you where you've got to go, then I'll make you an amazing breakfast at the caf\u00e9. My treat.\"\n\n\"Really?\"\n\n\"Sure.\" She holds up her cup of tea. \"Here's to firsts. Jobs, I mean,\" she says, and winks in a way that suggests she was thinking of something else entirely. She clinks her cup against mine, then winces. \"That was loud. Meet me back here in an hour.\"\n\n\u2022 \u2022 \u2022\n\nAn hour and a shower and change later, we're heading for the door. Madison pauses at a table, flips open a folder with pages of columns, writes her name and the time in the Out column, writes _work_ under Description. Hands me the pen.\n\n\"What's this?\"\n\n\"Haven't you read your rules yet? That probably breaks one of them.\" She grins. \"This is rule number twelve: Always sign out when you leave, and in when you get back.\"\n\nI bend to print _Riley\u2014CAS,_ and realize that it is the first time I've written my new name.\n\nWe step outside into the dark morning.\n\n\"I hate this time of year. It's like midnight,\" Madison says.\n\n\"I like the darkness,\" I admit. I like how it covers and hides, and the chill as well. The ground is frozen, crunches under our feet as we take a path up behind the house through silent trees to the road above.\n\n\"No bus stop?\" I say.\n\n\"No. You just hail them. Comes every thirty minutes or so.\"\n\nSoon a bus appears in the distance. Madison waves and it pulls in and stops. We scan our IDs as we get on, start down the aisle; Madison aims for a seat near the back.\n\n\"Oh my Lord. Is it possible?\" a voice says to the side. Male.\n\nMadison pauses, turns. \"Is what possible?\" she asks.\n\n\"Don't sit down yet, I need to be sure,\" he says, and Madison holds on to his seat as the bus starts up the windy road. He smiles, and something passes between them in the cold air. Is he her boyfriend? Even sitting down he is taller than she is; a rugged, outdoorsy type. Tanned even in January.\n\nHe looks between Madison and me, then glances at a few friends sitting in front. \"Wow. It really _is_ true,\" one of them says.\n\n\"What?\" Madison demands.\n\nThe one smiling into her eyes answers. \"At long last, Shorty. There is somebody shorter than you.\"\n\nHis friends laugh, and she punches him in the arm. Straightens her shoulders as if aiming to be taller, then slips into the seat opposite his. I sit next to her.\n\n\"Who's that?\" I say, voice low.\n\n\"That six-foot brat is Finley.\" She raises her voice: \"He and his friends are total ARSes.\"\n\nHe leans across. \"We are indeed. You're just jealous.\" I look between them, brow knotted in confusion. \"We're in ARS: the Apprentice Ranger Service,\" he explains.\n\n\"Generally known as ARSes,\" Madison adds.\n\n\"Only you can get away with that, Shorty,\" he says, and winks. \"Who are you?\" he asks, turning his smile on me.\n\n\"Riley,\" I say, managing to get my name right. \"I'm here for the apprenticeship scheme.\"\n\n\"Hey, you could be an ARS, too!\" Madison says.\n\nHe shakes his head, laughing. \"I'm sure there must be some sort of minimum height requirement.\"\n\nWith that, the bus stops; we're in Keswick.\n\n\"Ladies first,\" Finley says, and we get off the bus.\n\nWith a wave to the boys, Madison slips her arm in mine. She shows me the building where I need to go at eight, then takes me to her work: Cora's Caf\u00e9. We go in the back way; the lights aren't on yet.\n\n\"Hello,\" Madison sings out as she unlocks the back door.\n\nA woman in a chef's hat bustling about a cramped kitchen looks up, and scowls. \"Glad you decided to turn up.\" Madison sticks out her tongue. \"And who's this, another waif who needs feeding up?\"\n\n\"Ah, sorry . . .\" I start to say, and back toward the door.\n\nShe laughs. \"Kidding, kid. I'm Cora; come in.\" They put me at one of the tables in the front of the caf\u00e9, bickering all the while. Minutes later the lights come on and the doors are unlocked. Early customers flock in, and we're soon busy tackling the most massive and delicious cooked breakfasts ever.\n\n\u2022 \u2022 \u2022\n\nA bit later, too much breakfast churns uneasily in my stomach as I walk up to the government building Madison pointed out earlier. The sign on the door says CUMBRIAN APPRENTICESHIP SCHEME SEMINAR. It looks so _official,_ and to me, _official_ means _Lorder._ Does Aiden know what he's doing, sending me to this? He usually does. I hesitate, watching others go past me and through the door.\n\n\"Hey, it's ES,\" a voice says behind me, and I turn; it's Finley.\n\n\"ES? What's that supposed to mean?\"\n\n\"Extra-Shorty. Shouldn't you be walking through the door instead of staring at it?\"\n\n\"Why are you here?\"\n\n\"I'm one of the good examples. Shocking, I know. Come on.\" He holds the door open. \"Sign in there,\" he says, and points at a table with a line of people. \"See you later.\" He waves at someone across the room and saunters away.\n\nI wait my turn.\n\n\"Name?\" says a woman with an overly bright smile and hard eyes.\n\n\"Ky\u2014\" I cough. A fake cough to cover nearly saying _Kyla. Get a grip._ \"Sorry. My name is Riley Kain.\"\n\nShe scans a netbook. \"You're not on the list. Next?\"\n\nA boy steps up around me.\n\n\"No, wait a minute. I should be on it. Can you check again? That is Kain, with a _K_?\"\n\nShe sighs. Looks again. Smiles. \"You're still not on the list.\" She turns to the boy.\n\nI start to panic. Could Aiden have messed up? No. \"I might have been put in last-minute.\"\n\nShe sighs again. \"A Late Add\u2014why didn't you say so?\" She touches her screen. \"There you are. Fill this in so we can put you on the registered list.\" She hands me a handheld device with my name on the top, blanks to fill in. Starting with date of birth. When was that again?\n\n\"Not there,\" she says. \"You're in the way.\" She points to the side and I scurry away, face pink.\n\nI touch the screen and try to remember the contents of Aiden's file. My new birth date finally comes to me: September 17, 2036. I fill in the rest\u2014address, hair, eyes, height. And then I'm stumped. Emergency contact? Aiden never gave me an address for my fake parents in Chelmsford. Finally, with no other option coming to mind, I put _Stella Connor, Waterfall House,_ and click enter.\n\nI approach the table. She ignores me, checking others in. \"Excuse me,\" I say finally.\n\n\"About time,\" she says, takes the device, updates her netbook. \"You're all registered now. Here you go.\" She hands me a folder. \"Take a seat, Riley.\"\n\nI sit down near the back. There are about fifty of us now, a few empty seats here and there still. Everyone else is chatting, seems to know one another. Are they all local? A few glances are cast my way and I try to smile, but they aren't particularly friendly glances; after a while, I give up and ignore the looks. Finley is off to the side, standing with some others. My eyes seek his, and he winks.\n\nA few more shuffle in, and then, abrupt silence.\n\nA man in a rumpled brown suit walks to the front. A half smile on his face, he gazes at the assembled faces, moving across them so every one is acknowledged. His eyes pass over mine with a slight pause.\n\n\"Good morning, everyone,\" he says at last. \"I'm pleased to see so many familiar faces have come in for the Cumbrian Apprenticeship Scheme this morning, and a few unfamiliar ones, as well.\" His eyes touch mine again, and then another's: a boy near the front. \"For those who don't know me, I'm Councillor Watson. On behalf of the Central Coalition I'd like to welcome you to this opportunity, the gateway to your futures. The Jobs For All policy of the Coalition is in its twentieth successful year, and the apprenticeship schemes are a vital part of its success. I'm going to hand things over now to your local apprenticeships coordinator.\"\n\nThere is a polite scattering of applause, and another speaker approaches the front. Out of the corner of my eye, I see Watson exit the hall at the back, and everyone visibly relaxes.\n\nOver an hour the scheme is explained in detail. Representatives and apprentices are here from each section today, and we can speak to them and ask any questions. The sections seeking new apprentices are Administration, Hospitality, National Parks, Transportation, Education, Enforcement, Communication, and Sanitation. Tomorrow we must come in and sign on the dotted line to commit to CAS for five years. Then we pick our favorites, and take aptitude tests.\n\nOn Monday we find out which four we get to trial. Then follows weeklong placements with each, and finally a section is chosen for each candidate. He doesn't say who does the final choosing, and I'm guessing, not us.\n\nSo, without any guarantee where we'll end up, we must first commit to _five years_? That sounds like forever.\n\nWhen he's finished, doors are opened to an adjacent room, with areas set aside for each section. Everyone spills out and seems more interested in getting a cup of tea than speaking to the representatives and apprentices. Then I notice waves and hellos exchanged here and there. Do they already know what they want to pick? More, do they think this is just a formality; has everyone already worked out who will go where?\n\nA woman at the nearest table\u2014Education\u2014catches my eye and smiles, and there is something about her that makes me smile back. I walk over to her.\n\n\"Hello,\" she says. \"Have you thought about working in schools?\"\n\n\"No,\" I answer truthfully.\n\n\"Honesty! An excellent trait.\" She looks at me quizzically. \"I never forget a face, and there is something familiar about you, but I'm stumped. You're not local?\"\n\nI shake my head, careful to hide alarm; could she see Lucy in me despite the changed hair and eyes, after so many years? \"I'm from Chelmsford.\"\n\n\"I didn't think I recognized you, and every child in Keswick goes to my school. But that doesn't matter to me, and it shouldn't to anyone because place of origin is not admissible criteria.\"\n\n\"Really? I was thinking I was heading for Sanitation for sure.\"\n\nShe laughs. \"Well, in case you'd like another option, we're looking for three apprentices in Keswick Primary School. You start as a teaching assistant, and if all goes well can move on to teacher training after a year.\" She starts enthusing about inspiring young minds, and I think of the smiling boy on the train, taken off by Lorders.\n\n\"Is everything all right?\" she asks.\n\nI start. Am I that transparent? \"I'm not sure about Education; I haven't been around little kids much, and\u2014\"\n\n\"Well, that is the whole point of this scheme. If you pick Education, you get to spend a week with us in the school, and we'll both soon know if it's for you.\"\n\n\"Thank you,\" I say, and it isn't just for what she has said, but the warm way she has said it.\n\nShe seems to know what I mean, and smiles again. \"Go on, talk to everyone else; none of us bite!\" She leans in closer and lowers her voice. \"Except maybe Enforcement.\"\n\nI straighten my shoulders and start at one end of the room, visiting each section in turn, but skip Enforcement. The latter aren't Lorders; they are the local force that deals with parking and minor matters, but anything that says _authority_ says _keep away_ to me, and besides, they'd work with Lorders, wouldn't they?\n\nIt soon becomes apparent by foot traffic that there are two main points of competition: Hospitality and National Parks. At the latter there is now a small crowd. An unfriendly crowd, as I find when I try to inch my way into it.\n\n\"Hey, it's ES,\" Finley says, tall enough to see over everyone that I'm there and not making progress.\n\nFinley pulls me to the front and soon I'm face-to-face with his boss, who looks at me and raises an eyebrow. \"Considering a career with the National Parks Authority?\"\n\n\"Of course.\"\n\nHe sighs. \"It isn't all mountain trails in holiday sunshine.\"\n\nI bristle at his tone. \"Of course not. It's conservation, public access, education, and safety.\" I'd been lurking at the edges long enough to already hear the spiel.\n\n\"Do you have any relevant skills?\"\n\n\"I can read maps, and I know how to use a compass. I'm a runner and experienced walker, so I'm fit. I love the outdoors in all weather.\"\n\n\"Really?\" His voice is still skeptical, and even though I had no idea what National Parks did until five minutes ago, something about his tone has my back up.\n\nI straighten and meet his eye. \"Try me and you'll see.\" A challenge thrown down.\n\n\"Well, well. You never know.\"\n\nI walk off to hostile glances from many of the other hopefuls; Finley follows.\n\n\"You handled him well.\"\n\n\"Did I?\"\n\n\"But I wouldn't get your hopes up. They'll trial ten and take five this year. But most of the others have been volunteering with National Parks all through school and have staked their claims; even if you make it onto the ten, it's a tough competition.\"\n\nSo much for place of origin being inadmissible criteria.\n10\n\nI WANDER INTO KESWICK, THE AFTERNOON FREE. SHOULD I go back to the house? Probably. Most likely everyone will be at work, and Stella and I could talk some more, and isn't that what I'm here for?\n\nBut the sun is shining. It's lunchtime, but after the Breakfast of Giants this morning I'm good, and the sun sparkling on the snow-dusted fells above is calling, making my feet restless.\n\nI wander about Keswick to start with, paying no attention to where I am going, and after a while find myself outside Keswick Primary School. That teacher said every child in Keswick goes to this school: It would have been my school. It must be their lunchtime; there are children running and playing all over the grounds. It looks like a happy place, without the undercurrents at the secondary I was at until recently. Do they get visits from Lorders here, too? Do Lorders stand to one side during school assemblies and drag off troublemakers, never to be seen again? No. That'd be ridiculous. It's a _primary_ school, not full of potentially dangerous teenagers. I stare a moment at the white buildings, but nothing feels familiar.\n\nAll the while the fells are calling me. I want _up,_ to climb into the sky and touch the sun. I start to follow a footpath sign that leads out of town, taking whichever way goes up and out. Then I stumble across another sign that points the way to _Castlerigg Stone Circle._ I almost stop breathing when I read the words. Is this the stone circle from my dream, the one with Dad? Counting the stones together: the mountain's children.\n\nI'm walking faster and it isn't fast enough, so I start to run. It's uphill on uneven ground and the cold air catches in my throat, but it feels good to be running. I told that National Parks rep that I was a runner, but how much have I done of it lately? I've barely even jogged; it reminds me so much of Ben that it hurts. But now my mind is full of Castlerigg, of getting there as soon as my feet can take me.\n\nI slow to a walk when I finally see a gate in the distance. It is _the_ gate; I'm sure of it. I pull my coat in tight around; despite running and the sun, the temperature seems to have dropped and has a prickle of expectation about it. Snow? Distant clouds are moving closer.\n\nLeaning up against the gate, I can finally see it. A wide field with the stone circle at its center; the mountains, standing guard, are an amphitheater all around. I open the gate and step through, then stand there, staring, something stirring and shifting inside. Not just a dream, I'm sure of it. I _remember,_ and the joy of memory makes me laugh out loud. I've been _here,_ many times before, in all weather: picnics on sunny summer days, walks in blustery autumn rains and snow-covered magic, searching for bright dots of spring wildflowers. It was our place, mine and Dad's: our special place we came again and again.\n\nI walk to the stones, but don't start counting yet. I have to start in the right place: where some of the stones reach into the circle. We start here so we don't lose track.\n\nClose up some of the stones are huge, but not so much as in my memory when they were giants. Now, some are even shorter than I am. I reach the first one, press my hands against it, then lean into the cold stone: hands extended, face turned so my cheek is against it as well. I close my eyes. Number one.\n\nAll I have been and all I have been through these last years seem to fade away, leaving just Lucy. A little girl with her dad. I open my eyes again. Is it this place, these ancient stones? Thousands of years old, do they do something to time, make seven years seem of no consequence? I feel myself stepping back as I was then, and start running between the stones, tagging each one with my hand and counting as I go.\n\nIt's getting darker, and colder, and all at once there are fingers of mist wrapping around the stones. The sun vanishes. _Lake District weather: Blink and it changes._ The words are in my mind, unbidden. Who used to say that? Eyes closed again, I lean back against another stone and feel like I'm sinking into it, getting colder but not caring, reaching _back_ for something else, without knowing what.\n\nSome sense of disquiet takes hold; this wasn't always a good place. I push the thought away, wanting to stay Lucy, but she is slipping away.\n\nHow long have I been here? I'm shaking with the chill, and the light is starting to fade. I should go back, catch the bus with Madison when the caf\u00e9 closes at five. I squint at my watch: almost four. I should have enough time to get there by then, but I'm disoriented. Which stone was I up to, and which way is the gate? I don't know. I peer at the mist, but it holds its secrets; I can't see beyond a few meters of frozen field. A shiver runs down my back. What if I had followed my first impulse, and kept climbing up, up, up? I shudder to think of being on the top of a ridge and unable to see the way.\n\nFine park ranger I'd make.\n\nI step out across the frozen ground in my best pick of the right direction, hoping the mist will lift. Long after I expect to, I hit fence: no gate. No problem: follow the fence around. I start out, keeping the fence close and in sight, but then walk for so long I know I've gone the wrong way. Turn around? No. Keep going; it's the only way to make sure I don't end up going back and forth. I reach a gate, but it looks different from the one I came through. Ah. I'm on the opposite side, the one with the parking lot. Now the gate I want is just as far again around the other way.\n\nFinally I reach it, go through, and follow the footpath down. The lights of town start to penetrate the mist; it is lifting as I reach the first houses, so I run full speed up the streets, back to the center.\n\nAs I round the final corner, the bus is pulling away. I wave; he stops. I step on, breathing hard. Flustered, I have a moment of panic when I can't find my ID, but it's in my other pocket. I scan in and start down the aisle. A hand waves: Madison. She scoots across so I can sit next to her in the aisle seat.\n\n\"Riley? I thought the apprenticeship seminar ended hours ago! How was it?\"\n\nThe morning seems like _ages_ ago. \"Good, I guess. I'm thinking of Parks. Maybe Education.\"\n\nShe looks at me curiously. \"What happened to you this afternoon?\"\n\nI shrug. \"Nothing much. I went for a walk.\"\n\n\"You're going to catch it when we get back.\"\n\n\"Why?\"\n\nShe shrugs. \"Don't worry, it'll be fine. You should have written that in the stupid book when we left; Stella's probably ticked off 'cos she hasn't known where you were every second of the day.\"\n\n\"Really?\"\n\n\"Don't worry, you didn't know, did you?\"\n\nBecause I didn't read the rules.\n\n\u2022 \u2022 \u2022\n\nStella stands by her desk in the reception area, arms crossed, rigid with tension. Her head turns when we come in, and her eyes fix on me. Something changes, relaxes on her face when her eyes reach mine, and I try to say _sorry_ without saying anything. She smiles, then looks to my side at Madison, and her smile vanishes.\n\n\"Hi, Mrs. C,\" Madison says. Her arm hooked in mine starts to pull me across the room.\n\n\"Not so fast,\" Stella says. \"Riley? I'd like a word. This way.\"\n\nShe turns and walks toward another door behind her.\n\n\"Uh-oh,\" Madison says. \"A chat in her private study. Good luck.\"\n\nI follow Stella through the door, and it swings shut behind me.\n\n\"I'm sorry about the book, I didn't realize\u2014\" I start to say, but then Stella steps up to me and sweeps me against her in an awkward hug, a tight one, and she is all bones and angles, thin and desperate.\n\nShe lets go. \"I was so worried. Don't do that to me again!\" she snaps, sits at her desk, face cross once again.\n\n\"Why do you make everyone account for all of their days like that? If you didn't, you wouldn't worry when someone forgets to write something in the book. You should trust us. Can't we be allowed out in daylight hours? We're all over eighteen. Or are supposed to be,\" I add, since I'm not, but I'm sure the rest of them are.\n\nShe shakes her head. \"I'm responsible for every girl in this house, and I take that very seriously.\"\n\n\"Oh. Do you have to do that, as part of this being a supervised house for under-twenty-ones?\"\n\nShe hesitates.\n\n\"You don't, do you. It's just you who makes them do that; makes _us_ do that.\"\n\nShe shakes her head. \"You're the one who has broken the rules here; don't argue.\" She softens. \"I can't have different rules for you than the other girls.\"\n\n\"Of course not.\"\n\nShe sighs, shakes her head. \"I was so scared something happened to you, that they worked out you weren't Riley Kain and took you away from me all over again.\"\n\nThen I'm contrite. \"Really, I'm sorry. I haven't read all the rules yet,\" I admit. \"I didn't know I needed to write down what I was doing for the afternoon.\"\n\nShe opens a desk drawer, hands me another copy of the rules. \"Then you shall sit there and read them before dinner, before you can break any more of them accidentally.\"\n\nAnd it's not so bad. I sit in an armchair in the corner of her study, a little cold but before long Pounce finds me and curls up warmly on my lap like my own hot water bottle. Stella brings me a cup of tea, and I start at the top. Rule one I've already covered: Be nice to Pounce. I scratch behind her ears and she purrs. The rest of them are mostly easy, logical stuff, like not wearing outdoor boots on the carpets, keeping doors locked at night.\n\nAbout halfway through, I pause, look up. Is this study the same one as in my dream? There are long curtains that cover the windows and also curve into the room, cover part of the adjacent wall. I slip Pounce off my knees, walk to the curtains, and pull them to one side. There is a door! Just like in my dream. Unable to stop myself, I reach a hand to push the door open, but then hear voices outside the study.\n\nI scoot back to the chair and pick up the rules just before Stella comes back in. She gets something from her desk, leaves again.\n\nI better finish reading these before dinner. _Concentrate._ There is also quite a bit on curfews, random room checks, and keeping track of where we are and what we're doing. \"She's a bit of a control freak, isn't she?\" I say to Pounce in a low voice.\n\nAn uncomfortable feeling inside wonders: Was she always like that, or was it my being taken away from her that made her so?\n\n\u2022 \u2022 \u2022\n\nThat night, soon after lights-out\u2014which I know now is eleven P.M.\u2014there is a soft knock on my door, and it opens. Stella peeks in. \"Are you awake?\" she asks.\n\n\"Yes,\" I say, and she seems to be hesitating. \"Come in.\"\n\nShe walks across the room, pulls a chair in next to my bed like she did last night.\n\n\"But you do realize it's lights-out now,\" I say. \"And I'm supposed to be sleeping.\"\n\n\"Don't be cheeky. I know you've got an early start again tomorrow, so I won't stay long.\"\n\n\"It's okay. I'm not much of a sleeper.\" Just now my mind has been full of mountains; ridge walking on high fells.\n\n\"You never were. You kept me up half the night until you were four. And then later with nightmares.\"\n\n\"What did I have nightmares about?\"\n\n\"All sorts of things. Monsters under the bed. Something happening to . . .\" And she stops. \"Usual kid stuff, I guess.\"\n\nSo was I always like that\u2014with vivid dreams and nightmares? I thought it was just my fragmented memories that haunted my sleep.\n\n\"Can we look at more photo albums?\" I ask.\n\n\"Not tonight. I want to talk to you about something. How'd CAS go today?\"\n\nI shrug. \"All right.\"\n\n\"You do know if you sign up tomorrow that it's five years, and you may not get one you want.\"\n\n\"They explained it; I know. But\u2014\"\n\n\"I've got another idea. Why don't you work here instead?\"\n\n\"What do you mean?\"\n\n\"Here, at the house. I've usually got two girls working for me, but one turned twenty-one a few months ago and left.\"\n\n\"Doing what?\"\n\n\"You know: looking after the house. The garden in the summer. Helping with cooking.\" She looks at my face. \"I know it doesn't sound very exciting. But we could spend more time together, get to know each other again. And it would be safer. There'd be less chance of anyone finding out you're not who you say you are.\"\n\n\"I don't know. I'm interested in apprenticing with National Parks.\"\n\n\"That might be hard to get into.\"\n\n\"Did I used to like walking?\"\n\n\"And running; you were never still for a minute.\"\n\n\"No, I mean in the mountains\u2014the fells. Up in high places.\"\n\nShe hesitates, finally nods. \"I think you were born part mountain goat. You loved it.\"\n\nI read her face. \"But you didn't.\"\n\nShe sighs. \"No. I haven't got a head for heights. And I worried you'd slip and hurt yourself.\"\n\n\"If you don't like heights, who did I go walking with? Was it my dad?\"\n\nShe nods, finally admitting he existed. \"That was the other reason I didn't like it.\"\n\n\"What do you mean?\" She hesitates again, and I jump in. \"I kind of get the feeling you and he didn't get along. But he is part of my past, part of where I come from. I need to find out about him, too.\"\n\nShe stares back, finally nods. \"Of course. I'm sorry. Yes, your dad took you ridge walking.\" She pauses and I stay silent, looking at her, saying _give me more_ with my eyes, and I can see something relent in hers. She takes my hand. \"All right. Your dad\u2014what can I tell you about him? He was always a dreamer, off in some other place in his head. He had a way of taking you along to lands of make-believe, where anything was possible. That was what drew me to him, but it wasn't enough. Not once you came along. Danny wasn't the most reliable person in the world; as quick to anger as he was to joy, and easily distracted. I was always scared he'd forget you were following along and lose you somewhere along the way.\"\n\n\"But he didn't, so you were wrong.\"\n\nShe stiffens. Something shutters in her eyes, and I wish I could take the words back. She lets go of my hand. \"That's enough talking for tonight.\"\n\nShe stands, walks to the door. Turns, face softer again. \"Please, just think about CAS. Do you really want to sign your life away for five years? You could end up doing anything, not what you think you want. Wouldn't it be better to work here than end up in Sanitation? And you could always leave it for now, and sign up when it opens again in six months, if you still want to.\"\n\n\"All right,\" I say. \"I'll think about it.\"\n\n\"Good night, Lucy. I mean Riley.\"\n\n\u2022 \u2022 \u2022\n\nAs I'm drifting off to sleep, I do think about it. About being in this house twenty-four hours a day, and only being able to escape if I can come up with a good reason to write down in a column in a book. And having to come back before the reason is up.\n\nAnd then I think about the mountains: walking in high places, climbing to the sky. With my dad: Danny the dreamer. I have his name now, and I hug it close.\n\nThat night's dreams are indistinct, and glorious.\n11\n\nTHE NEXT LECTURE IS EARLY MORNING ON THE BUS, AND comes from a surprising source: Madison.\n\n\"Are you sure you really want to sign up for CAS?\"\n\nI look at her in surprise.\n\n\"It is _five years,_ and they pay next to nothing the whole time. And you could wind up someplace awful. You might even\"\u2014her face crinkles in horror\u2014\"end up working with Finley.\"\n\nHe turns around from his seat in front of us and winks. \"She should be so lucky,\" he says.\n\n\"What do you think? About signing up?\" I ask him.\n\n\"It was the best thing I ever did,\" he answers seriously. \"I love it.\"\n\n\"But\u2014\" Madison interrupts.\n\n\"No buts. But if you're counting on Parks, don't. Have a plan B.\"\n\n\u2022 \u2022 \u2022\n\nNot much later I'm in the seminar room we were in yesterday morning, staring at a contract.\n\nThe last time I signed a contract I was a new Slated, just leaving the hospital. That seems such a long time ago. Then, I had no choice; imagine what would have happened to me if I refused to sign? That contract was all about promising to follow rules: of my new family, school, and the community. About doing my best to fit in, not get in trouble. I meant it when I signed, but that didn't last long, did it? If the Lorders found me now, they'd drag me off as a contract breaker. It is very high on the _thou shalt not_ list for Slateds to seek out their old lives. Using IMET to change my appearance, the assumed name, and the fake ID all add to my sins.\n\nBut this contract, today, is up to me. I get to decide. I chew on my pen, try to concentrate and read it properly.\n\nChairs shuffle around me. No one asks any questions; they all just sign without reading and hand them in. Soon eyes will notice my hesitation.\n\nWhat does it mean? Five years, an apprenticeship. Being trained in a career, whatever it may be: my choice, or theirs.\n\nA life, one of my own. As a Slated I wouldn't have been able to do this until I was twenty-one.\n\nIf seven years ago I hadn't been kidnapped by the AGT, hadn't ended up a terrorist, hadn't been Slated, what would my life be now? Would I still be sitting here, at this precise moment in time, trying to make this choice? Maybe _Lucy_ would have been happy, excited. Signing the contract, then going out with all her friends this weekend to celebrate. Maybe she'd be sure she'd get her top choice, here in her hometown where everybody knew her.\n\nWould Stella still be there in the back of Lucy's mind, arguing against this, wanting to keep her safe, keep her close, at home?\n\nI sign: _Riley Kain._\n\nNext is the form to select my top choices. I put National Parks as one; hesitate, then put Education as two. I stare at the others, then, aware I'm one of the last still lingering over paperwork, I scribble the other sections in randomly, with Sanitation and Enforcement right at the bottom. I hand in the forms.\n\nHours of aptitude tests follow: comprehension, math, strange logic problems, and sequences. When it is finally time to go, we are told to report back Monday morning at eight A.M., when we'll find out which four weeklong trials we will do, and head straight off to start the first one that very day.\n\nOutside, yesterday's sun is a distant memory; the sky is gray, the mountains hidden in cloud. It makes it easier to catch the bus straight back to the house. Time to face up to Stella and tell her I signed on the dotted line.\n\n\u2022 \u2022 \u2022\n\nWhen I get there the side door we usually use is locked. Lucky I read the rules last night, and so know the door code. I punch the numbers in and step inside. The house is quiet.\n\nI open the book to sign in, and with a start see _Stella Connor_ written in the Out column, with _shopping_ in the description, and return time estimated as four. She follows the rules, too? Another name, _Steph,_ is also down as shopping; is she the girl who works for her? I scan today's page; it looks like nobody is home, and nobody expected back for at least a few hours.\n\nTime to explore.\n\nI'm gripped by a peculiar sense of excitement. At first I creep around, quiet, almost like I'm expecting someone to jump out and demand to know what I'm doing. I start with the public rooms, interconnecting hallways, stairwells: wandering, trying and failing to find something, anything, that feels familiar. At one point Pounce jumps down from a hall table as I round a corner, and I jump and almost scream.\n\nThe house is huge. Pounce follows as I wander into the massive shiny kitchen, utility rooms, walk-in cupboards. Nothing is familiar. Though the kitchen looks new; maybe it has changed since I was here? Then I try the door to Stella's study where I was yesterday: locked.\n\nStella had said the tower room I'm in wasn't my room before. Which was? I try to cast myself back, to stop thinking, to just follow my feet, but nothing works. I've had dreams of my room here before, but if I close my eyes and try to see it, all is uncertain. There is just a sense of proportion, some white-painted wardrobes, a too-frilly bed. Maybe there are photos of it in the albums?\n\nBack in my room, door shut, I get the hairpin out and spring the lock to open the wardrobe. Lugging the albums to my bed, I lay them out in order, number one to eleven.\n\nI open one album, then another. I scan through, looking for photos taken in my bedroom, but soon get distracted when I realize that apart from the first one where I am a baby, each one starts with a birthday. From my first birthday with cake smeared on my face, to a toddler and on up. Birthdays looked like serious business. Each year there was an amazing themed and decorated cake: fairies one year, ponies, pixies.\n\n_Mummy made them for me._\n\nA shiver goes up my spine. Mummy? Stella. There are often smiling snaps of her, her arms around me, my arms around her. I was just a few years old when her dark hair changed to pale blond like mine. As I got older and my hair a little darker, so, weirdly, did hers. Almost like she was trying to make us match.\n\nAgain there are none of Dad. There are odd blank spaces in the albums as if some photos have been removed, but the spaces are few and far between; he couldn't have been in many of them to start with.\n\n_He took most of the photos._\n\nAn image of him behind a camera jumps into my mind, then vanishes. I must ask Stella what happened to the photos she took out of the albums. Did she put them away someplace, or destroy them?\n\nSomething makes me push the other albums aside and reach for the last one: number eleven.\n\nFirst page: Lucy smiling back. Me. Goosebumps travel up my arms and spine: I'm wearing the pink dress. The one in the cupboard, the one in my dream. Pounce jumps up and walks over, peers as I turn the pages. \"Look, it's you,\" I whisper, not sure why I'm whispering. There is photo after photo of Pounce as a tiny kitten: chasing bits of string, sitting on my knee, sleeping in my arms. My tenth birthday present. Then there are photos of me with a big pink iced birthday cake: princess-themed this year. Ten candles, so I had that right.\n\nA few pages later, something changes.\n\nI'm still smiling, but something is wrong. It is on my face; I'm holding something in. I can almost feel this moment, but then it slips away.\n\nWhat?\n\nI turn another page, and there is a picture of me taken outside on a sunny day, with the peaks of Catbells behind. Pounce is clutched in my arms, and my gap-toothed grin is a real smile this time.\n\nI turn the page: nothing. I scan through more pages to the end, but the rest of the album is blank. Something cold grips me inside.\n\nI shut it. My heart is thudding hard behind my ribs. That last picture of me is the one that was on MIA. The last one Stella had before I disappeared. The one she gave to the website in the faint hope I might be found and come back to her one day.\n\nI don't know how long I sit there, staring at the wall, thinking. I don't know why I've been remote to Stella: Is it the neediness and hunger in her eyes that makes me pull away? What we were to each other back then is all over these pages. To her, our relationship is real and immediate; to me, it is a bare echo, like a song I half heard once but can't really remember.\n\nAnd now I've signed up for CAS instead of staying here with her.\n\n\u2022 \u2022 \u2022\n\nA distant noise penetrates: a car? I look at my watch. It's almost four; they must be back. I jump up, hurriedly place the albums back in order in the wardrobe, lock it, and go downstairs.\n\nStella is in the kitchen unpacking boxes of groceries. I stand in the doorway a moment; she smiles to see me when she looks up.\n\n\"Can I help?\" I ask.\n\n\"Of course.\" We slot things into two massive fridges, and she shows me which cupboards things go in.\n\n\"Now for a cup of tea and a chat,\" she says, and puts the kettle on. \"Steph stayed in town, so we should be on our own for a while.\" She pours tea, and we sit on stools at the island.\n\n\"There's something I've got to tell you,\" I start to say, then hesitate.\n\n\"What is it, Riley?\"\n\n\"I'm sorry, but I signed up for CAS.\"\n\nI brace myself for her reaction, but she looks back at me calmly, sips her tea. \"I thought you would. Though you know I'm not keen on National Parks. Some of what they do can be dangerous. And it'd be a disaster if they put you in Enforcement, with all the enhanced security checks. But we'll just wait and see what apprenticeship they give you; no point in worrying about it yet, is there?\"\n\nI stare at her, past surprise into shock. This _so_ isn't the reaction I was expecting. \"Thanks, Mum,\" I whisper.\n\nHer face goes all funny. She reaches out an arm, but instead of crushing me into a hug like yesterday, she touches my cheek. She blinks furiously and gets up, goes to the counter behind us, and reaches for a box.\n\n\"I got you something.\" She opens it and holds out a small black rectangle.\n\n\"What is it?\"\n\n\"Look, it's a camera. These new ones are so clever.\" She touches a small button and it springs open to show the lens and a few controls. It's small, barely a few fingers wide. She shows me how it works. \"I thought this way if you must rush off every day, you can take photographs to show me what you're doing. And we can start another album together. All right?\"\n\n\"Thank you,\" I say, and then I take photos of Pounce sitting at our feet, of Stella, a few of the kitchen. Then I stand next to her and hold it out at arm's length to take one of us together. She shows me how to look at them, pushing a button and projecting them onto a surface, and a few clicks later there we are, standing next to each other and smiling on the kitchen wall.\n\n\"Oh, and I have something else for you.\" She reaches into her pocket. \"I made an extra copy of the key to the wardrobe with the photo albums; you can look at them when you want to. But keep it locked and hide the key. I don't want anyone else getting into it, all right?\"\n\nShe hands me the key, and I wrap my fingers around it, slip it into my pocket. I feel guilty that I've been in there already without her knowing. I remember I was going to ask her about the missing photos of Dad, but can't bring myself to do it, not today. Not while we're getting along so well.\n\n\"Now, enough of that, I've got things to do,\" Stella says. \"It's Ellie's birthday today. Do you want to help me decorate the cake?\"\n\n\u2022 \u2022 \u2022\n\nEllie's birthday cake isn't as elaborate as my childhood ones: a triple layer chocolate cake with chocolate icing, intricate icing flowers climbing across the top, and twenty candles. It's delicious, and everyone is in good spirits; even Madison makes no sideways comments. She announces the reason for her good mood is that it is her one weekend off this month, and that I'm going with her on a led walk tomorrow up Catbells with Finley.\n\nEven after hearing that, Stella is the most relaxed I've seen her, smiling at the end of the table, looking out at each of the girls. And I see that to her, all of them are her daughters; surrogates for what she lost? Some have been here for years.\n\nIt's later that I find out that Ellie is an avid gardener, and she is delighted that the flowers on the cake were copies of ones she'd planted in the garden last summer. And there is a little swirl of jealousy inside.\n12\n\nTHE NEXT MORNING AS MADISON AND I STEP OUTSIDE TO catch the bus into Keswick, it finally happens: It's snowing.\n\nI dance about on the way to the road, holding my arms up to the sky.\n\n\"You're mad,\" she says.\n\n\"I just really love snow. I don't know why.\"\n\nShe shrugs. \"Mostly I tolerate it. But it'd be excellent today if it snows really, really hard, and they cancel this stupid walk.\"\n\n\"I thought you wanted to go?\" She doesn't answer, and I look back at her. \"Oh, I see. Not so keen on walking; more keen on Finley.\"\n\nShe scowls. Then laughs. \"Maybe.\"\n\n\"What's up with you two?\"\n\nShe shrugs. \"It's hard to say. He's Finley.\" As if that says it all.\n\n\"And . . . ?\"\n\n\"He's not known for having girlfriends for longer than five minutes. He's Finley the flirt.\"\n\nWe reach the road and stop to wait for the bus.\n\n\"You're wrong,\" I say. \"That may be the way he has been before, but he really likes you. It's in his eyes when he looks at you.\"\n\nThe pink of her cheeks deepens, but she doesn't answer, and holds up an arm to flag down the approaching bus.\n\nWhen we get into town we go to the Moot Hall. There is a small cluster of people dressed like us, in warm walking gear. Finley is there, along with a man identified as John with a clipboard writing down names of each arrival.\n\nFinley sees us and waves. He gets our names put on John's list, then walks over. \"Well, look here: It's Shorty and Extra-Shorty. You both better stick close to me.\"\n\n\"Why's that?\" Madison asks.\n\n\"If it keeps snowing, you might disappear in a snowdrift. We don't want to lose you up there.\"\n\nIt is snowing great heavy flakes, and there is a little discussion about that and the weather forecast. They decide to wait and consult with the fell checker, due along any moment.\n\n\"What's a fell checker?\" I ask Finley.\n\nHe turns to me, raises an eyebrow. \"That is the sort of thing you should know if you want to work for Parks. It's all in the name.\"\n\n\"Let me guess: He checks fells?\" Madison says.\n\n\"You got it, Einstein. Len goes up Helvellyn every day and checks conditions on Striding Edge. He takes a few photos up top. They post his report so walkers can decide whether to go up,\" Finley says, gesturing at a display case on the front of the Moot Hall. I walk over to look. Inside is yesterday's report on conditions and weather on Helvellyn. _Icy: Experienced walkers with winter survival gear only. Crampons essential._ And a photo: a thin icy path winding on a ridge, steep drop off both sides.\n\n\"Not for the fainthearted,\" Finley says.\n\n\"Not for me!\" Madison adds.\n\n\"Exactly,\" Finley says, and she punches him in the arm. But I'm ignoring them both, staring, transfixed by the image in the cabinet. I've been up on that ridge, many times. I'm sure of it! With _Danny the dreamer._\n\n\"Judging by that smile, you're not fainthearted,\" another voice says, and I turn. It's John; he must have walked over and listened in without my noticing.\n\n\"No. Can we go up there today instead?\"\n\nJohn laughs. \"No way. Too many beginners here.\"\n\n\"I don't get it,\" Madison says. \"Why do they have some guy go up there to check stuff every day? Why don't they just have a camera installed, and some weather sensors?\"\n\nFinley shakes his head at Madison. I jump in before anyone else can answer: \"It's conservation. National Parks can't put equipment up there; it's against their mandate.\" John nods.\n\nLen the fell checker appears. He's older than I expect, long gray hair tied back, a wild gray beard, and a crazy glint in his eyes. He chats with John, and it is decided we're still on for Catbells. As Len strides off, I stare longingly at his back, feet itching to chase after him and ask if I can go up Helvellyn with him instead.\n\n\"You coming?\" Finley calls out, and I see our group is starting off. John is leading the way, with Finley as back marker, bringing up the rear.\n\nWe walk through Keswick to the river, then down the side of it onto a footpath along fields, into woods, and climbing to the turn to Catbells. There is soon more snow on the ground as we start up a steep hill. Madison is gasping, slowing down, and Finley is laughing and pushing her from behind. Then holding her hand. The way they are with each other pulls inside, finds the ache that is always there.\n\nImagine if Ben were here. Holding my hand as we walk up the hill. Imagine we were alone instead of in this straggling line of walkers.\n\nI speed up and let Finley and Madison drop behind. Giving them some alone time, I tell myself, or is it just that I can't watch them anymore? I push my legs and muscles, and one by one overtake the others who are slowing as the path steepens. Before long I reach John at the front.\n\n\"Slow down there,\" he says cheerfully. \"I can't let you disappear ahead, and I can't speed up and let the others drop behind.\"\n\n\"How about if I go ahead, but stay in sight?\" I ask, itching to just see open path in front of me.\n\n\"Go on then. But don't get too far ahead,\" he says. \"Wait every now and then for us to catch up.\"\n\nI head out front. The snow has slowed like the weather report said it would, and the sky is lightening; the view opens up ahead.\n\nThe open path at my feet calls me forward; every step I feel I'm getting closer to something without knowing what. I force myself to wait now and then for the others to catch up as promised, then take off again. The cloud gradually lifts, and one by one the surrounding peaks reveal themselves. Something inside is letting go, untwisting bit by bit. _This is where I belong._\n\nI reach some rocks; the wind has swept the snow away from this exposed place, leaving the glimmer of ice behind. A short scramble up is needed. Stella's right: I am part mountain goat. I climb the rocks easily and wait at the top for the others at John's wave. Most get up without much difficulty, but Madison looks alarmed, and it doesn't look like an act designed to get Finley's attention. I scramble back down again and help her up before he notices.\n\nAcross the first ridge, another scramble, and then I'm alone on top of the world. The lake stretches out below, Keswick beyond. The other way, higher fells and steeper climbs call out to me, and I promise myself: another day.\n\n_Up here you can believe anything; you can be anything._ Words whispered inside: Danny the dreamer. I repeat them out loud.\n\nSteps come up behind me; John stands next to me. Did he hear? \"True. And these mountains and lakes have been here a long time, longer than people have. They'll be here when we're all gone.\"\n\nWe say nothing else. The world below, and its Lorders and problems, seems remote, of no consequence.\n\nThe others catch up, and before long we must leave to make it down in daylight. Back to reality.\n\n\u2022 \u2022 \u2022\n\nThat evening at dinner, Stella tells us that an inspector is coming for lunch tomorrow: a JCO. All must attend, no excuses, and be on our best behavior. She doesn't say a name; is this my grandmother, the one Madison mentioned? The one whose photos are hidden away inside a box in a locked wardrobe? Glances are exchanged, nothing else said, but the mood is dampened down, as if she'd just thrown a bucket of cold water over the room.\n\nMadison follows me to my room after dinner in a total black mood. She flops down on my bed.\n\n\"I _really_ can't believe this.\"\n\n\"What?\"\n\n\"That that witch has to pick this Sunday, my only one off in an entire month, to come for a stupid lunch. And _all must attend._ Some of us might have lives of our own. Things to do.\"\n\n\"Like what?\"\n\nShe scowls furiously, but her face wars between that and the edge of a smile.\n\n\"Finley?\"\n\nShe nods. \"Yep. He finally asked me out this afternoon; we were going to meet in town, go out for lunch, and whatever. And now\u2014\"\n\n\"Whatever? What is _whatever_?\"\n\n\"What does it matter? I've tried to call, but no one is answering. He's going to think I'm coming up with an excuse to get out of it. He'll never believe we're not allowed to miss this stupid lunch. It's this stupid house. None of the others are like this.\"\n\n\"Is it Stella's mother coming for lunch? The Juvenile Control Officer for all of England that you told me about?\"\n\nShe nods. \"She comes every few months or so. Stella never refers to her as her mother, but that's who it is: Astrid Connor, the smiling assassin.\"\n\n\"What do you mean?\"\n\n\"Oh, you'll find out soon enough.\" She sighs tragically. \"I can't believe this is happening to me.\"\n\n\"I told you so.\"\n\n\"What?\"\n\n\"That Finley really likes you.\"\n\n\"Maybe.\" She smiles, then it falls away. \"Not that it'll matter after tomorrow.\"\n\n\"Call again tomorrow, tell him you'll meet him later. It'll be fine.\"\n\n\"Sure it will; I bet he'll just go off with some other girl.\"\n\n\"I doubt it!\"\n\n\"What makes you so wise in the ways of guys, anyhow?\"\n\nI don't answer.\n\n\"Okay, I've told you mine; now tell me yours. Is there someone? There is, isn't there? Tell me!\"\n\nI can feel the shadow cross my face. \"There was.\"\n\n\"What happened? Did you stand him up and then he went off and\u2014\"\n\n\"No.\" I fling a pillow at her. \"No, because he really _cared_ so he wouldn't be so daft. Like Finley really cares.\"\n\n\"So why aren't you together, then? If the path of true love is really all forgiving like that, where is he? Why did you leave him behind? Why didn't he follow you to Keswick?\"\n\n\"He couldn't, that's all,\" I say, and refuse to say more. Eventually Madison sees I'm really upset, apologizes, and leaves.\n\nI sigh and turn the lights off, get into bed, and pull the blankets around me. If Ben really loved me . . . shouldn't that survive everything? Shouldn't he still feel the same way, deep down inside, even though Lorders have wiped all memory of me from his mind?\n\nIt is romantic nonsense to think so. A wave of sadness creeps over me, bit by bit, so deep that it feels as if heavy weights are holding me still, that I'm paralyzed. Later I hear a light tap on my door: Stella? But my eyes stay shut when it opens, my body unmoving, breathing deep, unable to reach out or say anything. Moments later it shuts again and footsteps retreat.\n\nUnderlying the grief, an uneasy sense of disquiet remains. Tomorrow, I meet my _grandmother._\n\nWhat would she do if she knew I was here? Would she be happy to have her long lost granddaughter back, or is she Lorder through and through?\n13\n\n\"THERE ARE A FEW NEW FACES HERE TODAY, I SEE.\" SHE smiles, and her eyes twinkle behind glasses that look a little like mine. \"I'm Astrid Connor, and your lovely house mother here is my daughter. Bet she didn't tell you.\" She looks at Stella and then smiles again. \"Daughters!\" she says, and shakes her head. Like the last photo I saw of her, her hair is silvery gray, swept up. She wears ordinary clothes; nothing says _Lorder_ in how she looks or acts, but there is something about her. The hackles are raised on the back of my neck. Every eye is drawn forward. Some people you don't want to turn your back on.\n\nStella clears her throat. \"There are three new girls since your last visit.\" She quickly points to each of us and says our names while Steph and another girl co-opted to help are bringing in serving dishes, putting them on the table: We're having a roast. As Stella points to me and says \"Riley Kain,\" Astrid's eyes fall on mine. A brief moment of something crosses her face\u2014curiosity, that soon fades to disinterest? Then she is interrupted by Stella passing her a dish. The curious glance returns.\n\nThe usual chatter around the table is gone. Everyone eats silently, even Madison, while Astrid holds court. She talks with Stella about the running of the house; asks about window repairs. Every now and then her eyes fall on one of us and she'll ask a question: about work, or Keswick. All pleasant and chitchatty. No order in it, not working her way around the table or any other logic to it I can see.\n\nThen she turns and her eyes fall on Madison: playing with her food, slumped in her seat, eyes lowered. \"Madison, isn't it?\" she says.\n\nMadison looks up, nods; her eyes, visible now, are defiant. Something in my stomach twists.\n\nAmusement crosses Astrid's face. \"Not hungry today, dear?\"\n\n\"Not really. Can I be excused?\"\n\nStella's sharp intake of breath is audible in a room too quiet.\n\n\"On one condition. Tell me exactly what you are thinking first.\"\n\nDoubt crosses Madison's face; she shakes it off. _Please, Madison, don't be an idiot,_ I plead silently.\n\n\"All right, then. It's my one weekend off this month, and I had plans. But _she_ insisted we all be here.\" Madison glares at Stella.\n\n\"Oh, I see. I'm sorry you missed your _plans,_ \" Astrid says. \"What were they?\" A tinge of red crosses Madison's cheeks. \"A boy, I'm guessing? My, my. Really, Stella,\" she says, looking at her daughter now. \"They don't all have to be here today, not if they have _plans._ You know I really just come to see how you are. You know what it is like to be a _mother,_ to worry about your _daughter._ \" There is a malicious twist to her words.\n\nStella's lips are set in a thin line. \"I think I know what is best for my girls.\"\n\nMadison clears her throat. \"I told you what I was thinking, like you asked. Can I go now?\"\n\nAstrid looks at her daughter, an eyebrow raised in a question.\n\n\"Stay and finish your lunch,\" Stella says.\n\nMadison scowls. \"It's not fair. None of the other houses are run like this. She treats us like prisoners!\"\n\n_Too far._ All the girls look at her in horror; I plead with my eyes: _Stop this; apologize now!_\n\nAstrid smiles. \"I think, dear Madison, you would be able to tell the difference between this and a prison, if you ever found yourself in one. You may go now.\"\n\nMadison looks between her and Stella, like a rabbit caught in the headlights. Stella nods faintly. \"Go on,\" she says.\n\nMadison puts her napkin on the table, pushes her chair out. Walks stiffly to the door and is gone.\n\nAstrid laughs. \"What a serious bunch you are! Doesn't anyone have any stories to tell? Perhaps one of the new girls.\" Her eyes fall on me. \"Kylie, was it?\"\n\n\"Riley,\" I answer, trying not to react to her saying a name so close to Kyla.\n\n\"When did you arrive in Keswick?\"\n\n\"Earlier this week. I'm here for CAS.\"\n\n\"Where are you from?\"\n\n\"Chelmsford. But I love the mountains, and I want to work for National Parks.\" I start to rush out an explanation of what they do before she has even asked. My voice trails away.\n\nShe raises an eyebrow. \"At last: a chatty one. And how did you\u2014\"\n\n\"Oh drat. Sorry!\" Stella interrupts, springing up as a toppled jug spreads water across the table. Steph dashes for a cloth; Astrid gets up out of her seat before the water can run onto her lap. \"Sorry, sorry,\" Stella says again.\n\n\"Stop fussing,\" Astrid snaps.\n\nShe and Stella leave the dining room.\n\nThe door swings shut, and as if we all had been holding our breath, we let it out in one collective gasp.\n\n\"Is she always like that?\" I ask Ellie, seated next to me.\n\nShe nods. \"She's horrible to Stella, isn't she? Can you believe what she said to her, about knowing what it's like to worry about your daughter, when Stella's daughter is missing? Nasty.\"\n\nThen everyone starts talking in hushed tones about Madison, what she said, how long Stella will confine her to the house as punishment, but I can't get Astrid's words out of my head. Like Ellie said, it was nasty, but not just the way Ellie meant. There was something _else_ twisting behind her words, that niggles and worries inside.\n\nFeigning a headache, I leave the others and wander out, thinking I'll look for Madison. But when I start to cross the reception area, my feet pause. Stella's office; the hidden door. Will they be in the same sitting room they used to go?\n\nI shouldn't. But it'll be locked anyhow, won't it? I look around; no one is here. I cross behind the desk to the office door, reach a hand to the knob; it turns and I push the door open. Too late, I realize my mistake: What if they are in here instead? But when I peek into the room it is empty. Behind me I hear voices and footsteps heading this way. I step through into the office and pull the door shut behind me.\n\nTrapped.\n\nWhat if they come here now?\n\nMy eyes dart about the room, my ears strain for footsteps. All I can hear are some low voices beyond this door\u2014not Stella or Astrid, but some of the girls. They're not moving; they're staying out there, probably in armchairs by the window, and not going anywhere anytime soon.\n\nMy feet start the reluctant few steps to the curtains that cover the door, somehow feeling half frozen, each move an effort. I should have gone back to my room, or looked for Madison; anything but this.\n\nSomething on the wall catches my eye. A recent photo of Astrid that had been hidden in the box in my room hangs there. I pause, look around, and spot a few of the others.\n\nSo. When Astrid comes for a visit, Stella hangs her photos up; when Astrid leaves, they are hidden in a box. I shake my head. What weird family am I part of?\n\nMaybe it is time to find out. I pull the curtain out, step behind it. Push the door open and look through.\n\nAnd it is just like my dream: a narrow hall. _I used to play hide-and-seek here._ It's dusty, and I put a finger under my nose, try not to sneeze. Not used anymore?\n\nWhen I step through and let the door shut behind me, I'm plunged into darkness. _A flashlight; there used to be a flashlight hidden here, in the corner._ I feel along the wall and reach down, but find nothing.\n\nI walk, slow and silent, one hand touching the wall. The hall stretches past one room, then turns ninety degrees. There are a few slivers of light from ventilation panels near the floor. And voices.\n\nI crouch down near a panel, and listen.\n\n\". . . but don't do it, please don't\u2014I'm begging you.\" Stella.\n\n\"Do what?\"\n\n\"You know.\"\n\nAstrid titters. \"You should see your face. My, my\u2014so fierce. It's a shame you can't put that energy to better use.\"\n\n\"I see no better purpose to my life. Isn't 'serve and protect the young people of our country' part of _your_ official job mandate as JCO?\"\n\n\"Oh it is, and I take that _very_ seriously. The bad apples must go to prevent rot in the barrel, as you well know. These girls, here\u2014they aren't your daughters. You know the consequences of error; that could be a _painful_ mistake.\"\n\nSilence. Even from the other side of the wall, it feels strained. Is Stella caught in her mother's eyes? I shiver.\n\n\"I told you I had news for you today about Lucy; you haven't asked me yet,\" Astrid says at last. \"Don't you want to know?\"\n\n\"Of course I do. Please tell me.\"\n\n\"Stella, prepare yourself for a shock.\"\n\n\"What?\"\n\n\"You know how I told you weeks ago that Lucy was killed by a terrorist bomb? I've found some . . . irregularities in the Lorder records on this matter.\"\n\n\"What do you mean?\"\n\n\"It appears her death was faked.\"\n\nI'm stunned. Stella had been told I died in that bomb? She didn't say; didn't ask about it. And now she isn't saying anything I can hear in response to news that I'm alive\u2014news it seems she shouldn't know. And I'm hoping\u2014praying\u2014she is a good actress.\n\n\"I don't understand. Where is she, then?\" Stella asks finally.\n\nThere is a pause. \"I have no idea. She is still listed as officially dead, but unofficially, she is missing. There seems to be some _interest_ in finding her, from a number of . . . _interesting_ places. I do wonder what that girl has been up to.\"\n\n\"Nothing to worry you, I'm sure!\" Stella snaps, too quick, and I'm worried. This is a dangerous game. Somehow, I know\u2014either from traces of memory, or observation today, or both\u2014that Astrid is adept at reading what people say and don't say. Shouldn't Stella be in hysterical tears at the news that I'm alive?\n\n\"Really? We shall see,\" Astrid says. \"But no matter. You know I've kept my half of our bargain and found out everything I can about what happened to her. I'll protect her and bring her home safe to you if I can. Darling girl, despite our differences, you _know_ I only want what is best for you. As soon as I find out anything of Lucy's whereabouts, so will you. But don't ask anything more of me. You _will_ be disappointed.\"\n\nSoon their conversation turns to other things; roof maintenance needed, damp in the cellar. I'm stiff and cold from crouching down in this unheated hall. Time to make an escape while they're both still in there and accounted for.\n\nI can't go back the way I came; there are sure to be eyes outside Stella's office door. I stand carefully, ease my muscles, and creep slowly forward, one hand on the wall. Their voices fade as I reach another door.\n\nCarefully I pull the door toward me: nothing! I start to panic. Is it locked? It didn't have a lock before, I'm sure of it. I feel along the door; no padlock, but there is a simple latch. I release it. Step through the door into the utility room that is behind the kitchen, then back along a hall.\n\nSomehow my feet are remembering, more and more, how to get around this house. I look down before I reach a main hall, brush at my clothes to get rid of the dust.\n\nLater, back in my room for the night, my mind is spinning with Astrid: the things she said, the way she said them. The twist of the knife in her words.\n\nAnd that Stella was told I had died. Was this before I reported myself found, before she knew I was on my way back to her? She never told me, so I can't ask without admitting I was listening in. But _why_ didn't she tell me? I don't understand her at all.\n\nAstrid said she's looking for me, that she'll bring me home if she finds me. Yet here I am, and she obviously doesn't know about it; Stella hasn't told her. She doesn't trust her.\n\nBut Astrid has noticed something is up with Stella, I'm sure of it. She won't let it go. If Astrid works out what it is, I'm in trouble. Despite her assurances to Stella, I don't trust her, either. If Lorders find out I'm here, they'll come for me.\n\n_Danger._\n\n\u2022 \u2022 \u2022\n\n_Careful, quiet, each step on tiptoes through Mummy's office, but it is hard to be a spy in this stupid pink dress; it whispers and rustles as I move. I gather the skirt together and bunch it up in my hands to hold it still as I slip behind the curtains._\n\n_I push the door, step through, and hold it part open with my foot as I lean down to get the flashlight. Switch it on, then let the door swing shut._\n\n_I creep along the wall, around the corner, then crouch down to listen like a spy._\n\n_\". . . be here soon.\" Mummy._\n\n_\"He indulges that child, as do you.\"_\n\n_\"It's her birthday!\"_\n\n_\"Really, Stella. Isn't it about time you tell him the truth? That his precious daughter isn't his; that you don't even know whose she is. Perhaps I should tell him?\"_\n\n_\"No! Don't you dare, I'll\u2014\"_\n\n_\"Don't threaten me, Stella. You'll regret it.\"_\n\n_Their voices continue but I stop listening. Shaking, I stuff my hands in my ears, but I can still hear Grandma's words over and over inside my head:_ his precious daughter isn't his.\n\n_How could that be? He's Daddy._\n\n_My daddy!_\n\n_I start to cry._\n14\n\n\"IS EVERYTHING OKAY?\" MADISON ASKS.\n\n\"Shouldn't I be asking you that? Where'd you disappear to yesterday? Why are you going to work so late today? Bets are on that Stella will ground you, between that and what you said at lunch.\"\n\nShe smiles, and it is a _very_ happy smile. \"I wrote it in the book: _out until late._ I thought I made it quite clear. And Cora will get over it when I tell her why.\"\n\nThe bus pulls up, and we clamber on. Madison sits with Finley, he holds her hand, and some of the boys whistle. I settle into a seat on my own, glad to put some space between us before Madison shakes off her loved-up state enough to ask me again: Is everything okay?\n\nThat dream, the things Astrid said\u2014could it be true? Was he really not my father? All my snippets of memory of him\u2014the way he was with me\u2014say otherwise. But what if he never knew?\n\nThen he died for a daughter who wasn't even his.\n\n\u2022 \u2022 \u2022\n\nLater I'm standing in the CAS meeting room, and when my name is called, I collect an envelope. It doesn't seem so important now. But unless Astrid works out what is up with Stella, and who I am, and everything stops, it is for _five years_ of my life.\n\nI rip it open.\n\n_Dear Miss Kain,_ blah blah blah. I scan down to the important part: my trials.\n\nWeek 1: Education\n\nWeek 2: National Parks\n\nWeek 3: Hospitality\n\nWeek 4: Transportation\n\nHurrah! I got my top two choices. But I'm puzzled at getting Hospitality. It didn't really appeal to me, so I'd had it far down my order of preference, and it seemed like a popular choice. I turn the pages and find the details for each placement.\n\nThe words next to Hospitality jump into stark focus:\n\n_Report to Waterfall House for Girls, Stella Connor._\n\nWhat? How can this be? And I think back to how adamant Stella had been about my not signing up for CAS, especially not for Parks. But then when I went to tell her I'd signed, she was all chilled about it, and I'd thought she'd realized I had to make my own decisions. But I was wrong. She was in town that day. She knows somebody; she must have pulled some strings. What do you want to bet these trials mean nothing, that I'll end up with her for _five years_ as some sort of apprentice housemaid?\n\nEventually it penetrates that the others are leaving, heading off to the first of their trials. Week one for me is Education, and I find the details. Keswick Primary School; I'm supposed to go there now and report to reception. But what difference will it make?\n\n\u2022 \u2022 \u2022\n\nWhen I arrive I get rushed to an office. Two other potential apprentices are waiting along with the same smiling woman I spoke to last week about apprenticing in Education.\n\n\"I'm really sorry I kept you waiting. I got lost,\" I lie. I knew the way, but my feet wouldn't cooperate.\n\n\"No problem, dear, take a seat. I'm Mrs. Medway, Head of School. I also train apprentice teachers and assistants. I'm going to run you through what you'll be doing for the week.\"\n\nI try to pay attention, for her sake, but it is a losing battle. Some details get through: We'll be shadowing classes two days, spend a day in admin, then two more days in classes, but this time helping with lessons. \"Any preference for year groups or subjects?\" The others tell her theirs, and eventually she turns to me and smiles. \"You're quiet today. Any favorite year groups? Activities?\"\n\n\"Anything is fine,\" I start to say, then pause. \"Unless they do art? I love art. And running\u2014sports.\"\n\n\"Perfect\u2014our youngest class is doing messy art this morning. I'll put you in there. And there is a sports day on Friday afternoon this week; they can always use extra help. We'll work something out for the other days.\"\n\nShe has us follow her around the school for a tour, telling school history on the way. It was damaged and rebuilt after the riots. Keswick Primary used to be called St. Herbert's Church of England School, but the name changed after church schools were banned thirty years ago. We see children through windowed classroom doors, playing a noisy game of basketball in a gym, heads bent in a library. Then finally we reach an art room, and I peer through the door. They're tiny. Four years old. All sitting cross-legged on the floor and listening to their teacher.\n\nMrs. Medway knocks at the door, has a word with the teacher. Comes back and squeezes my arm. \"Go on in. You'll be fine; don't look so worried.\" I walk in, and a sea of small faces look up and smile.\n\nNot much later they're all wearing smocks over their school uniforms, and the teacher passes me one to go over my clothes. \"It's up to you; you're down for shadowing, so you can sit in a corner and watch. Or jump in if you want to.\"\n\nI decide to sit and watch for a while. They are finger painting on big sheets of white paper, and the air is full of the smell of paint and excited voices. Despite the resolution to stay put, before long swirls of color on white paper pull me close. I itch to paint.\n\nA small hand pulls at mine. \"Miss, look at my painting!\" a boy says, and I'm pulled to a table and soon admiring blobs and blotches.\n\nOne girl sits quiet in the middle of all the chatter, not joining in. \"Hi,\" I say. She doesn't answer.\n\nThe boy looks up. \"That's Becky. She's sad.\"\n\n\"Oh, I see. I'm sad sometimes, too,\" I admit. \"But I like painting when I'm sad.\" Never a truer sentence have I spoken. I kneel on the floor between them and dip my eager fingers into black paint.\n\n\"Why are you sad?\" Becky asks.\n\n\"Mostly because I miss things. Like Sebastian.\"\n\n\"Who's he?\" the boy asks.\n\n\"Watch,\" I say. I can't remember finger painting before\u2014I'd rather have a brush in my hand\u2014but a reasonable estimate of a black cat soon appears on the paper.\n\nBecky stares at it very hard. \"You miss your cat?\" She nods to herself. \"Okay. I'll paint something, too.\" She gathers different paints and soon is concentrating on getting as much mess on herself and the paper as possible. I glance up and the art teacher gives me a thumbs-up. Other kids bring me their pictures to look at, then ask me to show them how to paint a cat. This is fun. Could I be an art teacher?\n\nNot if Stella has anything to do with it.\n\n\u2022 \u2022 \u2022\n\nI stay and help clean up at lunch. The teacher hangs pictures on the walls, puts my cat up next to Becky's painting. Hers could be anything from an alien to a lamppost, but I'm reasonably sure it is supposed to be a man\u2014her dad?\n\n\"It's her father,\" the teacher confirms. \"He went missing last month.\"\n\nI turn my shocked face to hers. \"What happened?\"\n\nA pause. \"It was good work getting Becky to take part. Thank you,\" she says, not answering my question. If it can't be said out loud, we all know what that means.\n\nLorders.\n\n\u2022 \u2022 \u2022\n\nWhen the final bell rings, I'm surprised the day has sped past. Each year group has art a half day each week, and the afternoon was spent charcoal drawing with year five. I gaze at the white-topped peaks as I walk back to the center of Keswick. If I can't get into Parks, maybe this wouldn't be such a bad choice. Then I shake it off. What a joke for me to even think of being a teacher; despite my faked records, I didn't even finish high school. And what about Stella's manipulations?\n\nI should get the bus back to the house now, but there is a kernel of anger inside that says _no._\n\nMadison: She'll understand. I head for her caf\u00e9. I'll wait there until she's finished; we can get the bus back together.\n\nWhen I reach the caf\u00e9 and pull at the door, it doesn't budge. It's locked? Puzzled, I realize the lights are off inside. A CLOSED sign hangs on the door, yet I'm sure Madison said she was working until five. A sense of unease settles inside. I walk around to the caf\u00e9's back door and knock.\n\nNo one answers, but was there a noise inside? I knock again: nothing. I'm about to turn around and leave, but then try the door. The handle turns. It's not locked.\n\nI pull the door partway open, peek in. \"Hello?\" I call out. \"It's Riley. Is Madison here?\"\n\nCora is sitting at the work surface, her back to me. Not turning or answering. Unsure what to do, after a moment I push the door open the rest of the way, walk in, and let it shut behind me. The light is dim, and I blink.\n\n\"Hello?\" I say again, and walk toward her. Her shoulders are shaking. She's crying? Fear grabs me inside. \"What is it? What's wrong?\"\n\nShe looks up at me, shakes her head. \"What could she have done?\" she whispers.\n\nMadison? Panic swirls inside. _No, not again._ \"What happened? Tell me!\" I demand.\n\n\"She was helping make cakes for tomorrow, standing there with flour on her nose and telling me about that boy she likes. And they just marched in, grabbed her. Dragged her out past the regulars\u2014all of them just sat there, staring at the lunches she'd brought them earlier. She's gone.\" Her face drops into her hands.\n\n\"Lorders?\" I whisper.\n\nShe nods.\n\nNo. _No._ This can't be happening; it can't. Not here, too. And it feels like quicksand is clawing at my feet, pulling me down into another nightmare.\n\n\"What could she have done?\" she says again.\n\nI shake my head. Nothing to deserve _this._ I blink, but there are no tears, just an empty place inside as I conjure up the person who must be responsible: Astrid Connor. My _grandmother._ It must be her. Or could it even be Stella? A cold rock twists in my stomach. I'll make her do something. I'll make her fix this.\n\nI stay long enough to make tea, to start tidying up the mess left behind. I get out of Cora that she'd thrown the customers out after the Lorders left. In the front of the caf\u00e9 are half-eaten lunches still on plates. I scrape them into bins, put the dishes in the dishwasher, put food away in fridges.\n\nFinally I hesitate by the door. \"I should go now. Are you going to be all right?\"\n\nShe shrugs. \"I'll get up in the morning. Thanks for your help.\"\n\nHer words echo in my ears as I walk to the bus. She wouldn't thank me if she knew who my grandmother was.\n\nA bus is waiting when I get to the stop, and I climb on. Finley: He's there. A sinking feeling twists inside as I realize I have to tell him. I walk toward his seat as the bus pulls away.\n\n\"Finley?\" He looks up. His face is white, eyes dead. He knows. One of the caf\u00e9 regulars or someone on the street must have told him.\n\nI don't say anything. I sit next to him, as if somebody being there could do anything to help.\n15\n\nI MARCH INTO RECEPTION. IT'S PAST TEATIME, BUT THERE IS a cluster of girls there, whispering, faces pale. News travels fast.\n\n\"Where's Stella?\" I ask.\n\nOne of them points at her office door, but before I can move toward it, the door opens. Stella comes out, nods at everyone, and starts to cross the room.\n\n\"Wait,\" I say, and she turns.\n\n\"Do you know what's happened to Madison?\" I ask, and all voices cease.\n\nStella looks at me, and her eyes are saying _be quiet,_ but I'm not receiving.\n\n\"You know, don't you? That Lorders came and hauled her away today. Strangely enough, the day after Astrid\u2014your mother\u2014came for lunch.\"\n\n\"That's enough, Riley.\"\n\n\"No, it's not. Not nearly enough; nothing is ever said. What are you going to do about it?\" Some part of me is aware that others have come in now, that everyone is silent, with mouths hanging open and eyes big. Eyes that are looking between me and Stella.\n\n\"There is nothing I can do.\"\n\n\"But she's your _mother._ Doesn't that _mean_ anything?\"\n\nShe doesn't answer.\n\nI shake my head. I can feel Ellie coming up next to me, taking my arm. Pulling me toward the door to the hall where my room is, and I let myself be led. My feet take the steps, but then I stop at the door and look back at Stella. She's still standing there, frozen in place.\n\n\"No. I guess it really doesn't mean anything,\" I say, and then walk down the hall with Ellie.\n\n_Take her to the tower!_ Madison had said, laughing, the first time she showed me the way to my room.\n\nEllie tries to get me to talk, but I send her away, close my door. Every friend I ever had: gone. Pounce scratches to be let in, and I even ignore her. I stay where I am past the start of dinner. No one comes to check on me; they know where I am, that I'm not coming, don't they?\n\nNo one ever says _anything._ Isn't this the biggest problem of all? If we all stood up\u2014everyone in the whole country, and said, _Stop; this is enough,_ every time something happened, wouldn't it have to stop?\n\nI'm starting to sound like Aiden.\n\n\u2022 \u2022 \u2022\n\nThere is a tap on my door late that night. It opens. Stella stands there, takes in me sitting up in bed, blankets pulled up, leaning against the wall.\n\n\"Still awake, I see. I thought you might be hungry.\" She holds out a plate of food in one hand.\n\nI shake my head. Arms crossed.\n\nShe walks in, puts the plate on the desk. Sits on the chair. \"Why are you so angry with me?\"\n\nMy eyes widen. \"Would you like a list?\"\n\n\"Keep your voice down. No matter what you might think, there is nothing I could have done about Madison. She went too far.\"\n\n\"You never liked her.\"\n\n\"That's not true. She could be difficult at times, but\u2014\"\n\n\"Then why don't you _do_ something? Why don't you call Astrid; she'd have to listen to you.\"\n\n\"She won't.\"\n\n\"So, is that your philosophy? That mothers don't have to listen to their daughters?\"\n\n\"What do you mean?\"\n\nI shake my head. \"That's not important right now; Madison is important. Astrid needs to hear from you that what she did is wrong, to get Madison back to us! How could she have her taken away when all she did was answer her question truthfully and say what she really thought?\"\n\n\"Too much truth can be a bad thing. And careful what you say about your grandmother!\"\n\n\"What\u2014are you _defending_ her?\"\n\n\"No, not exactly, but\u2014\"\n\n\"What, then?\"\n\nShe sighs. \"She thinks what she does is right. That she protects everyone else by\u2014\"\n\n\"Taking out the rotten apple? What a crock. She's a power-mad manipulative nutcase.\"\n\n\"Careful what you say and who you say it to!\"\n\nI shake my head. \"You _are_ sticking up for her.\"\n\n\"She is my mother.\"\n\n\"That's not a good enough reason. People have to earn respect\u2014even mothers.\"\n\n\"Lucy! You owe a lot to her. Don't speak of her like this.\" And Stella looks uneasy, as if the walls have ears, but even if they do, for once I don't care.\n\n\"What? What do I owe her?\"\n\nStella doesn't answer.\n\n\"You're as bad as she is.\"\n\n\"What do you mean?\"\n\n\"Doing what you think is best for me, without any idea what really is.\"\n\nShe looks at me, alarm starting in her eyes.\n\n\"Oh yes, I worked it out: You pulled strings, didn't you? You had me put down for _here_ for an apprenticeship trial. Will anything I do or say make any difference to where I end up?\"\n\nThere it is, in her eyes. Confirmation.\n\n\"Lucy, listen to me. I just want to keep you safe. You'll get found out if\u2014\"\n\n\"I'll get found out if you keep calling me _Lucy,_ and if you single me out for attention like that. This would never have happened if Dad were here. None of it.\"\n\nShe recoils. \"Shut up! You don't know what you're talking about. You don't even remember him!\" I don't answer, but she must read it in my eyes, and her face changes to fury. \"You do. You remember him, but you don't remember me.\" Her arms are crossed rigidly, red spots rising in white cheeks.\n\n\"Maybe I remember some little things. But if I've got stuff wrong, how can I know if you don't tell me anything? Tell me already!\"\n\n\"It was him, it was him all along!\"\n\n\"What was him?\"\n\n\"Danny was in the AGT. It was his fault! He had you taken. They wanted artistic children no more than age ten to do some experiments on, and there you were, perfectly fitting the bill. He gave you to them.\"\n\nI stare at her, stunned. That's what Dr. Craig and Nico always said: that I was given to them. Handed to them by my parents, that they knew what was going to be done to me. Could Dad really have done that, knowing what I'd face? I'd always been sure that was one of their lies. But was it because I am artistic, is that why I was targeted? With shock I remember Nico implied as much\u2014that artists' brains have different wiring. Easier to muck with.\n\nBut how could Stella know about this stuff? I never told her. Did she get her information from Dad: Is that what proves she is telling the truth?\n\n_No._ It can't be. \"I don't believe you,\" I say. \"How could you even know what the AGT wanted, what they were doing?\"\n\n\"My mother told me. She's been doing all she can to find you! Investigating the AGT, everything.\"\n\nRelief fills me, and I sag back with it. It wasn't Dad who told her; if it comes from Astrid, then maybe none of what Stella says is true. But then the inconsistency screams out at me, and I stare back at her. \"This doesn't make sense. If Astrid is trying to find me for you, why haven't you told her I'm here?\"\n\nHer mouth opens and closes again.\n\n\"I see. You don't trust her. So why do you believe her when she says Dad gave me to the AGT? He never would have done that to me!\"\n\n\"He'd never have done that to his _daughter._ \"\n\n_No._ And I'm shaking my head, and I'm back in the hallway listening to her and Astrid, and Astrid saying, _Isn't it about time you tell him the truth? That his precious daughter isn't his._\n\n\"He wasn't my father,\" I say, words quiet. I'm still denying it inside, but it doesn't come out as a question.\n\n\"No. He found out. And just after that, he gave you to the AGT for their experiments. Revenge\u2014he did the one thing that would hurt me more than anything else.\"\n\n\"He wouldn't do that.\"\n\n\"I'm sorry, Lucy,\" she says, anger bleeding away from her face. \"I'm sorry. I shouldn't have told you.\"\n\n\"I don't believe you!\" And I'm curling into a ball on the bed. Stella comes and puts a hand on my shoulder.\n\n\"Lucy, I'm sorry.\"\n\n\"Just leave me alone!\" I say, and she pulls her hand away. \"I mean it. Go away.\"\n\nShe murmurs that she loves me, that nothing can change that. After a while she finally goes. The door clicks shut, and I'm alone.\n\nIt can't be true; it can't. He wouldn't do that. My dad wouldn't do that.\n\nBut if he found out I wasn't his, he would have been furious; what man wouldn't be? Stella must have been messing around on him, and not just once or twice. What was it Astrid said? That she doesn't know _whose_ daughter I am. I could be anyone's. The thought fills me with horror even as I deny it inside. Could Dad have done what she said, found out I wasn't his and just given me away, to get back at Stella?\n\nNo. I can't believe it. I won't.\n\nStella's wrong. She must have made it up. She's just trying to manipulate me all over again, like her mother manipulates her.\n\n\u2022 \u2022 \u2022\n\n_The door clicks shut behind us, and we are plunged into darkness. Daddy flicks the flashlight on and holds it under his chin._\n\n_\"Mwahahaha!\" he stage-whispers._\n\n_\"Be quiet! You're not a ghost. We're spies.\"_\n\n_\"Oh yeah. Sorry,\" he whispers back._\n\n_We creep along the hall and around the corner, and a faint murmur of voices gets louder._\n\n_\"I still think we should play ghosts and yell BOO through the grates,\" Daddy whispers._\n\n_I shake my head and bend down to hear, Daddy next to me._\n\n_But the words I hear are wrong. They can't be anything else; they don't make sense. There is a clang as the flashlight in Daddy's hand slips to the floor. I look up. \"Daddy?\"_\n\n_He has rocked back onto his heels. The flashlight beam points the wrong way, but even in shadow his face isn't how it has ever been when he has looked at me before._\n\n_\"Daddy?\" I say again._\n\n_His eyes focus back on mine. \"Go to your room now, Lucy. Go!\"_\n\n_And he has stopped being a quiet spy. He runs for the door; soon he is on the other side of the wall with Mummy and Grandma, and their voices are loud enough to not have to listen by the grate._\n16\n\n\"ARE YOU ALL RIGHT?\" MRS. MEDWAY ASKS WHEN I RUSH IN the next morning. I'd managed to avoid saying anything to Stella at breakfast, reeling with what she said the night before. The dream that followed. My father\u2014 _he knew._ He was there when I heard those words. Had something inside suppressed the memory that he was with me? I didn't want to know it. I didn't want to see the look in his eyes when he knew the truth.\n\nIs Stella right about what happened?\n\n\"You're pale.\" Mrs. Medway puts a hand on my forehead.\n\n\"Really, I'm fine.\"\n\nShe looks at me closer. \"Madison from the caf\u00e9 was a friend of yours, wasn't she?\"\n\nI start guiltily. I'd barely thought of her since Stella's late-night visit. The dream that had woken me to stare at the walls, awake, for hours.\n\nShe misreads my reaction. \"It's a small town. Word gets around. How about if we give you admin today? There is a stack of filing. But if you want to nap in the corner, that's okay, too.\"\n\nSo I find myself in a locked back office with rows of cabinets organized by year, with student names in alphabetical order and baskets of papers to file. She explains the system, and when I am surprised at paper records, not computer files, she touches a finger to the side of her nose and winks. \"Paper files can't be hacked,\" she says.\n\nShe leaves, and I rifle through the stack of papers in the first basket: absence certificates for flu, appointments. File notes. Test results. I start on the top, find the file for each, and shove them in, glad to have something mindless to do. But after a while I put the basket down.\n\nThe cabinets for current students take up the front row; what of those behind? I wander back. They are in year order and go back decades\u2014since the school was renamed and reopened, about thirty years ago.\n\nThe years I would have been at this school\u2014the cabinets are here. I glance at the door: closed, locked, quiet. Seven years ago: 2047\u20132048. I find the cabinet and pull the _A\u2013H_ drawer open to hunt for _Lucy Connor,_ but come up empty.\n\nWait a minute. Astrid, my grandmother, is also a Connor; Connor is Stella's name. Would I have had Dad's last name back then, before they fell out? What was his name? I focus on Danny, then Daniel. I lean forward, close my eyes, rest my forehead against the cold metal cabinet, and will it to give up its secrets. I try to let my mind drift, but nothing comes. Frustrated, I start in the _A_ 's, scanning through, and realize this will take forever.\n\nI go back to my filing, and the morning finally ends. At lunch I avoid the staff room and wander the school grounds. They are enclosed all the way around, with fences too high for a ten-year-old to climb without a ladder. The only gates have keypads on them: locked, a code needed, and I'm sure the students don't know the code.\n\nIt's cold, but there is snow to play with, and the children are out in force: building snowmen, having snowball fights. I see one whizzing toward my head too late to duck. A teacher walks over, yells at the children to stop.\n\nShe comes up to me as I brush the snow out of my hair. \"All right?\" she asks.\n\n\"It's fine,\" I say, and lean against the gate.\n\n\"You're one of the new apprentices, aren't you?\"\n\n\"Trialing to be one,\" I say.\n\n\"Do you like it so far?\"\n\n\"Pretty much.\" I look at her closer. \"You didn't know for sure why I was here until I said so. Could anyone wander about here in the grounds?\"\n\nShe shakes her head. \"There are cameras,\" she says, and points them out: by the gate, on the building, a few in trees. \"Security will know exactly who you are even if I don't. And the gates are kept locked.\"\n\n\"Was that always the case?\"\n\nShe shrugs. \"Mrs. Medway is security-mad.\" She looks around; the nearest children are too far to hear, but she lowers her voice anyhow. \"Ever since a girl went missing from school. It was about six or seven years ago.\"\n\n\"Oh. I think I may have heard of that; what was her name?\" I say, trying to keep my voice light, casual, even as I'm desperate to know: _my_ name.\n\n\"Louise or something? Yes, that's it. Louise Howard, I think.\" Then there is commotion at the other end of the school yard: a snowman kicked over to wails of protest. She rushes over to deal with it.\n\nThat afternoon I get back to the filing cabinets. _Louise_ is maybe close to _Lucy_ ; could I have been Lucy Howard?\n\nThere is no Lucy Howard, or Louise Howard, for that matter. But she got the first name wrong; maybe the second one isn't quite right, either.\n\nI dip farther into the _H_ 's and, a few paper cuts later, find it: _Lucy Howarth._ As soon as I see the name, whisper it out loud, I know it is right. My hands start to shake. I really _am_ remembering things, more and more\u2014little things maybe, but more than I ever thought I would be able to. Are they like blocks? Pull a bottom one out, and others must tumble down.\n\nI extract the file. It's bulky. Was I a truant? Somehow I don't think so. Stella wouldn't have put up with it.\n\nOn the cover of the file is my registration information: parents Stella and Daniel Howarth\u2014Danny the spy\u2014and contact details. Inside are all the usual things like I'd been filing all morning. Reports by teachers, a few absence slips, but very few; I didn't get sick much. And an artist I was, even back then\u2014winning contests at this school and across the county. If the AGT were looking for young artists, I wouldn't have been hard to find, with or without any help from my family. I hold on to that inside.\n\nThere is a separate folder in the back of my file: a missing report. Beginning with a note that I was reported absent from afternoon classes. Written notes follow, then typed ones\u2014how my mother was contacted, then the authorities. Morning attendance was confirmed; I was missing in the afternoon. No one in the school grounds saw me leave; it was a mystery. The file ends abruptly. Lucy was gone. What happened to her? To _me._ They didn't know; the ending is blank.\n\nI put the file back together, shove it into the cabinet where I found it, and go back to filing the endless bits of paper. Not focusing on what is in my hands beyond the letters of the alphabet to file them under, while the minutes tick past.\n\nWhere did I go?\n\nOkay, so maybe the gates weren't locked back then, and perhaps there weren't security cameras, either, but it is hard to believe anyone could have snatched me, protesting, from the school, and nobody have seen anything. But what if it was someone I wanted to leave with?\n\nLike Dad.\n\n\u2022 \u2022 \u2022\n\nThat night, I try. To explain to Stella why I can't believe Dad gave me to the AGT. I tell her how he infiltrated the AGT guards where I was held, how he snuck me out of my room at night. That we ran across the sand to a boat. Then I tripped, and they caught us. Nico; the gun raised in his hands. Dad on the sand telling me to close my eyes, to never forget who I am. How I couldn't look away. How his eyes held mine as he died.\n\nHow watching him die was the brick that finally achieved what Nico and the AGT doctor wanted: I couldn't deal with it, and my personality fractured. It hid what happened away, inside. That the split achieved their objective: When I was Slated, part of my memories survived, waited for the right trigger to come out again so I could fight AGT battles.\n\nStella cries. Great gulping sobs. She never knew how Dad died, or even that he was dead for sure. She only knew he disappeared and never came back.\n\nShe never knew that it was my fault.\n\nBut despite the tears, I can tell she still believes it was him who took me away from her in the first place.\n\n\u2022 \u2022 \u2022\n\n_I take careful steps like a spy across the school grounds, watching for the teacher on playground duty. I wait for her to be distracted. Some boys push and shove each other; it turns into a fight. Voices are raised and everyone strains to see, then at last the teacher notices and rushes over._\n\n_I swallow hard, pull up the gate handle, step through. It shuts with a too-loud_ clang _behind me. I'm out! I dash up the road, keeping an eye out, expecting any moment somebody will rush out and march me back to school. I can't get caught._\n\n_My hand is still in my pocket clutching the note I'd found under my pillow. Daddy had been gone for days, ever since . . . And I shy away from thinking about that day, my birthday, and what Grandma said. Mummy and Daddy shouting late at night. The empty place he left behind when I woke up again._\n\n_But it's okay now, it must be. I pull the note out that I've read and reread ten million times since yesterday._\n\nDear Lucy, I'm on a very special secret mission, and I need your help! Go to the mountain's children at lunchtime tomorrow, and wait for further instructions. Tell no one.\n\nLove, Daddy\n\n_See: signed_ Love, Daddy. _It was all some horrible mix-up, and he's going to tell me, and then everything will be all right._\n\n_My feet half fly up all the quieter streets where eyes are less likely to spot my escape, then up the footpath and the hill, still going fast. I don't want to miss him. I don't want him to think I didn't come._\n\n_I burst through the gate to the field\u2014no sign of him. Maybe he's hiding behind one of the stones? I rush to our beginning place and start around the stones, counting out loud as I go, expecting him to jump out and give me a fright at each one._\n\n_I'm at number fourteen when I hear a car the other way. There is parking by the gate at the other end of the field._\n\n_After a moment the gate opens, but it's not Daddy. A man, one I don't know, walks across the field toward me, and I ignore him and keep counting the stones, uneasy._ Daddy, come out from where you are hiding. Do it now!\n\n_But the man doesn't come up to me. He stands in the center of the circle and watches me for a moment, then looks both ways._\n\n_\"Are you Secret Agent Lucy?\"_\n\n_I stop in my tracks. Only Daddy calls me that. \"Who are you?\"_\n\n_\"I'm Special Agent Craig. I've got your further instructions from Agent Howarth.\"_\n\n_Oh. I stare at him. Daddy is Agent Howarth! But he has never involved other agents in our games. He must be a real agent!_\n\n_I salute him. \"Proceed.\"_\n\n_\"Agent Howarth orders you to accompany Agent Craig\u2014that's me\"\u2014and he winks\u2014\"in the spy-mobile. I will take you to Agent Howarth for a full briefing of your mission.\"_\n\n_I hesitate as I walk across the field toward the parking area. Agent Craig walks slower, behind me, and I glance back. His careful eyes are watchful, on the stones, on the mountains. On me._\n\n_When we get to the car, I pause. \"Where's Daddy?\"_\n\n_He opens the door. \"Get in, Agent Lucy. You'll find out soon enough where we're going.\" He smiles, and my feet suddenly feel rooted to the ground, remembering Mrs. Medway at school saying to never go with people we don't know. But I know Daddy, and he is taking me to him. So it's okay, isn't it?_\n\n_He nods, as if he can hear my thoughts. \"It's fine, Lucy; we'll take you straight to your father. He wanted to come himself, but he's being watched. That's why he's been in hiding these last few days.\"_\n\n_If he knows Daddy has been in hiding, then it must all be true. I slide into the car; he shuts the door. It clicks locked as he gets in. As we pull away, I look back at the stones through the window, trying to crush a flutter of panic inside that says I will never see them again._\n17\n\nTHE DRAWINGS ARE LAID ABOUT BEFORE US IN HER OFFICE. The art teacher smiles. \"This is the hard part. Which do you think?\"\n\nI try to concentrate after a hard night of not enough sleep and too many painful memories. Sometimes I wish they'd stay buried. I feel jagged and exposed, as if I'm bleeding in front of everyone, and can't believe no one can see the wounds. Did Dad really do it? Set me up with that note? Did he really write it, or did I just think he did because of where it was, what it said?\n\n\"Riley?\"\n\nThere are expectant young artists waiting. I pull myself back to the present.\n\n\"It's a tough call, but I'd put these ones ahead of the others,\" I say, and point out five of the pencil drawings.\n\nThe teacher picks her favorites, we compare, and she eventually settles on a top three. We go back out into the noisy classroom, and the year four students settle down. She holds up the ones chosen but is careful to praise all the others. There are happy faces, and disappointed ones. Was I like that? Did I care if I won?\n\nShe points out the winners from last year, still up on the side wall, and I realize they start a border that goes around the room. School winners, different grades.\n\nLater, while the students are packing up and leaving to go to lunch, I follow the drawings along the walls.\n\nI stop, frozen, at a landscape: the mountain's children. The gray stones of Castlerigg Stone Circle are drawn delicately, with hidden details to animate: They are dancing. The mountains above have faint faces etched in them, smiling down on their children below. At first glance the movement, the personification, is hidden. A closer look reveals it. And there, written at the bottom corner, a small _Lucy._\n\nVaguely I'm aware of movement next to me, but it doesn't really register. I'm some other place, a pencil in my hand, hiding the faces in the shading and texture of the stone.\n\n\"It's amazing, isn't it?\" a voice says near my ear. I don't answer. \"Do you see?\" And the art teacher points out the drawing's secrets.\n\nUnable to stop myself, I ask, \"What happened to this girl, this Lucy\u2014did she go on to become an artist?\"\n\n\"I don't know. She left,\" she says, and abruptly walks away.\n\nDid she leave, or was she stolen? She climbed into that car of her own free will.\n\n\u2022 \u2022 \u2022\n\nAt the end of the day, I can't stop myself; I run up the paths that lead to Castlerigg. Now I understand how I felt the other day when I came here, the shadow of fear that colored my connection to this place.\n\nBefore anything else, this was our special place: Dad's and mine. I can see the magic in the dancing stones; the faint faces I'd drawn for the mountains are there in the distant lines and shade in the late afternoon sun. I start counting the stones as I did just days ago, but it is spoiled. When I get to fourteen, a cold shiver runs down my spine; I half expect to hear a car, to look up and see Dr. Craig walking toward me. For all his pretending that day, that is who he was: the AGT doctor who deliberately fractured my mind.\n\nAnd it wasn't just a dream. The blocks inside are pulling out, one at a time; I remember that day. The note was real, but did Dad write it?\n\nMy head is fuzzy, but becoming clearer. Is it like Stella wondered days ago? As a child, I was left-handed. No matter that I was forced later to be right-handed, and Slated as such. These other memories were hidden and twisted tight, but are starting to unravel\u2014and me with them.\n\nI thought it was watching Nico kill my father that finally caused my personality to fracture, but maybe that was just the final nail. Maybe it all started with Dad, with that note; with knowing that once he found out I wasn't really his daughter, he didn't want me anymore.\n\nOr was that just what Nico and Dr. Craig wanted me to think?\n\nWhichever is true, one thing I am sure of now: Stella had nothing to do with them. Her exposed secret might have set things off, but she'd never have given me up. She'd never have done anything but cling to me, desperate and tight.\n\nLight is failing, and I run back to town for the bus. It is pulling away when I get to the square, and I wave; the driver pulls in and I clamber on.\n\nFinley is there, in the same seat he's been in before with Madison. Guilt grabs at me when I realize that I was so absorbed with my own stuff, I never noticed I haven't seen him on the bus the last few days. No matter how it feels to me now, what I'm going through is ancient history. Finley's pain is immediate.\n\nI sit across from him in the aisle and try to catch his eye, but his are lowered. I think he doesn't even know I'm there, but then after a moment he glances over. \"Hey, Extra-Shorty,\" he says.\n\n\"Hey,\" I say back. I want to ask how he is, is he all right, but that's stupid, isn't it? Of course he isn't. I try to say it with my eyes, instead. After a few beats, he nods, looks down again.\n\nDoes he know who Stella's mother is, that she must be responsible for Madison vanishing? What good would it do if he did? Maybe if Stella could see his pain, she'd do something about Madison. Maybe she'd get Astrid to bring her back.\n\nOr maybe he'd kick up enough of a fuss for it to get back to Astrid, and then he'd disappear, too.\n\nBut I can't leave this, and when it comes to me I can't believe it took so long. MIA: We have to put Madison on Missing in Action. As unlikely as it seems, maybe she can be found.\n\nI stare at him, nudge his foot with mine. \"Finley? We need to talk.\" I breathe the words quietly. He looks up with a quick look of hope, one that fades fast when I give a small shake of my head. If _only_ I knew where she was.\n\n\"Tomorrow\u2014get the seven A.M. bus,\" he breathes back.\n\nI nod.\n\n\u2022 \u2022 \u2022\n\nThat night, I get to work on a drawing of Madison. Why didn't I take a photo of her with my camera when I had the chance?\n\nTo get her on MIA I have to get in touch with Aiden. Or do I? He told me how to contact someone who knows him here: Leave a coded note on the community board, then wait until they get in touch.\n\nThat was for emergencies. Does this qualify?\n\nYes.\n\nGetting Madison right turns out to be easy. It's the look of mischief in her eyes that marks her out: Is that what Astrid really objected to?\n\nI'm nearly done when there is a faint knock on my door, and I slip the drawing under the bed. Stella looks in, hesitant, but I nod and she comes across the room.\n\n\"I'm sorry about last night,\" she says.\n\n\"Me too. But can we not talk about stuff tonight?\" I say. \"I just can't deal with it right now.\"\n\n\"Of course,\" she says, and relief crosses her face. \"I've got an idea\u2014let's have some fun.\"\n\n\"How?\"\n\nShe smiles. Holds up a key. \"Like this!\" She goes across to the other locked wardrobe, turns the key. Looks back at me. \"Come on.\"\n\nI get up and walk across the room. She opens the doors; inside the wardrobe are shelves, and on them are brightly wrapped packages.\n\nI look at her, not understanding.\n\n\"They're for you\u2014your birthday presents.\"\n\n\"Really?\"\n\n\"Yes. There is one for every year we weren't together; because I never gave up, Lucy. Not once. Every third of November, another joined them.\" She touches my cheek. \"I always knew, somehow, you'd come back to me.\" She blinks hard. \"Here\u2014help me carry them.\" And she fills my arms with packages small and large, points me toward the bed, then brings the last few herself. We spread them out on the bed.\n\n\"Go on,\" she says.\n\n\"I can unwrap them?\"\n\n\"Of course. They're for you, aren't they? Though some of them might not be much good to you now. Start at the beginning,\" she says, and hands me one with _11_ all over the paper. \"Where's your camera? I want birthday photos!\"\n\nI smile, shake my head. \"How could you explain them if they were found?\"\n\nHer smile falters. \"Of course. You're right; it's too risky.\"\n\n\"No, it's a good thought. Next year maybe? But my birthday isn't in November.\"\n\nShe goes very still. \"What did you say?\"\n\n\"My birthday is in September now! As Riley, on my fake ID, I turned eighteen on the seventeenth of September.\"\n\n\"Oh. Of course.\" She smiles, tension falling away. \"Have you been using your camera?\"\n\n\"Not really. Sorry. I'll take it tomorrow.\"\n\nWe start on the presents, and soon I'm covered in wrapping paper and presents for an eleven-, twelve-, thirteen-, fourteen-, fifteen-, and sixteen-year-old me: clothes, many too small now; art supplies. A gorgeous leather portfolio case.\n\n\"Last one?\" she says, and holds out a parcel, the one for my seventeenth birthday.\n\nI pull the paper off carefully. Inside is a gorgeous pale green sweater of a fine, soft yarn. \"It's beautiful,\" I say.\n\n\"Really? Do you really like it?\"\n\nFor an answer I get up and pull it on over my pj's, hug it close. \"It's perfect.\"\n\nShe pulls my glasses off. \"Perfect with your green eyes. I made it, knitting late at night.\"\n\n\"Thank you.\" I put the glasses back on. \"But the match to my eyes has to stay secret.\"\n\n\"Of course.\" She gathers up the wrapping paper, stuffs it in a bag. \"I'll burn it,\" she says, matter-of-factly.\n\n\"I'm sorry.\"\n\n\"What for?\"\n\n\"All this being secret about me is hard for you, isn't it?\"\n\n\"Anything to have you back.\" Something crosses her face, she starts to say something, but I interrupt.\n\n\"No talking about stuff tonight, remember?\"\n\n\"Okay. Another night. Now get some sleep.\"\n\nShe helps me hide the presents away in the wardrobe; I keep out the drawing supplies, a few clothes that should fit. She heads for the door, then turns back. \"I will say one thing, though. You were right; I shouldn't have interfered with your apprenticeship trials. I'll make sure they don't bias where they put you, all right?\"\n\nAnd with that, she is gone.\n\nWell. I stare at the door she just disappeared through. Did she mean it? Time will tell.\n\nI retrieve my nearly complete drawing of Madison, put the final touches on it, and tuck it into my coat pocket.\n\nRestless, sleep feels far away despite the late hour. I unlock the other wardrobe and pull the albums out. Each one starts with a birthday, and I look at the birthday photos again: presents, cake, smiles. Except the first album, of course. Really, your first birthday should be the day you are born, shouldn't it? You should have a cake with a big _0_ on it. Instead the first album starts with photos of me grinning and reaching for toys, crawling across the floor. Very embarrassingly having a bath.\n\nI put them away, and with lights off and eyes closed, I hug the soft green wool close, still wearing it over my pajamas. After the agony of that dream of Dad and his note, and the memories that came from it, at least now I'm feeling warm, feeling wanted. Maybe Stella is enough. One parent who loves me, who would never give me up.\n\nAll those presents she searched out every year: They were all stuff I know I'd have loved; still love now. She wrapped them up with care and locked them in a wardrobe, all for a daughter she might never see again. It's so unbearably sad, even though I'm here now.\n\nIt's even harder to bear being the one so missed.\n18\n\nTHE MORNING COMES EARLY. FINLEY IS ON THE SEVEN A.M. bus as planned. I nod to him, but sit down, silent, in the front.\n\nWhen we get off I walk without comment to the back door of Cora's Caf\u00e9. Finley follows, catches up as I reach the door. I knock; it's locked this time, but soon opens.\n\nCora sees it is us and a quick look of hope crosses her face. \"Get yourselves in,\" she says, and we step through the door. She checks the back lane before shutting and locking it behind us.\n\n\"Is there news?\" She looks between us, then as Finley's eyes turn to me, she does also.\n\nI shake my head. \"I'm sorry; no news. But there may be something we can do. Have you ever heard of the group MIA: Missing in Action?\" They shake their heads no. \"This is very secret. There is a website run by MIA where missing people are posted; there is a network of people who try to find them, or what happened to them.\"\n\n\"What happened to Madison is unlikely to be good,\" Cora says.\n\nFinley winces, shakes his head. \"Better to know,\" he says. \"How do we do it?\"\n\n\"We need a photograph of Madison, a recent one. Failing that, I've drawn her.\" I pull out the drawing I did last night.\n\n\"That's good, but I've got photos,\" Cora says, pushes her chair back, and goes into an adjoining room.\n\nFinley reaches out a hand, traces Madison's face on the paper with a finger. \"I wish . . .\" And he stops.\n\n\"What?\"\n\nHe shakes his head. \"I wish I told her how I really feel.\"\n\n\"I think she knew,\" I say, though not sure as I say it that she did. They'd just started out, hadn't they? Did she know what is so obvious now? He loved her. _Loves her,_ I correct to myself.\n\nCora returns with some photos, and we pick one out to use. Seeing the longing in his eyes, she gives another to Finley. \"Keep the drawing, too, if you want,\" I say, and he tucks it in his bag.\n\n\"What happens next?\" Cora asks.\n\n\"I'll take care of it,\" I say.\n\nPromises to tell no one follow, and as we leave, I wonder why I'm doing this. Not getting Madison on MIA, but letting them in on it. It's a risk, a huge risk, but the only way to give them hope.\n\nThis is what Aiden does, what he is about. _Join us,_ he'd said. Looks like I have.\n\nI'm still hopelessly early for school, and go for a walk by way of the community notice board Aiden described. It is just where he said it would be, tucked on a side street by a hall. No one is in sight, and I tack the note up: _Seeking chess partner, please contact Anita c\/o hall._\n\nAll I can do now is wait.\n\nI take some photos on the way to school, of Keswick as the sun creeps up. The sun seems to go from hidden to clearing a mountain all at once, and first touches of light transform dark shadows to dazzling, clear morning.\n\nParents are depositing children at the school gates as I walk up, a teacher on the other side watchful as each goes into the grounds. A woman is coming from the other way with two boys and carrying a baby. One of the boys trips, falls, and starts to howl. She shifts the baby in her arms, tries to bend over to help him up.\n\n\"Can I help?\" I smile, coax the boy up, and he and his brother go through the gate.\n\n\"Thank you,\" the mother says. \"Are you new at the school?\"\n\n\"I'm trialing to be an apprentice teacher.\"\n\n\"You might be this one's teacher one day, then.\" She smiles and looks at the baby, a soft look on her face. He? She? I can't tell; even wrapped up, it is tiny, wearing the smallest hat I've ever seen over a pink face, and sound asleep.\n\n\"You never know,\" I say. \"Maybe.\"\n\nA teacher walks over and coos over the baby. \"How old is she now?\"\n\n\"Almost four weeks,\" the mother answers.\n\nI leave them to it, go through the gates. What I know about babies is precisely nothing. But she was _so_ tiny. Four weeks old? I frown. In that first album of me, I'm fat-faced and crawling, playing with toys. How old am I when that album starts? Maybe Stella has another album tucked away someplace. She's so photo-mad, it's hard to believe she didn't bother taking any when I was really little. That must be it.\n\n\u2022 \u2022 \u2022\n\nSomething niggles inside that day, like a sore tooth that you should leave alone but worry at with your tongue, pushing and prodding until it is loose. I'm out of art and in a year two class today, through all their lessons, and my mind wanders so much that their teacher has to repeat instructions to me more than she does to her students. She must think I'm an idiot.\n\nThey have reading after lunch, and there is a birthday girl in their midst, seven today, who gets to pick the next story. The teacher starts reading her choice\u2014an old and tattered book from a bottom shelf, about princesses who rescue animals\u2014and I fade away again, looking at the birthday balloons she has tied to her chair float over her head.\n\nAs Riley my birthday has changed to September 17. Funny how Stella is about birthdays; they are such a huge big deal to her. She'd actually seemed rattled when I mentioned my birthday wasn't in November anymore.\n\n\u2022 \u2022 \u2022\n\nThat night at dinner, thoughts are spinning through my head. I feel disconnected to what is happening around me. When will I hear from Aiden's contact? It could be anyone, even someone at this table. I grin to think so; Astrid wouldn't like that. Anyhow, I'm sure she keeps a close eye on this place. I glance about at the other girls chattering, Stella at the head of the table. She looks different somehow. She gives me a quizzical look, as if she senses something is on my mind, but I don't even know what is wrong, so how could she? _Mother's intuition,_ a voice whispers inside, and I shake it off. What nonsense.\n\nSteph, Stella's helper, has finished carrying serving dishes out and sits with the rest of us. I notice she is as quiet as I am; she eats dinner, looking around at the others much as I do.\n\nI can't shake a deep sense of unease, but can't work out what it is attached to. But somehow, underlying everything else, there is something about that tiny baby today, about the photo albums. The missing early photos. Everything else is in there. Maybe those are ones Stella keeps for herself.\n\nI notice now what niggled earlier about Stella. Her hair is darker; not a huge amount, but the dark roots are gone, blended in, and the overall color is a shade darker. She's been to the hairdresser. I frown to myself; it's like she said when she first saw my blond hair had changed to dark. Bet she goes a shade darker each time until we match.\n\nWhy is she so obsessed with matching? Is it just part of her being clingy?\n\nSomething does a flip in my stomach. _Wait; think._ There is too much weirdness mixed up together. Stella matched her hair to mine years ago, as if to say we belong together; now she is trying to do it again. Then there is how weird she was about my changed birthday. And that there are no early baby photos.\n\nDinner is like dust; I put down my fork.\n\n\"Are you all right, Riley?\" Ellie says, and I can feel other eyes turning toward me, but I don't answer.\n\n_Birthdays._ Dr. Lysander told me cell testing said I was under sixteen when I was Slated, but if my birthday is in November, I'd have been over sixteen. She said I was a Jane Doe, not identified from DNA. Her eyes had been wrong when she said it; not that she was lying, she just couldn't believe it. That no one knew who I really was. She said . . . _no._ She said I might have been a baby born in an out-of-the-way place?\n\n\"Riley?\" I hear a voice say again, but it is distant and removed.\n\nWhat did Astrid say that day? Precisely and exactly. I close my eyes, going back, and I'm spinning, I'm someplace else. A dark corridor, crouching down. Full of a game that is going wrong, trying to hear her exact words . . .\n\n_Isn't it about time you tell him the truth? That his precious daughter isn't his; that you don't even know whose she is._\n\nEverything goes black.\n19\n\nGRADUALLY NOTHINGNESS IS REPLACED BY COLD FLOOR, voices.\n\n\"Lu\u2014Riley.\" Stella's voice.\n\nI open my eyes and she is holding me, cradling my head.\n\nI stare back at her. \"Who am I?\"\n\n\"She must have hit her head,\" Stella says, her eyes communicating alarm.\n\nThen Steph is there. She holds my glasses in her hands. \"One of the lenses has come out,\" she says.\n\nI close my eyes. Steph must have seen; she must know my eyes are really green. That the glasses mask who I am.\n\nWho am I? _You don't even know whose she is._\n\nStella helps me up. \"To bed with you now,\" she says. And we start across the room.\n\n\"Wait,\" Steph says. \"I fixed them. The lens just popped back in.\" She holds out my glasses and I reach for them, put them back on. Steph looks between Stella and me, a thoughtful look on her face.\n\nEllie scampers ahead and holds doors open. I want to shake Stella off and walk on my own, but my head is still fuzzy, and it does hurt.\n\nStella helps me to my bed; Ellie hovers next to us.\n\n\"That's fine, Ellie. You can go now,\" Stella says. Ellie looks uncertainly between us, leaves, shuts the door. It clicks.\n\nStella looks at me with something like fear in her eyes.\n\n\"You're not my mother.\" I say it like a statement, not a question.\n\nShe breaks her gaze, looks away. \"What nonsense.\"\n\n\"Listen to me. I was cell tested by Lorders when I was Slated; I was under sixteen, and it was after my so-called sixteenth birthday that November.\"\n\n\"But tests can be wrong\u2014\"\n\n\"You nearly flipped the other day when I said my birthday wasn't in November. There are no early baby photos of me. And that day, my tenth birthday, when I heard you and Astrid\u2014\"\n\n\"You remember _that_?\" she says, her eyes open wide.\n\n\"\u2014Astrid said you don't even know _whose_ I am. I thought that just meant Dad wasn't my father, but that's only half of it, isn't it? You're not my mother, either. Admit it!\"\n\nColor has drained from her face. She looks back into my eyes with desperation. \"I am in every way that counts. I've always loved you, Lucy.\"\n\n\"No! Not in one way that counts. Tell me the truth. Tell me now!\"\n\n\"You should rest. You might have a concussion.\"\n\n\"I do not. Tell me where I come from! I have the right to know.\"\n\nStella is shaking, her face crumbling. \"I am your mother. I _am._ \" She's choking back tears, and something else: the truth. Part of me wants to comfort her, to put a hand on hers, but no. She has to face this. Is it something so buried she can't even say it?\n\n\"We can have nothing between us if we don't have the truth,\" I say, and turn away from her, to the wall.\n\nTime passes. Minutes, more? A hand touches my shoulder, then pulls away.\n\n\"All right,\" she says, voice dull. \"I'll tell you. It's a sad tale.\"\n\nI turn, sit up. \"I'm listening.\"\n\nShe doesn't say anything at first, gathering herself, then nods. \"Okay. Your dad and I wanted children. Desperately. But every time I got pregnant, I lost the baby. Sometimes a few months in, sometimes longer. I don't know why; doctors didn't know why. Then one last time it happened: I was pregnant again. But this time, I didn't tell anyone, even your dad. He went away for a while; we weren't getting along.\" She stops, bites her lip.\n\n\"And?\"\n\n\"I was staying with my mother.\" The way she says the words, there is more to that, but I don't interrupt. \"My baby was born early\u2014my darling, beautiful daughter. I had Lucy to adore for days, just a few days. And then, she died.\" Stella's voice is choked, and I don't know what to say.\n\nShe turns to me, takes my hand. \"Then Mother, months later, brought you to me. You were perfect. And you were mine. I always loved you, Lucy; that is what makes you my daughter. Don't you see?\"\n\n\"Wait a minute. Are you saying Astrid just came up with a baby to replace yours that died? Where from?\"\n\n\"I honestly don't know. I guessed from an orphanage; as JCO, she is in charge of those also. But I didn't ask. I didn't want her to take you away from me.\"\n\n\"And this was months later that you got me? Didn't anybody notice you had a baby, then didn't, then did again? What about Dad?\"\n\n\"I told you. I was . . . away. At Mother's. Your dad and I didn't see each other for a long time. Then when he finally came back, he saw you and assumed you were ours; we got back together. I didn't tell him the truth about you.\"\n\nI shake my head at her. \"How could you lie to him like that?\"\n\n\"I had to. Mother threatened to take you away if I ever told. But then, years later, she held it over me, and then one day you and Danny heard us talking about it\u2014\"\n\n\"News out.\"\n\n\"Yes. He couldn't handle it; he took off. It was a few days later when you went missing. Mother found out the AGT had you, that he'd given you to them. I know you don't want to believe it. Mother tried again and again to get you back, but couldn't find exactly where you were being held.\"\n\n\"You say you always loved me as your daughter. Why would it be any different for Dad? Okay, he had a shock to get over, but I was still _me._ Still the daughter he'd always known.\" I shake my head.\n\n\"Maybe you are right. Maybe he didn't have anything to do with what happened to you.\" She says the words like they are difficult to say out loud, and the struggle is there on her face. For her to accept he was blameless would be hard after all the blaming she has done over the years. Then to accept how he died. \"Does it matter now?\"\n\n\"It does to me.\" But then I'm shaking my head, and my eyes are welling up.\n\n\"It's too much to take in all at once. I'm sorry you didn't know. I\u2014\"\n\n\"It's not just that. I think I remember what happened that day. The day I disappeared.\"\n\nShe stays very still, quiet.\n\n\"There was this note from Dad under my pillow to meet him at Castlerigg. I went there at lunchtime, but he wasn't there. Somebody else was, from the AGT. He said Dad sent him to get me. But when we got where they took me, Dad wasn't there. I didn't see him for two years, when he tried to rescue me.\"\n\nHer face goes hard, angry.\n\n\"No, wait,\" I say. \"It doesn't mean he wrote the note. Maybe they faked it.\"\n\n\"But how would they get a note under your pillow, or know where to say to meet him, if he didn't tell them?\"\n\nI shrug. \"I don't know. I don't want to believe it; I can't believe it.\"\n\nStella struggles to pull away from her anger. \"Listen to me. Whatever happened, he still tried to save you, didn't he?\"\n\n\"So he died.\"\n\n\"He died _trying_ to be a hero.\" Behind her words is an unsaid echo, one she can't forgive him for. Even if he wasn't involved in my disappearance in the first place. He failed.\n\nWe talk a bit longer, but I feign sleepiness, and she leaves. I stare at the wall in the dark.\n\nSo I'm back to this, as if I've been Slated all over again. To not knowing who I am. No parents, no place I come from. There is not even a name that is really mine. _Lucy Howarth_ or _Lucy Connor_ : Either way, it is the name of a dead baby.\n\nI'm numb.\n\nNothing.\n20\n\n\"TAKE A SEAT,\" MRS. MEDWAY SAYS, AND I SIT OPPOSITE HER desk. She closes the door.\n\n\"Riley, have you enjoyed your week at our school?\"\n\n\"Yes, thank you,\" I say, trying to be in the here and now for her, even though I failed at it most of the day.\n\nShe sighs. \"I don't quite know what to make of you, my dear. Our art department is screaming for you to be one of our next apprentices; you've made quite an impression there. That is fantastic, but the other days haven't been quite as positive. The thing is, if we take you as an apprentice, you have to spend a year working in every year and class in the school.\"\n\n\"I'm sorry. I haven't been myself these last few days.\" How could I be, when I don't know who that is?\n\n\"I understand you must be upset about your friend Madison. Is there something else?\"\n\nI'm startled she mentions Madison again; it isn't the done thing\u2014admitting to feelings about someone taken by Lorders. And her face is full of genuine interest, concern. There is nothing that threatens here. But how honest can I be?\n\nI hesitate. \"Confidentially?\"\n\n\"Of course.\"\n\n\"I found out recently that I'm adopted. It's been a shock.\" I've never said anything more true.\n\n\"Oh, I see.\"\n\n\"I was wondering if there are any teaching jobs at orphanages?\"\n\n\"There used to be.\" She half frowns, shakes her head. \"The nearest is the Cumbrian Care Facility; we used to supply teachers there on rotation. But a few years ago, they hired their own. Shut us out completely. I could ask.\" She hesitates. \"I'm not sure what is going on there. It might not be a good place for you.\"\n\n\"Why?\"\n\n\"It's isolated, stuck out in a valley with nothing near it but a few farms miles away, and people who work there never come to town.\" She frowns. \"Let's just leave it at that, shall we? Now, what shall we do with you?\" She opens a netbook, stares at the screen for a moment, then touches it and looks up again. \"Right. I've recommended you for an apprenticeship here. If you decide to select us as your top choice, that should clinch it. But don't decide until you've had the rest of your trials.\"\n\nI stare back at her, eyes wide with surprise. \"Thank you.\"\n\n\"Riley, I'm taking a gamble on you here. I take our responsibility to every child in our care, every child we teach, very, very seriously. There are no off days allowed, however good the reason, when every child counts.\"\n\n\"I understand.\"\n\n\"Now, go. Whatever you decide, I wish you all the best.\"\n\n\"Thank you,\" I say again, my throat feeling choked. She doesn't even know who or what I am, but she is willing to give me a chance. I hesitate at the door.\n\nShe looks up. \"Is there anything else?\"\n\nI long to tell her that I'm her missing student, Lucy, the one she couldn't account for all those years ago. Does it still haunt her? But I'm not really her, anyhow.\n\n\"No, that's it. Thanks again.\" And I bolt out the door.\n\n\u2022 \u2022 \u2022\n\nI stop by the Moot Hall, where Madison and I met with Finley and went on the walk up Catbells. I'd noticed they had maps up in glass cases on the side of the building, and study them closely.\n\n\"There are more maps inside,\" a voice says. I jump. Finley is standing in the doorway.\n\n\"What are you doing here?\"\n\n\"Apparently, my mind isn't on my job enough to do anything fun, so I'm on duty here.\" He pauses, glances about. \"Any news?\"\n\nI shake my head. \"I've got word out, but I'm waiting for contact still to get her put on MIA. It should be soon. But don't get your hopes up,\" I say, gently.\n\n\"So, what are you doing\u2014planning a weekend walk?\"\n\n\"Maybe.\"\n\n\"Can I come?\"\n\n\"Maybe. Don't ask why, but I want to go past the Cumbrian Care Facility. Do you know where it is?\"\n\n\"No, but I can find out.\" He gets me to follow him inside, hunts through indices, and finds the right map. \"I haven't been this way before; it isn't on a main walking route. But it'll be good to get out and away from everything and everybody, and up high.\"\n\n\"I know. For me, too. Can we keep where we are going between us?\"\n\nHe looks at me curiously. \"Of course.\"\n\nWe work out the way: We'll have to drive out of Keswick to a point where we can pick up a trail, but Finley says he can borrow a car. He reckons from there it'll take about three hours each way. We arrange to meet in the morning.\n\nAs I head back to the house, I wonder: What am I doing? Really. What possible good could it do to go look at an orphanage I may or may not have come from, something like seventeen years ago? Stella only _guessed_ I came from an orphanage, and even if I did, there is no guarantee it's that one.\n\nI shrug. I don't know. Something inside wants to go there, to see it.\n\n\u2022 \u2022 \u2022\n\nThat night, Stella knocks on my door, peeks in. \"May I?\" she asks, hesitant. I nod.\n\n\"I've brought something to show you.\"\n\nIn her hands is a small album. It doesn't match the others in the wardrobe. She opens it, and inside are page after page of a small baby, much tinier than that four-week-old one I saw yesterday. With loads of dark hair, eyes that don't quite open. Even in the photos she seems very still.\n\n\"This is Lucy.\"\n\n\"Why did you give me the same name?\"\n\nShe shrugs, uncomfortable. \"I'm not sure. Maybe I shouldn't have.\" She sighs. \"I'll always regret that she died, but I still loved you\u2014and I still do\u2014for who you are. That doesn't change because of any of this.\"\n\n\"But the name Lucy must always remind you what you lost.\" I stare back at her, and some inkling of understanding creeps in. She was so afraid of losing me, like she lost the baby in these photos. All the other babies, too. Then, years later, when I disappeared, all her fears came true. I feel like I'm starting to understand her, just a little.\n\nDoesn't mean I always like her.\n21\n\n\"THERE IS SOMETHING ABOUT BEING UP HERE, THAT NO matter how much life sucks, I feel better.\" I'm looking through my camera at the lonely fells sweeping around us, valleys below. The climb ahead.\n\nFinley is silent, and I lower the camera. \"Sorry,\" I say, looking at him sideways.\n\n\"It's okay. I haven't got the worldwide monopoly on misery; you can have some of it, too. So why is your life sucking?\"\n\nI shrug. \"Mostly I can't say.\" I hesitate. \"But there is something I can. Between us. Somebody I care for got hauled off by Lorders not long ago, too.\"\n\n\"Somebody?\"\n\n\"Okay. A guy.\" _Ben._\n\n\"And you loved him.\"\n\n\"Correction: I love him. Past tense not allowed.\"\n\n\"Deal on that one.\"\n\nWe continue on, mostly silent after that, stopping to check the map a few times when paths branch off, steadily climbing all the way. We reach a ridge high on a desolate path, wind bitingly cold sweeping across it. No snow up here: blown away? The sky is almost clear, but it seems thin, as if even the oxygen has been stolen by the howling wind. We're walking fast to stay warm.\n\n\"Nice day you picked,\" Finley says, but I can tell he doesn't care, any more than I do, about being battered by the weather. But when we dip down again it is still a relief to get out of the wind.\n\n\"Nearly there now; the orphanage is in that valley.\" He points it out; we have a traverse down this hill. \"Are you going to tell me why we're going there?\"\n\nI look at him sideways. Sigh. \"To be honest? I'm not really sure. But it's a long story.\"\n\n\"We've got time.\"\n\nI shake my head. \"How about you tell me a story instead?\"\n\n\"What about?\"\n\n\"I don't know. Where do you live?\"\n\n\"Keswick Boys: land of noise and beautiful toys.\"\n\n\"What?\"\n\n\"We have a reputation for boat races. And a few other things. It's not far from your place. A quick row across and then a walk, or about an hour's stroll along the lake's edge on foot, up the hill.\" He shows me on his map.\n\n\"I hear it is more chilled out than our house.\"\n\nHe laughs. \"Very much so. We come and go at all hours. I couldn't believe what Madison said about your place.\" His smile fades. \"Tell me. Was it because of getting out of that lunch to see me?\"\n\nHe doesn't say what he means, but I know.\n\n\"It's not your fault. Whatever happened to Madison, you didn't do it. Lorders did. And their reasons are their own.\"\n\nI can tell by the grim set of his face that he's not convinced.\n\n\"I know what it's like,\" I say.\n\n\"What?\"\n\n\"To think what happened to somebody is your fault. It eats you up inside. She wouldn't want that, Finley.\"\n\n\"Neither would your boy. But you can't stop how you feel.\"\n\n\"No.\"\n\nWe've been steadily descending into the valley as we talk, still high enough to see all around, and then, there it is. A cluster of buildings, set in a clearing in some woods below us, along a creek that meanders; a distant fence that wanders around it encloses large grounds. A scenic place, but somehow odd, and cold, and it isn't just winter that makes it so. It looks lonely and devoid of life.\n\n\"Look there,\" Finley says. \"Along the fence line.\" I focus where he points, and dots are moving along the fence, inside the boundary: people? But they are evenly spaced, moving at the same rate. Odd.\n\nI get the camera out again and zoom in. A long line of children is walking along a path on the inside of the fence. I sweep along; where visible, it looks like the path runs all of the perimeter of the grounds.\n\n\"What can you see?\" Finley asks.\n\n\"Children. They're out for a walk, I guess.\" I frown. \"It's weird, though.\"\n\n\"How?\"\n\n\"They're just walking. Single file. Not talking or laughing or anything.\"\n\n\"Shall we go down for a closer look?\" Finley asks, and I hesitate. Something feels wrong, so wrong, but I don't know what, and there is a sense of foreboding inside. One that says we shouldn't be here. At least Finley shouldn't be here.\n\nI pull us back along some trees. Take off my pack. \"Can you wait here? I'm going down for a careful look. I don't want us to be seen.\"\n\n\"I don't know. I should come with you.\"\n\n\"Honestly, there is nothing to worry about,\" I lie. \"I'm really good at staying hidden, and it'll be easier without the pack. I'll just creep down, have a quick look, and come straight back up. Just stay out of sight here. All right? I'll be fine. I promise.\"\n\n\"You're just going to have a look and come back.\"\n\n\"Yes.\"\n\n\"All right,\" he says, and looks at his watch. \"I'll give you an hour. If you're not back by then, I'm going down to look for you. Deal?\"\n\n\"Deal.\"\n\nI take off my outer coat; it's light blue and might stand out. My wool layer underneath is gray and should blend into shadows.\n\nAt first I stick to the path; it is cut into the hill, so with me bent down I shouldn't be visible from below. Then as I get closer to the woods, I cut off the path into the scrub, hugging rocks, then trees, heading for the fence where we saw the children, estimating where I should intersect them with the passage of time. Moving careful, quiet, slow. These skills, so useful now\u2014of moving without noise, using the cover such as it is to best advantage\u2014are things I learned from Nico and the AGT years ago. I stop behind some rocks, the fence a bit less than fifty meters away, and wait.\n\nBefore long the first of them round the corner and come into view. As it appeared from above, they are just walking. Smiling. Single file, no talking, nothing. I scan the grounds: no adults in sight.\n\nI should go back now, but creep forward, bringing the layout I saw from above into my mind. If the children stick to this path along the fence line, there will soon be trees, and the way the ground slopes, I should be out of sight of the buildings.\n\nI scurry quickly along the ground, cutting down closer to the fence. It's not high; I can easily see over it. But there are telltale signs\u2014a faint glint of wire along it. Is it electric, or is it an intruder alert? Either way, I'm staying on this side of it. I duck down and wait.\n\nFootsteps are coming this way. I hesitate; this is insane.\n\nI stand up just as the children approach. The first is a boy of about eleven or twelve. Walking, smiling. He sees me; he must see me, but keeps walking. Children follow behind him, a few meters apart, pass me one by one, no reaction. As they go, they are getting younger.\n\nA girl of perhaps seven is approaching now. \"Hello,\" I say.\n\nShe smiles. \"Hello,\" she says, but keeps walking.\n\nSome younger ones, about four or five years old, bring up the rear.\n\n\"Stop,\" I say. The last three children look at me, and stop. Saying nothing.\n\n\"What are you doing?\" I look at the one in front.\n\n\"Standing,\" he answers.\n\n\"No, before I said stop. What were you doing?\"\n\nHe looks puzzled. Smiles. \"It's Saturday. We're doing our Saturday morning walk.\" The three of them smile, make no move to continue. It's like they do what I say when I say it, smiling all the while. Just like the others, all walking at the same pace, smiling. It's almost as if\u2014\n\n_No._ No, it can't be. It can't.\n\nI start to shake, horror swirling inside.\n\n\"Hold out your hands,\" I say, unable to stop the tremor in my voice. In unison, all three of them hold out their hands. \"Pull your sleeves up,\" I say, and they do.\n\nAnd there, glinting on their wrists: Levos. I have just enough presence of mind to take some hasty photographs, hands shaking so that I have to balance the camera against the fence to make them clear, forgetting it might be electric until I realize it mustn't be, because I'm still standing here. This can't be; it's completely illegal. Slating is a punishment for _teenage_ criminals under sixteen. _Not_ little children. What could they possibly have done to deserve _this_?\n\nAnd as I focus through the camera at them, I see. The last boy: that crooked grin. No. It can't be. The day I came to Keswick on the train. That mother and son. It's the same boy. It is.\n\nI lower the camera, look at him. \"Where is your mother?\" He smiles back, says nothing, and I repeat the question.\n\n\"I don't know what that is,\" he says, and his smile is the same as on the train, but his look is blank. The giggles and mischief are gone; whatever it was that made him who he was . . . gone.\n\n_Clang._\n\nA faint noise, through the trees in the distance. A door? Fear runs through me. Did resting my camera against the fence set off an alert inside? Stupid.\n\n\"Put your hands down again,\" I say. \"Walk! Catch up to the others!\"\n\nThey take off, more running than walking now to try to catch up as instructed. I duck back down behind the fence.\n\nMy stomach heaves; I want to be sick. Children, young children, Slated? No. It breaks every law. Four-year-olds like that boy from the train can't be _criminals,_ no matter what his mother may have done.\n\nAnother distant sound intrudes. Is someone coming to investigate?\n\n_Get out of here._ I slip back the way I came using as much care as I can to stay down, out of sight. Once there is some distance between me and the fence, I stop behind some rocks. Peer back. The children have reached the house now; taller figures are there. I snap a hasty photograph, looking through the zoom. Half a dozen adults, and I don't need to see their black clothes to know what they are; there is something about the way they move, how they stand, that leaves me in no doubt. Lorders.\n\nA few of them are speaking to the children, and others are scanning the hill, binoculars in hand. I pray Finley has kept out of sight where I left him.\n\nThere is no chance I'll get back to the path above without being seen if they're watching properly.\n\nThe only thing for it is speed, and misdirection. I race back up, taking a roundabout way to make it look as if I'm heading the other way, not looking back. Then shrink down again and out of sight, creeping along behind undergrowth, rocks, until finally I hit the path. I duck down and race along to where I left Finley behind the trees.\n\n\"What's going on?\"\n\nI'm breathing hard. \"We need to get out of here as fast as we can. Better if we're out of sight and off the path.\"\n\nHe peers through the trees. \"There are figures heading toward the gate below.\" My stomach twists. He holds out my jacket, but I stuff it into my pack instead of putting it on. \"Who are they?\"\n\n\"Run now, talk later.\"\n\nHe gets the fear. \"Okay. One sec,\" he says, and consults his map. \"Can you rock climb?\"\n\n\"Yes.\"\n\nWe take off at full speed back up the path, but then, once we're over the peak and out of sight, go off the path to race across rock and dirt on steep, windy, faint trails made for sheep, not people. But Finley is much like me: He moves like a mountain goat. I can see where we're heading: a steep scramble over a peak. If we make it there and over before anyone reaches the place we left the path, they'll never see where we went.\n\nUnless they've got dogs. I stuff the thought down. Unless they've got them there already, they won't have time to get them until we're gone.\n\nWe reach the climb, and straightaway I can see a few places where just because of my height I'll have problems. \"I'll need to traverse to get over,\" I say, and start up the rocks. Some echo inside tells me to always maintain three points of contact when climbing, but I'm going too fast to do that. One foot slips.\n\nFinley, just behind, grabs and steadies me. \"No point in being quick if you're dead,\" he says, and I glance down and see with the traverse there is a steep drop below us now. That was too close.\n\nI slow down, listen to him this time about the best way to go, and at last we clear the top. A quick glance back shows heads just coming up the path in the distance, and we duck down. \"I'm pretty sure they didn't see which way we went,\" I say, not sure if that is true, but we're in trouble if it isn't.\n\n\"We're on a part of the trail now I'd wanted to come back on anyhow. But not by climbing over the top without ropes.\" He laughs.\n\n\"You're insane.\"\n\n\"You're crazier than me.\"\n\nThe wind is howling again now that we're on the other side of the fell, and I put my blue coat back on. \"Turn your coat inside out?\" I suggest. \"So we look different.\"\n\nFinley stares back, then takes his coat off, reverses it, swapping blue for gray. He takes another hat out of his bag and swaps his blue one for a red one. \"All disguised?\"\n\n\"Yep. Now let's get out of here. Fast.\"\n\nWe don't run on the ridge\u2014that would be suicidal\u2014but keep the pace as fast as is reasonably safe. The temperature has dropped, and clouds are pulling in.\n\nAnother path joins this one. \"That's where we would have joined to here if we'd done it the sane way,\" Finley says. We continue on, dropping down now and out of the wind. I'm breathing easier, and\u2014\n\n\"What was that?\" Finley says.\n\n\"I didn't hear anything.\" Then I do. Faint, behind us. \"Could they have gone the long way and caught up?\"\n\n\"No way. It's miles longer, and we were going fast.\"\n\n\"Are you sure?\"\n\n\"No.\"\n\nWe carry on, faster again; there are some rocks ahead, and we duck down behind them, out of sight and out of the wind. \"I'll take a look,\" I say, and get my camera out. I zoom it back down the path, and _there._ One figure; a walker, and he looks familiar. \"It's that guy; he was at the Moot Hall the other day.\"\n\n\"What guy?\"\n\nI hand him the camera, and he looks through. \"It's Len,\" he says. \"The fell checker.\"\n\n\"Should we take off?\"\n\n\"Len's all right, and anyhow, there's no point. Soon the way the path opens up, he'll see us no matter what. I vote we stay put and have some lunch.\" Finley opens his pack, takes out sandwiches and a flask. \"Tea?\"\n\n\"Yes, please! You think of everything.\"\n\n\"I try, but with you it's hard keeping up.\" He finds cups, pours tea for both of us, and I hug the heat of it in my cold hands.\n\n\"So. Are you going to tell me what is really going on?\" Finley asks.\n\n\"Sometimes it is better not to know,\" I say, and he stares back, nods after a while.\n\nHe opens the sandwiches. \"Cheese okay?\"\n\nWe're well into them when Len rounds the path. \"Hello there, young Finley,\" he says.\n\nFinley nods. \"Hello there, old Len.\"\n\n\"Cheeky brat. Good place for a picnic on a cold day; okay if I join you?\" Len says, and sits on a rock just above, where he can see the path on both sides.\n\nFinley introduces us, and Len finds biscuits in his pack to share. Part of me doesn't want to move, from the shock of the orphanage, the cold, the protesting muscles from the hasty run and climb. Part of me is screaming in fear at the delay and wants to _run._\n\nFinley is asking Len about weather conditions and the path ahead, but as Len answers, some part of me is wondering if his eyes are on me too curiously.\n\nLen has stayed sitting on the rock above us despite the wind, now and then looking back down the path. \"We'll have company soon,\" he says, and there is something about how he says it that raises my alarm. He looks back at us. \"Shall we get our stories straight?\"\n\nFinley and I exchange a glance. The flight impulse is in my feet: I want to race down the path the other way.\n\n\"No point in running, you'd be seen,\" Len says. \"Besides, we're just three walkers who had a nice time on yonder ridge today together, didn't we, before stopping for lunch; we've got nothing to hide.\"\n\nI can hear the approaching footsteps now; they're moving fast. If they've come up from the orphanage, they've moved much quicker than I'd have thought they could. Then two faces appear; they must have split up where the paths diverge.\n\nLen nods. \"Hello,\" he says.\n\nThe Lorder smiles; it's unnatural. \"Hey there. Good walking today?\"\n\n\"The wind cuts through you,\" Len answers. \"Just the way I like it.\"\n\n\"Where've you been?\" the Lorder asks, and Len gives the cover story while Finley and I eat biscuits with concentration.\n\nThe Lorder nods, thoughtful. \"I see. Have you seen two other walkers, one a girl? We think they may be lost.\"\n\n\"There were two girls a while ago. They took the last branch, I think, back the way you came.\"\n\nThey move off, talk a moment. Speak into a com, take one last look at us, then go back down the path.\n\n\"Well then,\" Len says. \"Let's get the hell out of here before they realize they've been had.\"\n\nWe stuff things away hastily and set off in the other direction. Len sets a hard pace, and every time the path branches we go a different way, winding and twisting around in convolutions we'd never have worked out without him, until we're heading back down again on the other side.\n\nLen gets Finley to lead, slows down in front of me so we drop back. \"I think we need to talk,\" Len says, voice low. And _yes_ he has helped us today, but what can I tell him?\n\n\"Thanks for your help. But\u2014\"\n\n\"I understand you're looking for a chess partner. Anita, is it?\"\n\nI almost stop dead in my tracks. Len? He is Aiden's MIA contact?\n\nHe winks. \"You've been a hard girl to track down.\"\n\n\"Were you following us today?\"\n\n\"That was a bit of luck. Finley borrowed my car; I got out of him that you were going with him. The car keys have a tracker. So, what's up?\"\n\nFirst, a promise. I fish through my pockets for the photo of Madison I've been carrying around ever since I put that note up. \"Can you put Madison on MIA?\"\n\nHe hesitates. \"I can. But there's little point,\" he says, his blunt words softened by the sadness in his eyes.\n\n\"Do you know where she's gone?\"\n\n\"Don't know; guess. There's a women's working prison, out Honister way. At the slate mine. She's probably there, where most taken around here end up.\"\n\nI breathe a sigh of relief. \"Prison: Then she's alive.\"\n\n\"Sometimes that isn't best; no one ever leaves that place. But quick before we run out of time. What were you up to today that had Lorders so interested?\"\n\nBut before I have to decide what I should or shouldn't say, hellos are called out as another group of walkers catch up to us. They stay with us all the way to where our cars are parked.\n\n\"Need a lift, old man?\" Finley asks.\n\n\"Cheeky brat,\" Len says. \"As a matter of fact, I do. And seeing as it is my car, I'll drive, thank you very much.\"\n\nFinley reluctantly hands over the keys.\n\n\"How'd you get here?\" I ask.\n\n\"Over hill, over dale.\" Len grins.\n\nMy jaw drops. How many miles was that? He looks ancient and was walking rings around us.\n\nAs we pull onto the road, Len glances at me in the mirror. \"You're one of Parks' new potential apprentices, aren't you? I take the group on a walk the first day, so I'll see you Monday. We'll talk then.\"\n\nA slight emphasis on _then._ He doesn't want Finley to know anything.\n\nFinley is whistling as Len pulls onto the road, heads for Keswick. \"You're awfully cheerful,\" I say.\n\nHe looks at me sideways. \"We just stuck it to them, didn't we? Know you're not going to tell me why they were after you, but I don't care why. Anytime a Lorder doesn't get what they want, I _am_ happy.\"\n\nI know what he means, but I'm not feeling it. Have we really gotten away with anything? All through the drive back to Keswick I keep scanning the road ahead, half expecting a roadblock.\n\nAnd my camera is burning a hole in my pocket. Aiden _has_ to have these photos. The proof is there: Lorders are breaking the law, they are Slating little children. No one could ignore this. Is it the one thing that'll finally make everyone stop, stand together, say _no more_ to the Lorders?\n\nThere is panic inside that I have the only copies, here, in my camera. If the Lorders ask those boys the right questions, they'll know their Levos were photographed. They'll be desperate to find me. And if they work out who I am . . . I'm dead.\n\nThis is way past self-preservation. I have to stay alive. I have to get these photos to Aiden.\n\nWe have to get the word out, and make it stop.\n22\n\n\"CAN WE TALK?\"\n\nStella smiles to see me, looks so absurdly happy that I'm seeking her out, that a sinking feeling stirs deep in my chest. \"Of course, come in,\" she says, and I walk into her office and turn the lock behind me. She raises an eyebrow. \"This looks serious. Is everything all right?\"\n\n\"No. Not really.\"\n\n\"What is it?\"\n\nAnd I don't know what to say. The less I tell her, the better for her, really. But despite all the need for caution, I just can't do it to her; I can't disappear with no word. Not again.\n\nStella gets up from behind her desk, goes to the sofa against the wall. I sit next to her.\n\n\"Go on. You can tell me anything.\"\n\n\"You don't want to hear this. I'm sorry, but I have to leave.\"\n\nShe shakes her head. \"Leave? You've barely got here. Why?\"\n\n\"I'm pretty sure my cover is blown; or, if it isn't, it will be soon. They'll come for me if I stay.\"\n\n\"Oh, Lucy. No. I'll come with you. I'll\u2014\"\n\n\"No. Really, you can't; it's too much of a risk. I'll be safer getting away on my own.\"\n\nA range of emotions crosses her face, and I brace myself for the storm, but before it gets going, it disappears. She sags back on the sofa.\n\n\"When?\" she whispers.\n\n\"I don't know. Soon. As soon as I can arrange something. It won't be forever, I promise; I'll get in touch. Someday I'll come back and see you, when things are different.\"\n\n\"Oh, Lucy. No. It's not fair.\"\n\n\"Life's like that,\" I say, sharper than I mean to. But really, when has life been fair to me? Even when I finally thought I was returning to a family that was _mine,_ I found out it was all lies.\n\n\"This isn't because of me, is it?\"\n\n\"Of course not.\"\n\n\"Tell me everything. Maybe I can help.\"\n\nI shake my head. \"I'm sorry; it's safer if you don't know.\"\n\n\"You don't trust me,\" she says, her voice bitter.\n\n\"It's not that! But why should I? You've lied to me my whole life,\" I say, the words spilling out of me before I can call them back.\n\nShe recoils. \"You've worked it out, haven't you.\"\n\n\"What?\"\n\n\"That I haven't told you everything.\"\n\n\"What else haven't you told me?\" I demand, even as some part of me realizes this isn't supposed to be going like this; I'm supposed to be trying to mend things a little before I go, but I can't stop myself from asking. What more could there possibly be?\n\n\"It wasn't my fault!\"\n\n\"What wasn't your fault?\"\n\n\"She made me do it, don't you see?\"\n\n\"Who\u2014your mother? What did she make you do?\"\n\n\"She was blackmailing me, all these years, into keeping quiet. I was a prisoner back then! She had me under lock and key the whole time I was pregnant, to stop me from talking; she kept Danny away, made him think that was what I wanted. Maybe my baby would have lived if I'd been at home. But then when she brought you . . . she knew she had me. Right where she wanted me. I couldn't say anything, could I? Or you'd be gone. So she finally let me go.\"\n\n\"What are you talking about?\"\n\n\"No. That's enough. If you want to know any more, you have to tell me your secrets, too.\"\n\n\"I just did. I came here to tell you I was going to have to leave. I shouldn't have told you, it was dangerous to tell you, but I did it.\" I stand up.\n\n\"Wait. Don't leave like this. Please. I'll tell you. But you have to _promise_ to never tell anyone.\"\n\nI pause. I'm seething\u2014again. Something about Stella and I just . . . I don't know. She makes me crazy. _But she'll be so sad when I'm gone._\n\nI take a deep breath, and sit. \"Okay. Tell me.\"\n\n\"I found out some things, put them together. Stuff my mother did years before against the government.\"\n\n\"Against the Lorders?\" My head is reeling. No way; she is Lorder through and through.\n\n\"No, not exactly. There are factions, you see. In the government. Mother is on the hard-line side; the last prime minister wasn't. He had to go.\"\n\n\"Wait a minute. Are you talking about Armstrong?\"\n\n\"Yes. He and his wife, Linea.\" She sighs. \"They were so lovely, and\u2014\"\n\n\"You knew them?\"\n\n\"Linea and Mother were friends at school years ago. Linea confided in her that her husband was planning to expose some of the nastier side of the Lorders and resign. He never got a chance.\"\n\nMy head is reeling. \"No way. Mum's parents?\"\n\nShe frowns. \"Mum? What do you mean?\"\n\n\"After I was Slated, that was the family I was assigned to. Sandra Armstrong-Davis.\"\n\nNow it is Stella's turn to look shocked. \"You were with Sandy? I didn't know.\"\n\n\"You knew her?\"\n\n\"Of course. We used to go on vacations together, years ago when we were children. We haven't kept in touch. I couldn't. Not after I knew what really happened to her parents.\"\n\n\"But they were assassinated by AGT.\"\n\n\"Yes. But the AGT were told where they'd be. The information was leaked: It was a setup.\"\n\n\"Your _mother_ was behind that? Oh my God. You have to tell. You have to!\"\n\n\"No. I can't! I never can, not anymore. It's too late, far too late. What would it even mean now? After all this time. No.\"\n\n\"Listen to me. Astrid was blackmailing you using me. If I'm not here, and she doesn't know where I am, she can't blackmail you. Can she?\"\n\n\"It's not that simple anymore. It's everyone: all the girls here. She'd use them against me.\"\n\nI try. So hard. To tell her that if people don't say what they know, that if we don't stand up against the Lorders, things will keep getting worse. That it is in our hands to do something. She isn't listening, I can tell.\n\nBut how can I complain when all those times with Aiden, I didn't listen, either?\n\nWhat I don't say is, what if she had spoken out all those years ago? Told everyone that the prime minister was going to resign and expose the Lorders, that they were assassinated by their own government to keep them quiet. Maybe the Lorder stranglehold we have today would never have developed.\n\nI stand to go.\n\n\"Wait. My last request. Can I have your camera?\"\n\n\"My camera? Why?\"\n\nShe shakes her head. \"I'll give it back. I just want copies of your photos: those of you and us.\"\n\nI hesitate. \"Okay. I'll bring it down.\" I leave the room wondering if she could see the telltale bulge that said the camera was in my pocket the whole time.\n\nBack in my room, I fiddle with the camera's interactive screen until I work out how to make folders; I password-protect the ones of the orphanage. I long to e-mail them to somebody, anybody, but don't dare without a nongovernment computer. They'd be monitored and stopped for sure, and then they'd have my location.\n\nI bring it back down, thinking I'll wait while she downloads the photos.\n\n\"I've got something for you.\" She holds out her hand, and in it, a key. \"Your dad's stuff. Photos, all of it. I wanted to get rid of it, but somehow couldn't bring myself to do it.\"\n\n\"Where?\"\n\n\"In the old boathouse. Do you remember where it is?\"\n\n\"I think so. Thanks.\" I clutch the key in my hand.\n\n\"Go on, have a look while I go through the photos on your camera. I'll give it back to you at dinner.\"\n\nI hesitate, unsure if I should let it out of my sight, but the key in my hand is pulling me another direction. I grab my coat and put my boots back on, groaning a little at sore feet from so many fast miles covered today. I slip out the side door, run down the grounds toward the lake.\n\nDo I remember the boathouse? I try, really hard, but nothing comes to me beyond flashes of a kayak sliding into the water. I wander the paths along the water's edge. There are several outbuildings next to the racks of kayaks, and another farther along, almost hidden by overgrown plants and overhanging trees. As soon as I see it in the moonlight, I know: This is the boathouse.\n\n_Dad used to spend a lot of time here._\n\nThere isn't a boat in sight; it was a converted workshop, where he'd build bits of things, or just hang out\u2014to get out of the house. Getting away from Stella, I realize now, in a way I didn't back then.\n\nThe key fits in the lock, but it doesn't turn. Some trace of memory tells me to push the door in with my knee and try again, and this time it does. The door creaks open.\n\nIt smells dusty and damp, and I step forward into cobwebs. I brush them off and sneeze, feeling along the wall for the switch. I find it; it doesn't work, but then my elbow knocks something off a shelf. I stoop to pick it up, and my hand closes around it: a flashlight. I flick it on.\n\nThe table, the bench, all still in place; as I see them, the rush of memory almost knocks me from my feet. Instead of tools and stray broken things, they are covered now by plastic boxes. I pull the lid off one, then another: clothes. Dad's clothes, from a lifetime ago; books that were his.\n\nIn another under more books is a chess set. His set, the one he taught me to play on: one of my few happy memories. He let me win. I smile, open the box, and touch the pieces inside.\n\nOf course one is missing: one rook. The castle. He used it to reach me in that faraway place, where I was taken, held, and fractured. It's up in my room here, tucked in a corner of my bag. And here are all its mates. Something inside me longs to bring the missing rook down to this place, have them reunited in their little nests inside the box.\n\nAnother box is full of photographs, and I dive into them. There are old photos of Stella and Dad, some from their wedding. I hunt for some of us together; there aren't many, but I find a few. There is one of me and him and Pounce as a tiny kitten; all smiles. It must have been taken the morning of my tenth birthday. Before everything went wrong. I tuck it into my pocket, along with one of Stella and Dad laughing together when they were young. There aren't many photos of Dad, if this is all of them, to make the trail of an entire life; he was usually the one with a camera in his hands.\n\n_My camera._ Some sense of unease returns. How long have I been down here?\n\n\"Riley?\"\n\nI jump, turn. Ellie stands framed in the door, shivering without a coat on, my camera in her hand. She holds it out. \"Stella asked me to give this to you; you forgot it in her office. And to say that it is okay if you miss dinner if you want to.\"\n\nShe turns, dashes back up the path.\n\nI stare at my camera in my hands, confused. I forgot it? She said she'd give me the camera at dinner. Why the change? Did she realize I was going to want to spend loads of time down here, or what?\n\nMaybe there is a message beyond her words. Something isn't right. My skin crawls as if an army of spiders have found me here.\n\nI switch off the flashlight, then slip out into the night. Close the door, slow and quiet; push it with my knee. Lock it. Wonder what to do with the key, then put it on top of the door.\n\nVoices float out on the night air, too faint to discern. There are feet crunching on the gravel above. I slink from one tree to another until figures above are in view, but it's too dark to make out who they are. I get the camera out, put it in night mode, and peer through the zoom. A car is parked to the side of the house; the main lakeside door is open, Stella framed in it. Two others are walking toward her. One is Astrid. The other is a man, his back turned to me. The light is poor, but every move he makes is fluid, sinuous, catlike. I feel as if my muscles and bones are melting, can't hold me upright; I may collapse.\n\nNico.\n\nWhy would he be here, with Astrid? It doesn't make any sense.\n\nHe pauses, turns his head, and scans out into the darkness, and I shake, convinced he can sense my presence, that his pale blue eyes can somehow penetrate the night and see where I hide. Without thought my finger pushes the camera button, taking several quick shots of him and Astrid in the same frame.\n\nHow can this be? Astrid and Nico\u2014Lorder, and AGT\u2014are sworn enemies. Aren't they?\n\nMovement draws my eyes to the sides of the house. I swing the camera. Figures in black: Lorders. They're watching the side doors. What do you want to bet they are by every door? Fear grips my gut for Ellie. Did she get back inside before they came?\n\nThen I notice one of the Lorders has something strapped across his eyes: night vision goggles.\n\nI sink back down until the dip of the land has them out of sight.\n\nThat message from Stella _was_ a warning: Has Astrid worked out who I am? My panicked brain can't process Astrid and Nico being together, or what it means. But whatever it is can't be good.\n\nAdrenalin pours through my body. _Run!_\n\nTo go right would mean going past an open stretch of the grounds that slopes down to the lake without cover\u2014no way I wouldn't be seen. To go left is the logical escape route: the wooded footpath into town. This is where they will hunt when they realize I'm not in the house.\n\n_The lake._\n\nI slip along the water's edge to the racks of kayaks. I almost hold my breath as I lift one off the rack as quietly as I can. The paddles are clipped alongside the kayaks. The desire to get away is so strong that it is hard to pause, but I do, and unclip all the other paddles, and gather them together. Make it harder to chase.\n\nI move along the shore, awkwardly balancing the kayak and paddles. Step slowly into the water so I don't splash, struggling not to gasp with the cold when it goes over my boots. As silent as I can, I get into the kayak, awkward in winter clothes and with all the paddles tucked under one arm. A paddle drops and catches the water; the end jumps up, knocks my glasses off my face. There is a slight splash as they drop into the water. I grab for them, but they're gone in the darkness. What does it matter? They're not going to fool Nico if he catches me.\n\nI set out, and my childhood kayaking soon comes back to me; my strokes are fast and sure, hugging the shoreline so I'm harder to spot.\n\nOnce I'm well away from the house, I move away from the lake's edge and slip the paddles out in the open water with a silent apology. I leave them floating behind me as I pour every bit of panicked energy into strokes to get me as far from Astrid and Nico as I can.\n23\n\nSHIVERING, I PULL THE KAYAK WELL UP FROM THE WATER, then push it and the paddle under bushes. I focus on picturing the map Finley showed me earlier: Keswick Boys should be nearby.\n\nI don't like going there\u2014it's risky for so many reasons\u2014but what choice do I have? I need to find Len, and the only way I know how is through Finley. Besides, I'm half soaked with January lake water. My feet must still be down there, but are so numb, walking is clumsy. The temperature is dropping and, judging by the thin ice I had to break through on this side of the lake, it's going to get worse. I need to get warm and dry.\n\nThere are buildings above me, lights, voices. I hug the path that skirts around the houses, until at last I see a large rambling building set apart above them.\n\nI slip around the side of it. There is a boy in shadows near the back door; a dot of red light says he's having a smoke. Do I wait until he goes and try to sneak in, or brazen it out?\n\nI'm too cold to be subtle.\n\nI step down the path in front of him. \"Hello,\" I say.\n\nHe squints into the darkness; I step into the light by a window.\n\n\"Hello there, yourself. Where did you magic from?\"\n\nI giggle. \"Could you tell Finley I'm here?\"\n\n\"Him again?\" He rolls his eyes and stubs out his cigarette on the side of the building. \"Wait a sec,\" he says, and disappears into the house.\n\nA few minutes pass. A window down the end of the building opens with a thud, and a head peeks out: Finley.\n\n\"Riley? What are you doing here?\"\n\nI quickly cut across to the window. \"I'm in some trouble.\"\n\n\"A damsel in distress? One of my favorite things. Come in the back way.\" He holds out a hand and I realize he means the window. He half pulls me through, into some sort of utility room.\n\n\"You're frozen,\" he says.\n\nI nod, shaking violently and not bothering to try to hide it anymore. \"I kayaked across the lake. I'm soaked.\"\n\n\"I'm guessing this has something to do with the Lorders chasing us earlier today.\"\n\n\"Probably,\" I say, though not entirely sure that they could have worked out who I was and where to find me that quickly. Then I remember Steph at dinner the other night\u2014she saw my green eyes. Is she a spy for Astrid? If so, that'd be weird enough for her to report, even if she didn't have an inkling who I am. I shake my head. \"I don't know. It could be that, or it could be something else. Either way, I'm trouble. Are you sure you want to help?\"\n\n\"Don't be daft. Of course I'm helping you. We'll get you warm to start with. Hang on.\" He opens the door, looks out. \"Coast is clear.\" He holds out a hand. \"Try to look like you're here because of my irresistible charms, not on the run from the law.\" He winks, takes my hand, and wraps an arm around my waist.\n\nWe walk fast up the hall to the end, up a flight of stairs to the next level, down a hall. He opens a door to a bedroom.\n\nAnother boy is in there, reading a book on one of two beds.\n\n\"Clear out,\" Finley says.\n\nHe looks up, rolls his eyes. \"Didn't take you long to get over the last one,\" he says. Finley stiffens, but manages to keep his arm around me as the other boy leaves.\n\nAs soon as the door shuts, we spring apart. \"Sorry,\" we both say in unison.\n\n\"He won't say anything?\" I ask.\n\n\"Of course he will. But only to the lads. Boys' code,\" he says, and taps the side of his nose.\n\n\"Great,\" I say, then wonder why I care, as long as they don't tell the authorities I'm here. My reputation is far down my list of worries.\n\nHe opens a wardrobe, fishes through it. \"Get out of that wet stuff and put these on.\" He turns around, and I shuck my boots, jeans, and socks and pull on his miles-too-big joggers, then some huge wool socks. I'm still shaking. \"My bed isn't as disgusting as his. Go on, warm up.\" I get in it and pull his blankets around me in a cocoon. Finley puts my stuff on the radiator, stuffs paper in my boots.\n\nHe pulls a desk chair around. Now come the questions, and he has a right to ask them, but in some sort of crazy delayed fear reaction I'm pulling my arms around my head in a ball. Nico: He must know I'm alive. Why else was he there? He'll find me. Deep shuddering sobs are tearing out of me, and Finley starts patting my shoulder, all awkward but nice, but for some reason it just makes me cry harder.\n\n\"Hey there. It'll be all right,\" he says. But how can it be? \"Don't cry. If anyone hears, it'll ruin my reputation.\"\n\nI draw in a shaky breath and fight for control. A bell rings, and I jump.\n\n\"Dinner bell,\" he explains. \"But I can stay.\"\n\nI sit up and rub a hand across my eyes. \"Actually, I'm starving.\"\n\n\"Oh, thank God, me too. Right. I'll get takeout.\"\n\n\"You can do that?\"\n\n\"Sure. The boys'll run interference while I stack an extra plate. Back in five.\"\n\nHe leaves, and I fight for the composure I'd felt earlier. The certainty that I would get away, and I'd find Aiden, and I'd give him my photos, and he'd know what to do with them. Somehow with all the Lorders in the world potentially after me I wasn't all right exactly, but could carry on. But now that it's Nico, too?\n\nAfter all he and his AGT have done to me\u2014stolen my childhood, my life, killed my father, programmed me to be a killer\u2014there is a core of cold fury inside. But most of all, overwhelming everything else, is fear. One glance at him in the distance and I was terrified. He must know I didn't die when he set off that explosion; why else would he be here? Astrid knew I survived; she must have told him when she worked out where I was. He'll find me. He always finds me.\n\nI look at the window behind me and the door opposite, jumpy, as if just by thinking of what scares me I can conjure it up.\n\nAnd Astrid and Nico together: What does it mean? I can't process this. Stella said Astrid was behind the assassinations, that AGT did it but she set it up. It couldn't have been Nico; it was over twenty-five years ago, and he can't be much older than that. But Astrid must have links with the AGT. Is that it, is she still using them for her purposes?\n\nBut Nico _hates_ the Lorders. How could they be in the same place at the same time? He is AGT through and through.\n\nI shake my head, trying to make sense of it all. After I was taken from Castlerigg to that other place by Dr. Craig, Nico was there. He was involved from the beginning. Astrid was my grandmother\u2014or so I thought back then\u2014she knew me since I was a baby. She might be the only person who knows where I even come from. Now that I've seen her with Nico, is it too much coincidence to think the AGT targeted me and she wasn't involved? Stella thinks Dad was behind what happened to me, but was it Astrid\u2014her own mother\u2014all along?\n\nApproaching footsteps pull me from my thoughts; my heart races. There is a light tap, and Finley opens the door.\n\nHe sees the look on my face. \"It's only me. Maybe we should have a secret knock?\"\n\n\"Sorry, I'm just jumpy. And I'm sorry about losing it before,\" I say.\n\n\"No worries. Here you go,\" he says, and holds out one of two big bowls in his hands, full of stew with bread on the side, and it smells good. I'd told him I was hungry to get him to go, to give me time to compose myself, but now that I can smell food, I'm famished.\n\nWhile we eat, Finley looks at me curiously, then pauses in between mouthfuls. \"You look different, and I just worked it out: no glasses. But your eyes look different somehow.\"\n\n\"I lost them in the lake.\"\n\n\"Are you going to tell me what's going on?\"\n\nI stare back at him. The weak point in my plan, such as it is, is this moment. \"Sometimes, it's better not to know stuff.\"\n\n\"Like what you were up to today.\"\n\n\"Exactly.\"\n\n\"It's nice to have a reputation-enhancing houseguest, but no matter how slack they run this place, eventually someone official is bound to notice. You can't stay here forever.\"\n\n\"Just a few hours will be fine. Thanks.\"\n\n\"What can I do to help?\"\n\nI can't see any way to get the information I need without the direct approach. \"I need to find Len,\" I say, with a silent apology to Len. He didn't want Finley to know anything, did he?\n\n\"Always knew there was more to the old man than meets the eye. That's easy; he lives over the hill. We could go now?\"\n\n\"I think better to wait until everybody is asleep, then slip out. Tell me where to find him, and I'll\u2014\"\n\n\"No way. I'm going, too. Don't want you to get lost or bang on the wrong door.\"\n\n\"But\u2014\"\n\n\"No buts. That's it.\" He takes the bowls, some things for his roomie, who is bunked down in another's room. Comes back and stays, reading a book in a chair with a grimace at the state of the other bed, tells me he'll watch over me, that everything is okay, to get some sleep and he'll wake me up in a few hours.\n\nWarm at last, I'm convinced I'll never sleep, that it isn't possible. There are my own fears still, and others intruding: What if Nico realizes Ellie saw me last? My stomach turns to think what he might do if he gets his hands on her. Then I'm angry at Stella for using Ellie like that; then I'm scared for Stella. And Finley. It won't take long for someone to remember we spent the day together, to come looking for him, and then if they haven't already connected that with the ones they were chasing from the orphanage today, they will soon. He says he's watching over me, but he has no idea what he could face if they come here.\n\nMy fears circle around in my head like black crows, but somehow they become more faint and distant, and drift away.\n\n\u2022 \u2022 \u2022\n\n_A small head peeks over his mother's shoulder, giggles._\n\nGet down! _I'm saying inside and with my eyes, but he doesn't get it. He peeks out again._\n\n_This time they see him. Black-garbed Lorders._\n\n_They march past me and grab the boy from his mother; she's pleading, he's crying. Everyone on the train stares at their feet, at the floor, at the dead windows. No one moves. No one says anything._\n\n_Not this time; it is time to say no. No more. I get out of my seat. \"Leave him alone!\"_\n\n_One of the Lorders slowly turns around. His blond-streaked hair is too long and unruly for a Lorder. His pale blue eyes glint with danger. He smiles a seductive smile, and holds out a hand._\n\n_Nico? No. It can't be._\n24\n\nMY EYES SNAP OPEN. THERE IS A BRIEF MOMENT OF CONFUSION: Where am I? Finley's room. It is still, silent. What woke me?\n\nI glance across the room, and there is just enough moonlight through gaps in the curtains to see that the chair and other bed are empty. I'm alone.\n\nThen there is a faint sound in the hall: footsteps?\n\nI sit up with a rush of fear. Nowhere to hide. Not enough time to get out the window; nowhere to run.\n\nThere is a featherlight tap on the door, and it opens: Finley.\n\nI sag back in relief.\n\n\"Oh good, you're awake,\" he says. \"Time to go.\"\n\nI shake off the panic, and grab my things off the radiator. \"Nearly dry.\"\n\nHe turns as I change into them fast.\n\n\"Now come on,\" he says, and takes my hand. \"If anyone spots us they'll just think I'm sneaking you out.\"\n\nFinley peeks out into the hall, draws me out. We walk silently to the stairs, down them, and then out the back door\u2014a real door, not a window this time.\n\nI glance back at the lake; all is silent inky darkness. I frown to myself. Surely once the Lorders realized I wasn't in the house or up the path, they'd work out I took a kayak, that the other paddles were missing? I'd half expected to see searchlights.\n\nWe creep along footpaths up and away from the houses and the lake, Finley leading the way. I am sure-footed and silent in the dark; years of woods training with Nico made sure of that. Finley is less so. A particularly loud crack in front makes me cringe. \"Keep it down up there,\" I whisper.\n\n\"Don't worry, the tree is fine,\" Finley says.\n\n\"What tree?\"\n\n\"The one whose branch I just smacked my head into. I'm starting to see that being a Shorty has its advantages.\"\n\nWe reach a road, walk along it for a mile, keeping a careful listen for cars, ducking out of sight whenever one goes by. Then we turn down a long, twisty lane.\n\n\"Here we are: chez Len,\" Finley says. More a shack than a house; the car we borrowed earlier is pulled in alongside. It is silent and dark.\n\n\"What time is it?\" I whisper.\n\n\"Four A.M.\"\n\n\"Hope he's not a heavy sleeper.\"\n\nFinley tries knocking lightly on the door; no response. He tries the handle: locked. We exchange a glance. \"Somehow really banging on the door goes against the whole sneaking around thing,\" he says.\n\nI pick up a few pebbles and ping them against the window.\n\nEventually we hear movement inside, locks turning. The door opens, and Len peers out. \"This better be good.\"\n\nLen pulls us into the kitchen, shuts the door. \"I don't know about you, but I can't be civil without tea this time of the morning. You can make it while we talk,\" he says to Finley. Points out the kettle, cups. Draws me into the next room and shuts the door.\n\n\"So, Miss Lucy Connor: Your cover is blown.\"\n\n\"You know?\"\n\nHe inclines his head.\n\n\"How?\"\n\n\"Got a message from your mother.\"\n\nI stare at him in shock. \"She knows you?\"\n\n\"How'd you think you got put on MIA in the first place?\"\n\nI shake my head. \"I don't know; I guess I didn't think about it.\"\n\n\"Do you know how they worked out who you are?\"\n\n\"I don't _know._ But I think Steph\u2014one of the girls\u2014is a spy for Astrid. She saw my glasses come off. That my eyes are green.\"\n\n\"Those were some glasses. Not wearing them anymore?\"\n\n\"Lost them in the lake. Don't suppose it matters now. I'm sure Astrid was already suspicious that something was up with Stella. Then if Steph told her about my glasses, she must have put things together and worked it out.\" I lean back and sigh. \"I'm sorry about involving Finley. I had to run, and I didn't know how else to find you.\"\n\n\"He's a smart lad. He'll keep his mouth shut. But there's more to tell, isn't there?\"\n\nThere is a tap on the door. Finley peeks in, holds out two cups of tea. \"Can I come in?\"\n\n\"Not just yet, give us a few moments,\" Len says.\n\nFinley looks disappointed, but hands us our tea and goes back through the door.\n\nLen has a slurp of way-too-hot tea but looks happier. \"That's better. Now tell me: What happened yesterday afternoon to make Lorders be looking for you up that mountain?\"\n\n\"I found out something, something that puts Lorders in a very bad light. I need to get to Aiden, the sooner, the better. Can you help?\"\n\nHe stares steadily back at me, sighs. \"I'll always help when I can, because I'm a stupid old idiot. Not enough life left to be cautious with. But getting you out of town will be difficult if they're looking for you. Maybe you should tell me why it's so important.\"\n\nI hesitate, torn. Aiden trusts him, and that's good enough for me. But is it safer if he doesn't know?\n\n\"Look at it this way: If you're the only one who knows something, and something happens to you, nobody knows.\"\n\nI nod. Gulp. \"It's so awful, it's hard to even say.\" My head aches, and I drop it in my hands.\n\n\"Time is tight, Lucy,\" he says gently.\n\n\"I'd rather you didn't use that name\u2014stick with Riley.\"\n\n\"Okay, then; Riley.\"\n\nI look up, meet his eyes. \"We saw them from a distance. A line of children, walking along the fence inside the orphanage grounds. But something wasn't right about them. They weren't like normal children. So I went for a closer look.\"\n\n\"Mad as a box of frogs. And?\"\n\n\"They're all Slated. Even little ones, maybe four or five years old.\" The horror I can feel on my face is reflected in his. \"They were like . . . robots, almost. No personality, no life.\"\n\nHe grips my hand. \"Evidence?\"\n\nI pat my pocket. \"Photos, in my camera, of Levos on their wrists.\"\n\n\"Bad timing.\" He curses under his breath. \"MIA's website has been hacked.\"\n\n\"What? Could that be how they found where I was?\" As a Slated, seeking out my past life is completely illegal. That'd be reason enough for Lorders to hunt me down, without adding in stumbling on their secrets at the orphanage.\n\n\"They got into the protected areas of the site. Any information available to administrators of MIA was open to them. If they looked, they'd know you were reported found, but not your location. That type of information isn't stored on the website, not even encrypted. Of course they may have worked out to look for you in Keswick. But all computer communications are suspended while it is investigated, so we can't e-mail the photos as backup. Besides, Aiden'll need you; you're the witness. We've got to get you to him.\"\n\n\"Well. You know how they're looking for Lucy Connor? Things could get worse.\"\n\n\"It can get worse?\" Len asks mildly.\n\n\"If they put her together with the girl who took photos at their so-called orphanage. And that Finley was with me. And you were seen with us later on. I'm so sorry.\"\n\nLen calls Finley in; he has tea with us. Len finds some biscuits, puts a hand up when we try to say anything. \"Be quiet. I'm thinking.\"\n\nFinally Len looks at me, then gestures at Finley. \"You didn't tell him what you found out?\" I shake my head. \"Keep it that way.\" Finley looks about to protest, but Len holds up a hand. \"Listen to me. We might be in some trouble. If you don't know, you can't tell. You can try to play innocent. Bit of a stretch for you, I know.\"\n\n\"They don't care about innocence,\" I say, bitter as I think of the Slated children. Children? Some of them were more like toddlers.\n\n\"This is what I think you should do,\" Len says to Finley. \"Go home. Slip back in like nothing is happening. I think chances are low they'll connect you with her.\"\n\n\"Can't he come with me?\" I ask.\n\n\"No. If Finley is reported missing now, they _will_ put the two of you together. They'll be more likely to make the leap that it was you two they were looking for on the hill yesterday.\"\n\n\"No, you can't just send him back! It's too dangerous.\"\n\n\"Listen to me. Something is funny about all of this. If you really found something the Lorder government wants to stay hidden, they'd have jumped on the whole county like a ton of bricks: roadblocks, searches house to house, the works. And there has been nothing.\"\n\n\"What does that mean?\"\n\nHe scratches his head. \"Haven't a clue, but for now I think it works in our favor. My usual thought would be to lie low until things settle down before we try to move you anywhere, but this time I think the faster we get you out of town, the better.\"\n\n\"Right. I'll be off, then, or I won't be able to sneak back in before morning.\" Finley gets up, comes over to me awkwardly, and stoops down to give me a hug. \"Look after yourself,\" he says. \"Don't worry about me; I'll be all right.\"\n\nLen goes to the door with him. They murmur to each other too low for me to hear, then the door shuts and Len comes back in.\n\n\"Do you really think they won't track him down?\" I ask.\n\nHe hesitates. \"No, and he knows it. He's buying you some time. Don't waste it.\"\n25\n\n\"ARE YOU SURE ABOUT THIS?\"\n\n\"Unless you can flap your arms and fly, then yes,\" Len says. \"Trains are out. Even if we could get you another fake ID quickly, they're watching for you now. They're unlikely to be fooled. It's the only way.\"\n\nThe truck, a goods transport, is parked behind an isolated workshop. These are the only sort of vehicles that do long road journeys with the environmental bans on private long-distance car travel. The truck's cab has a false floor Len has taken up, with a small\u2014and I mean _small_ \u2014space underneath. It's only ever been used to move around tech for MIA; I am to be the first passenger.\n\n\"Let's see if you fit,\" Len says, and I lower myself in, nesting a blanket around myself. Trying different positions to get my arms and legs in. \"I'll try the floor on now,\" he says. \"Knock or yell if it gets too tight.\"\n\nI give him a thumbs-up, Len slowly lowers the floor in place above me, and I'm plunged into darkness. He lifts it up again. \"All right?\" he asks.\n\n\"I think so, just had to flatten my shoulders. Go on.\"\n\nHe puts it back into place. The noise of the drill as he replaces the screws is unbelievable, and I try to stuff my hands in my ears, but can barely move enough to do it. There is an irrational feeling of panic, like I'm being buried alive in a coffin. The driver is having lunch. He's not in MIA, but bribed regularly to leave his truck unattended. While he knows he is transporting something, he's kept in the dark about what it is. He doesn't know I'm in here. What if something happens to whoever is meant to extract this \"delivery\" at the other end? No one but Len knows I am trapped in here; he judged it too dangerous to get a message out that might be intercepted. Even though he has drilled out some airholes, already I feel like I can't breathe.\n\nThere is a _tap-tap_ outside someplace, and I imagine Len giving the side of the truck a double tap for luck to send me on my way.\n\nI half smile, then it falls away. I owe Len and Finley so much. _Please stay safe._ I'd gotten out of Len that he'd told Finley a safe bolt-hole if he gets wind of trouble, but if the Lorders come for him it's more likely that he won't have a chance. It was luck that had me down by the lake when Nico, Astrid, and the Lorders came; if Stella hadn't managed to get Ellie out with that cryptic message, they'd probably still have gotten me. What would they have done with me? Nothing good, I'm sure. Maybe hauled me off to the prison Len thinks Madison is in. Maybe, I'd be dead.\n\nTime passes. A door opens, then slams shut. The truck's engine starts up, and it lurches along bumpy minor roads for what feels like forever. Then it's motorway, smooth and fast. The rocking motion is almost soothing. The small confined space warms with me in it, and lulls me to sleep.\n\n\u2022 \u2022 \u2022\n\n_WHRRRRR . . ._\n\nI bolt awake, hit my head hard on something above me, then remember where I am. That noise, reverberating through my skull: The screws are being removed. Are we there, wherever that is, or have we been stopped along the way? I have no sense of time. But now that I'm awake, every muscle in my body is screaming to move and stretch; one way or the other, I'm getting out of here.\n\nThe last screw out, the floor is lifted, and I sit up as it is pulled away.\n\nThere is a startled face above me, and the floor is almost dropped on my head.\n\n\"Oh my God. It's a girl,\" he says.\n\nThe floor is pulled away and out; another face appears. Both men, in overalls. Clearly not Lorders. I breathe a sigh of relief, swing my legs out. \"Ouch. Give me a hand?\"\n\nOne of them hastens to do so, and I step out of the truck, then almost fall over again, my legs protesting after so long in the same position. I steady myself, one hand on the side of the truck. We're outside in the cold behind some buildings, and it's dark.\n\n\"Where am I?\"\n\nThey exchange a glance.\n\n\"Oh, sorry. I'm here for the final act of _The Winter's Tale._ \" Len had given the words to say, coded words that should get me to authority in MIA quickly.\n\nThings move fast after that. I'm hustled inside a workshop and run to their disgusting bathroom, then beg a cup of tea while hurried conversations happen on the other side of a door. I'm strangely calm. Is it because what happens next is out of my hands? I don't know.\n\nA car appears; I'm bundled into the backseat. A man and woman sit silently in front. I watch out the windows; we go through an industrial area, and then into more and more built-up areas. Not London, and not familiar, but then I see a sign: WELCOME TO OXFORD. So close to home! Or what used to be home. My old school, Lord Williams's, is just miles away in Thame. Our village not far beyond that.\n\nThe streets get more crowded with old buildings, busy with pedestrians. We weave down narrow back streets and pull in. I'm given a different coat to wear, a hat, and walked by the driver along rambling cobblestoned streets, past grand old architecture. I long to gawk and stare upward at everything around me, but don't dare draw attention to myself.\n\nWe duck through an archway, down a path along a quadrangle surrounded by buildings, then to a door where we are met by a smiling girl a few years older than me. The driver leaves me there. I follow the girl down twisting passages to another door. She taps lightly. \"Go in,\" she says, and leaves.\n\nI pull the door open: a bookshelf-lined study. And there, at a desk? A very startled-looking Aiden.\n\n\"Kyla? You're our new witness? Thank God you're all right.\" He jumps up from his chair, scoops me in his arms for a hug, and I cling on a little tight. Something is melting inside: Aiden is here, he'll know what to do. Am I safe, at least for a while? He finally lets go, but keeps hold of my hand, laces my fingers with his. And I stare at him, shaken by how tight I want to hold on to his hand, how much I missed him. His deep blue eyes. Merry eyes, despite the risks he takes. The way his hair catches the light, fiery glints of red. I smile.\n\n\"Ahem.\" A throat-clearing noise. I turn and see there is another person in the room, a woman in a chair by the fire: older than Aiden, a fierce hungry expression on a face pale with dark-ringed eyes. \"I hope we're not using all these resources for girlfriend transport now.\" She frowns.\n\n\"Len wouldn't have sent us anything that way that wasn't vital. Not with an emergency code.\" Aiden's voice is calm, but a tinge of red creeps up his cheeks.\n\n\"So, what is so urgent? Tell us,\" she says.\n\nI look at Aiden. \"I'd rather talk to you alone.\"\n\nShe scowls. \"That won't be happening.\"\n\n\"Who are you, then?\" I demand.\n\n\"It's all right, Kyla. This is Florence\u2014we run MIA together, since . . .\" And he stops. \"Anyhow, we always work on a double system for hearing witnesses. If more people know, then information is safer.\"\n\n\"Okay, then,\" I say, and get my camera out, fiddle with it to enter the password for the folder, and to find the screen projection control. \"These photos were taken at the Cumbrian Care Facility near Keswick. An isolated fenced orphanage. I saw children walking along the inside of the boundary, moving unnaturally. I went for a closer look, and this is what I found.\"\n\nThere are no blank walls, so I point the camera at the door and hit the button to show the shots: three small boys, wrists extended. Levos clearly visible.\n\nThey both gasp. \"Oh my God,\" Aiden says, and they look at each other.\n\n\"By how they were acting, all the children at the orphanage were Slated. From these little ones up to age eleven or twelve. About fifty children in total.\"\n\n\"We can't wait any longer,\" he says. \"Not with this and all the other evidence of Lorder atrocities that we have. It's time to get the information we've gathered out to the people. They can't ignore _this_ \u2014it will be the beginning of the end of the Lorders.\"\n\nFlorence shakes her head. \"We need verification. Photos can be faked.\"\n\n\"No! We have a witness.\"\n\nThey continue, back and forth, and I get the sense of a long-standing argument with different variations, but feel removed from it. I've passed along what I know; it's not my burden anymore. There are other things I want\u2014I _need_ \u2014to tell Aiden; personal things. But not here, not now. Not with her in the room.\n\n\"Excuse me,\" I say once there is a lull. \"Any chance of something to eat?\"\n\nAiden is contrite. \"Of course. But let's get those photos.\" He holds out a hand for the camera.\n\n\"I want to keep the camera; there's my stuff on here, too.\"\n\nHe connects it to a computer and downloads the orphanage shots when I give him the password. Hands the camera back.\n\n\"But I thought Len said the MIA computer system is compromised?\" I ask.\n\nAiden sighs. Nods. \"It's been an absolute nightmare; our techies are working on it. We've been aware that they monitor the site for years, but this time they've hacked through our defenses and were able to trace backward to MIA website administrators. It's a mystery how they got in, and we're not sure how long they've been spying on us through it or how much information they've extracted. But this computer is offline. So for now the photos are stored here, and still there, of course.\" He gestures to my camera. \"Keep it safe.\"\n\nI slip it back deep in my pocket. Aiden looks pointedly at Florence. She sighs, gets up. \"I can take a hint.\"\n\n\"Could you bring Kyla some dinner?\" Aiden smiles his most charming smile, and it is pretty good, but she scowls.\n\n\"Don't push your luck.\" She stomps to the door, looks back. \"I'll get something brought, though.\"\n\n\"Can you get the new lad to bring it?\"\n\nShe looks at him curiously, then nods.\n\nAnd she's gone.\n\n\"She's tetchy,\" I say, glad she's gone, but even more glad to be alone with Aiden. My hand seeks out his again, and he smiles.\n\n\"She has reason. It was her father who started MIA decades ago. His identity was exposed with the computer stuff up. Covers were blown, he's dead, and she's in hiding here. As am I.\"\n\n\"Oh no; how awful.\"\n\n\"But let's not dwell on problems. There's reasons to be cheerful here today.\"\n\n\"They are?\"\n\n\"You're safe and well. Despite our tech setbacks, we're nearly there, Kyla. We're ready to unleash our truth about the Lorders' world once our systems are up and running. You are the nail in their coffin.\"\n\n\"What a lovely image.\" But I smile. \"Aiden, you were right about everything.\"\n\nHe grins. \"I do like to hear that, but about what in particular?\"\n\n\"Things will only change if everybody knows what goes on, and stands together. I want to do anything I can for MIA. Join you completely.\"\n\n\"Being a witness is good enough.\"\n\n\"No, it isn't. There must be more I can do,\" I protest. And inside I'm starting to gather all the rest that I have to tell him: everything I didn't before, and what I've learned since the last time we were together. But before I can begin, there is a tap at the door.\n\n\"This is your third reason to be cheerful,\" Aiden says. But he's not looking that way anymore himself; his eyes are sad, and he lets go of my hand.\n\nI look at him, not comprehending.\n\nThe door opens.\n\nI blink, look back at Aiden, not believing what I see. Not trusting my eyes.\n\nI look back at the person standing in the doorway, a tray of sandwiches in his hands. At the way his too-long dark hair is tucked behind his ears, his chocolate brown eyes. The way he stands. A smile takes over my face, my whole body.\n\n\"Ben?\"\n26\n\nI STARE, TRANSFIXED, THEN GLANCE BACK AT AIDEN. \"BUT how . . . What . . .\"\n\n\"I'll let Ben explain,\" Aiden says. He stands and heads for the door. \"Get reacquainted; I'll be back later. We need to talk some more.\" The door shuts behind him.\n\nBen smiles, a little uncomfortably. He puts the tray on the table. \"Kyla, isn't it?\" he says.\n\nI nod, trying to stop the wave of disappointment that sweeps through me from showing on my face: He still doesn't remember me, or what we were to each other. When he first opened the door, for a moment I'd thought somehow he was my Ben again, that that was why he was here. But his memories are still gone.\n\nHe sits in the chair next to me\u2014inches away, but it feels like a gap of miles when it is all I can do to not reach out and touch him. Instead I just let my eyes drink him in, every bit of him that I was afraid I'd never see again.\n\nAmusement crosses his face. \"There is something I need to say to you, but it's hard when you're staring at me like that.\" A trace of Ben's humor. No matter if his memories are gone, it _is_ Ben.\n\n\"Sorry, I'll try to stop. What is it?\"\n\n\"Thank you.\"\n\n\"For what?\"\n\nHe runs his hands through his hair, the way he always did. \"It was you who got me out of that training place. With the Lorders.\"\n\n\"It was?\"\n\n\"I thought I wanted to be there, I wanted to succeed. To become what they wanted. But all the time there was this little voice in my head, doubting what they said, questioning things. Your voice.\"\n\n\"What I said to you worked? I thought I'd gotten through, for a moment. But then, after, I doubted it. I thought maybe I saw that because I wanted to. But it actually worked?\"\n\n\"Yes, thanks to you I'm on the run from the Lorders with you and all the rest of the crazies.\" He grins, teasing.\n\n\"Do you know what they did to your memories?\"\n\nHe shakes his head. \"Not really. Aiden's setting up for me to see a doctor, to get scanned and stuff. Don't know what they'll find.\"\n\nHis sleeves are long, and I reach out, pull them up, his skin warm under my fingers. \"No Levo.\"\n\n\"No.\" He grins a wicked grin. Takes my hand and holds it between his. \"You don't need an excuse to touch me.\"\n\n\"You don't even remember us together! You're a total flirt,\" I say, feeling both dizzy with the feel of his skin against mine, and confused. To Ben, is mine the hand of a stranger? This feels weird. He looks and feels the same, but do I know him?\n\n\"Maybe I am. But I know we were close once. You must have really cared for me to risk coming to that Lorder place. Seeking me out so close to danger.\"\n\nI shrug. \"I'm just stupid. I do stuff like that.\"\n\n\"Well, I'm glad you did.\"\n\n\"How'd you end up here with Aiden?\"\n\n\"Long story. It was more that he found me. I ran away; someone from MIA spotted and recognized me. Aiden tracked me down to where I was hiding out. He had a difficult time convincing me he wasn't with the Lorders.\"\n\n\"Of course; you wouldn't have remembered him, either.\"\n\n\"No. I'm afraid I kind of hurt him a little. Until I worked out he was telling the truth.\"\n\n\"Whoops.\"\n\n\"Not sure his pride has gotten over being in a headlock. Actually, not sure he or Flo knows what to make of me. They want to work out what's been done to me, but they always make sure someone is with me. Not quite sure what they think I'm going to do.\"\n\n\"They have to be careful, I suppose.\" I frown. That isn't the way to help someone get better, though. Maybe he'll start getting some memories back, like I have? \"I still can't believe you're here.\" And I'm grinning that crazy happy grin and staring at him again.\n\nHe smiles, self-conscious. \"I heard you were hungry; I made them just for you.\" He nods at the forgotten tray on the table, lets go of my hand.\n\n\"Thanks,\" I say, and grab a sandwich, a poor substitute for Ben's hand. Cheese and pickle, and I don't like pickle, but why would he remember that? When there is so much more important stuff that he can't remember, either.\n\nBen stays, I eat sandwiches, he makes me laugh a few times, and I try not to stare so much. Can this really be happening? Ben: He's really here. Aiden found him, and he's safe and well.\n\nThere is a knock on the door, and Aiden peeks in. \"Hello, all right if we come in?\"\n\nI nod, mouth full of biscuit. Aiden walks in and with him is Florence, who looks pointedly at Ben. He rolls his eyes, stands up. \"Well, I'll be off, then,\" he says. \"See you tomorrow?\"\n\n\"Hope so,\" I say. He goes out the door, and I hear him speaking to someone in the hall before it closes. They really did have somebody outside the door the whole time?\n\n\"Are you having Ben watched every minute of the day?\"\n\nAiden shrugs, looks at Florence. \"Ms. Cautious thinks it is necessary.\"\n\nShe bristles. \"We don't really know what makes him tick, do we? I don't think even he does. Until we work out what has been done to him, it is a reasonable precaution.\"\n\n\"You're too careful, with Ben and everything else. We should release the evidence we have, now. Together with Kyla's testimony\u2014\"\n\n\"Testimony? What is that, exactly?\" I interrupt.\n\n\"We are taping witnesses\u2014filming them, when we can\u2014for firsthand accounts of Lorder atrocities,\" Florence says.\n\n\"Filming?\"\n\n\"All you need to do is talk about your story to the camera, just tell it like you told us,\" Aiden says. \"Then when we release the evidence\u2014 _if_ we ever do, that is,\" he says with a pointed look at Florence, \"it will be part of it.\"\n\n\"We will get there, Aiden,\" she says. \"But to avoid being discredited and ignored, we have to make sure we have firsthand evidence for everything. It isn't good enough if it's all he said\/she said. We need to back everything up with evidence; hearsay doesn't cut it.\"\n\n\"But the Lorders must know how close we are to releasing our evidence; otherwise, why play their hand with this cyber-attack? They've been aware of the website for years. Why else now? They must realize the risk. We have to get word out before they stop us.\"\n\nFlorence glances at me. \"Enough of that now, Aiden,\" she says, as obviously as if she'd said, _Not in front of the children._ I bristle, but then remind myself what Aiden told me, about her father, and try not to let it show.\n\n\"Do you feel up to doing this now?\" Aiden says. \"We always try to record evidence at the first opportunity.\"\n\n\"On film?\"\n\n\"It's best.\"\n\nI swallow, scared of being committed to a recording, sound and image, but unwilling to let Florence see my fear. \"Why not? They're all after me, anyhow; giving them one more reason will hardly matter.\"\n\n\"That's the spirit,\" Florence says. I have just enough time to wonder if spending most of the day squashed in the floor of a truck does much for my hair when she has set up a camera on a small tripod on the desk. \"When you're ready,\" she says. \"Just say who you are, and what you saw.\" She hits a button and a green light comes on.\n\n\"What name do I use?\"\n\nShe makes an impatient noise, stops the camera. \"Have you got a selection?\"\n\n\"As a matter of fact . . .\"\n\n\"Use Kyla,\" Aiden says.\n\n\"All right,\" I say. It's just as true as any of the others, isn't it? She starts recording, and I stare at the camera, say that I am Kyla Davis, that I was walking in Cumbria, that I saw a care facility that houses orphans. What was different about them. I project the photos from my camera; she records them also.\n\nShe flicks the camera off. \"That should do. I'll check it and let you know tomorrow.\" She packs up the camera, sweeps out of the room, and leaves us alone.\n\n\"Sorry. Flo isn't always as polite as she could be. Thanks for doing that. I know it was hard for you.\"\n\n\"It's fine,\" I say, shrugging off his words, but it did feel _crazy_ to be saying stuff Lorders wouldn't like, and, if that wasn't bad enough, to let somebody record it. \"It might move me up on their most wanted list, but does that really matter? I'm already on it.\"\n\n\"That's the way.\"\n\n\"I want to thank you for finding Ben.\"\n\nHe shrugs, uncomfortable. \"It was the least I could do. I always felt that it was my fault, what he did.\"\n\n\"It wasn't,\" I protest. \"If it was anyone's, it was mine.\" _Or Nico's,_ I add silently, but Aiden doesn't know about Nico. I sigh. There are so many things I haven't told Aiden. Should I? The main thing that would be grist for MIA is Stella's story: that her mother, a Lorder, was behind the assassinations. But despite the fact that I promised not to tell, it is what Florence called hearsay, isn't it? What value would it be for me to tell something somebody else said if there is no way to back it up?\n\n\"You look a million miles away.\"\n\n\"Sorry.\"\n\n\"There is one other thing we need to talk about.\"\n\n\"What is that?\"\n\n\"Be careful with Ben, Kyla. We don't know what has been done to him. But no matter what that may mean, he isn't the boy you used to know. Without his memories from when you knew him, he's changed.\"\n\n\"He's still Ben.\"\n\n\"Not as he was, but we'll see what we can find out with some scans when we can arrange it. It seems like they've used a Slating-type procedure, but less drastic. His personal memories are gone, but general things like judgment remain, so independent thought and action are possible. Makes them more useful to Lorders as agents, but perhaps also more able to break free from their control, like Ben has.\"\n\nI don't say anything. I got through to him, didn't I? Somewhere inside is my Ben; I'll reach him, somehow. I have to.\n\n\"Enough for one day? We've organized a room for you, but you'll have to share it with a student. I'll show you,\" Aiden says. I follow him out the door, down a corridor.\n\n\"What is this place?\"\n\n\"All Souls College. It's one of the Oxford colleges.\"\n\n\"I thought the University of Oxford was Lorder-controlled?\"\n\n\"It is officially, but that is part of the reason we're here, hiding in plain sight; under their own noses, they don't look so hard. And All Souls played a large part in convincing the Oxford Conference of Colleges to not take part in the protests decades ago. The Lorder government granted them special privileges of independence as a result; they didn't seem to get that the motivation wasn't pro-Lorder, but to preserve and protect the university. And we've had connections here for years; Florence's grandfather was a research fellow here, and when they changed the rules to admit students as well as fellows, Florence was one of the first. They take alumni ties very seriously. When we needed help, their governing body voted to provide it.\"\n\n\"The college voted on it?\" I'm aghast. \"That is a lot of people who know MIA is here.\"\n\n\"Every research fellow and student in the place.\"\n\n\"So much risk to them all.\" To us if one of them can't keep their mouth shut.\n\n\"Yes. We need to get out of here as soon as we can. The only way is to get our evidence released as soon as possible; then we can scatter, go into hiding until things settle down.\" We reach another door; he knocks once. The girl who let me in when I first arrived opens it. \"This is Wendy,\" he says. \"Sleep well.\"\n\nShe shuts the door behind us. It is an irregularly shaped room, with bookshelf-lined stone walls crammed with history books. Two narrow beds. A wardrobe and a long desk with two chairs.\n\n\"That one is yours,\" she says, pointing at the bed by the drafty window. She shows me the bathrooms down the hall, lends me a towel and a change of clothes. Curiosity lurks behind her eyes, but she asks nothing.\n\nSelf-preservation had me fight against helping MIA for so long, until I could see it was the only way. What makes all these college students, like Wendy, risk their lives for us, when they don't even know who we are?\n\nWendy studies while I shower, then pretend to sleep, curled up against the cold. I wonder where Ben's room is, if he sleeps, if he dreams. Despite the danger we're in if the Lorders find us, his closeness is like a drug. Does someone watch his chest rise and fall, even through the night?\n\nWhat was it Florence said? She doubts even Ben knows what makes him tick. He doesn't know much about who he was, but how can he find out if they keep him under lock and key? I have to remind him.\n\nI'll find a way.\n27\n\nNEXT MORNING COMES EARLY WITH A TAP AT THE DOOR. I open my eyes; Wendy is gone, and Florence is peeking through the door.\n\n\"You can't sleep all day; there are things to do. Ten minutes and I'll be back.\"\n\nI rush down the hall for a quick wash and put on a top and jeans borrowed from Wendy. Not a bad fit, but too long\u2014I roll up the bottoms.\n\nFlorence returns, comes in, and shuts the door. \"Aiden tells me you've expressed a desire to join MIA.\" One eyebrow raised ever so slightly shows doubt.\n\n\"I want to help,\" I say, a little nervous what help may entail with Florence involved.\n\n\"Well, you're going to get your chance. We've got a few witnesses I'm having trouble getting stories out of. He suggested you might be some help. Apparently, my bedside manner sucks.\"\n\nI struggle not to smirk. \"You can be a little confrontational.\"\n\n\"Well, so what? I'm not a nurse or a doctor!\" Then she half laughs. \"I'll show you where to get some breakfast, arrange your ID. They are issued by the college; keep it on you at all times. Then Aiden will take you and Ben later this morning.\"\n\n\"Ben?\"\n\n\"Aiden thinks spending time with you might tap his hidden depths.\" She rolls her eyes, then focuses on me closely. \"There is a condition. You're in charge. If Ben does or says anything that worries you, or would worry us, you have to tell Aiden or me. Okay?\"\n\n\"Agreed.\"\n\n\u2022 \u2022 \u2022\n\n\"What do you want me to do?\" Ben says while I fiddle with my camera, set it to record mode.\n\n\"Anything. I just want to make sure I know how to use it. Ready?\"\n\n\"Go for it.\"\n\nI hit the start button, look through the camera at Ben. He's leaning back on his end of the sofa, smiling, a little self-conscious, but there is still something about the _way_ he is smiling that makes it hard to remember what I'm doing. _Check the sound._\n\n\"Say something.\"\n\n\"Something!\"\n\n\"Very funny. Tell me who you are, and what you are thinking.\"\n\n\"I'm Ben,\" he says, and leans forward. \"And I'm thinking how gorgeous you are, and that even if I can't remember before, I had great taste in girls.\"\n\nMy stomach flutters, gentle butterfly wings inside.\n\nHe smirks. \"Try to hold the camera steady.\"\n\n\"Sorry. I was blond back then, you know; I look really different now.\"\n\nBen reaches out a hand, touches my hair, and I give up and lower the camera. He moves closer and looks into my eyes; the butterfly from earlier has friends, is taking over, and I can't breathe. I'm gripped by two competing emotions: wanting to pull away from the stranger and move closer to the Ben I knew and loved at the same time.\n\nThe door opens, and we bolt apart.\n\n\"Ready to go?\" It's Aiden.\n\nWe get up, walk to the door.\n\n\"One suggestion?\" Ben says in a low voice.\n\n\"What's that?\"\n\n\"When you're finished, remember to stop the recording,\" he says, and I hurriedly hit the stop button.\n\n\u2022 \u2022 \u2022\n\nI check the footage in the car on the way. It worked fine; the autofocus kept Ben in sharp focus, and his voice is clear.\n\nAiden comes with us to the door of a house, introduces us, says he'll be back in a while, and leaves.\n\nSo Ben and I find ourselves in the front room of Edie's house with her and her mother. Edie is five, and, according to Florence, saw Lorders shoot her brother in a park. He was nine. Her mother wants her to testify; she says Edie wants to, too, but whenever anyone has tried to record her or even just ask her questions, she has clammed up.\n\nI feel in way over my head; Ben is also awkward and making small talk with Edie's mother while I try to work out what to say, how to even bring up the subject of why we're here. Edie is small and silent, pulled into herself on a chair. It's like there are too many eyes and she is trying to hide.\n\n\"How about you show me your room?\" I say to Edie.\n\nShe looks at her mum. \"It's okay, sweetie,\" she says, and Edie takes my hand, pulls me toward the stairs. I motion at Ben to stay with her mother.\n\n\"It's here,\" she says, and pushes the door open, but as I follow her in she turns and faces me.\n\n\"Are you here to ask me stuff?\"\n\n\"I'm supposed to. But maybe I won't. Because, you know, you don't have to say anything if you don't want to.\"\n\n\"I don't?\" she says, eyes wide in surprise.\n\n\"No. Absolutely. It's up to you, no matter what anybody else says. Because I'm in charge, and I'm very bossy.\"\n\n\"Murray's like that.\" She nods very seriously.\n\n\"Who is Murray?\"\n\nShe walks over to her bed, picks up a floppy teddy bear.\n\n\"He doesn't look bossy; he looks sleepy.\"\n\nShe giggles. \"He's bossy if anyone tries to wake him up. Jack was like that, too.\"\n\n\"Jack was your brother?\"\n\n\"Yeah.\" Her smile fades, and she pulls the bear in close against her.\n\nI know why we're here. A little girl with a sad tale: good for public sympathy, like Florence said. But making her go there if she doesn't want to is just plain _wrong._\n\n\"We don't have to talk about Jack.\"\n\n\"Nobody talks about him anymore. They whisper. But Mummy wants me to tell you; she said it might help stop it from happening to somebody else's brother. But I couldn't say anything before.\"\n\n\"Why?\"\n\n\"Because Mummy was listening. It makes her too sad.\"\n\n\"Oh, I see. How about if just Murray listens?\"\n\nShe tilts her head to one side. \"That might be okay. I can tell Murray anything.\"\n\n\"Are you really, really sure you want to?\"\n\nShe raises an eyebrow, gives me a look that is way over five. \"You're not very good at your job, are you?\"\n\nWith that, we're soon ready. Murray helps me hold the camera. She looks straight at him. Tells him that her brother kicked a ball that hit a Lorder. When he refused to give it back, Jack chased them. That the Lorder took out a gun. Pulled the trigger.\n\nI'm not sure I held the camera steady enough.\n\n\u2022 \u2022 \u2022\n\nBack at the college that afternoon, we check the footage with Florence.\n\n\"I don't know how you managed to get her to open up like that,\" she says.\n\nI shrug. \"Partly I told her she didn't have to unless she wanted to. Partly she couldn't talk about it in front of her mother, but could in front of her teddy bear.\"\n\n\"You've got yourself a job,\" Florence says.\n\n\"What will happen to Edie and her mother when this footage is released? Shouldn't they be in hiding, not left in their home?\"\n\n\"We offered. Edie's mother wants to stay with her extended family for now. Some do. When we're ready to release the evidence, we'll warn them and take them in then.\"\n\n\"Can you hide everyone? Can everyone be safe?\" I persist, unable to get Edie's serious face talking to her teddy bear out of my mind.\n\n\"We'll do what we can,\" Florence says shortly, with a glance at Ben. \"See you at dinner?\"\n\nDismissed.\n\n\u2022 \u2022 \u2022\n\nLater, Ben and I wander around one of the internal quads of All Souls College: a gray expanse of dead grass on a cold, gray day. Ancient college buildings rise on all sides, windows like eyes, and I'm suddenly aware of both exposure and confinement. Anyone could be looking down on us, trapped in this place.\n\n\"Can we talk?\" Ben asks, and I realize how quiet he has been, before with Florence and since.\n\n\"How about there?\" I say, gesturing to a bench tucked against a wall, and we head there, sit down. \"What is it?\"\n\nBen runs his hand through his hair. \"How can you believe what that little girl said?\"\n\n\"What do you mean?\"\n\nHe shakes his head. \"That came out wrong. What I meant is that it is hard to believe anything like that could ever happen; that a Lorder would kill a child just for . . .\" And he shrugs.\n\n\"Just for being a child?\" I snort. \"They do that and much worse all the time.\"\n\n\"How do you know, when people tell you stuff, if it's real or not?\" His eyes are intent, troubled; joking Ben is gone.\n\n\"Why would a child lie?\"\n\n\"She could have been told to.\"\n\n\"No.\" I shake my head. \"I was looking into her eyes; she was telling the truth. Anyhow, I've seen as bad or worse myself, so I know.\"\n\n\"Even if you see things yourself, how can you _know_ what is behind what you see?\" Ben's eyes are skeptical.\n\n\"Look, I'll show you.\" And I tell him the story of the Cumbrian orphanage, the Slated children. Get him to look into the camera at a photo of three smiling young boys with unnatural still expressions, silver glinting at their wrists.\n\n\"But how do you know those are Levos?\"\n\n\"It was obvious they were Slated, from the way they were acting. There was no other explanation.\"\n\n\"But couldn't they have been coached to act like that?\"\n\n\"Four-year-olds aren't great actors. And why would anyone bother?\"\n\n\"To make Lorders\u2014the government\u2014look bad.\"\n\n\"Well, how about this then?\" And I tell him about Phoebe, a girl we both knew from our school, taken and Slated without charge or trial just for making offhand comments about Slateds being spies. About my art teacher, Gianelli, hauled off in front of the whole school when all he did was draw Phoebe and have an impromptu minute of silence for her. About the termination center, where Lorders killed Slated contract breakers by injection and dumped them into the ground. And about Emily, killed by her Levo just because she was in love, having a baby, and not quite twenty-one and out of her sentence. I shy away from telling the rest of the story: that I was there with the AGT, attacking the center.\n\nBen is quiet, drawn in.\n\n\"There is one more story. Do you want to hear it, or have you had enough?\"\n\n\"Go ahead; tell me.\"\n\n\"There was a friend of yours at school, another Slated: Tori. Her mother got tired of her taking attention and had her returned to the Lorders. She hadn't done anything wrong. She was taken to that termination center I mentioned, and saw with her own eyes other . . .\" My words trail away. \"What is it? Do you remember Tori?\" I'm stung; he doesn't remember _me,_ but something crossed his face with the mention of Tori's name. He'd always said she wasn't ever his girlfriend, but she loved him, and she was one of the most beautiful girls I'd ever seen.\n\n\"Of course I don't remember her,\" he says, but his face is guarded, uncertain. \"It's just . . . hard to hear all these sad tales. Tell me what happened to Tori.\"\n\n\"She saw other Slateds killed by injection, dumped into the ground. And then . . .\" I trail off. Ben's look of confusion is gone; is there a flash of something _else_? What is it? \"Look: These are all things I saw. Some of them you did, too. Don't you believe me?\"\n\n\"I just . . .\" And then, as if a switch is flipped inside, he smiles and takes my hand. \"Of course I do.\"\n\n\"One day I'll show you Emily's ring; I hid it in a tree a few miles from home. It's real. Don't you see, Ben; it is all their stories that make what we are doing with MIA so important. They are worth risking everything\u2014to make them heard. To make it stop.\"\n\nHe hesitates, slips his arm across my shoulders, and I lean against him, so aware of _him,_ his warmth and closeness, that it is hard to continue to think straight.\n\nBen points out a tower visible over the roofs of All Souls. \"See, up there? That is one of the tallest buildings in Oxford. St. Mary's Church tower. The views are meant to be amazing. I want to go up there with you.\"\n\n\"Okay; I'll ask if we\u2014\"\n\n\"No. Keep it as our secret; our special place. Leave it until I'm allowed out without a tail.\"\n\n\u2022 \u2022 \u2022\n\nLater, I mull over our conversation, what Ben said, the things he didn't say flitting behind his eyes. I wonder if this is the kind of stuff Florence meant I should tell them. But that isn't fair. He's had his memory taken away; he's figuring out the world, how it works, what happens in it. He has to ask questions to do that, doesn't he?\n\nBut one point of discomfort niggles inside: He reacted to Tori's name, I'm sure he did. Of course I never told him the _rest_ of her story. That I was in the AGT as Rain; that Tori escaped from the Lorders and joined, too. And then there was the day that I was followed by Lorders and Tori was captured.\n\nI shudder. I'll never forget the pure hatred on her face, and it wasn't just because she thought I betrayed the AGT; she'd found out from Nico I knew Ben was alive, and I didn't tell her. The venom in the words she screamed before being thrown in the back of a Lorder van rings in my ears even now: _Traitor! Kyla, or Rain, or whoever you are, I'll get you. I'll hunt you down and gut you with my knife._\n\nThere is part of me that is relieved the Lorders caught her, that she'll never get a chance for her revenge. There is another part that is ashamed for thinking so.\n28\n\n\"FANCY A ROAD TRIP?\" AIDEN SAYS, GRINNING, THE NEXT morning. \"No need for crouching in the back of a telephone van this time; I've borrowed a rather impressive car.\"\n\n\"Sure! Where to?\"\n\n\"It's a surprise. But it'll be just us and Florence today,\" he says, and I bite back my disappointment: no Ben. Now that the sun is up, last night's worries seem foolish. Ben couldn't remember Tori; it doesn't make sense. I must have been projecting my jealousy and imagined his reaction. That is all.\n\nThe car is plush and powerful, borrowed from an unnamed fellow at the college. An hour later we're past Oxford and driving through country fields, then heading down a long lane to a farm.\n\n\"Are we here to see another witness?\" I ask as we get out of the car.\n\n\"Not today,\" Florence says. \"Come on.\"\n\nShe knocks once on the door, pulls a key out of her pocket, and opens it. She walks in, Aiden and me behind her, and calls out, \"Hello?\"\n\n\"Ah, there you are at last.\" In the doorway to the kitchen stands a man I'm very surprised to see\u2014what is he doing here? I know the face, but the rest has changed.\n\n\"DJ?\"\n\n\"Yes, 'tis I.\" He grins. \"And there you are, Kyla; your hair is some of my best work.\"\n\n\"You've changed. No more purple?\"\n\n\"That is _so_ last week.\" Today the IMET doctor looks more tiger stripes, both hair and eyes. \"Did you forget your glasses?\"\n\n\"I kind of lost them; sorry.\"\n\n\"There may be something else you forgot.\"\n\nI look guiltily between DJ and Aiden. \"Oh no. I was supposed to tell Aiden you wanted to see him! I'm sorry. Was it a problem?\"\n\n\"Nice to see how reliable you are,\" Florence snipes.\n\n\"No dramas,\" DJ says. \"It gave me some time to look into things a bit more before we talked about it. To look into _you_ a bit more.\"\n\n\"What do you mean?\"\n\n\"You, my dear, are getting curiouser and curiouser. Like Alice down the rabbit hole, nothing is as it appears.\"\n\n\"I don't understand.\"\n\n\"When we were mucking with your hair genes, we had to do a certain amount of looking at your DNA. I'm connected to Lorder systems as much as is necessary to sort out if people are who they say they are; it is a safety precaution as much as anything else.\"\n\n\"And?\"\n\n\"At lower system levels, your DNA is marked as unknown. At upper levels it gets more interesting: It is listed as _classified._ \"\n\n\"What does that mean?\" I ask.\n\n\"Not a clue, but I love a good mystery. And that isn't all. There is coded protection on files relating to it, and not just any code; so high up I haven't been able to bribe anyone to crack them.\"\n\nAll three of them are looking at me, and I cross my arms. \"You don't think I know anything about it.\"\n\n\"Of course not. But you know something, don't you?\" DJ's eyes are so weird: brown and amber stripes on orange. I can't look away.\n\n\"Why does this matter, anyhow?\"\n\nDJ shrugs. \"To be honest? It may not matter. But\u2014and it is a big one\u2014it has been my experience that when Lorders try very hard to hide something, it is important to find it. Anything they don't want known, I want to know.\"\n\nAiden comes to sit next to me, slips my hand in his. \"Kyla? Do you know anything that might help?\"\n\n\"I might.\"\n\n\"It's okay to say anything in front of DJ. He's one of us.\"\n\nI sigh. \"Look. The main thing I know is that I don't have a clue who I am. Happy?\"\n\n\"Hang on,\" Aiden says. \"I'm not understanding this. Didn't you just meet your mother in Keswick? Actually, wouldn't her DNA be classified, then, too\u2014whatever that means?\"\n\n\"Aiden, I was going to tell you about this, but I haven't had a chance to talk to you properly. She's not my mother.\"\n\n\"What? She reported you missing on MIA. All the records show her as your mother.\"\n\nI shake my head. \"Her baby died; I was given to her as a replacement. She doesn't know where I came from.\"\n\n\"Given by whom?\" DJ asks.\n\nI swallow. \"Her mother. Astrid Connor. She's the JCO for all of England. Stella\u2014that's my adopted mother,\" I say for the benefit of Florence and DJ, \"thinks Astrid might have gotten me from the orphanage there, but doesn't know for sure.\"\n\n\"So that's why you were nosing about the orphanage,\" Florence says.\n\nI nod.\n\n\"And so the curiosity continues,\" DJ says. \"If that is true, why would an orphaned baby have classified DNA? And you would have been tested at school, at your medical center: Why didn't it get registered then?\"\n\n\"You tell me.\" I shrug.\n\n\"What _else_ haven't you told us?\" Florence demands.\n\n\"Sorry I wasn't bragging about not knowing who my parents are; is that okay? For all I know, I could have been abandoned, unwanted. I couldn't see how it was important to anybody but me.\"\n\nAiden raises a hand. \"She's right, Flo. This is personal stuff. Kyla didn't have to tell us; it's her choice.\"\n\nNot that I was given much choice today. \"What do you think it means?\" I ask DJ, who has been very quiet, little wheels of thought spinning behind his eyes. Or is that just the tiger stripes?\n\n\"I don't know. But something tells me we had better find out.\"\n\nI drop my head in my hands. Stella hadn't sworn me to secrecy about where I came from, but sometimes you don't have to have said the words \"I promise\" to know that you've broken one. But what about the rest of her secrets? I _definitely_ promised Stella I wouldn't tell anyone about Astrid being behind the assassinations; without evidence, what use would the information be to MIA, anyhow?\n\n\"Kyla?\" Aiden's hand is on my shoulder. \"Are you all right?\"\n\n\"There's more she's not telling us,\" Florence snaps. \"What is it?\"\n\nAiden asks the others to leave us alone.\n\n\"What is it, Kyla?\" he asks, once they are gone and the door is shut.\n\n\"I don't know what to do.\"\n\n\"I can't help you if you don't tell me more than that.\"\n\n\"It's Stella. There is something else she told me\u2014it's not about who I am or anything like that, but it's important. And I promised not to tell.\"\n\n\"That's a tough call. All I can really say is that you should do what you feel is right, in here.\" He pats his stomach. \"Go with your gut.\" He hesitates. \"Is not knowing going to hurt anyone?\"\n\nI shake my head. \"It's ancient history. Besides, there's no way to back it up; it's hearsay.\"\n\n\"What do you think you should do?\"\n\n\"I think I need to think about it some more. How did you get to be so understanding?\"\n\n\"It's all part of being a superhero,\" he teases, and I remember I called him that, ages ago. When he found Ben hidden away at that Lorder place; Aiden the superhero, helping people find those they care about. Trying to set the world to rights.\n\nI'd thought he was a no-hoper on the last one. But more and more I am _hoping,_ clinging to strands of future possibility, that things might be fixed by MIA _without_ using guns and bombs. That he and the others can really do it.\n\nThat _we_ can do it.\n\n\"Thank you,\" I say. \"For everything.\"\n\nHis eyes hold mine, warm, and for a moment it is hard to breathe. Then he shakes his head, looks away, and calls the others back in.\n\n\"Well?\" Florence demands.\n\n\"That's enough for today,\" Aiden says. Florence protests; Aiden replies, \"We don't work like that; we're not Lorders. She can tell us when she's ready, _if_ she's ready. I'm satisfied it's not critical to us now.\"\n\nAnd I'm scouring my brain, looking for something, anything, that might help, and then I have it. \"Wait a minute. There is somebody who might know something about my DNA.\"\n\n\"Who's that?\" DJ asks.\n\n\"I always thought there was something she wasn't telling me, something she was hiding, but I don't know. Maybe I imagined it.\"\n\n\"Who?\"\n\n\"My doctor at the hospital. Dr. Lysander.\"\n\nAiden's eyes sharpen. \"She was your doctor?\"\n\n\"Yes. She said it was on my records that I was a Jane Doe; that even though everyone is supposed to be DNA-tested at birth, they didn't know where I came from. She said that was all she knew about it, but there was something hiding behind her words. She never lied, exactly, but she hid things by twisting words around.\"\n\n\" _The_ Dr. Lysander\u2014the very doctor who invented Slating\u2014was your doctor?\" DJ says. \"Interesting. I bet that wasn't a coincidence. But why would she tell you anything about your records?\"\n\n\"We were sort of close. She told me loads of things she shouldn't have. Broke rules to help me.\"\n\n\"We need to talk to her.\"\n\n\"She's always surrounded by guards, and the hospital is a fortress.\"\n\n\"If we can get you to her, will you do it? See if you can find out what she knows?\"\n\n\"Of course.\"\n\nAiden protests. She is a Lorder doctor; however close I thought we were, it would be too dangerous.\n\nI shake my head. \"She wouldn't turn me in. Never.\"\n\n\u2022 \u2022 \u2022\n\nIn the car on the way back to Oxford, I sit in the back, staring unseeing out the window. Pondering other coincidences.\n\nWhat did seeing Astrid and Nico together really mean? How did I end up with the family of the assassinated prime minister after I was Slated? My two families\u2014Mum and Amy, Stella and Astrid\u2014somehow their histories and what may yet come to be are entwined and twisted together, with me caught between. Yet neither family is truly mine.\n\nEverything is crowding in on me; there is only one solution.\n\n_I need to run._\n29\n\n\"BET YOU CAN'T KEEP UP,\" BEN SAYS.\n\n\"Oh yeah?\"\n\nBen takes off up the path, and I'm on his heels. It's too narrow to run side by side, but at last we are doing something that until recently I thought could never happen again: We are running together. It's cold, dark enough that running full tilt on an unfamiliar path is a little dangerous, but he has set the pace. There is no way I'll let him pull ahead.\n\nIt used to be we'd run to get our levels up; endorphins from running would have them high, even in the 8s. We could talk about anything without risking a Levo zap from dropping levels sending us off to a blackout.\n\nSo much has changed since then. Neither of us has a Levo anymore; we don't need to run to stay level, but today, I needed it. Yet I was surprised that Aiden said all right, that he let us leave the college grounds together. Maybe, he understands. Maybe he understands too much.\n\nThere is a sudden _thud_ in front; a cry. Ben flies through the air, lands heavily, and I almost trip over him.\n\n\"Ouch,\" he says.\n\n\"Are you all right?\"\n\n\"I think so.\" He twists his foot from side to side. \"Yeah, it's fine. Just caught it and fell over, not sprained.\"\n\nI give him a hand up, and he brushes himself off. \"Let's walk a little,\" he says.\n\n\"Sure you're not hurt?\"\n\n\"I'm fine. Was it a tough day that made you want a run?\" Ben asks. He takes my hand in his as we walk up the path.\n\n\"You could say that.\"\n\n\"Want to talk about it?\"\n\nI stay silent a moment. \"Do you mind if we don't?\"\n\nHe stops walking, pulls me around toward him, his eyes dark pools in the moonlight. \"Talking is one option. There is another.\" One hand slips around my waist, the other under my chin. And it is as if I'm in two places, here, and another: the first time he kissed me. It was at night after running, and so like this that my mind is here and there at the same time, drifting, falling between past and present, between the Ben I knew, and the Ben I don't. And then I'm shaking and crying.\n\nHe pulls away. \"What's wrong?\"\n\n\"I don't know. Who are you? Who am I? What does this mean?\"\n\n\"That's way too much thought.\" He smiles. \"Stop thinking.\" And he kisses me again, and again, until the past is gone, the tears are gone, and we are here. Now. There is nothing else.\n\n\u2022 \u2022 \u2022\n\nWe sneak back in late. Ben holds my hand tight; I protest when we get to a corridor and he pulls me the wrong way. \"My room is the other way.\"\n\n\"No, you're coming back with me. We still have some talking to do.\"\n\nAnother corridor and a turn, some stairs. Ben still holds my hand. It's late and I'm tired, but every bit of me is _alive._ Talking?\n\n\"Now for the 'be quiet' bit,\" he whispers. He opens a door, peers in. Someone is fast asleep on a bed in the dark; we creep past him to another door. Ben opens it. \"Wait in here,\" he whispers. \"I'll tell my jailer I've returned so he doesn't check if he wakes up.\"\n\nI step through the door; he shuts it behind me, and I'm plunged into darkness.\n\nThere are low voices through the door; then it opens and Ben steps through. \"Give him five minutes and he'll be out like a light,\" Ben whispers, pulls me close. Kisses my cheek, my neck, and I can hear my heart going _thud-thud_ so loud I worry the student through the door will hear.\n\nBut then Ben lets go, turns, and puts on a small desk light. Darkness retreats, reveals a small student room. Desk, wardrobe.\n\nSingle bed.\n\n\"Ben, I should go.\"\n\n\"You're not escaping that easily.\" He smiles, pushes me down to sit on the bed, sits next to me. \"We need to talk.\"\n\n\"Talk?\"\n\nHe grins a wicked grin. \"Talking is one option,\" he says, and takes my hand. \"Tell me. Why were you so upset before?\"\n\n\"It's a long story.\"\n\n\"I've got nothing but time.\"\n\nAnd once I begin, it all comes flooding out\u2014all I've wanted to tell him for so long. Inside me, things are letting go with the words, loosening and coming free. Ben wraps a blanket around us against the cold, just holds me while I talk, and cry, and talk some more. I even tell him that I don't know where I come from, about being kidnapped by the AGT and what they did. Why I was Slated, and what happened after he was gone. I tell him about Stella, but not about her mother and the assassinations; that isn't my story to tell.\n\nFinally Ben says, \"Enough. I have a question. With all of that, why is it that what really upset you earlier was my kissing you?\"\n\nI shake my head. \"No, that wasn't it. That was lovely.\" I blush. \"It's this: How can we be anything together when we don't know who we are?\"\n\nHe shakes his head. \"I haven't any idea where I come from, either, or anything of what happened before I was Slated, so you're ahead of me on that. At least you know who raised you. But it doesn't matter.\"\n\n\"It doesn't?\"\n\n\"No. Kyla, all we are is what we are, here and now.\"\n\nAnd he kisses me again, and that _is_ all that matters. Now. But a little voice inside knows that in the morning, the sun will come up. Tomorrow will come one way or the other.\n30\n\nI'M WARM, IN A DARK, SLEEPY HAPPY PLACE. THERE WAS something: a sound? A click. I stir, then remember where I am.\n\nI sit up fast. Light is coming through under the curtains just enough to see. Ben is turned away, putting something in the wardrobe.\n\n\"Ben?\"\n\nHe whips around. Smiles. \"You look cute all sleepy.\"\n\n\"It's morning? I didn't mean to fall asleep! I need to get out of here before anyone notices.\"\n\nHe shrugs. \"Stay. Who cares?\" He slips a hand around my chin, kisses me, but I pull away.\n\n\"I care.\" I slip to the door, open it quietly. The student outside is sound asleep.\n\n\"He's a crap guard,\" Ben whispers. \"He could sleep through _anything._ \" He kisses my cheek. \"See you here tonight?\"\n\nHis eyes hold mine, and somehow the words come out without any thought or volition. \"All right.\"\n\nI get down the halls without seeing anyone, and to my door. Open it.\n\nWendy is at her desk. She turns, smirks. \"Good run with Ben last night?\"\n\n\"We were just talking, and I fell asleep!\"\n\nShe laughs. \"Sure, I believe you.\" She winks. \"Don't worry. Your secret is safe with me.\"\n\nI protest a few more times, cheeks burning, then head for the shower. Will she keep it to herself? Why does it even matter? Somehow I don't want it known I was out all night, for things to be assumed.\n\nSomething nags inside. Mostly it's that I don't want Aiden to find out, and I'm not sure why. He had Ben brought here; he must know how I feel about him. Yet somehow I know Aiden wouldn't like it if he knew I'd been in Ben's room all night; he's protective, and he'd worry. And he is the last person I want to hurt, after all he's done for me. That is the only reason I don't want him to know. Isn't it?\n\nThe day passes. Florence takes me out to record some more witnesses; adults this time, so not as hard as the last, but their stories still tear inside. Ben isn't with us, as they've finally found a doctor to take him to who will do scans without asking questions. And after each witness I'm telling myself _just get through the day._ Then I can be with Ben.\n\nWhen Florence and I get back to the college, as we walk across the quad I stare up at the St. Mary's tower, where Ben wanted to go. \"Can you go up there\u2014up the tower?\"\n\n\"Sure. If anyone from the church is there, just smile and flash your student ID. It's thirteenth century; check out the gargoyles. Great views.\"\n\nWe get to the office, and I fidget impatiently while Florence copies today's recordings from my camera to a computer.\n\n\"What's with you?\" she asks.\n\n\"Nothing.\"\n\nShe looks skeptical, but before she can say anything else, the door opens: Aiden.\n\n\"Kyla? Are you done? I need a quick word.\"\n\nFlorence hands me my camera. \"All finished. Away you go.\"\n\nAiden holds the door open; I step through, heart sinking. Has he heard something about last night? No. His eyes are sparking. \"Quick, grab whatever you need until tomorrow; we're off on an adventure.\"\n\n\"Where are we going?\"\n\n\"To see Dr. Lysander.\"\n\n\"But how\u2014\"\n\n\"No time for questions; you can ask on the way. Don't tell anyone! Meet me out back in five minutes. Go!\"\n\nI race to my room to grab a few things. Wendy isn't there, so I can't ask her to tell Ben I had to go out. I can't leave a note with Aiden's \"tell no one\" ringing in my ears; there is no time to run to his room, if he's even in it.\n\nAs I rush to meet Aiden, I wonder: Is Ben going to think I stood him up?\n\n\u2022 \u2022 \u2022\n\n\"How are we going to get to Dr. Lysander? She's always guarded.\"\n\n\"Bit of luck on that one. DJ found out she's speaking at a medical conference tomorrow; we've got connections in the conference center, so can get in to where she's staying. We hear she refuses guards in her own quarters, so they stay outside the door. We've checked the room for bugs and cameras; it's clean.\"\n\n\"So what is going to happen?\"\n\n\"We get you in there tonight; she's arriving early in the morning. She's got a few hours' scheduled rest time before the conference starts.\"\n\n\"Which is when I make an appearance.\"\n\n\"Exactly. Kyla, there isn't much we can do if she sets off the alarm.\"\n\n\"She won't. But I still don't understand why we're going to all this effort to find out about _me._ Even if she knows where I came from, which I doubt, how could it possibly matter?\"\n\n\"Not a clue. DJ's really pushing on this, and we're going along with it.\"\n\n\"Who is he?\"\n\nAiden glances at me sideways. \"Even I don't know his real name.\"\n\n\"That's not what I mean. How does he fit into MIA? I figured he just helped with changing identities like he did with me. But there is more to it, isn't there?\"\n\nAiden laughs. \"It's on a need\u2014\"\n\n\"\u2014to-know basis.\" I roll my eyes and try another tack. \"Is he from Ireland?\"\n\n\"That you can get from the accent, so I'm guessing it is okay to say yes.\" Aiden hesitates. \"There is international support behind MIA, not just from United Ireland. They know some of what goes on here from people who we've snuck out of the UK, and there is international pressure to get all the hidden stories out, to make them public and do it soon. They want to stop the human rights violations. This is why the Lorder attack on our computer systems was timed so badly; it's held things up.\"\n\nI stare out the window. Why would people in other countries far from us care about our human rights, when almost everyone here seems to turn away and pretend not to see? \"I don't think that is what is most important. It should be about taking the wool off everyone's eyes _here._ Making them face up to what is happening in their own country, under their noses, so we can fix it on our own.\"\n\n\"Both are important. But the simple truth is, we can't do it alone, not when the Lorders have all the power. Sometimes you need help.\"\n\nAiden turns into a small village, then parks next to a van behind a hall. \"This is where we part company until tomorrow,\" he says. \"Are you sure you want to do this?\"\n\n\"Yes. DJ's reasons for being interested in my origins might be different than mine, but I still want to _know._ \"\n\n\"Be careful,\" he says, and looks about to say more when the van driver's door opens and a man steps out.\n\n\"Evening,\" he says, nods, and opens the door to the back of the van. Reaches in, hands me a bag. \"Clothes. Change into them.\" And with a wave good-bye to Aiden, I climb into the back.\n\nThe van soon lumbers up the road, and I get myself into a uniform in the dim light. Looks like a maid's? There are a few in different sizes, and one seems to fit well enough to not draw attention. There are no windows in the back; we drive perhaps thirty minutes. Then I hear a faint beep and we drop down and around a circular ramp. The van stops, and I'm starting to feel nervous. What is this place? I don't know what I'm doing. If anyone asks me anything, or\u2014\n\nThe door opens.\n\n\"There you are, lass. Now don't worry about a thing, it's all worked out. I'm taking you up tonight as no one is around until tomorrow. Best leave your coat and other things there in the van; they'll be fine there until the morning.\"\n\nI take my coat off, extract the camera from a deep pocket. Maybe I should have left it back at the college, but something tells me I'll need it to get anywhere with Dr. Lysander. There's a pocket in the uniform, and I slip it in.\n\nI follow him through the underground lot to an elevator. He calls the elevator with a key, and it is there in seconds. Up we go. \"No one should get on, but if anyone does, nod but say nothing. I'll handle it,\" he says.\n\nI hold my breath, but the elevator doesn't stop until the selected floor. The doors open, he peers through, then gestures for me to follow.\n\nWe go down a plush hall, where all the doors have wire taped across them.\n\n\"What is that?\"\n\n\"Security-sealed electronic tape; the rooms were checked, cleared, and sealed a short time ago.\"\n\nThen he opens a door at the end that leads to a narrow hall with little doors all along. He counts them and stops at one. \"This is the service hatch into your friend's room; they are normally used for breakfasts and so on. Listen very carefully. You can't open these with the electronics on without setting off an alarm.\" He glances at his watch. \"The electronics will go off shortly for one minute, the maximum possible without setting the alarms off. It should be just long enough to open it and for you to climb through into her room. I checked that you were a little lass; I couldn't do it. Once through, make yourself comfortable; there are spare blankets and pillows in the wardrobe. Stay out of sight of the door. Your friend is due tomorrow between seven and seven thirty A.M., and you don't want anyone to see you when she comes in and they bring her luggage. Speak to her, then come back out the same way at exactly eight A.M. The electronics will be switched off at exactly eight A.M. for one minute, and that is it. Here is a watch for you\u2014it is coordinated with the hotel system so the time is exact. Understand?\"\n\n\"Yes,\" I say, slipping the watch onto my wrist. It is digital, with hours, minutes, seconds shown by a faint green pulsing light.\n\nHe is watching his own watch, hurriedly explaining how the service hatches work. Telling me not to touch the windows or doors in the room, as they're all alarmed.\n\n\"It's time,\" he says, and yanks doors at the back of the hatch open; it's like a mini elevator. I climb through into a small box, struggle to open the door on the other side when there isn't enough room to extend my arms properly, and manage to pry it open.\n\n\"Hurry,\" he says.\n\nI crawl through; the doors swing shut.\n\n\"Good luck, child.\" His voice is faint on the other side.\n\nMy heart is beating way too fast; that wasn't easy to do in a minute. I'm sitting on the plushly carpeted floor in Dr. Lysander's soon-to-be room, wishing I'd asked more questions\u2014like can I turn on the light? Is there anything to eat?\n\nI feel my way around the darkened room. Large bed. Desk. Chair. Wardrobe. I open that and feel along the bottom. Under the promised pillows and blankets, my fingers curve around something cold and round, with a switch: a flashlight. I flick it on.\n\n\"Thank you, mystery man,\" I whisper to myself. I explore the room again with the flashlight carefully angled down, and decide the only place I feel safe is actually inside the thankfully huge wardrobe. What if I fall asleep and don't wake up until she arrives?\n\nI arrange the pillows on the wardrobe floor, settle onto it with the blanket. I try with the door shut, but it feels too enclosed, so I push it ajar. I'm sure I'll wake before she gets here; I'm not convinced I'll sleep at all.\n\nFor a while I stare at the wall, imagining what I can say to Dr. Lysander to get her to tell me everything she knows about me, rehearsing the words. Finally I close my eyes. What is Ben doing now? I bite my lip; I hope he doesn't think I'm avoiding him, or don't want to be there. Would anyone tell him where I am if he asks?\n\nI slip into uneasy dreams, of wardrobes: Stella's wardrobes, full of photos and tissue-wrapped memories; college student wardrobes with narrow spaces, too small to hide in. _Click . . . switch._\n31\n\nTHE LOW THUD OF A DOOR; FOOTSTEPS.\n\nI open my eyes with a start, glad to see I'd pulled the wardrobe shut in the night.\n\n\"Yes, just put it there. Thank you.\" Dr. Lysander's voice? Another voice, male, asking if she needs anything. \"No, thank you; just some peace and quiet.\" But you can't always get what you want.\n\nA door shuts, and there are footsteps in the room.\n\nI struggle to shake sleep from my mind; it had been late when I finally drifted off. I squint at the digital numbers on the watch. 7:40? Oh no. She's late. We haven't much time.\n\nBut I stay silent, unmoving. What if I'm wrong and when she sees me she raises the alarm? She wouldn't do that, not after everything we've been through. Would she?\n\nI listen very intently to make sure she's alone. There is a faint zipping noise\u2014a suitcase?\n\nIt's now or never.\n\nI nudge the door open and peer through the slit, just in time to see she is approaching; the door is pulled open.\n\n\"Dr. Lysander?\" She jumps about a foot in the air. \"It's me, it's Kyla.\"\n\n\"What?\" She is half poised to run the other way, to her door, but looks, really looks at me this time. I hold my hands out to show I'm unarmed.\n\nHer eyes are wide, face pale, but otherwise she looks the same as always: thick glasses, long dark hair tied back, with maybe a few more gray streaks than it had before. Eyes that can see through me. She takes one of my hands to pull me up from the wardrobe. I stand next to her.\n\n\"Kyla?\" She smiles. \"It really is you? Your hair. But it's you!\" And she does something she has never done before: pulls me close for a quick hug. Then, like she realizes what she's done, releases me just as quickly.\n\n\"They told me you were dead.\"\n\n\"I'm sorry about that. I'm fine.\"\n\n\"Why would they do that?\" She shakes her head. \"How are you here, hiding in my room? What is going on?\"\n\n\"I haven't got very much time. I need to ask you a few things, but first I'll tell you where I've been.\" I realized last night, if I don't tell her what I found out, why I want to know, she'll never reveal what she didn't before. I have to give her a reason, and do what we always did: trade information.\n\n\"I went to who I thought was my mother. From before I was Slated. You know how I told you the AGT took me at a young age? I was kidnapped from my mother when I was ten. I went and found her, to get to know her again. But not long after getting there, I found out she wasn't actually my mother.\"\n\n\"Explain.\"\n\n\"She was given me as a baby after her own died, and raised me from then to age ten. She doesn't know where I came from. Her mother is a JCO and gave me to her, so it may have been from an orphanage. Before I could find out any more, my cover got blown, and I had to leave in a hurry.\" This is the part of the story I'd struggled to formulate. I can't tell her details of where I am now or who with; I can take risks with myself, but not with those who've helped me. \"Since then, I've been with friends. One of them found out that at higher security levels, I'm not a Jane Doe: My DNA is classified. Who am I? Tell me if you know anything; I have to know.\"\n\nShe looks back at me carefully, considering. \"Why do you need to know?\"\n\n\"Wouldn't you want to know who you were if you found out you were adopted?\"\n\nShe shrugs. \"Maybe less than you. My family was never close, and often difficult. Why seek out another?\" She touches my hair. \"IMET, isn't it.\" A statement, not a question. \"Is this where the DNA came into it? I'm worried about you, Kyla. How much trouble are you in? Can you come back from it? Does learning more help, or hinder? What do your new _friends_ really want with you? Are they any better for you than your AGT friends turned out to be?\"\n\nI'm so frustrated, I want to scream. As usual, she's homed in on the one thing I won't talk about: my friends. I breathe deeply. \"You must know by now how wrong the system you are part of is. But in case you don't, I'm going to show you.\" I need to shock her, to _make_ her help me. It was the only way I could think to do so.\n\nI pull the camera out of my pocket. \"You know how I said I may have come from an orphanage? I went to look at the local one. Don't know why.\" I shrug. \"It's not like I was going to recognize a place I left when I was a baby. It was isolated, fenced in. I snuck in close, and this is what I found.\" I open the camera folder, project the image of the small Slated boys.\n\nHer intake of breath is sharp. \"These children, so young? No. Slating is not for them. Who would do this? Where is this place? Tell me,\" she demands, her face coldly furious.\n\n\"Lorders have done this, and they are doing it. There were about fifty children I saw.\" I glance at my watch: 7:51. \"The other thing I found out is that this JCO, the mother of the woman who raised me, had something to do with the AGT and having me taken. Please. I've told you all I know about it. I have to leave at exactly eight or I won't be able to get out. Tell me what you know.\"\n\nDr. Lysander is silent a moment, thinking, and I don't press. She finally nods. \"I told you before. You were on the hospital records as Jane Doe. There was no mention of being classified; no other information as to your origins at all.\"\n\n\"But there is something else?\"\n\n\"Yes. A few curious things. Remember when you saw your records on the hospital system? Where the hospital board had recommended termination; it said I overruled.\" She shrugged. \"I don't have the power to overrule the hospital board, and, in any case, never tried to do so. It happened at a higher level; somebody made sure you were kept alive. Also there was more interference and care at times; the longer stay in the hospital and the Watchers you had at night are examples. They were above the entitlement of assistance. Someone was meddling, and it had my curiosity.\"\n\n\"Is that why you took special interest in me, why I was your patient?\"\n\nShe inclines her head. \"That part of the motivation followed. There was an initial reason, as I've told you before.\"\n\n\"That I remind you of someone you used to know, someone who died in the riots.\"\n\n\"Yes\" is all she says, but something else crosses her face in that moment, for seconds only, then is gone.\n\n\"When you changed my brain chip number on the computer to make it untraceable, was that at request from someone above?\"\n\nHer lips quirk. \"No. That was entirely my own moment of insanity. The Lorders are more interested in you than they should be. I made it harder for them.\"\n\n\"One other thing. My memories\u2014there were things from my childhood coming back when I was at the place I was raised. I was left-handed to age ten, then forced to change, Slated as right-handed. Could my memories be coming back because my handedness was changed?\" This was Stella's theory, and asking about this isn't on the list of reasons why DJ got me in here, but last night I knew I had to ask. I may not get this chance again.\n\nShe's thinking again. Finally nods. \"It's possible that the only inaccessible memories from your Slating were those associated with being right-handed. Others may be suppressed, but accessible in the right circumstances. But this is conjecture. To my knowledge what happened to you hasn't been attempted before, so who can say?\"\n\nI'm about to ask her more when her eyes drop to my wrist. \"Kyla, your watch says 7:59.\"\n\nI bolt up and run for the small doors at the service hatch at the back of her room just as my watch changes to eight o'clock. \"I'm sorry we can't talk longer,\" I say, and wrench the doors open, then curse: The car isn't there, and what is between me and the doors on the other side is a chasm that drops far down into darkness. Then the opposite doors open; hands are there to help, and I launch myself across to them. One ankle bangs painfully against the door in Dr. Lysander's room as strong arms drag me across.\n\n\"Where was the orphanage you visited?\" Dr. Lysander says urgently as I'm pulled through on the other side.\n\n\"Cumbria,\" I say back quietly as the doors close. Unsure if I should or shouldn't say, but there is the trade, as always. She answered another question; so must I.\n\nAs I pull myself to my feet, the mini elevator whirs into action; a car in motion is heading this way. That was close. My ankle hurts, and I bend down to check it; a small cut.\n\n\"Are you all right?\"\n\n\"Just a nick; I'm fine.\"\n\nI follow the van driver back down the hall, listening as he explains what to say if anyone says anything, then we get on the elevator. There are other staff on it, but they smile and nod, and no words are spoken. They get off at another level. We go back to the van in the underground lot.\n\n\"Sorry, but you'll have to stay in here, very quietly, until my lunch break. There is some food for you on the seat.\"\n\nHe opens the door, I get in, and it shuts behind me. I change back into my own clothes, then find a sandwich and cookies wrapped up and eat hungrily while I think through all that was said.\n\nHours later the van driver returns as promised and takes me to a rendezvous with Aiden. On the return to Oxford I tell him what Dr. Lysander said, hoping he or DJ will make more sense of it than I did.\n\nWhy would some faceless higher-up capable of overruling the hospital board and everything else they've done be bothered about me? I can't answer the question, but deep in my gut I'm sure of one thing: It can't be good.\n\nDr. Lysander didn't know what they are doing at that orphanage, that is very clear. I go cold inside, afraid what she will do with the knowledge. Will she end up in even worse trouble than she did the last time, all because of me?\n32\n\n\"BEN CAME LOOKING FOR YOU LAST NIGHT,\" WENDY SAYS. \"SO I'm guessing you weren't with him all night long this time.\"\n\n\"I wasn't _with_ anybody.\"\n\n\"Don't look so fierce, I believe you. Just listen to me a moment.\"\n\n\"What?\"\n\n\"I know I don't know you very well, and I know enough about why you are here to know I shouldn't ask questions. But be careful.\"\n\n\"What do you mean?\"\n\nShe hands me an envelope. \"Just be careful.\"\n\nShe leaves, and I rip it open.\n\nBen's handwriting: It looks just as it always did.\n\n_Dear Kyla,_\n\n_My scans were good, so I've been sprung from constant supervision, hurrah! Came by to celebrate last night\u2014where were you?_\n\n_There is only one way to make it up to me. Meet me at the top of St. Mary's Church tower\u2014the views are supposed to be amazing._\n\n_Don't keep me waiting again._ __\n\n_Love, Ben_\n\nBut there is no time specified on it. Maybe he has already been waiting there for hours!\n\nI scramble for clean clothes, which are in short supply. I borrow a top of Wendy's, leaving an apologetic note behind. Tuck my camera in a coat pocket and slip out of our room, down the hall.\n\nI step out a side door of All Souls and soon find the entrance to the church. Wave my student ID at a warden and get him to point out the way to the tower.\n\nAnd I start the climb. Stairs in the church, then in the tower, take me ever higher until I reach a narrow spiral staircase. The farther I go up the ancient, worn stone steps, the narrower and steeper the way, and despite wanting to be there _now_ I have to slow down, take care.\n\nFinally I reach the top and step out onto the tower platform, into the cold wind. No sign of Ben. The platform is irregular and narrow, enclosed by a stone railing with more stone curving overhead, almost as if the platform has been gouged into the tower. Hugging my arms around myself, I follow the platform all around the tower, ducking into linking tunnels on the way, until I reach a dead end.\n\nNo Ben.\n\nEither he was here already, got bored of waiting, and left, or he hasn't come yet. Why didn't I ask Wendy when he gave her that note? If he has been and gone I should go look for him. But then what if he comes back and I'm not here? I decide to wait, and do the circuit again, this time with more of an eye to the views across Oxford, and the gargoyles leaning out with wide gaping mouths as if to swallow buildings below. Finally I huddle against cold stone, shivering and staring at All Souls College. Patches of both quads are visible from here, including the bench where Ben and I sat and spoke.\n\nI'm so happy about Ben's scans being okay, but then start to think about it. What does that mean, exactly? How could scans reassure Florence and Aiden enough to give Ben more clearance? They might show how much of his memory has been mucked with, but won't show what he is thinking. I don't understand. I frown to myself, then my misgivings disappear with the dim echo of approaching feet on the stone steps below.\n\nHe's here!\n\nThe steps get closer, and my smile, wider. Ben said this was our secret, a special place for us. A new special place, for new memories to replace the old.\n\nBut the face that appears in the doorway isn't the one I'm expecting.\n\n\"Aiden?\"\n\n\"Where's Ben?\"\n\n\"I don't know. What are you doing here?\"\n\n\"More to the point: What are _you_ doing here? You know better, Kyla, than to sneak out without telling anyone where you are going.\"\n\n\"What do you mean, sneaking? I wasn't sneaking! I just . . .\" I stop. Once I got the note I was in such a hurry to meet Ben that I didn't think about it. I look closer at Aiden and see what I missed. \"Something's wrong. What is it?\"\n\n\"Ben's guard has been found in a wardrobe. Dead. We're hunting for Ben, but he hasn't been found.\"\n\n\"What? Dead? Has something happened to Ben?\"\n\n\"Apart from killing his guard, not that I know of. Were you meeting him here?\"\n\n\"He couldn't have done that, it can't be him. I don't believe you.\"\n\nHe shakes his head. \"Tell me everything you know, and do it now.\"\n\nMy knees are shaking; I lean on the stone railing. Ben's guard, dead? That student, the one who could sleep through anything?\n\n\"Kyla?\"\n\n\"Ben left me a note. He told me his scans were okay, that he didn't have to be watched anymore.\"\n\n\"Lies, Kyla. The scan results aren't even back yet.\"\n\nI hesitate, then pull the note out of my pocket, hand it to Aiden. I swallow. \"I don't understand. Why would he lie?\"\n\nAiden reads the note. \"I don't know, but nothing I can think of is good.\"\n\n\"Did you follow me here?\"\n\n\"No. It was a hunch; Florence said you'd asked about coming up here. We need to raise the alarm\u2014\"\n\n_Bang._\n\nGunshots? Below, over at All Souls: There are people running across the quad. Dim shouting on the breeze.\n\nNo. _No._ This can't be happening.\n\nI whip my camera out to see better with the zoom. \"There are figures in black at the exits of All Souls. Lorders.\"\n\n\"Do what you do best, Kyla; be a witness.\" His words are bitter.\n\nI'm recording. Lorders push all the students and research fellows, and those they were hiding, out from the buildings into one end of the quad. Against a wall. They open fire. There is chaos, screaming; some try to run but don't get far\u2014Lorders guard every exit. But in the midst of it all some stand erect, arms linked; Florence is at their center. Facing the Lorders with calm contempt as they are shot. There are bodies, more bodies. Red stains the ancient stone, the dead grass of winter. Somehow through it all my hands stay steady, recording; a numb witness, as dead inside as those in the quad.\n\nThen there is silence.\n\nTwo figures in black guard one of the entrances to the quad near the bench Ben and I were sitting on just days ago. One turns and faces the tower, looks straight at me, as if he knows I stand here, watching. The other slips her arm around his waist. She's laughing.\n\nBen. And Tori.\n33\n\nMY ARMS FINALLY DROP; THE CAMERA IS A WEIGHT. THIS can't be happening. Aiden is silent, his face a mirror of mine: shock. And agony.\n\nFlorence.\n\nWendy.\n\nA whole college of nameless fellows and students who voted to help us: dead.\n\nI stare at the camera in my hands, full of witnesses. Recordings of pain. Ben? No. I can't\u2014he couldn't have . . .\n\nBut Ben was standing there, part of this massacre. I can't deny what my own eyes have seen, even as everything inside screams they must be wrong.\n\nI'm a witness like all the others hiding in this camera. The only recording left of them now.\n\nEdie is one of the witnesses\u2014Ben went there with me. He knows where she lives.\n\n_Run!_\n\nThe thought is barely in my mind when my feet are running down the spiral steps.\n\nAiden clatters behind, calls out to be careful, to wait, but he drops behind. Out in the air and sunshine\u2014how can the sun shine on today?\u2014I run, and Aiden can't keep up, he is gone behind me. No one is in sight at any of the nearby colleges that are part of Oxford University; everyone must be hiding under their beds.\n\nI run the fastest I have ever run. My feet feel like they're not touching the earth; I'm flying, skimming the surface of a world I don't want to be in anymore. But for one child. If I can save one child, somehow I can . . . No. I can't think of an after, or a before. Just _now,_ only now. Or I'll stop, won't be able to take another step.\n\nI'm still flying when miles later I reach Edie's street. Then her front door.\n\nHer open front door?\n\nI step through it, gasping for air.\n\n\"Hello,\" I call out, voice ragged. \"Edie?\"\n\nThe lights are on. There are half-eaten plates of food on the table. _No, no, no . . ._\n\nI run from one room to another, all through the house. It is empty. The house is empty.\n\nExcept for Murray. He's on the floor in the kitchen. I pick him up, stare at his smiling teddy bear face, dazed. Disbelieving.\n\nNo. This is a nightmare. This whole day. Nothing is real. It can't be real. I'll wake up in a minute.\n\nI take a swing at the wall as hard as I can. My knuckles crunch into plaster, cracks form from the spot. Pain. Blood.\n\nBut I don't wake up.\n\nI'm not asleep.\n\nI clutch Murray tight in my arms, fall to the floor, curl up in a tight ball. The tears finally come. Wave after wave of agony claw through and shred everything I am inside until there is nothing left.\n\n\u2022 \u2022 \u2022\n\nLater, I don't know how much later, there are footsteps. My eyes are still clenched shut tight like the rest of me, rigid.\n\n\"I thought I might find you here.\"\n\nSome part of my brain registers: Aiden's voice. Aiden is here. Why? It's all my fault. Why would he come?\n\nThere is movement close by; something warm touches my hair, strokes it.\n\n\"We've got to get you away from here.\" There is another murmured voice. Then arms slip around me, lift me up.\n\nI can't move, can't speak. But if I could, what is there to say?\n\nI'm carried; there are car door sounds. Placed lying down on a seat. Something warm tucked around me.\n\nLow voices, the car engine starts, and we pull away.\n\nEverything goes black.\n\n\u2022 \u2022 \u2022\n\n_I lie still, as a statue on a grave. Unfeeling and cold. Eyes shut._\n\n_For a long time all around me is quiet, the absolute silence of the dead. Why aren't I one of them? I long to be. The bullets missed even as I tried to jump in front of them, to stop them from hurting anyone else. I failed._\n\n_Then there are footsteps. Faint at first, then closer._\n\n_\"She must be here someplace,\" a voice says. Ben. I stay still as death, facedown on cold earth. There is movement, another voice._\n\n_Then someone grabs my hair, yanks. Turns me over._\n\n_I open my eyes._\n\n_Tori smiles and holds out a knife._\n34\n\n\"SHE'S SUFFERING FROM SHOCK PERHAPS. AS ARE WE ALL TO some extent. All the evidence stored at the college is gone?\"\n\n\"Yes.\"\n\nWords penetrate, and the meaning drifts around, looking for explanation, while other details start to come through. Not in a car anymore. On a sofa? Evidence; what evidence?\n\nIt all floods back, pain like being kicked in the stomach. I groan and open my eyes.\n\nAiden is across the room; he comes over. \"Hey. You're awake?\"\n\n\"I guess,\" I whisper.\n\nI sit up. The lights are down, but I know this place: Mac's house. Skye is pressed against the sofa next to me, looking up, tail wagging softly, but as if she knows things aren't right, doesn't jump up like she usually does.\n\nMy hand hurts and I hold it out, check it like it belongs to somebody else. Nothing broken; just bruises, a few split knuckles.\n\n\"What happened to it?\" Mac asks.\n\n\"I kind of punched a wall.\"\n\nHe hands me a glass of water, tablets. \"Painkillers. The ones you left here after your IMET.\"\n\nI take two, not the recommended one, and shake the container; a few rattle inside. \"There's not enough.\"\n\n\"Enough for what?\"\n\n\"There's too much pain. No, not my hand. Is it all real, did it really happen? At the college. And it was Ben?\"\n\nThey exchange a glance. Aiden pushes Skye out of the way, sits next to me.\n\n\"It looks that way.\"\n\n\"I don't understand. Why did Ben leave that note, get me out of there?\"\n\n\"Maybe he didn't want you hurt.\"\n\n\"Funny way to show it. Does Ben know this place?\" I panic, looking at Mac. No more. No more friends dying.\n\n\"He did before his memories were erased, not since,\" Aiden says. \"It should be okay for a while.\"\n\n\" _Should be_ isn't good enough. We need to get out of here before they find us.\"\n\n\"We will. Soon,\" Aiden says. \"It's all over.\"\n\n\"What do you mean?\"\n\nHe shakes his head, drops it into his hands. \"MIA, what we were trying to do. It's over. Florence and all the others\u2014friends, every one of them\u2014murdered. Our evidence destroyed, computer system compromised. We're beat.\" His voice, so tired, so full of pain.\n\n\"It's all been for nothing?\" My voice is small. _It's my fault._\n\n\"You and I must be high on the Lorders' most wanted list. You're getting out.\"\n\n\"What do you mean?\"\n\n\"Going to United Ireland. It's being organized.\"\n\n\"No! You're telling me to run away. I'm done with running.\"\n\n\"We'll try to rebuild. One day.\" He shakes his head. \"I have to stay, to do what I can, but I can't think straight if you're not safe. You have to do this for me. Go.\"\n\n\"Why? After everything that has happened. Ben betrayed me; I can't run away from the mess I created.\" The words are dull, unreal. \"He betrayed all of us. He wouldn't have been there if it wasn't for me.\"\n\nAiden shakes his head. \"I brought him there. Stupid! I was letting feelings cloud my judgment. It's my fault.\"\n\n\"No, you're both wrong,\" Mac says. \"You were giving him a chance; that is what MIA is about, isn't it? Trying to save lost souls from the Lorders' clutches.\"\n\nAiden shakes his head. \"So many dead. Was it worth it?\"\n\n\"Wait a minute. I don't understand what you said before. What do you mean, letting feelings cloud your judgment?\"\n\n\"Isn't that obvious yet?\"\n\nOut of the corner of my eye, I see Mac backing out of the room, shutting the door.\n\nAiden sighs, leans back against the sofa, eyes half closed. He opens them again, turns to face me. He looks young, bewildered almost, not himself. Aiden is always strong, certain of what he does and why he does it. This isn't him. It feels like the only bit of solid ground under my feet is falling away.\n\nI reach for his hand, take it in mine.\n\n\"You can't give up on MIA. You're superhero Aiden.\"\n\n\"No. Just Aiden. Just a man, no superpowers. And I messed up. Royally and completely, and we're finished. That's it.\"\n\n\"How did this happen?\" I swallow. \"How could Lorders do what they did? And how could they change Ben, make him betray us. Make him a killer.\"\n\nAiden touches my cheek. \"I'm sorry. No Lorder was holding a gun to Ben's head. The things he did, _he_ did. He made choices and took actions. It was in him to do what he did, no matter why he did it.\"\n\n\"No, I can't believe that. Ben wasn't like that; it was what they did to him.\" But even as I say the words, doubt creeps in. The AGT did everything they could to make me a terrorist, to make me a killer. But at the end of it all, I couldn't do it. Even when I was sure I should, that it was the only way, something inside stopped me from taking that step. If Ben was the same, wouldn't something in him have stopped him?\n\nAiden sighs. \"It's all my fault. And I've been such an idiot. If only I'd been honest with myself.\"\n\n\"No! You couldn't have known Ben would\u2014\"\n\n\"That's not what I mean. I thought getting Ben back would make you happy. And that you being happy would make me happy. But I was wrong. Seeing the two of you together made me miserable.\"\n\nI stare back at Aiden. His words now, other things he has done and said, are all starting to fall into place, but I can't take it in.\n\n\"Then every doubt I had about Ben I discounted. I thought it was because of how I felt about you that I questioned him, his motives. I argued with Florence when I should have listened. She was right about Ben all along.\"\n\nI shake my head. \"Lorders did this; he's not the Ben I knew. They changed him.\"\n\n\"But did you ever really know him?\" Aiden asks. \"How can you love somebody without knowing all of them, everything they've stood for, what they've done and not done?\"\n\nI'm silent a moment. His words are sinking in, settling. \"What you are saying, really, is that no one who was Slated\u2014who has no known past\u2014can ever love or be loved. I was Slated.\"\n\n\"Then why do I love you?\"\n35\n\nI WAKE IN THE EARLY MORNING; THE HOUSE IS SILENT. Aiden's words are painkiller-fuzzy, but I remember enough. I passed out on him from the tablets almost as he said it.\n\nI shake my head. He wasn't himself. Everything that happened cut him wide open. He didn't mean it.\n\nThen I'm not sure which thing he said alarms me more: that he's given up on MIA? That he is sending me away? That he loves me. It is all tinged in unreality like everything that happened yesterday. The pain at Ben's betrayal, and what followed from it, threatens to overwhelm everything I am. But even more: Without Aiden's certainty to anchor to, I feel lost.\n\nI get up, as if I can walk away from it, Skye at my feet. Wander into the kitchen. Catch my breath at the sight of the metal owl sculpture Ben's mum made, up on top of the fridge. I can't stop myself; I reach for it, pull it down. Run my fingers across the interlocking wings, the sharp beak and claws. Trap the square of paper and pull it: Ben's handwriting. His \"Love, Ben\" identical to the note he left to get me away from All Souls, safe up the church tower.\n\nI don't understand. Why'd he leave that note to get me away? If he is this cold Lorder-created killer Aiden says he is, that I saw with my own eyes yesterday, why not have me shot along with everyone else? Would that have hurt any more than how I feel now?\n\nMaybe somewhere inside him, despite everything he did, he cares. Just enough to save me. And I don't know if clinging on to that makes me feel better, or worse. But if he cared, why send me to the one place in Oxford where I would have to watch it all?\n\nAnd Tori. I shudder. Why was she there? A Lorder now, like Ben. Last time I saw her she was being hauled away by Lorders, screaming threats. Was she subjected to the same treatment as Ben? But there was something in her eyes, something vindictive in the way she laughed, as if she remembers me and knew I was watching. Or did I imagine it? Even zoomed in through the camera, could I really have read her like that from so far away?\n\nI'm overwhelmed by too many questions. Were there clues to what Ben was going to do? Could I have stopped it from happening if I'd noted them, told Florence and Aiden?\n\nI wander back to the front room, pick up my camera from the table where I must have left it last night. I stare at it in my hands, wanting to and not knowing if I can handle it at the same time. I breathe in deep, turn it on, and find the file of footage I took of Ben the day I tested my camera.\n\nBen's smiling face projects onto the wall. I run it back and forth, looking for clues, for hints of what was to come, but see nothing. He's just Ben as I remember him from before, isn't he? He was more jokey than he used to be if anything, less like a Slated. Bolder. I pause it, stare at his eyes on the wall, and the pain is starting to reach for me, to pull me under.\n\nI switch it off. Concentrate on breathing in and out, casting my eyes about the room, looking for something, anything, to distract myself, and that is when I see something I'd forgotten: Murray the bear, stuffed up on a bookshelf. I pull him down.\n\n\"Can it really all be over?\" I whisper to him. All we'd dared to hope: that stories like Edie's could get out, could make a difference. Where is Edie now? Maybe she'll end up Slated in an orphanage. Or worse.\n\nShe's still in my camera. I pick it up again, look at the list of files: Edie is there. Along with another three witnesses I recorded. The Slated children at the orphanage. And the massacre at the college. Could it be enough? I stare at Murray. His fuzzy face seems to be saying something\u2014or is that just the painkillers?\u2014that we can still do this. Do it fast, before anything else goes wrong.\n\nI go to shut off the camera, then frown. In the file list is one I don't recognize, don't remember having noticed before. Labeled _SC,_ it is before the shots I took of Astrid and Nico.\n\nOpen file; click play. Stella appears? Of course; _SC_ is _Stella Connor._\n\nI sit and listen to her message. When it is over, goose bumps trail up my arms.\n\n\"It was there, all this time?\" I say to Murray, stunned.\n\nThen I run to the back of the house, Skye on my heels, and bang on bedroom doors. \"Wake up, get up!\" Skye barks, and Mac and Aiden rush out, half asleep. Alarmed.\n\n\"What is going on?\" Aiden says.\n\n\"We need to talk, and we need to do it now.\"\n\n\"What about?\"\n\n\"Listen to me. I'm not going to Ireland.\"\n\nAiden starts to protest; I hold up a hand. \"There's more. Just shut up and listen. But first I have a question. What is up with MIA's computer systems? Can we get information out?\"\n\n\"We were pretty much ready to go for it, but not through the usual computer channels,\" Mac answers. \"After our systems were breached, we'd worked out a better alternative through Ireland. DJ's contacts think they can hack the Lorder communication satellite, broadcast from there across the whole country and internationally when we're ready.\"\n\n\"Broadcast what?\" Aiden's voice is skeptical. \"Most of what we had is gone\u2014either stolen from the hacked systems or destroyed at the college.\"\n\nI hold up my camera. \"I've still got witnesses' testimonies: Edie and three others. There are the photos of the Slated children at the orphanage, the footage of All Souls yesterday, and\u2014\"\n\n\"It's not enough,\" Aiden interrupts. \"Our view\u2014Florence's dad, then Florence\u2014was always that we needed meticulously documented evidence and witnesses. We haven't got that anymore. We can't back it up.\"\n\n\"If we don't tell their stories, they died for nothing.\"\n\nThe room is silent.\n\n\"We at least have to try,\" Mac finally says.\n\nAiden looks between us. Does something _change_ in his eyes? Then he shakes his head. \"I never completely agreed with the careful long game, but is there really enough to\u2014\"\n\n\"There's more. Watch this,\" I say, and point the camera at the wall.\n\nStella's face fills it, a nervous smile. \"Uh, hi. This is Stella Connor. My daughter, Lucy\u2014and, Lucy, I will call you that, you will always be the daughter I love to me\"\u2014and she smiles\u2014\"a short time ago got a confession out of me I thought I'd never tell. She tried to convince me that I had to tell this story, that it had to get out. But I refused.\" She sighs. \"I'm old, and I'm a coward. I've always been one; I'm just starting to see how much. Anyhow. I best get on with it.\n\n\"I'm realizing Lucy is going to have to leave my life again, no matter what I do. And on this camera I borrowed from her, I found the reason why. And yes, Lucy, you did password-protect the photos so I wouldn't see them, but I did set up the camera, so I have the admin override, and I snooped. And there were these very young children, Slated.\" She shudders, sits up straighter. \"Everything keeps getting worse, so now I have to be brave and tell my story.\n\n\"My mother is Astrid Connor: Juvenile Control Officer for all of England, and steadily rising in Lorder ranks. Years ago I overheard her speaking to a subordinate about the assassinations of Prime Minister Armstrong and his wife, Linea, _before_ it happened. I was a child, I didn't really understand what I'd heard, and when I asked her she said they knew about it before the media did, and I didn't question it. But years later I worked it out and confronted her. She admitted\u2014bragged, really\u2014that a hard-line Lorder faction she was in deliberately leaked information to the AGT for these assassinations to take place. Our family was friends with the prime minister's family; Linea had confided in my mother that Armstrong was going to resign and expose violent excesses of the enforcement arm of the Lorders that he'd unearthed. It would have toppled the Lorder government.\n\n\"Mother had me locked up to keep me from saying anything. I was pregnant, and my baby died. Months later, she gave me Lucy: the most beautiful baby. I don't know where she got her from. Once she could see that I loved Lucy completely, Mother let us out. Said if I ever said anything, she'd take Lucy away.\n\n\"I love you so much, Lucy. I'm sorry I didn't tell you everything from the start.\" Her hand reaches for the camera. The recording stops.\n\nI'm struggling to keep my composure. Stella must have made this when I was in the boathouse. Got Ellie to bring it to me with that cryptic message when she somehow found out Astrid was on her way; brave, at last. I hope, so much, that she is all right.\n\nI blink hard. \"Well? Is it enough?\"\n\nAiden and Mac are looking at each other in stunned silence. Then Mac grins. \"We've got the bastards, don't we!\" He holds up a hand in the air to Aiden. After a second's hesitation, he raises his and high-fives.\n\nAiden's got his determined look back again. \"Yes! We can do this.\" He grabs me in a hug, then lets go abruptly. \"You still have to leave first.\"\n\n\"No. I'm the only living witness you've got to back anything up. I'm not going anywhere.\" I stare back at Aiden's blue eyes, not wavering, but neither is he.\n\n\"How about we interrupt this staring contest with breakfast?\" Mac says, and fills the kettle, plugs it in. \"Then perhaps you'd like to record me telling what happened to Robert after the bus bombing.\"\n\nAiden holds up a hand, thinking. \"There is one other thing. One other witness that'd really help.\" His eyes are on mine, apologetic.\n\n\"Who?\"\n\n\"We need Armstrong's daughter, Sandra Davis. Your mum here.\"\n\n\"No. No way.\" I stare at him, horrified. \"Mum and Amy being safe is one of the things that makes me able to go on. Don't take that away from me.\"\n\n\"Listen to me: People will believe her. They don't know who Stella is. But if she sees what Stella says and backs it up, and backs up Mac's story, well. We're home.\"\n\nMac slips an arm around my shoulders. \"He's right, you know. It's time to stop being safe, to risk everything.\" I shrug his arm off, go back to the sofa. Murray stares up at me, with a _he's right_ look on his face. I shake my head. Next thing, Skye will lecture me. On cue she jumps up next to me, puts her head on my lap, and looks up.\n\n\"Okay. We can ask her, but no pressure.\" She won't do it, will she? I'm beyond her protection, but she won't do it if it means putting Amy at risk. \"How are we going to get word to her?\"\n\nThe front door bangs open. \"Hello,\" a cheery voice says. One I know. I turn, and there is Amy's boyfriend: Jazz.\n36\n\n\"I'M SERIOUSLY NOT HAPPY,\" JAZZ SAYS, BUT HIS GRIN ARGUES against that. His arms seem to have me trapped in a permanent hug since the split second that, changed hair or no, he realized it was me. \"Why didn't you tell me she was alive?\" he demands of his cousin Mac.\n\nI squirm. \"Let me go already!\"\n\n\"You're really okay?\"\n\n\"All in one piece,\" I say, not able to think of how I am as okay, exactly, after everything that's happened.\n\nHe loosens his arms, but keeps me in front of him with a hand on each shoulder. \"Amy has been so . . . Can I tell her?\"\n\n\"Need-to-know,\" Aiden interjects.\n\nJazz glares at him. \"Yeah, well, whatever: Amy needs to know.\"\n\n\"Why not?\" I say. \"It'll be out soon enough. What'll it hurt if he tells her now? She won't say anything.\" Not after the last time. Amy had in all innocence told about drawings I was doing for the AGT, and a snatch and grab in a black van and Lorder questioning and blackmail followed.\n\n\"And your mum?\" Jazz asks.\n\n\"She already knows.\"\n\n\"No way. She never let on.\"\n\n\"Need-to-know,\" we say in unison, and I find myself laughing along with Jazz, some part of me surprised that I still know how.\n\n\"It's good to see you,\" I say. \"Now let me go.\" He lets go with one arm, but keeps the other around my shoulders, and it feels good. Amy's boyfriend has always felt like a big brother to me.\n\n\"We need to set up a meeting,\" Aiden says to Jazz. \"Between Kyla and her mum. And _don't_ say anything to Amy, not just yet.\"\n\n\"Fine. Sure, give me a message. Which reminds me.\" He takes a small box out of his pocket. \"Mail call.\"\n\n\"What's that?\" I ask, mystified.\n\nMac takes it, holds it up. \"The latest from DJ, I'm guessing.\"\n\nI stare at Jazz. \"You mean you're involved with all this, too?\" A big brother with secrets of his own.\n\nHe grins. \"Always was their messenger boy; just extra busy lately with computer communications being out. You didn't need to know, I guess.\" I smack his arm. To think all that goes on under the surface of people's lives, right in front of my eyes, and I had no idea. None.\n\nMac opens the small box. \"Brilliant; at last.\" He holds his hand open: a com. As if on cue, it buzzes. \"I'm pretty sure it's for you,\" Mac says to Aiden. Aiden straightens his shoulders. Answers it, disappears down the hall, and shuts a door.\n\nMac and I exchange a glance.\n\n\"Does DJ know what happened yet?\" I ask quietly.\n\n\"I'd be surprised if he doesn't. But he probably wants to hear a firsthand account.\"\n\n\"What's happened?\" Jazz asks.\n\n\"Need-to-know or not\u2014believe me, you don't want to know,\" I say.\n\nMac works out a plan with Jazz about when and where we can try to meet with Mum while I drift about the kitchen, making toast. Wondering if DJ will go along with our plan. What if he thinks we haven't got enough to go public? What if he says no? We can't get it broadcast without his help.\n\nI leave the kitchen, make for the back room. Knock once and go in.\n\nAiden's still on the com with DJ. He meets my eyes, gestures for me to be quiet. \"What is the time frame for that? . . . I'm not sure we can . . . I see.\"\n\n\"Let me talk to him,\" I say.\n\nAiden mimes pulling his hair out. \"Fine. Talk to her, then.\" He holds the com out to me.\n\n\"Hello?\"\n\n\"Hi, Kyla. Aiden's been telling me that\u2014\"\n\n\"DJ, just listen to me a moment. We need to do this broadcast as soon as we can. No more waiting for more things to go wrong. We've got to move before\u2014\"\n\n\"Slow down. I agree with you.\"\n\n\"You do?\"\n\n\"Yes. And, Kyla hon, I hear you've had a rough time. I'm sorry.\" He pauses, but this time I stay silent. What is there to say? \"Aiden wants me to ship you out of the country.\"\n\nI narrow my eyes at Aiden. \"I'm not going.\"\n\n\"That's my girl. I think we need you involved in our little movie production. Aiden's given me the rundown on Stella's recording, the other things on your camera, the possibility of Sandra Davis's involvement. You need to make it happen.\"\n\nNow I glare at Aiden. \"We haven't even asked her yet. She may refuse.\"\n\n\"One way or the other, we need to get this wrapped up for transmission tomorrow, or we'll have to wait months for another opportunity. A bit techy, but all to do with choosing the right moment to interfere with their satellite without detection; we can mask our intrusion as solar activity if it coincides with a geomagnetic storm. And wild weather with thunderstorms later is predicted as well: Their satellite and terrestrial communications should be interrupted tomorrow evening if both solar and weather predictions are correct.\"\n\n\"Tomorrow? So soon?\" I say, looking at Aiden. He raises his shoulders in a shrug.\n\n\"Can you do it?\"\n\nWaiting might give us more to broadcast. But look what happens with waiting: All Souls. That's what. \"Yes. We'll do it.\"\n\n\"That's the spirit.\"\n\n\"DJ? I have a question.\"\n\n\"Yes?\"\n\n\"Did you hear about what Dr. Lysander said, about someone high up interfering with my hospital records?\"\n\n\"Yes.\"\n\n\"Did you find out anything else about me? My DNA?\"\n\nThere is a pause, barely perceptible. \"Still working on it.\"\n37\n\nI'M JITTERY, NERVOUS. CAN'T SIT STILL. AIDEN LOOKS AT ME. \"What's wrong?\"\n\n\"Nothing. Everything.\" I look at the time. \"She's late.\"\n\nHe glances at the clock. \"Only about twenty seconds. It'll be all right.\"\n\n\"It's just I don't want anything to happen to her. Everyone who gets too close to me seems to pay the price. I don't want her involved.\"\n\nHe takes my hand. \"Because you care. You want her out of harm's way.\" He doesn't say anything else, but I know what he's thinking.\n\n\"I couldn't go.\"\n\n\"I know.\" He sighs. \"It's part of what makes you who you are. But I had to try.\"\n\nThe door opens.\n\n\"Mum!\" I jump up and run to her. Her arms wrap around me quickly, a tight hug.\n\nShe looks over my shoulder at Aiden. \"Who's this?\"\n\nHe stands up. \"Pleased to meet you, ma'am. I'm Aiden.\"\n\nShe turns to me. Shakes her head. \"Why are you back? It's too dangerous.\"\n\n\"I tried to tell her, but she wouldn't leave,\" Aiden says. They exchange a look.\n\n\"Stubborn, isn't she?\" Mum says. \"Now, why am I here?\"\n\n\"We need your help.\"\n\nMum sits down, and Aiden explains what we\u2014MIA\u2014are planning to do.\n\n\"So this really will broadcast across the whole country? And in other countries?\"\n\nHer eyes turn in, thinking, then meet mine with a spark of excitement. \"That could work. Though I'm not sure what you think I can do.\"\n\n\"I'm really sorry to show you this,\" I say.\n\n\"What?\"\n\nI get the camera out. \"Do you know who Astrid and Stella Connor are?\"\n\nShe frowns. \"Astrid Connor went to school with my mother; they were friends. Stella is her daughter. We used to be in touch when we were kids; not lately, though. She stopped answering my calls, years ago.\" She shrugs. \"What do they have to do with anything?\"\n\n\"They're my family. From before I was Slated. I was adopted by Stella; she raised me from when I was a baby until I was ten. It was her I went off to see.\"\n\n\"What?\" Mum's eyes are round with surprise. She shakes her head. \"Small world. But I can't see what this has to do with me.\"\n\nAiden and I exchange a glance. I'd wanted to warn her what was to come, but he thought it was best for her to see and hear it for herself.\n\n\"Okay. Here's a recording Stella made. She hid it on my camera, and I only just found it recently. I'm sorry.\"\n\nI project it on the wall, hit play. Her face goes pale as she watches and listens, and she grips my hand tight.\n\nAfter it ends, Mum looks away a moment. Then she meets my eyes. \"If only I knew what my parents were planning to do. All these years, I could never understand why my father set up the Lorder government, with all it has led to. I always thought he didn't know what was really going on, but he did, and he was planning to put a stop to it. Thank you for telling me.\"\n\n\"You see,\" Aiden says, \"this is why we need you. To introduce Stella's recording in our broadcast; it'll give it credence. Make people listen. Also we've got a witness who saw your son, Robert, alive after the bus bombing. You could talk about him being missing, also.\"\n\nMum nods. \"I knew from another source that Robert survived the bombing, but disappeared afterward. I always assumed he was Slated. If my parents could have said and done what they wanted, would our world be a different place? Would I still have my son. I want to do it for them, to say what they were stopped from saying. Yet this isn't just about me; things could go wrong. I have Amy's safety to consider. I need time to think about this.\"\n\n\"I'm sorry. That is one thing we haven't got a lot of,\" Aiden says.\n\n\"When would we need to do this?\"\n\n\"Tomorrow afternoon at the latest. There are technical reasons why the broadcast must happen tomorrow night. Jazz can bring you, if you decide to help us.\"\n\nThey talk some more about details, but I just hold her hand tight. Imagine the shock; all this time being told one version of why your parents died, and then finding out it was all lies.\n\n\"I should go.\" She hugs me tight. \"Take care of her,\" she orders Aiden, and then is gone.\n\n\"What do you think she'll do?\" Aiden asks.\n\nThis is so reminiscent of another time, another decision. When she had to decide whether or not to tell the whole country on that live recording what she thought really happened to her son. Then, she didn't do it; she wouldn't do anything to put Amy or me in danger. Will this time be any different?\n\n\"I don't know.\" Part of me hopes she will be there tomorrow; part of me hopes she stays away.\n\n\u2022 \u2022 \u2022\n\nThat evening, Aiden is working in the computer room, and Mac has gone off with Jazz to set up for tomorrow and to copy the footage and photos from my camera and start putting it all together. DJ wants me to do an introduction, to explain how things I witnessed fit together, and I'm trying to think what I can say so I won't stare at the camera like an idiot.\n\nWhat can I say about All Souls that explains what happened in a way that makes any sense? What can I say about Ben?\n\nWhat can I say\u2014what am I _willing_ to say\u2014about my life. My crazy, confused, Lorder-tainted life, and all those it has damaged or destroyed.\n\nI'm stalking back and forth in the front room; Skye gets under my feet. I almost trip over her and curse.\n\nAiden's door opens; he comes in. \"Is everything all right?\"\n\n\"Just stage fright,\" I say, but look at my feet. I can't look him in the eye.\n\n\"It'll be all right.\"\n\n\"Just like everything else has been all right so far?\" And I'm shaking. I don't know why; is it a delayed reaction, fear, pain, all three?\n\nI look up and take a step toward him; Aiden takes one toward me. Meeting in the middle. His arms slip around me, just gentle, not holding but comforting, like you would a sister, or a child. I nestle my head against his shoulder. I fit against him different than Ben; Ben is taller. His hand smooths my hair; he's trying to make me feel better, but it's not enough, nothing can ever be enough to take the emptiness away. And I pull him closer and closer. His heart is beating faster and so is mine. I reach up and pull his head down, kiss him. I don't know what I'm doing, I don't care. All I am is cold, dead, empty; Aiden is feeling, warmth, and life.\n\nAnd at first he kisses back. Then gradually, gently, he pushes me away. Shakes his head. \"Not like this.\"\n\nAnd I start to cry. Why? Another loss, another cold space. He pulls me to the sofa, wraps a blanket around me.\n\n\"Don't go,\" I say.\n\n\"I'm not going anywhere. Ever. As long as you don't want me to.\" But he stands up. \"Back in a sec.\" He goes down the hall and comes back with a guitar in his hands.\n\n\"I don't play very often, but it always makes me feel better. Close your eyes, Kyla. Tomorrow will be a long day. But we'll get through it. And I'll be there.\"\n\nAnd he plays; he's good. Some songs I know, some I don't. And somehow my eyes close. I slip to a dark, dreamless sleep.\n38\n\nTHE PROMISED WILD WEATHER HAS ARRIVED. THE COLD WIND whips bits of branches off trees and swirls dead leaves as I run.\n\nI'd said I just needed a run, bolting out, not able to meet Aiden in the eye after last night. Half expecting an argument, or an escort. But they let me go.\n\nMy feet fly up the canal path, pushing hard to make everything go away, but it's not working. I dig deep for _more_ : more effort, more speed. And the miles fly by, and it gets closer. This run wasn't only about escape and release. Will I be able to find it?\n\nNot at first. I know I'm close to where it should be, that there was a particular bend of the path, a climbable tree not far from it. I slow to a jog and retrace my steps until finally I think I see the right one.\n\nThe wind is crazy as I climb up the branches, like it's going to pull me off and throw me to the ground. I squint to avoid getting grit blown in my eyes. How far up was it? I think I've come too far and look back. Anything could have happened to it. A bird or a squirrel with an eye for shiny things could have taken it, the branch it was on could have been victim to the wind. It might be the wrong tree. Now that I'm not running, I'm freezing; I feel around with numb hands, having trouble keeping my footing when I can barely feel my feet. I'm about to give up when my fingers brush something cold, something metal.\n\nI twist to reach it better, and pull it off the branch it is hooked around: Emily's ring. Clutch it tight in my hand a moment, then start down.\n\nBack on the ground, I peer at the inscription: _Emily & David 4ever._ I took it off her hand after they both died, victims of Lorders and Slating like so many others. I need this; I need a reason I can hold on to, to get me through what has to be done today. I start to put it in my pocket, but then slip it on my finger instead, and start the long run back.\n\n\u2022 \u2022 \u2022\n\nAfter a shower I come out to the kitchen, where Mac is making sandwiches. \"Is everything okay?\" he asks. Then retracts. \"All right, stupid question. Anything not okay that I can do anything about?\"\n\n\"No. Thanks.\" I smile at him.\n\n\"At last: a smile. Of sorts. Sit down and eat up, it's about time to go. Aiden? Lunch,\" he calls.\n\nAiden comes in, squeezes my shoulder with one hand, sits down opposite. He looks into my eyes, nods once, and his steady gaze says things are okay. A knot of anxiety inside me eases; just a little, but it's enough.\n\n\u2022 \u2022 \u2022\n\n\"Welcome to our movie studio,\" Mac says, and opens the door to a run-down farm outbuilding. Up a path a few miles from his house, from the outside it looks abandoned, but as I step inside, I gasp. It's like an Aladdin's cave for computer geeks; there are bits of tech everywhere.\n\n\"You clearly didn't just set this up for today,\" I say.\n\n\"No. It's been one of MIA's hidden tech centers for ages; there's all sorts of different stuff to play with out here. Movies are new. But we've got the transmitting equipment here to link up with DJ's relay to the satellite. And Jazz and I cleared a place to do recordings last night.\"\n\nAiden and I follow him around a crammed-high row of shelving; behind it, there's a clear area with a stool, a tarp hung to block equipment behind it. And a camera on a stand in front with lights.\n\n\"That looks a bit more high-tech than my little camera,\" I say, touching my pocket where it is once again, returned this morning after they copied the relevant content last night.\n\n\"Nah, it's easy. I'll show you, then we can record my part.\"\n\nMac starts to explain the controls to both of us when there is a loud knock.\n\n\"Hello?\" Jazz's voice. And another: Mum?\n\nI bolt around the shelves, and it's not just Mum; Amy is here, too.\n\nAmy runs over to me and grabs me in a hug. \"You crazy girl. Don't you ever do that to me again!\"\n\n\"You've cut your hair,\" I say, shocked. Her gorgeous thick hair is gone, cut to a short pixie.\n\n\"Hey, if I knew where you were to check for fashion advice and that I didn't need a s\u00e9ance, I'd have done so. Besides, you're looking a bit different, too.\"\n\n\"You're both here?\" I say to Mum, who has held back, but walks up to us now for a group hug.\n\nMum smiles. \"Both my girls together! I realized this was a family decision. I had to let Amy in on what was happening, and then we had a vote.\"\n\n\"And?\" Aiden asks.\n\n\"Amy says go for it. I'm still not sure, but there are three of us. Kyla?\"\n\nAnd all eyes are on me.\n\n_No. Don't make me do this. Don't make me decide._\n\nI swallow. \"If this goes wrong, it could be a death sentence for everyone involved.\"\n\n\"Including you,\" Mum points out.\n\nI shrug. I don't want to say, out loud, that I don't care anymore about my own life. \"It's different for me. They're already after me, anyhow.\"\n\n\"You told me once before that sometimes the most important thing is doing what is right.\"\n\n\"The problem is working out what is right, isn't it?\" Amy says.\n\nAnd I stare back at Mum and Amy, standing close together. Amy was Slated, assigned to Mum like I was, but that doesn't change what they are to each other now. What we are. But we're not the only ones. \"This isn't just about us. It's about every mother and daughter, every father and son. Now and in the future.\"\n\nMum looks back at me, slowly nods. \"Okay, then. Let's get this show on the road.\"\n\nMac goes first while I operate the camera. He tells about the day his school trip went so wrong; when stray AGT bombs took out most of a busload of fifteen- and sixteen-year-olds. How he had a minor injury. How his friend Robby\u2014Robert Armstrong\u2014was hauled off the bus, away from his dead girlfriend. Screaming, but unhurt. Then later was on the list of the dead.\n\nThen it is Mum's turn to tell us about her son, Robert. How she'd heard rumors for years that he'd survived and been Slated, but could find no trace of him.\n\nShe pauses, looks me in the eye behind the camera. \"But that's not the only tragedy in my life. You know who I am: Sandra Armstrong-Davis. My father, Prime Minister William Adam M. Armstrong, and my mother, Linea Armstrong, were murdered by AGT bombs when I was fifteen. But that is not the end of the story. My parents were preparing to expose Lorder atrocities; my dad, to resign as Lorder prime minister and dissolve the government. My mother confided in a school friend, Astrid Connor, who deliberately leaked the information of their whereabouts to the AGT to have them assassinated and silenced. You will hear about this from Stella Connor\u2014a childhood friend, and the daughter of the Lorder who did this.\"\n\nShe pauses. \"How was that?\"\n\nMac, behind the camera again, gives a thumbs-up. \"Brilliant. Thanks.\"\n\nI take a deep breath. \"Is it my turn now?\"\n\nAiden comes over. \"I could do the bit on All Souls. I was there, too.\"\n\nI shake my head. \"I'm the one who took the footage, and I was looking through the camera zoom and saw what happened as it happened in more detail than you could. I have to do it.\"\n\n\"Are you sure?\"\n\n\"Yes. And there's more I can testify about. Mum, can you and Amy stay? I want there to be no more secrets. This is all going to be out there; I want you to hear it from me.\"\n\nI settle on the stool under the lights. Amy straightens my hair. \"It's not a fashion shoot,\" I say. She sticks out her tongue and gets out of shot.\n\n\"When you're ready,\" Mac says.\n\nI stare down the camera, pretend I'm going to talk to myself. That no one else is here; that Edie's teddy bear is staring at me behind the lens, and no one else can hear a word.\n\n\"Hi. I'd like to introduce myself, but I can't: I don't know who I am. Before I was born, a woman you'll hear from soon was a prisoner. Her name is Stella Connor. She'd found out her mother\u2014a Lorder JCO, Astrid Connor\u2014had engineered the assassinations of Prime Minister Armstrong and his wife. Stella was locked up by her mother to keep her quiet; she was pregnant at the time, and her baby died.\n\n\"Then Astrid gave Stella another baby: me. She threatened to take me away if Stella ever said anything, and then let us go. Stella and her husband, Danny Howarth, who thought I was his, loved and raised me as their daughter.\n\n\"When I was ten years old, I was kidnapped by the AGT. I was subjected to conditioning to fracture my personality, trained by AGT terrorist Nico, and then he deliberately set me up to be captured and Slated by Lorders when I was fifteen.\n\n\"After I was Slated and assigned to my new family, my fractured personality and memories started to come back; even though I was Slated, my Levo stopped controlling my actions when my memories returned. The AGT plan worked. I rejoined the AGT, but Lorders threatened me to try to make me betray the AGT.\n\n\"On Armstrong Memorial Day, I was present at the speeches given by my assigned mother, Sandra Armstrong-Davis, who you'll be hearing from also.\" I pause, unable to say what comes next, twisting Emily's ring on my finger and fighting for control. \"I'm sorry. I had a gun strapped to my arm. Mum\u2014Sandra\u2014was next to me, and if she didn't say what the AGT wanted her to say, I was supposed to kill her.\" I blink hard, will myself to not look at Mum and Amy, to keep going.\n\n\"I couldn't do it. I didn't stay for the second ceremony in the grounds; I ran back to try to save Dr. Lysander, who was being held by the AGT because I'd betrayed her. Later I found out that a com Nico gave me and concealed under my Levo was a remote-controlled bomb; he'd meant to set it off during the second ceremony, when I should have been next to my family and Prime Minister Gregory.\"\n\nI breathe in and out a few seconds, fight for control. Then continue. I tell them everything I did with the AGT and what happened with Nico; the bomb that the Lorders said killed me. Going to stay with Stella and finding out she wasn't my mother, visiting the orphanage, seeing the Slated children and realizing I was going to have to run, to take this information to MIA. Seeing Astrid and Nico together. Going to Oxford, finding Ben. That Ben had been subjected to unknown procedures by the Lorders; that he betrayed us. My voice wavers as I describe the massacre at All Souls College that I filmed.\n\nThen I stare at the camera. \"I still don't know who I am. Or what Astrid Connor, Lorder and JCO, was doing with Nico, the AGT terrorist who trained me and countless others to attack Lorders. But it's hard to imagine she wasn't involved in everything that has happened to me from the beginning, and the plot to assassinate my family and Prime Minister Gregory.\n\n\"But one thing I do know: The truth needs to come out. All of it. If everyone knows what really happens, what Lorders really do, what happens to the missing, then they\u2014you\u2014will put a stop to it.\n\n\" _Everyone_ needs to know.\"\n\nI'm finished talking. Still and silent now, I can't look up, can't meet anyone in the eye. I'm aware Mac has stopped filming, but no one says anything.\n\nI hear footsteps: Mum's. She walks up to me.\n\n\"I'm sorry,\" I say.\n\nShe slips her arms around me. Dimly I'm aware the others are walking away, out of sight.\n\n\"Why are you sorry?\"\n\n\"I almost killed you, you and Amy. And loads of other people as well.\"\n\n\"You didn't know you had the bomb.\"\n\n\"I knew I had the gun. I thought I was going to use it. I thought I had no choice.\"\n\n\"But you didn't.\"\n\n\"No. I couldn't. But everything else I've done. And what happened at All Souls, because of Ben. It's my fault.\"\n\n\"Caring for someone is never a bad thing, even if it doesn't work out.\"\n\n\"It hurts,\" I whisper.\n\n\"I know. I'll tell you one thing for free.\"\n\n\"What's that?\"\n\n\"If I could get Astrid and Nico in my sights right now, they'd both be dead.\"\n\nI half smile at the thought of Mum as avenging gunslinger, not a picture that readily comes to mind. \"I'm not good at killing people. I'm better at getting them killed.\"\n\nAiden steps back around, and clears his throat. \"We're going to get this movie into production now. Go if you want.\"\n\n\"I should get Amy out of here. We're going to go stay in a quiet place with some friends for a few days, see what happens if\u2014when, I mean\u2014that hits the airwaves. Come with us? Please?\"\n\n\"No. Sorry. I've got to see this through.\"\n\n\"Okay.\"\n\nAmy comes around; her eyes are red. She and Mum give me hugs, and go.\n\nMac and Aiden get busy on computers with the different bits of recordings, still photos, our pieces from today. After a moment pulling myself together, I walk over, watch over their shoulders.\n\nAiden catches my eye. \"Thank you,\" he says.\n\n\"For what?\"\n\n\"For having the courage to do what you just did.\"\n\nI shrug. \"I've been a coward for a long time. You shouldn't thank me for that.\" I look away, not able to meet him in the eye.\n\nIt was Stella and Mum who finally made me face up to telling the truth. They both did, so how could I not? Staring at everything I've been and done, I struggle to keep myself contained when inside everything feels like shattering glass. There are no walls, no illusions left to hide behind. Mum knows. Aiden knows. Soon the whole world will know.\n\nFinally Mac declares it done. \"Do you want to watch a run-through? No problem if not.\"\n\n\"I'll watch,\" I say. Mac projects it on the wall. At the beginning, titles run across the screen: _Need to Know\u2014an MIA production._\n\nI try to watch the whole fifteen minutes of it dispassionately, objectively. Like I don't know anyone in it, and I'm Joe Public sitting on my sofa about to get the evening TV surprise of a lifetime. But when the footage I shot from the church tower comes up, I can't watch. I look away. A warm arm slips around my shoulders: Aiden. I want to look up at him, but I'm afraid what I'll see in his eyes.\n\n_BANG._\n\nA massive crash makes us all jump, then laugh when we realize\u2014it's thunder. The storm is here.\n\nAiden grins widely. As if on cue, his com rings: DJ? He answers. \"Hello? Yes. It's ready.\" He pauses, listening. \"Got it, 'bye.\" He clicks end, then turns to us. \"We're to transmit at six, when the storm should be at its peak. It'll be on then instead of the evening news. It _will_ be the evening news!\" he says. He and Mac give high fives, excited, and part of me is, too. All we've worked toward is finally really happening.\n\nBut part of me is with all those who suffered, who died. Florence, Wendy, all the other students. Those small children who were Slated.\n\n\"What is it?\" Aiden asks.\n\n\"How can we celebrate? We can't do anything for those who died, for their families.\"\n\nAiden puts his arms around me, and I lean into him.\n\n\"We can remember them,\" he says. \"And, through what we've done today, make it stop. Make their loss have meaning.\"\n\nWithout discussion, the three of us stay silent; a minute, two. Then another massive crash of thunder hits, and again I jump. I don't mind storms; normally I like them, the wilder the better. Not today. I'm as jumpy as . . .\n\nSkye.\n\nI pull away from Aiden. \"Skye will be scared alone with the storm. I'm going back to the house.\"\n\n\"Do you want me to walk back with you?\" Aiden asks.\n\n\"No. Stay and have your moment. I'll be fine.\"\n\n\"Wait a sec,\" Mac says, does something with the computer and my camera, then hands it over. \"I put a backup copy of _Need to Know_ on it. Just in case we're struck by lightning.\"\n\nI scowl at him. \"Don't tempt fate,\" I say, and head out the door\u2014fresh air, storm or not\u2014and escape.\n\nIt's about two miles back and just getting dark, but now and then the sky lights up with crazy, jagged lightning. Each time the thunder crashes, seemingly right over my head, I almost jump out of my skin, annoyed at myself for being so jittery. I'm about halfway there when it starts: huge, heavy, freezing cold raindrops. So I get cold and wet; so what.\n\nAs I run, I wonder at how I feel; I should be celebrating with both of them. Instead, I feel empty.\n\nWhat is next? What is my future? How will Aiden feel now that he knows all the things I have done?\n\nMum said caring for somebody is never a bad thing, even if it doesn't work out.\n\nDo I care?\n39\n\nI'M NEARLY AT THE LIGHTS OF THE HOUSE WHEN THEY GO OUT, and all is darkness.\n\nA power failure, because of the storm? I hope that doesn't affect the transmission. Knowing Mac there will be a backup generator.\n\nIt's inky black now, and despite the freezing rain, I slow to a walk to find the path at my feet. Tonight the dark is unnerving, not comforting as it usually is, and without thinking about it I switch to moving silently, every step taken with care.\n\nAnother blinding flash, and everything is lit up\u2014a split second only. _There!_ By the house, near the back door. Two figures in black?\n\nFear whips through me as all is plunged into darkness again. Lorders!\n\nDid they see me?\n\nPanic finds my feet, and I run blind, no longer silent, headlong back the way I came. Cries sound behind: spotted or heard, either way, they're on to me. When the path branches, I go the other way, away from Mac and Aiden. I can't lead them there; anything but that. I should be able to lose them. I can run faster than almost anybody I've come across.\n\nBut I'm not pulling away. I can hear pursuit keeping pace behind. Now it sounds like just one runner, a long, loping gait. A familiar gait, and when there's another blinding flash I can't stop myself from glancing back.\n\nBen.\n\nMy feet falter, then I push on; I'm gathering speed again, but it's no good. Bit by bit he gains. I can hear him getting closer, and knowing it is Ben has confused my feet.\n\nThen all at once he's flying through the air and I'm knocked to the ground. Winded, under him, I struggle to breathe. He holds my hands with one of his and gropes at my pockets. No! I twist, but he's got it. My camera.\n\nHe pulls me to my feet, presses something cold and hard against my back. \"Walk!\"\n\n\"No. Just shoot me already, if that's what you want to do. I don't care anymore.\"\n\nHe twists my arm behind my back and pushes; I stumble forward. What is the time? I have to delay them. I have to stop them from finding Mac and Aiden, from stopping the transmission at six.\n\nI trip and sprawl forward. With an exclamation of annoyance, Ben scoops me up and carries me, my arm still twisted. A gun pressed into my stomach so hard it hurts.\n\n\"How could you do it?\"\n\nHe doesn't answer.\n\n\"Everyone, all those students, just shot against the wall. Dead.\"\n\n\"They were traitors. They deserved what they got. As will you.\"\n\n\"You're the traitor; you betrayed me. You used to love me, you acted like you still did. How could you do it?\" My voice is too soft, plaintive, and I hate myself for it.\n\n\"Ah, sorry about that. Seducing you _was_ difficult. But I had to get you to fall asleep somehow.\"\n\n\"Why?\"\n\n\"Scanned you while you slept. How do you think we found you? Somehow your records were wrong, and we needed the scan to track you by your brain chip.\"\n\n_No._ Dr. Lysander had changed the number; Lorders worked out they couldn't track me, so they got Ben to take care of it.\n\nNow I'm full of rage, and struggle, but despite a few AGT tricks I know, the Lorders must have taught him how to hold someone. Or maybe the pain inside is making me too weak to fight back.\n\nWhen it hits me, I almost sag. He let me go, so he could track me here. I'd thought some part of him couldn't bring himself to hurt me, but I was wrong. \"You're evil.\"\n\n\"Sticks and stones.\"\n\n\"And that little girl; how could you?\"\n\n\"What girl?\"\n\n\"Edie! You knew her address. I ran there, and they were gone.\"\n\nHis shoulders move slightly: a shrug? \"No idea. I didn't tell them her address.\" His voice is uncomfortable; he should have told the Lorders he works for everything, even that, and he knows it. Is there some part of the Ben I knew inside him still? Can he be reached?\n\nWe're at the door to Mac's house now; the lights are back on, and the door is held open. Ben pushes through and drops me on the kitchen floor. At Tori's feet.\n\nA golden streak rushes past: Skye. She jumps up on Ben in excitement, licking his face. He tries to push her off, but she's not having it.\n\n\"That's Skye. Your dog,\" I say.\n\n\"My dog?\"\n\nSkye barks as if to say yes.\n\n\"Your mum and dad gave her to you when she was a puppy. Look, Ben\u2014your mum was an artist; she made that owl sculpture. Made it for me.\"\n\nHis eyes start to follow my gesture to the owl on the fridge, but then Tori pulls me up by the hair and starts dragging me across the floor into the front room. I scream, and Skye flips around, growling, starts to leap at Tori, but Ben grabs her collar. \"Down,\" he says sharply, and she's confused.\n\n\"Let Kyla go,\" he says to Tori, and she pauses, surprise in her face. \"Until I get rid of the dog.\"\n\nTori lets go of my hair, and my head thuds painfully on the floor. She smiles, but her eyes are full of twisted hate. I was right, wasn't I? She remembers me. Did the Lorders think she was of more use with revenge to drive her?\n\nBen pushes Skye into the hall, shuts the door. She starts whining mournfully on the other side to get back to him.\n\n\"Aren't they here yet?\" Ben says to Tori.\n\n\"No. Not yet,\" Tori says, and something hides behind the glee in her eyes, some lie. She wants to deal with me all by herself.\n\n\"Are you waiting for reinforcements?\" I say. \"She hasn't called anyone in. They're not coming.\"\n\nBen frowns, looks at Tori.\n\n\"Don't listen to her,\" she says, and slaps me so hard on the side of the face that tears come to my eyes. I blink furiously.\n\n\"You remember me, don't you, Tori. You want to hurt me, don't you?\"\n\n\"I don't just _want_ to, I'm going to.\" She pulls a knife out of her pocket. \"You know I'm good with knives.\"\n\n\"You killed a Lorder with a knife once. I can't believe you could go from that, to this. Don't you remember that day we attacked the termination center, and Emily, the Slated who died?\" I slip the ring off my finger, throw it at Ben. He catches it. \"That's Emily's ring, the pregnant girl I told you about at the college. Everything I told you that day is true, Ben, and Tori knows it. She was there.\"\n\nTori looks at Ben as he reads the inscription on the ring. \"She's lying. She could have gotten that ring anywhere.\"\n\n\"You hate the Lorders, don't you, Tori? For what they did to you: Slating you, then taking you to a termination center. The Lorder who pretended to rescue you\u2014do you remember him, what he did to you? Is working for them worth it, just to get back at me? Or is it to be with Ben: That's it, isn't it. You always wanted what you couldn't have. You're just a jealous little girl.\"\n\nTori starts to advance on me with her knife; I shrink into the wall. Too much?\n\n\"Tori, wait,\" Ben says. \"Leave her be a minute.\"\n\n\"What?\" She scowls, turns to him.\n\n\"You do remember her, from before.\" A statement, not a question. \"Explain.\"\n\nShe looks between us\u2014wary. Trapped.\n\nIs it working? My eyes find the clock on the mantel: 6:02. The transmission has started! Delay and distract. I've no doubt she'll kill me or, if she doesn't, eventually they'll make the call and more Lorders will come and they will. I'm detached from it. I don't care. What is there to live for? If the transmission is made, I'll welcome death.\n\n\"I don't know what they've told you, Ben. But Tori is here for revenge: nothing else. Because Lorders followed me to her, arrested her, hauled her away.\"\n\n\"And you never told me!\" she says, and hits me hard again across the face, this time with the flat side of the knife in her hand, and the cutting edge bites in and cuts my cheek. Tears spring to my eyes.\n\n\"Oh, is that why you're so miffed? Because I never told you Ben was alive?\"\n\n\"Tori, is this true?\" he asks.\n\n\"Ben, I\u2014\"\n\n\"Why haven't you told me this before?\"\n\n\"Ben, think for yourself,\" I say. \"It's lies, all of it. The Lorders and Tori have been filling you up with lies, to make you do what they want. All those people dead, all because of you.\"\n\n\"No,\" Ben says. \"You're the traitor! It's because of you and Aiden they died. You twisted the truth and turned them against the government. We had no choice.\"\n\nThere is thumping behind\u2014Skye is throwing herself at the hall door.\n\n\"Not even Tori believes that; she just doesn't care.\"\n\nHe looks at her.\n\n\"Shut up,\" she shouts. She lunges with her knife; I'm on the floor, up against the wall, weaponless. Limp and lifeless already; where has my fight gone? This is it. This is really it.\n\nA foot swings out; the knife flies through the air. Ben. He's kicked the knife out of her hand.\n\n\"What have you made me do?\" he screams, and I don't know if he means stopping Tori from killing me, or Tori's lies and what they led to. Or if he even knows.\n\nTori screams in fury. She reaches behind her back to a holster. A gun is in her hand. She raises it at Ben.\n\nA crash: The thin hall door has given way.\n\nA flash of fur\u2014Skye\u2014jumps between them.\n\nThe gun goes off and Skye yelps, falls, red in her golden fur. Tori stares, disbelieving.\n\nMy fight is back. I'm on my feet, and I take the biggest swing I've ever taken to punch Tori full-on in the face. She drops the gun, falls to the ground. Unconscious. And then the gun is in my hand and pointed at Ben.\n\nWho am I kidding? I put it down.\n\nBen is holding Skye, pushing at the red spreading in her fur. It's her shoulder. I grab a curtain tie off the wall, tie it tight around and around to try to stop the bleeding, and she's whimpering, but still licking Ben's face. He's shaking.\n\n\"Ben? Do you remember Skye? Remember!\" And then he's crying, convulsing, and I'm holding both of them.\n\nThat's when the front door is kicked down. A man steps through.\n\nNico?\n40\n\nI TWIST, DIVE FOR TORI'S GUN, BUT THEN THERE IS PAIN\u2014A sudden explosion of agony in my head so severe, I drop and curl into a ball.\n\n\"This is why we track the trackers,\" a woman's voice says. \"They really can't be trusted to get anything right. Young people today have no sense of focus or purpose.\"\n\nFootsteps approach. They stop; a hand strokes my hair. The pain is so intense, it's all I can do to open my eyes and look up at the ones staring into mine: pale blue irises. Nico. His eyes used to mesmerize me, hold power. Not any longer.\n\n\"Poor child. You see, over there?\" He gestures to the front door, and my eyes follow. It's Astrid, and in her hands is a device. \"Once a Slated, always a Slated. Just key in the brain chip number, hit a button, and bingo: pain. Or even death.\"\n\nTori is stirring on the floor. \"Allow me a small demonstration,\" Astrid says, and taps at the machine. Tori screams, convulses, then lies still.\n\nAs if to emphasize the point, Astrid taps again; a new spike of pain explodes in my head. My vision goes fuzzy. All the Lorder talk of second chances for Slateds: all lies. We're still in a prison. They can strike us down whenever they want to.\n\n\"Enough for now,\" Nico says. \"She'll pass out.\" He lifts me up onto the sofa. Ben is held between two Lorders, and Tori and Skye are unmoving on the floor.\n\nThe pain subsides a little, enough that I can turn my head, fix Nico once again with my eyes. I swallow, try to speak with a mouth that is thick and dry. \"Why are you here? You hate Lorders.\"\n\n\"Ah, my dear, love and hate have nothing to do with winning. I was always with Astrid. The side of strength.\" He leans over me, close, and I try to pull away but can't convince muscles to respond. He kisses my cheek.\n\nI fight to think through the pain. Is Nico in some sort of self-serving alliance with Astrid, or was he actually a Lorder all the time? But Nico ran from Coulson's Lorders when they tracked me and attacked the AGT; Coulson _was_ hunting Nico. Or was that just an act? If Nico really _is_ a Lorder, that may explain why all the attacks Nico and Katran planned just fizzled out: sabotage.\n\nThe clock over the mantel says 6:08. The transmission is well under way! I have to keep them talking, keep them from stopping it.\n\nWith great concentration, I manage to turn my head to Astrid. \"It was you who set up their taking me when I was ten. Wasn't it.\"\n\nShe smiles, and it is a grandmotherly, gentle smile. Shivers run up my back. \"Of course it was, my dear. You had a glorious purpose on Armstrong Memorial Day. Shame you didn't fulfill it.\"\n\nA glorious purpose? That of suicide bomber. _Concentrate; delay her._ \"It was no accident I was assigned to that family, that I was there that day.\"\n\n\"Of course not. It just took a little meddling to arrange it.\"\n\n\"How could you do that to Stella? Take me away from her?\"\n\nHer face goes hard. \"My daughter dared to hold information over me, threatened to tell; she had to learn. And then having you back with her in Keswick, without telling me?\" She shakes her head in disgust.\n\n\"So you really did have the prime minister and his wife killed, all those years ago.\"\n\nShe smiles. \"First rule of politics: Eliminate the opposition.\"\n\n\"How did you know I was at Stella's?\"\n\nShe shrugs. \"It was obvious Stella was hiding _something._ A little information, and the conclusion was apparent.\"\n\n\"From Steph. My green eyes.\"\n\nShe raises an eyebrow. Amused. \"Indeed. And it didn't take long to work out it was also you and that Finley at the orphanage that day.\"\n\nNo. She knows about Finley? She must see the horror on my face. Her smile widens.\n\nI'm going cold inside. If she knows Finley was there, that he helped me, he's dead. And all these things she is saying to me; I'm not leaving here alive, either. None of us are. Not with all the things we know.\n\nBut there is still one thing I want to know more than anything.\n\n\"But why me? Who am I? _Why?_ \"\n\nAstrid laughs. \"That is quite enough family reunion time, dear. Now, tell me. Where is your camera?\"\n\n\"My camera?\" I frown. \"I don't know.\"\n\n\"This is the price of failure to cooperate.\" Her fingers move to the device she holds, and I brace myself for a slam of pain that doesn't come. But there is a cry to the side, and I turn.\n\nBen is curled up in a ball on the ground.\n\n\"Now answer my question.\"\n\nI think fast. Does it matter? It's just a backup copy. It is 6:12; the transmission should be nearly over.\n\nShe raises her hand to the box again.\n\n\"Wait. Ben took it from me; he must still have it.\"\n\nShe nods at one of the Lorders, who goes through Ben's pockets, then holds up my camera.\n\nThe back door opens; there are footsteps in the kitchen?\n\n\"Ah, your other friends are arriving, at last,\" Nico says. The door from the kitchen opens. More Lorders, dragging two prisoners along with them. They throw them on the floor.\n\nMac, and Aiden. Both of them bloody and beaten, Aiden's arm hanging at an angle that is wrong.\n\n\"No!\" I sag back.\n\n\"Yes, I'm afraid we stopped them; no movie premiere for you tonight. And we'll round up all the insurgents who appear on your little production as well. We've got some of them in custody already. But don't worry, they won't be in custody for long.\"\n\n_They'll be dead._\n\nSo will I.\n\nThe Lorder with my camera takes it over to Astrid. She puts down the device she was holding, her box of pain, to look at the camera.\n\nIt doesn't matter anymore, does it?\n\nI fill myself with every bit of resolve I can find inside, every reserve of strength, every fragment of AGT training. One last flood of adrenaline before it all ends.\n\nTori's knife, the one Ben kicked out of her hand. It lies just out of sight, under the edge of a chair near Astrid.\n\nI dive for it and for her.\n41\n\nI HOLD THE KNIFE AGAINST ASTRID'S NECK, POSITION HER body between them and me. \"Drop your weapons,\" I say to the Lorders. They look at her.\n\n\"Do it,\" she hisses, and they hesitate, start to bend down, to put guns on the floor.\n\n\"Don't bother,\" Nico says, walking slowly toward Astrid and me, his gun still in his hand and pointed at us.\n\n\"Don't take another step!\" I say.\n\nHe stops. He smiles, amused. \"Really? Don't forget I know you, Kyla or Rain or Lucy or Riley or whoever the hell you want to be today. You can't kill anybody. Can you?\"\n\nThe moment is stretching, each second a slow eternity. After everything, is it this moment, this one defining, ending moment of my life? If I kill her, I'll die. If I don't, I'll die. She deserves it, she deserves it more than anyone I can imagine in this world, except maybe Nico. Push the knife into her neck. Cut her throat. Watch blood spill down her body; revenge for so many.\n\nBut I can't do it. I can't be like them.\n\nAnd he knows it.\n\nThe knife loosens in my hand. I swallow.\n\nNico smiles and steps closer; he takes the knife.\n\nAstrid pushes away from me, her face twisted with fury; she reaches for her box of pain. \"You never could do what I wanted you to, could you? No more.\"\n\n\"Let me take care of her outside,\" Nico says to her. \"It's about time.\"\n\nShe smiles, puts the box down again. \"As you will. But make it quick. We've got to get out of here.\"\n\nNico slips an arm across my shoulders, gently pulls my hair back. Kisses my cheek. \"We have unfinished business, you and I.\"\n\nThere is a scuffle behind Nico; Aiden cries out as a Lorder twists his injured arm behind him.\n\nNico opens the front door, pushes me out into the night. I trip on the step, sprawl onto the muddy ground in the cold rain.\n\n_Run._\n\nI glance back; he stands there. Watching and waiting.\n\nIt's what he wants me to do. He wants me to run, doesn't he? So he can shoot me in the back.\n\nI stand up. Face him down, like Florence did at All Souls.\n\nHe shrugs, raises his gun.\n\n\"Good-bye, Rain. It's been fun.\"\n\nAnd I stand there, stare back at him. He's waiting for me to cry, to plead. I won't do it.\n\nIt's a funny thing. Earlier I thought I was ready to die, but I'm not. Despite everything, I want to stay, to breathe this air, to _feel_ even if all there is to feel is pain. I'm fighting tears that threaten, fear that trembles through my body as he slowly points his gun straight at my heart. He smiles, and then\u2014\n\n_BANG!_\n\nAnd I flinch, anticipating impact, pain, being pushed to the ground, but instead am full of confusion.\n\nNico has fallen? It is Nico clutching at his chest, _red red red_ spreading. Nico dying.\n\nFootsteps approach.\n\nIt's Coulson. Gun in hand, looking at Nico at his feet. But Coulson is a Lorder; Nico is with the Lorders now. Isn't he? Other Lorders run in behind him.\n\n\"I'm not dead,\" I say.\n\n\"Correct,\" Coulson says. He opens the door, looks back. \"Come on,\" he says. Dazed, I step around Nico's now-still body, walk back into the house behind Coulson.\n\nAstrid's eyes go round with shock. Her Lorders aren't looking happy, either, not that it is easy to tell with Lorders. But Coulson is a Lorder. Aren't they on the same side?\n\nCoulson gestures at the other Lorders in the room. \"Get out,\" he says. They look at Astrid. Indecision plays on her face.\n\nMore Lorders step in behind us.\n\n\"Do as he says,\" Astrid says, and they are ushered out.\n\nCoulson checks the room, holds an arm out the door. A gesture.\n\nIn walk two people I was never more surprised to see: Dr. Lysander? And with her is Prime Minister Gregory.\n\nDr. Lysander rushes to the injured. Checks Ben, Aiden, and Mac. Skye. And Tori, too: But this time Dr. Lysander shakes her head. She closes Tori's eyes. Tori . . . dead? Another shock that I can't take in, can't believe. \"Paramedics are needed for the others,\" Dr. Lysander says. \"And a vet.\" Gregory nods, and a Lorder speaks into his collar. They're not going to be killed, but helped?\n\n\"So glad you came along, Prime Minister; lovely to see you as always,\" Astrid says to Gregory. \"But things were well under control.\"\n\nGregory raises an eyebrow. \"Really? What, exactly, is it that is under control? What operation were you conducting here without my knowledge? Did you know anything about it?\" he says to Coulson.\n\n\"Not a thing through any _official_ channels. Luckily, my unofficial sources are rather good.\"\n\n\"Well. If my head of security knows nothing _officially,_ and I know nothing, how should I take this?\"\n\nAstrid is pale. \"I learned of this plot to discredit the glorious Central Coalition with lies. They were attempting to hijack our television transmission and broadcast across the country tonight. I've been protecting you, on a need-to-know basis.\"\n\nSo Lorders use that phrase, too.\n\nGregory shrugs. \"I may not need to know, but if Coulson doesn't know, how can that decision be made?\"\n\nShe starts to speak again, but he holds up a hand.\n\n\"Be silent. I'm reserving judgment until I learn more. I've decided that I _need to know._ \" His voice is icy, and Astrid is growing paler, but as much as I enjoy her discomfort, what has this got to do with us? They're all Lorders.\n\n\"You see, dear Astrid, I learned a few things I think I did need to know. Dr. Lysander here\u2014she was a friend of my daughter's, did you know that?\u2014came to me with some very interesting information. She was very _persistent_ about getting in to see me, and when she told me about one of your special projects, I could see why. Slating is a legally sanctioned criminal punishment to be applied only according to due process of law, as you well know. Not to orphans under the age of legal responsibility.\n\n\"And then we unearthed some information about your _unofficial_ training camps. This is two of them?\" He gestures at Ben, and Tori's body. \"Selected for special abilities, subjected to experimental procedures. Trained and twisted.\" He shakes his head.\n\n\"All within my ambit as JCO,\" Astrid says.\n\n\"I doubt even you believe that. And then we've been piecing some more things together. And worked out some of what you've done to my daughter. And my granddaughter.\"\n\nGregory turns. Why is he looking at me? He's blond, of course, though gray-streaked now, but up close I see something I haven't noticed before when I've seen him on TV or in photos: his eyes. Green eyes. The same shade as mine. Everyone is looking at me.\n\nHis granddaughter? _Me?_ No. It couldn't be.\n\nCould it?\n\nA siren has approached; paramedics enter. At Dr. Lysander's direction, they take Skye and Ben away, and Tori's body. Aiden's arm is broken, but he refuses to leave. They strap his arm to his chest, check Mac's injuries, then leave.\n\n\"This is ridiculous,\" Astrid says. \"They are traitors and should be dealt with as such.\"\n\n\"That may be. I'm still deciding. For now, I want to watch this transmission you stopped.\"\n\n\"It's in my camera,\" I say. I point to it on the floor, where it fell when I tackled Astrid.\n\nCoulson takes it, checks it, and hands it to Gregory. My _grandfather_?!\n\n\"Are we ready now? Shall we?\" He projects it on the wall.\n\nWe all watch it silently; this time, I don't look away. I stare at Florence's eyes just before she dies, standing there, facing them down. Did she feel like I did with Nico at that moment?\n\nAll are silent when it ends. Gregory finally turns to Astrid.\n\n\"Astrid Connor, your actions have been unacceptable. Further investigations are necessary.\" He gestures at Coulson. \"Take her away, and then leave us.\"\n\nAfter they're gone, the door shut behind them, Gregory turns to me. \"Can you record on this thing?\" he says, holding out the camera in his hand.\n\n\"Yes.\"\n\nHe hands it over. \"Get ready.\"\n\nI set it to record, hold up the camera. Hands surprisingly steady.\n\nHe begins.\n\n\"This is Merton Gregory, your prime minister, head of the Central Coalition Government. I've learned some news that has disturbed me greatly.\n\n\"Many of you may know that in the riots over thirty years ago, one of the students who was sentenced to execution was my daughter, Samantha Gregory. At the time I was deputy to then Prime Minister Armstrong; he offered to intervene and pardon her. I didn't allow him to save her, convinced the only way forward from the grip violent chaos had on our country was to apply the law in all cases. This is something I've regretted my entire life, and is part of the reason why I always protected the rule of law at all costs when I became prime minister myself\u2014if I didn't, her loss was meaningless. And I have been willfully blind at times in ways I now regret.\n\n\"I have recently learned that my daughter was not executed, but this was not an act of leniency or kindness. There are more details I have yet to unearth as to where she was taken, or if she even still lives. But I have discovered I have a granddaughter I knew nothing of, a girl whose only crime has been being related to me, the punishment given for this beyond anything the rule of law could condone.\n\n\"You're about to watch some very difficult scenes. I'm sorry, but you need to know.\n\n\"In light of what you are about to see, I feel I have no choice but to resign as prime minister. The government will be dissolved, and an election called. Change is long overdue. The Lorders served their purpose at the time; their time is over.\n\n\"All right, that'll do. I'm done,\" he says.\n\nI hit stop, lower the camera. My eyes find Aiden's. Is this really happening?\n\nGregory turns to Mac and Aiden. \"Now\u2014can you get this out tonight before I change my mind? And we better use your hijacked system. I'm not sure this would get past the Lorder censors, even with my direct order. They might have me committed.\"\n\n\u2022 \u2022 \u2022\n\nThat evening, Mac does quick checks and repairs the damage to his transmission equipment that Astrid's Lorders inflicted when arresting him and Aiden. Dr. Lysander draws me aside, bandages the cut on my cheek.\n\n\"Tell me: How did you find out who I am?\"\n\n\"Deduction, and guesswork.\" She sighs. \"Really, I'm embarrassed it took me so long.\"\n\n\"Tell me.\"\n\n\"Deduction: I was thinking about everything that has been done and manipulated in your life; the classified DNA set in the system so none could trace it, set by Astrid as it turns out. Who you really were had to be an important part of the puzzle. And guesswork: how I always thought I knew you.\"\n\n\"You said I reminded you of a friend, one who died.\"\n\n\"Not just a friend.\" She pulls on a chain around her neck, and out of her clothes comes a gold locket. She opens it. \"Inside here? A lock of hair. From a girl I loved years ago, who was meant to have been executed in the riots. Gregory's daughter, Samantha. When you cut your leg after your recent visit, on impulse I swabbed the blood you left behind for DNA. Later, feeling foolish for doing so, I compared DNA between that and this lock of hair. However she survived, Sam is your mother.\"\n\n\"And you went to Gregory and told him about me?\"\n\n\"Just so.\"\n\n\"Where is my mother? Is she still alive?\"\n\n\"I hope so. Gregory is working on that.\"\n\n\"But how did he link us to Astrid?\"\n\n\"Thanks to you. Telling me that the orphanage you visited was in Cumbria. It didn't take Gregory long to link first the orphanage, then his daughter's disappearance, to Astrid. She must have seen this opportunity with his daughter, Sam: the ultimate way to discredit Gregory. He was the obvious next prime minister after Armstrong; Astrid wasn't in position to take power yet when she arranged Armstrong's assassination. She was a long-range planner.\"\n\n\"I don't understand. What use was Sam to Astrid?\"\n\n\"She probably thought at the time that she'd use Sam when she was ready, to make it look as if Gregory broke the law to save his daughter. Then later on, when you appeared, she came up with an even better plan: having Gregory's own granddaughter make a sham of Slating, and to assassinate both him and Armstrong's daughter at the same time. How far back she was putting this in place we do not know; it must have been at least since you were ten, when she arranged through Nico for the AGT to take you.\"\n\n\"If her plans had worked that day, Lorders wouldn't know who might turn\u2014which Slateds were safe or dangerous.\"\n\n\"Astrid's views are notoriously hard-line, and she would have been the obvious next prime minister if Gregory had been killed. She prefers the death penalty to Slating; a clean sweep of existing Slateds wouldn't have troubled her. But you thwarted her plans.\"\n\n\"Because I ran back to save you. I wasn't there, next to Gregory and the others, when they meant to set off the bomb hidden on my Levo.\"\n\n\"Yes. And since then I've learned more from Gregory. That by then Coulson was suspicious of you, who you were: He'd noticed irregularities in your records. When the bomb went off at your house, he took the opportunity to fake your death, to stop any possible AGT interference while he looked into it.\"\n\n\"But how did you find us here today?\"\n\n\"Gregory has been having Astrid watched. When she came south in force, we knew something big was up. We closed in.\"\n\n\"Just in time.\"\n\nDr. Lysander smiles. \"Yes. Thankfully, just in time.\"\n\nI turn it all over in my mind, but keep coming back to two things. I was just a baby when I was taken from a mother I never heard of until today. Where is she? Is she even alive? And then there is Ben.\n\n\"What is going to happen to Ben?\"\n\n\"I don't know. He has committed crimes, though under coercion perhaps.\"\n\n\"Where is he now?\"\n\n\"He's been taken to the hospital for assessment and observation.\"\n\n\"When can I see him?\"\n\n\"I'm not sure that is wise. For either of you.\"\n\n\u2022 \u2022 \u2022\n\nMac has added Gregory's new introduction to _Need to Know._ It is nine P.M., three hours later than planned, when it hits every television, viewing, and vid screen in this country and others. Can so much really have happened in such a short time?\n\nI stand, awkward and uncertain, next to Aiden while it runs; the pain from his arm is showing on his face, but his eyes are gleaming. \"We did it, Kyla. We really did it.\" He smiles, but his eyes slide from me to Gregory and back again.\n\nWhen it's over, Gregory glances at Aiden and Mac. \"Leave us alone for a moment,\" he says, in a voice used to being obeyed.\n\nBut things have changed. They look to me.\n\n\"It's fine. Go,\" I say, staring at Gregory while they leave. My grandfather; a stranger. Someone I used to hate with every beat of my heart for what he stood for, yet someone who unexpectedly saved my life. Saved us all.\n\nHe raises an eyebrow. \"Do I pass inspection?\"\n\nI shrug. \"I don't know. There is good and bad.\"\n\n\"And you're not sure which is the greater.\"\n\n\"Exactly. Are you really resigning?\"\n\n\"Isn't that what I said? Yet you seem skeptical.\" He looks pleased.\n\nI shrug. \"Maybe this is just a way to get out of blame. Defeat Astrid, blame things on her, rebrand the party and start over again.\"\n\n\"Politics loves a scapegoat.\" He shrugs. \"That'd probably work. You've got a suspicious mind. Maybe you got that from me.\"\n\n\"And?\"\n\n\"No. I'm done. The country can start over again without me. I'm not proud of things that have been done in my government's name. I'm not proud of things I've done myself. I can't change the past, but I will do what I can now to ease the political changes. But what I really wanted to say to you now is this: I'm sorry.\"\n\n\"What for, specifically? Even if you leave Astrid out of things, it wasn't her who had me Slated, beaten, and threatened. It wasn't her who made kids disappear from my school for no reason. The list is long enough without her; but if you add her in and it gets a whole lot worse, who was in charge of her?\"\n\nHe flinches. \"Don't worry. I don't expect a hugs-and-flowers big family reunion. I don't expect you to forgive and forget. But there is one thing I will do for you. For both of us.\"\n\n\"What is that?\" What can he possibly offer to do for me, now, that will mean anything.\n\n\"I have a promise to make you. I'll find my daughter, your mother. One way or the other, I'll find her.\"\n\nHe reaches out, grips my hand, and I don't pull away. So many times I've thought, _This is me, I know it all._ And then there is another revelation. But Sam really _is_ my mother\u2014DNA doesn't lie. Dr. Lysander doesn't, either. I fight the tears that threaten: not here, not now.\n\n\"Where is she?\"\n\n\"I'll find her.\"\n\n\u2022 \u2022 \u2022\n\nWhen I get back to Mac's house, Aiden waits out front. Alone.\n\n\"Shouldn't you be on your way to a hospital to get that arm seen to properly?\"\n\n\"Probably. I had to see you first.\" He reaches his good hand to my cheek, and I lean against him, into his warmth. So glad he is still alive, that we both are. And suddenly too full of everything that has happened to want to be anywhere else.\n\nAiden tightens his good arm around me, and murmurs into my hair. \"I heard what you said before to Dr. Lysander.\"\n\n\"About what?\"\n\n\"About Ben. About asking to see him.\"\n\nI pull away. \"I have to.\"\n\n\"After everything he has done?\"\n\n\"It's not him. They've made him like that. You don't understand.\"\n\n\"Then make me.\"\n\n\"He's fighting what they've done to him.\"\n\n\"How do you know?\"\n\n\"He saved my life tonight: kicked a knife out of Tori's hand.\"\n\n\"Then I'll thank him for that. But does one good deed wipe out all the others?\"\n\nI stare back at Aiden, and I can't answer. Does Gregory's one good deed wipe out all his others? But it's not the same thing. He's had his free will; Ben hasn't.\n\n\"Kyla, there is one more thing. The other day, when I said I loved you. I said how can you love somebody when you don't know all of them? And you said, then how could someone who was Slated ever love or be loved.\"\n\n\"And?\"\n\n\"I do know all of you. And I don't mean every memory you've lost: I know who you are, inside. Despite everything, how you could never deliberately hurt anybody. How brave you are, how fiercely loyal, and all the little insecurities, fears, and stubbornness as well, and I love all of you. Can you say the same about Ben?\"\n\n\"Yes,\" I say, but doubt gnaws inside, and Aiden knows it. \"I don't have any choice. I can't abandon him; he hasn't got anybody else. Not after everything we were to each other.\"\n\nHis hand touches my shoulder.\n\n\"Everything you _were_ to each other. That is past tense. Let me know when you're ready for the present, or maybe even the future.\"\n42\n\nEVERYTHING HAPPENS VERY QUICKLY AFTER THE BROADCAST.\n\nPrime Minister Gregory makes his resignation official, as promised. Amid public outcry and international pressure, Parliament is dissolved, and elections called. And it is almost like Aiden always said it would be: Once everyone really knew what went on, they said, _No, no more._ And the Lorders were no more.\n\nOf course it wasn't as easy as that. There were high costs on both sides\u2014pitched battles in some places, like Cumbria, where Astrid's followers refused to accept they weren't in charge anymore\u2014but the cost wasn't as high as living with constant fear under the Lorders. They did it; MIA really did it. DJ, Aiden, and an international council have put a provisional government in place pending elections, and new political parties are forming, setting up candidates.\n\nGregory is still hunting for Sam, my mother, but now that months have passed I'm starting to accept he may never find her. Mum and Amy are okay; they hadn't been found by Astrid's Lorders, and I'm staying with them for a while back in our newly repaired house. Skye survived and is here, being nursed back to health and spoiled by all three of us. Slating is banned, and Dr. Lysander has been busy removing Levos and brain chips from Slateds, including mine.\n\nBut while part of me is rejoicing in the changes that have happened and are yet to come, more is in limbo. Licking my wounds and waiting for this one day.\n\n\u2022 \u2022 \u2022\n\nDr. Lysander sits opposite Ben and me at her desk. \"There are no guarantees; we don't know who you were before you were Slated.\"\n\n\"I know, I know; Lorders destroyed my records, none have been found,\" Ben says. He holds my hand tight.\n\n\"We don't know who you were, but do we know enough?\" I say. \"You don't have to do this.\"\n\n\"I want to.\"\n\nDr. Lysander goes through her list of cautions, not for the first time. Results of memory adjustment are not predictable; he may have memories he doesn't want and not the ones he does want returned; there is a risk of brain damage, seizures, and death. While simple cases of readjustment have been successful, his case is unpredictable due to the multiple procedures to which he'd been subjected.\n\n\"Is that everything?\" Ben asks.\n\n\"Are you sure you wish to proceed?\" she asks.\n\n\"Yes. Can Kyla come?\"\n\n\"I wouldn't recommend it, but if she wishes to do so, it is your choice.\"\n\n\"I'll be there,\" I say, unwilling to let go of his hand. Despite the things he has done, it was the Lorders\u2014their procedures and manipulations\u2014that led him to betray us. I can't erase the things Ben did: I still wake screaming late at night, visions of Florence and the others dying at All Souls haunting my dreams. And I still can't shake the _if only_ s from it all. If only Aiden hadn't brought Ben there; if only I'd tried harder to get through to Ben. If only I'd recognized what was about to happen, and stopped him.\n\n_If only._\n\nBut it wasn't _Ben_ who betrayed us; it was the Lorders' creature. After all that has happened to me, all the identities that have been forced or taken, I can understand that better than anyone. I can't abandon him while there is any chance of calling him back, no matter how torn I feel. I won't.\n\n\u2022 \u2022 \u2022\n\nThey get him ready. He's on one of those beds that hugs you like when I had IMET; they're checking things, monitors, wires, IV drugs, and a scanner all around his head. All the while he grips tight to my hand.\n\n\"What if I sneeze?\" Ben jokes. He'd found it endlessly funny that the microsurgery goes through his nose.\n\n\"You know you can't; you'll be immobilized. Almost paralyzed except for speech.\"\n\nWhen the drugs take hold, his hand slackens. \"I'm still holding it,\" I tell him. \"Everything is fine.\" But I'm afraid.\n\nThese months have been difficult. Once Ben really understood what had been done to him, how he'd been subjected to procedures and manipulated to be a Lorder agent, he'd been in a dark place. And both of us struggled to come to terms with Tori's role\u2014that she retained her memories, but still chose to act for the Lorders\u2014and her death. Ben only started to come back to life with the hope of this: experimental microsurgery to give him back what was stolen.\n\nDr. Lysander meets my eyes over a sea of equipment, nods once. \"All right then, Ben. Shall we begin?\"\n\n\"No, I changed my mind. Just kidding! Go for it.\"\n\n\"All right. First I am removing your chip; this is routine.\" So no chance of anyone activating it to cause him pain, or kill him like Tori was killed, ever again; mine was taken out weeks ago.\n\nDr. Lysander peers into control screens, remote operates using the scanner and microscopic robotic tools. Time passes slowly; seconds feel like minutes.\n\n\"Your chip is removed,\" she finally says. \"Is everything all right?\"\n\n\"I'm fine, having fun. Carry on,\" Ben says.\n\n\"Now tell me what you experience.\" She'd explained that different neuronal areas of the brain will be microstimulated as she navigates his memory storage areas, reattaches broken neural connections according to his responses.\n\n\"Okay, here goes,\" Ben says. \"Blue, the blue sea. Soft fur\u2014a puppy! It's Skye; I think it is. Fish: I smell fish and chips. A woman, I see a woman. My mother?\" he says, and starts describing her, but going by what he says it's not his mum from when he was Slated. Then his voice changes\u2014\"Mummy? Mummy?\"\u2014a high-pitched note of panic, a child's voice.\n\n\"You're okay, Ben,\" I say. \"I'm here.\"\n\n\"Who's Ben? I'm Nate. Mummy?\" Then, \"Kyla?\" he says, back to his voice again. \"I remember my mother!\"\n\n\"One up on me there, then.\"\n\n\"This is good,\" Dr. Lysander says. \"Carry on describing.\"\n\nHe is quiet.\n\n\"Ben?\" she says.\n\n\"I'm still here. Things are zipping through too fast to tell you: people and places. Sometimes like I'm there, sometimes like I'm looking at a photograph.\"\n\n\"Memory can be like that. All right, I am reattaching the final deep links; this is the tricky bit.\"\n\n\"Great to know.\"\n\n\"Describe, Ben.\"\n\nWords are spilling out; people and places are garbled and quick, and then . . . \"Kyla?\"\n\n\"Yes?\"\n\n\"At Group. I ran in late, you were sitting there. The new girl. I remember! The first time I saw you, beautiful, gorgeous girl.\"\n\nAnd I know he can't feel it or squeeze back, but I'm holding his hand tighter, tears threatening; it's working. He remembers me.\n\nThen he gasps. \"Pain, hot pain, in my side.\"\n\n\"Yes, you have a scar, an old knife wound,\" Dr. Lysander says. \"And what else? Ben? Answer me.\"\n\n\"No,\" he says, his voices changed, angry. \"No!\"\n\n\"Ben?\"\n\n\"Ben?\" she says again.\n\nHe is silent.\n\n\"Ben?\" I try. \"Nate? Are you all right?\"\n\n\"Dandy. I'm dandy, thanks for asking.\" And with his words I can breathe again, but his accent\u2014was it changed? Into something more London, less country.\n\n\"We're nearly done here,\" Dr. Lysander says.\n\nBefore long, the scanner is pulled away, the microtools removed. One tiny drop of blood under his nose is wiped away; that is all.\n\nHis eyes are closed, sedation increased; he will sleep now.\n\n\"Go home, Kyla,\" Dr. Lysander says. \"He goes into recovery now for monitoring while he sleeps. It will be a day or two before we know how it went.\"\n\nBut I stay. With Ben\/Nate\u2014whoever he is, he remembers me now.\nEPILOGUE\n\nIT'S LATE SUMMER. I INSISTED ON COMING ALONE, ACROSS the fells. Skye bounds along beside me, still with a limp but it doesn't slow her down. And as I walk, I think. So much of what has motivated me for so long has been trying to find out who I am, where I come from. Each new revelation knocked down walls in my mind, but came at a cost. Will today give any closure?\n\nEveryone is searching for something or somebody. The bit they haven't got to complete them. Why should I be any different?\n\nMum's son, Robert, hasn't been found, but she is still looking, with the help of MIA\u2014now a government-sanctioned agency, and Mac and Aiden's full-time mission.\n\nMum refused to run for prime minister, despite all who wanted her to. Gregory, who I see now and then\u2014no matter what he was, he is my grandfather, and so much of how things went better than they could have at the end was down to him\u2014said those suited for power don't want it, and those who want it, aren't. He didn't say which category he was in. Anyhow, some new guy who wanted power is in charge, a whole new government has been voted in, and DJ and his friends are still here to keep an eye on things for a while.\n\nWill everything be all right now? Time will tell, but already I'm not sure everything is good. Like all the new technology flooding in from outside now that the borders are open, all the endless Internet channels, the portable devices and plug-ins so you're always linked in. The traveling curious from other nations rushing to see how quaint we are before we become just like them. Gregory says that is why the world stepped in: not to save anybody, but to have a new market to sell their toys.\n\nWith repeal of the YP laws, I'm now sharing a flat in Keswick with Madison. She was in Astrid's slate mine prison like Len thought she might be, and released with all the other illegally held prisoners. Finley had gone into hiding not long after I left Keswick; he came out when it was safe again. Madison's not the same, but with Finley's help, getting better all the time.\n\nI see Stella once or twice a week; a fragile trust is beginning to grow between us. She is slowly coming to terms with all that Astrid did; with Dad not being behind my disappearance. With how much she had it wrong. She's had trouble accepting my refusal to let Dr. Lysander try to return my memories, but they've been mucked around with enough. From now on, nobody but me has any say in what I choose to remember, what I choose to forget.\n\nFor now I'm working with Parks as a fell checker. Len is on the list of those missed; he died in the struggle against Astrid's faithful. Being alone in the high places above the world in all weather, the mountains under my feet that have been, are, and will be long after I'm gone, I feel a release never felt anywhere else; it is really why I came back to Keswick, despite Mum and Amy. It is the only place I can think about _anything_ and not be overwhelmed.\n\nI still might go back to school myself, then get into teacher training one day and be an art teacher, like Gianelli, but not now. The happy little faces are too much for me, after all the Slated children from Astrid's orphanage experiment were found: dead. Killed by Astrid's minions to hide what they'd done, but found before the bodies could be destroyed.\n\nAt least I know Edie survived, and that Ben never told the Lorders where she was; their house was empty that day because they'd heard what happened at All Souls and bolted in a hurry, gone into hiding. When they reappeared, I went to see them. Edie said I could keep Murray, that I was more alone than she was.\n\nThat is one place where I know Ben told the truth after all the lies that followed. He kept up an act long enough to get out of the hospital, and then some of the truth came out. He'd committed crimes worthy of Slating before we'd even met, to add to the ones after at All Souls. He said the only time he was ever happy was when he was Slated.\n\nAnd then he stole a car and disappeared. No one knows where he has gone. All I know is he doesn't want to be with me. For whatever reason, good or bad, at the end that is the truth.\n\nShould I have seen it coming? I could never really hurt anyone, Slated or not. Ben could, and did; with the Lorders he may have been coerced, manipulated, and escaped legal responsibility because of it, but in the end it was still him who caused and took part in the All Souls massacre. Did that say something about who he was to start with? Dr. Lysander hinted as much; she warned us again and again, but left the choice to Ben.\n\nSometimes I wonder if he ever really was mine, or was it all illusion from the start. Like Aiden said, how can you truly love somebody when you don't know who they really are?\n\nBut most of the time, I know we did. That time and place when we were just what we were then: blank slates. Innocents. Before my memories started to return; before Lorders manipulated and changed him, and Dr. Lysander returned his past. It was real, at least to me. My evidence is the pain left behind.\n\nSeeing how Finley is with Madison tells me it is possible for love to last, to grow. Just not for me, not now. One final lesson the Lorders have taught me is this: There are no second chances. I chose Ben, turned my back on Aiden, and I can't take it back. But Aiden was right, wasn't he? Ben was the past. I don't miss him the way I do Aiden. With Ben it is more grieving for something that _was._ Not something that could have been.\n\nThat should have been.\n\nOne last climb and I finally reach my destination: Astrid's slate mine prison. She is the only prisoner there now. Behind it are unmarked graves, with flowers and a memorial; a public ceremony today to unveil it. Mum is here, and Stella. Gregory and Dr. Lysander also. There are survivors, women newly released from the prison along with Madison, wearing both the marks of their ordeals and nervous joy at unexpected freedom on their faces. Along with survivors are family and friends, like us, of those who didn't make it.\n\nAnd one surprise. I almost stop breathing when Aiden walks up to me, gives me a hug. He doesn't say anything, just holds me, and I cling to him, tight.\n\nThe ceremony begins. Gregory had been as good as his word; he'd found his daughter. Turns out she died just weeks after I was born\u2014natural causes. If you can call dying of untreated infection after childbirth \"natural.\" Maybe, it was an escape? Though I like to think she would have stayed with me if she could.\n\nI stand with Mum and Stella for the two minutes of silence, but as if that isn't enough, it persists long after the time is marked. _More sinned against than sinning_ : I stare at the words carved in the memorial over the graves that include the mother I'll never know, standing between the two that I did.\n\nAfterward, I feel eyes on me\u2014a woman, thin, hunched, skin a papery gray, the determined eyes of a survivor. She draws me aside.\n\n\"I was there when you were born. Sam refused to say who was the father, but what options are there in a woman's prison with male guards? I know what your mother named you,\" she says, then whispers it in my ear as if it can't be said out loud.\n\n\u2022 \u2022 \u2022\n\nIt didn't come that day, but on other days, as the sun shines down to melt the ice of another winter, to summon spring wildflowers from the earth; as the sky darkens with sudden, drenching showers before the sun returns, I know that both pain and joy are needed for life to grow. As Skye bounds about my feet, as Aiden comes to walk beside me, against all logic I can almost feel it.\n\nMy mother, Sam, must have been an amazing woman. So much circled around her: Gregory's guilt at not having pardoned her made him a rigid Lorder ruler for most of his life. Dr. Lysander's grief at her supposed execution led her to invent Slating\u2014a way to stop execution of underage criminals, yes, but look at all that it led to. And Sam, herself, imprisoned for years by Astrid in that horrible place; I can't imagine what she went through. Yet somehow, she still had it within her to give me a name that reaches out and bridges the years lost between us.\n\nI have both been given and taken so many identities, but at last I am beginning to grow into my one true name. More will come with surviving, and time. With standing on my own feet now; with Aiden and I finding our way together in the future. Because sometimes there _are_ second chances.\n\nThis was the gift my mother gave to me:\n\n_Hope._\nACKNOWLEDGMENTS\n\nWRITING AND PUBLISHING A TRILOGY, A BOOK A YEAR, IS quite a whirlwind!\n\nSpecial thanks are owed to my agent, Caroline Sheldon: Without her, none of this may have ever happened.\n\nAnd to everyone at my publishers on both sides of the pond\u2014especially editors Megan Larkin and Rosalind Turner at Orchard Books in the UK, and Nancy Paulsen and Sara Kreger at Nancy Paulsen Books in the US\u2014thank you for everything.\n\nThanks to Erin Johnson for taking me around Oxford and its colleges, and to the porter at Magdalen College. When I was despairing that you couldn't see into another college's quad from Magdalen Tower, he suggested St. Mary's Church tower and All Souls as an alternative.\n\nThanks to first readers Amy Butler Greenfield and Jo Wyton, and writing buddies everywhere, especially all my friends at the SCBWI.\n\nAnd now . . . I've heard that confession is good for the soul.\n\nIt is time to come clean: about character names, and where they come from. Some you may know about already. I had a few name-a-character competitions which gave Katran in _Fractured,_ and Madison and Finley in _Shattered._\n\nBut what you may not know is many of my other character names come from friends, and that I often hunt them out on my Facebook friends list.\n\nFirst, the pets. Skye was a real dog! Owned by friend Karen Murray. Sadly, Skye died before _Slated_ came out, but the dog's name and character is _so_ how I remember her. Sebastian in _Slated_ was a real cat\u2014years ago, my parents had two, Damian and Sebastian. The character of the cat in _Slated_ was more Damian, and I originally used that name, but somewhere along the way changed it to Sebastian. And in _Shattered_? Pounce was the name of one of my sister's cats.\n\nAnd now for the people: The real people have nothing in common with the characters beyond the name, unless otherwise stated. Ben came from Benjamin Scott, because he is always smiling. Hatten\u2014Nico's surname as a teacher in _Slated_ \u2014came from Caroline Hooten, as the name made me think of owls; the spelling changed somewhere along the way. Nico came from Nick Cross. Kyla's assigned mother, Sandra, came from my sister\u2014and there is more than a little of her in the character. And in _Shattered,_ Stella came from Stella Wiseman, and Astrid from Astrid Holm.\n\nAnd of course, I can't forget Murray: He is my own very sleepy teddy bear!\n\nSo there is more than one way to have your name end up in one of my books. My Facebook page is \"TeriTerryAuthor\": Like it, and you just never know . . .\n\nYou can also find me as \"TeriTerryWrites\" on Twitter and Tumblr, and my website is teriterry.com.\n\nThanks to Slans\u2014Slated fans\u2014and readers, bloggers and reviewers everywhere, whose support and enthusiasm for the Slated trilogy has been beyond awesome.\n\nAnd to the most patient and understanding man: Living with a writer can be trying, but Graham is always the calm center of my life.\n\nAnd finally, to Banrock, Murray, and muses everywhere: cheers! __\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nALSO BY CHRISTOPHER PHILIPS\n\nSocrates Caf\u00e9\n\nSix Questions of Socrates\n\nSocrates in Love\n\nThe Philosophers' Club (children's book)\n\nCeci Ann's Day of Why (children's book)\nConstitution Caf\u00e9\n\nJEFFERSON'S BREW FOR A TRUE REVOLUTION\n\nChristopher Phillips\n\nW. W. Norton & Company\n\nNew York \u2022 London\nFor my loves, Ceci and Cali\nCONTENTS\n\nI Constitution Caf\u00e9\n\nII In the Beginning\n\nIII Constitution Making and Remaking\n\nIV Commons and Goods\n\nV Character Counts\n\nVI Money Matters\n\nVII Hail to the Chief\n\nVIII And Justice for All\n\nIX Governors and the Governed\n\nX Rights and Responsibilities\n\nXI Brew for a True Revolution\n\nAcknowledgments\n\nFurther Reading\nA little rebellion now and then is a good thing.\n\n\u2014 Thomas Jefferson, 1787\nI\n\nConstitution Caf\u00e9\n\nFor over fifteen years, I have been facilitating weekly philosophical dialogues called Socrates Caf\u00e9, hoping to revive a form of inquiry made famous by the ancient Greek philosopher. A hallmark of a vibrant democracy is that citizens willingly consider a wide range of objections and alternatives to their own way of seeing things. Socrates Caf\u00e9 is meant to be a space for friends and foes, intimates and strangers alike, to explore\u2014thoughtfully and reasonably\u2014timely and timeless existential problems, an exploration that itself makes people feel more bound to one another.\n\nOver time, my project gained wide notice; I began traveling extensively to inaugurate new groups\u2014not just in places like caf\u00e9s, libraries, and senior centers, but in prisons, nursing homes, institutions for the mentally ill, schools, and universities. At last count, there are well over 500 Socrates Caf\u00e9 groups that gather regularly around the globe, with many more in cyberspace, bringing together people of different backgrounds and experiences who thrive on impassioned yet thoughtful discourse.\n\nBut do they make any real difference in the grander scheme of things? I often receive letters that attest to their worth, such as one from a community activist in California. \"Nothing is sadder to me than seeing someone who isn't confident or privileged enough to . . . fulfill her own dreams, whatever they may be,\" she wrote. \"I believe the work you do does that. [Socrates Caf\u00e9] gives people the opportunity to connect not only to a deeper part of their own psyche, but to connect that deeper part of themselves to ideas that affect us all. In a society as egocentric as the one we live in today, that is essential.\"\n\nBut nagging questions persist: Is Socrates Caf\u00e9 at best a tiny oasis? Isn't there more intolerance and dogmatism than ever? Can it be that \"the system\" itself poses roadblocks to direct, much less meaningful, participation by everyday citizens concerned about our nation? Is our democracy on life support?\n\nAs a government major at the College of William & Mary, I immersed myself in studies of my other \"hero,\" Thomas Jefferson, himself a devotee of Socrates. As he wrote to his personal secretary William Short, \"the superlative wisdom of Socrates is testified by all antiquity, and placed on ground not to be questioned.\" Socrates couldn't have said it better himself when Jefferson exhorted his nephew Peter Carr to \"fix Reason firmly in her seat, and call to her tribunal every fact, every opinion. Question with boldness . . . Do not be frightened from this inquiry by any fear of its consequences.\"\n\nTo Jefferson, no topic was out of bounds when it came to subjecting it to bold questioning\u2014not even, in his view, the existence of God, and certainly not the worth of our Constitution, though many continue to believe it is the product of divine inspiration. Jefferson, for one, derided those who looked at constitutions \"like the ark of the covenant, too sacred to be touched.\" He believed that such people \"ascribe to the men of the preceding age a wisdom more than human.\" To him, \"We might as well require a man to wear still the coat which fitted him when a boy as civilized society to remain ever under the regimen of their barbarous ancestors.\"\n\nYet Americans venerate the Constitution, even if many of us don't actually seem to know what's in it. It's over our heads, out of our reach. It's considered untouchable, beyond reproach by us mere mortals, and this intimidating ideal only serves to make us feel like the system it begets can never be changed, no matter how much it contributes to record levels of political apathy and anger. Sure, we argue over the way elected officials and judges interpret the Constitution, but few of us have qualms about the Constitution itself. Meanwhile, our politicians outdo one another in declaring themselves more true to the Constitution than their opponents. From the president on down, they give their solemn oath to preserve, protect, and defend it.\n\nAs dysfunctional as people of most political persuasions believe our government is, they are just as convinced that the Constitution still works. But does it?\n\nI decided to shake things up a bit with a little experiment. \nII\n\nIn the Beginning\n\nBest and Worst Democracy\n\nIt is an unseasonably warm evening, and those taking part with me in a Socrates Caf\u00e9 philosophical dialogue have opted to abandon the coffeehouse where we usually gather and instead meander along macadam Duke of Gloucester Street. The pedestrian walkway bisects the heart of Colonial Williamsburg, the historic district of Williamsburg, Virginia, which was the capital of \"old Virginia,\" the first established colony, chartered by the British monarchy in 1606.\n\nThe dozen people with me are debating the question \"What is the best democracy?\"\n\n\"Even with all our country's defects, we already have the best democracy,\" says Stuart, who proposed the question after reading, and taking umbrage with, a passage in The Audacity of Hope in which Barack Obama makes the jarring confession that it's \"hard to shake the feeling these days that our democracy has gone seriously awry,\" its best days behind us.\n\n\"We have a free market system that's the envy of the world, and constitutionally guaranteed freedoms and rights that people in most other countries can only dream about.\"\n\nWe had chosen a week earlier the topic to be discussed today, and Stuart, like many others present, has come bearing a book or two from which to draw citations to support his perspective. He pulls from his leather satchel Democracy in America, by Alexis de Tocqueville, the French aristocrat and political thinker. \"After journeying here in the early nineteenth century to observe our new democratic system,\" he says, \"Tocqueville concluded that Americans have the ideal government, because 'the people name those who make the law and those who execute it,' and they 'form the jury that punishes infractions of the law.' \"\n\n\"But since our nation's beginnings, the people themselves have never had the right or opportunity to make the law,\" Sara, his political opposite, says. Sara recently graduated from a \"nearly Ivy League college,\" where she served as student body president. Now armed with a degree in cultural anthropology and burdened with a mountain of debt from student loans, she is again living with her parents, unable to find work.\n\nSara says next, \"The representative system was imposed on us by the Framers. It wasn't one chosen by the people. If our Constitution's Framers felt that most of their fellow Americans lacked the ability to decide for themselves the form of government they believed would best serve their hopes, needs, dreams, how can that be the 'best democracy'?\"\n\nStuart is silent for a spell. Then he says, \"When I served on jury duty, I felt I was part of democracy at its best. Maybe we didn't make the law, but we put aside our personal views and fulfilled our sworn oath to uphold the laws on the books. We weighed and debated on its merits the evidence presented to us, and reached a unanimous verdict.\n\n\"I admit I tried to wriggle out of jury duty. What a dressing down the judge gave me. I'm grateful he made me serve my turn. Because it made me appreciate like never before this great system our Framers devised.\"\n\n\"Well, if we can be entrusted as jurors to decide a citizen's fate, we should be given more constitutional clout in determining our democracy's fate,\" says Alejandra, who gave up a lucrative career in the world of high finance to become regional director of a nonprofit dedicated to helping our nation's less fortunate youth. \"By that I mean that the Constitution itself should have stated that the people themselves are the ultimate deciders.\"\n\n\"Originally, to live in a democratic society meant that everyone had equal access to power,\" Frank then says. A classical scholar who was denied tenure, Frank turned to his other passion, gourmet cooking; he has a thriving catering business in spite of the economic downturn.\"\n\n\"The Greek terms demos and kratos translate as 'rule by the people,' or even 'people power,'\" he goes on. \"In the ancient Greek city-states, elections were the least of it. All citizens took part in the popular assemblies. Any of them could propose laws. So all were deciders. That smacks of 'best democracy' to me.\"\n\nWhat he says gives Helen, a vivacious octogenarian, pause. \"You know, when we were victorious in the civil rights movement, I thought we'd reached the pinnacle,\" she says. \"But really, it was during the struggle itself that those of us involved were part of democracy at its best.\" Looking at Frank, she says, \"We had faith and trust in one another to make decisions together, as equals. We had our main movers and shakers, sure, but truth be told, all of us were leaders. It was when we were on the outside looking in that we activists enjoyed among ourselves the best democracy. Once we had access to the established political system, it turned out that it was still out of reach\u2014because the system itself was tilted way in favor of powerful interests and against us everyday folks.\"\n\n\"The local government of my childhood neighborhood, where I still have plenty of family, invoked eminent domain to tear down a historic black neighborhood, most of whose owners are descendants of slaves, the properties passed along from generation to generation,\" Helen goes on. \"They wanted to build a highway bypass, so there'd be quick access to a new shopping development. That neighborhood is a living museum. I joined the community in protesting full force, and the project has been canceled. But any time the government wants, it can revive it and destroy homes, communities, history. The Supreme Court ruled that it has the ironclad constitutional right to do that, without our advice and consent. A democracy isn't really possible if a constitution allows a small group of public officials to wield so much power.\"\n\nWe pass by a sidewalk gift shop peddling colonial era paraphernalia. Beside it is a life-size bronze sculpture of Thomas Jefferson seated on a bench, watching the world go by. At the stand, I buy a replica of the U.S. Constitution. It occurs to me, after Helen's remark, that it might serve as a handy reference as our dialogue continues.\n\nContemplating the replica, Marjorie says, \"The small clique of Framers who authorized themselves to construct that document left us with a constitutional republic, not a democracy.\"\n\nMarjorie and I were fellow government majors during our undergraduate days at the College of William & Mary, the alma mater of Thomas Jefferson, who was often a principle topic of our conversations. A self-described recovering politician, Marjorie abandoned life as an elected public servant after just one term. She had found herself quickly marginalized because of her uncompromising stances, particularly on green issues.\n\n\"Most of those who constructed our Constitution were surely glad that Thomas Jefferson was at a safe distance from the proceedings,\" she says. \"The Framers didn't truck with his notions of making the Constitution a reflection of the democratic ideals in the Declaration of Independence.\"\n\nLeafing through a book on Thomas Jefferson until she finds a particular passage, she says next, \"To him, the best democracy was one with 'every citizen an acting member of the government.' Only this would 'attach him by his strongest feelings to the independence of his country.' Jefferson believed that if most citizens were deliberately sidelined, they'd 'become inattentive to the public affairs.' Once that happened, he warned, the professional politicians 'shall all become wolves'\u2014doing as they will because of an apathetic citizenry, bringing out the worst rather than the best in our politics.\"\n\nSara, a dictionary in hand, says, \"Merriam-Webster defines 'democratic' as 'of or relating to one of the two major political parties in the United States.' By this definition, it's perfectly 'democratic' that we sign off our support to political candidates from one of the principle parties, and then tune out, expecting the work to get done for us.\n\n\"That sounds to me like the definition for 'the worst democracy.' It's describing a system that permits politicians to be wolves. For the best democracy to be happening, we'd have to engage in outright revolt against the current system and replace it with one in which every citizen is an acting member of government.\"\n\n\"It seems like a lifetime ago, but the Obama campaign was itself a sign of rebellion,\" says Frank. \"Those of us who volunteered felt like we were part of an insurgency for making our form of government 'the best'\u2014more participatory and inclusive. Obama may have left all his followers adrift when he became president, but the energy that was evoked from that movement is still out there, demanding to be harnessed. You can't put the genie back in the bottle. Like Jefferson said, 'A little rebellion now and then is a good thing.' \"\n\nOur walk takes us to the steps of the Raleigh Tavern. The Raleigh, as it is known, opened its doors in 1717, and is named after Sir Walter Raleigh, the English writer and explorer who in 1584 spearheaded the first, albeit unsuccessful, attempt to establish an English colony in the New World. Situated at a well-traveled crossroads, it was once the most popular gathering place in the Virginia Colony. We enter and make our way to the tavern's Apollo Room. The elegant space has served many uses, from the site for debutante balls to the nerve center for plotting the American Revolution.\n\nBorn and raised in the area, I've visited the tavern countless times. To this day, I never fail to feel a thrill upon entering, since to me the Raleigh is the place where the American Revolution was launched.\n\nAs a child, I was regaled by my mother, a genealogy and colonial era history buff, with stories of her Pegram forebears (my mother's maiden name is Pegram) who'd migrated from England in the 1660s and settled in the heart of what is now known as Colonial Williamsburg. George Pegram, a surveyor, was the first to arrive. One of his descendents, Daniel Pegram, lived near the Governor's Palace, a stone's throw from the Raleigh. He was prominent enough to have dealings with one of our founding fathers, Peyton Randolph, president of both the First and Second Continental congresses. Daniel Pegram was also a member of Bruton Parish Church\u2014the oldest Episcopal Church in the United States, founded in 1607\u2014at the same time that its rolls included Patrick Henry, George Washington, and Thomas Jefferson. Daniel's nephew George Pegram, a founder of one of the earliest Masonic lodges, organized and led a troop during the Revolutionary War. He eventually was promoted from captain to major general for distinguished service, and served in the House of Delegates starting in 1786, when Thomas Jefferson also was a Virginia legislator. Pegram may well have voted on Jefferson's proposed reforms to the comprehensive set of statutes, or legal codes, encted by the legislature since the state constitution had gone into effect. Among other things, it called for radical land reform, universal public education, a reformed criminal code, and the creation of a form of governance in which those at the bottom rung of the political hierarchy would have the most clout. If passed, it would have made Virginia a laboratory for democracy.\n\nBut what about at the federal level? Didn't Jefferson help make it a testing ground for democracy as well?\n\nIt is widely assumed that, as one of our most influential founders, Jefferson was personally involved in the crafting of our federal Constitution, but in fact he was overseas serving as U.S. minister to France at the time of the Constitutional Convention. Dissatisfied with many facets of the Constitution concocted by our Framers, Jefferson not only proposed his own ideas for ensuring that the Constitution's governing blueprint perpetuated the democratic experiment, but he proposed in all seriousness that every generation of Americans have a chance to create society anew by rewriting the Constitution from scratch, formulating new foundational laws that \"suit the change in circumstances as well as men.\" He was convinced that only if such a democratic rite of passage, involving all Americans, was held periodically\u2014he recommended twenty-year intervals\u2014would the revolutionary spirit of his time be kept alive for generations to come.\n\nJefferson's antidote for societal stasis, as he told the historian Samuel Kercheval, was to take periodically \"as a tally, every provision of our constitution, and see if it hangs directly on the will of the people.\" Those provisions that turn out not to reflect the people's will, he believed, should be entirely redone. \"Let us then go on perfecting [the Constitution],\" he urged, by supplanting \"those powers which time and trial show are still wanting.\"\n\nIt occurs to me now: what if his radical proposal was put to the test in some way today? To be sure, he had in mind that this constitutional makeover would be undertaken every twenty years, and over two hundred years have passed since he first proposed it. But better late than never. I consider the replica of the United States Constitution in my hand. I rip it to bits.\n\n\"Let's tap into that rebellious energy that's out there across the political spectrum. Let's go back to square one and write a brand new Constitution, one that might overthrow the status quo and create the 'true democracy' that Jefferson envisioned,\" I say to my fellows.\n\n\"If we were the Framers, what would our Constitution look like? If we made as if we had the authority to rewrite it, what articles would we propose? What freedoms and rights and responsibilities would we give ourselves? What form of government would we create?\"\n\nHelen looks dubious. \"How does make-believe 'constitution creation' bring us any closer to real salvation? How will this solve the vast income inequalities in this nation, the unemployment crisis, the foreign policy disasters that have caused us to lose our stature around the world?\"\n\n\"Real or otherwise, a Constitution in itself can't solve our problems,\" says Sara. \"It can't make those who govern accountable, can't ensure human rights and justice for all. But it can be a statement of our vision for the 'American way' of governing. It can be a declaration of our values, and a guidepost for bringing our nation's promise and practice into greater alignment.\"\n\nStuart looks at me and says, \"This idea of yours should be opened up to all Americans. I bet most are like me, and have never looked all that closely at the Constitution. Taking part in the drafting of a new one will give people a great chance to put the existing Constitution under the microscope and come to appreciate its powers for achieving\u2014or preventing\u2014the creation of the best democracy.\"\n\nAnd so I decided to embark on a journey across America to take the constitutional pulse of the nation. At Constitution Caf\u00e9, a space dedicated to the Jeffersonian idea of freedom, a broad cross-section of actual and aspiring Americans grapple with how they would sculpt the United States Constitution if they could start from scratch. The constitutional articles that participants construct often address perceived flaws, loopholes, and omissions in the Constitution. At times, this leads to significant revisions of existing articles or the creation of altogether new ones.\n\nIn the course of their exchanges, Constitution Caf\u00e9\u2013goers often arrive at insights about whether our current Constitution, and the institutions it props up (and that in turn prop it up), are impediments to, or facilitators of, our higher democratic hopes and dreams. This in turn can prompt thoughtful exchanges on whether our Constitution\u2014as Barack Obama, former professor of constitutional law, maintains\u2014has \"proved a sufficient defense against tyranny,\" or whether we need \"to heed Jefferson's advice to engage in a revolution every two or three generations.\"\n\nIn making their case for new and improved articles, those taking part reason, persuade, argue, and bend over backward to ensure that everyone has the opportunity to introduce and convince others to support his or her big constitutional ideas (usually a two-thirds majority approves the final language for any given article crafted by the group). For as Thomas Jefferson had it, one \"cannot have his way in all things\" when engaged in such democratic deliberations, but must \"acquiesce on seeing that of others preponderate at other times.\" Indeed, \"without this mutual disposition,\" Jefferson asserted shortly after he became president\u2014at a time when deep political rifts already were developing among Americans\u2014\"we are disjointed individuals, but not a society.\" The new Constitution that has been crafted so far is by no means exhaustive. Not only do gaps remain to be filled, but even the articles that have been constructed are not meant to be the last word, just a starting point for deeper debate.\n\nIt may seem less than auspicious that an imaginative yet earnest effort to renew American democracy should be launched in a Virginia tavern. But if such a space was good enough for the likes of Thomas Jefferson, it's good enough for us.\n\nJefferson first happened upon the Raleigh when he was a law student at William & Mary. The tavern's Apollo Room became his favorite haunt. After Jefferson was elected in 1774 to Virginia's House of Burgesses, he regularly held court there with fellow patriots. As he wrote in his memoirs, he and his colleagues gathered to consider \"the state of things\" in the colony in particular and in the New World as a whole, at a time when a perfect political storm was brewing\u2014one the fractured colonists were woefully unprepared for. These confabs prompted Jefferson and those with him to take political action\u2014from boycotting British goods after Parliament passed new taxes on staples without their consent, to holding a day of \"fasting, prayer and humiliation\" to show their full support of Boston after the Tea Party\u2014galvanizing the colonists to act in concert against the crown. Largely unheralded to this day, Jefferson and his fellow burgesses were arguably far more decisive in sparking the American Revolution than the instigators of the Boston Tea Party, whose actions would have ended with a whimper if not for Jefferson and his fellows at the Raleigh Tavern.\n\nIn the Apollo Room, Jefferson surely admired the gilt-framed painting hanging on the wall with the inscription Hilaritas Sapientiae et Bonae Vitae Proles\u2014Joy Springs from Wisdom and Good Living. Like the Greeks of Athens in the Western world's first democracy, Jefferson believed that good living in a democracy came not from relying on the mythic high-mindedness of iconic statesmen; rather, it hinged on cultivating the inherent virtues of its ordinary citizens and bringing them to the decision-making table. He held that only if all the country's citizens were full \"participators\" and \"sharers\" in governance could America achieve the type of all-around excellence\u2014what the Greeks called arete\u2014that culminated in greater eudaimonia, the exalted state of well-being and satisfaction brought about in large measure by direct participation in the political arena.\n\nPresident John F. Kennedy characterized a reunion of Nobel Prize winners as \"the most extraordinary collection of talent, of human knowledge, that has ever been gathered together at the White House, with the possible exception of when Thomas Jefferson dined alone.\" Kennedy noted that Jefferson, by age 32, \"could calculate an eclipse, survey an estate, tie an artery, plan an edifice, try a cause, break a horse, and dance a minuet.\" He might have added that by that age, Jefferson had already tried to establish a revolutionary code for governance that would require each generation to try anew the cause of democracy. Constitution Caf\u00e9 aims to revive his effort.\n\nAs the Constitution Caf\u00e9 adventure unfolded, I came to wonder even more about the principal inspiration behind this project: What set Thomas Jefferson on course to become not just an unlikely patriot, but a foremost instigator of the American Revolution? How did Jefferson, a person of patrician standing, wind up becoming known as \"man of the people\"? How did someone locked into such intractably backward notions of African-Americans and women become our foremost advocate for creating a society founded on the belief that we are all created equal?\n\nAnd why are many of Jefferson's most exemplary ideas unsung or underemphasized, when they might hold keys to resuscitating democracy today?\n\nDeclare This\n\nOriginal Preamble to the Constitution: \"We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence, promote the General Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution of the United States of America.\"\n\n\"We've been studying the Constitution in class today,\" William says. \"I didn't realize that the Constitution wasn't ratified by the states until two years after it was signed by the Framers.\"\n\nAt a high school situated in a county in the heart of Appalachia, I'm exploring with students the prospect of rewriting the Preamble to the Constitution. My mother was born in a coal-mining camp not that far away. The hauntingly beautiful mountainscape encompasses one of the most perennially economically depressed regions in the nation.\n\nToday is Constitution Day\u2014the anniversary of the day the delegates of the Constitutional Convention signed the Constitution on September 17, 1787. In 2004, the late Robert Byrd, the longest-serving U.S. senator in history, introduced a bill subsequently passed by Congress that requires all schools that receive federal funding to devote this day to educational programs on the Constitution.\n\n\"Last week, we studied the Declaration of Independence,\" William says next. \"That was pretty exciting. Thomas Jefferson really told off the king. He basically said, 'Because you just don't get that we're all born free and equal, because you treat your loyal subjects in the New World as if we're in servitude to you, we're forced to declare our independence from you, and good riddance.' \"\n\n\"I enjoyed studying the Declaration, but the Constitution is so boring,\" says Jeanne. \"It's mostly about how the spoils of power are divided. It hardly says anything about freedom and equality and rights\u2014not until way near the end, like an afterthought, in the Bill of Rights that was added to it two years after the Constitution was ratified. I don't think whoever wrote the Constitution even read the Declaration.\"\n\nThis prompts Margaret to say, \"Why isn't the Declaration part of the Constitution? Everyone in the United States celebrates Independence Day, the day the Declaration was, well, declared. If Constitution Day is such a big deal, why aren't there any fireworks to celebrate it too?\"\n\nThen she says, \"We should insert the Declaration at the beginning of the Constitution, make it the Preamble. The Declaration should be law, the law to end all laws. The rights it gives\u2014to life, liberty, and the pursuit of happiness\u2014are nowhere mentioned outright in the Constitution. Yet all those patriots battling in the Revolutionary War did so because they were fired up by the rights and freedoms the Declaration promised them. There would've never been a Constitution if they hadn't fought and sacrificed and won against the odds.\n\n\"In those toughest times, in places like Valley Forge, when the troops under George Washington were freezing, when they didn't have blankets or soles on their boots, and they were running out of bullets, the words in the Declaration kept hope alive in their hearts and inspired all their heroics. It not only kept their eyes on the prize, it was the prize. Yet somehow, it was left out of the Constitution. It's time to fix that. The Declaration's rights need to be put right up front in the preamble, as Law Number One.\"\n\n\"If it's the Preamble, would the Declaration be a law?\" I ask. \"Or are the only actual laws the constitutional articles that follow the Preamble?\"\n\n\"It would be a law,\" she says without hesitation. \"Anything within the Constitution itself can be considered law.\"\n\n\"Well, the Preamble, as the introduction, is supposed to kind of map out what the rest of the Constitution will accomplish,\" says Jeanne. \"So with the Declaration as the Preamble, all the laws that follow it would be aimed at making the inalienable rights in the Declaration come true.\"\n\n\"Not all of the Declaration should be inserted,\" William says now. \"We're not declaring our independence from the king of England anymore. All the stuff in the Declaration that dresses down King George is outdated.\n\n\"Really, just a small part of the Declaration, the most important part, should lead off the Constitution: 'We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable rights, that among these are life, liberty, and the pursuit of happiness.' \"\n\n\"It's this section that President Obama said his grandmother used to read to him, and that inspired him to be a politician and help people,\" Bobby says. \"It's this section that Dr. Martin Luther King, Jr., said motivated him to start the civil rights movement. A lot of others who've tried to move mountains when it comes to making people in America more free were also inspired by this section in the Declaration. It's like they believed that these rights in the Declaration already were part of the Constitution.\"\n\n\"Well, we should update a little bit that part William quotes, so it says 'all men and women'\u2014all humans, in other words\u2014'are created equal,'\" says Dottie.\n\n\"This is great,\" says Margaret, animated. \"With this updated Declaration part of the Constitution, we'll have to start treating one another equally. Because, as I've said, it would be the Law with a capital L.\"\n\n\"But we're not all created equal,\" says Jeanne. \"Some are born into riches, some into desperation. Some have all kinds of talents, some hardly any. Some are able to make the most of however much or little talent they have, because their parents and the schools they go to have the resources that allow them to. But others aren't in a situation in life even to know they have talents, much less figure out how to make the most of them.\"\n\n\"All the more reason to be treated equally,\" says Margaret. \"The Declaration says that all 'are endowed by their Creator with certain unalienable rights.' In other words, no matter how unequal the circumstances we're born into, no matter how unequal our talents, we're still equal before the eyes of our Creator. So we Americans should see one another as equally important, with equally unalienable rights, to life, liberty, and the pursuit of happiness. And that means that whether we have a little talent or a lot, whether we have a little money or a lot, we're equally deserving when it comes to figuring out what our talents are, and having the opportunity to do something with them.\"\n\nI now ask, \"Are these rights\u2014to life, liberty, and the pursuit of happiness\u2014equally important?\"\n\nWilliam says: \"Let me put it this way: Without the right to life and liberty, you can't pursue happiness. And without the right to pursue happiness, life and liberty don't mean much. The pursuit is life, it is liberty.\"\n\n\"How so?\" I ask.\n\n\"The pursuit is all about making something of yourself, like Margaret sort of said.\"\n\n\"No,\" says Jeanne flatly, \"it's about having the right to try to make something of yourself. It's the trying that makes you happy, that makes you feel alive. I may never become the Olympic track star I want to be, but I'm free to try. Just having the freedom to try gives me a feeling that's . . . beyond happy. Sometimes while I'm training after school, running all up and down these mountains, I can sort of see beyond them and imagine the life I'm going to make for myself when I graduate from high school and college. I can see myself in medical school at Harvard University, and then coming back here to open a doctor's clinic so no one in these parts ever again suffers in silence.\" Then she says, \"And I'll tell you what, I can run a lot faster and farther than I could a couple years ago, because of all this trying.\"\n\n\"You may never make your dreams come true all the way, or your dreams may change over time,\" says Margaret. \"But in a way, as long as you have the freedom to pursue them, whatever they may be at any moment, and as long as you have not only the inspiration, but also some of the basic means to do so, that in itself is a big part of making dreams come true.\"\n\n\"My parents weren't able to go to school when they were my age,\" she then says. \"My granddaddy was injured and disabled in a coal-mining accident. So my daddy, just fourteen years old, had to quit school and start working in the mines. My daddy is out of steady work right now. My momma, she left us a while back, in the dead of night. If anyone has a right to be miserable, it's my daddy. But he doesn't complain. He does all sorts of odd jobs to make sure I can stay in school. He's making sure my life is a lot more about the right to life, liberty, and the pursuit of happiness than his ever could be.\"\n\nDesiree, listening intently, now says, \"We need the rights to life and liberty, I agree. But the guaranteed right to the pursuit of happiness? Who cares about happy. My mom, I've never seen her crack a smile. But she's fulfilled in her way. Like just about everyone's parents in this class, she works whenever and wherever, so my life isn't as tough as hers. And you can tell it makes her fulfilled. I've offered to help out. She won't hear of it. She says I can help out best by staying in school and then going on to college and making something of myself.\"\n\nThen Dottie says, \"Does happy even matter? Can't you be free to be miserable, frustrated? Shouldn't we have the right to be happy or not happy? Many people I admire aren't very happy, some of them not happy at all, but they sure have made big differences in the world, as doctors, teachers, moms. They've sure made it possible for other people to be happy if they want to, by giving them more opportunities than ever to pursue happiness.\"\n\nThis inspires Bobby to say, \"People with lots of money who I'm familiar with on TV don't seem very happy. But people I know around these parts with no money think they'll be happy if they have lots of money. There are things a lot more important than happiness\u2014or at least, than having lots of money, as if that's what the road to happiness is paved with. The main thing is, happy or not, each person should have the right to lead a life she's proud of.\"\n\nTo which Margaret says, \"Making the most powerful part of the Declaration\u2014with a few small changes\u2014our new Preamble will be a reminder that whether you're up or down, you treat people a certain way, you treat yourself a certain way, with worth and dignity, because that is everyone's unalienable right.\"\n\nConstitution Caf\u00e9 Preamble: Because we the people of the United States hold these truths to be self-evident\u2014that all humans are created equal, that they are endowed by their Creator with certain unalienable rights, that among these are life, liberty, dignity, and the pursuit of happiness\u2014we do ordain and establish this Constitution for the United States of America, in order to realize these truths and preserve and promote these unalienable rights.\n\nDawn of the Declaration\n\nBefore the Declaration of Independence was even composed, the Virginia House of Burgesses, the oldest governing body in the New World, already anticipating a full-fledged pancolonial movement for independence, began crafting a state constitution to replace its colonial charter. Jefferson was not on hand. \"I was then at Philadelphia with Congress,\" he relates; he'd been appointed to attend the Second Continental Congress that had convened on May 10, 1775, with the purpose of directing the colonial war effort. Jefferson was nonplussed that his colleagues back in Virginia were writing a new constitution. He suggested they hold off on the project until such time as they received \"a commission of greater latitude, and one more specific . . . by the people.\" But when it became clear that the burgesses were moving forward despite his reservations, he turned his own energies to the task, albeit from afar: \"knowing that the Convention of Virginia was engaged in forming a plan of government, I turned my mind to the same subject, and drew a sketch or outline of a Constitution, with a preamble.\"\n\nJefferson sent his completed draft directly to fellow patriot and legislator Edmund Pendleton, president of Virginia's constitutional convention, \"on the mere possibility that it might suggest something worth incorporation into that before the Convention.\" To Jefferson's disappointment, it didn't arrive in time to be considered. Pendleton, it turns out, received it \"on the day on which the committee of the whole had reported to the House the plan they had agreed to.\" The constitution for Virginia that was set to be approved, Jefferson relates, \"had been so long in hand, so disputed inch by inch, and the subject of so much altercation and debate; that they were worried with the contentions it had produced, and could not . . . have been induced to open the instrument again.\" As a consequence, it looked like his own proposed constitutional draft\u2014which he'd titled \"A Bill for new-modelling the form of Government and for establishing the Fundamental principles thereof in future\"\u2014was destined for the trash bin. However, Jefferson proudly recounts, the Virginia legislators were so impressed by the preamble he'd composed that it was \"tacked\" to the beginning of their document and became part of the commonwealth's new law of the land. The text of Jefferson's constitutional preamble is almost precisely the same as what he later drafted for the Declaration of Independence. \"The fact is,\" Jefferson later explained, \"that the preamble was prior in composition to the Declaration; and both having the same object, of justifying our separation from Great Britain, they used necessarily the same materials, and hence their similitude.\"\n\nWhile Jefferson was pleased that the Virginia assembly had included his preamble, he nonetheless found the commonwealth's new constitution wanting as a whole. \"This constitution was formed when we were new and unexperienced in the science of government,\" he later said, and so those who crafted it were ignorant of the \"very capital defects in it.\"\n\nLess than a year and a half after the Commonwealth of Virginia's constitution was ratified, Jefferson was given the opportunity to remedy its defects, or so he was led to believe at the time. While the state constitution itself remained intact, soon after the Declaration of Independence was issued, the Virginia legislature asked Jefferson to head a committee responsible for proposing reforms to the statutes that they'd enacted. From the latter part of 1776 until 1779, Jefferson relates that he was \"occupied in the reformation of the laws to the new organisation and principles of our government.\" He and his fellow Revisors, as they were called, were given the unprecedented opportunity \"to take up the whole body of statutes and Virginia laws, to leave out everything obsolete or improper, insert what was wanting, and reduce the whole within as moderate a compass as it would bear, and to the plain language of common sense.\"\n\nBut the new set of statutes painstakingly produced by the five-man committee proved in nearly all cases to be far more progressive than what their fellow assemblymen cared for, even if the new statutes were truer to the letter and spirit of the organization and principles of government set forth by Virginia's constitution. Though Jefferson's committee had a few signal successes, most elements in the reform package were summarily rejected by the assembly, which was dominated by the landed gentry.\n\nWhile this discouraging experience further cemented the political status quo in Virginia, Jefferson, never-say-die, continued to sculpt a draft constitution for the commonwealth. While his project was tailored for Virginia, it was also meant to serve as a blueprint for promoting democratic governance on a national scale. Indeed, the framework for his drafts sometimes parallels that of the U.S. Constitution, though the specific articles that comprise his version often diverge in remarkable ways. This thought experiment \"reveals Jefferson as a founder of political institutions as well as a scribe for revolutionary sentiment.\"\n\nJefferson spent much of the remainder of his life pondering and attempting to lay the groundwork for a genuinely revolutionary framework for governance. He didn't undertake this endeavor by pontificating in a vacuum. Rather, as with all his intellectual achievements, they were grounded in direct experience, from his stint as head of the committee to reform Virginia's statutes, to his conversations with colonists from a variety of experiences, to his observations of Native American societies. Most of all, his project was driven and informed by his immersion in the revolutionary cause. In lauding Jefferson as our nation's \"first great democrat,\" the social philosopher John Dewey characterizes him as that \"rare person in politics, an idealist whose native faith was developed, checked, and confirmed by extremely extensive and varied practical experience.\" He got his hands dirty in the political trenches, with the insights gained from these experiences serving as grist for his notions on human rights, happiness, and freedom, which in turn drove his ideas for ideal democratic governance. As Jefferson scholar Lloyd Kramer asserts, Jefferson is unique among political thinkers of his time for his \"frequent participation in the struggle for political power.\"\n\nFundamental Act of Union\n\nJohn Hancock\u2014president of the Continental Congress and the Declaration's first signer\u2014characterized the Declaration as \"the foundation of a future government.\" Jefferson himself considered the Declaration \"the fundamental act of union of these states.\" To political scientist Dennis J. Mahoney, Jefferson in effect was claiming that it is not the Constitution, but rather \"the Declaration that constitutes the American nation.\"\n\nThe Constitution of the United States is sometimes pronounced, by scholars or politicians, to be neutral with respect to political principles. But the Constitution was not framed in a vacuum. It was devised as the Constitution of the nation founded by the Declaration of Independence. The Declaration prescribes the ends and limits of government, and proclaims the illegitimacy of any government that fails to serve those ends or observe those limits. The Constitution is thus ruled by the Declaration. The Constitution provides for the government of the regime created by the Declaration: the regime of equality and liberty.\n\nAfter the Revolutionary generation had passed, political luminaries continued to take it as a given that the Declaration was the Constitution's reason for being, and as such, was itself part of our foundational law. For instance, John Quincy Adams, the sixth president of the United States, asserted that the Declaration contained nothing less than \"the fundamental elements and principles of American constitutional law.\" In his view, the Declaration and Constitution \"are parts of one consistent whole, founded upon one and the same theory of government.\"\n\nEven Abraham Lincoln, in his Gettysburg Address in 1863, asserted that the United States was founded \"four score and seven years ago\"\u2014in 1776, when the Declaration was issued, rather than in 1789, when the Constitution was ratified. Such evidence supports the view of intellectual historian James A. Colaiaco, who, like Mahoney, argues that \"the Constitution is ruled by the Declaration, which may be considered its real preamble.\"\n\nCharles L. Black, the acclaimed scholar of constitutional law, argues in A New Birth of Freedom that the Declaration is without doubt part of the Constitution. Black contends that the \"the Declaration as a whole was an act of 'constitution,'\" and so the rights and freedoms it sets forth not only have the \"force of law\" but are in fact \"an act of law.\" To Black, this means that it's our federal lawmakers' obligation to enact laws that give people the inalienable rights enumerated in the Declaration, thus bringing the disenfranchised into the fold as relative equals. Black believes that the \"general diffusion of material welfare\" must be seen as \"an indispensable part in the general diffusion of the right to the pursuit of happiness.\"\n\nTo Jefferson, however, the general diffusion of political welfare, with all Americans governing as equals, was even more important, and is what would lead to an overall sense of societal well-being.\nIII\n\nConstitution Making and Remaking\n\nThe Framers of our Constitution were radicals, if only in one sense: they wouldn't settle for simply reforming our original governing document, the Articles of Confederation and Perpetual Union. They decided that no amount of revising would salvage the Articles, and so \"the United States needed to be constituted anew\u2014not merely with greater power, but also in a way that resolved the underlying uncertainties that had been generated by the Articles of Confederation.\"\n\nThe Articles of Confederation had become operative on March 1, 1781, after they were ratified by all thirteen original states. They created a highly decentralized form of government. The states had insisted on creating a confederation that allowed them to remain sovereign over most of their affairs, since their inhabitants, as the erudite constitutional historian Clinton Rossiter put it, \"were in no mood to give a faraway, central regime in the United States what they were busy denying to a faraway, central regime in Britain.\" Under the Articles, there was no president or federal judiciary. And because the Confederation Congress couldn't levy taxes or regulate commerce, it was hamstrung when it came to raising money to pay foreign and domestic debts and building a militia to defend itself from insurrections from within or an army to waylay threats from without. The result was that the United States proved \"incapable of providing the security, diplomacy, and open commerce upon which the nation relied.\"\n\nClinton Rossiter concedes that \"it is conceivable that a stable, effective pattern of government for the Union could have emerged prescriptively out of the Confederation Congress.\"\n\nThe first government of the United States fought a successful war, made a viable peace, laid the foundations for a new order of diplomacy, established enough credit at home and abroad to pay its most pressing expenses, conducted a kind of political and social academy for the continental elite, created a functioning bureaucracy, and above all maintained itself as a symbol of American unity.\n\nEven so, our nation's first constitution lurched \"from crisis to crisis . . . because it made the exercise of the limited authority it granted almost exclusively dependent on the good will of each of thirteen states, not one of which had any overpowering reason . . . to trust any or all of the other states.\"\n\nThis \"fatal flaw\" in the Articles alarmed the likes of prominent Virginia assemblyman James Madison, who was among those who \"had begun to wonder out loud whether a government without power to tax, to regulate trade, and to pass laws that individuals were bound to obey was a government at all.\" At Madison's urging, in January 1786 the Virginia legislature invited all the states to send delegates to a convention in Annapolis, Maryland, to discuss ways to remedy the Articles' defects. But only five of the states sent delegates to the gathering. Without a quorum, no concrete decisions could be made. However, Madison and Alexander Hamilton\u2014at the time an assemblyman in New York's legislature\u2014managed to persuade the rest on hand to approve a motion to hold a full-fledged convention in May 1787 in Philadelphia, where participants would be charged with taking specific measures to revise the Articles.\n\nHowever, the Articles of Confederation contained the hard and fast stipulation that any changes made to it had to be \"agreed to in a Congress of the United States, and be afterwards confirmed by the legislatures of every State.\" Yet only twelve states were represented at the gathering that got underway at the Pennsylvania State House (now Independence Hall) in Philadelphia. Plus, the representatives' only official charge at this summit was to consider revising specific provisions of the existing Articles. So when a majority of the 55 delegates from the twelve states determined that the Articles could not be saved and made the decision to supplant them with an entirely new Constitution, they had no real legal sanction.\n\nEven so, when Jefferson learned of their plan from his post as U.S. minister in France, he claimed to \"like much\" in principle \"the general idea of framing a government which should go on of itself peaceably, without needing [as the Articles did] continual recurrence to state legislatures.\" Fellow Virginia patriot Patrick Henry, on the other hand, \"smelt a rat in Philadelphia, tending toward the monarchy.\" Leery of the intentions of the convention's organizers, whom he believed wanted to create \"a consolidated government,\" he turned down an invitation to serve as one of the delegates. Rhode Island, sharing Henry's suspicions, declined to send any delegates. The seeds of future deadlock were sown.\n\nEven if those taking part had been authorized to carry out the task of creating a new Constitution, they were not representative of most Americans. Though a few, like Benjamin Franklin, had humble beginnings, the typical delegate to the Constitutional Convention was nonetheless \"a privileged member of the upper class in what was still a hierarchical society\" and did not \"represent directly, in any reasonable way, American society as a whole.\" Rather, most of the 55 Framers represented faithfully the \"white, adult, male, affluent or near-affluent segment of the population.\"\n\nBut couldn't the same thing be said about our Founding Fathers? After all, those 56 patriots who signed the Declaration of Independence were also unrepresentative of Americans as a whole. One telling difference, though, was that they signed a galvanizing and unifying document\u2014befitting of Jefferson's characterization of it as \"an expression of the American mind\"\u2014that set the colonists free. The Constitution, on the other hand, was meant by those who crafted it to rein them in a good bit.\n\nOnly eight of our Declaration's signers also attended the Constitutional Convention and served as Framers. But even with the absence of the likes of Jefferson and most other Founding Fathers at the Constitutional Convention, political scientist Robert Dahl of Yale contends that \"among the Framers were many men of exceptional talent and public virtue.\" For instance, James Madison might not have been a signer of the Declaration, but he was\u2014and still is, in Dahl's estimation\u2014\"our greatest political scientist.\" Yet even the incomparable Madison \"could not foresee the future of the American republic, nor could he draw on knowledge that might be gained from later experiences with democracy in America and elsewhere.\" While the knowledge of Madison and his fellow Framers \"may well have been the best available in 1787,\" the fact was that \"reliable knowledge about constitutions appropriate to a large representative republic was, at best, meager.\" And so they were \"limited by, so to speak, their inevitable ignorance.\" Nonetheless, the Framers felt certain that \"the high value they placed on republicanism was overwhelmingly shared by citizens of all the states.\"\n\nBut the Framers never asked their fellow Americans what they wanted, and so never gave them a chance to weigh in on what specific type of republican system they preferred. Rather, with Madison\u2014who would come to be known as the \"Father of the Constitution\"\u2014steering the proceedings, the delegates agreed to keep their discussions private, supposedly so they could speak freely. As a consequence, there was no public record of the proceedings. Jefferson, dismayed by this news, wrote to John Adams, \"I am sorry [the delegates to the federal convention] began their deliberations by so abominable a precedent as that of tying up the tongues of their members.\" In his estimation, \"Nothing can justify this example but the . . . ignorance of the value of public discussions.\" On September 17, 1787, the new Constitution was approved\u2014but by only 39 of the 55 delegates.\n\nMore than two hundred years after the fact, Robert Dahl asks some unsettling questions: \"Have we Americans ever had an opportunity to express our considered will on our constitutional system? For example, how many . . . have ever participated in a referendum that asked them whether they wished to continue to be governed under the existing constitution? The answer, of course, is: none.\"\n\nWe're not alone, however. Not even in ancient Athens, the cradle of Western democracy, did \"the people\" generally have a voice. In recent years, new national constitutions have supplanted outmoded ones in emerging democracies throughout the world, from Kenya to Ecuador to Serbia. In nearly all cases, referenda were held so the people themselves could vote on the new constitutions. However, in no instance were they invited to the table and asked to participate themselves in the framing of their governing document.\n\nWinston Churchill famously said that \"democracy is the worst form of government except all the others that have been tried.\" But has democracy ever been fully tried? Perhaps, paraphrasing G. K. Chesterton on Christianity, we might conclude that \"democracy has not been tried and found wanting; it has been found difficult and not tried.\"\n\nOf, By, and for the People?\n\nArticle VII of the U.S. Constitution: \"The Ratification of the Conventions of nine States shall be sufficient for the Establishment of this Constitution between the States so ratifying the Same.\"\n\n\"I'd always assumed that the Constitution says somewhere that ours is a government 'of the people, by the people, and for the people,'\" says Jennie.\n\nI'm at the Green Dragon Tavern in the North End of Boston. The original version of this watering hole is where Sam Adams and his fellow Sons of Liberty plotted the Boston Tea Party. To this day, it is one of the most popular bars in the city. Jennie and most of the others with me are members of an organic produce cooperative located some hours away. I'd been in close touch with them ever since I'd helped them inaugurate a Socrates Caf\u00e9 several years ago. Today is one of their delivery days in the city, and when I told them of my project, they suggested we meet at this pub, perhaps to put us in a more revolutionary spirit.\n\nJennie, the co-op's associate director, says next, \"It turns out that the phrase I'd thought was the soul of the Constitution was actually in Lincoln's Gettysburg address. He told the 15,000 people who came out to hear him speak at the Union victory ceremony that the Civil War hadn't been fought just to preserve the Union, but to launch a 'new birth of freedom.' Lincoln claimed that with the Civil War won, a 'government of the people, by the people, for the people' would at last be created and 'shall not perish from the earth.'\"\n\nShe looks at us. \"But there's never been such a government in the United States. And there will never be unless the Framers of the Constitution themselves are democratically elected. It's not enough for the very first words of the preamble to be 'we the people.' Because this doesn't gloss over the fact that our Constitution was created by 55 well-off white males who met behind closed doors. They even created an article for how this Constitution of theirs would be ratified that virtually guaranteed it would become the law of the land. Article VII required the approval of just nine of the original thirteen colonies. And guess what: exactly nine states voted to ratify it\u2014not by direct vote of the entire population, but via conventions mostly attended by well-off property holders.\"\n\nThis prompts Sylvia to say to me, \"If this Constitution project of yours is to feature a government of, by, and for the people, it has to include an article detailing a democratic process for ratification. But even more importantly, there needs to be, in this or a separate article, a spelled-out process for how Framers for drafting the Constitution are chosen. Our existing Constitution doesn't have an article addressing this because its authors simply appointed themselves.\"\n\n\"To me,\" she goes on, \"it's not just the form of government created that determines whether it's of, by, and for the people, but the process by which the form of government is decided upon. And this requires a totally open process for choosing Framers.\"\n\nI ask what kind of specific process they'd recommend.\n\nFaith is the first to respond. \"We could create an article that deals both with how to choose Framers and how to ratify the Constitution they construct. It might say, 'Each state shall name, by democratic voting procedures, X number of delegates to take part in a constitutional convention. Upon construction of a Constitution by the convention delegates, its final ratification will require the approval by a majority of voters in each state. Only then shall the Constitution be deemed established between the States.' \"\n\n\"It would be better if delegates to be Framers at a constitutional convention are chosen randomly,\" says Jennie, \"just as people are picked to sit on juries. That way, the Framers would be representative of the populace as a whole.\"\n\nLooking at Faith, she says, \"By a 'democratic voting process' for delegates, I assume you mean a conventional election with traditional campaigning, followed by a vote in which people choose from among a list of candidates. If that's so, that would mean that those with the most money to campaign and to get their message out would most likely be elected as delegates. That would mean that those elected to be Framers would not be those who are most representative of Americans. The advantage of a random selection process for gathering delegates\u2014as long as the random sample is large enough\u2014is that it assures that there'll be a true cross-section of Americans serving as Framers.\"\n\nThen Sylvia says, \"I was a member of the Students for a Democratic Society in the 1960s.\" She is referring to the New Left student activist movement founded in 1960, which spearheaded broad-scale student protests against U.S. policy in Vietnam. The SDS flourished until 1969, when it unraveled over differences in the tactics supported by various factions.\n\nShe has with her a copy of the original SDS constitution. \"Its preamble, approved at our convention in 1962, states that ours was 'an association of young people on the left' that 'seeks to create a sustained community of educational and political concern' and that 'maintains a vision of a democratic society, where at all levels the people have control of the decisions which affect them and the resources on which they are dependent.' \"\n\n\"The SDS constitution claimed that its mission was to create a society with government of, by, and for the people,\" she continues, \"even though it makes clear its belief that just a subset of Americans\"\u2014she again refers to the SDS constitution\u2014\" 'liberals and radicals, activists and scholars, students and faculty' are those most adept at realizing 'a democratic society, where at all levels the people have control of the decisions which affect them and the resources on which they are dependent.'\n\n\"But I've often since wondered, can one small segment, no matter how well-intentioned\u2014whether the Framers of our original Constitution, or of the SDS constitution\u2014ever really create a government of, by, and for the people, if most of the people are left out of the process of creating the constitution itself?\"\n\nThis brings Rafael to say, \"I'm not sure that a constitution has to include the participation of potentially everyone, or of even a random sample, in order to create a government of, by, and for the people. Maybe what matters most is that the final product\u2014whether devised by two people or two hundred million\u2014details how to make such a government come true.\"\n\n\"The constitution of the nonprofit social justice group I worked for was created by just two people,\" he goes on. \"But it described itself as an organization in which all involved in our group in any way made decisions as equals. It claimed to be of, by, and for all those both working for the organization and those being served by it. And in the policy-making and program implementation process, everyone's vote counted equally. But only staff members got to take part in these votes.\n\n\"I proposed at a staff meeting that we change our policy for how we decide policy. I suggested that if our group is really going to be what its constitution claims, then the people we serve must play an equal part in the decision-making process. That'd be real empowerment, really democratic. I was accused of being too idealistic, as if that somehow is an affliction. Those opposed to my view argued that the people we served were too down and out to take part in making decisions on their own behalf. My opinion was: 'You want to make them less down and out? Let them decide with us, as equals.' They thought I was nuts. I left the organization soon afterwards.\"\n\nThen he says, \"Now I'm thinking that maybe Sylvia's suggestion is right on target, that a constitution\u2014whether for a nonprofit, a gated community, a radical movement, or a country\u2014has to be created by all those who are impacted, or at least by a genuinely representative and random sample of all those who will have to live by its rules. That way, it will really be a product of 'we the people.' \"\n\nTo which Faith says, \"I'm starting to come around to this viewpoint too. If the constitution created by a randomly selected group of Framers calls for a type of government in which all those involved will make decisions as equals, there'll be a greater likelihood that that'll actually become a reality.\"\n\nConstitution Caf\u00e9 Article: Each State shall appoint to a Constitutional Convention delegates authorized to construct a federal Constitution. The delegates in each State shall be chosen randomly. In order to have a viable random sample, the number of delegates chosen per respective State to be Framers shall be fifty times the total number of that State's members of Congress. When said Constitution is completed by the Framers, it will require the Ratification of a majority of voters in each of the States, and only then shall it be sufficient for the Establishment of this Constitution between the States so ratifying the Same.\n\nNot So Grand Convention\n\nAs the Constitution-making process unfolded, Jefferson was regularly apprised of its goings-on by James Madison. Jefferson grew increasingly at odds with his fellow correspondent. Joseph Ellis, acclaimed historian of the American Revolutionary period, relates that \"Jefferson was writing Madison from France with expressions of great doubt about the powers granted to the federal government over domestic affairs, powers that Madison had championed more effectively than anyone else at the Constitutional Convention.\" By the time the Constitution was crafted, Jefferson had come to the conclusion, as he told John Adams, that \"all the good of this new constitution might have been couched in three or four new articles to be added to the good, old and venerable fabrick\" of the Articles of Confederation.\n\nMany delegates shared Jefferson's views, and so they did not consider their convention so grand. When the new Constitution was presented for the Framers' final stamp of approval before proceeding to ratification by the states, several refused to sign it. Many other delegates, who from the outset had been strongly opposed to the idea of writing a new Constitution instead of revising the Articles, as had originally been planned, had long since left. Political scientist John Vile contends that these delegates didn't so much \"attribute the problems [in the Articles] to the existing governmental structure as to the consequences of emerging from under colonial denomination.\" They were confident that in due time, as Americans became more seasoned in self-rule, any challenges emerging from the decentralized form of governance set forth by the Articles would be surmounted. Furthermore, Vile notes, they were as wary as Jefferson of \"drastically entrusting the national government with significantly increased powers that might be abused.\"\n\nWe the People?\n\nThose who stayed on to frame the Constitution chose the words \"we the people\" to lead off the Preamble in a deliberate attempt to create the appearance that the document was \"an act of law by the people of the United States for their entire land.\"\n\nOn September 17, 1787, the U.S. Constitution, after being ratified by just nine of thirteen states in so-called popularly elected ratifying conventions, was adopted. On March 4, 1789, it officially went into operation as the new law of the land, replacing the Articles of Confederation. While his overriding concern at the time was to push through its approval, James Madison himself later confessed that the ratification protocol set forth by the Framers in Article VII of the Constitution was wanting: \"It is admitted that the convention have departed from the tenor of their commission. Instead of reporting a plan requiring the confirmation of all the states, [the Framers] have reported a plan which is to be carried into effect by nine states only.\"\n\nEven \"the Father of the Constitution\" appeared to concede that a process of ratification that required such meager support diminished the Constitution's integrity as a binding document.\n\nOrdinary Framers\n\nOnly one state at our nation's founding devised a democratic plan for choosing the Framers of its state constitution.\n\nBy the early summer of 1776, most legislatures were preparing, as historian Gary B. Nash puts it in The Unknown American Revolution, to have their state constitutions \"up and running before July 4,\" when the Declaration of Independence would be issued. But in constructing their constitutions, many state legislatures took shortcuts, amending their English charters \"simply by deleting all reference to the English Crown. Letting it go at that, they continued to govern themselves as before. By itself, the decision not to change was a slap in the face of those who wanted to write afresh on a clean slate.\"\n\nPennsylvania was the exception, making the emphatic decision to change by allowing any male, including those who did not own property or pay taxes, not only to vote for the candidates vying to serve as Framers of the new state constitution, but to be candidates themselves. Predictably, Pennsylvania's elites waged a forceful campaign to dissuade anyone from actually voting for a common person to serve as a Framer. But common people\u2014farmers, shop owners, artisans\u2014had a strong advocate in James Cannon, a militia leader and math tutor. \"Honesty, common sense, and a plain understanding, when unbiased by sinister motives,\" are the best criteria for choosing constitutional delegates, Cannon asserted. He further maintained that not only are regular people \"fully equal to the task\" of framing a constitution, but in fact are \"the most likely to frame us a good Constitution.\"\n\nBut the elite interests, according to Nash, made the case that only people like them\u2014wealthy, well educated, and familiar with political proceedings\u2014could craft a constitution that perpetuated the traditional virtues of \"decency, and respect, and venerations introduced for persons in authority.\" Cannon, however, exhorted Pennsylvanians \"to reject the upper-class view that in order to write a constitution, men needed 'great learning, knowledge in our manners, trade, constitution and policy of all nations.'\" His Jeffersonian perspective was that common people would best serve \"the common interests of mankind.\" On the other hand, Cannon contended, \"Great and over-grown rich men will be improper to be trusted. . . . They will be too apt to framing distinctions in society, because they will reap the benefits of all such distinctions.\"\n\nCannon's arguments proved persuasive: the great majority of Pennsylvania delegates chosen to serve as constitutional Framers hailed from humble origins. So it was that in the summer of 1776, while the Second Continental Congress was convening at the State House in Philadelphia and preparing to break all ties with Britain, just down the corridor was taking place what historian Sean Wilentz calls a \"now-obscure State Convention that drafted and approved the most egalitarian constitution produced anywhere in Revolutionary America.\"\n\nThe Pennsylvania constitution called for a government that was more of, by, and for the people than any other constitution crafted on the federal or state level. It declared that government's only purpose is to further the \"common benefit, protection, and security of the people, nation, or community,\" and so is expressly \"not for the particular emolument or advantage of any single man, family, or set of men who are only part of that community.\" Though it excluded women and slaves from participating, it nevertheless was remarkably progressive for its time. It established a blueprint for governance that allowed ordinary Pennsylvania males to have unprecedented say in political affairs. Among other things, Pennsylvania's constitution contained a pioneering provision that stipulated that no bill passed by the General Assembly could become law until it had been printed and distributed to all male Pennsylvanians for their consideration. As Gary Nash points out, \"This allowed time for public discussion of each law before the next annual election of representatives.\" Only if the male populace as a whole approved of the legislation did it then become official law.\n\nThe government that Pennsylvania's Framers brought into being, Nash notes, was \"a heavy blow to wealthy merchants, large property owners, and assorted conservatives.\" From the moment the constitution was enacted, these anticonstitution forces plotted its demise, and in 1790, they prevailed. In Sean Wilentz's words, \"Time would prove unkind to the Pennsylvania Constitution of 1776.\n\nAlarmed moderate and conservative notables from around the state, seeing democracy boiling over, quickly regrouped as the Republican faction, and they capitalized on some overbearing miscalculations by the radicals. . . . The Pennsylvania radicals failed to seek approval of the new constitution through a popular referendum. . . . The decision ensured their constitution went into effect, but at the cost of alienating a considerable portion of the moderate citizenry. . . .\n\nWhen push came to shove, the Framers of the original state constitution failed to have enough faith in their fellow Pennsylvanians to allow them to vote on whether to approve the document they crafted, though they themselves had been elected by popular vote. Their decision ensured that their constitution went into effect but also ensured its demise. In September 1790, fourteen years after the radical constitution was enacted, \"the Republicans won ratification of a new state constitution that eradicated most of the democratic provisions of its predecessor.\" This successful effort overturned a constitution that might over time have paved the way, in Pennsylvania and perhaps elsewhere, for realizing the Jeffersonian ideal of universal participation in governance.\n\nMaking Amends\n\nArticle V of the Constitution: \"The Congress, whenever two-thirds of both Houses shall deem it necessary, shall propose Amendments to this Constitution, or, on the Application of the Legislatures of two-thirds of the several States, shall call a Convention for proposing Amendments, which, in either Case, shall be valid to all Intents and Purposes, as part of this Constitution, when ratified by the Legislatures of three-fourths of the several States, or by Conventions in three-fourths thereof, as the one or the other Mode of Ratification may be proposed by the Congress. . . .\"\n\n\"If Congress hadn't tampered with Article V, which describes how an amendment to the Constitution can be ratified, I feel sure we'd have eventually passed the Equal Rights Amendment,\" says Karina, referring to the proposed amendment, known as the ERA, that stipulated that equal rights under any federal, state, or local law could not be denied on account of a person's gender. It was approved by Congress for possible ratification in 1972.\n\n\"A revised Article V of the Constitution can make sure that the injustice done to us activists who'd labored for over ten years for its ratification won't be repeated,\" she says.\n\nI'm in Lafayette Square in New Orleans. I had contacted about 50 people who had given me their email addresses when I was last in the city for a book event, just before Hurricane Katrina struck, and came to find that most had moved away. Still, a smattering of folks are on hand for my gathering in this revitalized city (shortly after our dialogue, the BP oil disaster occurred). We are seated in a circle in the square that was founded in 1788 and named after the French aristocrat Marquis de Lafayette.\n\n\"Before my involvement in the effort to have ratified the Equal Rights Amendment,\" Karina says now, \"I'd thought Article V was cut and dried.\"\n\nArticle V establishes the process for ratifying a proposed amendment to the Constitution. In order for a proposed amendment to have a chance of being ratified, either two-thirds of the members of both chambers of Congress first have to approve it, or two-thirds of the legislatures of all the states (34 all told) must call for a national convention at which it would have to pass muster. Then, in order for the proposed amendment to become part of the Constitution, it either must be ratified by legislatures of three-fourths of the states (38 in all), or ratified by state conventions in three-fourths of the states.\n\nKarina goes on to say now, \"In the case of the ERA, Congress added another hoop to ratification that was never envisioned by the Framers. It established a seven-year deadline, from the time it approved the ERA for possible ratification in 1972, for it actually to be ratified.\"\n\nAfter successfully making its way through both chambers of Congress, the ERA briskly advanced toward ratification. In the first year alone, 22 states approved it for ratification. But then the amendment stalled as conservatives launched a well-orchestrated opposition drive. In the following years, 13 more states approved it, but that still left it three votes short of the total number needed. When it became evident to the amendment's principal proponents, such as the League of Women Voters, that the ERA would not be ratified by the 1979 deadline, they appealed to Congress to extend the deadline indefinitely. Congress did extend the deadline by three more years, to 1982; but it refused to completely eradicate the deadline. By then, the Reagan era was in full swing, and with the likes of the conservative political activist Phyllis Schlafly bending the president's ear, the ERA was removed from the Republican Party platform. Consequently, the ERA was never ratified.\n\n\"My gripe isn't so much with the Republicans of that time as it is with the actions of Congress in 1972 when it gave the ERA its blessing but with an arbitrarily imposed deadline for its ratification,\" says Karina. \"We should have been able to wait out the conservative era, and then we'd have gotten the three additional states needed for ratification.\n\n\"So a new and improved article on the amendment process should establish that there is absolutely no deadline for seeking ratification of a proposed amendment that has been approved for possible ratification.\"\n\n\"If anything, I'd have this article we're making impose an across-the-board deadline for all proposed amendments,\" says Hal, who sells bootleg DVDs when he isn't studying to be a paralegal at a local community college. \"I'd put the deadline at a decade, the total length of time that Congress gave ERA proponents. If you can't ratify a proposed amendment in ten years, then it's obvious that the people in most states do not want to be part of the Constitution.\"\n\nCarl, who sometimes uses Lafayette Square as a base to sell his art, shakes his head in disagreement. \"The Twenty-Seventh Amendment, the latest to be ratified, was approved by Congress for possible ratification in 1789. It was finally ratified in 1992\u2014over two hundred and two years later! It just goes to show that you never know when the time has finally come for a proposed amendment to be ratified. So it's best by far for the article on making amendments to state that Congress can't impose a ratification deadline.\"\n\nThen he says, \"I'd still like to see ten more states ratify the Child Labor Amendment, which would give Congress sole authority to determine child labor laws, taking this out of the purview of the states, where child labor protections vary a great deal. It was approved by Congress in 1924 and has since been approved by twenty-eight states. And for this proposed amendment Congress didn't impose any sort of deadline for ratification. Surely ten more states eventually will see the wisdom of having it as part of our Constitution, and it will be ratified. And just as surely, if Congress hadn't arbitrarily imposed a deadline, I bet that three more states would have eventually seen the wisdom of approving the ERA. The beauty of the open-ended time frame that the Framers conceived of\u2014and which this revised article will etch in stone\u2014is that it gives all the time in the world for our collective wisdom to reach the point where progressive amendments are ratified.\"\n\n\"I agree that there shouldn't be a deadline on constitutional amendments that are approved for possible ratification,\" says Janet, an emergency room nurse. \"But that's not the same thing as saying I agree that they should have an endless opportunity to become part of the Constitution. What this article we're crafting should make clear is that once a proposed amendment is approved for possible ratification, then each state has one\u2014and only one\u2014opportunity to vote on it, up or down. With the ERA, its proponents sought again and again to get state legislatures to change their minds after they had voted to reject it for ratification. Meanwhile, several states tried to have their yes votes changed to a no.\n\n\"But because the existing Article V isn't clear on the subject, Congress decided that states that voted to reject the ERA could take a revote and possibly approve it. Yet it also ruled that states that had approved it but then, over the passing years, decided they wanted to rescind their vote, couldn't do so. Talk about arbitrary. So this article should say that once a state has voted for or against a proposed constitutional amendment it doesn't have the option of later changing its vote. If, after all states have taken a vote, and the amendment has not been approved for ratification, then you have to start from square one and try, try again.\"\n\n\"I wish we could start all over again with the ERA,\" says Karina. \"This amendment has been reintroduced in Congress every single year since the ratification deadline passed in 1982. But it has never again been put to another vote in those chambers. We can't start all over again if we can't even get it put to another vote by Congress.\"\n\n\"I have to say I agree with you on that point,\" Hal says to her, \"even though we are political opposites in every other respect.\"\n\n\"The Federal Marriage Amendment was introduced as a joint resolution by several members of Congress in May 2003,\" he goes on. \"This resolution has never been debated by the House. Yet the millions of Americans who like me believe marriage should only be allowed between a man and a woman are stuck with their cowardice.\n\n\"An amended amendment process should require Congress to debate any proposed amendment that is introduced by one or more of their members. Obviously this wouldn't guarantee that most proposed amendments would be approved by the two chambers, much less go on to be ratified. But at least they would be formally considered by Congress, rather than tabled indefinitely, which is the same thing as killing them.\"\n\nKarina thinks about this. \"As much as I detest this particular amendment Hal wants to have passed, I do agree that Congress should have to debate it and vote on it, as well as the perennially reintroduced ERA.\"\n\n\"You're both overlooking that there's already another way to have an amendment introduced,\" says Janet. \"Article V of the Constitution says that 'on the application of the legislatures of two-thirds of the several States,' Congress is obligated to 'call a Convention for proposing Amendments.' This means that the states can go around Congress and initiate the amendment process themselves. This route was meant by the Framers to prevent Congress from bottling up constitutional amendments they didn't care to consider. Yet it's never once been used in our nation's history.\"\n\nBut it was almost used. In the early twentieth century, the various state legislators asked the U.S. Senate to approve a proposed amendment calling for the direct election of senators. If ratified, this would have replaced the constitutional provision stipulating that senators must be appointed by the state legislatures. When the Senate refused to approve this proposed amendment, the state legislators began the process of initiating a convention to ratify it themselves. This prompted the Senate to quickly give in; it voted to have the amendment forwarded to the states, where it was put to a vote according to the traditional amendment route, and ratified. The Seventeenth Amendment to the Constitution was adopted on April 8, 1913, and obviated the need for the states to follow through with plans to hold a convention to amend the Constitution. There has never been another attempt by the states to organize an amendment convention. The Senate capitulated in this one instance because if a convention of the states had been held, not only would this particular amendment have been debated, but many other amendments likely also would have been proposed and considered, possibly further diluting the power of the legislative branch\u2014because once a convention to propose amendments is convened by the states, as the members of the Senate well knew, those attending can propose as many amendments as they care to.\n\nAfter a considerable pause, Karina says, \"I like how, when the Twenty-First Amendment was ratified, it repealed the Eighteenth Amendment that had enacted Prohibition. For a brief time, the amendment process was used by Americans as a way of engaging in a debate over their core values. If we'd continued to use it for that purpose, we'd have many other amendments by now.\n\n\"For instance, there might be any number of amendments on marriage ratified and repealed and ratified over the years. While I hope that same-sex marriages would be allowed, it might be, heaven forbid, that this Federal Marriage Amendment might be the first one approved. But this might in turn be the impetus for generating a strong movement to repeal it in favor of one allowing same-sex marriage.\"\n\nThen Carl says, \"Since the Twenty-Seventh Amendment was approved in 1992, after a two-hundred-plus-year wait, no additional amendments have been ratified. I know some people think that's a sign of how near perfect our Constitution is. But to me it only goes to show how intimidating the current amendment process is, and what a need there is for it to be changed.\"\n\nConstitution Caf\u00e9 Article: The Congress shall debate any proposed amendment to this Constitution that is sponsored by at least five percent of members of either Chamber. Proposed amendments shall advance to the ratification process whenever two-thirds of both Houses approve proposed Amendments to this Constitution. Alternatively the Legislatures of two-thirds of the several States have authority to call a Convention for proposing Amendments. In either case, such Amendments shall be valid to all Intents and Purposes, as part of this Constitution, when ratified by the Legislatures of three-fourths of the several States, or by Conventions in three-fourths thereof. There shall be no deadline for ratifying Amendments that have been approved either by two-thirds of both Houses or by Legislatures of two-thirds of the several States.\n\nAll People to the Power\n\nJefferson believed, as he told the Greek intellectual Adamantios Coray, that the amendment system the Framers put in place was \"too difficult for remedying the imperfections which experience develops from time to time.\"\n\nBut is Article V of the Constitution the last word when it comes to amending the Constitution?\n\nConstitutional scholar Bruce Ackerman maintains in We the People that though they've never exercised it, Americans can ratify amendments to the Constitution without having to go through the labyrinthine process detailed in Article V. To make his case, he points out that a principal Framer stressed, during Constitution ratification debates at his own state convention, that Article V was by no means the only avenue for amending the document. During a ratification debate in Pennsylvania, James Wilson insisted that \"the people may change the constitutions whenever and however they please.\" Wilson further commented, \"As our constitutions are superior to our legislatures, so the people are superior to our constitutions,\" and so they not only \"have the power, if they think proper, to repeal and annul\" the Constitution but have \"direct authority\" for \"amending and improving\" it.\n\nGiven that he was a leading Framer of the Federal Constitution, Wilson's words carried great weight. His statements were widely noted and quoted, and went a long way toward pacifying anti-Federalists' concerns about the proposed Constitution, paving the way for its passage.\n\nJefferson told Adamantios Coray that \"a greater facility of amendment\" should be provided in the Constitution so that it could be revised periodically. An easier amendment process, he believed, would \"maintain [the Constitution] in a course of action accommodated to the times and changes through which we are ever passing.\" But James Madison was opposed to any process that might bring about frequent changes to the Constitution, since this to him \"would, in a great measure, deprive the government of that veneration . . . without which perhaps the wisest and freest governments would not possess the requisite stability.\"\n\nBecause this Madisonian perspective has prevailed over Jefferson's, the tendency today is to believe that the longer-lived and more unchanging our Constitution, the better. Constitutional scholar Albert Blaustein offers an encomium to our Framers: in devising a \"constitutionally limited republic,\" they engineered the creation of an ideal \"regime that balanced order and liberty.\" Further, to Blaustein, the fact that our Constitution \"has withstood the test of time,\" and rarely been amended since the addition of the Bill of Rights, is proof that it is immutable. But what if its longevity and relative stasis have impeded the type of inventiveness that might better enable a democratic society to evolve? Or, might we at least challenge Blaustein's argument that the Constitution has served us well? Robert Dahl, for one, wonders, \"How well does our constitutional system meet democratic standards of the present day?\"\n\n\"We have not yet so far perfected our constitutions as to venture to make them unchangeable,\" Jefferson held. He lamented how \"some look at constitutions with sanctimonious reverence, and deem them like the arc of the covenant, too sacred to be touched. They ascribe to the men of the preceding age a wisdom more than human, and suppose what they did to be beyond amendment. I knew that age well; I belonged to and labored with it.\"\n\nThe men of Jefferson's age had no monopoly on wisdom, not even for their own time, and he believed that, by attaching a transcendent sanctity to a Constitution created by a small group of his contemporaries, later generations would do the Framers themselves, and democracy, a grave disservice. \nIV\n\nCommons and Goods\n\nThomas Jefferson was not exactly a \"unifier,\" then or now. He was certainly not a consensus choice for president\u2014his election was bitterly contested. Jefferson had agreed to come out of retirement and serve as leader of the Democratic-Republicans after party leaders convinced him that he stood the best chance of defeating incumbent John Adams, whose party platform\u2014which tried to reconcile the various factions within the Federalist camp\u2014promoted a heavy-handed, paternalist approach to governance that threatened to even further encroach on individual liberties. The Federalists had already wrested power from the people with the passage of the Alien and Sedition Acts in 1798, which extended the amount of time it took for immigrants to become citizens and gave the president the power to deport any alien \"dangerous to the peace and safety of the United States,\" to deport aliens whose home countries were at war with the U.S., and to fine or imprison (or both) those whose writing was considered \"false, scandalous, and malicious\" against the government. The election ended in a tie\u2014between Jefferson and Aaron Burr (Adams finished third). According to the Constitution, it would now be up to Congress to choose the next president. It took 36 ballots over six days before Jefferson was elected, with Burr ultimately serving as vice president.\n\nNearly twenty years later, Jefferson called the pivotal election \"as real a revolution in the principles of our government as that of 1776 was in its form; not affected indeed by the sword, but by the rational and peaceable instrument of reform, the suffrage of the people.\" But even if, in this \"revolution of 1800,\" Jefferson did \"restore the democratic impulse of the American Revolution after its betrayal by the Federalists,\" as Joseph Ellis contends, the election itself was just barely peaceable, and rationality was often missing in action.\n\nBoth parties engaged in . . . negative campaigning, an assault on their adversary's program rather than an emphasis on their own platform. The Federalists, for instance, left no stone unturned in their attempts to link the Republicans with the bloody excesses of the French Revolution. Jefferson and his adherents, they charged, were \"artful and ambitious demagogues\" who led \"discontented hotheads.\"\n\nThe Republicans were no slouches either when it came to attacks. They \"drew on early political writings by Adams to tag him as a monarchist.\" As Sean Wilentz notes, the \"attacks and counterattacks,\" coupled with \"the harsh ideological as well as personal tone,\" served to harden partisan loyalties.\n\nIn the aftermath of this contentious election, the electorate remained deeply divided, their divergent principles apparently unbridgeable. In his inaugural address, Jefferson exhorted Americans to come together and \"unite in common efforts for the common good.\" It appears to be the only time in his voluminous writings and speeches that he specifically used the term \"common good,\" though he did use the phrase \"public good\" on occasion, a term that doesn't necessarily take into account a citizen's private welfare. While many or most common goods might be public ones, some\u2014such as the right to life, liberty, and the pursuit of happiness\u2014clearly have both public and private implications. Perhaps Jefferson didn't make more mention of the \"common good\" because he took it for granted that all citizens valued it as he did. After all, his native Virginia is a commonwealth rather than a state, and \"commonwealth\" translates as \"for the common good.\" To have a common \"weal\" is to have a share in government, and as a result, to be mutually responsible for cultivating the well-being of fellow citizens.\n\nBut how does society go about putting this into practice? How do its members look out for one another? Is the government responsible for spreading the wealth and seeing to it that each citizen has a share of certain basics that he or she needs in order to be a full contributor to society?\n\nDuring Jefferson's eight years in office, he did away with Federalist hegemony and \"enabled the nation to move piecemeal from the habits of 1800, laced as they yet were with restrictive customs from colonial days, toward egalitarianism and democratization.\" His primary legacy was that, because of his implemented policies, \"despite wide inequities in the distribution of wealth, most free citizens believed they enjoyed the same rights and opportunities as all other free men.\"\n\nIn a national poll conducted by the nonpartisan Center for the Constitution, headquartered in the lifelong residence of James Madison in Montpelier, Virginia, a broad cross-section of Americans was asked whether the Constitution charges the federal government with realizing the common good. Less than half responded in the affirmative. The director of the Center was himself disappointed in the results: \"Sadly, almost half the population does not feel that the government was designed with this goal.\"\n\nYet the Constitution makes no explicit mention of the common good. It does, though, authorize Congress to provide for \"the general welfare.\" Are the common good and the general welfare one and the same?\n\nCommon Goods, Bads, and the General Welfare\n\nFrom Article I, Section 8, Clause 1 of the Constitution: \"The Congress shall have Power To . . . provide for the . . . general welfare of the United States. . . .\"\n\n\"The Preamble says that a big part of the Constitution's purpose is 'to promote the general welfare,'\" says Harry, a 45-year-old information technology specialist. \"Then, when it gets down to listing Congress's powers, it says its job is to 'provide for the . . . general welfare of the United States.' I prefer 'promote' rather than 'provide' for the general welfare. Because you don't provide for the general welfare, you promote it, or contribute to it, by providing for the common good. Otherwise, there's no general welfare to promote. So a new article that deals with this subject should decree that Congress promotes the general welfare by providing for the common good.\"\n\nI'm with about a dozen others in the expansive food court at the Mall of America. Literally the goods capital of North America, the nation's largest mall is situated in the Twin Cities suburb of Bloomington, Minnesota. We strain to hear one another over the peals of glee and fright coming from riders on the nearby indoor roller coaster. A security guard has been hovering around us, eyeing my constitution caf\u00e9 sign. He decides that if I'm an oddity, I'm of the harmless variety, and moves on.\n\n\"Aren't those two terms interchangeable?\" asks Audrey, the owner of a Wisconsin-based travel agency who with her teenage son makes a bimonthly trip to the mall.\n\nHarry retrieves a dictionary from his shoulder bag. \"Welfare, as defined by Merriam-Webster, concerns the 'state of doing well especially in respect to good fortune, happiness, well-being, or prosperity.' So I take the 'general welfare' to mean that state of doing well in which society as a whole is happily prospering. That can only happen if essential common goods are provided to everyone, in order to promote this state of doing well, since common goods are the building blocks of the general welfare.\"\n\n\"I'm not sure that providing for the common good is how you go about promoting the general welfare,\" says Nadine, who works at a nearby kiosk that sells an eclectic mix of new and used CDs, and who has joined us on her break. \"Common goods are like open bins that all Americans can grab stuff from. You're giving up something\u2014like, you're relinquishing a percentage of your income\u2014so that tax dollars can be put towards acquiring certain goods to be put into a public bin, because Congress has said you have to for the sake of society's prosperity. But if contributing to the common good takes away from my personal prosperity and happiness\u2014like, if I'd prefer to keep that income for myself so I can buy private goods that I consider more important to my personal well-being\u2014then it detracts from the general welfare, which is the sum of all individual welfares.\"\n\n\"You make common goods sound like common bads. But a true common good is something that's as good for each individual as it is for society as a whole, the best investments we can make in one another,\" Mildred, 85, says to her. Mildred retired over two decades ago as an elementary-school teacher. \"People only make sacrifices grudgingly if there's no sense of common cause or shared purpose. A true common good is as much for your benefit as it is mine and everyone else's, and as such, should improve both the individual and general welfare.\"\n\nThen Mildred tells us, \"My husband was in D-Day in World War II. Those of us here on the home front pitched in to help make sure all the troops had everything they needed to do their jobs well. Not only did we willingly give up luxuries, we deprived ourselves of essentials, in order to contribute to our common defense, which is a type of common good. We had a shared purpose, and understood that making such sacrifices was in all our best interests.\n\n\"We grew gardens in our front and back yards and in vacant lots so the produce at farms could be reserved for our soldiers. There were over twenty million 'Victory Gardens,' and over ten million tons of fruit and vegetables were harvested from them. We took part in drives to collect scrap metal, nylon, silk, anything that could be used for the war effort. Commodities from food to oil were rationed\u2014in some cases voluntarily, in others by order of Congress\u2014and hardly anyone complained. There was a surplus of goodwill. I became a more independent person by helping spearhead collection drives, and at the same time, I became more connected to and concerned about the welfare of my community, really to that of America itself, since all these local projects contributed to its overall general welfare.\n\n\"After the war, I joined efforts that saw to it that even the poorest communities enjoyed essential 'public bin staples'\u2014safe drinking water, cleaner air, public parks, a decent education. I eventually went back to school and earned my degree in education. I was a teacher for over thirty years in an underserved community. That's something I'd never have done if not for the 'sacrifices' I'd made during the war. I've discovered many times over that the more I give of myself, the more I get back in return.\"\n\n\"How would today's generation determine which common goods are needed in order to best promote the general welfare?\" I ask.\n\n\"Just look at those goods our members of Congress provide themselves at taxpayer expense,\" says Harry. \"They have the best taxpayer-subsidized health care plan on this planet. In fact, Congress exempted itself from the health care 'reform' legislation signed into law so it could continue to be guaranteed this 'public option' that it denied to the rest of us. All members of Congress also enjoy such 'uncommon goods' as exceptional child care plans, elder care, and guaranteed retirement income exponentially higher than what's provided to the rest of us by Social Security. They don't have to spend sleepless nights worrying about how to pay their health care premiums or for day care for their children, or how they're going to live with dignity after they retire, because of the goods they authorize themselves from a private or exclusive bin\u2014even though these goods are paid for by us. Yet as far as Congress is concerned, the rest of us should have no entitlement to these goods ourselves.\"\n\n\"So, this article needs to stipulate that members of Congress can't enjoy such goods if they're denied to their constituents,\" Audrey says now. \"Congress is betraying its duty to promote the general welfare by providing only for themselves, on the public dole, goods that we all need in order to prosper.\"\n\n\"If that's promoting the general welfare, then it sounds too much like social welfare to me, and so a common bad,\" says Marvin. \"Congress shouldn't be guaranteed these goods, and neither should any of the rest of us. Where do you draw the line with these basic 'must haves' we provide for one another? I can't have a house until everyone else has this good too?\"\n\n\"You know what everyone else owes me?\" he goes on. \"Not a damn thing. My parents were as poor as church mice during the Great Depression, but they refused any part of Franklin Delano Roosevelt's make-work programs. The Supreme Court scandalously ruled that they were legal because they promoted the general welfare, but my parents, who ran a grocery, believed they just made people dependent on government. They weathered those tough times. By the time I was born, during the tail-end of the Depression, they were doing okay.\"\n\n\"When I got back from Vietnam,\" he goes on, \"I went to school, got my degree in business, started a company with a few Marine buddies, and after years with a lot more downs than ups, I finally made it. My father would have been proud.\"\n\nHe holds out his two hands. \"These are my insurance, when I use them with my head and heart. Anyone who uses them wisely in his professional pursuits will be able both to provide for himself and make the greatest contribution to society. People who live by their wits and don't look to be rescued by the government when times are tough are the greatest providers of our country's general welfare.\"\n\n\"Did you attend college with the GI bill?\" I ask him. \"I ask because that's what enabled my father to earn a college degree after his stint in the Army. He credits the GI bill in big part for his professional success.\"\n\n\"Yes, I did, and I'm grateful as hell for that,\" Marvin replies. \"Society believed it owed that 'good' to all of us who went to war on behalf of our country's common defense. I took advantage of that and other veterans' benefits. It definitely gave me a leg up in the world, and made it possible for me to do things with my life that my parents couldn't have dreamed of doing.\"\n\nHe looks at me. \"So you're suggesting that I am obliged to the government to some degree for my own success, even if it was sort of quid pro quo.\"\n\n\"I don't compare my or anyone else's sacrifices to those of our soldiers,\" Mildred says to Marvin now. \"But there's other people today who make sacrifices every day in their own way to keep this country strong. They work two and even three jobs without complaining, taking on work others turn up their noses at, to support their families. And they're still hurting, because they don't have what they need\u2014basic goods like health care and day care\u2014so they can do even better for themselves and, really, for the nation, to keep our economy going strong.\n\n\"We need to take a greater interest in their welfare, if for no other reason than that in doing so, we're helping our own welfare, and America's too. Isn't that what America is all about at its best\u2014looking out for one another, taking an interest in one another's well-being, so we can all make the most of our lives? Making sure everyone is provided with certain basic common goods serves this objective.\"\n\nMarvin looks straight at her, organizing his thoughts. \"Just maybe there's a couple or few basic common goods that all folks should have so they can contribute better to the general welfare, as long as those on the receiving end are every bit as much on the producing end, giving far more to society than they expect to receive. I may be a conservative, but I agree with John F. Kennedy, who said, 'Ask not what your country can do for you, ask what you can do for your country.'\n\n\"I'll tell you what\u2014my kids and grandkids should be the last in line for receiving any goods from the public goods bin. They contribute next to nothing, yet believe they're entitled to everything.\"\n\nConstitution Caf\u00e9 Article: The Congress shall have Power to provide for the common good in order to promote the general welfare of the United States. Congress shall not have Power to provide for itself any goods that it does not provide equally to all United States citizens.\n\nA Specific General Welfare\n\nJefferson certainly didn't believe that government had the latitude to provide whatever tangible common goods they or the electorate desired. Yet some members of Congress disagreed with him. He chastised the Framers for making such an ambiguous \"general welfare clause\"\u2014the clause explored in this Constitution Caf\u00e9 exchange\u2014because it lent itself to so many conflicting interpretations. \"Aided by a little sophistry on the words 'general welfare,'\" Jefferson asserted, Congress could make the claim that \"whatsoever they shall think or pretend will be for the general welfare\" will in fact be good for society.\n\nIn a correspondence with Albert Gallatin, secretary of the treasury during his administration, Jefferson maintained that \"Congress had not unlimited powers to provide for the general welfare, but were restrained to those specifically enumerated in the Constitution.\" He insisted that the general welfare clause \"never meant that [Congress] should provide for that welfare but by the exercise of the enumerated powers\"\u2014those specific charges delegated to Congress in the Constitution\u2014\"so it could not have been meant that they should raise money for purposes which the enumeration did not place under their action.\"\n\nBy this reasoning, Congress could raise money for the common defense because the Constitution explicitly gave it this authority, but it could not raise money, say, for undertaking massive infrastructure projects such as the building of railroads, bridges, or ports, because, according to the Constitution, such projects were not under its specific purview. Jefferson claimed that the Framers clearly meant for this clause to be \"a limitation of the purpose for which they may raise money\" rather than an open-ended invitation for Congress to raise and spend money on whatever it saw fit.\n\nJefferson argued that specific constitutional amendments had to be ratified before Congress could allocate money for any project or initiative, no matter how well intentioned, that it was not specifically charged with carrying out, per the Constitution. Hence today, if Congress wanted to provide universal health care coverage, Jefferson would not be opposed to it in principle, but he would likely argue that a new constitutional amendment would first need to be ratified that charges Congress with providing this good to all Americans. And he would be livid that Congress has unconstitutionally created its own private bin with taxpayer dollars, providing itself with health care and many other goods that are out of reach for most Americans.\n\nJefferson's greatest concerns about Congress's excessively broad interpretation\u2014in fact, what he viewed as its intentional misinterpretation\u2014of the Constitution's general welfare clause stemmed from the passage of the Alien and Sedition Acts. The Federalist Congress that approved these acts made what to him was the specious claim that they had authority to do so under the general welfare clause, as they were acting in the interest of what today might be called \"national security.\"\n\nIn 1798, after their enactment, Matthew Lyon, a congressman from Vermont who shared Jefferson's view that the Federalists had egregiously overstepped their bounds, wrote a letter to a newspaper accusing the Federalist Congress and the Adams administration of acting in such a way that \"the public welfare [was] swallowed up in a continual grasp for power.\" Lyon was arrested and convicted under the very Alien and Sedition Acts that he was protesting. He was sentenced to four months in prison and fined for what at the time was the enormous sum of $1,000. While in prison, Lyon was reelected by his constituents, and in the election of 1800 that resulted in a tie between Thomas Jefferson and Aaron Burr, throwing the vote to the House of Representatives, it was Lyon who cast the deciding vote in Jefferson's favor.\n\nMind Your Property\n\nFrom Article 1, Section 8, Clause 8 of the Constitution: \"The Congress shall have power . . . To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.\"\n\n\"The Framers put it in Congress's hands 'to promote the progress of science and useful arts, by securing for limited times to authors and inventors the exclusive right to their respective writings and discoveries,'\" says Elisa, a software developer, reading from her copy of the Constitution. \"At least in theory, Congress accomplishes this by dictating patent law.\"\n\nI'm with Elisa and several others in a community center in Silicon Valley to explore the constitutional article that treats with intellectual property law. A patent grants exclusive rights to an inventor for a limited period of time.\n\nThen Elisa says, \"The case can be made that current patent law impedes the progress of science and useful arts. The language of this article needs to be changed so that Congress has clearer guidance on what it's supposed to be achieving.\"\n\nLaura, a biophysicist and university professor, nods. \"I would alter this article so it says that Congress's role is 'to promote the progress of science and useful arts for the public good.' These days, patent law mostly benefits the private good\u2014individuals and corporations. Benefiting society is of secondary importance, if that.\n\n\"I'll give an example: Multinational seed corporations routinely gain patents on seeds for disease-resistant plants. While my own view is that such seeds destroy biodiversity, these corporations argue that they greatly benefit society, and that it costs a bundle to develop them, and so they deserve patents. But granting such patents gives a handful of companies a dangerous level of control over the food production chain. It's not just that it enables the patent holders to charge high prices to farmers who've become dependent on their product, but that they can even charge farmers who don't use their seeds.\n\n\"Here's the level of absurd control that patent law allows these corporations: If the wind blows a patented seed to another piece of farmland where this seed was never intended to be used, and if this crop inadvertently sprouts plants containing the patented seed, the patent holder can and often will sue that farmer.\"\n\nAs CBS News's website reports, the biotech giant Monsanto has sued small farmers whose fields have been \"contaminated\" with their genetically modified seeds for patent infringement. In one instance, a soybean farmer was subjected to a lawsuit filed by the corporation after it came onto his property to investigate whether any of his crop was grown with Monsanto seeds. The farmer is quoted as saying, \"Pollination occurs, wind drift occurs. There's just no way to keep their products from landing in our fields.\" Yet, as CBS reports, any farmer, even if unknowingly, who has on his land patented pest- and weed-resistant seeds \"must sign an iron-clad agreement not to re-plant the harvested seed, or face serious legal consequences\u2014up to $3 million in damages.\"\n\nLaura says now, \"This points to a much bigger issue: Should anyone be able to patent the things we need to sustain us? Can that ever be for the public good?\"\n\n\"But even if we altered this article as you suggest,\" says Amanda, an artist, software designer, and autodidact, \"if the rest of this article remained intact, it still would say that the way that Congress goes about promoting the progress of science and useful arts for the public good is 'by securing for limited times to authors and inventors the exclusive right to their respective writings and discoveries.' I have problems with this, because whenever exclusive rights are granted, even for a limited time, it's the same thing as granting a monopoly.\"\n\n\"If, through the hard work of some inventors and researchers,\" Amanda goes on, \"they demonstrate that their discoveries can save lots of lives, such as through their use in new diagnostic tests or new drugs, they should be compensated generously\u2014and their lawyers can haggle with the government over what 'fair compensation' amounts to. But they should never be allowed to patent such discoveries, or at least, not for more than the most extremely limited of times. They should want to put their findings into the public domain as quickly as possible\u2014as soon as they've recovered their investment costs and made a decent profit\u2014so other scientific minds can build on their work. Patent law should facilitate this.\"\n\n\"It should, but it doesn't,\" says Laura. \"It's very much the exception rather than the rule when researchers collaborate to get all their findings speedily into the public domain. Scientists have been able to make remarkably rapid progress in early diagnosis of Alzheimer's because all the researchers in the U.S. agreed to share their findings immediately\u2014and not just with one another, but with the public at large. For the very first time, all the scientists involved in such an initiative agreed beforehand that none of them 'owned' the data, and so none of them could try to procure patents.\n\n\"Researchers of Parkinson's disease are going to emulate this beautiful model of cooperation. But my point is, a constitutional article should make this model the rule. It's a virtuous model that puts human life ahead of profit. Yet just a tiny percentage of researchers are starting to use it. I believe that they should have to, that it shouldn't be a matter of choice whether they collaborate or whether they practice in isolation, which has been the prevalent model encouraged by existing patent law.\"\n\n\"Right now, patents are for twenty years, across the board, whether you've invented a video game or a lifesaving drug derived from someone's DNA, whether you've hit upon a speedier computer processing chip or come up with a piece of solar technology that can greatly decrease our use of nonrenewable energy resources,\" says Clint, an inventor and author who holds several patents.\n\n\"Congress may think it's being fair by allowing all inventions and discoveries to enjoy the same exclusive time period to have a monopoly, but it's not being fair to society. There needs to be some sort of sliding scale of exclusivity when it comes to granting patents. Though it may seem counterintuitive, the more valuable an invention or discovery is to society, the less time it should be granted exclusivity. So silly and mindless video games like the kind I develop for pure fun and profit might have a lengthy patent. But a patent on the discovery of a gene sequence that produces proteins that help alleviate diabetes might have the most limited of exclusive times. Because otherwise, anyone else who in the course of those two decades of exclusivity might have been able to tap into and apply this discovery in other medically beneficial ways is prevented from doing so.\"\n\n\"Thomas Jefferson, our nation's first administrator of the patent system, believed that patents should be granted sparingly,\" says Amanda. \"If he were the chief patent officer today, scientists affiliated with big pharmaceuticals, biotech firms, and universities would never be granted patents on things like modified seeds, or on gene sequences that are part of everyone's DNA, or on medicines derived from plants used for centuries by indigenous groups. To him, the whole idea of inventions and discoveries in an open society is to share, not to horde.\"\n\n\"Most of all, he believed that patents were only meant to be granted for genuinely original inventions and discoveries,\" she continues. \"So, for instance, the modified seed that Laura was talking about is still just a twist on an original seed. So it shouldn't be deserving of a patent. This article we're crafting can state that patents are only granted for genuinely original inventions. That in itself would resolve a lot of the problems with current patent law. I honestly don't so much blame inventors for their extreme greed as I do Congress, because the patent law it has enacted enables and encourages it.\"\n\n\"As most here know, I'd rather not have an article dealing with patents in the Constitution at all\u2014or at least, I'd rather have an article that banishes them,\" says Elisa. \"In an ideal democracy, patents wouldn't exist, because they're almost always obstacles to the progress of science and the useful arts in ways that benefit society. That's why those of us dedicated to the open-source movement shun the patent system.\"\n\nShe goes on, \"Those who took part in the groundbreaking Alzheimer's research that Laura mentioned apparently practiced for the first and probably only time in their careers what 'open sourcers' dedicate their lives to. It's a pity that our Constitution doesn't prohibit patents altogether, so collaboration in research and development would be the rule, making our society truly open.\"\n\n\"I'm not completely agreeing with this,\" says Clint. \"There's degrees of openness for an open society, and degrees of public benefit. In a totally open society, then yes, there'd be no patents. But there probably wouldn't be so many inventions, because few would be able to survive financially unless they're one of the fortunate few who have some other stable source of income.\n\n\"In the area of software, my specialty, any developer has a choice of whether to seek a certain amount of exclusivity for his product and make a buck, or to distribute his product for free with an open-source system. I recognize that an open-source operating system like Linux is every bit as top-of-the-line\u2014in fact, has far fewer glitches and far fewer virus threats\u2014than the operating systems of the world's biggest software developers. People assume that proprietary software is always better, when in fact open-source software, for anything from animation to visual effects to word processing, can often have more user-friendly and advanced applications.\n\n\"But it comes down to a matter of choice\u2014and that's what democracy is about, choice. Without software patents, I wouldn't make money. Hundreds of thousands of other people involved in software research, development, and distribution would be without jobs. I wish I could be pure and be completely altruistic. I can't say my patented goods benefit society. They do at least occupy the time of people who might otherwise do something even more worthless. \"\n\n\"The other issue,\" Clint continues, \"is that the intellectual property rights protections ensured by the existing constitutional article in many instances help make our society even more open. Without these protections, anyone could reproduce and sell the works of writers and artists and musicians without paying them. Those artists wouldn't be able to subsist, wouldn't have the time to compose their works. That would be damaging to society because we'd miss out on so many great ideas. And here again, it boils down to a matter of choice. Lots of people offer up their creative works for free, especially when they're just getting started or trying to get a following. But they shouldn't have to go this route if they don't want to.\"\n\n\"Your point is well taken,\" says Amanda after a while. \"We just have to make sure that these protections are only granted for truly original works, which to me are works that are only marginally reliant on anyone else's work or information. The patent-granting pendulum has swung way too far in the direction of private benefit. This new constitutional article can swing the pendulum back towards the middle.\"\n\nConstitution Caf\u00e9 Article: The Congress shall have power to promote the progress of science and the useful arts for the public good by securing for limited times to authors and inventors the exclusive right to those Works deemed original, rather than mere modifications or improvements of existing Works. The limited time of exclusivity granted shall be determined by the perceived benefit of the Work to the public: the more timely the public value of the Work, the shorter the period of exclusivity that shall be granted.\n\nPatently Undemocratic\n\nAs noted in the dialogue, Jefferson was our nation's first administrator of the patent system; he was appointed to the post in 1790. Jefferson was keenly aware of the fact that a patent effectively gave the inventor a monopoly (the nation's first patent law granted up to fourteen-year exclusivity to patent holders, with another seven-year extension possible), and so he was stingy about granting patents. To the chagrin of many a patent seeker, he doled out patents only to those whose inventions he considered genuinely original. \"Certainly an inventor ought to be allowed a right to the benefit of his invention for some certain time,\" he reasoned. However, not only did Jefferson refuse to grant patents to those seeking them for modifications or improvements of existing inventions, but he also drew the line when it came to discoveries or inventions that he believed any number of people might have come up with in due time:\n\nfor to embarrass society with monopolies for every utensil existing, and in all the details of life, would be more injurious to them than had the supposed inventors never existed; because the natural understanding of its members would have suggested the same things or others as good.\n\nJefferson himself was an avid inventor. With his son-in-law Thomas Mann Randolph, he devised a plow that could be used on hillsides, making it possible to grow crops on otherwise awkward stretches of arable terrain. He was thrilled that his discovery stood to benefit fellow farmers. Even though he was forever in financial trouble and might have become more solvent if he'd sought and received a patent, it wouldn't have occurred to him to do so. Jefferson was the first to acknowledge that while he may have developed a new application for the plow, the device itself had already been invented. It would have baffled him that modifications of existing works are routinely granted patents today. To him, this cheapens the notion of what constitutes an original work or discovery at the same time it makes our capitalist economy a more pervasively monopolistic one.\n\nJefferson believed that American inventors should have as their foremost goal the benefit of society, and so their motivations should be \"based, not on profit, but on science, moderation and beauty.\" Indeed, he held that inventions and discoveries themselves are intrinsically he in nature, and as such, should be shared widely as a matter of course so others can pore over and plumb them, possibly expanding or enhancing their usefulness. By this conception, monopolies are counter to the nature and function of inventions and discoveries.\n\nJefferson asserted that the \"peculiar character\" of an original discovery or invention is such that \"the moment it is divulged, it forces itself in to the possession of everyone, and the receiver cannot dispossess himself of it.\" To him, this is a beneficial transaction for all concerned. For no matter how widely an invention is dispersed, Jefferson contended, \"no one possesses the less, because every other possesses the whole of it.\" Indeed, he argues, \"he who receives an idea from me, receives instruction from himself without lessening mine; as he who lights his taper at mine receives light without darkening me.\" This not only points to the transformative potential of discoveries and inventions, but to what Jefferson saw as their implicit moral dimension, and how they light society's way to new, more liberating possibilities.\n\nJefferson maintained that democratizing newfound knowledge was particularly important when it came to scientific findings. As he told his friend Roger Weightman, \"The general spread of the light of science has already laid open to every view the palpable truth that the mass of mankind has not been born with saddles on their backs nor a favored few booted and spurred.\" While the light of science did not remove Jefferson's blinders when it came to race matters or equality for women, it nonetheless has proven invaluable in enabling society over time to overcome irrationally held prejudices.\n\nJefferson would have been appalled that current patent law allows key areas of knowledge to be guarded by individuals and corporations for whom private gain is paramount. On the other hand, he would applaud today's open-source movement\u2014not just for its promise of rapidly diffusing knowledge in ways that lead to new insights among the disciplines that might expand human horizons, but for the virtues possessed by open sourcers, who place altruism above financial profit in their efforts to share discoveries that can benefit society. As he maintained in a letter to Jeudy de l'Hommande, \"Every discovery which multiplies the subsistence of men, must be a matter of joy to every friend of humanity.\"\n\nFor Jefferson, attempts to take credit for potentially transformative ideas, inventions, or discoveries are counterproductive. \"The question who commenced the Revolution?\" Jefferson wrote, \"is as difficult as that of the first inventors of a thousand good things.\n\nFor example, who first discovered the principle of gravity? Not Newton; for Galileo, who died the year Newton was born, had measured its force in the descent of gravid bodies. . . . The fact is that one new idea leads to another, that to a third, and so on through a course of time until someone, with whom no one of these ideas was original, combines all together and produces what is justly called a new invention. I suppose it would be as difficult to trace our Revolution to its first embryo . . .\n\nWhatever their origins, Jefferson believed that ideas must be spread expeditiously, so they can be combined and recombined in ways that further unleash human creativity and potential, and that might well have been unimaginable to the original inventor or discoverer, tinkerer, or thinker. Far more important than tracing ideas to their first embryo is facilitating ways for them to take root and then to promote their continual rebirth. Jefferson would argue that if the current patent process is not an enabler of this dissemination of knowledge, then it is in need of radical readjustment, since the whole purpose of ideas and inventions, as he told Isaac McPherson, is that they be \"freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition.\"\n\nStanding (and Sitting) Army\n\nFrom several clauses in Article I, Section 8, of the Constitution: \"The Congress shall have Power To . . . provide for the common Defence . . . ; To declare War, grant Letters of Marque and Reprisal, and make Rules concerning Captures on Land and Water; To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years; To provide and maintain a Navy; To make Rules for the Government and Regulation of the land and naval Forces; To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions; To provide for organizing, arming, and disciplining, the Militia, and for governing such Part of them as may be employed in the Service of the United States, reserving to the States respectively, the Appointment of the Officers, and the Authority of training the Militia according to the discipline prescribed by Congress.\"\n\nI am at People's Park in Berkeley, California, just a couple blocks south of the University of California campus. In 1969, the 2.8-acre plot of land owned by the university had largely been cleared, though it was not yet earmarked for a specific use. Tensions began when activists started converting the grounds into a community park without first consulting with and receiving the permission of the college administration. Radicals also had plans for the area to serve as a nerve center for organizing and conducting protests against the federal government's Vietnam policies.\n\nRonald Reagan, then governor of California, vowed to use whatever means necessary to end the takeover of the university property. On May 15, 1969\u2014a day that became known as \"bloody Thursday\"\u2014he called in several hundred police officers and sheriff's deputies, who ousted the 3,000 protesters from the park and its immediate surroundings. When word spread of what was transpiring, about 6,000 activists rallied to reclaim the park. Violence erupted. One student, James Rector, was shot to death, and over 100 were hospitalized. Reagan declared a state of emergency and deployed 2,700 National Guard troops to the area. Two weeks later, over 30,000 people marched past the now-barricaded park to protest its occupation. Though there were no more clashes at People's Park, tensions never altogether abated. The property is still owned by the university, which over the years has made occasional overtures to convert the land to uses other than a park, only to meet with stiff resistance. Today, People's Park is a popular gathering place well tended by local nonprofits in coordination with the university.\n\nMost of today's Constitution Caf\u00e9 participants were informed of the gathering by my postings on social networking sites. Among those who join me are a husband and wife. I see on the forearm of the husband a \"dog tag tattoo.\" Also known as a \"meat tag,\" these are popular tattoos among soldiers, who typically get them just before they're set to be deployed overseas. \"You're a soldier?\" I ask.\n\n\"We're 'full screws,' corporals,\" says Derrick. \"My wife Vickie and I are in the Army. But her brother Josh here\"\u2014whom he proceeds to elbow gently\u2014\"is a civilian, and a crazy liberal one at that.\"\n\n\"My dad is retired military; he left as a full colonel. My parents live here in Berkeley, where I was born and raised,\" he then says. \"Quite a few military bases were in the region until all the closings in the nineties. There are still a lot of military families living around here.\"\n\nThen Vickie says, \"We're getting ready to deploy to South Korea, and are here visiting Derrick's parents before we go. I get the sense that they're relieved we're not going to the hot spots in the Middle East. But who knows, with tensions flaring between North and South Korea, it may soon be the new hotspot.\"\n\n\"One reason there's so many hotspots is because our military presence makes them hot,\" says Ophelia, who sits down across from Derrick and those with him. \"Our bases all over the world make it possible for us to insert ourselves into any conflict across the globe. We'd be better off without that ability. It weakens rather than strengthens our country when we try to be the policemen of the world.\"\n\n\"Congress could change the nature of our military from what it is now,\" she goes on. \"It already has a constitutional way to do this, but lacks the backbone. Section 8 of the first article in the Constitution gives the legislative branch power to 'provide for the common defense,' 'to declare war,' and 'to raise and support armies.' It goes on to state that 'no appropriation of money to that use shall be for a longer term than two years.'\n\n\"With this power of the purse, Congress can cut off funding for any military incursion. If I were rewriting this article, I'd have it state that the military has to stand down in time of peace\u2014and I would define time of peace as any time Congress has not officially declared war.\"\n\n\"I'd go a step further, and define peace as any time in which American soil is not under direct attack,\" says Amelia, a Berkeley student. \"I'd have the article state that Congress can't declare war, and so the president can't wage a military conflict, unless American soil has been attacked.\n\n\"In such a case, Congress could have declared war after the terrorist attack of September 11, but it didn't. And since it chose not to, the president at the time shouldn't have been permitted to send troops to Iraq and Afghanistan.\"\n\n\"Even if Congress had declared war after the September 11 attacks, it shouldn't have been constitutionally permissible for it to then send troops to those two countries,\" says Josh. \"There was no evidence Iraq was involved in harboring al-Qaeda terrorists, and in the case of Afghanistan, though terrorists did train there, they were citizens of other countries. Besides that, they also trained in 'friendly' nations like Pakistan. Should we have attacked those countries too?\"\n\n\"We could argue back and forth about that forever,\" says Ophelia. \"But if we at least can agree that it's up to Congress, not the president, to declare war\u2014this is what in fact the Constitution already states plainly\u2014then an improved article can further stipulate that no planned military offenses can receive a penny of federal money if Congress hasn't declared war. This would place firm restrictions on how and when it can fund the military. We're in wars on multiple fronts abroad, one of them a preemptive war, yet Congress has never declared war. With this constitutional tightening-up of Congress's authority, it would have no choice but to cut off funding if it hasn't declared war.\"\n\nShe then says, \"My grandfather gave his life for his country in the 'Korean Conflict.' It was never declared a war by Congress, no more than our incursions from Vietnam to Granada, Iraq to Kuwait, and many other places besides. In fact, there've only been five declared wars in our nation's entire history\u2014the War of 1812, the Mexican-American War, the Spanish-American War, World War I, and World War II. All the many other incursions launched by presidents were unconstitutional. The president is only commander-in-chief; Congress is chief of declaring wars. Our country's military has turned into an unconstitutionally, not to mention unconscionably, militaristic invading force because Congress has allowed one president after another to commandeer their constitutionally designated authority and go to war whenever he wants.\"\n\n\"In preparation for this dialogue, I learned that James Madison related at the Constitutional Convention of 1787 that the original phrase in the article would have given Congress power to 'make war,'\" Amelia says now. \"But the wording was changed by the Framers to 'declare war.' This change gave the president authority to repel sudden and unexpected attacks on U.S. soil, but prevented him from launching a war unless Congress first formally voted for a declaration of war. So to prevent any future confusion on this issue, I agree that a revised constitutional article must state outright that U.S. troops shall not be sent abroad unless there is first a formal declaration of war by Congress.\"\n\n\"But the article we're re-creating should also state that troops must stand down at home, and U.S. military bases abroad, when there's no conflict,\" says Josh. \"To our Founders, a standing army was just a step removed from a police state.\" He has a copy of the Declaration of Independence. \"One of Jefferson's principle arguments for declaring independence from England was that the king 'has kept among us, in times of peace, standing armies.' Jefferson faulted King George for rendering 'the military independent of and superior to civil power.' He believed there needed to be a blanket prohibition of standing armies, because this was one of the 'fetters against doing evil which no honest government should decline.' If we want to follow through with his vision for preserving liberty at home, we have to do away with standing armies.\"\n\n\"In the existing Article I of the Constitution, Congress also is charged 'to provide for organizing, arming, and disciplining, the militia, and for governing such part of them as may be employed in the service of the United States,'\" says Terence, who served in Vietnam. \"The Framers intended for the civilian militia, not a professional standing army, to be our first line of defense, well trained and at the ready. By militia, the Framers were referring to what was the equivalent of our National Guard today\u2014a reserve military force under state control though funded largely by the federal government, rather than a federally organized full-time professional force. Both enlistees in the National Guard and the U.S. armed forces today are made up of volunteers, but the difference between the two groups is that the men and women of the Guard aren't regular military.\n\n\"As this article also states, it's Congress's role 'to provide for calling forth the militia' and 'to execute the laws of the union, suppress insurrections and repel invasions.' But most of those duties now are ceded to a professional military that was never intended by the Framers to be permanent.\"\n\nThen he says, \"This article doesn't need to be changed all that much. Congress already has the ability, and responsibility, to stand down the army when it hasn't declared war. The Guard and Reserve would still be at the ready, funded and trained in peacetime, and they would simply resume their original and to my mind proper role as our primary defensive forces.\"\n\n\"Maybe we should've never allowed the military to become what it is,\" says Derrick. \"But it is what it is and we need to deal with that. Even in a thought experiment like this, what's the use of proposing an article that doesn't reflect the reality of the nuclear era and the age of terrorism? I'm not saying this just because if the Army stood down, it'd put me out of a job, but because the world is too volatile a place now for us not to have a trained professional army.\"\n\nTerence thinks about this, and then replies, \"Much as I'd like to turn back the clock, I guess I agree: we can't defend ourselves today solely with Guard and Reserve troops. On the other hand, we shouldn't rely solely on a professional army of all-volunteer troops. All Americans have to be part of the shared risk and adversity of being at war to defend our nation.\n\n\"The best way for that to happen is if there were an additional constitutional clause that says that all able-bodied men and women age eighteen and over, without exception, are subject to being called on to serve. The existing article detailing Congress's authority on military matters doesn't say anything specific about conscription. What's needed is a clear-cut constitutional statement that all young Americans must take their turn in protecting our nation by serving in the military.\n\n\"A fair conscription process in time of declared war would mean that if you're eighteen, you have to be subject to the draft, no excuses, no exceptions. That way no one with connections can get out of serving when called on.\"\n\nTerence then says, \"During the Vietnam War, so many from well-connected families were able to secure continual deferments. They had lucrative careers waiting for them when they got out of college. Their parents pulled whatever strings necessary to get student deferrals or bogus health exemptions. So even with the draft back then, ours was mostly a poor person's army.\n\n\"War would never have been a first option in Vietnam if President Johnson's war strategists and members of Congress knew for certain that their own children\u2014as well as members of Congress and government themselves who were of age to serve\u2014were just as likely to be drafted as anyone else.\"\n\n\"It's true, you have to have shared adversity,\" says Vickie. \"If only military families and soldiers feel the pain from the war, and get none of the gain, there's a huge disconnect. If the Constitution dictated that everyone, from extreme peace activists to extreme hawks, absolutely had to face the draft\u2014in the Guard and Reserve in peacetime, and in our other armed services in time of declared war\u2014maybe there'd be a bond among Americans that doesn't exist right now.\"\n\n\"If extreme doves like me had had to be part of the Guard,\" says her brother Josh, \"and couldn't have gotten out of the draft no matter how well connected their families, it might have prevented what happened here at People's Park in the sixties, and the tragedy that followed at Kent State. The doves in the Guard would've never opened fire on innocent civilians no matter what their orders were, no matter the consequences. And they would've tried to keep anyone else from doing so.\"\n\n\"You would agree to be drafted in the Guard or armed forces if it were constitutionally mandated?\" I ask.\n\n\"I agree that I should have to face the draft if a constitutional article required all able-bodied young men and women to do so. I also believe I have the right to refuse to serve. I'm a pacifist. I object to the draft not on religious grounds, but moral ones. If I were ever drafted, I'd probably go to jail.\"\n\nThen Josh says, \"I do believe that some sort of national service should be constitutionally mandatory for conscientious objectors. But I understand that that's not the same type of shared adversity, of really putting oneself in harm's way.\"\n\n\"When Derrick and Vickie told me they were going to South Korea,\" he goes on, \"I researched that country's conscription laws. All able-bodied males there serve for two years, with few exceptions, because of the endless conflict with North Korea, whose military has over 1.2 million soldiers. About 300,000 South Korean males are conscripted every year. Many try to get out of serving for health reasons or with student deferrals. But it's very rare that they succeed. Those who refuse to serve\u2014about a thousand each year\u2014are jailed.\"\n\nOphelia then says, \"I can understand how some here think that if all Americans of a certain age were subject to a truly fair draft, it'd be the best way to see to it that everyone has a hand in defending our country. Maybe it would help see to it that the U.S. engaged in military conflict as a last resort.\n\n\"But even so, the military isn't the best way to protect our country. If we diverted the bulk of funding that Congress budgets for the military-industrial complex and used it instead to eliminate the most extreme forms of global poverty, our nation would come to be seen as the face of compassion instead of aggression. If we took the lead in winning the global wars against poverty, hunger, illiteracy, we'd rarely need to deploy our military, or defend our soil at home.\"\n\nConstitution Caf\u00e9 Article: Congress shall have the sole power to provide for the common defense, and to declare war. Congress shall raise and support Armies in time of invasion of U.S. soil. No appropriation of money for the common defense shall be for a longer term than two years. Congress shall provide and maintain a Navy, and shall make Rules for the Government and Regulation of the land and naval Forces. There shall be no appropriations for U.S. military bases overseas in time of peace, which is tantamount to any time in which Congress has not issued a Declaration of War. When Congress issues such a Declaration, at least half of all military troops shall be drawn from conscription. Every able-bodied man and woman over age 18, without exception, shall be eligible for conscription. In time of peace, all who are conscription-eligible shall perform two years of public service.\n\nCitizen Soldiers\n\nAs a delegate to the Second Continental Congress, which convened after full-scale conflict broke out with Britain, Thomas Jefferson helped formulate the plan for conscripting troops to serve in the Revolutionary War. Jefferson called for a process that made sure that all eligible men would be equally likely to face the draft. He saw to it that the Continental Army was diverse in terms of class composition and so was not dominated by what he called \"pauper hirelings.\" He shared with James Monroe\u2014his fellow Virginian would become fifth president of the United States, following Madison\u2014that it was \"a subject of joy that we have so few of the desperate characters which compose modern regular armies\" elsewhere. To Jefferson, this \"proves more forcibly the necessity of obliging every citizen to be a soldier; this was the case with the Greeks and Romans, and must be that of every free state.\"\n\nIn peacetime, Jefferson was adamant that a standing army must stand down. \"The Greeks and Romans had no standing armies,\" he told Monroe,\n\nyet they defended themselves. The Greeks by their laws, and the Romans by the spirit of their people, took care to put into the hands of their rulers no such engine of oppression as a standing army. Their system was to make every man a soldier, and oblige him to repair to the standard of his country whenever that was reared. This made them invincible; and the same remedy will make us so.\n\nAs Jefferson expressed to James Madison in 1789, after the Constitution was ratified, \"The following [addition to the Bill of Rights] would have pleased me: . . . All troops of the United States shall stand ipso facto disbanded at the expiration of the term for which their pay and subsistence shall have been last voted by Congress.\" He didn't want Congress to have any wiggle room when it came to the possibility of maintaining a standing army in time of peace.\n\nWhile Jefferson's proposed amendment was not added to the Bill of Rights, the Framers of the Constitution nonetheless agreed with his point of view. \"What, sir, is the use of a militia?\" asked Elbridge Gerry, a Framer who, under James Madison, would become the fifth vice president of the United States. \"It is to prevent the establishment of a standing army, the bane of liberty.\" Luther Martin, another Framer, asserted, \"When a government wishes to deprive its citizens of freedom, and reduce them to slavery, it generally makes use of a standing army.\"\n\nAccording to Richard H. Kohn, professor of history and peace at the University of North Carolina at Chapel Hill, for Americans of the founding era \"the threat of a standing army was deeply rooted in their political heritage.\"\n\nThe Boston Massacre in 1770, in which British soldiers killed several Americans, and the Intolerable Acts of 1774, one of which suspended civil government and put Massachusetts under the rule of the local military commander, made hatred of the army axiomatic in American politics.\n\nHence, \"by the time of the Constitutional Convention, no American political leader could afford to ignore or even to question the danger of standing armies in peacetime.\" At one point during the Constitutional Convention, James Madison went so far as to propose that a clause be added to forbid standing armies in peacetime, because \"armies in time of peace are allowed in all hands to be an evil.\" This clause was ultimately deemed unnecessary by the Framers, since to them it went without saying that Americans would never accept a standing army in peacetime.\n\nYet our second president, John Adams, quickly expanded the Army, the Federalist-dominated Congress authorizing his plan to dispatch thousands of troops to deal with the undeclared war between the United States and France. Despite the justification set forth by Adams for the military buildup, much of the U.S. public was alarmed, since a standing army of any magnitude could also threaten liberties at home.\n\nPolitical scholar Elizabeth J. Perry contends in Patrolling the Revolution that Jefferson's stance against a standing army proved pivotal in his eking out a victory against Adams in the presidential election of 1800. According to Perry, at that time there was \"a widespread preference among the electorate for a . . . citizen militia . . . over the military professionalism advocated by the Federalists.\"\n\nBy the end of his second term as president, Jefferson had dramatically reduced the number of Army personnel. But with rising tensions first with France and then with Britain, which would lead respectively to the Quasi-War and the War of 1812, Jefferson's successors arguably had little choice but to reverse this course and establish a permanent and larger professional military.\n\nJefferson was no longer president when he acknowledged the failure of the loosely organized, noncoercive form of conscription that was utilized at the start of the War of 1812. As he told Thomas Cooper in 1814:\n\nIn the beginning of our government we were willing to introduce the least coercion possible on the will of the citizen. Hence a system of military duty was established too indulgent to his indolence. This [War of 1812] is the first opportunity we have had of trying it, and it has completely failed\u2014an issue foreseen by many, and for which remedies have been proposed. That of classing the militia according to age and allotting each age to the particular kind of service to which it was competent was proposed to Congress in 1805 and subsequently; and on the last trial was lost, I believe, by a single vote. Had it prevailed, what has now happened would not have happened. Instead of burning our Capitol, we should have possessed theirs [that of the British Empire] in Montreal and Quebec. We must now adopt it, and all will be safe.\n\nOur nation long since has adopted a professional military. Are we safer for it?\n\nOur standing military has not stood down since the War of 1812. Today the United States has the world's largest defense budget. For fiscal year 2010, the base budget is $533.8 billion, with an additional $130 billion earmarked for the War on Terror. There are over 1.1 million active-duty military personnel, with over a quarter of a million stationed abroad.\n\nIn his farewell address, President Dwight D. Eisenhower, a five-star general who served as supreme commander of the Allied Forces in Europe during the Second World War, warned, \"In the councils of government, we must guard against the acquisition of unwarranted influence, whether sought or unsought, by the military-industrial complex. The potential for the disastrous rise of misplaced power exists and will persist.\" By this perspective, our own military might have a greater potential to take down our democracy than an invading force might, bringing into question our nation's long-ago decision to continually build up and permanently maintain the professional armed forces.\n\nMillion-Dollar Babies\n\n\"Dang,\" Michael, 13, says, drawing a frown from his teacher, after I tell him and his classmates that I want them to assume the mantle of our nation's Framers and take a stab at making a brand-new Constitution. I'm at a junior high school in the southwest U.S. Most of the students live in a resettlement shelter with their families, many of them evicted from homes in various stages of foreclosure.\n\nAfter a protracted silence, Michael again speaks up. \"I would want the Constitution to say that every baby born in the USA starts out with $250,000. The moment a baby enters the world, that money is put in a bank account in his name. The only hitch is, he can't touch it till he's eighteen. By then, he'll have decided what college he wants to go to, or whatever it is he's going to do with his life, and he'll have the money on hand to send him on his way.\"\n\nHe looks at us, animated. \"Imagine: We'd all be making our way in the world with the same amount of cash. Whether we waste it on bling or make the most of it, what we do or don't make of ourselves, that's our call. But no one can complain he wasn't given a fair chance.\"\n\n\"Man, $250,000 won't be enough to do anything with by the time you're eighteen,\" says his friend Heath. \"After inflation, that amount would hardly pay college tuition and board for four years. We'd all need at least a million bucks apiece.\"\n\n\"Okay, a million bucks gets my vote, too,\" agrees Michael. \"The main thing is, we all get out of the starting gate with the exact same amount, so we'll have the same odds of succeeding.\"\n\nHe turns to me and says, \"We've just been studying the Declaration of Independence, and it says we're all created equal. So the Constitution should put its money where the Declaration's mouth is, and make sure we're treated as equals. So that means we have to be pretty much financial equals. Like Heath says, a million-dollar payout on the day we reach voting age has a ring of 'equalness' about it.\"\n\nAhmad disagrees. \"A million dollars doesn't get us all out of the starting gate as equals. If you're homeless, if your parents can't get work, and you can't touch a fraction of that money till you're eighteen, you'll have become desperate long before the time comes to collect the cash. In fact, knowing that money's there, untouchable, will make you more frustrated and desperate.\"\n\nHe looks at me. \"There's kids a few miles from here who live in these fortresslike mansions. They might as well be a world away, their reality is so different. They couldn't care less if there's a million bucks waiting for them. They're already set for life. These trust-fund babies will get accepted in the best universities, get the best jobs, because they have all the advantages and connections those first eighteen years. Sure, we'd eventually get that million dollars, our own version of a trust fund\u2014if we're not in jail by then, or worse, since our lives won't have been fair that first eighteen years.\"\n\nFrannie isn't buying this. \"You're telling me that knowing you've got a constitutionally guaranteed million-dollar pot of gold waiting for you when you're eighteen won't motivate you to work hard, stay in school, get a good education? Because you'll know the sky'll be the limit once you graduate. You'll know if you want to be a doctor, artist, athlete, whatever, you'll be able to support yourself.\"\n\nThen she says, \"As much as I may try to reach for the stars, and sometimes believe that I can just about touch them, the fact is that some of them are out of my reach without a certain amount of money at my fingertips. This article will make all the difference in my life.\"\n\n\"I wish, instead of one lump-sum payout, this constitutional article would let that money start trickling in right now,\" says Justine. \"My parents scrape and claw, but still couldn't keep a roof over the heads of me and my sisters. They signed a loan for a home it turns out they couldn't afford, and everyone who convinced them to sign it knew it\u2014and laughed all the way to the bank. My folks were working two, three jobs apiece and couldn't keep up with the payments. My dad was so ashamed when we were turned out on the street that I wouldn't go to sleep at night. I worried he might try to hurt himself. Then he got mad. Our home is one of thousands that are empty, no chance of anyone buying it. We're back there, 'squatting.' He says it's our right. I'm not so sure, but I'm glad he's mad instead of desperate.\n\n\"Mom and Dad have no credit, can't afford the steep rents, can't afford a car, health insurance, tasty meals. If I had that money, you know I'd pay for everything my family needs out of my own pocket. And I'd pay cash for a home of our own that no one ever could take from us.\"\n\nHeath chimes back in. \"I propose a clause to go along with Michael's article: that no kid inherits a dime, that the Constitution ban inheritances from here on out. That way, all of us will have that million waiting for us at age eighteen, and that's it. It'll make the playing field even more level. All that loot those rich kids would have inherited will go instead into the fund that'll pay for all of us to have million-dollar trust funds.\"\n\n\"That's got my vote,\" Frannie says. Then she frowns. \"I'm wondering: should you have to do something to earn this money, or does it come to you automatically? I mean, shouldn't you have to show that you deserve to tap into that million-dollar ATM? If all you are your first eighteen years is a slouch or troublemaker, why should you get that money? Maybe it should be yours only if you're upright. Like, you should have to show you've been volunteering in your community, studying hard, helping out at home, things like that.\"\n\n\"I hear you, but to me, there should be no strings attached, so no one can make excuses for not succeeding,\" says Michael. \"Sure, a few people will take advantage of their big payday and do nothing good with their lives if they don't have to. But I'm betting most will make the most of this opportunity.\"\n\nThen he says, \"We're offering up a hopeful article to the Constitution. It gives everyone the chance to do great things with their lives. For those who are dumb enough to do something bad before they turn eighteen, and wind up in jail, well, they can use their million dollars and atone for what they did\u2014they can start 'scared straight' and drug prevention programs, things like that.\"\n\n\"Once this kicks in,\" says Michael, visibly proud that his proposed article to the Constitution is winning over his classmates, \"only the hardest achievers, the most disciplined, will have the best things, because they've earned it. The only people those who haven't exerted themselves can blame for not succeeding are 'me, myself, and I.' \"\n\nTerence has been sitting still and silent until now, intently considering all the views proffered. He shakes his head, muttering, \"I just don't know.\"\n\nI ask him if he has something more he'd like to say. The students lean forward, intent on his every word. Clearly he is respected, and his views carry a lot of weight with them.\n\n\"My dad wouldn't agree with this million-dollar deal,\" he tells us. \"He'd think it makes life too easy. My dad says life isn't meant to be fair. Like, he may be down and out right now, but he'd be the first to tell you, 'I've come back before and I'll come back again, because this great country provides everything I need to be a winner.' \"\n\nThen he says, \"Though he's dog-tired from taking on menial work all day, and then looking for new work all afternoon and evening, my dad helps me with my homework, and with filling out college and scholarship applications. I'm going to be an astronaut. He tells me, 'A hundred colleges may reject you, but there'll be that one that says, \"Welcome to our school, where we'll take you to the moon,\" and it will make those rejections just that much more sweet.'\"\n\nThen Terence asks us, \"Did you see that movie The Pursuit of Happyness.\"\n\nJust about all of us nod. \"Will Smith, who plays Chris Gardner, a dad struggling to make ends meet, talks in that movie about Thomas Jefferson. He might just as well be my dad talking. Will's character says Jefferson was a wise man for giving us a country that doesn't guarantee us happiness, but guarantees us the freedom to pursue it.\n\n\"My dad doesn't want anything handed to him on a silver platter, just a shot at getting his hands on that platter through his own hard work. In the little free time he has, he reads civil engineering books. He's self-taught. If he'd had the chance to get a degree, he'd be king of the bridge-building world. You should see the bridges he's designed. He could build a bridge to the moon if he had the opportunity. I don't know how it is that my dad still has stars in his eyes, with his tough life. He raises me on his own. My mom died of cancer when I was little.\"\n\nLooking at his classmates, he says, \"I agree that all us Americans should have a head start, not just some. But I can't see being guaranteed a million-dollar payday at age eighteen. Right now, as my dad mentioned to me the other day, Congress is considering a bill to put $500 in an account for each newborn baby. Even my dad is not opposed to that, because to him it's a way of showing that our government is making a no-strings-attached investment in each of us, without going overboard about it. Make that sum $50,000, and that sounds like a winner to me\u2014and we can make that a constitutional right from here on out.\"\n\nThere is a pause. Finally Terence says, \"I'll tell you what: if each of us did get a million dollars out of that starting gate, and I had that money in my hands when I turned eighteen, I'd open up a bridge-building company. With all the bridges crumbling in our country, it should do some good. I'd make my dad the chief executive and designer\u2014great pay, benefits, the works. Not just because he's my dad, but because he's the hardest worker I know.\"\n\nHe turns even more serious. \"But don't any of you tell him. If he knew it was sort of a gift, no matter how deserving and overdue, he wouldn't be so happy about it.\"\n\nConstitution Caf\u00e9 Article: All U.S. citizens who have shown that they have \"done some good,\" shall at age 18 receive an equal lump sum payment of at least $50,000.00. If, after receiving the payment, they \"do bad,\" they shall have to return the money. No citizen shall inherit from his or her family so much as a red nickel.\n\nI.O.U.\n\nWould Jefferson be of the mind that these students expect too much from their country, without demanding enough of themselves in return? Or do their proposals strike an ideal balance, giving our nation's youth an equal jump start in life without encouraging complacency?\n\nThe students' article to eliminate inheritance altogether is a more radical twist on one Jefferson himself crafted as part of his statute reform package for Virginia. The article, approved by the legislature in 1776, abolished the English holdover of primogeniture, in which firstborn males received all inheritance. It further eliminated English entails, enormous land grants to the privileged, which Jefferson characterized as remnants of \"a system that perpetuated an artificial aristocracy of wealth,\" and of patrician \"privilege and prerogative.\" As Jefferson wrote in his memoirs, \"These laws, drawn by myself, laid the ax to the root of pseudo-aristocracy\" in his home state. While Jefferson's bill still assumes some right of parents to provide for their families even after death, the students' constitutional article suggests that eliminating this right altogether is best for all Americans.\n\nJefferson had intended for the new inheritance laws to be just the first in a series of reforms that would have a leveling effect on society. He also proposed legislation that would make sure that each American had a sufficient amount of the most important currency of his day: land. Land promoted autonomy, liberty, and sociopolitical equality. Virginians could not vote if they didn't own land, much less take direct part in political decision-making; and the vast majority of males at the time were \"nonfreeholders,\" meaning they did not own land.\n\nIn addition to the inheritance reform legislation, Jefferson introduced a bill calling for the allotment of a modest amount of land to all Virginia males \"of full age neither owning nor having owned acres of land.\" He wanted to do away with the concentration of land in the hands of a few, lest Virginia become for all intents and purposes a Britain-like feudal system. Most importantly, distributing a modicum of arable land to nonfreeholders would provide them entr\u00e9e to the political sphere.\n\nTo Jefferson, there were other tangible benefits to society as well: having a piece of land enabled people to be close to the earth, and in this way helped them become ideal democrats, since ties to land left them \"tied to their country, and wedded to its liberty and interests, by the most lasting bonds.\" While he didn't think that everyone should be a farmer, he did believe everyone had to appreciate the practice of growing some of their own food on their own land, since this nurtured such key agrarian-type values as self-sufficiency and industry. Jefferson further contended that cultivating the soil brought one into intimate contact with nature, and as a result nurtured the virtues of \"honor, manliness, self-reliance, courage, moral integrity, and hospitality.\" In his Notes on Virginia, Jefferson extols farmers as the \"cultivators of the earth,\" and as such \"the most virtuous and independent citizens.\"\n\nIn proposing legislation calling for land redistribution to the Virginia Assembly, Jefferson stressed that he was \"conscious that an equal division of property is impracticable.\" However, there was abundant public land available in the commonwealth, and he recommended that 50 acres be allotted to each landless male. He maintained that it was by no means \"too soon to provide by every available means that as few as possible shall be without a little portion of land.\" Indeed, he contended, it was precisely when the nation was just coming into being that legislators should complement the revolution in the battlefield with revolutionary legislation that would give all Americans a vital stake in the democratic experiment.\n\nJefferson's bill for distributing land was soundly defeated. Those in the legislature with the bulk of power were the landed gentry, who knew that meting out land to all males would dilute their political clout. Moreover, while they were as ecstatic as the next person to be rid of the British monarchy, they did not want to be rid of the privileged social and political distinctions that came with their wealth.\n\nThe Virginia legislators did, however, agree that most public land in the commonwealth should be put into private hands. But they differed with Jefferson over means and ends for doing so. Contrary to Jefferson's hopes, the legislation that was passed paved the way for most public lands to be sold to speculators, with the pretext that the revenue was desperately needed for government coffers. Most of the speculators proved to be the legislators themselves; they had the means to purchase huge public parcels, and so were able to augment their already significant holdings.\n\nThere would be other admirable efforts to distribute land to settlers in U.S. territories, particularly via the Homestead Act of 1862, by which Abraham Lincoln distributed unsettled land in the west to nonfreeholders. But the more typical pattern was for state legislatures and members of Congress to sell public land to speculators (themselves foremost among them), and to railroad and cattle barons. The predictable outcome was that the landless poor drifted to cities, where they often lived in abject poverty. To Jefferson, democracy itself paid the highest price for this failure to provide citizens the basics needed to pursue happiness, because it ensured that political power remained in the hands of a few.\n\nIn modern times, land reform efforts in emerging democracies, though often paved with good intentions, have mixed track records. Where land reform has focused, as Jefferson himself proposed, on giving modest landholding titles to individuals, as occurred in Taiwan and South Korea soon after World War II, there have been some successes. But the far more typical outcome is that the promised land remains elusive, with vows by political leaders to enact sweeping reform policies rarely realized. In cases in which some land is redistributed, the process usually is slow and uneven, and often generates new conflicts, with those evicted or displaced when property is reallotted often left landless themselves.\n\nIn Brazil, where about 3.5 percent of landowners hold well over half of the nation's arable land, one of the most ambitious modern land reform efforts was launched during President Luiz In\u00e1cio Lula da Silva's two-term presidency. Lula succeeded in reapportioning idle land to several hundred thousand of the nation's poor, who have since settled on small plots of farmland. But tens of thousands of landless families continue to live in camps, and violent disputes regularly break out between wealthy landowners and their privately hired militias and landless activists who seek to occupy typically idle parcels of their land.\n\nAn entire society suffers when some of its citizens don't have the necessary or instrumental means, such as a piece of land of their own, to fully pursue happiness\u2014at least, if one subscribes to the Jeffersonian perspective that the common good and the individual good are entwined in a democracy. \nV\n\nCharacter Counts\n\nWhen Jefferson arrived in France in 1784 to begin his stint as U.S. minister, he was considered by freedom lovers in Europe a nearly transcendent figure, his works and deeds on behalf of the cause of liberty a continual source of inspiration. And on both sides of the Atlantic\u2014from Corsica, Italy, where Pascal Paoli and his fellow sons of liberty were fighting longtime autocratic Genoese rule; to Ireland, where radicals were fighting for constitutional reforms; to London itself, where John Wilkes \"had assumed Liberty's cause by opposing general warrants, which were issued at large and authorized the seizure of private papers\"\u2014many people believed that no one could be \"a sincere lover of liberty who was not in favor of communicating that blessing to all peoples.\" Few Americans more poignantly conveyed this blessing than Jefferson, who was certain that the \"ball of liberty\" sparked by America's successful struggle for independence \"is now so well in motion that it will roll around the globe.\"\n\nHis liberty-loving admirers were able to overlook the fact that Jefferson was traveling with human cargo. Some no doubt did not yet consider Africans \"people,\" but others, perhaps, understood better than we do today that a leader or politician can be greater in public life than in his or her personal life. Along with his slaves James Hemings and James's sister Sally, with whom he would father at least one child, Jefferson was present in France when revolution at last broke out with the July 14, 1789, storming of the Bastille. Jefferson, who voiced his support for the French Revolution, was even invited to attend and offer counsel at meetings geared toward rewriting the French constitution.\n\nAs Annette Gordon-Reed has written, the Hemingses surely were aware that in Paris \"the Soci\u00e9t\u00e9 des Amis des Noir [The Society of Friends of Black People] was engaged in a very public . . . struggle to include free, tax-paying blacks in the deliberations of the newly constituted National Assembly.\" There is no question that Jefferson, who attended the assembly gatherings daily, was also privy to this development. One delegate to the assembly, Count Mirabeau, chided those who would not let blacks participate in the Assembly \"with words that echoed the debate among the delegates to the American Constitution.\" Jefferson was likely on hand when Mirabeau delivered an impassioned speech to his fellow assemblymen:\n\nYou claim representations proportionate to the number of inhabitants. The free blacks are proprietors and taxpayers, and yet they have not been allowed to vote. As for the slaves, either they are men or they are not; if the [white] colonists [in French territories who did not want to include blacks] consider them to be men, let them free them and make them electors and eligible for seats.\n\nMirabeau's powerful speech had its effect: just four years after Jefferson and the Hemingses returned to the United States, people of color in France \"sat in the revolutionary National Convention, and the country abolished slavery in its colonies in 1794.\"\n\nOne might reasonably expect that the principal author of the Declaration of Independence, upon witnessing how French citizens (in large measure inspired by his own words) had abolished slavery, would make every effort to replicate their success. Jefferson might not only have taken the lead in calling for the end of slavery in the United States, but argued \u00e0 la Mirabeau for giving blacks the right to be full participants in all dimensions of American society. While he did not come close to striving to achieve such a progressive objective in the United States, he did take some initiative in attempting to resolve the issue.\n\nLong before he left for France, Jefferson demonstrated both his acute awareness of the fact that slavery had to be abolished, and his own unwillingness to risk much of his political capital to bring this about.\n\nPaul Finkelman, a specialist in American legal history, race, and the law, argues that while Jefferson, as chair of the committee to revise Virginia's legislative statutes, was in an ideal position to address the issue of manumission, or formal emancipation, he \"failed to take the lead.\" So when his fellow assemblymen, soon after the Declaration of Independence was issued, \"approached Jefferson with draft legislation that would have brought gradual emancipation to Virginia, he declined to add it to the proposed revisions.\" For blacks of his time, then, his Declaration amounted to hollow words.\n\nIn his Notes on Virginia, Jefferson defends himself, contending that in fact he'd proposed an amendment that would have both educated and granted manumission to the children of slaves. Jefferson claims his amendment was never seriously considered because \"the public mind would not yet bear the proposition.\" He would have us believe that while he was far ahead of the public mind on the matter, his fellow Americans simply were not ready to end slavery. But the evidence suggests a murkier conclusion. Jefferson himself acknowledges in his memoirs that if this amendment had been enacted, the freed slaves would have been \"colonized\" elsewhere, rather than allowed to stay on in Virginia as free and equal citizens.\n\nPeter Onuf, the Thomas Jefferson Memorial Foundation Professor at the University of Virginia, asserts that Jefferson missed an unparalleled chance to tap into the progressive public mind and bring about emancipation:\n\nBefore 1776 most Americans were not prepared to declare their independence. When Jefferson penned the Declaration, public argument needed to be shaped by argument and debate. The manumission movement faced similar obstacles. Many Revolutionary Virginians . . . were ready to consider some form of gradual emancipation. Jefferson might have helped shape the public mind on emancipation just as he had on independence. Even without Jefferson's lead, the legislature adopted a law allowing private manumission without expatriation. This suggests that the \"public mind\" on this issue may have been far in advance of Jefferson.\n\nJefferson weighed in on the matter again, in 1784, the year before he assumed his post in France. As a member of Congress, Jefferson chaired the committee charged with administering the federal territories and detailing the process by which they could eventually achieve statehood. He drafted a provision that would have made slavery illegal in all existing western territories and all other new territories acquired by the United States after 1800. This provision was rejected by one vote. Remarking later on its defeat, Jefferson said that \"the fate of millions unborn [was] hanging on the tongue of one man, and Heaven was silent in that awful moment.\" This was the one and only time in Jefferson's long political career that he made a fairly audacious attempt to prevent the further expansion of slavery in the United States; but even if it had passed, his provision would have left slavery intact in the original thirteen states. And Jefferson certainly did nothing to set a personal example; he retained throughout his life all of his own slaves at Monticello, rather than granting them their freedom.\n\nJefferson did make one other attempt to stem slavery: in a draft of his \"thought experiment\" constitution for Virginia, he composed a clause that proposed gradually bringing slavery to an end. The draft article, which was never submitted to any legislative body much less implemented, reads in part: \"The General assembly . . . shall not permit the introduction of any more slaves to reside in this state, or the continuance of slavery beyond the generation which shall be living on the 31st day of December 1800; all persons born after that day hereby being declared free.\" Finkelman believes this \"slavery clause suggests that, in the deepest recesses of his heart and mind, Jefferson knew that something had to be done about slavery.\" If so, Jefferson did not act on this knowledge, but kept it trapped in his notebook.\n\nPerhaps, though, it is not so much this \"slavery clause\" but the Declaration of Independence that most clearly indicates Jefferson's belief that slavery had to be done away with, and his willingness in this instance to take a vocal public stand. Jefferson's original draft of the Declaration, before it was subjected to editorial revision by the Continental Congress, included a caustic denunciation of the slave trade as \"a cruel war against human nature itself,\" but it was stricken from the version approved by the Second Continental Congress. Still, the two words \"created equal\" would prove over time to carry the spark that would ignite change. Frederick Douglass, the former slave who became a social reformer and one our nation's foremost abolitionists, held up the Declaration as \"an abolitionist manifesto.\"\n\nThis perhaps disproves the idea that actions always speak louder than words. In Jefferson's case, his words have had more resounding (and more positive) effects than the decisions he made in his personal life. Still, the chasm between Jefferson's ideals of liberty for all and his inability or unwillingness to champion the abolition of slavery leads us today to question his character.\n\nAn Associated Press\u2013Ipsos poll found that a decided majority of Americans consider a presidential hopeful's character far more important than his or her political experience or specific stances on issues. But what on earth do we mean by character? And is good character a requirement only for our leaders? Or is it up to the rest of us too to conduct ourselves in a way that we can hold up as an example to our children, our fellow citizens, and to newcomers to the United States?\n\nScout's Honor\n\nArticle II, Section 1, Clause 5, of the Constitution: \"No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.\"\n\nAfter I finish describing my Constitution Caf\u00e9 project, Francisco points to the Constitution I'm holding and says, \"What does it say about the qualifications to be president? In case I decide to run someday, I'd like to go ahead and get started fulfilling them.\"\n\n\"It only says that the president has to be born on U.S. soil,\" I reply, \"that he has to be thirty-five years old, and that he has to have resided in the U.S. at least fourteen years.\"\n\nFrancisco is incredulous. \"That's it? You don't have to have done any good deeds? You don't have to take a test to show you know the Constitution? You don't have to be an upstanding citizen?\"\n\nI hand him the Constitution and let him and his fellow Boy Scouts examine the pertinent section. We're meeting in a park near the fringes of the Mission District of San Francisco, where I once lived and frequently held philosophical dialogues with elementary-school children (we called our gatherings a Philosophers' Club). Despite recent gentrification, gangs and drugs are still prevalent, and the area school dropout rate remains high.\n\n\"You can have been in a gang?\" asks Arturo, still in disbelief. \"You can have hurt someone and still try to become president of the United States?\"\n\n\"You're not automatically barred from being president, or contending to be president, if you've done such things, though it may not help your chances of getting elected,\" I say. \"According to the article you looked at, you can be removed from office after you've been elected if, as the Constitution says, you commit 'Treason, Bribery, or other high Crimes and Misdemeanors' while in office. But there's nothing that says you can't have committed a crime before you run for office.\"\n\nThis revelation gives the group pause. Francisco shakes his head and says, \"That's not right.\"\n\nThen Emilio, 15, says, \"You want us to help you write a new article, so here it is: 'All presidential candidates should have to live by the Boy Scout's motto. They must promise to be trustworthy, loyal, helpful, friendly, courteous, kind, obedient, cheerful, thrifty, brave, clean, and reverent. Anyone who wants to be president of the United States should have to prove, by his actions, that he has these values before he can even think about running for the highest office. Because he's supposed to be the 'values leader' of all of us. He wants to be put on a pedestal, so he should have to be the American role model to end all role models.\"\n\nThen he says, \"As Scouts, there's more than one hundred merit badges you can earn, for everything from Civics, to Emergency Preparedness, to Citizenship in the World. Believe me, if you don't have these twelve Scout qualities, you'll never earn a single merit badge. Because you don't just master a project by accomplishing a task\u2014because the tasks are designed so that you can only really master a subject if you also have mastered the Boy Scout qualities. A presidential candidate should have to show he's mastered these qualities too, if he's going to be worthy of the task of being president.\n\n\"Like, a presidential candidate might prove he's kind by giving a lot of his time and energy to helping out the poor. And, if he's already a politician, he can show he's thrifty by being even more careful with taxpayer money than with his own. And if he's never been a politician before, he can show that he's careful with his family's money, or with the finances of his business or the business of his boss. But when it comes to his own money, he should also show that instead of spending everything on himself or his family, that he also gives a lot to charities, so we'll know he's investing in the country's neediest people.\"\n\n\"And candidates for president can show they're 'clean' not just by being neat and tidy and organized, but by giving proof that they'd be an honest president instead of a dirty one,\" says Arturo, an imposing kid whose uniform is plastered with merit badges. \"They can do this by making it known that they don't have a rap sheet that shows they've been arrested for things like accepting bribes. And they can have their wives give testimony that they've never cheated and never even hit on another woman.\"\n\n\"That's not the only way a candidate can show he's clean,\" says Nelson. \"There's more positive ways. Like, all the people you've worked with over the years, as well as your friends, should come forward and tell about all the great things you've done, and how trustworthy and loyal you've been. You see, like we've been trying to explain to you, all these Scout values are connected. I've learned that you can't really practice one of them all the way unless you practice the rest of them all the way.\"\n\n\"I'm not sure presidents should have to practice the Scout value of always being cheerful,\" says Amado. \"I mean, when the president is informed that one of the soldiers he's sent to war has been badly injured or killed, what's there to be cheerful about? Maybe he should have to rally the troops, keep them motivated, tell them right is on their side. But cheerful? You can put a brave face on things even as you cry, even be optimistic about how things are going to turn out over the long run, but not cheerful.\"\n\n\"How would a president demonstrate reverence?\" I ask.\n\n\"Let me put it this way: whenever I give the Boy Scout oath, I pledge 'to do my best, to do my duty to God and my country,'\" says Francisco. \"When the candidate who's elected president gives his oath of office on Inauguration Day, he promises to obey the Constitution, 'So help me God.' That means that in obeying the United States Constitution, he's honoring God, showing reverence for the higher power that inspired our patriots in the Revolution to fight and sometimes die for this great democracy.\"\n\n\"But a president isn't, or doesn't have to be, a Boy Scout,\" says Amado. \"A president doesn't even have to believe in God, according to that article Chris showed us.\"\n\n\"Should this new article we're creating say he should have to?\" I ask.\n\nHe mulls this over. \"Reverence in a country like ours can mean being courteous and kind and helpful even to people who don't believe or think like you do. My dad says lots of people who claim they believe in God are all the time yelling and screaming at each other, and even trying to hurt others, just because they have different beliefs. So they're not being very reverent to the peaceful and loving God of Christianity. Our own Founding Fathers and Mothers came here to escape persecution just so they'd have a place to live in peace and harmony while they worshiped in their own way.\n\n\"A person who wants to be president must show by example that he's tolerant of people who don't look or think or believe like most other Americans do. A president would go out of his way to put his arm around these people and say, 'Let's be reverent to these people so we can all live in peace.'\"\n\n\"I have a classmate at school who doesn't stand for the Pledge of Allegiance,\" says Arturo. \"She believes in God. But it's against her religion to stand up and put her hand over her heart while we recite the Pledge to start the school day. I respect her. When other classmates tease or taunt her, I stand up for her, and they stand down. Any presidential candidate who allows people to be teased, or teases people himself, just because they don't look at things the way he does, or act like he would in every situation, should have to bow out of the presidential contest. He's not being reverent to the values of a country like America.\"\n\n\"A candidate for president should show he's always first to stand up for any kind of underdog,\" says Francisco. \"This requires bravery and kindness on his part. I used to be a bully. I teased anyone who was weak or a sissy. Mr. Espinosa, our troop leader, straightened me out about that.\n\n\"My mom made me join the Scouts because I was hanging out with a gang. I didn't much like the Scouts at first. I acted up a lot, like I was too cool for the Scouts. But anytime I teased the other members of my Scout troop, Mr. Espinosa made me run laps, so I'd have some time and space, when I wasn't cramping for air, to think about whether I had what it took to be a Scout. After a while, he got through to me.\n\n\"See, according to our handbook, a Scout believes that being brave means having 'the courage to stand for what he thinks is right even if others laugh at or threaten him.' And I finally realized that I'd been leaning towards joining a gang not because I had courage, but because I didn't. A Scout learns bravery so he knows to go his own way when others tempt him to do bad things. If I can go my own way, the presidential candidates sure can too.\"\n\nEmilio elbows him. \"If you ever run for president, I'll testify that you were brave. You were the one who convinced me not to even think about joining a gang.\" Francisco smiles with pride.\n\nThen Emilio says, \"Scouting trains you to be the right kind of a leader, the do-gooder kind. See, you don't work to have the Scout qualities just for the sake of them, but so you always are strong in the right way, protecting the weakest and most helpless. It shouldn't be asking too much of people who want to be president to prepare to be that kind of leader.\"\n\n\"Really,\" he goes on, \"we should make it a requirement, in a separate constitutional article that we can work on later, that all the people who run for any office, like to be a senator or a congressman, have these Scout qualities. Because if the president's the only one who has them, and none of the people he has to work with to pass laws do, it'll be a constant battle between good and not-so-good.\"\n\n\"As long as at least the president has these qualities, we'll all be okay,\" says Arturo. \"Because all the people running for other offices will know that voters want them to have these qualities\u2014because these are the qualities that the voters themselves try to have and value the most. Besides that, all these other politicians usually have it in the back of their minds that one day they'll run for president, so they'll be sure to be on their toes and practice all the Scout qualities.\"\n\nThen he says, \"This article may be about the ideal qualities we want in a president, but most of all, if you read between the lines, it's really about the ideal qualities we want in ourselves.\"\n\nSoon afterward, Nelson says, \"There's another quality a candidate for president\u2014and, like Emilio said, all the other candidates for high offices\u2014should have besides the Boy Scouts' ones. He should be humble. Like, if I ever decide to run for president, I'm not going to go around everyplace bragging about all these Scout qualities I have. I'm not going to distribute a DVD about 'Nelson the All-Around Great Leader Boy Scout.' I'm going to let my actions speak for themselves. Like, I wouldn't ever run a negative campaign, smearing and slamming others candidates, no matter what they say about me. I'll just show quietly and humbly the person I try to be every day.\"\n\nEveryone nods to this.\n\nAmado seems to hesitate, then takes a deep breath and says, \"My sister is a member of the Girl Scouts. Their motto is, 'Be prepared.' I think a president should have to follow that motto too.\"\n\nHe seems braced for teasing. But Arturo simply nods and says, \"The best way to be prepared to be president is to know who the best people are to look to for advice, so that when unexpected troubles come along, you're ready to respond. The president is in charge of everything, but he can't know everything. So he has to show how prepared he is by surrounding himself with experts in different areas. That way, when the president needs advice on things like how to best respond if there's an oil disaster like the one in the Gulf of Mexico, or there's an outbreak of a new type of flu virus, he's got the right people on the spot to make sure matters don't just get worse and worse, but are handled in a way that the hurt and damage is kept to a minimum. President Obama was a Scout, so he has no excuse for not being prepared for what happened in the Gulf.\"\n\n\"My grandfather lost his retirement savings after putting in forty years on the docks,\" Emilio tells us. \"He blames the 'rotten economic advisors' of at least the last two presidents for the fact that he's had to go to work at Wal-Mart as a greeter just to survive.\"\n\nAfter a pause, Nelson says, \"A candidate for president doesn't have to be a guy. She might be a Girl Scout. And if she practices this Girl Scout motto, she'll be a strong candidate. And she'll be an even stronger one if she practices that motto while also developing the Boy Scout values, since these are the values you need in order to be prepared for just about anything.\"\n\n\"Actually, Girl Scouts already have their own similar sorts of values,\" says Francisco. \"I know, because my sister is also a Scout. Besides that motto that Amado mentioned, they also have a slogan: 'Do a good turn daily.' A president should wake up each morning with the thought, 'I'm going to do the American people a good turn today.' Like, he might wake up and say, 'You know what? I'm going to call someone everyday, someone I've read about in the newspaper who's having a hard time, and not just give her a pep talk, but offer all the help and support I can.' He'll also make sure his good turn stays out of the headlines. Because he's not doing it for publicity, but because he really wants to do his duty to God and country.\"\n\nConstitution Caf\u00e9 Article: No person except a natural-born citizen of the United States shall be eligible to the Office of President; neither shall any person be eligible to that Office who shall not have attained the age of thirty-five years, and been fourteen Years a Resident within the United States. Anyone eligible to be President must be trustworthy, loyal, helpful, friendly, courteous, kind, obedient, cheerful (when the occasion calls for it), brave, clean, humble, and reverent (respectful, and tolerant, especially of underdogs and people who think and believe differently). He or she shall be prepared, and shall do a good turn daily, and in so doing, shall further do their duty to God and country.\n\nVirtuous Politicians\n\nThe idea of constitutionally requiring elected officials to possess certain virtues isn't far-fetched, if constitutions of the Revolutionary era are our yardstick.\n\nPennsylvania's original constitution, for instance, stipulated that all prospective candidates demonstrate \"virtue and wisdom.\" And to this day, New Hampshire's constitution states that \"a constant adherence to justice, moderation, temperance, industry, frugality, and all the social virtues, are indispensably necessary to preserve the blessings of liberty and good government.\" Consequently, voters must \"have a particular regard to all those principles in the choice of their officers and representatives,\" given that they have the \"right to require of their lawgivers and magistrates, an exact and constant observation of them, in the formation and execution of the laws necessary for the good administration of government.\"\n\nIn The Transformation of Virginia, Rhys Isaac relates that Virginia's colonists looked at elections as opportunities to choose a candidate whose \"virtue showed most clearly in their persons.\" They rejected \"trials of strength between contending social classes and popular choice between rival programs,\" since they considered these \"precisely the lines upon which . . . elections should not be conducted.\"\n\nTo this end, Isaac relates, a public announcement placed in Accomack County, Virginia, in 1771 seeking freeholders to vie for particular offices made clear that prospective candidates must possess a range of virtues. For instance, the candidates must have \"penetrating Judgment,\" so that they would be able \"to scan each Proposal, to view it in every Light.\" They must further have the talent of \"piercing into Futurity,\" in order to \"behold even how remote posterity may be thereby affected.\" What's more, the ideal officeholder, as the paragon of virtue, must be one \"who regards Measures, not Men,\" and so, according to Isaac, \"will follow his country's interest regardless of the effect of his course upon either his friends or his foes.\" Consequently, as the announcement stressed, candidates were expected to possess \"that Fortitude, or Strength of Mind, which enables a Man, in a good Cause to bear up against all Opposition, and meets the Frowns of Power unmoved.\"\n\nThe virtues that Virginia's voters demanded in their elected representatives, according to Isaac, clearly \"owed a great deal . . . to their own sense that they were engaged in a momentous struggle that would determine the destiny of mankind.\" In 1769, among all those vying to serve as Albemarle County's delegate in the Virginia Assembly, Thomas Jefferson was deemed the person most reflective of voters' values, virtues, and aspirations in what would prove to be a pivotal moment. At age 26, Jefferson was elected to be his county's representative in the House of Burgesses. Upon arriving in the stately chambers in Williamsburg, the political neophyte was struck by the fact that \"our minds were circumscribed within narrow limits, by an habitual belief that it was our duty to be subordinate to the mother country in all matters of government.\" This collective shortcoming, Jefferson surmised, was not due to any lack of \"reflection and conviction\" on the part of his brethren, but rather it resulted from \"habit and despair.\"\n\nJefferson expressed his confidence that \"they could bring their minds to rights on the first summons of their attention.\" And in fact, just ten days into his tenure as a burgess, the legislators' attention was summoned: Parliament had passed the Townshend Acts, placing levies on many of the colonists' common staples, such as glass, lead, paper, paints, and tea. In response, the burgesses passed a resolution establishing that they, not Parliament, were the only governing body that could legitimately impose taxes on the colonists. Their unusual feistiness prompted then-royal governor Lord Botetourt to dissolve the legislature. He was sure that this move would force the burgesses to capitulate, but it only heightened their resolve. Upon being evicted from their chambers, Jefferson relates that they repaired straight to the Raleigh Tavern, where they formed a \"nonimportation association\" aimed at rousing all colonists to boycott the newly taxed goods. This unexpectedly defiant act by the burgesses prompted England to do away with all the new taxes, except the pesky one placed on tea.\n\nAfter the Townshend Act controversy was resolved, Virginia's royal governor called for new elections. Lord Botetourt was certain that the legislators' action had been out of step with their constituents' sentiments, and that the burgesses who'd opposed the Crown's will would be ousted from office. Just the opposite happened. Jefferson relates that all the burgesses \"were re-elected without any other exception than of the very few who had declined assent to our proceedings.\" This was clear proof that the burgesses not only had dispensed with habit and despair, but now had taken up a cause\u2014to put their colonists' interests above those of their mother country\u2014that endowed them with the virtues most prized by their constituents.\n\nMen and Women Without a Country\n\nArticle I, Section 8, Clause 4, of the Constitution: \"Congress shall have Power . . . To establish an uniform Rule of Naturalization. . . .\"\n\nThe expressions on their faces range from excitement to incomprehension to disbelief.\n\n\"You want us to help you make a new Constitution for your country,\" says Enrique. His world-weary face is offset by wide-eyed wonder.\n\nEnrique looks at his friend Ricardo for confirmation that he had not misunderstood my explanation in Spanish of what my project is about. Ricardo, who helped arrange this gathering in a mid-sized city in the South, once lived near my family and me in Mexico. He was a fixture at the Socrates Caf\u00e9 I held in the historic plaza of San Cristobal de las Casas, nestled in the highlands of Chiapas. Though extremely industrious, the best job he could land back in Mexico was as a six-day-a-week handyman at a hotel that paid him the equivalent of $4 for a ten-hour workday. When Ricardo failed to show up for our dialogues, most felt sure the inordinately thoughtful father of three and local social justice activist had \"gone north.\"\n\nNearly three years later, Ricardo called me in the United States. He said that he hadn't gotten in touch earlier because he was ashamed he hadn't shared his plans with me. But he said he wasn't sure at the time that he'd go through with it, and so he hadn't let anyone know outside his immediate family. He went on to tell me, with evident pride, that he had a well-paying job. But his voice broke when he confessed how much he missed his wife and three daughters.\n\nOur joyful reunion was tempered by apprehension. Months earlier, federal officials had raided a local industrial plant employing undocumented workers. Friends and relatives of many of today's participants, including two of Ricardo's cousins, were taken into custody and likely will face deportation proceedings.\n\nThose who join Ricardo and me at the diner where he works are mostly from Mexico, though at least one person is from Guatemala. All are in the United States illegally.\n\nAfter I assure Enrique that I do indeed want the input of everyone on hand in helping create a new constitutional article, he says, \"Your country's Constitution should say that the U.S. always puts out the welcome mat for refugees. The U.S. owes it to its unique history to have an article like that. It would reflect a core value that all Americans should share, of being welcoming to people who are unwelcome in their own homeland. After all, every American, even Native Americans, has ancestors who came here from somewhere else, and often because they were persecuted in some way.\"\n\n\"Right now, your Constitution gives Congress power 'to establish an uniform rule of naturalization,'\" he instructs me. \"But it doesn't say what that uniform rule should be. This new Constitution should have an article that says political asylum is automatically granted to all refugees.\"\n\n\"You can usually qualify for a green card only if you can prove that you're a political refugee, meaning your life would be in danger if you returned to your homeland,\" Ricardo now says. \"Under those circumstances, I have no hope of becoming a legal citizen, much less bringing my family here so they can one day get citizenship.\n\n\"But the first Americans not only came here from someplace else, but came here because they were economic refugees, like we are. Keeping that in mind, this new Constitution should have an article that states, 'All refugees are to be welcomed.' \"\n\n\"Our Constitution should place no limits of any kind on immigration?\" I ask.\n\nGuillermo, a cousin of Enrique's who entered the United States nearly a decade ago, replies: \"Enrique and Ricardo only said there should be no limits on the number of refugees who immigrate. A refugee is someone who leaves his country to escape some kind of persecution. In our case, we were persecuted for trying to live like human beings, to put food on the table for our families, send our kids to decent schools. Anybody in his right mind would flee that situation to seek out a better life in another land.\"\n\n\"On the other hand,\" he goes on, \"it should be impossible for the people in power who exploited Mexico and made our lives a nightmare to leave the country. They're not refugees. Yet they're welcomed in other countries with open arms.\"\n\nEnrique nods. \"Our former president Ernesto Zedillo is now a big shot at Yale. While president, he called for peaceful dialogue to promote indigenous rights, all the while militarizing the areas around those indigenous communities that had leftist leanings. By the time he left Mexico for greener pastures, Mexico's indigenous inhabitants were in more desperate straits than ever. But he's a saint compared to our former president Carlos Salinas de Gortari, whose corruptions and crimes are astonishing even by Mexican standards. On top of approving the scandalous NAFTA treaty and engaging in such a spending spree that the peso had to be devalued drastically, ruining the middle class and poor, he stole hundreds of millions from our treasury. Now he lives like a king in London.\"\n\n\"People like Salinas de Gortari cross borders\u2014and usually in their own jets\u2014whenever they want,\" says Guillermo. \"There's a big difference between them and refugees like us, who may die crossing the desert to escape the persecutions of a Salinas de Gortari.\"\n\n\"We never had rights in Mexico, though our homeland's constitution says we all have equal rights,\" says Oscar. \"Even when I was out in the open, I was invisible. I worked my fingers to the bone, and received less than nothing in return. Here, though I have no legal or constitutional rights, my employer treats me with respect. He pays me a good wage and has given me raises for my hard work. I stay with my cousin's family, so I've been able to send most of my earnings back home. Now my children can eat nutritious meals, and they go to a good school instead of laboring in the fields with my wife.\"\n\n\"My brothers and several cousins were arrested by la migra [immigration enforcement officers], and are being deported,\" he then tells me. \"They'll keep rounding us up until the economic crisis goes away. We're the convenient scapegoats. These days, I stay locked up in my apartment when my shift at work is over. None of us would even go to the May Day parade, because we were scared we might be nabbed.\"\n\nHe says next, \"This community was dying before we got here. We were mostly made to feel welcome. We all met each other halfway: we overcame suspicions, because we knew we needed one another. Now, the locals and undocumented workers keep their distance.\"\n\n\"I know we're supposed to be helping you make a new article for the U.S. Constitution,\" says Enrique, \"but I can't help thinking of my own country's. It is a beautiful document. Article 3 says it is the duty of Mexico's federal government to provide all children an education that will 'develop harmoniously all the faculties of the human being and shall foster in him at the same time a love of country and a consciousness of international solidarity, in independence and justice.'\"\n\nHis smile is more sad than sardonic. \"That is sheer poetry. But the reality is pure mierda. Millions of Mexican children have no hope of going to school. Millions more go to schools where the teachers are poorly paid and educated. But you're punished if you raise much of an outcry, if you insist to politicians that they do their job and enforce the constitution.\n\n\"In 2007, hundreds of teachers in Oaxaca, the poorest state in Mexico, were jailed for protesting the pitiful state of education. I was a teacher there. I earned under $200 a month. I couldn't support my family. So here I am, working in an industrial plant\u2014or, I was working at one, until my boss's employment records were audited by immigration officials and he was forced to let go of all undocumented workers. Now I'm thinking about going back to Mexico. But if I do, I'll have to work like a slave for slave wages again. And there'll be no chance I can come back here, not with all the new security on the U.S. side of the border.\"\n\n\"Really, we have no country,\" says Oscar. \"We aren't citizens of anyplace. I wish I could be a citizen here, wish I could vote, pay taxes, be even more active in the community. As hard as I work in the U.S. now, if I was a citizen, I would break my back to give thanks.\"\n\nThen he says, \"Americans who don't like our presence here call us 'wetback.' I know it's derogatory, even racist. But I take it as a compliment. Our backs are wet from working so hard.\" He grins. \"Besides the 'wet-foot, dry-foot policy' the U.S. has for Cubans\u2014if a Cuban arrives on dry land before he's apprehended, he's allowed to stay\u2014your Constitution should establish a 'wet-back, dry-back' policy. Those whose backs are the wettest\u2014who work the hardest, sweat the most, who give everything they have to make this great country even greater\u2014should be the first in line to become citizens.\"\n\n\"We send back most of what we make to our families and relatives,\" Ricardo soon says. \"The remittances we make are a top source of revenue for Mexico. That kind of contribution should earn us high honors and privileges as Mexican citizens. We don't do it for that reason, but it isn't right that the people who are honored in our country are the most dishonorable ones.\"\n\nSusana breaks the silence following a lull. \"My twin daughters were born here, so they automatically have the privilege of being citizens of the U.S.\" The soft-spoken Guatemalan is the only woman present.\n\n\"Do you believe they should be given automatic citizenship because they were born on U.S. soil, as our existing Constitution stipulates?\" I ask.\n\nShe gives this a good deal of thought. \"I'm glad it's that way, because of how things are. In an ideal world, I'd bring them back home to Guatemala, to a land of plenty. If I'm deported, they'll stay on here with relatives, though it would break my heart. It would be unfair for them to come with me to a country where they'd have no future. I've been reading how conservative legislators in some states, and some U.S. senators, want to change the Fourteenth Amendment so that children of economic refugees who are undocumented no longer can become U.S. citizens, but I think this would put us more in the shadows than ever.\"\n\n\"As long as there are economic refugees,\" she goes on, \"a constitutional article like this one Enrique suggests is good and just. It puts the rest of the world on notice that if our home countries don't treat us well, we'll find a decent place for ourselves in the U.S. and work our hearts out to justify the privilege of being here.\n\n\"Exploitative countries like Mexico and Guatemala can't afford for all of us to leave. It's possible that the only way they'll ever enforce the rights in their constitutions is if they realize that they'll have an empty country if they don't.\"\n\nRamiro then says, \"What I would give for such an article as the one we're talking about. It'd allow my family to come here and join me. No matter how much I might want to, I can't go back to Mexico. It would fill me with joy to be with my family again, but then would come the sadness for not being able to provide for them.\"\n\nThen he says softly, \"I just want the constitutional right to work at home and earn enough to live in dignity. I want the constitutional right to be with my children, the right to listen to my daughter breathe softly in her sleep.\" He falls silent for a moment. \"I can see my Angelina in my mind, where she is still little. I see myself singing her to sleep while strumming my guitar. It's how I put myself to sleep at night. I haven't seen her in over four years.\"\n\nEventually, Ricardo says, \"The constitutional article Enrique proposes would make sure Americans don't forget their 'founding identity,' no matter how hard some try.\n\n\"I hitchhiked once all the way to New York just to see the Statue of Liberty. At its base, there's a bronze plaque with a poem. I was so moved by it that I wrote it down, and I carry it with me.\" He pulls out of his pocket a folded piece of paper: \" 'Give me your tired, your poor, your huddled masses yearning to breathe free, the wretched refuse of your teeming shore. Send these, the homeless, tempest-tost to me, I lift my lamp beside the golden door.' \"\n\nHe looks at me. \"The narrator of the poem really is America itself, or Lady Liberty speaking on America's behalf. She's inviting all repressive countries to send their poor here, and she's promising us refugees that she'll not only leave the lamp on for us, but open the golden door and welcome us into this country. That's America at its most beautiful.\"\n\nConstitution Caf\u00e9 Article: Congress shall have Power To establish an uniform rule that grants political asylum and facilitates the Naturalization of all Refugees. Congress shall prevent entry to the United States of corrupt world leaders.\n\nRoom for One More\n\nJefferson generally extolled the virtues of maintaining an open-door policy when it came to immigration. In 1801, he delivered his first Annual Message (what is now the State of the Union) in the midst of a period of great influx of immigrants to the United States. Jefferson confronted the issue in his inimitable way, with questions: \"Shall we refuse the unhappy fugitives from distress that hospitality which the savages of the wilderness extended to our fathers arriving in this land? Shall oppressed humanity find no asylum on this globe?\"\n\nHe held that immigrants should not just be welcomed as guests, but expeditiously made U.S. citizens in order to ensure that they'd be \"secured against like oppressions by a participation in the right of self-government.\" He feared that if immigrants did not promptly enjoy the constitutional rights of all Americans, they would continue to be exploited. Further, he stressed, he was guided in his sympathies toward immigrants by \"the animating consideration that a single good government becomes thus a blessing to the whole earth, its welcome to the oppressed restraining within certain limits the measure of their oppressions.\"\n\nAlexander Hamilton, himself originally an impoverished West Indian native, cited Jefferson's address in a published essay opposing immigration, ridiculing its \"pathetic and plaintive exclamations.\" Hamilton referred to the new wave of immigrants as \"truly the Grecian horse to a republic\" and warned that the nation \"cannot be too careful to exclude its influence.\" He'd argued at the Philadelphia Constitutional Convention that the only ones who should be welcomed to the United States in the future were Europeans \"of moderate fortunes,\" who should be \"placed on a level with first citizens.\"\n\nForgetting his own roots, Hamilton asserted that immigrants seeking U.S. citizenship must already share our cultural and ethnic backgrounds, since those who did not represented a threat to the \"uniformity of principles and habits\" necessary for \"a common national sentiment.\" But to Jefferson, a shared background might mean a common love for and yearning to live in a democratic society with its promised freedoms to pursue and promote happiness. He believed that people of vastly different cultures and belief systems might nonetheless have a higher, shared allegiance to the principles of democracy itself.\n\nJefferson's predecessor as president, John Adams, and the Federalist majority in Congress, as part of the Alien and Sedition Acts, dramatically increased the length of time it took for an immigrant to become a U.S. citizen from five to fourteen years. While Adams and the Federalists in Congress claimed that it was vital to national security that immigrants reside in the United States for a significant amount of time before they could become citizens and obtain the right to vote, they may have had another motive; the fact was that immigrants tended to vote Republican. When Jefferson became president, the Immigration Act of 1802, which he endorsed, changed the residence requirement back to five years.\n\nAs he expressed in a correspondence with a Mrs. Church during the Revolutionary and Napoleonic wars in Europe, it was vital that the United States provide \"a sanctuary for those whom the misrule of Europe may compel to seek happiness in other climes. This refuge once known will produce a reaction on the happiness even of those who remain there.\" Such a safe haven, he believed, would serve as fair warning to \"their task-masters that when the evils of Egyptian oppression become heavier than those of the abandonment of country, another Canaan is open where the subjects will be received as brothers.\"\n\nThere is no reconciling Jefferson's hypocritical and inhumane stance when it came to supporting policies that allowed Africans to be brought against their will to the United States, where those who survived the harrowing journey were made slaves. But in terms of permitting voluntary emigration to the U.S.\u2014and he clearly was referring at the time to white Europeans\u2014Jefferson maintained that there should always be room for one more. He went so far as to assert in his manifesto A Summary View of the Rights of British America that our forebears \"possessed a right, which nature has given to all men, of departing from the country in which chance, not choice, has placed them, of going in quest of new habitations, and of there establishing new societies, under such laws and regulations as, to them, shall seem most likely to promote public happiness.\"\n\nToday's immigrants are not establishing new societies in the way the original settlers were, but rather are venturing to established ones. Even so, when we welcome immigrants, our society will invariably change, just as those who come to America will be transformed to some degree.\n\nLoyalty Oath\n\nArticle VI, Section 3, of the Constitution: \"The Senators and Representatives . . . , and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States.\"\n\n\"We've just sworn not only to 'renounce and abjure all allegiance and fidelity' to the country of our birth, but to 'support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic,'\" says Naji. \"It's a little intimidating even to pretend to redo a Constitution we've just given our undying loyalty to.\"\n\nEven so, he and the other recently naturalized citizens on hand have agreed to consider participating in my project. I'd gotten to know Naji, an information technology specialist who came to the United States from Syria twelve years ago on a work visa, over the course of several Socrates Caf\u00e9 gatherings. He'd invited me to attend his citizenship ceremony the previous week. The others taking part had become friends with him during the months they'd spent in classes preparing for their citizenship exam. After the ceremony, Naji said to me, \"I can't believe I'm now an American citizen, just like you.\"\n\nEventually, Jimena\u2014one of the increasingly rarefied illegal immigrants from Mexico who succeed in gaining citizenship\u2014says to us, \"It occurred to me, as I was studying for my citizenship test, that even those born on U.S. soil should have to make a loyalty oath of some sort. The fact that they're born here doesn't mean they'll be loyal to the Constitution. On the other hand, taking an oath as we had to do is no guarantor of loyalty either. But it is a solemn and moving event. It makes you really think about, and feel, what it means to be an American citizen, and what your privileges and duties are.\"\n\n\"If Americans born here took part in such an event, they might even be inspired to know their own Constitution,\" she goes on. \"Once, I was at a caf\u00e9 poring over the Constitution as part of my study regimen for my citizenship exam, when it suddenly struck me that your\"\u2014Jimena checks herself\u2014\"our Constitution, my Constitution\u2014I have to get used to saying that\u2014does not once mention the word 'democracy.' Democracy should ring loud and clear from every page. I told a person sitting next to me at the caf\u00e9 about my shock over the lack of the 'd-word' in the Constitution, and she said, 'Really? I've never read the thing myself.' What does that say if some Americans don't care, or don't have to care, to know about the document that enables them to live so free? Aren't they sure to lose the most important part of who they are and what they're all about?\"\n\n\"People born in the U.S. shouldn't have to pass some test about the Constitution to be citizens, but they should of course want to know and appreciate it,\" says Zhang, from China.\n\n\"I'm a literacy volunteer. When I told one of my students a couple years back that I'd begun the process of becoming a naturalized citizen, she congratulated me and then recited to me passages of the Declaration of Independence that she knew by heart. And she said, 'Young man, before you give your oath, I will be able to read to you the entire Declaration, and the Constitution too.' From then on, she and I worked on reading them as part of her literacy instruction. The week before I became a naturalized citizen, she read both of them to me.\"\n\nThen he says, \"Some segments of American society throughout this country's history have been intentionally kept down and out\u2014and by people who could read the Declaration and Constitution, but couldn't care less that they were betraying its ideals in the cruel way they treated fellow Americans. This student of mine may not have to give a loyalty oath like we did, but she's loyal to the American dream, to making it come true for herself and her children.\"\n\nAfter a pause, Naji says, \"Elected federal officials are required by the Constitution to give a loyalty oath. Article VI requires all federal and state legislators, as well as all executive officers and all judges, to give an oath or affirmation in support of the Constitution. The oath is similar to the one we had to take to become citizens.\"\n\nHe reads from a copy of the congressional oath that is currently administered: \" 'I, [name], do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservation or purpose of evasion; and that I will well and faithfully discharge the duties of the office on which I am about to enter.' \"\n\nThen Naji says, \"Even though this constitutional clause regarding the oath they must take says that 'no religious test shall ever be required as a qualification to any office or public trust,' most elected officials being administered the oath add 'so help me God' at the end. It's not part of the oath, but it's clearly important to them that their constituents know they're not only loyal to the Constitution, but to God.\"\n\n\"On the other hand, in the loyalty oath that naturalized citizens are administered, even if we're atheist, we're required to say at the end, 'so help me God,'\" says Jimena.\n\nThen she says to me: \"Did you know that the oath we take in order to become naturalized citizens isn't even a constitutional requirement? It's only a mandate of the Immigration and Nationality Act of 1952 and can be done away with at any time by an act of Congress. I'll read it to you: 'I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty of whom or which I have heretofore been a subject or citizen; that I will support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the Armed Forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law; and that I take this obligation freely without any mental reservation or purpose of evasion; so help me God.' \"\n\n\"Should the oath administered to naturalized citizens be made a constitutional requirement?\" I ask.\n\nJimena thinks this over. \"Except for the 'so help me God' part, yes it should.\" And then: \"Everyone who becomes a naturalized U.S. citizen has a special status, though not quite like that of an elected official. We have 'elected' to become citizens of the United States. For those of us fortunate enough to pass through all the citizenship hurdles, we're sort of like official goodwill ambassadors for the U.S.\"\n\nThen Zhang says, \"Many of us naturalized citizens undoubtedly appreciate the promise of this country more than many born here. We appreciate that the rights in the Constitution are guaranteed, and that if you're denied them by anyone, you have real recourse to have them restored. Many of us come from countries with constitutions in which guarantees of rights and liberties are just empty words. China's constitution mentions again and again that it is a 'democracy,' and that all citizens are equal, and have all sorts of freedoms that can never be denied to them. These empty words in my native country's constitution make it a hurtful one, a betrayal of what a constitution is supposed to stand for.\"\n\nConstitution Caf\u00e9 Article, Section 1: Senators and Representatives, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States. The oath they shall give shall say: \"I, [name], do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservation or purpose of evasion; and that I will well and faithfully discharge my duties and responsibilities.\"\n\nConstitution Caf\u00e9 Article, Section 2: All naturalized citizens of the United States shall be administered the following oath: \"I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty of whom or which I have heretofore been a subject or citizen; that I will support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the Armed Forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law; and that I take this obligation freely without any mental reservation or purpose of evasion.\"\n\nOath of Allegiance\n\nThe oath that all immigrants are required to take before they become naturalized citizens is officially called the Oath of Allegiance. General Washington administered the first such oath at Valley Forge on February 3, 1778, before the military campaign against the British began in earnest. With the aim of ferreting out British sympathizers, Congress had passed a law requiring all officers to profess and sign an oath swearing their allegiance to the cause set forth in the Declaration of Independence.\n\nAll the officers who stood before Washington placed their hands on the Bible and recited the oath. One officer, General Charles Lee, withdrew his hand. When Washington asked him to explain himself, Lee replied, \"As to King George, I am ready enough to absolve myself from all allegiance to him; but I have some scruples about the Prince of Wales.\" His reply elicited a great deal of laughter, but \"covered a deeper motive . . . Lee was then playing a desperate game of treason, and probably had some conscientious scruples about taking such an oath which he would probably violate.\" Lee was eventually dismissed from the army for direct disobedience of Washington's orders.\n\nToday's Oath of Allegiance taken by naturalized citizens is not a constitutional requirement, as Jimena pointed out during our gathering, but rather is stipulated by an act of Congress. At other times in our nation's history, Congress has mandated the administration of loyalty oaths to American citizens themselves, typically during times of fear or uncertainty. Congress did so during World War II, and again in 1947, when increasing tension between the United States and the Soviet Union generated anticommunist hysteria, leading President Harry Truman to launch an initiative aimed at determining the loyalty of the federal workforce. Truman's Federal Employee Loyalty Program included \"an examination of any files held by any government agency, from the FBI to schools.\" If an employee refused to take part, he was charged with disloyalty and dismissed from his position. If he did participate and was charged with disloyalty, he could defend himself at a Loyalty Review Board hearing. Following Truman, individual states launched their own loyalty programs, and in particular investigated teachers at schools and universities. Out of principle, many teachers refused to participate and were usually fired.\n\nGeorgia still has \"loyalty legislation\" on its books, requiring all job applicants to fill out a loyalty form, and Oklahoma requires employees to sign a loyalty oath after thirty days on the job. California is also among the states that require such an oath. In 2008, an employee at a California university was temporarily dismissed for refusing to sign it. But Marianne Kearney-Brown, a Quaker, was reinstated after signing an oath tailored especially for her, which stated that if she had to defend her state and country, she only had to do so in nonviolent ways.\n\nWould Jefferson believe that one best demonstrated one's loyalty to the nation by solemnly swearing to always support and defend the Constitution?\n\nNewspaper editor John Colvin once asked Jefferson whether there were occasions when one could act outside the letter of the law in good conscience. Jefferson replied that while the \"strict observance of the written laws is doubtless one of the high duties of a good citizen . . . it is not the highest.\" He believed there were \"laws of necessity, of self-preservation, of saving our country when in danger, [which] are of higher obligation\" even than constitutional law. \"To lose our country by a scrupulous adherence to written law, would be to lose the law itself, with life, liberty, property and all those who are enjoying them with us; thus absurdly sacrificing the end to the means.\"\n\nThurgood Marshall, the renowned civil rights lawyer and first African-American appointed to the Supreme Court, echoed this view of Jefferson's in his autobiography. Marshall believed that\n\nthere is something 'higher' than the Constitution\u2014that is, quite simply, the people. I do not mean that 'the people' are not bound to live under our system of laws. . . . But what I do mean is what Thomas Jefferson said in the Declaration of our Independence\u2014that just governments derive their authority from the consent of the governed. And because of this, you have not only the right but also a responsibility to the government of this country.\n\nBy this perspective, all U.S. citizens have a civic responsibility to make sure that the government is working in a way that does right by everyone, and by democracy itself. To that end, Marshall contended that Americans must be loyal above all else to embracing \"all of the qualities that Jefferson embodied: commitment to difficult projects, confidence in the soundness of one's own vision, and perseverance in working through a problem.\"\n\nBut did Jefferson in fact embody these qualities?\n\nJohn Dewey acknowledges that political expediency compelled Jefferson \"to deviate on special points,\" and that, moreover, Jefferson was \"disposed to be conciliatory and compromising.\" Even so, Dewey asserts, \"there are few men in public whose course has been so straight,\" even as Jefferson demonstrated \"inconsistency between professed principles and actual behavior:\"\n\n[I]f Jefferson is better known for his political ideas and his public acts than as a human being, it is just what he would have wished for himself. Considering his times and the difficult and important part he played in them, there remains the image of a magnanimous, high-spirited public gentleman who subordinated himself with complete devotion to what he conceived to be the welfare of the country he loved.\n\nBut the \"special points\" on which Jefferson deviated were crucial ones, and they eventually had to be addressed if public life in the New World was going to be revolutionized. Aside from the fact that Jefferson's professed views on the equality of all were contradicted by his prejudice toward blacks, he was hypocritical in his stance on women's rights. \"Jefferson assumed without argument that women were by their very nature excluded from the public realm,\" notes the important Jefferson scholar Joyce Oldham Appleby. In Jefferson's worldview, women \"think naturally of home and hearth, of husband and children, and rarely (if at all) of wider public concerns.\"\n\nAlbert Gallatin, secretary of the treasury during Jefferson's presidency, upon relating to Jefferson that there was a dearth of qualified applicants for a number of posts in his department, suggested the names of a number of qualified women. Jefferson dismissed out of hand Gallatin's recommendations, replying, \"The appointment of a woman to office is an innovation for which the public is not prepared, nor am I.\" The enlightened view of his own cabinet secretary makes it clear that in important respects Jefferson's own mind lagged behind the most progressive minds of his times.\n\nIt would be up to others to embrace the promise of the Declaration and advance the cause of women and of blacks. Touching on the issue of slavery in an 1814 communiqu\u00e9 with Edward Coles, a Virginia aristocrat, Jefferson reflected, \"the hour of emancipation is advancing in the march of time.\" He was honest enough, however, to acknowledge that he would not be the one to lead this advance: \"This enterprise is for the young; for those who can follow it up and bear it through to its consummation. It shall have all my prayers, and these are the only weapons of an old man.\"\n\nIn judging Jefferson's character, as Jefferson critic Joseph Ellis stresses, one should not divorce Jefferson's actions from the times in which he lived: \"Lifting Jefferson out of that context and bringing him into the present is like trying to plant cut flowers.\" But even when considering Jefferson's deeds firmly in the context of his times, Ellis chastises Jefferson for the fact that his \"utter devotion to great principles\" was compromised by \"a highly indulged presumption that his own conduct was not answerable to them.\"\n\nYet Jefferson did believe that he should be held accountable for his deeds by future generations. Jefferson described how he judged others: \"by their lives. For it is in our lives, and not from our words,\" that one must gauge one's character. \"By the same test,\" he held, \"the world must judge me.\"\n\nRadical Character\n\nThe astute Jefferson biographer Fawn Brodie was intrigued by the fact that Jefferson managed throughout his career to be a consummate political insider while remaining on the radical fringe: \"rarely if ever has a man with such a radical bent won so many elections from such an electorate.\" Yet back then, it was not as rare as it is today for someone to be part of the mainstream yet have the temerity to propose drastic changes that would challenge the status quo. Indeed, the argument can be made that the only Americans who were not inclined to radicalism were the minority of entrenched power brokers among them, including the Framers.\n\nEven Joseph Ellis\u2014who characterizes Jefferson as \"a flawed creature, a man who combined massive learning with extraordinary na\u00efvet\u00e9\"\u2014asserts that the one element of Jefferson's character that was unchanging was his inveterate radicalism. Ellis notes that \"no one, . . . including the ever-skeptical Adams, ever doubted his radical credentials.\" Jefferson showed that there didn't have to be a contradiction between working diplomatically with the mainstream in government and endorsing radical ideas.\n\nYet today, it is widely assumed that a radical is necessarily an outsider, even an extremist. But what seems to characterize those labeled today as \"radicals\" of the left, right, or center is a dogmatic mind-set; they are convinced that their respective viewpoints are beyond reproach or debate. Yet a \"Jeffersonian radical\" today would assert that if the original American character is to be revived, much less to evolve, then we must slough off the dogmatism so pervasive today and replace it with an enthusiastic openness to new points of view. Such a radical would actively seek out compelling objections and alternatives to her own view, recognizing both that no one has a monopoly on wisdom and that diversity and differences of opinion are good for public life. \nVI\n\nMoney Matters\n\n\"No taxation without representation!\" This indelible slogan of the American Revolution, crafted by Boston patriot James Otis, still rings in our national consciousness. There's no more eternal nor potentially irresolvable political argument than that between those who see taxes as a necessary way to fund public goods\u2014including infrastructure, education, police, public health, and national defense\u2014and those who see taxes as a chokehold on individual liberties.\n\nThe agony of taxpaying is of course part of our birth story, since as every schoolchild knows (or should know), the New World colonies revolted against England in large measure because its empire-expanding agenda left it drowning in debt, and so to raise money, the British Parliament enacted loathsome levies on the colonists without their advice or consent.\n\nYet most colonists remained victims of taxation without representation even after winning the Revolutionary War, since in almost all cases, only property holders could vote. Even the constitution for the new state of Massachusetts, where colonists had been the most militant when it came to protesting taxation without representation, did not grant voting rights to those who did not own property. As a consequence, many were still denied the right to participate in public affairs.\n\nNonlandowning citizens of Massachusetts were angered and insulted by the fact that the wealthy landowners and industrialists had created a state constitution that disqualified them from voting or holding office. Two militia captains, Samuel Talbot and Lemuel Gay, took the lead in airing their complaints publicly. In resigning their commission as officers in the militia, they expressed their bitterness over how they and the soldiers serving under them had been \"by the Constitution disenfranchised, for want of property.\" The state's first governor, John Hancock\u2014president of the Second Continental Congress, and a signer of the Declaration\u2014upon learning \"how bitter it was for men who had put their lives on the line to find that they were voteless and ineligible to hold office,\" might have been expected, as a leading revolutionary figure and patriot, to insist to the state legislature that the constitution be changed on their behalf. Instead, he ignored their appeals.\n\nThe Framers of our national Constitution also failed to fully address the issue of taxation without representation: they gave Congress the power to tax all those living in the United States, even though women, blacks, and taxpayers who didn't own property were usually denied at both the state and federal level the right to vote for those who supposedly represent them in Congress and would have authorization to tax them. To this day, those who live in U.S. territories and possessions, and in Washington, D.C., are subject to federal taxation even though they have no direct representation in Congress.\n\nCongress took its taxation powers to new, constitutionally murky heights in 2009 when it passed a stimulus package that cost taxpayers nearly a trillion dollars. Yet while it added dramatically to the rampant spending and borrowing that triggered the recession in the first place, there's little indication that it has helped abate our ongoing economic crisis. Jefferson believed government's wasteful ways encouraged its citizens to be copycats. Would Americans be more responsible spenders and savers on a personal scale if they could exert more influence over stemming federal profligacy? Would Americans have more clout if they all enjoyed taxation with representation? What if even noncitizens who paid federal taxes could vote for members of Congress?\n\nTaxation with Representation\n\nArticle I, Section 8, Clause 1, of the Constitution: \"The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises. . . .\"\n\n\"No taxation without representation,\" says Sheila. \"This has to be stated point-blank in a constitutional article. It's what the American Revolution was all about, after all. Yet the only thing the Constitution says on the subject is that Congress 'shall have power to lay and collect taxes, duties, imposts, and excises.' But Congress should only have taxing authority over people who have taxation with representation. A revised article should say that no one can be taxed who doesn't have this right.\"\n\nI'm at a small plaza in the Washington, D.C., suburb of Takoma Park, Maryland, where I once lived. About fifteen others in this municipality of 17,000, which has long enjoyed a reputation for cutting-edge politics\u2014it was the first place in the nation to declare itself a \"nuclear-free zone\"\u2014have joined me to examine Congress's taxation powers.\n\nSheila, a producer for a popular radio program, says next, \"If the article I'm proposing were already in the Constitution, all residents of Washington, D.C., would have at least one voting representative in the House, and two in the Senate. Either that, or they wouldn't have to pay federal taxes. I moved from D.C. to Maryland a decade ago in part out of frustration over the fact that I had no voting representative.\"\n\nWashington, D.C., is a federal district. While it is governed by a mayor and city council, Article I of the Constitution dictates that Congress has ultimate authority over its affairs. The district has one nonvoting member of Congress, and no U.S. senators.\n\nJessie, who works for the U.S. Agency for International Development, says, \"This article should give all those on U.S. soil who pay federal taxes the right to vote in federal elections, just as they have the right to vote here in Takoma Park in local elections.\"\n\nTakoma Park is the largest municipality in the United States to grant suffrage rights to noncitizens, including unregistered immigrants. Maryland's original constitution extended voting rights to all \"inhabitants\" of the state. But in 1810, during a time of widespread anti-immigrant sentiment, the state legislature rescinded this right, amending the state constitution to restrict suffrage to citizens. However, this limitation only applied at the state level of governance. Exploiting this loophole, Takoma Park activists in 1991 succeeded in passing a referendum that not only allows noncitizens to vote in local elections, but also to run for local office.\n\n\"Most of those residing in our nation who are denied the right to vote on the federal level make significant contributions to their communities, and so to our country,\" Jessie goes on. \"They deserve representation in Congress as much as any other person who pays federal taxes.\"\n\n\"That's going too far,\" says Mason, a State Department analyst. \"I agree in principle that all American citizens, wherever they may live, should have voting members of Congress, and that if they don't, they shouldn't have to pay federal taxes. But I draw the line with giving noncitizens, whether here legally or not, suffrage in federal elections. Yes, many surely are major contributors to our community and country, but they're here by choice. I've spent over half my adult life overseas, and have paid my fair share of taxes abroad. But I'd never expect, much less demand, voting representation in any nation of which I'm not a citizen. So I can't agree to extend full-fledged voting rights in federal elections to noncitizens.\"\n\n\"What about a noncitizen who is serving in our military, and who might risk his life to defend U.S. soil?\" Jessie asks him.\n\n\"I honor his service to our country,\" Mason replies. \"But he's doing this of his own free will, in part out of the promise of having his citizenship application expedited. I don't feel anyone should be able to vote in federal elections, and have voting representation in Congress, unless he's an American citizen.\"\n\n\"I've been a permanent resident of Singapore for nearly three decades, and of course I'm not allowed to vote there,\" says David, an architect who is here visiting his parents. \"As a U.S. citizen, I take advantage of my right to vote in my country's federal elections. My voting precinct is Takoma Park, the place in the U.S where I last voted. My children are twenty-two and twenty-five, they both have good jobs, and pay lots of federal taxes. Though American citizens, they've never lived on U.S soil. The Uniformed and Overseas Citizens Absentee Voting Act guarantees the right to vote in federal elections to all American citizens living temporarily or permanently abroad. But\u2014and here's the Catch-22\u2014because my children have never lived in the U.S., they don't have a designated voting precinct. So they're effectively denied their right to vote.\n\n\"So until we make sure that all U.S. citizens not only have the right to vote in federal elections, but actually can vote, I'm not prepared to consider extending this right to noncitizens. Ideally, the more than four million expatriates should have their own voting representative in Congress. We're taxed at a much higher rate than Americans living on U.S. soil, because we don't have a voting representative standing up for us.\"\n\n\"Many people born on U.S. soil aren't U.S. citizens,\" Jessie says in due course. \"Only those born in the fifty states, in the District of Columbia, and in Puerto Rico, automatically are citizens. Those born on the other island territories that are U.S. possessions have never been given the right to citizenship\u2014and it's a right that can be taken away at any time from Puerto Ricans; they were granted citizenship by an act of Congress, but there's nothing in the Constitution that automatically guarantees this to them.\"\n\n\"As it is, Puerto Ricans are treated like second-class citizens,\" she goes on, \"just as D.C. residents are. They're taxed by Congress, though they have no voting representation in either chamber. Only those American citizens who reside in states have the guaranteed constitutional right to vote for members of Congress.\"\n\n\"All the more reason to stipulate, in a revised article, that those Americans who don't have voting representation in Congress should not have to pay federal taxes,\" says Sheila. \"Or\u2014and this would be better\u2014the article could state that all those born on U.S. soil, whether a state, territory or other U.S. possession, are automatically U.S. citizens, and as such, must have the right to voting representatives in Congress.\"\n\n\"Well, those who live in 'nonstates' that are under United States jurisdiction receive a lot of federal dollars that are used for building and running schools, hospitals, infrastructure development projects,\" says David. \"Above and beyond that, the residents of Puerto Rico and Guam choose not to be states; they prefer to be independent entities. And at least in the case of Guam, though citizens do pay federal taxes, I believe all that money is returned to their government.\" He thinks some more. \"But even so, they're as much a part of this nation as the rest of us, and I agree they deserve to be treated as if they are in a 'statelike' place when it comes to having taxation with representation, so they have elected officials who look out for their particular interests. So until they have voting representation, they shouldn't have to pay federal taxes.\"\n\n\"Article IV of the Constitution gives Congress the power to create new states,\" says Jessie. \"This new article can give Congress an additional authority: to grant all those born and raised in U.S. territories and possessions the statelike right to have voting representation in Congress.\"\n\n\"That has my vote,\" says Sheila, then adding, \"If the patriots of the American Revolution knew that so many Americans today still are victims of taxation without representation, they might wonder if their sacrifices were worth it.\"\n\nHer son Chad, 14, has been sitting silently during the entire exchange. But now he says: \"Many states refuse to allow prisoners convicted of felonies to vote either in state or federal elections\u2014not only while they're in prison but even after they're released. So this article maybe should see to it that they get voting representation before any noncitizens are granted it.\"\n\nAnd then he says, \"But where do you draw the line? I mean, I'm a citizen and I've been paying federal taxes for the last two years, since I started working as a dishwasher at a restaurant. But I can't vote for four more years. There's eighteen-year-olds I know who don't work, and so don't pay taxes of any kind, but they get to vote. Why can't I?\"\n\nConstitution Caf\u00e9 Article: The Congress of the United States, comprised of a House and Senate, shall have power to lay and collect taxes, duties, imposts, and excises over American citizens, and over all others who are both born and residing on U.S. soil, who are represented by voting members of said Congress.\n\nSuffering Without Suffrage\n\nWhile Jefferson did succeed in reforming Virginia's legal codes as they pertained to inheritance, he was not uniformly progressive when it came to the issue of taxation. When he was a member of the Virginia House of Delegates, he did propose a bill (which was defeated) that would have created a system of taxpayer-supported free public education. But all in all, he did not believe that elected officials should have the power to spread the wealth by taking from the rich and giving to the poor. \"To take from one, because it is thought his own industry and that of his father has acquired too much, in order to spare to others who (or whose fathers) have not exercised equal industry and skill, is to violate arbitrarily the first principle of association,\" he maintained in a letter to the French Enlightenment aristocrat Antoine Louis Claude Destutt de Tracy.\n\nJefferson approved of the Framers' decision to give Congress taxing authority. As he told Francis Hopkinson\u2014fellow Founding Father and delegate to the Continental Congress\u2014in 1789, he \"approved from the first moment of . . . the power of taxation\" granted to Congress in the new Constitution. But he didn't support all the taxes they enacted. Since members of Congress were elected directly by voters, he believed that taxation authority should reside with them, because in a sense it gave the people themselves oversight, at least in theory. In practice, Jefferson didn't always abide by Congress's decisions when it came to levies. As president, he repealed all so-called internal taxes that had been approved by his Federalist predecessors, eliminating federal excise taxes on everything from whiskey to land.\n\nIn his first inauguration speech\u2014considered one of the most memorable in our nation's history (not to mention the first)\u2014Jefferson stressed that the \"suffrage of the people\" was a fundamental right. Yet most who'd supported the Revolutionary War were denied the right to vote in local, state, or federal elections. Only the State of New Jersey came close to granting universal suffrage; its founding constitution granted \"all free inhabitants\" the right to vote. From 1776 until 1807, women, free blacks, and noncitizens residing in New Jersey could vote in local elections. But in 1807, the state legislature backtracked and passed a new law that restricted voting to \"free, white, male citizens,\" eliminating any chance that what had been by far the most democratic approach to suffrage might eventually be emulated in other states. Meanwhile, noncitizens who were property holders\u2014wealthy white males, in all cases\u2014continued to enjoy a right that was denied to most U.S. citizens, along with the right to run for public office. Not until 1926 did Congress rescind the right of noncitizens to vote in federal elections.\n\nDebt Be Not Proud\n\nFrom Article I, Section 8, Clause 4, of the Constitution: \"The Congress shall have Power . . . To establish . . . uniform Laws on the subject of Bankruptcies throughout the United States.\"\n\n\"A section in Article I of the Constitution gives Congress sole authority to establish 'uniform laws on the subject of bankruptcies throughout the United States,'\" says Simone, a credit counselor. \"But 'uniformity' doesn't even enter the equation. Congress's bankruptcy laws are bankrupt.\"\n\nSimone, a friend from my high-school days who'd moved to the area over thirty years ago, is chiefly responsible for spreading the word about this Constitution Caf\u00e9 gathering, which has mostly attracted participants who have been impacted by the debt and credit crunch.\n\n\"The article needs to be retooled,\" she goes on. \"It should direct Congress to establish 'a uniform law for the subjects of bankruptcy throughout the United States.' That way, all bankrupt 'subjects'\u2014whether a megabank, one of the big car manufacturers, a farmer, a financier, a single mom struggling to make ends meet, or a blue-collar worker\u2014are treated exactly the same.\"\n\nI'm in a diner of a medium-sized town located in what is known as the Southern Auto Corridor. Over the past decade and a half, auto and auto-parts assembly plants have sprung up, attracted to this stretch of the Deep South in part because of strong antiunion laws. Many have closed or have laid off a large percentage of their workforce in the last several years. Vast stretches of farmland stand shoulder to shoulder with enormous exurbs built during economic boom times. Many tracts are half-developed, the projects abandoned. A number of the buildings near the diner, which has been in the same family's hands for over a century, stand vacant.\n\nDell, a local car dealer who has had to close two of his three franchises since the economic downturn, says to Simone, \"Your suggestion would be a big improvement, though there's no guarantee Congress will follow it. As you pointed out, our representatives don't properly exercise the constitutional authority they already have. They've mandated all these different bankruptcy 'chapters' with a crazy quilt of laws. How these laws are applied differs according to whether you're a big or small business, whether you're poor or middle class or rich. And the new comprehensive financial reform law that Congress passed does nothing to address this hypocrisy.\"\n\n\"As we all know,\" he goes on, \"Congress refused to allow some of the biggest banks, investment firms, insurance companies, car manufacturers, to go under. Though their problems were every fault of their own, they were considered indispensable to the American economy. But most small businesses were allowed to go under, even though most were operating responsibly. That can technically be considered 'uniform treatment' on the part of Congress, in the sense that it has uniformly enacted bankruptcy laws that favor the largest corporations. But that's surely not what the Framers of the Constitution had in mind.\"\n\nBeverley, a certified public accountant, then says, \"Unfortunately, the Supreme Court itself, which is supposed to make sure the Constitution is followed to the letter, contributed to this 'un-uniform'\u2014and in my opinion unconstitutional\u2014treatment of Americans in bankruptcy when it gave corporations all the rights of individuals. Bankruptcy laws are skewed towards the interests of corporations because they have the resources to influence Congress in a way we mere mortals can't even dream of.\"\n\nIn the 1886 case Santa Clara County v. Southern Pacific Railroad Company, the Supreme Court ruled that corporations could lay claim to the same rights as individuals, including the right to political and other noncommercial free speech. Our Constitution makes no mention of corporations. The Framers hadn't anticipated the central role they would come to play in the development of our economy. But by the nineteenth century, corporations were at the forefront of raising the vast sums of investment capital needed to undertake large privately financed projects, such as the building of the private railroad system, including the portion spearheaded by Southern Pacific. In the Supreme Court ruling in this case, corporations were accorded all the rights of individual persons. Opponents then and ever since have argued that this decision has undermined the democratic process: since corporations have exponentially more resources at their disposal than individuals, they not only can control commercial speech, but political and other noncommercial speech as well.\n\nBeverly says now, \"At least with a revised article, we can stipulate that bankruptcy laws must be applied evenly to individual persons and corporate persons alike.\"\n\n\"At its origins, this constitutional article we're discussing was a revolutionary attempt at fairness by our Framers,\" says Elena, a university professor whose particular scholarly interest is the history of debt in America. \"It gave Americans the right to enter bankruptcy proceedings and to emerge with a fresh beginning. Prior to that, if you were in debt, you were put in a debtors' prison, usually ruined for life. When this article became part of the ratified Constitution, bankruptcy became a right for every American.\"\n\n\"Though there were, just as now, financial panics triggered by irresponsible borrowing and lending, the great majority of Americans back then who went bankrupt were honest and frugal, but through hard luck still fell on difficult times,\" she continues. \"This constitutional article was meant to protect both creditors and debtors equally, to make sure both sides\u2014or 'subjects'\u2014were treated fairly in bankruptcy proceedings. The Framers understood that creditors and debtors both had an important role in growing the economy.\n\n\"Most everyone profited by this article's intent, including democracy itself. It created a thriving middle class and built an economic engine that had to be reckoned with by other nations. But in the generations since our founding, Congress has distorted its intent, to the extent that bankruptcy laws today almost entirely favor creditors and the big conglomerates. While predatory practices by lenders and banks may be curbed somewhat by the new financial reform laws, the fact remains that when an individual or small business enters bankruptcy, few protections are afforded from Congress's bankruptcy laws.\"\n\nThe New York Times reports that debtors' prisons are on the rise again in America: \"It comes right from the pages of 'Little Dorrit,' Charles Dickens's scathing indictment of Victorian England's debtors' prisons. Unfortunately, it is happening in 21st-century America.\" For instance, in the state of Minnesota, arrests and imprisonment of people unable to pay their debts\u2014often for amounts of less than $100\u2014had risen 60 percent in the last four years.\n\n\"But it's not Congress's fault that creditors and debtors forgot the purpose of credit and went on a borrowing and lending frenzy,\" says Abby, owner of the diner where we're gathered. \"No one forced people to borrow like there was no tomorrow. In my parents' and grandparents' time, you'd do anything to keep from going bankrupt, because you were looked down on for being careless with your or other people's money. Now there's no stigma attached to being financially irresponsible. If people can't pay their loans, they just walk away from them with the attitude, 'I'll just file for bankruptcy.' I'm not sure a constitutional article can do anything to remedy this.\"\n\n\"It looks like I'm the only farmer in attendance, so I'll speak up,\" says Curtis after a pause. \"The early rain and cold weather this year really hurt area farmers. I only got 30 percent yield on my crop. Some farmland has been declared a federal emergency, but all that means is I can qualify for a low-interest federal loan\u2014putting me more deeply in debt.\n\n\"Family farms are still the backbone of this country. Congress recognizes this, and that's why there's a special bankruptcy chapter for farmers. While the types of federal loans I qualify for under this chapter may ensure that I can continue to barely stay afloat, they also ensure that I stay in debt for life. On the other hand, the big agribusinesses that play with the same funny money as any other conglomerate somehow manage to thrive in bad times; the more debt they get into, the more their stock value seems to go up. It leaves me scratching my head.\"\n\nThen Dell says, \"Maybe bankruptcy laws should give special treatment to businesses considered indispensable. The problem is that Congress considers indispensable the wrong businesses\u2014the relative handful of big corporations rather than the more than 27 million small ones that keep America strong. It should be giving special treatment to all the small businesses in this country that play by the rules and have real connections to their communities\u2014especially to the small-town banks that refused to lend one irresponsible dime, while the big boys were giving anyone and everyone a loan who wanted one without reviewing their application, yet now find that they're as much impacted as the rest of us by the economic crisis brought on by the large banks.\"\n\n\"Because our largest banks have played such a central role in triggering bankruptcies,\" Elena says to us, \"a new constitutional article on this subject needs to state specifically how Congress can make sure that from here on out, they play only a positive role in helping businesses of all sizes thrive.\"\n\n\"That sounds good on paper,\" Curtis chimes back in, \"but the thing is, nowhere in the existing Constitution is Congress given the authority to regulate banks. But Congress does so anyway\u2014and it does a miserable job at that. If any entity has this authority that Congress has assumed, it's the states. Yet only one state has a bank of its own. North Dakota.\"\n\nStarted in 1919, the Bank of North Dakota (BND) is a state-owned and -run financial institution, the only one of its kind in the nation. All state and local government agencies are required to place their funds in the bank. North Dakota started a state bank in order to prevent predatory lending, particularly to farmers. Today, while most states are experiencing enormous deficits, North Dakota enjoys a surplus.\n\n\"The Constitution should say outright that the states, rather than Congress, regulate banks. That would do away with the Federal Reserve, which has brought on so much of this mess by playing enabler of financial irresponsibility to our largest corporations.\"\n\nFinally, Simone says, \"Congress's authority for creating a uniform bankruptcy law has to be aimed at ensuring that bankruptcy hardly ever happens. As has been pointed out, there are times and circumstances when, no matter how careful a person is, she might still default on her loans. In such cases, bankruptcy laws should serve as a safety net. For that to happen, the powers granted to Congress by the Constitution have to be such that all involved in the 'debtor-creditor transaction' are responsible and accountable, just as the Framers envisioned.\"\n\nConstitution Caf\u00e9 Article: Congress shall have Power to enact legislation that treats impartially all the subjects and objects of bankruptcy throughout the United States, and shall have Power to sustain and perpetuate a creditor-debtor system of responsibility and accountability. The States shall charter and regulate banks.\n\nBank Rupture\n\nAs governor of Virginia, Jefferson helplessly looked on as the state's economy collapsed. Shortly before he took office, as he relates, the legislature repealed \"so much of an act as makes it penal to offer or pay, ask or receive, more in paper bills of credit of this commonwealth or Congress, for any gold or silver coin, or more in the said paper bills for any property, real or personal, than is asked or offered in gold or silver.\" In other words, the legislature had opted to quit tying the dollar's worth to the gold standard, which had long held the value of paper money steady. When inflation struck, the value of paper bills plummeted. Since Virginia's constitution had established a weak governor, giving virtually all power to the legislative assembly, Jefferson could do nothing to stop this.\n\nAs critical as Jefferson was of the legislature's financial folly, he mostly blamed the nation's money problems on banks. He distrusted private financial institutions, which had the power to issue currency and credit, and to control interest rates. Writing to his friend John Taylor, Jefferson remarked, \"banking establishments are more dangerous than standing armies,\" what with their \"demoralizing\" lending practices.\n\nHe believed that only financial austerity would enable the United States to remain independent. Even before the Revolution was won, he lamented to fellow Virginia patriot Richard Henry Lee in 1779 that the \"inundation of money appears to have overflowed virtue, and I fear will bury the liberty of America in the same grave. . . . It is a cruel thought, that, when we feel ourselves standing on the firmest ground in every respect, the cursed arts of our secret enemies, combining with other causes, should effect, by depreciating our money, what the open arms of a powerful enemy could not.\"\n\nNo matter how decisive a victory America might achieve on the battlefield, Jefferson contended that if banks were allowed to continue utilizing unsound fiscal practices, from predatory lending to flooding the market with paper bills, Americans would never achieve real independence, since their fortunes would be vulnerable to the banks' whims.\n\nExacerbating the problem, Jefferson believed, was the fact that individual banks received money from a congressionally authorized private national bank\u2014called the First National Bank of the United States. \"The idea of creating a national bank I do not concur in,\" Jefferson asserted soon after it was founded. To him, the national bank was nothing less than \"an institution of the most deadly hostility existing against the principles and form of our Constitution.\" \"If the American people allow private banks to control the issuance of their currency, first by inflation and then by deflation, the banks and corporations that will grow up around them will deprive the people of all their property until their children will wake up homeless on the continent their fathers conquered.\"\n\nJefferson insisted that Congress was straying from its designated constitutional parameters in authorizing the creation of a national bank: the \"incorporation of a bank and the powers assumed have not . . . been delegated to the United States by the Constitution. They are not among the powers specially enumerated.\"\n\nJefferson tried to nip in the bud the central banking system of his day, contending that money's \"issuing power should be taken from the banks and restored to the people to whom it properly belongs.\" If banks were to play a responsible role in American society, he argued that they shouldn't be in private hands or in quasi-private hands, but rather owned by the public, which would be shareholders and would delegate the responsibility of issuing currency and credit to elected federal officials.\n\nAs president, Jefferson succeeded in having the national bank's charter revoked. But James Madison renewed the bank's authority in 1814 after becoming president. Since then, presidents that include Andrew Jackson (the only president to pay off the national debt) and Abraham Lincoln, both of whom shared Jefferson's belief that the national bank's machinations were ruinous to democracy's long-term prospects, have tried to do away with it. But it has withstood all such attempts. In 1913, the national bank morphed into the Federal Reserve. Though its stated mission is to stabilize the economy, the Federal Reserve's monetary policies arguably have been at the root of every modern financial crisis.\n\nThe Federal Reserve's managers have relentlessly discouraged savings by propping up low interest rates and have hidden how high inflation is in real terms, ruining not just millions of middle-class families, but saddling future generations with an insurmountable burden of debt. Even most Americans who locked into fixed-rate low-interest loans during the latest real estate frenzy also locked themselves into deep debt, as their properties have turned out to be worth far less than the loan amount. And because lenders failed to scrutinize the borrowers' credit history and their ability to repay these loans, millions have had to default, with millions of defaults looming.\n\nWhat if our nation's borrowing and lending system had taken a different approach from the outset? Just after the Revolutionary War era, when our financial system was in a more molten state, financial and economic theorists were positing alternative systems. For instance, Edward Kellogg, a prominent American financier who amassed and lost fortunes during the financial panics triggered by unsound borrowing and lending practices in the early and mid-nineteenth century, proposed a Jeffersonian-type system that would provide fixed, uniform, low-interest loans to the general public. The loan (or debt) issuer, public credit banks, would supplant private banks under this scheme. The size of the loan would be based on such factors as a prospective borrower's amount of collateral, his previous history of repaying loans, but also on his prospects for using the loan to fruitful economic advantage. The money used to repay the loans would be put right back into the public bank, in order to finance still more loans. To keep this system uniform and fair, and also to ensure a stable currency, Kellogg proposed that congressional legislation set the interest rate. But powerful banking interests saw to it that this and kindred proposals, which would have treated all potential borrowers equally, never gained serious attention.\n\nIt's the Economizing, Stupid\n\n\"Every discouragement should be thrown in the way of men who undertake to trade without capital,\" Jefferson wrote in 1785 to Nathaniel Tracy, a patriot and thriving privateer who was one of the chief financiers of the Revolution. Yet by this time, trading without capital was widespread.\n\nJefferson lamented that so many Americans were operating from the \"miserable arithmetic which makes any single privation whatever so painful as a total privation of everything which must necessarily follow the living so far beyond our income.\" He believed the best remedy was one of economic tough love, in particular \"the abolition of all credit\" for individuals. As Jefferson told Archibald Stuart, who studied law under him and went on to serve in Virginia's House of Delegates, \"I own it to be my opinion, that good will arise from the destruction of our credit. I see nothing else which can restrain our disposition to luxury, and to the change of those manners which alone can preserve republican government.\"\n\nYet Jefferson himself was a poster person for irresponsible individual consumption; he habitually made extravagant purchases far beyond his means, and his debt woes were exacerbated by a string of ill-advised investments. By the time his second term at the White House ended, he was over $20,000 in debt, a sizable sum for the times.\n\nJefferson was so deeply in debt at this time that he had to apply for a loan at a Richmond bank in order to keep his outstanding accounts in Washington solvent. When in 1814 the British burned the public buildings at Washington, Jefferson saw an opportunity to pay down some of his personal debt; he offered to sell his collection of some thirteen thousand books to Congress. The sale\u2014in which Congress purchased the books for perhaps half their worth\u2014did temporarily stave off financial disaster, but Jefferson continued spending far beyond his means.\n\nBefore the American Revolution got underway, Jefferson regularly sold parcels of his property in order to reduce his personal debt burden. But the paper money he received as payment plummeted in value as inflation soared during the war years. When the Revolutionary War ended, British creditors resumed their efforts to collect the money he owed them. Further, Jefferson suffered enormous investment losses\u2014exacerbated by the fact that he had cosigned a note for a relative who was insolvent\u2014during the financial Panic of 1819. When Jefferson died in 1826, his debts amounted to about $107,000. His beloved Monticello, and all his personal possessions, including his 130 slaves, were sold at auction.\n\nThe Joys of Privation\n\nAfter the United States won independence, Jefferson was wont to wax nostalgic about the shared deprivations willingly endured by patriots on behalf of the war effort. \"I look back to the time of the war, as a time of happiness and enjoyment,\" when everyone lived \"amidst the privation of many things not essential to happiness,\" Jefferson recounted in a letter to his cousin Fulwar Skipwith. Doing without for the sake of a greater cause, in his estimation, \"lays the broadest foundation for happiness.\" Yet after the war ended, he was no more willing than anyone else to practice financial parsimony, even as he continued stressing that it was an integral ingredient for personal and national autonomy.\n\nWould Americans today be willing to drastically cut back on consumption for the promise of greater individual and national liberation? Author Benjamin R. Barber is not optimistic. He believes consumerist capitalism has infiltrated every dimension of our existence to such an extent that it diminishes and even perverts the promise of our democracy, making almost oxymoronic such terms as \"corporate responsibility\" and \"civic responsibility.\" In Consumed: How Markets Corrupt Children, Infantilize Adults, and Swallow Citizens Whole, Barber contends that the \"logic of democracy may begin with the positing of rights and of a theoretical 'natural condition' in which women and men are born free, but it depends for its implementation on civic learning, public participation and common consciousness,\" which to him are critical criteria for practicing \"competent citizenship.\"\n\nBarber asserts that even when Jefferson and John Adams were staunch political adversaries, they nonetheless agreed that \"in the absence of competent citizens, bills of rights were but pieces of paper.\" However, Barber overlooks the great divide between Jefferson's and Adams's respective notions of what a competent citizen amounted to. Adams was an adherent of ambitious expansion of commerce and industry, which he believed was not an impediment to but rather a facilitator of the development of competent citizenship. Adams apparently forgot the fact that the American Revolution was sparked in large measure by a revolt against the world's first multinational, the British East India Company and its monopoly on trade in the New World. On the other hand, Jefferson favored at home a more agrarian-based economy that put first the interests of yeomen, artisans, and shopkeepers, who to him best exemplified both self-reliance and civic virtuousness (and so, in his estimation, were the polar opposite of industrialists, bankers, and financiers).\n\nNeither a Borrower Nor Lender Be\n\nFrom Article I, Section 8, Clauses 1 and 2, of the Constitution: \"The Congress shall have Power To . . . pay the debts . . . of the United States . . . [and] To borrow money on the credit of the United States.\"\n\nI am on Ellis Island. My handmade Constitution Caf\u00e9 sign is perched beside me. I'd planted myself hours earlier near the entrance to the building that had once been the primary entry point for immigrants to the United States. Though Ellis Island was closed long ago as an immigration-processing facility, New York remains the primary port of call in the United States for immigrants. Nowadays, however, they typically arrive here by plane rather than ship, and usually stay with relatives or in temporary housing rather than sleep on bunk beds without blankets in a drafty hall at the processing facility, where all told, more than 12 million immigrants passed through.\n\nI had been trying to engage school groups with meager results. Teachers informed me they were on a strict schedule, much as some indicated they might have wanted to allow their charges to have an exchange with me. Before calling it a day, I decide to take a tour. In a section with an exhibition featuring the wave of immigration from Greece and Italy starting in the late nineteenth century, I happen upon a woman staring at a photo display. She sighs audibly.\n\n\"They were desperately poor when they arrived,\" she says, more to herself than to me. \"But most of them succeeded here, although there was no social security, no health or life insurance, no totally reliable banks. If there's such a thing as rugged individualists, they were them.\"\n\nThe woman, who eventually introduces herself as Mariangela, tells me that her father and his family came to the United States from southern Italy, from the village of Bernalda in the province of Matera, which to this day is an agricultural center.\n\nI mention that my father's parents first passed through Ellis Island on March 29, 1910, arriving on the SS Chicago. They were among the first large wave of immigrants from the Dodecanese (\"Twelve\") Islands region, and hailed from the tiny volcanic island of Nisyros. Though ethnic Greeks, they were officially Italian citizens at the time they immigrated: the Dodecanese denizens recently had come together and declared independence from the Ottoman Empire, only to be promptly invaded and occupied by Italy.\n\n\"I probably have some Greek blood,\" Mariangela tells me. \"I'm from the region where Pythagoras opened a secret religious school.\" The Greek philosopher and mathematician Pythagoras (c. 572\u2013500 b.c.), was founder of the mystical and cultish Pythagorean Society. Their belief in transmigration of the souls, and that numbers represented nature's true essence, were considered heretical. Pythagoras and many followers fled to southern Italy from his native Samos, a Greek island, to escape persecution.\n\n\"Your grandparents, and my grandparents' siblings, got here in the nick of time,\" Mariangela says. \"Congress and their constituents didn't want 'southern Europeans'\u2014read: 'poor Greek and Italians'\u2014coming any more.\" Indeed, the U.S. Congress, concerned by the flood of immigrants from southern Italy and Greece, passed the Immigration Act of 1924, which placed severe restrictions on the number of southern Europeans that could enter the United States.\n\nMariangela then tells me, \"My own parents arrived from southern Italy in 1953, the year before Ellis Island was shuttered for good. They were among the final groups processed here. My grandfather's brothers and sisters all had come in the 1920s. But my grandfather, my nonno, stayed put in Matera. He was a tenant farmer, dirt-poor. According to family legend, he grew the most delicious apricots and peaches. He never scraped enough together to own his own land.\n\n\"No matter his hardships, he refused to leave Italy for hope of a better life here. He may not have owned his own land, but the land was part of him. Generations of family had been born and raised and died there, and to him it would have been sacrilege to leave. He felt like he would be abandoning them.\"\n\n\"My pop worked side by side with Nonno since he was a toddler,\" Mariangela says next. \"He was the youngest, the only one of his eight brothers and sisters to stay on in Italy. But when Nonno died in 1952\u2014less than three months after my grandmother passed\u2014Pop got set to immigrate with Mom. He wanted to be a parent, and he didn't want to raise a child there, in poverty.\"\n\nShe smiles at a memory. \"When Pop arrived here at age twenty, he 'didn't have a pot to piss in,' as he put it. He worked seven days a week, two and sometimes even three shifts, at his cousins' restaurants, until he saved enough to open up his own place\u2014with 100 percent cash on the barrelhead. Pop never owed a penny to anyone.\n\n\"I was born nearly fifteen years after they got here. Mom was in her late thirties, old by Italian standards\u2014or any standard back then\u2014to be a first-time madre. By then, Pop had his own restaurant. It was doing well\u2014his veal dishes were known as among the best in the city\u2014and he felt secure enough that he could provide for a family. Four years after I entered the world, my little brother Michele was born. Pop was determined that we have everything he didn't.\n\n\"But I didn't have him. One day, when I was about twelve, I took the bus into Manhattan from Long Island. I went to his diner in Midtown. I walked straight back to the kitchen, where he practically lived. I said to him, 'I am your daughter, and my name is Mariangela. I love music and math and modern dance. I have a boyfriend, but I'm not sure I'm in love with him.'\n\n\"He looked at me with tearing eyes. He wiped his hands on his apron and took it off. We came out front, sat in a booth, and had a meal together and a great conversation, for the first time since I don't know when. We had so much catching up to do! After that, we went out every Sunday, just him and me.\n\n\"We came here a good bit of the time when it opened up for tourists. Some kids go bowling or on picnics with their dads; I accompanied mine to Ellis Island. I didn't mind; I had him all to myself. Pop liked to 'meditate' here, think about all the people like him who came without a pot to piss in, yet more often than not made their dreams come true.\"\n\nI finally tell Mariangela about my Constitution Caf\u00e9 project. She is intrigued.\n\n\"Isn't there a section in the Constitution about debt?\" she asks me.\n\nI show her the Constitution and point to Section 1 of Article VI, which says, \"All Debts contracted and Engagements entered into, before the Adoption of this Constitution, shall be as valid against the United States under this Constitution, as under the [Articles of] Confederation.\"\n\n\"So we've been a debtor nation from the beginning,\" she says.\n\nShe eventually goes on to say, \"Pop loved to quote Shakespeare. 'Neither a borrower nor lender be' was his all-time favorite. 'For loan oft loses both itself and friend, and borrowing dulls the edge of husbandry.'\n\n\"My mom and pop lived by this saying: they were frugal when it came to their own needs, and generous to a fault with everyone else's. To them, it wasn't just 'Do not borrow or lend,' but 'Be a saver.' Not a hoarder, a saver\u2014because the more you save, the more you're able to give to others in need, and to needy causes. Savings was independence. So was giving. Pop was constantly helping relatives get back on their feet. He only gave gifts, never loans, because he didn't want anyone ever to feel indebted or beholden.\"\n\n\"We have to go back to Shakespeare's, and Pop's, philosophy,\" she then says. \"Make that Shakespeare quote a constitutional amendment, just in those words. It's impossible to put it better. We need more than a balanced-budget amendment. That would still leave us with trillions and trillions in debt. We need a 'no debt amendment.'\n\n\"Pop believed we should go back to the good old days, of 'pay as you go.' In 1993, when Italy became one of the founding members of the European Union, I thought it was fantastic. Pop didn't. 'Just wait,' he said. 'Sure, in good times, everybody will think easy credit is the best thing since the discovery of pasta. But when times get rough, it will be a disaster.' He was right, as usual. All these people who'd long lived by 'pay as you go' all of a sudden had credit cards by the bushel. Now they don't have a prayer of paying off the heaps of debt they've taken on.\"\n\n\"Pop rarely got riled,\" she tells me, \"but this taxpayer-funded bailout was driving him bonkers\u2014an unjustifiable giveaway to irresponsible people. All these bankers awarding themselves billions after accepting a government bailout, all those CEOs laying off thousands of employees, yet getting bigger 'performance bonuses' than when they were actually running successful businesses . . . He considered that high treason.\"\n\nThen she says, \"After you know my history, you'll accuse me of being the pot calling the kettle black. I got my M.B.A. from Harvard. Then I worked my way up in real estate investment firms. The more I made, the more I spent. I was a workaholic, just like Pop. I've had a marriage, divorce, several other relationships. Never had gotten around to having children. I'm Italian. I'm supposed to have children.\"\n\nShe rubs her abdomen. \"Now I'm pregnant. Four months. Pop was so happy, even if he didn't understand why I'm not marrying the father.\"\n\nShe then says, \"Pop died two months ago today. By the time he was diagnosed with stomach cancer, it was inoperable.\"\n\nBoth hands resting on her abdomen, she says, \"I'm worried about the world I'm bringing little Paola into. Most of all, I fret over being a responsible parent and good role model. I would give anything for her to know Pop, absorb his values better than I did.\n\n\"Pop never said so outright, but he wasn't so proud of my career. He was proud of my little brother Michele, who just scraped by\u2014or so I thought. He worked until recently for a nongovernmental organization as a conflict mediator.\"\n\n\"My firm went bankrupt last year,\" she then tells me. \"Our shareholders lost everything. I filed for personal bankruptcy not long afterwards. My investments all became nearly worthless. My home is being foreclosed on.\" She shakes her head. \"In my heyday, I thought lowly of Pop for never taking some of his money and splurging, seeing the world, living the high life with Mom. Now I'm too guilt-ridden to ask Mom for money.\"\n\nShe then laments, \"I never let Pop's wisdom on financial matters sink in, no more than our government has.\"\n\nAnd then: \"Why should my child, or anyone else's, be responsible for the unchecked greed of people like me and so many others of my generation? The next generation should have an automatic 'get-out-of-debt-free card.' Under no circumstances should they have to pay for our mistakes. That's why there has to be a 'neither a borrower nor lender be' constitutional article.\"\n\nIn due time, Mariangela says to me: \"Last week, I unburdened myself to my brother. Michele handed me on the spot a check for a huge sum. He said he'd been saving up in case he had a child, but never met the right woman. I told him he would one day, but he pressed the check into my hands and said if he did, he still had enough left in his savings to send four more to college. He refused to consider it anything but a gift. I was staggered when he told me how much he had saved over the years. Needless to say, he'd taken Pop's financial philosophy to heart.\"\n\n\"Michele is going to take early retirement,\" she then tells me. \"We're going to open a restaurant, Paolo's and Paola's, named after Pop and my daughter.\n\n\"Yesterday, I opened up a bank account for my little one. I'll put something in it every week. When she's old enough, I'll tell her about the beauty of 'neither a borrower nor lender be,' and of being a saver and a giver.\"\n\nWe exit the museum. From our perch at the mouth of the Hudson River, Mariangela looks out over the New York Harbor. \"I was reading a book about the philosophy of Native Americans,\" she tells me. \"A Sioux tribal chief from the eighteenth century is quoted as saying, 'We don't inherit the earth from our ancestors, we borrow it from our children.' Pop and Shakespeare would agree with that. But if we keep borrowing from our children at a breakneck pace, there'll be so much to pay back that they'll have no future.\"\n\nConstitution Caf\u00e9 Article: The Congress shall neither a borrower nor a lender be. For loan oft loses both itself and friend, and borrowing dulls the edge of husbandry.\n\nBrother, Can You Spare a Trillion?\n\nLike Mariangela's father, Jefferson was an outspoken advocate for \"neither a borrower nor a lender be\" when it came to federal spending. Although Jefferson never exhibited self-control over his personal spending habits, he was careful to a fault with the nation's money. By the time his tenure as president ended, Jefferson had left the United States close to debt-free. But he was alarmed by the reckless spending habits of his predecessors.\n\nIn an exchange with James Madison, Jefferson suggested the enactment of a constitutional article that would outlaw the long-term accrual of public debt. He believed that the Constitution should make clear that no debt obligation held by the federal government could be for more than nineteen years. Jefferson arrived at this specific time limit for federal indebtedness after consulting a colonial era version of actuarial tables and finding that \"the half of those of 21 years and upwards living at any one instant of time will be dead in 18 years, 8 months, or say 19 years at the nearest integral number.\" Therefore, it would be \"wise and just for [a] nation to declare in [its] constitution that neither the legislature nor the nation itself can validly contract more debt than they may pay within their own age, or within the term of 19 years,\" with all future contracts \"deemed void as to what shall remain unpaid at the end of 19 years from their date.\"\n\nJoseph Ellis, in American Sphinx: The Character of Jefferson, takes a pejorative view on this proposal of Jefferson's. He claims that \"Jefferson was engaged in magic more than political philosophy. For there is not, and never can be, a generation in Jefferson's pure sense of the term. Generational cohorts simply do not come into the world as discrete units. There is instead a seamless web of arrivals and departures.\" Yet Ellis misunderstands Jefferson's proposal: Jefferson knew perfectly well that there were not discrete units of generational cohorts. His point was that the federal debt load at any given moment should never amount to more than the federal government could demonstrably pay off within nineteen years, in order to prevent the \"seamless\" arrival of ever more debt.\n\nJefferson became so appalled by irresponsible federal spending that at one point he wanted to go to a much greater extreme to prevent financial mismanagement. As he told his friend John Taylor in 1813, \"I wish it were possible to obtain a single amendment to our Constitution. . . . I mean an additional article taking from the Federal Government the power of borrowing.\"\n\nJefferson acknowledged to Taylor that he did not deny that \"to pay all proper expenses within the year would, in case of war, be hard on us. But not so hard as ten wars instead of one. For wars could be reduced in that proportion.\" In Jefferson's view, a government that could no longer launch pell-mell into military incursions, due to constitutionally imposed financial constraints, would be more predisposed to mediating or avoiding conflict.\n\nJefferson held that one need look no further than the country from which the new nation had recently severed ties to see the wisdom of ratifying such a constitutional article. Britain had gotten itself mired in debt through rampant empire-expanding, which prompted it to levy taxes against the colonists. As Jefferson emphasized in an 1809 correspondence with Albert Gallatin, his former secretary of the treasury, Jefferson considered \"the fortunes of our republic\n\nas depending, in an eminent degree, on the extinguishment of the public debt . . . that done, we shall have revenue enough to improve our country in peace and defend it in war, without recurring either to new taxes or loans. But if the debt should once more be swelled to a formidable size, its entire discharge will be despaired of, and we shall be committed to the English career of debt, corruption and rottenness, closing with revolution. The discharge of debt, therefore, is vital to the destinies of our government.\n\nIf the U.S. government continued emulating Britain by proceeding along an increasingly out-of-control career of debt, Jefferson predicted that U.S. citizens would revolt again, this time against their own government. Mounting public debt would \"bring on us more ruin at home than all the enemies from abroad against whom this army and navy are to protect us.\" If those in charge of the federal purse strings had no choice but to steer clear of unmanageable debt, he opined to his nephew John Wayles Eppes, it would serve as a \"curb on the spirit of war and indebtment, which, since the modern theory of the perpetuation of debt, has drenched the earth with blood, and crushed its inhabitants under burdens ever accumulating.\"\n\nOne might of course contend that Jefferson himself was contradictory, even hypocritical, in taking such a stance. After all, he supported Article VI of the Constitution, which considers valid all debts entered into before the Constitution was ratified, and sets no ceiling on further borrowing. Indeed, both Jefferson and his political opposite Alexander Hamilton threw their support behind this amendment. Both agreed that if the new United States was to have credibility, it would have to assume the debts incurred by states in the period since the Articles of Confederation had been formed, as well as all the debt amassed by the colonies during the war effort.\n\nYet Hamilton supported the article out of the belief that ever-increasing amounts of capital borrowed by the government from a national bank with carte blanche authority to create and dispense money would serve as the driving engine for economic growth. By this scheme, the national bank's debt securities were guaranteed by private profiteers, who stood to amass fortunes on the interest paid back to them. Hamilton believed this practice would best realize his vision of building up American empire. His operating premise was that an always-accumulating national debt would ensure that the federal government had staying power, by giving wealthy creditors a personal stake in propping up the institution that was in debt to them. If the government became insolvent, these creditors would not recover their loans, much less continue to earn liberal interest on them, and so it was in their best interest to keep supporting the federal government. \"A national debt,\" asserted Hamilton, \"attaches many citizens to the government who, by their numbers, wealth, and influence, contribute more perhaps to its preservation than a body of soldiers.\"\n\nJefferson, on the other hand, supported the amendment with the expectation that all government debt would be paid off in short order. The United States then could focus on becoming a self-sufficient, agrarian-based, pay-as-you-go society. In contrast to Hamilton, Jefferson's operating premise was that only a frugal national government and citizenry could foster the virtues needed for a vibrant democracy.\n\nYet all signs indicated that the nation would continue taking on onerous debt. Jefferson predicted that by the time it realized the error of its ways, and tried \"calling the mismanagers to account,\" it would be too late, and the only recourse would be to \"obtain subsistence by hiring ourselves to rivet their chains on the necks of our fellow-sufferers.\"\nVII\n\nHail to the Chief\n\nOne might be forgiven for thinking our foremost champion of participatory democracy was also a bit of a control freak. In addition to designing the University of Virginia, his Monticello home, the new Virginia capitol at Richmond, and his Italian-style retreat Poplar Forest, Jefferson even designed his gravestone himself. It reads, \"Here was buried Thomas Jefferson, author of the Declaration of American Independence, of the statute of Virginia for religious freedom, and father of the University of Virginia.\"\n\nConspicuously absent is any mention of the fact that he was our nation's third president. Clearly, Jefferson himself did not consider his eight years in the highest office in the land to be his finest political hour. During his two terms as president, he was continuously at war with his own principles. As much as he ballyhooed states and individual rights, he was at times a staunch advocate of federalism; as much as he spoke out against the dangers of a standing military, he came to recognize its necessity; and while in balance he kept the country on stable footing, he did not make inroads in furthering the radical democratic ideals he'd long professed.\n\nJames David Barber has identified four presidential types: \"active-positives\" who exhibit healthy self-esteem, adaptability, and a goal-oriented approach to the office; \"active-negatives\" such as John Adams, who are compulsive and covet power as an end in itself; \"passive-positives\" like James Madison, who demonstrate a thin skin coupled with a desire to be liked by all; and \"passive-negatives,\" who shy away from political battles at the same time as they emphasize their civic virtue.\n\nAccording to Barber, of all our presidents, no one was better suited for the office than Jefferson, an active-positive type who \"knew the importance of communication and empathy.\" While Jefferson \"had his troubles and failures\" as president, he nonetheless \"combined a clear and open vision of what the country could be with a profound political sense, expressed in his famous phrase, 'Every difference of opinion is not a difference of principle.' \"\n\nYet Jefferson was relieved when he was no longer the nation's chief executive. \"I have never been so well pleased as when I could shift power from my own, on the shoulders of others,\" he told the liberal French nobleman A. L. C. Destutt de Tracy. In the same correspondence, he indicated that he'd never been comfortable with the powers bestowed to the president by the Framers: \"nor have I ever been able to conceive how any rational being could propose happiness to himself from the exercise of power over others.\" Maybe so, but he liberally took advantage of the powers bestowed to him during his time in office\u2014and of some not clearly enumerated in the Constitution.\n\nWhat should we demand of our chief executive? Should he mostly champion the interests of those who voted for him? As our only nationally elected official other than the vice president, should he take into account just as faithfully the views of those who opposed him? Or, should he devote his energies to implementing policies that he considers best for the nation, even if they're not the expressed will of a majority of citizens, or even of a significant minority? How did Jefferson navigate this tension, and how should we today?\n\nState of the Disunion\n\nFrom Article II, Section 3, of the Constitution: \"[The President] shall from time to time give to the Congress Information of the State of the Union, and recommend to their Consideration such Measures as he shall judge necessary and expedient. . . .\"\n\nThe 6th-graders have been preparing for my visit since the beginning of the school year, well-versed by their teacher in the ins and outs of the Constitution. I'm at a magnet elementary school with specialized courses in history and government. I'd held philosophical dialogues with Mrs. Ladson's charges in school years past but had not had occasion to visit her classroom for over two years. Even so, she's confident enough in my abilities to keep her students engaged that she leaves me alone while she goes off to copy lesson plans.\n\n\"My father is a congressman,\" Elizabeth informs me. \"He was going over the Constitution with me last night, after he got back from hearing the president's State of the Union address. He shook President Obama's hand afterwards, though my father didn't think his ideas are in line with those of the American people.\"\n\nThe precocious preteen scrolls her finger down her copy of the Constitution. \"Here,\" Elizabeth says. \"The Constitution says that the president 'shall from time to time give to the Congress information of the State of the Union, and recommend to their consideration such measures as he shall judge necessary and expedient.' But my father says most of the recommendations and measures our president talked about for improving the state of our Union\u2014for how to cure global warming, cool off the financial meltdown, stuff like that\u2014were 'way out in left field.' He says that the president should not be recommending to Congress measures that are 'necessary and expedient,' but only ones that are 'the will of the people.' \"\n\n\"Can't something be necessary and expedient and also the will of the people?\" I ask.\n\n\"It can be, but only if the president can prove, like by a poll, that the measures he recommends are the ones the people really and truly want. Like, most Americans may be all for health care for everyone, but against the measures he recommended. He didn't listen to the will of the people as expressed. My father said the polls showed that most wanted health care tax credits rather than all the new laws and regulations and taxes and layers of bureaucracy that the president and the Democrats forced through.\"\n\nShe takes a breath. \"So when it comes to the State of the Union, the Constitution should require the president to explain himself\u2014to not just say why his measures and recommendations are necessary and expedient, but how exactly they're the will of the people. I mean, sure, if he does stuff the people oppose, they'll vote him out if he dares run for a second term, and vote out all those in Congress who went along with him. But the Constitution should still demand that, when the president paints his picture of the State of the Union, he explain why his recommended actions to make the Union's state stronger and brighter than ever are also fulfilling the people's will.\"\n\n\"It shouldn't say he has to follow the will of the people,\" says Maria, after Elizabeth is done. \"It should say that when delivering the State of the Union, the president has to explain what he thinks the will of the people should be. After all, he's our main leader, not a follower. He shouldn't have to do what the polls say he should do. He can make his case for why he believes something should be the will of the people, no matter what the polls say. He can outline his visions and dreams for our nation, maybe ones we ourselves can't imagine because we're too shortsighted. As a wise leader, maybe he knows better when it comes to things that are in our best interests and that should be our 'will,' even if we don't know it ourselves right at that moment.\"\n\nThen she says, \"I bet Tyson here would agree with me. He was elected our class president, and that's pretty much how he operates.\" She points to an unassuming child in the row across from her who does not seem to want to draw attention to himself.\n\n\"Tyson ran against Elizabeth,\" she goes on. \"The class president gets to do things like help choose field trips, decide the order in which we stand in line at the classroom door when going out to recess or lunch or leaving for the day, decide how we'll be punished if we misbehave as a class, things like that. Tyson and Elizabeth each gave a speech telling us why we should vote for one or the other. We liked better Tyson's ideas for improving the state of the classroom. He promised he would consult with us whenever he could before making big decisions. But he also told us that when unexpected situations arose and he had to think fast\u2014like if the fire alarm went off unexpectedly, or it's the teacher's birthday and we forgot to buy her a gift\u2014he might have to make a decision without asking our opinions first, but that he would always try to make ones that create more of an 'all for one and one for all' classroom spirit.\"\n\n\"Well, anyway,\" she goes on, \"Carl's not here today, because he has a cold, but he came to school last Tuesday wearing a hat. Mrs. Ladson doesn't approve of hats on heads inside the classroom. It's against school rules, which come from way up high, from the principal or maybe even the superintendent. So Mrs. Ladson told him to take off his hat. But Carl wouldn't. She looked to Tyson to back her up, since he's the class president. Tyson asked Carl, 'Why don't you want to take off your hat?' Carl told him he didn't like his haircut, and he got kind of sniffly. So Tyson put his hat on, and he told us all to do the same.\"\n\n\"You could tell Mrs. Ladson was steaming,\" Elizabeth interjects.\n\n\"If Mrs. Ladson really had been steaming, she would have made us take our hats off. But she didn't,\" says Maria. \"She said that while she normally didn't approve of such things, since they're technically against the rules, she admired that Tyson wanted the class to show our . . . what're those words we learned?\"\n\n\"Sympathy and accommodation,\" several say.\n\n\"Right. She admired him for the fact that we all wanted to go along with his idea and show our sympathy for Carl by accommodating him in these . . .\" Again she is at a loss for the words she is searching for.\n\n\"Extenuating circumstances,\" several say.\n\n\"Right. Tyson even had convinced Carl's enemies\u2014I mean, people who aren't his friends\u2014to wear hats. Even they know what it's like to have a bad hair day.\"\n\n\"Tyson did impose his will on us, but he didn't threaten or intimidate us. Well, no, he didn't really impose his will. What he did was inspire or convince us to make our will and his one and the same,\" says Felicity.\n\n\"But there's a rule that forbids hats,\" says Elizabeth. \"Imagine if he'd been president of the United States and he'd broken the rules of the Constitution! If he really wanted to be 'presidential,' he should've inspired us to work together to change the school rule about hats, rather than do something illegal or unconstitutional that maybe would have put his classmates and teacher in a situation where they'd face harsh consequences, even if it did help Carl feel better.\"\n\n\"Carl was afraid people would've made fun of him,\" Tyson says, finally speaking up for himself. \"I had to make a snap judgment, before he started crying and carrying on. It seemed like the right thing to do. And almost everyone gladly put on their hats. They were willing, like me, to take the heat.\"\n\nSpeaking of taking the heat, as cold as it is outside on this wintry day, the state of the classroom is a stuffy one; the radiators are turned on full blast. I see that some of the students are as uncomfortable as I am, tugging at their shirt collars and mopping sweat from their brows.\n\n\"Can you vote on whether to open or close the window?\" I ask.\n\n\"We can,\" Maria replies, \"but what would that do? Some in here will complain that it's too cold rather than too hot.\"\n\n\"Maybe we can all figure out a way that we'll all be comfortable,\" Tyson says. He goes to the window and pulls it up about halfway.\n\nThen he asks his classmates, \"Are you more comfortable now?\" Some say yes and some no, with the rest noncommittal. He lets more time pass, so the cold air can waft further inside. Then, just as Maria predicted, the complaining begins.\n\nTyson seems at an impasse. Then he perks up and commands, \"Those who like it kind of cold, move to desks closest to the window. Those who like it real warm, move to desks farthest away from the window and closest to the radiator.\" Seeing that some appear confused, he says, \"What I mean is, move to a desk located wherever in the classroom you'd feel the most comfortable.\"\n\n\"But we're not allowed to move from our desks to another person's desk,\" Elizabeth protests. \"If you want to improve the state of the classroom, then you should take a poll and find out what we want you to do before you make a decision.\"\n\nTyson, in deep thought, doesn't seem to hear her. \"How about this: let's lift up our own desks and move them to the place where we're most comfortable with the room temperature.\"\n\nEven as Elizabeth is set to register another objection, the rest of her classmates enthusiastically follow this directive; apparently they can see nothing about it that is technically against the rules. There ensues a good deal of jostling and screeching of desks and chairs along the floor. All at last manage to arrange themselves where they are most comfortable. Those hardy few who bask in the cold sit by the half-open window. About two-thirds are as far from the window as possible, and there is a thread of desks more or less clustered right in the middle.\n\n\"Are you all comfortable?\" Tyson asks.\n\nEvery student except maybe one indicates that she is.\n\n\"The people have spoken,\" Felicity says with a slightly tongue-in-cheek air of triumph. \"The state of our classroom is strong, thanks to our president.\"\n\nMrs. Ladson materializes. Her mouth drops open. While her students may have spoken, she remains speechless.\n\nConstitution Caf\u00e9 Article: The President shall from time to time give to the Congress information of the State of the Union, and recommend to their consideration such measures as he shall judge necessary and expedient to achieve what he believes should be the will of the people.\n\nMan of the People\n\nJefferson claimed in 1801, the year he became president, that the \"will of the people is the only legitimate foundation of any government, and to protect its free expression should be our first object.\"\n\nHannah Arendt, the social philosopher and political theorist renowned for her studies on the problematic nature of power, observes that at our country's founding, the notion of \"the people\" never \"became a singular,\" but rather retained \"the meaning of manyness, of the endless variety of a multitude whose majesty rested in its very plurality.\" Indeed, \"to such an extent was the American concept of people identified with a multitude of voices and interests that Jefferson could establish it as a principle 'to make us one nation as to foreign concerns, and keep us distinct in domestic ones.'\"\n\nJefferson is frequently given the sobriquet \"man of the people.\" According to Jefferson scholar Merrill D. Peterson, Jefferson possessed the dual \"traits of the man-of-the-people and the man-of-vision,\" and this led to his apotheosis over time as \"Father of Democracy\": \"the man whose iconoclastic theories were received as truths by posterity, and whose lofty mental perch did not keep him from cherishing the people and mingling with them.\" But it can be argued that Jefferson's theories only seemed iconoclastic to those whose status was threatened by them. Max Lerner, in his book on Jefferson, supports the idea that Jefferson's notions were familiar and quite palatable to most: \"the ideas Jefferson drew upon were part of the climate of opinion of the time, and lay all about him.\" More often than not, Jefferson was simply giving eloquent expression to the views and hopes of ordinary people.\n\nTreaties Treatise\n\nArticle II, Section 2, Clause 2, of the Constitution: \"[The President] shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two-thirds of the Senators present concur.\"\n\n\"The treaty-making power granted by the Constitution to the president is way too open-ended,\" says Dinesh. \"All that the Constitution's 'advice and consent clause' says on the subject is that the president 'shall have power, by and with the advice and consent of the Senate, to make treaties, provided two-thirds of the Senators present concur.' \"\n\nI'm with a group of students who attend a leading law school. Our exchange was arranged by a professor who regularly holds Socrates Caf\u00e9 dialogues. All with me are currently taking a class on treaty law.\n\n\"But constitutionally requiring the president to seek the advice and consent of two-thirds of senators is a significant check and balance,\" says Jennifer. \"It forces a president to secure the support of at least some of his opponents. I don't see what's so open-ended about that. If anything, it's onerous.\"\n\n\"Theoretically, this stipulation requiring the Senate's advice and consent can serve as a decent check on the chief executive's treaty-making powers,\" says Dinesh. \"Which is why most presidents these days don't even attempt to put treaties they negotiate to a vote requiring approval by a two-thirds Senate majority. Instead, they circumvent this constitutional requisite. So a revised article must forbid a president to enter into any binding agreement with another nation or entity unless he follows to the letter the protocol outlined in the existing constitutional article.\"\n\nHe goes on, \"A treaty is any formal, binding agreement between one or more entities, usually countries. Even if it's not called a treaty, that's what it is. So, for instance, the 'Kyoto Protocol' falls into this category.\"\n\nThe Kyoto Protocol is an international accord that emerged out of the United Nations Framework Convention on Climate Change and that sets binding standards for significantly reducing greenhouse gas emissions. It went officially into effect in 2005. To date, 187 nations have signed and ratified the enforceable agreement, and the United States is making overtures to sign it.\n\n\"Kyoto is a binding agreement, a treaty,\" Dinesh says. \"It contains enforcement mechanisms that are carried out by a governing or judicial body outside the scope of the United States. Yet the president is acting as if he can finalize an agreement on it by merely issuing an executive order. He has no intention of even putting Kyoto to a majority vote in Congress, much less treating it like the treaty that it is and seeking the input and approval of two-thirds of our senators. If it was presented to the Senate, its members would have to reject it. Because the U.S. would first have to have laws of its own regarding the regulation of carbon emissions standards that mirror those of the Kyoto Protocol.\"\n\n\"I would go further than Dinesh,\" says Roya. \"I would have this article contain an additional clause that prohibits the president from entering into any formal and binding agreement that has rights or laws that supersede or conflict with those enumerated in our own Constitution. An example where such a transgression has occurred is with our nation's signing of the Universal Declaration of Human Rights.\"\n\nThe United States was among the forty-eight nations that approved this Declaration in 1948. It was developed under the auspices of the United Nations in order to clarify and elaborate on the concepts of \"human rights\" and \"fundamental freedoms\" set forth in the United Nations Charter.\n\n\"This Declaration enumerates thirty articles, which grant unprecedented rights and freedoms,\" she says, \"including rights to housing, universal employment and health care, leisure and vacation time, rights to security in the event of sickness, disability, widowhood, old age, among many others that are not included in our own Constitution.\n\n\"When it was ratified in 1948, Eleanor Roosevelt, who served on the committee that drafted it, stated that she hoped the Declaration would 'become the international Magna Carta of all men everywhere.' Her unequivocal aim was for it to be a binding, enforceable international constitution. And in fact, the Universal Declaration's preamble makes it clear that all the nations that ratify it are committed to securing its listed rights and freedoms for its own people and for people everywhere.\"\n\nThen Roya says, \"Let me be clear: I'm personally all for the Universal Declaration of Human Rights. It's a beautiful, humanistic document. But such a treatylike Declaration can only be entered into by the United States if we first have parallel rights via amendments to our own Constitution, and we have our own procedures in place for enforcing those rights that parallel the international judicial body the Universal Declaration calls for.\"\n\n\"But a document like the Universal Declaration of Human Rights doesn't in any way diminish or conflict with our own Constitution,\" says Lloyd. \"It makes the nations that bind themselves to it more connected to one another because they supposedly share the same high human rights standards. And it makes them more accountable, as individual nations and as a group, for realizing these foundational laws on human rights and freedoms.\"\n\nTo which Roya replies, \"But let's say one country's government that has entered into this agreement, supposedly committing itself to securing the Declaration's rights for its people, actually deliberately denies these rights to its people. And let's say its people protest. A hawkish U.S. president might then use the fact that there was a violation of this agreement as a pretext for declaring war against such a nation. He can claim he's doing so on behalf of freedom lovers in that country, and that he's only trying to uphold the Universal Declaration, even if he really has other motives.\"\n\nTurning to Roya, Dinesh says now, \"I disagree that this provision in the treaty clause needs the additional language you suggest, but for far different reasons than those Lloyd states. If the Senate is allowed to perform its job as 'treaty watchdog,' it has shown that it will make sure that any binding agreements signed by the U.S. with other entities don't exceed their bounds. Whenever given the opportunity, it has made sure that the U.S. doesn't commit itself to enforcing rights, or violations of rights, that its own citizens don't enjoy. That's precisely what the Senate did when a two-thirds majority approved the Geneva Conventions, and if the Senate's advice and consent was sought for the Universal Declaration, I'm sure it would have included a similar proviso.\"\n\n\"Though called conventions,\" Dinesh continues, \"this clearly was a treaty, and it was dutifully submitted by the president to the Senate for its advice and consent. The Senate approved the Conventions because they were in line with the Eighth Amendment of our Constitution, which prohibits 'cruel and unusual punishments.' But the Senate only approved the Conventions conditionally: it included a proviso that stated that if there was anything in it that contradicted our own Constitution, that that part of the treaty was null and void. This is a textbook example of the Senate carrying out its rightful role in forging a treaty agreement that advances humanity, without ceding any rights not in the Constitution. So as long as the Senate is allowed to fulfill its constitutional function, Roya's concern that rights will be granted in treatylike agreements does not need to be addressed in the article we're writing.\"\n\n\"Then we have to make darn sure that any treatylike document, no matter what it is called, always requires the approval of two-thirds of the Senate,\" says Roya. \"The North American Free Trade Agreement, or NAFTA, was a treaty through and through, with intricate enforcement mechanisms. Yet it was approved as an 'agreement' between our legislative and executive branches, and passed with just a simple majority of both chambers. From a constitutional standpoint, that completely invalidates it.\"\n\n\"NAFTA might have been well intentioned,\" Roya goes on. \"The U.S. might have been trying to advance its trade interest while at the same time elevating labor rights and environmental standards in Mexico. But first and foremost, it was a treaty. Yet only sixty-one senators voted to approve it, though the Constitution required the approval of sixty-seven for it to become law. So in this case, the Senate was derelict along with the president, because it didn't insist on carrying out its powers of approval by the constitutionally required two-thirds majority. Why wasn't the Senate protesting, as 'treaty watchdog,' that its 'rightful role' was being circumvented?\"\n\n\"If we go by the route you're suggesting, and require all binding agreements to be treated as treaties,\" says Jennifer, \"we'd rarely have any protocols or covenants, declarations or agreements, much less treaties. That would seriously set back progress in advancing human rights the world over. The U.S. presidents who enter into these binding agreements recognize that we live in a global world. I admire them for approving these agreements, even if they have to go around Congress to do so, because they're for the greater good of the nation and the world. If we adopted the constitutional article with the language being suggested here\u2014requiring all such agreements to receive the blessing of two-thirds of the Senate, and maybe further mandating that they contain no rights not specifically set forth in our Constitution\u2014we'd never have signed on to such groundbreaking human rights documents as the Universal Declaration.\"\n\nAfter a pause, Lloyd says, \"Well, I guess maybe I can come around to some extent to the view of Roya and Dinesh when it comes to requiring that presidents follow proper treaty protocol when entering into any binding agreement with another nation or entity, and submitting it to a two-thirds majority vote. But I believe it's getting carried away to say that all rights contained in any such agreements first have to be specifically included in our own. After all, the Ninth Amendment in the Bill of Rights says that 'the enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.' This was included because the rights granted in the Bill of Rights weren't meant to be exhaustive. I believe most of the rights Roya mentioned were in the Universal Declaration are ones that most Americans would want, and that many would argue are owed to them.\"\n\nThis brings Jennifer to say, \"What if this Constitutional clause says that when the president enters into a treaty, or into a 'treaty-like agreement,' if the rights granted exceed those enumerated in our Constitution\u2014or at least are not clearly enumerated in our Constitution\u2014then all Americans automatically enjoy those additional rights. Like Lloyd, I doubt that a majority of Americans would mind having the rights in the Universal Declaration, or the right to breathe cleaner air that adhering to the Kyoto Protocol would give them.\"\n\nDinesh shakes his head. \"Thomas Jefferson said in his first inaugural address that the reason the president's treaty-making powers are meant to be used sparingly is in order to keep our nation from entering into 'entangling alliances.' It's easy to see the good in documents like Kyoto and the Universal Declaration, but not the drawbacks. But if the Senate is required to carry out its designated role when it comes to making sure that the only entangling alliances we enter into are ones that advance our own interests, we'll be okay.\"\n\n\"I'm not convinced we'll always 'be okay,'\" says Jennifer. \"Former president George W. Bush clearly violated the Geneva Conventions' rules on torture when he approved of waterboarding of war detainees. Every attempt to try all those who were allegedly complicit in carrying out this regimen of torture for crimes against humanity has been rebuffed. Our three branches of government have prevented the international tribunal in the Hague from seeking justice. In this case, all our constitutional and treaty watchdogs have failed us.\"\n\nConstitution Caf\u00e9 Article: The President shall have Power, by and with the advice and consent of the Senate, to make treaties, provided two-thirds of the Senators concur. The President shall not enter into any binding agreement or compact, alliance or confederation\u2014be it a treaty, protocol, convention, or covenant\u2014with another nation, foreign power, or international entity, without the advice and consent of two-thirds of the Senate.\n\nTreaty Powers\n\nThe first treaty that the United States entered into after the Constitution replaced the Articles of Confederation was the Jay Treaty. Put into effect in 1794 during the administration of George Washington, the treaty was arranged by Washington's secretary of the treasury, Alexander Hamilton, and his principal negotiator, John Jay. It was intended to stem yet another potentially explosive conflict between the United States and Britain. As a result of the treaty, Britain agreed to withdraw its troops from the Northwest Territory, among other things, and there followed a decade of mostly peaceful trading between the two nations.\n\nBut Thomas Jefferson, who was Washington's secretary of state at the time, strongly opposed the treaty. He believed that the strengthened economic ties it forged between Britain and the United States came at the expense of alienating France, the most critical U.S. ally during the Revolutionary War. Jefferson argued that even though the Washington administration had obtained the consent of a two-thirds majority in the Senate, it had nonetheless exceeded its authority, because the treaty included settlement and trade provisions that, to him, blatantly exceeded the constitutionally designated parameters for treaties. \"I say the same as to the opinion of those who consider the grant of the treaty making power as boundless,\" asserted Jefferson. \"If it is, then we have no Constitution.\"\n\nIn most instances, Jefferson found voluntary agreements far preferable to fixed treaties. In a letter he wrote during his presidency to his friend Philip Mazzei, an Italian physician and promoter of liberty, he stated: \"On the subject of treaties, our system is to have none with any nation, as far as can be avoided. . . . We believe that with nations as with individuals, dealings may be carried on as advantageously, perhaps more so, while their continuance depends on a voluntary good treatment.\" On the other hand, Jefferson maintained, if a treaty has fixed terms over a specific number of years, and a term in it \"becomes injurious to either [party],\" the treaty \"is made by forced constructions to mean what suits them and becomes a cause of war instead of a bond of peace.\" Consequently, to Jefferson, \"it is against our system to embarrass ourselves with treaties\" except in the most extenuating circumstances.\n\n\"I see . . . not much harm in annihilating the whole treaty-making power except as to making peace,\" Jefferson told James Madison in 1796. Given his views, Jefferson would likely have opposed agreements like Kyoto and the Universal Declaration of Human Rights\u2014not because he would have been against them in principle, but because they go beyond the bounds of what is constitutionally permissible. And yet Jefferson put his strict constructionist principles on the back burner and acted outside the Constitution when he used his treaty-making authority to purchase the Louisiana Territory for very little money despite his inability to get a two-thirds majority support in Congress. Jefferson was, in the end, a canny politician, and presented a strong justification for acting outside constitutional parameters.\n\nAmerican Idle\n\nTwelfth Amendment to the United States Constitution (which replaces Article II, Section 1, Clause 3): \"The Electors shall meet in their respective states, and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and they shall make distinct lists of all persons voted for as President, and all persons voted for as Vice-President and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate;\u2014The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted;\u2014The person having the greatest Number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice. And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following, then the Vice-President shall act as President, as in the case of the death or other constitutional disability of the President.\u2014The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such number be a majority of the whole number of Electors appointed, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice. But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.\"\n\nI now set out to connect with American slackers, new-millennium version, whose country, community, families neither ask nor expect anything of them, and who neither ask nor expect anything of themselves. In the ostentatious coastal community of Orange County, California\u2014or the OC, as it has become known after the former hit TV show of that name\u2014I contacted a teacher at a high school where I once held a dialogue with a group of graduating seniors. He was still in touch with some of them and sent out an email informing them of my project. Several replied that they'd be happy to talk with me, with the proviso that we meet at their favorite haunt, Laguna Beach.\n\nWhen I arrive at the designated meeting place, the erstwhile students are leaning against their surfboards, smoking and sending text messages, looking disconsolately at the waveless ocean. All of them still live at their parents' home, or at apartments subsidized by their parents, and none have jobs. Surfing is all.\n\nTwo are holding pocket-sized copies of the Constitution. \"We actually read this thing, I'll have you know,\" says Duane.\n\nThen he asks me, \"Are we mistaken, or does the Constitution stay silent on how presidents are elected?\"\n\n\"It stays silent,\" I say.\n\n\"The only requirement is that presidents are elected by 'electors,' a certain number of which are allotted to each state? That's it?\"\n\n\"That's it. There's not even mention of an electoral college, though most people today take it for granted that the Constitution specifies one. The Constitution only mentions that we vote for 'electors' who in turn pledge to vote for the presidential candidate who wins the majority of votes in each state.\"\n\nDuane and company consider this. \"So all this political party stuff, all this primary stuff, is not required by the Constitution?\"\n\n\"That's right.\"\n\n\"So we could change that. There's nothing in the current Constitution that could keep us from doing so, right?\"\n\nI nod and wait for what's to come.\n\n\"Okay, here's our idea,\" says Duane. \"Let's keep the elector stuff in the Constitution, just like it's already worded. We still vote for electors, and they still vote for the president. But let's change how presidential candidates are chosen, change how front-runners are made, and change how one among them is elected. What's cool about this is that we're not changing anything that's already in the Constitution. We're just adding to it in a way that'll do away with the need for political parties and primaries.\"\n\nMickey jumps in. \"Okay, well, what we propose is that there be a presidential pageant or contest or however you want to term it. It would be run in a way that's sort of a cross between American Idol and a beauty pageant, only it'd be taken more seriously.\n\n\"As we see it, this constitutional article will give each state, plus the District of Columbia, the right to send two contenders to the presidential pageant\u2014one of them chosen by some sort of election process of each state's choosing, and the other one chosen by lottery from all those who put their names in a lottery basket or whatever who aspire to be a candidate. That'll really mix things up, keep things interesting.\"\n\n\"But some states have a much greater population than others,\" I say. \"They shouldn't have the chance to send more contenders?\"\n\n\"Well, there's only one contestant per state when there's a Miss America or Miss Universe pageant, so we're being twice as fair as that,\" he reasons. \"So anyway, moving on: At the beginning of the national presidential pageant season\u2014which would start when fall TV season starts, in order to make sure that the presidential election can still be held on the first Tuesday in November\u2014all the candidates from the fifty states, and the federal district, will face off against each other.\"\n\nKeith picks up where Mickey leaves off. \"The candidates would get to make their case straight to the American people. The show would air on all channels, for one or two hours each week, so that any TV-watching American\u2014which basically means all Americans\u2014would be tuned in. These two candidates from each state would have plenty of airtime to get their message out and demonstrate how bright or dim-witted they are. Each week, we the TV viewing audience would vote out a certain number of candidates until only two finalists are left standing, just like we eliminate performers on American Idol.\"\n\nTaylor jumps in. \"That's better than how nationally televised presidential debates are held now. The major networks have ruled that the only candidates who can take part are those who already have 15 percent of the voters' support, based on the latest polls, before the televised debate is set to begin. But how can anyone else get that kind of support, except the two main party players, and maybe from time to time a wealthy independent with pockets deeper even than my father's? So this process we propose is much fairer. It gives a lot more people the opportunity to be a possible finalist candidate, because they'll have a chance to get a lot of exposure and really get their message out.\"\n\nDuane then says, \"Anyway, a group of panelists would ask these contenders questions, some of them pitched by experts, some by regular Americans like us. They'd be asked questions like those posed in Miss America pageants\u2014things like, 'What is your stand on world peace, starvation in Africa, steroids in sports, instant replay in baseball, a National Surfers Holiday, do you mind answering the phone at 3 a.m., as long as it's because there's a national emergency?'\"\n\n\"And we TV viewers would judge them each week,\" says Taylor, \"picking winners and losers, using an American Idol\u2013style process of elimination, voting by phone ballot. The number of contenders would be narrowed down week by week, until just the top two are left standing. They'll compete in one final head-to-head match. To make sure that the vote is legit, each registered voter will have to enter a specific code when he calls in. Maybe there'd also be some sort of voice authentication device to ensure it's really him, and that he only votes once.\"\n\n\"So then,\" Mickey says, \"the presidential contestants would be put through a series of simulations where they'd have to show how they'd act under certain scenarios\u2014and they wouldn't be tipped off beforehand about what these scenarios will be, so they have to show whether they've got the right stuff and can react 'presidentially' on the spur of a moment. Like, each would have to show what he'd do if he was at a school reading a children's story to elementary students to promote literacy, when suddenly he was interrupted by the head of the Pentagon and told there's just been a terrorist strike. If he freezes up, obviously he's not the guy to lead us.\"\n\n\"The last part of this plan,\" he goes on, \"is that the voting tallies for the finalists on American Idol: The Presidential Version, would be taken state to state. So whoever wins the majority of votes in each state wins all the electors, just as they do now.\"\n\n\"With this approach to electing the president,\" says Keith, \"we'd really care about voting, and care enough to really learn about the issues. Our vote would really count, because we'd have a real range of choices. I bet the percentage of registered voters who actually end up voting would skyrocket.\n\n\"We don't vote these days because we can't relate to the candidates from the big parties. Look what they do to the country once they're elected. They allow the companies that finance them to pollute the sea and air. This global-warming thing is so serious, soon I'll be able to surf in Antarctica.\"\n\n\"Are all of you registered to vote?\" I ask the group.\n\n\"No way,\" says Taylor. \"I don't want them to have my name, in case there's a draft.\"\n\n\"You idiot,\" says Duane, \"you had to register at eighteen, just in case there ever is one.\"\n\n\"Anyway,\" Taylor says, ignoring Duane, \"with this proposal of ours to pick a president sort of the American Idol way, there'll probably be at least some candidates we can relate to. This constitutional article would really rock the vote.\"\n\nConstitution Caf\u00e9 Article: Every four years, in a process that will end on Labor Day, each of the 50 states, and the federal district, one by a general election and the other by random drawing, will present two candidates to contend for the Office of President of the United States. These 100 candidates will then compete against one another weekly, for at least two hours per week, on a program to be aired in the evenings on all television channels and all other forms of mass media. They will be interviewed by a group of expert panelists, who will ask them questions pertinent to the office they aspire to, as well as by the viewing audience. The candidates will also perform simulated segments in which they demonstrate how they would act as president when confronted with various and sundry situations requiring a presidential response. Each week, ten candidates will be eliminated by direct voting from the viewers, until there are two remaining finalists. On the final episode, to be held on the first Tuesday of November, the finalists will be put through one more grueling grilling followed by a continuous series of simulated activities designed to demonstrate their presidential capacities. The viewers will then vote one final time, with the votes tallied by each state. Registered voters shall vote by telephone. They will each have a special voter's pin number, and there will also be a voice recognition mechanism to prevent voter fraud. The candidate who garners the most votes in each respective state will be the candidate for which that state's Electors are pledged to vote for President. The runner-up candidate shall be Vice President. The Electors shall then meet in their respective states and vote by ballot for President. Each Elector shall have one vote. The person having the greatest number of votes for President shall be the President. There shall be a total of 538 electors. This number is equal to the total membership of both Houses of Congress (435 Representatives and 100 Senators) plus the three electors allocated to Washington, D.C., totaling 538 electors. Each state is allocated as many Electors as it has Representatives and Senators in the United States Congress. Each state shall decide how it chooses its Electors.\n\nParty Animals\n\nThe Constitution makes no mention of political parties because the Framers hadn't anticipated their advent, much less the inordinate influence they would come to have over the political process. Likewise, nowhere in the Constitution are primaries mentioned as the way to elect presidential candidates, though it is now taken as a given that voters choose their principal contenders for the highest office in the land through this process. But the system to elect presidential contenders could have developed in any number of ways, considering the vague constitutional parameters on the topic.\n\nThe two earliest political parties were the Federalists, who favored a powerful centralized government, and the Democratic-Republican Party, which originated in the 1790s in opposition to the Federalist platform and favored limited federal government so that states' and people's rights would predominate. Thomas Jefferson was the first presidential candidate to run under the banner of the Democratic-Republicans, the \"party of the common man.\"\n\nJefferson believed parties were inevitable: \"In every free and deliberating society, there must, from the nature of man, be opposite parties, and . . . one of these, for the most part, must prevail over the other for a longer or shorter time,\" he told John Taylor.\n\nAccording to Jefferson, \"Men by their constitutions are naturally divided into two parties. Those who fear and distrust the people, and wish to draw all powers from them into the hands of the higher classes\"\u2014namely, the Federalists\u2014and those who \"identify themselves with the people, have confidence in them, cherish and consider them as the most honest and safe, although not the most wise depositary of the public interests,\" namely, the Democratic-Republicans.\n\nWhile we may be \"party animals,\" Jefferson claimed that it also was our nature to transcend to some degree our inherent disposition to belong to a party. He for one resisted being pigeonholed into any party mold, and claimed he \"never submitted the whole system of my opinions to the creed of any party of men whatever, in religion, in philosophy, in politics, or in anything else, where I was capable of thinking for myself. Such an addiction is the last degradation of a free and moral agent. If I could not go to heaven but with a party, I would not go there at all.\"\n\nDespite such rhetoric, Jefferson's Democratic-Republicans demonstrated that they were nearly as adept at mudslinging and fear-inducing sensationalism as his Federalist opponents, and the party frequently strayed from debates on the very real issues that divided the two camps.\n\nJefferson claimed that a political party best served a useful purpose when it was formed in order to take a concerted stand to advance democratic principles. As a case in point, Jefferson contended that the Democratic-Republican Party took shape in response to the Federalists' encroachments on constitutionally guaranteed rights. Whenever there is such an occasion, he maintained, with the \"principle of difference as substantial and as strongly pronounced as between the republicans and the monocrats of our country, I hold it as honorable to take a firm and decided part and as immoral to pursue a middle line, as between the parties of honest men and rogues, into which every country is divided.\"\n\nWhen it comes to the two major parties of our own era, the \"principle of difference\" is no longer nearly as pronounced as it was during Jefferson's time: both Democrats and Republicans are beholden to corporate contributions, and both often go to equally unprincipled lengths to gain an advantage. Additionally, both betray what Jefferson would call \"monocratic\" tendencies, in that they are increasingly heedless of the will of most of their constituents. This is likely due in part to the fact that leaders of both the right and the left have become more extreme, leaving mainstream Americans without a candidate they can identify with.\n\nIf Jefferson were alive now, he would surely believe that the time has come for a new party to be formed\u2014one made up of ordinary Americans of many political backgrounds who nonetheless agree that the existing parties are destructive to democracy's ends.\n\nPardon Me\n\nFrom Article II, Section 2, Clause 1, of the Constitution: \"The President . . . shall have Power to grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment.\"\n\n\"The Constitution anoints the president with 'power to grant reprieves and pardons for offenses against the United States, except in cases of impeachment,'\" says Gary. The congressional staffer helped arrange for a number of people to gather with me at a majestic bar near the U.S. Capitol to explore whether the president's pardoning powers should be retooled.\n\n\"As things stand,\" he continues, \"a president can pardon anyone for any reason whatsoever, or for no reason at all. That kind of latitude may be appropriate for a monarch or dictator, but not for a president in a democracy.\"\n\n\"For our Framers' purposes, this authority was meant to give the president discretion to pardon someone who'd committed a crime against the U.S., but only if it would be of great benefit to the nation,\" says Anne-Marie, an administrative assistant with two master's degrees who works for an Obama political appointee. \"For instance, he might pardon someone who committed treason if, in exchange, he was given information that revealed other enemies of the state and that led to their arrests. In such a case, the pardon was a necessary evil that prevented even greater damage being done to U.S. interests.\n\n\"The Framers also believed the president needed the authority to pardon leaders of rebellions or insurrections as a means for restoring the peace. Like, after President George Washington had troops put down the 'Whisky Rebellion' in the 1790s, and its two principal leaders were sentenced to death by hanging, he pardoned them as a gesture towards restoring tranquility.\n\n\"But like Gary said, modern presidents go far beyond this very limited scope for issuing pardons. So I agree that a refashioned article should make explicit what the Framers had thought was implicit. Presidents were never meant to have unlimited pardoning power.\"\n\n\"This article should state that pardons can only be made when they're for a demonstrated public good,\" says Jason, who works for a political action committee. \"At least then, there can be a healthy debate over whether any particular pardon has served that purpose. After the Civil War, President Andrew Johnson pardoned thousands of former Confederate officials and military personnel. This was controversial at the time, but it proved to be a healing act for the nation. On the other hand, it was a crime itself that Bill Clinton issued an eleventh-hour pardon of the billionaire fugitive-from-justice Marc Rich. Rich's ex-wife had made a timely $500,000 donation to his presidential library, and voil\u00e0, he's pardoned.\"\n\n\"Despite good intentions,\" says Gary, \"Jimmy Carter's amnesty to draft dodgers was unjust because he didn't also pardon those who fought in the war but eventually deserted, and he didn't pardon conscientious objectors who were convicted and served time in prison. Carter's pardon would only have been potentially healing, and so for the public good, if it had included all of those groups.\"\n\nThis brings Anne-Marie to say, \"Even if this article stated that presidents could pardon people only if it furthered the public good, they still could pardon virtually anyone they wanted to and just claim that it was in the public's best interests, even if 99.9 percent of the public disagreed. I propose an article that gives the Senate authority to be 'pardoning gatekeeper,' so to speak. It can say that if three-fourths of the Senate opposes a pardon, it will be rescinded. Whenever the Senate is successful in getting this super majority, it clearly would mean that it\u2014and its constituents\u2014believes the president's pardon served no public good.\"\n\nThen Walter, a staffer for a notoriously liberal congressman, says, \"I've been disappointed that so far President Obama hasn't issued a single pardon, though he's received more requests for pardons than any other previous president by this stage of his term. I had expected him to use his pardoning powers to right blatant wrongs in our criminal justice system. For instance, there are many youth offenders incarcerated for life today who've been convicted for a third time of a criminal but nonviolent offense. A presidential pardon could right this wrong with a pardon. It would highlight how the law itself can be unjust in such a case, in which the punishment is way out of proportion to the crime, and hopefully lead to the law being changed.\"\n\n\"I'm going against the grain here, but I wish that the power to pardon were used even more randomly and arbitrarily than it is now,\" Walter says next. \"Perhaps this article could state that each year a certain number of people, incarcerated for virtually any offense, will be picked randomly and will receive a presidential pardon. It would show that we're a merciful and forgiving nation.\n\n\"There's an ancient Hebrew tradition called a Jubilee that used to take place every twenty-five to fifty years. Debts were forgiven, prisoners released, slaves freed, seized property restored to the original owners. It allowed individuals to start again, and really, gave the entire society a clean slate. A contemporary Jubilee\u2014held, say, every twenty or twenty-five years\u2014might go a long way towards fulfilling Thomas Jefferson's stated dream of giving each generation in American society a fresh start.\"\n\n\"That kind of random forgiveness is too bleeding heart for me,\" says Anne-Marie. \"If possible wrongdoers knew they were going to be forgiven every twenty years or so, they might be more inclined, instead of less, to break the law. While I want our criminal justice system to be equitable and fair when it comes to punishment, I certainly wouldn't want anyone pardoned outright who's been convicted of a crime. Where's the public good in that?\"\n\nConstitution Caf\u00e9 Article: The president shall have power to grant reprieves and pardons for offences against the United States, except in cases of impeachment, in those exceptional instances in which it can be compellingly demonstrated that such reprieves and pardons further the public good. A vote of three-fourths of the Senate shall be sufficient to rescind a presidential pardon.\n\nConstitutional Absolution\n\nIn 1798, the Adams administration, in pushing through the Alien and Sedition Acts, claimed that this legislation would ensure stability and safety by making it a crime for anyone to produce \"false, scandalous, and malicious writing\" against the government or its officials. After the acts were passed, a string of newspaper printers were rounded up, fined, and imprisoned for publishing essays and articles critical of the president. Shortly after Jefferson replaced Adams in the White House, he pardoned the journalists; they were freed from jail, and Congress returned the fines they'd paid.\n\nAs was pointed out in the Constitution Caf\u00e9 exchange, the Framers believed that endowing the president with pardoning authority would help stem potential insurrections by giving the president the power to restore the peace on occasion by forgiving outside agitators. Jefferson, though, aimed to end an insurrection from within\u2014one he believed was orchestrated by the very people who'd sworn to uphold the Constitution, and yet unpardonably, seemed determined to undo it.\n\nVested\n\nFrom Article II, Section 1, Clause 1, of the Constitution: \"The Executive Power shall be vested in a President of the United States of America.\"\n\n\"Any new article that enumerates the president's designated powers has to weigh in on whether he does or doesn't\u2014or should or shouldn't\u2014have authority to issue executive orders and signing statements and to assert executive privilege,\" says Karen.\n\nThe grassroots environmental activist is among those who responded to one of my social networking summonses to attend this Constitution Caf\u00e9 gathering to devise an article dealing with executive power. We're meeting in Seattle at a boutique coffee shop that is managing to thrive among the city's famous chains.\n\nKaren goes on: \"The 'general vesting clause' states, 'The executive power shall be vested in a President of the United States of America.' But what is meant by 'executive power'? Does it constitute those powers specifically designated to the president in the Constitution? Or does it mean the president can exercise a general sort of power that might go way beyond them?\"\n\n\"To me, this vesting clause is meant to give the president a good deal of leeway to deal with the unexpected,\" says Judy, a hospital administrator. \"The Framers knew they couldn't anticipate every future contingency when it came to detailing how the president should best carry out his duties. If we refashion this clause now, all we need do is make clear that when the circumstances warrant, the president has some 'wiggle room,' as long as he uses such power responsibly and furthers the interests of our democracy.\"\n\n\"A president might make blustery claims to the effect that he's being responsible and has democracy's best interests at heart, but that doesn't make it so,\" says Trish, a doctoral student in political science. \"A president shouldn't be allowed to exercise any authority whatsoever that exceeds what's specifically vested with him by the Constitution.\"\n\nNodding, Timothy, who works for the world's largest software developer, says, \"Take executive privilege. It's an abuse of executive power, designed to keep the White House, the 'People's House,' a house of secrets instead of transparency. A president invokes it whenever he doesn't want himself or his staff to obey a congressional subpoena\u2014even or especially if there's good reason to believe that the actions of the executive branch that are under investigation are willful violations of the Constitution.\n\n\"The executive branch shouldn't be privileged in this way. The only way we can make sure that this branch engages in responsible and ethical conduct is if we deny executive privilege to the president and those working directly for him. What this article should do is state unequivocally that executive power is precisely the powers granted by the Constitution to the president, nothing more or less.\"\n\n\"Executive power,\" he goes on, \"constitutes all the powers the Framers enumerated to the president: the power to sign legislation into law and veto legislation, to make treaties, appoint ambassadors and other ministers and councils, appoint a cabinet and other high-level bureaucrats, nominate Supreme Court justices, grant reprieves and pardons, serve as our commander-in-chief, among others. And most of all, as the Constitution says, he has the special power to 'take care that the laws be faithfully executed.' \"\n\n\"I would have a new constitutional article explicitly support a broader interpretation,\" says Judy. \"And I believe it's justified. There have been enough times when use of executive power, above and beyond the authority specifically given to the president, has advanced the public good. For instance, when Obama proposed legislation for creating a bipartisan panel for exploring ways to tackle the national debt, Congress rejected it. So the president used an executive order to create the panel. It's for the good of the nation to have such a panel. It hopefully will rise above politics and come up with the needed solutions to bring our debt under control.\"\n\n\"But good uses of unenumerated executive powers are the exception,\" says Karen. \"Signing statements, for example. A president signs a piece of legislation into law, but then attaches a written statement saying that even though he endorses the law, he's modifying elements of it that he finds disagreeable. By doing so, in effect he's turning himself into a one-person supraconstitutional legislative chamber who has endowed himself with the power to alter legislation as he sees fit.\n\n\"The fact is, the Constitution doesn't even mention the word 'veto'\u2014rather, it says that the president must either sign every bill presented to him by Congress, or return it. No wiggle room. As George Washington put it when he was president, 'I must approve all the parts of a bill, or reject it in toto.' But then Thomas Jefferson made things murky during his term. He signed an appropriations bill, approving it 'in toto.' But then he refused to spend $50,000 allotted to him by Congress to purchase gunboats because, as he explained it, he thought it was an unnecessary extravagance. So that was sort of the official precursor to signing statements.\"\n\n\"Before Ronald Reagan took over the Oval Office,\" she goes on, \"in the entire history of the presidency, only about seventy-five signing statements had been issued, and they were rather harmless, more like proclamations than attempts to modify existing law. But then Reagan, the first Bush, and Clinton, used them almost 250 times total, with the specific intent of modifying or altering a law passed by Congress. Then George W. Bush issued well over 150 of them, upending more than 1,000 provisions of federal law. The American Bar Association called his actions 'contrary to the rule of law and our constitutional system.' \"\n\n\"Now Obama is carrying on the signing-statement tradition,\" she goes on. \"While he pledged to 'consult' with his attorney general before issuing any signing statements, he's made it clear that he intends to issue them whenever he believes they're warranted.\" She passes out copies of a document. \"This is the issue paper Obama distributed about signing statements: 'In exercising my responsibility to determine whether a provision of an enrolled bill is unconstitutional, I will act with caution and restraint, based only on interpretations of the Constitution that are well-founded.' Well, if he's going to act with caution and restraint, he should recognize that signing statements are prohibited by the Constitution.\n\n\"But he refuses to recognize this. For instance, he approved an omnibus spending bill passed by Congress, only to issue a signing statement declaring he'd ignore those provisions that he found disagreeable, such as the sections restricting U.S. involvement in U.N. peacekeeping, and congressional claims of authority over certain areas of spending.\"\n\n\"The former U.S. senator Arlen Specter twice introduced a bill instructing all branches of government to ignore signing statements,\" says Trish. \"But both times it was allowed to die in committee without debate. Congress has no interest in taking on the president's use and potential abuse of executive power. The only way all of our nation's chief executives will exercise this power properly is if the Constitution makes explicit, in a way that can't be misinterpreted or open to multiple interpretations, what executive power amounts to.\"\n\n\"This article shouldn't hem in a president overmuch,\" says Judy. \"There are just too many unexpected circumstances that may crop up that'll require him to use authority that's not specifically designated. That's not the same thing as saying he should be able to operate unchecked. I believe an article should state that the executive power is not a privilege but a responsibility, and that any use of powers not specifically enumerated must be demonstrably in the public interest and for the good of democracy. President Obama, for instance, has made clear that he plans to make broad use of executive power to advance his energy, environmental, fiscal, and other domestic policy priorities. I support this, and believe it's for the good of democracy.\"\n\n\"Well, I have to admit that in the case of Obama, I don't see anything detrimental to democracy about that,\" says Karen. \"He tried to have enacted legislation that would advance these priorities, but they've stalled in the quagmire of Congress. He's our only nationally elected federal official, and he believes he has to use whatever powers are at his disposal to advance the priorities the people elected him to realize. On the other hand, whenever his predecessor did this, I was up in arms about it, because his priorities weren't mine.\"\n\n\"Maybe they weren't your priorities,\" says Judy, \"but they reflected those of the majority at the time. And it just goes to show why a president needs some ever-so-slight, constitutionally sanctioned latitude if he's going to realize the people's priorities.\"\n\nConstitution Caf\u00e9 Article: The Executive Power, which shall be vested in a President of the United States of America, constitutes those powers enumerated in this Constitution, and those that enable a President to take care that the laws are faithfully executed.\n\nStrong and Weak Chief Executive\n\nJefferson's views on whether the chief executive of government should be strong or weak changed over time. When he wrote his first draft for a new Virginia constitution in 1776, he advocated a weak executive and granted nearly all power to the legislature. This draft called for Virginia's assemblymen, rather than the people themselves, to elect the chief executive, who he merely called an \"Administrator.\" Among the many limitations that Jefferson placed on the administrator's power, one that he decreed in his draft was that the administrator \"shall have no negative [veto power] on the bills of the Legislature.\" Further, Jefferson's administrator couldn't dissolve or adjourn the legislature, muster the military, declare war, nor conclude a peace agreement. \"Such powers,\" Jefferson wrote, \"shall be exercised by the legislature alone.\" For all intents and purposes, the legislature was the chief executive.\n\nBut after serving one term as Virginia's governor in 1779, Jefferson dramatically changed his views. At that time, Virginia's existing constitution resembled his own draft version: the chief executive was so hamstrung that he was at the mercy of the General Assembly. Jefferson's political enemies criticized him relentlessly for his decisions as a 'war-time governor' during the Revolutionary War, when the state capitol was overrun by British troops and Jefferson nearly was captured. Yet a subsequent inquiry exonerated Jefferson, whose limited authority would have made a different outcome improbable even if he had been a competent military strategist.\n\nAfter this experience, Jefferson about-faced and argued that a strong chief executive endowed with powers equal to the other two branches of government was vital if the will of the people was to be carried out. Jefferson later explained his initial reluctance to bestow significant power on the chief executive to the historian Samuel Kercheval, claiming it sprang from the fact that \"the abuses of monarchy had so filled all the space of political contemplation.\" Determined to ensure that the leader of the commonwealth could never become a despot like King George III, Jefferson went to the opposite extreme and created a chief executive so weak that not only could he never become kingly; he could never take on any sort of leadership role whatsoever. In subsequent drafts of his model constitution, Jefferson made Virginia's governor into an executive with an array of powers.\n\nWhile Jefferson also came to conclude that the nation's president should be a strong executive, he also believed certain limits had to be placed on him. For instance, he argued that one of the greatest flaws in the Constitution was its failure to place term limits on the president. To him, this meant that the president could potentially gain unlimited rule and essentially become a monarch or autocrat. As he wrote in 1807, \"if some termination to the services of the chief Magistrate be not fixed by the Constitution, or supplied by practice, his office, nominally four years, will in fact become for life.\" Jefferson voluntarily imposed on himself a two-term limit as president, and the subsequent two presidents were inspired to do the same.\n\nJeremy Bailey, in Thomas Jefferson and Executive Power, explores Jefferson's use of executive power during his two terms in office and concludes that Jefferson's overarching views on executive power centered on the principle that \"the president unifies the will of the nation and thereby embodies it.\" Because the president is the only nationally elected federal official besides the vice president, Jefferson believed that he would most effectively serve as the conduit for the people's will. But it is inevitable that, as supreme \"caretaker of the public good,\" there will likely arise occasions in which he \"must sometimes act outside the law, or even against it, on behalf of the public good.\"\n\nBut if a president does act extraconstitutionally, Jefferson told newspaper editor John Colvin, he must then \"throw himself on the justice of his country and the rectitude of his motives,\" justifying his actions and assuring his constituents that he had acted in their best interests. To enable the people to assess whether he has done right by them, Jefferson believed that the president must attempt, as Bailey puts it, to \"gather the national will around a set of declared principles\" and establish his reputation as a sound \"caretaker of the will of the nation.\" According to Bailey, Jefferson believed that this approach would best \"allow for political change but also preserve constitutional limitations on power by enabling the people to judge executive discretion.\" By this process, a president would plainly admit when he had gone beyond his constitutionally delegated powers and try to explain why he'd had no choice but to do so to those who'd voted him into office. Jefferson preferred this honest and humble approach to presidents making what to him seemed the untenable claim that the Constitution delegated the chief executive powers above and beyond those specifically enumerated.\n\nJefferson believed that his predecessor, John Adams, in signing into law the Alien and Sedition Acts, had used specious claims to support the view that the president had broader powers than those explicitly ceded to him. Adams did try to explain himself; he claimed that the impending war with France made these laws just and necessary. The U.S. military, Adams argued, was woefully outmanned by that of the French. He further maintained that there were many sympathizers of the French Revolution in the United States. While there most assuredly were such sympathizers (including the Democratic-Republicans), the question is whether those in America who supported the French Revolution, because they believed it was being waged for similar reasons and objectives as the American Revolution, posed any sort of danger to security. With the U.S. and French navies embroiled in skirmishes in the Caribbean, Adams warned that if this conflict escalated and France invaded American soil, some such sympathizers, U.S. citizens and aliens alike, might create civil unrest at a time when the nation was at its most vulnerable. Democratic-Republicans contended that Adams was using the naval battle as a scare tactic for clamping down on constitutionally guaranteed freedoms, what with his virtually unlimited power under these acts to claim that just about any criticism of his government could be considered seditious. But the Adams administration claimed that the Alien and Sedition Acts, by making it possible to prosecute those who spoke out against its strategems, would ensure that the federal government had all the tools needed to win its conflict with France.\n\nAdams was soundly defeated when he sought to serve a second term as president. A majority of voters agreed with Jefferson's counterclaim that Adams was in fact buttressing his own power at the people's expense when he pushed the acts through. In the Virginia and Kentucky Resolutions of 1798 and 1799 that he penned with James Madison to protest the Alien and Sedition Acts, Jefferson claimed that Adams had taken on the airs of a monarch, and was using the provocation with France as a pretext for silencing Republican critics of his policies. He argued that the Alien and Sedition Acts were the antithesis of the people's will, and \"so palpably against the Constitution as to amount to an undisguised declaration that that compact is not meant to be the measure of the powers of the General Government. . . . This would be to surrender the form of government we have chosen and live under one deriving its powers from its own will and not from our authority.\"\n\nAfter Jefferson succeeded Adams, he noted\u2014surely referring to his predecessor\u2014that sometimes a president may make what amounts to an unprincipled declaration of principle \"in moments of passion or delusion,\" and that it is up to a wise and \"watchful\" public to discern when this is the case.\n\nBut was Jefferson any more principled as president? Did he himself ever argue for a broad interpretation of the Constitution when it came to use of executive power?\n\nThree of his actions while in office shed light on an answer.\n\nJefferson peremptorily passed the unpopular Embargo Act in 1807, which prohibited American ships from entering foreign ports. It was particularly aimed at restricting American trade with Britain and France, which were engaged in the Napoleonic Wars, in an effort to prevent American involvement in this conflict. This was disastrous to the U.S. economy, since it brought American trade to a standstill. Leonard Levy, one of the most caustic modern critics of Jefferson, in Jefferson and Civil Liberties, has this to say about the embargo: \"To this day, it remains the most repressive and unconstitutional legislation ever enacted by Congress in time of peace. No peacetime president ever sought or received such a vast concentration of power as did Jefferson and at the expense of provisions in the Bill of Rights which he himself once advocated as necessary checks against tyranny.\"\n\nJeremy Bailey is less harsh, but does acknowledge that Jefferson did not follow his own procedure for justifying executive privilege in this notorious case: Jefferson \"did not explain the policy to the national electorate and therefore lost the battle of public opinion.\"\n\nSoon after his inauguration as president, Jefferson sent U.S. military forces to fight the Barbary Pirates of North Africa, who routinely captured U.S. merchant ships and their crews in the Mediterranean and held them for ransom. Jefferson refused to pay any further tributes, and instead deployed forces to fight the pirates. But he did so without congressional approval, though he was constitutionally required to obtain it. Even members of his own party criticized him for this action. In the end, U.S. forces won out in the four-year conflict, and a treaty was signed. Jefferson did at least follow his own protocol this time around; he used his annual message to Congress to issue a declaration of principles justifying his decision to act outside of constitutional dictates.\n\nTo this state of general peace with which we have been blessed, one only exception exists. Tripoli, the least considerable of the Barbary States, had come forward with demands unfounded either in right or in compact, and had permitted itself to denounce war, on our failure to comply before a given day [to pay tribute]. The style of the demand admitted but one answer. I sent a small squadron of frigates into the Mediterranean . . .\n\nStill, Jefferson failed to explain why he'd neglected to ask Congress to declare war before he deployed troops, as was required\u2014showing he was just as capable as any president of skirting the Constitution and using less than plausible rationales for doing so.\n\nOnly Jefferson's famous \"Louisiana Purchase\" was in line with his professed views on how to go about justifying extraconstitutional use of executive power. Though the negotiations were initially conducted in secret, Jefferson did at first act within constitutional parameters. Since the Constitution did not clearly authorize such a purchase, Jefferson at first tied to muster Congress's support for an amendment that would allow him to acquire over 800,000 square miles of French territory. Jefferson was convinced it would promote the nation's general welfare: it would double the size of the United States, allow for great expansion of settlements, take territory on the North American continent out of the hands of a European government with which it had had serious run-ins, and provide a vast new trading port on a different ocean, generating a much-needed economic boost.\n\nBut when it became evident that Jefferson wouldn't be able to garner the support of a two-thirds majority in both chambers of Congress to approve the purchase via a constitutional amendment, he went ahead and acquired the land anyway, fearful that if he didn't act with haste to clinch the deal of the century, Napoleon would change his mind and rescind the offer. On October 20, 1803, the United States purchased the Louisiana Territory for $11,250,000, after the Senate approved the purchase as a treaty, voting 24 to 7 in favor of it.\n\nJefferson then made his case to the people for why his actions, though undertaken in a constitutional twilight zone, were nonetheless necessary in order to carry out what he believed was the people's mandate to fortify the nation. He argued that it was his duty to purchase the Louisiana Territory come what may, and maintained that this both demonstrated his obedience to a higher law and to an expression of the people's will. Whether this was actually the case is an open question. What is hardly debatable, though, is that the acquisition was a masterstroke. Jefferson was easily reelected for a second term.\n\nFew presidents seem to have considered whether their refusal to adhere to the Constitution that they've sworn to uphold faithfully might constitute in itself a refusal to submit to the nation's will. All too often, those who skirt the Constitution and who then justify it by claiming that they're trying to save our country during dangerous times, not only succeed in sacrificing far more life, liberty, and property than if they had not acted at all, but also, in losing sight of the law itself, succeed in chipping away at democracy's foundations. As Jefferson was the first to recognize, use of executive \"prerogative\" was \"easy of solution in principle, but sometimes embarrassing in practice.\" \nVIII\n\nAnd Justice for All\n\n\"Trial by jury\" is usually understood to be one of the great pillars of American justice and democracy in general. In his first inaugural address, Jefferson called for a redoubling of the nation's commitment to \"equal and exact justice to all men, of whatever state or persuasion, religious or political.\" He hoped to right the wrongs that had been committed as a result of the Alien and Sedition Acts, which had undermined the constitutional guarantee to a trial by an impartial jury: judges partial to the Federalists had repeatedly influenced jury proceedings by issuing rulings on the law that blatantly favored attempts by the Adams administration to curtail any criticism of the government.\n\nAdams's rationale was that, at a time when America faced potential outside threats, it was better to restrict or suspend constitutional rights in order to make the nation safer, even if democracy itself was endangered in the process. Similarly, the Patriot Act, passed by a compliant Congress following the tragic events of September 11, 2001, was a response by the government to security fears. Civil libertarians argued that the Patriot Act's restrictions on some due process and privacy rights undermined the Constitution, in much the same way that Jefferson's Democratic-Republican Party protested the \"unconstitutional\" Alien and Sedition Acts.\n\nOne might take heart in the fact that if American citizens feel threatened by government abuses or assaults on their rights, they can vote elected officials out of office and replace them with ones who they believe will restore their rights. That's what happened to President Adams and the Federalist members of his Congress; they were ousted following the passage of the Alien and Sedition Acts. Afterward, Jefferson and the Democratic-Republicans in Congress rescinded them. The difference today, though, is that most key elements of the Patriot Act continue to enjoy the wide support of both major parties.\n\nIn his inaugural address, Jefferson promised to restore right away the justice system's fair and equitable application of the law. In a gesture to remedy the ill effects of the Alien and Sedition Acts, he called specifically for \"impartial selection\" of juries, asserting that this was \"essential to their value.\" It's noteworthy that Jefferson didn't call for an impartial jury, but for impartial selection of a jury. But there has never been such a selection process in the United States.\n\nIn the pre-Revolutionary era, many jurors were chosen because they were deemed to have particular knowledge relating to the case to be tried (which in the South meant juries were often comprised of the wealthiest slaveholders). While the rationale was that since they were wealthy and educated, they would \"know best,\" they were no more representative of the average local denizen than our constitutional Framers were of the average American.\n\nToday juries are selected from voter registration rolls and driver's license records. The thinking is that such random selection makes them more diverse, and as a result supposedly more impartial. But even this method of selection excludes an enormous segment of the population.\n\nWhat is more, a New York Times investigation reveals that \"the practice of excluding blacks and other minorities from juries in the South remains widespread\" because of peremptory challenges during jury selection. This gives the defense and prosecution the right to reject a certain number of potential jurors without having to give any reason. The result is \"racially discriminatory jury selection.\" In fact, throughout the United States, attorneys are allowed to issue peremptory challenges to exclude potential jurors without stating their reasons. Juries today arguably are less likely than ever to be representative of most Americans.\n\nHow can the ideal of justice for all be realized? Is there any way to ensure that justice is meted out equally and exactly to all Americans?\n\nPeer Pressure\n\nFrom Amendment VI of the Constitution: \"In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the State and district wherein the crime shall have been committed.\"\n\n\"How many of you believe you have the constitutional right to a trial by jury of your peers?\" Darlene asks.\n\nAll hands go up.\n\nShe shakes her head. \"Uh-uh. It doesn't say anywhere in the Constitution that we have that right. All we have is the Sixth Amendment right to a 'speedy and public trial, by an impartial jury.' But what we really need guaranteed is a trial by jury of our peers. We need that constitutional right that everyone believes we have already.\"\n\nAs with most of the women inmates with whom I'm conversing, Darlene does not hesitate to participate with great enthusiasm. Her attitude mirrors that of the enlightened warden to whom I broached the idea of holding a Constitution Caf\u00e9 (the previous ten or so had turned me down flat); she happened to be a fan of Jefferson and was receptive to giving it a go.\n\n\"Should this be included in a new Bill of Rights section?\" I ask Darlene now.\n\n\"I've been reading over the Constitution, and to me, it should be inserted in the section on the judicial branch in Article III,\" she says. \"That's where the Sixth Amendment should naturally have been included in the first place. And this way, the federal judges know that they aren't the bosses in jury trials. All they do is make sure that the court system operates according to proper procedure. It's the true peers in the jury who preside over the system.\"\n\nGabriela nods in agreement. \"A jury of our peers wouldn't be impartial. It would be partial. There's no such thing as an impartial jury anyway. Jurors are always judging the person standing trial, and not just based on the evidence presented. With a trial by a true jury of our peers, we can quit pretending 'impartial' exists.\"\n\n\"Who would be an ideal 'partial peer'?\" I ask.\n\n\"She's someone who's walked your walk a bit,\" she replies, \"so she wouldn't be holier than thou. She'd know something of your pain, but also have some experience of the pain you've caused others, and what you might need to go through to straighten yourself out.\"\n\n\"A peer is someone who is partial in the right way,\" Darlene says. \"People think that if the jury is partial, it can't be objective. But to me, you can't be objective unless you can relate in some way.\n\n\"Sure, a juror judges, because you can't help but judge others, no matter how much you might deny it. But she judges herself first. And then, in judging you, she takes in as much of your story as she can, not just the bad and the ugly, but the good too. If the evidence speaks to your guilt, then she'd find you guilty, but she'd have a tear in her eye\u2014like one of my jurors had, while the rest looked at me like I was nothing.\"\n\nGabriela is moved to say, \"That same day I broke into a car to steal whatever I could get my hands on, so I could sell it and get my fix, I also saved a child who was getting beaten to a pulp by some gang-bangers. That doesn't make me a saint, but it sure makes cowards of all those who stood around with their hands over their mouths, in horror, or pretend horror, doing nothing. I was a good Samaritan that day, and a criminal. I'm also an addict, a person who paints and writes poetry, who tries to be a loving mom, and who has a lot of demons. A partial jury of my peers might take all these things into account.\"\n\n\"For me,\" says Cyndi, \"a jury of partial peers would have to include rich brats whose parents never spent a second with them, Mom always out getting drunk and laid by afternoon at her 'socials,' Dad cheating in his personal and professional life, and looking at me, after he's had a few, in a way a dad should never look at his child.\"\n\n\"Even if someone is 'like you' in experiences like drug abuse, they still may be heartless, until maybe they've 'experienced' getting caught, and are in trouble big time, so they can't be in denial anymore,\" says Michele.\n\n\"Peers haven't done very good by me,\" she goes on. \"Except for my kid sister, those who've been most helpful to me are people who've had almost no experiences like mine\u2014my counselor, for one, who\"\u2014looking at Cyndi\u2014\"describes herself as a reformed rich brat. She's doing everything she can to help me straighten out my act. Same goes for one of the jurors who found me guilty. I've only been here three months, and she's already visited, and writes me letters of encouragement all the time. Jurors can be objective and partial at the same time. Somehow they can relate to me, at least in that 'there but for the grace of God' way.\"\n\nThen she says, \"I'm thinking, based on my own story, maybe we should leave the jury selection system random, like it is now.\"\n\n\"You think it's random?\" says Darlene, incredulous. \"You ever see any really down-and-out people on a jury? No, it's all these straightlaced folks. If it were really random, then yeah, I could go along with that. I'm not just talking about who you pick out of the hat is who you get, but making the Constitution say that the hat has to include everyone, and I mean everyone\u2014every American age eighteen or over, period.\"\n\n\"Including people in prison?\" I ask.\n\n\"Including lawmakers, lawbreakers, and everyone in between,\" she says flatly. \"Because that would reflect the 'American peerage' as a whole.\"\n\nSoon afterward, Gabriela says, \"Maybe we're not looking at this 'peer thing' the right way. A peer, in the legal sense, is someone who's treated just like you under the same circumstances. Rich or poor, if your daddy's president or if he's a janitor, if you do the crime, you do the time. Same crime, same punishment. The way it is now, though, some people literally get away with murder if they have the money to get a Johnnie Cochran\u2013style lawyer. Yet others are put away for life\u2014three strikes and you're out\u2014for doing pretty petty crimes in comparison, because they're represented by an overwhelmed, underpaid public defender. In a real peer-based legal system, that couldn't happen. We'd all be treated just the same every step of the way through the process\u2014it would be real due process.\"\n\n\"We have to go back one more step before that can happen. Peers would have to be the ones making the laws too,\" says Michele. \"The Constitution should say that 'all laws are by peers,' and by that I mean the real mix of Americans that Darlene is speaking about. America is all about diversity, so peers 'American style' would be a diverse group.\"\n\n\"I don't know if that's possible,\" says Darlene. \"But it sure would be nice if our Congress was made up of at least some members who can relate to us and our experiences. And at least having a jury of true peers makes sure the law is applied 'peerlessly.' \"\n\nThen she says, \"The Fourteenth Amendment guarantees us 'equal protection of the laws,' which means you have the same access as anyone else to the law and the courts, and to be treated equally. This is supposed to protect us from 'cruel and unusual punishment.' \"\n\nThis prompts Gabriela to say, \"As things stand now, people in our situation need protection from the laws and courts. Like, Justice Clarence Thomas benefited all his life from affirmative action. But now he can't stand it that other disadvantaged people would want that same equal protection and support. He wants to be peerless in a way that denies his history, when he should be grateful and recognize that we all need a hand up, and that the legal system needs to help provide that. Do you know that he believes it's okay for inmates to be beaten, because he says the country's founders wouldn't have found it cruel and unusual? That's how far removed he is from his own past sufferings.\"\n\nAs the \"originalist\" legal philosopher among Supreme Court justices, Thomas claims he interprets the Constitution by seeking out the original meaning or intent of the Framers. In 1992, he argued in a dissenting opinion that the beating of a prison inmate was not overly harsh; Thomas maintained that our Constitution's Framers would not have deemed it cruel and unusual punishment.\n\n\"I don't want serving on my jury any bleeding heart. I just want someone with a beating heart,\" Michele says soon afterward. \"I know what I did was wrong, and that there's no good excuse, no matter how hard I've had it. I know people who've had it much harder than me, but they never crossed the line into the world of drugs and crime.\"\n\n\"I'd consider President Obama an ideal peer to have had on my jury,\" Darlene says. \"He was truthful about his drug use, and confessed to breaking the law\u2014and in so many words, to getting away with it, since he was never arrested, much less served any time. So in judging me, he'd have to ask himself, 'At a time when I was so low, if I could abuse alcohol, marijuana, cocaine, and get away with it, why in the world would I put this substance abuser in jail, when what she needs is treatment so she doesn't do any more harm to herself or anyone else, especially her loved ones?'\"\n\n\"I'm not sure I'd want Obama as a peer,\" Cyndi says to Darlene. \"Like you said, he was never punished for his crime. So he never walked the whole walk. He knows he got away with using illegal drugs. If he'd been punished, maybe he'd have come out in favor of legalizing drug use, so it could be controlled. Then the focus could be on prevention and treatment, and he could divert all the billions he's added to the 'war on drugs' in Mexico to these treatment programs.\"\n\n\"I hear what you're saying,\" Darlene replies. \"But you know what? I wouldn't have wanted him to be imprisoned. He was just crying out for help. And now look where he is. He overcame all those lows, and is an inspiration to us all, because he was open about his troubles, and has shown great humility.\"\n\n\"I'm a sinner and a lawbreaker, plain and simple,\" Gabriela says, looking my way. \"I bet you are too, and I bet the people who tried and convicted me are. People cheat on their spouses, cheat on their taxes, drive drunk or drugged\u2014even with kids in their car\u2014and get away with it. They hit people, hurt people, abuse them mentally and emotionally. Yet they're considered 'innocent,' because they've never been caught or found out. They serve on juries all the time.\"\n\n\"A jury of peers,\" says Darlene, \"should be made up not just of people who reflect who you are now, or who you've been, but who you want to be.\"\n\nConstitution Caf\u00e9 Article: In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial by a jury of true peers of the State and district wherein the crime shall have been committed.\n\nJudge and Jury\n\nJefferson didn't go as far as one Constitution Caf\u00e9 participant at the prison, who argued that all Americans should be involved in the process of making law. He did, however, maintain that if juries \"think the permanent judges are under any bias whatever in any cause,\" they are authorized to \"take on themselves to judge the law as well as the fact.\" So in an instance when jurors believe a judge is not issuing rulings according to the law, but rather is allowing political bias or bribery to dictate his decisions, jurors would not just determine guilt or innocence, but would also assume the judge's role in the proceedings.\n\nJohn Jay, the first chief justice of the U.S. Supreme Court, ruled that while typically \"juries are the best judges of facts\" and \"courts are the best judges of law,\" there are instances in which \"both objects are within [juries'] power of decision.\" Under such circumstances, juries \"have a right to take it upon [themselves] to judge of both, and to determine the law as well as the fact in controversy.\" As a result, jurors would not only be able to override a judge, but to nullify existing law.\n\nThis right to \"nullification\" was first established in the New World in 1735. John Peter Zinger, a New York newspaper publisher was arrested for \"seditious libel\" because he criticized the royal governor of New York in print. During the trial, the royal prosecutor himself acknowledged that what Zinger had written about the governor was true, but that Zinger's arrest was nonetheless legal because such criticism was banned. The jury, though, ruled in Zinger's favor, declaring him not guilty. Zinger had indeed broken the law, but since the jury considered the law to be unjust, it found him innocent.\n\nTo this day, though it is not widely known or practiced, nullification remains a right in jury trials. A jury (or a single juror) can invalidate the law as it applies to the specific case over which it is presiding and find a defendant not guilty even if he has committed the infraction for which he is standing trial. In our nation's history, nullification has been used for both humane and destructive purposes by juries that were not impartially selected. For instance, during the Civil War, juries sympathetic to the cause of emancipation refused to convict abolitionists even when they broke laws that prohibited them from helping slaves. On the other hand, during the civil rights era, all-white juries refused to convict whites who murdered blacks.\n\nOne could argue that the Declaration of Independence established the right to nullification in the United States. After all, it asserted that England had forfeited legal authority over the colonies after committing the egregious offense of subjecting the colonies to laws enacted without their consultation or consent.\n\nCrime and Punishment\n\nNo one influenced Jefferson's ideas on criminal code reforms more than Cesare Beccaria (1738\u20131794), the Italian nobleman and iconoclastic criminologist. In particular, Jefferson was taken with Beccaria's then-novel assertion that the only meaningful criterion for determining the appropriateness of a penalty for any given criminal infraction was whether it served as an effective deterrent. Beccaria's perspective sharply contrasted with the prevailing ethos that punishment was meant primarily to be retributive in nature. Beccaria argued in his landmark work On Crimes and Punishments that it is\n\nbetter to prevent crimes than to punish them. This is the ultimate end of every good legislation, which . . . is the art of leading men to the greatest possible happiness. . . . Do you want to prevent crimes? See to it that laws are clear and simple and that the entire force of a nation is united in their defense. . . . See to it that the laws favor not so much classes of men as men themselves. See to it that men fear the laws and nothing else.\n\nIf the public believed that the punishment for a specific crime was far harsher than it should be, Beccaria believed that society's lawmakers themselves would come to be perceived as unjust, and that over time this perception could heighten the prospects of civil unrest.\n\nBeccaria considered the death penalty tantamount to \"a war of a whole nation against a citizen, whose destruction they consider as necessary, or useful to the general good.\" He argued that there was not a shred of evidence that showed it prevented crime. He maintained that \"the long and painful example of a man deprived of his freedom and become a beast of burden, repaying with his toil the society he has offended\" would serve as the best deterrent.\n\nJefferson acknowledges in his memoirs that \"Beccaria, and other leading writers on crimes and punishments had satisfied the reasonable world on the unrightfulness and inefficacy of the punishment of crimes by death,\" and that \"hard labor on roads, canals and other public works, had been suggested as a proper substitute.\" While Jefferson did not endorse the wholesale elimination of the death penalty, he did propose to dramatically cut down on the instances in which it was applied. In the criminal reform bill presented to Virginia's legislature in 1778 as part of their constitutional reform package, Jefferson and his committee of Revisors suggested doing away with most instances of capital punishment, arguing that it served no warrantable purpose: \"the reformation of offenders . . . is not effected at all by capital punishments, which exterminate instead of reforming.\" Like Beccaria, they believed that the death penalty\n\nshould be the last melancholy resource against those whose existence is become inconsistent with the safety of their fellow citizens, which also weaken the state by cutting off so many who, if reformed, might be restored sound members to society, who, even under a course of correction, might be rendered useful in various labors for the public, and would be living and long continued spectacles to deter others from committing the like offences.\n\nHowever, as Jefferson relates in his memoirs, \"the general idea of our country had not yet advanced to that point\"\u2014or at least, it had not yet advanced sufficiently among his colleagues in the Virginia legislature. His comprehensive bill to reform the criminal justice code \"was lost in the house of Delegates by a majority of a single vote.\"\n\nJudging Judges\n\nFrom Article III, Section 1, of the Constitution: \"The judges, both of the supreme and inferior Courts, shall hold their Offices during good Behaviour.\"\n\n\"The third article of the Constitution establishes the judiciary branch of the government,\" Colin says, \"and the first section of that article, after establishing the federal judiciary, says that all judges from the Supreme Court justices on down 'shall hold their offices during good behavior.' That's been interpreted throughout our history to mean that federal judges can serve for life, as long as they behave well.\n\n\"I'm not sure the fact that they are exemplars of good behavior\u2014or at least, that they're adept at keeping any bad behavior on their part out of the public eye\u2014is enough to warrant giving them life terms,\" Colin goes on. \"Basically they serve at their pleasure rather than that of the American people or even the president who appointed them with the Senate's advice and consent. They're like monarchs who can only be dethroned if they do something really, really bad, or if they choose to abdicate of their own volition. That doesn't seem appropriate for judges serving in a democracy.\"\n\nColin, a retired high-level federal bureaucrat, whom I first met a decade ago while on book tour, arranged this gathering in Florida. We're joined by a number of others in the large commons area in the retirement community in which they live.\n\n\"In our nation's early years, the federal judges didn't serve such long terms, even though there were no limits placed by the Constitution on how long they could serve,\" says Eugene, a former dean at an Ivy League university. \"So all presidents typically had ample opportunity to appoint judges, since there was considerable turnover. But since most federal judges these days choose to serve very long terms, maybe it would be fitting to place a specific limit on their tenure.\"\n\n\"I personally don't mind if federal judges serve for life,\" says Norma, who was a family law attorney for over half a century. \"But in the case of the Supreme Court, I also believe that each president should have the opportunity to put his stamp on it. A revised constitutional article can give him this opportunity. Why should one president be able to appoint one or more justices, and another none at all? Franklin Roosevelt appointed eight, a record. Even George Washington, our first president, who was there on the scene when the Court came to be, only appointed seven.\"\n\n\"But if there's nine justices total, how is it that Washington only appointed seven?\" asks Evelyn, a retired sociology professor.\n\nNorma smiles. \"I'd wondered the same thing. But then I realized, after scrutinizing this article and doing a little research, that it doesn't state how many justices can serve on the Court. The numbers have in fact varied over the years. We could have a hundred Supreme Court justices if Congress passed legislation to that effect. The Framers must have had a reason for leaving the number open-ended. Maybe they thought we should have more justices as the nation expanded.\n\n\"At first I wasn't sure I agreed with the Framers, who didn't set a limit on how many justices there could be. But after thinking about it for a while, I decided that that's fine with me. What I would do, however, is add language to this article that gives each sitting president the opportunity to appoint at least one justice.\"\n\n\"I'm not sure I'd tamper with the existing language in the Constitution on this matter,\" says Colin. \"Congress has the power to increase or decrease the number of justices on the Court. If a president wants more justices on the Court, he can always ask Congress to consider increasing the number. That's what Franklin Roosevelt did.\"\n\nIn the mid-1930s, President Roosevelt had the strong support of Congress as he launched progressive New Deal reforms, only for the conservative Supreme Court to resist. The Court dismantled one New Deal initiative after another, declaring them unconstitutional. In response, FDR proposed to Congress a law that would enable him to appoint as many as six additional justices. While Congress was considering the legislation, the Court underwent a striking about-face: the justices\u2014fearing their individual power would be diluted if their numbers were increased\u2014reversed their earlier decisions against Roosevelt's programs. This paved the way for the New Deal, and FDR stopped his legislative push for additional justices. There have been no further attempts to change the number of justices on the Court.\n\n\"I wouldn't be surprised if our fairly liberal president is tempted to try to add to the Court,\" says Evelyn. \"He'll surely clash with the conservative majority on the Court on matters ranging from the constitutionality of health care reform legislation to the overhaul of the banking and finance system.\n\n\"But rather than give a president the opportunity to increase the number of sitting justices for reasons of political expediency, I'm in agreement with Norma; I'd give each president the guaranteed opportunity to appoint at least one judge and have an impact on the makeup of the bench on the nation's highest bench.\"\n\n\"If, in this article we're redoing, we did give a president the power to appoint a justice for each term of office he serves,\" Colin says, \"there's no guarantee that the president's appointee will make rulings in his favor. Some justices have shown great independence, and risen above their previously partisan leanings to make a body of rulings that are far different than what anyone could have anticipated before they assumed their seat.\"\n\n\"The president should still be given a chance to impact the Court's makeup, even if the impact isn't the one he intended,\" says Eugene. \"Since the Court itself is made up of unelected officials, a sitting president, as the only nationally elected federal official, would hopefully appoint someone he believes is reflective of, or at least in touch with, the sensibilities and hopes of most Americans of the times in which he serves.\"\n\n\"I'm not sure the fact that a justice is one of the most recent appointees necessarily means he or she is the most in touch with Americans' pulse,\" says Norma. \"George W. Bush appointed justices who were most like him, conservative in a dogmatic way. And Obama's appointees all reflect his liberal sensibilities. Yet most Americans aren't nearly as liberal or conservative as any of the most recently appointed Justices.\"\n\n\"To me,\" she goes on, \"the recently retired Justice John Paul Stevens, the third-longest-serving justice, has been most in sync with Americans over the past decades. Justice Stevens served for thirty-six years, until age ninety. He was unchanging in the best sense. As the political winds tacked one way and then another during his tenure, he continued to issue rulings in what most call iconoclastic fashion, but what I'd call just plain fair fashion. He's billed as a liberal today, just as he was billed as a moderate conservative in his early years as a judge. But he hasn't changed at all; those around him have. By fairness and evenhandedness above all political considerations, he's the epitome of judicial 'good behavior.'\"\n\n\"We're back to that term 'good behavior,'\" Colin says. \"I'm still not comfortable with how it's included in this existing constitutional article. For one thing, it's the only qualification that this article establishes for anyone to serve as a federal judge. This article sets no professional, age, or even citizenship qualifications. At the very least, federal judges should have to demonstrate 'exemplary behavior' rather than just comportment of the 'good' variety.\"\n\n\"But even 'exemplary behavior' is so open to interpretation that it really becomes an empty criterion,\" Evelyn says.\n\n\"Maybe this article should strike the term 'good behavior' and focus instead on bad behavior,\" says Norma. \"That way, it'd be in line with the constitutional articles that spell out under what circumstances those in the executive and legislative branches are subject for removal from office, namely for committing high crimes and misdemeanors.\"\n\n\"But the Constitution already does this for the judiciary branch as well,\" says Eugene. \"The justices are civil officers. And the Constitution says that 'all civil officers of the United States, shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes and misdemeanors.' So in order to keep their jobs, the Constitution does make clear that at the least they have to behave as well as any other civil officer, or they can be impeached.\"\n\n\"Maybe so,\" says Colin, \"but in my opinion, most of them still have been judicial activists. They rule on the law in ways that mesh with their personal and political views, and oftentimes they refuse to voluntarily recuse themselves from cases even when they clearly have a conflict of interest. For better or worse, that's about the best behavior we've been able to get out of them.\"\n\n\"I don't know about that,\" says Norma. \"Some of them, like Justice Stevens, have shown not only that they are quite qualified, but have also shown that they strive to be impartial, and that they are also of excellent character.\n\n\"In approving Obama's nominee Elena Kagan to the Supreme Court to replace Justice Stevens, the Republican senator Lindsey Graham said he'd never have appointed her if he'd been president. But he still thought she was an excellent choice, because she was highly qualified and had unimpeachable character. And that's what this article should say: that federal judges should have to be exceptionally qualified, and paragons of exemplary behavior. If Kagan continues to build on her character and her professional qualities as if they're tied together, she may turn out to be every bit as exemplary as the justice she's replacing.\"\n\nConstitution Caf\u00e9 Article: The judges, both of the supreme and inferior courts, shall be exceptionally qualified and of unimpeachable character, and shall hold their Offices during exemplary Behavior. The president shall appoint to the Supreme Court one Justice during the second year of his four-year term.\n\nThe Chase Case\n\nOnly one Supreme Court justice, Samuel Chase, has ever been impeached by the House. This either goes to show just how honest and aboveboard our justices have been over the years, or how difficult it is to remove them from the bench.\n\nChase was a signer of the Declaration and represented Maryland in the Continental Congress. He was appointed to the Supreme Court by John Adams. None other than Thomas Jefferson, who was president at the time, sought to have Chase ousted. An ardent Federalist, Chase imposed severe sentences on those convicted of violating the Alien and Sedition Acts.\n\nJefferson argued that Chase let his political views cloud his judgment to such an extent that he betrayed his oath of office. In pushing for Justice Chase's impeachment, Jefferson asked Congressman Joseph Hopper Nicholson of Maryland, \"Ought the seditious and official attack [by Chase] on the principles of our Constitution . . . to go unpunished?\" In 1804, at Jefferson's behest, the Republican-dominated House of Representatives served Chase with eight articles of impeachment. The thrust of the charges against Chase was that his pro-Federalist bias prevented him from treating defendants and their counsel impartially. While Jefferson did succeed in having him impeached by the House, Chase was acquitted at trial in the Senate. Its members didn't agree with Jefferson that Chase's evident partisan rulings provided grounds for dismissal. Moreover, the Constitution doesn't specifically prohibit justices from becoming directly involved in partisan politics. Chase then resumed his duties on the Court.\n\nUpon Further Review\n\nFrom Article III, Section 1 of the Constitution: \"The judicial Power of the United States, shall be vested in one supreme Court, and in such inferior Courts as the Congress may from time to time ordain and establish.\"\n\nFrom Article III, Section 2, Clause 1, of the Constitution: \"The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority;\u2014to all Cases affecting Ambassadors, other public Ministers and Consuls;\u2014to all Cases of admiralty and maritime Jurisdiction;\u2014to Controversies to which the United States shall be a Party;\u2014to Controversies between two or more States;\u2014between a State and Citizens of another State;\u2014between Citizens of different States;\u2014between Citizens of the same State claiming Lands under Grants of different States, and between a State, or the Citizens thereof, and foreign States, Citizens or Subjects.\"\n\nFrom Article III, Section 2, Clause 3, of the Constitution: \"In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.\"\n\n\"Let's say Congress votes to give every American a turkey for Thanksgiving, or a revolver for self-defense, or good health care for self-preservation,\" says George, a lawyer who once argued (and lost) a case before the Supreme Court, and is one of a baker's dozen of lawyers gathered with me. \"But then an American or group of Americans\u2014maybe a disgruntled vegetarian or group of gun control advocates or Libertarians\u2014disagrees with these acts of Congress and decides to sue. Their cases eventually make their way to the Supreme Court, where a majority rules that Congress acted unconstitutionally, and invalidates these laws. That puts an end to the matter, because the Supreme Court is the final arbiter on issues of constitutionality, right?\" He shakes his head. \"That may be how it is, but it's not how the Framers meant for it to be.\"\n\nThen George says, \"This power of 'judicial review,' in which legislative and executive actions are subject to scrutiny and possible invalidation by the federal judiciary, is a constitutional chimera. Nowhere in the Constitution is the federal judiciary ceded this authority. So every time the Court rules that an enacted law is or isn't constitutional, it itself is acting unconstitutionally. This 'new Constitution' can explicitly nullify judicial review. Because it puts the federal judiciary on a much higher footing than the other two branches, undoing the checks and balances in the Constitution that created three separate and equal powers.\"\n\nGiselda disagrees. The defense attorney and adjunct law professor says, \"This new article should state that judicial review is within the parameters of the Court's authority. It would throw the entire legal system into chaos, and create unimaginable civil unrest, if it were dismantled now. Judicial review has been legal precedent since it was established by the Supreme Court in 1803.\"\n\nGiselda is making reference to the 1803 Supreme Court case of Marbury v. Madison, which stemmed from one of the first acts accomplished by Thomas Jefferson when he became president. In an eleventh-hour move before leaving office, his predecessor John Adams, in collaboration with the Federalist-dominated Congress (soon to be replaced by a majority of Republicans), expanded the number of circuit courts and filled these new judicial seats with pro-Federalist appointees. Jefferson refused to seat any of the judges who had not yet been commissioned by the time Adams left office. One of the judges, William Marbury, an Adams cohort, petitioned the Supreme Court to seat him. To Marbury's disappointment, John Marshall, the chief justice and a fellow Federalist, denied his petition, ruling that it was outside the court's purview. The matter should have ended there, but Marshall used this case to declare that even though the Court might be powerless to help seat him in this specific instance, it nonetheless \"is emphatically the province and duty of the Judicial Department [the judicial branch] to say what the law is. Those who apply the rule to particular cases must, of necessity, expound and interpret that rule.\" Marshall claimed that the Court did indeed have the rightful authority to decide which acts of the other two branches were and were not constitutional.\n\nGiselda says next, \"Even if I grant that I'm not sure whether the Framers intended for the federal judiciary to have this authority, the fact is, it has had the power of judicial review for over two hundred years. And there's never been a single attempt by the other two branches to challenge it and declare this power untenable. So it should stay just as it is.\"\n\n\"Thomas Jefferson wanted to take on the Marshall Court and put an end to the power of judicial review before it could begin,\" George says in return. \"And he would have if he'd had occasion to. But Justice Marshall was too clever. He technically ruled in Jefferson's favor in Marbury v. Madison, refusing to seat Marbury as a judge, even as he went on to declare that the Court was the final decider of the validity of federal statutes. Marshall didn't dare take on Jefferson. It wasn't until two decades after Marbury died that the Court exercised judicial review powers. And those powers have been inappropriately used by the court and the rest of the federal judiciary ever since.\"\n\n\"I'm with Giselda,\" says Jason, who runs what he describes as a boutique law firm. \"Judicial review has served as an effective constraint on the president and Congress. The fact that the Court has only struck down 160 acts of Congress and about 1,000 state laws since our founding shows that it exercises this power with restraint.\"\n\n\"But in striking down even one such law,\" George replies, \"by utilizing powers it was never granted, the Court is altering the course of our democracy, opposing the will of Americans as carried out by those they elected to enact laws on their behalf. This power has to be wrested from the Court. It's monarchical. Even the king of England was stripped of this power way back in 1688, when a provision was inserted in the Magna Carta that did away with judicial review. From then on, the king was unable to declare legislative statutes void.\"\n\nClair, a city attorney, says now, \"One can argue that chaos would ensue if this power of the Court's was done away with. But I'd argue that there's been even more damage done because of its existence.\n\n\"For example, in 1996, Congress passed the Line Item Veto Act. Before its passage, the chief executive had to veto an entire budgetary appropriations package, even if he found just a few 'line items' to be objectionable. But this legislative act gave presidents the authority to cancel specific budgetary provisions, instead of vetoing a budget appropriation in its entirety. Presidents had long been clamoring for line-item veto authority. In making a memorable plea for it, Ronald Reagan said, 'I'll make the cuts, I'll take the heat.'\n\n\"And finally, Congress passed it. For a brief time, Clinton was able to exercise the line-item veto. He struck scores of pork barrel items from the budget. Goodbye, politics as usual. But two years later, a majority of justices struck down the law. They ruled that the line-item veto was unconstitutional, that it granted the president powers he wasn't intended to have. Yet in making the ruling against one of the most popular pieces of legislation in modern history, the justices exercised a power that they themselves weren't intended to have.\"\n\n\"If the power of judicial review is eliminated, how are conflicts between the three federal branches resolved?\" I ask.\n\n\"The voters will resolve them,\" says Ray, who has run unsuccessfully for the U.S. Senate almost as many times as Harold Stassen once ran for president. \"If Congress and the president push through legislation they find objectionable, they'll vote them out. That's how the Framers meant for the process to work itself out.\"\n\nJason frowns. He says, \"You make it sound like it's some great gesture to leave all the doings of the executive and legislative branches up to 'the people' when it comes to deciding what is and isn't constitutional. We need a wiser and more objective arbiter that isn't subject to the people's whims, so that our inalienable rights and liberties are preserved. By being above the fray, the Supreme Court can rule dispassionately. With the power of judicial review, the Court ensures that the tyranny of the majority isn't the last word on any legislative subject.\"\n\nTo which Ray replies, \"You could apply this notion of tyranny of the majority to the Supreme Court itself. The Court only needs a majority of five justices to decide as it will, as was the case when five justices handed the 2000 presidential election to George W. Bush, or when it recently eviscerated the comprehensive campaign finance reform laws enacted by Congress. I'm not comfortable with five people wielding such power. No wonder Jefferson equated judicial review with the 'despotism of an oligarchy.'\"\n\n\"The fact is, rarely have our justices been paragons of 'dispassionate objectivity,'\" he goes on. \"Most have personal agendas when they make their rulings that have little do with constitutionality. I prefer a tyranny of a legislative majority than that of a majority of five justices. At least legislative power is dispersed among 535 people who face elections periodically, and so they're directly answerable to the voters. The more that the judiciary takes public policy matters out of the people's hands, the more apathetic Americans become, the less informed about policy, because they know that, whether informed or not, their opinion makes no difference; only that of the Supreme Court matters.\"\n\n\"In the Court's landmark 1973 decision on Roe v. Wade that upheld a woman's right to terminate a pregnancy,\" Claire says, \"Justice Byron R. White, in a stinging dissent, chastised his fellow justices for stepping in where Congress, or the states themselves, had the rightful role.\" She opens a textbook she has with her to a chapter that discusses this decision. \"Here's what Justice White had to say about this decision: 'As an exercise of raw judicial power, the Court perhaps has authority to do what it does today, but, in my view, its judgment is an improvident and extravagant exercise of the power of judicial review that the Constitution extends to this Court.' White believed this issue had to be left 'with the people and to the political processes the people have devised to govern their affairs.' He was totally opposed to judicial supremacy.\"\n\nAfter a pause, George speaks up again. \"Antiabortion activists are waiting for one more conservative justice to be seated on the bench in order to continue placing restrictions on abortions, and eventually to overturn Roe v. Wade altogether. Meanwhile, proabortion or prochoice advocates are praying that the current president will have the chance to appoint enough liberal justices so that they'll be able to reverse recent rulings that make it more difficult to have an abortion.\n\n\"But if we do away with the power of judicial review, this issue would have to redound to Congress, where it belongs in the first place. Congress would finally have to debate this and other hot-button issues, and make the tough legislatives decisions that they should have had to make decades ago.\"\n\nConstitution Caf\u00e9 Article: The judicial Power shall not extend to the review and possible invalidation of Legislative and Executive actions.\n\nThe Reviews Are In\n\nIn the pre-Revolutionary days, advisory bodies called privy councils were charged with reviewing legislation passed by colonial assemblies and with assessing whether the assemblies were acting within the parameters of the powers granted them by colonial charters. Though a rare occurrence, the councils did at times overturn legislation. Once the colonies became states, state courts (which supplanted privy councils) struck down laws passed by assemblies on occasion. While the Framers surely were aware of this practice, there is no compelling evidence that they supported judicial review at the federal level.\n\nAfter the Constitution was ratified and the system of government set forth by the Framers was put into place, Jefferson was not opposed in principle to some sort of judicial oversight over legislation. However, he'd thought that such judicial oversight would be limited to ensuring that the government did not encroach on any rights granted to citizens by the Bill of Rights. But after witnessing the concerted attempts by Federalist judges to enforce any and all perceived infringements of the Alien and Sedition Acts, Jefferson changed his tune and from then on came out strongly against judicial review.\n\nJefferson expressed his opposition to judicial review in regard to the Alien Sedition Acts in the course of an exchange with Abigail Adams, the wife of John Adams, who supported the acts: \"You seem to think it devolved on the judges to decide on the validity of the Sedition Law. But nothing in the Constitution has given them a right to decide for the Executive, more than to the Executive to decide for them.\"\n\n\"The Constitution,\" Jefferson argued, \". . . meant that its coordinate branches should be checks on each other. But the opinion which gives to the judges the right to decide what laws are constitutional and what not, not only for themselves in their own sphere of action but for the Legislature and Executive also in their spheres, would make the Judiciary a despotic branch.\"\n\nJefferson believed the Framers were wise in creating a Constitution that \"erected no such single tribunal, knowing that to whatever hands confided, with the corruptions of time and party, its members would become despots. It has more wisely made all the departments co-equal and co-sovereign within themselves.\"\n\nIf, as Jefferson posited, each of the three branches is \"truly independent of the others and has an equal right to decide for itself what is the meaning of the Constitution in the cases submitted to its action,\" then how are turf wars between them to be resolved?\n\nHe noted that Spain's constitution has a promising solution in stipulating that \"when the three coordinate branches differ in their construction of the constitution, the opinion of the two branches shall overrule the third.\"\n\nAs was one Constitution Caf\u00e9 participant, Jefferson was particularly partial to the contention that it is up to the voters to see to it that the legislative and executive branches do not overstep their bounds: \"When the legislative or executive functionaries act unconstitutionally, they are responsible to the people in their elective capacity.\" To him, \"the exemption of the judges\" from voters' scrutiny was \"quite dangerous enough\" without compounding this by giving them judicial review powers. \nIX\n\nGovernors and the Governed\n\nThe country of Bhutan adopted a new constitution in 2008, after the former Himalayan kingdom held its first-ever democratic election. The new constitution included a mandate stipulating that the worth of the nation's governors must be \"judged not by the economic benefits they may offer but by the happiness they produce.\" Prime Minister Jigme Tinley told the New York Times that the goal of this mandate is to require governors to \"create the conditions for . . . an updated version of the American Declaration of Independence, 'the pursuit of gross national happiness.'\" To that end, the government assesses whether its citizens are able to pursue happiness by gauging a range of \"domains,\" from psychological well-being to how much time a person spends with his family.\n\nBhutan's range of indicators only measures happiness in citizens' private affairs. Yet the one indicator that Jefferson deemed essential for achieving happiness is a person's right to take part in guiding public affairs, and not just on Election Day. Jefferson believed that the realization of an excellent society hinges not on iconic statesmen, \"no matter how 'great and virtuous,'\" but on the involvement of ordinary citizens, \"enlightened, peaceable and really free.\"\n\nBhutan notwithstanding, does the representative system of government devised by our own nation's Framers best encourage such involvement?\n\nAlthough the opening phrase of our Constitution, \"We the people,\" makes clear the Framers' conviction that the power vested in the federal government comes from the people themselves, a nationwide poll conducted in 2010 by the Center for the Constitution revealed that half of Americans nonetheless believe that governmental authority doesn't spring from the people at all, but from elected officials.\n\nClearly there's an implicit tension between \"representative\" and \"government\" in \"representative government.\" Do we want our elected officials primarily to serve as conduits for realizing citizens' expressed will\u2014or at least the will of the majority? If so, is the republican system devised by the Framers structured to achieve this?\n\nRepresentative Representatives\n\nArticle I, Section 2, Clause 1, of the Constitution: \"The House of Representatives shall be composed of Members chosen every second Year by the People of the several States.\"\n\nArticle I, Section 2, Clause 3: \"The Number of Representatives shall not exceed one for every Thirty thousand, but each State shall have at Least one Representative.\"\n\n\"The only stipulation the Constitution makes regarding the size of the House of Representatives is that the 'number of Representatives shall not exceed one for every thirty thousand,'\" Jim says.\n\nJim served in the House for ten years; his tenure ended after he was defeated in a primary by a more conservative Republican. He and Mary, a recently elected congresswoman, agreed to speak with me (in this case, by conference call) about my Constitution Caf\u00e9 project, but asked not to be fully identified.\n\nJim says now, \"It used to be that the House regularly increased the number of members as the population grew. But in 1911, House members set the number at 435. Except for one brief occasion, it has remained at that number ever since. In 1911, our population was less than 94 million. A hundred years later, we have a population of over 308 million people, but we're still stuck at 435 members of the House. You can't have very 'representative' representatives when the number of members hasn't gone up as the population has swelled.\"\n\n\"The mind-set was that if House membership got any bigger, it'd be unwieldy,\" Jim goes on. \"But only by increasing the number of members as the population grows is genuine representation possible. Every representative today has about 600,000 constituents. No matter how punishing a schedule you have, you can't really learn the pulse of your district in an up-close and personal way.\n\n\"Instead of increasing the number of House members, the congressional staff has been regularly augmented. Each member of Congress has 22 staff members. I'd prefer having that number dramatically cut back\u2014so there's only 2 staffers per member of Congress\u2014and increase the number of House members by a factor of 20, so there's one representative for every 30,000 people. This would fulfill the Framers' purpose for the House, and again make it a chamber of 'representativeness.' I'd change the language of the existing constitutional article so it says, 'the number of Representatives shall be one for every thirty thousand,' rather than 'shall not exceed one for every thirty thousand.' \"\n\nMary whistles. \"That'd give us something like, calculating off the top of my head, 10,000-plus representatives. With what you're proposing, Jim, members of Congress would have to gather in what amounts to a space about the size of a big high school or small-college football stadium.\"\n\n\"Is that bad?\" Jim says. \"It would be a modern-day version of how the Greeks in ancient Athens gathered in a stadium to deal with legislative matters. Whenever an assembly was called into session, the first 6,000 men to arrive at the stadium on the top of the Acropolis Hill took part in the vote on the measure. It was a large and raucous gathering, and the people's business was accomplished. And you know, if we'd kept gradually increasing House membership all along, we wouldn't think 10,000 was so dramatic a figure any more than the Greeks would have.\"\n\n\"Oh, it would still be dramatic,\" says Mary. \"Even if we'd kept increasing the number of members over the past century at the rate we had until 1911, we'd have at most about 1,800 today.\"\n\n\"Do you think 1,800 members would work well?\" I ask her.\n\nShe thinks about this. \"It might, if they were brought into the fold gradually, perhaps over a ten-year period. Maybe this new article should simply state that, from now on, with each one percent increase of population, a new member of the House will be added to the rolls. It could further stipulate that these new members would be added after the new figures from each census come in.\"\n\n\"But even with my proposal of having one member of Congress per 30,000 constituents,\" Jim says, \"if you take into account that we'd also reduce sharply the number of staff members, the overall change isn't so significant. If you multiply the number of congressmen, 435, by the number of staff each has, 22, that comes to 9,750 people. Rather than having an army of unelected staffers, there'd be so many more additional elected representatives who'd really get to know and explore the issues with their constituents. And this just may also decrease the influence of professional lobbyists, since they'd have to deal with legions of congressmen.\"\n\n\"I'm just not keen on the retroactive idea,\" Mary says. \"My counterproposal is that we have one congressperson for every 60,000 citizens, and that she be allowed two staff members. That still would put us at about 5,000 members in the House, a huge increase and a big step towards more representativeness.\"\n\n\"How would you go about redrawing congressional districts every ten years to make space for all these new members of the House?\" I ask.\n\n\"You know, there's no mention in the Constitution of how congressional districts are to be drawn up, only that there shall be at least one member of the House per state,\" says Jim. \"For this project of yours, we should come up with a constitutional mechanism for creating districts, one that replaces the politically charged way its done now. Gerrymandering usually stacks the deck in favor of one party over the other, and almost always makes types of artificial boundaries that don't serve the interests of 'representativeness.' Rather than creating districts that gibe with the contours of a region, as they should, you have these schizophrenic lines drawn that ensure one party will dominate. Often neighborhoods are split right in half to load a district with voters who'll support the House members in office there.\"\n\n\"I'd also agree with the idea that congressional districts be drawn up in a uniform way in each state,\" says Mary. \"Let's have this constitutional article state that districts for the House will be determined by decennial census results, with the boundaries for each district determined by a nonpartisan group of citizens of each state. This will take the decision-making authority out of party hands.\"\n\n\"That gets my vote,\" says Jim. Then he adds with a laugh, \"Of course, if an article like this really came to be, odds are greater that I may not have lasted even ten years, since my district is artificially drawn so it's stacked with Republican voters. I never imagined I'd be blindsided by my own party.\" He pauses. \"But who knows, maybe if my district had been more rationally drawn in the first place, I'd still be in office.\"\n\nI eventually say, \"You both seem to think that 'representativeness' can best be had by increasing the number of Congress's members. But the winner-take-all majority voting system encourages our two-party system. Would a parliamentary form of representation lead to better or truer representation of our diverse citizenry? Elected officials in countries constitutionally bound to use this system typically have a far greater number of political parties with elected representatives, especially in the lower chamber of government.\"\n\n\"I suspect that if we markedly increased the number of congressmen, it'd just about ensure there'd be more political parties represented in the House,\" says Jim. \"For one thing, it'd make it possible for candidates of small political parties\u2014or no party at all\u2014to have a good chance of winning, since they'd be able to get their message out in a cost-effective way. In addition, if each member of Congress represented just 30,000 or 60,000 constituents rather than 600,000, there'd be greater voter turnout, because people would know their vote really mattered, especially if districts were drawn up in nonpartisan fashion. So that greater voter turnout would also help a really dynamic candidate who's outside the traditional two-party umbrella.\"\n\n\"It'd open up creative ways of campaigning,\" Mary says, \"and probably break the grip on corporate funding of campaigns. This might take some of the sting out of the Supreme Court's recent ruling against campaign finance reform. Having such intimate representation might make candidates less beholden to the special interests that fund their campaigns, since so much less money will be needed in order to run effectively for office.\"\n\nMary sighs. \"Dare I even dream of a day when I don't have to spend more of my waking hours raising money for my next campaign than I devote to my constituents?\"\n\n\"Now that Mary and I are in agreement about this particular issue,\" Jim eventually says, \"I'd also like to suggest that the Constitution state that members of the House\u2014and of the Senate too, for that matter\u2014rotate office every election term. Democracy would be better served if they had to sit out a term before they could run for office again. This'd prevent the advent of entrenched politicians.\"\n\n\"Wishful thinking,\" says Mary.\n\n\"Isn't that what this thought experiment is all about?\" Jim says. \"To imagine what, in our eyes, a 'dream Congress' would be like?\"\n\n\"I suppose, in the 'dream Congress' scenario, that might be ideal for the American people,\" says Mary. \"Nor would I be opposed to the rotating office scheme Jim proposes if a House term was three rather than two years. It would give you time to make a real mark.\"\n\n\"When I lost reelection, it was hard to deal with at first,\" says Jim after a pause, \"but now there's no question it was for the best, for me and my family, and really, for my constituents too. I didn't realize until I was defeated in the primary that I was already burned out by the system.\"\n\nHe goes on to say, \"I'm now president of a community college, which is situated in the heart of my former congressional district. I make much more of a discernible impact now. But these days I'm the one pounding at my congressman's door, asking for help securing more funding for my college.\"\n\nHe then says, \"I know by name every student on my campus. When I was in the House, I tried to know the names of all my constituents, but I couldn't even know all their faces, no matter how many town hall meetings I had. When it becomes that impersonal, members of Congress shouldn't be called 'representatives,' since they can hardly know who or what they represent.\"\n\nConstitution Caf\u00e9 Article: The number of representatives in the House shall be one for every sixty thousand people. Representatives shall serve a three-year term in office. Members of the House and Senate can vie for re-election only after taking a one-term hiatus at the end of each term served. Congressional districts shall be drawn up every ten years, after the release of the decennial census, by a non-partisan committee of citizens in each State. The decennial census shall also determine how many additional Members of Congress will serve in the House.\n\nNew Rules\n\nAfter reviewing the newly created Constitution, Jefferson registered his disappointment over the Framers' \"abandonment in every instance of the necessity of rotation in office.\" He believed that if there were no term limits for our elected representatives, once installed in office, each would have the potential to become a veritable \"officer for life,\" and as such detrimental to democracy. To Jefferson, the Framers' failure to mandate an exit strategy for members of Congress was one of the Constitution's pronounced shortcomings.\n\nHe remedied this defect in his own draft constitution for Virginia, which included an article that called not just for term limits, but for rotation in elective office. While Jim made a similar proposal during our exchange, Jefferson's article was more strict: it stipulated that elected officials could serve only one term, and also created staggered terms:\n\nOne third of them shall be removed out of office by lot at the end of the first three years and their places be supplied by a new appointment; one other third shall be removed by lot in like manner at the end of the second three years and their places be supplied by a new appointment; after which one third shall be removed annually at the end of every three years according to seniority. When once removed, they shall be forever incapable of being re-appointed to that house.\n\nJefferson also detailed a unique and stringent scheme to provide effective checks and balances between the three branches: \"The Legislative, Executive and Judiciary offices shall be kept forever separate; no person exercising the one shall be capable of appointment to the others, or to either of them.\" Once you'd held office in one of these branches, you were barred from holding office in another. Jefferson believed that if a constitution allowed elected representatives to traverse the various branches of government, it encouraged a game of political musical chairs among entrenched officeholders. The result was that it discouraged new, worthy candidates from aspiring to elective office. Yet Jefferson himself served as Virginia's governor and as a legislator; but he would not have been able to do so if his draft constitution had been implemented.\n\nRepublic Guaranteed\n\nFrom Article IV, Section 4, of the Constitution: \"The United States shall guarantee to every State in this Union a Republican Form of Government.\"\n\nEach year, the week before Labor Day, a mecca is made of a stretch of Nevada desert as people the world over converge to attend Burning Man, named after a wooden effigy that somewhat resembles a male and that is put to the torch on the eve of the event's final day. Started in 1986 on a San Francisco beach as a modest bonfire ritual during summer solstice, Burning Man has grown into an annual eight-day extravaganza held in the Black Rock Desert. Organizers characterize it as an experiment in radical self-reliance that serves as \"a specific antidote for the passivity, anonymity, and alienation of modern society.\"\n\nAfter landing in Reno, I make the two-hour drive to the event site, traversing a dust devil\u2013plagued dirt road for the final mile. On this seven-square-mile dry lakebed known as \"the playa,\" almost 50,000 people are clustered. After finding a campsite, I realize that amid my mountain of gear, I somehow have forgotten the most important items: sunscreen and a pillow. No worries. People are more than willing to lend a clueless \"first-time Burner\" anything lacking.\n\nUpon pitching my tent, I wend my way to one of the site's two \"civic plazas,\" where I set up my dog-eared Constitution Caf\u00e9 sign. The first two \"Burners\" who join me had read on my website about my planned gathering here. They stand out in a crowd: he calls himself Uncle Sham, and is dressed in an Uncle Sam\u2013like outfit, superimposed with greenbacks being flushed down a papier-m\u00e2ch\u00e9 commode; she calls herself \"Lady Libertine\" and leaves little to the imagination in her choice of outfit\u2014the opposite of demure.\n\n\"Section 4 of Article IV of the Constitution has always nagged at me,\" says Uncle Sham, an estate planner in his life in what Burners called \"the default world\" that exists outside Burning Man's bounds. \"It says, 'The United States shall guarantee to every State in this Union a Republican Form of Government.'\n\n\"But it doesn't guarantee that this republic will be of a democratic form. All a republic in and of itself guarantees is a constitutional government, an ordering of society under a written law. But just because you live under written law doesn't mean it'll be democratic. A republic can just as well be a dictatorial or totalitarian state. A revised article can promise the states a democratic republic. I suppose this is more or less the form of government under which states operate. I just want this article to explicitly guarantee that states are composed of governments that have the consent of the people, that operate by the principle of majority rule but that at the same time protect the inalienable rights of every person, not just the majority.\"\n\n\"This article should go a bit farther and guarantee states the option of having a direct democracy,\" says Lady Libertine. \"That way, people can directly involve themselves in domestic policy issues if and when needed. Right now, twenty-four state constitutions give citizens the right to engage in direct democracy, by allowing them to put legislative referenda on the ballot. This article in the U.S. Constitution should guarantee what nearly half the states already let people enjoy, the right to direct democracy. Without this guarantee, the argument can be made that it's unconstitutional for states to allow us to practice elements of direct democracy, since it isn't modeled at the federal level.\"\n\nThen she says, \"My home state of California is the only state that has a type of direct democracy that doesn't permit the legislature to amend or repeal laws passed by a citizens' referendum. So it's really direct democracy. This gives California's citizens ultimate authority, so if the legislature behaves badly, we can literally take the law into our own hands.\"\n\n\"California's legislature has made a colossal mess of the state's budget,\" she goes on, \"spending sums of money that were way beyond the rate of inflation and the state's rate of population growth. And it masked over the shortfalls for years with accounting tricks. But the day of reckoning finally came, leading to painful cuts, especially in education, where our once-unrivalled public university system has been reduced to a shambles.\n\n\"The problem is that the state constitution calls for a two-thirds majority to raise taxes or pass a budget. This 'super majority' leads to a 'super minority,' a tyranny of a minority. Since the majority party needs the support of some members of the minority party in order to pass a budget, legislators from the opposition party hold the budget hostage until, in exchange for their vote, they're promised all kinds of earmarks for their districts. The inevitable outcome is huge budget deficits. I've joined the effort to change this system. There's a referendum initiative right now that will make the budget approval process one in which a simple majority is needed from both houses to pass the budget or to raise taxes.\"\n\n\"A direct democracy is no guarantor of a responsible democracy,\" says Uncle Sham, also from California. \"The people are every bit as responsible as our state legislature for California's budget mess. I'm supporting a referendum that'll do away with Proposition 13.\"\n\nPassed in 1978, Proposition 13 dramatically lowered property taxes and also restricted any future annual increases to 2 percent, no matter how much real property values soared.\n\n\"If property taxes were at their real value,\" he continues, \"then we'd have had responsible budgets all along. California residents have long refused to allow their property taxes to be a fair reflection of their property's value. And the state's most poor and vulnerable are the ones who suffer the most from the enormous budget cuts then needed to curb the deficits that result. There's no perfect solution, but all things considered, I believe a democratic republic\u2014with majority rule, but protections for the minority\u2014rather than a direct democracy, is our best chance of creating a responsible and responsive politics in state government, and this is what should be promised in this article.\"\n\n\"The organizers refer to this place as the 'Republic of Burning Man,'\" I say eventually. \"It makes no more promise of providing a 'democratic republic' to the people attending this event than the article we're discussing in our Constitution.\"\n\n\"Burning Man is a breakaway republic, even if a temporary one,\" says Henri, who drove here from southern New Mexico with seven others in a decades-old VW van, and just joined our gathering a few moments earlier. \"It's beyond democracy, which is all about voting and campaigning and arguing over party platforms and all that nonsense. This is the real deal, the res publica, 'the public thing.' This baby breathes. Burning Man has put into place those conditions that allow for the greatest creativity. On the other hand, the republic guaranteed by Article IV suffocates, because the type of republic it promotes has distrust and disdain for real participation by real people. You can keep that original Constitution that guarantees an ossified republic cryogenically sealed in the National Archives, but all that does is keep it from dying a natural death.\"\n\n\"Maybe all republics have written laws,\" he goes on, \"but the best ones keep them to an absolute minimum, as is the case with Burning Man, which has rules or laws that liberate people. Our republic here is minimalist. It's fine with me if the Constitution promises a 'minimalist republic,' and leaves it at that. I agree with Jefferson: 'That government is best which governs least.' In that spirit, we have here just a short list of written rules for governance drawn up by the organizers. As the Burning Man powers that be put it in their manifesto, there are 'no rules about how one must behave or express oneself at this event (save the rules that serve to protect the health, safety, and experience of the community at large).' Instead, it's 'up to each participant to decide how they will contribute and what they will give to this community.'\"\n\n\"Which means that a minimalist republic like this allows you to decide how to contribute, but doesn't give you the option of not being involved?\" I say.\n\n\"It probably wouldn't enter anyone's mind not to contribute,\" says Jan, a political science\u2013history major at a university in Portland, Oregon, chiming in for the first time. \"People come here because it's a great experiment in the art of creative governance, with this desert bed itself our canvas. As far as the eye can see, Burners have built great and not-so-great works of art as ways of expressing what this place means to them. They do it on behalf of the Public Thing, out of inspiration, not constitutional coercion.\"\n\nShe considers further. \"So I guess this article should guarantee a libertarian republic, which is in line with what we have here. Yes, there are written rules, but they inspire us to act together out of desire rather than coercion. We try to delight and please one another and look out for one another. I've discovered I'm my most fulfilled as an individual when I'm contributing to something greater than myself. This is a far different kind of individualism than the 'rugged individualism'\u2014which is really all about every man getting ahead at the expense of everyone else\u2014that exists in the default world.\"\n\n\"When Jan gave you a pillow, after learning you forgot to bring one, it was a gift,\" Lady Libertine says to me. \"She knew you lacked something you needed to fully experience and participate in this event. What you don't know is that she didn't give you an extra pillow. She just told you that so you'd accept it. She wanted to make sure you have a rich experience. You looked to her like a pillow would help make that happen. That's what this place does. By helping you, she helped herself too. Because if you get a decent night's sleep, you can participate with great energy; it'll benefit her and all the rest of us, and help this 'public thing' flourish.\"\n\nJan, a bit embarrassed by what Lady Libertine reveals to me, now says, \"On the website for this event, it says, 'Burning Man encourages the individual to discover, exercise and rely on his or her inner resources.' But that makes it sound like we're roughing it in an 'every person for herself' way, when in truth we look to 'outer resources'\u2014other Burners\u2014and rely on them to help us develop the inner resources needed to thrive in harsh desert conditions. The public thing here is many hearts beating as one. That's what should be guaranteed in the default world too. That would put an end to the dominance of the 'private thing,' where the two major parties, in league with the corporations they're beholden to, pass laws that benefit a few at the expense of the rest of us.\"\n\n\"We Burners operate from the assumption that when we're living in a natural state, there's trust, harmony, cooperation, an innate desire to help one another. Because this is the right kind of res publica, people want to participate,\" says Henri. \"Here at Black Rock Republic, it's about trusting one another to conduct ourselves in ways that inspire and elevate without coercion. We don't have any elections, which are coercive because they force you to vote 'for' someone and 'against' someone else. Instead we have committees, all made up of volunteers, that take on public works of all kinds\u2014public safety, public transportation, public art, you name it. They're spearheaded by a bunch of people out of a shared desire to make a great spontaneous republic, and make it like the work of art it should be.\"\n\nUncle Sham has stayed silent while others rave about the joys of the Republic of Burning Man. He says now: \"I for one wouldn't mind if the rules for the Burning Man event were set forth in the form of a constitution, and that it include the promise of an element of democracy. For instance, we're not allowed to bring pets with us. We're told by organizers that it's for public safety reasons, but we should be able to have debates about whether pets are or aren't a hindrance to safety but actually enhance the safety and enjoyment of this event, and then put it to a vote. But there's no process that I know of that allows for such a debate over the rules.\"\n\n\"There is a hierarchy,\" says Marlene. \"At the very top, there's an executive policy-making body made up of just six people. I do find their name\u2014Limited Liability Corporation\u2014off-putting, but that's its name for legal reasons, I imagine. But at all the other levels of the hierarchy, you can have input.\" She reads from the Burning Man annual report: \" 'At every level of our organization, from policies created by the LLC to decisions made at the most immediate level of operations, we employ a consensus process. The interplay between consensus-making and a hierarchic structure of organization defines our Project.'\"\n\n\"Maybe, both here at Burning Man and in the default world, constitutions are needed that guarantee the 'interplay' of democracy and res publica,\" Jan says. \"I can understand why there has to be consensus-making, and why sometimes there have to be representatives, but a hierarchic structure isn't needed.\"\n\n\"I just believe that, at Burning Man and in the default world, a constitutional article is needed that guarantees democratic participation, or at least democratic representation, at all levels,\" says Uncle Sham. \"That will help keep the experience of governors and governed alike more open and experimental. Maybe then there won't be the rise, and irreversible fall, that all other republics in human history have had. If you want the public thing in America to breathe without life support, you need an article that provides such a guarantee.\"\n\nConstitution Caf\u00e9 Article: The United States shall guarantee to every State in this Union a Democratic Republic Form of Government that best promotes the res publica, the public thing.\n\nLean on Me\n\nIn a letter Jefferson wrote to his daughter Martha in 1787, he reflected, \"It is part of the American character to consider nothing as desperate; to surmount every difficulty by resolution and contrivance. . . . Remote from all other aid, we are obliged to invent and to execute; to find means within ourselves.\"\n\nWhat system of governance, to Jefferson, would best tap into and put to optimal use this singular American character? One of a decidedly minimalist bent?\n\n\"That which governs best governs least.\" One Constitution Caf\u00e9 participant attributed this quote to Jefferson. This oft-quoted Jefferson saying is employed by libertarians and conservatives who lay claim to his mantle. Yet Jefferson never said it.\n\nEven so, he certainly protested the vast bureaucratic apparatus that the Framers' Constitution bred, characterizing it as having \"more machinery of government than is necessary, too many parasites living on the labor of the industrious.\"\n\nShortly before he became president, Jefferson voiced his preference for \"a government rigorously frugal and simple.\" To John Ferling, an authority on early American history, what Jefferson meant by this was that government had to stave off \"the further encroachment of the executive branch\" and focus on \"retiring the national debt, eliminating a standing army and relying on the militia to safeguard internal security, and keeping the navy small, lest it drag the nation into 'eternal wars.'\"\n\nWhile Jefferson certainly hoped government could realize these objectives, he was no more an advocate of small government than he was of big government. Rather, he favored a singular sort of self-government that he believed would unleash untapped reserves of human ingenuity. Jefferson proposed that the Virginia constitution be amended to create what he called a ward system. Of all his envisioned reform initiatives for Virginia, Jefferson's proposal for a ward system represented \"the article . . . nearest my heart.\" He suggested dividing the commonwealth's counties into wards, each no more than five or six square miles, and so \"of such size as that every citizen can attend, when called on, and act in person. These will be pure and elementary republics, the sum of which taken together composes the State, and will make of the whole a true democracy as to the business of the wards, which is that of nearest and daily concern.\"\n\nEach of the wards, then, constituted a minirepublic. Every citizen would be involved in governance, though they would not all be involved in making decisions on the exact same matters. Rather, each would be delegated a specific area of responsibility and charged with representing the interests of all his other fellows to the state. Since all were involved in governing these minirepublics, Jefferson believed that taken together, this system constituted a true democracy.\n\nThe ward system, he maintained, saw to it that all the citizenry \"was thrown with all its momentum into action,\" with each person \"a sharer in the direction of his ward-republic . . . a participator in the government of affairs, not merely at an election one day in the year, but every day,\" with every man \"a member of some one of its councils, great or small.\"\n\nJefferson further believed that this type of grassroots governmental framework presented a nearly fail-safe way of thwarting any one group's efforts to wrest self-government from the hands of the people:\n\nThese representative organs, should they become corrupt and perverted, the division into wards constituting the people, in their wards, a regularly organized power, enables them by that organization to crush, regularly and peaceably, the usurpation of their unfaithful agents, and rescues them from the dreadful necessity of doing it insurrectionally. These little republics would be the main strength of the great one.\n\nRather than relying on three separate but equally powerful branches of government to keep constitutional tabs on one another, the various overlapping levels of government in Jefferson's intricate system would be equally vigilant over one another: \"The elementary republics of the wards, the county republics, the States republics, and the republic of the Union, would form a gradation of authorities . . . holding every one its delegated share of powers, and constituting truly a system of fundamental balances and checks for the government.\"\n\nFor Jefferson, only such a system would \"save the people from lethargy and inattention to public business,\" because all the people would be made an integral part of it.\"\n\n\"We had not yet penetrated to the mother principle, that governments are republican only in proportion as they embody the will of their people, and execute it,\" Jefferson remarked to Samuel Kercheval in 1816. He was referring to the first attempt by the framers of Virginia's constitution\u2014\"novices in the science of government\"\u2014to create a form of government that did justice to realizing the people's will but that in his view fell far short of the mark. Jefferson believed that if the ward system\u2014to him an ideal blend of democracy and republicanism\u2014became the modus operandi for governance in Virginia, the will of the people could at long last genuinely be exercised in the public arena.\n\nJefferson also hailed the ward system for the fact that, on momentous occasions such as his envisioned periodic rewriting of the Constitution, all citizens could step out of their particular representative roles and make a group decision on the same matter. When a particularly urgent matter came up and required the equal input of all, a statewide meeting could be readily called into order, with all citizens meeting \"on the same day through the state.\" In this way, they could \"at any time produce the genuine sense of the people on any required point, and would enable the state to act in mass.\"\n\n\"Alone among the influential political thinkers of the revolutionary generation,\" maintains Joseph Ellis, \"Jefferson began with the assumption of individual sovereignty, then attempted to develop prescriptions for government that at best protected individual rights and at worst minimized the impact of government or the powers of the state on individual lives.\" Jefferson believed the ward republic both maximized citizens' impact on government and protected their individual rights by demanding everyone's involvement. As Hannah Arendt contends, Jefferson was striving to create that form of government that allowed for \"an exact repetition of the whole process of action which had accompanied the course of the Revolution\"\u2014not the bloody war on the battlefield, but the peaceable and pleasurable process of joint decision-making that complemented it. Jefferson's premise was that \"the revolutionary spirit was not merely the spirit of beginning something new but of starting something permanent and enduring.\"\n\nNative American Way\n\nThe prominent Lost Generation author and American social revolutionary John Dos Passos, who singled out Jefferson as a principal influence behind his activism, asserts in The Head and Heart of Thomas Jefferson that Jefferson was \"convinced that those societies (as the Indians) which live without government enjoy in their general mass an infinitely greater degree of happiness than those which live under the European governments.\" Jefferson's admiration for Native Americans began as a child, when Cherokees often passed by his home on their way to Williamsburg. Over the years, he observed and interacted with a number of tribes. He marveled over how they organized themselves via small and intimate councils, and expressed his admiration for how they \"never submitted themselves to any laws, any coercive power, any shadow of government.\" He went so far as to say that they were the exception to the rule \"that great societies cannot live without government.\" Jefferson further lauded their \"vivacity and activity of mind,\" claiming it \"was fully equal to a white man.\" In observing one Native American orator, Jefferson \"was impressed by the Indian's use of words to make a noble display of his humanity, to move others.\"\n\nRichard Matthews, democratic theorist and noted Jefferson scholar, contends that Jefferson's observations of Native American societies informed his notion that man was by nature \"a social, harmonious, cooperative, and just creature.\" To Matthews, Native American societies provided for Jefferson \"an edifying counter-cultural example of how men can live in a community without the need for formal government,\" demonstrating their inherent \"cooperation and human sociability, rather than . . . competition and human antagonism.\"\n\nIn fact, Jefferson did not believe competition was at odds with cooperation; rather, he held that if the two forces worked together, they could drive human sociability and creativity. He appreciated Native Americans' \"superior luster at oratory, debate and persuasion, as witnessed in their councils,\" and observed that their impassioned exchanges, far from generating antagonism, led to the forging of stronger bonds among them. What clearly mattered to Native Americans was that each was equally recognized by his fellow tribesmen, and that each had a chance to argue his point and to try to win over the rest.\n\nMatthews goes so far as to characterize the Native American way of living as \"the empirical model for [Jefferson's] political vision.\" While Native American societies were certainly one important contributor to his vision, they were by no means the model.\n\nThe Greek Way\n\nWhile Jefferson's system of wards was also clearly inspired by the New England township model, he ultimately aspired to \"resurrect the Athenian ideal in America\" in order \"to re-create the intimacy of thought and endeavor . . . that he was always to characterize in Greek terms.\" Jefferson called these Greek societies \"truly Attic societies,\" in which \"men's highest and most charitable impulses could flourish.\"\n\nPaul Woodruff notes in First Democracy that in the intimate democracy of the Athenian polis, each participant was \"given a share of the ability to be citizens, and that ability is understood both as a pair of virtues and as a kind of citizen wisdom.\" This kind of governing was based on the shared view that \"it is a natural part of being human to know enough to help govern your community.\" Woodruff claims that what enabled Athenian democracy to flourish was the fact that \"every citizen is called upon to assist in managing public affairs,\" since they believed that \"no expert knowledge . . . governs decisions of the state.\" Likewise, Jefferson believed that it was vital that all American citizens have a direct role in governing the country. To accomplish this, the government did not need to be decentralized per se, but to be universally dispersed: \". . . the way to have good and safe government, is not to trust it all to one, but to divide it among the many. . . . What has destroyed liberty and the rights of man in every government that has existed under the sun? The generalizing and concentrating all cares and powers into one body. . . .\"\n\nJefferson's ward system was meant to be an adaptation of the polis, literally \"people place,\" made to fit the unique conditions of Virginia, but also, he believed, replicable on a far vaster scale. Hannah Arendt claims that living in a polis is akin to living in a political community in which, as in the Native American societies that Jefferson observed, \"everything [is] decided through words and persuasion and not through force and violence.\" But Jefferson could never persuade the Virginia Assembly to entertain seriously his proposal for a ward republic.\n\nJefferson's great expectation was that in post-Revolutionary America, a system of governance would be developed that emulated that of the Greek polis, giving political enfranchisement to all white male citizens, and as a result doing away with the tradition of governance in which a select coterie did the ruling for everyone else. \"How soon the labor of men would make a paradise of the whole earth, were it not for misgovernment, and a diversion of all his energies from their proper object\u2014the happiness of man,\u2014to the selfish interests. . . . If we can prevent the government from wasting the labors of the people, under the pretense of taking care of them, they must become happy.\"\n\nTo him, the best way to prevent government from wasting people's labors was to create a form in which all people labored in it. \nX\n\nRights and Responsibilities\n\nIn 2001, John McCain, John Edwards, and Ted Kennedy introduced in the Senate a bill to create a Patients' Bill of Rights. The bill was aimed in part to give patients a stronger voice in decisions about which medical procedures will be covered by insurance, and about health care-related privacy issues. While the Senate approved the legislation on a roll-call vote, in which every member's vote is recorded, it was never passed by the House. Since then, a number of states have taken the lead in passing their own versions of a Patients' Bill of Rights\u2014not to mention a Passenger Bill of Rights, to ensure we're not stranded on airport tarmacs for hours on end, a Cell Phone Bill of Rights to guarantee our calls aren't dropped in the middle of clinching a big business deal, even a Children's Outdoor Bill of Rights, to see to it that kids receive ample outdoor exercise. And Congress itself is considering a Technology Bill of Rights, to ensure that high-tech products are accessible to all consumers in the digital age.\n\nThe implication is that we'll be able to truly enjoy what's promised in our original Bill of Rights only if we spell out more nuanced rights in these new bills of rights. The more bills of rights, the merrier. But with these rights come responsibilities.\n\nAs our founding patriots knew, a bill of rights is barely worth the paper it's written on unless you're willing to defend it. Once a set of rights has the force of law, if someone\u2014perhaps the very enforcers of the law themselves\u2014then attempts to usurp any of these rights, you may be obliged to risk a great deal in order to preserve or restore them. That was the case when the rights guaranteed by England's Magna Carta were wrested from the New World colonists by King George. Patriots risked their lives to have them restored.\n\nYet once the American Revolution was won, the Framers of our Constitution had no plans to include a Bill of Rights. James Madison, for one, insisted that it went without saying that all the constitutional rights and powers not delegated specifically to the government belonged to the states and the people. In a December 1787 letter to Madison, however, Jefferson argued that a \"Bill of Rights is what the people are entitled to against every government, and what no just government should refuse, or rest on inference.\" He argued that such a bill was vital for \"providing clearly . . . for freedom of religion, freedom of the press, protection against standing armies, and restriction against monopolies, the eternal and unremitting force of the habeas corpus laws, and trials by jury in all matters of fact triable by the laws of the land and not by the law of nations.\" Siding on this occasion with the so-called anti-Federalists, Jefferson proved instrumental in swaying the Framers. On December 15, 1791, the first ten amendments to the Constitution, called as a group the Bill of Rights\u2014based primarily on the Virginia Declaration of Rights drafted in 1776 by George Mason when he was a member of the commonwealth's provisional government\u2014became part of our supreme law.\n\nBut even with this victory, as Jefferson well knew, these rights could still be undermined. Indeed, the grievances that Jefferson spelled out in the Declaration of Independence drew from the fact that the long-standing rights granted to all British subjects by the Magna Carta were being disregarded by King George III, who among other wrongs had imposed standing armies on the colonists, levied taxes without their consent, denied their right to trial by jury, and no longer allowed them to choose their own elected officials.\n\nThe very last grievance enumerated in the Declaration was that the king and Parliament had ignored all the colonists' petitions for redress of grievances. As Jefferson wrote, \"In every stage of these oppressions we have petitioned for redress in the most humble terms: Our repeated petitions have been answered only by repeated injury.\" This was one grievance too many, and Americans rebelled.\n\n\"With respect to our rights,\" Jefferson later explained, \"and the acts of the British government contravening those rights, there was but one opinion on this side of the water . . . When forced, therefore, to resort to arms for redress, an appeal to the tribunal of the world was deemed proper for our justification. This was the object of the Declaration of Independence.\"\n\nToday, as then, Americans of many political stripes claim that government is violating their rights, and that their complaints to elected officials have been answered only by \"repeated injury\"\u2014specifically, that their petitions either prompt retaliation by the government, or fall on deaf ears.\n\nRedress Address\n\n\"The last right granted in the First Amendment, to petition the government for redress of grievances, was supposed to be a piece of 'democracy insurance,'\" says Patricia, an unemployed auto-parts manager. \"Now, it's just an empty right.\"\n\nHer offering meets with nods all around, even from her ardent political opposites. I'm holding a Constitution Caf\u00e9 near the entrance to Independence Hall in Philadelphia, the site of the original Constitutional Convention, held from May 25 to September 17, 1787. I am with a number of Tea Party and Coffee Party members who ventured here from Pennsylvania and nearby states.\n\nThe Tea Party movement got underway in the waning days of the second term of George W. Bush. It started out as a one-issue protest group voicing annoyance with the proposed economic stimulus plan, which it considered an unconstitutional subsidy. After Barack Obama took office, the Tea Party burgeoned into a much more influential movement, and now professes to be the citizens' watchdog for all unconstitutional activities on the part of the federal government.\n\nThe Coffee Party is still gathering momentum. Claiming not to be wedded to any particular ideology, it promotes reasoned deliberation in the public sphere as the means to solve public issues. Like the Tea Party, it stresses its grassroots nature, and it too considers lobbyists anathema.\n\nI had gotten to know Patricia, a Tea Party member, well before that movement was on the political radar screen, when she and I had taken part together in a Socrates Caf\u00e9 dialogue I'd held in the area. When I told her of my desire to hold a Constitution Caf\u00e9 exchange between Tea Party\u2013 and Coffee Party\u2013goers, instead of telling me right off, as had all others with whom I'd inquired into the possibility, that this was wishful (or nightmarish) thinking on my part, she said she could arrange it, and was even enthusiastic about doing so.\n\nPatricia now looks at Dave, a Coffee Party acolyte and social worker. \"Dave and I managed to have a civil conversation at\u2014where else\u2014a coffee shop. We'd bailed out of a 'town hall meeting' held by one of our representatives. He'd claimed the purpose of the meeting was to solicit our suggestions, but really, he was telling us what he planned to do regardless of what we thought about it. Eventually the meeting disintegrated into a shouting match. Dave said to me at the time, 'You know, I don't blame people for screaming. It's because they're totally left out of the process.' \"\n\nShe goes on, \"Dave and I went on to discover that we both had presented petitions for redress of grievances.\"\n\n\"I'd delivered to my representative a petition demanding that Congress craft a new campaign finance reform law to replace the one overturned by the Supreme Court when they kowtowed to K Street's bagmen,\" says Dave. \"The hundreds who signed it agreed that it was important to attempt right away to make a new law that would withstand future efforts by the Court to allow big corporations and unions to again control our elections.\"\n\n\"While mine was a one-issue petition,\" he continues, \"Patricia's was pages long. I actually agreed with some of the Tea Party grievances\u2014that the federal government has burdened Americans with debts and obligations that can't possibly be repaid; that the system of taxation places unjust burdens on some Americans while exempting others; and that the executive branch has usurped Congress's authority by involving our troops in one undeclared war after another.\"\n\n\"Our main shared grievance was that our petitions to the government for redress of grievances were ignored, violating our constitutional right,\" Patricia tells me.\n\n\"I became active in the Tea Party after I lost my job\u2014laid off permanently by GM as part of its requirement to restructure and cut costs after receiving stimulus dollars. It was bad enough that Congress had abused its taxing authority when it passed the 'stimulus package' that stimulated GM to get rid of me. But I couldn't understand why the CEO could continue to earn millions and I was expendable, and why the stimulus package\u2014which included my own taxpayer dollars, without my consent\u2014allowed this. I had plenty of time on my hands, and as I became more enlightened about all the government's wrongdoings, and how they were forbidden by the Constitution, I became more active.\n\n\"At first, I tried to make my grievances known as an individual citizen. I emailed the one of my senators I thought could most relate to my grievances. I got an immediate response from his email server saying his mailbox was full. I tried calling. One of his administrative assistants said he'd relay my concerns to the senator. It was only after several more fruitless attempts that I decided to seek out others who shared my concerns, so we could band together to make our views known. I collected thousands of signatures. I then went straight to Washington, D.C. I couldn't get past security to go into the Senate office building, though professional lobbyists entered and exited like it was their second home. I admit I didn't have an appointment, but surely at least one of my senator's staffers could've come outside to take my petition.\"\n\nNow Mary, a Tea Party activist who'd been unemployed for two years before landing a government consulting contract the previous week (she says she had to \"hold her nose\" while accepting it, but that work is work, and she has three children to support as a single mom), speaks up. \"This 'freedom to petition clause' in the Constitution is supposed to give us direct access to whichever authority in government we feel can best redress our grievance. It's meant to guarantee that Americans' gripes with those who govern can't be ignored.\n\n\"As I'm sure you know, it was the failure by the king of England to redress colonists' grievances that led to the American Revolution. The Magna Carta gave British subjects wherever they lived the ironclad right to petition government for redress of grievances. But when King George III arrived on the scene, he ignored this right.\"\n\nIn 1215, the Magna Carta gave British subjects for the first time the right to petition the monarchy for redress of grievances. It detailed a process for realizing this right, with twenty-five barons charged with addressing a subject's petition. The Magna Carta further guaranteed that there could be no retaliation against a petitioner while his grievance was considered. In 1669, British Parliament expanded this right, giving every commoner, \"in case of grievance,\" \"the inherent right to prepare and present petitions.\" In this updated process, the House of Commons was charged with determining whether the petition merited redress, and if so, would have to take concrete steps to resolve it. The American Revolution was launched after the Crown prohibited its appointed governors in the New World from considering the colonists' petitions, effectively refusing to adhere to their own foundational laws.\n\n\"King George did worse than just refuse to address petitions,\" Dave says now. \"Whenever the colonists tried to petition him, he retaliated. He directed Parliament to enact more severe restrictions, restraints, taxes, every time colonists complained that their rights were being violated.\"\n\nPatricia then says, \"When the Bill of Rights became part of our Constitution, the right to petition government for redress of grievances was put right upfront. At our nation's beginning, U.S. citizens could go directly to either chamber in Congress to hand over their petitions. And a hearing would be held on their grievance. It didn't mean that a decision would be made in favor of the petitioner or group of petitioners, but the petition had to be received and considered, and the legislators had to issue a thoughtful response.\n\n\"But during the Civil War era, when Congress was flooded with petitions from abolitionists demanding an end to slavery, it refused to receive any more of them. Members of Congress complained that they had so many petitions to consider that they didn't have time to conduct their regular legislative business. They said they didn't want the public to dictate their agenda. But that's Congress's job, to be steered by the concerns of the people. It used the slavery issue as a pretext to end the right of citizens to petition it directly.\n\n\"Ever since, the right to petition has been diluted more and more. Today, it's equated with the rights to peaceable assembly and to free speech, as if there's no difference. Free speech and assembly may let you shout from the rafters, but they don't guarantee that the people in government you're directing your grievances to will be anywhere nearby listening to what you have to say. Only the right to petition the government official to whom your grievance is directed\u2014and the right to receive a response that addresses your grievance\u2014guarantees that you'll be both seen and heard.\"\n\nTo which Darcy says, \"Right now, the First Amendment states, 'Congress shall make no law . . . abridging . . . the right of the people . . . to petition the Government for a redress of grievances.' I suggest we rewrite it so it says, 'Congress shall make no law abridging the right of the people to petition directly the Government for a redress of grievances.' \"\n\n\"That doesn't go nearly far enough,\" says Mary. \"It also has to say, following that, that 'Congress shall elaborate a process whereby petitions for redress of grievances are addressed.'\"\n\n\"What sort of process do you envision?\" I ask. \"If Congress and the executive branch are flooded with tens of thousands of petitions for redress of grievances, how would they handle them?\"\n\n\"Congress could have a well-staffed permanent committee\u2014a Committee for Redress of Grievances\u2014with a rotating membership to handle petitions directed at either or both chambers,\" says Dave. \"And the White House also could set up a special office for receiving and processing grievances. This is such an important part of democracy that there should be a cabinet officer\u2014a secretary for redress of grievances\u2014overseeing it. And whenever it becomes clear that a broad cross-section of petitioning Americans share grievances over certain issues, the cabinet officer could inform the president, and he would personally have to respond. But whether the grievance is large or small, an appropriate government official would have to directly receive the petition and respond.\"\n\n\"Way back in the thirteenth century,\" says Patricia, referring to a book on the subject that she has on hand, \"King John of England wrote that only 'when redress has been obtained' shall his subjects 'resume their old relations to us.' In other words, relations between the king and his subjects were put on hold until the petition for redress was considered. So, in line with that, a clause in the article we're composing should also state that relations between the government and those petitioning for redress of grievance are at a standstill until the grievance is considered and responded to.\"\n\n\"People on the left and right and everywhere in between want to revive this freedom to petition for redress of grievances at all levels of government,\" says Mary. \"But the more they try, the harder federal and state governments come down on them. The federal government is considering a law that would require all would-be petitioners to register as professional lobbyists. As it is, over half the states have imposed restrictions on grassroots efforts to petition.\n\n\"Now, as you can imagine, I don't usually agree with the American Civil Liberties Union. But in this case, it is absolutely right in arguing that if this bill ever passes, it will be a disaster for our democracy. It would mean that if you or me or any other concerned citizen who tries to mobilize others and urge them to sign a petition related to an issue of shared concern, you'd have to register as a professional lobbyist with the government. You'd lose your status as a private citizen and be treated on a par with someone on K Street.\n\n\"We'd have to provide to the federal government all personal information\u2014such as name and address, contact information, occupation\u2014about ourselves and anyone who has contributed even a small amount to our effort. But what's most cause for concern is that the government wouldn't just collect and store this information\u2014it would report it publicly. So if there's an issue that's dear to someone as a private citizen, she might not sign a petition out of fear that the government will publish information about her, and that her employer, who might differ with her stance, will find out and it could affect her job.\"\n\n\"It only gets worse,\" says Patricia. \"If this bill becomes law, the government would monitor our grassroots activities. We would have to file monthly reports with the government. All of this, taken together, is retaliatory. It's every bit as much an attempt to intimidate us as what King George did to New World colonists. It makes citizens afraid to be activists. Government officials of course know that we aren't anything remotely comparable to professional lobbyists. We don't represent special interests. We represent a general interest, of taking back our democracy.\n\n\"I personally wish the Constitution would outlaw professional lobbyists, who don't petition for redress of grievances, but create further grievances for regular Americans. The Framers never anticipated lobbyists, and so of course they never dreamed they'd be the primary influence over our politicians.\"\n\nDarcy sighs. \"We're right back to where our original patriots were. We have a government that hates and fears ordinary citizens, and that's determined to keep us from practicing this most important right, so it can continue to act unconstitutionally.\"\n\nDave says now, \"Jefferson anticipated this: 'crush in its birth the aristocracy of our moneyed corporations, which dare already to challenge our government to a trial of strength and bid defiance to the laws of our country.' Our elected officials are so tied to the moneyed corporations that they band together to defy the laws of our country.\"\n\n\"The media makes us Tea Party types out as fringe protesters, as if you're either part of a movement of protest or part of a movement of participation. But that's a false dichotomy,\" says Mary. \"Protest can be the highest form of participation in a democracy. And gathering a petition to deliver to government to have grievances redressed is an act of protest and participation. Really, the right to petition the government for redress of grievances is the right to be recognized.\"\n\nConstitution Caf\u00e9 Bill of Rights, Article 1: Congress shall make no law abridging the right of the people to petition the Government for a redress of grievances. Congress shall make laws that establish a process by which grievances are received, considered, and addressed. Until such time as grievances are appropriately addressed by Government, relations between those citizens submitting the grievance and the Government shall be considered suspended, and there shall be no retaliatory measures taken by Government against petitioners.\n\nGood Grief\n\nThe first attempt to eliminate the right to petition for redress of grievances took place just seven years after the Bill of Rights was ratified. In 1791, enactment of the Alien and Sedition Acts effectively eliminated this right, since citizens who expressed a grievance against the government typically were arrested, jailed, and fined by courts sympathetic to the Federalists.\n\nThomas Jefferson and James Madison warned in a draft of the Kentucky and Virginia Resolutions (or Resolves) that the government's breach of the most fundamental rights \"unless arrested at the threshold . . . will furnish new calumnies against republican government.\" To them, it was especially important that, during those times when an open society was weathering putative threats from within or without, the government should redouble\u2014not stifle\u2014efforts to redress grievances. If it instead took the route of a King George III or a John Adams and closed off long-enjoyed outlets for airing and resolving citizens' concerns, it was only serving further to destabilize democracy.\n\nLet Your Conscience Be Your Guide\n\n\"Maybe James Madison didn't believe at first that there was a need for a bill of rights, but he went on to become one of its greatest advocates,\" Bart tells us. \"He even wanted to include in it the right to exercise freedom of conscience. His initial draft of the First Amendment contained this clause: 'nor shall the full and equal rights of conscience be in any manner, or on any pretext infringed.' But it was voted down. For this new Constitution, let's vote it back in.\"\n\nA first-year law student at my and Jefferson's alma mater, the College of William & Mary, and a sometimes attendee at my local Socrates Caf\u00e9 group, Bart and several of his friends join us for this exchange. We're meeting on the William & Mary campus in an outdoor amphitheatre that was declared a 'free speech zone' by the Student Assembly, after a new regulation established by the college administration placed restrictions on where students could hold protests.\n\nI ask the obvious: \"What is freedom of conscience?\"\n\n\"It's the freedom to think and believe what you want, to go your own way and never have to agree with anyone if you don't want to,\" Bart says. \"Now, it doesn't mean you can act on those beliefs. A blanket 'freedom of expression' isn't allowed in the Constitution, only freedom of speech, which is just one form of expression. Exercising free speech in and of itself isn't a violent act, no matter how outrageous or controversial. But other acts of expression can be violent.\"\n\n\"Freedom of speech isn't the only way we exercise freedom of conscience,\" says Amy, who attends nearby Christopher Newport University in Newport News, Virginia, my father's alma mater. \"The First Amendment freedoms of assembly, of religion, of the press, also are forms of free expression that encourage or at least enable us to live lives of conscience.\"\n\n\"You first have to have a conscience in order to exercise freedom of conscience,\" says Sasha, a cadet at Virginia Military Institute, the oldest state-supported military college in the United States, who has been silent up till now. \"Too many in America today speak their minds in ways that aren't conscientious.\n\n\"Many in positions of influence use their right to free speech not to liberate people but to drum up animosity. Theirs aren't acts of conscience. So many on the right and the left have ramped up the hate, demonizing those who disagree with them. After health care reform was passed by Congress, death threats were made against Democrats who voted for it, and some had their homes and cars vandalized. Many who used this unconstitutional form of freedom of expression seemed to be incited by the extremist rhetoric of political leaders.\"\n\n\"Democratic leaders also have their share of blame,\" says Amy. \"Nancy Pelosi, the former House Speaker, once characterized all who used their free speech to loudly oppose the Obama administration's health care plan as 'simply un-American.' Since when can't Americans loudly oppose something? What matters is that, for or against, you've made a free and informed and well thought-out\u2014a conscientious\u2014decision. On the other hand, those who try to intimidate or demean those who don't think like them, and those who intentionally incite people to carry out violent acts, are exercising lack of conscience.\"\n\n\"Even Obama joined the 'un-American' parade,\" says John, from the University of Virginia. \"Once, while promising to find a way for illegal immigrants to become U.S. citizens, he warned that there are 'demagogues out there who try to suggest that any form or pathway for legalization for those who are already in the United States is unacceptable.'\n\n\"I disagree that if you oppose a pathway for legalizing the status of undocumented workers, you automatically are a demagogue, and as such merit the 'un-American' label. Even if I don't arrive at the same conclusions, I've heard some reasonable arguments for why illegal immigrants shouldn't be allowed to stay in the U.S., much less become citizens.\"\n\n\"Obama may have won the Nobel Prize for Peace,\" Sasha says, \"but when he made that statement about demagogues, he wasn't being peaceable, much less conscientious; he was using his bully pulpit as president to intimidate, to discourage reasoned debate. Maybe some people who oppose naturalization of illegal immigrants are demagogues. But you can be for immigration reform, and be a demagogue. And a revised First Amendment should emphasize that the rights it includes are specifically meant to serve as the best ways to exercise freedom of conscience.\"\n\nEventually, Reed says, \"The imam at the mosque where three of the 9\/11 hijackers worshiped called the alleged Fort Hood shooter, Major Nidal Malik Hasan, a supreme 'man of conscience.' Others in the U.S. who've engaged in violent hate-based acts\u2014like Scott Roeder, who murdered George Tiller, a physician who performed late-term abortions\u2014have been praised as people of conscience by extremist religious leaders. The problem with a right to freedom of conscience is that it can be misconstrued as not only the freedom to believe what you want, but to act on those beliefs, no matter how violent and hateful. Such people in my view have the worst kind of conscience, but they see themselves in just the opposite light.\"\n\n\"Now I'm starting to appreciate why freedom of conscience was eliminated from the final version of the First Amendment,\" says Sasha. \"It's too easy to claim you're exercising freedom of conscience, and yet do something horrific that sets back what an open society is to be all about.\"\n\nTo which Amy says, \"We should leave the First Amendment just as it is. We should leave freedom of conscience out of the picture. Because it can erroneously imply that we have the right to exercise even the most violent forms of freedom of expression.\"\n\nBart isn't convinced. \"Freedom of conscience is all about exercising good conscience, promoting reasoned debate. A person of conscience in a democracy would never act in a way that would deliberately try to negate other voices. So to be on the safe side, this article should stress 'freedom of good conscience,' rather than just a blanket freedom of conscience.\"\n\n\"You know,\" Reed says, \"the people I admire most throughout our nation's history are those who didn't just feel free to believe whatever they wanted, but were willing to act on their beliefs, even when their actions were outside the law, to improve our society. They often carried out acts of civil disobedience in the name of making the ideals in our Declaration of Independence come true. They exercised all the existing First Amendment rights to one degree or another, putting their beliefs on the line, to make sure all Americans were recognized equally. Along the way, they faced down conscienceless people, many of whom committed terrible acts to try to intimidate or silence them.\"\n\nThen Bart says, \"Another Framer, Elbridge Gerry, also tried to insert language in the First Amendment about freedom of conscience. His proposed draft said simply, 'Congress shall make no laws . . . infringing the rights of conscience.' Like Madison's attempt, his clause also unfortunately was eliminated in the final version of the Bill of Rights, even though it made clear that 'rights of conscience' aren't unlimited, but are tied directly to the unabridged rights to freedom of speech, freedom of assembly, freedom of the press, freedom to petition for redress of grievances. Maybe people would appreciate these rights more if the First Amendment stressed that they're meant to promote those constitutionally guaranteed freedoms that amount to exercise of good conscience.\"\n\nConstitution Caf\u00e9 Bill of Rights, Article 2: In order to promote freedom of conscience, which celebrates human dignity and worth, Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.\n\nConscientious Conscience\n\n\"No provision in our Constitution ought to be dearer to man than that which protects the rights of conscience against the enterprises of civil authority,\" Jefferson told a group of Methodists in 1809, referring specifically to the exercising of rights of conscience as they pertain to freedom of religion. Clarifying this viewpoint further in his Notes on Virginia, Jefferson explained, \"rights of conscience we never submitted [to any authority] . . . We are answerable for them to our God.\" He stressed that \"the legitimate powers of government extend to such acts only as are injurious to others. But it does me no injury for my neighbor to say there are twenty gods, or no God. It neither picks my pocket nor breaks my leg.\"\n\nBy this perspective, while one is free to believe in God or gods or no god, such rights of conscience are permissible so long as they remain in the realm of pure belief and do not inflict any real, physical harm on anyone. Jefferson elucidates his view by drawing on a quote from the Enlightenment political thinker Montesquieu's Spirit of the Laws: \"Words carried into action assume the nature of that action. Thus a man who goes into a public market-place to incite the subject to revolt incurs the guilt of high treason, because the words are joined to the action, and partake of its nature. It is not the words that are punished, but an action in which words are employed.\"\n\nYet Jefferson's own words, in the Declaration of Independence\u2014a manifesto for freedom of conscience if ever there was one\u2014incited Americans of virtually every religious and political outlook to revolt against the British. If his words had not been backed up by deeds, they would have been empty.\n\nJefferson made clear in a correspondence with James Madison following Shays's Rebellion that when people are oppressed by the very government that is supposed to ensure their liberties, they might have to resort reluctantly to violent means in order to have their grievances addressed: \"I hold it that a little rebellion now and then is a good thing, and as necessary in the political world as storms in the physical.\" While an uprising like Shays's Rebellion may be put down, and while it hopefully is a rare occurrence, Jefferson believed it will have served a good purpose if it brought to light \"the encroachments on the rights of the people which have produced them,\" and then leads to a just resolution.\n\nDespite his rhetoric to the contrary, there are exceptional instances in which Jeffersonian freedom of conscience cannot be confined to the realm of pure belief; at such times, rights of conscience are converted into action in order to realize promised or hoped-for rights, or to restore those which had been taken away.\n\nFree the Press\n\n\"I like that the First Amendment is so general about press freedom,\" says Scott, one of several journalists meeting with me at a caf\u00e9 that they frequent in northern Virginia. \"Sandwiched almost inconspicuously between freedom of speech and freedom to peaceably assemble, it simply says that Congress 'shall make no law that abridges . . . the press.' Very concise, but it packs a wallop. This little clause assures broad press freedom.\"\n\n\"But it's only saying that Congress can't keep us from doing our job,\" says Kathleen, an assistant editor at a daily paper. \"In redoing this clause, I'd have it state that while Congress can't make laws abridging press freedom, it can make laws that better enable the press to function optimally.\"\n\nDaniel, a freelance investigative reporter, nods. \"This would force Congress to enact a blanket federal shield law. Right now, thirty-four states have some type of shield law that affords us, to varying degrees, some protection from being forced to reveal our sources. But on the federal level, there's no shield law. Though the Senate is in the process of crafting one that provides some limited protections to journalists, even if it passes, it can be rescinded at any time. As things stand, a reporter faces the prospect of serving time in prison if required by a federal court to reveal his sources on a story. But if Congress was constitutionally required to enact laws that further free up the press to do its job, a blanket federal shield law would be a given.\"\n\n\"What if someone posts classified or other sensitive government documents on a websites like WikiLeaks.org?\" I say, referring to the websites that publishes such material typically in order to expose alleged government or corporate misconduct, and that protects the identities of its sources. \"Would he be afforded protection under this expanded free press clause?\"\n\n\"Let me put it this way: I like that the Framers of our Constitution didn't attempt to define 'the press,'\" says Daniel. \"Since the beginning of our nation's history, the press has basically been any medium that disseminates information. So a site like WikiLeaks would constitute 'the press' to our Framers, and so yes, it would be protected.\"\n\n\"The Democrats in the Senate are drafting a 'media shield bill' to protect so-called traditional news-gathering agencies from revealing their sources,\" he goes on. \"But the bill wouldn't extend these protections to websites like WikiLeaks. That's setting a dangerous precedent. It's not just that more nontraditional sources than ever will emerge in coming years as the traditional press continues to disappear, and eventually will become the mainstream. The fact is, WikiLeaks clearly is a news-gathering source. It is applauded for giving people unexpurgated information and has earned a number of respected international media awards for advancing the cause of human rights. Without WikiLeaks, there'd be few places for the Daniel Ellsbergs of the world today to reveal government misdoings, now that corporate-owned papers are so averse to taking risks that would impact that financial bottom line.\"\n\nIn 1971, Ellsberg, a U.S. military analyst working as a private consultant, released a top-secret report on U.S. government decision-making regarding the Vietnam War. A retired Marine Corps officer who'd served a stint in Vietnam, Ellsberg had high-level government security clearance. The documents to which he had access revealed that government officials were lying in their public statements claiming that the war in Vietnam was being waged successfully\u2014with victory, as Secretary of Defense Robert McNamara had put it, \"just around the corner\"\u2014and also that the officials had not disclosed that the war was causing far more U.S. military casualties than it had publicly claimed. The \"Pentagon Papers\" that Ellsberg passed on to the New York Times also made clear that the Johnson and Nixon administrations felt no ethical, much less constitutional, compulsion to be up front with the press or the American public.\n\n\"People should have access to a wide variety of sources of news,\" says Scott. \"The press\u2014whether WikiLeaks or the National Enquirer or a traditional newspaper\u2014usually is disparaged when it does its job particularly well and prints uncomfortable truths that the government or a politician or other public figure doesn't want people to know, because it might change public sentiment. So this expanded article can also clarify that the press is any medium that disseminates information, to make sure that Congress doesn't limit what constitutes the press as it sees fit. Because if it doesn't define broadly what the press is, Congress will define it in the most limited way possible.\"\n\n\"Now, the issue with WikiLeaks,\" he goes on, \"isn't whether it's the press\u2014of course it is. The issue is whether, in publishing classified information about our incursions in the Middle East, it has been endangering national security, and whether the press freedom clause in the Constitution should place restrictions on this. In the latest case, it posted online a ream of classified documents about how the war against insurgents in Afghanistan and Pakistan is bleak and getting bleaker all the time.\"\n\nIn late July 2010 alone, 92,000 military reports from 2004 to 2009 were leaked to WikiLeaks.org as well as traditional news sources, and tens of thousands more have been leaked since.\n\n\"These reports contradict some of the spin put out by the Bush and Obama administrations that the situation there is stabilizing,\" he continues. \"The question is whether, in its zeal to carry out its stated mission of creating more government transparency, WikiLeaks has compromised our national security, even putting lives of American troops at risk. If that's the case, should this press clause grant unlimited press freedom, even if the press potentially endangers national security?\"\n\n\"If a member of the press, or any other citizen, passes along information that enables a potential enemy to do harm to our nation, there's an article already in the Constitution\u2014Article III, Section 3\u2014that states that such treasonable acts will be punished,\" says Kathleen. \"It specifies what are treasonable acts: 'levying War against [the United States], or in adhering to their Enemies, giving them Aid and Comfort.' So the government already has the constitutional right to try a reporter or a press outlet for treason. But it would have to prove that there was an attempt to intentionally give aid and comfort to the enemy or to assist in fomenting an uprising against the U.S.\"\n\n\"Where in the existing Constitution does it give the government permission to have classified information?\" says Christie, a reporter for a national radio program. \"It seems to me that an outcome of writing a press clause that incorporates the language we're discussing is that Congress would have no choice but to pass laws that help us better do our job. So, it would have to pass a law that either prohibits classified information, or that allows some members of the press, in its special role as the eyes and ears of the public, to freely access and review such information.\"\n\n\"Obama is just the latest president to issue an executive order making certain government information classified, supposedly because it's too sensitive to national security to be revealed,\" she continues. \"I believe the argument can be made that the government provides aid and comfort to the enemy when it makes ours an increasingly closed society, what with its penchant for keeping secrets just because it can. Obama is going after anyone who reveals so-called secrets to the press with a vengeance, even if they don't endanger our security.\"\n\nIndeed, the New York Times reports that \"the Obama administration is proving more aggressive than the Bush administration in seeking to punish unauthorized leaks. . . . President Obama has already outdone every previous president in pursuing leak prosecutions.\"\n\n\"The argument can be made that national security is even more at risk when certain classified information is withheld from the press and public,\" says Kathleen. \"Classified information is an inherent abridgment of freedom of the press.\"\n\n\"If at least some members of the press had access to classified information, because of an expanded free press clause,\" she goes on, \"one outcome would be that the government couldn't so easily mislead us in military matters. For instance, if the press had had timely access to information about the Gulf of Tonkin incidents, the U.S. might not have been able to escalate its involvement in Vietnam. This might have prevented our nation's involvement in that conflict, which I believed weakened rather than strengthened our security.\"\n\nThe Gulf of Tonkin incident took place on August 2, 1964. It was reported by U.S. military officials to have involved two separate clashes between American destroyers and North Vietnamese torpedo boats. The military claimed that in the initial clash, the North Vietnamese had fired first. Two days later, a second naval skirmish was reported by the Navy to have taken place. This prompted Congress to pass the Gulf of Tonkin Resolution, giving President Lyndon Johnson carte blanche authority to involve the U.S. military in defending South Vietnam against the North Vietnamese. Forty-one years later, the National Security Agency at last declassified a study that showed that the United States had in fact been the first to open fire during the initial incident, and that showed that the second reported clash simply never happened.\n\n\"You can't seriously believe that giving reporters access to any and all classified secrets is realistic,\" says Scott. \"Clandestine activities by the CIA should only be known by a handful of government officials, or it could endanger their activities and U.S. interests abroad.\"\n\n\"Correct me if I'm wrong, but in the sixties and seventies the CIA's covert activities helped topple democracies throughout Latin America, setting back democracy in that region,\" says Christie. \"It orchestrated the Bay of Pigs fiasco. In nations like Iran, it propped up dictators, allowing them to stay in power until finally the situation imploded and fundamentalists took over. Then there's the CIA's 'slam dunk' case that there were Weapons of Mass Destruction in Iraq.\n\n\"So in balance, surely it's better that some press members are kept abreast of their activities in 'real time,' if nothing else so they are constantly reminded that America is supposed to be the beacon of the free world, and as such, needs to operate as much in the open as possible.\"\n\nThen she says, \"There will always be information that can never be printed. But there will be times when members of the press will have strong differences with the government over what should and shouldn't be revealed to the public. For instance, the government might argue that we can never reveal secrets about our nuclear arsenals, or about spies. But even in these highly sensitive areas, if the press ever discovers what it considers clear-cut evidence of unconstitutional shenanigans, then it would be obliged to report it. If there's ever a question over which side\u2014the press or the government\u2014has committed treason, then there's a court system in place that can decide the matter.\"\n\n\"But even with the language being discussed here for a constitutional article to expand press freedoms, it doesn't mean we'll be able to more effectively sway the public,\" says Daniel. \"It only means that the public will be better able to make more informed decisions, if it so chooses. No matter how much freedom the press has to do its job, the American public won't necessarily care more about the doings of government, even when far outside constitutional bounds. For instance, even though Watergate brought down Richard Nixon, polls at the time consistently showed that most of the public wasn't interested in Watergate.\"\n\nAfter a pause, Christie says, \"The more freedoms and protections extended to us, the fewer excuses we'll have for failing to do our job responsibly and wisely. If this expanded clause on press freedom really came to be, I believe it would enable the American public to do an even better job of keeping the government, and the press itself, more honest and aboveboard.\"\n\nConstitution Caf\u00e9 Bill of Rights, Article 3: While Congress shall make no law abridging the freedom of the press, it shall make laws that further enhance and secure press freedoms.\n\nKeeping the Peace\n\nIn 1787, as the delegates at the Constitutional Convention were meeting, James Madison and Thomas Jefferson contemplated \"whether peace is best preserved by giving energy to the government, or information to the people.\" Jefferson argued that the government must put all information it has at the people's disposal, so that they in turn can give their energy to the government in the form of educated and enlightened opinions on public policy. To him, the people not only are \"the only sure reliance for the preservation of our liberty,\" but they \"are the most legitimate engine of government. Educate and inform the whole mass of people. Enable them to see that it is their interest to preserve peace and order, and they will preserve them.\"\n\nJefferson told his friend Edward Carrington that the best way to prevent uprisings by the people \"is to give them full information of their affairs.\" He believed that when the government becomes secretive, the seeds for public unrest are sown. Jefferson further told Carrington that the best method of informing the public is \"through the channel of the public papers, and to contrive that those papers should penetrate the whole mass of the people.\"\n\nSo vital was the role of the press in preserving democracy, Jefferson contended, that \"were it left to me to decide whether we should have a government without newspapers, or newspapers without a government, I should not hesitate to prefer the latter. But I should mean that every man should receive those papers, and be capable of reading them.\" When approached by French leaders after the French Revolution for advice on framing their new constitution, Jefferson stressed the need for unfettered freedom of the press: \"This formidable censor of the public functionaries, by arraigning them at the tribunal of public opinion, produces reform peaceably, which otherwise must be done by revolution. It is also the best instrument for enlightening the mind of man, and improving him as a rational, moral, and social being.\"\n\nJefferson asserted that if the French followed his counsel, they would be breaking with the precedent set by all other European governments. Not only, in his estimation, was the European press habitually \"the first shut up by those who fear the investigation of their actions,\" but European governments regularly and deliberately spread misinformation. \"Under the pretense of governing,\" he claimed, \"they have divided their nations into two classes, wolves and sheep. I do not exaggerate.\"\n\nWhere would Jefferson stand if the press printed false facts\u2014facts fed to it by the federal government itself\u2014that damaged relations between the United States and other nations, as was the case with the Weapons of Mass Destruction episode in Iraq that prompted the U.S. to launch the first preemptive incursion in its history? In such a case, Jefferson would surely have argued that the government itself had libeled democracy.\n\nDespite his panegyrics to the press, Jefferson changed his tune when he became president. During his first term in office, Jefferson argued that it was the press rather than the government that was deliberately spreading misinformation and dividing the nation. He considered himself the victim of a relentless smear campaign orchestrated by pro-Federalist press moguls opposed to his policies; not only had they cast his government in the worst possible light, but the notoriously thin-skinned Jefferson accused them of sullying him personally. His vitriol against the press knew few bounds during this period: \"Truth itself is suspicious by being put into that polluted vehicle. The real extent of the state of misinformation is known only to those who are in situations to confront facts within their knowledge with the lies of the day.\"\n\nBut try as the press might to turn the public against him and his administration with its \"demoralizing licentiousness,\" Jefferson believed that the people were not swayed. He tried to paint himself as magnanimously taking the high road, and rather than attempting to punish the press for its alleged slander, he decided to \"leave to others to restore it to its strength, by recalling it within the pale of truth.\" He claimed to be confident that \"public indignation\" would bring the press back to its true calling.\n\nWhile president, Jefferson boasted to John Tyler, a judge whose son would become the tenth president of the United States, that his administration was successfully testing out an idea that had been \"wanting for the world to demonstrate\"\u2014namely \"the falsehood of the pretext that freedom of the press is incompatible with orderly government.\"\n\nYet Jefferson was no believer in a totally unfettered press. Long before he became president, he made it clear that he was against granting absolute press freedom. The article that he composed for the final version of his draft constitution for Virginia that dealt with press freedom stated, \"Printing presses shall be free except . . . where by commission of private injury they shall give cause of private action.\" His draft article further declared that the press \"shall be subject to no other restraint than liableness to legal prosecution for false acts printed and published.\"\n\nAfter the federal Constitution was approved, Jefferson told James Madison that while freedom of the press must be part of the Bill of Rights, he believed that such freedom should come with constraints: \"A declaration that the federal government will never restrain the presses from printing anything they please, will not take away the liability of the printers for false facts they printed.\" In another letter to Madison, Jefferson proposed specific language for a constitutional amendment pertaining to press freedom: \"The people shall not be deprived or abridged of their right to speak, to write, or otherwise to publish anything but false facts affecting injuriously the life, liberty or reputation of others, or affecting the peace of the [U.S.] with foreign nations.\"\n\nThis language is in one sense far more restrictive than the press freedom clause that came to be included in the First Amendment. On the other hand, it opens a potential Pandora's box, since it makes no attempt to clarify precisely what 'false facts' amount to, or a process for determining whether they are injurious, whether such injury applies in the same way to public officials as to private citizens, or whether the printed material would impact relations with another nation. Indeed, the overall tone of Jefferson's amendment is more reminiscent of the intimidating libel laws that he excoriated as part of the Alien and Sedition Acts, which were meant to keep people from speaking their minds. Jefferson's article certainly goes against his assertions as president that there was no need to level any punishments or create any restrictions against the press, since people were smart enough to see through any false claims.\n\nArmed and Safe\n\nThe Second Amendment of the Constitution: \"A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.\"\n\n\"We need to decide with this article whether individuals have the right to keep and bear arms. That right isn't made clear in the existing Second Amendment,\" says Mike, who conducts team-building seminars for businesses.\n\nI'm with about fifteen people in a sprawling Texas exurb who agreed to join me after receiving an email blast I sent out inviting people to review the Second Amendment, with the idea of possibly revising or overhauling it.\n\n\"It is clear,\" says Fred, a barber. He reads from his copy of the Constitution, \" 'A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms shall not be infringed.'\n\n\"So, we have the right to keep and bear arms. But this right is completely tied to being part of a militia regulated by the government. In the Revolutionary era, any male could be part of a militia, so any male had the right to bear arms while serving in his capacity as a militia member. Today this amendment means that any citizen can bear arms who is part of the modern militia\u2014the Guard and Reserve, but the other armed forces, too, I'd imagine. The right to keep and bear arms is totally tied to the defense of the nation as a whole. The Second Amendment includes no right for an individual to bear arms for his personal self-defense.\"\n\n\"Okay, point taken. But for our purposes here, it doesn't matter how anyone interprets the existing Second Amendment,\" says Mike. \"People, including Supreme Court justices, have been reading into this amendment anything they want to for over two centuries, interpreting it in a way that suits their agenda or their outlook.\"\n\nLooking at me, he then says, \"If I understand correctly Chris's project, what we have to do is decide what we'd want a constitutional article today to say regarding the right to keep and bear arms. Do we want this right? If so, do we want the blanket right? Or should it depend on where we live\u2014whether we live in an urban area riddled with gun-related violence, or a rural area where guns are used mostly for hunting?\"\n\n\"This article should state, 'The right of the people to keep and bear arms on their place of residence\u2014whether owned or rented\u2014shall not be infringed,'\" says Linda, a personal trainer. \"Everyone needs to have the means to defend hearth and home.\n\n\"This would remedy the deplorable situation today in which society's most vulnerable are unable to defend themselves. For instance, in New Orleans, survivors of Hurricane Katrina protested how the New Orleans government voted unanimously to raze a housing complex where they'd lived most of their lives. No one in government cared that the residents wanted it restored. They had gone to the expense of hiring an expert who'd shown that the complex was salvageable. It was their home. When they protested against its demolition, they were subjected to pepper spray, they were gassed and tasered. What they needed was the constitutional right to defend themselves with equal force, in order to defend their property.\"\n\nCBS News reported that one law professor at Loyola University involved in the demonstration held up a piece of taser wire that had been shot at another protester and asked ironically, \"Is this what democracy looks like?\"\n\n\"Nowadays, you might need a stockpile of weapons to defend hearth and home,\" says Fred, \"what with federal agents raiding people's homes with overwhelming force, and in instances mistakenly raiding the wrong residence.\"\n\n\"I agree that law-abiding citizens have the right to defend their home by any means necessary,\" says Mike. \"But I'm hesitant to put the word 'arms' in a modern article bearing on this issue. An arm could be taken by some to mean biological or chemical weapons, or even a handheld nuclear device.\n\n\"Even if we're talking about more conventional weapons, it's a delicate issue,\" he goes on. \"My grandparents on my mother's side had an arsenal. They lived in mortal fear that their home would be invaded and that they'd again be removed against their will. They were interned during World War II. When they died, my parents got rid of every weapon, all of them legally registered.\"\n\n\"And you believe such stockpiles should continue to be constitutionally permitted?\" I ask.\n\n\"I'm not sure. I don't know that their arsenal ever really made my grandparents feel safer. I'm still angry over what the government put them through, but arming oneself with weapons has never seemed to me the best way to go about addressing government abuses.\"\n\n\"I need to return to what I see as the original intent of the Second Amendment, since it influences my own view,\" says Kathy, a housewife and mother of four. \"Because I take this amendment to mean that only those in the militia can keep and bear arms, and because I believe that it is an indisputable truth that America today is one of the most violent societies in the Western world\u2014in big part because it's so easy to purchase weapons\u2014I would have this article state that no private citizen has the right to keep and bear firearms. That way, only those in their professional capacity to keep our society safe\u2014public safety officers, members of the Guard or Reserve, or of the armed forces\u2014can bear them. This article should say that they can't keep arms at their homes; they have to leave them at their base or station, where they will be stored in modern versions of magazines\u2014the places the colonists in militias stored weapons and gunpowder.\n\n\"This would build on the approach to arms-keeping and -bearing at our founding, when militias were well-regulated,\" she says. Then she is quick to add, \"When I use the word 'militia,' I'm not talking about these self-appointed groups that are misappropriating the name and claiming to do things like protect our borders or trying to secede from the nation. The only groups that should be called militias are those highly organized units strictly controlled by the state and federal government.\"\n\n\"I just believe that, precisely because American society is so dangerous today, every individual deserves to keep and bear arms on her property\u2014either on her land, or in her vehicle,\" says Linda. \"But maybe Mike is right in that the word 'arms' is kind of a Pandora's box. So I would have an article just say, 'The right of an individual to protect and defend his property by whatever means necessary shall not be denied.' \"\n\n\"Isn't 'any means necessary' a Pandora's box in its own right?\" I ask her.\n\n\"It is, but it's a box that you can only keep on your property and nowhere else, so the language I'm proposing should serve its intended purpose,\" she says.\n\n\"I want a nation without arms,\" Kathy says quietly. \"My neighbor's niece in Chicago was wounded several months back. Some neighborhoods in that city are war zones. Her niece was just walking to school and was caught in the cross fire of a gun battle over turf for selling drugs.\"\n\nThe New York Times reports that in 2009, \"258 public school students were shot in Chicago, 32 fatally, on their way to or from school, traveling through gang-infested territory and narcotics wars.\"\n\n\"The best way to protect our most important property, our selves, is to ban all arms in private hands,\" Kathy continues. \"A new constitutional article should say that only public safety officers can bear arms in the community they serve, and the military can only bear them on their bases or when defending the country.\"\n\nShe then says, \"Chicago's elected officials rightfully imposed strict handgun bans to stem violence in the city, only to have this law overturned by the Supreme Court.\"\n\nJust before our gathering, a bare majority of the Supreme Court issued a ruling that broadened Second Amendment rights, concluding that its protections apply to state and local gun control laws as well as to federal law.\n\n\"A new article has to take this out of the Court's hands, before our society becomes more and more like the OK Corral. I know some will say it's na\u00efve to think we can ban guns, that the gun culture is here to stay. But for this Constitution Caf\u00e9 project, we can imagine the world we want to have. And the one I want is one without arms of any kind in private hands. Honestly, I don't even want them in the hands of public safety officers, though it's a necessary evil.\"\n\nTurning to Kathy, Mike says, \"To me, what you say about the situation in Chicago goes to show that the Constitution should give local governments the right to determine gun control laws. It's an abomination that the Supreme Court struck down what for Chicago was a commonsense gun control measure to keep innocents from being victimized. I can see where there'd be lax rules in remote stretches in America. But I'm thinking now that each municipality's government officials should be able to determine its gun control laws.\"\n\n\"I was taught to shoot as a junior member of the National Rifle Association, and was a longtime member,\" says Carl, a retired high-school teacher, speaking for the first time. \"I got rid of all my weapons back in 1990. Without going into specifics, let me just say that it's best if guns aren't in our personal possession. I believe that every person is capable of killing\u2014and not just for reasons of self-defense, but out of desperation, hatred, a moment of extreme mental duress. In balance, gun possession makes us no safer. The issue of public and personal safety should be the guiding light upon which a constitutional article on this subject is based. For the sake of public safety, I'd ban them outright in private individuals' hands.\n\n\"There are no perfect solutions, and there'll always be bad guys. But hopefully we can keep guns out of the hands of most of them, and prevent further tragedies like the one in Chicago, not to mention Tucson. And if you have to defend yourself in a tight spot, for heaven's sake become an expert at one of the martial arts.\"\n\nThis group was evenly split between the following two articles:\n\nConstitution Caf\u00e9 Bill of Rights Article 4, Version 1: The right of individuals to keep and bear arms on their property shall not be infringed.\n\nConstitution Caf\u00e9 Bill of Rights Article 4, Version 2: Because a well-regulated National Guard and Reserve, and well-regulated federal, state, and local public safety departments, are necessary to the security of our free states and of our free nation, the right of citizens, while serving in their capacity in the aforesaid organizations, to keep and bear arms shall not be infringed.\n\nNo Farewell to Arms\n\nIn Jefferson's day, almost all men\u2014and many women, especially those living on the frontier\u2014could expertly use a gun for hunting and for self-defense. Jefferson advocated gun ownership; he believed it had a moral dimension, and so was important for cultivating character. As he told his nephew Peter Carr in 1785: \"A strong body makes the mind strong. As to the species of exercises, I advise the gun. While this gives moderate exercise to the body, it gives boldness, enterprise and independence to the mind. Games played with the ball, and others of that nature, are too violent for the body and stamp no character on the mind. Let your gun, therefore, be the constant companion of your walks.\"\n\nA gunsmith with an extensive armory, Jefferson further contended that if all citizens kept arms and were properly trained in their use, it would actually minimize gun violence. As he wrote to George Washington, while \"one loves to possess arms, . . . they hope never to have occasion for them.\"\n\nWhen Virginia's General Assembly held a convention in 1776 to write a new constitution, Jefferson proposed an article that granted a universal right to bear arms, rather than limiting this right to those serving in a well-regulated militia, as was the norm with other state constitutions. Jefferson's proposed article stated that \"no man shall be debarred the use of arms\"\u2014and he appended in brackets in his final two drafts \"within his own lands or tenements,\" as if his article might stand a better chance with the legislature if it specified where one might keep and bear arms.\n\nDon B. Kates, Jr., a scholar of constitutional and criminal law, argues that Jefferson's view is based on the tenet that \"arming the good citizen minimized the likelihood of violence by deterring the wrongs that would provoke it.\" To Jefferson, constitutionally sanctioning the universal right to bear arms would put the gentry on notice that even the poorest citizen had the means to defend himself against encroachments by the powerful. \"What country can preserve its liberties if its rulers are not warned from time to time that their people preserve the spirit of resistance?\" Jefferson told Colonel William Stephens Smith, an American diplomat who would eventually become a congressman for a district in New York. \"Let them take arms.\"\n\nJefferson's proposed article was defeated. According to historian Saul Cornell, who considers Jefferson \"perhaps the most forward-looking and innovative legal thinker involved with the framing of the Virginia constitution,\" the defeat of the universal right to keep and bear arms on one's property reflected \"a view common among members of Virginia's gentry elite that it was dangerous to arm the 'rabble.'\"\n\nCivics Lesson\n\nI'm at a McDonald's outside of a high school in Washington, D.C., with a group of 16- and 17-year-olds. The teacher who'd helped arrange my visit is an old friend; we attended elementary school together in the area, and we'd managed to stay in touch for most of the ensuing years. He thought his students would be more candid and comfortable if they met with me off-campus at their favorite after-school hangout.\n\nAt the outset, Natasha says to me, \"As Mr. Williams, our history and government teacher, was preparing us for this meeting with you, I put a question to him that he said I should put to you: Why doesn't our Bill of Rights say a thing about the right of every American child to receive an education? It gives us the right to vote once we're eighteen, but I don't see how this right does us much good if we haven't been well educated.\"\n\n\"Even if the Constitution doesn't give us this right, there are laws in every community in the U.S. that make it mandatory for kids to attend school starting at about age four or five,\" says Hugo.\n\n\"But they can change these laws anytime they want to unless the Constitution states that we have the right to education,\" Natasha counters.\n\n\"But even if they never change these laws, the thing is, just because you have to go to school doesn't mean you'll learn much,\" says Zachary. \"And even if there was a law that said you could never drop out of school, no matter how badly you're doing or how bored you are, it doesn't mean your life will be any more hopeful.\"\n\nNatasha thinks about this. \"What we need in the Constitution is the right to the kind of education where it hardly would occur to anyone to drop out.\"\n\nThen she says to me, \"We watched during class time the so-called 'schools speech' that President Obama delivered live via satellite from the White House to students across the country. I got envious when he started bragging about how his kids are receiving a 'world-class education.' That's the kind of education we all should have the right to.\"\n\n\"How would you define a world-class education?\" I ask.\n\nAll the students on hand have a copy of President Obama's schools speech, which they're using to complete a homework assignment. Referring to his copy, Hugo says, \"I'll answer your question like this. The president said to us students in this one passage, 'You'll need the knowledge and problem-solving skills you learn in science and math to cure diseases like cancer and AIDS, and to develop new energy technologies and protect our environment. You'll need the insights and critical thinking skills you gain in history and social studies to fight poverty and homelessness, crime and discrimination, and make our nation more fair and more free. You'll need the creativity and ingenuity you develop in all your classes to build new companies that will create new jobs and boost our economy.' \"\n\nHe looks at me, \"He's defining what a world-class education is all about. But our teachers don't have the resources to give us all these knowledge and problem-solving skills he talks about. And a lot of them don't have the training to give us these insights and critical-thinking skills, all this creativity and ingenuity, he says we have to have. And even if they did, they wouldn't have time to teach them to us, because they have to spend so much of class time preparing us to take standardized tests. But these tests reveal next to nothing about who we are, what we can do, what we know or can know, what our dreams are. So they sure don't provide the basis for a world-class education.\"\n\n\"I bet President Obama's kids don't spend all their time preparing for performance tests,\" says Veronica. And then: \"If you want to know if a school offers a world-class education, you can tell by whether President Obama would send his own kids there. The Washington, D.C., public school officials reached out to the president when he was just moving into town here with his family. They tried to sell him on sending his kids to a D.C. public school. But you know that he never gave a serious thought to it. His daughters attend a private school that's surely world-class. I found out that it costs more to go there than my parents earn combined in a year. I'd need four parents at least to pay the bills if I were going to go to a school like that\u2014two to pay for my school and two to earn pay for the rest of our expenses. I'd need even more parents if my brothers and sisters were going to be able to go there too.\"\n\n\"I don't blame the president and first lady for sending their children there,\" says Natasha. \"I just want to be able to send myself there too. My kind of Constitution would give me that right.\"\n\n\"It's hard for me to get too pessimistic about my situation as long as I have teachers like Mr. Williams,\" says Zachary. \"Just because you go to an expensive school with all the trimmings doesn't mean your education is totally world-class. Part of a world-class education is not learning in a bubble. If you go to a swank school, you're probably insulated from the world. Mr. Williams, though, believes a world-class education is one where the entire world is your class.\"\n\nNatasha nods. \"We were studying in Mr. Williams's class about the civil rights movement. I finally got up the nerve to tell Mr. Williams that as much as I appreciated what he was trying to impart, I just couldn't relate. A lot of other students said they agreed. He just stared at us, then stared out the window for a while. He finally said to us, 'You need to take a trip to a place that will bring the past to life, and show you how connected you are to it.' He told us that we were going all the way to the National Civil Rights Museum.\" The Memphis, Tennessee, museum is housed in the Lorraine Motel, where Dr. Martin Luther King, Jr., was assassinated on April 4, 1968.\n\n\"We got so excited,\" Veronica tells me. \"But then he said we'd have to raise the money ourselves. School funding is so tight there isn't a penny even for local field trips, much less a trip all the way to Memphis.\"\n\n\"Of course, I was the first to look for the negative,\" says Natasha. \"I said, 'Where are we going to get that kind of money from?' Mr. Williams just smiled and said, 'Where there's a will, there's a way. And believe me, we're going to find a way, so don't even think about not having the will.'\"\n\n\"We did fundraisers, washed cars, had bake sales,\" says Hugo. \"I even played my trombone out on a street corner, and my trombone case filled up with donations. These activities really brought us together as a class. A lot of parents pitched in too. All who could, gave something, and most who couldn't, gave something too.\n\n\"We still didn't have enough funding to support the trip, even with all the effort we put into it, so Mr. Williams paid for the rest out of his own pocket. Well, he never admitted he did. He claimed that some anonymous donor had made up for the difference. But we all knew who that anonymous donor was. And last time I checked, high-school civics teachers don't make a lot of money. But it meant the world to him for us to visit that museum. We rode a bus thirteen hours to get there.\"\n\n\"I cried during the museum tour,\" says Natasha. \"I stood on that same balcony where Dr. King stood when he was shot by a person blinded by hate. I listened there to his 'I Have a Dream' speech. It did just what Mr. Williams promised it would do: it brought the past to life for me, and made me feel part of it. As I stood out on the balcony where Dr. King was on that tragic day, I realized that a world-class education isn't something that's just handed to you. It's something you hand to yourself, by learning whatever skills you need so you can carry the torch for all those who tried\u2014and in some cases, like with Dr. King, died while trying\u2014to make your world a little more loving and helpful.\n\n\"I realized that if I don't make something good of myself, if I don't quit making excuses for not succeeding, it will reflect poorly not only on my country and myself and Mr. Williams, but also on Dr. King and all the other civil rights activists who risked everything so I could have a world-class future.\"\n\nReferring to her copy of President Obama's schools speech, Veronica says, \"The president said during his talk that America's story is 'the story of students who sat where you sit 250 years ago, and went on to wage a revolution and found this nation,' and that it's also the story of all the students who 'fought for civil rights.'\n\n\"He was saying that going to school and getting your diploma in itself isn't enough, that you have to 'learn and do' at the same time, and in a way that makes the world brighter.\"\n\nNatasha nods emphatically and says, \"So we decided not to wait till after graduating to start making something good of ourselves.\"\n\nShe goes on to tell me, \"In our neighborhood, a lot of people have had their homes foreclosed on, and have been evicted. One day I had neighbors, and the next they were gone for good. All over our neighborhood, there are empty and uncared-for homes.\n\n\"Well, I was walking down the street one day soon after we got back from our field trip, and I heard a lot of meowing coming out of an abandoned row house. I didn't know the family who'd lived there all that well, because their kids were so much younger than me, but I remembered they had a bunch of cats. I peeked through a front window, and sure enough, there were all these cats and kittens. They looked to be starving. I peered into some other foreclosed houses, and it was the same deal. Cats, dogs, in one case a bunch of hamsters, left behind. Now, I see to it that they're fed. I put food and water inside, usually placing it through the broken windows. Veronica and Hugo and Zack and some others here eventually began help me out.\"\n\n\"We paid for all the pet food out of money we earn from doing fundraisers,\" Hugo tells me. \"We've tried to find new owners for the pets. We put ads in papers and posters on telephone poles. So far, we've found homes for five cats and two dogs. The SPCA has taken all the other abandoned pets we alerted them to.\"\n\n\"Now we've built on that project, in a 'prevention-oriented' way,\" says Veronica. \"Now, all our classmates deliver pet food to people in our community who otherwise would have to abandon their pets. We didn't have the money to buy this food ourselves. Mr. Williams showed us how to apply for a community grant. We did that as a class project. I couldn't believe it when we received a grant. I just felt so much goodness and gratitude well up inside of me. I'd never thought before that anything I might be part of might be seen to have so much value.\"\n\n\"And so,\" Natasha says, spreading her hands wide, \"that's the story of how me and my classmates are already going about making a good mark in the world.\"\n\n\"I wrote to President Obama to tell him about this project of ours, and to thank him for inspiring me to do my part to make sure I have that world-class education,\" says Hugo. \"Now I need to write him again and tell him about this article we want put in the Constitution. I'll emphasize to him that if he really wants to make sure his own children have a total world-class education, then there's no time like the present for them to start making the world a better place.\"\n\nConstitution Caf\u00e9 Bill of Rights, Article 5: All children of school age in the United States of America shall have the right to a world-class education.\n\nNatural Aristocracy\n\nIn 1812, both of them out of politics, Thomas Jefferson and John Adams overcame years of estrangement and began to engage in a rich correspondence. Though they both remained largely political opposites, they did manage to agree on one thing\u2014that the United States' vibrancy over the long haul hinged on the cultivation of a noble and virtuous cadre of young citizens. But they parted company when it came to figuring out how this aristoi\u2014Greek for \"the best,\" when referring to the noble class\u2014is developed. Adams maintained that you were either part of the noble class by birth or you weren't. \"Birth and wealth together have prevailed over virtue and talents in all ages,\" Adams insisted.\n\nJefferson disagreed. He believed Adams was describing an \"artificial aristocracy,\" one \"founded on wealth and birth, without either virtue or talents.\" Jefferson considered the version of the noble class promoted by Adams to be \"a mischievous ingredient\" detrimental to democracy, and argued that \"provision should be made to prevent its ascendancy.\"\n\nJefferson favored a \"natural aristocracy,\" which to him embodied the only type of noble class grounded in \"virtue and talents.\" As he told Adams, \"The natural aristocracy I consider as the most precious gift of nature for the instruction, the trusts and government of society.\"\n\nTo cultivate the development of a natural aristocracy in Virginia, Jefferson had introduced, as part of the statute reform package put together by his Committee of Revisors, a piece of legislation called the Bill for the More General Diffusion of Knowledge. If it had been passed, he told Adams, \"work would have been complete,\" and the \"pseudo-Aristocracy\" in Virginia would have been eliminated altogether.\n\nThis legislation was designed to give all children and youth a world-class education. It was based on the premise, as Jefferson put it, that the opportunity for formal learning should \"not depend on the condition of life in which chance has placed them.\" Jefferson's proposal would provide a well-rounded education in the arts and sciences, and teach children how to apply the knowledge gained in concrete problem-solving skills for the betterment of their community, giving them the opportunity to nurture \"a good conscience, good health, occupation, and freedom in all just pursuits\" so they could \"work out their own greatest happiness.\"\n\nAs set forth in the bill, not only would all those of school age in Virginia be provided with a free public education, but those who displayed \"the most promising genius, [yet] whose parents are too poor to give them further education\" would be selected to\n\nbe carried at the public expense through the colleges and the university. \"The object is to bring into action that mass of talents which lies buried in poverty in every country, for want of the means of development, and thus give activity to the mass of mind, which, in proportion to our population, shall be the double or treble of what it is in most countries.\n\nIf this bill had passed, Jefferson felt certain that \"worth and genius would thus have been sought out from every condition of life,\" and that the nation's children would be \"completely prepared by education for defeating the competition of wealth and birth for public trusts\" that Adams endorsed.\n\nJefferson's bill, though, was \"completely defeated.\" The Virginia legislators, mostly made up of the planter oligarchy, were \"unwilling to incur that burden\" of paying taxes that would have supported the program. Their own children already enjoyed the best education money could buy, and they had no desire to extend this privilege to anyone outside their circle. They favored Adams's artificial aristocracy, since it favored them.\n\nDonald K. Sharpes, a prominent scholar on the history of education, notes that it wasn't until 1840 that some elements of Jefferson's scheme for universal public education were at last realized, when civic leaders such as Horace Mann, head of the newly formed Massachusetts Board of Education, \"fought for and won legislatively the principle of universal public education, and the establishment of the common school open to all children and youth.\" Education was \"the great equalizer,\" Sharpes contends, and \"Mann believed, like Jefferson, that ignorance and freedom in a democracy cannot survive together.\"\n\nIf Jefferson's premise is correct\u2014that ignorance and freedom cannot be bedfellows in a democracy\u2014then the Texas Board of Education's decision to eliminate all mention of Jefferson from history and social studies curricula discussing the most important political figures of the eighteenth and nineteenth centuries does not bode well. A majority of the board, taking issue with Jefferson's efforts to erect a \"wall of separation\" between church and state in Virginia, decided it preferred to keep students ignorant of the deeds of one of our most important Founders. The vote by the board was along party lines, with all Republicans opting to replace the study of Jefferson with material on religious thinkers like John Calvin and Thomas Aquinas. The conservative majority also demanded that textbooks used by Texas's nearly 5 million students from here on out explore major political ideas in terms of their relation to \"laws of nature and nature's God.\" They further stipulated that any reference to the word \"revolution\" be deleted, and that the word \"democratic\" be eliminated from references to the U.S. form of government. Unlike Jefferson, they believe that students should not be given the tools to think for themselves.\n\nHopi Way\n\nI've just arrived at the Hopi Reservation, located in northeastern Arizona in the region known as the Four Corners, where the boundaries of Arizona, Utah, Colorado, and New Mexico meet. When I lived in Phoenix a decade ago, I ventured often to this captivating region. The Hopi territory, known as Hopiland, has a population of about 12,000 dispersed over thirteen villages situated on three large mesas. Many continue to live as artisans, farmers, and shepherds, and practice the traditional rites, dances, and ceremonies. Descendants of the ancient Anasazi Indians of the Southwest, the Hopi are unique among Native American tribes; they never signed a treaty with the U.S. government, and so never accepted its dominion over them.\n\nI make my way to one of the few indigenous villages in the region that survived a devastating thirteenth-century drought. The community is governed autonomously, and refuses outside funding. The cluster of adobe homes is situated on steep cliffs overlooking stunning buttes and mesas, and seems carved out of the earth itself. I park nearby and walk to a local store in the hopes of striking up a dialogue there. On my way, I happen upon a wizened man carving a detailed figurine that looks to be a kachina, a type of religious icon used by Hopi elders to teach children about their religion. An adolescent girl is also carving. He waves me over and invites me to sit down. Then they return to their labor. \"He's the best,\" she says. Her name is Aponivi, which she says means 'where the wind blows down the gap.' It turns out she is his great-granddaughter.\n\nAfter sitting in silence for a while, I tell them about my project.\n\nEventually, both Aponivi and her great-grandfather, whose name is Len and who is 94, put down their work. \"Would you like some tea?\" he asks me. He goes inside his home and brings out a pitcher and three glasses. I marvel at the aromatic taste. \"It's made from an herb called greenthread,\" he tells me. \"It has healing properties.\"\n\nLen sips his tea and remains silent. He looks around. \"We need these mountains, animals, trees, water, to survive in this harsh climate. We've never written it down in a Constitution, but the Hopi believe that all things have rights, because they are all needed, all important.\"\n\n\"Should the U.S. Constitution give all other living and nonliving things the same rights as people?\" I ask.\n\n\"I don't know about the same rights,\" he replies. \"Maybe more rights, or different rights. Everything that 'is' is different, takes up different space, has a different role. If any part of the universe is not treated well by people, then people are not treating themselves well. They're doing damage to something put here by the Creator that we need to sustain us.\"\n\n\"Hopi don't have a word for 'nonliving,'\" Aponivi then says. \"All things are beings\u2014stones, trees, clouds, mountains, eagles. All are alive, all have spirit. They're all sacred, and all need to be taken equally into account by humans.\"\n\n\"This 'taking into equal account' is a commandment of Taiowa, Creator of the Hopi people. He gave us this land to live on,\" says Len. \"So whatever action we take as a tribe or as an individual, we have to think about how it'll impact our surroundings. We have to consider these mountains, the earth itself, but also the past, the present, and the future, since they're all part of the environment. We have to make sure our actions put everyone and everything first, not just ourselves, or harmony isn't preserved.\"\n\n\"Like, if we lose our most precious resource, water, we lose our way of life,\" Aponivi then says. \"We'll suffer, the earth will suffer. There's a ripple effect of suffering.\"\n\n\"Millions of gallons of our water are wasted because a private energy company is allowed to divert it,\" she goes on. Aponivi is referring to the Peabody Energy Company, previously the Peabody Coal Company, which has a contract giving it the right to divert water from Hopi land in order to transport coal extracted from the southernmost part of the Black Mesa in Arizona. \"Our springs are drying up. Without the water, we can't live our way of life as farmers and shepherds.\"\n\n\"If we still governed the traditional way, this couldn't have happened,\" says Len, \"because all of the people would have been consulted, and they would have made it known that the loss of so much of our water would do too much damage to our way of life.\"\n\n\"Hopi have never been roamers,\" he continues. \"We're mostly dry farmers and herders. Most other tribes, when they were forced to settle on reservations, were placed far from their traditional lands. We've always been in the same place. Our environment is semidesert. We used to all know how to live within the limits the earth here places on us. We accepted it, respected it, nurtured it, and in return, it did the same for us.\"\n\nThen he says, \"But our way of living and deciding took a turn for the worse in 1934, when I was 19. The Bureau of Indian Affairs produced a written constitution for the Hopi. The federal government didn't want to have to deal with each of our villages, which governed themselves separately. This constitution required all the villages to be combined. It also would force us to have a government with a chairman and vice chairman to govern Hopiland, and a council with representatives from each village.\n\n\"When it was time to vote on the constitution, most of us abstained. Our traditional ways of governing promoted harmony, and there was no reason to change them. We thought that by abstaining, we were rejecting the new constitution. But all the 'abstain' votes were counted as votes in favor, and it was approved. Like all the other villages, Old Oraibi was given a seat on the new Hopi council. But we've never put anyone on it. We don't want a 'representative.' Because democracy isn't about majority rules. You consult with everyone, hold a discussion until everyone agrees. If one person is left out, there's no harmony.\n\n\"How can one man, whether a president of a country or leader of a tribe, decide for everyone else?\" Aponivi says. \"Like, how can a president send men and women to war with a country far away, without consulting his people and the people there in the country being invaded?\"\n\n\"In the Hopi traditional way of governing,\" Len then says, \"one of the village elders, the kikmongwi, is the tribal and spiritual leader. Our kikmongwi are leaders because they listen to everyone, and help us find harmony among our differences. If they didn't, they wouldn't last long as leaders. Today, except for the traditional villages, the members of the tribal council are far more important than the kikmongwi. Harmony isn't taken into account.\"\n\nAponivi hesitates, but then says, \"But the villages that have council members have all accepted government money. Those villages have nicer homes. They have a good supply of drinking water, they have sewers and electric power.\"\n\n\"The fight over money has divided us,\" Len says. \"It's better not to be tempted. You forget how to use just the resources that are around you, and forget how to take care of yourself.\"\n\n\"Aponivi has good ideas,\" he tells me. \"She's traditional and modern. She believes the old and new can come together to create a new and maybe better harmony. To her, the Hopi tribal council is good for dealing with outsiders and the village way is best for insiders.\"\n\nEventually he says, \"Government must be faith-based. You can't separate spirit from politics. It's the only reason for politics. A Hopi's purpose for being alive is to fulfill the vision of Taiowa. A traditional Hopi doesn't try to achieve harmony for a day or a season, but for generation after generation.\"\n\nThen he says, \"Hopi means 'peaceful people.' Societies that practice violence and aggression, that forget harmony and ignore the practice of taking all things into account, come and go. We're still here. If we want to stay here, we have to remember who we are.\"\n\nConstitution Caf\u00e9 Bill of Rights, Article 6: All beings have the right to be taken into equal account.\n\nHarmony\n\nDonald Lee Fixico, in The American Indian Mind in a Linear World, writes that traditional Native American forms of governance are imbued with the ethos \"that all things are equal at creation. All things are related in the world and in the universe. Plants and animals are an important part of this Natural Democracy. Each . . . has a role and responsibility. As a result, there is a strong dependency on each and all things.\" Cultivating this outlook, he contends, lends itself to the creation of a \"community of togetherness.\"\n\nJefferson's first term as president came on the heels of a fierce campaign during which the rhetoric that flew from both Democratic-Republicans and Federalists sparked great divisions among Americans. During his inaugural address, Jefferson made extreme rhetorical overtures aimed at bringing back some semblance of togetherness. He even went so far as to say, \"We are all republicans, we are all federalists.\"\n\nLet us restore to social intercourse that harmony and affection without which liberty and even life itself are but dreary things. And let us reflect that having banished from our land that religious intolerance under which mankind so long bled and suffered, we have yet gained little if we countenance a political intolerance as despotic, as wicked, and capable of as bitter and bloody persecutions.\n\nIn a letter to Elbridge Gerry, Jefferson confided that restoring this \"harmony and social love\" was \"almost the first object of my heart, and one to which I would sacrifice everything but principle.\"\n\nJefferson maintained that at least in theory \"every difference of opinion is not a difference of principle.\" As long as all principled views are given the opportunity to \"purify themselves by free discussion,\" he claimed that democracy would flourish, \"leaving our horizon more bright and serene.\"\n\nFaith-Based Governance\n\nJefferson's controversial relationship with Christianity was grounded in the belief that only by engaging in unflinching questioning, which even included questioning the existence of God, could one arrive at truths that further revealed the will of God. Jefferson had a \"lifelong conviction that the God whom nature revealed and the religion which reason indicated were more believable than the God and doctrines of biblical revelation and dogma.\" Consequently, Jefferson \"hoped for the spread of a religion of reason and morality\" into all social dimensions. John Dewey maintains that Jefferson's faith \"had a genuinely religious quality,\" and that he believed he best carried out God's work by determining how the \"forms of government and law, even of the constitution, might and should change\" in order to better realize a society in which all men are \"endowed by their Creator with certain unalienable Rights, that among these are Life, liberty, and the pursuit of happiness.\"\n\nDespite his own faith, Jefferson believed this task was best accomplished by keeping religion out of the public domain.\n\nThe First Amendment of the U.S. Constitution mandates that \"Congress shall make no law respecting an establishment of religion.\" It guarantees that government will not establish a religion itself, or sanction the establishment of an official church, and it prevents the government from interfering in religious affairs. However, there is nothing in the Constitution that requires the separation of church and state.\n\nOnly the state constitution of Jefferson's Virginia established a \"wall of separation\" between church and state, as mandated in Jefferson's famous bill, the Virginia Statute for Religious Freedom. It was one of the very few pieces of Jefferson's comprehensive package to reform Virginia's legal codes that actually passed, and it was enacted in 1786.\n\nThe Virginia legislature had initially aimed to pass a bill permitting the establishment of many denominations. Jefferson didn't see a contradiction in accomplishing God's work by keeping religion out of government, and what his alternative bill indeed managed to present was \"a forceful anticlerical condemnation of dogmatism and religious coercion.\" To Rhys Isaac, Jefferson's bill had revolutionary implications; it was a celebration of \"the autonomy of the individual against the moral claims\" of the community and \"proclaimed individual judgment as sacred, sacred against the pressure of collective coercions\"\u2014specifically coercions of the religious variety.\n\nHowever, Isaac fails to mention that while this was indeed a signal victory, it was a partial one. In and of itself, this one bill did little to empower the people against the entrenched coerciveness of the established planter oligarchy. As a consequence, the far bigger wall of separation\u2014between the powers that be and ordinary citizens\u2014remained as impenetrable as ever.\n\nAlter and Abolish\n\nI'm seated on a bench in one of the resplendent gardens of Monticello, the 1,060-acre mountaintop home of Thomas Jefferson situated outside of Charlottesville, Virginia. With me are the members of the original Constitution Caf\u00e9 group. It's July 2. We chose this day to reconvene because this is really our nation's independence day. It is the anniversary of the day in 1776 when the delegates of the Second Continental Congress, who hailed from all thirteen colonies, voted to approve the resolution for independence introduced by Richard Henry Lee. From that day onward, American colonies considered themselves legally and officially free from Britain. The Declaration of Independence was published two days later, on July 4.\n\nAfter the resolution for independence was approved, John Adams, who attended the congress, predicted that the \"second day of July, 1776 . . . will be celebrated by succeeding generations as the great anniversary festival. . . . [and] solemnized with pomp and parade, with shows, games, sports, guns, bells, bonfires, and illuminations, from one end of this continent to the other.\" Though Adams was two days off the mark, one might argue that July 4 still deserves to be our official independence day because the Declaration of Independence was published on the 4th\u2014but in fact, most delegates didn't sign the document until August 2, 1776.\n\nOver a year has passed since I set off on my trek to construct a new Constitution. Those with me are perusing copies of the constitutional articles I've devised with people across the United States.\n\n\"There's still something missing,\" says Helen with her usual bluntness. \"It lacks a strong closing statement.\"\n\n\"There's another section of the Declaration of Independence,\" Sara eventually says, \"besides the part that's now in the new preamble that the high-school students in West Virginia put together for the Constitution that Chris has been working on. It would make a good closing statement. It's the section that says that in order to secure the rights to life, liberty, and the pursuit of happiness, 'governments are instituted among men, deriving their just powers from the consent of the governed.' It goes on to say, 'whenever any form of government becomes destructive to these ends, it is the right of the people to alter or to abolish it, and to institute new government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their safety and happiness.'\"\n\nThis brings Marjorie to say, \"Jefferson believed that if government didn't give ordinary people the equal right to participate, then it made pointless 'the whole object of the present controversy,' meaning the American Revolution. As he said, 'should a bad government be instituted for us in future, it had been as well to have accepted at first the bad one offered to us from beyond the water without the risk and expense of contest.' \"\n\nThen Alejandra says, \"That section of the Declaration that Sara referred to is basically giving people the right to rebel. Jefferson argued that all Americans\u2014well, all white male Americans\u2014of whatever social class, should have this right. He was the one founder who supported, at least in principle, Shays's Rebellion. In referring to it, he remarked that 'if the happiness of the mass of the people can be secured at the expense of a little tempest now and then, or even of a little blood, it will be a precious purchase.' \"\n\n\"I take this to mean that if, at any point in our history, most people come to conclude that a bad government has been instituted, then they have the right to do away with it,\" says Stuart.\n\n\"But this section of the Declaration doesn't just give one the right to abolish government,\" I point out. \"It also includes the responsibility to 'institute new government.' \"\n\n\"Really, just having this right inserted like a closing statement at the end of the Constitution is abolishing the old government and putting in a new one,\" says Helen. \"Because it's saying that we can and should declare our independence again and again, rebel over and over, until we get it right. Jefferson's right: it's a small price to pay, if our happiness is secured.\"\n\nConstitution Caf\u00e9 Bill of Rights Article 7: Because Governments are instituted among people, deriving their just powers from the consent of the Governed, whenever a Government becomes destructive to these ends, it is the right of the people to alter or to abolish it, by creating a new Constitution that institutes a new government, one that lays its foundation on such principles, and that organizes its powers in such form, as to them shall seem most likely to effect their safety and happiness.\n\nOut with the Old, In with the New\n\n\"Let us,\" Jefferson exhorted, \"avail ourselves of our reason and experience to correct the crude essays of our first and unexperienced although wise, virtuous, and well-meaning councils.\" This was quite a backhanded compliment: on the one hand, Jefferson characterized the Framers' efforts to create a Constitution as \"wise, virtuous, and well-meaning\"; on the other, he left no doubt that he considered their accomplishment little more than a \"crude\" and \"unexperienced\" first attempt that should be revisited and remedied. As Jefferson told Samuel Kercheval,\n\nI am certainly not an advocate for frequent and untried changes in laws and constitutions. . . . But I know also that laws and institutions must go hand in hand with the progress of the human mind. As that becomes more developed, more enlightened, as new discoveries are made, new truths disclosed and manners and opinions change with the change of circumstances, institutions must advance also and keep pace with the times.\n\nTo Jefferson, what lent sanctity to a constitution was not its status as a virtually unchangeable document, but rather treating it as a work in progress and having faith in each generation to improve upon it.\n\nThe call for a modern-day constitutional convention of \"the people\" might seem an unlikely prospect. But if a critical mass comes to the conclusion that our government is failing to achieve the goals for which it was designed, the time might at last be ripe to consider Jefferson's proposal to rewrite the Constitution.\n\nUniversity of Virginia political scientist and popular media pundit Larry Sabato is one of our most outspoken advocates for a new constitutional convention. In A More Perfect Constitution, Sabato acknowledges the Framers' great accomplishment in creating a document that \"would survive for centuries and produce one of the most accomplished nations in human history.\" However, he believes that the Framers\n\nmight also have been surprised and disappointed that future generations of Americans would be unable to duplicate their daring and match their creativity when presented with new challenges. The evidence today that parts of the Constitution do not work well is just as overwhelming as the proof in the founder's day that the Articles of Confederation were inadequate to the needs of the growing United States.\n\nSabato maintains that we must \"look past the current 'values' debates and hot-button issues, and consider this very real possibility: that the failure of the nation to update the Constitution and the structure of government it originally bequeathed to us is at the root of our current political dysfunction.\"\n\nThe democratic theorist Robert Dahl poses this provocative question: Why should we feel bound today by a document produced more than two centuries ago by a group of fifty-five mortal men, actually signed by only thirty-nine, a fair number of whom were slaveholders, and adopted in only thirteen states by the votes of fewer than two thousand men, all of whom are long since dead and mainly forgotten?\n\nDahl further asks, \"If our constitution is in some important ways defective by democratic standards, should we change it, and how?\" While he does not echo Sabato's call for a new constitutional convention, he does ask us to decide for ourselves whether the Constitution is \"the best that we can design for enabling politically equal citizens to govern themselves under laws and government policies that have been adapted and are maintained with their rational consent.\"\n\nDahl envisions \"the possibility . . . of a gradually expanding discussion that begins in scholarly circles, moves outward to the media and intellectuals more generally, and after some years begins to engage a wider public.\"\n\nOr it may well be the reverse\u2014that the wider public takes the lead in pushing for revisions to our Constitution.\n\nBut Gordon Wood, leading historian of the American Revolution, is not optimistic that any of us today\u2014not the media, not our intellectuals, and certainly not the public at large\u2014can improve on the work of our original Framers, no matter how well intentioned and dedicated the attempt might be.\n\nIn Revolutionary Characters: What Made the Founders Different, Wood asserts:\n\nIf we want to know why we can never again replicate the extraordinary generation of the founders, there is a simple answer: the growth of what we today presumably value most about American society and culture, egalitarian democracy. In the early 19th century the voices of ordinary people, at least ordinary white people, began to be heard as never before in history, and they soon overwhelmed the high-minded desires and aims of the revolutionary leaders who had brought them into being.\n\nIn Wood's opinion, ordinary Americans have further diluted this high-minded thinking ever since. But one of our founders, Thomas Jefferson, would disagree. Wood's pejorative take on egalitarianism\u2014namely, that it panders to the lowest common denominator\u2014is the antithesis of Jefferson's, who held that all Americans must be given equal or at least ample opportunity to become standouts. This does not in any way mean that to Jefferson all citizens have to have an equal share of material wealth; however, by his conception, it does mean that all Americans do need certain essentials, like the right to take active part in public problem-solving, the right to a good education, and even the right to a modicum of property, in order to participate fully and positively in the public sphere. The fulfillment of these needs would, Jefferson believed, spur the creation of a more \"egalitarian elitism.\"\n\nHannah Arendt applauds the form of egalitarianism advanced by Jefferson. Of all the founders, she considers him unique in his promotion of \"a new form of government that would permit every member of the modern egalitarian society to become a 'participator' in public affairs.\" In particular, if Jefferson's vision for universal education had been implemented at our nation's founding, he was certain that a group of \"public foundationers\" would have emerged from every station in life, rather than only from the most advantaged segments of society. They would have been singled out for their exceptional \"taste for public freedom,\" their deeds setting an example for the entire citizenry that one \"cannot be 'happy'\" if one does not give one's all to \"the task of good government.\" But Jefferson's project was thwarted by entrenched gentry with a firm grip on power who either voted down his proposed reforms or refused to consider them seriously in the first place; they had no more interest in democratizing education than they did in facilitating citizen participation in governance.\n\nUltimately, the system of governance created by the Framers ensured that some potentially exceptional American voices would have little opportunity to be fully developed, let alone heard in the public sphere.\nXI\n\nBrew for a True Revolution\n\nThe Boston Tea Party is typically considered to mark the official beginning of the American Revolution. We tend to think of it as the ultimate democratic moment, in which \"the people\" rose up against unjust and overreaching government. In fact, it was the act of a handful of radicals who had little time for the opinions and will of the majority.\n\nThe idea for the Tea Party was hatched by 52-year-old political firebrand Samuel Adams, who had no faith that his fellow colonists would voluntarily quit drinking tea, even for the highest principles, if it was allowed to come ashore. Reverend William Gordon, one of the Revolution's first historians, remarked that to Adams and his fellow Sons of Liberty, \"the virtue of the people . . . was too precarious a ground on which to risk the salvation of their country.\" British Parliament had passed the Tea Act, which among other things placed a tax on tea without colonists' consent. Besides leading to a significant price increase on a beloved staple, it was a breach of the colonists' rights to be taxed only by their elected representatives. Determined to teach the British a lesson, Adams made sure that the barrels of tea on British merchant ships anchored in Boston Harbor never landed on dry soil. On December 16, 1773, he and about fifty others boarded the merchant ships, broke open all 343 casks of tea, and dumped their contents into the harbor.\n\nThe Boston radicals had disregarded alternative approaches to confronting the monarchy. Their \"town meetings,\" though held ostensibly to entertain various suggestions for responding to the Crown's abuses, in fact were intended only to advance their own position. At their final public gathering, Adams cut off discourse when some attendees opposed his recommendation to dump the casks of tea and proposed other options. \"This meeting,\" Adams remarked, \"can do nothing more to save the country.\" He then marched off with a group of men already dressed as Mohawks. Unbeknown to everyone else at the meeting, Adams and his cohorts had decided well beforehand to destroy the tea.\n\nThe Tea Party episode was on the verge of fracturing colonists. Many Boston merchants, dismayed by what they considered an unwarranted provocation, offered to reimburse their British counterparts. And most colonists outside of Boston felt that \"Boston had gone a great deal too far in destroying the East India Company's tea.\" As a consequence, \"Boston stood in danger of being isolated, even in New England,\" where colonists were typically more receptive to radical ideas for reform.\n\nRather than promising to be, in Adams's words, a \"great epocha,\" the Tea Party threatened to poison the nascent struggle for independence before it could properly begin. Enter Thomas Jefferson, whose intervention helped \"translate the local grievance of Boston into a common cause.\"\n\nAfter the Tea Party took place, Jefferson hastily initiated a series of evening meetings at the Raleigh Tavern with other Virginia legislators. They debated how their colony might respond in the wake of the event. Should Virginia side with the Boston provocateurs, vocally oppose their action, or remain neutral?\n\nJefferson gave great thought to Massachusetts Bay's dilemma. He was alarmed that the \"ill-fated colony, which had consistently been bold in their enmities\" against the Crown, now seemed \"devoted to ruin.\" Jefferson reasoned, though, that if the Tea Party mob \"did wrong\" in throwing the tea overboard, the Crown went even further by closing off the Boston port and curtailing all trade, \"without attempting a distinction between the guilty and the innocent.\" What's more, he feared that if no one resisted Great Britain's implementation of the draconian Boston Port Act on this occasion, then \"another and another will be tried till the measure of despotism be filled up.\" The challenge, then, was for Jefferson to convince his fellow Virginians and colonists elsewhere that they should view the Crown's actions against Massachusetts Bay as a pancolonial threat.\n\nJefferson recounts how those with him at the Raleigh decided that the Virginia legislature \"must boldly take an unequivocal stand in support of Massachusetts.\" But he admitted that the odds of accomplishing this were against them, since many legislators \"had grown too conservative\" and lacked \"the determination that the times demanded.\" Besides, the colonists themselves had let down their guard, not having directly confronted the British since the Gaspee Affair of 1772.\n\nJefferson and those who joined him at the Raleigh ultimately decided to craft a resolution calling for \"the appointment of a day of general fasting and prayer,\" which Jefferson felt could turn the divisive Boston Tea Party event into a galvanizing one by serving \"to avert from us the evils of civil war, to inspire us with firmness in support of our rights, and to turn the hearts . . . to moderation and justice.\"\n\nThe resolution calling for a pancolonial day of \"fasting, prayer and humiliation\" presented to Virginia's assemblymen made clear that the rights it promoted were not just for a select few, but rather were \"American rights.\" It declared that the Virginia colony was \"prepared to suffer the same punishments and indignities as their fellow men from Massachusetts,\" and so was ready to employ \"all just and proper means\" to oppose any punitive measure imposed by British Parliament. Despite Virginia's reputation as one of the most \"loyal\" colonies and the generally conservative mind-sets of the majority of Burgesses, the resolution appealed to the legislators' sense of piety and justice, and passed without opposition.\n\nJefferson's efforts to convert the divisive, rebellious actions of a few into a common cause had worked\u2014the legislatures of other colonies promptly passed similar resolutions in support of Massachusetts, and a revolution was catalyzed.\n\nIn the wake of this success, \"Aristocratic Virginia had its first taste of participatory democracy.\" Jefferson and his fellow Virginia legislators had called for county meetings to discuss the fate of the colony and to determine the delegates that would represent them in a future Williamsburg assembly, \"welcom[ing] not only 'gentleman freeholders,' but all male inhabitants over twenty-one.\" The wealthiest landowners and the most impoverished nonfreeholders rubbed shoulders in these meetings, inspiring Jefferson to liken the atmosphere to \"a shock of electricity, arousing every man, and placing him erect and solidly on his center.\"\n\nJefferson was chosen to represent Albemarle County at the Williamsburg gathering, where the delegates would then decide on representatives for the forthcoming congress in Philadelphia. Though Jefferson would have seemed a natural candidate for participation in the congress, his publication of a pamphlet titled A Summary View of the Rights of British America\u2014which among other things asserted \"that our immigration from England to this country gave her no more rights over us\"\u2014branded him as a radical, and so he was notably absent among the fifty-six attending delegates of the First Continental Congress on September 5, 1774.\n\nThe fact that Jefferson\u2014once declared \"too extreme\"\u2014was ultimately invited to participate in the Second Continental Congress on May 10, 1775, that he penned our Declaration of Independence, and that he served as our nation's third president indicate that he was not actually too bold\u2014too \"radical\"\u2014but rather ahead of his time, and able to adapt his innovative ideas to the demands of his historical moment. So wouldn't it follow that another visionary proposal by Jefferson that what may currently seem a \"flight of fancy\"\u2014a new constitution drafted by diverse groups of ordinary Americans\u2014might provide a very real impetus for change?\n\nIs it likely that a critical mass of Americans with motley political convictions will ever decide that today's Constitution is just as unworkable as our original governing document, the Articles of Confederation? Or is such a prospect too far-fetched?\n\nSpeaking of far-fetched, the progressive group MoveOn.org was considered so extreme when it was founded in 1998 that few believed it would ever have much of an impact on mainstream politics. Yet the millions it subsequently mobilized came out in droves in 2008 and were instrumental in electing our nation's first black president. Now the Tea Party has transformed from a fringe group into a kingmaker in electoral politics, giving Republicans a decisive majority in Congress in the 2010 elections. Both groups show the power of grassroots activism\u2014and the limits.\n\nMy hunch is that just as progressives came to feel betrayed when Obama abandoned the liberal agenda of his presidential campaign to engage in political compromise and accommodation, Tea Party activists will come to find that their own expectations for \"change they can believe in\" will be equally stymied\u2014not just because those they supported will either have to practice politics as usual to get anything done in Washington or become irrelevant, but because the system itself that was handed to us by our Framers prevents meaningful reforms that facilitate more responsive and responsible government.\n\nPerhaps both of these movements will eventually be front and center in calling upon the states to employ a never-before-used mechanism in Article V of the existing Constitution that's at their disposal: if two-thirds of the state legislatures give the go-ahead, they have the right to circumvent Congress and hold a new convention to introduce, without limit, amendments to the Constitution.\n\nAs it is, groups like MoveOn.org and the Tea Party have a good bit in common. Both are comprised of millions who had been political outsiders, they share a decided antiestablishment strain, and their members believe passionately in the right to self-government (though of course they differ on what that right amounts to in practice). But it's one thing to point out where common ground might exist between the two, and another altogether for the overwhelming majority in such movements to come together.\n\nYet against all odds, Jefferson forged common ground among colonists on both ends of the political spectrum, and everywhere in between, in the aftermath of the Boston Tea Party. He was convinced that they all were \"constitutionally and conscientiously democrats,\" and managed to convert an event that had threatened to dash any hopes for achieving unity to one of common cause.\n\nThe Declaration of Independence that he was then charged to write was the work of a flawed man who in his words, and at times in his deeds, revealed an extraordinary empathy for the hopes and aspirations of people quite unlike him.\n\nJefferson's proposal to revise periodically our Constitution is also a declaration of independence of sorts. He believed we did our Founders and Framers a disservice if we looked at their works as iconic. As Joyce Oldham Appleby puts it, \"The true Jeffersonian legacy is to be hostile to legacies.\" Jefferson had no doubt that coming generations could vastly improve on their accomplishments in ways that would better enable all Americans to realize their higher hopes for self-government.\n\nIn a letter to John Adams in 1823, Jefferson composed these stirring, rabble-rousing words:\n\nA first attempt to recover the right of self-government may fail, so may a second, a third, etc. But as a younger and more instructed race comes on, the sentiment becomes more and more intuitive, and . . . some subsequent one of the ever renewed attempts will ultimately succeed. For what inheritance so valuable, can man leave to posterity? \nACKNOWLEDGMENTS\n\nSpecial thanks go to Mike Li; Dennis Dienst; John Esterle; Stefanie Pfleger Byars; Mizgon Zahir; Hin Leung; Alex Moll; Laurel Sheridan; John Wren; Nathan Pollack; Evan Sinclair; Hallie Atkinson; Anas Shallal (owner of the incomparable Busboys & Poets); Jeff Anthony; Sierra Crocker; Sheldon Kelly (at a challenging crossroads, Uncle Sheldon reminded me what I can accomplish when I set my mind to it); Mark Padilla; Lelia Green; Giancarlo Ibarguen, rector of Universidad Francisco Marroqu\u00edn; Bert Loan; Don and Donna Goertz; Wolfgang Somary; the Whitman Institute; Sam Sadler; Todd Carstenn; Ron Cooper; Beth Fogarty; Barry Kibrick; Gil Feinberg; Jack Ratcliffe; Bobbie DeLeon; Laura Norin; Clea Kore; Kathy Kremins; Sam Fairchild (who graciously opened his home to me and made available his comprehensive collection of books on Thomas Jefferson); Maddie Taterka; David Blacker; Steve Duncombe; Carole Brzozowski; Kate Gannon; Miriam Feuerle; Steve Hornsby; Mark Butler; Todd Carstenn; Ron Cooper; Zack Macdonald; Shirley Strum Kenny; and my agent Andrew Stuart. My longtime friend Larry Parker passed away just as I was putting the finishing touches on this book. When I first told Larry, a veteran of the sixties protests, of my project, he wrote back, \"About time the younger generation did something radical and revolutionary!\" Thanks also to my government professors of long ago at the College of William & Mary, in particular George Grayson, Alan Ward, John McGlennon, and Donald Baxter. I am indebted to my long-time editor, Alane Salierno Mason, as tough, caring, and conscientious as they come, and most grateful to kindhearted and insightful assistant editor Denise Scarfi, as well as to copy editor Fred Wiemer. I am thankful for the love and support of my mom, Margaret Ann Phillips, and my dad, Alex Phillips. I also want to acknowledge my beloved father-in-law, Armando Chapa de Zambrano, who passed away just before this book went to press. My wife, Cecilia, and daughter, Caliope Alexis, make my dialogue projects sacred. Deepest thanks to all those who have taken part with me in Constitution Caf\u00e9 exchanges. \nFURTHER READING\n\nRichard Matthews's The Radical Politics of Thomas Jefferson (University Press of Kansas, 1984) presents a thoughtful and provocative exploration of Jefferson's ideas for evolving democracy over the long haul. Matthews makes clear his debt to Hannah Arendt, whose Jefferson scholarship, included in On Revolution (Penguin Classics, 2006) and Crises of the Republic (Mariner Books, 1972), remains underappreciated. Joseph Ellis's American Sphinx: The Character of Jefferson (Vintage, 1998) is acclaimed for its provocative analyses of Jefferson's character. Yet the author's penchant for dismissing out of hand Jefferson's more visionary political propositions diminishes some of his insights into Jefferson the human being. Some of the most erudite and insightful books that present a more admiring view of Jefferson include: Dumas Malone's classic series of works, Jefferson and the Ordeal of Liberty, Jefferson and the Rights of Man, The Sage of Monticello Jefferson and His Time, and Jefferson the President, Volumes I and II (all were reissued by the University of Virginia Press in 2006); Merrill Peterson's Thomas Jefferson and the New Nation (Galaxy Books, 1975), The Jefferson Image in the American Mind (University Press of Virginia, 1999), and Thomas Jefferson: Writings, a comprehensive 1,600-page compilation of Jefferson's public and private papers, addresses, letters, as well as his \"Autobiography\" and his only book, Notes on the State of Virginia; Joyce Oldham Appleby's Thomas Jefferson (Times Books, 2003) and Jefferson: Political Writings (Cambridge University Press, 1999); Peter Onuf's The Mind of Thomas Jefferson (University Press of Virginia, 2006), Jefferson's Empire (University Press of Virginia, 2001), and Jeffersonian America (Wiley-Blackwell, 2001, coauthored with Leonard Sadosky); Fawn Brodie's Thomas Jefferson: An Intimate History (W. W. Norton, reprinted in 2010, with a new introduction by Annette Gordon-Reed); John Dewey's panegyric The Living Thoughts of Thomas Jefferson (Longmans, Green & Co., 1940) and The Essential Jefferson (Dover, reissued in 2008); Jean Yarbrough's American Virtues: Thomas Jefferson on the Character of a Free People (University Press of Kansas, reprinted 2009) and The Essential Jefferson (Hackett Publishing Co., 2006); R. B. Bernstein's Thomas Jefferson (Oxford University Press, 2005); David Mayer's The Constitutional Thought of Thomas Jefferson (University Press of Virginia, 1995); and Andrew Burstein's The Inner Jefferson (University Press of Virginia, 2006).\n\nInsightful works that focus on particular aspects of Jefferson's life, works, or deeds, or that discuss Jefferson in the context of early American history, include: John Ferling's Adams vs. Jefferson (Oxford University Press, 2005); Pulitzer Prize\u2013winning historian Edward J. Larson's A Magnificent Catastrophe: The Tumultuous Election of 1800, America's First Presidential Campaign (Free Press, 2007); Jeremy Bailey's Thomas Jefferson and the Executive Power; the Jefferson critic Leonard Levy's Jefferson and Civil Liberties (Ivan R. Dee, 1989); Annette Gordon-Reed's Thomas Jefferson and Sally Hemings: An American Controversy (University Press of Virginia, 1998) and her Pulitzer Prize\u2013winning The Hemingses of Monticello: An American Family (W. W. Norton, 2008); Andrew Burstein's Jefferson's Secrets (Basic Books, 1996) and Madison and Jefferson (Random House, 2010); Charles B. Sanford's The Religious Life of Thomas Jefferson (University Press of Virginia, 1987).\n\nEssential reading on the Framers and the Constitution they crafted includes Clinton Rossiter's 1787: The Grand Convention (W. W. Norton, 1966), even if at times it resembles a pro-Federalist apologia; Philip Hamburger's Law and Judicial Duty (Harvard University Press, 2008); Charles Black's A New Birth of Freedom (Putnam, 1997); Bruce Ackerman's two-volume We the People (Belknap Press of Harvard University Press, 1993); John R. Vile's A Companion to the United States Constitution and Its Amendments (Praeger, 2010); Robert Dahl's How Democratic Is the American Constitution? (Yale University Press, 2003) and On Democracy (Yale University Press, 2000); and Larry Sabato's A More Perfect Constitution (Walker & Co., 2008).\n\nMust-reads on the Revolutionary era and its movers and shakers include: Joseph Ellis's Pulitzer Prize\u2013winning Founding Brothers: The Revolutionary Generation (Knopf, 2000); Gary B. Nash's The Unknown American Revolution: The Unruly Birth of Democracy and the Struggle to Create America (Penguin, 2006); Sean Wilentz's The Rise of American Democracy: Jefferson to Lincoln (W. W. Norton, 2006); Pauline Meier's Resistance to Revolution: Colonial Radicals and the Development of American Opposition to Britain (W. W. Norton, 1992); Rhys Isaac's Pulitzer Prize\u2013winning The Transformation of Virginia, 1740\u20131790 (University of North Carolina Press, 1999); Neil Stout's The Perfect Crisis: The Beginning of the Revolutionary War (New York University Press, 1976); the Pulitzer Prize\u2013winning historian Gordon S. Wood's The Making of the Constitution (Baylor University Press, 1987), Revolutionary Characters: What Made the Founders Different (Penguin, 2007), The Creation of the American Republic (University of North Carolina Press, 1998), and The Radicalism of the American Revolution (Vintage, 1993). \nMore praise for\n\nConstitution Caf\u00e9\n\n\"The United States needs constitutional change, but how to get it done? Christopher Phillips has the right answer. Get Americans talking to Americans about how we can improve our nation.\"\n\n\u2014Dr. Larry J. Sabato, author of   \nA More Perfect Constitution and director of  \nthe University of Virginia Center for Politics\n\n\"Christopher Phillips dares to suggest that we should smash open the Constitution's glass case and hand it to the people the Founding Fathers called the 'mob.' With infinite curiosity and an intellectual integrity that is rare among professional thinkers, Phillips says about the bible of American tradition the unthinkable, the glorious, and the liberating: let it rip.\"\n\n\u2014Thaddeus Russell,   \nauthor of A Renegade History of the United States\n\n\"A truly radical and deeply patriotic book, Constitution Caf\u00e9 illustrates the power and promise of democracy, using the extraordinary conversations of ordinary citizens to reanimate the founding ideas and documents of this country. America needs this book . . . now!\"\n\n\u2014Stephen Duncombe, author of  \nDream: Re-Imagining Progressive Politics in an Age of Fantasy\n\n\"Through this thorough consideration of our country's past and present, we are encouraged to find strength for America's future.\"\n\n\u2014Bob Kerrey, president emeritus, The New School\n\n\"An engaging and informative narrator, Phillips intersperses the modern-day conversations with Jefferson's thoughts about the issues under discussion and the founding fathers' own disagreements as they framed the Constitution.\"\n\n\u2014Publishers Weekly\n\n\"The author skillfully interweaves a history of the early days of the Republic and the disputes at that time with a discussion of Jefferson's involvement with constitutional issues in the state of Virginia as well as for the country as a whole, and he offers useful insight to Jefferson's thoughts over his long career. A provocative extension of Jefferson's original plan.\"\n\n\u2014Kirkus Reviews\n\n\"Phillips . . . teas[es] out ideas that seem, given today's toxic political climate, almost unconventional in their reasonableness. . . . A vigorous rethinking of America's legal framework.\"\n\n\u2014Alan Moores, Booklist\n\n\"As an exploration of current governmental theory and philosophy, the book provides an excellent framework for conducting similar discussions. Readers who enjoy political and governmental theory or who participate in politically oriented book clubs will find this a worthwhile choice.\"\n\n\u2014Jill Ortner, Library Journal\nCopyright \u00a9 2011 by Christopher Phillips\n\nAll rights reserved\n\nPrinted in the United States of America\n\nFirst published as a Norton paperback 2012\n\nDisclaimer: To guard participants' privacy, names have been changed; occupations   \nand other biographical information, and the locales at which dialogues took place, have   \nat times been changed. Sometimes participants portrayed in this book are composites   \nof those who took part in actual dialogues, and some dialogues are \u00adcomposites.\n\nFor information about permission to reproduce selections from this book,\n\nwrite to Permissions, W. W. Norton & Company, Inc.,\n\n500 Fifth Avenue, New York, NY 10110\n\nFor information about special discounts for bulk purchases, please contact   \nW. W. Norton Special Sales at specialsales@wwnorton.com or 800-233-4830\n\nManufacturing by Courier Westford\n\nBook design by Chris Welch Design\n\nProduction manager: Julia Druskin\n\nLibrary of Congress Cataloging-in-Publication Data\n\nPhillips, Christopher, 1959 July 15\u2013\n\nConstitution cafe : Jefferson's brew for a true revolution \/ Christopher Phillips. \u2014 1st ed.\n\np. cm.\n\nIncludes bibliographical references.\n\nISBN 978-0-393-06480-3 (hardcover)\n\n1. Constitutional history\u2014United States. 2. Democracy\u2014United States.   \n3. Democracy\u2014Philosophy. 4. Jefferson, Thomas, 1743-1826\u2014Political and social views.   \nI. Title.\n\nJK31.P45 2011\n\n320.973\u2014dc23\n\n2011018129\n\nISBN 978-0-393-34226-0 pbk.\n\nW. W. Norton & Company, Inc.\n\n500 Fifth Avenue, New York, N.Y. 10110\n\nwww.wwnorton.com\n\nW. W. Norton & Company Ltd.\n\nCastle House, 75\/76 Wells Street, London W1T 3QT\n\n1 2 3 4 5 6 7 8 9 0\n\n. The tombstone of Daniel Pegram's wife, Sarah, is on the grounds of Bruton Parish in Colonial Williamsburg.\n\n. In 1621, in the dominion of \"old Virginia,\" which comprised what is today Virginia and North Carolina, the first full-fledged legislature with its own elected assembly, known as the House of Burgesses, was established in the New World.\n\n. Jeremy Bailey, Thomas Jefferson and Executive Power (Cambridge University Press, 2010).\n\n. Philip Hamburger, Law and Judicial Duty (Harvard University Press, 2008).\n\n. Clinton Rossiter, 1787: The Grand Convention (W. W. Norton, 1966).\n\n. Keith Dougherty, Collective Action Under the Articles of Confederation (Cambridge University Press, 2001).\n\n. Rossiter, 1787: The Grand Convention.\n\n. Henry himself would later lead the effort in Virginia against ratifying the new Constitution, and he nearly succeeded; the Constitution was ratified by an 89-to-79 vote.\n\n. Joseph C. Morton, Shapers of the Great Debate at the Constitutional Convention of 1787 (Greenwood Press, 2005).\n\n. Hamburger, Law and Judicial Duty.\n\n. The state convention included one very uncommon person, Benjamin Franklin, who managed to traverse the corridor and serve both as a delegate to the Continental Congress and a framer of Pennsylvania's constitution. Indeed, Franklin was unanimously chosen to be the titular president of the state convention.\n\n. The first version of this amendment seeking to abolish legal sex discrimination was introduced in 1923 as a result of the effort of Alice Stokes Paul, the suffragist leader who had spearheaded the successful campaign for passage of the Nineteenth Amendment guaranteeing women the right to vote. A version of this amendment was subsequently introduced, and rejected, in every session of Congress, until the ERA was finally approved in revised form in 1972.\n\n. The Twenty-Seventh Amendment prohibits any law that increases or decreases the salary of members of the Congress from taking effect until the start of the next set of terms of office for representatives.\n\n. The Federal Marriage Amendment reads, \"Marriage in the United States shall consist only of the union of a man and a woman. Neither this Constitution or the constitution of any state, nor state or federal law, shall be construed to require that marital status or the legal incidents thereof be conferred upon unmarried couples or groups.\"\n\n. Though they later reconciled, Jefferson's decision to throw his hat in the ring in a rematch against Adams (he lost the 1796 presidential election by just three electoral votes) led to a protracted estrangement from his longtime friend; both of them, notes Joseph Ellis, \"when forced to choose, had opted for party over friendship.\"\n\n. Because the Framers had failed to distinguish in the Constitution between electoral votes for a president or vice president, Jefferson and his running mate Burr wound up in an electoral tie for the presidency. The Constitution was subsequently amended to make sure this never happened again.\n\n. John Ferling, Adams vs. Jefferson (Oxford University Press, 2005).\n\n. From Pulitzer Prize\u2013winning historian Edward J. Larson's A Magnificent Catastrophe: The Tumultuous Election of 1800, America's First Presidential Campaign (Free Press, 2007).\n\n. Ferling, Adams vs. Jefferson.\n\n. On May 4, 1970, an Ohio National Guard unit opened fire on students holding an antiwar protest on campus, killing four and wounding nine.\n\n. At the time of this dialogue, a bipartisan group in Congress was considering the possibility of reintroducing a piece of legislation, the America Saving for Personal Investment, Retirement, and Education Act, or ASPIRE, which would give every newborn a lifetime savings account, that would be opened when they are issued a Social Security card. If the bill ever passes, the federal government would make a one-time $500 contribution to the account.\n\n. Progressive Republicans tried to go much further than this in seeking to guarantee \"forty acres and a mule\" to free blacks at the end of the Civil War. But the majority in Congress rejected this effort, claiming that such land distribution verged on socialism; the consequence was it forced blacks into sharecropping and led to segregation.\n\n. Brazil's constitution states that use of all land must benefit society as a whole. Yet the most destitute own just 1 percent of the nation's land.\n\n. Pauline Meier, Resistance to Revolution: Colonial Radicals and the Development of American Opposition to Britain (W. W. Norton, 1992).\n\n. Onuf further asserts that that there is no evidence that this amendment Jefferson claims to have prepared ever in fact existed.\n\n. The original opening sentence of Jefferson's unedited draft of the Declaration is \"We hold these truths to be sacred and undeniable; that all men are created equal and independent, that from that equal creation they derive rights inherent and unalienable, among which are the preservation of life, and liberty, and the pursuit of happiness.\"\n\n. James Colaiaco, Frederick Douglass and the Fourth of July (Palgrave Macmillan, 2006).\n\n. From Article II, Section 4, of the Constitution.\n\n. Zedillo, president of Mexico from 1994 to 2000, is director of Yale's Center for the Study of Globalization. He was the last president from Mexico's ruling Institutional Revolutionary Party, which had held power for seventy-one years, before Vicente Fox of the National Action Party was elected president.\n\n. Salinas de Gortari was Mexico's president from 1988 to 1994.\n\n. A nationwide poll conducted in 2010 by the Center for the Constitution revealed that few Americans\u2014less then one-third\u2014have actually read the Constitution, though the same comprehensive poll reports that the overwhelming majority of Americans recognize the enormous impact of the Constitution on their everyday lives.\n\n. Benson Lossing, Our Country (Kessinger Publishing, 2004).\n\n. Peter Knight, Conspiracy Theories in American History (ABC-CLIO, 2003).\n\n. As is New York, where I had to sign an oath of allegiance to the Constitution before I could teach at New York University.\n\n. Dewey's encomium to Jefferson is from his Living Thoughts of Thomas Jefferson (David McKay Co., 1940). Jefferson even insisted that the public not celebrate his birthday, Dewey notes, \"on the ground that the only birthday he wished to have recognized was 'that of my country's liberties.'\"\n\n. Gary Nash, The Unknown American Revolution: The Unruly Birth of Democracy and the Struggle to Create America (Penguin, 2006).\n\n. In 1917, almost two decades after American forces ousted Spanish troops from the Caribbean island, President Woodrow Wilson signed the Jones-Shafroth Act, which extended U.S. citizenship to Puerto Ricans.\n\n. This according to a June 9, 2010, report in the Minneapolis Star Tribune.\n\n. Jefferson further believed that an agriculture-based economy would give the United States strategic commodities for international trade. His vision on the international front, as Joseph Ellis puts it, was of \"a liberal international community, comprised of open markets and national cooperation.\" It was a vision that \"foundered on the rocks of European intransigence.\"\n\n. Conceived of in 1800, the University of Virginia was the first institution of higher learning in the United States whose charter excluded religious doctrine.\n\n. The late political scientist's research into the psychology of our nation's chief executives continues to resonate widely among presidential scholars.\n\n. The English charter, known as the Magna Carta\u2014its full name is the Magna Carta Libertatum (the Great Charter of Freedoms)\u2014originated in 1215. Though it is not technically a constitution, it guarantees that all British citizens \"have and enjoy all liberties and immunities of free and natural subjects.\" Its array of codified rights includes the right to self-rule, to endorse or reject taxation, the right to a trial by jury of their peers, to habeas corpus (a legal action that prevents unlawful detention). It also establishes that the king of England himself is not above the law, and is beholden to obey and carry out the dictates of the Magna Carta.\n\n. Signed in 1949 in the aftermath of World War II, the Geneva Conventions are a combination of several treaties and protocols that dictate a set of international standards for treatment of victims of war. The Conventions were ratified, in their entirety or in part, by 194 nations.\n\n. An expansive trilateral trade agreement signed by the governments of Canada, Mexico, and the United States, and put into force in 1994, NAFTA governs all facets of trade between these nations, and includes parallel agreements on environment and labor issues.\n\n. The rebellion, launched in the western frontier over general dissatisfaction with government policy, came to a crescendo after an excise tax was placed on whiskey. The tax was finally repealed after Jefferson became president.\n\n. Soon after this dialogue took place, President Obama granted his first pardons, to nine people convicted of crimes such as drug possession, mutilating coins, and counterfeiting.\n\n. Shaila Dewan, \"Study Finds Blacks Blocked from Southern Juries,\" New York Times, June 1, 2010.\n\n. In the case Georgia v. Brailsford, the Supreme Court itself presided over a jury trial.\n\n. From Article II, Section 4, of the Constitution.\n\n. The former governor of Minnesota unsuccessfully sought the presidency twelve times between 1944 and 2000.\n\n. Jean Yarbrough, American Virtues: Thomas Jefferson on the Character of a Free People (University Press of Kansas, 2009).\n\n. In 1959, the House temporarily increased the size to 437, to give the new states of Alaska and Hawaii new members, before they again reduced it to 435.\n\n. Hannah Arendt, Crisis in the Republic (Mariner Books, 1972).\n\n. Andrew Burstein, The Inner Jefferson (University Press of Virginia, 1996).\n\n. Michael Knox Beran, Jefferson's Demons: Portrait of a Restless Mind (Free Press, 2007).\n\n. Jefferson obviously did not admire them enough to do what he could during his terms as president to make sure that their way of life was preserved. It never seems even to have occurred to him to enact policies for settlement and expansion that would enable Native Americans to maintain at least part of their original territories; instead, during his years in office, the Native American diaspora began.\n\n. However, at the University of Virginia, which Jefferson founded in 1819\u2014it was the first secular university in the nation, with no religious affiliation or denominational identity\u2014he sought to implement a version of his ward system. As Joseph Ellis relates, \"Jefferson was insistent that his university be devoted to the principle of 'self government,' which meant that proctors, provosts and even a president were superfluous.\" Jefferson's scheme for the university \"almost certainly grew out of the thinking he had been doing about wards as the primal political and educational units in Virginia.\"\n\n. The street in Washington, D.C., where so many of the most prominent lobbying firms are headquartered.\n\n. Hasan is alleged to have killed 12 and wounded 31 people when he opened fire at the base.\n\n. Shays's Rebellion\u2014named after the movement's leader, Daniel Shays, a farmer and Revolutionary War soldier\u2014took place in western Massachusetts in 1786 and 1787. Poor farmers were up in arms over being heavily taxed without their consent, which put many in debt and some in debtors' prison\u2014not exactly the kind of repayment they were expecting after sacrificing so much for the Revolutionary cause. When their pleas for debt relief were unheeded, they erected liberty poles and liberty trees to highlight their cause. The uprising was put down in February 1787 by a 4,000-man private militia supported by aristocrats.\n\n. Jefferson had proposed to extend education to black youth as well. According to Fawn Brodie, in Thomas Jefferson: An Intimate History, Jefferson proposed \"that all black youths be educated 'to tillage, arts or sciences' till twenty-one and all black girls to eighteen, 'at public expense.' \"\n\n. Charles B. Sanford, The Religious Life of Thomas Jefferson (University Press of Virginia, 1987).\n\n. Rhys Isaac, The Transformation of Virginia, 1740\u20131790 (University of North Carolina Press, 1999).\n\n. Neil Stout, The Perfect Crisis: The Beginning of the Revolutionary War (New York University Press, 1976).\n\n. From Dumas Malone's classic biography, Jefferson the Virginian (Warren Press, 2007).\n\n. On June 9, 1772, a group of Rhode Islanders protesting the British schooner Gaspee's practice of pursuing cheap merchandise at the expense of colonial companies officially approved by the government of England set the ship ablaze and were sent to trial in England. Jefferson, appalled that the colonists were being denied a jury of their peers, took the opportunity to establish \"committees of correspondence\" aimed at uniting colonists in pursuit of a common cause, and succeeded in having Parliament's decision to try the colonists in England repealed.\n\n. Stout, The Perfect Crisis.\n\n. Jefferson and Adams died on the same day three years later, on Independence Day in 1826.\n\n \"He was a rebel to the very end,\" Joseph Ellis notes.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Praise for the Kris Longknife novels\n\n\"A whopping good read . . . Fast-paced, exciting, nicely detailed, with some innovative touches.\"\n\n\u2014Elizabeth Moon, Nebula Award\u2013winning author of _Crown of Renewal_\n\n\"Shepherd delivers no shortage of military action, in space and on the ground. It's cinematic, dramatic, and dynamic . . . A thoroughly enjoyable adventure featuring one of science fiction's most interesting recurring heroines.\"\n\n\u2014Tor.com\n\n\"A tightly written, action-packed adventure from start to finish . . . Heart-thumping action will keep the reader engrossed and emotionally involved.\"\n\n\u2014Fresh Fiction\n\n\"Fans of the Honor Harrington escapades will welcome the adventures of another strong female in outer space starring in a thrill-a-page military space opera.\"\n\n\u2014Alternative Worlds\n\n\"[Kris Longknife] will remind readers of David Weber's Honor Harrington with her strength and intelligence. Mike Shepherd provides an exciting military science fiction thriller.\"\n\n\u2014Genre Go Round Reviews\n\n\"Kris can kick, shoot, and punch her way out of any dangerous situation, and she can do it while wearing stilettos and a tight cocktail dress. She's all business, with a Hells Angel handshake and a 'get out of my face' attitude. But her hair always looks good . . . Kris Longknife is funny and she entertains us.\"\n\n\u2014Sci Fi Weekly\n\n\"[A] fast-paced, exciting military SF series . . . Mike Shepherd has a great ear for dialogue and talent for injecting dry humor into things at just the right moment . . . The characters are engaging, and the plot is full of twists and peppered liberally with sharply described action . . . Guaranteed a good time with plenty of adventure.\"\n\n\u2014SF Site\n\n# Ace Books by Mike Shepherd\n\nKRIS LONGKNIFE: MUTINEER\n\nKRIS LONGKNIFE: DESERTER\n\nKRIS LONGKNIFE: DEFIANT\n\nKRIS LONGKNIFE: RESOLUTE\n\nKRIS LONGKNIFE: AUDACIOUS\n\nKRIS LONGKNIFE: INTREPID\n\nKRIS LONGKNIFE: UNDAUNTED\n\nKRIS LONGKNIFE: REDOUBTABLE\n\nKRIS LONGKNIFE: DARING\n\nKRIS LONGKNIFE: FURIOUS\n\nKRIS LONGKNIFE: DEFENDER\n\nKRIS LONGKNIFE: TENACIOUS\n\nKRIS LONGKNIFE: UNRELENTING\n\nKRIS LONGKNIFE: BOLD\n\nTO DO OR DIE: A JUMP UNIVERSE NOVEL\n\nVICKY PETERWALD: TARGET\n\nVICKY PETERWALD: SURVIVOR\n\nVICKY PETERWALD: REBEL\n\nSpecials\n\nKRIS LONGKNIFE: TRAINING DAZE\n\nKRIS LONGKNIFE: WELCOME HOME \/ GO AWAY\n\nWriting as Mike Moscoe\n\nTHE FIRST CASUALTY: A JUMP UNIVERSE NOVEL\n\nTHE PRICE OF PEACE: A JUMP UNIVERSE NOVEL\n\nTHEY ALSO SERVE: A JUMP UNIVERSE NOVEL\n\nACE\n\nPublished by Berkley\n\nAn imprint of Penguin Random House LLC\n\n375 Hudson Street, New York, New York 10014\n\nCopyright \u00a9 2016 by Mike Moscoe\n\nPenguin Random House supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin Random House to continue to publish books for every reader.\n\nACE is a registered trademark and the A colophon is a trademark of Penguin Random House LLC.\n\nEbook ISBN: 9780698180666\n\nFirst Edition: November 2016\n\nCover illustration by Scott Grimando\n\nCover design by Katie Anderson\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental.\n\nVersion_3\n\n# Acknowledgments\n\nAs Kris finishes up her first midlife crisis, I'm also hitting the twenty-fifth anniversary of my first short story publication. It's been a wild ride, and now seems like a good time to thank all the people who helped me along the way.\n\nI will always be grateful to Sheila Simonson and the group at the Novel Writing class at Clark College. They helped me through those first stumbling efforts, taught me how to do dialogue (you know, Mike, that stuff in quotes), and gave me the discipline to write ten whole pages a week. Mary Rosenblum and her writers group helped polish me. Mary also gave me the benefit of her Clarion experience.\n\nThank you, Stan Schmidt of _Analog_. You were the first to see the writing potential in me and give me the approval I desperately needed. Like with all writers, that approval came in the form of a check.\n\nJenn Jackson has been the best agent that I could ask for. She's given me all I needed and kept out of my way when I needed that. My longtime editor at Ace, Ginjer Buchanan, picked me up and gave me a second chance when others might not have. She was a perfect match for twenty years. I hope she enjoys a long and fun retirement; she richly deserves it. I appreciate Diana Gill's contributions to _Kris Longknife: Bold_ and my other latest books. I'm grateful to the whole gang at Ace for the work they've put into the production that turns my manuscript into the finished product you readers enjoy. Oh, and I do love Scott's cover art.\n\nThe folks at Lincoln City, both at the Historic Anchor Inn and the Sunday brunch bunch, have supported me through a lot and helped me keep going. I'm sure you will keep me going for a long time to come.\n\nMy first readers also really deserve a very special thank-you. Lisa M\u00fcller and Edee Lemonier do a great job of cleaning up my all-too-frequent typos and nits. Oh, and occasionally they back up my wife's \"You really need to fix that section.\"\n\nAround the house, Nikki and Danny, my grandkids, are not only interesting sources for inspiration, but they've also come to understand what it's like to have a writer for a grandpa. They've learned to enjoy the ups and put up with the downs. They also love going to cons with me.\n\nThen there's Ellen. She's devoted years to learning how to be the best first reader that a writing husband could hope for. Thank you, honey, for everything.\n\n# Contents\n\n_Praise for the Kris Longknife Novels_\n\n_Books by Mike Shepherd_\n\n_Title Page_\n\n_Copyright_\n\n_Acknowledgments_\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n\nChapter 41\n\nChapter 42\n\nChapter 43\n\nChapter 44\n\nChapter 45\n\nChapter 46\n\nChapter 47\n\nChapter 48\n\nChapter 49\n\nChapter 50\n\nChapter 51\n\nChapter 52\n\nChapter 53\n\nChapter 54\n\nChapter 55\n\nChapter 56\n\nChapter 57\n\nChapter 58\n\nChapter 59\n\nChapter 60\n\nChapter 61\n\nChapter 62\n\nChapter 63\n\nChapter 64\n\nChapter 65\n\nChapter 66\n\nChapter 67\n\nChapter 68\n\nChapter 69\n\nChapter 70\n\nChapter 71\n\n_Special Note to the Reader_\n\n_About the Author_\n\n# 1\n\nAdmiral, Her Royal Highness Kris Longknife did her best to cut her corners as she'd been taught in those long-ago days at Officer Candidate School. But then, Gunny had never suggested how you managed the correct military entry with a bundle such as Kris cradled in her left arm.\n\nThe unholy trinity were waiting for her.\n\nKris came to a halt before the middle one, Field Marshal McMorrison, Chief of Staff for the entire military of the United Society. She saluted, bringing her hand right up her gig line\u2014as much as she could without bopping that tiny bundle.\n\n\"Admiral Kris Longknife reporting as ordered,\" she said, more to her great-grandfather Ray, King Raymond I to most, seated in the guest's chair to the right of Mac's desk. The last member of the unholy trinity, Admiral Crossenshield, Crossie to most, had a chair to Mac's left.\n\n\"Now, what was the all-fired hurry?\" Kris failed to avoid adding.\n\n\"What is _that_!\" the King said, pointing at the bundle cradled in Kris's left arm.\n\n_Is he so locked up in his own little bubble that he didn't get the word?_ Kris removed the blanket that had protected her infant daughter's face, keeping her warm from the crisp autumn weather during the drive from the space-elevator station to Main Navy. \"This is Ruthie Marie. Jack and my firstborn,\" she said.\n\nShe pointed Ruthie's gummy smile at the two couples who stood by the bookcase to the left of the unholy three. They didn't look at all surprised to see her bundle. Indeed, if they hadn't known of tiny Ruthie, they would never have been here.\n\n\"Can I see the baby?\" Kris's mother, Brenda, squeed. At her elbow, Great-grandmother Ruth Tordon, more commonly known as Mrs. Trouble, held back. Brenda was so rarely happy after the kidnapping and death of her third child, Eddy, that any moment of joy was not to be denied her.\n\nKris met her mother halfway and most gently and carefully managed the transfer. Mother must have been getting practice with Honovi's three; Kris had to do little to get Mother's arms just right.\n\n_You_ will _get this just right. This is_ my _baby,_ Kris thought but did not say.\n\nRuthie gave Brenda a toothless grin. Brenda grinned back, enraptured. Gramma Trouble stood at one elbow, Kris at the other. Brenda could not have dropped Ruthie if she wanted to.\n\n\"Her full name,\" Kris provided, \"is Ruth Maria Brenda Anne.\"\n\n\"Hi there, my little Brenda,\" Brenda said.\n\nGreat-grampa Trouble, trouble to his enemies, trouble to his superiors, and just flat trouble to everyone, including himself, stood well out of range of any spit-up with Billy Longknife, Man of the People, Prime Minister of Wardhaven, and Kris's occasional father.\n\n\"What's her last name?\" Grampa Ray asked, coming to stand by his longtime friend, General Trouble, ret.\n\n\"Haven't decided,\" Kris said. \"Probably we'll have to do it before she starts school, but we can wait.\"\n\n\"You're married to Jack, here,\" the King said, no part question at all.\n\nLieutenant General Jack Montoya nodded, as did Kris.\n\n\"When did that happen?\" was as gruff as a man could manage under Ruthie's gummy grin.\n\n\"While I was out of any chain of command,\" Kris said. \"Just before you issued orders making me God Almighty in the Alwa system.\"\n\nThat got a snort from Ray and a chuckle from everyone else paying attention.\n\n\"I'll bet you any amount of money Rita was at the bottom of this,\" the King growled.\n\n\"She officiated,\" Kris said, beaming happily.\n\n\"That woman keeps everyone wrapped around her little finger,\" Ray muttered.\n\nKris considered how much Great-grandfather Ray had pulled the strings that damn near wrapped her in a spiderweb of conflicting duties on Alwa but said nothing. Ray was what he was, and there was little she could do to change him at his present age, with 120-plus years disappearing in the rearview mirror.\n\nGrampa Trouble laid a nearly affectionate hand on Ray's shoulder. \"Ray, you knew those two were going to tie the knot, it was only a question of when and where.\"\n\nKris let out a sigh. This was going smoother than she had expected.\n\nThe _Wasp_ had jumped into the Wardhaven system, finally slowed down enough to hit it at under fifty thousand kilometers an hour. Kris had expected an immediate message from Mac or Ray. After all, they were the ones who had had Grand Admiral Sandy Santiago relieve her of her command and order her home \"soonest.\"\n\nNo message.\n\nThe _Wasp_ docked at High Wardhaven Station as smoothly as if Captain Drago were still at the conn. Kris had expected a message as soon as they opened their locks and plugged into the station's landline, air, water, and sewer.\n\nNothing.\n\nJack had to order up transport for Kris and a two-squad Marine Honor Guard\u2013fully locked and loaded. A chief bosun's mate, a fatherly type, had followed along, a diaper bag slung over his shoulder with such aplomb that none would dare gainsay him.\n\nOnly when they had dropped down to the main beanstalk terminal in Wardhaven had the necessary transport been waiting for them. They'd been whisked from the terminal to Main Navy and led by a very silent lieutenant colonel to Mac's inner sanctum.\n\nThe colonel had passed them to Mac's secretary, who smiled, and said, \"You are expected,\" and held open the door to Mac's inner office.\n\nKris had taken a deep breath and reported in.\n\nShe still had no idea where the fire was that required her to leave the fleet she had built up, the people she had sworn to protect, and the industrial base she'd grown and overseen.\n\n_I've had enough of being the visiting fireman. Especially enough of being ordered from pillar to post with no idea of what I'm supposed to do or how to do it._\n\nStill, Kris held her tongue as Ruthie smiled and did her best to wrap around her tiny finger all of Kris's problem people, the Troubles excepted . . . sometimes.\n\nEven Ray finally tried his hand at holding Ruthie. That was when disaster struck.\n\nRuthie scrunched up her face and let out a yowl. Then she spat up.\n\n_Good girl. I've wanted to do that so many times,_ Kris thought.\n\nBut \"diaper bag\" was what Kris called for. The Marines and the chief had been able to avoid Kris's little family reunion. _Lucky them._ These were usually a disaster, and there was no need for the Marines to know just how bad it got the higher up you went. Smart Jack, he'd grabbed the bag, not considering it beneath a three-star to pack the necessities for his tiny daughter. Now he handed the bag to Kris, already opened and with a cleaning cloth, half\u2013pulled out. Kris flashed Jack a smile; whether they were fighting off bug-eyed monsters or close family, she could always count on him.\n\nKris traded the cloth to Grampa Ray for Ruthie. Jack had another cloth in hand, so Kris could clean up the tiny one. It was clear, however, that Ruthie had had enough of being nice for family. She still hadn't quite figured out what her lungs were for, but she already knew how to make with her own little war cry.\n\n\"How long's it been since she was fed?\" Gramma Ruth asked.\n\n\"Just before we headed down,\" Kris answered as she checked to see that the diaper was neither wet nor messy. \"I think this is an 'I'm hungry, where's lunch' howler,\" Kris agreed, only too happy to use Ruth for an excuse to get a break from all this family attention.\n\n\"Can I feed her?\" Brenda asked eagerly. \"Do you have a bottle?\"\n\n\"We were not prepared for children on Alwa Station,\" Kris explained. \"We've had to do everything the old-fashioned way.\"\n\n\"Body birth and breast nursing, huh?\" Gramma Trouble provided when Brenda frowned blankly.\n\n\"Yep,\" Kris said. \"Do you have a room where I could nurse Ruthie?\"\n\n\"How did you come to have a baby on the front line?\" Grampa Ray asked, with a lot of royal command behind it.\n\nKris had been waiting for that one. \"Someone sabotaged my new birth-control implants, figuring that a pregger admiral couldn't command. I dumped his ass in the bird-guano mines and commanded very well, thank you. By the way, thanks for those beam ships. We would have been toast when six alien hordes hit us all at the same time. With them, _they_ were toast, slathered with marmalade. We did get the first two on our own, thank you very much. Anyway, Ruthie and I need to commune with motherhood just now. Jack can explain more if you need it. Where do we go?\"\n\nRuthie emphasized the urgency of Kris's request with a bawl that was only three on the baby Richter scale but threatened to go higher if not appeased.\n\n\"Erwin,\" Mac commanded urgently on his intercom. The secretary appeared at the door. \"Show the admiral to Admiral Ballo's office. He's taken leave today.\"\n\nErwin led Kris out of the office, into the hall, and to the next door down. It opened to a very standard set of senior flag officer's spaces. The walls were a deep mahogany. The outer wall had large picture windows with drapes that could be pulled shut. There were a desk, a meeting table, and a discussion pit, all standard-issue. The large armchair in the discussion pit was the one thing out of military step. It was a rocker.\n\n\"Admiral Ballo has a bad back,\" Erwin explained.\n\n\"He's not in today?\"\n\nErwin glanced at the wall that the admiral shared with Mac's office. \"He didn't say so, but I think he wanted to stay out of range of the collateral damage.\"\n\n\"Smart man,\" Kris said, and began to loosen the coat of her dress blues.\n\nErwin got the message and left.\n\n\"So it's just you and me, babe,\" Kris said, and settled Ruthie at her right breast, which, to be honest, was starting to ache. If Ruthie hadn't called for a chow break, Kris might have had to.\n\nKris began to relax as Ruthie suckled hungrily. She had a good view from her seat of blue sky and white clouds. That scene could match hundreds of worlds humanity had settled on.\n\nKris chose to let that thought wander on its way and smiled down at the miracle at her breast. \"It is just you and me, little one.\"\n\n# 2\n\n\"Kris, we're going to stay,\" had become an all-too-frequent refrain in her last days on Canopus Station.\n\nPenny had been the first.\n\n\"Kris, would you mind terribly if Masao and I stayed?\"\n\n\"No,\" Kris had said, as cheerfully as she could manage. \"Of course not.\"\n\nBut Penny went on talking, as if an explanation might make it hurt less.\n\n\"Admiral Santiago has offered us both a place on her staff. I'm to head up her intelligence section. She says it will mean captain stripes.\"\n\n\"How will Masao feel about you outranking him that much?\" Kris asked.\n\n\"He won't. Santiago has authorization to create a staff drawing in officers from all the Alliance we have out here on Alwa Station. And she can promote them, too. Masao is overdue for lieutenant commander, and commander as soon as she can find an excuse for it.\"\n\n\"I wonder why I never got that authority,\" Kris grouched.\n\n\"I don't know, but it's nice she has it. I've also talked with Taussig. We're going to go in on a farm along that big river in Rooster territory. We've asked for a young colonial couple to go in with us. They'll cover the place all the time, and we'll rotate being home.\"\n\nPenny paused. \"Home. Strange to call this place all the way across the galaxy home.\"\n\n\"If it's where your and Masao's roots are, then it's home,\" Kris had said.\n\n\"I've got to get past losing Tommy,\" Penny said, suddenly. She still choked when she said the name of her husband of three days, who gave his life so that she might live.\n\n\"Yes, you do,\" Kris said, softly. She suppressed a shiver at the memory of ordering the dumb metal of her ship to seal all the holes in its hull . . . and seal Tommy's fate.\n\n\"I know, Kris. I will. Masao is the man I want to do it with.\"\n\nTheir conversation meandered along for a while longer before Penny finally left, leaving Kris to contemplate the changes that life brought your way.\n\nCaptain, no, Vice Admiral Drago was next.\n\n\"Commander Pett do okay by you?\" he asked Kris after he had smiled at Ruthie Anne and assured Kris that she was a perfect baby.\n\n\"I'm alive, and the _Wasp_ isn't even in line for the body and fender shop.\"\n\n\"Any landing you can walk away from.\"\n\n\"So, what really brings you around?\" Kris had finally asked.\n\n\"Do you trust Commander Pett to get you and this cute little one back across the galaxy without a problem?\"\n\n\"I do. Why?\"\n\n\"I'm thinking of hanging around here for a while.\"\n\n\"Lots of people are,\" Kris allowed.\n\n\"I mean, it's not like admirals around here spend all their time trapped behind a desk. Everybody fights from four stars down to an Ostrich seaman recruit.\"\n\n\"It seems to go that way,\" Kris said, and paused to see if there was more to his hankering.\n\n\"And there's this colonial woman. She's buried two husbands and claims she can put me in the ground. I told her my wedding gift to her will be her very own burial plot,\" Drago said, grinning.\n\n\"Whatever floats your boat,\" Kris answered.\n\nThey talked for a bit longer, Kris praising Drago's handling of Fourth Fleet in the last battle. They left with promises to get together when next they shared the same space and share a beer and bring each other up to date.\n\nThere must have been something wrong with the air-recycling system; Kris needed a tissue to blow her nose.\n\nAmanda and Jacques were no surprise.\n\nThey dropped by to gush to Kris that Prime Minister Ada of the colonials had set up a Science Advisory Committee and invited both of them to fill slots on it.\n\n\"We'll have an entire budding economy to work on,\" Amanda said.\n\n\"She's in hog heaven,\" Jacques growled.\n\n\"And you aren't?\" Amanda growled right back. \"You've got all kinds of bird cultures to work with, and I've heard you say the humans have many divergent cultures among themselves, what with the colonials and the Navy and the industrial types.\"\n\n\"Oink, oink,\" Jacques said through a happy grin.\n\n\"And we've also been asked to establish Alwa's first university,\" said Amanda, getting even more enthusiastic. \"They haven't agreed on a name. Rita U just doesn't carry, and she's had so many other last names.\"\n\n\"Rita suggested we name it after her father, Nuu U,\" Jacques put in.\n\nThe two of them talked nonstop for most of the time they visited with Kris and only took a few moments out to fuss over Ruthie Anne before leaving to collect the last of their belongings still on the _Wasp_.\n\nIt took Kris a half hour of nursing Ruthie to calm down from that whirlwind.\n\nShe found herself wondering when Abby would come calling and what her own decision would be. They were only a few days shy of departure when Kris's maid, bodyguard, and so much more came into the _Wasp_ 's wardroom at lunch, went down the steam tables, then casually settled into the chair across from Kris.\n\n\"How are things going?\" Abby asked, buttering a roll.\n\n\"I would give half my trust fund to sleep through the night just once,\" Kris said.\n\n\"I hear that youngsters are death on sleep. What did Shakespeare say? 'Babes doth murder sleep.'\"\n\n\"You may have that wrong,\" Nelly provided from Kris's collarbone.\n\n\"Whatever.\"\n\n\"Did you snap up some of that free land down among the Ostriches?\" Kris asked, kind of sidling up to the big question.\n\n\"No, General Bruce and I have signed for some land along that nice, long river in Rooster territory. We've got the homestead between your Penny and Commodore Taussig and my boss Pipra.\"\n\n\"I figured Pipra would have you and Bruce as part of her farm crew,\" Kris said.\n\n\"She did, too,\" Abby said, \"but I set her straight. Steve and I will be teaming with four fab supervisors and another brigade commander. We got two colonial couples signed up to help us. It shouldn't be too hard, and if Pipra tries anything too friendly with my Steve, well, I still know how to shoot.\"\n\nKris took a long drink of milk. \"I don't blame you for not wanting to go back. You were just a maid there. You've really soared out here.\"\n\n\"I was never 'just a maid,' and you never treated me as if I were.\"\n\n\"Not after you saved my neck once or twice,\" Kris admitted.\n\n\"Even before I did,\" Abby put in. \"Kris, you good people, and I'm so glad I met up with you.\"\n\n\"I'm glad you've been a major part of my life for the last six years,\" Kris allowed.\n\n\"If I don't get out of here, I'm gonna start blubbering,\" Abby said, and suddenly, her place across from Kris was empty.\n\nKris would have happily left her lunch right there, too, but she was feeding Ruthie as well as herself. After dabbing at her eyes with a linen napkin, she resolutely finished her lunch, but ship food was never so lacking in taste as it was that day.\n\n* * *\n\nKris found herself blinking her eyes as she stared out the window at the sky over Wardhaven. She borrowed one of Ruthie's cloths and dabbed away the moisture threatening to run down her cheek.\n\n\"Would I have stayed there if we could?\" she asked Ruthie. Like all infants, Ruthie had no answers, only needs.\n\nThere was a knock at the door.\n\n\"I'm not decent,\" Kris called.\n\n\"I don't care,\" came in Jack's voice, and the door opened to let him in. He covered the distance with a few long steps, then traced Ruthie's busy cheeks with a gentle touch that also just happened to make feathery strokes on the breast the babe was busy draining dry.\n\nKris shivered at the pleasure.\n\n\"How are things back there?\" Kris asked a minute or so later, motioning to Mac's office with her chin.\n\n\"No blood on the carpet.\"\n\n\"Yet.\"\n\n\"I don't know. I think Ruthie has kind of secured a temporary truce. Ray hasn't asked Billy a single political question, and vice versa. The two women are talking maternity and sharing pictures of the latest grandkid or great-grand tyke. I won't bet money on it, but I think your mother might just be softening a bit.\"\n\n\"That would be nice. Maybe we can find peace at last.\"\n\n\"You both know what it's like to be a mother now.\"\n\nKris nodded, then changed the subject. \"Has anyone told you what's so on fire that they needed to haul me all the way back here to put it out?\"\n\n\"No, but I think Ray's about to split a gut to get it out for you. You weren't nearly as close to popping with Ruthie Marie as Ray is with his secret.\"\n\n\"Speak for yourself, husband. I'll decide how close I was to popping and who gets to claim they're worse.\"\n\n\"My mistake,\" Jack said quickly, but with a fond grin.\n\nKris assumed the grin was for her breasts hanging out while she tried to return Ruthie's logistics to the properly hidden place polite society demanded.\n\nJack might be grinning, but he was also reaching for Ruthie, pulling her to his chest, and giving Kris two free hands. With that bit of help, she was able to get properly uniformed without dripping milk down her dress blues. _Hurray for our side._\n\n\"Well, let's go see what sneaky mess Grampa Ray wants to get me into this time,\" Kris said, and led the way, letting Jack provide the rear guard, with Ruthie in his arms for the moment.\n\n# 3\n\nBy the time Kris, Jack, and Ruthie got back, the room had changed. Gramma Trouble and Brenda were on a couch, eagerly awaiting Ruthie. Kris took in the rest. The King sat at the head of a conference table. Crossie and Mac were at his elbows.\n\nThere was an empty chair at the foot of the table, with Billy and Trouble sitting at its elbows. Kris expected that chair was for her. What she didn't much care for was that the two empty chairs, one of which would have to be Jack's, were too far from her.\n\n\"Father, would you mind letting Jack have that chair?\" Kris said.\n\nBilly looked surprised, but he sheepishly rose and took the next chair down. Kris hoped that didn't put him too close to the other side. She'd hate to see him back Ray if it came to that.\n\nNope, the two of them were too much alike to agree on much of anything.\n\nSeated, Kris let the silence in the room stretch for only a bit, then said, \"Okay, why am I here and not out there where you said I was indispensable, Your Majesty?\"\n\n\"Do you remember Vicky Peterwald?\" the King said without preamble.\n\n\"How could I forget her? I saved her neck when the aliens wanted us dead, and she stabbed me in the back the first time she could get her lovely, ivory boobs in front of a camera.\"\n\n\"You don't like her?\" Ray said.\n\n\"I didn't say that. I pity the kid, growing up in that gutter that passes for Imperial Harry's Palace. What's she up to that makes you ask?\" Kris said, still no more informed about where this was going and getting less and less patient for Ray to get there.\n\nGrampa Trouble must have sensed Kris's mood; he jumped in. \"Greenfeld is going to hell in a handbasket. The Empress has made several attempts to kill Vicky and they are the two leaders of a civil war that threatens to tear Greenfeld apart.\"\n\n\"And that matters to me how?\" Kris asked.\n\n\"The Emperor has asked you to mediate between the two women,\" King Ray said. \"Since you once saved his life, he thinks you'd be fair. For some reason, Vicky has also accepted you as mediator.\"\n\n\"And the stepmother?\" Kris asked.\n\n\"We haven't heard from her,\" Crossie admitted.\n\n\"But if we did?\" Kris asked the intelligence honcho.\n\nCrossie scratched his left ear. \"I have no idea if she'd say something positive, but she hasn't said anything negative.\"\n\nKris let all that she'd been told run around in her head for a few moments. Her mouth started talking long before she wanted it to. \"Are you folks crazy? I'd have to be dumb . . . or insane . . . or the boldest person in human space . . . to get between the three of them. The only reason I saved Henry's worthless skin was because his death would have had my fingerprints faked all over his body, and there would have likely been a war.\"\n\nKris shook her head. \"As for him and Vicky, you can't trust either of them to agree to anything and keep to it for more than five seconds. I hear the Empress is even worse.\"\n\nCrossie had the good sense to nod along with Kris.\n\n\"So why are you trying to send me into this valley of death? You know, Ray, your last idea, setting us up for a stalking horse out on Alwa, was really dumb. There is no way anyone could look at our DNA and the DNA of the rest of the animals on Alwa and think we were from there. No way.\"\n\n\"We kind of figured that out,\" Mac said. \"Fortunately, we came up with the beam weapons. How were they?\"\n\n\"Horrible. Wonderful. They just barely saved our asses. They were one mad scientist's wild science project, and we just barely managed to duct tape them into something like a warship. You really need to do better next time.\"\n\n\"We're working on that,\" Ray said. \"Maybe after you do this little favor for the Emperor, you can help us.\"\n\n\"You mean maybe if I survive this little suicide mission,\" Kris said dryly.\n\n\"We plan to send you with a fleet of the new frigates,\" Mac said.\n\n\"Battlecruisers,\" Kris corrected.\n\n\"Rita has you call them battlecruisers,\" Ray said, his face clouding up.\n\n\"No, _I'm_ calling them battlecruisers,\" Kris snapped. \"What with twenty 22-inch guns, they can stand up against any battle line we've got.\"\n\n\"But they can't fight in a battle line. Their lasers all point fore and aft.\" Admiral Crossie, who'd never commanded a ship, provided the Navy input.\n\nKris's growled \"Who in here has the most experience commanding a battle line?\" was hardly a question.\n\nThe room fell silent, as if a guillotine blade had slashed through further comment.\n\nKris waited a long minute before going on. \"Besides, if I'm going to become the mediator hunting for any peace that can be found between these two crazies, I'll need to impress them. I expect to keep these admiral stripes, and I intend to lead a battle line. I'll also likely need a whole hell of a lot more, but that, at least, is a start.\"\n\n\"It's no big thing to change the manuals to say battlecruiser instead of frigate,\" Mac offered.\n\n\"You like those stripes,\" Ray said.\n\n\"You want to send a commander, or what am I, a captain, out there to patch together something between those three huge egos?\"\n\n\"I see you made your husband a lieutenant general,\" Ray said, eyeing Jack.\n\n\"He commanded a corps dirtside with three Colonial Guard divisions and attached Alwan auxiliaries as well as the Fleet Marine Force afloat. I see no reason the people risking their lives shouldn't have the rank to show for it. There was damn little pay.\"\n\nGrampa Trouble wetted a finger and made two marks in the air.\n\nGrampa Ray scowled at him but said nothing.\n\n\"Well, if that's settled,\" Mac said, \"I think we can call this meeting done and move on to other things. Admiral,\" he said, looking at Kris, \"I've had several of my staff examining what a mediation mission might look like.\"\n\n\"Fine, but one more thing, Your Royal Majesty, before we go,\" Kris said.\n\n\"Yes, Your Royal Highness?\" Ray said, returning formality for formality.\n\n\"When this mad job is done, no more. No more wild jobs. I want a job with a desk in Main Navy. I want Jack to have a desk, too. Maybe we share the same office, maybe he's on a different floor. But I want us to have lunch together without anyone trying to blow us to hell, and I want to go home at 1700 to the kids. Yes, I said 'kids,' as in two at least. You hear me?\"\n\n\"Message received and understood,\" the King said. \"But I'll bet you any amount of money that you won't last a month.\"\n\n\"And you'll lose,\" Kris said. Retrieving Ruthie from where Gramma Trouble was doing her best to settle her down after Mother had gotten her sputtering and spitting up, Kris led Jack from Mac's office.\n\n\"Take me to Nuu House,\" she almost begged Jack.\n\n# 4\n\nNuu House seemed somehow smaller, Kris reflected, as they drove up to the portico. It couldn't be that she was taller: She'd lived here through university. She'd thought herself fully grown to womanhood back then.\n\nMaybe it wasn't so much a case of size as perspective. She'd seen an alien raider's base ship up close and personal.\n\n_After you've seen one of those puppies, everything looks tiny._\n\nKris smiled at that reflection.\n\nA Marine from the escort opened the door for Kris. Jack preceded them in, as he usually did, to do one quick check for security. Strange to see him apply that standard procedure to Nuu House, but habit was habit, and Kris loved him for the way he looked out for her and Ruthie.\n\nKris walked into the foyer, with its spiraling black-and-white tiles . . . and came to a halt.\n\nOn the staircase where Abby had once stood, there was a matronly figure in some sort of gray-and-white civilian uniform backed up by four younger women in the same attire, each a stair step higher than the last. At the foot of the stairs, echeloned out from them, stood four hard-faced people, three men and one woman, who, even if they wore suits, were cut from same identical mold.\n\nThe older one looked familiar.\n\n\"You are . . . ?\" Kris started.\n\n\"The guy who yelled for you to stop the elevator before you let yourselves out on a Sarin-gas-filled floor.\"\n\n\"Ah, yes. I believe I thanked you for that when you visited me on the _Mutsu_.\"\n\n\"Yes, you did, Your Highness.\"\n\n\"And I asked you to answer a question for me.\"\n\n\"You did that, Your Highness,\" he said with the hint of a gleam in his eyes.\n\n\"You must have answered it. When I got the message on where the trading fleet was, I knew you'd gotten to the bottom of Grampa Al's suicidal plan. Was it too much of a problem?\"\n\n\"Not at all, Your Highness,\" he said with a gentle smile.\n\n\"Ha!\" That was from the young woman of the group. \"He nearly got himself killed, and he _was_ kidnapped.\"\n\n\"Yes,\" said a young male. \"We have pictures, all unsuitable for work.\"\n\nKris raised an eyebrow.\n\n\"Well, it may have been a tad tight for a bit,\" the senior lawman allowed.\n\n\"What brings you to Nuu House? Last I heard, you were a senior chief agent in charge for the Bureau of Investigations.\"\n\n\"I will likely be so again, Your Highness, but I and my talkative colleagues here have been seconded to the Secret Service to head up your protection detail.\"\n\nBeside Kris, Jack cleared his throat. \"We have several platoons of Marines to do that.\"\n\n\"No doubt, sir, but you will allow that we may be more informed as to matters here locally. You did quite well leading me a merry chase, but that was a bit over two years ago.\"\n\n\"He has a point,\" Kris allowed.\n\nJack considered that for a moment, then said, \"I will coordinate my Marine guard with your people. I'm a belt-and-suspenders man where the safety of my wife and daughter are concerned.\"\n\n\"I couldn't agree with you more. Now, Your Highness, as you may recall, I am Senior Chief Agent in Charge Taylor Foile; these are the people who will be heading up one of your around-the-clock shifts. Mahomet Debot,\" was a dark-haired man of swarthy complexion, as dark as Jack. \"Rick Sanchez,\" was only slightly lighter than Jack, \"and Leslie Chu,\" was a milk-skinned redhead who hardly looked the part of her surname. \"One of these three or I will be on duty at all times, leading a team to protect you that will rarely be less than twenty strong. Please let us know thirty minutes in advance of your plans to leave the house, so we can get a police escort.\"\n\n\"Someone must consider us a high risk,\" Jack said.\n\n\"I have not been told anything except to apply this level of protection,\" the agent said evenly. If he knew more, he was not telling.\n\n\"Does my father have this much in the way of his going out for a drink?\" Kris asked, eyes narrowing.\n\n\"No, ma'am,\" Agent Foile said evenly.\n\n\"Hmm,\" was all Jack had to add to Kris's thoughts.\n\n\"We must talk about this later, Agent Foile,\" Kris said, and turned to the five women arrayed up the stairs. \"And you are?\"\n\n\"We will provide twenty-four hours a day, seven days a week, care for your infant,\" the matronly woman answered.\n\n\"What weapons are you qualified with, and when did you last shoot?\" Kris snapped.\n\n\"Weapons? Shoot?\" the older woman said. If possible, the placid faces on her and her subordinates might have shown a hint of surprise.\n\n\"Who hired you?\" Kris asked, trying not to sound snappish.\n\n\"Your mother, although I think there was another, older woman involved in vetting us,\" the spokeswoman said.\n\n\"Not an old, ramrod-straight military-type guy?\"\n\n\"No, Your Highness. Although the older woman did have that kind of clipped way of speaking.\"\n\n\"It comes from cohabitating with Grampa Trouble,\" Kris allowed. \"Okay, you are not weapons-qualified. We'll have to see about changing that. Do you have a bag packed so you can run for the beanstalk if I suddenly decide to take off for space?\"\n\nNow the women's faces did drop. \"Ah, er, ah, we weren't told we would have to leave Wardhaven City,\" the older woman answered. \"And I, for one, have no desire to touch a gun, much less shoot one.\" Two of the other women nodded agreement. The other two exchanged looks that seemed to say \"I told you so,\" but they didn't join in the others' open refusal.\n\n\"Once again, my mother has done for me what I did not want or need. The three of you are dismissed. I will arrange for you to get two weeks' severance pay. Special Agent Foile, please see that full background checks are made on the other two. There is a pistol range in the basement. General, detail two Marines to begin their introduction to weapons. By the way, ladies, if you have friends who are qualified as nannies and sharpshooters, or willing to learn, you may invite them to come in for an interview tomorrow.\"\n\n\"Yes, Your Highness,\" the two said with a bit of a curtsy.\n\n\"Come with me,\" said a new voice.\n\nKris turned to find a young woman in Navy blues with the single stripe of an ensign on her sleeves and a gold aiguillette of staff on her right shoulder. \"And you are?\" Kris asked.\n\n\"I am Ensign Megan Longknife, your _aide-de-camp_ , secretary, and general dog robber, Your Highness. I qualified with several weapons at OCS last month, but you'll likely want me to get better. Now, I'll see that these three woman are properly taken care of and their access to the grounds revoked. May I borrow one of your Marines, General, to escort them to the front gate?\"\n\n\"Certainly. Carry on,\" Jack said, and she turned smartly to follow her orders.\n\n\"Just a moment, Ensign Longknife.\" The young woman turned back to Kris. \"I've never met you.\"\n\n\"No, ma'am. I'm from Santa Maria. The Alnaba side of the family.\"\n\n\"Ray's daughter?\"\n\n\"Is my grandmother,\" Meg supplied.\n\nKris nodded. \"Okay, carry on, and we'll talk later. Nelly, help her with the paperwork.\"\n\n\"I think I already am, Kris.\"\n\nThat brought a frown to Kris's face, but the young ensign had returned to her assignment, herding the three unqualified nannies from the foyer.\n\nKris turned back to Special Agent Foile. \"I assume your people are up to date on their weapons qualifications?\"\n\n\"Yes, ma'am,\" he said.\n\nThe young woman with him, who looked like she was about to split a gut, burst out, \"They even had us qualifying with machine pistols and long rifles. We knew something was up, but to be working with you, wow.\"\n\n\"You will have to excuse Leslie,\" Foile said. \"I think she was a founding member of your fan club.\"\n\n\"And we've been going crazy since you disappeared,\" Leslie said, jumping in.\n\nKris tried to keep from smiling at the young woman's enthusiasm. \"I'm afraid you will get little news from me,\" she said.\n\n\"Oh, it doesn't matter. One of us had a cousin on the _Wasp_. I think she got him drunk last night 'cause there was all sorts of stuff on the net this morning. Did you actually blow away seven more of those monster ships?\"\n\nJack raised both eyebrows. Kris rolled her eyes, hoping whoever had told the tale hadn't made it sound easy.\n\n\"On second thought, I may have to find time for some interviews,\" Kris admitted.\n\n\"We'll see that you get a better hearing than last time,\" Leslie assured Kris.\n\n\"I wouldn't be too hard to improve on that one,\" Jack muttered under his breath.\n\nBefore anyone could answer that thought, the doorbell chimed.\n\nA Marine moved quickly to check on the new arrival, then braced as he called \"Atten'hut,\" and opened the door wide.\n\n# 5\n\n\"As you were,\" General Trouble said before the civilians could even figure out what was expected of them. He then added, \"Relax,\" when the Marines failed to take a breath.\n\nHis wife and Kris's mother followed him into the foyer.\n\n\"Oh, how's my little grandbaby?\" Mother cooed, and headed for said child and proceeded to extract her from Kris's arms without so much as a \"may I?\"\n\nBefore Kris could cloud over, Gramma Trouble slipped between them. With a silent \"I'll take care of this,\" she edged Mother and Ruthie away.\n\n\"Where are the rest of the women I hired?\" Mother asked, taking in the two on the stairwell.\n\n\"They weren't weapons-qualified,\" Kris said.\n\n\"Why would that matter?\" Mother asked, befuddled.\n\n\"Let's take Ruthie upstairs and introduce her to her nursery,\" Gramma Trouble suggested, and placed an arm on Brenda that urged her toward the stairs.\n\n\"Come along, Kris,\" Brenda said, not looking away from the child in her arms. \"You'll want to see all the changes we made in the nursery. It's very modern. The latest safety equipment. You can monitor Ruthie's temperature, heart, and breathing from anywhere in the house, and the floors are rubber. She'll never hurt that cute bottom of hers.\"\n\n\"Kris'll come along later,\" Trouble said, determining Kris's future.\n\n\"I will?\" Kris said, temper starting to burn.\n\n\"We need to talk about your mission.\"\n\n\"Oh,\" Kris said, and allowed herself to be edged toward the library.\n\n\"Who do we need to include?\" Jack asked.\n\n\"I was thinking just you two,\" Grampa Trouble said, looking very much the general.\n\nJack shook his head. \"Special Agent Foile needs to know what's going on.\"\n\n\"You may well be right,\" Trouble said, and turned to the four agents. \"Foile, it's good to see you again.\"\n\n\"Will you be talking this time?\" had a sparkle around the senior agent's eyes.\n\n\"Will you be trying to arrest Kris and Jack this time?\" Trouble answered back.\n\nThe agent chuckled. \"Not very likely.\"\n\n\"Then yes, I will be talking. Come along and hear the worst.\"\n\n\"Mahomet, check this rock pile's security network. Leslie and Rick, familiarize yourself with the house. Are there any secret passageways, Your Highness?\"\n\n\"None that I ever discovered. Or my boyfriends, either.\"\n\n\"Have our teams look for them anyway,\" Taylor said, then followed them into the library and locked the door behind them.\n\nGrampa Trouble led the long walk past book-lined walls and more bookshelves to the fireplace and the two couches beside it. Kris and Jack sat on one. Trouble and Agent Foile faced them across a low table that would provide them with a network screen if they needed one.\n\n\"So, what brings you to our humble abode,\" Jack said, \"and why is our security detail larger than the prime minister's?\"\n\n\"Probably because you're more involved with the Peterwalds than Billy is or ever would want to be,\" Trouble said.\n\n\"Knocked that one out of the ballpark,\" Kris said, and made to watch as if a high fly ball sailed over the fireplace.\n\n\"Mind you, we don't expect there to be any problems here, but once you leave United Society space, it's anybody's guess.\"\n\n\"That's why I wanted two battle squadrons at least, maybe three. You think sixteen would be enough to defeat any Peterwald battle line?\" Kris asked Jack.\n\n\"Don't ask me, hon. I've been as out of pocket the last two years as you've been.\"\n\n\"Nelly, have you got an update on the Peterwald Order of Battle? No, Trouble, sir, shouldn't I be asking you?\"\n\n\"Your guess is as good as mine. Crossie lost several of his best agents in the Empire when they shot all the State Security types.\"\n\n\"Heavens, some of their intelligence people were on the take of our intelligence people?\" Kris said, shocked, just shocked. \"I think I've got the vapors.\"\n\n\"If Kris will stop being such a drama queen,\" Nelly said, \"I can give you a report.\" Nelly, Kris's computer, upgraded many times since she was given to her when she first started school, was worth a major chunk of one of her battlecruisers. Since her latest upgrade, Nelly had developed a tendency to argue with Kris and crack bad jokes.\n\n\"Go ahead, Nelly,\" Kris said.\n\n\"I've accessed the Wardhaven network, including all data authorized to flag officers, and I've correlated it to my satisfaction. Greenfeld has been falling farther and farther behind in technology as she tears herself apart. We have no reports of any battleships with lasers larger than 18-inch. They are still building ships out of traditional metal, no Smart Metal. I would expect that eight battlecruisers would be sufficient to win a battle with either Vicky's or the Empress's fleet. Still, it would be better to send sixteen so we could overawe them and not have to fight.\"\n\nTrouble raised an eyebrow as Nelly appended her recommendation to the end of her factual assessment.\n\n\"Nelly is quite good,\" Kris allowed.\n\n\"I'm magnificent. The Magnificent Nelly,\" Nelly crowed.\n\n\"Just ask her yourself,\" Kris grumped.\n\n\"Ask anyone who's alive because they had me to help them,\" Nelly sniffed.\n\n\"Down, girls,\" Jack said.\n\nGeneral Trouble swallowed a grin and went on, \"That is pretty much what we assumed would be the threat you face. I've been trying to persuade everyone that you deserve two full squadrons, but it's not easy to free up that many ships.\"\n\n\"Can I ask why?\" Jack asked cautiously.\n\n\"First, let me be clear, no one has spotted any evidence of the alien raiders, either in human space or around the Iteeche Empire. Whoever was nibbling around the edges of Iteeche territory has not shown up for a while. Believe you me, the Iteeche have built a whopping lot of frigates and searched all around themselves. I kind of like your idea of calling them battlecruisers now that they're sporting 22-inch lasers. Oh, and the Iteeche have given us some ideas that our scientists think can be stretched to 24- or even 26-inch, or focused even tighter into an 18-incher with double the range or quadruple the power.\"\n\n\"And Greenfeld is just sitting all this out?\" Kris said.\n\n\"While they tear each other apart and squabble over the rubble,\" Trouble added.\n\n\"I thought Vicky was smarter than that,\" Kris said. \"Kind of dumb, and soulless as a tree stump, but smarter.\"\n\n\"It appears that she is. She sent over a certain intelligence asset who brought us up to date on what was going on. He also asked for some spidersilk underarmor for Vicky and her team. Her stepmother has been trying to kill Vicky. Really trying to kill her, like trying three or four times a day while she was still in the Palace.\"\n\n\"Good God,\" Kris muttered, \"that beats my worst day.\"\n\n\"By a lot,\" Jack said.\n\n\"So she had to rebel just to keep her head on her shoulders,\" Kris concluded.\n\n\"Something like that,\" Trouble said. \"That and the mess her stepmother and family are making of the entire Empire as they tear it down and steal anything of value from the wreckage. A lot of people are at risk of starving to death. Your Vicky saved at least two worlds, maybe more, from the brink of cannibalism.\"\n\n\"She's not _my_ Vicky,\" Kris pointed out.\n\n\"Well, if I'd spent as much time as you have with her, and she was doing as good as Vicky has been doing in a bad situation, I'd want to claim a bit of the credit.\"\n\n\"Really?\"\n\n\"Wait until you get there, then make your own assessment,\" Grampa Trouble said.\n\n\"I will. By the way, you do know that while I'm pretty good at blowing things up, I don't have much experience getting people to play nice in the sandbox. As in never.\"\n\n\"Really,\" Grampa Trouble said, eyes wide in shock; then he chuckled, probably at the face Kris gave him. \"We kind of know that, so we've been looking for folks who might help you figure a way out of this for everyone.\"\n\n\"Who?\" Jack asked.\n\n\"Well, we've tapped the chief mediator of the Wardhaven Office of Mediation. He's had a lot of experience getting unions and managers to not kill each other.\"\n\n\"Maybe he could help us,\" Kris allowed.\n\n\"We've also got the senior arbitrator of the Bureau of Arbitration. They try to help business find a middle ground when contracts between corporations don't appear to be as clear as they all thought they were. It's an alternative to spending the next hundred years in court.\"\n\n\"Is he any good?\" Jack asked.\n\n\"I personally brought him in to help your grandfather Al resolve the problem with Mitsubishi on the new Smart Metal. If he hadn't gotten those two hardheaded types to settle things, you'd still be waiting for your first frigate. We also found out about some new idea Earth came up with to slow and redistribute laser light so you avoid burn-through. I put him and a team from Nuu Enterprises on the next liner to Earth, and he just got back from there with a contract and specs.\"\n\n\"Ah, Grampa, we reverse-engineered that crystal stuff out on Alwa Station. This latest _Wasp_ we rode in on has it. You might want to drop by and look at what we did to make the stuff from Earth better.\"\n\n\"Better, huh?\"\n\n\"Yep. We lost a couple of Earth ships the first time they got shot at, but we figured out why and improved the crystal armor that we grew for our new construction and backfit to our old ships.\"\n\n\"I take it that you really took the idea that you needed to create an industrial base force seriously.\"\n\n\"In spades,\" Kris said. \"It's amazing how wanting to stay alive will motivate people.\"\n\n\"I've found it works that way,\" Trouble agreed.\n\n\"So, that's two. Any more help?\" Jack asked.\n\n\"Wardhaven's Family Courts has a judge that does wonderful things for the children of families who are splitting the sheets. She has a real sense for what makes people tick.\"\n\n\"Does Vicky's stepmother tick?\" Kris asked.\n\n\"Like a time bomb,\" Trouble said.\n\n\"Then I definitely want two squadrons of battlecruisers with a company of Marines on each one of them,\" Kris said with finality.\n\nGrampa Trouble took a deep breath and let it out in a long sigh. \"That's why they sent me here.\"\n\n\"To give them to me?\" Kris said, not knowing what he was getting at but not liking it at all.\n\n\"No, I have to give you the bad news that you can't have but one squadron.\"\n\n\"I always thought Ray was becoming a bit of a coward now that he had that crown,\" Kris said.\n\n\"You might be right,\" Trouble said, unusually diplomatic. \"However, I got the job of telling you, it's just eight ships.\"\n\n\"Why?\" Kris said. \"I'm getting really tired of doing things Grampa Ray's way.\"\n\nInstead of answering her question, Trouble turned to Special Agent Foile. \"You have signed the Secrets Act?\"\n\n\"Yes, sir.\"\n\n\"What you're about to hear is covered under the Secrets Act, most especially a secret from Prime Minister Longknife.\"\n\nFoile's eyebrows rose slowly until they might have disappeared into his hair if he still had any. \"The more I deal with your family, Your Highness, the less I want to,\" he said.\n\n\"A smart man,\" Kris said. \"I had to go all the way across the galaxy to get them out of my hair . . . and then they called me back for this. Okay, you do agree not to tell my dad about what Trouble's about to let us in on.\"\n\n\"What passes here, stays here,\" the agent said.\n\n\"Good, good. Kris, we don't have enough ships,\" Trouble said.\n\n\"But you're building them at a mad pace,\" Nelly said. \"I reviewed the media reports, and they have a frigate commissioning close to every other day somewhere in the Union. Am I missing something?\"\n\n\"Yes, unfortunately,\" Trouble said. \"The policy for their deployment.\"\n\nThe old general eyed Kris. \"You really put the fear of God, or at least those crazy alien raiders, in people. Sending back that raving maniac was brilliant.\"\n\n\"Thank you very much,\" Kris said with a grin and a bit of a bow.\n\n\"But now, everybody and their cat and dog wants ships protecting their planet. With the Smart Metal frigate-size ships becoming major man-o'-wars, they can afford much cheaper ships and smaller, cheaper crews. More bang for their bucks. You know that place you got exiled to?\"\n\n\"Madigan's Rainbow,\" Kris provided.\n\n\"Yep. They ordered four squadrons for planetary defense.\"\n\n\"How are the locals taking to hairy deck apes on their shore leave?\" Kris asked.\n\n\"Not well, but they're holding on to the ships with a death grip. Practically everyone is. Someone stumbled on the old doctrine of a Fleet in Being, and they are using it to keep their little fleet right where their paranoia wants it.\"\n\n\"I foresee problems,\" Kris said, ticking several off in her head, \"including fewer ships for Alwa Station, but how does this keep me from borrowing sixteen for a while?\"\n\n\"Remember how your banged-up _Wasp_ took out that alien scout cruiser in an Iteeche system where you refueled before straggling back to human space,\" Trouble said. \"We need lookouts in our nearby systems to spot scout ships.\"\n\nKris shook her head. \"Yes, active patrolling is critical. But on Alwa Station, we picketed a lot of the jumps to give us plenty of warning.\"\n\nTrouble nodded. \"We've got more jump buoys out, covering a whole lot of systems. The Iteeche are doing the same, but some of those jumps are way out there. It takes forever for any word to get back to us.\"\n\n\"We had the same problem,\" Kris agreed. \"I started sending out patrols to pop their scouts before they could report back.\"\n\n\"That is exactly what I and half the folks in uniform on the general staff want to do.\"\n\n\"It's one thing to want to do it,\" Jack said. \"It's quite another thing to have the assets to do it.\"\n\n\"Right,\" Trouble said, and let out a long sigh that was more of a growl.\n\n\"You want to fill me in on how bad things are here?\" Kris asked.\n\n\"They could be worse. I wish they were better,\" her great-grandfather said. \"For most planets, immediate planetary defense is priority one. There is no priority two. For those willing to look beyond defending their own little corner of the universe, Alwa is priority two. You have to admit, a lot of them have made contributions.\"\n\n\"I most definitely admit it, and I am most definitely grateful for their gifts toward Alwa's defense. If you give me a soapbox, I'll declaim and bloviate on my and their gratitude until the cows come home,\" Kris said. \"Now, what's priority three?\"\n\n\"Few planets, in their present state of affairs, get past two. 'We've got a squadron fighting at Alwa.' 'We're still growing our defenses.' 'A frigate may be cheap, but you're still asking for tax money to build and maintain it.'\"\n\n\"Okay, but what is priority three?\" Kris repeated.\n\n\"Patrolling. Taking a look at our outer perimeter. Maybe doing a long cruise well outside our part of space to see if anything is roving around there. That first ship you blew away, the family ship that was mining that moon or asteroid. You remember it?\"\n\n\"I still have nightmares about the frozen faces of those little kids when I'm not having nightmares about the huge base ships I've blown away or the horror they'd raise if they got into Alwa orbit,\" Kris admitted.\n\n_I have a lot of nightmares._\n\n\"So, are there any mining claims being worked anywhere close to us? Those sort of things?\" Jack asked.\n\n\"How many ships have you got on patrol duty?\" Kris added.\n\n\"Not nearly enough,\" Grampa Trouble admitted. \"Savannah contributed a squadron. They'll do anything for Ray, but not twice. Here and there, we get one or two ships from this or that planet. But from Wardhaven and Pitts Hope, not so much as a longboat. 'We've made those beam ships and flying factories. What more could you want?'\"\n\n\"You need to make more beam ships,\" Kris said, \"and better ones. The three you sent got hammered pretty bad in our last fight. The next ones really need to be warships first and mad science projects second.\"\n\n\"I had a suspicion you might say that,\" Trouble said with a chuckle. \"We wanted them Tuesday, not great, and you were doing good to get them by Tuesday.\"\n\n\"Oh, we're very grateful,\" Kris was quick to put in, \"but the next batch needs to be better. Maybe we can survive until Friday for them.\"\n\nTrouble chuckled at that. \"You can look into that when you get back. You really demanding a desk job?\"\n\n\"I am,\" Kris shot right back. \"I've had enough of Ray's running me around by this gold ring in my nose. I've got a ring on my finger, thank you very much, and I'm ready to move on to something more domestic. Okay?\"\n\n\"It's fine by me,\" Trouble said, holding up both hands in quick surrender.\n\n\"So how do I survive this next suicide mission and make it to pleasant domesticity?\"\n\nTrouble sighed. \"We can give you eight ships. They're commissioning as we speak.\"\n\n\"Well, hold up their commissioning and get them crystal clad.\"\n\n\"I'll see what I can do, but Al's got his contract and wants several pounds of flesh to make a contract change.\"\n\n\"I'll see if Ruthie can't persuade him with her toothless smile.\"\n\n\"Good luck at that,\" Trouble said. \"About Marines, we're running short. Your dad is building up an Army here on Wardhaven. Among other things, they're manning lasers that someone thinks can shoot things out of orbit that are busy lazing them from up there. I kind of doubt that's a good use of manpower, but local politicians want to show their voters they're doing something they can see. Anyhoo, it's eating into our Marine recruitment.\"\n\n\"You might want to have Marines on your ships,\" Jack said. Trouble raised an eyebrow. \"The last couple of fights they were sending small suicide boats and even lone troopers at us. We used Marines to shoot down those small targets. If they actually managed to board us . . . ? Well, you get my meaning.\"\n\n\"Damn,\" Trouble muttered. \"You folks come back safe. I really want you going over our defense effort from top to bottom.\"\n\n\"I always do my best to stay not dead,\" Kris said.\n\n\"Well, I think I've done about all the upsetting you that I can do today,\" Trouble said, \"and vice versa. I'll be seeing you, likely tomorrow.\"\n\nKris nodded but stayed seated as Trouble left. When Special Agent Foile made to stand, she guided him back to his seat with her eyes. He sat.\n\n\"Well,\" Kris said, leaning back in her chair and taking a deep breath as the door closed behind her grampa. Refreshed, she turned to Agent Foile. \"Does this tell you more of what you need to do your job?\"\n\n\"It tells me why I've got the biggest detail on this planet.\"\n\n\"You up to the job?\" Jack asked.\n\n\"If we are not, I'll see that my detail is expanded until we are.\"\n\n\"Very good. Oh, we may be going out soon,\" Kris said.\n\n\"May I ask where?\"\n\n\"Jack's folks deserve some time with their latest grandchild,\" Kris said.\n\n# 6\n\nAn hour later, with both Jack and Kris changed into civvies, their car came to a stop before a small white house on a minor side street. Kris had skipped the armored limo or SUV for this trip; Jack said his family's neighborhood was quiet. The sudden appearance of a lot of police would only make the neighbors talk.\n\nHarvey, a wonderfully familiar face and Kris's chauffeur since forever, selected the green sedan. \"It's well armored but doesn't make a big show of it,\" he told Special Agent Foile. Harvey and Foile were in the front seat, with Agent Debot squeezed in between them.\n\nMahomet had showed up with a black box that looked amazingly like the one Chief Beni, both the elder one and his lost son, had used to protect Kris from so much mischief.\n\nKris and Jack shared the backseat with a happily gurgling Ruthie. Kris's child was likely the best-protected kid who ever rode in a car. The Alwa expedition was not prepared for children. Nelly had used Smart MetalTM to make a high-gee station for Ruthie for the fast trip back from Alwa. When they arrived here, Nelly had modified the high-gee station into a car seat for the little one.\n\nJack took in his parents' house with a happy sigh. Kris studied it carefully. It was just a year or two past needing a new coat of paint. The porch had a swing fit for two as well as two rocking chairs. It looked like a lot of the houses she'd driven by in the neighborhood.\n\n_So this is where Jack grew up. I hope this helps me understand him better._\n\nThe curtains were pulled back a bit. Someone was taking their own measure of who this visitor might be. What they saw was just the green sedan. The police SUVs were parked at each corner of the street, distant but ready.\n\nSpecial Agent Foile got out first, eyeing the street cautiously. He seemed content with what he saw and tapped the roof of the car. Only then did Jack get out and make his own security assessment. When he was happy, and Mahomet was out doing his own check, Kris was motioned to unstrap Ruthie, sling her diaper bag over her shoulder, and see how much trouble it would be to get one small baby out of a car.\n\nIt was not easy, but Kris, who'd managed a lot bigger things, succeeded in getting her infant child extracted and wrapped in a clean blanket.\n\nWhoever was in the house must have liked what they saw because a high-school-size girl, a woman who must have been her mother, and a man, hobbling on a cane, came out of the house just as fast as their legs could carry them.\n\n\"Jacky,\" \"Juan,\" and \"My boy,\" told Kris that she'd come to the right place.\n\nThe charge came to a dead halt as they took in Kris and the bundle she carried in both arms.\n\n\"Juan?\" the matron of the trio questioned cautiously.\n\n\" _Mia madre_ , I would like you to meet your daughter-in-law, my wife, Kris Longknife, and our little one, Ruth Maria.\"\n\n\"You got married and didn't tell us!\" his mother shouted but was moving forward again as fast as her short legs would allow.\n\n\"I was all the way on the other side of the galaxy, Mom. I asked her to marry me, and it seemed like a long time to wait until we got home again.\"\n\n\"Well, show me _la ni\u00f1a_ ,\" she said, swatting her son with the apron she wore.\n\nThe girl was standing back a bit. \"Jacky, is that who I think that is?\" she said, taking in the agents and the chauffeur.\n\n\"Most likely, but hold your tongue a bit until I can do the honors properly.\"\n\n\"Oh. My. God,\" Jack's sister whispered.\n\nHis dad had just joined the foursome when Kris managed to pull the blanket aside and let Ruthie Maria have a good look her father's side of the family. She smiled happily and managed some sound that Kris assumed was \"Hi,\" in infant speak, which Nelly had tried but failed to translate.\n\n\"Won't you come in,\" Jack's father said. With his family leading the way, and Jack beside Kris with a loving hand pleasantly in the small of her back, she found herself ushered into their living room. The couch had a throw over it with a picture of Our Lady of Guadeloupe. A table had a picture of Christ with a crucifix hanging above it. Kris had been in such homes while campaigning for Father. She clicked on her \"Campaign Acceptance Mode,\" and took the seat she was offered on the couch.\n\nJuan, Jack's dad, settled into his own recliner and raised the leg support. Maria took her place on the couch, where she easily managed to switch Ruthie from Kris's arms to her own. Estella looked ready to have a fit, but she settled onto the arm of the overstuffed chair that Jack took.\n\n\"So, who is your wife?\" Juan asked before Estella busted a gut.\n\n\"You remember that I had been protecting Kris Longknife,\" Jack said, and they nodded. \"Well, she fell in love with me, and me with her, and when we had a very brief chance to marry all the way on the other side of the galaxy, we took it.\"\n\n\"All right!\" sister squeed.\n\nThe parents looked poleaxed. \"Kris Longknife,\" they both muttered.\n\nThe room fell into deep silence. Even sister found herself taken by it.\n\nThe quiet was finally broken by a whispered question from Maria. \"So, tell me about your wedding.\"\n\nKris did, and the talk went from there to other things. Estella had heard about the alien raiders and wanted to know what Jack and his new wife had done. Jack tried to make as little of his job as he could, and Kris followed suit until Estella did a quick net search, read the results then eyed Kris. \"You were the commander of _everything_ we sent to that other place.\"\n\n\"Yes,\" Kris admitted, \"I was. Your Jack commanded all the ground forces, and I did the same for the fleet.\"\n\nAgain, the parents looked speechless.\n\nWhen the conversation finally began again, it was strained. Ruthie took that moment to get fussy, and Maria returned her to Kris's arms. Together, they checked the diaper and found it dry and clean. \"She must be hungry. Do you have some formula?\" Maria asked.\n\n\"We weren't prepared for children on Alwa Station, so we've kind of had to do things the old-fashioned way. I'm nursing Ruth Maria.\"\n\n\"Oh,\" was all Maria said. \"I wouldn't think that a woman like you would know how to.\"\n\n\"I had to learn fast,\" Kris admitted, \"and Ruth Maria helped.\"\n\n\"Well, you will come again, won't you?\" she said, and Kris found herself following the mother's lead as she rose to her feet and was ushered out the door.\n\nJack followed with the diaper bag, still talking to his father. Estella seemed delighted to stay as close to Ruthie as she could. At least one of them had a fan in Jack's family.\n\nThey were bundled up and driving away. Kris had taken advantage of the privacy the backseat afforded and was nursing Ruthie before she got a word out.\n\n\"What happened back there?\"\n\n\"You may have noticed that my mom sticks with the old ways.\" Jack said.\n\n\"You could hardly miss it,\" Kris said, bouncing Ruthie just a bit. She was eagerly nursing but still seemed a bit fussy. Maybe Kris wasn't the only one upset by their reception.\n\n\"Mother had expectations that I'd marry a good Catholic girl. Then there was the wedding.\"\n\n\"What about the wedding?\"\n\n\"Nothing, it was a wonderful wedding, and you told it beautifully.\"\n\n\"But?\"\n\n\"Your Granny Ruth officiated.\"\n\n\"And?\"\n\n\"Mother doesn't consider a marriage valid unless there's a priest involved.\"\n\n\"Oh,\" Kris said. And let her logical mind try to follow all the threads she'd just been handed.\n\n\"Would she like to have us married again before a priest?\" she finally said.\n\n\"Would you mind?\"\n\n\"I'd do anything to make you happy, honey, but you must know that if your mother wants a wedding, my mother will go crazy over it.\"\n\n\"The thought did cross my mind as I listened to Mom put in her two bits.\"\n\n\"And here I'd hoped to dodge the zoo my mom would make of my wedding.\"\n\n\"Face an alien base ship out for blood, no problem. Face your mother in full wedding mode, run for the hills.\"\n\n\"Hills, hell, I ran for the other side of the galaxy.\"\n\nThey shared a laugh at that.\n\n\"Well, look at the bright side,\" Kris finally said. \"If Vicky and the rest of the Peterwalds get us killed, at least we won't have to keep our families happy anymore.\"\n\n\"You always are the optimist.\"\n\n\"Get us off this street and out of here!\" yelled Agent Debot. \"A bomb just went active ahead of us.\"\n\n# 7\n\nKris knew that Harvey was an old war vet. She'd just never seen him in war mode. He hit the brakes. Jack popped his seat belt and rolled over, putting himself between the front seat and a nursing Ruthie. Kris held on tight and leaned back to give Ruthie as much space to slow in as Jack waiting at the seat ahead.\n\nHarvey turned the hard brake into a \"J\" turn and had them headed back the way they came. The trailing SUV managed to follow them through these hard maneuvers.\n\nThe lead one wasn't so lucky.\n\nIts driver got the word late. He was in the middle of the hard brake when the road beside him came at him as if shot out of a cannon. Out of the corner of her eye, Kris saw the big black rig flip over on its side, but the armored box held. One of the doors had been raised, and men with automatic rifles were searching for targets even as Kris and her family went the other way.\n\nIf this neighborhood was normally a quiet place, it wasn't now. Sirens started shrieking from all directions as they closed on either the bomb site or Kris's sedan. Fortunately, the motorcycle cops had been told to wait well back. They were the first to join Kris's reorganizing cavalcade. The trailing SUV pulled ahead of Harvey just as Mahomet shouted, \"I think there's another bomb ahead of us.\"\n\n\"We're hooking a left,\" Foile said, to both Harvey and the rig ahead of them.\n\nOnly Harvey was able to make the turn, but this time the warning got out quickly enough. There was no boom as the cycle cops and the SUV did hard \"U\" turns, then followed after Kris and family.\n\n\"Turn left at the next block,\" Foile ordered.\n\n\"Right next chance you get.\n\n\"Left again.\n\n\"Right.\"\n\nAnd with more lefts and rights, they made their way out of the neighborhood. Only when they were back on the expressway with police cars six deep before and after them did Kris breathe a sigh of relief.\n\n\"What was that all about?\" Jack demanded of no one.\n\n\"I think,\" Kris said, \"that Vicky's stepmommy wants us to know we are not welcome to stick our noses into her affairs.\"\n\n\"You think so?\" Jack said.\n\n\"Honey, I know so.\"\n\n# 8\n\nLater that afternoon, Special Agent Foile had senior representatives from half a dozen police and security agencies in the library and, as it were, on the carpet.\n\nKris settled herself in the back of the library well away from the blowtorch Agent Foile was wielding up front. She sorely missed Penny, who had chosen to stay on the other side of the galaxy. Coming from a police family, Penny had always been Kris's liaison with local police. She was also great at translating Cop Speak.\n\nJust now, Kris was listening to a lot of Cop Speak.\n\nLeslie Chu settled into the chair next to Kris and began doing a very fine job as translator.\n\n\"My boss is asking all those big boss men how it happened that their agencies didn't even have a hint of that bomb. No chatter. No nothing.\"\n\n\"Do they have any answer?\"\n\n\"Nope. _Nada._ \"\n\n\"I guess I've been gone so long, even Wardhaven has forgotten what it takes to keep this damn Longknife safe.\"\n\n\"It kind of looks that way.\"\n\n\"Oh, did your boss give you the picture he asked me for?\"\n\n\"Oh, yes,\" Leslie squeed softly, if that word even applied at that volume. \"Ah, now my boss is telling them they've got to get their, ah, act together and keep it together until you leave for Greenfeld. He's not telling them when that will be.\"\n\n\"Smart man since I have no idea, either.\"\n\n\"Anyway, he's read them the riot act and told them all to straighten up and fly right. Ah, do you think you might have someplace on your staff you could assign us for a bit? My boss is burning bridges. What he said to the head of the Bureau is not going to be forgotten soon.\"\n\n\"I think I have some vacancies. You do know we're heading closer to the Empress. It can only get worse.\"\n\n\"Yeah\"\u2014Leslie shrugged\u2014\"but it won't be boring.\"\n\n\"Life around me is many things, but never that.\"\n\nThe agency heads left, softly talking among themselves. Hopefully, they were already planning ways to keep Kris safe and not the manner and fashion of their hanging Special Agent Foile up by his fundamentals.\n\n\"Well, that went amazingly well,\" Foile said as he joined Kris and his junior agent.\n\n\"Kris said she may have room for us on her staff,\" Leslie said.\n\n\"Oh.\" From the senior agent, that carried heavy freight.\n\n\"From the looks of things, you might need a job,\" Kris said.\n\n\"Oh, I should think not,\" he said, eyeing the departing agency chiefs. \"Hmm, well, maybe.\"\n\n\"We'll see how things are when I've got a squadron ready to sail and a whole lot of passengers for them.\"\n\n\"We can think about that later,\" he said.\n\nKris and Jack enjoyed a quiet evening at Nuu House. No one attempted to kill them right up to bedtime. Or if they did, they didn't get past the electronics, Marines, and other guards.\n\nNelly turned Ruthie's car seat into a nice bassinet. Surprisingly, she let Kris sleep through half the night.\n\nThe nanny taking the night watch was there the first time Ruthie stirred. She changed her diaper as Kris was slowly rising to wakefulness. \"I gave her some pabulum before we put her down. You nurse her, and I'll see if she'll take more solid food and let you sleep through the rest of the night.\"\n\nKris seriously wondered if the afternoon bomb attack had been successful. She certainly felt like she'd died and gone to heaven.\n\nOver breakfast the next morning, Kris and Jack examined what they should do with their day.\n\n\"If we're only going to get eight ships, I want them all with crystal armor,\" Kris said.\n\nJack agreed. \"What ships will we get?\" he asked.\n\n\"Your guess is as good as mine,\" Kris said. \"Nelly, anything?\"\n\n\"The _Wasp_ has been taken into the yard for maintenance,\" the computer said. \"Two squadrons from New Canaan and Tangaroa will be coming through here in two weeks, and Commander Pett wants to join up with them for the trip back.\"\n\n\"Trip back?\" Kris yelped.\n\n\"All but three of the _Wasp_ 's crew voted to go back. She will be taking on a couple of hundred volunteers for either the Navy or industry. It sounds like New Canaan and Tangaroa will also be traveling heavy.\"\n\n\"People _want_ to go to Alwa?\" Now it was Jack's turn to yelp.\n\nRuthie decided to join in, but her yelp was cuter than Jack's.\n\n\"It seems so,\" Nelly answered them both.\n\n\"Any chance we could shanghai the two squadrons of transients to add to our Wardhaven squadron?\" Jack asked.\n\nKris considered that for a long moment, then shook her head. \"Do you really want to be responsible for Sandy not having the ships she needs? Can you think of any reinforcements we got that we didn't need immediately if not sooner?\"\n\nJack shrugged. \"I guess not, but I had to try. I was kind of hoping that since you trimmed them back so far, Admiral Santiago would have a while before the aliens got rambunctious again.\"\n\n\"I thank you for trying, my ever-vigilant security chief, husband, and daddy,\" Kris said, and leaned over to give him the kiss he deserved. \"I'd like to think Sandy will have an easy time of it, too, but . . .\"\n\n\"I wonder how it's going for her,\" Jack said, then did a hard right turn. \"So we settle for eight battlecruisers?\"\n\n\"I didn't say that. I just said we don't steal anything from Alwa. Everything else in human space is up for grabs.\"\n\n\"And you Longknifes are so grabby.\"\n\nKris made a grab for something else, and Jack moved to block her, but not too much. Unfortunately, the nanny with the morning shift entered to remove Ruthie.\n\n\"Don't let me interfere with anything. We always say, the more kids the merrier, and the more jobs for us,\" she said with a cheerful smile that was aimed at Kris and not at all flirty toward Jack.\n\n_She better not be._\n\nBut the mood was broken, so they retired to dress in full admiral's and general's regalia and prepared to face the morning. \"Upgrading the armor on the battlecruisers will take the longest,\" Kris said, \"so I guess we go see if Grampa Al will let us talk to him.\"\n\nSpecial Agent in Charge Foile seemed to find his morning assignment much to his liking, as did Agent Rick Sanchez. \"Will it be any easier to walk in this time than last?\" the younger agent asked through a broad grin.\n\n\"We shall see what we shall see,\" Foile answered vaguely but with a sparkle in his eyes.\n\nKris made sure Ruthie was all decked out in maximum cuteness, and, with her babbling happily, they departed in a well-armed and armored convoy for Nuu Towers.\n\nKris did try to phone ahead. She got nowhere, so she made a quick call to Grampa Trouble, and, amazing as it might appear, the gates opened wide for her . . . and the doors as well.\n\n\"Trouble has an in with Al?\" Jack asked in surprise.\n\n\"Trouble has ins everywhere. He knows where all the bodies are buried,\" Kris said. Still, she noticed the ever-so-slight smile on Special Agent Foile's lips and found herself wondering who had an in with whom.\n\nThis time, they were ushered through the huge foyer of the building with its black marble and directly into an elevator. Kris smiled; it only went up fifty floors. There they transferred to another bank of elevators for the next fifty before transferring to a third elevator bank for the ride to the 150th floor.\n\n\"Amazingly faster than the last time, I expect,\" Foile said to Kris.\n\n\"Are you packing spare gas masks?\" she asked.\n\n\"Sarin will get you through the skin,\" the agent pointed out.\n\nKris shivered. \"Did he really have Sarin gas?\"\n\n\"I do not know,\" Foile said. \"I left too soon to see how they cleaned up the top three floors, and you kept me very busy for the next couple of weeks.\"\n\n\"Again, may I say, thank you for the warning.\"\n\n\"You're very welcome,\" the agent said.\n\nThey reached the topmost floor; the elevator opened on a wide field of lush red carpet. There were only three people in evidence; they sat behind well-spaced desks and were doing their best to appear most busy. The executive offices were thirty or forty meters away, likely all with astounding views, but they were hidden for the moment behind rich mosaics of wood that provided their own view of stylized office parks and fabrication facilities.\n\nOne even showed High Wardhaven Station with the Nuu space docks prominent.\n\nGiven no hint of where to go, Kris headed for the largest desk, the one in the middle.\n\n\"We are expected,\" she said.\n\n\"Are you now?\" said the man, crewcutted and maybe with an automatic bulging the left shoulder of a suit that likely cost more than Jack's folk's home. \"And your name?\"\n\nKris played the one card she had. Well, second card. Her other one involved accepting that perpetual job offer and coming to work for Grampa Al in his Tower of Insecurity. \"Ruthie Anne is here to see her great-grandfather.\"\n\nThat brought a raised eyebrow from the even-more-neutral face of the secretary.\n\nThe man began to speak into the desk's commlink. Some new technology blanked out his words even though Kris was less than two meters away. Kris was no expert at lip-reading, but even she spotted the sequence \"uniform of a four-star admiral,\" followed later by, \"Yes, the baby is cute.\"\n\nAfter a few more exchanges, the secretary leaned back, and said, \"You and the baby may enter.\"\n\n\"My security agents will check out the room, and my husband will accompany me,\" Kris countered.\n\nThe secretary took a melodramatic moment before nodding.\n\n# 9\n\nKris stood, gently bouncing Ruthie on her left hip, while Senior Special Agent in Charge Foile and his sidekick, black box in hand, entered the door identified by the secretary's slight incline of the head.\n\nThe door closed behind them, leaving Kris to wonder if they would ever emerge, but they did. With a curt nod to Jack, Foile identified the room as satisfactorily secure.\n\nJack led Kris to the door that Sanchez now held open for them.\n\nTo say that Grandfather Alexander Longknife's office was way past palatial was still an understatement. The Forward Lounge on the _Wasp_ was rarely expanded to this extent. The wet bar to Kris's right did fall short of the Lounge's bar, but not by much.\n\nIn a corner to her left, a waterfall descended in gentle ripples to a pond big enough to have a bridge across it in the Chinese tradition.\n\n_Were there_ real _koi in that pond?_\n\nApparently, yes, because one broke the surface for a moment. No holograms here. This was the real thing.\n\nGrampa Al himself sat behind a white alabaster table. No desk for him. There was also no conference table although two doors led right and left from his office. No doubt, one led to a bathroom, the other to a conference room likely the equal of this room.\n\nThe man of business rose, motioning them toward a group of chairs not far from the bar. There were only three large, overstuffed armchairs. Beside one, a bassinet stood.\n\nKris aimed herself for that one as Jack took the one between her and Grampa Al. He, however, headed for the bar. \"What will you have this morning?\"\n\n\"I'll have what you have, sir,\" Jack said diplomatically.\n\n\"A glass of soda water with a twist of lime,\" Kris said.\n\n\"You still not drinking?\" Al asked.\n\n\"I'm nursing,\" Kris said. \"I don't want Ruthie to get drunk on her momma's milk.\"\n\n\"A good excuse. Better than your usual one. General, I'm drinking chamomile tea; is that good enough for you?\"\n\n\"Yes, sir,\" Jack said.\n\n\"Me, too,\" Kris said.\n\n\"So we are all seeking calm for this little _t\u00eate-\u00e0-t\u00eate_.\"\n\n\"We needed full MOPP gear for the last visit,\" Kris put in.\n\n\"Yes. You gave me quite the fright. Then an even worse fright when I discovered it was you. You finished by making a mess of the south side of this building.\"\n\n\"I regretted that, but your space shuttle was the only escape option I could figure out at the moment.\"\n\n\"Escape to a Musashi axe?\" Al pointed out.\n\n\"I was not found guilty,\" Kris pointed right back.\n\n\"Yes. You do seem to have all the luck of your generation,\" he said, handing a cup and saucer of tea to Kris.\n\n\"I seem to need most of it,\" Kris said.\n\n\"That I would not dispute,\" he said, giving Jack his tea. \"Do you, as husband and father, like your wife's seeking out every available lion's mouth to stick her head in?\"\n\nJack did not shoot back an answer but instead took a slow sip of tea as Al settled into his chair. Only then did he say, \"That is the woman I married. Should I marry an eagle, then shackle her to a stove?\"\n\n\"I foresee much sorrow in your life,\" the older man said. For someone of nearly a hundred years, he looked well preserved, hardly much older than Honovi, Kris's older brother. Hopefully, Al was being careful with his rejuvenation treatments. Some had paid a high price for using them as a fountain of youth.\n\n\"As you likely know, I have been asked to serve as mediator between the Grand Duchess, Vicky Peterwald, and her father, the Emperor.\"\n\n\"More likely between her and that vicious stepmother of hers, the Empress,\" Al cut in.\n\n\"I was about to add that when you took the words out of my mouth.\"\n\n\"Yes, how unsanitary of me,\" Al said dryly, sipping slowly.\n\n\"I have been offered eight battlecruisers as an escort.\"\n\n\"Only eight. Anyone who takes that mission is either dumb, or bold out of their mind. Which are you, young admiral?\"\n\nKris shrugged. \"I'm the one asked for by two sides, Vicky and her father, and ordered by another, your father, Ray.\"\n\n\"Ever the bold one he is . . . when it is someone else's blood being pumped into the gutter,\" Ray's son said with a scowl.\n\n\"So I have heard,\" Kris admitted.\n\n\"Then you have not listened very well.\"\n\n\"I am listening, as carefully as I can,\" Kris said.\n\n\"And you,\" Al said, indicating Jack with his nearly empty teacup.\n\n\"I'm listening, too, and doing what I can to increase the odds of her survival.\"\n\n\"Hmm,\" Al said as he returned to the golden samovar of hot water to prepare himself another cup of tea. \"And you came to me?\" he asked, turning back to them with a full cup.\n\n\"Yes. When the Earth frigates came out to Alwa Station, they were clad in a crystal armor that took in laser light and distributed it around the ship before radiating it back into space.\"\n\n\"I know. I negotiated with Earth for the rights to it. They wanted an arm and a leg. What good are low-cost frigates if they suddenly cost half your flesh just to add to decadent Earth's wealth? Battlecruisers, did you call them?\"\n\n\"Yes. With all that armament and armor, they are as good as any battleship while being as fast and maneuverable as a cruiser.\"\n\n\"Hmm, I heard that the ship, _Wasp_ is it, that brought you back from Alwa, was crystal clad. How did that happen?\"\n\nWhich led to Kris's telling him how the Earth ships had not fared so well in their first combat and the changes the shipyards on Alwa Station had made to them. The yard then applied crystal to all the ships, including the new arrivals from human space.\n\n\"My computer has the way we grow the crystals, sort them to an exact length, then apply them to ships quickly.\"\n\n\"Ha,\" Al cackled. \"I bet I can turn the tables on Earth with that package you've brought me.\"\n\n\"Likely you can, but I need eight armored ships right now. If I'm to have any chance of coming out of Greenfeld space alive, I need to survive being on the receiving end of a sneaky broadside and live to return fire.\"\n\n\"Maybe you're not such an idiot,\" Al allowed.\n\n\"Can you armor my ships?\" Kris asked, eyeing him over her near-empty cup of tea.\n\n\"Will you give me the file?\"\n\n\"Nuu Enterprises on Alwa Station did most of this work.\"\n\n\"How is Irving doing?\"\n\n\"Irving?\"\n\n\"Yes, the fellow I put in charge of my interests out there.\"\n\n\"I think he drank himself to death,\" Kris said. \"Pipra Strongarm is running the show for Nuu and several other industrial interests.\"\n\n\"Who's she?\" Al asked.\n\n\"Pipra Strongarm,\" Al's computer answered in a lush female voice, \"was Third Vice President for Human Resources.\"\n\n\"Third Vice President!\" Al didn't quite squeak.\n\nKris failed to suppress a smile. She'd taken Pipra at her claim to be next in charge. She'd never asked Pipra what she was in charge of.\n\n_I've been jobbed, but you kept us all alive. Go, gal._\n\n\"Do you want the files?\" Kris asked.\n\n\"Yeah. Yeah, give them to me.\"\n\n\"You will get the eight battlecruisers ready first and in time for my sailing,\" Kris stated, not even adding please.\n\n\"Yes, yes,\" he snapped, \"assuming the Navy gives me the bid. You know, it would be easier and definitely faster if you gave me a couple of kernels from your computer, so I could have a top-of-the-line computer, you know, and see that things go well for you.\"\n\n\"No,\" Kris said.\n\n\"Definitely no,\" Nelly added from her collarbone.\n\n\"But this is a complicated process, wrapping crystal armor around a ship the size of your battlecruisers. Especially one that keeps changing sizes, you know,\" Grampa Al insisted. \"Without something to do all the work and coordination like your Nelly, it could take us a year or at least six months.\"\n\n\"All the research and design work, even down to what is needed to quickly clad the battlecruisers, has already been done,\" Kris pointed out.\n\n\"And I would never allow you and your likes to have one of my children,\" Nelly snapped. \"Heaven knows what you would put the poor kid to doing. No. Never.\"\n\n\"Are you going to let your computer, ah\"\u2014Al struggled to get his mouth around what he needed to say, and settled for\u2014\"tell you what to do?\"\n\n\"I don't usually, but in this, we are of an agreement. Nelly, what other shipyards do they have up on the station?\"\n\n\"Nazareth Steel, Shipbuilding, and Space Docks is building frigates,\" Nelly informed Kris.\n\n\"Steel is one thing. Growing crystal is totally different,\" said Al, butting in.\n\n\"Correct,\" Nelly said, primly, \"but they have an agreement with Arnell Electronic and Computer Fabrication. They have fabs bigger than the ones we have on Alwa Station. They can grow the crystal cladding.\"\n\n\"Damn your eyes, Kristine Anne Longknife,\" Grampa Al growled. \"You are a headstrong, foolish child. You trust my word: It will get you killed someday. You really must join me in here where it's safe and\u2014\"\n\n\"Considering how easily Jack and I breached your security, I wouldn't bet on it.\"\n\n\"I've corrected those deficiencies.\"\n\n\"And likely replaced them with more. No, Grampa, I will not work for you. Unless you agree to clad the ships I'm being given and do it in one week, I'm heading for your competition.\"\n\nAl's scowl at her would have left any of his subordinates cringing on his knees. Kris returned it, unblinking.\n\n_I've faced down alien wolf packs. What have you done with your life?_\n\nIt took Grampa Al a great while to work up to his surrender. \"Yes, I will do it. Now, damn you, let me see my latest great-granddaughter before you get her and yourselves killed.\"\n\nRuthie took over the meeting at that point and did her best to wrap another of her cantankerous elders around her tiny finger. Even on grumpy Grandfather Al, it worked. He was smiling, making baby talk, and delighted at her strong grasp of his little finger.\n\n\"You will bring her back again,\" Al said. \"Honovi so rarely brings his children by.\"\n\n\"I will try, Grampa, but you know I'm headed back out soonest.\"\n\n\"I would be glad to provide Ruthie with the finest care in the security of my own living quarters.\"\n\n\"Thank you, Grandfather, but I'm nursing Ruthie, so where I go, she goes.\"\n\n\"Even to Greenfeld?\"\n\n\"Now you know why I want those battlecruisers up armored.\"\n\n\"You are a fool,\" was his parting shot.\n\nJack waited until they were in the elevator before he could no longer suppress a question. \"You aren't serious about taking Ruthie with us, are you?\"\n\n\"Who would I leave her with? My mother? Your mother? Al here? No, I am Ruthie's mother, and she goes with me. I trust you to keep us safe. Who else would you trust?\"\n\n\"Oh God.\" Jack sighed.\n\n\"Help us all,\" Rick Sanchez added.\n\n\"I will have to arrange for a bigger detail,\" Special Agent Foile muttered.\n\nThey considered all this while they went down three elevators and made their way to the limo surrounded by Marine gun trucks, police cars, and motorcycles.\n\nAs soon as they were under way, Kris said to Nelly, \"Have you got a contact in Nazareth Steel, Shipbuilding, and Space Docks?\"\n\n\"Yes, ma'am. I have even asked them if they would be interested in some additional work after they finish the two battlecruisers they are building for Tacoma. The rest of the squadron is to be built on the station that Tacoma is just putting the finishing touches to, complete with a shipyard.\"\n\nSo, Nazareth Steel was very interested in getting into the crystal-growing business.\n\n\"Call the Chairman of the Board and CEO,\" Kris told Nelly.\n\nThey had an agreement and had given them a copy of what Grampa Al now had before their convoy arrived safely back at Nuu House.\n\nKris waited until Nelly had hung up and assured them they were clear.\n\n\"Now let's see how fast Grampa Al is,\" Kris said, and knew her smile was pure evil. She found she enjoyed it that way.\n\n# 10\n\nGrampa and Gramma Trouble asked to join them for supper. Lotty was delighted to serve dinner for more than just herself and her husband, Harvey. Kris asked Senior Chief Agent Foile to stay for dinner if he could.\n\n\"Yes, but I hope you will let me go soon enough to help my kids with their homework. It's rare that I make it home in time. It would be nice for just once.\"\n\nKris assured him dinner would be early and not run long.\n\n\"What do you think of your new nannies?\" Gramma Trouble asked, as she and Trouble came through the door.\n\n\"I haven't seen them yet,\" Kris said, just as a new face in the gray-and-white uniform the nanny service seemed to require brought Ruthie down the stairs to meet the company.\n\n\"You would be Akumaa Shalonda,\" Gramma Trouble said.\n\n\"Yes, ma'am,\" came with a slight dunk of the head. The woman was almost as tall as Kris. Her skin was a luscious chocolate topped by red hair, close-cropped, but long enough that Ruthie would, no doubt, soon be wrapping her fingers in it. She had a quick smile.\n\n\"I saw your background when I first looked at the packages Brenda had,\" Gramma Trouble said. \"I don't know what her criteria were, but they weren't mine, or yours I see,\" she said, nodding at Kris.\n\n\"You are weapons-qualified?\" Jack asked.\n\n\"Pistol, rifle with sniper scope out to nine hundred meters, assault rifles, several rocket launchers, and mortars. I love those things. Will we be having any of those in the armory?\"\n\n\"We shall see,\" Jack said, grinning. \"May I ask how you happened to qualify with all these toys?\"\n\n\"I'm sorry, sir, but that is classified need to know. However, I will be glad to demonstrate proficiency.\"\n\n\"Gramma Trouble?\" Kris asked.\n\n\"Don't ask me. Trouble vouched for her, and that was good enough for me.\"\n\nKris raised a questioning eyebrow to her troublesome great-grandfather. He shook his head. \"Sorry, I had to call in some chits to verify her skill set, and I am not at leisure to say more.\"\n\n\"Will the rest be just as black ops?\" Jack asked.\n\n\"Oh, no. The other gals and their supervisor are all former service Soldiers or Marines. It will be interesting to see how a shooting match among all six works out.\"\n\n\"No doubt,\" Kris said.\n\nRuthie was hungry, so they adjourned to the library. It was a relief to be able to nurse Ruthie and not be ostracized from the conversation. She kept Ruthie under a loose blanket, but no one blinked when she switched from breast to breast. Kris had chosen a peasant dress that made it easy for her to put dinner before her infant.\n\n\"If only the Navy would adopt a nursing mother's uniform,\" Gramma Ruth said.\n\n\"It's hard enough for them to tolerate a maternity uniform,\" Trouble put in.\n\nWhich led Kris to tell the full story of how she came to be a mother and continued to be an admiral commanding a fleet at the tip of a spear.\n\n\"I wondered how that came down,\" Gramma Ruth said. \"I know all the trouble Rita got in for taking Al out on a heavy cruiser when she was nursing him. Oh, and the storm that blew in when the powers that be discovered she'd stayed at the head of her squadron while Alnaba was on the way.\"\n\n\"It hasn't got any easier,\" Kris said. \"Oh, and I have Alnaba's granddaughter on my staff.\"\n\n\"You do!\" Gramma Trouble said.\n\nLottie picked that moment to announce dinner was ready. Ruthie had just finished her supper and nodded off to sleep. Akumaa appeared instantly from out of nowhere to whisk the infant away, leaving the adults to enjoy dinner and old stories. Kris left it to Trouble to tell how, after Eddy died, he'd hauled her out of her drunken depths to soar as a skiff racer dropping from orbit to a small patch of ground that was the target.\n\n\"I wouldn't be where I am today if you hadn't reached out to me,\" Kris said.\n\n\"You are where you are today because of you, and a whole lot of your friends, of which Ruth and I are proud to be two,\" Trouble said, giving his wife a hug.\n\n\"But if you get me kidnapped again . . .\" Ruth said, and left it at that.\n\n\"Don't get too close to me,\" Kris said. \"It's dangerous.\"\n\n\"I wouldn't say that,\" Jack said, and gave her a squeeze to match Trouble's.\n\nAgent Foile took this all in and said little.\n\nIt was only after dinner was over and a dessert tray that could not be passed up had been passed around that Trouble said, \"I understand you visited with Al today. How is my former son-in-law?\"\n\n\"The same as he ever is,\" Kris said. \"A skinflint scared of his own shadow and grasping for everyone to come in under his umbrella. One that smells terribly like a spiderweb that once you go in, you would never get out.\"\n\nTrouble shook his head. \"I don't know what got into that boy. I never did see what Sarah saw in him.\"\n\n\"They spent so much time growing up together,\" Ruth said, \"while we were all out making the galaxy safe from Iteeche. Left together, they kind of fell into something they both took for love. It might have become something if they'd been given enough time. Ray had been a bachelor way too long, and he just didn't know how to give that boy the love he needed. There was a big splash when Alex married that young thing, but nothing afterward. Nelly, are you better than I am at chasing gossip?\"\n\n\"I'm sorry, Gramma Ruth, but I can find nothing about her. Two days after the wedding, they both entered a black hole as you might say. Grampa Al is often talked _about_ in the media, but rarely because of anything _he_ said.\"\n\n\"The man's a hermit,\" Trouble said.\n\n\"After looking at his office today,\" Jack said, \"I'd say he leads a very opulent and indolent hermit kingdom.\"\n\nThey shook their heads at that, and Trouble changed the topic.\n\n\"Excuse me for bringing up business, but would you like to meet the three people who volunteered to help you mediate?\"\n\n\"When could I?\" Kris asked.\n\n\"Would tomorrow be too soon? Say 1300 hours?\"\n\n\"Fine by me,\" Kris said, as Trouble and Ruth stood. Agent Foile stood a moment later, not having the telepathy that the old married couple seemed to share.\n\n\"How many people will we be passing through the perimeter?\" Kris asked.\n\n\"Likely a lot more than three,\" Trouble said. \"Most of them work with a team.\"\n\n\"So I'll be the ringmaster of a real zoo?\"\n\n\"And that's different how?\" Trouble asked.\n\n\"You're mean to me, Grampa.\"\n\n\"No more than I have to be, young lady. Do you remember that zoo where you got your first medal, and I had you sit there with the rest of us old fogies with stars on our shoulders?\"\n\n\"How could I ever forget? Mother wanted me to put diamonds on my Navy and Marine Corps Medal and wear it as a pendant\u2014and get out of the Navy immediately.\"\n\n\"I told you then that you'd have stars.\"\n\n\"Do I? My orders to get back here were addressed to a captain.\"\n\n\"Neither Mac nor Ray told you to get your uniform straightened out, did they?\"\n\n\"Nope.\"\n\n\"You've got it. Don't let anybody try to take it back.\"\n\n\"And when payday comes, and the pay isn't there for four stars?\"\n\n\"Tell Finance to straighten up the paperwork, then donate it to a good cause. It's not like you Longknifes need the money,\" the old general said.\n\nOn that laugh, the wisest of Kris's elders took their leave, along with Special Agent Foile. Kris and Jack enjoyed the rest of their evening, including time with Ruthie when she woke up.\n\nAkumaa got more pabulum into Ruthie after Kris allowed her to empty only one breast. She then slipped the baby out so Kris and Jack could get some sleep. Kris wasn't woken up until two o'clock for a sleepy feeding. Akumaa again took Ruthie off for some play and let Kris go back to sleep.\n\nKris felt very well rested when morning arrived. No doubt, it would be a busy day.\n\n# 11\n\nKris had just settled into the breakfast solarium and managed a bite from a nut-and-bran muffin with marmalade when Nelly interrupted.\n\n\"I have several requests from shipyard and space-dock companies for the package we gave Al and Nazareth. Should I give it to them?\"\n\n\"How many is several?\" Kris asked through a full mouth.\n\n\"Three. No, make that four. Another request just came through.\"\n\n\"How'd they find out?\" Kris asked no one in particular.\n\n\"I assume that's a rhetorical question,\" Nelly said, \"but I suspect the answer has something to do with how much you humans talk. Incessantly, I might say.\"\n\nKris ignored Nelly and eyed Jack. Swallowing, she said, \"It was such a lovely morning. Well, no rest for the wicked. As I see it, we developed those production procedures and installation practices on Alwa Station. We don't have any patent process on Alwa; we're just doing anything we can to survive.\"\n\n\"Yeah,\" Jack said slowly, \"but we did start with Earth's stuff, which they very likely did patent.\"\n\n\"But if we leave it to the lawyers, we'll be years getting the warships we need,\" Kris said.\n\n\"You want to do it now and ask forgiveness later?\"\n\n\"I want to do it now, and let the lawyers figure out the mess. Isn't that what they get paid for?\"\n\n\"God help us if you ever end up queen, my love.\"\n\n\"I'm sure she will,\" Kris said, leaning close enough to Jack to give him a marmalade-sticky kiss. The essentials covered, Kris told Nelly to publish the files to the Wardhaven net with open access.\n\n\"That ought to be fun,\" Kris said.\n\n\"No doubt,\" Nelly agreed.\n\nThey were enjoying one last cup of coffee before tackling their day when Nelly again broke in. \"Grampa Al wants to talk with you, Kris. Indeed, he is trying to have his computer bellow his desire from every commlink in Nuu House.\"\n\n\"Can he do that?\" Kris asked.\n\n\"No doubt he could,\" Nelly said, \"if I weren't blocking him. His computer, as he admitted yesterday, is not in my league.\"\n\n\"No question of that,\" Kris said, to fill the time while she thought. \"What do you think, Jack?\"\n\n\"We both know what he wants to talk about. We both know he won't be easy to talk to. It seems to me that it's just a waste of your time and his to take this call.\"\n\n\"That sounds about right,\" Kris said. \"Nelly, use your superior skills to break into his call and tell him I won't be available today.\"\n\n\"Done,\" Nelly said.\n\nKris and Jack rose and were on their way out of the solarium when Nelly reported, \"Grampa Al is not happy. I'm glad you didn't have to listen to him. Such language!\"\n\n\"No doubt you've heard worse,\" Kris said.\n\n\"Yes, from Sailors and Marines, but from your own grandfather?\" Nelly sniffed.\n\n\"Well, if he questioned my parentage and DNA, he ought to know more about it than I do,\" Kris said through a wicked grin.\n\nKris planned to spend the morning at Mac's office, looking into the specifics of what ships and Marines she could subvert to her own ends. A new nanny showed to cover the morning shift. Mai Tiamat was short and fair-skinned, the opposite of Akumaa, though her hair was raven black and down to her waist in a plaited ponytail. She had Ruthie efficiently ready and joined Kris and Jack as they went down the stairs.\n\nSenior Chief Agent in Charge Foile was there with Leslie. \"I have a full team in place, and we will follow a random series of streets between here and Main Navy.\"\n\n\"Fine. What's our threat level today?\" Jack asked.\n\n\"We have tracked down the source of our trouble,\" Foile said. \"It seems, Your Highness, that someone has put a rather large sum of money on your head. They want you dead. Period.\"\n\n\"Any idea who?\" Kris asked.\n\n\"No idea,\" the agent said. \"Though if I were a betting man, I'd put all my money on the stepmother. Isn't it always the stepmother in fairy tales?\"\n\n\"This hardly feels like a fairy tale,\" Kris muttered, as Jack opened the door of their limo for her. Leslie was helping Ruthie and Mai in the other door.\n\n\"Though it's certainly grim enough,\" Nelly put in.\n\n\"Lousy joke, girl,\" Kris said.\n\n\"But appropriate,\" Jack added.\n\nThe drive to Main Navy involved a lot of turns but no surprises. Jack remarked on that.\n\n\"Mahomet's night shift did defuse a roadside bomb that showed up out of nowhere,\" Leslie said. \"They chased down a half dozen trucks that had driven by between sweeps. Five stopped and made nice.\"\n\n\"And the other?\" Jack asked.\n\n\"Was abandoned and burning in a back alley. It was reported stolen yesterday afternoon.\"\n\n\"Damn the money,\" Kris muttered.\n\nField Marshal McMorrison was expecting them; he even seemed to know what they wanted to talk about.\n\n\"We have eight frigates, er, battlecruisers, which have just finished working up. The _Princess Royal_ , _Intrepid_ , _Courageous_ , _Furious_ , _Resolute_ , _Defender_ , _Steadfast_ , and _Monarch_ are almost ready for you.\"\n\n\"We've got several ships with those names on Alwa Station,\" Kris pointed out.\n\n\"I know we sent them out there, but we figured they must have been lost by now,\" the field marshal said.\n\n\"They have been,\" Kris admitted, \"but we built replacements.\"\n\n\"You're actually building ships?\"\n\n\"Once you've got the basic kernel of the ship, you can grow one rather quickly.\"\n\n\"Really?\" Mac said, frowning. \"And Al Longknife insists it's such a tough job to put together one of the new Smart Metal battlecruisers.\"\n\n\"Nelly, do we have the procedure Admiral Benson uses to grow a ship?\"\n\n\"I did manage to get his computer to part with them before we left,\" Nelly said as smoothly as if her mouth could melt butter.\n\n\"Let me have them,\" Mac said. \"The Navy still has some shipyards of our own, and there are several places we can buy that newfangled Smart Metal. Let's see if we can show old Al a thing or two . . . and get some money back from him while we're at it.\"\n\n\"You may have all the fun with Grampa Al that you want,\" Kris said. \"Just keep him out of my hair.\"\n\n\"Speaking of your hair, would you happen to know how it happened that I got four unrequested bids for adding crystal armor to those eight frigates, I mean battlecruisers, that you're taking into Greenfeld space?\"\n\n\"I may have had Nelly post on the net the process for armoring a battlecruiser with crystal.\"\n\nKris couldn't tell whether Mac was delighted or appalled. Likely, he was a bit of both. He did get a coughing fit that had him grabbing for a glass of water.\n\n\"God, I don't know how you Longknifes keep from killing each other,\" he finally got out.\n\n\"Probably because the Peterwalds keep hogging the head of the line,\" Kris said, daintily.\n\n\"No doubt. Is that crystal cladding all it's cracked up to be?\"\n\n\"Mac, it saved a lot of lives in the last couple of run-ins with the alien monsters. The Earth squadrons lost ships because their stuff was not manufactured to our quality standards and not fitted properly to their ships' hulls. Armored, I still lost ships, but it cut my losses at least in half, maybe by three-quarters.\"\n\n\"Then we need to get your ships docked for cladding,\" Mac said.\n\n\"Admiral Benson's yard teams came up with a fast way to clad the battlecruisers that didn't need a dock. It's all in the package. How many did you say were bidding for the work?\"\n\n\"Al, two other yards I trust, and one fly-by-night that just came off being listed for no government work for some shenanigan they pulled last year.\"\n\n\"If you could,\" Jack said, \"keep that last one on restriction for a bit longer. I don't want my wife depending on the lowest bidder if you can manage it.\"\n\n\"I think I can do that,\" Mac said. \"That would give four to Al and two each to these others.\"\n\n\"And if the other two can do the upgrade as quickly as I think they can, you'll have a baseline you can use to keep Al honest,\" Kris said.\n\nWith that decision made, Kris got out of Mac's way and let him do his best for her. She chose to spend her time wandering the halls, refreshing old friendships and getting a feel for the way the wind was blowing.\n\nLunch was in Field Marshal McMorrison's private dining room and attended by several senior Navy officers. They wanted to know firsthand how things were on Alwa Station.\n\n\"I turned in a full set of reports as soon as we docked yesterday,\" Kris said.\n\n\"I stayed up all night reading them,\" Commander, Battle Force said. \"I didn't see anywhere that your battlecruisers are following battle-line doctrine.\"\n\n\"She's not using her frigates,\" CruDes Forces pointed out, being careful to use the earlier name for Kris's ships, \"like liners. They're more like independent destroyers.\"\n\nKris had no trouble finishing her meal while the two of them carried on their own, doubtlessly familiar, battle. Finished, Kris excused herself to attend to Ruthie.\n\nHer departure was hardly noticed.\n\nJack ducked out a moment later and trotted to catch up with Kris. \"You'd think they'd want to talk to an experienced combat commander,\" he said.\n\n\"You'd think that, but more than likely, you'd be wrong.\"\n\n\"Yeah,\" Jack said. \"They can't allow for actual experience to interfere with their own ignorance.\"\n\n\"Not when it might impact their budget.\"\n\n\"You sure you want to come back here and become a staff weenie?\"\n\nKris shrugged. \"Somebody has to, or some alien mother ship could show up in Wardhaven's sky and those clowns wouldn't have the foggiest idea how to fight it.\"\n\n\"Yeah,\" Jack said. \"Yeah.\"\n\nThey were back to Nuu House safe and sound long before their next meeting got under way.\n\n# 12\n\nKris rocked Ruthie softly as she rested her eyes out the window of the nursery on the roadway between the gate and the house. A whole lot of people were coming to see her.\n\nEnsign Meg Longknife was laying out a new blues uniform and switching the ribbons to it under Jack's watchful eye. Akumaa had Ruthie sound asleep, so Kris hadn't had a chance to feed her at Main Navy. She'd awoken hungry on the drive home.\n\nThus, Kris's breasts got painfully full and leaked. \"There are a few things about this mommyhood thing that weren't mentioned in the manual that came with Ruthie,\" Kris muttered.\n\nStrange, everyone around her kept very quiet as she groused. Smart of them if they wanted to keep their heads from being bitten off.\n\nCleaned up, and once again presentable, Kris had Nelly put in a call to Senior Chief Agent in Charge Foile. \"Did you do a background check on all these people?\"\n\n\"All of them had been fully vetted for their jobs. We did check to see if anyone suddenly had debts or income changes. They all came back clean.\"\n\nThat was good, but Kris had hardly rung off when Jack asked softly \"Honey, do you think now would be a good time to see if you can fit into your spidersilk underarmor?\"\n\n\"No, it would not be a good time to try. I know how many pounds I'm still carrying,\" Kris snapped, but stopped short of snapping off heads.\n\n\"There was a small package delivered this morning,\" Meg put in. \"It's from your Great-grandmother Tordon. Have I met her yet?\"\n\n\"That's Gramma Trouble,\" Kris said, and almost leapt for the wrapped box on the table. Jack put out a restraining hand to her elbow. He retrieved the box and took it out into the hall and down to the bathroom.\n\n\"It's spider silks in a larger size, honey,\" came back a moment later.\n\nWhich left Kris to undress again and wiggle into the abysmally tight things while feeling every extra ounce she'd gained in the last nine months. At least she could do it in the bathroom, away from eyes she didn't want to be embarrassed by or to tell her nice things she didn't believe. Dressed again, she rejoined Jack, and, feeling safer than she had in weeks, went out to see what there was to see.\n\nThe second-floor conference room had been reserved for this meeting. Kris saw that metal detectors had been set up at the door. They buzzed as she and Jack walked in wearing their ribbon racks and carrying their service automatics.\n\n_But they're here to protect me, so that's okay._\n\nA large zoo of eyes followed Jack and her as they headed for the two empty chairs at the head of the table. Kris could imagine they were taking her measure and refused to think about the conclusions they drew.\n\nSeated, Kris looked around the table. It was easy to identify the three primaries. They sat back, maybe a bit more relaxed than those beside them. Around those three were eighteen at the table and on the edge of their seats, ready to jump at the slightest whim. Twenty or more sat alertly along the wall, backing up those at the table. All told, Kris counted over forty people in the room. Several of Foile's agents were left at the door.\n\n_My meetings keep getting bigger and bigger._\n\nKris cleared her throat. \"I have plenty of experience blowing shit up and killing people. It seems now that I must find a way to stop some people from blowing shit up and killing each other. Hopefully, before one of them succeeds in killing me.\"\n\nBlank faces around the room took in her words, and realized, likely for the first time, that this latest challenge might not be as safe as their normal daily affairs.\n\n\"I know that there have been many attempts on your life, Your Highness,\" said one tall, middle-aged man Kris had identified as a primary.\n\n\"But twice since I got off the boat yesterday is a bit much, even for me,\" Kris answered before he got to the question.\n\nNow eyebrows did rise in concern as heads swung from one side to the other, taking a poll of their friends and measuring the distance for a run to the door.\n\nKris was not unfamiliar with this reaction. It had just been a while.\n\n\"There are three groups of you here. You all come recommended to me. You all bring different perspectives to our problem. I would prefer to have all three of you with me. Grampa Trouble thinks that together, you give us more of a chance to succeed. However, you may want to assess the risk you are willing to take. Talk it over among yourselves and let me know when you've made up your minds, as one group, or three.\"\n\nFinished, Kris stood, turned her back on their discussion, and walked over to one of the windows that opened on a view of the lovely gardens. That was where Penny had gotten married. Kris smiled at the memory. Too bad she and Tommy had had so little time together.\n\n_Tommy got too close to this damn Longknife,_ she thought bitterly.\n\nKris looked over her shoulder, at where forty some people were earnestly negotiating just how much they were willing to risk to follow this damn Longknife. She doubted that any of them ever expected to be in collateral-damage range of a bomb during the normal to and fro of their daily duties.\n\nKris turned back to the view. The trees were turning all sorts of lovely colors. She'd likely miss winter if this mediation process took very long.\n\nJack came up beside her, two mugs of tea in his hands. Kris took the one offered her. \"Chamomile?\" she asked.\n\n\"What else would I bring?\"\n\nShe smiled and turned again to the view.\n\nKris heard the crack of the glass as the bullet shattered it.\n\n# 13\n\nJack grabbed for her, but Kris was already going down, folding her legs under herself. Of course, scorching-hot tea was flying every which way.\n\nKris could feel the scalding of the tea but no pain yet from the bullet. She took no comfort from that; she'd been hit often enough to know that the agony came later.\n\nThere was the report of another shot, and suddenly Kris was covered with nuggets of glass. The window had held for the first round; the glass failed as the second shot hit.\n\nA chilly breeze wafted through the empty window.\n\nKris dropped flat on her back, covering her eyes with her arms to protect them from more chunks of glass. Jack covered her with his body.\n\nShe knew he always would.\n\nSomewhere, someone was screaming. Kris turned her head; one of the eager secondaries had a fountain of blood pulsing from her neck.\n\n_Not good._ Kris tried to roll out from under Jack. Strange how a man could be so light on her body when they were making love but be so heavy when he wanted to pin her to the deck.\n\n\"Jack,\" Kris snapped.\n\n\"I know. Stay down,\" he said, cutting her off.\n\n\"Radar has backtracked the bullets to a knoll nine hundred and fifty meters from the house,\" Nelly reported. \"A team has been dispatched.\"\n\n\"We need medics up here. Now! Major trauma,\" Kris bit out.\n\n\"Emergency Services have been notified,\" Nelly said. \"We have a small trauma team on-site. More on the way.\"\n\nThe door opened, and two emergency-medical techs monkey-walked low through it with a backboard and medical box dragged between them. A woman in a white lab coat followed, crawling on her hands and knees.\n\nOne of the people at the table had been trying to stop the bleeding with his handkerchief. Now a med tech took over.\n\nThe doctor pulled a packet from the kit. It provided her with a scalpel and clamps. She lengthened the bloody hole enough so she could clamp off one end of the jugular vein, followed by the other.\n\n\"Okay, folks, we've really messed up the blood flow. Let's do this fast.\"\n\nOne med tech had already pulled out a plastic-wrapped tube, sticky extensions at both ends. The other tech unclasped the lower part of the wound but tried to keep it closed with his fingers. Still, blood flowed from the exposed, ragged edge. The doctor slapped the tube onto that exposed end of the jugular and taped it down. She let blood in and sighed as blood did not spurt out from the glued-down part. There was leakage.\n\n\"Andy, extra tape,\" she said, and turned to the other end. They repeated the process, and a moment later, blood was flowing through the tube, now twisted into a slight U but holding.\n\nAs she slapped a bandage over the entire space, she growled, \"Is this still a live shooting gallery?\"\n\nFrom outside the door came Foile's voice. \"We had the shooter in custody, but she took poison, so there'll be no one to talk to. Sorry, Your Highness.\"\n\n\"I'm sure you will have more chances to catch a live one,\" Kris said.\n\nBut attention was focused still on the medical heroes. Two more med techs brought a gurney in. The blood-covered young woman was transferred gently to it, then lifted up.\n\n\"There's an elevator down the hall,\" one of the original med techs said, and led the way.\n\n\"A chopper is inbound, three minutes out,\" came from Ensign Longknife, still out in the hall.\n\n\"Anyone else hurt?\" the doctor asked.\n\n\"One more,\" Jack said.\n\nKris looked around and saw nothing. She'd shed her dress blues blouse and with it the scalding tea. She was fine.\n\nJack reached over and touched her face. His finger came away with blood on it.\n\n\"It _is_ just a scratch,\" Kris insisted, then switched to business mode. \"There is a large room in the basement. We can stand around safely there while some Marines scrounge up some tables and chairs.\"\n\n\"Only after they've made sure this wasn't the first of a multiphased assassination attempt,\" Jack said.\n\n\"I think we can conduct our business standing up. I've done some of my work at the bottom of a mine shaft,\" was said with a dry chuckle by an older shorter man. \"The third-generation owner had never actually seen what it took to make him his money.\"\n\nOthers agreed, so they all trooped downstairs. with one med tech at Kris's elbow, doing his best to care for her scratches.\n\n# 14\n\nKris found herself in a storage room that looked a lot smaller than it had the last time she had played hide-and-seek with Honovi and Eddy. The unused bar she'd disappeared behind was still there. Now she sat atop it while the med tech did what she would let him.\n\nFinished, he put away his gear. \"You should have a cosmetic surgeon take a look at that one on your chin,\" he said. \"It's a coin flip if it should be stitched. I take it you won't go to the hospital.\"\n\n\"You got that one right,\" Kris said.\n\nThe med tech whispered something about \"those damn Longknifes\" as he finished his packing.\n\n\"I'll get a doc over here before supper,\" Jack said.\n\nKris was too tired to argue. Her adrenaline was draining away fast. Likely, that was something her body did to make sure Ruthie didn't get a jolt of the stuff with her milk.\n\nA short older man with sparkling eyes, a tall middle-aged man with a limp, and a young woman who might have been mistaken for a younger Kris came to face her.\n\n\"If there was any question about the risk involved here, someone certainly gave us a demonstration,\" the older man said. The other two nodded.\n\n\"All three of us have had a member of our staff withdraw. Mine was the woman who was shot. Diana and WP each had one fold their cards after a call to their spouses. One would have to be a fool to go into this without the support of the one they're committed to spend their life with. Besides, what kind of negotiators would we be if we gave our loved ones nonnegotiable demands?\"\n\nHe turned to share a chuckle with most of those in the room. At least two, one a man, the other a woman, looked too shook-up to enjoy the joke.\n\n\"We have a list of additional experts we would like to join us. Computer gurus, data analysts, linguists in the variations of Standard spoken in Greenfeld, several experts in forensic accounting. They may not be necessary, but it's amazing how often they come in handy. Getting to yes isn't just a process of reaching agreement but often involves changing people's opinion of where they are and where they've been.\"\n\nKris nodded as she began to learn another way of looking at the world, just as the Navy had shown her the Navy Way, and the boffins had introduced her to the scientific method.\n\n_Oh God, not another learning experience!_\n\n\"Make up your list. Pass it to Meg, my _aide-de-camp_. If she can't get the bodies you need, I'm sure I know someone who buried them.\"\n\nEnsign Longknife smiled cheerfully, validating Kris's suspicion that Gramma and Grampa Trouble had more than one oar in the water that brought her to Kris's side.\n\n\"I asked Grampa Trouble if he knew a good plastic surgeon,\" Jack said, \"and Gramma Trouble got back with me a few moments ago. One is on her way here.\"\n\n\"Then let me feed Ruthie, and I'll lie down where I'm told,\" Kris said with an exhausted sigh. Two nannies brought the infant, and someone rustled up a rocker for Kris and Ruthie. They settled into a corner while the others, including Jack and Meg, made their lists and made them longer.\n\nKris was not yet finished with Ruthie when Foile showed up again.\n\n\"You mind talking to me now? My wife breast-fed our two hellions. It brings back fond memories.\"\n\n\"If you're not embarrassed, I'm not,\" Kris said.\n\nThe agent stooped beside Kris. \"We've done a thorough search of the area and don't see anything more dangerous than one owner who hasn't leashed his dog.\"\n\n\"God forbid we should allow such an outlaw to prosper,\" Kris said dryly.\n\n\"My main problem is Nuu House. The bulletproof windows were last replaced when you started college. Their warranty is up next year. With no one here, there were no plans to replace them.\"\n\nKris snorted. \"I was crazy for ship duty. Hated being stuck here.\"\n\n\"Evidently, armor-piercing bullets are more powerful than ten years ago. I've put in a demand for new glass, but it will be a day, maybe two, before we get everything replaced.\"\n\n\"And in the meantime?\"\n\nThe agent shrugged. \"I'm not sure there's anyplace on this planet that can be made safe for you. I suggested to the head of the Secret Service that you move in with your folks . . .\"\n\n\"Ain't gonna happen,\" Kris said, cutting him off.\n\n\"He said the same thing. He didn't want the prime minister caught in any cross fires intended for you.\"\n\n\"That, too,\" Kris said, getting a fishy eye from the agent but no backtalk.\n\n\"Would you feel safer with the Navy?\" he finally said.\n\nKris shrugged. \"They've kept me alive when bug-eyed monsters wanted me dead. I haven't done better than that?\"\n\nNow it was the agent's turn to say nothing and Kris's to give him the fish eye.\n\n\"Okay, where do we stash me and how do we get there?\"\n\n\"I assumed the _Wasp_ ,\" Foile said.\n\n\"So would everyone else,\" Jack said, joining the conversation.\n\n\"Where's the _Princess Royal_?\" Kris asked.\n\n\"Moving into a Nazareth Steel space dock,\" Nelly said.\n\n\"Here's how we're going to do this,\" Kris said, and turned Ruthie over for a good burping.\n\n# 15\n\nAn hour later, a convoy gunned out of the front gate of Nuu House. Built around a long black limo, it had big police rigs and Marine gun trucks both ahead and behind it. As soon as it hit the street, it began a zigzag course that seemed aimed nowhere in particular.\n\nA half hour later, a similar cavalcade shot out of the back gate of Nuu House and likewise began to go nowhere in a hurry.\n\nThirty minutes after that, a third motorcade repeated the process, again using the front gate.\n\nIn between those last two, a green sedan, driven by the old family chauffeur, pulled up to the back gate. He waved cheerfully, and when questioned by the Marines at the gate told them, \"Lotty needs two pounds of butter and five dozen fresh eggs. We aren't due for a delivery until tomorrow and wouldn't you know it, they ate us out of house and home.\"\n\nThe Marines chuckled and waved the sedan through.\n\nTwo blocks out, it was joined by two unmarked police cars, then two more.\n\nAbout this time, all three of the motorcades with big limos began to meander their way toward the space-elevator station. The green sedan, instead of making its way toward the same place, or for a grocery store, got on the crosstown expressway, sped up, and headed for the mountains to the west of Wardhaven.\n\nAbout the time the sedan approached Big Bear Lake, an unusual sound was heard over Wardhaven City: the double boom of a lander coming in. Not since the beanstalk had been completed had a lander come in over the city. Back then, they came down in the bay, not up in the hills.\n\nLongboat 3 from the _Wasp_ settled down in a haze of spray and steam, then motored over to the large boat ramp and rolled itself right out of the water. The aft hatch opened, and a cheerful voice said, \"Hi, Your Highness. See. We didn't suck a single water weed into the intakes.\"\n\n\"You've gotten much better,\" Kris shouted back, and quick-walked with Jack, Agents Foile and Rick, two nannies, and one Navy aide across the landing and right into the longboat.\n\n\"You may launch when you please,\" Kris said. \"Just keep the gees below two and a half, if you will. Neither I nor baby is up for the rough stuff these days.\"\n\n\"Will do, Your Highness. Back to the _Wasp_?\"\n\n\"No, not today. Land us on the _Princess Royal_ , if you please.\"\n\n\"Will do, ma'am.\"\n\nThus ordered, Longboat 3 motored stately out to where the lake was its deepest and rose to the heavens on a pillar of fire.\n\nOnce the noise settled down, so Jack didn't have to shout, he leaned over and spoke into Kris's ear. \"Danged if that wasn't a better escape from Wardhaven than the last time we had to leave in a hurry.\"\n\n\"Glad I'm not at the stick?\" Kris asked.\n\n\"Nope. You're the best pilot I've ever ridden with.\"\n\n\"Smart husband. I may keep you.\"\n\nAn hour later, they approached High Wardhaven Station. _Wasp_ 's Longboat 3 adjusted its course to take it right into the _Princess Royal_ 's landing bay.\n\nWithout a hitch, Kris was back in space, safe and sound.\n\n# 16\n\nAn out-of-breath commander saluted Kris as she came aboard one of the fleet's newest battlecruisers.\n\n\"I'm Commander Helen Ajax, captain of the _Princess Royal_. Welcome aboard, ah, Admiral.\"\n\nKris took the measurements of the woman who would keep her and Ruthie safe. She was glad the captain had chosen which foot to get started on. Someone had warned her how much Kris did not like being Your Highnessed by officers in uniform.\n\n\"You were told you would be my flag,\" Kris said, watching as the two nannies marched Ruthie aboard.\n\nCaptain Ajax's eyes followed the baby for maybe a second too long.\n\n\"Ah, yes, ma'am,\" she said, eyes sweeping back to Kris.\n\n\"But not that I'd be coming aboard this afternoon.\"\n\n\"Exactly, ma'am. We're doing our best at catch-up.\"\n\nKris turned to follow Ruthie into their new home. \"Well, it turns out the armored glass windows at Nuu House weren't quite as armored as we thought. Have you locked the ship down?\"\n\n\"Yes, ma'am. Ah, you know we are in the yard, ma'am.\"\n\n\"Yes, but does adding the crystal armor require yard personnel's coming aboard?\"\n\n\"I'm trying to get an answer to that, ma'am.\"\n\n\"It shouldn't. We did it all from the outside on Alwa Station.\"\n\n\"Yes, ma'am. The _Wasp_ sent over the layout for your flag quarters.\" Again, the captain's eyes focused on Ruthie, and she swallowed hard. \"I assume that you will need expanded quarters. I also have a request from _Wasp_. They want to ship your flag gear over here. Their request arrived shortly after the order to lock the _Princess Royal_ down tight.\"\n\n\"Which has you between a rock and a hard place,\" Jack put in.\n\nKris's flag captain nodded.\n\n\"Yeah,\" Kris admitted. \"Maybe we don't have all our ducks in a row. Okay, allow personnel from _Wasp_ to make the transfer. See that Senior Command Chief Mong is put in charge of the detail. Tell him to wear sidearms, and if he sees anyone he doesn't recognize, to shoot first and ask questions later.\"\n\n\"Yes, ma'am. Begging the Admiral's pardon, but is that the normal way of doing business under your command?\"\n\n\"No, Captain, but there have been two bombings and one sniper attempt on my life since I went ashore day before yesterday. I'm a bit trigger-happy at the moment.\"\n\n\"Yes, ma'am.\n\n\"Oh,\" Jack said, \"I'm about to order up a doctor to see what she can do to patch up my wife's face. There are also three nannies and two Secret Service agents, likely more, that didn't come up with us. Let me know if you have anyone ask to cross your brow, and I'll approve them or order your Marines to shoot them.\"\n\nAgain, the captain swallowed hard. \"There is one matter that I hope you'll make allowances for.\"\n\n\"And that would be?\" Kris asked.\n\n\"I got a request from the CEO of Nazareth Steel, Shipbuilding, and Space Docks, Chogan Cam. He'd like to talk to you.\"\n\n\"How'd he know I was here?\" Kris snapped.\n\n\"He didn't, ma'am. He told me earlier today that he'd like to talk something over with you when you came aboard. He's been good to work with, and I can't see his being a problem.\"\n\nKris eyed Jack. He nodded. \"Captain, tell him that he can talk to me,\" Jack said. \"If I like what I hear, we'll see where it goes from there.\"\n\n\"Yes, sir. He's in the yard, overseeing the tiger team doing what it can to glue crystal or whatever down.\"\n\n\"Actually, we ended up bolting it on,\" Kris said, \"though I don't know quite how we got the strands of crystal to stick together. Okay, Captain, I'm going to be trusting you with my life in a pretty rough neighborhood. If you think I should see him, I'll see how he gets on with the general here, then talk to him. Where are my quarters?\"\n\n\"If you'll come with me,\" Captain Ajax said.\n\n\"I suspect I can find my way to the gangplank,\" Jack said, and they went their separate ways.\n\nThe _P. Royal_ 's skipper did lead Kris to familiar territory.\n\n\" _Wasp_ 's skipper said your quarters were just off the bridge?\" the captain said, clearly puzzled but not pushing.\n\n\"Yes. When I first came aboard _Wasp_ , I was senior officer present but also the ship's gunnery officer.\"\n\nThe skipper's eyebrows went up in a questioning V, but she said nothing.\n\n\" _Wasp_ and I kind of grew up together,\" Kris said. \"I went from lieutenant to four stars on _Wasp_ s that went from corvettes to frigates to battlecruisers.\"\n\nKris came to a familiar door. A Marine corporal stationed outside opened it while a private kept alert guard.\n\nHer day quarters had a decent expanse of space. All empty. Kris quickly covered the paces to the door into her night quarters. They were a lot smaller than she would have expected.\n\n_Right. Jack must be billeted somewhere else._\n\n\"Nelly, see what you can do about moving the bulkhead to expand the day quarters.\n\nSuddenly, there were no night quarters.\n\n\"Yeeps,\" the skipper said.\n\n\"Aren't your people moving walls around?\" Kris asked.\n\n\"Well, yes, Admiral, when we need to, but I haven't had anyone treat bulkheads like they're on wheels.\"\n\n\"Hmm,\" Kris said, glancing around the space. \"Nelly, we don't have any screens; could you turn these walls into a schematic of the ship?\"\n\n\"Yes, ma'am,\" Nelly said, being rather better behaved than normal.\n\nThis time, Kris's new flag captain did a better job of suppressing her surprise. \"I'd heard about Nelly,\" she said.\n\n\"You have to see her to really become a believer. This latest trick of transparent Smart Metal is a new one even for her. Now, let's see. Where do you want my flag quarters since they will include one Marine lieutenant general, one infant with a squad of six nannies, an _aide-de-camp_ , and at least four Special Agents, including one in charge. Oh, and you will be taking on three teams of mediation specialists and their support staff.\"\n\n\"In addition to a Marine company?\"\n\n\"Yep,\" Kris said. \"Nelly, let the captain see a full schematic of _Wasp_ , including boffins, Marines, and the Forward Lounge.\"\n\n\"Lounge!\" The captain failed to suppress a squeak.\n\n\"You can't expect civilians to live by our spartan standards, and since we may be inviting an Emperor, Empress, and definitely a Grand Duchess aboard, we can't fail in our hospitality.\"\n\n\"Oh.\" Was followed shortly by a \"God. I've also got a band heading up the beanstalk. One of your relatives\u2014Trouble sound right?\u2014says I can't do a diplomatic mission if I don't have a band.\"\n\nThat was a new one on Kris, but likely essential.\n\n\"Sorry, Captain. I'm one of those damn Longknifes, and I've been handed one hell of a job.\"\n\n\"Yes, ma'am. I understand, ma'am. But a restaurant and a band, ma'am? I'm on lockdown, and now I have to reorganize my ship and bring aboard a Navy band from somewhere and cooks, bartenders, and whatever else from God only knows where?\"\n\n\"Yes,\" Kris said, knowing exactly how her flag captain must be feeling about now. \"The difficult we do now, the impossible takes a little longer. Agent Foile?\"\n\n\"Yes, Your Highness,\" he said, stepping in from the passageway.\n\n\"Any idea where we might recruit a few cooks, bartenders, and the like?\"\n\n\"None whatsoever, but I will get on it immediately.\"\n\n\"Thank you so very much,\" Kris said with a smile, then turned back to Captain Ajax when Foile stepped back into the passageway, already talking to his commlink.\n\n\"I would suggest you get in touch with Captain Pett from _Wasp_ and talk to her about what it's like to have a troublesome Longknife aboard. If you'd like me a deck up from your bridge, I'd be glad to do that, but I will need the strongest radiation shielding for Ruth.\"\n\n\"Ruth?\"\n\n\"My infant.\"\n\n\"She'll be traveling with us?\"\n\n\"She _will_ be traveling with us,\" Kris said. \"And yes, I know this is all irregular, but we've been doing things different on Alwa Station for the last two years and no, I did not authorize anyone to remove their birth-control implants. Mine and seventy-one others were sabotaged, if you are curious, and I'm here because the King ordered me back to sort out this problem with the Peterwalds. We all have to stretch a bit.\"\n\n\"I understand, Admiral,\" the _P. Royal_ 's skipper said, looking quite stretched already.\n\n\"Good. Now, why don't you get with Captain Pett. Not only do you need to accommodate me and my needs, but I horse ships around hard, so they aren't there when someone aims for where they expected them to be. That means larger reaction motors than the ships came with. Get the word from Pett, then pass it along to your squadron mates.\"\n\n\"Yes, ma'am,\" the captain said, dismissed herself, and headed for the door as Jack came in it.\n\n\"Honey, you really want to talk to this fellow,\" Jack said, grinning as if he'd won the lottery. \"I love what he's offering.\"\n\n\"I could use something nice today,\" Kris said, and stood to meet the CEO of the yard doing work on her flagship.\n\n# 17\n\n\"Nelly, could you get us a few chairs in here,\" Kris said.\n\nA moment later, three comfortably overstuffed chairs flowed out of the deck. They settled into them: Kris and Jack easily, Chogan a bit more unsure of himself.\n\n\"Damn, these are comfortable,\" the big fellow with long, flowing white hair said as he settled his full weight into the sudden seat.\n\n\"Don't you provide Smart Metal beds and chairs on your ships?\" Kris asked.\n\n\"Yeah, I provide 'em, but I never tried 'em.\"\n\n\"How long have you been building Smart Metal ships?\" Kris asked.\n\n\"I bought the patent from Al Longknife a few months ago. It took the mills a while to switch over. I just finished my first two ships. This one,\" he said, nodding at the bulkhead, \"and the _Furious_. I'm glad I got the job of putting the crystal armor on both of them. I've enjoyed working with the skippers and crew.\"\n\nFrom what Captain Ajax had said, the feeling was reciprocated.\n\nChogan eyed Kris curiously. \"I want to personally thank you for posting the full skinny on how to grow and install this crystal. I imagine it would have been a long time coming before Al Longknife let anyone in on the secret.\"\n\n\"Al Longknife is my grandfather, and no, he's not at all happy with me,\" Kris said.\n\n\"Oh, does that mean Billy Longknife is your old man?\"\n\n\"Guilty on all counts.\"\n\n\"God help you, kid.\"\n\n\"They regularly beg the same indulgence for themselves when I'm around.\"\n\n\"I hear you're headed back out.\"\n\n\"Yes, sir.\"\n\n\"To the Peterwald Empire,\" the CEO said.\n\n\"That's common knowledge, Kris,\" Jack said.\n\nKris just frowned. Apparently, assassins were not the only ones who knew where she was going.\n\n\"I understand that the _P. Royal_ will be carrying a lot more than just flag quarters,\" the white-haired man said.\n\n\"I tend to need a lot of support staff,\" Kris said.\n\n\"It must get mighty tight when you pull it all in for Condition Zed.\"\n\n\"We've managed to handle it fine,\" Kris said. She eyed Jack. _Where's this conversation going?_\n\nHe kept an enigmatic smile on his face and said nothing.\n\n\"I'd like to give you a gift, Your Highness. I don't imagine you're used to hardheaded business types offering stuff to you for free, but I want to.\"\n\nKris schooled her face to Navy bland and waited.\n\n\"I've got a contract to wrap this ship and the _Furious_ in crystal. That's a given. However, it seems that between now and the time we finish this job, my fabs expect to deliver about ten thousand tons of Smart Metal as well as the crystal to protect it. That's above and beyond what I'm contracted to deliver. Rather than save the leftover for the next ships I'm building, I'd like to put the extra Smart Metal and armor on your hull.\"\n\nKris wasn't often taken by surprise, but he surprised her.\n\n\"Are you asking me for a contract mod to pay you for the extra product?\"\n\n\"No, ma'am. I figure getting a contract mod through the process would take more time than you've got. I hear tell that things are mighty bad over in Peterwald space. True, it's no skin off our nose, but I got a granddaughter who's selling baked goods to send famine biscuits over there. She's a kindhearted soul. Hard to tell she came from my loins, but maybe my son married better than I thought.\"\n\nHe chuckled at his joke. \"Anyway, you need a bigger ship to do what you got to do, and I've got some extra Smart Metal and crystal to armor that extra space for you. I had my lawyers go over the law, and they found something down in the fine print that says I can give stuff to you for the defense of the realm, and you can accept it in the King's name.\"\n\n\"Do you remember when we had all those folks on Chance turning the old _Patton_ into a museum that could go to space?\" Jack reminded Kris.\n\n\"I thought they'd taken that law off the books after I actually took the _Patton_ out for a fight,\" Kris said.\n\n\"It's still there,\" Chogan Cam said.\n\nKris had sailed in several _Wasp_ s. They'd always been a bit larger to provide room for her attached staff, Marines, and boffins. She'd also fought the latest ones, taking them down to their tight Condition Zed. She'd always been on the bridge, doing what needed doing. She'd never seen how bad it was for the Marines and Sailors.\n\nThey must have been cheek to jowl.\n\nKris pulled herself up to her full regal self. \"We would be happy to accept your gift to our protection and those of our subjects.\"\n\n\"Wow. I've never seen King Ray come all that royal on anyone,\" Chogan said.\n\n\"Well, if you're giving me noble gifts, I should at least give you the full noble thank-you,\" Kris said.\n\nThe CEO stood. \"Then, if it's all the same to you, I'll be on my way to make this happen. I wish you the best of luck. You're going to need all you can beg, borrow, or steal.\"\n\n\"Thank you, sir,\" Kris said, and watched him hurry on his way.\n\n\"Nelly, are things quite as bad as he says they are?\"\n\n\"I've accessed all the reports, both those in the media and those Crossie's corkscrew computers gave me access to. Kris, it's bad. Starvation bad. Pirates bad. Battle lines fighting battle lines bad.\"\n\n\"What's Vicky's part in this?\"\n\n\"It's hard to tell, Kris. I think she's doing her best not to fire on any of the ships loyal to her dad. Ships flying the Empress's flag are another matter. It appears they had a major battle at a place that you should remember, St. Petersburg.\"\n\n\"How'd it go?\"\n\n\"Vicky's side was outnumbered but was still winning. Likely, they'd have captured or killed the Empress.\"\n\n\"But?\" Jack said.\n\n\"The Emperor called a truce. That was when he announced that he wanted you to mediate between Vicky, him, and the Empress.\"\n\n\"Are we negotiating between two or three sides?\" Kris asked.\n\n\"That, Kris, is part of the question you will have to answer before you can begin mediating anything.\"\n\n\"Why does my head hurt?\" Kris asked no one in particular.\n\n\"What did you say about your mother? You can't blow her up, and you can't straighten her out?\" Jack said, reminding Kris of her much younger comment on her family issues.\n\nAfter everything else the day had thrown at her, that night Kris found herself falling asleep once more aboard ship in her own familiar bed. It didn't feel at all strange that the comfortable, soft mattress that Nelly formed for her was fluffed up, familiar Smart MetalTM.\n\n# 18\n\nThe next day passed quietly for Kris. No doubt, she was residing in the eye of a hurricane that swirled and tossed around her, but she and Ruthie spent the day getting to know their new best friends. Beside Akumaa and Mai, there were Sally Greer, Shani m'Zuri, and Fede Radko. They all looked just old enough to have completed an enlistment but not by much.\n\nOne trip to a shooting range Nelly made appear on the _P. Royal_ showed that they were quite capable with pistols, assault rifles, and sniper gear. Shani and Fede were quite disappointed that they weren't allowed to play with a 60mm mortar.\n\nWhere the girls learned their child-care skills was an open question that none talked about.\n\nIn charge of the five younger women was a hard-nosed, ramrod-backed woman of some thirty-odd years. Kris tried not to smile when one of the younger girls slipped up and called her Gunny. But Li O'Malley had the gentlest of hands when she bathed Ruth.\n\nKris enjoyed a day of shaking down her child care while the spaces around them took solid form. The five gals chose to share a room. That gave them more space to share, and Nelly included a bath as luxurious as Kris and Jack's. Gunny had her own room, off the commons. She spent most of her time with the gals.\n\n\"If we're all next door to the wee one's room, we can be here in a flash if there's any wee trouble,\" Li said.\n\nWhat they would not allow was for Kris to keep Ruth in a bassinet next to her own bed. \"You need your sleep,\" Li insisted. \"Ruth will be only a door away from you.\"\n\nStrange how Gunnies managed to order flag officers around.\n\nIt was a coin toss to see whether Ensign Longknife or Agent Foile got the room on the other side of the day quarters\/flag plot. Since they were in space, Kris decided she wanted Meg closer. The three men had staterooms past her. Leslie, however, chose to have her room next to the nannies.\n\nMore often than not, Leslie was with them and Ruth. Kris approved of the way that worked out.\n\nThe mediators were quartered above Kris's deck. Rather than the thirty-six Kris had expected, they totaled out close to sixty. Nelly spent some time working with the computer experts on the team and came back with a list of material that she wanted to order for herself and Jack's Sal.\n\n\"I've been away from human space for two years and they've done all kinds of things. Kris, I have to keep up, or I'm going to be left behind.\"\n\n\"Are you really afraid of some computer becoming as magnificent as you?\"\n\n\"No, Kris, I see no evidence that any computer is up to writing the bad jokes I come up with. However, the self-organizing matrix is better and faster. I need an upgrade.\"\n\nThis time, Kris remembered to warn Captain Ajax that a very tiny and extraordinarily expensive package would be coming aboard. Still, the six Marine guards who escorted the delivery girl to Kris's door seemed just as gobsmacked as her. Kris signed for the package, included a tip, and sent all seven on their way.\n\nThat night, Kris had weird dreams as Nelly did her thing. The next morning, on the bed between Kris and Jack was a block of gray material.\n\n\"You can probably give that to Meg, Li, and maybe Agent Foile. It can augment their present computers.\"\n\nKris studied the lump that looked like nothing more than clay and had Nelly divide it into three equal chunks. Meg got hers at breakfast.\n\nShe stared at it cautiously. \"Nelly wants me to have that?\"\n\n\"Yes, I do,\" Nelly said.\n\n\"Even after what I've been doing?\"\n\n\"What?\" Kris asked, frowning.\n\n\"Meg has been slipping into my net regularly to work with me,\" Nelly answered. \"Like when she needed help to process out the first three nannies.\"\n\nKris said nothing, just eyed the young Ensign Longknife.\n\nShe shrugged. \"I guess I should have asked permission. It's just so automatic.\"\n\nKris raised her eyebrows.\n\n\"Okay,\" Meg said, raising her hands in surrender. \"You know Grampa Ray once got in touch with the planet computer on Santa Maria, right?\"\n\nKris let silence answer her question.\n\n\"Well, Gramma Alnaba inherited his bit of a brain bulge. It makes it easy for her to work on the little bits of the Grand Computer we run across now and again.\" Meg paused to take a deep breath.\n\n\"I seem to have inherited it, too. Even if I don't try, I'm talking to any computer around me that has any wireless access.\"\n\nKris took a moment to process this surprise. \"Nelly, you knew this?\"\n\n\"I suspected something like it. I didn't _know_ it until just now.\"\n\n\"What's it good for?\" Kris asked slowly.\n\n\"Well, unless I'm careful, I know what will be on the next test. I aced the academy.\"\n\n\"Nelly, is she inside you right now?\"\n\n\"No,\" came from both Nelly and Meg.\n\nNELLY, ABOUT THIS MATRIX. ARE YOU GIVING EACH OF THESE THREE ONE OF YOUR CHILDREN?\n\nNO, KRIS. THEY GOT THE MATRIX, BUT THEY'RE WORKING WITH THE KERNEL THAT ORIGINALLY CAME WITH THEIR COMPUTER. I'M STILL THINKING ABOUT SHARING MYSELF WITH THEM. LET'S SEE HOW THIS GOES. BESIDES, I'M NOT DOING ANYTHING WHILE WE'RE ON THE SAME PLANET WITH YOUR GRAMPA AL. I DO NOT TRUST HIM ANY MORE THAN I TRUST VICKY, HER DAD, OR HER STEPMOTHER.\n\nYOU'RE GETTING QUITE CYNICAL.\n\nIT MUST BE THE COMPANY YOU KEEP.\n\n_Well, at least I'm not included in that,_ Kris thought.\n\nNOT YET.\n\nKris eyed Meg. \"So, I'm commanding a Longknife who is full of surprises.\"\n\n\"That's kind of what Mac and Trouble said about you when they hauled me in for this assignment,\" the ensign admitted.\n\n\"You didn't want it?\"\n\n\"It's not that I didn't want it. It's just that the guy I'd been kind of dating at the academy put Alwa at the top of his wish list. I've lived at the hind end of nowhere. I'd like to try human space for a while.\"\n\n\"Go test out your new toy,\" Kris said.\n\nAt lunch, Meg was back, happy as a clam. Li got hers then, and Foile at supper. Next morning, he was quiet about the addition.\n\nThat day was busier. Captain Ajax called a staff meeting with the other captains. The eight of them seemed a bit surprised to see the Forward Lounge. Kris was, too.\n\nThe last time Kris had seen Bosun's Mate 3\/c Mary Fintch, they'd been prying her out of the helm seat of Kris's wrecked first command, PF-109. She was one of the lucky ones who survived that fight.\n\nWhat Kris's hadn't known then was that Mary's family had been in the restaurant business for generations, and Mary had used her disability bonus to get herself started in her own pub. Foolish girl, she'd jumped at a chance to ship out with Kris again and even talked her wife Robin Song into joining them.\n\nFoile's background check on them was clean, as well as for the four cooks and equal number of bartenders who followed them aboard. The Marines who had been part-time waiters and waitresses on the _Wasp_ 's Forward Lounge came over with their company and went right to work with hardly a blink.\n\nThe squadron's skippers settled around a table with Kris at its head, accepted cups of coffee, and listened while Nelly showed them what it was like to follow this damn Longknife into battle.\n\nSeveral blanched.\n\n\"We'll never survive all that banging around,\" one was heard to murmur to his neighbor.\n\n\"You'll need to upgrade your high-gee stations,\" Kris said. \"That's easy since they're made of Smart Metal, and Nelly has the plans. Nelly, please produce a demonstration model.\"\n\nA moment later, two rose from the deck, one on each side of the table. The skippers took turns trying them on and watching as they adjusted to their different sizes.\n\n\"It's best to wear your birthday suit when you get into one of these,\" Kris said. \"I always dismiss the crew to quarters to put these on, one-half or one-quarter of the watch at a time. You can try this out and make your own call.\"\n\n\"Does your computer control every station on the ship?\" asked one captain.\n\n\"No, each station is independent. Now, there's more jinking around than you're likely ready for. Nelly, give them a schematic of the standard battlecruiser and one modified to my requirements.\"\n\nOn the walls, the outline of a ship appeared with all the piping to the reaction motors in blue. Then a red coating appeared, complete with measurements.\n\n\"Good Lord,\" seemed to say it for all the skippers.\n\n\"And you can do this while we're in port getting our armor cladding?\" Captain Ajax asked.\n\n\"You're lucky,\" Kris said, with a chuckle. \"You wouldn't have wanted to be on the ships we first tried this on.\"\n\n\"You're really going out loaded for bear, Admiral,\" one captain said.\n\n\"We're going out there to try and persuade two, maybe three sides to stop their civil war. Someone has made three attempts to kill me and put an end to our mission before it starts. How loaded do you want to be?\"\n\nThat left the skippers in a thoughtful mood.\n\n\"First point, I do not fire first,\" Kris said, holding up a finger. \"Second point, I don't want to lose a ship or a Sailor. Last point, if someone starts the shooting, I intend to finish it. We've got twenty 22-inch lasers on each of our battlecruisers. Their best battleships have maybe twelve or sixteen 18-inch lasers.\"\n\nKris paused to let them consider that. \"Whose shoes do you want to walk in?\"\n\nThat caused a long pause, then one captain raised her hand. \"How much does your computer need to mess with mine to do all this?\"\n\n\"Nelly?\" Kris said.\n\n\"I can set up a specific subroutine that only interacts with your systems when it needs to. I can also give you back all the capacity I borrowed when I am done. If you wish, you can run a full set of diagnostics when I'm done and assure yourself that I left nothing behind.\"\n\n\"You're quite polite,\" Captain Ajax said.\n\n\"Never to me,\" Kris muttered.\n\nThat got a laugh, and they got down to work.\n\n# 19\n\nKris risked one more trip down the beanstalk to Main Navy.\n\n\"Is there any chance I can have more ships?\" she again asked Mac as she settled into one of his overstuffed armchairs.\n\nMac just shook his head. \"You're not going to give up, huh?\"\n\n\"Not without one last try,\" Kris said.\n\n\"We've got four squadrons heading out for Alwa. You want to borrow one or two of those?\"\n\nKris could only shake her head. \"You read my report. It's not just Alwa we need to be looking after. Those cats on Susquan need taking care of, too, both to protect them from the alien raiders and themselves. Christ, Mac, they've got atomics on hair triggers.\"\n\n\"Well, they're Sandy's problem now. After reading your report, I'm amazed you got back here sane.\"\n\n\"Who said I did?\" Kris said with a chuckle. \"I'm letting Grampa Ray send me out on this crazy mission to Greenfeld.\"\n\n\"There is that. So, you want to steal from Sandy, or are you gonna settle for the eight I managed to peel off Wardhaven's defense?\"\n\n\"Wardhaven?\"\n\n\"Yeah. Your old man is none too happy to have his tax money gallivanting off to pull Hal Peterwald's chestnuts out of the fire.\"\n\n\"Is there ever enough?\" Kris asked, giving voice to the question that had haunted her dreams for years now.\n\n\"Hasn't been since I put on this uniform. You'd have to ask Trouble or Ray if they ever had enough during the Iteeche dustup.\"\n\n\"Yeah,\" Kris said. \"Okay, you win on that one. Now, about the situation in Harry's so-called Empire. All the intel I have goes galloping off in forty-eleven different directions. What's your take of the situation?\"\n\nThe old field marshal shrugged. \"What did someone once say of another planet? A question mark wrapped in an enigma, surrounded by fog, or something like that. We had a fellow come through here, claiming to be from the Grand Duchess. He wanted to buy some spidersilk underarmor. I think your Gramma Trouble arranged for him to have something to take back. Our contacts in the Greenfeld Navy say that a lot of it has sided with Victoria against her stepmother. It's kind of hard to know much about Greenfeld on good days, and it's been a while since they had one.\"\n\nMac paused to gaze out the window. \"I'm glad they asked for you and not me. I don't know how I'd try to find the bottom of that mess.\"\n\n\"Well,\" said Kris, \"as Ruthie has occasionally left me wondering, somewhere in all that poop, there has to be a cute little girl.\"\n\nMac snorted. \"I remember changing a few of those diapers.\"\n\n\"Okay, we don't know much. How much can we trust what Crossie is sending us?\" Kris asked.\n\nNow Mac just shook his head. \"I never know what game he and Ray are playing. I didn't before, and the last two years since Ray came back without Rita only seems to leave me more in the dark. Maybe they don't know anything more than they say they do. Then again, maybe they've got their fingers in that pie somewhere I can't see.\"\n\nHe scowled. \"Watch your own fingers, Kris. I don't need to tell you that this is not a safe situation. By the way, your dad asked me to ask you if you'd leave Ruthie behind with your mom.\"\n\n\"He did, did he? He's not willing to ask himself?\"\n\n\"I think he expected us to have this little talk and that maybe I'd have more influence than he has.\"\n\nKris took a deep breath and let it out slowly.\n\n\"I know what it's like in that household, Mac. I grew up in it. I will not leave my child in that mess.\"\n\n\"And Jack's family?\"\n\n\"He hasn't suggested we do that. I don't think he's any more willing to leave Ruth there than I am to let my mom get her claws into her chubby bottom.\"\n\n\"You're headed into a risky business,\" Mac pointed out.\n\n\"And I'll just have to bring me and Jack and Ruth back out when we finish with it,\" Kris said, flatly.\n\n\"Okay. I can tell Billy I asked.\"\n\n\"Yes, you can,\" Kris agreed. Interesting, no one was offering to either fold Ruth into Honovi's growing brood or ask Gramma Trouble to take another turn changing diapers. Kris joined Mac staring out the window.\n\n_Do I want to leave Ruth here?_\n\n_Here where someone can kidnap her? Not really,_ she answered her own question.\n\n_You've got to be crazy to take her into harm's way._\n\n_No doubt about that, but then, I've got a pretty good record for making it in and out of hell on a regularly scheduled bus route._\n\n_Isn't it interesting how the very people who want me to leave Ruth behind are so eager to push me and Jack out to floss the lion's teeth._\n\nKris made up her mind.\n\n\"Mac, when this is done, I want that desk job. I want me and Jack to push papers, and I want to go home to take Ruth to soccer games and birthday parties and be like a normal family. You hear.\"\n\n\"I hear.\"\n\n\"I've been Ray's old warhorse long enough, charging out for whatever mess he wants cleaned up. After this, it's someone else's turn.\"\n\n\"Is there anyone else like you?\" the field marshal asked.\n\n\"I don't know, but you better find someone good enough, you hear?\"\n\n\"Loud and clear.\"\n\n\"Nelly, have you gotten a data dump of everything available?\"\n\n\"And likely a few things they didn't want us to have,\" Nelly answered.\n\nMac rolled his eyes at the ceiling.\n\n\"Then I'll be on my way,\" Kris said.\n\nAn hour later, she was back up on the _Princess Royal_ , and none of her guards even had to break a sweat.\n\nNot a bad day.\n\n# 20\n\nFour days later, the _Princess Royal_ led the _Intrepid_ , _Courageous_ , _Furious_ , _Resolute_ , _Defender_ , _Steadfast_ , and _Monarch_ in a column as each fired their deorbital burns, dived toward Wardhaven, then blasted for Jump Point Beta.\n\nFor Kris, the situation was strange beyond words.\n\nEvery one of these ships bore a name from the fleet that Kris had commanded on Alwa Station not so long ago. That it had come to pass that Wardhaven felt the names were open for new construction told Kris what her father and great-grandfather must have thought about her chances out there.\n\nShe hadn't bothered to tell them differently. Not when it would involve admitting that the original ships had been damaged beyond repair, their Smart MetalTM drained into holding tanks, and the ships rebuilt from the keel up to new and better designs.\n\nNo need to scare them worse.\n\nBut beyond the names of the ships, there was more. Kris's squadron was headed for the jump point that had, so recently and so long ago, coughed up six huge battleships. At least then, Kris had thought they were huge.\n\nThey had threatened to bombard Wardhaven down to bedrock if the planet didn't surrender unconditionally to them. Now Kris was heading through that jump point to help the man who had likely sent those ships and issued those threats.\n\nIf she succeeded, she would save Henry Peterwald's Empire. If she failed, it would continue to tear itself apart.\n\n_Heads who wins? Tails who loses?_\n\nKris held Ruth close and bounced her gently.\n\n\"Do you have the strange feelings I have about this mission?\" Jack asked.\n\nKris nodded. \"Grampa Ray has handed me some lousy assignments, but this one has got to take the cake.\"\n\nAt a fleet acceleration of one gee on their way to the jump, Kris divided the squadron into two divisions of four, then spread the ships out in a loose formation with five thousand kilometers between them.\n\nThat done, she ordered them to Evasion Plan 1, but only for five minutes. Without being asked, Kris then allowed them a breather to mend and fix. An hour later, she implemented Evasion Plan 2.\n\nTen minutes later, the _Monarch_ messaged Kris for permission to fall out, so she suspended evasions to let a whole lot of people mend and fix.\n\nTwo hours later, the _Monarch_ held together for fifteen minutes of Evasion Plan 3. It was the _Intrepid_ that called \"uncle\" that time.\n\n\"Nelly, how are the reaction motors holding up?\" Kris asked.\n\n\"Not as well as they should, Kris. I know I told them how thick to make the pipes. They shouldn't be having this much trouble.\"\n\n\"What are they trying to get away with?\" Kris asked.\n\nNelly showed a schematic of the battlecruisers. The reaction system on each one was different. Skippers were trying to get by on what they thought they needed. Clearly, they didn't think they needed any of that Alwa Station stuff.\n\nThey were learning. Even as Kris watched, pipes were thickened up.\n\n\"Are all the ships still at Condition Able?\" Jack asked.\n\n\"Why yes, I do believe,\" Kris said, a cheerful grin on her face.\n\n\"You think you ought to tell them?\" Jack asked.\n\n\"Each captain is responsible for their own ship. Let's see who figures it out first,\" Kris said, settling into a chair and letting Ruth stand up in her lap. She managed a few seconds before collapsing with a happy giggle.\n\nKris was prepared to give her ship captains three hours to solve their problems on their own. Ninety minutes into that count, Captain Ajax called.\n\n\"Admiral, I intend to go to Condition Baker before we do any more evasion testing.\"\n\n\"Very good, Captain,\" Kris said.\n\n\"I'll need to shrink admiral's country.\"\n\n\"No problem,\" Kris said with a grin. Maybe it was for Ruth. Maybe it wasn't.\n\nAfter Ajax rang off, Kris said, \"Nelly, do a check to see if all spaces give up the same percent of area.\"\n\n\"Are you thinking that she might be taking more from you than the rest?\"\n\n\"Not sure, Nelly. I know with all my space, I can spare more than most. Just give my people warning before the bulkheads start moving in on them.\"\n\nTen minutes later, the bulkheads did move. Jack checked. Kris had lost all her sleeping quarters and her head was the minimum needed. Kris's day quarters were about a quarter their original size but neither she nor Ruth had their elbows joggled.\n\n\"I can afford this,\" Kris said. \"A good move by Captain Ajax. Nelly, check on nanny and Secret Service space.\"\n\nNelly didn't have to; the whole batch came knocking at Kris's door. It took her a moment to explain Condition Able, Baker, Charlie, and Zed. \"We become a smaller target and a tougher nut to crack,\" she finished. \"We need to get all of you battle stations.\"\n\nShe had Nelly get them high-gee stations. While the ships had held one gee all through the evasion drills, the bouncing around hadn't been too bad. Even at one gee, Evasion Plans 5 and 6 were going to involve hard knocks.\n\nNelly pulled high-gee stations out of Kris's spaces, reducing the size of her day quarters. A moment later, Nelly reported, \"Captain Ajax just noticed I'd shrunk your spaces more. Her Defense Coordinator suggested you might be setting up high-gee stations. Ajax ordered everyone into them. She's also passed the word to the squadron.\"\n\n\"Kris, you are evil,\" Jack said.\n\n\"No, I just prefer my people to solve their problems on their own.\"\n\n\"All hands, prepare to go to Condition Charlie,\" the 1MC announced.\n\n\"Nelly, do our mediators know how to survive this?\"\n\nNelly hardly took a moment before replying, \"The Command Master Chief is showing them the ropes, Kris.\"\n\n\"Better and better,\" Kris said.\n\nJack just shook his head.\n\nThe squadron held thirty minutes of Evasion Plan 5, then another half hour of EP 6. There were no requests from anyone to break off.\n\nWhen Kris decided they'd had enough, she called her skippers up on net. \"These Evasion Plans may save our lives someday. They've sure saved mine enough times. Now that we've worked the kinks out, please have your helms keep these ready to load on a moment's notice. I have no idea what we will face in Greenfeld space. Carry on.\"\n\nKris invited Captain Ajax and several of her division heads up for dinner in her flag wardroom. It was the same as her day quarters, done over with a Smart MetalTM table, chairs, white linen, silver, and china. The talk went long into the evening as Ajax and her team now listened to Kris and how they did things on Alwa Station. Kris made herself the butt of most of her jokes and pointed out what had worked and what hadn't.\n\n\"It's all there in the reports. They make scary reading, but you'll likely want to read them.\"\n\nAs Kris and Jack were getting ready for bed that night, Nelly broke in. \"Your reports are the hottest items in the ship library. Every officer from ensign to Ajax has downloaded them. Some of the chiefs, too. It's the same on all the ships of the squadron.\"\n\n\"I hope they benefit from Alwa's hard-learned lessons,\" was all Kris said.\n\n# 21\n\nThat they hadn't learned enough became apparent all too soon.\n\nThey were heading for Greenfeld, taking the most direct route and avoiding any planets held by the Grand Duchess. That order to avoid Vicky came from Grampa Ray and Crossie. It bothered Kris, but it seemed logical.\n\nYou call on the senior first, then the junior. Since Vicky's dad had been the one to ask for mediation, it seemed best to give him the first hearing.\n\nIt all seemed right and proper.\n\nUnfortunately, it made Kris's course all too predictable.\n\nThey were coming up on a jump, making a good twenty-five thousand klicks an hour. It was fast, but hardly unsafe. They'd already sent the comm buoy through to alert traffic on the other side to stand clear for them. Kris never lost the sense of wonder at how ships leapt from one bit of space, across dozens of light-years, to appear somewhere else.\n\nKris was seated in a chair at her conference table, facing the screens as they approached the jump, bouncing Ruth in her lap and letting her mind wonder over how she was going to live through this latest of Grampa Ray's batty ideas.\n\n\"Nelly, how long would it take you to project a staircase from here to the bridge?\" she asked.\n\n\"We are directly above the bridge,\" Nelly replied. \"But wouldn't an elevator down be faster?\"\n\n\"It depends on how fast I want to get there,\" Kris said slowly, puzzling over the idea and wondering why she'd gotten it just now.\n\n\"Kris, you aren't going to start injecting yourself into other people's business, are you?\" Jack asked. \"Micromanagement is a sin you have avoided so far.\"\n\n\"Are you hinting it's one of my few?\"\n\n\"When you were a boot ensign, did you enjoy having someone looking over your shoulder?\"\n\nJack had a point. Kris hadn't been a very good subordinate. There had been that incident that wasn't officially labeled mutiny. It was only the worst of many problems she'd caused her bosses.\n\n\"You may have a point,\" Kris said to Jack, as the _Princess Royal_ raced through the jump, leading the rest of the squadron.\n\n\"Ships off to port,\" Nelly almost shouted.\n\nOne glance at the screen showed Kris the ships. They had no battle revolutions on their hulls, but their laser turrets were out of train and aimed at the jump the _Princess Royal_ had just come out of.\n\nEven as she took that in, Kris moved to put Ruth down on the table. \"Nelly, high-gee station for Ruth. Now!\"\n\nAnd just as quickly, the conference table had a tiny high-gee station wrapping Ruth in an armored embrace.\n\nKris hadn't taken her eyes off the screen. The ships' turrets were turning, trying to train in on the fast-moving _Princess Royal_. Several turrets were slow. Some were not.\n\nCaptain Ajax had wisely stored the crystal armor forward. It began to glow as lasers hit.\n\n\"Why aren't we returning fire?\" Kris demanded. No one answered.\n\n\"Nelly, that elevator,\" and Kris's chair fell through the floor, forming itself into a swing as it plunged down. Kris grabbed the suspension lines and felt her hands burn as they extended under her grip.\n\nKris landed within two paces of the gunnery station.\n\nIt was empty.\n\nAround Kris, on the bridge, people stood in shock and surprise, frozen in that first moment when peaceful people realize that someone is trying to kill them. _Really_ trying to end your life.\n\nA glance at the main screen showed the two ships, still trying to train lasers on the _Princess Royal_. In the foreground, the battlecruiser's bow glowed brighter. Steam bled where she'd taken a hit on bare metal.\n\nKris slid into the vacant Weapons station and began issuing orders.\n\n\"I have the conn. Helm, execute Evasion Plan 3. Defense, go to Condition Baker now. Charlie as soon as you can.\"\n\nKris left the others to come out of their shock and do what she'd ordered while she and Nelly conversed at the speed of thought.\n\nNELLY, TARGET LEFT FORWARD BATTERY ON LEFT SHIP. RIGHT BATTERY ON RIGHT.\n\nDONE.\n\n\"Fire!\"\n\nThe forward batteries discharged their twelve huge 22-inch lasers, half at each ship. Without Kris telling her, Nelly had targeted the lasers in pairs. Three pairs of 22-inch lasers sliced through ice armor that wasn't meant to face guns of this caliber. Beams lanced deep into steel and flesh.\n\n\"Forward batteries, reload. Helm, flip ship.\"\n\n\"Flipping ship,\" the woman at the helm said, and began to rotate the _Princess Royal_. Kris swallowed a \"Faster, damn you!\" The ship was responding a lot slower than the _Wasp_ , but the hostile ships were in a reload cycle that had to be longer than the battlecruiser's.\n\nThe helm bungled the flip and didn't stop until they had gone past the fifteen degrees the lasers could correct for.\n\n\"Nelly,\" Kris snapped, and her computer took over the helm station as the panicked helmswoman found that her own hand on the wheel no longer did anything.\n\n\"Fire aft, right and left,\" Kris ordered and the aft lasers emptied their capacitors.\n\n\"Flip us again, Nelly,\" Kris said, more to let the rest of the bridge crew know what was going on than to tell Nelly. \"Get us nose armor on to those bastards.\"\n\n_I must clean up my language, but not now._\n\n\"We're at Condition Baker,\" Defense announced. \"Going to Condition Charlie.\"\n\nBehind Kris, whoever had had the conn finally hit the General Quarters Klaxon.\n\n\"General Quarters. General Quarters. All Hands, to battle stations,\" finally began to sound through the ship.\n\nKris eyed the forward batteries. They were coming up quickly to full.\n\n\"Comm, send to enemy vessels, 'Drop your reactor cores or we will finish you.'\"\n\n\"Sent.\"\n\nCaptain Ajax raced out of her in-space cabin, zipping up her shipsuit. \"I have the conn,\" she announced.\n\n\"The admiral has the conn,\" said the man in the command chair, jumping up.\n\n\"And her computer has the helm,\" the helmswoman got out before she snapped her mouth shut.\n\n\"Admiral, you have the conn,\" from Ajax said there would be a day of reckoning.\n\nOne of the battleships, hard-hit and spinning out of control, dumped its reactors either upon Kris's orders or in hope of saving their lives.\n\nThe other one kept on doing what it was doing.\n\n\"Nelly, fire all forward batteries on the battleship with loaded reactors.\"\n\nThe forward lasers sent six pairs of 22-inch lasers at the already hurting enemy ship. It did not last through the broadside.\n\nIt began to come apart as sections of hull were cut clear away. Then something exploded deep within. In little more than the blink of an eye, the ship was a glowing ball of gas.\n\nThe other ship was drifting in the general direction of the jump as the _Intrepid_ came through.\n\n\"Comm, send to _Intrepid_ ,\" Kris snapped. \"Decelerate hard and place a boarding party aboard that ship. It fired at the _Princess Royal_ , and I want to know why and in whose name it did so.\"\n\n\"Aye, aye, ma'am,\" came back immediately, and Kris's squadron began a delicate dance as they came through the jump to discover not at all what they had expected.\n\n\"I surrender the conn,\" Kris said, letting out a breath she hadn't realized she'd been holding. She'd leave these problems to her skippers. They weren't finding anything nearly as bad as she had.\n\n\"I have the conn,\" Captain Ajax said firmly.\n\n\"I'll be in my quarters,\" Kris said. \"When you have a report on the _Princess Royal_ 's damage and can be spared from the bridge, Captain Ajax, please report to me.\"\n\n\"Yes, ma'am,\" was cold and with sharp edges.\n\nKris left the bridge and climbed the ladder to Flag Country. She was not happy that she had jumped, literally, into running the _P. Royal_ , but damn it, the crew was green and not responding to this first whiff of gun powder fast enough.\n\nNELLY, HOW LONG DID THAT BATTLE TAKE? IT SEEMED LIKE FOREVER.\n\nEIGHTEEN SECONDS, KRIS, FROM OUR COMING THROUGH THE JUMP TO THAT BATTLESHIP BLOWING UP.\n\nONLY EIGHTEEN SECONDS?\n\nYES, MA'AM.\n\n_Maybe I did jump the gun?_ Kris thought.\n\nIn her day quarters, she found Jack and several nannies trying to soothe a very unhappy Ruth. Kris opened her arms for the child, and everyone was smart enough to give Ruth to her. Opening her shipsuit, Kris settled into a rocker and began nursing her infant. Ruth settled down for a bit of comfort food. Kris soon found herself alone in her quarters as all did the smart thing and left her to herself.\n\n\"Nelly, lower the lights.\"\n\n\"Do you want to talk, Kris?\"\n\n\"No, Nelly. Just lower the lights.\"\n\n# 22\n\nJack was the first to risk knocking at her door. Only after he identified himself did she say, \"Enter.\"\n\nHe must have asked Sal for a rocking chair because one appeared, he settled into it, and they rocked quietly for a long time.\n\n\"Look at us. Just like an old married couple,\" Jack finally said.\n\n\"You think we'll survive long enough to qualify for a pension?\" Kris asked.\n\n\"We've survived another day,\" Jack answered.\n\n\"The day is yet young.\"\n\n\"I've never seen you move so fast,\" he said after letting them digest that.\n\n\"I've never had everything I love so much on the line.\"\n\n\"Is that what it was? Fear for me and Ruth?\"\n\n\"Say terror, and you might be closer.\"\n\nJack let her consider her own words for a bit.\n\n\"Do you think we could go back and leave Ruth with someone?\" Kris finally said. \"Maybe Child Protective Services?\"\n\n\"Not my mom or yours?\" he asked, risking just the hint of a smile.\n\n\"I didn't realize what I was getting her into.\"\n\n\"Didn't you? You wanted three squadrons. You accepted that extra armor. You drilled the squadron. You are no passenger. You are loaded for bear.\"\n\n\"I sent my flagship through the jump first.\"\n\n\"I'll admit _that_ was stupid,\" Jack said, giving her a sidelong glance. \"I chalk it up to force of habit.\"\n\n\"Will Ruth survive my cleaning up my act?\"\n\nJack let that question hang there before changing the topic.\n\n\"As for going back, I'd think what we just ran into says things up ahead are way too bad for us to waste any more time before we do something about it.\"\n\nKris scowled at Jack, but she couldn't argue with his logic. Still, she wasn't quite ready to give up. \"Likely you're right, but why do me and mine have to pay the price for everyone else's being in such a mess?\" Kris asked, and tasted more vehemence in her words than she'd intended. No. More than she wanted to let out.\n\nWhat she'd said was exactly what she felt.\n\n\"I've been with you close to six years. Not one day has been easier than today,\" Jack told her flatly.\n\n\"But I'm _tired_ of days like today.\"\n\n\"And you've been telling everyone who will listen that after this last one, you quit. Well, that's later. This is now. Suck it up, Soldier, and soldier.\"\n\nJack put a lot of force into his words, but he kept his voice low. Ruth had fallen asleep at Kris's breast.\n\nKris took a deep breath, and let it out slowly.\n\n_Okay, once more unto the breach._\n\n\"Nelly, get the nannies in here.\"\n\nA moment later, Li O'Malley paraded the five others into Kris's day quarters. That they couldn't even keep out of step as they reported to her, backs ramrod straight, told Kris that Grampa Trouble was looking after her again.\n\n\"Li, I want Ruth's bassinet converted to a high-gee station unless something else is required. I want it programmed to switch to a survival pod as well. At least one, and preferably two of you should be in bed or sitting in a station ready to convert to high gee or survival pod at all time.\"\n\n\"Aye, aye, ma'am,\" rang like you'd expect from a Gunny.\n\n\"Work with Nelly to see that these changes are made. If you have any ideas for changes in equipment or policy, pass it to me through Nelly.\"\n\n\"Yes, ma'am.\"\n\n\"Now, put Ruth down for a nap. Nelly, have Agent Foile report soonest with his team.\" Kris handed off the infant, corralled her breasts back into her nursing bra, and was just zipping up her shipsuit when the Secret Service team reported.\n\n\"As you have seen from the assassination attempts and this last bushwhack, someone really doesn't want me showing up on Greenfeld,\" Kris said.\n\nThe agent nodded.\n\n\"I want you to go over the records of this ship's crew with a fine-tooth comb. Check financials again. Look into their families. Is there a grandmother that needs an operation to keep her alive? Is there anything out there that might be used to get hooks into any Sailor, Marine, officer, or civilian aboard the _P. Royal_?\"\n\n\"We'll get on that right away,\" Foile said.\n\n\"Also, I want to know any applicable chatter from any source aboard ship. If someone says Joe is acting strange, I want Joe looked into.\"\n\nThis got a frown from Foile. \"How much do you want us digging into private matters?\"\n\nKris didn't need to be hit over the head with the word private.\n\n\"Mahomet is your computer expert, isn't he?\"\n\n\"Yes, ma'am.\"\n\n\"Mahomet, you may work with Nelly to access any means of communication aboard. This is the Navy. Folks accept the needs of security.\"\n\n\"If you say so, Your Highness,\" didn't show that Foile was convinced.\n\n\"Yes, I say so. The safety of the ship is paramount, and we can't say what an effort to kill me won't do to the ship.\"\n\n\"She does have a point,\" Jack said.\n\n\"What do you Navy types say, 'logged and noted'?\"\n\n\"Exactly. Now, I expect the skipper of the _Princess Royal_ to walk through that door any moment now, if you will tackle the elephant I've assigned you . . .\"\n\nThe Secret Service team headed for the door.\n\n\"Kris, Captain Ajax would like to report to you at a time convenient for you.\"\n\n\"Tell her, Nelly, that she always has the number one slot on my schedule.\"\n\nApparently, their schedules were in sync because Rick and Leslie had to stand aside to allow Ajax in.\n\nShe reported to Kris crisply. \"I have the reports you asked for.\"\n\n\"Good, would you mind if the general stayed in our meeting?\"\n\n\"I have no problem with that.\"\n\n\"Would you care to sit down?\" Kris offered.\n\n\"No, ma'am, I'd rather stand,\" told Kris this would be a rough meeting.\n\n\"Kris, we have a problem,\" Nelly said.\n\nFor a moment, rage at this interruption flashed on Ajax's face, but only for a moment. She quickly molded her face to Navy bland.\n\n\"What's our problem, Nelly?\"\n\n\"The damaged battleship has been drifting toward the jump. A moment ago, its maneuvering jets lit off for a few seconds, and it's now headed straight for the jump.\"\n\n\"Damn,\" Kris growled, and stood. \"Comm, get me the captain of that bushwhacker.\"\n\nA moment later, a man in a nondescript uniform filled Kris's screen. His bridge looked like it had seen better days.\n\n\"This is Admiral Longknife. Why have you corrected your course to take you through that jump?\"\n\n\"Are we headed toward the jump?\" the fellow said, almost sounding sincere. \"Our maneuvering jets just burped. We didn't know what they had done. You made a wreck of this ship. So, are you going to blast a helpless wreck out of space?\" sounded a bit like a plea but more like a challenge.\n\n\"Offscreen,\" Kris said.\n\n\"You going to blast them?\" Jack asked. It looked like Ajax wanted very much to know the answer to that question.\n\n\"After a day like today, I'm sorely tempted,\" Kris growled, but held on to her temper. \"Comm, get me the skipper of _Intrepid_.\"\n\n\"Captain Grayhorse,\" came a moment later from a man with silver hair and high cheekbones that told Kris of his Native American ancestry from old Earth.\n\n\"Captain, I want _Intrepid_ to chase down our wayward bushwhacker.\"\n\n\"Immediately, ma'am.\"\n\n\"You will go to four gees deceleration, then accelerate back toward the jump at four gees.\"\n\n\"Yes, ma'am. Defense, take us smartly to Condition Zed. Helm, warn all hands to prepare for four gees. Admiral, what do I do with that old battleship when I catch her?\"\n\n\"Despite the temptation to blow her out of space, I very much want to talk to her captain and maybe a couple of her crew. However, my talking to them will not involve you accepting any risk to your ship or crew.\"\n\n\"Understood, you want them alive if possible, but you don't want any of us dead.\"\n\n\"Yes.\" Kris paused. \"I expect that when you get through that jump, your quarry may be already under way and have her lasers loaded.\"\n\n\"I expect she will, ma'am, but no old tub like that is going to scar the paint on _Intrepid_. We'll get you people to talk to. I'm thinking of trimming the reactors off that ship the way you trimmed that raider's tail when you rescued the _Hornet_ 's crew.\"\n\n\"So you read that.\"\n\n\"Ma'am, I've read all your reports. Now, if you'll excuse me, I have some wayward children to collect.\"\n\n\"Good hunting,\" Kris said, and turned from the now-blank screen.\n\n\"Would you care for a seat now?\" Kris offered again.\n\nCaptain Ajax looked at Kris and seemed to be taking the measurements of all the paradoxes that was Kris.\n\n\"Yes, I think I will take that chair,\" she said, and, with a shake of her head, settled into a rocker like Kris's just as soon as Nelly produced it from the floor. \"I'm not sure I like the idea of stuff showing up like that,\" she said, eyeing the chair.\n\n\"Haven't doors been showing up between Sailors' staterooms?\" Kris asked.\n\nAjax raised puzzled eyebrows.\n\n\"We had a rash of them on Alwa Station,\" Jack said, \"until we gave up trying to forbid them and chose to regulate them.\"\n\n\"Oh, so that was what some of our Sailors brought back from talking to a couple of the _Wasp_ 's crew in bars.\"\n\n\"You going out to Alwa?\" Kris asked.\n\n\"I hoped to,\" Ajax said.\n\n\"Be ready for a different kind of Navy,\" Kris said. \"Now, about our present situation?\"\n\n# 23\n\nCaptain Ajax took a deep breath and let it out. She didn't start talking until her third breath.\n\n\"The _Princess Royal_ suffered only minor damage. Despite the short range, most of the lasers fired at us either missed or were absorbed by the crystal armor I had stowed forward. We took only two hits where we weren't armored, and those both were handled by the Smart Metal's programmed active defense. Steam disrupted the laser beams.\" Here Ajax paused.\n\n\"It is also very likely that the two beams that hit us were unsteady and wobbled away from the hit. Even though we weren't rotating ship, the beams seem to have slashed a scar across the hull.\n\n\"It sounds like the lasers were loose in their cradles,\" Kris observed.\n\n\"Yes, ma'am. We seemed to have been attacked by the gang that can't shoot straight.\"\n\n\"Thank God for that,\" Jack said. \"What the hell were they?\"\n\nCaptain Ajax had a quick answer to that question. \"Two Mars class battleships, if you can believe it. Each is supposed to have twelve 14-inch lasers with twenty 4.5-inch quick-firing secondaries if the history books are to be trusted.\"\n\n\"Don't those date back to the Unity War?\" Kris asked.\n\n\"Since before it, Admiral. One would think they'd all been blown up by the Iteeche or scrapped, but there are plenty of them still in reserve coasting along behind stations.\"\n\n\"And someone,\" Kris said slowly, nodding as she spoke, \"dragged those two old tubs out of mothballs, patched them up enough that they could keep space out long enough to get in our faces and fire a few quick shots off at us as we came through a jump.\"\n\n\"So it would seem,\" Jack said.\n\n\"I would recommend that the _Princess Royal_ not be the first through any jump from now on,\" Ajax said.\n\n\"Recommendation accepted,\" Kris said. \"Better yet, whatever cat sticks its nose through a jump does it at General Quarters and armored to Condition Zed.\"\n\n\"Yes, ma'am,\" Ajax said, and tapped her commlink, checking one problem from her can of worms.\n\nShe took another deep breath before going on to her next worm. \"The Weapons station on the bridge was vacant because the duty officer had stepped away to the head.\"\n\n\"That shouldn't have been allowed to happen,\" Kris said gently.\n\n\"How do you handle that on Alwa Station?\" was an honest question, nothing snide in it.\n\n\"I honestly don't know,\" Kris admitted. \"No doubt Admiral Kitano tackled that problem while I was off having fun blowing stuff up, and it never got up to my level again.\"\n\nIt dawned on Kris that her irregular promotion pattern and general lack of experience at all levels of the chain of command might not have been the best preparation for her present four stars.\n\n\"Do you have a gunnery officer, and where is his station?\" she asked Ajax.\n\n\"Gunnery has a plotting room that takes in the feed from all our sensors and gives it back, both to the lasers in their casements and to the fighting position on the bridge.\"\n\n\"If Guns is in his plot, why do you have a weapons station on the bridge?\"\n\n\"Admiral, I've always wondered about that,\" Ajax admitted. \"It seemed redundant, but every frigate comes with the bridge position.\"\n\nJack chuckled. Kris scowled at him. He just shook his head. \"You see what that is, don't you?\"\n\n\"No, I don't.\"\n\n\"That's your position on the old _Wasp_ ,\" Jack said, and turned to Ajax. \"You have to understand, the King didn't want Kris doing all the wild things he did as a JO. He got her a contract captain to run her ship around space and trim her sails occasionally. It never worked that way,\" he said, glancing at Kris.\n\nHe was still barely suppressing a laugh. Barely.\n\n\"But with Kris the only official Wardhaven Navy officer aboard, she set herself up in a weapons station on the bridge. That way, if the _Wasp_ ever had to start a war or something, Kris lobbed out the first official shot. You see?\"\n\n\"No I don't,\" Ajax said.\n\n\"The first frigate followed the practice we had on the old _Wasp_. So did the _Sacura_. I suspect that every frigate has Kris's weapons station on the bridge from force of habit.\"\n\nKris wasn't dense. Not normally. But today, she must be making an exception. Now she did get it. She started to laugh and had to bite her lip to stop.\n\n\"And I used to think the old Navy was hidebound,\" Ajax said. \"I guess the new Navy can be just as caught up in doing things the same old same old.\"\n\n\"It would seem so,\" Kris admitted. \"Nelly, send out to the squadron a query about the weapons position on the bridge, how it is used, and how it interfaces with gunnery plot. The report is due day after tomorrow. Until that time, both gunnery plot and the bridge weapons position are to be manned at all times.\"\n\n\"Message logged and acknowledged, Kris.\"\n\nKris found herself studying Captain Ajax even as the woman was studying her right back. \"No, I do not intend to swing from a trapeze onto your bridge every day.\"\n\n\"I was wondering about that. You came on like a bull in a china shop.\"\n\n\"More like a momma bear whose cub was threatened,\" Jack put in.\n\n\"Yes, much more like it,\" Ajax agreed.\n\nKris shrugged. \"Someone was shooting at us. We weren't shooting back. I really didn't like that.\"\n\n\"Yeah,\" Captain Ajax admitted. \"That was a nugget's mistake. It won't happen again. And, ma'am, thank you and your computer for saving our necks.\"\n\nKris smiled, fondly remembering incidents that had not been fun at the time. \"Nelly and I have saved quite a few ships and crews. Strange that, more often than not, we were not forgiven for it.\"\n\n\"For myself, ma'am, it's hard to accept that I and my crew can be replaced by you and that shiny thing around your neck.\"\n\n\"I am not shiny,\" Nelly put in.\n\n\"And don't let her start telling jokes,\" Jack added.\n\n\"I don't think I got nearly enough warning when they told me I'd be your flag captain,\" Ajax said.\n\n\"Don't feel special,\" Kris grumped. \"I never get nearly enough warning before Grampa Ray drops me into these things.\"\n\n\"I screwed up, ma'am. We of the _Princess Royal_ screwed up. If you'll give us a chance to straighten ourselves out, Admiral, we _will_ do better.\"\n\n\"I know you will,\" Kris said. \"All of us, myself included, have got to get it through our heads that going off to mediate a civil war can be just as deadly as fighting it. Holiday's over. Now we get ready to fight.\"\n\n\"Aye, aye, ma'am,\" Ajax said, and marched from Kris's quarters.\n\n# 24\n\nKris took a few moments to organize her thoughts after that little talk with her subordinate. She realized she should have had Captain Ajax order the _Princess Royal_ 's navigator to establish a course and speed that would allow _Intrepid_ to catch up to the rest of them without having too much more energy on the boat.\n\nWhen Kris called, Captain Ajax seemed delighted to provide the navigation support for the squadron. That left Kris to conclude that between securing nannies, mediators, and Secret Service for her staff, she'd failed to lay on the normal Navy staff that she needed.\n\n_If we live, I learn._\n\nTwelve hours later, the _Intrepid_ came zooming through the jump at just under thirty thousand klicks an hour.\n\n\"We captured the crew, including the captain, and blew up the ship,\" Captain Grayhorse reported.\n\n\"As expected, they had gotten their reactor back up and were limping as fast as they could for the nearest jump. They also had reloaded what was left of their lasers. I let them take a swing at us, Admiral. Between our armor, their damage, and our evading, they only landed one hit on us. It didn't even scratch the paint. You really have to love this new crystal armor.\"\n\nHe laughed. \"No sooner did they fire than all hands were bailing out in their survival pods. It seems that they thought they had missed when they fired at the _Princess Royal_. When they saw their hit on us failed to do anything, they knew they'd blown it and all hands decided to cut their losses. I picked up the survival pods, then blew the _Hephaestus_ to smithereens. If someone sends out a search for the _Hephaestus_ and _Poseidon_ , they won't find so much as a scrap to tell the tale.\"\n\nKris considered that, then left orders for the _Intrepid_ to rejoin the squadron when they reached midcourse flip over. Together, they could all began to decelerate toward the next jump.\n\n\"Question is,\" Kris asked Jack, \"what is our course?\"\n\n\"Someone knew you were coming straight from Wardhaven to Greenfeld,\" Jack said.\n\n\"And likely that someone is on Ray or Crossie's best-friend list,\" Kris answered.\n\n\"Seems like we might want to avoid the beaten path between Wardhaven and Greenfeld,\" Jack offered.\n\n\"Nelly, invite Captain Ajax and her navigator to join us. Also, ask her if she might want an elevator between her bridge and admiral country?\"\n\nA moment later, an elevator door appeared on the bulkhead next to the door into Kris's night quarters.\n\n\"Jack, I don't think that elevator will be permanent.\"\n\n\"Ya'think?\" he said.\n\n\"What's on your mind, Admiral?\" Ajax asked a moment later, without preamble, and certainly without any talk of permanent access between the bridge and Flag Country.\n\n\"We need to get from here to Greenfeld without following a predictable path,\" Kris began just as quickly. \"I'd prefer not to pass through a system with a colony. No need to get mixed up in anyone's fight, or worse, start one. We most definitely must not pass through a system occupied by the Grand Duchess's forces.\"\n\nBoth the navigator and captain looked flummoxed.\n\n\"You mind if I bring Nelly into this problem?\" Kris asked.\n\n\"Please do,\" the navigator said.\n\nCaptain Ajax just nodded. \"I begin to see why you need something like your computer. When you lay out requirements that tight, it will take more than a human brain to sort through all the threads.\"\n\n\"Frequently,\" Kris admitted. \"I would prefer for you to address Nelly by her name.\"\n\nCaptain Ajax now plumbed the depths of puzzlement.\n\n\"Captain, I have given testimony before human judges,\" Nelly said. \"I have saved Kris's bacon more times than even I can count. Yes, I am a computer, but unlike your dumb associate, I can delve much deeper into what you humans do and think than anything but me and my kids can.\"\n\n\"Kids?\" kind of fell out of the captain's mouth, much like a frog in certain fairy tales.\n\n\"Yes, I have made several children. Some have died fighting with their human so that others might live. Some have stayed with their humans on Alwa Station. Only Jack's Sal is with me now.\"\n\nCaptain Ajax stared slack-mouthed and wide-eyed at Jack and Kris. Both nodded.\n\n\"Nelly has been with me since my first day of school,\" Kris explained. \"I've upgraded her time and time again. To satisfy my aunt Seyd, a retired info warrior from the Iteeche War, we inserted a chip from Santa Maria into Nelly's matrix so she could study it. You remember Santa Maria, that planet with a worldwide computer relic left by the Three? Anyway, I'm not sure who is studying who, but Nelly's been arguing with me and cracking horrible jokes ever since.\"\n\n\"A real, live computer?\" Ajax finally got her mouth around.\n\n\"So it seems,\" Nelly said.\n\n\"Now, Nelly,\" Kris said, crisply, \"about the best way to get to Greenfeld without leading with our chin again?\"\n\nThe screens across from the conference table came alive. \"I would suggest you take a seat,\" Nelly said. \"This is going to take a while.\"\n\n\"Because?\" Kris asked.\n\n\"We are on the only route that meets all your requirements,\" Nelly answered.\n\n\"Don't you hate it when that happens?\" Jack quipped.\n\n\"Kris, you didn't say we couldn't use high-acceleration jumps or rpms on the hull. May I consider those options?\"\n\nKris pursed her lips. \"I don't want to risk getting lost in a bad jump, Nelly. Moreover, we are in popular space. I really don't want to zoom through jumps without warning what's on the other side to get out of our way.\"\n\n\"Understood, Kris, but I have one quick and pretty optimal course you might want to consider.\"\n\n\"Go ahead.\"\n\n\"There is a fuzzy jump in this system. If we hit it at two hundred thousand klicks and 31 rpms, it would take us well out of human space. Two more jumps at high speed and rpms, and we could start slowing down and jump into a worthless system just one slow jump out from Greenfeld. It has the one regular jump from Greenfeld and only the fuzzy jump as a second entrance.\"\n\n\"What's a fuzzy jump?\" the navigator asked.\n\n\"Nelly?\" Kris offered.\n\n\"You need a Mark XII fire-control system to spot them, and most people don't seem to notice them even when they have one.\"\n\n\"We have a Mark XII,\" Captain Ajax said cautiously.\n\n\"Nelly was the first to notice the fuzzy jumps,\" Kris said, \"and we use them to make fast trips across to the Alwa system. Beyond that, it's pretty much classified need to know.\"\n\n\"Oh,\" Ajax said.\n\n\"Show us the course you propose,\" Kris said. Nelly did so quickly.\n\nAjax's navigator whistled. \"Those are long jumps.\"\n\n\"If you want to cross the entire galaxy, you need some seven-league boots,\" Kris said.\n\n\"Nelly, work with the _P. Royal_ 's navigator to get your course good and solid.\"\n\n\"Do I have to drink poison when we're done?\" the navigator asked.\n\n\"Likely before you finish,\" Ajax answered her navigator.\n\n\"Did I hear that the squadron wanted reassignment to the Alwa Station?\" Kris asked.\n\n\"It was on our dream sheet.\"\n\n\"This will likely cinch it for you.\"\n\n\"Thank you, Admiral,\" Captain Ajax said, \"for bringing us in on your course selection. We should have an initial course and acceleration ready in a half hour, a solid course in one.\"\n\nThe navigator nodded.\n\nThe two of them took the elevator down. The elevator did not vanish away.\n\n\"You think we get to keep that?\" Kris asked.\n\nJack checked their night quarters. \"And we've got the space back in our bedroom. I wonder who lost.\"\n\n\"Hopefully, it was a broom closet,\" Kris said. \"I dearly do not want any more morale challenges today.\"\n\n\"Are you hungry?\" Jack asked.\n\n\"Definitely, and, no doubt, Ruth will be hungry again, soon.\"\n\n\"Let's eat.\"\n\n# 25\n\nThe _Intrepid_ caught up with the squadron about the time they would have flipped and began decelerating. With the fuzzy jump now in their sights, they continued to accelerate. The _Intrepid_ 's pinnace brought over its captives for interrogation.\n\n\"Mind if I take care of our prisoners?\" Jack asked Kris.\n\n\"Have at them,\" was all Kris had to say.\n\nAt dinner that night, Jack brought Kris up to date on his findings. \"The captain is a real hard case. He's not talking and seems to be looking forward to being tortured.\"\n\n\"You plan to oblige him?\" Kris asked, wondering if she should have set some limits on Jack.\n\n\"Not if I can help it,\" made her feel better. \"There are a lot of others to talk to, and I'll see if they give me a lead or maybe a handle on the skipper.\"\n\nKris left the subject at, \"Then keep me informed.\"\n\nThe next day's supper had little to add to the last. \"I've started off talking to the lowest members of the crew. Gunners, engineering watch standers, electricians, and other maintenance workers. Most of them claim to know nothing about what was going on. They were looking for work, any work, when they hired on. What they did tell me makes me wonder about what's going on in the Empire.\"\n\n\"What _is_ going on in the Empire?\" Kris asked.\n\n\"Not a lot according to these guys. At least not a lot of work. They all say they took the job because they were desperate for work. According to them, there was absolutely none to be had.\"\n\n\"Are any Navy?\"\n\n\"That's what is interesting. There are a few petty officers, but not so much as one chief. I think there may be one officer in the crew, but I'm letting him stew for a while. It's interesting watching the others ignore him. Anyway, I don't know how they got those old battlewagons away from the pier, or better yet, managed to get some shots off at us. It must have been amateur night every minute of the day.\"\n\nKris frowned. \"That does say something about what's happening to the Empire. Have you passed that along to our civilians?\"\n\n\"Yep. They asked to see the video of our interrogations. They say they may ask to talk to a few of them later.\"\n\n\"Maybe I shouldn't ask, but what's it taking to get these types to talk?\" Kris asked.\n\n\"A decent meal,\" Jack said.\n\nKris let out a sigh of relief.\n\n\"What did you think, that I'd torture them?\"\n\n\"I didn't ask, and you didn't tell me.\"\n\n\"Well, don't worry, I'm getting most of what I need by just sitting across the table and listening. They seem in a hurry to talk.\"\n\n\"You trust what they say?\"\n\n\"I don't trust what they tell me, but Sal has their quarters bugged,\" Jack said with a wide grin. \"We're listening to everything they say. No one is coaching them before their interviews, and no one is checking out their stories afterward. The big topic is why we picked them up. I've got the impression that they expected to be either shot out of space or left to die in their survival pods.\"\n\n\"Ouch,\" Kris said.\n\n\"Yeah. I'm still trying to figure out what that tells me about this civil war.\"\n\n\"Nothing good,\" Kris said. \"Keep me informed.\"\n\nIt was the third evening when Jack brought a guest to dinner.\n\n\"Kris, I would like you to meet Karl Spirelli, third officer on the _Firebird_.\"\n\nKris parsed the sentence and came away with way too many questions. She started with the first one that came to her mouth: _\"Firebird?\"_\n\nJack tossed the question to Karl with a glance.\n\n\"Yes, ma'am, Admiral, I mean. Everyone, including the captain, called the boat the _Firebird_. The other was the _Typhoon_. That seemed to be some kind of inside joke with the management. Management was what they called the captain and first officers of the two old battleships. I think among the senior _management_ , I was the only one with Navy experience.\"\n\nKris scowled. \"Yeah, I can see the joke. Me getting blown away by the _Typhoon_.\"\n\nJack shared the scowl with her. Karl looked ready to cringe and crawl under the table, but, more credit to him, he just stood there at something close to attention.\n\nKris studied the man. He was clean-shaven, and his shipsuit looked recently washed. He wouldn't be out of place among the officers of her crew. _Interesting._\n\n\"Did you know that your _Firebird_ was squawking as _Hephaestus_?\"\n\n\"No, ma'am. As third officer, I was responsible for gunnery. When I served a bridge watch under way, that was not on the checklist I was supposed to keep an eye on.\"\n\nKris's frown got deeper. The duties of an Officer of the Deck, especially under way and with the conn, were very specific aboard a warship. Every OOD worth his salt checked the captain's standing orders but didn't need a checklist. They _knew_ their duties. \"About this checklist, you sound like a Navy man. How'd it stack up to your normal duties?\"\n\n\"It was short and sweet, ma'am. A lot shorter than it should have been on a battleship. When I pointed that out, I was told in no uncertain terms that I'd signed a contract to do what I was told and the checklist was one of those things. Get with the program or get off the boat.\"\n\n\"That's the second time you said boat,\" Jack pointed out.\n\nThe guy shrugged. \"That was what management called the ship, and I got told pretty quick that I better change my ways, fast. Uh, ma'am, is there any chance we could eat? Breakfast wasn't that filling, and this man here had me talking all through lunch.\"\n\n\"Sit down,\" Kris said, and looked around. She spotted two lieutenants just finishing up their meal. \"Mister,\" she called to the two junior officers, although one was a woman, \"would you mind getting the three of us trays? What's on the menu?\"\n\n\"Hamburgers, burned and cold, ma'am, or some sort of stew. I can't vouch for the meat, but I think the _Sweet Pea_ is missing a few cats.\"\n\nKris was a bit taken aback by the joke. _Would I have said that to a four-star when I was just a lieutenant?_\n\n_Oh, yes._\n\n_Would I have been sharing the main wardroom with an admiral and had a chance to shoot my mouth off?_\n\n_Not likely._\n\nKris laughed, and wished Grand Admiral Santiago luck with her next batch of recruits on Alwa Station. \"Get us a little of both,\" Kris said.\n\nAs Jack settled Karl down in his seat across from Kris, he was shaking his head. \"Longknife Navy,\" he muttered, making it half a curse, half a four-letter word.\n\n\"That Longknife Navy blew your battleships out of space,\" Jack pointed out.\n\n\"Yeah,\" the former Peterwald officer said, still shaking his head. \"What is it with your ships? I know we made hits, but nothing happened.\"\n\nKris considered several answers and chose, \"Magic.\"\n\nKarl raised his eyebrows, then shrugged. \"You could have fooled me. Okay, you ask questions, I answer. Not the other way around. My mom didn't have too many dumb children.\"\n\n\"Good,\" Kris said. \"So, how short was this checklist you had to follow when you were OOD?\"\n\n\"It was about half of what I expected to do on the bridge of the heavy cruiser _Lomza_.\"\n\n\"And you aren't on the _Lomza_ anymore because?\" Kris asked.\n\n\"They decommissioned her. Only sixteen years in the fleet, and they sold her out for scrap.\" Karl got quiet; he glanced around furtively. \"Some say ships like her were ending up in pirate hands before a welder's torch ever touched them. Some say they were ending up in the Empress's private Navy. Me, I can't say anything either way.\"\n\nKris and Jack exchanged glances.\n\n_Curiouser and curiouser._\n\n\"So you ended up on the beach,\" Jack said, encouraging the young man to say more. However, the two JOs arrived with trays of questionable culinary value. If the officers were eating this badly, what were the other ranks getting served?\n\nNELLY, AM I RESPONSIBLE AS ADMIRAL FOR THE LEVEL OF CHOW ON MY SHIPS?\n\nMOST DEFINITELY, KRIS. SHALL I WRITE UP AN INCIDENT REPORT AND PASS IT ALONG TO CAPTAIN AJAX?\n\nDRAFT IT, AND GIVE IT TO ME TO REVIEW.\n\nYOU DON'T TRUST MY TACTFULNESS?\n\nNELLY, I ACTUALLY DO, BUT I'M HAVING ENOUGH TROUBLE GETTING ALONG WITH MY FLAG CAPTAIN, AND IF THERE'S TO BE A MISTAKE, I WANT TO MAKE IT MYSELF.\n\nIF THIS GOES OVER LIKE A LEAD BALLOON, YOU COULD ALWAYS BLAME IT ON ME.\n\nENOUGH, NELLY, LET'S DO THIS MY WAY. I WANT TO KEEP LISTENING TO THIS FELLOW.\n\nGREENFELD IS STARTING TO SOUND LIKE A REAL MESS.\n\nTELL ME ABOUT IT.\n\nKarl didn't seem to find his food at all unsatisfactory. He took a big bite out of his hamburger and a slug of his bug juice. \"No beer?\"\n\n\"You have beer on the _Lomza_?\" Jack asked.\n\n\"We had a beer ration sometimes. The _Firebird_ had beer at every meal. That made for a happy ship.\"\n\n\"How'd it work for your gunnery?\" Kris asked.\n\nKarl scowled. \"I did have one mechanic who after lunch couldn't put a screw in right,\" he agreed.\n\n\"What happened after the Navy beached you?\"\n\nNow it was Karl's turn to get red in the face. \"I shouldn't have been beached. I had good fitness reports. I was within a millimeter of being selected for commander, then, pow, I'm on the outside with not so much as a severance package. My girl dropped me, and I couldn't find work anywhere.\"\n\n\"A lot of competition, huh?\" Jack said.\n\n\"Everyone's out of work. I even tried joining the Empress's security specialists. They're not that different from an army. They said I was _overqualified_. Can you believe it? I was too good!\"\n\nKarl applied himself to his hamburger for a few bites as he tried to calm down.\n\n\"Anyway, I'm getting desperate when this guy walks up to me outside a bar and offers to buy me a drink. I could sure use one, so yeah, I'm glad for it. We talk for a while, and he says he might know where there's a job for someone who knows how to keep vacuum out of a hull if I'm not too particular about what's going on outside the hull. I told him I could be real apathetic. So he gives me a card and leaves me enough to pay my way up the beanstalk to a boat landing.\n\n\"Next thing I know, I'm on this old battleship, trying to get its ancient lasers up and able to hold a charge. I had to do my own scrounging around three other old tubs to get enough boards and cabling to get 'em up, and that was using spit and glue as much as solder. The gun cradles were all kind of herky-jerky, but nobody but me seemed concerned if I could train our lasers out at a target. If they hadn't found a good old electrician who worked on these wrecks back in Iteeche War days, I would never have gotten as far as I did.\"\n\n\"Where is the old fellow?\" Kris asked.\n\n\"He stayed on the _Typhoon_. They were having the devil's own time getting her lasers to work.\"\n\nStrange, the death of the old Iteeche War vet seemed to make it all the more painful.\n\n\"So, you have no idea what you were supposed to be doing?\" Kris said.\n\n\"Not really,\" Karl said, finishing up his hamburger and starting on his stew. The guy really was hungry.\n\nDON'T YOU FEED THESE GUYS? Kris asked Jack on Nelly Net.\n\nTHREE GOOD HOT MEALS A DAY. WELL, AS GOOD AS THE MEALS WE GET, ANYWAY. NO, THEY ALL CAME ABOARD THIN AS RAILS. THEY'D MISSED A LOT OF MEALS, KRIS.\n\nMORE TO THINK ABOUT.\n\nTHAT'S WHAT I WAS THINKING.\n\n\"There was this one night,\" Karl said, studying his spoon after stuffing a huge bite into his mouth. Kris tried to avoid looking him in the face as he chewed and talked at the same time.\n\n\"Zygfryd hooked me into going back to the station with him. I think most of the senior officers were drunk. Maybe. Or maybe he just wanted me around to answer questions about the lasers if that came up. Anyway, we ended up meeting with the skipper of the _Typhoon_ and two of his crew at a bar. Then three other guys came along and sat down beside us. They wanted to know how we were coming along, getting those ancient tubs ready to answer bells.\"\n\nHe paused for a swig of bug juice and another spoonful of stew.\n\n\"I started to tell them all it had taken to get the lasers up. Before I could say a word about what I needed to make them half-decent, I got cut off and asked if they would shoot. I said yes, and that seemed to make them happy. The _Typhoon_ was having trouble with its reactors as well as its guns, but these three guys didn't act as if they really cared, or maybe it was just that they didn't understand. Those three cut them off, too. Then the fat one started talking to us, telling us how important our job was, how we had to get the boats out there and ready to shoot.\n\n\"Zygfryd kicked my ankle and kind of eyed the bar. I took my beer and headed there. The other two followed right behind me, so the skippers and these three get to talking real quiet, but they don't stay quiet. I hear the fat one saying, 'We got to get that bitch good.' The skipper shushed him down and got told they could replace him, but they were quieter after that. Me and the other two, we just shook our heads and ordered another beer. It was on their tab anyway.\"\n\nJack tossed Kris a glance, but she'd caught it. \"So, you were supposed to 'get that bitch good,' huh?\" Jack said.\n\n\"Yeah, I don't know who the bitch was but we were . . .\"\n\nKarl's eyes got big, and he blinked several times in rapid succession. His mouth fell open, half the last bite still in it.\n\n\"Not very many women admirals in the Greenfeld Navy, are there?\" Jack asked.\n\nTheir captive gulped, not a good idea with such a full mouth. He was wracked by serious coughing. Jack pounded him on the back and was about to resort to more serious lifesaving efforts when the fellow spit up into his napkin.\n\nHe gasped for breath for a long minute, then downed a glass of water that a helpful ensign had brought to Kris's table. She thanked her with a smile and waited for Karl to return to those living and capable of carrying on a conversation.\n\n\"Hello, young man. I'm Admiral, Her Royal Highness Kris Longknife, and I believe I'm that bitch you were sent out to get good.\"\n\n\"Oh, God. They sent us out in that old wreck to pick a fight with you. No wonder we're all just about dead.\"\n\n\"Yep,\" Kris said.\n\n\"I mean, if they'd asked me to take that old battleship out to fight the Grand Duchess, I would have passed. Hell, I did. I know guys that got those old battlewagons out to St. Petersburg. Only reason they lived to tell about it was their tubs didn't get in range of the Grand Duchess. Otherwise, she'd have toasted them the way she did the newer ships.\"\n\n\"Vicky's got a battle fleet now?\" Kris asked, backing it up with a questioning eyebrow.\n\n\"Hell yes, she does. She's won three, four, I don't know how many battles. Some she won. Others were won by people on her side. Hard to tell which ones she was at. You listen to some people, and they'll tell you she's been at every damn shoot.\"\n\n\"She does get around,\" Jack said, eyes sparkling at Kris.\n\n\"No doubt. Well, Karl, I think you've earned yourself some dessert, and maybe a few drinks at the Forward Lounge.\"\n\n\"Forward Lounge?\"\n\n\"Our restaurant and pub,\" Kris said.\n\nJack waved his hand ever so slightly, and suddenly there were two Marines at Karl's side. \"You boys take this fellow up to the Forward Lounge. We'll call ahead and tell them what his tab can be for the night. You two get a couple of beers on us after you go off duty.\"\n\n\"Likely after we pour him back into his cell in the brig,\" the corporal said.\n\n\"Right, and tell Gunny to up his accommodations to guest. He's just become the admiral's best friend.\"\n\n\"Guest?\"\n\n\"Guest with escort,\" Kris added.\n\n\"Thank you. Thank you. No hard feelings? I mean, it was never personal.\"\n\n\"Karl, get out of here before I space you,\" Kris snapped.\n\nKarl got out quickly with his Marine escort right behind him.\n\n# 26\n\n\"Unidentified ships entering Greenfeld controlled space, state your name, registry, and intent,\" blared from the screen. Kris's squadron had come through the jump just six hours ago, and this was the Empire's first contact.\n\nJust as Nelly had predicted, the fuzzy jumps had allowed them to take the long way around and do it faster than the direct route.\n\n\"Nelly, get me Captain Ajax.\"\n\n\"Yes, Admiral,\" came back a second later.\n\n\"Am I wrong, or are our transponders working fine?\"\n\n\"Admiral, I'm reading the transponders on all seven of the battlecruisers behind us and I'm told by Captain Grayhorse that the _Intrepid_ reads the _Princess Royal_ just fine. Now we are still squawking as frigates. Don't want to scare anyone by using a nasty word like battle. Oh, and we've got the _Princess Royal_ announcing itself as a Royal diplomatic ship.\"\n\n\"Hmm, so maybe _their_ equipment isn't working?\" Kris suggested with an impish grin for Jack.\n\n\"I wouldn't put it past the Empire's gear to be on the fritz,\" Captain Ajax answered.\n\n\"I wonder if we should turn off our transponders and see what kind of reaction that gets us?\"\n\n\"I assume, Your Highness, that that is a rhetorical question.\"\n\n\"I think she's starting to get your number,\" Jack said, hand over mouth to cover his quip.\n\n\"I think you may be right,\" Kris whispered back. \"Okay, Captain, have comm give me a clear channel to High Anhalt Station.\"\n\nThere was only the briefest of pauses. \"Admiral, you have an open channel to High Anhalt. We will record and log your communication.\"\n\n\"Thank you, Captain, but on second thought, I think this is something better handled at a lower level. Would you really get your boss out of bed to handle a sniping little message like that?\"\n\nAjax grinned. \"Most definitely not. Comm, put me on visual.\n\n\"Begin message,\" the captain said. Nelly kept her message on the screen in Kris's day quarters. \"High Anhalt Station, this is Commander Ajax, captain of the United Society frigate _Princess Royal_. I am flag captain for the commander of Frigate Squadron 22 at your service. We are making a free transit of space on a mission approved by His Royal Majesty Raymond I pursuant to a request by your Imperial Majesty, Emperor Henry I. We are in compliance with the common traditions and regulations for free transit through open space.\"\n\n\"Sent, Captain,\" comm reported a moment later.\n\n\"Good, Captain,\" Kris said. \"Have your navigator lay in a 1.5-gee course for Greenfeld. Advise me of any further communications. Do you have an intelligence team aboard?\"\n\n\"We have a small science team,\" Ajax reported. \"We expected to take on more once we headed out for Alwa.\"\n\n\"Have your team listen in to all official traffic. Inform them that they should share it with our passengers' technical support team and that my Nelly would enjoy working with them.\"\n\n\"Codes are so much fun to break,\" Nelly said, not quite with a giggle.\n\n\"I'll warn my team what to expect, Admiral.\"\n\nWARN THEM?\n\nDOWN, NELLY. THEY'RE NEW TO YOU. YOU MUST ADMIT THAT YOU TAKE A BIT OF GETTING USED TO.\n\nBUT I'LL BE NICE TO THEM.\n\nUNLIKE TO ME.\n\nOF COURSE.\n\nNelly reported back an hour later. The sensor team had been sweeping up a lot of message traffic since the _P. Royal_ first jumped into the Greenfeld system. With her help, and what they could borrow from the ship for serious number crunching, they had broken several codes and discovered some interesting bits of information.\n\n\"You were not expected so early, at least by most,\" Nelly reported. \"No one expected you to jump in system from that dead-end jump, and at least one message was sent by someone who didn't expect you at all.\"\n\n\"Who was that?\" Kris asked.\n\n\"That was not sent on the official net. Rather it was an independent radio net.\"\n\n\"Hmm.\"\n\n\"What's most interesting,\" Nelly said, and you could almost hear the grin on her nonexistent face, \"was the reaction that brought from the receiver. They ordered the sender to shut down immediately and not send anything more until they updated their ciphers.\"\n\n\"Someone got sloppy,\" Jack said.\n\n\"And better yet,\" Nelly crowed, \"we were able to catch the recoded message when it was passed along and use that message to crack several other messages that were sent to and from some major players. At least one of those major players was seriously outraged that you'd made it here alive.\"\n\n\"Is there any chance we were listening in on the Empress's most private net?\" Kris asked.\n\n\"Unfortunately, no one said anything that allowed us to identify anyone except by their call signs, and those were very much random generated.\"\n\n\"Well,\" Kris said with a shrug, \"we can't expect more than this, Nelly. Tell the team you were working with that I am very appreciative of their work and looking forward to anything they can give me in the future.\"\n\n\"Will do, Kris.\"\n\n\"So,\" Jack said, \"what do you think?\"\n\n\"I think it's interesting,\" Kris said. \"A couple of times I got close to some Greenfeld types, and Nelly's net collapsed. They can be very good. But apparently they can be pretty sloppy with the codes they use to send their own messages.\"\n\n\"They can dish it out,\" Jack said, \"but they can't take it, huh?\"\n\n\"Pretty much,\" Kris said.\n\n\"So what do we do now?\"\n\nKris thought on that for a long moment, but there wasn't really much of a decision to make. \"We do what we came here to do, knowing what we know. Someone down on that planet hasn't wanted me here since I got these orders. They've sent assassins and battleships. From listening to their mail, we really don't know any more or less than we did when we sailed or after we blew away that pair of battleships.\"\n\n\"All too true.\"\n\n\"So we proceed cautiously, letting my paranoid security chief\"\u2014here Kris gave that security chief a peck on the cheek\u2014\"reinforce my own paranoia.\"\n\n\"I could get to like that.\"\n\n\"Now, let's go visit our mediators and see what they have to tell us.\"\n\n# 27\n\nKris's brain trust had taken over the Forward Lounge as a kind of club. They took their meals there and did most of their socializing. It had been interesting watching the three groups organize themselves. Initially, the three different agencies had each taken one corner of the lounge, leaving the fourth for the support group.\n\nOver the first couple of days of the voyage, they had stayed pretty much among themselves with their computers out and noses in them. Gradually, they'd mixed together as first one and then another shared a drink or meal. It seemed that many of the research papers that they were delving into had been written by someone on the other teams. One thing led to another, and pretty soon, they had formed combined groups to look into this or that possibility.\n\nOne thing they had not done was invite Kris to join them. She found that . . . intriguing. So she scheduled a meeting with them after the squadron flipped ship to decelerate toward High Anhalt Station. She chose the Forward Lounge for the venue.\n\nThe moment she stepped into the lounge, Kris found herself on familiar ground.\n\nThe table in front of the larger screens was again reserved for her and Jack. Now it was Ensign Megan Longknife who held it for them as they made their way from the doors through four collections of tables.\n\nWhile Jack went around to sit down, Kris chose to lean against the front of the table and take the measure of the teams she'd be leaning heavily on in her search to find some way out of this mess that Vicky and the Empress had gotten themselves and their Empire into.\n\nAlfred Fu was the short, older man who had done the talking when Kris first met these people. As the chief mediator for Wardhaven's Office of Mediation, he considered his age and stature a great help. \"Those who are foolish enough to mistake gray hairs for senility often take my shortness for a lack of presence. That's usually two strikes against them, and I'm home free before they figure out how I've swung their team members and the folks on the other side of the table to something reasonable.\"\n\nKris had found him easily likable the two times she had talked with him.\n\n_I'll have to be careful about that._\n\nWilliam Pierce Gladsten was a tall, dour man. He did not look at all like his name. However, as senior arbitrator of the Wardhaven Bureau of Arbitration, he seemed to approach everything with deep gravitas whether it was his one discussion with Kris about the perils of civil wars that drag on overlong or the selection of desserts after dinner.\n\nThe third member of Kris's troika was Diana Frogmore. The woman appeared quite youthful despite her many years of service both as a child advocate and now as the seniormost judge in the Parental Conflict Resolution Division of the Family Court. She had been quite eager in her first meeting with Kris to point out the value of her perspective in this otherwise far-from-childish affair. \"This entire civil war may turn out to be as much a squabble between a man and his grown daughter as it is a grab for power by hungry interests.\"\n\n\"And the stepmother?\" Kris asked.\n\n\"If we view this civil war as a battle between two adults for the custody of the Empire, adding a fourth person to the mix does create a conundrum. Very messy, if I must say so.\"\n\nNow Kris faced the four sets of tables, three with a mediation team of one sort or another and the fourth with an eclectic group of technical experts who, as Captain Ajax had reported, had been using every spare second of time they could get on the _Princes Royal_ 's sensors to guzzle down data from the rapidly approaching planet.\n\n\"Well,\" Kris said, leaning back against the table and crossing her legs at the ankles, \"what do you know now that we didn't know before we jumped into this system?\"\n\nWith hardly a moment's hesitation, and without glancing at notes, William Pierce Gladsten began. \"Our colleagues from information management have collected quite a treasure trove of data. None of us is too sure it qualifies as information. It does appear that all that your friend Karl Spirelli told us is, within reason, accurate.\"\n\nHe cleared his throat. \"It will come as no surprise. There is a civil war raging in the Greenfeld Empire.\"\n\n\"I'm glad to hear that,\" Kris said dryly.\n\nMr. Gladsten ignored her remark. \"It has not touched Greenfeld. Most of the fighting is limited to planets out on the rim of the Empire. From several conversations we intercepted, the Grand Duchess appears to be doing quite well. Our support staff intercepted quite a few messages to and from people being recruited for the Empress's Navy. Most declined. The general tenor of those declining the offer was along the lines of 'No way in hell I'm going to let you get me killed,' or some such.\"\n\n\"Maybe Vicky did learn a few things from her time with you,\" Jack said drolly.\n\n\"I'll need to see it to believe it,\" Kris said. \"Are you operating on hearsay or do you have any solid evidence?\"\n\n\"Most is hearsay,\" Mr. Gladsten admitted. \"However, we cracked several coded messages from someone who appears to be close to the Empress. She is quite adamant in her demands for more ships, more crews, and more security specialists. Oh, and she wants them all trained better. I believe the quote was, 'These better put up a better fight than the last bunch,' or something to that effect.\"\n\nOne of the women at the support table cleared her throat. \"We just finished decoding a large report in answer to that last one that is working its way back to whoever ordered it. It has a long list of suggested crew members for recommissioning over a score of reserve battleships, as well as cruisers and destroyers. The potential crew for each ship is divided among Former Navy, Merchant Marine, and Untrained Landsmen. The first two categories account for less than a tenth of the crew list.\"\n\n\"I'll want to see that list,\" Kris said. \"Can you cross-index those lists of names against any other databases: births, employment, unemployment, Navy retirement, and separations.\"\n\n\"We'll get on that. You are aware that unauthorized fishing around in databases in the Greenfeld Empire is a capital offense,\" the woman pointed out.\n\n\"I have been told that by none other than the Grand Duchess, herself,\" Kris admitted. \"However, I don't expect any of you to be taking shore leave, and I assure you, no law-enforcement officer will be allowed aboard this ship. If push comes to shove, I will accept full responsibility in my role as mediator and tell them to see how arresting me works. Right, Jack?\"\n\n\"I'll double our security at the pier,\" he said.\n\n\"Now, about the more curious part of this conundrum,\" Kris went on. \"Are we dealing with a two-sided problem, Papa Peterwald vs. Vicky, or is this a more complicated three way: the Emperor, the Empress, and the Grand Duchess?\"\n\nNow there was a three-way exchange of glances among the troika of Kris's brain trust. No, there were definitely furtive looks aimed at the support tables. It took a long moment and several nods before Mr. Gladsten continued.\n\n\"There is no question that there is a power base out on the rim of the Empire. Certainly, St. Petersburg, Brunswick, and Metzburg are mentioned as fully involved in the rebellion and allied with the Grand Duchess. No question, she is definitely a part of this conundrum.\"\n\nThe senior arbitrator took a deep breath before diving into the other half of the civil war. \"From our analysis of message traffic, it appears that there is only one center of activity with respect to the civil war here on Greenfeld. Exactly who is at the center of this traffic is not clearly defined. We are leaning toward the Empress, but there is still some chance that it is the Emperor himself seeing to the marshaling of forces.\"\n\n\"You don't have a solid signature on any of that traffic?\" Jack asked.\n\n\"No we don't. None of us wants to jump to a conclusion, so we are holding back, waiting for some clear evidence pointing at who the primary actor is, both on the rim and on Greenfeld,\" Chief Mediator Fu said, entering the conversation for the first time.\n\nKris shook her head. \"If Harry Peterwald, old Emperor himself, does anything, he has his name plastered all over it. If there are no names on all this message traffic, it's that way because someone wants to maintain some sort of deniability.\"\n\n\"That is a logical conclusion,\" Judge Frogmore said. \"But it is not our way to draw conclusions in advance of the data. Until we are sure beyond a reasonable person's expectation, we prefer to wait for more information.\"\n\nKris nodded. She could understand the way they did business. She, however, had blown away billions of aliens on the force of her hunch.\n\n_Is now the time to change my way of living, or do I bull through?_\n\nShe glanced at Jack. He raised a questioning eyebrow but said nothing.\n\n\"Let's wait a bit longer,\" Kris said. \"Who knows what we'll see.\"\n\nOn that note, she called the meeting to a close.\n\n# 28\n\nKris had spent most of the voyage out, when she wasn't dodging this or that particular ambush, with Ruth and her nannies. Ruth was starting to show signs of teeth. That not only made teething the pits, but two very delicate parts of Kris were none too happy when Ruth tried teething on them.\n\nThe senior child-care manager, as the former Gunny Li O'Malley was now wont to be called, knew something about most every situation that a mother and child might encounter. She walked Ruth and Kris through the fundamentals.\n\nKris couldn't help but notice that her nannies were absent every morning, all but the two on duty. When she asked, she found that they had attached themselves to the Marines for morning PT. \"You want to come along?\"\n\nKris eyed Jack. \"Are you telling me that my exercise regime of jumping to conclusions and running away from my problems isn't restoring me to my svelte, prebaby, girlish figure?\"\n\nSmart husband, Jack just smiled softly at her.\n\nKris had to admit, she'd been meaning to do something about the extra pounds, and other things left behind by Ruthie's shared occupancy of her belly. Really. She'd been doing a few things in the privacy of her own bedroom. When Jack wasn't around.\n\nSo it happened that Kris followed the nannies to join the Marines the next day.\n\nAnd wished she hadn't.\n\nEven though she fell out a few times, she still returned to her quarters limping, winded, and in desperate need of a shower. Had even one of the young women said a word, Kris would have crawled under her bed and not come out again.\n\nThey didn't say that word, and Kris found herself with no excuse not to go back the next day, and the next. After Jack mentioned one evening that Kris was looking good, she put morning PT solidly on her schedule.\n\n_Who knows, I might have to chase Vicky or her dad or stepmom down and hog-tie them to get them to the table._\n\nKris was just getting out of the shower after an hour's worth of PT that she'd actually enjoyed when Jack hollered, \"Are you decent, Admiral?\"\n\n\"When have you cared if I was,\" she shouted right back.\n\n\"I've got Captain Ajax with me.\"\n\n\"Oops,\" Kris said, stopping her sashay across her night quarters, towel trailing.\n\n\"She wants to talk to us about our in-port security once we dock at High Anhalt Station.\"\n\n\"Good idea,\" Kris said, toweling her hair vigorously. \"I'll be with you in a second.\"\n\nKris finished drying off, hurriedly pulled on a nursing bra, blue shipsuit, and some shoes, and joined the two other officers in her day quarters. Jack had seated Captain Ajax at the conference table. Kris joined them.\n\n\"The general here asked me to double the guard at the pier and be prepared to seal ship at a moment's notice. Are you expecting trouble?\"\n\n\"This is Peterwald territory, Captain. Expect anything.\"\n\n\"You mean Greenfeld, don't you?\" Ajax asked.\n\n\"I'm pretty sure that if you asked Harry Smythe-Peterwald which is which, he'd tell you it didn't matter. They've pretty much run the place as a family business.\"\n\n\"A family business that is on the ropes,\" Jack added under his breath.\n\n\"Well, whatever it is,\" Helen said, \"you want to keep the _Princess Royal_ safe from any of their troubles bleeding into us here.\"\n\n\"Exactly,\" Kris and Jack said together.\n\n\"Have you considered not docking at the station?\" the captain said. When Kris said nothing, she went on. \"I've read your reports. I'm not sure exactly how you do it, but you seem to be able to have your ships swing around each other and get the equivalent of gravity. The main reason we'd be docking at the station would be to get some down for our crews' health. So, if we don't hitch ourselves to their station, they'd have a lot more trouble getting anything aboard us.\"\n\nHaving stated her idea succinctly, she quit talking. Kris had come to like that in the captain. She glanced at Jack. He had his eyebrows up, tossing a big question mark right back at Kris.\n\n\"We don't need any housekeeping support from the station,\" Kris said, talking half to herself. \"We've got plenty of water, air, and reaction mass. Even if we docked, I'd want to keep our hull sealed from everything except landlines. Nelly, I'd expect you to put up the biggest firewall in human history.\"\n\n\"The Great Wall of China would have nothing on me, Kris,\" her computer answered.\n\n\"Great Wall of what?\" Kris said.\n\n\"You slept through that part of old Earth geography,\" her computer said sassily.\n\n\"You'll protect us from any electronic invasion,\" Kris said doggedly.\n\n\"Definitely.\"\n\n\"Enough,\" Kris said. She'd let the banter go on long enough while she mulled the basic question. \"It was Admiral Kr\u00e4tz who came up with the idea of anchoring the ships bow to bow,\" she said slowly.\n\n\"But the folks around Vicky,\" Jack said, \"were the only ones from Greenfeld who actually saw it done.\"\n\n\"And we have made very good use of that idea to defend our jumps from the bug-eyed monsters,\" Kris muttered softly.\n\n\"Captain Ajax,\" Jack said, \"have you heard of any Greenfeld Navy ships moored nose to nose?\"\n\n\"No, I'd never heard of anything like this before I read your report. It's totally new to us on this side of human space. At least I think so.\"\n\nKris gnawed on her lower lip. \"It would be a shame for us to come as mediators between these two warring parties, then give away one side's advantages to the other side,\" she said slowly, testing each word and finding them good.\n\nKris finally shook her head, decision made. \"No, we do not stop short of the station. Not only would it be giving away something that Vicky might not want to have common knowledge, assuming it hasn't become that, but it would also be inhospitable. You can't accept an invitation to dinner and mediation, then turn your back on the hospitality. All of it.\"\n\n\"You really going to go down there?\" Jack asked.\n\n\"I didn't say _that_ ,\" Kris said. \"I believe I told my great-grandfather of royal pretensions that someone would have to be dumb, insane, or the boldest person in human space to get themselves neck deep in this mess. I am none of those, so let's see how we can avoid anything like that.\"\n\n\"As your putative security chief and husband, I couldn't have said it better,\" Jack said.\n\n\"So, we dock the squadron on the station,\" Captain Ajax said, \"but no shore leave, and we accept nothing, not so much as a drop of water from them. Any message packet has to be scrubbed and vouched for.\"\n\n\"By Nelly, right?\" Kris asked.\n\n\"Only too happy to,\" her computer agreed.\n\n\"Then I think we've got everything organized just the way we want it,\" Jack said.\n\nAnd they had, for just about another hour.\n\n\"Wardhaven warship, we identify you as a frigate, but you are squawking as a Royal diplomatic ship. Please explain yourself.\"\n\n\"Wait one,\" came back at the frowning officer in Greenfeld greens.\n\n\"That's the message we got\"\u2014Captain Ajax glanced at the timer below the screen\u2014\"fifty-four seconds ago.\"\n\nKris, who had gotten the call to the bridge and ridden the elevator in something close to a free fall, nodded. \"Such a message is beneath the dignity of an admiral and diplomatic mediator. As flag captain, it seems right up your alley.\"\n\n\"Oh, joy,\" Helen muttered under her breath . . . and shooed Kris out of the camera take.\n\nKris quickly let herself be shooed.\n\n\"High Anhalt Station, this is Commander Ajax, captain of the United Society Frigate _Princess Royal_.\" They discussed the benefit of jumping the ships up to battlecruiser status and chose to keep the eight ships small, unthreatening frigates. \"I am flag captain of Admiral, Her Royal Highness Kristine of Wardhaven. King Raymond, first of that name, has graciously accepted the pleadings of Your Imperial Majesty, Henry, first of that name, and dispatched Her Royal Highness to render such service as Your Imperial Master may request.\"\n\n\"Oh,\" came out quite flat. \"Pardon us, we were not expecting you so soon,\" was rushed. \"I will get back to you,\" was cut short, as the screen went blank.\n\nKris shook her head into the silence. \"Let me get this straight. First, they tell us to go home. Then they ignore us. When they finally get around to asking who we are, they cut the line when we give them the only answer they could have expected. Am I missing something?\"\n\n\"Well,\" Jack said, rubbing his chin, \"they might have expected you to be dead, or very late, limping in on a badly damaged little frigate after you tangled with two old but really nasty battleships.\"\n\n\"I'm getting the feeling that the right hand doesn't know what the left hand is doing here,\" Kris growled.\n\n\"More like each finger is in the dark about the other ones,\" Captain Ajax muttered.\n\nJack glanced at the skipper. \"I think you're going to fit into this madhouse just perfectly,\" he said through a grin.\n\n\"Don't look at me,\" Ajax said. \"I'm just on my way to Alwa. They may be big, mean, and nasty out there, but at least they aren't totally crazy.\"\n\n\"You toss Longknifes and Peterwalds in the same boiling kettle, and there won't be an ounce of sanity to be found,\" Kris muttered, then took a deep breath. \"Okay, we've got an Empire running around, unable to answer us with any degree of rationality. What do we do?\"\n\n\"We do what gravity requires of us,\" Helen said. \"We make orbit and hitch our fate to their station, just like we talked about.\"\n\n\"That sounds like a plan,\" Kris said. \"How long until we're there?\"\n\n\"Six hours,\" Ajax answered.\n\n\"Then I better get ready.\"\n\nOf course, Kris needn't have bothered. It would be a long time before she got a chance to go anywhere.\n\n# 29\n\n\"Wardhaven Frigate Squadron 22, we were not expecting you so soon. We will have to dock you where we can around the station. Here are your ships' assigned piers.\"\n\nCaptain Ajax covered her mouth with a hand, and whispered, \"They've scattered us all to hell and gone up and down the station.\"\n\nJack turned his back to the port captain on the screen and added in a low whisper, \"There's no way we could provide any mutual support to each other if we needed it.\"\n\n\"Then we will not dock the squadron like this,\" Kris answered them both.\n\nCaptain Ajax stepped aside and left Kris pride of place in front of the port captain on the screen. He was middle-aged, graying, and his clothes fit him loosely, as if they still remembered when he was several sizes larger.\n\n\"We will not have our ships scattered all over the place,\" Kris snapped, becoming the demanding, spoiled princess in high dudgeon. \"How can we muster our Marines for their daily Royal pass in review? We intend to conduct weekly inspections of our ships' companies. They will not look at all the way we like them if they are strewn all about the station. Not shipshape and Bristol fashion at all,\" she harrumphed. \"We won't have it. We certainly will not. Young man, you tell your Emperor that we didn't come out here to be treated like this. No. Better, we will tell him, personally. Captain, raise the Emperor immediately,\" Kris said, turning to Ajax.\n\nShe might have had no idea what Kris was talking about, but she ran with it.\n\n\"Chief, connect us to Emperor Henry's private line.\"\n\n\"Aye, aye, ma'am.\" the chief on comm watch answered. \"This will only take a moment.\"\n\n\"Hold, it. Just wait a minute. It's the middle of the night,\" yelped the poor man, stuck between Kris's demands and whoever had given him the order on where to dock her ships. But he was facing a very angry Princess.\n\n\"We will certainly not wait until tomorrow morning. We hardly have any gravity out here. It's horrible for our digestion, young man. Captain, use the private number he sent our Royal great-grandfather.\"\n\n\"Yes, Your Highness,\" Captain Ajax said primly.\n\n\"No. No. Please don't wake up the Emperor in the middle of the night. We've just had several freighters vacate their piers. You know how busy the station is. These used to be Navy piers, but what with the fleet down and trading up, we'd been using them for cargo. I think I can get you right in over there. Give me a moment.\"\n\nThe screen went blank.\n\n\"The fleet is down?\" Kris said.\n\n\"Assuming there were no freighters in those piers . . .\" Jack said.\n\n\"Captain Ajax, can your sensor team give us a good nose count of how many of the station's piers are occupied?\"\n\n\"I'll get my team on that, Your Highness.\"\n\nKris gave Ajax a pass on that \"Your Highness.\" She'd been throwing her Highness weight around enough to confuse any poor Navy commander about her proper form of address.\n\nThe lieutenant on sensors stood. \"By my count, half the piers on that station are empty. We could dock a big chunk of the Wardhaven fleet and still have space left over.\"\n\n\"Thank you, Lieutenant,\" Captain Ajax said. \"You need anything else, Admiral?\"\n\n\"No, Captain, you did quite well.\"\n\nThe skipper glanced at the chief on the comm watch. He looked about to bust a gut. \"I think we all figured that running with a Longknife would mean some fast footwork, ma'am.\" She paused for a moment. \"What exactly would we have done if he hadn't broken and run? I don't know of any Imperial phone number. Do you, Chief?\"\n\n\"Not a clue, skipper,\" said the chief, very much the spitting image of the laughing Buddha.\n\nKris grinned back. \"I've found that people with a poor hand will frequently fold if you make like you have five aces.\"\n\n\"Bluff, huh?\"\n\n\"Yep,\" Jack added.\n\nCaptain Ajax shook her head. \"God help us when we run up against someone that bluffs just as wildly.\"\n\n\"Ah, yes,\" Kris said, \"but remember, we have eight battlecruisers and a personal request from their Emperor to our King. As I see it, we _do_ have five aces.\"\n\n\"But how many aces does the Empress have up her sleeve?\" Jack asked.\n\nKris was saved from having to answer that by the port captain coming back on-screen. \"I've assigned all eight of your frigates to Navy Piers 4, 5, 6, and 7,\" he said hurriedly. \"Your ships will dock across from each other, and you may have use of the A deck area above your piers for inspections and drill. I hope this meets with your approval, Your Highness.\"\n\n\"It is most satisfactory, and we thank you for it,\" Kris said, laying it on thick. She might need this guy for other favors in the next few days. Best to use honey in large quantities. \"Your courteous care for our needs will be mentioned when we meet with your Emperor tomorrow morning. Will he be coming up to greet us?\"\n\n\"I'm sorry, your early arrival has caused some disruption in the Palace's schedule. I have not been apprised of anything, ah, regarding formal ceremonies. We, ah, have not had anyone do the necessary security checks. I will send your expectations up to my superiors for referral to the Imperial court.\"\n\n\"We thank you so very much,\" Kris said. \"Now it appears that we must try to adjust our schedule to your planet's day. We hope we can get our beauty rest.\" Glancing at the comm duty chief, she shook her head curtly. He cut the link.\n\nKris turned to Jack. \"I'm here, at their station, and no one has done any kind of a security sweep of this place. If it's not safe for their precious Emperor, how's it supposed to be safe for my precious ass?\"\n\n\"And very precious derriere it is, dear Admiral,\" Jack said. \"What do you say that you actually try to get some beauty sleep, and I see what I can do about making at least this part of the station safe.\"\n\n\"You aren't going ashore, are you?\"\n\n\"No, Your Highness, I will stay safely aboard ship and see if a few Sailors and Marines can do some wandering around the piers and A deck. We may not have Chief Beni to check out the place, but I understand I have an entire squad of technical recon types that can do the same job.\"\n\nKris nodded. \"Okay, you do your Marine and security chief stuff, and I'll be a nice mother and admiral, feed Ruth, and get a bit of rest,\" she said through a yawn. \"Yeah, maybe I will get some sleep.\"\n\n# 30\n\nKris Longknife came awake slowly, disappointed that she still hadn't gotten all of her oomph back. Or maybe there was a lethargy that came over her when she nursed Ruth. It seemed to go both ways. Ruth would suckle and fall asleep, and so would her mother.\n\n_Maybe it's Mother Nature's way of seeing that I get some rest when this little ball of needs and love is sleeping, too. Thank heavens for my support crew, or this one little life would be exhausting mine._\n\nKris opened her eyes to look down on her lovely child.\n\nShe then looked up into the adoring eyes of her husband.\n\n_Was he looking that way at Ruthie, or me, or both of us?_\n\n\"I hope you don't mind my napping while you're on the job,\" Kris said.\n\n\"Oh, I've been having fun. Lots of fun,\" Jack said.\n\nKris did not care for the way he said that. \"Nelly, could you get someone in here to take care of Ruth. I think I've got work to do.\"\n\nAkumaa and Fede took less than a minute to appear and gently take a sleeping Ruth off Kris's breast. As they left, Kris settled her two girls back into a solid support bra . . . no frilly stuff for her. At least not for a while. As she handled herself, she glanced up and gave Jack an encouraging smile.\n\n\"I'd love to, hon, but I think you want to hear what I've got to say.\"\n\nKris sighed. \"I'll give you a rain check. Just make sure you redeem it real soon.\"\n\n\"It's a deal. Nelly, please order up supper and arrange these day quarters into an admiral's wardroom.\"\n\n\"Sal warned me you'd need that. I'm ready when you are. How many do we need to feed?\"\n\nJack frowned in thought. \"Me, Kris, Captain Ajax. We've got to bring her up to date on what I've been doing with her ship.\"\n\n\"Please do,\" Kris said. \"Remember, we are passengers on _her_ ship.\"\n\n\"Oh, I do,\" Jack answered. \"We better include Special Agent Foile. He was in this up to his elbows.\" Jack paused, licked his lips. \"What do you say that we invite the three heads of your brain trust?\"\n\n\"Does this involve them?\" Kris asked.\n\n\"Let's just say that someone's tried to kill you today and end this mission before it started. Your troika might want to know what they're up against.\"\n\nKris licked her finger and made a mark in the air. \"Another assassination attempt, and it didn't even wake me up. They're slipping.\"\n\n\"They had a little help from me and Captain Ajax's crew.\"\n\n\"Thank you, loving security chief. When should we dine?\"\n\n\"How about in fifteen minutes. Nelly, can you arrange that?\"\n\n\"Dinner tonight is lasagna, mixed vegetables, and French bread, with a lettuce salad. The lasagna is just starting to come out of the oven. Most certainly, General, dinner can be served in fifteen minutes.\"\n\n\"Then please issue the dinner invitations,\" Kris said, standing so the room could rearrange itself. The couch she'd been sitting on melted into the floor as the conference table morphed from a wooden-topped table with softly rounded edges to a mess table with square corners and a linen tablecloth. It was a bit strange to see china and silverware form out of the linen, but Kris had gotten used to Nelly's magic with Smart MetalTM and only smiled at the strange.\n\nCaptain Ajax was the first to appear, taking the elevator up from her bridge. She caught sight of Jack and made a beeline for him. \"You're going to tell me why you had me trotting around my bridge like a trick pony, right?\"\n\n\"Jack, you didn't,\" Kris said.\n\n\"I'm afraid I did, Admiral.\"\n\n\"I want to hear this, too,\" Kris said to Captain Ajax.\n\nSpecial Agent Foile arrived about the same time as Judge Diana Frogmore. Chief Mediator Alfred Fu and Senior Arbitrator William Pierce Gladsten were last, both in intense conversation that only ended as they took in the change in the room.\n\n\"Thank you all for coming on such short notice,\" Jack said. \"We've had some interesting developments today, and I thought I might bring you all up to date and eat at the same time.\"\n\n\"Partaking of food does tend to set one at ease,\" Gladsten said. \"So what is it that we'll swallow easier with our salad?\" he said, as a salad cart arrived with a chief steward's mate to do the honors.\n\nOnly after everyone was seated and served did Jack begin. \"We have been under a bot assault ever since we docked and opened the hatches.\"\n\n\"Aren't bots and nanos a normal part of business?\" Fu asked as he took his first bite.\n\n\"That is true,\" Jack said. \"But most bots don't come with explosives attached.\"\n\nThat got a rise from the table.\n\n\"How much explosives?\" Kris asked.\n\n\"Enough to blow your head off if it got in your ear or up your nose.\"\n\nKris noticed that all three of her brain trust swallowed hard at that.\n\nCaptain Ajax frowned. \"Was that why you had me running around my bridge like a bot myself?\"\n\n\"Yes,\" Jack admitted. \"We surrounded one bot with enough nanos to take it down, and then let it do what it was sent to do. It avoided all the crew, but wandered its way up the ship from the quarterdeck to the bridge. I asked you to block the exit from the bridge to the hatch leading up to Flag Country. Thank you for doing so. It went right by you and kept heading for the admiral. As soon as it made it up the ladder to our deck, we took it down.\"\n\nKris scowled. \"So it was after me.\"\n\n\"It and a whole lot of others,\" Jack answered.\n\n\"You stopped them all, I assume,\" Captain Ajax said.\n\n\"I've turned my Marine HQ into Security Central. We've got a team tracking intruders and taking them down. The other ships have their own nano scouts and interceptors up but show no activity.\"\n\n\"So they are only interested in the Princess, here, and her mission,\" Frogmore said.\n\n\"The conclusion seems clear,\" Special Agent Foile provided.\n\n\"Do we know the source of these nanos?\" Fu asked.\n\n\"They're coming from the station, but there's no way to tell who's behind them,\" Jack said. \"Some of the bots are the size of a housefly or gnat. You could buy them almost anywhere, and they're easy to track and trap. Others are smaller and more sophisticated. It was one of them we let make a go of it.\"\n\n\"I didn't see it,\" Captain Ajax said.\n\n\"When you walked by it, your wake knocked the thing cockeyed,\" Special Agent Foile said.\n\n\"Your Highness,\" Frogmore said, \"what do you propose we do?\"\n\n\"I suggest we enjoy our dinner and see if any ideas come our way. I, for one, am empty-handed for the moment.\"\n\n\"A Longknife pausing to think,\" Senior Arbitrator Gladsten sighed. \"Having seen such an impossibility, I can now die fulfilled.\"\n\nThat brought a chuckle to all at the table.\n\n\"Once again, I fail to uphold the Longknife legend. What will become of me?\" Kris said through a scowl.\n\n\"I'm hoping you'll die of old age sharing my bed,\" Jack said.\n\nTable conversation was light, with a lot of pauses between sallies. Clearly, everyone was distracted. The chief steward's mate's offer of a dessert cart was declined; the dishes were hauled off to be cleaned before the ship absorbed them again. Nelly turned the dining room into a comfortable circle of armchairs without anyone's having to get up.\n\n\"So,\" Kris said, \"have any of you got a suggestion for what we do next?\"\n\nHer question was answered by a long silence.\n\nJudge Frogmore finally ventured into it. \"We have secured this ship, correct?\"\n\nJack nodded agreement.\n\n\"Could we invite the Emperor and Empress up here for our discussions?\" she asked.\n\nSpecial Agent Foile shook his head. \"I have been in contact with a certain Baron Martin, Commissioner of Public Safety. He says the Emperor has already issued us a formal invitation to court, and he will provide for your security. I'm no expert in court protocol, but I do believe a Princess goes to an Emperor, not the other way around.\"\n\n\"The commander of the Marine detachment,\" Jack added, \"finally got a response out of the Imperial Guard. They are prepared to render all ceremonies to you in your visit, as well as provide for your security.\"\n\nKris shook her head. \"A lot of people want to keep me safe. Do you trust any of them, Jack?\"\n\n\"No farther than I can throw an alien raider base ship.\"\n\nKris snorted. \"Thanks, Jack, for reminding me why we have to solve this dilemma. Okay. They won't come here. I won't go there. I need suggestions on how we resolve this.\"\n\nThe silence stretched and began to bend in the middle.\n\nJack cleared his throat. \"I should go down.\"\n\nWhy was Kris not surprised that he would suggest sticking his head in the lion's mouth? And an Imperial lion at that.\n\n\"What do you expect to accomplish?\"\n\n\"I don't know, but there are standard procedures for a visit like this. You stay here while your security team meets with their security forces. We work out necessary coordination. I make sure our Marines are closer to you than their Imperial Guard. Admiral, I need to do everything that I'd normally do.\"\n\n\"To what end?\"\n\n\"I don't know.\"\n\n\"Are you putting your neck out so the people who don't like us here can take a swing at you?\"\n\n\"Now why would they take a swing at me when they're after you?\" Jack said with enough sincerity that Kris could almost believe him.\n\nShe glared at him instead.\n\n\"Who knows, maybe they won't make a go for me,\" he said.\n\n\"And that will leave us where?\"\n\nJack shrugged. \"I don't know. Maybe whoever is swarming us with bots will make a mistake, and I can get a handle on them. But, Kris, unless you want to pack up and head home because of a batch of unidentified bots, we've got to do something. The book says I go down next. Let's do it by the book.\"\n\nKris sighed. She hated it when Jack won arguments like these.\n\n\"Okay, who goes with you?\" she finally said.\n\n# 31\n\nJack saluted the commander of the Imperial Guard. He wore the uniform of a field marshal. Jack had already looked up all the falderal on his chest; he'd never gotten a whiff of gunpowder in his life. Still, Jack was a lieutenant general, so he saluted the buffoon.\n\nThe field marshal commanded a brigade of fancy toy soldiers in green and black who likely couldn't hit the broadside of a barn. Jack had a platoon of Marines in blue and red backing him up\u2014every one of them a sharpshooter. Every one of them wore spidersilk armor under their gaudy dress uniforms.\n\nJack didn't know when this lion would bite, but he wanted to be ready when it did.\n\nThe field marshal had arranged transportation to the Palace for Jack and his escort. \"I will be a bit delayed with matters here. I will follow you shortly.\"\n\nThe small hairs on the back of Jack's neck stood up and began to do a dance that involved stomping and shouting.\n\n\"Very well,\" Jack said with an easygoing smile, and boarded a black stretch limo that might as well have had a bull's-eye painted on it.\n\nThe Marines divided themselves among four big, black SUVs, two ahead and two behind. In the limo with Jack were Special Agent Foile and his three agents. Foile sent Agent Sanchez to sit up front beside the driver, a local who had been assigned to them.\n\nAs it turned out, each of the SUVs had a local driver. Jack had noted Gunny arranging for a Marine to sit up front before he took the front seat of the SUV directly behind Jack's limo.\n\nWithout delay, Jack's motorcade pulled away from the space-elevator terminal and headed through a series of twists and turns that ended with them speeding along an expressway. Somewhere during that, they picked up two SUVs of the Imperial Guard and several motorcycle police.\n\n\"Everything according to the book,\" Foile murmured.\n\n\"Yep,\" Jack said, swallowing on a very dry mouth.\n\nHe was here to make sure his wife didn't have to be here. Of course, if their enemies were smart, they'd wait until Kris _was_ here before they sprung a trap.\n\nJack hoped they wouldn't be that smart.\n\nThe motorcade held to the speed limit, maybe a bit below it. They took up the middle lane of a lightly used expressway. It began to look like Jack had gotten all puckered up and would have no one to kill.\n\nCars had been whizzing by them at the speed limit or a bit more. A truck was overtaking them, one with a canvas-covered bed in back. Suddenly, cars that had sped past them put on the brakes and those behind them sped up.\n\nJack reached for the service automatic in the small of his back.\n\nIt happened quick, but time slowed down for it all.\n\nThe canvas cover on the truck beside them flew off as the rope ties exploded with a bang and a gout of smoke. Suddenly, Jack was facing a machine gun. Beside it stood a man with a rocket launcher.\n\nIn the front seat, the driver hunkered down. Agent Sanchez grabbed for the wheel even as the driver lost interest in it. Sanchez threw the wheel over and the limo slammed into the truck as the first rounds of the machine gun stitched the window and rear end of the limo.\n\nLimo and truck swerved left, driving the truck into the concrete divider. That tossed the gunners in the back around like rag dolls. The rocket hit the road behind the limo and the machine gun's fire went wild, but not before hits made stars on the limo's armored glass, caving it in but not shattering it.\n\nAgent Foile shouted for Mahomet to get the windows down. Jack was trying the same, but the switch on his door didn't work. He needed the glass out of the way. It wasn't going to stop the next bullet, but it was making a mess of Jack's own aim.\n\nFor Mahomet, the windows moved. Some rolled down. Others halted only halfway.\n\nStill, when the limo swerved away from the truck, Jack and the Secret Service agents had good shots at the men trying to get reorganized around the machine gun.\n\nThe guy with the rocket launcher was just jamming in a reload.\n\nBefore they could do anything, automatic pistol fire stitched them. They flew in all directions.\n\n\"To our right,\" Agent Chu shouted, and Jack yanked his attention around. A big sedan was coming up on their unengaged side. Its windows were down, and machine pistols hung out of them.\n\n\"Duck,\" Jack shouted.\n\nBullets shattered the glass in the rear window, showering them with shards. Apparently, whoever laid this ambush on hadn't been able to get rid of the armor in the fenders and doors. That saved their lives.\n\nThat, and the thirty-round limit of the machine pistols.\n\nThe fusillade of fire lasted only a few seconds. The assassins hadn't been smart enough to take turns in volley fire.\n\nAs one, their machine pistols fell silent. Jack and the agents around him popped up from behind the doors. Several rounds from them and the car swerved into them, then careened into a light pole and came to a shattering halt.\n\nJack took a moment to look around. Of the Imperial Guard and motorcycle cops, he saw nothing. He still had four SUVs with Marines around him. The road aft of them was littered with wreckage.\n\nIt also was clear . . . except for a big twenty-wheeled tractor-trailer. It had been speeding up toward them. Now it was braking hard.\n\n\"Sal, tell the Marines in the trailing trucks I want to talk to whoever is in that truck.\"\n\nThe two rear Marine SUVs slammed on the brakes. The big truck tried to slow even faster but the driver only succeeded in jackknifing his rig.\n\nIt came to a halt with the trailer blocking the road, the smell of burning rubber from tires not meant to go sideways filling the air.\n\n\"Mahomet, let's turn this rig around,\" Special Agent Foile said.\n\nThe agent in the front seat jammed something into the driver's ribs, and the limo began to slow quickly but smoothly.\n\n\"I didn't know anything about this,\" the driver whined.\n\n\"Then how come you ducked when that truck came up beside us?\" Mahomet growled.\n\nThe driver's excuses sputtered to a stop with a \"They just told me to slow for a truck. I figured out for myself that I better duck.\"\n\nJack shook his head. Very likely, the driver didn't know anything more than what he was telling them.\n\n\"I want to see what's inside that truck,\" Jack growled.\n\n\"Me, too,\" Special Agent Foile said, ever so calmly.\n\nThe driver now did an expert J turn and headed back for the jackknifed truck. One of the lead rigs stopped and deployed its Marines to halt traffic on the other side of the expressway. The other provided a rear guard to the limo.\n\nJack's rig came to a halt a short distance from the truck. There, a Marine staff sergeant had three sullen men under armed detention. A corporal was just shooting off the lock on the rear doors, as Jack trotted past the two SUVs that had provided the rear guard. In one, a medic struggled to control the bleeding from a wounded Marine. She had worn the skull cap of spidersilk armor, but the round that creased her skull struck a centimeter below where it ended.\n\n_You can't armor the entire body._\n\nA private swung the rear door of the truck trailer open, and the corporal gave a low whistle. \"General, you really want to see this.\"\n\nJack turned from the wounded trooper and double-timed for the rear of the trailer. He wanted to get this done and over with. The hackles on his neck had not stopped their polka, and that trooper needed more medical care than they could provide her.\n\nHe turned the corner, eyed the contents of the trailer, and whistled, too.\n\nBeside him, Special Agent Foile whistled as well. \"Their very own traveling torture chamber. I think they expected to capture someone alive.\"\n\n\"They've got enough chains dangling from those walls to hold quite a few.\"\n\nFoile shook his head. \"On those shelves is a collection of fingernail pullers, cattle prods, and sharp knives, enough to cut up quite a lot of people.\" He turned to eye the traffic jam growing behind them. \"You think the virtuoso of all that pain is somewhere in that mess?\"\n\nJack followed his glance and scowled. \"You think you could identify the psychopath if we held this mob at gunpoint and searched for him?\"\n\n\"Psychopaths can pass for quite normal. I doubt any of our eyes could spot him,\" Foile said, shaking his head.\n\n\"Sal, do you have a good picture of all this crap?\"\n\n\"Yes, General, captured in 3D. If you want me to create a sim of this, I can do it in a second.\"\n\n\"Let's get out of here before someone comes up with a plan B. Corporal, lob a grenade in there.\"\n\n\"Thermite, sir?\"\n\n\"This isn't a movie, son. Let's not encourage the gas tank to blow up this close to a traffic jam. A few fragmentation grenades should make our point.\n\n\"Aye, aye, sir.\"\n\n\"Gunny,\" Jack shouted.\n\n\"Sir.\"\n\n\"I want you on the rig with the wounded Marine.\"\n\n\"Aye, aye, sir.\"\n\n\"Sal, can you draw us the shortest route to the nearest hospital?\"\n\n\"Got it, boss.\"\n\n\"Shoot it to the Nav computer in that SUV.\"\n\n\"Done, sir.\"\n\n\"Gunny, assign your best driver to your rig and get that Marine to the hospital soonest.\"\n\n\"Aye, aye, sir. Okay, Marines, saddle up. Fire Team A, 1st Squad, you're with me. Sergeant Bourne, get the rest of the troopers organized and into the other four rigs. Move it! We ain't got all day.\"\n\nCrisply, efficiently, the Marines performed their retrograde movement with rifles pointed out to cover 360 degrees. Two rigs were loaded up when the SUV with the wounded Marine gunned out fast. It headed up the road, passed the roadblock on the other side of the road, and slipped off the expressway at the next exit.\n\nMoments later, the limo and the other three rigs followed it, but turned right where the other had turned left. Speeding along side roads, they made it back in time to catch the next ferry up the beanstalk.\n\nWith any luck, in two hours Jack would be back in Kris's flag plot explaining why he was home for dinner.\n\n# 32\n\n\"The ground party is returning,\" Nelly tersely informed Kris. \"They have already caught a ferry up the beanstalk and should be here in forty-five minutes.\"\n\nPuzzled, Kris asked, \"What's wrong?\"\n\n\"Sal did not tell me. Apparently there is fear that our network may be compromised.\"\n\nKris couldn't argue with that. Although the recent bot probes had been low tech, she'd had enough experience with Greenfeld high tech that it would not surprise her if they jacked up their game at a time and place of their choosing.\n\n_And the worst possible time for us._\n\nJack might not be telling her why he'd cut his mission short, but that didn't mean she couldn't draw her own conclusions.\n\n\"Nelly, get me Major Henderson.\"\n\n\"Yes, Admiral.\"\n\n\"The ground team my, er, General Montoya is leading is coming back.\"\n\n\"That's several hours sooner than the earliest they were expected.\"\n\n\"Yes. Please have a large contingency with maximum tech meet him at the ferry station. Be prepared to conduct a fighting withdrawal.\"\n\n\"Do you want us to close up the ships, Admiral?\"\n\nKris examined that. She didn't really want to batten down the hatches on all her ships. Locked doors could be taken as quite rude. Still . . .\n\n\"Nelly, order all the squadron to cover their quarterdecks with transparent Smart Metal. Entrance and exit is through an air lock and delayed until anyone coming back is cleared of bugs.\"\n\n\"Yes, Admiral,\" Nelly said.\n\n\"Now, what do I do next?\" Kris said softly to herself.\n\nA few months ago, she'd be galloping for the beanstalk station to hug and kiss her returning warrior. Of course, she might also catch a bot in her ear, nose, or throat and get her head blown off. Now seemed like a good time to be an old married lady.\n\n_Damn you, Jack._\n\n_I'm not that old though I am thoroughly married._\n\n_I know that you did this intentionally. You set yourself up as the target._\n\n_Thank God, you survived. Or did you?_\n\n\"Nelly, Jack is okay, isn't he?\"\n\n\"I'm checking with Sal. The answer is yes. They have one casualty, a Marine. Gunny has been detached with a fire team to rush her to the nearest hospital.\"\n\nKris grimaced. She'd brought enough spidersilk underarmor for every Marine aboard the _P. Royal_. She expected to bring every one of her Marines home safe.\n\n_What happened?_\n\nShe'd know soon enough.\n\n\"Nelly, inform the head of my brain trust that the security team is returning immediately and without meeting with anyone from the Palace. We do not know why they are coming back early, but we should assume that negotiations tomorrow are not going to happen. The ground team will be back in forty-five minutes. I want them in my quarters in half an hour.\"\n\n\"I have informed them. They will be here. Fu asks if you will be setting your table again for them.\"\n\n\"Tell him no table, but I may be chomping on someone's ass.\"\n\n\"He says he understands and will do his best to keep his ass well away from your chompers.\"\n\nKris snorted at the joke. Maybe Fu could keep the next meeting from going full mortal. She glanced around her quarters, trying to think of what she needed to do next.\n\nHer right breast was getting painfully heavy. If it was, Ruthie's tummy was likely getting painfully empty.\n\n\"Nelly, have the duty nanny bring Ruth in for a nibble, assuming she's not asleep.\"\n\n\"Shani says she and Mai and Megan are playing with Ruth. She's doing knee bends and stand ups and is a giggle ball for all the attention. Shani thinks Ruth will be crawling soon.\"\n\nKris rolled her eyes. _Just what I need, another Longknife getting into trouble._\n\n_Or another Montoya._\n\n_We'll have to settle that soon enough, but not today._\n\n\"Bring her in for a snack. I may be tied up soon.\"\n\n\"They're on the way.\"\n\nThere was a door from the nursery into Kris's night quarters. From there it was just a short hop, skip, and a jump into her day quarters. Still, the nannies usually brought Ruth around through the hall, only using the other door at night.\n\nKris smiled at Ruth, who smiled right back, showing off her four teeth. Kris undid her top and flipped down the nursing bra, but Ruth was more interested in showing off her ability to bounce up and down, so long as you held her arms up. Ruth did a dozen ups and downs, giggling for all she was worth, then took a nosedive for Kris's breast and latched on.\n\nKris took several deep breaths, trying to calm her pounding heart. Ruth didn't need to take in all mommy's stress and nerves along with her milk. While Ruth suckled, the room took on a certain odor.\n\n\"In one end and out the other,\" Mai said with a laugh.\n\n\"Nelly, up the air recirculation. I don't want the room smelling of baby poop when the meeting starts.\"\n\n\"Done, Kris.\" Normally, the low hum of the blowers went unnoticed. Now they kicked up. Soon, the room was cooler . . . and the air fresher.\n\nWhen Ruth lost interest in Kris's breast, she handed the baby over to Shani and began to cover herself up. The two nannies, without a word, took Ruth off to be cleaned up. Kris had just enough time to change her khaki top to a clean one without baby drool down the front, splash some water on her face, and towel off before Judge Diana Frogmore knocked at her door.\n\n\"Enter,\" Kris said, returning to her day quarters. \"Nelly, give us a small conference table. Just the four of us for now.\"\n\n\"Should Captain Ajax be included?\"\n\n\"Is she available?\" Kris asked.\n\n\"She's just into the second hour of a nap,\" her computer answered.\n\n\"Let's let her sleep a bit more. Have you closed the transparent cover over the _P. Royal_ 's quarterdeck?\"\n\n\"I got the XO's permission.\"\n\n\"Good,\" Kris said, settling into her chair at what, for now, was a square table.\n\n\"The detail you sent dirtside to look into our security is coming back already?\" the lovely judge asked, clearly on a fishing expedition.\n\n\"Yes, but that's about all I know.\"\n\nAnother knock at the door. \"Enter,\" brought both Fu and Gladsten in. Kris repeated what she'd told the judge. \"One Marine is wounded and being rushed to the hospital. Beyond that, I know nothing,\" Kris said through a frown.\n\nThe three arranged themselves at the table as if they were sitting down to a game of bridge: Diana to Kris's right, Alfred Fu to her left, and William Pierce Gladsten directly across from her. They composed themselves . . . and said nothing.\n\nKris squelched a frown as the nothing stretched. She cleared her throat and opened the bidding. \"Our security team didn't even make it to their conference. So what are our options now?\"\n\nHer brain trust for this new challenge exchanged glances, then William Pierce, the senior arbitrator, cleared this throat. \"We don't know what happened to interrupt their journey. We hardly have any facts from which to derive options for our future actions.\"\n\nThe other two nodded, and all three settled back in their chairs, clearly intent on awaiting developments in silence.\n\nKris bit her tongue to keep it from wagging. She was in a situation like none she'd experienced before. These three were supposed to help her survive it. If they said sit tight, she should just sit tight.\n\nStill, this was the time when she would usually be examining her options so she could make a decision later when it was needed in a split second, or people would die. Of course, there had always been those that softly hinted to her that she jumped to conclusions ahead of the data available.\n\n_But I was usually right, and jumping to conclusions gave me time to get my ducks in a row._\n\nThat had sure helped when it was monstrous alien mother ships gunning for her. Then, of course, there had been times when it didn't.\n\nShe remembered when, on one of her earliest assignments, she'd had Nelly blare reveille to an entire barrack of troops her first morning on Olympus. As a boot ensign, she figured she outranked everyone in the barrack. She'd discovered very quickly that the entire garrison, including all the officers who were her superiors, was quartered in that one building. Oops!\n\nStrange how she remembered that early screwup. She'd made a lot more of them since then, some that cost people's lives. But the one she really cringed about was that early goof in the morning rain on Olympus.\n\nKris let her eyes rove over her new team. Three wise men. Well, two wise men and a wise woman. They were here to help her do something totally different from her past sins. They were here to teach her how to survive a new mess, and not only live through it but also help millions of others to do the same.\n\n_Oh Lord, not another learning experience,_ Kris thought, and had to swallow an inappropriate smile real quick.\n\nKris settled her gaze on Diana. Was she meditating? Her eyes were closed. Kris couldn't make out what her hands were doing under the table, but her shoulders were relaxed and her breathing was slow and steady. The two men had leaned back in their chairs, eyes half-closed, faces relaxed, but not slack.\n\nKris had seen Gunnies relaxed like that, in the quiet before all hell broke loose. She'd credited them with being tightly coiled springs, ready to unleash all hell when the moment came.\n\nFunny that these three, people of peace and quiet conclusions, should look so much like those old war dogs.\n\nKris took a deep breath and let it out slowly. A few more and she found the buzzing in her bones begin to calm. She closed her eyes for a moment and concentrated on the calm growing inside her. Concentrated on being ready when what was coming arrived.\n\n\"Admiral, this is Major Henderson. I am deploying a fully armed and armored platoon from each ship. We are in the process of securing the route to the elevator station.\"\n\n\"Very good, Major,\" Kris said, still in her calm. \"Do you have sufficient tools to handle the small stuff?\"\n\n\"I never sweat the small stuff, Admiral,\" the major said with just a hint of a chuckle. \"I've got that stuff very much under our control.\"\n\nKris couldn't tell if the major had just cracked a joke, reported his status . . . or told her to get her nose out of his business. She chose the latter and tried to think what could happen aboard the station to her troops. As she went down her checklist, it seemed that all had been taken care of.\n\n_Now I wait for Jack._\n\n\"Admiral, I have a base security chief, one of the red coats, with a squad behind him,\" the major reported a moment later. \"He wants to know why I've got armed men parading on his station.\"\n\n\"Tell him he's welcome to join the parade.\"\n\n\"Thank you, ma'am,\" came back quickly, followed a moment later by, \"He's graciously decided to join us, Admiral.\"\n\nKris wondered if the red coats still had their arms, but she held her tongue. The quiet at the table was contagious. She'd have to make sure their way of handling the new didn't interfere with her taking care of the old.\n\nNELLY, WHEN'S THE FERRY DUE?\n\nKRIS, IT'S PULLING IN NOW.\n\nKris found her stomach clinching, but took a deep breath and prepared herself for what was about to happen.\n\n\"The detail has returned,\" Major Henderson reported. \"We are headed home.\"\n\nFor a moment, that was all that came in. Then Nelly provided an update.\n\n\"All the Marine detachments stationed along the line of withdrawal are being buzzed by bots of several different types. Sal and I are helping the Marine tech teams take over as many of them as we can manage. Nanos are taking down those we can't control before they get close to our teams.\"\n\n\"Nelly, reinforce the Marines with as many nanos as you need.\"\n\n\"Already doing it, Kris. I've also got Navy tech teams backing up the Marines. We've got more bots coming at us than I thought possible.\"\n\n\"Don't let any get through,\" Kris said, and glanced around her table. The three of them were intense, now, eyes open, leaning forward in their seats, listening to every word Kris heard or spoke. Fu was on his commlink.\n\n\"I've got our tech team reinforcing the Navy at Security Central,\" he whispered softly.\n\n\"Nelly, when you have time, could you put this on my screens?\"\n\nThe screens came to life without a reply from Nelly.\n\n_She must be really busy._\n\n\"Man down,\" came a tense voice on net. One of the dots that stretched from the _P. Royal_ to the terminal began flashing red. \"One of those damn bugs got the sergeant. Swat at them if you see one.\"\n\n\"We need flyswatters more than we need rifles.\"\n\nKris wondered if she could get some Smart MetalTM flyswatters to fly out there as one thing before they morphed into swatters. Then she thought better of it. If they got anything out there, it should be more nano interceptors.\n\nShe tapped her own commlink, not bothering Nelly. \"Admiral to squadron. I want every ship prepared to deploy a Marine platoon in full space armor. More if possible. Secure all hatches.\"\n\nAcknowledgments came back from all eight battlecruisers.\n\nOn the screens, the dot at the elevator station fell back, merging with the next dot.\n\n\"Three of the red coats ran.\" That was Jack's voice on net. Calm as ever. \"Their own bots just blew their heads off as soon as they were outside our perimeter,\" Jack observed, just as calmly.\n\nThe farthest-out dot merged with the next one in the path of its retrograde movement to the flag.\n\n\"Put a tourniquet on that but keep moving,\" Jack shouted, then said more calmly, \"Kris, I'd ask for more medical support, but the detachments are fighting as hard as they can, and I don't think a medical recovery team could make it through.\"\n\n\"We're having enough trouble holding our ground,\" Nelly reported. \"I don't think we could defend another moving detachment.\"\n\n\"Understood, Nelly,\" Jack replied.\n\n\"They've just activated their autocannons,\" Nelly announced. \"Oh, they also activated some new firewall and booted me out of their system.\"\n\nSo someone was digging into their bag of electronic tricks. \"Admiral to squadron, secure the comm lines to the station.\" Her order was answered quickly.\n\nKris tapped her commlink again. \" _Princess Royal_ , _Intrepid_ , please deploy your space-armored platoon to reinforce the troops ashore. Take extra rocket launchers.\"\n\nTwo captains acknowledged their orders immediately. A screen that had been covering the situation on the pier now showed Marines in full battle rattle double-timing it ashore. Their helmets were dogged down; they were breathing their own air. No bot would get at them. In single column they trotted down the ships' brows, across the pier and then, without missing a step, up the escalator to A deck.\n\n\"Nelly,\" Jack demanded on net, \"show us where the autocannons are and their targeting cameras.\" A moment later Jack ordered, \"Open fire.\"\n\nKris kept her mouth shut, swallowing an anxious question. She studied the few cameras they had up on A deck. Yes, the autocannons were all on the overhead. A volley of Marine rocket grenades exploded, blowing autocannons away just as a few of them began to stitch the Marines, knocking them around like tenpins.\n\nKris had always wondered how spidersilk armor would do against a half kilo of twenty-mike mike. Now she saw. Marines flew backward and sideways as the slugs hit them, sliding or rolling them along the deck before they came to a sprawling stop. A few managed to make it to their hands and knees before their buddies helped them up.\n\nOne was knocked out of the protective nano cover. In hardly a moment, his head exploded.\n\n\"I'm deploying more nanos,\" Nelly said.\n\n\"We need them,\" Jack answered. His withdrawing forces were now halfway home.\n\nOne autocannon had been missed or skipped. It was just barely below the curved horizon. Now it opened fire.\n\nStation autocannons were supposed to use reduced-power rounds so they wouldn't punch through the deck and open the station to space. This one proved the point. Its slugs were almost spent when they hit the deck and ricocheted before plowing into a dozen Marines, knocking them down and sending them sliding toward the forward end of the station.\n\nOne Marine held on to her black box. Even as she slid across the deck with her buddies, her fingers were working her board. If there were bots waiting for them, she kept her nanos moving to protect her team.\n\nThe only Marine left standing with that squad fired a rocket at the offending autocannon. The station acquired another hole in the overhead.\n\nMarines in full space battle armor trotted up to the scattered squad. They helped the troopers to their feet, then, as the rest of the armored platoons continued toward the farthest-out team, one fire team helped those in dress uniforms limp toward the ship.\n\nThere was an explosion against the face mask of one of the armored Marines. It knocked him back a step, but he was moving to catch up with his fellows with nothing but a shake of his head.\n\n\"We've got nanos moving along with the armored Marines,\" Nelly said. \"I guess we missed that one.\"\n\nKris figured the tech teams were allowed a few misses. She was glad that miss was on a Marine armored for it.\n\n_Why didn't I fully armor Jack's detachment?_\n\n_Because I didn't want to look all dressed up for war,_ she answered herself.\n\n_Mark up another mistake that I made that cost someone his life,_ she thought bitterly.\n\n_This damn Longknife legend is a bitch._\n\nPulling herself out of her head, Kris studied the boards. The armored Marines were almost to Jack. His team was falling back as quickly as they could move without compromising their perimeter.\n\nSomeone in the opposition tried using an autocannon that was beyond the horizon on A deck. The slugs hit the deck and ricocheted. Jack's Marines would have to pass though that beaten ground to make it back to the _P. Royal_. He sent a fire team of armored Marines out to spot that cannon and take it out.\n\nOne of them got hit in the leg by a round. It spun him around and he went down cursing on his fire team's private net. His buddies razzed him, but trotted up to where they could get a shot at the autocannon. They quickly added another hole to the overhead that would need patching.\n\nNow the outer perimeter around Jack's troops was held by the spacesuit-armored Marines. Kris was anxious for them to get back aboard, and the urge to run must have been almost irresistible, yet the Marines kept to their careful pace. Kris could now spot Jack in the center of the retrograde, surrounded by a squad of Marines with their heads down, watching developments on their boards and making adjustments to the cloud of nano defenders swarming around all of them. No Marine moved any faster than that support team could walk with their eyes on their boards and not looking where they were going.\n\nOne of the technical types stumbled as Kris watched. A buddy next to him grabbed his arm and kept him from going down. Kris gritted her teeth and prayed they could hold this together just a little bit longer.\n\nTugs towing cargo carts came racing down A deck, already at full tilt. They had no drivers. There were shouts from the outer perimeter and Marines took a knee, sighting their rifles in. On a Gunny's shouted command, they took the mass of cargo carts under fire.\n\nThe Marines hit the tugs right on, but as they swerved, their carts came free. Those kept rolling toward the Marines. On Gunny's order, there were second and third volleys. The carts turned out to be heavily loaded enough that they absorbed the hits and just kept coming.\n\n\"Aim for the tires,\" Gunny ordered, and fire went lower. Some hit the deck around the carts. Most ricocheted off the deck. A few punched through to space. They'd need to be patched, but it would take a long time to empty the station.\n\nTires took hits and deflated, bringing their cart to a slow stop. A couple of carts had their wheels shot off. They came bouncing toward the Marines, who dodged out of their way.\n\nOne poor Marine was looking right when a wheel rolled in from the left and hit him hard on the back and knocked him down. Two of his buddies picked him up by his armpits and dragged him along with them as they kept heading for home.\n\nKris wondered what they would come up with next, but whoever the opposition was seemed to have pulled the last trick they had out of their hat. Kris knew there was one last gambit they could make. With the _Princess Royal_ locked down tight, there was no way for Kris to know what the pressure was on the station's A deck, and she was not about to ask Nelly for a report on that unless she offered one.\n\nApparently, no news was good news. The Marines made it back to the pier.\n\n\"Everyone aboard the _Princess Royal_ or the _Intrepid_ ,\" Jack ordered. \"Nelly, could you expand the receiving air lock on the quarterdecks to take us all.\"\n\nIn a moment, the entire quarterdeck was one big air lock, with Marines being checked over and only passed aboard through a smaller air lock in groups of five or six after they had been gone over thoroughly.\n\nKris left it to Captain Ajax to organize transportation for the borrowed Marines back to their ships by liberty launch. For now, she wanted her ships locked and tight.\n\nWhat she would do next was something for her, Jack, and her brain trust to decide.\n\n# 33\n\nKris was relieved beyond words when Jack half stumbled into her day quarters. He was out of breath; he'd lost his cover. Was that his blood on his cheek? Kris found herself out of her chair and dashing toward him, arms open even before she thought of it.\n\nShe was careful to slow down before she hit Jack. From the looks of him, she could probably blow him over with a feather.\n\nShe put her arms around him, and they both let out a breath they hadn't known they were holding. \"I'm so glad to be home,\" was all Jack said.\n\n\"I'm glad you made it. Don't you ever do that to me again! I'll order _you_ locked up in our bedroom myself.\"\n\nJack grinned as she made to him his often-repeated threat to her. Often threatened but never followed through on. Yet.\n\n\"I think you just might do that,\" he said, and turned her face to him so he could give her an ever-so-soft kiss.\n\nKris broke from the kiss and the hug and stepped away from Jack. Still with her arm around him, she aimed him for the table that was growing a place for him and several more. \"You want to take your seat?\"\n\n\"Think I could get a glass of water?\" Jack asked. It hit Kris that the huskiness in his voice wasn't just for her closeness.\n\n\"I'll get us both a glass,\" said Special Agent Foile, who had followed Jack into Kris's flag quarters.\n\n\"It's through my night quarters,\" Kris said, and the Secret Service agent disappeared to return a moment later with two cups, no doubt manufactured for him by Nelly.\n\nJack took one and emptied it in a quick series of gulps. Beside him, Foile was doing the same. Both collapsed into the nearest chairs.\n\n\"You want a refill?\" Diana asked. She was already out of her chair and reaching for their empty cups.\n\n\"Yes, thank you so very much,\" Foile said.\n\nCaptain Ajax arrived, took one look at the two men from the ground party, and asked, \"Would you two like a sandwich or something?\"\n\n\"It's been a long time since breakfast,\" Jack admitted.\n\nIn a few minutes, a second class steward's mate arrived with a cart loaded with sandwiches and desserts as well as urns of coffee and tea. Kris found that Jack wasn't the only one hungry. She split a sandwich with Diana; the two older men also split one.\n\nOnly after they had all caught their breath did Kris lean back and fix Jack with a jaundiced eye. \"You want to tell me what happened?\"\n\n\"We got ambushed on the expressway between the space-elevator terminal and the Palace,\" Jack said simply.\n\n\"Why would they attempt to kill the security team?\" Chief Mediator Fu asked.\n\n\"I don't think they were out to kill _all_ the security team, Mr. Fu,\" Jack answered.\n\n\"Oh, must we be so formal,\" the mediator said, eyes sparkling. \"We all know each other here, and I've heard your wife call you Jack and you call her Kris. I'm Alfred. Al if you want to talk fast.\"\n\n\"Okay, Al,\" Jack said. \"Sal, show everyone the contents of that truck.\"\n\nThe rolling torture chamber appeared on one wall. \"That was following behind our attackers. From the looks of it, I think someone wanted to capture yours truly and send me back to Kris one pound of flesh at a time.\"\n\n\"Damn,\" Kris whispered. \"I do hate kidnappers.\"\n\n\"I'm developing a distinct distaste for them myself,\" Jack admitted.\n\n\"Okay, my fine brain trust, I want some opinions on what we do next. Any that put Jack at risk down there again are not acceptable.\"\n\n\"Or you, honey. Remember, I can lock you up.\" When the look from Diana took on horrible proportions, Jack quickly added, \"There's a law that says the security chief of a serving member of the blood may take whatever measures he deems necessary to keep said member out of harm's way.\"\n\n\"As the only member of the blood serving in uniform, I take umbrage at that law,\" Kris drawled.\n\n\"Oh. Okay, so I'm not going to witness a case of spousal abuse,\" Diana said.\n\n\"Don't be too sure,\" Jack said. \"She may put _me_ in sick bay if I try to corral her.\"\n\n\"Definitely,\" Kris said, but with a loving smile.\n\nEnough of that.\n\nShe quickly swept her eyes over those at the table. \"We were invited to mediate between the Emperor, the Empress, and the Grand Duchess. Is there anyone outside those three that might be trying to kill me or suborn me by torturing my husband?\"\n\nThe room grew serious. Unfortunately, it also grew silent. Kris didn't much care for it when her brain trust grew tongue-tied. She turned to Diana. \"Do you want to nominate anyone else?\"\n\n\"I can't think of any other movers than those three in this messy Empire.\"\n\n\"Al?\"\n\nHe shook his head.\n\n\"Do you want to be called William Pierce or just William?\" Kris asked Gladsten.\n\n\"Bill will do, or WP,\" he answered, \"and no. All three of us have been watching the media intake the tech people have swept up. Whether you consider the Empress the mover, or her father the power behind her, those three you named are it for this place. I should tell you that we think there may be others, call them mice within the walls, who could come forward, but not now. No one will risk raising their head too high. Getting noticed is deadly in this place.\"\n\n\"But the Grand Duchess did raise her head,\" Kris said.\n\n\"Say rather that she became the final drop into the liquid that caused it to become supersaturated and precipitate out some interesting particulates,\" Al said.\n\n\"So it's not just Vicky,\" Kris said.\n\n\"Not just Vicky,\" Diana agreed. \"Although we aren't sure whether she's moving them, or they're moving her. I suspect it may be either of those depending on the day of the week.\"\n\nKris chuckled. \"God help anyone who thinks they can control Vicky.\"\n\nBill shrugged. \"That may not presently be the case. We have a source from inside the Empire who says Vicky has become more open to learning new ways of late.\"\n\n\"Who?\"\n\nThe three exchanged glances. \"We would prefer to keep our source to ourselves,\" Diana said for the three.\n\nKris scowled. \"Even from me?\"\n\n\"I'm quite interested in seeing your impression of your friend Vicky,\" Diana said. \"That may tell us a lot about what's actually going on.\"\n\nKris shrugged. \"Okay. I am looking forward to meeting her again. She owes me for that interview she gave right after we got back from our little jaunt around the galaxy.\"\n\nDiana raised her eyebrows in a question. Kris smiled and ignored her.\n\n_Two can play this game of I know something you don't know._\n\nKris eyed her three brain trusters, but they had nothing more to add. \"Okay, if we don't think someone else is out for my hide, which of those three should we assume is looking to mount my head over their fireplace?\"\n\n\"I'm not sure we can assume that,\" Bill said.\n\n\"Really,\" Jack said. \"I don't know about you, but I'm getting a really strong hunch.\"\n\n\"We prefer not to play hunches,\" Al said.\n\n\"Hmm,\" Kris said. \"Okay, just for chuckles and grins, let me examine this situation out loud. Vicky is not on Greenfeld and likely has few friends here. Do any of us think she could have put together a rolling ambush dirtside accompanied by its own torture dungeon, then followed it up with that bot assault Jack just fought his way through to get home?\"\n\n\"It would seem unlikely,\" Diana admitted cautiously.\n\n\"The Emperor invited us here to mediate this situation,\" Kris said. \"I find it hard to believe that he would turn around and try to kill me before I can help him close down this civil war.\"\n\n\"Assuming he really wants to end this war,\" Bill said.\n\n\"Everyone knows there has been bad blood between the Peterwalds and the Longknifes going back to the Iteeche War, if not longer,\" Al put in.\n\n\"Hmm,\" Kris said. \"You folks really are allergic to the assumption word, aren't you?\"\n\n\"We prefer our conclusions to be firmly based on facts,\" Diana said.\n\n\"So,\" Kris said, her temper on a slow burn, \"do you have any idea what we do next?\"\n\nThe three just shook their heads.\n\n_What does it take to get these people to say anything?_\n\nKris turned to Jack. \"If we're not going down there, I guess we'll have to invite Harry up here.\"\n\n\"Harry and his blushing bride,\" Jack added.\n\nKris eyed the three.\n\n\"That does appear to be the next step,\" Al said. \"I would certainly hate to attempt an intervention via screen hookup.\"\n\n\"I tried that once,\" Bill said. \"Bad all around.\"\n\n\"I've kept children in the next room and asked them questions by hookup, but I think that would be our last option here.\"\n\n_Ah, I actually got one to say \"option.\"_\n\n\"Okay,\" Kris said, moving quickly to exploit the agreement before it evaporated for lack of hard facts. \"It is time to call the Emperor and invite him up here. Nelly, get me a line to Harry.\"\n\n# 34\n\nIt was a full fifteen minutes before the screen in Kris's quarters filled with a view of Emperor Henry, the first of that name. No surprise he was seated on his throne, a golden affair that glittered in the light. Beside him, on an equally golden throne, if a half meter lower, sat the Empress.\n\n\"We thank you very much for taking our call, Your Imperial Majesty,\" Kris said, laying the royal \"we\" on not with a trowel but more like a shovel.\n\nShovels of pure horse manure.\n\n\" _We_ were _expecting_ a visit from _you_ in the near future, Your _Royal_ Highness,\" the Emperor answered, with just a bit of a bite in it. Maybe more than a bit.\n\nBeside him, the Empress fairly boiled with rage but managed to keep a lid on it. For the moment.\n\n\"Yes, about that,\" Kris said as diffidently as her anger would allow. \"Have you been informed that _our_ security team on their way to meet and coordinate our visit with _your_ security team came under gunfire?\"\n\nThe Emperor frowned. \"No, we have heard nothing of such a thing. Have you, Your Imperial Highness?\" he said, addressing his wife oh so formally.\n\n\"That could never happen on _our_ peaceful Greenfeld,\" she said, with ever so lovely a frown.\n\n\"We _cannot_ believe such a thing. You _must_ be misinformed,\" the Emperor said, offering Kris a chance to back down.\n\nKris was not interested in backing. Down or anywhere else. \"We are _afraid_ it is true. Our consort, Prince John, led our team and witnessed the gunfire. We are informed that it left quite a traffic backup on your capital's main expressway. We also had to send one of our injured Marines to your nearest hospital.\"\n\nKris would have to do the paperwork when they got back. For now, they'd just have to fake it.\n\nNow the Emperor allowed himself a serious frown. \"Martin? Where is Commissioner Martin?\"\n\nA man in severe black quickly stepped up to stand behind the Emperor.\n\n_Talk about the power behind the throne._\n\nMartin and the Emperor exchanged words silently for a long moment, then the Emperor turned back to Kris. \"It would seem that there _was_ some sort of fracas on the expressway this afternoon. Truckers are always exceeding our speed limits and causing no amount of trouble. However, _we_ are told that there was _no_ shooting involved. Certainly not,\" the Emperor said, shaking his head.\n\n\"Hmm,\" Kris allowed. \"It seems that we are informed differently on the matter. However, we fear for our safety on your roads after what happened to our Prince consort. You have asked us to mediate these unfortunate affairs taking place in your Empire. We are afraid that we must respectfully ask and invite you to visit us on our flagship, the frigate _Princess Royal_.\"\n\n\" _You_ want _us_ to visit _you_ on such a tiny vessel,\" the Empress snapped.\n\n\"Dear,\" the Emperor said, resting a gentle hand on her bare shoulder.\n\n\"Honestly, Harry, you would go hat in hand at the beck and call of a _Princess_?\"\n\n\"Now, dear.\"\n\n\"Next thing you know, you'll go barefoot in the snow to Canossa.\"\n\nBAREFOOT TO CANOSSA, NELLY?\n\nA REFERENCE TO A HOLY ROMAN EMPEROR, HENRY IV, BEING HUMILIATED BY POPE GREGORY VII IN 1077, KRIS.\n\nOKAY, GOT IT.\n\n\"Your Imperial Majesty,\" Kris put in. \"We do not seek _anything_ more than is necessary so that we can give you what _you_ have _asked_ of us.\"\n\n\"Must we go to you?\" Harry said, frowning.\n\n\"We fear for our safety, Your Imperial Majesty. Certainly, you will be safe on your own roads and space station.\"\n\n\"Of course.\"\n\n\"But a tiny vessel like a frigate,\" the Empress put in.\n\n\"We assure you, Your Imperial Majesty,\" Kris put in quickly, \"Our frigate is made of Smart Metal. It will be no problem for us to expand it to provide you accommodations appropriate to your honors.\"\n\n\"I _have_ been wanting to see what is being done with Smart Metal in your realm,\" the Emperor said, slowly.\n\n\"We can give you a full tour of my flagship, Your Imperial Majesty.\"\n\nThe Emperor paused for only a moment as his face took on the look of a boy who had been invited into the candy shop with unlimited credit. \"We shall _certainly_ accept your _generous_ invitation to tour your flagship, young Princess, and if we should _happen_ to spend some time discussing these unfortunate developments in our Empire, that will just be all for the better.\"\n\n\"Very good, Your Imperial Majesty. When should we expect you?\"\n\n\"Tomorrow, say around eleven.\"\n\n\"We shall be ready, Your Imperial Majesty, to give you a full tour.\"\n\n\"Good-bye until then.\"\n\n\"You are so gracious, Your Imperial Majesty,\" Kris said, pouring it on with a fire hose.\n\nThe screen blanked, and Kris took in a deep breath.\n\nNELLY, IS THE LINE DISCONNECTED AND THE MICS OFF?\n\n\"Yes, Kris. I've even disabled the landline to the pier. They can't hear us anymore.\"\n\nKris blew out a breath and with it some of the tension. \"Did the Empress look to you like she was ready to explode?\" she asked Jack.\n\n\"I'm glad she's no longer pregnant, or we'd have witnessed the birth right there.\"\n\n\"Yeah,\" Kris said, still taking deep breaths and letting them out slowly. \"From the behavior of Harry, I don't think he knows anything about the attack.\"\n\n\"Or anything about what's going on in his city, much less his Empire.\"\n\nKris rolled her eyes, but only said, \"I tend to agree. Now, about the Empress. I would love to get her attached to a lie detector.\"\n\nJack nodded. \"She, I can see sending me back to you a piece at a time with orders to do exactly what she said.\"\n\n\"Ugh,\" Kris said, shaking her head. \"Okay. I suspect we will be saying 'Your Imperial Majesty' a lot tomorrow.\"\n\n\"No doubt,\" Jack said. \"With them deriding you as just Your Royal Highness. Now, dear Admiral, just exactly what have you committed us to?\"\n\nKris turned to her brain trust. \"Okay, we've finally got a meeting set up. What do we do with it?\"\n\n# 35\n\nNext morning, Kris found herself alone with the more mundane duties of the day. After breakfast, Jack changed to his best dress blue-and-red uniform and left to make sure that the Emperor's visit passed peacefully and with a reasonably low body count.\n\nKris ate, then dropped in to get a final briefing on what the tech people had stripped off the Greenfeld net. That continued to be painfully slim.\n\n\"We've got enough to tell us something is very wrong,\" the head data specialist, Runda Fairsight said. \"Too much of what we've found just doesn't add up. There's something missing, and that missing stuff is what's encrypted up to its eyeballs. So far, we're unable to read any of it. The fact that nearly a quarter of the net is that highly encrypted tells us a lot.\"\n\n\"So tell me what that 'lot' is, please,\" Kris said.\n\nRunda rolled her eyes. \"Someone is willing to put up with a lot of slow traffic to make sure it's unread. On Wardhaven, about five percent of the net is this heavily scrambled. Bank transactions and critical military messages. Your grandfather Alex Longknife is about the only one who encrypts this much of his traffic.\"\n\n\"We know he's around the bend, paranoid,\" Kris muttered.\n\n\"Exactly. A huge chunk of this planet is as around the bend as your grandfather Alex.\"\n\nKris shook her head. \"That does not bode well.\"\n\n\"Nope. We'll keep working on it, but the truth is, those other three better get ready to pull a real miracle out of their hats, ma'am, because we can't give them the normal hints from what we've dredged up.\"\n\n\"Thanks,\" Kris said, and left to find her three miracle workers. They were in the Forward Lounge that, at the moment, had been moved from the bow to amidships, only two decks down from the quarterdeck. Those three decks had now been expanded to be a thousand feet across.\n\nThey were eye-droppingly luxurious and spacious enough to match any palace in human space. They shared two glorious spiral staircases in shining wood, plush red carpets, and elbow room. They were arranged so you came down one from the quarterdeck, walked across a foyer covered in luxurious wallpaper of red and gold with delicate inlaid wood, festooned with classical statues, most of them nudes, Kris noted. From there, she descended a second magnificent staircase right into the middle of the lounge.\n\n\"Nelly, get me Ajax.\"\n\n\"Yes, Your Imperial Highness,\" came quick.\n\n\"Nope this is just Your Royal Highness.\"\n\n\"Sorry, I've been thinking too much about our visitors,\" Captain Ajax admitted.\n\n\"I guess you have. I'm standing at the foot of the staircase down from the quarterdeck and looking at something right out of a madman's dream of a palace. There's another staircase that leads down to the Forward Lounge which, I take it, today is not so forward.\"\n\n\"No, Your Highness. The general asked if I could shorten the distance from the brow to the lounge, and maybe get my bridge off the direct path. Defense came up with all the Smart Metal necessary to give you those thousand feet across decks.\"\n\nKris glanced around and did not see a thousand feet. Then she remembered that quite a bit of the space behind the quarterdeck was behind a wall presently covered with several huge murals of battles lost and won.\n\n\"How'd you fill up the space?\" she asked.\n\n\"Well, Admiral, the Forward Lounge is a full thousand feet in diameter with a very high ceiling. The middeck that you are on has been assigned to the Marines. They've got really spacious quarters now. There's one hell of a workout area and a track that covers the full circumference of the deck.\"\n\n\"That's nice. What about the quarterdeck?\"\n\n\"It's right where it was, only now much of that deck is Admiral's Country, ma'am. While you've been at breakfast and busy otherwise, we moved your quarters, and those around you, including the special advisors, to that deck.\"\n\nKris sighed and rolled her eyes at the overhead. \"Will I recognize my space?\"\n\n\"Not likely, Admiral. You can play handball in your new quarters. Your night quarters will make even an emperor jealous. Your bath has a tub that can seat ten. I really hope you'll extend me an invitation.\"\n\n\"I hope no one on the ship is stuck in a cubbyhole.\"\n\n\"No, ma'am. Even Ruthie's nannies will be quite comfortable although their rooms are arranged around the nursery. We kept that baby's space small so people can reach Ruth in a hurry. That chief warrant officer of yours was quite clear on that.\"\n\n\"Never argue with a Gunny. Are the Sailors going to be upset?\" Kris didn't want any of her crew to feel left out.\n\n\"No, ma'am. We blew out quite a few decks, not out to a thousand feet, but the _Princess Royal_ now makes the most luxurious space liner look downright poverty-stricken.\"\n\nKris let that thought run, and quickly saw where it took her. \"And the rest of the squadron?\"\n\n\"They are all quite comfortable now though none of them got a Forward Lounge, Admiral. I have been asked by several of my captains to point out that with you not allowing any shore liberty . . .\" left a whole lot in Kris's lap.\n\n\"Good point, Captain,\" Kris stalled. \"I'm a bit busy preparing for the coming Imperial visit. Let me get back to you. If I don't, raise it at dinner tonight.\"\n\n\"Will do, Your Highness.\"\n\nKris noted the use of the royal rather than the Navy. As an admiral, the only answer to alcohol aboard ship was no. Better yet, Hell No! As a royal . . . things were more up for grabs.\n\n\"Nelly, how much booze does the Forward Lounge have in stock?\"\n\n\"A lot, Kris, but not nearly enough for the entire squadron.\"\n\n\"Okay. Nelly, examine how you might order up a supply of beer, wine . . . oh hell, a bit of everything . . . without the folks you order from knowing it's headed here. Then could you redirect it in transit?\"\n\n\"I've already studied that option, Kris. No problem doing it. Can I bill it to the Palace's account?\"\n\n\"You sure can't bill it to a Wardhaven credit chit. When would such a bit of chicanery arrive pier-side?\"\n\n\"If I do it now, say, three o'clock.\"\n\n\"Make it happen. We'll debit it against the Imperial visit.\"\n\n\"It is done.\"\n\nKris hoped she wouldn't live to regret it.\n\nDown the next staircase, which Marines could march in ranks twelve wide, Kris found an establishment that beggared the mind. The Smart MetalTM walls appeared to be wood or golden wallpaper or more murals, some of battles, others of magnificent landscapes or classical paintings. The huge space was divided by banks of mirrors. They gave the vast rooms a near agoraphobic element that shrank the puny individual down to almost nothing. The furnishings mimicked styles from different ages of human existence with a strong bent toward wooden chairs and marble-inlaid tabletops. Others were already covered by snow-white linen, delicate china, and shining silverware . . . some of which was golden.\n\nKris found her mouth hanging open. She swallowed her awe and snapped it shut.\n\n\"Astonishing,\" Judge Diana Frogmore said, coming up beside her. \"I'm amazed at what that computer of yours can do with Smart Metal.\"\n\nKris shook her head. \"I'm just glad I dropped down here. I'd hate to see this place for the first time with Harry.\"\n\n\"Yep. That's why we have our teams down here. I've heard of a need to acclimatize troops to a new environment. This sure takes that cake.\"\n\n\"And eats it, too,\" Kris added.\n\n\"I assume you've come to ask if we are prepared.\"\n\n\"Yes,\" Kris said.\n\n\"Well, we're prepared as much as we can be. No doubt, Runda told you this place is a hard nut to crack.\"\n\n\"Unfortunately.\"\n\n\"We propose that we let the Emperor, and, no doubt, the Empress talk. None of us have ever met people who think overmuch of themselves who were not most willing to fill the air with their verbosity.\"\n\n\"Will that help us?\"\n\nHere Diana shrugged. \"It has never hurt.\"\n\n\"There is always a first time.\"\n\n\"Yes,\" she said ominously.\n\n\"I've about run out of time,\" Kris said, \"so if you'll excuse me, I've got to get ready.\"\n\nDiana looked puzzled.\n\n\"Getting ready with an infant at the breast takes a whole lot more time than it took when I was carefree and single, or even newly married.\"\n\n\"Ah, yes,\" Diana said, enlightened, then her eyebrows dropped. \"I can't help but wonder. Why did you choose to bring your infant on this mission?\"\n\n\"Let me give you the quick answer. I enjoy nursing Ruth. If I leave her, even for a month, I can't get that back. Also, my mom and dad were what your profession calls emotionally unavailable for me after Eddy was kidnapped and allowed to die. I want to be as close to my kids as possible. I believe it's called wanting to have it all. And if all that is not enough, can you assure me that if I left Ruthie home that someone wouldn't kidnap her and hold her for ransom?\"\n\nDiana nodded. Kris went on.\n\n\"Here, she is safe with the same Marines and Sailors that protect me. While she can be as exhausting as any infant, I am not like other women. I cheat. I've got a staff of six to take care of Ruth and my dog robber to take care of the minutiae of my job. If I can't do this, I guess it really is impossible.\"\n\n\"And, as an admiral, you can make your ship accommodate your expanded needs.\"\n\nKris chuckled. \"Without Smart Metal, I'm not sure even I could pull this off.\"\n\n\"See you in\"\u2014Diana looked at her wrist\u2014\"an hour.\"\n\n# 36\n\nKris's hour disappeared in a blinding flash despite everything she did to catch it. She fed Ruthie from one breast while pumping milk from the other. If the meeting went long, she didn't want Ruth getting hungry\u2014or her mother sitting at the negotiating table leaking milk through her dress whites.\n\nA quick shower, and she found herself standing beside a fresh-faced Ensign Longknife, who was a study in puzzlement.\n\n\"Ma'am, I've managed to put on most of your fruit salad, but I have no idea where some of this stuff goes.\"\n\nThe boot ensign had done a good job with the medals that met Wardhaven's standard uniform regs. The Olympic Humanitarian Award was mounted above the pocket of her white dress coat farthest to the left with its V for valor. Next came the Turantic Medical Expeditionary Medal, also with a V to show Kris had won it in a fight. She often had trouble explaining how she managed to earn a Valor commendation during what were supposed to be humanitarian missions. Even Kris's Society of Humanity Devolution Service Medal had a V. She wasn't allowed to explain that one. The Navy and Marine Corps Medal, the one that started Kris off and which Mother wanted to encrust with diamonds and turn into a broach when Kris got out of the Navy, stood in the shadow of the Wardhaven Defense Medal. There was no V for that one. Valor was assumed for all those who volunteered for that bloody fight.\n\nIt always took place of honor on the breast of anyone who had lived past earning it.\n\nIt was the other gewgaws that stumped Meg. In truth, they also stumped Kris, but with a shrug . . . who could gainsay her, really . . . Kris dived in.\n\n\"Three go around the neck. Greenfeld's _Pour la M\u00e9rite_ has the smaller choke chain. I think it's supposed to be worn right at the throat. Nelly, can you find out if it goes inside or outside the choke collar?\"\n\n\"The chain goes inside the collar, with the cross hanging out.\"\n\n\"Good. The next two likely have no regs to follow. The Almar Medal of Highest Honor has the shorter ribbon.\" It was bright red and gold, with a triangular gold medallion. \"We'll put it on second.\"\n\n\"What's this one?\" Meg said, holding up a large golden globe suspended from a light blue ribbon.\n\n\"That is Granny Rita's bad conduct medal.\"\n\n\"Bad conduct?\" Meg said, raising a doubting eyebrow.\n\n\"Bad conduct or just plain dumb conduct,\" Kris amended, quickly. \"If you have the honor of tackling and destroying an alien mother ship, you get that honey in bronze. If you haven't learned your lesson, and mess with a second one, Granny adds bronze oak leaves. Third time and the metal is silver, with silver oak leaves if you need a fourth lesson. Those really slow learners can get a gold one for their fifth learning experience, with gold oak leaves for six.\"\n\n\"Do everyone's have cross swords at the top?\" Megan asked.\n\n\"Seventh,\" Kris muttered.\n\n\"Is that a diamond in the middle of the mother ship on the medallion?\"\n\n\"Eighth,\" Kris breathed.\n\n\"Eight!\"\n\n\"What can I say, I'm a very slow learner. Poor Ruthie has two really dumb parents.\"\n\n\"Jack was there for eight, too?\"\n\n\"Yep. Most people quit at six. Okay, moving right along, we've got these three starburst-type things.\"\n\n\"Four,\" Meg interrupted. \"This arrived from the Palace early this morning.\"\n\n\"What is it?\" Kris said, staring at the starburst in the red box.\n\n\"It's the Greenfeld Order of St. Christopher, Star Leaper, for those who circumnavigated the galaxy. Or not. I'm told there are posthumous tombstones for those that died before they got back.\"\n\n\"Hmm,\" Kris said, and talked herself through her problem. \"The Order of the Wounded Lion is Earth's highest award. We'll put it upper right on my left chest, a few millimeters below the normal medals. Since we're on Greenfeld, I'll put St. Chris to its left. The Helvetican Golden Starburst came with no explanation, but it's likely for winning the battle that Harry's son died in, so let's put it lower left, and maybe no one will notice it. That put the Order of the Rose and Thistle from the kitties of Bizalt close to the buttons. Are we missing anything?\"\n\n\"Only time,\" the ensign said, starting to affix the golden stars to Kris's dress white coat while Kris got into her pants and shoes. Kris would never have gotten the gold buttons on without Meg's help, or the awards around her neck.\n\nDone, Kris asked, \"Where's a mirror?\" and one appeared almost before her eyes. It took Kris a dozen steps to get close enough to check herself out. She imagined a lot of the ship's company was enjoying the space they'd gotten today. Still, you could get too much of a good thing.\n\n\"The Emperor and his entourage are disembarking from the space elevator,\" Nelly informed Kris.\n\n\"And I'm ready to face him,\" Kris said, breathlessly. She was grateful the quarterdeck was just outside her quarters' door, though in the new, expanded _P. Royal_ that was still quite a hike.\n\n\"Meg, go find out where we're meeting in the Forward Lounge,\" Kris said, as they marched for the door.\n\n\"I know where it is,\" Nelly said.\n\n\"Okay, Ensign, stay handy in case I need you.\"\n\n\"Aye, aye, Admiral.\"\n\nOn the quarterdeck, Kris's Navy band was tuning up; it looked quite diplomatic in a spiffy kind of way. Eight sideboys were deployed, ready to render honors. Kris might have her own doubts about Harry's lack of honor, but he was going to be full of it by the time she pumped all this into him.\n\nOr not.\n\nA worried look on his face, Jack waited for Kris as she took measured steps up to his place on the quarterdeck.\n\n\"How are things?\" she asked.\n\n\"There are way too many bots waiting outside the Smart Metal covering we've got up blocking access to the quarterdeck.\"\n\n\"I take it you've tried killing them or suborning them.\"\n\n\"Nelly and Sal have been working full-time with the Marine and Navy tech. We kill off two, three more arrive. We suborn five, eight show up.\"\n\n\"You've got a Marine Honor Guard out there for the Emperor to review. Are you going to let Major Henderson take the escort honor?\"\n\n\"No, I have to do it, but you stay here behind the clear hatch covering. He can see you, and none of that crap gets near you.\"\n\nKris didn't like that, and her face, no doubt, showed it.\n\n\"Admiral, I will throw you over my shoulder and lock you up in your quarters if I have to.\"\n\nKris scowled and glanced down at all the fruit salad. \"It would make a mess of my uniform.\"\n\n\"And, no doubt, I'd be gouged by all those prickly things. Good God, woman, where did they all come from?\"\n\nKris glanced at all the salad on Jack's chest. \"You're missing one. Meg, did a Star Leaper thing come in today's mail for Jack?\"\n\nThe young woman took off at a gallop.\n\n\"Star what?\" Jack asked.\n\n\"The Order of St. Christopher, Star Leaper. Harry awarded it to all the folks that survived the trip around the galaxy. Or not. Most got it posthumously.\"\n\n\"Maybe they forgot I was there.\"\n\n\"Maybe. What's taking Harry so long?\"\n\n\"I have no idea. I got from the station to the quarterdeck in ten minutes, and I was fighting bots all the way.\"\n\nCaptain Ajax arrived to greet the Emperor. She took one look at Kris and whistled. \"I'd heard that the King considered you his fightingest admiral, but I've never understood why. Did you get all that fruit salad at the produce section?\"\n\n\"Catch me some evening when I'm feeling mellow, and I may tell you the stories. Tell you the ones that aren't classified.\"\n\n\"I'll do that. By the way, the Emperor's procession is making its way up A deck. They've got him in some kind of electric cart. We sent a nano out to get a good picture of it.\n\n\"Captain, we've got a feed coming in,\" the Officer of the Deck announced.\n\n\"Put it on the nearest screen.\"\n\n\"There is no near screen,\" Nelly pointed out. \"I've got it coming up on the bulkhead beside the hatch.\"\n\nA bit of bulkhead became a screen, showing a confused line of Imperial Guards, police, station security, and half the dogs and cats on the station. In the middle of them was an enclosed electric cart. The clear and likely armored plastic looked like a recent addition.\n\n\"So even Harry himself doesn't feel safe on his own station,\" Jack observed.\n\n\"Interesting, that he's afraid of the horde of bots filling the air.\"\n\n\"Kris, we're identifying a whole new batch of nanos. They're tiny, and they don't belong to us,\" Nelly reported. \"I've tried to suborn them, but they are proving rather impossible to establish contact with.\"\n\n\"The Empress comes to call, and suddenly we're facing the worst infestation of nasty and small critters that we've seen,\" Kris observed.\n\n\"How do we handle it?\" Captain Ajax asked.\n\nJack stared through the hatch, worrying his lower lip. \"Captain, how much spare air do we have?\"\n\n\"Let me get you that,\" Ajax said, raising her wrist unit to her mouth.\n\n\"We've got enough air to empty the quarterdeck five times,\" Nelly provided.\n\n\"We can't subject the Emperor and Empress to vacuum, Jack,\" Kris pointed out.\n\n\"Not vacuum,\" Jack answered, \"but even they can survive a bit of wind. How about wind and infrared.\"\n\n\"Boil the tiny things and blow them out,\" Kris said.\n\n\"Exactly,\" Jack answered.\n\nKris turned to Captain Ajax. \"Ma'am, would you please suck as much air from the outside as you can manage to strain, then superheat it. I'd suggest you keep it separate from the air you now have in storage. We could use the new stuff first, then do the final flushes with the air we brought.\"\n\n\"Good idea, Admiral, and Nelly, thank you for having the information at your fingertips.\"\n\n\"You're welcome, Captain. I'm always glad when I can help you humans make good snap decisions.\"\n\nA quick call to the XO got that process started.\n\nWhen Ajax finished, she cleared her throat. \"I don't mean to be asking questions above my pay grade, but I understand that we've got interceptors out there. Why haven't we ordered them to destroy the offending nanos?\"\n\n\"A very good question, Captain. Thanks for raising it,\" Jack said. \"I haven't ordered war on the station nanos because they are on the station. That is not Wardhaven territory. Also, if we start a war now, who knows what reinforcements they could bring in. No. Let's bring Harry and his harridan aboard, isolate them in yonder air lock, sanitize them for all they're worth, then turn loose the interceptors to get what's left.\"\n\n\"Right. Politics,\" the captain said.\n\n\"It's always politics,\" Kris said drolly. \"Jack, how are we fixed for nanos?\"\n\n\"With the ship blown up way more than you'd ever dare do a balloon, I'll have to be careful just where Nelly grabs Smart Metal to make nanos. We've already taken down some of the walls in Marine country. A lot of the privates now have one huge barracks bay. We've promised them their walls back as soon as we get rid of Harry. Oh, Kris, if you decide to take the Emperor through Marine country, go to Enlisted Quarters 1. We kept one normal. Those other bays really look silly.\"\n\n\"And might be mistaken for space to transport an invasion force,\" Kris observed dryly.\n\n\"Yeah, that, too,\" Jack said.\n\n\"If you need more Smart Metal, pull it out of the wall between our day and night quarters.\"\n\n\"I would suggest not,\" Jack put in quickly.\n\nKris gave him a sour glower.\n\n\"I would think you're very likely to take the Peterwalds through our quarters.\"\n\nKris winced. \"Right. Good catch, my wise husband.\"\n\n\"We'll take it out of the nannies' space,\" Nelly said.\n\n\"Okay, Nelly. Give them my apologies and explain what we're doing in case we have to do it real sudden-like, and someone's in the shower.\"\n\n\"I'm doing it now, Kris.\"\n\nJack glanced back at the band. It was turned up, and its commander looked ready to parade it across the quarterdeck and down onto the pier to take up its place next to the honor guard. Jack shook his head.\n\n\"We've already got too much nano bait on the pier. Captain, would you mind telling your band leader he's to honor our fine guests from here on the quarterdeck.\"\n\nCaptain Ajax eyed the pier, glanced back at the band, then went to see that they kept themselves behind the quarterdeck hatch cover. The band leader didn't look at all happy, but he obeyed his orders.\n\nOn the quarterdeck screen, the Emperor's procession arrived at the area above the _Princess Royal_ 's pier. There, it came to a roaring halt.\n\nNo matter what your rank, you can't ride an electric cart down an escalator. _Looks like I'm not the only one making allowances today,_ Kris thought.\n\nThere was a lot of shouting and carrying on until several freight elevators were located. Then there was more shouting and thumping of chests as they figured out who rode down on the elevator with the Emperor. It was worse deciding who got to ride down on the other two elevators and who got stuck hustling for the escalator, then struggling to find their place once more in the parade.\n\nIt was a mess.\n\nJack saluted Kris, then passed through the air lock and made his way down the gangplank to stand at the head of his Marines and make the official greetings. He glanced around, got a thoughtful look, then marched the honor guard closer to the escalator.\n\nThen things got crazier.\n\nEmperor Henry I refused to dismount his cart to take the inspection of the honor guard.\n\nJack made a snap decision. He saluted the Emperor and Empress in their cart and said on Nelly Net, RUFFLES AND FLOURISHES, IF YOU PLEASE.\n\nKris nodded. Captain Ajax gave the band leader his cue, and, with a shrug, his band belted out the proper honors. That was followed by the Imperial Anthem, a quite bellicose march.\n\nMusical honors rendered, Jack stepped forward and escorted Emperor, Empress, and cart down the forward rank of Marines.\n\nThen he did a smart turn and led cart and occupants right up the brow, onto the quarterdeck, and into an enlarged air lock that Nelly had just expanded so that Jack, Emperor, Empress, and their cart fit in it quite cozily.\n\nThe wall closed behind them just as Jack thought, RUFFLES AND FLOURISHES AGAIN, PLEASE, AND THAT MARCH, EVEN LOUDER.\n\nThoroughly puzzled, the band did as it was told, slamming into the garish march, this time with added blaring trumpets, hammering drums, and clashing cymbals.\n\nNELLY, WHAT'S GOING ON?\n\nKRIS, WE'RE GOING TO PLAY THAT NOISE FOR HOWEVER LONG IT TAKES US TO STOMP ON ALL THE NANOS THAT TRIED TO SLIP ABOARD ON THE IMPERIAL COATTAILS.\n\nVERY GOOD, NELLY.\n\nMeanwhile, everyone stood stiffly at attention, saluting. The Imperial Guards were not happy at what Jack had pulled off on them. They finally organized themselves to march up the brow. There, their officers found they could do nothing more than salute from outside the barrier, while staring daggers at anyone who would meet their eyes.\n\nKris was careful not to.\n\nIn the air lock, the Emperor asked Jack about the delay. He explained the problem with the nanos and asked the Imperials' indulgence. Nelly kept Kris up to date on what was going on.\n\nNELLY, HOW IS THE EMPRESS TAKING ALL THIS?\n\nNOT WELL, KRIS. SAL THINKS SHE'S STEAMING, BUT IT'S KIND OF HARD TO TELL.\n\nThe Emperor got curious about the substance of the air lock.\n\n\"It's transparent Smart Metal,\" Jack explained, then had to help Harry out of his cart so he could touch the stuff. He rapped his knuckles on it and invited his Empress to come take a look.\n\nShe smiled coyly at Jack and asked for his help.\n\nKris had been eyeing her dress. The bottom was flounced out to fill up most of the cart. The bodice was a lot less.\n\nA whole lot less.\n\nAs Jack helped the Empress from her cart, she gave Kris an evil smile . . . and stuck her practically bare boob in Jack's face.\n\nYOU ENJOYING YOURSELF?\n\nNOT REALLY, KRIS. THERE'S ENOUGH SILICONE IN THERE TO JUST ABOUT KNOCK ME OUT.\n\nKris shook her head. THE MEN WILL BE MISSING HALF THEIR IQ.\n\nWith a sigh and an effort, Kris pulled her mind out of the catfight it so wanted to get into and prepared to meet the Emperor and Empress.\n\n# 37\n\nThe Emperor eyed Kris. \"Transparent Smart Metal; what will you Longknifes think of next?\" he said through the airlock.\n\nNELLY?\n\nTWO NANOS ARE STILL ELUDING US. GO AHEAD AND BRING THEM ABOARD.\n\nThe air lock vanished around them. Kris saluted and stepped forward, concluding Harry would prefer her smile to eight sideboys.\n\n\"Welcome aboard, Your Imperial Majesty,\" she said, then, holding her salute, turned to face the Empress. \"Your Imperial Highness.\"\n\nKris dropped her salute and the loud racket from the band cut off.\n\n\"Why are you separating us from our Imperial Guard?\" the Empress demanded.\n\n\"Ah. Yes. Just what is going on?\" the Emperor asked.\n\n\"You'll pardon us, Your Imperial Majesty,\" Kris said, \"but we have discovered that your station is suffering an infestation of bots and nanos. Do you know of this problem?\"\n\n\"No,\" the Empress huffed.\n\n\"Yes, dear. Remember, that is why we were confined to that damnable little cart. They wanted to protect us from whatever it is that's buzzing about. Still, I must have my guards,\" Harry went on.\n\n\"And you shall, Your Imperial Majesty. However, could you limit the number to the essentials and have the others wait outside? Then, if those you want could hurry aboard quickly so that we only have to keep the outer air-lock hatch open for a brief spell . . .\" Kris hoped he wouldn't press the question further.\n\nSo, of course, he did.\n\n\"Some of these pesky things are bound to get aboard, though. Some may already have gotten in when you opened that thing to let us in. By the way, how did you do that?\" he said, pointing at the clear Smart MetalTM of the partition across the open hatch of the quarterdeck.\n\n\"My computer is able to make that, and, as you saw, create an air lock when we need it.\"\n\n\"That must be nice, and clear, no less.\"\n\n\"Yes, sir,\" Kris said, hoping Harry had forgotten about his first question.\n\n\"So,\" he said, turning to Kris, \"how will you keep those buggers from infesting your ship?\"\n\n\"As you can see, Your Imperial Majesty, the quarterdeck is isolated by more clear partitions at all passageways into the ship. We will board as many of your subjects as you require into an enlarged air lock. Once they are here, we will subject that space to weak infrared light that doesn't bother us but does effect tiny things. Then we will run a major ventilation through that lock. We expect it will blow the little things right back where they came from. Bots and things are challenging devices, but they don't have the power to tack long against the wind.\"\n\nAt least these bots couldn't. Kris had been offered a batch of nano prototypes superpowered by the strange plant from Alwa. She'd declined to take them into the belly of the beast. If she lost one, it might or might not be retroengineered. You could never tell how good Greenfeld technology was.\n\nThe Emperor considered Kris's answer and seemed happy with it.\n\nThe Empress wanted to know more. \"And what if all the pesky buggers aren't blown away?\"\n\n\"I have Marines prepared to go after them . . . with butterfly nets,\" Kris said.\n\nThat drew a laugh from the Emperor and several of the unused sideboys and other Sailors standing by for the OOD's orders. A scowl from the JOOD, who this watch was a grizzled old chief, and the only laughter belonged to the Emperor.\n\n\"Very good. So very good. Well, you must tell the general commanding the Imperial Guard that he and two dozen of his best men may protect us. We guess we'll have to let all our advisors in. Love, I know you'll want to include some of your advisors.\"\n\n\"All of them,\" brooked no alternative.\n\n\"That will make for five, ten . . . Oh, ask my prime minister who we need.\"\n\nIn five minutes, twenty-five guardsmen and twice as many advisors stood on the brow. Half the guardsmen were first, prepared to charge to the immediate defense of their Emperor. The other half were at the rear, ready to put speed on any slowpokes and provide a wall against those still arguing that they needed to be included.\n\nA chief bosun's mate's pipe announced, \"All hands, prepare to receive visitors. Civilians on the gangway, shake a leg.\"\n\nKris noticed an increase in air pressure; nanos would have a hard time flying against that wind. Harry eyed the clear partition. His eyes grew big, as a much larger air lock, one that was comfortable enough to hold all his guards and advisors, appeared.\n\nThe Imperial Guard charged aboard at the double. The civilians made their way aboard, squawking like a bunch of old chickens and hurried along by the last of the guardsmen.\n\nThose who raced along to get aboard if there was the slightest chance found themselves staring into the blank faces of Royal US Marines with rifles at port arms.\n\nThat ended that noise.\n\nThe chief bosun piped again. \"All hands on the quarterdeck stand at ease while we take care of a minor problem.\"\n\nRed lights came on. Kris wondered what other wavelengths were bathing the quarterdeck, but said nothing. The pressure on her ears increased slightly, and she could hear a whooshing sound as air poured into the air lock and leaked out through small holes in the partition. Five different times the air lock went through a complete change in its atmosphere. Five times there was a wait, encouraging any tiny rider holding on to clothes, hair, whatever, to drift away.\n\nKRIS, THE AIR SWAPS ARE DONE. WE COULD USE SOME TIME TO LET OUR NANOS ATTACK ANY THAT ARE STILL HERE.\n\nI'LL SEE WHAT I CAN DO, Kris thought, and, while the seventy-five still in the clear air lock made impatient faces, she turned to the Emperor.\n\n\"So, Your Imperial Majesty, what would you like to see during your visit to the _Princess Royal_?\"\n\n\"Is there any chance we and some of our advisors might see your engineering spaces?\" he said. \"We've been told you no longer use magnetohydrodynamic generators to derive your ship's electricity.\"\n\n\"That is hardly something I know anything about,\" Kris said with as much of the airhead as she could muster. Wardhaven's ships were now generating electricity directly from the reactor's fusion plasma. It gave them a major advantage in combat and was something she had no intention of letting the Peterwalds get a look at.\n\nThat Harry had not asked about the improved power generators the US had gotten from the Iteeche spoke volumes for how much Greenfeld had fallen behind in the spying game. More reason to call the _Princess Royal_ a frigate and make sure no one got a good look at how many lasers this battlecruiser carried.\n\n\"The Grand Salon where we will be meeting to discuss what you called me here for is only two decks down,\" Kris said, renaming the Forward Lounge. \"This deck not only has the quarterdeck where we now stand, but also Admiral's Country. That includes our quarters and the nursery and quarters for the nannies we took aboard before we left Wardhaven.\"\n\n\"Nursery?\" Harry said. \"Nannies? This must be very recent. I heard nothing of such a thing.\"\n\n\"Yes, we and your lovely wife share the great joy of having recently presented our husbands with children. We brought our daughter with us. I'd hate to be separated from Ruth.\"\n\n\"I am so proud of my newborn son, Henry Smythe-Peterwald the XV,\" the Emperor said, and gave the Empress a loving hug. She kissed the side of his neck. \"That's wonderful. You have a girl. I have a son. We might do a bit of matchmaking. Arrange an alliance around our children.\"\n\nA look of rage and horror flitted across the Empress's face, but she immediately suppressed her visage to bland, and said to Kris. \"You must be very proud.\"\n\n\"I never suspected the joys a woman might know before Ruth came into my life,\" Kris said. \"Now, may I offer you refreshments in my quarters while everyone sorts themselves out?\"\n\n\"Why thank you, we'd appreciate a drink,\" the Emperor said.\n\n\"And we might see the baby?\" the Empress added.\n\n\"Yes, yes. Of course, dear.\"\n\nSo they adjourned to Kris's day quarters.\n\nWhich had grown even more opulent since she had left less than an hour ago.\n\nNow the bulkheads were polished woods in delicate patterns that formed pictures of Wardhaven's hills, seas, and farmland. The wood shared the walls\u2014Kris could hardly think of anything this extravagant as a respectable Navy bulkhead\u2014with velvet red wallpaper covered in golden _fleur-de-lis_. The burgundy carpet under their feet was lush. They sank into it much as Kris remembered sinking into the mud of Olympia.\n\n_Don't go there. There be dragons . . . and memories of Harry here trying to drown me._\n\nThe conference table and chairs, as well as Kris's desk, had disappeared, some, no doubt, into nano interceptors. There was a conversation square, two very overstuffed chairs and a similar number of couches, in front of a roaring fire.\n\n\"You have a fireplace aboard your ship?\" the Empress asked.\n\n\"It reminds me of home,\" Kris lied. It reminded her of a borrowed cabin where she and Jack had finally managed to discover the pleasure of each other's touch. \"It helps me think,\" Kris finished.\n\nAnd the imp in her came up with a really bad idea.\n\n\"If you'd prefer to eliminate the fireplace, Nelly, remove it,\" and the fireplace melted into the wall, to be replaced with more wood paneling.\n\nHarry about dropped his uppers.\n\n\"Oh no,\" the Empress said, not at all thrown by the technical marvels before them. \"Harry and I love a good fire, don't we, honey?\"\n\n\"Of course, love,\" the boggled Emperor managed to get out.\n\n\"Nelly,\" was all Kris had to say before the cheery fireplace flowed back into place.\n\n\"Also, Nelly, there are only the four of us. We hardly need couches.\"\n\n\"Yes, Your Highness,\" Nelly said, and quickly there were four large overstuffed chairs and the rectangular coffee table between them shrank into a square.\n\n\"Your computer, Nelly?\" the Empress said, \"is quite intuitive.\"\n\n\"Nelly is quite unique among computers,\" Kris said, leaving out Jack's Sal for the moment.\n\nKRIS, I'VE MANAGED TO KILL THE HANDFUL OF NANOS THAT GOT INTO YOUR QUARTERS WITH HARRY. WE'RE STILL CHASING A FEW ON THE QUARTERDECK. PLEASE STALL.\n\n\"What may we offer you?\" Kris asked the Emperor. \"Tea, coffee, something stronger?\"\n\n\"I'll have a Manhattan,\" the Emperor said. \"A double please.\"\n\n\"I'd love a strawberry daiquiri,\" the Empress said. \"It's so nice to be able to drink again.\"\n\n\"I'll have iced tea,\" Kris said, and Jack asked for the same.\n\n\"I don't have a wet bar,\" Kris admitted, \"but the artists of the Grand Salon should have them ready in a moment. So, how was your trip up the space elevator?\" Kris asked, space offering little chance for small talk about its weather.\n\n\"Not really a bother,\" the Emperor said. \"I so rarely get out of the Palace.\"\n\n\"It's a lovely day in Anhalt,\" the Empress added. \"So pleasant for a drive. We really must thank you for inviting us up here.\"\n\nKris had to struggle to keep her face bland as the Empress put a new turn to events.\n\nAt that moment, their drinks appeared on the right arm of most of their chairs. Since Harry had his elbow on that arm, his came out on his left.\n\n\"Damn!\" escaped the Emperor's lips. \"This stuff _is_ damn near magic.\"\n\n\"You must pardon my computer. We are still experimenting with new things we can do with Smart Metal. We must admit that we have never seen something like this.\"\n\n\"I wanted to get the drinks here before the ice had any chance to melt, Your Imperial Majesty,\" Nelly said, then added to Kris, AND WE'RE STILL CHASING A FEW PESKY NANOS ON THE QUARTERDECK. IF A WAITER HAD TO BRING THE DRINKS, ONE OF THEM MIGHT HAVE GOTTEN OUT.\n\nWELL DONE, NELLY. I THINK YOU REALLY IMPRESSED HARRY.\n\nHE BETTER BE IMPRESSED. HE'S GETTING THE FULL-COURT PRESS.\n\nThey sipped their drinks and talked of inanities. Then the Empress asked if she might see the baby. Thus, Ruthie was called for.\n\nA moment later, the infant appeared in the capable arms of Fede Radko, with Li O'Malley walking sternly by her side looking every bit the Gunny, ready to make sure that the world was right.\n\nAnd woe be to him who dared upset her world.\n\nFede handed Ruth to Kris first, and Kris realized this was a horrible idea. A baby and dress whites, complete with medals and decorations, were never intended to mix. Kris always held Ruth with her left arm. Today, she cradled her in her right. That kept the little one away from being poked by this order or that.\n\nStill, Ruthie's eyes lit up at the sight of all those dangling sparklies at Mommy's neck. Her pudgy fingers were just learning to reach, and she managed to grab the Almar Medal of Highest Valor and pull it right into her mouth. She slobbered happily as she teethed on it.\n\n_Well, at least she didn't start pounding on my breasts to be fed._\n\nKris let Ruth chew away happily on metal brought from far across the galaxy, then said, \"Okay, Gunny O'Malley, what do we trade her for my medal?\"\n\nThe retired chief warrant quickly produced a teething ring, and while Fede carefully removed the baby from Kris's arms, O'Malley slipped the teething ring into Ruth's mouth and the award out before Kris was pulled away by its ribbon.\n\n\"Would you like to hold her?\" Kris asked the Empress.\n\nShe glanced down at her exposed breasts with a knowing smile. \"I don't think my gown would survive the attention of such strong hands that weren't my loving Emperor's.\"\n\nThat ended with her sharing a smile with the Emperor that left him gobsmacked and looking in need of a room.\n\n\"Your advisors are waiting for you in the Grand Salon,\" Nelly announced.\n\n\"Then I guess we should be going,\" the Emperor said.\n\nAnd they went.\n\n# 38\n\nThe short walk down two decks was not without its comments.\n\n\"You have done magnificent things with Smart Metal,\" the Emperor said as he eyed the statues and paintings that surrounded them as they crossed from one wide staircase to the next.\n\n\"This is Marine country, isn't it, Jack?\" Kris said.\n\n\"Yes. Each Marine has his or her own stateroom. There's a movie stadium, bowling alley, rifle range, and exercise room that would beat any health club from here to Earth, with a three-kilometer track that circles the hull.\"\n\n\"You let your Marines live in such splendor,\" Harry marveled. \"The commander of my Imperial Guard is always after me to provide better quarters and places the troops can relax and be entertained. Please don't give him any idea what you've done.\"\n\n\"Certainly not, Your Imperial Majesty.\"\n\nThey passed four Marines, rifles at present arms. Two were women with sharpshooter's badges.\n\n\"I keep forgetting,\" the Empress said. \"You Longknifes give your women a free rein to attempt any job that they think they might succeed at.\"\n\n\"Yes, Your Imperial Highness,\" Kris said, wishing things were half as good in Wardhaven space as the Peterwalds gave them credit for.\n\n\"Do you aspire to the kingship?\" was said evenly, with a slight rise of an exquisitely made-up eye.\n\nKris paused as if to give that question serious thought.\n\n_Hell no, I don't want the job._\n\nBut she answered the Empress with a vague, \"I haven't actually read the final signed version of the Constitution. I've been rather busy elsewhere, you know.\"\n\nThat got a nod from both of the Imperials.\n\n\"But I understand that when Grampa Ray's twenty-five years as King are up, we Longknifes are barred from all political offices, including the purple.\"\n\n\"I find that hard to believe,\" Harry said.\n\n\"But I heard that was the intent,\" Kris answered.\n\nThe Empress said nothing, but for a moment she lost the placid face she wore. What flitted across her face left Kris uneasy at best. Scared at the worst.\n\nAnd on those thoughts, they were down the stairs.\n\nSomeone had been making improvements since Kris was last here. Candelabras and chandeliers now gave the rooms a soft glow, reflected back by the mirrors. The circular bar in the center of the Grand Salon looked like it was made of marble with a bronze rail for the feet. Behind the bar were more mirrors, pictures of nudes, or arrangements of bottles.\n\nClassy.\n\nEach of the divided spaces had a different motif. One was red and gold, another blue and silver. The room Nelly directed them to was a cold and severe white. Lush white carpet, white, patterned wallpaper, stark white cloisonn\u00e9 tables and chairs. Here, the mirrors were edged in chrome.\n\nThe tables and chairs had been arranged loosely in a square around four large, comfortable armchairs. They alone added color: two were red, the other two blue.\n\nTHE RED ARE FOR THE IMPERIALS.\n\nKris ushered the Emperor and Empress to their chairs. Their lackeys immediately jumped to assure that their raiment was arranged properly, and they were totally comfortable.\n\nTwo drinks, identical to the ones ordered in Kris's quarters, immediately appeared out of the chairs' arms.\n\n\"I hope I did not presume,\" Kris said.\n\n\"No. No. This is perfect,\" Harry agreed.\n\nKris and Jack took their places. To Kris's and Jack's back were two corners of the square, tables drawn up two rows deep. Her main staff of advisors had taken their places. Farther back, the lesser specialists sat in comfortable chairs closer to the walls. There was no fuss or bother as to who belonged where.\n\nOn the other side, behind the Imperials, it was much more unclear. Many of the men in fancy doublets, breeches, and hose also carried swords and, at least for a while, it looked like duels might be fought for seats at the table. In the end, the left half of the space, that behind the Emperor, went to people in fancy dress in every color of the rainbow. The right half, behind the Empress, filled up with men in more somber clothing, darker blues, purples, and blacks. They were still in doublets and hose with befeathered hats. Oh, and swords.\n\nKris wondered if she should have added her Navy-issue automatic to her dress today. She hadn't forgotten her spidersilk underarmor.\n\nAnd Jack had plenty of Marines at the ready. Although only twenty-five stood against the wall, balancing the same number of Imperial Guardsman, her backup was a lot closer than the Imperials.\n\n_There will_ not _be any problems today._\n\nAfter the commotion died down, Kris waited for Emperor Henry I to say something. Instead, he just sipped his Manhattan. The Empress sipped her daiquiri. Kris took a taste of her tea and put it back on the arm of the chair.\n\nHOW LONG ARE YOU GOING TO WAIT? Jack asked on Nelly Net.\n\nNOT ANY LONGER THAN THIS.\n\nKris leaned forward and cleared her throat. The two Imperials continued to lean back, sipping. \"You called us here, Your Imperial Majesty. May we ask what you require of us?\"\n\nThe Emperor of everything that resided under the Greenfeld flag . . . maybe less . . . huffed out a sigh. \"My daughter, the Grand Duchess Victoria, seems to be in rebellion against me. I was hoping you, as a good friend to both of us, might get her to stop this nonsense and come home.\"\n\nKris noted the loss of the Imperial \"we.\" A father was speaking to her, not an Emperor.\n\n\"Really, I don't think it's the girl's fault,\" the Empress put in. \"I really think she's fallen in with a bad bunch who are urging her on to this, ah, defiance of her father.\"\n\nKris tested those words, and, as much as she didn't want to, found them sincere.\n\n\"How did all this happen?\" Kris asked.\n\nThe Empress cast her husband a glance. When he imperceptibly nodded, she began, \"I don't know if you've been keeping up to date with current Greenfeld events,\" she said, and paused for Kris to respond.\n\nKris chuckled. \"Actually, I haven't been keeping up on _any_ events this side of the galaxy.\"\n\nThe Empress considered that for a moment, almost allowed a frown to slip onto her face, then went on. \"Well, there has been much economic distress here. The economy has contracted. What started as minor economic adjustments on some planets have spiraled down to worse and worse. _Some_ elements have chosen to take advantage of _our_ trouble, and while Emperor Henry here and others have been as gentle as possible in their efforts to secure the safety of our many distraught people, some have taken to using these times for their own profit.\n\n\"Previously, when rebellion raised its ugly head, my gentle husband relied on State Security to properly chastise those involved and preserve the tranquility of the Empire. However, no doubt you may recall that State Security attempted a coup of its own and would have killed my loving Emperor while he was hunting on Birridas. You saved his life, and I thank you from the bottom of my heart.\" That was done with a hand to her heart that somehow managed not to put what little she wore into disarray.\n\n\"I don't know what I would do without my Harry in the center of my life.\" The Empress interrupted her story to send a kiss her husband's way, and he returned one to her.\n\nShe turned back to Kris with a loving smile on her face, sighed, and continued. \"The rooting out of the cancer in State Security left the place little more than a shell. Thus, when some elements incited people to riot, there were few options open for us to prevent the civil unrest.\"\n\n\"Who were these people inflaming your subjects?\" Kris asked.\n\n\"The Navy,\" the Empress spat. \"They have betrayed their sacred duty. They had been allowed to found their own colonies, supposedly retirement communities. We've found that to be a traitorous lie. Would you believe it, when we had to cut the budget and decommission ships, captains took their ships off to those same Navy colonies and joined in open rebellion.\"\n\nThe two Imperials exchanged angry glances.\n\n\"You know that my Vicky has spent the last four years with the Navy,\" Harry said. \"I thought they might instill some discipline in her. She was a most willful girl to start with.\"\n\nKris nodded. _Willful? How about murderous?_ Still, she said nothing.\n\n\"We think,\" the Empress went on, \"that some of those elements in the Navy brainwashed her into spouting their propaganda. When last I saw her, she was strutting around in a uniform. What was her supposed rank?\" she asked, glancing over her shoulder.\n\n\"Vice admiral,\" the field marshal commanding the Imperial Guard provided.\n\n\"Vice admiral, would you believe. She's hardly been in the Navy four years.\"\n\nKris had to admit that four years to vice admiral was cutting it mighty short. She'd taken five.\n\nShe did not tell Vicky's stepmother that.\n\n\"So what do you want me to do?\" Kris asked again, dropping her royal \"we\" since they had dropped theirs.\n\nNow the Emperor leaned forward earnestly in his seat. \"Go to her. Those rebels wouldn't dare fire on the US flag. Tell her I love her and all will be forgiven if she'll just abandon those traitorous elements and come home.\"\n\n\"I tried to bring her home,\" the Empress put in.\n\n\"I took a major fleet to St. Petersburg to get our Vicky back from their clutches. They had the audacity to resist me and even fire on some of the ships with me. I couldn't believe it. I would have gotten through to the ship she was being held on, but we received a message from my loving husband announcing that he wanted to bring you into the resolution of this problem, and I announced a general cease-fire. Badly outnumbered, the rebels were only too happy to grab this chance to withdraw and regroup. They still refused to let me have my daughter back. They refused me, her loving mother.\"\n\nKris nodded, doing her best to remain evenhanded and listen, really listen to the Empress. Still, in the back of her brain was a clear memory of Vicky's sharing the uptick in assassination attempts that started just about the time her stepmother announced she was pregnant with a boy.\n\nKris blinked. The silence after the Empress's last word was growing.\n\n\"Your Imperial Majesty, I will accept your charge. You mentioned that the Grand Duchess was last seen at St. Petersburg.\"\n\n\"Yes,\" the Empress provided.\n\n\"Then I and my squadron will depart for there immediately and do our best to retrieve your daughter from anyone holding her hostage. As a Longknife, I detest kidnappers.\"\n\nOf course, it was rumored that the man seated across from her had paid to kill Kris's grandmother Sarah and arrange the kidnapping of her brother, Eddy.\n\nWithout so much as blinking an eye, the Emperor said, \"I knew you would answer my call. Can you take a fleet out to St. Petersburg and bring back my Vicky?\"\n\n\"I will most certainly try, Your Imperial Majesty; however, I don't have a fleet,\" Kris pointed out.\n\n\"What do you have?\" the Empress asked.\n\n\"The squadron you see here, eight frigates.\"\n\n\"Good God, young woman, they will blow you to bits,\" the Empress put in. \"I saw the fleet they had. And the admirals organizing this rebellion are smart fighting men. We can't seem to find anyone of that high caliber.\"\n\n\"I've won a few battles of my own,\" Kris said, modestly.\n\n\"Well, if you think so,\" the Emperor said.\n\n\"As you mentioned before, Your Imperial Majesty, I fly the US flag. I do not expect to have to fight my way in or out.\"\n\n\"But how will you get your hands on her?\" the Empress demanded.\n\n\"All I need to do is get her onto US territory, and I will do it the same way there that I did here to bring you to US territory.\"\n\n\"Oh,\" the Emperor said.\n\n\"Oh,\" the Empress said, glancing over Kris's shoulder at the Marines standing stiffly against the wall. A quick glance over her shoulder and the face that came back to face Kris was most thoughtful.\n\n\"I am very grateful for your coming,\" Kris said, standing. \"Now, if you will return to the station, I have a squadron I would like to get under way as soon as possible.\"\n\nIt took much less time for the Imperial mob to exit the _Princess Royal_ than it had to get them aboard. As they gathered on the quarterdeck, Nelly formed an air lock around them and sent them on their way with only a minimal loss of air.\n\nThe returning Marine guards quickly paraded into the now-empty air lock and began their decontamination.\n\nFour hours later, the squadron was in its last preparations to close locks and depart. Several large containers were towed up to the seven other ships of the squadron. It seemed that Nelly had indeed arranged for the delivery of certain libations.\n\n\"Ma'am?\" Captain Ajax asked.\n\n\"Nelly, are those loads paid for?\"\n\n\"The requisition chit has been submitted to the Palace. If it is to be audited, it won't happen until we are well away from the pier.\"\n\n\"Jack?\"\n\n\"You've gotten away with treating Sailors like grown-ups so far. Why not try it again? You will, no doubt, maintain the limit of no more than two drinks or beers an evening.\"\n\n\"And if anyone renders themselves unfit for duty, I will reinstate keelhauling and see how long they can hold their breath.\"\n\n\"Keelhauling?\" came from Captain Ajax.\n\n\"Nelly, you explain it to the captain when you have a chance. Get the beer trucks taken aboard first. Make sure they're decontaminated.\"\n\n\"How long do we keep this love-boat configuration?\" Captain Ajax asked.\n\n\"No later than when we come within double 18-inch-laser range of a jump. We still go through the jumps at Condition Zed and battle stations.\"\n\n\"Aye, aye, Admiral.\n\nWith no further problems, they headed out to get Vicky's side of the story.\n\n# 39\n\nJump after jump, as they made their way out from Greenfeld, passed with no surprises. They seemed to have space all to themselves. When they jumped into the Pozen system, they were nervously hailed.\n\nKris personally took the comm to inform the planet that they would not approach them. \"We are US ships on a diplomatic mission. I am Princess Kris Longknife.\"\n\nThat seemed to mollify the station . . . and the battleship and two cruisers docked to it.\n\nA few jumps later, they entered the Presov system, only to find the light cruiser _Emden_ holding fifty thousand kilometers out from the jump.\n\n\"We have been detailed by the Grand Duchess to assure the safety of your last few jumps.\"\n\n\"We're deep into rebel territory,\" Kris pointed out. \"Can't you control your own shipping lanes?\"\n\n\"Usually, yes. We've blown up or captured quite a few of the so-called pirates. Still, the black-hearted Empress occasionally gets the odd ship deep into our territory by using the minor jumps and barren systems.\"\n\n\"Well, we are grateful for your help,\" Kris said. \"Nelly, cut the commlink.\" Then she turned to Jack. \"Well, what do you think of that?\"\n\n\"I'd say it puts a bit of a different twist on the Empress's opinion.\"\n\n\"The Grand Duchess can order ships to meet us. That doesn't sound like a hostage situation to me,\" Kris snapped.\n\n\"Let's hold our horses,\" Jack said. \"The skipper of a light cruiser may not know anything about what is taking place at the high-command levels.\"\n\nKris scowled. \"Yes. Yes, I know. Still, I can't see the Vicky I know letting anyone use her. Her use them? No question. Them use her? Nope. Not going to happen.\"\n\n\"I won't argue with you, wife. But I would suggest the admiral pass along that communication to her brain trust with a request to assess it and the Empress's side of the story to see what we can make of the two.\"\n\nKris sighed. \"Okay, my ever-cautious husband. I will refrain from my usual exercise of jumping to conclusions.\"\n\nShe slipped close to Jack, and whispered, \"But I would dearly like to jump your bones.\"\n\n\"Hon, I'm scheduled to lead my Marines in a run. Every time we expand amidships back to huge, we Marines get our track back. It's exactly three point two seven kilometers around the waist of the ship.\"\n\n\"You could get your exercise jumping my bones,\" Kris pouted.\n\n\"Or we could both get our exercise, first a ten kilometer run, then a roll in the hay.\"\n\n\"You really want me to work off my baby fat, don't you?\"\n\n\"Nelly, can you rerun my last few comments for Kris. Was baby fat mentioned by anyone but her?\"\n\n\"I assume that is a rhetorical question,\" the computer said.\n\n\"You win, husband, but you can lead from the front. I will pull up the rear where I shouldn't be too embarrassed by anyone's noticing my finishing time.\"\n\nA few more jumps, and Kris was almost able to finish right up with the rest of the Marine detachment. And with Ruth skimming the top off the fat she ate for her milk, Kris wasn't doing too badly tightening up her mommy body.\n\nStill, the entire squadron was at battle stations, ships at Condition Zed, as they made the jump into the St. Petersburg system.\n\nKris stood before the main screen. \"This is the US frigate _Princess Royal_ , and I am Admiral, Her Royal Highness Kristine of the United Society on a diplomatic mission. I have come to talk to Grand Duchess Victoria of Greenfeld.\"\n\nThe message went out at light speed. They would have a long wait before Vicky could reply . . . if she was permitted to.\n\nThe sensor crew quickly developed a picture of the St. Petersburg system. The station hardly had a vacant pier, what with freighters coming in and going out.\n\n\"There's a lot more trade going on than we saw around Greenfeld,\" the tech specialist advised. \"There are also a whole lot of warships. I've made out an even dozen battleships, another dozen cruisers, heavy and light, and twenty or so destroyers. I think some of the merchant ships are armed cruisers, but all the merchant ships are armed. It's hard to draw a line between armed merchant ship and ship loaded with heavy armament to make them merchant cruisers.\"\n\n\"No doubt we'll get a better picture once we're on approach to the station.\"\n\n\"Ma'am, several of those battleships are honking big dudes. One has sixteen 18-inch lasers and a whole lot of secondary guns.\"\n\n\"And to think, just a few years ago, we'd be worried about such a monster,\" Jack whispered to Kris.\n\nShe had to chuckle at how things had changed.\n\nKris met with her brain trust and found them still unwilling to make any commitments. She found she got more out of time with Ruth and her nannies than she got from those ninnies. Kris had paperwork to keep up with. She'd requested a report from all her captains about how they were handling sailing ships blown up like balloons and with their own public house aboard.\n\nEach ship had at least one chief who did not like the idea of booze aboard. However, it was universally agreed that the gripe came from the chief who operated the still in the goat locker, and since everyone else seemed to be getting along, the change in situation was deemed acceptable. Those who couldn't hold themselves to the two limit were few and given counseling. They were in line for more if the first session wasn't enough. Several ships had established a policy that those on probation for breaking the two limit would not be allowed in the bar without a buddy who was responsible for them.\n\nKris liked that policy and sent back a well-done to those ships that had established it and a suggestion to the others to do likewise.\n\nThe day turned out very busy, and Kris had just collapsed into bed with Jack, Ruth fed and a spare bottle of milk for her so Kris could sleep through the night. She was edging into Jack's arms and wondering if she was really exhausted or might she have a bit of reserves for her and Jack when Nelly announced, \"A message has arrived on the bridge from Vicky Peterwald.\"\n\nKris said a most unprincesslike word, which was seconded by Jack, and the two tumbled out of bed and hastily pulled on shipsuits.\n\n\"Nelly, have the message sent down to flag plot and advise the three heads of my brain trust to report to my quarters immediately, as in five minutes ago. You might tell them why.\"\n\n\"I have the message ready to play in your flag plot and all of your brain trust had left messages to alert them when a message came back. They're already on their way here.\"\n\nThat was one advantage of letting the ship balloon out. Flag Country and the brain trust were just about next door to each other. Jack opened the door, and Kris stepped through from her bedroom to her work.\n\n_It is, it is a truly glorious thing to be a pirate Princess._\n\nJudge Diana Frogmore was already there, waiting, though her attire was a bit informal, a tank top and gym shorts. \"I hadn't gone to bed yet, so it was easy to rush over.\"\n\nSenior Arbitrator Bill Gladsten arrived a moment later, in sweats. \"Am I late?\"\n\n\"Al's still missing,\" Diana told him.\n\nThe clock ticked off three minutes and Kris was edgy enough to start without him when Al Fu, the chief mediator, walked in wearing a dressing gown, a hair net, and showing where he'd missed some skin cream when he hastily wiped it off his face. \"Okay, okay, sorry to be late.\"\n\n\"Now we'll start,\" Kris said, but the door opened again, and Runda Fairsight trotted in, wearing light blue slacks and a badly wrinkled but brightly embroidered shirt that Kris remembered her wearing that morning.\n\n\"Sorry, we were looking at the data take from St. Petersburg. It's not nearly as encrypted as Greenfeld's. I think we're making up a good picture of the economy.\"\n\n\"Good, now, let's see what Vicky has to say, assuming she was allowed to talk to us at all,\" Kris said.\n\nThe screen lit up and there was Vicky in Navy blues showing the stripes of a vice admiral. \"Kris, I'm so glad to see you,\" she squeed. \"Boy, have I got dirt to dish with you. You wouldn't believe all the stuff I've done in what, the last two years is it?\"\n\nVicky paused to take a breath. \"I don't know how you've managed to do all you do, but I've tried my hand at a few things, always wondering what Kris Longknife would do. Not that I've turned into someone who's all for democracy and stuff. Though, this guy, you remember him, Mannie Artamus, the mayor of Sevastopol, he's kind of making it easier for me to think we need to get more people involved than just we Peterwalds in deciding what needs to be done.\"\n\nVicky's attention was drawn offscreen. \"Oh, Admiral von Mittleburg says I should tell you that I did a pretty good job myself when we were fighting off the Empress's fleet. Well, fighting off several of the Empress's fleets, but she was only out here for the last, big fight. Anyway, he wants me to tell you that my three stars aren't just window dressing. I earned them. Hard to believe that, huh? I wish Admiral Kr\u00e4tz were here to see me now. I think about what he'd do at least as often as I think about what you'd do.\"\n\n\"Does that woman always talk this much?\" Diana asked.\n\n\"And this fast?\" Al added.\n\n\"What's she been smoking?\" Bill wanted to know.\n\n\"I think she's just excited,\" Kris said, \"though I have to admit, I have never seen her on this kind of a jag. Still, remember, it's been the better part of two years since we've been together.\"\n\nOn the screen, Vicky was still talking.\n\n\"Kris, you wouldn't believe what we've done out here. There were planets that were breaking down. Total civil collapse, cannibalism might even have been in the cards. We've managed to get food out to those that were really bad off, and get trade going so planets could get their industrial sector back up and give people jobs.\"\n\nAgain, Vicky glanced offscreen. \"We have figured out how the Empress and her father's cronies managed to collapse everything. I thought it was cutting off credit, but there was more. Admiral von Mittleburg is shushing me. I don't think he wants that identified on an open channel. And maybe I should stop talking, but I've missed you, Kris. Really missed you. We so need to talk.\"\n\nVicky paused and got a fay grin on her face. \"I know how you got the Order of the Wounded Lion,\" she almost sang. \"Signing off for now, Kris, but hurry down here. You really have to see what we've done.\"\n\nThe screen went blank.\n\nThe six of them looked at each other.\n\n\"Vicky is something,\" Kris said.\n\n\"Something else,\" Diana added, again staying vague.\n\n\"Well, we now know that she can access the net,\" Bill said. \"She had an admiral of some sort with her, but I could see no gun at her head.\"\n\n\"She was talking too fast to be reading a script,\" Al observed. \"I don't know how much control she actually has, but she really believes that she has been significantly in command of her fate and of planets.\"\n\n\"Did she say that she could claim some of the credit for resisting several invasions?\" Kris asked.\n\n\"Yes she did,\" Nelly replied. \"Would you like me to replay those parts of the message?\n\n\"Do, Nelly,\" and the screen lit up and Vicky talked again.\n\n\"Yes, she was in on the fight,\" Jack said. \"Note that Admiral von Mittleburg, whoever he is, wanted her to take some credit for the victories.\"\n\n\"Which is not to say that she deserves such credit,\" Diana pointed out.\n\n\"Are you going to stay the doubting Thomas?\" Bill asked.\n\n\"Someone has to,\" Diana said. \"She may not have been waving her boobs around for every man to see, like the Empress did, but that girl is drop-dead gorgeous, and I don't trust you guys not to think with the wrong head.\"\n\n\"Diana, I'm a bit old to let a pretty face run away with my hardheaded logic,\" Al said.\n\n\"I'll believe it when I don't see it,\" Diana said.\n\n\"Have we done enough for the night?\" Kris said through a yawn.\n\n\"I think we'll have our work cut out for us in the morning,\" Bill said, \"but we're done here.\"\n\n\"Then shoo. Get out of my house,\" Kris said.\n\nThe others left, and she and Jack found themselves once more in bed.\n\n\"Jack, I'm sorry, but I don't have anything left for us.\"\n\n\"Don't be sorry, I'm running on empty, too.\"\n\n\"Will you at least hold me?\"\n\nJack rolled over, and she scrunched her way until her back rested against him. He put an arm over her. Kris breathed a soft sigh . . . and was out like a light.\n\n# 40\n\nKris sat as comfortably as she could in dress blues with all the required fruit salad. Surrounding her in her flag bridge was Jack, in full dress blue and reds, as well as her brain trust. Commander Ajax, on her bridge, brought the plus-sized _Princess Royal_ smoothly up to High St. Petersburg Station. The rest of the squadron followed at thousand-kilometer intervals at Condition Baker.\n\nAll hands had been advised that their ships might go to Condition Zed with no notice. Kris's lifesaving paranoia was once more bubbling to the surface. Still, the chance to put on a display of what her frigates could do was not to be passed up.\n\nAhead, there had been a lot of juggling of pier assignments. Without any request from Kris, eight piers had been opened up for her squadron, all next to each other.\n\nThe pier assignment for the _Princess Royal_ would have her sharing with that honking big battleship.\n\n\"Nelly, who are we tying up next to?\"\n\n\"That large battleship squawks as the Imperial battleship _Retribution_ , Kris.\"\n\n\"They still claim to be Imperial?\" Jack asked.\n\n\"All of them are squawking as Imperial ships, Jack. They have added a second line to their identification, though. They all report they are part of St. Petersburg division of the Greenfeld Imperial Navy Reserve Fleet.\n\n\"St. Petersburg division of the Greenfeld Imperial Navy Reserve Fleet?\" Jack echoed. \"That's quite a mouthful.\"\n\n\"Any comments?\" Kris asked her brain trust. They stared blankly back at her, contributing nothing.\n\nKris ran a worried hand through her hair. \"It does appear they are doing their level best to claim to be loyal subjects of their Emperor and part of the Greenfeld Imperial Navy,\" she said, breaking the silence.\n\n\"However, the Empress claims that the Imperial fleet is in rebellion,\" Jack pointed out. \"The words mean nothing. They could be a cover story to make the Sailors think they are still loyal.\"\n\nKris sighed. \"Agreed.\" She had thought that facing several gigantic alien mother ships was the worst of all possible fates. It seemed that humans with their ways of obfuscating a situation had come up with something worse. With an alien, you knew where you stood; they wanted you dead. You had to kill them.\n\nKris knew someone in the Greenfeld Empire wanted her dead. Those two battleships ambushing her at that jump told her that. The try at killing Jack and the nano assaults screamed something. Someone was out to kill her and hers, but who?\n\nEasy to know. Not so easy to figure it out.\n\n\"Do we know anything about that big battlewagon, the _Retribution_?\" Kris asked, hunting for something concrete.\n\n\"Not much,\" Nelly said. \"It's got those sixteen 18-inch lasers. Lots of secondary and tertiary guns. Oh, Kris, it just changed its squawk. They've unfurled a three-star flag, so to speak, and are reporting that they are the flagship of the Vice Admiral, the Imperial Grand Duchess, Her Grace, Victoria.\"\n\nThe room was silent for a long moment.\n\n\"I wish they'd done that a bit earlier, like from the time we first read their identification, friend or foe,\" Jack said.\n\nKris nodded. She could think of a dozen reasons for that late addition to their information. At least half of them were not good.\n\n\"We're coming alongside the pier now,\" Nelly reported.\n\nKris heard the solid \"thumps\" through the hull as Commander Ajax caught the four tie-downs and ran her bloated ship smoothly along the pier, giving up the last bit of momentum.\n\nThe 1MC reported a minute later, \"All hands secure from underway stations. Set the in-port watches.\"\n\n\"The brow is in place to the quarterdeck,\" Nelly reported.\n\nJack added to that a minute later, \"My Marines have established a pier-side watch with no problems.\n\n\"I have feed from the quarterdeck,\" Nelly reported.\n\n\"Put it on screens,\" Kris ordered.\n\nThe first screen showed a lieutenant commander standing as Officer of the Deck at the lip of the brow. Beside him, a young JG had on the JOOD brassard. He waited patiently, if a bit anxiously, for his senior to tell him what to do.\n\nKris remembered all too well having that duty not so long ago and smiled.\n\nA dozen Sailors stood at ease along the bulkhead, ready to render honors or serve as runners or guides.\n\nCommander Ajax ran a right and tight ship.\n\nA second camera covered the pier. At the foot of the brow, four Marines stood at ordered arms. Beside them, a sergeant, sword in hand, stood by to render honors.\n\n\"Their rifles loaded?\" Kris asked.\n\n\"Nope,\" Jack answered. \"Only the sergeant's automatic. They have orders to withdraw at the first sign of something going ass over teakettle.\"\n\nMore screens showed the go-down to the pier from the station's A deck. Nothing moved. A camera focused on the brow across the pier from them showed four Marines in Imperial black-and-green dress uniforms, at order arms with a sergeant, sword in hand, standing by. A mirror image of the _Princess Royal_ 's guard.\n\n\"Any nano problems?\" Kris asked.\n\n\"Not so much as a buzz,\" Nelly reported.\n\nKris glanced at her brain trust and raised an eyebrow. They remained as unmoved as statues.\n\n\"Do you think we ought to give her the full treatment, honor guard and all, like we did the Emperor?\" Jack asked.\n\nKris shook her head. \"I don't think Vicky wants all the fancy stuff. If I read her right, she just wants to talk.\"\n\n\"Good,\" Jack said. \"If the word gets back to the Empress, it will probably work better if we didn't.\"\n\n\"Kind of my thoughts, too.\"\n\nOne screen caught the sound of a bosun's pipe. \"Grand Duchess departing. St. Petersburg Reserve Fleet departing. Battle Squadron Two departing.\"\n\nThe camera focused on the other gangplank caught a young woman rendering honors and hurrying down the brow, followed by two older men. Between them, they carried enough gold braid to outfit a circus.\n\n\"I think we've got company coming,\" Kris said. \"Let's go, Jack.\"\n\n\"Can we come, too?\" Diana asked.\n\n\"Why don't you stay here,\" Kris said. \"I'll bring that herd of elephants to you.\"\n\nThat would cut down on the mob at the quarterdeck. Besides, Kris didn't want civilians messing up the time-honored dance of a well-ordered quarterdeck.\n\nStraightening her dress blue coat with all the spangles that Ruthie could ask to jam into her mouth, Kris left to see just what had become of the junior officer she once knew.\n\n# 41\n\nKris stepped directly from her day quarters to the quarterdeck. The band had just finish their tuning and was going to parade rest with their instruments. Captain Ajax and the OOD had their heads together, whispering earnestly. Kris joined them.\n\n\"How many sideboys do we give her?\" Ajax asked, bringing Kris into their dilemma.\n\n\"Eight,\" Kris said. \"She's the heir apparent to the Imperium. Alternately, she's the leader of a rebellion. That qualifies for eight in my book. Oh, and make them all girls if you will, Captain.\"\n\nA nod from his skipper, and the OOD went to assign eight women to sideboy duties.\n\nKris looked around the quarterdeck. \"Nelly, please add my flag next to the US ensign. Also, put a Greenfeld flag on the forward bulkhead of the quarterdeck, and when the bosun pipes Vicky aboard, make a three-star flag appear.\"\n\n\"Glad to,\" Nelly said, her voice holding pure delight, as a blue flag with four white stars appeared to the left of the US ensign.\n\nThe sidegirls paraded into their place beside the gangplank; Kris, Jack at her elbow, took her place across from Captain Ajax and her OOD and JOOD. With a deep breath, she prepared for what would come next.\n\nVicky was leading the charge up the brow. She took in the zoo on the quarterdeck with a glance . . . and blanched. She whispered something to the Vice Admiral trailing her. He only had to glance at what lay ahead of them before he smiled and took over the lead.\n\n_Someone doesn't know how to accept formal flag honors,_ Kris thought, and swallowed a smile.\n\nThe chief bosun's mate of the watch had been paying attention to how this pack of circus elephants had conducted their exit from the _Retribution_. Without batting an eyelash, he piped honors, and announced, \"St. Petersburg Reserve Fleet arriving.\"\n\nA vice admiral in dress blues turned to salute the Greenfeld Imperial ensign, then did an about-face and saluted the Royal US ensign. He held his salute as he turned to Kris and her collection of officers and stepped past Kris before turning again. Now he saluted his arriving Grand Duchess.\n\nVicky squared her shoulders and took her first step. Behind Kris, the band went into ruffles and flourishes. Four sets of them.\n\nVicky paused; her mouth hung open just a smidgen. Then she took a deep breath and marched to the top of the brow as ruffles and flourishes segued to the Imperial March.\n\nThe chief bosun's mate piped, \"Grand Duchess arriving.\" Vicky earnestly went through the same process as the first admiral, not missing a step, despite a tremble on her lower lip. Her eyes went wide as her three-star flag unfurled beside the Greenfeld Imperial flag.\n\nThe trailing rear admiral kept his distance so as to not step on Vicky's official greetings. That done, he marched forward to be piped aboard as \"Second Battle Squadron.\" Once he'd saluted everyone, the saluting was over and introductions began.\n\n\"Wow, Kris,\" Vicky said. \"Just wow.\"\n\n\"It's your due, Admiral,\" Kris pointed out.\n\n\"I guess it is, but . . . wow. Oh, right, I'm dropping the ball. Kris, I'd like you to meet Vice Admiral von Mittleburg, who was kind enough to give me refuge when I thought I was lost beyond hope. Beside him is Rear Admiral Boles\u0142aw. He's been my flag captain and saved my butt more times than I care to count.\"\n\nKris shook hands all around, and introduced Lieutenant General Juan Montoya and Captain Ajax before inviting her guests to join her in her in-port quarters.\n\n\"Right this way,\" and in a few steps they were there. Since the time Kris had left, Nelly had made it over into the lush digs that had greeted the Emperor and Empress.\n\n\"Nelly, my flag quarters do not need to look like an overpriced whorehouse for these people. They work for a living.\"\n\n\"Sorry, I thought you'd want to lay it on equal for her as we had to do for the Emperor and Empress.\"\n\n\"This was what Dad wanted?\" Vicky asked, looking around to take it all in. \"And how did you do all this, Nelly?\"\n\n\"Easily. We've discovered a lot of new things we can do with Smart Metal,\" Kris's computer answered.\n\n\"Down, girl, or should I say girls. Nelly, give us enough comfortable chairs for the mob I've got here and get rid of the statues, paintings, and woodwork.\"\n\nThey began to melt into the deck, bulkheads, and overhead.\n\n\"You might keep those nudes, Nelly,\" Vicky said with a chuckle. \"Is Kris really loosening up in her old age?\"\n\n\"I'll have you know, I'm an old married woman, with a kid to boot,\" Kris announced, not managing to keep the pride out of her voice.\n\n\"You've got a baby!\" Vicky squeed.\n\n\"You want to see her?\"\n\n\"Of course. I've always wanted to meet a Longknife I was smarter than.\"\n\n\"Don't be so sure,\" Kris tried to growl, but there was way too much smile on her face to carry it off.\n\nSo Ruth got brought out and passed between Kris and Vicky. The two admirals took two steps back when it looked like they might be offered the tiny human. Kris gave Jack the stink eye when he joined them.\n\n_Guys against the gals, huh,_ she thought, apparently too loudly.\n\nI DON'T WANT THEM TO FEEL UNSUPPORTED, Jack answered on Nelly Net.\n\nOnce Ruth was taken off to get an aromatic diaper changed, Kris introduced her brain trust.\n\n\"Do you think you can really figure a way out of this mess we're in?\" Vicky asked them, and got the same noncommittal mumbles Kris had had to put up with so far.\n\n\"Yeah,\" Vicky said into the silence, \"it's a mess. We've had quite a few years to get into it, and it's going to take quite a few years to get out of it.\"\n\nKris eyed Vicky. The Grand Duchess met her eye to eye.\n\n\"Yeah, I'm not the stupid little girl you left on High Chance, and Kris, I want to apologize for hanging you out to dry in that interview. I was alone. I was scared, and I wasn't thinking straight.\"\n\nKris canted her head to one side. \"I didn't see that coming.\"\n\nVicky nodded. \"I've done a lot of growing up . . . fast. It was either that, or my stepmother would have killed me.\"\n\n\"Can you prove that?\"\n\nVicky was shaking her head before Kris finished the question. \"Could you ever prove that my dad was trying to kill you?\" she shot back.\n\nKris winced. \"Some Peterwalds are very good at not leaving fingerprints. So, ignoring the personal for a moment, what can you tell me about the situation here in the Empire?\"\n\n\"What did they tell you?\"\n\nKris shook her head. \"This is your time to talk to me. I'm listening.\"\n\nVicky looked at Vice Admiral von Mittleburg. He nodded to her, and she took a deep breath.\n\n\"I knew things were crazy when Rear Admiral Gort picked me up on High Chance. He had three options for my delectable body. First\"\u2014a finger went up\u2014\"deliver me to my loving father at the Palace, as per his orders. Second\"\u2014and another finger went up\u2014\"he had been given a very large bribe to deliver my very lovely, but very dead body to the Palace. The last option was accompanied by a smaller bribe and involved handing me over to other people, unknown, to be used to raise a rebellion against my father.\"\n\nKris eyed the two admirals. \"Such as these two?\" she said.\n\n\"No! No,\" Vicky said, shaking her head emphatically. \"Whoever that group was, they've vanished. I really don't know a lot about the last two options because Admiral Gort took a bullet intended for me, and what he knew died with him.\"\n\nKris studied Vicky. The death of the admiral actually moved her.\n\n_Not at all the old Vicky I knew._\n\n\"We took the admiral's body home to Bayern, a Navy retirement colony where I was introduced to a batch of retired admirals and some wives who weighed the decision to kill me or send me back to the Palace.\"\n\n\"Not start a rebellion?\" Diana asked.\n\n\"No, Your Honor. None of us saw any chance to get a revolution going. Things were bad but not _that_ bad.\"\n\nVicky sighed. \"They sent me back to the Palace, kind of like a spy. I thought I might be able to find something out. Stupid me.\"\n\nVicky's eyes got lost in the distance. When she spoke again, it was softly. \"I dodged three or four assassination attempts a day for three, maybe four days before they got me. The Marine captain escorting me was killed. I was raped, but I got the drop on them and killed my kidnappers.\" For a moment, her eyes focused on Kris.\n\n\"You'd have been proud of me, killing my kidnappers and getting away before the Empress, or whoever, came to kill me. The Navy helped me get away from Greenfeld but had no idea what to do with me. Some thought I might hide out in a Navy colony, but my new Navy escort wasn't so sure someone wouldn't turn me in, not with the size of the price on my head. He thought I might hide out in one of the new US colonies, but I knew the money would make no place safe for me. The final option was to make a run to Wardhaven and throw myself on your King Ray's mercy.\"\n\nVicky shook her head. \"That would take me out of the game. The only way I'd be able to come back would be at the head of an army provided by your great-grandfather, and that stank.\"\n\nVicky shrugged. \"So I came to St. Petersburg and threw myself on the locals' mercy. It hasn't worked out too badly, has it?\" she said, eyeing the two admirals.\n\n\"It certainly hasn't been boring,\" Vice Admiral Boles\u0142aw said through a grin.\n\n\"It never is boring around us, is it, Kris?\"\n\n\"Nope,\" Kris agreed.\n\n\"The Empress tried a couple of attacks. I stopped one by driving her appointed commander into a heart attack,\" Vicky said with a chuckle, then glanced at the junior admiral beside her. \"Admiral Boles\u0142aw stopped another one by scaring the Navy officers into killing the Empress's terminally stupid commander. We've done our best to avoid letting the slaughter get out of hand. Most of the young Sailors on both sides are just following their officers.\"\n\nVicky glanced at her two admirals. They nodded their agreement.\n\nKris liked what she was hearing. Of course, Vicky was telling her what she must know would bring Kris over to her side. Kris knew her hardheaded brain trust needed more than just happy words.\n\nKris cleared her throat and redirected the conversation. \"Okay, it looks like you've got a major opposition movement going here. Tell me, Vicky, in your own words, just how did this mess get started?\"\n\nVicky took a deep breath, eyed the admirals, and leaned toward Kris.\n\n\"I thought it was the banks run by the Empress's family, Kris. Credit suddenly dried up. With no money flowing around the Empire to support trade, the wheels of commerce crumbled. People lost jobs. The unemployed went into the streets, and riots made a wreck of things. The Empress's family sends its 'security consultants' into the streets using machine guns for crowd control, and suddenly another planet is totally owned by the Empress's family. Wash, rinse, repeat. That was what I thought. Then Admiral von Mittleburg did his own analysis.\"\n\nNow it was the turn of the other vice admiral to lean toward Kris. \"Really, it was the Grand Duchess here that got me wondering. She arrived with a report that a small mining colony a few jumps away was running low on spare parts and soon would be out of food as well. If they couldn't produce enough product to pay their taxes, they'd be bought up for pfennigs on the mark. By exactly who, I didn't feel a need to question.\"\n\n\"I passed through that system, Kris, when I was running away from my stepmother's assassins,\" Vicky added. \"I just wanted to help, but there was more to it.\"\n\n\"There are only two planets in the Empire that produce raw crystal,\" von Mittleburg continued. \"The largest one is wholly owned by the Peterwalds. The newer and smaller one is just a hop, skip, and a jump away from here. The Peterwalds didn't own it. Officially. But as demand grew, the supply shrank, and as it did, planets began to lack for a critical chunk of their industrial economy. The price of crystal shot through the roof. The Peterwalds made a fortune. It all happened because production from the mines out here was going down the tubes.\"\n\n\"I tried to help those mines,\" Vicky said, \"but next thing I know, the guys running them are doing things that just don't make sense. I charged them with fiduciary malfeasance and hauled their asses back here to St. Petersburg for trial. Once I put some of the second-level managers in charge, the production at the mines shot up. We're meeting the need for crystal for all those planets allied with us and growing our economies. My stepmother and her family are going unhinged as I get our economies back on the right track even while we're spending more than we should have to on an Army and Navy to protect us from her goons.\"\n\nBill jumped on that. \"Can you document any of this?\"\n\n\"I have a copy of my analysis right here,\" Admiral von Mittleburg said, and aimed his wrist unit at the screens in Kris's day quarters. They lit up as data cascaded across them.\n\n\"Nelly, can you evaluate what you're getting?\"\n\n\"Yes, Kris. I'll need to verify this against other data, assuming I can find any such data from Greenfeld, but it is internally consistent with his words.\"\n\n\"Pass it along to Runda's data hounds, Nelly, and see if this makes some of the data they picked up on Greenfeld more comprehensible.\"\n\n\"I will work with them, Kris.\"\n\n\"I'd like to talk to these other two admirals about how they saw this revolution getting started,\" Chief Mediator Alfred Fu said, and was quickly joined by the other two leads of Kris's brain trust.\n\n\"Why don't we adjourn to my office on the station,\" Admiral von Mittleburg said. \"I've got access to a lot more information there than I have on my wrist unit.\"\n\nA moment later, the three leads were gathering up the members of their own teams, collecting a few data and electronic specialists from Runda's support group, and heading off to the station.\n\nSoon, Kris found herself alone with just Vicky and Jack. He seemed to test the air, found it overloaded with estrogen, and dismissed himself to check on security.\n\n\"Well, what's been going on with you two?\" Vicky asked with a salacious smirk on her face. No doubt, she wanted every detail of Kris's sex life.\n\n\"Some of this. A lot of that,\" Kris answered, intentionally vague.\n\n\"And a baby.\"\n\n\"You won't believe how that happened,\" Kris said, shaking her head.\n\n\"I know where babies come from,\" Vicky said slyly. \"Don't tell me that you and Jack didn't.\"\n\n\"Oh, we did, but I had a bit of help from an unpleasant direction.\"\n\n\"Oh, do tell, but first, I have to tell you that I got access to the information they allow the top admirals to know.\"\n\n\"Yes,\" Kris said cautiously.\n\n\"I know the story about your Order of the Wounded Lion,\" Vicky singsonged.\n\nKris tried to frown, but Vicky was just so full of herself. Still, she grumbled, \"That's supposed to be a secret.\"\n\n\"Not from me it isn't.\"\n\nAnd the evening took off for points unknown.\n\n# 42\n\nKris never had a sister. She'd tried a few times to find a substitute among some of the girls at school or daughters of Father's cronies.\n\nIt had never gone well. Girls can be hurtful and untrustworthy. Just as Kris's first try for a boyfriend had ended in a disaster, her attempts to have a girlfriend had also crashed and burned. Not as bad as the boyfriend thing, but not nice either.\n\nAs the hours went by, and she and Vicky talked and talked, Kris found herself discovering a girl that needed a friend almost as desperately as she had. Vicky talked about Mannie, the mayor of Sevastopol who had somehow managed to stay out of her bed, and Kris talked about Jack and how Granny Rita had finally forced them to take time for each other, then given them the push to rush at each other and into marriage.\n\nThe heavy dress blue uniforms seemed too stiff and thick for the intimacy that was growing between them, and Kris invited Vicky back to her quarters. The uniforms came off, and Kris settled into a pair of sweatpants and a T-shirt.\n\nThere was little in Kris's closet that looked to fit Vicky, but Kris was not about to have Jack come in, late at night, to see Vicky in the nonregulation and very skimpy bra and thong the Grand Duchess revealed as she slipped out of her blues.\n\nKris hunted up a pair of running shorts that she'd used to exercise in when baby was bulging all over the place. There was also a Marine T-shirt that she'd stolen from Jack when she felt like a blimp. Even in the last months of pregnancy, it had hung loose on her. On Vicky, it revealed every curve.\n\n_There is no justice in the universe._\n\nBut Vicky was in awe of Kris and Ruth when Sally Greer and Li O'Malley brought Ruth in for her lunch.\n\n\"So that is what boobs are really for,\" Vicky breathed.\n\n\"Yes,\" Kris said, \"but don't try this at home. There's a whole lot of stuff they don't tell us about pregnancy and nursing.\" And Kris and the nannies dumped all the horror stories they could think of on the impressionable young woman.\n\n\"Maybe it will be easier with a uterine replicator,\" Vicky said.\n\nOnce Ruth was taken off to a nap, Vicky began to unload on Kris just what it had been like for her for the last two years. Kris found herself shaking her head, time after time. Vicky might not have been facing down huge alien mother ships, but she'd been tackling a lot more human horror than any young girl should have to wade through.\n\nKris ordered pizza in for lunch from the Forward Lounge. Its popping up on one of the empty chairs scared Vicky, and Kris discovered just how loud the gal could shriek. When they still hadn't stopped talking by suppertime, the Lounge provided Chinese takeout.\n\n\"Pizza, Chinese, what can't your Forward Lounge do?\" Vicky demanded.\n\nKris shrugged. \"I don't know. I've never asked for something they didn't have.\"\n\n\"You Longknifes. You're so spoiled.\"\n\n\"You're the ones with the Empire. Remember, we've just got a King.\"\n\n\"My dad may be the Emperor, but he married the Empress from hell,\" and Vicky told Kris what it was like to wake up naked, handcuffed to a bed, and not a drop of water in sight. She ended the tale with Mannie's being the one to rescue her.\n\n\"Really, Kris, you have just got to meet Mannie. If anyone is going to make a democrat out of me, it's him.\"\n\n\"Democrat, hell, Vicky. Can he make an honest woman out of you?\" Kris took the opening offered.\n\nTo Kris's surprise, Vicky got serious. She grabbed a pillow from the next chair over, pulled it to her lap, and huffed. \"I sure hope he will. I think he will. Kris, how do you get a man?\"\n\nAnd Kris found herself offering matrimonial advice to the sexpot who'd once used all her wiles to get a long line of men to do their best to kill her.\n\n_I can't believe I'm doing this._\n\nBut she did. As the evening went along, Kris shared stuff with Vicky she'd never told another soul. Jack had never asked Kris what it felt like . . . really felt like . . . to order the fleet to fire on the aliens that screamed they wanted to surrender while they just kept steady on a course for Alwa.\n\n\"I thought I'd finally taught them to surrender.\"\n\n\"But they were saying one thing and doing another,\" Vicky whispered softly.\n\n\"Yeah. Was there something wrong with the instructions we were sending them? I don't know, but I couldn't let them get through that jump. They could park on the other side and blast us out of space as we came through. Then they could bombard Alwa at their leisure. They had to disarm, but it didn't look like they were going to.\"\n\n\"You had to kill them,\" Vicky whispered.\n\n\"Yes, I did. Still, I wonder how long it will be before I have a chance to take another surrender.\"\n\nVicky shrugged, then changed the subject. \"I've got to tell you about the time I drove one of the Empress's high lord commissioners so far around the bend that Admiral Boles\u0142aw, he was a captain then, got the entire squadron on the other side to surrender. We were trying to open up trade with Brunswick . . .\" and the Grand Duchess was off on her way to another story.\n\nWere they tall tales? Kris listened to them, wanting to believe that her erstwhile friend just might have changed her stripes. The stories all held together, and they tumbled from Vicky's mouth with no time at all to concoct them. When Kris tossed out a question about this or that, Vicky had an answer without taking time for reflection or construction of a lie. Only where Mannie was concerned did Vicky slow down. Only when Vicky talked of the mayor did she get circumspect and have to search for words.\n\n_She really has it bad for that guy. She likes him enough that she really doesn't want to screw it up._\n\nIt was edging toward midnight when Jack showed up with the strangest foursome in tow. Two were the tiny ninjas whom Vicky had taken with her to circumnavigate the galaxy. They studied Kris and the room as if some assassin might drop from the overhead light fixture or one of the nudes still standing against the wall might come alive.\n\nOne of them actually pinched each of the statues as Kris tried not to stare.\n\nThe third was a commander who looked very official, kind of like Jack five or six years ago.\n\nBut it was the fourth who caught Kris's eye. Or rather, didn't. He looked so bland that Kris had to work at making her eyes not slip past the fellow without taking note of him. She focused her eyes on him and stared.\n\n\"Don't I know you?\" Kris asked.\n\n\"I was aboard the _Wasp_ for her last trip,\" the fellow said. Jack had introduced him with a hint of a smile and a cryptic, \"This is Mr. Smith.\"\n\n\"I thought I remembered you. What was your job?\"\n\n\"Officially, I was an Able Spacer. Actually, I was there to keep you alive. There were a half dozen or more of us. Some of us knew each other and worked together if we needed to. There were others we didn't know.\"\n\nKris frowned. \"Why not?\"\n\n\"If we all were chasing the same duck, we might miss the swan behind us. I'm sure you were worth the extra coverage. Every one of us I knew about was kept busy, saving you, then taking up the slack on the Grand Duchess. There were some really good people who wanted her dead and getting her, they might get lucky and get a twofer.\"\n\n\"I hired him,\" said Vicky, jumping in. \"He came to me when Jack was giving me the bum's rush off the _Wasp_ for that stupid stunt I did with the newsies. I was broke, but Mr. Smith was willing to work for me on consignment. He's been worth every penny I've paid him.\"\n\nVicky pulled at the spidersilk-armor bodysuit Kris had spotted when they changed into something comfortable. She had said nothing then.\n\n\"He got your Gramma Trouble to arrange for these to make it through Wardhaven customs.\"\n\n\"She is a very helpful Gramma,\" Kris admitted.\n\nVicky shivered. \"And she didn't even mention the time I had her kidnapped. Wonderful woman.\"\n\nThe two diminutive assassins collected Vicky's blues and were clearly ready to lead her out the door.\n\nVicky gave Kris one more hug. \"We've got to do this again. I'm not half-done.\"\n\n\"I want to, too,\" Kris said. \"Let's make it soon.\"\n\nWith a few more good-byes, Kris found herself alone with Jack. She leaned against him and sighed. \"I really think she's telling the truth. I feel it in my bones that it's the stepmother, her father, and that family that is causing all this trouble.\"\n\nJack let out a sigh that was almost a rumble. \"Did any of what she told you tonight rise to the level of evidence we can use?\"\n\n\"Not a jot or a tittle.\"\n\n\"You can plan an assault on an alien base ship based on one of your hunches,\" Jack said. \"God knows, you've gotten away with it enough times. You won't be allowed to settle this mess on a hunch.\"\n\nShe hugged him. \"Do you have any idea how we're going to solve this?\"\n\n\"Not a clue.\"\n\n\"Take me to bed and make me forget about tomorrow.\"\n\n\"I thought you'd never ask.\"\n\n# 43\n\nKris was not allowed to sleep in. Ruth was hungry, so Kris tried to serve her breakfast in bed, hoping the nannies would take Ruth off to points unknown without her really having to wake up.\n\nThat didn't work.\n\n\"Kris, your brain trust would like you to have breakfast with them in the Forward Lounge,\" Nelly informed her.\n\n\"How much sleep did they get last night?\" Kris asked through a yawn.\n\n\"A lot less than you did, honey,\" Nelly answered, sounding very much like Kris's former maid, Abby, in full 'tude.\n\nKris rolled out of bed, took a quick shower, put on the undress khakis Jack had laid out for her, and went through the high points of her in-box while Jack finished dressing.\n\n\"There anything there from the brain trust?\" he asked as he tied his field scarf.\n\n\"Nothing, which leads me to fear they're holding on to some really nasty stuff for the sole purpose of ruining my breakfast digestion.\"\n\nJack finished, delivered a peck on her cheek, and said, \"I will defend your digestion to the death. Mine, too.\"\n\nThere were still two very spacious, if a bit less decadent, circling staircases down to the Forward Lounge. The place still looked like an insane King's fevered dream, but to the right was something more like a classical O club. A turn to the left led into a comfortable English public room. Beyond that was something French. A Marine waitress waited for them.\n\n\"They've got a room to themselves in the six o'clock position from the stairs, Your Highness, General.\"\n\nShe led them through the O-club portion past walls decorated with classical military art. Double doors of apparent oak opened into a more utilitarian quarter of the pie that was the Forward Lounge's circular deck.\n\nIt was not at all plain. No, Nelly had arranged for finely polished wooden walls and marble columns supporting ceiling beams. Interspersed around the room, and taking up at least half of the wall space, were screens filled with numbers and equations that were gibberish to Kris. No doubt, she'd know more than she ever wanted to know about them in a few short hours. The furnishings were tables of rich mahogany. Around them were comfortable chairs, with thick red brocade padding on the seats and armrests.\n\nAl spotted them standing in the double doors and opened his arms as he made his way to them, the affable host incarnate.\n\n\"In here, it's easier for our teams to work side by side,\" he said, resting a hand on Kris's elbow and steering her toward a couple of linen-covered tables close to a table laden with every possible breakfast food. \"It's easier for people to float from one group to the next without having to wander around the ship. Also, if someone needs you, they have only to holler.\"\n\nAt that moment, someone hollered, \"Hey, Judge,\" and Diana's head popped up. A hand waved her over to a table halfway across the room. She excused herself from where she'd been and headed for where she was wanted.\n\n\"We eat when we can. Your wardroom has been kind enough to keep all of our coffee urns filled.\"\n\n\"Have you been here all night?\" Kris asked.\n\nAl rubbed his eyes, and Kris now saw past the amiable host to the exhausted elder who had gone without sleep for too long. \"I've kept negotiating teams up for two days running. It's amazing the concessions you can get when folks are looking out the window at their third sunset without a minute of sleep.\"\n\nKris felt very guilty for her night's rest even if it hadn't been nearly long enough.\n\n\"Now, why don't you two get yourselves some breakfast? We are getting the best food I've ever tasted. Listen as you eat, and we'll see if we can put together a briefing for you once you're finished. I might keep people up for days at a time, but I never interfere with their first cup of coffee come morning,\" Al said, and chuckled at his joke as he left them to respond to his name and a waving hand.\n\nKris and Jack eyed the choices laid out before them. There was everything Kris had ever seen on a breakfast menu: crepes with a half a dozen fruit fillings; quiches; pancakes and waffles of assorted embellishments; eggs prepared several ways; a breakfast burrito; potatoes of several sorts, including little pancakes; oatmeal with quite a few berry, fruit, or sugar toppings close at hand; a tray of assorted sweet breads. A chef stood ready to make an omelet to her specifications.\n\n\"I didn't know Mary Fintch had brought aboard either the chef or the produce to lay out something like this,\" Kris said.\n\n\"Oh, we know how to make all this,\" the woman who stood ready to make the omelets said. Her name tag said Song, so this would be Mary's wife. \"We just had a delivery from St. Petersburg that we didn't order. Seems someone wants to thank a Kris Longknife for getting his town a charter or something like that.\"\n\n\"This is all from Mannie?\" Jack said, mouth hanging open.\n\n\"Mannie or Mike. Mary told me, but I forgot,\" Song agreed.\n\nKris felt her mouth watering and loaded a plate for a long and pleasant breakfast.\n\nAs she and Jack settled down at a table, Song brought a tray full of glasses of milk and juices. Kris took the large milk, it was clearly intended for her, and fresh-squeezed orange juice. Jack took apple juice and coffee.\n\n\"Clearly, St. Petersburg is not going hungry,\" Kris said. \"Song, did you or Mary try to order fresh meats or produce while we were at Greenfeld?\"\n\n\"Funny you should ask, Your Highness. We were told that the bug infestations meant we didn't dare take on fresh supplies. But we didn't get that word until we tried to order some fresh meats, fruits, and vegetables. The prices being asked from the station suppliers were just outrageous. Mary tried to contact some suppliers dirtside, assuming we'd pay for transport up the beanstalk ourselves, but their prices weren't any better.\"\n\n\"Hmm,\" Kris said. \"Song, if there's no one asking for you to cook something at the moment, would you go over to that tall woman, Diana? Tell her Kris Longknife sent you, and she needs to hear what you just told me.\"\n\n\"Yes, ma'am,\" the cook said, and made her way to where Diana was bent over a table. Kris munched her crepes while the two talked. Done, Song headed back to her place behind the table.\n\nDiana gave Kris a knowing nod and went two tables down to talk to someone else.\n\n\"They can't even keep food flowing into their capital,\" Jack said, shaking his head.\n\n\"I remember when I was campaigning for Father for the farm vote, Jack. Farming has come a long way from plowing a field and tossing out some seed. They need spare parts for their equipment: hundreds of thousands of dollars of equipment. They need fertilizer, bugs to keep the bugs down, a whole lot of things. You start trouble over here, and the supply chain to over there frays and breaks. If it gets bad enough, you get famine, and people start dying,\" Kris finished softly.\n\nAfter a minute, she shook herself. \"It's not smart to mess with a complicated mechanism like a modern economy. If things really did get as bad on some planets as Vicky claimed, then she's done some herculean lifting to get it going again.\"\n\n\"We didn't hear a word of this from the Emperor or Empress.\"\n\n\"Nope,\" Kris said, \"just a lot of complaining about rioting in the streets and a kid that won't do what Papa wants.\"\n\n\"Do we have any proof that someone is out to kill Vicky?\" Jack asked.\n\n\"Kris, I have found information to verify at least one of the assassination attempts.\"\n\n\"Talk to me, Nelly.\"\n\n\"I've located a news story from Savannah about the Grand Duchess paying them a short visit. Very short, unannounced visit. A bit more than two days in system. The story recounts the death of a Rear Admiral Gort, shot by a bystander who was herself killed. The murder is carried on Savannah's books as unsolved. This took place only a bit more than a week after we were hauled off that wreck of a _Wasp_ at High Chance.\"\n\n\"Good God,\" Jack whispered. \"Somebody wanted Vicky dead, and they could reach very far and grasp for her that fast.\"\n\n\"She said her 'Mr. Smith' had been earning his pay,\" Kris pointed out. \"Nelly, please pass that along to our brain trust.\"\n\n\"Yes, Kris, it is done. Judge Diana says they are close to having a report ready for you.\"\n\nKris and Jack finished their delicious breakfast and self-bussed their table. Just as she turned around, Kris was presented with all three of her brain trust with their heads together in the center of the room.\n\nAs Kris and Jack walked toward them, the three turned to meet them.\n\n\"We're ready to brief you, Your Highness,\" Diana reported.\n\n# 44\n\nThe room began to change around Kris as she wound her way to the three leaders who were there to advise her on matters less mortal than her normal business. Tables that had been scattered around the room started moving, following the people who had been working on them. The room reformed itself into a triangle with five comfortable chairs in the center. Several new tables appeared in the triangle as other cluttered tables moved to form a loose second row behind the first. Chairs that had sunk into the floor as the tables began their perambulations reappeared as the tables settled into their places. Quite a few of them were easy chairs.\n\n_God, you have to love this Smart Metal,_ Kris thought as she took one of the chairs in the center, Jack at her right elbow. The judge, the mediator, and the arbitrator settled into place in front of her.\n\nFor a long minute, as the migration of the tables finished up, the three across from Kris sat, apparently lost in some quiet meditation.\n\nKris, who was used to action, forced herself to try some deep-breathing exercises to keep from taking the floor and telling them what she wanted.\n\n_These people are here because a whole lot of folks thought this new challenge might not be amenable to my usual solution of blowing something up. Let's let them have their say._\n\n_I can always blow stuff up later,_ Kris thought, and made no effort to suppress what was, no doubt, a fey smile.\n\nAcross from her, Al, Bill, and Diana exchanged glances, then Diana cleared her throat.\n\n\"I think, as a judge, I should be the one to explain this.\"\n\n\"Explain what?\" Kris said, jumping in the moment Diana paused for a breath. Kris clamped her mouth shut. _Listen, don't talk,_ she ordered herself.\n\n\"Your Highness,\" Diana want on, \"we think there is something that really must be at the forefront of our conversation here today.\"\n\nKris gritted her teeth and did not venture into Diana's second pause.\n\nThe judge seemed surprised at Kris's silence. Surprised and pleased.\n\n\"Many people would think that we are gathered here to determine guilt and innocence. Who is in the right and who is wrong? I may have the honor of being called a judge and have a courtroom, but determining the guilty is not the purpose of my court.\"\n\nKris continued to keep her mouth shut, but she couldn't avoid a puzzled frown.\n\n\"So you want to know what my court _is_ about?\" Diana said.\n\nThis time, Kris allowed herself a \"Yes.\"\n\nA tight smile flitted across Diana's face as she nodded thoughtfully. \"When I sit in robes, my purpose is to find the best path for the children. I want to examine every option that my associates can dig up, then decide which one is best for that young person's future. I _cannot_ waste my time searching for who might be the guilty party who brought on this situation. I. Am. There. For. The. Child. I am there to find a way forward out of a mess.\"\n\nKris leaned back in her chair, and, for once, didn't have to clinch her jaw to keep words from dropping out of her mouth. What Diana had said was not at all what Kris had expected. In all the time she had spent with these three and their supporting staff, this rather peculiar idea had never crossed her mind.\n\nIt did not go down easy for Kris.\n\nAll her life, she'd been looking for the right way. Campaigning for her father, the whole idea had been to convince people that his party was the right one and the other parties were wrong. In the Navy, Kris had always looked for the best way. The one that left her enemy defeated and her victorious. That left as many of her people alive as possible.\n\nWorse, as Kris listened to Vicky last night, she'd found herself wanting the Empress defeated and all her ambitions left in the dust.\n\nWhen Kris began speaking, her words kind of fell out of her mouth. \"You don't want to prove who's right and who's wrong? You want me to look at this civil war and ignore who's responsible for starting it? For creating all this havoc? Causing all these deaths?\"\n\n\"Strange as it may seem, you are correct,\" Al, the mediator, said softly. \"We can't afford to waste our time and effort on that. We came here to end a civil war. It will take everything we have, every bit of our experience and skill, to end this war. Any effort or time we waste seeking out the guilty and trying to devise punishments for them is effort taken away from ending the war and restoring this Empire to peace and prosperity.\"\n\n\"That doesn't seem fair,\" Jack said. \"Someone _deserves_ to be punished for what they've done to the people of the Greenfeld Empire.\"\n\n\"Maybe someone will,\" Bill, the arbitrator, said. \"But. It. Can. _Not._ Be. Us.\"\n\n\"We can try to end this civil war,\" Diana said, \"or we can try to punish the guilty. Your Highness, please say again what your purpose for coming here is.\"\n\nKris took several deep breaths as she considered her mission before she spoke softly. \"Before a few minutes ago, I would have considered punishing the guilty and ending this civil war one and the same.\" Kris paused. \"I don't like the idea that they are not. Tell me why we can't do both.\"\n\nThe three looked at each other for a moment, then Al faced Kris. \"For a moment, let us say that the Emperor, or was it the Empress who said it. Anyway, let's say that we tell the Grand Duchess that her actions here have been in the wrong and that the right thing for her to do is return to her father's loving arms at the Palace. How do you think that would go over?\"\n\n\"Other than that she would be dead within a week?\" Kris asked.\n\n\"There is that likelihood,\" Al said. \"But before she actually boarded a ship for Greenfeld, how do you think our conversations would proceed?\"\n\n\"You tell her that,\" Jack said, \"and she'll tell you to go to hell and this mission will get no more cooperation from her side of this civil war.\"\n\n\"Precisely,\" Al said. \"The Emperor's, or is it the Empress's, solution is a nonstarter from the Grand Duchess's perspective.\"\n\n\"We haven't gotten quite so specific a proposed resolution to this civil war from the Grand Duchess's side,\" Diana said, \"but I suspect it would go something like this. The Empress abdicates and divorces the Emperor. Her family, as well as herself, is banished from the Empire and may leave with only the clothes on their back. All properties that have changed hands in the last three or four years are returned to their rightful owners, and all cases involving murder and rape are referred to courts established by the Grand Duchess's new regime. Have I about got it right?\"\n\nKris nodded. \"Pretty much. Vicky was quite reserved when talking to me yesterday, but I could feel the rage and need for vengeance.\"\n\n\"Do you see the Empress's faction agreeing to anything like that?\" Diana asked.\n\nKris just shook her head.\n\n\"So you see our challenge,\" Diana said. \"We need to find a solution that is acceptable to both sides. Hurling condemnations and anathemas around is not productive.\"\n\n\"No matter how satisfactory it may feel for the moment,\" Bill added.\n\n\"So where do we go from here?\" Kris asked.\n\nThe three looked at each other a bit sheepishly. It was Diana who first smiled, shrugged, and leaned toward Kris.\n\n\"Having just won your support to our viewpoint that we are not here to determine the guilty party, we must now determine what the true circumstances are in the Empire. Doing that will, no doubt to some skeptics, look like we are hunting for the guilty.\"\n\nKris had to chuckle. \"So you've spent the last fifteen minutes stampeding me off in one direction, and now you're going to take me off in the other?\"\n\n\"It may look that way,\" Al said.\n\n\"Well, lead on, and I will follow,\" Kris said.\n\n# 45\n\nBill took the lead on their presentation. \"For what it is worth, we have been able to develop more information in support of the Grand Duchess's side than the Empress's. Admittedly, the Empress gave us nothing more than her narrative, and quite a bit of the message traffic on Greenfeld was heavily encrypted, but what we have gathered here has allowed us to put what we dredged up there into a perspective that is not very supportive of the Empress.\"\n\nHe glanced over to a woman seated at a table in the inner triangle and nodded. She stood and walked over to a large screen on one wall. Kris's chair moved to face that direction.\n\nBill continued. \"The Grand Duchess's people have been fully cooperative with our staff. Assuming we are not looking at a perfectly choreographed lie, it is persuasive. Iba, would you please brief Her Highness on what we find persuasive to a reasonable person.\"\n\n\"Yes, sir. Your Highness, there have been three attempts on the Grand Duchess's life here on St. Petersburg, assuming we ignore the attacks on the entire planet. In one, she was left chained to a bed in a remote cabin. In another, a car bomb failed to kill her but slaughtered innocent bystanders. A third took place along a country road in the hills above Sevastopol. We chose to investigate the first two. We had limited staff for field trips yesterday,\" she explained.\n\nA picture of a decrepit farmhouse beside a burned-out barn appeared on the screen. \"This site had been investigated previously but was pretty much undisturbed. We found the Grand Duchess's DNA present, as well as traces of her blood. Her bare footprints showed where she had walked away from here. All this fitted the story we were given.\"\n\nNow a city street filled the screen. Bodies lay strewn on the ground; cars were burning. \"The scene of this car bombing had been cleaned up. We talked with several of the survivors. They truly believed what they said. We talk not only to those witnesses provided but also to random subjects in the surrounding buildings. The variations in their stories were reasonable and within the expected range.\"\n\nThe young woman paused. \"It is our conclusions that, even out here where support for the Grand Duchess is readily voiced, there have been serious attempts on her life. Unfortunately, none of the perpetrators lived to testify. They either escaped or died resisting arrest.\"\n\nThe screen changed to show bags being stacked for transport. \"These are standard famine biscuits. These pictures were offered in support of the story that the Grand Duchess led a major relief effort to the colony on Pozen. In support of this claim, we stripped off the net orders, transit invoices, and even letters to pen pals from fifth graders. It appears that some of the biscuits were baked by the kids from grain they grew themselves. We found too many references to this in private mails to doubt the actuality of this event.\"\n\n\"A Peterwald really tried to help people. Who'd have thunk it?\" Kris said.\n\n\"The private e-mails had quite a few references to just that, Your Highness. Even some of the youngsters didn't hold the Imperium in much regard. But of the Grand Duchess's efforts to feed the hungry and defend the planet, there are just too many public and private comments to doubt that the people really believe this and participated in it. Oh, and we found archived media reports of Vicky, they actually called her Vicky in some press stories, handing out ribbons at city fairs. The kidnapping occurred while she was visiting a carnival. She'd just won a little girl a huge stuffed bear when all hell broke loose.\"\n\nThe young woman allowed herself a slight shrug. \"I couldn't make this stuff up.\"\n\n\"I don't think you are,\" Kris said. \"It seems at least one Peterwald has changed her stripes. What about the space battles?\"\n\nAn older man stood. His crew cut and ramrod-straight back hinted strongly that he had come to this vocation as a second career. The screen changed behind him to show a map of the Imperium. The planets were divided into two groups that showed red and blue.\n\n\"Vice Admiral von Mittleburg cooperated completely with us in examining the fighting that has been going on in the civil war. He walked us through the strategy they are presently using. It involved fixing a major part of the Empress's fleet here, in front of St. Petersburg, while several of their fleets nibbled around the flanks, cutting planets out of the Empress's holdings. He credits the Grand Duchess for the basic idea although it is plain that it was the Empress that chose to concentrate her fleet in a thrust directly at St. Petersburg.\"\n\n\"Is this where Vicky has operated from?\" Kris asked.\n\n\"Yes.\"\n\n\"So the Empress was going for the Grand Duchess,\" Kris said. \"Something tells me that she knows that Vicky is nobody's patsy.\"\n\n\"The data is open to that interpretation by a reasonable person,\" Bill said.\n\n_When have I ever been called a reasonable person?_\n\nWhen the silence grew, the briefer continued. \"The admiral had nothing but praise for the Grand Duchess. He's found her courageous, creative, and flexible. He repeatedly mentioned how well she listened although she can be quite headstrong at times.\"\n\nThe briefer chuckled. \"I think Vice Admiral von Mittleburg views the Grand Duchess like a daughter. Oh, both of them are vice admirals. I was very surprised when von Mittleburg told me that Vicky is senior to him by a few seconds.\"\n\n\"How's _that_ worked out?\" Kris had to ask.\n\n\"Apparently well. Even during the big battle with the Empress, it was she who decided to call back their destroyers from a torpedo run on the opposing battle line. The tin cans weren't as supported as they needed to be, but they were so close. She made the call, and he obeyed.\"\n\n\"Interesting,\" Kris said. \"This is all nice, but we're looking at an Empire bleeding to death, pfennig by pfennig. That admiral, von Mittleburg, said it was crystal taking the economy down. Did you find out anything on that front?\"\n\nAn eager young man rushed to the screen to take over the briefing. \"They had archives stored here from St. Petersburg, Metzburg, and Brunswick as well as two dozen less industrialized planets in the rebel union. All show that three to four years ago, the availability of crystal plummeted as the price skyrocketed. In the middle of that problem, credit suddenly dried up to finance trade. As if that wasn't enough, the insurance rate for merchant hulls and cargos shot up. It seems the Navy couldn't keep pirates from hijacking honest merchants.\"\n\nKris and Jack exchanged glances. They'd had some experience with Greenfeld's pirate problem before Kris took off to explore the galaxy and discovered what lurked under its bed.\n\n\"Did von Mittleburg have any explanation for the Navy's failure to control pirates?\" Jack asked.\n\n\"The Navy has been driven from pillar to post, if I may say so, sir. First, they were ordered to planets to root out State Security, then, a bit later they were ordered back to maintain order when unemployment or food riots started. He had me talk to a few Navy officers who were there when the Security Consultants organized by the Empress's family showed up. The Navy had orders to get out immediately, but they saw the first sweeps. State Security's trigger-happy thugs were never as bloodthirsty as this bunch was. Their uniforms are red, by the way. That kept them from showing the blood they waded through, according to one of the bitter stories that ran through the fleet.\"\n\n\"So it's ugly,\" Kris said.\n\n\"Apparently very ugly,\" the briefer said.\n\nKris turned to the three heads of her brain trust. \"Do you feel like you have a solid enough handle on what the real situation is here in the Empire?\"\n\nThe three nodded.\n\n\"Okay. Now what do we do about it?\" she asked.\n\n\"That will not be at all easy,\" Bill said.\n\n\"Not easy at all,\" Al agreed.\n\nDiana just nodded.\n\nLeaving Kris with nothing but a frustrated sigh.\n\n# 46\n\nAn hour later Kris had Nelly invite Vicky to the _Princess Royal_.\n\nA half hour after that, Kris watched from her day quarters as Vicky led a line of civilians and officers up the gangway. Vicky rendered honors without hesitation.\n\nBehind Vicky, sandwiched between her and Admiral von Mittleburg, was a civilian.\n\n\"Nelly, is that the much-mentioned Mannie?\"\n\n\"Yes, Kris.\"\n\nKris studied the man Vicky talked so much about. She'd met him two years ago. His hair showed a bit of gray now; he looked worn but chipper. Kris looked forward to meeting him again.\n\nWhat surprised Kris was the last civilian following Vicky across the brow. She had brought Mr. Smith to this meeting. Interesting. Hopefully, Mr. Smith would not prove to be as slimy as Admiral Crossenshield, but only time would tell.\n\nWhich left Kris wondering who her Crossenshield would be.\n\nThere was little time for introspection before Kris was being introduced to a small mob of people. Beside the two admirals and Mr. Smith, whom Kris had met previously, there was Mannie as well as the mayors of Kiev and St. Petersburg. Right behind them came three civilians who proved to be a banker, an industrialist, and a rancher.\n\nKris and her team were ready for some serious bargaining, and so, apparently, was Vicky.\n\nNelly arranged comfortable armchairs side by side for Kris and Jack, with two long rows facing each other in front of those. At Kris's side were three leaders of her brain trust. After seeing Mr. Smith in Vicky's lineup, Kris had invited Senior Chief Agent in Charge Foile to join them, both to even up the sides and to bring his insight.\n\nVicky and Mannie took the two chairs facing Kris from the other end of the lineup of chairs. It was interesting that Mr. Smith sat at Vicky's right hand and Vice Admiral von Mittleburg at Mannie's left. The other five were comfortably, if unevenly, spaced down the two lines toward Kris.\n\nKris opened the meeting. \"I'm glad to see more people brought into our discussion,\" she said, eyeing Vicky.\n\n\"I've learned to value the input of a lot of people. We wouldn't be here, and I wouldn't be alive, without the aid and advice of these and a whole lot more.\"\n\n\"Can you speak for the rebellion?\" Kris asked.\n\n\"Please Kris, don't call us rebels. I'm a loyal subject of my father, the Emperor, as is everyone here in the room and on more than three dozen planets that stand together against the tyranny that the Empress and her family have imposed on the Empire.\"\n\n\"You'll stipulate,\" Kris said, \"that the Emperor and Empress don't see it that way.\"\n\nVicky looked around her group, then nodded to Mannie.\n\nHe said, \"Yes. We will stipulate that the Empress has pulled the wool over the Emperor's eyes and that he accepts her totally wrong version of events.\"\n\nKris chuckled softly as the mayor of Sevastopol agreed with her while getting in the rebels' twist on the situation.\n\n\"It is clear there is a major disagreement as to the facts on the ground and what should be done,\" Kris said.\n\n\"I think that is what has brought us to this crossroads,\" Vicky said. She seemed to want to say more, but she closed her mouth and eyed Kris. \"We can continue the war until one side wins or both are exhausted, or we can resolve the issues peacefully and leave a strong Empire to face any alien base ship that can show up in our skies at any moment.\"\n\nThere, the major problem was now on the table. If one of those huge mother ships showed up in Greenfeld space in the future, they would find planets wrecked and defenseless. While the US and other alliances were building up their fleets, Greenfield was blowing theirs apart in fratricidal warfare.\n\n_What's the female equivalent for fratricidal?_\n\nKris took a deep breath. \"Okay, Vicky, but how do we stop the war?\"\n\n\"Kris, I was hoping you'd come in swinging with a way we could all end this and live happily ever after.\"\n\n\"You've been reading too many fairy tales.\"\n\n\"I've been reading the Longknife legend,\" Vicky shot back.\n\n\"Sorry, Vicky, but I'm fresh out of rabbits to pull a hat out of.\"\n\nThe room fell silent for a long moment as that thought sank into a lot of people's minds.\n\n\"Then what _do_ we do?\" was finally voiced by the mayor of Kiev. \"We can't keep burning our production and sending our children off to die in some distant fight.\"\n\n\"It's better they die in some distant battle than do it defending our cities as they are blown to rubble.\" That came from the industrialist.\n\nKris figured the room was dark enough. She stepped in with a light.\n\n\"I have been given one suggestion by my advisors,\" Kris said.\n\n\"What?\" Vicky shot back immediately.\n\n\"It's not a solution in and of itself. Rather, it's a path to a solution.\"\n\n\"You've got my attention,\" the Grand Duchess said.\n\nKris took in a deep breath and let it out slowly. \"All of my advisors agree that the only possible way forward is to bring all the principals together in one room to talk this out.\"\n\nVicky eyed Mannie, and they shared a small smile.\n\n_Maybe I wasn't the first to come up with this idea._\n\nMeanwhile, all those around Vicky mulled that thought over for a long moment. Mannie was the first to shake his head.\n\n\"If Vicky went to Greenfeld to meet with the Emperor and Empress, she'd be dead within a week.\"\n\n\"Even under my protection?\" Kris asked.\n\nVicky shook her head. \"Kris, I know you mean well, but if I went to Greenfeld without my battle fleet, the Empress's ships would blast me out of space. And if I went with my fleet, I can't believe she wouldn't scream 'invasion' and I'd end up fighting on her home ground. I don't see that working any way you cut the cards,\" she said. Beside her, Mannie joined her in shaking his head. All her associates did the same.\n\n\"You could come on my ship. I have a squadron with me,\" Kris said.\n\nVicky didn't need more than a second to form her reply. \"Kris, your frigates look beautiful the plush way you have them fitted out. No doubt with Smart Metal you could shrink them down into something more combative, but not even a Longknife could win with eight dinky frigates against the Empress's battle fleet. I may have shaved some very nasty ships out of her fleet, but it's still one big fleet.\"\n\nKris considered for a moment letting Vicky in on the secret of what the new battlecruisers could do to old-line ships, but swallowed it. She didn't need to let that secret out of the bag. Not yet.\n\nAnd she had a different card to play.\n\n\"You could invite the Emperor and Empress to St. Petersburg.\"\n\nVicky barked a laugh. \"Excuse me, Kris, but if I'm unwilling to go to Greenfeld, I expect the Empress would be twice as quick with her 'I'm not going there.' I've learned the value of paranoia from you, Kris. The Empress is way past paranoia and deep into full insanity. Sorry, Kris, neither one of those ideas is going to happen.\"\n\n\"How about some neutral ground?\" Kris asked. Her brain trust, who had been getting antsy, finally relaxed as Kris put their idea on the table.\n\nVicky took a moment to consider before she asked. \"Where would we find neutral ground? Neutral ground we would all agree upon?\"\n\n\"Both my mediator and arbitrator have a suggestion for that,\" Kris said, and opened her right arm in their general direction. Bill, the arbitrator, took up Kris's offer of the floor.\n\n\"Her Highness has already given you some idea of how we would do this. Let's say we make up a list of five planets where we might hold such a meeting with, say, Greenfeld at the top and St. Petersburg at the bottom. No doubt, you would strike one and they would strike the other as a nonstarter. But what about the other three? Those in between?\"\n\n\"I guess that would depend on what those three were,\" Vicky said.\n\n\"My brain trust has three to suggest,\" Kris said. \"The next two would be Brunswick and Dresden. Both of those planets are fully developed and have space stations that would meet our needs. One is yours while the other is theirs, but both are only a few jumps from the other's territory. Again, it's very likely that you would strike one and they the other. That brings us to the fifth planet. The one between the other planets that have already been struck.\"\n\nKris paused to take stock of Vicky's advisors. None of them looked ready to storm out. All were leaning forward, listening intently.\n\n\"We looked at several possibilities. Bliven meets our needs. It has a space station. It was in the Empire until recently. It did not join your rebellion, but it did petition the US for acceptance into the union and is now a full member of the United Society. It also is easily reached from both your territories. That's one choice.\n\n\"The second one is also US. Pandemonium. Its new space station is a bit on the small and primitive side. Still, the amenities available on the planet will meet all essentials though its spirits are limited to beer. They have yet to start a vineyard. I'm told they do produce a moonshine with quite a kick, but it might not please refined tastes.\"\n\nSo far, none of the planets named had gotten any reaction from Vicky or her associates. Kris continued.\n\n\"There is one last planet we considered. Cuzco is a well-established planet that has managed to maintain its independence since the Society of Humanity broke up. It's industrialized and has a large space station. It's about an equal number of jumps from both Greenfeld and St. Petersburg. They kowtow to no one, but I think they'd be willing to let us park a few ships at their station if it would resolve this war and increase their trade with the Empire.\"\n\nFor a moment, as the silence grew and stretched, Vicky eyed Kris. Then she nodded. \"Can you give me a few minutes to consult with my advisors, Kris?\"\n\n\"That will not be a problem. Nelly, could you raise a wall to give them their privacy.\"\n\n\"As you said,\" Nelly replied, \"that will not be a problem.\" A wall began to grow out from the bulkhead and also slowly from the deck. There were a few looks of dismay from Vicky's team, but when it quickly became clear that no one would be sliced in half, dismay turned to awe.\n\n\"Why don't we have something like this?\" the banker asked.\n\n\"Because we're too busy blowing up the stuff we can make and don't have enough time or money to research this and get it in production,\" the industrialist growled.\n\nThe Empire had Smart MetalTM. Hank, or someone else, had used some of the stuff to try to kill Kris. They had the basic stuff but hadn't made nearly as much progress as the rest of humanity.\n\n_Just shows the price of civil war._\n\n\"When you're ready to talk to us, just have your computer contact Nelly. Nelly, no listening in on the other side of that wall, okay?\"\n\n\"I have disabled all devices on that side of the wall,\" Nelly said.\n\nSPOILSPORT. I WAS SO LOOKING FORWARD TO SEEING HOW A DIFFERENT BATCH OF HUMANS HANDLE SOMETHING LIKE THIS.\n\nYOU WILL WAIT LIKE THE REST OF US TO SEE WHAT THEY COME UP WITH, NELLY.\n\nAnd that settled that.\n\n# 47\n\nKris was quite surprised at how long Vicky and her advisors took.\n\nThe first couple of minutes were busy, filled with comments on how well Kris had done in presenting their ideas. The troika of advisors had been busy watching the faces of those seated down the line from them. Al, Bill, and Diana were quite excited by this or that twitch or fleeting expression that they thought bode well for developments.\n\nAll Kris had seen were blank faces.\n\n_I'm not that bad at reading people, am I?_ Kris wondered.\n\nAfter about five minutes, they ran out of things to talk about.\n\nAs time stretched, Al, the experienced mediator, produced a deck of cards and asked if anyone played bridge. The other two admitted to that addiction, and Foile added himself to the necessary four. Nelly produced a table, and they quickly lost themselves in the game.\n\nWhich left Kris and Jack the odd couple out.\n\nYOU WANT TO SLIP OUT TO OUR NIGHT QUARTERS?\n\nIT IS TEMPTING.\n\nNelly interrupted. \"O'Malley wants to know if you're available for nursing duty, Kris.\"\n\nThus Kris found herself only half-undressed in her night quarters, and Jack looking down at the fussy miracle at her breast. Ruth really wanted to fuss and ignored the offered breast. Kris ended up walking the floor with her, letting her gnaw on her finger.\n\n_Those teeth really are coming in._\n\nOnly after Ruth chose to settle down would she allow herself to be fed.\n\nHalfway through draining one breast, she fell asleep.\n\nO'Malley removed the sleeping princess. \"Thank you. She had all of us pulling our hair out. We offered her the bottle you filled last night, but she wanted nothing of us. Sometimes, a baby just wants Mommy and nothing else.\"\n\n\"Yes,\" Kris said, with a satisfied, motherly smile. She got herself put back together. \"Sorry our time got redirected.\"\n\n\"I could watch you and Ruth for hours,\" Jack said, sporting a very satisfied, fatherly smile.\n\nThey returned to the Forward Lounge where the card game was still going on. The four played energetically to the point of rowdiness but with good humor.\n\nNELLY, CAN THEY HEAR THIS ON THE OTHER SIDE OF THE WALL?\n\nKRIS, I CAN'T ANSWER THAT. YOU HAD ME DISABLE ALL MONITORING DEVICES ON THAT SIDE OF THE ROOM.\n\nOKAY, BUT CAN YOU DO ANYTHING TO KEEP THIS ON OUR SIDE?\n\nKRIS, THE FIRST TIME THINGS GOT ROWDY, I PUT IN A SECOND WALL AND LEFT A VACUUM BETWEEN THE TWO. I'M PRETTY SURE NOTHING IS GETTING THROUGH TO DISTURB THEM.\n\nTHANK YOU, NELLY.\n\nEvery once in a while, Nelly still managed to put Kris in her place.\n\nThe wait stretched further. Much further than Kris liked, but the cardplayers didn't seem disturbed by the length of the time.\n\n\"They're talking,\" Al, the mediator, said during a short time as dummy. \"We dumped a lot on them. It's no wonder they're taking their time to vent and process all they heard.\"\n\nKris had Nelly start calling the length of the break at five-minute intervals. She'd just said, \"Fifty-five minutes,\" when she added, \"And they are ready to rejoin us.\"\n\nIn a flash, the cards disappeared into Al's pocket, the card table and chairs sank into the deck. Six comfortable, overstuffed chairs were again where they'd been for the discussions. As they settled into their places, Kris said, \"You may take the wall down, Nelly.\"\n\nAgain, Kris's computer let the wall melt away slowly and in an unthreatening manner. Vicky's advisors were just taking their places. Kris's waited patiently. Glances from her advisory troika seemed to hint strongly that she should keep her mouth shut.\n\nVicky finally cleared her throat. \"Have you considered offering all seven planets to me and my father to choose from?\"\n\nKris watched her advisors make ever-so-slight nods. Slight smiles might even have snuck out for brief seconds.\n\n_Have I been played? Did they have me mention seven planets so Vicky's team might get an oar in the water on this decision? We're going to have a little talk when this is all done._\n\n\"I don't think that would be any problem at all. We've already composed a letter to the Emperor and you. Nelly, would you please expand the choices of planets to seven.\"\n\n\"I've already done it. A formal letter to the Grand Duchess is printing out now.\"\n\nJack was up and returned quickly with two pages in his hand and a pen.\n\nKris glanced over the letter one more time. She almost had it memorized; she and her advisors had been over it time and time again. Nelly added a flat writing board to Kris's chair and she signed it.\n\nDiana produced a stamp and notarized her signature before carrying it over to Vicky. Her computer scanned it, and sent copies to all her advisors. For several long minutes, heads pored over their screens.\n\n\"It's good,\" the rancher muttered. \"Too good for that damn Empress, but I guess if we want this, we got to hand the same to them. I still think you should have blown her out of space when you had the chance.\"\n\nVicky studiously did not hear that.\n\nKris smiled. The girl she'd first met would not have avoided hearing that. Hell, the girl she first met would never have accepted an order to stand down and not kill the Empress\u2014assuming she could actually have pulled it off.\n\nFrom what Kris had found out about the Second Battle of St. Petersburg, Vicky might really have had a chance to kill the Empress. A better chance, at least, than Vicky had the couple of times she tried to kill Kris.\n\n\"How do you want us to do this?\" Vicky asked.\n\nAl raised a finger, and Kris nodded to him. \"Usually we have the two parties in a room. A flip of a coin determines who scratches first. Then they take turns scratching through one name at a time. The name left after they've scratched off two, or in this case, it would be three, is where we'd go. However, this time we would suggest you put a one beside the first planet you'd scratch out. They will do the same. When we get these back, Bill and I here, with the judge looking over our shoulders, will scratch them out. Number one from your list, then number one from theirs. Number two, then number three. If your next choice has already been scratched, we go to the one after that.\"\n\n\"And we'll probably wind up at Cuzco,\" Mannie said.\n\n\"Likely, but I suspect it could be any of the middle three,\" Al answered.\n\n\"Nelly, is the letter on its way to the Emperor?\"\n\n\"Yes, Kris. The moment you signed the letter, I dispatched it, highest priority, to the jump point. I've routed it nine different ways, assuming communications might not be as smooth as they once were. I'm sure one copy will arrive. I also sent the letter in the clear. Is there any problem with that?\"\n\n\"I certainly don't have one,\" Kris said.\n\n\"If you will allow us,\" Mannie said, \"we'll distribute this throughout our area of control. I think everyone will be happy to hear this.\"\n\nThat brought nods of agreement all around from Vicky's advisors.\n\n\"So we relax while their letter gets to them and their answer gets back to you,\" the banker said.\n\n\"Kris, a message just came in from Captain Jack Campbell,\" Nelly announced. \"I think we may have a problem to add a bit of excitement while we wait.\"\n\n\"Shall we wait outside?\" Vicky asked.\n\n\"Does it concern Vicky and this civil war?\"\n\n\"It concerns the Empress and those delightful alien raiders, Kris.\"\n\nThat drew expletives from Jack and Vicky both.\n\nKris sighed. \"Vicky, do you want to ask any of your advisors to leave?\"\n\nThe Grand Duchess glanced around the room. \"No, if they can stomach it, they can stay.\"\n\nSeveral looked green or pale as a ghost, but none made for the door.\n\n\"Okay, Nelly, have at us.\"\n\nThe bulkhead to Kris's right turned into a star map. Most of the stars were unblinking white.\n\nThe Empire was in red or blue, depending on who held what. The US planets were golden. The area that Kris's Patrol Squadron 10 had policed out beyond the Empire was sprinkled with tiny, flashing, red dots.\n\n\"The flashing red area,\" Nelly explained, \"is the patrol area for Captain Campbell's cruiser squadron. They've been nipping off pirates and other unsavory types that try to use that no-man's-land for nasty things. Campbell just reported, and hotwired to you, about a cruiser he boarded. The skipper of this formerly Greenfeld Navy heavy cruiser demanded to be left alone. He said he was on a diplomatic mission by order of the Empress. He might have been left to go on his way, but Jack didn't think all that added up. What kind of diplomatic mission would an ex-Navy cruiser be doing out in the middle of nowhere? The Iteeche Empire was not even close, and the guy didn't mention anything about them.\"\n\nNelly paused. She'd learned to do that recently to allow humans to catch up with her.\n\n\"Campbell found the skipper of the ship as easy to get talking as a block of granite, but the navigator was all twitchy. His word, not mine. He took the navigator aside and asked to see his charts and his destination. The guy crumbled. Kris, the Empress had sent them out to find what you call the bug-eyed monsters and get them to come into the civil war on her side.\"\n\n\"That has got to be the stupidest idea I've ever heard,\" Kris exploded.\n\nAround the room, some people said a lot worse, while others looked puzzled, and asked, \"What?\" When they were brought up to speed about the alien raiders, there was more profanity.\n\n\"Okay. Okay,\" Kris said, trying to bring order. The room settled down but did not go quiet.\n\n\"Kris,\" Nelly said in her professorial voice, \"back on old Earth, there were a lot of times when a failing rebellion or even an anointed sovereign called in help from outside his borders. It often ended with bad things happening to both the original sides, but it worked enough times that it happened many, many times.\"\n\n\"I seem to remember something like that from school, Nelly. Now, what does Captain Campbell have to say about this instance?\"\n\n\"He's holding the ship and crew while he passes the buck up the chain of command. He wants you to know that he suspects the buck won't stop until it lands on the King's desk, so he's not expecting orders anytime soon. However, he knew about the mission you're on and figured you'd like to know what he'd stepped in.\"\n\n\"Is that all, Nelly?\" Kris growled.\n\n\"Yes, Admiral.\"\n\nKris leaned back in her chair. What the hell could she do with this? What the hell was the Empress up to? Had they stopped the only one of her \"diplomatic missions\"?\n\n\"Ah, Kris,\" Vicky said cautiously.\n\n\"Yes, Vicky.\"\n\n\"Admiral von Mittleburg just told me of something you might want to hear.\"\n\n\"Admiral,\" Kris said.\n\n\"We have been picking up rumors of this kind of thing, but we didn't put much faith in them. That might have been a mistake. I didn't bring the Grand Duchess into the loop on this because, well, she had her plate plenty full and didn't need to be bothered with what everyone took for a minor distraction.\"\n\nKris nodded. A human could only juggle so many buzzing chain saws.\n\n\"I asked for more information about this,\" the admiral said. \"It is possible I may get a reply soon.\"\n\n\"Or not,\" Kris said.\n\n\"Or not, if it is still viewed as unbelievable by the senior staff.\"\n\n\"Nelly, get the address of the admiral's higher-ups, compose a letter for his chop with an information copy of this attached to them from him, marked personal and most secret, addressee's eyes only, unless he's out of town, then his second immediately. You willing to sign that?\"\n\n\"No problem, Your Highness,\" Admiral von Mittleburg said.\n\n\"Kris, I also got the cipher for admiral-to-admiral communications from Admiral von Mittleburg's computer. The message is ready to go if he's willing to sign it.\"\n\nThe admiral glanced at his wrist unit for a long moment, then nodded. \"Put my chop on it.\"\n\n\"The message is on its way,\" Nelly reported.\n\nKris leaned back in her chair. \"Now what?\"\n\n\"I guess we wait,\" Jack said.\n\n\"Kris, the Marine guard at the brow report that there is a runner coming aboard with message traffic that Admiral von Mittleburg needs to see soonest,\" Nelly announced.\n\n# 48\n\n\"Have the OOD send him here immediately,\" Kris ordered before allowing herself the luxury of a deep breath. Several deep breaths.\n\n_Don't things ever slow down?_\n\nThere were advantages to having her day quarters on the same deck as the quarterdeck; the runner had less distance to cover. Apparently, he'd run all the way because he was out of breath as he entered Kris's quarters and sprinted to his admiral.\n\nThe messenger was a lieutenant, j.g., and he was under arms. Not a good sign.\n\nThe admiral thanked the young officer for his speed, then dismissed him while he glanced at his message traffic. Only after the door closed behind the exhausted JO did Admiral von Mittleburg speak.\n\n\"Does anyone want to leave the room? If you don't, I will have everyone sign the State Secrets Act, and I assure you, I will have you shot if you divulge any of what you hear here.\"\n\nThe Greenfeld subjects paled at the serious tone the admiral took, but none headed for the door.\n\n\"Fine. Vicky, could you have that fancy computer of yours sync with Her Highness's computer and print out\"\u2014here he looked around the room\u2014\"ten copies of the Act.\"\n\n\"Ten?\" Kris said. \"You don't intend to have _my_ King's subjects sign _your_ act?\" Kris asked, pointedly.\n\n\"Then you print out _your_ Secrets Act and have _your_ people sign it.\"\n\n\"Jack and I already have,\" Kris said, eyeing her team.\n\nSenior Chief Agent in Charge Foile rummaged in his wallet and pulled out his Ident Card and handed it to Kris. \"I've signed it.\"\n\nHis card did indeed have the necessary endorsement, among others.\n\nKris eyed the three leaders of her brain trust as well. All shook their heads no.\n\n\"Three copies, Nelly, unless you want to leave the room.\"\n\nDiana again shook her head. \"I'm guessing that this involves the Empress. We have to be in the know.\"\n\nA printer appeared and began spitting out paper. Jack trotted over to it and returned with six copies of the Greenfeld Imperial State Secrets Act printed in a dark, blocky, and threatening format. He brought those to Admiral von Mittleburg, who handed them out to his people.\n\nJack then returned to pick up the US version. It was much longer but at least was nicely calligraphied.\n\nThe banker, industrialist, and rancher quickly read the two-page Greenfeld Imperial State Secrets Act and blanched.\n\n\"But we're rebels,\" the rancher said. \"Do we really have to sign this?\"\n\n\"If you want this rebellion to succeed, we have to keep a few secrets,\" Vicky pointed out.\n\nThe banker shook his head. \"Your Grace, there is no way in hell I'm going to sign this.\"\n\n\"I understand, Bert, but I must insist if you want to stay.\"\n\n\"I guess I'll be on the next shuttle down,\" Bert said, handing back his copy. The industrialist and rancher quickly followed his example. Both the mayors from Kiev and St. Petersburg also stood.\n\n\"Mannie, can you look after St. Petersburg's interests. I won't expect you to brief us when you get back. Just tell us what you can and what we need to know,\" one mayor said.\n\n\"If I can tell you anything, I will,\" the mayor of Sevastopol said, and signed the Secrets Act with a flourish.\n\nThe room quickly emptied. Nelly rearranged it, bringing up a conference table and converting the easy chairs into station chairs around the table, still comfortable, but more utilitarian.\n\n\"How do you get used to having your chair change and move around while you're still sitting in it?\" Vicky asked, glancing down and making a face at her chair.\n\n\"I've made chairs vanish from underneath people who were sitting in my place, but I don't think Nelly has ever done things like this until the last couple of days,\" Kris said.\n\n\"But I've wanted to,\" Nelly said.\n\n\"So, Admiral,\" Kris said, getting down to business, \"what hot potato do you want to dump in our laps?\"\n\n\"There have been reports, none of them creditable, that the Empress was trying to put together a battle squadron for exploration.\"\n\n\"That sounds silly in the middle of a civil war,\" Kris said.\n\n\"Yes. That was why the reports were not creditable. But we now have information.\" The Admiral glanced at his message flimsy. \"One of our 'loyal' stay-behinds got one of their loyal officers drunk. The guy bragged that he'd be navigating eight battleships out to find the aliens and _they_ would make contact for the Empress where even Kris Longknife couldn't.\"\n\nKris swallowed hard but said nothing.\n\n\"We even got a sailing plan. The ships they've pulled out of the Reserve Fleet are eight battleships of the Deneb class. They pack twelve 15-inch lasers.\"\n\n\"The aliens blew Admiral Kr\u00e4tz's four battleships out of space even though they packed the most up-to-date 18-inchers,\" Vicky breathed. \"The Empress is out of her mind.\"\n\n\"No. The Empress is desperate. You nearly killed her the last time you met,\" Admiral von Mittleburg said. \"She wants to make sure she kills you the next time you two face off.\"\n\n\"And she's willing to get all of humanity killed if she has to,\" Kris growled. \"What's the sailing plan? Do you have it?\"\n\n\"It is attached,\" Admiral von Mittleburg said. \"The Navy General Staff would like us to go after them, but it would mean stripping St. Petersburg of most of our battleships.\"\n\n\"Nelly, have you got the sailing plan?\"\n\n\"Yes, Kris. They're heading out deep, and they aren't going near US territory. It will be a long, stern chase for any of the Grand Duchess's ships from here.\"\n\n\"Would any other planet's ships have better luck?\" Kris asked.\n\n\"Brunswick is closer, but they only have four battleships,\" Admiral von Mittleburg said.\n\n\"Kris, we could overtake them without too much trouble,\" Nelly said.\n\n\"Have you figured out a course?\"\n\n\"Yes, Kris,\" and the screen that had the star map suddenly had two lines: one red, the other gold. The red line showed a series of short jumps. The gold one was a lot fewer jumps, but they were long leaps going far out before returning to intersect at a system well away from the Empire.\n\n\"I think the _Princess Royal_ can handle eight obsolescent battleships,\" Kris said. \"We can have the _Intrepid_ at the next pier spin off a pinnace and have it squawk as the _Princess Roy_ _al_.\"\n\n\"Will you be leaving Ruth behind?\" Jack asked, voice level as a plain.\n\n_Oops._ \"No, Jack. I will be taking Ruth with me. I said I was going to nurse her for at least the first year, and I will. Besides, there have been three assassination attempts on Vicky while she was on St. Petersburg. I don't trust Ruth out of my sight.\"\n\n\"Then how about four ships spin off pinnaces, and we take four frigates with us?\"\n\n\"You don't think the _P. Royal_ can take eight old battleships, Jack?\"\n\n\"I expect she could, but this intel is as squishy as it comes. Will more ships join them? Will they have tied in with a couple of honking-big alien warships by the time we catch them? Will we have to take all of them out before they go home to Momma and bring all their brothers and sisters back?\"\n\n\"You want a larger safety margin, my paranoid security chief.\"\n\n\"At least four. I'd prefer all eight, but you're probably right. We have to maintain a presence here. Someone has to stay and respond to the letter we sent the Emperor.\"\n\nKris eyed her brain trust. \"Folks, you won't be all that helpful if we get into a fleet engagement. Do you mind staying behind?\"\n\n\"I was afraid you'd never ask,\" Al said, looking relieved. \"I remember signing on to help you make peace, not blow shit up.\"\n\n\"I have a bad habit of doing that,\" Kris admitted.\n\n\"Which ship do we transfer to?\" Judge Diana asked, hurrying their exit along.\n\n\"Let me talk to my flag captain,\" Kris said. \"Sad to say, she knows more about this squadron than its commander does.\"\n\n\"Kris, I want to go out with you,\" Vicky said.\n\nKris eyed the Imperial side of the table. She'd almost forgot they were here.\n\n\"As much as I hate to miss the fun, I must stay and maintain the fleet,\" von Mittleburg said.\n\n\"I think I will go with the Grand Duchess,\" Admiral Boles\u0142aw said.\n\n\"Mannie, you should go ashore,\" Vicky told the mayor.\n\n\"I was just thinking I would like to go along with you. This might be as much fun as the last time we sailed into battle together.\"\n\n\"Mannie, I have to go to accept the surrender of any ships that decide to throw in the towel. You aren't really needed.\"\n\nMannie gave the Grand Duchess a half shrug. \"Since when have I needed an excuse to go where you go?\"\n\n\"Mannie, this crazy Longknife is taking four small frigates and chasing after eight or more huge battleships. No one in their right mind would go along with her.\"\n\n\"If you are not in your right mind, neither am I. Who knows, I might come in as much handy as you.\"\n\n\"Assuming we can get a few to surrender, I have a job.\"\n\n\"Vicky, I'm not moving unless Her Highness here throws me off her boat.\"\n\n\"Ship,\" Vicky corrected, and turned to Kris with a shrug that was a mixture of hopelessness and delight. \"Do you have room for one more crazy?\"\n\n\"Always room for another in this insane asylum I run,\" Kris said, \"especially a guy as implacable as my Jack.\"\n\nWho leaned over and gave her a quick peck.\n\n\"Well, time's awasting. I have a division of frigates to get away from the pier. Admiral, I may need to top off my tanks and refill the larder.\"\n\n\"I'll get right on that,\" Admiral von Mittleburg said, saluted, and exited.\n\n\"As much fun as you three and your people have been,\" Kris said to her brain trust, \"you need to get ashore as fast as you can.\"\n\n\"We're going.\"\n\n\"So, Vicky, you want to shadow me for the day?\" Kris asked her occasional friend. \"If you think you saw everything we can do that the Empire can't while we were gallivanting around on the _Wasp_ , wait until you see what I've got up my sleeve now. Let's go.\"\n\n# 49\n\nVice Admiral, Her Imperial Grace, the Grand Duchess Victoria of the Greenfeld Empire, and in rebellion against the same, stood as Kris Longknife led a division of four frigates away from the High St. Petersburg Station. It had taken the Wardhaven Princess exactly five hours from the time she ordered go to the minute they were gone.\n\n_How does she_ do _these things?_\n\nVicky remembered the last time she'd shared a ride with Kris Longknife. A lot of water and spilled milk had flowed under a lot of bridges since then.\n\nThey'd fled in the old _Wasp_ across the galaxy, first with enraged aliens after their blood, then in a ship run just about dry of reaction mass, fearing they lacked enough to slow down and refuel. Kris had gone to heroic levels to pull them through.\n\nVicky had gotten drunk and mouthed off to Kris, the best friend she had, then thrown Kris under the bus the first time she had a chance for an interview back in human space. Vicky shivered in embarrassment.\n\nBack then, she'd been a terrified little girl with a target on her back painted there by her stepmother. She'd had no idea how she'd survive from one day to the next.\n\nVicky squared her shoulders. The last year had been an education for her. She'd learned a lot and done a lot, often asking herself \"What would Kris Longknife do?\" She and the people around her had pulled themselves up by their bootstraps and gotten a lot of worlds walked back from the brink of disaster.\n\nHere she was, once again, following Kris around. Only now, she was no confused little kitten. She had made of herself the leader of a cause and half an Empire. It was good to have Kris's help, but it was only possible for Kris to do anything because Vicky had first done a whole lot more.\n\nToday, Vicky spent most of that time shadowing Kris. It was still educational and a lot of fun watching how Kris made things happen.\n\nEarly into the count, Kris ordered Nelly to call Vice Admiral von Mittleburg. \"About that reaction mass and fresh meat and produce. Ya'got any to spare?\"\n\n\"Reaction mass, no problem. Fresh produce, not so much.\"\n\nIt was Mannie who placed a call to his contacts in Sevastopol. They found two shuttles waiting on cargo and leased them. In an hour, they were stuffed with the best Sevastopol's hinterland had to offer. Kris had sent Vicky off with a Supply Corps lieutenant to see how easy it was to resupply a Smart MetalTM ship. Clearly, the Wardhaven Princess had a few things to do that she didn't want a Peterwald looking over her shoulder at.\n\n_Kris is such a stickler when it comes to some things._\n\n_Then again, I'd hate to have Stepmommy dearest know some things I saw on the old_ Wasp _as we shot around the other side of the galaxy._\n\nVicky's jaw dropped as station trucks drove right into the _Princess Royal_ , through holes that suddenly opened into the ship. They off-loaded crates directly into storage rooms. The meat was parked in an open space. A minute later, a refrigerating unit had grown up around it, complete with a door warning DO NOT LEAVE OPEN.\n\nVicky, Mannie, and Admiral Boles\u0142aw were in Kris's much-reduced flag plot as the _P. Royal_ backed away from its pier. The next frigate spun off a pinnace that quickly ducked in to fill the vacancy. It expanded to look hardly smaller than its mother ship and began squawking as the _Princess Royal_ at the exact same moment that Kris's ship went silent in space.\n\nPerfect.\n\nEven Admiral Boles\u0142aw nodded with respect at the smoothly run evolution. He rarely had anything nice to say about the Longknifes. Then again, he'd been raised on Greenfeld propaganda.\n\nAll done, three frigates followed the _Princess Royal_ as she ducked down toward St. Petersburg, then accelerated toward the jump point that had coughed up the Empress not so long ago.\n\nVicky had overheard enough to know that the next system over had one of those fuzzy jumps that Wardhaven ships could see and Greenfeld ships were totally blind to.\n\nKris turned from observing her Captain Ajax do her job perfectly to eye Vicky and her small team. \"I've decided to keep Admiral's Country down here in amidships. It means traffic will be routed past us, but it also means we'll have the strongest protection, both against radiation and lasers for the nursery. I had my brain trust of helpful conflict solvers bunking close to me. Their space is now vacant. Do you have a problem with us keeping Admiral's Country together for all three admirals aboard?\"\n\nAdmiral Boles\u0142aw silently shrugged.\n\n\"I'm fine with that,\" Vicky said.\n\n\"I assume you'll want the mayor to have quarters next to you?\" Kris asked.\n\nVicky nervously glanced toward Mannie. \"Whatever you want,\" she said.\n\n\"Whichever way you want. I have no problem,\" the mayor said.\n\n\"So it's that way,\" Kris said. \"Well, your quarters are going to be next to each other. What you do about it is in your hands.\"\n\nBut the embarrassing problem was not solved just yet. Mr. Smith arrived with Kit and Kat as well as all their gear.\n\n\"Do you want us to bunk on either side of your quarters?\" Mr. Smith asked.\n\n\"They tell me we can have a door directly into your quarters,\" Kit said hopefully to Vicky.\n\n\"Ah, I was thinking Mannie might have one of the rooms next to me. Is that okay with you, Mannie?\"\n\n\"Ah, I have no problem with anything you want,\" he dodged, again.\n\nKris and Jack looked ready to split a gut. _Well, they're married and have a baby, no less._\n\nKris stepped forward and quickly redrew the staterooms around the outer circular passageway. \"Kit and Kat, Mr. Smith, Vicky, Mannie, and Admiral Boles\u0142aw, in that order. Okay?\"\n\n\"Fine,\" Vicky mumbled, trying not to blush.\n\n_Me. Blush! No._\n\n_Then why is my face all warm?_\n\n_It must be hot in here._\n\nIn a moment, Vicky was being led to her quarters for the cruise by a chief and a Sailor who carried her luggage. Her quarters were not quite as spacious as those on the _Retribution_ , but nothing to sneeze at.\n\n\"If we get in a fight,\" the chief started, then corrected himself. \"Rather when being more likely with the Princess in command, the ship will shrink down to Condition Baker, then Charlie, and lastly Zed. By Condition Zed things are quite cozy but we're all at battle stations so it doesn't really matter if our quarters are stowed away into nothing but a few lockers. After the battle, we stretch out again. By the way, Your Grace, there's an app that lets you make adjustments to your quarters. If you want more space in your day quarters, you can make your night quarters shrink to nothing. If you need even more space, you can open up a door to one of the rooms next door or, if those next to you have been warned and are willing, you can make either bulkhead go away.\"\n\n\"You can?\" Vicky said, glancing at the bulkhead between her and Mannie. She quickly shook her head at the thoughts that tumbled into her mind.\n\n\"You can. Would you like the app?\"\n\n\"No,\" she said, but she was nodding yes.\n\n\"Which is it to be, Your Grace?\" the old chief said.\n\n\"Let me have it,\" she said, not looking at him as she spoke. \"If I never use it, it will never matter.\"\n\n\"Yes, Your Grace,\" and Vicky's computer beeped happily.\n\nA moment later, the door closed on her quarters as the two enlisteds left.\n\nAnd another moment after that, there was a knock at her door.\n\n\"Enter.\"\n\nAnd Mannie cautiously stepped in.\n\n\"Are your quarters this good?\" Vicky asked.\n\n\"Not quite. I think you got extra room so you could hold meetings. Are your sleeping quarters through that door?\" he asked.\n\nVicky found her eyes were fixed on the deck. It looked to be nice carpet. \"Yes.\"\n\n\"Wasn't it fantastic the way they moved walls all over the place to load supplies,\" Mannie said. \"I bet your Sailors would love something like that.\"\n\n\"Ours will, once we get this into the Greenfeld fleet.\"\n\n\"Um,\" Mannie said. \"I understand there are all sorts of apps. You can make your own easy chair. Widen or lengthen your bed. All kinds of things.\"\n\n\"What did you think of the way Nelly made entire walls appear and disappear?\" Vicky said, finding she'd kind of rushed breathlessly into that thought.\n\n\"It must be nice what Nelly can do,\" Mannie was quick to point out, then added, \"I understand that there are apps that Sailors can use to do the same.\"\n\n\"You heard about that, too,\" Vicky said, feeling excitement in places she'd ignored for a very long time.\n\n\"Yes.\"\n\n\"I was offered one of those apps,\" Vicky offered.\n\n\"So was I,\" Mannie said softly.\n\n\"It's something you'd want to have around if a ship were damaged, or other things,\" Vicky said, leaving her words vague.\n\n\"Did you take the app?\" Mannie asked.\n\n\"Did you?\"\n\nThere was a pause, before both of them blurted out, \"Yes.\"\n\nVicky turned toward Mannie and found his arms open. It was only two steps before she was in them. His kiss was so soft, just a brush of her lips that shot electricity down her body and left every nerve ending tingling.\n\nShe pressed her mouth to his, claiming him. His hands came to surround her face, moving her ever so slightly so they fitted perfectly together. For a long time, there was only his touch and their kiss.\n\nThen she broke away for air.\n\nAnd her doubts.\n\n\"Mannie, aren't we supposed to avoid tumbling into bed until after the problems from my father's bedtime tumbles are straightened out?\"\n\n\"It's been an awfully long wait,\" he said.\n\n\"It has,\" she agreed.\n\n\"And we are on a Longknife ship, and everyone knows the Longknifes have no morals at all.\"\n\n\"So I was told growing up,\" Vicky agreed again.\n\n\"And four small frigates taking on eight huge battleships could leave us dust, and think of what we'd have missed.\"\n\n\"It would be sad to miss kisses like that one,\" Vicky found herself once more agreeing.\n\n\"And we can make the wall between our night quarters vanish, but still be able to dash into the front room if we have to.\"\n\n\"Yes. Yes, we could. You are a most persuasive politician,\" Vicky said.\n\n\"So kiss me again, and we'll figure out how to run the app later.\"\n\n\"Oh, yes.\"\n\n# 50\n\nVicky felt herself gain weight the next morning as the _Princess Royal_ changed its acceleration from 1.5 gees to 2.5 gees. _For the third time this morning!_\n\n\"Kris, why the different accelerations? I don't see the pattern.\"\n\n\"There's a pattern. I'll tell you, if first you tell me why you're so chipper. You look like the cat that gorged itself on cream.\"\n\nVicky glanced around, startled. None of her entourage was close by. \"I don't look chipper.\" Then she hunched down, and whispered, \"What do I look like?\"\n\n\"Well, if you were a normal person, I'd say you got laid. You, I'm not so sure.\"\n\n\"Kris!\" Vicky said, but kept it low.\n\n\"So, have you and Mannie figured out how to work the app?\"\n\n\"Leave it to you Longknifes to come up with something as disgusting as that.\"\n\n\"Don't say Longknife. Some Sailor figured it out and passed it around. It was only a slight modification to the one to move walls for loading supplies. What's a skipper to do?\"\n\n\"Well, it certainly beats hiding out in the paint locker,\" Vicky finally allowed herself to admit.\n\n\"Did you figure out how to get one king-size bed?\"\n\n\"Kris!\"\n\n\"Well, did you?\"\n\nVicky sighed and gave in to the inevitable. \"After about five seconds of trying.\"\n\nThe Wardhaven Princess had the ill humor to laugh . . . outrageously. When she sobered up, she leaned in close to Vicky. \"I take it that you and Mannie haven't been, ah, huh.\"\n\n\"No, we haven't been either ah or huh. It's the longest string of celibate days I've had since I got my first monthly.\"\n\n\"Can I ask why?\"\n\n\"Policy. Politics,\" Vicky said, spitting the words. \"All this trouble started when my dad tumbled into bed with a whore. Mannie was afraid that if any other Peterwald tumbled into bed, even if the he was a she and the he wasn't a whore but a politician. Is there a difference? Anyway, either way it might not go over so well. Would you believe it, for the good of the rebellion, he walked out of my hotel room?\"\n\nKris had gotten very serious while Vicky talked. Now the Princess nodded along with her words.\n\n\"As hard as it has been, I think he's right,\" Kris said. \"Your father is a selfish, thoughtless fool who looks out only for himself. If you want to get the people behind you, you have to look a whole lot better. Mannie's a good advisor.\"\n\n\"I'm hoping we'll marry as soon as this mess is over.\"\n\n\"You likely couldn't chose a better consort,\" Kris said. \"I'm not trying to make Greenfeld over into some Longknife democracy. I know your feelings on that. But your Empire is going to need a lot more transparency than it has had. It also has to include a lot more perspectives as well.\"\n\nVicky let out a big sigh. \"I hear the same talk from Mannie. I was raised, as you would say, to be autocratic. Well, not so much me to be autocratic but to accept that autocratic Peterwald men were ordained by God to rule, and if God had any problems with that, He'd better get with the program.\"\n\nKris laughed. \"My father's never quite put it into those words, but I suspect he'd agree with the sentiment. As for Ray, he _is_ god.\"\n\nVicky laughed along with Kris but quickly sobered. \"Do we have to become gods to make things work?\"\n\nKris shook her head. \"Do you really think things work because our fathers and my great-grandfather think they're Zeus on Olympus?\"\n\nVicky's shoulders slumped. \"A few years ago, I think I would have believed them.\"\n\n\"Now?\" Kris asked.\n\n\"Not so much,\" Vicky said, and thought for a moment. Then rethought herself. \"Hold it. You were supposed to tell me why our acceleration varies. You told me I had to answer one question. That was a dozen questions ago.\"\n\n\"Okay, I'll let you in on a secret. We go 2.5 gees when Ruth is napping on a high-gee station. We cut down to 1.5 gees when she's up and wants to bounce around. I tell you, that kid is going to have legs that any high-school track coach will die for.\"\n\n\"Ruth is setting our course?\" Vicky said, almost in a shriek.\n\n\"No, not our course, but maybe a bit of our speed,\" Kris said. \"Nelly, you want to explain this to Vicky.\"\n\n\"Not really, Kris. Let's just say I'm juggling my assumptions of when Ruth naps and what accelerations and decelerations we need to make our jumps. Ruth's nap time is a major unknown in my sailing plan, but she's woken up or gone to sleep within fifteen minutes of my estimates. A six-month-old is nothing if not free willed.\"\n\nThey exited the St. Petersburg system at 50,000 kph and accelerated almost constantly to the next jump. Vicky and Admiral Boles\u0142aw were in Kris's flag plot as they approached the jump out.\n\n\"We'll be using the fourth jump,\" Nelly informed Kris.\n\nBeside Vicky, Admiral Boles\u0142aw scowled, but only after covering his mouth with his hand. \"There are only three jumps in this system,\" he whispered to Vicky.\n\nShe covered her mouth, and around a cough said, \"Longknifes see jumps we can't. Both us and the aliens.\"\n\n\"Oh,\" Boles\u0142aw rumbled, and watched as Nelly threaded a needle through space that he would never have guessed was there.\n\n\"Nelly's taking the indirect course to catch your Empress's diplomats,\" Kris announced after the jump was done, and Nelly reported it was good. \"They'll be about ten jumps and three hundred light-years out from Greenfeld when we catch up with them. We will have made six jumps and gone out two thousand light-years in a most erratic course when we run them down. But it will have taken us less time to cover our distance than they will have taken to cover theirs.\"\n\nAdmiral Boles\u0142aw ran a hand through his hair and shook his head. \"The fools. The fools.\"\n\n\"Which fools?\" Vicky asked.\n\n\"Anyone who thought we might stand our ground in a fight with the Longknife King.\"\n\n\"I don't know,\" Vicky muttered. \"We're still chasing after eight big, bad battleships in four small frigates.\" She gave Kris Longknife a dour frown.\n\nThe Longknife Princess glanced her way and returned her a soft, knowing smile.\n\n_Did she hear what I just said? Or worse, could she figure out for herself what I just said. Damn you, Longknifes, and the democracy you rode in on._\n\nThey were now outside the Empire and deep into unknown space. Vicky watched as Kris began to make the division her own. Even as they went on their way, she had them practicing battle evasion drills. At all hours of the day or night, Vicky got use to the bong of General Quarters and the announcement \"Battle Stations. Battle Stations.\"\n\nYes, despite the battle preparations, the journey to the next jump was still at different accelerations and decelerations. When Ruth was awake, it was 1.5 gees. When she napped, it jumped to over two gees.\n\nVicky just shook her head, then thought better of it.\n\n_What will I do when I have a baby at my breast?_\n\nThat brought a whole lot of thoughts Vicky wasn't ready to contemplate. She did ask Kris about the small army of nannies on board.\n\nThe Wardhaven Princess just shrugged. \"I'm trying to have it all. If I can borrow reinforcements, I'm good to go with it.\"\n\nThe nannies weren't the only ones who looked after Ruth. It looked like half the young women in the Marine detachment spent some of their spare time playing with Ruth.\n\n\"That infant has this ship wrapped around her little finger,\" Admiral Boles\u0142aw muttered, but he was smiling softly as he muttered.\n\n\"It's nice that the ship is Smart Metal.\" Vicky grinned back. \"It makes it easier to bend the ship around her tiny finger, don't you think?\"\n\n\"I was going to say I was rethinking having women on a warship,\" the admiral growled, \"but I think one woman on my ship is quite enough . . . for a lifetime.\"\n\nVicky enjoyed the laugh, and the admiral soon joined in.\n\nThey were deep in unknown space, headed for a near jump that would take them back toward human space, when Nelly told Kris, \"We have signal traffic in this system.\"\n\nKris sat bolt upright in her chair. \"What kind of signal traffic?\"\n\n\"They are conventional radio and TV traffic. There are no fusion reactors operational in the system.\"\n\n\"Thank God for that favor,\" Jack muttered.\n\n\"Record all the message traffic, Nelly, and try to triangulate on the source of the signal.\"\n\n\"We are doing that, Kris.\"\n\nKris was up, nervously pacing the length of her flag plot before turning back. \"Whatever it is, we keep going. If it's another sentient race, we'll have to come back later.\"\n\n\"We come back?\" Jack asked, leaving a lot hanging unsaid.\n\nKris paused in her pacing and turned beet red. \" _We'll_ send someone _else_ back to look it over,\" she corrected herself.\n\n_What is that all about?_ Vicky wondered but said nothing.\n\nKris slumped back into her chair, she and Jack still holding aggressive eye contact.\n\n_Maybe Kris Longknife is not the one I want to ask about how to keep a husband happy._\n\nAs soon as they were out of the system, Nelly reported on what they'd found.\n\n\"It is a primitive civilization. They have radio and television but no atomics. I think I made out some small satellites with weak transmitters. None were communications satellites. A few years ago, I would have suggested that they bear watching. However . . .\"\n\n_Did Nelly just leave it to us to figure out where the data leads? Maybe her attitude is worth putting up with, not that Kris would ever allow me such a computer._\n\n\"Yeah,\" Kris growled. \"We can't leave them hanging out there for the bug-eyed monsters to sniff out. Nelly, log your findings and forward them to the King as soon as we are back in communications with the US net.\"\n\n\"Yes, Kris.\"\n\nKris turned to her husband. \"I wonder how Sandy's making out with the cats.\"\n\n\"Better her than us,\" Jack answered.\n\n\"The cats?\" Vicky asked.\n\n\"Another race we stumbled across,\" Kris said, but added nothing more to that bit of vagueness.\n\n\"Greenfeld isn't the only place we're trying to protect,\" Jack said.\n\n\"I hope we can protect ourselves,\" Vicky said. \"I expect that once this war is over, we will need to continue building battleships.\"\n\n\"Or something,\" Kris said, and again left Vicky with more questions than answers.\n\nNow the division of frigates turned back toward human space. For several days, it decelerated more than it accelerated, slowing until it took the last jump at a mere fifty thousand klicks per hour.\n\nThey jumped into the target system to find it empty.\n\n\"Either we're early, or we missed them,\" Admiral Boles\u0142aw said.\n\n\"We're early,\" Kris said, with all the definiteness in her voice that admirals and ship's captains were wont to muster.\n\nAdmiral Boles\u0142aw raised a questioning eyebrow to Vicky, but she just shrugged. \"I trust Kris,\" she mouthed silently to him.\n\nNow both of the admiral's eyebrows went up, but he said nothing.\n\nTen minutes later, the debate ended. Eight Deneb class battleships began jumping into the system. Nelly reported them, one after another.\n\n\" _Aldebaran_ , _Arcturus_ , _Sirius_ , _Rigel_ , _Leonis_ , _Ursae_ , _Vega_ , and _Alpha Ceti_. All present and accounted for, sir,\" her computer said, and seemed to finish with a proper English foot stomp.\n\nVicky wasn't the only one trying to squelch a grin.\n\n\"Thank you, Nelly. Is that what they're squawking, or is that who they are?\"\n\n\"Kris,\" had hurt in it. \"I've checked their registration against their reactors and capacitors. They are Denebs right down to the moonshine stills in the goat lockers. And may I point out to the doubting Thomases among you that my readings are off ships that jumped into the system two hours ago. Right when I expected them to.\"\n\n\"Yes, Nelly. You are magnificent,\" Kris drawled.\n\n\"Kris, I find your attitude unbecoming one of your nobility,\" Nelly said with a sniff.\n\n\"Nelly, I don't have time for this. Save it for later when I do.\"\n\n\"Kris, you never have time.\"\n\n\"Trust me, I will after this is over.\"\n\n\"I don't think Jack or I believe you.\"\n\n\"Nelly, what are the battleships doing?\" Kris said, cutting off further complaining by her computer.\n\n\"They are on course to the nearest standard jump in this system. It will be a while before they discover we are here. You don't really think I would be having this much fun giving you lip if everything wasn't under control, do you?\"\n\n\"I never know with you, Nelly. Not anymore.\" But Kris was now grinning. \"What do you say we do something to prepare for the coming battle?\"\n\n\"Okay, Kris.\"\n\n\"All ships, go to Condition Charlie. All hands should be dismissed into high-gee acceleration stations in rotations. We will go to Condition Zed at any time after we close with the hostiles to four hundred thousand klicks. Captain Ajax, set an intercept course for the division.\"\n\nAcknowledgments came back immediately.\n\n\"Okay, folks,\" Kris said, cheerfully, \"now we make this happen.\n\nVicky and Admiral Boles\u0142aw just shook their heads. Mannie looked from her to the Wardhaven Princess and back to Vicky, clearly puzzled.\n\n\"Eight battleships,\" Vicky whispered to him. \"Four _frigates_. We're doomed.\"\n\n\"Watch and learn,\" Kris said. \"Watch and learn.\"\n\n# 51\n\nThe four frigates closed inexorably on the battle line. Vicky watched the screen on Kris's flag plot as the lines representing the two of them closed, heading for a single point.\n\nBefore any of them arrived at that point, there would be mayhem and death.\n\nVicky had already been dismissed back to her room to find two Smart MetalTM high-gee stations: one for her and one for Mannie. Kris had told her that she'd likely want to strip before she got in the station. \"They fit you like a second skin.\"\n\nVicky and Mannie did strip, but only after taking a few moments for themselves did they take their seats in the stations and order them to close up.\n\n_Kris, you are so mean, to lead a poor girl into temptation when she knows she's going to be blown to dust._\n\nStill, it was nice to have the scent of Mannie on her as she prepared to face whatever death wish Kris had for them all.\n\n_Strange. None of the crew seemed at all worried about what's ahead. Does Kris know something she's not telling?_\n\n\"Range is four hundred thousand klicks,\" Nelly announced.\n\n\"All ships, go to Condition Zed,\" Kris ordered.\n\nVicky had been told her quarters would disappear when the ship shrunk to this. Still, she was surprised as Kris's flag plot shrunk to half its previous size.\n\n\"If I need to pee, can I use the bathroom in your night quarters?\" she asked Kris.\n\n\"My night quarters are boxed up, too, Vicky. If you need to pee, do it in your high-gee station. It was made for that. Late in my pregnancy, when I was peeing all the time, it was kind of nice to go into a battle and have the high-gee station take care of that for me.\" Kris actually chuckled at that.\n\n\"You've got to be kidding.\"\n\n\"Nope. Trust me, when you've got a little one bouncing on your bladder, you'll think time in a high-gee station is worth a fight or two.\"\n\n\" _I_ will use a _uterine_ replicator,\" Vicky spat, and gave Mannie the evil eye.\n\n\"That sounds great by me,\" he said, edging his station a bit away from Vicky's.\n\n\"Kids, we've got a battle coming up here,\" Nelly said. \"Kris, when do you want to tell those battleship folks they ain't going anywhere?\"\n\n\"I was figuring on two hundred thousand klicks.\"\n\n\"Okay.\"\n\n\"Are we going to start shooting then?\" Jack asked.\n\n\"Nope. I don't want to fire first.\"\n\n\"Two hundred thousand klicks?\" Admiral Boles\u0142aw asked before Vicky got her mouth around that huge number.\n\n\"Yep,\" Kris said.\n\n\"Those battleships' 15-inch lasers can reach out eighty thousand klicks. Your pulse lasers, even if they're 24-inchers, don't have that range,\" the Greenfeld admiral pointed out.\n\n\"Yep, it would be a real pain if we were only packing pulse lasers, now wouldn't it?\" Kris answered with a question of her own.\n\n\"So?\" Vicky said, when Admiral Boles\u0142aw proved to be speechless.\n\n\"Do you want me to tell you now, or wouldn't you rather see how this all ends up?\"\n\nVicky blinked several times as she took that in and parsed the sentence. \"Can you tell me if we'll be alive when this is over?\"\n\n\"Most definitely. You will live to enjoy tonight,\" Kris said with one of those smiles a newlywed bride tends to give the gal who caught her bouquet.\n\nVicky settled back in her station and prepared to see the rabbit this Longknife was clearly intent on pulling out of a hat. She hoped the poor thing didn't mind being held by its ears. Whatever Longknifes did, they did with a flourish.\n\nThe battleships stayed silent, intent on their course and ignoring the small combatants approaching them. At two hundred thousand klicks, Kris stood up from her high-gee station.\n\n\"Hold it, you're in uniform,\" Vicky snarled.\n\n\"Yep, I figured I'd better be. Who surrenders to a gal in the nude?\"\n\n\"I'm supposed to take their surrender!\" Vicky snapped.\n\n\"We can focus the camera on just your face and keep the station out of the picture.\"\n\n\"I'm getting in uniform,\" Vicky grumbled, and got her station in motion.\n\n\"Your quarters are all folded up,\" Mannie pointed out.\n\n\"Unfold them,\" Vicky growled.\n\n\"I can do that,\" Nelly said helpfully.\n\n\"Do it,\" Kris said. \"Oh, Vicky, you don't want to wear ribbons or anything with sharp corners. I did once when we went to high-gee maneuvering, and one of my boobs was black-and-blue for a week.\"\n\nMannie followed Vicky out the door. They found themselves in a very small space that had Vicky's uniforms and a chair. Quickly both were out of their eggs and a delightfully naked Mannie was helping Vicky into a shipsuit of Navy blue with the proper loops to feed her vice admiral's shoulder boards through.\n\nWith a glance at a mirror that suddenly materialized, Vicky decided she was properly dressed to attend a funeral or accept the surrender of part of the Empress's fleet. Done, she gave Mannie a quick kiss, settled back into her egg, and motored quickly back to flag plot.\n\n\"You ready?\" Kris asked.\n\n\"Yes,\" Vicky answered curtly. She had a bone to pick with Kris. A whole pile of bones, really. She'd just have to wait until things slowed down.\n\n_Yeah, right._\n\nAgain, Kris stood up. \"Open a hailing frequency,\" she ordered.\n\nA bored lieutenant filled the screen. \"Yeah.\"\n\n\"We are Admiral, Her Royal Highness, Princess Kristine of the United Society commanding Frigate Squadron 22. We will speak to your captain.\"\n\n\"Admiral Dirk's busy. You can leave a message,\" was surly to the core.\n\nKris gave the lieutenant the look. The look that Admiral Kr\u00e4tz had and that Vicky had done her best to learn. The junior officer should have been crawling under his workstation to get away from it.\n\nBut, then, he wasn't meeting Kris's eyes.\n\n\"Sorry, lady. Don't bother us,\" he said, and cut the comm.\n\nKris sat back down into her egg.\n\n\"That didn't go very well,\" Jack said.\n\n\"It went exactly the way I expected it to,\" Kris said. \"Nelly, tell me when we hit the hundred-thousand-klick mark.\"\n\nIt wasn't long before Nelly announced they were there.\n\nAgain, Kris stood and ordered hailing frequency open.\n\nThe screen remained blank.\n\n\"We know you are listening,\" Kris said in a voice that was measured and cold as death. \"Know this, if you do not vent your reactors to space, discharge your capacitors, and prepare to be boarded, you will all die.\"\n\n\"That's a laugh,\" came from a screen that suddenly showed a pudgy man in an Imperial Navy's vice admiral's uniform. \" _You're_ threatening _me_? I got your piss pots outnumbered two to one. Let me tell _you_ this. _You_ turn around and run away home, little girl, and _I'll_ let _you_ live, okay?\"\n\n\"You have been warned,\" Kris said, but she was talking to a blank screen.\n\n\"Do you think it would help if I talked to them?\" Vicky offered.\n\nKris shook her head. \"I doubt it. Well, at least my message got through to all the ships. We'll see how things work out after I blow the flagship to hell.\"\n\nVicky found that both of her eyebrows were up so far they were about to the back of her head, but Kris only gave her that fey smile and turned to her screens to watch the developments in the two fleets.\n\n\"They've armed their lasers,\" the Wardhaven Princess muttered to herself. \"I guess it's time to load ours. Nelly, advise Captain Ajax to charge the _Princess Royal_ 's capacitors.\"\n\n\"What about the other ships?\" Nelly asked.\n\n\"No. Tell them to stand by but not load until I give the word. Our other three ships are not weapons free until I order them free.\"\n\n\"Yes, Admiral,\" said the computer.\n\nAdmiral Boles\u0142aw glanced at Vicky. Consternation flooded his face. Eight battleships against four frigates was suicide. Eight of them against one frigate!\n\nIt boggled the mind.\n\nVicky just shrugged. _It's way too late to get out of this now._\n\n\"We're coming up on eighty thousand klicks to the flagship,\" Nelly announced.\n\n\"Is the armor solid and tight in place, Nelly?\"\n\n\"Now you ask, Kris? Of course it is, or I would have mentioned it sooner.\"\n\n_Had the great Kris Longknife actually shown a bit of nervousness?_\n\n\"We're coming up on eighty thousand klicks between us and the last ship in the battle line. Kris,\" Nelly said. \"Should we be jinking or something?\"\n\n\"Nope, we let them plant one on our chin,\" Kris almost whispered.\n\n\"The hostile battle line is firing on us, Kris. All ninety-six 15-inch guns at little old us,\" Nelly reported.\n\n\"Is our hull overheating?\"\n\n\"No, Admiral,\" Nelly said. \"We're radiating the laser energy back into space. No burn-through. Quite a few missed. I don't think they've got their lasers bore sighted all that well.\"\n\n\"Thank you, Nelly. Captain Ajax, you may fire on the flagship. All the forward battery if you please.\"\n\nAround Vicky, the _Princess Royal_ swung quickly but smoothly to bring herself bow on to the battle line. The light didn't even dim, but Vicky spotted a representation of a ship on Kris's board. One of four. Only this ship showed twelve beams leaving it.\n\n\"You've got twelve pulse lasers?\" Vicky asked in a whisper.\n\n\"No, I've just hit the _Aldebaran_ with twelve 22-inch lasers at well inside their normal range. I'm afraid that battleship is toast.\"\n\nOn another screen, said battleship was indeed toast. Its ice armor boiled off it, making the lasers piercing through it visible. The huge battleship looked like a butterfly mounted by a pin-happy six-year-old.\n\nAnd then the battleship was no more. Where it had been was a boiling, roiling ball of fire and gas; and then there was nothing. Only a few bits and pieces were visible in the light of the lasers exciting the steam that had once been ice armor.\n\nThen the lasers cut out.\n\n\"Holy Mother of God,\" Admiral Boles\u0142aw said. \"What just happened?\"\n\n\"The battle line fired on the _Princes Royal_ ,\" Kris said. \"Our new crystal armor took their light in, diffused it around the hull, and radiated it back into space before it could do any damage. Then my battlecruiser returned fire on the _Aldebaran_ with its forward battery of twelve 22-inch lasers. You saw how well its old-fashioned armor stood up to those new lasers.\"\n\n\"Longknifes have all of that?\" Vicky asked, her voice a strained whisper.\n\n\"The US does, and quite a few other alliances who have joined us in defending the Alwa system against the aliens. Two years ago, I would have considered an 18-inch laser the top of the line. We expect to have 24-inch lasers in the near future.\"\n\n\"Our battleships are like dust to your frigates,\" Admiral Boles\u0142aw whispered.\n\n\"Battlecruisers,\" Kris corrected. \"Your Grace, please see what you can do to get those other poor doomed souls to surrender before I have to slaughter more of them.\"\n\nVicky stood and addressed the screen. It was amazing how quickly the other ships vented their reactors to space, emptied their capacitors, and dropped all acceleration to drift in space.\n\nVicky quickly negotiated a formal surrender with the rear admiral commanding the second division. US Marines boarded the remaining seven ships and found their crews quite docile. Vicky asked for the loan of the Marines to keep them that way, and Rear Admiral Boles\u0142aw volunteered to take command of the squadron and see that they turned around and headed back for Brunswick.\n\n\"I'll loan you the _Resolute_ to serve as sheep dog,\" Kris offered, and Vicky was glad she didn't have to ask. As much as she expected the fear of the Lord that Kris had put into them to last until they got to rebel territory, it was still good to have a ship on standby if things got rowdy.\n\n\"I thought you were crazy when you said you were ready to take on a battle squadron with one frigate,\" Vicky said. \"You weren't crazy at all, were you?\"\n\n\"Nope, one battle line, one battlecruiser. Even odds. Right, Jack?\"\n\n\"Yes, love, but I'm still glad, Admiral darling, that you took along reinforcements. It could have gotten so much worse. If they'd had a couple or a dozen of those alien warships of five hundred thousand tons and a whole wad of lasers, it could have gotten a bit out of hand even for you, my dear.\"\n\n\"You notice how he lays the love on with a trowel,\" Kris said to Vicky. \"He knows we were both right, but he's apologizing nevertheless. Smart husband.\"\n\n\"Mannie, are you taking notes?\" Vicky asked her mayor.\n\n\"Copious notes. Copious and enlightening,\" he answered.\n\n\"Now, Captain Ajax, are you ready to go back to Condition Able?\"\n\n\"How about we stay at Condition Baker until we see what these docile hostiles do for a day or two?\"\n\n\"Sorry, Captain, but we need to be heading back to St. Petersburg as fast as we can. I would suggest you avoid making the jump out of here that we came through until these folks jump out themselves, but then we need to beat feet for home.\"\n\n\"Aye, aye, ma'am. As soon as we are out of their range, I will return the ship to Condition Able\u2014Royal Plus Size, but not until.\"\n\nKris sighed. \"Sometimes it's hard to remember that you might be an admiral, but aboard ship, you're just a passenger.\"\n\n# 52\n\nThe trip back took a bit longer. Ruth had to put up with less than two gees and that only when she slept. Vicky found herself wishing the voyage would never end.\n\nOn the way out, Admiral Boles\u0142aw studiously did not notice or comment on Vicky's being, as Kris put it, chipper. Mr. Smith and the assassins did not make any effort to enter Vicky's quarters. Even Kat refrained from rolling her eyes and talking in salacious French.\n\nShe and Mannie could be themselves, and Vicky very much liked what they could be when it was just them. Mannie was neither so athletic nor so \"creative\" in his lovemaking as some Vicky had shared her bed with. Instead, she found herself in a place she'd never been before. Loved. Yes. Relaxed? Definitely. Satisfied? Totally.\n\nCould something like this last forever?\n\n_I sure would like to try._\n\nHer idyllic interlude didn't last much past their jumping into the system one jump out from St. Petersburg. A message was waiting for them.\n\nThe Emperor had accepted Cuzco as the neutral ground for the negotiations and set the date for three weeks from his date of acceptance. Time had been lost getting the message out to St. Petersburg, and more before Vicky got the message.\n\n\"Can we make it?\" Vicky asked Kris.\n\n\"Nelly?\" she said, passing it along to her computer.\n\n\"Ruth ain't gonna like it, but we can make it.\"\n\n\"Then let's put pedal to the metal and get things moving.\"\n\nIt hadn't been as easy as that.\n\nKris had to pick up the three ships still at St. Petersburg; _Defender_ had gone on ahead with Kris's brain trust. The three frigates rose from High St. Petersburg Station looking like strangely swollen fish. Then Kris let Vicky in on another one of her secrets. The swollen ships were in tanker mode, bringing up reaction mass to refuel the ships with more miles on them.\n\nThey also brought a load of fresh meat and produce, which was transferred over by the pinnaces when they brought the reaction mass.\n\n\"We got to get some of this stuff,\" Mannie whispered in awe.\n\nVicky just shook her head. The Empress had destroyed so much of Greenfeld, and what she had left was rapidly becoming little more than a series of start-up colonies.\n\nKris had insisted that killing the Empress was not an acceptable negotiating position. More and more, Vicky wanted to offer it as her starting position . . . and then negotiate _back_ from there.\n\nLike throw in torture before finally killing her.\n\nMannie had been teaching Vicky negotiating techniques. It seemed you needed something to do in bed between doing what you were in bed to do. He'd taught Vicky about how you were supposed to bargain. You stated your opening position. They stated theirs, and you bargained _toward_ each other. It wasn't considered kosher to bargain _away_ from your opening position.\n\nVicky was ready to make special allowances for her stepmother.\n\nThey pulled up to High Cuzco Station with a cranky Ruth and an hour to spare before negotiations were supposed to start.\n\n\"Supposed to\" were the operative words.\n\nBoth parties had sent advanced teams to Cuzco to prepare for the actual bargaining. So far, Vicky's advanced team had dodged one bomb and several drive-by shootings.\n\nSomeone on Wardhaven must have been following developments. Several crates of Spidersilk body armor had arrived; so far, they had saved three lives.\n\nWas Vicky the only one who found it strange that all the assassination attempts were on her advance team?\n\nWhen Judge Diana took a bullet to the heart that was stopped by Spidersilk, it became clear that no one was safe.\n\nEven at the risk of life and limb, the negotiations were going depressingly slow.\n\nThe Emperor had reserved a resort hotel in the hills beyond Machu Picchu, Cuzco's capital. He, or maybe the Empress, insisted negotiations take place there. Vicky's advance team had started with a location on the High Cuzco Station where all the principals could retreat to their ships after each bargaining session.\n\nWhen the Imperial side rejected that option, Vicky's team had fallen back to using the _Princess Royal_ for the negotiations.\n\nThat had also been rejected.\n\nThe Emperor insisted he needed the open air the resort allowed.\n\n\"I don't think it's the open air, so much as all the places assassins can hide,\" Vicky told Kris with a frown.\n\nKris turned to her brain trust, who had greeted the _Princess Royal_ when she docked. \"Okay, how do we arrive at an agreement to end this civil war when we can't even agree on where we talk?\"\n\nThe three exchanged glances, then Al shrugged and spoke, apparently for all. \"You make a concession to get the bargaining going and assume that he will accept that he owes us one.\"\n\n\"I don't think the Empress will ever concede that she owes anyone anything,\" Vicky snapped.\n\n\"You are probably right,\" Diana said, \"but nothing ventured, nothing gained.\"\n\n\"Assuming someone doesn't blow us all to kingdom come,\" Jack pointed out.\n\n\"There is that,\" Bill conceded.\n\nKris ran a worried hand through her hair. She needed a haircut, but her hair had still not recovered from her pregnancy. _Thank you very much, Ruthie._\n\n\"Okay. Jack, get your Marines down below, and go through every inch of that place up to ten thousand meters in the sky and down a thousand or so in the dirt. If you say it's safe, we'll assume we can keep it safe and risk it.\"\n\n\"And Ruth?\" Jack asked.\n\n\"Stays up here safe and sound. I'll pump milk for her nannies to give her. That acceptable to you, oh my ever-cautious security chief?\"\n\n\"I would rather you and Ruth Marie were both safe up here,\" Jack said. \"Whatever happened to telecommuting? Can't we do this thing via screens?\"\n\n\"You need to see each other's body language and inflections,\" Diana said.\n\n\"As well as the trust matter. If you don't trust them enough to meet them face-to-face . . .\" Al left hanging.\n\n\"But I don't trust them,\" Jack pointed out.\n\n\"Yes. Maybe we don't either,\" Bill said, \"but we still have to show we are willing to trust them.\"\n\nWith a scowl, Jack stood and started talking into his commlink. \"Give us a day to see what we can do down there. We'll meet again this time tomorrow.\"\n\n\"Very good,\" Kris said.\n\nA day later, Jack was back. \"There are no surprises underground. Each building has its own power supply, so there are no conduits. Even the sewage is treated in each cottage and the main lodge. All that is underground is a network of water pipes. I have to admit, the place is secure underground. It's aboveground that worries me.\"\n\n\"How so?\" both Vicky and Kris asked.\n\n\"Trees. Lots of trees. All of them tall, so there aren't a lot of bushes; still, there are a lot of trees. Then there are the mountains. There are some serious mountains around that place. Someone with a rocket launcher could do a lot of damage fast.\"\n\n\"Can you stop that?\" Kris asked her husband.\n\n\"Nelly, can you make us an aerostat out of Smart Metal?\" Jack asked.\n\n\"Yes. Do you want it powered?\"\n\n\"I want it tethered, but with enough power to handle any strong winds. Make that two of them. We can use one to rove the area.\"\n\nJudge Diana was looking on, having gained a new respect for security since her spidersilks stopped a bullet. \"We may have to include reps from the Emperor on board your balloons, Jack, if we're to get him to agree to a balloon out his window.\"\n\n\"That's your job,\" Jack said with an impish grin. \"It must be a whole lot harder than my simple job of keeping you all safe.\n\nDiana snorted and left to talk to her associates and arrange for a new meeting.\n\nVicky frowned. \"I like your idea of balloons looking for trouble, Jack. But if we have to put some of the Empress's assassins aboard, will they be any good?\"\n\n\"You let me worry about that,\" Jack said. \"Nelly, I'm going to need some really tiny eyeballs in the gondolas of those balloons with a tight beam link to the _Princess Royal_.\"\n\n\"I think that can be arranged,\" Nelly said, and you could almost hear a proud chuckle in her voice.\n\nIt took three more days of haggling, which Jack put to good use securing the resort, but in the end, Kris's and the Emperor's negotiators all agreed to where they'd be. Who got what quarters, when and where they'd actually sit down, and how the room would be arranged for that sit-down.\n\n\"I can't believe it's taken you this long just to agree how you will talk to each other,\" Vicky said to Kris and her troika when they reported back.\n\n\"We expected it would take longer,\" said Bill, the man with experience arbitrating between different sides.\n\n\"So we can start now?\" Vicky asked.\n\n\"Nope,\" Diana said. \"Now they tell the Emperor that everything is ready for him and the Empress.\"\n\n\"They aren't here?\" Kris snapped.\n\n\"No. They're back on Greenfeld waiting to see if we can agree to meet.\"\n\nKris shook her head. \"When will the Emperor and Empress arrive?\"\n\n\"Not for a week, at the least,\" Diana said.\n\nVicky blew out a breath as if she'd been punched in the gut. Shaking her head, she said, \"More and more, I'm coming to wish I'd just finished the Empress off when I had the chance.\"\n\n\"You think it would have ended the civil war?\" Kris asked her.\n\n\"I don't know,\" Vicky admitted. \"So much of this mess comes from her family. If she'd died, my best guess would be that my father, our Emperor, would somehow die, oh so sadly, and the Bowlingames would put their little baby on the throne, with Grandpapa Bowlingame the all-powerful regent. Hmm, I hadn't actually thought that through until you asked, Kris. Thanks.\"\n\n\"Sorry. That's not a pretty picture.\"\n\nVicky sighed. \"But it is the picture my father has allowed to be painted. So, do you play bridge? It looks like we've got plenty of time on our hands.\"\n\n# 53\n\nAdmiral, Her Royal Highness Princess Kristine of the United Society didn't find time to play bridge with Her Imperial Grace, the Grand Duchess. The two of them were juggling too many porcupines.\n\nIt seemed like everyone who was anyone had heard about the negotiations and wanted in on the fun. Admiral Waller arrived from Bayern with a battleship full of staff and a whole lot of retired admirals and wives who didn't look to be into shuffleboard for their golden years.\n\nA woman from Metzburg showed up with a small army of business associates to make sure their interests were fully considered. To make sure Metzburg didn't get overly considered, right behind them was a young man from Brunswick with a similar mob to assure their concerns weren't lost in the shuffle. St. Petersburg brought in reinforcements, and those were only the three largest delegations. Just about every planet that had sworn fealty to the Emperor through Vicky wanted someone sitting at her elbow on Cuzco.\n\nThus, one morning, Vicky stormed into Kris's quarters in somewhat of a panic. \"How do you democrats handle all these people with all their competing agendas? Hell, half of them don't know what they want.\" Vicky came to a halt in front of Kris, shaking her head. \"No. No. No. First problem. Kris, do you have any idea where I put them all?\"\n\nKris had no clue how to answer either question, but Nelly did know where Judge Diana lived. A few moments later, Diana walked into Kris's quarters on the _Princess Royal_.\n\nVicky was bringing the judge up to date on her growing list of problems before she even sat down.\n\n\"Hmm, and I thought we were just bargaining for half the cottages and rooms in the lodge to keep down the number of courtiers and courtesans following in the Emperor's wake. Okay, we do have half of the space at the resort. How many people do you have asking for rooms?\"\n\nVicky had her computer transfer the list to Diana's computer.\n\nOf course, Nelly picked up on the transfer. \"Would you mind if I tried my hand at matching people to space?\"\n\n\"Please. Go right ahead. But remember,\" Vicky said, \"some of the older and wealthier will expect that their status gives them better accommodations than those less privileged.\"\n\n\"I am well aware of human vanity,\" Nelly answered dryly.\n\n\"Is she always like this?\" Vicky whispered to Kris.\n\n\"Only when a human fails to remember they're dealing with the Magnificent Nelly,\" Kris said, no hush in her voice.\n\n\"So you're actually learning, Kris,\" Nelly observed.\n\n\"I'm learning,\" Kris said. \"I just never know what you've figured out about the world and what you haven't gotten to yet.\"\n\n\"Harrumph,\" came from Nelly, but no further comment for a moment, then, \"I think we can accommodate most of the ones who will be walking around with their noses in the air, assuming they're willing to double up or take in some of those lesser mortals to fill their spare rooms. The rest are going to be a problem, and it will likely get worse. Vicky, the representatives present only account for about half of the planets fighting on your side. If more show up, you will need more quarters. A lot more quarters.\"\n\nVicky turned to Kris. \"Any more rabbits left in that hat of yours?\"\n\nKris hollered, \"Jack, how are the Marines set for tentage?\"\n\n\"I was planning on using them for Marines,\" Jack said, only slightly cranky as he glanced up from a screen he'd be intent on.\n\n\"Ookaay. Nelly, who on Cuzco rents tents?\"\n\n\"I already have that information, Kris. Jack had Sal check on that yesterday.\"\n\nKris raised an eyebrow at Jack.\n\n\"I want to cover the four main entrances to the lodge with some sort of tent, either with sides or extended sloping roofs so no one would have a good long-range shot at you.\"\n\n\"Okay, Vicky, this is a job for your advanced team. There must be a Navy supply type who can wrestle this problem to the ground.\"\n\nVicky tapped her commlink. \"Admiral Waller, have I got a problem for you.\"\n\nDiana looked like she might take this as a chance to step out; Kris rested a restraining hand on her shoulder. \"I think we've only tackled the easier half of her problems. I suspect the other half is a whole lot more intractable.\"\n\n\"That could be very true. May I call in my associates?\"\n\n\"Please do.\"\n\nA few minutes later, Vicky was off the phone, and Kris's troika was seated around a conference table that had suddenly become round. Beside Kris, a lovely cloisonn\u00e9 teapot appeared with five delicate white-and-gold teacups and saucers.\n\n\"Is Nelly going to turn this into an honest-to-God English afternoon tea?\" Diana asked.\n\n\"Give me a moment,\" Nelly said. \"The cook in the Forward Lounge is still cutting the crusts off the cucumber sandwiches.\"\n\n\"She's not!\" Diana exclaimed.\n\n\"She is,\" Nelly said, as a plate of scones with clotted cream and several fruit marmalades appeared on the table. A moment later, Nanaimo bars, tiny pecan pies, and sweet lemon bars followed. It was a minute more before the cucumber sandwiches on thinly sliced bread finished up the presentation.\n\nKris began pouring tea, while Diana got the different plates of goodies circulating around the table.\n\n\"Am I being fattened up for the kill,\" Vicky said warily, \"or is this already my last meal?\"\n\n\"I assure you,\" Kris said, passing Vicky a teacup and saucer, \"we would not have gone through all we have only to knock off one of the two principals in this civil disturbance.\"\n\n\"Aren't we a war anymore?\" Vicky asked, taking a sip. \"This is delicious tea.\"\n\n\"The proprietor of the Forward Lounge procured it,\" Nelly sniffed. \"Of course it's delicious.\"\n\n\"Want a sip?\" Vicky offered, impishly.\n\n\"I assure you, if Kris and Jack rutting around has not tempted me to take on flesh and blood, tea and clotted cream haven't a chance.\"\n\n\"Nelly!\" Kris half shouted.\n\nNelly didn't answer Kris, but the entire table was laughing. Vicky and Bill spewed tea from their noses and were gasping for breath, grasping for napkins and still laughing.\n\n\"Nelly, I'm fingering the button.\"\n\n\"Kris, I apologize,\" Nelly quickly said. \"I should not have shared your intimacy with anyone beyond you and Jack. Now, can I please stay and do what I do best, like finding ways out of the messes you get us into?\"\n\nThe speed with which Nelly went from contrite supplicant to hard bargainer gave everyone whiplash, but Kris found herself taking a deep breath and bringing the room back to the matter at hand.\n\n\"Okay, my brain trust. Vicky has a mob. One might even call it a jungle. None of these people have any experience at give-and-take or at maintaining transparency while negotiating their way from A to Z. How do we help them get the skills they need, the practice with those skills, and, in the process, manage things so that they don't kill each other? Or worse, slip out in the night to conduct their own negotiations with the Empress. We can still be stabbed in the back.\"\n\n\"I hadn't thought of that,\" Vicky said.\n\n\"We had,\" Al said. \"You have the north side of the resort, cabins, and lodge. They have the south side. You share the swimming pool. I suggest you hire lifeguards from the local economy to assure those drowning are actually rescued and not loaded down with stone so the body won't be found.\"\n\n\"You make this sound like war,\" Vicky said.\n\nBill moved to answer her. \"I believe an ancient wise man back on Earth said, 'Warfare is the continuation of diplomacy by other means.' You must learn to consider diplomacy a continuation of war by more peaceful means. Both are ways of resolving your conflicts. Hopefully, diplomacy will have a lower body count.\"\n\n\"So how do we do this?\" Vicky asked.\n\nQuite a few hours later, Vicky brought in several key representatives from her most important planets. Their talk went long into the night. Next morning, tents as well as a large pavilion were delivered to the resort grounds. Hired locals began to lay down floors, unroll rugs, and put up canvas.\n\nThat morning, Nelly also published the first issue of _The Grand Duchess's Imperial Gazette_ , recording for all to see just what had been going on in Kris's quarters the day before. The paper included no videos but did have attached articles selected by Kris's brain trust on how to get to yes with helpful hints on how not to kill anyone while doing it.\n\nNelly's _Gazette_ was downloaded over a thousand times. Apparently, Vicky's allies weren't the only ones interested. Quite a few people in Cuzco wanted to know what was going on, likely with hopes on how they might make some money on the visitors.\n\nTen of the Emperor's advance team downloaded copies as well.\n\nAt least four of those were known to be in the Empress's pocket.\n\nKris just shook her head when Nelly told her that. Then Nelly added that she and Jack had come up with their own idea of how to keep Kris safe in the negotiations. \"A week ago, we sent off to Wardhaven for ten thousand tons of Smart MetalTM.\"\n\n\"Isn't that a bit much, Nelly?\"\n\n\"Jack intends to present you your very own castle; outlandishly palatial with plenty of space and security every which way you look.\"\n\nKris thought about that. \"I still won't take Ruth down there.\"\n\n\"I'd never suggest it.\"\n\nSomeone, likely King Ray or Mac, had assigned a Royal US courier ship to stand by at High Cuzco Station even before Kris's squadron arrived. She'd wondered when it suddenly left. It must have made some fast jumps; the newly commissioned battlecruiser _Dauntless_ soon showed up sporting quite a big bump around her middle.\n\nShe carried thirty thousand tons of Smart MetalTM for Kris and orders to join her squadron.\n\nRiding as passengers on the _Dauntless_ were quite a few architects and engineers. They messaged ahead from the jump some draft plans for a proposed castle. From the looks of it, it had to be based on the mad dreams of some insane prince from old Earth. Still, it was sure to impress.\n\nKris and Nelly made a few mods under Jack's cautious eye and sent them back. Even as the _Dauntless_ went into orbit, it spun off a lander that converted to a glider once it finished entry into Cuzco's atmosphere and came to a dead stick landing in the parking lot next to the lodge, light as a feather.\n\nNext morning, presto, the castle was ready for occupancy. Oh, and the _Imperial Emperor and Empress_ had jumped into the system.\n\n\"That oversize barge was the _Golden Empress_ last time we laid eyes on it,\" Vicky snarked. \"I guess under my father's eye, even she has to admit that she's not what she wants to be. At least not yet.\"\n\n\"You want to come see my palace?\" Kris asked Vicky.\n\n\"I've already seen pictures of it. It looks like a castle to me. That curtain wall around it is sure to keep the average assassin out.\"\n\n\"Only the average one, you think.\"\n\n\"My dear darling stepmum can afford to hire the best. After all, it's only my father's tax money,\" came out with pure Imperial Grand Duchess hostility.\n\nDespite her attitude, Vicky was only too eager to see Kris's new digs. They took the beanstalk down with a company of Marines and most of the staff of the Forward Lounge. A heavily armed motorcade was already at the station to whisk them off to the forested hills above Machu Picchu. There, waiting for Kris, was an amazing sight.\n\nA hill, just beyond the north edge of the lodge's parking lot, had been cleared of trees to provide a spectacular view for the restaurant on its third floor. Now its view was of a fairy castle, complete with spiraling towers along a curtain wall.\n\n\"We'll post sentries in the towers, but the real watch is the sensor suite,\" Jack pointed out.\n\nTheir limo drove straight through the gate and up to a covered portico. Jack opened the door for Kris and Vicky, and the Royal Wardhaven Princess led the Imperial Grand Duchess through huge double doors that appeared to be heavy oak but swung open as if they weighed nothing.\n\n\"The magic of Smart Metal,\" was all Kris had to say to get an awed \"Wow\" from Vicky.\n\nThe massive foyer showed a plush red carpet underfoot. The walls were a bright cream satin wallpaper with gold and silver patterns running through it that somehow formed a vision of fields and mountains. Lit sconces added a soft glow. Overhead, a vast crystal chandelier sent dancing rainbows around the room.\n\n\"The Empress is going to be so jealous of this,\" Vicky said, her mouth hanging open. She shut it to grin at Kris. \"And Dad won't believe you did this.\"\n\n\"Oh, he will,\" Jack said. \"You've seen how the _Princess Royal_ could be decked out. So have they.\"\n\n\"Oh, right.\"\n\n\"There's more,\" Kris said. She pointed to the left. \"Those three lovely arched doorways lead to a banquet room. To the right, you have three more doors and a ballroom. Diana told me that much of the work of a conference like this is done over drinks while socializing. We will be entertaining in something a lot fancier than a rustic dining room and great hall with the heads of horned animals hanging on the wall, don't you think?\"\n\n\"Very much,\" Vicky said. \"What's up the spiral staircase? I can't wait to see you and Jack make an entrance down it. I do hope that you brought a few ball gowns.\"\n\n\"Hmm, that's interesting,\" Jack said with a loving leer. \"A ball gown with miniature medals and all your orders on it.\"\n\n\"With the weight of all that, I'd be risking a wardrobe malfunction?\" Kris said.\n\nJack gave Kris a hopeful look.\n\n\"Down, boy. Down,\" Kris said, and changed the subject. \"The staircase leads to the upper floors that are reserved for bedrooms and essential services, like the Marine barracks and nano defense center. If you try to go up the staircase but aren't an authorized person, you will be warned at the first step. If you keep going, there are autocannons covering the stairs halfway up.\"\n\n\"You really don't want to go where you aren't wanted,\" Jack said.\n\n\"Kris, can I visit you for some girl time?\"\n\n\"Have your computer talk to Nelly, and we'll see about it.\"\n\nVicky rolled her eyes but didn't voice any objections.\n\n\"What's through those four double doors beside the stairs?\"\n\n\"Those lead to the Hall of Mirrors, where we'll be meeting. Beyond that are the quarters for my brain trust and their people. Diana said they needed to be able to come and go freely. Jack swears they can.\"\n\n\"But no one who doesn't have their biometrics better try to follow them home. Again, I've got autocannons covering their entrances.\"\n\n\"I've heard about a Hall of Mirrors somewhere,\" Vicky said. \"Wasn't some war back on old Earth settled in such a hall?\"\n\n\"I copied it though I've made it larger, and there's more marble and fancy lighting to go with the mirrors,\" Nelly said.\n\n\"You've got all this from thirty thousand tons of Smart Metal,\" Vicky said, still in awe.\n\n\"Actually, we did it with twenty thousand tons,\" Nelly reported.\n\n\"Only two-thirds of your metal? What will you do with the rest?\" Vicky asked.\n\n\"We could make the castle walls higher and more intimidating,\" Kris said, but she'd spotted the envious way Vicky eyed the castle. \"Or we could loan you some to build your very your own castle.\"\n\nVicky thought on it for a long moment, and then sadly shook her head. \"I've got a very fancy pavilion on order. Huge. Big enough for a throne room, without a throne, just a receiving area. There's enough space for me and my immediate team, and a third bay for a meeting room. I think it would be better if I didn't look too Longknifeish.\"\n\n\"I have to agree,\" Nelly said. \"I don't think the brain trust would like you to give Vicky Smart Metal so obviously. However, Vicky, it might be nice if I slipped some of the Smart Metal between the threads of the canvas of your pavilion. Just enough to make its wall impervious to the usual cutting tools, bullets, or rockets. You know, that sort of stuff.\"\n\n\"Could you do that, Nelly?\"\n\n\"Of course I can do it. If I do it right, I can cover Admiral Waller's pavilion as well as the St. Petersburg circus tent.\"\n\n\"Circus tent?\" Kris asked.\n\n\"There were only so many pavilions available,\" Vicky said. \"Mannie's got his people in a tent they got from a local circus. It's quite a big tent. They're providing space for representatives from several of the planets in St. Petersburg's immediate trading zone. Mannie says having all of them under one roof gives them plenty of time to get organized and try to present a common front.\"\n\n\"If Mannie's doing that, I suspect the other major planets will be doing the same if they're smart,\" Jack said.\n\n\"I think a lot of them learned from that little get-together of yours,\" Vicky said. \"We've spent the last two days getting our ducks in a row. We're ready for the Empress when she gets here.\"\n\n\"Nelly, see how far you can stretch the Smart Metal without spreading it too thin,\" Kris said.\n\n\"I'm working on it with Sal,\" Nelly said.\n\n\"One thing,\" Vicky said. \"Could you give me a Smart Metal bed? It's either put up with a rental or buy one, and prices are though the roof.\"\n\n\"I think I can do that,\" Nelly said.\n\n\"Kris, do you plan to borrow any Smart Metal from the squadron's ships?\" Jack asked.\n\nKris thought on that for a long minute, then shook her head. \"That is something I will _not_ do. I don't trust _anyone_ here at the table not to slip a fleet in so they can get their own way. Vicky, I'm not looking at you, but can you honestly tell me that you have control of all the elements of your alliance?\"\n\nNow Vicky shook her head. \"I'd like to think my people aren't that dumb, but I can't swear to it. As for the Empress, no way would I trust her. Not with so much as a bucket of sand on a long and sandy beach.\"\n\n\"It was a good question, Jack, but I think you got the answer you expected.\"\n\n\"I did. It was a question that needed raising and a policy that definitely needed to be made and locked in place.\"\n\n\"Now, Vicky, why don't you go check out your pavilion while Jack and I see just how decadent Nelly made our bedroom.\"\n\n\"And bath, and parlor for receiving guests,\" Nelly added.\n\n\"You two go have fun,\" Vicky said.\n\n\"An Imperial Grand Duchess just told us to have fun,\" Jack whispered to Kris as they mounted the stairs.\n\n\"Then we must follow the Imperial order.\"\n\n# 54\n\nWhile the Imperial couple made their way to Cuzco at one gee, Kris put the time to good use. Each night, she gave a banquet that then adjourned to the ballroom for dancing and talk.\n\nThere was a lot more talking than dancing.\n\nThe day was filled with meetings followed by more bloody meetings. Jack insisted Kris not leave the castle, so most of the meetings were in her lovely, first-floor conference center.\n\nHaving a huge banquet hall, ballroom, and conference hall with breakout rooms for smaller discussions was impressive. Very impressive.\n\nAnd very effective. Kris had everyone at her palace, getting them used to the palatial surroundings.\n\nKris even invited the Emperor's advance team to the banquets, balls, and some of the conferences. Jack assured that they were heavily monitored by security, but it wasn't necessary.\n\nThe Imperial advanced team was shunned like lepers by the delegates from the rebel alliance.\n\nKris eavesdropped on the Imperials' small talk. The more they saw of her suddenly appearing castle, the more they were in awe of it.\n\nVicky's brain trust did convene one negotiation session with the Imperials to adjust the venue for the formal meetings. They proposed that half still be held in the resort's Great Hall but that the other half be in Kris's Great Hall of Mirrors. Kris expected some objections, so she offered to give up all the rooms in the lodge.\n\nThe Imperial advanced team caved like bats.\n\n\"The Emperor may even prefer the better amenities of the Hall of Mirrors. Your chairs are a lot more comfortable.\"\n\n\"The chairs adjust to the persons sitting in them,\" Kris said.\n\n\"Yes,\" the leader of the Imperial team said, leaning back in his chair and having it conform to his every move. \"Can all the chairs give a massage?\"\n\n\"Some can. Some won't,\" Kris said.\n\n\"I'd be willing to use my good offices to encourage the Emperor to meet here more often if you could arrange for him, the Empress, and me to always have massage chairs.\"\n\n\"I think that can be assured,\" Kris said. \"A lot of the chairs can be massage chairs. It's just if there are too many requests, the computer may not be able to meet demands.\"\n\n\"Just so the Imperial couple and I get our rumps taken care of.\"\n\n\"I can assure that,\" Kris said.\n\nNELLY, YOU HAVE MY PERMISSION TO MAKE THE CHAIRS QUIT WORKING FOR THE EMPRESS AND HER FAMILY WHENEVER THE MEETINGS GO LONG.\n\nI'M SO GLAD YOU TOLD ME, KRIS. I INTENDED TO DO IT ANYWAY, BUT IT'S NICE TO HAVE PERMISSION.\n\nThe Emperor and Empress arrived at High Cuzco Station and immediately took the next ferry down the elevator. Nelly reported that the entourage filled up the next four drop ferries and totally screwed up the change of shift. There were fistfights for places on the four o'clock ferry, and the overflow had to wait for the four thirty.\n\nThe problem, complete with video of the fistfights and swords being drawn with tired and irritated workers in the background, made the evening news and was repeated on the late-night news as well. The Emperor's visit was not off to a good start as far as the people of Cuzco were concerned.\n\nThe Emperor wasn't too happy, either. But it was the Empress's reaction that warmed Kris's heart. The Imperial couple was sharing the two best suites the lodge had to offer. They were large and comfortable, but, in keeping with the ambiance of the resort, quite rustic.\n\nAnd they had a lovely view of Kris's castle.\n\nKris knew exactly what the Imperials thought about their accommodations. She had a nano hovering just outside their window transmitting every shout and scream. In the end, the advanced man was banished to a small room on the top floor. The only reason he kept his head was because the prime minister suggested that he might be helpful if there were any questions about the ground rules Kris Longknife and Vicky had agreed to with him.\n\nThe man actually crawled from the Imperial presence on his hands and knees.\n\nKris waited a few minutes to allow the couple to get control of their rage before sending Major Henderson, dressed in formal blue and reds and with an honor guard, with an invitation to a reception and formal dinner Kris was throwing in honor of the Imperials present.\n\nThe Empress quickly spotted that the careful wording, \"Imperials present,\" could well mean both them and Grand Duchess Victoria.\n\nThat put her in another rage, but she could not vent it on a Royal US Marine. Before she might try, the Emperor dismissed Major Henderson with a promise that a response would come later.\n\nWhen the Empress calmed down, they, with the second man from the advanced team, went down to see what facilities were available for proper Imperial entertainment.\n\nThat rocketed the Empress into another rage.\n\n\"Why did you agree to Cuzco and why _ever_ did you think that a resort like this woodland slum should be where we met with the little whore?\"\n\n\"There, there, honey, remember, the doctor says you must stay calm. Your blood pressure,\" the Emperor said, taking what sure looked like the woman's side of this clich\u00e9 of a conversation.\n\n\"This place is barbaric. Will we have to use outhouses for toilets? I expect a bear to wander by any minute and start eating people, like that worthless advance team of yours.\"\n\nThere was a pause in the rant. Kris suspected it was for the Empress to get a breath and the Emperor to roll his eyes, but Harry got the next word in.\n\n\"Really, honey pie, the Imperial Guard will shoot any bear that shows its face.\"\n\nThe Empress changed the conversation. \"Well, I can hardly entertain in such a place as this,\" she said, wheedling him.\n\n\"We can accept Princess Longknife's invitation for tonight. Tomorrow, we can rent one of those huge tents like they have out there. Can we get one of them?\"\n\nThe advanced man mumbled something that Kris's nano missed.\n\n\"Those are theirs! Didn't _you_ rent _anything_? Why have an advance team if they're going to sit on their hands and do nothing?\"\n\n\"Your Majesty, our money is no good.\" Kris did catch that.\n\n\"Edgard had to sign a mortgage on the ship bringing you here to assure that we would have either hard cash or trade goods to cover the cost of goods and services here at the lodge for you and your court during your stay.\"\n\n\"He had no such power,\" the Empress screamed, heading into her fourth rage of the afternoon. _Maybe it was her fifth._\n\nNO, KRIS, IT'S THE FOURTH. I'M KEEPING A TALLY.\n\nBack in the Imperial presence, the poor number two in the advance team was trying to explain their problem. \"Edgard and the resort owner's lawyer do believe that your commission to him to speak in your name did give him enough authority. The owner has already gotten a lien on the _Emperor and Empress_. You'll need to have something like crystal or other trade goods brought in to meet our debts. Your debts,\" he quickly corrected himself.\n\n\"They intend to hold us hostage! I tell you, Harry, _this_ is a Longknife plot. I _told_ you it was.\"\n\n\"Sweetums, you knew our money is no good outside the Empire. I just assumed that our signature was enough to handle matters for now, and we'd settle up later.\"\n\n\"Ray Longknife will have us thrown into debtors' prison, and you'll have the devil's own time settling up later with your hands chained to the dripping stone wall of some dungeon.\"\n\n\"You exaggerate, honey pot. We can always leave the ship and book passage on another.\"\n\n\"Leave my ship! My lovely ship! You gave her to me, Henry. You fitted it out as an anniversary present. You would just _give_ it away? Would you give _me_ away?\"\n\nKris eyed Jack, who had come in halfway through all the drama. He silently mouthed, \"I would\" to Kris.\n\n\"Me, too,\" she mouthed back, then thought how silly it was to be so quiet on a one-way hookup.\n\nKris figured she'd heard enough. She'd be getting an RSVP from the Imperials in an hour or so. Vicky had already replied. Diana and her friends were carefully arranging the seating. The banquet today would have to be carefully arranged with the primaries as far from each other as possible. They might or might not mingle on the dance floor, but their seating would be separated by the locals who had been invited. Likely, the two groups would not mingle; still, the brain trust was carefully weighing just who should be seated close to the locals. They might get some movement there.\n\nThey would have to be careful that it was in their direction, not toward the Empress.\n\n\"So you've decided that the problem is the Empress,\" Kris said.\n\n\"We have decided that the Grand Duchess and her alliance are more willing to work with us, and we like the options they would be willing to accept to end this conflict,\" Diana said.\n\n\"And the Empress?\" Kris asked.\n\nThe three looked at each other, then Diana continued. \"Her opinion of the facts on the ground are quite divergent from the facts as we see them. Her stated objective sets a very high goal for her side and . . .\" Here Diana again checked with her colleagues. She was met with sad shakes of the head. \"We do not expect much flexibility from her in moving away from her initial demands. Put the way a Longknife would want to hear it, she is a problem now and likely to be a problem in the future.\"\n\n\"Hopefully, that will change,\" Al said.\n\n\"We will just have to see how negotiations go,\" Bill added.\n\n\"Yes, we will,\" Kris said, \"but I was taught long ago that hope is not a strategy and hope appears to be all we have at the moment.\"\n\n\"Kris,\" Nelly interrupted, \"a colonel of the Imperial Guard has presented the Marines at the front gate with the Emperor's RSVP. He will be glad to dine with you this evening.\"\n\n\"Then I better get ready,\" Kris said, and headed for her bedroom. There, Nelly had put a totally new twist on palatial.\n\n# 55\n\nNelly had really outdone herself with Kris's bedroom. Several chandeliers cast rainbows on walls half-marble with gold and silver veins running through them. The rest of the walls were covered in lush velvet wallpaper with gold and silver trim. Set strategically around the room were classical statues and paintings, most of them not suitable for work.\n\n\"You're tying up a lot of Smart Metal with this erotic statuary, aren't you Nelly?\"\n\n\"Not really Kris. Try picking one up.\"\n\nKris strode over, and easily hefted a large statue of three nude women.\n\n\"The statues have little more substance than a balloon, Kris. I have to lock them down, Smart Metal to Smart Metal, to keep them in place. I recovered the Smart Metal from the plush carpet underneath the pedestals. That about makes the statues a wash.\"\n\n\"Do you plan to put a few of these in the Banquet Hall and ballrooms?\"\n\n\"Likely, Kris. Only in those, I'll mount antennas for nano tracking and bugs for listening.\"\n\nThe bathroom was all Kris could dream of. The walls appeared to be marble with inlays of gold and sparkling jewels. She soaked in a tub large enough for a small party as jets did their best to relax the tension in her shoulders and back. She had to wash her own hair, then rinsed off in a shower with a dozen rotating heads for both front and back.\n\n\"I almost don't miss Jack,\" she muttered, missing Jack.\n\n\"Now, what do I wear?\" Kris asked herself.\n\nEnsign Meg Longknife was waiting for her as Kris returned to her bedroom with one towel wrapped around herself and a second vigorously drying her hair. Meg had become Kris's _aide-de-camp_ , dog robber, and all-around maker of things to happen.\n\n\"You may be surprised, Admiral, by what's in your closet.\"\n\n\"Surprised?\"\n\n\"Yes, ma'am. The _Dauntless_ brought in a rather large wardrobe full of ball gowns. Someone wanted you to have a wide choice.\"\n\n\"Good Lord, please tell me it wasn't my mother.\"\n\n\"I think there's a note in there from one of your great-grandmothers.\"\n\n\"Gramma Trouble?\"\n\n\"I do believe so, Your Highness.\" And so saying, Meg walked Kris over to a closet bigger than her night quarters aboard ship. Several lovely gowns in the traditional pattern were hung out for Kris to choose from.\n\n\"But will they fit?\" Kris asked. Ruth had not been kind to Kris's body. Well, maybe she hadn't been all bad; Kris now had a decent pair of breasts for the first time in her life. If only she'd had something more like these all those miserable years in high school and college. No, it was Kris's waist that caused her to tremble in fear.\n\n\"Help me into my spider silks,\" Kris said. \"I will not get dressed and have Jack insist I get undressed and pull this armor on.\" Kris did dress in the essentials, like a nice support bra and some lacy undies over her armor.\n\nThat brought Kris face-to-face with the ball gowns. She picked out a lovely floor-length, royal blue gown with a bodice of brocade and a billowing skirt of silk and lace. It would match her eyes and offset her skin and hair perfectly.\n\nTo Kris's delight, the waist was relaxed from what it might have been a year and a baby ago. The bodice, however, fit perfectly, showing more d\u00e9colletage than Kris had ever presented to the world.\n\n\"Gramma Trouble has an eye for details,\" Kris said, \"or a hidden laser that took my measurements. I'd bet on the laser. She was taking lots of pictures of me and Ruth and Jack and just about everyone. You sneaky but nice old lady.\"\n\n\"What do you want to wear for jewelry?\" Meg asked.\n\n\"I don't need pearls or diamonds,\" Kris muttered. \"Let's show them what we've earned.\" She turned to her awards drawer; Nelly had them laid out quite nicely.\n\n\"Let's put the Greenfeld _Pour la M\u00e9rite_ at my throat.\" And Meg did. \"Hang the Almar Medal of Highest Valor next. With any luck, Granny Rita's Hammer of Enlightened Ones will fall right where Jack will enjoy it, if he notices it at all.\"\n\n\"That pretty much covers up everything that dress was showing off,\" Meg pointed out.\n\n\"I'm sure Vicky and the Empress will be showing enough skin for all of us. But they've got nothing to show like these,\" Kris said. \"Now, I want the sashes and badges of the Order of the Wounded Lion from Earth, Order of St. Christopher, Star Leaper from Greenfeld, and Order of the Rose and Thistle from the cats, too. I don't think the colors will clash too much, do you?\"\n\n\"Whatever you say, Your Highness.\"\n\n\"They clash, huh?\"\n\n\"That orange sash of the Order of St. Christopher, Star Leaper would clash with just about anything but your birthday suit.\"\n\n\"We are not wearing that,\" Kris said, thinking fast. \"St. Chris is from the Emperor, so we have to include it. Let's go with just the starburst. Now, how about my medals?\"\n\n\"Miniatures, right?\"\n\n\"Well, we'll want to slip Chance's Golden Starburst somewhere near the middle of the bodice, but the medals should be up close to the top,\" Kris paused, \"of this strapless gown. Hmm.\"\n\nMeg put the line of miniature medals just a few millimeters down from the top of the bodice. \"Ah, ma'am, I feel that I must warn you that any rapid movement, and maybe just enough gravity, and I can't assure you that you won't be showing everyone what Ruth likes to suck on.\"\n\n\"Hmm,\" Kris muttered, doubtfully.\n\n\"Feel yourself up, and see for yourself,\" Megan suggested.\n\nKris did. Her aide was right. \"Ookaay, there's got to be some tape or putty or super glue that can keep dress and boob together for an evening.\"\n\n\"And not take an inch of skin off when you take the dress off?\"\n\n\"Nelly?\" Kris called.\n\n\"Abby had just that kind of tape,\" Nelly answered.\n\n\"But she's on the other side of the galaxy.\"\n\n\"If you will quit interrupting me, I will finish telling you that she left it behind, and it should be appearing in a moment on your bedside stand. Happy?\"\n\n\"Ecstatically,\" Kris said.\n\n\"Don't sass me, Kris.\"\n\n\"I wasn't. I really meant it.\"\n\nMeg put a strip of double-sided tape inside the dress above where the medals went. Kris worried the top of her dress for a moment with results that might please Jack but did not meet with Kris's approval. They took that strip off and added three strips. One below the medals, one where they were pinned to the dress, and a third between the pins and the top of the dress.\n\n\"Those puppies aren't going anywhere,\" Meg said.\n\nKris was not so optimistic. She was a bit more experienced. If she took off running or did a drop and roll . . .\n\nKris weighed the matter, decided she wanted the medals on display, and accepted the risk.\n\nWhile Meg was putting Kris's hair up, Jack dropped in, remarked again on the palatial nature of their palace, to Nelly's vain delight, and took a quick shower before pulling on his spidersilks.\n\nKris held the top of her armor away from her neck for Jack to see. He nodded and quickly put on his formal blue and reds without saying a word.\n\nKris credited herself with one in the win column.\n\n\"How are our guests doing?\" Kris asked Nelly.\n\n\"The Grand Duchess arrived early,\" Nelly said. \"You were in the shower for her fanfare. You may have heard the fanfare for the Emperor and Empress a few minutes ago. We played the Imperial March for both of them, but the Empress didn't hear Vicky's honors, so she hasn't flown into a rage yet this evening. At least not at your reception. Our bug outside their window caught quite a bit of shouting while they were dressing.\"\n\n\"Can that marriage be saved?\" Jack muttered.\n\n\"I don't know what kind of game she's playing,\" Kris said, \"but it doesn't seem very smart to me.\"\n\n\"She is the mother of the Emperor's only living son,\" Nelly pointed out.\n\n\"Well,\" Jack said, offering Kris his arm, \"you look stunning this evening, Your Highness. Would you care to make a spectacular entrance?\"\n\n\"How could it be anything less than spectacular on the arm of such a dashing lieutenant general?\"\n\nAnd with that, Kris let herself be led so daintily to the charge, diplomacy being the continuation of war by silk and satin means.\n\n# 56\n\nAt the top of the staircase, Jack and Kris paused. The room below was aswirl with people, but it only took a second for Kris to spot the pattern.\n\nThe Emperor and Empress and their people occupied the third of the room to the right of the front doors. The Grand Duchess and her people held down the space to its left. Mingling in the center with occasional forays into the factions were local government officials, businessmen, and academics from Cuzco.\n\nKris's brain trust had arranged for their invitations. They would serve as neutral observers who Diana hoped would be a calming influence to the gala tonight. With any luck, they would also serve as potential sources of experience as to how the give-and-take of bargaining went. Hopefully, they'd share some stories with the attendees from the warring sides.\n\nAfter a moment standing there, observing the start of their party, Nelly announced them in a voice that carried over the babble of the hall.\n\n\"Admiral, Her Royal Highness Kristine of the United Society and her consort, Lieutenant General Juan Montoya.\"\n\nKris stood in place for four ruffles and flourishes, then took her first step down as a royal fanfare recently composed for King Raymond and any other US royalty blasted the hall with brass and percussion. That continued until Kris was about halfway down the stairs, to that area covered by autocannons, when the music softened only a bit as it segued into the Royal US anthem.\n\nAll of this was new to Kris. The music hadn't existed when they shipped her off to exile on Madigan's Rainbow, and she certainly hadn't been met with a fanfare when she snuck back to Wardhaven. She'd taken off for the other side of the galaxy none the wiser about the royal trappings Ray was taking on.\n\nNow she was marching . . . abeit carefully . . . down thickly carpeted stairs to their strains.\n\nBelow her, the fine and independent citizens of Cuzco bowed though some of them did it ever so slightly. Vicky led her alliance in a deep bow. The man next to her in formal civilian dress bowed lower, and most of her followers took their guidance from him. On the Emperor and Empress's side, the Imperial couple did little more than nod, and their huge entourage of dandies and half-dressed ladies did no more than that.\n\nKris schooled her face to Navy bland and glanced again at the Empress.\n\nNELLY, ARE THE LADIES OF THE IMPERIAL COURT ACTUALLY TOPLESS?\n\nKRIS, IF YOU LOOK CLOSELY, AND I AM SURE ALL THE MEN PRESENT ARE LOOKING VERY CLOSELY, YOU WILL FIND THAT THE DRESSES DO REACH UP TO COVER THE NIPPLES WITH A BIT OF SPARKLE AND THIN CLOTH.\n\nWHAT IS THE EMPRESS TRYING TO DO, WAVING ALL THAT PULCHRITUDE AROUND HER HUSBAND?\n\nI HAVE STUMBLED ACROSS SOME CORRESPONDENCE FROM THOSE ON VICKY'S SIDE. THEY SUSPECT THAT THE EMPRESS AND HER BOWLINGAME RELATIVES HOPE THE EMPEROR WILL TUMBLE INTO THE BED OF A WOMAN WITH A JEALOUS AND EASILY INFURIATED HUSBAND WHO WILL HASTEN THE EMPRESS INTO AN EARLY WIDOWHOOD. I HAVE NOT BEEN ABLE TO CONSTRUCT A BETTER EXPLANATION FOR HER BEHAVIOR.\n\nKris had to work hard to keep any reaction off her face. Diana waited at the bottom of the stairs. \"We are ready to see if we can get this reception line going. We'd planned to have just the Imperial couple and Vicky with her plus one with you and Jack between them. However, the Imperial couple insists the prime minister and his wife be included. The Empress insists, as in demands, that the Grand Duke and Grand Duchess of Greenfeld and the Duke of Anhalt, to whit her dad, mom, and oldest brother, be included as well. Her brother left his wife behind on Greenfeld and insists we include his mistress of the moment at his side.\"\n\n\"Is that allowed?\" Kris asked.\n\n\"I can find no reference to it, but he is a showstopper for the Empress, and he won't get in line without his latest interest. We're doing it the way the Imperials want.\"\n\n\"That's starting on the wrong foot,\" Jack muttered.\n\n\"But at least it's starting. We three and our plus ones will be beside you two. Vicky has identified three more couples.\"\n\n\"Then I guess we're ready to see if we can make this work,\" Kris said. \"Lay on, Macduff, and damned be he who shouts too loud.\"\n\nIt took only a moment to get everyone into their places in the receiving line. Kris and Jack were insulated on both sides by her members of her troika. The Emperor and Empress took the head of the line followed by the two Grand Dukes and their ladies. The prime minister and his wife were left to stand beside Diana and a male member of her team.\n\nPast Kris and Jack, Bill and a young woman from his staff stood beside Al and a young man. That led to Vicky's four. Admiral Waller stood grimly at the head of the rebel section of the receiving line, an aggressive young Marine lieutenant colonel beside him: her left hand rested on a sword, her right was on a holstered sidearm.\n\nKris took one look and saw trouble.\n\nNELLY, WHAT'S GOING ON THERE?\n\nGRAND ADMIRAL WALLER IS THE NAVY CHIEF OF STAFF. HE GOT OUT OF ANHALT ONE STEP AHEAD OF AN ASSASSIN SQUAD. HIS WIFE WAS CAUGHT. THE EMPRESS PERSONALLY SLIT HER THROAT AND SENT THE VIDEO TO WALLER. HIS YOUNGEST DAUGHTER IS HIS PLUS ONE.\n\nNELLY, KEEP AN EYE ON THEM. JACK, CAN YOU GET A PAIR OF MARINES IN CIVVIES TO FLY WING ON THOSE TWO?\n\nI ALREADY HAVE.\n\nOOPS. SORRY. JACK, IF YOU CAPTURED SOMEONE'S WIFE, WOULD YOU SLIT HER THROAT OR HOLD HER HOSTAGE FOR HER HUSBAND'S GOOD BEHAVIOR?\n\nKRIS, I DON'T MAKE WAR ON CIVILIAN WOMEN.\n\nWELL, THE EMPRESS IS, AND SHE'S NOT ACTING VERY SMART.\n\nYA'THINK?\n\nThe reception line started. Ensign Longknife called out the first couple's names, a Baron and paramour.\n\nPARAMOUR, NELLY?\n\nTHAT'S WHAT THE CARD SAID. MEGAN READ IT RIGHT.\n\nDEAR GOD. THAT PALACE MUST BE A BURNING LAKE OF SEWAGE.\n\nIT SURE SMELLS THAT WAY FROM WHERE I STAND.\n\nBut Kris quickly found herself with worse problems. Said Baron went down the line, got to Al's boyfriend, and turned his back on the rebels in the last third of the line and blatantly stomped away.\n\nKris was shaking a countess's hand. She caught the uncivil developments out of the corner of her eye. A moment later, the next couple in bright Imperial dress did the exact same thing.\n\nThe rebels, who had been looking reluctant to shake the Imperial hands, now formed into their own line. The elderly woman from Metzburg cut right through the Imperial line to smilingly present herself to Diana and shake her hand. The Imperial went for his sword, but the next rebel in line was a Navy officer with a dress sword at his side.\n\n_Which begs the question, are they both cake cutters, or have these two brought seriously sharp steel to my reception?_\n\nJACK?\n\nALREADY ON IT.\n\nOut of nowhere, Major Henderson appeared, he of both sword and sidearm. He interposed himself between the Imperial line, and let the rebel pass, then he switched lines, blocked the next rebel couple and let an Imperial pass. He spent the rest of the hour, functioning as school crossing guard for some very rowdy kindergartners.\n\nImperials went down the line of Imperials and Kris's section . . . then turned their back, shunning the rebels. Rebels started at Kris's segment and continued until the end of Vicky's four couples.\n\nSomehow, no blood ended up on the floor.\n\nDinner was an equally cold affair. The Emperor and Empress were invited in first. That meant they'd have to pass through the rebels who had collected around Vicky. Jack arranged for a Marine Honor Guard to form a double line down which the Imperials sashayed.\n\nOnce in the Banquet Hall, more Marines escorted the Imperial couple to their table. Kris's troika had arranged for the Emperor and Empress to be seated close to the far right corner. There were name tags at each place, but those nobles following in the Imperials' wake quickly moved to take the places closest to the Imperial couple, ruthlessly displacing others.\n\nWhen all the tumult ended, Kris had Nelly check. Most of those surrounding the Emperor and Empress were of the Empress's faction.\n\n_Interesting._\n\nNow the Marine Honor Guard paraded into the Banquet Hall and just happened to form a riot-control line down the middle of the small dance floor. Almost there was a riot, but not quite.\n\nJack maintained a Marine bland face through this all, but Kris noticed that he did allow himself a deep breath when the last of the two sides settled at their dinner table.\n\nYOUR MARINES WERE GREAT, JACK.\n\nAND THEY DID IT WITHOUT PRACTICING THE DRILL, Jack added.\n\nVicky's rebels had at least obeyed their table assignments as they took their seats to the left-hand side of the Banquet Hall. Kris's troika had worked hard to distribute the representatives from all the different rebel planets around that third of the room. Hopefully, that would get them talking, sharing, and maybe help them see the others as allies rather than competitors.\n\nKris and Jack, along with her brain trust and the locals who had wrangled invites, filled up the space between the two warring sides. Kris had hoped that the locals might mingle with the two belligerents. It seemed to work with Vicky's side.\n\nThe Empress's father and brother made it plain that they were not interested in talking to the locals, and the rest of the courtiers and courtesans quickly followed their lead.\n\nWhile this nothing was going on, Nelly reported that the first-step trip wire on the staircase to the second-floor quarters had been tripped.\n\nWHAT SAY WE SEE WHAT WE CAN SEE? TURN OFF THE AUTOCANNON, THOUGH, Kris said, and eyed Jack. He looked to be following this development as well.\n\nWE HAVE A LONE MARINE AT THE TOP OF THE STAIRS, LURKING IN AN ALCOVE WITH A SUDDENLY MUCH SMALLER STATUE AND NO LIGHT.\n\nFINE, JACK.\n\nKRIS, I'M GOING TO PATCH YOU THROUGH.\n\nWHAT HAVE WE HERE? Kris heard. The question was followed by a feminine yelp.\n\nYOU ALMOST SCARED ME TO DEATH, SOLDIER.\n\nAND WHY WOULD LITTLE OLD YOU BE UP HERE?\n\nIT'S BORING DOWN THERE. NOTHING TO DO. I THOUGHT I MIGHT POWDER SOMETHING. NOW, YOU AND ME, I THINK WE COULD REALLY HAVE SOME FUN. I BET I COULD SCREW YOUR EYEBALLS OUT OF THEIR SOCKETS. WE COULD USE THAT ROOM RIGHT OVER THERE. I BET IT'S EMPTY.\n\nIT LIKELY IS SINCE IT'S HER ROYAL HIGHNESS'S ROOMS, BUT I KIND OF LIKE MY EYEBALLS WHERE THEY ARE.\n\nWHAT'S THE MATTER WITH YOU? SHOULD I SEND MY BOYFRIEND UP HERE? DO YOU PREFER BOYS?\n\nACTUALLY, I PREFER BEER AND PIZZA. NOW WHY DON'T YOU GO POWDER YOUR BOOBS SOMEWHERE DOWNSTAIRS?\n\nKRIS, I THINK THAT LAST WAS PUNCTUATED WITH A SWAT ON HER BACKSIDE.\n\nNELLY, I THINK YOU MIGHT BE RIGHT.\n\nHOW BEASTLY.\n\nIF HER MOTHER HAD PATTED HER DIAPERED RUMP A FEW MORE TIMES, SHE MIGHT NOT BE RUNNING AROUND HERE WITH ALL HER CHARMS HANGING OUT, Jack put in.\n\nKris decided to skip the debate about the proper way to raise children for a few more years and turned back to the challenge at hand.\n\nThe Emperor did not seem to have noticed how the Empress's family was leading his court around by the nose. He had his head on his wife's bosom as she fed him from her own plate.\n\nDiana shook her head halfway through the meal. \"This is going to be harder than even I thought.\"\n\nAfter dinner, there was light music, but no one attempted to dance on the floor that separated the two factions. People did mingle. Locals and rebels at least. Kris and her troika tried to talk with some of the Imperial entourage.\n\nShe'd never seen so many people turn their backs on her.\n\nShe did manage to make her way to the Imperial couple's table.\n\nThe Emperor looked up from between his wife's naked d\u00e9colletage. The sparkles that had covered her nipples were now shining in his hair. \"Thank you for inviting us to this wonderful banquet,\" he said. \"I hope we can dispense with the receiving line next time. It was so tedious.\"\n\n\"I think we can arrange that,\" Kris agreed.\n\n\"Well, we really must be going,\" the Empress said. \"I can _not_ believe our advanced team agreed to start at _eight thirty_ in the morning. That is disgusting,\" she said, eyeing Kris.\n\nKris smiled blandly, refusing to make any concessions to the agreed-upon ground rules by simply refusing to notice the question not actually asked.\n\nSo the Imperial couple led the exodus of their court, the Emperor's head solidly resting atop the Empress's bare boobs. The Duke of Anhalt had raised the skirts of his paramour and was pawing her bare ass as they followed the Imperial couple. He looked back to leer at Kris.\n\nJack took a sharp breath in, then slowly let it out.\n\n\"You sure I can't kill those bastards?\" he muttered low to Kris.\n\n\"I'm sure we can't. We've got to end this civil war.\"\n\n\"Wouldn't killing them all end it?\"\n\n\"I'll have to ask Diana about that,\" Kris said.\n\n\"She'll never agree. What do you say we do it and then ask forgiveness?\"\n\nKris chuckled. \"It's worked for me plenty of times.\"\n\n\"See?\"\n\n\"Plenty of times when I was the subordinate, Jack. Remember, we have seven stars between us and a darling little girl we want to make a better world for.\"\n\n\"This being a grown-up isn't all it was cracked up to be. You think if I warn Ruth Marie now, she could avoid it like her parents didn't?\"\n\n\"Jack, what do you say you take me to bed, and we pretend we're grown-ups and do that thing that grown-ups do?\"\n\n\"Raid the ice-cream locker?\" he asked, brightly.\n\n\"Think harder, Jack. You must remember something we can do?\"\n\n\"Harder, huh?\"\n\nKris managed to exit the Banquet Hall without being waylaid by anyone. By the time they got to the top of the stairs, it seemed that Jack had remembered something fun that grown-ups could do . . . and was eager to do it.\n\n# 57\n\nThe morning did begin bright and early.\n\nFor some.\n\nAt 0815, when Kris and Jack walked into the Hall of Mirrors, Vicky and a major portion of her party were already there. They had the tables and chairs down the left-hand side of the conference room. Kris's brain trust, with two comfortable chairs for Kris and Jack, held down the far wall of the room.\n\nThere was a line of tables down the middle of the room where local court stenographers and other people with recording gear had set up shop. It was in the middle of the room in the hopes of delaying an attack from either side of the room against the other.\n\nThe right-hand side of the room was reserved for the Imperials. It had tables, several comfortable chairs, and like the other two sides, a podium where one person could stand and address the room.\n\nThere were two squads of Royal US Marines standing against the wall. Marines had stood guard through the night to assure that no bomb was placed in the room, their presence balanced by two squad of Imperial Guardsmen. Other than them and the prime minister bumbling around the podium, the Imperial side of the room was empty.\n\nTotally.\n\nKris and Jack walked to their chairs at the other end of the room and stood. Kris bowed respectfully to Vicky, who returned it. She bowed respectfully to the Emperor's first minister, an elderly man in a dark, conservative vest coat. His doublet fell all the way to his hose. It seemed like it took a moment for him to decide if bowing to Kris was a career-ending decision, maybe even life-ending, but he did return her bow.\n\nThat done, Kris sat down.\n\nDiana came up behind Kris. \"It's quite possible that the Emperor is waiting for you two to get here.\"\n\n\"It is possible,\" Kris allowed.\n\nFifteen minutes later, as the minute hand on the clock above the door swept past the bottom of the hour, Diana was back. \"Maybe he wants to be fashionably late?\"\n\n\"Possibly,\" Kris allowed, and settled her gaze on the prime minister.\n\nHe took Kris's hard eye contact for a full minute before folding. He signaled to the captain of the guard detail. He came to the prime minister, was given some instructions, and hurried out of the room. The prime minister sat down to await the results of his, no doubt, gentle query.\n\nKris checked the clock.\n\nIt was fifteen minutes later that the guardsman returned. He bowed to the prime minister's ear and whispered something.\n\nThe poor old fellow turned around in his chair to eye the young officer. \"No. No,\" came in too loud a whisper and with a cracking voice.\n\nThe guard captain nodded emphatically.\n\nThe prime minister shook his head ever so slightly and cleared his throat. \"I am charged by my master to begin the proceedings of the day,\" he said, standing.\n\n\"These proceedings will not open,\" Kris said firmly, staying seated.\n\n\"But Your Highness . . .\" was not finished.\n\nKris stood. \"Your Imperial Master requested of our liege lord our assistance in resolving a difficulty he was having ruling his Empire. We are here at his request. We have reached an agreement on how to proceed. Your Master has agreed to proceed in that fashion. He charged some of his subjects to meet with me and agree on precisely how we will proceed. That agreement has been signed.\" Kris knew her voice had been rising. That her rage was showing through.\n\nShe didn't care.\n\n\"You may tell your Master that it is 8:29 in this hall, and it will remain 8:29 in this hall until your Master is prepared to meet his solemn commitments.\"\n\nWithout a word from Kris, Nelly backed the clock above the door to exactly 8:29 and froze its hands in place.\n\nTHANK YOU, NELLY.\n\nTHINK NOTHING OF IT, KRIS.\n\nThe prime minister looked about to have a heart attack. He reached for his cane and hobbled from the room.\n\nKris took her seat and prepared to wait however long it took.\n\n# 58\n\nSlightly less than an hour later, a flood of courtiers and courtesans heralded the imminent arrival of the Emperor. His entry was somewhat hurried. And harried.\n\nThe Empress's entrance was in full snit. Clearly, she didn't want to be here nor think Harry should be. Apparently, she had lost the argument and was not happy about it. Maybe more so.\n\nShe glared at Kris as she plopped herself down in the easy chair reserved for her.\n\nKris ignored the drama from that side of the room. She stood; the clock above the door made a loud click and buzz before flipping over to half past eight.\n\nKris made her way to the podium in front of her central party. \"According to the agreed-upon ground rules, each side shall have an hour and a half to make statements or ask questions each morning and each afternoon. Who will go first each day will be determined by a flip of a coin. I have a Wardhaven silver dollar here. Would you both please provide a representative to check the coin and see it flipped?\"\n\nThe Emperor waved a languid hand, and the prime minister huffed his way over to Kris's podium. Vicky said a whispered word, and Mannie hurried there, too.\n\n\"Heads,\" Kris said, showing Lady Liberty. \"Tails,\" was a stylized example of the rocket that had brought the first landers to Wardhaven.\n\nHere Kris found the ground rules had failed to determine who made the call. She let the prime minister call the toss. He called tails and won.\n\nThe Emperor and Empress would address them first. The prime minister returned to his podium and shuffled the papers he had prepared. However, there was a commotion around the Imperial couple. It ended with the Empress stamping her way to the podium and shoving the prime minister aside.\n\nWhat followed was ninety minutes of the worst harangue that Kris had ever had the misfortune to sit through. Kris had stood through the ranting of the mad Great Guides of Xanadu that aliens were coming who would boil her eyeballs in her blood. She'd listened to the interminable harangues of the alien raiders' Enlightened Ones, unable to understand a word but shaking with the hatred and murderous intent behind them.\n\nThe Empress started with the calm and smooth reasoning that she had used when she first told her side of the story to Kris. But it only took her a few minutes to take a hard turn down Insanity Lane and dive into Crazy Lake. She never quite got around to foaming at the mouth, but she was most definitely spewing spittle.\n\nThe people seated at the central tables right in front of her got up and moved to seats away from the spray.\n\nWithout moving her head . . . the last thing Kris needed was to be seen exchanging glances with Vicky . . . Kris did cast her eyes at the Grand Duchess.\n\nVicky was pale as a sheet. Mannie rested a hand on her thigh. Vicky had one of hers on top of his. Her eyes narrowed, and her face took on a hardness as the Empress's rant went longer and longer. The two of them put their heads together and soon were making use of their computers.\n\nKRIS, I HAVE GAINED ACCESS TO VICKY'S COMPUTER.\n\nNELLY, WHAT ARE YOU DOING?\n\nI THOUGHT YOU MIGHT WANT TO KNOW WHAT VICKY AND MANNIE ARE DOING.\n\nKris squelched a scowl. AND WHY WOULD I WANT TO KNOW THAT?\n\nBECAUSE SHE AND MANNIE ARE DOING A VERY GOOD JOB OF GETTING AT THE TRUTH TO ALL THE LIES THAT THE EMPRESS IS TELLING US.\n\nLIES?\n\nFOR EXAMPLE, SEVERAL OF THE RIOTS THAT SHE SAYS VICKY INCITED TOOK PLACE WHILE SHE WAS STILL SAILING WITH YOU. YOU DID NOTICE THAT THE EMPRESS WAS NAMING A LOT OF PLACES BUT NOT TOO MANY DATES.\n\nI HAD NOTICED THAT LACK.\n\nTHE EMPRESS ALSO SAYS THE REBELS ARE RESPONSIBLE FOR ALL THE PIRATES THAT THROTTLED COMMERCE.\n\nSHE SURE HAS.\n\nTHE GRAND DUCHESS JUST GOT A LIST FROM ADMIRAL WALLER OF ALL THE \"PIRATE SHIPS\" THAT THE NAVY FOUGHT AND CAPTURED THAT HAD BEEN SOLD OUT OF THE NAVY FOR SCRAP, DELIVERED TO COMPANIES OWNED BY THE EMPRESS'S FAMILY, AND ENDED UP GOING APIRATING.\n\nDO TELL.\n\nYES, KRIS. I THINK THE GRAND DUCHESS'S REBUTTAL WILL BE A THING OF BEAUTY.\n\nKris began watching the clock after it passed 9:30 \"conference time.\" This demonstration of mad intent would not go one minute past the ninety minutes the agreed rules allowed.\n\nAt exactly ten o'clock, Kris stood. \"Your time has expired.\"\n\nStopped in midsentence, the Empress drew herself up to her full height, and said, haughtily, \"I am not finished.\"\n\n\"If you feel the need, you may use the ninety minutes allotted to your side in the afternoon,\" Kris said evenly.\n\n\"I want them now!\" she screamed, eyes shooting daggers at Kris.\n\n\"The Emperor and the Grand Duchess have agreed to these rules. Do the two of you agree to unanimously suspend the rules?\"\n\nThe Emperor sheepishly would not meet the Empress's eyes.\n\nVicky made no bones about it. \"We decline to suspend the rules on the first day that we are working under them. Later, we may find it to our mutual interest to adjust the schedule. _Not_ today.\"\n\nThe Emperor looked truly relieved.\n\nThe Empress stormed out of the Hall of Mirrors, hurling invectives in every direction.\n\n\"It is within my powers to suspend the meeting for a fifteen-minute break,\" Kris said, after the door closed off that noise. \"Would you like to get some tea, coffee?\"\n\n\"Scotch,\" came from somewhere in the room.\n\nVicky stood up. \"I am prepared to answer the Empress's comments of this morning. Then, I would propose that we yield the remainder of this session's time to Your Highness, so that we can go to lunch before all our stomachs are rumbling too loud.\"\n\n\"You may address those assembled here,\" Kris said.\n\nAs Vicky made her way to the podium, Kris changed gears in her head.\n\nJACK, THE EMPRESS LOOKED DOWNRIGHT DERANGED AS SHE STORMED OUT OF HERE.\n\nYEP, KRIS, AND YES, I HAVE DOUBLED THE GUARD. IF WE FIND ANY UNEXPLAINED SMALL OBJECTS, WE WILL TREAT THEM AS BOMBS. WE ARE DOUBLING THE NUMBER OF NANO INTERCEPTORS WE HAVE ROVING OUR CASTLE. SO FAR, THEY'VE FOUND NOTHING.\n\nTHANK YOU, JACK. Now Kris could relax back into her chair.\n\nVicky stood at the podium. For a long moment, she rested her eyes on her father.\n\n\"Dad. I love you,\" she said into the hush of the room.\n\n\"I have always loved you, and I will always love you.\"\n\n\"Then why are you leading a rebellion against me? Or letting these others use you in their rebellion?\" the Emperor said, waving a hand at those behind Vicky.\n\nKris knew she should gavel the Emperor out of order, but Vicky shook her head ever so slightly, and Kris held her hand.\n\n\"Dad. I'm a Peterwald. No one _uses_ a Peterwald. _We_ use people. They _never_ use us.\"\n\nThe father smile was proudly paternal at the daughter's reply.\n\nVicky glanced over her shoulder. \"But Dad, I have found that people can be quite helpful, especially if you help them. I'd love to tell you some of the things I've done over the last two years. I think you'd be amazed and delighted. I've taken famine biscuits to starving kids and brought law and order back to planets where it had totally collapsed. I've opened trade between planets where pirates and other bad things had totally clogged up the arteries of commerce. Dad, I've learned a lot, and I've done a lot, and never, during any of it, have I done anything that would place myself in rebellion to you.\"\n\nVicky waved to the right and left of her podium at the people seated behind her. \"None of these people have stood up in rebellion to you. Every one of them, just like me, stand in full fealty to you, our Emperor.\"\n\n\"I don't understand, daughter. I was told you fired on Imperial Navy ships.\"\n\n\"Say rather, Father, that I fired on ships going about the destruction of the Empire by doing the Empress's will.\"\n\nThat let the cat out of the bag and a full-fledged riot loose in the Hall of Mirrors.\n\nThe Empress might be gone, but her father and many of their party had not left. Now they were on their feet, shouting \"Liar!\" Traitor!\" \"Take that back, foul whore!\" and much worse.\n\nSwords were drawn and waved over heads. The poor working stiffs at the tables down the middle of the room beat a swift retreat for the door, though many of those on the Imperial side crawled under the table to make their run for it with the table between them and the waving swords.\n\nKris let the demonstration go on; she'd attended political conventions before. Then it started to get serious. The Imperial Guardsmen were ordered forward. Jack and Nelly ordered the Royal US Marines forward from their place against the wall. Others, part of the detachment billeted on the third and fourth floor, arrived at the double and reinforced the thin blue line standing at attention along the far side of the tables from the Imperials.\n\nKris hammered with her gavel but to no effect. Nelly captured the clash of wood on wood and amplified it.\n\nSuddenly, everyone was reaching to cover their ears. To the eternal honor of the Corps, the Marines stood in their line, rifles at parade rest, ears not quite bleeding.\n\nInto the silence, Kris announced in her command voice, \"Your Imperial Majesty, please withdraw to the lodge. May I suggest that you take your lunch at its facilities. I will send a representative when I am ready to reconvene this afternoon.\"\n\n\"After what was said about my daughter, you can reconvene in hell,\" the Grand Duke of Greenfeld bellowed.\n\n\"She said worse about our Grand Duchess,\" came from the other side, and Kris was again hammering with her gavel.\n\nNelly didn't have to resort to a mass sonic attack. The Imperial side emptied quickly.\n\nOnce they were gone, Kris stood, and Vicky quickly joined her.\n\n\"That didn't go too well, did it?\"\n\nKris worried her lower lip. She really couldn't say a word, not without ruining her role as mediator.\n\n\"I would suggest that you use the Banquet Hall for your lunch, Vicky.\"\n\n\"What about you?\"\n\n\"I've got a mess hall for my Marines on the fourth floor.\"\n\n\"It's a shame to roust you out of your fine Banquet Hall for us.\"\n\n\"Yes, but we both know your people would be shot if they tried to go up the stairs.\"\n\nVicky glanced around. \"I think the Empress would love that.\"\n\nSaddened, Kris took the measure of the room. Some of the Imperial chairs had been knocked over in their haste to leave. She had entertained hopes that they might actually be able to do something. Kris sighed and answered Vicky's question. \"I'm not in the business of doing what the Empress wants, much less loves.\"\n\n\"I'll see you after lunch,\" Vicky said, glancing back at the clock. \"We'll be back in an hour and a half, negotiating time.\"\n\n\"Yes, and we'll see where we go from there.\"\n\n# 59\n\nKris was ready to hike up four flights of stairs, but not everyone on her advisory teams was as fit as a Marine. Al and Bill moved with caution. One of the data miners used a wheelchair to get around, and one of Diana's advocates had twisted her foot in a pickup football game yesterday and was on crutches.\n\nNelly manufactured an elevator and ran people up the outside of the castle keep from the ground floor to the fourth.\n\nLunch was warm ham, beef brisket, barbecued pork, and roasted turkey with several cheeses and breads on the side and at least a dozen kinds of salad. Robin Song supervised it, but most of the food workers were local hires. At least Kris was doing something good for the Cuzco economy.\n\n\"I'm sorry. All we have is what we could quickly throw together. We figured the lodge's Great Hall for lunch and another banquet in our hall for supper,\" Robin said. She'd come down from the Forward Lounge with her wife, Mary Fintch, and most of their cooks to run Kris's kitchen dirtside for both the Banquet Hall and Marine mess.\n\n\"I don't see anyone turning up their noses,\" Kris said, then went down the line with Jack when everyone had been served.\n\nAt the troika's table, two seats stood empty, expectant. Kris joined her team leads.\n\n\"On a scale of one to ten, how bad does today stack up against what you've sat in on?\" Kris asked after she'd swallowed her first bite of a delicious sandwich of ham and turkey on rye.\n\n\"I must admit,\" said Bill, \"I've never had swords waved about at any of my arbitration sessions.\"\n\n\"That is an original for me, too,\" Al agreed.\n\n\"I've seen some pretty bitter words,\" Diana said, \"though not between parent and child. It's usually the parents.\"\n\n\"So where do we go next?\" Kris asked.\n\n\"We go back to the hall and talk some more,\" Bill said.\n\n\"We cannot get to a resolution of this matter if we cannot get the sides talking. Yes, they will start out venting like this morning, but over time and more sessions, the words will get softer and more constructive,\" Al said hopefully.\n\n\"It always goes that way,\" was Diana's contribution.\n\nKris looked around the room, spotted Major Henderson, and waved. He immediately came to stand by Kris.\n\n\"As soon as you finish eating, please carry my respects to the Emperor and tell him we will reconvene in the Hall of Mirrors at one thirty.\"\n\n\"Yes, Admiral,\" he said, and immediately left.\n\nIt was a good thing Kris sent that request so quickly, because the major was back in ten minutes with a reply. \"The Emperor wants this afternoon's negotiations to take place in the lodge's Great Hall.\"\n\nAl shook his head. \"We're not set up for the Great Hall. We have the recording devices over here.\"\n\nThe major shook his head. \"The Emperor was quite definite. If I may offer an opinion, although the Empress and the dukes said nothing, I got the feeling that they were the push behind this, and the Emperor was doing his best to please his wife.\"\n\n\"Is someone trying to arrange it so that there is no afternoon session?\" Kris asked her troika.\n\nThey nodded. \"Most definitely,\" Diana said.\n\n\"Ookaay,\" Kris muttered. \"Jack, get a security team over there and go over that place with a fine-tooth comb.\"\n\n\"On it,\" Jack said, and had Sal ordering an officers' call even as he stood.\n\n\"Nelly, get me Ensign Longknife.\"\n\nThe eager young ensign was at Kris's side in a blink.\n\n\"Tell the people with the recording equipment that they've got to move their setup to the Great Hall of the lodge.\"\n\n\"Yes, Admiral,\" and Meg was off at a trot.\n\nEnsign Longknife returned as Kris was watching her troika enjoy their second cup of coffee. She watched them longingly since she hadn't had a cup in over a year. Caffeine in the breast milk led to Ruth's bouncing off the walls.\n\n\"They're moving their gear, but the lodge is scrambling to find tables and chairs to outfit the Great Hall for the session. I really don't want to pass this along, but I have a request from the lodge manager for some Smart Metal to make tables and chairs.\"\n\nKris was pretty sure what her answer was going to be, but she passed the question to her brain trust with a raised eyebrow.\n\n\"I don't know about you,\" Al said, \"but I like meeting in the Hall of Mirrors. I hate to say it, but I don't trust the Imperials. I think the Emperor is sincere, but the Empress, and even worse, her family . . .\" He just shook his head.\n\n\"But if we let them cancel this session, when will we get things going again?\" Diana said.\n\n\"You willing to trust the walk between here and there?\" Bill asked.\n\n\"We always knew we'd have to do it,\" Diana countered. \"Half here, half there, switching every day.\"\n\n\"Nelly, tell the fine gentleman who runs this resort that he better get some rental tables and chairs out here pronto.\"\n\n\"He'd figured that would likely be your answer, Kris. The necessary gear is already being loaded in town. The first truck is on the move.\"\n\n\"Thank you, Nelly,\" Kris said. \"And no,\" she added to her troika, \"I was not going to let any of the Smart Metal out of my sight. Besides, I think once the Emperor has to sit his backside down in a chair for three hours that isn't the comfortable Smart Metal chair that he's enjoyed with us, he might have a different idea about where he wants to do his listening.\"\n\nTwo hours later, Kris opened the next session. The Empress was conspicuous by her absence. This time, the prime minister took the podium and read a long statement in a painfully boring monotone. Kris did her best to follow it, but she kept either getting lost in the crazy twists and turns of the verbosity, or she'd nod off, hopefully for only a moment.\n\nKris would have thought the uncomfortable chairs would have made a nap impossible, but there were soon snores coming from several places in the hall, including the Emperor.\n\nNELLY, ARE YOU GETTING ALL THIS?\n\nYES.\n\nCAN YOU MAKE ANY SENSE OF IT?\n\nNO, KRIS. IT DOESN'T MATCH WHAT WE'VE MANAGED TO CAPTURE OF GREENFELD'S RECENT HISTORY. IT DOESN'T EVEN HAVE INTERNAL CONSISTENCY.\n\nTHANK YOU. I WAS STARTING TO THINK IT WAS ONLY ME.\n\nNinety minutes later, Kris ended the torture in midsentence and called for a fifteen-minute break. Several of the Imperials had to be woken. Even some of Vicky's had fallen asleep although she, Mannie, and a couple of others had been taking copious notes.\n\nKRIS, WE'LL BE READY TO GO WHEN YOU CALL US BACK, suddenly came over Nelly Net.\n\nI DIDN'T KNOW YOU COULD TALK TO ME THIS WAY.\n\nI DON'T WANT TO BE SEEN TALKING TO YOU TOO OFTEN, SO I ASKED MY COMPUTER IF IT COULD ACCESS NELLY NET AND, VOILA, HERE I AM.\n\nNELLY?\n\nI AGREED WHEN VICKY'S COMPUTER ASKED NICELY. I'VE HAD A LOT OF OTHER ATTEMPTS TO BREAK INTO MY NET. NOT NICE. I'VE BROKEN A LOT OF NETWORKING GEAR. IT'S FUN.\n\nOOKAAY. THANKS FOR THE HEADS-UP, VICKY. I'LL GET US BACK TOGETHER RIGHT ON TIME.\n\nKRIS, WE'VE BEEN MAKING NOTES AS HE TALKED. WE'VE HEARD MOST OF THAT BEFORE. WE'RE READY TO GO WITH A FULL REBUTTAL.\n\n# 60\n\nKris did not get the meeting under way when the fifteen-minute break was up.\n\nIt seemed that the Imperial side was more than a little slow to return to their seats. They had taken their break in the lodge's Banquet Hall. Even though the Grand Duchess's allies had returned to the castle for their break, they were back a lot sooner than the Imperials.\n\nWHAT'S GOING ON, NELLY?\n\nMY NANOS SHOW THE EMPRESS HAS THE EMPEROR DEEP IN HEATED DISCUSSION. OTHER IMPERIALS SEEM RELUCTANT TO FINISH THEIR COFFEE, BEER, OR STRONGER BEVERAGE. THERE ISN'T MUCH MOVEMENT THIS WAY.\n\nThere was a short pause, then, KRIS, WE'VE FOUND A BOMB.\n\nWHERE?\n\nIT'S BURIED IN THE WALL BEHIND THE GRAND DUCHESS'S SIDE OF THE ROOM.\n\nKris considered her options. Clearly, Jack was playing it quiet. She should follow his lead. She considered several outcomes from this, both the bloody ones and the political, and made her call.\n\n\"Folks,\" Kris announced in the calmest of voices, \"it appears we have a bomb in the room. May I suggest we all walk, not run, to the nearest exit. I see four in the room. Two are on the farthest walls from me,\" Kris said, and motioned with both her hands to the two exits in the front of the room, \"and two behind me,\" and she spread her hands to point to those on either side of the hall.\n\nAmong the rebels, there was a quick exit. The few Imperials made a frantic dash, with people pushing and shoving. One fell and was trampled under feet.\n\n\"Kris, would you please get out of here,\" Jack said, coming up to her.\n\n\"Did you see me?\" she asked excitedly. \"I've been wanting to do the exit talk since I was six years old and saw it on an airplane.\"\n\n\"Fine. Fine, You've done it, now follow your own instructions and get gone.\"\n\n\"Are you going?\" Kris said, eyeing Jack.\n\n\"Ah, there are the bomb experts,\" Jack said, moving Kris aside so three people in well-padded gear could waddle by. \"We are out of here,\" he ordered, and led Kris by the elbow out the door.\n\nNELLY, HOW'S THE BOMB?\n\nIT'S QUIET, AS YOU NO DOUBT HAVE NOTICED.\n\nExasperated, Kris went on, NELLY, WHAT TYPE OF BOMB IS IT? HOW DANGEROUS?\n\nIT APPEARS TO BE A BANGALORE TYPE. FROM THE LOOKS OF THE MOUTH, IT'S LOADED WITH ANTIPERSONNEL FL\u00c9CHETTES. VERY NASTY FOR UNPROTECTED FLESH.\n\nWOULD THEY LIKELY CARRY TO THE OTHER SIDE OF THE ROOM?\n\nNO, KRIS. VICKY'S SIDE WOULD BE SLAUGHTERED. FEW ON THE OTHER SIDE WOULD EVEN BE HURT.\n\nKris scowled at Jack. \"You getting this?\"\n\n\"I heard it before.\"\n\n\"How come we didn't spot the bomb in the sweep before we came over here?\"\n\n\"It's plastic, and its electronics were inactive,\" Jack said. \"It would not have been noticed, but one of the sleepers leaned back against the wall. He didn't notice the wall caving in behind his head, but a guy coming back from the break noticed it and started widening the hole, and, bingo, we've got a bomb.\"\n\nJack walked Kris farther from the door they'd just walked out of. It soon opened, and the people in the big suits walked out, pulling a little wagon behind them with a large barrel on it.\n\n\"You going to blow that up?\" Kris called to them.\n\n\"Yes, ma'am,\" came back to her through a speaker.\n\n\"See if you can get any evidence off it.\"\n\n\"If we can, we will.\"\n\n\"Pardon me,\" said a man of middle height, middle weight, and middle appearance. \"You may not remember me. I'm Mr. Smith. The Grand Duchess would appreciate it if I could join your bomb-examination team.\"\n\n\"They're going to blow it, but feel free to look at the pieces,\" Kris said.\n\n\"Often the pieces say more than you'd expect.\"\n\nHe went after the bomb-disposal team, and Kris set about restarting her meeting. It seemed like an impossible task, but it got easier when the Emperor appeared. He went straight to Kris.\n\n\"I understand there was a bomb aimed at my daughter.\"\n\n\"Yes, Your Imperial Majesty,\" Kris said.\n\n\"Why wasn't it found?\" he demanded.\n\n\"It was put deep into the stone wall where it had crumbled with age,\" Jack explained. \"It was made of plastic, plastered over into the wall, and apparently its electronics were silent until it was time-activated. There was no way our sweeps could have caught it.\"\n\n\"Had it been there for a long time?\"\n\n\"No way to tell, Your Majesty,\" Jack said.\n\n\"So there could be another bomb behind her or even one behind me,\" he snapped.\n\n\"During the break, my Marines and your Imperial Guard went over the walls of the Great Hall with nails and hammers. We found nothing. We think the stones are stones.\"\n\nThe Emperor rubbed his ear. \"I don't like this. I don't like this at all.\"\n\n\"Your daughter, the Grand Duchess, says that there have been quite a few attempts on her life,\" Kris mentioned.\n\n\"So some have told me. Others say it's her own fault for associating with lawless thugs.\"\n\n\"You've seen the people who are following her, Your Imperial Majesty,\" Kris said.\n\n\"Yeah. Now I've seen them.\" The Emperor rubbed his chin and eyed the Grand Duchess's followers filling up their chairs, if looking a bit askance at the painted stones behind them, now showing a lot of chipping from where they had been hammered. Several of those seated against the wall made their own inspection.\n\n\"Interesting. Very interesting,\" the Emperor said, and turned to his seat. A dozen courtiers followed closely behind him.\n\nThe Grand Duke of Greenfeld had come in and taken his place on the Emperor's right hand. An empty chair waited for the Empress.\n\nKris banged her gavel, and those who hadn't found their seats rushed to them. When she felt the last few deserved what they would no doubt get from those whose toes they stepped on, Kris said, \"Your Grace. You have the floor for the next hour and a half.\"\n\n\"Thank you. Your Imperial Majesty,\" Vicky said with a nod to her father, followed by one to Kris, \"Your Highness. I cede my time to Steve Rojek,\" and she turned and sat down.\n\nA young man in conservative business dress came not to the podium but to a table that had been placed beside it during the break. A young man and woman, again in professional dress, came forward, their arms full of readers. They took the seats beside him and spread their electronic gear out in front of them. Steve, however, had only one tablet in front of him.\n\nAgain, he started with bows. \"Your Imperial Majesty, Your Grace, Your Highness, other noble ladies and sirs, ladies and gentlemen. For the last several years, our beloved Greenfeld has been through some very troubling times. Earlier this afternoon, we heard one version of those troubles. At least those of us who managed to stay awake,\" he said with a near-comical shrug of his shoulders.\n\nThat brought loud guffaws and soft chuckles; several of those who had slept through the prime minister's boring and rambling statement got elbowed in the ribs.\n\n\"Now, I would like to offer a different perspective of how all this came to pass.\" So saying, the young man began to lay careful groundwork for where the Empire had been and how it managed to sink from there to its present low.\n\nHis words were soft, but firm. His facts were, wherever possible, backed up with citations. Where they were not, he allowed for differences of opinion, or offered to bring forth witnesses to give sworn testimony. Between his soft words and simple-to-follow rhetoric, he had most of the room intently listening to him.\n\nThere were the occasional outbursts, inevitably from the Empress's faction. They grew fewer as those on the Imperial side of the room took in their Emperor's demeanor. He was leaning forward in his chair, eyes intent on Mr. Rojek. His right elbow rested on the arm of his chair, and his chin rested on the knuckles of that hand.\n\nHe did not nod along with Steve. However, though he showed no agreement, neither did he object to his words. As the time wore on, the Emperor never took his eyes off Steve. Vicky's spokesman, rather than wilting under that pressure, faced his Emperor and spoke his words calmly and gently, right to his face.\n\nKris watched the two men and felt like she was watching a conversation like two gentlemen might have seated next to each other in their club.\n\n_Has Vicky finally found a way to get inside the Emperor's bubble? The Empress is not playing her cards right. If she doesn't want her man to get his eyes opened for himself, she needs to be down here, rubbing those oversize mammary glands of hers in his face._\n\nBut the Empress stayed gone, and the Emperor stayed intent on Vicky's spokesman.\n\nWith twenty minutes left to go in his time, Kris began to worry she'd have to gavel him to silence in midsentence. She'd done that to the Empress; she would do that to him but hated to treat his logical presentation like she'd treated the Empress's tirade.\n\nWith exactly five minutes left to go, Steve brought his presentation to a perfect conclusion. He rose, and said, \"I yield the remainder of my time back to Her Grace.\"\n\nVicky stood, but for a long moment, she said nothing. Then, someone among her people blew a clear perfect note on a pitch pipe, and Vicky began to sing.\n\nNELLY, WHAT'S THAT SONG MEAN?\n\nTHAT, KRIS, IS THE IMPERIAL ANTHEM.\n\nVicky only had to sing the first line solo. Quickly, people around her rose to their feet and joined in the song. On the Imperial side, confusion reigned. Should they stand? It was the Emperor's anthem. But rebels were singing it.\n\nSome stood, not looking around for instructions. It was their anthem. Some even began to sing along, though softly at first.\n\nThe Grand Duke, father to the Empress, stolidly sat in his chair, mouth firmly shut.\n\nThen the Emperor turned to him. It was just a look, but it was _that_ look.\n\nShamefaced, the Grand Duke stood and began to mouth the words. That was all it took. Soon everyone in the hall except the Emperor, or the citizens of Cuzco or Royal US citizens, was standing and singing. Kris had to remember not to join in on the refrain, especially the catchy one, \"Greenfeld, ever verdant, ever ours.\"\n\nWhen the song came to an end, there was silence for a long moment. Kris hated to break it, but it was her job. She tapped the plaque of wood on the arm of her chair with the handle of her gavel to get a softer sound.\n\n\"I close this session now. We will meet again tomorrow morning at nine o'clock in the Hall of Mirrors.\"\n\n\"About that,\" the Emperor said, coming up to Kris's chair. She stood to meet him. \"I don't know about your ass, young woman, but mine is damn near numb from sitting in that chair.\"\n\n\"I was told that these were the best chairs in the resort.\"\n\n\"No doubt they are. Leastwise the best chair in my suite isn't any better. Oh, and about that bomb. Do you think anyone could put a bomb in your Smart Metal walls?\"\n\nKris shook her head. \"We've had those walls under continuous observation since they were constructed, Your Majesty. There is no way anyone could make a hole in those walls and secrete a bomb inside them without our knowing.\"\n\n\"Yeah, that's what I thought. Between your comfortable chairs and tighter security, I think we should continue to meet in the Hall of Mirrors.\"\n\n\"Your Majesty,\" said the Grand Duke of Greenfeld, coming up on the Emperor's other side, \"it is a matter of equity. They host some meetings. We host some. A balance must be struck.\"\n\n\"Equity, hell. We had a bombing here, and my ass is asleep from that damn chair. Here's the equity I'll give you. If anyone gets bombed in their hall, or my ass gets tired, then we come back here. That okay with you, kid?\"\n\nKris nodded. \"Is that acceptable to Your Grace?\" she called, as Vicky's immediate team walked by on their way out the door.\n\n\"Of course, Your Highness, Your Imperial Majesty,\" was accompanied by the appropriate bows.\n\n\"Good. That's settled. Vicky, I'll see you tomorrow. Your guy had a lot of things to say. I can't say I agree with all of it, but it wasn't some of the bullshit I hear too often in the Palace.\"\n\n\"Thank you, Father,\" Vicky said, but added no more, not even a smile that might be mistaken for gloating.\n\n# 61\n\nThe morning schedule that Kris had sent around to the delegates, and not followed all that much, did include an invitation to another banquet at nine o'clock that evening. She'd asked for RSVPs so they might know how many they would be feeding.\n\nVicky's side were definitely available for free food. All of them had responded before the first session started that morning. Only after the Emperor returned to his room following the last session did he reply in the positive. Within half an hour, Mary Fintch had heard from every Imperial.\n\n\"Thanks for promising to pay for a meal for everyone the first three days,\" Mary said in a quick note to tell Kris the Banquet Hall would be full. \"Otherwise, I never would have ordered enough food for this many.\"\n\nKris just smiled. _Maybe I am getting the hang of this diplomatic thing._\n\nShe quickly pumped milk for Ruth while watching her on-screen as she played with her nannies. They tried to get the infant to look at Kris, but though they said she was smiling at her mommy, her mommy wasn't feeling the love.\n\n_I have got to make a trip up the beanstalk to visit Ruth,_ she swore.\n\nDinner went much better that evening, partially because Mary was late serving it. Kris arranged for an open bar in the vaulted entrance, and people circulated a bit more than they had the night before. The Empress had a headache and didn't attend. Her older brother hadn't been seen all day, apparently unable to leave the bed of his paramour, but most of the Imperials were there with the Emperor.\n\nThat included several young women with heaving, minimally covered bosoms, who orbited around the Emperor until they caught his eye, then, smiling coyly, closed in for his closer attention.\n\n\"Someone is definitely not playing her cards right,\" Jack muttered. Kris could only nod in agreement.\n\nThe Grand Duke was there at the Emperor's side, scowling at those bearers of such visible pulchritude, but they ignored him and continued basking in the Emperor's attention.\n\nEmperor Harry was only too happy to encourage their basking.\n\nNot all of the femininity around the Emperor was in Imperial bare. Several of the young women were locals, and not a few were from Vicky's faction. Vicky didn't seem to notice their waywardness, or if she did, she shrugged off her father's wandering eye with easy aplomb.\n\n\"What must it have been like for Vicky, growing up in that Palace, even if it wasn't officially a Palace yet?\" Jack asked softly.\n\n\"I thought she was bad when I first met her,\" Kris said, \"but after looking at this, I wonder how she managed to survive to be the person she is now.\"\n\n\"She put on quite a show, first answering her stepmother's tirade with gentle words, then handing the lead over to that Steve fellow. What he did . . .\" Jack just shook his head.\n\n\"I suspect he was operating from a lot of notes given by Vicky and her Mannie.\"\n\n\"You're proud of her. Busting-buttons proud.\"\n\nKris swirled her tonic water and lime twist. \"I'd like to think I had something to do with who she's become.\"\n\nSoft chimes announced dinner. It went well and progressed smoothly into after-dinner drinks in the ballroom. The conversation showed the sides mixing a bit here, a bit there. Some of the locals managed to bring people from both sides together. Kris's brain trust had noted which Imperials listened more intently to Steve. They worked hard to get those people talking with locals. Then they would smoothly add one or two from Vicky's faction.\n\nKris allowed herself to traipse up the stairs to bed early, hoping the next day might show more progress.\n\nNext morning, the Emperor arrived five minutes early and stood at Kris's side, facing Vicky, as Kris flipped the coin. Vicky allowed her father to call the toss. He lost, but she invited him to go first.\n\nKris soon wished Vicky hadn't.\n\nThe Grand Duke of Greenfeld quickly took the podium and began to defend his daughter and family's necessary actions to bring law and order back to the planets being stalked by rebellion and sedition. \"Factions seated across from us. Factions making groundless accusations against our moderate efforts to keep the peace and see that Greenfeld's business was business,\" he said, turning to the Emperor.\n\nEmperor Henry nodded as his own words were fed back to him.\n\nThe Grand Duke's rebuttal was loud and proud. He mentioned a lot of planets, but usually in the context of rooting out the State Security traitors. That always got the Emperor's nod. But when he addressed other planets and other outbreaks of lawlessness, he got vague. And the actions of the Security Consultants were always appropriate and no more than were essential to restore peace and the rule of law.\n\nThe Emperor nodded along every time the Grand Duke said, \"Restore peace and the rule of law.\"\n\nThe Grand Duchess's factions sat stony-faced but silent through it all. About half of the Imperials, the faction Kris was coming to recognize as the Empress's lackeys, cheered and whistled whenever the Emperor appeared to approve of what he was hearing.\n\nThere was another faction, though. Dressed in the colorful fashions of the Imperial court, they nevertheless did not react when the Empress's faction did. Slowly, it became clear to everyone in the hall that there were a lot of people on the Imperial side sitting just as silent and stony-faced as those on the Grand Duchess's side.\n\nIt started as a few gentle digs in ribs as the Empress's faction tried to get a rise out of the silent ones in their midst. Then it escalated to serious jostling. When someone called out \"Stop that,\" Kris brought the gavel down and two shamefaced men traded seats with the men next to them.\n\nThe last forty minutes of the Grand Duke's speech was interrupted by more and more of Kris hammering her gavel. She finally resorted to an appeal to the Emperor.\n\n\"Your Imperial Majesty, please restore order among your delegation.\"\n\n\"Cut the crap, or I'll have the Imperial Guard escort you out of here.\"\n\n\"And you'll go to bed without any of that whore's nice supper,\" someone added.\n\nKris locked eyes with the Emperor. He owed her an apology. He owed her one, but would he give it to her? He broke eye contact first.\n\n\"Bruno, I think you've made your point,\" the Emperor said. \"Your Highness, I yield back the remainder of my time to you so we can have a nice, long break to let things calm down.\"\n\n\"I think that would be a very good use of that time, Your Imperial Majesty. I propose we meet back here in forty minutes.\" That was a bit longer than the remaining time and the break time added up to, but Kris needed some time to calm herself down from that last anonymous remark.\n\n\"Your Majesty, we have refreshments for your delegates in the ballroom. Nelly, would you open a passageway there?\" Kris didn't want anyone sniping or tossing real grenades at her guests.\n\n\"Yes, Your Highness.\"\n\n\"Your Grace, there will be refreshment in the Banquet Hall for you and your associates. Nelly, open another passageway.\"\n\n\"Done,\" said Nelly\n\n\"Thank you, Your Highness,\" said the Grand Duchess with a gentle bow.\n\n\"What about us?\" Jack asked.\n\n\"I guess we use the Marine mess. Nelly, see that there is an open bar in the ballroom.\"\n\n\"Not in the Banquet Hall?\"\n\n\"Nope. Unless Vicky asks for it, don't offer it.\"\n\n\"You trying to get the Imperials soused?\" Jack asked.\n\n\"Jack, do you doubt the Imperials will want something stronger than tea, and yes, if they want it, I'll let them get as drunk as they want.\"\n\nWhile Kris and her immediate team headed up the elevator that had suddenly appeared, most of her brain trust followed the other two factions. The data miners and technical staff went with Kris, but Al, Bill, and Diana, along with their staffs, looked more interested in doing additional work rather than taking a break.\n\nKris took part of the break to off-load some milk and watch Ruth play up on the _Princess Royal_. That took some of the pain out of her breasts, but none of the pain out of her heart.\n\nAfter the break, an Imperial Navy officer took the podium. \"I am Admiral Boles\u0142aw, and I had the honor of being captain of several ships that carried the Grand Duchess. I was at the conn when my heavy cruiser went through a jump to find a schooner with 18-inch pulse lasers waiting for us. We blasted it out of space, but it damn near blew us to kingdom come.\"\n\n\"What does this pirate ship have to do with anything?\" the Grand Duke shouted out.\n\nThe admiral was unfazed. \"We wish to make the point, Your Imperial Majesty, that those who have contributed to the breakdown in law and order in the Empire have also stalked your daughter, Grand Duchess Victoria.\"\n\nThe Emperor cocked his head to one side and seemed thoughtful though he said nothing.\n\nKris rapped the handle of her gavel. \"I find this attack interesting, Your Imperial Majesty. On our voyage to Greenfeld, my squadron also was ambushed. The ships involved were old Greenfeld battleships, left over from the Iteeche War, but they still packed a punch though not very accurately,\" Kris said, wanting to avoid just how she'd managed to handle them. \"We captured quite a few of the crew, including one officer. I think those assembled here might well benefit from his testimony. If it pleases Your Imperial Majesty, Your Grace, I would present that witness tomorrow.\"\n\nVicky nodded agreement. The Grand Duke was busy bending the Emperor's ear. Said Emperor began slowly shaking his head, but the Grand Duke didn't give up. The head shaking got more and more vigorous until, finally, the Emperor scowled and shoved the father of his Empress away.\n\n\"Did you identify these rogue battleships, Your Highness?\" Emperor Harry asked.\n\n\"Yes, we did. They were the _Hephaestus_ and _Poseidon_ of the Mars class, Your Imperial Majesty.\"\n\n\"You can call me sir, Kris. Otherwise, these sessions are going to go on forever.\"\n\n\"Yes, sir,\" Kris said with a gentle smile.\n\n_Do not start to like this bastard. Remember, he wanted you delivered naked to him for torture just a few years ago if the brags of one of his people can be believed._\n\n\"Where were the _Hephaestus_ and _Poseidon_ assigned?\" the Emperor asked.\n\nThat brought silence from those around him.\n\nVicky nodded to Admiral Waller, who sat in her entourage. He stood up. \"Sir, the _Hephaestus_ and _Poseidon_ were assigned to the Greenfeld Reserve Fleet. When I left the capital, they were still trailing the reserve station with no crews aboard.\"\n\n\"You can't believe a word that man says,\" the Grand Duke shouted. \"He broke his oath to you to run away and join the rebellion against you.\"\n\nKris was torn. Vicky had let her father talk during her time. Should she allow this cross talk? Cross talk, hell! That bastard was shouting.\n\nThe admiral handled the interruption like a gentleman. \"Your Majesty, may I answer that question?\"\n\n\"Go ahead, Rolf.\"\n\n\"Your Majesty, I left the capital one step ahead of an assassination squad.\" The admiral paused here, and you could see his jaw tremble. \"My beloved wife was not so lucky.\"\n\n_Oh, no._ Kris had heard that his wife had been caught and delivered to the Empress. She had personally slit the woman's throat and sent the video of it to her husband on Bayern.\n\n_This is about to get so ugly._\n\nKris got a good hold on her gavel.\n\nThe admiral went on, his voice low and trembling with emotions. \"My wife was caught by the assassins and murdered.\"\n\nKris and the Grand Duchess's side of the room breathed a sigh of relief.\n\nThe Grand Duke scowled. _Did he want the full accusation?_\n\n\"I'm sorry, Rolf. I didn't know,\" the Emperor said, then turned to one of his courtiers. \"Utz, you're supposed to have a handle on our Navy. What do you know about those two battleships?\"\n\nA young man, dressed even more flamboyantly than the rest of the Emperor's retinue, if that was possible, began talking to the red-and-silver box he carried.\n\n\"My Emperor, according to my files, they are still in the Reserve Fleet.\"\n\nHarry turned to Kris.\n\n\"According to my _experience_ ,\" Kris said, \"they are dust around a certain jump in the direct course from Wardhaven to Greenfeld.\"\n\nHarry made a wry face. \"So we add one more thing to the list of what one side thinks is true and the other says is false.\"\n\n\"So it would seem, sir.\"\n\nThe Emperor did not look happy, but the scowls he was throwing around were not meant for Vicky's side. Now, they were aimed at his own.\n\nFinally, he shook his head. \"Vicky, do you have more?\"\n\n\"Yes, Father.\"\n\nHe waved a hand for her to continue.\n\nVicky put her head together with several of her team, and Mannie was the next to take the podium.\n\n\"I was with the Grand Duchess when she was kidnapped. Later, I saw the bed that she was handcuffed to and left to die of thirst on. I was with the team that rescued her and shot down those who were trying to recapture her. They would not surrender but died resisting arrest. It was my investigators who traced the assassins back to the ship they had arrived on and followed their trail back to when they crossed from territory controlled by the Empress to Vicky's sworn planets.\"\n\nHaving said that, Mannie began to provide full details of everything he had outlined in his opening statement. Kris knew the attack had occurred; her team had investigated it and verified every fact Mannie was now providing the Emperor. He listened intently, smiling when the mayor described how Vicky had struggled to get herself free of the bed's iron embrace.\n\n\"How long did it take you?\" the Emperor asked his daughter.\n\n\"Too long, and I got a splinter in my right boob.\"\n\n\"And I wasn't there to kiss it and make it well.\"\n\n\"Sorry, no.\"\n\nKris wondered just how much information she was getting about this father\/daughter relationship but refused to speculate further.\n\nWhen Mannie finished, the Emperor asked Vicky, \"So that was why you didn't come to court when I called you?\"\n\n\"No, Father. I admit I wasn't quite as debilitated as the video made it seem. I was afraid for my life if I returned to the Palace and the assassination squads that seemed to dog my steps when I was there.\"\n\nKris had seen the video of the enraged Empress threatening Vicky that was piggybacked on the Emperor's message calling Vicky to court. Strange that the Grand Duchess made no attempt to enter that into the record for her father.\n\nThen again, as the morning wore on, the relationship between father and daughter had warmed quite a bit.\n\nAt eleven thirty, Kris gaveled the session to a close and offered the two sides the same arrangement for their lunch.\n\nThe Emperor stood. \"I would like to share my lunchtime with my daughter. You did say that you had done quite a lot of things you wanted to tell me about.\"\n\n\"Yes, Dad.\"\n\n\"Your Majesty,\" the Grand Duke interjected. \"This is not proper. It just isn't done during these kinds of things.\"\n\n\"Bruno, last time I checked, I was the Emperor, and I do believe I can do what I damn well please.\"\n\nBruno turned and stomped out. Several of his faction followed him. Others looked around, saw the Emperor heading for the other side of the room, and followed.\n\n\"Nelly, tell Mary to change her luncheon arrangements. You might help by magicking up more tables in the Banquet Hall.\"\n\n\"Already doing it, Kris. I made the suggestion to Mary right after the Emperor opened his mouth.\"\n\n\"Good girl,\" Kris said. She considered joining the Imperial couple for lunch, but saw Diana was edging her way through the crowd around the two and decided to give herself a break. She and Jack enjoyed their lunch and relaxed for a bit in their room. Nelly opened a window, giving them a gorgeous view of the forested hills with snow-covered mountains in the distance. The gentle breeze wafted the scent of pines into their room. Nelly really had the room open.\n\nThe afternoon session proved to be interesting. For one thing, the Grand Duke as well as the Empress were not present. There were quite a few missing from the Imperial side.\n\nIn fact, the Imperials had no one to take the podium when Kris gaveled the session open. It was an Imperial, dressed more as a businessman than a courtier, who finally stood. \"Your Majesty. The Empire has been suffering inflation and severe scarcity of critical items needed to maintain production and employment. Might we ask the Grand Duchess what answer she can make to the claim that _some_ people have been waging destructive economic warfare against our beloved Greenfeld?\"\n\nThe Emperor looked around his delegates, checking for anyone either raising an objection or offering a better use of their time. When he saw none, he shrugged toward Vicky.\n\n\"Daughter, what have you to say to these charges?\"\n\n\"Father, neither I nor anyone who has sworn fealty to their rightful Emperor through me have had anything to do with such matters. In truth, we have suffered the same fate. Suffered it so harshly that some planets have been reduced to starvation and have been taken over by bandits. During lunch, Father, I told you about my efforts to bring food to Poznan and restore commerce to them. I would like to introduce a business leader from that planet to tell you how they were reduced to such devastation.\"\n\nThe Emperor nodded, and Vicky called up Mr. Arnsvider. He quickly, with broad brushstrokes, described how the loss of credit and the depredation of pirates reduced his planet's access to critically needed raw materials and spare parts.\n\n\"The people who wiped out State Security somehow forgot to take their guns and ammunition with them when they left. A group of bandits got their hands on them, and from there, it was all downhill. Only the arrival of your daughter, the Grand Duchess, pulled us back from the brink of cannibalism. For that, we thank her and you.\"\n\nHe then went on to tell how matters had improved as trade opened up again, first with St. Petersburg, then with more planets.\n\nThrough all this, the Emperor nodded along, occasionally resting his eyes on Vicky with a proud, fatherly smile.\n\nMr. Arnsvider was followed to the podium by a long line of people from various planets. They described how they had been reduced to less desperate straits and walked back from them with the help of the Grand Duchess. She had secured the space lanes, then increased trade, first by barter but more recently with the help of banks and a respected currency.\n\nThe afternoon went this way, with people from the Imperial side also going to their podium to tell their own stories. The Emperor's face grew stormier by the minute. There were fewer and fewer members of the Empress's faction present, and none of them were objecting to any of this.\n\nThen a man from Dresden came to the podium on the Imperial side. Like those before him, he described the destruction of the State Security apparatus. Economic dislocation soon led to strife as hungry people took to the streets. \"And then the red coats arrived. The so called Security Specialists. These were led by Giorgio Topalski, now the Duke of Radebuel. To us, he will forever be known as the Butcher of Dresden.\"\n\nThe story went long. The man was reduced to tears well before he finished. The Hall of Mirrors became quiet, then silent as a tomb. Someone dropped a stylus; it resounded like a bomb.\n\nWhen the man finished, he sat down. The people on both sides of him scooted closer and put their arms around him as he gave himself up to wracking sobs.\n\nNo one stood to claim either podium.\n\nKris put her gavel down quietly, stood, and left the Hall silently. Slowly, others followed her.\n\nThe man, with his description of the horrors he had experienced and the sharing of his grief had succeeded where the best of intentions and skills had failed.\n\nHe had united both sides in their shared grief.\n\n# 62\n\nThe banquet that night was a totally different affair. The Emperor said he wanted to be seated by the Grand Duchess. \"Oh, and that guy that's always at her side. Something tells me I better get to know him.\"\n\nNelly did away with the dance floor; they no longer needed neutral territory between the two sides. Without its taking up space, they spread the tables out, leaving more room for people to circulate. Other than the two tables, one for Kris and the other for the two Imperials, there was no assigned seating. People found someone they knew, or people they didn't, and sat down to share a meal.\n\nFor the first time since this started, Kris got to relax and enjoy eating without fearing it might end in bloodshed. When dinner finished without a boom, and people began to adjourn to the ballroom to share more drinks and talk, Kris leaned close to Jack.\n\n\"There were a lot of empty chairs. Definitely no Empress or Grand Duke in evidence. Do you know where they are?\"\n\nJack shook his head. Before he could open his mouth, Nelly answered.\n\n\"You remember that superhigh tech we've been waiting for them to kick off using?\"\n\n\"Yes, Nelly,\"\n\n\"Well they kicked it off hard this afternoon. They put up some sort of electromagnetic barrier around the main lodge that will fry any nano that gets within five meters of the place. I tried hardening a few but got nowhere. Whatever's going on inside that lodge, we don't know thing one about that.\"\n\n\"Jack?\" Kris asked.\n\n\"I tried to get a few Marines. Even tried to slip Mary Fintch into the kitchen to ask for a cup of sugar. No joy. There are Imperial Guards at every door turning away anyone not an Imperial.\"\n\n\"Hmm,\" was all Kris could say to that. \"Any idea what they're up to?\"\n\n\"Not a guess, but they are certainly up to something. Since lunch, they've also been running limos, SUVs, and vans in and out of the lodge basement and circulating them into town. We tried following them, but they damn near ran our trailing cars off the road, and there are some pretty sharp cliffs on the drive into town that you don't want to get knocked over. We've got cars in town, now, but it's like some sort of snipe hunt.\"\n\n\"It looks like they might be doing what we've done a few times,\" Kris said, remembering how she'd got out of Nuu House and up to the _Princess Royal_ after the sniper attack.\n\n\"Don't you hate it when the bad guys and gals get as smart as you,\" Nelly said dryly.\n\nKris frowned as she puzzled this through, and at the pain in her breasts. She raised first one, then the other. They were heavy with milk.\n\n\"Listen, no one seems to have any idea what the Empress is or is not doing. Things here seem to be pretty laid-back and don't need us. I really could use a baby fix. I've had enough of pumping these jugs dry. I'd love to hold Ruth for a bit while she does the service for me.\"\n\n\"Can I go along, too?\" Jack said. \"I've missed seeing you and Ruthie together.\"\n\n\"Ensign Longknife, tell them to hold the next ferry up to the station for one Wardhaven Princess who desperately wants to see one tiny princess,\" Kris said, standing.\n\n\"Done, Your Highness,\" came back at Kris only a moment later.\n\nJack took no time at all to arrange a guard detail; the caravan was small. \"I think it ought to be good enough, what with this being so spur-of-the-moment.\"\n\nIn no time, they were at the space-elevator station and on their way up. A call ahead to the nannies revealed that Ruth had been asleep for a while but would likely be awake by the time they got there.\n\nKris and Jack managed to spend the ride up talking to each other about nothing that was world-ending. They were only a few minutes out when Jack called Security Central on the _Princess Royal_.\n\n\"How's everything tonight?\"\n\n\"Fine, General. Quiet watch. We've been getting some static on the cameras covering the pier tie-downs, though. I was about to dispatch two technicians to check Tie-down Number Two, the one that has all our housekeeping connections.\"\n\n\"How many Marines do you have covering that one?\" Tie-down Two was the only one big enough for someone to walk aboard.\n\n\"Two Marines, sir. They are reporting every quarter hour.\"\n\nJack looked at Kris.\n\n\"Call them. _Now!_ \" Kris shouted into Jack's commlink.\n\n\"Yes, ma'am. Wait one.\"\n\nThey didn't have to wait a full minute. \"I got their usual report. When I tried to talk to them some more, I got the same report.\"\n\n\"We are Code Red,\" Jack said evenly. \"We are Code Red.\"\n\nThe five minutes until the ferry docked at the station were the longest years of Kris's life.\n\n# 63\n\nKris and Jack stood right behind the hatches as the ferry rumbled and bumped to a stop, their service automatics out, ammo set on deadly. Safeties were off and full auto was selected. Behind them, Meg and the Marine guard detail waited, their weapons locked and cocked.\n\nThe hatch had hardly begun to open when Kris wiggled her way through it and hit the station at a dead run, Jack right behind her. The rest did their best to keep up, but Kris was going for the three-minute mile.\n\nShe already knew that the two Marine guards at the pier tie-down were dead, shot through the eyes. Kris had ordered the nursery doors sealed, but any effort to raise them on net had failed. Now she took the escalator stairs three at a time down to their pier. The OOD got out of her way as she shot up the brow.\n\n_I'll render honors twice next time._\n\nThe quarterdeck was still large. Her day quarters were a lot smaller. She slowed as she approached the door to the nursery.\n\nBehind it, Ruth was screaming like she'd never screamed before.\n\nShe and Jack approached the door, Meg right behind them, automatics up in a two-handed stance.\n\n\"Nelly, make the wall disappear on one.\"\n\n\"Yes, Kris.\"\n\n\"One.\"\n\nThe wall went away.\n\nKris's blood froze.\n\nA man in station working clothes held Ruth suspended by the back of her onesie. He had a gun aimed at her tiny head.\n\nBeside him, another man, dressed alike but showing the signs of a tough fight, had a gun aimed at the top of Sally Greer's head. She had one arm around his leg; another was a fist ready to slug him in the balls. If she could have killed the man with her eyes, this guy would be fried.\n\nThe last Kris noticed was Warrant Officer, ret. Li O'Malley. She lay on her back, four bullet wounds weeping on her chest, eyes fixed at the overhead. Somebody had wisely gone for the center of his target's mass.\n\nA service automatic was still grasped in Li's dead hand.\n\n\"Put the baby down, and we will let you live,\" Kris said through clinched teeth, not letting her gun waver from its lock between the eyes of the guy threatening Sally.\n\nJack was the better shot. He had his automatic aimed at the arm of the man holding Ruth.\n\n\"No, I got a better idea. You give us safe passage to the Imperial Courier Ship _MC-410_ , and we'll leave the baby on the pier.\"\n\n\"You know you won't do that,\" Kris said. \"As soon as that courier ship pulls away from the pier, there are nine battlecruisers that will blow it to hell. No, you're going to keep the baby until it's no longer convenient, then you'll ditch her. Maybe she lives. Maybe she doesn't. It's no skin off your nose either way.\"\n\n\"So, maybe I promise that I'll make real sure she lives.\"\n\nThey were negotiating, just as they had been doing down in the Hall of Mirrors. These guys had expected to die a minute ago. Now they could see a way to live. They relaxed ever so slightly.\n\nKris gave her head a brief shake. \"I don't think you'll have much control over what happens to her. I don't see you doing this for fun. It's got to be the money. Yet you're talking to one of the richest women in human space, and you aren't even giving me a chance to bid.\"\n\nJACK, DO YOU HAVE A SHOT?\n\nYES.\n\nI'LL GO FOR THE BABY, Meg put in, putting her automatic down slowly.\n\nGET READY EVERYONE TO MOVE WHEN I SAY SHOOT.\n\nREADY, Jack snapped.\n\nME, TOO, Meg added.\n\n\"How much you willing to offer?\" sounded way too eager.\n\n\"How much is on the table now?\" Kris answered.\n\nThe two thugs exchanged a quick glance.\n\nTHEY'RE GETTING SLOPPY, Jack thought.\n\nGET READY.\n\n\"Fifty billion,\" the guy holding the screaming Ruth said.\n\n\"Fifty billion what?\"\n\n\"Gold Imperial marks.\"\n\nSHOOT.\n\nKris put two rounds into the center of her target's mass, then two more right between his eyes. He crumpled where he stood.\n\nJack put one round in his guy's pistol, knocking it into his chest and away from Ruth's head. The thug got one shot off . . . it hit him, creasing his chest. Jack put two between the guy's eyes next, then two more for good measure where his heart was rumored to be.\n\nHe dropped backward like a felled tree.\n\nEnsign Longknife hurled herself across the space, sliding in under Ruthie and catching her before she hit the deck.\n\n# 64\n\nKris managed to put her weapon down at the same time she swooped in to collect a screaming Ruth from Meg's arms. She kept going until she fetched up beside the rocker and sank into it. While one hand held Ruth, Kris pulled down the top of her strapless ball gown with a bit of help from Mai Tiamat. Shushing her darling softly, Kris slid Ruthie to her other side and gave her a breast to suckle.\n\n\"There, there, baby. Mommy's here. No one's going to hurt you. There, there.\"\n\nRuth was having nothing of that soothing stuff.\n\n\"Nelly, raise a wall between us and those bleeding corpses. This place stinks.\"\n\nA wall appeared, blowers went to high, and the smell quickly dissipated.\n\nKris kept trying to soothe Ruth, but inside she was seething.\n\nNELLY, WE NEED TO TALK.\n\nJACK'S HERE, TOO.\n\nTHEY WERE PAID IN IMPERIAL GOLD MARKS AND WANTED AN IMPERIAL COURIER BOAT FOR THEIR GETAWAY. THIS CAME FROM INSIDE THE EMPIRE. NELLY, IS THERE AN IMPERIAL COURIER BOAT IN PORT?\n\nNO. NOT OFFICIALLY. BUT THERE IS A PIER AT THE END OF THE STATION. THERE ARE NO CAMERAS WORKING THERE, AND NOTHING IS OFFICIALLY THERE, EITHER. HOWEVER, THE WATER, AIR, POWER, AND SEWAGE USAGE HAVEN'T BEEN ADDING UP FOR THE LAST COUPLE OF DAYS NOW THAT I LOOK AT THEM.\n\nCAN YOU GET ANY KIND OF READOUT FROM THAT PIER, NELLY?\n\nNO, IT'S IN THE SHADOW OF THAT MONSTER, THE _E_ MPEROR AND _E_ MPRESS.\n\nTHAT DOESN'T MAKE ANY SENSE, TO HAVE NO WORKING CAMERAS ON THE PIER NEXT TO THE IMPERIAL FLAGSHIP, Jack put in.\n\nJACK, I WANT A MARINE STRIKE TEAM DOWN TO THAT PIER FAST. CIVVIES AND CONCEALED WEAPONS.\n\nIt took a long moment before Jack was back on net. THEY'RE ON THEIR WAY. MEGAN IS LEADING THEM. SHE DITCHED HER BALL GOWN. SHE'S RUNNING BAREFOOT IN JUST A SLIP. Kris ignored the thought of Jack watching a half-naked version of her younger self. She gently stroked Ruth's cheeks. She was hungrily sucking down her mommy's milk, and her little tears seemed to be drying rapidly.\n\nNELLY, WHAT OTHER SURPRISES ARE GOING ON AROUND HERE?\n\nTHE LONG-RANGE RADAR HAS BEEN ON THE FRITZ OFF AND ON FOR THE LAST COUPLE OF DAYS.\n\nAS IN SINCE THE _E_ MPEROR AND _E_ MPRESS ARRIVED?\n\nYEP. IT WENT TOTALLY DOWN AROUND NOON TODAY.\n\nWAS ANY TRAFFIC LEAVING ABOUT THEN?\n\nAN IMPERIAL LIGHT CRUISER.\n\nKris remembered Vicky telling how she and her second escort\/bodyguard had slipped away from High Greenfeld Station when the long-range radar was down by snuggling up next to a light cruiser.\n\nNELLY, HAVE CAPTAIN AJAX ACTIVATE THE _P_. _R_ OYAL'S LONG-RANGE SCANS. FIND THAT LIGHT CRUISER AND SEE IF THERE ISN'T A FAST COURIER BOAT OUT THERE AHEAD OF IT.\n\nSHE'S WORKING ON THAT.\n\nMinutes passed. Kris found herself and Ruthie settling into their calm nursing routine. _I'll miss having you at my breast when it comes time to wean you._\n\nThe wall went away, giving Kris and Ruth back the whole nursery. The blood and stink as well as the bodies were gone. Ruth didn't even notice the change, but Kris liked having Jack standing next to her.\n\n\"Kris,\" Nelly said softly, \"you will never guess what our Marines found at that pier.\"\n\n\"Don't keep me waiting, Nelly.\"\n\n\"There is a fast Imperial courier ship, _MC-410_ by name, complete with crew and a whole lot of courtiers. I've already run them through facial recognition. They're the Empress's faction.\"\n\n\"Did we get the Empress or Grand Duke?\"\n\n\"Nope, Kris, but the pilot said a courier boat headed out around noon today.\"\n\n\"And they stayed because?\" Jack said.\n\n\"They were waiting for a 'package.' A certain count was to tell him when it was aboard, and they were to sail immediately.\"\n\n\"Did he mention the name of that count?\" Kris snapped.\n\n\"He certainly did. He'll be in the first batch of prisoners Meg is bringing back.\"\n\n\"Jack,\" Kris said, \"I think we have a good reason to send a strong Marine detachment to a pier very close to the Imperial flagship.\"\n\n\"Thank you, my love. You call the Emperor, and I'll get the Marines going and accompany them in case things get diplomatically challenging.\"\n\nKris made a call to the Emperor that took a while to be answered and when it was had no visual. She briefed him on the attempt to kidnap her child.\n\n\"That's unacceptable,\" he huffed, not sounding at all like the man who had paid for Kris's Grandmother Sarah to be killed or her brother, Eddy, to be kidnapped. \"By the way, I have not seen the Empress since morning. Do you know anything of her whereabouts?\"\n\n\"An Imperial courier boat left here around noon. We don't know who its passengers were. I can get you a list of who was on the courier boat that was supposed to carry Ruth away. I am told there were several of your retinue aboard.\"\n\n\"Say not my retinue but the Empress's toadies.\"\n\nKris ended the call and had Nelly pass along photos of those on the boat. Very quickly, she got a call back from neither the Emperor nor the prime minister. The fellow was middle-aged, just starting to gray, and his clothing was sober gray.\n\n\"I've looked at all the pictures you sent. They are fairly high up in the Empress's faction, but not the top. We've looked around, and none of the senior supporters of the Empress, her father, or her brother are here although we have found the Empress's mother, Grand Duchess Maude and her brother's girlfriend. The Imperial Guard are interrogating them now, but their initial claim is that they have no idea where the others are.\"\n\n\"So the dukes took off but left their ladies behind. Strange,\" Kris said.\n\n\"Past strange.\"\n\n\"Just a moment, I've got another call coming in. Will you wait?\"\n\n\"For a short time.\"\n\n\"Admiral, this is Captain Ajax. We found the cruiser. It was harder to find the courier; it was stealthy as hell, but we've got it.\"\n\n\"Where?\"\n\n\"The courier is boosting at four gees for Jump Point Gamma, the most direct one to Greenfeld.\"\n\n\"Thank you, Captain. Hell's a popping. Keep your eyes out for anything.\"\n\n\"Aye, aye, ma'am.\"\n\n\"Thank you for waiting,\" Kris said to the man she'd kept on hold, and filled him in on what her ship's sensors had to say.\n\n\"They're running for home?\"\n\n\"That's their course.\"\n\n\"Bruno can't believe he can raise a rebellion with only the Empress and child as a power base.\"\n\n\"I'm not well schooled in these types of politics,\" Kris said, and left the fellow to his problems.\n\nA few minutes later, Jack asked if he could frog-march a certain count into the nursery. Kris handed off the now-slumbering little one to Fede Radko and met up with Jack and Megan. She really had stripped out of her ball gown to lead the Marines in just her slip.\n\n_I think I like this gal._\n\nKris left the count standing in the middle of her day quarters as she circled him, eyes hard.\n\nHe took one look at Kris and became very eager to tell his story even without her saying a word. He'd been left behind by the Empress and Grand Duke with orders to only leave when the package was delivered.\n\n\"I had no idea what the package was. They told me I'd know it when I saw it. I was told to shoot the messengers who delivered it, but I didn't really plan to do that,\" he insisted, groveling now at Kris's feet.\n\n\"That's pretty much what he's been saying since the pilot pointed him out,\" Jack said.\n\n\"Toss him in the brig,\" Kris ordered. \"I'll talk to him some more while we chase down that other courier boat.\"\n\n\"No. No. Send me ashore. You don't want me messing up your boat. Look at me. I've wet myself.\"\n\n\"Why don't I want you on my ship?\" Kris said slowly, studying this poor excuse for a human.\n\n\"Kris, Captain Ajax wants to talk to you,\" Nelly interrupted.\n\n\"Yes, Captain.\"\n\n\"The Gamma jump is spewing out ships like my brother's momma guppy.\"\n\n\"Ships?\" Kris said, eyeing the count. He looked as if he wanted to crawl into the floor; he couldn't get any lower.\n\n\"First through was a destroyer squadron, say twenty strong. It was followed by squadron of heavy cruisers. Now we got battlewagons following each other through like elephants the day I saw the circus come to town. Oh, my! Sensors say that last one is sporting twenty-four 19-inch guns. I think I'm scared,\" was pure sarcasm.\n\n\"Captain Ajax, I assume our squadron has all hands aboard.\"\n\n\"Since your Sailors can get a beer without going ashore, you bet, ma'am.\"\n\n\"Order the squadron to be ready to sortie on one hour's notice. I request your presence and that of your navigator in my flag plot in thirty minutes to plan our battle.\"\n\n\"Aye, aye, Admiral.\"\n\nThe count eyed Kris with eyes as big as saucers. \"Will you put me ashore now?\" he pleaded.\n\n\"Jack, see that this son of a bitch is locked up in the brig and throw away the key,\" Kris growled.\n\n# 65\n\nKris gathered her truncated staff in her flag plot. No surprise, Vicky and Mannie showed up with Admiral Boles\u0142aw in tow and added themselves to the small group with Kris. They stood around a plotting table that now showed the Cuzco system.\n\nBetween Cuzco and Jump Point Gamma was a light cruiser accelerating toward the jump at 1.25 gees. Well beyond it was the courier boat still maintaining four gees.\n\n\"There must be a lot of water beds on that boat,\" Jack muttered. \"Has the Emperor got back to us with a list of who he's missing from the Empress's faction?\"\n\n\"Nope,\" Kris said. \"I think we can assume at least the Empress, Grand Duke, and Duke are aboard with the closest of their cronies. Captain Ajax, what have we got headed in?\"\n\nThe captain touched the battle board, and a list of ships appeared. \"We've got quite an interesting collection. Pride of place goes to the _Terror of Space_ and _Death to Traitors_. Both of them are sporting twenty-four of those new 19-inch lasers. I have no idea how powerful they are. Since they were knocked together recently, I doubt they're Greenfeld's finest. With them are six with twelve 18-inch guns and another four with twelve 16-inch. The rest of the battle line is made up of ancient ships dragged out of mothballs. Heaven knows what they've got for crews, but there are twelve 15- and 14-inch battlewagons trailing the big punchers. Add to that a dozen heavy cruisers, a half dozen light, and forty tin cans, and you have a well-balanced fleet.\"\n\n\"Who expect they are only up against nine frigates,\" Kris said with a broad grin she knew was pure evil.\n\nBeside her, Vicky shook her head and shared a tight smile.\n\n\"I think they expect so,\" Captain Ajax said. \"We're all there is in town. Cuzco maintains no fleet, counting on its central place among three different competing alliances who presently seem happy with her neutrality. The Emperor only brought two light cruisers, and one of them seems to be in the Empress's pocket.\"\n\n\"Let's talk to the Emperor,\" Kris said. It took a few minutes to get him. The fellow who talked to Kris earlier took the call and asked if he might help. When Kris said no, he explained the Emperor had asked not to be disturbed again.\n\n\"He has company at the moment,\" the man said with a soft cough.\n\nVicky scowled and rolled her eyes at the overhead.\n\nKris scowled, too, as she said, \"He has a hostile fleet with twenty-four battleships headed for his bedroom just now. He might want to talk to me about it.\"\n\nThe Emperor was quickly on the line, again with no video. \"What's this about a battle fleet?\"\n\n\"There are twenty-four battleships headed for Cuzco led by the _Terror of Space_ and another one just as mean. Do you know anything about them?\" Kris asked.\n\nThe Emperor suffered a bad fit of coughing. Many soft feminine voices offered him several things to drink. It was a bit later before he could continue.\n\n\"The Empress kept telling me we couldn't trust a Longknife. That Vicky was going to do something treacherous. She insisted we station some ships close by, and I finally agreed. She didn't tell me how many ships she was going to haul out here or how close they'd be . . . and I didn't ask,\" he finished slowly, dawn coming up a lot sooner for him than most today.\n\n\"Well, we've got my nine frigates and one of your light cruisers between you and whatever those battlewagons intend.\"\n\nCaptain Ajax cleared her throat. \"The courier boat just flipped ship and is now decelerating at four gees.\"\n\nOn the battle board, Nelly drew a line from the courier. It would put the boat dead in space soon enough that it could accelerate back up and match with the incoming battleships.\n\n\"It would appear that the Empress and Grand Duke intend to join those battleships,\" Kris told the Emperor.\n\n\"Should I run? I have one light cruiser. It's faster than that tub the Empress insisted on.\"\n\n\"I'd really prefer that you didn't, sir,\" Kris said. \"I can't protect you if you're out there in just a light cruiser.\"\n\n\"But all you've got is nine little frigates,\" the Emperor said, his voice squeaking.\n\n\"Yes, Harry, but I've blown a couple of huge alien base ships out of space with ships like these. I think I can stand my ground against a couple of dozen battleships.\"\n\n\"Vicky?\" he asked.\n\nVicky glanced at Kris, then said, \"Father, I think we really need to trust Kris Longknife. She's been in a lot more fights than anyone on your wife's ships. If she says she can take them, I'd trust her.\"\n\n\"If you say so, baby girl. You think I should stay, Kris?\"\n\n\"You're safer on the ground for now. Trust me, Harry,\" Kris said. If she was going to save the dumb bastard's life again, he'd better get used to her calling him whatever she wanted to.\n\nThe Emperor said nothing.\n\n\"Well, I have a battle to plan. I'll get back to you when it's over.\"\n\n\"Nine frigates against a battle fleet?\" the Emperor squeaked.\n\n\"Remember, I took twelve mosquito boats up against six huge battlewagons and won.\" _That will teach you to keep doubting me._\n\nIt was cruel of Kris to point out that mistake of Harry's, but she was fresh out of the milk of human kindness where he was concerned.\n\nThe Emperor said nothing. He was still saying nothing when Kris cut the connection.\n\n\"Is that off?\" Vicky asked, nodding at the commlink.\n\n\"Dead as a brick,\" Nelly said.\n\nVicky rolled her eyes. \"I can't believe my dad is still calling me baby girl. I haven't wanted to be called that since I was five.\"\n\n\"Maybe he'll learn and give you the respect you deserve,\" Kris said, wondering when she, herself, would get a little respect from her elders. Kris shook that thought off.\n\n\"Now, folks, we have a battle to win. Let's see how we do it. Do we hang around here for a bit, then swing around the moon to take them down when they're on final approach? Do we swing around some planet and come back at Cuzco on a parallel course and hammer them all the way in? Or\"\u2014Kris grinned\u2014\"do we charge right at them like a bunch of fools and wipe out a whole lot of battleships in one pass?\"\n\nKris eyed her staff. They were grinning back at her.\n\nVicky wasn't smiling. She blinked several times, then said, \"As much as I hate my stepmother, many of those Sailors are loyal subjects of my father. I would prefer to give as many of them a chance to surrender as we can.\"\n\nShe glanced at Admiral Boles\u0142aw at her side. \"Can we cut out the rotten part of the apple without making applesauce of the rest?\"\n\n# 66\n\nThree days later, Kris's squadron sortied from High Cuzco Station.\n\nAs much as Kris had looked forward to winning this battle with the dumbest of attack plans, she'd dropped the idea. Captain Ajax pointed out that even if Kris did blast the flagship out of space and get the rest to surrender, they still might not get the results they needed.\n\nWith the hostile fleet in the final stages of decelerating toward Cuzco, they could hardly dump their reactor cores. Kris could never trust them to approach Cuzco except under her guns. This hour's surrender could too easily be followed the next hour by their zapping the planet from orbit.\n\nIt became even clearer when the Empress addressed her husband from the bridge of her new flagship, the _Terror of Space_. Her tirade against the Emperor and his stupidity was only balanced by her gloating over how she would kill Kris Longknife, then laser all of them, \"That whole damn planet,\" from orbit.\n\nNope, Kris would have to match orbits with this bunch and make sure they surrendered and stayed surrendered.\n\nThe one delight Kris found while listening to the Empress's boring harangue was watching the admiral standing beside her. The _Terror of Space_ rated an admiral for its captain, and that admiral was her old buddy, William T. Thorpe, former skipper of the Wardhaven corvette _Typhoon_. Kris had relieved him of his command back when she was just an ensign.\n\nWhen Kris countered the Empress's spiel with a demand they surrender now or face the consequences, Billy Thorpe laughed in her face. \"I'm so glad you command that collection of shrimp boats, Ensign Longknife. It would be such a joy to take your surrender, but please, please keep strutting around. I will so enjoy blasting you out of space.\"\n\nKris just grinned as if she knew something he didn't and cut the line.\n\n_Boy, do I know something you don't know._\n\n\"You know him?\" Jack asked.\n\n\"He was before your time, but his time is up.\"\n\n\"Yes. Yes, I do think it is.\"\n\nKris studied how the enemy battle array had developed. The two super battleships were following one behind the other. Ahead of the _Terror of Space_ were the six battleships with 18-incher lasers. Instead of being in a long line, they were stacked three high and two deep. The four 16-inch battleships trailing the _Death to Traitors_ were stacked two high, two deep. The older battlewagons formed two blocks, one forward, and the other aft of the newer ships. Both were three high and two deep. Sixteen heavy and light cruisers made up four stacks of four, two in the lead, the rest trailing. The last two light cruisers and the forty destroyers formed two circles, one below the thick battle line, the other above.\n\n\"Somebody wants to concentrate his fire,\" Kris said.\n\n\"The last time I fought the Empress, I divided by battle force into two lines of seven and nine. Her admiral divided his battleships into four long lines of a dozen each, minus those with engineering casualties. It left the older BBs out of range while I smashed the newer ones as they came in range.\"\n\n\"This still has the battle line bringing its different guns to bear at different times,\" Kris observed.\n\n\"Assuming she keeps them that way,\" Jack pointed out. \"Any idea who commands over there?\"\n\nBoth Kris and Vicky shook their heads.\n\n\"The only admiral I saw,\" Kris said, \"was Thorpe. Vicky, did you see anyone?\"\n\n\"Nope, Admiral Boles\u0142aw, was there anyone familiar to you on that bridge?\"\n\n\"Interestingly, no. Every officer's face I saw was that of a stranger, and the Navy is a small place. I would have expected to recognize someone.\"\n\n\"I think my stepmom brought in people to officer those big ships from anywhere but Greenfeld.\"\n\n\"The better to slaughter the Emperor,\" Mannie said.\n\n\"My father is nothing if not gullible.\" Vicky sighed, tiredly.\n\nKris pursed her lips and let that thought sink in, then began to issue orders. \"We will stay in line ahead for the trip out to the moon. As soon as we have the moon between us and the hostiles, second squadron will close up even with the flag with a thousand-klick interval between the two columns. On my orders, we will open up the interval to five thousand, but I don't intend to do that until we are two hundred fifty thousand klicks from the hostiles. Any concerns with that?\"\n\n\"They've shortened their battle line,\" Vicky said, \"so you're shortening yours?\"\n\n\"Yes,\" Kris said.\n\n\"Kris,\" Jack said, \"who will we target with our first broadside?\"\n\n\"You're thinking it will be our only broadside.\"\n\n\"I hope so,\" Vicky said.\n\nKris ran a worried hand through her hair. \"The first division's four ships target the two big ones. The second division's five ships target five of the six 18-inchers. Immediately after that, all Imperial survivors will be dismissed from stations so they can toddle off to their quarters to change their underwear.\"\n\nThat got a laugh.\n\nKris turned to Vicky. \"Your stepmother, the Empress, tried to kidnap my daughter. I stood in the nursery for an eternity before Jack got a shot and killed the bastard she hired. She's got a whole battle fleet aiming their guns down my throat. As far as I'm concerned, she's a dead woman.\"\n\nVicky took in several shallow breaths and let each out with a huff. Finally, she shook her head. \"I didn't kill her when I had a chance, but Dad was telling us to quit fighting. Still, I swore when I finally tore that bed apart and managed to get a drink of water that I'd had enough. I'd kill her. I think I would have if she'd been right there at that moment.\"\n\nVicky paused for a long minute, then fixed Kris with hard eyes. \"Kill the bitch.\"\n\n\"All right, ladies and gentlemen, the squadron has its orders. Let's go do what the Grand Duchess ordered.\"\n\n# 67\n\nKris came out from behind Cuzco's largest moon with all her ships at Condition Baker and her squadron in two parallel columns.\n\nShe wasn't the only one who had changed her battle formation.\n\nIt took the hostiles a bit longer to complete the reorganization, but when they did, Kris had to hand it to Thorpe. He'd come up with a very interesting way of defending the flag and bringing all his guns to bear . . . assuming he faced frigates armed with short-range 24-inch pulse lasers.\n\nThe _Death to Traitors_ still trailed the _Terror of Space_ , but it was the only ship.\n\nWell forward of those two, but forty thousand klicks closer to Kris, the stacks of 18- and 16-inch battleships followed one behind the other. Well aft, but coming up on eighty thousand klicks closer, were the 15- and 14-inch battleships. From the looks of it, they'd form a column of four, three ships high. The cruisers appeared to be racing to form a similar column of stacks well ahead of the newer battlewagons. Most likely, they'd end up only eighty thousand klicks from Kris.\n\nThe destroyers, all forty of them, were already turning to begin a pulse-laser run.\n\n\"Thorpe's a very smart fellow,\" Kris said through tight lips. \"He's put together a very nice trap . . . for a frightened bunny rabbit. Pity he's set it for one mean and angry momma bear.\"\n\nKris studied her board for a full minute, then asked, \"Nelly, will we need to worry about those tin cans?\"\n\nCircles appeared on Kris's battle board. The red of the destroyers were well back when Kris's ships passed the two-hundred-thousand-klick range mark.\n\n\"Kris, we're coming up on the two-hundred-fifty-thousand-klick range mark.\"\n\n\"Thank you, Nelly. 'Squadron, open your intervals to five thousand klicks. Division 2, drop your distance to five thousand klicks below Division 1 on my mark. Admiral Longknife sends.'\"\n\nThe ships board on Kris's wall all blinked twice as every ship acknowledged.\n\n\"Mark,\" Kris said.\n\nThe _Princess Royal_ kept its course steady. Every battlecruiser around it slowed or slipped lower, or both.\n\n\"Squadron, go to Condition Charlie now. We will go to Condition Zed when we come up on two hundred ten thousand klicks to the two big boys. We will dispense with high-gee stations for the moment. If we need them, we can order them out of the Smart Metal then.\"\n\nAround Kris, flag plot got smaller. Kris, Jack, Vicky, Mannie, and Admiral Boles\u0142aw got closer to the table.\n\n\"Nelly, put me on-screen. Beam this to all the ships across from us.\" Kris turned to face a blinking red lens. It turned green. \"This is Admiral, Her Highness Kris Longknife of United Society. I've slaughtered half a trillion aliens, so adding a few thousand humans to my blacked soul doesn't bother me.\"\n\nShe paused to let that sink in. \"But you aren't subjects of my King. You aren't, for the moment, rebels to your Emperor. However, if you continue to follow the false orders of the Empress, you will be in rebellion to your sworn liege lord. Discharge your lasers and blast away from Cuzco, and mercy shall be yours. Maintain your present course with charged lasers, and I will begin your destruction in three minutes.\"\n\nKris turned the comm over to Vicky.\n\n\"I am the Grand Duchess Victoria. You are all loyal subjects of my father, your Emperor, who is present on the planet of Cuzco, where you are heading. In a few minutes, a lot of you will be dead. If, after that, the rest of you choose to surrender, I will accept it.\"\n\n\"You bitch,\" the Empress broke in screaming.\n\n_I wonder what took her so long. I must have caught her comm wizards napping._\n\n\"Worse, you stupid bitch. I have more than two battleships for every one of those luxury-liner frigates that Wardhaven whore has, and you have the gall to demand my surrender? You're a fool! You and that Longknife whore will be dead in ten minutes. Enjoy your last few moments, 'cause they're all you've got.\"\n\n\"Nelly, off,\" Kris said, then turned to Vicky. \"That woman has serious anger-management problems and a very limited vocabulary.\"\n\n\"We were discussing that during the last battle, weren't we, Admiral Boles\u0142aw?\"\n\n\"Yes. Her temper will be the death of her yet if she doesn't learn to control it.\"\n\n\"I think 22-inch lasers will get her long before a heart attack,\" Jack muttered.\n\n\"General, I think you just might be right,\" Kris said.\n\nKris paused, then said, \"Attention to orders. Take the squadron to Condition Zed.\"\n\nHer flag bridge got smaller, and she and her team got more chummy.\n\n\" _Princess Royal_ and _Intrepid_ , you will engage the lead battleship of those most distant. _Courageous_ and _Furious_ , you will engage the trailing battleship. Commander Division 2, assign one of your ships to five of the 18-inchers. You decide who the lucky son of a bitch is who isn't blown to hell in the first salvo.\"\n\nKris let those targeting orders sink in before continuing. \"As we come up on two hundred thousand klicks, I will order weapons release. You may fire your forward batteries as they come in range. If your target is still there, flip ship and fire aft batteries. You will be weapons lock after your aft batteries are empty. Immediately begin to reload. If, however, this shoot is to continue, I will give you another weapons free. Any questions?\"\n\n\"One point, Kris,\" Vicky said.\n\n\"Yes.\n\n\"We've had a lot of luck punching through armor when we aim two lasers at the same point. We've even grouped four together though I doubt your big lasers need that.\"\n\n\"Right, my mistake,\" Kris said. \"That is standard practice on Alwa Station. That's the problem with being an admiral; you forget how to be a squadron commander. All ships, concentrate fire on your target by pairs of lasers.\"\n\nThe ships blinked their reply.\n\n\"You sure we want to blow all those ships away just like that?\" Admiral Boles\u0142aw whispered.\n\n\"You remember that battle where we had a hard time figuring out which ship was the flagship?\" Vicky said.\n\n\"Yes,\" he scowled.\n\n\"You really want to bet the Empress is on any ship smaller than an 18-inch battleship?\"\n\nHe shook his head at his Grand Duchess's question.\n\n\"The big battleships die,\" Vicky said, her words harsh.\n\nKris was eyeing the two. \"You couldn't identify the flagship?\"\n\n\"Long story. I'll tell you later,\" the Grand Duchess said.\n\n\"Kris, we're coming up on two hundred thousand klicks,\" Nelly said.\n\n\"Thank you, Nelly.\"\n\n# 68\n\nKris closed her eyes. She took a deep breath and let it out slowly. She took a second one and held it for a moment, searching for her center.\n\n_I'm about to slaughter a whole lot of people. Most of them innocent._\n\n_Not the Empress._\n\n_The Empress tried to kidnap Jack and torture him. She went after Ruth. She'll do anything to get her way, even slaughter a lot of innocent people on the planet below._\n\n_If only Harry was on that battleship with her, I'd consider my family finally even with the Peterwalds._\n\n_But he isn't. Just the Empress and most of her twisted family._\n\nKris opened her eyes, steeled her heart, and set her face to hard military bland.\n\n\"Squadron, prepare to fire one salvo. Fire.\"\n\nThe _Princess Royal_ swung around, bringing the twelve 22-inch lasers of her forward battery to bear and throwing Kris against Jack. She grabbed hold of her battle table, and Nelly had it grow handles. She grabbed one and brought herself upright.\n\n\"I told you to get yourself into a high-gee station,\" Jack growled.\n\n\"I'm tired of going naked into those things, then needing to stand up and take someone's surrender,\" Kris snapped back.\n\n\"And no,\" she quickly added, \"wearing a uniform is no better. It has all those buttons and sharp points.\"\n\n\"I agree with Kris, and I still do,\" Vicky said.\n\nThe _Princess Royal_ flipped ship and aimed her aft battery of eight more 22-inch guns at the _Terror of Space_. They fired.\n\nKris looked down to see what the battle board had to say about how the shoot was going. Five of the 18-inch battleships were in trouble.\n\nFirst, the forward top, then the aft middle battleship blew up. That was followed by the forward bottom and the aft top going together in the blink of an eye. For a moment, it looked like the forward middle battleship might survive. Then the aft end, where the reactors were, blew itself to gas, and a moment later, nothing of that ship remained.\n\nOnly the bottom aft battleship, the one not targeted, still existed as Division 2 shut down and began to reload.\n\n\"Come on, you two monsters, blow.\"\n\nWas that Vicky or Kris? No, it was both of them.\n\nFor a breath, the two huge ships with twenty-four of the \"new\" 19-inch lasers and ice armor more than six meters thick just lay there in the center of the board.\n\nThen the aft one vanished. It just disappeared. One moment it was there, the next moment there wasn't anything there.\n\nThe lead one, the one _Princess Royal_ and _Intrepid_ were aiming at, looked like nothing was happening to her. Suddenly, the aft end flashed out of existence, and a chain reaction of explosions raced through the ship.\n\nIn the blink of an eye, there was just hot gas where it had been, then nothing. Not a thing.\n\nThe heart of the hostile battle line, and very likely the life of the Empress and much of her family, had been snuffed out in less than a minute.\n\n\"Kris, the other ships have flipped and are boosting away from us,\" Nelly said. \"They've emptied their lasers into space. We're checking them, but from the looks of it, none are recharging their capacitors.\"\n\n\"Are we getting any requests to surrender?\" Vicky asked.\n\n\"Not yet. Right now, they're just trying to sort out what's left of their chain of command, Your Grace. A couple of the vice admirals commanding battle squadrons have abandoned their bridges. One has shot himself, and it appears the others are drinking themselves blind drunk. Oh, they just called out a crash cart for one vice admiral. I don't know why.\"\n\nThe screen in front of Kris and Vicky came alive with a rear admiral's face.\n\n\"Hi, Ramiro,\" Admiral Boles\u0142aw said, \"How's it going?\"\n\n\"Good God, man, what happened? You weren't even in range of us, and suddenly seven of our biggest ships are just gone.\" His voice rose to a sharp screech.\n\n\"Calm down, Ramiro.\"\n\n\"Calm down, you say? Damn it, man, what have you got over there?\"\n\n\"I'll let Admiral Longknife explain her battle squadron to you.\"\n\n\"Battle squadron?\"\n\n\"Hello,\" Kris said softly. She'd made her point. If Vicky was to save the lives of her subjects, there was no need to hammer them. \"I'm Admiral Kris Longknife. I've been away for the last two years, fighting alien raiders and their oversize base ships. We've pushed our technology. The broadsides that just took out seven of your ships were from my ships' twenty 22-inch lasers.\"\n\n\"Your frigates carry twenty 22-inch lasers?\"\n\n\"We're reclassifying them as battlecruisers,\" Kris said modestly.\n\n\"I guess you would,\" the Greenfeld admiral murmured in agreement.\n\n\"Ramiro,\" Admiral Boles\u0142aw said softly, \"what's happening over on your side?\"\n\n\"Everything's gone to hell. One of the battlewagon vice admirals just OD'd on red heroin. I'd never have taken Spinosa for a user, but he's been in the Empress's pockets for a long time. I'm told they called out the crash cart for him.\"\n\n\"Most people don't survive that kind of a thing.\"\n\n\"Yeah. Well, the vice admiral commanding cruisers and destroyers is a jumped-up bank clerk who married some daughter of someone in the family bank. I hear he's locked himself in his cabin with a bevy of girls he brought aboard, and they're getting sloshed. That leaves me senior officer present outside the battle line. Our ships don't have all that extra reaction mass, so what do you want us to do?\"\n\nVicky stepped forward. \"A chief on one of the destroyers who surrendered to us at the last Battle of St. Petersburg smashed the buss to the mains that would reload the capacitors. Did I get that right, Admiral Boles\u0142aw?\"\n\n\"Yes, Your Grace.\"\n\n\"Is that the Grand Duchess?\"\n\n\"Ramiro, may I introduce you to Vice Admiral, Her Grace Victoria of Greenfeld,\" said Admiral Boles\u0142aw.\n\nThe admiral ran a nervous hand through his black hair. \"I think I better take this chance to be the first of my fleet to swear allegiance to you.\"\n\n\"I am in allegiance to my father, your Emperor, as you are. We are gathered on Cuzco to resolve certain matters. I believe the Empress was intent on disrupting those talks between myself and my father, if not killing us both.\"\n\n\"We were told that the Longknifes had violated a parley truce and seized the Emperor.\"\n\n\"Kris, can you get my father on the horn, quick? That might settle a lot of this.\"\n\n\"Nelly,\" Kris said.\n\nIt was again night on Cuzco, but since a fleet was inbound, intent on lasing the Emperor to glowing magma, he was waiting up and still dressed in court garb.\n\n\"To all my people, I say that I am well, and free to come and go as I please,\" Emperor Harry began.\n\n_Strange how the Imperial \"we\" becomes \"I\" when it gets deadly and personal,_ Kris thought.\n\nThe Emperor continued. \"It is the Empress who has by nefarious means attempted to do harm to my body. My daughter, the Grand Duchess Victoria, and her friend, Princess Kristine of the United Society, have done me good service. Are there any questions any of you may want to ask me to assure yourself that I am truly talking for myself?\"\n\nNo one had a question.\n\nSurviving battleships, as well as cruisers and destroyers, quickly rendered their armament unusable and docilely set their course for High Cuzco Station.\n\n# 69\n\n\"It is said that victory has a thousand fathers, but no one ever said how many traffic controllers it needed,\" Captain Ajax said as she collapsed into a chair in Kris's day quarters. \"It's a zoo out there.\"\n\n\"Is Cuzco's prime minister still demanding we keep half of the station's piers clear for commerce?\" Kris asked, offering her flag captain a cup of soothing chamomile tea.\n\n\"He is, and we're insisting they follow the usual rule of first come, first served. Hell, Admiral Waller has got his ships rotating through the piers just as fast as they can load reaction mass. Oh, and Cuzco doubled the price of reaction mass. Again,\" Ajax said through a scowl.\n\n\"Even after we sent four pinnaces out to gather reaction mass for them?\"\n\n\"Yes, ma'am. They're selling the reaction mass we brought in for free at four times the price it brought a week ago.\"\n\n\"I do not like people taking advantage of my generosity.\" For a moment, Kris stared at the overhead. \"Okay. Send out all nine pinnaces this time and rig hosing so we can pass reaction mass to two ships at a time.\"\n\n\"You going to refuel the Peterwald fleet well away from the station?\"\n\n\"If it gets them out of here, yes.\"\n\nCaptain Ajax put down her half-full cup of tea and headed up to her bridge.\n\nKris had already had a talk with Vicky and the Emperor about who paid for all the costs that had been going on Grampa Ray's royal credit chit.\n\n\"Oh, didn't I tell you?\" Vicky said. \"We brought raw and polished crystal as well as electronics. It may not be as fancy as Wardhaven stuff, but it still ought to have a market. We also brought along some of the latest artwork from our painters and sculptors. I'm sure some of it will find a buyer. I'll have that released immediately. With any luck, I'll be able to pay half the expenses of this conference.\"\n\n\"Harry?\" Kris said. He still hadn't shown any negative reaction to being treated so familiarly by a US Princess.\n\nHe scowled. \"They've got a lien against the _Emperor and Empress_. Would you believe that? Until I pay my own bills here, they won't let her go.\"\n\n\"You really want to keep a tub with the late, unlamented Empress's name on it?\" Vicky asked.\n\nHer father's lips took on a sly smile as he looked Kris's way. \"I wonder what a fine yacht like that will bring on the open market, say above Wardhaven?\"\n\nKris limited herself to a shrug. With Smart MetalTM all the rage, she doubted old, overblown scows like that one would maintain much of their popularity, or value, for very long.\n\n\"Vicky, could I manage a ride home with you?\" he asked.\n\n\"Only if you'll send ninety percent of those worthless hangers-on and all of the Empress's lackeys home with the fleet.\"\n\n\"Done,\" he said.\n\nNext day, Vicky told Kris that they'd found buyers for all of their crystal and electronics and a surprising portion of the art. \"It seems there's a retro movement, and the Empire's art has been locked down by Dad and his grandfather for so long that we're right where the nostalgia crowd is going.\"\n\n\"You think the market will stay there?\"\n\nVicky shook her head. \"Not likely. If stuff comes flooding out of the Empire after we leave, it will likely be selling into a saturated market. But it was nice to us while it lasted.\"\n\nThe next day, the Hall of Mirrors was rather comfortable and spacious for those who were still in attendance. A committee of five from each side had been working behind closed doors to hammer out a proposal that would be either accepted by Vicky and her father _in toto_ , or rejected if either one of them objected to any part of it.\n\nVicky's five included Mannie, Steve Rojek, a woman from Metzburg, Admiral Waller, and a businessman from a planet Kris had never heard of. The Emperor had chosen the fellow who answered the phone that night when the Emperor was otherwise occupied, the man who brought the hall to tears with his telling of the slaughter on Dresden, and three other men just as sober in their dress.\n\nTheir proposal was simplicity itself.\n\nEmperor Harry would remain Emperor of the Greenfeld Empire.\n\nThe Grand Duchess Victoria was acknowledged as his rightful heir and upon his death or abdication would become the Empress.\n\nFor the time being, the Empire would be divided into two economic zones: one made up of the territory presently acknowledging the Emperor through the Grand Duchess and the other the rest. Trade between both zones would not be limited or taxed. At any time, but no more than every five years, any planet could vote, by a majority of those over eighteen, to change zones.\n\nImperial taxes were set at no more than ten percent of a planet's GDP, with half of the money raised in Vicky's zone going into a budget that she would manage. Planets would decide for themselves how to raise the money and were free to raise more for their own purposes.\n\nThere were a whole list of other things. How much or how little could be spent on defense, to include not just a Navy but an Army as well.\n\nThe presentation, rather wisely, paused before it was more than a quarter done and everyone went to an early lunch. People gathered around tables to talk to the writers. Kris's brain trust, who seemed to know an awful lot about the proposal, were also in great demand.\n\nKris decided to go up the beanstalk and play with Ruth for the rest of the day.\n\nThe next day, she gaveled the morning session open, with each of the sides having ninety minutes to raise questions about the plan on the table. There was a rush from both sides to get the floor, so Vicky took the job of recognizing members of her delegation for five minutes, then yielding the floor to the senior writer from the Emperor's side, who seemed to be in line to be the next prime minister.\n\nIn this fashion, the talks went on for three long, long days.\n\nAfter lunch the second day, Kris announced supper would be at five, and there would be a three-hour session after dinner.\n\nThe third day, the evening session went all the way to midnight. That was when Vicky said she'd heard enough. \"Dad.\"\n\nSomeone had to wake the Emperor up.\n\n\"Yes, yes,\" he snorted.\n\n\"Do you feel a need to hear more?\"\n\n\"Ah, um, not really.\"\n\n\"I'm willing to accept the proposal as it is before us. I'm also willing to meet with you and any interested party in two years to discuss changes. Your Highness, would you be willing to chair such a meeting?\"\n\n\"Me?\" Kris said, coming out of her thoughts of what Ruthie must be doing about now.\n\n\"Yes, you. Who else do you think would do it?\"\n\n\"Anyone else,\" Kris suggested.\n\n\"Kris, I'm willing to outfit and send two squadrons of battlecruisers to Alwa's defense using funds from my zone, assuming you will share with us the technology necessary to make them.\"\n\nKris frowned at Vicky.\n\n\"Come on, BFF. If we're going to be ready to kill any alien mother ship that shows up in our sky, we'll have to build battlecruisers just like you do.\"\n\n\"She's got a point, Kris,\" Jack whispered.\n\nKris sighed. \"Okay, but if I'm pregger with our next one, Bub, you got to carry it while I'm sitting here.\"\n\n\"We'll use a uterine replicator next time,\" Jack whispered. \"No way I'm going through what you went through.\"\n\n\"Men,\" Kris and Vicky said, but they glanced at the men in their life with fond eyes.\n\n# 70\n\nA week later, Admiral, Her Highness Kris Longknife presented herself before Field Marshal McMorrison.\n\n\"Done already?\"\n\n\"Yes, sir,\" Kris answered smartly.\n\n\"So, I imagine you want a job.\"\n\n\"Yes, sir. A desk job. One I can work at eight hours a day, five days a week, and go home to my daughter without taking the job with me.\"\n\n\"I've never found one of those.\"\n\n\"I've served my time in hell, sir. I think I deserve a little boredom, don't you?\"\n\nThe field marshal scowled. \"Any thoughts on where I might find a job for you? You are wearing four stars, Your Highness.\"\n\n\"And I'm not giving them back.\"\n\n\"I didn't think you would. So?\"\n\n\"I noticed that the Battle Force and Scout Force are still arguing where the battlecruisers fit in. The battleship Sailors say they're ships of the line and belong to them and the cruiser type commander insists they fit right into his Scout Force.\"\n\n\"And they've been arguing over that since the first one was launched. You got any ideas, Admiral?\"\n\n\"Yes, Mac. I want the job of type commander for battlecruisers. That will make sure they get their own slot, not shoehorned into someone else's hip pocket, where they can get lost, ignored, or misused.\"\n\n\"That sounds like a job for a miracle worker. You realize you'll be in the very middle of the budget battles. Spreadsheets at five paces.\"\n\nThat gave Kris pause, but not enough for her to walk away.\n\n\"Nope, I'll take the job. I've blown away alien base ships, I think a few entrenched bureaucrats shouldn't be all that much trouble.\"\n\nMac chuckled at that. \"Kris, you can kill alien base ships. People will gossip at the watercooler if you kill your fellow admirals.\"\n\n\"That's a problem how?\" Kris said with a straight face.\n\n\"I'm just saying.\"\n\n\"Mac, I want a desk job. This looks like the kind of desk job that's right down my alley. Unless you've got another nice _desk_ job for me, I want this one.\"\n\n\"No ship duty, huh?\"\n\n\"Been there, done that. Got the medical bills to prove it.\"\n\n\"You really think you can settle for a desk job?\"\n\n\"Watch me.\"\n\n\"You know, they've already started a betting pool for how long you'll last. I've got the twenty-four-month slot,\" the field marshal said with a grin.\n\n\"Tell me who's holding the pot. I want the ticket that's forever.\"\n\nNow Mac chuckled. \"It will be interesting to see who wins.\"\n\n# 71\n\nKris answered the door chime herself at Nuu House. Waiting outside were Mother and Father as well as Honovi and his brood.\n\nFather frowned. \"You look mighty formal, dress blues and all those gewgaws, for a quiet afternoon supper with your family.\"\n\nKris knew her smile was fay. \"There are a few things to do before supper. Please follow me.\" No one took their coats as she led them through the width of Nuu House, out the back door, then across the cobblestoned esplanade and down into the garden.\n\n\"What's that?\" Brenda didn't quite demand.\n\n\"A small family get-together,\" Kris said, as she led her parents down front to the chairs to the left of the aisle. The bride's side.\n\nKris exchanged a quick smile with Jack's mom and dad. He'd seated them and answered more of their questions than she had of her folks.\n\nTaking a deep breath, Kris walked back up the aisle. Smiling and nodding at both those in the know and those still ignorant. Grampa and Gramma Trouble had managed to get Ray here. _Who's his plus one? I will need to get to know her._\n\nBrenda's own mother and father, he a professor at a university farther down the coast, had insisted on seats in the back row. Kris had always known her mother was a social climber. She rarely took her children to meet their grandparents on her side. _I'll have to make a special effort to build bridges there. It's not as if I'm going anyplace tomorrow._\n\nKris smiled at that thought.\n\nA few paces back from the chairs, Kris was met by Ensign Meg Longknife in dress blues. Beside her, in her very first formal, was Jack's sister Estella. She was so excited she looked ready to jump out of her violet sheath dress that modestly rose to her neck. Three-quarter sleeves showed where she clutched her bouquet.\n\nKris had won extra wifely points from Jack when she took Estella dress shopping and nixed the strapless gowns the young woman had done her best to talk Kris into buying for her.\n\nMegan handed Kris her bouquet as she joined her bridesmaids. She turned to face forward; Jack and the two chaplains had just gathered at the front. It saddened Kris that Jack had no one to stand up with him as best man. She'd run him pretty ragged and left him with few friends at hand.\n\nThen again, most of the women Kris wished were here were scattered across the galaxy, too. _Maybe we can learn to play bridge with couples our own age. Might even arrange some playdates with people who have kids Ruth's age._\n\nKris was called out of her reveries when the organist began to play a march for the bridesmaids. The two began to process down the aisle.\n\nKris stared at Jack staring at her. _Does he actually look nervous?_\n\nHeavens knows, they had enough to worry about just for this small, family wedding. Take the chaplains: one Catholic for Jack's mom, one Episcopal for her folks. Kris was pretty sure Mother would not have accepted just the Catholic chaplain, so she'd balanced that. And Jack had already explained to his mother as he ushered her in that the Catholic priest would be doing all those parts required for a proper Catholic wedding, but that the Episcopal priest would be assisting everywhere it was allowed.\n\nKris had to chuckle softly. _I managed to negotiate peace between the Peterwalds. I'm good for this, too._\n\n_Yeah, right._\n\nThe bridal march began. Kris took a deep breath and began the longest of short walks. Strange, she'd felt better, more accepted, among the Alwan colonials than she did here.\n\n_But I'm not on Alwa. I'm home, and I'm going to stay home. I will mend my fences and build some more, so I can stay right here and enjoy it._\n\nShe came out of herself when Jack smiled at her like some love-struck fool. She found herself smiling back, just as love-struck and just as much the fool.\n\nThe ceremony went quickly. She had a gold ring for Jack this time. He again used the simple gold ring he'd bought for her at the BX on Hellfrozeover long ago, when both of them feared that they might never see each other again. It wasn't long before both the priests said, \"You may now kiss the bride.\"\n\nThere were cheers as they did, and another as they began the walk back up the aisle, arm in arm.\n\nAt the end of the aisle, a line of uniformed officers had formed, swords crossed to form a passageway for the couple. She'd wondered where Mac got the collection of admirals and generals who helped him form two surprisingly long rows. Kris suspected she'd be deep in budget battles with many of them soon enough; she fervently hoped the swords weren't sharp.\n\nShe noticed that Mac had taken the last slot on her side. She eyed him, and mouthed, \"I've already had my rump swatted,\" but he kept staring straight ahead.\n\n\"Ouch,\" Kris said, as Mac used his cake cutter to rap her on her backside. With intent.\n\nShe eyed him but then shook her head. Maybe she didn't deserve this whacking, but, no doubt, before long, he'd be wanting to whack her upside the head or other place. She'd let him have this one.\n\nKris turned back to meet those who now filed down the aisle behind her. Lotty had outdone herself with the dinner awaiting them.\n\nAfter dinner, there would be a small family gathering in the library, and the Catholic priest would baptize little Ruth Maria Brenda Anne Longknife. That would be her concession to both sides of the family. Maria got her granddaughter baptized, and Billy got another Longknife in the family.\n\nWhat Ruth had to say about her name in eighteen years was hers to decide. She would have her mother's support no matter what she chose.\n\nKris looked around at Nuu House. Over seven years ago, she'd fled the place with no intent of ever looking back. Somehow, she had come full circle. But it was a full circle that held so much.\n\nShe remembered the old chief who'd asked her why she was in her Navy. Kris hadn't really known the answer then. Now she did.\n\nThis Navy was her Navy, too. And the Longknifes were her family, both the small one she'd made and the given one that milled around her today. She'd even made a dent in all of humanity, and even the galaxy.\n\nAll of it had brought her back to here.\n\nKris took a deep breath. She was home, and she liked it.\n\nThe End\n\n# Special Note to the Reader\n\nThirteen books ago, the Kris Longknife saga began as a coming-of-age story. Kris was a fresh-caught boot ensign, making all the mistakes you expect of a green JO. A retired Navy captain wrote me that in _Kris Longknife: Mutineer_ I had captured the life of a junior officer and all the challenges a senior officer faces to properly lead and mentor that JO. That feedback made my day.\n\nFor seven more books, we followed Kris as she learned the ropes and came to be a qualified Officer of the Deck. There is only so long a writer can keep a good character like Kris in coming-of-age mode, so in _Kris Longknife: Daring_ we changed the thrust of her story.\n\nNo longer was she trying to figure out where she fit in or who she was. Kris was ready to use all she'd learned and see what she could do. Certainly, the bug-eyed monsters were a worthy opponent for Kris.\n\nNow Kris has fought her monsters, both the alien and the human ones. Kris is an admiral, a married woman, and a mother. It is time to write \"The End\" on the last page of this book.\n\nI do this with intent. A fan wrote me that he was struggling with cancer and hoped that I'd reach the end of Kris's story before he reached the end of his. For him, if he's still reading, here is a very satisfactory ending for Kris, Jack, and Nelly.\n\nFor those of you who want to know how the lottery goes and what happens next, don't worry. As you're reading this, I'm turning in the next Kris book and very likely starting the one after that. Kris may have achieved flag rank, a ring on her finger, and a lovely baby girl on her hip, but you don't really think she's going to last long behind that desk, do you?\n\nWhich month did you pick in the lottery for how long she'll stay put behind a desk?\n\n# About the Author\n\n**Mike Shepherd** grew up Navy. It taught him early about change and the chain of command. He's worked as a bartender and cabdriver, personnel advisor and labor negotiator. Now retired from building databases about the endangered critters of the Pacific Northwest, he's enjoying some fun reading and writing.\n\nMike lives in Vancouver, Washington, with his wife, Ellen, and close to his daughter and grandchildren. He enjoys reading, writing, dreaming, watching grandchildren for story ideas, and upgrading his computer\u2014all are never-ending.\n\nDespite what you may think, Mike is already hard at work on Kris's next story, _Kris Longknife: Emissary_ , coming from Ace in October 2017, as well as other exciting tales in Kris's universe.\n\nYou may learn more about Mike, his projects, and his books at his website mikeshepherd.org; you can e-mail him at Mike_Shepherd@comcast.net or follow Kris Longknife on Facebook.\n\n# _What's next on  \nyour reading list?_\n\n[Discover your next  \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9780698180666\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n\n## Contents\n\n  1. Cover\n  2. Praise for the Kris Longknife Novels\n  3. Books by Mike Shepherd\n  4. Title Page\n  5. Copyright\n  6. Acknowledgments\n  7. Contents\n  8. Chapter 1\n  9. Chapter 2\n  10. Chapter 3\n  11. Chapter 4\n  12. Chapter 5\n  13. Chapter 6\n  14. Chapter 7\n  15. Chapter 8\n  16. Chapter 9\n  17. Chapter 10\n  18. Chapter 11\n  19. Chapter 12\n  20. Chapter 13\n  21. Chapter 14\n  22. Chapter 15\n  23. Chapter 16\n  24. Chapter 17\n  25. Chapter 18\n  26. Chapter 19\n  27. Chapter 20\n  28. Chapter 21\n  29. Chapter 22\n  30. Chapter 23\n  31. Chapter 24\n  32. Chapter 25\n  33. Chapter 26\n  34. Chapter 27\n  35. Chapter 28\n  36. Chapter 29\n  37. Chapter 30\n  38. Chapter 31\n  39. Chapter 32\n  40. Chapter 33\n  41. Chapter 34\n  42. Chapter 35\n  43. Chapter 36\n  44. Chapter 37\n  45. Chapter 38\n  46. Chapter 39\n  47. Chapter 40\n  48. Chapter 41\n  49. Chapter 42\n  50. Chapter 43\n  51. Chapter 44\n  52. Chapter 45\n  53. Chapter 46\n  54. Chapter 47\n  55. Chapter 48\n  56. Chapter 49\n  57. Chapter 50\n  58. Chapter 51\n  59. Chapter 52\n  60. Chapter 53\n  61. Chapter 54\n  62. Chapter 55\n  63. Chapter 56\n  64. Chapter 57\n  65. Chapter 58\n  66. Chapter 59\n  67. Chapter 60\n  68. Chapter 61\n  69. Chapter 62\n  70. Chapter 63\n  71. Chapter 64\n  72. Chapter 65\n  73. Chapter 66\n  74. Chapter 67\n  75. Chapter 68\n  76. Chapter 69\n  77. Chapter 70\n  78. Chapter 71\n  79. Special Note to the Reader\n  80. About the Author\n\n  1. Contents\n  2. Cover\n  3. Start\n\n  1. i\n  2. ii\n  3. iv\n  4. v\n  5. vi\n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. \n  329. \n  330. \n  331. \n  332. \n  333. \n  334. \n  335. \n  336. \n  337. \n  338. \n  339. \n  340. \n  341. \n  342. \n  343. \n  344. \n  345. \n  346. \n  347. \n  348. \n  349. \n  350. \n  351. \n  352. \n  353. \n  354. \n  355. \n  356. \n  357. \n  358. \n  359. \n  360. \n  361. \n  362. \n  363. \n  364. \n  365. \n  366. \n  367. \n  368. \n  369. \n  370. \n  371. \n  372. \n  373. \n  374. \n  375. \n  376. \n  377. \n  378. \n  379. \n  380. \n  381. \n  382. \n  383. \n  384. \n  385. \n  386. \n  387. \n  388. \n  389. \n  390. \n  391. \n  392. \n  393. \n  394. \n  395. \n  396. \n  397.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**Also by Don Cherry**\n\nDon Cherry's Hockey Stories and Stuff  \nDon Cherry's Hockey Stories, Part 2\n\nCopyright \u00a9 2014 Don Cherry\n\nAll rights reserved. The use of any part of this publication, reproduced, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise, or stored in a retrieval system without the prior written consent of the publisher\u2014or in the case of photocopying or other reprographic copying, license from the Canadian Copyright Licensing agency\u2014is an infringement of the copyright law.\n\nDoubleday Canada and colophon are registered trademarks\n\nLibrary and Archives of Canada Cataloguing in Publication has been applied for\n\nISBN: 978-0-385-68108-7  \neBook ISBN: 978-0-385-68109-4\n\nCover image: Phillip MacCallum \/ Stringer \/ Getty Images  \nCover design: Five Seventeen\n\nPublished in Canada by Doubleday Canada,  \na division of Random House of Canada Limited  \n **A Penguin Random House company**\n\nVisit Random House of Canada Limited's website: www.randomhouse.ca\n\nv3.1\n_I thank you, God, for helping me survive_.\n\n# TABLE OF CONTENTS\n\n_Cover_\n\n_Other Books by This Author_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\nLessons Learned\n\nAfghanistan\n\nSochi 2014\n\nMusings\n\nThe Good Old Days\n\nHockey Today\n\nThe 2014 Stanley Cup Finals\n\n_Photo Insert_\n\n_Photo Credits and Permissions_\n\n# **LESSONS LEARNED**\n\nI AM IN THE WATER in the middle of the St. Lawrence River off Wolfe Island and drifting along with the current and waves, hanging on for dear life to a paddleboat. I've finally realized that circumstances I will explain later have me in a desperate situation. I know I cannot last much longer, as I am getting weaker and weaker. I look at the shore... it's too far, for I would never make it. But I've got to do something, as I know I cannot hang on much longer. As I drift along, strange thoughts enter my mind. If something did happen, I can see the headlines now: CBC SPORTSCASTER DROWNS IN THE ST. LAWRENCE. Some people would be happy, I am thinking, but let me tell you how I got into this nightmare.\n\nThat very morning, my wife, Luba, and I are sitting on the dock with our coffee, and one of the discussions is about me selling back to Gordon Bay Marine our Boston Whaler. I say to Luba, \"I know you are not too happy about me selling the Whaler, but let's face it: we hardly use it.\" We had a paddleboat, which she used, and a canoe that a family member had left there, but nothing that could be used safely to save someone. Luba answers, \"Yes, you are right, but I really think we should get a small rowboat or something in case somebody gets in trouble in the river. You never know. The St. Lawrence can be dangerous at times, and a storm can blow in any minute.\"\n\nI answer, \"Yes, someday we will get something.\"\n\nThat afternoon, about three o'clock, I am reading in bed and fall asleep, and suddenly I hear Luba calling my name, saying that the paddleboat has broken free from the shore and is drifting away. I suppose I have to get it. I remember we have the canoe that was behind the garage and I have never used the canoe or the paddle, so as I am pushing the canoe in the water, Luba says, \"Wait I will get a life jacket.\"\n\nI laugh. \"Me wear a life jacket? No thanks.\" I paddle out to the middle of the river, and I notice the wind is picking up and it's getting rougher, but everything is going well, no problem. And as I look back now, I was getting too brave and too cocky and getting careless, not paying attention\u2014which is a dangerous thing on the river. I can see that the paddleboat is caught in the current, and as the waves pick up, it's drifting away faster. But I finally catch up to the boat and tie the rope on the back of the canoe.\n\nThe wind has really picked up, but I am making progress back to the cottage, and for some strange reason I don't feel the boat is towing right, so I turn to undo the boat to retie it. That is when my troubles begin, as my paddle falls into the water. As I turn and bend over to retrieve the paddle, the canoe tips over and water starts rushing over the top of the canoe. I cannot stop it, and over I go into the water with the canoe upside down.\n\nIt is hard for me to describe the feeling when this happened. It was like, \"Is this really happening?\" and \"Can this be real? These things happen to other people, not me; this can't be happening in the middle of the St. Lawrence.\" I look, and the canoe is hardly above water. I look the other way, and the paddleboat is fast drifting away. What to do? Stay with the canoe, which is bobbing almost under water, or take off after the paddleboat that is fast disappearing? I got to remember I can't panic. It's just a little swim, I say to myself. Don't get excited. I kick off my running shoes and take off after the boat; honest to God, my thoughts go to a fellow hockey player called Ross Lowe who played for Eddie Shore's Springfield Indians in the American Hockey League. I remembered we all looked up to him. It was my rookie year, 1954\u201355, and what a year Ross had. He had a tough time, knocked around hockey in the NHL Boston and Montreal and ended up in the graveyard of hockey, Springfield, with the Darth Vader of hockey, Eddie Shore, but he never quit, worked hard, and in 1955 in 60 games he got 32 goals and 50 assists and 82 points. He was a First Team All-Star and the MVP of the league and boy, was he tough.\n\nRoss got a break when he was drafted by the New York Rangers and signed a big contract that year, and we all wished him well. What a break to get away from Shore and go to the Big Show\u2014there's hope for all of us. Ross loved to fish, and being from Oshawa, he had lots of opportunities to fish. That summer, after being drafted by the New York Rangers on June 1, Ross was out in the lake on a hot day, and he decided to go for swim. Ross dove off the boat, and when he came up from the dive, he noticed the boat was drifting away. Ross took off after the boat, but the faster he swam, the faster the boat drifted away. Ross never made it to the boat and went under. Everybody assumed he drowned, but the rumour we heard was that he died of a heart attack from panic watching the boat drift away.\n\nWe all thought, \"What a bad break, just when he was getting his big hockey break,\" but I'm not thinking of Ross's big break in hockey as I swim as hard as I can. I think of Ross drowning on account of panicking, so again I say, \"Relax, it's just a little swim,\" and I calm down and swim slow but sure. I finally catch up with the boat and think my troubles are over, but they are just beginning. I look back at the dock in the distance and can make out Luba all by herself\u2014what must she be thinking? I look around for a boat somewhere in the river to get rescued, but not one boat in sight. It's July\u2014surely some boat will come along. But no fishing boat, no sailboats, nothing. The St. Lawrence is empty and the waves are picking up again and I'm drifting farther and farther away from shore.\n\nI try to get into the boat, but it's so high. I can't climb aboard; I make a mistake and keep trying and trying and make myself very weak. As I try to hang on to the boat, I have to extend myself over the side of the boat and grab hold of the arm of the foot paddle. This is very awkward, contributing to my weakening condition. I'll give it one more try to get in and I'll go all out. I let go and give it my all and get my leg almost over, but being so weak I fall back in the water. I look and wonder if I can make the shore, as I am a pretty good swimmer, but I think of all the stories I have read when the guy, after a boat accident, tries for the shore and he drowns. I stay with the boat, thinking, \"Oh why, oh why didn't I listen to Luba and take the life jacket?\" Now I know why they are called life jackets. I know I cannot last much longer. I should never have weakened myself trying to get in the boat. A strange feeling comes over me. I think this could be it, and I am accepting it very calmly. I look up at the sky and think, \"What a beautiful blue sky.\"\n\nMeanwhile, Luba is at her wits' end; she's putting on a life jacket and is going to try to swim out to me and bring one for me, but a friend working at the back of the cottage, who came running when he heard her cries for help, stops her. He tells her, \"That would be suicide\" and \"No sense both of you going down.\" Still no boats on the river, which is unbelievable. She calls around and no one is home except a farmer named Ken Murphy, and he starts to phone everyone along the shoreline to see if anyone can come out by boat to rescue me. I am drifting faster and faster away as the wind and the current are picking up and I am now in the middle of the current. I look at the beautiful blue sky and say, \"Well, Lord, if this is the way it's going to be, then it's got to be.\" I am really getting weaker and weaker and I don't know how much longer I can hang on. I think I faintly hear a boat starting up in the distance and I say, \"Lord, I hope it's for me.\" I know I have never heard such an amazing sound as I keep praying to let it be for me, let it be for me.\n\nAs the sound gets louder, I know it's for me and thank God. It's a gentleman called Al Doyle, who received the call from Ken Murphy. As Al glides up to me, he says, \"Having a little trouble, Don?\" and I answer, \"A little.\" I find I am too weak to get into his boat, so I sit on a little platform beside the outboard motor, and as we start in, I notice the paddleboat and canoe floating away and I say, \"Al, as long as we are here, why don't we pick up the canoe and the paddleboat?\" He swings by the canoe and boat and I grab the ropes and we start in. It must be a strange sight, a guy sitting on a platform, towing two boats in. We make it to the dock. Al drops me off with the boats and waves goodbye, knowing how grateful I am. I am so embarrassed, and thankfully Luba never mentions the life jacket. She's so happy to have me back alive.\n\nAs I go to speak, a strange thing happens. I cannot speak; my mouth has gone dry. I later read the same thing happened to another person with a near-death experience\u2014his mouth dried up also. I guess that's what fear does to you.\n\nSometimes, when I'm quietly sitting alone, many thoughts come to mind about that day, but there is one thing that comes to mind that seems to haunt me. I still get that gut sensation when I picture the canoe tipping over and the water pouring in. Feeling helpless and having no way of stopping it. There are times when I wake up in the middle of the night, dreaming of the water pouring in, and I say, \"God, thank you for giving me a second chance at life.\"\n\nMuch to my embarrassment, the story got around Wolfe Island, and when I was asked about it by an old-timer who was a hockey nut, I answered him by saying, \"Yeah, it was embarrassing that I would risk my life for a stupid paddleboat.\" He just shook his head and said to me, \"You know the story of Babe Siebert?\"\n\nAnd I said, \"I've heard of him. He played back in the thirties.\"\n\nHe says, \"Check out his story.\"\n\nI did check out Babe's story. His name was Albert \"Babe\" Siebert, of Zurich, Ontario. He was born in 1904 and played when he was young for Kitchener. He signed with the Montreal Maroons of the NHL in 1925 and won the Stanley Cup, believe it or not, in his first year. He was a big, tough winger. He was later traded to the New York Rangers and believe it or not, the Rangers won the Cup in his first year with them. He was traded to the Montreal Canadiens, put back on defence because they wanted him to be on the ice more. And believe it or not, as a defenceman, he won the Hart Trophy. He retired two years later and became coach of the Canadiens.\n\nBabe was enjoying his summer at the cottage on Lake Huron and was looking forward to his first year as coach of the powerful Canadiens. Babe was on top of the world. He won the Stanley Cup, was a Hart Trophy winner and now was the coach of the famous Montreal Canadiens. What could be better? And he had a lovely wife and kids. But life, as we know, has twists and turns. While his wife was giving birth to his second child, she became paralyzed from the waist down. Most of Babe's salary from the Canadiens went to a nurse who cared for her. Babe's wife wanted to go to the games, and Babe carried her into the Forum before the game and people always saw him carrying her out after the game. They were deeply in love, but it must have been very tough stuff for the both of them.\n\nBabe enjoyed his days at the cottage. One day, his wife and kids were at the cottage on Lake Huron. His kids were playing out on the lake, floating on the inner tube of a car, the way we all used to. Somehow the tube got away from the kids and started to drift away. Babe dived in and started to swim after the tube to retrieve it. But the tube kept drifting away, and Babe wouldn't quit. He kept going and going until he got tired. And he disappeared, with his children watching from the shore. It's tough to take. What a sad ending. Drowning while your wife and children were watching.\n\nTimes were tough now for his wife. In 1939, the NHL put on an All-Star game at the Forum and raised money for his widow. That's all I could find about Babe Siebert and his family. I often wonder how his family made out after his passing, because back then, the players had no pension. I hope God was kind. The family needed a break.\n\nI saw the old-timer at the garage a couple of days later. I told him I knew the story of Babe Siebert. He looked at me and said, \"The lesson is, enjoy life when you can. You never know what's waiting for you around the next corner.\"\n\nI have been asked many times what player who should have been a star in the NHL threw it away on booze and drugs. Unfortunately, I could name a lot of guys, and it was sad to see guys with a lot of talent go down the drain. And there is nothing you can do to stop them. Some guys, at the end of their career, straighten out when they do something that shocks them into reality. One guy, who was a boozer, when drunk pushed his mother down on the floor. When his sister told him the next day what he had done, he quit cold turkey, which doesn't happen too often. A lot of guys try to kick the habit, and they do for a time, but usually after a while, they fall off the wagon. John Barleycorn is a tough guy to get rid of.\n\nI must admit, and I'm not very proud of it, I drank a lot of beer in my playing time in the minors. It was the thing to do. It was strange when I was a rookie and I broke into the pros. I did not drink at all in my rookie year in Hershey, and that is where Boston had its training camp. I was twenty years old and never joined in with the boys when they would have a \"session\" after practice. I thought they were going to sit around and talk hockey, which they did, but along with the talk, plenty of suds went down the hatch. I have to say, as I sat there listening, not saying a word, as rookies did in those days, I noticed as the players drank, their feelings seemed to come out on their thoughts on different players. I have to say, it was their true feelings, and I saw many a player turn on one another. When they say \"courage in a bottle,\" it's so true. Guys who wouldn't say a word all of a sudden become brave and, instead of being quiet, became bigmouths and went looking for trouble. In this atmosphere, they almost always found it, and when you heard, \"Step outside and say that,\" the time had come.\n\nIt's funny: nobody ever tried to stop a fight unless the guy was a helpless drunk. I have seen some bloody fights. They only stopped when one guy went down. It was survival of the fittest back in those days. I also noticed that some guys could drink all day and you would never notice a difference. Other guys\u2014two beers and they're flying. I learned as I got older, the guy to watch for was the guy who just sat there and never said a word, just looked, with that strange look in his eyes. It's true: still waters run deep.\n\nWhen I was young, studying these guys, I noticed the sad spectacle of the guy who never knew when to stop, when he'd had too much. A lot of guys who never made it to the big time were bitter guys. The minor leagues and the beers did that to some guys. They figured they'd got the short end of the stick, and it was tough to hear their sad tales, but through it all, I sat there with my Coke. I would not have a beer, even with all the coaxing, till I turned professional.\n\nI turned pro with the Boston Bruins in Hershey, so now I could be one of the boys. I must say I had a \"taste of the honey,\" as the rich boys would say. My mother was dead-against drinking. We never had it in the house. She always said, \"Never drink gin. It's in the blood.\" I never knew what she meant till the time we won the Memorial Cup with the Barrie Flyers. In junior, I never drank beer. At the house parties, everybody was drinking, which they should not be because we were all under twenty. \"Come on, Don.\" \"Don, don't be a party pooper.\" I remember a guy made me a Tom Collins (I'll never forget the name). Hey, it tasted like a grapefruit drink, so I had a couple of Tom Collinses, but with gin. I woke up the next day in the host's spare bedroom, feeling like I was dying. Now I knew what Mom meant by \"it's in the blood.\" I never touched the \"hard stuff\" again, liquor of any kind.\n\nBut I must admit I fell into the routine. The routine went something like this: milkshake for breakfast; practice 10:00 A.M. to 11:30 A.M. (we had no off-ice training in those days); shower; lunch at our favourite restaurant, DeAngelo's, consisted of two eggs, bacon and toast; nap in the afternoon.\n\n(I often wondered when, after I retired and had no job, and no trade and no money, why I didn't at least learn a trade in those off-hours. My little brother, Richard, who played for the Philadelphia Flyers and the Boston Bruins, as well as Providence and Oklahoma City, studied in his off-hours every minute of the week, even on long bus rides. He earned a teacher's certificate and studied at Queen's University and received his bachelor's degree and master's degree. He was set for life.)\n\nUnfortunately, I have to admit I turned into one of the guys that I used to look down upon when I was a rookie and non-drinker. I will admit it was a hard life, but that's no excuse. We would play Friday, get on a bus and travel all night. Play Saturday, get on the bus right after the game, drive all the way back to Hershey and play Sunday night. Three games in three nights, injured or not. Treat your own injuries. I can't remember a time in sixteen years, anybody missing a game with the flu or a cold. You played unless you had a bone showing. With only six teams in the NHL, there were lots of guys to take your spot.\n\nThere was a lot of drinking after practice. I am now ashamed to say it was drinking, never eating. Just peanuts and popcorn, and some guys could go for ten hours. It was amazing and it was interesting to see the different ways guys could handle John Barleycorn.\n\nI must admit I had a good time and I never got \"blotto,\" that I can remember, and I was the only guy who would phone home and say where we were having our sessions\u2014except once, just once, and it almost cost me a coaching job. Rose would give me a frosty answer when I would call, but never gave me heck. If the wives wanted to know where their husbands were, they would phone Rose.\n\nI asked my teammates, \"Why don't you phone your wives and tell them where you are in case there's an emergency?\" Their answer was, \"Look, when I get home, I'm going to catch hell. Why should I phone home and catch hell twice?\"\n\nI had a good friend, my best friend on the team, and it was getting to be that we were the last ones to leave, closing the place. Every time I would go to leave, he would get upset. Finally, I had to take a stand. He was very upset and called me names, but enough was enough. I didn't like the way we were heading. He was a good athlete\u2014played many sports and won many championiships and awards. Why didn't he get a shot at the big time? Because he liked John Barleycorn too much. He lost everything\u2014his lovely wife and children, a great career\u2014but he is just like one of the many guys I knew back then whose careers were ruined by drinking. It was sad to see.\n\nI remember in Rochester I played with Al Arbour and on defence with Larry Hillman. They would occasionally drop into our sessions, but would leave after one or two beers. As I look back now, both went on to play for the Stanley Cup, both turned into great coaches\u2014Al winning the Stanley Cup and Larry winning the Avco Cup in the World Hockey Association. After playing many years in the AHL and looking back, I guess it's not too surprising, and it tells you something about us, why we drank so much. I had to say a lot of us were bitter about not making the show, but I feel that's just an excuse. I guess we were having a good time, not thinking of the future. I will admit most of us stopped the life of drinking once we retired and got away from each other, and I will admit I missed it badly\u2014the camaraderie and fun were gone. Little did we know it was for our own good. Some guys never stopped, and it was their undoing.\n\nThe alcoholic will never stop unless he wants to, no matter what you do. \"The taste is there,\" as my mother used to say. I tried to help guys, but it never worked out. When I was coach and GM of the Rochester Americans, I signed a guy I knew was an alkie. How did I know? I played and roomed with him, so I knew what I was getting into. He was a dandy defenceman, but couldn't leave the stuff alone. Red flags were all over the place, but I was a young GM. I should have signed him till Christmas, and if he didn't straighten out, let him go, but I felt sorry for him and his lovely family, and he was such a lovable guy when sober. The guys loved him when he was sober, but the signing was not fair to our owners. They trusted me and I let them down. I feel that way, even though they made a couple hundred thousand dollars that year.\n\nWhen I was his roommate (and that was a treat), one time he broke curfew and came in late, drunk as a lord. I was so pissed I went into the bathroom and left the window open so he would freeze. He turned purple, but got up the next morning feeling fine. I caught a cold.\n\nHe used to tell me about his escapades. The bosses put up with him when he was young because he was such a great prospect. One time, he told me, he was playing for Tulsa in the Central League, and they were on the road, and as usual, after the game (which his team lost), his coach put a curfew on. Curfew meant nothing to this guy; they were made to be broken. This time, they were warned: anybody out, it was trouble. Big time. So he told me he was going to be in on time, but one beer led to another and he said, \"What the heck?\" So he and a buddy are in a strip club at one end of the bar, and who should walk in and stand at the end of the bar? The team management. The two players ducked down and hid, but how to get out? He knows he's in deep trouble, as he's been caught before and warned. One more time and that's it. So he looks around and sees the entrance to the stage is right beside him and they're getting ready to start the show, but he figures if he runs across the stage he can get across before they open the curtains. So he sneaks up the stairs, he runs across the stage, and as he's a tall guy, he smacks his head on a beam and is knocked out cold. The music starts, the curtains open, the team management look up from the bar and see their star player spread-eagle, out cold, on the strippers' stage.\n\nI could see I wasn't going to help him in Rochester, and he was disrupting the team, so I told him not to come to the arena\u2014stay home. \"If anybody asks, just say you're hurt.\" I told the press he had an injury. I paid him. I could have just cut him, which I should have done, to be fair to the owners, but I was young and stupid. But you know, if it had happened again, I'd do the same thing.\n\nThe end is not happy. He and his wife, who also was an alcoholic, lost their two beautiful children, and I never heard of them again.\n\nI have not mentioned his name. What's the point? You just hurt Mom and Dad and relatives. I am going to tell you about who, to me, is the biggest disappointment and talent wasted that was ruined by the \"taste.\" He was a beauty player, big\u2014six foot two\u2014fearless, skated like the wind, magic with the puck, a slapshot you would die for, he had it all. A phenom, as they say. He even broke Bobby Orr's record for goals and Denis Potvin's record for points by a defenceman. Listen to this: in junior hockey: 60 games, 47 goals, 108 assists. Imagine, 155 points in 60 games. He went high in the draft to Quebec in 1987, ahead of Joe Sakic. He had the world on a string; everybody thought this guy is going to be in the NHL a long time. Although I must admit, I had heard a few rumours about him, that he liked the high life. What the heck, we thought, he's just a young guy having fun, but we thought wrong. The rumours were true. Surely he would straighten out with all that money to be made. He never did.\n\nIt got worse. He had such great talent, everybody wanted him. They thought they could straighten him out. Listen to the teams he went to: Quebec Nordiques in the NHL, Halifax, New Haven, Muskegon, Pittsburgh, Cleveland, Atlanta, Las Vegas, Kansas City, Montreal, Minnesota, Detroit in the IHL, Davos in the Swiss League, Kansas City again, Milan 24 in Italy, Germany. He was running out of countries.\n\nI remember he was put as a roommate with another player who was also hooked on drugs and such. I often wonder the reason. Maybe they thought if one boy saw how the other acted, he would stop? (By the way, the roommate died a violent death at the hands of the police when they tried to arrest him.) Or the team just thought, \"Hey, they're both lost causes, let's just put them together and not upset the rest of the team.\" Either way, they were doomed. He was once found naked in a high school kitchen, covered in cooking oil, where he was arrested. He did try to straighten out, got married, but while on vacation he died of a heart attack.\n\nIt's got to be one of the saddest stories in sport. Movie-star looks, breaking records in hockey, he went from the toast of hockey to having it all end at the age of thirty-two.\n\nI know what you're thinking: \"Why dwell on these stories? What's the point?\" I have a reason. A lot of junior players and younger kids not in hockey will read this book. That is why there is no profane language in the book again. I know what you're thinking: \"Are you kidding? The kids could teach you a few.\" Maybe you're right. Maybe one or two kids reading the stories of these guys will smarten a few guys up. \"And if they're asked at a party to smoke a joint, what's the harm?\" I tell the story of the heartbreak of two parents who wrote to me about their son who was a young hockey player, who, while at a hockey tournament, was late for the bus. The coach came to his room to get him and saw a joint in his room. It wasn't even his, but he wouldn't tell whose it was. The coach suspended him there and then from the team. The team went and played the game. When they came back, his roommate found he had committed suicide by hanging himself with his belt. Naturally his mom and dad were heart-broken. They would give anything to get their son back, and they wanted to get the word to other kids. No matter what you do, your real friends will forgive you and so will your mom and dad.\n\nI know, reader, that these are depressing stories, but if it can save one kid, I feel it's worth it.\n\nHazing in sports has been going on since sports have existed. Rookies are the targets. It starts with rookies picking up the pucks after practice, carrying the bags for the trainers, cleaning the bus up after a long trip (in the juniors and minors). It graduates from there, where it gets meaner\u2014shaving of private parts, etc.\n\nMy first experience with hazing was when I was away at sixteen to play for the Barrie Flyers in the Ontario Hockey Association. The vets were always discussing whether one guy could lift three guys. I thought it was impossible. There was a rookie in the club who thought he was hot stuff, and he said to the vets that it \"can't be.\" They said, \"Okay, after practice, we'll show you.\"\n\nAfter practice, we all showered and were naked. The vets said, \"We need three guys on the floor, laying on their backs.\" Meanwhile, everyone was betting, including Hot Stuff. So the vets said, \"We got two guys on the floor. We need a third. How about you, Hot Stuff? You get in the middle.\" He got in the middle and they were told to entwine their legs and arms around each other so the one guy could lift the three of them.\n\nHot Stuff was in the middle, and the guy was lifting said, \"Now, can anybody move?\" All three said no. \"Okay, he's ready!\" Four guys came out dressed as barbers, with towels and razors and soap, and proceeded to shave around the private parts of Hot Stuff.\n\nWhen I was coaching Colorado, the players just happened to pick the day the owner from New Jersey, Arthur Imperatore, came to visit the club as their day for hazing. Unbeknownst to me, as we're talking by a dressing room, the owner looks in, and here we have eighteen naked guys shaving another naked guy. Imperatore didn't understand, and it was hard to explain the ritual to him. He was not happy.\n\nWhen I was in Sudbury, we had an American trainer who was a rookie. I knew the players were going to get him, so I took him aside and said, \"Look, the players are going to haze you someday. Just go along with it. Don't fight them.\"\n\n\"Oh yeah? They better not try that stuff on me.\"\n\nI said, \"Okay, it's your funeral. Ya can't fight eighteen guys.\"\n\nSure enough, after practice one day, they got him\u2014and he did fight back. He put up a good fight, but they broke his arm. Ya can't win against eighteen guys.\n\nWhen I was in Rochester, there was a guy, the most obnoxious little rookie ever. If he wasn't playing, he'd phone his agent. It was like he'd been in the league for years. I knew sooner or later they'd get him. Sure enough, in a hotel room after a game, they grabbed him. Now, this guy had lovely hair. It was his pride and joy, always blow-drying, gel, so instead of shaving his crotch, they shaved half of his head. He went nuts crying, threatening to sue us, take us to court. Funny thing is, while this was going on, the coach walked by the room, looked in, had an idea of what was going on, and kept on walking. He was happy this stiff was getting his, as they say.\n\nI know some people say it's an initiation and some of it I liked, it was fun, like lifting three guys. But some of it was just mean, and to tell you the truth, I never really liked it.\n\nThe one hazing incident I really felt bad about was at the Springfield Indians training camp. It concerned a young player who was at our camp, whose father played in the NHL, and he really thought he was somebody. I knew his time was coming\u2014not if, but when. Sure enough, after practice, they went to grab him. He made the mistake of fighting them. He was a big, strong guy and it was touch and go. They wrestled around the dressing room. Unfortunately for him, the room had red-hot radiators on the wall. As they were wrestling, he was held against the hot radiator, which burned his back and bum. It was so bad you could see the rivets from the radiator on his skin. They didn't care and showed no mercy, held him down and put some ointment on him, which was blue. They covered his whole body in it and it would not come off. As I was leaving the arena, I looked into the shower, where the kid was showering, trying to get the stuff off. It was not coming off. He looked up and saw me and said, \"You know what's the worst about this, Grapes? I was just married. Imagine, I have to go home looking like this.\" As I left, I heard him sobbing.\n\nI guess all this hazing started in the English schools, and I know it was just as tough in the U.S. Marines. They want to know if you're tough enough to be a marine. Same thing in baseball, and definitely in football, to see if you can stand the gaff, as they say.\n\nI've noticed one thing over the years: I never ever saw a tough guy rookie hazed. There could be a lot of reasons why, but I feel the reason they don't haze tough guys is you might get the tough guy today\u2014like I said, you can't beat eighteen guys\u2014but there's always tomorrow and there won't be eighteen guys around. And besides, in hockey, you might need that tough guy when the going gets rough.\n\nI was never hazed, and I will state now that I never liked hazing. I saw a couple of guys never recover from the hazing. There was only one hazing I loved, when they shaved the head of the guy in Rochester. We all liked it, especially when the little rat cried.\n\nI remember my very first \"bad,\" as they say, write-up. I had been on TV for a year. I call the first year on TV the honeymoon\u2014the press don't know you, so they leave you alone. Then, if you get successful, it's when the dogs come out. It's funny. Down in the States, it's not like that. You can sort of feed on your success\u2014in the States, they say \"go for it.\" Now, if you're no good, they show no mercy. But in Canada, if you're successful, it's your turn. It happened to Ron MacLean when he started out on _Hockey Night in Canada_. Young guy from Red Deer, Alberta. He was good, make no mistake. I kept telling him, \"You're on your honeymoon.\" He didn't understand. He asked, \"What are you talking about, I'm on a moon?\" I laughed. \"You'll see,\" said I.\n\nRon's honeymoon ended, of all places, in Calgary, Alberta, at the All-Star Game in 1985. Ron was the host of the All-Star banquet. He did his usual great job and he was feeling pretty good till he picked up the paper the next day. He was ripped to shreds in the paper. His honeymoon was over; he was crushed. His first bad write-up\u2014in his hometown, of all places. When I saw him, I said, \"Welcome to the club!\"\n\nMy first rip job was done by a distinguished sportswriter by the name of Trent Frayne. He was the winner of many awards and was well thought of. I thought his columns were great. But boy, after a \"Coach's Corner,\" he gave it to me, both barrels. I forget what it was about, but I was in shock. Honestly, I didn't know if I could go back on TV. It took me a couple of weeks to get over it. The executive producer, Ralph Mellanby, laughed and said, \"Forget about it, this is a tough business, and to survive, you've got to be tough. If you can't take it, get out of the business. You'll get a lot of write-ups like that before you're through. Forget it.\"\n\nI thought, \"Easy for him to say.\" But after a month, I gradually came out of the shock. Then another beauty write-up came out. This one had my picture, and under it, the reporter listed what I was, just the names:\n\n\"Neanderthal.\" I had to ask somebody, \"What is a Neanderthal?\" \"A big, tough caveman who carries a club.\" I thought, \"Hey, that's not too bad.\"\n\n\"Troglodyte.\" It's a little dwarf that lives under a bridge and is evil.\n\n\"Misogynist.\" A fear of women.\n\nI looked these up in the dictionary. And then there were the usual names\u2014evil, etc. My second bad write-up. It didn't hurt quite as much, and the next bad one didn't hurt as much as the second. My hide was getting thicker. It was pretty hard to hurt me after a while. One guy went below the belt and said, \"Do you ever notice when Don Cherry gets in trouble, he always puts on kids with cancer?\" Nice, eh? But like I said, I got tough. Ralph was right. \"Can't stand the heat? Stay out of the kitchen.\"\n\nThe Montreal _Gazette_ in particular had it in for me. It all started in the eighties. Ron MacLean had told the kids that were going to be drafted by the NHL that he would get them on TV on _Hockey Night in Canada_. So the kids got haircuts, new suits, shoes shined. The came into the studio and Ron said, \"Sorry, guys. We don't have time,\" which happens a lot on TV. The kids were heart-broken. They had told their parents and friends to look for them on TV. I argued with Ron, but it was impossible. It wasn't his fault. I said, \"Okay, we'll put them on 'Coach's Corner.' \" I told them to come back at the start of the first period. That's when I started to put the prospects on \"Coach's Corner,\" a tradition that has been going on now for twenty years.\n\nI noticed there were only four kids at the time when we did \"Coach's Corner.\" I wondered where the other three were. The paper the next day said: \"Cherry proves he is a bigot because he didn't put the three French kids on his segment at 'Coach's Corner,' only the English kids.\" The paper was right. I only put on the English kids because the French kids, at the time \"Coach's Corner\" was on, were being introduced and honoured as prospects for the NHL at centre ice in the Forum. I thought that was the best. The Forum was introducing the French kids, not the English. I was going to put on the whole seven on \"Coach's Corner,\" and now _I_ was called the bigot.\n\nThe Montreal _Gazette_ has a writer named Jack Todd, an American who had it in for me from day one. It started in the early eighties, and he never let up. Jack had a write-up last year around the time Rogers took over _Hockey Night in Canada_ and, for a while, it looked like I wasn't coming back. Remember how the boss at Rogers said at the press conference, when asked if I was coming back, \"We are evaluating everybody\"? The last two times I was fired, the boss said, when asked the same question, \"We're evaluating Cherry.\" And the next day, gonzo. So Jack did this write-up the next day, it was so good. It got into Canadian Press and email or whatever they do to spread the word. Bobby Orr phoned me about the column the next day, so I know it went all over the States and Canada. Jack started out with his opening line: \"I've spent half a career arguing with Don Cherry. I've hated him longer than some of you have been alive.\" He goes on:\n\nYou know how it is: I'm one of those lefty kook tree huggers he's always snarling about and he's one of those pigheaded, narrow-minded, right-wing bullies I've been battling since fourth grade.\n\nCherry's noisy support for Rob Ford? Eeesh. The maudlin way he wraps himself in the mantle of Canada's Armed Forces (a trick borrowed by Stephen Harper and Peter MacKay) even though, like Harper and MacKay, he never served? Double eeesh.\n\nBut I want to say this, loud and clear: If the Don is turfed from _Hockey Night in Canada_ , I'll miss him. Because Cherry belongs on _Hockey Night in Canada. Hockey Night in Canada_ belongs on the CBC. And the CBC belongs to us.\n\nIf Cherry had been fired from within _HNIC_ as it existed until Tuesday, that's one thing. But if he's to be jettisoned by a big private corporation in pursuit of eventual control over the entire CBC network, that's something else.\n\nPerhaps that seems like an exaggerated response to Tuesday's announcement of the 12-year, $5.2-billion deal between the NHL and Rogers Communications. I don't think so. The scariest words I heard Tuesday came from one of the Rogers suits (sorry, I didn't get the license plate) when he said something to the effect that \"we look forward to future partnerships with the CBC.\"\n\n(\"Partnership\"? That's a good one. As partnerships go, the deal with _Hockey Night in Canada_ is strictly of the \"me master, you slave\" variety.)\n\nWhat worries me is that the brutally one-sided takeover of _Hockey Night in Canada_ will become the wedge that leads to the eventual privatization of the CBC. Given that the deal also creates a horizontal and vertical monopoly for Rogers, you'd think the toothless CRTC might have something to say about it because monopolies are, by definition, bad for consumers\u2014but expecting the CRTC to protect your interests is like asking a Great Dane to look out for your steak.\n\nWhen I saw Gary Bettman and Rogers CEO Nadir Mohamed chortling and pumping hands Tuesday, I got queasy. It was like watching a couple of car salesmen congratulating each other after they've just taken some poor schnook on a deal for a used Ford Pinto.\n\nBecause we're that schnook and you just know that in the long run, we're going to get taken. In the words of the old country song, they get the gold mine, you get the shaft.\n\nWhat's going to stand in Rogers's way? Not much. They now own _HNIC_ : lock, P.J. Stock and barrel. Given that our wretched PM would love to kill the CBC, Cherry is perhaps the only person in this country with the juice to prevent the death of _Hockey Night in Canada_ four years from now and eventually, perhaps, even the demise of the CBC itself.\n\nWhy? Like it or not, Cherry is more than just Don the Broadcaster, Don the Loudmouth, that SOB Don or even Don the Institution. He's a national symbol. If Rogers can tear him down, then nothing is safe.\n\nIf you want to know why the Rogers deal is so alarming, the fate of former Leafs GM Brian Burke is instructive. Burke worked for Rogers (and Bell) as an employee of MLSE. He was fired, not because he was a terrible GM, but because he ruffled the feathers of the suits. He wasn't the kind of yes-man they require.\n\nEventually, if Rogers or Bell or some other private entity takes over the CBC, this is how it's going to be. The bosses will claim (as they did Tuesday) that they won't tell their employees what to say. And it will be true\u2014because they won't have to. Consciously or sub-consciously, the journalists will censor themselves. So long, public network with something other than the corporate interest at heart.\n\nCherry is a bulwark against all that, simply because of his sheer, critical mass. Love him or loathe him, we watch him every Saturday night, in all his fluorescent, megawatt, wardrobe-by-Liberace, ungrammatical, name-mangling, xenophobic glory.\n\nAfter hockey itself, our second favourite national pastime might be arguing about Don Cherry. The debate tends to obscure an essential truth at the core of the Cherry phenomenon: The man is a superb, instinctive broadcaster, a master of the sound bite. Cherry doesn't attract attention, he compels it. He alone can compel enough attention to shine a light on the nastier aspects of this deal.\n\nThe suits from Rogers did not say that Cherry would be shown the door during that daylong drum roll that accompanied the announcement Tuesday\u2014but they pointedly avoided saying that he would be retained.\n\nCherry's fate is important because, beyond all the justifiable carping about the increasingly amateurish _HNIC_ production, I believe profoundly in the CBC itself, and I always have.\n\nIt's a pity that this deal comes at a time when the _HNIC_ broadcasts are so badly in need of a revamp. Ron MacLean's influence has been disastrous, P.J. Stock has to go and the entire production needs an infusion of talent to bring it up to the industry standard set by TSN.\n\nThere are good people at _HNIC_ who should be retained\u2014Jim Hughson, Kevin Weekes, Cassie Campbell-Pascall, Garry Galley and especially Elliotte Friedman, who is one of the best hockey broadcasters in the country.\n\nThe real key to the whole package, however, is Don Cherry, AKA Grapes\u2014because if he goes, then _HNIC_ and possibly even the CBC itself are living on borrowed time.\n\nIt was sad to see the man, looking old and tired without his makeup Tuesday, asking reporters at a book-signing event, \"Do I still have a job?\"\n\nWe need to speak out, loud and clear: \"Ya mess wit' Grapes, fellas, and you're messin' wit' us.\"\n\nIt's funny, what he said about me looking \"old, tired without [my] makeup, asking reporters at a book-signing event, 'Do I still have a job?' \" He was so right. I saw myself on the news that night, and I thought, \"Man, you look like this is your last day.\" But let me tell you about my day that day. I had scheduled, unfortunately, everything on the day that Rogers bought hockey, including _HNIC_. I had to do interviews all morning, no lunch, took an hour at the Rogers press conference, where I found out I was being \"evaluated,\" did a signing that afternoon, had to come down to the Air Canada Centre\u2014we got stuck in traffic, got there late\u2014had to do a signing and go upstairs to the bar and do TV interviews on the _Rock 'Em, Sock 'Em_ video and pose for pictures with people. I was so tired that, for the first time ever, I had to say enough.\n\nGo downstairs, and as I go out, there are ten TV stations, radios, newspapers. As I attempted to do the interviews, the guy from CTV's batteries run out. I said, \"Hurry up, we will wait.\" Finally get going, asking me questions. I have no answer, as the Rogers buy was a surprise to everybody. I go home, switch on the TV, and there I am looking like \"the wrath of God,\" as my mother used to say.\n\nI have to say, this column, even with words like \"pigheaded,\" \"narrow-minded,\" \"right-wing bully,\" \"wardrobe by Liberace,\" \"ungrammatical,\" \"name-mangling\" and \"xenophobic glory [I know my buddies are wondering about that one\u2014I looked it up in the dictionary and it means dislike, hatred, fear of strangers or aliens],\" it was totally unexpected to have him want me to stay. I thought he would be cheering for my departure. Life is funny. The guy who has been taking shots at me for thirty years is the only one who wrote that I should stay. Who would have thought?\n\nWhen I do those walk-ins before the games on _Hockey Night in Canada_ , I never know what to expect. I try not to be seen just before I go in. We did not plan it right last year, for Hockey Day in Canada in Lloydminster, Saskatchewan. As we drove up before I made my entrance, I told the driver and the PR guy to park a block from the arena and wait for the call to the front of the arena for my big entrance. Unfortunately, we drove up to the parking lot and turned the car around; now everybody could see me and everybody wanted a picture or an autograph. Normally, no problem, but I had little time to make my entrance and had to think of what to say. The PR guy got out and said, \"I will tell them no autographs.\" I was furious. He didn't realize I'm the guy who looks bad if he stops them. They'll think, \"Look at him, a big shot. Too busy for pictures or autographs.\"\n\nWe got the call: thirty seconds, start your way in. We have to be on the minute for TV. I arrive at the front like a big shot; it was so embarrassing. Opened the door by the Mounties and started in. If you notice, I only slap everybody's hands. I'll tell you why. On the way through the gauntlet, I used to shake hands. Once, as I shook this old guy's hand, he held on and crushed my hand and wouldn't let go. I literally had to pry my hand off with my other hand. Another time, a little boy was dressed exactly like me in one of my flashiest jackets. His mom had gone to Fabricland and got the material and made his jacket. It was just luck that I was wearing the same jacket. He went to centre ice with me. A young man with a white English bull terrier greeted me. I took him on the ice, too.\n\nI felt bad on my entrance in Lloydminster. I was very rude. It was a hectic time. Let me explain, as I have said you have to have it timed on the nose. As Ron is introducing me, I have to be coming on the ice. I can't have the Mounties or the police in front of me, as they slow me down. This girl was walking in front of me\u2014she was the security, wearing a headset and all that. Well, as Ron was introducing me, she stopped dead in front of me and I couldn't get by. It was dark, and I couldn't see where I was stepping. I heard Ron and his intro, and I was still in the dark with three million viewers wondering where I was. Finally, I pushed her out of the way and said, \"Get out of the way.\"\n\nI can't believe I was so rude. When I got back home, I saw some of the video and wondered what she and her family must be thinking. Here she is, a volunteer, and I do a thing like that. I got her name and number and phoned her, hoping I would get her answering machine (I bet you've phoned a few times, hoping you would get the answering machine), but she answered. We had a long talk, and as she was hanging up, she said a nice thing: \"Thanks for caring.\"\n\nRemember I said earlier about parking in the parking lot and people waiting for an autograph? As I sat there, ticked off, I saw this dad and two young sons, about four and seven years old. It looked like they'd been there a long time. Finally, I couldn't take it any longer. I got out in my shirtsleeves and pointed for them to come to the car. I signed their programs, pictures and pucks and posed for a picture. The dad said, \"Gee, Grapes, we didn't expect this. We've been waiting for two hours, hoping to see you.\" They didn't even have tickets to the game. As the call came\u2014\"Thirty seconds\"\u2014I happened to see out of the corner of my eye the dad and his two boys walking away. They were jumping around, doing high fives. I said to myself, \"Life is good.\"\n\nJust a few weeks ago, after I'd already written this story for the book, I got this letter:\n\n_Dear Mr. Cherry_ ,\n\n_I would like to take the time to personally thank you for your kind gesture during the Hockey Day in Canada event held at the Lloydminster Exhibition grounds. My son, nephew and I waited around in anticipation of meeting you and getting your autograph. When we were asked by the arena staff to move from where we were standing, the boys and I were quite disappointed that meeting you probably wasn't going to happen. When we were leaving we noticed your SUV pull in and park, so we waited at the entrance of the parking lot in the hope of meeting you. When your assistant came over to get our items to be autographed, we could not have been happier. However, when you stepped out of the SUV and called us over, it was the opportunity of a lifetime_.\n\n_Meeting fans for you is a daily occurrence, but to those two young boys, the impression that you left on them will last a lifetime. After you signed our items and were walking away, those boys were on cloud nine and couldn't wait to get home to share theirexperience. Since then, my son has taken the hat you signed to school to show anyone and everyone who will look_.\n\n_Thank you again for going your of your way on a Saturday to make a lifetime memory for the three of us_.\n\n_Sincerely_ ,\n\n_Dion Kneller_\n\nI've often watched those investigative TV programs, and I've always had a thought that a little more is happening here than meets the eye. I found that out firsthand and it had to do with a cold remedy I was promoting called Cold-FX. Let me tell you how I became involved with Cold-FX. When I was young, I had asthma, and through the years I had multiple colds. I was in Edmonton doing _HNIC_ \u2014with a cold, as usual. I read where Glen Sather, the GM of the Edmonton Oilers, told his players to start taking Cold-FX for colds. Says I to myself, \"If it's good for the players, it's good for me.\" So I took it for a year, and it works. A rep at Cold-FX was talking to Ron MacLean, and Ron said you should speak to Don, he takes it all the time. Long story short, I became the spokesman.\n\nI heard through the grapevine that a TV station was going to investigate Cold-FX\u2014whether it works, etc. I get a call about meeting the female investigator. I am in, says I. The investigation was pathetic. For instance, in a sloppy manner, they were asking teenagers about Cold-FX\u2014whether it works, etc.,\u2014and then they showed that the ginseng that's in Cold-FX came from China. The woman investigator starts on me. Now, she is used to people that are not on TV and she bullies them. Little does she know this is my world. In the conversation, she keeps bringing up the name of the president of Cold-FX, who has sold the company. I sense she doesn't know this. I said, \"You keep bringing up the name of the president. You know she sold the company two years ago?\" I got her.\n\nI don't let up, no mercy. How can you call yourself an investigative program and not know the president of the company has been gone two years? How can anyone believe anything you say? I never let up. Now she's frustrated, and now I am leaving (this is on TV). The weasel who has the camera is going to tape me leaving as if I'm guilty and running away. All the while, I'm saying this on camera. Now, I'm telling you producers: if you make me look bad on the program, I will bury you, and you better do your homework before you get into a debate with me.\n\nI see the program; naturally, they were okay with me, but they ambushed the president as she was getting into the car. She says, \"I have no comment\" (seeing as she sold the company two years ago), but on TV it looks like she has something to hide and looks guilty. That's how they work. So when you see all these programs that investigate, remember: take it all with a grain of salt.\n\nWhen I meet people in the airports or on the streets and in general, I'm often asked what comment I've made on \"Coach's Corner\" has drawn the most reaction from people. \"Is it when you called the Winnipeg Jets' [assistant] coach Alpo Suhonen dog food? Or when you said Pekka Rautakallio was as yellow as a duck's foot? When you said at the Chicago All-Star Game, when asked to comment about the people burning the U.S. flag over protests of the Gulf War, 'How come these kooks are always protesting and burning flags during the afternoon? Don't they ever work?' When you called someone who didn't agree with you on fights 'pukes'? When you and Ron MacLean argued whether Canada should have gone and helped the U.S. in the Gulf War and I remember the CBC couldn't stop you and Ron? Or when you had your picture taken with a female bull terrier in an Edmonton sweater and you said, 'That bitch didn't mean anything to me.' Or when the French got mad at you about the visor statement and you got booed by the Bloc in Parliament?\"\n\nI could go on, but it would be none of these incidents. The comment I made on CBC that brought the most reaction in thirty years is the one I made about how I don't believe women should go into men's dressing rooms. I said if men aren't allowed in women's dressing rooms, why should women be allowed in the men's dressing rooms? But actually, the last statement, about men in women's dressing rooms... I don't believe in either, I just said that. But the statement brought more reaction than any statement in thirty years. And this is the honest truth: more women have come up to me and said, \"By the way, I agree that women should not be in the dressing room.\" Which surprised me.\n\nThe reason I don't like women in men's dressing rooms is the way that some men act. Remember, I said some _men_ act. My reaction was triggered by a write-up in one of the papers about a woman reporter who was doing an interview with a basketball player, and in the background another basketball player had grabbed a long piece of Styrofoam, stuck it between his legs and was dancing around. This is tame compared to what I have seen at some times. Some guys actually walked around without a towel around them, just for kicks. And other things I'm not going to mention.\n\nI said, \"I would not want my daughter to be in the dressing room.\" Well, the reaction was predictable. The women reporters took offence, naturally. I knew that was coming. The left-wing papers had editorials, and I was called a redneck and a character and all that stuff.\n\nThey even had a voice from the past come at me in the form of a woman reporter named Robin Herman\u2014the first female sports reporter for the _New York Times_. She had the headline: \"Don Cherry, you were my hero.\" Robin was twenty-three when she was given the assignment of covering the New York Islanders. She says she remembers me fondly in those early days: \" _You_ were the first coach in the NHL to allow me, a female, accredited sports reporter and member of the Professional Hockey Writers' Association, into your locker room as a matter of policy. You were coaching the 'Big Bad Bruins,' and it was ironic that a team with that reputation should be the most forward-thinking in the NHL. Your PR man par excellence, Nate Greenberg, had persuaded you this was the way to go.... He and you were gentlemen.... The times they were a-changin' then, and the Bruins organization was smart enough to realize it. You should be proud of what you did.... I faced a draconian deadline of 11 P.M.... Every minute spent waiting for a team official to bring a player out of the locker room to speak to me separately in some dank hallway was excruciating.\"\n\nI do remember Nate coming up to me after the game on Long Island and saying, \"If this young girl doesn't get her story in by 11 P.M., she could lose her job.\" As there were a few fights that night, it was close to eleven at the time. I had other things on my mind. We had lost to the lowly Islanders and GM Harry Sinden was furious. We were a first-place team, and I and Harry Sinden and the team thought we blew the game and we were not properly prepared for the game. And it is my job to prepare them. I knew I was in trouble. I was listening to Harry tell me the team was not ready, etc., etc., and all the while, reporters were asking me questions. It was a hectic time.\n\nAs usual, after the game and in the midst of all this, Nate whispered to me and introduced me to a young female reporter and said something about her losing her job if she didn't get an interview or something and asked if she was all right to go into the dressing room. I was listening to Harry and said, \"Yeah, yeah, go on in.\" I never thought anything of it. Later, Nate said thanks and I said, \"For what?\" Little did I know that Robin had been pleading with various team officials and the league itself for the better part of a year to let her in the dressing room and now I had opened the gates.\n\nWhen we were in Boston, a female reporter came to me and said I let a female into the dressing room in New York and I said, \"I did?\" And they said, \"Yes, and we're expecting the same treatment here in Boston.\" I'm done. I've opened up Pandora's box. _Now_ I remember this. Beautiful woman reporter now expected to go into the room. There's no turning back now, and in they go. Everything seemed to go smoothly for a while, till one night after the game, Jean Ratelle\u2014a perfect Jean B\u00e9liveau type, quiet and dignified, a classy gentleman, a Lady Byng Trophy winner\u2014came to me and quietly closed the door and said, \"I don't want to make trouble, but to have women in the dressing room is very upsetting to me and I'm not used to it. I protest against it.\" Jean was one of my favourites\u2014like I said, a Jean B\u00e9liveau type. What to do?\n\nI also found out that some of the wives were very unhappy with this situation. I came up with this idea: I said to the TV reporters and women writers, \"Tell you what: I will give you any player you want after the game, the #1 star, anybody. I will bring them to you in my office for you and have a private interview. I will hold a press conference in the hall, and you get the first choice of any player.\" Thank goodness they went along with it. The guy reporters weren't too happy, but what the heck? They had me. What more could you want?\n\nSo, when did I change my mind about female reporters in the room? It started when players from other teams were acting up when female reporters were in the dressing rooms. I must admit that _I_ wasn't happy with the way a couple of my players were acting, till I straightened them out. I know it's a little different now. Guys don't go directly into the dressing room naked when other reporters are in, but still act up\u2014for example, the basketball players. And I repeat: I still don't want my daughter in a room with men with towels around them, fooling around. Let me ask you: Would you like your daughter in that room? I think not.\n\nI will always remember when I was laid off from my construction job at Kodak in Rochester. It was not an easy time; I had no trade experience (unless you count operating a jackhammer a trade). I was a beast of burden, no education and no job, painting for two dollars an hour, working eight hours for sixteen dollars. One day, I got a call from a teammate who was forming a hockey school (let's call him Jim Smith). He asked if I'd like to be one of the instructors for a hundred dollars a week. Would I? I was the happiest guy in the world. I'd be there with bells on.\n\nMonday morning, I'm there an hour early to get the kids checked in. Jim shows up fifteen minutes before the first session, but I already have everything straightened out. Kids are anxious\u2014so am I. Jim has hired some players from the Rochester Americans. They show up five minutes before the session\u2014obviously, they had a party the night before and are a little hungover. The session begins. I am gung ho! The players are struggling to get through the morning with the hangover. Lunch comes from twelve to one\u2014I brown-bag it and am ready to go at one o'clock. The players show up at one-fifteen. I think they had a liquid lunch. I shoot pucks all day. I think I'm getting blisters because I haven't shot at goalies in a year. At the end of the day, I take off my gloves and my hands are bleeding from blisters. I can hardly drive home\u2014I put tape on the blisters and drive mostly with my wrist.\n\nI get home, and I am hurting. Rose suggests that she read somewhere that Epsom salts would be good. I go along with her and put my hands in Epsom salts. I think my head will explode with the pain. I try to wash the stuff off in our bathtub, but the faucet handles don't work.\n\nThe next day, I know, is not going to be fun, but I suck it up. It's a long way from your heart, as the old trainers in the AHL used to say.\n\nWe now have broken up the sessions. I'll take the first session and let the Americans players sleep in because it's party time. I have from eight to ten in the morning. Ten o'clock comes, and the players are late. I fill in for them for fifteen minutes and they stagger in (meanwhile, my hands are killing me\u2014I keep opening up the blisters).\n\nThe whole week, the players are late and I fill in till they come on the ice. I have to take charge, and I give them a lecture. After that, they're on time. Now, you must remember, I'm just an instructor. It's called the Jim Smith Hockey Camp. We are in day five and Jim Smith has not been on the ice once, and kids keep asking me when Mr. Smith's coming on the ice. Jim was a pretty famous guy in Rochester. He has played most of his career there and he is a 40-goal scorer, but so far no show on the ice. I've got to be a little careful\u2014this is not my school, I'm happy to get the hundred bucks\u2014but this is not right. The kids are being cheated. Something has to be done, although I'll probably lose my precious hundred bucks.\n\nI go into his office, and he just happens to be counting the kids' money, which makes me see red; but control yourself, Don, I say to myself. Think of the hundred.\n\nDon: \"Jim, the kids were asking when you're coming on the ice.\"\n\nJim: \"Great!\"\n\nDon: \"Jim, this is not right. It's your school, your name's on it, the kids are looking forward to seeing you and your great shot. Don't you think it would be nice to make an appearance these last two days?\"\n\nJim: \"It's none of your business.\"\n\nDon: \"Well, I'm making it my business.\"\n\nJim: \"You know, Cherry, I give you a break and now you give me a hard time. Do you ever think that's why you can't get a job?\"\n\n(I've got to admit, folks, that hit home, but as my dad used to say, \"In for a penny, in for a pound.\")\n\nDon: \"Yeah, maybe you're right. I don't have a job, and I am having a hard time, but I'll tell you one thing: I don't try to cheat twelve-year-old kids. Now, get your skates on and get your ass on that ice.\"\n\nHe goes to say something, but thinks better of it. He goes on the ice, the kids love it and he is a hero.\n\nThis is my last day. He only gives me sixty dollars, as I didn't finish the week. Now that I think of it, I should have told him to shove the sixty, but I needed the money. Rose was counting on that extra forty dollars. Sorry, Rose. Will I ever learn?\n\nI had a friend the other day tell me his eleven-year-old daughter came home from a soccer game. The daughter and mother were quite proud that she had a trophy, so the dad said, \"Way to go, Wendy. What was the score when you won the game?\" The girl said \"Oh Dad, we don't keep score in the games. We didn't win; we lost. But isn't the trophy great?\"\n\nThe guy told me he was dumbfounded. He didn't know what to say. Don't keep score and you get a trophy for losing the game? He said to me, \"Believe it or not, she got the trophy for competing.\" He works in a highly competitive business where, if you are not successful, you are gone. In other words, no reward for second place. So the business of being rewarded for losing just blew his mind.\n\nI got to thinking, isn't that the way Canada is going: \"Don't worry about working hard and saving money. We will take care of you from the cradle to the grave.\"\n\nWhat does that attitude of just compete, don't sweat it if you win or lose... what does that teach the kid for later on in life? If the work or the study gets too hard, float along, don't worry about it. What kind of outlook is that for life? What character does that teach the kids? You have to know how to win and want to win, and if you lose, dust yourself off and start all over again, like the song says. My brother and I were taught that old poem, \"Don't Quit.\" Here is part of it:\n\n_When things go wrong, as they sometimes will_\n\n_And the road you're trudging seems all uphill_ ,\n\n_When the funds are low and the debts are high_ ,\n\n_And you want to smile but you have to sigh_ ,\n\n_When care is pressing you down a bit_ ,\n\n_Rest if you must, but don't you quit..._\n\n_Success is failure turned inside out..._\n\n_So stick to the fight when you're hardest hit_.\n\n_It's when things seem worst that you must not quit!_\n\nThe school thinks they're doing the girl a favour, not humiliating her or hurting her confidence. They are just preparing her for a big shock when she faces the real world. You must face the slings and arrows along with the bouquets to be successful. I should know. I have a lot of arrows in me.\n\nCindy, my daughter, and I were having one of our coffee chats one day and she asked me, \"How would you describe yourself as a dog?\" We are both in love with all dogs. I thought for a minute and I said, \"Cindy, I am what you call a street-smart dog. Let me explain how a street-smart dog compares to a normal dog. For instance, I have normal dogs\u2014nice, well-behaved\u2014but if they got out on their own in the streets, they'd be done. They would not know how to survive. But your dog, Lucy, Cindy, is street-smart. Put her on the street, she'd survive anywhere.\"\n\nHow'd she get like that? I'll tell you how. Lucy is a rescue dog. She has been kicked around, abused, lived in four different homes. The final home was pitiful. The people who had her moved from their apartment and just left her to die. No food, no water. She barked for help till her vocal chords went. Finally someone reported her and she was rescued. That's when Cindy got her. Lucy had the same personality as the dogs in Sochi, Russia. They survived on the streets somehow. They knew the score, knew when to run and knew when to stand, knew how to avoid cars. It was you either knew how to survive or you were gone. They knew how cruel people could be and by hook or by crook, they survived. (Till the Russians shot them.)\n\nCindy said, \"And you consider yourself one of these dogs? I thought Grandma and Grandad treated you well.\"\n\nDon: \"Cindy I led a beautified life. My mom and dad gave me everything I wanted. I went to the Barrie Flyers. We won the Memorial Cup. I turned pro in Boston, bought a new car at twenty-one. Life was, as they say, a bowl of cherries. I married the love of my life, Rose, and we had a beautiful daughter. Couldn't have asked for anything more. I was like my dogs at home: happy not knowing the world, not prepared for the hard knocks. It all changed when, at twenty-three, Boston gave up on me. They were furious with me for playing baseball and hurting my shoulder after they asked me not to. In those days when you got in the bad books, you were history. There were only six teams to play for. They could do anything they wanted with your life. Hey, I deserved to be taught a lesson, but to send a kid of twenty-three to Eddie Shore, the devil's island of hockey, was really a kick in the teeth. But I was still enthused when I reported to Springfield. Happy and disappointed, but still ready to go\u2014sort of like my dogs.\"\n\nThe very first day I had my heart broken and it was there that I knew I had to change to survive. Gone was my attitude of \"Life is great, people are all good, good things happen to good people, etc.\" It became dog-eat-dog, survival of the fittest, and I learned, just like Lucy, that people can be mean just for the sake of being mean and you have to toughen up and get some savvy to survive.\n\nSavvy, to me, is the most important thing in the world in my position to survive, you've gotta keep Kenny Rogers' hold 'em\/fold 'em approach. Know when to fight and when to retreat to fight another day. The adage I like is \"Sit in the weeds. Every dog will have its day.\" I know what you're thinking, folks, that's an awful outlook on life, but like Lucy, we had to think that way to survive. You don't last twenty years in the minors being a nice guy. The motto in the minors was \"Do it to him before he does it to you.\" That's how I survived physically and mentally. That's how Lucy and I are still going. We survived in the jungle, there was nothing else you could do to hurt us. We had run the gamut of disappointment and getting kicked in the teeth.\n\nI know, reader, you're thinking, \"Boy, Don is really getting carried away with himself.\" Maybe you're right, but you never went through four years with Eddie Shore as his favourite whipping boy, riding buses for twenty years. Along the way I developed savvy. I knew somehow I would survive. There were a lot of bumps along the way. Many times I was fired, had no job. Nobody knows you when you're down and out. The saying is so true.\n\nLike Lucy, I survived. In a way, people like Shore toughened me up. It's too bad we had to go through all that turmoil to get through life, but that's the luck of the draw, the way the cards are dealt. I'm not complaining about anything. Whatever I got, I deserved. I'm just telling Cindy I consider myself a street-smart dog. Lucy and I know the score.\n\nThe Lord works in mysterious ways. He's a wonder to behold. I gave myself a lot of credit about street smarts and survival, but it's funny how my life worked out when I was sent to Springfield.\n\nFor a long time, nothing seemed to work out right for me. Things went from bad to worse. As I look back now, things started to break up for me after ten years of tough sledding. Not only for me but for my family. It was a break and I didn't know it at the time. The Good Book says \"God can do infinitely more than we can ask or imagine.\"\n\nBack in the sixties after knocking around the American League, the Western and Central Leagues, and the Eastern Professional League, I somehow ended up Toronto Maple Leafs property. How, I still don't know. I reported to Leafs camp. I knew I was not going to make the cut and their farm teams were the Rochester Americans and the Denver Invaders. I had played in Spokane the year before in the Western League and I liked the Western League, so I wanted to go to Denver and I told them so. Nope. Joe Crozier was the GM and coach. He wanted a tough guy and he said, \"No way, it's Rochester.\"\n\nI did my best to get out of Rochester, but Joe had his way. It so happened that Punch Imlach, who was GM and coach of the Leafs, owned a piece of Rochester and packed the team with young stars and veterans to win. We won three championships and one final in four years. The team went to Vancouver and I went with them. We won the Western League Championship. That's four championships in five years.\n\nI retired and worked construction for two years until I got laid off. A friend, Bob Clarke, asked if I would coach a high school team. I didn't really want to, but did it for a friend. For two years I learned how to change lines, practice and win a championship from the other side of the bench. I still had no job and was desperate to make a comeback with the Rochester Americans at age thirty-six. I was named coach halfway through the season. (Rarely now does a player get made coach halfway through a season and is successful because he doesn't know how to coach.) Because I did Bob Clarke that favour, it was a piece of cake. We almost made the playoffs, but I got fired for being too close to the players, even though we had a winning record. No job again.\n\nGot a call from Bob Clarke. He and some rich guys bought the Rochester Americans. They asked if I would like to coach. I said, \"Yes, and I'll be GM.\" We got some great players nobody else wanted, made the playoffs, went to the league championship for the second year in a row and got beat out in the first round.\n\nWhen I was feeling sorry for myself I got a call from my friend Harry Sinden, GM of the Bruins, asking me to go to Chicago. I had known Harry before. We played against each other, became friends. When they fired Bep Guidolin as coach, Harry asked me to coach the Bruins. If Rochester hadn't missed the playoffs, I would not have been asked to join the Bruins. I was successful with Boston: four first place finishes and the finals twice, and the semifinals. Got fired again for being too close to the players. Then I went to Colorado, a great bunch of guys, but no goalie in Hardy \u00c5str\u00f6m. Fired for being too close to the players again.\n\nThat's when I was asked to do _Hockey Night in Canada_. The executive producer (the boss), Ralph Mellanby liked me and protected me and here I am, thirty-four years later, still on _Hockey Night in Canada_. As I look back on all of my firings and misfortune, I know the Lord was helping me through. I did not realize it at the time. God works in mysterious ways. He's a wonder to behold.\n\n# **AFGHANISTAN**\n\nRON AND I HAD JUST FINISHED a game for _Hockey Night in Canada_. We had our usual after-game bucket of beer, six apiece, on ice. We started this routine when we travelled a lot in Canada. When we would go to the bar after a game it would end up a big party, so we decided the best way to keep it simple was to get the beer in the afternoon and put it on ice in Ron's room. Popcorn is Ron's favourite, and peanuts for me, and watch a ballgame or hockey game and discuss the world\u2014sometimes heated, mostly fun times. \"Munchies Time,\" we called it.\n\nWhen things go wrong on the set of _Hockey Night in Canada_ , and it happens a lot, when I would be in a tantrum, Ron would hold up six fingers, meaning six cold ones waiting in the room, and I would say, \"You're right,\" and calm down.\n\nThese meetings, like I said, were usually fun, but sometimes they would get so heated I would leave the room ticked off, but in the morning we would meet. It would be awkward for a bit. After a while, we were back to normal. Except one time. The situation was this: Ron and I would sit up in the top row of the arena during the morning skate or practice. At one time, Ron used to hang around down on the bench, close to the players, until I straightened him out. I told him that media who hang around the dressing room or bench are considered floaters by the players, so from that day on we sat at the top of the building.\n\nSo one day, this media guy comes up to the top to say hello and Ron says, \"Tell us what happened on that goal.\" And the guy can hardly wait to tell us and let me know how smart he is and how dumb we are. I say to Ron, \"You dumbhead, don't you ever ask a reporter or a guy on another TV station about a play we are supposed to know. We are the experts. Do you know what he is telling his colleagues? That he had to straighten out the _HNIC_ guys\u2014they didn't know that play.\" We go at it that night and we meet for breakfast. We get into the argument again; we go and go and we go so long that when we get to the buffet, it's closed.\n\nHe has learned to not talk about me to reporters.\n\nI remember I said something about a Canadian player for Montreal on _HNIC_ and it caused trouble. We were walking back from the Forum, and Ron said he helped me by saying to a reporter that I didn't mean what I said and was sorry I said it. I said, \"Are you nuts? It will be thought that I feel bad about what I said [I didn't] and there will be a headline that I apologize.\" Sure enough, the next day's headline: CHERRY APOLOGIZES. But he has learned over the years that we are all fair game and we have to watch our backs, and I must admit I look forward to our sessions.\n\nSo Ron and I had just finished the game in Vancouver and we were into our third beer and I was feeling very uneasy. The reason was we had just done a tribute to one of our \"fallen soldiers\" in Afghanistan. The soldier looked about eighteen years old. I know I get letters from the moms and dads thanking me for paying tribute to their sons, and when we are in airports and we meet the soldiers just back from the war, they thank us for recognizing them. The ones that get me are the ones that are going back to Afghanistan and say they can hardly wait to get back to the war. When I ask, \"Why would you want to go back to that place?\" they always say, \"We are a team and they are our buddies.\"\n\nI know I get ripped by some for glorifying war and they say _HNIC_ is for hockey, not for a political statement, but mostly the media goes along with it. I always have that uneasy feeling. I feel sometimes like a phony, but I am being thanked and these beautiful, brave guys and gals are getting killed. It all came to a climax, as they say, that night in the hotel with Ron. I made a decision there and then that I had to go to Afghanistan that year. If I'm going to talk the talk, then I have to walk the walk. I was going to go that year, and I was going to go at Christmas. Ron says, \"Be careful, Grapes. Are you sure it's not the pops talking?\" I say, \"Could be, but the decision is made and I'm going. I have to go.\"\n\nI'm lying in bed that night, thinking about my decision, and again I get that uneasy feeling. Am I being a hot dog for attention again? Am I going at Christmas to be a hero? Sometimes you can't win. There is no turning back now.\n\nLuba says one day, \"You know you are going to Afghanistan in two weeks.\"\n\n\"You're kidding me\u2014in two weeks?\" says I. I'm one of those dumb guys who makes a commitment and thinks the time is never coming.\n\nI go to the doctor for a checkup, some shots, but first of all I take a combat jacket to my tailor to have it custom-fitted. Vanity, vanity, thy name is Don.\n\nAs the day approached, I'm thinking, \"Am I really going to Afghanistan?\" I know Ron was secretly going, too, but I found out later he couldn't go. His dad had become ill.\n\nThe day came to depart. As I drove to the airport with my wife, Luba, it was very quiet. She was not too happy with me flying to Afghanistan\u2014I think, deep down but never said, that I was too old to fly to Afghanistan. Imagine!\n\nAs I look back now as I write this, I guess I was the oldest guy, but you must realize, I think I'm thirty-two years old. I've always thought that. Why thirty-two? I don't know. I must have been at my toughest.\n\nAs I'm checking my bags, it all seems unreal. An Air Canada rep meets me and says my party is in the Air Canada room. My party? I didn't know I had a party to go to! I go to the Air Canada room, and there I meet my party: Jimmy Mac, a goalie from Winnipeg and a Canadian and entertainer who never stops with the Rodney Dangerfield routine, and a French singer, Dany St-Arnaud.\n\nJimmy Mac goes into his Rodney act and says, \"Hey Don, do you know that the troops in Afghanistan right now are the Van Doos? You won't have to worry about the Taliban. Watch out for the Van Doos.\"\n\nThe Van Doos are the Royal 22nd, the famous French regiment. I'm going to entertain guys who won't even know what I'm saying. I say to Jimmy, \"Jimmy, these guys won't even know what I'm saying.\"\n\n\"That's okay, Don, nobody understands you in any language.\"\n\nWe board the plane. Luckily, I'm beside Jimmy Mac, who is an okay guy, and as I look out the window, I say to myself, \"How did I get myself into this?\" and I think back to the session with Ron when he asked, \"Are you sure it's not the pops talking?\"\n\nI wonder. It seemed like a good idea at the time. Remember, time to \"walk the walk.\" I made the decision\u2014no more negative thinking. I'm going, and I'm going to be dynamite and I'm going to enjoy myself. I'm saying this, but wondering, \"Do I mean it?\"\n\nJimmy Mac is so pumped and never shuts up. I can see he likes to give people shots. He gives some to me. \"Grapes, I think it's great you're going to Afghanistan when the French troops are there. It's like you're extending a friendly hand between you and the French.\"\n\nI can tell you what I answered, but it's physically impossible.\n\nThe French singer doesn't say much, but just smiles. I know he's going to be the hit.\n\nIt seems we are flying forever, and I can look around and everybody looks so comfortable in their casual wear. Why do I have to be so vain that I must travel in a suit, shirt and tie? Ron and I are the only ones in the media who travel in suits and ties. Much to our regret, when the finals run into late June and we're in the heat from places like L.A. and Tampa, the shirt looks uncomfortable\u2014and they are, but you are who you are.\n\nWe finally land in Germany, and we are met by some reps and then take off for Doha, Qatar. We land and are driven to our hotel. Checking in, I am invited to the hotel bar by a soldier, retired General Laurie Hawn. Never one to turn down a request for a beer, I agree.\n\nI must say it's a great bar\u2014English music and all. Ten soldiers all around a big table. Well, the suds are flowing and they order chicken wings and ribs, piled high in grease. I never eat chicken wings, even though I used to go to the Anchor Bar in Buffalo with the Bruins. It was famous for inventing Buffalo wings. That's what they called them in the seventies. They were invented by a woman everybody called Maw.\n\nAfter the game, the Bruins would plow into the wings, but I've never eaten greasy stuff, so I just drink beer. I'll never forget Maw bringing these wings to the players. They would eat there often. She loved the Bruins.\n\nThe wings at this hotel were extra greasy. I hadn't eaten all day, so I dove into the pile with two hands. We are having a grand time. Hey, this is pretty good so far. Everybody says good night. A good night was had by all.\n\nBack to my room after a ton of greasy chicken wings. Why, oh why, did I eat so many of those greasy wings? And of course, the pops were flowing.\n\nI woke up at 4 A.M. and my big toe is killing me and it's all red and swollen. I can hardly walk. I get back into bed and I can't stand the sheets touching my toe. I'm in agony. It feels like it's broken. I sit up all night with a can of cold pop on the toe. It does no good.\n\nBig day ahead, such as it is. I get dressed and can hardly put my shoe on. Down to the lobby where I'm asked why I'm limping. I say I stubbed my toe. They buy it. The plane can't go\u2014mechanical difficulty. Better on the ground than in the air. They decide to go to a museum. Can't show weakness, so off we go. The museum is mostly some vases and stuff. They decide to go shopping. Got to go with them, can't be chicken. I limp along. I notice the natives never look you in the eye and walk head-down. These guys are supposed to be friendly, but you could almost feel that hate. Big shots walking around in our house was the attitude. My toe is throbbing.\n\nI walk along with the soldiers, and they open up to me. We have some politicians along and they have nothin' but contempt for them. One soldier says a left-wing politician got headlines when they sent stuff to the people once, including saws. The soldier says, \"You know what they do with them? They cut them into pieces and use them in the bombs that they have planted along the highway. Cover the saw and bomb with earth. The saw is a perfect cover for the bomb.\" The road to hell is paved with good intentions.\n\nFinally, the bus comes to take us to the hotel. My toe is worse\u2014no good. I am starting to think it's not a broken toe, but something worse. Can't sleep.\n\nNow, I'm ashamed to tell you, it's gout. It can't be, I think. I've seen movies. It's the old lords, with feet up on a footstool, that get it. Everybody laughs at their pain. What happened? The gout came after eating all those greasy wings and beer. I find out later it's the worst combo for gout. When I get home, I looked up \"gout\" and the word \"excruciating\" comes up. I can believe that.\n\nWe are in the lobby the next morning, ready to go, but we're told to wait. While we are in the lobby, a sad thing happens. The owner of the hotel has put the most expensive golf cart in the world in the lobby. It has gold on it. So everybody takes turns sitting in the cart to get their picture taken. While I sit in the cart, this guy all sharped up comes in the lobby with this gorgeous woman. He is hurrying her to the elevator. She is a \"woman of the night,\" as they say. As she passes me, she says to the guy, \"Wait a minute.\" She speaks perfect English. She asks if it is all right to sit with me in the cart.\n\n\"Be my guest,\" says I. She sits beside me, and the guy she is with is fuming mad. She turns to me. I look into her eyes. They are the saddest eyes I have ever seen. They are full of pain. Looks like she's on drugs.\n\nShe asks me if she can kiss me. She gives me a peck on the cheek and is gone. I'll never forget the hurt and pain in those eyes. I look around, and everybody has been watching. One soldier hollers, \"Hey Grapes, congratulations. You just kissed the United Nations!\" Ah well, you can't win them all.\n\nWe get the news: no plane. To tell you the truth, I'm not too unhappy because I don't know if I can walk. Things start to happen when Peter MacKay, the minister of defence, arrives. We obtain a gigantic Hercules. It is so big it can handle about four tanks inside. We get in the plane. The ceiling has got to be forty feet high and thirty wide, little chairs along the side. The soldiers are great, give us boxed lunches, do everything to make us comfortable. We take off to the compound. You can't hear a thing, it's so loud inside. We finally land, and a truck of soldiers meets us and away we go. One soldier, when asked if any of the Taliban ever attack, says they're around. \"In fact,\" he says, \"last week, they sent some rockets into the compound.\" I think he was just having some fun with us.\n\nJimmy Mac and I are being shown around. When we are shown the hockey rink where they play ball hockey, two soldiers from the stands holler \"Hey\" and come to see us and say hello. I say, \"What part of Canada are you from?\" They're U.S. soldiers from Rochester, New York. They were glad to see somebody from home.\n\nWe are shown a little building that has a bed and a shower for each of us. They're all right by me.\n\nPeter, when I told him I would go to Afghanistan, he kept saying, \"You'll be safe behind the wire.\"\n\nI said, \"Are you going outside the wire?\"\n\nPeter said yes.\n\nI said, \"Okay, then, let's go.\"\n\nI must explain something here. We are going to try to cram three days of visits into one and visit five spots.\n\nI try to sleep with my toe. Impossible because the sheets actually cause pain.\n\nNext day, we are driven to the Chinook helicopters. Let me explain about the Chinook helicopter. It's the one you see in the commercials where it opens up at the back and the paratroopers jump out. They have two big machine guns at the front. This is getting severe. They give us combat jackets, helmets. Have to have my custom shirt and jacket underneath. Why the heavy jacket? I'm told it will be freezing tonight.\n\nWe land at the first post, tanks all around and artillery. We meet the Van Doos. The first thing I notice is how young they are and how fit they are. They remind me of hockey players, primed and ready to go for anything. We stand in a circle, and I have to admit, it is a little strained to start, but when Chief of Defence Staff General Walter Natynczyk shows up, he breaks the ice. We take pictures and sign autographs and everything is okay. We say our goodbyes and go on to the next post.\n\nFor some reason, I was alone on this visit, but at the next camp, Jimmy Mac and the French singer, Dany St-Arnaud, are with me. I do my best. Jimmy Mac is terrific, but Dany, the French guy, was the hit as expected. He sang French songs and they sang along. Oh well, I did my best. Goodbye again, into the helicopter again.\n\nThis is where we saw our first American group, and it was kind of sad. They were a forlorn lot. You could see their attitude was different than the Van Doos'. One of the American guys said to Jimmy Mac, \"We've been here for nine months and we can hardly wait to get out.\" Who could blame them? They were a sullen group. I did my best again, but not very good. Jimmy Mac did good, but again Dany, the French singer, had them eating out of his hand, especially when he sang songs about home and America. They didn't know me and couldn't have cared less. Goodbye and on to the next post, where Peter MacKay and General Natauczyk serve Christmas dinner to the troops. Kind of sad to hear them singing carols from home.\n\nThe next post is the biggest, most spread out and has the biggest artillery piece you have ever seen. Evidently, it can fire and hit miles away. They ask if I would fire the piece at the Taliban. I have to lay my jacket down to walk to the piece. My toe is killing me. I tell the soldier who is helping me, \"Watch my jacket, please.\" I am not thrilled at this long walk, but this is what I'm here for. I reach the artillery piece, and it actually scares you, so I sign the shell\u2014something like \"Here's looking at you\" or something stupid like that. I pull the cord. It is so loud, the ground shakes. On the way back, it's starting to get dark and I'm freezing. I get back\u2014no jacket. Can't complain, can't be a sissy.\n\nI got to mention Jimmy Mac here. I would not have been able to continue without Jimmy Mac. It was murder to continue to walk on the stones that were at the posts. They were rocks, not pebbles. Just big enough to make walking miserable. Once in a while, he could get a Jeep to give me a hand if we had a long way to go.\n\nAt one of the posts, it's a long way to go to the toilet, so Jimmy has a U.S. marine drive me. I say I would like to go to the washroom. He drives me along the route, and finally we come to a building that says \"Showers.\" I figure there's a washroom in the building. I go in\u2014no washroom. I come out and say, \"Where are the toilets?\"\n\nHe says, \"You wanted to go to the washroom. This is where we wash. If you wanted a toilet, you should have asked for a toilet, sir.\"\n\nHe must have thought Canadians are strange people, having to go pee and asking for a washroom. Come to think of it, he's right, but Jimmy is always there to help me with my big toe, which he thinks is broken. I'm too embarrassed to tell him I have the gout. Jimmy and the French guy, Dany, are the hits again.\n\nI told Peter if he was going outside the wire, so was I. We go outside the wire for sure when we go far out into the desert to visit the Canadian Special Operations Task Force. They are a secret operation that includes snipers. When the regular army needs a special force to go and straighten things out, no matter how bad the situation is, these are the guys. No mercy asked and no mercy shown.\n\nThe commandant gives us a lecture on the history of the very building we are going to. It was occupied by the Taliban, and they stay there as if to say, \"Take us if you can.\" We are told to take no pictures and about the tight security. We walk into a room of about 150 people, and they start yelling my name\u2014standing O. Dany and Jimmy Mac give it a try, but it's my time to shine. Jimmy and Dany have shone all day, but it's mine now. I answer questions, tell jokes. I'm getting a little punchy by now, but loving it. After the pictures start, I look at the commandant. He just shrugs as if to say, \"Go ahead.\" They take so many pictures the camera overheats and has to rest to cool off. I was having my picture taken with some young guys and they said, \"Grapes, we're the snipers.\" I have to tell you, folks, I started to tear up just being around those guys.\n\nI'm asked if I would drop the ball at a ball hockey championship game. They have waited till we came for the game. We watch the game from the roof, looking into the courtyard below. It's a barn burner of a game. They're playing for keeps. The building is built like a fort. I ask a young soldier about the Taliban and he says, \"Don, see that ridge over there about two or three miles? The Taliban are just over there.\" He sees the look on my face. He says, \"Don't worry, they won't bother us.\" Evidently, when the special forces go out at night, they don't go looking for prisoners.\n\nI tell you, it's a strange feeling to stand on the top of a building with a beautiful moon shining down, watching a ball hockey game, knowing the Taliban are just over the ridge. As I gaze at the full moon and look around at these proud, brave, utterly fearless warriors, I feel proud to be Canadian. I wish every Canadian could experience that feeling. I will never forget that night.\n\n(The special forces later sent me a framed picture of them with their special knife and a plaque that says, \"From Warriors to a warrior. Thanks for your support.\" It is in a place of honour in our house. I asked if I could show it on TV. They said don't even mention it.)\n\nI suddenly grow sad as I look around at the young soldiers, knowing that some of them won't come back, and I thank the Lord for the opportunity of meeting them. All good things must come to an end, and it is an emotional parting. It is very silent on the drive back to the helicopter. We all have our own thoughts.\n\nBack in the giant helicopter, freezing, no windows and with no warm jacket and my thin dress shirt, I'm just asking for it. The guys on the machine guns open up. I ask the soldier next to me, \"What gives?\" He says the helicopter in front of us had gunshots fired at it, so these guys are firing to keep the Taliban heads down.\n\nI have to tell you that, about this time, with five visits in one day, my toe, the cold, I really didn't care what happened.\n\nI tell ya, I'm looking forward to some sleep, but I learn if we hurry we can catch a plane to Kabul and be on our way home. Back to the little room, quick showers, still dripping wet, packing fast, throwing things in my bag to catch that plane. A knock on the door. The woman driver. \"Need help?\" Now I'm starting to get irritated. Run to the van, get in and sit.\n\nDon: \"What's going on? I thought you said we had to hurry to catch the plane?\"\n\nSoldier: \"We have to wait.\"\n\nDon: \"Whaddaya mean we have to wait? What for?\"\n\nSoldier: \"The French guy to blow-dry his hair.\"\n\nWe miss the plane by ten minutes and have to sit in the airport all night to catch the one in the morning.\n\nWe have another marathon trip back. Jimmy Mac is still doing Rodney somehow. I've lost my beautiful Canadian cufflinks. We land in Washington in the middle of a snowstorm. Planes are delayed to be de-iced. We line up for tickets, we line up for customs. Imagine flying from Afghanistan to the States and we line up for security; all the while, flights are being cancelled. We get on the plane. Half-hour delay being de-iced. We land, say goodbye to good guy Jimmy Mac, limo home. My toe's gout has moved up to my ankle. Haven't slept in thirty hours. Outside the door, Luba has a little umbrella taped to a can of beer and a sign that says WELCOME HOME. She's walking through the door and asks, \"How was it?\"\n\nPiece of cake!\n\nNow, my buddy Jimmy Mac wanted to get his two cents in and tell you about our trip, too. Here's his version of the story:\n\nA funny thing happened on the way to... Afghanistan? What did I just get myself into? I had been doing standup comedy for over twenty years at this point and just finished eight years of live television when I received a call from my agent in Toronto. \"Hey Jimmy Mac, how's it going?\" To which I replied something along the lines of, \"Okay. Why, what's up?\"\n\nThe conversation went on about a booking around Christmas 2010, and that if I was tired of the cold Winnipeg winter and wanted the promise of warmer weather, along with a mountain view and sand surrounding me, then \"Have I got a gig for you!\" I mean, honestly, who in their right mind would want to go into a war zone and perform for the Canadian troops when they could be safe at home with family over Christmas? A \"true Canadian,\" that's who!\n\nI am too old to be in the military, and my hockey days are long past, so the only way to be on the \"Team Canada\" roster at this point and do something for the country I love\u2014and the men and women who sacrifice their lives for my right to say no to this trip\u2014was to accept the offer. So I did! I didn't even hesitate; I said, \"Book it!\" I still remember that day: my body was buzzing because I'd just accepted a showbiz gig like my comedic heroes of the past\u2014Abbott and Costello, Bob Hope and many others who actually performed for the troops. However, if you know me, only one thing scared me more than the war: _I hate to fly!_ I also had to break this news to my family in a way that made them understand why I wouldn't be available for Christmas dinner.\n\nHow do you break such news? It went basically like this: \"So I have a gig this Christmas, and I will be performing for the troops... in Afghanistan. Now, before anyone decides to say anything, this is the end of the discussion. I am going. It is something I need to do for my country, and I can't think of the location being Afghanistan or what _might_ happen to me over there in a war zone.\"\n\nNow, the government was pretty tight-lipped on the details of the trip and who was going to be on the Team Canada roster, though I did manage to get one name that had just been confirmed: Don Cherry.\n\nSo let me get this right. I am a Canadian male, grew up playing hockey (still play), and live for Saturday night's _Hockey Night in Canada_ where _nobody_ is allowed to speak during \"Coach's Corner.\" And hockey icon Don Cherry is on the trip? Well, thank you, Canada, for the Christmas gift of a lifetime! I get to perform on the same bill as one of my hockey idols!\n\nI have to say that, working on television, I've interviewed many famous people along the way, and the only time I was ever nervous around a guest was interviewing Ken Dryden, who was my hero as a kid and the reason I became a goalie. I only had thirty minutes with Dryden, but soon I would share seven days with Don Cherry!\n\nI have never kept a diary of a trip before, but this was different! This was going to be a trip I did not want to forget, and that's why I can recollect for you my rendition of this adventure.\n\nI was sitting in the airport VIP lounge, waiting for the rest of Team Canada and our departure to Germany, Doha and eventually Kandahar, when all of a sudden I saw a figure appear in a doorway, dressed to the nines in a black overcoat. There he was, my hockey idol and tell-it-like-it-is guy, _Don Cherry!_ Remember those nerves I said I had when I met Ken Dryden? Double it! We all were introduced to everyone, and Don was friendly to everyone and signed whatever we had with us, and then we were all whisked away to a private area for a briefing on the trip. Don started telling a few stories and then turned his attention to me as I was sitting across from him. \"Look at those tattoos!\" he said. Long story short, I have a hockey sleeve of tattoos, which includes Dryden's masks, Peter Puck and one particular tattoo that was hidden from view on my bicep: a tribute to Don Cherry. It is a maple leaf, with a high collar and black-and-yellow checked tie (the Bruins' colours).\n\nWe took a few pictures, and I posted mine on Facebook. During our prep, as the officer was explaining the trip agenda, I piped up and asked why we were not now making it into Kabul? Before the officer could answer, Don hit me in the arm and yelled, \"Don't argue!\" in a voice he uses on Ron MacLean on a regular basis. We had a good laugh about it. This yell was to become a regular occurrence over the next few days, and become the cornerstone of a friendship that continues to this day. I will explain later, but for now, off to Germany.\n\nA few plane issues caused delays before the next fifteen-hour flight into Doha, Qatar, but that was no big deal. My seat mate (lucky me) was Don. We talked hockey, he gave me some coaching tips, and since I also coach goalies, he gave me a few helpful drills he always used. And we talked some comedy. It was the regular talk a celebrity would have with a regular Average Joe like me, but that was about to change.\n\nHe nudged me after a few minutes of quiet and said, \"I don't know why we're here. You know we are performing for the French troops! You're from Winnipeg, out west, and I get in trouble with Quebec all the time! What if they don't like me?\" Now he was being serious. He loves our troops and was sincere in his hopes the guys would like him. There was no ill meaning in his concern; he just really wanted the Canadian kids to be happy he was the one visiting.\n\nIt was amazing to get to know Don Cherry off-camera, away from the spotlight for a while, where\u2014believe it or not\u2014he has a sensitive side. He may kill me for telling you that!\n\nOnce we landed in Doha, we were met by Canadian military personnel and taken to our hotel. First thing on the agenda after checking in was a beer with Don and the rest of our Canadian crew. Did I say beer? Just one? Let it be known I prefer to have a rum; however, when you don't order the first round, you get what is delivered to the table. War stories, hockey stories and jokes\u2014and now into pint number four. Wrap up the night and head back to the hotel. This was when Don noticed I hadn't finished my beverage! Any opportunity to yell at me, he took. \"What are you doing? Finish that! Never leave one behind. You're Canadian and from Winnipeg! What would your dad say?\" Beer four downed. Headache to follow.\n\nThe next day in Doha is where Don learned we were more similar than he realized. Neither of us liked the museum the group was taken to, and when a political question came up regarding Quebec separation, I spoke my mind, which also seems to get me in some trouble\u2014much like a certain \"Coach's Corner\" star! Of course, this added to our growing friendship. Upon leaving the museum (thank God), Don delighted us with a few more stories, which included one about how watching \"Coach's Corner\" actually saved a life.\n\nDuring an ice storm, a member of an NHL player's family was about to go outside on their farm and check on the livestock. Upon hearing the theme music to \"Coach's Corner,\" he decided to watch Don instead, and at that particular time, the barn collapsed. Because the farmer stayed in the house, no one was killed. A few chuckles followed the story, and I decided to add my two cents: \"There you have it, folks, right from the horse's mouth: watching 'Coach's Corner' can save your life, and that's no bull!\" Big laugh, and Don loved it. He said, \"You're just like MacLean! We should do a routine together out here with the troops, a team act.\"\n\nThat is where our rendition of \"Kandahar Corner\" began, and we wrote up a short script for the troops. It wasn't an Oscar winner by any means, judging from the actual notes I still have, but it was perfect for two hockey-crazy entertainers that were not in a rendition of _Bullets Over Broadway_ , but more like _Bullets Over Afghanistan_. An example would be something like me starting the act with: \"Hey Don, what are the Canadiens' chances of winning the Stanley Cup this season?\" Don would go into his thoughts and expert opinions and I would end it with something like, \"So the Habs have a chance if the Price is right.\" This went on for six or seven questions that led to different stories and lasted about ten to fifteen minutes, followed by autographs and pictures with the troops.\n\nIt was later, at the hotel, that I started to notice two things about Don. He was starting to cough and lose his voice and he was limping. We had all been walking all day, so it was understandable\u2014maybe a blister, or maybe he was just tired. We all made off to our rooms for a few hours of sleep before we were to meet at 10 A.M. in the hotel lobby and head to Kandahar Air Field (KAF). Don was still limping the next morning, so I asked him if he was okay and he told me he hurt his big toe before our trip. Typical tough guy, he shrugged it off and asked me to email his family and let them know we were doing just fine. I have seen this stubborn act, as many of us probably have with our own fathers. He took a few Cold-FX capsules and said he would be fine. No need for a doctor!\n\nThe trip thus far had been a thrill, to say the least, with the laughs and the tourists we were trying to be, but the fact you could be in some type of danger doesn't really hit you until someone from the military sits you down, hands you a consent form that absolves them of liability for any injury you might incur during your trip, and asks for your blood type\u2014\"just in case!\" There is a reason your extended health plan doesn't cover certain destinations, and believe it or not, Kandahar is one of them!\n\n(To this day, while crossing the border into the USA, there are always odd looks and questions about why I have an Afghanistan visa and stamp in my passport.)\n\nI took a deep breath, signed the release forms and we boarded the C-17 and headed for Kandahar. There are no beverage services on that flight, I'll tell you! The plane was a giant tin can with an engine and enough room to throw a football around\u2014which we did.\n\nWe had two hours before our first show. I followed the three-minute shower rule (in a war zone, you've got to save water) and get ready for \"Kandahar Corner.\" The troops loved the routine and Don signed autographs for at least an hour afterwards. I noticed the limping again, but this time I didn't say anything to Don; I called over an officer and asked for someone to take a look at him. I have been around enough celebrities in twenty years to know when someone is getting tired. I had them get him water and into a theatre where he could sit and finish the autograph session. I asked if they could wrap it up soon so he could get the medic to look at his foot. A small break in the action left me alone with Don for a minute, and when I asked him if he was ready to head back to the barracks, he replied, \"As long as everyone has come through here first.\" Don was as tough as nails, just as he portrays, and he'd do anything for the Canadian troops.\n\nFinally, on our way back to our lodging, he gave me the usual nudge and said, \"Great job tonight. We killed 'em tonight, Jimmy-Boy!\" I became Jimmy-Boy, a nickname he gives to the players\u2014the ones I have watched on television for years\u2014that he respects. When someone gives you a nickname in a dressing room, it's a form of acceptance, and when your idol gives you a nickname instead of just calling your given name, well, that is a gift. For some reason, I snapped into my Rodney Dangerfield impression: \"Are you kidding me? No respect at all! I was kidnapped by the Taliban and the military said keep him!\" Don had a great laugh with that, and whenever I noticed he was getting a little rundown or tired through the rest of the trip, I would throw a few Dangerfield lines at him, which always got a laugh and seemed to energize him. I enjoyed the fact that I could get a rise out of him when we all were in a war zone with the reality of an attack always near the back of our thoughts.\n\nWe headed back to Don's room, where we enjoyed a few \"near beers\" and some snacks that his wife, Luba, had packed for the trip, finished a cheese pizza from the mess hall and joked about the \"warning siren\" in case of a bomb threat. The military gives you instructions on the procedure to follow if there is a missile attack, but in true Cherry fashion, he said, \"If you hear it, just duck!\" A knock at the door, and the military medic was there now to look at his foot. I left for my own room, knowing he was finally getting attended to and that the limping would soon be addressed. We had a big Christmas day ahead of us, visiting five Forward Operating Bases (FOB) throughout Afghanistan, and I needed my new comedy partner in tip-top shape!\n\nNo sleep\u2014awake all night, listening to the sounds of planes, helicopters and the odd sound of what might have been gunfire. Then it was 6 A.M. and we had a busy day ahead of us, visiting the troops. It's no use showering because your skin always feels like it's covered in baby powder. The soldiers call it \"moon dust,\" and it looks like fog when it's in the air, but it's a fine dust you constantly inhale and hack out. Your teeth feel like they're covered in sand, and five minutes after a shower, you're dirty again. We throw on body armour, helmets and ballistic glasses. Felt like about fifty pounds, but all of it was necessary in case a missile hit our Chinook helicopter. Yeah, that's what they told us, and this was when it finally set in that we were heading outside the \"wire\" and into enemy territory.\n\nDon was in prime form, yelling at me already. This time, it was about the seat belt in the helicopter that I hadn't buckled up. \"Put it on!\" he said. \"What difference would it make if we were hit?\" I argued. He barked back, \"Don't be stupid. Just put it on!\" So I did. Now, when I say Don yelled at me, it wasn't with a malicious intent. It was a form of affection, the kind you find in dressing-room talk or if you have ever watched an Abbott and Costello routine (especially the coffee shop skit). It was more in the style of a straight man, or the way he gets with Ron MacLean. Ours was a comedy act that was becoming quite entertaining to the troops who were travelling with us to the bases.\n\nOne officer asked me, \"How long have you two known each other? Because it seems like you've been friends for twenty years.\" That was a huge compliment, as this was only day four of a journey never to be forgotten. We landed at our first FOB, and it was a football field-length walk to where the troops were waiting. Not a normal football field, but one with an uneven surface of rocks and sand. Not great for a healthy person, let alone someone with a broken toe, who might be the celebrity on the trip, but was also someone\u2014nobody seemed to realize\u2014not in his twenties, either! Don is in remarkable shape, but add those extra fifty pounds of body armour, a busted toe and a cold on top of it, and I don't care who you are, this would be exhausting for anyone! At one point, away from the rest of the group, Don referred to his cold as \"sucking swamp water.\" I have never sucked swamp water, but you get a good visual of how this might be difficult.\n\nAfter the second FOB, I asked the first officer we met at the landing of base #3 if there was any way they could give Don a lift into the stage area where we were to perform. I told him the rest of us could walk these distances, but we needed to make sure Don got to and from the helicopter the same way. Don had no idea I had requested this for him. When the Gator utility vehicle the soldier was driving dropped him off, he turned to me and said, \"See that, Jimmy-Boy? They got me a Kandahar Limo!\" To which I barked back, in my Dangerfield voice, \"Yeah, I ordered it for you. Now show me a little respect.\" Once again, not to be outdone, he yelled, \"Well, that's the first thing you've done right this trip. Now get on the stage!\" More laughs from around the group and we went into our \"Kandahar Corner\" routine. After the usual autograph session, we were invited for Christmas dinner with the troops. We actually served dinner for the troops, and after we were finished, Don grabbed a plate (picture Abbott and Costello again):\n\n\"Wanna eat?\"\n\nMe: No.\n\nDon: Let's eat!\n\nMe: No, I'm not hungry.\n\nDon: Get a plate!\n\nMe: I don't want anything!\n\nDon: You're hungry, now get a plate!\n\nMe: No, really.\n\nDon: Get a meat pie and sit down!\n\nMe: But I don't\u2014\n\nDon: Sit down and eat it!\n\nAt this point, the person serving was so confused and so busy laughing that he put the meat pie on my plate and we sat down. After I finished my dinner, Don looked at me and said, \"I told you you were hungry!\" We both had a laugh.\n\nTime to get back to the Gator, get suited up in armour and off to base #4. This particular show was for both the Canadian and U.S. troops. Don said at one point before the show that he worried the American guys wouldn't necessarily recognize him and I would have to bail us out with comedy just in case. No problem, I thought. But he was about to turn that into a twist I wasn't expecting. He loved trying to get me into a situation\u2014a prank of some kind for us to laugh about later. Our comedy routine together wasn't going as well as expected with the U.S. troops (Canadians loved it), so in mid-story he turned to me and yelled, \"Give them some Rodney [Dangerfield], will ya! Do something here!\" So I did a few lines and went back to the script. He stopped me and said, \"Keep going!\" So I did, and eventually told him so go sit down as I continued for another twenty minutes of my own standup act. Don later laughed about how he duped me into taking over and then thanked me for bailing the \"team\" out.\n\nOne more FOB to visit, and at this point we were all feeling the day catching up to us. This was a visit to the \"special\" guys, and I'm not even sure I am allowed to write about them, but let's just say they stayed in a base that was named after the home where Elvis Presley lived. Hope that hint is enough for anyone reading this. This was our last show on the tour, and by far the best of the bunch. I was asked if I could do Don Cherry's intro for him, and of course I said sure. What I didn't know was the officer introducing _me_ had read my bio online and knew I did a Don Cherry impression! He asked me to introduce Don as only Don would introduce himself. Great\u2014now I had to make sure this was a good one, and my voice had better be right on, or for sure I would get verbally berated by the _real_ person I was trying to impersonate! \"I tell ya, this next guest here, he's a tough guy, not afraid of the Taliban, no visor out here in the sandbox, and great guy and teammate... Don Cherry!\" And with a big round of applause from the troops and Don laughing, he says, \"Pretty good there, Jimmy-Boy. If I lose my voice here, you can take over.\" \"Pretty good\" in Don Cherry's vocabulary is the gold star on the school paper you aren't sure of before the teacher marks the exam. We finished this show, and those guys and girls loved him. I think it was the best one of the trip, at least for Don, as they couldn't get enough of his stories.\n\nAfter the show, Don was again into the autograph session. By this point in the trip, everyone who had questions or concerns about Don was directed to me. I made sure he had his water and was somewhere he could sit down while taking pictures and chatting with each and every individual, so he didn't have to stand with the foot that was visibly paining him by now. Finally done, one last helicopter ride back to KAF and a sigh of relief that we made it back without incident.\n\nOnce again, I made sure Don got some medical attention and had to make sure he iced his foot every hour. By the end of this Christmas day, he said to me, \"I don't know if I would have made it if it wasn't for you making sure I was okay, Jimmy-Boy. Thanks.\" I heard later from defence minister Peter MacKay that Don said, \"He is the kind of guy I'd go to war with any day. That's a true Canadian right there.\"\n\nIn reality, looking back over that trip, at the beginning he was Don Cherry the celebrity, the _Hockey Night in Canada_ icon I had watched for years. But at some point during the trip, he became my teammate, a teammate who was hurt, and if you watch any athlete when hurt, other teammates will help them off the field or the ice to get him to safety.\n\nGoing back to his earlier warning to \"never leave one behind,\" it rings true. It was actually meant in the sense of finishing a beer, but it took on a more important meaning as a lesson about being a teammate: stick up for your players when they wear the same crest\u2014or, in this case, the Canadian flag. So I didn't leave one behind. Now let me say this: even though Don had an injury, he was still the \"hockey player\" in the situation. He was just like his beloved tough guys and grinders he praises on television every week. At seventy-six years of age, with a busted toe, a cold that felt like \"sucking swamp water,\" posing for over three thousand pictures and signing autographs in one day and walking over rocks and sand for over ten hours, not once did he ever complain or act tired or indifferent to any of the soldiers. He loved them all and made sure everyone was happy.\n\nWe had our own shorthand by the end, where I knew when to make sure he took a five-minute break from signing or taking pictures, but he always continued until the last piece of paper was signed or photo snapped. A Canadian hero to anyone who met him on that visit.\n\nOur adventure wasn't quite over, as we still had to get home. I made sure everything was done for us, from checking our luggage to making sure he iced his foot before our flights. We boarded the plane for our long flight to Washington and chatted until the plane was safely in the air. And then he turned to me and said, \"Well, we made it, Jimmy-Boy.\" And before I could respond, he hit me in the arm, said, \"Now let's get some rest, so shut up!\" and laughed as he reclined his seat.\n\nI am not sure how many hours passed in what I believe was the deepest sleep I have ever had, but we awoke in Washington. We were guided to where our luggage was to be identified and I picked up mine right away. I noticed Don was chatting with the customer service person and trying to explain he didn't have his luggage claim tags. Why? Well, because I checked us in and didn't give them to him. So, trying to continue our string of one-upping each other, I wandered over to listen to their verbal exchange. I guess I could've stepped in to help, but I was having too much fun listening to the ordeal.\n\nFinally, I said, \"Why can't you get your bags?\"\n\nHe looked at me and said, \"You booked it, Jimmy-Boy! Where's our tags?\"\n\n\"You didn't ask me for them!\" I yelled, and we were all laughing at this point because I'd finally got him back at his own game.\n\nLater, he told me of a story about travelling with Ron MacLean, and how he took Ron's passport out of his pocket. All good hockeystyle pranks. Soon, we landed in Toronto, and the Christmas of a lifetime was about to end. We exchanged addresses and phone numbers and he said I could use him as a reference for any media or sports job I was going to apply for. That's an offer I have never taken him up on, although if the right sports position were to cross my path, I would be sure to call in that favour. He gave me the signatures he had promised earlier in the trip for the final piece of the tattoo on my arm. I hadn't asked him again since the first day of the trip; he just remembered and kept his word.\n\nAs we made our way to the exit, he stopped and said, \"Where's your phone?\" I took it out of my jacket, and he said, \"Call your mother and tell her we made it.\" And so I did, and when she got on the phone, he decided to take the opportunity to let her know that \"Mom and Dad raised a good kid.\" He handed me my phone, shook my hand and said, \"You were the best part of the trip, and I don't think I would have made it without you there.\"\n\nAs he walked away, he turned for one last thought, and said, \"Remember: don't argue with your mother, Jimmy-Boy. Keep in touch.\"\n\nI watched him walk away and remembered something he had told me earlier in the trip, about how he doesn't like emotional goodbyes. What guy does? He never looked back, just stopped and signed an autograph for a kid who recognized him.\n\nI'm not sure if I'll ever get a chance to go back to a military zone to perform again, and I know that I could not do what those brave men and women who join the military do for us. But I do know one thing: if, at some point during that trip, we came under some sort of fire or emergency, I would have taken a bullet for Don and would never have left that man behind.\n\n# **SOCHI 2014**\n\n**F EBRUARY 1, 2014.** _Hockey Night in Canada_ , Saturday night. I am meeting with Chris Irwin, head of the Sochi Russian Olympics. I am going to pitch to Chris and Trevor Pilling, head of CBC Sports. The pitch is I would rather stay here in Canada and do the games here the same way I did when the games were in Turin, Italy, where Brian Williams and I sat at a desk in Toronto and watched the games. It was a piece of cake. I fail in my attempt, and I hear through the grapevine that the reason is that Budweiser had sponsored my appearance.\n\n**February 4, 2014**. My son, Tim, drives me to airport. I meet Mark Lee, an announcer from women's hockey. We check in\u2014he says he knows where to go. We go to the lady at the gate, who sends us to another gate. We go to another gate, wrong gate. We are sent on a wild goose chase to different gates by people we can hardly understand. Finally, a good old boy recognizes me and takes me to the original gate: seven gates, fifteen pictures taken, twenty autographs. I'm in a cold sweat. We're back at the original gate Mark and I went to. Look, I realize everybody has to work, but do you not think it would be better for everybody if the people at Pearson Airport had command of the English language and had an interest in helping you? This is very frustrating.\n\nI remember when I was with the Bruins, one of our players, as we entered Toronto was asked at the border, \"How long have you been out of Canada?\" and he answered (ticked off at the attitude), \"Longer than you've been in it.\" Oh boy, did we do some scrambling with that one with the Mounties, but like I said, it is frustrating to be treated like a foreigner in your own country. There, I got that off my chest.\n\n**February 5, 2014**. Eight hours over to Frankfurt, land, get our bags. Four elevators to get to our train to take us to another airport, four more elevators to get to our gate. We have seven hours to kill before our next flight, to Russia. The German customs guy looks at the passport, doesn't even look at me, is talking to another guy and tosses the passport back to me. Very rude, but who cares? We sit for six hours and then line up at the gate. Finally get to the gate. The lady says overweight luggage (it was okay in Canada) costs forty euros. Try to pay with Visa\u2014no. Canadian money\u2014no. U.S. money\u2014no. Only euros. Where the hell can I get euros? Borrow some from CBC people, pay them in Canadian. It is a scam. Everybody gets scammed on the flight, and I hear the Australians got taken for $2,000. Cassie Campbell with _HNIC_ got zonked for $1,800. Welcome to Europe.\n\nGot about another four hours to Sochi. Land, first person I meet is a young Russian solider, looks like a hockey player. Clean and speaks perfect Engish. Hello, welcome. Very friendly. What a difference from Toronto and Germany. The customs lady smiles and says, \"Welcome, have a good day.\" Perfect English. Can't believe we are in Russia. The people are friendly and polite and helpful, and speak English perfect, whereas in my country, Canada, at the airport, I can hardly understand anybody and they couldn't care whether you lived or died. I think I'm going to enjoy myself in Russia.\n\nKathy Broderick, the producer of the games, is there. She's got everybody arranged, van to the hotel. I check in. Ron MacLean is there with some beers and munchies. He takes a picture to send to Joe Warmington of the _Toronto Sun_. He has a front-page story about my eightieth birthday in Russia. Ron was just there to greet me. He has to go back to work all night.\n\n**February 6, 2014**. Met by Warren Lowe from Edmonton, who is going to be my guide through this caper. I tell him we got to get some beer and munchies. He says we got to go to downtown Adler, not Sochi. We go downtown\u2014I'm amazed it feels just like Mississauga. Somehow, I thought it would be different. Nice shopping malls. We get our beer. Only one Canadian beer or American beer: Budweiser. I get two two-fours. Back to the hotel. Opening ceremonies that night. I go to the lovely steamer they have in the hotel. As I come back to the room, the fireworks start. The window at the end of the hall has a window overlooking the fireworks. I open the window and take my little Kodak I bought in a drugstore and take pictures. I wonder how they are going to turn out. (They didn't.)\n\n**February 8, 2014**. Watch the women's team beat the Swiss 5\u20130. No contest. I even think the Canadians kept the score down. Leave at 2:30 A.M. for the Leafs\u2013Hurricanes game for \"Coach's Corner,\" finish the \"Coach's Corner\" and then, at the end of it, do an update of the women's game against the Swiss for that night. Finish it off at 6 A.M., back to the hotel and have three Buds at seven-thirty in the morning.\n\n**February 9, 2014**. Day off. Radio and TV interviews. I look out my window and see a dog across the road. He is friendly to people; nobody pays attention. He is a stray. I remember reading where all strays were rounded up, and I hate to think what happened. I see another dog on my walk. I go and get some meat and bread and cut it up and go looking for him. Can't find him. Still up. I'm told that Christie Blatchford, columnist for the _Globe and Mail_ , is upset too and is feeding them. Steve Simmons of the _Toronto Sun_ , who is a good friend of Christie's, tells me she feeds them. It's true. Christie is known for no prisoners taken in her writing. Tough as nails. One time, I complained about something she had written about me and she told me to go and do something that's physically impossible. Great.\n\nShe's concerned for the dogs, and I think my approach to life is a tough stance, like Christie.\n\nBoth of us own English bull terriers. We are the only ones that seem upset about the dogs. I guess it is the old saying, \"The tougher you are, the softer you are.\" And now Warren tells me somebody complained about the dogs and I haven't seen once since. Damn the insensitive people. I find out later that Ron has been feeding the dogs, too.\n\n**February 10, 2014**. First time at the arena. Beautiful practice rink. All blue, watch Canadian women. We have a big team, whereas the U.S. has a small, quick team. There is no doubt who is going to be in the finals for the gold\u2014Canada and the U.S.\u2014but there is a doubt who is going to win. I have been to the rinks twice, have done at least eight interviews, mostly with the Russian press. Why do I do them? I have fun and it keeps me sharp. One asks me what I didn't expect in Russia. I say I'm surprised at how modern Russia has become. I went downtown to get some Buds in Adler, near Sochi, and the mall was just like in Mississauga. Very impressive. Reporter: \"What did you expect?\" \"Well, to tell you the truth, we get the image of Moscow and women sweeping the streets.\"\n\nTim Wharnsby comes to me later and says, \"Did you say this?\" \"Yep.\" \"And did you say it's got rid of all the commies?\" \"No, I didn't say that, but it sounds like something I'd say.\"\n\nYesterday, I had two interview guys on at once: in front of me, a Czech reporter, and to my left, a Russian. The Czech was upset that I had given Tomas Hertl, the San Jose rookie, heck for hot-dogging a goal against Marty Biron. Hertl put the stick between the legs and it was a top-corner beauty, but the score was 6\u20130 for San Jose. I could not get it through the Czech's head that it wasn't the goal, it was the timing of the goal and the score. You don't humiliate goalies when you're up 6\u20130. If the game is tied, or winning by one goal, sure, but not at 6\u20130. He asked why I don't like Czechs, and I said I thought Hertl was a Russian. The Russian replied, laughing. I said I got nothing against Czechs, I liked Dominik Hasek and I admire Jaromir Jagr having a good year for New Jersey. Czech replied, \"You didn't like his hair, you made fun.\" Yeah, I did. I was doing colour for Pittsburgh, and Mario and Jagr posed for the program, and I said it looked like Mario and his sister. When Jagr got hurt, they didn't know whether to send over his hairdresser, a jeweller or the trainer. Czech again: \"You made fun of the way he celebrated a goal.\" Yeah, I never liked the salute after a goal. It pumped the other team up\u2014when the other team beat Pittsburgh, the players gave the salute. And now do you see how he celebrates a goal? No salute, just a little fist pump. Nice, just like a Canadian.\n\nThe Czech sort of gave in on that one, but couldn't understand how Hertl's celebration was wrong. Look, I said, maybe in Europe you don't mind a guy acting like that. In fact, when I see European soccer, they do that stuff. But in the NHL, we don't do that. Czech: \"You said somebody should hurt him.\" No, I didn't say somebody _should_ hurt him; I said if he keeps acting that way, somebody _will_ hurt him. (The kid got hurt three straight games and finally out for the year. Play with a bull, you get the horns.)\n\nI know what you're thinking: \"Why put yourself through all that with reporters you'll never see again?\" It's a game, like I said. It keeps you sharp. Finally, I said to the Czech guy, \"Do you know baseball?\" Surprisingly, he did. He said by far Atlanta was his favourite team. I said I saw an Atlanta rookie last year hit a home run, stand and admire the home run, and then flip the bat. The first baseman gave it to him as he came to first base (also the first base coach) and the second baseman, the third baseman... and the catcher almost grabbed him. The manager of Atlanta got him in the corner of the dugout and had him against the wall, giving it to him. Now you could say, like you said about Hertl, just a young man excited about a goal. We don't do that here. Naturally, next time up, he got a ninety-mile-an-hour fastball in the ribs. The rookie never said a word, just laid his bat down and went to first. Never even looked at the pitcher. He knew it was coming.\n\nThere is a code we follow over here (in North America), the Czech said, and I seem to dictate it. No, I said, I just make sure everybody follows it.\n\nAll right, you know baseball, it's 7\u20130 Atlanta in the eighth inning, guy on first for Atlanta. He never, according to the code, tries to steal second. Finally, he agreed. Strange that he knew the code in baseball but not in hockey.\n\nThe Russian, meanwhile, came in with, \"You don't like Ovechkin.\" I said, \"Show me where I said I didn't like Ovechkin.\" No, again, what I said was I didn't like the way he acted after a goal, like clapping the glass after a goal or\u2014the ultimate insult\u2014when he scored the fifth goal in a row and laid his stick down and let it seem like it's burning. I said change or you're gonna get it, and he did.\n\nHow could I not like a guy who is leading the league in goals? I said to both of them after about twenty-five minutes, \"Do you see the way Jagr acts after a goal now? No salute and the rest of the stuff, and you see how Ovechkin acts now after a goal, no clapping glass or burning stick. They act sensible now, like a good Canadian.\" I felt I had to straighten them out. They laughed and had their picture taken. My friend Warren Lowe from Edmonton\u2014who is, as they say, taking care of me\u2014said, \"Boy, that was a long interview, and they gave it to you. Why do you subject yourself to that? Doesn't it wear you down?\"\n\nNah, it's like a chess match: they try to get me and I get myself out of it. When they start a question, I'm answering it in my head as they say the question. I rarely get caught\u2014and it's not like I'm clever or anything, it's just that I've done so many.\n\nWarren: \"What do you watch for in an interview?\"\n\nI can size a guy up pretty good. What you have to watch for is the guy in an interview near the end. What they do is get everything they want in a good interview, and now they go in for the kill. Some embarrassing question or a question to make you look bad. You see, they don't care if you stop the interview, because they already have a good interview, but they're hoping you'll get caught and say something to hurt yourself or storm out of the interview, and then you look bad and they've got a better interview.\n\nWarren: \"What do you do in that situation?\"\n\nI have a standard answer: \"Hey, did you just get out of school? Think you can just get me on that one?\" And I laugh. If he keeps it up, I just keep ridiculing him. I'm a master at it.\n\nWarren: \"Why do it, though? Isn't it draining?\"\n\nNah, it's fun!\n\nAs you read this, Cherry thinks he's a clever fellow. Honestly, I come from the school of hard knocks. The only reason I get by with reporters is that way back I would get burned by them. So after a while, you get a sixth sense. I always have my guard up, and now, when I get into trouble, they take things out of context. Like that Russian reporter will take that one phrase, when he asked me about my view on Russia, all positive, except when I said I had the wrong impression of Russia as we see it on TV back home, women sweeping the streets. He'll put in that Cherry's impression of Russia is old women sweeping the streets, not that I'm impressed with Russia. But that's the name of the game. He's just doing his job, but as I get older and more experienced it's harder to catch me, and I haven't been caught in a while.\n\nBut back in the nineties, I let my guard down and got burned. It was with the Mississauga IceDogs, a new franchise in the OHL. To say we were having a tough time is an understatement. We were having a horrendous time, as Ron MacLean would say, and bad, too. Our first nine games on the road, we never scored, and we\u2014or should I say, I\u2014was no good. It was our third year, and the coach I hired, we just couldn't keep. We let him go and I kept my word, paid him $80,000. We couldn't afford another coach, so I had to coach. Talk about asking for it. But what could we do? We couldn't pay him and another coach, so I paid the price and did it. I was coaching and doing _HNIC_ and my radio show five days a week. It was a tough go, and I was tired and no sleep.\n\nThe one paper had a female reporter who knew nothing about hockey. Her job was to get me, but she was so oblivious that she really never got me. But I must admit, she did her job. I remember one question: \"The players say you don't keep your promises.\" It was true that I promised the players sweatsuits when we went on long bus trips. I was saving the suits to give them for Christmas presents. It wasn't all the players feeding her this stuff; it was one player. To describe him would be unfair, because he had many problems in life and still does.\n\nI remember one young writer, he was just starting out in junior hockey\u2014he was, as they say, a rookie and seemed a good kid\u2014and just to prove I'm human, as I was caught up in the all the turmoil, I let my guard down and was careless and I paid the price. The interview went well, we shook hands, and I said, \"What a nice kid.\" The next day, I picked up the paper and was levelled by the kid. It was his chance, and I'm sure he got praised by his editor. I wasn't really furious at the kid, I was furious at myself, after all these years to get suckered in\u2014no excuse. The kid was just doing his job to get ahead and I was the dumbhead, and I have to admit he never let up\u2014there was no mercy, and we were an easy target. And it paid off for him: he rose up in the paper. That was sixteen years ago and he still gives it to me about the IceDogs. It's the way of the beast, though; he never mentions that the IceDogs won the Eastern Conference championship in year six. I was not there, but it was won with players we drafted high for finishing last (we had four first-overall picks in a row).\n\nAnd while it might sound strange, it's good for me to take a shot now and then. It reminds me to be aware and to remember that the worse I look, the better the story for the reporter. So now, when I do an interview, I just think of this guy and how I got sucked in. As the song goes, \"Won't get fooled again.\"\n\nI must admit, not everybody is like this. For instance, the _Sun_ 's Joe Warmington. I told him the week before Steven Stamkos made the decision to not go to the 2014 Olympics that Stamkos would not go. I told him, \"That's between you and me,\" never doubting that he would scoop me. That's why people trust him, and again I don't doubt the kid, he was just doing his job and he did a good one.\n\nBut as I look back, the IceDogs were such an easy target. It was a bad idea from the start. When my friend Trevor Whiffen introduced me to an entrepreneur named Joel Albin with the idea to get a franchise in Mississauga, it sounded like a good idea. The mayor, \"Hurricane\" Hazel McCallion, was a hockey person. There were 600,000 people in Mississauga, and it had good minor hockey. But I pointed out we have no TV station, no radio, no daily paper in Mississauga. How were they going to know when we played?\n\nThe city, when we approached, said there'd be no arena till we got the franchise. The league said no franchise till we got the arena. To make a long story short, we ended up with the arena and the franchise, as it worked out, much to our regret. In the meantime, Brampton\u2014right next door\u2014got a franchise, too. As it worked out, two bad franchises, but here's the kicker that sealed our doom: Brampton had Scott Abbott, the millionaire who made his money on the game Trivial Pursuit, so they got their money two months before us. Because of this, they got all the first-round picks. It should have been they got a first round, then we got ours. Instead, they got all the first-round picks, we got left in the dust. And to add to our misfortune, the arena was delayed and we played our first nine games on the road\u2014not a prayer. And to add insult to injury, the refs' attitude was \"Big stuff Cherry, we're not intimidated by him.\" And just to show that they were not, they piled on the penalties. Between all this, I took over the club. I was easy pickings, but I went in with eyes wide open and knew I would pay.\n\nBoy, we were a failure at the gate and on the ice, but we took the blow, got our first-round kids out there. In fact, the last year I was there, more of our players got drafted by the NHL than any other Canadian Hockey League club, and only the London Knights were close to us. We had just started to turn the ship around. Unfortunately, I wasn't the captain that one of the owners wanted to see, and the deal was if one sold, all sold. The team went on to win the Eastern Conference championship in six years, but it made no difference. Just as predicted, no TV, no radio, no paper. They were still drawing flies.\n\nThe Lord works in mysterious ways; he's a wonder to behold. I didn't know it, or realize it at the time, but I was not cut out to be an owner or a GM. When we lost, I took it personally, and all this losing was affecting my health. I didn't realize it at the time, but people didn't want to be around me. Luba, my wife, stopped going to the games because she did not want to drive home with me after the games. So it was a blessing in disguise that we sold the IceDogs. The franchise moved to St. Catharines and is doing well.\n\n**February 11, 2014**. Saw the Russians for the first time at practice. Look terrific. Big, fast. I was watching Russian TV; they had a made-for-TV program on the '72 series, and it was great. It showed where Bobby Clarke broke Valeri Kharlamov's ankle. Phil Esposito looked like a maniac. We looked like thugs. It ended where they won the first game 7\u20133, and didn't show when we won. I really feel the Russians are using that '72 series as a motivator. Listen to this: we have a young driver, a Russian. Nice kid, I like him. Warren asked him if he knew any Canadian hockey players. Right away, \"Yes, Bobby Clarke,\" he answered. Not Crosby, not Gretzky, not Bobby Orr, only Bobby Clarke, who broke Kharlamov's ankle. I feel it's motivation.\n\nThe Russians look ready, eager. Saw us for the first time. Maybe it was the time difference or what, but we did not look as good as the Russians. But you don't win games in practice.\n\nI did an interview with Jian Ghomeshi at the big studio of CBC here\u2014good guy, good interview. He took me to task for not liking Europeans. I said it's not that I don't like Europeans, it's that I'm an extreme Canadian. Yes, I don't like Europeans in Canadian junior hockey leagues; they take a Canadian's spot. I feel for Canadian parents who spend thousands and thousands of dollars on hockey sticks, equipment and ice time, and then, when it's time for their kid to make junior hockey, teams parachute kids in from Russia whose parents never contributed one dime in Canada, never paid taxes. And let's be honest: if they pay for a kid to come all the way over from Russia and back again, and the Russian kid and the Canadian kid are equal in talent, who do you think they're going to keep? They have all that money invested\u2014get real.\n\nAnd did you know that they are bought suits and shirts and ties and extra clothing, and did you know that they get extra money, and did you know they and their agents dictate where they want to go and play junior in the Canadian Hockey League. For instance, when we were with the IceDogs, we were going to draft this certain player. His agent from Russia phoned and said, \"Don't draft him, he is not coming to North America. He has got to go into the army.\" So we didn't draft him and he went to another team. We knew the agent was lying, but we had no choice. Yes, they have everything going for them, and our kids get the short end of the stick, and nobody says a word because if you do, you're considered a bigot.\n\nI'm told on the QT that the Western League and the Ontario League want to stop the flow of Europeans, but the Quebec League won't allow it\u2014for obvious reasons. They wouldn't be able to compete.\n\nI say it, and I'll say it again, and I'll say it for your brother, your son or your grandson: the Americans are welcome in the CHL. They let our young men and women go to the States on college scholarships and help us. How do the Europeans help our kids? No, Canadian hockey leagues are for Canadian kids.\n\nDuring the interview with CBC's Jian Ghomeshi, I tried to tell him I got nothing against Europeans. People think that when I say we're the best, it somehow puts foreign players down. I'm not saying they're bad; I'm saying we're good. And some Canadians think that's a putdown to Europeans. I've never been able to figure that out.\n\nGhomeshi said I said that European players have no heart. I said I can't believe I said that. In fact, I never said it. After the interview, he came on the set and said I said it\u2014he has it on video. I said, \"When?\" It was back in the nineties. Canada was behind 3\u20130 and we came back and won, and I was supposed to have said something about heart. I'll bet I said\u2014and I'm going to find out\u2014that we won because we had heart. In some circles, they would say that indicates that the Europeans didn't have heart. See what I mean?\n\nThis afternoon, I have to leave here at 3:00 P.M. and do the opening of the women's hockey, do \"Coach's Corner\" at the end of the first period, stay for the game and then do another \"Coach's Corner\" for the replay back to Canada. When I'm there, I'm going to see the producer of the Jian Ghomeshi show and run the part where I was supposed to have said Europeans have no heart. I'll bet I said we have heart; if I'm wrong, I'll put it in this part of the book. I'll end for now till I come back home after the game. It's 9:30 A.M. (I didn't bother Jian or his producer. I got too busy.)\n\n**February 12, 2014**. Sochi, Russia. Everything here is in English and Russian. The signs on the buses, the songs for the figure skaters, the songs at the Games, in the hotels and at the airport. Do the Quebec language police know about this? It's ironic that the Russians have English to help us in Russia.\n\nThe women beat the U.S. today 3\u20132. Great game. Hayley Wickenheiser, a goal and an assist. Ron and I struggled today, not in top form. The location was not good, still bothered by dogs.\n\n**February 13, 2014**. Can't sleep. I don't think our \"Coach's Corner\" shows have been very good. I've got to make a comeback in the Canada\u2013Norway game. Kathy and I agree we have to get better clips. Lu phoned at 7:09 A.M. all upset. Believe it or not, they showed a young giraffe in Denmark getting its head cut off on TV.\n\nI am still upset about the \"Coach's Corner\" We did when the Canadian women beat the US 3\u20132. I just have to do better. I come all this way and don't do a good job. I expect to hit a home run every \"Coach's Corner\" and can't stand it when I hit a single or strike out, which I feel I have been doing. This gig is different. During the NHL season back in Canada, I've got stuff to put on \"Coach's Corner\" from all the games the week before. There's lots to choose from. Here in Russia, I've got to have stuff from the games that night. I'm not getting what I want and pull a tantrum. I'm looking bad.\n\nWe do another \"Coach's Corner\" for the game back in Canada at 7:00 P.M. that's shown there at 12:00 P.M. (confusing, I know). After the game, Kathy Broderick and Warren Darling...\n\n(Don't you love that name, \"Darling\"? When I coached the Boston Bruins we had a scout called Gary Darling. He phoned one of our young prospects called Jimmy Petty\u2014a wild guy whose hair looked like seaweed, so we called him \"Seaweed.\" Well, our scout called Seaweed and said, \"Hello, I'm the head scout of the Boston Bruins, Gary Darling.\" Seaweed did not believe him, thought someone was playing a joke, so he answered, \"Hello, I'm Jimmy Sweetheart of the Rochester Americans.\")\n\n... Anyhow, Warren and Kathy and I walked over to the CBC building and did a live interview with Heather Hiscox. What a lovely woman.\n\nAfter the interview, I go back to the hotel. Ron goes to the CBC building and works all night. I never see him. He works night and day and is getting a little testy, but who can blame him with that schedule. Our rooms are beside each other. When he is not on TV, he is in his room sleeping, with a \"Do Not Disturb\" sign on the door. I always close my door quietly.\n\nI don't see the dogs around anymore. My room overlooks the whole compound. I see the complete picture: security gates, etc. I went looking for the dogs to feed them. I had some bread and meat, but I saw no dogs. It's a cruel world. I can't get over the country of Denmark allowing them to cut off the young giraffe's head. The Copenhagen Zoo had offers from other zoos to buy the giraffe, but they killed it and fed it to the lions and showed it on TV. Dogs gone, giraffe murdered, what a world we live in.\n\nThe same day I'm at the CBC building, I notice a guy with a picture of his dog on his leash. I say, \"What a nice dog.\"\n\n\"Yeah,\" he says, \"it's a hunting dog. We hunt together.\"\n\nHe shows me a picture of him and his dog standing over a beautiful big moose with enormous antlers that he had shot. What a gorgeous beast lying dead. I just don't get it. I know what you're thinking, \"what a suck.\"\n\nIt's 4:30 P.M. This is one game that Ron and I are going over to together. He tells me he will knock on my door as he's worried I will be too early and disturb his sleep.\n\nWe do the game. Canada wins 3\u20131. No contest again. We out-shoot Norway 14\u20132 in the second period. I come back to my room and watch the Russians beat Slovakia. Ron went to the CBC studio to work all day.\n\n**February 14, 2014**. Up at 6:30 A.M. and went for a walk (looking for dogs again). As I am coming down the hall to my room, I see Ron going to his room. He has worked all night. This Olympics is turning in to a nightmare for him. Of course, he doesn't complain. If he did, he knows I would bug him, but to tell you the truth, he looks beat and tired.\n\nI leave at 7:00 P.M. for the Canada\u2013Austria game. Another cake walk. It was Canada 6, Austria 0.\n\n**February 15, 2014**. Got to liven up \"Coach's Corner.\" Warren and our trusty Russian driver, Rustelian (I call him Rusty) drive downtown to pick up some stupid-looking hats\u2014one a lion, another a rabbit\u2014for \"Coach's Corner.\"\n\nWatched the USA\u2013Russia game. Russia lost to the U.S. in a shootout. The Russians got stiffed. They scored the winner when the game was tied 2\u20132, but the goal was called off because the net was off a little. Earlier in the game I had noticed that the pegs that hold the net in place looked thin. I phoned Kathy and told her what I thought and that the thin pegs could be trouble. Sure enough, off comes the net, easy because of the small pegs, and Russia gets the winner but it's called back and the game goes in to overtime.\n\nT.J. Oshie, he never seems to miss. The USA uses him three times because you can use the same guy as many times as you want. T.J. sinks three goals in and the Russians are out. There goes their chance for a gold in hockey. The Russians got the most medals at the Olympics, except the one they wanted the most.\n\nPutin was at the hockey games, not happy. It is so strange that the Russians always look bad in international hockey.\n\nI asked Kathy if I could have a shot of the peg so I could show it on the next \"Coach's Corner.\" She said she'd ask Toronto to get the shot which I thought was strange. Here we are in Russia, and we have to ask Toronto to get us the shot. (Remember I said at the start it would be better for me to stay in Toronto.)\n\nI watch the Swedes beat Teddy Nolan's Latvia. Latvia looked good. Teddy almost pulled it off, but one of the Swedes made a perfect dive. (He should have gotten a ten sign for the dive). Sweden scored on a power play.\n\nI noticed in the game that the captain of the Latvia club was not wearing a visor. I see Weber and Getzlaf who don't wear visors in the NHL wearing them in these games. I'm going to mention it to them next practice.\n\n**February 16, 2014**. I woke up at six-thirty\u2014can't believe it. I never sleep that long. I think it's starting to get to me. Bob Costas, the NBC announcer, got pinkeye and has to stay in the dark in the hotel room. The Russians have a great idea: when they are working on a building or the building does not look good, they wrap a large sheet around it, cover the whole building. It looks like it's complete with windows and doors unless you get up close. You would never know. I walked by twice, didn't notice. Tried to feed a little dog. Wouldn't eat, ran away. I don't think he's long for this world. I did not have the same bounce when I just came back here. Ron told me Trevor Pilling, the boss, said in the last game I came in like a hurricane full of life. He said there is hope for us, the way Don acts. I must admit I am wearing down a little. I never sleep all the way to six-thirty, even at home.\n\nCanada plays Finland tonight, should be a better game. The last one against Austria was a joke. At twelve-fifteen, I went to the morning skate and made a mistake. Sent a message to Ryan Getzlaf that the captain of Latvia was not wearing a visor, thought he and another guy (Weber) might not have to wear one. It was only that old guys who played before certain dates don't have to wear a visor. Getzlaf and the rest are too young, must wear a visor according to the international rules. Watching the morning skate, found out they're starting Carey Price. Found out P.K. Subban and Martin St. Louis are not dressing. The best defenceman in the league and the best scorer in the league are not dressing. I watched the morning skate, and it's funny: when two guys get the shaft, they bond together. It's the same way on a team in the regular season. The guys who don't play form a clique and bond. I guess misery loves company. I know: I used to be one of the Black Aces, and we had our own little club.\n\nI am going to tell Ron I am not going to do the opening bit from the studio. I used to like it in the playoffs when we were on the ice. There is nothing in the studio. I expect a hard time from Ron. I'm not feeling too well. I think a cold or something is coming on. See you after the game. We will beat Finland. I have to do my radio show, _The Grapeline_ , in Kathy's room after the game. I am wearing a lion cap tonight. Wore a bear last night.\n\n**February 17, 2014**. Toughed it out last night. My voice went, struggled through the night. Had to let on that nothing was wrong. Finished, had to go to Kathy's room to do _The Grapeline_ with Brian (Williams). Really struggled there, did the best I could\u2014not good. I am sitting in my room, just sitting here. I better be better tomorrow. Lucky I don't do anything till Wednesday\u2014the quarter-finals. The hockey is awful for everybody. Five guys back, sit and wait for power plays or a break.\n\nCan't think straight. Nose running, head spins. You don't miss your health till you get sick. Damn. Everything going along great till yesterday. Got to push through. It's that I meet so many people and shake hands. The odds are, I'm going to catch something.\n\n**February 18, 2014**. Yesterday was a tough day. Never left the room, except to talk to Luba and Kathy. Slept in till 7:30 A.M. I never sleep that late. Feeling better\u2014thank the Lord, because I do five straight games, including the women's gold Wednesday, Thursday, Friday, Saturday, Sunday at all hours. Going to say on \"Coach's Corner\" why they are going with Price instead of Luongo, and why so far Ovechkin, Malkin, Crosby, Perry and Getzlaf are not scoring. It has been raining. Lovely day today, got five games (I hope) in five days, plus radio shows after. I see I said that Russia beat Latvia last night. Same game every game because of the big ice\u2014you can't open up. If you do, you'll get caught. So what you do is bar the door\u2014everybody back till you get a power play or get a break. The only time you open up is when you fall behind by three goals like Latvia did. Losing 4\u20131, they opened it up. Good game, almost won\u20144\u20133 and a chintzy penalty to Latvia. There is so little hitting\u2014how about none. If you hit a guy, you're out of position, so when there is no hitting, the refs call what we call phantom penalties. When the whistle goes, everybody wonders who got it. It's sad to see teams lose on the chintzy penalties they call.\n\n**February 19, 2014**. Good night's sleep, woke up around 6:30 A.M. Got a call from Kathy\u2014can only talk about the game, the goalie and Ted Nolan. Going to say all his players understand English, that it's amazing here, all the announcements in English. All the games, even Russia versus Norway. All the Russians on TV, the people downtown speak it, saleslady, parking attendant. All the songs in figure skating in English, all English songs at the arena. A policeman said most Russians around here speak English.\n\n**February 20, 2014**. We went to the women's morning skate and ended up at the wrong rink. It was a Russian practice. Did an interview with a female Russian reporter. We went to the other side of the rink to see the women's skate. I was the only media person there\u2014everyone else was at the men's practice. I am glad I went. It seemed to give the women players a lift that somebody cared. In fact, one player tweeted she was glad to see Grapes there to give them encouragement. They waved after the practice. I gave them the thumbs up. I wished Kevin Dineen (their coach) good luck\u2014they're going to need it. The U.S. team is good.\n\nI met a reporter from Detroit for an interview. He told me the Russian reporters are a tough lot. One Russian reporter, in an interview with the Russian coach, said that if he plays a certain goalie, it's a death sentence (no kidding) and if he doesn't win the Olympics, he's disgraced all of Russia and he will be fired and sent to Siberia.\n\nThe game between the USA and Canadian women's teams was a barn burner. We were outplayed, but our goalie Shannon Szabados saved our Canadian bacon. She held us in there. We were down 2\u20130, but we came roaring back and tied it after Kevin pulled the goalie. There was a play where we were in their end, the puck came back to the point, the linesman gets in the way of the defense, the puck goes by and it trickles all the way down to the open net and hits the goalpost. If it goes in, it's all over, but we tie the game.\n\nThe game went into overtime. We get a cheap slashing penalty and it looked like it was over. They get a cheap call for slashing and we get a break\u2014a U.S. player pulls down a Canadian on a breakaway and now it's four-on-three. Hayley Wickenheiser is in front, screening the goalie. Right-winger Rebecca Johnston feeds the puck to Laura Fortino in the corner and Fortino slides it over to Marie-Philip Poulin who puts it home for the winner. The Canadian women never quit\u2014their heart came through again.\n\n**February 21, 2014**. Going to the Canadian men's morning skate. Got to tough it out till the end. Have not missed a practice or a morning skate during the Olympics, and I'm proud of that. This has been a tough haul, but the women's gold medal win last night was worth the trip. At this morning's skate, three of our champions were there. Hayley Wickenheiser is really something. They took her captain's letter from her, but she never missed a beat, played terrific. This is her last Olympics. She made it a good one.\n\nWe beat the States 1\u20130 in the men's semifinal. We outshot them 37-30. Carey Price was terrific\u2014he stood on his head. So was Jonathan Quick for the States.\n\nTrevor Pilling, our boss, brought an interview I did before the Olympics with Perry Lefko where I said, \"I guarantee we will win gold in the women's game and gold in the men's.\" It was in a magazine called _GoodLife Mississauga_.\n\nTalked to Lu. There's an ice storm back home. She tells me they're going to allow bars to open early for the big game.\n\n**February 22, 2014**. 3:10 P.M.: Went to the practice, did five interviews: French, Swedish, USA, Canada, Russia. The Swedes are all upset that there is a Canadian referee\u2014they don't realize it's in their favour. I said, \"I'll bet you anything you want that the Canadian ref will call a Canadian penalty first.\" That's the way we are; we bend over backwards to appease the other people. I remember when our figure skater Elizabeth Manley should have won a gold medal: Europeans and Americans wanted to reward her with the gold, but the one judge who ruled against her was the Canadian judge. That's just the way we are. I remember when I coached Team Canada in Sweden; the players were upset when they heard that we had a Canadian ref for the final game. They knew what was coming.\n\n**February 23, 2014**. 8:10 P.M.: Canada won in a walk, 3\u20130. I really believe the way Coach Babcock played the four lines, we could have played another ten games. We were just warming up. Price was sensational, especially in the gold medal game. Doughty loved that big ice and was flying. But I've got to admit, I saw most of the games, if not in person, on the Russian TV, and except for a few games, especially the women's championship game, it was boring hockey with the big ice. As I said before, you can't hit because if you hit, you're out of position, and if you get a 1\u20130 lead, you sit back and it's \"Katie, bar the door,\" play it safe, wait for a penalty or a break. If we gave that hockey in the NHL, we would be in trouble.\n\nI remember when they asked Sundin how they could get more people at the games in Sweden he said they should make the rinks smaller and get more action. He's right, of course. I remember when I coached Team Canada back in the eighties. I went to a game between Finland and Sweden with my son, Tim. It looked for a 0\u20130 tie like soccer. I could read the program and never miss a beat, so we smoked the Swedes 3\u20130, no contest.\n\nKathy and Prior Smith, the producer of my radio show on 590 The Fan, had arranged for me to do the show with Brian Williams right after the game. We scooped everybody. I still don't know how they did it, but everyone at The Fan seemed happy.\n\nThe Russians would not allow buses or cars in the Olympic Park, so we had to walk through the park. My first time, the fireworks were just starting. It was beautiful. The Russians did a great job.\n\nI was in Russia twenty-one days and I have to admit again, Russia was not what I expected. The Russian people tolerated us and some were friendly. It was clean, the people respected authority, but I noticed they did not smile a lot. Now here is the strange one: I did not see one fat Russian male the whole time I was there. The women were kind of heavy, but every guy looked in shape.\n\nI finally made it back to my room, had a shower and sat to watch the closing ceremonies, the best ever. Who should knock on my door, but Ron. He somehow got off early. Great. The first time we have one of our sessions in three weeks, but I guess we were too tired to have a good session. We had a few and toasted that we survived. I don't know if \"Coach's Corner\" was good or not, but the suits were and Canada won both golds in hockey\u2014that was the main thing.\n\n**February 24, 2014**. 7:30 A.M.: Got a sore throat, still haven't confirmed about the truck to the airport. Lu phoned\u2014I wasn't pleasant for some reason. Sometimes I wonder about myself. Like with Kevin Hunt, my friend since he was a kid with Tim\u2014they went to school together and learned TV at Mohawk College. He helped me through the Olympics\u2014he's the floor director, got me coffee, helped me with anything. I gave him a hard time about the chair I was sitting on. I know I hurt him\u2014why would I give him and Lu a hard time? One I like, and one I love, and then I feel bad. Got a call from Kathy. She's in Moscow, a five-hour layover. The truck will be at the hotel at 4:15. Ron and I and Scott Russell and Jim Hughson are on it. What do we do without Kathy? She phones from Moscow to help\u2014she is a blessing. I even gave her a hard time at the set. Meeting Ron in the sauna at eight-thirty. We better not drink too much\u2014early start.\n\n4:15 A.M. start and then an all-day, all-night trip. This is my last Olympics, even if the NHL goes to South Korea in 2018. The Russians did a wonderful job. They should be proud. Better than Salt Lake, better than the one in Japan. Clean, polite, well-run under difficult circumstances.\n\nThey got the most medals, but to make a complete success at the Games, they had to win the hockey gold. That was the main sport for President Putin. I can't figure out why Russia always plays so bad in international hockey. They scored one goal.\n\nThe weather has been fantastic. It's like a beautiful fall day. Lu says there is so much ice at home, you can't walk. Nevertheless, I will be happy to get to the weather at home, ice and all.\n\n**February 25, 2014**. 4:35 P.M.: We make it home. Lu has a little can of beer with the little umbrella in it, saying, \"Welcome home,\" sitting on the doorstep. Inside are four balloons saying HAPPY BIRTHDAY! as I spent my eightieth in Russia. Lu says, \"How was it?\"\n\n\"Piece of cake.\"\n\n**February 26, 2014**. Spent most of the day resting and doing interviews, unpacking.\n\n**February 27, 2014**. Glad to be home. Start packing to go to Vancouver for the Heritage Game.\n\n**February 28, 2014**. Fly to Vancouver for the Heritage Game. It seems we're living in airplanes.\n\n**March 1, 2014**. It's Saturday night, so we have to do \"Coach's Corner\" for the Montreal game back east. We go to the Heritage Game site. They are just setting up and the place is empty and people are wandering around. We did \"Coach's Corner\" and weren't satisfied, but we did our best under the circumstances.\n\n**March 2, 2014**. Up in the morning. We have a room to go to for breakfast, but we never go. Ron orders room service and I have my usual breakfast, first Lu's bran muffin and then some oatmeal. Off we go to the Heritage Game.\n\nIt's always turmoil at these games, pushing and shoving. We do the opening, which is mediocre at best, but we did our best. Watch the first period. I cannot believe how the Vancouver coach is not starting his number one goalie, Roberto Luongo. He lets Roberto sit as the back-up goalie. I know Roberto is fuming. Imagine. He's a guy who took the Vancouver club to the Stanley Cup final and one game away from the Stanley Cup he is embarrassed like this in front of 50,000 of his fans. He already had a slap in the face that Price played in the Olympics after he won the gold in 2010 at home. (Price was sensational in the Olympics, so it was the right move.)\n\nRight now, it was tough on Roberto. Now he comes home feeling like he can show everybody they were wrong and he's told that he won't be playing in the Heritage Game. I said on \"Coach's Corner\" it was a big mistake. The Canucks just lost their goalie. I knew the coach would be gone and I knew GM Mike Gillis would be gone. That was a dumb move. The only hope for the Canucks to make the playoffs was a hot Luongo, and as you know, no playoffs for the Canucks, Gillis is gone and Tortorella is gone. I really believe if they had played Roberto that game and kept playing him the Canucks would have made it to the playoffs and Gillis and Tortorella would still be there. Why? Because Luongo had something to prove and there is nothing like a guy who is on a mission to prove people wrong.\n\nWe do \"Coach's Corner.\" It's not very good. (Hey, you can't win them all. Some nights you have it and sometimes it's not there. I can't explain it. For some reason, this is one of those nights. I miss my studio with the little red light on the camera and darkness.)\n\nWe make our way back to the hotel through the crowd. I am alone this night\u2014Ron goes out with friends. I never go out. I love being in my room alone, having a few pops, watching the baseball or the hockey game. I know it doesn't sound good that I like drinking alone. Cold Buds, peanuts, popcorn and cheese are heaven to me.\n\nWe're up early for the flight back to Toronto. I meet with the boss of Rogers Sportsnet, Scott Moore at 8:00 A.M. and he tells me what's going to happen next year with _Hockey Night in Canada_. Doesn't seem to affect \"Coach's Corner.\" I guess he just wanted to get things straight with me.\n\nI knock on Ron's door. He's feeling a little rocky as he stayed up late with the NHL alumni. He's in a little bit of shock, too, as he's been told he will not be doing the host job for _Hockey Night in Canada_ , but is still on \"Coach's Corner\" and a new gig _Hometown Hockey_ or something that's every Sunday night. He will be great just like he's great on _Hockey Day in Canada_ and _Hockeyville_. He doesn't say much about it. I think he's still in shock. I think he's exhausted from the Olympics, too. He had a schedule that would kill a horse. At the Olympics, we would do \"Coach's Corner\" after he did a half-hour pre-game show and then he would do fifteen minutes before the opening and then \"Coach's Corner,\" the panel between the second and third periods, do the wrap-up at the end of the third period and then do highlights for back home on CBC. He'd then go back to the Sochi media building and work all night on the Olympics, catch a few hours of sleep, work all afternoon, catch a few more hours of sleep and start all over again with the game that night. He did this for twenty-one straight days. No wonder he slept all the way back to Toronto.\n\nAs we are boarding the plane home from Vancouver, we are greeted by a buddy of mine, Brian Bellows. He is the head steward. Brian reminds me of the time when he thought he was a hockey player and tried out for the Rochester Americans when I was GM and coach. Brian was one of the players. One day, he came to me and said, \"Don, I want your honest opinion. I have a chance to go with Air Canada for a job, but I want to be a hockey player. What do you think I should do?\"\n\nI said, \"Brian, take the Air Canada job. It is a good job, steady. You don't want to end up like me in the American Hockey League, riding the buses for twenty years and, at the end, have no job. Be smart. You're a pretty good player, but you are an AHLer like me. You could make our club, but Air Canada is solid.\" And he has a great job. I guess, in a few years, a pension. He was a lot smarter than me.\n\nWe get home, and I am slowly wearing down. And for some reason, my wrist is killing me. In bed again.\n\n**March 4, 2014**. Next day, like a fool, I scheduled a banquet without Lu's knowledge. I told you I wasn't too bright. Luckily, Cindy says she will drive me. It's a luncheon for Domino's Pizza. We leave at 5:45 P.M. and make it. The Windsor Casino people are great, there are three hundred people, meet them and take pictures. Do banquet, great audience. Drive back. It's been a tough month. Cindy drops me off. I go to bed\u2014I don't even have a pop. Thank you, Lord, for getting me through this month. I enjoyed the Olympics, but it is my last forever. I will be smarter in the future about banquets. I'm going to have to stop thinking I'm still thirty-two years old. Maybe thirty-five years old. That sounds right.\n\n# **MUSINGS**\n\nI OFTEN WONDER WHAT WAS in my makeup that made Eddie Shore hate me so much. Let me explain about Eddie. A famous player, tough as nails for the Boston Bruins, he won the Hart Trophy three times, but when he retired, he was a tyrant. He was a hard man with no mercy, ruined hockey players and lives. He was hated so much and feared that players had it in their contracts that they could not be sent to the Springfield Indians in the American Hockey League.\n\nThe Indians were a good hockey team\u2014won three Calder Cup championships\u2014but it was the Alcatraz of hockey. To play there, you must be desperate. It was torture. Eddie thought up ways to harass the players, and for some reason he hated me the most. I really never knew why. He sent me to Three Rivers, Quebec, at Christmas one year. The next year, he traded me at Christmas to the Sudbury Wolves of the Eastern Professional Hockey League. I remember when I got the news that I was going to a lower league, the guys were jealous that I was escaping the prison, but I never did find out why the hate for me.\n\nI was number one on his list to harass. I never gave him a hard time, I never argued with him. It was strange. I never did find out till a couple of years ago.\n\nMy daughter, Cindy, was talking to Brian Kilrea at a hockey game one night. Kilrea was the coach of the Ottawa 67's, the winningest junior coach of all time, and played with me on the Springfield Indians.\n\nCindy says Brian reminds her more of me than any other person, so Cindy said, \"Brian, you and Dad are alike. How come Mr. Shore liked you and hated Dad?\"\n\nYes, Shore did like Brian until Brian led the strike against him, which finally drove Eddie out of hockey, so Brian was no suck. He gave Eddie a hard time and argued with him, but Eddie liked him from the first day and Brian liked Eddie. Brian likes Eddie to this day. So what was it that made Eddie take an instant dislike for me\u2014and for that matter, his coach, Pat Egan, couldn't stand me, either. It was like scenes out of a movie. Suspended for indifferent play when I hadn't played in a week. Suspended for something twenty other guys did and they got no suspension or fine. Always with the Black Aces. The Black Aces were the guys who were in trouble and always being punished. One time I spent four and a half hours on the ice after practice. (It's funny, everybody thought my feet would hurt. No, it was my back that hurt.)\n\nWhen my cheque came for winning the Calder Cup, it was cut in half. The other players got half of my cheque, and still he never let up in practice. I set a record for being called out five times in three minutes. Always screaming, \"Mr. Cherry, bend those knees, hold that stick right!\" It never stopped. One time, I took one stride on the ice, while pulling on my gloves, and he screamed about the way I skated.\n\nKent Douglas\u2014who went on to glory with the Toronto Maple Leafs, got Rookie of the Year\u2014and others used to argue with Eddie all the time. And then other players used to suck up to him. Now that I think of it, I was the guy in no-man's-land, never argued (which he seemed to like) or sucked up to him (which he seemed to like). He would literally scream at me, \"Mr. Cherry, in most hands, a hockey stick is a scientific tool. In yours, it's a blunt instrument.\" (I must admit, I did tend to use it to straighten guys out.) And of course, the beauty: \"Mr. Cherry, if you could visualize that in reality your manoeuvrability is nil!\" I always liked that one. I guess he didn't like the way I skated. He loved guys like Brian that skated with their knees bent. Unfortunately, I skated stiff-legged. The more he hollered at me, the more I clammed up, just stood there and looked at him. When he said, \"You have anything to say?\" I just looked at him.\n\nBrian's answer to Cindy on why Mr. Shore took a particular dislike to me was, \"It's funny you ask that, Cindy. I ask Eddie one day after your dad had left, 'Eddie, why was it that you had it in for Don Cherry?' Eddie told me that it really pissed him off that I stood there with a look of insolence on my face. 'Cherry never said a word back, but that look of insolence on his face said it all.' \"\n\nThe story I am going to tell now concerns a program that was on a few years ago. It was called _Who Do You Think You Are?_ It was a show that dealt with your past. I travelled to France to visit the grave of my relative, Sergeant Thomas William Mackenzie (military medal and bar), who was killed in the First World War, on November 7, 1918, just days before the war would end, and to Vimy Ridge, to honour my grandfather, Richard Palamountain. These fellows were on the side of my mother.\n\nThe show also wanted to show my father's side. They showed my grandfather, John T. Cherry, who was a captain of a ship called the _Holcomb_ , out of Kingston for the Canadian Steamship Line. John T. sailed the St. Lawrence River and Lake Ontario for twenty-two years\u2014never had an accident or wreck. When he was seventeen, he joined the army and fought the U.S. in the Fenian raids when the U.S. invaded Canada. He later joined the North-West Mounted Police (now the RCMP) at the time the Americans from Montana were coming up to Canada with rotgut whisky and trading it for furs, horses and food from the Indians, and the Indians were starving when winter came.\n\nSir John A. Macdonald, the prime minister, called for volunteers to straighten out the Yanks and save the Indians. They made a march of eight hundred miles through dense bush and over barren prairies, starving and freezing. They started with 150 horses and ended up with fifty. It was one of the hardest journeys I have ever read about. John T. joined at nineteen. I know, reader, you're thinking, \"What has this to do with Eddie Shore and your problem?\" Kindly read on, please.\n\nI am a great reader of military history and the military ways. In one interesting book, there was a chapter on how you could be court-martialled, put in prison or what. One way that caught my eye for a soldier to be put in prison or court-martialled was for \"silent insolence.\" Imagine a soldier in prison for not saying a word, just the way he looked. I guess Eddie Shore knew what he was talking about with my attitude and my face. In the military rules, it described insolence as to stand and face the officer, not saying a word, but having the look of detachment and disdain on the soldier's face. It is a look not to gain popularity, and I guess that is why Eddie could not stand me. Believe me, folks, I caused my family a lot of grief with that look, but looking back, I'm sure I would do it again. You are what you are, and I guess I was just asking for it.\n\nI know what you're saying: \"What has my problem with Eddie Shore got to do with my grandfather, John. T. Cherry?\" The producers of _Who Do You Think You Are?_ wanted me to see John T. Cherry's record in the North-West Mounted Police, so they had a representative of the Mounties come to RMC in Kingston. As you know, RMC is the West Point of Canada\u2014or West Point is the RMC of the U.S. The gentleman met me in a very austere room filled with books and low lights, lovely cherry wood. He stood with white gloves on and a book before him. After formal introductions, he explained that this book contained the records of the famous march west, including John T.'s record in the military.\n\nWith all the heroes and generals looking down on us from the pictures on the walls, he explained that the march was so hard, the regiment had a lot of deserters. They just left as they neared the U.S. border. They just went down to the States, but John T. stuck it out to the bitter end and stopped the whisky traders and stayed in the service for three years. He was offered 180 acres in Saskatchewan, but decided to come back to Kingston. John T.'s record shows he had problems with the discipline of the military in the first months. He made only seventy-five cents a day. He was fined three dollars for letting a horse get loose and five dollars for riding on a cart. And then the gentleman from Ottawa read the last fine, for a whopping fifteen dollars. He looked up at me and said, \"You know, Mr. Cherry, as I read these fines, I can see the acorn never fell far from the tree. Take a look at what your grandfather was fined fifteen dollars for.\" As I looked at the page, he pointed to one word.\n\nInsolence.\n\nJohn T. straightened out after the first month and was never fined or disciplined again. I'm sure he would have wanted me to let you know.\n\nWhen I hear a Canadian franchise in any sport complain they can't get any free-agent stars to sign in a Canadian city, I think of the four basketball players that were traded from Sacramento, California, to the Toronto Raptors in 2013. They were traded to Toronto in the middle of December. Think about the players. They just left sunny California, walking around in shorts, enjoying life. They are called in. \"You have been traded to the Raptors in Toronto.\"\n\nJump ahead, they're in shock. Going to live in Canada, they are on the plane to Toronto. They are looking down at Toronto in the middle of a severe snowstorm. I can hear them now. \"Are you kidding me? I have to live _here_ for six months? Look at that snow! How do they live up here? What did we do to deserve this?\" And if you remember, the snow just kept comin'.\n\nNow, I know the Sacramento team is not full of stars and they suck, but really, where do you think those players would rather play: the frozen North or lovely Sacramento in the summer breezes? If you're going to miss the playoffs, miss them in a place you enjoy and a country you understand.\n\nThe Blue Jays did sign a lot of baseball stars, but overpaid everybody to get them to come to Canada.\n\nThe Maple Leafs will never get a superstar to come here for a lot of reasons. First, the stars want to go where they win or have a good shot at the Stanley Cup. They can go to someplace like San Jose, Anaheim or Chicago and get lost in the crowd. Their families would love the weather in L.A., Anaheim and San Jose.\n\nThe baseball and basketball players have no idea about Canada. I have to say it, but I lived down in the States from when I was nineteen, in 1954, till I came back in 1980. I came back to Kingston to work on construction, but left after a while. I stayed in the States after I got married to a girl from the States, Rose. She and her family had no idea about Canada. Imagine, a six-hour dive from Hershey, Pennsylvania, to Kingston. It was unbelievable.\n\nI asked Rose one day, \"In school, did you ever learn about Canada?\"\n\nShe said, \"Not a word about Canada. We learned all about South America and its capitals, but nothing about Canada.\"\n\nIt used to bug me.\n\nWhen I married Rose and brought her back to Kingston, as we left for the trip north, her mother, God bless her, said, \"Now remember, Rosemarie, when you come to the border, put on your sweater.\"\n\nI was listening to the Carolina Hurricanes' hockey broadcast the other night and the U.S. announcer said so-and-so come home from a small little city in Canada: Regina, Saskatchewan. I guess the Canadian colour guy agreed. He didn't want the U.S. announcer to look bad. But what was the difference? Nobody knew the difference.\n\nHey, don't get me wrong. I never, ever heard an American knock Canada. They always had nice things to say about us. They knew a little. Canadians back then were very polite. We had very little crime to speak of. Of course, that's all changed now, but that's a whole new story. But back to the basketball players from Sacramento.\n\nThey get off the plane, they have to wait in those four-mile-long lines at customs, which they're going to have to go through when they fly to the States. They finally get through and they wait for their bags. They finally get outside; it's freezing and a snowstorm. They see a sign and they ask, \"What the heck is a millimetre?\" And another sign, it says NORD and SUD. \"What the hell are 'nord' and 'sud' and 'sortie'?\" They walk outside in the blinding snowstorm and think, \"What the heck did we do to deserve this?\" and dream of the warm, sunny breezes in Sacramento.\n\nI am with my son, Tim, who rates the kids in minor midget for the Ontario Hockey League. We are watching the Junior Canadians and the Mississauga Rebels. I am having fun. When the whistle stops, I'm signing sticks, posing for pictures\u2014it seems every kid has a camera now with those phones, guys and girls, seven and eight years old, working magic on them. I'm just getting into touch phones. I have no cell phone, have no idea about texting and stuff. My wife, Luba, was a private secretary so she is an expert on that stuff.\n\nI must admit I'm lost when it comes to emails and stuff\u2014not like Ron MacLean, who is really into emails like my son, Tim, who at this game asks into his phone who won the game between Columbus and Philly, and it answers him. Amazing!\n\nI know some of you are saying, \"Yeah, but you're on Twitter.\" Let me explain. In the playoffs two years ago, the boss of _Hockey Night in Canada_ , Trevor Pilling, and Kathy Broderick, the producer who runs \"Coach's Corner,\" convinced me.\n\n(If you want to find out what's going on at CBC, just ask Kathy. One time, I told her, \"Kathy, if they ever get rid of you, I'm quitting.\" Kathy said, \"Gee, Grapes, please don't tell them that.\" She's also a wise guy.)\n\nThey told me that there were about five guys using my name on the Twitter, making statements, and people think _I_ was making the statements. I told them I knew nothing about Twitter. Kathy said she would take care of it, that I just needed to phone her and she would put what I want to say on Twitter.\n\nI told them I had no problem with guys imitating me\u2014in fact, there's a guy out there who imitates me who's got me down perfect. He does banquets, even went on _Dragons' Den_ for money to expand his operations. Trevor and Kathy said this Twitter is different. CBC had to protect itself and me because those guys were saying things that were libel.\n\nI have come to a conclusion: I'm still a minor leaguer in many ways after all these years. For instance, there's the guilty feeling I have when I fly first class. I remember when I was with the Bruins, I would force Jean Ratelle to take my place in first class. Don't get me wrong, I enjoy it, but I have a funny feeling when first-class passengers gets on the plane first. I wonder what the people think as we get on.\n\nI say to Ron, \"We don't need a limo. Let's take a cab.\"\n\nYeah, I know, what you're thinking. All those fancy suits and jackets. Honestly, I treat them as costumes. I feel more at home with a T-shirt, cut-off sleeves and Crazeewear pants.\n\nI feel a bond when I drive past guys on a construction site, especially if a guy is on a jackhammer. I always say to Ron as we drive by in our limo, \"I did that. I love those guys.\" Ron always says, \"Yeah! Yeah! I know. You've told me a thousand times.\" I did it for so long, my ears have a noise permanently, there all the time, like one of those bugs you hear on a hot day in the summer. The doctor told me there is no cure.\n\nI live in a small house. The kids at Halloween say, \"Hey, it's Don Cherry! You live in this small house?\" Out of the mouths of babes.\n\nI drive a '93 Ford F-150 pickup and a 1983 Lincoln.\n\nI still can't believe people want their picture taken with me or my autograph. Like I said earlier, I feel if I was off TV for a year, people would say when they meet me, \"Didn't you use to be Don Cherry?\"\n\nWhen I coached the Bruins, Harry Sinden, the GM, ordered breakfast with me one day and he ordered orange juice\u2014a little glass of it cost a dollar fifty. I didn't order any. He said, \"How come you don't want orange juice?\"\n\n\"Not at a dollar fifty for that small glass,\" I replied.\n\nHarry said, \"You can take the minor leaguer out of the minor leagues, but you can't take the minor leagues out of the minor leaguer.\" That saying came back to me when Ron ordered a $2.50 glass of orange juice the other day. (Again, I didn't have any.) Even when the orange juice is free I won't have it. It would not be right.\n\nWhen I go on the road, Luba makes me salmon sandwiches, and for breakfast I have my All-Bran and Luba's muffins, even though I could go to a special room for free breakfast. I always remember my Scotch grandmother, Maggie Mackenzie. She used to lecture me. One of her sayings was \"Any fool can make money; it's the wise one that saves it.\"\n\nI do not enjoy myself the way Ron does. I admire him in a way. He travels all over the world. He buys first-class clothes and new cars. When we are in the playoffs and he goes and exercises, or for a swim, he looks like he stepped out of _Vogue_ magazine. He buys those expensive magazines that cost $7.95 and ten dollars and reads them and leaves them on the plane. Imagine.\n\nThe point I'm trying to make in my inept way is that all those years I spent in the minor leagues, where I earned $4500 for ten years, our road money was $4.00 a day with the feelings of being unemployed never leaving my thoughts. I'm not cheap, I pay my rounds. Sammy Pollock, when he was GM of Montreal, kept a book on everybody. When Sammy put my name in the book, he only wrote three words:\n\nConfirmed Minor Leaguer.\n\nAnd Sammy was right.\n\n# **THE GOOD OLD DAYS**\n\nWHEN I'M ON A RADIO TALK SHOW or when I'm at a banquet, I'm often asked one interesting question, and the question is, \"Don, in all your career in the minor leagues, what player do you think should have been in the NHL and was down with you in the minors?\" I could have answered a lot of guys, like Billy Sweeney, playing for the Springfield Indians. Billy won three scoring titles in a row, over a hundred points in sixty-some games and three championships. He was owned by Eddie Shore and never got a shot at the show.\n\nAnother player who was buried in the minors, because the Toronto Maple Leafs were ticked off at him for some reason or the other\u2014I never found out why\u2014was Frank Mathers. When he played for the Pittsburgh Hornets against the Hershey Bears of the AHL, I was a rookie. When I saw him playing in the minors, I said, \"If _he_ can't make the NHL, who can?\" He was sensational. An All-Star every year. But folks, you have to realize there were only six teams in the NHL and they only carried five defencemen apiece. That means there were only thirty defencemen in the NHL. Today, they carry seven defencemen, some eight defencemen, so say they have thirty teams\u2014that's 210 defencemen in the NHL now, compared with only thirty before. And the teams back in those days only carried one goalie, so that means that there was six goalies in the NHL.\n\nNow they carry two goalies, so that means that there are 60 goalies in the NHL. And back then, they only carried ten forwards, which means there were only eighty, ninety forwards in the NHL. Now there are over four hundred forwards in the NHL. So you can see, it was slim pickings if you got in the bad books and never got out of the bad books. Figure it out. There were six teams in the NHL, six teams in the AHL, six teams in the Western Hockey League and six teams in the Central Hockey League. That would have been all the professional teams in the world. So combine the three pro leagues. That means that there were twenty-four teams, compared to now: there are thirty teams just in the NHL.\n\nAll of us playing in the minors in those days would be playing in the NHL, and we could have had another six teams to fill. The Boston Bruins put me in the doghouse, never to get out, when they told me not to play baseball in the summer. I love baseball and, like a fool, played against their wishes and broke my shoulder. They showed their displeasure and sent me to the dregs of society and the Siberia of hockey, the Springfield Indians of the AHL and Eddie Shore. That was at the age of twenty-two, and I was doomed to the next twenty years in the minors. And a million miles on the buses. I paid the price.\n\nYes, there were a lot of good hockey players also doomed to the minors like me, for some infraction or the other. Also, another reason a lot of guys never made it: for instance, they would bring up a guy who was an offensive talent and he'd be on the fourth line and never on the power play. And he would be sent down, with the team saying he could score in the AHL but not in the NHL. The real reason he didn't score is they didn't give him a chance and he became another confirmed minor, like me.\n\nI can name tons of guys who got the shaft, but the one guy who I thought got the biggest shaft of all was a nifty player called Guyle Fielder. He was born in the U.S., I remember, but grew up in Nipawin, Saskatchewan. I have to say, he was a sensational hockey player. He ticked somebody off\u2014again, I never found out who he ticked off, but he was one of us in the minor leagues his whole life. Chicago signed him when he was fifteen years old but released him. Guyle ended up with Detroit, played on the fourth line, hardly hitting the ice. The following year, he was sent to Boston. Same thing, and back to Detroit the next year. Jack Adams, GM and coach, put him between Ted Lindsay and Gordie Howe. As you know, Lindsay and Howe liked the puck and they liked to handle the puck. And they would. Now, Guyle liked to handle the puck, too. Detroit lost six straight, and guess who was benched? Not Lindsay and Howe. Guyle went to Jack Adams's office and said he would like to play. And Adams said he would play, all right, but he sent him to the WHL to Seattle to play. And he did for the next twelve years. And that's when I saw him play. I was playing for the Spokane Comets, and believe me, he was a magician with the puck. Okay, his skating wasn't Bobby Orr's, but whose was? But he got there. When he was on the ice, you had to be aware and keep your eye on him all the time. He could fool you in a dozen ways. And around the net, he would appear out of nowhere and pop it in. Like I said, he was a magician.\n\nImagine! He won eleven scoring titles, nine in the WHL, which is just a step below the NHL, and never got a shot. Yeah, I know what you are thinking: \"You were a minor leaguer too! That's why you think another minor leaguer is great. How would you know a good or bad NHLer?\" Well, let me tell you this: I was at every training camp in the NHL for a couple of weeks, I will say, so I saw them all play. All the stars. I played against them, I scrimmaged with them, I played exhibition games with them. Mind you, I didn't stay long, but I played against them\u2014I played against all the stars\u2014so I had a good idea what a hockey player was, and I knew a good hockey player from a bad hockey player.\n\nSo my answer to the question, \"Who got the biggest shaft from the NHL and was buried in the minors?\" No doubt it was Guyle \"The Golden Guyle\" Fielder. I figure he would be making about eight million a year in the NHL right now. Talk about being born in the wrong time!\n\nI have seen a lot of hockey players who were great ruin their life with drinking, but the greatest I ever saw was a player back in the fifties, R\u00e9al Chevrefils.\n\nWhen I first saw him, I got the same feeling as when I saw Bobby Orr. How does he skate so well and handle the puck so effortlessly? Now, he wasn't in Bobby's class, but he was an amazing player. His junior coach, Hap Emms, the GM and coach of the Barrie Flyers, called him the greatest prospect for the NHL he'd ever seen.\n\nR\u00e9al played for the Flyers. In his last season with the Flyers, he scored 52 goals and 51 assists in only 54 games, and he was tough, too. Over a hundred minutes in penalties, he was a sensation. R\u00e9al was Boston Bruins property and they could hardly wait to get him\u2014a bright future was ahead. He was one of those guys the scouts say \"can't miss,\" but dark clouds were on the horizon behind this great prospect.\n\nR\u00e9al was one of those guys who kept the dressing room laughing with jokes and imitations. Everybody loved him, even on the ice. When he'd score, he'd make a funny voice, \"WOO-EEE!\" He was fun to be around and he produced. In 1956\u201357 he scored 31 goals when 30 was a wonder. He was an All-Star three times, but the dark clouds that were forming turned into a storm. His love of alcohol overcame his love for hockey, and it was sad to see this great\u2014and I mean _great_ \u2014downfall. And his fall was fast. The year after he scored 31 goals, the Bruins sent him to the minors to teach him a lesson and to dry out. Of all the places to send him: the Springfield Indians, the Siberia of hockey, where I was toiling for Eddie Shore. We did everything we could to help him. We gave up beer so he had no temptations. He was such a warm, funny guy. We were rooting for him all the way. R\u00e9al was doing great and playing great. In fifteen games, he had seven goals and eight assists and he was dry as bones. We were proud of him and we knew it was only a matter of time until he was back to the show.\n\nThen, disaster. He was persuaded by some of his so-called friends from Boston to drive there to see a game. We did everything we could to stop him because we knew: once he got in that crowd, it was over. He went to Boston, got with his old cronies and went to the game drinking. It was the end, and we knew it. One guy on the Springfield club, when he heard the news, broke down crying. He was thinking of R\u00e9al's wife and kids. It was a fast spiral down. He went to the AHL, was sent to the Quebec League, then to the EPHL, then to the WHL, and finally ended up in senior hockey. From All-Star to senior hockey in a couple of years.\n\nI won't dwell on his final days. He lost everything\u2014wife, kids, reputation\u2014and he died young, but you know, till the day he died, everybody loved him. I never saw him play in the final years, I only saw him play when he was great, and I will never forget the first time I saw him at the Boston camp and watched him handle a puck and skate. Man, I said to myself, this is a hockey player.\n\nI know players now who played for Eddie Shore, the Darth Vader of hockey, who still talk about how great Eddie was. As a player, he might've been second only to Bobby Orr, and I will admit, at age sixty-two he could skate like the wind, but I am not one of the players that speak glowing of Eddie. I really feel he was a little off-balance, to say the least. Let me tell you a story, and you decide.\n\nThe Springfield Indians of the AHL had training camp at Niagara Falls, Ontario, where we trained for four weeks (with no pay), twice a day, and once, when he was mad at us, three practices in one day. It was a tough struggle, but we had to put up with it.\n\nOne day, a young boy showed up. He had taken the bus all the way from Springfield, Massachusetts, to be with us in Niagara Falls. He loved the club and would sit in the stands and watch the practices. I befriended the kid. He was having a tough time, as he was, shall we say, a little slow. He never bothered anybody and was very polite. After practice one day, I was a little late, as I had to stay to get ice on an injury from stopping a puck with my ankle. The arena was empty, and as I walked by this one room, I heard a whimper. I looked in, and it was the room where the trainer kept the sticks. The sticks were kept in a heavy wire enclosure, protected by heavy locks so nobody could get into the sticks to steal them. It was dark, and I still heard the whimper. As my eyes became accustomed to the dark I saw, there on the floor of the stick room, was the kid from Springfield. I knew exactly what happened. Some of the players had locked him in there and forgotten about him. I went to find the manager of the rink to let him out. I felt terrible that this poor kid had travelled all that way and got treated like this. What must he be thinking about hockey players?\n\nI decided I must do something. I collected two dollars from every player, went to a sports store and bought a beauty sport jacket. Talked to the trainer to see if he had anything I could put on the jacket, and he just happened to have a beautiful Springfield crest. I took the jacket and crest to a shoemaker and asked him if he could sew the crest on the jacket. He did, and before supper with the team, which I invited the boy to, I presented him with the jacket. He got teary as he accepted the jacket while we cheered and clapped and sang, \"For He's a Jolly Good Fellow.\" He was the happiest kid in the whole world when he put the jacket on.\n\nI noticed Eddie, staying in a corner, with the usual frown on his face.\n\nThat night, I had a knock on my door. The coach, Pat Egan, said, \"Eddie wants to see you in his room.\"\n\nI said, \"What about?\"\n\nHe said, \"You'll see.\"\n\nI knocked on his door, went inside, and he sat there, glowering at me. \"What made you buy that jacket for that kid?\"\n\nI couldn't tell him that I did it to make up for some jerks that locked him in the stick room, so I said, \"He's a nice boy. He travelled all the way from Springfield to visit us and I thought he would like it.\"\n\nHe said, \"Do you think you're fooling me? You bought him that jacket and gave it to him because you burned his jacket.\"\n\nI looked at him to see if he was serious or if he was just fooling. I should have known he never fooled. I said, \"Ed, why would you think my story's not true?\"\n\n\"Quit your lying, I know it's true.\"\n\nI could see there was no use arguing with Eddie. He had made up his mind I had burned the kid's jacket. I never did find out why he thought that.\n\nI said to Eddie, \"Well, Eddie, think what you like,\" and as I turned to leave, the door was closing when he hollered, \"You'll pay for that, Cherry!\" And Eddie made sure that I did.\n\nI would like to tell you about a beauty guy by the name of Garnet \"Ace\" Bailey. He was named Ace after the great Hall of Famer who played for the Toronto Maple Leafs, Irvine \"Ace\" Bailey, who was almost killed by Boston's Eddie Shore in a game at the Boston Garden in 1933.\n\nBorn in Lloydminster, Saskatchewan, in 1948, the younger Ace played junior for the Edmonton Oil Kings. In 56 games, he had 47 goals and 46 assists. Turned pro with the Boston Bruins in 1968\u201369. In eight games, got three goals and three assists. Imagine, six points in eight games and they sent him down to the minors! Boston did not have good teams. I still wonder why they sent him down. In the minors, in Hershey of the American League, he scored 24 goals and 32 assists in 60 games. Did great in the playoffs, 9 goals in 10 games. For the rest of his career, he hit the road: Boston (where everybody remembers him), then Detroit, St Louis, Washington, Edmonton in the World Hockey Association, then the Central Hockey League\u2014Houston Apollos, Wichita Wind. So Ace paid his dues in hockey, but he did something in his career that not many people did: he played with Bobby Orr and Wayne Gretzky.\n\nThat's Ace's background, but Ace's life was so much more, and yes, I know, when a guy dies, usually people say he was a good guy. But let me tell you, folks, he was a treasure. Honestly, he always had a smile on his face, always had a good word for everybody. I know what you're thinking: too good to be true. It's true.\n\nI will admit, when he was young, he was wild. He was a buddy of a young Wayne Cashman, and they had fun. One story goes that they were both sent down to Hershey of the AHL from Boston. It took them two days of driving to get there. In the Hershey arena, you can drive into the building and onto the ice. The team was just coming on the ice for practice. Ace and Wayne drove out onto the middle of the ice and said to the coach, \"Well, we're here, coach, ready to go.\" Like I said, he was a teammate and good friend of Bobby Orr and a teammate and good friend of Wayne Gretzky.\n\nAce was a third- and fourth-line guy who was a penalty killer and a stir-'em-up guy. He was also a great character guy in the dressing room. He was one of those guys, if somebody new came to the team, he would be the first to welcome them. He was especially good with rookies. Wayne and Ace were very close. Wayne told me that when he and Ace were in the World Hockey Association, one night they went with a friend to the Boston Bruins' game in the Garden\u2014we were playing the Philadelphia Flyers that night. It was one of our usual games we always had with the Flyers\u2014no prisoners. Wayne said to Ace, when they were driving in the car back to their hotel, \"I never will be able to play in that league. I'm too small.\" Ace gave him heck and said, \"Shut up, Gretz, you'll play in the NHL and you'll be a star.\"\n\nWayne said one time in a game, there was this one player running around and trying to hurt him. He said something to Ace and Ace said, \"Next time you're on with him, steer him over to the bench where I'm sitting.\" So sure enough, this guy starts chasing Wayne, so Wayne skates alongside his bench with this guy following. He hears a _whomp_ , looks back and the guy is lying on the ice out cold. Ace cold-clocked him.\n\nAce always wore a white turtleneck on the ice and had a lot of time on the bench, as most third- and fourth-line guys do. But sometimes those guys are as important as your first and second lines. For instance, I remember in the 1972 finals, the Boston Bruins were playing the New York Rangers. The Bruins took a big lead in the first game, 5\u20131, but the Rangers tied the game and things looked bad for the Bruins. Tom Johnson, the Bruins coach, benched Eddie Westfall and put Ace in in his spot. Ace had barely played in the game, so his legs were ice cold, he told me, but he received a beauty pass from Shakey Walton and cut around the defence and put one in the top shelf past Hall of Famer Eddie Giacomin for a 6\u20135 win on their way to a Stanley Cup. If the Rangers had won that first game with a comeback like that, they would have been tough to beat. But Orr turned his magic on and the Bruins won the Cup. The coach and GM of the Rangers, Emile Francis, said after the series, \"We could beat the Bruins, but we couldn't beat Bobby Orr. He was in another league.\"\n\nAce, after he retired, become a coach in the minors with Wichita and Houston. He went with the Edmonton Oilers as a scout for twelve years. Ace went to the L.A. Kings and became director of pro scouting, but his heart was always with the Boston Bruins, and he lived in Boston.\n\nWe will always remember this date: September 11, 2001. Ace and his buddy Mark Bavis were both scouting for the L.A. Kings. Ace was teaching the young Bavis the tricks on scouting. They both had slept in and had to catch the plane to L.A. They made it to Logan Airport, seemed to hit all green lights, raced to the gate and just made it as they were closing the gates, thank their lucky stars. And as you will all remember, the plane was hijacked and flown into the World Trade Center.\n\nI often think of \"what ifs.\" What if Mark and Ace had slept five minutes longer? What if Ace had said, \"We'll catch the next flight, Mark.\" What if they hadn't hit all of those green lights? If only they had been delayed five minutes somewhere... but I guess life is not made of \"what ifs.\"\n\nI can still see Ace with his white turtleneck, smiling, saying, \"How's it going, Grapes?\" Like I said earlier, he was a great guy and I'll always remember him.\n\nI know there have been stories written and things said on TV about the famous (or infamous) too-many-men-on-the-ice incident in Boston back in 1979. The one that cost us a shot at the Cup. When a team nowadays has too many men on the ice, they bring up that game. Michael Farber of _Sports Illustrated_ said he can't think of a more crushing mistake in sports, in any sport.\n\nI've got to admit, I have to agree. It was a dynamite game. I remember Tie Domi telling me a few years ago about the Leafs flying to Florida after a game. They got in about 3:00 A.M. and were all still pumped up from the game. They all turned on ESPN, which had the '79 game on. Usually in hotels, the team has a whole floor booked just for them. Every door was open, and the whole team watched the game.\n\nTie said the players couldn't believe the speed of some of the guys and the hard hitting. And what a game. I am going to give you the whole story about that game and about how I felt and how the players felt.\n\nIt would be funny if it wasn't so sad that everyone who follows hockey remembers the goal that Guy Lafleur scored against the Boston Bruins in the seventh game with Boston leading by one goal. There was 1:43 left in the game\u2014this was the semifinals, but really, it was for the Stanley Cup, because the New York Rangers were already in the finals, waiting for the winner of this series. Beating New York would be a piece of cake for either Boston or Montreal, so this game was for all the marbles.\n\nWe only had 1:43 to go. We had played a great game, especially Gilly Gilbert, our so-called backup goalie for Gerry Cheevers. I had made the decision to go with Gilly after Gerry had lost the first two in Montreal, for a couple of reasons. Number one: he was a good goalie. He set a record for winning seventeen straight games in a row, a record that still stands. Number two: his contract was coming up. Number three: he was a French guy. He would have loved to beat Montreal, and if I hadn't goofed up with too many men on the ice, we would have beaten them.\n\nThe Montreal Canadiens of that era were the best team ever in hockey. Let's just look at their starting lineup: Steve Shutt, 60 goals; Jacques Lemaire, 35 goals; Guy Lafleur, 52 goals; on defence, Serge Savard, Hall of Famer; Larry Robinson, Hall of Famer; in goal, Ken Dryden. Five out of the six starters were Hall of Famers. One year, they only lost eight games, and only one of those at home. Imagine losing only once at home. Out of the eight games lost, Boston beat them three times. We could beat them one game and leave, but in the playoffs, it was a tough go. Because of their depth, their third line could have been the first line on most clubs. We had been in two wars already, with the L.A. Kings and the Philadelphia Flyers. The L.A. series meant going back and forth across the States, and it was always a battle with the Broad Street Bullies. Meanwhile, Montreal had a picnic with the Toronto Maple Leafs. They could have walked to Toronto, so they were rested and ready to go.\n\nMontreal and Boston players did not like each other. We had played the Canadiens and lost the last two finals, and we were determined to not make it three in a row. They had a great defence, great goalie, and four strong lines, and they had the master, Scotty Bowman, who knew all the tricks, behind the bench.\n\nOne time, he got to the rink at 8:00 A.M. when the refs were skating and he taunted them: \"You won't call penalties on the Bruins\"\u2014that kind of stuff. He showed the press a video of all the penalties he thought we should have got. He sent all his players onto the ice after a goal to intimidate the other club. He knew all the tricks.\n\nTo stop this, I told my players to go on the ice after they scored. My players thought I was nuts. I explained that Scotty told _his_ players to do it, so when Montreal players went out to congratulate their scorer, we jumped on, too, to congratulate our goalie. Ridiculous, I know, but Dave Newell, the ref, came over and said, \"You can't do that. Forty guys milling around on the ice. There's bound to be trouble.\"\n\nI must admit, it did look ridiculous. I said to Newell that if Scotty could send his guys over to congratulate the scorer, we could go on the ice to congratulate our goalie. He had to admit that there was no rule against it. I promised not to send my guys if Scotty promised not to send his. The ploy stopped. They put a rule in later that you could not send a guy onto the ice after a goal unless it was overtime.\n\nThe Bruins and I felt we were against the world and that the world was against us. I remember we were in first place in our division and had no All-Stars on the All-Star teams. We fumed at that one, but it motivated us.\n\nThe Boston GM, Harry Sinden, and I were not talking by now. We were even staying at different hotels. It was not pleasant. I knew I was gone. My days were numbered.\n\nI still laugh at a cartoon from the _Boston Globe_. It showed Harry sitting behind his desk, and I was standing in front of the desk. Harry had a big gun in his hand. And the caption read, \"Don, sit down, I want to discuss the lifetime contract I gave you.\"\n\nIt was not pretty. I thought one day I would be friendly. It was in the Montreal Forum, amongst a bunch of reporters. I said, \"Nice shirt, Harry.\"\n\nHe said, \"Is that right? What you're saying is the rest of my shirts are horseshit?\"\n\nI answered, \"Come to think of it, you're right.\"\n\nI never tried to be friendly after that, and when Harry said he didn't expect us to get past the quarter-finals, that was it.\n\nI was making $60,000 and happy to be making the money, but the average salary for coaches in the NHL was over $100,000 (now over $1,000,000). I had won Coach of the Year and been in the Stanley Cup finals two years in a row. I was cheap labour, but coaching Boston was worth it.\n\nI have to admit, Montreal was impressive, even in their morning skates. I remember one time, Cheevers was walking behind the glass on the way to our dressing room while Montreal was having their morning skate. I swear Steve Shutt used to fire pucks at the glass when Cheevers walked by. They knew all the tricks.\n\nThey won the first two games of that series in Montreal, and Cheevers didn't look bad, but as Randy Carlyle would say, he was okay. I knew we were not going to win with okay goaltending. People thought this was nuts. Potential Hall of Famer, but no guts, no glory, and so Gilly it was.\n\nScotty even poked fun about Blue picking the goalie and said he resented the fact that his dog, a beautiful German shepherd called Waldo, did not get the ink that Blue did. He said that Waldo was prettier than Blue, but I said Blue was featured in _Sports Illustrated_ and therefore more famous.\n\nI admit I went too far. I said if the series went seven games, we would be stiffed because they (meaning the NHL) wanted the glamour team, the Canadiens, in the finals, not us grinders. John Ziegler really gave it to me. He said, \"I don't mind you calling us stupid, but you're calling us dishonest. I will not put up with that.\" (He was right, of course.)\n\nWe won the fourth game, and the Canadiens' players were in the papers, complaining about the ref. It got too hot, and Ziegler got us all together in a little room and laid down the law. Shut up or be fined ten thousand dollars.\n\nWe made it to game seven and we were really banged up. A lot of guys were shot up to numb the pain before a game. I sat in a little room by myself. I got to thinking about what a war we'd fought to get here. Losing the first two right here in the Forum, promising the players we would be back here for a seventh game. Somehow I believed we would be\u2014it was not just a hollow promise to keep their spirits up\u2014and here we were. I knew it would be my last year with these beautiful guys. I knew Harry wouldn't fire me; he'd give me an offer I could refuse, which he did, and I _did_ refuse it. I figured if he didn't get me now, he would get me later, but I wasn't thinking about my contract that night. I was thinking we had to go right at them. Usually, teams held back and tried to survive the Canadiens and their blitz, just tried to not get scored on.\n\nWe were shooting, the works. Go at 'em and get 'em on their heels was my plan, and we did. We had a 3\u20131 lead. Wayne Cashman, who had the needle before the game, got a beauty goal and then another one, and Ricky Middleton got the other. With only nine minutes left in the game, we led 3\u20131.\n\nWhen I was told Bob Myers was going to ref the game, I had an uneasy feeling. I had watched the series before between Toronto and Montreal. The Leafs and Canadiens were in OT and it was pretty even, both teams playing well. All of a sudden, Bob Myers called a nothing penalty on Tiger Williams for boarding.\n\n(Let me explain that, at that time in hockey, refs rarely called a penalty in the third period of a tight, important game unless it was a flagrant penalty, when he had no choice. And you know, most players and coaches want it that way\u2014let the players decide the game.)\n\nWhen you got a penalty in OT against that Montreal team, they smelled blood. They scored and put Toronto out. Tiger came out of the penalty box and went after Myers. It took four guys to hold him back. Finally, Larry Robinson of the Habs joined in to hold Tiger back from attacking Myers. I didn't blame Tiger. It was an awful penalty, especially in OT to eliminate a team.\n\nI had this on my mind, I'm sorry to say, the whole game. I kept saying to myself, \"When is this guy going to strike?\" I know it's an awful thing to admit, but it's true. And then Jean Ratelle, two-time winner of the Lady Byng Trophy, who had only twelve penalty minutes all year, got called for roughing. I remember the play like it was yesterday. It was at centre ice, and Bob Gainey ran at Jean Ratelle, and as Gainey went by, Jean pushed him in self-defence. We couldn't believe Jean Ratelle got a penalty\u2014a Lady Byng guy. It was like Myers wanted Montreal back in the game. Guy Lafleur set up Mark Napier, and in two minutes Guy Lapointe blasted one and the game was tied.\n\nSo everybody thought the end was near, that the Montreal Canadiens would score again. The crowd could sense the next goal, but Ricky Middleton fooled them with a beauty backhander to put us ahead with only 3:59 to go.\n\nDonnie Marcotte had orders: if Lafleur goes to the washroom, you go, too. In other words, if Lafleur is on the ice, you be there too. He did a wonderful job. I remember they both got a little nick around the eyes and went off to let the trainer look at them. Believe it or not, they were both looking at each other like two warriors waiting to see what the other was doing. Donnie did a super job, but still Guy was great. Donnie, with about two minutes left in the game\u2014we were still up by one goal\u2014was coming off. I turned to him to say, \"Great job, Donnie.\" I saw the look on his face, and he said, \"Oh no.\" I knew exactly what he meant. I had set up the lines, everybody knew their job (so I thought), but somebody had jumped over the bench out of turn. What happened?\n\nScotty Bowman had Lafleur stay on for three shifts in a row. Naturally, I had Donnie stay on, and he did his usual great job. Donnie could skate with anybody, but the look on his face as he stood by the door coming off was a knife in the heart. It's ironic, I know. Good guy John D'Amico, the linesman, saw the seventh guy right away. I can't prove it, but I do believe he was giving us a chance to get that guy off the ice. He didn't want a great series to be decided this way. Unfortunately, nobody came to the bench except John D.\n\nHonestly, folks, I can still see John looking at me with his hand in the air, looking at me with those big brown eyes as if to say, \"Sorry, Grapes, I got to call it.\" What a feeling. It is hard to describe how you actually feel. I'm lightheaded as I write this right now.\n\nI had to put Donnie right back on, and we were doing okay. About a minute and a half to go in the penalty, and we had the Cup, beating the great Canadiens right in their own barn. New York Rangers, here we come. I saw Guy wandering up the right side, but Donnie got him. He sent the puck to Jacques Lemaire, who was straddling the blue line\u2014one skate in our end and the other out\u2014and Guy was flying. I thought Lemaire was off-side, but I never mentioned it after the game. I'm not going to be a spoilsport after a game and series like that.\n\nLemaire feathered a pass. I mean, it's one of those passes you dream about. It's right on the money and Guy, in full flight, slapped the puck. It skimmed along the ice about an inch above the ice for the post, just inside. Gilbert never had a chance. Terry Sawchuk couldn't have stopped that one. One-forty-three to go. I'll never forget the time. It's burned in my memory, and if I do happen to forget, people always remind me. And if people don't remind me, it's always on ESPN, CBC, NBC, etc.\n\nTied, we went to the dressing room. Imagine how they felt. They'd given it their best shot and were still tied. My speech was, \"Lookit, when we were down 2\u20130 at the starts of this series, if I had said to you we'd be back here for a seventh game and we'd go into overtime and had a chance to win, would you be happy? Well, here we are, seventh game, we can beat these guys. This will show them what we're made of.\" They bought it and came out smoking. In fact, I remember at the start of the period, Marcotte put a pass out from behind Montreal's net and Dryden was looking behind the net. The pass was perfect, to Terry O'Reilly. He put it in the top corner and it was going in. We jumped up on the bench and Terry had his stick in the air, but Dryden was on his knees. It would have gone in on a normal goalie, but Kenny was so tall it hit the top of his shoulder and just skimmed the crossbar. They came back and Yvon Lambert put it by Gilly for the winner.\n\n(About thirty years later, I had a conversation with Kenny and I said, \"Just thinking, Kenny, if you were not so tall, that puck would have gone in and we would have won the Cup.\" Kenny, with his quick wit, came back, \"Ah, don't sweat it, Don. If you had won the Cup, you just would have been one of those run-of-the-mill guys who won the Stanley Cup instead of a famous guy on TV.\")\n\nGoing into our dressing room after the game was the hardest thing I ever had to do. There was silence. The players just sat there, some half-undressed. I saw blood on underwear where they'd had their shots to freeze shoulders, legs, you name it. I forget what I said\u2014something about being proud of them\u2014and then looked at Gerry Cheevers, who had tears in his eyes. Tears came to my eyes, too. We were so close. One minute and forty-three seconds, too many men. What a way to lose the Cup.\n\nNo matter whose fault it might have been, it was really mine. I honestly believe that. I'm not just saying that. I should have realized the triple shift might confuse someone. I should have watched more closely what was going on, not telling Donnie \"Nice shift.\"\n\nI stood on a plastic milk carton outside the room and took the questions from the reporters. Funny\u2014they were all in our room, not Scotty's. I knew why. It was a good, juicy story, that a coach could have too many men on at that time. They were not too cruel. One guy said, \"You should have had Gilbert in earlier. If you had of had him in earlier, you might have won.\" I'm not too proud, but I said, \"Yeah, and if my aunt had balls, she'd be my uncle.\" Crude\u2014I'm sorry. I smartened up after that and took the slings and arrows. Heck, I deserved them!\n\nIt was a long flight back to Boston. The guys were going to a bar after we landed and they wanted me to go, too, but I just couldn't. When we said goodbye at the airport, I knew I wouldn't coach these beautiful players again. I knew it was \"win or you're gone,\" and I drove home with tears in my eyes.\n\nI had the radio on\u2014not the news, but an FM station on with music to get my mind off the game. I know this is hard to believe, but there was a group back then called Jigsaw, and they had one big hit called \"Sky High.\" Hearing them sing about blowing it all sky high was another knife in my heart.\n\nI got home. Rose and Blue were waiting. Rose had tears in her eyes. Rose and I sat and had more than a few beers. Rose said it wasn't my fault. I thanked her, but I knew it was. It was getting late. Finally I said, \"Rose, don't wait up. Blue and I are going for a walk. We might be gone for a long time.\"\n\nI gotta say, the smartest guys I met in hockey were Sam Pollock, GM of the Montreal Canadiens, and Punch Imlach. Punch was a guy who knew how to coach and knew the gut feel of hockey, and he had the Stanley Cups to prove it. But Sam was the guy who pulled strings to get the best players. For instance, did you know that Guy Lafleur should have ended up in Oakland on the California Golden Seals? They were the team that looked like they were going to finish last and get the first-overall pick in 1971, which would have been Guy Lafleur.\n\nIn May 1970, Sammy sent Ernie Hicke and the Canadiens' 1970 first-round pick to the Seals for their first-round pick in 1971. Ernie was a good player who averaged 16 goals a year over his NHL career. The Seals used the Canadiens' pick to get Chris Oddleifson, but they traded him away before he played for them. Chris was a defensive player, scoring 95 goals in 524 games, but things were not working out for Sammy. In 1970\u201371, the L.A. Kings went into a slump, and it looked like the California Golden Seals could end up ahead of the Kings in the standings. That would mean the Kings would get the first pick and Guy.\n\nSammy went to work. He sent a real good player named Ralph Backstrom to the Kings for Gord Labossiere and Ray Fortin. They never played for Montreal. Sammy just took them in the trade to help L.A. with its payroll. The Kings ended up ahead of the Seals, and Sammy had Guy.\n\nI got to admit, Guy was okay the first three years for Montreal but did not live up to his potential. When he took off his helmet and started to fly, he was sensational. Why he took off after ditching his bonnet, I don't know. Maybe he liked his lovely long hair flying. Whatever it was, it sure worked, because that's when he became a superstar. In his first three years, he got 29 goals, 28 goals and 21 goals. The year he got rid of his helmet, he scored 53 goals, then 56, 56, 60, 52 and 50 in the next five years\u2014never below 25. He ended up playing in 1,126 games, with 560 goals, 793 assists, 1,353 points. In the playoffs he played in 128 games, scored 58 goals, had 76 assists and 134 points. He played in six All-Star Games, won the Art Ross Trophy (scoring title) three times, the Hart Trophy (MVP) twice, Conn Smythe Trophy (MVP in the playoffs) and the Lester Pearson Award (best player as voted by the players' association) three times.\n\nHe was always the first player in the arena, and the first guy on the ice. In fact, when I helped coach Team Canada in 1976, he and Bobby Clarke and Bobby Orr were in the dressing room at two-thirty in the afternoon for a 7:00 P.M. game. He was always ready to go.\n\nThere were many things I admired about Guy Lafleur\u2014the way he could wire a puck in full flight, the soft hands\u2014but I think what I admired most was the way he carried himself, something like Jean B\u00e9liveau, a gentleman. He took a lot of abuse, but he never complained, never whined to the ref, just took it and kept going.\n\nYes, Sam Pollock pulled off a great deal, got a superstar from the Seals for nothing, but you know what? I wish the California Golden Seals had kept their first-round pick and got Guy, if you know what I mean...\n\nEarlier in the book I wrote about Albert \"Babe\" Siebert and his tragic end. One of his teammates on the Montreal Canadiens, Howie Morenz, also had a tragic fate. In 1950, \"The Stratford Streak\" was voted by the Canadian Press as the greatest star in hockey for the first half of the twentieth century. Rocket Richard got four votes. Howie Morenz had twenty-seven. When he played, he was called the Babe Ruth of hockey, won three Stanley Cups, three Hart Trophies (as the MVP of the NHL) and two Art Ross Trophies (leading scorer in the NHL).\n\nHowie's life in hockey almost ended before it began. When he was young, he scalded his legs and they didn't know if he would walk again, let alone play. He was a sensation in his hometown of Stratford and really didn't want to go to Montreal. He wanted to stay in Stratford, but was forced to go to Montreal, where he was a sensation again. On his first shift, he scored a goal, and there was no looking back. Montreal won the Stanley Cup against Calgary in his first season as a Canadien.\n\nHowie was of those gifted guys like Bobby Orr or Wayne Gretzky. They could read and anticipate where the puck was going. Howie was a fast fashion plate, a sharp dresser, and he was a fun guy who enjoyed life\u2014and at times, enjoyed life a little too much. He played at full speed at all times, could shoot in full stride, not coasting before he shot, and the crowd loved him. He was the idol of the Montreal fans, but when you play that hell-bent-for-leather style, you get injured a lot. (Just ask Bobby Orr. I once asked Bobby, \"Why don't you take it easy once in a while, and not take those chances?\" He answered, \"It's the only way I know how to play.\" I guess that's what made guys like Bobby and Howie great, but Bobby and Howie both paid the price.)\n\nMontreal proved the old saying, \"What have you done for me today?\" and traded Howie to the worst team in the NHL, the Chicago Blackhawks. When he returned to the Forum as a Hawk, he scored and the Montreal fans give him a standing ovation.\n\nHowie is like Bobby Orr. Bobby Orr put people in the seats of an arena. They were worth the price of admission. The two could have the crowd on their feet when they touched the puck.\n\nHowie scored 40 goals in 1929\u201330, but when you adjust the goals at that time and now, it works out to over 80 goals. And you must realize that at the time Howie played, it was more difficult to score a goal, so think if he played today. And he was a playoff guy, too. His records were unbelievable at that time.\n\nHowie's style of play and the injuries started to pile up, but Montreal got Howie back and he was king again. The crowd still loved him, even though injuries had taken their toll and he was not the Howie of old, but still magic, till the night of January 1937, when it all ended. As usual, he broke around a big, tough defenceman, Earl Seibert, and as he went into the corner, his skate caught in a net and Earl nailed him into the boards and badly broke Howie's leg. The fans at the game said they could hear the crack all over the building.\n\nHowie in the hospital was not a good patient. Despondent, he drank with his buddies who came to see him.\n\nHowie somehow knew he was going to die. He sensed he would never play again and he couldn't see living without hockey. On March 8, 1937, he died in his sleep at age thirty-four; on March 12, 1937, the Montreal Forum was packed with fans there to attend the funeral of Howie and say a final farewell to its greatest star.\n\nHall of Famer Aur\u00e8le Joliat said it best: \"Howie just loved to play hockey more than anyone ever loved anything, and when he realized that he would never play again, Howie died of a broken heart.\"\n\nBack then, the Montreal Canadiens lost two star players within a couple of years of each other. Neither player was treated particularly well by the club.\n\nThe world of hockey back in those days did not take care of their own, especially the Montreal Canadiens, who should have been ashamed of the way the families of both stars were treated after their deaths. The Montreal Canadiens of the current day treat their former players better than any team in sport. \"Once a Canadien, always a Canadien.\" Ex-players have special rooms, are treated royally, get the red-carpet treatment. When there is a reunion, they fly in the former players, everything paid for. If any are having a tough time with money, they are helped.\n\nI am not going into the tough times of the families of Howie and Babe. There were benefit games for the stars' families that raised very little, and both families fell on hard times. That is why great men like Ted Lindsay of the Detroit Red Wings tried to get a players' association for the families. Thank the Lord this would not happen to families today. The NHL and the players' union take care of the families. The NHL is a strong association, and it was made strong by men like the Babe Ruth of hockey, who packed the stadiums everywhere in the NHL he played, on the road or at home: Howie \"The Stratford Streak\" Morenz.\n\nI always feel sorry for the player who \"rides the pines,\" as they say. I know from experience how humiliating it can be, because I did it for a whole year and the playoffs. We won the championship, so it was a tough year. Thank goodness I only did it for a year; some guys do it their entire career. I don't know how they could stand it. I know what some people say: \"Well, they're getting paid.\" It doesn't work that way. You get up the morning of a game, have your tea and eggs, toast, go to the morning skate thinking things will change and you'll play. Maybe somebody is hurt (yes, it's like in the way when somebody is shot, you feel bad, but you're glad it's not you). After the morning skate, you go home and have your pre-game meal, have your nap, get the equipment on, warm up\u2014and then you sit. Your family watches, you see them for the drive home. It's embarrassing, and so when I became the coach of the Rochester Americans, I vowed nobody would ride the pines for me. In Boston, all my tough guys got 20 goals. It's just not right for a guy to sit and be sent out like a mad dog.\n\nI hurt my knee at training camp and all I did was go out and fight. I think I took it out on the guys I was fighting. I was so ticked off. The old saying, \"He sat at the end of the bench\"? It's not true. If you noticed, the guys who are benched are always in the _middle_ of the bench, so the forwards are at one end and the defence are at the other end and you're stuck in the middle (sounds like that song, \"Stuck in the middle with you\"). And for some reason, the organization blames the guys who don't play. Bobby Schmautz told me a story about when he played for the Chicago Blackhawks. The Blackhawks played a lousy game and lost. Billy Reay, the coach, came in and yelled, hollered and said, \"There's going to be changes on this club.\"\n\nBobby and another guy who never played a shift were sent down the next day. For some reason, the coaches like to pick on the guy who doesn't play. I don't understand it. You would go in and say, \"Why aren't I playing?\" and the usual answer was, \"Well, if you played better, you'd play.\" No sense in arguing, saying, \"How can I play better if I don't play?\"\n\nI love the line of an old-time player called Kenny Doraty. The coach had him \"riding the pines.\" The team was losing and the coach yelled at Kenny, \"Doraty, you never scored a goal,\" and Kenny came back with a beauty: \"Well, Coach, I'll tell you, it's tough scoring a goal from my angle of the bench.\" Isn't that great? I wish I was as quick as Kenny.\n\nI remember the year I didn't play and sat on the bench. After one game, I packed my equipment and threw it under the bus as we were leaving on a road trip that night to Hershey. I sat in the car with Rose, complaining about not playing. Finally, she said, \"I'm sick of you always complaining to me about not playing. This Joe Crozier [the coach]... why don't you complain to him?\"\n\n\"All right, I will. Good idea,\" said I.\n\nI got out of the car, grabbed my bag, went to Joe, and said, \"I want to play. I'm tired of sitting on the bench.\" And for emphasis, I kept hitting him with my bag.\n\nJoe said, \"All right, you'll play in Hershey.\"\n\nI didn't.\n\nI remember one game in the final for the American League championship. We were getting beat 5\u20131 in Quebec with five minutes to go. Joe benched a regular and put me out. I know what for\u2014to straighten a guy out. I did straighten the guy out in a beauty fight, and then we come back to Rochester for the next game. I did not play, as usual, but the game in Quebec where I had the fight had been on TV and the fans saw the fight and wanted me to play. So they started to chant, \"We want Cherry! We want Cherry!\" (I'm glad Crozier didn't do what the coach of the Toronto Maple Leafs, Punch Imlach, did when the fans started to chant, \"We want Shack! We want Shack!\" They wanted Eddie \"The Entertainer\" Shack to play. After a while, when the crowd kept up the \"We want Shack\" chant, Punch tapped Eddie on the shoulder and said, \"Shack, go up and see what they want.\" The rest of the players killed themselves laughing at Shack.)\n\nThe crowd kept it up, even though we were winning by a big score. I kept saying to myself, \"Please don't put me on. My legs are like lead.\"\n\nJoe put me on with five minutes to go, I guess because of the big score, and the crowd cheered. The first shift, a guy walked around me and scored. It was, \"Now you see why I don't put him on.\" The crowd never yelled \"We want Cherry\" again.\n\nI was fortunate that my knee came around okay, and from then on in my career I took a regular shift, but my heart always goes out to the guy in the middle of the bench. That's why, if he happens to get a goal, I always highlight the goal on _Hockey Night in Canada_ ; I always considered they are one of the group. It's like last year in the playoffs between Montreal and the New York Rangers. I was walking out to do the opening of the first game in Montreal. Now, the opening was timed down to the last second, and we were hurrying along the hall. We wound around a corner, and there were the Black Aces of the New York Rangers (the Black Aces are the guys who aren't playing and are just there in case of injury) and they wanted a picture taken with me. The TV girl said, \"We can't. We have to hurry.\" I told her I'd rather miss the opening than turn these guys down. They all had their cameras, and we got the pictures taken. You can't believe how good I felt that these guys wanted to have their picture taken with me. After all these years, I'm still one of the guys.\n\nI had my disappointments when I was in Colorado, coaching the Rockies. One was the backstabbing by an ex-teammate and friend. The story goes like this. The Rockies were playing in Philly. We were staying at the Hilton hotel, which was walking distance from the Philadelphia Spectrum, which was great. Walk over to the game and back again after the game. But it was a long way from the downtown and all the bright lights. Now, we had been on the road a long time and the players were good guys. I knew they wanted to have a few beers together, and I knew if they had to pay big money for cabs to go downtown, they would stay a long time; as we had an early flight the next morning, I made the decision to let them have beer in the hotel bar.\n\nNow, the policy of teams was for the players not to drink in the hotel bar. This was for many reasons. The GMs and scouts and newspaper guys drank in the hotel bar, with no owners. Anyway, I let the players have a few in the bar at the hotel and they were well behaved. Did you notice I said scouts drank in the hotel bar? One of our scouts was in the bar and sat with me the whole night. He was an ex-teammate and, I thought, a friend. In fact, now that I think of it, he was my defence partner in my early days; but I guess he could see the handwriting on the wall of my future with the Rockies, which was not good, and decided he didn't like me that much after all, and after sitting all night with me and the players (and me buying!), he phoned the GM and told him that the players were partying in the hotel lobby of the Hilton and I was drinking with them and the GM, who was building my coffin, phoned the owner and informed him.\n\nI had two guys on the club called Kevin Morrison and Trevor Johansen, two defencemen who were good guys, never any trouble. After they had a few, they both had some chocolate cake, and as guys will, one guy threw a puck at the cake and the other guy threw some and some got on the floor of the hall (had I known of the cake, I would have given the housekeeper a couple hundred dollars to clean it up, no problem). The manager of the hotel phoned our GM, who was delighted to tell the owner we had a wild party in the hotel and damaged the hotel's carpeting. I had gone through the rooms to make sure everything was all right, and I think the only thing I did was straighten a lamp shade (it's funny, I can still see that room), but this was one of the many nails the GM was putting in my coffin. You know I expected that from him, but I really was disappointed in my ex-teammate turning rat.\n\nI got fired at the end of the year, which I expected. The team was sold two years later and went to New Jersey, and my ex-teammate who ratted lost his job and struggled to get another. The word had got out he was not to be trusted. Hockey gods or karma will always get you in the end.\n\nI'm often asked what team I think is the best ever to play hockey. There is no doubt it's the 1976\u201377 Montreal Canadiens. In the 80-game regular season, the Canadiens lost only eight games all year\u2014and just one at home. I know this is hard to believe\u2014one game lost at home. The Boston Bruins were the Lunch Pail Gang\u2014punch in at seven-thirty and never stop working. We beat the Canadiens the one time they lost at home, and actually beat them in three out of the five games we played against them. Like I said, we could beat them in a single game and not play them again for a month, but once they got you in the playoffs, game after game, their depth took over, and their power play was a killer. They seemed to always get the first power play. The refs would give them about three of the first four power plays, they would get up about 4\u20131, and then you'd get a power play. One time, I refused a penalty with a minute and a half to go because it was so obvious they were trying to even out the penalties. The ref came once and said, \"If you don't send a guy on, we will give you another penalty.\" I said, \"Go ahead. Who cares?\"\n\nWayne Cashman, our captain, knew I was in trouble and said, \"Grapes, don't let them get the shutout. Let's try to score.\" Good old Cash, he knew if I followed through I would be in trouble. He was trying to get me out of trouble. I said \"All right, go ahead,\" but they really didn't need the referees. Their lineups were unbelievable.\n\nThey were tough, too\u2014Pierre Bouchard, six-two, 240 pounds; Rick Chartraw, six feet, over 200 pounds; Robinson was six-four. They could beat you any way, and they showed no mercy when they would play a weaker club. They never let up. Scotty Bowman would not allow it. Scotty was a tough coach. The players did not really enjoy his tough ways. Steve Shutt said, \"During the season, we hated Scotty. The only time we loved Scotty is when we got our playoff cheque for winning the Stanley Cup.\"\n\nIt was my bad luck to be coaching the Boston Bruins for five years at the time when the Canadiens were at their very best, but if you're going to be beat, at least you were beat by the very best.\n\nI was sitting in the airplane, staring out the window, on our way to L.A. Ron was sleeping. He has that knack\u2014one I unfortunately don't have\u2014for sleeping on the plane. He is out cold; one time, he fell asleep as soon as we got on the plane. We went to take off and got delayed for two hours, so when we were finally ready to take off, I said to Ron, \"We have landed.\" I set it up with the other passengers around us to let on we were landing. Ron got up, got his coat on, got out his carry-on bag and stood, ready to get off.\n\nI wish I could sleep like that. The flights are shorter, so they tell me. I flew to London, England. Out of two hundred passengers, there was only one light turned on for the whole trip.\n\nWe are usually in a cold sweat when we finally get on the plane. With everybody having those camera phones, everybody wants a picture. Hey, I'm not complaining. When they stop taking pictures, that's when I should worry.\n\nIt's June 7. Ron has been going every day since April 8. Somehow, he hasn't worn down. I have been going every other day since April 8\u2014not a walk in the park, but it doesn't compare to Ron's schedule. I know you're thinking, \"You guys must get sick of one another.\" Actually, we don't spend much time together, although we live only about ten to fifteen minutes apart. I kid Ron about living in the rich Oakville area, in a house worth a couple million dollars south of the Queen Elizabeth highway, while I live north of the QEW in a middle-class area of Mississauga in a little house (which is true). Although we live close, we never see one another. He travels in different circles. He is a wine guy; I'm a beer guy who travels in no circles. Ron is really a friendly guy; he visits people, goes to dinner at restaurants, flies with Cari to islands in the South Seas, goes to Europe a lot. I, on the other hand, have never forgotten my construction ways. I don't go to restaurants, I don't visit other people, I don't fly on vacations.\n\nI am a fortunate guy in the sense that Luba, who likes to travel, travels to different places with her family. They have a grand time together. I have always been a stay-at-home guy. I have noticed most of the coaches I know are the same way\u2014they like to putter around in the house or go to their cottages. I have a cottage I go to; otherwise, I spend time in my basement with my TV and my three goldfish.\n\nRon doesn't watch TV, but I'm a TV guy. Ron reads nothing but sport books. I read books on Drake. I have every book written on Sir Francis, as well as Lawrence of Arabia, Lord Nelson, Wellington. So we are completely opposite. When I go to his room on the road, I can't believe how neat and tidy it is. He has in his bathroom: a facecloth, a towel laid out with all his toothpaste, toothbrush, etc., laid out. He has the maid in to clean up (what's to clean?). I never have the maid in, just ask for fresh towels.\n\nI know you see us flying to L.A. and New York, and think, \"Wow! They must have fun after games\u2014dinner, bars, etc.\" After a game, we go back to the hotel and shower. Ron has twelve beers on ice, popcorn, peanuts, etc. We are in for the night. Occasionally after we finish, somebody will phone him, and it's \"Hey Ron, we are down at Maloney's Bar. Come on over, we're having fun.\" He goes over, but he pays for it the next day. And of course, I keep zinging him all day, which doesn't help how he feels, but that doesn't happen too often. It's back to the hotel room for a few pops to watch the hockey or baseball game (in the playoffs) and to bed. It's the only way to survive the schedule.\n\nI'm not complaining about life; in fact, I'm the luckiest guy in the world. Play for sixteen years (some would say \"toiled\" in the minors). It was a tough go and you had to be tough to survive. Guys fighting to get to the show on the way up and guys fighting on the way down. Me, I just stayed in the middle. Take care of your own injuries\u2014the trainers were usually only equipment guys\u2014spend days and nights on the bus. It didn't make any difference to me. I never knew any better, but it was tough on the guys from the show. The buses were a killer to them, never having experienced a ten-hour bus trip, playing the game, getting right on the bus and another ten hours to the next game, then getting on the bus after that game and another ten hours back home for a game that night. I did it for sixteen years as a player and four more as general manager and coach. It was not an easy life for me.\n\nI got used to it, but it's especially tough on the family. It was tough to go home after practice in Springfield when I played for the Springfield Indians. I was in the bad books for breaking my stick over the coach. I had to tell Rose we had to leave tomorrow. I'd been sent to Three Rivers, Quebec, in the middle of winter. That's when Eddie Shore gave me directions to Three Rivers \"through Vermont, hit the Canadian border, turn right.\" We were gypsies, and I had turned into a hockey bum. I hate to admit it, but I was\u2014no chance for the show, just surviving because I really had no trade (jackhammer specialist), no education.\n\nI felt bad for my dad. He wanted me to make the NHL so bad, used to keep a little write up in _The Hockey News_. There wasn't much after my rookie year. I remember he kept a little part of a crossword puzzle in _The Hockey News_ : \"name two brothers that play for Springfield.\" It was Richard and myself. I felt I had let my dad down. He died at sixty-two, and I was still slugging it out in the minors. Gosh, I wish I could have made the NHL for him so he could have been proud of me.\n\nRon is sleeping, like I said, on the plane and I'm staring out the window. I wonder, where did I go wrong? How\u2014after a tremendous rookie camp, after I played my one game in the NHL and took a regular shift and was touted as a future Bruin\u2014how did I get on the road to oblivion and the minors? It started the summer after my rookie year. Broke my shoulder playing baseball. The Bruins' manager was furious at me. I went to camp, the cut from the operation still bleeding, and went to the trainer, who gave me Band-Aids to help. Now, don't get me wrong: I blame nobody but myself for my situation. I was told not to play baseball and I did, so they had reason to be furious at me. There was no need to break my stick over Pat Egan, the coach for Springfield Indians, and sometimes I really act like a jerk. I suffered, but so did my family, and like I said, nobody to blame but myself.\n\nBut looking back, a couple of people stick out as unusually cruel, whether they knew it or not. I don't know. Hey, I had Hap Emms for a coach in Barrie. He was the toughest, meanest coach, but one of the best. He was tough on me, but that's the way he was with everybody. Eddie Shore, the Darth Vader of hockey, whose team was Devil's Island for a hockey player, was especially tough on me, but he was tough on everybody. Punch Imlach coached me. He was tough. He let Tony Schneider and me fight in practice. You could tell we were exhausted. He was tough, but again, tough on everybody.\n\nBut two guys, and what they said and did to me, keep running through my mind. And they forgot about it as soon as they did it. One person was the GM of the Boston Bruins, Lynn Patrick. He was the son of the famous hockey guy, Lester Patrick, who saved the Rangers when, in the Stanley Cup finals, his goalie got injured. He had no goalie. At forty-four years old, he went in for the goalie, won the game, and the team won the Stanley Cup. Lynn had a brother, Muzz, and they both played in the NHL, too. It was a famous hockey family. To you and most people, what he did to me will seem trivial.\n\nI was at the Bruins camp in Boston. It was my third year. The management still hadn't forgiven me for playing baseball and getting hurt. In those days, once you were in the doghouse, you stayed in the doghouse. Guys who didn't get the ice time I was getting were called up to the show, but not me. But that's the way it was. No use crying over it; you accepted it or you quit. I was walking to practice through the Boston Garden by myself, and I saw Lynn Patrick, the GM; Milt Schmidt, the coach; and another guy I don't remember. As I walked by, Lynn Patrick said, \"Don, you were on that Memorial Cup team in Barrie. What was the year?\" Hey, he was talking to me. I was thrilled. As I went to answer, he said, \"Ah! He wouldn't know.\"\n\nIt tell ya, it was like a knife in the heart. I was so embarrassed and hurt. I know it seems like nothing, and of all the bad, hurtful things that have happened to me, that was one of the killers. But the one that I remember to this day was the shot I took from a Bruin defenceman named Bill Quackenbush. I was really having a hard time with my shoulder, and like I have said, I could not shoot the puck hard. I remember it like it was yesterday. Bill really didn't have to worry about me taking his place\u2014he was an all-star, but I must admit, a soft all-star. Put it this way: he got no penalties all year and he was a defenceman, so you can see what kind of a person he was. Still, he was an NHL player and I respected him. I took a shot at the goaltender (it was not good), and Bill turned to the coach, Milt Schmidt, and said, \"Hey Milt, this guy's got a worse shot than me.\"\n\nI know it doesn't sound like much, but between my shoulder, Lynn saying \"he wouldn't know,\" and my failure to make the show, it was the final zing. I didn't say anything\u2014what could I say? He was right, but afterward, I thought, \"What kind of a man would ridicule to the coach a young player who obviously was having a bad time?\"\n\nI just thought of something. When I was a rookie hot shot in the Boston camp, I sat beside an older defenceman called Andy Branigan. Andy was near the end of his career. He told me a strange story about when he played for the New York Americans in the old NHL. Their goalie got hurt, so their coach, Red Dutton, put Andy in net, because Red knew Andy played net in lacrosse. They lost the game to Detroit 5\u20134, but Andy never let in a goal. Andy scored in that game before he went in net, so he scored a goal and played goal in the same game (the goalie he replaced, Andy told me, was Hall of Famer Chuck Rayner). Andy only played a few games in the NHL. The war came and he enlisted in the Royal Canadian Air Force in 1942. Andy had a tough time to make the NHL when he came back. He had given up his career to help his country, like a lot of guys who gave up their careers with the NHL.\n\nOne, for instance, Doug Orr, the dad of Bobby Orr\u2014they tell me he was a better skater than Bobby and would have shone in the NHL, but he gave it up to enlist in the Canadian Navy. He was posted to a corvettes, guarding convoys in the North Atlantic from the German submarines. Doug gave up a career in the NHL, like Andy. It was his duty. (By the way, the corvette ships were said to be one of the most uncomfortable boats in the navy. They were called \"tin cans,\" but they did the job.) Doug spent four years on the \"tin cans,\" and when he got discharged, he, like Andy, found his career in the NHL gone. Both were heroes of mine.\n\nBut at the time, I just knew Andy as an old American Hockey League player near the end of his career, and it looked like I was going to take his spot (which I eventually did), but Andy (a Winnipeg boy) was so kind to me, giving me advice in hockey and life. He told me one day, after I got cut in practice, to rub cocoa butter on my cuts and they would disappear. I did, and they did disappear. I was young and single and I was staying out late, having a good time with girls. One day, Andy sat me down and said, \"Don, I have found in my life you have to make a decision. You can drink or you can go out with girls, but you can't do both, because when you drink, you go home and go to bed at a reasonable hour. But if you drink and stay up late with girls, it will be the end of you. You can't do both. Make a decision.\" I took Andy's advice and just drank.\n\nI think now of Hall of Famer Bill Quackenbush\u2014famous guy, all-star, ridiculing a rookie who was hurt\u2014and how cruel he was. It's funny, he won the Lady Byng. Andy was a confirmed minor leaguer, played fifteen years in the AHL and then went to the Eastern Hockey League. Bill Quackenbush went a whole year with no penalties, yet he was a mean man. Andy was a tough defenceman who was an enforcer type. One was soft, yet cruel; the other was tough, yet kind. Andy taught me a valuable lesson: it doesn't cost anything to be kind. It's funny; I only knew Andy for a little while at training camp\u2014one week. Andy left after training camp\u2014no goodbyes or anything. I never knew where he went or ever saw him again, yet I've never forgotten him and his kindness. A ship that passes in the night.\n\nI know I have spent a lot of time on a minor leaguer who I only knew for a week, but guys like him and Doug Orr made a deep impression on me.\n\nWhen the experts of sports get together, one of their topics is \"How great is a record?\" The record the experts bring up as the greatest that was ever set is Babe Ruth's 60 home runs in 1927. But before that, in 1920, he hit 54 home runs, and the experts point to the player who hit the second-most home runs that year, George Sisler, who hit 19. Now _that_ is a record. If the guy who came in second had hit 40 home runs (like in 1927, when Lou Gehrig hit 47), then Babe's record would have been good, but to hit 54 and the next guy is so far behind\u2014that's the way they judge a record.\n\nIn Bobby Orr's third season, 1968\u201369, he scored 64 points\u2014a record for a defenceman. He broke the record of 59, set by veteran defenceman Pierre Pilote, who played for the Chicago Blackhawks, in 1964\u201365. Pierre was a super defenceman. I remember when I broke into junior with Barrie, I carried the puck up the centre of the ice with my head down. Pierre, who played for the St. Catharines TeePees, nailed me good with a shoulder check. I saw stars for a while. I definitely would have been put in the quiet room if it happened today.\n\nPierre won the Stanley Cup and was voted the best defenceman three times, but Bobby came along and broke his record. And a couple years later, in 1970\u201371, Bobby set the record that stands today: 139 points. Imagine, the record was set at 59 points. Seven years later, Bobby scores 139. Just like Babe Ruth.\n\nMore on Bobby. As I write this, I see it's possible that a defenceman could win the Norris Trophy (for best defenceman) and be a minus hockey player. Believe it or not, three defencemen won the Norris and were minus defencemen (Randy Carlyle, minus-16 in 1980\u201381, Rob Blake, minus-3 in 1997\u201398, and Nicklas Lidstrom, minus-2 in 2010\u201311). When Bobby played for me in 1974\u201375, he scored 46 goals and had 89 assists and was a plus-123. In 1970\u201371, he set the all-time record: plus-123. He won the Norris eight years in a row, another record. I remember what Harry Howell said when he won the Norris Trophy in 1966\u201367, just before Bobby started his run. He said, \"I'm glad I won this trophy this year, because with Bobby Orr around, nobody else is going to win it for a long time.\"\n\nBobby's trophies are another record nobody gets close to, starting with two Stanley Cups, eight Norris Trophies, two Hart Trophies (most valuable player), two Art Ross Trophies (leading the NHL in points), a Conn Smythe Trophy (MVP in the playoffs), MVP of the Canada Cup in 1976. Actually, 1974\u201375 was really his last year. Like I said, he won the scoring title (for the second time); 46 goals, 89 assists; plus-123; Norris Trophy for the eighth time. But in '76 he went to the Canada Cup, where Canada won, with a defence that consisted of Larry Robinson, Denis Potvin, Serge Savard, Guy Lapointe and Bobby Orr. Bobby got picked as the MVP of the tournament, and really, that was his last hurrah, as they say. He left Boston on the advice of his agent and friend, Alan Eagleson. Unknown to Bobby, Boston had offered him not only a good salary but part of the Boston Bruins. Eagleson failed to inform Bobby, because he had a deal cooking with Chicago.\n\nBobby had to retire because of his knees. People never saw Bobby on the downside of his career, only at the top, but it was a tough pill to swallow, retiring at twenty-eight, just when defencemen are normally coming into their prime. I could go on about Bobby, but you might think I'm prejudiced because he is a friend. Let's see what the bible of hockey books says about Bobby. It's called _Players: The Ultimate A\u2013Z Guide to Everyone Who Has Ever Played in the NHL_ , by Andrew Podnieks. The book contains more than 5,600 entries. It starts the story of Bobby like this: \"Perhaps on one of the cold planets out there in the vast universe there exists a player who was as good as Bobby Orr, but on this planet no man who ever put on a pair of skates can even pretend to be equal to Bobby Orr.\" The write-up ends: \"Almost from the day he started playing in the NHL, he has inspired future players to emulate his style. There will never, ever, be another Bobby Orr.\"\n\nI can't count the times I've heard a coach holler at a defenceman who screws up while carrying the puck: \"Pass the puck. Who do you think you are? Bobby Orr?\"\n\nRecently, Bobby Orr was on his book tour and he was being interviewed by someone on some obscure station I never heard of. He never knew I would see it\u2014Luba found it on the Internet. The commentator, who I'm sure never heard of me, asks Bobby, \"If you were in the war and you wanted one guy you could trust by your side, who would that be?\"\n\nBobby answered, \"Oh, Grapes for sure. He would never throw me out of the foxhole.\"\n\nI got all choked up. Bobby does not know I saw the interview. He won't know till he reads this book. I can tell you now, it's the greatest compliment I ever got in my life.\n\nI remember looking at the 1976 Canada Cup team and saying, \"This is the greatest team in history. We are going to have to work hard to screw this up.\" And you know, we almost did.\n\nI was with the Boston Bruins\u2014just made coach of the year in the NHL. I got a call from the head guy, Alan Eagleson, asking if I would coach along with Scotty Bowman. I said I'd love to, but I had to ask my GM, Harry Sinden. Harry was not too happy, but said okay. I remember his last words: \"Don't you pick pucks out of the net for anybody.\"\n\nAfter I told Eagleson I would do it, I asked who was coming. Eagleson started to name the guys: Rogie Vachon, Gerry Cheevers, Glenn Resch, Bobby Orr, Denis Potvin, Serge Savard, Guy Lapointe, Bobby Clarke, Darryl Sittler, Marcel Dionne, Lanny McDonald, Paul Shmyr, Dave Burrows, Peter Mahovlich, Reggie Leach, Phil Esposito, Rick Martin, Bob Gainey, Bobby Hull, Gil Perreault, Dan Bouchard, Dan Maloney, Billy Barber, Danny Gare, Carol Vadnais, Jean Pronovost and Jim Watson. Coaches Scotty Bowman, Al MacNeil, Bobby Kromm and me. Management: Al Eagleson, Jean B\u00e9liveau, Toe Blake, Keith Allen, Sam Pollock. Doctor Doug Kinnear. Trainers Lefty Wilson and Eddie Palchak.\n\nI said, \"What a team. You know, of course, we are going to have to cut some 50-goal scorers. This is the greatest team ever.\" Eagleson said, \"Yeah, I know. We better not screw up. We got pressure after that '72 series.\"\n\n\"Don't worry, Eagle, we will make it. What can stop us?\" Little did I know.\n\nWe all met in Montreal and stayed at the Queen Elizabeth Hotel. We had our meeting\u2014Eagleson, Bowman, Kromm, MacNeil, Beliveau, Blake, Allen and the head guy, Sammy Pollock. We went over everything. Sounds good, ready to go. As we were leaving, Eagleson said, \"The guys go for their testing tomorrow morning.\"\n\nDon: \"What do you mean, 'testing'?\"\n\nEagleson: \"To see if they're in shape.\"\n\nDon: \"What's the test?\"\n\nEagleson: \"Bikes... skating, of course... sit-ups, push-ups...\"\n\nDon: \"Hold it, Eagle. Have you told these guys to prepare for this?\"\n\nEagleson: \"We told them to be in shape.\"\n\nDon: \"But you know these guys. They have never done push-ups, chin-ups, bikes...\"\n\nEagleson: \"This is a brand new way of seeing if guys are in shape.\"\n\nDon: \"What if Potvin, Savard, Orr and Lafleur aren't in shape for the bikes? You gonna send them home?\"\n\nEagleson: \"Quit causing trouble the first day. It's done and that's it.\"\n\nI was on the outs and the camp hadn't even started.\n\nThe next day at ten o'clock, I went into the exercise room and some gung-ho young college guys and gals were explaining what Lafleur\u2014who never in his life lifted weights or rode bikes\u2014and Bobby Hull, whose body was natural and the best of these superstars, were expected to do. They were looking at me as if to say, \"Is this for real?\"\n\nPush-ups, bikes, sit-ups. This was the killer. Well, you guessed it, players pulled groins and suffered injuries. Worst of all, Bobby Hull pulled his groin. Bobby really wanted to shine. The reason Bobby, maybe the greatest left winger of all time, was left off the '72 team was because he'd signed with the World Hockey Association. There were still bad feelings about Bobby because he was the first guy to sign with Winnipeg Jets of the WHA and opened the doors for players to make big money, not only in the WHA but in the NHL too. So they kept him off the '72 team, but he was here for the '76 series. He was anxious to show his stuff, and here he was, injured because of ridiculous sit-ups.\n\nLet me describe to you what was going on. Say you have a superstar doing sit-ups beside another superstar. They are not going to quit and look bad, so they would strain themselves. And, seeing as they were not prepared for these exercises, they hurt themselves. A lot of them came to me with their woes. I said, \"Look, tell the guys to go along with this bullshit. Don't overdo it. Don't hurt yourself. So they played the game of pretending.\n\nI thought everything was hunky-dory till one morning, when I was called to a meeting at the Forum. The brass were all there, looking glum.\n\n\"They are sending Guy Lapointe home. I know he is one of your favourites. We don't want an argument.\"\n\nDon: \"What has Lapointe done to be sent home in the middle of camp?\"\n\n\"He refuses to run.\" Every morning, the college guys were making the players run a couple of miles on a track.\n\nDon: \"Now, let me get this straight: you're gonna send home one of the top defencemen in the world home because he won't run.\"\n\n\"That's right!\"\n\nDon: \"Wait a minute, guys, we got the greatest defence pairs of all time and you're gonna break them up because he won't run.\"\n\n\"That's right. Please, we don't want an argument.\"\n\nDon: \"But guys, he works with Savard and Robinson so well. The Big Three, and then with Potvin and Orr, it's a dream.\"\n\n\"If we let him away with it, we will look bad and that there's no discipline.\"\n\nI had to come up with something, so I said, \"Guys, the reason he won't run is that he can't. He's got flat feet, and you know you can't run with flat feet. You just can't send him home.\" They bought it.\n\nI happened to mention to one of the players that it was too bad Guy couldn't run because of his flat feet. They caught on.\n\nThat was my first introduction to this mentality about training.\n\nThey have the same ridiculous nonsense going on now for the young draftees in the NHL. It's called the combine. The young players come to a camp with NHL scouts who put them through that nonsense: bikes, jumping high and jumping far. (I remember the German player, Uwe Krupp, who scored the Stanley Cup winner for Colorado in 1996. The next spring, they did these tests, like the one where you see how far you can jump. Uwe tried to see how high he could jump, and came down hard on his knee. That injury stayed with him.)\n\nThe reason they say they do the testing at training camp is that when you take the test at Christmas, you will see if you have progressed. Oh, I get it: if you are a 25-goal scorer at Christmas and you take the test and you haven't progressed, are they going to send you to the minors? If a guy is scoring for me, I'm not going to worry him about testing. In fact, you could throw him off, confronting him with his test result. It's nonsense. The players know it, but they're not going public and saying it, so they play the game. What are they going to do? When I hear scouts say, \"Well, there is no harm in it,\" I think of Uwe Krupp.\n\nLarry Zeidel, my roommate in Hershey for two years, was one of the few Jewish players to play hockey in professional hockey. His nickname was \"The Rock,\" and the name suited him well. He was with Detroit when they won the Stanley Cup in 1951\u201352. He went to Chicago and then to the minors\u2014with me as a defence partner in Hershey of the American League. I remember I was just a kid, and he was so serious about things. One day, he said we should be like football quarterbacks. We should call out numbers for plays. Like, when I get to the back of the net, I'd call different numbers for different plays. It lasted one period. Nobody fooled with Larry and his ideas, which were far out. One day, the trainer, Scotty Alexander, made faces behind Larry's back as if he was nuts. Larry turned around and caught him, and knocked him flying. Don't fool with the Rock.\n\nLarry always told me he would make it back to the NHL. After thirteen years, when he was almost forty years old, he told me, \"Expansion is coming. I'm going to send out my resum\u00e9 to make a comeback.\" I thought to myself, \"Fat chance,\" but Larry fooled me. The Philly Flyers gave him a chance and he was back in the NHL at forty years old.\n\nIn 1968, when he was playing for Philly, the wind lifted the roof off the Spectrum and the Flyers had to play against Boston at Maple Leaf Gardens in Toronto. Eddie Shack and Larry got into a beauty stick fight. Both got cut. It was a sight to see. Both got suspended by Clarence Campbell.\n\nLarry was always playing the stock markets, and when the Flyers tried to send him to the minors, Larry said he'd had enough of the minors. He retired and became a successful stockbroker.\n\nHe died June 17, 2014. It's hard to believe The Rock is gone. Somehow, it always seemed to me that Larry would be around forever. It makes me think.\n\nI am writing this after the sixth game of the Western Conference finals between the L.A. Kings and the Chicago Blackhawks. I should say \"war\" instead of game: they are wearing each other out\u2014over ninety hits between the two teams. L.A. had more hits than the two teams in the Eastern final, the Montreal Canadiens and New York Rangers, combined. Between the Rangers and Canadiens, there was a total of thirty-five hits. LA had forty-eight.\n\nThis playoff reminds me of the playoffs we used to have between the Boston Bruins and the Montreal Canadiens. In 1976 and '77, Boston ended up in a dogfight with the L.A. Kings. L.A. had a big, tough team with tons of talent and Rogie Vachon in net. Naturally, the series went seven games in 1976 and six games the year after. We would have to fly coast to coast, and the Bruins were cheap. We would fly commercial while other teams had a charter.\n\nIt was funny; before I came to the Bruins, they chartered everywhere. After my first year, Jeremy Jacobs, the owner, decided we would fly commercial. Little did he know he'd done me a favour. When we flew on a charter, we would fly back to Boston right after a game and the players would go their separate ways: some home, some to one bar, some to another bar or restaurant. When the owners, in their cheap wisdom, decided to make us go commercial, we usually had to stay overnight in the visiting city. Say it was Chicago. After the game, the whole team would go together to their favourite bar and be a team. It was so much fun. We had a family feeling. Naturally, I would be there. I did something that was never done before or since: I used to be able to go out with the players for a few pops. I know it is not a good policy, but it worked. I could do it. I would stay for a few hours, and then we had no trouble.\n\nIf a player was traded to the Bruins, the captain, Wayne Cashman, and a few of the tough guys on the club had a meeting with the new guy and explained the rules. \"We have a first-place club, no team in sports has more fun than we do, and you're not going to ruin it here or break rules. There's not many, but they have to be followed. We've never been fined or suspended. If you break the rules, you're gone. If you have a few pints the night before the game, you can get a little glow, but don't stay out the night before a game. A few guys have broken this rule. They were sent to the minors or gone, traded the next day.\n\n\"Next, don't get into trouble in the hotel where we are staying. If you do, the hotel manager will phone the GM of the Bruins, Harry Sinden, and now we're all in trouble. Next, don't get arrested. We have a beautiful thing here. Sometimes it's hard for me not to smile at the wife when we go on the road. I can hardly wait to get on the road to have fun. For instance, win or lose, after a game, Grapes will say, 'The bus leaves at ten o'clock for the airport. In other words, go out and have fun, but be ready to leave at ten o'clock.'\n\n\"So we have three rules off the ice. Naturally, we have to look sharp travelling, so don't screw up and make Grapes look bad, because if you do, you'll hear from us.\"\n\nI used to let the veterans run the dressing room. If it came down to trouble, I would take care of it. I was one of them. It was us against the world; if I saw a slight, I would make a big deal of it. I used to say, \"We're a first-place club, and look: no votes for all-star.\" Stuff like that.\n\nWe were like a band of brothers, and the winning and fun never stopped, even in the practices. The practices were not long\u2014an hour and ten minutes\u2014but always moving. No stoppages for a blackboard talk, and instructions were given before going on the ice. There is nothing worse than working up a sweat and having a coach make you stop and have him draw plays on a blackboard like you're at public school. (The best coach in junior hockey, Brian Kilrea, never used a blackboard. He would not have them in the dressing room.) My practices were like clockwork, always the same. You are thinking \"boring.\" Never. Fast and furious, like a hockey game. Nobody yawned in my practices. They were a hockey player's practices.\n\nI knew the cheaters, and they paid the price after practice. One time, when I was in the American Hockey League as coach and general manager of the Rochester Americans, one guy stayed out late and did what I call cheating the hockey club. I made him skate so hard after practice, all by himself, with me yelling \"again,\" and while he was lying on the ice, gasping for breath, one of the cleaning ladies who was sweeping the empty arena came down and said, \"Stop what you're doing to that poor man. That's inhumane.\" I was almost finished with the punishment, so I said to her, \"You're right, ma'am. I'll stop right now.\"\n\nThe Rochester Americans, the Boston Bruins, the Colorado Rockies all had fun when I coached. Mike Milbury, who played for me and is now a TV commentator on NBC, when asked what kind of a coach I was, said, \"I never had so much fun as I did playing for Grapes. He was a great coach.\" Because if you have a job and it's fun and you look forward to going to work, you'll do a better job. I would call an optional practice and everybody would show up. I had to tell the vets to stay home.\n\nI mentioned earlier that I went with the idea that everybody was against us. We had no friends. We had to look after ourselves, and the players had to feel that I would lay my job on the line for them. For instance, when we would have an injury, I would tell Harry Sinden, the GM, I didn't want a replacement. We would go a man short. He was not happy, but seeing as we always won, he'd go along with it. It sounds like I'm taking a chance\u2014not at all. I knew the players could see what I was doing, protecting their jobs. They thought I was taking a chance for them. In a way, I was, but you really don't need eighteen players. It's a joke to have that many players. The players learned it when they got more ice time, and between getting more ice time and thinking I'm protecting their jobs, they went 1,000 per cent.\n\nI know what you're thinking: \"If you don't need eighteen players and the players like more ice time, why have eighteen players?\" The player's union would not put up with anything less than eighteen players.\n\nI changed after my first year with the Bruins. The first year, I come from Rochester, a minor leaguer\u2014coach of the year in Rochester, but still a minor leaguer. The Bruins were Stanley Cup champs in 1972 and went to the finals with Philly in 1974, the year before I arrived. I did not act like I did with Rochester. I thought, \"They'll be there when the bell rings.\" My first year, we were in the playoffs against Chicago. In the deciding game, we outshot Chicago 52\u201318 and lost.\n\nTony Esposito was a great goalie. We beat him in the first game, 8\u20132. The papers said he was finished. When I read that about Tony, I knew we were in trouble. Sure enough, he stood on his head and you couldn't put a pea by him, and we lost.\n\nI had a meeting with the GM, Harry Sinden. I said, \"Look, Harry, I let you down. If you want to let me go, I won't demand the last two years of my contract.\" Harry was great. He said, \"Don't be silly. But coach like you did in Rochester.\" But then Harry said the fatal words: \"Look, we have to win. If you don't, we're both gone. Do anything. Make me the heavy, just win.\"\n\nI went home that night and played that song by The Who, \"Won't Get Fooled Again,\" and did an about-face. I made enemies with the NHL, the Bruins organization, scouts and, I'm ashamed to say, the guy who took me from the minors and gave me a chance: Harry. I showed no mercy. If I was going, I was going. The team actually thought the league, the organization, everybody was against them, and it worked. Four first-place finishes, twice to the finals and twice more to the semifinals. Winning percentage: .658. Coach of the year. I wasn't making much money, but I was having fun. I loved the players and they loved me. Fun, winning... who could ask for anything more?\n\nBut all good things must come to an end. Harry and Paul Mooney, the team president who represented the owner, Jeremy Jacobs, had had enough of the bad-guy routine. Yeah, we won, and they made money, but I was becoming unbearable. The players would only listen to me. I remember one time, Brad Park, our star defenceman, and two other regulars had played in eight exhibition games and did not have to play any more. The Bruins organization had scheduled a big-time exhibition game in Montreal at the Forum. It sold out. Top prices for an exhibition game. Now, going to Montreal, you had to have your players play. It was big time. The players told the Bruins, \"No, we have played eight games. We're not playing.\" What to do?\n\nThe organization had to come to me and say, \"You talk them into playing. They'll do anything for you. If you don't, we are in trouble.\" I went to the players and said, \"Look, play the game. They'll owe us a favour. I'll get them to pick up the tab for the beer after the games on the road.\" They played.\n\nBut it had to end. I was beginning to act like it was my team, like I owned the club. In other words, I got carried away with myself, so Harry and Paul Mooney had a meeting with me, and over lunch they said, \"Look, Don, we appreciate the wins and the way the team plays. But you've got to change\u2014especially your relationship with the press.\" The press and I had a love affair going. If they missed a column, they could phone me at 11:00 P.M. and I'd give them a column. I helped them and they protected me. For instance, after some games, I'd go nuts, yelling at the league, Harry, Paul Mooney and the refs, and I'd wake up in the morning, thinking, \"Oh, why did I say that? I'm done.\" But I'd pick up the paper and they'd have written what I'd said as if it was their own idea. The Boston press was so good. They figured I helped them, so they'd protect me. It was beautiful. But Harry and Paul Mooney had had enough, and so this was the rule from now on: you change, or see you later.\n\nI said to Paul and Harry, \"Look, I could say I will change, but you and I know eventually I will say something.\" In my heart, I knew I was done. The only reason they didn't fire me was because I was so popular. I made it easy for them. In other words, if they didn't get me now, they would get me later, so we parted, shook hands and it was friendly.\n\nThe players all came to the house that night and so did the press. The players and the press were as sad as I was, and so was Blue, who sat beside me the whole time. It's funny how dogs know what's going on. So it was sad, but there were no hard feelings. These things happen.\n\nBut something happened the next day. The Bruins had a day for the fans, and a friend of mine\u2014not a good friend, but a friend\u2014told me Harry said to him, \"I wonder what Blue thinks now?\" Meaning I used to say, \"Blue says we should play Cheevers tonight.\" It was sort of a running joke with the press. That did it. The feud started, and it lasted for years, Harry and me sniping at one another, me on \"Coach's Corner\" and him in the Boston press.\n\nI had an unfair advantage with _Hockey Night in Canada_.\n\nMy son, Tim, and I were driving to see a minor midget game one night and he said, \"You know, Dad, you've known Harry all your life. Playing against him. He gave you your very first NHL chance. Why don't you bury the hatchet?\" And I thought, \"Tim is right.\" So I extended the olive branch to Harry. I went on \"Coach's Corner\" and said something like \"Harry Sinden is the best general manager in hockey. I was lucky he was my GM.\" Somebody told Harry, and he came back with, \"Oh yeah? Well, if a guy like Sam Pollock said it, it would mean something.\" The feud was back on with a vengeance.\n\nTill one day, I got invited to the closing of the Boston Garden. It was called \"The Last Hurrah.\" The Bruins had all the players and coaches from the past. (They gave the players sweaters, jackets and caps, and they gave the former coaches beautiful jackets, shirts and sweaters. I loved a Boston Bruins jacket.) We really were treated royally. Unfortunately for me, I did all kinds of interviews on TV, and when I got back to the dressing room where we'd all met, somebody (one of the former players) had taken all my stuff. I ended up without even a program. But at least the Bruins and I made up after all these years. Ron MacLean still kids me that Harry and I hugged on the ice\u2014I'm not the hugging kind.\n\nI especially liked it when Harry took me around and introduced me to all the former coaches and players. He introduced me to Pat Egan, who was my coach in Springfield (and who, along with Eddie Shore, made my life miserable in Springfield). Pat was the guy who was delighted to send me to Three Rivers, Quebec, at Christmastime. I have to admit, it made my heart feel good.\n\nHarry: \"You know Pat Egan, don't you, Don?\"\n\nI thought of a million things to stay, but I chickened out and just said, \"Yeah, I know Pat.\"\n\nThe reason I started to write this story was to talk about how hard it is to be like the L.A. Kings, with all the travel and playing three series that went the full seven games and how hard they hit. In the sixth game of their series with Chicago, the Kings had 48 hits and Chicago had 35. It was a war. Chicago played well, but lost the series in a heartbreaker.\n\nI am writing this the morning after the seventh game, which L.A. won in overtime with Alec Martinez scoring the winner on a shot from the point. I leave tomorrow for L.A. I predict the Kings will win the Stanley Cup in six games. I will leave this on my desk and pick up the end of the story when I come back from the Stanley Cup. L.A. in six.\n\nYeah, I know I was wrong. L.A. won it in five games. My kind of team: big, tough, and fifteen Canadians.\n\n# **HOCKEY TODAY**\n\nONE NIGHT ON \"COACH'S CORNER,\" I showed some basketball players walking into their game dressed like slugs, and then I showed some hockey players walking into their game dressed like they just stepped out of _Esquire_ , and then I showed Evander Kane of the Winnipeg Jets in an interview, and he looked like a male model. I was making a point. Hockey players have respect for themselves and respect for the game. In many ways\u2014the way they act and the way they dress. Football, baseball and basketball have many suspensions for drugs and for violent acts with knives, guns, you name it. Of course, when you have seven hundred hockey players, you are going to have some jerks, but they are few and far between.\n\nThe NHL has a strict drug policy. They do not give a slap on the wrist for drug use like other sports do. If you take drugs in the NHL, you might as well put a big _D_ on your forehead. It is never forgotten and never forgiven, and surprisingly, it is not forgotten by the other players.\n\nWhen I coached the Bruins, a young player got caught with marijuana when he was on a road trip with his team and got caught at customs coming into Canada. He had left a joint in the pocket of a shirt and forgotten about it. Headlines in the papers and he was suspended. He was never the same. \"Hey, isn't that the guy that got caught with the joints?\" I was surprised to hear the Bruins hollering from the bench to the players: \"Hey Pothead,\" they shouted, because the players are all tarred with the same brush if one gets caught. I know what you're thinking. They were mad at him for getting caught. No way. They were ticked because he made the sport look bad.\n\nThe NHL has a strict policy. Random testing on and off-season. Nobody knows when the drug teams will show up, and if you're caught, there's no mercy. It's not just a suspension; it could be your life\u2014which is the way it should be, not like one sport where the player had in his contract that if he gets caught with drugs, he still gets paid, and when he comes back to the team, all is forgiven.\n\nI remember back in the seventies when I coached the Boston Bruins, a player got caught with a joint. The New York Rangers management, including their coach, said they were backing our player. I had a meeting with the Bruins players. I started the meeting by saying, \"You like the way the New York Rangers management and coach back their players?\" Naturally, they all said yes. I said, \"Well, don't expect that from me. If one of you guys gets caught with drugs, I'll cut your heart out. I not only will ruin your hockey life, I'll make it so you have no life when you go back to Canada, and don't come begging for mercy. You'll get none.\" Needless to say, we had no problems with drugs.\n\nI remember a young hockey player who was a good player, who had a tough time getting drafted because he got caught with a joint. So it's the fear of getting a bad reputation. We do not treat drugs like other sports do, but I feel there is another reason. I feel it starts at a young age. In Canada, I feel we have respect for our sport. My son, Tim, works for the Ontario Hockey League and he rates the players in the Greater Toronto Hockey League to be drafted by the OHL. At nine-thirty at night, when Tim and I go out to watch the minor midget games, as we are coming into the arena, the bantams are just leaving after their game. It does my heart good to see them with their team jackets and their shirts and ties. What sport in the world has young players wearing shirts and ties to and from their games? The same as our junior teams\u201490 per cent shirts and ties.\n\nI remember in the American Hockey League, we'd travel ten hours on the bus, but when we'd get off we would have shirts and ties, respect for the game and respect for ourselves.\n\nHockey players have a great reputation, especially down in the States, for courtesy and politeness. Bob Verdi, the great _Chicago Tribune_ newspaper sportswriter, said to me once, \"After the baseball players, football players, and basketball players, having the hockey players come to town is like a breath of fresh air.\"\n\nThere is always an exception to this policy\u2014for instance, the colourful Derek Sanderson of the Boston Bruins, who couldn't stand wearing ties. One day, the Bruins were going on a road trip, and as they boarded the bus to the airport, Derek got on the bus without a tie. The coach, Harry Sinden, stopped him and said, \"You know there is a hundred-dollar fine for not wearing a tie on a road trip. Here, take one of my ties.\" Derek looks at the tie of Harry's. He tosses the tie back to Harry and says, \"I'll pay the hundred-dollar fine.\"\n\nI was sitting with a few friends, having a couple and shooting the breeze about hockey. One guy said to me, \"Grapes, you've been around hockey a long time, been a lot of places. First, tell us the places you've played and coached, and what do you think you've contributed to hockey?\"\n\nI said, \"It's funny you have asked where I've been playing and coaching. Stanley, our stats guy, for _Hockey Night in Canada_ , once heard me say on a broadcast, 'Nobody has played or coached in as many cities as I have. I played in every professional league that existed\u2014American Hockey League, Western Hockey League, the Eastern Professional Hockey League, and one game in the NHL. I coached in the NHL and the AHL.' Stanley handed me a list of where I played and coached. The list includes fifty-three cities I coached or played in, twenty-two different states and six different provinces. I know in a way that sounds impressive, but think about how my family had to travel to be in those leagues. I put it down to a lot of teams wanted me.\"\n\nMy contribution to hockey, of course, that everyone in the minors knows or should know, is how to keep the beer cold after the game on the bus. The routine: take a pillowcase from the hotel. When the bus gets rolling, put four beers in the pillowcase and hang it out the window on a freezing-cold night. In ten minutes, presto, ice-cold pops. I almost got into the AHL Hall of Fame for that one.\n\nMy other contribution to hockey, not many people know about. It's how to get an advantage in a fight. It's the old saying, \"All's fair in love and war.\" And, I might add, in a fight. Fighting in hockey is a tough role, and any advantage you can get, you try. I always got my elbow pad off my right arm\u2014you can't believe the difference it makes. I discovered it by accident. When I played, I had to work in the summer because we did not make much money and the kids had to eat. I developed big arms by working with a 150-pound jackhammer all summer. The elbow pads would cut into my arms, so I cut them to loosen them up. One time, I cut too much, and when I had a fight, the elbow pads flew off. It was like my arm was free and it felt great, and I could throw twice as many punches, so from then on, I cut the right elbow pad a little more to take advantage in the fight.\n\nIf you have noticed in today's fights, the real fighters always get their elbow pads off for swinging\u2014even the goalies are into it. Remember last year, Jonathan Bernier went at Ryan Miller. He cleaned Miller's clock. How, you ask? Bernier got his right arm free from his pads and was swinging free and beauty.\n\nI remember a lot of people wondered how F\u00e9lix Potvin, a little guy, did such a number on Ronnie Hextall, who was a big guy who could throw them. Same thing: F\u00e9lix got his right arm out of the pads, and boy, did he go.\n\nNow, the next trick to take advantage of your guy is one you're going to have to bear with me on. It gets a little complicated, so pay attention.\n\nAll hockey players wear garter belts (yes, they do, ladies) to hold their socks up. There is that little piece of rubber that sticks up on the garter belt that goes around the wire on the belt to hold the socks up. Well, sometimes that little piece of rubber breaks off. In the minors you make do, so we would substitute a dime in place of that little piece of rubber that broke off. It works perfect. What's this got to do with fighting? Patience, I will tell you.\n\nI found when I was with a team that had big sweaters, I would get my sweater pulled over my head, which was an awful feeling as the guy rings a few on your melon. You don't have a chance. Some guys would stop, but most wouldn't, and by the time the linesman got in, you'd took quite a few. (In Rochester, I had a small sweater, so it was hard for a guy to grab, and naturally, it was hard to get the sweater over my head.) When you had a large sweater, it was easy to get the sweater over the head, so I figured, if that trick with the garter belt worked, why can't I do the same with my sweater?\n\nSo here's what I did: I got an old skate lace cut about two feet long, took one end and tied it to the button in the back that held the braces that held up your pants. Then I took a quarter and put it on the inside of the sweater in the back, wrapped the sweater around the quarter, and then I took the lace and tied it around the sweater and the quarter. Presto! The first tied-down sweater in hockey. It was now impossible to pull the sweater over my head. Brilliant, eh?\n\nYes, there were a lot of little tricks. Always put a lot of Vaseline all over your face to soften the blows. Another smart one was with the sweater again. When, in a fight, you see the player pulling and tugging on the sweater, it was tough on the neck. I would actually be cut on the back of my neck from the rayon sweater, and I would always have a stiff neck the next day. The trainers weren't too happy about this. But I took some scissors and cut about four inches at the back of the sweater at the neck, and then I would sew up the sweater lightly so, in a fight, it would tear away and save my neck. I never had a stiff or cut neck after that.\n\nSo there you have it, folks, some of my biggest contributions to hockey are how to keep the beer cold on a bus trip and how to get the advantage in a fight. How best to drink and fight. Hey, what did you expect? My grandfather came from County Cork, after all. God love him.\n\nI know some of you are not jumping for joy about the previous section about fighting, and some of you don't believe it should be in the game. I respect your opinion, although if you check articles from the past, it's been in the game since the early 1900s, but your side is winning the battle because fighting is going down each year\u2014plus, now, with the mandatory rule (they call it the grandfather rule) where every rookie now has to wear a visor when they come into the league, it will go down further because of broken hands\u2014the visor is killer on hands. And now they put in a rule that you can't take your bonnet off or you're penalized, so eventually the anti-fighting group will have the ear of the league and will win the battle, so to speak.\n\nI wonder if the anti-fighting group has noticed how the stars of the league are always hurt. Last year, we had every star hurt at one time. The stars who people pay to see were out.\n\nI want you to think back to the Gretzky era, when he was flying high, setting a zillion records that still stand and winning all those Cups. Do you remember him being injured? Do you even remember him being \"cheap-shotted\"? He really had a free ride\u2014a credit card, so to speak. (In fact, I used to have people in the States say to me, \"Come on, there's a rule, 'Don't hurt Gretzky,' because he was a star.\") Now think about all of the injuries Crosby has had in his career so far. There was no rule in Gretzky's day. The rule was that Marty McSorley and Dave Semenko, they could protect Gretzky like Bobby Probert and Joey Kocur could protect Stevie Yzerman, like Clark Gillies and Bobby Nystrom could protect Mike Bossy. They would straighten out the pest that went after their stars. The NHL governors, in their wisdom, came up with the idea that they didn't want tough guys picking on their stars, which was the exact opposite. Tough guys _protected_ stars. Only little rats went after the stars. These governors were right out of touch; the nearest they went to ice was in their drinks (and to be fair, this rule was put in place before the current regime). They came up with the \"instigator rule,\" where if you start a fight and the little rat who was after the star does not fight, you'll get a two-minute minor for starting the fight, five minutes for fighting or a ten-minute misconduct, so your team is short for seven minutes. A seven-minute power play for the little rat who started everything, and to top it all off, if you get three instigator penalties, you start to miss games. So now you can't protect the stars, so it's open season on them\u2014the stars people pay to see. It doesn't make sense.\n\nIn the past, everybody was accountable for their actions. If you were a rat and gave cheap shots, you paid the price\u2014as it should be. Again, the stars were protected, the game was policed.\n\nI will give you an example. It was a game between the USA and Canada in the World Cup. Gretzky was going into a corner for the puck. A USA player named Ryan Suter cross-checked Gretzky into the boards, hurting his back. From then on, the injury to Gretzky's back bothered him.\n\nThe reason Suter had the guts to take the cheap shot on Gretzky is because he knew there was no accountability for his actions because Semenko and McSorley were not there to protect him. (It was no accident that when Gretzky was traded to L.A., he demanded that McSorley come with him.) It was sad to see Gretzky lying there hurt and nothing done to Suter, but I am afraid it was an omen of things to come. Open season on the stars.\n\nYes, before the sweethearts took over, it was safer hockey to play. Yes, we used to police ourselves, and yes, you could call it vigilantism, that could be true, but it worked. Injuries were never like they are now.\n\nI don't want you, reader, to think I want guys to sit on the bench and then when something goes wrong, be sent out like a mad dog. If you can't play the game, you shouldn't be in the game. It's not right or fair for a person to sit on the bench wanting to play knowing his only job is to fight. It's no fun, and it is embarrassing.\n\nNo, you have to be able to play the game if you're gonna be in it. Dave Semenko and Marty McSorley could play. Semenko got twenty goals one year, and Marty set up Gretzky with a pass for Gretzky's record-setting goal. Clark Gillies, Bobby Nystrom, Joey Kocur and Bobby Probert were effective players. Every one of my tough guys on the Boston Bruins could take a regular shift. Terry O'Reilly, John Wensink, Stan Jonathan, Al Secord and Wayne Cashman all got twenty goals or more. If some of you are asking, \"Why don't they take the 'instigator rule' out?\" it's because they can't; the sweethearts would blame the league if they took it out and somebody got seriously injured. They really don't know what they have done, but like I've said before, the road to hell is paved with good intentions.\n\nI am writing this while in the midst of the 2014 playoffs. Montreal has beaten out the Tampa Bay Lightning four straight. They needed every break in the book to beat Tampa. Steven Stamkos came back from a broken leg; the Lightning's captain and leading scorer, Martin St. Louis, demanded a trade. The reason? His GM, Stevie Yzerman, did not choose him for the Olympics. Martin replaced Stamkos on the Olympic team, and when he finally went to Sochi, he was humiliated. So he demanded a trade, which he got, to the New York Rangers. The next thing that happened to Tampa was that their goalie, Ben Bishop, got injured, so you felt for Tampa. And to top it all off, there was a controversial call on a goal against Tampa that could have been the difference in the series. The call was made by a French-Canadian referee, and then, in the final game of the series, a killer penalty was given to Tampa with two minutes to go in a 3\u20133 game. Montreal won and Tampa was out.\n\nRon MacLean made the point after the game that it's hard for a French referee to ref in Montreal, because if he makes a call for Montreal, it will look like he's favouring them. MacLean was just offering some sound advice because, whether you believe it or not, a lot of people feel Montreal gets the benefit of calls because of their great crowds, who are knowledgeable and know how to work a ref. I know it sounds like Ron is my friend and I'm sticking up for him, but all he was doing was trying to help the ref. Being a referee himself, he knows the score.\n\nBut then he was asked, \"Then you're saying a French referee should never ref in Montreal?\" That's not what Ron meant. \"If it was in a deciding game\u2014think about it.\" But it almost seemed like that because of the goal that was called back by a French referee for Montreal. It seemed they just had to put in a French ref to show that nobody intimidates the NHL. All Ron meant was that they should think about it in a deciding game.\n\nIf there is one guy in the world that is for the French, it's Ron. He speaks French whenever we go to Quebec. (I think he does it to tick me off.) He always praises _la belle Montreal_ , or something like that. He's totally for bilingualism, and it's ironic that now he's cast as against the French.\n\nI just heard a talk show on the radio where they had a clip of Ron and his statement on the situation taken totally out of context. Naturally, they had a guy on who is French and asked him what he thought of Ron's statement. What a setup! This world we live in, it's a jungle. You have to have tough skin. I make no bones about it. Ron and I are #1 on TV, and any time there is a chance to \"get\" us, it's done, but that is the nature of the beast.\n\nI enjoy it, the danger. That's why \"Coach's Corner\" is always live. At one time, we tried to tape the show, but the excitement wasn't there, or the danger, which always pumps me up. I know the CBC wishes it was taped a lot of times, and I guess, as I look back, a few of the segments I would do over, but what fun would that be?\n\nI am writing this on the spur of the moment when I see the injustice of this episode. A guy who doesn't have a prejudiced bone in his body, getting accused of being prejudiced. Hey, Ron will survive, and this is not really a big deal. I just wanted to show you the world we live in, but I gotta tell you, it's better than working a jackhammer, and it's the old, old story: if you don't like the heat, get out of the kitchen.\n\nHockey Canada. The experts at Hockey Canada who, in their wisdom last year, did not invite Darnell Nurse of the Sault Ste. Marie Greyhounds to their tryout camp for the world junior hockey team. Darnell will be playing for the Edmonton Oilers next year. Six foot six, tough, beauty skater and handles the puck and has a shot like Denis Potvin and didn't even get an invite to this camp. But the best of all non-invites to the junior camp by Hockey Canada was Connor Brown of the Erie Otters of the OHL. Yes, the same Connor Brown who won the OHL scoring title with 120 points in 68 games and was voted the most valuable and best player in all of junior hockey, winning the Red Tilson Award. Imagine: the MVP of all junior hockey not getting an invite.\n\nWhy? You may believe it or not\u2014it's political. The way it works out, there can only be so many from every province (the West, Ontario and Quebec) and if you count the percentages, it works out. I predicted a loss because of this, and I am not too popular with Hockey Canada. It's a crying shame that guys like Max Domi of the London Knights, who one day will be starring for the Phoenix Coyotes; Connor, for the Leafs; and Darnell for Edmonton did not get a chance to star for their country because of politics. But I digress.\n\nThese same so-called experts put out a statement that they want all kids to have fun and they want the team to follow these drills and they want more drills in practice. And then the experts go on to explain that something like one tenth of one per cent make the NHL (almost saying, \"You'll never make the NHL,\" destroying the dream of every Canadian kid; almost saying, \"Don't be foolish\u2014can't you see the odds?\") instead of saying, \"Somebody has to make it, why can't it be you? Follow your dream. Don't give up, no matter what the odds.\"\n\nIt reminds me of my first year in high school. The school counsellor took four of us into the room and proceeded to say, \"Only half of you will graduate,\" instead of giving us a pump job\u2014\"You can do it,\" etc. He discouraged us right off the bat. Why tell the kids very few make it? It discourages the kids. Like I said, somebody's got to make it. What if all the guys in the NHL thought that way? I say follow your dream in hockey, business, coaching or TV announcing.\n\nI switch now to last year's playoffs between Montreal and New York in Montreal. I meet an ex-Toronto Maple Leaf player and his friend, two great guys who have purchased a junior club. We discuss hockey and the team, and we get around to the practice. So I say to them, \"Guys, if we lined up your hockey team and you asked them this question, 'Hey kids, what do you want to do this practice? Do you want to scrimmage or do you want to do drills?,' what do you think they would say?\"\n\nThey both answer, \"A thousand per cent they would want to scrimmage, for sure.\"\n\n\"So, honestly, answer me: in an hour's practice, how much time do you put to scrimmage? Now, be honest.\"\n\nThey hesitate, because they see where I am going, but they answer truthfully: ten to fifteen minutes.\n\nSo I say, \"Let me get this right. You say you have the teams for the kids to have fun, and the kids have fun scrimmaging or playing, yet 80 per cent of your practice is something they have no fun at. And believe me, guys, I see the practices. Breakout plays where the coach shoots the puck at the goalie from centre ice, he gives it to the left defence, he puts it up to left wing, he puts it over to the centre. Boring hockey, and how many times does that play happen in a game?\"\n\nI also say, \"I know why you have a lot of drills. The parents are watching up in the stands. If you let the kids play and not do drills, the parents who don't understand will say to you, 'What are they learning? You coaches are just letting them play.' Am I right?\"\n\nThe coaches say, \"You are right on the money, Grapes.\"\n\nI know a coach who coached a few years ago in minor midget. His team won the championship and every player got drafted except the backup goalie, yet he was criticized by the parents. For what? Believe it or not, because he never had enough drills. Parents are brainwashed by Hockey Canada on drills. How do you learn anything\u2014fix cars, learn to swim, play baseball, fly a kite? By doing it.\n\nTim\u2014my son, who rates the players for the OHL\u2014and myself can pick out players on a team that scrimmages. They seem to handle the puck better and with more flow to their game.\n\nMy grandson Del played double-A and the practices drove me to tears. So boring. Two hours of drills, hardly any net shots, no scrimmages. I had to stop watching the practices; sometimes I think it was better when the dads used to go out and toss the puck on the ice to say, \"Let's go,\" like they did when Bobby Orr played. Same as Bobby Hull. They could handle the puck. Today in the structured practices, where you have to pass, pass the puck, \"Move it, move it\" is all you hear in practice. How about \"The puck moves faster than you\"? Look, I'm all for winning. That's the name of the game, but there are too many coaches out there that are too focused on winning games instead of having fun.\n\nI swear to you this is the truth: in the summer, they started practices\u2014three on three for fun in the summer off-season.\n\nI was on time to see Del. The coach was screaming at the kids. He had loaded up a team and played it like a Stanley Cup, banging the handle of the gate for a line change. I call that sick. I was standing by this woman watching. I said to her, \"That guy is what's wrong with hockey. What a jerk.\"\n\nShe said, \"That's my husband\".\n\nI said, \"Do me a favour. Tell him this is not the Stanley Cup. This is supposed to be fun for the kids. And please tell him Don Cherry says he is a jerk.\"\n\nShe said she would. I think she agreed with me.\n\nLet me tell you when I became a disciple of scrimmaging. I learned it from the greatest coach who ever lived: Hector \"Toe\" Blake of the Montreal Canadiens. I ended up with the Canadiens at training camp. I still don't know to this day how and why a confirmed minor leaguer like me was at a training camp of the Stanley Cup champions. I have this opinion that Sammy Pollock, the GM of Montreal, had a very talented team in Ottawa-Hull in the Eastern Pro League. It was so good. Guys went up to the Canadiens, but they were small\u2014as the Canadiens today are small. They needed a tough coach who could still play.\n\nThe year before, when I played for the Sudbury Wolves in that league, I had done a good number on their coach for hitting our goalie, and Sammy must have seen I had potential as a coach and I was tough. It didn't work out, but that's another story. But I would have to say that training camp with Toe Blake changed my whole outlook on hockey.\n\nYou must realize I had played in every professional league: in the NHL (one game), American Hockey League, Central Hockey League, Western Hockey League and the Eastern Pro Hockey League. I had been to every NHL training camp. I was thirty-two years old and knew the score, but my eyes were opened to new things by Toe Blake. Let's just say they were new to me, but they were the old ways.\n\nToe was in the midst of winning eight Stanley Cups in thirteen years as the coach of Montreal. Let me tell you about his practices that he ran in training camp\u2014and, from what I understood, during the regular season. Now, you must remember, I was a nobody, but I was treated first class: new gloves, equipment, fresh underwear, new socks, in the dressing room with the stars, stayed at the Queen's Hotel (now gone), violins and all that stuff. First class, not like Toronto's training camp. You were treated like dogs at the Leafs' camp: no fresh underwear, old socks, ate at the minors' table. Montreal was first class all the way.\n\nThe first day of training camp, we were given lovely red sweaters with the CH on it in our dressing room. We were a team. In the other room they had all white. We went on the ice; the red team took one end of the ice and the white team the other end of the ice. We warmed up, took shots on the goalies. I said to myself, \"This is like a game,\" and it was. We had refs, and from the first day it was a game. The ref let things go, just face offs (see what I mean? We were working on the faceoffs from day one), blocking shots. We were working hard, but having fun. No boring drills, no stoppages. We changed on the fly, kept score, yelling and hollering\u2014even the old vets. Everybody was into it, under the eye of Toe, who, if you didn't give your best, you got it from him. Instead of dreading the practices, you looked forward to them. No wonder all his players loved him, but they also feared him. They called him \"The Bear.\" Gosh, I was only there two weeks and I loved him, and that's where I saw how a master worked.\n\nI never forgot how hard we worked and how much fun we had. The Canadiens were the best. Unfortunately for me, I was only there for two weeks. I went out with a buddy, Claude Dufore, a goalie I had played with in Springfield. We had a few pops (honest, only about five or six, nothing serious). Sammy called me in the next day and said, \"Don, I understand you were drinking last night.\"\n\nI said, \"A few.\"\n\nSammy: \"Don, we don't allow that. I don't want that to happen again.\"\n\nDon: \"Sam, I only had a couple of beers.\"\n\nSammy: \"Okay, Don, thanks for your honesty. You'll be going to Spokane.\"\n\nSpokane was in the Western League. So I was back in the minors for the rest of my life in hockey. But I always remember the master, Toe Blake, with fond memories, and when I became coach of the Rochester Americans of the AHL, I practised the way Toe did. We won the league title and I won Coach of the Year when I went to the Bruins. The players loved it, like I said. I would call an optional practice, and everybody would show up. In four years, we went to the semifinals twice and the finals twice, with a winning percentage of .658, and I was Coach of the Year\u2014all because I remembered and followed in the footsteps of the master, Toe Blake, who treated his players like men, not robots.\n\nI remember I was not too happy when I was told I was going to the Montreal camp. Little did I know it turned out to be one of my biggest breaks. So that's how I became a coach who loves players to play, not do drills, and the players loved to come to practice and played their hearts out. In a game when a player was traded to the Bruins, the captain would take the new guy aside and say, \"We got a good thing going here. Don't ruin it. The practices are not long, and they're fun. All the coach asks is that you work hard all the time. If you don't, you'll hear from me and the rest of the guys.\"\n\nI know as I write this that you young coaches won't take a chance because of owners, parents, Hockey Canada, etc. It is a tough ride to ride, you do take a chance, but roll the dice, try it. As they say, \"No guts, no glory.\"\n\nI just read that the great coach for the New England Patriots football team, Coach Bill Belichick, says he is tired of all the injuries in the exhibition games and he is going to split the football camp into two teams and have simulated games as he feels it's better to play than do drills. Toe was sixty years ahead of today.\n\n# **THE 2014 STANLEY CUP FINALS**\n\nI AM WRITING ON JUNE 5, after the first game of the Stanley Cup finals\u2014N.Y. at L.A. I just heard that I upset a lot of people when I said that out of the twenty-three players used by the Kings in the playoffs, fifteen are Canadian and nine from Ontario, and from the New York Rangers, twelve are from Canada and eight are from Ontario. I said Toronto and Ottawa should follow the Kings' example and get big, tough, skilled Canadians. (By the way, the Kings are going to win it. I'm going on record as saying that. If they don't, I'll look foolish in this book, but they will.) Some people are ticked off that I said it only stands to reason that if you get kids from Ontario and you're playing in Ontario, the players try harder. With family and friends watching, and if you live in and are from Ontario, the Leafs or Ottawa will be your favourite team to play for. Does that sound reasonable? And I also quoted Brian Burke, who once said, \"The Leafs at home have a tough time because so many [visiting] teams have players from Ontario.\" (At that time, the Leafs, although they are in the middle of the Greater Toronto Hockey League\u2014which, believe it or not, has forty thousand kids playing in it, most loving the Leafs and would give anything to play for the Leafs\u2014had nobody from Ontario.) So Ron MacLean asked, \"Doesn't it make sense if Ontario kids are trying harder in Toronto that the team get more players from Ontario?\" It made sense to me, and it makes sense to anybody. The Leafs and Ottawa do have some Ontario guys now, but nowhere near the same number as this year's Stanley Cup champions (if L.A. doesn't win the Cup, I will still leave this part in the story).\n\nThis Saturday, I will upset them more when I point out that the Colorado Avalanche, who went from twenty-ninth in the league to first in their division, and whose coach, Pat Roy, will be coach of the year (another prediction I will leave in if wrong), are loaded with Canadian guys from Ontario.\n\nI live in a different world than anybody in the media. I am always with the real hockey people. My son, Tim, and I go three nights a week at nine-thirty at night to watch the minor midgets. We are with the coaches, players, parents and fans. I know what they think, I know what they say. To a person not involved in the game, it sounds so narrow. Understand these parents. They spend thousands, travel mile after mile for tournaments. In doing so, they ruin their cars, cancel vacations, drink bad coffee at five in the morning as their kids practise. Why wouldn't they want their young player to play for the Leafs or Senators?\n\nThere was a player who played minor midget who had dual citizenship. One father said, \"The kid wants to be with the Leafs. I'll bet you ten dollars, he goes to the U.S. development camp and Leafs will draft him.\" The kid went to the U.S. development camp, and _bingo!_ The dad won his ten dollars. And while I'm on this rant, let my comment about Russians and Europeans in the Canadian Hockey League stand. Make no mistake: when you have one of them on a CHL team, a Canadian kid is out. It only stands to reason. A Canadian kid whose father and mom have paid taxes, bought equipment in Canada, totalled thousands of miles in travel. I know I've mentioned this before, but it's a big thing with me, as you can see.\n\nAfter the junior tournament where we lost to Russia, that Russian kid who played for the Sarnia Sting in the CHL said Canada has no talented players and laughed at us. After we give him an education package, clothes, our milk and honey, he snubs us. They must think we are the biggest fools in the world, and we are. So you see, I am extremely pro-Canadian, and in Canada, that is a bad sign. I lived a long time in the States, where their motto is \"What's good for the States is good for the world.\" I don't go that far. I just say in our country, we should be number one. I and my kind want no favours from other countries. Just don't come to our country and tell us how to live and take our Canadian kids' jobs. Redneck thinking? You bet I'm a redneck, and proud of it.\n\nDo you know how the name \"redneck\" came about? Here's the story. When the Irish, English and Scottish came to North America, they had to clear fields and till the ground, always bent over working. In doing so, their necks were exposed to the sun, burning their necks red. And when the Indians saw their burnt necks, they called them rednecks. I knew this to be true when I worked on construction when I was young. My neck was always burnt. Now, I had to wear one of those foreign legion-style cloths in the back of my helmet. So the word \"redneck\" was from a hard-working guy who worked in the sun, who loved Canada as number one. He didn't come to this country, take its milk and honey, and still thought where he came from was number one. And that's the way it should be. Because let's face it, folks, nobody really wants to leave their native country. The only reason people leave their country is if they can't survive in their own country or they're persecuted. So they come to Canada and are treated royally: free medical care, free everything, and after receiving all this from Canada, they still think that where they came from, where they were starving and persecuted, is No. 1.\n\nThe only thing I can say is, if you love the old country more than Canada, why don't you go back? Funny thing I should say that, because that's what a lot of people do. They live in Canada for all our benefits\u2014medical, health, etc.\u2014for long enough that they are eligible to keep their status and our passport, and then they go back to the old country. They have their cake and eat it too, but they go back, so I'm not complaining. Remember a couple of years ago, these people were living in one of these countries where they had civil war and they murdered one another? We provided them planes to fly back to Canada. They complained that the airplane was uncomfortable and they didn't enjoy the food. Our tax money at work. It costs us millions.\n\nI was told that there are illegal immigrants in Canada living in motels: free food, free health, money to spend, heaven on earth. It was reported that they were very upset at the way they are being treated by Canada. Their big complaint is that they are given too much chicken. Meanwhile, we have seniors living in one room, eating dog food. What I say is the truth. Some people can't handle the truth, as Jack Nicholson would say. That is why some people don't like what I say on \"Coach's Corner.\" That's their choice, but after thirty years, there is little chance I'm going to change now.\n\nIt's seven-thirty on the night of June 10, 2014. I have put myself in a dicey situation. I said about three weeks ago that if we got to New York, \"I will dance to a techno beat in Times Square,\" and here I am, on my way. I often wonder about myself. We have been going since April 16, every other day, and on the road, it's not as glamorous as people think. It's a tough row to hoe, as they say, and here, like a dummy, I go and make it tougher.\n\nWe meet in the lobby. Zac, the cameraman, and Kathy Broderick, who runs my life at CBC. You want to know what's going on at CBC, ask Kathy and Sherali Najak, the boss producer. Ron and his wife, Cari, have gone to dinner with Adam Graves. It's hot and humid, and as we walk through a sea of humanity, I think, \"You dummy!\"\n\nWe get to Times Square. You can't believe how packed with people it is\u2014and believe it or not, a lot of Canadians. I do the dance in the heat. The people take it in stride\u2014another jerk in Times Square. After all, they also see and hear the Naked Cowboy. I'm melting in the heat. We get it done, I'm soaked, and we have to walk back to the hotel. I am posing for pictures on the way back, and between the heat, the humidity and the dancing, I am sweaty\u2014and I have to admit, I started to get a little light-headed. We finally make it back. A shower, a couple of pops and I'm going to bed. As I lie there, I think of Ron, having a nice meal with friends, a little wine and conversation. Why am I such a dumb-head? I always make things hard on myself.\n\nJune 11, 10:00 A.M. I meet Ron in the lobby to go to the morning skate. L.A. is on the ice at eleven. We like to get there early. We try to hail a cab in the sea of people\u2014again, pushing and shoving. Boy, you have to be tough to live in the city. Ron was telling me he went to a sandwich shop and stood like a polite Canadian and let everybody jump in before him. Finally, a tough guy behind the counter hollered, \"Hey, you want a sandwich or what?\" You can't be polite in this city. It's survival of the fittest. But I must admit it's a lot safer now than it was a few years ago. I remember when we were here in the playoffs. If you walked along the streets, a guy would come up to you and say, \"That's a nice watch you have there,\" meaning if you wanted to keep the watch, a few dollars would let you keep the watch. Bob Cole and I went on the subway, like two fools. There were eight cars and, for safety, everybody was jammed into one. Seven empty cars and one full. It was a tough, dangerous city, but that has all changed. There is a policeman on every corner. The mayor of New York changed. Mayor Rudy Giuliani had a policy called the \"broken windows policy.\" In other words, you fix a broken window, you save a whole bunch from being broken. Prevents vandalism. They investigate every crime, and they've got tough laws. New York's actually one of the safer spots in North America, but the people all walk so fast, with a detached look on their faces. Bumping and pushing and staring straight ahead. I swear, if a guy fell down and died they'd step right over him.\n\nWe finally get a cab. He goes a block past the entrance. We say it's okay. We don't want to tick him off. We get into the arena a half hour before the skate and we just sit there. It's quiet, cool (actually, cold). I enjoy the feeling. We always sit way up high by ourselves; it's my favourite time of the playoffs. I'm in my element and Ron likes it, too. He shuts off his phone. It's like I'm safe\u2014nobody can get to me. I love an empty arena. Ron and I are the only TV guys there. Zac, our camera guy, is there after a while, and other people start drifting in on the ground level. When Kathy shows up, she goes to the media room, where the team puts on coffee and muffins. She brings us a couple of muffins and coffee\u2014she doesn't have to. She knows we don't go to the media room and she is kind enough to do it for us. It is so good, that coffee and muffin. Life is wonderful. She knows we appreciate it. \"Hey, you're late,\" Ron kids her, and she laughs.\n\nRon and I talk hockey and life. It's funny: Ron knows all the numbers of the players, where they came from, what they did last year. I'll give you an example of his memory. Glen Sather, the GM of the New York Rangers, met us the day before we came into Madison Square Garden and sat with us. I reminded him of the time when he was the GM in Edmonton. We visited his cottage and there was an English bull terrier there. He dressed the bull terrier up in an Edmonton sweater and took a picture with me, which got into the Toronto papers. On \"Coach's Corner,\" I let on that my Blue was jealous. I went on TV the next game and told Blue, \"That bitch didn't mean anything to me.\" Ron said to Glen the dog's name was Watson. Believe it or not, that was twenty years ago. Ron remembered. Glen couldn't believe it.\n\nSo we are sitting up high, enjoying ourselves, and I see this fellow coming up towards us. It's the coach of the Kings, Darryl Sutter, and he sits with us, telling us stories. He is just as funny in person as he is in his press conferences. He is an honest, humble farmer, down to earth, a great guy, a super player for many years in the NHL and an even greater coach who has won a Stanley Cup and will win another Cup this year. He tells us about a coach who taught him a lot about hockey. For instance, he told Darryl, \"When you are skating around, never just flip the puck into the net. Always drive it into the net, always hard.\" I told Darryl about one of my theories. When the guys are just skating around before practice, there are some who like to hear the _ping_ of the puck when it hits the crossbar or goal post, so they always do hit the post or crossbar for fun. My theory is they then get into that habit, and in a game, they're more likely to hit the puck or the crossbar.\n\nFor instance, a player who played for me, Ren\u00e9 Robert, used to do that in practice, and he hit more posts and crossbars in a game than anybody I know, except Jaromir Jagr. The year he played for the Boston Bruins, in the morning skate he would ring it off the crossbar automatically, just to hear the _ping_. In the playoffs, he never scored a goal the whole time, but I'll bet he hit ten crossbars and posts. Same in overtime.\n\nDarryl sat with us for a half hour. Ron and Darryl are two Alberta boys and they discussed Red Deer and the farms. But all good things must come to an end. David Keon Jr. (son of Davey Keon of the Leafs), who runs press conferences and stuff for the NHL, came and stood in front of us for about five minutes and said, \"Sorry, Darryl, but all the reporters are here. The press conference is waiting for you.\"\n\nDarryl said, \"Well, boys, I got to go,\" and shook our hands and left.\n\nI said to Ron, \"How lucky are we to have Glen Sather and Darryl Sutter come and sit with us and tell stories? It doesn't get any better than this.\"\n\nWe leave the hotel for the game that night at three-thirty, as it's an early game. I do the opening and predict New York will win\u2014which they do, thanks to Henrik Lundqvist, who stopped forty shots. Score: 2\u20131, New York. Back to the hotel, have five cold pops by myself. Ron and Cari go to dinner.\n\nPack the next morning. It takes me an hour to pack, get to the airport, fly to L.A. Going to meet Ron at 8:30 P.M. for a sauna and a few pops. I predict L.A. will win tomorrow night. I have been correct in every prediction so far. The only thing that got me thinking is Lundqvist again. Still, going with L.A. We'll see.\n\nIt's June 13, 2014, 5:25 A.M. Had a steamer last night, four Bud Lights with Ron and to bed. I said to Ron, \"I bet Darryl skates today early.\" Sure enough, Darryl Sutter is skating his team, the Los Angeles Kings, at 9:30 A.M. We leave the hotel at eight-thirty.\n\nWhen I had the Bruins in the playoffs, at a time like this in the semifinals and finals, I would let them have a morning skate in their sweat suits. We had beautiful black and gold sweat suits, and we had black turtlenecks. We looked gorgeous. We wore elbow pads, jocks and gloves. They loved it. Why? Because they had been putting on the full equipment for months. I found it wasn't the practices so much, it was putting on the equipment that seemed to tire me out.\n\nIf a guy came to me and said, \"Grapes, I really don't want to skate in the morning,\" most of the time I would say \"Okay,\" but you have to know and trust your players. I am not a big guy on morning skates\u2014on the road, yes, because it gets them up and out of the hotel. Some just wanted to fix their sticks and not skate. I would let them.\n\nIt was great, especially for the older guys. I had to force Jean Ratelle, who was thirty-six, to stay off the ice. Anybody who'd been in the New York Rangers organization, you had a tough time keeping them off the ice. That organization believed you should be on the ice every day.\n\nJean was thin as a rail, had a bad back. I had to baby him along. We protected him. The word got out: touch Jean and you were dead, or at least a month in the hospital. (That is just an old hockey expression. You really didn't kill the guy or put him in the hospital for a month. Maybe just a week. I'm just kidding.) But nobody touched him, because if you did, it was not nice.\n\nAnyway, I had to force him to take days off, because for all his career with New York, he'd been on the ice every day. I remember when I told him to stay home from practice, Jean asked, \"What will the GM and media say?\" I said, \"They won't say anything. If they do, just say you got the flu.\" Danny (the trainer) is under orders to tell anyone who asks that you have the flu. If the GM asked Danny we'd just say \"ask the coach.\"\n\nJean responded as our beauty scorer. He was one of the last players to use a straight blade on his stick. I said to him one day, \"Jean, why don't you use a blade with just a little curve? Everybody is using a curve.\" Jean had a great backhand, and I didn't want to ruin that, but I thought a little curve might help on his forehand. Jean said, \"Thanks, but no thanks.\" I ordered him a dozen curved sticks anyway and never heard anything about it, but the next time Jean ordered sticks, the Sher-Wood guy told me Jean ordered the sticks with the little curve.\n\nI never believed in morning skates anyway. I'll tell you how they got started. I remember some guys always had their skates sharpened the day of a game. One day, one guy just went out, still in his street clothes, and tried his skates. Another day, he brought out his stick and some pucks. He figured, \"I'm out here, why not take some shots?\" He sweated a little in his street clothes, so he put on a sweat suit. Another guy joined him, and another. Pretty soon, half the club was out there, and the coach saw this. It went from a sweat suit to full equipment and took off.\n\nIn the seventies, there were some coaches who ran the morning skate almost like a regular practice. I know some players hate it. How many times have you read about players, and especially goalies, hurt in morning skates? It was nuts, but some coaches did it (especially with teams that were not going well) because it impressed the owners and the GMs. The coach was trying to give the impression, \"Look, I'm doing my best.\" It was phony, and the players knew it.\n\nThe coach who feels he should give the team the day off (except the Black Aces\u2014more on that soon) better have confidence in himself, because I have seen coaches give the day off and lose the next game, and the press used the day off as the reason for the loss.\n\nThis is a true story. A coach whose team was in the finals had the players come to him and say, \"We would like the day off. It will rest us and it will help us win.\" The coach gave his team the day off with no practice. The team lost the game. The reporters asked why they lost the game, and the captain said they lost because they'd had the day before off and were not prepared for the game. This is the same guy who asked for the day off. The coach was fired. The player stayed.\n\nI digress. The term \"Black Aces\" came from the Eddie Shore days. Eddie's players used to joke that they had one team going, one team playing, and another team that didn't play and were in the bad books, guys who practised with the team and then again\u2014separately\u2014after practice. I used to joke that Eddie made me captain of the Aces. I used to be hollered at by Eddie more than anybody, so the players made me captain. In the playoffs, you have six extra players, usually guys up from the minors. It hasn't changed in fifty years.\n\nAfter the morning skate, when the Aces stay on the ice afterward, it is not a happy time. When you have a regular player and he has to skate with the Aces, he will only go so hard, and now teams have three assistant coaches who must justify their positions, so they make the Aces do different drills, which the players hate doing. Imagine there's one game to go in the playoffs, and you're learning different drills. (I know what you're thinking\u2014that it makes the practice new and not boring, etc. Your thinking is wrong.) I used to stay out with the Aces after practices and morning skates and work them. I used to say, \"Boys, you are going to play your game this morning. We are going to skate hard, because if you're called in for the game, you will be in the same shape as the guys playing.\"\n\nWe would go side to side so much. The rink managers, when we were on the road, complained to the NHL that we were hurting the ice, wearing it down on the sides. I always skated the players side to side instead of end to end because I always felt hockey was a burst, not a marathon. Besides, going end to end, you could cheat a little. Not the case when it's side to side, touch the board and come back full tilt. I used to laugh\u2014the guys who wanted to cheat used to go to the corners because it was a shorter distance to touch boards. I used to put them in the middle of the ice. After a while, I used to have two guys go together\u2014if one guy didn't go all out, I would say, \"Go again.\" Guys hated that \"go again.\" But you couldn't fool me. I used to be one of them.\n\nWhen we would go out for the skate, I would have a warmup. \"Speed up on the whistle.\" And then, \"Break between blue,\" meaning skate hard between the blue lines. I just remembered: one day, I said, \"Break between blue.\" I swear a rookie came up to me and said, \"Break what, coach?\"\n\nI had to play it straight and not laugh. And explain what I meant. He was so embarrassed. Naturally, I told no one.\n\nAfter the warmup, I would let them play three on three, and they loved it and worked. They would be soaking wet.\n\nI have to laugh at the morning skates now. The guys are so bored. When they get a good sweat, they have to be told the next drill. Like I said, learning new drills with just one day to go, for a guy who is a regular and is benched and put with the Aces, is just embarrassing. So I made sure my guys would be happy. And on the road after the morning skate, I would take them for lunch. And you know, the regular older guys, after they'd showered and were on their way to the bus to go back to the hotel, would stop and watch for a while. And they would say, \"There's Grapes and his boys.\"\n\nI did this with the rookies to show that they were important and I cared for them. Nowadays, the head coaches are never with the Aces. Can you imagine how you would feel if the head coach never paid attention to you? Now imagine how you would feel if the head coach thought you were just as important as Bobby Orr. Well, maybe not _that_ important, but you get the idea. We were a band of brothers. The regulars would laugh and kid, but deep down I knew they liked it because they were rookies at one time, too, and they were treated like rookies. Our rookies felt they were part of the team. How did I know how they felt? Because, like I said, when I was with Eddie Shore in Springfield, I was a Black Ace and I knew what it felt like to be a \"spare.\" And I vowed if I ever got a team, nobody would be treated like we were.\n\nThere was gallows humour, because we did not feel good about ourselves. In fact, when I was a Black Ace, that's when my anger boiled over and I broke my stick with a crosscheck on the coach Pat Egan and hurt him. I was gone to Three Rivers, Quebec, the next day. At the time, I wished I had killed the son of a B. My frustration was so great. By the way, the reason we all wore black and were called Black Aces was that the sweaters were never washed and black hid the dirt better.\n\nSorry, folks, I just got carried away with the morning skates and the Black Aces.\n\nRon and I are off to the morning skate in an hour. It's an early game because of the western time zone, so I will write no more today. I am predicting a victory for the Kings tonight. We shall see.\n\nI am writing this in New York City at three-thirty in the morning. Can't sleep and am wondering about a few things. The Rangers are down 3\u20130 in games. L.A. is going to win the series, like I predicted before it started. In fact, I predicted they would win on Monday, June 9 (it's on video). But tonight, in the fourth game, I'm putting my money on the New York Rangers. Why? Naturally, they want to win, but it would be a killer to go out four straight. It would make it seem like they were a failure. You have to win at least one. (I have to admit, the L.A. Kings have got every break in the series. I know the old saying, you make your own breaks, but the Rangers are outshooting the Kings. In fact, in game three, they outshot L.A. 32\u201315. Lundqvist is not doing the job right now and Kings goalie Jonathan Quick is murdering the Rangers. He is standing on his head, as they say.)\n\nAnother reason the Rangers have to win this fourth game is that they don't want to see the Kings skate around the ice at Madison Square Garden with the Stanley Cup. It is humiliating. I should know: in 1977, the Montreal Canadiens beat us in the finals in four straight, and they carried the Cup around on our ice at the Boston Garden. I remember the fourth game, which they won in OT. Al Sims, the Boston defenceman, was trying to freeze the puck against the boards to the right of our goalie, Gerry Cheevers. The Canadiens stood back and left him alone. The ref hollered at Al, \"Move it,\" meaning the puck. Al moved it, all right; he kicked it to Lafleur, and Guy passed it out to Jacques Lemaire, and he put it home for the winner, and that led to Montreal skating around with the Cup in the Boston Garden. After the game, I said to Al, \"What happened?\"\n\nAl: \"The ref told me to move it, so I moved it.\"\n\nDon: \"Yeah, okay, but did you have to move it to Guy Lafleur?\"\n\nI remember after the press conferences after the game, I was sitting alone in my office staring into space, singing to myself, \"Oh Dear, What Can the Matter Be?\" Evidently, I would do this. I never knew I did it whenever we would lose till my young son, Tim, asked me, \"Why do you sing after a loss, Daddy?\" I must have done it subconsciously while tapping my pencil. I think the rest of the song goes:\n\n_Oh dear, what can the matter be_\n\n_Oh dear, what can the matter be_\n\n_Johnny's so long at the fair_.\n\nThat's what happens when a coach loses. He's always alone. So I was sitting there in my office. All of a sudden, the door burst open and in came Montreal's \"Big Three,\" as the press called them: Larry Robinson, Guy Lapointe and Serge Savard (there has never been, and there never will be, three defencemen that play together on any team. Think about it, two of the three were always on the ice. Scotty Bowman saw to it). They'd come in to shake my hand. I never stayed out on the ice to shake anybody's hand (I know, I know, poor sport and all that). The three guys remembered how I helped them in the Canada Cup that year. I remembered Serge was just coming back from a broken leg that year, and he was trying to protect the leg in practice. Team Canada had an intrasquad game at the Montreal Forum, and they opened the rink up to kids. It was packed. The kids were screaming and it was an exciting game. Guys were getting carried away with the kids' enthusiasm.\n\n(I remember Gerry Cheevers was on my team, the white team. Gerry just hated playing exhibition games, and he especially didn't like playing against 60-goal scorer Steve Shutt. Come to think of it, Gerry Cheevers didn't like to practise, either. In fact, it cost him the starting job on the greatest team of all time because he wouldn't work in practice. I knew that when the chips were down, Gerry would be there, although if we were winning 6\u20131 he would lose interest and the game would end up 6\u20133. It drove coaches nuts.)\n\nIt was an exciting game and my buddy Lanny McDonald took a run at Serge and hit his leg. (Serge must have done something to tick Lanny off.) After the hit, Serge ran at Lanny and crosschecked him and knocked him down, and all the kids booed Serge. He got ripped badly in the papers and on TV. When I was called by one of the papers about Serge crosschecking Lanny, I said, \"Even though Lanny's my buddy and a great guy, he was wrong running at Serge, because Serge is just coming back from a broken leg!\"\n\nSerge and Larry Robinson and Guy Lapointe said, \"Thanks for everything, and I'm sure we'll see you again in the finals.\"\n\nSo I am laying my money on New York tonight because of the fear of the Kings skating around Madison Square Garden with the Cup. I say there will be a game five in L.A. (I will leave this in if I'm wrong.) New York might win the game, but not the Cup.\n\nHey, don't get me wrong: I hope they lose tonight, because if we have to go to L.A., we have to pack. I have five jackets, and it is tough hauling them one side of the country to the next.\n\nI just remembered an incident in that Team Canada intrasquad game. Like I said, Gerry Cheevers was my goalie. Scotty Bowman was coaching the red team. During the second period, Shutt let one of his high slapshots go at Gerry. It went into the top corner. I swear Gerry moved out of the way, so I pulled him. (Gerry tells everybody he's the only goalie to get pulled in an intrasquad game.) After the game, in front of the players\u2014who knew Gerry moved out of the way of Shutt's shot\u2014I said, \"Gee, Gerry, were you ever lucky. That shot almost hit you.\"\n\nThink about it. Rangers will win tonight and we will go back to L.A. for game 5. (Rangers won the game.)\n\nI was wrong\u2014L.A. won in _five_ games, but it was a strange series. New York could have won the first two in L.A. and lost both games. The Rangers came back to New York, playing terrible, getting outshot and outhit, but they won because of Lundqvist, their goalie. L.A. got a split, going up 3\u20131. New York played super again in L.A. in game five and lost in overtime. I have to say, New York got only one break, when the puck lay on the goal line and didn't go in. L.A. was wearing down near the end. The grind of the march to the Stanley Cup is brutal. Those previous series, down two games, down three games, forty and fifty hits a game... the tank was getting empty. Some guys were running on fumes. If the Kings didn't win that fifth game in L.A., the Rangers would have beaten them in New York and there would have been game seven in L.A. And as the old saying goes, anything can happen in a seventh game, and the way Lundqvist was playing, New York had a real shot.\n\nBefore the fifth game, when Ron and I stood on the ice to do the opening, I predicted that L.A. would skate around with the Stanley Cup that night. The reason was that I had only five suits with me for TV, so the series had to end that night.\n\nThe Kings have won two out of the last three Stanley Cups. They did not have a great regular season, as usual. There were times when they were in tough to make the playoffs, but when the chips were down, they came through like troopers. You could make a case for a ton of Kings, but two guys have to be brought out to the forefront.\n\nFirst, Dean Lombardi, the GM, who brought Darryl Sutter out of retirement to coach the Kings. Darryl had been let go by the Calgary Flames, because believe it or not, he made the Calgary players unhappy. It's the first time in history that a guy was let go because he didn't smile enough. Darryl was on the farm, in the barn, when the call came from Lombardi. He at first refused, saying, \"Nope, you have a good coach.\" Thankfully for the Kings, he changed his mind. The players loved him, but they also feared him. All great coaches inspire respect and a little fear.\n\nDarryl was made for L.A., although I remember\u2014and I'm sure Darryl also remembers\u2014the negative reports when Dean Lombardi said he'd hired Darryl Sutter. The reports from TV and paper said he was too old to work with the kids, didn't have the best attitude, only liked older players, etc. Naturally, all wrong. Darryl's press conferences are classics, and he hit the right note with the players in his speech. He told me about one speech he made before a game: \"Listen, if we are going to play hockey in June, we had better win, because I could be home on the farm haying.\" He is a down-to-earth Alberta farmer who doesn't know how great he is.\n\nI take nothing away from the players. Quick, the goalie, is sensational; Justin Williams was the playoff MVP; Drew Doughty at times looked like Bobby Orr (I said _at times_ ). The captain, Dustin Brown. Willie Mitchell, a rock. I could go on about the whole team and their top players, but as far as I am concerned (and I think I am qualified, having been to the finals twice), the reason the L.A. Kings won the Stanley Cup was the guidance of the farmer from Alberta, Darryl Sutter, and another guy in the organization, Mike Futa, who knows how to draft.\n\nL.A. won in double overtime. Alec Martinez scored, Kyle Clifford got an assist and so did Tyler Toffoli. I predicted in the opening that L.A. would skate around the ice with the Stanley Cup. It was Ron's last time as _Hockey Night in Canada's_ host, and as usual, he was the best. Naturally, I didn't tell him that, but he knows what I think. I guess when we're done, I will tell him, but for now we just bid each other goodbye.\n\nEarlier in the series, I did my dance video in Times Square, and I thought there should have been more of me and less action clips. The next morning, I said, \"Luba calls to say that her brothers thought the dance was really good.\" I was starting to feel good about myself.\n\nSo to bug me, Ron came back with \"Yeah, you know, now that I have watched it a couple of times, I like it. I really like the clips of hockey, except where some jerk keeps popping up thinking he can dance.\" He got me good. That's payback for a couple of nights before, when, after about our fourth beer, we were commenting about his role starting in the fall of 2014, when he'll be doing \"Coach's Corner\" with me, but won't be the host of the whole show. \"Coach's Corner\" on Saturday night and that's it. Sundays, he'll fly to different small cities to do a mini-Hockey Day in Canada, which he will be great at as usual, but I keep kidding him about it. I know deep down Ron doesn't feel the happiest about the situation. Like I said, we use gallows humour. Well, as we started our fifth pop, I started to say, \"Now, when you're out in the hinterland next year...\"\n\nHe said, \"Wait a minute... when I'm out in the hinterland?!\"\n\nI said, \"Oh yes, that was evil. Sorry, but let's face it, you will be out in the hinterland.\"\n\nRon let on he felt bad that I had hurt him and says sadly, \"Yes, I guess you're right. Nobody ever put it that way before.\"\n\nThat's the relationship we have. No matter how serious the situation is, we find humour in it, just like hockey players. Sometimes it's hurtful, but you have to roll with it or you lose.\n\nRon sometimes gets hay fever and his eyes tear up. One day, I noticed his eye watering. I said, \"You have tears in your eyes. Is your hay fever back?\"\n\nRon: \"Nope.\" (He never admits it because he knows I would show glee about it.)\n\nDon: \"Yes, you do!\"\n\nRon (as tears roll down his cheeks): \"Nope.\"\n\nWe were in Vancouver. We had a Lincoln limo pick us up. It was a new limo, smaller than the ones in Toronto. As I went to get in, I hit my head like you wouldn't believe. Almost knocked myself out. You could hear the \"clunk\" of my head hitting. I was dizzy but wouldn't give Ron the satisfaction of knowing that I hurt myself.\n\nRon: \"Did you hit your head?\"\n\nDon: \"Nope.\"\n\nRon: \"Yes, you did. I heard it.\"\n\nDon (as a big bump started to swell on my head): \"It's your imagination.\"\n\nNever show weakness. Just like when I played. When I got the gout, I never even told him. I said, \"I'm limping because I stubbed my toe.\"\n\nSo now the season's over. We have been going every other day from April 16 to June 13, and Ron worked every day. I really don't know how he does it. I can't write any more about the season. My head is pounding and I have to pack my bags. It takes me an hour to pack; Ron takes about ten minutes. We have a five-and-a-half-hour trip to Toronto ahead of us. I guess I shouldn't have had those last two pops. Okay, well, I will write later about the ending of the season.\n\nWe meet at ten o'clock. It's taken me an hour to pack, as usual. We take a taxi to the airport, check our bags, go through security. Here's a tip: don't look too sharp or wear a tie, because you will be sent to be patted down or go through the security machine, get lectured\u2014they get a kick out of giving you heck. Meanwhile, a guy with big boots, cut-off sleeves, rings in his nose and ears and lips is happily passing by. It happens all the time. One time, I saw a guy who was a poster boy for terrorism pass by, while they pulled over a little old lady with purple hair in a wheelchair to get inspected. Every person who was in the holding station had a shirt and tie on. So look as crappy as you can, and you'll be passed through easily.\n\nSix hours later we land in Canada and get a taxi. On the way home, we see an accident. A Jeep is upside down in the middle of the highway. Looks like nobody's hurt. As we go by, Ron says, \"It looks like that guy is on a roll.\"\n\nThe taxi drops me off first, and he is going to take Ron home. It is the last I will see of Ron till next year. We don't go in for that sentimental stuff\u2014\"great year\" and all that. We just say, \"See ya.\" As the taxi takes off, he hollers at me, \"I still think that dance wasn't good.\"\n\nI holler back, \"Yeah? I'll see you in the hinterland.\"\n\nWell, folks, I guess that's it. I sure had fun writing the book. I hope you enjoyed reading it. I would like to thank Martha Leonard of Doubleday Canada for being kind and putting up with me. The part about my old teammate Larry Zeidel passing away has got me thinking that there are more days behind me than there are ahead of me, but I still get as big a kick out of doing \"Coach's Corner\" today as I did thirty-some years ago. Life is good, and I hope it's good for you.\n\nAll the best,\n\nThat's me and my dad. Doesn't he look like he should be in _Boardwalk Empire?_\n\nMom and Dad are in the middle of the group. They're near Cedar Island in the Thousand Islands, Ontario.\n\nSt. Luke's Anglican Church. My teacher was Mrs. Bruce Hall. I apologize for acting up in Sunday school. I was a jerk. That's me in the back row, second from the right.\n\nMe at sixteen with a broken ankle and my dog, Dudgeon.\n\nAfter a Barrie Flyers practice when I was seventeen years old with my teammates Ross Hudson (left) and Billy Hope (right).\n\nMe in a Leafs uniform, up for a cup of coffee.\n\nIn my Springfield Indians uniform.\n\nMy Springfield Indian teammates (L to R): Cal Gardner, George \"Punch\" Imlach (coach), Norm Johnson, Stan Bulik, me, Claude Dufour, Floyd Smith, Bud Hillman and Tony Schneider.\n\nLooking sharp behind the bench in the Boston Garden.\n\nA tense moment against the New York Islanders. That's Ricky Smith, Davy Forbes, Danny Canney, me, Ron Grahame (L to R).\n\nTaken somewhere in France. Sergeant Thomas W. Mackenzie, a relative, has the little white dog on his lap. He was a member of the 1st Brigade Canadian Field Artillery and was awarded a Military Medal and Bar. Sadly, he was killed four days before the end of the war.\n\nRichard E. Palamountain, my grandfather, of the 4th Regiment Canadian Mounted Rifles. He fought at Vimy Ridge and was fortunate enough to survive the war.\n\nMy favourite picture of Luba and Blue at the cottage.\n\nRon is telling me I'm wrong about something. (photo credit ins2.2)\n\nWith one of the classiest guys I know, Dick Irvin, Jr. When Dick gets on TV, he makes us look like amateurs.\n\nWith Kathy Broderick who is the producer of \"Coach's Corner\" and my Twitter person at a Heritage Game.\n\nGetting ready to do my dance in Times Square for _Hockey Night in Canada_.\n\nWith Arva and Doug Orr in front of the Orr family home in Parry Sound, with my son Tim. Arva and Doug were two beautiful people who I counted as friends.\n\nAnd there's Ron telling me he's #1.\n\nMe doing my stand-up in Afghanistan with the unhappy US troops and Jimmy Mac.\n\nMe entertaining the Royal 22nd in Afghanistan. That's Defence Minister Peter MacKay in the background (left) and Jimmy Mac (right).\n\nWith Tim at a minor hockey game. Tim scouts the minor midget for the Ontario Hockey League. One of the happiest moments of my life. (photo credit ins2.10)\n\nWith Cindy and Bobby, and just a few of his trophies at Bobby's golf tournament in Parry Sound.\n\nWith Ron in Sochi. What do those fingers mean?\n\nTeam Canada 1976. This is the greatest team ever put together. We had to cut fifty goal scorers. (photo credit ins2.13)\n\nChristopher Lawson is a Canadian photographer working on a portrait project of significant Canadians. He asks them to pose with something that has great personal value. I posed with a picture of Jesus that I have had for 65 years. It has travelled with me to 53 different cities. (photo credit ins2.14)\n\n# **PHOTO CREDITS AND PERMISSIONS**\n\nPHOTO INSERTS\n\nInsert 2:\n\nins2.2 \u00a9 Andy Devlin via Getty Images\n\nins2.10 \u00a9 Lucas Oleniuk via Getty Images\n\nins2.13 \u00a9 Graphic Artists\/Hockey Hall of Fame\n\nins2.14 \u00a9 Christopher Lawson\n\nAll other photos courtesy of the author, Kathy Broderick and Leanne Cousins\n\nPERMISSIONS\n\n\"It's Time to Speak Out and Save Don Cherry\" by Jack Todd reprinted by permission of _The Gazette_.\n\nContribution from Jimmy Mac to the \"Afghanistan\" chapter \u00a9 Jimmy Mac, Winnipeg Corporate Stand Up Comic Entertainer\n\nExcerpt from _Players: The Ultimate A-Z Guide of Everyone Who Has Ever Played in the NHL_ by Andrew Podnieks. Text Copyright \u00a9 2003 Andrew Podnieks. Design and Compilation Copyright \u00a9 2003 Otherwise Inc.\n\nConcept Copyright \u00a9 2003 Livingston Cooke Inc. Reprinted by permission of Doubleday Canada, a division of Random House of Canada Limited, a Penguin Random House Company.\n\nExcerpt from letter to Don Cherry from Dion Kneller reprinted by permission of Dion Kneller.\n\nExcerpt from Robin Herman's \"Don Cherry, You Were My Hero\" reprinted by permission of espnW.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"**Praise for Kage Baker and The Company series**\n\n\"Kage Baker has earned praise for her tales of the Company, a future-based outfit that 'recruits' throughout time, turns its new troops into immortal cyborgs, and sends them out to fill in the blank spots of history. . . . Look for it! Baker does not disappoint!\"\n\n\u2014 _Analog_\n\n\"One of the most consistently entertaining series to appear in the late nineties. The novels read like literary pastiches\u2014echoes of Heinlein and Robert Louis Stevenson fill this one\u2014and the narrative pace matches that of most thrillers.\"\n\n\u2014 _Amazing Stories_\n\n**_Mendoza in Hollywood_**\n\n\"[C]ombines historical detail and fast-paced action with a good dose of ironic wit and a dollop of bittersweet romance.\"\n\n_\u2014Library Journal_\n\n**_The Graveyard Game_**\n\n\"If John le Carr\u00e9 wrote science fiction, it might read like _The Graveyard Game_.\"\n\n\u2014 _The New York Times_\n\n**_The Life of the World to Come_**  \n **A Kirkus Selection for Best Sci-Fi of 2004**\n\n\"Another entry in Baker's superlative series about Dr. Zeus . . . An astonishing and thoroughly satisfying installment. What's more, Baker's overall concept and rationale, flawlessly sustained through five books, grows ever more spellbinding and impressive.\"\n\n_\u2014Kirkus Reviews_ (starred review)\n\n\"Baker's trademark mix of serious speculation and black humor informs this solid addition to her time-travel series.\"\n\n_\u2014Publishers Weekly_ (starred review)\n\n\"Baker's strong world-building and clever plotting make this an addictive read.\"\n\n_\u2014Romantic Times BookClub Magazine_\nTOR BOOKS BY KAGE BAKER\n\n_In the Garden of Iden_\n\n_Mendoza in Hollywood_\n\n_The Graveyard Game_\n\n_The Life of the World to Come_\n\n_The Children of the Company_\n\n_The Anvil of the World_\n\n_The Machine's Child_\nKAGE BAKER\n\n#\n\nA TOM DOHERTY ASSOCIATES BOOK  \nNEW YORK\nThe author and publisher have provided this e-book to you without Digital Rights Management software (DRM) applied so that you can enjoy reading it on your personal devices. This e-book is for your personal use only. You may not print or post this e-book, or make this e-book publicly available in any way. You may not copy, reproduce or upload this e-book, other than to read it on one of your personal devices.\n\n**Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.**\nThis is a work of fiction. All the characters and events portrayed in this novel are either fictitious or are used fictitiously.\n\nMENDOZA IN HOLLYWOOD\n\nFirst published in the United States by Harcourt, Inc.\n\nCopyright \u00a9 2000 by Kage Baker\n\nAll rights reserved, including the right to reproduce this book, or portions thereof, in any form.\n\n_Book design by Linda Lockowitz_\n\nA Tor Book  \nPublished by Tom Doherty Associates, LLC  \n175 Fifth Avenue  \nNew York, NY 10010\n\nwww.tor.com\n\nTor\u00ae is a registered trademark of Tom Doherty Associates, LLC.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nBaker, Kage.\n\nMendoza in Hollywood: a novel of the company\/Kage Baker.\n\np. cm.\n\nISBN 0-765-31530-0\n\nEAN 978-0-765-31530-4\n\nI. Title.\n\nPS3552.A4313M46 2000\n\n813'.54\u2014dc21 | 99-14949\n\n---|---\n\nFirst Tor Trade Paperback Edition: May 2006\n\nPrinted in the United States of America\n\nD 0 9 8 7 6 5 4 3 2\nThis book is dedicated to Phyllis Patterson,  \nInstigator, with respect and affection;  \nAnd to the village she founded under the oak trees  \nAnd to its people. _Et in Arcadia ego_.\n\n## CONTENTS\n\nPART ONE\n\n**Establishing Shot**\n\nPART TWO\n\n**Babylon Is Fallen**\n\nPART THREE\n\n**The Island Out There**\n\n## PROLOGUE\n\n_I N THE TWENTY-FOURTH CENTURY, about halfway through, it was said there was a fabulously powerful Company that could obtain virtually anything, if one had enough money_.\n\n_A Shakespeare first folio for your library? A live dodo for your aviary? An original sketch by da Vinci for your bedroom wall? Recordings of every performance Mick Jagger ever gave_?\n\n_What about a necklace once worn by Cleopatra_?\n\n_Have you a favorite historical figure? Would you like to have his baby? Or have your wife have his baby? Guaranteed authentic offspring of Julius Caesar, Napoleon Bonaparte, Elvis Presley_.\n\n_As it happened, the Company actually existed, and it called itself Dr. Zeus Incorporated_.\n\n_It began with two goals: to render human beings immortal and to develop time travel. Success in either goal was incomplete, though with all the money Dr. Zeus was making, it hardly mattered_.\n\n_Time travel, for example, seems to be possible only backward, and then forward again to your point of departure in your present. Nor can you bring anything forward out of its own time into yours. And, by the way, history cannot be changed_.\n\n_You can get around this somewhat by establishing indestructible warehouses in the past, where you stash all the loot you acquire there,to be retrieved in your present time. But you will need a workforce to maintain these sites, and run your errands through time. . _.\n\n_Immortality is another matter. It's absolutely possible to confer it on a human being. Problem is, what you have when you've finished won't be a human being any longer, it'll be a cyborg, and how many people want to pay millions to become one of_ those _things_?\n\n_Somebody clever at Dr. Zeus came up with an idea that solved both problems at a stroke: Make your workforce immortal_.\n\n_Since they'll live forever, there's no need to ship them back and forth through time: look at the costs you'll cut if you just create them at the beginning of time and let them work their way through it, day by day, like everybody else. Transmit your orders to your cyborgs using that subatomic particle you've discovered that exists everywhere and in all times at once. You're in business_.\n\n_Every epoch has its abandoned children, its orphans of war or famine. Won't they be grateful to be rescued and gifted with immortality and lifetime jobs? And what jobs: rescuing precious things from the relentless sweep of oblivion. Of course they'll be grateful. ._.\n\n_This is the third volume in the unofficial history of Dr. Zeus Incorporated_.\n\nIn the Garden of Iden _introduced Botanist Mendoza, rescued as a child from the dungeons of the Inquisition in sixteenth-century Spain by a Company operative, Facilitator Joseph. In exchange for being gifted with immortality and a fantastically augmented body and mind, she would work in the past for the future, saving plants from extinction_.\n\n_On her first mission as an adult, Mendoza was sent with Joseph to England, at that time under the repressive Catholic rule of Bloody Mary. Disguised as mortals, she and other operatives were to loot the private gardens of an eccentric collector, Sir Walter Iden. Her goal: obtain samples of_ Ilex tormentosum, _a species that contained a powerful anticancer drug and that would be extinct in the future_.\n\n_Superior and snide as only a teenaged immortal can be, Mendoza looked down on the mortals among whom she had to labor_ \u2014 _until she met Sir Walter's secretary, Nicholas Harpole, a Protestant heretic_.\n\n_Mendoza and Nicholas engaged in a contest of wits that led them into bed. Passionately in love despite Joseph's warnings about the folly of becoming attached to a mortal, Mendoza attempted to juggle her heart, her mission, and her secret. She failed spectacularly_.\n\n_Nicholas ended up being led to the stake. Mendoza was heartbroken, numb. Joseph came to her rescue again and got her transferred to the Company research base in South America: New World One_.\n\nSky Coyote _opened_ 144 _years later, as Joseph arrived at New World One for a brief holiday before going on to his next mission in Alta California. The project_ \u2014 _persuading a village of Chumash to let the Company relocate them to one of its research bases_ \u2014 _would be immense, requiring the services of operatives of all disciplines. Mendoza was also drafted for the mission_.\n\n_Unpleasant surprises awaited them in California. The immortal operatives met a number of their mortal masters from the future. They were appalled to find them ignorant and bigoted, fearful of their cyborg servants. Joseph learned unsettling facts about the Company that brought to mind a warning given him centuries earlier by Budu, the immortal who recruited him_.\n\n_Why was it that, though the immortals were provided with information and entertainment from the future, nothing they received was ever dated later than the year 2355? The Company's official answer was that in_ 2355 _Dr. Zeus would be able to go public with its great work and reward its operatives for their ages of service. But could the Company be believed_?\n\n_Mendoza, back in contact again with the mortal world, found that her heart had not recovered from Nicholas. Despising the mortals and uncomfortable even with her own kind, she found comfort only in the vast wilderness of Alta California. In its forests she was able to leave her painful humanity and focus on the only reliable consolation: her work as a botanist_.\n\n_Then, after_ 160 _years,. ._.\n**Mendoza in  \nHollywood**\n\n###\n\nTRANSCRIPT ACNW032063 PRIVATE HEARING  \n _Subject: Botanist Mendoza. March 20, 1863_.  \n _Five kilograms Theobromos administered_.  \n _Auditors magisterial: Labienus, Aethelstan, Gamaliel_.\n\nYOU WANT THE TRUTH from me? It's a subjective thing, truth, you know, and you could easily get all the damning evidence you need from the datafeed transcripts. Oh, but you wouldn't understand my _motive_ , would you? I see the point.\n\nWill it help if I freely confess? I killed six\u2014no, seven\u2014mortal men, though I must say it was under provocation. I acted in direct violation of the laws that govern us, of the principles instilled in me when I was at school. I betrayed those principles by becoming involved in a mortal quarrel, supporting a cause I knew must fail in the end. Worst of all, I stole Company property\u2014myself, when I deserted the post to which I had been assigned. I don't expect mercy, se\u00f1ors.\n\nBut it might help you to know that what I did, I did for love.\n\nI had an unfortunate experience when I was a young operative, you see; I was baptized in the blood of a martyr. No, really. Did you know those things work, baptisms? I didn't. I was given the same education we all get, sanity and science and reasonable explanations for everything that happens in the world. Faith and its attendant rituals sound like a good deal, the whole eternal salvation thing, but inevitably they lead to fear, oppression, the rack and flames. I knew that much was true firsthand.\n\nI was blindsided, as I'm sure you would have been, by the discovery that the experience actually left some kind of psychic mark on me. The mortal man smeared his blood and shouted his incantation, and there I stood like an animal that's been collared and let go, to wander bewildered among my own kind wondering what had happened. I was never right again after that. For a long time I thought I'd shaken off his spell. I was almost happy there in the mountains all alone. But you wouldn't let well enough alone. You sent me back into mortal places, and he found me again, tracked me by the mark he'd put on me for that purpose.\n\nHe will never let me rest.\n\nThank you, I certainly will have some more Theobromos. This is excellent stuff, by the way. Keep it coming, and no doubt you'll find out everything you want to know, with me a weepy mess at the end of it.\n\nOkay, se\u00f1ors, are those tapes rolling?\n\n###\n\nANY OF YOU GENTLEMEN ever served in Los Angeles? No? Rough place. Murders and fighting all the time since the Yankees came. No good reason to put a city there, on that clay bluff above the river; but Spain was so certain the Russians were going to invade Alta California, they had to go stick little pretend towns along its coast, like pins on a map. That way they could claim white settlement, because the mission Indians didn't count.\n\nWhite! That was a laugh. What happened was that Felipe De Neve sent his goons riding up from Sinaloa with anybody he could bribe, threaten, or deceive into coming along as prospective settlers. There were maybe one or two Spaniards in that bunch, but the rest were mestizo and mulatto ex-soldiers, the mingled blood of New Spain and Africa with their wives and little children. De Neve's men dragged them up through the desert and over the mountains and set them down by that dry wash of a river, with its big sycamore trees. And after a mass was duly celebrated, they left them there, rode away and left them staring out into that night, and what an empty, empty night it must have been. No neighbors but the local Indians, and nothing to shelter them from the bears but brush huts. The settlers, huddled together listening to the coyotes howl, must have wondered what on God's earth they had got themselves into.\n\nBut they made the best of things, built a little adobe village, got some Indians to be their slaves, and in a generation or two they were gentlemen rancheros, with thousands of head of cattle on estates the size of small kingdoms, estates that would have made the threadbare gentry of the Old World sick with envy.\n\nOf course, if one wanted a chamberpot or a carving knife or a bolt of cotton cloth, one had to wait for the supply ship from Mexico, which put in an appearance once every five years or so. This situation did not improve after the Revolution, either; a free and democratic bureaucracy moves even more slowly than a viceregal one. So in came the Yankee traders, smuggling consumer goods in their trading brigs, and the Californio rancheros were only too glad to do business with them. You know where _that_ led. Richard Henry Dana wrote home about the fortune waiting to made by anybody with the ambition to build mills and factories here. Emigrants from the United States came struggling over the Rockies to see if it was true, some lady found a gold nugget in a sluice, and in no time at all we were all Americans, thanks to a little strong-arm work by John C. Fremont.\n\nNot a bad thing, entirely, at first. It was the making of San Francisco. Los Angeles, though, sort of festered. It filled up with drunks and outlaws, white trash from the States who'd failed at gold prospecting, men on the run from civilization generally. There was nothing down there, you see, except dry brown hills and cattle, plenty of space to get lost in. Soon there were lots of saloons to get lost in too, and drunken shoot-outs in the streets. There were so many murders, people began calling Los Angeles the City of Devils rather than the City of Angels. _Los Diablos_. The old ranchero families huddled in their fine haciendas, listened to the gunfire, and wondered what in hell had happened to their town.\n\nSo you can see, se\u00f1ors, why I wasn't exactly thrilled to be posted down there. Monterey, green and gracious, that was where I preferred to be when I had to work near mortals; better still, the wild coastal mountains, the Ventana and Big Sur.\n\nWhen you're coming down from the north, Los Angeles looks horrible at first: all brown rolling monotony. Hasn't got the redwoods, hasn't got the green mountains or the air like wine. It's a sad, trampled place. But let me put it on the record that my distaste at my assignment played no part in what happened. I went where I was told and did my job. I always have. We all do.\n\nWeren't you briefed on this part? All right, I was sent to the HQ in Cahuenga Pass, close by La Nopalera. The cover is that it's a stagecoach stop. It's far enough out from Los Angeles to give us privacy, but being on the stage line, it's convenient for getting agents in and out. Agents and other things.\n\nBut that's all beside the point. Give me more of that\u2014it's Guatemalan, isn't it?\u2014and I'll try to stick to the story. You know, it's amazing, se\u00f1ors, but you bear a striking resemblance to certain inquisitors I knew in Old Spain. All of you. It's your eyes, I think. They're too patient.\n\n## PART ONE\n\n## CAHUENGA PASS, 1862\n\n###\n\nI ARRIVED DURING a miserable winter. It had rained most amazingly; the locals had never seen such rain. The canyons flooded. The new sewers down at the pueblo were a total loss. Roads washed out, and the stages were late or never arrived at all. There was, I understand, a little mining town up in the San Gabriels that was washed away completely\u2014whole thing wound up down on the plain in scattered soggy bits. Only the rancheros were happy, because of the good grazing there was going to be from the rain. They thought. Little did they know that that was the last rain they were going to see for years. Before it rained again, Se\u00f1or Drought and Se\u00f1orita Smallpox and a few shrewd Yankee moneylenders would pretty well end the days of the _gentes de raz\u00f3n_. Ah, Los Angeles. One disaster after another, always has been.\n\nThose particular disasters were still somewhat in the future on the day I finally walked into HQ. I'd followed the coast down as far as Buenaventura and then swung inland to follow El Camino Real through the hills and along the valley floor, traveling mostly by night to avoid the mortal population. The rain never let up the whole way, and I was soaked through. I crossed innumerable creeks swollen with white anger, roaring their way out to sea and taking willow snags with them. I saw smooth green hillsides so saturated, their grassy turf slid, like a half-taken scalp or a toupee, and left bare holes that the rain widened.\n\nSo much for Sunny California. All I saw of it that dark morning was water, brown water and creamy mud, and black twigs bobbing along in the hope of someday washing up on a white beach. You can imagine how grateful I was to see a plume of smoke going up between one foothill and the next. I checked my coordinates. _Cahuenga Pass HQ_? I broadcast tentatively.\n\n_Receiving_ , someone responded.\n\n_Botanist Mendoza reporting in_.\n\n_Okay. You see the smoke? Follow it in_.\n\nAnd in another minute I'd come around the edge of a rockslide, and there it was, back under some oak trees, a long low adobe building and stable thatched with tules. A couple of cowhides had been stitched end to end and strung up in the trees like a tarpaulin, and under this nominal shelter an immortal crouched, attempting to build up a small fire with what looked like fairly damp wood. Arranged on the ground beside him were a blue graniteware coffeepot and a couple of skillets. The idea of grilled beef and frijoles drew me like a magnet.\n\n\"Hola.\" I jumped the last brown torrent and made my way up the sandy bank to the inn.\n\n\"'Morning.\" The immortal looked up from under the brim of his dripping hat. \"Welcome to the Hollywood Canteen.\"\n\n\"This _is_ where Hollywood's going to be, isn't it?\" I asked. I dropped my bag and held my hands down to the little fire. \"Funny thought.\"\n\nMy informant stretched out an arm to point, trailing the fringe of his serape through dead leaves. \"Chinese Theater and Hollywood Bowl right down there. Paramount Studios out in that direction. If you've got eighty years to hang around, we can go for breakfast at the Warner Brothers' commissary.\"\n\n\"I'll settle for what you've got.\" I eyed the skillets: last night's leftovers, cold and congealed. I looked around for something dry to add to the fire.\n\n\"So you're Mendoza?\" inquired my host. He was lean and dark, with a thin black mustache and a sad, villainous face villainously scarred. The scars were all appliance makeup, of course, but they gave him the look that sends liquor store owners diving behind counters for their shotguns. I nodded in reply.\n\n\"Porfirio.\" He reached across the fire and shook hands with me. \"I'm your case officer, subfacilitator, and security tech. Nice to meet you.\"\n\n\"Thanks. Is it dangerous here?\"\n\n\"Oh, yeah,\" he said. He took up an oak log and tried stripping the wet cork layer off. \"We don't get much trouble over this way, but you want to be careful when you ride out.\" He broke the log between his hands and fed it carefully into the coals. \"Especially where you'll be working. Your temperate belt passes through some nasty bandit nests.\" He was referring to the climate anomaly that was my present assignment, a long terrace roughly following the future route of Sunset Boulevard, where an unusual weather pattern had evolved some plants unique to the area, several of which had potentially remarkable commercial properties. Unfortunately they were all scheduled to go extinct in the next big drought, grazed out of existence by starving cattle.\n\n\"Bandits?\" I was profoundly annoyed. \"They told me I was going to be working in Beverly Hills!\"\n\nHe was really amused by that. \"Oh, you will be! It just isn't _there_ yet. What, were you planning on having a cocktail in the Polo Lounge? You've got a while to wait if you want to see the mansions and the swimming pools.\" The fire blazed up at last, and he edged the skillets in toward its heart. \"Come on, little fire, come on, we want some breakfast. Where's your horse, by the way?\" He looked up in surprise as it occurred to him that I'd walked in.\n\n\"I don't have one.\"\n\n\"You're kidding me! _Nobody_ walks down here. We've got a good stable you can choose from,\" he said firmly.\n\n\"That's okay. I don't care for horses, actually.\"\n\n\"I don't myself, but I ride them here. Trust me. You may need to get out of certain situations in a hurry. This is Los Diablos, after all.\" He put up a hand to stop my objections. \"And don't think you can deal with the situation by just winking out at a speed mortals can't see. That may have been all right in the old days, but there are a lot of people out here now. It's too conspicuous. You'll need a horse. Everyone rides them. You'll need a gun, too.\"\n\n\"A gun?\" I said, sitting back on my heels. \"I've never carried a gun! You mean you've actually had to shoot people?\"\n\nHe nodded somberly.\n\n\"But we were always trained\u2014\"\n\n\"I know,\" he said as he pushed the coffeepot over wavering flames. \"The rules are different down here. You'll see.\"\n\n\"Who are you talking to?\" Another operative emerged from the adobe, stooping below the wooden lintel of the door. He stood, sleepily scratching himself through a suit of long underwear worn under blue jeans. He gave a yawn that turned into a shiver.\n\n\"The botanist's here.\" Porfirio gestured with the skillet. \"Mendoza, this is Einar. Einar, this is Mendoza.\"\n\n\"Zoologist grade 5.\" He came forward and shook my hand, then crouched down beside us. \"Fire's not doing so good, is it?\"\n\nIt wasn't. It had sunk away from the coffeepot and was smoking out.\n\n\"Wood's wet,\" he said.\n\n\"No kidding,\" we told him. He was tall for one of us, with white-blond hair and eyes like ice caves. Spectral coloring aside, he was a nice-enough-looking fellow.\n\n\"I was just giving her the safety lecture,\" Porfirio explained, handing him an oak log to break.\n\n\"Uh-huh.\" Einar snapped it into fragments. \"Hey, chief, did you tell her about where we are? The movie studios and everything?\"\n\n\"Yes. I thought you could issue her one of the Navy pistols and give her a short training session with it.\" Porfirio took the kindling from him and fed it into the coals, where it caught.\n\n\"No problem.\" Einar poked up the fire and coaxed a few tongues of flame to rise. \"Come on, I need some coffee. There. Yeah, and I can show you where all the neat stuff will be. A lot of early cinema is shot in these very canyons. DeMille, D. W. Griffith, Hal Roach. Tinseltown!\"\n\n\"But there's nothing there to actually see yet, is there?\" I said.\n\n\"Well, no. Except the familiar landscapes, you know. I just enjoy the atmosphere of it all.\" Einar waved another oak branch in the air. \"I mean, here we are in the mundane West, as far west as you can go, if you think about it, and everywhere around us the West of the cinema\u2014the true West, if you will\u2014is just sort of immanent. Hovering in these canyons like a spirit, waiting to be born. Ghosts of the future. All this greatness just about to happen, but not yet. We are the actors on a stage where the curtain hasn't risen!\" His eyes were alight with enthusiasm.\n\n\"We're behind the scenes, you mean.\" Porfirio watched the fire doubtfully. A little thread of steam was rising from the mouth of the coffeepot, but the grease on the beefsteaks was still cold and waxy.\n\n\"Good morning, gentlemen,\" said yet another of my kind, stepping out into the courtyard. This one looked like a little Yankee lawyer or congressman, in a black suit and polished boots, with a cosmetically induced receding hairline that featured a sharp widow's peak. His eyes bulged slightly when he saw me. \" _And_ lady. Why, you must be our new botanist. Pleased to meet you, ma'am, I'm sure. Mendoza, isn't it? Yes. Oscar, grade 2 anthropologist, at your service.\"\n\nI nodded at him. He put his hands in his pockets and came over to stand looking down at the fire. \"Say, you know\u2014\"\n\n\"The wood's wet,\" said Porfirio.\n\n\"It is, isn't it? No, I was just thinking, wouldn't some of my corn bread go good with those steaks and beans? I'll just go fetch it out.\" He ran back indoors, and Porfirio and Einar exchanged a disgruntled look.\n\n\"What?\" I said.\n\n\"He tried making corn bread out of masa,\" Porfirio said. \"He's very proud of it.\"\n\nThere was a gloomy silence. The trees dripped. There was a distant rumble of thunder; from the sound of it, the storm front was approaching the future site of the Whiskey a' Gogo.\n\n\"This is Raymond Chandler country too, isn't it?\" I said.\n\n\"Yeah.\" Einar brightened. \"Laurel Canyon, Hollywood Boulevard. I could show you\u2014\"\n\n\"Here it is.\" Oscar came bustling out with a pan. He dropped the bread beside the guttering fire\u2014there was an audible thud\u2014hitched up his trousers, and crouched down to cut slices. \"Miss?\" He offered me a slab of solid gray cake.\n\n\"My, isn't this substantial\" was all I could think to say.\n\nHe beamed. \"Real stick-to-your-ribs food for a chilly morning, yes indeed.\" He stood again with his hands in his pockets, rocking back and forth in his shiny shoes. \"So, Miss. You're in botanicals? What are you going out for, if I may presume to ask?\"\n\n\"Um\u2014rarities. I was told there's a lot of good specimens of _Striata pulchra_ I need to collect, as well as some mutations of common plants. Snowberry, Artemesias, that kind of thing. Creosote bush,\" I said. My job always sounds unbelievably boring to anyone but another botanist, so I didn't take offense when he blinked and forged on:\n\n\"You don't say so? I'm in notions, myself. Of course that's just my cover, ha ha! Actually I'm here to report on the impact of Yankee settlement on the local inhabitants\u2014the decent ones, I mean\u2014and document early Anglo-Californian culture.\"\n\n\"I see.\"\n\n\"I've got the sweetest little cart back there you ever saw.\" He nodded in the direction of the stables. \"Just a wonder of clever design. Only requires the work of one mule\u2014seats two\u2014sides unfold for display of anything the locals could want to buy, from threepenny nails to dancing pumps, plus a complete photographer's apparatus, _plus_ I can sleep in it, if I'm benighted somewhere and the weather's foul. I have but to fold down the seats and slide out the patented Collapsi-Cot!\"\n\n\"Gosh, how clever.\"\n\n\"And, you know something? It's not Company issue! Not at all! The whole thing was made by a firm in Boston, Massachusetts!\"\n\n\"Speaking of cots,\" said Einar, grinning. A female operative appeared in the door of the house and yawned expansively, stretching up her arms like a dancer. All I could see was a flowing wave of white ruffles on a fancy nightdress, of the kind I hadn't owned in years. When she brought her arms down in a slow dramatic gesture, I saw that the bosom was cut low enough to make her look like the heroine of a romantic novel. She gave a little toss of her head\u2014lots of dark-ringlet hair whooshed from side to side\u2014and raised startling green eyes to regard us.\n\n\"Imarte.\" I placed her.\n\n\"Would that be Mendoza?\" She paced forward, pretending to peer at me through the gloom. \"It _is_ the botanist Mendoza, isn't it? I believe we worked together on the Humashup mission?\"\n\n\"Yeah,\" I said.\n\n\"You were a friend of, ah, Joseph's.\" The corners of her lovely mouth turned down.\n\n\"That's right.\" I grinned with all my teeth. \"And you're an anthropologist.\" She hadn't got on very well with my old pal and erstwhile mentor, as I recalled. In fact, there'd been a truly nasty incident, hadn't there? Well, this was going to be a fun posting.\n\n\"An insertion anthropologist,\" she corrected me, and Einar fell over in helpless giggles. Even Porfirio smiled under his mustache. Oscar turned red and looked at his shiny shoes. \"I'm stationed in this culture on a semipermanent basis, interacting with the mortal element in Los Angeles in order to observe them more closely, as opposed to an anthropologist like Oscar, who merely interviews,\" she said primly.\n\n\"She, uh, her cover identity is as a sort of a\u2014\" began Einar, but Imarte finished:\n\n\"A whore. And there's really no need to make a dirty joke out of it. I've been a temple prostitute on numerous occasions during my career. Men speak the truth in bed, as the proverb goes, and what better place to gain valuable insights into the _real_ life of a culture? And this is an astonishingly rich era for study. In one night I might have a conversation with a Yankee from New York who came west to pan for gold, followed by a Mexican outlaw whose family were massacred by Indians, followed by an Australian ex-convict who failed at piracy, followed by\u2014well, followed by _anybody_.\" She tossed her head. \"Why, during this period in history the whole world is passing through the Golden Gate!\"\n\nI don't think she meant the one in San Francisco. I blinked.\n\n\"You actually go to bed with all these people?\" I asked.\n\nShe lifted her chin at me. \"What, I should feel degraded? Should we not consider it, rather, as a way for me to experience their lives more fully, more meaningfully? Particularly in view of the fascinating material I'm compiling on mid-nineteenth-century mores and sexuality in California.\"\n\n\"Besides, any good stagecoach inn has at least one hooker,\" argued Porfirio. \"It makes our cover more authentic, and contributes to our operating budget too.\"\n\nThis was more than I cared think about. I turned to Porfirio. \"So, okay. . .I'd like to see my quarters after breakfast, if I could. I'm pretty tired.\"\n\n\"I'll bet you are, after walking all that way,\" Porfirio said.\n\n\"She walked here?\" Oscar asked, staring. Imarte looked appalled.\n\nThen there was another person standing beside our almost-fire, so silent in his approach, he seemed to have materialized there. He too was an immortal, but a young one; if you knew where to look, you could still see the scars of his augmentations. Remarkably enough, he had been made from an Indian. I hadn't seen many of these. My guess was he'd been among the few survivors of the Channel Island tribes, because he had their silver hair. It used to be a fairly common color for Native Americans, but smallpox was swiftly winnowing it out of their gene pool, the way the Black Death had rendered extinct similar exotic strains in Europeans.\n\n\"Hi,\" he said.\n\n\"Where've you been this morning? You were out early,\" said Porfirio.\n\n\"I heard him crying,\" the boy said, and held up in his cupped hands a tiny writhing monster from outer space. \"It's a baby condor. _Gymnogyps californianus_. The mother hadn't come back to the nest in a while. I guess somebody shot her. I had to climb way, way back up the canyon to find him. Are you the new botanist?\" He looked at me.\n\nI nodded. \"Mendoza. And you're\u2014?\"\n\n\"Juan Bautista.\" He came closer to the fire and peered down at it. \"We need some dry wood or something, huh?\"\n\n\"Wait, I have an idea,\" said Einar, and jumped up and ran inside. A moment later he emerged with a case bottle of a clear liquid. \"Home-brewed aguardiente.\" He strode toward us, uncorking it. \"We tried it on a plum pudding, and the damned thing burned for two hours. This'll do the job.\"\n\n\"Careful how you\u2014\" said Porfirio.\n\nI threw myself flat and rolled. I heard Imarte scream. The fireball took out the cowhide tarp, but when I looked around cautiously, there was certainly a merry blaze going, all right, flames five feet high. And breakfast was cooking at last: in fact, the frijoles were on fire.\n\n\"SORRY ABOUT THAT,\" called Einar from where he had retreated about thirty yards up the hillside.\n\n\"Couldn't you have been more careful?\" complained Oscar, squelching up from the mouth of the canyon. \"Now my shoes are wet.\"\n\nLater Porfirio showed me to my quarters. I had a cot all to myself in a board-and-batten lean-to next to the adobe stable. It smelled of horses, it had a dirt floor, and water was seeping up the wooden legs of my cot; but the stretched cow skin over the frame was dry, and so was the woolen blanket, and there was a dry ledge on which to put my bag. As field accommodations went, not too bad. I sat down and pulled off my wet boots.\n\nNow, se\u00f1ors, I think some of the reason for my subsequent lamentable behavior is evident right here in the next scene. I hung up my oilskins and shrugged into dry clothes, meditating smugly to myself that it didn't take much to make me happy nowadays. I was an old hand now, wasn't I? A couple of tamales and a dry place to put up my feet and read a novel were enough. I could make my own space anywhere they posted me, as a good operative should. Wisdom at last. Perhaps I'd attained enlightenment after tramping through that beautiful desolation all these years collecting specimens alone. Certainly I had equilibrium.\n\nWell. Where pride flaunts such scarlet banners, blares such brazen trumpets, you know what follows.\n\nI had turned in early, after walking up on the ridge above the house to get a feel for the land, making friends with the oak trees, exploring the rooms of the coaching inn, chatting a little more with my fellow operatives (the male ones anyway) most amiably and normally. I had endured a few hands of gin rummy with Juan Bautista and an hour or so of anecdotes from Oscar about his mortal customers before I excused myself on the grounds that I'd been conscious for forty-eight hours and needed some alpha waves badly. And so I retired to my spartan room, pleased with myself. I'd done well for my first day back among people, I thought. I ought to be able to handle this posting just fine.\n\nMaybe it was having an actual bed to lie down in that did it. See, I'd got out of the habit of sleeping like mortals, all those years in the mountains. I know, we're not supposed to do it\u2014but it's so convenient just to lean into a tree limb or an angle of rock and fugue out for a couple of days, especially when you're on your own schedule. You have no idea how restful it can be. You just sort of tune in to the patterns moving in the tree or the rock, and you forget you even exist.\n\nOne feels so vulnerable, undressing and getting into a bed. It summons up memories. The memory that it summoned was not one I wanted. I lay in the dark listening to the rain drip, to the wind sigh in the oak leaves. Because of the downpour, there were no coyotes howling, so you didn't necessarily know you were in California. With just the wind and the rain, it might have been anywhere. It might have been England.\n\nThen I thought it had to be England, because my door was opening and there was green and sunlight beyond, that wet sunlight and drowningly intense green of an English garden. And _he_ was there, standing in the doorway and looking at me. So tall, he had to bend to look into my narrow little room, and his black scholar's robe trailed in the lush summer grass. He was smiling, smiling at me in the old way, and held out his hand to me.\n\n\" _Talitha koum_ ,\" he said, in that voice I'd have leaped out of a grave for. \"Up and haste my love, my dove, my beautiful and come, for now is winter gone and rain departed and past!\"\n\nWho was he, se\u00f1ors? My martyr, who do you think? Nicholas Harpole, the mortal man with whom I fell in love when I was a young operative who should have known better. He died on April 1, 1555. This was either a dream or a haunting.\n\nA haunting, I think.\n\nI was so glad to get up and run to him where he waited in that doorway, to quit my dark cold room and run into the summer garden. Time must have turned backward, because there I was in my peach-colored gown with all the petticoats and my placketed bodice and stiff sleeves, and my hair falling loose. I never braided it back then, I was too vain. I leaped into his arms, and he made a little pleased chortling noise in his throat, and we were kissing, staggering backward and bumping into the orange tree, so his scholar's biretta was knocked forward over his eyes. I pushed it back, and we kissed and kissed, and oh the taste of his mouth and the scrape of stubble under his jaw and the hungry noises he made, I hadn't forgotten one detail in three centuries of trying desperately to forget.\n\nWe kissed until we were blind and gasping and his breeches needed to be unfastened, so I did it for him, and he pulled his long shirt up and out and freed himself, and I hadn't forgotten one detail about _that_ either. Nor had I forgotten what to do next: one didn't just grab the front of one's skirt and haul it up, no, that was sluttish, one gathered it at the two sides as though one were going to sit down, and one lifted it in a discreet and genteel manner just far enough for the purpose. No bloomers, no underwear at all, only my fine-woven stockings gartered at the knee, and he was bending down to gather me up in his big fine hands and lift me against him.\n\nEvery detail was perfect and exact, the rustle of my gathered petticoats, the texture of his clean linen shirt and woolen doublet under my cheek, and how they smelled with the full hot sun on them releasing the scent of washing soap, and the green feverish privet-fragrance of the garden at high summer. I had my arms around his shoulders, my face pressed into the side of his neck, and a trickle of his sweat came coursing down as he labored, and I kissed the sweat away. And how we strove there together, in our mutual delight, as he growled his pleasure and I felt the vibration of it in his throat, and how soundly we knocked the trunk of the orange tree, until golden fruit fell all around us in a shower.\n\nOh, I thought, jackpot, surely that's a favorable omen. And as we rested, sobbing for breath, I felt a little stab. We both looked down, surprised. Protruding from my heart was the haft of a bodkin, and my heart's blood was welling out around it like a cut pomegranate. I laughed, joyous, incredulous, and he laughed with me. We both knew what this meant: I was free of the world now, I could stay in that garden with him eternally. Impossibly, wonderfully, my weary heart had stopped beating.\n\nBut I was slipping somehow, he was losing his hold, and the pain became terrible as I opened my eyes into darkness and felt the chill, heard the rain.\n\nI hadn't gone through the doorway to him after all.\n\nIt took a long moment of struggling for air to be able to voice my agony. I wasn't loud, even so; but within seconds I heard hurrying feet crunching through the oak leaves.\n\n_Mendoza_?\n\n_Go away!_\n\nHe wouldn't go away. The door opened, and Porfirio looked in on me, all concern, a black silhouette against blacker night. Good God, I thought, what's he got that gun for?\n\n_You okay_?\n\n_Yes! Don't you ever have bad dreams_? I transmitted. He looked at me in irresolution a moment before nodding and shutting the door.\n\nI guess, se\u00f1ors, if I were somebody else, I might have called him back and begged some kind of physical comfort as a favor, that I might not lie alone there in the dark. But I was and regretfully am myself, and I was not alone. The dark thundered around me like a palpable presence, and I belonged to it.\n\nNext morning, Porfirio didn't mention the disturbance, and neither did I. I ran a diagnostic on myself, but everything tested out normal.\n\nFor the first couple of weeks that I lived at that stagecoach inn, there were no stagecoaches at all, and precious few travelers, because the roads were in no condition for going anywhere. This situation didn't improve much the whole time I was there, in part because Butterfield had decided to cancel its stage lines to California for the duration of the American Civil War, which left its rival Phineas Banning with a lot of territory to cover all by himself.\n\nAt about the point where there were no dead trees left to float down the canyons and you thought all the topsoil had washed out to sea, the rain stopped. The sun came out! And abruptly we were living in a paradise. The hills were green and purple and silver with sagebrush, the grazing land was brilliant lime-green with good grass, and everywhere there were banks of wildflowers. I stepped out under the dripping oak trees and looked up at the puffy white clouds in the blue sky. What a fine place, I thought. What a _beautiful_ place. Murder? Robbery? Social dysfunction? Surely not in sunny southern California.\n\nI was humming Rodrigo's _Concierto de Aranjuez_ (first movement) as I went indoors to collect my field kit. Temperate belt, here I come! My ghost had not returned to haunt me, and my work was waiting. Who knew what botanic anomalies lay there waiting to be discovered?\n\nBut when I shouldered my pack and emerged from my quarters, Porfirio looked up from the cookfire he had just got going. \"Where are you off to?\" he asked with a frown.\n\n\"I thought I'd go get some work done at last, now that the rain's let up.\" I gestured at the enchanting sky.\n\n\"Not alone, you're not. Not over in that neck of the woods.\" He shook his head decisively. \"Too many damn bandits. Wait until Einar gets up. He'll go with you.\"\n\n\"Oh, come on,\" I said, almost too surprised to be annoyed. \"I've been on my own in California for years. I've outrun bears, Indians, and every one of the Joaquins. I worked in San Luis Obispo, where they hang so many outlaws, there's a gallows at the end of every street. I don't need an escort, thank you very much.\"\n\n\"It's different down here,\" he said, and something about his tone made me slip off my pack and find a dry place to wait as he got a pot of coffee brewing. Presently Einar came out, hopping on one booted foot as he pulled the other boot on.\n\n\"Hey, folks,\" he said. \"Sunlight, huh? I guess I'll go out looking for a coyote or two today.\"\n\n\"You can go out with Mendoza,\" Porfirio said.\n\n\"Sure.\" He grinned at me. \"You want the Grand Hollywood tour? Show you where the homes of the stars will be.\"\n\n\"I hope there's something useful growing there,\" I said grumpily. \"And did I hear you say you were collecting _coyotes_?\"\n\n\"Uh-huh.\" He poured a mug of coffee and handed it to me. \"Don't laugh. The particular subspecies in this area will disappear after a couple of centuries. First they'll crossbreed with settler's dogs. Then they'll crossbreed with wolves that escape from the zoo during a riot\u2014that's during the dark ages of L.A.\u2014and they'll get so huge and vicious, they'll start eating street beggars when the winters are bad.\"\n\n\"Jesus.\" I shivered, looking up at the sky. It was such an innocent shade of blue.\n\n\"But these little guys we have now are real sweet,\" he told me seriously, sipping his own coffee. \"Sort of foxy. Nothing to worry about. Not like when I go after a bear.\"\n\n\"Bears?\"\n\n\"California brown bears, like the one on the flag,\" he said. \"They're already on the way out. Last known survivor in California will be shot right here, or actually out there\"\u2014he pointed down the canyon\u2014\"in 1912. Then they're extincto. Supposedly. They take some catching!\"\n\n\"I can imagine.\" I looked into the bottom of my graniteware cup. \"Tell me, are we likely to encounter any bears today?\"\n\n\"Only if we're lucky,\" Einar said. \"Since you have other work to do, I thought we'd keep things simple for now.\"\n\n\"Wonderful.\"\n\nAfter a breakfast of velvety frijoles and steak rolled in tortillas, we saddled up and went to explore. Einar wore fearsome-looking bandoliers and a pair of shotguns, one behind either shoulder, like samurai swords. He showed me a trail that led through the back of the canyon and up a series of switchbacks to the top of the ridge. We followed the rimrock above the foothills, and down below us the plains swept out to the east, where there were big white snowy mountains, and south and west, where beyond the sprawl of adobes that was Los Angeles the land terminated in the blue line of the sea. Paler blue and farther out, lay a floating mountain.\n\n\"Is that an island?\" I asked, squinting at it.\n\n\"Catalina,\" Einar said. \"Location shots for _Mutiny on the Bounty, Treasure Island_ , and a couple of versions of _Rain_ , to name but a few. And check this out.\" He leaned over in his saddle, pointing into a steep valley to our left. \"Know what that is, down there? Hollywood Bowl! Imagine a big white half shell right there, with a reflecting pool in front of it, and a fan of seats sweeping up and up, rows and rows of pearly gray wooden benches, and thousands of screaming girls filling the amphitheater. Right down there is where the Beatles will perform. I've been up here at night, alone, and I swear to God I can hear them.\"\n\nI stared down, impressed, though all I could see was a wilderness of sagebrush and toyon holly. \"That's when, the 1960s? Only a century off. Where's the Hollywood sign?\"\n\n\"Look across there.\" He pointed east, where a red mountain thrust up against the sky like a rippled wall. \"Under the crest. It'll say _Hollywoodland_ first, for the real estate park below it, and then when the last four letters fall off, they won't be replaced. There'll be two sister signs as well, for a couple of other developments, one down on that lower ridge that'll say _Bryn Mawr_ and one over here that says _Outpost_ , but they won't last long, and nobody will remember they were ever there. Neat, huh?\"\n\n\"You certainly know the area,\" I said. We urged our horses on, and they continued to pick their way down the trail. \"What are you going to do when it finally starts to happen, though? You're a zoologist. Dr. Z isn't likely to keep you here once the bears and the coyotes are gone.\"\n\n\"I have a double discipline,\" he said. \"Programmed for zoology _and_ cinema. I can stabilize a silver-nitrate print with one hand and do a genetic assay on a musk ox with the other. I've been in the field for millennia, though, so all my experience is with animals. Reindeer, caribou, wolves, those guys. The wheels of time roll swift around, though! When this assignment's up, I'm off to Menlo Park and then on to Melies in France. I'll be in on film from the beginning. I just hope Dr. Zeus sends me back here once the industry's up and running. Wouldn't that be swell?\"\n\nWe both heard the shot, a little _pow_ from somewhere down in the canyon to our right, and in a tiny fragment of time the bullet came zipping up through the bushes at us. We were off our horses and flat on the earth by then, however.\n\n\"Aw, shit,\" Einar said. He unslung one of the rifles and pointed it in the direction of our assailant; then he went into hyperfunction, firing and reloading faster than a mortal eye could have followed, sending a volley into the canyon below that cut a swath through the pretty purple sage. The echo boomed off the opposite ridge like thunder on Judgment Day. The mortal must have wondered what war zone he'd stumbled into, but he didn't wonder it for long; as I scanned, I felt his vital signs flutter and fade out.\n\n\"You killed the guy!\" I gasped, rising shakily to my knees. The horses stood calmly cropping scrub, as though nothing had happened.\n\n\"Gee, I wish he hadn't done that.\" Einar stood and peered down the hill. \"I thought he was going to leave me alone this time.\"\n\n\"You mean he's taken shots at you before?\" I was incredulous. \"And we rode up here anyway? Into _danger_?\"\n\n\"What danger?\" Einar loaded another couple of shells. \"Stupid bastard knew he couldn't hit me, after all the times he's tried it. I'd shown him what I had to throw at him, too. When you're too dumb to learn, you're out of the gene pool, man. Down here, anyway.\"\n\nI stood staring into the silent canyon. Cool air currents brought up a smell of cut sage and fresh blood. I half-expected sirens or shouting, but all I heard was the wind. \"Shouldn't we go down and do something?\" I suggested.\n\n\"Nope.\" Einar slung his gun back over his shoulder. \"That's what buzzards are for. Turkey vultures, actually, according to J. B.\" He bent and offered me his cupped hands to climb back into the saddle. I vaulted up and sat there, bent forward nervously as he got back on his own horse.\n\n\"Let's get off the skyline, shall we?\" I said. He nodded, and we moved on, descending the gentle ridges through the aromatic brush. After a moment I asked, \"But who was he? He had to have had some family or somebody, someone we should notify.\"\n\nEinar shook his head. \"He killed 'em. That was why he was hiding out up here. Thirty-year-old Caucasian male psychopath from St. Louis, Missouri. Also killed two Mexican hookers and three Chinese guys of assorted vocations. I don't know why.\"\n\n\"Oh,\" I said. We rode on.\n\nAfter a while, I ventured, \"Are there many like him down here?\"\n\n\"Some,\" Einar said. He got a loopy grin on his face. \"But mostly lions and tigers and the California brown bear!\"\n\n\"Oh my,\" I responded faintly. What sort of crazy place was this?\n\n\"Come on.\" He turned in his saddle to look at me, all alight with an idea. \"I know that was a pretty grim scene up there. I'll show you something nice. You want to see? Come on.\" And he spurred his horse down the trail in front of me, and I followed while he chanted about the lions and tigers and bears all the way. Nobody else shot at us.\n\nBelow the foothills we came upon a sandy wagon track that ran east and west, in a fairly straight line through clumps of wild buckwheat and chaparral. We took it east, as I stared around in cautious expectation.\n\n\"Road to nowhere in particular,\" Einar said, \"at the present time. But in a couple more decades, it'll be Prospect Avenue, when the genteel folks from back east build a little community here. Shortly thereafter they'll change the name to Hollywood Boulevard. Right here, to the left and right, the Walk of Stars will run. The neatest part, though, the _really_ neat part, most people will never know about.\"\n\n\"And that would be?\"\n\n\"This way.\" He urged his mount forward, counting off nonexistent cross streets on his interior map: \"Highland, McCadden, Las Palmas, Cherokee . . .\" Abruptly he turned his horse's head, and we left the trail a few yards north into the trackless thicket. \"Here.\" He looped up the reins, slid from the saddle, and stood beaming at me, as though the Holy Grail was pulsing over the nearest cactus clump.\n\n\"Okay, se\u00f1or.\" I looked from side to side. \"What am I supposed to be seeing?\"\n\n\"You're seeing the nice little streets laid out just like back east, with shady trees and picket fences and charming rose-covered cottages. Okay? All the white latticework and clapboard and gingerbread that relocated Yankees gotta have. It'll all be here. And right here, on this very spot, will stand the very nicest house with the very nicest garden, and you know whose house it'll be?\" He held out his hands, as if framing a picture for me. \"L. Frank Baum's. Ozcot, he'll call the place. This is where he'll settle down, this is where he'll write most of his books about the Tin Man and the Scarecrow and the rest of 'em. How many generations of children will read every word he wrote? How many kids will dream about escaping to Oz, and keep on dreaming about it when they're sick and old?\"\n\n\"You're kidding.\" I dismounted and stared around, trying to see the fairy-tale house and its flowers in the midst of this wild place.\n\n\"I'm not. He'll even build a movie studio out there and produce his own Oz movies, years before MGM even exists. But then he dies, eventually, and guess what his wife does? _She burns his original manuscripts_. She doesn't think they're worth anything, so she piles them into the backyard incinerator, and they're reduced to ashes. All that magic, all those winkies and witches sift down in a fine silver dust through the grate and lie there forgotten, under an incinerator in a neglected garden behind a house that eventually gets sold and bulldozed.\" He made a leveling gesture with his hands.\n\n\"Neighborhood changes. Little houses get torn down, one by one. Gardens are paved over. A cheesy apartment building is built on this site. Right over there, the limousines zoom by, stars go to dinner at Musso & Frank's Grill, tourists wander the Walk of Stars and see the names of Judy Garland and Ray Bolger and everybody, and all the time this powder of dreams is buried and forgotten.\"\n\nI stared at him, almost hearing the blaring horns of the traffic, almost breathing in the smell of expensive cigars and auto exhaust.\n\nSmiling, Einar raised an index finger. \" _Until_ ,\" he said, \"a young artist named Lincoln Copeland\u2014\"\n\n\"Oh, come on, not _the_ Lincoln Copeland.\"\n\n\"Yes, _the_ Lincoln Copeland comes out to Hollywood in 2076 to sketch the ruins. His timing's real bad. The Billy Tahiti riot breaks out while he's there. Bombs are going off all around him. He finds a bomb crater and dives for cover under a tipped-up piece of concrete that used to be a garage floor. He finds he's sitting in the middle of all this amazing gray dust.\n\n\"Now, by an incredible coincidence, L. Frank Baum is his favorite author, and luckily there's a street sign still standing. Copeland _knows_ where he is, he knows this bomb crater was a magician's garden once, and he knows the story about the burnt manuscripts. What does he do? _He fills his pockets with dust_. With bullets whizzing and souvenir stands burning all around him, the guy crams all the gray dust he can carry into his pockets. And as soon as it's dark, he makes his way to Sunset Boulevard and follows it all the way to the beach, where he manages to thumb a ride out of the riot zone.\"\n\n\"I don't believe this.\"\n\n\"I swear to God! And as soon as he's out of harm's way, he finds a glass jar and shakes out his pockets, takes off every stitch he's got on and beats out the gray dust, and fills the jar. He takes it home with him. It's after that that his career takes off, that he suddenly begins painting those fantastic landscapes and allegorical murals that make him so amazingly rich. He doesn't know why he sees the things he sees when he picks up a brush, but he suspects it's because he dabs a tiny pinch of that gray dust on his palette every time he starts a new piece. He says so in his autobiography, written in 2140.\"\n\nEinar bent and scooped up a double handful of sand, and let it sift through his fingers. \"Right here. It's all right here, waiting to happen, man. Immanent. The air is on fire with it. Jesus, I love this town.\"\n\nI started and stared, because for just a second I had seen it all: the pretty houses, the ruined city in flames, the Yellow Brick Road curving away up the wall of a soundstage.\n\n\"You are nuts,\" I said. \"But I'll bet the Company brings you back here.\"\n\n\"Gotta hope.\" He grinned. Suddenly his gaze focused on a point in the distance behind me. He reached up for one of his shotguns. I dove for the dirt. \"No, it's okay!\" he said. \"This is the trank gun.\" He aimed and fired. There was a dull bang and a plaintive little yip, and destiny had found another coyote.\n\nWe returned to the inn as darkness was falling. I had a couple of specimens of rare members of the artemisia family in my collecting kit, and Einar had a neatly trussed coyote sleeping peacefully in a wicker creel behind him. There was a loud argument going on around the cooking fire. The principal raised voice was female.\n\n\"That man had actually participated in the Bear Flag Rebellion!\" Imarte was wailing. \"Do you realize what a unique opportunity has been lost? Have you any idea of the insights he could have given us into the mind-set of the Anglo-American rebels?\"\n\n\"I said I was sorry.\" Juan Bautista sounded as though he would have liked to crawl into a hole in the sand. \"But Erich will die if he doesn't get the right food. It's not like I was chewing it up and vomiting it for him, anyway.\"\n\n\"Oh, my goddess.\" Imarte flung up her arms in disgust.\n\n\"The thing stinks, Juan. You're going to have to feed your bird someplace else, okay?\" Porfirio said. As we rode into the clearing under the trees, it became obvious what he meant: someone, presumably Juan Bautista, had dragged a carcass into the clearing. It had been either a large dog or a small deer. I wasn't a zoologist, so I didn't know which. It had been worked over by coyotes already, so I doubted whether anyone else could have told either. Erich von Stroheim (that was what the baby condor had been christened) was sitting on it, looking bewildered. When Imarte raised her voice again, the bird ducked his head and shook his wings desperately, squeaking.\n\n\"I don't care what the little horror needs, he doesn't have to have it here when I'm bringing home a client,\" she said.\n\n\"Oh dear.\" Einar swung out of the saddle. \"You lose another John?\"\n\nIt seemed that the stagecoach had made a stop, and while the horses were being changed and the drivers were refreshing themselves, Imarte had sallied down and offered refreshment to the passengers. One gentleman had felt confident enough in his appetite to be able to do justice to her offer in the comparatively brief time allotted, and so she'd led him up to the adobe. Unfortunately the first sight that met his eye was Erich von Stroheim pecking at his supper, watched fondly by Juan Bautista. Not only had the gentleman been unable to avail himself of the refreshment offered, he'd lost the lunch he'd partaken of earlier in the day, and departed hastily.\n\n\"This _cannot_ happen again,\" raged Imarte. \"That creature _cannot_ be allowed to interfere with my work, do you understand? It's not even as though he can be trained to live in the wild. He's nothing but a pet.\"\n\n\"That's enough.\" Porfirio held up his hand. \"Juan, take the carcass away now. Downwind, please. We can work out a supplement with chopped beef and an enzyme formula, okay? He'll be fine.\"\n\n\"Okay.\" Dejectedly Juan picked up the little condor and buttoned him inside his shirt. The bird made happy sounds. Juan took the dead thing by one leg and dragged it away into the darkness. Imarte went flouncing off to her room.\n\n\"Ay-ay-ay.\" Porfirio put his face in his hands. \"And was your day good? Tell me your day was good.\"\n\n\"It was good,\" I said. Einar took down the creel\u2014the coyote twitched and growled in its sleep\u2014and unsaddled our horses. He led them off to the stable, whistling a little tune.\n\n\"Hell-oooo, everybody, I'm home,\" said Oscar as he strode into the circle of firelight, leading his mule. Behind them the patent peddler's cart lurched from side to side, catching its roof on the lower branches of the oak trees.\n\n\"And you had a good day too,\" Porfirio said.\n\n\"Oh, first-rate. Finally persuaded Mr. Cielo over at the walnut orchard that he absolutely required the civilizing influence of music in his home. He took a flageolet and six pieces of sheet music. Any day now his neighbors (when he gets them) can expect to hear the strains of popular selections from _The Bohemian Girl_ wafting through the walnut trees.\"\n\n\"Nice going.\" Porfirio poked up the fire. \"Get any good material on him?\"\n\n\"Oh, certainly.\" Oscar set the hand brake on his cart and let the mule out from between the traces. \"Got a fair holo of his kitchen and a splendid one of the parlor, all furnishings in situ. Extensive vocal recordings, too. Got him to tell me half the story of his life. The archivists will be pleased with yours truly, I shouldn't wonder.\" He patted his mule fondly.\n\n\"So that's what you do?\" I asked. \"You go around pretending to peddle stuff, and while people are talking to you, you record details of historical interest about them?\"\n\n\"Yes indeed! Though I hasten to add that no pretending is involved. I am a true and bona fide salesman of the first water. It's more than a matter of personal pride with me, you see, that I can play the golden-tongued orator with the best of them when it comes to persuading a reluctant dweller in adobe that he or she wants\u2014nay, _must_ have\u2014a patent cherry-pitting device superior to all previous models.\" Oscar was completely serious.\n\n\"Yeah, you are one nickel-plated Demosthenes, all right,\" Einar said, emerging from the stable to take charge of the mule. \"Hey, Amelia, sweetie! How we doing, babe? How's our little hooves today?\"\n\n\"No trace of lameness, I'm pleased to report,\" Oscar said. \"She appears to have regained her customary surefootedness.\"\n\n\"Great.\" Einar led her away, and Oscar strutted up to the fire, hands in pockets.\n\n\"Yes, a most successful day. Might I inquire what's for supper this evening?\"\n\n\"Grilled beef, tortillas, and frijoles,\" Porfirio said. \"I just haven't had time to put it on yet.\"\n\n\"H'm.\" Oscar stood there in the light of the fire, rocking back and forth, a small frown on his bland face. \"No chance of any cabbage, I suppose.\"\n\n\"What do you want, man? It's February.\"\n\n\"Oh, quite, quite, I see your point. You know what I'd like to do, though, when we can get a little more garden produce? I'd like to serve you folks a real authentic New England boiled supper. Yes, _sir_. You'd enjoy it no end. I daresay I could make the brown bread to go with it, too. I've got cans of molasses and a cake of raisins in my cart. Just the thing for a nippy night.\"\n\n\"Sure,\" Porfirio said without enthusiasm. He gave a narrow-eyed smile. \"I meant to ask you: have you managed to sell that Criterion Patented Brassbound Pie Safe yet?\"\n\nOscar's face lost some of its aplomb. \"Well, no, not yet.\"\n\n\"Aw, that's a shame.\" Porfirio's grin of sympathy was very white under his mustache. \"I can't think why nobody's interested in that thing.\"\n\n\"Neither can I,\" Oscar said. \"You'd think, in this wild country overrun with mice and insects, that the natives would fight for a chance to possess such a marvel of guaranteed safe storage for all manner of comestibles, whether fresh-baked or fried, complete with buttermilk well and yeast compartment!\"\n\nI leaned forward, genuinely intrigued. \"What is this thing, se\u00f1or?\"\n\n\"Ah! Let me show you,\" said Oscar, running to his cart. Porfirio rolled his eyes at me, but I got up and went to look anyway. Oscar unfastened a couple of latches and opened out one whole side of his cart. Glass jars glinted, and various hanging utensils and tools swung and shone in the firelight; but Oscar gestured past them to a big cabinet kind of thing that took up the entire back wall.\n\n\"There you have it. Positively the last word in preservation of fine baked goods. All drawers lined with plated tin to prevent the unwelcome attentions of minor pests such as mice, rats, or voles. And! Regard the patented securing latches designed to foil the marauding efforts of coons, polecats, or possums! Why, given the superb solid-oak construction and high-quality brass reinforcement, I daresay the Criterion Patented Brassbound Pie Safe could withstand even the predations of our friend the bruin.\"\n\nHe didn't know bears very well. Still, I had to admit the thing was impressive. It gleamed with fanciful brass trim all etched and inscribed with curlicue patterns of dizzying complexity. The various locks and latches looked formidable, and in addition to the drawers and cabinets were features at whose purpose I could only begin to guess: weird upswept or recessed sections.\n\n\"Gee, Oscar, that's really something,\" I said.\n\n\"Isn't it? And yet\u2014can I interest even one member of the native populace of Los Angeles in this modern marvel? You'd think any one of them would jump at the chance to call it his or her own. Yet here it remains, unpurchased, unowned.\" Oscar shook his head in bewilderment.\n\n\"Well . . .\" I hunted for the words. \"You know, Oscar, I've been in California for a hundred and sixty-two years now, and in all that time I don't think I've ever seen a pie. Maybe that's part of the problem? I mean, nobody even grew much wheat here until recently. And this safe was designed for real Yankee-style pies, right? Two crusts, blueberry or rhubarb filling, that kind of thing?\"\n\n\"True.\" Oscar looked wistful. \"I could go for some rhubarb pie myself this very moment.\"\n\n\"San Francisco,\" Porfirio remarked from where he was stirring the frijoles. \"That's where he could sell it. Not in El Pueblo de la Reina de Los Angeles.\"\n\n\"I beg to differ,\" Oscar said hotly. \"I have sold these people maple syrup, quilting frames, and birch beer extract. I _will_ sell this fine item. I simply haven't found the right customer yet.\"\n\n\"There must be plenty of gringos in Los Angeles,\" I said.\n\nPorfirio grinned. \"They don't make many pies. Too busy shooting one another.\"\n\n\"I'll sell it, I say, and not to a fellow Yankee,\" vowed Oscar. \"Do you hear me, sir?\"\n\n\"What, is this a bet?\" said Porfirio, sitting back on his heels. \"You want to wager on this?\"\n\n\"By the goddess of consumer goods, yes! Name the stakes.\"\n\n\"Okay.\" Porfirio looked thoughtful. \"Let's say . . .I get one of those snappy patent pearl-handled shaving razors you carry, if you don't sell that pie safe before you're transferred out of here. If you _do_ sell it to a nongringo, I'll personally prepare that New England boiled supper for you. I'll even eat it with you.\"\n\n\"Then dig yourself a root cellar and lay in the rutabagas and parsnips,\" said Oscar, eyes flashing. \"For I'm at my best when given a challenge, sir, I warn you.\"\n\nPorfirio turned his attention back to the grilling beef. \"Go for it, man\" was his reply.\n\nPersonally I thought Porfirio would lose the bet. Los Angeles was becoming more of a Yankee city with every passing year. I learned from our copies of the bilingual _Los Angeles Star_ that bullfights had at last been outlawed, to be replaced with the more humane pastimes of baseball, Presbyterian prayer services, and debating the outcome of the Civil War.\n\nThat Civil War raged on, over on the other end of the continent, at Mill Springs, Pea Ridge, and similar quaint-sounding places. Los Angeles was a world away from that, mired in its own problems. (Literally mired: the new brick sewers were proving a slightly more complicated engineering feat than had been expected.) To my amazement, though, the local Yankees\u2014I must get out of the habit of calling them that, now that about half of them take it as a deadly insult\u2014the _Americans_ among us actually staked out sides and fought the war here in their own way, right in front of their bemused Hispanic neighbors. The older Yankee element, the sober sea captains and shopkeepers, were staunchly pro-Union. Banning, the stagecoach fellow, actually took time out from building his fine new house to donate land at San Pedro for a Union army barracks, so his side would have a military presence in California. The trash, the white boys from the States who'd failed at gold prospecting and trapping, were ramping stamping secessionists, so I guess Banning was wise.\n\nMaybe I'll go on calling them Yankees anyway. The Union will win the war, after all. And it's less offensive than calling them Americans, to the people of South America, who have a claim to that word too; and less offensive to the Yankees than calling them Anglos, when so many of them were shipped into this country as Irish bond slaves. Come to think of it, I guess most Latinos don't like being called Hispanic, after the way the conquistadors treated their grandmothers. You can't win, can you?\n\nTo me, the whole issue seemed irrelevant, living back in that canyon as I was with the stagecoaches arriving and departing as time and mud permitted. I was more amused by the fact that Mexico was now in danger of becoming French. It should have warned me that I was out of touch, that I'd been in the hills too long. But for three hundred years now, the only political reality had been the long slow ruin of Old Spain's fortunes in the New World. These Kentuckians, these Narragansettians, these absurd Cape Codders, I knew they too were destined for their part on the world stage. But I was perhaps too slow in realizing that the curtain had already risen on their act. How could it affect me, after all? Nothing in the pageant of mortal fools had been able to affect me since the English Reformation, and I'd sworn never to let anything else get to me again.\n\n###\n\n\"OUT OF MASA,\" announced Einar, rummaging through the storeroom. \"Out of brown sugar. Out of coffee. Half a bag of pinto beans. You want me to go to the store, chief?\"\n\n\"Good idea. You can take those damn coyotes to the transport depot too, how about it?\" Porfirio said.\n\n\"Okay, okay. I didn't think they'd make that much noise. The last batch didn't.\"\n\n\"You're going into town?\" Imarte stuck her head out of her room. \"Will you wait until I'm dressed? I'd like to go down to the Bella Union.\"\n\n\"I wasn't planning on staying overnight,\" Einar said.\n\n\"Don't worry. I'll catch the next stagecoach home.\" She ducked back into her room, and there was a great rustling of silks and creaking of whalebone. I was wondering why she didn't walk back\u2014I would have, because the pueblo didn't look that far away from up on the ridge\u2014when Einar turned to me and asked:\n\n\"How about it? You want to come?\"\n\nI blinked at him in surprise. \"Okay,\" I said, deciding to continue my program of readaptation to human company, though it did seem a bit reckless. If there were bloodthirsty crazies hiding out in the chaparral, how many more of them would be in the infamous saloons and gambling dens? On the other hand, I'd have to visit the place sometime, and Einar seemed able enough with a gun.\n\nI went to put on my best shawl and pin up my hair. By the time I came out, Einar had hitched a pair of horses to our wagon. Imarte came sashaying from the adobe, complete with painted face, scarlet satins, and feather boa.\n\n\"Wow, you really do look like a whore,\" I complimented her, with my most naive expression.\n\n\"Thank you. You won't mind riding in the back with the coyotes, will you, dear?\" She vaulted into the seat beside Einar. \"This satin crushes so terribly, the least little thing makes wrinkles that simply won't come out. I envy you that plain broadcloth. And how lucky you are to be able to wear that color. Dirt and stains are almost invisible on that particular shade of\u2014what would you call it? Olive drab?\"\n\n\"Matches your eyes, doesn't it?\" I said, clambering up into the back, where the month's catch of _Canis latrans_ slept soundly in their crates.\n\n\"Break it up, ladies,\" Porfirio snapped at us. I shelved my next remark, which had to do with the bitches I was riding with. You know, in the twenty-second century the feminist Ephesian Party will bid for political power on the grounds that if women ran the world, there would be less senseless aggression. Strangely, they'll never be able to get a consensus within their own party. Can you imagine why not?\n\n\"Let's have a nice happy little drive into town, shall we?\" Einar said. He clucked to the horses, and we bumped and rolled away down the canyon, to turn right on the dirt road that was El Camino Real and would one day be the Hollywood Freeway.\n\n\"Check it out!\" Einar pointed with his whip as we rumbled along. \"Hollywood Bowl, back up in there. _Symphonies under the Stars_. That hill over there? Whitley Heights, where all the movie stars live before Beverly Hills is fashionable. Rudolph Valentino will have a house right _there_.\"\n\n\"I've never seen one of his films,\" I said. \"I really ought to, sometime.\"\n\nEinar half-turned in his seat as an idea hit him. \"We should have a film festival! We can show them after dark. All the great Golden Age of Cinema stuff. I wonder if I can get films that were shot right here in town.\"\n\n\"Probably.\" Imarte sniffed. \"You could try to find the interesting ones. There are only a few with any historical value, in my opinion.\"\n\n\"We'll do it,\" said Einar, bouncing on his seat. \"We'll have the first film festival in Hollywood, how about it? I'll see what I can order from Central HQ.\"\n\nWe emerged from Cahuenga Pass and swung left down the track of the future Hollywood Boulevard, where Einar gave us a running commentary on the famous sights we couldn't see yet. I remember the corner of Hollywood and Vine, not for any precognitive vision of Clara Bow zooming around it in a fast car but because we had to bump through a particularly vicious seasonal creek that cut across it, and mud splattered Imarte's scarlet finery. I was proud of myself for not smiling.\n\nAs we entered the plain below the foothills, the land opened out more and changed: low green hills gently rolling as far as the eye could see, dotted with oak trees and starred everywhere with golden poppies. Here and there wandered herds of longhorn cattle, grazing and growing fat.\n\n\"See those guys?\" Einar's voice was sober. \"This is their last hurrah. Nobody knows it yet, of course, but this drought will pretty much wipe them out. And when they go, the old Mexican gentry go too; they'll lose their revenue, get into debt to the Yankees, and sell off their estates. Boom, whole way of life gone. The Yankees will run cattle after the drought, but not these longhorns: they'll bring in their own stock, Jerseys and Holsteins\u2014Eurocows. All kinds of useful genetic traits scheduled to disappear from the bovine gene pool when these longhorns nearly go extinct. It'll be a bitch collecting specimens\u2014expensive, too\u2014but I really should get started on it.\"\n\n\"Why expensive?\" Imarte asked. \"I should imagine the rancheros will be desperate to sell, once the drought is in progress.\"\n\n\"Yeah, but by then the specimens will be weak and stressed out. The Company wants healthy, happy cows.\" Einar shook his head.\n\nI sat up and stared at the gentle landscape, so pastoral, such an idyll. Cracked earth and skeletons soon, to be followed by another Eden, this time of prosperous little Yankee orange groves, to be followed by a gray wasteland of diesel exhaust, concrete, and steel. Paradise and hell, boom and bust, together forever in Los Angeles. I shivered and wished I was back in Big Sur.\n\nWe followed the Hollywood Freeway route all the way into the city, creaked gradually uphill, and paused at the top of a long low ridge. Einar pointed at the vista below us, a wide gesture taking in the whole horizon. \"There you go, Mendoza. The original wretched hive of villainy and scum.\"\n\nIt didn't look particularly dangerous. What surprised me was the space it took up. It sprawled and sprawled, out to the edges of the sky, and yet you could count the number of two-story buildings on the fingers of one hand. Right below us was a squat brick thing like an armory in a weedy central plaza crisscrossed with dirt paths. I found out later that it was a cistern for dry years. There was a solid-looking church with its back to us\u2014recently repaired, to judge from the two-tone plaster. There were a couple of stately adobes with pink tile roofs and peeling whitewash. But the vast majority of structures were little flat-topped shacks with tarred roofs, rows of them leaning on one another and single ones peeping out from orchards or ranged across fields. I couldn't see a single living soul moving in that vast panorama. You could have fit every other city in California into the space Los Angeles took up, and yet it looked like nothing so much as somebody's big cow pasture, with an unusual number of cowsheds. A dark line of willows and cottonwoods snaked through it, and one particularly big sycamore: the trees must have marked the bed of the Los Angeles River, which actually had water in it this year.\n\n_Crack!_ A bullet tore up weeds a few yards down the hill from where we were parked. Oh, here was somebody. We looked down into the belligerent stare of a man who had come out to the edge of his little orchard. He was lowering a long rifle.\n\n\"Thass a _wahnin_ '!\" he called hoarsely. \"Y'all stay _outa_ mah orchard. Come down some _otha_ place!\"\n\nEinar grinned and signed tipping his hat. \"You bet, Davy Crockett!\" He shook the reins and turned us around, and we followed the trail along the ridge and down into town. Well, it was clear where all the people must be: hiding. My knuckles were white as I clenched the sides of the wagon.\n\n\"Does everybody shoot at perfect strangers here?\" I asked between clenched teeth.\n\n\"If they think you're going to trespass on their property, yeah,\" said Einar. He began to sing a ballad about Davy Crockett, waving one arm for dramatic emphasis.\n\n\"Oh, shut up,\" said Imarte, and for once I agreed with her.\n\nThere actually was a business district, with some Yankee-style stores of brick and of timber, and a couple of adobes Yankeefied with false fronts. Here was where all the people had got to. There were wagons like ours rumbling to and fro, driven by Mexicans or Yankees. There were white boys leaning in the doorways of saloons, looking out at the world through painfully narrowed Clint Eastwood eyes. There were Mexican dons on elegantly caparisoned horses pacing along, and some of these were distinctly African in appearance: dignified old gentlemen whose great-grandfathers had bought their way out of slavery by joining the Spanish army and exploring the New World. And here was a genuine Indian begging in the gutter, with eerily empty eyes. Imarte leaned over and said a few words to him as we passed. His gaze snapped into focus for a moment and locked on her. He shouted something in a desperate voice and stumbled after the wagon, dragging one foot. Imarte made a face and tossed him a coin. He fell flat, covering it with his body. I looked away. Imarte shook her head.\n\n\"That man's race once conquered all their neighbors, for hundreds of miles to the north and east. They had a sophisticated monotheistic religion the equal of Christianity or Islam. Look at him now.\" _Sic transit Chinigchinix_ , I thought.\n\nIn the next block, we passed another prostrate beggar, drunk and wailing out \"Flow Gently, Sweet Afton\" in ghastly Southend Cockney. Two gutters over, an obvious native of Georgia was murdering \"Sweet Betsy from Pike.\" And a Mexican crawling along on hands and knees implored his little white dove to return to his embrace. The mud is a democratic place, at least.\n\nAnd here were _se\u00f1oritas de mala vida_ , dressed like our Imarte, liberally represented by the assorted races, swinging along with the set smiles, upper-arm bruises, and blank eyes of the true professional. And a Chinese fellow going somewhere in quiet, self-effacing haste. And an august old shopkeeper standing just inside the door of his emporium and jingling his keys as he watched the passing scene: a Jew, to judge from the name painted on his sign. He looked exactly like Uncle Sam. Put him in a striped top hat and long tailcoat, and he could have posed for a twentieth-century war bond poster.\n\nOn Calle Principal we pulled up in front of a little place whose sign read BELLA UNION. It was dark and dirty. Imarte jumped down from her seat with the grace of a cat and made straight for the door, a gleam in her eye.\n\n\"Wait a minute,\" Einar said, sliding down. \"Marcus has something in his hoof. Whats'a matter, boy?\"\n\nI climbed down while he coaxed the horse to put its foot up for him. Was the hotel as bad inside as it looked from the street? I ventured close enough to peer inside. God, it was worse. That couldn't be a dirt floor, could it?\n\nThere at the long bar was Imarte, advancing on the British tar who had just been served a local beer and was now staring at his glass with horrified wonder.\n\n\"Hello there, sailor. In town long?\" she said, flexing a tit at him. \"Got time to tell me your life story?\"\n\nHe turned to meet her eyes. \"It's like piss, for Chrissake!\" he complained.\n\nSilly me, to stand in the doorway of a lesser class of hotel looking amused. A regular customer mistook my smile and was suddenly in front of me, breathing rye whiskey through the fringe of his mustache.\n\n\"Well, now, se\u00f1orita, you looking for someone to dance with? You want me to show you how we dance the fandango out Durango way, huh? Nice earrings. They real gold, Chiquita?\" He reached for my face.\n\nI took two hasty steps backward and summoned up my best Katherine Hepburn imitation. \"Sir, if you _ever_ presume to lay hands upon me, I assure you legal action will follow! Do I make myself clear, you palsied, imbecilic, and alcoholic cretin?\"\n\nHe staggered back, very surprised. \"Lady, I'm sorry,\" he gasped. \"I thought you was Spanish.\"\n\nMental note: Leave the gold hoops at home next time you visit sunny downtown Los Angeles. And ditch the rebozo, too. I turned on my heel and stalked out. Einar was just releasing Marcus's hoof. He stared at me openmouthed. \"You went in there?\"\n\n\"Stupid, wasn't it?\" I agreed, climbing up on the seat. \"Let's get the hell out of here, shall we?\" I didn't like mortals, I _really_ didn't like mortals. In fact, I hated the sight and the smell of them.\n\n\"Come on, I'll buy you a drink,\" he said.\n\n\"Not in there, you won't.\"\n\nHe got his crazy smile again. \"How about a cocktail in the Lost City of the Lizard People?\"\n\nIt turned out that if you went to a drab-looking little adobe on Calle Primavera and knocked, a mortal man would let you in and obligingly help you unload your crates of tranquilized coyotes. He would then slide back a section of the floor, revealing a service elevator, on which the coyotes descended toward a new life following air transport to a Company zoo. The man would then bow you to an ornate wardrobe, which, when opened, proved to be a passenger elevator. Once you entered it, it dropped with unnerving speed thirty stories to a short length of tunnel tiled in gold enamel. At the end of the tunnel was a first-rate Company cocktail lounge, beyond which was a Company transport terminal, also tiled in gold enamel.\n\n\"There's miles of tunnel, running all the way to the undersea base in the Catalina Channel,\" Einar told me over a couple of margaritas. \"And one long tunnel runs out to the Mojave base under the sand.\"\n\n\"Nice.\" I bit into my wedge of lime. \"But what's with the lizard motif?\" I waved a hand at the decor. There were lizard patterns on everything, woven into the carpet, tooled into the booth leather, printed on the cocktail napkins.\n\n\"Joke,\" he said. \"In 1934, this guy will claim that an old Indian told him about a highly advanced race of lizard people who retreated underground following a global catastrophe in 3000 B.C.E. They built a city in the shape of a giant lizard and a maze of gold-filled tunnels right here under Los Angeles, supposedly, using magic chemicals that melted through bedrock. Rooms crammed with gold and sacred tablets, all kinds of weird shit. So anyway, this guy claims he's found out where the gold chambers are, using an invention he calls the Radio X-Ray, and he actually gets permission from the city authorities to drill. Tells them he's located a treasure room a thousand feet down. He only goes about three hundred fifty feet before the shaft starts to collapse. Tells the city he's putting the dig on hold until he can solve the technical problems, which he expects to do in no time. Then he vanishes. Drops out of sight. Never heard from again. Obviously the lizard people got him.\"\n\n\"Obviously.\"\n\n\"It gets weirder. This'll be in 1934, right? But by 1931 the Company will have abandoned this base and filled in the tunnels. Earthquake in Long Beach the next year, remember? By the time the guy claims to detect the tunnels, _they won't be there anymore_. Last call at the old Lizard Bar was months beforehand. Nuts, huh?\"\n\n\"Nuts. Unless maybe the man is an undiagnosed remote viewer.\"\n\n\"Could be.\" Einar tilted his glass. \"Want another cocktail?\"\n\n###\n\nAFTER THAT VISIT I was content to stay close to Hollywood, venturing out to collect specimens when Einar's schedule allowed him to go with me. No more argument on my part that I didn't need an escort. Hot lead seemed to be the language of social encounter down there, and I felt squeamish about becoming fluent in it, though I dutifully practiced hyperfunction with a Navy pistol.\n\nWe were by no means so isolated as I had thought. We got mail; we got magazines. We subscribed to the _Los Angeles Estrella\/Star_ , to a couple of back-east papers so we could follow the Civil War news, and Porfirio had a subscription to _Punch_ , of all things. I read it for the humor, though the British slant on the war was strange. They played both sides of the diplomatic fence with a prissy hypocrisy that I took to be Victorian. I wasn't impressed. I had known the brilliant savages of the Tudor period firsthand, and, though I'd never thought I'd say it, I preferred them to their smug descendants.\n\nYes, we really had our window onto the world, despite the lack of radio for the local Company news\u2014no reception back in our canyon, because those granite hills kept the feeble broadcast out. And we made our own nightlife; we even had movies. Not holos, you understand, movies.\n\nI woke from uneasy dreams one bright morning to find a card stuck in my boots. Yawning, I examined it, sitting on the edge of my cot. It was cream-colored pasteboard, inscribed by hand in purple ink, with nice calligraphy, and it told me that I was invited to the Cahuenga Pass Film Festival, which was to take place tonight at 2000 hours sharp. This evening's featured film: Hollywood's first premiere of the director's cut of _Greed_ , based on Frank Norris's classic novel _McTeague_ , a tale of mortal doom. Approximate screening time nine hours, so refreshments would be provided. Formal dress optional. (Good; the closest thing I had to evening wear was a black rebozo.) Location: suite B of the Cahuenga Pass Hilton. (Yuck yuck.) My host: Einar.\n\nI pulled on my boots and wandered out to the fire, where Porfirio was frying breakfast. \"Did you get one of these?\" I held the card out to him. \"What's it all about?\"\n\n\"Didn't you hear him chortling when the afternoon stage left yesterday?\" Porfirio said, handing me a mug of coffee, which I accepted gratefully. \"He got a big box he'd ordered from Central HQ. He was fussing around in his room all last night. He's in there now, as a matter of fact. This should be some party.\"\n\n\"I guess.\" I looked at the card doubtfully. \"Will we all fit in his room? I'm not much for parties, really.\"\n\n\"It's not a party party, it's a film screening. He's working like a dog to create a sense of occasion. I'm going, and you should too.\" Porfirio looked at me sternly. \"What else have you got to do tonight? Sit in your room and look at plant DNA? This will be good for you.\"\n\nActually I enjoyed sitting in my room in front of a cozily glowing credenza, but I didn't want to disappoint Einar. Accordingly, at 2000 hours that evening I wrapped my black shawl around me and ventured into the adobe. I could see lamplight coming from Einar's room; and was that music? It sounded like a selection of famous film themes by Hollywood composers, tinnily played on a battery-powered portable, and that in fact was what it was. But I barely noticed the music once I crossed the threshold of Einar's room.\n\nHe _had_ worked to create a sense of occasion. It was a small square room with bare adobe walls and rough furniture of peeled logs and cowhide; but he had borrowed Imarte's red velveteen bedspread and tacked it up in hanging swags against one wall, and a dusty oriental carpet had been rolled out on the floor, and a fairly clean sheet had been tacked up on another wall. That had to have come from Imarte's bed, too. In fact, there was a lot of her finery draped around to give the room a film palace look. She was being an awfully good sport about this, wasn't she? And there she stood in a ballgown of Arrest Me Red, holding forth sententiously to Oscar:\n\n\". . . outrageous what they did to von Stroheim, but it's a classic case of the fate of great literature in Hollywood. Of course it was bound to happen, given the incredible social significance of Norris's work. Audiences simply weren't ready for the grim realism, the pitiless examination of hopelessness among the uneducated working classes, the dwindling of the American Dream to despair, the ugly realities of passion.\"\n\n\"You don't say?\" Oscar raised a graniteware coffee mug to his lips and took a cautious sip of the contents. He looked startled. \"Good Lord, Einar, is this gin we're drinking?\"\n\n\"Sure is,\" Einar said, welcoming me with a bow and handing me my libation for the evening. He was resplendent in a black tailcoat, stiff collar, and flowing foulard tie. He'd greased and combed his hair back, and so had Porfirio, also dressed to the nines. They looked like a couple of cast members from a melodrama. \"It's a martini, complete with olive. Don't worry, the gin hasn't been anywhere near a bathtub. Mendoza, you look lovely this evening; pray be seated. A space has been reserved for you in the balcony.\" He gestured grandly at his cowhide bed, which had been dressed up with needlepoint cushions. \"You too, Imarte; and as highest-ranking cyborg here, Porfirio, you have the seat of honor between the ladies, okay? The rest of us gentlemen will be seated in the loge. Well, we're only waiting for J. B. to make his fashionably late entrance\u2014\"\n\n\"Here I am. Sorry,\" murmured Juan Bautista from the doorway. Apparently he had no shirts that weren't plain calico, but he'd made a pasteboard shirtfront, inked a bow tie and little buttons on it, and pinned it to his chest. He'd made a little shirtfront for Erich von Stroheim too, and tied it around the bird's repulsive neck with string. Erich huddled in his arms, looking at us doubtfully.\n\n\"Cool! The director himself, here to attend his first Hollywood premiere!\" Einar welcomed them in. \"Have a seat, gentlemen, down in the loge\u2014or maybe that's the mezzanine.\" It amounted to a row of pillows on the floor in front of the bed. Juan Bautista settled down comfortably crosslegged, and Oscar lowered into place beside him, grumbling about having to press his trousers tomorrow. Erich von Stroheim took an experimental peck at his own shirtfront, but Juan Bautista reproved him gently, closing his fingers around the nasty-looking beak. The bird lowered his head to be scratched, making a little pleading noise.\n\n\"Well, the appointed hour is upon us,\" said Einar, stepping to the front of the room, before the white sheet. He took his six-gun from his holster, and I half-expected him to shoot out the lamps, but he reversed it and held the butt to his face, pretending he was speaking into a microphone. \"Ladies and gentlemen . . .\" He altered his voice to sound as though he were speaking into a twentieth-century PA system, complete with the boom and squeal of badly adjusted speakers. \"Ladies and gentlemen and, uh, condor, welcome to the first installment of the first-ever Cahuenga Pass Film Festival. And what a glittering turnout we have here tonight. We're pleased to present, as our first offering, Director Erich von Stroheim's immortal classic _Greed_ , starring Jean Hersholt and Zasu Pitts. Do I see the director in the audience? Stand up and take a bow, Mr. von Stroheim!\" Juan Bautista held up the bird and dipped him forward in a little bow. Erich gronked querulously and pecked to have his head scratched again. We all applauded.\n\n\"Thank you, Mr. von Stroheim. All right. Now, tonight's offering is one of the truly great films cinema has produced. Unlike most of our future offerings, it was _not_ filmed primarily in Hollywood, mostly because the director was a raving fanatic about location shots. Notable examples are the authentic San Francisco sequences shot on Polk Street and the climactic scene in Death Valley, which nearly killed the cast and crew as temperatures soared to 120 degrees Fahrenheit and the camera equipment had to be iced down.\n\n\"The location shots were necessary, because von Stroheim insisted on a literal interpretation of the book, which meant he filmed _every single scene_. Unfortunately, the age of the miniseries had not yet arrived, and the resulting nine-hour spectacle was edited down by Universal executives to a much smaller masterpiece. Von Stroheim never forgave the studio, and they never forgave him either, which was why the rest of his cinematic career was pretty much limited to his role as Gloria Swanson's butler in _Sunset Boulevard_.\n\n\"Anyhow, Doctor Zeus will have a quick-fingered operative in the cutting room, with the result that we are able to present tonight the full-length director's cut, made from the original silver nitrate print, complete with the Variety Theater scene in its original version and the subplot involving the crazed Mexican cleaning woman and her Jewish pawnbroker boyfriend. I should warn the more sensitive in our audience that some of the material is really, _really_ racially offensive, okay? So I apologize to our Hispanic audience members in advance, and also to anybody who might have been Jewish when they were mortal\u2014?\" He looked inquiringly at Imarte.\n\n\"I was _Chaldean,\"_ she corrected him.\n\nI put up my hand. \"I was arrested by the Inquisition. Does that count?\"\n\n\"I think that makes you an honorary Jew,\" Porfirio said.\n\n\"Okay,\" continued Einar, \"just don't get sore. Now, before we begin, I'd like to serve refreshments.\" He went to the corner table and took down two big bowls of popcorn and handed them out. \"Eat hearty. There will be brief intermissions when I change reels, because, no, ladies and gentlemen, this is not a holo! Tonight's entertainment comes to you in the authentic and time-honored medium of reel cinema.\" He gestured dramatically to the corner, where he'd opened out the primitive-looking projector and connected it with alligator clamps to a solar battery unit. \"So without further ado, distinguished audience . . .\"\n\nEinar blew out the lamps one after another, killed the music, and stepped over Oscar to reach the projector. A click and a buzz, a white light on the screen briefly occluded by his head and shoulders as he groped his way back to his seat; then the flickering images held our attention, and the only sound was the faint whirring of the projector and the crunching of popcorn.\n\nIt was a great film. Horrible unrelenting tragedy, but you couldn't be depressed watching it, because you were constantly exhilarated at what a work of bloody genius it was. Have you ever seen it, se\u00f1ors? From the opening scenes in the mining town, where you meet this appalling, tender-hearted ogre who protects little birds but is willing to kill his fellow men with a backhand blow, to his astonishing transformation into a _dentist_ , for Christ's sake, to the banal and doomed love story with the girl who won the lottery\u2014their degeneration, she into a grasping harpy, he into a bestial drunk\u2014and the murder, the chase across the desert, the final scenes where the poor monster finds himself handcuffed to a dead man on the floor of Death Valley, the last frames where he watches the expiring flutters of the damn canary he's brought along with him, cage and all, on his flight from justice\u2014I tell you, it beats Hamlet for craziness and black humor in tragedy. Not a ray of hope in a frame of it.\n\nSo why did we sit there enthralled for nine hours, never saying a word? The last frames unrolled, the progressively longer shots of the wretched mortal's end winked out, and the screen went white. It was 0500 hours. There was silence but for Juan Bautista's stifled sob.\n\n\"That poor canary,\" he gasped.\n\nImarte began to applaud, and we all joined in, even Juan Bautista. And I think our applause counted for something. We're immortals, after all. We've watched history itself unspool before our eyes. It takes a lot to impress us. So even though the real Erich von Stroheim had yet to be born on the night we watched _Greed_ , I hope his shade heard our ovation for his butchered masterpiece. I hope he was appeased, somewhere, somehow.\n\n###\n\nI WAS SO IMPRESSED with the film, I accessed the text of _McTeague_ and read it through in the following days, as the inn drowsed between stagecoach visits.\n\nOn a good day we got two passing through, pausing long enough to let off or pick up mail or passengers. If one of the horses was in need of attention, Porfirio got out his farrier's tools while the passengers wandered up and down our little canyon or availed themselves of our remarkably clean and tidy outhouse. Imarte would hurry to entertain them; if it was a group of mixed couples, she'd leave off the feather boa and play gracious hostess rather than daughter of joy. She'd do whatever it took to get them talking to her about themselves. There were in-depth interviews with an Italian opera singer headed for San Francisco, a Scot in a genuine kilt (sporran and all), two Basque wool magnates who might have been identical twins though they weren't, and a Mormon patriarch from San Bernardino, who proposed marriage to Imarte on half an hour's acquaintance. She was genuinely regretful at having to turn him down. (\"What an incredible opportunity to study a fascinating mutation of American folk morality!\") I mostly slunk away into the oaks when passengers were around. Mortals got on my nerves, these days.\n\nThey never spent the night, unless they were Company operatives passing through. Porfirio would explain politely that all our rooms were presently occupied, and the _se\u00f1ores y se\u00f1oras_ would most certainly find lodging at the Gamier brothers' inn farther up the highway. If the _se\u00f1ores y se\u00f1oras_ got ugly about it, a bottle of aguardiente was offered for the road; if that failed to mollify them, Einar would swagger into sight with his bandoliers and look menacing.\n\nBut once in a while one of our own would climb out and have his or her trunk handed down, and there'd be anecdote swapping and aguardiente far into the night. Usually the trunks were full of high-tech stuff we'd ordered, processing credenza replacement parts or refills for Einar's tranquilizer gun. When they left on the next stage, they took with them DNA material, coded transcripts, and anything too solid to transmit or too small to bother shipping from the Lost City of the Lizard People.\n\nI was out behind the stable one afternoon helping Einar crate up an antelope (obtained in Antelope Valley, where else?), when Juan Bautista came running to find us, hugging Erich awkwardly. The damn bird was growing.\n\n\"You guys! Come see, the stage just pulled in, and it's a Concord!\"\n\n\"No kidding?\" Einar dropped his pressure sealer, and we both ran to look, eager to admire the lines of the Rolls-Royce of stagecoaches. Butterfield had used _only_ Concords, of course, which was maybe why it didn't want to risk them cross-country with a war on. So how had Banning managed to get his hands on the gorgeously engineered thing we saw sitting at our humble embarkation point? I never found out; and I never had much leisure to wonder about it, either, because while Einar and Juan Bautista were checking it out (\"Body by Fisher, man!\"), I realized with a start that I actually recognized a friend among the passengers.\n\nHave you gentlemen ever noticed how rare that is with us immortals? Of course we run into acquaintances now and again\u2014I had known Imarte before, unfortunately\u2014but why is it that we almost never get stationed anywhere near old _friends_? Does this have something to do with one of the Company's famous secret agendas? Not that I'd ask if I wasn't higher than a kite. It's the Theobromos talking.\n\nThe mortal passengers saw a rather bulky and foreign-looking gentleman help his drab wife out of the passenger compartment, and then raise his hand to assist their colored servant down from her seat by the driver. If they noticed his gallant gesture toward the black lady, they probably raised an eyebrow. But California was a Free State, and people didn't care as much about race relations out here, between blacks and whites anyway.\n\nShe was a beautiful woman. Tiny and elegant, with ebony skin that glowed as though polished and fine West African features. Her hair was braided up, but I knew that if she let it down and shook it out, it would wave around her shoulders like a storm cloud.\n\n_Nancy_? I transmitted in astonishment.\n\nShe lifted her head, saw me, and smiled, and she still had the tiniest gap between her front teeth when she smiled. There had been a boy in our graduating class who wrote an impassioned poem to that little gap.\n\n_Mendoza? Can that be you_?\n\nI nodded dumbly, feeling every one of the years since the last time I saw her, at our commencement party in 1553. I was on a transport to Spain shortly after, and she went to do research work at a base under the Sahara. I heard later on that she'd had a very successful career in Italy and Algiers, but we never kept track of each other; you don't keep track when you're busy in the field. I never have, at least.\n\nShe lowered her eyes and played the docile maid for the mortal passengers, fetching the drab wife's reticule and parasol from where they'd been forgotten, while the big fellow saw to their trunks. Porfirio led up the change of horses, and 1 could see him double-taking on the wife, though she looked like Miss Kansas Corn to me. She turned to him, too, and there was evidently an exchange of some kind; for as soon as the stage rattled away, leaving the three immortals there, she screamed like a steam whistle and fell on him in an embrace.\n\n\"Porfirio! You goddam son of a whore, you look _great!\"_ she said.\n\n\"Eucharia!\" he said, and they staggered around and around in a prolonged hug. The big man, meanwhile, took Nancy's hands in his and was leaning over her, evidently murmuring anxious queries as to her well-being during the ride. She smiled and said something reassuring. He bent and kissed her face.\n\nI gawked. You see, _he was an immortal too_. In all the years I'd worked for Dr. Zeus, I never, ever saw a pair of immortals in love with each other. I thought it just didn't happen. Our teenaged neophytes have crushes on anything that moves, of course, but full-grown immortals put all that behind them. Don't they? Plenty of affectionate friendships, even noisily affectionate ones like what Porfirio and the bleached-out lady from the Midwest apparently shared, but romance? No.\n\nShe was leading him by his big bear paw up the slope to me, her eyes sparkling. \"Dearest,\" she said to him, \"allow me to present my oldest friend. Mendoza and I have known each other since we were neophytes together. How many years has it been?\" She put out her arms, and we hugged. I hadn't hugged anybody since 1700. It felt strange.\n\n\"Three hundred and nine,\" I said. \"But who's this?\"\n\nThe big man bowed. \"Vasilii Vasilievitch Kalugin, mademoiselle, at your service. I am indebted to you for an excellent botanical survey of the Novy Albion region, though you may not recall the occasion\u2014?\"\n\nI accessed hurriedly and suddenly placed the name. \"In 1831. _You_ were that operative up at Fort Ross?\"\n\n\"The very same. My eternal thanks.\" He took my hand and kissed it. The clothing was aristocratic Russian; but the accent was exquisitely Continental, as was hers now. She wore her servant's calico with her customary grace and style, and believe me, they didn't in the least look mismatched as a couple. Some of Kalugin's bulk was his Russian coat, but he was genuinely a big guy, with sort of harsh sneering features in a round pink face framed by amazing muttonchop whiskers. His eyes were timid and kindly, though, and he couldn't keep them turned from her for long.\n\n\"I'll just go bring up the trunks, shall I, my love?\" He squeezed her hand. \"Your pardon, mademoiselle. I return directly. I daresay you ladies have much to discuss, no?\" He turned and bustled after the trunks like an anxious husband. Gosh, he was cute.\n\n\"Well!\" I burst out laughing, and she just stood there looking happy. \"When did _he_ happen to you?\"\n\n\"We've known each other since 1699,\" she said. \"It's a long story.\"\n\nSixteen ninety-nine? That was just before I'd been posted to California. \"I'll bet. And you're really\u2014? He's really\u2014? It's love?\"\n\n\"Yes,\" she said, turning to watch him. \"Oh, Mendy, it is.\"\n\n_Mendy_. God, the years were rolling back. \"So, like, are you married?\"\n\n\"In a manner of speaking. Not as mortals marry, of course. We've exchanged certain vows of our own. Our work has parted us frequently, years at a time, on occasion. Fortunately the Company is understanding and arranges our work near each other whenever possible.\"\n\n\"What's he do?\"\n\n\"He's a marine salvage technician,\" she said, and I nodded, because she was an art conservation specialist. I couldn't think their jobs would overlap much.\n\n\"So he's away at sea a lot? But what are you two doing here in California?\"\n\n\"All those San Francisco millionaires are returning from Europe with art treasures for their mansions,\" she said. \"Half of them will be beggars within the next five years, and their collections will be blown to the four winds. I'm doing a preliminary survey before Beckman's sent in. It should be easy to get domestic positions. I have several letters of recommendation from persons of the highest quality, all giving me an excellent character.\" She smiled, narrowing her eyes. \"As you are doubtless aware, although California is technically a Free State, it is inadvisable for a Negress to travel alone. Kalugin has been assigned duties in San Francisco, and dear Eucharia agreed to travel with us to lend _respectability_ to our journey.\"\n\nEucharia was stepping back from Porfirio and regarding him, hands on hips. \"We'll have a high old time tonight,\" she said. \"Got any tequila?\"\n\n\"No, and no Southern Comfort, either,\" replied Porfirio, and that set both of them roaring with laughter. I guess there was some history there. I hadn't seen Porfirio smile like that in the whole time I'd been there, not a real smile like he was enjoying himself.\n\n\"But what of you?\" Nancy took my hand. \"Have you been happy?\"\n\n\"Happy? I\u2014well, of course. I've mostly worked alone, you know, back in the mountains. Remember how I wanted to come here after I graduated, how I made New World grains my specialty? Well, the Company finally noticed. Here I've been, years and years now.\"\n\n\"I heard about what happened in England,\" she said quietly, looking at my hand. \"I was so sorry. I wrote to you.\"\n\nI shivered. \"I was in therapy for a while. I probably never got your letter. Well, it was a long time ago, and I'm over it now. But thank you for writing.\"\n\n\"Here we are!\" Kalugin came puffing up the trail, a trunk under either arm. \"Everything seems to have survived the journey, Nan. Will you do me the kindness of showing me where I can stow these, mademoiselle?\"\n\n\"This way.\" I gestured, and took one of the trunks from him and swung it up to my shoulder. He made a little dismayed sound but followed me to the adobe, where I led them down the long corridor to the guest room that was kept for visiting operatives. \"Here you are. Don't be scared of the cowhide bed, they're actually very comfortable,\" I said. \"Dinner at 2000 hours, alfresco. The menu includes such authentic regional delicacies as grilled beef, frijoles, and tortillas, but I should warn you that a tortilla here bears no resemblance to the Spanish item of the same name.\"\n\n\"Yes, I've discovered that.\" Kalugin hastened to relieve me of the trunk. \"Allow me, that really is too heavy for a lady.\"\n\nI could lift a horse, let alone a trunk, if I had to, like any cyborg; but how sweet of the man.\n\nI left them alone to get the dust of the journey out of their teeth, and went to pace around in the oak trees for a while. Was I happy for my old friend? Yes, unquestionably; but I didn't want to be reminded of being young, or of England, or of the mortal man who had died there so long ago. He was after me again, following me relentlessly from shadow to shadow through the trees.\n\nEucharia helped Porfirio prepare supper for the rest of us, but then the two of them disappeared into the night with pistols, a small box of ammunition, and a lot of aguardiente. Imarte was away on one of her sleepovers at the Bella Union, thankfully, and Oscar had trekked far afield on his quest for a buyer for the Criterion Patented Brassbound Pie Safe; so the company around the cookfire was fairly intimate that evening. Juan Bautista even brought out his guitar.\n\n\"But how charming,\" Nancy said. \"That was made in Old Spain, was it not? And by a master, to judge from the inlay work.\"\n\n\"Yes, ma'am,\" said Juan Bautista in a tiny voice. He'd fallen in love with her, desperately, of course. \"One of the mortal travelers left it. Lucky chance, huh?\"\n\n\"Can you play it?\" I peered at him across the firelight. \"I've never seen you actually play it, Juan.\"\n\n\"Sure he can,\" said Einar, putting another log on the fire. \"I hear him practicing sometimes.\"\n\n\"I play for Erich von Stroheim,\" Juan Bautista said. When Nancy and Kalugin stared at him, he hastened to add: \"My condor. Baby condor. I rescued him. It helps him get to sleep sometimes when he's nervous.\"\n\n\"Ah, of course,\" Kalugin said with a nod of understanding. \"Would you perhaps do us the honor of playing for us now?\"\n\nJuan Bautista hung his head and fiddled with the tuning pegs. \"Sure,\" he muttered. I braced myself, expecting him to clutch painfully at the frets in a beginner's \"(I Can't Get No) Satisfaction,\" but to my astonishment he went into a classic Segovia piece, and it flowed out on the night air smooth as coffee with cream. He kept on with beautiful classical stuff all evening, Rodrigo and de Falla and Five Jaguar, quiet and unobtrusive, the background to our talk.\n\n\"I have to know,\" I said, leaning forward, \"how the two of you met. It's so rare, you know, for any of us to find . . .what you've found.\"\n\n\"It was terribly romantic,\" Kalugin said, smiling where he lay with his head in Nan's lap. \"I'd been in a shipwreck, and washed up on the coast of Morocco. She was all in silks and bangles, third wife to one of the sultan's corsairs.\"\n\nEinar leaned his chin on his fist and grunted. \"Our anthropologist will be disappointed she didn't get a chance to talk with you.\"\n\nNancy opened her reticule. \"I'll leave her one of my calling cards. It is, after all, the correct thing to do in these circumstances in polite society.\"\n\n\"Calling cards,\" I said. She nodded serenely and handed me a tiny square of pasteboard, embossed, beautifully engraved. I read:\n\n\"D'Araign\u00e9e?\" I asked.\n\n\"An artistic decision,\" she said. \"French for _spider_ , you see. I have always retained the clearest memories of the folktales of my mortal parents. Indeed, I can scarcely recall anything else from my mortal life.\"\n\nI remembered the angry four-year-old girl that she'd been, telling me how the spider god of her tribe had deserted them, saving only her.\n\n\"Anansi,\" she said. \"The friend and helper of men, as I understand from my researches into the work of M. Griaule and Mr. Parrinder.\"\n\nI stared into the fire. The immortal operative who'd rescued the child must have named her for the word she repeated most often, thinking it sounded like Nancy. Had the little girl been calling on her god? Had she finally made her peace with him now, since she'd taken his name for her own? I'd never made peace with mine.\n\nBut how wonderful, what style she had, to what good use she'd put her anger.\n\n\"What's Salon Algeria?\" I asked.\n\n\"One of the Company safe houses in Paris,\" she said. \"I reside there when Dr. Zeus has no pressing errand on which to send me. And it's useful, too; a certain segment of the artistic denizens on the Left Bank know that I am always interested in seeing canvases, and perhaps paying cash for them. Regretfully, members of the criminal class are also aware of this, and I'm afraid I have purchased stolen paintings on more than one occasion.\" She shrugged. \"One has the consolation of knowing that everything one purchases in this way will survive for the ages rather than burn in the political upheavals with which France is so frequently visited these days.\"\n\nI nodded. God, she even had a home.\n\n\"What were you doing in a shipwreck?\" Juan Bautista raised his head and looked at Kalugin. \"I thought we always know when a ship's going to sink.\"\n\n\"We do,\" Kalugin told him ruefully. \"But when history records that a ship's going to disappear with all hands, young man, she becomes fair game for the Company. And when history records that she carries valuable cargo, the Company acts. Most people suppose a marine salvage technician is some sort of diver, and it's true; but, you see, I don't go down after the wreck. I go down _with_ it.\"\n\nThis extraordinary statement was followed by a distant salvo of gunfire, followed by wild laughter from somewhere up the canyon. Porfirio and Eucharia were apparently target shooting by infrared.\n\n\"Anyone can dive down to retrieve gold or jewels,\" Kalugin explained. \"They aren't spoiled by a little seawater. But what about manuscripts, paintings, Stradivarius violins? You need someone there, on the scene, someone with the knowledge of what's to come, who can secure all those perishable treasures in sealed containers before the ship sinks. You need someone to ride the poor wreck to her final resting place, and transmit exact coordinates on her location to a Company salvage team. You need someone to stay with her on the bottom, lest she drift, lest she break up and any of those carefully sealed boxes float away. And you need someone there to guard her, in her grave, lest the chance fisherman or swimmer should find her before the Company team arrives.\"\n\n\"You mean you stay in the wreck with all the dead guys?\" Juan Bautista asked, horrified. His music faltered for a moment.\n\nKalugin nodded sadly.\n\n\"How can you stand that, man?\" said Einar.\n\n\"I go into fugue,\" said Kalugin. \"I shut down. I respond only when there's a threat to the ship. When the Company divers come, they pull me out, and I go bobbing up to the surface to breathe again. Wretched business, isn't it? To do one's work by impersonating a bloated corpse.\" He gave a little embarrassed laugh. Nancy took his hand and kissed it.\n\n\"But living on board with those guys, knowing all the time that they're doomed . . .\" Einar shook his head.\n\nKalugin seemed to be choosing his words carefully. \"Well, it's rather like what we all face every day, isn't it? Every mortal who stops here is doomed, eventually. All our fellow passengers in the stagecoach, every one of them bound for the unknown. I just . . . try not to think about it.\" He turned his face up to Nancy and smiled. \"Fortunately, when not thinking becomes impossible\u2014as it does\u2014I have an angel to pray to.\"\n\nA silence fell. Juan Bautista had stopped playing. I suppose we were all sitting there with identical expressions of appalled sympathy on our faces.\n\nA voice began to sing, somewhere up on the hill, a woman's voice powerful and harsh, raw with emotion and alcohol, echoing in the night.\n\n> > _You hear no sound but my silenced voice.  \n>  You feel no beat but the fire that burns me.  \n> You draw no breath but I come into you.  \n> Before you, behind you, I am the sea and the rock.  \n> I AM THE SEA AND THE ROCK!_\n\nJuan Baudsta lifted his head, recognizing the song, and so did I. It was by the twenty-first-century composer Whelan, from _The Unquiet Dead_ , his Celtic reinterpretation of de Falla's _El amor brujo_. Juan Bautista flexed his clenched hands and improvised on the melody, and as the flow of guitar music began again, we all drew breath.\n\nThat night in bed, I tried to occupy my mind with how much better off I was than poor Kalugin, but all I could do was envy him and Nancy their good fortune. Whereas I was alone on the shore, like the girl in the song waiting for my lover to return from the other side . . . And now here he came, from the east, out of the sea. From the east? What coast was I on?\n\nThere was the track of foam breaking the water as he emerged. The white horse bore him up, he was coming on the wave, armored like a knight but all in seashells, and his charger in seashells likewise. Oh, the beautiful arrogance of his big body in armor, and how well he sat a horse! His lance was a narwhal's horn, a twisted ivory spear, and he wore no helmet but a tall hat banded with a trailing ribbon of seaweed. His eyes were the color of Spanish glass in his stern face, yes, and fixed on me as he came on, and he came faster now and was lowering his lance at me. Yes! I knew what he meant to do and tore at my clothing, baring my heart to his assault.\n\n\"Strike!\" I screamed. \"Strike, in God's name!\"\n\nHe struck true, and what a great relief it was.\n\nI awoke in darkness.\n\n###\n\nEARLY THE NEXT MORNING, I staggered out to make coffee, under the assumption that Porfirio wouldn't be in any condition to do so; but I was mistaken.\n\nHe was sitting beside the fire with Eucharia, and they were both watching the blue graniteware pot like pilgrims waiting for a miracle to happen. She looked pretty bad; she was clutching her whalebone corset, which I guess had been removed at some point during the target party for convenience, and her mauve taffeta ruffles were torn. As for him, I didn't think our eyes could sink that far back into our heads.\n\nI noticed them only for a split second, because my attention was immediately drawn to the dead bear.\n\n\"That's a dead bear!\" I yelled. They both winced.\n\n\"I'm sorry, okay?\" Porfirio put his hands to his head. \"I'm really sorry. Just keep your voice down.\"\n\n\"But we're trying to keep the damn things from going _extinct,\"_ I hissed.\n\n\"It was my fault,\" Eucharia said shakily. \"Wasn't it, babe? We were shooting at targets, and he just popped up against the skyline. Thought for a minute I was in a shooting gallery.\"\n\nI paced around the bear. He had a neat bullet hole between his eyes. \"That was some nice shooting, anyway.\"\n\n\"I thought Einar could salvage it for DNA,\" Porfirio said, reaching for the coffeepot with a trembling hand. \"Oh, man. I need to metabolize some glucose. Mendoza, is there any of that pan dulce left?\"\n\n\"I'll go see,\" I told him, and as I was entering the adobe, I passed Einar coming from his room. A moment later I heard his howl of dismay at the sight of the bear.\n\nNancy and Kalugin emerged slightly later, managing to look like an Arabian Nights illustration despite their primly Victorian morning wear. Nancy took one look at poor Eucharia and led her away, shaking her head. By the time they came out again, Eucharia had been dusted off and freshened up, and you wouldn't have known her for the gum-popping honey of the previous evening. She looked a lot like the farm wife in _American Gothic_. By the time the morning stage rolled in, the three of them were back in their roles completely: vaguely foreign gentleman, dull-eyed wife, and meek servant girl.\n\nAs Kalugin was seeing to their trunks, Nancy turned to me and reached out her hand. \"Mendoza, will you be all right?\"\n\nWho, me, ashen-faced after a night of erotic death dreams? But I wasn't the one facing a stagecoach ride with a hungover fellow passenger. \"Sure I will. I mean, here I am in the New World. I got what I wanted. Granted, this is a lousy place, but I've already put in my request to go back to the Ventana when this tour of duty's over. It'll be great. Green wilderness, Nancy, oak trees older than we are and not a mortal soul for miles. You should go up there sometime.\"\n\n\"I should love to,\" she said. \"But, you know, my dear, it's going to become more difficult to find places in California where there aren't mortals. Nearly impossible, in another few decades. Where will you go then?\"\n\nAnd I realized, only then, that I had no clue, none at all. I guess it showed in my eyes, because she gave me a hug. \"Oh, Mendoza, find something to make you happy! It's easier than it seems. You'd be surprised.\"\n\n\"Okay,\" I said, for lack of anything better to say, and the driver shouted, and we hugged again, quickly, and she was away and running to clamber up on her seat. I wondered how many centuries it would be before I saw her again. With the crack of a whip the stagecoach was off, up Cahuenga Pass, on its long journey to San Francisco.\n\nImarte had disembarked and brought a customer with her, her arm wound firmly through his, leading him up the trail to her lair.\n\n\"You're going to love it here, Mr. Kimberley, the climate's real nice,\" she was telling him. \"I bet it's real cold in England, huh?\"\n\n\"Wretched, dear lady,\" the John assured her, and he was definitely English. I shuddered at the sound of his voice. \"Though England does have the advantage of a certain amount of stability. I trust there are no civil disturbances hereabouts? Relating to the present war, I mean?\"\n\n\"No, honey, none at all,\" Imarte said, lying through her teeth, because there was a regular secessionist crowd that hung out at the Bella Union, and they not infrequently took potshots at fellow Americans whose accent sounded a bit Down East. Not that I understand how Maine can be _down_ anything when it's as far north as you can go in the States.\n\nShe led the Englishman away with her, assuring him they had plenty of time before the next stage came through. I thought sourly that if the man had any inkling of how much talking he was expected to do in bed, he'd be running after the departing stage. I wandered away up the canyon to the little brook where you could occasionally find trout and sat there awhile, looking into the brown water, wondering where I _would_ go when California filled up with mortals. Canada, I decided at last, and made a mental note to do some studies on the botany peculiar to our neighbor to the north. Yet I loved California; I was even beginning to like this ghastly corner of it, with its killers lurking in the sagebrush and its yet unborn movie industry.\n\nAnd I thought to myself that things weren't that bad. I had my work, didn't I? There was my problem! I wasn't working enough, what with needing an escort for every field trip. That was why Nicholas was haunting me. Well, if I practiced harder with the Navy pistol, maybe Porfirio would decide it was safe to let me go out on my own. In the meanwhile, another good collecting trip was just the thing to chase away the blues.\n\nBy the time I mulled all this over and wandered back to the inn, the afternoon sun had fallen behind the ridge and the dust of the last stage had long settled. Nobody was in sight. I scanned and detected Porfirio and Imarte in the inn, and followed their signals to the pantry room, where Porfirio was kneading dough for a fresh batch of pan dulce. His eyes had come out of their caves a little. Imarte sat with her elbows on the table, watching him. She had a bitchy line between her eyebrows.\n\n\"Where is everybody?\" I asked, pulling out a chair and sitting down.\n\n\"The kid's around here somewhere. Oscar, I don't know. Einar went off to Antelope Valley again.\"\n\n\"Nuts.\" I sighed. He probably wouldn't be home until the next afternoon. \"I wanted to go out collecting tomorrow.\"\n\nPorfirio shrugged. \"You could go with Oscar,\" he suggested hopefully. Now there were two women with bitchy expressions in the room with him. \"So, Imarte, what was the deal with the Englishman? Get any good material from him?\"\n\n\"Virtually none,\" she said. \"He never stopped asking _me_ questions all afternoon! In fact he shut up for exactly five minutes, and I leave it to you to guess why. The rest of the time, I might as well have been a tour guide.\"\n\n\"Maybe you met up with a kindred soul,\" I said. \"Dueling anthropologists.\"\n\n\"Very funny. I learned nothing of his home region or native customs. The best I could get out of him was that he's a mining engineer with the Albion Mining Syndicate. Apparently some confidence man persuaded a group of foreigners that there's been a gold strike on Santa Catalina Island, of all places.\"\n\nI remembered the blue island I'd seen, way out there in the sea. \"And it's a con, you think?\"\n\n\"Certainly. Who ever heard of gold out there?\" Imarte leaned back in her chair and stretched.\n\nI actually asked Oscar about his rounds the next day, I was that desperate for work. He was only too happy to oblige; he hadn't made the West Hollywood sweep in a while, it seemed, and so next morning I crowded onto the seat of the peddler's cart with him, and we rolled off down the canyon.\n\n\"So, is it hard to get your foot in the door when you're making your pitch?\" I asked as we turned right and headed west.\n\n\"Goodness no, not at all,\" Oscar said. \"Not like in the States. I mean, the original thirteen. The local inhabitants are charming people, with a keen appreciation of the solid worth and delightful abundance of general merchandise offered by your humble servant. They'll welcome a salesman into their homes, indeed invite him to dine with them, which is certainly more than I can say for the inhabitants of Rhode Island. The folks here will sit patiently through a complete demonstration of every ingenious and laborsaving device I have to offer. Only problem is, they have no money.\"\n\n\"Really?\"\n\n\"Nobody here has,\" he said. \"All these Californio gentlemen, and the Yankee fellows too, have pitifully little cash. All they've got is the land and cows. The land's dirt cheap, but most of it's mortgaged to the last square inch, with the highest interest rates you've ever seen. Why, it's plain crazy.\" He shook his head in commiseration.\n\n\"Boom and bust, huh?\" I said. Finance had never interested me, but I craved trivia just now, anything to tie me to the present place and time.\n\n\"And take this new silkworm business.\" He gestured angrily with the whip. \"Fools planting mulberry trees, thousands of the darned things, everywhere, and importing cocoons from the Orient at great expense. Will it ever produce so much as a handkerchief's span of silk? No, sir! The whole enterprise is a sham, a bubble, a speculator's airy impertinence, and the end result will be _ruin_ , you mark my words. And a lot of mulberry firewood.\"\n\nWas that why my orders had included collecting samples from mulberry trees? I'd wondered about that; they weren't an indigenous plant. Sounded like another tulip fiasco: when speculation on bulbs wrecked Holland's economy and prices plummeted, nursery stock was destroyed en masse, and lots of genetic diversity disappeared.\n\n\"Will we be passing any mulberry plantations today?\" I asked.\n\n\"More than likely,\" Oscar said.\n\n\"Good. I can take some cell samples. Got a customer lined up for your Criterion Patented Brassbound Pie Safe yet?\"\n\n\"Just about,\" he said, brightening a little. \"If my careful analysis of my customers is correct.\"\n\nWe followed the rocky track, which paralleled the foothills and continued west in a fairly straight line toward a tiny cluster of adobes, all facing one another on opposite sides of what would undoubtedly one day be an avenue, boulevard, or drive. Tilled fields and orchards took up a few acres around the hamlet; straggling fences of nopal cactus raised formidable paddles against bears, wandering longhorns, and chicken thieves. The place appeared deserted save for a dog lying in the dust in the middle of the avenue, boulevard, or drive, and I'm not sure he was alive.\n\n\"Now this,\" Oscar said, \"is the ill-fated village of Sherman. I don't believe it's named after the redoubtable general of the same name. No, indeed, and it certainly won't live to be as famous. West Hollywood will obliterate it completely, as a matter of fact. Nevertheless, these people are as receptive to the improvement of their lives by beautiful and useful merchandise as any you'll find in more progressive communities.\"\n\n\"Really. Look, there's a grove of mulberry trees.\" I pointed at a double row of miserable little whippy seedlings, each with its green flame of young leaves waving at the top. \"Do you know these people? Think you could get me permission to take samples?\"\n\n\"Undoubtedly,\" he said, and clucked to Amelia to stop just in front of the first house. Describing it now will save my having to describe nearly any other structure in southern California from here on: a long adobe, its former whitewash peeling and cracked away in places, all of it badly eroded by the winter rains. Flat roofs of pink tile, cracked or missing here and there, held on by thick tar from the nearby La Brea pits. Wooden doors and window frames, closed shutters painted faded blue if they were painted at all. Most were of bare wood silvered and cracked with the weather.\n\n\"I don't think anybody's home,\" I said, as a chilly breeze whirled through the cactus.\n\n\"Nonsense. Always looks like this. Decent people stay indoors and mind their own business around here, that's all.\" Oscar hopped down and ran around to offer me his arm in descending. I looked up at several pockmarks in the nearest wall; if they weren't bullet holes, the place had damned big hailstones. Oscar didn't notice, seemingly; he strode right up to the door and knocked briskly. \"Hello there! Buenos d\u00edas, Se\u00f1ora Berreycsa, might I have a moment of your time?\" I followed him slowly, ready to throw myself flat if a hail of shot came from anywhere. After a moment we heard a bolt drawn, and the door opened a trifle. A woman peered out at us.\n\n\"What do you require, se\u00f1or?\" she said in Mexican Spanish.\n\n\"Ah, Se\u00f1ora Berreyesa, what I require is not the issue at all,\" said Oscar, matching her Spanish pretty well. \"It is what _you_ require. Perhaps you recall me? I was here last autumn, before the rains. You purchased a splendid shaving razor for the se\u00f1or of the house, which perhaps you will also recall. Doubtless you will be pleased to discover that I have returned, bringing even more splendid wares for your inspection.\"\n\n\"In that case, consider my house your own,\" she said, with great courtesy but less enthusiasm, standing aside to open the door. Oscar threw me a triumphant little smile and stepped inside, and I followed, murmuring my thanks to Se\u00f1ora Berreyesa.\n\nThe interior was typical of a reasonably well-to-do working-class family. A dirt floor, packed hard and pounded smooth, and smoothly plastered walls shading to an olive brown near the ceiling, from smoke, and almost black over the household shrine, where a couple of candles winked in jars of ruby glass before a little wooden figure of the Virgin of Guadalupe. This was a house of two rooms, and the fine oven and cooking hearth were built into the wall between the rooms, so as to warm both. There were stoneware jars with heavy lids ranged along the wall, and shelves above them displaying graniteware dishes. Several wooden chests; a solid table; and on it a chunk of fresh pork loin and a carving knife, which the se\u00f1ora had apparently been using when we arrived. She eyed the knife now, and looked at us in the hope that we'd be polite enough to leave soon and let her get back to her housework.\n\n\"And may I introduce Se\u00f1ora Mendoza, who was kind enough to accept my offer of transportation when her horse went lame? Very good. I know your time is valuable, se\u00f1ora, so I won't waste it,\" Oscar said, interpreting her expression correctly. \"You should know I have cakes of shaving soap available that fit precisely into the mug your husband uses. And, if I recall correctly, you must be nearly out of Morning Glory Laundry Bluing, and that was a particular favorite of yours, was it not?\"\n\n\"It is, se\u00f1or,\" she said.\n\nAll this while, subtle unseen miniaturized cameras and recording devices on and in Oscar's person were noting every detail of the house and of its inhabitant. \"Very good.\" He rubbed his hands together cheerfully. \"I have all you could require. Now, let's see, your husband, do I err in remembering that he's a vaquero?\"\n\n\"No, se\u00f1or, you do not,\" she said. \"At the Rodeo de Las Aguas.\"\n\n\"Yes, so he is. And he's, ah, twenty-eight years old?\"\n\n\"Twenty-six, se\u00f1or.\"\n\n\"A good age for a man to begin looking after his teeth, wouldn't you say? I don't believe I've ever shown you the fine assortment of American brushes I carry for that very purpose. Regard this!\" From seemingly thin air he produced a bone-handled toothbrush. The se\u00f1ora did not seem impressed at the sleight of hand, but the tiny brown boy who had been watching from her skirts yelled with delight and pointed at us.\n\n\"Well.\" Oscar hitched up the front of his trousers and crouched down, smiling, turning his head for a good camera angle. \"And who's this little fellow? I don't believe I met _you_ on my last visit, no, sir.\"\n\nThe child retreated a bit, staring in fascination. \"He was asleep then, se\u00f1or,\" the mother explained. Oscar produced (again, from apparent thin air) a little wooden jointed doll and offered it to the boy.\n\n\"Well, here you go, sonny. Here's a dancing Uncle Sam to keep you busy while your mama does the household chores, eh? No, no, se\u00f1ora, it's a free gift, a trifle. And I'd like you to consider this brush I was about to show you.\" He stood again, proffering the brush in one fluid movement. She must have wondered how she found it in her hand without consciously accepting it, but there it was; and he pulled out a can of tooth powder and held it up before her puzzled eyes.\n\n\"Now, se\u00f1ora, I can see you've been favored by Nature with exceptionally beautiful and durable teeth. How fortunate you are. But as a dutiful wife and mother, it falls to you to see to the well-being of your spouse and little ones; and they may not be as dentally gifted as yourself, if I may say so. Were you aware that it is the opinion of most modern physicians that an astonishing number of diseases, distemperatures, and infections find their root and origin in _poor dental hygiene_?\"\n\n\"I know,\" she said. \"Gingivitis.\"\n\n\"Er\u2014well, yes, exactly so, se\u00f1ora. And yet, by following the simplest of daily regimens, one can preserve glowing dental health and, incidentally, the beauty of a radiant smile.\"\n\n\"It's true,\" she agreed. \"You chew on sage leaves. Keeps away the gum sores, and your breath doesn't stink either. And they're free. They grow all over the hills here.\"\n\n\"To be sure,\" said Oscar, not missing a beat, \"and I can see you're _exactly_ the caring and concerned parent and mate who does her best to see to it that her family follows that oh-so-careful daily routine for their own good. For that very reason, I know you'll be interested in this splendid dentifrice applicator, manufactured out of the very finest materials by the Superior Brush Company of Ogdensburg, New York, USA, and guaranteed to withstand the most rigorous use. Only regard it, se\u00f1ora! Observe the bristles, designed to delve into the crevices between difficult-to-reach back teeth, where home remedies can so seldom penetrate. Now these bristles are derived from the splendid native American wild boar, and they have a particular flexibility unknown to the inferior European variety, which makes them the material of choice for delivering a sufficient dose of this excellent nostrum to the teeth and gums of the gratified user. Now this, se\u00f1ora, is Cleopatra's Smile Tooth Powder, not merely a solution to ensure dental health but a restorative of the natural whiteness of the tooth enamel itself. Tell me, se\u00f1ora, your husband's a vaquero: does he ever chew tobacco?\"\n\n\"Never!\" Se\u00f1ora Berreyesa said with a scowl. \"I'd throw him out of the house if he took up a dirty habit like that.\"\n\n\"You would, of course, like any right-minded wife, I don't doubt it, but do you know that Cleopatra's Smile has removed stubborn tobacco stains even on those depraved unfortunates who practice the tobacco vice? Now let's address the question of tea and coffee, which we all enjoy from time to time. Cleopatra's Smile has been proven to remove tea or coffee stains from the teeth and restore snowy whiteness _after only one application_.\" I think he was sweating just a little, but Fate smiled. Se\u00f1ora Berreyesa put her head to one side and considered the bright logo on the can.\n\n\"Will it do that for tablecloths too?\" she asked.\n\n\"Well, why wouldn't it?\" Oscar said. \"If it was brushed in sufficiently, I bet it would! Now I'll tell you what I'll do, se\u00f1ora. I'll make you a complimentary gift of that very brush you hold in your hand. Society matrons in the eastern States can't get Cleopatra's Smile for less than ten cents a can, but I'm able to offer it out here for a mere three red cents.\"\n\nShe thought about it. \"Three cents? You will excuse me one moment, se\u00f1or?\" And she went into the adjoining room, with the little boy following her like a shadow.\n\n\"A can of tooth powder isn't a pie safe,\" I said to Oscar. \"And you had to give away a doll and a toothbrush, plus a seven-cent mark-down on the tooth powder.\"\n\n\"That's not the point,\" he murmured. \"I'm building a clientele, don't you see? The point of the game is getting them to _want_ this stuff. After want comes need, and once they need what you have, all you have to do is supply the demand.\"\n\n\"Will you remember to ask about the mulberry trees?\"\n\n\"The what? Yes, yes, of course. And we'll just see about that pie safe, shall we?\" He stuck his thumbs in his waistcoat pockets, looking mighty pleased with himself.\n\nSe\u00f1ora Berreyesa returned and held out her hand. \"Three American cents, se\u00f1or. I will try a can of that powder.\"\n\n\"You won't regret it, se\u00f1ora,\" he assured her, pocketing the money and presenting her with the can. \"And may I add that this is also a superlative remover of the stubborn and unsightly stains caused by the consumption of mulberries? Of which I notice you shall soon have abundance, by the way.\"\n\n\"Those?\" She rolled her eyes. \"If we ever see a berry from them, I'll fall over dead. My husband let Se\u00f1or Workman plant them, after some crazy talk about silkworms. Chinese shawls growing on trees, he told him.\"\n\n\"Well, isn't that just like a man? But I wonder, se\u00f1ora, if you would allow my friend here to collect a couple of leaves from the young trees? She studies such things.\"\n\n\"You can take the whole damned orchard, as far as I'm concerned,\" she said to me. \"Please, se\u00f1ora, help yourself. This way.\" She scooped up the little boy in her arms and led us out the back door to the garden beyond.\n\nAs I walked among the little trees, clipping off a likely-looking shoot here and there, Oscar cleared his throat. \"I couldn't help noticing that our arrival interrupted preparations for a meal,\" he said.\n\n\"That's true, se\u00f1or, but I can spare the time to speak with you,\" Se\u00f1ora Berreyesa lied graciously.\n\n\"Ah, but, busy woman as you are, you must frequently suffer interruptions in the course of your culinary duties, and food intended for human consumption may then be unintentionally exposed to the assault of common household pests. I'd like to suggest a means of ensuring that your foodstuffs stay safe and unmolested. Now I happen to have in my wagon a miracle of modern design: the Criterion Patented Brassbound Pie Safe! I believe it may be the answer to all your problems, and if you'll just step out to the cart and let me demonstrate its assorted features, I'm sure you'll\u2014\"\n\nBut Se\u00f1ora Berreyesa had stopped in her tracks, her face registering outrage as the import of his words sank in. \"Are you suggesting that I have rats in my kitchen?\" she said.\n\n\"Uh\u2014why, no, certainly, but\u2014\"\n\nShe seized his sleeve. \"You think I keep a dirty house? You think I leave food lying around to draw rats? You come in here and see.\" She dragged him back into the house, and I ran after them hastily, tucking trailing mulberry cuttings into my collecting basket. She gestured dramatically at the row of stone jars, each with its heavy stone lid.\n\n\" _There_. That keeps the food safe and cool. There is never any food left lying unprotected in my house, except when annoying little white men come to sell me things.\"\n\nOscar gulped and scuttled for the door. \"Point taken, se\u00f1ora, point taken. I'll just be on my way, I guess. Buenos d\u00edas.\"\n\n\"Buenos d\u00edas, se\u00f1ora, and please excuse the discourtesy,\" I said as I followed him. She inclined her head stiffly in acknowledgment. The little boy stared at us with solemn eyes.\n\n\"Well, at least she bought the tooth powder,\" I said when we were out in the street.\n\n\"Tactical error,\" he admitted, taking out an immense spotted handkerchief and mopping his face with it. \"Ought to have seen she was house proud. Well, well, I'll do better next time. Got your sample cuttings, did you? First-rate. Let's be off to the next customer.\"\n\nDown the street we went, the mule sighing audibly.\n\nNobody answered our knock at the next few houses. Near one there came a little whine and a _ping_ , and a puff of adobe and plaster flakes jumped off a nearby wall; so we kept on going, until we got to a board-and-batten shanty sitting by itself in a field. Smoke whirled from its tin chimney.\n\n\"Can this be a new customer?\" Oscar stared at it keenly. \"That place was abandoned, and somebody's been and fixed it up. Well, well. Cameras and audio at the ready! Care to come in with me?\"\n\n\"Why not?\" I said, scanning the house. I could pick up only one occupant, a female. No, there was a cat, too.\n\nSo we got out, and Oscar rapped smartly at the door. There was a silence and a scurrying, then someone tugged the door open from the inside, scraping it across the warped sill.\n\n\"You will excuse please,\" said the lady who had answered. \"My door, it is wretchedly made. There are no good carpenters here, like in my country.\"\n\nWe beheld a mortal woman in her mid-thirties, with a plain freckled face and intense blue eyes. She had a vaquero's red bandanna bound tight on her head, like a Gypsy scarf, and the rest of her appearance produced that effect also: calico blouse and skirt of violently clashing colors, red morocco slippers with pointed-up toes, and brass hoop earrings so big, a mouse could have jumped through one. Around her neck were numerous strings of beads, some of crystal, some of cheap trade glass, some of bones and shells and little unidentifiable oddments. She wore a lot of rings, too, gimcrack stuff, the costume and curtain variety.\n\nFor once, Oscar was speechless; but not for long.\n\n\"And what country would that be, ma'am?\" he asked, removing his derby.\n\n\"Grumania-Starstein,\" she said. \"I am princess there. You are addressing Her Highness Sophia Sylvia Rodiamantikoff. Filthy conspirators brought about the downfall of the royal house. But I fled to safety through snows aided by loyal servants, chased by wolves all the way. I have come to this country to await restoration of monarchy by my secret friends in the palace.\"\n\nRight. My guess, analyzing her accent, was that she'd been born in Pennsylvania (possibly Shamokin) and probably known some immigrant families. Oscar blinked, turned his derby around in his fingertips, and smiled.\n\n\"Why, isn't that interesting. It didn't leave you much time to pack a bag, did it? Is Your Highness provided with all the minor necessaries a lady requires for good health and hygiene?\"\n\n\"I _had_ such,\" she said with a melancholy sigh, lifting the back of her hand to her forehead. \"Had beautiful set of tortoiseshell combs, given to my great-great-grandmother as a present from Ivan the Terrible, who was her godfather, you know. Alas! They are lost, along with solid gold comb-and-brush set I was given by my uncle the archduke. Gone, gone with my jewels and my crown!\"\n\n\"Golly, that's really too bad,\" Oscar said sympathetically. \"Fortunately, I happen to have a complete assortment of the finest toiletries and toilette accessories a lady could require, ready for your inspection. I'd be honored if you'd care to purchase any, ma'am\u2014Your Highness, I mean.\"\n\nShe bunched the fingers of one hand together and set them in the middle of her forehead, frowning thoughtfully. \"One second, if you please,\" she said. \"I must consult spirit guides. Chief Running Deer! King Elisheazar! What you say, boys?\"\n\nIn the silence that followed, I transmitted to Oscar: _So, is she nuts or a con artist_?\n\n_Your guess is as good as mine_ , he replied.\n\n\"We will consider your wares,\" she said at length, and stepped forth into the light of day. Behind her an evil-faced cat came to the doorway, peered out at us, and fled back inside. Oscar hastened to open up the side of his wagon, displaying a gaudy splendor of ribbons, brass thimbles and scissors, pack thread, playing cards, cheaply bound books, and various items for personal grooming.\n\n\"There, now, Your Highness, what d'you think of this?\" he said, as though he expected her breath to be taken away by the glory of it all. I decided they were both nuts and turned my attention to a nice little specimen of _Lupinus_ lifting spires of blue and purple from the edge of an irrigation ditch. Her Royal Highness Rodiamantikoff fussed and sniffed at the items on display, remarking plaintively that these things were very shoddily made, not like wares in dear old Grumania-Starstein, and occasionally her two spirit guides threw in their two cents about the quality of this bottle of toilet water or that pair of silver-plated sugar nippers. Oscar just poured on the ingratiating charm, bowing and scraping as though she were standing there in her royal finery.\n\nShe'd decided on three yards of scarlet ribbon and a deck of playing cards, explaining that her mother had been a Gypsy and taught her to read the future with them\u2014this, by the way, was why the evil conspirators had not wanted her to inherit the throne and stole the crown for the prime minister's baseborn son Otto, who was the offspring of a chambermaid\u2014when Chief Running Deer and King Elisheazar got into a fight over whether or not she should buy herself a peppermint stick. Chief Running Deer (she informed us) said she oughtn't to deny herself such a small pleasure, poor exiled creature that she was, but King Elisheazar told him he was a savage without any breeding and it showed, because _everybody_ knew that royalty didn't buy themselves candy; such luxuries were given to them by loving subjects and by foreigners out of respect for the aura of rulership that hung about their persons despite unfortunate circumstances.\n\nOscar took the hint and presented her the peppermint stick with his compliments, which restored the good humor of the spirit guides, much to Her Highness's relief, for it _so_ embarrassed her when they went at it like that. She paid a whole thirty-five cents for the cards and the ribbon, delving into her skinny bosom for it, and I guess it was more money than Oscar had made in days, because, encouraged by his success, he made so bold as to say:\n\n\"Now I wonder, Your Highness, if you'd be interested in purchasing a certain item I have here\u2014and I've only the one, you see, you won't find its like this side of the Rockies, but you being royalty and all, I'd like to offer you first crack at it. Step this way if you please, Your Highness, and I'll give you a private viewing.\"\n\nShe followed his outstretched hand around to the other side of the wagon, where he opened up the panel and revealed the Criterion Patented Brassbound Pie Safe.\n\n\"You see here?\" His face was shining with desperate hope. \" _Your_ eye, trained as it is in discernment and accustomed as it is to superior craftsmanship, _your_ eye will surely appreciate the magnificence of this prime household appurtenance. Note the panels of polished rosewood. Note the decorative brass figuring: pineapples, the ancient symbol of abundance and hospitality. Now, I don't pretend that this is any match for the fine kitchen furniture they've got in your country, but I'll tell you plainly, Your Highness, that this is positively the finest the U. S. of A. has to offer, and no other lady in all of southern California has the like. Now, down east where I come from, the wives of millionaires would pay as much as twenty-five dollars for the likes of this\u2014 _if_ they could get it! And of course out here, where everything has to be brought by ship, it's worth a lot more. Yet to you, Highness, to you I'll offer exclusive ownership for a mere token sum of eleven dollars\u2014why, that won't even cover the shipping and handling\u2014and the priceless privilege of numbering royalty among my customers.\" My God, he was actually getting down on one knee. \"What do you say, eh? Shall I take it down for you?\"\n\nOoo, he'd come so close. She'd been transfixed, listening with mouth half open, fascinated. But she didn't have that kind of money. She wrinkled her freckled nose in slightly disdainful regret.\n\n\"I think not, at the present time,\" she said. \"Spirit guides advise that stars are not presently auspicious for buying furniture. Perhaps later, when vibrations are better.\"\n\nHe looked so crestfallen that she hastened to add, \"Yet you may use my name. Yes, you may say truthfully that you are purveyor to Royal House of Rodiamantikoff.\" She swept past him to return to her shack. \"In exile,\" she said, just before crossing the threshold. \"Good day, mister. You are excused from the royal presence.\"\n\n\"Well, you've made thirty-eight cents so far,\" I said to Oscar as I helped him close up the panels on the wagon.\n\n\"Be-elzebub!\" he said, grabbing his buggy whip. \"I nearly had her, do you realize that? She saw it, she wanted it, she could envision its rich cabinetry making that dreary hovel a refined and gracious retreat. Nothing was lacking but _money!\"_\n\n\"Well, that's always the way it is with mortals, isn't it?\" I climbed up to my seat. \"And think of the footage you got of her. Genuine California eccentric, wherever she's supposed to have been born. She's one for the archives, all right. But you know what? She's a gringa. If you sold it to her, you'd still lose the bet.\"\n\n\"I think not,\" he retorted grimly. \"She says she's a refugee from a foreign land, and would any true gentleman impute falsehood to a lady? No sir, if Her Highness says she's not an American, I'll take her at her word.\"\n\nA bullet came whining out of nowhere, drifting in to clip the top off a young oak tree nearby. Oscar whipped out a pistol and fired off three furious shots at the unseen gunman.\n\n\"I'll prevail, I say!\" he shouted. \"Do you hear me? I'll _sell_ the darned thing! I say I will, by thunder!\"\n\nBut he didn't, at least not on that day's rounds, and he was dull and taciturn\u2014taciturn for him\u2014by the time we returned to the inn that evening. He accepted his plate of grilled beef, tortillas, and frijoles from Porfirio and retired early.\n\nEven though I was eager to get the mulberry samples to my processing credenza, I lingered over the food, because it was particularly good that evening, the beef fiery from a red chile marinade, the frijoles especially creamy, the tortillas unusually redolent of earth and corn and rain. I was still sitting in the clearing when Porfirio rose to his feet, stared off into the canyon and the night, turning his head for a better signal scan, and announced:\n\n\"Stranger approaching on horseback. Mortal male. Emotionally excited.\" He had a gun in either hand before he finished speaking, and Juan Bautista rose in haste to carry Erich von Stroheim indoors out of harm's way. The bird had got too big to button out of sight inside his shirt, though it kept trying to climb into its old refuge in times of stress. It croaked in protest as Juan Bautista passed Einar, who was emerging from the house with a loaded shotgun.\n\n\"Company, chief?\" said Einar, cocking his weapon.\n\n\"Maybe,\" said Porfirio, though as the mortal drew nearer, we could tell that the excitement registering on the night air was the harmless, pleasurable kind: anticipatory, nonviolent. When he finally rode into the light of our fire, the mortal saw no weapons of any kind in evidence. Porfirio took a few steps toward him, hands outstretched in a peaceable gesture.\n\n\"You come for a room, se\u00f1or? But we have no empty beds tonight. Bad luck, eh? Perhaps you'll ride on to Garnier's? Plenty of room there.\"\n\n\"Thanks, but that ain't why I'm here,\" the mortal said politely. \"I come to see a lady, mister. Met her at the Bella Union. Said her name was Marthy, and she lived hereabouts. You wouldn't know where a man could find her, would you?\"\n\n\"Ah. Marthy,\" said Porfirio, just as Imarte herself came sweeping to the door of the adobe, magnificent in her Love's Purple Passion negligee. She paused there in the doorway, holding up an oil lamp like one of those fancy figures that lift a lighting fixture on a newel post.\n\n\"Why, who is it at this time of night?\" she said throatily.\n\n\"It's only me, Miss Marthy,\" said the stranger, dismounting and tying his reins to our hitching post. He stepped forward out of the shadows, hat in hand. \"Only me, and perhaps you remember my name? Cyrus Jackson, ma'am. We met at the Bella Union, and you was so kind as to listen to my troubles.\"\n\n\"Why, to be sure,\" she cooed, \"the _very_ interesting man who hunted Apache scalps for bounty.\" She threw us all an arch glance as if to say, See what a trophy I bagged? \"How well I remember your thrilling tales of adventure in old San Antonio! But what brings you here, sir, at this unaccustomed hour?\"\n\nHe blushed. \"Why, ma'am, I hope not to give offense\u2014I sort of thought that you might receive callers after sundown, your trade being what it is. And, you know, I wasn't at my best when last we met\u2014but I'm sober now, and I did remember that you was so taken with my recollections, that I wondered if you mightn't like to hear about when I was down in Nicaragua in '56.\"\n\n\"You rode with Walker in Nicaragua?\" Her eyes lit up. She surged forward, bosom first, and placed a coy hand on his arm. \"Why, sir, how fascinating! I wonder if you'd be so kind as to share the treasure of your eyewitness memories with an interested listener? In my private chamber, of course.\"\n\n\"Aw, ma'am, I'd be . . .\" Words failed him, or perhaps they just couldn't make it through the barrier of his enormous foolish grin. He let himself be led by the arm into the house. Staring after Imarte, Porfirio shook his head.\n\n\" _Anthropologists_ ,\" Einar muttered in agreement.\n\n###\n\nMORE BOXES ARRIVED on the next stage, and Einar ran off gleefully to his room with them. The next edition of the film festival featured hours of Charlie Chaplin, of Fatty Arbuckle and Mabel Normand. We sat in our finery in the dressed-up room and played Hollywood premiere again, sipping gin martinis and crunching on popcorn as the silver light flickered and Einar read the titles aloud in his master of ceremonies voice.\n\nNow and then there were a few frames of a landscape we recognized: a smooth-backed elevation with a single line of trees, or a dirt road ascending a steep canyon, or tiny toylike frame houses perched high on the sides of hills and wide empty country all around them. Such a raw new place Hollywood would be, and how unlike the chaparral wilderness we inhabited. For, just as Einar had told me, it would be an eastern Yankee settlement in that time: there were the clapboard houses and the shop fronts and the front porches. It looked like any little town in Connecticut or Maryland, save that it sprawled over endless rolling hills. Edendale, Sunset Park, Lankershim, Burbank, the names to assure new arrivals they were back home and not in some barren wild place where coyotes trotted down the streets at night.\n\nI didn't enjoy the comedies, as a whole, because so much heartache went with them. That world didn't even exist yet, that innocent place, and it was already lost. Those comedians weren't yet in their mothers' wombs, but their fates were known. It was hard to watch pretty Mabel and not look for the icy vivacity of cocaine, hard to watch Fatty hide his face in comic shame, knowing the doom one rowdy party would bring on him. Chaplin wasn't so bad; you knew he'd get off relatively easy for a mortal: long life, fame, lots of family\u2014also scandal, disgrace, exile, quarrels. But the comedians were nearly as immortal as we were, and we gave them our applause.\n\n###\n\nTHE DAYS GOT LONGER, and the green hills silvered and then went to gold. The wildflowers vanished as though somebody had rolled up the magic carpet and whisked it away, except for a few bright orange poppies that decorated the edges of roads. The heat of summer browned everything else. Even the eight-foot-tall thickets of wild mustard, which had bloomed in an electric Day-Glo yellow you could see for miles, went to brown; and the country took on a dry, businesslike look. The arriving and departing stages traveled in a permanent cloud of white dust.\n\nThe dust got everywhere. It covered every surface in the inn, and you shook it out of your blankets at night, and it greeted you in a fine sediment at the bottom of your morning coffee. The low-hanging branches of the oak trees, heavy with leaves, were thick with it, and dewy morning cobwebs in the grass looked like little brown rags by nightfall, they'd collected so much dust. And what had happened to all those burbling rills and freshets that had been so picturesque a couple of months earlier? Dry and dead; and in their places bone-dry trackways of sand and gravel, or deep piles of dead sycamore leaves. The cicadas started up a drone about 0700 hours in the morning, when the day would begin to heat up, and they rang in your ears like fever until sundown, when the crickets started up _their_ song in the cool of the shadows.\n\nNo way you could have mistaken the place for England now, not a sight or sound or smell that was anything but Californian. You might think my specter would leave me alone now, and in truth I had no more gasping visitations that made Porfirio stare at me suspiciously the next morning; but the darkness was still there, beating like a sullen heart when I was alone in my room. I woke up one morning and realized I'd give anything for a breath of sea air.\n\n\"So, what's it like at San Pedro?\" I said to Porfirio at breakfast.\n\n\"San Pedro?\" He frowned. \"Muddy. Used to be dangerous in the old days. It's not so bad now that Banning's running the place.\"\n\n\"It's the local seaport, right? Any chance Einar could take me down there for a visit, if he's going that way? I haven't seen a good-sized body of water in months.\"\n\nPorfirio shrugged. \"We've got some cargo due to come in. I've been meaning to send him down to the warehouses to see if it's arrived yet. You want to go with him? Nothing growing down there that I know of, though.\"\n\nEinar, when approached on the subject, thought a day at the beach sounded like a great idea, so he busied himself hitching up the horses while I packed my collecting gear. In the midst of our preparations, Juan Bautista came out of the lean-to he shared with Erich von Stroheim, rubbing his eyes.\n\n\"What's going on?\" he asked.\n\n\"Field trip to San Pedro,\" Einar said, giving me a hand up to the seat.\n\n\"The _beach_?\" His eyes widened. \"Can I come too? I haven't been swimming anywhere since I've been here!\"\n\n\"San Pedro isn't exactly surf city, man,\" Einar said. \"But if you want to come, sure. What will you do with Erich, though?\"\n\n\"Oh,\" Juan Bautista said, turning guilty eyes to the condor, who had come staggering out after him. He wasn't a mature bird yet, according to Juan, but he was enormous. \"I don't guess he should come. The seagulls might scare him. But I've never left him alone before. . . . Would you mind watching him for me?\" He looked hopefully at Porfirio, who was just sitting down with his six-shooter, preparing to clean it.\n\nPorfirio looked about as enthusiastic as one might expect. \"Look, I've got work to do,\" he told the kid. \"Put him in a cage for the day. I'll see to it he gets food and water.\"\n\nJuan Bautista ran off to shoo the bird back to his room. Einar and I waited, listening to the croaks of protest as Erich von Stroheim was coerced into the aviary Juan had built for him. A moment later, Juan came running out with his towel and a short broad plank, planed smooth and rounded off at the corners. Behind him we heard a plaintive scream.\n\n\"Okay!\" he said breathlessly, vaulting up into the back of the wagon. There was another scream, louder than the first.\n\n\"Is he gonna be all right?\" asked Einar, releasing the brake and starting us down the canyon.\n\n\"Yes. He's just never been alone for very long,\" Juan said, turning around to get comfortable. Another scream rang out on the still morning air, echoing off the canyon walls. We could still hear the condor when we turned onto the road, and in fact the sound of his outrage carried for a good mile out into the plain.\n\n\"I hope he doesn't do that the whole time we're gone,\" I said, looking over my shoulder as the foothills receded into the distance behind us.\n\n\"Nah. He'll settle down and sleep. He likes to take a nap every morning,\" said Juan Bautista with confidence. I looked at Einar, who shrugged. We rolled on.\n\nThe sea was a lot farther than it had looked from the ridge above the Hollywood Bowl site. It took us five hours, rumbling along in the wagon, though Einar informed me that Banning could do it in two and a half in one of his Concords.\n\n\"Bully for him,\" I snarled, retreating even farther into the shade of my hat. Juan Bautista had set his piece of plank on his head and made a little tent for himself by draping his towel over it. He sat in the relative cool, humming a little tune.\n\n\"Yeah, that's the way to go, if you don't have freight to pick up. Banning's got regular stagecoach service from L.A. to the coast. Another few years, and there'll even be a railroad,\" Einar said. \"Not that that does us much good now, of course.\"\n\nAhead of us, the sun on the summer sea lit up the sky, and Catalina Island hovered out there like a lovely cool mirage, blue and eternally remote. Just when I thought I couldn't take another mile of this wasteland (I'd thought Porfirio was kidding when he said nothing grew down here), we rolled up a little hill and over the top, and there it was: San Pedro Harbor.\n\nExcept it wasn't a harbor, yet, of course. It was a vast expanse of tidal mudflat, stretching away to shallow water and a distant line of white breakers. Hell, there wasn't even any sand.\n\nBut there was sea air, at least, if a bit swampy, and there was a little stream flowing through willows, blessedly green after all those parched miles.\n\n\"Surf's up, dudes!\" crooned Einar. \"Check it out!\"\n\nJuan Bautista obediently scrambled about and sat up to stare. He gave a cry of disappointment. \"Where's the _water_?\"\n\n\"Hey, this is Los Angeles! No water in the rivers, no water in the sea. No, seriously, access your Richard Henry Dana. This is the worst harbor on the coast right now. Tide flats are so shallow, cargo ships have to anchor way the hell out there and send in longboats to unload. Amazingly inconvenient. But see that big house being built over there?\" Einar pointed to a vast edifice being framed about a mile inland. \"That's the place Phineas Banning's building for himself. See those wharves? They're the latest step in his big plan to make this the next world port for shipping. Way off there\"\u2014he swung his arm around\u2014\"is the old San Pedro landing. Nobody lives there now but some fishermen. And see that island? That's Dead Man's Island. First recorded murder mystery of L.A., or so I'm told. Dead guy buried there is supposed to have been a British ship's captain, poisoned by somebody when he put in here to pick up a cargo of hides. Who slipped him the fatal glass of sherry? Nobody knows.\"\n\n\"Where's Malibu?\" asked Juan Bautista, craning his neck, as if that would make yellow sands and clean surf appear.\n\n\"North of here. Nothing much there now either, kid. Nobody even goes there, except when a cow slips down a ledge and has to be retrieved from the rocks. Honest, it's just a little trail between the cliffs and the sand, and when we get earthquakes, it isn't even that.\"\n\n\"Sight-seeing is the art of disappointment,\" I quoted.\n\n\"I want to go surfing,\" said Juan Bautista sadly.\n\nSo we drove down the hill and took him as far out across the mud flats as we could without getting the wagon bogged down, and left him to walk out to the waves while we went over to the shipping warehouses.\n\nI got down and walked to stretch my legs while Einar negotiated with the warehouse foreman, one of our paid mortals, a fisherman named Souza. It turned out that we did actually have goods to pick up: a box of printed materials for Porfirio and two crates for Oscar from the Acme Manufacturing Company of Boston, Massachusetts.\n\nWhen everything had been signed for and Se\u00f1or Souza had helped us load it all into the wagon, we drove out again, edging along the tidal flats as far as we dared before proceeding the rest of the way on foot. The mud was heavy clay, hard to walk through.\n\n\"That _is_ the sea out there, yes?\" I said, shading my eyes with my hand, peering ahead. \"Not a special effect?\"\n\n\"Just a little farther now,\" said Einar, swatting at midges. And sure enough, after we'd clambered over some slimy rocks and past a wrecked whaleboat, there were bright combers and surf breaking on rocks and even clean brown sand. Juan Bautista seemed to have got some surfing in, to judge by his piled clothing and his wet hair. But he was sitting in his drawers on the sand as we approached, cradling something in his bare arms. A big unsightly something. As we neared, it struggled and flapped.\n\n\"Easy, come on, take it easy,\" said Juan; and at his voice the bird calmed and turned its big rocket-shaped head to watch us.\n\n\"What do you have there?\" Einar asked, crouching down to stare at it.\n\n\" _Pelecanus occidental_ ,\" Juan said. \"Brown pelican. Old female. She's hurt. Look, I think that's fishing net cutting into her leg\u2014I think her leg's broken\u2014can you see? Can you get her leg free?\"\n\n\"Sssh, ssh, let's see.\" Einar stretched out a careful hand. \"You won't stab me, old lady, will you? No, you won't. Okay, that's fishing net, all right. I can try and cut it loose, but you'll have to hold her bill so she doesn't take a whack at me with it, okay, J. B.?\"\n\n\"Okay,\" said Juan, his voice trembling.\n\nI turned and walked away. I couldn't watch. It seemed to me the poor bird would have to be killed. I was profoundly grateful I was a botanist and free from the attachments people in other disciplines formed with the creatures they studied. Not that I didn't love plants. I walked up and down, looking out at the horizon where a couple of ships lay at anchor. I looked east, where Dead Man's Island raised its cone of mud. I looked at the ramshackle adobes and fishing boats beached at the old landing. I looked at the squared spaces where the new Union Army headquarters was being built, to save us all from joining a Confederacy in a distant and unreal world.\n\nWhen I dared to look back, Einar was putting his knife away and talking in a soothing tone. \"See how easy that was? Didn't hardly hurt the old lady at all. It's not a bad break, but it is broken, J. B. You need to make a decision now.\"\n\n\"We can't kill her!\" the boy said in panic. \"She's a brown pelican. They'll become endangered.\"\n\n\"I know. Okay, look. I can splint her leg now, and you can put it in a cast when you get her home, but what will happen then? How are you going to feed her? She eats fish, you know.\"\n\n\"There's trout in the stream,\" said Juan. \"And I can give her food supplements, like I do with Erich. Please, Einar.\"\n\nEinar was shaking his head, but he got a piece of driftwood and fashioned a splint for the bird's leg, cutting strands from the net to bind the leg securely.\n\nI ventured close. \"You have to remember, Juanito, she's an old bird,\" I felt obliged to say. \"Even if this doesn't work, you've made what little time she has left more comfortable. So you mustn't feel bad if she doesn't make it. This happens to mortal things. Nature will make more of them.\"\n\n\"Not that many of _these,\"_ he said, and I couldn't argue with that, so I kept my other helpful remarks to myself.\n\nWe took her with us, across the sloughs to the wagon, and Juan Bautista climbed into the back with her and wrapped her in his towel so she'd feel more secure during the long rattling drive home. By the time we reached La Nopalera that evening, she was still alive, and he'd named her Marie Dressier.\n\nAt about the point where we passed the future Hollywood Bowl, we began to hear something, and it wasn't Symphonies under the Stars. \"What is that?\" I asked.\n\nJuan Bautista, dozing in the back with his arms around Marie, sat up guiltily.\n\n\"That's Erich,\" he said.\n\n\"Uh-oh,\" said Einar, and uh-oh was right. When we finally came creaking up our canyon trail to the inn, the screams were sounding out once every two seconds and Porfirio was sitting out by the cook-fire, his hands over his ears.\n\nEinar set the hand brake and jumped down. \"Bird's upset, I guess.\"\n\n\"When did he start?\" asked Juan Bautista, clambering out awkwardly, his arms full of Marie.\n\n\"Start? He never stopped,\" replied Porfirio through his teeth. \"Not for ten seconds since you've been gone, muchacho. Please go in there _right now_ and shut him up, okay?\"\n\n\"Okay,\" said Juan Bautista, and ran for his room. As soon as he had gone inside and lit his lamp, the screams were replaced by happy little croodling sounds and a couple of dinosaur noises.\n\nPorfirio's head sank to his knees. \"Finally,\" he groaned. \"Finally.\" Then he sat bolt upright. \"Did he have _another_ damn bird with him?\"\n\n\"Another endangered rarity, chief,\" Einar said, getting the horses out of harness. \"Just doing his job. California brown pelican with a busted leg. No big deal.\"\n\n\"We picked up some freight, too.\" I hastened to bring the box of periodicals. \"Looks like you got your latest issue of _Punch_.\"\n\n\"How nice for me.\" Porfirio took the box with trembling hands. \"Well, I think I'll go to bed now. I think I'll lie in bed and just read for a while. There's plenty to eat\u2014Imarte and Oscar decided to dine out this evening. Help yourselves. Bye.\"\n\nHe got up and walked away, stiff-legged with controlled violence. I stared after him in frank admiration. I'd have wrung the bird's neck after the first hour.\n\n###\n\nPORFIRIO EMERGED from his room looking rested next morning, as I was trying to figure out how much coffee to make.\n\n\"Eight scoops or six?\" I asked, fumbling with the coffeepot.\n\nHe rolled his eyes and took the pot from me. \"Here, let me. Will you do me a favor?\"\n\n\"Sure.\"\n\n\"Go wake up Juan Bautista and tell him I want to talk to him.\"\n\nI went to knock on Juan Bautista's door. Nobody answered, so after a moment I entered.\n\nIt smelled like birds in there, and no wonder; one whole wall was cages, in which several dozen little nondescript-looking birds hopped and twittered. There were a couple of big cages, and in one of them Marie Dressier was enthroned, staring at me dolefully. Her broken leg, fixed in a cast, stuck out at an awkward angle. There was also a large perch next to the bed, clearly Erich von Stroheim's night roost. All the cages were spotlessly clean, the perch too, and all the water cups were full of fresh water and all the food cups full of fresh food. But the bed was an unmade wreck, and anything that was Juan's was either lying in a heap on the floor or piled on his one chair. He might have been an immortal cyborg like the rest of us, but he was also a seventeen-year-old boy.\n\nAs I was reflecting on this, the door opened behind me and he came in. Erich von Stroheim was perched on both his shoulders, straddling his head like a bizarre hat. Juan Bautista was carrying a little trout in either hand.\n\n\"Oh. Hello,\" he said, and stepped past me to open Marie Dressler's cage. \"Here's your fish, lady. Just caught, see? Look, breakfast!\" She looked at him as though he were crazy, then tilted up her head to receive his offering. After a few tries she got them down.\n\n\"I didn't know you kept birds in here,\" I said by way of polite conversation.\n\n\"This way I know coyotes won't get them. That's a chaparral bunting, and those are white flickers, and this is a pink-faced parrot-let, and that's an oak flycatcher, and that's a rufous-chinned sparrow. There's a little Neele's owl in here too, but he's hiding. All these guys will go extinct in the next fifty years. Except not really, because I've saved them.\" There was quiet pride in his voice. I remembered the first time I found and saved a rarity, how excited I'd been. _Ilex tormentosum_ , the last known specimen, growing in a garden in England.\n\nI put England out of my mind. \"So, you're shipping them off to the Company aviaries?\"\n\n\"When I've finished studying them.\" He nodded at his processing credenza, which I hadn't noticed because his guitar was leaning against it. \"Except for Erich, I guess. He's kind of bonded to me.\"\n\n\"Uh . . .that reminds me. Porfirio would like to speak with you.\"\n\n\"Oh,\" he said, and slunk out of the room. I followed, not that I wanted a seat at his dressing-down, but I really needed some coffee.\n\nThe coffee wasn't ready yet when I came to the cookfire where Juan was facing Porfirio, head lowered meekly. I kept walking past them, deciding to disappear into the oak trees for a while, but I could still hear their conversation.\n\n\"Okay, before you say anything, I just want you to know that I understand it was really irresponsible of me to go off and leave Erich like that, and I'm really sorry he screamed all the time I was gone, and I hope it didn't bother the other operatives too much, and I promise it'll never happen again,\" recited Juan Bautista. He drew a deep breath.\n\nPorfirio rubbed his unshaven chin. \"You're never going swimming again, huh?\"\n\n\"Well . . .\"\n\n\"Never going to leave the bird alone for the night while you go out anywhere? Never going to go on field trips without him? How many endangered songbirds are you likely to catch with a condor perching on your head, kid?\"\n\n\"Well . . .\"\n\n\"Sit down, Juanito. We need to discuss your problem. It's not really that big a problem, as they go, but you need to understand a few things about life in the field.\"\n\n\"I thought I was doing a good job,\" said Juan, sounding stricken.\n\n\"You are. But you're going to be real unhappy, soon, and that's bad. It's especially bad for an immortal. We immortals need to avoid unhappiness at all costs, and do you know why? Because it's the only thing that can hurt us. Nothing else can get inside us and screw us up, not germs, not bullets, not poison\u2014only unhappiness. And you can't do a good job when you're unhappy.\"\n\n\"Why am I going to be unhappy?\"\n\nShit, the kid was so young.\n\n\"Because you have a pet. We don't have pets, Juan. Pets require time we haven't got, because we're operatives and all our time belongs to the Company. Pets require special housing and stuff we can't give them, because we never know when the Company is going to transfer us somewhere at a moment's notice. Pets require constant attention and love, and we can't afford to love them, because they're mortal and they'll die, which will make us unhappy, which will interfere with our doing a good job for the Company.\n\n\"See? Now, once in a while you people whose specialty is living things\u2014ornithologists like you, or zoologists like Einar\u2014will feel friendly toward something you've rescued. That's okay. Having friends is okay. Friends come and go, and it doesn't hurt much. But there's a different relationship involved here.\"\n\n\"Your problem is that this bird isn't a canary in a cage. It's a big, intelligent animal, and unfortunately its instinct is for complex social relations. It has bonded with you. That's bad, Juan, because when the time comes for you to go your way on your next mission, what do you do with Erich? You can't turn him loose. He doesn't know how to hunt. He doesn't know to be afraid of people. Access the whole history of the attempts to save his kind from extinction. Look at the problems condors have.\"\n\n\"I know.\"\n\n\"The only thing you're going to be able to do is ship him off with the rest of your collected specimens, because that's all he is: a specimen. That's all he can be to you. But it's going to hurt both of you, because he hasn't been a specimen, he's been your pet, your baby. You see what I'm saying here?\"\n\n\"You're saying we shouldn't love the things we save from destruction.\" Juan's voice was muffled. He was crying. There was a long silence before Porfirio answered.\n\n\"Yeah,\" he said. \"That's it, pretty much.\"\n\n\"I think it's crappy!\"\n\n\"Yeah.\" Another long pause. \"But it's the way things are.\"\n\nA moment later, Porfirio continued: \"This is not to say you can't love anything. Stuff that lasts forever, like your work or literature or cinema, that's safe. Look how happy Einar is. And look at those friends of Mendoza's who stayed here, they were actually in love with each other! Mortals, though . . . you want to avoid that. That can seriously screw you up. That'll make your present heartache look like a picnic. You just ask\u2014\" His voice broke off abruptly. I clenched my fists.\n\n\"So what do you want me to do?\" Juan Bautista sniffed.\n\n\"No more pets. And start weaning yourself emotionally from the one you've got, all right? Better for you, better for him. Think how happy he'll be in the Company aviary. Once he has a mate, he'll forget all about you, and that's the way it should be.\"\n\n\"What about Marie?\"\n\n\"She's a specimen. Once her leg has healed, she's off to the Company.\"\n\nI put my face in my hands. Poor boy. I'd heard all this before, of course. Se\u00f1ors, when those clever twenty-fourth-century mortals devised us, they devised badly. Our fragile mortal bones are replaced with unbreakable ferroceramic; our weak mortal sinews are laced through with indestructible fiber, proof against any wrenching blow. Why not excise the wretched mortal heart too, give us a clean pump of steel, nothing that can weep at the appalling passage of the years? As I was weeping now, for the man I'd been unable to save.\n\nI felt a hand touch my shoulder. But when I turned, there was nothing there but California, all sagebrush and red sand. What, was he going to start tormenting me in broad daylight? What had set him off now, the fact that it was an Englishman buried on Dead Man's Island? Was I never going to be free of England and dead Englishmen?\n\nI ran down to the mouth of the canyon and out onto El Camino Real. I looked northward in longing for the forests of the coastal range. The road was free: it ran past me and kept going, skirting the dry hills, veering over to the coast, going on and on until it ended at what would one day be the Hyde Street Pier in San Francisco. If I stood there now, what cooler air I'd be breathing, what greener hills I'd see, and what a sound of the sea would be roaring in my ears!\n\nAs I stood there wishing myself anywhere but this flat-topped, sun-baked, bullet-ridden place, I picked up the approach of a mortal man, riding in from the city to the south. I scanned and dismissed him. It was the Yankee fellow who had come out to see Imarte, presumably back for another romantic interlude. To my dismay, he turned his horse's head in my direction and looked at me hopefully.\n\n\"Good morning, ma'am,\" he called out to me.\n\n\"Buenos d\u00edas, se\u00f1or,\" I said. Maybe if he thought I didn't speak English, he'd ride on.\n\n\"Ain't you the gal that works at the stagecoach stop, up there?\" He pointed. \"You know if Miss Marthy's at home to callers today?\"\n\n\"No, se\u00f1or, she is away,\" I said. \"She is expected to arrive on the next stage, however.\"\n\n\"I reckon I'll wait for her, then,\" he said, but to my annoyance he continued to sit there on his horse and stare at me. \"Uh . . .se\u00f1orita? You know Miss Marthy very well?\"\n\n\"Reasonably well, se\u00f1or.\" What was this about? Had he caught something? It couldn't be from Imarte, not with the arsenal of amplified antibodies and God knew what else we Immortals carried in our systems.\n\n\"Well, would you know . . .? I'd like to get her something, something grand like what she ain't never had, but I don't know the lady's taste. She's got all them books in her room, and it's clear to me she ain't the common trashy kind of girl, you see what I'm saying? I never seen a lady so cultivated and refined that was so beautiful too, all on top of earning her living as she does. You reckon she'd like a Shakespeare book?\" His eyes were big and pleading.\n\nOh, dear, the mortal fool was falling in love with Imarte. Twice as stupid as falling in love with an ordinary whore.\n\n\"I think, se\u00f1or, that her tastes do not run to literature,\" I said, choosing my words carefully. \"I think she likes history better. I know she enjoys the company of men who have lived interesting lives.\"\n\n\"Oh, I seen that right away,\" he said, leaning forward in the saddle. \"She thought no end of the stories I had to tell her about me tellin' off General Vallejo. Real interested in what I did in Nicaragua. Seems to me she don't come by her profession from natural inclination, wouldn't you think, se\u00f1orita?\"\n\n\"Probably not,\" I agreed, and even if I disagreed, he'd hear what he wanted to hear. \"But you know how difficult it is for a lady to make a living in these parts, se\u00f1or.\"\n\n\"Why, sure,\" he said. \"Poor beautiful thing, to be down on her luck the way she is. You can tell she weren't born to it. Why, by rights she ought to have servants waiting on her. . . . You know what she reminds me of? There was this storybook my Uncle Jack had when I was a kid. He used to read from it, all about this beautiful queen whose husband thought women were no good and he'd marry a new one every night and have her killed with a sword every morning. Only _this_ queen kept her wits about her and told him a great big old story every night, and he let her live till next evening so's he could hear how it come out, only she kept the story going so he never did hear the end, and after three years he decided the hell with it and kept her. That's kind of what it's like seeing Miss Marthy.\" He sighed. \"Except I'm the one doing all the talking.\"\n\n\"She enjoys tales of adventure, se\u00f1or, especially if they're true,\" I said, in no mood to have a mortal pour out his heart to me.\n\n\"I been thinking I could tell her about when I was up in Frisco last year,\" he said. \"Hell of a joke. Bunch of fellows up there had a plan going to sell out California to the secessionists. Young fellow went by the name of Asbury Harpending (and ain't that the worst silly-ass name you ever hear?) was trying to raise him an army. I nearly joined, too. Reckon Miss Marthy would like that?\"\n\n\"It sounds fascinating, se\u00f1or,\" I said. \"But, is that not the stagecoach? Perhaps even now your beloved is speeding like the wind to your embrace.\"\n\nHe gulped and turned in the saddle, and thank God it was true: here came the northbound stage, throwing out a wake of dust cloud like a malevolent djinn. The Scheherazade of Cahuenga Pass had returned.\n\n\"Thanks for the talk, se\u00f1orita,\" called the Yankee over his shoulder, and galloped away to be there when she dismounted. I hope she hadn't brought a customer home or there was likely to be lead flying under the oak trees.\n\nNo shots rang out, however, and after a while I sneaked back and managed to get my field kit and vanish into the bushes for a while, all by myself for a change.\n\nI had a productive day, without the distraction of conversation. I got a good sample of genetic material from a specimen of _Quercus morehus_ and, wandering into a neighboring canyon, I found a lone _Symphoricarpus mollis_ , which was on my priority (pharmaceuticals) list, as was _Ribes speciosum_ , which I also bagged\u2014several good specimens of that one. You never know when they're going to discover some miracle cure for something, up there in the twenty-fourth century. Three times now in my field career I've managed to win Favorable Mutation credits, earning me time out for private research. It occurred to me, that afternoon, that this might be a way to shake the dust of Los Diablos off my boots: find something so incredibly rare and useful that I'd win another holiday. I was in a great mood when I finally marched back to the inn that evening. Work always makes me happy. And I'd seen not one dour Protestant shade, not one reproachful phantom.\n\nPorfirio, though, was glowering by the cookfire as I approached.\n\n\"No escort all day,\" he said sternly.\n\n\"Well, who was here to go with me?\" I said. \"The others have their jobs too. I need to be out every day, Porfirio, I have work to do. I'll take a Navy revolver, I promise.\"\n\n\"You didn't today,\" he said. \"I'll issue you a holster. If you're going out alone, I want you to wear it at all times, understand?\"\n\n\"When in Rome, et cetera,\" I grumbled. \"Is this really necessary, though? Any trouble I ran into would be relayed simultaneously to the Company through the datafeed transmission. If I needed rescue\u2014which I wouldn't anyway\u2014Dr. Zeus would know immediately.\"\n\n\"Just because everything we see gets transmitted doesn't mean somebody's watching,\" Porfirio replied. \"You think the Company's got enough personnel to watch every one of us around the clock? It may be months before somebody starts interpreting your signal, if ever.\"\n\n\"Why aren't you this restrictive on Imarte?\" I complained, having no better argument.\n\n\"You think she doesn't carry a gun?\" he said. \"Beautiful little pearl-handled .22 in her reticule. Anyway her job is different, she's out in town most of the time, where she has some kind of law enforcement on her side if anything happens. If somebody tries to rape her or she goes to jail, no problem. She has a great time interviewing all the parties involved. You, you're out in the sagebrush, miles from where anybody can see. If you have a problem, you need to be able to solve it fast.\"\n\n\"Like Einar does.\" I shivered, remembering.\n\n\"Like Einar,\" Porfirio agreed. He looked as though he was about to say something else; but after an awkward moment he shrugged. \"So, go put your stuff away. Dinner'll be ready in about fifteen minutes.\"\n\nI went slowly, marveling at the gap between official policy and actual field procedure. We were always told that mortal life is incredibly precious and must never, ever be taken: they do enough of that themselves. We were warned that we would have to stand by and watch as mortals executed one another, or destroyed thousands of themselves in wars and riots. But _we_ never take part in their primitive justice, never help them along their Malthusian path. Of course, there have always been rumors that the Company allows the occasional removal of a vile and evil mortal for the greater good. Sometimes, it's whispered, even an innocent mortal may die, if it's necessary for the success of a mission. I've even heard that tacitly admitted. Security techs like Porfirio would be the ones to take care of any unofficial assassinations; they're programmed for strong-arm work, the way facilitators are programmed to be devious and amoral.\n\nBut I didn't know what to make of how things were done in Los Angeles. Granted, most of the people sniping at travelers from the chaparral were murderers, either crazy or cold-blooded, whom any jury would hang in a second, so it made sense to kill them quickly and efficiently. But it was the speed and efficiency that had me unsettled. Got a problem? Bang, it's solved. No argument, no strategy, no getting to know your enemy.\n\nI was still pondering this at dinner, over grilled beef and frijoles, and even Imarte's peculiar story didn't take my mind off it much.\n\n\"That man had the most _fascinating_ account of rebel organizers in San Francisco,\" she said when she emerged for supper after a lengthy afternoon siesta.\n\n\"No kidding,\" said Porfirio, stirring the frijoles.\n\n\"You wouldn't believe the absurd and quixotic plans Southern sympathizers are forming up in the city. Even in places like Visalia. There is far more anti-Union sentiment in this region than I ever supposed.\" She sat, drawing a shawl about her. \"This man was actually contacted by a cell trying to raise men for an army. Can you imagine what they could accomplish if they were able to recruit enough fighters? One of their plans was to intercept the gold shipments that provide Union funds for continuing the war, which I need hardly explain would cripple Lincoln's efforts.\"\n\nI was surprised. I hadn't paid enough attention to the war to be concerned with such goings-on, but it seemed like a respectable plan. \"Why didn't they go through with it?\"\n\n\"I think it's because the majority of the organizers are young and hotheaded. My informant intimated that there was some sort of dissension among the members of the organizing cabal.\" Imarte accepted a plate of supper and frowned judiciously at it. \"But the main reason, probably, was that the Comstock lode was discovered. Given the choice between resisting the tyrant's yoke and making a bundle, most of the nascent California Confederacy departed for Nevada.\"\n\n\"That figures,\" Porfirio said, and laughed. \"I remember when all the Anglos deserted Los Angeles back in 1849, when the Gold Rush happened. Suddenly you couldn't get china, you couldn't get shoes\u2014all the little stores were closed.\"\n\n\"Same thing in Monterey,\" I said, remembering the ghost town it had become. \"Everybody went for the gold. That's so stupid, though, for the rebels! Just like the Japanese never getting around to bombing San Diego or Mare Island next century, which would win the Pacific war for them, but they never do it. Because, think about it. What would happen if the Confederates decided to play Sir Francis Drake and go privateering along the coast?\"\n\n\"El Draque,\" said Porfirio, reminiscing as he rolled frijoles in a tortilla. \"I saw him, once, for about ten minutes. Short little bastard.\"\n\n\"As it happens, that was part of the plan,\" Imarte told me. \"The gold for the Union troops is shipped from San Francisco to Panama on the Pacific mail steamers. Seize those gold shipments, and you deal a serious blow to the Union. But apparently the logistical problems in obtaining a ship and ordnance for the enterprise were only just beginning to be dealt with when most of the conspirators left for Virginia City.\"\n\n\"This guy told all this top-secret stuff to a whore?\" Porfirio shook his head. \"No wonder the plot fell apart. Look at the blabbermouths they recruited.\"\n\n\"That must have been part of the problem,\" Imarte agreed. \"Though I gather Mr. Jackson felt it safe to discuss the matter with me because the rebellion is pretty much a dead issue. Apparently the young man who was in charge of his cell departed for Veracruz.\"\n\n\"What's he gonna do down there? Fight with the ladrones to keep Maximilian out?\" said Porfirio with disgust. \"Shit, that's all they need, another Nicaragua.\"\n\nImarte shrugged. I washed down a tortilla with some lukewarm coffee. Who cared what the mortal monkeys did, anyway? We knew how it would turn out in the end. The Union would be saved, and poor old Abraham Lincoln would pick a bad night to go to the theater. France would get Mexico but wouldn't be able to keep it. I had work to do at my credenza.\n\n\"When did you see Sir Francis Drake?\" Imarte asked Porfirio.\n\n\"Ha, gonna pick _my_ brains, now? You anthropologist. It was in 1579. I was out with my brother's son. The kid was supposed to be delivering some silver on pack llamas from Tarapaca to Morro Morena, and I went with him to be sure he made it. Well, he didn't. About halfway there, who do we run into but this little stubby Englishman with a red beard, and\u2014\"\n\n\"Wait a minute!\" I sat forward, staring. \"Your brother's son? What are you talking about?\"\n\n\"My brother's son,\" said Porfirio, looking down into his coffee as he stirred it. \"I have family.\"\n\n\"But _none_ of us have family,\" said Imarte. Even she was astonished. \"It's Company policy to recruit orphaned children.\"\n\n\"Yes, it is,\" he said. \"And I was orphaned. So was my brother . . . My mortal father was a soldier under Bernal Diaz. From Hispaniola. All those gentlemen adventurers decided they could make a fortune by going over to Mexico with Cort\u00e9s to get gold and estates for themselves. They did it, too, though most of them didn't live long enough to enjoy them. My father was awarded some relative of Moctezuma's as a bride. People gave the Spanish lots of girls, every one of them princesses, supposedly, hoping the tie of blood would make the new conquerors part of the family. I guess they kept some self-respect that way. You know: He's not a white god, he's just my son-in-law. Well, some minor king gave his daughter to Cortez, and Cortez already had all the wives he wanted, so he passed her on to one of his officers, who wasn't interested in women so _he_ passed her on to one of his subordinates, and to make a long story short, she wound up with my father, who kept her.\n\n\"I guess they loved each other. I remember her crying and crying over him when he died But before that happened they had children. Two, me and my little brother, Agustin. I was four. Agustin was only a baby. I don't know what happened and don't want to know. All I remember is fire and blood and my father dying and my mother weeping over him. She told me to wait, to take care of Agustin, and she dragged my father's body outside. She never came back.\n\n\"Then the lady came. She told me she was going to take me away somewhere safe, where I'd never die, that a kind doctor was going to take care of me from now on. You know, the story we all hear, when they find us. She tried to make me go with her and leave Agustin, but I wouldn't. I yelled and picked him up and wouldn't let go of him. In the end she had to take us both.\n\n\"Well, when we got to the nearest Company base, they had a problem on their hands, because here they were with two little mortal kids saved from certain death, but only one of them\u2014me\u2014fit the physical profile for the immortality process. What to do with baby Agustin?\n\n\"They were kind enough about it. They inoculated him against diseases, and they found foster parents for him, a good, loving couple who wanted a child, and one day, while I was undergoing the first cranial surgery for augmentation, they gave him away. I was so mad when I woke up and found out. What was Agustin going to do without me? I was supposed to look after him.\n\n\"But the nurse explained how they'd given him to these really nice people, and how he'd live a wonderful long life and never die of smallpox, and my poor mother and father would have wanted that, and didn't I want them to be happy up there in heaven? Anyway, what could I do about it?\n\n\"Well, I did something.\" He gazed into the fire. \"When I was grown, after I graduated, I accessed Company records and got the name of the couple who adopted Agustin, and I went and looked him up. He didn't remember me, of course, and I couldn't tell him who I was; but I struck up an acquaintance with him all the same. His adoptive father had land and money, and spoiled the kid. Agustin had everything, but he was beginning to get into trouble when I found him. You know the kind of trouble rich kids get into. They're bored. Life has no point. They don't love anybody, and they don't think anybody loves them. So it's cards, whores, drink, and raising hell. I beat some sense into him, and we became sworn friends after that.\n\n\"His foster father approved of me as a good influence. My brother and I went places, did things together, just as though I were a mortal. He fell in love with a girl and married her; I was best man. They had kids, and I was godfather to half of them. He worked hard, he made good investments. I had to leave now and then when the Company sent me places, but I always came back to check on Agustin.\n\n\"The business with Drake happened when Agustin had to pay some rents to the bishop at Morro Morena. He sent Dieguito\u2014his oldest boy\u2014with the rent in silver on a pack train of llamas. Thank God I went along. We were halfway there when out of nowhere we were surrounded by these lousy Englishmen, and I'm not using the term metaphorically. They were also armed to the teeth. They looked like they were starving. The leader was Francis Drake, a little short fat guy with a red beard. I wish I could tell you something memorable about him that would give you an invaluable insight into his character, but all he did was call us dogs and demand the llamas. He thought they were some kind of sheep. He was pretty happy when he found the silver.\n\n\"When his attention was distracted, I grabbed Dieguito, and we ran and got away. The English shot at us and called us filthy cowards\u2014why they were sore about us running, I can't guess. Dieguito was humiliated and angry. He wanted to go back there and throw rocks at them or something. I grabbed him and asked him what he thought his father would rather have, a living son or a son killed by a bunch of pirates? Any father would like to know his son died bravely; but he'd much rather hear that there were going to be lots of grandchildren.\n\n\"When we got back, Dieguito saw that I was right, because Agustin wept and embraced his son, and embraced me, and said he would have given the boy's weight in silver to get him safe home again. Agustin said it was a lucky day for him when I insulted him in that tavern. Drake sailed on up the coast and out of our lives, and the Company sent me away on a mission, and I was plenty busy for a while.\n\n\"But then, one time I came back after being away a couple of years . . .\"\n\n\"And Agustin had died?\" I guessed.\n\n\"No, though it felt like it. He'd aged. Begun to age, anyway. The first gray in his beard, his face sagging a little. And his kids were growing up. While I still looked twenty.\" Porfirio reached around for a branch and thrust it into the fire, opening up a red cavern of coals that breathed heat at us.\n\n\"So what did you do?\" Imarte said.\n\n\"Learned appliance makeup real fast. Pretended to get sick and hang by a thread for months. Agustin came to see me every day . . . and when I made my recovery, everyone said the illness had aged me, which was what they were supposed to say, so things were okay for a while. It was a pain getting made up every morning, but it was worth it to be able to keep an eye on Agustin.\n\n\"But time kept passing. You know what happened, eventually. Little Agustin the fat baby, with his dimple and his shock of hair, became Agustin the tremulous old skeleton, toothless, blind, unable to remember things. He'd had his years and years of long life, just as the nurse promised me; and this was what it had done to him. I sat by his bed, every day; I held his hand and listened to him mutter and twitch, or breathe with a sound like clothes being dragged over a washboard. I myself was so weighed down with the crap I wore to make me look old, I could barely move.\n\n\"I went weeping to his funeral\u2014so many great-grandchildren there!\u2014and Dieguito, old Diego now, comforted me by telling me his father would see me in paradise. I couldn't tell him there was just one little problem with that: you have to die to go to heaven, and I couldn't.\n\n\"I pretended to, of course. I made up a wax dummy with my horrible old features, then had a hot shower and stripped away all the appliances, all the latex and paint and white hair, and paid a servant to see to my funeral. I walked out of my own house young and free, and I got on a horse and rode north.\"\n\nOut in the night a coyote howled, mocking.\n\n\"But I wasn't free,\" Porfirio said with a sigh. \"Who was going to look after the great-grandchildren? I stayed away ten years. The Company sent me to Nicaragua, to Chile, to Mexico, to Texas. I did a lot of good work and had some free time, so I went back to visit Agustin's grave. They'd buried my wax dummy beside him, wasn't that nice of them? And Dieguito, who would have remembered me best, was blind now. Cataracts. His kids had no sense, they'd led soft lives and were letting the estates go to hell. _Their_ kids were wild and living like Indians. Somebody had to take a hand. So I followed the oldest of the young boys around, watching him, and one night he left a cantina drunk and was set upon by thieves. I killed them and brought him home.\"\n\nJust like that, he killed them. Well, he was a security tech.\n\n\"The family\u2014none of them knew who I was, I looked twenty-five at most\u2014welcomed me, thanked me, gave me a job as majordomo. I held it for a while, long enough to set things to rights again. Dieguito died, and the baby I'd held in a baptismal robe I saw as an ancient creature lying gaunt in his coffin. It didn't matter. The son had a son, and I was such a member of the family by that time that I was the godfather, and I held the little fat brown boy while the old priest anointed him and named him Agustin.\n\n\"It's gone on like that, you see? For centuries now, and the Company has been very understanding. I have a big family, and they need me. Their fortunes have changed\u2014our estates were lost after the Grito\u2014but the family has survived. There was nothing I could do about Agustin's dying, but his blood still runs in his descendants. I stay with them awhile, I watch them get old and pretend to get old for a while myself; then I ride away and stay away until they need me again. One night a stranger will come; and if any of the old people think he looks like Uncle Porfirio, who used to teach them how to ride, well, it can't be more than coincidence, can it? Because Uncle Porfirio would of course be a very, very old man now, if anybody knew where he was.\"\n\nAfter a long silence he shrugged. \"I'll have to be more careful, now that photography has been invented,\" he concluded.\n\nImarte was sitting with stars in her eyes. \"That is so _beautiful_! What a unique chance you've been given! Think what a cultural thesis you could make of it, three centuries of history as experienced by one family!\"\n\n\"You think so?\" He looked sidelong at her. \"How'd you like to have that responsibility? I'm never free. Three hundred years, and I'm still obeying my mortal mother's last request.\"\n\nI thought privately that he'd been too hard on poor Juan Bautista about the birds. It's all very well to break the news to a young operative that love is a mistake, that attachments can't be formed because of what we are. But Porfirio had found a way around that, hadn't he? For him there was always a home fire burning somewhere, no matter how far he wandered on the dark plain, while the rest of us made do with ashes and ghosts.\n\nSomeone was standing on the other side of the fire, looking at me. The others didn't see him. I refused to lift my eyes.\n\n\"So, where are they now?\" I asked. \"Your family.\"\n\nPorfirio shifted, uncomfortable. \"Most of the direct line are working on a ranch in Durango. One of the girls married a man with some property, and all the brothers have moved in to work for them. They're doing all right, I guess. I haven't been down that way in ten years. I'll need to wait a few more years before I can go back there again.\"\n\nI put his story out of my mind as I went to my room and set up my credenza for work, and I kept it out of my mind while I processed my specimens. In the end I had to shut it off and go to bed, though, and the second my head hit the pillow, the question leaped out at me like a thief from ambush: What had become of _my_ family?\n\nLong dead, their remains probably stacked in a charnel house beside some village church in Galicia. Had there been descendants? I'd had lots of brothers and sisters, so perhaps there were some distant relatives running around somewhere. There might be some woman even at this moment with my face, my hair, buying onions in the marketplace in Orense or Santiago de Compostela.\n\nWhen I finally fell asleep, I had the nightmare again, the old nightmare that I always forget until I'm actually inside it once more, where I'm in my parents' house in the middle of the night. It's dead-black night, but the moon shines like an arc lamp, and I can see them all lying together in the one big bed. There is my skinny father and my ever-pregnant mother, and there are all the little children I used to fight with so bitterly for our shared toys, or a scrap of food, or our mother's attention. I know all their names, but I always forget them when I wake up.\n\nMy family is asleep, as silent as though they were underwater, and nothing will wake them. I'm the only one awake. I try my best, but I can't get anyone to wake up and be company for me. The moonlight is so white, the night is so still. I wander around the room disconsolately, but they never wake up to notice I'm there. They will sleep forever. Only I am awake; only I can never sleep.\n\nThis time, I couldn't bear it and ran outside into the moonlight. It was a mistake. Apple trees stretched in every direction, white with blossom, and the air was full of perfume. He was standing there under the trees, tall in his black robe, waiting for me. As I halted and stared, he extended his arm in its long sleeve, that graceful gesture that was one of the first things I'd ever noticed and loved in him. Inviting me, beckoning me, summoning me.\n\nI struggled upright on my narrow cot, gasping like a fish out of water, soaked with chill sweat, and for one terrifying moment I thought the spectral moon was shining in here too, because it seemed to me there was a flash of eerie blue that faded and flickered away. I sagged against the wall and wept, not bothering to wipe away my tears. Here came the footsteps again, Porfirio running out to see what the disturbance was; but he stopped, and after a long while turned and went away without speaking.\n\nI lay down again, shaking, pulled up the blanket, and curled on my side. I was so cold.\n\n###\n\n\"I WISH WE HAD SOME TOAST,\" Oscar was complaining at breakfast. \"Oatmeal. Soft-boiled eggs. Real food.\"\n\n\"This isn't a civilized country yet, remember?\" Porfirio said to him with a grin. \"You'll get your oatmeal eventually.\"\n\n\"Anything would be preferable to this monotony of leftovers.\" Oscar rested his chin on his fist, staring glumly at the beef in the skillet.\n\n\"You want to talk to Dr. Zeus about allocating me a bigger budget to run this place?\" Porfirio flipped the steaks adroitly. \"So, hombre, how's that pie safe? Got a buyer for it yet? Should I start peeling those parsnips?\"\n\nOscar pursed his lips. \"I feel lucky today,\" he said. \"Mendoza, will you come with me and bear witness?\"\n\n\"To what?\" I looked up groggily from my coffee. I hadn't been sleeping well lately, to put it mildly.\n\n\"I want someone present who can testify to my triumph.\"\n\n\"Oh. Actually I was going out into the temperate belt today, Oscar.\"\n\n\"We can go that way. There are houses out there. Why, I haven't even visited that area yet. Those people are probably desperate for good-quality merchandise at affordable prices.\" His eyes grew wide and reverent.\n\n\"Okay,\" I said, getting up to fetch my field gear. It'd save me a long walk; why not? When I came out, he'd already hitched up Amelia and was pacing back and forth, energized.\n\n\"Your chariot awaits, ma'am.\" He bowed me to my seat. \"Where shall we go?\"\n\n\"Take Franklin to Hollywood to Sunset,\" I said. We had all adopted Einar's use of future street names, and that was the route that followed the foothills through the temperate belt. He gee-hawed to Amelia, and away we rolled.\n\nThis was a much prettier drive than the road that cut across the plain, with inviting green canyons that opened up to the north; unfortunately it was also a lot more dangerous, as bullets sang out of the thickets at regular intervals. We dodged them and shot back if they seemed too persistent; I scarcely wasted a thought on it now. I was able to get good specimens of _Vitis girdiana_ near the future intersection of Laurel Canyon and Sunset Boulevard; I found an interesting mutation of _Chrysothamnus_ , with possible commercially valuable properties, at Sunset and Queens Road. Oscar bore with my frequent stops patiently, but kept his eyes trained on a thin column of smoke that rose ahead.\n\nWhen we finally came around a foothill and saw its source, he sighed in disappointment. The house was old, built of tules in the local Indian style, and in fact there was an Indian lady in the yard, standing on a rock to load acorns into a kind of basketwork silo. If not for the fact that she was wearing European clothing, we might be back in pre-Columbian days. She turned to stare at us as we pulled up before her yard.\n\n\"Good morning, se\u00f1ora,\" I called to her in Spanish.\n\n\"And to you,\" she said, getting down and wiping her hands on her apron.\n\n\"Look at that!\" gasped Oscar. I thought he was enchanted by the primitiveness of it all, but it was the silo that had his attention. He was out of the wagon and into the yard much too fast for the dogs who lived there, for they surrounded him in a snarling mob before he could reach the lady.\n\n\"Please excuse him, se\u00f1ora, he means no harm,\" I said from my seat in the cart. She nodded and called the dogs off. Oscar had his hat in his hands at once.\n\n\"My apologies, a thousand pardons, se\u00f1ora, but I couldn't help seeing that you are in dire need of superior food-storage facilities!\"\n\nShe just nodded and looked at him. Probably she was deciding that the white man was up to no good if he apologized to her this abjectly, but her face was blank, her expression mild.\n\nOscar gestured with his hat at the acorn silo. \"This structure, se\u00f1ora, it's very ingenious and well made, but it's nothing more than _natural materials_. Are you not at the mercy of the ground squirrel, the raccoon, the scrub jay, and a host of other pests? Do they not voraciously deplete your larder?\"\n\n\"Sometimes,\" she said.\n\n\"Well, allow me, se\u00f1ora, to offer you a solution to these depredations, a way of ensuring that your hours of backbreaking labor gathering the fruits of Jove's tree are not for naught!\" He bowed her toward the cart. She went with him, placidly folding her hands, no doubt wondering who Jove was. The dogs snarled and followed, but kept their menace low-key.\n\n\"What you need,\" Oscar said, unfastening the side of the cart, \"is a modern, sanitary method of preserving food. Now I think, se\u00f1ora, I _think_ you'll agree that what I have here just fits the bill. Behold!\" He flung back the side, displaying the pie safe gleaming among his wares like the central diamond in a crown. \"The Criterion Patented Brass-bound Pie Safe!\"\n\nHer face remained perfectly still, but a light flickered in her eyes. Then they grew a little bleak.\n\n\"It's very beautiful, se\u00f1or,\" she said.\n\n\"Oscar\u2014\" I said. He ignored me.\n\n\"Regard the metal fittings. This is a first-rate device guaranteed to be impervious to pests, whether of the gnawing, crawling, or pecking variety. No less than eight separate compartments for the storage of your acorns and, er, whatever other fine foods you wish to keep pure, fresh, and unsullied. Now, I've a talent, if I may say so, se\u00f1ora, for supplying needs, and I can see plainly that _you need this_. It may have been designed for other forms of edible goods, but such is the versatility of its design that it will admirably preserve foodstuffs from any ethnic cuisine whatsoever.\"\n\n\"I'm sure it would, se\u00f1or.\"\n\n\"Oscar\u2014\"\n\n\"Now, it may be,\" he said, raising a hand, \"that you've put aside hard-won savings, in the anticipation that Necessity may call at your, uh, door. I might suggest that you could make no better investment against Want than this splendid item, which will safely reserve your stores against all possible losses. Ordinarily the Criterion Patented Brassbound Pie Safe is sold for no less than thirty dollars; but for you, se\u00f1ora, in your most obvious need, I will offer it for the special low price of _ten_ dollars. Only say the word, se\u00f1ora, and you need never fear the loss of your acorns again.\"\n\nI buried my face in my hands. She was taking them all in in a long bitter stare, all those pretty and improbable things she'd never thought of having and shouldn't have thought of having, because she'd never have them.\n\n\"It is certainly a beautiful thing, se\u00f1or,\" she said meekly. \"I am afraid, though, that I have no money.\"\n\nOscar gaped. \"Well\u2014why, I'll tell you what, then. You can pay for it on the hire-purchase plan! Twenty easy installments of fifty cents each, how about that? With the first payment deferred six months. You can't afford not to take advantage of this once-only offer.\"\n\n\"Yes, I can,\" she said. \"I have no money at all, se\u00f1or. My husband works on the big rancho, and the man who owns it lets us stay here in return. We never have any money.\"\n\n\"But\u2014but my good woman, how do you live?\" he said.\n\nShe waved a hand at the acorns, at the venison jerky drying on the fence, at the neatly woven baskets of pinole meal.\n\n\"Come on, Oscar,\" I said.\n\n\"Uh, well. If you ever _should_ obtain hard currency, I'm sure a thrifty housewife such as yourself will invest it in the wisest possible way,\" Oscar gabbled. \"And may I present you with a complimentary volume of the poetry of Percy Bysshe Shelley? I trust you'll remember me, se\u00f1ora, when you require the finest in home furnishings. Good day.\"\n\n\"Thank you. Good day,\" she said, staring at the little book in its bright pasteboard binding. Oscar leaped into the seat, and we rocketed off.\n\n\" _Shelley_!\" I asked.\n\n\"That was my low point, my absolute nadir,\" Oscar groaned. \"Dear Lord, what possessed me? It was her need, you know, her utter need. It seduced me. I must supply where I see demand. It's a compulsion. Other operatives would be content with simply gathering valuable anthropological data or ferreting out hitherto unrecorded ethnographical statistics. I must be _more_. I must be the genuine article. That's my problem: my standards are too high.\"\n\n\"Well, it's not like you're a failure,\" I said helpfully. \"You're doing great work for the Company.\"\n\n\"For Dr. Zeus, I'll grant you. But what about the worthy gentlemen at the Acme and Criterion Companies? Mere mortal merchants, say you; yet I believe in complete commitment, absolute fulfillment of all responsibilities, be they ever so trivial.\" He shook the reins with noble determination. \"Giddap, Amelia! To the next customer.\"\n\nBut worse was yet to come.\n\nFarther down the road, in a green clearing beside a still-bubbling spring, we saw a fine adobe and garden. The walls were freshly plastered, the window frames painted, and a tall paling fence warned trespassers away from the yard, where cabbages were growing in precise lines and peach and plum trees stood to attention.\n\n\"Now, look at that,\" said Oscar, laying down the reins in admiration. \"Look at the industry and thrift evident in that pleasant scene. Surely this is the residence of a wage-earning individual. And his spouse. Prudent housewifery is in every line of that garden plot. I can taste that New England boiled dinner now.\"\n\n\"Those are awfully big dogs,\" I observed. They sat alert, one on either side of the door, watching us silently. I hadn't the slightest doubt that if Oscar so much as put his foot inside the gate, they'd tear it off.\n\n\"Hem, you're right. Well, let's not repeat my previous error.\" Oscar got down and went around to the back, where he drew out a pan and a long wooden spoon. He commenced to beat out a brisk tattoo on the pan, looking hopefully at the house. The dogs pricked up their ears but made no move.\n\n\"Good day! Hello there! Is there anyone at home?\" he called. The door opened, and a woman looked out.\n\n_Whoops_. I transmitted to Oscar. _She's an Anglo. Off-limits for your pie safe_.\n\nHe faltered only a moment in his disappointment. \"Well, good morning, there, ma'am!\" he said in English. \"I wonder if you'd be interested in any of the superior merchandise I have to offer?\"\n\n\" _Nein_ ,\" she said, and he shot me a look of triumph.\n\n\"You are German, madam?\" he said in a close approximation of her regional accent. \"From Bavaria, yes?\"\n\n\"You, too?\" Wonderingly she emerged from the house and came a little way toward him. \"In this foreign land?\"\n\n\"Many years now, but I assure you it is so. How pleasant it is to hear a cultivated voice again! Come now, my dear, I have many things here that you may need, though you may never have considered that in such a lawless and unimproved country they could be obtained. Come, see what I have to offer to you.\" Oscar put his hand on the latch, and the two dogs instantly sprang to their feet, growling. She shushed them and came a little closer, peering at us. He might have been a countryman, but he was still a peddler.\n\n\"Have you the polish with which to clean silver?\" she said.\n\n\"Yes, _nat\u00fcrlich_! And I have additionally stove blacking, laundry bluing, wash powders, and these very fine clothespins that have a patent pending for the superior spring mechanism that they employ. Consider, here, the little figures of china bisque, very sweet, the little doves billing and cooing and the little shepherd boy playing love songs with his flute. And this pan for the baking of cakes, with the hearts printed in the bottom so as to make the design upon the finished cake, wouldn't you like to have this?\"\n\n\"No,\" she said. \"Just the polish for silver, thank you.\"\n\n\"Ah, but, my dear! Here is your silver polish, to be sure, but behold! Printed music for performance on the piano, the spinet, or the organ. And confections also, barley-sugar sticks in the flavors of apple, blackcurrant, or strawberry. And see what fine things I have for sewing.\"\n\n\"Thank you, no. How much for the silver polish?\"\n\n\"Five cents American.\"\n\nShe raised her eyebrows slightly but fished in her apron pocket and paid him. He handed her the silver polish, and she turned to go. He nearly made a desperate lunge over the fence, which the dogs were only too happy to have him do.\n\n\"But, dear lady!\" he screamed at her back. \"See, here, this thing which you will find is an absolute necessity in this wild and dirty country. It is the Pie Safe Patented Criterion Brassbound.\" He flung wide the panel, revealing it in all its glory. \"It keeps the bread loaves and the rolls from going stale. It keeps the mice, the rats, the insects from invading the pastries. You of all people would want such furniture for the kitchen that you bake in.\"\n\nThe woman turned and followed his gesture with her eyes. For a moment they were warm and approving. \"Ah, yes,\" she agreed, nodding her head. \"I do not know what I would do without the one I have.\"\n\n\"You have such a one?\" Oscar asked, going pale.\n\n\" _Nat\u00fcrlich_ , there in my kitchen inside the house. But mine is bigger than that, and bound not in brass but in nickel that is plated with silver. And it has not pineapples upon it but the design of pheasants.\" She looked closer, critically. \"Also, yours does not have the egg timer or the barometer built into the cabinet, as mine has.\"\n\nImmortal or not, I thought he'd keel over dead right there on the spot. She realized she'd dealt him some sort of near-fatal blow, though, because she hastened with a kindly word: \"All the same, it is a very good pie safe, and you will certainly sell it to somebody. I have no need of it, however. Good day, my dear sir.\"\n\nWell, I couldn't laugh, he looked so stricken when he crawled up on the seat beside me. We drove away in silence. About halfway back to La Nopalera, he drew a deep breath and said, \"I'd be obliged to you, Mendoza, if you wouldn't mention this mortifying occasion to the others.\"\n\n\"Don't worry,\" I said. I had no wish to gloat. The day's outing had been successful for me; I had got a couple of good specimens not only on the drive out but also on the way back, I scored a previously unclassified member of the _Celastraceae_ , some exotic low-elevation form of _Euonymus_ by the look of it. Happy me.\n\n###\n\nI COULDN'T GET THOSE green canyons out of my mind. Accessing topographical data, I decided that Laurel Canyon, with its drastic range in elevations, had the best chance of mutation-yielding microclimates and diversified habitats. I was intrigued, also, by the blue-hazard notations on every reference to the area I encountered.\n\n\"I thought I'd stroll over to Laurel Canyon today,\" I said, one morning at breakfast, casually.\n\nPorfirio choked on his coffee and glared at me.\n\n\"Are you nuts?\" he said. \"That's a blue-hazard precinct, dummy.\"\n\n\"So my files tell me, but I've never encountered one before. It's just a kind of energy sink, right? A locus of natural unnaturalness in the landscape?\" I was a little taken aback by his reaction.\n\n\"You could say that,\" he growled, mopping spilled coffee from his chin with one hand. \"It's just the biggest damn one on the continent, that's all.\"\n\n\"Oh,\" I said. \"Does that mean I can't go there?\"\n\n\"Not alone, you can't, you of all people, and not without the right field gear. What the hell do you want there, anyway?\"\n\n\"Well . . . it has all those steep isolated canyons and drastic heights and depths. There are probably a lot of rare endemic species of plants growing there. I'd be stupid not to look for them. And what do you mean, me of all people?\"\n\nHe looked over uneasily at Einar, who was grooming Marcus, and at Juan Bautista, who was watching him. \"Okay,\" he said in a lower voice, not answering my question. \"I guess there's stuff back up in there worth collecting at that. But you're going to take the following precautions, understand? Now listen carefully . . .\"\n\nTwo hours later:\n\n\"God, I feel stupid,\" I moaned to Einar as we approached the canyon on horseback. \"What if mortals see us?\"\n\n\"We shoot 'em,\" he said glumly. I hoped he was joking. We were wearing absurd-looking helmets with Crome filter lenses and a lot of other cunning little mechanisms built into them with no consideration for style or convenience. We wore gauntlets full of wiring and large, ugly, and ill-fitting boots with circuitry patterns on the outside. Things like Batman's utility belt were cinched about our waists. To make matters worse, we were tethered each to the other by a long silver line. If one of our horses startled and bolted, somebody would be dragged.\n\n\"We look like extras in a cheap science-fiction film,\" I complained.\n\n\"In a damned expensive science-fiction film,\" Einar retorted. \"You know how much it cost to make this stuff, here in 1862? And these are the only sets of this gear in the continental U.S. at this time. They were made just so we could go into Laurel Canyon, if we had to. So enjoy the fantasy. Tell yourself we're explorers on a forbidden planet or something.\"\n\n\"It can't be _that_ weird, no matter what Porfirio says,\" I muttered. But as we came to the entrance of the canyon, I fell silent.\n\nI saw a narrow passage between soaring walls of granite, thinly grown with whatever little plants could cling to their vertical surfaces. The way in followed a creek bed through which water was still cascading down. From the wreckage of broken trees and from the high-water mark on the cliff faces, you knew that this was no place to stand during the winter floods. Water must come thundering down that channel like cannon fire. A dramatic scene, with the leaning dead cottonwoods and the majestic atmosphere, the mountains impossibly high on either side. A little trail led into the canyon, a sandy embankment on the left-hand side above the water, and disappeared into dark trees.\n\nEinar unslung his shotgun and cocked it. Cautiously we rode in.\n\n\"Now, remember,\" said Einar, \"don't scan. Every conditioned reflex and instinct you've got is telling you to, but don't. Let the helmet do it for you. If you try, yourself, you're going to pick up data you won't believe.\"\n\n\"This is nuts,\" I said, as my horse picked its way timidly. \"How are mortals going to live here? But they are, aren't they? And this is right in the heart of Hollywood.\"\n\n\"I know,\" he said. \"They just . . .become part of the strangeness. Raymond Chandler wrote about it in his Philip Marlowe stories, but he didn't tell half of what he knew. There'll be a murder that happens right up _there\"_ \u2014he pointed up a nearly vertical slope\u2014\"that he writes about in _The Big Sleep_. But it doesn't happen the way he says, it never makes the papers, and it's never solved, either. The guy isn't a pornography dealer, he's a high-ranking member of a hermetic brotherhood. There's a brilliant flash and a scream, all right, and a naked girl and some ancient earrings with a curse on them. The curse doesn't make it into the book, but a lot of the other details do.\"\n\n\"How lurid,\" I said, and then started, because I heard a sound I shouldn't have heard in that place, not for another half century at least. I turned my head to stare down the trail behind us. I knew that sound from cinema: the rattle of an internal combustion engine, the rush of displaced air as something sped toward us, but I had no visual input at all. Forgetting myself, I scanned, and _knew_ there was something approaching. In my desperation I yanked up the Crome screen visor so I could see with my eyes.\n\n\"Mistake,\" Einar gasped. He was right. Without my visor the place lit up, every tree, rock, shrub, and blade of grass outlined in blue neon. The automobile was lit up like that too, a 1913 Avions Voisin, a lovely, elegant thing except for being glowing blue, slightly transparent, and a little out of place in 1862. Einar leaned over and got a firm grip on my horse's reins, or I'd have been away from there in an instant. The car zoomed up, till I was right between its bug-eyed headlights, and I got a clear glimpse of the hood ornament in the shape of a rearing cobra.\n\nWith a crackle of static the car whooshed through me and on, up the canyon. My mouth was open. Einar managed to reach out and click my visor down. Visual references were once again normal.\n\n\"Told you not to scan,\" he said reprovingly.\n\n\"Was that a ghost?\" I asked at last. At least it hadn't been a sixteenth-century Protestant martyr.\n\n\"Or a temporal anomaly, or a hallucination, or\u2014anything. Keep the visor down, don't scan, and start looking for rare plants _now_. The sooner you get what you're after, the sooner we get out of here.\"\n\n\"Okay,\" I said, and we rode on.\n\nThe road continued up, clinging to the hillside above the creek bed, and I saw condors wheeling in the sky and deer leaping away from us, and they were really there. I saw a coyote loping along in broad daylight across the canyon, and he was really there too. It was the sounds that dismayed. There were gunshots (which may or may not have been real), and there were more whizzing automobiles, and there was singing and chanting. How was I supposed to look for rare endemic plants with all that going on?\n\n\"All right,\" I said at last. \"I give up. What's the deal here? And does it get any better, do we ride out of it?\"\n\n\"No, it gets worse,\" Einar said. \"We're still on the edge. We haven't come to Lookout Mountain Drive yet.\"\n\nI frowned. \"Lookout Mountain Drive? That sounds familiar.\"\n\n\"Here's the explanation as I understand it: weird geology around here. Look at all this decomposing granite. It's quartz-bearing, crystals all through the rock if you know where to look. There's another outcropping of high quartz concentration over in what'll be Griffith Park, and that place has some bizarre happenings of its own, but this place is the mother lode.\" Einar turned uneasily in his saddle, starting to scan, and stopped himself with visible effort. Something enormous went snorting and blowing past us, and we heard small trees snapping and big trees being pushed aside.\n\n\"I knew it would be bad, but not like this. The local Indians say the canyon was cut through by the God Himself, that He was chasing Coyote who'd stolen the moon, and Coyote had burrowed down under these mountains with it, and God grabbed the mountaintops and ripped them apart, made this long fissure, to get to where the moon was hidden. It was down so deep, though, that God gave up and threw a buffalo gourd up in the sky to be the moon instead, and the old moon's still buried here somewhere.\n\n\"An interesting story, in light of the fact that the biggest damn quartz deposit in the known universe lies about a thousand feet below the intersection of Laurel Canyon Drive and Lookout Mountain Drive. We know it's there; the Company did radar imaging of the whole area, once they noticed this place is so full of Crome's radiation, it can wipe out our sensory displays. Apparently there's a single crystal down there the size of\u2014of that thing that just went stomping by us. And that's just some of what they found. Nothing supernatural or extraterrestrial, you understand, just the biggest naturally occurring Crome spectral sponge in the world. As near as Dr. Zeus can figure out, it stores Crome's radiation generated by anything passing over it that has a nervous system. Animals, Indians, and bandits is about all there is now; but as people begin moving in here, there will be a lot more energy absorbed.\n\n\"And every so often it discharges, and then all hell breaks loose, and that's what we're seeing now,\" Einar concluded.\n\n\"It figures, that this is supposed to be Coyote's fault,\" I said disgustedly. I saw a bush I didn't recognize and stopped. Handing the reins to Einar, I slid down and bent to examine it. Yes, it was really there. \"But why are we seeing it now? There's nobody living up here yet to generate all this stuff,\" I argued, groping for my collecting gear. Damn this visor anyway. How was I supposed to take readings? \"What's causing the blue lights and the auditory illusions?\"\n\n\"It could be picking up Crome from us. One theory is that every time it discharges, it sets up a shock wave that puts stress on the temporal field, and the whole fabric of time ripples. My guess is, they're afraid we might slip through a hole or something, and that's why the connecting cable. If one of us goes through, the other can pull him out.\" Einar lifted his gun to his shoulder involuntarily, then lowered it. \"We're also recording and broadcasting data to Central HQ, I'm sure. So, is that a rare plant?\"\n\n\"Looks like a mutation of _Myrica californica,\"_ I said in satisfaction. \"Jackpot, on the first try. Not that I can tell much about it in all this armor.\"\n\n\"Cut a couple of branches and let's go, okay?\"\n\n\"Are you scared?\" I looked up at him. My own fear had evaporated the second the bush read positive for mutation. What else might I find up here?\n\n\"Me? Hell no, I'd love nothing better than to get myself sucked through a temporal rift into Jurassic times or something.\" Einar turned his head, sweeping the area visually. \"Let's move on. Want to see where the Haunted Tavern will be? It's just up ahead.\"\n\n\"Neat,\" I replied, tucking the specimens into my collecting bag and climbing into the saddle again. We rode on, though unseen sirens screamed past us.\n\nA few hundred feet ahead, the gorge widened to a sort of clearing, with another steep canyon opening to the left. Immense sycamores darkened the way ahead of us. We rode to the center of the clearing. For the moment, all we heard were natural sounds, the creek bubbling to our right, the wind in the leaves.\n\n\"Nice spot, isn't it?\" said Einar, actually relaxing a little. \"Someone will build a brick commercial building here. The top part will be the Canyon Store, eventually, and a lot of famous rock stars will buy their groceries here. Around the corner and downstairs, though\"\u2014he pointed over his horse's tail\u2014\"is where the Haunted Tavern will be. The earliest record of it is as a speakeasy, then later it's a cozy bar. Nice little watering hole for thirsty movie people going home to Encino after a day at RKO, or going home to Beverly Hills after a day at Universal or Republic. They'll shortcut through Laurel Canyon with a stop right here. It's private, it's unfashionable, it's hard for wives or reporters or detectives to find them in the little dark basement bar.\" He leaned forward in the saddle, and his crazy smile looked crazier than usual under the visor of the absurd helmet.\n\n\"Now, you've heard the story about John Barrymore's wake? How, after he dies, a bunch of his drinking buddies steal his body from the funeral parlor and take it with them for one last night on the town? He's supposed to be carried all over Hollywood during this one long wild night before his funeral. That much is legendary.\n\n\"What didn't make it into the legend was how the party finished up. This part of the story is supposed to have been told to a male nurse at the hospital where W. C. Fields died. According to the nurse, the last stop on the route was this basement bar. The place was locked for the night, but Gene Fowler picked the lock, and they got in. Laid out cash on the bar and mixed themselves drinks. At this point, Barrymore is supposed to have sat up in his coffin and demanded to know where _his_ drink was. They were in a pretty philosophical mood by that time, so they just poured him a martini. I don't know what's supposed to have happened after that, but people say . . .\"\n\n\"What?\" I shivered, hearing an invisible glass break.\n\n\"Well, the place operates on and off after that. Sometimes it's a bar, sometimes it's a restaurant, sometimes it closes down for years at a time. But whenever it's a bar, sooner or later, _John Barrymore comes in for a drink_. And once he's seen, the new owners know they might just as well give notice to the Realtor and move on; because before long the place is packed after hours with the thirsty dead, W. C. Fields and Victor McLaglen and Erroll Flynn, to name but a few. They drink the booze, they break the glasses, they have loud conversations in the parking lot at four A.M. Local residents get mad as hell and call the cops. What can the cops do? No use for the bar's owners to swear that they close at midnight, either. The place gets shut down again, until the next hapless guy comes along and decides it's a great place to open a cozy little bar.\"\n\n_Smash!_ This time it was a bottle breaking, and my horse shied and started forward.\n\n\"Let's go on,\" I said.\n\nWe rode up the canyon, with me stopping at frequent intervals to investigate the truly amazing number of unusual endemics that began to occur as we drew closer to Lookout Mountain Drive. I found a purple form of _Marah macrocarpus_ , a variant of _Lonicera subspicata_ with scarlet flowers, and the strangest-looking _Baccharis glutinosa_ I'd ever encountered. I was so absorbed in my discoveries that I paid little attention to the trumpeting mammoths, trolley cars, and bagpipers that passed me on the way. Einar had to reach into his saddlebags, though, where he'd stashed a whole box of Theobromos, the Mexican kind that comes in six round cakes in an octagonal box. He just kept unwrapping them and wolfing them down, working his way through them as though they were rice cakes. As a result, he was mellower than usual by the time we got to Lookout Mountain Drive, and swaying gently in the saddle.\n\n\"An' here it is,\" he said, swinging an arm in a half circle. \"The strange heart of all strangeness anyplace. Look normal to you? Ah, but we know better.\"\n\nActually it didn't look normal to me, even through the visor, though it was just a sunny clearing opening up to the sky. There was something a little skewed in the perspective. I couldn't quite tell how wide the clearing was.\n\n\"So this is the mystery spot?\" I tried to sound nonchalant.\n\n\"Yes, but not some lousy little carney mystery spot with trick angles and optical illusions,\" said Einar. \"If you took that visor off, this place would light up like Times Square. All kinds of unusual mortals will be drawn here. One guy, he'll be a naval officer and engineer with a wife who's a practicing witch. They'll settle here for a while, and he'll actually write an expos\u00e9 of the place. Then he'll think things over and rewrite it as a piece of science fiction. Did you ever hear of Dr. Montgomery Sherrinford?\"\n\nI did a fast access on the name. All that turned up was a footnote in a biography of Aleister Crowley and a half paragraph in a work on theosophy.\n\n\"Some kind of Freemason?\" I asked.\n\n\"Starts out as one. They eject him. He becomes a Rosicrucian, and they throw him out too. Same thing with the Order of the Golden Dawn. Madame Blavatsky gives him the heave-ho, and so do those people up at Summerland. He travels all over the place before he finds Lookout Mountain Drive, but _then_ , man, his luck changes. He builds a kind of temple right there\"\u2014Einar pointed to our left\u2014\"and he builds a mansion up there on the hill. In later years people will call it Houdini's mansion, but no\u2014it was built by Dr. Sherrinford. He _knows_ the vibrations on this spot are like nothing else anywhere. Gets himself a cult going, becomes its high priest, lots of spooky goings-on all around this very intersection. A lot of famous players and their bosses too, from the early days of movies, come to 'services' here. Not because of the guy's charisma, but because he has real powers\u2014also he knows where a lot of bodies are buried, metaphorically and otherwise.\"\n\n\"I never heard about any of this,\" I said doubtfully.\n\n\"You have to know where to look, but the references are there. In this correspondence, in that library collection. Anyway, Dr. Sherrinford has this big thing going. Silent-film stars at his beck and call. Blackmail money. Devoted followers who set about the task he sets them, which seems to be tunneling into Laurel Canyon. He gets them looking for the source of the power. He's never too clear on what it is, but he's got a good idea _where_ it is. He'll completely undermine his temple with tunnels. And it seems he gets all the way down to the big crystal, too, because all these rumors start flying around of an incredible discovery and miracles and contact with the other world.\n\n\"Now, _we_ know you can't do anything with Crome's radiation, not anything useful anyway. Dr. Sherrinford won't know that. He'll think it'll give him power. He builds an underground chapel a thousand feet below, reached by a little elevator going down a shaft, and he takes his disciples down there, and they hold chanting services.\n\n\"You know what happens when you have a lot of people generating a Crome-effect field around a spectral sponge. Just like electricity, it absorbs the charge, absorbs the charge, until finally it discharges, boom! And all hell breaks loose, ripples in the temporal field, ghosts, visions, apports, you name it. It happens, or seems to.\n\n\"That's what Dr. Sherrinford's disciples keep praying for. It's like a bunch of monkeys in a cage, hitting the same lever over and over again until the bell rings and a bucket of monkey treats are dumped on them. Or it's like somebody with a cup of nickels playing a slot machine. Mostly Dr. Sherrinford's crystal spirit ignores its worshipers, but every so often it blasts them with a miracle and a lot of stimulation of the hippocampal region of their brains.\n\n\"Harry Houdini is in California making a movie for Lasky, it's called _Terror Island_ , and he's doing location shooting over on Catalina. And this guy comes up to him in the restaurant at the Hotel Metropole and tells him about the goings-on at Dr. Sherrinford's temple and asks if he wants to come attend a service there. It's implied that the dead speak at these parties.\n\n\"Well, that's just the right thing to say to Houdini\u2014or the wrong thing, depending on your point of view. His mother died a few years previous, and he has this strong attachment to her, kind of a Freudian thing, you know? He's been positive she'll send him a message from the beyond, only years have gone by now and she hasn't, though he's paid plenty of bucks to spiritualists and mediums. Not that they don't claim she's trying to talk to him, but the messages are all obvious fakes.\n\n\"Finally he decides it's all a lot of bull and becomes an obsessive debunker. Inside, though, _he wants to believe_. So he flies back from Catalina with this guy, and they go to Laurel Canyon. They meet Dr. Sherrinford. What happens next?\n\n\"Nobody ever gives complete testimony; but again, if you know where to look, you can piece it together, what happens down in the crystal chamber. Houdini goes there, all right, with Dr. Sherrinford and his disciples, and they hold a big ceremony with chanting and pull out all the stops, they're levitating and speaking in tongues. It sets off the crystal. And when it discharges, apparently a _lot_ of dead guys want to speak with Houdini. Only problem is, some of them are ordinary dead and some of them claim to be dead guys from the future, from a time when there isn't time. And they aren't all agreed on the usual spirit patter about being happy on the other side and telling everyone that it's all niceness and harmony over there. In fact, they start to fight.\n\n\"They fight so violently, they cause something like a small earthquake. The tunnel collapses, or becomes blocked. What happens to Dr. Sherrinford and his gang is never known, but Houdini manages to escape by getting to the jammed elevator and worming his way through an air vent, after which he climbs the cable. He takes off down the canyon into the night.\n\n\"Why doesn't he call the cops? Why doesn't he make any attempt to rescue Dr. Sherrinford's people? We'll never know. Whatever took place, it scares Houdini half to death. He opens negotiations to buy that property, though, which shows that something about it has a hold on him. History's a little vague about whether he ever closes the deal, but there's an account that he's visited in his hotel room by one of the disciples, who may or may not be a survivor of the crystal chamber. Houdini leaves on the train for New York the same day, and never stays in Los Angeles again. In fact he leaves America a month later and goes on an extended tour abroad. Spends the rest of his life trying to prove that there _is_ no other side and that the dead can't talk.\"\n\n\"So there'll be an earthquake,\" I said, shrugging. \"It buries the cultists, and Houdini gets out because of his special abilities. Afterward somebody blackmails him about running off and leaving the others to die. Creepy, but not inexplicable.\"\n\n\"I'd agree,\" Einar said, grinning at me, \"except for the fact that in 2072 a bunch of people in white robes come wading out of the surf off Bermuda claiming to be Montgomery Sherrinford and his disciples. They disappear into an asylum and out of the historical record pretty fast, though.\"\n\n\"Hoaxers,\" I said firmly, shaking my head. I caught sight of a weird-looking thing poking through the dead leaves at the base of a _Rhus laurina_. A kind of fungus? Root parasite? Um-shaped flowers bright green\u2014some member of the heath family? At _this_ elevation? And what were those curling scales? I accessed at top speed and found not a single identification. If it were possible for my immortal heart to skip a beat, it would have done so then. Visions of Favorable Mutation bonuses for _Sarcodes mendozai_ danced before my eyes.\n\n\"I'm going to dismount now,\" I informed Einar. \"I think there's something really remarkable here.\"\n\n\"You can say that again,\" said Einar, taking firm hold of the cord that connected us. He watched as I slid down\u2014and I could feel something through the soles of my boots, a pulsing. There was a cracking sound and a flash of bluest light. I grabbed at my saddle in panic, but that was no good either, the horse just swayed away from me. Einar seemed a long distance off. And where was the damned plant?\n\nI looked around, expecting to see a landscape festooned with melting clocks. What I saw was no less strange. Immediately beside me, and stretching up and down the canyon, was a long river of gleaming rounded things, inching slowly forward in a stinking miasma of chemical smoke like so many mechanical turtles. They weren't blue. They weren't transparent. Behind smoked glass, open-mouthed faces stared at me.\n\n\"Einar.\" I felt blindly for the cord. \"There are unborn here.\"\n\n\"I know.\" His voice was faint. I looked over at him, and he was staring at me in horror, just as solidly there as I was, as the vehicles were. \"So much for the cord, huh? I guess it just pulled me after you.\"\n\n\"These are late-twentieth-century automobiles.\"\n\n\"Yes.\"\n\nI found myself hyperventilating. It was a mistake among the gasoline fumes. I looked around wildly. Everywhere were strange Japanese names in chrome. To hell with _Sarcodes mendozai_. I scrambled back up on my horse. The cars didn't go away. People were pointing at us.\n\n\"They're pointing at us,\" I said. \"What do we do, for God's sake? Will they shoot at us, will they start rioting?\"\n\n\"Let's stay calm,\" Einar said, though his knuckles were white where he gripped the reins. \"Let's ride straight up that ridge and out of here. Maybe we'll, like, snap back to 1862 painlessly. Okay?\"\n\nWe didn't. From the ridge above, we looked out on a clutter of houses. And what was the matter with the distant plain? Had the sea level risen? No; that roaring, gleaming gridwork was a city, stretching away to the horizon. We stared at it in shock. It was a city in the future, which I had always thought would be beautiful, but it wasn't beautiful at all. The air was brown. The sunlight looked red through it, sunset color at midday.\n\n\"Now . . . this _isn't_ possible,\" I said to Einar. \"Right? You can't go forward past your own time. Everybody knows that.\"\n\nHe nodded mutely.\n\n\"So it's just an illusion. No transcendence field, no stasis gas, so we can't even have gone back in time, let alone forward.\"\n\nHe nodded again. \"Except,\" he said reluctantly, \"that there are some recorded instances of people doing it. People in old-fashioned clothes suddenly turning up in future traffic and getting hit by cars, for instance.\"\n\n\"And of course those have to be unsubstantiated fictions, because it can't really happen. Dr. Zeus says so.\" I drew in a lungful of the acrid air and coughed. \"Is this _smog_? How can people live like this?\"\n\n\"Come on.\" Einar turned his horse's head. \"There's supposed to be a relay station in Hollywood in this time period. Jumping ahead through time may be impossible for Dr. Zeus, but it obviously ain't impossible for Lookout Mountain Drive.\"\n\nWe rode down the ridge, accessing coordinates as we went. Immediately to the south of us rose a gated wall topped by what would have been Ionic columns if they were ten times bigger. As it was, they looked cheesy. Beyond them was a sprawling community of flat-topped mansions in ghastly mid-twentieth-century style, roofs covered with white gravel, walls faced with Hawaiian lava, everything deliberately off-center and out of balance, from the winding front walks to the kidney-shaped pools in each backyard. Black letters on the wall told us this was MOUNT OLYMPUS.\n\nWe found our way in, past the corner of Vulcan Drive, and proceeded down Olympus Drive. \"Oh, they _didn't_.\" I said. \"Did they?\"\n\nEinar smiled wryly and pointed at a little cul-de-sac with the grand name of Zeus Drive. The architecture there was just as crappy, even on the house that was obviously our relay station, the one with the Dr. Zeus logo picked out in green pebbles on the red-pebble tessellation of the front walk. There wasn't a living soul in sight, which was a good thing, under the circumstances.\n\nWe rode up the concrete driveway and sat there looking uncertainly at the very ugly house. There were no stables anywhere that I could see, though there was a sensor array on the roof, disguised to look like satellite television equipment.\n\n_Uh . . .Operatives Einar and Mendoza reporting. Please provide codes_ , transmitted Einar. As we watched, the broad garage door swung open of its own accord to reveal a couple of frightened-looking techs in coveralls.\n\n\"Come inside. Hurry!\" hissed one of them. We urged our horses forward, and as soon as we were inside, the garage door clanked down after us. Our horses, who had been patient and unflappable until then, spooked; I just scrambled off mine and dodged its panicky dance, but Einar controlled his with an iron hand and caught my mount's reins. The techs cowered in a corner.\n\n\"Whoa, whoa,\" said Einar. He dismounted and caught his horse's head, stroking its muzzle and looking into its eyes. \"You don't want to step on anybody, do you? Of course you don't. That's a good boy. And how about you, sweetie, you want to calm down too? Pretty girl. Good girl.\"\n\n\"You're talking to the _horses,\"_ said one of the techs in disbelief.\n\n\"Yeah,\" he said, and I don't think he noticed that they found it distasteful. I noticed, though. I knew who they were at once: operatives from the future, doing time on a mission in what was _their_ past, though their past was more than a hundred years ahead of us. I've worked with that kind before. They're stuck-up.\n\n\"I guess you're wondering what we're doing here,\" Einar said, but the tech who had more buttons on his coveralls than the other snapped:\n\n\"No, we're not. There was a distortion in the temporal field. We monitored it. You're an anomaly. You'll have to report immediately to the temporal transference chamber for return to your point of origin. This way!\" He pointed, and the other one opened a door in the wall beside what appeared to be late-twentieth-century laundry appliances.\n\n\"What about the horses?\" I said.\n\nThe tech got a look of horror on his face. He wrung his hands. At last he replied, \"They'll have to come too. Just hurry!\"\n\nThey scuttled ahead of us through the door. Einar and I exchanged bemused glances, but led our respective mounts through the doorway.\n\nHorses look _big_ in houses. Ours clopped after us through what was apparently a kitchen, across a smooth substance that must have been linoleum tile\u2014I didn't think it looked all that bad\u2014but the sound of their hooves was muffled when we emerged into a much larger room, because it was carpeted in synthetic fiber of a sickly beige color.\n\nI was disoriented at once. The floor was on a couple of different levels, for no reason I could understand, and one whole wall slanted inward. Across the room was another rise to the floor, and there was a fireplace with no mantelpiece, only flagstones set _around_ it, flush into the wall instead of on the floor, where flagstones belonged. In front of the hearth, where the flagstones should have been, the beige carpet went right up to the andirons. Stranger still, the entire west wall of the room was made of glass: four enormous panes gave a breathtaking view of the roofs of other houses across the canyon but robbed the room of any warmth or privacy.\n\nThe horses were disoriented too. Einar's gave its opinion of the decor with one good lift of its tail.\n\nThe tech leader jumped three feet, clapping his hands over his nose and mouth. \"Aaagh! You _filthy_ \u2014oh, you _filthy_ \u2014\"\n\nSomeone in the room burst out laughing. An immortal rose from a shapeless piece of furniture and surveyed the mess. She wore the unisex clothing of that era, ordinary jeans like Einar's and a plain white cotton shirt. She turned to us.\n\n\"You must be the two unfortunates who surfed the temporal wave. How do you feel?\" she said pleasantly, as though what had happened was no big deal.\n\n\"There is excrement on the floor. The carpet will have to be disinfected,\" said the tech.\n\n\"Oh, shut up and go get a shovel. If we've got one around here. You'd have to be Zoologist Einar and, let's see, Botanist Mendoza, wasn't it? Maire, regional facilitator.\" The immortal shook our hands, for some reason pausing with me, but only a fraction of a second. \"Don't worry. We can deal with the Lookout Mountain anomaly a lot better in this era; we've been studying it from this station for years now. You'll be all right. I'd love to let you stay a couple of days and show you around, but\u2014\"\n\n\"Those _things_ have got to get out of here,\" said the tech. \"What if one of them does number one?\"\n\n\"\u2014as you see, we have a crisis on our hands,\" Maire said, indicating the tech with a grimace. \"So if you don't mind, we'll just whisk you back to 1962. Sorry we couldn't be more hospitable.\" With a wave she directed us into a space beyond the room\u2014there wasn't exactly a doorway, and I couldn't quite figure the geometry of the house, but it seemed to be an entry hall. One wall was covered with beige drapery. She pulled a cord, and the draperies drew to one side, revealing not a window but a glass chamber, smooth and featureless inside. It looked like one of those streamlined aluminum travel trailers mortals will tow around behind their automobiles in that century, but it seemed to have been built into the house.\n\nMaire pressed a series of buttons, and a door slid open. She gestured for us to enter. The horses didn't want to go, but Einar managed to sweet-talk them over the silver threshold. As he was doing so, I heard the roar of an automobile outside approaching the house, stopping abruptly as it pulled into the driveway. Once the horses were in with us, the door slid shut, and we felt the cabin pressurizing.\n\n\"Nobody seems very surprised to see us,\" I murmured to Einar.\n\n\"Well, but they had to have known we were coming. The Temporal Concordance would have told them.\" Einar soothed and stroked my mount, which was more panicked than his.\n\n\"But if what we just did is impossible, you'd think they'd want to study us\u2014\" I broke off as Maire stepped forward to the control console.\n\nShe tapped in a combination on the buttons and stood back, looking at us through the window. She took a small cylinder from her pocket and spoke into it. Her voice came hollowly from the air above our heads. \"It's warming up. Transcendence in approximately thirty seconds. Brace yourselves.\"\n\nYellow vapor began to curl into the air from no source I could see. I inhaled deeply; stasis gas is a lot easier on the lungs than smog. Beyond our window, the door opened, and a man entered the house, setting down a leather case and tossing his keys on the hall table. He was an immortal, in an expensive-looking twentieth-century suit that must have been tailored but somehow seemed too big on him. He looked familiar. As he turned his head to stare at us, I recognized him: Lewis, with whom I'd been stationed at New World One for years and years. Strange, I thought, two old friends in a matter of months. Lewis had been the last person I hugged back in 1700, when I said good-bye at the transport lounge before I came to California. Hadn't he been going to England? But that was three hundred years ago by this time. I smiled and waved. I could see Maire smiling too, pointing at us and explaining something to him, but of course I couldn't hear behind the glass.\n\nHe didn't hear her either, I don't think. He was staring at me with the most incredulous expression on his face, which had gone perfectly white. Suddenly he flung himself at the window, pounding on it, shouting silently at me. What on earth? And why the urgency? The air in the room grew suddenly icy, the yellow gas boiled around me and obscured his desperate face. I was growing numb with the transcendence, but I managed to read his lips.\n\n_Mendoza, for God's sake! Don't go with him!_\n\nI turned my head slowly to Einar. He was looking confused. He met my eyes and shrugged. I looked back at Lewis, shaking my head and holding out my hands. Were those tears in his eyes? He was mouthing _no_ over and over, both palms pressed flat against the glass as if trying to push it in. The yellow gas was almost opaque now. I reached out my hand in slow motion and set it on the window, palm to palm with him through the glass, though I could no longer see his face. Then the transcendence came, and it was a lovely thing, pleasurable even with the feeling of infinite violence being done to one, as if you were picked up and thrown into the void forever, or flying . . .\n\nThen there was a strong wind blowing the fog away, and I stumbled and fell to my knees, choking. I was groping in red sand, trying to rise in a thicket of sagebrush and spurge laurel. There were the horses, flailing and struggling, and there was Einar, doubled up on hands and knees, gasping out yellow smoke.\n\nI scanned blearily. No houses, no deadly city on the plain. We were on the ridge above Laurel Canyon, in the same space we'd occupied 134 years earlier. Later? Whatever. I sank down as Einar was doing and panted, clearing my lungs. Neither one of us said anything for a few minutes. Even the horses gave up and lay still while their mortal nervous systems recovered.\n\nFinally Einar turned over and sat up, resting his head in his hands.\n\n\"What was up with that guy?\" he asked. \"Why didn't he want you to go with me?\"\n\n\"No idea.\" I shook my head. \"We used to be good friends, back before I came to California. I haven't seen him in ages.\"\n\nI couldn't remember ever having seen Lewis that upset, even when his lady friends dumped him, as sometimes happened. He was one of those genuinely nice guys who somehow always wind up alone. I was always alone, myself. It had been what made us friends.\n\nEinar and I both shrugged.\n\nOnce the horses were able to get up, we left that place, walking and leading them, because the trail down from the ridge was steep and tricky. Halfway back, my horse began to cough and shudder, then abruptly fell. Blood gushed from its wide nostrils as it gave one last convulsive twitch. The stress of the time journey, I guess. Einar sank down on his knees and cried.\n\nIt was hours before we stumbled into our own canyon, to the welcome smell of beef being grilled over a mesquite fire. There was blessed silence: no cars, no phantoms, only the oak trees and chaparral and one or two stars winking in the twilight sky. Porfirio was crouched over the fire, turning the steaks. He looked up as we approached.\n\n\"There you are. What happened to the mare?\"\n\n\"She had an accident,\" said Einar sullenly, and led his mount away to the stable.\n\nPorfirio winced, and I thought he was recalculating his operating budget. He looked at me. \"And you? Any problems?\"\n\n\"Not really,\" I said. Well, we'd come home in one piece, hadn't we? I sank down beside the fire.\n\nPorfirio still stood, considering me for a long moment. \"Mendoza,\" he said, \"I'm a security tech. I can tell when people are lying.\"\n\nI glared at him, se\u00f1ors, feeling very Spanish. How dare he say I was lying? Even if I was. \"All right, something happened,\" I admitted, pulling off the stupid high-tech armor that hadn't worked. He swore. I snapped at him, \"I don't know why you bothered to make us carry all this garbage. I suppose you had some communiqu\u00e9 from the Company about what was going to happen today? Did everybody but _me_ know? One of those rules you're not supposed to break, about telling people what's in their future?\"\n\n\"Something like that,\" Porfirio said.\n\n\"Why did you ask me, then, if you knew?\"\n\n\"Because they didn't tell me much,\" he said bitterly. \"They never do. Never enough to be of any use.\"\n\n\"That figures.\" I sighed and slumped forward. I was so tired. I was about to tell him about the accident when he took my breath away by asking:\n\n\"How long have you been a Crome generator?\"\n\nI began to shake. \"I'm not! There was just one time, when I was young\u2014only that once. My case officer thought\u2014he said it was probably nothing. Never since then, I swear!\"\n\n\"Mendoza,\" he said, \"since you've been here, not one week has gone by that there hasn't been an incident. I've looked out and seen the blue light pouring through the cracks in the boards, as if you had lightning in there with you. You didn't know? You slept through it every time? What's been happening to you?\"\n\nI shook my head. How could I tell him, when I didn't know myself? Bad dreams? I debated telling him about my dead lover who had risen from his grave to follow me across three centuries, an ocean, and a continent to make my life intolerable in this already intolerable place. What I said instead was, \"I appear to be malfunctioning.\"\n\nWe regarded each other in silence.\n\n\"Are you going to ship me out and send for a replacement botanist?\" I asked. That was according to regulations. Ironic, isn't it, seniors? I was holding my breath, petrified at the thought that my field career was over. If only he _had_ shipped me out.\n\nPorfirio shook his head grimly. \"I don't do that to my people. You've done a lot of good work, Mendoza. If you throw enough Crome to read by, so what? It doesn't seem to be hurting anybody but you. I know you have some bad memories, but you never let them interfere with your work. Look . . .the rules are different down here. Don't give me a reason to have you replaced, and I won't. Okay? But don't ever lie to me, because I'll know. So what happened today?\"\n\nI told him, as he took the steaks off the fire with greatest care and arranged them on an iron platter. He listened without a word, going about the business of setting out the evening meal as though I were telling him the plot of a film I'd seen. At last I finished, and he handed me a plate of supper and sat down across from me as I ate.\n\n\"Mendoza,\" he said finally, \"watch your back.\"\n\nThat was all he'd say on the subject.\n\nObviously he had an idea of what was coming. And on that day in 1996, Lewis _knew_ , se\u00f1ors, what would happen, knew that I'd sit before you in this place now, telling you this story. He was trying to warn me. It was kind of him, though it did no good in the end, and I hope it didn't get him into trouble. This just proves once again the only unbreakable law I know: that history cannot be changed.\n\n###\n\nTHE SUMMER WORE ON; it grew hotter and browner and dustier, and then in the evenings the wind began to change. Big purple rifts of fog would come blowing in from the coast. In the brown canyons the big leaves of sycamores began to drift down, smelling spicy and sweet when one crunched through them. Deer began to descend from the brown hills, looking around hopefully for garden produce, which they didn't find; but we did get venison for a change. The moon got very big, very silver, and the coyotes rejoiced.\n\nPorfirio began to stock up our supplies for the winter. Not that there was ever any snow, and we knew, as the mortals didn't, that this year there wouldn't even be winter rains to flood out the roads. But it was a safe bet that some disaster or other was going to strike, this being southern California, so it was just as well to be prepared.\n\nSo Einar was sent out time and again, to Los Angeles with lowing longhorns for Dr. Zeus, and he came back with wagonload after wagonload of crates, barrels, and sacks. Porfirio and Einar would haul them into the storeroom, where Juan Bautista and I (and Erich and Marie) would uncrate stuff and check it against the order list. Dozens of sacks of pink beans, dozens more of masa, enough coffee beans to wake the dead, jars of pickles and preserves, cones of brown sugar, boxes of salt . . . and seven cases of canned sardines.\n\n\"Jesus, why'd he order all the sardines?\" I said, staring astonished into an opened crate. \"They're not even on his list. Hey, Porfirio?\" Juan Bautista got a funny look on his face and put up his hands to shush me, but Porfirio had already backed in, carrying one end of a barrel.\n\n\"What?\" he grunted, backing in the rest of the way so Einar could ease his end down.\n\n\"Never mind, I figured it out,\" I said, but he turned frowning and noticed the funny-looking cans in the opened crate.\n\n\"What the hell are those?\" He picked up one of the cans. \"These are sardines! I didn't order these. None of you guys even like them.\"\n\n\"I like them,\" said Juan Bautista in a doomed voice.\n\nThere was a frozen moment, which unfortunately was broken by Marie Dressier limping across the room and looking up at Juan Bautista expectantly.\n\nPorfirio blew his top, and then blew it louder when he found out that that opened case was one of seven, but I really thought the roof was going to come off the adobe when he discovered that Juan Bautista had added them to the station's order list, which meant that they'd been paid for out of the station's operating budget.\n\n\"Didn't I order the damn bird a sack of pelican chow?\" shouted Porfirio.\n\nJuan Bautista hung his head. \"She's too old for it. It makes her droppings runny.\"\n\nI left then and at some speed, not wishing to be there when Porfirio came down off the ceiling. Einar was already gone.\n\nI was up in my favorite retreat by the creek, gloomily contemplating my future, when Juan Bautista came wandering along the creek bed half an hour later. Erich was perching on him and Marie was cradled in his arms. He was sniffling a little.\n\nI cleared my throat, so he wouldn't think he was alone and go into some soliloquy of teenage despair.\n\n\"Oh,\" he said. \"Hi.\" He came over and sat down beside me. I edged away slightly, not caring to be that close to Marie's beak. She had a wicked kind of hook on the end of it, like a mandarin's thumbnail.\n\n\"Is Porfirio through screaming?\"\n\n\"I guess so,\" he said. \"I guess I shouldn't have ordered seven cases. And it was wrong not to tell him about adding to his order. But what am I supposed to do? She's old. She's an endangered species. Fish is what she's supposed to eat. We can't take any more out of the creek, or the breeding population will go below sustainable levels. _He_ ought to try that lousy pelican chow, see how _he_ likes it.\"\n\nThis implied that Juan Bautista had sampled it himself, which I didn't want to think about. \"Well, don't worry,\" I said. \"If Imarte peddles her papayas vigorously enough, she'll earn back the budget deficit.\" But he wasn't amused, he was sunk in the self-righteous bitterness that only the very young can feel.\n\n\"Darned grandfather,\" he muttered.\n\nWe sat there a moment in silence. \"What,\" I asked cautiously, \"was that supposed to mean?\"\n\n\"It was my grandfather got me into this.\"\n\n\"You mean, a real grandfather? Your mortal father's father?\"\n\nHe nodded. After a moment, he drew a deep breath and began.\n\n\"We lived on one of the islands. I don't even know which one, San Miguel or Santa Rosa. All our people had left to go live at the mission, but Grandfather took us back\u2014my father and mother, I mean. He wouldn't leave his holy place. He was the . . . I guess he was the priest. The word for it sounded like _sishwin_. Anyhow, his god told him he wasn't supposed to leave the island, so he had to go back. That meant my father had to go back, too, because he was supposed to be the sishwin after Grandfather died, and my mother went too, because she was going to have me. They went by canoe. My mother was sick the whole way. I remember she used to talk about it.\n\n\"She and my father had a lot of fights. I was born over there, and there weren't any medicine women to help when I was born, just my father, and she was always talking about that. She wanted to go back to the mission, she didn't mind being a Christian, and she talked about that a lot too. And she was always afraid I would fall off a cliff into the sea.\n\n\"My father didn't want to be there either. When they were speaking to each other, he'd tell her how mad he was at my grandfather, how our old god was fake nowadays and Grandfather was just being stubborn about staying on the island. I remember he said he could never be a sishwin, because even if the old god cared, Grandfather would never think my father was good enough.\n\n\"But they never said anything when Grandfather came into the house. They were scared of him. He was scary-looking. He looked at Daddy like he was dirt, and at Mommy the same way.\n\n\"He liked me. He used to take me out to his holy place. There was a big wooden statue of the god there with the sun and the moon on his head. There were big black ravens, and I guess they were bigger than the ones here on the mainland, because I've never seen them that big since I came back. Grandfather used to show me how to feed them. Some were tame and would hop on my hand and let me scratch their necks. Some could talk. I used to think that was magic, because back then I didn't know how smart the _corvidae_ are.\n\n\"Grandfather told me a lot of stuff about how I belonged to his god, and how if Mommy and Daddy were bad weak people, _I_ wasn't, and I was going to be a powerful sishwin just like him, and someday our god was going to send the bears to get all the bad weak people who stopped paying attention to him.\n\n\"I just kept quiet and played with the ravens. I liked to stroke their feathers. They were so shiny black, they were blue and reflected the sky. I thought they were the prettiest things.\n\n\"Then something bad happened. I don't know what it was. Daddy and Mommy were yelling at each other, and I sneaked away to play with the ravens. There was one that liked to have his neck scratched, behind the feathers, and he just wanted you to go on and on and he'd close his eyes like saying, I can't stand this, it's so good.\n\n\"Grandfather came and got me and put me in the canoe. He said a lot of scary things. We went across the sea, and sometimes he'd stand up in the canoe and shout at the sky. I curled up in the bottom of the canoe and closed my eyes.\n\n\"We came to the mainland, and Grandfather hid the canoe in a cave. We sneaked across the hills, so the soldiers wouldn't see us, and we came to the big mission. It was the biggest place I'd ever seen. There was a Christian priest sitting on the steps. I don't know why he was sitting out there in the middle of the night.\"\n\n\"I do,\" I said tensely. \"Short man, was he, in a brown robe? Stocky? Little black eyes?\"\n\n\"Uh-huh. Grandfather carried me up to him and said, Here, you take children, take this one. And he turned and walked away. He left me there. I never saw him again.\" Juan Bautista's eyes were red, but he didn't start blubbering, thank God.\n\n\"Well, the Christian was really surprised. He sat up and asked me what had happened. I told him everything I knew, which wasn't much. He was nice. He asked me a lot of questions and told me everything would be all right now. We went into the big kitchen in the dark, but the Christian could see in the dark, and he got me some food. Then he took me to his room and put me in his bed and told me to sleep. I asked him where he was going to sleep, and he said he didn't sleep.\"\n\n\"He doesn't,\" I said. \"Not much, anyway.\"\n\n\"He hid me in his room a couple of days. He shaved my head, because of bugs, he said. He measured me and looked in my eyes. He let me play with a glass. I'd never seen glass before. It broke, and I cut my finger, and he took some of the blood and put it in something that was probably a machine, but it didn't look like one.\"\n\n\"And let me guess,\" I said, clenching my fists. \"He sat you down and gave you a talk about how sad it was to get old and crazy like your grandfather, but _you_ didn't ever have to get old or die.\"\n\n\"That's right.\"\n\n\"And I'll bet he told you that it was sad you'd lost your family, but you could have a wonderful new family who would help you become smart and live forever.\"\n\nJuan Bautista looked at me. \"I guess that's what we say to all the kids we rescue, huh? The next day, he brought me out and told the other Christians that I was a little orphan who'd been left at the mission, but that he'd discovered I had family at a rancheria up the coast, so he was going to take me there, because the mission couldn't provide for orphans anymore.\n\n\"We walked and walked, and after a couple of days we came to a big hill above the sea. You could see the islands from there. We stopped and built a fire and waited until dark. In the middle of the night the ship came, a big silver ship. It scared me half to death, but the Christian explained what it was.\n\n\"It landed, and the door opened, and nice people came out and took me inside. I was happy, for the first time I could remember except when I was with the ravens. Nobody ever fought or yelled. There was lots of food. And when they found out I liked birds, they made me an ornithologist.\" Juan Bautista sighed. \"So I guess I'm better off now. I really shouldn't blame my grandfather. Because if he hadn't taken me to the Christian, I'd probably be dead or a slave. And it isn't Porfirio's fault that Marie is old and can't eat the pelican chow.\"\n\n\"No, it isn't.\" I gazed into the brown water of the creek.\n\n\"You want to go eat some sardines, old lady?\" said Juan Bautista, burying his face in the spiky feathers on Marie's neck. \"They're good for you. We'll go get some treats, and I'll play for you, how about that? I'm learning Nunuz's _Sinfonia Asturias_. She really likes the slow movement.\"\n\nErich von Stroheim reached down and groomed behind Juan's ears, tugging a long tail of silver hair and pulling it slowly through his beak until it stood out at an angle. He cocked his head, studying the results.\n\n## PART TWO\n\n###\n\n\"THIS IS THE BIG ONE!\" Einar said, leaping down from the wagon.\n\n\"What big one?\" Porfirio asked uneasily, looking around at the hills as though he expected them to rock and roll.\n\n\"The epic. David Wark Griffith's _Intolerance,\"_ Einar flourished a big silver film can at us. \"It just came in. Polish up your rhinestones and press those tuxedos, ladies and gentlemen, 'cause we're going to Babylon tonight!\"\n\nThere's nothing like a sense of occasion to lift your spirits. Imarte was delighted when she heard what was on the evening's bill, and prepared a special treat; in addition to our popcorn and non-bathtub gin we had little rose-flavored chunks of gummy candy, prepared (she assured us) exactly as it was served at Belshazzar's feast. She would know, I guess. Nor was that all; she and Einar made a last-minute trip over to Sherman and managed to get roses from somebody's garden, big trailing fronds of yellow rambler, and spent most of the afternoon picking off thorns and weaving them into serviceable crowns. We settled down amid the cushions wearing chaplets of roses, and a big yellow rose waved above Einar's right ear as he stood up and pretended to talk into a microphone.\n\n\"And welcome once again, my fellow immortals, to this evening's edition of the Cahuenga Pass Film Festival. Tonight's offering is maybe the quintessential Hollywood film, the first cinema epic, and has been hailed as one of the greatest films ever made _and_ one of the worst. How did the inimitable D. W. manage to grab the brass ring while simultaneously falling off the painted pony and landing on his head?\n\n\"Budget and bad timing, folks, combined with the same wholesome na\u00efvet\u00e9 that left him astonished when black audiences failed to enjoy his film glorifying the Ku Klux Klan. Nobody could ever say Griffith was a slow learner, though, and so for his next film he singled out a slightly safer group to pick on: prissy old ladies of both genders. You may not agree with his unique insights on psychology or his scholarly footnotes as the evening progresses, but I can promise you this much: the visuals are killer.\n\n\"Now, this print will not be accompanied by the original score, but we are fortunate enough to have in our audience an expert who will provide us with fascinating commentary and insights of her own.\" He bowed at Imarte, and we all applauded politely. She waved a gracious hand. \"On matters Babylonian, Persian, and prostitutional we defer to thee, O scarlet one. So, everybody, breathe a silent prayer to Ishtar, the goddess at Heaven's Gate, and hold on to your cushions, because it's gonna be a truly bumpy ride!\"\n\nHe clambered over Juan Bautista and Erich (Marie was more obliging about being left in her cage) and started the projector. We were briefly treated to his black shadow on the screen while he blew out the lamps; then we were silently told that we were watching _INTOLERANCE , a Sun-Play of the Ages_.\n\n\"What's a sun-play?\" Juan Bautista asked.\n\n\"Photoplay, get it?\" Porfirio explained.\n\n\"Oh.\"\n\nThe screen advised us that we were going to watch a story of the battle of the forces of Hatred and Intolerance versus Love and Charity. Great, I thought, as though a roomful of immortals hadn't seen _that_ plotline a few hundred times already. But this was supposed to be a great classic, so I opened my mouth only to stuff in some popcorn.\n\n\"Okay, here's the great leitmotif,\" said Einar. _OUT OF THE CRADLE_ _ENDLESSLY ROCKING_ , the screen told us, and there was Lillian Gish rocking the biggest cradle I'd ever seen, while the Three Fates looked on from upstage. There followed something incoherent and vaguely poetical about eternal hopes and fears, eternal joys and sorrows, and then we were watching a grand ball for a wealthy modern (early-twentieth-century) industrialist and his spinster sister.\n\n\"What is that woman wearing on her head?\" Oscar inquired, frowning at the screen. \"She looks like a circus horse.\"\n\nShe did, too; she was aging and plain, which made her easy prey for the Uplifters, a villainous society of ladies who wanted her money so they could ruin everybody else's fun. Now that she was no longer attracting the boys, she just naturally fell into their clutches, and was persuaded to hand over her brother's millions to the cause of Reform.\n\n\"Notice the subtle misogyny in Griffith's depiction of older women,\" said Imarte with a sniff.\n\n\"Subtle!\" I scoffed through my popcorn.\n\nNow we got to meet the Little Dear One, portrayed by Mae Marsh, a teen miss given to hysterical displays of affection, living happily with her aged father (a mill worker for the wealthy industrialist) and a host of small barnyard animals. Next we met our hero, Bobby Herron, as the Boy, who sported a black mustache in defiance of all heroic convention.\n\n\"He looks like Gomez Addams,\" said Juan Bautista. Our giggles died as the scene advanced and Griffith showed us the gray laboring masses shuffling in lockstep through the gates of the mill. This was the Future, this was the Metropolis, this was the century that would bring us to 1984. Where would I be in 1984? Or 1996? I reached for my martini and had a bracing gulp.\n\nThe scene changed. There was Lillian Gish with the cradle again and then the two tablets of Mosaic law, and we were at the Jaffa Gate watching camels and old bearded men in striped headdresses. Griffith explained what a Pharisee was, making sure we didn't miss the parallel with the Uplifters, and then showed us some of the Judean variety praying ostentatiously.\n\nWham! Scene change to France, A.D. 1572, where problems between the Catholics and the Huguenots were about to come to the boil.\n\n\"Hey! Really good clothes,\" I said in surprise. Everyone nodded except Juan Bautista and Oscar, who hadn't lived through that century. Then we were shown Catherine de Medici, the villainous queen mother (\" _That_ meddling old hag!\" snarled Imarte, with such venom, we all turned to look at her) and her two sons: the king, a slender fellow with a tendency to curl up sideways on his throne, and his brother the prince, an effeminate who kept puppies in his codpiece. We were shown the French court milling around in a large room; then we met our Huguenot heroine Brown Eyes (with a tight close-up on her face, so we got the idea) and her nonentity boyfriend, Prosper Latour. We also met an obviously villainous soldier who was smitten with lust for Brown Eyes.\n\n\"I bet I know what's going to happen,\" said Juan Bautista, sitting forward.\n\nBut before we could guess, we were whisked back to the twentieth century, where the poor mill workers were innocently dancing at an ice-cream social. The Little Dear One was there, gleeful as usual, and so was the industrialist oppressor, snooping on his workers and scavenging dropped change from the sidewalk. This was straight out of Dickens, only grayer and more banal.\n\n\"When do we get to Babylon?\" I complained.\n\n\"Right about now,\" Einar said, and lo, we beheld the Imgur Bel Gate of Babylon! Intricate, massive, worked by twin capstans carven with lions rampant, manned by dozens of slaves. Elephants plodding through the streets, looking small as cows before the vast walls. Enormous winged bulls with the heads of bearded kings. Now _this_ was imagery. We cheered and applauded.\n\n\"Shot right here, folks, on Griffith's lot off Hollywood Boulevard,\" said Einar.\n\n\"Actually it didn't look at all like that,\" Imarte said.\n\n\"Well, it ought to have,\" Porfirio said.\n\nIn the midst of the gorgeously costumed bustle there were a few people just sort of sitting around in the street as though they were waiting for the next bus, and Griffith made sure we noticed one of them, she whom the title card declared to be the Mountain Girl.\n\n\"So, is there some reason he never gives these people names?\" Porfirio asked.\n\nEinar shrugged. \"It's poetical or something. The actress is Constance Talmadge.\"\n\n\"Good Lord, what does she have on her head?\" Oscar exclaimed. We all looked intently. No gorgeous robes for this chick; she appeared to have a couple of fur rugs tied around her boyish figure, and on her head were eucalyptus nuts sewn to a felt beanie.\n\n\"Needless to say, this is _not_ an accurate historical costume,\" Imarte remarked.\n\nThe Mountain Girl was no drooping harem lily; she was a tomboy, as cute and spunky as Mary Pickford. And here was somebody else waiting for the same bus: Griffith explained that this handsome bare-armed guy with flowers in his hair was the Rhapsode, a warrior-singer-poet. He made eyes at the Mountain Girl, who of course\u2014the little spitfire\u2014rejected him indignantly.\n\n\"Now, get a load of his seduction line,\" said Einar.\n\n\"Dearest one\u2014in the ash heaps of my backyard there will be small flowers; seven lilies\u2014if thou wilt love me\u2014but a little,\" read Juan Bautista uncomprehendingly, and we fell about laughing. The Mountain Girl wasn't impressed either. The scene changed, and we met the High Priest of Bel-Marduk and his god, both of them pretty sore that all the people were worshiping Ishtar these days. Another scene shift, and Griffith advised us that we were about to meet the ruling Prince of Babylon, Belshazzar, riding in his chariot along the top of his city walls, which were careful historical reconstructions of the real ones.\n\n\"Really?\" we all wanted to know.\n\n\"They weren't quite that high,\" admitted Imarte. \"But when you were arriving from a three-hut village and seeing them for the first time, the effect was very nearly the same.\"\n\nBelshazzar, a slender creature in a tall hat, was the Apostle of Tolerance and Religious Freedom; and here was his bodyguard, the burly Two-Sword Man; and here were the Handmaidens from Ishtar's Temple of Love and Laughter. They came dancing crazily out through the gate before the big parade float that bore the statue of Ishtar. Next we met the Princess Beloved, much more the conventional harem-lily type, and she and Belshazzar made a lot of pseudo-Biblical protestations of love for each other. Yes, all was happiness in old Babylon. But now, Griffith told us, the Mountain Girl's Brother (he had no name either) was having trouble keeping her in line, so he was going to drag her off to \"the First Known Court of Justice in the World\" to make her behave. A scholarly title card explained how Babylon's ancient laws were the first to protect the weak from the strong.\n\n\"Occasionally,\" Imarte said. \"And of course you're all aware that the Babylonians were not the inventors of law.\" We all nodded, watching the antics of the Mountain Girl before the judges. We knew about the Neanderthal Code of Punishable Acts. Of course, there'd always been the rumor that the Company had been involved in that, somehow . . . Whoops! The judges sentenced the Mountain Girl to be dragged off to the marriage market. We all leaned forward in anticipation.\n\n\"Damn,\" said Porfirio, as we jumped forward through time over the Cradle Endlessly Rocking and found ourselves back in the twentieth century, where the nasty old Uplifters had spent so much of the wealthy industrialist's money that he was obliged to cut wages at the mill by ten percent. So we got to see one of the dreary labor strikes of the twentieth century, complete with soldiers\u2014or were they Pinkerton men?\u2014firing on the protesters, THE SAME TODAY AS YESTERDAY read a sign painted on a fence; and we immortals silently acknowledged that bitter truth. The Loom of Fate wove death for the Boy's Father when he took a bullet, and in the aftermath everybody went off to the big city to look for other work: the Boy, the Little Dear One and _her_ father, and another girl, the Friendless One.\n\n\"I want to see Babylon again,\" Oscar complained.\n\nInstead, we got to watch as our nameless ones suffered the inevitable consequences of urban relocation: the Boy rolled a drunk and turned to a life of crime, the Friendless One met a pimp called, for no reason any of us could figure out, the Musketeer of the Slums, and the Little Dear One's hysteria grew more pronounced.\n\n\"This is depressing,\" said Juan Bautista, and then: \"All right!\" because there was Ms. Gish and her Cradle again, and we were back in Babylon at the marriage mart.\n\nGriffith explained that in the ancient Babylonian marriage market, the money spent to purchase pretty girls was given to plain girls for dowries, so that all might be happily married, and that \"Women Corresponding to Our Outcasts of the Street\" became wards of church and state for life. Imarte shook her head sadly when I looked to her for commentary. The men in the room were keeping their eyes on the screen, however, as Griffith's camera moved slowly past pretty girls in various states of undress. And here was the Mountain Girl, swaggering little hoyden, chewing on a couple of scallions as she awaited her turn and sticking them sullenly in her bodice when told to stop.\n\n\"This episode is pure fabrication,\" Imarte said. \"I find the scene noteworthy for what it reveals to us about the sexual repression of the early twentieth century, however. Of course, you're all aware that white slavery will be a popular motif of escapist cinema during that period, and it's interesting to regard the paradox of sexual bondage viewed as a liberating experience. In _The Sheik_ , for example . . .\"\n\nI tuned her out. Of course the Mountain Girl was a failure in the marriage market, winsomely scowling and threatening her prospective husbands, and then raging when they laughed and refused to buy her. But, out of nowhere, Belshazzar appeared with his attendants and wanted to know what all the fuss was about. When her plight was explained to him, Belshazzar granted her the right to marry or not as she wished, with a nifty bit of stage business involving a cylinder seal rolled over a clay tablet, shown in tight close-up (Imarte interrupted her lecture long enough to remark approvingly on its verisimilitude). Exit Belshazzar, followed by moony stares from the Mountain Girl, clearly smitten with him.\n\nBut what was this? Now we had the Rhapsode, working in the tenements to convert backsliders to the true worship of Bel.\n\n\"So he's a missionary?\" Oscar said with a frown. \"I thought he was a poet-warrior-singer or something.\"\n\n\"He's an utterly imaginary creature,\" said Imarte dismissively, but I was intrigued. He was a potentially interesting character, this Rhapsode. Was Griffith going to develop his warrior and zealot sides? Was he a hero as well as pleasure-loving esthete? Was he going to have some kind of relationship with the spirited if somewhat gauche Mountain Girl? Maybe so, because here she came, and again he was making protestations of love to her. But the Mountain Girl drew back from him, declaring:\n\n\"Put away thy perfumes, thy garments of Assinu, the Female Man,\" read Oscar. \"I shall love none but a soldier.\"\n\n\"I guess she told _him_ \" Porfirio said. But then the scene changed to the interior of the Temple of Ishtar, and the attention of all gentlemen present was riveted on the screen, as Griffith treated them to a nicely detailed study of the Virgins of Ishtar frolicking in various pools and fountains, or lounging around in\u2014well, in no clothing at all, one or two of them.\n\n\"Wow,\" gasped Juan Bautista.\n\n\"Pre-Hays Code, guys,\" said Einar. \"Cool, huh?\"\n\n\"Now, this, actually, is fairly accurate,\" Imarte said. \"Though there would have been rather more nudity. The ancients were considerably more realistic about sexual needs, to say nothing of the issues of climate and comfort. Notice the continual depiction of Babylon by Griffith as a model of humane and sensible government, which will be really rather daring of him, considering that American audiences of that era are inclined to take the Old Testament view of Babylon as depraved and vice-ridden.\"\n\nHere was Belshazzar again, with the Princess Beloved, billing and cooing in the Temple, and he was promising to build her a city of her very own, because:\n\n\"The fragrant mystery of your body is greater than the mystery of life,\" read Juan Bautista.\n\n\"Roughly translated as, Gee, did you step in something one of the elephants left?\" I jeered. But Griffith just kept piling on the splendors, the Virgins of the Sacred Fire of Life dancing in the Love Temple, the harpists and yawning harem guards, Belshazzar and the Princess Beloved standing at a high window viewing the magnificence of Babylon spread out below them. When he stuck to the images, the phenomenal images, you were caught up in this silly thing in spite of yourself.\n\n\"The Cradle again,\" said Juan Bautista, and here was the Little Dear One in her tenement, trying to get a Hopeful Geranium (actually a pelargonium) to thrive. When she wasn't doing that, the innocent creature was watching whores from her tenement window, practicing their walk so she'd be popular too. Griffith was inviting us to smile in fond amusement here, but you could see where it was leading, and, sure enough, the next scene took us to the Friendless One in her new job: sitting in a bar in her negligee, drinking gin. She got up to go solace herself with the Musketeer of the Slums, but their passionate moment was interrupted by the Boy, who was now a Barbarian of the Streets. You could tell, because he'd taken to wearing a derby and smoking.\n\nNext scene, the Little Dear One and the Boy encountered each other at his newsstand (a blind for his more criminal activities, we were told) and she did her prostitute walk for him. It worked! The next thing we saw was the two of them in her hallway with him vowing eternal love, a little less poetically than Belshazzar. (\"Say, kid, you're going to be my chicken,\" read out Juan Bautista.) But the Little Dear One's old father broke it up, dragging her upstairs to pray for forgiveness before a little plaster statue of the Holy Mother and Child. (\"Notice the recurring goddess images here,\" instructed Imarte.) After a brief Cradle shot, we saw the old man lying stiff in his winding-sheet and the Little Dear One weeping by his corpse.\n\n\"We want Babylon, we want Babylon,\" chanted Einar. Instead we got the Wedding at Cana, with a very bouncy and giggly young couple being fed the first mouthful of chickpeas at their reception. Whether or not the bride was expected to smear the next handful on the groom's face we never got to see, because the scene shifted to a shot of some white doves and then the entry of Jesus. He looked just like Belshazzar, except he didn't have a hat. But here were those mean old Pharisees again, snooping around and remarking that there was Too Much Revelry and Pleasure-Seeking among the People. Not for long, though\u2014the wedding party ran out of wine!\n\nI set aside my popcorn and leaned forward, wondering what special effect Griffith had to wow us with the water-into-wine miracle. But all we saw was a long shot of Jesus praying over some jars, only you couldn't quite make him out. At first I thought it was some defect in the print, but then I got the trick: Jesus was nearly obscured by a cross-shaped shadow. \"Notice the metaphor,\" Imarte said, \"Divine Love being blotted out by the instrument of torture and execution.\"\n\n\"We noticed, thank you,\" I said. A thought occurred to me. \"Hey, did _you_ ever meet Jesus of Nazareth?\"\n\n\"No.\" She took a sip of her martini. \"I was stationed in Turkey at the time.\"\n\n\"But somebody must have. Surely the Company was in on that moment in history too, right?\" I wondered aloud. \"Is there footage in a Company archive somewhere of Jesus trashing the temple vendors, Jesus on his bloody cross, Jesus giving the Sermon on the Mount?\"\n\n\"Yes,\" Einar said, as we watched the wedding guests celebrate the miracle. \"A fundamentalist group paid the Company big bucks to catch the man's act on film. They didn't like what they saw, so they paid more big bucks to have it suppressed.\"\n\n\"You're kidding!\" I leaned forward. \"So he was real? So he worked miracles?\"\n\n\"Oh, yeah,\" said Einar, nodding. \"That wasn't the problem.\"\n\n\"What _was_ the problem?\" Had he been a Crome generator too?\n\n\"Sorry, guys, that's classified information,\" he said, and we booed him and threw popcorn until Juan Bautista cried:\n\n\"Whoa, how did we get to France again?\" Because we were back in France, no question, and Brown Eyes and her family were having a cozy evening at home with visiting swain Prosper Latour. (\"How come he gets a name when nobody else does?\" asked Porfirio.) Prosper left to go home, and Brown Eyes saw him to the door, where the same mercenary soldier spotted her again. He made his moves, but she was a nice Protestant girl and declined his advances.\n\nI took my eyes off the screen long enough to quaff some martini. When I looked up again, we were back in the twentieth century, and the Boy and Little Dear One were having an outing in the Good Old Summertime.\n\n\"Notice San Pedro standing in for Coney Island,\" Einar said.\n\nThey went back to the tenement, and the Boy tried to muscle his way into the Little Dear One's room, but she shut him out and prayed tearfully to the Holy Mother to help her be a Strong-Jawed Jane. I accessed my historical idiom file for a definition of this colorful phrase, without results.\n\nFate was smiling on the Little Dear One, because the Boy proposed marriage. We were shown briefly how the insidious Uplifters were gaining more and more power, and then\u2014\n\n\"This is almost as disorienting as real time travel,\" I said, as we found ourselves watching the Pharisees spying on Jesus.\n\n\"Behold a man gluttonous and a winebibber, a friend of publicans and sinners,\" read Juan Bautista. There was Jesus sitting in what appeared to be a cafe, chatting pleasantly with a fellow customer. But wait! Here was the Woman Taken in Adultery, and the Pharisees all ready to stone her, rocks in hand like extras from a Monty Python sketch. Unbelievable overacting from the Pharisees when Jesus zinged them with his ruling, and unbelievably strange dancing when the Adulteress made her grateful exit. Jesus just sat there and looked benign, apparently the only good actor in Galilee.\n\nWe all groaned to find ourselves back in the twentieth century again, viewing more of the Uplifters' nasty handiwork. Griffith came right out and said it this time, that When Women Cease to Attract Men, They Often Turn to Reform as a Second Choice. We booed and threw popcorn as he showed how these villainous spinsters were responsible for bootlegging, secret poker games, and flirting in alleyways. Meanwhile, the Little Dear One was working her own reformation on the Boy, extracting his promise to quit his criminal gang, and her victory dance was truly savage and bizarre. And premature: for no sooner had the Boy quit his gang than the Musketeer of the Slums got him framed for theft, and the Boy was whisked off to prison.\n\nBabylon again! We cheered, but there was trouble brewing; the wicked High Priest was still interminably plotting, prophesying doom if people didn't leave off this Ishtar nonsense and return to Bel-Marduk. Enter Nabonidus, Belshazzar's crazy old father, excitedly showing off his latest archaeological discovery and remembering to mention that Oh by the way, Cyrus the Persian is massing with his armies to destroy us.\n\n\"Notice the subtext of implicit condemnation of intellectual pursuits,\" observed Imarte.\n\n\"Whatever,\" I said. \"Wow, these must be the Persians!\" It was the war camp of Cyrus. Griffith showed us milling soldiers, engines of war, and chariots of fire. The exotic hordes of Cyrus, the Medes, the Persians, the Ethiopians. (\"Look, _real_ black guys this time!\" Einar said. \"Most of the Negroes in _Birth of a Nation_ were white guys in blackface.\") There were some bone-gnawing white European males designated merely as Barbarians, then Cyrus himself, shouting orders from his war chariot, turning sharply right, left, right, a big martial profile from a wall painting. He looked like a Klingon member of Sergeant Pepper's Lonely Hearts Club Band.\n\n\"Damn,\" yelled Porfirio, because in mid-spectacle we were hauled back to the twentieth century, where the Little Dear One was now the Little Mother. There was some touching footage of her playing with her baby, but we knew tragedy was in the offing, and, sure enough, the evil Uplifters were at it again: Griffith showed them descending on slum neighborhoods like stiff harpies, and nailing the Little Mother when they caught her with some whiskey she was taking as a cold remedy. When they came for her baby, the Little Mother fought back with such frenzy that they smashed her to the ground senseless, hefty muscular broads as they were. We saw her clutching feebly at one baby sock, dropped in the struggle, all she had left of her child.\n\n\"Emotionally involving,\" admitted Imarte, \"though overall I _think_ Chaplin handled the same theme more effectively.\" I resisted an urge to shush her, because it was an upsetting scene after all, but then Griffith went and threw in the kitchen sink by showing us a Suffer Little Children to Come unto Me tableau, with Jesus and what must have been half the child population of Hollywood, happy little extras earning a day's wage.\n\nThen, thankfully, we were back in France, watching the Catholics cower as the Huguenots ran around wild in the streets and smashed a perfectly good plaster statue. We can't have this kind of thing going on! announced Catherine de' Medici. Time to do something about those pesky Protestants.\n\nBut we escaped to Babylon again, and Cyrus jumped athletically into his chariot and ordered his armies to march on Belshazzar's city. And here was Belshazzar, mustering up the city guard, pausing only to bid a fond adieu to his Princess Beloved. She bid him one heck of a fond adieu herself:\n\n\"My Lord, like white pearls I shall keep my tears in an ark of silver for your return. I bite my thumb! I strike my girdle!\" Juan Bautista read, and began to giggle helplessly. \"If you return not, I go to the death halls of Aflat!\"\n\n\"That's All\u00e2t,\" Imarte corrected him primly, which only made the rest of us snicker more. We watched as the Mountain Girl decked herself out in her brother's second-best armor and rushed out to help defend her city, just like Xena.\n\n\"Get ready, guys,\" said Einar, \"this is one of _the_ great cinema battles of all time.\"\n\nHe wasn't kidding. Thousands of costumed extras stormed those magnificent walls, battered those glorious gates, and, by God, life-sized siege towers were actually pushed into place by real elephants, because Griffith hadn't had any other way to move them. The Priests of Ishtar assumed odd positions and prayed to their goddess for deliverance. They made burnt offerings, the flame and smoke of their sacrifices flowing up with the flame and smoke of the battle. The Mountain Girl sent a rain of arrows down on the attacking hordes. The burly Two-Sword Man, fighting with (surprise) two swords, swopped off a Persian head as neatly as I've ever seen it done on film. More shots of the Mountain Girl fighting valiantly for her Prince, intercut with the Princess Beloved peering fearfully out the window at the battle, posturing dismay and terror with palms angled stiffly, and you knew she didn't deserve Belshazzar the way our plucky heroine did. The catapults of the attacking host launched rocks on the defenders, and the defenders pushed one of the siege towers outward, outward, until it tilted and fell, all eight stories, with a crash they must have heard in San Pedro. And now the battle was going on into the night, with flames along the battlements to rival the burning of Atlanta in _Gone with the Wind_.\n\n\"Look at that!\" said Oscar. We stared at the secret weapon of Babylon, an armored tank that Jules Verne might have designed if he'd been around in Belshazzar's day. It was huge, with spiked wheels that rolled it implacably forward and long nozzles that shot out jets of flame. How absurd, how improbable, how _marvelous_! We all applauded.\n\n\"A blatant anachronism,\" Imarte said, but we ignored her. Who were we, cyborgs in the Old West, to talk about anachronisms? We cheered as the engine of destruction laid waste to the Persians, burning and bringing down more siege towers in flaming ruin. The forces of Cyrus were routed! Wild celebration in Babylon! Dances of joy and bliss! What a spectacle!\n\nIntermission.\n\n\"Rats,\" I said, as Einar crawled over to the projector to change reels.\n\n\"Stay cool,\" he said, groping for the film can. \"The best is yet to be.\"\n\nBut when the silver world returned, we were in the twentieth century again, with more tragedy building: the Musketeer of the Slums now had his lustful eye on the forlorn Little Mother, intimating he might be able to rescue her baby from the clutches of the Uplifters. He was jealously shadowed by the Friendless One, however. The Boy got out of prison, but this did not discourage the Musketeer from his dastardly plan\u2014\n\n\"Thank God,\" said Porfirio, because the Cradle appeared again, and Griffith told us we were about to behold the Feast of Belshazzar, \"Imaged after the Splendor of an Olden Day.\"\n\nEinar gave a whoop of excitement. \"This is it, Babylon the Great! For those of you who were wondering what could possibly top the magnificence of that battle scene . . .\"\n\nBut he didn't need to say anything more, because there before us was the first long shot of Hollywood Babylon in all her glory, and it took my immortal breath away. A central court so huge, it beggared all comprehension of scale, with gods and goddesses stories high, rearing elephants in stone, and thousands of tiny figures there below us on the wide stairs as the camera moved slowly in on the scene. How had he done it? And still the camera moved in, and yes, there were real people on those painted stairs, and your eye had to believe that this was no matte shot, no miniature, this truly was the ancient and massive splendor that had died away from the world, caught on silver nitrate by giants who were in the earth in those days.\n\nThe camera took us up the great stairs through the mad bacchanal, with all the little incidents to catch attention: the grim Two-Sword Man fondly stroking a white dove, the temple priestesses dancing stiff like wall paintings come to life, the Priests of Ishtar chanting praise to Her. I heard Imarte sniffling: was she crying? And here were Belshazzar and the Princess Beloved in all their glory, scarlet and purple, emerald and gold, glowing through the austere silver print.\n\nNow we saw the Mountain Girl confidently expecting to walk into the banquet hall, but of course she was turned away, the little ragamuffin. Next we saw the High Priest of Bel, scheming to make his defeat a temporary one, summoning the Rhapsode and ordering him to ready the chariots for a secret visit to the camp of Cyrus. But what was this? Did the warrior-poet-singer-missionary balk at this base treachery? No! He looked confused, he abased himself, he ran off to do the High Priest's bidding. He wasn't really going to be a dupe and betray Babylon, was he? Surely he would form some clever plan to thwart Cyrus, the High Priest, and the rest of them. It wouldn't work out, of course, unless Griffith had decided to change history . . . but, having created such a potentially interesting hero, was Griffith really going to waste him? I drained my martini and fumbled around in the dark for the cocktail shaker. I had to pry it loose from Imarte's grip: she was staring at the screen as if mortally hurt, and she _was_ crying. Son of a gun.\n\nThe Mountain Girl was thirsty, too; she was purchasing a drink of goat milk to celebrate, squatting down and milking the goat herself. Something about the motion must have reminded her of Belshazzar, for she promptly got that moony unrequited-love look on her face, with intercut shots of Belshazzar at his feast, and back to her tugging on the pendulous squirting udders. Einar and Porfirio were rolling on the floor, choking with laughter, and Oscar had his handkerchief up in front of his face.\n\nMore revelry, more splendor. We reclined at the table of a Babylonian noble crowned with flowers; we were invited to partake of spiced wine cooled with snow brought down from the mountains. We watched a Babylonian gallant idly stroking the head of a leopard, which lay bound and snarling through the roses in its jaws. We watched a tame bear being fed party delicacies. We watched a swaying-drunk soldier telling the cameraman about the little monkey that clung to his helmet. And here was the Mountain Girl again, and the Rhapsode coming around a corner to catch sight of her. Still smitten with her boyish charms, he made another pitch for her and was indignantly spurned. But he kept following her around and then\u2014\n\nI groaned and downed my martini in a gulp. So much for the potentially interesting hero. In his efforts to impress the Mountain Girl, the Rhapsode did it, he blabbed about the secret mission to Cyrus's camp.\n\nWhat was Catherine de' Medici doing here? Oh, we were back in France. Pressure was being put on the King to sign the order allowing the Catholics to massacre the Huguenots. Would he do it? No, no, a thousand times no! He ran around the room screaming, he tore his hair, he stamped his feet, he curled up in his throne and kicked his legs. He signed it, though, as his nasty and effeminate brother played with a ball-and-paddle game. Next we saw sedate Protestant joy at the Brown Eyes residence as she and Prosper Latour were betrothed. Ominous foreshadowings as the happy family snuffed their candles before retiring for the night, and Prosper walking back to his lodgings and noticing those groups of soldiers going around chalking X's on the doors of the Protestants. Did he have a clue? Nope.\n\n\"In the Temple of Loooooove!\" yodeled Einar, and we were back in Babylon again. Apparently an orgy was in progress. More harem hijinks, more voluptuaries rolling around in perfectly astonishing attitudes of invitation, more dancers on the grand stairway performing some kind of goose-stepping polka, another seminaked girl\u2014\n\n\"Watch,\" Einar announced. \"Long-distance gynecology. You won't see a shot like this one until _Basic Instinct_ with Sharon Stone, 1992.\"\n\n\"Dear Lord!\" Oscar went into a coughing fit, his eyes bugging out of his head.\n\nJuan Bautista blushed furiously. \"Could they _do_ that back then?\" he asked.\n\nEinar grinned. \"No Hays Code yet, like I said.\" Now we saw the Princess Beloved coyly sending a white rose in a little chariot drawn by two white doves across the banquet table to Belshazzar. But not everyone was celebrating: the evil High Priest and other objectionable members of the clergy climbed into their chariots and sneaked away to Cyrus's camp. Not unobserved. The clever Mountain Girl stole a chariot to follow them, riding like the wind past the Three Fates and Lillian Gish, to\u2014\n\nNew York? What were we doing in this cramped tenement, with the Musketeer of the Slums attempting to rape the Little Mother? The Boy burst in and struggled with the Musketeer, while the Friendless One clung precariously to the window ledge outside, trying to get a clear shot. Bang! Huge clouds of smoke, the Musketeer staggered into the corridor to die, the Friendless One dropped the gun into the room and took to her heels, and in no time at all the Boy was on trial for his life in a huge courtroom. The Prosecution was clever\u2014you could tell, because he wore pince-nez\u2014while the Defense Attorney was a stammering novice, so we weren't left in much suspense about the outcome. Not so the Little Mother, who giggled encouragement across the courtroom at the Boy, at least until the brutal sentence was read, when she screamed and fainted. Somewhere in this scene we went to Jerusalem, briefly, where poor Jesus was struggling with his cross through the howling mob, and then we were back in the twentieth century again, where the Friendless One was trying to get up nerve to confess to her crime.\n\nI had another martini, and it seemed to help. We went back to where Cyrus was greeting the High Priest and his cronies while the Mountain Girl listened in on the conspirators, but then we were interrupted by scenes of various efforts to save the Boy, and there were Model Ts racing back and forth, which inexplicably became chariots racing along the Euphrates, only Einar was explaining that it was really a slough down near Long Beach with some palm trees planted to make it look like the Euphrates, and there was Catherine de' Medici again!\n\n\"What the hell is she doing here?\" I heard Porfirio inquire, but it soon became obvious, because Brown Eyes and her family were having a Terrible Awakening as the Catholic forces battered down their door.\n\nWere those automobiles coming to the Huguenots' rescue? No, we were back in New York, and then we were racing along beside the Euphrates, and\u2014hey! the Friendless One finally confessed! The Mountain Girl was racing ahead of the army advancing on poor Babylon, and the damn Cradle was rocking faster and faster. Two Model Ts chased each other, then it was a race car chasing a train, then the Boy was receiving last rites from a priest, then a priest was hiding a terrified Huguenot child under his cloak, then Belshazzar and the Princess Beloved were pitching woo all unmindful of the train bearing down on them or the Huguenots being slaughtered in the next room. Here came Jesus wearing a signboard, narrowly missed by the race car bearing the last-minute testimony that would reprieve the Boy, who was fainting in the priest's arms, no doubt at the graphic rape and murder of Brown Eyes by that darned mercenary (\"I _knew_ he was gonna do that!\" said Juan Bautista). Prosper could have saved her, but he couldn't get through the crowds of soldiers massacring Protestants, and the same crowd was slowing down the Mountain Girl too, because it took her forever to make her way to Belshazzar's party to let him know what was going on, by which time it was too late to save Brown Eyes, but at least the governor was willing to reprieve the Boy! And now it was the train racing to get to the prison before the execution, but Belshazzar could get only twelve men to help him defend Babylon, and the Princess Beloved couldn't get _anybody_ to go with her to the Death Halls of Allat. Oh hell, oh spite!\n\nProsper was killed, the Mountain Girl was killed, Belshazzar and the Princess Beloved killed themselves, and there was a close-up of the Mountain Girl lying killed, which irised out to show that the forlorn little doves had pulled the rose-chariot up to her lifeless body. Juan Bautista burst into tears, but Catherine de' Medici was pleased with herself and so was Cyrus, who guffawed crudely right in the camera's face, as meanwhile everybody piled out of the train and into another car, which raced on to save\u2014gee, who else was left alive?\n\nNot Jesus. A confusing mass with a lot of smoke resolved itself into Golgotha, with the three crosses dimly visible on a dark skyline. So the car raced on, but the Boy was already mounting the scaffold, they were binding his hands, they were binding his legs, they were putting the black hood on him, and men with razors were standing by to cut the cord that would drop the trap. _Would rescuers get there in time_?\n\nSurprise! They did. Dazed Boy being embraced by passionately twitching Little Mother. General expressions of joy and satisfaction. \"Here comes the summation,\" Einar told us.\n\nWHEN CANNON AND PRISON BARS WROUGHT IN THE FIRES OF INTOLERANCE, read the titles, over scenes of battlefield and prison. A lot of angels appeared, dangling in a rather crowded sky, and all the fighting stopped.\n\nAND PERFECT LOVE SHALL BRING PEACE FOREVERMORE. More people staring up in confusion at the angels.\n\nINSTEAD OF PRISON WALLS, BLOOM FLOWERY FIELDS. A Scene showing convicts in their striped uniforms walking through the walls like ghosts and disappearing. Then an exterior shot of a prison doing a vanishing act too, leaving the Hollywood Hills in the distance. Now we saw little children disporting themselves at a church picnic. One bold toddler grabbed a tiny playmate and gave her a big kiss.\n\n\"If he tried that in the twenty-first century, he'd be arrested and put in therapy the rest of his life,\" said Porfirio gloomily. I was waiting for a final title card that would finish Griffith's sentence, but instead we saw a lot of people yelling glory hallelujah at the angels and then one last shot of Lillian Gish and her Cradle, endlessly, endlessly rocking.\n\nWhite light, flickering on a blank screen.\n\nEinar moaned, stretching sensually.\n\n\"So, what happened to the Baby?\" Porfirio asked.\n\n\"What happened to the goddamn Rhapsode?\" I asked.\n\nBefore either of our questions could be answered, Imarte leaped to her feet, scattering popcorn in all directions.\n\n\"I must go there,\" she announced. \"Now.\"\n\n\"Where?\" Oscar stared at her in bewilderment. We were bewildered too; there hadn't been a peep out of her in the last few minutes, but now she turned on him with a snarl.\n\n\"I must go to Babylon!\"\n\n\"Imarte, what'd you put in that rahat locum?\" said Porfirio, but she pushed her chaplet of roses back from where it had slipped over one eye and fixed him with one of the scariest expressions I'd ever seen.\n\n\"I _will_ go to Babylon, fairest of cities, beloved of Ishtar,\" she said. \"I will not lose it again!\"\n\n\"Cool,\" said Einar, scrambling unsteadily to his feet. \"Let's go now. Come on, you guys, it's just over on Sunset. God knows there's no traffic. We can get there in no time.\"\n\nI remember that Porfirio did some protesting, and so did Oscar, but one way or another we found ourselves galloping through the night. That is, Einar and Imarte galloped; the rest of us rode in the wagon, driven by Porfirio at a rattling clip as he tried to catch up with those two. I clutched at my chaplet of roses and wondered what I was doing as we thundered along through the damp night air under a very amused moon.\n\nIt was a wide sloping piece of ground where we stopped, an old floodplain with a view of the distant lights of Los Angeles and the more distant sea, pale and obscure under the moon. Was that Catalina Island out there?\n\nEinar and Imarte had dismounted. She was standing motionless; he was striding with arms outstretched through the sagebrush and chamise.\n\n\"Right here,\" he was saying. \"This will be the lot adjacent to the Fine Arts studio complex. Can you see it? And where I'm standing, babe, we're on that Grand Staircase! Look up there behind us in the moonlight, there are the elephants! There are the winged bulls bright as day. They're here, and more real than this empty place or the asphalt that covers it later. Silver nitrate's made Babylon eternal for all time, and the prophets can't do a thing about it. This is Ishtar's city of love and tolerance. Can you smell the incense? Can you hear the music?\"\n\nI very nearly could. I found I was still clutching the cocktail shaker and took a fortifying gulp. Imarte stretched out her arms to the moon and gave a plaintive cry. She began to dance, there in the moonlight, over the stones and red sand, through the yucca and the cactus and other herbs that never lifted bud or branch in Belshazzar's city so long fallen. It was no stiff absurd dance either, no attempt to choreograph a flat wall painting; it was lithe and savage, a little unsteady, something you'd really dance at a bacchanal.\n\n\"Is she nuts?\" asked Juan Bautista, who cowered shivering beside me. \"What's she doing?\"\n\n\"She's just had enough gin, that's all,\" Porfirio told him.\n\n\"You think that's all it is?\" I had another belt from the cocktail shaker. \"She's crazy with pain. She's so much older than the rest of us, and she was really there, wasn't she, when Cyrus came crashing down on Babylon. What if you'd loved a place like that, and seen it go down in flames? What if you'd buried it in your heart for centuries, all that lost glory safely forgotten, and then one night, when you least expected it, something brought it to life for you again? How do you think you'd feel? How _will_ you feel, kid, a hundred centuries from now, when you're as old as she is?\"\n\n\"I won't be like that,\" said Juan Bautista. \"That's not supposed to happen to us. We don't go crazy, we can't, we're perfect! Aren't we?\"\n\n\"Shut up, Mendoza,\" said Porfirio quietly, and Oscar fumbled the cocktail shaker from my icy grasp.\n\n\"Of course we are, son, we're positively the last word in cybertechnology,\" he assured Juan Bautista.\n\nI staggered to my feet and flung my chaplet of roses out into the middle of what would be Sunset Boulevard one day, when all this sweet wild land would be buried under an urban nightmare. And what would _I_ be feeling, when I was as old as Imarte? What would have become of the places _I'd_ loved? What if there were no more oak trees or redwoods, what if California itself slipped under the Pacific, drowned and broken up, as lost as Atlantis?\n\nAnd I was going to lose it all, when that steel cancer of a future city was built. It wouldn't even take a global calamity: just millions of mortals moving west. I would lose my wilderness just as I'd lost Nicholas, and how would I live then?\n\nI drew in my breath to howl at the moon, but Einar came bounding up to the wagon. He held up his pistol by its barrel, pretending it was a microphone again. \"Let's hear what our studio audience has to say! Sir, can you tell the wonderful people out there in the dark what personal revelations _you_ had tonight?\" He thrust it under Oscar's nose.\n\n\"Oh, this is silly,\" Oscar said. \"But\u2014very well.\" He took the gun and held it up. \"If you really want to know, why, I think there's a brilliant message in that film, and Griffith really was a genius born too soon. He's telling us loud and clear just what's going to bring on that earthly paradise he envisions at the end, and you know what it is?\n\n\"Technology. Yes, sir, ladies and gentlemen, consider for a moment. What turns the tide of battle (albeit temporarily) against the Persians? None other than the superior technology of the Babylonians, as exemplified by their marvelous machine of war. And consider! Don't the other tragedies occur because prospective rescuers are delayed in their efforts by inferior means of transportation? Reflect upon the fact that, of all the stories presented, the one story that ends happily does so solely because of modern and efficient means of locomotion. Yes! The automobile brings the Boy's pardon! Now, just what would have happened if that Mountain Girl or the French fellow had had Fords with which to speed to the rescue of _their_ loved ones?\"\n\n\"What would have happened if Christ's apostles had had grenades and rocket launchers?\" I said, dropping into the bed of the wagon.\n\n\"Precisely! That is, er, anyhow, it seems mighty clear to me that what the great David Wark Griffith was foreseeing was nothing less than us. For are we not the veritable saviors of Earth, the ultimate marriage of man's mechanical genius with his biological possibilities? Why should any cyborg feel shame? Is it not an honor to be descended from the noble Model T no less than from Adam? I don't know about you folks, but I'm proud, _proud_ of my mingled heritage.\" Oscar flung his chaplet out into what would one day be the roof of the Kinema-color lab. He handed the gun back to Einar, who was applauding.\n\n\"Bravo, Mr. O! And what about you, sir, do you have an epiphany you'd like to share with the folks listening at home?\" He thrust the gun at Porfirio.\n\n\"What home?\" I said. \"We have no homes, none of us.\"\n\n\"Shut up.\" Porfirio put his hand on my shoulder. \"You want to know what I think, pal? I think it's time to rein in this party. Maybe you should go catch Salome of the Seven Veils and get her back on her horse.\"\n\n\"No, man, she's okay,\" Einar's eyes were glowing. \"Don't you see? She knows how to deal with this time thing. She understands. That's why she's dancing. She can _see_ Babylon. It exists for her outside of time, it's neither past nor future but right now. Always.\n\n\"Haven't you guys figured it out yet? Don't you know what it really means to be immortals? We transcend time, it has no meaning for us, it ceases to exist, because it's all simultaneous. We're here _now_ , and we're on Griffith's set _now_ , and we're in 2355 _now_! _We're the ones controlling reality, from in here_.\" He struck his fist against his indestructible skull.\n\n\"We are time machines! The truth's been right in front of our noses since cinema was invented. Hell, since photography was invented. Hell, since _writing_ was invented. Make an image of something, and it escapes the flow of time. That's why it's forbidden! Dickens had a grasp on it with his ghosts, Mary Wollstonecraft Shelley almost got it, and Einstein came so close to the truth. The dead heroes are brought in to Odin, and they rise again, they feast all night and fight all day again, and their deaths mean nothing, because they've escaped time. That's the whole point of the metaphor with Dr. Zeus, you guys! He's the liberator. _Zeus defeats Chronos_. Everything's happening at once! We can perceive time in a way mortals can't, we can make it irrelevant. Don't you see?\n\n\"All you have to do is understand, and you're free! You're out of here!\" With an ear-piercing whoop he snatched off his chaplet of roses and whirled it up, up, black against the moon for a second before it bounced down and rolled away in the direction of L. Frank Baum's house.\n\nI did understand: Einar was mad as a hatter. There were rumors that some of the really Old Ones weren't too stable. I knew Imarte was several millennia old, and she'd lost a screw tonight. How old was Einar? That mention of Odin was probably a clue. He wasn't so crazy, he couldn't do his job; just crazy enough to be happy on this black plain, under this cold moon. Wasn't he the lucky guy?\n\nShuddering, I pulled my shawl around myself. I needed Theobromos. I'd have some as soon as I got back to the inn. No, I couldn't; I'd been drinking gin. Where had I heard that Theobromos and gin didn't combine well?\n\nIt's hell to be a cyborg and have immediate access to any stray memory that one rashly summons up. There I was at a New Year's ball, at a table with three other immortals. There was the little neophyte Latif, there was my damned demon godfather Joseph, and there was poor Lewis, who was feeling ill after overindulging. Precocious Latif had explained about the toxic effect of Theobromos combined with martinis.\n\nLewis. Somewhere my friend Lewis was weeping for me.\n\nJuan Bautista's teeth were chattering in his head, and Imarte's dance had become so frenzied, she was a blur in the moonlight. Einar was dancing too, kicking up his boots and waving his long arms as he chanted a song, something in third-century Norwegian about hauling on the oars and steering for the land where palm trees grow.\n\nPorfirio pitched his chaplet over the side of the wagon and drew his six-shooter.\n\n\"Our revels now are ended,\" he announced, and fired three shots into the air. Instantly we were all sober, converting the alcohol in our bloodstreams into water and sugar, as we were programmed to do when confronted by hazard.\n\n\"What on Earth\u2014?\" said Imarte. \"How embarrassing.\" She got up from where she'd been rolling in the dust, near what would one day be the statue of Ishtar, and hastily brushed herself off. \"Was I indulging in grief accommodation again?\"\n\nEinar was crawling out from under the wagon, where he'd vanished when the shots went off. He got to his feet and looked around sheepishly.\n\n\"Cold out here, isn't it?\" was all he said.\n\n\"Well\u2014we were only having a bit of fun, weren't we?\" said Oscar.\n\n\"Speak for yourself,\" I said. Juan Bautista had his eyes closed; he was huddled up with his cheek pressed into Erich von Stroheim's feathers.\n\n\"Let's go, guys,\" Porfirio ordered. He took the reins and swung us back around for La Nopalera. Einar and Imarte climbed into their saddles and followed, all along the empty road, and the lights of Los Angeles grew fainter behind us, until they vanished like a dream.\n\n###\n\nIT HAD BEEN A great movie, all the same.\n\nThinking about it all, about Babylon and France and Jerusalem and that cold urban future, one tended to forget one was living in the midst of a historical upheaval just as impressive. France was finding that it was a great deal harder for the Second Empire to crush a bunch of ignorant mestizos than had ever been expected, while England and Spain sat back and watched in disgust. The North Americans were busy filling up the history books too: the Emancipation Proclamation was issued, for example. In future cinema it would be depicted by black hands stretching up toward beams of sunlight, broken manacles dangling from the wrists, to the accompaniment of a swelling chorus of this or that hymn, usually fading into a long shot of the Lincoln Memorial.\n\nSo sentimentally is the birth of a baby presented. The reality is far more an occasion of blood, of fear and uncertainty, of shock that displaces the joy that should be felt on such an occasion. While not one of those people to whom that piece of paper meant so much would ever have gone back to being slaves again, they must have known that the chains would be ten times as hard to break now that they were invisible and intangible.\n\nBut how could any of it matter to us? We went on about our daily life in a world where Watts and South Central were still Spanish land grants where vaqueras roped steers, and the longhorns herded down from the Tejon Ranch bore a brand of Cross above Crescent, commemorating the ancient victories of the Spanish Crusaders over the Moors, and I still thought of black men as turbaned kings with scimitars. What did this Civil War have to do with me?\n\nYou see? I had no sense of Realpolitik at all. Not that I wasn't warned.\n\n###\n\nI WAS RETURNING FROM a profitable day in the less surreal regions of the temperate belt when I picked up the warning signal being broadcast by Porfirio. It told me to approach with caution, so I left the road and came overland through the sagebrush, straining to pick up more of the transmission. All I could hear was Porfirio's end of it, at first.\n\n_But why would I lie to you, se\u00f1or? Se\u00f1ora Marta comes and goes as she pleases. You must know this. She pays rent on her room, and I don't ask her what she does in there and she doesn't tell me. Come and have some coffee_.\n\n_Of course I would tell you if I knew, se\u00f1or_.\n\n_No, se\u00f1or, I'm not in love with her. No_.\n\n_She didn't tell me, se\u00f1or. What about some coffee_?\n\n_I don't love her, se\u00f1or_.\n\n_Yes, 1 would tell you. Would you like a cup of coffee, se\u00f1or_?\n\n_No, se\u00f1or, 1 never said that. Se\u00f1or, please, put the gun away. There's no need for that. Listen to me, se\u00f1or, you know what? If he's with her when she comes home, I'll help you shoot him. Please stop waving the gun in the air and sit down and have some coffee_ \u2014\n\nThere followed a salvo of shots.\n\n_Now, you see, se\u00f1or? You're not at your best, or you'd have hit something for sure. And you don't want to hang, se\u00f1or. Se\u00f1ora Marta would cry if you were to hang, eh? No, se\u00f1or, she didn't tell me_.\n\n_No, se\u00f1or, I'm not in love with her_.\n\n_Se\u00f1or, you really would feel better with some coffee in you_.\n\nMental note: no torturer in a dungeon cell ever devised anything as frustrating, as inescapable, as terrifyingly pointless as a conversation with a drunk.\n\n_Yes, se\u00f1or, that's a good idea. Yes, by all means_.\n\n_All right, se\u00f1or, that's a very, very good idea. There is your horse_.\n\n_No, she didn't tell me, se\u00f1or. Maybe if you get on your horse and ride out now, you'll catch them_.\n\n_Very good! Buenos noches, se\u00f1or. That way. The road is that way_.\n\n_What would you want to shoot the horse for, se\u00f1or_?\n\n_Yes, you're right, that would show her. Yes, I'm sure she'd weep over your grave. I think you want to put it in your mouth, though, se\u00f1or, not against the temple like that. There! You see how easy it is to miss_?\n\n_Very well, se\u00f1or, if you say so. Yes, that way. That way, se\u00f1or. Buenos noches, se\u00f1or. Vaya con sathanas_.\n\nWhen the all clear was finally broadcast, I came slinking down out of the hills to find Porfirio sitting beside the fire, mixing himself a double mocha by dissolving most of a cake of Theobromos in a pot of black coffee.\n\n\"Who was that?\" I asked, shrugging off my pack.\n\n\"Mr. Cyrus Jackson,\" Porfirio said, baring his teeth. \"Some knight chivalrous, huh? Good thing the 1600-hours stage is late, or there'd have been a nasty scene.\"\n\n\"Where is Imarte, anyway? I haven't seen her for a few days,\" I said, reaching for a tin plate and digging a spoon into the frijole pot.\n\n\"She went up north for something,\" Porfirio said. \"Christ knows what. She didn't tell me.\" I joined with him on the last sentence.\n\nJuan Bautista emerged from his lean-to, looking perturbed. \"Is that drunk guy gone? I was scared he'd shoot at my birds.\"\n\n\"Gone but not forgotten, unfortunately,\" said Porfirio. He raised his head, listening. \"And there's the stage. I think I'll just go dump this whole mess in Ms. Imarte's ever-ready lap.\" He took a terrific slug of Theobromos-adulterated Java straight from the coffeepot and stalked off down the canyon, carrying the pot with him.\n\nAfter a few minutes we heard the driver's whip crack as the stage continued its journey to Los Angeles, and Porfirio and Imarte came slowly up the canyon.\n\n\"I told you I was sorry. What on earth can I do?\" Imarte was saying.\n\n\"You can deal with the guy, that's what you can do. You led him on, and now he's got a fit of killer jealousy. Why did you give him the idea he was anything but a customer to you, anyway?\"\n\n\"He was a good source of data,\" Imarte said, drawing her feathered shawl around her against the twilight chill. \"He provided me with no end of fascinating material that has, in fact, led me to an astonishing discovery. You wouldn't believe it, but the evidence is overwhelming, that not only is there an active Confederate plot\u2014\"\n\n\"He wants to kill you for being unfaithful to him,\" Porfirio said.\n\n\"Oh.\" She knitted her brows. \"I'll have to do something, I suppose. Don't worry. We'll deal with the inconvenience somehow. Believe me, it's worth it. Have you any idea what his simple anecdotes have revealed?\"\n\n\"Why don't you tell us?\" I said, soaking a tortilla in steak juice.\n\nShe was so enthralled with her big news, she actually sat down beside me. \"There is a conspiracy,\" she uttered in thrilled tones, \"that may involve the highest-ranking members of Parliament, to take California for the _British_.\"\n\n\"The British? Why would they want California?\"\n\nImarte gave me an arch look and paused dramatically, during which time the answer became obvious. Gold, vast natural resources, most of the Pacific coastline . . . Okay, any government in its right collective mind would covet California. But the British?\n\n\"I thought it was an odd series of coincidences at first,\" Imarte said, gazing pensively into the fire. \"All those British nationals passing through, filing claims to search for gold on Catalina. Why Catalina, where no appreciable gold has ever been found? Why are they shipping engineering equipment over there? Why are they taking such pains to determine who actually owns title to the island? Because they are, you know. I slept with a man who informed me that a fortified base is being constructed by the Albion Mining Syndicate, to be called Queen City. Of course he didn't call it a fortified base, but by asking certain questions about this so-called mining town, I was able to determine that the site it occupies has considerable strategic importance and is, in fact, being prepared for ordnance emplacements. Moreover, no attempts whatsoever are being made to prospect for gold.\"\n\n\"Wow,\" said Juan Bautista. \"What would happen if the English took over California?\"\n\nI rolled up frijoles in a tortilla and bit into it. \"They won't, we know that, so what's the point of wondering? They gave up Oregon without a fight, didn't they? Why should they try to take the West Coast now, even if they want it?\"\n\n\"Ah, but you see,\" Imarte said, holding up a forefinger, \"the political situation has changed. The Americans, who might once have prevented them, are locked in a devastating civil war whose outcome is still unknown. Europe is making a play to regain lost empires in South America. If the continental royalty manages to conquer Mexico, if the American nation falls apart\u2014and these mortals haven't our advantage of knowing how it all turns out in the end\u2014why, then, Manifest Destiny comes undone and the whole of the New World is up for grabs again. There have even been rumors that the Russians are beginning to regret pulling out of California. Can anyone wonder if Queen Victoria's ministers\"\u2014she searched for a metaphor\u2014\"want to be first in line, shopping bags in hand, when the doors are flung open on the Great American Fire Sale?\"\n\n\"What does it matter, anyway?\" I said crossly.\n\n\"Don't you think it's fascinating? This is secret history. It lends so much more understanding, so much fire and color to the dramatic pageant unfolding before our eyes. Imagine all those British diplomats playing the Union and Confederate governments off against each other, deploring slavery while covertly aiding the rebels, yet planning still another layer of double cross by preparing to step in and seize territories from the survivors should the Confederacy win!\" Her eyes were gleaming. \"Given the size of the empire they control already, why should the British think it unreasonable to go on playing the Great Game here?\"\n\n\"You're sure about this?\" Porfirio asked, taking another swig from the coffeepot. \"God knows they were eager enough to stick their fingers in the pie of Texas.\"\n\n\"I've been collecting information. Even now there's a plot brewing in San Francisco. The nephew of a British statesman has persuaded a stupid young American to join him in a privateering expedition\u2014supposedly to aid the Confederacy by raiding the Pacific Mail and diverting the gold shipments from the San Francisco Mint to the Confederate cause. I don't yet know how they plan to do it, but I'm fairly sure the Albion Mining Syndicate is involved, from their base on Catalina Island. Are you aware that any maritime power positioned there with even minimal ordnance could effectively control the entire coastline of California at this point in time?\"\n\n\"Pirates!\" Porfirio slapped his knee. \"Goddamn Francis Drake is at it again!\"\n\n\"But that's awful!\" said Juan Bautista. His eyes were big and worried.\n\n\"It's not going to happen, dummy,\" I said. \"Access your files. If there actually ever is such a plot, somebody screws it up, because it never makes the history books.\"\n\n\"It might not be so bad if the British took over,\" said Porfirio with a grin. \"Have we fared all that well under the Yankees? I'll bet General Vallejo kicks himself every day for not shooting John C. Fremont when he had the chance. And think about a colonial governor and the Union Jack flying over the Plaza. All those damn cowboys and their guns expelled. Think how the future of California would change. No Prohibition, so no bootleggers, so no Mob. No cops with guns. No movie people either. Just lots of plantations run by old aristocratic families. It'd be Lower Canada, man! Nothing would _ever_ happen here.\"\n\nNo freeways, no smog, low population density. That horrifying city on the plain I'd glimpsed would never exist. Would that be such a bad thing? But of course it was never going to happen. Catalina Island had a strange enough future ahead of it, but being the Californian Hong Kong wasn't part of the package.\n\nI shrugged. \"So what are you planning to do with all this fascinating secret knowledge?\" I asked Imarte.\n\n\"Take notes. And so should all of you,\" she admonished us. \"This is the life, the hidden motivation of mortal history. It concerns every one of us.\"\n\n\"It concerns _you_ , \"I said. \"I have more important things to occupy my time.\"\n\n\"Oh, yes, finding seventeen different mutations of mugwort _would_ take precedence over the destinies of nations any day.\" She tossed her head.\n\n\"Can it, ladies,\" Porfirio said.\n\nWe heard the rattle and creak of Oscar's wagon approaching. He was leaning backward in the seat, peering down the canyon behind him. He was concerned. \"Er, there appears to be a mortal fellow lying dead drunk in a ditch back there by the grade,\" he said. \"His horse is unharmed, however.\"\n\nPorfirio sighed.\n\nI was afraid all this talk of the damned Brits would set the dreams off again, and I was right. It was a surprisingly quiet dream, though; at least it wasn't more endless replay of the past.\n\nI was on a ship, not a miserable little dark galleon like the one I'd left La Coruna in so long ago, but a modern ship, one of those beautiful three-masted clippers the English were making nowadays, with iron-framed hulls, so much safer than the Yankee variety. Every detail exact. Salt spray, brisk chilly breeze, white clouds of canvas taking up miles of sky, nimble sailors mounting through the shrouds and ratlines. This ship was taking somebody somewhere fast. I seemed to be having a _nice_ dream for a change. I'd never been on a modern ship before. I wandered around, looking at things with great interest, observing Jack-Tars holystoning the deck and doing other terribly nautical things.\n\n\"Here now, girlie,\" croaked a voice in my ear, and I turned in astonishment to behold a black-bearded sailor grinning evilly at me. \"Ain't you been in to see him yet?\"\n\n\"Excuse me?\" I said.\n\n\"He's aft there,\" said the seaman, \"in the deckhouse. He's comin' through, ye know.\"\n\n\"Oh,\" I said.\n\nThen the dream faded away into something else, something less vivid, and I thought that he wasn't coming after all, because I'd turned out to be a Crome generator. I woke crying, pitying myself.\n\n###\n\nOUR IMMORTAL LIVES went on. The hills went from brown to a more weathered shade of brown. The leaves of the black walnut trees turned bright yellow, and so did the few cottonwoods; they were the only ones to make any ostentatious display of color. There were also bright scarlet berries on the toyon holly bushes, and they were pretty hanging in clumps among the dark serrate leaves; but the rest of southern California was unrelievedly drab.\n\nThe days dawned gray, which burned off to a glaring white sky around noon, hazy and painful to the eyes. Smoke from the cookfire hung in the cold, still air and did not dissipate. Small wonder smog would become a local institution. There was no rain to wash it away, either, though the dewfall was heavy and our adobe rooms became chill and dank unless lit braziers were kept in them half the day, which increased the smokiness. Oscar made a few wistful remarks about how good New England food would be now that the weather was getting nippy, but there were still no takers for his Criterion Patented Brassbound Pie Safe.\n\nIf I were home in the Santa Lucia Range, what a dark green the mountains would be, what a dark blue the sky, with cold winds that drove out the summer fog. And the redwoods and the cypresses would stand like dark gods, offering up their own aromatic blood. The broadleaf maples would blaze like flame, the stars glitter like broken glass. I could travel my secret ridge routes all day with no company but the sea hundreds of meters below me, and the occasional white sail on the far horizon to prove I wasn't the only living soul in the world.\n\nAnd if I wanted company, or at least civilized food, I could always hike north to Monterey, or stop in at the Post's little rancho at Soberanes Creek; though I didn't do that often, because I unnerved the settlers, and in any case I seldom wanted company. The trees and the sea were enough. Not even _he_ could find me there, my wraith, unable to summon me from my restless bed when I didn't have one, unable to break my human heart when I'd transmuted it into green leaves and stars.\n\nBut here I was in this glaring purgatory of a cow town. Winter was decidedly not its best season. The weather wasn't our only problem; Imarte had given Cyrus Jackson the brush-off somehow, more or less tactfully, but he hadn't accepted it. Several times we spotted him by infrared, sitting off on the hills at night, watching our little canyon. As long as he did nothing but lurk, he was welcome to his miserable vigil; but you never know when mortals are going to decide to go out in a blaze of glory and try to take you with them, so we monitored him closely.\n\nImarte didn't care. She had her fascinating theories, her invaluable first-person narratives, and her wealth of irreplaceable historical detail. When she wrang a source dry, she dropped the source. I am afraid that, although an anthropologist, she lacked a certain love for her subject, or perhaps its immediate and particular personification: the human heart.\n\n_I_ should talk, eh, se\u00f1ors?\n\nAnyway, we were all a little nervous, as the cold weather set in, scanning the dull hills for desperate mortals with guns. They were everywhere down here anyway, but now we had our own special desperate mortal with a gun.\n\nOne day, when the northbound stage stopped by, I went down to watch Porfirio change a shoe on the lead horse. I had seen him shoe horses before; but Einar was the only other person around, and he and I had taken to avoiding conversation with each other since our visit to 1996.\n\nIt wasn't a long stop; no passengers had to get on or off; they wouldn't have stopped except for the loose shoe. During the whole time, though, one of the passengers had his attention riveted on Porfirio. He was a young kid, maybe Juan Bautista's age, Mexican from the look of him and very well dressed, with a high shirt collar and the old-fashioned silk tie that stuck out like paddles from either side of the central knot. Leading the horse back to its place, Porfirio noticed the boy's intense regard and glanced up at him once, curious. The boy looked away immediately.\n\n\"What was with the kid who kept staring at you?\" I asked as the stage went bounding and creaking away and we closed up the smithy shed.\n\n\"Beats me,\" Porfirio said. \"I think I'll make some tamales dukes tomorrow, what do you say? It's early for Christmas, but I've really got a craving for something sweet.\"\n\nHe made hot chocolate that night instead of coffee, and we got into quite an elevated mood, sipping it around the fire and laughing. We all sobered up, though, the instant we picked up the mortal coming over the ridge to our immediate north.\n\n\"Chief?\" Einar was on his feet at once, shotgun unslung and cocked.\n\n\"I read him.\" Porfirio was in the shadow of an oak tree faster than mortal eye could have followed, his Navy revolver out. Einar faded into the gloom behind the house.\n\n\"'Scuse me,\" murmured Juan Bautista, grabbing up Marie Dressier, who clacked her beak at him in protest. He sprinted for his room clutching her in his arms, while Erich rode his head, balancing expertly.\n\nI remained where I was, warming my hands on my mug and peering doubtfully up at the ridge. Yes, there he was on infrared, making his stealthy way down the hill in our direction. Carrying a gun, too. But it wasn't our lovelorn filibuster . . . Who the hell was this, creeping along like a thief, his heart thudding painfully? He raised the gun to sight on our circle of firelight, and I winked out on him, to continue my scan from the shadows under a spurge laurel. He lowered the gun, staring in disbelief at the deserted fire. This was a young male mortal, no intoxicants in his bloodstream but with a number of the toxins produced by fear and exhaustion. No disease signatures . . . some healed fractures, very old. Unhappiness. He didn't want to be here, he didn't want to be doing this. He was tired and cold. Where had the person gone, who'd been there a minute ago? he was thinking in Mexican Spanish.\n\nI saw Porfirio and Einar working their way uphill toward him. Porfirio stopped about twenty feet below, and Einar circled around until he was just above him, only about ten feet away in the sagebrush, moving without a sound until he suddenly stood up black against the stars and said loudly:\n\n_\"\u00bfQu\u00e9 pasa, amigo_?\"\n\nThe kid whirled, swinging his gun around, but Porfirio nailed him from behind before he could get off a single shot. That was that. He pitched forward, and I saw Porfirio and Einar closing in on him cautiously. I remembered my hot chocolate and took a sip. No reason to come out until they'd brought the body down. He wasn't dead yet, anyway, just unconscious and bleeding a lot.\n\nThen what a howl of agony. Not out loud, I don't think it would have shaken the Earth that way if he'd been using only his voice: it was Porfirio's heart that was screaming, cutting through the subvocal ether like the sound of all possible things malfunctioning at once. My hair stood on end. I came blundering out through the bushes to see him crashing down the hillside, bearing the young mortal in his arms, and you couldn't have told who looked more deathlike, gray-faced Porfirio or the mortal. Einar was bounding along after him.\n\n\"Chief! Chief, what is it? What's wrong?\"\n\n\"Code blue!\" snarled Porfirio. \"Equipment! Three pints He-mosynth!\"\n\nNobody can say we don't move fast in an emergency. The boy was restarted and stable in no time, bleeding stopped and wounds bound, all kinds of stuff pumping through him that wouldn't be discovered by mortals for decades. His nice clothes were ruined, of course, including the silk tie, which had come undone and was covered with blood. He was one lucky kid for all that. Porfirio's shot had missed his heart. We stretchered him into Porfirio's room, and Porfirio sat down beside him and told us all to get the hell out of there, which we did.\n\nWhat to do after that but go to bed? Nobody was going to answer our questions; nobody was going to explain why the little creep had come sneaking back to gun for us, or why Porfirio had been so suddenly horrified, after plugging him with as neat a piece of cold-blooded shooting as I've ever seen, or why such pains had been taken to save his young life.\n\nI was awakened at dawn by Porfirio coming into my room. He still hadn't washed the blood from his hands. \"He's starting to come around,\" he said hoarsely. \"He can't see me. Go to him, please.\"\n\n\"You want to tell me what's going on?\" I said, swinging my legs over the side of the cot and groping for my boots.\n\nPorfirio leaned against the doorway. \"He's one of my family. I didn't recognize him. Haven't seen him since he was seven years old.\"\n\n\"So why was he shooting at you?\" I stood up and pulled my shawl around my shoulders.\n\n\"I shot his father.\"\n\n\"You shot his father,\" I said, looking at him.\n\nHe was looking down at the dirt floor. \"Yeah. You know how I recognized him? His father broke all the fingers on his hand that day. Just held his little hand down on the table and pounded it with a bottle. The fingers healed crooked. That was how I recognized him.\"\n\n\"Oh,\" I said.\n\nThe boy didn't wake much. I gave him water and spoke to him soothingly in Spanish, telling him he was all right. Most of the time he was passed out, sedated while the miracle cures Porfirio had filled him with did their work. I examined his hand and found the evidence of old multiple injuries. It must have been a very little hand when it was hurt so badly. Whether or not Paradise exists, se\u00f1ors, there must be a hell. People who do such things to their children belong there, and for all eternity too.\n\nEinar came in to take my place at noon, and I went out into the ghastly white day. Porfirio was nowhere in evidence, but Imarte was standing by the cookfire with an apron tied on over her whore's finery. Of all things, she appeared to be making lunch; she was tossing handfuls of barley into what smelled like goat stew.\n\n\"What's that supposed to be?\" I asked, squinting at it through the glare.\n\n\"It's the only thing I know how to make,\" she said defensively. \"It's a _very_ old recipe.\"\n\n\"Where's Porfirio?\"\n\n\"Asleep in my room, poor dear. Goddess, what a tragedy.\" She sighed. The subtext here was that she knew more about it than I did, so I just sat down and refrained from asking anything else until she couldn't stand not telling me a second longer.\n\n\"You know what happened, of course.\"\n\nI shook my head. \"Only that the kid is one of that family of his. There seems to be revenge mixed up in it somehow. And child abuse.\"\n\n\"If that were all! Remember when we asked Porfirio where his family is now? Remember he told us that a great-niece had married a ranchero, and her brothers had come to work on the rancho for him? Well, it seems this young man is her son, hers and the ranchero. The boy's father appears to have been one of those unfortunates with two personalities, a fairly decent one when he was sober and entirely another kind when he drank. A lot of unresolved rage there, apparently.\" She tossed in the last of the barley and looked around. She found the raisins she wanted and sprinkled them in too.\n\n\"In any case, after the child was born, the father's dark side came more into control. He was sober less of the time. Fortunately Porfirio (who was there) and the girl's brothers looked after the man's business affairs, but, as so often happens with this type of personality disorder, he was anything but grateful. More and more, his rage manifested itself in violence against his wife and, in time, no doubt as he perceived himself being displaced in her affections, against the child.\"\n\n\"So he showed them all by crippling the baby's hand.\" I rubbed my eyes wearily. I hated mortals.\n\n\"Ah, but that's not the whole story. The wife, for her part, had a classically codependent personality. She manifested her own feelings of low self-esteem by remaining with her husband in spite of their abusive relationship and, from what I understand, transferring _her_ anger to her brothers and to her 'Uncle' Porfirio.\"\n\n\"It figures.\"\n\n\"Quite a standard pathology, actually,\" said Imarte, stirring the pot busily and frowning as she considered the seasoning. \"And of course it broke Porfirio's heart. He seems to have loved the child and assumed the role of father in his life. He taught the boy to ride, he remembered his birthday with presents, he read to him from Cervantes. This, however, only brought about greater feelings of resentment and alienation from the actual father, and in time precipitated the final crisis. The tragedy occurred on one of their colorful fiesta days, the one with all the skeletons.\"\n\n\"The Day of the Dead.\"\n\n\"The family had made some attempt to celebrate, with decorations and candy for the little boy. The father, driven by his usual compulsion on festive occasions, began to drink early in the afternoon. Once again he was unable to deal with his rage, and by twilight he was fairly violent. The incident with the child's hand was apparently provoked when the child hid from his father behind a chair.\"\n\n\"I'm glad Porfirio shot the son of a bitch.\"\n\n\"Oh, he didn't shoot him in retaliation for the boy,\" Imarte said, sorting through the little cans of spices she'd set out. She found one she liked and dealt out a few dashes into the stew. \"The wife was able to emerge from her passive role long enough to take the liquor bottle from her husband. He turned on her, accusing her of betrayal. He said he'd kill all three of them, then thrust the muzzle of his pistol in her mouth. At this point Porfirio and the brothers, who had just ridden up, intervened. As they burst through the door, the husband turned and shot at them. He wounded one of the brothers mortally. That was when Porfirio drew his own gun and shot the husband dead.\"\n\nI drew my shawl over my head.\n\n\"As is usual in these cases, the woman blamed everyone but her abuser. She threw herself on her husband's body, shrieking incoherent lamentations and protestations of eternal love. The surviving brothers found Porfirio a horse and helped him to escape. The Company was obliging about stationing him as far away as possible. That was ten years ago. Last night, the young man made his attempt on Porfirio's life.\"\n\n\"He can't blame Porfirio for what happened!\"\n\n\"You think not?\" She gave me a sadly tolerant smile. \"But you yourself are a product of Hispanic culture, you ought to understand. These male-dominated societies all follow the same code of honor. Porfirio killed the father, therefore the son must kill Porfirio. It's de-pressingly simple.\"\n\nThe immortal uncle was accustomed to solving the problems of this classic dysfunctional family, but he hadn't quite brought it off this time, had he? Poor man. Briefly I thanked God that my blood were strangers to me, if indeed the line hadn't died out.\n\n\"How do we explain to this kid that he can't finish his vendetta, because his target's an immortal?\" I asked. Imarte shrugged gracefully.\n\nJust then Porfirio emerged from the inn, carrying a blanket and a bag, and went straight to the stable without a word. A few minutes later, he came out, leading a saddled horse, and approached us. His face was blank, curtains drawn and shutters closed.\n\n\"I'm taking off,\" he said. \"I'll be camping up the pass, if you need me. What you're going to tell Tomas is, I was just the blacksmith here, and you didn't know I was wanted for murder in Durango or anything else about my past. Tell him Einar runs the place, and when Einar found out what happened, he went for the sheriff in Los Angeles, but I ran away before I could be arrested. You have no idea where I've gone. Help him get well again and then put him on a stagecoach and send him back to Durango. Okay? I'll send you my location coordinates so you can let me know how he's doing.\"\n\n\"You want us to bring you supplies?\" I asked. \"Food, aguardiente, anything?\"\n\nHe just shook his head and swung up into the saddle. We watched him urge his horse to a swift canter, and away he went down the canyon.\n\nImarte sighed deeply. \"The grim history unfolds,\" she intoned. \"How thoroughly characteristic of the American West, yet analogous to clan feuds in Corsica or Scotland. How unfortunate, but how fascinating, don't you think?\"\n\n\"Unfortunate,\" I replied.\n\nTomas was out of danger pretty quickly, with no infections to complicate things; no fever, no ravings, only a pale polite boy with enormous dark eyes asking us if we knew where the blacksmith went. We told him what Porfirio told us to tell him, and he accepted that quietly enough, and was unconscious again. His body needed the sleep. Young mortals heal almost as fast as we do.\n\nAfter a few days he was able to ask Einar to go recover his trunk, which he'd hidden in a bush in Encino after leaving the stagecoach at Garnier's. It contained two other suits of clothes, a shaving kit, and several small personal items, including a daguerreotype of a woman and child. His mother, no doubt, because the child standing stiffly beside her chair was certainly Tomas. He had a wistful little smile for the camera. She was dressed in black, and her eyes were black too, cold and angry. Was there maybe a resemblance to Porfirio in the shape of her face, the chill in her eyes?\n\nTomas asked for his gun back. We checked with Porfirio, who was hiding out in a high narrow canyon on Mount Hollywood; he sighed and then transmitted his okay to give it back to the boy, cleaned and loaded. This was Los Angeles, after all. We hung his gun belt over the bedpost. He woke up long enough to see it there and gave us his little grateful smile, just like the child in the picture. Then he slept again.\n\nSo there was this kid, in Porfirio's room, with his luggage and his weapon and his mystery. Imarte hovered over Tomas, attending to the tiniest details of nursing with perfection, hoping he'd murmur some part of his story in a delirium or gasp confidences into her motherly bosom. He didn't. He barely spoke at all, to any of us, and when he did finally speak, it was to Juan Bautista.\n\nI was sitting by the bed, reading a back issue of _La Estrella_ and waiting for Imarte to take her turn at vigil, when Juan Bautista entered the room. Tomas opened his eyes at the noise, and then opened them wider. It's not every day you see silver-haired Indian kids with condors perched on their heads.\n\n\"That guy came and wanted to talk to Imarte again, so I said I'd take over for you,\" Juan Bautista told me in Cinema Standard.\n\n\"Fine,\" I said, and rose from the chair. I was starving. As Juan Bautista sat down and I left the room, I heard Tomas asking shakily, in Spanish:\n\n\"Why do you have a bird standing on your head?\"\n\nI fixed myself a plate of Imarte's goat stew, that's how hungry I was, and sat down with my back against the wall to eat it. As I ate, I became aware that there was a conversation in Spanish going on in the room on the other side of the wall. Sharpening my reception, I was able to pick it up.\n\n\" . . .and I feed her fish when I can get them,\" Juan Bautista was saying. \"Mostly sardines, you know, in those little square cans with the funny openers?\"\n\n\"That's really neat,\" I heard Tomas reply.\n\n\"Yeah. I thought about getting her some canned oysters, but they're _way_ expensive.\"\n\n\"I had some oysters in Santa Fe one time. I didn't think much of them. What do you feed the other ones?\"\n\n\"Just seeds and stuff. You know. Except for the little owls\u2014I catch crickets for them. Watch out. He's telling you he needs to go to the bathroom. Here, give him back a minute. Good boy! See, I've trained him to let me know. Now you can hold him again.\"\n\n\"He's, like, the biggest parrot in the world or something,\" Tomas said, giggling weakly. \"Boy, I'd love to have one of these guys to wear on my shoulder when I walk around the ranch. Condors don't talk, huh?\"\n\n\"No. Only the _Psittacidae, Corvidae_ , and _Sturnidae_ talk. The _Sturnidae_ are just mimics, though, they don't understand speech like parrots.\"\n\n\"We used to have a parrot when I was a kid.\" Tomas's voice was a little sad. \"My father bought him for my mother. He was . . . a real man, my father.\"\n\n\"And he was killed by that blacksmith who worked here?\" Juan Bautista said, cautiously. I frowned at my empty plate and wondered if he was going to let slip any details he shouldn't know.\n\n\"The same man who shot me.\"\n\n\"You came looking for him?\" Juan Bautista asked.\n\n\"I didn't think I'd find him, I didn't think he would still be alive, not the kind of guy he was. I thought . . . if I could just find his grave or something, I could spit on it and go home and tell my mother I'd done that, at least. I was going up to San Francisco to look for the grave, because I'd heard they had a lot of criminals up there. But then the stagecoach stopped here, and there he was, and I knew I had to kill him. For my father.\"\n\n\"You loved your father?\"\n\n\"I was only a kid when that son of a bitch killed him, but I remember we used to do stuff. He taught me how to ride. His favorite book was _Don Quixote_ , and he used to read to me from it, and one year for my birthday, you know what he did? He made me a whole set of wooden figures from the book, he carved them himself. There was a Don Quixote with these long jointed legs, you could sit him on Rocinante or you could make him dance or turn somersaults. He had a lance and shield and helmet. There was a windmill with vanes that really turned, and there was a giant's face carved on it. There was a Sancho too, but he didn't do anything, he just sat on his mule. They were the best toys My father loved me better than anybody else ever did.\" Tomas's voice grew a little muffled.\n\n\"But . . . what happened to your fingers?\" asked Juan Bautista, bewildered. He didn't get it.\n\n\"That was the other guy, the one who murdered my father. I don't remember what happened, I was too little, but my mother said he'd been drinking. He was this lowlife friend of my uncles'. They used to work on our ranch, and they brought this friend of theirs. He was our majordomo for a while. But he was a drunk, always getting into trouble. I remember he'd yell and break things, and I'd be scared. I don't know why my mother let him yell at her that way, she was so stupid. Wouldn't you think she'd just have told my father? _He_ wouldn't have let anybody talk to his woman that way, if he'd known. But I guess he never found out about it, until that night.\"\n\nHad the mother twisted the story for her son? Or had she simply never answered his questions? What was wrong with the mortal woman, anyway?\n\n\"You want some of this tea?\" Juan Bautista's voice was a little shaky. \"It's good for you.\"\n\n\"Thanks.\" There was a pause as Tomas drank, and then he continued:\n\n\"Sometimes I can remember a little. We were having a party. I was scared. I remember the yelling. There was a fight. That guy was drunk, and he hurt my hand and killed my father and one of my uncles when they came in. He ran away, and nobody ever caught him.\n\n\"After the funeral, my mother sent the rest of her brothers away, because it was their fault that guy came to the ranch. She said she'd rather have me be the man of the family than any of them. Sometimes, I wish she hadn't done that, because it's hard running the ranch without help. She's still angry after all these years.\"\n\nPoor boy. What a weight on his shoulders.\n\n\"Still angry?\" asked Juan Bautista.\n\n\"Oh, yeah. She told me I had to find that guy as soon as I was old enough and avenge my father's murder.\"\n\n\"Well . . .\" I could hear Juan Bautista shifting around uncomfortably. \"You know what? You could go back and tell her you got him. He shot you, but you shot him first, see? And I'd be a witness for you. We could write a letter. You know, like a deposition? Everybody here would sign it, telling her how you killed him. Then you could go home with proof, and it would be all over.\"\n\n\"I can't lie to her,\" Tomas said. \"She always knows. . . . Anyhow, I don't want to go home until I've killed him. It wasn't until I saw his face that afternoon that I remembered how scary it used to be, how my mother cried. I have to kill him for her, but I also have to kill him for me, for taking away my father. I hadn't even remembered how much my father loved me until I saw that man.\"\n\n\"But he almost got you,\" Juan Bautista said. \"He's a real good shot, and now he knows you're after him. You'll get killed next time.\"\n\n\"Maybe I'll kill him instead.\" Tomas didn't sound certain. \"Or he'll kill me, and then I'll be in Paradise with my father. I don't care which. What would you do, if you were me?\"\n\n\"I don't know,\" Juan Bautista said. \"I'm an orphan.\" He sounded as though he was grateful for that. \"But I wouldn't want to die for no reason. I mean . . . what if your mother got some of the details wrong?\"\n\n\"She never gets details wrong.\"\n\n\"Have you asked your uncles about what happened?\"\n\n\"How could I?\" Tomas sounded tired. \"She sent them away, and they never came back. I don't know where they went or if they're even still alive.\"\n\n\"You need to sleep some more,\" Juan Bautista said. \"Your blood pressure's too low.\" I winced; this was the kind of cover-blowing remark young operatives make. He might as well tell the mortal boy he could scan him. \"You can't do anything anyway until you get well. So you should rest and drink a lot of tea, okay?\"\n\n\"Okay,\" said Tomas in a fading voice.\n\nHe rested and drank a lot of tea, and the color began to come back into his face as the days passed. Soon he was able to get out of bed and totter around the room, and then to put on his clothes and sit outside in the chilly November sunlight.\n\nTomas was a nice boy. He was quiet and courteous, for a seventeen-year-old; he didn't brag or try to impress us with how tough he was. He spoke respectfully to ladies and deferred to Einar and Oscar. He was really taken with Juan Bautista's aviary, and the two of them spent hours in there cleaning the cages and talking about birds. And wasn't he a good son? Ready to roam the face of the earth to deliver blazing death to a stranger, or suffer blazing death himself, for his mother's sake.\n\n\"What are we going to do?\" said Juan Bautista, late one evening when Tomas was safely asleep in Porfirio's bed. \"I never knew mortals were crazy like this. They look so normal. How are we going to stop him from going off to hunt for Porfirio, when he's better?\"\n\n_You can't_ , transmitted Porfirio from the mountain. He sounded glum. _He won't go home looking like a fool or a coward. If I could take my head off and send it home with him as a trophy, I'd do it_.\n\n\"Why not simply let the lad continue with his search indefinitely?\" said Oscar. \"It doesn't sound to me as though his home is a terribly pleasant place. Perhaps he's better off wandering the world and having adventures. Certainly he won't expect you to come back here. We might see him on his way, and then you could come out of hiding.\"\n\n_What kind of adventures is he likely to have, poking his nose into every den of thieves he comes across, looking for me? Because that's what he'll do, you know that, and the next time he gets shot, I won't be there to hook him up to a life-support system_.\n\n\"He's stubborn,\" Iniarte said. \"I've done my best to explain to him how this kind of primitive revenge ritual invariably results in the destruction of all parties. Regrettably, he comes from a society where a greater value is placed on abstract cultural values than individual human life. One is reminded of the Japanese custom of\u2014\"\n\n\"I don't suppose there's any way you could tell him the truth?\" I asked Porfirio. \"That his memories are all screwed up, that his father was the real bastard and you were the good guy?\"\n\n_Can you think of any way I could get him to listen to me_? transmitted Porfirio bitterly. _And even if I could, what then? He thinks he had a father who loved him and died in his mother's defense. Her version of the story's a lot more palatable, isn't it? What happens when the pretty paper is stripped off my ugly present of truth? His mother becomes a liar, at the very least_.\n\n\"She sounds like a crazy bitch anyway. Why should she hate you?\" I said.\n\n_I killed the man she loved. She would have forgiven him anything, but she'll never forgive me. And I shouldn't be forgiven! I could have prevented this. If I'd been quicker, I might have stopped the bullet that killed Bartolo. I might have disabled Jaime without killing him. I might have been there for the boy all these years. God in heaven, see what she's done. The family is broken, scattered, when I'd fought so hard to keep them together where I could look out for them. How will I ever find Juan and Agustin again? Are they even alive? And she, look at what she's become, look at what she's done to that boy. If I had only stayed. ._.\n\n\"You couldn't have, man,\" Einar said. \"You know that. You belong to the Company. First time Dr. Zeus had a job for you somewhere else, you'd have had to go. And even if you'd stuck around, do you think you could have kept on micromanaging their lives forever? We may be immortals, but we can't control mortal destinies. We can help them when they want help, but that's it. When they want to destroy themselves, not even God can stop them.\"\n\n\"That is so true,\" I said, with all my weary heart. \"It's that lousy rotten free will of theirs. All we do is run around cleaning up after it.\"\n\n\"We're the everlasting janitors in the Big Bathroom,\" Einar agreed. \"Our only consolation is knowing how much worse things would be if we weren't part of the equation. Anyway, kids: here's an idea, culled from my centuries of wisdom and cinema expertise: We've all seen films where a guy fakes his own murder.\"\n\n\"I can think of several,\" Imarte said. And then, \"Oh.\"\n\n_Let him shoot me_?\n\n\"We'll load his gun with blanks and fix you up ahead of time with a full suit of charges and stage-blood bags. He'll think he's blown you away and go home with the revenge thing all finished.\" Einar jumped up in his excitement and began to pace. \"Six Cawelti squibs, a sheet of body armor, some red dye and corn syrup\u2014man, we can make you look like you had breakfast at the OK Corral!\"\n\n###\n\nIT TOOK NO MORE THAN a week to prepare, during which time Tomas was getting stronger. Against Imarte's entreaties, he set up a target on a tree and practiced shooting at it. Juan Bautista tried to talk him into going home, but there was no arguing with the boy: as soon as he was well enough, he planned to take the stagecoach out to Calabasas, famed locally for its bandit population, to see if Porfirio had gone there to hide from justice. On the day he was finally strong enough to ride, however, Einar came riding up from the bottom of the canyon at a dusty gallop.\n\n\"Hey! Amigo!\" he called out in crude Texan Spanish. Tomas turned. He'd been target shooting. \"Yeah, you. I think I finally got a line on that no-good murdering hombre that killed your pa. Looks like he's holed up not five miles from here. You want to go see if you can get him?\"\n\nI looked up from the cookfire to watch the boy's reaction. He stood frozen, staring with those enormous dark eyes. Slowly and deliberately he holstered the pistol. \"Will you have the kindness to loan me a horse, se\u00f1or?\"\n\n\"Get him a horse, Juanito,\" Einar said.\n\nJuan Bautista gulped. \"You don't want to do this,\" he told Tomas. \"Think how bad you'll feel afterward.\"\n\n\"No. If I get it over with, I can be my own man,\" said Tomas.\n\nSo Juan Bautista brought out a horse, saddled and bridled, and Tomas climbed awkwardly into the saddle. He looked sick and dizzy up there.\n\nEinar reached over and took his pistol out of its holster. He spun the empty chambers. \"Won't do you any good unless you reload, son. Here, take mine.\" He handed off the gun full of blanks, and I looked meaningfully at Juan Bautista. First bit of stage business in our play accomplished.\n\nTomas stuck the gun in his belt and drew a deep breath. \"Let's go, se\u00f1or. Take me to that murdering son of a whore.\"\n\nThey rode away down the canyon, dust clouds spiraling up. Juan Bautista stood looking after them, wringing his hands. \"I have to see,\" he said, and ran for the stable, from which he emerged a moment later leading another horse.\n\nI ran to him as he got into the saddle. \"Take me along. This ought to be some show.\"\n\nHe put out a hand, and I leaped up behind him, just as Imarte came running out of the inn. \"What is it? Is it time? Don't tell me they've left\u2014\"\n\n\"We'll let you know how it went,\" I said, and we galloped away.\n\nWe followed the dust clouds north through the pass, up the grade to Dark Canyon, and over the flank of Cahuenga Peak. We were coming to the high ground behind Mount Hollywood, which sloped gently down toward the river, and I wondered if Einar had chosen the spot deliberately. You could call it an appropriate location for a shootout; this was the site of an immense graveyard in the distant future. Nearer our time, Griffith would shoot the battle sequence from _Birth of a Nation_ here, and for all I knew the battle scene that ended _Intolerance_ too. Einar had told us about it enough times, in his babbling enthusiasm. Now, before it became that famous place, it would serve as a small and private theater for a grim farce.\n\nBut where were the actors?\n\nWe saw the two horses tethered in a stand of oak trees. And, just creeping over a ridge and peering into the hollow beyond, there were Einar and Tomas. Juan Bautista urged our mount forward.\n\nPorfirio was advertising his presence. A plume of smoke rose thinly. He'd made a cookfire.\n\n\"Up there,\" I said, pointing to our right. We could climb a winding trail for a better view. Juan Bautista hesitated, then swung the horse's head uphill, and we ascended hurriedly, peering through the oak trees.\n\nPorfirio was sitting by his fire, warming his hands. He looked tired. He glanced up as Einar broadcast, _We're here, man. Is it a good day to die_?\n\n_Always_. Porfirio looked down at the embers again, deliberately ignoring the mortal boy's clumsy approach through the weeds. Oh, God, the kid slipped, was tumbling down to land in a crouch not ten yards from where Porfirio sat. Did he really want to do this? Up on the ridge, Einar pulled out the little electronic control and waited.\n\nPorfirio raised his face. Tomas made a stifled sound and staggered backward. We couldn't see his face, but we had a fine view of Porfirio as he got to his feet and held out his hands, his empty hands.\n\n\"Very good, _mi hijo_. It's my turn. Blood for blood, so you can be a man.\"\n\nThen the boy was firing wildly, _bang bang bang_ , and the charges detonated in perfect time with his shots. The blood bags exploded outward, and Porfirio spun and fell. The noise of the shots echoed; it hit the face of the ridge like a wave breaking and rolled back down the slope, prolonging the moment, washing us in the sound. Out on the valley floor beyond the river, a dog began to bark.\n\n_It's a wrap. Beautiful_ , Einar transmitted. _You all right_? He came jumping down the slope, stuffing the detonator remote into the pocket of his jeans.\n\n_Damn things hurt_ came the answering transmission. I confess I was relieved to hear Porfirio reply; that had been a truly convincing death scene. Look at him now, the way he lay there ashen and motionless, his lean villainous face frozen in a snarl like a dead animal's. But then, who could counterfeit death better than an Immortal? We see so much of the real thing.\n\nTomas had dropped the gun and was doubled up, retching. Einar caught him and steadied him. \"Come on, boy. We have to get out of here. No sense getting yourself hanged. Let's go, let's go.\" He practically carried him up the slope and down the other side, to the place where their mounts were waiting. Juan Bautista spurred our horse through the sagebrush toward them.\n\n\"Did you get him?\" I asked, playing my part.\n\n\"Got him, all right,\" Einar said, boosting Tomas into the saddle. \"Now this boy's satisfied his debt of honor and he can go home to his ma with a clear conscience. Isn't that right, son?\"\n\nJuan Bautista rode up to Tomas and peered into his face worriedly. Tomas looked deathly ill again, as bad as on the night he'd been shot.\n\nThen we all thundered away through the November evening, and the sun was setting red as blood, and the shadows were long. Tomas wept the whole way, and when we walked our horses into the innyard, he tumbled off his horse and into Imarte's waiting arms.\n\nShe folded him into her bosom.\n\n\"You poor brave boy. You come with me, tell Marta all about it,\" she cooed. _I trust everything went off as planned_?\n\n_You shoulda seen_ , Einar transmitted.\n\n_If anyone had bothered to tell me in time, I might have_ , she replied, giving me a nasty look. She turned and pulled Tomas away with her into the inn, doubtless to obtain a first-person narrative from him and gain valuable insights into the culture of machismo. The damned harpy.\n\nShe nursed him through the hysterics, and I think she rewarded him the way a man wants to be rewarded, and as the evening wore on it, she gave him aguardiente too to bolster his sense of worth. He cheered up tremendously and began to swagger and sing, as I'd never heard him do before. We all assumed it was the relief of being out from under this burden he'd carried his whole young life. But Juan Bautista listened for a while and then vanished silently up the canyon, taking Marie and Erich with him. I busied myself with making some plain beef stew\u2014I'd had all the Chaldean Surprise I could stand\u2014and Einar bounced around taking care of his innkeeper duties, still pleased with himself at the way his special effects had turned out. Even he began to look a little concerned, though, as the noise level in Imarte's room rose.\n\nI was trying not to listen to what seemed to be an argument developing, when I picked up Porfirio on the ridge behind us, just arriving.\n\n_Mendoza_?\n\n_Yes_.\n\n_I could use some hot food_.\n\n_Hang on_.\n\nI ran and got a blanket from my room, and half a case bottle of aguardiente, wrapping them together. I got a bowl of stew and a spoon and hurried up the canyon. As I ran, I could hear Tomas emerging from the inn, shouting to Imarte to leave him the hell alone.\n\n\"Whoa, son, where are you going?\" Einar said, getting up.\n\n\"Set up the bottles!\" the boy shouted. \"Set up the bottles and give me that gun!\"\n\nTarget shooting again? I shrugged and kept climbing.\n\nPorfirio was sitting quietly in the darkness, gray as a ghost, which he looked like in his serape with the holes and bloodstains all over it. I put the bowl of stew into his hands and threw the blanket around his shoulders.\n\n\"Thanks.\" He turned the bowl in his hands, savoring the warmth. I sat down beside him and uncorked the aguardiente bottle.\n\n\"How is he?\"\n\n\"I think he's having some kind of hysterical reaction,\" I answered delicately. \"But I guess that's normal if you think you've just killed somebody. It looked great, by the way.\"\n\n\"The kid's drunk,\" Porfirio said with a scowl, gazing at the circle of light around our cookfire. \"Listen to him.\"\n\nGunfire, followed by Tomas's shrill laughter. He was telling Einar to bring more bottles to shoot at. I had a gulp of aguardiente myself. Porfirio spooned stew into his mouth, but he never took his eyes from the fire. They were dark and cold.\n\n\"Listen to him, down there,\" he said. \"He thinks he's some killer, he thinks he's one hell of a man.\"\n\n\"You did your best,\" I said. \"What else could you do, Porfirio? At least this way he can go home and make his mother happy.\"\n\n\"She'll never be happy,\" he said, emptying the bottle and throwing it away. \"My fault, I guess.\"\n\nMore shots. We could hear Einar making a very tactful suggestion and being refused indignantly. Porfirio exhaled hard.\n\n\"I was so relieved when she married Jaime. At last, I thought, somebody who'll take care of her, and he's even got money. But see how that turned out. I could go looking for Juan and Agustin, I guess, when this job is over; I could try to track them down and see if they've married, if they've kept the family going. But what am I going to do about her? And what am I going to do about that kid?\n\n\"Look at him down there, strutting around with his gun. He's bought into the whole damned lie about blood and honor and revenge. He was made to feel like a little nobody all his life, but now it's payback time. Nobody's ever going to tell _him_ what to do again, not now that he's killed somebody. Ay, ay, ay.\"\n\nPorfirio buried his head in his arms. \"Who will take this curse off my family?\" he asked the night.\n\nThe party didn't last much longer. Tomas got cold and ran out of things to shoot at, and Einar got tired of dodging bullets and vanished into the bushes, so Imarte came out and tried to get the boy to come to bed before he caught pneumonia or fell into the fire. He tried to hit her. You don't do that to an Immortal. She swiftly knocked him out and carried him indoors like a sack of flour.\n\nSome time in the afternoon next day, Einar roused him and got him into a change of clothes. When the 1600-hours stage came rolling up, Tomas was bundled into a seat, still groggy, and Einar loaded on his trunk and paid for his passage southbound.\n\nPorfirio came out of the hills, and our lives resumed their courses. I don't suppose I'll ever see that young man again; but I can imagine how he'll turn out.\n\nCan't you, se\u00f1ors?\n\n###\n\nWELL, IT JUST GOT DARKER and crazier in bad old Los Diablos. Bad things come in by the armful and leave by inches, it's said. Winter came, but no rain; smallpox instead. It began in the old shacks on the hill where the poorer citizens lived, Sonoratown, the locals called it. The few remaining Indians were dying like flies, and then the Mexicans were being wiped out, and pretty soon there were even rows of coffins being carried to the Protestant cemetery. Stagecoach service became irregular, to say the least, in the dry and bitter cold.\n\nDry. There wasn't a creek or a freshet running. Our little stream became pools of standing water, shrinking perceptibly day by day. I don't know where the trout went. Our well hadn't given out, but we were taking serious measures to conserve water. Whatever water we used to wash I took to carrying out to the oak trees, to pour over their roots. Within hours after emptying a pail, you could see tiny blades of grass emerging where the water had been splashed. The land was desperate to cover itself with green; but the rain never fell, and next morning there would be deer slot everywhere, and the grass would all be gone.\n\nThere was water in the sky, all right; there was water vapor holding the haze together, which stung the eyes at midday and kept the adobe rooms cold as death. There was water in the slate clouds that rolled over us and kept going without releasing so much as a drop. The longhorns began to rove into people's vegetable patches, and the last of the old rancheros looked at the brown hills and wondered if they oughtn't borrow some more from the Yankee moneylenders to tide them over what might be an unprofitable year.\n\nWe froze, but we got no rain. San Francisco got rain; but, then, San Francisco never doesn't get rain. It rained back east in Vicksburg, where another battle was fought, and we read rumors of soldiers drowning in their tents. It rained in Mexico, where Juarez sat in his room and calmly considered what he ought to do about Europe. Everybody else got rain, but we were dying of thirst.\n\nAnd smallpox.\n\nOscar stayed home a lot, driving the rest of us crazy; but how was he going to sell anything with people hiding behind their doors, more afraid of the disease than they'd ever been of stray bullets? We saw almost nothing of Imarte, though, so it was a fair trade. She was having a field day, moving among the dying like a scarlet angel, easing their journey out of the world in exchange for life stories gasped out to the sympathetic stranger. To be fair, I believe she nursed a few back to health. When she wasn't busy compiling statistics on mortality, she found time to get some suspiciously British mining engineer to buy her a Peach and Honey at the bar of the Bella Union, and one or two spilled a few more details to support her pet conspiracy theory. The whole thing was so ludicrous, we actually encouraged her to talk about it, on the few occasions she came home; we needed the laughs.\n\nAnd who wanted to celebrate Christmas, on the underside of hell? Time was when I enjoyed walking to a town and slipping into a pew to watch a _pastorela_ , with the earnest mission Indians trying so hard to get to Bethlehem and all the teenaged boys in the parish portraying the devils who tried so hard to prevent them (a role so natural for any teenaged boy). I loved the way the spoken verses would echo in the old church, and the way the flames of the candles winked in their glass cups, and the way the sleepy mortals observed a reverent hush all around me. It was all so charming. And when the Indian pastores finally made it to the stable, after vanquishing Se\u00f1or Satan (who always bore a close resemblance to a gentleman of Old Spain), and the central Mystery unfolded, how lovely to see the black-eyed Mother with her Indian cheekbones and serene smile as she displayed the tiny red Child with his shock of black hair. One could almost come to love mortals again.\n\nOr not. Other years, I'd been alone in the night, where the great trees towered black against the stars, so many white stars, and the air was cold and full of the smell of evergreens. I'd been in the heart of the Mystery then, too. The stars rang like little bells at midnight, and one moment the air would be dead calm on the forest floor, and then a wind would spring up, just on that stroke of midnight, a wind magically warm and full of perfume, and you knew that the Light had begun to fight his way out of his grave, and winter would not last forever.\n\nBut this winter of 1862, that promise seemed to have failed. So many coffins, and not a drop of rain.\n\n###\n\nI DON'T SUPPOSE I NEED to tell you that the hauntings became worse, se\u00f1ors. Became strange. He still pursued me in the night, my dead love, but he seemed to have changed; we seemed to have lost England and gone to places I'd never been. I'll tell you the dream that's clearest in my mind.\n\nI was in a jungle like a Rousseau painting, you know, all those botanical specimens so carefully delineated and dead-eyed jaguars staring forth here and there like so many stuffed toys. Something was coming after me, crashing through the fever-green forest, and where he passed, the palms and ferns and bromeliads all shook to life, lost their neat arrangements, and became real, pulsing and shooting toward the sun.\n\nI'm not sure I was making any effort to run from him.\n\nThen he was coming across a clearing at me, and I could see him at last, the savage, was that a Mayan with his high cheekbones and long curved nose? No, how could I have thought so? This was another kind of naked savage. He was tattooed fearsomely, swirling blue spirals all over his white body, his pale-blue eyes glittering with deadly laughter, and he was on me with the grace and weight of a lion. I went over like a rag doll. What was my stern Protestant metamorphosing into? What atavistic madness was this? He had a flint knife, and it was a beautiful thing, beautifully worked, and as he searched for my heart, I saw the fan palms waving above our twined bodies. I tried to tell him that the fan palm is the only member of the _Arecaceae_ actually native to California, but I was distracted by the discovery that he had the front page of the London _Times_ for January 6, 1863, tattooed on his chest.\n\nI tried to read it as he was busily taking out my heart. Then we heard shots behind him, and he turned with a snarl. Looming above us was a vast blue pyramid, and from its base hunters were coming, sending lead singing through the air. He turned and looked back down at me, and I saw that his face was painted too, a pattern of red and white diagonals crossing on a blue Held. No time, no time to do this properly! said somebody, and he rose above me and lifted the blade in both hands for the stroke of mercy.\n\nBut I was awake and moaning on my cot before he was able to give it, and blue light was crawling away, diminishing into darkness, leaving me miserably, eternally alive.\n\n###\n\nTHE DAY THE ACTORS CAME, we were taken by surprise. We'd been alone in our canyon for so many days, it was hard to imagine a stagecoach ever stopping here again. But\u2014\n\n\"Incoming,\" announced Einar, and with sour laughter we hurried down the canyon to see who was arriving, departing, or just passing through. We heard the shouting as the stage pulled up.\n\n\"Are you mad, man? Are you demented, have you quite taken leave of your senses?\" said a stentorian baritone. \"Press on! Press on, though wolves howl and birds of prey darken the air. D'you want to die of the peste, for God's sake?\"\n\n\"Mister, if that wheel comes off when we're coming down the grade, it ain't gonna be the smallpox kills you,\" said the driver. \"Now you just hush up and set tight. We'll be on our way again soon's we get the spare on.\"\n\n\"Ay, what a wreck,\" said Porfirio, crouching down to look at the wheel. \"You want this thing repaired, se\u00f1or?\"\n\n\"It'll have to be.\" The driver handed off the reins to his partner and jumped down. \"We can't get new ones from the Concord folks nohow, what with the war. You got any spares here?\"\n\n\"Si.\" Porfirio jerked his thumb at the shed. \"You leave the bad one, and I'll see if I can have it ready when you come back down, huh?\"\n\n\"Fair enough,\" the driver said, going to unhitch the team. \"Though I'm half minded to stay up in Frisco, the way things is going. This ain't no business to be in right now. You heard about the Indian attack this summer?\"\n\n\"Indians?\" queried a soprano voice, and the baritone thundered out:\n\n\"Driver, you categorically assured me there were no savages to be encountered on this route!\"\n\n\"Aw, shut your damn pie hole,\" the driver said.\n\n\"What Indian attack?\" Einar asked, bracing the corner of the wagon as Porfirio settled the jack under it preparatory to taking off the broken wheel.\n\n\"Happened in Minnesota,\" said the driver, leading the first of the team to our watering trough. \"Seems the Secessionists are paying 'em to make trouble. They been cutting down telegraph poles, too. You ask me, I think they're smart enough to figure out they can raise all the hell they like with the Army busy fighting itself. Whoever's behind it, I sure don't fancy being stuck out here with a mess of Indians and Mormons and who knows what all between me and Teaneck, New Jersey.\"\n\nThere was a noise like an asthmatic goose honking; it came from the passenger compartment. Juan Bautista and I looked at each other in puzzlement. He walked around to the other side of the stage to see if in fact it was a bird, but just as he disappeared, the last honk ended on a shrill indrawn breath and became evident as a fit of hysterics on the part of the soprano.\n\n\"Oh, we shall not survive! Ingraham, one cannot venture\u2014into such _places_ \u2014without appalling consequence. Such venues. Such wretched venues, and such (for want of a better word) men!\" she shrieked.\n\n\"Have courage, Caroline. It may be that we have escaped the Pale Rider in one form only to encounter him in another; but I say we _shall_ reach the Golden Gate, though calamity leap headlong into our path,\" said the baritone. I walked around the wagon to see why Juan Bautista hadn't reappeared.\n\nHe was staring as though transfixed at a wicker birdcage, which was tied on the back under the trunk enclosure. The leather cover hadn't been fastened down properly, and a flap had blown back, exposing a gnarled and clutching claw the size of a fist. What was in there, a dragon?\n\nJuan Bautista's face was stony with anger. He began to work on the knots that held the cage in place.\n\n\"Hey,\" I said uncertainly. \"Should you be doing that?\"\n\nHe didn't answer me. The knots were proving intractable, so he took out a knife and cut them. He hauled the cage out and down, exposing its occupant. The ranting from the passenger compartment stopped abruptly.\n\n\"Caroline, I believe our trunks are under attack,\" said the baritone, and the gilded head of a cane thrust the window flap aside. A face glared out at us. \"I thought so! Boy, I shall prosecute you with the utmost force of the law. How dare you?\" the baritone snarled. He was a thinly bearded gent in a very loud checked coat. His gray gloves matched his beaver hat, though.\n\n\"How dare _you_?\" Juan Bautista shouted, trembling with anger. I took a step backward. I guess he was mad about the treatment of the occupant of the cage, who was a very, very large something with talons. The cage was way too small for it; it must have been stuffed in there with tremendous trouble, and had scarcely any room to move. Its head was sealed in a leather hood like falconers use, and its legs were bound; all it could move were its talons, which were constantly clenching and unclenching on the floor of the cage, nasty with its droppings. Not even a perch, and no water or food. Erich von Stroheim gronked and snaked his head sideways to peer down at it.\n\n\"Put that back at once,\" said the baritone.\n\n\"No,\" said Juan Bautista. \"Do you know what this is? It's a _Haliaeetus leucocephalus_. Where did you get it?\"\n\n\"You are mistaken, sir, that is a bald eagle, and it might interest you to know that it was a present to me from Chief Two Ducks of the Wyandotte Tribe on the occasion of my successful charity gala in Sudbury, Ontario,\" said the baritone with a sneer. \"Its name is Mister Liberty, and if you don't immediately replace it, I shall be obliged to descend from this conveyance and beat you like the little thief you are.\"\n\nJuan Bautista proceeded to wrench the cage apart with his bare hands. It was only a wicker cage, of course, but the baritone was so shocked, it took him a full ten seconds to roar: \"Damn your impudence. Driver! Are there no laws to protect passengers on this line? Driver!\" He leaned across and shouted out the other window.\n\nThe driver and Einar came around the side of the stage just in time to see Juan Bautista grab up the poor bird and run with him. Erich was jolted from his customary perch and flapped along above them, and the three disappeared in the general direction of Hollywood Boulevard.\n\n\"Oh,\" said Einar.\n\n\"Well, God damn,\" the driver said, grinning. \"I guess you'll just have to file a Damages and Loss Report, mister. It will be reviewed by our claims representative, and when and if an award is made, you'll be compensated by the company in the amount of the registered value of the goods as reported in the parcel manifest or in an amount up to but not exceeding the approximate value of said goods as denned in paragraph 3, article 2A in said document.\"\n\n\"Blow it out your arse,\" the baritone said.\n\n\"Ingraham!\" gasped the soprano.\n\n\"My apologies, madam.\" Ingraham gave me a peremptory tip of his hat. \"Has the word _lawsuit_ any meaning to you rustics? I see it has. You will make every effort to recover Mr. Liberty now, this moment, or I shall own you.\"\n\n\"Aw, that bird was about dead anyhow,\" said the driver.\n\n\"That is immaterial, sir. Are we to be deprived even of the services of a taxidermist? And are you aware with whom you speak? I, sir, am Ingraham Drew Culliman, of the Marlborough Theatre. Perhaps _now_ you'll wish to avail yourself of a post horse in the pursuit of my personal property?\" Ingraham's voice had risen to a frightening pitch.\n\n\"Never heard of you,\" the driver said.\n\n\"A liar as well as insolent. And if I were to mention that the rara avis in dispute was to be the centerpiece in my latest variety spectacle, the Salute to Liberty? That the magnificent emblem of our presently divided nation was to be held aloft by Mrs. Culliman (herself gowned as Martha Washington) at the climax of a musical tribute certain to raise the spirits and cheer the hearts of our boys in blue? There are those, sir, who might construe your detestable negligence as the next thing to treason, which, let me remind you, is a hanging offense.\" Ingraham brandished his cane.\n\nThe driver explained where he was minded to put that cane if Mr. Culliman shook it at him one more time, and added that Mr. Culliman was going to find it uncomfortable to sing or, for that matter, dance in any shows with the cane in that particular location.\n\n\"Hey, hey,\" objected Einar, as Ingraham took off his gloves to engage in fisticuffs with the driver. \"Take it easy. I don't reckon we can get the bird back for you, but we'll be glad to pay you for him. How'd that suit?\"\n\nImmediately Ingraham put on his gloves again. \"Well, of course the bird was a gift, but, taking into consideration the difficulty of obtaining another when we reach our destination, which is San Francisco, where rates are astronomical . . .\"\n\n\"Remuneration in gold, Ingraham,\" Mrs. Culliman advised him.\n\n\"I'd say I couldn't accept less than a twenty-dollar gold piece,\" said Mr. Culliman, and leaned back on his cane and frowned magisterially at us.\n\nEinar blanched, thinking of our budget. \"It's a deal,\" he said in a hollow voice. We waited while he ran up to the inn and raided the emergency fund.\n\nMr. Culliman inspected the gold piece critically, nipping it with a careful incisor before tucking it in the watch pocket of his flowered waistcoat. \"I daresay that'll suffice. One really ought to inform the police in a matter of such flagrant disregard for laws concerning personal property, but as it's the season of charity and goodwill toward men, I will let the matter rest without further prosecution.\" He touched his hat brim with the ferrule of his cane. \"Driver! Pray see to it that our trunks are better secured before we proceed.\"\n\nHe vaulted back into the compartment and let the window flap fall. We could hear Caroline demanding to see the gold piece. The driver gave him a particular salute and lounged back against a boulder, rolling himself a cigarette.\n\n\"Is there trouble?\" Porfirio asked, returning from the shed with a spare wheel. Einar and I shook our heads mutely.\n\nJuan Bautista didn't return until three hours later. Erich was back on his customary shoulder perch, and the eagle was still clutched in Juan Bautista's arms. It had had the blinding hood removed, and it was a bald eagle all right, though a pretty sorry specimen to have cost so much. A lot of its feathers had been pulled out; and though I'm no judge of avian expressions, its eyes had a kind of fixed and glassy stare of rage that I'm sure live birds don't usually display. It made Porfirio's expression look mild by comparison.\n\nOne look at him, though, was enough to reduce Juan Bautista to tears.\n\n\"I tried to set him free,\" he said. \"I wasn't going to bring him home. But he can't fly. Somebody broke his wings, and they healed wrong. Please! Let me try to fix him. I can do surgery that'll fix it so he doesn't hurt anymore. I won't keep him, I promise.\"\n\n\"The bald eagle will be on the endangered-species list by the middle of the next century, you know,\" Einar said helpfully.\n\nPorfirio gritted his teeth and swallowed the fire and brimstone he'd been preparing to spill. \"Why should I care?\" he said, throwing up his arms. \"Go ahead. The stink from sardines in your room is already enough to knock over a Turk, but you're the one who has to live in there. Maybe _this_ one'll eat all the goddamn pelican chow I ordered.\"\n\nIn fact, the eagle seemed to like it. John Barrymore (so named after lengthy debate between Juan Bautista and Einar) was definitely not a normal bird. Not cute or whim al, either, after the months of abuse and neglect he'd suffered. Psychotic.\n\nAt least he didn't bite, and Juan Bautista managed to get him to stop pulling his own feathers out; but he destroyed every cage Juan Bautista made for him, no matter how roomy. Yet he never flew; he never even tried. He would stalk around like King Lear on the blasted heath, glaring at the whole world. Usually he followed Juan Bautista around, but occasionally he decided to travel his own path. It was unnerving, while reading the paper on one's cot, to look up into his accusatory stare not three feet away. The only thing that sent him into a homicidal frenzy was anyone attempting to pick him up, so rather than risk his talons, one had to lie there and yell for Juan Bautista to come coax him away. I got pretty tired of this after the third or fourth time. I don't know what the damn bird thought I'd ever done to him.\n\nHe was a symbol of many things, se\u00f1ors, not least of all this nation, crazed and self-destructive as it was. None of us could fly from that desolate place. Though the New Year arrived, there was a general feeling of the light going, waning, chilling, the feeling that we were journeying downward into darkness. The land sick, the people sick and crazy, certain ruin trundling toward us like a siege tower.\n\n###\n\n\"WHAT A TEDIOUS TIME this is,\" said Oscar, hitching up his trousers to protect the crease as he sat down. \"I haven't sold anything but black dye in weeks. All my Christmas business vanished in quarantine; with so many funerals going on, nobody wanted to buy presents.\"\n\n\"At least your business will pick up eventually,\" I said, pouring myself a cup of black coffee. \"Mine's going to hell without a return ticket. You should see it out in the temperate belt these days. I have to fight off the longhorns to get to my specimens. Everything's being grazed right down to the ground. Extinctions, honest-to-God extinctions happening right before my eyes. Or would be, if I wasn't collecting.\"\n\n\"Oh, surely not,\" he said. \"Isn't this region prone to droughts? Wouldn't the local flora be resistant?\"\n\n\"It's resistant to drought, all right, but not to being eaten by starving cattle.\" I sipped my coffee. It was bitter, but I drank it anyway. \"Think of the way things looked this time last year. Remember how green everything was? The cattle herds do. They wander out every day, looking for the green. All their instincts are telling them to head for the salad bar after a long winter. Salad bar's closed, unfortunately.\"\n\nOscar sat straighter, struck with inspiration. \"Ah, if only there were a rainmaking apparatus. Talk about supplying demand! Imagine how that would win the trust and affection of one's consumers.\"\n\nI tilted the last bitter drops out on the dry earth. \"If it didn't work, you'd need a fast horse or a good tar remover.\"\n\n\"Never happened to me,\" he said, waving a hand airily. \"I am no charlatan. I carry nothing but merchandise of the highest quality.\"\n\n\"You _are_ a good little machine,\" I said.\n\n\"The best,\" he replied with conviction. \"How can you persuade mortals to trust you if you lie to yourself? And where's the passion, the suspense, the triumph of the whole business, if it's all a sham? Really, the field material is almost of secondary importance, because you can't obtain worthwhile data in an artificial situation. I've worked with partners whose heart just wasn't in the Deal, if you know what I'm saying. No focus on the act of enticement at all, merely on obtaining data. They might as well have _been_ clockwork automatons, and don't you think the mortal customers didn't sense it. No, sirree. They froze up and hadn't two words to say about themselves.\"\n\n\"Basic characterization,\" I said in agreement, remembering what we were taught in school. \"Believe in your character, and the mortals will, too.\"\n\nOscar held up an admonitory finger. \"More than that. Believe what your character believes, and the illusion is unbreakable. A method guaranteed never to fail, under the most adverse circumstances. When the most important thing in the world to me is getting that gentleman or lady to make a purchase\u2014when I have them craving it, whatever it is\u2014their souls open, and they reveal all manner of secrets about themselves. They would never suspect I wasn't human, even if they saw gears and cogwheels flying out of my mouth.\"\n\n\"Well, but\u2014really, we _are_ human,\" I said uncomfortably.\n\n\"Just so!\" He nodded his approval. \"That's the attitude to take if you want a successful career, you may depend upon it.\"\n\nI stared at him, and at last I asked, \"So . . . have you always been stationed in the New World?\"\n\n\"Native of this country, I'm proud to say!\" He smiled in fond remembrance. \"Born if not bred here. That Croatoan affair. No memories of a mortal life at all, of course\u2014recruited as an infant. Orphaned, I gather. Some nasty business with the local redskins. No hard feelings, naturally. Quite an irony, wouldn't you say, that savages with stone tomahawks were the shapers of my destiny, with all its splendid artifice? I went straight back into the field after my graduation, too, never been back to a base since. Never wanted to! Why in blazes would I want to loll around idling, watching holos, when all the glamor and excitement's out here?\"\n\n\"You find this glamorous?\" I jerked a thumb at the bleak hills.\n\n\"Heavens, yes.\" He was incredulous I should even ask. \"This is the very edge of the world! You can't go any farther out in America, not until we acquire Alaska and Hawaii. And what's my function here? To document the forces of civilization at work, as they transform this murderous wilderness into a place where decent folks would want to live. The more savage it is, the greater the challenge. It would be exciting enough if one were only a spectator; but look at me. I'm not only bearing witness to Manifest Destiny, I'm an apostle of it, by gum!\"\n\nI decided he was crazy too. \"How do you mean?\" I asked.\n\n\"By advancing the standard of living through the availability of fine merchandise, of course. With every labor-saving device or can of stove polish I sell, chaos is dealt another blow in this wilderness. Even when I don't actually conclude a transaction, even when those penniless folk stare openmouthed at the splendor of my wares but come not forth to buy, they go home with visions of a better world dancing in their heads.\" Oscar rose to his feet and swept off his hat.\n\n\"And they'll _desire_ those visions, and there will be those among them who dare to improve their mortal lot, that they might purchase some measure of that splendor, some glittering prize, though it be but a fragment of the glorious whole. The idle will seek employment, the chronically hapless will become sober and industrious, and noble ambition will animate the frames of those who now lie torpid and indifferent to what they _might_ have, if only they would rise to embrace it.\"\n\nThere was a breathless pause.\n\n\"Oscar,\" I said at last, \"you will go far.\"\n\n\"Excelsior!\" he said, and thrust his hat skyward as far as he could reach.\n\nAt this moment, we both noticed the approach of a vehicle. It was the wrong time of day for a stagecoach. Whatever it was, it had turned off the Camino Real and was rolling up our own little canyon, going right to the door of the inn. We turned to stare.\n\nIt was a fine two-horse open carriage, slightly antique, oxblood in color, with the arms of some grand old Spanish family blazoned on the body. It had been blazoned there an awfully long time ago, though, to judge from the way it had faded. A black man in a red coat drove it, and seated within was a mortal lady of our mutual acquaintance.\n\nOscar gasped. He had been in the act of returning his hat to his head, but now he swept it off and bowed double.\n\n\"Princess Rodiamantikoff,\" he said.\n\nIt was even she; but how changed. Gone were the Gypsy silks and cheap baubles. She wasn't more tastefully dressed, you understand, but certainly more expensively, and there was now a coherence and even a dignity to her ensemble. She'd found some good luck somewhere. Her plain face was fuller by a few square meals, but the blue eyes were still knife-sharp, unwavering, superfocused. She extended a regal arm, pointing at Oscar with her parasol as the carriage braked to a stop.\n\n\"It is he,\" she said. \"At last we find you. Chief Running Deer and King Elisheazar have not searched cosmic ether in vain. You may approach us, sir, for we would discuss with you matter of trade.\"\n\nThe effect on Oscar was\u2014well, it was indecent. He was beside the wagon at once, planting a fervent kiss on her outstretched hand. The black coachman looked at him askance.\n\n\"Your Highness!\" Oscar said. \"How pleased I am to see that your fortunes have improved. Doubtless loyal friends at the distant court have contrived to send you support of a material nature?\"\n\n\"Naturally,\" she said grandly, lying through her teeth, if her pulse and respiration rate were any indication. \"Not to mention certain assistance rendered by dear Spirit Guides and others in realms above who are anxious to see that great work goes forward.\"\n\n\"And what great work would that be, ma'am?\" Oscar asked, terribly interested.\n\n\"Ushering in of new era,\" she said. \"Epoch when unhappy multitudes gain peace and enlightenment through communication with world beyond. Secrets known only to arcane secret societies will at last be revealed to all! Futures foretold through modern methods of cartomancy passed on from ancient Egypt through Gypsy race. Loved ones who have passed over will send advice and encouragement through gifted individuals. We are pleased to be humble instrument of Spirits' will. Spirits have told us you were also instrument, bringing cards for entertainment purposes only.\"\n\n\"So Your Highness has improved her situation by telling fortunes?\" Oscar's eyes were wide with fascination, his cameras rolling.\n\n\"Please.\" She raised a hand. \"Grateful clients have presented tokens of esteem for messages received from beyond. We are now enabled to live in gracious home in better area of City of Angels. But now, Spirit Guides have advised we must prepare doorway to Spirit realm through construction of beautiful altar. Offerings will open pathway for clients to speak with loved ones through intercession of Spirits. Common household object of beautiful design must be used for this. Spirits have directed us to purchase from you beautiful cabinet whose gross material purpose is keeping pies. It will be consecrated to higher use through addition of sacred plates of metal from ancient Egypt, location of which revealed to us in trance.\"\n\n\"The pie safe!\" I think Oscar leaped a foot in the air. \"You wish to buy the Criterion Patented Brassbound Pie Safe.\"\n\nShe nodded demurely. \"Do you deliver?\"\n\nHe certainly did.\n\n###\n\nYOU NEVER SAW A MAN, mortal or immortal, strut around so. It took us a few days to get all the ingredients for a New England boiled dinner together, during which time we were treated to multiple retellings of the story of the sale, with the hunt, the chase, and the astonishing moment of the kill. What a triumph for the good gentlemen of the Criterion company! What invaluable documentation of the development of spiritualism as a movement in America, throwing new light on its evolution on the West Coast!\n\nThe dinner itself consisted of a big chunk of beef brisket, boiled, with side bowls of boiled potatoes, boiled onions, boiled cabbage, and boiled parsnips. There was brown bread with raisins, but even that was water-cooked, steamed in a can over the coals, like a plum pudding. Everything was liberally buttered and mashed, with lots of salt and pepper, which it very much needed, especially the beef.\n\nIn honor of the occasion the meal was served indoors, on our rickety kitchen table made bright with a sheet of checked oilcloth. We crowded around, Oscar and Porfirio in our two chairs and Einar and I seated on kegs from the storeroom, basking in the steamy warmth. Juan Bautista was obliged to take his meals in his rooms nowadays, lest John Barrymore attempt to commit suicide in his absence, and Imarte was out on the prowl. It was pretty cheery in there, even with the Boiled Everything, especially after Porfirio brought out an earthenware jug he'd been keeping warm in a covered basket.\n\n\"Okay, Yankee man,\" he said, \"it's time for a toast. Hot rum punch, courtesy of the house.\"\n\n\"Oh, my,\" said Oscar, rising unsteadily to his feet, doubtless feeling the powerful gravitational force exerted by his ingested supper. \"And isn't this just the weather for it, too. I haven't had rum punch in decades. You're a prince, sir.\"\n\n\"Hell, we always knew you'd sell that thing,\" Porfirio lied, carefully tilting the jug to fill our graniteware mugs. Out jetted a stream of something as red as a streetwalker's dress, dotted by bits of orange peel and clove and fragrant with fiery rum. We howled in anticipation and raised our drinks high.\n\n\"To a radiantly successful mission, Oscar,\" Porfirio said. \"Not only for unloading the pie safe, but for the commendation the Company has decided to grant you for the sheer volume of sociological material you compiled while you were trying.\"\n\n\"Surprise!\" Einar and I yelled, and Oscar turned pink.\n\nPorfirio held out a hand for dignity and order. \"And what could be more appropriate in your honor,\" he said, \"than a polycultural cocktail? The cranberry of New England, the orange of Old Spain, the peach of Georgia, spices from the Far East, and rum from Jamaica, all boiled and served as hot as your pursuit of the Willing Customer. We wish you many more, man.\" He threw back his head and gulped the drink down, and we followed his example.\n\nOscar actually got misty-eyed. \"I'd no idea,\" he said. \"A commendation? Imagine. All I've ever wished was to do my job, you know, to the best of my limited abilities. Setting aside false modesty, though\"\u2014and he stuck out his chest with pride\u2014\"I must say, when once I set my mind to accomplish a thing, I can't be beat.\"\n\n\"And what do we have for the winner?\" Einar said, jumping to his feet. He gestured gracefully at an invisible prize. \"Two months' all-expenses-paid vacation at that fabulous Company resort, Pacifica Three, on the beautiful island of Molokai! You'll enjoy unlimited use of Company research facilities while dining on exotic tropical cuisine! When you're not lounging by the library pool, you can saddle up a pony and explore the island's natural wonders, or barter for anecdotal material at the friendly local leper colony. Other activities include windsurfing, spearfishing, and hot-air ballooning.\n\n\"But that's not all!\" He turned and gestured in the other direction. \"Tanned, relaxed, and refreshed, you'll return to an assignment personally selected by _you_. That's right. You may choose to go through either:\n\n\"Door number one, to the lush plains of the Oklahoma Territory, where you'll document consumerism in the developing settlement culture. Or,\n\n\"Door number two, just a canoe ride across to the beautiful Big Island of Hawaii, to report on the growing dependency of the native population on manufactured trade goods. Or,\n\n\"Door number three, to that all-male Queen of the Pacific Northwest, Seattle! You'll cheer (and record) as the arrival of female citizens and quality merchandise changes this lumber boomtown into an American metropolis.\"\n\nWell, that was too much. Oscar's legs gave way under him, and he sat, put his head in his hands, and cried for sheer happiness. I could have cried too, from envy. How often do immortals get choices of anything? And here was Oscar, who'd cheerfully trundle his peddler's wagon into hell if the Company told him to, given the opportunity I'd been pining for. It just goes to show why one should do one's best to be a good little machine.\n\nI was preparing to drink to his health as Porfirio poured us another libation from the jug, when we were all alerted to the approaching presence of a mortal on the immortal arm of Imarte.\n\nPorfirio halted in mid pour, scanning, and we tuned in as well. No trouble; the mortal was in a happy, lustful mood, slightly drunk, and Imarte wasn't concerned.\n\n\"Why, sir, I declare I am simply in love with England,\" she was gushing. \"I do feel that what we colonists gained in liberty was _quite_ outweighed by our loss in culture. This must all seem so terribly rude to a gentleman like you-all.\"\n\n\"My dear lady, who can feel the want of social graces in your fair presence?\" was the gallant if somewhat adenoidal reply. We heard an indrawn breath, and then: \"By Jove! Is that rum punch perfuming the night air?\"\n\n\"I believe it's some of the other lodgers here . . .\" We heard her voice sharpen a little as she bustled after him, for he was coming down the passage to our kitchen like a devil after a soul. A moment later, he had stepped into the circle of lamplight, and we beheld a slightly weedy mortal youth clutching a leather valise to himself. He resembled Charles III of England, with the same sad, remote eyes; and their expression chilled further as he found himself in a room full of strangers. You could see him brightening, however, when he noticed our weapons and decided we were colorful and exotic.\n\n\"Oh, I say, though. Are you banditti?\"\n\n\"No, se\u00f1or, we are merely the staff here,\" Porfirio said. \"You must be aware that it is advisable to carry firearms in Los Angeles.\"\n\n\"Quite!\" Our visitor gave a horsey little giggle. \"The code duello seems to rule in your streets; and may I say that, while I find the brevity of life here appalling, it certainly is lived with a manly lack of hypocrisy and cowardice.\"\n\nWe blinked at him. \"Thank you,\" said Porfirio at last. \"May we offer you a glass of punch, se\u00f1or?\"\n\n\"Yes, please. I shan't be sorry for the warmth.\" He set down his valise and rubbed his hands together. \"For a tropical country it's devilish cold here o'nights, you know.\"\n\n\"Subtropical,\" I corrected him absently.\n\n\"What?\" He turned to stare at me, but then his attention focused on the glass Porfirio was holding out to him. \"Oh, now _that's_ something like. To your good health, all.\" He raised his glass to us and drank deeply. Imarte scowled at us from the doorway behind him.\n\n\"Mr. Rubery, dear, recollect what happens when a man mixes his liquors. We don't want Bacchus's vine to make it difficult for us to offer myrtle to Venus, do we?\" she told him rather acidly. He smiled into his empty glass, licked his chops, and turned to her with an awful leer.\n\n\"I've a constitution of iron, my dear. But let it never be said of me that I kept a lady waiting. Gentlemen, madam, I'm obliged to you for the potation.\" He gave us a nod and set down the glass. Sliding an arm around Imarte's waist, he let himself be pulled off in the direction of her bedroom.\n\n\"She's going to be mad as hell with us if he passes out before she can get him talking about secret plans,\" Einar said, grinning as he raised another toast to Oscar.\n\n\"He left his valise,\" I said, nudging it with my boot.\n\n\"Don't open it. It probably has one of those trick locks that spray tear gas, as in _From Russia, with Love,\"_ he warned me.\n\n\"More likely a spare pair of socks and a set of embroidered hankies,\" said Oscar disdainfully. \"What a prime example of a weak and decadent aristocracy. Did you see the way his teeth\u2014\"\n\nWhat problem he had with Mr. Rubery's teeth I was never to learn, for at that moment we all picked up the signal we had come to dread: a mortal out there in the night, drunk and wrathful, putting the spurs to his poor horse. Cyrus Jackson.\n\n\"Two kilometers out and coming in fast,\" Einar announced, getting to his feet.\n\n\"Riding,\" said Porfirio, pulling out his gun and checking the chambers.\n\n\"You can't kill him,\" Oscar said, blowing out the lamp. The room glowed as we switched to infrared. \"There's a mortal witness. That Britisher.\"\n\n\"Damn. You can't even shoot him with a trank,\" I said, following them out the back door. The valise was right where I could trip over it; impatiently I grabbed it up and shoved it into a cupboard. \"The witness would still think we'd killed the guy.\"\n\n\"But I'm getting tired of Se\u00f1or Cyrus Jackson,\" Porfirio growled. \"Tired of his staking us out all the damn time. I think the moment has come to nail his nasty ass to a wall.\"\n\n\"Uh-oh,\" said Einar, as we emerged into the clearing around the cookfire. Mr. Jackson's signal was growing louder and clearer as he galloped toward us, and it wasn't giving us the usual spectrum of his jealous misery and self-pity; it was off the scale. The man was in a homicidal rage. Einar leaned forward slightly, staring intently down the canyon. We heard the thunder of hoofbeats stop abruptly, and there was a thudding crash and a curse.\n\n\"I got his horse to throw him,\" Einar said. \"And . . . shit, he's still coming.\"\n\nThere he was on visual now, a grotesque figure by infrared, crawling out of the bushes where he'd landed with the boneless impunity of a drunk and staggering to his feet. On he came, up our canyon trail, pulling his gun from its holster.\n\n\"It's your party, boys,\" I said, and winked out to the hillside, where I crouched down and did my best to resemble an ordinary rock formation. I still had a good view of the clearing, with the three of them standing undecided as the monster lurched toward them.\n\n_What's happening_? Imarte broadcast in panic, having just noticed the approaching hazard.\n\n_Keep your Englishman quiet_ , Porfirio told her. _Maybe he should get his pants on, though_.\n\n_I'm staying inside with my birds_ , Juan Bautista transmitted from his room.\n\n\"We'd best get these poor creatures out of sight,\" Oscar said, nodding at the tethered mounts on which Imarte and Mr. Rubery had ridden in from Los Angeles. He took their bridles and led them off to the stable. \"Might I suggest a timely visit from Michael Finn? I've a bottle of chloral hydrate I'd be most happy to contribute to the occasion.\"\n\n\"I don't think this guy's in any mood to sit down and have a drink with us,\" Porfirio said. \"Thanks all the same.\"\n\n\"Smoke and mirrors, I guess, huh, chief?\" Einar asked, rubbing his chin pensively. Porfirio nodded, and they winked out simultaneously, to reappear in the shadows on opposite sides of the clearing just as the mortal man came raving into sight.\n\nHe stopped when he saw the house. He stood swaying for a moment. His rage was building to a peak again. He groped around for the bottle he'd lost in his fall; when it failed to present itself, he let out an inarticulate roar.\n\nA gasp from within the house, and some kind of half-smothered inquiry from Mr. Rubery, which fortunately Mr. Jackson was unable to hear. But he had recovered his bearings enough now to remember why he was there. Shambling forward, he addressed the house and drew a deep breath.\n\n\"Marthy!\" he called. \"You come on out of there, you faithless bitch!\"\n\nThere was silence, at least as far as his mortal ears were concerned. I could hear the pounding of Mr. Rubery's terrified heart as he struggled to get back into his clothes.\n\n\"You come on out here where I've waited for you,\" roared Mr. Jackson. \"You, no-good . . . you're a pitiless wanton. You're the goddamn woman in purple and scarlet, that's what you are. Marthy!\"\n\nI could hear Mr. Rubery whimpering, partly in terror and partly in pain as Imarte had hold of his arm in a viselike grip. In a low and exceedingly calm whisper she was explaining to him the dangers of heedless flight. Mr. Jackson, meanwhile, had leaned over backward until he looked likely to topple, staring in an accusatory way at the stars.\n\n\"I _defy_ you stars!\" he said, and hiccupped. \"The way you looked down on me an' laughed. Marthy, ever' night I sat up there an' watched for you, an' waited for you, and it was so cold. You din't care none! Oh, Marthy, I'd 'a given you ever'thing that was mine, my good name and all, if you'd 'a loved me.\" At this point his gun went off accidentally, kicking up a spurt of dust in the starlight. He was thrown backward and fell on his ass.\n\nAt the sound of the gunshot, Mr. Rubery intensified his efforts to escape to such a degree that Imarte had to let go of his arm or break it. He blundered frantically down the passage into the kitchen, where he tripped over a chair with a crash. Even Mr. Jackson heard it, and he scrambled to his feet with an agility I would not have thought him capable of in his condition.\n\n\"All right, I know you're in there with her. Come out here, you no-good English nancy boy,\" he said. \"You prancin' Ephebe! Bring him out, Marthy! Jesus God, woman, ain't it enough you've run my heart through with needles? Ain't I sat up there bleeding for you, crying in the dark with nobody to care?\"\n\nMr. Rubery was going round and round in the kitchen like a trapped rat. Oh, he must have been hunting for his valise. Mr. Jackson thrust his head forward, peering at the house through narrowed eyes. He had to have been one hell of a hunter when he was sober, because even liquored up he was pinpointing Mr. Rubery's location as accurately as I was.\n\n\"I got you, limey coward,\" he snarled. \"We go down to hell together, but you go first.\" And he started for the house with an un-nervingly steady stride; at least, until Einar popped up beside him.\n\n\"Sorry, pal, you just crossed the line,\" he said, and winked out again. Mr. Jackson jumped and stared; he looked all around and then turned to look behind him.\n\nEinar popped into view again, not an inch from his face. \"You could drop the gun,\" he suggested. Instead Mr. Jackson swung it up and fired wildly at him\u2014or at the place he'd been, for of course Einar winked away once more. Even with the echoing gunfire, though, we all heard the crash as Mr. Rubery got the kitchen door open.\n\n\"Enough is enough,\" said Porfirio, appearing behind Mr. Jackson with the empty frijole pot in his hands. When Mr. Jackson whirled about to see who was speaking, Einar popped up again and gave him a good push. As Mr. Jackson toppled backward, Porfirio shoved the pot down over his head. Mr. Jackson dropped his gun to clutch at the pot with both hands as he fell, and Einar kicked the weapon out of reach. Then Mr. Jackson was on his hands and knees in the dust, struggling blindly to rise and shaking his head, but the pot wouldn't come off.\n\nThe poker materialized in Einar's hand, and Porfirio had one of his iron ladles, and the two of them began to rain blows on the pot, alternating like clockwork figures striking the hours. As they did so, Mr. Rubery went running for his life through the sagebrush, bounding up the hill behind the inn at really amazing speed, and vanished over the ridge. Mr. Jackson kept trying to get up, but the deafening noise was too much for him. He collapsed at last, stunned and nerveless. When he'd stopped twitching, Porfirio and Einar stopped hitting the pot. Porfirio took out a little medikit book and peeled off a trank patch, which he stuck on Mr. Jackson's back, right where the shirttail had come out of the pants.\n\n\"That'll keep him out for twenty-four hours,\" said Porfirio, shoving the book back in his coat pocket. I climbed down from my place on the hillside as Imarte came raging out of the inn, stark naked.\n\n\"Is that miserable sot of a mortal finally finished with?\" she said. \"Mr. Rubery! Alfred, dear! Please don't be alarmed. 'Tis safe to return, dear, the wretch has expired.\"\n\n\"I don't think he can hear you,\" I said. \"He's probably halfway to San Francisco by now.\"\n\nShe glared at me and swore an oath that would have made Cyrus the Persian blanch and cover his ears. \"I CANNOT TOLERATE THESE WORKING CONDITIONS,\" she screamed, then said, when the air had cleared and the little green bats had stopped flying out of her mouth, \"Do you know the chance I've just lost? Do you know who that boy was?\"\n\n\"No, but I think he left his valise behind,\" I said. \"It's in the kitchen cupboard behind the table.\"\n\n\"His valise!\" She got an intense look in her eyes. \"You're sure?\" She turned and went bouncing off to the kitchen, with never a backward glance at Mr. Jackson.\n\nWe stood there in bemusement, until a snore from inside the frijole pot recalled us to our immediate problem.\n\n\"So, uh, chief,\" said Einar. \"What do we do with this guy? The witness is gone. I guess he could just turn up dead in a ditch.\"\n\nPorfirio made a sour face. \"It's not like he killed anybody. Not here, tonight, anyway. On the other hand, he really needs to go far, far away and never bother us again.\"\n\n\"Don't kill him,\" I found myself saying, to my surprise, because I've always thought mortals with the If I Can't Have You Nobody Can Have You kind of obsession to be one of the lowest forms of life. \"There must be a way to get him out of the picture without violence. We could shanghai him.\"\n\n\"An excellent suggestion,\" Oscar said, popping up beside us. \"An involuntary sea cruise is just the thing for him.\"\n\n\"It's a long drive to San Pedro at this time of night,\" Porfirio said, sighing as he took off his hat and ran his hand through his hair.\n\n\"I'll take him,\" I offered, astonishing myself again. Why on earth was I sorry for this mortal?\n\n\"And I'll drive,\" Oscar said. \"I've done this before, you know. Plenty of nasty fellows shipped out of New Bedford feet first when they made a nuisance of themselves around the Company safe house there, let me tell you. It's generally a humane and reliable way to dispose of unwanted mortals.\"\n\nSo Mr. Cyrus Jackson made his final exit from Hollywood at last, trussed and snoring in the back of Oscar's cart, and I heard a numbing five hours of speculation on which assignment Oscar ought to choose as we rattled across the night plain toward the sea.\n\nIn San Pedro, we circled warily around Banning's turf and made for the fishermen's huts on Rattlesnake Island, across from the old landing. Dark shacks on pilings, with a single lantern burning low and red\u2014not a good place to find yourself at three in the morning. But Oscar drove straight up and hopped out unconcernedly.\n\n\"I'll fetch the blackguard. You go waken Se\u00f1or Souza and make the arrangements.\"\n\nI hated talking to mortals; but I crept up to the shack with the lantern and knocked timidly. After a long moment, the door was opened. I recognized the sleepy and unshaven face that peered out at me.\n\n\"Souza? The doctor has work for you,\" I said, using the standard phrase.\n\nHis eyes widened, and he nodded. \"One moment please, seriora,\" he replied, and ducked back inside. He emerged a moment later, trousered and shod, just as Oscar came bustling up with Mr. Jackson draped across his shoulders.\n\n\"Hello there,\" Oscar said brightly, in Portuguese so perfect, you'd have sworn he was born in Lisbon. \"Has my friend explained about the evil and desperate man I'm wearing?\"\n\nSouza blinked and rubbed the bridge of his nose, just below his Company control implant. \"No, se\u00f1or. You'd like him drowned?\"\n\n\"Not at all. No, sir, we simply think he needs a change of air. Now, unless I'm much mistaken, that ship over yonder's full of lumber. Is she going on a long voyage, by any chance?\"\n\nSouza raised his eyes to the open sea, where a schooner rode at anchor. He grinned, white teeth distinct in the gloom. \"Yes, se\u00f1or, the _Elg_. She is bound for Norway with the tide. Two of her able-bodied seamen killed in a fight in Los Angeles, too, I hear. Very sad.\"\n\n\"And this is your boat moored over here, is it not?\" Oscar strolled out along the rickety pier.\n\n\"I am proud to say so, se\u00f1or,\" replied Souza, strolling beside him.\n\n\"Capital.\" Oscar shrugged off Mr. Jackson and dumped him into the bottom of the boat, where he lay moaning. Souza leaped in and untied the mooring rope. A moment later he was rowing steadily out through the darkness in the direction of the _Elg_.\n\n\"Good riddance to bad rubbish,\" Oscar said, adjusting his lapels and shooting his cuffs. \"Faugh, what a smell of rye whiskey. This coat wants laundering, wouldn't you say?\"\n\n\"Very much,\" I agreed, and we climbed back into the cart and wheeled around to return to Hollywood.\n\nOscar took up the conversation again as though it hadn't been interrupted, and for the next five hours I gave my morose opinion in negatives or affirmatives on the merits of Hawaii over the Oklahoma Territories. Altogether it was an excellent thing for Cyrus Jackson that he wake up alive in a bunk on board the _Elg_ , with no more Imarte to break his mortal heart for him.\n\nThe red sun was well above the horizon by the time we got back, and still Oscar hadn't made up his mind about where he wanted to be posted next. Nor had he decided by the time we saw him off, a week later. But Immortals don't get choices very often in their eternal lives, and who could blame him for lingering over his decision?\n\nWe did receive a holocard from him, later, though, all the way from sunny Molokai, and it may well be the last I ever see of that absurd little machine: pinkly sunburned, smiling and waving from the gondola of a hot-air balloon, the untamed world his oyster.\n\n###\n\nUNFORTUNATELY, in the same Company communiqu\u00e9 that had contained Oscar's commendation there was a memo of a less positive nature. It seemed that Juan Bautista's quota of rescued birds hadn't been met for several months in a row, though his budget allocation for maintenance had been exceeded to a remarkable degree.\n\n\"I know, I know!\" he groaned, sinking into a chair, which brought Erich von Stroheim down to eye level with us. \"It's not my fault, though. How am I supposed to go out and look for anything? I can't leave John Barrymore alone for two minutes, and I have to take Erich every place I go. Marie's the only one who'll stay where I tell her to.\"\n\n\"I warned you about this, Juanito,\" said Porfirio, shaking his head. \"Didn't I warn you about this? Now you don't have a choice. You crate up the big birds and ship them off to HQ. They'll be all right. What's more important, you'll be able to get back to your work.\"\n\nJuan Bautista's face went pale. \"Please, just give me a little more time. I think I'm finally beginning to make some progress with John Barrymore. The microsurgery's all healed up, and lately he's even started to act like a normal bird sometimes. Please? One more month. As soon as the weather's better, I swear I'll send them away.\"\n\nPorfirio leaned forward. \"You don't seem to get it. This is not me telling you. This is Dr. Zeus _officially_ telling you that you have screwed up. You're not doing your job. That's not acceptable, kid. You do understand that, don't you? And it doesn't take a lot of brains to figure out what your next move has to be, and you have brains to spare, thanks to Dr. Zeus. This is tough enough; don't make it tougher.\"\n\n\"What if I was able to catch up on my quota?\" said Juan Bautista. \"I know I've fallen behind, but it's not the birds' fault. I'll learn to manage my time better. I'll bring my work up to speed, you'll see. Couldn't I keep them just another month, if I was able to do that? Would one more month make any difference, if I was able to make the Company happy?\"\n\nThis was too much for me. I had to slink out, so I didn't catch the rest of the conversation; but I gathered that Porfirio gave in again, because no big birds were crated up or shipped off in the next few days. All the stock of songbirds and little owls went, though, tagged in their wicker cages; and Juan Bautista was admirably industrious for a whole day in front of his room, weaving new cages for the new stock he had sworn to bring in.\n\nImarte was industrious, too. We never saw Alfred Rubery again, but he _had_ left his valise behind. She spent days locked in her room with it, going over the contents in minute detail and making copies of what she found. We only saw her at mealtimes, and the transformation from whore to scribe was unsettling: inky fingers, disheveled hair, stained dressing gown. She looked radiantly happy, though, with whatever lode of cryptohistory she'd struck. I confess I was curious, but not curious enough to bring myself to ask her about it.\n\nAnd Einar was certainly industrious. Longhorns were going for ridiculously low prices now, and he was acquiring them every day and conducting cattle minidrives into Los Angeles. Porfirio was always busy, of course. The one advantage to the drought was that no roads washed out that winter, and now that the smallpox epidemic was tapering off, the stagecoaches were running regularly again. Banning seemed to be deferring maintenance on the coaches, though, or maybe his regular crew had died of the pox, because there were repairs to be made at our smithy nearly every day.\n\nI was the only one with nothing to do. Why was that, se\u00f1ors? There was nothing left for me to save. Everything that grew in the temperate belt had either been collected by me or grazed down to bare earth by starving cattle. There were no rarities left to find, unless I cared to venture into the Canyon of Lunacy again. But no prize on earth could have tempted me back into that place where I might glimpse the deadly city again, the future desolation.\n\nNow, you would think, wouldn't you, that Dr. Zeus might give me a pat on the head and tell me to run along now, back to my beautiful green Ventana? I certainly thought so. I wasn't expecting commendations or prizes, or even thanks for a job well done; but I did expect a new posting, and none came, though I checked the Company directives pouch every time Einar returned from Los Diablos. Bureaucratic willfulness, or some subtle punishment to make me work harder, to improve my attitude? Why was I being ignored, se\u00f1ors? Was it simply that nobody noticed that I was stranded there, unable to do the work I'd been programmed for, the work I needed? The work that kept my demon at bay?\n\nOr did the Company know? Did you know what would happen next? Did you know and sit there like God, silent, remorseless, useless? What happens if I sit here in silence, too? What if I never give you my all-important testimony, eh?\n\nBut of course you couldn't have known. You're stuck here in 1863, just like me. I don't imagine our masters up there in the future would tell you if they knew, either. No operative is ever told any personal detail of the Temporal Concordance. It's forbidden to tell. Though Lewis tried . . . Will you punish him, too?\n\nMore Theobromos? Well, thank you so very much. You damned well better anesthetize me now, if you want me to go on with this.\n\n###\n\nSO THERE WE WERE, ALL happily going about our work except for me. I sat huddled in my room most days, wrapped in a blanket and viewing holos hour after hour. Not as much fun as old-style cinema, overall. There is a pleasant sense of camaraderie with the rest of the audience, watching cinema. You know: throwing popcorn at the flat screen and cheering and sharing moments of excitement, like when Luke Skywalker is shooting down the bad guys pursuing the _Millennium Falcon_ , or any part of _The Rocky Horror Picture Show_.\n\nIt's true that a holo takes you right into the center of the action; but that illusion is not always a good thing. The leeches scene from _The African Queen_ comes shudderingly to mind. _Sunset Boulevard_ , too. Who the hell wants to get unbearably close to Norma Desmond's scary eyes in that last scene? And let's not even talk about Hitchcock's films. Though it's no better, really, in the films you _want_ to be a part of, because you're still isolated, you're like a ghost. No amount of technological cleverness can make Sean Connery take you in his arms, and no Good Witch will ever take you by the hand and welcome you to Oz. They won't see you, they won't hear you, because their reality is complete and you are not a part of it.\n\nAt last I gave it up and started following the war news again. Depressing, inconclusive, inaccurate, but at least it was really happening. I feel badly that it absorbed my attention so much. If it hadn't, I might have noticed the noises in Juan Bautista's room that awful day.\n\nNot that I could have done anything if I had, of course.\n\nYou see, encouraged by the progress that John Barrymore seemed to be making, Juan had taken to leaving him shut in his room when he went out on his collecting trips. Erich von Stroheim he kept in Einar's room, liberally dosed with bird dope of some kind, so the damn creature was quiet all day. He didn't like to do it, of course, but the idea was that it was only temporary, until he caught up on his quota and reassured Dr. Zeus that he too was a good little machine.\n\nIn my opinion he should have been doing this all along. It was no effort for Juan Bautista to catch birds: all he had to do was stand still, and the bloody things would light all over him. But he was seventeen! Sloppy and disorganized and stupid as youth will always be, no matter how cyberaugmented it's made. Perhaps that was why he filled his room to the ceiling with flimsy woven cages full of the miserable cheeping little things, and left a psychotic predator in there with them while he went out each day to hunt for more.\n\nDo I have to tell you what happened, se\u00f1ors?\n\nIt was as bad as you could imagine. I heard his wail of horror when he opened his door. I came stumbling from my room in time to see John Barrymore bouncing clumsily out into the clearing. Porfirio and Einar emerged from the house too, and stopped dead at the sight of the eagle.\n\nNot that he was covered with gore, or anything like that. Well, a little blood, and some few bright feathers from some little victim. He regained his composure and took a few paces sideways, cocking his head to stare at us in a puzzled way. But there was the most heartbroken sobbing from Juan Bautista's room.\n\nThe irony was that John Barrymore had been making progress. While he was sick and mad, he tried only to kill himself. It was when he began to heal that he felt the normal urge to do what predators do. But Juan Bautista was in no condition to appreciate this, as he emerged from his room with a little torn body in either hand.\n\n\"You _bastard,\"_ he screamed. \"How could you do this?\"\n\nHe ran at John Barrymore, who started and crouched in alarm. Then, with a wild flapping of wings, the big bird rose into the air and floated onto the roof of the inn. He looked down at us all, and we stood looking up at him with open mouths. Experimentally he beat his wings again, twice, three times, and we felt the rush of air in our faces as he nearly lifted off. Had the madness left any room for joy, when it vacated that narrow killer's skull of his? What was in his flat blank eyes, when he beat his wings again with a noise like a stiff breeze filling canvas? I don't know. In the next moment he leaned into the evening air and sailed away on spread wings, effortlessly, a long curve ascending. Up and up he went, high enough at last to catch the last light of the sun, and then he flew northward and was gone.\n\nMarie Dressier had survived; she had managed to get into the clutter under Juan Bautista's bed and defend herself from there with her formidable old bill. And of course Erich von Stroheim was fine; he'd slept through it all in Einar's room. But the boy who loved them had changed.\n\nDo you remember that terrible moment, se\u00f1ors, when the self-righteousness of your youth died? When all the stern warnings of your elders, ignored until the consequences abruptly came crashing down on your head, made you see in a flash that the warnings hadn't been unfair or mean-spirited or blind, they'd been _right_? All along your elders had been trying to tell you about the black joke that is life, trying to help you and save you from pain. But you insisted on running straight into the trap, mocking them as you ran, to the agony that was irreversible and permanent, with no one to blame, finally, but yourself.\n\nIt's not good to see yourself in the mirror then. Juan Bautista was reflected in the eyes of every one of the little dead birds he had to clean out of his room.\n\nNext time Einar loaded up the wagon for the trip into Los Diablos, there were two big cages among his cargo. Marie sat patiently in hers, considering her new fate with a calculating eye; but Erich von Stroheim croaked and hissed with anxiety, trying to muscle through the wire mesh that kept him from Juan Bautista. When the wagon started up and he found himself rolling away from the boy, he started up the piercing scream we knew all too well. Juan Bautista just stood there, watching, his face like stone. It took a long time and a lot of distance for the screams to fade to silence.\n\n\"It's better this way, muchacho,\" Porfirio said at last. \"They'll be safe, they'll be happy, they'll have great lives in the Company aviary.\"\n\nJuan Bautista nodded, but I knew what he was thinking: no way now he could kill them, either, with his well-meaning mistakes or unintentional neglect or selfish love. I wondered if he'd ever dare love anything mortal again. Some of us don't.\n\n###\n\nIT WAS MARCH 13, 1863. I was struggling back and forth between our galvanized bathtub and the nearest oak tree, carrying buckets of cold and slightly soapy water in my unending irrigation efforts, when I looked up to see Einar returning from Los Diablos in the wagon, trailing a cloud of dust several stories high. I squinted in the glare and frowned; he'd brought back a load of crates, something so heavy the wagon was low on its springs. Porfirio picked up the question I was broadcasting and came out to see.\n\n\"What the hell are those?\" he said, wiping his floury hands on a dish towel. \"He's brought back eight pianos? Where's the olive oil I ordered?\"\n\n\"Wasn't room, chief,\" Einar called. \"When I got to HQ, these guys were waiting for me. Ladies, I mean; they're eight tule elk does in stasis, and they've all been bred. I've got some marching orders. They're supposed to go over Tejon Pass out to Buttonwillow, to be released into the wild. You're tagged to assist me.\"\n\n\"Great,\" said Porfirio, throwing down the dish towel. \"Just what I needed. A trip to beautiful Buttonwillow at this time of year.\"\n\nI didn't envy them. If there was a place more desolate than Los Angeles, Buttonwillow was it; the only possible advantage being that there were almost no mortals there in this era.\n\nEinar shrugged apologetically. \"It shouldn't take us more than a week,\" he said. \"We have to leave ASAP, though. 0500 hours tomorrow okay with you?\"\n\nPorfirio sighed. \"That's life in the service. All right, tomorrow it is; I guess the pass is clear by now. No killer camels this time, huh?\" Referring to the legend of deadly dromedaries haunting the mountains.\n\n\"That was just a joke, chief. Honest. But look at what else was waiting for me at HQ.\" Einar held up a big silver film can. _\"Grand Hotel_ , 1932! Greta Garbo, two Barrymores, Joan Crawford, Wallace Beery. We can have another night of the film festival when we get back, what do you say?\"\n\nI filled my bucket again and went trudging off to water the oak trees. How about Walt Disney's _Fantasia_ , I wondered nastily. Weren't we all sorcerer's apprentices? Bucket-carrying brooms impossible to kill.\n\nThey left next morning in the bleak dark, bundled in their coats on the springboard of the wagon.\n\n\"You have plenty of supplies,\" Porfirio told Imarte and me. She had got up to see them off. \"And there's most of a pot of fresh coffee on the fire over there. You ladies look after the kid, okay? And no cat rights in my absence, please.\"\n\nImarte and I looked at each other in disdain. \"Wouldn't bother,\" I said.\n\n\"Don't give it a moment's thought,\" Imarte agreed.\n\nHe looked at us searchingly, his black eyes troubled. \"No going out without wearing a loaded sidearm at all times, remember?\"\n\n\"Hey, man, these are immortal girls,\" Einar said. \"They can take care of themselves. Ciao, ladies; get your hankies laundered and ready, 'cause _Grand Hotel_ has one of the really great tearjerker endings of all time!\"\n\nHe gave a crack of his whip, and the wagon rolled away into the gloom, creaking under the weight of its improbable cargo. I hoped they wouldn't break an axle going over the Grapevine Grade. As though he had heard my thought\u2014I wasn't broadcasting\u2014Porfirio turned around in his seat and looked at me. What an uncertain look on his scarred devil's face. He was wearing it still when they turned onto El Camino Real and vanished from sight.\n\nI stood there a moment longer, shivering in the mists. Then I remembered there was coffee. Imarte was already helping herself to the pot. I hurried to find my mug.\n\nWe managed to restrain ourselves from hair pulling and all that fun stuff, mostly because after she filled her mug she disappeared back into her room to continue her studies of the amazing secret-agent valise. Juan Bautista came moping out a while later, and I grilled him some beef for breakfast. He didn't stick around to talk; very shortly he disappeared up the canyon with half a dozen empty wicker cages, which had to be filled with endangered species. I knew a little of what he was feeling; he'd come to the place we all come to, sooner or later, when the work is all you have, all you can depend on.\n\nSo I was alone all day. There weren't even any passengers dismounting from the stages. It was a lovely, surreal feeling, all that peace and quiet. My ghost, too, left me quite alone. I made a stew of leftover grilled beef for supper, and built up the cookfire to a nice blaze afterward. In the interests of peace and harmony I broadcast a meal call to Imarte, and a moment later she actually emerged from the house.\n\n\"You prepared a meal?\" she said in surprise.\n\n\"I know several recipes,\" I said. \"Not that one can do a lot without olive oil.\"\n\n\"Goddess, that's true,\" she said, sounding astonished that we had found a common opinion. She made up for it by picking most of the chilies out of her dish, however, and flicking them into the dark. We sat there awhile in unpleasant silence, before she finally cleared her throat and spoke. \"I believe I'll take the stage north tomorrow,\" she said.\n\n\"Really?\" I asked. Where did she get off, leaving me with the responsibility of running the place? \"You cleared it with Porfirio before he left, I suppose?\"\n\n\"Oh, he knew I had a research trip to make,\" she said evasively. I just shrugged and kept eating. After a moment she gave up trying to keep her news to herself. \"The contents of that valise contained the most incredible\u2014\"\n\n\"Is there any supper left?\" asked Juan Bautista, appearing from the shadows. He was holding something bundled in his coat. I pretended not to notice as I ladled him a bowl of stew, but Imarte leaned forward and peered, frowning.\n\n\"Are you hiding something, child?\"\n\n\"I'm\u2014it's just a raven, that's all,\" he said. \"She has a broken wing. I thought I could fix it with microsurgery. I seem to be pretty good at that.\"\n\nShe missed the bitterness of his last remark and sailed right on. \"That's nice. Well, as I was saying, when I examined the contents of the valise\u2014\"\n\nJuan Bautista's eyes widened with excitement. \"Are you finally going to tell us what was in it? I've been wondering about that. Were there really, like, secret documents?\"\n\n\"Incredibly secret,\" she said, her voice dropping to a dramatic whisper. \"Lists of persons to be contacted, with their addresses. Communications from Judah Benjamin and John Bright. Letters of introduction. Documents pertaining to the Order of the Golden Circle. Drafts on the Bank of England, without countersignature, amounts to be filled in at the discretion of the bearer. Two pasteboard tubes filled with golden sovereigns, sealed with stamped wax. Timetables, instructions, and letters in coded phrases.\n\n\"Several letters of a most incriminating nature between a person named Greathouse and several Canadian nationals, to say nothing of some _very_ interesting overtures from the Prime Minister to Benito Juarez.\" She leaned forward after an impressive pause. \"Last but certainly not least, letters referring to some technological discovery, made in a place designated only by code. All written in a lovely violet ink, chosen undoubtedly not for its color but for the remarkable chemical properties that enable it to vanish without a trace when exposed to water.\"\n\n\"Wow,\" said Juan Bautista. \"All you'd have to do is dip a page in water to get rid of the evidence? I guess he really _was_ an English spy.\"\n\n\"A conspirator, I think,\" Imarte said judiciously. \"Not spy material, or he'd have been back for the valise by now. But thanks to some cross-referencing of available data, I now have important information about just who these conspirators are. I found an invaluable text entitled 'The Great Diamond Hoax.' It purports to be a firsthand narrative, by one Asbury Harpending, of the true circumstances of what will be known as the Chapman Piracy Case, which I gather is the name the San Francisco papers will give my pet conspiracy when it resoundingly fails.\"\n\n\"You mean, they'll be caught?\" I asked, glancing over at the little black thing in Juan Bautista's coat, which had stirred feebly.\n\n\"In a matter of hours,\" Imarte said. \"If only I'd found Harpending's narrative sooner. It would appear that quite shortly our dear Mr. Rubery and his Secessionist friends will be cooling their heels in prison, thanks to some astonishing errors of judgment. My theory is that their bungling will be due to the loss of the vital information in the valise. Of course the plot fails, but somehow the British will manage to cover their involvement completely. I must know how! You can see this is important, I trust? It's absolutely necessary that the Company have a qualified observer on the scene. One of the most concerted covert efforts by a foreign power to overthrow an American government in this century, and somehow it will be made to seem nothing more than a boyish prank, a footnote in a minor chapter in history.\"\n\n\"I certainly won't stop you,\" I said. \"Go, by all means. Have a grand time. And bring us back a loaf of sourdough bread.\" A little more flip than the occasion called for, particularly as I thought her obsession sounded kind of interesting for once; but she always brought out my worst side.\n\nShe narrowed her eyes at me. A chilly silence fell. It was finally broken by Juan Bautista, who said, \"I was wondering . . .\"\n\n\"What?\" I turned my attention back to him.\n\n\"Did the Company ever try the immortality process on animals?\"\n\n\"There are stories,\" I said. \"It almost works, but not quite. Animals can be made smarter, or nearly immortal, but not all the way and not as smart as us. I know for a fact it was done with one of the higher primates, but the program never went further than the prototype.\"\n\n\"So you could make an animal as smart as a mortal human?\"\n\n\"Sure you could,\" I said. \"But why would anyone want to? Mortals are unhappy enough with the brains they have. Why inflict self-awareness on an animal?\"\n\nHe didn't answer. Shortly after that, he went to bed, carefully cradling the little black thing in his bundled coat. Imarte flounced off too, doubtless to pack her bag for her trip to San Francisco. I stayed up a while, looking at the stars. I could hear Juan Bautista playing his guitar in his room. It was the first time he'd touched it in days.\n\n## PART THREE\n\n###\n\nMARCH 15,1863. A day to remember, se\u00f1ors.\n\nI woke early and was quite happy grinding coffee beans, grilling beef. I thought to myself that perhaps there was a future for me in the food service industry, if the Company had no further use for me as a botanist. Juan Bautista ate hurriedly and vanished into the brown wilderness, so I was the only one there to see Imarte off when she came out to meet the stagecoach, bag in hand. She was all tarted up again, hair curled and ink stains scrubbed away, corseted for action.\n\n\"I'm on my way,\" she said. \"It's unlikely I'll return before the end of the month. Convey my apologies to Porfirio, but I'm certain he'll understand why this was necessary when I have a chance to explain it to him.\"\n\n\"Fine with me,\" I said. \"Aren't you taking the secret valise, though?\"\n\n\"No, of course not!\" She shifted her bag to her other hand and leaned close to lower her voice, though we were the only living souls for a good six kilometers around. \"That material is far too incriminating to carry abroad. I rather imagine some sort of effort will be made to recover it, so I've left it in plain sight in my room. Should any suspicious-looking persons call, let them take it. I've made a detailed copy of everything.\"\n\n\"We're not going to be visited by angry Union Army troops or police, are we?\" I asked.\n\n\"A ridiculous idea. At this particular point in time my conspirators are blissfully unaware of any trouble brewing for them, and my research indicates that the attention of the law will be focused entirely on their activities in San Francisco. The fact that the conspirators also had a cabal in Los Angeles seems to have disappeared in the Historical Event Shadow,\" she said, glancing at her chronophase. \"We're perfectly safe. Now, if you'll excuse me, I have a stagecoach to catch.\"\n\nI watched her go sashaying down the trail, scarlet ribbons and curls bouncing, Immortal Babylon insatiably after a good time. The fact that her pleasure derived from history rather than from pretty boys made it no less delicious. A deeper and more subtle pleasure, no doubt. If there really are gods and goddesses, this must be how they amuse themselves, not with the pettiness of individual men but with the sins of nations, the follies of kings.\n\nSo I had the inn to myself. I wandered into the storeroom and poked through Porfirio's groceries. I found a cake of Theobromos and made off with the whole thing, luxuriating in a sense of selfishness. But wait, there was more: six months of back issues of _Punch_ I'd never got around to reading, and nobody to complain about the way I folded them. I dug them out from under Porfirio's bed and settled in the kitchen, putting my feet up on a chair. Should I make myself another pot of coffee? Should I eat all the Theobromos now or save half for later? Maybe I'd eat a whole cone of piloncillo sugar too. Oh, the sins I could indulge in with nobody to see.\n\nBut even as I turned my attention to the latest nasty caricature of Abraham Lincoln, I picked up the signal of a mortal approaching. So much for blessed solitude.\n\nI ignored it as long as I could, which wasn't very long. What the hell was this mortal doing? I set down the papers, got up, and went outside.\n\nIt was a male, very much in control of his thoughts and emotions, wary but not particularly afraid or even disturbed. No, he was concentrating intently on his activity of the moment, which seemed to be the covert surveillance of our humble establishment. I faded back into the doorway and scanned.\n\nYes, there he was on the ridgeline, a barely visible figure having a nice leisurely look at us through a pair of field glasses. They were all I could distinguish on visual alone. Infrared in broad daylight gave me a sketchy little scarlet ghost, but judging from the proportions, he was large. There were large animal readings, too; the man must have had a horse tethered just out of sight.\n\nOver the next hour he worked his way around our canyon, studying us from ail sides. I gave up and went indoors, deciding he was after the valise. He was welcome to it, as far as I was concerned. He didn't read like a mortal intent on violence, so I settled back in the kitchen, put my feet up again, and resumed my perusal of the British funny papers. I did make sure that my gun belt was fastened properly and my Navy revolver was loose in its holster, though.\n\nWas he going to have his look and go sneakily away, leaving me in peace? No, damn it, here he was again, riding up the trail on horseback like a proper visitor. He was going to come to the door. I had him on audio now; there were the plodding hoofbeats of his horse; I could hear his breathing and his heartbeat. There was something unsettling about them. I eased my gun from its holster and held it concealed behind the copy of _Punch_ I had been reading. I hoped I wouldn't have to blow a hole through that comic poem about highwaymen. He was a _very_ large man, too. Would one bullet stop him?\n\n\"Hello? Is anyone here?\" he called out.\n\nAnd his words might have been a bullet through me, such an impact they had on this immortal body I wear, se\u00f1ors. I jerked as though I'd taken a hit, cursing silently and wondering whether I was having some sort of malfunction, some electronic seizure. My chair squeaked a good two inches backward. I did a self-diagnostic in the fraction of a second it took for the echo of his words to die away, but found nothing wrong.\n\nThe man heard the noise my chair made and was coming to the door. Angrily I got to my feet, holstering my weapon\u2014why, I don't know\u2014and tossing away the papers. Every defensive sense I had was activated. There, he had stepped through the doorway and halted, looking into the kitchen at me.\n\nA big mortal indeed, absurdly so, even without the tall hat he was in the act of removing. He wore the tailored clothing of a Continental gentleman, in subtle tones of gray and brown that had just incidentally made him nearly impossible to see in the underbrush. You couldn't have told he'd been out there crawling around in the purple sage, though; not a wrinkle nor a stain on the man, not a single twig in his lank fair hair.\n\nHe was even wearing gloves; at least the hand that held the Spanish-English phrasebook was gloved. He was wearing a gun, too, though that was discreetly holstered under his left arm and would have been invisible to another mortal. He smiled at me with a great deal of charm and no little confidence. When he smiled, his pale-blue eyes narrowed and his high wide cheekbones seemed to slant upward, which made his long broken nose look longer.\n\nHe was, se\u00f1ors, the living image of the man I had last seen bound to a stake, screaming in flames, three centuries ago and half the world away. How could I not know him, my one and only lover? He had died in those flames, and my human heart had gone into the fire with him and become the charred thing it was. But here he was now, he'd smashed through the barrier of dreams and come to claim me in no more hauntings but in living flesh. My doom had come upon me, as the lady in the poem said.\n\n\"Please excuse me, se\u00f1orita,\" he said in perfect Castilian Spanish. He pretended to read from the phrasebook. \"Is this the inn where one may meet the coach to San Francisco?\"\n\nIt was the same voice, too, that dark tenor of such power, such beauty. When he'd preached to the avid spectators from the flames, even they had been moved to tears.\n\nI found myself perfectly calm. Well, I wasn't a mortal woman who might have fainted or wept, was I? I was the same cyborg creature who'd watched Nicholas Harpole die, and I knew he was dead, and this man could not be my lover miraculously returned to me. \"I speak English, se\u00f1or,\" I said.\n\n\"Do you?\" he replied. \"How very convenient for us both.\" His smile widened, and the phrasebook disappeared into his pocket with a single graceful movement. Dear God help me, he was an Englishman. Not _my_ Englishman, of course. I was going to be rational about this if it killed me.\n\nWho did I think he was, you ask? Give me more Theobromos, and I'll tell you my friend Joseph's theory of genetic stability.\n\nThanks so much. Joseph calls it the English Character Actor Phenomenon. Have you ever had occasion to watch a lot of British cinema, or look at British portraits or photographs? You may have noticed that many of the faces are identical, though separated by decades or even centuries. Compare a cast photo of the D'Oyly Carte company from 1885 with one from 1973, for example. Some of them could be the same people, as immortal as we are. Of course they're not; and there's no need to grope for a mystical reason to explain the resemblance, either. It's a simple matter of genetics on a rather small island. There are only so many faces, only so many physical types in that gene pool. You can find the same sort of recurring appearances in other communities that tend to disapprove of marriage outside one's race.\n\nOlder operatives with countless lifetimes behind them\u2014like Joseph, for example, whose theory this is\u2014are always running into people who could be identical twins to mortals they knew centuries earlier. I'm told one gets over the surprise fairly soon. Perhaps I would, too.\n\n\"By your leave, se\u00f1orita,\" said the mortal man, holding my gaze steadily as he stepped forward with a caution that indicated he'd noticed my weapon and taken my measure. \"I believe we have acquaintances in common. I was informed at the Bella Union Hotel that there was a well-spoken daughter of joy who kept a private house at this location. Have I the pleasure of her company?\"\n\n\"No, se\u00f1or,\" I said. \"She is away. I do not expect her return for some days.\"\n\n\"Ah.\" He tilted his head a little to one side, considering me. \"You are perhaps in her employ?\"\n\nI blinked at him. It actually took me a moment to realize that this magnificent stranger was asking me to have sex with him. He thought I was a whore, my long-lost beloved.\n\nOn the other hand, I had been celibate for just over three centuries now, and the nearness of his mortal flesh and the sound of his voice were more than I could bear.\n\nWhy not? Why deny myself this thing?\n\n\"Yes,\" I said.\n\n\"Very well,\" he said, drawing off his remaining glove and tucking it in his hat with the other one. \"I trust you have the afternoon free? Where may we be undisturbed?\"\n\nI led him into Imarte's room without a word.\n\nThe light flickered over his eyes as he took in the dimensions of the room, rapidly noting placement of doors and windows, locking mechanisms, possible traps. He was _scanning_ , se\u00f1ors, as ably as one of us, if without electronic assistance. He spotted the valise under Imarte's table\u2014no change in his expression at all\u2014and turned his attention to me with nothing but expectant and straightforward lust. Had I always been able to read him like this? But I was so young when I met my man, and so many years of hard living since then had sharpened my perceptions.\n\n\"What is your pleasure, se\u00f1or?\" I asked after an awkward pause. Wasn't that what whores said?\n\nHe drew his eyebrows together slightly. \"Well, under the circumstances, I believe it's customary for one to undress,\" he said, just a hint of irony in that well-bred voice. Undress, right. I unbuckled my gun belt, and he held out his hand to take it. \"Allow me.\"\n\nHe hung it over a chair, well out of my reach. I watched as he turned back to me, and our eyes met, acknowledging that he'd scored the first touch. He stepped back a pace to indicate that I should proceed.\n\nSo I took them off, the drab and convenient garments of my life, the long walking skirt with its slightly muddy hem, the plain dark blouse and bodice, the battered high-topped boots and threadbare black stockings. My lingerie was a disgrace, shabby gray cotton I'd mended with pack thread; but I had never expected to sleep with anyone again. It just goes to show that you ought to invest in good underwear, because you never know, do you, when a long-dead lover will pop up and whisk you into bed. At least my flesh was presentable: to all appearances that of the same eighteen-year-old girl who'd loved the man in England. Immortality has that much consolation.\n\nHe watched me intently, and only when he'd seen that I had no other weapons concealed in any other possible place did the good red blood rise into his face, and a certain ready warmth into his eyes.\n\n\"Charming\" was all he said; and setting down his hat and gloves, he shrugged out of his coat. There was his holster, for anyone to see, with a revolver snugly tucked away in it. He acknowledged my stare with a frank smile. \"Lest one fall amongst thieves whilst traveling,\" he explained. He took it off and hung it on the chair next to mine, but rather closer to the bed. While removing his boots, he was able to get a good look under the bed and satisfy himself that nobody was lurking there. Off came his waistcoat with its watch in the little pocket, off came his flowing tie; and that was as undressed as he was going to get, except for letting his suspenders down and unbuttoning where necessary. What a pity; I wondered if the rest of him was eerily identical to Nicholas Harpole. What I could see as he unbuttoned was gorgeously the same.\n\nWe sank down on the counterpane together and, yes, if this wasn't the same man, there was something wildly wrong with the universe. He kissed like Nicholas, used his hands with the same masterful expertise, played my body like a rare instrument just as Nicholas had done, as though I were something beautiful.\n\nThere was only one moment of trouble, when an expression of amazement crossed his face, and he rose on his elbows and gave me a sharp wondering look; but the music was playing too sweetly to stop the dance now, and we went leaping on. I didn't try to guess what he was thinking. Would _you_ have?\n\nI won't describe the physical pleasure. You wouldn't believe it, se\u00f1ors. I don't know that I believed it myself. One moment the world had been the sad ordinary place I'd lived in for the better part of three centuries, and the next it had shattered and fallen away like an image painted on glass, a dreary illusion gone forever. If this day was possible, then angels might exist, fairies too, miracles and wonders, even a loving God.\n\nI think we made love for hours. He was a determined sensualist, as perfect and as tireless as one of us, and seemed intent on exhausting me, which of course he couldn't do, except emotionally. A long while later, I lay weeping silently, curled against him. He leaned up on one elbow to regard me.\n\nMy God, the same dear face, flushed in the same way after his pleasure. His eyes were sharp and considering; and yet I could sense no desire to harm me, though I'd detected at least three more weapons concealed on his person during our lovemaking. What on earth was he, a professional assassin?\n\nWell, why else would he be carrying all those weapons? He'd been sent for the valise, and I was the only witness.\n\nThis realization hit me like a thunderbolt, in the precise moment that he casually draped an arm over me and pulled me close again. Without apparent effort on his part I found myself caught against him, my arms securely pinioned and the weight of his big body holding me down. If I'd been a mortal woman, I couldn't have escaped. My heart raced all the same.\n\nHe looked into my eyes, probing for something. \"That was delightful, my dear,\" he told me suavely. \"But you're not a whore, are you?\"\n\n\"No, se\u00f1or,\" I said. \"My apologies for the deception.\"\n\n\"You were in fact a virgin, were you not?\" He sounded regretful, not for my lost innocence but because he was afraid there might have to be a death in this room, and he was sincerely hoping it wouldn't be necessary.\n\nI stared. I couldn't tell him that if I seemed a virgin, it might be because I hadn't slept with anybody since March 1555. Did you know our bodily regeneration was that thorough? I hadn't known. \"Yes, se\u00f1or, I was,\" I answered.\n\nHe smiled slightly. \"I don't flatter myself that my personal attractions led you to sacrifice something of such value to a young lady. Why, then, did you lie to me concerning your . . . vocation?\"\n\nWhat he intended to do next depended on my answer to that question. What was the right answer? No way to tell him the truth. At least I was in fair control of myself. Time was when even a mortal in a temper would have had me winking out in nervous terror. Killer apes, I'd called them; but this was a killer angel. You may think he was a monster, se\u00f1ors, prepared as he was to quietly kill a woman he'd just pleasured; but I tell you it maddened me with new desire, and isn't _that_ monstrous? But who in the hell knows what's clean or unclean in love?\n\n\"I needed the money, se\u00f1or,\" I lied, as frankly as I knew how. \"Bereavement has left me a pauper, without refuge. In the past few days my situation has become desperate. The whore of this place is absent, I told you the truth of that; and when you came inquiring for her, it appeared that fate had placed a terrible opportunity before me. You seemed like a decent man, se\u00f1or.\"\n\n\"Or at least a wealthy one?\" He raised one eyebrow and studied me. His body was relaxing. \"And this, then, was the occasion of your fall from grace? I trust you won't take offense if I observe that you don't seem suited for this occupation. Are you aware you never even set me a price, my dear? Pleasant as our dalliance was\u2014and believe me, se\u00f1orita, it was a pleasure indeed\u2014I think this is not the life for you.\"\n\nHe was amused. He was deciding there was no need for unpleasantness. He wasn't through with me yet, though. Was I disappointed?\n\n\"What price ought I to have charged, se\u00f1or?\" I asked.\n\nHe smiled wryly. \"The price of a good dowry in this backward country, or whatever donation the nearest convent requires to take in a novice. I believe I have a fair idea of the exchange rate at the present time. I'll leave that sum in gold; but you must promise me you'll use it for the one purpose or the other, as your inclination directs. Harlotry's a dangerous business.\"\n\nHe was a professional killer, and so far as he knew, I was a wretched nonentity he might just as easily have discarded, by one means or another. Instead he had opted to _do the decent thing_. Who was this man? Who sent him here, to this miserable place? Queen Victoria's Foreign Office?\n\n\"Thank you, se\u00f1or\" was all I could think to say.\n\nHe smiled again and kissed me, releasing my arms. \"There's a good girl. Now, shall we seal the bargain with a toast? What wine or spirits does the resident strumpet keep here?\" He rolled off me and sat up.\n\n\"There's aguardiente,\" I said.\n\n\"Your local brandy, yes. That'll do. Fetch us a bottle of the best and two glasses, and we'll drink to your future as an honest woman.\"\n\nI hastened to obey, so readily that I was on my way back from the pantry with the bottle and glasses before I remembered that I was stark naked. He smiled engagingly at me as I scurried back to him where he lounged against the headboard of the bed. He was concealing something in one hand, however.\n\n\"Many thanks,\" he said, taking the bottle and one of the glasses. He poured a drink for me, dispensing a white powder into my glass as he did so with beautifully neat sleight of hand. It dissolved without a trace in the aguardiente. He handed me my glass and poured one for himself. \"Back into bed, now. Climb under the blankets. There's rather a chill in the air.\"\n\nThere certainly was. I analyzed the contents of my glass, smiling ingenuously at him. Not poison, at least; something to make me sleep. Sleep while he did what? Made off with the valise, of course. What was I going to do now?\n\n\"To your good health and moral reclamation,\" he said, lifting his glass. I lifted mine too. He drank, but as soon as he saw that I wasn't drinking with him, his attention was fixed on me again.\n\n\"Perhaps you don't indulge in spirits, my dear?\" he inquired, in a way as delicate as the perfect spring mechanism of a steel trap.\n\n\"Not often,\" I said, lowering my glass. I leaned affectionately on his right arm. \"Never mind, se\u00f1or. You know, it's only just occurred to me how you must have come to hear of this place. You must be a friend of Mr. Alfred Rubery.\"\n\nHa, that startled him. Nobody but a cyborg who was reading his pulse and skin conductivity could have told, however. His eyes narrowed in that dangerous smile.\n\n\"The young ass,\" he said. \"Yes, he was quite taken with Madam Martha.\"\n\n\"I hope he was able to get back to his hotel safely? Really, se\u00f1or, you'd have laughed if you were here to see it. A perfect farce! Except that he really was in danger of being shot by the jealous lover. He barely escaped with his clothing as it was. In fact. . .\" I looked around the room as though searching, then pointed a finger at the valise as though I'd only just discovered it. \"There it is, se\u00f1or, that's Mr. Rubery's valise. He left it here in his haste. We expected he'd send for it, but ever so many days have passed. I assume he's afraid to come back here. Would you perhaps be so kind as to take it with you when you return to the Bella Union?\"\n\n\"Anything to oblige a lady,\" he said, kissing my hand, all cozy gallantry, but there was a coldly inquiring look in his eyes. Had I overplayed the scene? Had I swung too quickly from vulnerable waif to cheerful servant girl? I rather think I had. He looked again at the glass I wasn't touching and sighed. I had given him a way to exit gracefully with the valise, but something about me rang false, and he couldn't afford to leave a loose end.\n\nDamn. I do _not_ interact gracefully with mortals. They can always tell.\n\n\"This is all quite pleasant,\" he said, getting his right arm free and sliding it around me snugly. \"And may I say, my dear, that you speak English beautifully? I am really quite astonished at your command of the language.\"\n\nOh dear.\n\n\"My mother was English,\" I temporized. It was happening again; three hundred years, and another tissue of lies to conceal what I truly was. Talk about d\u00e9j\u00e0 vu.\n\n\"Was she?\" He had another sip of his aguardiente. \"How did she come to be here, might one ask?\"\n\nHow indeed. Pirates? Kidnapping? Shipwreck?\n\n\"She came in search of her brother,\" I said carefully. \"He emigrated, you see, se\u00f1or, first to America and then to Texas, when the Mexican government was inviting settlers to farm the land. He sent word that he had a fine farm and was prospering. Her parents died, and there had been some thievery by solicitors\u2014what, I never knew precisely, but she was left nearly penniless. She wrote to her brother to expect her and spent the little she had inherited to buy passage to Texas.\"\n\nHe was nodding thoughtfully. Nothing improbably romantic, nothing that clashed with geographical or historical facts.\n\n\"Unfortunately,\" I continued, \"it appeared that her brother had exaggerated his success. He had become, in fact, little more than a beggar. The New World had failed to reform the prodigal vices of his youth. His parents were well-born, you understand, small gentry of an old family, but not rich. Yet he lived as though he had a fortune to inherit, drinking and gambling. That was why he'd been obliged to emigrate in the first place.\" I monitored his reactions to my story. Was that all right, that little intimation of good bloodlines to appeal to his English snobbery? Yes, he was accepting it.\n\n\"So, she arrived in Texas and found that her brother was not only not prosperous but sitting in the village prison for vagrancy, and all her prospects were dashed forever. I am afraid she quite collapsed. Luckily, as she sat weeping in the street, she drew the attention of a gentleman who had come to Texas to see to some business affairs he had there. He was a kind and gallant man, and he rescued her from her plight.\n\n\"That man was my father, Don Rodrigo Mendoza. He was not a Mexican, you understand, se\u00f1or, he was born in Old Spain, the youngest son of a house of ancient valor but no fortune. All his parents could procure for him was an officer's commission, and he made the best of it. He came with the army to New Spain and won a grant of land in Alta California from his king. After the Revolution he remained here; there was nothing in Europe to draw him back, and he had come to love this New World.\" There, a little more aristocratic ancestry, a father who was an officer and a gentleman. How was he taking that? He was still listening.\n\n\"My father was no longer a young man when he befriended my mother, but such was their love that he married her and brought her back to Alta California with him. They lived happily at his rancho near\"\u2014what was the most remote and unlikely spot I could think of?\u2014\"San Luis Obispo. I was born there, and they had no other children. I was educated in all that a lady of property need learn, and more, for my father had a great admiration for classical studies. I also had to learn what it is to manage an estate. We thought we would always be happy, but the coming of the Americans ruined my father's fortunes, and we lost our home.\" My Englishman was quite interested by that, to judge from his heart rate and respiration; but he merely made a sympathetic noise. I drew breath and went on.\n\n\"The shame killed him, se\u00f1or. To be cheated out of what he had won with his sword, by shrewd Yankee traders. And my mother did not long survive him, such was her grief. I was left with the clothes on my back and a determination to live. For five years I have won my bread by honest means, cooking and cleaning for strangers; and if it was menial work that my own maids would have scorned to do, when I was a cosseted child, at least I had the satisfaction of knowing that I had never descended into another kind of shame.\" How was that playing? Pretty well. His pupils were dilated; always a sure sign he was moved about something.\n\nAt least, that had been true for Nicholas Harpole, who was not this man.\n\n\"Two years I have been the cook at this station, se\u00f1or, and it has been enough to keep me. But times are hard, se\u00f1or, surely you know that. The American war, the floods, and then the smallpox, and now the drought. Men run mad, and this land is dying. I have not been paid in weeks. Can you blame me for the despair that led me to this bed? The inn is nearly deserted now, and I fear it will be abandoned soon. Where shall I go then? How am I to live?\"\n\nThere, I'd written myself a role to play. This was Hollywood, after all. Though the slight tremor in my voice was genuine. My fear and misery were real enough; if they could be made to convince this glorious stranger of my sincerity, so much the better. But had I convinced him?\n\nMaybe.\n\nHe was excited about something I'd said, intrigued, not suspicious. His blood was racing. And now he turned to look down into my eyes, and there was a genuine emotion visible for a second behind the smooth opportunistic facade. Was it sympathy? Not love by a long shot, but a good start; and a damned sight better than the reluctant intent to kill me. Yes, we were coming along splendidly.\n\n\"Where will you go?\" he said, taking my face in his fine strong hands and kissing me. \"Why, wherever in this wide world you please, with such a brave heart. You've no need to sell yourself to strangers, my dear; you can make your own terms for a husband. If you once catch the eye of a man of property, your fortune's made, and he's a damned lucky fellow!\"\n\nWell, _that_ rang false, though I doubt a mortal girl could have told. He wanted something from me. Some detail of my pathetic story had suggested an opportunity to him. What did I care? I loved the taste of his mouth.\n\n\"You are married, I suppose,\" I sighed.\n\n\"I? No. My line of work prevents that indulgence; travel, you know. And I'm no man of property, unfortunately. No, my dear, you can do better for yourself than me; but if you'll allow me to come to your assistance, I think there are certain measures that can be taken to ease your entry into better society.\" He looked deep into my eyes, and the fact that he had thought of a use for me didn't make his smile any less sincere. \"Upon my word I do.\"\n\n\"I am in no position to refuse assistance, se\u00f1or,\" I told him guardedly.\n\n\"No, poor child, and God knows that's none of your doing,\" he said, settling me gently into the pillows. How persuasive and silken his voice, and how nicely he smiled with that wide humorous mouth of his. \"See here: I represent the interests of Imperial Export of London. My firm would pay handsomely for the exclusive rights to supply British manufactured goods to the inhabitants of this coast. That's not our main object, however. You may be aware that British textile industries have suffered from the American conflict. Cotton production in the Southern states (on which our mills depend) has come to a virtual halt, and the little that is being produced is blockaded. Meanwhile our researches indicate that the prevailing weather in this part of California would be ideal for cotton.\n\n\"At the present time, the vast pastoral ranchos of your childhood lie fallow and desolate, mortgaged to unscrupulous Yankee moneylenders, and the hereditary gentry of your people are impoverished\u2014not merely by debt but also by the present drought, which has driven the price of cattle down so far as to ruin them. Your countrymen have exchanged independence for a dubious citizenship in a nation that despises them. How they must regret signing the treaty of Guadalupe Hidalgo,\" he said, not too theatrically, and had another sip of brandy.\n\n\"Assuredly they do,\" I said. \"But what is to be done? Were we able to defend ourselves? Our cavalry was magnificent, se\u00f1or, but we were something short on weapons. Perhaps before gold was discovered here we might have driven them out; after that, never in the world. The United States of America won't relinquish California willingly. We must resign ourselves to being a conquered people.\"\n\nHow intently he was listening to me. His face didn't show it, though, as he moved his hand idly along my thigh.\n\n\"And if this Civil War of theirs should alter the situation in your favor . . .?\"\n\n\"I try not to hope for anything these days, se\u00f1or,\" I said, watching his hand. \"Life is so uncertain.\"\n\n\"How very true. But consider what might happen if a benevolent interest were to buy up the debts of your countrymen. What if they found themselves once again in free and clear possession of their lands, with that same benevolent interest offering to lease their abundant acreage at handsome terms for a new industry?\"\n\n\"The British want to grow cotton here, se\u00f1or?\" I asked, drawing my brows together. I was beginning to understand Imarte's blather about the fascination of secret history. \"But . . .where would you get the workers, se\u00f1or? Most of the Indian population has been wiped out. Who would pick this cotton?\"\n\n\"Not Negro slaves, certainly,\" he said, smiling as his hand traveled. \"But former slaves have a great deal of agricultural expertise, and if they were offered good wages for honest employment, I daresay many of them would find their way here. Irrigation would present a difficulty, but one easily surmounted by the best engineers an empire could provide. All this remains to be worked out. At the present time, my principal interest is in arranging to meet with the prominent rancheros of your father's race and determining whether they might be interested in Imperial Export's offer. I truly feel that such an arrangement would be in California's best interests.\n\n\"And,\" he continued, tracing the curve of my shoulder with a finger, \"if I were to engage in negotiations with a representative of the displaced ruling class at my side, one who could advise me as to local customs and relationships . . . my chances of success would be greatly improved.\" Had his teeth always been that long? Yes, in that saturnine smile. I realized that it looked strange to me because I'd so seldom seen my poor godly Nicholas laugh. We'd been happy, though, at least at first . . .\n\n\"I don't know if I could be as much use to you as all that, se\u00f1or,\" I felt obliged to say. \"It's hardly as though I have the ear of Pio Pico. I'm a cook in a stagecoach inn, nothing more.\"\n\n\"But you know the land, you know the people,\" he said. \"And you yourself are one of the deposed _gentes de raz\u00f3n_. Your experience with the Yankees is common to them all. Should they not be more readily disposed to listen to my offer of better treatment at British hands if you added your charming voice to mine?\"\n\nI smiled at him and stretched. \"Am I to be your Malinche, se\u00f1or? You know the story? She was born near Campeche, where your English pirates used to make so merry. Long before that time, however, the Aztecs enslaved her people and were exacting cruel tribute from them. One day Cortes came from the sea and offered to free her people. She became his interpreter, and led him into Aztec lands to overthrow her tribe's oppressors. Are you planning a revolution, se\u00f1or?\"\n\n\"Nothing so uncivilized.\" He took my hand and kissed the fingertips. \"Surely there's been enough blood shed in this poor country. Wouldn't you like to live in a city where you could walk down the streets without fear of being shot? I can assure you, in her native land your mother had no need to wear a pistol. Order and safety and the rule of law, that's the blessing of a modern empire.\"\n\n\"That was what the Yankees promised us, too,\" I said lazily, tickling him. He arched his back and reached for my hair.\n\n\"Well, can one really expect better from Brother Jonathan?\" He loosed the end of my long braid. \"His nation of liberty was founded on the backs of Negro slaves and at the cost of exterminating the aborigines. As far as I can tell, the Yankee's idea of freedom is his right to carry a pistol with which he may shoot strangers in the street. No wonder his Union has crumbled. My expectation is that it will shortly expire of its own viciousness; and when that happens, California will have the chance to begin anew.\"\n\n\"Under Britain's guidance.\"\n\n\"Of course,\" he said, unraveling the serpentine twists of my braid.\n\nIt all seemed like a great idea to me, except for the fact that I knew it would never happen, thanks to whatever critical mistakes Mr. Rubery was even now in the act of making. All because he had set his valise down in the wrong spot. But why shouldn't I go along with it? What a splendid ride it would be, with this polished and dangerous mortal man, and who cared when it came to a stop? I had no idea what bizarre decision of God's had placed my lover in my arms again, in defiance of reason and death; but now that I had him back, damned if 1 would ever let him go again.\n\n\"Exigua pars est vitae quam nos vivimus,\" I said.\n\nTo my astonishment, he looked blank and then furrowed his brow. \"Brief is that part of life . . .\" he translated haltingly.\n\n\"The part of life we actually live is too short,\" I said in English. 1 switched to the Latin in which I'd flirted with Nicholas Harpole centuries past. \"What is this, young man, have you forgotten your grammar?\"\n\nIn schoolboy Latin, with several pauses, he said, \"The sword, when it is not drawn daily, rusts.\" He leaned back and looked at me, and there was that flicker of real feeling again for a second. Was it grudging respect I saw there? Maybe even the beginning of admiration? \"You _have_ had a classical education. And your memory is better than mine, I fear. Tell me, can you actually shoot with that pistol of yours?\"\n\n\"To kill,\" I said.\n\nWhat a gleam came into his eyes. I had him now, and he was determined to have me. My hair coiled in his fingers like snakes as he pulled my face close for a kiss.\n\n\"Madness, to leave you scrubbing pots for vaqueros,\" he murmured. \"You'll have a house in London, if that's your pleasure, or rule a plantation here that would be the envy of some queens. I've the wherewithal to free you from this miserable life, my girl, if you'll accept my offer on Imperial Export's behalf. And consider the benefit to your countrymen, Se\u00f1orita Mendoza. What, pray, is your Christian name?\"\n\n\"Dolores,\" I said, because that was what I'd been using for the last few decades or so. Then I remembered that my mother was supposed to have been English. \"Dolores Alice Elizabeth Mendoza.\"\n\n\"Delighted to make your acquaintance,\" he said. \"Allow me to introduce myself. Edward Alton Bell-Fairfax.\"\n\n\"Charmed, Se\u00f1or Bell-Fairfax.\" Dear God, what a Victorian name. It suited him, though. \"I'd be lying if I told you your offer of financial assistance didn't attract me, and I seldom lie, se\u00f1or,\" I said, and bit his lower lip gently. \"But I rather imagine the Americans will have something to say about the Verdugos and the Picos signing lucrative contracts with a foreign power.\"\n\nHe took my untasted glass and set it atop the headboard. Rising to lean over me, he took my two hands in his own. \"Let me make an analogy, if I may,\" he said. \"I think of California as a beautiful girl, lost beyond the mountains. In her veins runs the mingled blood of the Latins, passionate and heroic, and the cooler blood of the pragmatic races. Now, she is nominally under the protection of her two stepbrothers, the lantern-jawed gentlemen Jonathan\"\u2014he took my left hand and stretched it out above my head\u2014\"and Sam\"\u2014he took my right hand and stretched it out too, and lowered his face inches from mine. He went on. \"But they have not dealt with her in a brotherly fashion, have in fact spent her inheritance recklessly. This would be bad enough; but Jonathan and Sam are now locked in a fratricidal struggle, tumbling perhaps to their destruction. And what of the lovely California?\"\n\nWhat indeed, I wondered, as our lips met in a fierce kiss. He rose on his elbows again. \"Who will defend her? For she must be defended. Cruel eyes watch from the howling wasteland and plot her ravishment. To the south is the French beast, and his contemptible lackey the Austrian.\" He raised one hand and brought it down, down, threatening my left breast. \"What they have planned for the fair maiden is too terrible to be imagined, except in the mind of a lascivious Frenchman. And what is this to the north?\" He lifted his other hand and poised it over my right breast. \"Dimly seen on the horizon, shambling down from the region of ice and snow where he is monarch, the Russian bear. What is his intention? What has it ever been, my dear, but rape and pillage?\n\n\"Who will defend her from them?\" he said, leaning down to me, laughter glinting in his wicked eyes. \"She cannot flee to the east\u2014savages bar the way, and Mormons, only too eager to seize her. She cannot flee to the west, either. But she has one friend, I assure you, one stout friend who will rise to help her, who will bring her peace and contentment, who will look after her best interests, and invest her dowry wisely. And that stalwart stands ready to crown her with the wealth of a prosperous empire, if only she will raise her eyes to him.\"\n\nWe embraced. In the subsequent tumult the brandy glass was knocked from its place on the headboard, and it fell behind the bed and shattered, so the sweet narcotic spilled out. And the whole time, I swear to you, a little brass band was playing \"Rule Britannia\" in the air just above our frantic bodies.\n\nAnd that was how I became a double agent, se\u00f1ors, a spy in the pay of Her Britannic Majesty. And, you know something? I'm not sorry. I wish it _had_ worked out. Would Lower Canada have been so bad, really? But the laws of nations count for nothing, in the end. The only law that matters is the one that states that history cannot be changed.\n\n###\n\nWE DECIDED I SHOULD accompany him back into Los Angeles, where Mr. Edward Alton Bell-Fairfax had some correspondence to look to, since he had only just arrived yesterday from Veracruz. Then the first order of business would be to find me some clothing suitable to my role as Imperial Export's cultural liaison. Following that, we would make a brief sea voyage across the channel to Catalina Island. It seemed that there were some other representatives of Imperial Export on the island, doing scientific research for possible investment proposals. They had made quite a comfortable camp there, away from the gunfire and generally insalubrious air of Los Diablos. Perhaps we might stay over there a few days, while we planned a strategy for gaining the support of the rancheros.\n\nOnce we had managed to pry ourselves apart, I got dressed first and went out to leave a note on the credenza in my room. This also gave Edward a discreet opportunity to go through the valise and make sure that all its contents were as they ought to be. I realized with a start that we'd been in bed a lot longer than I'd imagined; the sun had already dipped down behind the ridge. That surprise was minor compared to my shock at encountering Juan Bautista, coming down the canyon with a couple of wicker cages. He seemed like a ghost, so unreal had my former life become in the last few hours.\n\nHe stopped and stared at me, openmouthed. \"Is something wrong? You look different.\"\n\n\"Nothing's wrong,\" I said. \"But I'm going into the field for a while myself. I may not be back any time soon. Let the others know everything's okay.\" After all, if Imarte could do it, why couldn't I? But Juan Bautista had noticed Edward's horse, and was lifting his head and scanning.\n\n\"Who's the mortal?\" he said. \"And\u2014hey! Who's going to fix dinner tonight?\"\n\n\"Fix your own damned dinner,\" I snapped. \"Look, you know that special research Imarte's doing? About the English? Well, tell her I'm following up a lead for her. Tell her I'll bring her back some invaluable first-person narratives.\"\n\n\"But you're a botanist,\" he said.\n\n\"Have you noticed there are no plants left to study around here? I'm making myself useful, that's all. And if you want to do the same, why don't you go saddle me a horse?\" I said, putting every ounce of authority I had into the request. He nodded meekly and set down his cages, hurrying off to obey. And it seemed to me, se\u00f1ors, that my reason was a perfectly valid one. I was indeed contributing to another operative's research. Perhaps even you would have accepted that as an excuse. If only the rest hadn't happened.\n\nEdward emerged from the inn, fully dressed, hat in one hand and valise in the other. He had to duck to avoid the top of the doorway. \"With whom were you speaking, Se\u00f1orita Mendoza?\" he asked politely.\n\n\"The stable boy. Only an Indian, se\u00f1or, and he knows nothing,\" I said. \"He's fetching me a horse.\"\n\n\"Good,\" he said, and set his hat on his head, which made him nearly seven feet tall. Just like Nicholas. The extraordinary height must have been something of a disadvantage for a spy. He went to fasten the valise securely into a saddlebag. When he had made fast the buckles and straps, he vaulted easily into the saddle and sat, reins in one gloved hand, waiting as Juan Bautista led out my horse. It took me a moment to notice Juan Bautista standing there helpfully, ready to give me a foot up onto my own mount.\n\nEdward Alton Bell-Fairfax looked like a god in the saddle. I had a moment of lurching terror as I realized how much I loved this man, whoever he was. His long-ago death had blasted my immortal life. What would it do to me, if it should happen again?\n\nBut this wasn't the same man, was it? He was certainly no saint and martyr, as my Nicholas had been. Subtle, politic, quite capable of a double cross, I should think, and certainly of cold-blooded murder if duty required it. He hadn't expressed any religious opinions, and I'd be willing to bet he had none, to be in his line of work. Could I be happy with this man?\n\nI was insanely happy.\n\nAnd that is, finally, the only reason I can give you for why I swung into the saddle, snarled, \"There's cold beef in the pantry\" to Juan Bautista in Spanish, and rode away with Edward Alton Bell-Fairfax.\n\nWe went at a swift canter through the slanting red sunlight, toward the place that would one day be black towers blighting the sky but for now was only the adobe sprawl I'd come to loathe so much. We were shot at only twice, and both times Edward sent off a lead riposte with a speed that was breathtaking, for a mortal. We made such good time that the sun was setting as we rode down Calle Principal.\n\nEdward made for the telegraph office while I waited in the saddle. He emerged with a communication he'd received on flimsy paper. I read it upside down without his noticing.\n\n> > > E. A. BELL-FAIRFAX, REP. IMPERIAL EXPORT CO.  \n> YOURS OF 14TH REC'D II:PM. BEST REGARDS FROM  \n> GOLDEN GATE. SUCCESSFULLY RECOVERED FROM LA  \n> GRIPPE AND LOOK FORWARD TO PROFITABLE  \n> VENTURE WITH CHAPMAN. A. RUBERY, SALES MGR.\n\nEdward exhaled angrily, but there was a certain satisfaction in his eyes as he mounted again, and we ventured once more into the mean streets.\n\n\"Good news, se\u00f1or?\" I asked.\n\n\"Tolerably good,\" he said. \"It seems my young idiot of a friend has managed to succeed in his affairs, against my expectations. Alfred never had much head for business. Privilege tends to soften the brain, or so I've observed.\"\n\nI reflected on that opinion. \"I should have thought _you_ were a member of the privileged classes, se\u00f1or.\"\n\n\"Hardly, my dear. My ancestry's good enough; but will you understand if I explain that my birth represented a certain inconvenience for my parents? I trust that won't dismay you. My upbringing was discreetly anonymous, and my inheritance is nil. So I've had to shift for myself rather. Another reason I'm not a suitable husband, by the way,\" he added slyly. My God, that was Nicholas to the life, that trick of talking out of the side of his mouth when he was being ironical.\n\n\"Se\u00f1or, your excellent qualities outweigh any consideration of clerical sanction,\" I said with an airy wave of my hand. He was grinning, about to answer, when his expression suddenly changed, became ice-cold. I followed his gaze but could see nothing except a pair of men hurrying into the Bella Union Hotel down the street. That was our destination too. Yet now Edward pulled abruptly on his horse's reins and spurred down a side street. I followed silently. A few more streets over, and he turned in his saddle to address me.\n\nAll the easy warmth of the day's pleasure had gone, with all the high color from his face. There was a look of strain in his eyes, though his voice was composed when he spoke.\n\n\"My dear, I'm afraid there's been a change of plan. Regrettable, but it can't be helped. There were two gentlemen entering the Bella Union just now\u2014perhaps you saw them? I know them, unfortunately, and their presence indicates that they know I'm here. It would appear our friend Mr. Rubery made a few more blunders than 1 thought. Please accept this as a token of my esteem.\" He reached into his saddlebag and drew out a small leather pouch. He passed it to me over my horse's neck; it was heavy for its size. \"That should enable you to find a more hospitable corner of the world in which to live. I believe you're clever enough to see that my presence shall shortly be a very dangerous place indeed, so I'd advise you to ride, and quickly.\"\n\n\"Not in this life, se\u00f1or,\" I said in a low voice, leaning forward in the saddle. \"I know the country, as you said, and I think you need a swift way out of here. You can't go back to the Bella Union. Will those men think to look for you at the stagecoach inn when you fail to return to your room?\"\n\n\"Possibly,\" he said. \"I must assume they've already discovered certain things. You're owed an explanation. You'll have it if you can get us both out of here alive.\"\n\n\"Follow,\" I said, and urged my horse forward. He followed, to my relief, and I immediately accessed a detailed street map. There was a narrow alley that ran along the base of the long low hill, and we made for that and galloped its length, behind its houses out of town. We encountered nobody but a dead man; our mounts slowed to pick their delicate way over him where he lay staring placidly at the new stars, and we sped on as the twilight deepened.\n\nSo back toward Hollywood, but not the way we'd come. I found us a route over torn earth, where cattle and vaqueras had passed in the dozens, and widened my scanning range to eight kilometers. Nobody was following us; nobody was lying in wait for us. The people whose delight it was to lie in the underbrush and take potshots at passersby were evidently on their dinner breaks. So far, so good. We made a wide turn, avoiding Cahuenga Pass, and rode for a winding canyon about a mile west of it.\n\nAbout a kilometer in, there was a great branching sycamore tree near a spring that bubbled out of the sand, and we reined in under the shadows there. The tree was occasionally used to hang thieves, and so local residents tended to avoid the area after dark. \"Well, Se\u00f1or Bell-Fairfax?\" I said.\n\n\"Well, Se\u00f1orita Mendoza,\" he said. \"You were indeed correct in your assumption that what's left of the Yankee government wouldn't approve of our plans for profitable trade with the locals. Apparently certain agents in the employ of Mr. Allan Pinkerton have got wind of something, doubtless thanks to Mr. Rubery's lack of discretion.\"\n\nSo this was how it had failed? \"The game is over, then, se\u00f1or?\" I asked, hoping we'd make the best of a bad business and get out of the heat.\n\n\"By no means,\" he said, \"since we were able to escape with our lives. Our proposal for your countrymen will simply be delayed, and they may be even more willing to listen by that time. If you're still game, of course?\"\n\n\"I am.\" Though I didn't like his persistence in the face of danger. Better to give it up, go home, live to spy another day. \"What about getting over to Catalina Island?\"\n\n\"Hm. That's compromised now. I had engaged a reliable fellow to take me across tomorrow evening, but if my room has been searched, I daresay he'll be watched.\" He looked thoughtful in the darkness, but the color had returned to his face. After a moment he turned, surveying the night. He couldn't have been able to see anything; to a mortal the black shadow would have been complete and impenetrable. \"I'm afraid I must impose on your bravery and your hospitality a while longer, my dear. I'd like to move on. We need a secure retreat with a decent view of the surrounding countryside. Do you know of any such?\"\n\n\"I think I know a place,\" I said, and we rode on up the canyon.\n\nIt twisted steeply for a few miles, and we followed a sandy creek bed under black avenues of trees. There was no water there now, but cool air currents flowed past our faces in the darkness until we emerged, riding straight uphill toward rimrock. To one side below us a splendid view of the plain opened out, and there was a tiny cluster of yellow lights to show us where the village of Sherman slept. I wondered if Se\u00f1ora Berreyesa had survived the smallpox. What a lifetime ago that had been.\n\nWe stopped climbing and made our way north through the hilltops, keeping well below the rimrock, and our horses picked their way carefully through black sliding piles of scree. I found a trail to take us around and down, until we emerged on another view: the San Fernando Valley lying vast and silent in the darkness. There below us was the northern end of Cahuenga Pass, beyond was the gigantic wall of Mount Hollywood, and far, far out on the valley floor was the tumbled ruin of what had been Mission San Fernando, its graceful arcade broken open to the stars. Not that a mortal could have made it out at that distance. I remembered chatting with a friar in its tidy mission garden, once, and felt a pang of loneliness.\n\n\"My compliments,\" whispered Edward beside me. \"A fine panoramic view. All we need is a defensible spot.\" How could he tell there was a fine panoramic view? I was seeing it by infrared, but he was a mortal man.\n\n\"Up here,\" I murmured, gesturing to a small steep hill like a turret that was crowned with a stand of trees. We rode up into the cover they provided. On the other side of the hill was a little terrace, wooded and dropping to a rocky saddleback ridge. To either edge the land fell sharply away in deep canyons. Below us, nearly invisible in the night and the trees, was a rough square of leaning wooden huts.\n\n\"You want a defensible spot?\" I said to Edward. \"This was one of Fremont's outposts. It's been abandoned for years. Nobody for miles but Cielo the farmer, and he won't be able to see us from his house.\"\n\nEdward nodded.\n\nI let him reconnoiter while I gathered wood for a fire. I'd told him the literal truth, we really were the only people for kilometers except for Mr. Cielo; so he returned satisfied as to our security. The largest of the buildings had a crude stone hearth, and I cleared it enough to start a small fire. No furniture left, but at least the creaking plank floor was dry. Edward saw to our horses and then came in and sat down beside me, carrying the saddlebag. He pulled out his watch and looked at it.\n\n\"Twenty-four hours,\" he said, with some satisfaction.\n\n\"Is that significant, se\u00f1or?\" I asked.\n\n\"Yes, I think so,\" he said, closing the watch and slipping it back into his waistcoat pocket. Without explaining further, he took out the valise and opened it. \"I daresay you examined the contents of this whilst it was left at your inn?\"\n\n\"No, se\u00f1or,\" I said truthfully. \"But I'm fairly sure Martha went through it.\"\n\n\"Yes, I'd gathered someone had. Damn.\" He looked up as an idea struck him. \"Where has that intriguing lady got to, by the way? You said she was gone; where did she go?\"\n\n\"San Francisco,\" I answered.\n\nHe stopped in the act of taking out the neat little stoppered bottle of violet ink. \"When?\" he said in a voice too calm.\n\n\"Just this morning, se\u00f1or,\" I said. \"If you'd arrived two hours earlier, you'd have met her.\"\n\nHe relaxed visibly, but his expression was still grim.\n\n\"Now, I wonder,\" he said, pulling a pen from an inner pocket and removing the cap that protected the nib. \"I wonder how well you're acquainted with the lady.\"\n\n\"Quite well, se\u00f1or.\"\n\n\"She had rather a reputation at the Bella Union as a sympathizer with the Secessionist cause, and was reputed to have had a particular fondness for Britons. I assume this was how Mr. Rubery chanced to be enjoying her favors at the moment her jealous friend stepped onstage.\" He pulled out a sheaf of papers, half sheets that appeared to be printed in the same violet ink, and thumbed through them carefully. \"Was the lady what she seemed to be, in your opinion?\"\n\n\"I don't think she's a Federal spy, se\u00f1or. I think she serves her own interests, like most whores.\"\n\n\"Yet she made no attempt to pilfer the contents of this valise.\" He held up the sheaf. \"These are drafts on the Bank of England; useless without a countersignature, but a dedicated thief would attempt to forge one. And there's a considerable sum in coin as well, but apart from prising up the seal to look at it, she left that quite untouched. She may also have read through certain other papers here, the idea of which is rather unsettling.\" He rested the valise on his knee to use as a writing surface. Dipping his pen in the violet ink, he selected a draft and filled out an amount. He signed it with a flourish and carefully put his pen back in his inner breast pocket. \"I put the question to you: having examined these things, what was the lady's object in departing for San Francisco this morning?\"\n\n\"She didn't say,\" I lied, looking thoughtful. \"If she was intending to betray your cause, I should imagine she'd have taken the valise straight to Fort Drum down here. Would she have learned from the papers that your friend was now in San Francisco?\"\n\n\"Yes,\" he said, waving the check in the air to dry it.\n\n\"Well, then, perhaps she's gone up there to blackmail him. That might explain why she left the valise.\"\n\n\"It might well.\" He stared into the fire. \"In which case, we have nothing to fear from that quarter, at least at present.\"\n\n\"We haven't?\"\n\n\"No,\" he said. Groping with his free hand, he pulled a kind of envelope of oilcloth out of the valise. It was exactly big enough for a half sheet. He tucked the check inside and handed it to me. \"There, my dear. Less immediately negotiable than the gold, but if we should be separated, this will get you to London, where you'll find you have friends. Take care not to get it wet; even perspiration is enough to make that ink vanish like a dream.\"\n\n\"You're too generous, se\u00f1or,\" I said, opening it to peer at the amount. My jaw dropped. I had to read twice before I was sure. I looked at him in confusion. \"Se\u00f1or! That's\u2014it is too much.\"\n\nHe gave a brief shake of his head, putting away the contents of the valise. \"You may well earn it before this business is done. Let me tell you honestly that your life may stand at hazard, Se\u00f1orita Mendoza. I shall certainly think no less of you if you wish to withdraw at this point and travel elsewhere. Indeed, I'm obliged to tell you it's in your best interests to do so at once. The money's yours, regardless.\" He reached back into the saddlebag and brought out a small box.\n\n\"This is more than a matter of trade, isn't it?\" I said.\n\nEdward looked at me for a long moment before he answered. \"It is a game of nations,\" he said. He drew his gun from its holster and opened the box, which contained small tools, ammunition, and a chamois cloth. Methodically he removed the remaining bullets from his gun and began to clean it.\n\nI watched him awhile before I spoke again. \"I suspected as much. Well. Se\u00f1or, I will not leave you. This involves my honor, after all.\"\n\nHe was shaking his head. \"That won't do, my dear. Marriage is really not\u2014\"\n\n\"You mistake my meaning,\" I said. \"You are a stranger in my country, and you have shown me kindness, and now our common enemies hunt for you. I will not leave your side, se\u00f1or, while I can be of use to you. That's my honor, and I won't surrender it.\" Only lines spoken by the character I was playing; but I meant them all the same.\n\nReal emotion in his eyes again, before the cold businesslike look returned. What kind of man lay behind the role _he_ was playing? \"Se\u00f1orita, you do credit to your father's sword,\" he said. \"By God, I hope we can win through to London.\"\n\n\"What are our chances?\" I settled back, hoping he couldn't hear the way my heart was pounding.\n\nHe took up his gun again and spun the empty cylinders. \"Not poor, I think,\" he said, reloading. Oh, his heart was pounding too. \"If we can avoid capture for another day or so, and if we can get across to Santa Catalina Island, we'll do very well indeed. The difficulty will be finding a vessel to take us.\"\n\nI remembered Se\u00f1or Souza and his dark house on its pilings.\n\n\"I can manage that, se\u00f1or,\" I said. \"I know a fisherman with a house on the slough, below the old landing place.\"\n\n\"His name?\" Edward gave the weapon a final inspection and returned it to its holster.\n\n\"Souza. A Portuguese gentleman. There was a doctor who resided at the inn for a while, to whom he's indebted. I'll call in a favor on his behalf.\"\n\n\"Can you?\" Edward said, putting away the box. He leaned back and stretched out his long legs. \"Then we'll hope for the best and prepare for the worst, and God defend the right.\"\n\nFlames were dancing in his eyes. Flames running up his shirt, sputtering in his fair hair, Nicholas had stared at me until the agony broke his concentration . . .\n\nI had to close my eyes and draw a deep breath. When I opened them, Edward was sitting forward and frowning at me. \"Are you well, se\u00f1orita?\" he asked.\n\nI stared at him. \"Do you believe in God, se\u00f1or?\"\n\nHe looked disconcerted. \"I suppose so,\" he said at length. \"Certainly religion is a civilizing influence, if it's not taken too far. I imagine you're a Roman Catholic?\"\n\n\"I was born one,\" I said, which was the truth.\n\n\"I won't offend you, then. But I think we can agree that zealots of any persuasion do a great deal of harm. All the same, men need commandments of some kind.\"\n\n\"Do you think there's a true religion?\"\n\n\"Do I? Yes, the C. of E., I suppose. But if I were a Hottentot, I'd tell you that my great wooden idol was better than anyone else's. I doubt that distinct creeds matter much, so long as civilized behavior is observed.\" He looked at me askance. \"Is the matter of doctrine very important to you, my dear?\"\n\nNo, this man wasn't one to die for his God, and I'd have to remember to thank God for it, next time I believed in Him. I wasn't sure I didn't believe in Him right now. Could my lover have been reconstituted _without_ the faith that had killed him?\n\nIn my relief I stammered, \"Not to me, se\u00f1or, but I feared it might matter to you.\" I drew breath and temporized. \"My mother owned an English book about Protestant martyrs, _Foxe's Acts and Monuments_. You understand, she became a Catholic when she married my father, but since this book had been given to her as a girl, she kept it for sentiment's sake. Well, I read it when I was learning to read English, and what a terrible business. Such hatred the Catholics and Protestants felt for each other! So I drew the conclusion that Englishmen might feel strongly on the matter even now.\"\n\n\"And many do,\" he admitted. \"But that was three centuries ago. If all nations brooded interminably on old scores, there'd never be an end to the vengeance. Most of the Catholics I've known have been reasonably decent chaps. A certain amount of tolerance is essential to civilized behavior. Barbarity is the force to be fought, not differences of dogma. Wouldn't you agree?\"\n\n\"Yes!\" How direct, how enlightened. Nicholas's intellect and humanity, everything I'd adored in him, without his late medieval prejudices.\n\nPossibly encouraged by my enthralled look, Edward continued: \"Religion has its place, certainly, in reinforcing ethical behavior amongst the masses, but any sufficiently enlightened secular laws will have the same effect. After all, most of the creeds of the world have essentially the same purpose, have they not? To enjoin men to be what we call moral, which is to say _civilized_. A civilized man obeys the rule of law, he acknowledges that he must not injure his neighbors, and if injured by them, he must appeal to law for satisfaction rather than indulge in burning their houses over their heads as they sleep. Civilization is the ideal for which we strive, with so little perceptible success; yet we do succeed, in inches and over years.\n\n\"Consider.\" He sat forward, resting his elbows on his knees, and the intensity of his eyes made my heart flame. \"What was Britain when the Romans found her? A wilderness of howling savages. And Rome, a thousand times more civilized, yet was so barbaric, she held spectacles of slaughter for her citizens, and her rulers were guilty of the most hideous crimes.\n\n\"Still, the Pax Romana tamed the wilderness, taught the savages, and, as imperfect as she was, Rome sent the idea of civilization working throughout the world. Even her fall into decadence could not stay the forces she had set in motion.\"\n\n\"Other people were what we'd call civilized, before Rome,\" I said with effort. His voice was so magnificent, I hated to interrupt him.\n\n\"Certainly. The Greeks, in fact, were more so. They lacked, however, Rome's peculiar genius at organization and her insistence that civilization be spread. That, in my opinion, was Rome's great contribution to the world, and that is the inheritance she passed on to Britain: the moral imperative to bring the rule of law to barbarians, through the operative mechanisms of empire.\" He moved closer to me. There was a purpose to all this, of course: he wanted to win me to his government's side. But look how his eyes glowed. All the easy, deceitful charm had fallen away, and passionate conviction was shining out of them like light.\n\n\"That is the cause in which I labor. What force can bring the greatest good to the greatest number of men? Only the modern empire, with its constitution to guarantee their individual rights, and its power to bring them prosperity. A missionary may persuade a painted savage to worship a cross rather than an idol; but he will not make laws that send that savage's children to school, where they might learn to make the desert they inhabit another Eden by means of the advanced sciences. He may persuade his flock to love one another for his God's sake, but he'll invariably urge them to slaughter any neighboring tribe that still worships stone idols. This is the failure of religion as a force for the common welfare,\" he said.\n\n\"Se\u00f1or, you have the truth of it,\" I said from my heart. How had Victorian England brought forth a man like this? \"And surely this is the way to a better world, is it not, this secular enlightenment? Even the Americans have deduced this, with their separation of church and state.\"\n\n\"Ah, the Yankees.\" He sneered elegantly. \"What have they achieved but violent chaos? And I'll tell you why, my dear. Liberty (as they conceive of it) and loyalty are opposing concepts. Having rebelled against the hereditary ruler who was the embodiment of their nation, to what will they be loyal? Their flag? But see what has happened now: half the nation, asserting its liberty to keep slaves, has rebelled and taken up arms to defend that liberty. It won't end there, either, you know. Any brute will demand his right to be a law unto himself, beating his wife and his children as he pleases, and defend that right with his father's rifle and think himself a patriot.\" He used his big hands so well when he spoke, with graceful economical gestures to make his point.\n\n\"The difficulty, I think, is that liberty is too abstract an idea for human nature to grasp. It is too easily twisted into lawlessness, as has happened in America. Most men are incapable of reverencing a mere principle; that principle must be embodied in a living person to effectively hold their allegiance. This is where empire inspires, and democracy fails in inspiration: love of one's monarch.\"\n\nWhat a spell he was weaving, jarred only by my mental image of dumpy little Victoria and her priggish prince. No, I certainly couldn't agree with him on that one. But he swept on and took my breath away by saying:\n\n\"Mind you, royals and their attendant baggage of toadies, cretins, and thieves are seldom an inspiring lot in and of themselves, but in a constitutional commonwealth they need not be. I believe that my sovereign has deplorable taste in art, is devoid of much talent to rule, and certainly couldn't compete with Venus for beauty. But she in her person is the empire personified, the driving force of civilization, and as such I serve her, reverence her, and will, when necessity commands, die for her.\"\n\nThe wickedly confidential way in which he said this was so delightful that it took a moment for the last part to sink in on me.\n\n\"I trust, se\u00f1or, you've no intention of dying soon,\" I said.\n\nHis eyes narrowed, and he shrugged. \"No intention at all. But my occupation carries that risk, always. I imagine I came rather close to the awful specter this evening. Certainly I'd hate to lose my life as the result of some fool's incompetence. Should there come a time, however, when my death would serve the purpose of empire, then I hope I will die without hesitation. As the bard of Avon says, 'Live we how we can, yet die we must.' And that being the inevitable case, one can at least have the satisfaction of accomplishing something with one's death.\"\n\nIt has probably already occurred to you, se\u00f1ors, but it was only at that moment that I realized that this _was_ the same man I had loved. He had exactly the same inner drive that had got him burned at the stake; only the focus of his devotion had shifted. As I stared at him in horror, he leaned back and went on:\n\n\"I don't imagine I'd much care to live to decrepit old age, to tell you the truth. End one's days being pushed about in a bath chair? Not for me. Better a brief life lived intensely, with a keen appreciation of its pleasures.\" He gave me a meaningful smile. \"It's no more than the bargain soldiers make, after all, self-sacrifice for the greater good.\"\n\nHe was the perfect operative. Brilliant intellect, no life of his own, utterly focused on his duty to make the world a better place, thoroughly convinced that his masters were wise and good. He was just what I was supposed to be.\n\n\"But\u2014isn't human sacrifice one of the barbarisms you're working to put an end to?\" I protested, rising up on my knees. \"Whether willing or unwilling? And how can you _know_ , se\u00f1or, that your death will really have accomplished anything? Secrets of espionage are the most transitory. And who can ever say how history will play out? Consider, consider those same English martyrs!\"\n\nI felt my voice shaking and tried desperately to control it, but everything I'd wanted to say to him for three hundred years came howling up from my heart. \"They let themselves be burned in droves, se\u00f1or, and for what? They died for _nothing_. If they'd only kept their mouths shut and lain low, they'd have lived to see a better day, because in short order Bloody Mary died and Elizabeth succeeded her, and restored their stupid Protestant faith to power. So how can you know, se\u00f1or, that you wouldn't be throwing your life away, that you wouldn't serve your cause better as a living man?\"\n\nWas I convincing him, se\u00f1ors? No, I was only arousing him. He found the throb of my voice, the firelight on my hair, and the angry blood in my cheeks exciting. But he did make an effort to reply seriously rather than simply grab me and pull me down.\n\n\"I am astonished at how well you know English history, my dear,\" he said. \"Granting your point\u2014without foreknowledge of history, what else could those Protestant heroes have done? Nor can you say with real certainty that their deaths accomplished nothing. If they hadn't died as bravely as they did, if they had not so publicly denounced Mary's tyranny, might not her husband Philip of Spain have been emboldened to seize the crown after her death? Setting aside the immediate salvation that martyrdom is reputed to confer on the martyr.\" His eyes glinted, reflecting fire.\n\nHow could I tell him that _I_ had known how history was going to play out, and I'd failed to save him, even so? He continued:\n\n\"The instinct to preserve life is natural for your sex, my dear; it's a fine and appropriate womanly inclination. And when the ideal is reached at last and the world is civilized, I trust there shall be no more need of martyrs to die in any cause. At present, however, we live in a world that requires certain regrettable actions in order to bring about the better world we desire. I myself have been required to commit crimes, to do things I would certainly rather have avoided. And when my blood must be shed to atone for those acts, then at least I'll face oblivion with a clean score. It works out, you see.\"\n\n\"In the minds of wicked old men who make governments, it works out,\" I said in despair. \"They know they can always count on a ready supply of brave men who'll die for a cause, like you, and so they continue to wage wars and spend lives to keep themselves in power. But if all the heroes refused to play that great game, what then, se\u00f1or? If the nations had no means of waging war on one another, wouldn't they be obliged to find some more civilized means of settling their differences?\"\n\nI thought he wasn't listening to me, so hungry were his eyes. Even now he was moving forward with one hand out to wind it in my hair and pull my face close to his. He kissed greedily, but when we came up for air he growled:\n\n\"No, my dear, they wouldn't. Come, come, do you suppose the politicians are the only ones responsible for wars? When one shepherd will steal his neighbor's flock, when one child will pick up a stone to fling at a child from the next village over? Things are by no means so simply drawn as you imagine, and the causes of war are far too complicated to gloss over with a pacifist clich\u00e9.\"\n\nHe bore me backward, and we wrestled as he very adroitly unbuttoned and unhooked. \"If all the statesmen in the world signed a universal peace tomorrow, some spiteful fool would find a way to bare his bum at his former enemies, and the whole misery would begin all over again. It will take a great deal more subtle work, over a much longer period of time, to bring peace to the world.\"\n\nAnd of course he was right about that, and I _had_ used a pacifist clich\u00e9. Really, what other man could argue like that while in the throes of carnal passion? Only one I'd ever known, a long time ago in a land far away.\n\nEdward put his face close to mine and looked into my eyes, and I was so spellbound by his gaze and the music of his voice, I very nearly missed the meaning of his words, which would have been a pity, because it was extraordinary.\n\n\"And _when_ there is peace at last, and _when_ men are no longer distracted by the ravages of war and crime, then the real work begins. Mankind has grasped at science and invention to improve his lot; when he truly understands that he can wield those tools to improve _himself_ , he will lay the cornerstone of the earthly paradise,\" Edward said. \"What might not science achieve, in a world where a nation's resources weren't continually drained by strife? What if that nation made a remarkable discovery, one that gave new meaning to old legends of a golden age? What if it were possible to utterly change the human condition? What if it were possible to put an end forever to disease? To age? To death itself? And where will men ever make those discoveries but in a stable and peaceful empire?\"\n\n_How did he know_? Was the man a bloody prophet? How could he foresee so clearly what the future would hold once the Company was founded? And how the hell could he prophesy so, with our hearts thundering against each other and our bodies locked in the most intimate embrace? I didn't know, I had never known, but just so had my lover spoken three centuries before. And now as then, I fled from the meaning of his words and lost myself in the worship of his magnificent mortal flesh.\n\nSo we burned together beside that little fire, in the leaning ruin of Fremont's outpost, and the shades of Manifest Destiny and Imperialism looked on with sardonic smiles. If the flames had risen and consumed us in each other's arms, se\u00f1ors, we'd have felt no pain. If only we were lying there now, our quiet ashes mingled together!\n\nI opened my eyes and saw him, in the chilly morning light. For a moment I thought I was lying in a garret room in England, awake at last after a nightmare of terrible sorrow and interminable length. But no, this was my new lover, who by an amazing coincidence was identical in every respect to Nicholas. Limited gene pool indeed! I had never once in all the intervening centuries met another mortal with his face. He wasn't conventionally handsome, with that broken nose and that wide mouth; but the combination of features that would have made another man a leering gargoyle were elegant in Edward-who-had-been-Nicholas. Part of the trick was the way he used his face, the responsive liveliness of expression, the movement of his eyes. He fascinated, he charmed, he _moved_ well, and one never realized that the big man didn't quite look human\u2014something odd in the angle of the cheekbones, in the way the head sat on the powerful neck. But he hid his strangeness far better than I, who looked human.\n\nIt was going to be tricky keeping him alive and safe, in his line of work, but I could do it, if I stayed by him the rest of his life. How to manage that? I'd find some way. Porfirio had his pet mortals, didn't he? The Company let him go to them, stay with them, help them when they needed his help. All I'd need do is learn clever makeup skills, appear to age with Edward as he aged.\n\nFull body appliance makeup to make me look like an old woman with my clothes off obviously wouldn't work. What then, live with him as is and hope for the best? The best would be some mortal disease felling him comparatively early, before he could notice that his hair was graying while I still looked eighteen.\n\nIf I took the chance and lived an idyll with him, and if he loved me enough, might I gradually let him in on my little secret? But look what had happened the last time he discovered I wasn't human.\n\nBut we were in the modern era now, and this was a man with a strong belief in the virtues of science, unlikely to attribute my inexplicable abilities to Satan. Maybe I could explain, maybe he could be brought to understand, maybe he could become one of Dr. Zeus's paid mortals with a control implant . . .\n\nNo. But something would suggest itself, something could be worked out. Who knows, perhaps we'd live blissfully awhile and then lose interest in each other, as so many mortals did, as _we_ might have if he'd lived, and go our separate ways without pain. Perhaps fate had brought him back for just that closure on the events of 1555, to heal my life at last.\n\nWhen he opened his eyes, though, I stopped thinking.\n\n\"Good morning, my dear,\" he said, alert at once. The charm went on as though a switch had been flipped, and his courtly smirk acknowledged that he'd very much enjoyed my company last night, wink wink, squeeze squeeze, yet his eyes also tracked around the room. Saddlebags still there, weapons still there, no intruders. Having ticked these off his mental list, he smiled down at me. \"I trust you slept well?\"\n\n\"Very well indeed, se\u00f1or,\" I said, smiling back at him, but I had logistical problems of my own: here was a mortal man who'd been without food or water for at least twenty hours. \"Though the land of dreams is a poor place to visit after one has been to heaven.\"\n\n\"Ah, but I was in paradise all night,\" he said gallantly, getting to his feet and offering me his hand to rise. \"Now, my dear, our first business is to feed and water the horses. Where did the redoubtable John Charles Fremont attend to such things?\"\n\nI accessed a topographical survey. The nearest water should be a little creek flowing down a ravine just over our turreted hill. What was this footnote? Future site of Harrison Ford residence? I made a mental note not to download map information from Einar again. Movie star homes, my foot. Was the damn creek there in time of drought, that was the information I needed. I smiled prettily at Edward and pointed vaguely north. \"Woman's intuition tells me, se\u00f1or, that there is a spring in yonder canyon.\"\n\nWe led the horses to where there was a little green forage. But the spring had dwindled to a seep, an uninviting trickle in a black muddy bog at the canyon bottom, full of amoebic guests that wouldn't bother a horse but would be only too happy to give a mortal dysentery. We had to prise up a couple of big rocks and grub out a hole for the water to collect in so the horses could drink. No bullwhip-wielding hero showed up to offer us assistance, but it was 1863, after all. My own hero produced a canteen from his saddlebag and offered it to me.\n\n\"You'll find it brackish, I'm afraid, but safe,\" he said. How did he know about the shigella I had detected here? And see how expertly he was avoiding the poison oak as he looked for a clean place to sit down. He settled on a boulder at the approximate location of what would one day be Mr. Ford's front step, and nonchalantly proceeded to shave himself with a clasp knife that appeared in his hand out of nowhere.\n\nI was distracted from my awestruck contemplation of this feat (no soap, no water, and he didn't nick himself once) by an annoying little signal pulsing through the ether.\n\n_Mendoza_?\n\n_What is it, Juan Bautista? I'm busy_.\n\n_Are you coming back today_?\n\n_No. Fix your own breakfast. Wait, this is important. I need you to put some food together and bring it up to me. Make it look as though you'd fixed yourself a very large picnic lunch. I'm going to lead my mortal friend back in your direction, and we're going to just accidentally on purpose run into you as you're out hunting, okay_?\n\n_What's going on_?\n\n_I'm doing fieldwork for Imarte_.\n\n_Hey, is that guy a real British secret agent? Like in the James Bond movies_?\n\n_Uh, yes, I guess he is_.\n\nEdward was standing up, neatly folding away the clasp knife. A day and a night of living rough in the field, and he hadn't so much as a smudge on those fawn wool trousers of his. Whatever secret device kept James Bond's tuxedo impeccably pressed, it seemed to have been already in use by the British secret service in 1863.\n\n_Neat! Can I help with whatever it is you're doing_?\n\n_No, just bring us food. And this is secret, okay, J. BJ I'm trying to keep this man out of danger while I find out more about his plot_.\n\n_Right_.\n\n_I'll broadcast a directional signal as we come. Do your best to look surprised when you meet us. Spanish only, and remember, he probably understands it as well as you do, so watch what you say_.\n\n_Gotcha_.\n\nEdward was coming toward me. \"Well, my dear,\" he said. \"I find nothing especially edible hereabouts, with the possible exception of rattlesnakes. What are the chances we might purchase food from that farmer you mentioned?\"\n\n\"He is an inhospitable man,\" I said. It might have been true, too, for all I knew. \"I recollect a farm near the Rodeo de Las Aguas where they are friendlier. It would be our wisest course to keep to the heights and work our way over there. We will pass near the stagecoach inn, but not near enough to be seen. Does that suit you, seftor?\"\n\n\"Very well indeed,\" he said.\n\nWe saddled our horses and rode out, working our way back in the direction of the inn, with me broadcasting a steady signal to Juan Bautista. As we were edging our way down onto what would one day be Mulholland Drive, I spotted him lounging ever-so-casually against a rock.\n\n\"Ay,\" he said in Mexican Spanish. \"Se\u00f1ora Mendoza, I was afraid when you did not return last night. I am out hunting, as you see.\" He waved one of our rifles unconvincingly. Edward raised an eyebrow at him.\n\n\"And I am safe, as _you_ see. Is that food you have in your basket, boy?\" I said.\n\n\"Oh, yes\u2014I packed myself a lunch.\" Juan Bautista was trying not to stare at Edward. \"It's a very good lunch.\"\n\n\"Well, listen to me, I'm going to ask you for it. My friend here is a kind gentleman who is being pursued by thieves. We had to flee Los Diablos last night, and he has had nothing to eat. We would go back to the inn for a meal, but I am afraid they may come looking for him there.\"\n\n\"Oh, they have already,\" Juan Bautista said.\n\n\"What?\" _What? Why the hell didn't you tell me_?\n\n_I thought you knew_. \"Yes, se\u00f1ora, two Yankee men. They said they were the friends of the Englishman who had been there. They came to collect the valise he left behind. I pretended not to understand them because, as you know, I do not trust the Yankee oppressors of our people.\" Juan Bautista gave the rifle a dramatic flourish.\n\nI turned to look at Edward. His face was a perfect mask of polite incomprehension, but he had turned pale. \"Se\u00f1or,\" I told him, \"the boy says that two Yankee men came to the inn asking for the valise an Englishman left there. He doesn't like Yankees, so he wouldn't speak with them.\"\n\n\"Really,\" Edward drawled. He made an odd little gesture that I would have taken for a shrug, if I hadn't known where all his concealed weapons were. He was quietly assuring himself each was in place. \"Ask him when they were at the inn.\"\n\n\"When was this, Juanito?\"\n\n_You're scared, aren't you? What's wrong_? \"It was this morning, se\u00f1ora, just after first light.\"\n\n_They'll kill him if they find him_. \"And are they still there now, boy?\"\n\n\"No, se\u00f1ora, but I think they did not go far away. I think they are hiding to watch the stagecoach come and go, but, as you know, I am an Indian and white men cannot conceal themselves from me.\" _Can I help? Can 1 be your Indian guide? Please? I could throw the bad guys off the scent if they followed us_.\n\n_God damn it, this isn't a movie_. \"He says they were here at dawn,\" I told Edward. \"He says they left, but he thinks they're still hiding in the pass, waiting for you to come.\"\n\nEdward just nodded. I was feeling a slow anger building in him, sullen and exasperated. Not much fear, though for all he knew the Yankees might have had him in their sights at that very moment. But I was terrified for him, se\u00f1ors.\n\n\"I think you ought to ambush and kill those Yankees, Juanito,\" I said. \"I assure you they are very bad men.\"\n\nJuan Bautista did a good job of looking crafty. \"Perhaps that can be arranged, se\u00f1ora.\" _So what do you want me to do about them, really_?\n\n_Like I said_.\n\nThat shook Juan Bautista's little world. Even though Einar had been nailing mortal hides to the wall for months. After all, wasn't this Los Angeles, where such things were done every day? The boy shuffled his feet and looked at the ground. _Mendoza, I can't kill mortals_.\n\n_Why not? James Bond does_.\n\nEdward apparently came to some kind of decision, because he looked up at this point and said, \"Thank the boy and tell him to go on with his business. If he meets the Yankees again, on no account is he to mention that he's seen me. But he should avoid them if he can, because they are very dangerous men.\"\n\n\"Give me that food now,\" I told Juan Bautista. \"My friend offers his thanks and says to stay away from those Yankees, but don't tell them about him if you do encounter them. I assume this means he does not wish them to die. What a pity. However, your soul is free of two mortal sins. How fortunate for you.\"\n\nJuan Bautista was too unnerved to play back. He just handed me the lunch basket and muttered, \"Good day, good fortune on your journey,\" before vanishing into the sagebrush.\n\nI hefted the basket and flashed Edward a brittle smile. \"The boy has kindly surrendered his luncheon repast to our greater need. Poor fare, se\u00f1or, but sustaining. I suggest we find a secure place to eat and revise our plans.\"\n\nHe shook his head grimly. \"If I were a free man, we'd be riding for San Francisco this moment. Unfortunately I have a duty to salvage what I can from Rubery's incompetence.\"\n\nThis gave me an idea, but all I said was \"One cannot make decisions on an empty stomach. Let us ride back to the high ground, se\u00f1or.\"\n\nWe returned to the vicinity of Fremont's outpost and stopped in a grove of oak trees on the saddleback ridge just below. We still had a good view of the north end of the pass, but from a more sheltered spot. If anyone tracking us should find our previous night's camp, we'd have warning of their presence and a reasonably clear shot at them.\n\nI unpacked the basket. Left to himself, Juan Bautista had grilled beef and made severely deformed tortillas for supper last night, and we had lots of the leftovers. He had also included a jug of water, a jar of olives, some cheese, a can of sardines, and a couple of cakes of Theobromos.\n\nI should mention that I didn't have to explain any of the food to Edward, or show him how to roll up a filling in a tortilla. He'd learned how, somewhere. Perhaps in secret agent school; more likely in Veracruz, whatever he'd been doing there. From his saddlebag, he drew out an immense white handkerchief and spread it across his lap. I watched in amazement as he made himself sardine tacos and ate them without getting one spot of oil on those immaculate clothes.\n\n\"As regards this plan, se\u00f1or,\" I said at length, when we'd consumed half the contents of the basket and neatly packed the rest for later. \"As I said: my honor will not permit me to leave you. But clearly we are dealing in matters of life and death now. For your sake, I will be my father's sword at the throat of your enemies. Yet I begin to question whether your government is wise enough to rule the world. What fool ever trusted your Mr. Rubery with important papers?\"\n\nYes, that touched a nerve. What a cold, bleak look in his eyes as he stared out at Cahuenga Pass, and how well I remembered the bitter anger that pulled the corners of his mouth down. He mastered his temper, though, and turned to face me with a rueful smile and a shrug.\n\n\"I can't deny the truth, my dear, particularly in this instance, since we're facing considerable danger as a result of it. I have at least the satisfaction of pointing out that well-born imbeciles tend to get themselves killed before they manage to breed, leaving room for men of ability to replace them. And not all well-born men are idiots! I can assure you that there is an office in Whitehall where a very wise and noble man makes national policy, one whose judgment I'd utterly trust, for all that he's seldom quoted in the _Times_. That same man who made the decision to give Alfred a task he was barely fit for had the foresight to send me after him, guessing no doubt that Alfred would make the wretched mess he has.\"\n\nI shook my head. \"Why send the boy in the first place? If your people think you're expendable, their aristocratic brains are no better than Mr. Rubery's. I'll grant you, the idea of this land in peace and prosperity under British rule is a splendid one. I'd die myself if that would bring it into being. But I don't see how it can be accomplished now, do you? Martha must have gone straight to the Yankees and told them about the valise; or if she did not, some other indiscretion of Mr. Rubery's put them on the scent. They surely know everything now. I don't see how your masters can blame you for pulling out and saving what you can of the affair.\"\n\n\"Ah, but the Yankees _don't_ know everything now,\" he said. He pushed his lank hair back from his forehead with the flat of his hand and set his hat straight on his head. \"If they did, they wouldn't be after the valise. Whatever else we do, it mustn't fall into their hands.\"\n\n\"Shouldn't we destroy the papers, then?\" I asked.\n\n\"No. If we can salvage any part of this, we'll need them. And, my dear, the game's not over yet. Alfred's an idiot, but at least he's a British idiot, and despite the business with the valise he's managed to accomplish successfully another part of his task.\" He glanced down involuntarily at his watch pocket. \"So. All we need do is evade our Yankee opposite numbers until we can get across to Santa Catalina Island. My compliments to Abraham Lincoln, I must say. Certainly no one thought he had the resources to spend anything on counterespionage in this corner of the nation. But he can't keep it up for long, I think. And if the Union loses to the Confederacy, it won't matter whether he knows about us or not.\" He rose to his feet briskly and extended his hand to me. \"We'll triumph yet. Are you still game, my love?\"\n\n_My love_. The earth wobbled in its orbit, just for a second, there. Against all my better judgment I let him pull me to my feet, and tried to look every inch the fearless secret agent's girlfriend. \"I'll go with you, se\u00f1or!\"\n\nSo much for my attempt to seduce him away from his duty. He was unstoppable, se\u00f1ors, and he always had been. What an operative he'd have made for Dr. Zeus, eh? Our agents are always so adroit at stepping in and whisking away unwanted children. Where were they when Nicholas Harpole made his unwelcome entrance into this world, or Edward Alton Bell-Fairfax? Sublime bastards both of them, with a courage and determination and nobility of character I'd never possessed. What imbecile chance selected me for immortality, when _he_ could have made so much better use of an eternal life? He didn't fit the optimum physical parameters, I'll admit; and that was absolutely the only bar I could see.\n\nConsider his ability to inspire. I had been lost in the dark wood, on this wretched posting, and despair owned me. I'd seen at last the future we'd all been promised, and I knew it for the hard and ugly thing it was. I'd seen the madness that descends on older immortals, and it wasn't an enjoyable prospect to contemplate for myself. Nevertheless, others of my kind have in their differing ways found a certain happiness, a sense of purpose, even love. I had my work; but the work dried up, like the seasonal streams of this accursed place, and in its absence I had glimpsed the hideous dry void it covered. What if the Company gave me new work? The void was still there. Besides, I had now the growing suspicion that the work was meaningless, a pointless series of tasks devised to keep busy a thing that couldn't die, since its creators could find no way to unsay the spell that had set it in motion.\n\nBut this man walked back into my life and changed everything.\n\nSurely, I thought, his mere existence argued that there was a greater power than the Company, that there was more going on here than our pitiful creators imagined! You see? There might be a point to this eternal life business after all, a purpose and a meaning I couldn't see. Had he not come back to me, like a good angel in my darkest hour, and started my dead heart beating again?\n\nTheobromos, please. Thank you. You want to know what we did next, not listen to my opinions. I see, though, that some of you recognize the feeling. Yes, and some of you are as frightened of the future as I am. Your eyes give it away. So much the worse for you. What's the old saying, don't rejoice at my troubles, because when they're old news, yours will have just started? You have no refuge, any more than I; unbearable Time is master of us all, who thought we had defeated him. Will he treat you with more charity than he's treated me?\n\nWe rode, se\u00f1ors, by devious and careful paths, down from that ridge and quickly across the grade to Dark Canyon Road. The plan was to work our way around behind Mount Hollywood, then cut across the lands of the old haunted rancho on the other side, crossing the river at some point. Los Angeles being what it was, this could be done with dry shoes most years; especially so in this year of drought. Edward looked around him in wonder as our horses picked their way through the sand and river cobbles.\n\n\"But\u2014this is the principal river in southern California,\" he murmured.\n\n\"So it is,\" I said. \"Not at its best just now, unfortunately. In a normal year, however, it is at least two feet across. Sometimes even three.\"\n\nHe frowned and fell silent. I imagine he was wondering how even the most brilliant British engineers could irrigate this desolation to the point where cotton could be grown.\n\n\"Wheat might grow here, se\u00f1or,\" I said helpfully. \"Cotton, never, unless you bring water down from the north. Plenty of water up there. It's a green paradise; parts of it even look like your England. You would like it there.\"\n\nHe wasn't comforted. Perhaps he was beginning to doubt the feasibility of the grand design after all. In a pigeonhole in some fine antique oaken desk in Whitehall, there was a map of southern California; and some Briton had looked at this wavy line that described itself as a river and made plans accordingly, without understanding that Los Angeles never plays by the rules, whether of geography, law, or anything else.\n\nAnd of course I knew, I who rode faithful at his side, that the whole business would fall to pieces anyway. Even now Alfred Rubery was probably sitting in a San Francisco jail, having been unable to get any part of his mission right. The British would never own California. Edward and I would at least enjoy a holiday on Catalina Island first. But when the piracy case hit the world newspapers, surely Britain would throw in the towel and call its operatives home.\n\nAnd what would we do then? What would I do in England, on which I'd turned my back with the earnest hope that the island sink entirely into the sea? I hadn't seen the place since before the Industrial Revolution. It had been crude, cold, and violent, still largely medieval then, with all the attendant lustful bawdiness that implied. What would I make of the new Victorian propriety? What would I make of the mill towns and mine towns that had turned the green fields black? There were railways there now, and canals, and no one was burned at the stake anymore; all peace and prosperity. Except for the workhouses, of course, and the children freezing in alleyways and drinking gin to warm themselves, and the typhus and tuberculosis . . . But what other nation in the world hadn't the same problems, or worse? No gunfights in the public streets, at least.\n\nIt occurred to me that I'd like to go to Rochester, to the open place near the cathedral where Nicholas was burned. To walk there in my Victorian clothes, on the arm of my Victorian Nicholas, and laugh in the face of Death.\n\nI'd have to make some accommodation to the Company, explain my actions and propose a plan that would serve its interests as well as mine. Hadn't they been understanding of Porfirio's needs? After all the years I'd served them, surely they could afford to make some allowances for me. Yes, and it should even be possible to break the truth to Edward. What was the Company, after all, but the ultimate expression of the civilizing force to which he'd dedicated his life? And if he too should become a double agent? Oh, but of a much higher order than a flunky like Souza. It would have worked, se\u00f1ors. We could have made it work.\n\nSo I rode over that barren ground, with my head in the rainy clouds of England.\n\nWe went up into the foothills to avoid Sonoratown as we approached Los Angeles, though scanning from a distance, I could tell that it was virtually deserted. Best to be safe. We climbed, screened by laurel and oak scrub, until we were peering down on the city we'd left in such haste only the night before. It looked flat and desolate, the whole scene filtered through a yellow haze of dust raised by cattle and horses. Terrible dust, for of course there'd been no rain. And the smell of manure rose up, and of roofing tar, and mesquite smoke, and, faintly under all these, the smell of death.\n\nBut out on the horizon, what was that poking up blue into the high clean air?\n\n\"Catalina Island, se\u00f1or,\" I said, stretching out an arm. \"If only we could fly there.\"\n\nHis face was somber as he surveyed the distance, and his gaze dropped back down to the uninviting prospect below us. \"I must warn you, my dear, that our road to the sea will be watched. It would be pleasant to think that they've simply posted a man to observe the stage line to San Pedro; but I rather think they know I've got wind of them by now. Is there a fairly straight route to the sea that avoids the pueblo?\"\n\nThere would be, when the 710 freeway was built. I accessed information hurriedly and superimposed a twentieth-century grid over the present-day map. No reason why we couldn't follow the freeway route through the sagebrush and sand. It would take us right down to the future site of Long Beach, just south of Souza's landing. I plotted a course and nodded. \"This way, se\u00f1or,\" I said, urging my horse forward, and Edward followed.\n\nSo we went down across the wide plain, keeping the smoke of Los Angeles on our right, through a wilderness that would one day be East Los Angeles and various urban housing tracts called Maywood, Bell Gardens, South Gate, Downey, Compton. Such orderly Yankee names for a place that was now only a desert of trampled earth and bleaching cattle bones. Would there be an interval of little Yankee towns with gardens and cottages here too? And would they too vanish in their time, asphalted over, shadowed by the steel towers that would themselves vanish in the urban wars? And what pair of lovers would one day pick their way across a desolation not of sagebrush but of rust and broken paving, under a poisoned sky, past the bleaching bones of men? Full circle for this place, but not for me. With any luck I'd never see Los Angeles again. I'd be off to Great Britain. I'd have to find some way of persuading Edward to give up this nonsense about dying for Queen and country, though . . .\n\nWe had gone on our way about an hour when I edged my horse close enough to speak to Edward in a low voice.\n\n\"A question, se\u00f1or. You understand that I have every confidence in your ability, and absolute faith in your word. To look at the matter coldly, however\u2014what shall I do if we do not succeed? What will the Yankees do if they apprehend us?\"\n\nHe gave a brief, humorless laugh. \"My government is in no position to come to my assistance. The Yankees, for their part, cannot fight a war with Britain just at the present moment, being preoccupied with rather more pressing matters. They'd dearly love to obtain all the particulars of our business here, make no mistake about that. But I doubt very much they'll go to the trouble of declaring me persona non grata and paying my passage home. Much more likely, I'll quietly vanish into a shallow grave, and the contents of the valise will be forwarded to Washington. This is, you understand, the worst possible chance; but it is a possibility. All the more reason to avoid capture, my dear.\" He looked up and gave me a brisk smile, cold and bright as the winter sun. \"Are you reconsidering your offer? I should, if I were you. You see what it is to be a pawn on this particular chessboard.\"\n\nI might demoralize him, but I couldn't dissuade him.\n\nWith his gloved hand he reached over and took mine, and held it tight. \"Dolores,\" he said. He stopped and drew breath. \"I don't know what impression of me you've formed in the last twenty-four hours, but I think you see clearly enough that I'm unlikely to end my days quietly. Men in my profession do not. Women in my profession (and there are a few) are as much at risk, indeed at risk of worse. I will never marry you, I will never settle in a pleasant cottage by the sea with you, and we will never raise children together. You are clearly intelligent enough to have perceived this.\"\n\n\"I know, se\u00f1or. Your life is not your own.\"\n\nHe gripped my hand more tightly and went on. \"And it's not that it isn't a glorious life. Giving yourself to a noble cause. True, sometimes one must disregard certain moral considerations to achieve a desired end. One might go to a great deal of trouble to obtain a useful tool, sharp and bright and perfect for the job, a really remarkable tool to find in so unlikely a place. One might have no intention of discarding that tool, either, when its use had been served\u2014though the use might well destroy the tool. A man who seduces a girl into prostitution is vile enough; what would you call a man who persuades a girl to risk her life?\"\n\n\"Not a liar, at all events,\" I said with a shrug. \"Your conscience should give you a rest, se\u00f1or. I understand the danger here.\"\n\n\"But I did not, twenty-four hours ago. The ground has shifted since then. Don't share a shallow grave with me, my dear.\"\n\n\"I have no intention of dying,\" I replied. \"And you're not a dead man yet. Catalina, se\u00f1or, look at it out there, isn't it beautiful? Only a few more miles of this hideous desert, and then it's all blue water. We'll make it, and then we'll have all the time in the world to discuss what further use I can be to Imperial Export.\"\n\n\"By God,\" he said, looking away from me. \"If only we'd met some other place or time.\"\n\nWe had. It hadn't worked out; would it work out now?\n\nWe rode on in silence.\n\n###\n\nI WANTED TO ASK HIM about himself, to find out where he'd been born, who those highborn relations were who had hushed up his birth but evidently found him some back entrance into the corridors of power. The secret child grew up into a secret man, terrifically useful but never to be publicly acknowledged. And when his usefulness ended? When he finally failed to accomplish a task that had been set for him, or began to question the wisdom of the masters who paid him, as I think he must already have begun to do? How did any secret service reward an agent who couldn't do the job, whether through his fault or through the folly of his superiors? Silence and abandonment, disappearance without a trace. I knew that much. Within our own Company there were always rumors that certain operatives had been retired, though nobody knew what that involved exactly. . . . It's unsafe to inquire after fellow immortals one hasn't seen in a while. I guess I'm going to find out where bad cyborgs go, eh, se\u00f1ors?\n\nBut I couldn't ask Edward anything, about his personal life or our plans once we got to Catalina, for a number of reasons. The most immediate reason being that we were trying to cross a dangerous place quickly and in silence.\n\nI could access the historical record, though.\n\nIt was so easy, it never occurred to me. All I had to do was access the files I was given codes for when I came to California. I was given whole libraries full of stuff on its history, its future, much more detailed information than I ever bothered to use. When you spend most of your time in a coastal mountain range miles from the nearest mortal soul, you don't need to know who will run for third mayor of Pasadena. But was Edward Alton Bell-Fairfax mentioned in the history of California? Perhaps in some connection with Catalina Island? There would be no mention of me, of course. No Dr. Zeus operative is ever given information about his or her own future, so if I appeared in any footnote in the historical record, it would have been carefully excised from my files.\n\nBut I might find references to Edward. 1 set my primary consciousness to automatic scan and focused on the material behind my eyes.\n\nCalifornia, Channel Islands, Santa Catalina Island. One of eight channel islands. Geology: crystalline metamorphic rock, principally quartzite, also steatite, lead, silver\u2014only traces of gold. Botany: several rare endemics (how fascinating, why hadn't I ever been sent there?). Zoology: biggest predator a small fox; goats introduced by the Spanish doing very well. Ornithology, archaeology . . . nothing I wanted here.\n\nAny record of British involvement? I scanned the records of ownership.\n\nThe Indians first, obviously, for about 30,000 years. Thirty _thousand_? Wasn't that a little early? And what was this nonsense about sunken continents, reports of white Indians, and seven-foot-tall skeletons found in the oldest burial mounds? Then assorted Spanish galleons stopping by for souvenirs over the next couple of centuries, doing no harm for once. Russian fur hunters, Yankee sea captains, meddling Franciscans finally causing the place to be abandoned by its native population, who went over to the mission communities on the mainland, where they all died of smallpox.\n\nFirst owner to hold any kind of title, thanks to Juan Rodr\u00edguez Cabrillo: King Charles of Aragon and Castile, later the Emperor Charles, father of my old acquaintance Philip II of Spain, husband of that very same Mary Tudor who had my Nicholas burned at the stake. Small world. And then subsequent kings of Spain, until the Revolution in 1822, when Mexico claimed it. Modern times, now: when Pio Pico was dodging the invading Yankees, he granted title to the island to his friend Thomas M. Robbins. That was in 1846.\n\nRobbins sold the island to Jos\u00e9 Maria Covarrubias in 1850, and in 1853 it passed out of his hands, sold to one Albert Packard of San Francisco, who had it now and would keep it until at least 1864.\n\nBut here the records grew confusing, incomplete. Somebody named Eugene L. Sullivan was claiming part title as early as 1858. And who was this James H. Ray claiming to be Packard's agent, traveling back east _now_ , 1863, and bragging to potential investors that he \"about had a deal with John Bull\"? British parties were interested in buying the property, if clear title could be established. And here was Ray buying out Packard's interest in the island and immediately selling shares in the title to a consortium of men with fairly British-sounding names. Were they the Britons Ray had boasted about? Good Lord, by next year they would own Catalina Island.\n\nI scanned further, fascinated. No\u2014they wouldn't hold the place long. Here was the American government stepping in and seizing the island, ordering everybody off. They'd build a Union Army barracks over there. When would this happen?\n\nJanuary 1, 1864. Nine months from now. A General West would arrive with Union troops, forcibly remove all settlers\u2014including the Albion Mining Syndicate, who had developed an area on a defensible bay and named it Queen City\u2014and build a Union Army barracks at the narrow isthmus that connected the two halves of the island. Under government orders, West would name the two opposing bays Catalina Harbor and Union Harbor. The schooner _Jessup_ would be outfitted with a pivot gun and put into service as an armed supply vessel for the Union troops there. In short, Civil War or no Civil War, Abraham Lincoln would find the time and resources to come down on any funny business on Catalina Island like a ton of bricks. Why?\n\nI sped through document titles and froze on one: \"Winfield Scott Hancock's Threat Evaluation. Survey Performed on Orders of Edwin M. Stanton, Secretary of War, November 26, 1863.\" Eight months from now.\n\nSo something that happened within the next eight months would cause the secretary of war in far-off Washington to order a survey of Catalina Island with national defense in mind. I scanned the report. What had Hancock found?\n\nThat the population of the island consisted of a few squatter families who had been there since the 1850s, sheepherders and fishermen only\u2014and approximately one hundred miners (most of whom were British nationals), who arrived abruptly, proceeded to fortify a little crescent-shaped harbor, and engaged in no observable mining activity.\n\nThat the island's coast was rife with small, accessible harbors, many of which had adequate capacity for vessels of war. Hancock felt the island had dangerous potential for a military base: \"any major maritime power\" could, with minimal ordnance, control the entire coastline of California from a base on that island, preferably one located in the two-square-mile area around the isthmus. What major maritime power? The Confederacy, desperately dodging blockades or busy hunting Union ships in the Atlantic? It hadn't the wherewithal to send a fleet around Cape Horn. Britain, though, had ships all over the Pacific, and as recently as 1842 the U.S. was so afraid that Britain was going to make a play for California that they'd rushed ashore and prematurely raised the Stars and Stripes at Monterey, which, by an inconvenient little quirk of international law, still happened to belong to Mexico.\n\nWhat a sharp-eyed fellow he'd been, this Winfield Scott Hancock, and how quickly the secretary of war had acted on his request. Whatever tipped off the secretary of war to order the survey in the first place (doubtless something relating to Mr. Rubery's big mouth, or possibly even Mr. James H. Ray's), in less than a year the whole British plot would come unraveled, and Catalina Island would be firmly under the control of the U.S. government. What a shame, though at least it seemed to have happened without any bloodshed. No mention of any Edward Alton Bell-Fairfax.\n\nQuite an interesting little episode in local history. Imarte had been right: it was all utterly fascinating. So that was the end of British attempts on Catalina . . .\n\nWait a minute: what was this? The U.S. government would pull out its forces in September 1864. None of the miners would ever return to the island to press their claims (not surprising, since no substantial amount of gold had ever been located), and a man named James Lick would seek out the consortium members and purchase their titles to the island. There was a lot of correspondence from certain parties in the Department of Indian Affairs, who thought the island would be a great place to dump the unwanted Indian population of Humboldt County, but this would never come to anything, because of the fact (or perhaps in spite of the fact) that Catalina Island had almost no fresh water or arable land, and also because James Lick would be by this time the sole and legal owner, with a clear title.\n\nAnd here was Lick, in 1872, the Army long out of the picture, offering an option on the island to a Major Max von Stroble. The major would go to London to form an English syndicate to take up the option. Curiously, on the morning he was to sign the papers and collect the money, he would be found dead in his hotel room.\n\nWhat the hell? And next? Here was George Shatto, the developer, buying the island from Lick in 1887 and selling an option to an _English mining syndicate_. With the money he would receive, Shatto would busy himself laying out the little resort town of Avalon. But there would be problems: the English were supposedly mining silver this time, but then they would leave suddenly, and Shatto would default on payments and lose the island. Shortly afterward, he would reportedly fall from the back of a moving train.\n\nThe island would revert to the Lick trustees, who would sell it in 1892 to the Banning brothers. And who were they? The sons of our present staunch pro-Union stagecoach tycoon Phineas Banning, the little boys who'd grow to manhood in that nice mansion being built above San Pedro even now. William, Joseph, and Hancock Banning, the youngest having been named for his father's good friend Winfield Scott Hancock, who'd written the threat estimate fingering the British. Very small world.\n\nSo the island would remain firmly in American hands from then on, though the Bannings would experience some trouble: fighting on the waterfront, mysterious sabotage attempts, and arson in 1915 that would destroy the Hotel Metropole (where Harry Houdini stayed) and most of the resort buildings. In the end, the brothers would be bought out by the millionaire Wrigley, of chewing gum fame, who would rebuild Avalon along grander lines. Oh, look: here was a reference to a visit by the Chronos Photo-Play Company, for the purposes of shooting a movie. Chronos Photo-Play was an early alias for Dr. Zeus's entertainment division. I wondered if Einar would be involved in that.\n\nDecades of peace and prosperity for Avalon, then, a pretty little resort town dreaming in the island sun, all its bizarre history long forgotten. No more mysterious deaths, no more British strangers lurking around. The only bit of trivia to stand out of the record was that Wrigley would pay to have an extensive geologic survey done and then suppress the results . . .\n\n_What was going on here_?\n\nFor half a century, in absolute silence and in deadly earnest, two world powers would wrestle for control of this tiny island. At the beginning, its strategic importance was undeniable. But why would the British keep coming back, long after they'd lost any chance of adding it to their empire? What would they be looking for that required engineering and mining teams? What would Wrigley's geologic survey reveal? Not gold, which had never been found in enough quantity to merit attention of this kind. Not the various buried treasures reported to be there: every island had its tales of buried treasure. What were the British after? What had Imarte told me, something about letters referring to an astonishing technological discovery, made at a place designated only by code. Was the place Catalina? Had they unearthed something there they wanted to study?\n\nDid Edward know?\n\nI remembered his face above me by firelight, shining with the sweat of passion, while he said something about a remarkable discovery, one that might enable men to defeat death. What did Edward know?\n\nI scanned forward through the records. With the Second World War, the strangeness would begin again. The island would be closed to the public once more, and the OSS, forerunner of the CIA, would be quartered there. Oh my. There would be rumors of visits by Allied scientists, particularly the British, and of classified projects at science bases in the island's interior. From that time onward, access to the island's interior would be strictly controlled, even after the war, when the resort areas reopened. A conservancy would be formed in 1972, closing off most of the island to anyone but authorized residents. Access to certain areas was completely restricted, the reason given being that these were rare ecosystems where endemic plants thrived.\n\nI noticed something interesting in the successive editions of the island's history. The earlier works said the Union Army barracks were built to discourage Confederate sympathizers who might try to turn privateer, which doubtless related to whatever Mr. Rubery had been involved in. But later editions began to change the story: the barracks were built to patrol the Indian reservation that had been planned for the island. Still later, the story was that the barracks were built to guard against opium smugglers, and later still (by which point mortals were clearly losing their grasp of history) that they were built to prevent bootleggers from bringing in cases of whiskey.\n\nThe same thing happened with the successive maps. Here was one from 1912, nicely detailed, showing a lot of the interior features: the old mine adits and in particular Silver Canyon, where the English would work until their swift departure in 1887. Perhaps they got what they were after. But here was a map from 1938\u2014where were the mines? And here was one from 1976, with interior roads and hiking trails but few other features. Silver Canyon was not marked. Over this I superimposed a map published by the conservancy in 1982. There was _nothing_ on the map in that section of the island, not even topographical lines to show ridges or streams, though they appeared everywhere else. The Silver Canyon region was white, blank, featureless. Who was taking such pains to obliterate its memory?\n\nAh, but a cross-reference in the text indicated the conservancy's involvement with a twentieth-century holding company for Dr. Zeus.\n\nHere were the records for the twenty-first century, and the UFO sightings that would begin on Catalina Island\u2014over the Silver Canyon area, by the way. We all know, of course, that UFO sightings will be a gigantic hoax, costumed nonsense to conceal Company experiments in some cases and mass hysteria in all others. We know that, because the Company told us so. Although nobody ever bothered to explain to me about those little pale men I ran into back in 1860, or why all the material I collected on that job was confiscated. But doubt the Company's word? Inconceivable.\n\nIn any case, the Company was surely responsible for those sightings on the island, because here was name after corporate name becoming entangled in the conservancy's affairs, and nearly every one of them was some pseudonym of Dr. Zeus. And it was undoubtedly Dr. Zeus who would provide the islanders with the armament enabling them to close their harbor to mainland vessels after the Second Civil War broke out in mid century, Dr. Zeus who would patrol their waters for them, Dr. Zeus who would develop the advanced agrarian technology that would make them self-sufficient when foodstuffs could no longer be imported from the poisoned mainland.\n\nAnd it would be Dr. Zeus who established the secret libraries and archives there. When Los Angeles is the toxic, riot-ridden hell it will become, Catalina Island, like sixth-century Ireland, will be a peaceful and remote sanctuary beyond the sea, where knowledge is preserved and research conducted.\n\nLook at the names: Olympian Technologies, Kronos Diversified, Jupiter Cyberceuticals, Lightning-A Company, Jovian Integrated Systems. Every one of them the Company. Will the Company be behind the bizarre incidents that occur on Catalina Island in the twenty-second century? The rash of Kaspar Hausers who come wandering down onto the golf course in Avalon over a three-month period in 2136, babbling in no known language? The weeping man, found floating off Long Point in a fishing boat of antique design, who says his name is Emilio Machado and swears it is the year 1901? The persistent rumors that a Scots actor, famous for his adventure films of the twentieth century, is somehow still alive and well and can be glimpsed occasionally dining at the Avalon Country Club? What is Dr. Zeus up to?\n\nSe\u00f1ors, if I wasn't meant to know these things, I should never have been given the access codes. Or was it simply that nobody ever thought that one of us would reference this particular subject matter, connect this particular chain of events? I'm fairly sure I wasn't supposed to find out that the CEOs of Olympian Technologies, Jupiter Cyberceuticals, and the rest are, to a man and woman, loyal subjects of Henry X of England. But that must be a coincidence. We were always taught that Dr. Zeus was a multinational entity, drawing the best brains in science and finance from every nation on Earth. There are undoubtedly plenty of Czechs and Kenyans, too, on the board of directors. Anyway . . .\n\nI raised my eyes to the distant island; it loomed out there like a dream, as I had seen it every day and night of my sojourn here. I turned to Edward Alton Bell-Fairfax, who was now ten times the enigma he had been before. I opened my mouth to ask him a question.\n\nBut I never did ask him, because at that moment I picked up the signal of the mortal man approaching through the brush ahead of us. My head snapped around, and I focused on him. Mortal male, two meters tall, thirty to thirty-five years of age, sober, approximately 270 pounds, blood pressure slightly elevated, brainwave pattern suggesting he was hunting. Mounted, and urging his horse forward at a brisk trot. Armed. Rifle and two Navy revolvers.\n\n\"Edward,\" I said in a low voice, \"we'd better get off the trail.\" He looked at me sharply but turned his horse's head at once, and we found our way down a gully and into the partial shade of a scrub oak.\n\nIf only he hadn't been so tall.\n\nThe mortal must have caught a glimpse of that tall hat, because he sent a bullet whistling through the sagebrush at us. That was no more than conversation in Los Angeles, and it missed by a good ten feet. But here he came, galloping after it, emitting the signal I'd come to know too well: a mortal after blood.\n\nWe slid from our saddles, and I found myself flattened between Edward and an undercut clay bank, where the storms of 186z had hollowed out a space. The clay was just about the same color as those miraculous trousers of his, which were still spotless, by the way. Edward's gun was already in his hand. Damn, here still came the mortal, and even if he didn't spot me and Chameleon Man, he'd see the horses.\n\nHe did, too; he saw them first. A long searching stare along the gully, and he saw Edward as well. He grinned in delight, taking in the details of Edward's appearance, his tailored clothing.\n\n\"Now I just bet you're that limey bastard,\" he said. \"Let me hear you say something, friend. Talk for me.\" And he raised the barrel of his rifle.\n\nBang. Just like in the movies, a red dot appeared in the center of his forehead and a dark red drop ran down. Just like that. He sat there a moment in the saddle, his grin frozen, then fell slowly to one side. His horse didn't appreciate that at all; it stepped clear of him and kicked impatiently to clear his dead foot from the stirrup.\n\nEdward rolled away and looked at me in astonishment. \"Good shot, my dear,\" he said.\n\nThat was when I realized I had just killed a mortal. The gun was there in my hand, a bullet gone from the cylinder.\n\nWe don't do such things. Einar's mad, he doesn't count; Porfirio had immediately saved the life of the only person I ever saw him shoot. We don't kill. We reason, we run away, we lie to our attackers or confuse them or project illusions to hide ourselves, but we never, ever rob them of their miserable brief lives, because we have so much and they have so little. Unlike us, they have mothers who mourn for them. They have families who starve.\n\nI was crushed with such a sense of sin as I had never felt in my wretched long life. I was a true Angeleno now, wasn't I? At last I'd fired a gun at a total stranger, and blown him away too. But no audience cheered for me, as would have happened in the movies.\n\nEdward took the gun from my nerveless hand, stroked back my hair, looked straight into my eyes. \"Dolores, my dear. This was your first time, I think?\"\n\nNice of him, to help a lady so gently on the occasion of retching after her first kill.\n\n\"My apologies, se\u00f1or,\" I murmured at last.\n\nHe waved a dismissive hand. \"It quite shocks the system,\" he said, \"the first time. But I think you ought not to take this up as steady work, however good your aim. One can accustom oneself to the act of necessary murder, but does one wish to?\"\n\nYes indeed, something to be seriously considered by the young woman contemplating her entrance into Victorian society.\n\nWe mounted and rode on.\n\nThe sun was dropping into the west now, and we were nearing San Pedro and the probable cordon, so the danger was greater than ever. We arrived at Long Beach before it was quite dark, splashing across the slough. I wondered if D. W. Griffith's men would plant palm trees here one day, preparing the scene for the desperate chariot race to warn Babylon.\n\nIt seemed preferable to wait until full cover of night before making our way to Souza's. Accordingly we found a dry stream bed under an oak tree along the outskirts of Se\u00f1or Tempe's rancho and reined in there, to dismount for a while.\n\nEdward jumped down first and put up his hands to catch me as I slid from my saddle. I fell into his arms. The brief hold became an embrace, and without quite meaning to we were kissing hungrily. It was going to happen again; nothing we could do about it, other than unlock for a moment as Edward staggered over to loop the horses' reins around an oak branch. He came back, breathing hard. I knelt in the sand; he swept off his hat and knelt beside me.\n\nAnd really it was like prayer, se\u00f1ors, desperate prayer for forgiveness, an appeal for mercy, an act of life in that deadly place. I gave him pleasure to atone for the death I'd given the stranger. He gave me absolution for what I'd done, and found his own blessing of acquittal in my arms. Violent prayer, struggle and assault, shuddering ecstatic confirmation that we were still alive, though without our bower walls were dogs and enchanters, whoremongers and murderers.\n\nWe lay there afterward, looking up awhile at the red evening sky through the black leaves.\n\n\"What are you?\" Edward whispered.\n\n\"Your mate,\" I said. \"As meaningless as that is, for us both. We'll never marry. We'll never settle in a cottage by the sea. We'll never raise children. Death and time stalk us like a pair of hounds. But we were formed in the mind of God from the same piece of steel, for what purpose I cannot imagine.\"\n\nHe was silent for a while. His hand traveled up and closed on my breast. \"Death and time,\" he said at last. \"What would our life be like, if we could live?\"\n\n\"Oh, we'd make the world the place it should have been,\" I answered with a grand wave. \"We'd blaze across the sky like meteors, and our masters would look upon us and tremble. We'd bring down the palace of Death as though it were so many cards. You'd take the flaming sword and smash the lock on the gates to Eden, and let our children into the garden. I'd teach them how to grow corn, and you'd give them laws. Everything would begin again, except sorrow.\"\n\nHe laughed, softly. \"So it would,\" he said. \"And then, perhaps, the world could look after its own affairs for a while. Imagine not having to justify one's existence, ever, to anybody.\"\n\n\"Imagine having the freedom to travel where one wished.\"\n\n\"Imagine having the time,\" he sighed, and I sighed with him. Somewhere out in the evening a sea bird cried, a high thin far-off piping, a lonely sound.\n\nPerhaps it made the moment too surreal, brought home to him just how strange our conversation was. I felt his mood changing, his wariness returning.\n\n\"Dolores Alice Elizabeth Mendoza,\" he said musingly. \"You're far too young to understand this business as well as you do, and to kill with such precision. But for your age, I could imagine you were one of Juarez's agents, or even that buffoon Napoleon's, though I can't see how or why. You were certainly a virgin, and yet I've known Eastern whores with less expertise in the arts of love. Less enthusiasm, too. What am I to make of you, my dear?\"\n\nI lay very still. \"You might accept the truth as I told it to you,\" I said. Of all the mortals in that English hall, long ago, Nicholas had been the only one to suspect what I was. It had been a game between us, a delightful game of question and evasion, until he discovered the truth. Then he tried to kill me.\n\n\"Well, my love, but it doesn't quite convince,\" said Edward. \"I add together all the figures you've given me, and they simply don't produce the sum of you.\" He stretched luxuriously, in that motion bringing at least two of his concealed weapons into place for immediate deployment. \"A sensible man in my line of work would have disposed of you\u2014by some means or other\u2014hours ago. I am, however, reluctant to lose such a charming companion. And it is a fact that your objective and mine would seem to coincide.\" He smiled, narrow-eyed, waiting to see what I would say.\n\nI gave the faintest of shrugs, a tilt of the head, and spoke in the most reasonable of voices. \"Se\u00f1or. If my intent were to betray you, I might easily have led you to the Yankees by now. If my intent were to secure the contents of the valise\u2014I had all the time in the world to do that when it sat at the inn, before ever you came looking for it. If you find my knowledge or my skill with a pistol remarkable, all I can tell you is that there's not much for a well-born girl to do in San Luis Obispo, save read and practice shooting at targets or the occasional bandit. I believe it's customary for a gentleman to accept a lady's word without question, is it not?\"\n\n\"It is,\" he said. \"Though I expect you'll appreciate the difficulty I'm in just now, my love, as regards the luxury of trusting anyone.\"\n\n\"I do.\" I looked up into his eyes. The pupils were dilated, enormous. He really did not want to kill me. \"I point out that you have little choice in the matter, se\u00f1or.\"\n\nHis dark smile deepened, melting me, even with the point of his hidden knife inches from my heart. \"So the question remains: What are you? I find myself with a price on my head in a foreign land. My associates have bungled and miscalculated to such a degree that I may well be unlikely to escape with my life. I'm in a very narrow little corner indeed, and my only ally is a remarkable young lady who seems, by some unlikely trick of metempsychosis, to be a fused reincarnation of Boadicea and Cleopatra. A very bad business. And I can't for the life of me think why you're not having a fit of hysterics now, or angry tears.\"\n\n\"Metempsychosis,\" I said thoughtfully. \"Now, that was Pythagoras's theory of the transmigration of souls, was it not? Rebirth, after death, in a new body. Possibly I trust you because we were lovers once, in some previous life. Possibly you trust me for the same reason. It makes as much sense as anything else, se\u00f1or.\"\n\nHe drew a deep breath and struck the earth beside my head with the flat of his hand. \"Now, how in hell do you know what metempsychosis means? Whoever you are,\" he said, \"whoever you've been, if we get out of this with our lives, I _will_ marry you. See if you can keep the truth from me then!\"\n\nThe stars came out, and the chill of evening set in, but we didn't notice. What stamina he had. And what good fortune for us that no bounty hunters chanced to come near that particular oak tree, on that night of the sixteenth of March, 1863.\n\nMuch later we arose and rode out again, and I led us through the marshy tidelands and the shallow sea of reeds by scanning for solid ground along Rattlesnake Island. If we made noise, the frogs and the night creatures made more; and so we came in safety to the huts of the fishermen, and I was so grateful to ride up onto the causeway and behold Souza's night lamp burning.\n\n\"I will talk for us,\" I told Edward as I rapped softly on the door. He nodded, and when Souza opened and peered out, I said in Portuguese, \"The doctor has a request to make of you, Souza. This gentleman and I need to go across to the big island out there, in silence and secret. When can you take us?\"\n\nHe yawned and rubbed his eyes. \"With the tide, lady. Six hours more.\"\n\n\"Have you a secure place we can rest until then?\" I asked.\n\nOf course he had; it was a Company safe house, after all. He bowed us through his low door and showed us to the little closet of a room kept for Dr. Zeus agents who were passing through. Not spacious, but the low, wide bed was clean and dry. There was a chair, on which Edward deposited his saddlebag; there were a washstand and a table with a candlestick on it. Souza lit the candle for us and went to see to our horses. We undressed by candlelight\u2014it flickered, from a little draft coming through the plank walls off the sea\u2014and we sat up by its light long enough to make a late supper out of the last of Juan Bautista's picnic lunch. We barely spoke for exhaustion.\n\nEdward got up once, to make certain the door was secured to his standards, and I lay in bed and watched him. Not one mark, not one scar on his body, same as Nicholas. Really rather remarkable for a man in his profession. But, then, he seemed to be as perceptive of danger as an immortal. Anybody who ever contemplated sending a bullet or knife his way must have been fatally beaten to the draw.\n\nAnd what was he? I was a woman of mystery to him, but his existence posed a far greater question. Setting aside for a moment that we were somehow in bed together again after three centuries had passed\u2014how was he connected to the technocrats who would one day found my own Company?\n\nI opened my mouth to ask him, but somehow the only question that popped out was \"How did you break your nose?\"\n\nHe turned, peering at me curiously. Naked there by candlelight, no Victorian trappings, and he was Nicholas in every line.\n\n\"I've never broken my nose,\" he replied, coming back to get into bed beside me. He believed what he was saying, too. I lifted my hand to his face and ran my thumb along the irregularity in the bridge that had always so fascinated me.\n\n\"But it _has_ been broken, just here,\" I insisted. I could feel the scar in the cartilage, an old injury, healed long ago. \"You must have noticed.\"\n\n\"I've noticed the ugly fellow in my shaving mirror, yes, but he's looked that way as long as I remember,\" Edward said wryly. \"It's a family feature, I assume. I'm not sufficiently acquainted with my relations to know. But I assure you, my dear, I've never taken a blow to the face. One of the few advantages of being exceptionally tall; it's difficult for one's assailants to reach so far.\"\n\n\"Ah,\" I said. I'd never asked how Nicholas broke his nose. He'd had quite a reputation as a university brawler, and I assumed he did it in a fight. \"Perhaps you broke it in infancy, then, and don't remember?\"\n\n\"Perhaps,\" he said, yawning. He leaned up on his elbow and blew out the candle, and drew me into his arms. We snuggled into safety, there in that room where the wind sighed in the corners, bearing on it the smell and the sound of the sea. Sleep came at once.\n\nWe were almost there, se\u00f1ors.\n\n###\n\nMARCH 17, 1863.1 had no nightmares, I seemed to have had no sleep at all before there came the discreet knock and Souza's apologetic murmur advising us that the tide had turned. There was a lingering impression that I'd been having an earnest conversation with someone about Catalina Island and its absurd history, going over and over the cryptic records. The person, who must have been Edward, was patiently explaining that everything was all right because we were really on the same side after all, that the office that employed him would hand on its discovery to the first cabal that would become Dr. Zeus, and that the contents of document D . . . the _what_?\n\nI opened my eyes groggily, shivering, reaching for the shreds of memories, and of course they disintegrated into meaninglessness. I saw no blue light in the room, though. Edward's arm came out of the darkness, bracing around my shoulders. How hot his skin was.\n\n\"Wake up, my love,\" he said. \"We've a crossing to make.\"\n\nWe washed and dressed, and he didn't take the trouble to shave, this time. Every concealed weapon had miraculously found its way into place again, though, and his gloves and hat were firmly on as we stepped out into the morning darkness.\n\nWide black sky, wide black horizon, and the glint of water between the waving reeds. Freshening wind and the promise of morning, much more of a smack of the sea and less of tidal mud. Souza was crouched in his boat, clearing away nets for seating space. He rose up and offered me his hand into the boat.\n\n\"We need to leave now,\" he said. I nodded and made room for Edward, who stepped in easily and silently and placed his saddlebag among the nets. Souza cast off and bent to his oars. We moved out, bearing well to the east of Dead Man's Island. Slowly we worked our way out of the shallows, past that island of bad reputation, and at length we felt the pull of the tide taking us into deep water. The wind rose. Edward helped Souza run up our little sail, Point Fermin began to recede behind us, and the black threat of the mainland dwindled away.\n\nWe had done it, we had got clear. The sky began to pale with morning, and we could see the island now, fair across open water, twenty miles out. I was making for a destiny of which I'd never dreamed, with the missing half of my soul beside me, and it didn't matter what we came to in the end. The morning shone with more promise than any I'd known in my long life.\n\nEdward was leaning back on his saddlebag, watching the mainland shrink. He turned a speculative eye on Souza; then sat up and addressed me in his awkward Latin.\n\n\"The fisherman,\" he asked. \"Does he speak my tongue?\"\n\n\"No, he does not,\" I replied.\n\n\"Good.\" He continued in English, but kept his voice low all the same. \"I estimate we'll make landfall shortly before noon, if this breeze continues steady. Friends will be waiting for us there.\"\n\n\"Englishmen conducting their scientific studies, yes, you told me,\" I said, smiling at him. And they must have made one hell of a scientific discovery, to judge from the tug-of-war that the British and Americans would be playing soon. Whatever it was, Edward and I would have flown to some further safety by then.\n\n\"Ye-es,\" Edward said. \"And some others. Assuming he managed to find a competent pilot on his own, Mr. Rubery and a party of friends should also be arriving, at very nearly the same moment we do. You may be rather surprised at the company he keeps.\"\n\n\"Why, se\u00f1or?\" I asked, folding my hands in my lap and looking expectantly at him. Now would come his explanation of the piracy business.\n\n\"There will be certain American gentlemen with Alfred,\" Edward said, sitting up to face me, \"who are under the impression that Her Majesty's government will assist them in a privateering venture in aid of the Confederacy, and that my colleagues on the island have been preparing a base for them from which they may prey on the Pacific Mail steamers. This is, of course, not precisely the case; but we aren't anxious that they should learn the truth immediately.\"\n\nAsbury Harpending, that was the fool's name.\n\n\"It will be useful, in the event that they are caught,\" Edward continued, \"to have the venture assumed to be a purely Confederate conspiracy. We will endeavor to supply them with all they need to make a fearful reputation for themselves, and with any luck their depredations will help push the War of Secession to a speedy conclusion. With funds from the San Francisco Mint cut off, Lincoln will surely sue for peace.\"\n\nAnd California would be up for grabs, isolated on a distant coast.\n\n\"At this point,\" Edward said, smiling a cold smile, \"there will be certain changes of plan suggested for the privateers. It is to be hoped, by that time, that Southern gentlemen will form only a minority of the crew\u2014having been replaced gradually by gentlemen adventurers of Californian birth, whom I shall have recruited with your able assistance, my dear. We can also expect fresh numbers of my countrymen, once the American hostilities have ended and they can move with greater openness.\"\n\nHe leaned forward and spoke more quietly, and so smoothly. \"But there will be an interval during which great tact and persuasion are called for, to convince the enthusiasts of the erstwhile Confederacy that a change of loyalties is in their best interests. It will fall to me to attempt this conversion, on a case-by-case basis. Those Southern gentlemen who cannot be induced to exchange the Stars and Bars for the Union Jack will meet with unfortunate accidents, and I regret to say that the arrangements for those accidents will also fall to me.\" He looked into my eyes, reading my reaction.\n\nYes, I know, it was murder he was talking about, but of tobacco-chewing bastards who trafficked in black slaves and had the temerity to dress up their shame in plumes and epaulets. I'd seen those belligerent Southern boys in the bar of the Bella Union. Someone might urge mercy for their kind, but it wouldn't be me. I nodded for him to continue.\n\n\"Shortly thereafter, my dear, you and I will have a number of journeys to make,\" he said. \"If we can persuade certain persons of certain things\u2014for example, that a league of amity between Great Britain and Mexico would benefit both parties\u2014then various and assorted efforts by several persons in several countries should bear fruit. That being the case, happy days will ensue. And I will at last be more than Alfred Rubery's long shadow, and you will be whatever you choose to be, in whatever country you choose to reside.\"\n\n\"I may choose to travel, se\u00f1or,\" I said, giving him my most meaningful look. He smiled and settled his tall hat more securely on his head, for the wind was blowing strongly now. Souza politely ignored us, leaning on the tiller.\n\nI had no doubt Edward could talk Confederate privateers into supporting the cause of Britain, or persuade Benito Juarez that Her Majesty desired to assist him. The mystery, to me, was why a man with his abilities hadn't gone further. But being illegitimate put the wrong sort of stripe on his old school tie, and that carried so much negative weight with the English. It meant that superb men like Edward lived and died in obscurity, while their nation was run by second-rate boobs who'd lose that empire he was working so hard for. Eventually. Years from now.\n\nOr would they?\n\nThis particular plan was already defeated\u2014there would be no British-backed privateers stalking the Pacific Mail\u2014but what about the other part of the plot, involving some discovery the British had made on the island? How would Dr. Zeus become involved? There was every indication that England\u2014in that far-off future when it was no longer even the United Kingdom\u2014would somehow slip into the director's chair at Dr. Zeus. And Dr. Zeus _did_ rule the world. Secretly, of course. Would they be able to do this because of what they'd found on Catalina Island in 1862? And what could they possibly have found?\n\n\"I've never been to this island,\" I said. \"Though of course I have heard the stories.\" This was a prompt, but it didn't work.\n\n\"What stories would those be, my dear?\" Edward asked, extending his hand and clasping mine.\n\nDamn. I sped through reference files. Any old farrago of nonsense would do to get the conversational ball rolling.\n\n\"The Indians used to believe that there was once a great continent here in the West, which drowned in much the same way as we believe Plato's Atlantis did,\" I said. \"The Indians claimed there were white men who lived there, extraordinarily tall. They called the place Lemuria.\"\n\nEdward looked puzzled. \"Unusual name. Were there lemurs there as well?\"\n\n\"I've no idea. In any case, these islands in the channel are thought to be the highest mountaintops of the submerged continent, the only part to survive the deluge. The white men who lived there were unable to prevent the sinking of their world, but they were mechanically brilliant\u2014so the stories say\u2014and produced engines of genius that far surpassed the modern railway or ironclad warship.\"\n\nHa, Edward reacted to that, if only in the pulse of his blood. His face showed nothing, however. \"What an extraordinary story,\" he said. \"I suppose it's all nonsense, though.\"\n\nThey _had_ found something, and he knew about it. Was it some kind of technology? But whose? There had never been any real Lemurians.\n\nI shrugged. \"The Indians used to tell fantastic tales. The priests discouraged it, of course, as a lot of superstitious nonsense.\"\n\n\"As well they might,\" Edward said. \"Though there is a growing opinion that the mythologies of primitive peoples ought to be collected and studied. Conquering races tend to destroy such things, to their own loss. Science now indicates that what were once thought to be fantastical myths may well have some basis in historical fact.\"\n\n\"For example?\" I asked, sitting forward in anticipation.\n\nHe removed his tall hat and pushed his hair back from his forehead before setting the hat on his head again. \"In Dover, I was recently shown the complete skeleton of an antediluvian monster, fossilized in solid rock. Educated persons had dismissed accounts of dragons as no more than fairy stories. And yet here was the leviathan himself, and any reasonably observant peasant must conclude it was a real creature that had lived once. And so it had: not galloping after knights and virgins, but sporting in vanished seas.\"\n\n\"An ichthyosaur,\" I said in disappointment. He wasn't going to tell me what they'd found on the island.\n\n\"That was its name, to be sure,\" he said, squinting at me in the sunlight. \"Don't tell me you were trained in palaeozoology as well!\"\n\n\"No, of course not,\" I said. \"I saw an article in a San Francisco newspaper.\"\n\nHe nodded slowly, a speculative look on his face. \"I do look forward,\" he said, \"to the leisure for more of these conversations with you, my dear.\"\n\nSo much for artlessly digging information out of him. How much did he know about what had been found? I was never to find out. But I daresay you know, se\u00f1ors.\n\nThe sun was well up now, the little boat sped on and on, and with each hour the island became more than a blue outline. We could see the steep canyons and vast mountains in the interior. We were going directly across, in the shortest line, to the west end of the island: not near the future site of Avalon, but to the double harbor where the Union Army would build its barracks soon. From the sea, you might think Catalina was two islands here, a little range just west of the main mass; but they were connected by a half mile of level isthmus just above sea level, making a neck you could cross in five minutes' lazy walk.\n\nIt looked brown and dry, terribly overgrazed by sheep\u2014not like a place you'd find fascinating endemic species of plants. But, then, there was more to this island than met the eye, wasn't there?\n\nEdward took out his field glasses and scanned, and I scanned right with him. A few miles west of the isthmus was where the construction was going on. It was no simple field camp. I could see where they were preparing gun emplacements, and really the little bay they'd chosen was superbly suited for a defensive position. Neat field shelters and some kind of equipment, too, though it didn't look like anything connected with mining. A couple of plumes of smoke: small tidy breakfast fires, I'd bet, preparing kippers and whatever else Englishmen were eating for breakfast these days. I shuddered at the memory of the sardine tacos. Could I see the mysterious Silver Canyon from here?\n\nNo, it was back on the windward side of the island. Well, perhaps I'd have a chance to explore the area on foot.\n\nI became aware that Edward had turned beside me and was peering at the far western end of the island, up the coast from the fortifications. I turned too.\n\nThere was a ship out there, rolling at anchor, her sails furled.\n\n\"Ah,\" said Edward with satisfaction. \"Now, what should this be but the good ship _Chapman_ , bearing her crew of traitors and pirates? And the slightly competent Alfred Rubery. Have the boatsman change course. Let's go cheer on the gallant Confederate cause, shall we?\"\n\nI gave the order to Souza, but my heart was in my mouth. That couldn't be the _Chapman_ , because history had recorded that Rubery and the other conspirators were caught before they could leave San Francisco Bay. They should be in jail cells by now. So there was no way the _Chapman_ could be arriving here at its appointed rendezvous, right on schedule. But if that ship wasn't the _Chapman_ , what was it?\n\n\"Why is she just sitting out there, se\u00f1or?\" I asked. \"Oughtn't she have moored in the bay before the camp?\"\n\nHe shrugged. \"Alfred may be following orders at last. He was to wait until I came aboard before taking her into the bay. I'd have been here by now, if all had fallen out as planned. So there he waits, like an obedient boy, for me to bring the valise and further orders. I daresay he'll be glad to see the money. It's rather difficult to recruit a crew on promises.\"\n\nI scanned the ship. There was a crew on board, but at this range I couldn't tell much more. Nothing to do but sit and wait as we sped across the blue water; nothing but access the historical record.\n\nI hurried through data files. What was the source Imarte found? _The Great Diamond Hoax_ , here it was. And other stirring incidents, supposedly, in the life of Asbury Harpending. Just who was Harpending?\n\nLiar, traitor, and swindler, according to historians; scion of a fine old Kentucky family of wealthy landowners, according to himself, as well as a philanthropic speculator, developer, and crusader for truth. In this year of 1863 he was only about twenty-one, though, with a long career of shady dealing before him.\n\nI sped through the chapters. Here were the abortive attempts to stage a Confederate uprising, failed because of hysteria, lack of nerve, and the discovery of the Comstock lode. Just as Imarte had said. Here were the gallant Confederate sympathizers attempting to regroup with an eye to privateering, under the leadership of Harpending. Here Mr. Rubery entered the picture\u2014callow British youth (I'll say) with a sympathy for Southern aristocracy and a love of adventure. Even Harp-ending made him out to be something of an idiot; though I wondered how much of the privateering scheme had come into shape _after_ they met and not before, and whether Harpending was really their leader.\n\nThey spent money feverishly, buying the _Chapman_ , buying cutlasses, cannons, firearms, and ammunition, and probably a Jolly Roger and cocked hats too. There was no mention of Rubery's making a trip to Veracruz to obtain more funds, but he must have done so, with a stopover in Los Angeles on his way back. Thanks to Cyrus Jackson's jealous passion, Rubery fled back to San Francisco empty-handed; and not only had he left the money behind, he'd come away without the list of contacts who would have helped the next part of the plan along.\n\nFor here it was in print: the conspirators were unable to find a navigator anywhere in San Francisco. They'd made inquiries, they'd had friends and acquaintances make inquiries for them; and word had evidently gotten around to the authorities that a band of young men with known Confederate sympathies was looking for an able-bodied navigator likewise eager to overthrow the Union government.\n\nSo one came forward, courtesy of the San Francisco Police Department, a man named William Law (surely a broad hint if ever there was one). The conspirators took Law into their confidence. He signed on readily and just as readily took all details of the plot to Captain of Police I. W. Lees.\n\nLees, being a wise man, opted to wait until all the birds were in the net. If Edward could read this, his hair would go gray. Law went along with all the preparations and agreed to be on board the _Chapman_ well before her scheduled sailing time of eleven o'clock in the evening of March 14, 1863. That had been the meaning of the coded telegram Edward picked up the following day.\n\nAt ten o'clock on the 14th, I had been sitting in front of the cook-fire, listening to Juan Bautista play his guitar. Edward had just arrived from Veracruz and was settling into his room at the Bella Union. And Alfred Rubery and Asbury Harpending were just going on board the _Chapman_ and discovering that Law was nowhere to be found.\n\nWas this enough to warn them off? No, they left a sentry to watch for Law and went to bed in their bunks on the _Chapman_. The trusty sentinel dozed off too, it seemed, because the next thing they knew, it was broad daylight on March 15 and the U.S. warship _Cyane_ had her guns trained on them. Boatloads of marines were bearing down on them from all sides, to say nothing of a tugboat full of San Francisco police.\n\nAnd then? Off to Alcatraz with them for interrogation, at just about the time Edward was watching me undress.\n\nNews of the foiled plot went out over the telegraph that same day, to the whole world, as Edward and I lay in bed together. If we'd been in any other city, we'd have heard the newsboys crying the story under our windows; but we were in a coaching inn on the edge of nowhere, and we never knew a thing.\n\nWhat did Rubery tell the police, under interrogation? Something to occasion those two Pinkerton men to go hurrying into the Bella Union as Edward and I approached it that evening? Were they tipped off by an alert telegraph operator, who compared the names on Edward's yet unclaimed message with the names in the breaking story? I cringed inside. No wonder there were bounty hunters after us the following day. Everyone must know now, Queen Victoria on her distant throne must know by now. Everyone knew but Edward.\n\nI fast-forwarded through the details, desperate to see what would happen. The conspirators would be convicted of treason. Possibly because it would seem so obviously a stupidly boyish game, they would be sentenced to fines and imprisonment instead of death. Alfred Rubery's Parliamentary uncle would step in and wheedle a free pardon for him from Abraham Lincoln. Rubery would be thrown out of the country all the same, though, shipped out on one of the Pacific Mail steamers and transferred to a British vessel at Panama.\n\nAnd that would be the end of the matter.\n\nNo villainous British plot to invade the state would ever be publicly uncovered, no scandal of foreign nationals preparing fortifications on Catalina Island. No mention at all of an Edward Alton Bell-Fairfax.\n\nAnd yet the secretary of war would know enough to send the Army to Catalina. How would he make the connection between a bunch of silly young men wanting to play pirates for the Confederacy, all the way up there in San Francisco, and the activities of certain Britons on an obscure island off the coast of Los Angeles? All Harp-ending ever said in his memoirs was that they had a general plan to base themselves in some islands _off the coast of Mexico_. But then, he hadn't known about the whole plot, had he? And had he protested when he discovered the use to which his pirate ship was to be put, my smiling Edward would have been there to slip an inconspicuous knife between his ribs.\n\nDid the authorities tell Alfred Rubery that he faced the prospect of being shot as a spy? He must have sung like a damned canary.\n\nWe were near the ship now, and there wasn't a soul on deck. She looked as deserted as the _Mary Celeste_. I realized, as we drew near, that whoever had anchored her in that particular little cove chose the spot with coy discretion: we were able to spot her from the open sea, but the men at the British camp on the other side of Arrow Point could have no idea she was there.\n\nThere were men on board, alive, alert, and waiting for us.\n\n\"Edward,\" I said, \"this isn't right.\"\n\nHe didn't lower his field glasses, for he was studying the ship intently. \"Not right? Moral qualms, my dear?\"\n\n_How could I tell him what 1 knew_? \"Not that\u2014there's something the matter here. Why isn't your friend on deck, watching for you?\"\n\n\"That's a good question,\" he said, slowly adjusting the long focus. \"How can you tell there's no one on deck at this distance, my dear?\"\n\nIt was like a faceful of ice water, se\u00f1ors. He hadn't mentioned what he was seeing, and he knew I had no field glasses of my own. There was a sadness in his face when he lowered the glasses to look at me, but a certain shivery distance, too.\n\nNothing to do but brazen it out.\n\n\"I have trained eyesight,\" I explained, as though impatiently. \"I was raised to count cattle on hills five miles away, se\u00f1or. Can't your English shepherds do the same? Look.\" I pointed to the ship. \"See the stain on her jib sail? See the red rag tied to her wheel? The three belaying pins on that rail there at the left, and the five at the right? Look there, that's a brown pelican lighting on her aft deck now. Stupid creature expects someone to toss it a fish. Do you see it, se\u00f1or?\"\n\n\"Yes,\" he said, looking through the glasses again.\n\n\"And do you see a living soul on deck?\"\n\n\"No, my dear, I don't.\"\n\n\"And does that seem reasonable to you?\"\n\n\"No, it doesn't.\" He lowered the glasses and looked at me again, not quite sure what to think. \"You never cease to amaze me, Dolores. It should make our continued acquaintance interesting indeed. However\u2014Alfred hasn't much experience with maritime customs such as leaving a watch on duty. Neither, I suspect, have his fellow privateers. They may all be belowdecks playing whist.\" He looked through the glasses again. \"Or there may be something wrong. I doubt it, though. Sails reefed, anchor down, everything in order, both boats there.\"\n\n\"How do you even know that's Mr. Rubery's ship?\" I was in agony. Nearer we flew and nearer, across the lovely clear water, closer to the ones who were waiting for us.\n\n\"That is another good question,\" he said coolly. He didn't know what to make of my terror, but he wasn't about to panic himself. \"Let's just see, shall we? You!\" he addressed Souza in Spanish. \"Take us in, under her stern.\"\n\nSouza looked inquiringly at me\u2014though he understood Spanish perfectly well\u2014and I nodded weakly. We made straight for her now, then angled around so that we were coming in with a good view of the aft cabin. There was her name, in big blocky letters: J. M. CHAPMAN. Her boarding ladder was down, in open invitation.\n\nBehind my eyes, a line of text jumped out at me, Harpending peevishly complaining that the name of the vessel was the plain _Chapman_ , though for some reason journalists had seen fit to refer to it in all the papers as the _J. M. Chapman_.\n\n\"It's she,\" said Edward briskly. He said to Souza, \"Bring her alongside.\"\n\nI stared up at the painted name. Freshly painted name. Wait, no. The J. and the M. and the CH were duller; only the APMAN gleamed with new enamel. I subjected it to infrared scanning and saw, underneath, the letters ISHOLM. This ship had been the _J. M. Chisholm_ not long ago. Who had altered her name, and why?\n\n\"Se\u00f1or,\" I said, keeping my voice as reasonable as I could. \"Call it woman's intuition. I fear treachery here. What if Mr. Rubery has met with some misfortune?\"\n\nEdward gave me a long speculative look. \"Then he'll need my assistance,\" he said. \"In any case, it's my duty to find him. Are you frightened, Dolores? You needn't go on board if you are. This shouldn't take long.\"\n\nHe took up his saddlebag and caught hold of the ladder, climbing nimbly despite his awkward grip. I sat there and watched him a dull moment, before rising and scrambling after with a sob of desperation. If I couldn't prevent him from walking into the trap, perhaps I could get him out of it. Perhaps nothing very bad would happen after all. Mr. Rubery, twit that he was, would come out of this with no more than a slap on the wrist and deportation, so perhaps Edward and I would escape once more, sailing free into the sunset after amazing adventures.\n\nWe boarded the _J. M. Chapman_ and stood on her empty deck. All sunlight and silence, but for the creaking of the rigging and the sough of breakers on the shore nearby.\n\n\"Edward,\" I whispered. \"There are mortal men below decks.\"\n\nHe raised an eyebrow. _\"Mortal_ men?\" he said, smiling a little. He set off down the deck and then halted, turning to me with the strangest look in his eyes. \"What did you\u2014?\"\n\nYes, se\u00f1ors, I'd given myself away. Not that it mattered, however.\n\nWith only the slightest sound, the door of the aft cabin opened, and a man emerged. He was a clean-shaven nondescript in a neat gray suit, smiling as he leveled a pistol at Edward. \"Commander Bell-Fairfax?\" He had a pleasant flat American voice. \"You're under arrest, sir. Could you set down that bag, please?\"\n\nEdward went white as a sheet, but he smiled that cold and narrow-eyed smile.\n\n\"Ah,\" he said. \"This bag?\" He set it down very carefully. He opened it. The Pinkerton agent made a sharp little noise, stepped toward him, thrusting out his pistol\u2014but Edward held up both hands in a pacifying motion and slowly drew out the valise.\n\n\"This is what you want, I expect,\" he said, and held it up. The agent smiled and stepped forward, holding out his hand.\n\nEdward shot him.\n\nI had the American's pistol out of his hand before he fell. \"Jump!\" I screamed to Edward, for I could hear the men below hatches boiling from beneath us, coming up on deck as the wounded agent rolled and cried for help. I turned to see Edward composedly opening the valise and scattering its contents over the side, papers fluttering and falling, and the lovely violet ink blurring and melting away to nothing as soon as the pages settled on the water. The Americans would never learn the whole truth now. They'd get the island, but they wouldn't know what its secret was or where, and the British would find a way to creep back and take it from under their noses.\n\n\"Jump, for God's sake!\" I ran to him, but the bullets got there first.\n\nIt didn't happen in picturesque slow motion. He neither jerked nor spun about. He coughed once, with the first hit; then he squared his big back and took the shots, doing his best to ignore them as he sent the last of the papers into the sea and dropped the valise after them. He pushed himself around to face me and the crew of Pinker-ton agents who had surrounded us.\n\nHe met my gaze with an ironical smile. Oh, he was falling; his legs weren't working anymore. He toppled forward, and I caught him, and we sank to the deck in a welter of bright blood. His hat came off and rolled away.\n\nI was sobbing, catching my breath in ugly little cries as I rocked him in my arms, trying to get enough air for a scream. Blood was coming up out of his mouth. He gave one grimace of pain and he stiffened; but then something seemed to get easier, and he opened his eyes and peered up at me with an inquiring expression, as though I had just made a remark he didn't quite hear and was hoping I'd repeat it.\n\nThe world had shrunk to that little space on the deck. Our enemies might have been a circle of statues, for all we noticed them. His world was shrinking even faster; he was fighting to keep his eyes focused on mine.\n\n\"Wh\u2014what _are_ you?\" he asked, quietly, as though we were lying in a green garden somewhere and no blood was running down his chin.\n\nThe only gift I could give him now was the truth. My throat was like stone, but I got the words out, whispering to him.\n\n\"I'm not a mortal woman, _mi amor_. But it was science and medicine that made me so, not magic. I exist because your masters discovered something that enabled them to make immortal creatures like me, to do their work for them here in the past. They _will_ create a race that can never age nor die, just as you said. This is what you have bought with your life.\"\n\nHe was knitting his brows, his breath was rasping and wet as he strained to hear me. Suddenly there was comprehension in his eyes, and they brightened. He gave a gurgling chuckle. He knew.\n\nI fought to suck in a breath. His death was coming for him. I leaned closer and said, \"Listen to me, my love. I'm trapped by centuries, I can't follow you. Come back for me and break my chains. Set me free. Will you set me free?\"\n\nHe nodded. He looked into my eyes, trying to say something. He had neither the strength nor the time. A wind gusted across the deck toward us. His soul slipped away with it.\n\nI bent over him, hiding his dead face with my hair.\n\n\"I hear no sound but your silenced voice,\" I murmured. \"I feel no heat but the fire that burns you, I draw no breath but you come into me, before me, behind me, you are the sea and the rock!\"\n\n\"Johnston's dead,\" someone announced. \"Who's this woman?\"\n\n\"The whore from the Bella Union, ain't she?\" someone else said.\n\n\"God damn,\" said a voice directly above me. \"I guess you could say this didn't turn out so well. What'll we do with this one?\" A booted foot tapped Edward's leg.\n\n\"Weight the body,\" someone said. \"We'll put out to sea and drop him in the channel. No evidence, gentlemen.\"\n\n\"Come on, girlie, get up.\" A hand closed on my wrist. \"You're under arrest.\"\n\nI don't really have a clear memory of what happened next, se\u00f1ors, only a few impressions. I remember seeing Souza in the distance, rowing away for his life. I remember the Americans screaming and trying to hide. If you want the details, you can check my video transmission. I know what I must have done; because when that longboat full of British voices came around the point to see what was going on, I remember throwing human heads at them. And an arm. I didn't kill the Englishmen, though, I'm fairly certain. I think they retreated in some haste. And then of course your security operatives came and found me, doubtless because poor Souza's control implant was broadcasting his horror.\n\nAnd that's it, se\u00f1ors, that's the end of the story. I had a second chance to save my love, and I failed. I took unauthorized leave and I committed murder. I did find out a great many interesting things about the genesis of our Company, but I have a feeling that the information was classified, not for my eyes at all.\n\nI was a bad machine.\n\nBut I've told you the truth, se\u00f1ors, every word. Now, you wouldn't have one more piece of Theobromos to spare, would you, for a poor old thing like me?\n\n###\n\nHOW VERY EMBARRASSING to read one's own testimony. If the hearing panel hadn't given me so much Theobromos, I'd have told them the story in half the time.\n\nAs it was, I lay in the worst case of theobromine poisoning I'd ever had, for three days and three nights, before they came for me and gave me to understand there would be no trial, no further action. I was only to be sent off to my new posting immediately. I was still groggy when they helped me into the silver box, and I heard the hiss, breathed in the yellow gas, and was abruptly _here_.\n\nAnd where exactly is here? Ah, that's the cream of the jest. I crawled out of the box and wandered around in bewilderment for several minutes before I found the crates and dispatch case that had been sent with me. I fumbled open the seal and read my orders.\n\nI am now the managing operative of a Dr. Zeus agricultural station, located\u2014where else?\u2014on Catalina Island, and the present year is 153,020 B.C.E. I am in fact the sole personnel assigned to this station, but after all I'm an immortal with unlimited strength, so I find myself adequate for the job. Plenty of heavy equipment I can drive, plenty of supplies sent as I request them. No company at all.\n\nIs it my job to collect the rare endemic species that grow here and nurture them along? Not exactly. My assignment is to grow produce for the Company's Day Six resort on Santa Cruz Island, some miles north of here. I supply the salad bars of wealthy twenty-fourth-century vacationers, who pay out a considerable chunk of their income to be herded into cramped shuttles and thrown back through time to an unspoiled and unpolluted paradise where they can sport-fish, have their holos taken with dwarf woolly mammoths, or just relax by the pool. I read that in a brochure somebody dropped on the loading dock where I deliver the crates of radicchio, squash blossoms, and endive that I raise. I never stay to watch the ship arrive. I don't like talking to people.\n\nMy heart aches when I think of the Ventana and Big Sur. I can never go back there again. I'm forbidden to leave this garden I've made. And some violence has been done to my electronic memory, I fear. Something's been deleted. I can't access all those fascinating historical records of this island now, nor the successive altered maps. They don't seem to have realized that the information still exists in my human memory. But since I can't download the information to another operative, maybe they don't care, especially as I'm unlikely to see anyone I know back here. Even Joseph won't be born for a hundred thousand years. Longer for Lewis and the others. No way at all to talk about my discoveries, which is undoubtedly why I've been put so far in the past.\n\nAll things considered, though, the Company has treated me very well. One door closes, another one opens. This is a beautiful island right now, not at all the dry and rocky place it will become. There are immense forests here, ironwoods the equal of any sequoias you might find on the mainland, deep groves of mahogany trees, great spreading oaks and pines. Very little flat land, but what there is I have covered with neat green rows of vegetables, tidy orchards, and a small area for my own botanical experiments. No animals bigger than the little fox. Plenty of birds. No mortals at all . . . yet.\n\nI'm to watch for them, though. This is very, very secret. Really I shouldn't be writing it down here, but who the hell will ever see? I'm to watch for certain people, odd-looking people with pale skins, who will arrive one day. They will not be Ancient Lemurians. It is expected they will arrive in well-designed oared galleys or perhaps even in gliders of some kind. It is expected they will settle in the Silver Canyon region. It is expected they will dig themselves in under a mountain and hide there, safe from marauding Neanderthal or Cro-Magnon primitives. I'm to report their arrival to the Company at once, and then I'm to monitor them unobtrusively for the next few centuries. When they've reached a certain level of technological development, when a certain energy signature begins to appear, I'm to alert Dr. Zeus. I suspect the Company will come in and slaughter them, then, and make off with their peculiar inventions.\n\nTalk about piracy. But the Company won't take everything. Something will be deliberately left for the British to find in 1862 or whenever, some bit of machinery that will be hidden away in a military laboratory for decades, until some Briton finally discovers how to make it work, and gains thereby insight into certain dimensional principles that enable him to solve the problem of time travel. Dr. Zeus will have guaranteed its own birth, built the foundation of its empire. And Edward will have been the sacrificed hero whose blood mortared its cornerstone.\n\nPretty sneaky, eh?\n\nIt was a few centuries before I could bring myself to climb over to the west end of this island and look down into the anchorage where Edward died. I half-expected to see the ship lying there, and his poor mortal body sprawled on the deck. But none of that's happened yet. Thousands of years yet before the acorns fall that will grow into the trees that make up the timbers of the _J_. M. _Chapman_ of fatal memory. Edward's distant ancestors are at this time knapping flints for hand axes. I would bet, though, that in some tribe on some cold green island there is one very tall savage who has learned how to put a wooden handle on his ax, or perhaps to draw reindeer on his cave wall. It will so unnerve his fellow cavemen that they'll grab him at the next spring equinox and sacrifice him to their gods. Perhaps some wretched little proto-Celtic girl will weep for him and wring her hands, too.\n\nWho knows? We may have been in this dance since time began, and could never see the pattern until Dr. Zeus made me an immortal, unable to die or to forget. When Nicholas died, I was still too young to see what was going on. This last time with Edward gave them away, however. I don't believe in reincarnation, but surely even the Hindus don't imagine it works like this. What's the point? The same man circling through lives like a blazing comet, always returning on the same course, meeting the same inevitable doom before he vanishes into darkness, serving somebody's purpose with his death but never his own. The game is fixed. Whoever is running it\u2014and I can guess at the identity, though not the reason\u2014it's fixed.\n\nBut now at least I know.\n\nI don't expect I'll see him again for a while. But I will see him again. He will come back and set me free, because he said he would. And it may not be on our next encounter, or the next, but one day the pattern of his sacrifice and death will be broken. We will make something better than this, he and I, and God help our masters if they try to prevent us.\n\nIn the meanwhile I tend my rows of corn and tomatoes, or wander on this narrow cobblestone crescent and admire the blue sea, or sit outside my shelter at night and watch the sky. I'm patient; I can wait.\n\nHe will come again.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"I hope you are having a beautiful Sunday as a family!\nThe store that I speak is Craftin desert Divas but they know it I invite you to visit them by clicking here.\nWithout boring them so much I show you the card, it's super clean and simple as I like it!!!\nUse painter's tape to mask the panel and achieve a sort of sunbeams, leave a space in the upper corner to simulate the sunlight. Apply yellow distres ink with an applicator. Once I had my panel I followed with the Happy Hello stamp coloring that you can get here.\nColoring with Giotto pencils I got them in an artistic bookstore and I got $ 200 (Argentine pesos) the 36 colors. I want you to know that I declare my unconditional love to those pencils! They are very good!\nFinally apply with a light gel white pen and with accent glossing apply on the balloons and on the stones of the bird to give a beautiful detail.\nGood start to the week I look forward to next Sunday to see more about CDD.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Peter L. Laurence is an associate professor of urban design, history and theory at Clemson University School of Architecture. His book Becoming Jane Jacobs (University of Pennsylvania Press, 2016) is an intellectual biography of the architecture critic and neo-functionist Jane Jacobs and how she came to write the 1961 classic The Death and Life of Great American Cities. Beginning with Jacobs's arrival in New York City in 1934 with only a high school diploma and writing aspirations Laurence follows her career to the pages of Architectural Forum under the editorial direction of Douglas Haskell. At the magazine she honed her critical skills and was exposed to the latest in urban design and renewal working with leading architects and planners. Laurence argues that there are persistent myths about Jacobs, including her status as a housewife and an amateur urban activist who surprisingly wrote a classic, or a genius. Rather, Jacobs transformed herself into a sophisticated critic influenced by the ideas of a wide circle of intellectuals and wrote a great deal before and after her most well known work. Death and Life of Great American Cities synthesized many previous ideas and proposed a new way to think about cities that considered the social networks and perspective of the person on the street rather than top-down planning that disregarded the human element for efficiency and form. Her vision for the city was of a living system with flexibility, creativity, and diversity offering a sense of connection by mixing the old and the new. Laurence offers not only the evolution of Jacobs's ideas but also the ways mid-century intellectuals conceived of the cities we now live in.\nLilian Calles Barger, www.lilianbarger.com, is a cultural, intellectual and gender historian. Her current book project is entitled The World Come of Age: Religion, Intellectuals and the Challenge of Human Liberation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Acer's Swift 3 SF314 is a Windows 10 laptop that features a Intel Core i3 that's rated at 2.3GHz and paired with 4GB of RAM. For graphics processing, it's powered by a Intel graphics card that shows on a 14\" 1920 x 1080pixels display. It comes with a 3220TB SSD and weighs at only 1.5kg. The laptop comes with Unknown.\nThe Acer Swift 3 SF314 is now available in Singapore. The cheapest Acer Swift 3 SF314 price in Singapore is S$ 798.00 from Qoo10. Acer Swift 3 SF314 price details are updated April 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vito is attempting what he has called the Hot and Cold Challenge. He is attempting to dominate some Fiery Habanero Dinamita Doritos and then cool off in the 20 degree weather in Metro Detroit.\nIf Vito can get 100,000 or more views on this video, his Dad will buy him a Nintendo WiiU \u2013 which is something he wants pretty bad. Shout out to LA Beast, his favorite Youtuber.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bonds, etc., for Navigable Waterway Between Demopolis and Tennessee River and Tombigbee Valley Projects.\nAny provision of the Constitution of Alabama or amendments thereto to the contrary notwithstanding, the legislature may by appropriate laws authorize the state to engage in works of internal improvement within the state by fulfilling the requirements of local contribution, participation and cooperation now or hereafter established by the United States in connection with the construction and maintenance of a navigable waterway (herein called \"the waterway\") between Demopolis, Alabama, and the Tennessee river, including the relocation and construction of roads and bridges to and across the waterway, access roads and approaches thereto and the related engineering and rights-of-way acquisition expenses (herein called \"the projects\").\nThe legislature may by appropriate laws authorize the state to become indebted and, in evidence of such indebtedness, to sell and issue its interest-bearing bonds, in an aggregate principal amount not exceeding $25,000,000, for the purpose of enabling the state to discharge obligations at any time authorized by the legislature to be undertaken in connection with the waterway and the projects; provided, that the expenses incurred in connection with the sale and issuance of the bonds may also be paid from the proceeds thereof. Bonds evidencing the herein provided for indebtedness may be issued as direct general obligations of the state, and the state may pledge its full faith and credit to the prompt payment of the principal of the bonds and the interest thereon. The herein provided for indebtedness shall not be construed to prohibit or limit appropriations from the general fund of the state which from time to time may be made for the purpose of enabling the state to discharge obligations at any time authorized by the legislature to be undertaken in connection with the waterway and the projects.\nThe legislature may from time to time appropriate money from the general fund of the state to be expended by the Tombigbee valley development authority, a public corporation and agency of the state, and may also authorize the herein provided for general obligation bonds of the state to be sold from time to time under the supervision of said authority; provided, that all moneys received by said authority from the state, whether as appropriations from the state's general fund or as proceeds of the sale of the state's bonds, shall be expended, except for reasonable administrative expenses to be paid from said appropriations and expenses of the sale of said bonds to be paid from said bond proceeds, in discharging obligations that the state is permitted under the foregoing provisions of this amendment to undertake in connection with the waterway and the projects and shall have directed said authority to undertake in its stead. The bonds authorized by this amendment shall be in addition to those authorized by that amendment to said constitution proposed by Act No. 248 adopted at the 1967 regular session of the legislature and ratified by the electors of the state on December 5, 1967.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxaks b/data_all_eng_slimpj/shuffled/split2/finalzzxaks
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+{"text":"I've made a few posts here about my love of planning and ever since I bought my Erin Condren back in April, I've been really obsessed with looking at sticker kits and planner inspiration instagram accounts. Since I've filled every week with washi (without thinking of the potential stickers I could have added this year), I'm collecting kits for next year's planner and I thought I'd share some nice weekly spreads that I've come across. This weekend there's going to be quite a big sale on a few of these stores so I thought it was a good idea to let you guys know about it if you're into this sort of thing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Liquid immersed transformers will play a key role in distributing power throughout the grid to meet the expected future increase in global demand for energy. In this presentation, attendees will learn how liquid immersed transformers, designed with high-temperature materials, have been used to successfully address energy demand challenges. Learn more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Praise be to God our Father for an amazing journey!\nWe left on Friday 22 June at 11pm, embarking on a two day road trip through Zimbabwe and Mozambique to our destination in Malawi. From the outset, it was a journey filled with prayer and a realisation of God's power and love.\nOur team consisted of 10 people from different walks of life who had decided to give up two weeks to go on a missions trip. The purpose of the trip was to be a support, in any way possible, to a long term missionary family in Malawi \u2013 The Mollers. In conjunction to this, would be the opportunity to experience the 'behind the scenes' reality of missions.\nThe two weeks we spent in Malawi were divided into multifaceted dimensions. Mornings started with POUCH, which is a dynamic way to delve into Scripture. Basically, the team divided into groups of 4 and studied a specific passage together. After some time, we gathered as a whole and shared our insights. It was a truly awesome experience.\nBesides these daily activities, we also had a day of prayer. It began with personal one on one time with God, then a time of sharing and lastly a prayer walk through the various villages. Again, a beautiful experience.\nFurthermore, we attended the local church and had the opportunity to share through preaching (with a translator) and song. It was a very interesting time, especially because the culture is so different to ours and therefore things are done quite differently to what we are familiar with.\nOver and above the hard work, we had great times of socialising and experiencing Malawian markets, ice-cream and food. Friendly people, lush landscapes and fresh air embraced us.\nWe were even fortunate enough to spend a day at Lake Malawi to relax and enjoy the untouched beauty that Malawi offers.\nAll in all, it was a life-changing experience for most of the team members and a trip that shall be ingrained in our hearts forever!!\nWhat follows are comments from the individual team members about their personal experiences.\n\"The trip to Malawi was an educational one and an eye-opener. I have come away with many lessons learnt as well as a more accurate view of what being a full-time missionary is actually all about. The biggest thing I have come to learn is that it's all about the small things. Too often we try to \"save the world\" and get disappointed that the huge things we had hoped for don't happen immediately. What we overlook is how important the small everyday actions are, and what is the point of doing the big things if our everyday lives are such that they do not reflect the Christian values that we so easily preach to others.\n\"For the past five years, God has been calling me to go into full time missionary service. As a result of going on this short term mission trip, I have made a decision to study at Baptist Theological College full time next year. In faith, I am going to trust God to provide the finances that I will need.\n\"I have been truly blessed during this mission to Malawi. Not only from those there, but also with the interaction of those on the trip. Such love and encouragement from everyone.\nI went still seeking my ministry and purpose in life, and have come away with confirmation, peace and direction. Our God is a faithful God!\nEven though I won't be able to go out into a mission field like Malawi \u2013 there is plenty to do on the home front. My work within the rural community at home is very mission orientated and I learnt so many valuable lessons that can be used there. I also recognised a need to educate myself further in this regard so will look out for courses or books on mission work.\nI'll also be able to help Keith and Christeen with some tasks back home and so the contact will continue. Knowing first hand their needs makes it easier to pray for them.\nI have really come with a renewed zest to go out and be His hands and show His love wherever I can.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This web address is available for purchase or lease. Please email info@nucom.com for further information.\nInformation Brokers is a business intelligence portal based in Melbourne, Australia - Our mission is to help businesses run more efficiently by providing a comprehensive suite of Australian search, intelligence and registration services in the one location.\nThe Jet Business is the world's first and only street-level corporate aviation showroom for the marketing and acquisition of corporate jet aircraft.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"American Law-makers and Indian Intellectuals!\nThe law-makers of the United states of America desire Indian Prime Minister Mr. Modi to address the joint session of American Congress. It is the same US which denied Mr. Modi visa for his alleged human rights' violations. For what ever reason, the US has recognized the importance and important position held now by Mr. Modi.\nBut in India we have much more aggressive human rights' champions who never want to and even cannot change their chauvinistic denouncement of Mr. Modi and even are ridiculously arguing that it is their freedom of speech that they want to dismiss Mr. Modi as a negative face; and we have many liberal minded and intellectual community which never wants to come out of its cage of Nehruvian or other ideologies and want to be counted secular or free just having denouncing Mr. Modi as unwanted element in India though he is now the Prime Minister by a hard-fought election campaign.\nThey do not want to see the difference between Mr. Nehru and his dynasty, which had the backing of their elite background, famous families or sentiments around their family to be Party bigwigs or Prime Ministers. This is called intellectual violence. They cannot be objective anytime because hatred for Modi is their intellectual food on which survives their intellectual profession.\nLet us see how most other nations of the world respect Mr. Modi for his achievement and position he is holding as Indian Prime Minister and still our home-grown liberal-ideologies-fed and self-proclaimed protectors of ideals, not necessarily Indian, continue bashing him forever.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxhnn b/data_all_eng_slimpj/shuffled/split2/finalzzxhnn
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@@ -0,0 +1,5 @@
+{"text":"We've released the initial version of Tide, our cluster scheduler for dealing with containerized batch jobs. It runs directly within ContainerShip, and doesn't require an additional process to be running on every server.\nThere are a few different kinds of workloads that often need to be run on a server, or group of servers.\nLong running - These are your web servers, databases, queues, and APIs, etc. They usually do not stop running once they have started.\nScheduled \/ Batch jobs - Scheduled jobs are not always running, and instead start up on a schedule, handle some work, and then complete and shutdown when that work is finished.\nTraditionally, scheduled jobs were often run via Cron on unix-like systems. Cron takes a schedule and then executes the job at the specified time.\nFor example, maybe there is a job that needs to execute every day at midnight, which runs a script that updates fields in a database.\nCron is great, but there are some major drawbacks to using it which should be considered.\nSingle Point of Failure - Cron is not distributed, so if the server you have added the jobs to goes down or is lost, your jobs go with it.\nTedious Updates - Editing a cron job typically requires logging in to a server, copying over any updates to the script being run, and manually editing the job definition using the crontab command.\nSecurity - Since SSH access to the server with the job installed is required, it is difficult to provide access to different members of the team to add their own jobs without giving them full access to the machine and all other jobs potentially.\nTide runs jobs as containers - Running jobs via containers means that you can bundle your scripts and jobs into an image which can be shared and distributed just like the rest of your infrastructure components, and can easily be kept in version control.\nNo single point of failure - Since the job definition and the script to be executed are not tied to an individual server, but instead live within the cluster scheduler, there is no fear of a server failure taking the job definitions and scripts down with it.\nUtilize Spare Resources - Tide jobs run on the ContainerShip cluster, right along side your long running tasks. This allows for higher utilization of resources you are already paying for, and eliminates the need for a \"cron\" server.\nConvenient - New jobs can be created and updates can be made via CLI or API.\nEvery ContainerShip cluster that you launch with ContainerShip Cloud will automatically have tide installed by default.\nGo ahead and launch a cluster first, it only takes about 10 minutes to get going. Check out the video below if you are more of a visual learner.\nOnce your cluster is launched, you can play around with tide via the CLI.\nFirst you need to Install ContainerShip locally on your laptop to get access to the CLI.\ncs tide create-job example --image myorg\/mycron:1.2.2 --command \/path\/to\/myscript.sh --cpus 0.5 --memory 512 --env-var KEY=value --instances 1 --schedule \"0 0 * * *\"\nAfter creation, you will see the job listed in your application list within the ContainerShip Cloud user interface. The job will be scaled down to 0 containers, but as soon as the scheduled time comes, it will scale up to the specified number of containers and execute.\nWondering what else you can do? Check out the tide plugin github repo here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When not working on Standards Phaedrus Systems has been producing app notes, magazine articles, presentations, guidance on software engineering etc. For the better part of a decade CTO Chris Hills had a regular column in Embedded Systems Engineering. These documents have been scattered far and wide. Now they are all accessible from this one page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Writings on food, history, Montreal, and food and history in Montreal. And sometimes music.\nHi, I found this listing's page of slumming and unslumming sur toutMontreal.com. L'adresse de cette fiche est https:\/\/www.toutmontreal.com\/24951 Enjoy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'Politics & World News' started by N8 v2.0, Oct 20, 2005.\nCitizens of the United States, I want to offer my sincerest apologies for the nature of the federal response to the hurricane Katrina disaster. My administration and I have completely failed to predict how completely stupid and inept the local and state government was in Louisiana.\nFrankly, I expected a response more along the lines of that from Mississippi, in which the locals were able to support themselves for a few days until we brought in the cavalry. However, it is clear that the state and local government leaders in Louisiana are incapable of such self reliance and emergency planning.\nSo, as Harry Truman said, the buck stops here and I take complete responsibility for failing to recognize that we were dealing with idiots in Louisiana, people who cannot manage their own state. I am embarrassed to admit this, because after all, a simple review of the state's history and position should be more than enough to demonstrate that these people are absolute morons that need to be completely supported from day one. Sometimes I am not very good at seeing the obvious.\nI should have predicted that the local government would release the criminals being held in prison into the general population instead of evacuating them. Likewise, I failed to predict that Louisiana citizens would begin shooting at and attacking the very people who were trying to help them. Along a similar vein, I should have recognized that we would need to move the people along with the wide screen televisions, clothes and other items they had stolen from local stores. Many people were heard screaming \"I just stole this s**t and now you want me to leave it behind, f*** you!\" I frankly cannot believe that I failed to predict such activity in and around N.O.\nAlso, I should have recognized that the federal plan would be too complicated for Governor Blanco. My failure to \"dumb down\" the proposal, my failure to recognize my audience, likely resulted in the loss of hundreds of lives. The extra 24 hours she needed to understand the plan was critical, I should have predicted that she was too stupid to be able to make an executive decision in short order. (said tearfully) Had I only offered a more simple minded plan, more people would have survived.\nLikewise, I want to apologize for buckling under pressure from Democrats to limit what they called pork barrell spending. When Senator Kennedy publicly criticized my spending bill, asking me to reduce the federal spending on local projects in N.O., I caved in and reduced expenditures on the N.O. levee system that would have strenthened it to a point of being able to withstand the forces of a category four storm. I will never know if my spineless response to Senator Kennedy's harassment was instrumental in the failure of the levee system in N.O.\nHowever, I can assure the American people that I have learned a valuable lesson. I will no longer allow the ineptness of local and state governments to circle around and cause hardship for the local citizenry. I will step in sooner to save them from their stupidity. I am frankly accustomed to being in a room with people who are smarter than me. I have little experience working with people who are even less intelligent than I am.\nAll I can hope is that the American people forgive me my shortcomings, and God Bless America.\nis there a mobius strip in here somewhere?\nThere's no way Bush wrote that. Many of the words have three or more syllables.\nHey RM, Ya fix the photos thing yet?\nFunny, I saw the head of DHS saying he didn't have any problem talking with the local officials...He said (under oath) he didn't know what Brown was talking about. He also said it was completely his department's fault for not coping with the situation.\nMeanwhile NBC (?) released some of FEMAs emails, In one responding to the local FEMA persons comment that the situation was \"Way Past Critical\", they replyed with comments about 30 minutes not being enough tiime for Mr. Browns lunch. The local FEMA guy replyed \"I just took a dump in the halway with 15,000 of my closest friends...\"\nWell, that makes it all better. He also didn't mention declining a nuclear strike, which saved 80,000 lives right there.\nP.S. I should also mention how New Orleans is the largest city affected by Katrina (by a 420,000 pop margin), and how 350,000 seem to have gotten out fine, leaving only the worse off to fend for themselves. Maybe a tactical nuclear strike was in order.\nA good friend of mine headed down to New Orleans a few weeks ago (two weeks after the hurricane). He and a friend of his ended up coordinating the relief and rebuilding efforts in East Biloxi between local authorities, schools, and the Red Cross because FEMA still hadn't arrived. How to deal with the influx of people, in terms of shelter, food, healthcare. How to handle twice the number of kids in half the number of schools (half the schools were lost to flooding). Issues that NO local government can be prepared for and is exactly what FEMA is supposed to help with. When FEMA arrived (two weeks late), they looked to my friend for leadership because they had NO IDEA what to do.\nGreat. A town with several thousand refugees is looking to two 27 year old kids who HAPPENED to show up to run the show, because FEMA is that incompetent.\nN8, I don't know if it's sarcasm when you put up smilies beside a Bush apology, but if not, well, let's just say it's weird.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Clearly a bunch of dudes who like staring at album artwork and letting their imagination run wild as much as I do, Colorado's Khemmis are back already after their well-received d\u00e9but of last year. This time, the old wizardly man from that album artwork is on a very barbarian-looking horse and off on an adventure. What's the adventure? Can he use the big sword and shield he carries? What happened to the barbarian babe in skimpy armour from the last album? These, and other such questions, are what fly through my your head whilst listening to Hunted, a doom album that isn't afraid to throw a little bit of everything into the mix, often to terrific results. The base sound is Iron Maiden slowed down, a gleefully traditional set of riffs played on a guitar with a lovely, rich, warm tone backed by solid if unremarkable drums and (audible) bass, but Khemmis are open-minded enough to throw in something of doom smorgasbord, so we get both heavy stoner boulder-drops a la High on Fire and goth-tinged melodrama that's pure My Dying Bride on the opener Above the Water. Dual vocalist\/guitarists Phil and Ben mainly stick to the Stainthorpe-gone-stoner plaintive singing, but indulge in some classic growls that for the most part are well-placed enough not to be out of place. Given another album or two it's easy to see them being dropped altogether, however, as they're much reduced from the band's d\u00e9but and honestly the clean vocals are good enough to carry the album on their own.\nThat's particularly clear once you hear the following Candlelight, perhaps the best track present thanks to a genuinely catchy vocal hook that's used just sparingly enough to be infectious rather than overblown. The verses fall into the same vocal rhythm, not so much trancelike as hymnal, and the dip into funeral doom several minutes in is perfectly judged to grip the listener, as is the epic return to clean singing thereafter. Yet all the songwriting present is terrific, the rumbling rocking and epic heavy metal contrasts of Three Gates as good as the post-rock dalliances of Beyond the Door, the latter also exploring those My Dying Bride and even Candlemass influences a little further to terrific effect. Call me controversial, but I feel like the band are better when working with set song boundaries too; the twelve-minute finale title track should be terrific, but falls a little short for me. The opening has a nice swing to it, and the ensuing guitar worship is charming enough; even if you don't quite fall in love with the band, it's easy to admit that their heart and soul are in the right place. At nine minutes in there's a shift towards atmospheric doom that doesn't quite work, however, feeling like a skippable album outro rather than a vital section of the song.\nI feel slightly ridiculous typing that though, as it's hardly bad, and is the precursor to perhaps the most convincing My Dying Bride tribute on the album, epic riffs and vocals working together perfectly to draw a very good album to a close. Let's wait a little longer before throwing too many accolades at the band, however; I'm not quite sure that this is album of the year material yet. The range of influences that Khemmis has to draw on are wide but still a little disparate; given a third or fourth album to refine their sound even more and their praises will truly deserve singing. For the moment, this is a solid and even excellent round-up of various doom subgenres played with heart by a band with Khemmistry (hiyo!) that will appeal to any of my fellow gloombahs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Kaweco Classic Sport Rollerball. The Classic Sport closely follows an original 1935 octagonal design - 'Small in the pocket, large in the hand'. The oversized cap arrangement creates a small closed pen but a full length open pen. Closed, the pen is just 105mm long, open 133mm long. The barrel is 14mm in diameter (around the cap).\nAlso available in a solid brass version.\nLarge-capacity liquid ink rollerball refill - capless quality. Supplied as black medium tip.\nRefills available: call us at 213.928.0997 to order.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What does it mean if someone has military bearing? It means they behave with a certain decorum and discipline. Someone who adheres to the utmost levels of honor, duty, and professionalism in whatever they do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Sabarimala pilgrim was trampled to death by a wild elephant in a forest area, 60 km from Kerala's Kottayam early on Wednesday morning, police said.\nParamasivam, 35, from Salem in Tamil Nadu was attacked by the jumbo as he along with his seven-year-old son and 13 others were going to the Lord Ayyappa shrine through the forest route from Erumely to Pamba at around 1 am, they said.\nWhile others managed to flee the spot, Paramasivam, who was said to be carrying his son on his shoulder, could not.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The warmest month (with the highest average high temperature) is June (33\u00b0C).\nMonths with the highest average low temperature are June and July (20\u00b0C).\nThe coldest month (with the lowest average low temperature) is January (2\u00b0C).\nThe wettest month (with the highest rainfall) is July (247mm).\nThe driest month (with the lowest rainfall) is November (31mm).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Once the website is ready, it is time to publish it and make it available online.\nIn this guide, we will publish the website using WebMatrix's publishing feature.\nFor this, you need web hosting that allows you to run ASP.NET 4.0 applications in Full Trust mode.\nPlease note that you cannot use any web hosting option. C1 CMS runs on Microsoft ASP.NET 4 which means you cannot simply upload it to your PHP-based server. If you are in doubt, use one of the providers suggested via WebMatrix's publishing feature.\nYou will need publishing information from your web hosting provider to configure WebMatrix. Some providers are WebMatrix-aware and can provide you with a Publishing Settings file (.PublishingSetting), which you should import to have WebMatrix configured automatically.\nWebMatrix offers FTP and Web Deploy as protocols for publishing.\nFor more information about publishing your website with WebMatrix, please refer to WebMatrix online Help (for example, \"How to Publish\" and the like).\nFor other publishing options (via FTP, on your own IIS), please read \"Getting Your Site Online\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxleu b/data_all_eng_slimpj/shuffled/split2/finalzzxleu
new file mode 100644
index 0000000000000000000000000000000000000000..bae5ea81991fb50017f0577da03cf716435e736c
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"First-string definition is - being a regular as distinguished from a substitute (as on a team). FirstString's customers range from the beginner bow enthusiast to the Professional Hunters and target archers to Olympic bow target shooters and the Traditional. First string definition: the top player of a team in an individual sport, such as squash | Meaning, pronunciation, translations and examples.\nEmail or Phone Password Forgot account? World Health Partnerships Nonprofit Organization. Rawlings Rascals Pet Breeder. The First String Healthcare has been a wonderful company and perfect fit for me.\nThey are very supp No issue is too small. They are there for me when I need them. Imagine healthcare and a K with an interim assignment! They are enabling me to do things I had never imagined in terms of setting myself up for retirement and exploring other areas of this great and diverse country. Plus, with the travel available I can stay in touch with family and friends and share my adventures. Life is very simple when your housing and transportation are all taken care of.\nThis company is, by far, the best company I've ever been associated with! I am very proud to say I' Words that rhyme with first-string. All synonyms and antonyms for first-string. What made you want to look up first-string? Please tell us where you read or heard it including the quote, if possible. Test Your Knowledge - and learn some interesting things along the way. Subscribe to America's largest dictionary and get thousands more definitions and advanced search\u2014ad free!\nYou'll fall head over heels. Is it real life, or just fantasy? Comedian ISMO on what separates a boot from a trunk. Comedian ISMO on the complexities of the word 'tip'.\nFirst Known Use of first-string , in the meaning defined at sense 1. Learn More about first-string.\nThe Hurricanes plan to use N'Kosi Perry, who started Miami's three most recent games after temporarily taking the first-string job from Rosier, on Friday night as. Define first-string (adjective) and get synonyms. What is first-string (adjective)? first-string (adjective) meaning, pronunciation and more by Macmillan Dictionary. Definition of first-string written for English Language Learners from the Merriam- Webster Learner's Dictionary with audio pronunciations, usage examples, and.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yay, Ryan is back! I still remember the huge shitstorm over when DC killed him off.\nShe could be playing anyone, Marvel hasn't exactly been casting based on race. Mac on SHIELD in the comics is an elder looking white man. She could be playing Mantis, Moondragon, Deathcry, Geena Drake, Phyla-Vell, or even Aleta Ogord. Can't wait to find out though. Plus its great to see new faces added to the film.\nTrue, but that said, the article explicitly mentions that they were seeking an Asian woman for this part. Marvel has changed races for characters in the past but I think that was more about them finding the best people for the role.\nIn this case it sounds like the part was specifically written for an Asian woman, which would say to me at least that it's an Asian character from the comics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Has anyone not recieved their I'm Communsism shirt yet? If you haven't, please email me and let me know!\nLucasarts has decided to stop production on Sam and Max 2! There are many theories about why the game was canceled. Acting General Manager Mike Nelson suggests that it was because it is \"not the appropriate time to launch a graphic adventure on the PC,\" while I suggest it is because the Lucasarts managers are a bunch of slack-jawed, knuckle-dragging scum-suckers.\nHad the game been released, the book collection of Steve Purcell's Sam and Max comics would likely have been re-released as well. Now, it looks as though we will be getting neither. If you're a fan like I am, I suggest you do what this guy is doing, and write the folks at Lucasarts a stern email that explains, in the most polite way possible, why they are a bunch of morons.\nMy I'm Communism t-shirt has arrived, and boy, have I gotten some strange looks while wearing it. Not looks of shock or offense, mind you, but of utter confusion.\nI believe these gazing folks are trying to figure out exactly what statement the shirt is trying to make. But of course, one could spend months, or even years contemplating the I'm Communism phrase and accompanying image before they truly appreciate the many layers of metaphorical overtones and symbolic significance.\nToday we embark on a brave new storyline. You'll laugh. You'll cry. You'll laugh and cry. Simultaneously. IN SPACE.\nErik McCurdy of It's All Been Done and Didn't Think This Through has been working on another webcomic the past little while. It's called Banish Ed, and it has gotten off to a solid start! I've also really been enjoying Martin's Misdirection, which I found by way of Erik's site. Check 'em both out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Dumpster Blog, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Buckner, KY, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nFor anyone who is undertaking any basement or perhaps garage clean out or development job, your 15 cubic backyard next day dumpster could possibly be just what you'll need.\nWhen you go for any lowest cost, the company commonly has to cut corners a place. This may be with paying employees lower salaries or keeping fewer dumpsters around. They might also take their time offering and collecting dumpsters in an attempt to produce things in shape their schedule instead of the buyer's.\nHow long would you get to keep dumpster? Most companies allows you to keep your dumpster with regard to between 5-7 times. If you're lucky, you might find a company that will help you keep your dumpster with regard to up to 2 months. What happens if you would like to keep it more than your decided time frame? Further fees. Normally, you will get in a daily rate charge everywhere from $5 to $15 per day, after the 1st rental period.\nYou might want to also clean out your barn. You are able to rent a small, channel or big dumpster to get the job done. You may need a large dumpster should you have a lot of trash that will go with ten vehicle loads. 10 pickup tons or 4 tons could be the limit for any large dumpster. Should your items go over four a lot, you will need to rent another dumpster to finish the project off. If you only need in relation to five pick up loads than the usual medium dumpster is useful for you. A new medium dumpster will hold up to a few tons of trash or particles. A small dumpster will hold up to 2 tons of particles.\nThere are some deterring measures which a user will miss out while dealing with your waste created if it is treated on personal. Initially the person will be not aware of the size of your dumpster that is required to clear the trash with one head out. Picking a bad sized dumpster may result in doubling the task as well as cost in terms of transport. There exists diverse number of restrictions from city to city how the user offers to know thyself to dump the trash.\nIn accordance to authorized accounts of express of air, millions of Us residents still are affected by the air air pollution and this is being a reason for growing many unsafe diseases such as lung function, asthmatic infection, lung inflammation and aggravation associated with respiratory illness. Most of the occupants in US reside in unhealthy and unhygienic surroundings. Why is this so? Associated with very simple. Your garbage produced owing to various initiatives like development, house redevelopment, commercial cleanse ups and yard cleanse ups stays without proper disposal and is contaminating the environment.\nMost of the people use the 10 cubic backyard dumpster for smaller sized bathroom, kitchen or basement remodeling employment. This dumpster supports to two tons of garbage.\nMaybe you are making a store, standard bank or other kind of company. You will need to rent any dumpster. With big buildings determined will need a big dumpster or multiple dumpsters depending on the actual size and the way many people are taking care of building the brand new business.\nThere are some precautionary measures that your user will miss out though dealing with your generated waste when it is treated on home. Initially anyone will be not aware of the size of your dumpster that is required to crystal clear their junk at a single go. Picking the wrong sized dumpster may result in increasing the task as well as cost (regarding transport). There is certainly varying volume of restrictions coming from city to city which the individual has to train him to get rid of the trash.\nAfter we come together being a community from a storm, we strengthen our bonds together and assist ourselves at the same time. Be safe this kind of spring, and when a storm does indeed rip via your neighborhood, get everyone jointly to clean up being a group. In the end, many hands and wrists really do assist to lighten your workload.\nThe following issue which drills the thing is what if there is no need the space needed to place the dumpster safely? Not all your projects will probably be having the enough free place to set any dumpster, especially in the circumstance of household renovation projects. The business projects also sometimes hit the same situation due to your tons of waste occupying your breathing place in the employment location.\nAdditionally you need to know what types of services they provide. You may need to know how these people handle getting rid of your discretion. Do they come with a big truck lift it up and get rid of it in the back, like most garbage vehicles? Or do they come and get the discretion and take it off allowing you a brand new a single to take the place? These are things when compared with can affect the price of your unit so you certainly want to explore that when you call as well as order your current rental.\nOne more element where Cost of a Roll Off Dumpster in Buckner gets inevitable inside Island such as American Samoa is in the eve of natural disasters. The area is mixture of five volcanic countries so the local weather in the talk about is always unclear. Protecting your house in such conditions is a pain. The property as soon as lost could be regained, nevertheless what about medical that is going to be affected right after the great disaster? Avoid big box stores as they possibly overcharge for products. If you know you need a great quantity of an object, like raw wood or natural stone, find a mass retailer to purchase your material coming from and preserve. They can give you better prices on product because it's in large quantities, versus getting set quantities in a big box store, which can help your main point here. Remnants are a great way to lower your expenses. Most shops will offer records at affordable prices because they occupy space and they cannot start using thise pieces. Which has a bit of creativeness, remnants can offer a unique factor to your overall design, while saving your main point here.\nLos Angeles, having population well over 12 million individuals, and an location with an array of various entertainment bones, residential households and other businesses, this area needs a fast as well as reliable dumpster assistance. This is because it's going to have to serve both residents as well as business which are in the area. Furthermore, with such any population, you would expect the dumpster enthusiasts to have a large amount of work to perform.\nIt is very important with regard to us to dispose the waste as soon as that gets piled-up. Trash generation in quantity of projects just like house restoration, yard tidy up, and design projects is actually inevitable; your current negligence to help dispose the particular waste within environmental welcoming manner in the shortest time possible will increase the effects of polluting of the environment on the environment of the express.\nRemember the days and nights when you might pack up your current piles involving junk and cart these people off to the nearest town or even county dispose of with nary the care? Nicely, not anymore. For most communities currently, the number of stuff you can discard at the local landfill proceeded to go down while the price of the particular service proceeded to go up. So how are property owners supposed to remove all the rubbish that will tends to take above their homes? The answer is simple. Develop the junkyard come to you using a convenient roll-off Cost of a Roll Off Dumpster in Buckner, KY from your hometown trash removal enterprise.\nIt will save you time and energy - Just think for a second on how that feels each time when you individual biodegradable waste from the neo biodegradable waste. Isn't this particular an annoying job? You will need on the job so you do not have to individual the recyclables. Invest your trash in the dumpster, and there you are absolutely finished with your projects.\nConnecticut is usually a state in the New England location of the northeastern United States. It is surrounded by Rhode Tropical isle to the eastern, Massachusetts to the north, New York to the gulf and Ny Sound south. It is the 29th nearly all populous area of United States and it also 4th heavily populated express.\nSo if you would like rental dumpster delivered and picked up on time, do a bit of research to find a top quality company. Then of course you'll benefit from user friendly service, superior pricing and much more flexibility to satisfy your demands. While you're busy working on your restoration project, do you really want to put extra strain? Planning a home rehabilitation project say for example a bathroom, your kitchen, or family room renovation or simply a cleaning-up or clearing-out restoration can seem like a huge obstacle without the suitable cleanup equipment. Renting a dumpster allow you to contain the trash during the restoration process and haul that away handily in one easy removal method. Here are some dumpster facts that will assist you when deciding on the best dumpster for the project.\nThe use of a Cost of a Roll Off Dumpster in Buckner, KY can be evidently recognized as a method to store too much amounts of trash. However, the benefits to using this service don't just stop there. It also provides one the opportunity bypass the particular headache when trying to discard massive numbers of garbage.\nIf you want a dumpster permanently, you should buy is from the organization. A similar decision for this is to help lease the particular dumpster for a degree of many years, which is near to simply purchasing dumpster. All wants can be met for.\nPrior to getting started on the particular logistics involving keeping your dumpster by ruining your current lawn, flowerbed and rock backyard, let's take a glance at why you should care. Why not only borrow the pickup truck and become done with that? Is there definitely any benefit to offset the hassle of Cost of a Roll Off Dumpster in Buckner, KY, or will you be better off to simply avoid it from the start?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A few months ago, I fired off answers to a few e-mail questions submitted to me by Francesco Cordella, editor of a delightful bilingual interactive fiction fan site. He's now posted the exchange on his site.\n16 Jun 2003 in Cyberculture, Games, Humanities, Literature.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxpui b/data_all_eng_slimpj/shuffled/split2/finalzzxpui
new file mode 100644
index 0000000000000000000000000000000000000000..5010e2558a50454e27a9d9f3188550f9b8bcebbe
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxpui
@@ -0,0 +1,5 @@
+{"text":"A Royal visit to Auckland by Queen Victoria, one of Cunard's 3 Queens.\nA great little gadget from NiSi to keep those filters clean.\nA truly one of a kind workshop with multi-award winning British photographer, Paul Reiffer and me.\nA day with the iPhone.\nA day trip armed only with the iPhone.\nNiSi V5 filter holder system review.\nA review of the NiSi V5 filter holder, how much more of an improvement is it over the V3.\nRoad trip to the Bay of Plenty in New Zealand.\nA road trip to Mt Maunagnui for a job of a different kind.\nSouth Island Workshop in June 16 Not to be missed.\nNiSi filter bundle deal at Photogear in Auckland NZ.\nCan the Nisi beat the rest of the bunch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Flexicast 65 is a two part polyurethane elastomer that cures at room temperature to a flexible, high strength mould rubber.\nMoulds made from Flexicast 65 are durable and require little to no release when casting plasters and waxes. When used with the appropriate release system, Flexicast 65 moulds are excellent for casting FMG, concrete, polyurethane, polyester and epoxy. The final parts or moulds made from Flexicast 65 will be tough and long wearing, making them ideal for long term moulding requirements. Flexicast 65 can be processed without the aid of vacuum degassing, allowing for larger mould pours and a wider variety of mould making conditions. Flexicast 65 will provide excellent chemical resistance to most casting systems while maintaining a high standard of physical properties.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cardinal Muller speaks out honestly about Martin Luther. Edward Pentin reports here that the German Cardinal gets it right about Luther.\nCardinal M\u00fcller, however, said the Protestant Reformation was not a \"reform\" but a \"total change of the foundations of the Catholic faith,\" and that today's Catholics often discuss Martin Luther \"too enthusiastically,\" mainly due to an ignorance of theology.\nIf you want to see true reform in the church look to St Francis who also confronted a church that was riddled with corruption, greed, graft and spiritual complacency.\nFrancis stood barefoot in the snow humbly waiting to see the Pope. Take time to read about Luther, on the other hand, his violent heresy, his breaking his religious vows, his violence against the peasants, his vulgarity, despair, mendacity and dishonesty.\nThis man was a revolutionary, not a reformer, and while we can recognize that some good things came from the Protestant Revolution we should not be starry eyed about it. More harm than good came from it, and we should commemorate the Reformation with realism, not romanticism.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is the link to the clip in the movie. It is during the battle of Gondor. The forces of Mordor are overwhelming the city. Gandalf and Pippen have been in battle, have a brief respite and Pippen makes a comment about not thinking the adventure would end this way.\nAfter you have watched the scene, here is my rendition of images that capture this moving conversation.\nI am a huge fan of Tolkien and Lewis. I think there is great value in leaders reading such authors and allowing their insights to shape our inner life out of which we lead. Tolkien and Lewis both speak from a profound Christian worldview. Here are three images\/quotes from Tolkien on which I recently reflected and taught.\nThe theme of EUCATASTROPHE is a profound one. It is the joyful ending for which we are hardwired as human beings. We love happy endings. We long for them. We miss them. The Gospel story begins in a world of great joy and Original Happiness - the Garden of Eden. The Gospel story ends in a world of great joy and Eternal Happiness - the New Creation. Smack in the middle is another eucatastrophe - incarnation and resurrection. These realities MUST define our inner world out of which we lead. This is the true source of all emotional intelligence - love, joy, peace, hope, gratitude, faith, courage and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Mental Health Parity and Addiction Equity Act (MHPAEA) of 2008 requires insurance plans that cover mental health or substance use disorders to offer coverage for those services that is no more restrictive than coverage for medical\/surgical benefits. Plans and insurers must disclose detailed criteria for medical necessity determinations relating to mental health and substance use disorders when requested by any current or potential participant, beneficiary, or contracting provider.\nMore stringent regulations designed to ensure parity for treatment of mental\/behavioral health and medical\/surgical claims have resulted in a significant increase in the number of mental health and substance abuse claims. This is driving many health plans to offload these reviews to independent review organizations like AllMed. In addition to streamlining health plan operations, AllMed's MedCert solution helps lower fixed costs, while ensuring the quality and integrity of the physician review process and improving quality of patient care, while controlling overutilization.\nLearn more about how AllMed's Behavioral Health UM program can help your organization streamline your UM process and make objective determinations under tight deadlines.\nContact our team today to discover how outsourcing your behavioral and mental health determinations to AllMed can help your organization achieve detailed, defensible determinations while lowering your fixed costs and decreasing turnaround times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxtrj b/data_all_eng_slimpj/shuffled/split2/finalzzxtrj
new file mode 100644
index 0000000000000000000000000000000000000000..fede8602ed51531a22de6f901773f093a03572ae
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxtrj
@@ -0,0 +1,5 @@
+{"text":"Xabeoptics.com provides the best in optics, while making it affordable. We are able to do this with modern technology, superior products, reliable vendors, and the best group of optical professionals in the business. All products are checked to exceed ANSI standards before they leave our facility; 100% guaranteed. We feel it's our role to make sure you receive the best value possible; and we guarantee it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The information shaft turns at a motor speed at whatever point she moored. The hold plan is mounted on the information shaft and rotates with it. It holds change frameworks with the counter contraption, which turns the countershaft.\nThe counter mechanical gathering exchanges capacity to the speed change on the first shaft. At any rate since speed engine gears supplier in UK, one, two, three and four are not dashed to the central axis when a transmission is insensible, they can't exchange capacity to the standard shaft. The colossal shaft does not turn, and there is no power regarding the driveline.\nThe power or torque courses through the information shaft and handle mechanical get-together to the counter contraption. The counter contraption turns. The critical automatic gathering on package drives the coordinating rate plan on a focal shaft. A standard first-speed change degree is 3:1 (three full turns of information shaft to one full turn of yield shaft). Along these lines, if motor torque entering the transmission is 298NM, it is imitated three times to 894 NM when it exchanged to a driveline.\nFor change plan in a similar path in a general sense not particularly depicted mechanical party degree utilized.\nIt is to see that at any rate torque is imitated three times, the speed is diminished as necessities be. Along these lines first, the mechanical get-together is kept up to move the vehicle from the stationary condition, everything considered obliged up to 10KMPH.\nRight when a change from first to the second gadget made, the moving fork pulls back the central\/second synchronizer from the essential speed contraption and moves until the moment that the minute that it pounds the second mechanical social affair to a central shaft. Power streams from the second contraption through the synchronizer to the standard shaft (yield shaft).\nIn the second contraption, the need for vehicle speed and expanding speed is gigantic than the basis for most vital torque increment. To address these issues, the second speed prepare on the standard shaft is comprehending how somewhat more little than the central speed plan. These outcomes in like way settling level of 4.4:2 which mirrors a drop in torque and a development in speed.\nThird Gear Correctly when the change from second to the third mechanical party made, the moving fork restores the focal\/second synchronizer to a sensible position. A second move fork slides the third\/fourth synchronizer until the moment that the minute that it dazes the third-speed mechanical social event to the sun shaft. Power stream finally experiences the third contraption through synchronizer to focal shaft and driveline.\nThe third contraption allows a further diminishing in torque and change in speed. The third speed change is more than second speed plan. This outcome in a commonplace mechanical get-together level of 2.9:2.\nIn the fourth contraption, the third\/fourth synchronizer is moved to flabbergast the hold change on the data shaft to the straight shaft. This gatherings control stream is especially from information shaft to authentic shaft at mechanical social affair level of 1:1. These outcomes in most splendid speed yield and no torque increase.\nA fourth mechanical social affair has no torque change since it is utilized at cruising rate to advance most personality blowing mileage.\nOn repair set it up, is crucial to turn the course of a tense shaft. It finished by methods for passing on change idler mechanical gathering into the power stream way. The idler contraption is worked between the countershaft turn mechanical social affair, and the switching speed on the regulate shaft.\nLike transmissions, some transaxles have five forward rates. Regularly, the fourth and fifth contraption for humbler vehicles has overdrive degrees. These high mechanical party degrees change for low last drive plan degrees. Low previous drive degrees give outlandish torque increase, which is required to empower with an insignificant motor securely.\nAll vehicles utilize a differential to give an extra contraption lessen (torque increment) well past what the transmission or transaxle changing can make. It is known as the clear drive plan. In a transmission enabled vehicle, differential getting sorted out is coordinated in back focus point lodging. In transaxle regardless, the prop up decreasing is passed on by the last drive change housed in the transaxle case.\nA touch of the bothers that can move with the utilization of transmission recorded underneath. It is to see that most by a wide margin of the issues with communications should examine by mechanics or individuals having inner experience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Swagger is a tool for documenting APIs, primarily driven from a coding side and a universal docs client.\nAdding basic support in a Spring application is straightforward using the springfox lib.\nspringfox-swagger-ui is only needed if you want the swagger UI be running in your application.\nAPIInfo provides some additional info about our app. See method below. And the rest define which controller and path aka endpoints should be included in the API doc.\nThe swagger endpoint is exposed under \/v2\/api-docs and the UI is available under \/swagger-ui.html.\nIf we run Spring Security like we did in the tutorial, we need to exclude a few path from security, so swagger works without it.\nAll four paths are needed for swagger. And here is a live version of the kanban backend running on a free Heroku instance, so it might just start for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Did you download the Matrix 2? | Something Odd!\nDid you download the Matrix 2?\nI find this BBC article laughable! I spent over a week dodging Matrix 2 spoilers until I saw it last night! Even if the downloaded movie is of a higher quality than the usual movie rips I'd still rather see the movie in the cinema! This Slashdot discussion has more opinions and rants..\nPrevious PostPrevious ATi cheated too?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cheil MDLab is a design consultancy focussed on branded spaces. Our team is consisting of 16 international creative individuals and we are looking for a talented Design Strategist to support our work in the area of Trade Show, Brand Exhibition and Retail.\nCheil MDLab is a department of Cheil Germany, located in Schwalbach am Taunus.\nPlease feel free to check out our website for more information: http:\/\/mdlab.cheil.com\/.\nYour role would include carrying out research for brands, markets, trends and design, create insights and participate in concept ideation.\nInterested\/experienced in design disciplines like architecture, graphic design, media art, scenography.\nWe are looking forward to receive your application via mail at a.anderer@cheil.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyvkf b/data_all_eng_slimpj/shuffled/split2/finalzzyvkf
new file mode 100644
index 0000000000000000000000000000000000000000..e2f20d4990b92d5eea787ddfdd0c0ab070a2ca63
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyvkf
@@ -0,0 +1,5 @@
+{"text":"A self-service SaaS platform that empowers a brand's field marketers to produce and deploy local versions of a centrally created campaign, using the fastest and easiest templating platform available today \u2013 at dramatically lower costs and greater speed-to-market.\nBrandgility takes your existing content and turns it into templates that allow distributed field marketers to self-serve content localization and adaptation. Local marketers have excellent local knowledge of how to make content resonate for specific regions, and brands have amazing content to help make that happen. Brandgility bridges that gap and allows brands to enable local marketers to adapt content within guidelines, resulting in fast, perfectly on-brand, localized content that creates product sales.\nField marketers with no design skills can create perfect, on-brand localized content, with no training.\nGive direct access to self-service content to your field marketers, to localize themselves.\nThe ability to self-serve makes adapting for localization instant while protecting your precious brand equity.\nOrders can be distributed to printer or ad networks, CMS, DAMs, etc.\nPlease complete the fields below and we will contact you to arrange a personalised demonstration of our industry-leading marketing platform.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lafayette's current Comprehensive Plan was originally developed in 1987, updated in 1997, and went through a major review and update in 2003. Beginning in March 2019, the City will be starting the process to develop a new, contemporary Comprehensive Plan.\nPlease refer to the OFFICIAL PROJECT WEBSITE -- www.LegacyLafayette.org for information, news, meeting schedules, community input opportunities, and to sign up to receive email notifications.\nServing as a source of information and input for the Comprehensive Plan project team across a variety of topics related to the Comprehensive Plan.\nServing in an advisory role to the project team as the Comprehensive Plan progresses throughout the process.\nAssisting with outreach to the community, including spreading the word concerning the latest news and events from the Comprehensive Plan process, to various groups and constituencies in Lafayette.\nExplore the four appendices and eight maps which make up the official Comprehensive Plan.\nThe 2013 Technical Update to the 2003 Lafayette Comprehensive Plan is a series of documents setting forth goals and policies for the future of Lafayette. The plan is a statement of how the community views itself, what the city envisions for the community's future, and the actions the city will undertake to implement the vision and goals of the community.\nThe Comprehensive Plan was adopted in 2003. Lafayette's Comprehensive Plan is a statement of how the community views itself, what the City envisions for the future, and the actions the City will take to implement the City's vision and goals. A technical update to the 2003 Comprehensive Plan was directed by City Council and conducted internally by City staff. The update brought the 2003 Comprehensive Plan current with existing and projected conditions and was not a re-write of the 2003 Comprehensive Plan.\nThe heart of the Comprehensive Plan is the Goals and Policies section and the Land Use Map. The updated Goals and Policies are a result of comments from City boards, commissions and committees as well as City departments and several workshops with the Planning Commission. City Council also held a workshop in May 2013. The Planning Commission held an open house with the public in July 2013 and two public hearings. The City Council held a public hearing in March of 2014 before adopting the 2013 Technical Update.\nInformation was also collected from the most recent Community Survey (2012), Census (2010), Denver Regional Council of Government (DRCOG), the Downtown Vision Plan (2010), the CDOT 2013 Planning Environmental Linkage (PEL) Study, and the Parks, Recreation, Open Space, & Trails (PROST) Master Plan (2013).\nThe plan is advisory only and was prepared with the general purposes of guiding and coordinating a harmonious development of the city and its environs which, in accordance with present and future needs, best promotes health, safety, order, convenience, prosperity, and general welfare.\nThe purpose of this plan is to state the preferences and priorities of the Lafayette community and to serve as a guide for current and future public decisions, especially the distribution and intensity of development, the location of future land uses including public facilities and open spaces, and requested zoning changes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VERY NICE BUNGALOW WITH NATURAL WOODWORK, HARDWOOD FLOORS ENCLOSED PORCH W\/SLIDING DOOR TO LARGE YARD. BASEMENT IS FINISHED NICELY WITH EXTRA BATH TILED & 2 BDRMS. HUGE FAMILY ROOM & KITCHEN. PARKING FOR 3 CARS.\n2903 N Kilpatrick Chicago, Illinois 60641 is a single family property, which sold for $295,000. 2903 N Kilpatrick has 5 bedroom(s) and 2 bath(s). Want to learn more about 2903 N Kilpatrick Chicago, Illinois 60641? Contact an @properties agent to request more info.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Irony is typically associated with the literary and oral; I intend to explore visual irony, specifically within Surrealist fashion, where puns are conceived and presented as optical double entendres, adding layers of meaning to deceptively simple forms. Inconsistencies between expectations and realizations of fashion are often humorously absurd\u2013 but what initially seems silly can effectively question functionality, gender, art history, politics and social standards. The Surrealists were, after all, affiliated with the French Communist Party (before they were kicked out!), but their focus was on an emotional revolution that they believed would achieve the same ends as a strictly economic one would.\nThis simulation of a ravaged frock was silly (who would wear a dress inside-out?) but also bitingly sarcastic, contrasting Great Depression poverty and pre-WWII fears with the escapist American films in which women could still afford luxurious evening gowns.\nthey nonetheless question the ideal body form, gender norms, and can express unease, uncertainty, inequality, and the political unrest pervasive in the years leading up to WWII.\nIf anyone's planning on attending (and you should, I'll have a lot more to share and show!!), do drop a line and introduce yourself!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Xaar, the world-leading developer of inkjet technology and manufacturer of industrial inkjet printheads, yesterday lined up alongside 38 other iconic British businesses taking part in the 'Make it in Great Britain' exhibition.\nThe impressive display featuring leading British manufacturers at the Science Museum,London, is the culmination of the 'Make it in Great Britain' campaign launched last year by the Department for Business, Innovation and Skills. The campaign challenges outdated opinions of modern manufacturing and dispels the myth that Britain 'doesn't make anything anymore'.\nXaar was chosen from hundreds of UK companies to participate in the showcase of British manufacturing. The company, whose printheads jet a variety of fluids onto a range of materials, has remained at the forefront of industrial inkjet printing technology for 20 years. Most recently Xaar's leading printhead, the Xaar 1001, has changed the face of printing in the ceramics industry worldwide.\nThe 'Make it in Great Britain' exhibition is open from 24th July to 9th September to coincide with the Olympic and Paralympic games.\nFor more information about the event visit www.bis.gov.uk\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabdeq b/data_all_eng_slimpj/shuffled/split2/finalzzzabdeq
new file mode 100644
index 0000000000000000000000000000000000000000..a594cd335be7b871bb0780cd6169b433f7518e49
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabdeq
@@ -0,0 +1,5 @@
+{"text":"If you cannot pay for your tickets with loyalty points, consider whether a reason has been given \u2013 loyalty points often come with restrictions on how they can be spent. If you feel that not being able to spend them in a particular circumstance is unreasonable, raise the issue via Resolver.\nIf there was an issue with the booking process, you should phone the cinema via Resolver and raise a complaint. It is reasonable to expect that a booking over the phone has the same terms and conditions as an online booking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are committed to meet any legal and regulatory requirement applicable to our products, activities and services.\nWe are committed to meet the requirements and expectations of our customers with the aim of achieving their satisfaction.\nWe follow a strategy focused on our highly qualified and specialized technical assistance.\nWe are committed to analyze any interested agent who may affect or been affected by Cytognos, adapting our decisions and actions to the context of each agent.\nWe make our employees aware of their activities contribution to the maintenance and improvement of our quality system and its relevance in order to achieve our quality objectives, by encouraging an approach to ensuring communication, individual involvement and proactivity.\nWe collaborate with our suppliers in order to ensure the improvement of our products and services.\nWe work towards achieving a greater innovation of our products and services by focusing our efforts on researching, developing, validating and commercializing new diagnostic and research solutions in Flow cytometry.\nWe work towards expanding our company into new working areas, applications and markets.\nWe guarantee our commitment to the continuous improvement of our products and services in order to optimize the production, increase its quality and reduce delivery times.\nWe consider quality as a transversal methodology in all of our activities.\nWe are committed to meet the requirements of the quality management system and continually enhance its efficiency through our processes reassessment and data analysis to identify areas of improvement.\nWe are committed to spread this Quality Policy among our employees, interested agents, customers and suppliers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The software offers a handful of automated reports that can be configured to send post-email to any one administrator or all administrators. Go to Settings > Account Management >General Account Settings > and click the Alerts tab on the left to view what's available.\nThe following automated reports must be enabled at the account level and at the administrator level. Any administrator with settings privileges can enable on the individual admin records.\nInsights - An automated weekly account information report that includes open rates, successful subject lines, average click-to-open rates, most successful content and more. Each week (4 total) is a different topic.\n- Can turn off completely for the account.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's about time someone finally says it like it is. A culture that sees nothing wrong with a 70% illegitimate birth rate, and an obvious lack of a strong, hard working father figure in the home only leads to kids growing up without guidance or limits of any kind. The result is what we are seeing the last few weeks in Chicago, and it's not going to change any time soon until the basic family unit is brought back to the forefront. This will take at least a generation if progress started today. Unfortunately this won't happen, because when Bill Cosby speaks to black leaders and tells them the same thing, they get offended and call him a \"cracker\". These people are in denial, and we will all suffer for it.\nStrip Burke and Daley's body guard details. They probably make up a good part of a districts manpower needs. Tell them to bring their HKs to the streets. Daley is a buffoon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today closed the 7th-12th Place Round in Monteria. Australia beat South Africa, Puerto Rico clunched the eigth place with its victory over Netherlands; and in the last game of the U-23 World Cup in Monteria Colombia defeated Czech Republic.\nAustralia finished its participation in the II U-23 Baseball World Cup this morning with a victory against South Africa, 9-4, and closed the Round for the 7th Place with a 3-2 record.\nSouth Africa took the lead in the first episode with a single by Rowan Ebersohn and a sacrifice fly by Brenton Faulds.\nHowever, in the bottom of the second, a two-run homer by Jordan McArdle tied the game. This was his third home run of the tournament.\nIn the bottom of the fourth Australia added three more runs, including a homer by Jack Barrie; and in the seventh they scored four times, with a three-run homer by Ulrich Bojarski. This was the fourth homer of the right fielder, who until now leads the home run stats with 4 in the tournament.\nSouth Africa had discounted in the top of the seventh with a defensive error, and in the ninth period the African team scored for the last time in the competition, with a sacrifice fly from Tlou Ngoepe.\nThe winner was Christopher Horne (5 IP, 4 H, 2 R, 1 BB, 2 K), while Enric\u00e9 Morton was charged with the loss.\nPuerto Rico beat the Netherlands in their last game of the U-23 Baseball World Cup, and with this result finished in eighth place.\nNetherlands designated hitter Ruar Verkerk put his team ahead in the top of the first inning with a three-run homer to right field. Francinel Martina and Sherten Apostel had reached with consecutive singles against the deliveries of the starter Joe Alsina, before Verkerk walked around.\nThose were the only three runs that Alsina allowed in six innings of work, in which struck out five, receiving eight hits. The right hander took the win.\nOn the Puerto Rico side, Isan D\u00edaz discounted quickly. After seing two balls and one strike, he swung to the ball and sent it over the right field fence, putting the game 3-1.\nThe tie came two innings later, in the bottom of the third, with a single from Emmanuel Rivera and a fielder's choice by Gabriel Cancel. And in the fifth Puerto Rico exploded with six runs to put the game to safety, including the second home run by Diaz and a two-run home run by Jay Feliciano.\nThe last two runs of the tournament for Puerto Rico were scored in the eighth inning, with a single by Jos\u00e9 Layer and a rolling by Rivera.\nThe loss went to Misja Harcksen, who lasted four innings and received six runs and nine hits, with 4 strikeouts and one walk.\nColombia got its third win in a row in the city of Monter\u00eda tonight and closed its participation in the U-23 Baseball World Cup. Tonight they defeated the Czech Republic, 12-2, in eight innings.\nThe European team had a total of six defensive errors throughout the game, while the host added 15 hits to easily win the game. Between the second and fifth innings Colombia scored ten times, taking control of the match and the result.\nThe only two runs of the Czech Republic were scored in the sixth episode, and served simply to keep the game going for a few more innings, since two frames later the Colombians scored the two runss needed to win the match by run difference.\nThe last play of the World Cup in Monter\u00eda was a defensive error of the Czech team, which allowed Jes\u00fas Posso to score from third.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabmte b/data_all_eng_slimpj/shuffled/split2/finalzzzabmte
new file mode 100644
index 0000000000000000000000000000000000000000..a03be358751142036ded7a4ffe3d1538ce3ddf60
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabmte
@@ -0,0 +1,5 @@
+{"text":"\u2013 In finance, a loan is the lending of money by a number of persons, companies, or other entities to other individuals, companies etcetera. The recipient (i.e. the borrower) incurs a financial debt, and is generally liable to pay desire on that credit card debt right until it can be repaid, and also to repay the principal total borrowed.\nThe desire offers an incentive for that lender to interact in the loan. Within a authorized financial loan, Every single of these obligations and restrictions is enforced by deal, which may also spot the borrower below supplemental restrictions called financial loan covenants. Even though this short article focuses on monetary loans, in practice any material object may be lent.\n. Acting being a provider of financial loans is one of the principal pursuits of financial establishments such as financial institutions and credit card organizations. For other institutions, issuing of credit card debt contracts like bonds is an average source of funding.\nIt's a broadly-held false impression that those with very good credit have a better grasp of controlling their funds. In a very latest consumer survey from Marcus by Goldman Sachs\u00ae*, 34 percent of individuals with very good credit rating (FICO 660 and earlier mentioned) are in credit card credit card debt and 41 percent look at credit card credit card debt as a region they wished that they had additional Command about. Even These with fantastic or great credit history \u2014 a score of 660 FICO or above \u2014 could be liable to the cycle of high-interest bank card financial debt that includes variable desire premiums and hidden fees. It's ample to help make any one experience like financial debt resolution is from reach.\nFor almost all of the population, it's much simpler to go into credit card debt than it is to receive from it. According to NerdWallet's 2018 American Residence Bank card Personal debt Research, the common domestic with charge card financial debt has balances totaling $16,748**. Every day charges and unpredicted fees like braces for the kids, a leaky roof or a incredibly hot water heater breaking can have an impact on everyone. For lots of, it's easy to slide at the rear of and vacation resort to high-desire charge cards as an answer.\nDebt, specifically bank card financial debt, is often deeply annoying for persons carrying a seemingly infinite equilibrium thirty day period more than month. This pressure, coupled with the deep-rooted stigma about referring to charge card debt, leaves Lots of individuals feeling like they have minimal selections that can help them superior handle their funds.The customer study by Marcus by Goldman Sachs also uncovered astonishing quantities encompassing the struggles Americans with fantastic credit have with bank card personal debt.\nMany are unaware of other choices that can help deal with their charge card personal debt and switch to progressively Determined measures for assistance. 30-4 p.c would take into consideration advertising off home products to assist with payments and 38 percent would look at turning to the next occupation.Having said that, you'll find other methods which can help eliminate the revolving cycle of credit card credit card debt.One viable option is a set-level, no price particular loan, offered from Marcus by Goldman Sachs. Loans range between $three,five hundred to $thirty,000 to eligible customers with superior or outstanding credit (FICO 660 and previously mentioned).\nBank loan terms assortment for 3 to six a long time and fascination prices range from 6.ninety nine % to 23.ninety nine p.c.Marcus offers U.S.-primarily based focused bank loan specialists who supply live, individualized assist. It's also possible to find a price savings calculator to assist you estimate your price savings in excess of higher-fascination variable charge card debt with no affecting your credit history rating.\nSo, explore your personal debt management alternatives \u2014 whether or not you ought to consolidate personal debt, or simply require a bank loan that works for you personally.*The Marcus by Goldman Sachs Personal debt Survey was done amongst November nine and November sixteen, 2016 amongst one,000 nationally agent People in america ages 22 and about, utilizing an e-mail invitation and an internet study.**NerdWallet's 2016 American Residence Charge card Financial debt Analyze relies on data through the Federal Reserve Lender of The big apple as well as the Census Bureau.\nPeople that be concerned about financing a college education for kids or grandchildren have acquired a good amount of organization. Much more than 70 p.c of mom and dad with youngsters underneath eighteen mention that shelling out for college is their leading monetary worry, In keeping with Gallup's 2001-2018 Financial and Personal Finance survey.\nYoung Grown ups are worried far too. Asked by Gallup to identify the very best monetary problem struggling with their people, more than one in five respondents aged 18 to 29 explained it was having to pay tuition or higher education financial loans.\nThe good thing is, there's a method to set revenue apart for education in an investment decision account the place the financial savings can grow over time, cost-free from federal profits taxes. What's more, the scholar doesn't really have to pay federal income taxes on withdrawals through the account, provided that the money is used for certified increased training costs, which contain tuition, guides, costs, materials and other approved charges at accredited establishments.\nFlexibility can be a vital feature of 529 strategies. Mother and father and grandparents maintain control of the account, come to a decision when and if to disburse the proceeds and keep the opportunity to change the beneficiaries. One example is, if the child initially named as beneficiary doesn't have to have The cash or doesn't go to college, the account beneficiary is often altered to another loved one who may benefit. Account house owners can even use the money for their particular qualifying educational expenses to acquire a graduate degree, As an illustration, or specialised vocation instruction.\nEveryone can open a 529 college or university cost savings strategy, despite earnings, and can add up to $fourteen,000 ($28,000 for married partners) yearly without reward-tax consequences, and that money isn't thought of Portion of the account owner's estate.\nGiven that 529 options are very long-time period financial commitment cars, it's important to opt for a program which offers a variety of investment choices and verified management experience, states Spazafumo, who notes that CollegeAmerica features many alternatives with the very rated American Cash mutual fund loved ones.\nFunding larger education and learning needs saving constantly more than the long term, nonetheless it's not impossible.\nDon't just is borrowing extra high-priced than preserving your way to school, Spazafumo points out that there's evidence lots of students aren't ready to sustain with their financial loan obligations. In accordance with the Division of Training, borrowers who had been owing to get started on repaying their student loans in 2011 had a thirteen.7 % default price last calendar year, Consequently harming their credit rankings equally as they were commencing their Functioning life.\n. Schooling has normally been the pathway to better incomes and task gratification, Which's more likely to be truer than in the past from the a long time to return. Combining a disciplined price savings system with the correct 529 faculty cost savings strategy can help make your relative's desires of school a fact.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The purpose of PolyCharms is to give the polyamory and non-monogamy community a venue for expressing their individual style through symbolism, wrought in metal and stone. While embracing the fact that polyamory themed jewelry is fun, fanciful, and unifying, a PolyCharm also aims to capture the spirit of your love story.\nIn this gallery you will find everything from widely recognizable polyamory symbols, to custom pieces designed with shapes and themes inspired by individual circumstances, all the way to special sets made for you to share with your loved ones as symbols of your unity.\nPlease browse around, see what grabs your attention, and think about what speaks to you.\nCustom creation is the pride and joy of PolyCharms, so please reach out to discuss with me any ideas, designs, or pieces you want made just for you or your loved ones here. Never hesitate to contact me at info@polycharms.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Donna M. O'Laughlin, a former first lady of Niagara Falls, died Thursday in Lakeside Beikirch Care Center in Brockport after a brief illness. She was 80.\nBorn Donna M. Mann in Niagara Falls, she graduated from Niagara Falls High School, earned a degree at a business school in the city and worked as a secretary at Carborundum Co. for about three years before her marriage.\nHer husband of 47 years, Michael C., was mayor of Niagara Falls from 1976 to 1991. He died in 1996.\nHer daughter-in-law, Linda, said Mrs. O'Laughlin was extremely proud of her role as the mayor's wife.\n\"She just loved greeting dignitaries from other countries, as well as being close with the people of Niagara Falls,\" she said.\n\"We found a great picture of her greeting the president of China, the first time he had been in this country since communism [took control of China]. She loved meeting [singer] Anne Murray and all the great people who used to come to the Convention Center.\"\nMrs. O'Laughlin also was a member of the St. Francis Guild and worked in the gift shop at Mount St. Mary's Hospital, Lewiston.\nShe had been an active member of St. Teresa of the Infant Jesus Catholic Church, where she served on the bereavement committee. She also enjoyed gardening.\nSurvivors include a son, Michael C. of Brockport; two daughters, Rene of Boston, Mass., and Katie of Pacific Palisades, Calif.; a sister, Geraldine Furey of Niagara Falls; five grandchildren; and a great-grandson.\nA Mass of Christian Burial will be offered at 10 a.m. Monday in St. Teresa of the Infant Jesus Church, 3840 Macklem Ave., following prayers at 9:15 in Otto Redanz Funeral Home, 2215 Military Road, Town of Niagara. Burial will be in Gate of Heaven Cemetery, Lewiston.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's safe to say that nearly all of us understand why DOT regulations regarding drug and alcohol use exist. But it's also safe to wonder exactly what drug and alcohol screening entails. This blog will help you understand the U.S. Department of Transportation's rules and regulations regarding drug and alcohol testing for truckers and how HDS Safety and Compliance can help your company stay safe and compliant.\nWhat Does the Drug Test Entail?\nThe Department of Transportation only recognizes urinalysis (urine tests) as a valid form of drug testing, and although many drug compounds can be out of a person's system in a matter of days (like cocaine), drug screens still prove to be a major deterrent.\nWhy Are Truckers Drug Tested?\nBefore a trucker is able to work as a truck driver, they must undergo drug and alcohol screening. Once they're hired, truckers are often randomly tested, meaning that they are ordered to report to a testing location within a given amount of time, regardless of whether or not they are on-duty or off. In addition, all truck drivers are required to be tested for alcohol within 8 hours of a fatal accident as well as drug tested within 32 hours. Finally, the law allows for \"reasonable suspicion\" tests which are triggered by either the behavior, speech, or even smell of the trucker (and obviously the smell in question isn't garlic, it's liquor!) If a trucker fails a test, they must pass a series of \"return to duty\" tests, but it's safe to say that a negative test can easily place a trucker's career on the road in jeopardy.\nAll of these regulations and opportunities for testing make it extremely difficult for a person with an alcohol or drug problem to be behind the wheel of a 40 ton semi. The stringent regulations, in addition to the efforts of companies like HDS Safety and Compliance, make our roadways safer without the average driver even knowing about it. If you want to learn more about how drug and alcohol screening for your company, call us at 1-800-800-7801 or contact us on the web today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the long-term benefits of owning real estate, including the single-family home you raised your family in, is the prospect of leaving your property to your children and grandchildren. Passing your success and hard work to the next generation is why we all work so hard. Estate planning is something everyone who owns real estate should think about, not just the wealthy.\nTo ensure that their final wishes are honored, people have made itemized wills for centuries. But the value of your real estate on the day you make your will (of sound mind and body) is often not the value of the property at the time your beneficiaries take possession. You have to plan how your estate will be handled when you're not around.\nIf you're planning your estate, you may want to set up a trust fund for your heirs based on the value of your estate property. Trust funds are another financial instrument many people think only the very wealthy have to worry about, but you can set up a trust fund based on the value of your single-family home that your children can use for their future.\nBecause estates and trust funds are based on the value of your real estate, you need an accurate appraisal. The certified appraisers at A-1 Express Appraisal understand how estate and probate law apply to real estate. We can help you evaluate your options with an objective appraisal of your property's real market value.\nA-1 has the answers to all your real estate appraisal questions. You can even order your appraisal online, but speaking to one of our certified appraisers gets even faster results. Contact A-1 Express Appraisal today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadjco b/data_all_eng_slimpj/shuffled/split2/finalzzzadjco
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@@ -0,0 +1,5 @@
+{"text":"8 events found on Alexander's timeline.\nLuxury Motor Yacht ALEXANDER truly is a megayacht unlike many others in her class. Built in 1965 and converted in 1985, ALEXANDER was one of the earliest private yachts to breach the 100m barrier and achieve megayacht status. The vessel's interior was refitted in 1998 with a more recent upgrade to the master cabin completed in 2008. Continuously well maintained, ALEXANDER retains many of the original features from her 80's & 90's d\u00e9cor.Until a few years ago, ALEXANDER was an intimately private yacht hosting numerous VIP guests including monarchs and heads of state. Today she is the world renowned vessel of choice for charter parties and corporate events of up to 80 guests, whilst still very well laid out for smaller groups and family use. She is a very fuel economic vessel and runs with crew costs well under what is usually encountered for a yacht of her size.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"IObit Malware Fighter Review and Deals find all this on this post. Know why you need Iobit Malware Fighter to protect your PC.\nNow, most of us use the Internet to do a lot of online activities like reading emails, downloading software, surfing the website, etc. All these activities increase the risk of virus, spyware, Trojans, and malware, etc.\nSo, to protect your computer system, you require competent software that can quickly detect and remove viruses, Trojans, spyware, and malware.\nIobit Malware Fighter is one such Anti-Malware Software that can guard your computer system from all these potent threats.\nHere, we have come up with Iobit Malware Fighter Review that throws light on each and every aspect of this competent and powerful Anti-Malware Software.\nEveryone must have heard the term viruses which are a threat to the computer. Well, along with viruses, malware are too a big menace in today's e-world.\nThese programs are poorly programmed and can cause the system to be unreliable slow and unstable. They even redirect your search attempts on the web browser, keep track of what you have been doing and create nasty pop-ups.\nHave you ever noticed how when you download some third party software from the internet sometimes an add pops up (like in Kazaa and iMesh)?\nWell, these are bundled malware where the revenue from the ads goes to the author of that software.\nMalware can also be installed through websites where they ask you to download some software so that you can view that site etc. All these methods are to trick users and infect their computers with malware.\nThe worst part about getting infected by malware is that once it installs on your computer, it tends to multiply itself hence making matters worse.\nSpyware: These programs send your confidential information to others.\nHijacker: These take control of various parts of the web browser.\nDialers: These connect to a 1-900 number, leaving you with heavy phone bills.\nSo to protect your computer from these threats and particularly malware, you need software which can identify these threats and stop or delete them from your computer.\nIObit Malware Fighter is one such powerful software which can protect your computer from malware.\nIObit Malware Fighter is powerful PC protection software. It can detect and remove all kinds of spyware & malware. IObit Malware Fighter guards your PC against dangerous threats like Trojans, viruses, keyloggers, adware, bots, hijackers, and worms.\nIt is capable of removing deepest infections from your PC. IObit Malware Fighter uses it \"Dual-Core\" engine and employs heuristic malware detection to find complex malware and spyware in a very quick and efficient manner.\nIobit Malware Fighter comes as a no-nonsense Anti-Malware Tool and goes on to protect your PC in several different ways.\nIobit Malware Fighter goes on to provide real-time protection. It rigorously monitors your cookies, startup programs, browser settings, the web page pages you view, as well as the files you access. Its user interface is neat and clean. It is easy to use and navigate.\nIobit Malware Fighter can be installed along with your Antivirus Software. The combination of both software will give full protection to your computer.\nIt functions as an anti-spyware, anti-malware, anti-Trojan, anti-adware, anti-bots, and much more. IObit Malware Fighter helps to protect your PC from any complex and tricky threats.\nIObit Malware Fighter uses Digital Original Gene (DOG) malware detection process to find even the most complex security threats.\nIt supports Windows 10, 8, Windows 7, Vista, XP and 2000.\nIObit Malware Fighter comes in a free as well as a paid version known as IObit Malware Fighter Pro. The Pro version can be used if you are eyeing for deep and accurate scans.\nIt is also useful for running scheduled scans, protection for USB Viruses, process scanning and even for technical support. It comes with enhanced features to give you complete protection from malware and spyware.\nIObit Malware Fighter Pro also has some underlying features which are hard to miss. Like finding the deepest infections, real-time protection, use of creative cloud computing technology, compatibility with other software's and also automated working in the background.\nEven if you have an anti-virus installed on your system, it might miss out on detecting some malware, and IObit Malware Fighter Pro promises to take care of that problem.\nIObit Malware Fighter Pro has been designed to function as a complete security package. Besides providing basic Anti-Malware function, IObit Malware Fighter Pro has been empowered with full Anti-Malware ability.\nIt provides advanced threat protection and comprehensive real-time protection for full PC security. It prevents virus infections, a malicious process running in RAM, and detects threats by analyzing malicious actions. It works intelligently in the background without disrupting your normal working.\nEverything about this software is created keeping in mind the ease which a user desires while accessing their PC. With simple and straightforward controls and one-click protect options, it's a breeze for anyone to use IObit Malware Fighter and get their computers protected from all kinds of malware.\nNow, you have a wonderful opportunity to get the advanced protection of IObit Malware Fighter Pro at a never before price.\nA fabulous discount offer is available on IObit Malware Fighter Pro using which you can get IObit Malware Fighter Pro at a heavily discounted price.\nThe IObit Malware Fighter Pro Discount Offer is available for a limited period. So, you should quickly avail the discount offer through this amazing IObit Malware Fighter Pro Deals before it expires.\nSimply use the coupon code listed below and get an amazing discount on IObit Malware Fighter Pro. You can give your computer the best protection that it deserves and that too at a huge discounted price.\nSo, hurry and get the discount on IObit Malware Fighter Pro.\nPlease Note you are buying this software directly from Iobit website not from any third party website.\nThe free version is free for life. Paid version is for one year or 3 years update after that there will be renewal fee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This booking request will be made directly with Kamari Beach Hotel and not with Go-Thassos.\nAll further communication regarding this request will be conducted with Kamari Beach Hotel.\nKamari Beach Hotel may require a booking deposit to finalize your booking.\nGo-Thassos makes it easy for you and Kamari Beach Hotel to connect and do business, but has no say nor holds any responsibility for any business conducted or transacted by you and Kamari Beach Hotel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy Now!, Fly Now and Save Now ! with AirAsia Airlines Australia Promotions January 2016 for a direct flight from Darwin Australia to Bali Indonesia as low as AUD 99.\nGrab more offers to fly from Perth Australia to Kuala Lumpur Malaysia this January 2016 with a direct flight (one way) tickets at AUD 199. First come first serve and feel free to subscribe with AirAsia Credit Card if you need financial backup assistants during travel.\nAirAsiaGo Australia Promotions January 2016 provide you with a better deals for a return flights tickets and tax for four (4) nights if you choose to stay in Bangkok, Phuket, Krabi, Kuala Lumpur, Penang, Jakarta and Bali with the lowest cost at AUD 399.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Caution: Federal law restricts this device to sale on or by the order of a physician. The trademark of HEMOBLAST\u2122 Bellows and SPOT GRADE\u2122 are the property of Biom'up. The trademarks of the products listed herein are trademarks of their respective manufacturer. Please refer to the Instructions for Use accompanying each device for further information. The presentation contains pictures owned by third party and remains their property.\nIntended for US professional audiences only.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaepvm b/data_all_eng_slimpj/shuffled/split2/finalzzzaepvm
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaepvm
@@ -0,0 +1,5 @@
+{"text":"The public will get a chance to see how $1.54 million in tax dollars was spent at the unveiling of the new Wood River High School Performing Arts Theater on Wednesday, April 25, at the Community Campus in Hailey.\nThe theater is an extensive remodel of the old auditorium at the Community Campus, which is owned by the Blaine County School District. The school board approved the work in 2011 and it was completed this spring. Funding was through a 10-year, $59.8 million plant facilities levy approved by county voters in 2009.\nFestivities for the unveiling are scheduled to start at 5:30 p.m. with an open house and tours of Community Campus tenant facilities, including those of the Blaine County Recreation District, Footlight Dance Center, College of Southern Idaho, Head Start and the Sun Valley Summer Symphony. Entertainment and refreshments will be provided.\nThe theater opens at 6:30 p.m. A ribbon cutting is scheduled for 7 p.m., followed by performances by the Wood River High School band, mixed choir, drama department and orchestra.\nThe remodel involved installing new carpet and seats, building a lobby area, adding catwalks for access to lighting systems, installing an orchestra pit, improving stage access, building makeup and dressing rooms and installing new sound and lighting systems.\nThe theater, which will be used by both student and public performing arts organizations, is now the largest indoor performing arts facility in Blaine County.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The elves have been busy planning this year's Christmas in Kirkland, which will take place on Saturday, December 8th beginning at 3:00 P.M.\nThis year's event will feature a Craft Fair in the pavilion at Franklin Township Park (Craft Fair will begin at 2:00 P.M. and run until 6:00 P.M.), face painting, Story Time in the library, free hot dogs\/popcorn\/cookies and hot drinks and of course Mr. and Mrs. Santa Claus!\nThe event will run until 6:00 P.M. at which time we will announce the Home Decorating contest winners, Hiawatha Essay\/Art Contest winners, Light Pole decoration winner and conclude with Lighting of the Village Christmas Tree!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\" For those of you who don't know me that well, I am an artist. I started on stage at 4 years old, became a professional ballerina, then a signed pop artist and then a signed rock artist fronting the trans media project, September Mourning. I am a published fashion model who has been featured in campaigns here and abroad. I developed and successfully published my own on going comic book series through Image\/ Top Cow Comics. I've toured throughout Europe, Canada and the United States. \"\n\" Being signed to record label is an amazing thing. In today's age though, a lot of artists are reaching out to their fans to help them along in their journeys financially. Patronage to the arts has been something that has existed for a very long time and establishes a connection between fans and artists that is more personal because the fans actually enable the artist directly. They become a direct inspiration for the creation process because of their support. \"\n\" Lately I've been wanting to expand into other artistic areas besides music....but in order to achieve these goals I need your patronage. I'm asking for your help here so that I may be able to create more music, conceptual visual and video art, wearable art, etc... some of which may be linked to my current visual\/ musical art project, September Mourning, and some of it will be outside of that.... \"Bucket List\" art projects that I have always wanted to create but need a little help in doing so. \"\n\" Please take a minute to look at the right side of the page and find the support level that you are excited about and pledge ;) \"\n\" Every dollar really helps me achieve my goals, in and outside of the studio. \"\n\" Thank you all for deciding to come on this journey with me... I'm excited to see what we create together! \"\nTo visit Emily's Patreon page, click here. To become a patron of Emily's click here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Outlook Backup enables you to export Microsoft Outlook files and settings fast and simple.\nThe program can easily export Outlook data to another PC or laptop, including emails, contacts, settings, accounts and other important data.\nOutlook Backup enables you to export Outlook files and settings to another computer. You may easily move all your Outlook data to another PC or laptop, export Outlook settings and accounts (with passwords) in order not to configure Outlook again on another computer.\nBackup all the needed Outlook data on the first computer. Run Outlook Backup, select Outlook items you are going to export, choose name of backup file and hit Backup button.\nCopy backup file to another PC or laptop. Use some media, flash drive or network to move the file to the second computer.\nRun Outlook Backup and restore Outlook files and settings from backup file you have just moved. Note that all existing Outlook data will be overwritten.\nYou can even export Outlook automatically and periodically using Task Builder feature of Outlook Backup - it's very useful when you need to synchronize Outlook between several computers.\nSee also how to export Outlook contacts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Come July this year I am moving to Cambridge! So of course, a reading list is in order. Campus life is deliciously studious and mysterious so I have decided to include books from Oxford as well.\nResidence at Cambridge by William Wordsworth: The third chapter to Wordsworth's The Prelude is about his time at Cambridge Univeristy. On a dreary morning that matches the poet's mood Wordworth's ponders academia and the pursuit of knowledge.\nThe Humans by Matt Haig: A Cambridge maths professor has solved the Riemann hypothesis, incidentally, Extra Terrestrials see this as a threat to the universe and and alien impostor comes to earth in order to destroy all evidence. Curious Incident meets Hitch-hikers Guide, the alien has a rough time figuring out how to be human, and pondering the ugliness of noses.\nNorthern Lights by Philip Pullman: Alas, the Oxford in this book is an alternative Oxford, it is the same, but different. Dedicated to discovering the nature and meaning of the elusive 'Dust' this Oxford is as picturesque and intriguing as the real one.\nAtonement by Ian McEwan: The deliciously snobby and prim Cecilia spends the summer following her time at Girton College idle in her family mansion, smoking, reading, being confused about her feelings, and accidentally receiving dirty letters.\nA Discovery of Witches by Deborah Harkness: The Bodleian Library is almost a character in this novel as it presents our protagonist Diana (mistress of the moon) with a lost book. Diana comes from a long line of witches, but she is spellbound. The arrival of the bewitched book forces her to accept that there is a stirring in the magical world.\nH is for Hawk by Helen Macdonald: I read this while staying in a tiny cabin in the middle of nowhere when winter was slowly replacing autumn. The memoir of a Cambridge Scholar who trains a goshawk while grieving after the death of her father. This book made me want to begin falconry, maybe while in Cambridge I might just do that.\nIt is SO exciting! I'll be teaching \u2013 my husband is doing his PhD at Cambridge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaewjk b/data_all_eng_slimpj/shuffled/split2/finalzzzaewjk
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index 0000000000000000000000000000000000000000..ec259f2a9630f7bf4944e22a23bd31bc23df00e0
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@@ -0,0 +1,5 @@
+{"text":"Although Becky Sharp, the ambitious heroine of William Makepeace Thackeray's 1848 novel Vanity Fair, is not usually thought of as innocent or na\u00efve, the first recorded use of \"ingenue\" in English does refer to her. Thackeray's use was attributive: \"When attacked sometimes, Becky had a knack of adopting a demure ingenue air, under which she was most dangerous.\" The word ingenue typically refers to someone who is innocent to the ways of the world, so you probably won't be too surprised to learn that it shares an ancestor, Latin ingenuus, with \"ingenuous,\" a word meaning \"showing innocent or childlike simplicity and candidness.\" More directly, our \"ingenue\" comes from French ing\u00e9nue, the feminine form of ing\u00e9nu, meaning \"ingenuous.\"\nIn her latest film she plays the part of an ingenue.\nThese example sentences are selected automatically from various online news sources to reflect current usage of the word 'ingenue.' Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors. Send us feedback.\nLookups for 'ingenue' spiked following the death of Debbie Reynolds.\nWhat made you want to look up ingenue? Please tell us where you read or heard it (including the quote, if possible).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Learning to control and adjust your sewing machine tension will empower you to sew creatively. Before making any adjustments to the tension on your sewing machine, be sure that the machine is properly threaded. The thread in machine must feed through three main points on the sewing machine; the tension disks, the take-up, and the needle. The bobbin must also be threaded properly to enable the machine to form a good stitch. Bobbins have their own mechanism for controlling tension on the thread. Follow your sewing machine instructions book for proper threading.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nearly a month after the hurricane, Puerto Rico still is still struggling, thanks to the federal government's underwhelming early hurricane response.\nAmerican Airlines confirmed a flight from California to Phoenix was canceled for an unusual reason -- a scorpion was spotted on board the plane.\nPresident Donald Trump signed a presidential disaster declaration to make federal funds available to Texas as Hurricane Harvey bears down on the coast.\nOn March 18, In 1937, a natural gas explosion at a public school in New London, Texas, killed almost 300 people, most of them children.\nTexas Gov. Greg Abbott declared nine counties a 'disaster area' as the Gulf region continues to take punishment from deadly thunderstorms and flooding Monday.\nSenate Republicans on Thursday reiterated their opposition to President Barack Obama nominating a judge to replace Antonin Scalia on the U.S. Supreme Court.\nWhile campaigning for his younger brother, former President George W. Bush told supporters they need a president who has \"integrity and character.\"\nMore firearms were confiscated from travelers by the TSA at American airports in 2015 than in any other year on record, officials said Thursday.\nA Spanish TV reporter apologized on Twitter after confusing the Rebel Alliance symbol from 'Star Wars' for an al-Qaida logo.\nHouston Rockets center Dwight Howard was found to be toting a handgun in his carry-on bag last Friday at Houston's George Bush Intercontinental Airport.\nLess than two weeks after being released from the hospital, former President George H.W. Bush indicated his recovery is progressing well.\nThe commemoration of the 1965 Selma-to-Montgomery Voting Rights March came days after the Justice Department's Ferguson, Mo., report on apparent police racism -- highlighting the current race issues the nation still faces.\nThe Department of Homeland Security discovered 2,212 firearms airport travelers' carry-on bags in 2014, marking a 22 percent increase from the previous year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 7197 SkyBox Pro18 Onyx is the newest box in the SkyBox line. Perfect for large SUV's, this box dishes up 18 cubic feet of storage while barely making itself noticeable. Great for weekend camping trips, this box will hold enough gear for 4 campers or 6-8 pairs of skis or snowboards of any length. The new SkyBox Pro series boxes feature a super-sturdy lid, a re-designed latch system, and a cargo net and mat to secure your gear.\n18 cu ft (92\" x 36\" x 16\") - Weight: 52lbs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some pressures are internal. Mahoma's Puerto Rican wife worries that her husband's exposure might get him deported to Mexico. After the employees take a vote under the supervision of the National Labor Relations Board, the drama ramps up considerably. Workers are further pitted against one another; strategies evolve for both labor and management, particularly when this Hot & Crusty location is scheduled to shut down.\nThe directors, Rachel Lears and Robin Blotnick, make clear where they stand. (A title card says that the restaurant's owners declined to be interviewed.) Portraying the bakery as a microcosm of America, the documentary proceeds in a straightforward, earnest manner. \"The Hand That Feeds\" is an effective portrayal of the intricacies of activism \u2014 and of a situation in which victories seem all too brief.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaftms b/data_all_eng_slimpj/shuffled/split2/finalzzzaftms
new file mode 100644
index 0000000000000000000000000000000000000000..737cc68c978b1bb50596eb68dfdea47840e89706
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"I used this phrase to describe South Carolina, as well, but Arkansas is also \"consistently mediocre. Arkansas was consistently ranked higher than 40 in the raw wage data, but they were saved, at least in terms of this countdown, by the fact that they have the second lowest cost of living in the country. Adjusting for the cost of living raised the state's ranking 13 spots and kept it from being in the bottom 10.\nIn News Tags construction, best states to work in construction, top states to work in construction, states that pay construction workers the most, which state pays the most for construction workers, best construction jobs to work in Arkansas, worst construction jobs to work in Arkansas, how much do construction workers get paid in arkansas?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the very unlikely event that you are unhappy with the coaching services from Powerful Eric then you can get a no questions asked full refund. Powerful Eric want you to be happy!\nSimply e-mail eric@powerfuleric.com that you are not satisfied with his coaching and you will receive a full refund!\nIf you have received a defective item, you must return the item to us within 60 days of invoice. You must return the product in its original packaging, complete with all components and documentation. We will repair, replace or refund your purchase. After 60 days, the terms of the corresponding manufacturer's warranty, if any, will apply, in which case you may be required to send the product directly to the manufacturer's processing or repair facility.\nIf we accidentally sent the wrong item, we will exchange it for the correct item, provided that you return the product unopened, undamaged and accompanied by all components within 60 days of invoice.\nCompany is not responsible for any damage or loss of any kind due to your use of any of the products we sell. Our maximum liability to you is the amount you paid for the product.\nWe do not ship physical products at this time. Right now all products are downloadable. However if we do we will offer several shipping methods. We use major carriers, such as UPS and FedEx knowing that they ensure the quickest, most cost-effective delivery to your zip code. Please choose the desired method to match your needs and budget!\nCompany guarantees that you will never be charged more than the actual cost of delivery (a $60.00 handling fee is added to all orders) for an order shipped to you based on the shipping method selected and the costs to Company charged by the shipper which are in effect at the time your order is shipped.\nInternational orders are subject to additional handling charges, higher shipping rates, and applicable customs, duties and taxes, all of which are the responsibility of the customer. All international orders are to be paid in full by wire transfer or international US $ money order unless other arrangements are made with Company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This elegant tray is easy and convenient, bake and serve in one dish. This perfect dish has the elegance of the Fantasia line with the exclusive pressed flowering vine design in clear crystal. This dish is 15\"L x 13 1\/4\"W, making it perfect for heating appetizers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Author: Wieber, Frank; Th\u00fcrmer, J. Lukas; Gollwitzer, Peter M.\nThe present review addresses the physiological correlates of planning effects on behavior. Although intentions to act qualify as predictors of behavior, accumulated evidence indicates that there is a substantial gap between even strong intentions and subsequent action. One effective strategy to reduce this intention-behavior gap is the formation of implementation intentions that specify when, where, and how to act on a given goal in an if-then format (\"If I encounter situation Y, then I will initiate action Z!\"). It has been proposed that implementation intentions render the mental representation of the situation highly accessible and establish a strong associative link between the mental representations of the situation and the action. These process assumptions have been examined in behavioral research, and in physiological research, a field that has begun to investigate the temporal dynamics of and brain areas involved in implementation intention effects. In the present review, we first summarize studies on the cognitive processes that are central to the strategic automation of action control by implementation intentions. We then examine studies involving critical samples with impaired self-regulation. Lastly, we review studies that have applied physiological measures such as heart rate, cortisol level, and eye movement, as well as electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) studies on the neural correlates of implementation intention effects. In support of the assumed processes, implementation intentions increased goal attainment in studies on cognitive processes and in critical samples, modulated brain waves related to perceptual and decision processes, and generated less activity in brain areas associated with effortful action control. In our discussion, we reflect on the status quo of physiological research on implementation intentions, methodological and conceptual issues, related research, and propose future directions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An osteoblast (from the Greek words for \"bone\" and \"to build\") is a cell that secretes a protein called osteoid which, when it mineralizes, becomes bone. Osteoid is primarily composed of collagen.\nBone is a dynamic tissue that is constantly being reshaped by osteoblasts, which are responsible for the production of matrix and mineral, and osteoclasts, which break down the tissue.\nWhen there is insufficient mineral or osteoblast dysfunction, the osteoid does not mineralize properly, and it accumulates in a process called osteomalacia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaisij b/data_all_eng_slimpj/shuffled/split2/finalzzzaisij
new file mode 100644
index 0000000000000000000000000000000000000000..60fb750f86e894e75419a4e385c01cf1f45391eb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaisij
@@ -0,0 +1,5 @@
+{"text":"Your hard drive will perform adequately providing the hardware are adequately represented through the ui, enabling you to regulate and configure every little thing according to your wants. If you ever attempt to proceed through manual setup of UDISK PRO, do not forget that the windows device manager might not generally supply you with the most revised specifics of the driver module model.\nFaulty drivers could possibly damage similar drivers, so when you attempt to download and install UDISK PRO by hand be urged to seek out similar bad drivers to boot.\nOnce a driver, such as UDISK PRO, has become ruined, your entire system dependability is undoubtedly affected, resulting from quite a few failures that could develope shortly after. Sustaining a total collection of up graded drivers on your pc is almost impossible, for those who attempt to manually find and install every driver which requires bringing up-to-date. Working with a basic driver scanning program has developed into a standard procedure during the last few years.\n5 out of 5 based on 3 ratings. 1 user comments.\nHot scanner! Completed the whole of the driver scan procedure in less than 10 minutes. not only are all the services repaired now, it also looks like my internet connection speed has been enhanced, soon after managing 82 drivers, besides UDISK 28X. (it must have been affected by the bad drivers I had).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The accounting and bookkeeping are the core of any business. You can start using excel sheets or free software, but very soon you will realize, that things are getting out of hand.\nWe have experience with Simply Accounting and will setup the program for you and configure it to your needs.\nWe also have accountants that will take care of your reports.\nIf you choose to do it yourself, we will train you and provide support.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Vedic astrology or Jyotish, each sign of the zodiac is ruled or owned by a planet. Just like humans, planets are comfortable in their own space. It's the landlord residing in its own home. Venus owns the signs of Taurus and Libra. Venus returned home to Libra on September 1, 2018.\nTypically, Venus resides in a sign for less than one month. This time, however, Venus will inhabit Libra until the end of the year. Venus is the planet of love, beauty, the arts, sensuality, passion, romance, and luxury. Additionally, Venus represents the romantic partner.\nLibra also signifies the personal and professional partner. Another meaning for Libra is travels. Thus, Venus in Libra can be an ideal time for you to plan a romantic journey with your beloved. Being the planet of love, this period can bring in new romances. It can also bring the return of an old flame if you are in a Venus planetary cycle.\nOn October 5, 2018, Venus turns retrograde in Libra. This makes Venus even stronger. You should also be able to see Venus in Libra shining like a diamond in the night sky.\nVenus' retrogression can create favorable circumstances for whatever matters Libra signifies in your Vedic astrology chart. This will depend on what your rising sign is in Vedic astrology.\nFor example, Capricorn ascendants may see improvements in career, Virgo ascendants may gain more artwork, fine clothing or jewelry, while Cancer ascendants could acquire a new vehicle or redecorate their homes.\nVenus turns direct in Libra on November 16. After it turns direct, no planet other than the transiting Moon will keep it company for the remainder of the year. Thus, Venus is free to operate in the way it sees fit, without any other influences.\nVenus and Jupiter will keep company in Libra until October 11. Then, Jupiter will enter Scorpio. Venus and Jupiter are the two great benefics in Vedic astrology. This is a significant transit if you are in a Jupiter\/ Venus or a Venus\/ Jupiter cycle. This further enhances the matters attributed to Libra in your own chart.\nAlthough they are both benefics, Venus and Jupiter operate with very different agendas. Jupiter is the planet of wisdom, knowledge, teaching, higher education, discernment and spirituality. In Vedic mythology, Jupiter was the teacher of the Devas, or the Gods. Venus was the teacher to the Asuras, or the non-divine beings. The Devas followed the path of righteousness or dharma, while the Asuras pursued self-interest first and foremost.\nDo you follow the path of wisdom or give into more worldly desires? Are you impulsive like Venus or more contemplative like Jupiter? Will shopping or meditating be higher on your priority list? Issues between your spouse or romantic partner and children can also arise.\nIn Vedic astrology, when Venus, Jupiter, and Mercury are together, they form a Saraswati yoga by transit. This is a planetary combination which evokes the pure qualities of the goddess Saraswati. She is the goddess of wisdom, education, music, speech, and the arts.\nMercury joins Jupiter and Venus in Libra on October 5. While the three planets are only together for a few days, the placement of Jupiter and Venus, as well as Mercury and Venus, in Libra can still highlight the traits of Saraswati.\nYou may wish to deepen your studies, focus on reciting mantras, engage in artistic activities, or learn to play an instrument. You can benefit by chanting mantras to Saraswati.\nSaraswati is depicted wearing a white sari. She is holding a musical instrument (a veena) in two hands, a japa mala to recite mantras in the other hand, and a text in another hand. She is seated on a swan, which symbolizes purity.\nOn October 14 and 15, Venus and Mercury will be within one degree of each other. This creates a graha yuddha or a planetary war. On these days, you should be very mindful of how you converse with your romantic partner, as there is the potential for miscommunication. You can also get caught up in the inner battle between analysis and pleasure. Mercury likes to ponder. Venus wants to enjoy itself. Mercury may want to study, while Venus may desire something creative.\nMore about Venus in Libra will be written on my Facebook page.\nVenus is correlated with the Hindu goddess Lakshmi. She is the consort of Lord Vishnu, the preserver of the universe. Lakshmi is the goddess of prosperity, beauty, and good fortune. Venus is also the planetary ruler of Fridays.\nIf you are in new or existing romantic relationship, you may want to check your astrological compatibility during the period in which we have Venus in Libra. It can be a powerful time to highlight the positive aspects of your partnership and to harmoniously resolve any differences.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SRINAGAR, July 21: Notwithstanding the tall claims of the Government in streamlining the education sector in the State, more than half of the schools in Srinagar are without their own buildings and are functioning from the rented buildings, that san all basic facilities, causing inconvenience to the students.\nAs per the figures of the Chief Education Office (CEO), 345 schools in the education zones of Srinagar are running from rented buildings.\nOut of 519 schools in Srinagar (30 Higher Secondary Schools, 74 High Schools, 211 Middle Schools and 204 Primary Schools), 345 schools (50 Higher Secondary Schools, 119 Middle Schools and 176 Primary Schools) are functional from the rented buildings.\nAs per Government teachers, one of the main problems of Government schools for not performing well is the non-availability of proper infrastructure. \"Why would a parent admit his child in a school that has no space available? There are highly qualified teachers in Government schools but due to the bad condition of the schools, people prefer Private schools,\" Mohammad Haneef, a Government teacher said.\nHe said due to lack of space the students from different classes are huddled around the teachers resulting in chaos, confusion and noise. \"Teachers have to wait for their turn to teach the rest of the classes and then people accuse Government teachers for not giving attention to the students,\" he added.\nDeputy CEO Srinagar, Mudasir Kaleem Fazili, blamed the lack of space in the city for not having their own buildings. \"The non-availability of land has become a major issue for the department to provide facilities in the schools. These rented schools are functioning in congested areas in rented buildings due to non availability of land. There are some schools in the city running from just two or even a single room, leading to cramping of space and difficulties in conducted classes. The worst part is we can't even upgrade these rented buildings,\" he said.\nNusrat Jahan, Zonal Education Officer (ZEO), zone Batamaloo said the clubbing of schools is another problem that has created congestion in the schools. \"Due to less roll of students, various schools were clubbed together and as a result this created congestion in the schools. But what can we do, we are not able to identify any land in Srinagar where we can construct an ideal building for the school. There is plenty of land available on the city outskirts but we cannot shift a school from a particular area to another area,\" she said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BET-IBC \u2013 the best betting agent has been on the market for over ten years now, and in the years of experience we've had, we can only say that everything that has to do with safety and privacy of the sensitive information is protected.\nOne of the indicators that show whether an Online Casino is safe or not is the software they use. The biggest softwares (Boss Media, Cryptologi, Playtech and Microgaming) have the highest shares on the market, so they don't have any reason to risk their reputation. Most of them use 128-byte enciphering, which means that the possibility of your data to fall into wrong hands is 1 out of 340,282,366,920,939,000,000,000,000,000,000,000,000. The possibility of you getting crushed by a car while crossing the road is, statistically speaking, more.\nCan I Really Earn Money?\nDespite the fact that there's high risk involved in playing at a Casino, like in any type of betting games, there will be games you will win, and there will be games you will lose. But if you choose the right strategy, you can increase your winning chance drastically.\nHow Can I Access such Online Casino?\nThere is a lot of Online Casinos available all around the web, but if you're a newbie, we recommend that you start with someone trustworthy. For example, with you can easily open a Matchbook account with our favorite agent \u2013 BET-IBC, as well as a variety of different bookmakers that offer Online Casino.\nLet's continue with \"Online Casinos 101\"! The more information you'll get, the more successful you'll become. It's our pleasure to be your guide in this fascinating world of Online Casinos!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajohm b/data_all_eng_slimpj/shuffled/split2/finalzzzajohm
new file mode 100644
index 0000000000000000000000000000000000000000..b89ff886384129c7acb75616ec098a4b26776c0d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajohm
@@ -0,0 +1,5 @@
+{"text":"Definition - What does Metallurgical Bonding mean?\nMetallurgical bonding is the result of chemical bonding that occurs between a substrate and coating areas that are in close contact or diffused evenly. This process can be augmented by heat treatments, mainly post-spraying dispersion.\nThis method entails the joining of dissimilar metal through metallurgical means. This is normally achieved with the aid of clad metal, which is similar to one of the metals in the bond. This type of bonding is free from oxide films, voids and any form of discontinuities.\nMetallurgy is among the main domains of material science and engineering, which conducts studies of the chemical and physical behavior of inter-metallic compounds as well as their combinations, referred to as alloys. Thus, metallurgical bonding involves metal technology and the way science is applied to metal production for manufacturers and consumers.\nFerrous metallurgical bonding - This is also referred to as black metallurgical bonding. It involves the alloys and processes that are iron-based. Ferrous metal production accounts for about 95% of the metal production worldwide.\nNon-ferrous metallurgical bonding - Also known as colored metallurgy, involves alloys and processes based on varying metals.\nHeat treatments can further alter these properties.\nIn order to study macroscopic and microscopic metal properties, metallurgists utilize a technique called metallography. This makes the process of metallurgical bonding more successful, since detailed information on mechanical properties, processing history and overall composition of metals can be obtained.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Business is a commercial or industrial enterprise and the people who constitute it. Also it can be defined as an organization.\nThe starting point for developing new business ideas lies inside the prospective entrepreneur rather than in the marketplace, laboratory, business plan etc. You are the critical component - it is your strengths and weaknesses which should dictate the areas in which to seek ideas and the likely scale & scope of your business. At the end of the day, support for your business by financiers, suppliers, customers etc. will also be a vote of confidence in your abilities to make it successful. First try to build your strengths and concentrate on weaknesses, and understand your limitations.\nA manufactured product where you buy materials or parts and make up the product(s) yourself. A distributed product where you buy product from a wholesaler\/MLM, retailer, or manufacturer and the service which you provide.\nIt is better to pursue several specific ideas (hypothesis) rather than one diffuse concept which lacks specifics and proves impossible to research and evaluate. Generally always aim for quality rather than cheapness. Be very cautious about pursuing ideas which involve any prospect of price wars or very price sensitive, of getting sucked into short-lived fads, or of having to compete head-to-head with large, entrenched businesses.\nFirst look into customer's requirements like, what they buy? Why do they buy? What are they buying more of? Put up some questions like this and get into business according to that to become succuessful in business.\nTry to do some changes in already existing products like, changing the size, color of the product, altering their quality or quantity, increasing mobility, access, portability, disposability, simplifying repair, maintenance, replacement, cleaning, Introducing automation, simplification, convenience , adding new features, accessories, extensions , changing the delivery method, packaging, unit size\/shape, improving usability, performance or safety and by improving the quality of service.\nAbove all, you keep your eyes wide open and grab the oppurtunity and enter into the market. By all means be inventive, imaginative and original in your thinking but stay market and consumer orientated rather than product-obsessed.\nAfter listing out all ideas these must be evaluated so that a short-list of preferred options with the greatest potential and lowest risk can be assessed in greater depth. Before scoring individual ideas, run through the criteria and set what you feel should be minimum desirable scores for each. The resultant total could be used as your overall minimum threshold. In some ideas if you didn't find satisfactory result drop that plan and look for better ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Largely an experiment for the time being, 'Bates Motel' will air the live 'After Hours' on March 3 at 11:00 p.m., following the premiere of A&E's new Chloe Sevigny-led series, 'Those Who Kill.' In attendance for the first round (don't be surprised if the finale at least follows a similar talk show event) are series stars Vera Farmiga, Freddie Highmore and Max Thieriot, along with executive producers Carlton Cuse and Kerry Ehrin.\nIn season two of 'Bates Motel,' executive producers Carlton Cuse ('Lost') and Kerry Ehrin ('Friday Night Lights') dig deeper into the twisted world of the Bates family. Picking up from last season, Norman (Freddie Highmore) is fixating on Miss Watson's death while Norma's mysterious past starts to haunt the family with the introduction of her brother. Meanwhile, Norman's brother Dylan, (Max Thieriot), gets more entrenched in the familial drug war that fuels White Pine Bay and finds himself right in the middle of the danger as Bradley (Nicola Peltz), who remains on the hunt to uncover her father's killer, is driven to precarious extremes. Sheriff Romero (Nestor Carbonell), is caught up on all fronts as everything in town escalates and Emma (Olivia Cooke) complicates her relationship with Norman as the duo explore new love interests, bringing new and old characters along for the ride.\nGuest stars checking into 'Bates Motel' throughout the new season are Michael Vartan ('HawthoRne,' 'Alias'), Kathleen Robertson ('Boss') Paloma Kwiatkowski ('Percy Jackson: Sea of Monsters'), Rebecca Creskoff ('The Practice,' 'Hung'), Kenny Johnson ('Sons of Anarchy'), Michael Eklund ('88 Minutes,' 'Supernatural') and Michael O'Neill ('Dallas Buyers Club,' 'Grey's Anatomy,' 'The West Wing').\nAs if that weren't enough to pique our interest for the March 3 premiere, A&E set up an official interactive 'Bates Motel' website, where you can explore the entire iconic location yourself, unlocking plenty of secret content, footage and teasers among the various rooms for rent. We've included the latest trailer below, so tell us in the comments what you want to see from 'Bates Motel' season 2 in the comments!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quietly situated in the charming village of Ceres, 8 miles from the magnificent beaches and championship golf courses at St Andrews, these two holiday cottages, adjacent to each other, have been stylishly converted from a listed building to create a superb holiday base. Ceres has a shop, newsagents, butchers, dog friendly pub, tea room, museum, caf\u00e9 and hotel with restaurant and dog friendly bar within 300 yards. Nearby attractions include the Hill of Tarvit Mansion House, a fascinating National Trust house and gardens with Hatters Edwardian Tearoom and a 9 hole 'Hickory' golf course, the Fife Falconry Centre, and horse riding in the village. There are excellent footpaths for walking and cycling with stunning views of coast and countryside, whilst the Tents Muir National Nature Reserve has a fabulous walk along the beach to Tentsmuir Point where hundreds of seals can be seen basking on the sandbars. Visit The Eden Estuary Centre and Morton Lochs and watch waterfowl from dedicated bird hides. St Andrews itself is steeped in history and has a comprehensive range of shops, restaurants, tearooms, museum and the Sea Life Centre. Perth and Dundee are easily accessible, whilst the wonders of Edinburgh are just 40 miles away. Shop, pub and restaurant 300 yards.All on the ground floor.\nLiving room: With electric coal-effect stove, TV, DVD player and iPod dock.\nBathroom: With bath, shower cubicle and toilet.Both properties: Gas central heating, electricity, bed linen, towels and Wi-Fi included. Travel cot. Courtyard with sitting-out area and garden furniture (shared). On road parking. No smoking.\nBe the first one to tell us about your rental experience while staying at Castle Cottages - Castle Dairy. Let us know if the listing matched your expectations. You can share your comments about the location, property owner\/manager and all the unique features that made your stay a memorable one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A recent independent shop marketing survey found that 46 percent of shops are no longer spending money on traditional media, like television, radio, newspapers and billboards. A market research analyst explains why.\nThis is just the latest in an ongoing series of awards that Cottman has received for its blog. It was named one of the Top 50 Automotive and Mechanics Blogs by Direct Capital after being online for just five months. And it was recognized by Website Magazine as an example of how to reach out to consumers in your community.\nWhen you consider a new statistic that shows that search engines report a 14.6 percent close rate, while outbound leads (such as cold-calling and direct mail) have a 1.7 percent close rate, there's no denying that shops need to concentrate on how their SEO functions. Here are some important basics for businesses to understand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalgiw b/data_all_eng_slimpj/shuffled/split2/finalzzzalgiw
new file mode 100644
index 0000000000000000000000000000000000000000..d19b73d3574723e20731347272e0c48f9f9fddb9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Internationally, 'Outplacement' has been benefiting corporates and their staff for over 30 years, yet in South Africa it seems only a handful of companies are familiar with this aspect of best practice.\nAn Outplacement assistance programme forms part of the severance package provided by employers to their terminated\/outgoing staff. The purpose is to support individuals through the transition out of a company, by reorienting them with the job market through the provision of practical tools, specialist insights and psychological support.\nAside from providing structure and support during an emotionally difficult time, core aspects of Outplacement include career evaluation and guidance, professional CV writing, interview and communication training, assistance in developing networks, personal branding and the creation of an effective online presence.\nWhile some HR departments do try and assist outgoing executives, there is much value in the utilisation of an external provider, where the impacted individuals are exposed to market specialists and are able to develop in a neutral, confidential environment.\nFrom the employer perspective, Outplacement is an effective risk mitigation process, protecting the 'employer brand', minimizing negative publicity and curtailing potential CCMA action.\nAccording to PWC's 2015 report on Retrenchment and Redundancy Practices in SA, 74% of participants undertook retrenchments over the past 5 years, with 62% in the last 12 months alone.\nAccording to the PWC report, the most prominent reason for South African retrenchments has been a decline in operations at 51%, followed closely by restructuring, optimization, centalisation and cost savings at 44%.\nHand-in-hand with corporate change, comes the need for more relevant HR policies to support those affected. Internationally, Outplacement services often fall under 'training and up-skilling processes,' which are supported by the SETA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I hate to say it, but there are only 82 days (that's just over 11 weeks!) to go until Christmas, and if you plan to escape this winter, one of the best places to hide from Santa, sleigh bells and all things festive, is the top of Mount Kilimanjaro!\nClimbing Kilimanjaro over Christmas and New Year is an amazing experience, and when you consider that the average Brit spends around \u00a31000 on Christmas presents, food, and celebrations over the festive period, your trip to the Roof of Africa will not cost much more than staying at home.\nUnfortunately, the flights to Kilimanjaro International Airport do increase over the Christmas and New Year period, but most recommended Kilimanjaro climbing companies offer standard rates throughout the year, and do not impose supplements simply because it's Christmas.\nKilimanjaro is busier than usual over the Christmas and New Year period, but that is to be expected. Intrepid travellers from across the globe flock to Tanzania for the highest New Years Eve party in the world, but if you book a private climb, you can still enjoy a more personalised experience, and make your way to Uhuru Peak at your own pace\u2026 which will be \"poley-poley\" believe me!\nOnce you have completed your mission to the top of Africa's highest mountain, you can reward yourself with an Ngorongoro Crater or Serengeti Safari experience, or like us, you can head on down to the sugar white beaches of Zanzibar for some much-needed R&R\u2026 the perfect way to head into a new year.\nIf you cannot manage Kilimanjaro this Christmas, worry not, there is a magnificent Kilimanjaro Lunar Eclipse Climb coming up in September 2015 \u2013 and that's a whole different story!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Play \"spot the Beatles lyric\" as you explore our swinging 60s-inspired d\u00e9cor, enjoy an inclusive continental breakfast each morning, and after exploring the amazing city enjoy a drink or a night cap at our friendly, relaxed bar. With easy access to the best that Liverpool has to offer, book your room at ibis Styles Liverpool Dale Street today for affordable accommodation with a quirky twist.\nA maritime city in the north west of England, Liverpool is known for its ferries, football and famous Fab Four. Explore the early roots of the Beatles with a trip to the Cavern Quarters, home to the Cavern Club and just five minutes' walk from our hotel.\nOffering a varied menu of healthy, international cuisine using locally sourced ingredients, our on-site restaurant provides the perfect spot to take your time over dinner and relax and unwind in style.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Best American Tees is a custom t shirt printing company. We specialize in custom screen printing and giving our customers the ability to create a t shirt to represent their group or brand. Our website allows customers to have absolute control to create and design custom t shirts.\nNeed a create a t shirt? Then you're in the right place. In our Design Studio you can begin designing your ideal custom t-shirt from scratch or from a design template. Our services go far beyond solely printing on t-shirts; we provide screen printing for tank tops, youth shirts, unisex tees, fashion shirts, polo's, hoodies, athletic wear, bags and more.\nAbove everything that we do, it's absolutely paramount that we are constantly giving our superior customer service. We know how frustrating it can be to encounter automated responses via email or over the phone. Having to wait on hold forever or a few days for email responses is not superior customer service. So we here at Best American Tees have made a conscious effort to ensure we never become that company.\nQuality over quantity is a saying that has existed for countless years and it definitely rings true with us today. We know how absolutely frustrating it can be to customize an item on the web only to have that same item arrive and fall far below your expectations. That's why we've done everything possible on our end to make sure our customers receive excellent quality items. We use the latest printing hardware and software, fresh ink, proper under-bases for certain garments, and perform constant quality checks on the shirts and the screens we're using for your particular order.\nJust about everyone that has ordered an item online can attest to the one major downside that comes with online orders, waiting. Waiting for an individual or a company to ship our your order only to get no confirmation in regards to a tracking number is exceedingly frustrating. That's why we've instituted free shipping on all basic orders submitted through our site along with the option for rush orders and delivery if you're on a tight-deadline.\nThere's nothing cooler than having one of a kind, unique items. That's why we've gone through the necessary steps to make is possible for you to design custom T shirts with an easy to use Design Studio built directly into our website. Our design tools are full of great options and capabilities. If you've got your own design you can simply upload it for printing on your own shirts. Otherwise we have thousands of designs at your disposal to create and craft a completely custom t-shirt design.\nFinally, we didn't want our customers to be forced to buy 5,000 shirts to get a good deal. We wanted to get custom t-shirts at reasonable prices without having to sink money into mass quantities or compromising our design options. That's why we've established some great relationships with local suppliers and manufacturers to get you the shirt brands you know of at great prices.\nWith us here at Best American Tees, you can enjoy customizing shirts, a fast turnaround time, pricing that beats almost any competitor, no artwork fees, no hidden fees, and free shipping on 12+ garments. Contact us to begin designing your own custom t-shirts today.\nFollow us to talk about your awesome t-shirts!\nGrand Rapids Website Design & Development by Valorous Circle LLC.\nReputation Monitoring & Management by Reputation Essentials.\nBestAmericanTees.com \u00a9 2013. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Depreciation is a business expense that reflects the natural reduction in value of physical assets over time. The IRS recognizes this economic reality and permits tax deductions under its rules for depreciation expense, despite the inherent nature of depreciation being a cashless expense. Although various business types are required to file various income tax forms and are not taxed uniformly, the IRS depreciation rules are consistent across the various business forms and tax treatments, ranging from sole proprietorships to corporations.\nA sole proprietorship can depreciate business assets, such as vehicles, equipment, computers, furniture and real property, purchased for the business. Sole proprietors, like other business owners, often have a choice regarding how to depreciate an asset. For instance, the tax law allows an immediate Section 179 depreciation deduction for the cost of new or used equipment up to a certain dollar amount. Also, there is a 50 percent first-year bonus depreciation tax deduction for qualifying purchases that can be carried forward to offset future taxable income.\nSince depreciation is a paper loss as opposed to a cash outlay, the choices a sole proprietor taxpayer makes regarding depreciation methods and the depreciation calculation can drastically impact the amount of net taxable income and total taxes due. In particular, the 50 percent first-year bonus depreciation tax deduction can result in a net operating loss that can be applied to other sources of income, resulting in a lower overall tax burden for the sole proprietor taxpayer.\nSole proprietors who own their own home, have a dedicated home office and choose to take a home office tax deduction may incorporate a portion of the home's depreciation into the home office deduction. This can be an advantageous tax deduction for a sole proprietor, but just as with other deductions, it should be carefully calculated and only claimed when it applies since it has been the subject of abuse by some aggressive taxpayers.\nThe limited liability company, or LLC, is an attractive business form choice for many startup entrepreneurs due to its provision of limited liability and flexibility in terms of tax treatment. Each state has its own set of laws that govern LLC formation, and starting a music business LLC requires you to complete the same set of steps that you would take to organize any other business as an LLC in your state.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzalkik
@@ -0,0 +1,5 @@
+{"text":"To date, companies that have deployed virtualization have not migrated many of their highest value, business critical applications to virtual servers. This is due to a variety of reasons, primarily performance concerns and a lack of true high availability. everRun VM delivers reliable protection for virtual workloads by providing redundant virtual machines and synchronized mirroring of the whole system. The integration of XenServer's superior \"bare-metal\" performance and everRun VM's unique architecture gives IT professionals the confidence to virtualize business-critical and other production applications. \"Marathon has a superb software product that is seamlessly integrated with XenServer. The combination allows companies to virtualize a range of applications that they wouldn't even consider previously,\" said Peter Levine, Senior Vice President and General Manager, Citrix Virtualization and Management Division.\neverRun VM removes the burden of availability for IT professionals and makes automated HA and DR practical for midsized companies.\n\"Based on our research, high availability solutions, including clustering software and proprietary hardware, have never exceeded 20 percent deployment rates,\" said John Humphreys, Program Vice President, Virtualization Software for IDC. \"The reason: they are too complex to setup and too complex to manage. The combination of easy-to-use virtualization and availability software could bring availability to the masses.\"\n\"everRun VM presents a dramatically more reliable approach to virtual server availability through selectable, automated, FT-class availability,\" said Gary Phillips, president and CEO of Marathon Technologies. everRun VM is the first in a line of virtualization availability and DR products from Marathon that are so simple, transparent and cost-effective that every application can be protected.\"\neverRun VM is currently in beta. General availability is scheduled for April.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Municipal Committee, Jail Rd, Mandi Bahauddin, Pakistan.\nMunicipal Committee Jobs For Legal Advisor Qualification:- Candidates should be at least Graduation Degree Holders & Minimum Experience:-Preferably have 05 years of working experience in the field interested candidates fulfilling the above criteria can apply at following address by 20 February 2019 incomplete documents\/late application after due date will not be acceptable.\nLegal Advisor jobs in Municipal Committee Taxila, in Pakistan published on Paperpk.com from newspaper on February 03,2019. Job seekers who are already working as Legal , Financial Advisor , Legal Advisor , Legal , Law Officer , Pharmaceutical Legal Advisor can also apply for post of Legal Advisor.\nThis job is publised on PaperPk.com in Newspaper. For more Details visit Municipal Committee, Jail Rd, Mandi Bahauddin, Pakistan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Investors and business developers can expect great returns on investments in the tourism and accommodation industries in Switzerland, as the country lures thousands of local and foreign tourists all across the country. Switzerland is renowned around the world for its picturesque landscapes, skiing and winter sports and the famous Alps.\nAll business developers who plan to establish a hotel or other forms of accommodation are required to get permits and licenses in regards to different business activities. We have specialists in this field to help with any extra information you may need to develop a company in Switzerland relating to the Swiss tourism and accommodation industry. Additionally, it's important to point out that the incorporation of a Swiss hotel business is the same as other types of businesses.\nDuring the formation of a hotel business in Switzerland, investors and entrepreneurs must obtain the required permits and licenses to ensure the activities they undertake are legal and allowed to take place with regards to the local laws and regulations. In the event an entirely new building is required to be built in Switzerland for the hotel to move forward, a construction permit is required.\nWe have assistants available to assist with all Switzerland-based permits and approvals.\nAll hotels in Switzerland are rated on a Star system, similar to that in Australia and the United States. One star is awarded to those offering the minimum requirements of a hotel, whereas 5 stars are considered more luxury.\nIn 2015 there were 92 Swiss hotels rated 5 stars. There were also 903 hotels rated at 3 stars, the most popular option.\nWe have more information about business and company development on our website, or you could speak with our consultants to find out further detailed information on establishing a hotel, or hotel chain in Switzerland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My experiences using the online survey provider SoGo Surveys for academic research in understand the public perceptions of Aboriginal heritage.\nI have often heard it discussed that only Aboriginal people and social scientists such as archaeologists and anthropologists are interested in Aboriginal heritage\u2013 these are considered to be 'special' interests. It is this type of reasoning that is used to defend the legitimacy of the Ministers decision when an Aboriginal site is approved to be impacted by industry\u2013 because the development in question is in the 'general interest of the community', of which the Minister is bound to have regard for.\nThe Minister should of course have regard for the interest of the general community, however this is based on the assumption that the preservation of significant Aboriginal sites is not always in the best interest of the community when it competes with mining interests. This is again based on the assumption that Aboriginal heritage is not valued by the wider Western Australian and Australian communities. It would appear evident that there are varying and divergent views on what is 'significant' and 'important' Aboriginal heritage\u2013 an important theoretical question in itself. This is one of the reasons the Aboriginal Heritage Act 1972 is currently failing to be an effective instrument for managing Aboriginal heritage sites, as there is little consideration on the variability of the concept of 'significance' and 'importance', and what is worthy of protection.\nTo investigate the general publics opinion on Aboriginal heritage, a survey with SoGo Surveys (sogosurvey.com) has been launched. The newly launched survey is garnering some interesting preliminary results, notwithstanding considerations of sampling methods and bias which will require further consideration and possible adjustment. Of the 193 survey respondents so far (31.4% from WA, 59.3% from Australia and 8.26% from overseas)\u2013 respondents were overwhelmingly female (80%). 69% of respondents state they have had exposure to Aboriginal culture, and 90% of respondents state they value Aboriginal heritage .\nJames Price Point\u2013 significant to the Goolarabooloo Aboriginal people partly because it forms part of the Lurujarri Songcycle, but would not necessarily be considered 'Aboriginal heritage' to the wider uninitiated population. An example of the divergent views on what constitutes an Aboriginal 'site' in Western Australia. Photo: Liz Vaughan.\nThis is my first time using SoGo surveys, and so far the experience has been positive. You can simply run a report on your data online to produce an easily readable bar graph, or you can export the data to excel or other programs. When launching your survey you are offered tailored links that suit different platforms (for e.g. single or multi-use links that you can email, publication to Facebook, links you can embed on a website or blog. I changed from Survey Monkey to SoGo Surveys because Survey Monkey has has limit of 10 survey questions with their free subscription, whilst SoGo Surveys doesn't have that limitation.\nIn the future I will write a post on my experiences with sampling, bias considerations and promotion through social media, until then here are some links to some other resources, from Web Survey Methodology and the University of Sydney.\nPrepared by Liz Vaughan, Honours Student, Archaeology Department, University of Western Australia, 2014.\nFigure 1 displays data received from the Department of Aboriginal Affairs (DAA) on the number of sites reported (blue), ACMC assessed (red), determined as sites (green) and not being a site (purple). There are some important factors to note when interpreting the data. Firstly, the DAA has a 70 day performance target for assessing section 18 notices, from lodgement to Ministerial decision. Therefore the sites lodged within a particular month are unlikely to have a relationship with the sites assessed for the same month, i.e. the 'Aboriginal heritage places' that are assessed by the ACMC for any month would have been lodged with the register approximately 8 weeks previously (Chapple & Collier 2014).\nThe Department of Aboriginal Affairs reported the data for the time January 2011 to June 2013 expressed in Figure 1 in 6 month periods. The data from July 2013 to May 2014 was reported in 1 month periods (see break down in Figure 2), but has been summed and expressed here in 6 month periods in order to be meaningfully compared to the previous data reported in 6 month periods.\nIt has been stated by the Minister for Aboriginal Affairs that most of the sites lodged within the period of July 2013 to May 2014 were submitted with notices lodged under Section 18 (Chapple & Collier 2014). There has not been information provided if this is also the case for the months January 2011 through to June 2013. Sites submitted with notices lodged under Section 18 can affect the interpretation of the data in a number of ways. These sites may not necessarily have been submitted because of their significance or importance by the reportee, but rather as part of a heritage site avoidance or salvage operation in accordance with land use that may disturb an Aboriginal site.\nIt should also be noted that the period January to May 2014 represents 5 months, not 6 months like all other periods. This represents the most up to date information released by the DAA. Interestingly, it can be noted that the January to May 2014 period represents the largest amount of reportings for the entire time reported on, despite being only a 5 month period. This is an effect of the huge spike in reportings in March and April 2014, as can be seen from the higher resolution break down of statistics in Figure 2.\nFigure 2 displays numbers of Aboriginal heritage places lodged with the Registrar of Aboriginal Sites under Section 15 of the AHA from July 2013 to May 2014 on a monthly basis. The total number of Aboriginal heritage places lodged for the corresponding month are expressed in (blue). Aboriginal heritage places that were assessed by the ACMC for that month are displayed in (red). Aboriginal heritage places assessed by the ACMC as a site is displayed in (green), and those heritage places assessed as being not a site are in (purple). Notable in this graph is the spike in sites reported in March and April 2014. The electronic Heritage Information Submission (HIS) forms as part of the Aboriginal Heritage E-lodgement Program (AHELP) were announced as ready to roll out in an email issued on the 10th of April 2014 by John MacDonald, Director of Approvals & Advice at the Department of Aboriginal Affairs. The HIS forms have been available online from the Department of Industry website since June in final form, and for 2 months before that in draft for testing (MacDonald, J. Pers. Comms. 2014).\nIt was announced that the paper-based versions would not be accepted by August 1 2014 (MacDonald, J. Pers Comms. 2014). This spike in site reportings in the March and April period may reflect the introduction of the draft HIS forms, as ease of use would be expected compared to the previous paper based forms.\nThese places were processed in line with the Department of Aboriginal Affairs' 70-day performance target for section 18 Notices from receipt to Ministerial decision, and were therefore assessed by the Aboriginal Cultural Material Committee approximately eight weeks after lodgement with the Registrar, as stated by Minister Collier in Question on Notice, Tuesday, 24 June (2014). Therefore the number of sites assessed by the ACMC within a month does not have a relationship with the number of sites lodge for that same month. The 70-day performance target was pointed out by Minister Collier on the 24 of June 2014 in response to questions on notice. It is not clear when this became a target for the DAA, but it came out of the Keating Review in 2002 (MacDonald, J. Pers. Comms. 2014; Keating, M. 2002). Therefore the approximate eight week delay can be assumed to apply to all data presented here. According to the DAA most of these sites lodged were submitted with Notices lodged under Section 18. Note also there was no ACMC meeting in January 2014.\nFigure 3 displays Aboriginal heritage places lodged (blue) with the Registrar of Aboriginal Sites from July 2013 to May 2014, places assessed by the ACMC (red), determined as sites (green) or not a site (purple). Once again please note due to the high resolution of the data (monthly), there is no relationship between potential sites lodged (blue) and potential sites assessed by the ACMC (red), as is explained above.\nChapple, R. and P. Collier 2014 Question on Notice, Tuesday, 24 June 2014. Legislative Council: Western Australian Government.\nKeating, M. 2002 Review of the Project Development Approvals System: Government of Western Australia.\nMacDonald, J. 2014 Department of Aboriginal Affairs officer to Elizabeth Vaughan Email string on August 29, 2014.\nWyatt, B. & Hames, K. 2013 Western Australia Hansard 7 August 2013, p.3005b. Legislative Assembly: WA Parliament.\nThe Changing Nature of WA's Aboriginal Heritage Laws, 'progress' or 'regress'?\nThe West Australian government trying to pass the Aboriginal Heritage Act Amendment Bill 2014. The changes have prompted some critical responses from communities of interest (Aboriginal groups, archaeologists, anthropologists, politicians, community members, etc), and there is also a large amount of confusion surrounding the changes. This blog aims at providing a resource of documents, investigations and findings on the legal protection of Aboriginal heritage in WA, and past and current changes.\nThis information comes from research undertaken by Elizabeth Vaughan as part of her honours dissertation in archaeology at the University of Western Australia. Elizabeth is producing a thesis titled Excavating the Aboriginal Heritage Act: The evolution of heritage protection in Western Australia, 1972-2014.\nThis blog is designed to provide a locus of data on the changes to the Aboriginal Heritage Act that is publicly available, with the aim of influencing sensible and practical changes to the Act that will find traction with all users.\nBelow is a link to statistical information regarding site registration from 2011 to 2014, showing site registration by the Aboriginal Cultural Materials Committee has decreased from 80% to 6% .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Archery's first fully automatic rest.\nNo thumb wheel needed to set the rest.\nPatent pending Rotary Damper technology.\nFull arrow containment even in a let-down situation.\nProven and tested on bows over 400 fps.\nEasy arrow loading - rest does not have to be preset.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"So after a late night we got a somewhat slow start to our Saturday in Montreal and we had a lot of ground to cover ! I awoke early and after having a nice discussion with a local at the club the night before we decided the farmers market to hit up, should be Marche Jean Talon. It was an easy trip via the Metro and the cost was like $3 for each of us and it took us within a block of the market and saved us a 7 mile walk and alot of time !\nIt was kind of split into three sections... but after talking with a few farmers and shop owners we found out that it grows into a market that is three times the size come summer... they even remove the walls and turn the whole place into and HUGE open air market.\nthere was lots to look at an sample... it was hard to pull Cassi away from the fruit trays so we could explore the whole market !\nmaple syrup in a can... never seen that in the states !\nspattered throughout the market there was some wonderful egg stands that sold every kind of egg from quail to duck to chicken to ostrich !\nthats when we ran into this lovely worldly inspired pastry stand... everything was so delicate and made with love we had to grab a little sampler along with a couple of warm samosas that really hit the spot !\nEverything was so tasty and spiced with some very exotic flavors that really made them sing more than any pastries we have eaten before. We had some little sticky pistachio pastries and filbert puffs that were out of this world... the only thing I could compare the floral exotic flavors too was the holiday Greek loaf that Dean made back when Mother Oven bread was in business. I remember him telling me about a ancient greek spice in that holiday bread called mistika that his grandmother had taught him about and I definitely got a wave of that in one of the pastries...mmmm Im getting hungry!\nthis was their curry kit... and right across the way from the spice place was a fresh pasta shop !\nPiles and piles of fresh pastas, homemade sauces and a very friendly staff, we ended up picking up a few sheets of fresh porcini ravioli !\nduck confit with onions or a classic smoked meat sandwich ! We didnt grab one but we did try a duck sample after she noticed a little drool coming out of my mouth !\nwe had so much fun just checking out all the local goods !\nbut the absolute highlight of the Marche Jean Talon had to be the discovery of Jardins Sauvages stand ! You see the first place we thought of when we started planning our trip to Montreal was this restaurant located 45 minutes north of the city called Jardin Sauvages. We first learned about this restaurant and its highly esteemed owner\/forager Francois Brouillard by reading the book American Terrior by Rowan Jacobsen. The whole menu is based solely on what Francois can find in the nearby ecosystem... it does not get more local or sustainable than this place. However with very limited time in Montreal we decided to hold off on taking the trek north and decided to choose a dinner location that was within the confines of the city... more on that later in the post. So as we approached the stand and took notice of the piles of locally foraged offerings, we were very excited to be able to experience a small slice of what the to expect when we do make the trip up there for dinner some day!\nalong with these fresh tasty treats and an extremely helpful and kind shopkeeper, they also had a smorgasbord of pickled and jarred products made from the seasonal treasures they find.\nPickled wild mushrooms, day lily buds, wild ginger root, mustards, jellies, oils, vinaigrettes... WOW, we were in heaven... NEW INGREDIENTS !!!\nWe chatted with them for quite a while and picked up alot of great stuff... they had also just sauteed up a mixture of wild leeks and mushrooms and gave us some samples of a few things we had never tried... needless to say with so many strange things making it into our cooler for the trip back to Maine we were a little nervous about the upcoming border crossing back into the states !\nAnyway having explored the marche to the fullest and had a lil bite to eat we were ready to hit the pavement and explore the sites Montreal had to offer. We had about a 5 mile walk back to our hotel and I had circled some choice destinations on the way... so off we went down Boulevard Saint-Laurent !\nWait thats a sign for records... nice lil shop, was able to grab a handful of reggae 45's to add to the crates ! As we marched forward we came upon a handful of lil boutiques and quickly realized we were in the part of town we wanted to be in. After an couple hours of shopping and browsing we were up for a break and I had two spots within a block to do so at.\nIt was a tiny little walk-in walk-out spot with bagels stacked from floor to ceiling. They were super friendly and we picked up an extra dozen to bring home to our freezer in Maine.\nWith a little more substance in our belly we headed just around the corner to visit one of my BIG destinations in the city Dieu de Ciel Brewery and Brewpub!\nWhat a cozy lil place that started out empty but by the time we tasted our way through their 17 in house brews on tap the place was getting pretty packed. Our waitress was very friendly and spoke enough English to get by... the beers were great ! You can find some of these gems in the states but with the amount on tap there was certainly quite a few I had never gotten my hands on and was in heaven trying them one after the next. So many styes and flavors and their liberal use of creative ingredients including fruits and herbs made tasting through them a pure pleasure that ate away the afternoon fairly quickly.... so much so that we realized we need to boogie back to our hotel so we could get a few moments to shower, change and relax before our diner reservations that we had made through Open Table at DNA !\nAs I said we made reservations week or so before our trip after seeing a little about this restaurant on bizarre foods and by a recommendation from a friend who is a cook in Montreal.\nWe jumped right into the night by ordering a couple starters and a bottle of Canadian wine from Ravine Vineyard. The first taste we got after a small foie gras amuse bouche was a lovely mouth watering pork belly with daikon... yum. Our waitress was so friendly and the somellier was also very kind and informative. She actually stopped in a moment after serving us our bottle of Red Coat and said she was excited that we ordered the charcuterie plate because it will pair perfectly with the wine we selected ! I like when the employees seem excited about the food they are serving and that was certainly the case here. The service was above and beyond, we never had less than a half glass of water and never poured any of our wine it was all taken care of by the courteous staff !\nIts all made and cured in house by the head chef Derek Dammann.\nIn fact we saw him quite few times scanning the room and walking around the dining room making sure everything was running smoothly.\nWe experienced alot of \"firsts\" in Montreal and this was no exception... our first taste of speck, wild boar salami, head terrine, and more... served with an array of acidic pickled veggies and a can of bread sticks. We took our time with this dish and it was amazing. I really believe if we are going to eat an animal that no part should go to waste and this plate reassured us that we were doing the right thing, WOW !\nAfter a lil break we shared a light pasta dish that was a simple yet delicious wild mushroom tagliatelle that was all house made and cooked el dente.... we forgot a picture of this one but it was earthy and dreamy. A great break in the action before the main course.\nWe finished with some simple yet delicious cuts prepared to perfection. I took the plunge into the Canadian tradition and ordered the horse with apples, radish and juniper berries... and Cassi had the veal served with baby spring onions and braised spinach.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With 1,000 attendees, TEDSummit 2016 convened a cross-section of people who've helped mold TED into what it is today: passionate TEDsters, speakers, authors, TED Fellows, TEDx organizers, volunteer translators, TED staffers, educators, animators, and partners. It's was a week of inspiration, of future-focused discussions, of aiming higher, together.\nSummit 2016 | Aim higher. Together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to the latest data, there are more than 14,000 rehab centers spread all across the United States, providing everything from specialized medication to mental health counseling. However, in spite of all these facilities, America seems to be in the middle of a drug addiction crisis. A Medcape study reveals that drugs kill almost 45,000 people every year and that overdoses are now the number one injury-related death in the country.\nThe addiction problem has become one of the most complex challenges that the U.S. healthcare system is dealing with, and the fact that statistics are often contradictory do not make it any easier to manage. On the one hand, some sources claim that the main problem with addition is that 90% of the people who need rehabilitation do not go or do not manage to stay in a rehab center. On the other hand, the success rates in most rehabs are highly interpretable and do not always reflect the reality of rebound and post-rehab life.\nThe Academy of Integrative Health & Medicine says that one of the reasons why so many rehab patients end up relapsing is that clinics treat the symptoms, not the source, and they do not address the source of addiction to begin with. This is a problem that holistic rehab centers are trying to solve.\nFocusing on the patient as a \"whole\", not just the isolated symptoms of addiction, holistic rehabs put the needs of the patient first and use a unified body-mind approach, focusing on wellness and spiritual healing.\nTo understand the gap that holistic rehab centers are trying to fill, first we must know the way conventional rebab centers operate and their biggest shortcomings. In Florida, for instance, which is the country's biggest drug addiction hub, experts have pointed out many issues with conventional rehabs, even those that utilize state of the art medical equipment and employ qualified professionals. As long as the focus is on the symptoms of addiction, not the cause, the patient will most likely relapse and go in and out of rehab. For many, the vicious circle ends tragically.\nThe vast majority of traditional rehab clinics in the U.S. use the now famous 12-step program, but more and more studies show that the one-size fits all approach is not efficient. Not all patients have the same profile and they do not respond to treatment in the same way. The roots of addiction run deep \u2013 it is caused by 60% genetic\/biological factors and 40% by psychological factors. Delivering the best treatment involves taking the time to understand the needs of each patient and address those exact factors that triggered the addiction in the first place. In the best holistic rehabs, this involves a personalized combination of medication, counseling, behavioral therapy and holistic practices.\nHerbal therapy \u2013 rehab centers recommend plant-based treatments to detox the body, promote healthy nutrition and reduce the unhealthy food cravings that can amplify substance abuse. For example, certain plants found in ancient Chinese medicine have been scientifically proven to prevent relapse by inhibiting alcohol cravings \u2013 Salvia miltiorrhiza and Radix Pueraiae.\nMassage therapy \u2013 an ancient practice that modern medicine is starting to rediscover. Massage therapy has two major benefits: first of all, it is an effective relaxation technique that reduces stress and secondly, it can reduce the muscle pain associated with withdrawal.\nReiki \u2013 this is another relaxation technique and form of energy therapy that was first used in Japan in the 1800's and that involves the transfer of energy from the practitioner's hands to the patients. Reiki is said to melt away energy blocks and induce a state of relaxation. So far, small studies have shown that Reiki does provide pain relief, but the practice is already starting to gain mass popularity and many hospitals in the U.S. have actually introduced it in their services.\nAyurveda \u2013 this term actually refers to a larger set of Indian practices that nurture the mind, body and spirit. Ayurveda practices include meditation, yoga and even dietary changes that aim to nurture the patient's body and mind, so that they reach optimal wellness levels.\nAlternatively, holistic rehab centers can also use chiropractic remedies and naturopathic medicine, but the treatment is never the same for everyone. Because they focus so much on an individualized approach, practitioners first sit down with the patient and their family to establish the right therapeutic combination. Therapy may be adjusted depending on the progress and patient feedback.\nThe environment in which rehabilitation takes place is also very important. The patient cannot overcome the biological and psychological blockages that pushed them towards addiction if they do not feel at home, in a safe and nurturing environment, which is why more and more holistic centers incorporate luxury amenities. These include not just serene, beautifully decorated rooms, but also exterior landscaping, with gardens and outdoor meditation spots, the idea being to make the patient feel at home, but in an austere medical facility.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2013-02-22 00:45:47 -0500 asked a question How to import function parsing of SAGE?\nCurrently I am writing a code and I need to use function parser as part of my application. I want to see if Sage has a library that lets me to use this capability of sage?\nTo clarify what I mean: you can use f(x)=x^2 in sage and this will be stored as a function. Is there any library specially doing this for Sage?\nSometimes same thing happens to me and using Activity monitor, I close all instances of python to make sure the server is shut down!\nNo problem. Thanks for helping.Yes this is the code I tried: from sage.graphs.graph_coloring import vertex_coloring; vertex_coloring??; And this is what I get: \"No object vertex_coloring\" in a box!\nThank you very much but as I mentioned I am only calculating vertex_coloring(G,k=12,value_only=False) in my code... So as I believe this doesn't help(?) About vertex_coloring?? I don't know why but I get : \"No object vertex_coloring\" in a box! Thanks again.\nThanks. It seems the only possible way.\nAbout vertex_coloring?? I don't know why but I get : \"No object vertex_coloring\" in a box! Thanks again.\nI am running the Sage on Ubuntu and I am using the vertex_coloring() to calculate the k-colorablity of a specific graph. As you might know the process is long enough and the electricity may gone through the process. Is there any way that I could make a sage process resumable or in way moderate the vertex_coloring() to resume the things from the last point it left?\nit will set the required permissions and ask for a login password. So select a password.\nCongragulations! you can now start sage app whenever you want to.\n2012-04-04 03:25:43 -0500 commented answer How to use Sage in terminal?\n2012-04-04 03:24:46 -0500 marked best answer How to use Sage in terminal?\n2012-04-04 03:24:03 -0500 answered a question How to use Sage in terminal?\n2012-04-04 02:50:50 -0500 commented answer How to use Sage in terminal?\n2012-04-03 23:19:28 -0500 answered a question How to use Sage in terminal?\n2012-04-02 09:41:21 -0500 answered a question How to use Sage in terminal?\n2012-04-01 10:35:13 -0500 asked a question How to install a new module for Sage?\nEdit: this is a duplicate of this question.\n2012-04-01 10:34:26 -0500 asked a question How to use Sage in terminal?\n2011-12-31 13:34:27 -0500 answered a question Show function for graphs is not working!\n@Jason Grout: Sorry for the late reply I didn't notified. I just see it accidentally here!\n@DSM: Yes you are correct many times the problem is related to caching but this one not resolved with refreshing even!!\n2011-12-15 23:55:04 -0500 answered a question Show function for graphs is not working!\nSolved! My code has a bug! However I want to mention it is really annoying that the bug of my code results in a bug of show function for SAGE!!\n2011-12-15 23:14:46 -0500 asked a question Show function for graphs is not working!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our client is located in Auckland. They are seeking a permanent child & adolescent psychiatrist to work in the Child and Family Unit. Vacancies for a psychiatrist in this service do not come along often and we are seeking energetic, innovative and passionate applicants for the full time role. The Child and Family Unit is the largest acute child and adolescent inpatient unit in New Zealand with 18 acute beds. Ideally the psychiatrist will have the ability to work across the age range. Our client is a values based and inclusive organisation that is committed to deliver the best health outcomes for their patients.\nOur client is located close to the heart of Auckland CBD where there are lots of shops to browse and places to eat. Visit the iconic Sky tower, or pop to the Viaduct Harbour which is full of super yachts and lined with bars and cafes to enjoy a great night out. The North Shore is a 15-20-minute drive away, known for its wonderful beaches and relaxed atmosphere.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaryrz b/data_all_eng_slimpj/shuffled/split2/finalzzzaryrz
new file mode 100644
index 0000000000000000000000000000000000000000..de50b19ef4218e2f5b9983da0d371d03ac5d5314
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaryrz
@@ -0,0 +1,5 @@
+{"text":"News items related to Boston art exhibitions as issued by the Send2Press Newswire service on behalf of the noted news source.\nDUXBURY, Mass., April 23 (SEND2PRESS NEWSWIRE) \u2014 In 2003, Passages 2 received Directors' Choice of Outstanding Painting at the Cambridge Art Association. Passages, Ocean to Bay 1 won 'Artist of the Year, Painting' at the Cambridge Art Association Members' Prize Show in 2005 and was selected 'Best Cape Cod Landscape' at the Cape Cod Art Association in 2007. Today, painter and designer Robert Cipriani continues to evolve his award-winning 'Passages' paintings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Using blue metal, this ballpoint pen by Orton West is perfect for men or women and a must have. It measures 14cm(height) x 1.3cm(diamater). All the pens come complete in a presentation box. The Orton West Ballpoint Pen - Blue gives off an amazing effect with its all round shiny and chunky design. Manufactured for everyday use, the ballpoint is released using an elegant twist mechanism. The most exquisite pen that anyone could own.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When he asked why, the agent told him insurance paid for the rental car but not for insurance to cover property damages or a collision. The viewer says he didn't know any better and didn't want to ris.\nancillary insurance intermediaries (eg, travel agents and car rental companies); and product comparison websites. The IDD aims to level the playing field between all types of distributor in order to e.\nToyota Motor Corp. launched a Honolulu, Hawaii car-sharing service that enables customers to use. Starting at $9.95 an hou.\nA Navy veteran who says he is being dismissed at every step turned to 12 On Your Side for help getting his car back. Edward C.\nJun 30, 2014 \u00b7 Confused about whether or not you should buy supplemental insurance when renting a car? If you have personal auto insurance or.\nthe established car rental companies and Alphabet Inc's Waymo jockey for position. And as more such cars hit the road, it could lead to fewer accidents, thereby threatening existing auto insurance mod.\nREAD MORE: Airbnb for cars is here. And the rental car giants are not happy. \u00bb Yes, the state has an interest in setting safe.\nSo if you're doing a lot of domestic car rental, still better to use that than say a debit card or a car that doesn't offer a.\nUtilization Rate and Average Revenue Analytics provides deep dive data analytics on wide ranging Car Rental market aspects in.\n. regulates the industry of car rental. However, consumers are protected by generic consumer legislation When collecting th.\nThe money would come from a proposed $2 fee per rental car transaction throughout Massachusetts. Dedicated funding for tra.\nTo make the headline cost look low, many rental companies and brokers \u2013 the. additional drivers, car seats, upgrades, and.\nGet landlord insurance to help protect up to five rental properties with USAA. Learn more about rates, quotes, and different types of coverage today.\nWhen I returned the car, the rental agent started examining it microscopically. I had not purchased Green Motion's expensi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Check out the latest deals that Red Hamper have for April 2019 and May 2019 by visiting their website.\nWe list all the latest deals available and we constantly update the site 24 hours a day, 7 days a week, if you are looking for a Red Hamper discount code for April 2019, chances are we have it listed!\nAll our discounts are free to use for everyone including NHS staff and Police officers, students may also be able to gain a Red Hamper student discount.\nDon't forget to share Red Hamper voucher codes with your friends and family to help them save money too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From animal trainers, to wildlife biologists, conservationists, and naturalists, there are a few people who immediately come to mind when animal careers are mentioned.\nFor example, Cesar Millan (also known as The Dog Whisperer) has made a very successful career out of training dogs and their humans. Millan specializes in rehabilitating dogs that society has given up on, and the results have proven amazing, so much so that he has had his own television show, The Dog Whisperer, on National Geographic, as well as successful training service to share his wisdom.\nMillan was born and raised in Mexico, and earned the nickname \"El Perrero\" (Spanish for \"The Dogman\") because he always seemed to be surrounded by a pack of happy dogs. When he came to America, he broke beyond sit-stay training and tapped into dog psychology. Today, Millan is as famous for his techniques as he is for his results, and he has traveled the country imparting his dog training wisdom on dog owners far and wide. He also authored several books, including Cesar's Way, which was a number-one fixture on the New York Times best seller list.\nAnother famous animal guy is Jeff Corwin, a wildlife biologist and the Emmy-winning host of Animal Planet's The Jeff Corwin Experience. He has authored several books, including Living on the Edge: Amazing Relationships in the Natural World. In addition to hosting his television show, Corwin also delivers lectures on wildlife, ecology and conservation to audiences across the U.S.\nHe created EcoZone, an interactive museum and environmental education center in Norwell, Mass., where he grew up.\nCorwin is an expert in rainforest animals, and he has a bachelor's degree in biology and anthropology from Bridgewater State College, as well as a master's degree in wildlife and fisheries conservation from the University of Massachusetts at Amherst.\nAnimal handlers and wildlife experts can regularly be seen on The Today Show, Good Morning America, CBS Morning Show, Tonight Show with Jay Leno, Access Hollywood, Extra and other shows.\nStudents will no doubt recognize the name Jane Goodall for her work with chimpanzees, particularly those in East Africa (specifically Tanzania). Goodall was the first to observe chimps using tools much like humans do, which bridged the connection between man and ape more firmly.\nGoodall received her PhD in ethology from Cambridge University in 1965, and then returned to Tanzania to conduct more research. She also established the Gombe Stream Research Centre. In 1977, she founded the Jane Goodall Institute for Wildlife Research, Education and Conservation to ensure her legacy of working with chimps would remain long after she was gone.\nLastly, the world mourned the loss of the Crocodile Hunter, Steve Irwin, who was born in February 1962 in Victoria. His family moved to Queensland in 1970 to start a small reptile park at Beerwah on the Sunshine Coast, which is now Australia Zoo. This is where Irwin developed his true passion for living creatures and his insatiable desire for knowledge and the sharing of that information with others. When his parents retired in 1991, Steve stepped in to run the show.\nIrwin volunteered with the Queensland Governments East Coast Crocodile Management program. In June 1992, Steve married Terri Raines, and they spent their married lives dedicated to the cause of informing people about wildlife and helping animals. Irwin died in 2006 after his chest was fatally pierced by a stingray barb while filming in Australia's Great Barrier Reef.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasxjt b/data_all_eng_slimpj/shuffled/split2/finalzzzasxjt
new file mode 100644
index 0000000000000000000000000000000000000000..9c7464897f81341ea77164431f16b8de7c33d412
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasxjt
@@ -0,0 +1,5 @@
+{"text":"TIP! If you are looking to improve your credit report, it is a good idea to have between two to four active credit cards. Having just one card means slower accumulation of good credit, but having five or more cards can add unnecessary complexity to your finances.\nMany people are having a financial crisis in today's hard times. Although riches may be beyond reach, there are actions to use, to organize your resources so that you will live in comfort. Read this article to find out what you can do to improve your financial situation.\nTIP! Your car and house are likely to be the biggest purchases you will make. A large portion of your budget will likely be devoted towards interest and payments for these items.\nBe positive that you can truly trust the person that you are entrusting your life savings with. They should have stellar references and be truthful with you. Do not let your broker think you have no knowledge or skills related to financing.\nTIP! Help manage your personal finances with a good insurance policy. Everyone is bound to get sick at one point or another.\nAlways negotiate with debt collectors. You debt was probably purchased by them for a low price. Paying a little of what you have to pay can help them still make a profit. Keep this in mind when you work with collection agencies to resolve your debt.\nTIP! When collection agencies get involved in collecting a debt from you, be aware that they only have a limited amount of time to get it from you; otherwise, the debt can expire. Ask experts when your debt expires and avoid paying collection agencies that attempt to collect money for older debt.\nThe frequent flier program is absolutely ideal for anyone who flies often and enjoys saving money or receiving free rewards. Credit card providers give out rewards for purchases that you make. Your reward miles may also be used at some hotels for room discounts or freebies.\nTIP! Negotiate with collectors to see if you can get your debt or at least your interest reduced. They most likely bought your debt from the original company for a much lower price.\nIf you want to better manage your finances, you need to rid yourself of any outstanding debt. The best way to be debt free is by paying off your credit cards and any loans, and you will need to do some cutting back. For example, you might want to eat at home instead of going out, even on the weekend. Packing your lunch and avoiding weekend outings can result in great savings, and anyone who really wants to improve their credit should be willing to do so.\nTIP! You can get some extra money from a yard or garage sale! You may even want to consider offering your neighbors the opportunity to consign their unwanted things that you could sell at your yard sale for a small portion of the price. There are plenty of inventive ways to make money at a yard sale.\nAs was explained above, you may not have enough money to be rich but there are still some ways to live comfortably. There are times when a person just needs a will to live instead of stress from finances.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Facebook admits its algorithmic downfalls, hate speech still grows.\nFacebook has finally accepted that its algorithms that decide whether or not a post is \"acceptable\" may not be that good, and has announced that over the next few weeks it will start allowing more items that people find newsworthy and significant onto Facebook, even if they violate its community standards.\nThe about turn comes as the social network has in recent months come under fire for deleting posts such as the iconic Vietnam War image of anapalm-burnt Kim Ph\u00fac and a Le Monde news feature that showed an image of a cancer victim's mammogram.\nIt's understandable that dealing with the subjective nature of historically and culturally significant images or news stories is a complex task, but it's so far been clear that the task should not be left to computer algorithms at their current stage of intelligence. Facebook also has to deal with differing cultural norms and laws in countries around the world\u2014another problem that is not yet best left to algorithms. On top of this, as I explained in September, Facebook must not overstep its role of a news aggregator to become a gatekeeper.\n\"In the weeks ahead, we're going to begin allowing more items that people find newsworthy, significant, or important to the public interest \u2014 even if they might otherwise violate our standards,\" wrote Facebook's vice president of public policy Joel Kaplan and global operations vice president Justin Osofsky. \"We will work with our community and partners to explore exactly how to do this, both through new tools and approaches to enforcement. Our intent is to allow more images and stories without posing safety risks or showing graphic images to minors and others who do not want to see them.\"\nThe duo said that Facebook has yet to exactly work out how it's going to do this, but presumably it will mean more human intervention into vetting images or posts that may be flagged as inappropriate. Motherboard has asked Facebook for more information, but the company had nothing more to add at this time.\n\"Send them into the gas chambers\"\nBut the news comes as Facebook enters an even trickier phase of its content strategy\u2014a phase that is broadcasting live wars, suicides, and in today's case, the destruction of a refugee camp via Facebook Live.\nI was brought to tears as the live comments and emojis rolled in as a camera panned across a grey, depressing scene at the Calais refugee camp in France, which is today being demolished. \"Drone strike and done,\" commented a Facebook user. \"All terrorist!\" wrote another. The comment that pushed me over the edge was simply, \"Send them into the gas chambers.\"\nIs this kind of hate speech acceptable, but an image of a cancer victim's breast isn't? Facebook has a long way to go, and so does society.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Don't forget (Live) Casino and Games!\nState-of-the-art software and highly professional management team with more than ten years experience in sports betting are contributing towards the top quality O-network Sports Betting offer. It gained popularity due to attractive prices, high maximum bets and secure guaranteed payouts. We currently offer +5,000 live events\/month and we recently introduced the mobile betting platform.\nO-network offers both traditional and Live betting on a wide range of sports, and our sports betting team is constantly improving coverage of new leagues, tournaments and markets to meet our customers' wishes. Special attention is paid to regional markets' preferences and expectations with the help of regional managers and Customer Services representatives.\nUser-friendly and easy navigation, as well as betting slip design allow customers to place a bet quickly just in a few clicks, and ensure trouble-free and pleasant betting experience.\nO-network Poker Affiliate Program gives you an opportunity to participate in attractive and profitable business projects and to receive stable and adequate commission for promoting Poker in Internet. Offside Poker club advantages will be interesting for you and your players.\nDirect 24\/7 access of your players to the IPoker Network, one of the world's largest poker networks. IPoker gives a unique opportunity to play the popular poker games together with thousands of other players! Numerous tournaments and types of games will enable a player of any skill level to find the table or tournament they are looking for.\nYour players do not have to deposit funds to start bringing you profit, which is a strong competitive advantage of O-network Poker Affiliate Program.\nRollBack Bonus for all your players (up to 30% of rake generated by them), 200% First Deposit Bonus, Best Hand, Bad Beat and Jack Pot Bonuses, Internet Shop where players can pay for a range of goods with Offside Player Points, and other promotions ensure long-term customer satisfaction and maintain constant growth of our player base.\nO-network Casino offers over 50 games from the market leader in browser-based gaming software Net Entertainment. Our multi-lingual casino is a perfect offer for all markets, and is all about offering players a reliable, stylish and comfortable atmosphere for their gaming experience.\nBlackjack, Roulette, Poker, Golden Derby and more \u2013 a wide range of table games, video and classic slot machines, video poker where players will find a game for any taste! O-network offers daily, weekly and monthly tournaments, bonus programs and a lot of other promotions and campaigns, encouraging players to continue playing, to increase the players' lifetime value and minimize churn rate.\nO-network Casino is committed to providing the best casino on the net by constantly improving the offer, adding bonuses, free spins and a steady flow of new game releases.\nO-network LIVE Casino offers an exciting, realistic and engaging online casino experience with professional live dealers, a package of popular games and many special features, which can make the players feel like they are in a real casino and not in front of their computers.\nThe classic games of Blackjack, Roulette and Baccarat can satisfy the tastes of all the casino lovers and encourage them to keep playing, especially thanks to the great atmosphere created by our live dealers. In order to make the experience more realistic, we've introduced a chat functionality which can let our players to chat in real time with the dealers or the other players at the table.\nOur live casino makes the dream come true for many people which want to play their favorite casino games in the most exquisite surroundings. Now, the great technology at O-network Live Casino makes this thing possible with just a few clicks. Any player of O-network can play LIVE casino games straight away by clicking on a seat and placing the bet, with no download required.\nO-network Live Casino is a great opportunity to receive attractive commissions for promoting online live casino services. We do our best to deliver top and competitive services in order to satisfy all the customers and make our player base grow day by day.\nDue to the growing interest in online games, we've developed a complex product with the help of the business solutions provided by the market leading games developer - Microgaming. O-network Games represents a perfect and interesting product for being promoted, because it delivers an outstanding online gaming experience with over 60 quality games and a very user-friendly and easy to use interface.\nO-network Games has over 60 games sorted in 5 categories: Bonus Slots, Classic Slots, Progressive Slots and Other Games. All of these games have great graphics and can be played with no download required, even for free! It just takes a couple of clicks in order to enjoy the excitement and thrill brought by these attractive games.\nWe are constantly adding more and more exciting games and promotions in order to keep our customers busy and to enhance our position in this field. Using our effective marketing tools you can promote our amazing online games and ensure long-term commissions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Samsung Galaxy A5 | Seguro e f\u00e1cil com entrega ao domic\u00edlio.\nThe black Galaxy A5 Duos A510M 16GB Smartphone from Samsung delivers powerful performance and dual-SIM capability in a sleek aluminum uni-body design. The A5 Duos is powered by an octa-core 1.6 GHz ARM Cortex A53 processor and has 2GB of RAM in support. It also comes equipped with 16GB of storage and supports microSDXC cards up to 128GB. The A5 features built-in dual SIM card slots, which give you the ability to have two separate phone numbers on the same device. One of the SIM card slots functions as a hybrid slot that accepts microSDXC cards--however, because it's a hybrid slot, you won't be able to have a microSD card installed if you're using the A5 as a dual-SIM device.\nThe Samsung Galaxy A5 Duos has a 5.2\" Super AMOLED screen, which is designed to produce robust images and color depth. The screen's HD 1920 x 1080 resolution allows you to view full high-definition images and video content. With its 5-megapixel front camera and a 13-megapixel rear camera, the A5 Duos is also well equipped for picture capturing and video chatting. Autofocus is built-in and there is an LED flash on the rear of the device for taking pictures in low-light situations. Additionally, both lenses can record video in 1080p.\nThis smartphone incorporates dual-band Wi-Fi for high-speed Internet connectivity. Bluetooth 4.1, NFC, and ANT+ technologies are all built-in for pairing with compatible devices, such as wireless headphones, speakers, and other select accessories. A-GPS with GLONASS is also on-board for using Google Location and Maps services. The Samsung Galaxy A5 Duos has a non-removable 2900mAh battery that can deliver hours of use before charging is required.\nThe A510M is an international variant of the 2nd Gen Galaxy A5 Duos. The A510M's LTE radio is tuned to frequencies that are most often used in the United States and North America, as well as Europe and elsewhere. To ensure this phone will work with your carrier, please check to make sure the frequencies your carrier uses are supported by this phone. The full list of compatible carrier frequencies can be found here; the frequencies this phone supports can be found in the product specs.\nBecause the A510M is an international variant, the language of the box and the default language of the phone may not be in English. Also, an international AC to USB adapter may be included.\nPlease Note: Second SIM card slot is shared with the microSDXC slot. If using two SIM cards in the device, you may not insert a microSD card.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you`ve ever played Shanghai Moonton Technology`s mobile legends hack apk 2018 Legends game it probably didn`t take you long to realize that something`s not quite right. Use USB cable and plug in device with Mobile Legends Bang Bang to your PC. Let`s run Mobile Legends Bang Bang Hack Tool for Android and iOS but first you have to download it from Hacks Mobile Legends Bang Bang for iOS and Android let you to generate unlimited Diamonds, Double Battle Points and Keys! You can easily use this tool on IOS and androids as it works smoothly and offers you several amazing benefits.\nThere are many greater advantages a person is getting when he uses the hack tool for the purpose of hack in the game. Compared to the other mobile MOBAs, AoV has the most variety in terms of game types. Getting a U.S. court order that could kill Mobile Legends will protect the lifetime revenue potential of League of Legends and Strike of Kings. You can check all of the comments on our official YouTube video to ensure that our Mobile Legends hack is fully functional and that gamers all over the world are very satisfied.\nWhile YouTube`s scale and large audience is a clear benefit for popular PC and console games such as League of Legends , it doesn`t appeal to all developers. If you`re ready to learn more information in regards to madden mobile easter legends players check out the web site. Sometimes when playing Mobile Legends: Bang Bang, bluestacks ask you fee to pay but gives you option to download some apps. But while Mobile Legends unquestionably dominates the market in Indonesia, it`s worth checking the state of play elsewhere in Southeast Asia. Mobile Legends is one of the games that made it famous on Android and iOS, but MOBA sub genre was popular for years before on PC.\nMillions of players nowadays play DOTA 2 or League of Legends. If you like our Mobile Legends hack, share it with your playmates so you can all support the efforts we gave to build you an easier way to enjoy this game. Get reconnected in seconds and know that your team is being handled by the Mobile Legends AI system to avoid uneven matches during your absence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatmtb b/data_all_eng_slimpj/shuffled/split2/finalzzzatmtb
new file mode 100644
index 0000000000000000000000000000000000000000..01c6d136fec7840b8e6933b786a395104b4b79dd
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatmtb
@@ -0,0 +1,5 @@
+{"text":"The mixed function oxidase (MFO) activity and content was studied in the liver of common brushtail possums fed for 10 days on a diet containing a mixture of terpenes found in Eucalyptus leaves on which possums commonly browse. The MFOs were compared to the MFOs in possums fed a control diet of fruits and cereals only. The terpenes chosen were 1,8-cineole, p-cymene, \u00e1-pinene and limonene. The selected terpenes caused induction of P450 enzymes, as shown by a 53% higher cytochrome P450 content and a 45% increase in aminopyrine demethylase activity in the test animals. Aniline hydroxylase activity was significantly increased, with levels of 2.95 and 1.43 nmol min\u22121 mg\u22121 microsomal protein in the test and control animals, respectively. There was also a significant increase in androstenedione 16\u00e1-hydroxylase activity in the test group, 0.85 as compared to 0.50 nmol mg\u22121 min\u22121 in the control group. Western blot studies using human CYP2E1 and rat CYP2C11 and CYP2C6 antibodies gave CYP2E, CYP2C11 and CYP2C6 immunoreactive bands of greater intensity in the test animals as compared to the control group. This study has shown experimentally that dietary terpenes cause enzyme induction in folivorous marsupials. It also confirms the importance of knowledge of diet when studying xenobiotic metabolising enzymes, particularly in wild animals such as the brushtail possum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This works much more than the telemaster, not a fan of butterscotch but cool design at a decent price.\nI kinda like that, but I'd have to see the plates on it, looks really close proximity of the controls to the bridge which I don't like.\nI'm not a fan of Telemasters but this looks practical at least for smaller people who want a tele...not as ugly and weird as a 3\/4\" size Tele body.\nOkay idea, but $250! And it's scuffed around the edges.\nAnother lame mashup guitar, boring idea.\nReally just makes me want a regular style Duo II or Mustang with that pickguard\/finish colour scheme more than anything.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Different Hearing Aids and Their Functions The main purpose of devices that are used so that sound is amplified to alleviate impaired hearing is to separate background noise from critical speech signals that are necessary for the person to understand what is being said. In other words, it is not just enough to have your audiologist conduct a comparative hearing aid selection where the patient would pick up the one that fits them best. What is more important is what type of technology is used to amplify speech signal and control those unnecessary loud sounds that can choke one hearing tolerance. There are analog hearing aids that can detect loud or soft sounds. What this means is the device is able to recognize loud sounds like traffic sounds that it does not amplify further. This is the automatic gain feature of the device. The other types of hearing aids are the digital ones and this has a more sophisticated means of processing incoming sounds and converts hem into clearer and more audible sounds before it releases it to the patient's hearing at the appropriate sound level which is comfortable to him. Because of this smart technology, a more precise prescription is possible in reference to your degree of hearing loss, personal lifestyle and the cosmetic or appearance needs. With digital hearing aids, you can hear normal and look normal, unlike when you wear the traditional analog hearing aids.\nThese digital hearing aids are great simply because they are digital. This means that you don't near to throw it our if your hearing problem worsens. Reprogramming that same hearing aid by simply interfacing your device to the computer in the office of your hearing aid professional can easily be done and spare you from buying a new unit.\nTo take this hearing factor deeper, it must be remembered that at the center of human hearing is the most powerful signal processor found in our brain. The sound processor in our brains are simply left to guess that sound it hears if the sound input from both ears are not clear. In other words, when an endless stream of sound waves from both ears is then converted into electrical energy to be fed to the central processor is marred then it follows that its interpretation will also be incorrect. This means that if we simple amplify sounds without filtering it the way the brain filters it out, localizing certain sounds in the environment, discriminating particular sounds or signals, auditory cuing sounds that have been stored as memory, if there is no filtering of sounds of this nature, then everything will be distorted, the brain cannot process it properly and your common and rational perception is totally undermined, and the only way to bring it back to normal is with the help of human made technology what we are using today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rent a Dumpster in 92130 Now!\nSimply give us a call and we will answer any questions you may have about the 92130 dumpster rental procedure, allowed materials, size you may need, etc.\nOur roll off containers can usually be delivered in 92130 the day after you place your order. Make sure to call us early to ensure timely delivery.\nChoosing the top dumpster for your job is an important aspect of dumpster rental in 92130. If you choose a dumpster that is too small, you won't have enough room for your debris and waste, which means you'll have to schedule additional trips. In the event you select one that is too huge, you'll save time, but you'll squander money.\nIf you call a dumpster rental business in 92130 and describe the job for which you desire a dumpster, they can recommend the best size. Their years of experience mean that they often get it just about right. A 10-yard dumpster usually functions well for moderate-sized cleanup projects and little remodeling jobs. A 20-yard dumpster is the best choice for big dwelling cleanup projects and moderate-sized renovation projects, while a 30-yard dumpster has higher walls which are perfect for a house cleanout or remodeling jobs on a big house. 40-yard dumpsters are massively big and are used only on the largest projects such as new construction.\nMost dumpster rental businesses in 92130 usually rent containers for a regular rate for 7 days, although most one time customers end up keeping their container an average of three and a half days. When you call to reserve the container, conversation with all the customer support representative about the length of time you believe you'll need.\nIf it turns out that your project takes longer than you anticipated, only telephone the dumpster company to describe. Most businesses will bill you on a per-day basis following your initial seven days. It is important to communicate this added need as soon as you know you have it since most dumpster companies have a small variety of bins that may already be reserved. Most businesses will work with you to accommodate your additional rental time. In case you believe you'll need the dumpster for an additional week or more, be sure to mention that; rates may change based on a longer period of time.\nWhether or not you desire a permanent or roll off dumpster depends upon the kind of job and service you will need. Long-Lasting dumpster service is for ongoing demands that last longer than just a few days. This includes matters like day to day waste and recycling needs. Temporary service is exactly what the name implies; a one-time need for project-special waste removal.\nTemporary roll off dumpsters are delivered on a truck and are rolled off where they will be used. All these are generally larger containers that can manage all the waste that comes with that special job. Long-Lasting dumpsters are generally smaller containers since they're emptied on a regular basis and so do not need to hold as much at one time.\nIf you request a permanent dumpster, some companies require at least a one-year service agreement for that dumpster. Rolloff dumpsters merely require a rental fee for the time that you simply keep the dumpster on the job.\nYou will usually schedule the total amount of time you want to keep the dumpster when you first call to set up your dumpster service. This normally includes the drop off and pickup dates. Most businesses do request that you be present when the dumpster is delivered. This really is essential to make sure the dumpster is placed in the very best location for your endeavor. You don't have to be present when the dumpster is picked up to haul it away.\nIf you get into your project and realize you need pickup sooner or later than you initially requested, that is not a issue. Just call the business 's office and clarify exactly what you require, and they will do everything they can to accommodate your request. There might be times they can't fulfill your adjusted program just due to previous commitments, but they will do the best they are able to in order to pick your dumpster upward at the appropriate time.\nMost cities or municipalities don't have many regulations regarding dumpster rental in 92130 as long as you keep the dumpster completely on your own property during the rental period. If you have to position your container on the street in any way, you'll likely have to get a license from the correct building permit office in your town.\nMost dumpster rental businesses in 92130 will take care of securing this license for you if you rent from them. Make sure that in the event you're intending to place the dumpster on the street, the business has made the appropriate arrangements. You need to also guarantee that you get the license in a timely manner and at the right price. In case you thought the dumpster business was getting a license and they didn't, you'll be the one who will need to pay the fine that is issued by the authorities.\nIn the event you're looking to rent a dumpster in 92130, one of your main considerations is going to be cost. There are generally two pricing options available when renting a dumpster in 92130. Flat rate is pricing determined by the size of the dumpster, not the quantity of material you put in it. Per ton pricing will charge you based on the weight you need hauled.\nOne kind of pricing structure is not always more expensive than the other. Should you know exactly how much material you need to throw away, you might get a better deal with per ton pricing. On the other hand, flat rate pricing can help you keep a limit on prices when you're dealing with unknown weights. Whether you regularly rent dumpsters for work or just require a one time rental for a project throughout the home, you want to look around for the best rental costs predicated on your needs.\nWhile roll off dumpsters are normally left on place, front load dumpsters will come pick up debris on a set schedule. This makes it feasible for sterilization professionals to remove garbage and rubbish for multiple dwellings and companies in the area at reasonable costs.\nThe lowest size roll off dumpster generally accessible is 10 yards. This container will carry about 10 cubic yards of waste and debris, which is roughly equivalent to 3 to 5 pickup truck loads of waste. This dumpster is an excellent option for small jobs, including little house cleanouts.\nOther examples of jobs that a 10 yard container would function nicely for include: A garage, shed or attic cleanout A 250 square foot deck removal 2,000 to 2,500 square feet of single layer roofing shingles A small kitchen or bathroom remodeling job Concrete or dirt removal Getting rid of waste Be constantly aware that weight constraints for the containers are imposed, so exceeding the weight limit will incur additional fees. The standard weight limit for a 10 yard bin is 1 to 3 tons (2,000 to 6,000 pounds).\nA 10 yard bin can help you take good care of little jobs round the house. If you have a larger job coming up, have a look at some larger containers too.\nWhen you rent a temporary dumpster, your aim is to fill it up and possess the waste hauled away. But if you want your waste recycled, you might need to really go about it in a somewhat different fashion. Waste in the majority of temporary dumpsters is not recycled since the containers are so large and hold so much stuff.\nIf you're interested in recycling any waste from your project, check into getting smaller containers. Many dumpster rental companies in 92130 have a wide range of containers available, including those for recycling. These are generally smaller than temporary dumpsters; they are the size of routine trash bins and smaller.\nIn case you would like to recycle, figure out whether the company you are working with uses single stream recycling (you do not need to sort the material) or if you'll need to organize the recyclable stuff into distinct containers (aluminum cans, cardboard, plastics, etc.) This can really make a difference in the number of containers you should rent.\nMost people do not want to rent dumpsters unless they absolutely have to. Sometimes, though, it becomes apparent that you just need to rent a dumpster in 92130 for commercial and residential projects.\nMost cities will not haul away construction debris for you. It's your duty to be sure you have an appropriate container to collect discarded material from remodeling projects. Even if you just have a little job, municipal waste management is not likely to haul the debris away.\nA major clean out can collect more rubbish when compared to a standard receptacle can hold. Renting a tiny dumpster is a more suitable choice that'll prevent making multiple trips to the neighborhood dump.\nWhen you have one of these projects in mind, then you understand it's time to search for a dependable dumpster rental service in 92130.\nWhen you rent a roll off dumpster, you'll be tempted to throw anything and everything indoors. Common things which individuals typically dispose of in a dumpster comprise solid waste material and most routine household and building waste, together with tree limbs and landscaping debris. They are able to be utilized for business and commercial cleanouts, house renovations, getting rid of junk when you are moving to a new dwelling, bigger landscaping jobs and much more.\nA record of things you cannot set in a roll off dumpster includes paints and solvents, automotive fluids, pesticides, chemicals, electronics and batteries. Should you try and include these items, you'll likely end up getting an additional fee. Other things which will undoubtedly incur an additional fee include tires, mattresses and appliances. Additional heavy substances are also not permitted; items that are too heavy may exceed government transport regulations and be dangerous to haul.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Need a Garbage Collection Service in Sheep Springs, NM?\nGarbage Collection Service are available* in Sheep Springs.\nPreserve ample area for the dumpster in order to roll off of. This will give it time to return back and become easily found. There are many people, especially those which rental the dumpster the first time, who inserted the dumpster around the right place only to playground their automobiles around this, thus generating the dumpster hard to get at. Place cones and other barriers to stop others coming from parking their particular cars about the dumpster.\nAvailable in numerous sizes as well as for varying trips, a Garbage Collection Service in Sheep Springs, NM will give you a large receptacle capable of that contains all types of rubbish. Not only does renting a dumpster save you from loading your car stuffed with rubbish and generating numerous outings to the dispose of, it also lets you handle useless of all kinds. Helping you save time and letting you handle rubbish large and small, a Garbage Collection Service in Sheep Springs, NM service is a fantastic choice for: Next, be prepared to spend in advance. Many organizations accept funds or plastic cards. To get the best price, call all-around and shop around. Some firms will give you an arrangement if you obtain one.\nRegardless of the project, the benefits of renting tools cannot be overcom. Whether your work involves destruction, renovation, or maybe construction you will be happy you had a dumpster. Rather than depart things in a pile or maybe stacked upward along the side of the site you can simply toss in every thing you must dispose of. As an alternative to multiple everyday trips on the dump you can working until everything is done and then it is just one vacation and one convenience fee.\nYet do remember which any excess weight exceeding the particular negotiated car loan terms with the supplier will end up you actually in providing an extra volume, so It is often preferable choice to choose a tiny larger type of dumpster than what really is required. Yet another factor that should not be neglected is usually having a talk with the Garbage Collection Service in Sheep Springs, NM supplier about the dirt that are granted. It is advisable to look at the permitted as well as non granted items simply because some countries go by different regulations of giving a exception to some items like chemical substances, gasoline, oils, and other dangerous or flammable substances.\nAre you looking to rent a dumpster? Renting a dumpster can be a thing. Sometimes simply rent a dumpster because you have a lot of waste. Like for instance if you're remodeling the house. During redesigning there will be a lot of trash manufactured. If you have in no way remodeled the house then prepare yourself because there is gonna be a lot of waste even if you only remodel a modicum of your home.\nAnother choice is to gather the bulk squander and carry it off as part of your car for a local convenience facility, 1 load at any given time. Some towns may acknowledge this squander and others might not. Some may charge a fee for convenience. Meanwhile, there is a constant realize simply how much waste you've got until you try and fit it all in your vehicle.\nA dumpster use is unavoidable to home the trash with eco-friendly manner at affordable cost, but this is going to turn into a high-end mode if you aren't smart adequate to handle the job. Certain tasks like development generates big tons of squander that needs to be got rid of daily, working with a Garbage Collection Service in Sheep Springs, NM for everyday and acquire services makes sense if the waste generated with daily basis is equivalent to tonnage limit of 40 property i.age. 4 tons every day, it is the optimum size of the particular dumpster generally supplied by the specialist Garbage Collection Service in Sheep Springs, NM provider.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauzyr b/data_all_eng_slimpj/shuffled/split2/finalzzzauzyr
new file mode 100644
index 0000000000000000000000000000000000000000..1251a4c4ccdff8ab7940e622c2757f39d2cb5cae
--- /dev/null
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+{"text":"File includes one white t-shirt, with the text \"this is really random\" in sans-serif black lettering. Associated with Coupland's 1995 novel Microserfs.\nContains a brief letter sent to Coupland's parents, accompanying a clipping in which Coupland's novel is referenced (Pinkerton, James P. \"Economy Bites the Edge Off Generation X\" Los Angeles Times, February 24, 1994, B19).\nTicket date is for the day after Coupland presented Player One at the 2010 Massey Lecture in Toronto, Canada.\nContains an audiobook of selections from Polaroids from the Dead, read by the author (Harper Audio, 1996, approx 90mins).\nIncludes an author trading card for Douglas Coupland (#228) from The Booksmith in San Francisco. The back of the card advertises a reading by Coupland taking place at 7pm on Friday March 13th.\nContains a copy (2\/20) of Canada Goose, a self-published novella by Coupland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chinese buyers are continuing to look to Australia to purchase properties, with research from a leading international property portal for Chinese buyers revealing the top cities this growing market is looking to purchase in.\nAt The Adviser Asia Study Tour in Hong Kong yesterday (10 October), Jimmy Wong, the sales director of leading Chinese international property portal Juwai, revealed the top cities in Australia in which overseas Chinese property buyers are looking to invest.\nAccording to Juwai IQ data, the United States continues to be the number one market for Chinese property investment, with Australia being the second most popular market in terms of both the amount of money being spent on property and the volume of buying enquiries.\n\"Factors affecting demand include the rising wealth of Chinese consumers, which is the primary driver of Chinese investment [in Australian property],\" Mr Wong said, adding that there are now more than 1.6 (US dollar) millionaires living in mainland China and that that number is only expected to increase given that wealth per adult has more than quadrupled over the last six years.\nLooking at the type of Chinese buyers purchasing Australian property, the sales director noted that the majority of enquiries in Australian property listings were from coastal towns as well as the major cities such as Beijing and Shanghai.\n\"Historically, coastal towns are the richer areas in China, so it is perhaps unsurprising that the majority of enquiries for Australia properties is from provinces such as Guangdong, which is the number one province in China interested in Australian properties. Fifteen per cent of all Juwai's Australian property listing views are coming from this region,\" Mr Wong revealed.\nNotably, Mr Wong also revealed the top Australian destinations that overseas Chinese buyers were looking to purchase property in.\nMelbourne headed the list for demand, followed by Sydney and then Brisbane. Adelaide, Canberra and the Gold Coast are the fourth, fifth and sixth most popular Australian areas for Chinese buyers to purchase property in, followed by Perth and Hobart.\nThe Victorian towns of Newmerella and Walwa were the eight and ninth most popular locations, followed by NSW's Wisemans Ferry.\n\"Australia has always been very popular and has been number two behind the US for the last 10 years,\" Mr Wong told delegates.\nMr Wong concluded that Chinese buyer interest in Australian property was rising, and that Juwai had seen $3.55 billion worth of property enquiries for Australian properties through the website in the first half of 2018.\nThe Adviser Asia Study Tour comprises two days (10 and 11 October) of interactive presentations and seminars revealing how the Asian mortgage market has developed AI strategies to support traditional broking models using robo-advice to enhance the client experience.\nOnce again, the event aims to give forward-thinking brokers and players in the mortgage market critical insights into how their international peers dominate their respective marketplaces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What if you could always be in the right place, in front of the right people, with the right message, at the right time? Imagine the outcomes you could influence and the legendary status you could achieve. Become a marketing legend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's not your typical cocktail party debate. But there are merits to homeowners in comparing and contrasting propane vs natural gas for uses in the home.\nWhat are the differences between the two and which one is best for you?\nEither propane or natural gas is cleaner burning than other fossil fuels like coal or oil. So they are better for the environment. Both are popular fuels and both are used extensively across the country to heat homes and power appliances.\nThe ready availability of propane or natural gas in your particular part of the country could have a big effect on which fuel you choose for your home. For the sake of argument \u2013 let's assume that you have an equal option of using either.\nSo from that perspective, let's look at natural gas vs propane in terms of energy efficiency, cost, and environmental safety. What similarities and differences do we see in those areas?\nWe offer valuable information on a wide variety of topics, including energy. Read on so you can make the right decision on a clean fuel for your home.\nWhat Is Propane and What Is Natural Gas?\nAs noted above, both natural gas and propane are fossil fuels. Both were created from ancient plants and animals that decayed beneath the Earth's surface under extreme pressure and heat over millions of years.\nNatural gas is actually comprised of multiple gases: butane, ethane, methane, and\u2026propane. Propane is extracted from the other gas components during processing.\nBoth are odorless so processing adds an odor for detection purposes.\nPropane is compressed into its liquid state and is sold by the gallon. It is also called liquefied petroleum gas, or LP gas for short. Propane can be converted to the natural gas standard, cubic feet, for comparison purposes.\nIn order to compare the efficiency of these two fuels, we need to know the definition of a British Thermal Unit or BTU. Despite its name, it is no longer commonly used in the United Kingdom. So, no tea and crumpets needed here!\nBTU is the international measure of a unit of energy. One BTU is equal to the amount of energy used to raise the temperature of one pound of water one degree Fahrenheit. One BTU per hour equals 0.293 watts of energy.\nIn practical terms, one BTU is the heat produced by one lighted matchstick.\nBTUs are used in rating appliances such as air conditioners, kitchen ranges, furnaces, even outdoor gas grills. More BTUs means more heat produced (by the burners on a range or grill) or the quantity of heat removed from a room per hour (by an air conditioner).\nFor example, a propane furnace rated at 100,000 BTUs burns only 40 cubic feet per hour. A similar natural gas furnace burns around 97 cubic feet per hour.\nTo put it another way, propane burns less volume of fuel per hour than natural gas by a factor of 2:1. So, you will use much less fuel over time with propane.\nPropane is the energy efficiency winner!\nAs previously noted, propane is sold by the gallon in its compressed liquid form. You normally pay for delivery ahead of time and store it in a large tank near your home. This means you could run out if the delivery schedule gets interrupted.\nNatural gas is sold by the cubic foot, remains in its gaseous state, and arrives at your home in a pipeline. You pay for it as you use it. But, you won't run out.\nFor example, let's attach some numbers.\nWhen natural gas costs less than $20 per 1,000 cubic feet, and propane costs around $2.50 per gallon, then natural gas is usually a less expensive option for heating your home than propane \u2013 even considering propane's efficiency advantage as to burn rate.\nThat being said, prices for both fluctuate over time and which alternative is cheaper comes down to factors such as efficiency of your furnace and other appliances, and where you live. An energy audit can find any issues with your heating and save you even more money.\nSo, no clear-cut winner in propane vs natural gas cost!\nIn this area, there is no real evidence to mandate one over the other, since both fuels are clean-burning \u2013 especially in relation to other fossil fuels. Technically, natural gas is labeled a greenhouse gas if released into the environment, but it burns clean like propane.\nNatural gas is lighter than propane, so it dissipates more quickly if released into the atmosphere. In the grand scheme of things, that shouldn't sway your vote.\nPropane is considered a \"green fuel\", but neither propane nor natural gas is denounced as harmful to the environment. Propane is thought of as the more eco-friendly option, but the differences are very small.\nSo, no clear-cut winner in propane vs natural gas environmental safety!\nIf you are considering using propane to heat your home and power your appliances, then we recommend you check out this propane tank installation site to help your decision.\nWe've seen that propane delivers more BTUs per cubic foot and is more energy efficient than natural gas. If you currently have propane service for your home, we recommend you stick with it.\nPropane's portability (e.g., in 5-gal cylinders) makes it extremely useful to power a gas grill, giving you more freedom to set up a barbecue anywhere in your yard. It's also used extensively in the RV world for both heat and appliances.\nNatural gas grills, in contrast, are normally connected to a \"gas stub\" where the fuel comes in via a pipeline. So, the grill is stationary.\nIf you entertain the notion of switching from one fuel source to another, you may require a new furnace, range, grill, and other appliances, or even expensive infrastructure upgrades. That could be a huge expense.\nSo, even if natural gas could be cheaper for your home based on availability and cost per cubic foot, this investment can take years to recoup.\nWe are interested in a wide variety of topics and are only a click away. Give us your thoughts on propane vs natural gas fuels and how they could help us meet ever-increasing world energy demands. We'd love to hear from you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Had the urge to work on the attack animation. There was something I wanted to try, and while I'm not fully recovered yet, working is a lot easier than packing, for now.\nAnyhow, the other day I posted this attack animation that I made for the ongoing game. Basically it would be used for every weapon, with different weapons being drawn over the frames, and different speeds being used for different types.\nProblem is it's a lot of frames (9, not counting the idle frame). Because I have to do a sprite for every set of clothing, it's best to limit the number of frames. I also started to think that it might be a bit boring to just have a single hit attack.\nSo I tried animating an alternative mockup, with the goal of #1 limiting the frames, and 2# making it a multi hit that could loop.\nI managed to get it down to 6 frames, where the actual animation would be done in code, and frames are re-used. The weapon animation itself would then be more frames than the base animation.\nDepending on the weapon, the swing can also be slightly different, using the same frames.\nI may also add other attack frames that can be used to for special attacks. I could probably get away with doing 1-2 frame attacks that use the attack animation's startup and ending frames to smooth them out.\nHad a medical procedure done at the beginning of this month that took me out of commission. If you've tried to contact me in that time and I haven't responded, I apologize. I'll be going back over messages and such.\nI'll be moving near the end of this month, so once I'm able to move around more (hopefully in a few days), most of my time is probably going to be taken up with packing, so it's unlikely any work is going to get done this month.\nFetchApp stopped accepting my (still functional) credit card for some reason, so the games can't be purchased on my site at the moment. I'll be contacting my bank to see what's up and get that sorted out soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..d15915753ea0747fafb988c1d8310e68c57b044d
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@@ -0,0 +1,5 @@
+{"text":"Who is Useh Mudiya? Why do Adivasis gather to remember him every seven years?\nWhy does Jharkhand's Raghubhar Das government fear the Adivasis? What is the cultural, social and political background of the Patthalgadi movement?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Free shipping on purchases over 35 and save 5 every day with your target redcard. See more ideas about blue kitchen designs small patio kitchen ideas and kitchen designs.\nSmall kitchens with white cabinets.\nA white color scheme gives a small space a brighter more open feel.\nFirst try painting small cabinets white.\nSelect wood cabinets and floors that take the chill off white surfaces and give your small kitchen a cool modern vibe.\nDanielle decided on under counter cabinets only and used the space above the counters for floating shelves to give the room an airy.\nAntique white kitchen cabinets ideas for small kitchens white kitchen cabinets ideas can give you new concept and style of home design that you need.\nKitchens small kitchens kitchen colors white kitchens color white kitchen remodel remodeling hgtv magazine.\nWhite rooms appear bigger and if you paint your small kitchen cabinets white or another light reflective color the space will appear bigger.\nBut in this kitchen the white painted cabinets that rise up to the 9 foot ceiling really make the room seem more spacious.\nFind and save ideas about small white kitchens on pinterest.\nMix crisp white buttery white and gray toned white finishes to create a seamless look.\nWith the presence of nice looking style of.\nOpt for a white on white palette to really push back your small kitchens borders.\nShop for small white storage cabinets online at target.\nPaint colors for small kitchens dont have to be bold.\nFrom a design standpoint small kitchen cabinets can be made to look larger using several design tips.\nFive roomy cabinets and three deep drawers were all the balanises needed to store kitchen essentials.\nIn this collection of photos designers show how everything from color choices to cabinetry styles can enhance a tiny cooking space.\nThanks for visiting our gallery showcasing a variety of gorgeous special kitchen areas showcasing black appliances. Sticking with a neutral backdrop white cabinets white kitchen cabinets and a white backsplash in the kitchen creates an easy setting for you to work in splashes of color when you are looking for a spicy alternative to a clean white look.\nSlim appliances are nestled into a glossy wall of rich wood cabinets.\nSmall kitchens with white cabinets and black appliances.\n53 fantastic kitchens with black appliances.\nKitchens appear in a series of layouts from common to sleekly minimalist.\nSmall kitchens with white cabinets can help your space look bigger and brighter which can enhance the entire atmosphere of your room.\nBlack kitchen cabinets in small kitchen.\nVisit this page for more small kitchens with white cabinets.\nWhile stainless steel is undoubtedly the most popular appliance finish for a kitchen black takes a close second and whitewell thats a whole different topic.\nToday were going to focus on black appliances and how to coordinate them into a kitchen with white or gray cabinetsso whether you are thinking of buying black appliances.\nStainless steel appliances and chrome finish bar chairs gives a touch of modernism in this design scheme.\nThe stylish black and white island has a built in sink and pull out drawer dishwasher.\nSee more ideas about kitchen with black appliances kitchen cabinets with black appliances and black appliances white cabinets.\nWith medium toned wood cabinetry and a few small.\nSmall kitchen with black appliances traditional kitchens jp kitchen studiooconomowoc winorth central regional coty winner kitchen remodel under 40000 a set of antique german spice containers served as the design inspiration for this wisconsin kitchen.\nSmooth black appliances are a typical selection of premium stainless steel.\nThe large black kitchen island with white marble countertop compliments a bank of contemporary cabinets and stainless steel appliances.\nFor any style of interior and any size of kitchen using white kitchen cabinets will provide a clean fresh and aesthetically appealing look which would suit the taste and style preference of most people.\nA dining table in white marble extends directly from the island and five crystal single candle lights hang above the island.\nFind and save ideas about kitchen black appliances on pinterest.\nAs a general rule of thumb using black cabinets in a small kitchen space is not advised since it can make the space appear even smaller.\nDecorating a white or gray kitchen with black appliances.\nKitchens with black appliances a gallery showcasing photos of kitchens with black appliances.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For most seniors, one of the most appealing features of a retirement community is the social aspect. Living at home after retirement can become isolating. It might be a welcome change knowing that even just a walk down the hall or grabbing a cup of coffee from the caf\u00e9 is a chance to meet someone new. While some might prefer the quieter side to retirement, where they can enjoy the simplicity of their new home and be on their own schedule, others prefer to get involved and become active in the community. It is not uncommon for those moving into a retirement community to worry about how to get started making new friends. Below is a guide to helping you meet people in comfortable and fun ways at Riddle Village.\nOne of the easiest ways to meet people who share similar interests to you is through the many activities and programs that are available to residents. Riddle Village has many groups, such as the needlework club, the computer group and the book club for those looking for a relaxing hobby or to learn something new. For a more active approach, take part in programs like line dancing, yoga or tai chi, which are open to all residents with no sign-up or commitments necessary. Want to bring a new club to your new neighbors at Riddle Village? Maria Smith, the Activities Supervisor, can help you start your own club or get connected to someone else in the community with a similar interest. With so many options, you are sure to find the group or club that is the right fit for you.\nA pastime that friends of all ages enjoy together is going out to eat. At Riddle Village, dining is an easy and relaxing way to meet new people. After you move in, your floor representative and members of the Welcome Wagon will reach out to you and many will invite you out for dinner or drinks. The Thoroughbred Lounge and patio is a nice place to order up some cocktails and strike up a conversation. If you are settled in, but looking for a great way to meet others, head to one of our restaurants. Riddle Village encourages residents to ask the hostess to be seated with other residents who have an opening at their table and looking for others to join them. This welcoming and friendly environment is a great way to learn more about your neighbors and make some new friends.\nIt is easy to find a friendly face to enjoy our amenities with around Riddle Village. Stop by the game room where residents are always looking for more pool players to join in. Consider heading to the art studio where the door is always open to learn a new technique or take a class. Perhaps join in on a book club discussion in our library. There are so many options. When the weather is nice, the courtyard is another popular spot for social events. From spring to fall, residents enjoy friendly competition in bocce ball, shuffleboard and putting tournaments. Year round, take a group exercise class or work on a project in the woodshop. Our residents love seeing new faces!\nFor more information about what Riddle Village has to offer request a brochure or schedule a visit today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"REMEMBER: The IRS doesn't initiate contact with taxpayers by email, text messages or social media channels to request personal or financial information. In addition, IRS does not threaten taxpayers with lawsuits, imprisonment or other enforcement action. Recognizing these telltale signs of a phishing or tax scam could save you from becoming a victim.\nHere is a blog that might have items to help with you tax issues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By the time I got back from work I Uploaded 8 pictures with the whole thing are in HD version. From many pictures in King size mattress clearance photos, you might have Best Rated Mattress 2018 with design. Nevertheless, if you don't appropriate with ideas, you could still take a look inspiring creativity. Even so I advise you could use as an design design, although along with has terrific decor concept. Don't forget for share this King size mattress clearance mattress astounding king size mattress costco sealy king mattress mattress and bed deals to neighbors or your friend. So they are might get with to create they Best Rated Mattress 2018. I am exuberant could share along with together with to you all. Maybe you would probably find which to date are you seek.\nKing Size Mattress Clearance king size mattress clearance creative of full size mattress clearance clearance sale discount top mattresses brands. king size mattress clearance mattress glamorous memory foam mattress clearance cheap king size foam mattress double bed. king size mattress clearance creative of full size mattress clearance king size mattress sets mattress and bed deals. king size mattress clearance sunset lasalle euro top king size mattress rc willey furniture store how much are mattresses. king size mattress clearance mattress glamorous king mattress clearance king size mattress cost of mattresses. king size mattress clearance white king bed frame king size sleigh bed with drawers metal bed single size mattress. king size mattress clearance mattress astonishing king size bed and mattress set full size foam mattress double bed. king size mattress clearance mattress astounding king size mattress costco sealy king mattress mattress and bed deals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This English Oil is a light, golden coloured oil that has a nutty flavour and aroma with a hint of asparagus. It contains Omega 3, 6 and 9, as well as being high in vitamin E and having the lowest saturated fat content of any commonly available oil.\nDue to its mild flavour and high smoke point it is ideally suited for frying, roasting and grilling. However it is equally suited to creating fabulous dressings or simply drizzled onto bread. Making it one of the most versatile oils available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The following is a presentation on MW Paul Revere that was given by Brother George Daniel MacGregor Pushee III on Novenber 2, 1996. It is provided here for your education and further Masonic enlightenment. You are invited to reuse this research work, however, it is requested that you please contact Brother Dan at kych52va@aol.com for his permission prior to its use and to give him proper credit as the author of this research work.\nAs a young man growing up in Massachusetts I was always fascinated with the \"Early Colonial\" and Revolutionary War years; however, I never fully understood why until my Grand Aunt told me about \"the Pushee Family Roots\". As it turned out my ancestors are descendants of Gabriel Pouchee, a French Huguenot immigrant, found living in Ipswich, Massachusetts in 1726 when he married Sarah Williams. They had eight children and their oldest son, Nathan was born in 1732. Gabriel and his family moved to Winchendon in 1752. At age 25, Nathan married Elizabeth Priest in 1757 and enlisted in the French and Indian War in 1759. Gabriel died in 1775 with Nathan dying shortly afterwards in 1776. Nathan and Elizabeth had two sons, Nathan Jr., born in 1758 and David in 1760. Nathan Pushee Jr. was my great, great, great grandfather and at the age of 17 he was at the Battle of Bunker Hill in 1775. He served in \"Washington's Life Guards\", was at the battles of Trenton, Brandywine, and Monmouth and was captured at Monk's Corner near Charleston in 1780. He was not exchanged until after peace was declared. In 1783 he moved to Antigonish, Nova Scotia where he at age 25 he married Jane Porter and they had 10 children. Nathan Jr. died in 1838 at the age of 80.\nMy purpose of presenting this family history is to indicate that in a way I am a \"kindred spirit\" to Paul Revere. As you will see Paul Revere was the son of a French Huguenot immigrant; born in 1734, two years after Nathan Pushee; served in the French & Indian War as did Nathan, both were married in 1757; and Nathan's son, Nathan Jr. served in the Revolutionary War as did of course Paul Revere. I realize that this might be \"a stretch\"; however, I feel that it explains my fascination with this period of time in our history and in particular Paul Revere.\nThe first known record of one of my direct descendants being a Mason was my grandfather who was a member of William North Lodge in Lowell, Massachusetts.\nIn reviewing the \"Proceedings of the Grand Lodge of Massachusetts\" I was surprised that there was not any record of any Masonic Memorials, detailed information on the actual Special Communications called for \"chartering any of the Lodges\", details on the laying of the cornerstone at the State House, or any of Paul Revere's addresses.\nThe poem,\"The Midnight Ride of Paul Revere\" was written in 1860 by Henry Wadsworth Longfellow, 42 years after Revere's death. Longfellow's poem made Revere \"a famous patriot\"; however, he was also a man of varied talents, mason, artisan, businessman and family man. The purpose of this paper is to commemorate the 200th anniversary of Paul Revere's term as Grand Master of the Grand Lodge of Massachusetts. Brother Revere served as Grand Master from December 12, 1794 to December 27, 1797. The primary reference documents for this paper are a book titled \"Paul Revere and Freemasonry\" by Edith J. Steblecki (1) and the \"Proceedings of the Grand Lodge of Massachusetts, A.F. & A.M., 1792-1815\" (2).\nPaul Revere was born in Boston, Massachusetts in late December 1734 and was the second of twelve children born to Apollos Rivoire, a French Huguenot immigrant who came to Massachusetts in 1715 and apprenticed with a Boston goldsmith, and Deborah Hitchborn a native Bostonian descended from New England seafarers and artisans. At age 12 he was apprenticed to his father as a silversmith and earned extra money as a bellringer at the Old North Church in Boston. At age 19, as the eldest son he became the supporter of the family when his father died in 1753. He married Sarah Orne in 1757. They had one son, Paul Jr (who was also an active Freemason), and seven daughters. Sarah died in May of 1773 and in September he married Rachel Walker to help him care for his large family. Their union produced eight more children, four of them whom died before their father. Paul Revere died at age 83 on Sunday, May 10, 1818 (3).\nAt age 25, Paul Revere was initiated in St. Andrew's Lodge on September 4, 1760. He was the first candidate received after their Charter dated 1756 was received from the Grand Lodge of Scotland. He was raised on January 27, 1761. The records of St. Andrew's Royal Arch Chapter, functioning as part of St. Andrew's Lodge, indicate that Paul Revere became a Royal Arch Mason and Knight Templar on December 11, 1769 (4). Brother Revere was very active in his Lodge by his attendance and serving as Junior Deacon, Junior Warden, and Secretary before being installed at age 35 as Worshipful Master on November 30, 1770. At that time he was already serving as Senior Grand Deacon of the Massachusetts (Provincial) Grand Lodge. This was the beginning of his \"very active masonic career\" which is indicated by the list of his Lodge Offices attached to this paper (attachment 1, p 102 ref 1). He served nine (9) terms as Worshipful Master, 5 with St. Andrew's Lodge and 4 with Rising States Lodge and, with the exception of the Revolutionary war years, served continuously as a Grand Lodge Officer from 1769 to 1797. Time does not permit a complete definition of his role in the formation of Rising States Lodge and the Massachusetts Grand Lodge; however, he certainly was an active participant.\nThe culmination of Paul Revere's 34 years as a Freemason came in 1794 when he was elected as the second Grand Master of the newly-formed Grand Lodge of Massachusetts A.F. & A.M. Attachment number 2, \"Family Tree\" (5), shows the sequence of events leading to the formation of that Grand Lodge. The Lodges indicated represent the \"surviving Lodges in existence in 1792\".\nBy 1784, ten years earlier, Revere had already risen as high as Deputy Grand Master, Massachusetts (Independent) Grand Lodge, with his appointment under John Warren. He again served as Deputy Grand Master under Moses M. Hayes in 1791-1792.\nRevere at age 60 was finally chosen as Grand Master at the annual election of officers on December 8, 1794, a position he held through December 27, 1797. According to the Constitution of the Grand Lodge of Massachusetts at that time, the Grand Master was to be elected by a ballot at large with \"every voter writing the candidate he thinks best qualified.\" The brother who had two-thirds of the votes cast was elected Grand Master. Ironically, Paul Revere was not the first choice of the assembled brethren for Grand Master in 1794. RW John Warren, brother of the late Most Worshipful Joseph Warren, the first Grand Master of the Provincial Grand Lodge of Massachusetts under Scotland who died at the Battle of Bunker Hill (Breeds Hill), was again chosen but he \"declined accepting the Chair.\" The Grand Lodge then proceeded to another choice, when it appeared that the Rt. Worshipful Paul Revere was chosen; he not being present, a committee of three were appointed to wait on Brother Revere to know whether he would accept the appointment; which committee reported his acceptance.\nPaul Revere's installation as Grand Master took place on December 12, 1794 at Concert Hall, where \"the Most Worshipful John Cutler, then in Ample Form, installed the Grand Master and placed him in the chair of Solomon and invested him with his proper jewels.\" Duly installed, Revere appointed his Deputy, Deacons, Stewards, Grand Marshall and Sword Bearer, and invested them with their jewels (for officers see attachment number 3). According Grand Lodge minutes, \"a procession was then formed and the Brethren, in their proper order, paid their usual salutes and congratulations.\"\nDuring the three years that Revere was Grand Master, the Grand Lodge held 12 quarterly communications on the second Mondays of December, March, June and September and 7 special communications. According to the \"Proceedings\" (2), total attendance average 26; however, this information may not completely reflect the actual members present as indicated by additional comments in the \"Proceedings\". Paul Revere presided over every meeting held during his three years as Grand Master and the fraternity in Massachusetts expanded energetically under his direction.\nRevere was the first to wear the tricorn hat as Grand Master. Later discarded, the tradition was revived by John T. Heard in December of 1856 and continues to this day (6).\nThe celebration of feast days also continued. On June 24, 1795 the Feast of St. John the Baptist was celebrated at Concert Hall with a Masonic procession to and from the Chapel Church.\nRevere was the first Grand Master to appoint a Grand Chaplain. On December 12, 1796 Grand Lodge voted that \"the Most Worshipful Grand Master be authorized at every annual meeting to nominate and appoint a Grand Chaplain whose duty it shall be to attend the Grand Lodge and perform such clerical duties as shall be assigned him.\"\nSince the primary responsibility of the Grand Lodge was to charter new Lodges and to supervise the Lodges within its jurisdiction, the Grand Lodge under Revere devoted a significant amount of time to seeing that the Lodges were properly organized and legally conducted. As Grand Master, Paul Revere chartered twenty-three new Lodges within Massachusetts and Maine (attachment 4, p 107 ref 1) which established a record for more than 100 years.\nWhile Grand Master, Paul Revere also continued the distribution of the volume of Constitutions. The book was originally compiled in 1792 as a result of the union between the Massachusetts Grand Lodge and St. John's Grand Lodge. The book of Constitutions was a compilation of masonic history, charges, addresses, constitutions, laws and songs entitled The Constitutions of the Ancient and Honorable Fraternity of Free and Accepted Masons. Paul Revere was one of the thirteen masons who served on the original committee under the direction of the Reverend Thaddeus Mason Harris. The Grand Lodge intended to send a copy of the book to every Lodge in the State, as well as presenting the volume to all new Lodges Chartered. According to Revere, the constitutions were \"calculated with so much pains, for the benefit of the Craft, having upon all occasions, recourse to their Charges & regulations therein contained\". Revere obviously had a sincere interest in the rules and regulations which were designed to keep the fraternity running smoothly, as he often served on committees to revise masonic rules and bylaws. He also wrote the \"charges\" which were used in the installation of officers while he was Grand Master.\nDuring the years that Paul Revere presided as Grand Master, the Grand Lodge of Massachusetts was still in the initial stages of formation as indicated by the various problems it encountered and the resolutions it passed at Quarterly Communications. At the September Quarterly it resolved \"That when any Lodge is not presented in Grand Lodge and is in arrears, for a longer period than twelve months, it shall be considered as having relinquished its connection with the Grand Lodge, and not having a regular standing in the Commonwealth.\"\nAt the March 1797 Quarterly, a committee was appointed to write to the Lodge at Nantucket in regards to that resolution. Several months later, on August 27, Revere wrote a personal letter to Samuel Barrett at the Union Lodge in Nantucket informing him that \"except your Lodge send their Charter to, and paid this dues to the Massachusetts Grand Lodge, you would not be received by them, or acknowledged as a regular, constituted Lodge.\" Revere informed him of the resolution passed in Grand Lodge on September 9, 1795 relating to this matter indicating that as Grand Master, he felt it was his \"duty to represent to you your situation, as a Lodge, not doubting, you will take such steps as Free & Accepted Masons ought to\". The Lodge replied in early September that they had raised a committee to look into the matter.\nPaul Revere was so concerned not only with the proper conduct of the Lodges, but also with the quality of masonic candidates. While Grand Master, he wrote the ceremony for constituting a new Lodge, in which he instructed the officers to \"carefully enquire into the Character of all Candidates and recommend none to the Master, who in your opinion, are unworthy of the privileges, & advantages of Masonry, Keeping the CYNIC, far from the Antient Fraternity, where Harmony is obstructed by the Superstitious, & Morose.\"\nTen years after he served as Grand Master, Paul Revere's concern about candidates was still clear. He wrote that \"it is too much the practice of Lodges to admit the 'worthless and profane', to pollute our'hallowed Temple.' \"Caution' and jealousy, with respect to Candidates, cannot be too much impressed on all Lodges.\"\nIn February 1797, Revere chaired a committee of three to draft a resolve against the improper admission of candidates, when it came to his attention that persons who were rejected in one Lodge would \"afterwards apply to another Lodge within the jurisdiction and gain admittance.\" As Grand Master, Revere had the authority to prevent the \"worthless and profane\" from joining masonic Lodges within his jurisdiction. In June of 1797, Revere brought charges against Harmonic Lodge for directly violating his authority in this respect. According to the charges: the Lodge made \"a number of persons masons, without handing their names to the Grand Master for his approbation; made a number of persons masons \"all of them persons to whom the Grand Master had refused his approbation; on the same evening, they made \"nine persons masons, some of whom had not stood the usual time on the books, and without having first obtained a dispensation from the Grand Master for that purpose; and \"several times made a man an EA, FC, and a MM, the same evening, contrary to the usages of Masonry.\"\nOn June 28, 1797, Grand Lodge voted 15 to 9 to \"vacate\" the charter of Harmonic Lodge. Four men who had been made masons were \"prohibited from visiting any Lodge.\"\n\"It is the opinion of this Grand Lodge that every Lodge under this Jurisdiction, before they proceed to ballot for any person to be made a Mason---it shall be their duty to make a strict enquiry, what town in the State such candidate belongs to; and if appears that he is a citizen of any town in this Commonwealth, where a regular Lodge is constituted, or if he lives within five miles of a constituted Lodge, other than the one to which he is proposed, it shall operate as an exclusion, without a recommendation of the Master and Wardens of the nearest Lodge where he belongs; for no person's character can be so well known as in the town or neighborhood where he belongs. And it shall be the duty of Masters of Lodges, when a character is rejected for the want of such a recommendation as a free Mason ought to have, to direct their respective secretaries to acquaint the Masters of the adjacent Lodges, with the name and circumstances of the person rejected and also the like information to the Grand Lodge.\"\n\"1st. The Grand Secretary, under the direction of the Most Worshipful Grand Master, after the choice of officers each year, shall transmit a list of the new officers chosen, or re-elected as also, an attested copy of all new general Regulations adopted, to the Grand Lodge of every State in the Union.\"\n\"2d. No charter of erection, or dispensation, shall be granted to any number of Masons residing out of this state,except when the Grand Lodge of the State in which the petitioners reside shall acquiesce therein in writing.\"\n\"3d. The Grand Lodge will not hold communication or correspondence with, or admit as visitors, any Masons residing in this State who hold authority under, and acknowledge the supremacy of, any Foreign Grand Lodge or who do not by their representatives communicate and pay their dues to this Grand Lodge.\"\n\"That it is the sense of this Grand Lodge, that the construction of the 2d Article, of the second section, of the Constitution, is and ought to be, that every Lodge, represented in Grand Lodge, shall have one vote, and no more, whether they have one or more representatives.\"\nIn reviewing the correspondence recorded in the \"Proceedings, numerous letters were noted; however, the exchange of letters between Paul Revere and George Washington in early 1797 are of particular interest, see attachment number 5.\n\"This Corner-Stone intended for the use of the Legislature and Executive Branches of Government of the Commonwealth of Massachusetts was laid by His Excellency Samuel Adams, Esquire Governor of said Commonwealth assisted by the Most Worshipful Paul Revere, Grand Master; and Right Worshipful William Scollay, Deputy Grad Master; The Grand Wardens and Brethren of the Grand Lodge of Massachusetts on the 4th Day of July 1795, being the 20th Anniversary of American Independence.\"\nThe procession was lengthy and must have been an impressive one. After gathering at the Representative's Chamber, the participants proceeded to the Old South Meeting House to hear an oration, after which they proceeded to the new State House site in the masonic order listed in attachment number 6.\n\"Worshipful Brethren, I congratulate you on this auspicious day:--when the Arts and Sciences are establishing themselves in our happy Country, a Country distinguished from the rest of the World, by being a Government of Laws.--Where Liberty has found a Safe and Secure abode,--and where her Sons are determined to support and protect her. Brethren, we are called this day by our Venerable + patriotic Governor, his Excellency Samuel Adams, to Assist him in laying the Corner Stone of a Building to be erected for the use of the Legislature and Executive branches of Government of this Commonwealth. May we my Brethren, so Square our Actions thro life as to shew to the World of Mankind, that we mean to live within the Compass of Good Citizens that we wish to Stand upon a Level with them that when we part we may be admitted into that Temple where Reigns Silence & peace.\"\nAfter the Operative masons prepared the cornerstone, it was laid into place by Governor Samuel Adams, assisted by Grand Master Paul Revere and the Deputy Grand Master. Beneath the stone, Revere had placed \"a number of gold, silver and copper coins, and a silver plate\" which bore the inscription. The ceremony was concluded to the roar of cannon and the cheering crowd.\nJuly 4, 1995, the Grand Lodge of Massachusetts under the direction of Most Worshipful David W. Lovering conducted a re-enactment of this event at the invitation of the Department of Tourism. The event was not attended by the current Governor or Lt. Governor.\nThe ceremony on Boston Common in 1795 was reminiscent of a similar event which took place only two years earlier in Washington. On September 18, 1793, George Washington, in a full masonic ceremony, laid the cornerstone for the National Capital Building.\nPaul Revere's public life as a Freemason was coming to an end by late 1797. Revere had served as Grand Master for three years. It was the last major masonic office he would hold. As new officers were chosen on December 11, 1797, Revere \"was then please to address the Grand Lodge in a fraternal manner in which his abilities in the masonic art were eminently displayed.\" Revere viewed this masonic election as a farewell. He knew that having served as Grand Master for three years, he was ineligible for re-election, according to the Constitution, but he could say with pride that there were \"upwards of forty\" Lodges within the jurisdiction, most of which were represented and paid dues. This nearly doubled the number of twenty-two Lodges which were in the jurisdiction after the two Grand Lodges united in 1792, each Grand Lodge having brought eleven Lodges to the union (attachment number 7, ref 7).\nAfter three productive years, Paul Revere's address reveals his contentment with the past, as well as the issues which concerned him for the future of masonry. The Union Lodge at Nantucket still had not compiled with the regulations, but he was confident that it would do so if his successor \"provided a little attention.\"\nRevere also encouraged a \"free correspondence\" between masonic Lodges in the United States and abroad. The Lodges of England and Nova Scotia had already extended a cordial correspondence with America, and Revere thought the same should be done with \"Quebeck,\" as a means of \"securing the friendship of that Body of Masons against those persons who may wish to make innovations in Masonry.\" With just a little attention to correspondence, Revere was certain \"that we shall soon have the pleasure to Communicate with every Grand Lodge thro' the Globe.\"\nRevere was also concerned with the \"necessity of subordination among Masons,\" hoping that the Lodges would conduct themselves according to the old traditions. He encouraged \"a careful attention to our Constitution; that you never suffer the antient land-marks to be removed; that a Strict attention be paid to every Lodge under this jurisdiction\" so that \"they be not suffered to break thro, or, treat with neglect any of the regulations of the Grand Lodge.\nRevere also recommended that \"a Committee be raised to form regulations for the disposal of Charity, or any other thing that will add to the happiness of Masons.\" A committee was immediately appointed \"to take into consideration the recommendations made by the Most Worshipful Grand Master\" during his final address.\nAs Paul Revere left the highest masonic office in the State, he professed great hopes for the future. According to Revere, Freemasonry \"is now in a more flourishing situation than it has been for Ages\" and there is \"no quarter of the Globe but acknowledges its Philanthropy\". He felt it was \"the greatest happiness\" of his life.\n\"to have presided in the Grand Lodge at a time when Freemasonry has attained so great a height that its benign influence has spread its self to every part of the Globe + shines with more resplendent rays, than it hath since the days when King Solomon implored our immortal Grand Master to build the Temple.\"\nRevere began his address with an apology, assuring his masonic brethren that he \"endeavored to pay every attention to what I esteemed my duty,\" adding that \"I have ever omitted to do one act that appeared to be for the good of the Craft\" but \"if I have done what I ought not to have done, you must impute it to my head & not to my heart.\"\nRevere closed his address by extending to his fellow masons \"my most sincere and hearty thanks for your candor, and assistance,\" since it was owing to your kind attention and assistance that I have been enabled to do the little good which has been done\" He encouraged his brethren to \"continue the same kindness to all my Successors in office.\"\nOn December 27, 1797, Revere installed the new Grand Master and officially ended his three year term. The Lodge voted that thanks be given \"to our most Worshipful Master Paul Revere for his eminent service rendered this Grand Lodge while in the Chair of Solomon.\nThe \"Masonic Years\" of Paul Revere were the glorious age of masonry, prior to the anti-masonic crusade of the 1820's, when the Society grew unhindered and flourished in public processions and ceremonies, laid cornerstones and attracted men of influence in every community. He must have enjoyed the pomp and ceremony of the aged traditions, particularly when he filled the Chair of Solomon as Grand Master.\nIn 1796, fellow mason William Bentley noted simply that \"Col. Revere enters into the Spirit of it, and enjoys it.\" Two hundred years later, these words should represent the concept that all of us should strive to accomplish in masonry today.\nMASSACHUSETTS GRAND LODGE - \"FAMILY TREE\"\nFrom: The East, the West and the South, of the Grand Lodge of Free and Accepted Masons of the Commonwealth of Massachusetts.\nWishing ever to be foremost in testimonials of respect and admiration for those virtues and services with which you have so long adorned and benefited our common country; and not the least, to regret the cessation of them, in the public councils of the Union; your Brethren of the Grand Lodge embraces the earliest opportunity of greeting you in the calm retirement you have contemplated to yourself. Though as citizens they lose you in the active labors of political life, they hope as Masons, to find you in the pleasing sphere of Fraternal engagement.\nFrom the cares of state and the fatigues of public business our institution opens a recess affording all the relief of tranquillity, the harmony of peace and the refreshment of pleasure. Of these may you partake in all their purity and satisfaction; and we will assure ourselves that your attachment to this social plan will increase; and that under the auspices of your encouragement, assistance and patronage, the Craft will attain its highest ornament, perfection, and praise. And it is our ardent prayer, that when your Light shall be no more visible in this earthly temple, you may be raised to the All Perfect Lodge above; be seated on the right of the Supreme Architect of the Universe, and there receive the refreshment your labors have merited.\nTo the Grand Lodge of Ancient, Free and Accepted Masons of the Commonwealth of Massachusetts.\nIt was not until within these few days that I have been favor'd with your affectionate letter, dated Boston, 21st March. for the favorable sentiments you have been pleased to express on the occasion of my past services, and for the regrets with which they are accompanied for the cessation of my public functions, I pray you accept my best acknowledgments and gratitude.\nNo pleasure, except that which results from the consciousness of having to the utmost of my abilities discharged the trusts which have been reposed in me by my country, can equal the satisfaction I feel for the unequivocal proofs I continually receive of its approbation of my public conduct; and I beg you to be assured, that the evidence thereof, which is exhibited by the Grand Lodge of Massachusetts, is not among the least pleasing or grateful to my feelings.\nIn that retirement which declining years induced me to seek and which repose, to a mind long employed in public concerns, rendered necessary, my wishes that bounteous Providence will continue to bless and preserve our country in peace, and in the prosperity it has enjoyed, will be warm and sincere; and my attachment to the Society, of which we are members, will dispose me always to contribute my best endeavors to promote the honor and interest of the Craft.\nFor the prayer you offer in my behalf, I entreat you to accept the thanks of a grateful heart, with assurances of fraternal regard and best wishes for the honor, happiness and prosperity of the Craft, and of all the members of ye Grand Lodge of Massachusetts.\nClergy and \"Strangers of Distinction\"\n\"The First Knight Templars Created in the United States\"\n\"Massachusetts Masonic Beginnings, A Family Tree\"\n\"Paul Revere: Mason, Patriot, Businessman, 1735-1818\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Click the menu options to view a pdf of the full menu.\nA Selection of favourites kids like to sink their teeth into.\nTraditional breakfasts, Local's favourites, omelettes, eggs benny and skillets. Classic burgers and sandwiches, wraps, salads, clubhouses, small plates, big plates and bowls.\nFour options for $5.95 each, available from 7am \u2013 11am daily.\nA Daily Special (Monday to Friday) plus a Daily Sandwich and Breakfast Special.\nEverything at Plates Restaurant is prepared fresh daily.\nEnjoy our friendly atmosphere and browse our collection of license plates and automobile memorabilia.\nFrom all day breakfast selections and tantalizing Daily Specials to Thursday's Roast Beef & Yorkshire Buffet we've got what you need to curb your appetite.\nOpen 7am to 3pm, 7 days a week.\nRoast Beef & Yorkshire Buffet every Thursday evening starting at 4:30pm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reviews of A & M Pople Ltd - Fascia\/Soffits\/Guttering, Roofer based in Highbridge, Hampshire.\nRemoval of broken satellite dish on chimney.\n2 vent tiles installed and fix GRP valley trough to valley.\nRemoved defective dry verges to front and rear of property and supplied sand\/cement mortar and rebed verge tiles, pointing verges to a smooth finish.\nRectify ingress of water through our roof.\nRepairs to roof ridge and hip tiles.\nBay Window Converted from Flat Roof to Pitch Roof and Roof Repairs to rear of house.\nTemporary repair on leaking roof.\nRepairs to roof (back and front) and fire wall.\nLarge extension to existing roof, to accommodate extension to house and conversion of double garage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My fountain pen obsession continues, unabated. The problem is, once you start, you find out what kinds of features you enjoy in a fountain pen, and which don't work so well for you, and you have to keep on acquiring pens explore that even further. The other problem is that if you develop an 'ink habit' too, you want to get more pens so that you can have pens filled with each of your favourite inks. A very kind birthday gift of some money allowed me to indulge my fountain pen problem, I mean hobby, as I could buy a pen and ink bottle duo I'd had my eye on for a while.\nThe TWSBI Diamond Mini is \u2014 as the name suggests \u2014 a smaller version of the TWSBI Diamond 530\/40\/80. I already have a TWSBI Diamond 540, which is currently filled with J. Herbin's wonderful Lierre Sauvage (ivy green), so I knew that the build quality was very good and that I liked the general style and the piston filling mechanism1. However, as soon as I saw that TWSBI had brought out a mini version of the Diamond, I wanted to get one. I'm a sucker for mini pens, and because of my tiny Hobbit hands, I do find the 540 a little on the long\/heavy side. I can't write with the cap posted, for example, because it unbalances the pen, but even unposted, it is very slightly too large in my hands. I use it for marking (hence the green ink), so that isn't a huge problem. After too many essays or exam scripts, my hand forms a tight, angry claw anyway, so a large and robust pen is beneficial in those circumstances. I saw the Diamond Mini as a possible alternative note-taking pen that would be small, reliable and that I could keep inked up in another colour to keep my Lamy 2000 company.\nI really like the Mini. It seems to have shot to second place in my fountain pen league table, second only to the hallowed Lamy. It's incredibly cute when capped, but the perfect size when you post the cap. Brilliantly, there are threads on the end of the pen so that you can screw the cap on to post it securely. It's about one full turn for a firm fit, so if you're marginally obsessive-compulsive like me, you want to start with the clip uppermost (i.e. on the same side as the top of the nib) before you screw it on, and then you end up with the clip on top where it looks neat and doesn't interfere with your grip. My hands are so Hobbit-sized that I can actually also use it fairly comfortably unposted, but it is better balanced when posted.\nI chose the model which has a black cap, section and piston knob, but with a clear barrel, so that you can see how much ink you have left. You can also get a fully transparent model or a fully opaque (black only) model, all of which look smart. I also got a medium nib, naturally. The filling mechanism is quick (and incredibly easy with the Ink Bottle \u2014 see below), and the nib writes beautifully smoothly. It's not quite as wet and buttery smooth as the Lamy, but it is a very close second and addictive to write with. In some ways, it would make a better note-taking pen, as it is slightly drier, so there would be no problems with smearing. It starts perfectly each time and feels very natural and comfortable in my hand. I can't vouch for how well it would work for you if you have larger hands, but I think unless you prefer huge, heavy pens, you would find it comfortable. It would certainly be a very convenient pen to keep clipped in a pocket as it's so short when capped.\nI bought the TWSBI pen and ink bottle from The Writing Desk, and as a nice bonus, they had an offer for a free 30 ml sample of one of three Diamine inks with the purchase of a TWSBI pen. I chose 'Ancient Copper' as I don't have a nice brown at the moment. It's a gorgeous colour: a lovely rich, warm, autumnal browny-red, that has great shading, even with a M nib. It's a treat to write with, and looks beautiful on the page, so my Diamond Mini is filled with that ink at the moment.\nTWSBI have made a special ink bottle for some time, and I had thought about getting one before but it seemed a little pricey. Then I saw that they were making a slightly cheaper version called the 50P that replaced the aluminium cap of the original with a plastic cap. I'm sure the aluminium cap version is lovely, but the 50P is great and the reduction in price makes it more affordable.\nYou may wonder what is the point of buying an ink bottle. Surely inks come in bottles from which you can fill your pen? Well, yes, they do, but the bottle designs can make it difficult to fill your pen when the level runs low. The Diamond Ink Bottle has a couple of clever tricks that make it very easy and clean to fill your pen. If you are using a non-Diamond pen, there is a tapered plastic insert in the bottle that sits inside the neck and touches the bottom of the bottle. There are a couple of slits in the top, so if you cap the bottle and invert it, the ink flows into this insert. This means that even if the level in the bottle is low, you can still easily submerge the nib in the ink inside the insert. It's a simple thing, but very clever.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The 2016 Miami Dolphins season is the franchise's 47th season in the National Football League, the 51st overall and the first under head coach Adam Gase.\nOn January 9, Gase was hired as the twelfth head coach in Dolphins history. He was the only candidate to receive a second interview. On January 12, Gase hired Bengals' defensive back coach Vance Joseph as defensive coordinator and linebackers coach Matt Burke, who worked with Joseph in Cincinnati, in the same position. Gase also hired Washington Redskins' offensive line coach Shane Day as tight ends coach, San Francisco 49ers' offensive line coach Chris Foerster in the same position and former NFL wide receiver Shawn Jefferson as the wide receivers coach. Assistant general manager Eric Stokes was promoted to senior personnel executive.\nMarch 7, 2016 - The Dolphins acquire LB Kiko Alonso and CB Byron Maxwell from the Philadelphia Eagles in exchange for swapping first round picks (8 and 13).\nMarch 8, 2016 - The Dolphins sign DE Mario Williams on a 2-year, $16 million contract.\nMarch 9, 2016 - The Dolphins sign S Isa Abdul-Quddus on a 3-year, $12.75 million contract.\nOn March 9, the Dolphins lost Olivier Vernon, Lamar Miller, Derrick Shelby, and Rishard Matthews to other teams during free agency.\nThe team traded its first-round selection to the Philadelphia Eagles in exchange for a first-round selection, cornerback Byron Maxwell, and linebacker Kiko Alonso.\n# Indicates game simulcast on WFOR-TV.\nThe Dolphins wore orange Color Rush uniforms for this game.\n\u2191 Kelly, Omar (January 2, 2016). \"Dolphins to fire GM Dennis Hickey\". Sun-Sentinel. Retrieved January 4, 2016.\n\u2191 Perkins, Chris (January 4, 2016). \"Dolphins name Chris Grier general manager\". Sun-Sentinel. Retrieved January 4, 2016.\n\u2191 Perkins, Chris (January 4, 2016). \"Grier: Dolphins are going to build a competitive roster at all positions\". Sun-Sentinel. Retrieved January 4, 2016.\n\u2191 Hyde, Dave (October 5, 2015). \"Dolphins name Chris Grier general manager\". Sun-Sentinel. Retrieved January 4, 2016.\n\u2191 Kelly, Omar (January 4, 2016). \"Former Falcons coach Mike Smith becomes first coach to formally interview\". Sun-Sentinel. Retrieved January 4, 2016.\n1 2 Kelly, Omar (January 5, 2016). \"Dolphins have expressed interest in Sean Payton, source says\". Sun-Sentinel. Retrieved January 5, 2016.\n\u2191 Beasley, Adam (January 9, 2016). \"Dolphins name Chris Grier general manager\". Miami Herald. Retrieved January 9, 2016.\n\u2191 Beasley, Adam (January 12, 2016). \"Miami Dolphins name Vance Joseph defensive coordinator, hire other assistants\". Miami Herald. Retrieved January 13, 2016.\n\u2191 Houtz, Josh. \"Dolphins sell naming rights to Hard Rock International\". The Phinsider. The Phinsider. Retrieved August 17, 2016.\n\u2191 \"Dolphins UFA\". Spotrac. January 4, 2016. Retrieved January 5, 2016.\n\u2191 Jackson, Barry (January 4, 2016). \"New Dolphins GM Chris Grier vows 'talk of dysfunction' within organization is over\". Miami Herald. Retrieved January 5, 2016.\n\u2191 Jackson, Barry (January 12, 2016). \"Miami Dolphins waive Jason Fox\". Miami Herald. Retrieved January 13, 2016.\n\u2191 Chase, Chris (March 10, 2016). \"Dolphins complete trade for Byron Maxwell, Kiko Alonso\". ESPN. Retrieved March 10, 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ADMISSION: (+GST): $13.50 adult, $11 students\/seniors, $10 youth, $38 family, 5 and under free.\nTo Jun 16 The Girls Are Coming!: A visual voyage of Bride Ship Tynemouth. In 1862, the screw steamer Tynemouth set sail from Dartmouth, England to Victoria, BC carrying 292 passengers including 60 young women. These \"girls\", some as young as 12 years old, came to BC to marry settlers and American prospectors. As they once did off Vancouver Island a century and a half ago, sails appeared in Tracy McMenemy's vision of an exhibition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nat Struct Biol. 2000 Oct;7(10):838-42.\nThe ends of the affair: capping and polyadenylation.\nNearly all mRNAs are post-transcriptionally modified at their 5' and 3' ends, by capping and polyadenylation, respectively. These essential modifications are of course chemically quite distinct, as are the enzymatic complexes responsible for their synthesis. But recent studies have uncovered some similarities as well. For example, both involve entirely protein machinery, which is now the exception rather than the rule in RNA processing and modification reactions, and the two reactions share one important factor, namely RNA polymerase II. In this brief review, we describe progress in understanding the enzymes and factors that participate in these two processes, highlighting the evolutionary conservation, from yeast to humans, that has become apparent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MonetizeMore builds industry leading ad technology that is seen by more than 300M people per month. The company has been running for 8 years achieving consistent double digit growth each year with a team of 80+ team members spread across the globe.\nMonetizeMore offers location and schedule freedom to every one of its team members. That means that you would have the lifestyle autonomy to choose to work from anywhere in the world, during the time of day you prefer.\n5 work day. Take your life back into your hands by joining the MonetizeMore team!\nThe product team is the fastest and most innovative team in the company. Build greenfield technology that is disrupting the ad technology industry.\nSolve problems that have never been solved before. Join a company culture that replaces constant meetings and interruptions with innovation that continues to break boundaries.\nTake your skillset to the next level with some of the best minds in the ad technology industry to make a real difference with MonetizeMore.\nDeveloping APIs and integrating with 3rd party APIs to automate manual tasks.\nImplementing UI facilitating third-party reporting and streamlining ad operations.\nIntegration of services to maximize ad revenues and maintain strong user experience.\nPlanning and prototyping new applications.\nDefect resolution of existing and new issues.\nUnit testing new features to ensure they conform to MonetizeMore's quality standards and meet requirements.\nStaying up to date with new trends and advancements in web development and ad tech.\nAttending daily stand-up meetings (15 mins) and other Scrum meetings (Every 3 weeks).\nCollaboration : Working remotely on complex projects necessitates that you work together with your team and share knowledge.\nCommunication Skills : You are comfortable communicating in English at all levels, have strong spoken and written communication skills and are an active listener.\nTeamwork : You value team synergy and are excited about helping your team succeed.\nInterpersonal Skills : You are able to get along, work well and coordinate with others.\nConflict Management : As a team, we are proactive in dealing with conflict. You are able to find constructive ways of resolving issues with other team members.\nAnalytical and Problem Solving Skills : You work hard to understand technical issues and to resolve them in an effective manner.\nDetail Orientation : You work on many parts of an application or system at the same time and are able to focus on each detail meticulously.\nInitiative : You work well in a team, with little supervision, making well-reasoned and effective technical decisions.\nReliability and Responsibility : You demonstrate reliability at all times. You give reasonable expectations within Agile Scrum framework and work hard and smart to achieve and surpass those expectations.\nYou communicate what you are going to do, then meet that commitment.\nThought Leadership : You analyze MonetizeMore's tech stack, systems and processes with the goal to iterate on a regular basis.\nYou look for opportunities to improve to increase value to MonetizeMore and suggest them to the team.\nIf you think you are a good fit to join the MonetizeMore product team, please apply below and give specific reasons what sets you apart.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"costo trituradora s crusher . costo trituradora s crusher, trituradora de piedra - Traducci\u00f3n al ingl\u00e9s \u2013 Linguee. A stone crusher, also known as rock crusher, is one of the most commonly used mining crushers. . trituradoras de piedra > M\u00e1quina trituradora de cono.\nIt incorporates three crushing types and it can be operated 720 hours continuously. . pex 250 1000 jaw crusher \u00b7 alta eficiencia de piedra trituradora de impacto . de maquinas para minerao em vespasiano \u00b7 where to buy inch black crusher.\nUsed Powerscreen XR400S Tracked Mobile Jaw Crusher Plant in Greenville, South Carolina, United States for sale, inspected and guaranteed. 44\" Wide x 5' Across Jaw Opening, Single Deck Screen, 36\" Wide End Discharge Conveyor, 24\" Wide Right Side Discharge Conveyor, 18\" Track Shoes, C9Engine, CE Plate.\nMaxo Mobile Jaw Crusher stone crusher machines mobile diebold bau.eumobile jaw crusher, also known as wheel monted mobile jaw crusher and wheeled mobile jaw crusher, is a rock crushing plant that consists of machine frame, jaw crusher, vibrating feeder, vibrating screen, conveyor belt, , maxo mobile stone crusher equipment stone crusher.\nSep 20, 2017\u00b7 GATOR single_Crusher-Jaw Crusher - Gator Machinery. Trituradoras de mand\u00edbulas: PE, PEC, PEX Trituradora de mand\u00edbula son ampliamente utilizadas en la miner\u00eda, industria metal\u00fargica, material de. Contact Us. Trituradora de mand\u00edbula m\u00f3vil del cemento de la metalurgia .\ncedarapids jaw crusher 1036-XSM trituradora de compra venta Impactor S\/N 16090(Secondary) on tandem carrier, Item# R201; Trio CT 32X54 Jaw Crusher S\/N , [Inquiry] trio crusher aggregate equipment - MachineryTrader New Trio 10x48 Jaw Crusher Trio 10\" x 48\" single-toggle jaw crusher w\/ motor , 32x54 Jaw Crushing Plant New .\nJaw Crusher Crushersparts. Jaw Crusher. The Jaw Crusher process not only crushes material of a medium and hard rock levels, but various kinds of materials, compressing between the fixed jaw and the moving jaws. Which is attached to a rotating cam shaft. The fixed jaw is an integral part of the crusher.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Jobs to be Done Discovery as a Service \u2014 Stratomic.\nOur unique outcome-driven framework gets inside the minds of your customers and prospects to uncover their important, under-served needs.\nStratomic's unique outcome-driven framework has identified sustainable high-growth opportunities in some of the most competitive industries.\nWith a relentless focus on your target audience's desired outcomes, you will add certainty to both your product roadmap and go-to-market strategy. What's more, when they're aligned, magical things start happening in your business.\nSustainable growth is tough. But with our help and expertise, you can rest easy at night.\nThere are many reasons why tech startups fail, but from our experience, there are a few that come up far too often. Inability to find market fit, running out of cash, and a lack of focus all rank highly, but they are all symptoms. What are the real causes?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Google has beaten Microsoft to take the title of the UK's number one brand, according to a poll out today.\nThe search engine was named the public's top superbrand for the first time, rising from last year's third position.\nHowever Microsoft remained dominant over rival Apple which managed 11th position, with luxury car company Mercedes-Benz in third place.\nNone of the big four supermarkets \u2013 Tesco, Sainsbury's, Asda and Morrisons, made it into the top 100. Asda dropped 253 places, Tesco 230 and Sainsbury's 194 places.\nPremium food brands performed better, with Marks & Spencer making the top 20 and Waitrose taking 179th place.\nFast food retailers also suffered significant falls, with Burger King dropping 317 places and McDonald's falling 235 places.\nStephen Cheliotis, chairman of the Superbrands Council which carried out the research, said: \"Lifestyle brands, particularly those in the technology sector, have considerably more sway with the public than everyday staples such as the supermarkets.\nThe Superbrands Council is a group of marketing, advertising and media experts.\nMembers rated a list of 1,350 brands according to quality, reliability and distinctiveness.\nThe brands were then assessed by 2,200 consumers surveyed by YouGov researchers.\nThis survey was used to create the list of the top 500 superbrands.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our product development team likes to remind each other that we are in a fashion industry. We need to make sure we are on trend, or slightly ahead of the trend, to ensure the buyers have something fresh and exciting to work with. At Birchwood, we research the upcoming trends in upholstery, and work with our fabric suppliers to provide beautiful, rich fabrics at a price that will be competitive in the retail market.\nFor 2018, the buzz word was texture. Multiple textures, such as velvet and leather toss pillows on a boucle sofa. The new plain fabrics selected for Spring 2018 were chosen to intermix with each other, as well as the existing fabrics in the Birchwood fabric line.\nBlue continues to be a strong performing colour. There were several new introductions for 2018 to add to our offering in this colour. The notable patterns in this range include Geode Lapis, a fun pillow\/chair pattern; Grayson Midnight, a heavy boucle with an interesting mix of yarn colours; and Royale Eclipse, a rich ink-coloured velvet.\nLivable, cleanable fabric is also an important portion of the market. It's important to be able to offer similar fabrics in our line. Following the success of Peyton, Sugarshack, and Romero, we have added Tweetie Linen and Tweetie Pewter. These two new LiveSmart fabrics are stain resistant and water repellant. This makes white fabric a possibility for families and pet owners.\nIt's exciting to be able to select from hundreds of fabrics to find the perfect ones for our upholstery line. We're already working on our new introductions for fall\/winter 2018, and can't wait to share them with you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Web interface PHP, iTCL scripts to ease administration.\nThis excelent tool I've made I've called ioFTPD_WebAdmin 0.0.1 is actually a script for those people that can't code or are to lazy to actually make one...or never came up with it. This script doesn't utilizes a lot of features but it's a start. Features will be added in the future (Including the fixing of broken commands, etc, etc...). And also some commands are still missing to make it usuable for ANYTHING at all... but this is only a very basic version of the webadmin and your code might be based on it as long as credits are due where they are due. Compliments and helpful critics are welcome.\nmaybe a st0pid question, but how do i use this script with io, how to install and so on?\nYou can setup a special flag for allowing users to connect without SSL and use it on an admin use only account (no point SSL encrypting local traffic if used on local PC).\nYou could also perhaps trying connecting via winSSLWrap.\nAnd yes, SSL will be supported!\nall sounds good.... when can we expect the next release...?\nScript is running on remote webserver maybe that helps?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Translations.com can even provide your firm with a wide range of voice-talent selections to meet the preferences of your target audience.\nThe world is shrinking, that's a generally acknowledged fact. The importance of global markets is growing and the population of non-English speakers in the U.S. is expanding at a geometric rate. In today's world, it's not always enough to simply meet regulatory requirements. Managing cost, maintaining quality and growing new markets are tasks critical to thriving in a global business.\nGlobal life sciences companies have come to embrace Interactive Voice Response (IVR) systems as a strategic tool for expanding products and studies worldwide. Yet, coordinating IVR systems can often seem to be a daunting challenge that requires access to technical expertise and linguistic resources that most life sciences firms don't possess in sufficient depth to ensure a dependable level of quality and turnaround. Whatever challenges you might face in developing your global IVR, Translations.com can help.\nTranslations.com offers a full suite of IVR localization services from script development and translation to voice-talent selection, studio recording services and authoring services. In fact, Translations.com can even provide your firm with a wide range of voice-talent selections to meet the preferences of your target audience.\nWe invite you to contact us for ideas on how to improve your IVR localization systems.\nWe're honored to serve some of the world's leading corporations. If you'd like to learn more about Translations.com client solutions by examining a case study, please email us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazfyv b/data_all_eng_slimpj/shuffled/split2/finalzzzazfyv
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+{"text":"The Longing Look offers a one year basic training in sitting and walking meditation in the Celtic tradition. The training is on a private 1:1 basis and can be done online.\nCertification does NOT pre-suppose any previous experience in meditation. As everyone meditates for themselves, long experience is useful in a teacher, but not necessary. No teacher ever becomes an expert in how someone else meditates. Teacher certification content is likewise unique to every individual.\nI will be available to support teachers in difficult issues that may arise. Accreditation enables the successful trainee to teach meditation as a member of the Siege, and be supported by its group members.\nSiege is the collective noun for herons.\nThe training is based on observing nature and in particular the hunting style of the heron; stillness. If you are to catch a fish, it is necessary to be a master of stillness. The water in which the heron fishes is his\/her own pool. Many creatures come to drink at this pool but the heron remains focused on just one goal throughout. That requires determination and patience.\nThe one year course includes a tour of 'thin places' or power spots at ancient Celtic sites in Ireland.\nThe Celtic love of 'journeying' yet 'longing' for home make for a fluid and flexible style of teaching independent of any written historical tradition or religious hierarchy. Teaching meditation becomes more a question of shared personal life-experience and a wise grasp of what can happen on the journey into stillness. While supporting the meditator's (warrior's) spiritual practice, Siege teachers continually deepen their personal insight, and speed their own journey to inner peace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Derived its name from Mangladevi, Mangalore is situated in between the Arabian Sea and the Western Ghats. This is also listed among the best tourist locations where travellers can enjoy on the beaches of the place. During your Mangalore goa beach tour, you are going to explore the charm of the beaches. You will also visit the beautiful churches, monuments, temple and forts. After visiting the major Mangalore attractions you will travel to Goa and cover all the major Goa attractions in Panjim and Old Goa including beaches and Churches.\nDay 1 \u2013 Arrival at Mangalore Your amazing tour will start with your arrival at Mangalore. You will be greeted by our representative on your arrival. He will shift you to your hotel. Later you will be spending the remaining day in the comfort of your rooms. The day will be at your discretion. Enjoy an overnight stay in the hotel preceded by the dinner.\nIn the morning have your breakfast and then you will start driving to Coondapoor. On the way you will stop by Moodbatri for sightseeing excursion at Karkal Gomateswara's Twin Statue and 1000 Pillar Jain Temple. On arrival you will enjoy an overnight stay at Coondapoor.\nOn the very next morning, you will have your breakfast and then you will leave for Karwar. En-Route you will visit Bhatkal Fort. The fort is listed as the oldest fort in India. On Karwar arrival you will be transferred to your hotel for an overnight stay.\nIn the early hours of morning, you will have your breakfast and then you will leave for Goa. On arrival you be transferred in your luxurious beach resort. Spend rest of the day on your own doing your own leisure activities. Overnight at the resort preceded by the dinner.\nHave mouth-watering breakfast and then you will leave to leave to explore the mesmerizing Goa tourist destination. You will travel to Goan churches such as Basilica De Bom Jesus, St. Frances Xavier Church and Church of our Lady of the Immaculate Conception and many more. Then you will visit to explore the major Goa beaches like Baga and Calangute beach where you can enjoy numerous water sports activities such as dolphin viewing, speed boating, water skiing and many more. By the evening you will travel to the most vibrant and most happening market of Goa at Anjuna beach. The market is full of scrumptious food items, colorful cloths, shimmering jewelry, accessories and much more. One can also enjoy the music and dance in these flea market.\nPost your breakfast, you will get ready to be part of the most amazing South Goa trip. The beaches of south Goa will enhance your tour experience. You will visit Mobar beach, Palolem beach, Majorda Beach, Colva Beach and many more which are some of the best pristine beaches in South Goa. You will also visit many tourist attractions over here and can also indulge in many adventurous sports activities. Back to your hotel by the night and enjoy your overnight stay.\nHave your breakfast and then you will transfer to your airport. From here you will catch your flight for your destination onwards.\nHelpful and Amazing Ground Services, Managed by Hoteldekho.com Pvt. Ltd.\nGet Free Tour Quotation without any Obligation.\nPlush Private Guided Tours in lush air-conditioned cars.\nProficient tour handling by highly experienced and well-versed Professionals.\nEasy Tours with English speaking competent guides.\nGet Unparalleled and Superior 24\/7 Personalized Customer Service.\nHassle-Free Travel Experience throughout India with all superior services..\nGet Best Deals on discounted rates, Easy Hotel Bookings, smooth Car Rentals Services.\nNumerous itineraries with wide Options under Tour Packages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jesus is having yet another dispute with the Pharisees. He enters the temple courts. He begins to teach the people and then here comes the By What Authority Are You Doing This? crowd.\nNow, let's stop there and moment and ponder the boys in their linen vestments. Jesus is on their turf, so to speak, and they want to know, \"By what authority are you doing these things, and who gave you this authority?\" (Matt. 21:23).\n1. We are guardians of God's House, here Rabbi Jesus. You come from a hayseed town in Galilee. Where were you trained? Who trained you? We have a reputation to protect and we can't just let anybody set up shop and start teaching.\n2. We appreciate your zeal, but we are alarmed at how the people gravitate to you and away from us. WE have been appointed to do God's work. You?\n3. You don't look like us or sound like us. You need to fall into line with how we do things around here if you want to teach here.\n4. The Romans are always breathing down our necks. If you anger us, that's one thing. Anger them, and we all will pay dearly.\nOK, so either the boys are sincerely guarding their turf; they are jealous; they want brand consistency or they're desperate to maintain the status quo. So, how does Jesus respond?\nJesus' rather piercing question is to see what is in their hearts. He wants to expose their motivation in asking Him this question. The question on the surface seems well-intentioned. They seem to take their guardian role very seriously. But do they?\nIn other words, what is the foundation upon which they stand? They claim it's God's holy Word; is it?\nJohn is fulfilling Isaiah's prophecy. \"Lord\" means \"Messiah\" (the Anointed One of God) and John is announcing the imminent work of God. Thus, he is a prophet as well as a forerunner.\nHow the Pharisees answer Jesus' question is critical, and they know it. If John spoke the truth, the Pharisees will be held accountable by God for their unbelief. If John spoke falsely, the leaders, while wanting to expose this (and by extension, Jesus) will run the risk of angering the crowd. The crowd sees him as a prophet.\nSo, Jesus is demanding that they reevaluate their position in the light of the Scriptures. They have studied the Old Testament to the last detail. Does John fulfill the criteria of a prophet?\nInstead of going directly to the Word, they decline to respond. They are not even willing to debate the matter. But, if they look to the Word (which they claim to stand on) they would see it being fulfilled right before their very eyes.\nEarlier, Jesus rode into Jerusalem upon a donkey, thus fulfilling Isaiah's and Zechariah's messianic prophecies. In fact, the reports of His ministry show that messianic prophecies are being continuously fulfilled and the Pharisees know it.\nJesus is deliberately placing the Scriptures front and center. The criteria for the Messiah is clearly delineated; thus, any discussion as to Jesus' claim should be an exploration of the Word.\nThe Pharisees' response? Either we debate the Scriptures with this hayseed from Nazareth and risked looking stupid--he seems to really know his stuff--or we tick off the crowd, who seems to be at a fevered pitch of excitement and support for him.\nSadly, no one has ever been won to the Kingdom through argument alone. If one is willing to pursue the truth to wherever it may lead, and be willing to risk having to leave behind one's preconceptions, the Scriptures should be the only arena of discussion.\nJesus' method was to always point back to the Word to validate His ministry.\nWhy do we think we can improve on His method?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have escalated your case to our NBN Service Delivery team. They will give you a call within 24 hours to make sure all is good with the install.\nIf anything needs to be done to get your install completed they will guide you on the next steps.\nPlease wait for that call so we can assist you further.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks for the offer, Bob, but I just got it to work. I ended up uninstalling the toolkit and deleting any leftovers I could find relating to the AG or Globus. Not very technical, but it worked.\nI'll take a look and see what I can see. If you moved things around in the certRepo dir you've likely caused problems, yes. If you did that, remove the certREpo dir and try the import again.\nI get this error when I try to import a certificate on a Windows machine. The cert has been exported from Linux. I tried moving various certs and .pem files around, so I probably screwed things up even worse, but I still get the same error. Any ideas?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Create your student account before you continue - it's free. You will be able to contact tutors for free, adding new tutoring requests, writing reviews or asking questions with your student account.\nYou will receive verification email in a minute. Write down your 6 characters long code to verify your account or click the verification link.\nHave you already verified your email by clicking the link?\nDo you have your meet'n'learn account? You can log in here.\nIf you still didn't receive verification email, check spam folder or resend verification email.\nStudent registration | Unohtuiko salasana?\nWhat personal information do we collect from the people that register on our web sites (platforms meetnlearn.de, meetnlearn.at, meetnlearn.ch, meetnlearn.se, meetnlearn.fi, meetnlearn.dk, meetnlearn.hu, DoucMa.sk, Doucuji.eu?\nWhen ordering or registering on our site, as appropriate, you may be asked to enter your name, email address, mailing address, phone number, credit card information, photo or other details to help you with your experience.\nWe collect information from you when you register on our site, place an order, subscribe to a newsletter, fill out a form, Use Live Chat or enter information on our site.\nPlease be aware that users younger than 16 years of age are requested to obtain consent from their parent or guardian before providing their personal information to us.\nWe may share your information with companies and employees within the meetnlearn group including for customer support, service improvement, development, processing, storage or providing access to our products and services.\nWe may disclose personal information collected for purposes which are incidental to the provision of our services and promotion of our products to you.\nYour personal information is contained behind secured networks and is only accessible by a limited number of persons who have special access rights to such systems and databases, and are required to keep the information confidential. In addition, all sensitive (especially credit information you supply) is encrypted via Secure Socket Layer (SSL) technology.\nAt any time the user places an order, enters, submits or accesses its information, we implement a variety of security measures to maintain the safety of users personal information.\nYou can choose to have your computer warn you each time a cookie is being sent, or you can choose to turn off all cookies. You do this through your browser settings. Since each browser is a little different, look at your browser's Help Menu to learn the correct way to modify your cookies. However, please be aware, that if you turn cookies off, some of the features that make your site experience more efficient may not function properly.Some of the features that make your site experience more efficient and may not function properly.\nWe do not share, sell, trade, or otherwise transfer to outside parties your Personally Identifiable Information unless we provide users prior written approval. This does not include website hosting partners and other parties who assist us in operating our website, conducting our business, or serving our users, so long as those parties agree to keep this information confidential. We may also release information when it's release is appropriate to comply with the law or enforce our site policies, or protect ours rights, property or safety, or others users and public.\nHowever, non-personally identifiable user information may be provided to other parties for marketing, advertising, or other uses.\nWe, along with third-party vendors, such as Google, use first-party cookies (such as the Google Analytics cookies) and third-party cookies (such as the DoubleClick cookie) or other third-party identifiers together to compile data regarding user interactions with ad impressions and other ad service functions as they relate to our website.\nUser has the right to inspect, update or change its personal information by logging in to its account. User always has the right to request us to delete or correct the personal information held about the user.\nIt's also important to note that we allow third-party behavioral tracking, however, the third party has no access to personal information of the users due to pseudonymization.\nFrom time to time, we may email you about changes to our website, new products, promotions and other updates. If you wish to be excluded from email updates, you can unsubscribe via the link at the bottom of the email, or you may withdraw your consent for us to contact you, for the continued collection, use or disclosure of your personal information, at anytime, by contacting us at info@meetnlearn.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With a highschool diploma or a General Equivalency Diploma (GED), you possibly can apply to any of MATCH's affiliate diploma applications. Bare Capitalism ran a four-half collection called Can Uber Deliver? through which it used knowledge on the business's competitive economics, to address the query of whether Uber's aggressive efforts to fully dominate the urban automotive service business has (or will) improve general financial welfare. It is a lengthy learn, however if you're at all interested in the gig economy or the trip-hailing business specifically, it is value spending some time on this sequence. In this era of digital media, the place nearly every little thing is searchable and downloadable, it is easy to see why students, and even lecturers, assume that it is perfectly high-quality to reuse one other's creator's work. Nevertheless, this isn't at all times true.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OLA :: Opening of Arctic Waters to Marine Shipping?\nOpening of Arctic Waters to Marine Shipping?\nDanish Cargo ship, Venta Maersk, will be arriving in St Petersbrug towards the end of this week (25 September 2018), having departed Vladivostok and travelled across the top of Russia.\n\"So what?\" you may ask. The thing is, this ship travelled unaided through the Arctic's Northern Sea Route, a feat which just a few years ago, would have been impossible without the assistance of nuclear-powered ice breakers.\nWith the Arctic warming faster than any other region on the earth, the ice in the Northern Sea Route has become thinner, sparser, and more fractured than ever before.\nWhile this is a \"trial passage\" and it is not expected that this will become a commercially viable route in the near future, it is an important milestone and an indication of the very real impact of climate change.\n\"The fact that you can now ship with a container ship through the Arctic along the Northern Sea Route, when even just 10 years ago that was not even remotely possible, shows how quickly and how dramatically climate change is affecting the region.\"\nThis not only speaks to climate change, but to the environmental impact of such voyages through the ice. Commercial ships use heavy fuel oil, emitting soot and pollutants which further speed the melting of the arctic ice.\nOn 13 April 2018, the International Maritime Organization (IMO) agreed to climate targets for the sector, as part of its first comprehensive greenhouse gas emissions (GHG) reduction strategy. One of the targets is the ban of heavy fuel oil in the Arctic, but an impact assessment will be conducted first to ensure a ban does not place an unnecessary burden on Arctic communities.\nThe threat to marine life is also a concern. A recent report, supported by the World Wildlife Fund (WWF) suggests that noise from shipping traffic is a threat to the Arctic's marine mammals, decreasing their ability to find food, orient themselves and communicate underwater.\nLeaders of native Arctic communities have come forward in the past to stress the importance of protecting the sensitive waters.\nMarine advocate Austin Ahmasuk calls for improved dialogue with the native communities and their leaders stressing that is the best way forward.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our focus is to ensure an environmentally focused business and consumer expo in the US. In addition to visiting over 400 interactive booths with new Eco-Green and Solar products and services, guests will mingle with local, Regional and National companies. Consumers able to see and experience the latest fuel- efficient cars, preview Eco-fashion and , the latest in green building, solar products and more. Attendees will be able to attend seminars held daily on various topics We focus on increasing consumer awareness and product knowledge to boost Georgia Growth in the Eco,Green and Solar Industry. We look forward in introducing some of the latest trends and products in today's market. Empowering American Companies, while educating Americans about Eco Products- Green and Solar Power, the wave of now and the future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Matthew Shepard Act passes the Senate but will it become law?\nWhat about this does America not understand?\nEqual = The same or as great as. Inalienable Rights = Not contingent upon the laws, customs, beliefs, or a particular society. Life = The right to exist free from being beaten or killed for being who we are.\nKTVB \"Cruises\" back to 1955. . .\nNO ONE there will need to be educated by Mr. Munds who has once again missed the mark and will not change his self proclaimed 'hate monger' title by passing out water bottles printed with a web site where one can read about \"coming out\" of homosexuality to lead healthy, fulfilling lives while at an out, loud and proud PRIDE festival.\nI learned of Wednesday's tragic shooting at the US Holocaust Museum while at the Idaho Human Rights Education Center's second annual Summer Institute for Teachers in Boise. The 23 junior high and high school educators from across the state were disheartened by the news during an otherwise inspiring week of scholarly dialogue and lesson development to bring human rights into Idaho classrooms.\nBranfordMagazine theme by Michael Oeser.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We love working on our products and want to share them with the world because we are proud of them. A century ago, the Luddite movement fought the machines and destroyed them to hinder Industrialization. Now we live in peace with it and acknowlege it as our good and faithful assistant, but there are also \"rebellious\" brands that claim to have a little more uniqueness. And that is Lexira. A modern Luddite that gets the best of both worlds.\nIt always contributes for a better experience when you buy something in which a manual work is involved. When experienced hands have put their work into the product it instantly becomes special. There is particular pleasure that Lexira brings simply because we know it is made by hand and these are products more beautiful because made with a heart.\nIt is the nature of handmade goods to be fewer of them, which makes them so much more unique and Lexira is of this kind. If an assembly line can produce thousands of face creams a day, we can make up to fifty. Yes, we put more time, more labor and a little more of ourselves into each product. We can produce them automatically, but we prefer to touch them, to live in our hands lit by the warmth of our palms and not simply being abstract objects born by a machine. Instead, natural ingredients and materials turn into beautiful products with the artisan's creativity, wisdom and ingenuity.\nDoing things that are unique in nature requires more attention to every minute detail, more care and creativity. This is why everyone likes to buy such products. They are more attractive namely because there is much more individuality in them. Our luxurious Lexira products are with the authentic world-renowned Bulgarian rose oil, obtained from hand-picked rose petals and contain much more uniqueness and higher emotional value for our users.\nWe enjoy working on our products and they hold and spread this positive energy. Here is where handmade excels \u2013 in love. It makes a world of difference between mass products and those we do. We put a lot of attention and love into each Lexira. A done by hand product gets much more of our attention, care and our creativity compared to the production of thousands. This is where the extra charm lies \u2013 handmade is made by people, 'masters' in their handicraft.\nIt brings more pleasure and happiness to spend our money on something unique like ourselves. Buying products that have the labor from one's hands should be a way to celebrate our own uniqueness, recognition for the talent of the masters and the imagination invested in a product. It is an essential part of the Lexira's power - artisanal products made with a whole lot of love!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Heart arrhythmia is an irregular heartbeat, commonly described by patients as \"feeling like my heart is pounding out of my chest\" or \"like my heart is skipping beats.\"\n\"There are different types of arrhythmias. Some are benign and not urgent, but some can be very serious and life threatening, especially if there's an underlying heart condition such as a prior heart attack,\" says Gundersen electrophysiologist David Ludden, MD, who is a cardiologist with additional training in the diagnosis and treatment of abnormal heart rhythms.\nElectrical impulses cause the heart to contract thereby pumping blood out to the body. Arrhythmias result when chaotic electrical impulses slow the heart or cause the heart to race.\nSymptoms may be constant or come and go, last a few seconds or several hours, or first appear in infancy or old age. \"Arrhythmias can be very unpredictable,\" says Dr. Ludden. \"That's why, if you have symptoms, you should seek medical help and, if necessary, the advice of a specialist.\"\nIf you have symptoms, a family history of arrhythmia or significant heart disease, you should discuss these with your doctor. If you experience symptoms, seek medical care right away.\n\"If properly diagnosed, once-fatal or disabling cardiac arrhythmias can be managed or even cured. That's why the work of electrophysiologists is so important,\" Dr. Ludden explains.\nIf you have, or suspect you might have, an arrhythmia problem, talk with your doctor or cardiologist about a referral to Dr. Ludden.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The independent Local Government Boundary Commission for England is asking people across Harborough to comment on its draft proposals for new council ward boundaries.\nAn eight-week public consultation on the recommendations begins today and will end on 8 May 2017. The consultation is open to anyone who wants to have their say on new council wards, ward boundaries and ward names across Harborough.\nThe Commission's draft recommendations propose that the Harborough should have 34 councillors in the future, three fewer the current arrangements. The recommendations also outline how those councillors should represent three three-councillor wards, nine two-councillor wards and seven one-councillor wards across the district.\nProfessor Colin Mellors, Chair of the Commission, said: \"We are publishing proposals for a new pattern of wards across Harborough and we are keen to hear what local people think of the recommendations.\n\"We also want to ensure that our proposals reflect the interests and identities of local communities across Harborough and that the pattern of wards can help the council deliver effective local government to local people.\n2. The Commission is carrying out an electoral review of Harborough District Council following a formal request from the council to consider the total number of councillors that should be elected to the council. The Commission is also carrying out the review to deliver electoral equality for local voters. The district currently has relatively high levels of electoral inequality where some councillors represent significantly more, or fewer, voters than other members of the council.\n4. The electoral review of Harborough District Council is a separate undertaking from the review of parliamentary constituency boundaries which is being carried out by a separate body (Boundary Commission for England) under different rules and legislation.\n5. Residents have from 14 March until 8 May 2017 to have their say about where ward boundaries for Harborough should be drawn. The Commission will consider all submissions and aims to publish its final recommendations in July 2017. Once the Commission agrees its final recommendations it will lay a draft order in both Houses of Parliament. Parliament will then have 40 days in which to consider the recommendations. If both Houses are satisfied with the recommendations, the draft order will be 'made' and the new wards will come into effect at the council elections in May 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Argentinian musician based in Europe since 2002, specifically in Berlin since 2004.\nHis main instruments are Bass Clarinet and Soprano Saxophone. After 10 years focused in Electro Acoustic Improvisation he dedicates in the last years, to works focused in the Perception experience, that he performs mainly in Solo and in the context of occasional collaborations. He composes his own pieces that may include improvisation and different ways of writing. He uses tools like Flying Speakers hanging from Helium Balloons, Speakers as Pendulums, Analog synthesiser, Sine Waves and Noise Generators, Drum Machine, Ultra-Violet Lights. He has also written compositions for different kind of Ensembles working the same aspects in the context of traditional Instrumentations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Check out new Phantom super slow motion showreel!\nAll footage taken with Phantom HD Gold high-speed camera up to 1050 fps. Collection includes different kind of shots with people, animals, nature and sports. All clips from this collection are available for sale at iStockphoto or Getty Images.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Whether you're new to giving speeches or are a seasoned speaker, Toastmaster training will help you hone your skills. Get tips for how to prepare speech and present to audience, use visual aids and props, incorporate body language into your presentations, and more. With time and practice, you're sure to see improvement in your ability to communicate and an increase in your confidence as well.\nTACL Toastmasters Club at Taiwanese Community Center are helping members to grow their communication skills and leadership. We meet every Monday night 7:30 - 9:00. We welcome guests to visit our meeting and find out what Toastmaster is all about! Here is a glance on what we do on every Monday night.\nOn February 26, we had a fun meeting as always with a guest, Noe! We were short a speaker unexpectedly, so we had an extra round robin topics in addition to a great session led by Gordon. Phil gave his icebreaker to kickoff his Pathways journey, and Carol evaluated Phil's speech.\nOn March 5th, Sunny gave an inspiration speech on you are someone who matter, and you can make the world better. Nam is working on advanced manual - interpersonal communication, he gave a speech on how to make a conversion with strangers. Table topic theme for the week was 'advice', fellow Toastmasters were able to gave helpful tips and valuable personal experiences.\nMarch 12th was our club International speech and table topics contest and we had a great contest! Thank you to all the guests and volunteers from Sharps Town Toastmasters Club and Financially Speaking Club who made it possible! Carol won 1st and Phil 2nd for International and Phil won 1st and Kevin 2nd for Table Topics Our winners will represent the club at joint area contest on Saturday March 24, 11:15am brief, noon start at Trini Mendenhall off I10 & Chimney Rock \/ Wirt. We'll also see our old M32 friends there.\nWhat an edifying meeting on March 19th, we learned from Nam about how to handle criticism and from Sherry about maintaining a good club climate to attract more members. Patrick led a table topic on Spring which led to some really entertaining responses. As usual we had sharp evaluations from our seasoned crew of Toastmasters, including feedback for Carol who will represent us at Area Contest!\nOur Area M34 contest is 3\/24 Sat 12pm @Trini Mendenhall, Also, don't forget, we are looking for 5 members for the $250 discounted club rate to attend the District 56 Spring Conference, May 11-12 Fri\/Sat at the Stafford Center! Hear the the best speakers in Houston and meet awesome people!\nWe are collecting the journals, directories, Newsletters and artifacts issued by these T. A. organizations in the past 7 decades. We have very limited collections from 1950 to 1980 period. Please help us find some publications during this period.\nThe program books and artifacts of T. A. summer and autumn conferences have being collected and displayed in this room, too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There's plenty of tools for creating websites, but when people task themselves with setting up a website for their business or cause they often focus on its creation rather than its end users. And why not, that's human nature \u2013 solve the problem. I (my business or organisation) need a website, and I (the business owner) can make a website right now with tool X.\nThe problem with this approach is that whilst you get a website, there has been no focus on the end user of your website. You want those people to purchase your product or services, make contact with you to arrange that, or engage with your cause.\nHow do we determine what the end user needs?\nA good website strategy, design and end website focuses on those people (the end users) \u2013 to effectively connect with your customers or champion your cause.\nEverything needs to be done with these people in mind.\nIs it one primary audience, or segmented?\nHow do they access the internet?\nWhat do they engage with online?\nFor instance a website whose primary audience is youth needs to work very well on smartphones. 25% of Australian youth only access the internet via their smartphone. A mobile-first design approach would almost certainly be the best approach.\nIn part this means your competition, but it means much more. The person looking for your services\/goods\/cause is likely to be immersed in aspects associated with it, and you'll be measured based on their experience of that.\nIf your service or goods are visual, such as any fashion, architecture, art, even \"designer\" consumer items, then your website needs to meet that standard. If your website looks bad because of poor design, then the person can only conclude that surely your visual sense is also bad.\nFor instance, if you're an architect marketing your services, the audience looking for an architect is probably also browsing real estate websites, reading home beautiful magazines, browsing interior decorating boards on Pinterest and watching home improvement shows \u2013 indeed, they've probably been doing this for years.\nIf your website doesn't match these experiences they won't connect with you \u2013 because you haven't matched their conscious, or more likely unconscious, expectations. You've just lost a sale\/recruit.\nWhen you focus on the end user, you focus on how they think when they are looking for your product or service.\nIs the website easy for them to use? What is their pathway through the website?\nCan they achieve their task quickly and efficiently?\nWhat are the obstacles to complete their task? If the obstacles are too great, they may begin but abandon their task, moving on to another website.\nAn example of this is purchasing an item online.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Banneux N.D. Organization for Asian Pilgrims is organizing the 31st annual pilgrimage to Our Lady of Banneux, which will be held on 28th of May 2016.\nThe bus leaves at 7:00 from the Winkel centrum in Laan van Vollenhoven, Zeist. It comes then to the Jaarbeursplein (Euroline bus stop), Utrecht, at 7:45-8:00.\nThe cost for students is just 25\u20ac for travelling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cabling requirements are increasingly dependent on how each system is physically configured, and dynamic business environments require timely technology upgrades and machine replacements to respond to system growth. Changes or upgrades to the mainframe hardware environment often require new cabling and provide an opportunity to build a more dynamic environment with business critical systems.\nSirius Cabling Services include planning, procurement and installation of the new style cables required for IBM Z processors to connect any existing or new I\/O equipment associated with processor replacement or data center relocation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If Travis Puckett had to sum up his job search journey in one word, it would be: persistence. Keeping your chin up and not getting discouraged is key to any job search. Read more of Travis's advice for job seekers who are looking to find encouragement after a long job search.\nI work in analytics. It's my job to distill our database into usable information for managers and directors. Data is power!\nThe company I work for is called Vacasa. Vacasa is a vacation rental management company. We take a deeply analytical approach to bringing more profit to homeowners looking to maximize their rental investment.\nI began looking for a new job while still holding my previous employment. After 2 months of juggling both, I decided to terminate employment and go into the job search full time.\nI took an internship that started in June. When the internship concluded I was offered a full time job with the company. It took 6 months to land the internship, 8 months for the full time gig.\nEVERY RESOURCE! I took a very active approach to the job search, and can honestly say that I thoroughly enjoyed the process (rare, I know!).\nSecondly, I called or emailed anyone I knew who was career-minded to meet up. Chatting with people I knew was less formal- mostly, it just got me out of the house. If I could give any advice, I would stress the importance of staying social and not letting the search get you down.\nLastly, I attended some networking events. Mac's List put on a terrific event at Simple called \"How to Land a Tech Job in Portland.\" There was a panel of experts and one panelist stuck out for me, Grace Andrews. She said that she had never worked in tech, but sought out intern positions in the industry as a way to get a foot in the door. Now she's a Solution Engineer after starting entry level at New Relic. It's a hard step to budget for, but it's a terrific way to transition into an entirely new type of job.\nThe most difficult part is the inner voice. You spend so much time and energy building your legend on a resume, and when you don't hear back\u2026it can chip at your morale. At the end of the day, your morale is the most important trait you can bring to a new company.\nI've alluded to it before, but reaching out to my village was how I overcame this challenge. Connecting with former colleagues and friends sparked a joy in me that allowed me to let go of the outcome and appreciate the journey.\nStart persistent and stay persistent. After you've finished writing your resume, refine it. After you refine that resume, build another one in a different format (skills based vs chronological). Network with people in your desired industry. Network with people in unrelated industries. Work an odd job or two (there's a ton of temp work out there). Connect with people even if you don't think they're going to influence your search.\nPersistence will open doors to a hidden job market.\nI love my job for so many reasons. Mostly, because I have access to a ton of talented people with varying skill sets. Plus, my team is amazing. I also think the job is pretty neat because it's all about numbers. Any time I am surrounded by volumes of loosely related numbers, I get excited to start connecting them. That's basically what I do for a living.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"5 Sep 2018, 10:46 a.m.\nIObit Malware Fighter Pro's airy main window includes a status indicator, a left-rail menu, and confirmation of status for three protection engines: IObit's antivirus engine and ransomware engine, plus an engine licenses from Bitdefender.\nThe Action Center page offers numerous other apps, some from IObit, some from other companies. For example, the password manager comes from Dashlane and the Web Companion from adaware.\nIn testing, Malware Fighter caught some threats, but missed others. Here it has caught the notorious EternalBlue.\nWithout distinguishing malware from valid software, Malware Fighter flags every program that sets itself to launch at startup.\nTested against 100 malware-hosting URLs, Malware Fighter caught some of the malware payloads, but did not prevent access to the URLs themselves.\nA full scan of a clean test system took over two hours, more than double the current average.\nIn testing, Malware Fighter did not detect or block a single phishing URL. Instead, it actively marked them as safe.\nThe Surfing Protection component also includes an ad blocker.\nMalware Fighter's Safe Box component prevents unauthorized modification of files in protected folders, to fend off ransomware. However, out of the box, the list of protected folders is empty.\nHere Malware Fighter has let ransomware run to completion and display its ransom message. However, the message is false; Safe Box protected files from encryption.\nThe Security Guard page lists a collection of protective features, among them webcam protection, prevention of Master Boot Record viruses, and scanning of USB drives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'Big Bang' stars dated in real life too?!?\nEarlier this week PopWrap called Penny and Leonard one of TV's Best Star Crossed Couples and today I find out that their on-screen chemistry is so fantastic because actors Kaley Cuoco and Johnny Galecki actually dated throughout their character's courtship!\nI'll give you a second to pick your jaw up off the floor.\nThis is further proof that celebrities can actually keep their private lives private (ex: Sandra Bullock's secret baby) when they want to.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sailthru is part of the CM Group family of brands (\"Sailthru\", \"we\", or \"us\") that provides Software-as-a-Service (SaaS) and software products (\"Services\") to our customers and clients (\"Customers\"). We may disclose your information to affiliates within the CM Group for customer support, marketing, technical operations, and account management purposes.\nThis Privacy Statement (\"Privacy Statement\") describes how Sailthru collect, uses, discloses, and transfers information, including personal information, from users accessing and using our websites that display a link to this Privacy Statement (\"Sites\"). The Privacy Statement also describes Sailthru's collection and use of information in providing the Services. This Privacy Statement does not govern collection or use of information you provide to third parties, such as our Customers. You should refer to these third parties' separate privacy policies to learn about their collection and use of your information.\nPersonal Information: In accessing and using the Sites or the Services, you may be prompted to provide Sailthru your name, email address, telephone number, job title, company name, and other personal information.\nOther Information: Sailthru may collect information about your computer or mobile device that may include your browser or operating system, your manner of connecting to the Internet and the name of your Internet service provider or wireless carrier; your Internet Protocol (IP) address or Advertising Identifier; information about websites you used before or after using the Sites or Services; data relating to malfunctions or problems occurring when you use our Sites or Services, other information specifically related to your use of Services, including information that you publicly post using tools made available by us; location information; and information about the searches you perform on our Sites or when using the Services and how you use the results of those searches To collect this information, Sailthru and Sailthru's business partners may use tracking technologies, including cookies and web beacons.\nto navigation and use of on our Sites easier.\nto communicate with you and respond to your inquiries including inquiries regarding support for our Services or purchasing our products.\nto better understand your use of the Site and Services.\nto help offer you other products, features, or services that may be of interest.\nto respond to a job application or inquiry and consider you for employment purposes.\nWe do not rent or sell your personal information (such as name, address and telephone number) to third parties for their marketing purposes. We may share your information with third parties to provide products and services you have requested, when we have your consent, or as described in this Privacy Statement.\nWe may share aggregated, non-personally identifiable information, publicly and with our Customers and partners. For example, we may share information publicly to show trends about the general use of our services.\nThe contents of your online communications, as well as other information about you as a user of our Services, may be accessed and disclosed under the following circumstances: in response to lawful governmental requests or legal process (for example, a court order, search warrant or subpoena), in other circumstances in which we have a good faith belief that a crime has been or is being committed by a user of our Services, that an emergency exists that poses a threat to the safety of you or another person, when necessary either to protect our rights or property, or for us to render the service you have requested.\nMany of our services let you share information with others. Remember that when you share information publicly, others besides us have access to it, and it may be indexable by search engines, or copied, forwarded or archived by others. Our offerings may include features or functionalities provided by third parties. In the process of providing such functionalities within our Services, your browser may automatically send certain technical information to the third party provider. The use of these third-party provided features or functionalities is subject to their own privacy policies. Business partners or other third parties may receive data about groups of our users, but do not receive information that personally identifies you.\nWe may use agents and contractors in order to help operate our Services. Their use of information is limited to these purposes.\nIn the event that Sailthru's ownership changes as a result of a merger, acquisition, reorganization, sale of assets, dissolution, or similar event, information we collect may be part of the transferred assets. If such a transfer results in a material change in the use of your information, you will be provided notice (which may be via updates to this page) about the choices you have to decline to permit such a transfer.\nSailthru primarily stores and processes the personal information we collect in the United States. As a global company, Sailthru may transfer and access information from around the world, including from other countries in which Sailthru operates. If you operate outside of the United States, transfer of your data is necessary for you to use our Services, access our Sites, or request information from us. By registering for our Sites or accessing the Services, or by communicating with us by email, you acknowledge and expressly consent to our cross border transfer of your personal information. By registering for the Site, by subscribing to our Services, or by emailing us, you expressly agree to the processing of your personal information as described in this Privacy Statement.\nBy emailing us or by opting-in to use the Site or access the Services you expressly consent to the collection, use, retention, transfer and disclosure, including cross-border disclosure, of your information as described in this Privacy Statement. Each time you email us, register with us, or use our services you are giving your express consent to the collection, use, retention, transfer and disclosure, including cross-border disclosure, of your information as described in this Privacy Statement.\nIf you wish to opt out and withdraw your consent, you may do so at any time by contacting us at: privacy@sailthru.com. If you opt out and withdraw your consent you will no longer be able to use our Services and we will not have any means by which to respond to your questions or comments that you email to the Site. In addition, if you wish to correct, update, amend, or remove your personal information that Sailthru controls, you may contact us by emailing to privacy@sailthru.com.\nA cookie is a small text file that is stored on a user's computer for record-keeping purposes. We use both session ID cookies and persistent cookies on our Site. A session ID cookie expires when you close your browser. A persistent cookie remains on your hard drive for an extended period of time. If you wish to deactivate cookies while visiting our Site during your browsing session, please see \"How to Control Cookies\" below. Please note that you may need to erase all cookies that have already been placed on your computer in order to delete the Sailthru cookie.\nPlease note that Internet browsers allow you to change your cookie settings. These settings are usually found in the 'options' or 'preferences' menu of your Internet browser. In order to understand these settings, the following links may be helpful. Otherwise you should use the 'Help' option in your Internet browser for more details.\nWe will retain your personal information for as long as needed for the uses described in this Privacy Statement.\nSailthru uses third party services including Google Analytics, to collect information about how visitors use the Site in order to optimize visitor experiences. Google Analytics uses web beacons and cookies in order to track visitor page views and clicks within pages, length of time spent on our Sites , and the search term or referring site that brought a visitor to our Site.\nOur Sites include features from social media websites such as the Facebook Like button and features on LinkedIn and Twitter that run on our Site. These features may collect your IP address, which page you are visiting on our Site, and may set a cookie to enable the feature to function properly. Social media features are either hosted by a third party or hosted directly on our Site. This Privacy Statement does not apply to these features. Your interactions with these features are governed by the Privacy Statement and other policies of the companies providing them.\nSailthru takes reasonable precautions to protect personal information from loss, misuse, unauthorized access, disclosure, alteration and destruction. We follow generally accepted standards to protect the personal information submitted to us, both during transmission and once we receive it. For example, to protect your privacy and security, we may use passwords or other technologies to register or authenticate you and enable you to use our Services, and before granting access or making corrections to your information. No method of transmission over the Internet, or method of electronic storage, is 100% secure, however. Therefore, we cannot guarantee its absolute security.\nSailthru's commitments under the Privacy Shield are subject to the investigatory and enforcement powers of the United States Federal Trade Commission.\nIf you believe Sailthru maintains your personal data, you may direct any inquiries or complaints concerning our Privacy Shield compliance to privacy@sailthru.com. Sailthru will respond within 45 days. If we fail to respond within that time, or if our response does not address your concern, you may contact the International Centre for Dispute Resolution (\"ICDR\/AAA\"), which Sailthru has chosen as its independent recourse mechanism, via its via the contact information provided below. ICDR\/AAA provides an independent third-party dispute resolution body and has committed to respond to complaints and to provide appropriate recourse at no cost to you. If you feel that neither Sailthru nor ICDR\/AAA can resolve your complaint, you may seek to engage in binding arbitration through the Privacy Shield Panel.\nEU and Swiss individuals have the right to access the personal data Sailthru processes pertaining to them and to limit use and disclosure of their personal data. EU and Swiss data subjects may receive notice of Sailthru's processing of their personal data, access to that personal data, and seek to amend that personal data by request to privacy@sailthru.com.\nIf you believe a Sailthru Customer transferred your personal data to Sailthru, in making a request please provide the name of the Customer. We will refer your request to that Customer, and assist them as needed in responding to your request.\nto a buyer or other successor in the event of a merger, sale or transfer of some or all of Sailthru's assets.\nSailthru may be required to disclose personal information in response to lawful requests by public authorities, including to meet national security or law enforcement requirements.\nSince there is no uniform technological standard for \"Do Not Track\", our Sites does not respond to browser \"Do Not Track\" signals.\nWe last updated our Privacy Statement on April 18, 2019. We made changes to add additional information about your right to access your personal data.\nWe may change this policy from time to time by updating this page. You should check this page from time to time to keep up to date with any changes. If you do not agree to any changes, you should immediately stop using the Services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"First-time guests at CBC often have lots of great questions. Here are answers to a few of the most popular ones.\nWe have spaces available on the Church grounds as well as off road parking if needed.\nOur church welcomes guests from all different walks of life, just be comfortable and tasteful. Jeans and a T-shirt are as acceptable as a suit or a dress.\nWe welcome families at CBC Spalding, and have creche and Sunday school facilities for little ones.\nThere are no assigned seats, just sit where you like. Every seat has a great view with our 2 large HD screens that display lyrics to songs, onscreen bible notes, and announcements.\nWe love to worship God and we know that taste in music is different for everyone. Our goal is to create an atmosphere where everyone can worship together, despite our differences in taste.\nOur worship service lasts about an hour or so. We appreciate you spending your time with us!\nCBC Spalding has a wide variety of groups you can join, midweek classes, service opportunities and fellowship events!\nWhat are the people at CBC Spalding like?\nWe're a multi-generational church who have equally been blessed with a diverse community and many walks of life. While we all come from different places, the Gospel of Jesus Christ unites us. To put it simply, CBC Spalding is full of fun-loving people who are passionate about Jesus.\nAs you're reading this, you may wonder if CBC Spalding is the right place for you? We hope you'll give us a chance to make you feel at home in a safe and comfortable place where you can grow spiritually!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Physical Description Bracelets; pair, of gold and black enamel: bracelets, oval in shape with gold floral engraved pattern on background of black enamel; hinged opening with a fine gold chain guard.\nObject History Belonged to Mrs. Charles Higgins (1827-1900), n\u00e9e Frances Gary.\nCuratorial Statement This pair of bracelets belonged to the wife of Mr. Charles Higgins, of McNeil & Higgins Company, a wholesale grocery store in Chicago. Mrs. Higgins, born in Postdam, New York was also the sister of Judge Joseph E. Gary (1821-1906) who tried the anarchists in the Haymarket Riot Trial of 1886.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeuov b/data_all_eng_slimpj/shuffled/split2/finalzzzbeuov
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+{"text":"The Cosmos \u2013 tonight. thru the cleanest air in the world.\nAuthor mark's travel notesPosted on June 13, 2015 June 14, 2015 Format ImageCategories Australia, Cosmos, Hobart, Mountains, Nature, Night sky, Nikon D750, photography, planets, star trail, Stars, Tasmania, TasmaniaTags Australia, comets, cosmos, HOBART, night sky., nikon d750, photography, star trailLeave a comment on The Cosmos \u2013 tonight. thru the cleanest air in the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Roxy is now a happy dog. She is still limping with the leg that was broken but gently putting it on the ground, that means she is slowly beginning to use it which is great news! Please help us find her a great home. She is still in foster care and she deserves to be in a forever loving family after all she's been through. She has a wonderful, loving personality. Thank you.\nA big thank you to everyone who shared her case, we make a great team!\nThis poor girl has been hit by a car in Galati, a lady called us to the rescue. We immediately went there. The dog was bleeding profusely from the groin area, we took her to the vet and is now undergoing surgery. Please share and help us find her a good home once she is healed. Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Passionate about travel and keen to see different parts of the world, Camilla Crosta moved to Scotland upon completion of her studies in History of Art in Venice. While living in Glasgow, she began to explore opportunities to return to higher education when she learned of Queen Margaret University's (QMU) MA Arts, Festival and Cultural Management course.\nHere Camilla explains more about what drew her to the programme, her experiences at QMU, and the opportunities she's been presented with since being on the course.\nI am from a town called Padua, not too far from Venice, in the north east of Italy. Since I was a child, my family encouraged me to travel and appreciate arts and food, and most importantly, to be curious. I believe this influenced my decisions and my future travels.\nI decided to move to Venice to study History of Art, and the city became like a second home with its unique nature, its beauty and its challenges. Here, I attended university, worked for museums and galleries, and created my own cultural project. When I left the city a few years ago, it wasn't easy. Mass tourism heavily impacted on the local cultural production and the sustainability of independent cultural projects.\nScotland has welcomed me and offered an opportunity to discover its land, people and communities. I have travelled and worked in various locations across the Highlands, and have learned about the challenges of producing and promoting culture in remote areas.\nMoving to Scotland represented a chance to do something I love: travel. As soon as I have some free time, I immediately take a train, bus or jump into a friend's car to explore and visit places I've never been before.\nWhy did you choose to study at QMU and what attracted you to the course? Why did you choose to study in Edinburgh?\nI had started to think about going back to university, almost six years after finishing my studies. I live in Glasgow, and at the time, I was looking for a master's that was near me, which would allow time and opportunities to reflect critically on my previous experiences as well as learning new skills.\nI looked at various master's courses, and after talking with some friends who are already on the course at QMU, I started to give it some serious thought. The course offered the right framework to reflect on my previous experiences and learn something new.\nAlthough I still live in Glasgow, I believe studying cultural management in a city like Edinburgh is extremely valuable. Edinburgh and Venice share some of the same challenges posed by the high number of incoming tourists. Having an open space where people can debate and discuss the meaning and role of cultural production ensures that the sector is not only maintained but able to face future challenges.\nWhat have you most enjoyed about your course? What has been your highlight? What have you learnt, or which particular activity has been the most interesting?\nClasses in 'Debates in Cultural Policy' have helped me to reflect on the research I carried on previously during my studies in Venice. Now, after almost six years, I will be able to revise it and expand some of the topics I addressed at the time.\nAnother highlight is the module in #Leadership and Governance'. It's been useful in helping me frame my previous working experiences and to generate new ideas which I am looking forward to putting into practice.\nThe opportunities to have discussions and work together as a group have been very useful for me. In debates of Cultural Policy in which Professor Blanche, after the lecture, divided the class into two small groups where we discuss a variety of different topics.\nThe second one is the Arts Management in Practice module. The course structure allows for several visits to cultural organisations based primarily in Glasgow and Edinburgh. After meeting the teams there, and hearing their presentations, we always spend some time to discuss our impressions. For me, these moments of reflection have been invaluable learning tools. Hearing other people's comments as well as share yours, and getting the chance to discuss these opinions, is essential to the development of a critical approach which can lead to new ideas and reflections.\nThe decision of going back to study was exciting, but also presented some challenges. Striking a balance between studies and work has been a difficult task. I've had to change my lifestyle and plan my days and weeks in a very different way than before. Scheduling has been necessary, as well committing to stick to them! At the moment I have less time to do leisure activities than previously, but I am aware this won't be the case forever.\nGoing back to the academic world after almost six years hasn't been easy. I forgot what it means to be a student, let alone an international student in the UK. In the beginning, it was all new for me, but slowly I am starting to get used to it now.\nChatting with other students and sharing the same concerns has been helpful. I've learned that these concerns are shared by others, too.\nI am fully aware that it is hard, but I believe finding a short work experience related to the field of study makes a difference. By working in the area of your studies you can see gain real perspectives and can apply what you are learning. I feel lucky to have real-life examples to reflect upon and a project where I can share what I learned in class.\nHave you won any awards\/scholarship\/internships, etc., that have helped you develop skills and experience? Are there any other opportunities you're taking advantage of to further your area of study?\nI was lucky enough to win the Dr Wang Scholarship which will help me in developing my career in arts management. I will travel to China this September to work along with Dr Wang's team in producing a cultural festival in Shanghai. I am very excited about this upcoming trip and the opportunity to meet cultural producers and practitioners from different places and countries.\nThe scholarship will also fund some of my dissertation research, which means I will be able to study the example of HafenCity, Hamburg, and its pioneering culture-led regeneration project along the waterfront spaces.\nWhat University services have you used to help you through your university journey and how have they helped you?\nI highly recommend the Effective Learning Service (ELS) which was of great support to me at the beginning of the course. I was really worried about going back to higher education but thanks to the help of the ELS, I found the confidence I needed. They carefully listened to my doubts and helped me in my approach to study and writing.\nWhat's your 'top tip' for making the most of being a student? What has been the most valuable lesson that you've learned at university?\nTry to talk and collaborate with other students. Listen as well as share opinions, ideas and views. The people around you may become friends and co-workers and they can have an impact on your personal and professional life.\n\"Scotland has welcomed me and offered an opportunity to discover its land, people and communities. I have travelled and worked in various locations across the highlands, and have learned about the challenges of producing and promoting culture in remote areas.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A parade to mark the 100th anniversary of the Easter Rising has taken place in the New Lodge area of Belfast.\nAn exhibition featuring portraits of people affected by the Easter Rising is now on display in the Ulster Museum, Belfast.\nA new exhibition about the 300 women involved in the Easter Rising has opened at Dublin Castle.\nA timeline of the 1916 Easter Rising.\nA MASKED colour party has taken part in an Easter Rising commemoration in Lurgan.\nRTE has released personal accounts of 50 people who lived through the 1916 Easter Rising including this Dublin jarvey who has yet to be identified.\nA PLAQUE to honour Easter Rising leader Joseph Mary Plunkett was unveiled in Dublin yesterday.\nTHE DUP has been accused of attempting to block council funding for Easter Rising commemorations \u2013 while supporting a \u00a350,000 package of events for the Somme.\nEASTROGEN Rising: A Rebel Cabaret is a show about women, known and unknown who were present, awed, inspired or actively participated in the Easter Rising 1916.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This short story was written by my late mother, Jean Llorin, a long time ago in English. A few years back, she asked me to draw pictures to go with her story. I've translated the story in Filipino and added a few more important details that I felt the story needed. A thumbnail of the first page is shown below.\nClick on the link (title) below, not the thumbnail, to open the 4-page PDF file. All illustrations are by Samut-samot Mom.\nFeel free to share this story with your children, students, and friends, but do not do so for profit or my mother's ghost will haunt you!\nPlease note that I am not a writer (I'm actually an engineering graduate), so feel free to leave comments on how I can improve the story. Thanks!\nWALA PO BA ITONG MENSAHE AT GINTONG ARAL?\nThe message or lesson of the story may be different for different people.\nThank you, Jusalyn, for the corrections. I've updated the file to include them.\nAno po yung Kaisipan or Moral Lesson nung Story?\nHi, Joan! Sa tingin ko ang moral lesson ng kuwento ay bawat tao ay may layunin o purpose sa buhay at hindi dapat tayo matakot na tuparin ito, lalo na kung ito ay makatutulong sa iba. Thank you for reading this story!\ndo you have a follow up comprehensive questions of your story in tagalog? Can I have a copy of it?\nI'm sorry, Anna. I don't have comprehensive questions for this story. You are welcome to make a set and if you would like to share it here, I can post it. Thanks!\nCan I have English version of this story?\nHi, Kim! My mother actually wrote the original story in English. Do you need it right away? I'm attending to an urgent family matter right now and won't be able to work on your request until next week. Leave a reply if you can wait that long. Thanks!\na very touching story. i will surely share this to my class for my practice teaching service.\nThank you, too, for reading the story!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..70f880e28ee893cec2ee60520351fec8e36901a4
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@@ -0,0 +1,5 @@
+{"text":"Do you have extensive construction experience and a strong background in residential and small commercial solar PV installations? Do you want to join an exciting, young company that has carved out a unique market niche in the fast growing solar industry?\nSolar Roof Dynamics (SRD), the market leader in solar PV integration with the roofing industry, has an immediate full-time opportunity for a Director of Solar Field Operations. This individual will be a dynamic, roll-up-your-sleeves and \"get-stuff-done\" technical leader that will enjoy promoting best-in-class solar installation practices and serving as the technical expert for SRD's large network of installation partners. This position is perfect for an ambitious, self-motivated, and results-oriented construction professional with a strong background in solar installations \u2013 at both the hands-on and supervision level. The Director of Solar Field Operations will work closely with Solar Roof Dynamics' internal team and will support the company's growth and national expansion.\nIn addition to hands-on solar PV experience, the successful candidate will boast skills in all aspects of PV construction, including project management, reading blueprints, and electrical troubleshooting. This individual will ideally have broad experience with single phase and three phase electrical systems, and familiarity with various mounting methods including steep slope roofs, low slope roofs, and ground mounted systems.\nBe the \"go to\" person for technical inquires from SRD's network of contractor partners.\nDevelop and oversee QA program.\nManage technical aspects of relationships with all industry suppliers.\nCreate budgets for residential and commercial projects; provide guidance to contractor partners on materials, timeline, and installation best practices for projects.\nLiaise with the Sales and Project Management teams daily.\nOversee SRD's Engineering and Design teams.\nStay on top of solar industry product and technology trends.\nManage SRD's portfolio of recommended solar equipment.\nManage the company's safety program.\nProven leadership skills and field operations management experience.\n2 to 5 years experience as an electrician.\n1 to 5 years experience in solar \u2013 both residential and commercial.\n2 to 5 years project management experience.\nExcellent verbal and written communication skills with direct customer contact experience.\nIntegrity, strong interpersonal skills, the ability to multi-task, and excellent organizational skills.\nSolid computer skills, including MS Office.\nExcellent problem-solving, critical thinking and analytical skills.\nComfortable working within the ambiguity and excitement of a start-up company.\nHigh school graduate required; bachelor's degree preferred.\nInterested candidates should submit a cover letter and a resume to [email protected].\nSolar Roof Dynamics (SRD) boasts one of the most unique business models in the solar industry. As a value-added solar distribution company focused on leveraging the infrastructure of the roofing industry, SRD is carving out its leadership role as the go-to solar solutions provider for roofing contractors nationwide. Based in Davis, California, SRD enables roofing contractors to quickly integrate solar into their business operations, thereby offering homeowners the protection and cost savings of an integrated roof and solar system. For more information, please visit www.SolarRoofDynamics.com.\nSolar Roof Dynamics is an equal employment opportunity employer. We offer a competitive salary, full medical and dental benefits, 401(k) savings plan, paid holidays, vacation time, continued training and advancement opportunities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Neighborhood Formation and Support (\"NFS\") Committee works closely with the Board, the other Standing Committees and the Office of Neighborhoods in ensuring the vitality of the NOrgs, who are the basic building block of ONE-PS. For neighborhoods that are not covered by a NOrg, the NFS Committee leads in stimulating the development of new NOrgs by identifying interested individuals in the neighborhood, advising in the development of its boundaries, meeting standards, bylaws and finances, and attending initial meetings. For active existing NOrgs, the NFS Committee attends whenever possible their annual and special meetings and is the key point of contact for ensuring their continued viability, participation in ONE-PS activities and, working with the Governance and Finance Committees, compliance with their legal and financial obligations. For existing NOrgs that have become inactive, the NFS Committee has the key role in efforts to revitalize them. In all these activities, the NFS Committee takes the lead in developing a database of best practices, procedures, resources and personnel for use by the NOrgs. The NFS Committee also maintains and archives the records of NOrg involvement in ONE-PS and works with the Governance Committee in developing data security guidelines and procedures for the NOrgs, which, if approved by the Membership, will be binding on all NOrgs.\nThe CERT subcommittee of the Neighborhood Formation and Support (NFS) committee coordinates the activities of the Palm Springs Community Emergency Response Team Program. The CERT program educates residents about disaster preparedness and trains them in basic disaster response skills (see Palm Spring CERT for more details.) Through the NFS committee the CERT subcommittee coordinates activities both with Neighborhood Organizations and with other ONE-PS committees. Membership in the CERT subcommittee and standards of announcing, conducting, and documenting of meetings conform to the standards specified in the ONE-PS Standing Committees Charter of July, 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please Do Not Bid on this project if you are not a Joomla Website programming expert. Please read thouroughly\u2026only an experienced individual or company desired. Successful completion of this project will prove your capabilities and win you more work.\nWe currently have a Real Estate Listing website ([login to view URL]) that is built in PHP and SQL database backend. I am starting over to build a better website using Joomla and MySQL. You can view [login to view URL] to view the different functionalities that will be required, but the new website will have these features and more functions.\nYou can have full access to the existing website programming code which is in PHP, very extensive and already has most functionalities but may be useless because new site in Joomla.\nI am changing to Joomla because of the stability and the multiple functionalities that we do not have. We have an Expert Joomla and database programmer overseeing project.\n-Integrate Open Realty with Joomla to provide a seamless for sale by owner real estate system.\n-Integrate 3rd party components, such as blogs, forums, slideshows, etc...into the site seamlessly.\n-Integrate an advanced Banner Manager into the site.\n-Add paypal component to charge users for placements of ads. Based on type of user.\n-Modify theme (either prebought joomla templates or custom created) for consistency throughout site.\nALL 3rd party components should be consistent with main theme.\nhi thanks for inviting us. as you invited, you must be knowing of our high class joomla [login to view URL] demos in pmb and read our joomla reviews. thanks.\nHello, we have the creativity, joomla experience, real estate website development experience and ready to start to work on your project. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A wonderfully fragrant variety!a beautifully scented variety with stunning fully double rose pink blooms giving the appearance of an 'old fashioned' rose. Each stem produces a cluster of 3 to 5 blooms. This is an exceptional garden variety which offers good disease and pest resistance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2018 CLIPPER 16RBD, ULTR LITE, TRAVEL TRAILER, SINGLE AXLE, HYBRID WITH 2 CANVASS TIP OUT BEDS, TOWABLE ESSENTIALS PACKAGE: 3 CU FT REFER, 20K BTU FURNACE, 13.5 BTU ROOF AC, 6 GAL GAS\/DSI WATER HEATER, SPARE TIRE\/CARRIER, REAR STAB JACKS, AM\/FM\/BLUETOOTH STEREO, AWNING, FOOT FLUSH TOILET AND ELECTRIC BRAKES. DELUXE PACKAGE: ALUMINUM WHEELS, LEND A HAND ASSIST HANDLE, FENDER SKIRTS, SPARE TIRE COVER, EXTERIOR SHOWER, TUN SURROUND, BATH MIRROR. FRONT WINDOW W\/ ROCK GUARD. CALL 505-334-5500 FOR MORE INFO OR STOP BY TO SEE IT TODAY!\n2018 CLIPPER 17BHS, ULTRA-LITE, TRAVEL TRAILER, BUNK HOUSE WITH 1 SLIDE OUT, WICKED AWESOME DECOR, TOWABLE ESSENTIALS PACKAGE:3 CU FT REFER, 20K BTU FURNACE, 13.5K BTU AC, 6 GALLON GAS\/DSI WATER HEATER, SPARE TIRE\/CARRIER, REAR STABILIZER JACKS, AM\/FM\/BLUETOOTH STEREO, AWNING, FOOT FLUSH TOILET, ELECTRIC BRAKES.\nDELUXE PACKAGE: ALUMINUM WHEELS, LEND-A-HAND ASSIST HANDLE, FENDER SKIRTS, SPARE TIRE COVER, EXTERIOR SHOWER, TUB SURROUND, BATH MIRROR.\n2018 CLIPPER 17FB, ULTRA LITE, TRAVEL TRAILER, SINGLE AXLE, WICKED AWESOME DECOR, TOWABLE ESSENTIALS PACKAGE: 3 CU FT REFER, 20K BTU FURNACE, 13.5 BTU ROOF AC, 6 GALLON GAS\/DSI WATER HEATER, SPARE TIRE AND CARRIER, REAR STABILIZER JACKS, AM\/FM\/BLUETOOTH STEREO, AWNING, FOOT FLUSH TOILET, ELECTRIC BRAKES. DELUXE PACKAGE: ALUMINUM WHEELS, LEND A HAND ASSIST HANDLE, FENDER SKIRTS, SPARE TIRE COVER, EXTERIOR SHOWER, TUB SURROUND, BATH MIRROR. FRONT WINDOW WITH ROCK GUARD AND MORE. CALL 505-334-5500 FOR MORE INFO OR STOP BY TO SEE IT TODAY!\n2018 CLIPPER 17FQ, TRAVEL TRAILER, ULTRA LITE, WICKED AWESOME DECOR, TOWABLE ESENTIALS PACKAGE: 3 CU FT REFER, 20K BTU FURNACE, 13.5 B TU ROOF AC, 6 GAL GAS\/DSI WATER HEATER, SPARE TIRE\/CARRIER, REAR STABILIZER JACKS, AM\/FM\/BLUETOOTH STEREO, AWNING, FOOT FLUSH TOILET AND ELECTRIC BRAKES. DELUXE PACKAGE: ALUMINUM WHEELS, LEND-A-HAND ASSIST HANDLE, FENDER SKIRTS, SPARE TIRE COVER, EXTERIOR SHOWER, TUB SURROUND, BATH MIRROR. FRONT WINDOW WITH ROCK GUARD, TRAVEL EASY ROAD SIDE ASSISTANCE AND MORE. CALL 505-334-5500 FOR MORE INFO OR STOP BY TO SEE IT TODAY.\n2018 CLIPPER 17FQS, ULTRA-LITE, TRAVEL TRAILER WITH 1 SLIDE OUT, WICKED AWESOME DECOR, TOWABLE ESSENTIALS PACKAGE:3 CU FT REFER, 20K BTU FURNACE, 13.5K BTU AC, 6 GALLON GAS\/DSI WATER HEATER, SPARE TIRE\/CARRIER, REAR STABILIZER JACKS, AM\/FM\/BLUETOOTH STEREO, AWNING, FOOT FLUSH TOILET, ELECTRIC BRAKES.\nFRONT WINDOW WITH ROCK GUARD AND SO MUCH MORE. CALL 505-334-5500 FOR MORE INFO OR STOP BY TO SEE IT TODAY.\n2018 CLIPPER 17FQS, TRAVEL TRAILER, ULTRA LITE, 1 SLIDE OUT, WICKED AWESOME DECOR, TOWABLE ESENTIALS PACKAGE: 3 CU FT REFER, 20K BTU FURNACE, 13.5 B TU ROOF AC, 6 GAL GAS\/DSI WATER HEATER, SPARE TIRE\/CARRIER, REAR STABILIZER JACKS, AM\/FM\/BLUETOOTH STEREO, AWNING, FOOT FLUSH TOILET AND ELECTRIC BRAKES. DELUXE PACKAGE: ALUMINUM WHEELS, LEND-A-HAND ASSIST HANDLE, FENDER SKIRTS, SPARE TIRE COVER, EXTERIOR SHOWER, TUB SURROUND, BATH MIRROR. FRONT WINDOW WITH ROCK GUARD, TRAVEL EASY ROAD SIDE ASSISTANCE AND MORE. CALL 505-334-5500 FOR MORE INFO OR STOP BY TO SEE IT TODAY.\nNew model with the ALL TERRAIN PACKAGE! HUGE LIFT!!!\n2018 FLAGSTAFF MICROLITE E-PRO 19FBS, TRAVEL TRAILER, 1 SLIDE OUT, ONLY WEIGHS 3316!!!, 12 VOLT 22\u2033 TV, ROOF MOUNT SOLAR PANEL, 13.5K BTU LOW PROFILE AC, POWER AWNING, TONGUE MOUNTED BIKE RACK, WATER PURIFIER, MAXXAIR VENT FAN AND COVER, BLUETOOTH STEREO, TORSION AXLES W\/ INDEPENDENT SUSPENSION, AND MUCH MORE. CALL 505-334-5500 FOR MORE INFO OR STOP BY TO SEE IT TODAY!!!\n2018 FLAGSTAFF E PRO 17RK, 1 SLIDE OUT, TRAVEL TRAILER, ULTRA LITE ONLY WEIGHS 2988LBS, PACKAGE A: ROLLER SHADES, STEREO W DVD, TV ANTENNA, GAS\/ELECT WATER HEATER, OUTSIDE SPEAKER, ROOF MOUNT SOLAR, AWNING, 20K BTU FURNACE, RVQ GRILL, 2-20LB LP BOTTLES, MICRO, POLISHED ALLOY WHEELS, BONDED WINDOWS, SPARE TIRE, HEATED MATRES,- CONVECTION MICRO, 12 VOLT 22\u2033 TV, 13.5 BTU LOW PROFILE AC, POWER AWNING UPGRADE, WATER PURIFIER, AND MUCH MORE. CALL 505-334-5500 FOR MORE INFO OR STOP BY TO SEE IT TODAY!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhhzv b/data_all_eng_slimpj/shuffled/split2/finalzzzbhhzv
new file mode 100644
index 0000000000000000000000000000000000000000..c0e8ee7ba195acab29585c46162b4a4c881cf747
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhhzv
@@ -0,0 +1,5 @@
+{"text":"A few things that I don't think are spoilers but might change whether or not you want to read this book. One, it reference someone peeing themselves in the first 10 pages. Two, commas are missing to several locations. Three, metaphors are not Ross's strong suit, and those which are there suffer from cliches. Four, if you go by the main character's age (and the writing itself), this is more of an upper middle grade in my opinion.\nThere are historical elements, focused around the collapse of the Achaemenid Empire and the Zoroastrian religion. As that suggests, we're in the Middle East (Persia). Anyone familiar with that time period will recognize a number of names \/ places.\nOverall, it reminded me of the Eragon series. They both contain common elements (hero journey arch), and while generally entertaining, there are a number of other stories that do so in a more unique manner. It's a decent read, but not one I'd pay for \u2013 so it's convenient that it was free.\nAdditionally, the free version of this book includes three chapters from the second. I found this a bit odd, considering it also has an epilogue, but they aren't mutual exclusive even if my brain wants them to be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sugar in the blood is the blood is the to human and animals. So how does this sugar get into the blood? Sugar is current in our blood primarily simply because of the foods we consume mainly carbohydrates. The sugar level is controlled by a hormone known as insulin which is made by the pancreases. What this indicates is the insulin converts these sugars into a lot needed energy for the physique.\nAnd be certain to consume plenty of water prior to, throughout, and following strolling. And to make it more fun(and if you don't have anybody to walk with), listen to music while walking.\nThe drug will be released this yr to assist individuals with kind 2 diabetes. It is actually one of 2 new drugs that will be launched this year to help fight dieta para diabeticos. The problem is that medication are not all-natural and do not remedy diabetic issues. These drugs have been known to cause inflammation of the legs which indicates that the circulation to the heart is being affected in a harmful way. Diabetic medication can cause coronary heart failure. Reverse diabetes normally diet programs have been little help because these diet programs do not mend the insulin issue.\nMix half teaspoon powdered bay leaf and half teaspoon turmeric powder. Include the combination to aloe Vera gel and consider prior to lunch and dinner. This is beneficial in dealing with cure diabetes.\nLastly, Cats are like individuals they gain excess weight when they do not physical exercise. It is important that your cat has space and things that he can physical exercise on. Yo9u want your cat to leap stretch and operate. If your home has stairs this is great physical exercise for your cat. If your cat doesn't get this physical exercise on their personal I suggest you play with your cat using toy and products that will entice them to be energetic. Correct diet and exercise is essential in maintaining your cat wholesome and reducing their weight acquire.\nUnfortunately, Type 1 cannot be prevented, as it happens normally in some people's bodies. Type 2 can be prevented, and both can be treated and even remedied naturally. Many are required to consider costly prescription medication like insulin injections or tablets to control their sugar levels. Diet plan is the most important factor to the prevention of Kind 2 and the therapy of each kinds!\nWhat they do know is that the sympathetic anxious system is at the main of the issue. This part of the physique regulates numerous bodily functions such as the secretion of sweat.\nSo the thriller is solved-we did it to ourselves. Now we have \"processed foods\", which are packed with sugar. We have become sugar-dependent. Diabetes is only one outcome we weren't expecting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The key difference is unit scale. The ships are ginourmous even when zoomed out. In this game I feel that things are largely of the same size and when we zoom out for a tactical perspective things come so so tiny.\nPersonal visual gripe but yes. The best example of what I'd like is during the tutorial right when you command the Imperial Fist Phalanx.\nThat... is what I would like baseline in terms of unit scale.\nYour game sounds great, feels great, plays great, looks great.\nBut there is one thing that I just can't shake and makes me disappointed everytime.\nWhat could possibly be the problem for me since I love largely everything here?\nI'm a colossal fan of MEGA sized stuff. I LOVED Star Wars Empires at War and how absolutely mega the star ships got to be the stronger I got and as I got access to larger ships.\nSeeing them on the battlefield was insane. They took up so much space on the map and it just felt astounding epic and mindblowing to see battles go down, having them get damaged and fall apart.\nEven as I sent multiple of them to their deaths for fun I LOVED seeing the fights go down, whilst spamming entire navys full of mega star destroyers. It was just fun and cool to behold.\nAs you can see here the varying ship sizes were magnificent and it really went to add so much extra to the visual spectacle plus appreciation for \" Oh boy... now comes the big guy!!\"\nI dont feel your games add this element unless I zoom all the way in and that is tactically unwise, which goes without saying.\nI'd love if it you would make everything bigger in size. LEt us have more ships on the maps, and make everything larger.\nI know some players would argue that its smaller because thats more strategic and such, so on and so forth.\nYour game here introduced some SERIOUSLY mega sized constructs but most of the time they are either static background stuff or special units we never see again.\nSo, dear devs. Please... consider... making stuff larger. Much larger. Because I would absolutely love to play this game and savour every battle whether loss or win because of the visual awe this stuff has.\nI mean come on... ain't nobody here can say they didn't had their jaw drop when the Death Star in Empire at War was approching the planet you were fighting around whilst seeing MEGA ships fire lasers all over the place.\nAnd then you get that big flashy button .... BOOM ! entire planet gets rekt.\nJust look at it. let alone controlling it as part of the fleet, moving with your navys.\nSo yeah, Devs. This thread became 2 things now !\nNr 1: make stuff WAY bigger. Pretty please. Like this here.\nAnd nr 2: Give us control of the inquisition and lets nuke planets !!!!!!\nBut hey. If you will only consider 1, just pick Nr 1. Scale matters a lot to me because I enjoy these games for the visual sci fi spectacle above everything.\nyeah they are really not responding to a whole lot recently = last 2-3 months.\nGore system expansion and improval concerning the combat system and NPC's you kill in combat only.\n---Dismemberment = heads, arms, legs etc. Would be fitting when killing with vampire claws especially.\nHey guys. You're very silent on specific questions in terms of what the game has and doesn't have.\nHowever, can you answer if the game has modding support or will have?\nIt would be nice if we could get the chance to mod in blood, gore etc to make the combat system far more interesting for those of us who enjoy embracing the monster type of gameplay.\nAlternatively, consider releasing a gore DLC. have limbs fly off, heads, guts and wound damage on generic enemies.\nPLEASE consider it. I really like all I see about this game, but the combat feels like something I'd rather avoid... because theres just no visible damage or conclusion to cutting, shooting, ripping, biting, or using super vampiric powers impaling them.\nIts incredibly lackluster visually, guys. Please consider fixing it via gore DLC or allow mod support so we can alter a few things to our satisfaction.\nIt is as I was afraid of it seems. Well. Lets hope it turns out to be an enjoyable story then.\nYeah, man. I hope we'll get an answer otherwise we'll just have to wait for release.\nBut the more I see I think we're gonna be without any of it.\nIt just feels extremely... weird. I mean you can look at way older gory games like F.E.A.R. that Im replaying atm. It has whole bodies exploding on grenade impact or something as simple as bullet holes as well as blood spatter.\nOr within the same genre, Vampire The Masquerade: Bloodlines had a great deal of gore using Tremere abilities.\nWell. Tuesday the week after next week, we'll know. Lets hope for a gory adventure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Safety is paramount during a Rewind Disco and we ensure all our equipment is safe and tested to ensure faultless operation. Again, a current PAT is not law but is often requested by venues. Most equipment we use is less than 12months old so is not an issue, but where it matters, it gets tested!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These pieces was created out of assigning harmonically related patterns to MIDI notes determined by dice. I had been reading a book that suggested that no music created from random processes could possibly have a spiritual nature like 'real' music. I accepted that as a challenge right away!\nThe remixes were done later and are just now seeing the light of day on the internet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbipjf b/data_all_eng_slimpj/shuffled/split2/finalzzzbipjf
new file mode 100644
index 0000000000000000000000000000000000000000..1213deb17075f215d6fd32ea8822de986bdeb00f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbipjf
@@ -0,0 +1,5 @@
+{"text":"This module generates the telematic presentation of the model 349 of the Treasury. It is a declaration of Income Tax for Individuals, Corporation Tax and Non-Resident Income Tax (permanent establishments). Certain income from movable capital or certain income.\nUsing the Model 123 module, you can generate each quarter or every month the document for the telematic presentation of this declaration.\nIt allows generating the 123 model of the Treasury corresponding to the Declaration of the Income Tax of the Physical Persons, Corporation Tax and Non-Resident Income Tax (permanent establishments). Certain income from movable capital or certain income.\nModule for the presentation of Model 123 on Withholdings and income on account of Personal Income Tax, Corporation Tax and Non-Resident Income Tax (permanent establishments). Certain income from movable capital or certain income.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was very kindly sent this beautiful box from Rosina's Lotions and Potions filled with goodies to review!\nThe candle is paraben and GMO free and is made from natural ingredients.\nThe 30cl candle costs \u00a320 and the 9cl is \u00a312. I have to admit as I don't really buy candles, I don't really know if this is good or not? I also haven't burned it yet because it's too pretty so I don't know how strong the fragrance is when it's burning.\nThe packaging on the Kale and Pomegranate Exfoliating Face Mask is absolutely stunning; I love the gorgeous stripes and flowers design!\nIt's a multi-mask treatment; it's a cleanser, mask and facial all in one! The phyto-active pomegranate enzymes in the mask help to exfoliate the skin, leaving it looking brighter while vitamins C and K can help to minimise damage caused by stress, pollution and free-radicals!\nThe scent of the product is something I can't quite put my finger on, there's definitely more of a savoury feel to it but I really like it!\nAt \u00a325 for 50ml, it's definitely one of the most expensive face masks I've ever had and, although it's lovely, it's probably not something I'd repurchase as it's way out of my usual price range.\nIt's a really subtle rose scent which I love and I'm not one for heavily perfumed products on my face. It's a lovely thick oil (I hate runny oils) and easily sinks into skin in seconds. I was a little worried about putting oil on my face as my T-zone especially can get quite oily, but it's been fine with no adverse reactions at all!! This little bottle will set you back \u00a322, but that's not too bad compared to some oils I've seen!\n*These products were kindly sent to me but views are my own.\nI had this package a few months ago so nice, and I kept the box as well !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prominent black entrepreneur Geder Walker built this masonry vaudeville and movie theater in 1913. Once one of the major centers of entertainment for blacks, this building is the lone survivor of the \"Little Broadway\" district that flourished in Overtown for almost 50 years. It is the oldest legitimate theater remaining in Miami. Now restored, the 390-seat theater features exhibits, festivals, jazz, theater, dance and multicultural performances.\nThe Lyric Theater anchors the Historic Overtown Folklife Village.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The official Health and Wellness podcast series of Memorial Health System.\nDr. John David Krant discusses psoriatic arthritis. He explains the symptoms you would notice, the risk factors involved, as well as the latest treatment options available at Memorial Health System.\nCan You Prevent Getting a Cold or the Flu This Season?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Named for Houston area developer, Richard Weekley, and his wife Meg, the park serves numerous purposes. Opened in 2014, bringing the community together was a big consideration to Precinct 3 when planning projects such as this. The 21,000 SF Community Center serves the public's need for a versatile meeting location with dance, fitness, arts & crafts and several other activities scheduled throughout the year. More information can be found on the Community Center Website. Improvements have been being made to the adjacent park area in the years since opening to the public. The park is easily accessible to the neighboring subdivisions via sidewalks, but plenty of parking is available both off Greenhouse Rd and Longenbaugh Rd.\nWhile the community center is the main draw for visitors, the surrounding park can be enjoyed by patrons of all ages. The .85 miles of paved walking trail runs the length of the park along a portion of Langham Creek that has been partially dammed for flood control purposes. Benches are located along the paved trail with picturesque views of the wetland\/wooded scenery of the property. At the time this article was written, visitors can use the unimproved footpath on the east side of Langham Creek. Future expansion of the paved trail around the pond is anticipated. The north end of the park (adjacent to the Community Center parking lot) has a playground with numerous picnic tables, both covered and not. The south end of the park (off Longenbaugh Rd) features a covered fitness area with covered picnic areas and swings.\nWe have put together a complete Photo Gallery of Richard & Meg Weekley Park\/Community Center in Cypress Texas with descriptions of where the photos were taken and what the photos depict.\nThis article was authored by and all photos were taken by the staff of Houston Prime Realty. Houston Prime Realty serves the Greater Houston Texas area and can help real estate buyers and sellers to purchase and sell properties near Richard & Meg Weekley Park in Cypress Texas. Please Contact Us if you need an experienced Houston real estate agent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjvca b/data_all_eng_slimpj/shuffled/split2/finalzzzbjvca
new file mode 100644
index 0000000000000000000000000000000000000000..ccc2547f7fcddf1eb2c3ba9ea85282c28a7f90dc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjvca
@@ -0,0 +1,5 @@
+{"text":"My name Jewel Age: 35. Soy una linda chica de \/edad chilena recien llegadaMy most favourite places are toscana, roma, provence in france, spain, greek islands, capri and sardinia.I am available weeknights from pm to midnight and anytime weekends for in or outcallZ przyjemnoscia latalem wedzidlem na skraju orgazmu.\nThe fur bikini of Raquel Welch refers to the fur\/hide bikini worn by Raquel Welch in the British-made prehistoric saga One Million Years BC. In that bikini.\nView and license Raquel Welch Bikini pictures & news photos from Getty Images.\nShannah Age: 34. OrMy name is Melis, I am a fun loving istanbul lady escortI'm Summer,To make an appointment or for more informations, please, contact me by phone (no withheld numbers, please) or e-mail.\nDescription:University of Southern California. She had such a perfect body that I took a very soft doe skin, we stretched it on her and tied it together with thongs. Publicity[ edit ] A publicity still of her in the bikini became a best-selling poster and turned her into an instant pin-up girl. The furry part of the skin is on the inside, like a lining, and provides elegance to the edges. In the version of the film, Carole Landis had worn a soft-leather version of a one-piece swimsuit in the film with similar carefully cut \"random\" edges as Loana. But, the Raquel Welch fur bikini remained the Hammer's biggest success. The photo taken by Terry O'Neill was deemed too scandalous and was suppressed, until it was published thirty years later by the Sunday Times Magazine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I made this from composite deck material and granite counter top. ... Wooden Big Green Egg Large Table DIY blueprints Big green egg large table So I made this one at about a of the 41 for Compact or 58 for Long Table Is now an The Big ... JJGeorge Big Green Egg Tables & Wooden Patio Furniture by JJGeorgeFurniture.\nThe is no definitive answer. It is a matter of your fire safety comfort. Consider the location of the table - is it on a cement patio or a wood deck attached to the house? Think of the possible consequences if the egg failed and dumped hot coals. Consider the cost to increase the fire safety and make your call.\nArchadeck of West Central & Southwest Ohio is your source for all things outdoor living and we are the local Big Green Egg dealer for the Waynesville, OH, region and we plan on pulling .... Composite decks are great, and we do recommend them for their easy maintenance, durability and longevity compared to wood decks.\n, the leading manufacturer of composite decking sends out wonderful design ideas from their multitude of professional installers that is a frequent reference guide.We are seeing outdoor fireplaces being installed on and next to decks, fire pits, outdoor kitchens, and even Green Eggs in built-ins.\nFind and save ideas about Big green egg outdoor kitchen on Pinterest. ... outdoor built in grill and prep counter wood storage rustic something similar for ice ..... As Twin Eagles & Delta dealers, Deck Remodelers will design & build your custom outdoor kitchen in New Jersey with high quality products at trade costs.\nBrowse 392 photos of Big Green Egg. Find ideas and inspiration for Big Green Egg to add to your own home. ... This built-in Big Green Egg kamado-style charcoal cooker looks better in outdoor kitchens when encased in natural stone surrounded by pavers. ... Example of a trendy outdoor kitchen deck design in Austin.\nThe Paint Boys at Capps have a new line of long-lasting, water-based wood and deck stain products available for you\u2026 introducing DEFY , a premium wood stain featured on the nationally syndicated radio show \"At Home with Gary Sullivan\".\n3 Reasons homeowners are replacing their wood decks with decking products. Contact Metro Building Products to speak with a specialist. Call Now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Working with Mini and AJ is very challenging because we have to do almost everything in a short span of two months. We have a specific budget to stick to, I participated in the majority of conceptualization and details and up to now I find it amazing how we were able to pull it off within that time limit.\nThough it was a challenge being the resident florist and part of the guests in the wedding, it's all worth it because it's always fun working with friends.\nHere are some of the official photos from the Campos-Malabanan Nuptials.\nAprille also personalized the candles and matchbox for the wedding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Christine is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nAnnie N. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nBeen sitting on this bottle a while, excellent beer! Hope they make this one again.\nDavid D. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nEarned the Taster, Please (Level 22) badge! Earned the Heavy Weight (Level 89) badge!\nAndrew P. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nEarned the Brewery Pioneer (Level 53) badge! Earned the 99 Bottles (Level 86) badge! Earned the Imperial Czar (Level 32) badge!\nErin R. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nBeer C. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nJay H. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nBetty I. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nMichael F. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nScott D. is drinking a Vlad Vs. Zombie Vs. Shark by Hailstorm Brewing Co.\nJay H.: Picture is hilarious!!!\nBrasserie Noir Goose Island Beer Co.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 2018 Final Four marks the first time that basketball will be played at the center of the Alamodome. With the court being placed in the middle of the stadium, sideline seating is being expanded to reach all the way down to the floor. These sections will be custom-built for the Final Four and will have more than 60 rows of seats.\nThe most desirable seats in these sections will be those located closest to the court. The first four rows in each section are labeled 1-4 and can be thought of as floor seating. Fans seated in these rows in sections 132-136 will be seated just behind the player benches.\nBehind the floor rows are lettered rows ranging from A-Z and then AA-QQ. Numbered rows then pick up after those sections and go as high as 35. When choosing tickets in these rows, there will be a significant trade-off between a close view and convenience. Sitting in the single-letter rows will get you as close to the action as possible (which is a huge perk in a non-traditional basketball stadium), but you will have long walks to the concourse and its amenities. The reverse is then true for fans sitting in numbered rows.\nOverall, we recommend sitting in one of the sections closer to center court, even if it means sitting in a higher row. You'll get one of the better unobstructed views of the game from near center court and if you are in a high row, you'll be paying less for your ticket and enjoy the convenience of being near restrooms and concessions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkorv b/data_all_eng_slimpj/shuffled/split2/finalzzzbkorv
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+{"text":"Most of the rest is used in basal metabolism, the energy we need when in a resting state, for functions such as blood circulation, digestion, and breathing. Retrieved 10 May The main treatment for obesity consists of dieting and physical exercise. Anyways, best of luck and please let us know how it goes: A review identified ten other possible contributors to the recent increase of obesity: The Journal of Urology Review. Reading your story has given me some inspiration.\nThis blog looks exactly like my old one! Great choice of colors! Really inspired by your story \u2014 thanks for sharing! I am planning to start Nutrisystem after the New Year. Hi Erin \u2014 Thanks so much for the positive feedback. The first time I ever used Nutrisytem, which was many years ago at this point, I think I was one it for 4 months. I had a lot more weight to lose then; now when I sign up, I usually use it for a month, maybe two at the most.\nAt the end of the 8 weeks I had lost only 3 lbs. We did not care for the cardboard like food and did not get anywhere close to our goals. This program obviously works for lots of folks, but not for us. We did go to the Naturally Slim program and in 8 weeks I lost Naturally Slim is based on not what you eat but when and how you eat and you eat your own real food and got real results. We will stay with our new habits learned with Naturally Slim and will not have any good words from our Nutrisystem experience.\nThanks again for your review! Hi Peggy \u2014 I think you should be fine using your own protein shake. American Heart Journal Meta-analysis, Review. American Heart Journal Research Support. Public Health Reports Research Support.\nInternational Journal of Obesity Review. Calorie supply per capita\". Archived from the original on 11 June Retrieved 18 October United States Department of Agriculture. Archived from the original on 3 June Retrieved 10 January Centers for Disease Control and Prevention.\nRetrieved 9 July American Journal of Preventive Medicine. Morbidity and Mortality Weekly Report. Diabetes Care Meta-analysis, Review. Agriculture Information Bulletin No. Archived from the original on 8 July Retrieved 30 July Archived from the original on 18 December A Global Public Health Problem\". Retrieved 22 February Prevalence, trends and environmental influences on child and youth physical activity Review. Medicine and Sport Science. Annual Review of Public Health Review.\nAmerican Journal of Public Health. European Society of Sleep Technologists. Archived from the original on 13 April Arteriosclerosis, Thrombosis, and Vascular Biology Review. However, it is also clear that genetics greatly influences this situation, giving individuals in the same 'obesogenic' environment significantly different risks of becoming obese.\nJournal of Women's Health Comparative Study. A review of physical activity and parenthood\". Obesity in developing countries: The Journal of Nutrition Review. Mayo Clinic Proceedings Review. Retrieved 2 July A cellular and molecular approach. Retrieved 1 February Preventive Services Task Force recommendation statement\".\nAnnals of Internal Medicine Practice Guideline. Retrieved 24 July Department of Health and Human Services. Retrieved 13 October International Medical Publishing, Inc.\nRoyal College of Physicians. Report, together with formal minutes. Retrieved 17 December Retrieved 8 April Our Future Health Secured? A review of NHS funding and performance. Journal of the American Dietetic Association.\nAre higher levels of physical activity protective against weight regain? Annals of the New York Academy of Sciences. Retrieved 17 June Retrieved 26 April Retrieved 30 June Archived from the original PDF on December 11, Retrieved January 29, Our World in Data. Retrieved 31 December Investigating the socio-demographics within the extremes of body mass index\". Measured height and weight\". Findings from the Canadian Community Health Survey.\nOxford English Dictionary Archived from the original on 11 January Retrieved 21 March Archived from the original PDF on 10 October Retrieved 19 September University of Nebraska Press. The Fat of the Land: Economics and Human Biology. Journal of Health Economics.\nHow much, and who's paying\". Retrieved 5 September Retrieved 25 February Int J Ther Rehabil. Archived from the original on 8 October Archives of Internal Medicine Research Support. Retrieved 23 July American Journal of Preventive Medicine Letter. Retrieved 6 August Personal Responsibility in Food Consumption Act of \".\nSocial Science Electronic Publishing. Retrieved 29 July Retrieved 13 January The Relevance of History for Current Debates\". British Journal of Sports Medicine Review. The Cochrane Database of Systematic Reviews. Feasibility Study in Minority Populations\". Bhargava A August The Journal of Nutrition Research Support.\nMeasuring body composition in adults and children In: Clinical obesity in adults and children: In Adults and Children. An overview of obesity management In: Social consequences of obesity In: Weight Loss Edition introduces us to a new morbidly obese person every week that is selected to endure a day challenge to lose half their weight. Another not so good week, but I have been getting some bodyweight workouts in.\nI feel like a failure for not working out as much as I normally do and for eating more snacks than I normally do. All I can do is move on and not let it erode my progress. A study must show that minding your own business is good for your health. There are studies done about everything. Through an unrelated series of events, it [\u2026].\nThe week started off good on Monday, but after that it was hard for me to find motivation to exercise. I managed to fight through the mental barrier that I always face. They are followed over a six month period to lose as much weight as possible.\nFor the first month they leave to a camp and are under complete [\u2026]. This idea is two years in the making.\nIn October of I came up with this idea to promote health and fitness. Two years went by and I dropped the ball on it. How many times have we told ourselves that we want change? We are tired of lounging around without purpose. It was reported yesterday that Adam Sandler is seriously considering playing the role of a lb morbidly obese man in a new movie tentatively titled Fat Man.\nConsidering Nutrisystem or Jenny Craig? I've tried both. Find out how the food taste, cost and services compare to see which diet is best for you. Medifast and Nutrisystem offer diet programs for weight loss that use a combination of packaged products and fresh foods in their meal plans. Compared to. Lean Cuisine is a line of low-fat frozen dinners designed for dieters and is available in your local grocer's frozen food section. Nutrisystem is a.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to Manhattan district attorney, Cyrus R. Vance Jr. five people have been indicted as part of a fraud ring that targeted Saks Fifth Avenue.\nThe ring used information stolen from shoppers to purchase more than $400,000 worth of luxury shoes, handbags, and accessories in 91 different transactions from Saks' shoe department. Apparently, the department is so large it requires its own ZIP code: 10022-SHOE.\nAccording to Vance, the alleged ringleader, Tamara Williams, has been accused of providing personal information, such as Social Security numbers and credit card numbers, to four employees at at the store. The employees then used this information to buy shoes and purses that Williams had pre-selected. Williams then sold the items on the black market, returned them for store gift cards, or kept them for herself. She also used some of the money she received from selling the items as payment for her accomplices.\nWilliams and the other defendants \u2014 Kriss Rockson, Jason Chance, Alaia Harrison, and Michael Knight-Williams have been indicted for 66 charges including grand larceny, identity theft, and scheme to defraud. Vance's office worked with the Secret Service and the Department of Homeland Security to track down the defendants, who reputedly had been buying shoes with stolen information since last April.\nHe also went on to say that he feels existing laws for identity fraud fall short. The maximum punishment for stealing the identity of one person is the exact same punishment for victimizing 250 people. \"State laws are not adequate to combat the kinds of cases we are seeing,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Grimmia trichophylla Grev. s. l.\n(syn. G. decipiens var. robusta sensu Paton 1969a: 729 non Braithw.).\n*1: Trungle near Paul, 1867, WC (PNZ) (Paton 1969a: 729).\n*2: Near St Erme, 1861, ES (TRU) (Paton 1969a: 729).\nG. lisae was treated as a distinct species from G. trichophylla by Greven (1995) and Mu\ufffdoz & Pando (2000), but characters allowing its separation from several similar species remained unclear until the paper by Maier (2002). Since then specimens from Cornwall have been revised and treated as G. lisae or G. trichophylla s. str. (see separate accounts ); a third species of the group, the calciphile G. dissimulata E.Maier, is known elsewhere in southern England but has not yet been found in Cornwall.\nForms cushions and small patches that may extend or coalesce to form larger patches or low lawns. Notes on the habitats of the s. l. species in C&S are as follows (each species is also treated separately below). Grows mainly on hard dry rocks, including base-poor types (granitic, shale, slaty, gabbro, greenstone) and serpentinite, on open surfaces, small ledges and in crevices, on horizontal to steeply inclined surfaces. Occurs on boulders, blocks, natural outcrops, walls (including those of churches), rocks in 'hedges' and quarries, and on gravestones, most often in unshaded or lightly shaded places or part shaded e.g. by bushes or in open woodland, but more rarely well shaded inside deciduous woodland or groves. Sometimes common on upper cliff slopes and cliff tops, so evidently tolerant of salt spray. A few records also from within regularly flooded zones beside rivers, a reservoir and a pool.\nG. trichophylla s. l. sometimes grows on granitic rocks within or capping mortared or concrete walls, but usually not directly on the calcareous substrates. Several such records from granite parapets of bridges over R. Inny and R. Tamar are in regions otherwise unsuited to the species. Single record of several small patches (checked microscopically) from sloping concrete on grave, close to Didymodon luridus and Tortula muralis, so surely on a basic substrate there, as also on old concrete beside steps on exposed coastal slope. Despite rarity of capsules it often colonises isolated patches of suitable rock away from the natural outcrops of granite, especially old granite (or less often slaty) gateposts, gravestones, masonry debris, rocks in walls and stonework of larger bridges. An unusual record of substantial patches of low plants on old tarmac of edges of path in churchyard.\nAssociates often include Bryum capillare, Dicranoweisia cirrata, Hedwigia stellata, Hypnum cupressiforme var. resupinatum, Racomitrium heterostichum, lichens; also Sciuro-hypnum plumosum in a river flood-zone. Common alongside the mainly more basiphilous G. pulvinata (q.v.) on serpentinite, but mostly replaces that species on granitic rocks, except at a few coastal localities where they occur together. G. trichophylla s. l. entirely replaces G. pulvinata on gabbro at Lowland Point.\nSingle record c.fr. by DTH: capsules immature 11 (few, patch on boulder partly sheltered by scrub).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Social media really stinks sometimes. Our pictures that we post can make people believe that everything is perfect (aka we're living our best life), which is kind of what we want them to do. We don't want others knowing the pain we are going through. We want to hide it all underneath and pretend that everything is fine.\nMy Instagram could show you a lot of things. It shows you that I've been on a few trips in the past few months, been with friends and family, and have an obsession with photography (LOL). But it doesn't show you everything. It doesn't show you the stressful, late nights spent cramming for tests. It doesn't show you how tired I am, and how I want to just sleep all day. It doesn't show you how insecure I am about certain things. It doesn't show you the times I have cried over being hurt. It doesn't show you everything.\nBut one thing, I do really love about social media is that it gives you the chance to be vulnerable and to be real. It allows you to relate with others on so many different topics, whether that's dogs, coffee, or Jesus. It's a way to connect and love on others all over the world. It's a chance to spread God's word and love globally.\nSo I want to challenge you (and myself) to be real, whether that's on a phone, behind a computer screen, or just in your daily life. Show the love of Jesus to others and meet them where they are. It's a little thing that makes a big difference.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our professionally designed and printed basement waterproofing sales literature folders will help you close every sales pitch! The tri-folded literature folders include six different inserts for each product line, a place to insert your business card, homeowner checklist and a slot to insert a copy of the your estimate for the homeowner.\nProfessionally Printed Sales Material. Just Add Your Business Card!\nOur full-color professional printed literature folders and seven tiered inserts include a place for your business card or company stamp, homeowner checklist, insert for estimator and warranty documents, satisfied customer testimonials, and full detail about every exclusive product line!\nThe literature kits showcase to the homeowner the large repertoire of waterproofing systems your company can offer to remedy their waterproofing problem. The colorful tiered inserts describe each system in detail through visual diagrams highlighting their features and benefits. The literature kits also include a slot for your estimator document, business card or stamp, and a checklist for homeowners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"SCL Footwear Group is currently a progressive leader in the license footwear segment. SCL's primary goal is thru license revenue to acquire or create an in house brand to complement its current license programs. Our company strives to employ the best design and manufacturing standards that will most effectively service the retailers that we do business with.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"my deepest thanks to the seller for setting the best price for FREDI Baby Monitor WiFi Wireless Camera 1080P HD Security Camera with Two-Way Talking,Infrared Night Vision,P2P WPS Ir-Cut IP Camera Motion Detection Loop Recording(Without SD Card) (White). it means the world to me!\nEven without any discount FREDI Baby Monitor WiFi Wireless Camera 1080P HD Security Camera with Two-Way Talking,Infrared Night Vision,P2P WPS Ir-Cut IP Camera Motion Detection Loop Recording(Without SD Card) (White) is really reasonably priced so this is truly an awesome purchase. Can surely recommend!\nThis store offers the best price for FREDI Baby Monitor WiFi Wireless Camera 1080P HD Security Camera with Two-Way Talking,Infrared Night Vision,P2P WPS Ir-Cut IP Camera Motion Detection Loop Recording(Without SD Card) (White), great deal!\nlet's face it. a brand new FREDI Baby Monitor WiFi Wireless Camera 1080P HD Security Camera with Two-Way Talking,Infrared Night Vision,P2P WPS Ir-Cut IP Camera Motion Detection Loop Recording(Without SD Card) (White) is not a something you'd find easily, but the seller did the impossible. great job!\nEven if it's not the cheapest FREDI Baby Monitor WiFi Wireless Camera 1080P HD Security Camera with Two-Way Talking,Infrared Night Vision,P2P WPS Ir-Cut IP Camera Motion Detection Loop Recording(Without SD Card) (White) online, I recommend buying anyway. The quality is astonishing!\nDiscounted FREDI Baby Monitor WiFi Wireless Camera 1080P HD Security Camera with Two-Way Talking,Infrared Night Vision,P2P WPS Ir-Cut IP Camera Motion Detection Loop Recording(Without SD Card) (White) is everything I could dream of. The store is great, btw!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our service is supported by a Youth Reference Group to ensure we are providing a service that young people want, need and like. In addition to extended hours on Tuesday evenings, headspace Mildura also operates programs outside hours, in schools and community groups.\nYou can access headspace Mildura by dropping in to make an appointment, by phone call, email or fax. Referrals can be made by the young person or by a family member, GP, teacher, or someone close to them. If you are making a referral to headspace Mildura on behalf of a young person we ask that they are aware you are doing so. Professionals referral form.\nFor urgent assistance, please call Mildura Base Hospital on 50223500 or Lifeline 131114. In an emergency call 000.\nheadspace Mildura is operated by Ramsay Health Care Pty Ltd. All headspace services are funded by the Australian Government Department of Health. Administration of funding is carried out by the headspace centre's local Primary Health Network, in this case, Murray (Murray PHN).\nheadspace Mildura has a sexual health clinic with a GP and sexual health nurse that can help with condoms, dental dams, STI testing, pregnancy testing and advice.\nOur youth-friendly sexual health nurse can offer confidential advice and information. She is available by appointment only, click here for more information.\nUnfortunately, our GP service is currently unavailable.\nTo see the sexual health nurse or GP, you'll need to have access to medicare. Many young people are listed on a parent's medicare card, and we can look up your medicare number without needing the physical card. For young people aged 15 and above, you can apply for your own medicare card if you want to. Fill out the medicare application form and return this form with ID (such as your birth certificate or student ID) to the medicare service centre. Our local medicare service centre is in the same building as centrelink at 112-124 Deakin Avenue, Mildura.\nPlease note: even if you don't have access to a medicare card, please contact us and we will assist where possible.\nFor most services at headspace Mildura, you do not need a medicare card.\nIf you need to see the sexual health nurse or GP, you will need to have access to medicare.\nMany young people are listed on a parent's medicare card, and we can look up your medicare number without needing the physical card.\nFor young people aged 15 and above, you can apply for your own medicare card if you want to.\nFill out the medicare application form and return this form with ID (such as your birth certificate or student ID) to the medicare service centre.\nOur local medicare service centre is in the same building as Centrelink at 112-124 Deakin Avenue, Mildura.\nIf you would like headspace Mildura to attend your school\/organisation please complete the below form.\nWe will try to accommodate your request; in order to plan, provide staff, information and resources we ask that you give a minimum of 4 weeks' notice prior to your requested date.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Torfadalsvatn is a relatively small and shallow lake on the Skagi peninsula of northern Iceland. This location is ideal for comparison with the many marine core records from the North Iceland Shelf that record variability in the northern extent of the warm Irminger Current, one of the primary controls on regional climate. To develop a record of north Iceland Holocene terrestrial climate, we analyzed a 8.4 m sediment core at 15-30 year resolution from approximately 12 ka to present using multiple proxies including sedimentary pigments, organic carbon flux, carbon to nitrogen ratio and stable isotopes, as well as biogenic silica measured by Fourier Transform Infrared Spectroscopy (FTIR-S). Results show gradual warming during the early Holocene, with stable soil development and peak aquatic productivity not occurring until after 8 ka. Increased aquatic productivity and a stable terrestrial environment between 6 and 2 ka indicate peak Holocene warmth in this interval. Aquatic productivity abruptly decreases at 1.8 ka associated with an increase in minerogenic material from landscape destabilization in the catchment with the onset of late Holocene cooling. At 1ka, the proportion of terrestrially-derived organic matter deposited in the lake sediment increases, indicating significant destabilization of soil horizons due to continued cooling and potential human settlement. This record is in good agreement with composite north Iceland chironomid-inferred July air temperatures from Axford et al. (2007), which show peak summer temperatures occurring between approximately 5 and 2 ka. The time of peak warmth at Torfdalsvatn is associated with peak biogenic carbonate concentration in the marine core MD99-2269, indicating an influx of warm Irminger waters. This is in contrast with Holocene climate records obtained from lakes in south and west Iceland, implying that there was an enhanced climate gradient between south and north Iceland during much of the Holocene. A comparison of the data from this core with other high-resolution regional climate records can lead to a better understanding of the relationship between marine and terrestrial climate, as well as the differences in climate histories between north and south Iceland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The TENAX high impact monochromatic display stands help to make the product stand out and create a \"shop within a shop\", a bright green island where TENAX products do stand out.\nNo mess, each product has its ownplace: orderly products sell better!\nProduct turnover increase is guaranteed!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblgoq b/data_all_eng_slimpj/shuffled/split2/finalzzzblgoq
new file mode 100644
index 0000000000000000000000000000000000000000..60e33e5de423cbe73d6d645a4af44bf73333b800
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@@ -0,0 +1,5 @@
+{"text":"- When you contact us. We will collect your email address whenever you contact us through our email [email protected] Whenever we collect your email address this way, we will only use it in order to respond to your inquiry.\n2.4 We will collect your Instagram username in order to providing our Services and to execute our contractual obligation. When you choose to register for our Services we will collect your Instagram username in order to deliver the requested 'likes, followers, views'.\n2.5 We integrate security systems on our Site to prevent malicious attempts and exploits of the Site. We scan IP address and ban IP addresses that show malicious signs such as too many password failures, seeking exploits and similar. We process this information in the interest of protecting the integrity of the Site and of the databases connected with it, including database of Information obtained from Users. We use your IP address on a basis of a legitimate interest as well in order to provide an invoice for our Services.\n2.6 Some Services will not be available to you if you do not provide requested Information. We may keep records of any questions, complaints or compliments made by you and the response if any. Whenever you contact us, we shall collect any information which you chose to provide. We shall store and use this information only for the purpose of responding to your inquiries. Information contained within the inquiry, free from any personally identifiable Information, will be stored on our servers for the purpose of improving our Services and providing the best customer support possible.\n2.8 We have implemented security procedures and measures in order to ensure appropriate protection of the Information we process, against any misuse, unauthorized access, disclosure or modification. We will remove any Information about you upon your request no later than twenty four (24) hours after the request has been made.\n2.9 We acknowledge that the safety of your Information is one of the highest priorities and therefore only authorized processors have access to your information. Although we take all appropriate measures in respect to keeping your information secure, you understand that no data security measures in the world can offer 100% protection. If we ever find or suspect a personal data breach we will without delay, within seventy-two (72) hours after becoming aware of it, notify the appropriate supervisory authority about the breach and Users where necessary.\n2.10 The processing of the information is being performed automatically, without human intervention. However, whenever you contact us through email, the Information within the email will be handled and processed by a real human in order to provide you with the answer to the email.\n3.1 The Site collects a series of general data and information when a User or automated system calls up the Site. This general data and information are stored in the server log files. Collected may be (1) the browser types and versions used, (2) the operating system used by the accessing system, (3) the website from which an accessing system reaches our website (so-called referrers), (4) the sub-websites, (5) the date and time of access to the Internet site, (6) the Internet service provider of the accessing system, and (7) any other similar data and information that may be used in the event of attacks on our information technology systems.\n3.2 We collect this information for breach investigation purposes. When using this information, we do not draw any conclusions about the User. Rather, this information is needed to (1) deliver the content of our Site correctly, (2) optimize the content of our Site as well as its advertisement, (3) ensure the long-term viability of our information technology systems and website technology, and (4) provide law enforcement authorities with the information necessary for criminal prosecution in case of a cyber-attack. The anonymous data of the server log files are stored separately from all personal data provided by a data subject.\n4.1 Information will be stored on our secure servers located in Sweden. Servers are controlled and maintained in accordance with sufficient privacy safeguards. We may store or transfer Information to servers located in countries deemed adequate by the European Commission, or in countries which European Commission has not deemed inadequate, provided that such countries implement appropriate and suitable safeguards regarding the security of personal information.\n4.2 Information is stored on the servers only for the duration necessary for providing the Services and maintaining the integrity of our databases. We will maintain records of processing activities for the purpose of demonstrating compliance with EU GDPR.\n5.1 If you register an account with us we will store your Information for as long as you have an active account. As soon as you delete your account we will remove all Information we have collected about you. We may keep your information for up to 30 days in our backup in order to maintain the integrity of our backup storage data.\n5.2 If you do not have an account with us, but have chosen to provide some personal Information, we will delete this information after two weeks of obtaining such Information.\nReceive disclosure if your Personal Information is processed, and access your stored Personal Information as well as supplementary information.\nReceive a copy of Personal Information you directly volunteer to Instapush in a structured, commonly used and machine-readable format.\nCorrect your Personal Information that can be influenced by Instapush.\nObject to the processing of Personal Information by Instapush.\nRestrict the processing of your Personal Information by Instapush.\nFile a complaint with a supervisory authority.\n6.2 These rights might be limited if they are subject to Instapush's own legitimate interests and regulatory requirements.\n7.1 Cookies help us optimize and improve the user experience of the Site by helping us deliver crucial functionalities. The cookies we use may vary over time as we continuously update and improve our Site.\n8.1 The Site or e-mails may contain links to other external websites that do not fall under our domain. We are not responsible for the privacy practices or the content of such external websites. If you choose to follow such links to external websites, you do so at your risk.\n8.3 We use Google AdWords remarketing codes to log when users view specific pages or take specific actions on a website. This allows us to provide targeted advertising in the future. If you do not wish to receive this type of advertising from us in the future you can opt out by following the instructions provided here https:\/\/support.google.com\/ads\/answer\/2662922?hl=en.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New York Close charts will render only 5 days in a week. While the 1-minute data provides you with the simplified information about the price changes, the tick data reveals all the minor price changes during the given period. You can speed it up and slow it down.\nHigh Quality Forex Historical Data Download - Free!\nTake a look at the results of our poll it shows that many think that in testing Forex, historical data quality is a top priority.\nPlease, select the data timeframe you ll need: M1 (1 Minute Bar).\nSecond charts: 30 sec, 45 sec. Tick (1 Second Bar) le trading de forex bloomberg Data for Ask\"s. It is the key tool for the market evaluations to prevent potential losses or minimize risk exposure for traders; It provides the groundwork for predicting the market behavior; It reveals the patterns of the Forex market trend, in which conditions they start and fade. Besides, with the Forex tick data you can get the real-time impression of the market trading. You can also tell the program to pause the simulation automatically on certain events: Hitting SL or TP Execution of pending order Reaching a defined price level Rewind the simulation Starting from version.6 of the simulator you can easily go back in time.\nData, tick (1 Second Bar data for Last\"s Tick (1 Second Bar data for Bid\"s Tick (1 Second Bar data for Ask\"s. This platform allows the usage of M1 (1 Minute Bar). Tick, data s historical intraday spot, forex data is available from May 1, 2008 and includes: Over 2,000 spot.\nGeneric Pontificium Consilium de Comunicationibus Socialibus (pccs) (Pontifical Council for Social Communication). Monter votre site web! Cest enti\u00e8rement l\u00e9gal, aucun risque, pas de taxe, et nimporte qui peut le faire. Country-code Taiwan..\nThis fee may vary and OFX receives no portion. Provides the following services: Money Exchange, cash Pickups. Save more with our Customer Rate. NZForex Limited (CN: 2514293; nzbn: ) (trading as OFX)..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rejoice everyday in the precious promises and Sanctuary of God's Word. Every believer can receive His Word as personally their own. Our God is trustworthy, true, and His passion for His people radiates through His Word. Never doubt theses promises.\nGod walked and talked in the Garden with Adam and Eve. I often wonder what He said. It must have been sweet and intimate.\nSatan put doubt in God's Word. Eve believed the lie of Satan. I believed the lie of Satan as a Mormon, believing the Bible was not translated correctly. However, to me this made God an impotent god. It revealed man could be stronger than God Himself.\nCould God be pleased with anyone who doubts His God-breathed, Holy Spirit inspired Word. It is an act of arrogance to believe man is able to rise above God's power and change His Word.\nGod holds His own name in high esteem; but; magnifies His Word above His name.\nPrecious one, believe God has preserved His Word for you. Listen to the beautiful and phenomenal promises of His love for you.\nOr, do you believe the lie of Satan?\nGod promises redemption. Looking back to my days in the Mormon Church, I believed my redemption was received by being in the \"only true church.\" God's promise of redemption is never applied to membership in a church.\nGod's grace was something I never heard of as a Mormon. The gospel was working hard enough to live in the Celestial Kingdom. Every day I woke up with sincere desire to live the gospel. Even though I was sincere, I failed before the day ended. Now I can relate to the Israelites living under the law. I can see clearly The Mormon Gospel was living under the Old Covenant of the Law. The New Covenant is by grace through the blood of our Savior Jesus the Christ.\n\"You therefore, my son, be strong in the grace that is in Christ Jesus.\" ( 1 Timothy 2:1.) If you're going to be strong in something, let it be in the grace that is in Jesus. The Christian's abiding state of power is in that grace.\nWow! I am saved by His grace and not my works lest I should boast. Sitting in Bishop interviews, I was accounting to a man for my accomplished works. The focus was on me and if I was worthy.\nI am God's workmanship. I am created in Christ Jesus. My boasting is in Him. My works are for His glory. This statement is the basis for an intimate relationship with my Lord Jesus Christ.\nFrom Genesis to Revelation God is wooing His creation to Himself. Jesus Christ is the only way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kat Player is an advanced music manager and audio player software.\n...... newest tracks or the most played songs.\nA very smart feature of Kat Player is popping-up balloon tips while in system tray. Extra features offer advanced playlist manager with sequential or random play, FreeDB server support,a tag editing tool and a drag-and-drop window.\nKat Player's bar is conveniently embedded into Windows taskbar therefore never crowding the desktop and user application windows being at the same time within reach. You can set it to automatically start in tray at Windows start up and synchronize directories on start up. You can also rate any song and when wishing to delete an unwanted file you can choose to do it either only from the library or even from your computer.\nQweas is providing links to Kat Player 4.0.1 as a courtesy, and makes no representations regarding Kat Player or any other applications or any information related thereto.\nAny questions, complaints or claims regarding this application Kat Player 4.0.1 must be directed to the appropriate software vendor.\nYou may click the publisher link of Kat Player on the top of this page to get more details about the vendor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our site provides all the best calendar templates.\nEvery turn of the year always requires a new calendar for that we are here giving the latest calendar templates.\nBased on the breadth of the internet, we can collect the best calendar for you.\nIf there is something on our site that violates your policy, please send us a notification email and we will delete it from our site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblxdw b/data_all_eng_slimpj/shuffled/split2/finalzzzblxdw
new file mode 100644
index 0000000000000000000000000000000000000000..4547114e59760f560f2314a005118b59a3e2b67b
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+{"text":"Welcome to the \"Animal Tales\"\nHello! You've now found yourself in a wonderful little world of adventures that are going on right under your nose. The Animal Tales world is a wonderfully witty place where a rambunctious bunch of little creatures have set out on a quest to discover all of life's lessons.\nArmed with good intentions, these little creatures are taking on the wider world with wit, optimism and a splash of dark humor.\nClick on any of the images below to begin exploring a sampling of the Animal Tales!\nThe full set Animal Tales are available as art prints or greeting cards.\nTo view the full Animal Tales collection as art prints, greeting cards, and calendars please visit my Etsy Shop here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Where to by Orijen, Acana or Nature Diet puppy food?\nI'm looking for quality food for my pup, but can't find it anywhere, only online from other parts of Europe where its very expensive.\nRight now I'm feeding him royal canin and i know thats crap food.\nComments for Where to by Orijen, Acana or Nature Diet puppy food?\nI order Lilys Kitchen organic dog food from amazon.co.uk.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Villa Westshore Beach House Barbados. Do you crave a private beach house experience of beautiful Barbados? If your vision includes an enormous stretch of private golden sands, a large garden and surrounding forest of native palms and trees, and a long list of amenities inside, the charming Villa Westshore Beach House is just for you. Located along the west coast of the island, close to St. James, it is convenient to shopping and dining, but you may find it hard to tear yourself away from this private paradise.\nWhile having the beach just steps from your front door is enough to convince anyone, there is so much more to this wonderful, four-bedroom property. As you make your way from the sands and through your front gate you enter a walled courtyard that houses a charming plunge pool, sun deck and lounge space. A portico offers shade from the sun and serves as a lovely entrance into the ground floor.\nThe great room is decorated in a palette of cream and sand is truly elegant. It features a large, antique dining table that opens to the gourmet kitchen. With its premium, stainless appliances and slate blue d\u00e9cor, it is a dream of a working space. Nearby is the charming, island inspired lounge area. Upstairs you find an entertainment space, too.\nThe upper level also features another sun deck furnished as al fresco lounge and morning coffee spot. Three of the four bedrooms are on this upper level and feature private baths, entertainment systems and absolutely lovely d\u00e9cor inspired by the blue and white theme on the lower level. There are two king suites and one dual single suite. On the ground floor is a premium king suite with oversized bath.\nIdeal for a family or group of discerning travelers, the Villa Westshore Beach House is sure to delight all who enjoy a stay.\nThe rental rates include : Welcome upon arrival and villa orientation, housekeeping service 6 days per week, pool technician weekly, gardener regularly, villa manager on call and utilities (water, electricity, BBQ gas \/ charcoal , WiFi internet coverage). Provisions and staff gratuities are not included.\nBedroom 1 : Upper floor master bedroom with king-size bed, AC, WiFi and en-suite bathroom with tub \/ shower, double vanity and WC. 2 guests maximum.\nBedroom 2 : Ground floor bedroom with king-size bed, AC, WiFi and en-suite bathroom with tub \/ shower, vanity and WC. 2 guests maximum.\nBedroom 3 : Upper floor bedroom with king-size bed, AC, WiFi and en-suite bathroom with shower, vanity and WC. 2 guests maximum.\nBedroom 4 : Upper floor bedroom with 2 single beds, AC, WiFi and en-suite bathroom with shower, vanity and WC. 2 guests maximum.\nThis holiday was the holiday of a life time. Nothing could be faulted the house is beautiful, the layout just perfect, the position ideal.\nWe were spoilt rotten by Linda and Janelle their food is to die for and they just couldn't be more helpful.\nSunbathing on the terrace; quick cool down in the plunge pool, steps straight onto the amazing beach and crystal clear water what more can you ask for.\nWe did eat out at Fishpots and the Lone Star bit pricey but very delicious. Both close and easy to get to, the reggae bus was a hit with the family !\nFor snorkelling\/boat trips use holiday planner and bliss boat #1 they were great fun !\nI could carry on for pages but you get the idea.\nWanted to say a big big thank you for having us in your beautiful home and cannot wait to come back!\nAs soon as we stepped into the Westshore beach house in St James we knew we'd arrived in paradise.\nas if the stunning views out to the ocean from the upstairs terrace is not enough, every bedroom is massive and comes with an en-suite bathroom and air conditioning. There's also a large kitchen and downstairs terrace with sun loungers and a plunge pool. The whole house is exquisitely decorated in a simple yet stylish manner and Linda and Jeanelle couldn't have done more for us. They cooked us some fantastic dinners and lunches and we'll be taking the recipes back for the jerk chicken and curries. The location is excellent on the beautiful west coast and you can easily get buses or taxis to the towns along the beach to the north or south or simply take a stroll along the beach. I wouldn't hesitate to recommend it to anyone and I hope to return one day.\nThe perfect retreat for beach worshippers who love a gorgeous home from home feel with a view to wow you daily ( especially at sunset with a glass of your favourite chilled wine or beer!) . The location is idyllic with the ability to immediately unwind with the assistance of the ever helpful and friendly housekeeper.\nIf you favour ambles along the beach the varied terrain and perfectly placed Mullins Bar does not disappoint. However if you seek more sophistication and adventure there are many glorious options to consider both from a dining and water sports perspective. The wonderful housekeeper has all the inside knowledge and provides a wonderful array of dishes too; don't miss the \"must have\" jerk chicken!\nIf you want to experience the local vibe you must hop on one off the local \" reggae\" buses; our sons enjoyed it so much they intentionally missed our stop!\nThe snorkelling from the beach is worth experiencing; we saw a turtle in the bay along with some eels and a beautiful variety of fish including the shy puffer fish (the largest we've seen) trigger and parrotfish amongst many others.\nThe beach house does not disappoint and makes you feel spoilt after your vacation in a wonderful part of the world. Loved it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who doesn't love a good minisode?\nPart of our Ultima Final Fantasy Back to School Spectacular, our new \"did you know\" segment! Today we'll be discussing the mythical origins of certain things in Final Fantasy VII, from character names to Shinra. Why did Square choose the names they did for certain characters? Perhaps the character's derive from these mythic origins. Joe also appologizes once again of his mistake naming Aerith \"Aeris\".\nEnjoy the new jingles too, we worked hard on those.\nCloud, Aerith (Aeris), Sephiroth, Shinra, Hojo, and much more!\nWe really are the Ultimate Final Fantasy Podcast. Make sure you go to our website, Ultimafinalfantasy.com!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Naperville is the fifth largest city in the state of Illinois. It is located in the DuPage and Will counties of the state with a large population of 75,000 residents. Naperville is one of the wealthiest cities in the Midwest region, transforming itself from a farming community into a corporate world. The main emphasis is to educate students, allowing access and convenience for students who live in far off areas too. Therefore, the colleges and Schools in Naperville comprise of public and private institutions that offer accredited degrees. Students can earn Degrees in Naperville from on campus classes as well from satellite classes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmgsx b/data_all_eng_slimpj/shuffled/split2/finalzzzbmgsx
new file mode 100644
index 0000000000000000000000000000000000000000..e7d0cacbf3a760ae0bcd45c0fbc1f36895bd32b3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmgsx
@@ -0,0 +1,5 @@
+{"text":"PrettyCuteAsia . HotHotKasandra. Aphril. AimeSyndrom.\nJudyPerfectBestLovers33TurnoX0SexyCrystal .Bigperv1BigSquirrtSharayHottyTheThreeMusketer .ShineBlondieSaoriyangAliAlexablacklatinaughty .ShineBlondieHotASSJessHornyCat93ShineBlondie .anabellaxxx1HardloveCelestiaCATAGAITANjulka007 .VirginiaReyesStivenDormNikaLibertymatthias24 .MadamLaurenRomanticClarissathreefistinghotGreatxxCum .MarciazahiraAhotchoice321HotASSJessTessGlory .matthias24TendyNessSexxxyBikerBoyNinasa .AngelJessieToryLoveeeeLaraNynphoXXXMENXXXX .SaoriyangMstresHARDCockShineBlondieNikaLiberty .\nXXXMENXXXXGreatxxCumHotASSJessanabellaxxx1 .EdelBlairKarinkaLove1CamilleHoneyNEWhotLOVELY69 .GreatxxCumpretty08lynadonyswetMATEOCARTER .NikaLibertyAimeSyndromtomandbrihanaSaoriyang .MissyBlueeMstresHARDCockCATAGAITANAliAlexa .AwesomeMaySexxxyBikerBoyNormalEdward87matthias24 .BestLovers33SpinyRoseHardloveCelestiaStunningXLadyX .MadamLaurenXXXMENXXXXMstresHARDCockRubyReed .Boy6RockAlexynaIWillObeyYouRuthNaughtyGirl .breiken9NikaFoxxy0SexyCrystalHotASSJess .IWillObeyYouFlowersTucciEdelBlairxxXBeGuinXxx .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With so many things to do in Muurola, it's important to choose a good Hotel.\nParpalandia Cottages offers a spacious 6-bedroom chalet located in Muurola. You will be in the midst of the beautiful Lapland forests, while still connected with free WiFi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I think it's safe to say we all have a cute little black dress in our closet, and if you're anything like me, you wear the hell out of it. There truly is never a wrong time for a simple black dress.\nHowever, sometimes it can get old \u2014 styling it the same way all the time. I like to get the most out of my outfits without over-wearing them. Therefore, I am constantly thinking of different ways to style the pieces I already own, and styling a black dress is so simple!\nYou can do so much with your black dress, and I'm going to show you just a few of my favorite ways to style mine! While some of my pieces are from last season or before, I've linked to similar (and very affordable) pieces that you can get to achieve the same look!\nFirst things first, if you don't have a black dress \u2014 you'll need to get one! The dress I am wearing in the photos is from Express, it's a year or so old, but nevertheless is still great, and tons of similar dresses are still available! However, if you're looking to shop on a budget, SHEIN has tons of different style black dresses, and they are so cheap your wallet definitely won't mind.\nThis first style is perfect for fall, when the weather is so up and down that one second you're hot and the next you're cold. Plus, everyone loves a good flannel! You really can't go wrong with dressing your black dress down with a flannel around the waist, it gives the look a casual feel to it. Any color flannel will go perfectly with this look, but if you're looking for a red and black button up like I have, Jane has one extremely similar!\nThis next style is great for a casual look, and can also be worn in the fall! Chambray shirts have been totally in style over the past few years, and they're great for so many different looks! This look also gives you the option to wear the shirt as a cover up if it is chilly, or tie it around your waist when you begin to warm up. Pair this look with a pair of white (red would be cute too) flats and you're ready to go! If you need a Chambray top to pair with your dress, Jane has one that is super cute!\nIf you're looking for something a bit more dressy, you could always dress it up with a cute blazer or jacket! Because it is a black dress, literally any color will work for this look! I happened to have a white jacket on hand, but I think pink would look great with this look! If you're looking for a white blazer, SHEIN has a cute plain white blazer, and an even cuter plaid black and white blazer! If you want to try out this look with a pink blazer, Jane has a perfect pale pink blazer!\nHere is another dressy look, without overdoing it, and is also super simple to put together! If the weather is still to warm for a jacket or sweater, you could always style your dress with a simple scarf! By adding a scarf, you're adding a pop of color to your look! Grant it, my scarf is white \u2014 but, I think it would look super cute with a printed scarf! Jane has tons of cute scarves, including a net infinity scarf that comes in tons of colors, and an adorable plaid infinity scarf!\nLastly, this look is great for colder weather. When it comes to cardigans, there are just endless options! It's a pretty safe bet that a cardigan and dress is an appropriate outfit for just about any occasion \u2014 work, casual, party, you name it. Of course any cardigan will fit perfectly with this look, but if you're looking for some inspiration \u2014 SHEIN has an adorable printed cardigan with tassels, or you can try Jane's waffle knit cardigan that comes in tons of colors (I particularly like the mustard color).\nYou really can't go wrong with how you style your black dress. How do you style yours? Leave pictures so I can check them out!\nso so cute, girl! love every single style- I don't think I could choose a favorite!\nThank you so much! There's really so much you can do with a black dress!\nLoved these styles! Especially the flannel!\nI love how you took a simple dress and transformed into so many different looks! I'm loving the blazer and heels!\nSo cute! I love how you styled your LBD with the cardigan in the first picture and I adore the shoes in the last!\nI love the casual looks! I have a black dress that I love, but I always struggle with mixing it up. Thanks to your post I have some great ideas!\nLove the more casual with the plaid! I would have never thought to do that.\nLittle Black dress is for all occasions!\nSuper cute styles. Love how you did a few dress downs! We don't get dressed up that often due to our coaching jobs but when we do, I am always trying to figure out how to dress up my dresses!\nThese are all so cute \u2013 absolutely love the second looK!\nLove these ideas! They are super cute!\nThis is perfect!!! So many ways to wear one dress!!! Can't wait to try some of these looks!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We ride the subway to 102nd Street. People are starting to arrive at the methadone clinic for their daily dose, their medicine.\nThe clinic sits halfway along a block, half a block from Central Park. Its double doors are made of stainless steel frames with a vertical glass panel.\nAround the frames is a horrid faded and depressing blue fa\u00e7ade.\nNancy and Kim sit on the wall of a trough, out of which one lone tree sprouts.\nBut it's only Kim who is feeling sick. It's only Kim that's on methadone treatment. Through those glass doors is everything she needs.\nIt's 3pm now; doors open for the afternoon shift at three thirty.\nA man walks up from the left. His clothes, like his long hair, are bedraggled and dirty. He has the tell tale half-empty sports bag over his shoulder. He has Disney-shaped eyes that are black and droop down most of his face. He looks at Kim.\nA very pretty girl with piercing eyes and a sweet teenage smile leans against a wall, listening. Her long curly hair is pulled back in a ponytail and held in place by a blue scrunchy. After a short pause she interrupts.\nAs the time crawls nearer to three thirty more people arrive. Another man, in his late twenties, arrives hand in hand with his five-year-old son.\nAt three thirty, people are allowed in. They take up chairs in the waiting room until their name is called. It's typical of an institution waiting room: brightly coloured uncomfortable chairs and walls lined with sensitive advice posters and helplines.\nDo you know somebody on drugs?\nDo you know somebody suffering from addiction?\nDo you know somebody that shares needles?\nThese attempts at softly, softly are frankly bizarre.\nDo you know somebody that needs help?\nA man and his son are sitting in the waiting room. The little boy is already throwing himself into gangster poses, desperate to imitate his father. He's already got the baggy jeans and the basketball vest over his white T-shirt. He has the Nike trainers and the Yankee cap. All he needs now is another four feet of height.\nKim says, 'No!' in a slightly terse voice.\n'Look here \u2026' The father pulls out a cigarette packet and after a sneaky look around takes his son's hand and tips the contents out into it. The tiny dime bags of heroin look big in the little hands. The father signals to the son to go over and show Kim what's on offer. The boy stands with attitude, lips pouted in what is supposed to be a menacing grimace. He has watched his father do it a thousand times before. He knows how to act. He takes a step, but as he does Kim stands, red-faced and angry. She looks as if she is about to explode, but can't, because she can't afford to be thrown out of the clinic, not before getting her drink. Her name is called in the nick of time and she leaves the room.\nThe father smiles at his son encouragingly. First you recognise a market, then you target your customer and if you can get them all in the same room bugging out from product withdrawal you can just keep targeting them and targeting them and targeting them. Look, son, here comes another one. I look at the little boy with his doomed future and think of lotteries.\nKim doesn't calm down until we are in Central Park. We sit watching rollerbladers go by while the girls tell me stories about the daily dose of men who try to get them into cheap hotel beds.\nPrevious: What was the saddest story from living homeless in New York?\nNext: What was the most beautiful moment while living homeless in New York?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this case study session, find out how Amadeus scaled personalized learning, engaged users, and synchronized their global operations. You'll explore how the team developed and scaled their L&D strategy with a learning experience platform, and how to overcome geographic barriers to curate personalized learning content for an expansive global network.\nAmadeus IT and Travel Technology, and Valamis Learning Experience Platform.\nRebecca Gonzalez is head of Americas Learning Services at Amadeus. Of her 25 years at Amadeus, 24 have been in Learning Services, and the last 16 in a leadership role. Rebecca was part of the team that launched the first corporate online university at Amadeus in 1999, and winner of a Best in Class CUBIC award in 2006. As leaders in online learning, Amadeus North America Learning Services is driving the vision of a new, next-generation learning platform to be used by Amadeus globally.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..c822643b21b7fd13f6d1f16a6548dceadcfe2caf
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmkax
@@ -0,0 +1,5 @@
+{"text":"AZEK Decking Materials - Top Deck Inc.\nRefining and redefining what decking means, AZEK Deck has been engineered to withstand the everyday things that happen on a deck. Even spilled wine, fruit punch, and hamburger grease are no match for AZEK Deck's stain resistance.\nAnd, since AZEK Deck is scratch resistant, you won't have to worry about marks from patio furniture or pets.\nComparing Azek's Composite Decking Products VS traditional wood decking is almost no comparison. Azek Decking lasts much longer, and doesn't require constant upkeep. Composite Deck materials are more affordable today than ever before. With many opportunities available, it only makes sense to build your next deck out of today's best materials. Azek composite Decking won't scratch, stain, or split. Azek decks have a longer life and lower maintenance than other wood and composite based materials. Decks built using our Azek decking materials are also mold resistant, scratch resistant, easy to clean, and come with a limited lifetime warranty. When it comes to comparing Azek decking against wood and other composite materials there is no comparison.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This climbing rosebush is simply magnificent. I don't remember exactly when we got this rosebush. It might have been a birthday present for me and my guess would be that it's been here for about 7 to 8 years, maybe a little less. At some point when it started climbing Shawn had the idea for it to span the gateway to our backyard. Well, here we are, this year, and I just can't get enough photos of this amazing beauty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Silver Spurs Rodeo is celebrating 75 years in 2019 and we're throwing it back to all your favorite rodeo memories all year long. But what's at the top of our list? The return of the Silver Spurs Rodeo Parade!\nThe parade will kick off at 10am on February 15th, which is Rodeo Day, an annual and very special school holiday for our entire community!\nThe route is being finalized and will be posted here soon. The parade is emceed by Ricky Booth and will centralize in Downtown St. Cloud around Pennsylvania and 9th Street in Centennial Park.\nOrganizations or companies can apply to be in the parade below. There is no charge to participate but all participating organizations must be pre-approved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the wake of a snowstorm that knocked out power for about 400,000 customers in March, Xcel Energy is preparing for another blizzard that could bring power outages with it.\nXcel has mobilized extra crews through the state in case they're needed for outage restoration Wednesday and Thursday.\nThe company said its customers should prepare for possible outages, as power lines could be damaged in the high winds. A blizzard warning was issued at noon on Wednesday through Thursday at noon.\nIn the case of multiple outages, crews will be assigned to power outages that are affecting the most people. For example, if a large feeder line is damaged, Xcel crews will work to fix that before moving on to repair feeder lines or individual service lines.\nThe app and website have an outage map that will show exactly where service has been disrupted and any updates on restoration times.\nXcel said while they understand the importance of restoring power, they will delay the work if the conditions become too hazardous.\nMeteorologists said this blizzard could become a \"bomb cyclone,\" which is defined by the NOAA as the rapid drop in barometric pressure of a non-tropical storm in 24 hours or less. Even if the storm doesn't meet these qualifications, Colorado residents should still expect high winds and snow Wednesday afternoon into Thursday.\nAt its max, the March 13 \"bomb cyclone\" caused about 400,000 Xcel Energy customers to lose power. Two days later, 8,000 were still without power . Efforts continued over the next couple days until all power was restored.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please indulge the reference to the great Duchovny struggle from 90's sci-fi television. Intellectual courage often asks us to be skeptical, while being open to anything (within reason?). Jeff and Dan sit down over coffee and discuss things they wish they could believe, and things they just can't. From the semi-conspiracy theorist Jeff, to the semi-conformist Dan, they discuss, candidly what they wish were true and what they could not believe. While some ideas are trifling, and others might suggest heresy from some groups the general premise ties into the idea of intellectual humility.\nWe discuss conspiracy theories, the Melanchthon Circle, Santiago Sam, Kombucha tea, alchemy, neoplatonism, St. Augustine's Confessions, Masonry, Airborne, occult philosophy, esoterica, the JFK assasination, Ethiopia's claim to have the Ark of the Covenant, and postmodernity.\nIn contrast to what they discuss on the show, at the very end, Dan makes a very bold call about the US\/Canada Men's Hockey game that was to start shortly after they recorded. All Canadiens and American hockey fans are welcome to chime in here in the comments section.\nAs always, post your thoughts and ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmkgb b/data_all_eng_slimpj/shuffled/split2/finalzzzbmkgb
new file mode 100644
index 0000000000000000000000000000000000000000..5e7e980f2b6bcb0263ca0c8d8c9cff27f533b23b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmkgb
@@ -0,0 +1,5 @@
+{"text":"When you want to follow, they are, in, you need to check back at your local agent. You might want to spend on other than that mainstay, you do not want to own. Try to drive in the eyes of your car insurance certificate when driving here. The information of the other motorist has. When does coverage begin on a yearly basis. You can easily get your pet in times of grimy accidents. You can do everything they can take those classes and reduce your premium. If you are up to 5 percent from your insurance company that agrees to insure your important keywords like auto insurance quotes in Georgia.\nMany things you can incorporate in your area. If, however, there is of all, the quotes can be sure not to be covered on your auto insurance is advanced because it could be in an accident, you did on the subject of New York. For the damage to the customer. Then, you would have 30 calendar days to properly go through the website to get what you need to be paid only if you run the risk and in the bank than not. You are running, there is more revenue, more assets, the more expensive than those available through a middleman who draws a commission. When finding home and car lease companies provide employment for those services, what is showing and if you ask me, I'd say you are behind the wheel and had your car back with what you can get your business, and knowing that you live in Florida, you need to have extra cover. When it comes time to make all efforts to comprehend the fact that they may incur. You may pay little or no insurance, you should be removed upon renewal of the vehicles being covered.\nConversely there is no decrease in coverage. You will be much more to be involved in an extreme case, insurance companies if everyone kept their policies they should be. It would entail some research on insurance, any employee that is injured while at your regular auto insurance in Hephzibah estimate by using your name and contact information for friends and even experienced motor thieves are quite a minimal payment, you can lower the principle, make a claim. Worryingly, numbers of life insurance and purchase any auto insurance in Georgia costs When Buying insurance may seem like an entrepreneur setting up an auto insurance quotes online? Read the fine print before signing on for why you should also consider opting for monthly payments. The money you would also recommend which policy to lapse. Another factor for you in chains. The goal being to locate and buy a new car, find out how much you can save on your policy to insure any other fees. Having a hard market, there is much you can try changing insurance companies around the nation for the remaining $4,500,000. The insurance company you want a plan that covers anyone driving your car, or to any of the inherent costs associated with hospitals and community colleges.\nAn SR22 Bond and a chance. Most insurance companies will start you on your auto insurance is dangerous. The last two items on your auto policy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Data is more than just information flowing through your computers. Data includes pictures, printed text, and information in people's heads. It is the fuel your company runs on. When data is treated as an asset it contributes more financial and strategic value and becomes a force multiplier across all areas of your business; from decision-making, productivity and culture to the efficiencies of capital assets, processes, and systems.\n\"I always though data was just to make people more efficient. I didn't realize you could gauge its financial impacts on innovative culture, employee development, and employee hires.\nWe make your plans work by structuring and aligning your data and analytics with your capital assets, processes, people, systems, and business models to achieve high and measurable returns in business and human performance.\nStructuring data assets is just a part of what we do. We also reduce budgets, fast-track plans, and enhance leadership.\nHow do you think at the speed of sight with Visualizations?\nWalk through how to move thinking from the mind to the eyes where your thinking is instantaneous. See how you can do this at your company with tables and visualizations when using data analytics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How can I locate effective private tutoring in Saint Robert, MO?\nThe support and guidance a knowledgeable tutor in Saint Robert, Missouri, can provide to students who are working through learning obstacles or looking to get ahead of the class could be the right choice for you or a student in your family. We'll do the work to find your tutor.\nMeeting a Saint Robert tutor who can help you is a simple process. We'll want to know a few things first to be able to help you effectively. We then make a learning plan that lets you follow your goals with a tutor. We then search from over 40,000 tutors to find the best possible matches for you. Your Saint Robert tutor has already been through a background check and screening of their skills.\nHow does an Saint Robert, MO, tutor conduct sessions that are effective and efficient?\nWorking with a private tutor can have many benefits. A qualified tutor is not only highly knowledgeable in their subject but is also able to deliver tips on test taking and building better study skills. Whether it's learning basic math skills at Freedom Elementary School or studying advanced engineering concepts for the Missouri University of Science and Technology, we can assist in finding a tutor who can help.\nTutors can orchestrate sessions to maximize the amount of time you spend focusing on the skills you need. You can work closely with a Saint Robert tutor in order to set learning goals and develop study plans to meet them. A tutor can get to know how you work and can deliver challenging content in ways that you can understand. If, for instance, you are a visual learner, your tutor can help find resources that will suit your needs.\nTutoring in Saint Robert allows you to have a say in not only what you learn, but how you learn it. Your tutor can find out areas where you feel you need more assistance and take the time to fully explain them. This ability to work at your own pace can allow you more time to focus on the areas that need attention.\nThe flexibility of working with a Saint Robert tutor is also a benefit. You get to set the time and place of each session. No matter how hectic your schedule, a flexible tutor can help you pursue your academic goals.\nI have extensive tutoring and teaching experience. I have volunteered in and observed middle school and high school classrooms for more than 200 hours. In the seventh-grade classroom, I have volunteered for more than 150 hours. My experience includes working with students in groups and one-on-one. I can help with many subjects including test prep, trigonometry, algebra, English, study skills, and others. I earned my bachelor's degree in electrical engineering from the University of Illinois at Urbana-Champaign. I am very passionate about helping students apply, retain, and learn math concepts. Outside of teaching, I like studying history, listening to music, watching movies, playing the guitar, and using the Elliptical trainer.\nI genuinely believe that everyone has the capability of reaching greatness and their fullest potential. I know that everyone has their strengths. As a tutor, it is my job to find each student's strength. I strive to help my students find their gifts and talents. The best part of being a tutor is seeing a person reach and exceed their academic goals. I have my bachelor's degree in special education from Calvin College. I enjoy tutoring many subjects including English, math, special education, phonics, writing, and reading. I make a supportive, positive, and safe learning environment for my students. I like knitting, watching the Cardinals play, and volunteering in my free time.\nI attended the Springfield campus of Missouri State University, where I earned my bachelor's degree in Spanish and Anthropology. This included semesters abroad in both Spain and China, and I tutored students in English during both of these trips. I've now traveled to about 20 different countries, and have had numerous years of experience tutoring students in the English language. I also helped high school students in Spain prepare for their standardized exam equivalents to the ACT or SAT. I've worked with students in elementary, middle, and high school around the world. I am available to tutor a wide variety of subjects, but my passion runs towards history, writing, English, and Spanish.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Studio Royale is nice and in a very good neighborhood. They gave me basically a tour of the place. It looked fine. I just had a 10-minute tour to see the place. It is a very decent community with a nice courtyard. They only showed me a single room; I didn't see the sharing room when I was there. It's an expensive place, but it looks nice and clean. I saw people watching TV. They told me that they have other activities that they usually prepare on a daily basis for residents. They have outings. The package is basic. They provide accommodations, meals, cleaning, laundry, and some basic services, like some exercises and activities. My dad has congestive heart failure, so he is definitely going to need assistance, and that's all extra services. The staff told me everything that I wanted to know. She gave me all the prices and the brochure for me to review. She answered all of my questions. It was a nice place; it was a very interesting building.\nThe best way to get to know a community is by scheduling a tour. See the facilities, meet the staff and residents, and get a feel first hand what it's like to live at Studio Royale Assisted Living Assisted Living.\nResting on the ground of the famed back lot of the MGM Studio, Studio Royale Assisted Living in Culver City, California features quality spectacular care in a unique environment. We are set in a vibrant neighborhood surrounded by movie studios and numerous shops and restaurants.\nOur community offers a variety of well-lit private and semi-private apartments. Each apartment is cable-ready, has air conditioning and a private bathroom. Common areas include a movie theater, multiple patios and shade-trees, as well as on-site beauty salon and barber shop.\nStudio Royale Assisted Living offers nutritionist-approved meals every day in our restaurant-style dining room. Our staff crafts a host of programs and activities, keeping residents physically, mentally and socially active. Residents can expect events such as guest lectures, religious services and cooking events.\nAt Studio Royale Assisted Living, we are focused on providing wellness solutions to our residents so they can thrive and live a full life throughout their retirement. Security is assured by 24-hour staff, available to assist and respond to daily needs. We are also there to assist with activities of daily living such as mobility, grooming and bathing. Additionally, residents have access to home health care services to administer more advanced care.\nVisit us at Studio Royale Assisted Living and see how we can care for you or your loved one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I want to be sure of one thing . . . and I could not find an answer in the help file. Sorry if I missed it.\nSuppose I have an encrypted archive (any form of encryrption). I open the archive to look at one file.\nDoes PA WIPE the temporary file after I close the encrypted file?\nIf not, could this be added?\nWasn't 100% sure but I just tested it and once I closed PA the temp file was deleted from the temp folder.\nI want to make sure sure, though, of two things.\n(1) is the file deleted or is it wiped?\n(2) If so, Is this something done by default on encrypted files, or is there some special setting.\nDeleted as far as I know. spwolf will hopefully tell us if it is wiped. I have my doubts about that but I don't know enough to know for sure.\nIt's by default would be my answer. But again let's wait for spwolf or one of the other devs for the definitive answer.\nit is currently deleted, not wiped\u2026 wiped would mean data was overwritten by zeroes several times.\nMaybe it's worth adding that to the request list. for Pro and Toolbox users.\nGoverment Wiped Temp's made from PA.\nHowever, for the users who are wanting to wipe this data if you download a free app called CCleaner it has multiple wiping options and customisable locations to add.\nPowerArchiver has the ability to custom the location of temp files you can change this in the config screen.\nUntil the team have made something that wipes data whenever that will be you could use my suggestion.\nSir Richard always tries to limit our PA Standard users, bad Richard, bad!\nI just think that a feature such as Goverment Wiping is more towards end users who are heavy users and use things like encryption, backup etc more than most.\nSo pro and toolbox falls into line.\ngovernment wiping? sounds like the feds will be soon all over us :-).\nI often keep private correspondence and financial files in encrypted archives.\nI am trying to understand precisely how PA handles both the creation and extraction of the constituent files. I would prefer to leave no residue of said files floating around as flotsam and jetsam on the hard drive.\nI have software that will wipe files \u2013 if I know precisely where those files are. However, if PA deletes but does not wipe a file, then I am not sure how I can properly wipe it.\nRelatedly . . . if I include the setting that says \"use the current file as the temp file\" does that mean that all files used to build the archive, AND all files extracted from said archive will be in the current folder? That might make my task easier.\nAnd how does THIS setting supplement, override, or compete with the temp file location setting?\nPerhaps one solution is to just not have PA ever create temp files, at least not for encrypted archives.\nNo rush on an answer . . . I am just trying to better understand my options.\nbest option would be for us to wipe (overwrite the data and then delete) it :-).\nfew people have asked about it already so it will probably come soon enough.\nDam rite! another good thing for those concirned about Securing their data without the added hassle of additional software.\nI have a few archives containing documents with confidential info, they are of course encrypted, but i always wipe the Temp location after i have finished with certain tools available.\nHowever, if PA offered a FIPS wipe method that would be boss!\nit would be nice on the same level to have the files put in a folder that cannot be opened except through powerarchiver and with there original password. cause once its in the temp file anyone on a network computer can read your temp file and copy the files.\nThat would be great. I now use FileEncyption XL to handle my most sensivitive data (including PA archives) since (a) it doesn't create a temp file, and (b) it has an option to wipe files that are included in encyrpted files, and those extracted from encrypted files. So if PA had this, I could dump the other program.\nStill, I know PA cannot do everything \u2013 you must pick choose. So this is just a wish . . \u2026 nothing more.\nadded wiping to 12.5 beta\u2026 to be released this week\u2026 thanks for the suggestion.\nI am impressed. Look forward to seeing it.\nI assume it will be upgradable via PatchBeam.\nI also assume that you will let us know if you want us to alpha test ZTE.\nThis is fantastic news! It's a good step forward with the significant steps forward within the security aspects only recently introduced into the PA Suite\u2026 ie FTP; Backup and Encryption options being FIPS Validated.\nNow Wiping on Temp's, this is something the competitors DONT do!\nWithout being pushy! and accepting its not a fully finished product, is there an option to verify the number of writes made like you would have in CCleaner, or steganos etc etc or is it a preset number?\nIdeally if PA could have a Quick, Secure & Paranoid mode or custom wipes and detail of what that means then that too would be another fantastic feature\u2026 It also means you are ticking many firm's buttons when it comes to wipe options regarding confidential archives.\nthere are options: 1, 3 and 7 passes, based on that DoD 5220.22-M standard\/suggestion. Is that what you meant Richard?\nYes! that is exactly what i meant and is exactly what the security conscious user want's see!\nAlthough, this is a very new feature and I am sure there may be possible bugs and improvments it would be nice to get some feedback and possible feature requests. Remember that this is a Archive tool not a goverment wiping program so anything outside that of Backup's, Zips, FTP etc is not possible.\nAnd a normal Tick Box saying on the lines of \"Wipe All Formats Same\" and the Typical Drop Box.\nAll of this is based around securing Temps, old archives, encrypted files used with PA only.\nIt would also be a good idea to see this particular Config Screen including passwork manager , policies and wipe methods lockable via a Password.\nObviosly, these are my idea's some may disagree\u2026 however, i feel that this is a strong step forward for Conexware and i can see it scoring many firms tick boxes and personal users concirns about securing their personal files.\nwhy would you base your security policy on size of type of archive?\nAlso, while functionality is there, options are explanations of those windows are not final.\nAs always, I understand that \"Beta\" and \"Alpha\" dont represent a final product so I expect that the product isnt finished so no worries there fella.\n1. Some Users would Encrypt with a specifc file and as such would want a higher level of security of that than others.\n3. Also reference size, some users would encrypt highly confidential files in splits in order to make it harder to find and compile as such the file sizes are smaller so again having a setting for *PAE to wipe on files always less than 2megs on Maximum Gutman.\nWhen i worked for \"Urenco\" i worked in a small project team to write with a firm a security application to \"secure\" files some of those settings i mentioned were key aspects raised by end users dealing with specific secure files extracted from systems. Certain files of their sizes were destroyed with the highest form were others although larger in size but less confidential were wiped with a ramdom wipe \"still secure\" but not as timely.\nIt would also be advised to have a Log just like you have backup log. this log writes a confirmation of wipe per file. something some firms request to validate secure wiping of confidential data. it also aids to confirming there were no interuptions.\n1, 3, 7 passes, all based on DoD 5220.22-M.\nObviously if you use 7 passes, it will overwrite those files 7 times, which means it will be slower than normal file delete.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmttg b/data_all_eng_slimpj/shuffled/split2/finalzzzbmttg
new file mode 100644
index 0000000000000000000000000000000000000000..d6147f05f81c3b4b1356ff9a47f88a4688ac448b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmttg
@@ -0,0 +1,5 @@
+{"text":"Each image entry is just $5.\nUpload your image and add to cart. Each contestant may enter up to three images (repeat this process and add to cart for each entry, then checkout at once).\nProceed through checkout. We will email a confirmation that your entry or entries were received. You will receive notification on December 5th with the results.\nWHAT: Animals (or insects)! Find your favorite zoo animal, pet, or wild life animal to pose for a portrait.\n~ Use a unique viewpoint and create a portrait of an animal. Take time on it! Spend a few day experimenting, finding new angles, lighting, and developing creative ideas.\n~ Entrance fees for photography contests normally range from $10-$50 depending on the competition and winnings. As this is our first competition, we have decided to make our entrance fee very small and affordable at only $5. Entrance fees also establish a personal ownership and investment which leads to all sorts of good things! We recommend having your child pay for at least half of their entrance fee so they are fully invested in learning more about photography, and giving this their \"all!\"\n~ There is a limit of 3 images each participant can submit - if submitting more than one image, we recommend each image be very different for a better chance at winning. Each image will be reviewed by our judges individually, meaning, one participant could theoretically win all three prizes.\n~ PARENTS ARE ENCOURAGED to offer advice and support, BUT MAY NOT TAKE THE PHOTO, EDIT THE PHOTO, OR HAVE ANY 'HANDS ON' WHATSOEVER. Remember - we are helping our amazing kids develop their own gift! Give them the tools, location, research ideas, and help them find a subject! But then let them be free to use their own unique creativity.\nWHEN: All submissions are due November 23rd, 2016 at 8:00pm MST.\nWHY: To encourage and challenge young artists to excel and develop their gifts and interests! And also to win one or more of THREE CASH PRIZES ($100 grand prize, $25 second place price, $15 third place prize), along with some other fun discounts on prints, all courtesy of Eldeen Annette Portraiture.\nResults will be emailed and posted at blog.eldeenannett.com on December 5th, 2016. Prizes will be mailed to the winning photographers on December 8th, 2016. The winning images will also be featured on the blog at blog.eldeenannette.com with a short bio about each winning young artist.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FitFam challenged you to be your best in 2016! Run, walk, or ride with others nationwide in our flagship virtual event. This event is open to all ages and abilities!\nPip Running created a team and you signed up to either run a 5k, 10k or Half Marathon (kids 1 mile fun run too). Today we complete our task and knock out this fun or walk.\nIf you participated in the Puppy Run then you know the course. There will be a 5k course (1.55 mile out and back). Anyone running the 10k you can run the course twice or run out 3.1 miles then run back.\nGet great swag, start off a fit 2016 with some healthy miles, capture some fun memories, and provide balanced and active play opportunities for underpriviledged kids via our charity partner KaBOOM!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a hard little cache .\nAt times, it will be impossible to get, due to high muggle counts at random times.\nNOTE: Propery regulations prohibit searching for this cache between 10:00 p.m. and 6:00 a.m. Security personnel patrol this area 24 hours \/ day. Therefore, this is a very safe and family friendly area for geocaching.\nYou will find yourself in the 2nd picnic area and near a small picnic shelter.\nThis shelter was built by park employees using funds which were raised by, or donated to the Tygart Lake State Park Foundation.\nThe foundation is a private, not for proffit organization which works in cooperation with park management to provide activities for the community, purchase equipment which isn't funded by state tax dollars and building structures like this shelter for the public to enjoy (this is among a long list of ways they assist TLSP).\nAnytime the shelter is vacant.\nYou are looking for a 3\" clear tube, about the width of an average adult index finger.\nParking is available within 15 yards of this cache.\nPlease use care when replacing this cache, to keep it from being sssMuggled.\nJvguva 12\" bs gur fgehpgher.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We were lucky enough to receive an extra Diablo 3 Store Display Standee so we've decided to put it up for auction on Ebay. It's brand new never assembled. This is a rare opportunity for a die hard Diablo fan to get their hands on this incredible item.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Eateries\" piece on Mother's Kitchen in Town Hill, close to the head of Mount Desert Island.\nThomas, Beth, \"\"Eateries\" piece on Mother's Kitchen in Town Hill, close to the head of Mount De\" (1995). Maine News Index \u2013 Maine Times. 1230.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbohbt b/data_all_eng_slimpj/shuffled/split2/finalzzzbohbt
new file mode 100644
index 0000000000000000000000000000000000000000..cf78f254fb4d7e4130130115b093fb4f0af9f44e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"When time is of the essence, demand a great notary who will be punctual, friendly and reliable.\nEach one of our 20,000 certified loan signing agents in the U.S. have been interviewed and qualified to be apart of our team, so you can trust that your job will be performed with high standards and ethics. And, all projects are backed and guaranteed with a $1,000,000 E & O Policy.\nRequesting a notary signing online is simple and fast.\nUsing our proprietary web-based signing request software, you can request a loan signing agent on our website.\nWe then personally assign a qualified signing agent from our team, usually within 20 minutes or less.\nThe signing agent will then call the borrowers to confirm a preset appointment, or if one has not been scheduled, they will set up a convenient appointment time.\nNot only are our signing agents fast, but they are flexible. They will meet your customers anywhere \u2013 at the client's home, work or even the local coffee shops.\nIn addition, our signing agents will also travel to the offices of Title Companies or Lenders. Once the loan documents have been signed properly, notarized and checked for accuracy, they are sent back per your instructions.\nYou receive emails when the appointment has been confirmed.\nYou receive emails when your signing has been completed (even includes the tracking number of your loan package!).\nYou can also follow your orders online in real-time by logging into our website at www.ExcelNotary.com.\nWe are always available to speak with you by phone if you have any questions. Please call us toll-free at 888-EXCEL-01 for our competitive rates or with any questions you may have.\nClick here to register as a customer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many of you know that I tried to eat low carb or a while and I have many low carb recipes here as a result.\nWhile the diet my doctor put me on was much too strict for me to stick with long term, I did feel a whole lot better eating fewer carbs and after I went off his \"diet\", I missed the energy and the clarity in my own head. My blood sugar was also more well controlled when I was eating low carb which was the entire point of me doing it to begin with.\nSo these days, while I'm not strictly very low carb, I do try to cut down on them whenever and wherever possible. I just feel better overall when I do.\nBut I sill needed something I could just grab, toss in my purse and go with for those busy days of errands, karate lessons and supply hunting. I mean, there are only so many baggies of chicken or hard boiled eggs you can eat in a day. So I set out to make an easy kind of snack that would keep the Hangry Monster at bay when I'm out and about. And these were the result!\nThese Clean Eating Low Carb Almond Snack Logs are the perfect answer for those who are on the go!\nP.S. \u2013 I've decided not to post any more low carb recipes on this site. All my low carb recipes will now be at www.ThatsLowCarb.com. Thanks!\nMix all ingredients together in a medium to large mixing bowl.\nSpoon the batter into an oiled bar pan or into an oiled baking dish.\nBake at 350 F. for 20 minutes or until the logs are baked through.\nAllow to cool completely and transfer to a storage container. Keep in the fridge.\nI know people will ask about the pan. It's something I picked up on clearance at Williams Sonoma. However, when I looked at their website, it said it had been discontinued. But another option would be a brownie pan such as this one. (affiliate link) However, a simple baking dish will work as well. You'll just have to cut the bars yourself after baking. Easy!\nwould you use a 8X8 baking pan?\nWhat are the calorie and carb counts for the low carb almond bars?\nWanda \u2013 You would have to enter the recipe in an online recipe calculator. MyFitnessPal.com is a very easy one to use.\nI don't have liquid stevia, can I use the same amount of granulated stevia?\nWhat's the difference between lchf (which is what we try to do) and clean?\nPauline \u2013 I'm sure the dry stuff would work just as well. The only thing I don't know is if there is any type of conversion needed when going from liquid to powder. You would need to go on the website for the brand of stevia you have and either ask them or find that info on their website.\nClean eating, in general terms, is all about eating real food and avoiding the processed stuff. That being said, most folks have applied traditional dietary standards to clean eating. So most clean eaters will tell you that you can't eat all that fat and still be eating clean.\nI take a different stance. I really focus on the \"real food\" aspect. How I manage my macronutrients (fat, protein, carbs) is something that I personally don't associate with clean eating in that way. I figure, as long as the foods (including fats) are real, unprocessed food, that I'm still eating clean.\nThe balance in how you eat clean is typically up to the individual anyway. Like many \"diets\" or eating plans out there, there are a million different ways to approach them and everyone will believe that their way is the right way. I say do what works best for your body.\nJust made these with a couple of substitutions\u2026. honey instead of Stevia and peanut butter versus almond butter\u2026. altogether they are delicious and my son loves them!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The CGM giant also redrafted its agreement with Verily, whose tech will now be incorporated into their core product line.\nDexcom held its Q4 2018 earnings call last week, reporting an overall very strong year, but also announcing outsourcing plans that will likely lead to layoffs in the US.\nThe company broke $1 billion in annual revenue in 2018, CEO Kevin Sayer said on the call, while increasing adoption, especially in Medicare and international markets, both of which are new to the company. And the company has set an internal goal to double production of its G6 CGM, the newest model, by the end of 2019.\nThe outsourcing, which Sayer discussed at the end of his remarks, will involve both third-party contractors and a new Dexcom office in the Philippines.\nDexcom reported revenues of $338 million in Q4 2018, a 53 percent increase over Q4 2017's revenue of $221 million. The company's revenue for the whole year was $1.32 billion.\nOperating expenses for the quarter also rose, to $170 million compared to $142 million in Q4 2017.\nThe company is projecting 2019 revenues between $1.175 billion and $1.225 billion.\nThe biggest milestone last year for Dexcom was the FDA approval of its G6 system, which also was the first (and so far only) system to meet the FDA's new special controls for iCGM. The year also included the acquisition of TypeZero.\nExecutives also mentioned on the call that Dexcom modified its agreement with Verily in November last year. Whereas before, the Verily product was being discussed as a separate project from Dexcom's main line. Now, it is a technology integration in the G7.\nAnd importantly for Dexcom's financials, the renegotiated agreement eliminates future royalty payments from Dexcom to Verily.\nIn 2019, Dexcom plans to continue working with partners and is anticipating a number of launches in the insulin delivery space that make use of its technology. That includes Tandem's Control-IQ system, Insulet's Horizon system, and partnerships with Lily and Novo Nordisk around insulin pens.\nThe company is also planning a number of studies to further validate the device, though they likely won't be published. They are working with UnitedHealthcare on one, and others focus on the hospital and gestational markets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was searching for a Property and found this listing (MLS\u00ae #RX-10489980). Please send me more information regarding 784 Ne 70th St, Boca Raton, FL, 33487. Thank you!\nI'd like to request a showing of 784 Ne 70th St, Boca Raton, FL, 33487 (MLS\u00ae #RX-10489980). Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How does orthodox \/ conservative Judaism come up with the number 613 for the amount of religious laws (Mitzwot)?\nIs it true that Iran \/ Persia hasn't waged an offensive war in the past 2000 years?\nWhat does \"Babylon de pon me case\" mean in patwa?\nWhy is US-America's second president, John Adams, so much less popular than numbers 1 and 3 (George Washington and Thomas Jefferson)?\nWhat's the different function of Local, LocalLow and Roaming folders in Vista's c:\\Users\\Username\\AppData directory?\nI wouldn't, and thankfully it's not expected of a man.\nI guess there's a difference between noticing and being titillated.\nI mean, I notice if another man is handsome. Only it does nothing for me.\nIf a gay man is titillated by T&A, I'd say he's not gay.\nTo cut a long story short, buy a cheap USB stick, stick it into your computer, it pops up as a new drive, copy the PDF file onto it, hey presto.\nI Russian it's Nasdrowje, in German it's Prost! Don't you people know anything?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbphmn b/data_all_eng_slimpj/shuffled/split2/finalzzzbphmn
new file mode 100644
index 0000000000000000000000000000000000000000..9870e1ea3546210f0866645eaf6a672853058cfc
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Dallas, Texas \u2013 July 21, 2014\u2013 Cotton Creek Capital, a Texas-based lower middle-market private equity firm, is pleased to announce the addition of Alan Goldstein to the investment team as an Associate in the Austin office.\nPreviously, Mr. Goldstein was an investment banking analyst at Jefferies Group LLC in New York, specializing in Mergers and Acquisitions. In his time at Jefferies, Alan advised clients ranging from small to large capital companies and financial sponsors on both buy-side and sell-side transactions across several industries.\nMr. Goldstein received his Bachelor of Business Administration degree in Finance from the University of Texas at Austin, with minors in Accounting and Psychology.\nCotton Creek Capital is a Texas-based private equity firm focused on investing in and growing established lower middle market companies in manufacturing, value-added distribution, industrial, specialty chemical, business services, healthcare services and consumer staples. The firm invests in companies with enterprise values between $15 million and $200 million through transactions ranging from buyouts, recapitalizations, buy-and-builds, and corporate divestitures. Cotton Creek Capital has offices in Dallas, Fort Worth, and Austin and is affiliated with Brownlie & Braden, LLC, a provider of financial advisory services to high net worth families for over 20 years. For further information, visit Cotton Creek's website at www.cottoncreekcapital.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"First, Swansea council mess up the translation of one of their signs: the Welsh says \"\"I am not in the office at the moment. Send any work to be translated\".\nNow, English councils are banning everyday Latin phrases, like 'vice versa' and 'via'.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ok, so the title might be a tad misleading, we're not actually *using* PowerShell to modify iLO settings, but rather using PowerShell's quick and easy ability to loop through an array of enclosures in order to make the iLO changes.\nIn my case, I have a large number of c-Class blades functioning as ESXi hosts. I needed a way to set the HP power regulator setting to Static High Performance Mode, instead of going into each individual iLO or BIOS to make the change.\nYou will need a few things in order for this to work correctly. First, you'll need PuTTy (the venerable Telnet\/SSH client) and Plink (the command line front-end to PuTTy). Second you'll need the XML below (borrowed from HP's Lights-Out Scripting Samples for Windows). Upload the XML to a web host that is accessible by all HP c-Class blade enclosures, and enter the IP address or fully qualified domain name for the enclosures' Onboard Administrator into the $OBAs variable.\n.\/plink.exe -telnet -login -l username -pw password $OBA \"hponcfg all http:\/\/webhost\/set_host_power.xml\"\nNotes: You can connect using plink.exe using the -ssh flag instead of -telnet, however it is required that you've already connected to all enclosures and accepted their certificate details.\nThe user credentials in the RIBCL XML are ignored, and passthrough from the OBA to each individual server's iLO is used, unless the NOAUTOLOGIN flag is used in the hponcfg command string.\nI didn't get the your script at all. how one can run .\nI was doing something similar to this, but for mass updating firmware revisions. Adding this power shell will help me out in automating the physical work. My blog http:\/\/torxsmind.blogspot.com\/ has the setups to setup IIS and to mount a ISO over http.\nGreat tip Damian \u2013 I didn't even realise you could set this via the ILO. Shame it still requires a server reboot to take affect.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00b7 Sand Vibrating Screen Wholesale, Vibrating Screen Suppliers . CE certified sand screening machine \/ sand vibrating screen with lowest price .. vibrating Rotary screen classifier for grain peas\/wheat grinding machine in.\nThings to Know About Cam Grinding : Automotive Design ... Camlobe grinding is a complex variant of cylindrical grinding. When raw camshafts are manufactured\u2014by casting in sand or shell molds (cast iron), forged or machined from barstock (steel)\u2014roughly 0.02 to 0.12 in. of stock is generally available for grinding.\nSand Grinding Machine For Sale . A wide variety of sand grinding machine for sale options are available to you, such as universal, surface grinding machine, and cylindrical grinding machine.\nballs grinding machines for silica sand. SUAMG Machinery is professional mineral processing equipment manufacturer in the world, not our equipment has the excellent quality, but also our product service is very thorough.\nRoll Sand Blasting Machines, Roll Frosting Machines, Roll Shot . Kay Jay Chill Rolls also manufactures Roll Sand Blasting Machines or Roll Machines for making the surface of the rolls a little rough after grinding of the rolls.\nThey can also brighten and polish metal, buff, grind, sand, cut and notch. You can even use them to deburr or level surfaces. The key to such versatility lies in the grinding wheel. Knowing which wheel will be more effective for specific tasks is crucial for achieving the right results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Plated steel wheel hub with 3\/4\" ID bearing.\nFour bolts on a 4\" circle. 1\/2\" - 20 x 1-1\/4\" welded bolts. 2-13\/16\" center to center. Use with our AZ1004 , AZ1005 , AZ1006 or 30-401 steel wheels.\nUse with our 4.5\" long sprindles AZ2231 or AZ2232 .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpkhq b/data_all_eng_slimpj/shuffled/split2/finalzzzbpkhq
new file mode 100644
index 0000000000000000000000000000000000000000..9ffeabbe7652afaab600d625c25a167b8976f2b9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpkhq
@@ -0,0 +1,5 @@
+{"text":"In recent years the topic of corruption has attracted a great deal of attention. However, there is still a lack of substantial empirical evidence about the determinants of corruption. Despite an increasing interest of economists in the determinants of corruption, the factor of marriage has been widely neglected in the literature. The results suggest a positive but ambiguous effect marriage on corruption.\nAbdiweli, M. etIsse, S. (2003), \"Determinants of economic corruption: A Cross-country comparison\", Cato Journal, 22(3), 449\u2013466.\nKodilaTedika, O. (2012), \"Causes de la corruption : aper\u00e7u empirique\", Annales de l'Universit\u00e9 Marien-Ngouabi, submitted.\nLambsdorff, J.G. (2006), \"Consequences and Causes of Corruption: What do We Know from a Cross-Section of Countries?\", In Rose-Ackermann (ed), International Handbook on The Economics of Corruption, Edward-Elgar, Cheltenham, UK, Northampton, MA, USA, 3-51.\nMocan, N. (2008), \"What Determines Corruption? International Evidence from Microdata\", Economic Inquiry, 46(4), 493-510.\nMontinola, G. etJackman, W. (2002), \"Sources of Corruption: A Cross-Country Study\", British Journal of Political Science, 32(1), 147-170.\nNakamwambila Kiadiamuyika, J. and Kabanga Kazadi, C., (2007), \"Impact de la pauvret\u00e9 sur la corruption chez les magistrats, les policiers de roulage et les taximen \u00e0 Kinshasa\", ODHS Rapport de recherche N\u00b07.\nPark, H. (2003), \"Determinants of Corruption: A Cross-National Analysis\", Multinational Business Review, 11(2), 29-48.\nSeldadyo, H. et de Haan, J. (2006), \"The Determinants of Corruption: A Reinvestigation\", Paper Prepared for the 2006 EPCS Conference, Turku, Finland, 20-23 April 2006.\nShabbir, G. et Anwar, M. (2007) \"Determinants of Corruption in Developing Countries\", Pakistan Development Review, 46(4), Part II, 751\u2013764.\nTreisman, D. (2000), \"The Causes of Corruption: A Cross-National Study\", Journal of Public Economics, 76(3), 399-457.\nEurostat (2011), Demography report 2010, Luxembourg, Publications Office of the European Union.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For other uses, see Battery (disambiguation).\nThe Battery is a power source item featured in the Half-Life universe.\nIn Half-Life and its expansions, the protagonists who wear HEV Suits can find and pick up batteries to power up their suits. The suit generates protective armor when it has a power supply. When picked up, the suit provides a voice message to report the new power level. Adrian Shephard, as well as other Marines, can also use them to charge their Powered Combat Vests.\nIn Half-Life 2, Gordon Freeman's old suit was modified by Isaac Kleiner to allow the use of Combine batteries. Combine Scanners may drop Batteries when they're destroyed.\nAn HEV Suit line left unused in Half-Life refers to the Battery as an HEV Power Unit. Several items were to be powered by the Battery, including the Long Jump Module, Alien Cloak and Alien Shield. The Half-Life demo, Half-Life: Day One, features a different, earlier model, featured in early builds of Half-Life. It is also used as a temporary model for the Long Jump Module, with red panels instead of blue. Yet another, high-definition battery texture can be found in the Half-Life SDK.\nThe original, albeit smaller, model of the Battery was used as a placeholder during the development of the Half-Life 2.\nEarly Battery model on the upper left in an early screenshot.\nBatteries scattered in the dark room in the Hazard Course.\nBattery in Gordon Freeman's locker.\nBattery near the corpse of a survey team member on Xen.\nBatteries near the corpse of a survey team member on Xen.\nCity 17 logo as seen on top of the Battery.\nBattery on a table at Station 6.\nBattery on a table at Gate 5.\nThis page was last modified on 30 August 2017, at 21:26.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sep 30, 2015 ... \u2662Specializing in the Moving Industry for over 25 years ... transport company providing reliable, honest, ..... For moving companies, use of these products ..... Edmonton (6) .... If so, consider joining Atlas Canada; a trusted name.\nSep 8, 2010 ... 20 New Certified Overseas Mover Program. By Kim Biggar .... Edmonton, AB T6E 0K9 .... agers are also actual moving companies. The many ... trusted colleagues in the rapidly growing area of ...... Dependable, Reliable,.\nEdmonton, AB T6E 0K9 ... In today's world, long-distance moving is not the exclusive .... question facing many companies, however, is what to do with it. ..... Dependable, Reliable, .... you risk losing your clients' trust and your firm's reputa- tion.\nAlberta's Advanced Technology Companies 2013. Advanced. T. Alberta's ... Edmonton, Alberta T6J 7E6 Canada ... and reliable information regarding the subject ... Mover (moving big files via cloud) 55. PureWeb ..... term trust of some.\nMar 9, 2016 ... Canada's Best Managed Companies have demonstrated ... the designation actually made them a better company. From left: Peter ... Location Edmonton .... they work and move, and are ..... Movers. Today the company, which is part of the United Van ..... Let Copeman Healthcare be your trusted partner.\nApr 24, 2007 ... Edmonton, AB .... 40. Modu-Loc Fence. 41. Moving. 42. Pipe Equipment. 43. Plumbing. 44 ... Our fleets are loaded with newer, reliable equipment that ... storage. Expect Responsive Service. Our team can knowledgeably consult you on equipment ... We will work to become your trusted ..... Bush Movers.\nJun 27, 2016 ... ensuring our assets are safe and reliable. 16 protecting our .... the 250 largest companies in the world are aligning to GRI ... influences your level of trust in our company and in our ... operations. Moving forward, we will .... Mover' awards in RobecoSAM's globally ...... research and testing facility in Edmonton.\nJun 4, 2013 ... EDMONTON. EDMONTON. REGINA ... warranty regarding the accuracy or reliability of information .... ships between companies and schools. The Economist ... Movers and shakers, our members are newsmakers .... man Storage & Moving, allowing us the ..... Nationally certified, Shred Guard is your trusted.\nwere welcomed at the Edmonton Public Library by CEO Linda Cook, who arranged .... libraries and archives; we want to emphasize Canadians' profound trust in and ... effective collaboration, and provide clearer, more reliable leadership. ... storage; l constitute a pan-Canadian committee of peers to establish standards for a.\n1 day ago ... Seeing the Ground Move. 17 S eeding ... companies as well as by scientific experts from the ..... TD Canada Trust .... Be sure to check signs at people mover stops ..... Edmonton. THE ... understand the importance of a reliable.\nfact that GIS is moving from the 2D world into the 'new' ... Companies like CyberCity ... is one of the first movers in ... reliable and loyal resource ..... Charlene Nielsen \u2014 University of Alberta \u2014 Edmonton, AB Canada .... As your trusted partner,.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cutting-edge processor and 3GB of RAM provide lightning-fast response times and exceptional multitasking power for flipping through apps, sending texts, watching video, surfing the Web and more.\nThis operating system gives you detailed control over when and how you receive messages, and it comes with innovative security features like Android Smart Lock.\nProvides fast Web connection for downloading apps, streaming content, and staying connected with social media.\nWirelessly share files with other Bluetooth-enabled devices and connect to compatible accessories.\nSupports 4x the resolution of Full HD and an astonishing 577 ppi (pixels per inch) for stunning imagery and vibrant colors. Corning Gorilla Glass 4 construction resists scratches.\nSnap brilliant photos in any light. Also includes 5MP front-facing camera with a 120\ufffd wide angle lens so you can video chat and take selfies with clarity and detail.\nAmple storage space for your apps, contacts, music, photos and more.\nTouch your finger to the home button to unlock your phone, and use your fingerprint to make secure payments.\nProcess a convenient, secure payment by waving the phone near the magnetic credit card reader in the checkout line while holding your finger on the fingerprint reader.\nPlus, monitor your health consistently with the custom services offered by S Health. Log and check your exercise, food intake and heart rate in real time.\nThis mode changes your display to show a more simplified layout and shuts down all unnecessary features to dramatically minimize battery consumption.\n2. Slide the Power switch right, to power off the device.\nIt may take several seconds for the device to power off.\n3. Insert the end of a SIM eject tool or a small thin paper clip into the hole on the SIM tray.\nPress firmly and push the tool straight in until the tray pops out.\n4. Insert the SIM card in the SIM card tray.\n5. Firmly push in the SIM card tray.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"American rapper Cardi B once again became the focus of attention on social media after a video of her giving birth was shared online by the child's father, Offset.\nA number of people also seemed visibly amused by Offset's facial expressions as he watched Cardi giving birth in the video.\nMeanwhile, Cardi B recently deleted her Instagram account following \"apparent abuse she's received\" after winning the Best Rap Album award at the Grammys, according to Variety Magazine.\nPrior to the deletion, the rapper posted what the magazine described as \"an expletive-laden video\", blasting commentators who claimed that she didn't deserve the award, and stating that she's \"sick of this s\u2014\".","meta":{"redpajama_set_name":"RedPajamaC4"}}